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+{"text":"\n\\section{Acknowledgements}\n\\label{sec:acknowledgements}\n\nThe authors would like to thank our colleagues: Fran\\c{c}oise Beaufays,\nAlex Graves and Leif Johnson for helpful research discussions and Mike Schuster for help with wordpiece models.\n\\section{Analysis}\n\\label{sec:analysis}\n\nWe observe that a large part of the improvements described in this work are from a reduction in substitution errors.\nUsing wordpieces instead of graphemes results in an absolute 2.3\\% word error rate improvement,\nof this 1.5\\% is due to fixing substitution errors. Inclusion of pronunciation and text data\nimprove voice-search word error rate by an absolute 0.6\\% and 0.6\\% respectively, all of these are\ndue to improvements in word substitution errors. Many of the corrected substitution errors seem to be from improved language modeling: words which\nmay sound similar but have different meaning given the text context. Some selected examples include\nimprovements with proper nouns: `barbara stanwick' recognized by a grapheme model is fixed when using\nwordpieces to the correct name `barbara stanwyck'. Similar improvements are found when including\npronunciation data: `sequoia casino' to `sycuan casino', `where is there' to `where is xur'\nand also when including text data: `soldier boy' to `soulja boy', `lorenzo llamas' to `lorenzo lamas'.\nWe also find that wordpieces capture longer range language context than graphemes in improvements like\n`tortoise and the hair' to `tortoise and the hare'.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nWe train end-to-end speech recognition models using the RNN-T which predicts graphemes\nor wordpieces and thus directly outputs the transcript from audio. We find pre-training the RNN-T\nencoder with CTC results in a 5\\% relative WER improvement, and using a deeper\n8-layer encoder instead of a 5-layer encoder further improves WER by 10\\% relative.\nWe incorporate pronunciation data using a pre-training hierarchical-CTC loss which\nincludes phoneme targets and find this improves the voice-search WER by 5\\% relative\nwith little impact on the voice-dictation task. To include\ntext-only data we pre-train the recurrent network in the decoder as LSTM language models resulting\nin a overall 5\\% relative improvement. We train wordpiece RNN-Ts with 1k, 10k and 30k wordpieces targets\nand find that they significantly outperform the grapheme-based RNN-Ts. For comparison we use a baseline\nspeech recognizer with individual acoustic, pronunciation and language models with state-of-the-art WERs\nof 8.3\\% on voice-search and 5.4\\% on voice-dictation. With a 30k wordpiece RNN-T achieving\nWERs of 8.5\\% on voice-search and 5.2\\% on voice-dictation we demonstrate that a single end-to-end neural model\nis capable of state-of-the-art streaming speech recognition.\n\n\\section{Experimental Setup} \\label{sec:experiments}\n\n\nWe compare the RNN-T end-to-end recognizer with a conventional ASR system consisting of separate acoustic, pronunciation and language models.\nThe acoustic model is a CTC trained LSTM that predicts context-dependent (CD) phonemes\nfirst fine-tuned with sequence discriminative training as described in~\\cite{lstm_am3} and\nfurther improved with word-level edit-based minimum Bayes risk (EMBR) proposed recently by Shannon~\\cite{embr}.\nAcoustic models are trained on a set of $\\sim$22 million hand-transcribed anonymized\nutterances extracted from Google US English voice traffic, which corresponds to $\\sim$18,000\nhours of training data. These include voice-search as well as voice-dictation utterances.\nWe use 80-dimensional log mel filterbank energy features computed every 10ms stacked every 30ms to a single 240-dimensional acoustic\nfeature vector. To achieve noise robustness acoustic training data is distorted as\ndescribed in~\\cite{lstm_am}. The pronunciation model is a dictionary containing hundreds of thousands of human expert transcribed US English word pronunciations.\nAdditional word pronunciations are learned from audio data using pronunciation \nlearning techniques~\\cite{pronlearning}. For out-of-dictionary words a G2P model is trained using transcribed word pronunciations.\nA 5-gram language model is trained  with a text sentence dataset which includes untranscribed anonymized speech logs:\n150 million sentences each from voice-search and voice-dictation queries, and anonymized typed logs including tens of billion\nsentences from Google search from various sources. The language model is pruned to 100-million n-grams with a target vocabulary of 4 million\nand the various sources of text data are re-weighted using interpolation~\\cite{interpolation} for the\noptimal word error rate performance. Single-pass decoding with a conventional WFST is carried out to generate recognition transcripts.\n\n\nThe RNN-T is trained with the same data as the baseline. The CTC encoder network is pre-trained with\nacoustic transcribed data and as with the baseline acoustic model the pronunciation model is used to\ngenerate phoneme transcriptions for the acoustic data. The RNN-T decoder is pre-trained on the text only data as a LSTM language model, \nroughly half a billion sentences from the text data are sampled according to their count and the data source \ninterpolation weight (as optimized in the baseline). All RNN-T models are trained with LSTM networks in the tensorflow~\\cite{tensorflow} toolkit with\nasynchronous stochastic gradient descent.  Models are evaluated using the RNN-T beam search algorithm\nwith a beam of 100 for grapheme models and 25 for wordpiece models and a temperature of 1.5 on the softmax.\nWord error rate (WER) is reported on a voice-search and a voice-dictation test set with roughly 15,000 utterances each.\n\n\\section{Introduction \\label{sec:introduction}}\nThe current state-of-the-art automatic speech recognition (ASR) systems break down\nthe ASR problem into three main sub-problems: acoustic, pronunciation and language modeling.\nSpeech recognition involves determining the most likely word sequence,\n$W=w_1,...,w_n$, given an acoustic input sequence, $\\mathbf{x}=x_1,...,x_T$,\nwhere $T$ represents the number of frames in the utterance:\n\\begin{equation}\n\tW^{*} = \\argmax_W P(W|{\\mathbf x}),\n\\end{equation}\nwhich is typically decomposed into three separate models, as follows:\n\\begin{align}\n\tW^{*} &= \\argmax_W \\sum_\\phi P({\\mathbf x}, \\phi | W) P(W) \\\\\n\t      &\\approx \\argmax_{W, \\phi} p({\\mathbf x}|\\phi) P(\\phi|W) P(W)\n\\end{align}\nThe acoustic model, $p({\\mathbf x}|\\phi)$, predicts the likelihood of the acoustic input\nspeech utterance given a phoneme sequence, $\\phi$; for conditional models that\ndirectly predict $P(\\phi|{\\mathbf x})$, the likelihood is typically replaced with a\nscaled likelihood obtained by dividing the posterior with the prior, $P(\\phi)$,\nin so-called hybrid models~\\cite{MorganBourlard95}.\nDeep recurrent neural networks with long short-term memory (LSTM)\ncells~\\cite{lstm} have recently been shown to be ideal for this\ntask~\\cite{lstm_am1, lstm_am2, lstm_am3}.\nThe pronunciation model, $P(\\phi|W)$, is typically built from pronunciation\ndictionaries curated by expert human linguists, with back-off to a\ngrapheme-to-phoneme (G2P) model~\\cite{g2p} for out of dictionary words.\nFinally, an N-gram model trained on text data may be used as a language model,\n$P(W)$.\n\nRecently, there has been considerable interest in training end-to-end models for\nASR~\\cite{las, baidu, BahdanauChorowskiSerdyukEtAl16}, which directly output\nword transcripts given the input audio.\\footnote{ In the context of this work,\nwe consider models that are all-neural, and directly output word transcripts from\naudio utterances as being end-to-end.}\nThus, these models are much simpler than conventional ASR systems as a single\nneural network can be used to directly recognize utterances, without requiring\nseparately-trained acoustic, pronunciation and language model components.\nA particular class of architecures known as sequence-to-sequence\nmodels~\\cite{seq2seq} are particularly suited for end-to-end ASR as they include\nan \\emph{encoder} network which corresponds to the acoustic model of a\nconventional system and a decoder network which corresponds to the language\nmodel.\n\nOne drawback of typical encoder-decoder type architectures (e.g.,~\\cite{las,\nBahdanauChorowskiSerdyukEtAl16}) is that the entire input sequence is encoded\nbefore the output sequence may be decoded and thus these models cannot be used\nfor real-time streaming speech recognition.\nSeveral streaming encoder-decoder architectures have been proposed previously,\nincluding the neural transducer~\\cite{nt}, the recurrent neural aligner\n(RNA)~\\cite{rna}, and the recurrent neural network transducer\n(RNN-T)~\\cite{rnnt1, rnnt2}.\nIn particular, these architectures allow the output to be decoded as soon as the first\ninput is encoded, without introducing additional latency incurred when\nprocessing the entire utterance at once.\nIn this work we only consider streaming recognition architectures, specifically\nthe RNN-T model.\n\nDespite recent work on end-to-end ASR, conventional systems still remain the\nstate-of-the-art in terms of word error rate (WER) performance.\nFor example, in our previous work~\\cite{rohit1} we evaluated a number of\nend-to-end models including attention-based models~\\cite{las} and\nRNN-T~\\cite{rnnt1, rnnt2} trained on $\\sim$12,500 hours of transcribed training\ndata; although end-to-end approaches were found to be comparable to a\nstate-of-the-art context-dependent phone-based baseline on dictation test sets,\nthese models were found to be significantly worse than the baseline on\nvoice-search test sets.\nEnd-to-end systems are typically trained using transcribed acoustic data sets,\nwhich are relatively expensive to generate and thus much smaller than text-only\ndata sets, which are used to train LMs in a traditional speech recognizer.\nA deficiency of end-to-end systems appears to be in their language modeling\ncapacity~\\cite{rohit1} which may be because large text-only data are not\nutilized in end-to-end systems.\n\nIn this work we explore a particular sequence-to-sequence architecure, RNN-T,\nand show how text and pronunciation data may be included to improve end-to-end\nASR performance.\nAnother contribution of this work is to investigate the use of\nwordpieces~\\cite{wp}, which have been explored previously in the context of\nmachine translation, as a sub-word unit for end-to-end speech recognition.\n\nThe paper is organized as follows: in Section~\\ref{sec:rnnt} we describe the RNN-T and\nhow it may be used for streaming recognition. Section~\\ref{sec:training} describes how the RNN-T\nis trained including the units, architectures and pre-training parts of the model. The experimental\nsetup including the baseline system are detailed in Section~\\ref{sec:experiments}. Section~\\ref{sec:results}\ncompares the word error rate performance of various RNN-T models and the baseline to show relative improvement.\nWe find that the techniques introduced in this work mostly improve the language modeling of the RNN-T,\nSection~\\ref{sec:analysis} shows some select examples of such improved recognition. A concluding summary and\nacknowledgements are in Section~\\ref{sec:conclusion} and Section~\\ref{sec:acknowledgements}.\n\n\n\n\\section{Results}\n\\label{sec:results}\n\nWe train and evaluate various RNN-T and incrementally show the WER impact with\neach improvement.\n\n\nA grapheme based RNN-T is trained from scratch (no pre-training) on the acoustic data\nwith a 5-layer LSTM encoder of 700 cells and a 2-layer LSTM decoder of 700 cells. A final 700 unit\nfeed-forward layer and a softmax layer output grapheme label probabilities. We compare this model\nto a model with identical architecture but with the encoder CTC pre-trained. We find CTC pre-training\nto be helpful improving WER 13.9\\%$\\rightarrow$13.2\\% for voice-search and 8.4\\%$\\rightarrow$8.0\\%\nfor voice-dictation.\n\nA model with a deeper 8-layer encoder is also trained with a multi-CTC loss at depth 5 and depth 8 where both\nlosses are optimized for the same grapheme targets. We found training 8-layer models without a\nmulti-loss setup to be unstable which we acknowledge may be addressed with recent advancements in\ntraining deeper recurrent models~\\cite{highway} but are not tested as part of this work.\nThe deeper 8-layer encoder further improves WER 13.2\\%$\\rightarrow$12.0\\% for voice-search and 8.4\\%$\\rightarrow$6.9\\%\nfor voice-dictation.\n\nTo incorporate the knowledge of phonemes and specifically the pronunciation dictionary data we train a 8-layer\nencoder with hierarchical-CTC with a phoneme target CTC at depth 5 and a grapheme target CTC at depth 8.\nIn this way the network is forced to model phonemes and is exposed to pronunciation variants in the labels where\nthe same word (and thus same grapheme sequence) may have different pronunciations (and thus phoneme sequences).\nThis approach does not address including pronunciations for words that do not occur in the acoustic training\ndata, which we leave as future work. We find that the pronunciation data improves WER 12.0\\%$\\rightarrow$11.4\\% for voice-search\nbut with little improvement for voice-dictation. Unlike voice-search the voice-dictation test set is \ncomprised of mostly common words, we conjecture that it may be sufficient to learn pronunciations for these\nwords from the acoustic data alone and thus may not benefit from additional human transcribed pronunciations.\n\nNext, to include the text data we pre-train a 2-layer LSTM with 700 cells as a language model with grapheme targets. The model\nis trained until word perplexity on a held-out set no longer improves, Table~\\ref{lmperp} shows the word preplexity and\nsizes of the various language models that were trained. Addition of text data in this way improves WER \n11.4\\%$\\rightarrow$10.8\\% for voice-search and 6.8\\%$\\rightarrow$6.4\\% for voice-dictation.\n\nWe explore modeling wordpieces, with 1k, 10k and 30k wordpieces, instead of graphemes and make several changes to the architecture. \nThe wordpiece encoder network is a 12-layer LSTM with 700 cells each, trained with hierarchical-CTC with phoneme targets at depth 5, graphemes at depth 10\nand wordpieces at depth 12. Since wordpieces are longer units we include a time convolution after depth 10 reducing the \nsequence length by a factor of 3. We find that this time convolution does not affect WER but drastically reduces training and\ninference time as there are 3 times fewer encoder features that need to be processed by the decoder network. Wordpiece language models\nare trained similar to graphemes, since the numbers of labels are much larger an additional input embedding of size 500 is used for wordpiece\nmodels. The wordpiece language models perform much better in terms of word perplexity (Table~\\ref{lmperp}) and the RNN-T initialized from\nthem also see significant WER improvements  (Table~\\ref{wer}). The best end-to-end RNN-T with 30k wordpieces achieves a \nWER of 8.5\\% for voice-search and 5.2\\% on voice-dictation which is on par with the state-of-the-art baseline speech recognition system.\n\n\\begin{table}[!h]\n  \\caption{The number of parameters (in millions) and word perplexity for LSTM language model trained with different units evaluated\n  on a held-out set.}\n  \\label{lmperp}\n  \\centering\n  \\begin{tabular}{lcc} \\toprule\n    Units & Params & Perplexity \\\\ \\hline \\midrule\n    Graphemes & 6M & 185 \\\\\n    Wordpieces-1k & 10M & 138 \\\\\n    Wordpieces-10k & 20M & 130 \\\\\n    Wordpieces-30k & 59M & 119 \\\\\n \\bottomrule\n  \\end{tabular}\n\\end{table}\n\n\\section{RNN-Transducer}\n\\label{sec:rnnt}\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.5\\columnwidth]{rnnt-model}\n\t\\caption{The RNN-T model. The model consists of an encoder network,\n\twhich maps input acoustic frames into a higher-level representation, and\n\ta prediction and joint network which together correspond to the decoder\n\tnetwork. The decoder is conditioned on the history of previous\n\tpredictions.}\n\t\\label{fig:rnnt}\n\\end{figure}\n\nThe RNN-T was proposed by Graves~\\cite{rnnt1} as an extension to the\nconnectionist temporal classification (CTC)~\\cite{ctc} approach for sequence\nlabeling tasks where the alignment between the input sequence, ${\\mathbf x}$, and the\noutput targets ${\\mathbf y}$ is unknown.\nThis is accomplished in the CTC formulation by introducing a special label,\ncalled the \\emph{blank} label, which models the probability of outputting no\nlabel corresponding to a given input frame.\nCTC has been widely used in previous works to train end-to-end ASR\nmodels~\\cite{baidu, e2e_ctc1, e2e_ctc2}.\nHowever, a major limitation of CTC is its assumption that model outputs at a\ngiven frame are independent of previous output labels: $y_t \\perp\\!\\!\\!\\perp y_j | {\\mathbf x}$, for\n$t < j$.\n\nThe RNN-T model, depicted in Figure~\\ref{fig:rnnt}, consists of an\n\\emph{encoder} (referred to as the transcription network in~\\cite{rnnt1}), a\nprediction network and a joint network; as described in~\\cite{rohit1}, the RNN-T\nmodel can be compared to other encoder-decoder architectures such as ``listen,\nattend, and spell\"~\\cite{las}, if we view the combination of the prediction\nnetwork and the joint network as a decoder.\nThe encoder is an RNN which converts the input acoustic frame ${\\mathbf x}_t$ into a\nhigher-level representation, ${\\mathbf h}^\\text{enc}_t$, and is analogous to a CTC-based\nAM in a standard speech recognizer.\nThus, as in CTC, the output of the encoder network, ${\\mathbf h}^\\text{enc}_t$, is\nconditioned on the sequence of previous acoustic frames $x_0, \\cdots, x_t$.\n\\begin{equation}\n{\\mathbf h}_{t}^\\text{enc} = f^\\text{enc}(x_t),\n\\end{equation}\n\nThe RNN-T removes the conditional independence assumption in CTC by introducing\na \\emph{prediction network}, an RNN that is explicitly conditioned on the\nhistory of previous non-blank targets predicted by the model.\nSpecifically, the prediction network receives as input the last \\emph{non-blank}\nlabel, $y_{u-1}$, to produce as output ${\\mathbf h}^\\text{dec}_u$.\n\\begin{equation}\n{\\mathbf h}_{u}^\\text{dec} = f^\\text{dec}(y_{u-1}).\n\\end{equation}\n\nFinally, the \\emph{joint network}, is a feed-forward network that combines the\noutputs of the prediction network and the encoder to produce logits\n($\\mathbf{z}_{t, u}$) followed by a softmax layer to produce a distribution\nover the next output symbol (either the blank symbol or one of the\noutput targets).\n\\begin{equation}\nz_{t,u} = f^\\text{joint}({\\mathbf h}_{t}^\\text{enc}, {\\mathbf h}_{u}^\\text{dec})\n\\end{equation}\nWe use the same form for $f^\\text{joint}$ as described in~\\cite{rnnt2}.\nThe entire network is trained jointly to optimize the RNN-T loss~\\cite{rnnt1},\nwhich marginalizes over all alignments of target labels with blanks as in CTC,\nand is computed using dynamic programming.\n\nDuring each step of inference, the RNN-T model is fed the next acoustic frame\n${\\mathbf x}_t$ and the previously predicted label $y_{u-1}$, from which the model produces\nthe next output label probabilities $P(y|t, u)$. If the predicted label, $y_{u}$, is non-blank,\nthen the prediction network is updated with that label as input to generate \nthe next output label probabilities $P(y|t, u + 1)$. Conversely, if\na blank label is predicted then the next acoustic frame, ${\\mathbf x}_{t+1}$, is used to update the encoder\nwhile retain the same prediction network output resulting in $P(y|t + 1, u)$. In this way\nthe RNN-T can stream recognition results by alternating between updating the encoder and the prediction\nnetwork based on if the predicted label is a blank or non-blank. \nInference is terminated when blank is output at the last frame, $T$.\n\nDuring inference, the most likely label sequence is computed using beam search\nas described in~\\cite{rnnt1}, with a minor alteration which was found to make\nthe algorithm less computationally intensive without degrading performance: we\nskip summation over prefixes in \\texttt{pref$(\\mathbf{y})$} (see Algorithm\n1 in~\\cite{rnnt1}), unless multiple hypotheses are identical.\n\nNote that unlike other streaming encoder-decoder architectures such as\nRNA~\\cite{rna} and NT~\\cite{nt}, the prediction network is not conditioned on\nthe encoder output.\nThis allows for the the pre-training of the decoder as a RNN language model on\ntext-only data as described in Section~\\ref{sec:training}.\n\n\n\\section{Units, Architectures and Training}\n\\label{sec:training}\n\\begin{figure*}[!t]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{rnnt}\n\\end{center}\n\\caption{The various stages of training a wordpiece RNN-T. The encoder network is pre-trained as a hierarchical-CTC network simultaneously\npredicting phonemes, graphemes and wordpieces at 5, 10 and 12 LSTM layers respectively. A time convolutional layer reduces the encoder\ntime sequence length by a factor of three. The decoder network is trained as a LSTM langauge model predicting wordpieces optimized with\na cross-entropy loss. Finally, the RNN-T network weights are initialized from the two pre-trained models, indicated by the dashed lines,\nand the entire network is optimized using the RNN-T loss.}\n\\label{rnnt}\n\\end{figure*}\n\nWe investigate the use of graphemes and sub-words (wordpieces) as output lexical units in RNN-T models.\nFor the graphemes, we use letters (\\texttt{a-z}), digits (\\texttt{0-9}), special\nsymbols (\\texttt{\\&.'\\%\/-:}) and a space symbol (\\texttt{$<$space$>$}).\nThe space symbol is used for segmenting recognized grapheme sequences to word sequences.\n\nState-of-the-art large vocabulary speech recognition systems recognize millions of different words,\ninference for RNN-T with that many output labels would be impractically slow.\nTherefore, as subword units, we use wordpieces as described in ~\\cite{wp}.\nWe train a statistical wordpiece model with word counts obtained from text data for segmenting each word individually into subwords.\nAn additional space symbol is included in subword units.\nAn example segmentation for the sentence \\texttt{tortoise and the hare} is\n\\texttt{$<$tor$>$ $<$to$>$ $<$ise$>$ $<$space$>$ $<$and$>$ $<$space$>$ $<$the$>$ $<$space$>$ $<$ha$>$ $<$re$>$}.\nWordpieces have be shown to benefit end-to-end recognition~\\cite{lsd} since they offer\na balance with longer context than graphemes and a tunable number of labels.\nSince the wordpiece\nmodel is based on word frequencies, more common words appear as a single label.\nA vocabulary of 1,000 generated wordpieces\nincludes words like `mall', `remember' and `doctor' while a vocabulary of 30,000 wordpieces also includes\nless common words like `multimedia', `tungsten' and `49er'. The wordpiece models may also output\nany word that the grapheme model may; we find that all the graphemes are included in the wordpiece vocabularies.\n\nFor the encoder networks in RNN-T models, we experimented with deep LSTM networks (5 to 12 layers).\nFor the decoder networks, we used a stack of 2 layer LSTM network, a feed-forward layer and a softmax layer.\nIn addition to training models with random initialization of parameters, we explored variations of initializing encoder and decoder network parameters from pre-trained models.\nIt has been previously shown that initializing RNN-T encoder parameters from a model trained with the CTC loss is beneficial for the phoneme recognition task~\\cite{rnnt2}.\nWe experimented with initializing encoder networks from models trained with the\nCTC loss and with initializing LSTM layer parameters in prediction networks from LSTM language models trained on text data.\nAfter initialization of encoder and prediction network weights from separate pre-trained models, the entire RNN-T model weights are trained with the RNN-T objective.\n\nWe show one example architecture for the RNN-T wordpiece model in Figure 2.\nThe figure also shows the pre-trained CTC LSTM acoustic model and LSTM language model architectures used to initialize the encoder and prediction network weights.\nThe dotted arrows indicate the pre-trained layers used to initialize specific layers in the RNN-T model.\nThe encoder networks in RNN-T models are pre-trained with the CTC loss using phonemes, graphemes and wordpieces as output units.\nWe investigate encoder architectures with multi-task training using hierarchical-CTC~\\cite{hctc} with various\n'hierarchies' of CTC losses at various depths in the encoder network. With hierarchical-CTC\nthe encoder networks are trained with multiple simultaneous CTC losses which was\nbeneficial for grapheme recognition~\\cite{hctc_g}. After pre-training all CTC losses\nand additional weights associated with generating softmax probabilities are discarded.\nFor the wordpiece models which have longer duration than graphemes, we employ an additional 'time-convolution' in the encoder network to reduce the\nsequence length of encoded activations which is similar to the pyramidal sequence length reduction in ~\\cite{las}.\nFor these models, we used filters covering 3 non-overlapping consecutive activation vectors, thus reducing them to a single activation vector.\nThe LSTM layers in decoder networks are pre-trained as a language model using the graphemes or wordpieces as lexical units.\nThe input to the network is a label (grapheme or wordpiece) in a segmented sentence represented as a one-hot vector.\nThe target for the network is the next label in the sequence and the model is trained with the cross-entropy loss.\nThe weights in the softmax output layer are discarded after pre-training and only the LSTM network weights are used to partially initialize the RNN-T prediction network.\nFor wordpiece language models, we embed labels to a smaller dimension.\nThese embedding weights are also used to initialize the RNN-T wordpiece models.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:level1} Introduction}\n\nMagnetic reconnection is a process that occurs on the order of Alfv\\'{e}nic timescales in which the magnetic topology is rearranged and the magnetic energy is converted into plasma kinetic or thermal energy \\cite{Finn1977,Bondeson1983}. Because of its broad applications in solar flares \\cite{Forbes1991} and corona \\cite{Chen2021b,Dong2022}, coronal mass ejections (CMEs) \\cite{Qiu2007}, Earth's and planetary magnetospheres \\cite{Chen2008,Le2017,Phan2018,Yang2022}, and also in laboratory plasmas \\cite{Yamada1994,Duan2018,Raymond2018}, reconnection has been of immense interest in recent years \\cite{Ji2022}.\n\nIt has been well established that collisionless magnetic reconnection occurs in the regime beyond ideal magnetohydrodynamics, where kinetic-scale physics becomes prominent in the formation and evolution of the thin current sheets near the reconnection sites \\cite{Zweibel2009,Yamada2010}. Thus, simulating collisionless magnetic reconnection typically requires kinetic approaches such as the particle-in-cell (PIC) method. In general, however, such kinetic simulations are computationally expensive and are hence unable to efficiently solve large-scale problems involving collisionless physics. In order to solve this issue with affordable computational costs, two broad approaches have been proposed for large-scale global simulations, i.e., the magnetohydrodynamics with embedded particle-in-cell (MHD-EPIC) model \\cite{Daldorff2014,Toth2016,Chen2017,Chen2019JGR} and the multi-moment multi-fluid model \\cite{Wang2015,Wang2018,Dong2019,Wang2020,Jarmak2020,Rulke2021}. Recently, some progress has been made to improve the fluid closure in the multi-moment multi-fluid model through machine learning \\cite{Qin2022,Cheng2023}, and meanwhile, MHD-EPIC has been improved to incorporate the feature of adaptively embedded PIC regions (MHD-AEPIC) \\cite{Shou2021,Chen2021,Wang2022}, which offers flexibility for the PIC code to capture the localized regions where kinetic physics is important.\n\nIt has been well demonstrated through a series of local studies \\cite{Wang2015,Ng2015,Ng2017,Ng2019} that the multi-moment multi-fluid model incorporating the higher-order moments is capable of reproducing some critical aspects of the collisionless reconnection physics from fully kinetic simulations. However, no such systematic local studies have been conducted using the MHD-AEPIC model, which motivates this study to employ the MHD-AEPIC model to investigate the magnetic island coalescence problem that is highly kinetic in nature \\cite{Stanier2015,Ng2015}.\n\nIn an MHD-AEPIC simulation, only part of the simulation domain, where kinetic effects are important, is simulated by the semi-implicit PIC code, Flexible Exascale Kinetic Simulator (FLEKS) \\cite{Chen2021} and the rest of the domain is handled by the MHD (or Hall MHD) model, Block-Adaptive-Tree Solarwind Roe-type Upwind Scheme (BATS-R-US) \\cite{Powell1999,Toth2012}. For the magnetic island coalescence problem, the embedded PIC regions are applied to simulate the regions with strong current density. As we will show later, the adaptive simulation comes in close agreement with the simulation employing the fully kinetic PIC code,  Object-oriented Simulation Rapid Implementation System (OSIRIS) \\cite{Fonseca2002,Hemker2015}, when analyzing their out-of-plane current densities, reconnection rates, O-point separations, pressure tensor elements, and agyrotropy (a measure of the deviation of the pressure tensor of a species from cylindrical symmetry with respect to the direction of the local magnetic field \\cite{Scudder2008}).\n\nIn Sec. II, we discuss the model setup for the magnetic island coalescence problem. In Sec. III, simulation results are presented and discussed by comparing the full PIC simulation results with the outputs from the MHD-EPIC and MHD-AEPIC models. Concluding remarks are given in Sec. IV. \n\n\\section{Model Setup and Methods}\n\nIn this study, we set a Fadeev equilibrium \\cite{Fadeev1965} as the initial condition where the initial magnetic vector potential is expressed as\n\\begin{eqnarray}\n\\textbf{A} =  B_0 \\lambda \\ln \\big[ \\epsilon \\cos(x\/\\lambda) + \\cosh(y\/\\lambda) \\big] \\hat{\\textbf{z}}\n\\label{eq:vecpot}\n\\end{eqnarray}\nwhere $B_0$ is the asymptotic magnetic field away from the x-axis, $\\epsilon = 0.4$ is a measure of the island size, and the initial density distribution is written as\n\\begin{eqnarray}\nn = \\frac{n_0 (1 - \\epsilon^2)}{\\big[ \\epsilon \\cos(x\/\\lambda) + \\cosh(y\/\\lambda) \\big]^2} + n_b\n\\label{eq:eqden}.\n\\end{eqnarray}\nwhere $n_b = 0.2n_0$ is the background number density. We define the global Alfv\\'{e}n time as $t_A = L_x\/v_A$, where $L_x = 4\\pi \\lambda$ is the length of the simulation box, $\\lambda = 5d_{i0}$ is the half-width of the current sheet (we also run additional simulations with $\\lambda = 10d_{i0}$; see Fig.~\\ref{fig:lambda10_rate}), $d_{i0} = (m_i\/\\mu_0 n_0 e^2)^{1\/2}$ is the ion inertial length, and $v_A = B_0\/(\\mu_0 n_0 m_i)^{1\/2}$ is the Alfv\\'{e}n speed. The width of the box is set as $L_y = 2\\pi \\lambda$. In the full PIC simulation with OSIRIS, we specify the ion and electron out-of-plane current densities as the following relationship $J_{zi0}\/J_{ze0} = T_{i0}\/T_{e0}$, where $T_{i0}$ and $T_{e0}$ are uniform. In the MHD-AEPIC simulations, the electron and ion bulk velocities are initialized from the MHD current and momentum \\cite{Daldorff2014}. In the following simulations, we use the same configuration described in \\citet{Stanier2015} and \\citet{Ng2015}, setting $T_{i0} = T_{e0}$, $m_i = 25m_e$ (so $d_i =  5 d_e$), $t_{max} = 2.5 t_A$, and $n_0 k_B (T_{i0} + T_{e0}) = B_0^2\/2\\mu_0$ (so $p = p_{mag}$). The relationship between the electron plasma and cyclotron frequencies is given by $\\omega_{pe} = 2\\Omega_{ce}$ where $\\omega_{pe} = (n_0 e^2\/m_e \\epsilon_0)^{1\/2}$ and $\\Omega_{ce} = |e|B_0\/m_e$. The initial perturbation follows the same form as in \\citet{Daughton2009_1}, with\n\\begin{subequations}\n\\label{eq:idealMHDpert}\n\\begin{eqnarray}\n\\delta B_x = \\delta B_0 \\cos \\bigg( \\frac{2 \\pi x}{L_x} \\bigg) \\sin \\bigg( \\frac{\\pi y}{L_y} \\bigg)\n\\label{magpert1}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\delta B_y = -\\delta B_0 \\sin \\bigg( \\frac{2 \\pi x}{L_x} \\bigg) \\cos \\bigg( \\frac{\\pi y}{L_y} \\bigg)\n\\label{magpert2}\n\\end{eqnarray}\n\\end{subequations}\nwhere we set the perturbation amplitude as $\\delta B_0 = 0.1B_0$.\n\nTo simulate the merging of two magnetic islands, we first perform two simulations using the MHD-AEPIC model, one with a large static, non-adaptive (or fixed) PIC region and the other with adaptive PIC regions. We also run a full PIC simulation using OSIRIS to validate and compare with the MHD-AEPIC simulations. In all simulations, we have used periodic boundaries in the horizontal ($x$) direction, and conducting electromagnetic boundaries and reflecting particle boundaries in the vertical ($y$) direction.\n\nThe OSIRIS simulation domain consists of 2000$\\times$1000 cells and each cell has 256 particles per species. In the MHD-AEPIC simulations, the MHD grid consists of 2048$\\times$1024 cells. The effective PIC grid resolution and the initial particle number per cell are the same as the OSIRIS simulation, and the Gauss's Law-satisfying Energy Conserving Semi-Implicit Method (GL-ECSIM) \\cite{Chen2019} is applied. In general, the MHD and PIC grid size does not have to be the same \\cite{Chen2021} (see Appendix A). In the magnetic island coalescence simulation, the two magnetic islands are associated with strong current density, which suggests a notable separation between the electron and ion velocities, and kinetic effects may play an important role when two islands move toward each other. Outside the islands, the current is generally weak, and neither the magnetic field nor plasma quantities show a strong gradient, so an MHD model is sufficient to describe this region. In the MHD-AEPIC run with adaptive PIC regions, the PIC regions only cover the areas of $J\\Delta x\/B > 0.01$, where the selection criterion $J\\Delta x\/B$ is dimensionless. The threshold 0.01 is carefully chosen based on numerical experiments such that the islands are well covered by the PIC region. The selection criterion is calculated from the MHD results. We usually run a pure MHD simulation first to estimate a proper threshold that would determine the desired PIC regions, then we run short MHD-AEPIC simulations with a coarse PIC grid to optimize the threshold. The small threshold for the island coalescence problem is due to the fact that it is highly kinetic in nature where the coupling between the macro-scale MHD and micro-scale kinetic physics is important \\cite{Stanier2015,Ng2019}. In general, the PIC region of an MHD-AEPIC simulation is selected based on the nature of a problem. For comparison, we also perform a simulation using the MHD-EPIC model with a relatively large static or fixed PIC region, which covers the majority of the entire simulation domain. \n\nIn MHD-AEPIC and MHD-EPIC simulations, when Hall MHD is used (which is the case for the current study), the Alfven and Whistler modes can travel through the MHD-PIC boundaries smoothly \\cite{Daldorff2014}. However, if there are kinetic modes reaching the MHD-PIC boundaries, they may be artificially reflected since they cannot propagate into the MHD regions. For such applications, we should increase the PIC regions so that the kinetic modes would have been damped near the PIC regions. The MHD-AEPIC model is initially designed for global magnetosphere simulations with a relatively small kinetic region. If the entire simulation domain is dominated by kinetic physics, a full PIC or hybrid-PIC \\cite{Dong2021,Winske2022} code is a better choice. \n\nWe note that the boundary conditions, with the exception of periodic boundaries, introduce numerical perturbations inevitably for all kinetic codes, which is a natural consequence of numerical discretization. The perturbations, however, would be reduced if the boundaries are far away from the kinetic regions. We, therefore, usually choose PIC regions where the solution is smooth at the MHD-PIC interface, such as the simulations presented in this paper.\n\n\\section{Simulation Results}\n\n\\begin{figure\n\\includegraphics[width=0.5\\textwidth]{Fig1.pdf}\n\\caption{\\label{fig:Jz-contour} Out-of-plane current density at $t = 0.9 t_A$ for the adaptive (top) and fixed (middle) PIC region runs and the full-PIC run (bottom) with $\\lambda=5 d_{i0}$, overlaid with the magnetic vector potential contours (black dotted curves) and with the boundary between the MHD and PIC regions (green solid curves).}\n\\end{figure}\n\n\\begin{figure\n\\includegraphics[width=0.5\\textwidth]{Fig2.pdf}\n\\caption{\\label{fig:pic_cell_ratio} Ratio of the active PIC regions over the entire simulation domain for the MHD-AEPIC simulation. The regions that are covered by the semi-implicit PIC code increase from $40\\%$ to about $50\\%$ during the simulation.\n}\n\\end{figure}\n\nFig.~\\ref{fig:Jz-contour} shows 2D plots of the out-of-plane current density at $t=0.9t_A$ from the three different simulations, where the $x$ and $y$ coordinates have been normalized over $d_{i0}$ and the current density over $|e| n_0 c$. In each panel of Fig.~\\ref{fig:Jz-contour}, the current sheet formation, which cannot be correctly captured in a Hall MHD model for this problem \\cite{Stanier2015}, are indicated as regions of positive out-of-plane current densities near the origin ($x, y = 0$). Also illustrated in Fig.~\\ref{fig:Jz-contour} are the boundaries (green curves) between the MHD and PIC regions in both the adaptive-PIC-region and fixed-PIC-region runs. \n\nFor the adaptive-PIC-region run, the variation of the active PIC cell number inside the entire simulation domain over time is shown in Fig.~\\ref{fig:pic_cell_ratio}. Initially, about 40\\% of the simulation domain is run by PIC code, then the ratio gradually increases to about 50\\%. In this simulation, the PIC regions are defined as the areas with large current density values and they do not vary significantly during the simulation. For other applications, such as modeling the Earth magnetotail \\cite{Wang2022}, the active PIC regions may change dramatically. We note that the smallest granularity to turn on or off PIC cells is a patch with two cells in each direction \\cite{Chen2021}. Since the MHD-AEPIC model adopts a semi-implicit PIC code, which requires an iterative solver to update the electric field \\cite{Chen2019}, it requires more computations per step. The computational efficiency also depends on the implementation and problem properties. For the simulations presented in Fig.~\\ref{fig:Jz-contour}, their computational cost (CPU hours) ratio is about $3:8:1$ among the adaptive-PIC region case, the fixed-PIC region case, and the full-PIC case. Given that FLEKS is a semi-implicit PIC code and the gird resolution between the PIC and MHD regions can be different, one can reduce the grid resolution of the PIC cells inside the MHD-AEPIC domain, which can lower the computational cost of the MHD-AEPIC run and thus the MHD-AEPIC run becomes computationally cheaper than the full-PIC case (see Appendix A). It is noteworthy that in three-dimensional (3-D) cases, MHD-AEPIC can be computationally even more efficient than a full PIC code. \n\n\\begin{figure\n\\includegraphics[width=0.5\\textwidth]{Fig3.pdf}\n\\caption{\\label{fig:ER} Reconnection rate as a function of time for the adaptive and fixed PIC region runs and the full-PIC run with $\\lambda=5 d_{i0}$.}\n\\end{figure}\n\n\\begin{figure\n\\includegraphics[width=0.5\\textwidth]{Fig4.pdf}\n\\caption{\\label{fig:O-sep} O-point separation of the coalescing magnetic islands as a function of time for the adaptive and fixed PIC region runs and the full-PIC run with $\\lambda=5 d_{i0}$.}\n\\end{figure}\n\nAn inspection of Fig.~\\ref{fig:Jz-contour} reveals that the simulation results (such as the current density and the magnetic flux) from three different runs present nearly identical features and the system evolves at the same rate, indicating the reconnection rates, $E_R$, from the three simulations are very similar. In order to verify this idea, we explicitly compare the reconnection rates. We normalize the reconnection rate to the maximum initial magnetic field between the islands such that \\cite{Stanier2015}\n\\begin{eqnarray}\nE_R = \\frac{1}{v_{Am}B_m} \\frac{\\partial}{\\partial t} \\big( A_{zX} - A_{zO} \\big)\n\\label{eq:three}\n\\end{eqnarray}\nwhere $v_{Am} = B_m\/(\\mu_0 n_0 m_i)^{1\/2}$, $A_{zX}$ is the out-of-plane magnetic vector potential, evaluated at the $X$-point and $A_{zO}$ is evaluated at the $O$-point. The time evolution of the reconnection rate for the three simulations are plotted in Fig.~\\ref{fig:ER}, where a maximum reconnection rate occurs around $t = 0.9 t_A$ for all the cases. After the reconnection rate reaches a maximum, it decreases with time before reaching a constant value at approximately $t = 1.6 t_A$. We note from Fig.~\\ref{fig:ER} that the case with adaptive PIC regions closely captures the behavior of $E_R$ for this problem, when compared to the full-PIC simulation.\n\n\nSince the O-point separation between the magnetic islands is largely dependent on reconnection rates, we demonstrate the O-point separation as a function of time in Fig.~\\ref{fig:O-sep}. In this plot, we have normalized the O-point separation by the initial separation between the islands $L_0$. It is evident in Fig.~\\ref{fig:O-sep} that for all three cases the island separation decreases at an increasing rate before $t = 0.9 t_A$, at which point the coalescence of the islands slows down, reaching an approximately constant rate at $t = 1.6 t_A$. Both Figs.~\\ref{fig:ER} and \\ref{fig:O-sep} show excellent agreement among the three simulation approaches, validating the use of the MHD-AEPIC model. \n\n\nAt this point, we are interested in analyzing the effects of ion kinetics, as its importance has been demonstrated previously \\cite{Stanier2015}. \\citet{Stanier2015} also pointed out that electron kinetics were not crucial for the island coalescence problem. To do this, we write the z-component of the normalized ion Ohm's law as \n\\begin{eqnarray}\nE_z^\\prime &=& \\frac{d_i}{n} \\bigg[ \\frac{\\partial}{\\partial t} (nv_{iz}) + \\nabla \\cdot (n\\textbf{v}_i v_{iz}) \\bigg] + \\frac{d_i}{n} \\nabla \\cdot \\big( \\bar{\\bar{\\textbf{P}}}_i \\cdot \\textbf{z} \\big)\n\\label{eq:four}\n\\end{eqnarray}\nwhere $E_z^\\prime = (\\textbf{E} + \\textbf{v}_i \\times \\textbf{B}) \\cdot \\textbf{z}$ is the z-component of the nonideal electric field, $\\bar{\\bar{\\textbf{P}}}_i$ is the ion pressure tensor, and the resistivity is neglected for collisionless reconnection. Near the X-point of the reconnection site, where the magnetic field is small, pressure tensor effects become important. Specifically, Eq.(\\ref{eq:four}) demonstrates that the non-ideal electric field is heavily influenced by the off-diagonal terms of the pressure tensor. Since it has been shown in \\citet{Stanier2015} that ion kinetic effects are of importance in this specific reconnection setup, we focus our analysis on the ion pressure tensor. Fig.~\\ref{fig:fig4} compares the off-diagonal ion pressure tensor elements from the three cases when the reconnection rate reaches its peak, once again showing spectacular agreement among the cases. Fig.~\\ref{fig:fig4} also demonstrates that these two islands should be covered by the PIC code, since the off-diagonal pressure tensor can not be described by the isotropic MHD used here. In general, if the ion pressure is anisotropic, but the off-diagonal terms are relatively small in the local magnetic field coordinate system, MHD-AEPIC supports coupling with an anisotropic MHD model \\cite{Daldorff2014}, and the region that requires PIC can be reduced. However, if the off-diagonal terms are prominent in the local magnetic field coordinates, which is the case here (see Appendix B), the PIC regions need to cover those areas, consistent with the previous study using a similar approach \\cite{Makwana2018}.\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{Fig5.pdf}\n\\caption{\\label{fig:fig4} Comparison of the off-diagonal ion pressure tensor elements at $t = 0.9 t_A$ for the adaptive (top row) and fixed (middle row) PIC region runs and the full-PIC run (bottom row) with $\\lambda=5 d_{i0}$. The left column plots $P_{xy}$, the middle column shows $P_{xz}$, and $P_{yz}$ is illustrated in the right column.}\n\\end{figure*}\n\nIn addition to the off-diagonal ion pressure tensor terms, we also compare the ion agyrotropies of the three simulations, employing the same formula for agyrotropy as in \\citet{Scudder2008}. We note that agyrotropy is a measure useful for characterizing the ion (or electron) diffusion region in collisionless magnetic reconnection. We once again focus only on ion agyrotropy as ion kinetics are required to capture the correct reconnection rates and describe the global behavior of the system for the island coalescence problem \\cite{Stanier2015}. A comparison of the agyrotropy from the three different simulations is shown in Fig.~\\ref{fig:agyrotropy}, indicating excellent agreement among different cases, especially near the current sheet formation where ion agyrotropy is of particular significance (indicated by the lighter colors inside the ion diffusion region). \n\\begin{figure\n\\includegraphics[width=0.5\\textwidth]{Fig6.pdf}\n\\caption{\\label{fig:agyrotropy} Ion agyrotropy at $t = 0.9 t_A$ for the adaptive (top) and fixed (middle) PIC region runs and the full-PIC run (bottom) with $\\lambda=5 d_{i0}$, overlaid with the magnetic vector potential contours (in white).}\n\\end{figure}\n\nIn Fig.~\\ref{fig:lambda10_rate}, we also present the reconnection rates for simulations with $\\lambda = 10 d_{i0}$ to demonstrate that reconnection in large systems becomes slower. The islands are unable to coalesce on the first approach due to the slower reconnection rates, and so bounce off each other \\cite{Karimabadi2011_1,Stanier2015}. All simulations reach nearly the same maximum reconnection rate and show two peaks around $t=0.9t_A$ and $t=1.2t_A$, respectively. Overall, all simulations show similar reconnection processes.\n\n\\section{Conclusion}\n\nIt has been shown that simulating collisionless magnetic reconnection requires kinetic approaches such as the PIC method. However, these simulation codes are generally computationally expensive and hence make it difficult to efficiently model large-scale global systems. In this study, we have presented an alternative method by utilizing the MHD-AEPIC model that uses the PIC treatment in regions where the current sheet formation is prominent and uses the computationally cheap MHD model in the rest of the simulation domain. We have shown that the case with adaptive PIC regions comes in close agreement with the full PIC simulation when analyzing reconnection rates and O-point separations as well as the ion pressure tensor elements and ion agyrotropy. These results demonstrate that the MHD-AEPIC model may accurately and efficiently simulate large-scale systems that involve collisionless reconnection physics. \n\nIt should be noted that the magnetic island coalescence problem that we studied here is highly kinetic in nature where the coupling between the MHD and kinetic scales is important \\cite{Daughton2009_2,Stanier2015,Ng2019}, so relatively large PIC regions are needed within the MHD domain. For large-scale simulations such as the case of the Earth's magnetosphere, \\citet{Chen2021} applied MHD-AEPIC to study the magnetopause reconnection while treating the magnetotail reconnection using Hall MHD, such that they drastically saved computational costs compared with the full PIC magnetosphere simulations.\n\nWe also note that in magnetic confinement fusion, near the X-point where the magnetic field is weak, gyrokinetic approximations begin to break down and it becomes necessary to include full kinetic physics at these locations. The MHD-AEPIC model may be useful for simulating FLARE \\cite{Ji2020} and NSTX-U \\cite{Ebrahimi2016} at Princeton Plasma Physics Laboratory, MAST at the UK \\cite{Tanabe2017} as well as TREX at Wisconsin Plasma Physics Laboratory \\cite{Olson2021}.\n\n\\begin{figure\n\\includegraphics[width=0.5\\textwidth]{Fig7.pdf}\n\\caption{\\label{fig:lambda10_rate} Reconnection rate as a function of time for the adaptive and fixed PIC region runs and the full-PIC run with $\\lambda=10 d_{i0}$.}\n\\end{figure}\n\n\\begin{figure\n\\includegraphics[width=0.5\\textwidth]{Fig8.pdf}\n\\caption{\\label{fig:lambda5_halfres_boundaries} Out-of-plane current density at $t = 0.9 t_A$ for the adaptive PIC region run (top), the adaptive PIC region run with half PIC grid resolution (middle), and the full-PIC run (bottom) with $\\lambda=5 d_{i0}$, overlaid with the magnetic vector potential contours (black dotted curves) and with the boundary between the MHD and PIC regions (green solid curves).}\n\\end{figure}\n\n\\begin{figure}[b\n\\includegraphics[width=0.5\\textwidth]{Fig9.pdf}\n\\caption{\\label{fig:lambda5_halfres_rate} Reconnection rate as a function of time for the three cases shown in Fig.\\ref{fig:lambda5_halfres_boundaries}.}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{Fig10.pdf}\n\\caption{\\label{fig:fieldaligned} Comparison of the off-diagonal ion pressure tensor elements at $t = 0.9 t_A$ for the adaptive (top row) and fixed (middle row) PIC region runs and the full-PIC run (bottom row) in field-aligned coordinates with $\\lambda = 5 d_{i0}$. The left column plots $P_{12}^\\prime$, the middle column shows $P_{13}^\\prime$, and $P_{23}^\\prime$ is illustrated in the right column.}\n\\end{figure*}\n\n\\begin{acknowledgments}\n\nThis work was made possible by support from the Department of Energy for the Summer Undergraduate Laboratory Internship (SULI) program. This work was supported by the U.S. Department of Energy under contract number DE-AC02-09CH11466 (through LDRD) and the DOE grant DE-SC0021205, the NASA grants 80NSSC21K1326 and 80NSSC22K0323, and the NSF grant AGS-2149787. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center. The authors acknowledge the OSIRIS Consortium, consisting of UCLA and IST (Portugal) for the use of the OSIRIS 4.0 framework.\n\n\\end{acknowledgments}\n\n\\section*{Data Availability}\nThe data that support the findings of this study are available from the corresponding author upon reasonable request.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\\vspace{-.5\\baselineskip}\nWe have performed extensive quantitative and qualitative evaluations of our newly proposed ViTCA on a variety of datasets under a denoising autoencoding framework. We have demonstrated the superior denoising performance and robustness of our model when compared to a U-Net-based CA baseline (UNetCA) and ViT, as well as its generalization capabilities under a variety of environmental changes such as larger inputs (\\emph{i.e}\\onedot, } \\def\\Ie{\\emph{I.e}\\onedot,  spatial interpolation) and changing inputs \\emph{during} cell updates.\n\nDespite the computation savings---owed to our circumvention of self-attention's quadratic complexity by spatially localizing it within ViTCA---there remains the same memory limitations inherent to all recurrent models: multiple recurrent iterations are required for each training iteration, resulting in larger memory usage than a feedforward approach. This limits single-GPU training accessibility. We have experimented with gradient checkpointing \\cite{chen2016training} but found its trade-off for increased backpropagation duration (and slightly different gradients) less than ideal. To fully realize the potential of NCAs (self-organization, inherent distributivity, \\emph{etc}\\onedot), we encourage follow-up work to address this limitation. Adapting recent techniques using implicit differentiation is one avenue to circumvent these issues~\\cite{bai2022deep,bai2019deep}. Also, as mentioned in our ablation (\\Sec{\\ref{subsubsec:ablation_study}}), we hope to further investigate the instabilities caused by increasing the depth of ViTCA.\n\n\n\n\n\n\n\n\n\\section{Background and related work}\n\\vspace{-.5\\baselineskip}\n\\label{sec:related_work}\n\n\\paragraph{Neural Cellular Automata.}\n\nCellular Automata are algorithmic processes motivated by the biological behaviours of cellular growth and, as such, are capable of producing complex emergent (global) dynamics from the iterative application of comparatively simple (localized) rules~\\cite{von1966theory}. \\emph{Neural} Cellular Automata present a more general CA formulation, where the evolving cell states are represented as (typically low-dimensional) vectors and the update rule dictating their evolution is a differentiable function whose parameters are learned through backpropagation from a loss, rather than a handcrafted set of rules~\\cite{mordvintsev2020growing,gilpin2019cellular,wulff1992learning}.\nNeural net-based formulations of CAs in the NeurIPS community can be traced back to the early work of \\cite{wulff1992learning}, where only small and simple models were examined. Recent formulations of NCAs have shown that when leveraging the power of deep learning techniques enabled by advances in hardware capabilities---namely highly-parallelizable differentiable operations implemented on GPUs---NCAs can be tuned to learn surprisingly complex desired behaviour, such as semantic segmentation \\cite{sandler2020image}; common RL tasks such as cart-pole balancing \\cite{variengien2021towards}, 3D locomotion \\cite{najarro2022hypernca}, and Atari game playing \\cite{najarro2022hypernca}; and image synthesis~\\cite{palm2022variational,niklasson2021self-organising,mordvintsev2021mu}. Although these recent formulations rely on familiar compositions of convolutions and non-linear functions, it is important to highlight that NCAs are fundamentally not equivalent to ``very-deep'' CNNs (\\emph{vs}\\onedot~\\cite{gilpin2019cellular}), or any other feedforward architecture (\\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  ResNets \\cite{he2016deep}), particularly, in the same way that a Recurrent Neural Network (RNN) is not equivalent: CNNs and other feedforward architectures induce a directed \\textit{acyclic} computation graph (\\emph{i.e}\\onedot, } \\def\\Ie{\\emph{I.e}\\onedot,  a finite impulse response), whereas NCAs (and RNNs) induce a directed \\textit{cyclic} computation graph (\\emph{i.e}\\onedot, } \\def\\Ie{\\emph{I.e}\\onedot,  an infinite impulse response), where stateful data can additionally be manipulated using (learned) feedback loops and\/or time-delayed controls. As such, NCAs can be viewed as a type of RNN, and both (N)CAs and RNNs are known to be Turing complete~\\cite{christen2021automatic,cook2004universality,siegelmann1995computational,wulff1992learning}.\\footnote{In the case of (N)CAs, a Turing complete example is the \\emph{Rule 110} elementary CA \\cite{christen2021automatic,cook2004universality}}\n\n\\paragraph{Vision Transformers.}\nVision Transformers \\cite{dosovitskiy2020image} are an adaptation of Transformers \\cite{vaswani2017attention} to vision-based tasks like image classification. In contrast to networks built from convolutional layers, ViTs rely on \\emph{self-attention} mechanisms operating on tokenized inputs. Specifically, input images are divided into non-overlapping patches, then fed to a Transformer after undergoing a linear patch projection with an embedding matrix. While ViTs provide competitive image classification performance, the quadratic computational scaling of global self-attention limits their applicability in high-dimensional domains, \\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  per-pixel dense processing. Recent developments have attempted to alleviate such efficiency limitations \\cite{ali2021xcit,hudson2021generative,arnab2021vivit,fan2021multiscale}, one notable example being Perceiver IO \\cite{jaegle2021perceiver,yifan2021input} with its use of cross-attention. We refer interested readers to a comprehensive survey on ViTs~\\cite{khan2021transformers}.\n\\section{Introduction}\n\\vspace{-.5\\baselineskip}\n\\label{sec:introduction}\n\n\\input{figures\/vitca_vs_vit} Recent developments at the intersection of two foundational ideas---Artificial Neural Networks (ANNs) and Cellular Automata (CA)---have led to new approaches for constructing Neural Cellular Automata (NCA). These advances have integrated ideas such as variational inference \\cite{palm2022variational}, U-Nets \\cite{zhang2020learning}, and Graph Neural Networks (GNNs) \\cite{grattarola2021learning} with promising results on problems ranging from image synthesis \\cite{palm2022variational,niklasson2021self-organising,mordvintsev2021mu} to Reinforcement Learning (RL) \\cite{najarro2022hypernca,variengien2021towards}. Transformers are another significant development in deep learning \\cite{vaswani2017attention}, but, until now, have not been examined under an NCA setting.\n\nVision Transformers (ViTs) \\cite{dosovitskiy2020image} have emerged as a competitive alternative to Convolutional Neural Network (CNN) \\cite{lecun1998gradient} architectures for computer vision, such as Residual Networks (ResNets) \\cite{he2016deep}. ViTs leverage the self-attention mechanisms of original Transformers \\cite{vaswani2017attention}, which have emerged as the dominant approach for sequence modelling in recent years. Our work combines foundational ideas from Transformers and ViTs, leading to a new class of NCAs: \\textbf{Vision Transformer Cellular Automata (ViTCA)}. \n\nAn effective and ubiquitous Transformer-based learning technique for Natural Language Processing (NLP) pre-training is the unsupervised task of Masked Language Modelling (MLM), popularized by the BERT language model \\cite{devlin-etal-2019-bert}. The success of MLM-based techniques has similarly inspired recent work re-examining the classical formulation of Denoising Autoencoders (DAEs) \\cite{vincent2010stacked}, but for ViTs \\cite{bao2022beit,dosovitskiy2020image,chen2020generative}, introducing tasks such as Masked Image Encoding \\cite{he2021masked} and Masked Feature Prediction \\cite{wei2021masked} for image and video modelling, respectively. This simple yet highly-scalable strategy of masked-based unsupervised pre-training has yielded promising transfer learning results on vision-based downstream tasks such as object detection and segmentation, image classification, and action detection, even outperforming supervised pre-training~\\cite{he2021masked,wei2021masked}. We examine training methodologies for ViTCA within a DAE setting and perform extensive controlled experiments benchmarking these formulations against modern state of the art architectures, with favourable outcomes, \\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  \\Fig{\\ref{fig:vitca_vs_vit}}.\n\n\\input{figures\/teaser}\n\nOur contributions are as follows: \\textit{first}---to the best of our knowledge---our work is the first to extend NCA methodologies with key Transformer mechanisms, \\emph{i.e}\\onedot, } \\def\\Ie{\\emph{I.e}\\onedot,  self-attention and positional encoding (and embedding), with the beneficial side-effect of circumventing the quadratic complexity of self-attention; \\textit{second}, our ViTCA formulation allows for lower model complexity (by limiting ViT depth) while retaining expressivity through CA iterations on a controlled state---all with the same encoder weights. This yields a demonstrably more parameter-efficient \\cite{mordvintsev2021mu} ViT-based model. Importantly, ViTCA mitigates the problems associated with the explicit tuning of ViT depth originally needed to improve performance (\\emph{i.e}\\onedot, } \\def\\Ie{\\emph{I.e}\\onedot,  we use a depth of 1). With ViTCA, we simply iterate until cell state convergence. Since ViT (and by extension, ViTCA) employs Layer Normalization (LN) \\cite{ba2016layer} at each stage of its processing, it is a fairly contractive model capable of fixed-point convergence guarantees~\\cite{bai2019deep}.\n\nIn relation to our first contribution, ViTCA respects CA requirements, most importantly that computations remain localized about a cell and its neighbourhood. As such, we modify the global self-attention mechanisms of a ViT to respect this locality requirement (\\Fig{\\ref{fig:teaser}}). Localized self-attention is not a new idea \\cite{chen2022regionvit,liu2021swin,chu2021twins,zhang2021multi}; however, because cells contain state information that depends on its previous state, over CA iterations the effective receptive field of ViTCA's localized self-attention grows increasingly larger until eventually incorporating information implicitly across all cells. Thus, admitting global propagation of information from spatially localized self-attention. Moreover, due to the self-organizing nature of NCAs, self-organization also manifests itself within the localized self-attention, resulting in a globally agreed-upon arrangement of local self-attention. Thus, circumventing the quadratic complexity of explicit global self-attention (w.r.t\\onedot the input size) through a linear amortization over time, and increasing the feasibility of per-pixel dense processing (as we demonstrate). This globally consistent and complex behaviour, which arises from strictly local interactions, is a unique feature of NCAs and confers performance benefits which we observe both qualitatively and quantitatively when comparing ViT and ViTCA for denoising autoencoding. \n\n\\input{figures\/overview}\n\n\n\n\n\n\n\n\\section*{Checklist}\n\\begin{enumerate}\n\n\\item For all authors...\n\\begin{enumerate}\n  \\item Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?\n    \\answerYes\n  \\item Did you describe the limitations of your work?\n    \\answerYes{See \\Sec{\\ref{sec:discussion}}.}\n  \\item Did you discuss any potential negative societal impacts of your work?\n    \\answerNo{Although we feel our work demonstrates the potential of NCAs as viable alternatives to common recurrent network architectures (ViTCA being our evidential contribution), our experiments intentionally tend towards the direction of optimizing model efficiency (and single-GPU training accessibility) rather than towards the increasingly popular direction of scaling upwards. However, as much as our work demonstrates the downward-scaling capabilities of NCAs, we also acknowledge that this similarly applies going upward, and as such, can be abused (\\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  creating a ``deepfake''-capable ViTCA).}\n  \\item Have you read the ethics review guidelines and ensured that your paper conforms to them?\n    \\answerYes\n\\end{enumerate}\n\n\\item If you are including theoretical results...\n\\begin{enumerate}\n  \\item Did you state the full set of assumptions of all theoretical results\n    \\answerNA{}\n        \\item Did you include complete proofs of all theoretical results?\n    \\answerNA{}\n\\end{enumerate}\n\n\\item If you ran experiments...\n\\begin{enumerate}\n  \\item Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?\n    \\answerYes{Code and instructions to reproduce results are included in the supplemental material.}\n  \\item Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?\n    \\answerYes{}\n        \\item Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?\n    \\answerNo{Given the combination of time and computational restrictions and our exhaustive list of experiments, we opted to prioritize experiment variety and dataset coverage as an implicit substitute for re-running experiments under different random seeds. For all experiments, we kept a fixed random seed, even pointing out (deterministic) differences caused by gradient checkpointing when used (see Appendix \\ref{sec:appendix}).}\n        \\item Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?\n    \\answerYes\n\\end{enumerate}\n\n\\item If you are using existing assets (e.g., code, data, models) or curating\/releasing new assets...\n\\begin{enumerate}\n  \\item If your work uses existing assets, did you cite the creators?\n    \\answerYes\n  \\item Did you mention the license of the assets?\n    \\answerNA{Licensed frameworks used such as PyTorch (BSD-style) and Hydra (MIT) will be mentioned in acknowledgements.}\n  \\item Did you include any new assets either in the supplemental material or as a URL?\n    \\answerNo{No new assets---aside from code and training our models---were created for the purposes of this work.}\n  \\item Did you discuss whether and how consent was obtained from people whose data you're using\/curating?\n    \\answerNo{We used publicly available datasets.}\n  \\item Did you discuss whether the data you are using\/curating contains personally identifiable information or offensive content?\n    \\answerNo{Although not discussed in the manuscript, we would like to point that the datasets we used that could potentially contain personally identifiable information (CelebA, CIFAR10, Tiny ImageNet) each have restrictions and\/or acknowledgements of such potential issues. Also, our work is not focused on classifying persons and ViTCA is not a generative model, \\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  it can not generate new faces.}\n\\end{enumerate}\n\n\\item If you used crowdsourcing or conducted research with human subjects...\n\\begin{enumerate}\n  \\item Did you include the full text of instructions given to participants and screenshots, if applicable?\n    \\answerNA\n  \\item Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?\n    \\answerNA\n  \\item Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?\n    \\answerNA\n\\end{enumerate}\n\n\\end{enumerate}\n\\section{Experiments}\n\\vspace{-.5\\baselineskip}\n\\label{sec:experiments}\n\nHere we examine ViTCA through extensive experiments. We begin with experiments for denoising autoencoding, then an ablation study followed by various qualitative analyses, before concluding with linear probing experiments on the learned representations for MNIST \\cite{deng2012mnist}, FashionMNIST \\cite{xiao2017fashion}, and CIFAR10 \\cite{krizhevsky2009learning}. We provide an extension to our experiments in Appendix \\ref{sec:appendix}.\n\n\\paragraph{Baseline models and variants.}\nSince we are performing pixel level reconstructions, we create a ViT baseline in which the class token has been removed. This applies identically for ViTCA. Unless otherwise stated, for our ViT and ViTCA models we use a patch size of $1\\!\\times\\!1$ ($P_H\\!=\\!P_W\\!=\\!1$), and only a single encoding block with $h\\!=\\!4$ \\mhsa\\ heads, \\embed\\ size $d\\!=\\!128$, and \\mlp\\ size of $128$. For ViTCA, we choose $N_H\\!=\\!3$ and $N_W\\!=\\!3$ (\\emph{i.e}\\onedot, } \\def\\Ie{\\emph{I.e}\\onedot,  the \\emph{Moore neighbourhood}~\\cite{weissteinmoore}). We also compare with a U-Net baseline similar to the original formulation \\cite{ronneberger2015u}, but based on the specific architecture from \\cite{lehtinen2018noise2noise}. Since most of our datasets consist of $32\\!\\times\\!32$ (resampled) images, we only have two downsampling steps as opposed to five. We implement a U-Net-based CA (UNetCA) baseline consisting of a modified version of our U-Net with 48 initial output feature maps as opposed to 24 and with all convolutions except the first changed to $1\\!\\times\\!1$ to respect typical NCA restrictions~\\cite{palm2022variational,mordvintsev2020growing}.\n\n\\vspace{-.5\\baselineskip}\n\\subsection{Denoising autoencoding}\n\\vspace{-.5\\baselineskip}\n\\label{subsec:denoising_autoencoding}\n\nWe compare between our baseline models and a number of ViTCA variants in the context of denoising autoencoding. We present test set results across six benchmark datasets: a land cover classification dataset intended for representation learning (LandCoverRep) \\cite{yeh2021sustainbench}, MNIST, CelebA \\cite{liu2015faceattributes}, FashionMNIST, CIFAR10, and Tiny ImageNet (a subset of ImageNet \\cite{russakovsky2015imagenet}). All datasets consist of $32\\!\\times\\!32$ resampled images except Tiny ImageNet, which is at $64\\!\\times\\!64$ resolution. During testing, we use all masking combinations, chosen in a fixed order, and we update cells using a fixed number of iterations ($T\\!=\\!64$). See \\Tab{\\ref{tab:denoising_results}} for quantitative results.\n\nBriefly mentioned in \\Sec{\\ref{subsec:update_rule}}, we employ a masking strategy inspired by Curriculum Learning (CL) \\cite{wang2021survey,bengio2009curriculum} to ease training. This schedule follows a geometric progression of difficulty---tied to training iterations---maxing out at 10K training iterations. Specifically, masking starts at covering 25\\% of the input with $1\\!\\times\\!1$ patches of noise (dropout for RGB inputs, Gaussian for grayscale), then at each shift in difficulty, new masking configurations are added to the list of available masking configurations in the following order: $(2^0\\!\\times\\!2^0, 50\\%), (2^0\\!\\times\\!2^0, 75\\%), (2^1\\!\\times\\!2^1, 25\\%), (2^1\\!\\times\\!2^1, 50\\%), (2^1\\!\\times\\!2^1, 75\\%), ..., (2^2\\!\\times\\!2^2, 75\\%)$. Masking configurations are randomly chosen from this list.\n\nWe initialize weights\/parameters using He initialization \\cite{he2015delving}, except for the final layer of CA-based models, which are initialized to zero \\cite{mordvintsev2020growing}. Unless otherwise stated, we train for $I\\!=\\!100$K iterations, use a minibatch size $b\\!=\\!32$, AdamW optimizer \\cite{loshchilov2019decoupled}, learning rate $\\eta\\!=\\!10^{-3}$ with a cosine annealing schedule \\cite{loshchilov2017sgdr}, pool size \\poolsize\\ $\\!=\\!1024$, and cell hidden channel size $C_h\\!=\\!32$. In the case of Tiny ImageNet, $b\\!=\\!8$ to accommodate training on a single GPU (48GB Quadro RTX 8000). Training typically lasts a day at most, depending on the model. Due to the recurrent iterations required per training step, CA-based models take the longest to train. To alleviate memory limitations for some of our experiments, we use gradient checkpointing \\cite{chen2016training} during CA iterations at the cost of backpropagation duration and slight variations in gradients due to its effect on round-off propagation. We also experiment with a cell fusion and mitosis scheme as an alternative. See Appendix \\ref{sec:appendix} for details on runtime performance, gradient checkpointing, and fusion and mitosis.\n\n\\input{tables\/denoising_results}\n\nAmongst baselines, ViTCA outperforms on most metrics across the majority of datasets used (10 out of 18). Exceptions include LandCoverRep, where UNetCA universally outperforms by a small margin, likely due to the texture-dominant imagery being amenable to convolutions. Notably, ViTCA strongly outperforms on MNIST. Although MNIST is a trivial dataset for common tasks such as classification, our masking\/noise strategy turns it into a challenging dataset for denoising autoencoding, \\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  it is difficult for even a human to classify a $32\\!\\times\\!32$ MNIST digit 75\\% corrupted by $4\\!\\times\\!4$ patches of Gaussian noise. We hypothesize that when compared to convolutional models, ViTCA's weaker inductive biases (owed to attention \\cite{yifan2021input,jaegle2021perceiver}) immediately outperform these models when there are large regions lacking useful features, \\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  MNIST digits cover a small space in the canvas. This is not the case with FashionMNIST, where the content is more filled out. Between baselines and ViTCA variants, ViTCA-32 (32 heads) and 32xy (xy-coordinate positional encoding) outperform all models by large margins, demonstrating the benefits of multi-head self-attention. We also experiment with a parameter-reduced (by $\\sim\\!60\\%$), inverted bottleneck variant where $d\\!=\\!64$ and \\mlp\\ size is 256, often with a minimal reduction in performance.\n\\vspace{-.5\\baselineskip}\n\\subsubsection{Ablation study}\n\\label{subsubsec:ablation_study}\n\\vspace{-.5\\baselineskip}\n\nIn \\Tab{\\ref{tab:ablation_results}} we perform an ablation study using the baseline ViTCA model above as reference on CelebA. Results are ordered in row-wise blocks, top-to-bottom. Specifically, we examine the impact of varying the cell hidden size $C_h$; the \\embed\\ size $d$; the number of \\mhsa\\ heads $h$; the depth (\\# encoders), comparing both ViTCA (used throughout the table) with ViT; and in the last block we examine the impact of various methods of incorporating positional information into the model.\n\n\\input{tables\/ablation_results}\n\nSpecifically, we examine the use of: (1) a xy-coordinate-based positional encoding \\emph{concatenated} (``injected'') to cells, and; (2) a Transformer-based positional encoding (or embedding, if learned) \\emph{added} into \\embed. These two categories are subdivided into: \n(1a) sincos5---consisting of handcrafted Fourier features \\cite{mildenhall2020nerf} with four doublings of a base frequency, \\emph{i.e}\\onedot, } \\def\\Ie{\\emph{I.e}\\onedot,  \\pe\\ $\\!=\\!(\\sin{2^0\\pi p},$ $\\cos{2^0\\pi p},$ $...,\\sin{2^{J-1}\\pi p},$ $\\cos{2^{J-1}\\pi p})\\!\\in\\mathbb{R}^{N \\times (4JP_HP_W)}$ where $J\\!=\\!5$ and $p$ is the pixel coordinate (normalized to [-1,1]) for each pixel the cell is situated on (one pixel since $P_H\\!=\\!P_W\\!=\\!1$); \n(1b) sincos5xy---consisting of both Fourier features and explicit xy-coordinates concatenated; \n(1c) xy---only xy-coordinates; \n(2a) handcrafted (our baseline approach)---sinusoidal encoding \\pe\\ $\\!\\in\\!\\mathbb{R}^{N \\times d}$ similar to (1a) but following a Transformer-based approach \\cite{vaswani2017attention}, and; \n(2b) learned---learned embedding \\pe\\ $\\!\\in\\!\\mathbb{R}^{N \\times d}$ following the original ViT approach \\cite{dosovitskiy2020image}. \nTo further test the self-organizing capabilities of ViTCA, we also include: (3) none---no explicit positioning provided, where we let the cells localize themselves.\n\nAs shown in \\Tab{\\ref{tab:ablation_results}}, ViTCA benefits from an increase to most CA and Transformer-centric parameters, at the cost of computational complexity and\/or an increase in parameter count. A noticeable decrease in performance is observed when \\embed\\ size $d\\!=\\!512$, most likely due to the vast increase in parameter count necessitating more training. In the original ViT, multiple encoding blocks were needed before the model could exhibit performance equivalent to their baseline CNN \\cite{dosovitskiy2020image}, as verified in our ablation with our ViT. However, for ViTCA we notice an inverse relationship of the effect of Transformer depth, causing a divergence in cell state. It is not clear why this is the case, as we have observed that the LN layers and overflow losses otherwise encourage a contractive $F_\\theta$. This is an investigation we leave for future work. Despite the benefits of increasing $h$, we use $h\\!=\\!4$ for our baseline to optimize runtime performance. Finally, we show that ViTCA does not dramatically suffer when no explicit positioning is used---in contrast to typical Transformer-based models---as cells are still able to localize themselves by relying on their stored hidden information.\n\n\\vspace{-.5\\baselineskip}\n\\subsubsection{Cell state analysis}\n\\label{subsubsec:cell_state_analysis}\n\\vspace{-.5\\baselineskip}\n\nHere we provide an empirically-based qualitative analysis on the effects ViTCA and UNetCA have on cell states through several experiments with our pre-trained models (\\Fig{\\ref{fig:analysis} (a,b,c)}). We notice that in general, ViTCA indefinitely maintains cell state stability while UNetCA typically induces a divergence past a certain point. An extended analysis is available in Appendix \\ref{subsec:extended_analysis}.\n\n\\textbf{Damage resilience.}\nShown in \\Fig{\\ref{fig:analysis}} (a), we damage a random $H\/2\\!\\times\\!W\/2$ patch of cells with random values $\\sim\\!\\mathcal{U}(-1,1)$ twice in succession. ViTCA is able to maintain cell stability despite not being trained to deal with such noise, while UNetCA induces a divergence. Note both models are simultaneously performing the typical denoising task. We also note that ViTCA's inherent damage resilience is in contrast to recent NCA formulations that required explicit training for it \\cite{palm2022variational,mordvintsev2020growing}.\n\n\\textbf{Convergence stability.}\n\\Fig{\\ref{fig:analysis}} (b) shows denoising results after 2784 cell grid updates. ViTCA is able to maintain a stable cell grid state while UNetCA causes cells to diverge.\n\n\\textbf{Hidden state visualizations.}\n\\Fig{\\ref{fig:analysis}} (c) shows 2D and 3D PCA dimensionality reductions on the hidden states of converged cell grids for all examples in FashionMNIST~\\cite{xiao2017fashion}. The clusters suggest some linear separability in the learned representation, motivating our probing experiments in \\Sec{\\ref{subsec:probing}}. \n\n\\input{figures\/analysis}\n\n\\subsubsection{Investigating update rule inductive biases}\n\\label{subsubsec:inductive_bias_investigation}\n\\vspace{-.5\\baselineskip}\n\nHere we investigate the inductive biases inherent in ViTCA and UNetCA by testing their adaptation to various environmental changes (\\Fig{\\ref{fig:analysis}} (d,e,f,g,h)).\n\n\\textbf{Adaptation to varying update rates.}\nDespite being trained with a $\\sigma\\!=\\!50\\%$ cell update rate, ViTCA is able to adapt to varying rates (\\Fig{\\ref{fig:analysis}} (d)). Higher rates result in a proportionally faster rate of cell state convergence, and equivalently with lower rates. UNetCA exhibits a similar relationship, although is unstable at $\\sigma\\!=\\!100\\%$ (see Appendix \\ref{subsec:extended_analysis}). For details comparing training with a synchronous \\emph{vs}\\onedot asynchronous cell grid update, see Appendix \\ref{subsec:extended_ablation}.\n\n\\textbf{Generalization to noise unseen during training.}\nViTCA is capable of denoising configurations of noise it has not been trained on. \\Fig{\\ref{fig:analysis}} (e; \\emph{left-to-right}): $4\\!\\times\\!1$ and $1\\!\\times\\!4$ patches of Gaussian noise at 65\\% coverage. In contrast, UNetCA induces a cell state divergence (see Appendix \\ref{subsec:extended_analysis}).\n\n\\textbf{Adaptation to changing inputs.}\nAt various moments during cell updates, we re-inject cells with new masked inputs (\\Fig{\\ref{fig:analysis}} (f)). ViTCA is able to consistently adapt cells to new inputs while UNetCA experiences difficulty past a certain point (\\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  at 464 iterations in the figure).\n\n\\textbf{Effects of not \\emph{vs}\\onedot completely masking input.}\n\\Fig{\\ref{fig:analysis}} (g; \\emph{left}): ViTCA is able to perform autoencoding despite not being trained for it. UNetCA induces a cell grid divergence (see Appendix \\ref{subsec:extended_analysis}). \\Fig{\\ref{fig:analysis}} (g; \\emph{right}): Interestingly, when the input is completely masked, ViTCA outputs the median image~\\cite{lehtinen2018noise2noise}. UNetCA does not exhibit such behaviour and instead causes cells to diverge (see Appendix \\ref{subsec:extended_analysis}). \n\n\\textbf{Spatial interpolation.}\nWe use ViTCA models trained at $32\\!\\times\\!32$ using various types of positioning to generate $128\\!\\times\\!128$ outputs during inference, assuming an identical cell grid resolution. \\Fig{\\ref{fig:analysis}} (h; \\emph{top-to-bottom of \\textcolor{vitcaPurple}{outputs}}): xy-coordinates, no positioning, Fourier features \\cite{mildenhall2020nerf}, Fourier features concatenated with xy-coordinates, and a Transformer-based handcrafted positional encoding (baseline) \\cite{vaswani2017attention}. Results are ordered from best to worst. The baseline approach is not capable of spatial interpolation due to being a 1D positioning, while, as expected, the 2D encodings make it capable. Surprisingly, removing Fourier features and using only xy-coordinates results in a higher fidelity interpolation. We believe this to be caused by the distracting amount of positional information Fourier features provide to cells, as cells can instead rely on their hidden states to store higher frequency positional information. Finally, with no explicit positioning, ViTCA is still able to perform high-quality interpolation---even exceeding using Fourier features---by taking advantage of its self-organizing nature. As a side note, we point attention to the fact that ViTCA is simultaneously denoising at a scale space it has not been trained on, exemplifying its generalization capabilities.\n\n\\vspace{-.5\\baselineskip}\n\\subsection{Investigating hidden representations via linear probes}\n\\vspace{-.5\\baselineskip}\n\\label{subsec:probing}\n\n\\input{tables\/linear_probe_results}\nHere we examine the learned representations of our models pre-trained for denoising.  We freeze model parameters and learn linear classifiers on each of their learned representations: converged cell hidden states for CA-based models, bottleneck features for U-Net, and LN'd tokens for ViT. This is a common approach used to probe learned representations~\\cite{chen2020generative}. Classification results on MNIST, FashionMNIST, and CIFAR10 are shown in \\Tab{\\ref{tab:linear_probe_results}} and we use the same training setup as for denoising, but without any noise. For comparison, we also provide results using a linear classifier and two 2-layer MLPs of varying complexity, all trained directly on raw pixel values. Correlations between denoising performance in \\Tab{\\ref{tab:denoising_results}} and classification performance in \\Tab{\\ref{tab:linear_probe_results}} can be observed. Linear classification accuracy on ViTCA-based features typically exceeds classification accuracy using other model-based features or raw pixel values, even outperforming the MLPs in most cases.\n\\section{Vision Transformer Cellular Automata (ViTCA)}\n\\vspace{-.5\\baselineskip}\n\\label{sec:vitca}\n\nBuilding upon NCAs and ViTs, we propose a new class of \\emph{attention-based} NCAs formed using a spatially localized---yet globally organized---self-attention scheme. We detail an instance of this class, ViTCA, by first reviewing its backbone ViT architecture before describing the ``pool sampling''-based training process for the ViTCA update rule (see overview in \\Fig{\\ref{fig:overview}}).\n\n\\paragraph{Input tokenization.}\n\nViT starts by dividing a $C_i\\!\\times\\!H\\!\\times\\!W$ input image \\gt\\ into $N$ non-overlapping $P_H\\!\\times\\!P_W$ patches ($16\\!\\times\\!16$ in the original work~\\cite{dosovitskiy2020image}), followed by a linear projection of the flattened image patches with an embedding matrix $\\mathbf{E} \\in \\mathbb{R}^{L \\times d}$ (\\Fig{\\ref{fig:overview}} \\embed), where $L\\!=\\!C_iP_HP_W$, to produce initial tokens \\pretokens\\ $\\!\\in\\!\\mathbb{R}^{N \\times d}$. Next, a handcrafted positional encoding \\cite{vaswani2017attention} or learned positional embedding \\pe\\ $\\in\\!\\mathbb{R}^{N \\times d}$ \\cite{dosovitskiy2020image} is added to tokens to encode positional information and break permutation invariance. Finally, a learnable class token is appended to the token sequence, resulting with \\tokens\\ $\\!\\in\\!\\mathbb{R}^{(N+1) \\times d}$. For the purposes of our task, we omit this token in all ViT-based models. In ViTCA, the input to the embedding is a flattened cell grid \\cells\\ $\\!\\in\\!\\mathbb{R}^{N \\times L}$ where $L\\!=\\!C_PP_HP_W+C_h$, $C_P\\!=\\!C_i+C_o+C_{\\peplain}$,\\, $C_h$ is the cell hidden size,\\, $C_o$ is the number of output image channels (one or three for grayscale or RGB), and $C_{\\peplain}$ is the positional encoding size when positional encoding is (optionally) concatenated to each cell rather than added to the tokens \\cite{mildenhall2020nerf}.\n\n\\paragraph{Multi-head self-attention (MHSA).}\n\nGiven a sequence of tokens \\tokens, self-attention estimates the relevance of one token to all others (\\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  which image patches are likely to appear together in an image) and aggregates this global information to update each token. This encodes each token in terms of global contextual information, and does so using three learned weight matrices: $\\mathbf{W}_Q\\!\\in\\!\\mathbb{R}^{d \\times d}$, $\\mathbf{W}_K\\!\\in\\!\\mathbb{R}^{d \\times d}$, and $\\mathbf{W}_V\\!\\in\\!\\mathbb{R}^{d \\times d}$. \\tokens\\ is projected onto these weight matrices to obtain Queries $\\mathbf{Q}\\!=\\!$ \\tokens$\\mathbf{W}_Q$, Keys $\\mathbf{K}\\!=\\!$ \\tokens$\\mathbf{W}_K$, and Values $\\mathbf{V}\\!=\\!$ \\tokens$\\mathbf{W}_V$. The self-attention layer output $\\sa\\!\\in\\!\\mathbb{R}^{N \\times d}$ is:\n\n\\input{equations\/sa}\n\n\\emph{Multi-head} self-attention employs many sets of weight matrices, $\\{\\mathbf{W}_{Q_i},$ $\\mathbf{W}_{K_i},$ $\\mathbf{W}_{V_i}\\!\\in\\!\\mathbb{R}^{d \\times (d\/h)}\\!\\mid$  $i\\!=\\!0,...,(h-1)\\}$. The outputs of $h$ self-attention \\emph{heads} are concatenated into $(\\sa_0,$ $...,$ $\\sa_{h-1})\\!\\in\\!\\mathbb{R}^{N \\times d}$ and projected onto a weight matrix $\\mathbf{W}\\!\\in\\!\\mathbb{R}^{d \\times d}$ to produce $\\mhsa\\!\\in\\!\\mathbb{R}^{N \\times d}$. Self-attention explicitly models global interactions and is more flexible than grid-based operators (\\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  convolutions) \\cite{perez2019turing,cordonnier2019relationship}, but its quadratic cost in time and memory limits its applicability to high resolution images.\n\n\\paragraph{Spatially localizing self-attention.}\n\nThe global nature of self-attention directly conflicts with the spatial locality constraint of CAs; in response, we limit the connectivity structure of the attention operation to each cell's neighbourhood. This can be accomplished by either masking each head's attention matrix ($\\mathbf{A}\\!=\\!\\texttt{softmax}(\\cdots) \\in \\mathbb{R}^{N \\times N}$ in Eq.\\ \\ref{eq:sa}) with a banded matrix representing local connectivity (\\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  \\Fig{\\ref{fig:overview}} \\localize), or more efficiently,\n\n\\input{equations\/localize_attn} Here, we assume top-left-to-bottom-right input flattening.\nInstead of explicitly computing the global self-attention matrix $\\mathbf{A}\\!\\in\\!\\mathbb{R}^{N \\times N}$ then masking it, this approach circumvents the $\\mathcal{O}(N^2d)$ computation in favour of an $\\mathcal{O}(N\\!M\\!d)$ alternative that indexes the necessary rows and columns \\emph{during} self-attention. The result is a localized self-attention matrix $\\mathbf{A}^{\\!\\star}\\!\\in\\!\\mathbb{R}^{N \\times M}$, where $M\\!=\\!N_HN_W\\!\\ll\\!N$. As we show in our experiments, ViTCA is still capable of global self-attention despite its localization, by leveraging stored state information across cells and their global self-organization during CA iterations (\\Fig{\\ref{fig:teaser}}).\n\nFollowing \\mhsa\\ is a multilayer perceptron (\\Fig{\\ref{fig:overview}} \\mlp) with two layers and a GELU non-linearity. We apply Layer Normalization (LN) \\cite{ba2016layer} before \\mhsa\\ and \\mlp, and residual connections afterwards, forming a single encoding block. We use an MLP head (\\Fig{\\ref{fig:overview}} \\mlphead) to decode to a desired output, with LN applied to its input, finalizing the ViTCA update rule $F_\\theta$. In our experiments, ViT's \\mlphead\\ decodes directly into an image output whereas ViTCA decodes into update vectors added to cells.\n\n\\vspace{-.5\\baselineskip}\n\\subsection{Update rule training procedure}\n\\vspace{-.5\\baselineskip}\n\\label{subsec:update_rule}\n\nTo train the ViTCA update rule, we follow a ``pool sampling''-based training process \\cite{palm2022variational,mordvintsev2020growing} along with a curriculum-based masking\/noise schedule when corrupting inputs. During odd training iterations, we uniformly initialize a minibatch of cells \\cells\\ $\\!=\\!(\\cellsplain_1,...,\\cellsplain_b)$ with constant values (0.5 for output channels, 0 for hidden---see Appendix \\ref{subsec:extended_ablation} for alternatives), then inject the masked input \\maskedinput\\ (see \\Sec{\\ref{subsec:denoising_autoencoding}}). After input injection, we asynchronously update cells ($\\sigma\\!=\\!50\\%$ update rate) using $F_\\theta$ for $T\\!\\sim\\!\\mathcal{U}\\{8,32\\}$ recurrent iterations. We retrieve output \\cellsoutput\\ from the cell grid and apply an $L_1$ loss against the ground truth \\gt. We also apply overflow losses to penalize cell output values outside of [0,1] and cell hidden values outside of [-1,1]. We use $L_2$ normalization on the gradient of each parameter in $\\theta$. After backpropagation, we append the updated cells and their ground truths to a pool \\pool\\ which we then shuffle and truncate up to the first \\poolsize\\ elements. During even training iterations, we retrieve a minibatch of cells and their ground truths from \\pool\\ and process them as above. This encourages $F_\\theta$ to guide cells towards a stable fixed-point. \\Alg{\\ref{alg:vitca_training}} in Appendix \\ref{sec:appendix} details this process.\n\\section{Appendix}\n\\label{sec:appendix}\n\n\\input{algorithms\/vitca_training}\n\n\n\\subsection{Training on high-resolution imagery with fusion and mitosis}\n\\label{subsec:high_resolution_training}\n\nAs an alternative to gradient checkpointing for reducing memory usage, we briefly experimented with a downsampling scheme inspired by cell fusion and mitosis when training on CelebA at $64\\!\\times\\!64$. Specifically, we split the $T$ applications of the update rule (within a training iteration) into multiple stages: 1) We apply the update rule twice so that cells will have, at minimum, some amount of knowledge of their neighbours. 2) We stash the masked input for a later re-injection. 3) \\emph{Fusion}---we apply a $2\\!\\times\\!2$ average pooling with a stride of 2 across the cell grid, combining $2\\!\\times\\!2$ groups of cells into singular cells. 4) We apply the update rule $T-4$ times at this $32\\!\\times\\!32$ downsampled cell grid resolution. 5) \\emph{Mitosis}---we perform a $2\\!\\times\\!2$ duplication of cells (each cell is duplicated to its right, bottom-right, and bottom). 6) We re-inject the stashed masked input. 7) We apply the update rule twice to adapt the cells to the $64\\!\\times\\!64$ resolution and to fill in any missing information.\n\nWe found that performing this fusion and mitosis scheme decreased training memory consumption to levels similar to our gradient checkpointing scheme ($\\sim\\!50\\%$ memory reduction) while having a $\\sim\\!70\\%$ faster backward pass. Loss-wise, we observed a $\\sim\\!33\\%$ increase in the average validation reconstruction loss during training, which can qualitatively be observed in the example provided in \\Fig{\\ref{fig:fusion_mitosis}} (\\emph{\\textcolor{unetcaBlue}{bottom}}). Although the results shown are not ideal---\\emph{i.e}\\onedot, } \\def\\Ie{\\emph{I.e}\\onedot,  we did not perform a hyper-parameter search here, for example, finding the optimal number of iterations preceding fusion and following mitosis---this brief experiment tests the feasibility of reducing memory consumption while maintaining denoising capability and avoiding gradient checkpointing. As shown in the figure, ViTCA with fusion and mitosis is able to successfully denoise the input despite applying updates at two different scales. This scale agnostic behaviour reveals potentially interesting research directions beyond the scope of this work, such as allowing an NCA update rule to dynamically and locally modify cell grid resolution based on a compute budget, which could see applications in signal (image, video, or audio) compression.\n\n\\begin{figure}[t]\n  \\begin{floatrow}\n    \\input{figures\/fusion_mitosis}\n    \\input{tables\/other_ablation_results}\n    \\vspace{-0.5\\baselineskip}\n  \\end{floatrow}\n\\end{figure}\n\n\\subsection{Extended ablation study}\n\\label{subsec:extended_ablation}\n\nHere we present an extension of our ablation study in \\Sec{\\ref{subsubsec:ablation_study}}, using the baseline ViTCA model as our reference. As before, the ablation examines the effects certain training configuration parameters have on test performance.\n\n\\input{tables\/attn_size_ablation_results}\n\\input{tables\/update_rate_ablation_results}\n\\input{tables\/gradient_checkpointing_ablation_results}\n\n\\paragraph{Pool size, cell initialization, and patch size.}\nIn \\Tab{\\ref{tab:other_ablation_results}}, we examine the impact of varying the (max) pool size $N_\\mathcal{P}$, cell initialization method, and patch size $P_H\\!\\times\\!P_W$ on CelebA. As shown in the table, it is difficult to correlate pool size with test performance. However, when pool size $N_\\mathcal{P}\\!=\\!8192$, there is a noticeable reduction in performance. Test performance also degrades when initializing cells such that their output and hidden channels receive random values sampled from $\\mathcal{U}(0,1)$ and $\\mathcal{U}(-1,1)$, respectively, as opposed to receiving constant values (0.5 for output channels and 0 for hidden). Finally, we see a consistent decrease in performance when the input image is divided into non-overlapping patches $> 1\\!\\times\\!1$, as well as an increase in the number of model parameters.\n\n\\paragraph{Attention neighbourhood size.}\nIn \\Tab{\\ref{tab:attn_size_ablation_results}}, we examine the impact of attention neighbourhood size $N_H\\!\\times\\!N_W$ on FashionMNIST. Interestingly, increasing the neighbourhood size past $3\\!\\times\\!3$ causes a degradation in performance. This is most likely attributed to the increase in complexity caused by incorporating more information into ViTCA's self-attention. One would expect explicitly increasing the receptive field of spatially localized self-attention to result in better performance, but it can also complicate the process of figuring out which neighbours to attend to. We believe this may be alleviated by increasing model capacity and\/or training duration. As described in \\Sec{\\ref{sec:experiments}}, we use the Moore neighbourhood ($3\\!\\times\\!3$) as it requires less computation while still demonstrating ViTCA's effectiveness.\n\n\\paragraph{Asynchronous \\emph{vs}\\onedot synchronous cell updates.}\nIn \\Tab{\\ref{tab:update_rate_ablation_results}}, we compare between training with asynchronous cell updates ($\\sigma\\!=\\!50\\%$) and training with synchronous cell updates ($\\sigma\\!=\\!100\\%$) on LandCoverRep, MNIST, CelebA, and FashionMNIST. Training with asynchronous cell updates provides a meaningful increase in performance compared to training with synchronous cell updates and comes with several benefits, such as not requiring cells in a neighbourhood to be in sync with each other and serving as additional data augmentation. Similarly mentioned in related work \\cite{niklasson2021self-organising}, this allows ViTCA to be used in a distributed system where cells need not exist under a global clock and can be updated at varying rates. Thus making it easier to scale up or down within a non-homogeneous compute environment. This was somewhat demonstrated in \\Fig{\\ref{fig:analysis}} (d) where ViTCA was able to adapt to varying update rates despite being trained on a fixed asynchronous update rate ($\\sigma\\!=\\!50\\%$).\n\n\\paragraph{Effects of gradient checkpointing.}\nIn \\Tab{\\ref{tab:gradient_checkpointing_ablation_results}}, we compare between training with gradient checkpointing disabled and with gradient checkpointing enabled on LandCoverRep, MNIST, CelebA, and FashionMNIST. Similarly shown in \\Tab{\\ref{tab:ablation_results}}, we see here that training with gradient checkpointing has an adverse effect on test performance. As mentioned in \\Sec{\\ref{sec:discussion}}, NCAs---during training---require all activations from each recurrent iteration to be stored in memory before performing backpropagation. This results in memory usage being proportional to the amount of recurrent iterations. As such, depending on ViTCA's configuration, gradient checkpointing may be required to be able to train on a single GPU. We make use of PyTorch's \\texttt{checkpoint\\_sequential}, which we use as follows: given the number of CA iterations $T$, we divide the sequential (forward) application of the update rule into $\\lfloor T\/2 \\rfloor$ segments of roughly the same length (depending on whether $T$ is even or odd). Then, all segments are executed in sequence, where activations from only the first and last segments are stored as well as the inputs to each intermediate segment. The intermediate inputs are used for re-running the segments without stored activations during the backward pass to compute gradients. This results in a trade-off between memory consumption and backpropagation duration since each intermediate segment's forward pass needs to be re-computed during its backward pass. Moreover, and not mentioned in the documentation of PyTorch at the time of writing, there exists a subtle yet meaningful side-effect which we have observed and confirmed through the use of GNU Debugger (GDB) and Python Debugger (PDB): Without gradient checkpointing, gradients are accumulated all at once at the end of backpropagating through the entire computation graph, resulting in the expected round-offs due to limitations in machine precision (\\texttt{float32} in our case). At this point, PyTorch may use a variety of numerical techniques to minimize round-off, such as cascade summation (verified to be used for CPU-based summation, see \\texttt{SumKernel.cpp} in PyTorch) which recursively sums two halves of a sequence of summands as opposed to naively summing them in sequence. \\emph{With} gradient checkpointing, gradients are accumulated at each segment. This means that round-offs are forced to (potentially) occur at each checkpoint\/segment instead of once at the end of the entire computation graph. Even if cascade summation is used when summing gradients within each segment, the segment-wise ordering may reduce its effectiveness. We verified this behaviour by observing an exact machine epsilon difference ($\\epsilon\\approx1.19\\!\\times\\!10^{-7}$ in IEEE 754 standard) in the gradient---when compared to the non-checkpointed scheme---of the final operation of the update rule at the second-last segment, once the loss started to diverge.\n\nIt is important to note that despite the difference in gradients, the accuracy of the forward pass remains unchanged between the checkpointed and non-checkpointed models. Also, we must remind ourselves that round-offs are unavoidable when performing floating-point arithmetic, meaning that gradients computed within a deep learning library such as PyTorch are always an \\emph{estimation} of the true gradient. Importantly, both checkpointed and non-checkpointed models exhibited the same spikes and dips in their validation losses over the course of training, also decreasing at similar rates.\n\n\n\n\\subsection{Extended analysis of cell state and update rule inductive biases}\n\\label{subsec:extended_analysis}\n\n\\input{figures\/unetca_extended_analysis}\n\nHere we present an extension of the analyses provided in \\Sec{\\ref{subsubsec:cell_state_analysis}} and \\Sec{\\ref{subsubsec:inductive_bias_investigation}}.\n\n\\paragraph{Adaptation to varying update rates (UNetCA).}\n\\Fig{\\ref{fig:unetca_extended_analysis}} (a) shows UNetCA capable of adapting to a slower ($\\sigma\\!=\\!25\\%$) cell update rate despite being trained with a $\\sigma\\!=\\!50\\%$ cell update rate. Interestingly, UNetCA experiences difficulty synchronously updating all cells ($\\sigma\\!=\\!100\\%$), producing a noticeably lower quality output compared to its outputs at asynchronous rates. This is in contrast to ViTCA (\\Fig{\\ref{fig:analysis}} (d)), where the quality of output remains the same across all update rates. Also, not shown in \\Fig{\\ref{fig:analysis}} (d), but is important to note, are the number of ViTCA iterations from left-to-right, which are as follows: 1, 8, 12, 16, 32. We point attention to the fact that UNetCA required 48 iterations to converge with $\\sigma\\!=\\!25\\%$, 24 iterations to converge with $\\sigma\\!=\\!50\\%$, and could not converge to a good solution with $\\sigma\\!=\\!100\\%$, while ViTCA required 32 iterations to converge with $\\sigma\\!=\\!25\\%$, 16 iterations to converge with $\\sigma\\!=\\!50\\%$, and 8 iterations to converge with $\\sigma\\!=\\!100\\%$.\n\n\\paragraph{Generalization to noise unseen during training (UNetCA).}\nAs shown in \\Fig{\\ref{fig:unetca_extended_analysis}} (b), UNetCA is incapable of generalizing to noise configurations unseen during training, inducing a divergence in cell states. This is in contrast to ViTCA as shown in \\Fig{\\ref{fig:analysis}} (e). ViTCA not only produces a higher fidelity output mid-denoising, but it also maintains cell state stability.\n\n\\paragraph{Effects of not \\emph{vs}\\onedot completely masking input (UNetCA).}\n\\Fig{\\ref{fig:unetca_extended_analysis}} (c; \\emph{top}): Although UNetCA is able to successfully autoencode the unmasked input image, it eventually induces a divergence amongst cell states. This is in contrast to ViTCA as shown in \\Fig{\\ref{fig:analysis}} (g; \\emph{left}). ViTCA not only produces a higher fidelity output mid-denoising, but it also maintains cell state stability. \\Fig{\\ref{fig:unetca_extended_analysis}} (c; \\emph{bottom}): Unlike ViTCA (\\Fig{\\ref{fig:analysis}} (g; \\emph{right})), UNetCA does not output the median image when attempting to denoise a completely masked input and instead causes cells to diverge.\n\n\\paragraph{Effect of masking heads.}\n\\input{figures\/attn_head_masking}\n\n\\input{tables\/runtime_profiling_results}\n\n\\Fig{\\ref{fig:attn_head_masking}} shows how ViTCA reacts to having its self-attention heads masked during autoencoding (no noise) an example from CelebA. The purpose of this experiment is to observe each head's contribution to the output. We can see that when none of the heads are masked, they attend to facial features and contours, and the output is as expected. However, once heads are masked, the unmasked heads stop attending to the features they once did and instead deteriorate. In some cases, the unmasked heads stop attending to anything at all. There are a couple of interesting cases: 1) When only the first head is masked, ViTCA is still able to successfully autoencode the input, although there is a slight degradation in quality. This is consistent with examples from the other datasets as well as when there is noise involved. 2) When certain heads are masked, the noise that the model was trained to denoise starts to appear (\\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  fourth column from left and fifth column from right).\n\n\\subsection{Runtime analysis of ViTCA}\n\\label{subsec:runtime_analysis}\n\nHere we provide a brief analysis of ViTCA's runtime performance and memory usage while training on a minibatch of random $32\\times 3 \\times H \\times W$ images through measurements of forward pass duration (ms), backward pass duration (ms), and training memory usage (GB), with and without using gradient checkpointing. We use $T\\!=\\!32$ ViTCA iterations and 16 checkpoint segments. Results are shown in \\Tab{\\ref{tab:runtime_profiling_results}}. Gradient checkpointing provides substantial memory savings at the cost of proportionally increasing the duration of the backward pass.\n\n\n\n\\section{Appendix}\n\\label{sec:appendix}\n\n\\input{algorithms\/vitca_training}\n\n\n\\subsection{Training on high-resolution imagery with fusion and mitosis}\n\\label{subsec:high_resolution_training}\n\nAs an alternative to gradient checkpointing for reducing memory usage, we briefly experimented with a downsampling scheme inspired by cell fusion and mitosis when training on CelebA at $64\\!\\times\\!64$. Specifically, we split the $T$ applications of the update rule (within a training iteration) into multiple stages: 1) We apply the update rule twice so that cells will have, at minimum, some amount of knowledge of their neighbours. 2) We stash the masked input for a later re-injection. 3) \\emph{Fusion}---we apply a $2\\!\\times\\!2$ average pooling with a stride of 2 across the cell grid, combining $2\\!\\times\\!2$ groups of cells into singular cells. 4) We apply the update rule $T-4$ times at this $32\\!\\times\\!32$ downsampled cell grid resolution. 5) \\emph{Mitosis}---we perform a $2\\!\\times\\!2$ duplication of cells (each cell is duplicated to its right, bottom-right, and bottom). 6) We re-inject the stashed masked input. 7) We apply the update rule twice to adapt the cells to the $64\\!\\times\\!64$ resolution and to fill in any missing information.\n\nWe found that performing this fusion and mitosis scheme decreased training memory consumption to levels similar to our gradient checkpointing scheme ($\\sim\\!50\\%$ memory reduction) while having a $\\sim\\!70\\%$ faster backward pass. Loss-wise, we observed a $\\sim\\!33\\%$ increase in the average validation reconstruction loss during training, which can qualitatively be observed in the example provided in \\Fig{\\ref{fig:fusion_mitosis}} (\\emph{\\textcolor{unetcaBlue}{bottom}}). Although the results shown are not ideal---\\emph{i.e}\\onedot, } \\def\\Ie{\\emph{I.e}\\onedot,  we did not perform a hyper-parameter search here, for example, finding the optimal number of iterations preceding fusion and following mitosis---this brief experiment tests the feasibility of reducing memory consumption while maintaining denoising capability and avoiding gradient checkpointing. As shown in the figure, ViTCA with fusion and mitosis is able to successfully denoise the input despite applying updates at two different scales. This scale agnostic behaviour reveals potentially interesting research directions beyond the scope of this work, such as allowing an NCA update rule to dynamically and locally modify cell grid resolution based on a compute budget, which could see applications in signal (image, video, or audio) compression.\n\n\\begin{figure}[t]\n  \\begin{floatrow}\n    \\input{figures\/fusion_mitosis}\n    \\input{tables\/other_ablation_results}\n    \\vspace{-0.5\\baselineskip}\n  \\end{floatrow}\n\\end{figure}\n\n\\subsection{Extended ablation study}\n\\label{subsec:extended_ablation}\n\nHere we present an extension of our ablation study in \\Sec{\\ref{subsubsec:ablation_study}}, using the baseline ViTCA model as our reference. As before, the ablation examines the effects certain training configuration parameters have on test performance.\n\n\\input{tables\/attn_size_ablation_results}\n\\input{tables\/update_rate_ablation_results}\n\\input{tables\/gradient_checkpointing_ablation_results}\n\n\\paragraph{Pool size, cell initialization, and patch size.}\nIn \\Tab{\\ref{tab:other_ablation_results}}, we examine the impact of varying the (max) pool size $N_\\mathcal{P}$, cell initialization method, and patch size $P_H\\!\\times\\!P_W$ on CelebA. As shown in the table, it is difficult to correlate pool size with test performance. However, when pool size $N_\\mathcal{P}\\!=\\!8192$, there is a noticeable reduction in performance. Test performance also degrades when initializing cells such that their output and hidden channels receive random values sampled from $\\mathcal{U}(0,1)$ and $\\mathcal{U}(-1,1)$, respectively, as opposed to receiving constant values (0.5 for output channels and 0 for hidden). Finally, we see a consistent decrease in performance when the input image is divided into non-overlapping patches $> 1\\!\\times\\!1$, as well as an increase in the number of model parameters.\n\n\\paragraph{Attention neighbourhood size.}\nIn \\Tab{\\ref{tab:attn_size_ablation_results}}, we examine the impact of attention neighbourhood size $N_H\\!\\times\\!N_W$ on FashionMNIST. Interestingly, increasing the neighbourhood size past $3\\!\\times\\!3$ causes a degradation in performance. This is most likely attributed to the increase in complexity caused by incorporating more information into ViTCA's self-attention. One would expect explicitly increasing the receptive field of spatially localized self-attention to result in better performance, but it can also complicate the process of figuring out which neighbours to attend to. We believe this may be alleviated by increasing model capacity and\/or training duration. As described in \\Sec{\\ref{sec:experiments}}, we use the Moore neighbourhood ($3\\!\\times\\!3$) as it requires less computation while still demonstrating ViTCA's effectiveness.\n\n\\paragraph{Asynchronous \\emph{vs}\\onedot synchronous cell updates.}\nIn \\Tab{\\ref{tab:update_rate_ablation_results}}, we compare between training with asynchronous cell updates ($\\sigma\\!=\\!50\\%$) and training with synchronous cell updates ($\\sigma\\!=\\!100\\%$) on LandCoverRep, MNIST, CelebA, and FashionMNIST. Training with asynchronous cell updates provides a meaningful increase in performance compared to training with synchronous cell updates and comes with several benefits, such as not requiring cells in a neighbourhood to be in sync with each other and serving as additional data augmentation. Similarly mentioned in related work \\cite{niklasson2021self-organising}, this allows ViTCA to be used in a distributed system where cells need not exist under a global clock and can be updated at varying rates. Thus making it easier to scale up or down within a non-homogeneous compute environment. This was somewhat demonstrated in \\Fig{\\ref{fig:analysis}} (d) where ViTCA was able to adapt to varying update rates despite being trained on a fixed asynchronous update rate ($\\sigma\\!=\\!50\\%$).\n\n\\paragraph{Effects of gradient checkpointing.}\nIn \\Tab{\\ref{tab:gradient_checkpointing_ablation_results}}, we compare between training with gradient checkpointing disabled and with gradient checkpointing enabled on LandCoverRep, MNIST, CelebA, and FashionMNIST. Similarly shown in \\Tab{\\ref{tab:ablation_results}}, we see here that training with gradient checkpointing has an adverse effect on test performance. As mentioned in \\Sec{\\ref{sec:discussion}}, NCAs---during training---require all activations from each recurrent iteration to be stored in memory before performing backpropagation. This results in memory usage being proportional to the amount of recurrent iterations. As such, depending on ViTCA's configuration, gradient checkpointing may be required to be able to train on a single GPU. We make use of PyTorch's \\texttt{checkpoint\\_sequential}, which we use as follows: given the number of CA iterations $T$, we divide the sequential (forward) application of the update rule into $\\lfloor T\/2 \\rfloor$ segments of roughly the same length (depending on whether $T$ is even or odd). Then, all segments are executed in sequence, where activations from only the first and last segments are stored as well as the inputs to each intermediate segment. The intermediate inputs are used for re-running the segments without stored activations during the backward pass to compute gradients. This results in a trade-off between memory consumption and backpropagation duration since each intermediate segment's forward pass needs to be re-computed during its backward pass. Moreover, and not mentioned in the documentation of PyTorch at the time of writing, there exists a subtle yet meaningful side-effect which we have observed and confirmed through the use of GNU Debugger (GDB) and Python Debugger (PDB): Without gradient checkpointing, gradients are accumulated all at once at the end of backpropagating through the entire computation graph, resulting in the expected round-offs due to limitations in machine precision (\\texttt{float32} in our case). At this point, PyTorch may use a variety of numerical techniques to minimize round-off, such as cascade summation (verified to be used for CPU-based summation, see \\texttt{SumKernel.cpp} in PyTorch) which recursively sums two halves of a sequence of summands as opposed to naively summing them in sequence. \\emph{With} gradient checkpointing, gradients are accumulated at each segment. This means that round-offs are forced to (potentially) occur at each checkpoint\/segment instead of once at the end of the entire computation graph. Even if cascade summation is used when summing gradients within each segment, the segment-wise ordering may reduce its effectiveness. We verified this behaviour by observing an exact machine epsilon difference ($\\epsilon\\approx1.19\\!\\times\\!10^{-7}$ in IEEE 754 standard) in the gradient---when compared to the non-checkpointed scheme---of the final operation of the update rule at the second-last segment, once the loss started to diverge.\n\nIt is important to note that despite the difference in gradients, the accuracy of the forward pass remains unchanged between the checkpointed and non-checkpointed models. Also, we must remind ourselves that round-offs are unavoidable when performing floating-point arithmetic, meaning that gradients computed within a deep learning library such as PyTorch are always an \\emph{estimation} of the true gradient. Importantly, both checkpointed and non-checkpointed models exhibited the same spikes and dips in their validation losses over the course of training, also decreasing at similar rates.\n\n\n\n\\subsection{Extended analysis of cell state and update rule inductive biases}\n\\label{subsec:extended_analysis}\n\n\\input{figures\/unetca_extended_analysis}\n\nHere we present an extension of the analyses provided in \\Sec{\\ref{subsubsec:cell_state_analysis}} and \\Sec{\\ref{subsubsec:inductive_bias_investigation}}.\n\n\\paragraph{Adaptation to varying update rates (UNetCA).}\n\\Fig{\\ref{fig:unetca_extended_analysis}} (a) shows UNetCA capable of adapting to a slower ($\\sigma\\!=\\!25\\%$) cell update rate despite being trained with a $\\sigma\\!=\\!50\\%$ cell update rate. Interestingly, UNetCA experiences difficulty synchronously updating all cells ($\\sigma\\!=\\!100\\%$), producing a noticeably lower quality output compared to its outputs at asynchronous rates. This is in contrast to ViTCA (\\Fig{\\ref{fig:analysis}} (d)), where the quality of output remains the same across all update rates. Also, not shown in \\Fig{\\ref{fig:analysis}} (d), but is important to note, are the number of ViTCA iterations from left-to-right, which are as follows: 1, 8, 12, 16, 32. We point attention to the fact that UNetCA required 48 iterations to converge with $\\sigma\\!=\\!25\\%$, 24 iterations to converge with $\\sigma\\!=\\!50\\%$, and could not converge to a good solution with $\\sigma\\!=\\!100\\%$, while ViTCA required 32 iterations to converge with $\\sigma\\!=\\!25\\%$, 16 iterations to converge with $\\sigma\\!=\\!50\\%$, and 8 iterations to converge with $\\sigma\\!=\\!100\\%$.\n\n\\paragraph{Generalization to noise unseen during training (UNetCA).}\nAs shown in \\Fig{\\ref{fig:unetca_extended_analysis}} (b), UNetCA is incapable of generalizing to noise configurations unseen during training, inducing a divergence in cell states. This is in contrast to ViTCA as shown in \\Fig{\\ref{fig:analysis}} (e). ViTCA not only produces a higher fidelity output mid-denoising, but it also maintains cell state stability.\n\n\\paragraph{Effects of not \\emph{vs}\\onedot completely masking input (UNetCA).}\n\\Fig{\\ref{fig:unetca_extended_analysis}} (c; \\emph{top}): Although UNetCA is able to successfully autoencode the unmasked input image, it eventually induces a divergence amongst cell states. This is in contrast to ViTCA as shown in \\Fig{\\ref{fig:analysis}} (g; \\emph{left}). ViTCA not only produces a higher fidelity output mid-denoising, but it also maintains cell state stability. \\Fig{\\ref{fig:unetca_extended_analysis}} (c; \\emph{bottom}): Unlike ViTCA (\\Fig{\\ref{fig:analysis}} (g; \\emph{right})), UNetCA does not output the median image when attempting to denoise a completely masked input and instead causes cells to diverge.\n\n\\paragraph{Effect of masking heads.}\n\\input{figures\/attn_head_masking}\n\n\\input{tables\/runtime_profiling_results}\n\n\\Fig{\\ref{fig:attn_head_masking}} shows how ViTCA reacts to having its self-attention heads masked during autoencoding (no noise) an example from CelebA. The purpose of this experiment is to observe each head's contribution to the output. We can see that when none of the heads are masked, they attend to facial features and contours, and the output is as expected. However, once heads are masked, the unmasked heads stop attending to the features they once did and instead deteriorate. In some cases, the unmasked heads stop attending to anything at all. There are a couple of interesting cases: 1) When only the first head is masked, ViTCA is still able to successfully autoencode the input, although there is a slight degradation in quality. This is consistent with examples from the other datasets as well as when there is noise involved. 2) When certain heads are masked, the noise that the model was trained to denoise starts to appear (\\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  fourth column from left and fifth column from right).\n\n\\subsection{Runtime analysis of ViTCA}\n\\label{subsec:runtime_analysis}\n\nHere we provide a brief analysis of ViTCA's runtime performance and memory usage while training on a minibatch of random $32\\times 3 \\times H \\times W$ images through measurements of forward pass duration (ms), backward pass duration (ms), and training memory usage (GB), with and without using gradient checkpointing. We use $T\\!=\\!32$ ViTCA iterations and 16 checkpoint segments. Results are shown in \\Tab{\\ref{tab:runtime_profiling_results}}. Gradient checkpointing provides substantial memory savings at the cost of proportionally increasing the duration of the backward pass.\n\\section*{Checklist}\n\\begin{enumerate}\n\n\\item For all authors...\n\\begin{enumerate}\n  \\item Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?\n    \\answerYes\n  \\item Did you describe the limitations of your work?\n    \\answerYes{See \\Sec{\\ref{sec:discussion}}.}\n  \\item Did you discuss any potential negative societal impacts of your work?\n    \\answerNo{Although we feel our work demonstrates the potential of NCAs as viable alternatives to common recurrent network architectures (ViTCA being our evidential contribution), our experiments intentionally tend towards the direction of optimizing model efficiency (and single-GPU training accessibility) rather than towards the increasingly popular direction of scaling upwards. However, as much as our work demonstrates the downward-scaling capabilities of NCAs, we also acknowledge that this similarly applies going upward, and as such, can be abused (\\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  creating a ``deepfake''-capable ViTCA).}\n  \\item Have you read the ethics review guidelines and ensured that your paper conforms to them?\n    \\answerYes\n\\end{enumerate}\n\n\\item If you are including theoretical results...\n\\begin{enumerate}\n  \\item Did you state the full set of assumptions of all theoretical results\n    \\answerNA{}\n        \\item Did you include complete proofs of all theoretical results?\n    \\answerNA{}\n\\end{enumerate}\n\n\\item If you ran experiments...\n\\begin{enumerate}\n  \\item Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?\n    \\answerYes{Code and instructions to reproduce results are included in the supplemental material.}\n  \\item Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?\n    \\answerYes{}\n        \\item Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?\n    \\answerNo{Given the combination of time and computational restrictions and our exhaustive list of experiments, we opted to prioritize experiment variety and dataset coverage as an implicit substitute for re-running experiments under different random seeds. For all experiments, we kept a fixed random seed, even pointing out (deterministic) differences caused by gradient checkpointing when used (see Appendix \\ref{sec:appendix}).}\n        \\item Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?\n    \\answerYes\n\\end{enumerate}\n\n\\item If you are using existing assets (e.g., code, data, models) or curating\/releasing new assets...\n\\begin{enumerate}\n  \\item If your work uses existing assets, did you cite the creators?\n    \\answerYes\n  \\item Did you mention the license of the assets?\n    \\answerNA{Licensed frameworks used such as PyTorch (BSD-style) and Hydra (MIT) will be mentioned in acknowledgements.}\n  \\item Did you include any new assets either in the supplemental material or as a URL?\n    \\answerNo{No new assets---aside from code and training our models---were created for the purposes of this work.}\n  \\item Did you discuss whether and how consent was obtained from people whose data you're using\/curating?\n    \\answerNo{We used publicly available datasets.}\n  \\item Did you discuss whether the data you are using\/curating contains personally identifiable information or offensive content?\n    \\answerNo{Although not discussed in the manuscript, we would like to point that the datasets we used that could potentially contain personally identifiable information (CelebA, CIFAR10, Tiny ImageNet) each have restrictions and\/or acknowledgements of such potential issues. Also, our work is not focused on classifying persons and ViTCA is not a generative model, \\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  it can not generate new faces.}\n\\end{enumerate}\n\n\\item If you used crowdsourcing or conducted research with human subjects...\n\\begin{enumerate}\n  \\item Did you include the full text of instructions given to participants and screenshots, if applicable?\n    \\answerNA\n  \\item Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?\n    \\answerNA\n  \\item Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?\n    \\answerNA\n\\end{enumerate}\n\n\\end{enumerate}\n\\section{Discussion}\n\\label{sec:discussion}\n\\vspace{-.5\\baselineskip}\nWe have performed extensive quantitative and qualitative evaluations of our newly proposed ViTCA on a variety of datasets under a denoising autoencoding framework. We have demonstrated the superior denoising performance and robustness of our model when compared to a U-Net-based CA baseline (UNetCA) and ViT, as well as its generalization capabilities under a variety of environmental changes such as larger inputs (\\emph{i.e}\\onedot, } \\def\\Ie{\\emph{I.e}\\onedot,  spatial interpolation) and changing inputs \\emph{during} cell updates.\n\nDespite the computation savings---owed to our circumvention of self-attention's quadratic complexity by spatially localizing it within ViTCA---there remains the same memory limitations inherent to all recurrent models: multiple recurrent iterations are required for each training iteration, resulting in larger memory usage than a feedforward approach. This limits single-GPU training accessibility. We have experimented with gradient checkpointing \\cite{chen2016training} but found its trade-off for increased backpropagation duration (and slightly different gradients) less than ideal. To fully realize the potential of NCAs (self-organization, inherent distributivity, \\emph{etc}\\onedot), we encourage follow-up work to address this limitation. Adapting recent techniques using implicit differentiation is one avenue to circumvent these issues~\\cite{bai2022deep,bai2019deep}. Also, as mentioned in our ablation (\\Sec{\\ref{subsubsec:ablation_study}}), we hope to further investigate the instabilities caused by increasing the depth of ViTCA.\n\n\n\n\n\\section{Experiments}\n\\vspace{-.5\\baselineskip}\n\\label{sec:experiments}\n\nHere we examine ViTCA through extensive experiments. We begin with experiments for denoising autoencoding, then an ablation study followed by various qualitative analyses, before concluding with linear probing experiments on the learned representations for MNIST \\cite{deng2012mnist}, FashionMNIST \\cite{xiao2017fashion}, and CIFAR10 \\cite{krizhevsky2009learning}. We provide an extension to our experiments in Appendix \\ref{sec:appendix}.\n\n\\paragraph{Baseline models and variants.}\nSince we are performing pixel level reconstructions, we create a ViT baseline in which the class token has been removed. This applies identically for ViTCA. Unless otherwise stated, for our ViT and ViTCA models we use a patch size of $1\\!\\times\\!1$ ($P_H\\!=\\!P_W\\!=\\!1$), and only a single encoding block with $h\\!=\\!4$ \\mhsa\\ heads, \\embed\\ size $d\\!=\\!128$, and \\mlp\\ size of $128$. For ViTCA, we choose $N_H\\!=\\!3$ and $N_W\\!=\\!3$ (\\emph{i.e}\\onedot, } \\def\\Ie{\\emph{I.e}\\onedot,  the \\emph{Moore neighbourhood}~\\cite{weissteinmoore}). We also compare with a U-Net baseline similar to the original formulation \\cite{ronneberger2015u}, but based on the specific architecture from \\cite{lehtinen2018noise2noise}. Since most of our datasets consist of $32\\!\\times\\!32$ (resampled) images, we only have two downsampling steps as opposed to five. We implement a U-Net-based CA (UNetCA) baseline consisting of a modified version of our U-Net with 48 initial output feature maps as opposed to 24 and with all convolutions except the first changed to $1\\!\\times\\!1$ to respect typical NCA restrictions~\\cite{palm2022variational,mordvintsev2020growing}.\n\n\\vspace{-.5\\baselineskip}\n\\subsection{Denoising autoencoding}\n\\vspace{-.5\\baselineskip}\n\\label{subsec:denoising_autoencoding}\n\nWe compare between our baseline models and a number of ViTCA variants in the context of denoising autoencoding. We present test set results across six benchmark datasets: a land cover classification dataset intended for representation learning (LandCoverRep) \\cite{yeh2021sustainbench}, MNIST, CelebA \\cite{liu2015faceattributes}, FashionMNIST, CIFAR10, and Tiny ImageNet (a subset of ImageNet \\cite{russakovsky2015imagenet}). All datasets consist of $32\\!\\times\\!32$ resampled images except Tiny ImageNet, which is at $64\\!\\times\\!64$ resolution. During testing, we use all masking combinations, chosen in a fixed order, and we update cells using a fixed number of iterations ($T\\!=\\!64$). See \\Tab{\\ref{tab:denoising_results}} for quantitative results.\n\nBriefly mentioned in \\Sec{\\ref{subsec:update_rule}}, we employ a masking strategy inspired by Curriculum Learning (CL) \\cite{wang2021survey,bengio2009curriculum} to ease training. This schedule follows a geometric progression of difficulty---tied to training iterations---maxing out at 10K training iterations. Specifically, masking starts at covering 25\\% of the input with $1\\!\\times\\!1$ patches of noise (dropout for RGB inputs, Gaussian for grayscale), then at each shift in difficulty, new masking configurations are added to the list of available masking configurations in the following order: $(2^0\\!\\times\\!2^0, 50\\%), (2^0\\!\\times\\!2^0, 75\\%), (2^1\\!\\times\\!2^1, 25\\%), (2^1\\!\\times\\!2^1, 50\\%), (2^1\\!\\times\\!2^1, 75\\%), ..., (2^2\\!\\times\\!2^2, 75\\%)$. Masking configurations are randomly chosen from this list.\n\nWe initialize weights\/parameters using He initialization \\cite{he2015delving}, except for the final layer of CA-based models, which are initialized to zero \\cite{mordvintsev2020growing}. Unless otherwise stated, we train for $I\\!=\\!100$K iterations, use a minibatch size $b\\!=\\!32$, AdamW optimizer \\cite{loshchilov2019decoupled}, learning rate $\\eta\\!=\\!10^{-3}$ with a cosine annealing schedule \\cite{loshchilov2017sgdr}, pool size \\poolsize\\ $\\!=\\!1024$, and cell hidden channel size $C_h\\!=\\!32$. In the case of Tiny ImageNet, $b\\!=\\!8$ to accommodate training on a single GPU (48GB Quadro RTX 8000). Training typically lasts a day at most, depending on the model. Due to the recurrent iterations required per training step, CA-based models take the longest to train. To alleviate memory limitations for some of our experiments, we use gradient checkpointing \\cite{chen2016training} during CA iterations at the cost of backpropagation duration and slight variations in gradients due to its effect on round-off propagation. We also experiment with a cell fusion and mitosis scheme as an alternative. See Appendix \\ref{sec:appendix} for details on runtime performance, gradient checkpointing, and fusion and mitosis.\n\n\\input{tables\/denoising_results}\n\nAmongst baselines, ViTCA outperforms on most metrics across the majority of datasets used (10 out of 18). Exceptions include LandCoverRep, where UNetCA universally outperforms by a small margin, likely due to the texture-dominant imagery being amenable to convolutions. Notably, ViTCA strongly outperforms on MNIST. Although MNIST is a trivial dataset for common tasks such as classification, our masking\/noise strategy turns it into a challenging dataset for denoising autoencoding, \\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  it is difficult for even a human to classify a $32\\!\\times\\!32$ MNIST digit 75\\% corrupted by $4\\!\\times\\!4$ patches of Gaussian noise. We hypothesize that when compared to convolutional models, ViTCA's weaker inductive biases (owed to attention \\cite{yifan2021input,jaegle2021perceiver}) immediately outperform these models when there are large regions lacking useful features, \\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  MNIST digits cover a small space in the canvas. This is not the case with FashionMNIST, where the content is more filled out. Between baselines and ViTCA variants, ViTCA-32 (32 heads) and 32xy (xy-coordinate positional encoding) outperform all models by large margins, demonstrating the benefits of multi-head self-attention. We also experiment with a parameter-reduced (by $\\sim\\!60\\%$), inverted bottleneck variant where $d\\!=\\!64$ and \\mlp\\ size is 256, often with a minimal reduction in performance.\n\\vspace{-.5\\baselineskip}\n\\subsubsection{Ablation study}\n\\label{subsubsec:ablation_study}\n\\vspace{-.5\\baselineskip}\n\nIn \\Tab{\\ref{tab:ablation_results}} we perform an ablation study using the baseline ViTCA model above as reference on CelebA. Results are ordered in row-wise blocks, top-to-bottom. Specifically, we examine the impact of varying the cell hidden size $C_h$; the \\embed\\ size $d$; the number of \\mhsa\\ heads $h$; the depth (\\# encoders), comparing both ViTCA (used throughout the table) with ViT; and in the last block we examine the impact of various methods of incorporating positional information into the model.\n\n\\input{tables\/ablation_results}\n\nSpecifically, we examine the use of: (1) a xy-coordinate-based positional encoding \\emph{concatenated} (``injected'') to cells, and; (2) a Transformer-based positional encoding (or embedding, if learned) \\emph{added} into \\embed. These two categories are subdivided into: \n(1a) sincos5---consisting of handcrafted Fourier features \\cite{mildenhall2020nerf} with four doublings of a base frequency, \\emph{i.e}\\onedot, } \\def\\Ie{\\emph{I.e}\\onedot,  \\pe\\ $\\!=\\!(\\sin{2^0\\pi p},$ $\\cos{2^0\\pi p},$ $...,\\sin{2^{J-1}\\pi p},$ $\\cos{2^{J-1}\\pi p})\\!\\in\\mathbb{R}^{N \\times (4JP_HP_W)}$ where $J\\!=\\!5$ and $p$ is the pixel coordinate (normalized to [-1,1]) for each pixel the cell is situated on (one pixel since $P_H\\!=\\!P_W\\!=\\!1$); \n(1b) sincos5xy---consisting of both Fourier features and explicit xy-coordinates concatenated; \n(1c) xy---only xy-coordinates; \n(2a) handcrafted (our baseline approach)---sinusoidal encoding \\pe\\ $\\!\\in\\!\\mathbb{R}^{N \\times d}$ similar to (1a) but following a Transformer-based approach \\cite{vaswani2017attention}, and; \n(2b) learned---learned embedding \\pe\\ $\\!\\in\\!\\mathbb{R}^{N \\times d}$ following the original ViT approach \\cite{dosovitskiy2020image}. \nTo further test the self-organizing capabilities of ViTCA, we also include: (3) none---no explicit positioning provided, where we let the cells localize themselves.\n\nAs shown in \\Tab{\\ref{tab:ablation_results}}, ViTCA benefits from an increase to most CA and Transformer-centric parameters, at the cost of computational complexity and\/or an increase in parameter count. A noticeable decrease in performance is observed when \\embed\\ size $d\\!=\\!512$, most likely due to the vast increase in parameter count necessitating more training. In the original ViT, multiple encoding blocks were needed before the model could exhibit performance equivalent to their baseline CNN \\cite{dosovitskiy2020image}, as verified in our ablation with our ViT. However, for ViTCA we notice an inverse relationship of the effect of Transformer depth, causing a divergence in cell state. It is not clear why this is the case, as we have observed that the LN layers and overflow losses otherwise encourage a contractive $F_\\theta$. This is an investigation we leave for future work. Despite the benefits of increasing $h$, we use $h\\!=\\!4$ for our baseline to optimize runtime performance. Finally, we show that ViTCA does not dramatically suffer when no explicit positioning is used---in contrast to typical Transformer-based models---as cells are still able to localize themselves by relying on their stored hidden information.\n\n\\vspace{-.5\\baselineskip}\n\\subsubsection{Cell state analysis}\n\\label{subsubsec:cell_state_analysis}\n\\vspace{-.5\\baselineskip}\n\nHere we provide an empirically-based qualitative analysis on the effects ViTCA and UNetCA have on cell states through several experiments with our pre-trained models (\\Fig{\\ref{fig:analysis} (a,b,c)}). We notice that in general, ViTCA indefinitely maintains cell state stability while UNetCA typically induces a divergence past a certain point. An extended analysis is available in Appendix \\ref{subsec:extended_analysis}.\n\n\\textbf{Damage resilience.}\nShown in \\Fig{\\ref{fig:analysis}} (a), we damage a random $H\/2\\!\\times\\!W\/2$ patch of cells with random values $\\sim\\!\\mathcal{U}(-1,1)$ twice in succession. ViTCA is able to maintain cell stability despite not being trained to deal with such noise, while UNetCA induces a divergence. Note both models are simultaneously performing the typical denoising task. We also note that ViTCA's inherent damage resilience is in contrast to recent NCA formulations that required explicit training for it \\cite{palm2022variational,mordvintsev2020growing}.\n\n\\textbf{Convergence stability.}\n\\Fig{\\ref{fig:analysis}} (b) shows denoising results after 2784 cell grid updates. ViTCA is able to maintain a stable cell grid state while UNetCA causes cells to diverge.\n\n\\textbf{Hidden state visualizations.}\n\\Fig{\\ref{fig:analysis}} (c) shows 2D and 3D PCA dimensionality reductions on the hidden states of converged cell grids for all examples in FashionMNIST~\\cite{xiao2017fashion}. The clusters suggest some linear separability in the learned representation, motivating our probing experiments in \\Sec{\\ref{subsec:probing}}. \n\n\\input{figures\/analysis}\n\n\\subsubsection{Investigating update rule inductive biases}\n\\label{subsubsec:inductive_bias_investigation}\n\\vspace{-.5\\baselineskip}\n\nHere we investigate the inductive biases inherent in ViTCA and UNetCA by testing their adaptation to various environmental changes (\\Fig{\\ref{fig:analysis}} (d,e,f,g,h)).\n\n\\textbf{Adaptation to varying update rates.}\nDespite being trained with a $\\sigma\\!=\\!50\\%$ cell update rate, ViTCA is able to adapt to varying rates (\\Fig{\\ref{fig:analysis}} (d)). Higher rates result in a proportionally faster rate of cell state convergence, and equivalently with lower rates. UNetCA exhibits a similar relationship, although is unstable at $\\sigma\\!=\\!100\\%$ (see Appendix \\ref{subsec:extended_analysis}). For details comparing training with a synchronous \\emph{vs}\\onedot asynchronous cell grid update, see Appendix \\ref{subsec:extended_ablation}.\n\n\\textbf{Generalization to noise unseen during training.}\nViTCA is capable of denoising configurations of noise it has not been trained on. \\Fig{\\ref{fig:analysis}} (e; \\emph{left-to-right}): $4\\!\\times\\!1$ and $1\\!\\times\\!4$ patches of Gaussian noise at 65\\% coverage. In contrast, UNetCA induces a cell state divergence (see Appendix \\ref{subsec:extended_analysis}).\n\n\\textbf{Adaptation to changing inputs.}\nAt various moments during cell updates, we re-inject cells with new masked inputs (\\Fig{\\ref{fig:analysis}} (f)). ViTCA is able to consistently adapt cells to new inputs while UNetCA experiences difficulty past a certain point (\\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  at 464 iterations in the figure).\n\n\\textbf{Effects of not \\emph{vs}\\onedot completely masking input.}\n\\Fig{\\ref{fig:analysis}} (g; \\emph{left}): ViTCA is able to perform autoencoding despite not being trained for it. UNetCA induces a cell grid divergence (see Appendix \\ref{subsec:extended_analysis}). \\Fig{\\ref{fig:analysis}} (g; \\emph{right}): Interestingly, when the input is completely masked, ViTCA outputs the median image~\\cite{lehtinen2018noise2noise}. UNetCA does not exhibit such behaviour and instead causes cells to diverge (see Appendix \\ref{subsec:extended_analysis}). \n\n\\textbf{Spatial interpolation.}\nWe use ViTCA models trained at $32\\!\\times\\!32$ using various types of positioning to generate $128\\!\\times\\!128$ outputs during inference, assuming an identical cell grid resolution. \\Fig{\\ref{fig:analysis}} (h; \\emph{top-to-bottom of \\textcolor{vitcaPurple}{outputs}}): xy-coordinates, no positioning, Fourier features \\cite{mildenhall2020nerf}, Fourier features concatenated with xy-coordinates, and a Transformer-based handcrafted positional encoding (baseline) \\cite{vaswani2017attention}. Results are ordered from best to worst. The baseline approach is not capable of spatial interpolation due to being a 1D positioning, while, as expected, the 2D encodings make it capable. Surprisingly, removing Fourier features and using only xy-coordinates results in a higher fidelity interpolation. We believe this to be caused by the distracting amount of positional information Fourier features provide to cells, as cells can instead rely on their hidden states to store higher frequency positional information. Finally, with no explicit positioning, ViTCA is still able to perform high-quality interpolation---even exceeding using Fourier features---by taking advantage of its self-organizing nature. As a side note, we point attention to the fact that ViTCA is simultaneously denoising at a scale space it has not been trained on, exemplifying its generalization capabilities.\n\n\\vspace{-.5\\baselineskip}\n\\subsection{Investigating hidden representations via linear probes}\n\\vspace{-.5\\baselineskip}\n\\label{subsec:probing}\n\n\\input{tables\/linear_probe_results}\nHere we examine the learned representations of our models pre-trained for denoising.  We freeze model parameters and learn linear classifiers on each of their learned representations: converged cell hidden states for CA-based models, bottleneck features for U-Net, and LN'd tokens for ViT. This is a common approach used to probe learned representations~\\cite{chen2020generative}. Classification results on MNIST, FashionMNIST, and CIFAR10 are shown in \\Tab{\\ref{tab:linear_probe_results}} and we use the same training setup as for denoising, but without any noise. For comparison, we also provide results using a linear classifier and two 2-layer MLPs of varying complexity, all trained directly on raw pixel values. Correlations between denoising performance in \\Tab{\\ref{tab:denoising_results}} and classification performance in \\Tab{\\ref{tab:linear_probe_results}} can be observed. Linear classification accuracy on ViTCA-based features typically exceeds classification accuracy using other model-based features or raw pixel values, even outperforming the MLPs in most cases.\n\n\n\n\n\n\n\n\\section{Introduction}\n\\vspace{-.5\\baselineskip}\n\\label{sec:introduction}\n\n\\input{figures\/vitca_vs_vit} Recent developments at the intersection of two foundational ideas---Artificial Neural Networks (ANNs) and Cellular Automata (CA)---have led to new approaches for constructing Neural Cellular Automata (NCA). These advances have integrated ideas such as variational inference \\cite{palm2022variational}, U-Nets \\cite{zhang2020learning}, and Graph Neural Networks (GNNs) \\cite{grattarola2021learning} with promising results on problems ranging from image synthesis \\cite{palm2022variational,niklasson2021self-organising,mordvintsev2021mu} to Reinforcement Learning (RL) \\cite{najarro2022hypernca,variengien2021towards}. Transformers are another significant development in deep learning \\cite{vaswani2017attention}, but, until now, have not been examined under an NCA setting.\n\nVision Transformers (ViTs) \\cite{dosovitskiy2020image} have emerged as a competitive alternative to Convolutional Neural Network (CNN) \\cite{lecun1998gradient} architectures for computer vision, such as Residual Networks (ResNets) \\cite{he2016deep}. ViTs leverage the self-attention mechanisms of original Transformers \\cite{vaswani2017attention}, which have emerged as the dominant approach for sequence modelling in recent years. Our work combines foundational ideas from Transformers and ViTs, leading to a new class of NCAs: \\textbf{Vision Transformer Cellular Automata (ViTCA)}. \n\nAn effective and ubiquitous Transformer-based learning technique for Natural Language Processing (NLP) pre-training is the unsupervised task of Masked Language Modelling (MLM), popularized by the BERT language model \\cite{devlin-etal-2019-bert}. The success of MLM-based techniques has similarly inspired recent work re-examining the classical formulation of Denoising Autoencoders (DAEs) \\cite{vincent2010stacked}, but for ViTs \\cite{bao2022beit,dosovitskiy2020image,chen2020generative}, introducing tasks such as Masked Image Encoding \\cite{he2021masked} and Masked Feature Prediction \\cite{wei2021masked} for image and video modelling, respectively. This simple yet highly-scalable strategy of masked-based unsupervised pre-training has yielded promising transfer learning results on vision-based downstream tasks such as object detection and segmentation, image classification, and action detection, even outperforming supervised pre-training~\\cite{he2021masked,wei2021masked}. We examine training methodologies for ViTCA within a DAE setting and perform extensive controlled experiments benchmarking these formulations against modern state of the art architectures, with favourable outcomes, \\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  \\Fig{\\ref{fig:vitca_vs_vit}}.\n\n\\input{figures\/teaser}\n\nOur contributions are as follows: \\textit{first}---to the best of our knowledge---our work is the first to extend NCA methodologies with key Transformer mechanisms, \\emph{i.e}\\onedot, } \\def\\Ie{\\emph{I.e}\\onedot,  self-attention and positional encoding (and embedding), with the beneficial side-effect of circumventing the quadratic complexity of self-attention; \\textit{second}, our ViTCA formulation allows for lower model complexity (by limiting ViT depth) while retaining expressivity through CA iterations on a controlled state---all with the same encoder weights. This yields a demonstrably more parameter-efficient \\cite{mordvintsev2021mu} ViT-based model. Importantly, ViTCA mitigates the problems associated with the explicit tuning of ViT depth originally needed to improve performance (\\emph{i.e}\\onedot, } \\def\\Ie{\\emph{I.e}\\onedot,  we use a depth of 1). With ViTCA, we simply iterate until cell state convergence. Since ViT (and by extension, ViTCA) employs Layer Normalization (LN) \\cite{ba2016layer} at each stage of its processing, it is a fairly contractive model capable of fixed-point convergence guarantees~\\cite{bai2019deep}.\n\nIn relation to our first contribution, ViTCA respects CA requirements, most importantly that computations remain localized about a cell and its neighbourhood. As such, we modify the global self-attention mechanisms of a ViT to respect this locality requirement (\\Fig{\\ref{fig:teaser}}). Localized self-attention is not a new idea \\cite{chen2022regionvit,liu2021swin,chu2021twins,zhang2021multi}; however, because cells contain state information that depends on its previous state, over CA iterations the effective receptive field of ViTCA's localized self-attention grows increasingly larger until eventually incorporating information implicitly across all cells. Thus, admitting global propagation of information from spatially localized self-attention. Moreover, due to the self-organizing nature of NCAs, self-organization also manifests itself within the localized self-attention, resulting in a globally agreed-upon arrangement of local self-attention. Thus, circumventing the quadratic complexity of explicit global self-attention (w.r.t\\onedot the input size) through a linear amortization over time, and increasing the feasibility of per-pixel dense processing (as we demonstrate). This globally consistent and complex behaviour, which arises from strictly local interactions, is a unique feature of NCAs and confers performance benefits which we observe both qualitatively and quantitatively when comparing ViT and ViTCA for denoising autoencoding. \n\n\\input{figures\/overview}\n\n\\section{Background and related work}\n\\vspace{-.5\\baselineskip}\n\\label{sec:related_work}\n\n\\paragraph{Neural Cellular Automata.}\n\nCellular Automata are algorithmic processes motivated by the biological behaviours of cellular growth and, as such, are capable of producing complex emergent (global) dynamics from the iterative application of comparatively simple (localized) rules~\\cite{von1966theory}. \\emph{Neural} Cellular Automata present a more general CA formulation, where the evolving cell states are represented as (typically low-dimensional) vectors and the update rule dictating their evolution is a differentiable function whose parameters are learned through backpropagation from a loss, rather than a handcrafted set of rules~\\cite{mordvintsev2020growing,gilpin2019cellular,wulff1992learning}.\nNeural net-based formulations of CAs in the NeurIPS community can be traced back to the early work of \\cite{wulff1992learning}, where only small and simple models were examined. Recent formulations of NCAs have shown that when leveraging the power of deep learning techniques enabled by advances in hardware capabilities---namely highly-parallelizable differentiable operations implemented on GPUs---NCAs can be tuned to learn surprisingly complex desired behaviour, such as semantic segmentation \\cite{sandler2020image}; common RL tasks such as cart-pole balancing \\cite{variengien2021towards}, 3D locomotion \\cite{najarro2022hypernca}, and Atari game playing \\cite{najarro2022hypernca}; and image synthesis~\\cite{palm2022variational,niklasson2021self-organising,mordvintsev2021mu}. Although these recent formulations rely on familiar compositions of convolutions and non-linear functions, it is important to highlight that NCAs are fundamentally not equivalent to ``very-deep'' CNNs (\\emph{vs}\\onedot~\\cite{gilpin2019cellular}), or any other feedforward architecture (\\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  ResNets \\cite{he2016deep}), particularly, in the same way that a Recurrent Neural Network (RNN) is not equivalent: CNNs and other feedforward architectures induce a directed \\textit{acyclic} computation graph (\\emph{i.e}\\onedot, } \\def\\Ie{\\emph{I.e}\\onedot,  a finite impulse response), whereas NCAs (and RNNs) induce a directed \\textit{cyclic} computation graph (\\emph{i.e}\\onedot, } \\def\\Ie{\\emph{I.e}\\onedot,  an infinite impulse response), where stateful data can additionally be manipulated using (learned) feedback loops and\/or time-delayed controls. As such, NCAs can be viewed as a type of RNN, and both (N)CAs and RNNs are known to be Turing complete~\\cite{christen2021automatic,cook2004universality,siegelmann1995computational,wulff1992learning}.\\footnote{In the case of (N)CAs, a Turing complete example is the \\emph{Rule 110} elementary CA \\cite{christen2021automatic,cook2004universality}}\n\n\\paragraph{Vision Transformers.}\nVision Transformers \\cite{dosovitskiy2020image} are an adaptation of Transformers \\cite{vaswani2017attention} to vision-based tasks like image classification. In contrast to networks built from convolutional layers, ViTs rely on \\emph{self-attention} mechanisms operating on tokenized inputs. Specifically, input images are divided into non-overlapping patches, then fed to a Transformer after undergoing a linear patch projection with an embedding matrix. While ViTs provide competitive image classification performance, the quadratic computational scaling of global self-attention limits their applicability in high-dimensional domains, \\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  per-pixel dense processing. Recent developments have attempted to alleviate such efficiency limitations \\cite{ali2021xcit,hudson2021generative,arnab2021vivit,fan2021multiscale}, one notable example being Perceiver IO \\cite{jaegle2021perceiver,yifan2021input} with its use of cross-attention. We refer interested readers to a comprehensive survey on ViTs~\\cite{khan2021transformers}.\n\n\n\n\n\n\n\n\n\\section{Vision Transformer Cellular Automata (ViTCA)}\n\\vspace{-.5\\baselineskip}\n\\label{sec:vitca}\n\nBuilding upon NCAs and ViTs, we propose a new class of \\emph{attention-based} NCAs formed using a spatially localized---yet globally organized---self-attention scheme. We detail an instance of this class, ViTCA, by first reviewing its backbone ViT architecture before describing the ``pool sampling''-based training process for the ViTCA update rule (see overview in \\Fig{\\ref{fig:overview}}).\n\n\\paragraph{Input tokenization.}\n\nViT starts by dividing a $C_i\\!\\times\\!H\\!\\times\\!W$ input image \\gt\\ into $N$ non-overlapping $P_H\\!\\times\\!P_W$ patches ($16\\!\\times\\!16$ in the original work~\\cite{dosovitskiy2020image}), followed by a linear projection of the flattened image patches with an embedding matrix $\\mathbf{E} \\in \\mathbb{R}^{L \\times d}$ (\\Fig{\\ref{fig:overview}} \\embed), where $L\\!=\\!C_iP_HP_W$, to produce initial tokens \\pretokens\\ $\\!\\in\\!\\mathbb{R}^{N \\times d}$. Next, a handcrafted positional encoding \\cite{vaswani2017attention} or learned positional embedding \\pe\\ $\\in\\!\\mathbb{R}^{N \\times d}$ \\cite{dosovitskiy2020image} is added to tokens to encode positional information and break permutation invariance. Finally, a learnable class token is appended to the token sequence, resulting with \\tokens\\ $\\!\\in\\!\\mathbb{R}^{(N+1) \\times d}$. For the purposes of our task, we omit this token in all ViT-based models. In ViTCA, the input to the embedding is a flattened cell grid \\cells\\ $\\!\\in\\!\\mathbb{R}^{N \\times L}$ where $L\\!=\\!C_PP_HP_W+C_h$, $C_P\\!=\\!C_i+C_o+C_{\\peplain}$,\\, $C_h$ is the cell hidden size,\\, $C_o$ is the number of output image channels (one or three for grayscale or RGB), and $C_{\\peplain}$ is the positional encoding size when positional encoding is (optionally) concatenated to each cell rather than added to the tokens \\cite{mildenhall2020nerf}.\n\n\\paragraph{Multi-head self-attention (MHSA).}\n\nGiven a sequence of tokens \\tokens, self-attention estimates the relevance of one token to all others (\\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  which image patches are likely to appear together in an image) and aggregates this global information to update each token. This encodes each token in terms of global contextual information, and does so using three learned weight matrices: $\\mathbf{W}_Q\\!\\in\\!\\mathbb{R}^{d \\times d}$, $\\mathbf{W}_K\\!\\in\\!\\mathbb{R}^{d \\times d}$, and $\\mathbf{W}_V\\!\\in\\!\\mathbb{R}^{d \\times d}$. \\tokens\\ is projected onto these weight matrices to obtain Queries $\\mathbf{Q}\\!=\\!$ \\tokens$\\mathbf{W}_Q$, Keys $\\mathbf{K}\\!=\\!$ \\tokens$\\mathbf{W}_K$, and Values $\\mathbf{V}\\!=\\!$ \\tokens$\\mathbf{W}_V$. The self-attention layer output $\\sa\\!\\in\\!\\mathbb{R}^{N \\times d}$ is:\n\n\\input{equations\/sa}\n\n\\emph{Multi-head} self-attention employs many sets of weight matrices, $\\{\\mathbf{W}_{Q_i},$ $\\mathbf{W}_{K_i},$ $\\mathbf{W}_{V_i}\\!\\in\\!\\mathbb{R}^{d \\times (d\/h)}\\!\\mid$  $i\\!=\\!0,...,(h-1)\\}$. The outputs of $h$ self-attention \\emph{heads} are concatenated into $(\\sa_0,$ $...,$ $\\sa_{h-1})\\!\\in\\!\\mathbb{R}^{N \\times d}$ and projected onto a weight matrix $\\mathbf{W}\\!\\in\\!\\mathbb{R}^{d \\times d}$ to produce $\\mhsa\\!\\in\\!\\mathbb{R}^{N \\times d}$. Self-attention explicitly models global interactions and is more flexible than grid-based operators (\\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  convolutions) \\cite{perez2019turing,cordonnier2019relationship}, but its quadratic cost in time and memory limits its applicability to high resolution images.\n\n\\paragraph{Spatially localizing self-attention.}\n\nThe global nature of self-attention directly conflicts with the spatial locality constraint of CAs; in response, we limit the connectivity structure of the attention operation to each cell's neighbourhood. This can be accomplished by either masking each head's attention matrix ($\\mathbf{A}\\!=\\!\\texttt{softmax}(\\cdots) \\in \\mathbb{R}^{N \\times N}$ in Eq.\\ \\ref{eq:sa}) with a banded matrix representing local connectivity (\\emph{e.g}\\onedot, } \\def\\Eg{\\emph{E.g}\\onedot,  \\Fig{\\ref{fig:overview}} \\localize), or more efficiently,\n\n\\input{equations\/localize_attn} Here, we assume top-left-to-bottom-right input flattening.\nInstead of explicitly computing the global self-attention matrix $\\mathbf{A}\\!\\in\\!\\mathbb{R}^{N \\times N}$ then masking it, this approach circumvents the $\\mathcal{O}(N^2d)$ computation in favour of an $\\mathcal{O}(N\\!M\\!d)$ alternative that indexes the necessary rows and columns \\emph{during} self-attention. The result is a localized self-attention matrix $\\mathbf{A}^{\\!\\star}\\!\\in\\!\\mathbb{R}^{N \\times M}$, where $M\\!=\\!N_HN_W\\!\\ll\\!N$. As we show in our experiments, ViTCA is still capable of global self-attention despite its localization, by leveraging stored state information across cells and their global self-organization during CA iterations (\\Fig{\\ref{fig:teaser}}).\n\nFollowing \\mhsa\\ is a multilayer perceptron (\\Fig{\\ref{fig:overview}} \\mlp) with two layers and a GELU non-linearity. We apply Layer Normalization (LN) \\cite{ba2016layer} before \\mhsa\\ and \\mlp, and residual connections afterwards, forming a single encoding block. We use an MLP head (\\Fig{\\ref{fig:overview}} \\mlphead) to decode to a desired output, with LN applied to its input, finalizing the ViTCA update rule $F_\\theta$. In our experiments, ViT's \\mlphead\\ decodes directly into an image output whereas ViTCA decodes into update vectors added to cells.\n\n\\vspace{-.5\\baselineskip}\n\\subsection{Update rule training procedure}\n\\vspace{-.5\\baselineskip}\n\\label{subsec:update_rule}\n\nTo train the ViTCA update rule, we follow a ``pool sampling''-based training process \\cite{palm2022variational,mordvintsev2020growing} along with a curriculum-based masking\/noise schedule when corrupting inputs. During odd training iterations, we uniformly initialize a minibatch of cells \\cells\\ $\\!=\\!(\\cellsplain_1,...,\\cellsplain_b)$ with constant values (0.5 for output channels, 0 for hidden---see Appendix \\ref{subsec:extended_ablation} for alternatives), then inject the masked input \\maskedinput\\ (see \\Sec{\\ref{subsec:denoising_autoencoding}}). After input injection, we asynchronously update cells ($\\sigma\\!=\\!50\\%$ update rate) using $F_\\theta$ for $T\\!\\sim\\!\\mathcal{U}\\{8,32\\}$ recurrent iterations. We retrieve output \\cellsoutput\\ from the cell grid and apply an $L_1$ loss against the ground truth \\gt. We also apply overflow losses to penalize cell output values outside of [0,1] and cell hidden values outside of [-1,1]. We use $L_2$ normalization on the gradient of each parameter in $\\theta$. After backpropagation, we append the updated cells and their ground truths to a pool \\pool\\ which we then shuffle and truncate up to the first \\poolsize\\ elements. During even training iterations, we retrieve a minibatch of cells and their ground truths from \\pool\\ and process them as above. This encourages $F_\\theta$ to guide cells towards a stable fixed-point. \\Alg{\\ref{alg:vitca_training}} in Appendix \\ref{sec:appendix} details this process.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{ACKNOWLEDGMENT}\nThis work is supported by the National Key program for S\\&T Research and\nDevelopment (No. 2019YFA0307700 and No. 2016YFA0401100), the National Natural Science Foundation\nof China (Nos. 11504215, 11774387, 11874246, 11834015, 11974383), the Strategic Priority Research Program of the Chinese Academy of Sciences (No. XDB21010400), and the Science and Technology Department of Hubei Province (No. 2019CFA035).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\nOwing to the lack of channel reciprocity in the frequency division duplex (FDD) mode, most proposals for FDD massive MIMO systems  \nenvision some sort of feedback from the user equipment for downlink channel state acquisition at the base stations \\cite{ad2013,dec2015,miretti18,miretti18SPAWC,xie2016overview,hag2018multi,dai2018}. If traditional channel feedback mechanisms are employed, and the number of antennas is  large, downlink channel estimation in FDD massive MIMO systems may incur a prohibitive feedback overhead. Therefore, approaches for reducing this overhead have received a great deal of attention in recent years. A promising approach is to design downlink pilots based on information about the downlink covariance matrix, which is estimated from the uplink covariance matrix \\cite{ad2013,dec2015,miretti18,miretti18SPAWC,xie2016overview,hag2018multi,dai2018}. In this study, we address this estimation problem, hereafter called the uplink-downlink conversion problem.\n\nAlthough channel reciprocity is lost in FDD massive MIMO systems, estimating the downlink covariance matrix from the uplink covariance matrix is possible by exploiting a different form of reciprocity, the so-called reciprocity of the angular power spectrum \\cite{miretti18,miretti18SPAWC,xie2016overview,hag2018multi}. The basic assumption behind this form of reciprocity is that the average receive\/transmit power at a unit of angle (hereafter called {\\it angular power spectrum}) at an antenna array is frequency invariant because, if the frequency separation is not too large, the scattering environments are the same for both the uplink and downlink channels. Building upon this characteristic of wireless channels, researchers have proposed to estimate the angular power spectrum from the uplink covariance matrix, with the intent to use this estimate in models of the antenna array to recover the downlink covariance matrix \\cite{miretti18,miretti18SPAWC,hag2018multi,dai2018} .\n\nIn particular,  the algorithms in \\cite{miretti18,miretti18SPAWC} estimate the angular power spectrum with set-theoretic methods that can easily include side information written in terms of closed convex sets in a Hilbert space.  Despite working in possibly infinite dimensional spaces, one of the approaches in \\cite{miretti18,miretti18SPAWC} have shown that good uplink-downlink conversion performance can be obtained with a very simple matrix-vector multiplication. In that scheme with remarkably low computational complexity, the matrix is computed only once for the entire system lifetime, and the vector is constructed by rearranging the components of the uplink covariance matrix to be converted. If additional information about the angular power spectrum is used, the studies in \\cite{miretti18,miretti18SPAWC} have also shown that the conversion performance of set-theoretic approaches can be further improved with simple fixed point algorithms that do not appeal to finite-dimensional approximations of the physical models. These approaches can easily take into account the polarization of antennas and real-world impairments (e.g., dissimilarities of the antennas in the array), but performance bounds on the conversion performance have not been considered in \\cite{miretti18,miretti18SPAWC}. \n\n\nMore recently, by using ideal antenna models, the study in \\cite{hag2018multi} has proved that, in uplink-downlink channel covariance conversion based on algorithms that first estimate the angular power spectrum (such as those in \\cite{miretti18,miretti18SPAWC}), some of the components of the downlink covariance matrix can be reliably reconstructed. Based on this observation, that study has derived a scheme in which the angular power spectrum is first estimated by using solvers for nonnegative least square problems (this first step can be interpreted as a finite-dimensional approximation of a particular case of \\cite[Algorithm~2]{miretti18}). This estimate is then used to reconstruct the downlink covariance matrix, and, based on a formal analysis of the reliability of the reconstruction, the authors of \\cite{hag2018multi} have proposed to set to zero the components of the downlink covariance matrix that are not guaranteed to be reliably estimated, in an approach called \\emph{truncation}. However, setting to zero these components is a somewhat heuristic approach that can actually decrease the performance of the reconstruction in some scenarios, as the simulations in \\cite{hag2018multi} have already shown. Furthermore, the reliability analysis does not seem easy to extend to realistic propagation and antenna models such as those in \\cite{miretti18SPAWC}, or to cases where the antenna array response is measured to mitigate modeling errors.\n\n\n\nInspired by the findings in \\cite{hag2018multi}, we derive novel bounds on the reconstruction error of each component of the downlink covariance matrix. Unlike previous results, the proposed performance bounds do not assume any particular antenna array or propagation model. They are based on elementary arguments in Hilbert spaces, and they can be easily applied to the realistic models in \\cite{miretti18SPAWC} (without any changes) or to cases where the array response is measured. Furthermore, the bounds provide insights to improve set-theoretic algorithms. In particular, we show that, if information about the support of the angular power spectrum is used appropriately, the simplest of the algorithms for uplink-downlink conversion in \\cite{miretti18,miretti18SPAWC} can be so effective that performance improvements obtained with more advanced conversion mechanisms are marginal at best (NOTE: the proposed bounds can also be used in the analysis of these complex mechanisms). This result is particularly appealing because that simple algorithm has no parameters to be tuned, and all steps of the enhancements we propose are justified by rigorous arguments.\n\nThis study is structured as follows. In Sect.~\\ref{sect.preliminaries} we introduce the main mathematical concepts, and we prove a simple result (Proposition~\\ref{prop.general_bound}) that is the main mathematical tool used to derive the novel bounds. The general result in Sect.~\\ref{sect.preliminaries} is specialized to the problem of uplink-downlink covariance matrix conversion in Sect.~\\ref{sect.error_bounds}, which also discusses how to exploit the proposed bounds to enhance set-theoretic methods (see Sect.~\\ref{sect.side_info}). To keep the presentation as general as possible, we do not assume any particular antenna array or propagation model in Sect.~\\ref{sect.error_bounds}. A concrete application of the theory developed here is shown in Sect.~\\ref{sect.ULA}.\n\n\n\n\n\\section{Mathematical preliminaries}\n\\label{sect.preliminaries}\nIn the following, we use ${\\mathbb R}_+$ to indicate nonnegative reals. The (coordinate-wise) real and imaginary components of complex vectors or matrices are given by, respectively, $\\mathrm{Real}({\\cdot})$ and $\\mathrm{Imag}(\\cdot)$, and $i$ is the imaginary unit, which is the solution to $i^2=-1$. Given a matrix $\\signal{M}\\in{\\mathbb R}^{T\\times N}$, $\\mathrm{vec}(\\signal{M})\\in{\\mathbb R}^{TN}$ is the vector obtained by stacking the columns of $\\signal{M}$, and $\\signal{M}^\\dagger$ is the Moore-Penrose (pseudo-)inverse of $\\signal{M}$. We denote by $\\mathcal{H}$ a real Hilbert space with inner product $\\innerprod{\\cdot}{\\cdot}$ and norm $\\|\\cdot\\|=\\sqrt{\\innerprod{\\cdot}{\\cdot}}$. Given ${x}\\in\\mathcal{H}$ and a set $C\\subset\\mathcal{H}$, we define $x+C:=\\{h+x \\in\\mathcal{H}~|~h\\in C\\}$. A {\\it linear variety} $V\\subset\\mathcal{H}$ is a set that can be expressed as $V=x+M$ for a vector  $x\\in\\mathcal{H}$ and a subspace $M\\subset\\mathcal{H}$; i.e., $V$ is a translation of the subspace $M$. If $C\\subset \\mathcal{H}$ is a nonempty closed convex set, the projection $P_C:\\mathcal{H}\\to~C$ maps an arbitrary vector $x\\in \\mathcal{H}$ to the uniquely existing solution to the optimization problem $\\mathrm{min.}_{y\\in C} \\|x-y\\|$. The orthogonal complement of a subset $C\\subset \\mathcal{H}$ is the closed subspace given by $C^\\perp:=\\{y\\in\\mathcal{H}~|~(\\forall x\\in C)~\\innerprod{x}{y}=0\\}$, and note that $M\\cup M^\\perp=\\mathcal{H}$ and $(M^\\perp)^\\perp = M$ for any closed subspace $M$. The closure of a set $C\\subset\\mathcal{H}$ is denoted by $\\overline{C}$. A set $S=\\{x_1,\\ldots,x_N\\}\\subset\\mathcal{H}$ is called {\\it linear independent} (respectively, dependent) if the vectors $x_1,\\ldots,x_N$ are linearly independent (respectively, dependent).  Given a function $f:\\Omega\\to {\\mathbb R}$ with  $\\Omega\\subset{\\mathbb R}^N$, we define its support to be the set  $\\mathrm{Supp}(f)=\\{x\\in\\Omega~|~f(x)\\neq 0\\}$. By $L^2(\\Omega)$, $\\Omega\\subset {\\mathbb R}^N$, we denote the space of real-valued square-integrable functions $f:\\Omega\\to{\\mathbb R}$  with respect to the standard Lebesgue measure. \n \n \n Below is a summary of standard results in convex analysis that we use throughout this study. The proof can be found in most standard references on convex and functional analysis (e.g., \\cite{yukawa2010,luen}).\n\n\\begin{fact} \n\t\\label{fact.basic}\n\t\\begin{itemize}\n\t\t\\item[(i)] Let $M\\subset \\mathcal{H}$ be a closed linear subspace. Then  $(\\forall x\\in\\mathcal{H})~ x=P_M(x)+P_{M^\\perp} (x)$.\n\t\tFurthermore, the projection $P_M:\\mathcal{H}\\to M$ onto $M$ is a bounded linear operator with operator norm given by $\\|P_M\\|_\\mathrm{o}:=\\sup_{\\|x\\|=1}{\\|P_M(x)\\|}\\le 1$, and the equality is achieved if $M\\neq\\{0\\}$.\t\t\t\t\n\t\t\\item[(ii)] Let $M\\subset\\mathcal{H}$ be a closed subspace. For a given $u\\in\\mathcal{H}$, consider the closed linear variety $V=u+M$. Then $(\\forall x\\in V)~V=x+M$ and $x=P_V(0)+P_M(x)$.\n\t\t\\item[(iii)] Let $V=\\cap_{k=1}^K\\{x\\in\\mathcal{H}~|~\\innerprod{x}{v_k}=b_k\\}\\neq\\emptyset$, where $(v_k,~b_k)\\in\\mathcal{H}\\times{\\mathbb R}$ for each $k\\in\\{1,\\ldots,K\\}$. Then we have $(\\forall u\\in V)~V=u+{M}^\\perp$, where $M=\\mathrm{span}\\{v_1,\\ldots, v_K\\}$.\n\t\t\n\t\n\t\\end{itemize} \n\\end{fact}\n\n\n\nIn the application described in the next section, we study the performance of algorithms producing an estimate $\\widetilde{\\rho}\\in V$ of $\\rho\\in V$, where $V$ is a closed linear variety generated by the translation of the orthogonal complement $M^\\perp$ of a finite  dimensional subspace $M$ (NOTE: the closed subspace $M^\\perp$ can be infinite dimensional). In that application, we are not directly interested in the error $\\|\\widetilde{\\rho}-\\rho\\|$, but in the approximation of $\\innerprod{\\rho}{y}$ by $\\innerprod{\\widetilde{\\rho}}{y}$ for a given  $y\\in\\mathcal{H}$. The absolute error of the approximation is given by $e:=\\left|\\innerprod{\\widetilde{\\rho}-\\rho}{y}\\right|$, and we show in Proposition~\\ref{prop.general_bound} elementary bounds for $e$ that decouples into the product of two terms. The first term depends on the choice of the algorithm. The second term is algorithm independent, and it depends on system parameters captured by the subspace $M\\subset\\mathcal{H}$. \n\n\\begin{proposition}\n\\label{prop.general_bound} Let $M\\subset\\mathcal{H}$ be a closed subspace, and consider the linear variety $V=\\rho+M^\\perp$ for a given $\\rho\\in\\mathcal{H}$. Suppose that an algorithm produces an estimate $\\widetilde{\\rho}\\in {V}$ of $\\rho\\in {V}$. Then each of the following holds:\n\\begin{enumerate}\n\t\\item[(i)] $(\\forall y\\in \\mathcal{H})$ \n\t\\begin{multline}\n\t\t\\label{eq.decoupled_bound}\n\t\t\\left|\\innerprod{\\widetilde{\\rho}-\\rho}{y}\\right|\\le \\|\\widetilde{\\rho} - \\rho\\|~ \\|y-P_{M}(y)\\| \\\\ = \\|P_{M^\\perp}(\\widetilde{\\rho}) - P_{M^\\perp}(\\rho)\\|~ \\|y-P_{M}(y)\\|  \n\t\\end{multline}\n\t\\item[(ii)] In particular, if $\\widetilde{\\rho}=P_V(0)$, then  $(\\forall y\\in \\mathcal{H})$\n\t\\begin{multline}\n\t\t\\label{eq.bound_algo1}\n\t\t\\left|\\innerprod{\\widetilde{\\rho}-\\rho}{y}\\right|\\le \\|\\rho - P_{M}(\\rho)\\|~ \\|y-P_{M}(y)\\| \n\t\\end{multline}\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\n\t(i) Let $y\\in\\mathcal{H}$ be arbitrary. By assumption, both $\\widetilde{\\rho}$ and $\\rho$ are elements of the linear variety $V$, so we have \n\t\\begin{multline}\n\t\\label{eq.error_x}\n\t\\widetilde{\\rho}-\\rho = P_{M^\\perp}(\\widetilde{\\rho})+P_V(0)-P_{M^\\perp}(\\rho)-P_V(0) \\\\ = P_{M^\\perp}(\\widetilde{\\rho})-P_{M^\\perp}(\\rho)= P_{M^\\perp}(\\widetilde{\\rho}-\\rho)\n\t\\end{multline}\n\t by Fact~\\ref{fact.basic}(i)-(ii).  From $y=P_{M}(y)+P_{M^\\perp}(y)$  (Fact~\\ref{fact.basic}(i)) and the definition of orthogonal complements, we obtain \n\t\\begin{multline*}\n\t\\left|\\innerprod{\\widetilde{\\rho}-\\rho}{y}\\right| = \\left|\\innerprod{P_{M^\\perp}(\\widetilde{\\rho}-\\rho)}{P_{M}(y)+P_{M^\\perp}(y)}\\right| \\\\ = \\left|\\innerprod{P_{M^\\perp}(\\widetilde{\\rho}-\\rho)}{P_{M^\\perp}(y)}\\right|. \n\t\\end{multline*}\n\tA direct application of the Cauchy-Schwartz inequality yields $\\left|\\innerprod{\\widetilde{\\rho}-\\rho}{y}\\right| \\le \\|P_{M^\\perp}(\\widetilde{\\rho}-\\rho)\\|~ \\|P_{M^\\perp}(y)\\|$. The result in \\refeq{eq.decoupled_bound} now follows from \n$P_{M^\\perp}(y) = y - P_{M}(y)$ (Fact~\\ref{fact.basic}(i)) and \\refeq{eq.error_x}.\n\n\t\n\t(ii) By Fact~\\ref{fact.basic}(i)-(ii) and $\\widetilde{\\rho}=P_V(0)$, we deduce $P_{M^\\perp}(\\widetilde{\\rho}) = 0$. Now use the equality $\\rho=P_{M^\\perp}(\\rho)+P_{M}(\\rho)$ (Fact~\\ref{fact.basic}(i)) in \\refeq{eq.decoupled_bound} to obtain the desired result.\n\\end{proof}\n\n\n Proposition~\\ref{prop.general_bound} has a very natural interpretation. If  $\\|\\widetilde{\\rho}-\\rho\\|$ is not too large, then \\refeq{eq.decoupled_bound} shows that the error $e:=\\left|\\innerprod{\\widetilde{\\rho}-\\rho}{y}\\right|$ is small if there exists a vector $u\\in M$ that is sufficiently close to $y\\in\\mathcal{H}$ with respect to the metric $d(u,y)=\\|u-y\\|$. In this case, the choice of the algorithm used to produce $\\widetilde{\\rho}\\in V$ does not play a decisive role in the minimization of the error $e$. Proposition~\\ref{prop.general_bound}(ii) shows the guaranteed performance bound of a simple algorithm for estimating $\\rho$. For this scheme, the error $e$ is small if $\\rho$ or $y$, or both, can be well approximated by vectors in the subspace $M$. \n\n\n\\section{Error bounds for uplink-downlink conversion in FDD MIMO systems}\n\\label{sect.error_bounds}\n\nIn this section, we apply the results of Sect.~\\ref{sect.preliminaries} to the problem of  covariance matrix conversion in FDD massive MIMO systems, which, as mentioned in the introduction, we call the uplink-downlink conversion problem. We first describe the problem in Sect.~\\ref{sect.system_model}, and then we proceed to tailor the bounds in Proposition~\\ref{prop.general_bound} to our particular application in Sect.~\\ref{sect.bounds_array}. In Sect.~\\ref{sect.side_info}, we show how the  bounds can be used to improve the approaches in \\cite{miretti18,miretti18SPAWC}. To keep the discussion as general as possible, we do do not assume any particular array geometry or propagation model in this section. \n\n\\subsection{The uplink-downlink conversion problem}\n\\label{sect.system_model}\n  We consider a single-cell flat-fading wireless system, in which a base station equipped with N antennas exchanges data with a single-antenna user. In the uplink, the base station first estimates the uplink channel covariance matrix $\\signal{R}_\\mathrm{u} = E[\\signal{h}_\\mathrm{u} \\signal{h}_\\mathrm{u}^H] \\in \\mathbb{C}^{N\\times N}$ from   samples of the uplink channel $\\signal{h}_\\mathrm{u}$ and any prior knowledge of this covariance matrix. In the uplink-downlink conversion problem, samples $\\signal{h}_\\mathrm{d}\\in\\mathbb{C}^N$ of the downlink channel are not available at the base station, and the objective is to obtain an estimate of the downlink channel covariance matrix $\\signal{R}_\\mathrm{d} = E[\\signal{h}_\\mathrm{d} \\signal{h}_\\mathrm{d}^H]\\in \\mathbb{C}^{N\\times N}$ directly from the estimate of $\\signal{R}_\\mathrm{u}$. The main challenge for the conversion in FDD MIMO systems is the lack of channel reciprocity. The uplink and downlink channels use different frequencies, so their statistics are also different, which in turn implies that $\\signal{R}_\\mathrm{u}\\neq\\signal{R}_\\mathrm{d}$. However, $\\signal{R}_\\mathrm{d}$ and $\\signal{R}_\\mathrm{u}$ are related. In particular, estimating $\\signal{R}_\\mathrm{d}$ from $\\signal{R}_\\mathrm{u}$ is possible by using the so-called reciprocity of the angular power spectrum, which we now formally describe.\n  \n  For typical frequency separation gaps, the real and imaginary parts of each component of $\\signal{R}_\\mathrm{u}$ and $\\signal{R}_\\mathrm{d}$ can be seen as the result of an inner product in an infinite dimensional Hilbert space $(\\mathcal{H},\\innerprod{\\cdot}{\\cdot})$, with the vectors and the inner product taking a particular form that depends on system parameters such as the antenna polarization, array geometry, and the propagation model, among others \\cite{miretti18,miretti18SPAWC}.  More precisely, let $r_{\\mathrm{u}, k}\\in{\\mathbb R}$ and $r_{\\mathrm{d}, k}\\in{\\mathbb R}$ denote one of the $k\\in\\mathcal{I}:=\\{1,\\ldots, 2N^2\\}$ components of, respectively, $[\\mathrm{Real}(\\signal{R}_\\mathrm{u})~\\mathrm{Imag}(\\signal{R}_\\mathrm{u})]\\in {\\mathbb R}^{N\\times 2N}$ and $[\\mathrm{Real}(\\signal{R}_\\mathrm{d})~\\mathrm{Imag}(\\signal{R}_\\mathrm{d})]\\in {\\mathbb R}^{N\\times 2N}$. It has been shown in \\cite{miretti18,miretti18SPAWC} that, for each $k\\in\\mathcal{I}$ and for a given inner product $\\innerprod{\\cdot}{\\cdot}$ that depends on the system model, we have \n  \\begin{align}\n  \\label{eq.upr}\n  r_{\\mathrm{u}, k}=\\innerprod{\\rho}{g_{\\mathrm{u}, k}}\n  \\end{align}\n  and \n  \\begin{align}\n  \\label{eq.dlr}\n  r_{\\mathrm{d}, k}=\\innerprod{\\rho}{g_{\\mathrm{d}, k}},\n  \\end{align}\n   where $\\rho\\in\\mathcal{H}$ is the unknown frequency independent function called {\\it angular power spectrum}, and $g_{\\mathrm{u}, k}\\in\\mathcal{H}$ and $g_{\\mathrm{d}, k}\\in\\mathcal{H}$ are known uplink and downlink functions related to the antenna array responses (see Sect.~\\ref{sect.ULA} for a concrete example). Intuitively, the angular power spectrum shows the average angular power density that an array receives from the user at a given azimuth (and possibly elevation) angle. In the literature \\cite{miretti18,miretti18SPAWC,xie2016overview,hag2018multi}, it is assumed to be the same for both the uplink and downlink channels, which is the phenomenon we call reciprocity of the angular power spectrum.\n  \n   With the above explanations, we can summarize the set-theoretic approaches in \\cite{miretti18,miretti18SPAWC} (and some of the approaches in \\cite{hag2018multi}) to the uplink-downlink conversion problem with the following two steps:\n  \\begin{itemize}\n  \t\\item[(i)] We first obtain an estimate $\\widetilde{\\rho}\\in\\mathcal{H}$ of $\\rho\\in\\mathcal{H}$ from the equations $r_{\\mathrm{u}, k}=\\innerprod{\\rho}{g_{\\mathrm{u}, k}}$ ($k\\in\\mathcal{I}$), and possibly known properties of $\\rho\\in\\mathcal{H}$, by using set-theoretic methods.\n  \t\\item[(ii)] With $\\widetilde{\\rho}\\in\\mathcal{H}$, we obtain an estimate $\\widetilde{r}_{\\mathrm{d}, k}\\in{\\mathbb R}$ of ${r}_{\\mathrm{d}, k}\\in{\\mathbb R}$ for each $k\\in\\mathcal{I}$ by computing $\\widetilde{r}_{\\mathrm{d}, k}=\\innerprod{\\widetilde{\\rho}}{g_{\\mathrm{d}, k}}$.\n  \\end{itemize}\n One of the contributions of the next section is to derive conditions guaranteeing that, given $k\\in\\mathcal{I}$, the estimate $\\widetilde{r}_{\\mathrm{d},k}$ of ${r}_{\\mathrm{d},k}$ in step (ii) is accurate even if the estimate $\\widetilde{\\rho}$ of $\\rho$ in step (i) is inaccurate. These conditions will be used to derive simple techniques to improve set-theoretic methods addressing the problem in step (i).\n\n\n \n \n \\subsection{Bounds for the error of UL-DL covariance conversion with general arrays}\n \\label{sect.bounds_array}\n  We now derive performance bounds for the set-theoretic approaches described above. To this end, we assume that the uplink covariance matrix $\\signal{R}_\\mathrm{u}$ (and hence $(r_{\\mathrm{u}, k})_{k\\in\\mathcal{I}}$) is perfectly estimated. By recalling that covariance matrices have structure (they are at least Hermitian), the number of different equations in \\refeq{eq.upr} and \\refeq{eq.dlr} is strictly less than $|\\mathcal{I}|=2N^2$ ($|\\mathcal{I}|$ denotes the cardinality of $\\mathcal{I}$). Therefore, many repeating equations can be removed, but for brevity this simple operation is not considered in this section. \n\nTo proceed with the bounds, we define  \n\\begin{align}\n\\label{eq.setSp}\nS^\\prime=\\{g_{\\mathrm{u},1},\\ldots, g_{\\mathrm{u},|\\mathcal{I}|}\\}\\subset\\mathcal{H}\n\\end{align}\nto be the set corresponding to the uplink functions in \\refeq{eq.upr}. The angular power spectrum $\\rho\\in\\mathcal{H}$ is related to $S^\\prime$ by the fact that, from \\refeq{eq.upr} and Fact.~\\ref{fact.basic}(ii)-(iii), we have  $\\rho\\in V^\\prime:=\\cap_{k\\in\\mathcal{I}}\\{x \\in\\mathcal{H}~|~\\innerprod{x}{g_{\\mathrm{u}, k}} = r_{\\mathrm{u}, k}\\}=\\rho+\\mathrm{span}(S^\\prime)^\\perp$. Intuitively, the linear variety $V^\\prime\\subset\\mathcal{H}$ is the set containing all angular power spectrum functions that produce the same uplink covariance matrix.\n\nTo include in the analysis any prior information about $\\rho\\in\\mathcal{H}$ expressed in terms of closed linear varieties or closed subspaces, we assume that\n\\begin{align}\n\\label{eq.Vdprime}\n\\rho\\in V^{\\prime\\prime}:=\\cap_{k=1}^Q\\{x\\in\\mathcal{H}~|~\\innerprod{x}{v_k}=b_k\\} \n\\end{align}\nand that the tuples $\\{(v_k, b_k)\\}_{k=1,\\ldots,Q}\\subset \\mathcal{H}\\times {\\mathbb R}$ used to construct the linear variety $V^{\\prime\\prime}$ ($V^{\\prime\\prime}$ is a subspace if $b_1=\\ldots=b_Q=0$) are  known. We now define $V:=V^\\prime\\cap V^{\\prime\\prime}$ and construct a new set $S\\subset\\mathcal{H}$ containing all vectors in $S^\\prime$ in \\refeq{eq.setSp} and all vectors in $S^{\\prime\\prime}:= \\{v_{1},\\ldots, v_{Q}\\}$; i.e.,  \n\\begin{align}\n\\label{eq.setS}\nS:=S^\\prime\\cup S^{\\prime\\prime} \\subset\\mathcal{H}.\n\\end{align}\n\nBy recalling that a nonempty intersection of closed linear varieties is a closed linear variety, the set $V=V^\\prime \\cap V^{\\prime\\prime}\\ni\\rho$ defined above is a closed linear variety $V$ that can be equivalently written as $V=\\rho+M^\\perp$, where $M\\subset\\mathcal{H}$ is the closed subspace $M:=\\mathrm{span}(S)$. With these operations, we are now exactly in the setting of Proposition~\\ref{prop.general_bound}. Before we proceed with the specialization of this proposition to the problem of uplink-downlink conversion, we first show that the projection $P_M:\\mathcal{H}\\to M$ is easy to compute numerically.\n\nTo simplify the notation, denote by $x_1, \\ldots, x_{L}$ the $L=|\\mathcal{I}|+Q$ vectors in the set $S$ in \\refeq{eq.setS}, and define the following matrix:\n\n\\begin{align}\n\\label{eq.matrixg}\n\\signal{G} = \\left[\\begin{matrix}\\innerprod{x_1}{x_1}&\\cdots&\\innerprod{x_1}{x_{L}}\\\\\n\\vdots&\\ddots&\\vdots \\\\\n\\innerprod{x_{L}}{x_1} & \\cdots &  \\innerprod{x_{L}}{x_{L}}\n\\end{matrix}\\right] \\in{\\mathbb R}^{L \\times L}.\n\\end{align}\n\n\nWith the above definitions, we can use arguments similar to those in \\cite[Ch. 3]{luen}\\footnote{Here we do not assume $S$ to be a linearly independent set.} to show that the projection from $y\\in\\mathcal{H}$ onto the closed subspace $M$ is given by:\n\\begin{align}\n\\label{eq.projm}\nP_M:\\mathcal{H}\\to M:y\\mapsto \\sum_{k=1}^{L}\\alpha_k x_{k},\n\\end{align}\nwhere $\\signal{\\alpha}=[\\alpha_1,\\ldots,\\alpha_{L}]^T \\in{\\mathbb R}^{L}$ is any solution to $\\signal{G}\\signal{\\alpha}=\\signal{z}$, and $\\signal{z}=[\\innerprod{x_1}{y} \\ldots \\innerprod{x_{L}}{y}]^T\\in{\\mathbb R}^{L}.$\n\n\nAs it will soon become clear, in the proposed performance bounds we are specially interested in the estimation error $\\|g_{\\mathrm{d}, k}-P_M(g_{\\mathrm{d}, k})\\|$ for each $k\\in\\mathcal{I}$, which can be easily computed as shown in the following standard result. We omit the proof for brevity, but it can be easily obtained by using \\refeq{eq.projm}.\n\n\\begin{proposition}\n\t\\label{proposition.error_qdk} Let $P_M(y)\\in M=\\mathrm{span}(\\{x_1,\\ldots,x_{L}\\})$ be the approximation in the subspace $M$ of an arbitrary vector $y\\in\\mathcal{H}$. Then the approximation error $\\|y-P_M(y)\\|$ is given by \n\t\\begin{align*}\n\t \\|y-P_M(y)\\|=\\sqrt{(\\|y\\|^2 - \\signal{z}^T \\signal{G}^{\\dagger} \\signal{z})},\n\t \\end{align*}\n\t  where  $\\signal{z}=\\signal{G}\\signal{\\alpha}\\in{\\mathbb R}^L$, $\\signal{\\alpha}\\in{\\mathbb R}^L$, and $\\signal{G}\\in{\\mathbb R}^{L\\times L}$ are as defined above.\n\\end{proposition}\n\nNow, let\n\\begin{align}\n\\label{eq.qmatrix}\n\\signal{Q} := \\left[\\begin{matrix}\\innerprod{x_1}{g_{\\mathrm{d},1}}&\\cdots&\\innerprod{x_1}{g_{\\mathrm{d},|\\mathcal{I}|}}\\\\\n\\vdots&\\ddots&\\vdots \\\\\n\\innerprod{x_{L}}{g_{\\mathrm{d},1}} & \\cdots &  \\innerprod{x_{L}}{g_{\\mathrm{d},|\\mathcal{I}|}}\n\\end{matrix}\\right] \\in{\\mathbb R}^{L\\times |\\mathcal{I}|}.\n\\end{align}\nThe proposed error bounds for uplink-downlink conversion are shown in the next corollary. \n\n\\begin{Cor}\n\t\\label{cor.general_bound} Denote by $\\signal{q}_k\\in{\\mathbb R}^{L}$ the $k$th column of the matrix $\\signal{Q}$ in \\refeq{eq.qmatrix}, and let $\\signal{G}$ be as defined in \\refeq{eq.matrixg}. Suppose that $\\tilde{\\rho}\\in V$ is an estimate of the angular power spectrum $\\rho\\in V$ obtained by a given algorithm, where $V=\\rho+M^\\perp$ is a linear variety containing, in particular, all angular power spectrum functions that produce the same uplink covariance matrix, and $M\\subset\\mathcal{H}$ is the closed subspace $M=\\mathrm{span}(S)$ with $S$ as defined in \\refeq{eq.setS}. Further, assume that $\\|\\rho\\|\\le B$ for some $B\\in{\\mathbb R}$. Let $\\innerprod{\\widetilde{\\rho}}{g_{\\mathrm{d}, k}}=\\widetilde{r}_{\\mathrm{d}, k}$ be the estimate of the $k$th ($k\\in\\mathcal{I}$ ) component $r_{\\mathrm{d}, k}$ of the downlink covariance matrix $\\signal{R}_\\mathrm{d}$. Then the estimation error $e_k := |\\widetilde{r}_{\\mathrm{d}, k}-r_{\\mathrm{d}, k}|$ for each $k\\in\\mathcal{I}$ satisfies the following: \n\t\\begin{itemize}\n\t\t\\item[(i)] If the algorithm used to produce the estimate $\\widetilde{\\rho}\\in V$ also guarantees $\\|\\widetilde{\\rho}\\|\\le B$, then \n\t\t\\begin{multline*}\n\t\t\t(\\forall k\\in\\mathcal{I})~ e_k \\le \\|\\rho-\\widetilde{\\rho}\\|~ \\|g_{\\mathrm{d},k}-P_M(g_{\\mathrm{d},k})\\| \\\\ \\le  2B \\sqrt{(\\|g_{\\mathrm{d}, k}\\|^2 - \\signal{q}_k^T \\signal{G}^{\\dagger} \\signal{q}_k)}.\n\t\t\\end{multline*} \n\t\t\n\t\t\\item[(ii)] Using $\\widetilde{\\rho}=P_V(0)$ as the estimate of the angular power spectrum $\\rho$, we have $(\\forall k\\in\\mathcal{I})$\n\t\t\\begin{multline}\n\t\t\\label{eq.bound_cor}\n\t\te_k \\stackrel{(a)}{\\le} \\|\\rho-P_M(\\rho)\\| ~ \\|g_{\\mathrm{d},k}-P_M(g_{\\mathrm{d},k})\\|\n\t\t\\\\ \\stackrel{(b)}{\\le} \\|\\rho\\|~\\|g_{\\mathrm{d},k}-P_M(g_{\\mathrm{d},k})\\| \\stackrel{(c)}{\\le} B \\sqrt{(\\|g_{\\mathrm{d}, k}\\|^2 - \\signal{q}_k^T \\signal{G}^{\\dagger} \\signal{q}_k)}.\n\t\t\\end{multline} \n\t\\end{itemize}\n\\end{Cor}\n\\begin{proof} \n\t\nThe proof of (i) is immediate from Proposition~\\ref{prop.general_bound}(i), Proposition~\\ref{proposition.error_qdk}, and the triangle inequality. To prove (ii), we note that the inequality in $(a)$ follows from Proposition~\\ref{prop.general_bound}(ii), the inequality in $(b)$ follows from $\\|\\rho-P_M(\\rho)\\|=\\|P_{M^\\perp}(\\rho)\\|\\le \\|P_{M^\\perp}\\|_\\mathrm{o}~\\|\\rho\\| \\le  \\|\\rho\\|$ [see Fact~\\ref{fact.basic}(i)], and the inequality in $(c)$ follows from  Proposition~\\ref{proposition.error_qdk} and the assumption $\\|\\rho\\|\\le B$.\n\\end{proof}\n\n\\subsection{Improving the performance of the conversion with information about the support of the angular power spectrum}\n\\label{sect.side_info}\n\nOne of the practical implications of Corollary~\\ref{cor.general_bound} is that, for a given  $k\\in\\mathcal{I}$, any algorithm producing an estimate $\\widetilde{\\rho}\\in V$ of $\\rho\\in V$ is able to approximate reliably the components $r_{\\mathrm{d}, k}$ of the downlink covariance matrix $\\signal{R}_{\\mathrm{d}, k}$ provided that the term $\\|g_{\\mathrm{d},k}-P_M(g_{\\mathrm{d},k})\\|$ is sufficiently small, regardless of how challenging the scenario for the estimation of $\\rho$ may be. By recalling that the projection $P_M(g_{\\mathrm{d},k})\\in\\mathcal{H}$ can be interpreted as the best approximation of $g_{\\mathrm{d},k}$ in the closed subspace $M$, adding to the subspace $M$ functions as similar as possible to $g_{\\mathrm{d},k}$  is a natural idea to decrease the estimation error bound $2B\\|g_{\\mathrm{d},k}-P_M(g_{\\mathrm{d},k})\\|$ of $r_{\\mathrm{d}, k}$. In the discussion below, we show a simple technique to design $M$ based on this simple principle.\n\n\n\nThe subspace $M$ is by definition the span of $S=S^\\prime\\cup S^{\\prime\\prime}$, where $S^\\prime=\\{g_{\\mathrm{u},1},\\ldots,g_{\\mathrm{u},|\\mathcal{I}|}\\}$ is the set of uplink functions and $S^{\\prime\\prime}=\\{v_1,\\ldots,v_Q\\}$ is the set of functions resulting from any prior knowledge about $\\rho$ (see \\refeq{eq.Vdprime}). Therefore, a simple means to include in the subspace $M$ functions that are close to each of the downlink functions $(g_{\\mathrm{d},k})_{k\\in\\mathcal{I}}$ is to make the uplink functions $(g_{\\mathrm{u},k})_{k\\in\\mathcal{I}}$ as similar as possible to the downlink functions $(g_{\\mathrm{d},k})_{k\\in\\mathcal{I}}$. Alternatively, we can also include in the set $S^{\\prime\\prime}$ functions that are as similar as possible to the downlink functions $(g_{\\mathrm{d},k})_{k\\in\\mathcal{I}}$ (NOTE: including $(g_{\\mathrm{d},k})_{k\\in\\mathcal{I}}$ directly while guaranteeing $\\rho\\in V$ is difficult).  The first approach, which corresponds to the design of uplink and downlink functions, may not be always possible because it typically entails changes in hardware (e.g., changes in the inter-antenna spacing) or other modifications in standardized system parameters (e.g., operating frequencies). Therefore, here we focus on the second approach; namely, the construction of an appropriate set $S^{\\prime\\prime}$, or, equivalently, the corresponding linear variety or subspace $V^{\\prime\\prime}$ in \\refeq{eq.Vdprime}. To derive the sets, we further assume the following:\n\n\\begin{itemize}\n\t\\item[(A1)] The angular power spectrum $\\rho\\in\\mathcal{H}$, the downlink functions $(g_{\\mathrm{d},k})_{k\\in \\mathcal{I}}\\subset\\mathcal{H}$, and the uplink functions $(g_{\\mathrm{u},k})_{k\\in \\mathcal{I}}\\subset\\mathcal{H}$ are functions in a Hilbert space of functions in $L^2(\\Omega)$, or, as in \\cite{miretti18SPAWC}, a Hilbert space $\\mathcal{H}$ of tuples in $L^2(\\Omega)\\times L^2(\\Omega)$ (NOTE: extensions to different Hilbert spaces is straightforward),~\\footnote{In these Hilbert spaces, which are used in Sect.~\\ref{sect.ULA}, we typically work with classes of equivalent functions, with the equivalence relation between two functions $f$ and $g$ defined by $f\\sim g\\Leftrightarrow \\|f-g\\|=0$. Equalities such as $f=g$ should be understood as equalities between the classes, not to the particular functions (in a pointwise sense) because $f$ and $g$ can differ, for example, in a countable set in their domains.}  where $\\Omega\\subset{\\mathbb R}^K$. \\\\\n\t\\item[(A2)] There exists a known measurable set $C_\\mathrm{S}\\subset \\Omega$ such that $\\mathrm{Supp}(\\rho)\\subset C_\\mathrm{S}\\neq \\emptyset$ for the angular power spectrum functions $\\rho\\in\\mathcal{H}$ that can be observed in the system. Intuitively, the set $C_\\mathrm{S}$ is a superset of $\\mathrm{Supp}(\\rho)$ for which $\\theta\\notin C_\\mathrm{S}$ implies $\\rho(\\theta)=0$.\n\\end{itemize}\nAssumption A1 is very natural. It is satisfied in many realistic models representing the angular power spectrum in practical systems. In these models, the set $\\Omega$ has the interpretation of azimuth and elevation angles \\cite{miretti18,miretti18SPAWC}. Assumption A2 is system dependent, but it may be valid in scenarios where signals of users impinging on the antenna array are not likely to have any significant power at certain angles, which are used for the construction of $C_\\mathrm{S}$.\n\nWe now proceed to show how support information of $\\rho$ can be used to design the subspace $M$ by using arguments that have a strong theoretical justification. To this end, consider the closed subspace\n \\begin{align*}\n\\mathcal{K}:=\\overline{{\\{x\\in\\mathcal{H}~|~(\\forall\\theta\\in  C_\\mathrm{S})~ x(\\theta)=0\\}}}.\n\\end{align*}\n The projection $P_\\mathcal{K}:\\mathcal{H}\\to\\mathcal{K}$ from $v\\in\\mathcal{H}$ onto $\\mathcal{K}$ is the function given by (we omit the proof for brevity):\n\\begin{align*}\n\\mathcal{H}\\ni P_{\\mathcal{K}}(v):\\Omega\\to{\\mathbb R}:\\theta\\mapsto\\begin{cases}\n0,&\\text{if }\\theta\\in C_{\\mathrm{S}}, \\\\\n v(\\theta) & \\text{otherwise}.\n\\end{cases}\n\\end{align*}\nSince $P_{\\mathcal{K}}(\\rho) = 0$ from the assumption $\\mathrm{Supp}(\\rho) \\subset C_\\mathrm{S}$ and the definition of the subspace $\\mathcal{K}$, we have $\\rho\\in \\mathcal{K}^\\perp$, and thus\n\\begin{align} \n\\label{eq.support_info}\n(\\forall v\\in\\mathcal{H})\\innerprod{P_\\mathcal{K}(v)}{\\rho}=0.\n\\end{align}\n  In particular, using the downlink functions as the function $v$ in \\refeq{eq.support_info} yields\n\\begin{align}\n\\label{eq.projgk}\n(\\forall k\\in\\mathcal{I})\\innerprod{P_\\mathcal{K}(g_{\\mathrm{d},k})}{\\rho}=0.\n\\end{align}\nWe have now reached the point to show the closed subspace $V^{\\prime\\prime}$ we propose to represent the prior knowledge about the support of $\\rho$. More precisely, in light of \\refeq{eq.projgk}, we use \n\\begin{align*}\nV^{\\prime\\prime}=\\cap_{k\\in\\mathcal{I}}\\{y\\in\\mathcal{H}~|~\\innerprod{y}{v_k}=0\\}\\ni \\rho,\n\\end{align*}\n where $v_k:=P_\\mathcal{K}(g_{\\mathrm{d},k})$ for each $k\\in \\mathcal{I}$. This choice is intuitively appealing because we add to the set $S$ in \\refeq{eq.setS} all vectors $(v_k)_{k\\in{\\mathcal{I}}}$ in $\\mathcal{K}$ that best approximate (with respect to the metric $d(x,y)=\\|x-y\\|$) the downlink functions $(g_{\\mathrm{d},k})_{k\\in{\\mathcal{I}}}$, and we recall from the above discussion that, for each $k\\in \\mathcal{I}$, the estimation error $|\\widetilde{r}_{\\mathrm{d},k}-r_{\\mathrm{d},k}|$ decreases as the ability to represent $g_{\\mathrm{d},k}$ with functions in $M=\\mathrm{span}(S)$ improves. Note that we could further improve the reliability of the conversion by repeating the above procedure to include in $S$ additional functions of the form $P_\\mathcal{K}(v)$ with $v\\in\\mathcal{H}$ [e.g., the functions $(P_\\mathcal{K}(g_{\\mathrm{u},k}))_{k\\in\\mathcal{I}}$]. Alternatively, we could also change the definition of inner products to consider only functions in $L^2(C_\\mathrm{S})$. These approaches can be numerically unstable in large antenna arrays if the information about the support of $\\rho$ is erroneous and appropriate mitigation techniques are not applied, but we leave this discussion to a future study because of the space limitation. \n\n  \n\n\n\nAll the above improvements are available for the simple approach using $\\widetilde{\\rho}=P_V(0)$ as the estimate of $\\rho$. This approach is particularly interesting because, as shown in the study in \\cite{miretti18}, which has not considered the enhancements discussed above, the whole process of estimating the angular power spectrum and using this estimate to reconstruct the downlink covariance matrix can be done with a simple matrix-vector multiplication. This important feature is not lost with the enhancements proposed in this subsection. More precisely, denote by $\\widetilde{\\signal{R}}_\\mathrm{d}$ the estimate of the downlink covariance matrix $\\signal{R}_\\mathrm{d}$, and recall that $\\widetilde{r}_{\\mathrm{d},1} = \\innerprod{P_V(0)}{g_{\\mathrm{d},1}},\\ldots,\\widetilde{r}_{\\mathrm{d},|\\mathcal{I}|}=\\innerprod{P_V(0)}{g_{\\mathrm{d},|\\mathcal{I}|}}$ represent the real and imaginary parts of the components of $\\widetilde{\\signal{R}}_\\mathrm{d}$. By ordering the functions  $(g_{\\mathrm{d},k})_{k\\in\\mathcal{I}}$ appropriately, and by mimicking the steps in \\cite[Sect~3.1]{miretti18}, we verify that uplink-downlink channel covariance conversion can be performed with the following simple linear operation:\n\n\\begin{align}\n\\label{eq.algorithm1}\n\\mathrm{vec}[\\mathrm{Real}(\\widetilde{\\signal{R}}_\\mathrm{d})~\\mathrm{Imag}(\\widetilde{\\signal{R}}_\\mathrm{d})] = \\signal{Q}^T\\signal{G}^{\\dagger}\\underline{\\signal{r}}=\\signal{A}{\\signal{r}},\n\\end{align}\nwhere $\\underline{\\signal{r}} = [{\\signal{r}}^T, 0,\\ldots, 0]^T\\in{\\mathbb R}^{L}$, ${\\signal{r}}:=[r_{\\mathrm{u},1},\\ldots,r_{\\mathrm{u},|\\mathcal{I}|}]^T\\in{\\mathbb R}^{|\\mathcal{I}|}$, and $\\signal{A}\\in{\\mathbb R}^{|\\mathcal{I}|\\times |\\mathcal{I}|}$ is the matrix obtained by keeping only the first $|\\mathcal{I}|$ columns of the matrix $\\signal{Q}^T\\signal{G}^{\\dagger}$. Note that these matrices depend on only the support information and the array response, so they need to be computed only once.\n\nBefore we finish this section, it is also worth noticing that, by increasing the subspace $M$ with support information about $\\rho$ as described above, we also decrease the algorithm error term $\\|\\rho-P_M(\\rho)\\|$ in the bound (a) in \\refeq{eq.bound_cor}, thus further improving the reliability of the conversion.\n\n \n \\section{Example: Uniform linear arrays}\n \\label{sect.ULA}\n\n \nWe now further specialize the results in the previous section to uniform linear arrays. This particular choice enables us to relate the analysis in the previous sections to existing results in the literature that, unlike our approaches, do not seem easy to extend to schemes exploiting information about the structure of the angular power spectrum or to systems where the functions $(g_{\\mathrm{u},k})_{k\\in\\mathcal{I}}$ and $(g_{\\mathrm{d},k})_{k\\in\\mathcal{I}}$ are determined by measurements instead of models.\n\n\\subsection{System Model and bounds without support information} \n \tIn a uniform linear array with $N$ antennas, under very mild assumptions \\cite{haghighatshoar2017massive,xie2016overview}, the uplink and downlink channel covariance matrices for typical frequency gaps are given by $\\signal{R}_\\mathrm{u}:=\\signal{R}(f_\\mathrm{u})\\in\\mathbb{C}^{N\\times N}$ and $\\signal{R}_\\mathrm{d}:=\\signal{R}(f_\\mathrm{d})\\in\\mathbb{C}^{N\\times N}$, where $f_\\mathrm{u}\\in{\\mathbb R}_+$ and $f_\\mathrm{d}\\in{\\mathbb R}_+$ are, respectively, the uplink and downlink frequencies; \n \t\\begin{align*}\n \t\\signal{R}(f) = \\int_{-\\pi\/2}^{\\pi\/2} \\rho(\\theta) \\signal{a}(\\theta, f)\\signal{a}(\\theta, f)^H\\mathrm{d}\\theta\n \t\\end{align*}\n \t(the integral should be understood coordinate-wise) is the channel covariance matrix for a given frequency $f$;\t${\\rho:[-\\pi\/2, \\pi\/2]\\to{\\mathbb R}_+}$ is the angular power spectrum; \n \t\\begin{align}\n \t\\label{eq.integral}\n \t\\begin{array}{rl}\n \t\\signal{a}:[-\\frac{\\pi}{2}, \\frac{\\pi}{2}]\\times{\\mathbb R}_+\\to&\\mathbb{C}^N \\\\\n \t(\\theta, f)\\mapsto&\\left[1,e^{i2\\pi \\frac{f}{c}d\\sin\\theta},\\ldots,e^{i2\\pi \\frac{f}{c}d (N-1)\\sin\\theta}\\right]\n \t\\end{array}\n \t\\end{align}\n \tis the array response for a given angle $\\theta$ and frequency $f$; $c$ is the speed of the wave propagation; and $d$ is the inter-antenna spacing. \n \t\n \tIn real physical systems, we can safely assume that $\\rho$ is an element of the Hilbert space $(\\mathcal{H},\\innerprod{\\cdot}{\\cdot})$ of Lebesgue (real) square-integrable functions $\\mathcal{H}=L^2([-\\pi\/2,\\pi\/2])$ equipped with the inner product $(\\forall \\rho \\in\\mathcal{H})(\\forall g\\in\\mathcal{H})\\innerprod{\\rho}{g}=\\int_{-\\pi\/2}^{\\pi\/2}\\rho(\\theta)g(\\theta)\\mathrm{d}\\theta$. As a result, by fixing $f_\\mathrm{u}$, in light of \\refeq{eq.integral} the functions $(g_{\\mathrm{u},k})_{k\\in \\{1,\\ldots,2N^2\\}}$ in \\refeq{eq.upr} are obtained from the equality $(\\forall\\theta\\in[-\\pi\/2,\\pi\/2])$\n \t\\begin{multline}\n \t\\label{eq.ordering}\n \t[g_{\\mathrm{u},1}(\\theta),\\ldots,g_{\\mathrm{u},2N^2}(\\theta)]^T = \\\\ \\mathrm{vec}\\left(\\left[\\begin{matrix}\\mathrm{Real}(\\signal{a}(\\theta, f_\\mathrm{u})\\signal{a}(\\theta, f_\\mathrm{u})^H)\\\\ \\mathrm{Imag}(\\signal{a}(\\theta, f_\\mathrm{u})\\signal{a}(\\theta, f_\\mathrm{u})^H)\\end{matrix}\\right]\\right).\n \t\\end{multline}\n \tThe downlink functions $(g_{\\mathrm{d},k})_{k\\in\\{1,\\ldots,2N^2\\}}$ in \\refeq{eq.dlr} are obtained analogously by considering the downlink frequency $f_\\mathrm{d}$ in \\refeq{eq.ordering}.\n \t\n \t In uniform linear arrays, the covariance matrices are Hermitian and Toeplitz \\cite{haghighatshoar2017massive,miretti18,hag2018multi}, so, with the ordering in \\refeq{eq.ordering}, we can consider only the functions $g_{\\mathrm{u},1},\\ldots,g_{\\mathrm{u},2N}$ responsible for the first column of $\\signal{R}_\\mathrm{u}$ because knowledge of this column is enough to reconstruct all elements of $\\signal{R}_\\mathrm{u}$. For the same reason, we use only the downlink functions $g_{\\mathrm{d},1},\\ldots,g_{\\mathrm{d},2N}$. By doing so, the set $S^\\prime$ in \\refeq{eq.setSp} is given by $S^\\prime=\\{g_{\\mathrm{u},1},\\ldots, g_{\\mathrm{u},2N})\\}\\subset\\mathcal{H}$, and we can redefine the index set $\\mathcal{I}$ accordingly; i.e., $\\mathcal{I}:=\\{1,\\ldots, 2N\\}$. \n \t  \n \t Without any information about the support of $\\rho$, we have $S=S^\\prime$, and we can use the results in \\cite[Sect.~4.1]{miretti18} to compute the algorithm independent term  $\\|g_{\\mathrm{d},k}-P_M(g_{\\mathrm{d},k})\\|$ ($k\\in\\mathcal{I}$) of the bounds in Corollary~\\ref{cor.general_bound} by using Bessel functions of the first kind, order zero, which we denote by $J_0:{\\mathbb R}\\to{\\mathbb R}$. In particular, the bound in the last inequality in Corollary~\\ref{cor.general_bound}(ii) reduces to\n \t\\begin{align}\n \t \t \\label{bound.specific}\n (\\forall k\\in \\mathcal{I})~ \t e_k\\le B\n\\sqrt{(\\|g_{\\mathrm{d},k}\\|^2 - \\signal{q}_k^T \\signal{G}^{\\dagger} \\signal{q}_k)},\n \t\\end{align}\n \twhere \n \t\\begin{multline*}\n\t \t\\|g_{\\mathrm{d},k}\\|^2=\\\\ \\begin{cases}\n\t \t\\dfrac{\\pi}{2} \\left(1+J_0\\left(4\\pi \\dfrac{f_\\mathrm{d}}{c} d (k-1) \\right)\\right)&\\text{ if } 1\\le k \\le N \\\\\n\t\t\\dfrac{\\pi}{2} \\left(1-J_0\\left(4\\pi \\dfrac{f_\\mathrm{d}}{c} d (k-N-1) \\right)\\right)&\\text{ otherwise, } \n\t \t\\end{cases}\n\t\\end{multline*}\n\\begin{align*}\n\\signal{G}=\\dfrac{\\pi}{2}\\left[\\begin{matrix}\n\\signal{G}_\\mathrm{r}& \\signal{0} \\\\ \n\\signal{0}& \\signal{G}_\\mathrm{j}\n\\end{matrix}\\right],  \\signal{Q}=[\\signal{q}_1,\\ldots,\\signal{q}_{2N}]=\\dfrac{\\pi}{2}\\left[\\begin{matrix}\n\\signal{Q}_\\mathrm{r}& \\signal{0} \\\\ \n\\signal{0}& \\signal{Q}_\\mathrm{j},\n\\end{matrix}\\right]\n\\end{align*}\nand the components of the $n$th row and $m$th column of the matrices $\\signal{G}_\\mathrm{r},\\signal{G}_\\mathrm{j},\\signal{Q}_\\mathrm{r},\\signal{Q}_\\mathrm{j}\\in{\\mathbb R}^{N\\times N}$ are given by $\\signal{G}_{\\mathrm{r},nm}=J_0(x_{nm})+J_0(y_{nm})$, $\\signal{G}_{\\mathrm{j},nm}=J_0(x_{nm})-J_0(y_{nm})$, $\\signal{Q}_{\\mathrm{r},nm}=J_0(p_{nm})+J_0(q_{nm})$,\n$\\signal{Q}_{\\mathrm{j},nm}=J_0(p_{nm})-J_0(q_{nm})$ with $$x_{nm}=2\\pi~d~\\dfrac{f_\\mathrm{u}}{c}(n-m),~y_{nm}=2\\pi~d~\\dfrac{f_\\mathrm{u}}{c}(n+m-2),$$ $$p_{nm}=2\\pi d\\left(\\dfrac{f_\\mathrm{u}(n-1)}{c}-\\dfrac{f_\\mathrm{d}(m-1)}{c}\\right),$$\nand $$q_{nm}=2\\pi d\\left(\\dfrac{f_\\mathrm{u}(n-1)}{c}+\\dfrac{f_\\mathrm{d}(m-1)}{c}\\right).$$\n\n\\subsection{Numerical experiments}\nFor a concrete example of the bounds, we use an antenna array with the configuration in Table~\\ref{table.parameters}. As discussed in \\cite{hag2018multi}, this configuration is particularly challenging for uplink-downlink conversion for two main reasons: (i) the uplink frequency is lower than the downlink frequency, and (ii) the antenna spacing is larger than half of the wavelength $c\/(2f_\\mathrm{d})$ of the higher frequency $f_\\mathrm{d}$, so we have the undesirable phenomenon known as grating lobes \\cite{van2004optimum,hag2018multi}. We show below that this challenging scenario for uplink-downlink conversion can be formally verified with the simple bounds in \\refeq{bound.specific}, and the problems for uplink-downlink conversion can be mitigated with information about the support of the angular power spectrum.\n\n\\begin{table}\n\t\\caption{Parameters of the uniform linear array}\n\t\\label{table.parameters}\n\t\\begin{center}\n\t\t\\begin{tabular}{cc}\n\t\t\t\\hline \\\\\n\t\t\tNumber of antennas $(N)$ & 30 \\\\\n\t\t\tUplink frequency $(f_\\mathrm{u})$ & 1.8~MHz \\\\\n\t\t\tDownlink frequency $(f_\\mathrm{d})$ & 1.9~MHz \\\\\n\t\t\tSpeed of wave propagation $(c)$ & $3\\cdot 10^8$~m\/s \\\\\n\t\t\tAntenna spacing $(d)$ & $1.05~\\dfrac{c}{2f_\\mathrm{u}}$ \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\n\n\\end{table}\n\nTo illustrate the theoretical gains that can be achieved with the technique discussed in Sect.~\\ref{sect.side_info}, we assume that $\\mathrm{Supp}(\\rho)\\subset C_\\mathrm{S}=[0,~\\pi\/2]$, and $C_\\mathrm{S}$ is known. For all simulations in this section, we use only the scheme in Corollary~\\ref{cor.general_bound}(ii) for the estimation of $\\rho$ because of its low computational complexity, as discussed in Sect.~\\ref{sect.side_info}. \n\nIn Fig.~\\ref{fig.bounds_theory}, assuming $B=1$, we show the bounds in the last inequality in \\refeq{eq.bound_cor} with and without support information (SI). For the computation of the former bound, we use the expressions in \\refeq{bound.specific}. For the latter bound, we construct the matrices $\\signal{G}$ and $\\signal{Q}$ in \\refeq{eq.matrixg} and \\refeq{eq.qmatrix} by computing integrals numerically, unless the integral falls into one of the cases computed  in \\refeq{bound.specific}. From Fig.~\\ref{fig.bounds_theory}, it is clear that, without any support information, the estimate $\\widetilde{r}_{\\mathrm{d},k}$  can be unreliable for many indices $k\\in\\mathcal{I}$, which is also in accordance with the results in \\cite{hag2018multi}. In contrast, with support information, all estimates $(\\widetilde{r}_{\\mathrm{d},k})_{k\\in\\mathcal{I}}$ of the components of the downlink covariance matrix are reliable, even if the estimate $\\widetilde{\\rho}=P_V(0)$ of the angular power spectrum $\\rho$ is not necessarily accurate. \n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=.9\\columnwidth]{bounds_theory.pdf}\n\t\t\\caption{Theoretical error bounds for the estimates of the components of the downlink covariance matrix.}\n\t\t\\label{fig.bounds_theory}\n\t\\end{center}\n\\end{figure}\n\n\n\n\nTo illustrate the above fact, consider the following example for $\\rho:[-\\pi\/2,\\pi\/2]\\to{\\mathbb R}_+$:\n\n\\begin{align} \n\\label{eq.aps}\n\\rho(\\theta)= n{e}^{-{|\\theta-.5|}\/{.05}} + 4n~{e}^{-{|\\theta-1.4|}\/{.05}},\n\\end{align}\nwhere $n\\in{\\mathbb R}_+$ is a normalizing constant chosen to guarantee that $\\|\\rho\\|=1$. This particular $\\rho$ can be interpreted as coming from a user with two multipath components at angles 0.5~rad and 1.4~rad. Note that we have violated the assumption of the support of $\\rho$, but the signal energy outside $C_\\mathrm{S}$ is small compared to the energy in $C_\\mathrm{S}$, so we can expect the bounds shown in Fig.~\\ref{fig.bounds_theory} to be accurate. This fact is illustrated in Fig.~\\ref{fig.bounds_practice}, which shows the absolute error $(|r_{\\mathrm{d},k}-\\widetilde{r}_{\\mathrm{d},k}|)_{k\\in\\mathcal{I}}$ (with $\\tilde{r}_{\\mathrm{u},k}$ computed by using \\refeq{eq.algorithm1}) of the estimates. Note that, by including support information, uplink-downlink conversion has been performed reliably for all components of the downlink covariance matrix, even though the estimate of angular power spectrum (APS) is not necessarily accurate, as depicted in Fig.~\\ref{fig.aps}.\n\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=.9\\columnwidth]{bounds_practice.pdf}\n\t\t\\caption{Estimation error of the components of the downlink covariance matrix with the angular power spectrum in \\refeq{eq.aps}.}\n\t\t\\label{fig.bounds_practice}\n\n\t\\end{center}\n\n\\end{figure}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=.9\\columnwidth]{aps.pdf}\n\t\t\\caption{Estimates of the angular power spectrum (APS) with the approach in Corollary~\\ref{cor.general_bound}(ii).}\n\t\t\\label{fig.aps}\n\t\\end{center}\n\n\\end{figure}\n\n\n\\section{Summary and Conclusions}\nRecent work has proved that, without side information about the angular power spectrum, existing algorithms in the literature may not be able to estimate reliably all components of the downlink covariance matrix. In this study we have introduced alternative reliability bounds that are based on elementary arguments in infinite dimensional Hilbert spaces. The main advantages of the proposed analysis are the simplicity and the generality. Unlike previous results, the bounds shown here can be straightforwardly used to analyze the performance of algorithms that exploit information about the support of the angular power spectrum in challenging scenarios that take into account the polarization of antennas and physical impairments of real antenna arrays. To illustrate a possible application of the bounds, we have improved a simple set-theoretic algorithm that does not require any parameter tuning. We have shown that, with coarse information about the angular power spectrum, all components of the downlink covariance matrix can be reliably estimated from the uplink covariance matrix with a simple linear operation. This result suggests that, in some scenarios, the main challenge may be the estimation of the uplink covariance matrix, not necessarily the uplink-downlink conversion problem.\n\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe $d$-dimensional semi-discrete wave equation is given as the following infinite system of second order ordinary differential equations: \n\\begin{equation}\\label{u0}\n\\frac{d^2}{dt^2} u(t)[k] \n-a^2 \\sum_{j=1}^d \\(u(t)[k+e_j] - 2u(t)[k] + u(t)[k-e_j]\\) =0,\\;\\;\n  (t,k)\\in \\R \\times  \\Z^d, \n\\end{equation}\nwhere $u(t)=\\{u(t)[k]\\}_{k\\in \\Z^d}$ and \n$a$ is a positive constant, $\\{e_j,\\ldots,e_d\\}$ is the standard basis of $\\R^d$. \n\\eqref{u0} is a discretization with respect to the space variables for the following $d$-dimensional wave equation: \n\\begin{equation}\\label{w}\n  \\pa_t^2 u(t,x) - a^2 \\sum_{j=1}^d \\pa_{x_j}^2 u(t,x) = 0,\\;\\;\n  (t,x)\\in \\R \\times \\R^d, \n\\end{equation}\nwhere $a$ describes the propagation speed of the wave. \nOne of the most important properties for the wave equation \\eqref{w} is the following equality which is called the energy conservation: \n\\begin{equation}\\label{ec}\n\\tilde{E}(t):=\\int_{\\R^d} |\\pa_t u(t,x)|^2\\,dx\n     +a^2 \\sum_{j=1}^d \\int_{\\R^d} |\\pa_{x_j} u(t,x)|^2\\,dx\n  \\equiv \\tilde{E}(t_0),\\;\\;\n  t,t_0\\in \\R. \n\\end{equation}\nHowever, if the propagation speed $a$ depends on time variable, then \\eqref{ec} does not hold in general. \nOn the contrary, the existence of a solution in the Sobolev space may not be valid if $a(t)$ has singularities like non-Lipschitz continuity or degeneration (see \\cite{CDS,CJS}); \nthus time dependent propagation speed is possible to give a crucial influence to the property of the wave equation. \n\nLet us consider the following Cauchy problem for the wave equation with time dependent propagation speed: \n\\begin{equation}\\label{wC}\n\\begin{cases}\n\\ds{\\pa_t^2 u(t,x) - a(t)^2 \\sum_{j=1}^d \\pa_{x_j}^2 u(t,x) = 0}, \n  & (t,x)\\in \\R_+ \\times \\R^d, \\\\\nu(0,x)=u_0(x),\\;\\; (\\pa_t u)(0,x)=u_1(x), & x\\in \\R^d, \n\\end{cases}\n\\end{equation}\nwhere $\\R_+:=[0,\\infty)$ and \n$a(t)$ satisfies \n\\begin{equation}\\label{a0a1}\n  a_0 \\le a(t) \\le a_1\n\\end{equation}\nfor some positive constants $a_0$ and $a_1$. \nSince the energy conservation does not hold for \\eqref{wC}, \nwe introduce the generalized energy conservation, which is abbreviated as GEC, \nas the following uniform equivalence of the energy with respect to $t$: \n\\begin{equation}\n\\label{tGEC}\n  \\tilde{E}(0) \\lesssim \\tilde{E}(t) \\lesssim \\tilde{E}(0),\n  \\;\\; t \\in \\R_+,\n\\end{equation} \nwhere $f \\lesssim g$ with positive functions $f$ and $g$ denotes that \nthere exists a positive constant $C$ such that $f \\le Cg$. \nWe will use the notations $g \\gtrsim f$ and $f \\simeq g$ if \n$f\\lesssim g$ and $g\\lesssim f \\lesssim g$ hold, respectively. \nMoreover, we will use $C$ to denote a generic positive constant. \n\nLet $a\\in C^m(\\R_+)$ with $m\\ge 1$ and $(u_0,u_1)\\in H^2\\times H^1$. \nThen a unique time global classical solution of \\eqref{wC} exists, and the following energy estimate holds: \n\\begin{equation}\\label{tE-a'L1}\n  \\tilde{E}(t) \\le \\exp\\(\\frac{2}{a_0}\\int^t_0 |a'(s)|\\,ds\\) \\tilde{E}(0),\n  \\;\\; t\\in \\R_+. \n\\end{equation}\nIt follows that GEC holds if $a'\\in L^1(\\R_+)$. \nOn the other hand, the success or failure of GEC depends on the properties of $a(t)$ and the initial data if $a'\\not\\in L^1((0,\\infty))$. \nIndeed, the following result is known: \n\n\\begin{theorem}[\\cite{RS}]\\label{Thm-RS}\n\\begin{itemize}\n\\item[(i)] \nIf $a\\in C^2(\\R_+)$, $|a'(t)|\\lesssim (1+t)^{-1}$ and $|a''(t)|\\lesssim (1+t)^{-2}$, then GEC is established for the Cauchy problem \\eqref{wC}. \n\\item[(ii)] \nFor any positive and monotone increasing function $\\nu(t)$ satisfying \n$\\lim_{t\\to\\infty}\\nu(t)=\\infty$, the conditions \n$|a'(t)|\\lesssim \\nu(t)(1+t)^{-1}$ and $|a''(t)|\\lesssim \\nu(t)(1+t)^{-2}$ \ndoes not necessarily conclude GEC. \n\\end{itemize}\n\\end{theorem}\n\n\\begin{remark}\n\\eqref{tE-a'L1} is derived by the estimate\n\\begin{equation}\\label{tE'}\n  \\tilde{E}'(t)=2a'(t)a(t) \\sum_{j=1}^d \\int_{\\R^d} |\\pa_{x_j} u(t,x)|^2\\,dx\n  \\le \\frac{2|a'(t)|}{a(t)} \\tilde{E}(t)\n\\end{equation}\nand Gronwall's inequality. \nWe observe from the first equality of \\eqref{tE'} that $\\tilde{E}(t)$ increases and decreases if $a'(t)>0$ and $a'(t)<0$, respectively. \nThat is, time dependent propagation speed causes increase or decrease for the energy. \nFurthermore, we notice that the second inequality is not taken account the sign of $a'(t)$. \nActually, \\eqref{tE'} is obtained with assuming both $a'(t)>0$ and $a'(t)<0$ increase the energy, but Theorem 1.1 (i) is derived with considering some cancellation of the energy which is caused by changing sign of $a'(t)$. \n\\end{remark}\n\nAccording to Theorem \\ref{Thm-RS}, the oscillation speed of $a(t)$, which is described by the order of $|a'(t)|$, is crucial for GEC, and the order of threshold is $(1+t)^{-1}$. \nHowever, GEC is not determined only the order of $|a'(t)|$. \nIndeed, under the additional assumptions to $a(t)$ below, GEC is possible even if $|a'(t)|\\lesssim (1+t)^{-1}$ does not hold. \n\n\\begin{description}\n\\item[(H1)]\nThere exists a positive constant $a_\\infty$ such that either of the following estimates hold: \n\\begin{equation}\\label{stbc-}\n  \\int^\\infty_t|a(s)-a_\\infty|\\,ds \\lesssim (1+t)^{\\al}\n  \\;\\text{ for }\\;\n  \\al < 0\n\\end{equation}\nor\n\\begin{equation}\\label{stbc}\n  \\int^t_0|a(s)-a_\\infty|\\,ds \\lesssim (1+t)^{\\al}\n  \\;\\text{ for }\\;\n  0\\le \\al \\le 1.\n\\end{equation}\n\\item[(H2)]\n$a \\in C^m(\\R_+)$ with $m\\ge 1$ and the following estimates hold for $\\be<1$: \n\\begin{equation}\\label{akc}\n  \\left|a^{(k)}(t)\\right| \\lesssim (1+t)^{-k\\be},\\;\\; k=1,\\ldots,m. \n\\end{equation}\n\\end{description}\n\nThen the following theorem is established: \n\\begin{theorem}[\\cite{H07}]\\label{Thm-H07}\nIf $m\\ge 2$, $a(t)$ satisfies {\\rm (H1)} and {\\rm (H2)} for \n\\begin{equation}\\label{albem}\n  \\be \\ge \\al + \\frac{1-\\al}{m},\n\\end{equation}\nthen GEC is established for the Cauchy problem \\eqref{wC}. \nOn the other hand, if $\\be<\\al$, then GEC does not hold in general. \n\\end{theorem}\n\n\\begin{remark}\nThe right hand side of \\eqref{albem} is smaller as $m$ larger or $\\al$ smaller. That is, the restriction on $\\be$ can be weaker if the differentiability of $a(t)$ is higher or the restriction of (H1) is stronger. \nActually, the Theorem \\ref{Thm-H07} does not conclude the optimality of the condition \\eqref{albem}, but the limit case $m=\\infty$ is nearly optimal for GEC. \n\\end{remark}\n\n\\begin{remark}\nSince the estimate \\eqref{stbc} with $\\al=1$ is trivial, \nTheorem \\ref{Thm-RS} can be considered a special case of Theorem \\ref{Thm-H07} without (H1). \n\\end{remark}\n\n\nLet us consider the following initial boundary value problem: \n\\begin{equation}\\label{wIBV}\n\\begin{cases}\n\\ds{\\pa_t^2 u(t,x) - a^2 \\sum_{j=1}^d \\pa_{x_j}^2 u(t,x) = 0}, \n  & (t,x)\\in \\R_+ \\times \\Om, \\\\\nu(t,x)=0, & (t,x)\\in \\R_+\\times \\pa\\Om,\\\\\nu(0,x)=u_0(x),\\;\\; \\(\\pa_t u\\)(0,x)=u_1(x), & x\\in \\pa\\Om, \n\\end{cases}\n\\end{equation}\nwhere $\\Om$ is a bounded domain of $\\R^d$ with smooth boundary $\\pa\\Om$. \nThen the following energy conservation corresponding to \\eqref{ec} for \\eqref{wC} is established:  \n\\begin{equation*\n\\tilde{E}(t):=\\int_{\\Om} |\\pa_t u(t,x)|^2\\,dx\n     +a^2 \\sum_{j=1}^d \\int_{\\Om} |\\pa_{x_j} u(t,x)|^2\\,dx\n\\equiv \\tilde{E}(t_0). \n\\end{equation*}\nMoreover, if $a$ depends on time variable, then the following theorem, which is corresponding to Theorem \\ref{Thm-H07} in bounded domain, is established: \n\\begin{theorem}[\\cite{H10}]\\label{Thm-H10}\nIf $m\\ge 2$ and $a=a(t)$ satisfies {\\rm (H2)} for \n\\begin{equation}\\label{albem-bdd}\n  \\be \\ge \\frac{1}{m},\n\\end{equation}\nthen GEC is established for the initial boundary problem \\eqref{wIBV}. \n\\end{theorem}\n\nThe condition \\eqref{albem-bdd} in Theorem \\ref{Thm-H10} corresponds to \\eqref{albem} in Theorem \\ref{Thm-H07} with $\\al=0$. \nThis implies that GEC is established without the assumption (H1) even though the oscillation speed is faster on the problem \\eqref{wIBV}. \nBriefly, (H1) and (H2) are required for the estimate of the solution in the time-frequency space for the low and the high frequency part, respectively. \nHowever, (H1) is not necessary for the problem \\eqref{wIBV} because the influence to the low frequency part can be neglected since the largest Dirichlet eigenvalue of the Laplace operator $\\sum_{j=1}^d \\pa_{x_j}^2$ is strictly negative. \nOne of our main interest of the present paper is how the properties (H1) and (H2) for the time dependent propagation speed $a(t)$ relate to the energy estimate for semi-discrete wave equation \\eqref{u0}. \n\nThere are many studies on the discretized wave equations with constant propagation speed, but not many results are known for time dependent propagation model, especially in the case that the propagation speed $a(t)$ is singular to collapse GEC. In \\cite{CR}, an approximation of the discretized wave equation with respect to time variable is studied in the case that $a(t)$ is degenerate and oscillating which was studied in \\cite{CJS}. The result is not directly related to the studies of the present paper; indeed, no approximation to the continuous model will be discussed, but discretization is a useful approach to study the influence of the time dependent propagation speed to the energy of the solution in time-frequency spaces. \n\n\n\n\\section{Discretization and discrete-time Fourier transformation}\n\nLet us consider a discretized model of the Cauchy problem for the wave equation \\eqref{wC} with respect to the space variables $x$. \n\nFor $f = \\{f[k]\\}_{k\\in \\Z^d}$ and $j=1,\\ldots,d$, we denote \nthe forward and the backward difference operators $D_j^+$ and $D_j^-$ by \n\\begin{equation*}\n  D_j^+ f := \\{f[k+e_j]-f[k]\\}_{k\\in\\Z^d}\n  \\;\\text{ and }\\;\n  D_j^- f := \\{f[k]-f[k-e_j]\\}_{k\\in\\Z^d},\n  \\;\\; k\\in \\Z^d,\n\\end{equation*}\nrespectively. \nThen the discrete Laplace operator in $\\Z^d$ is given by \n$\\sum_{j=1}^d D_j^+ D_j^-$, that is, \n\\begin{equation*}\n  \\sum_{j=1}^d D_j^+ D_j^- f[k]\n =\\sum_{j=1}^d \\(f[k+e_j]-2f[k]+f[k-e_j]\\). \n\\end{equation*}\n\nFor the solution $u(t,x)$ of \\eqref{wC}, we consider the $d$-dimensional infinite matrix valued function $u(t)=\\{u(t)[k]\\}_{k\\in \\Z^d}$ as a sampled of $u(t,x)$ on the lattice $\\Z^d$. \nThen a discretized model of \\eqref{wC} is given as the following initial value problem for an infinite system of ordinary differential equations, \nwhich is called the semi-discrete wave equation with time dependent propagation speed: \n\\begin{equation}\\label{u}\n\\begin{cases}\n\\ds{\n  u''(t)[k] \n -a(t)^2 \\sum_{j=1}^d D_j^+ D_j^- u(t)[k] =0, }\n  & (t,k)\\in \\R_+ \\times \\Z^d, \n\\\\[3mm]\n\\ds{\n  u(0)[k]=u_0[k],\\;\\; u'(0)[k]=u_1[k], }\n  &k\\in \\Z^d.\n\\end{cases}\n\\end{equation}\nThen we define the total energy for the solution of \\eqref{u} by \n\\begin{equation*\n  E(t):=\n  \\sum_{k\\in \\Z^d}\\left|u'(t)[k]\\right|^2\n +a(t)^2 \\sum_{j=1}^d \\sum_{k\\in \\Z^d}\\left|D_j^+ u(t)[k]\\right|^2. \n\\end{equation*}\nEvidently, the energy conservation $E(t)\\equiv E(0)$ is valid if the propagation speed $a$ is a constant, but it does not hold in general for variable propagation speed. \nTherefore, the following energy estimate corresponding to \\eqref{tGEC} can be considered: \n\\begin{equation}\\label{GEC}\n  E(t) \\simeq E(0).\n\\end{equation}\nHere \\eqref{GEC} will be also denoted by GEC. \n\nIt is usual to study the energy estimate of \\eqref{wC} and \\eqref{wIBV} in the time-frequency spaces by introducing Fourier transformation and Fourier coefficients with respect to the space variables of the solution rather than the solutions $u(t,x)$ themselves. \nTherefore, we introduce the discrete-time Fourier transformation to study the energy estimate of the solution for \\eqref{u}. \n\n\\begin{definition}\nFor $f=\\{f[k]\\}_{k\\in \\Z^d} \\in l^2(\\Z^d)$ we define the discrete-time Fourier transformation $\\cF_{\\Z^d}[f](\\th)$ by\n\\begin{equation*}\n  \\cF_{\\Z^d}[f](\\th):=\\sum_{k\\in \\Z^d} e^{-\\i k\\cd\\th} f[k],\n  \\;\\; \\th=(\\th_1,\\ldots,\\th_d) \\in \\R^d,\n\\end{equation*}\nwhere $k \\cdot \\th = \\sum_{j=1}^d k_j \\th_j$ and $k=(k_1,\\ldots,k_d)$. \nWe denote that $\\cF_{\\Z^d}[f]=\\hat{f}$ and \n$\\sum_{k\\in \\Z^d}=\\sum_{k}$ without any confusion. \n\\end{definition}\n\nSince the discrete-time Fourier transformation $\\hat{f}(\\th)$ is the $d$-dimensional Fourier series with the Fourier coefficient $\\{f[-k]\\}_{k\\in \\Z^d}$, \n$\\hat{f}$ is a $2\\pi$-periodic function in $\\R^d$. \nThat is, the following equality holds:  \n\\begin{equation*}\n  \\hat{f}(\\th+2k\\pi)=\\hat{f}(\\th)\n\\end{equation*}\nfor any $k\\in \\Z^d$ and $\\th\\in \\T^d$, where $\\T:=[-\\pi,\\pi]$. \nTherefore, we will restrict the domain of $\\cF_{\\Z^d}[\\cd]$ on $\\T^d$. \n\nHere we introduce some lemmas for the discrete-time Fourier transformation, \nwhere the proofs will be introduced in Appendix. \n\\begin{lemma}\\label{TDFT}\nIf $f \\in l^1(\\Z^d)$ then the following equalities are established \nfor $j=1,\\ldots,d$: \n\\begin{equation}\\label{TDFT2}\n  \\cF_{\\Z^d}[D_j^+ f](\\th) = \\(e^{\\i\\th_j}-1\\) \\hat{f}(\\th)\n\\end{equation}\nand\n\\begin{equation}\\label{TDFT3}\n \\cF_{\\Z^d}[D_j^+D_j^- f](\\th) \n  = -4\\(\\sin\\frac{\\th_j}{2}\\)^2\\hat{f}(\\th).\n\\end{equation}\n\\end{lemma}\n\n\n\\begin{lemma}\\label{lemm-Parseval}\nIf $f \\in l^2(\\Z^d)$ then\nthe following Parseval's type equality is established: \n\\begin{equation*\n  \\sum_{k\\in \\Z^d}|f[k]|^2\n =\\frac{1}{(2\\pi)^d}\\int_{\\T^d}|\\hat{f}(\\th)|^2\\,d\\th. \n\\end{equation*}\n\\end{lemma}\n\nFor $\\th\\in \\T^d$ we define $\\xi=\\xi(\\th)$ by \n\\begin{equation}\\label{xith}\n  \\xi(\\th)=(\\xi_1(\\th_1),\\ldots,\\xi_d(\\th_d)),\\;\\;\n  \\xi_j(\\th_j):=2\\sin\\frac{\\th_j}{2}\n  \\;\\;(j=1,\\ldots,d).\n\\end{equation}\nBy the discrete-time Fourier transformation, \\eqref{u} is reduced to the following problem: \n\\begin{equation}\\label{hu}\n\\begin{cases}\n\\partial_t^2 \\hat{u}(t,\\th) \n+a(t)^2 |\\xi(\\th)|^2 \\hat{u}(t,\\th) =0, &\n  (t,\\th)\\in \\R_+ \\times \\T^d, \n\\\\\n  \\hat{u}(0,\\th)=\\hat{u}_0(\\th),\\;\\;\n  \\(\\partial_t \\hat{u}\\)(0,\\th)=\\hat{u}_1(\\th), \n  & \\th\\in \\T^d,\n\\end{cases}\n\\end{equation}\nwhere $\\hat{u}(t,\\th)=\\sum_{k} e^{-\\i k\\cd\\th}u(t)[k]$. \n\nFor the solution $\\hat{u}(t,\\th)$ of \\eqref{hu}, we define the energy density function $\\cE(t,\\th)$ by \n\\begin{equation*\n  \\cE(t,\\th):=\n  \\left|\\pa_t \\hat{u}(t,\\th)\\right|^2\n +a(t)^2 |\\xi(\\th)|^2 \\left|\\hat{u}(t,\\th)\\right|^2.\n\\end{equation*}\nBy Lemma \\ref{lemm-Parseval}, the total energy $E(t)$ is represented by $\\cE(t,\\th)$ as follows: \n\\begin{lemma}\\label{lemm-E-cE}\nIf $u'(t), D_j^+ u(t)\\in l^2(\\Z^d)$ $(j=1,\\ldots,d)$, \nthen the following equality is established: \n\\begin{equation}\\label{EcE}\n  E(t) = \\frac{1}{(2\\pi)^d}\\int_{\\T^d} \\cE(t,\\th)\\,d\\th.\n\\end{equation}\n\\end{lemma}\n\nIn the Cauchy problem of the wave equation \\eqref{wC}, we shall call it a continuous model, not only the total energy $\\tE(t)$ but also the energy density $\\tcE(t,\\xi)$: \n\\begin{equation*\n  \\tcE(t,\\xi):=\n  \\left|\\pa_t \\cF_{\\R^d}[u](t,\\xi)\\right|^2\n +a^2 |\\xi|^2 \\left|\\cF_{\\R^d}[u](t,\\xi)\\right|^2\n\\end{equation*}\nis conserved if the propagation speed $a$ is a constant, \nwhere $\\cF_{\\R^d}[u](t,\\xi)$ denotes the Fourier transformation of $u(t,x)$ with respect to the space variable $x$. \nHowever, the energy density is not conserved for time dependent propagation speed in general. \nIndeed, time dependent propagation speed has the effect of changing the total energy with respect to $t$, and it considered to be a phenomenon that caused by transition of energy across frequencies. \nBasically, the behavior of $\\tcE(t,\\xi)$ is determined by the properties of $a(t)$, and the assumptions for $a(t)$ in Theorem \\ref{Thm-H07} ensure the estimate $\\tcE(t,\\xi)\\simeq \\tcE(0,\\xi)$. \nOn the other hand, the negative results for \\eqref{tGEC} are implied from the unboundedness of $\\tcE(t,\\xi)$. \nIn particular, the non-existence result in the Sobolev space is derived from the increase in the energy in high frequency part which is provided from the non-Lipschitz continuity with very fast oscillation of $a(t)$. \n\nThe energy density $\\cE(t,\\th)$ for the solution of the semi-discrete model \\eqref{hu} is also conserved if the propagation speed $a$ is a constant. \nOn the other hand, if the propagation speed is variable, then the estimate \n\\begin{equation}\\label{GECcE}\n  \\cE(t,\\th) \\simeq \\cE(0,\\th)\n\\end{equation}\ndoes not hold in general as in the case of continuous model, and thus GEC may not be established. \nIt will be natural that the estimate \\eqref{GECcE} is established if $a(t)$ satisfies the same assumptions of Theorem \\ref{Thm-RS} and Theorem \\ref{Thm-H07}. \nHowever, we may expect to prove \\eqref{GECcE} under some weaker assumptions to $a(t)$, because the range of $|\\xi|$ for $\\tcE(t,\\xi)$ is $[0,\\infty)$, but the range of $|\\xi(\\th)|$ for $\\cE(t,\\xi)$ is $[0,2\\sqrt{d}]$. In the other words, unlike the continuous model, the solution cannot have high-frequency energy above a certain level because the solution of semi-discrete model has a finite resolution. \nHere we note that the situation of high-frequency energy for the semi-discrete model is the corresponding to the low-frequency energy for the initial boundary value problem \\eqref{wIBV}. \n\n\\section{Main results}\nLet us generalize the properties \\eqref{stbc} of (H1) and (H2) by positive and monotone increasing functions $\\Th(t)$ and $\\Xi(t)$ on $\\R_+$ as follows: \n\n\\begin{description}\n\\item[(H1*)]\nThere exists a positive constant $a_\\infty$ such that the following estimate holds: \n\\begin{equation}\\label{stb}\n  \\int^t_0|a(s)-a_\\infty|\\,ds \\le \\Th(t).\n\\end{equation}\n\\item[(H2*)]\n$a \\in C^m(\\R_+)$ with $m\\ge 1$ and the following estimates hold for some positive constants $C_k$:\n\\begin{equation}\\label{ak}\n  \\left|a^{(k)}(t)\\right| \\le C_k \\Xi(t)^{-k},\\;\\; k=1,\\ldots,m. \n\\end{equation}\n\\end{description}\n\nFor $\\Th(t)$ and $\\Xi(t)$ we introduce the following conditions corresponding to the conditions $\\al\\le \\be$ and \\eqref{albem}: \n\\begin{description}\n\\item[(H3*)] \nThere exists a positive constant $C_0$ such that \n\\begin{equation}\\label{ThXi}\n\\Th(t) \\le C_0 \\Xi(t).\n\\end{equation}\n\\item[(H4*)]\nFor $m\\ge 2$, $\\Xi(t)^{-m}\\in L^1(\\R_+)$ and the following estimate holds: \n\\begin{equation}\\label{int_Cm}\n\\sup_{t\\ge 0}\\left\\{\\Th(t)^{m-1}\\int_t^\\infty \\Xi(s)^{-m}\\,ds\n\\right\\}<\\infty.\n\\end{equation}\n\\end{description}\n\nThen our first theorem is given as follows: \n\n\\begin{theorem}\\label{Thm1}\nFor the initial value problem of semi-discrete wave equation \\eqref{u}, GEC is established if any of the following \n{\\rm(i)} to {\\rm (iii)} is satisfied: \n\\begin{itemize}\n\\item[{\\rm (i)}]\n{\\rm (H1*)} with $\\sup_{t\\ge 0}\\{\\Th(t)\\}<\\infty$. \n\\item[{\\rm (ii)}]\n{\\rm (H2*)} with $m=1$ and $\\Xi(t)^{-1} \\in L^1(\\R_+)$. \n\\item[{\\rm (iii)}]\n{\\rm (H1*)}, {\\rm (H2*)}, {\\rm (H3*)} and {\\rm (H4*)}. \n\\end{itemize}\n\\end{theorem}\n\nIf we restrict ourselves $\\Th(t)=(1+t)^\\al$ and $\\Xi(t)=(1+t)^\\be$ with $m\\be>1$, then \\eqref{stb}, \\eqref{ak} and \\eqref{int_Cm} \nare the same as \\eqref{stbc}, \\eqref{akc} and \\eqref{albem}, respectively. \nMoreover, \\eqref{ThXi} is valid if \\eqref{albem} holds. \nComparing the assumptions of $a(t)$ in Theorem \\ref{Thm1} with \nTheorem \\ref{Thm-H07} we observe the followings: \n\n\\begin{itemize}\n\\item\nIf $\\al>0$, then GEC is established under the same assumptions for $a(t)$. \n\\item\nThough GEC does not hold for \\eqref{wC} in general if $a(t)$ is not Lipschitz continuous, Theorem \\ref{Thm1} concludes GEC without any assumption of the continuity for $a(t)$ if $\\al=0$. \n\\end{itemize}\n\n\n\\begin{example}\\label{Ex1}\nLet $\\chi \\in C^m(\\R_+)$ be a positive and periodic function. \nFor non-negative constants $p$, $q$ and $r$ we define $a \\in C^m(\\R_+)$ by \n\\begin{equation*}\n  a(t)=1 + (1+t)^{-p} \\chi\\((1+t)^q \\(\\log(e+t)\\)^r\\).\n\\end{equation*}\nThen we see the followings: \n\\begin{equation*}\n  \\int^t_0|a(s)-1|\\,ds \\le \\Th(t) \\simeq\n  \\begin{cases}\n  1 & (p>1), \\\\\n  \\log(e+t) & (p=1), \\\\\n  (1+t)^{-p+1} & (p<1).\n  \\end{cases}\n\\end{equation*}\nMoreover, for any $k=1,\\ldots,m$ we have \n\\begin{align*}\n  \\left|a^{(k)}(t)\\right| \n  \\lesssim \\: & \n  (1+t)^{-p}\\((1+t)^{q-1} \\(\\log(e+t)\\)^{r} \\)^k\n\\\\\n  \\le \\: & \n  \\((1+t)^{\\frac{p}{m}-q+1} \\(\\log(e+t)\\)^{-r} \\)^{-k}\n\\end{align*}\nfor $q>0$ and \n\\begin{align*}\n  \\left|a^{(k)}(t)\\right| \n  \\lesssim  \\: &\n  (1+t)^{-p}\\((1+t)^{-1} \\(\\log(e+t)\\)^{r-1} \\)^k\n\\\\\n  \\le \\: &\n  \\((1+t)^{\\frac{p}{m}+1} \\(\\log(e+t)\\)^{-r+1} \\)^{-k}\n\\end{align*}\nfor $q=0$. \nHence we set \n\\begin{equation}\n  \\Xi(t)=\\begin{cases}\n    (1+t)^{\\frac{p}{m}-q+1} \\(\\log(e+t)\\)^{-r} & (q>0),\\\\\n    (1+t)^{\\frac{p}{m}+1} \\(\\log(e+t)\\)^{-r+1} & (q=0). \n  \\end{cases}\n\\end{equation}\nBy Theorem \\ref{Thm1}, we have GEC if any of the following holds: \n\\begin{itemize}\n\\item $p>1$ (by (i)). \n\\item $m=1$ and $q<p$ (by (ii)). \n\\item $m \\ge 2$, $0<p=q$ and $r=0$ (by (iii)).\n\\item $m \\ge 2$, $p=q=0$ and $0<r\\le 1$ (by (iii)).\n\\end{itemize}\n\\end{example}\n\n\\begin{example\nLet $\\chi_\\ve \\in C^m(\\R_+)$ with $m \\ge 2$ be a $2\\pi$-periodic function with a positive small parameter $\\ve$ satisfying \n$\\chi_\\ve(\\tau)=0$ near $\\tau=0$ and \n\\begin{equation}\n  \\sup_{\\ve>0}\\max_{\\tau\\in[0,2\\pi]}\n  \\left\\{\\frac{1}{\\ve}\n  \\left|\\frac{d^k}{d\\tau^k}\\chi_\\ve(\\tau)\\right|\\right\\}\n  <\\infty\\quad(k=0,1,\\ldots,m).\n\\end{equation}\nFor a positive large constant $\\eta$ and positive constants $\\al$, $\\be$ and $\\ka$ satisfying \n\\begin{equation}\\label{pqka}\n  \\al \\le \\ka \\le 1\n  \\;\\text{ and }\\;\n  \\al+\\frac{1-\\al}{m} \\le \\be \\le \\ka+\\frac{\\ka-\\al}{m},\n\\end{equation}\nwe define the sequences $\\{t_j\\}_{j=1}^\\infty$, $\\{\\ve_j\\}_{j=1}^\\infty$, $\\{\\rho_j\\}_{j=1}^\\infty$ and $\\{\\nu_j\\}_{j=1}^\\infty$ by \n\\begin{equation*}\n  t_j:=\\eta^j,\\;\\;\n  \\ve_j:=t_j^{\\al-\\ka},\\;\\;\n  \\rho_j:=\\eta^{-1}t_j^\\ka\n  \\;\\text{ and }\\;\n  \\nu_j:=\\left[t_j^{-\\be+\\ka+\\frac{\\ka-\\al}{m}}+1\\right]. \n\\end{equation*}\nThen we define $a(t)$ by \n\\begin{equation*}\n  a(t):=\\begin{cases}\n  \\sqrt{1+\\chi_{\\ve_j}\\(\\nu_j\\rho_j^{-1}(t-t_j)\\)} \n    & \\text{ for } \\; t\\in [t_j-\\rho_j,t_j+\\rho_j]\\;\\; (j=1,2,\\ldots),\\\\\n  1 & \\text{ for } \\; t\\in \\R_+ \\setminus \\bigcup_{j=1}^\\infty [t_j-\\rho_j,t_j+\\rho_j].\n\\end{cases}\n\\end{equation*}\nwhere $[\\;]$ denotes the Gauss symbol. \nHere we note that \n\\begin{align*}\n  t_j + \\rho_j + \\rho_{j+1}\n= \\eta^j \\(1 + \\eta^{-j(1-\\ka) -1} + \\eta^{-(j+1)(1-\\ka)}\\)\n\\le 3 \\eta^j \\le \\eta^{j+1} = t_{j+1}\n\\end{align*}\nfor $\\eta \\ge 3$, it follows that \n$t_j + \\rho_j \\le t_{j+1} - \\rho_{j+1}$. \nLet $t\\in[t_{j-1}+\\rho_{j-1},t_j+\\rho_j]$. \nThen we have \n\\begin{equation}\n  \\int^{t}_0 |a(s)-1|\\,ds\n  \\lesssim \\sum_{k=1}^j \\ve_k \\rho_k\n  =\\eta^{-1} \\sum_{k=1}^j \\eta^{k \\al} \\simeq t_j^\\al\n  \\simeq (1+t)^\\al\n\\end{equation}\nand\n\\begin{align*}\n  \\left|a^{(k)}(t)\\right|\n  \\lesssim \\ve_j\\(\\nu_j \\rho_j^{-1}\\)^k \n  \\simeq t_j^{-k\\be-(\\ka-\\al)\\(1-\\frac{k}{m}\\)}\n  \\le t_j^{-k\\be}\n  \\simeq (1+t)^{-k\\be}, \n\\end{align*}\nit follows that \\eqref{stb}, \\eqref{ak} and \\eqref{ThXi} are established with \n\\begin{equation}\n  \\Th(t) \\simeq (1+t)^\\al\\;\\text{ and }\\;\n  \\Xi(t) = (1+t)^\\be.\n\\end{equation}\nMoreover, by \\eqref{pqka} and noting $\\al+(1-\\al)\/m>1\/m$, we have \n$\\Xi(t)^{-m}\\in L^1(\\R_+)$ and \n\\begin{align*}\n  \\Th(t)^{m-1}\\int^\\infty_t \\Xi(s)^{-m}\\,ds\n  \\lesssim (1+t)^{\\al(m-1)-m\\be+1} \\lesssim 1,\n\\end{align*}\nthus \\eqref{int_Cm} is established. \nTherefore, $a(t)$ satisfies (iii) of Theorem \\ref{Thm1}. \n\\end{example}\n\nIf $a \\not\\in C^1(\\R_+)$, then \\eqref{wC} is not $L^2$ well-posed in general, hence the energy estimate $\\tE(t) \\lesssim \\tE(0)$ cannot be expected. \nHowever, $\\tE(t)$ is not necessarily unbounded if \\eqref{wC} is not $L^2$ well-posed. \nFor example, if $a(t)$ is a H\\\"older continuous function, then \\eqref{wC} is not $L^2$ well-posed in general but the Gevrey well-posed. \nThat is, if the initial data are functions of the Gevrey class of suitable order, then the solution is a function of the Gevrey class, too; \nhence $\\tE(t)$ is bounded (see \\cite{CDS}). \nIf $a(t)$ does not satisfy \\eqref{albem}, then GEC does not hold in general. \nHowever, if the initial data is a function of the Gevrey class of suitable order and \\eqref{stbc-} holds, then there exists a positive constant $C(u_0,u_1)$, which depends on not only the initial energy $\\tE(0)$ but also a norm of the initial data in the Gevrey class, such that the total energy $\\tE(t)$ is bounded as follows (see \\cite{EFH}): \n\\begin{equation}\\label{tBE}\n  \\tE(t) \\le C(u_0,u_1). \n\\end{equation}\n\nFaster oscillation of $a(t)$, which is described as smaller $\\be$ not to satisfy \\eqref{albem}, may increase the high frequency part of the energy for the solution of \\eqref{wC} and cause GEC to collapse. \nHowever, since the high frequency part of the initial energy is small with the initial data in the Gevrey class, $\\tE(t)$ can stay bounded against the effect from the faster oscillation of $a(t)$. \nOur second theorem concludes a corresponding estimate to \\eqref{tBE} for the solution of the semi-discrete wave equation \\eqref{u} under some weaker assumptions than that for $a(t)$ in Theorem \\ref{Thm1}. \n\n\nFor a positive and strictly increasing function $\\La(t)$ on $\\R_+$ satisfying $\\lim_{t\\to\\infty}\\La(t)=\\infty$ such that \n\\begin{equation}\\label{La-infty}\n  \\frac{\\Th(t)}{\\La(t)\n  \\text{ is monotone increasing and }\n  \\lim_{t\\to\\infty}\\frac{\\Th(t)}{\\La(t)}=\\infty,\n\\end{equation}\nwe introduce the following conditions that are alternative to (H3*) and (H4*): \n\\begin{description}\n\\item[(H5*)]\nThere exists a positive constant $C_0$ such that \n\\begin{equation*\n  \\La(t)\\le C_0 \\Xi(t). \n\\end{equation*}\n\\item[(H6*)]\nFor $m\\ge 2$, $\\Xi(t)^{-m}\\in L^1(\\R_+)$ and the following estimate holds:  \n\\begin{equation*\n  \\sup_{t\\ge 0}\\left\\{\\La(t)^m \\Th(t)^{-1}\\int^\\infty_t \\Xi(s)^{-m}\\,ds\n  \\right\\}<\\infty.\n\\end{equation*}\n\\end{description}\n\nHere we note that if $\\La(t) \\simeq \\Th(t)$, then (H5*) and (H6*) are coincide with (H3*) and (H4*), respectively. \nOn the other hand, it is possible that (H6*) holds, but (H4*) does not hold if \\eqref{La-infty} is valid. \nOur second theorem is given as follows: \n\n\\begin{theorem}\\label{Thm2}\nLet $m\\ge 2$. \nIf $\\Th(t)$, $\\Xi(t)$ and $\\La(t)$ satisfy {\\rm (H1*)}, {\\rm (H2*)}, \\eqref{La-infty}, {\\rm (H5*)} and {\\rm (H6*)}, then there exists a positive constant $N_0$ such that the following estimate is established: \n\\begin{equation*\n  \\cE(t,\\th) \\lesssim \n  \\exp\\(2|\\xi(\\th)| \\Th\\(\\La^{-1}\\(\\frac{N_0}{|\\xi(\\th)|}\\)\\)\\)\n  \\cE(0,\\th). \n\\end{equation*}\nConsequently, denoting \n\\begin{equation*}\n  U(N,u_0,u_1):=\n  \\int_{\\T^d}\n  \\exp\\(2|\\xi(\\th)| \\Th\\(\\La^{-1}\\(\\frac{N}{|\\xi(\\th)|}\\)\\)\\)\n  \\cE(0,\\th)\n  \\,d\\th\n\\end{equation*}\nfor $N>0$, we have the followings: \n\\begin{itemize}\n\\item[{\\rm (i)}] \nIf $U(N,u_0,u_1)<\\infty$ holds for any $N\\ge 1$, then there exists a positiv constant $N_0$ such that the energy estimate \n\\begin{equation}\\label{Ebdd}\n  E(t) \\lesssim U(N_0,u_0,u_1)\n\\end{equation} \nis established. \n\\item[{\\rm (ii)}] \nThere exists a positive constant $N_0$, and the energy estimate\n\\eqref{Ebdd} is established for any $(u_0,u_1)$ satisfying \n$U(N_0,u_0,u_1)<\\infty$. \n\\end{itemize}\n\n\\end{theorem}\n\n\\begin{remark}\nDenoting $t=\\La^{-1}(r^{-1})$ for $r>0$, we have \n\\begin{equation}\\label{La-infty2}\n  \\mu(r):=r \\Th\\(\\La^{-1}\\(\\frac{1}{r}\\)\\)\n  =\\frac{\\Th(t)}{\\La(t)}\n  \\nearrow \\infty \\quad (r \\to +0)\n\\end{equation}\nby \\eqref{La-infty} since $\\La^{-1}(r^{-1})$ is positive and strictly decreasing with respect to $r$. \nIt follows that \n\\begin{equation*}\n  \\lim_{\\th \\to 0}\n  |\\xi(\\th)|\\Th\\(\\La^{-1}\\(\\frac{N}{|\\xi(\\th)|}\\)\\)\n =\\lim_{|\\xi|\\to 0}N\\mu\\(\\frac{N}{|\\xi|}\\) = \\infty. \n\\end{equation*}\nOn the other hand, if $\\Th(t) \\lesssim \\La(t)$, then we have\n\\[\n  \\sup_{\\th\\in\\T^d\\setminus\\{0\\}}\\left\\{\n  |\\xi(\\th)|\\Th\\(\\La^{-1}\\(\\frac{N}{|\\xi(\\th)|}\\)\\)\n  \\right\\}<\\infty, \n\\]\nit follows that $U(N,u_0,u_1) \\simeq E(0)$ by Lemma \\ref{lemm-E-cE}. \nTherefore, \\eqref{La-infty} is a reasonable assumption for the case that Theorem \\ref{Thm1} cannot be applied. \n\\end{remark}\n\nSince \\eqref{La-infty} provides \n\\[\n  \\lim_{|\\th| \\to 0} |\\xi(\\th)|\\Th\\(\\La^{-1}\\(\\frac{1}{|\\xi(\\th)|}\\)\\)=\\infty, \n\\]\nthe condition\n\\begin{equation}\\label{est_Thm2}\nU(N,u_0,u_1)<\\infty\n\\end{equation}\nin Theorem \\ref{Thm2} requires approximately that the $|\\xi(\\th)|\\hat{u}_0(\\th)$ and $\\hat{u}_1(\\th)$ degenerate at $\\th=0$ in an appropriate order which is determined by $\\Th(t)$ and $\\Xi(t)$. \nThe following examples of $\\Th(t)$, $\\Xi(t)$ and $u_1$ provide \\eqref{est_Thm2} for $d=1$ and $u_0=0$. \n\n\\begin{example}\\label{Ex1Thm2}\nLet $a(t)$ be defined in Example \\ref{Ex1} with $m\\ge 2$, $p=q=0$, \n$r \\ge 1$ and $|\\chi| \\le 1$. \nThen we have \n\\begin{equation*}\n  \\Th(t) = 1+t, \\;\\;\n  \\Xi(t) \\simeq (1+t)\\(\\log(e+t)\\)^{-r+1}, \n\\end{equation*}\nand (H4*) does not hold. \nLet us define $\\La(t)$ by \n\\begin{align*}\n  \\La(t):= \\: & (1+t)\\(\\log(e+t)\\)^{-r+1}\n = \\(\\frac{1+t}{(1+t)^{-m+1}\\(\\log(e+t)\\)^{m(r-1)}}\\)^{\\frac{1}{m}}\n\\\\\n \\simeq \\: &  \\(\\frac{\\Th(t)}{\\int^\\infty_t \\Xi(s)^{-m}\\,ds}\\)^{\\frac{1}{m}}. \n\\end{align*}\nThen \\eqref{La-infty}, (H5*) are (H6*) are valid. \nNoting that $\\La^{-1}(\\tau) \\simeq \\tau (\\log \\tau)^{r-1}$ for $\\tau\\ge e$, \nthere exists a positive constant $M$ such that \n\\begin{align*}\n  \\exp\\(2 |\\xi(\\th)| \\Th\\(\\La^{-1}\\(\\frac{N}{|\\xi(\\th)|}\\)\\)\\)\n  \\lesssim\n  \\exp\\(2M N \\(\\log\\frac{N}{|\\xi(\\th)|}\\)^{r-1}\\)\n\\end{align*}\nfor any $N\\ge 2e$ on $\\T\\setminus\\{0\\}$. \nFor $M_0>0$, we define $u_1=\\{u_1[k]\\}_{k\\in \\Z}$ by \n\\begin{equation*\n  u_1[k]:=\\frac{1}{2\\pi}\\int_\\T\n  \\(\\sin^2 \\frac{\\th}{2}\\)^{\\frac{M_0}{2}} e^{\\i k \\th}\\,d\\th, \n\\end{equation*}\nit follows that\n\\begin{align*}\n  \\cE(0,\\th)\n =\\(\\sin^2 \\frac{\\th}{2}\\)^{M_0}\n =\\(\\frac{N}{2}\\)^{2M_0} \n  \\exp\\(-2M_0\\log \\frac{N}{|\\xi(\\th)|} \\). \n\\end{align*}\nThen we have \n\\begin{align*}\n  U(N,u_0,u_1)\n  \\lesssim \\int_\\T \n  \\exp\\(2M N\\(\\log\\frac{N}{|\\xi(\\th)|}\\)^{r-1}\n       -2M_0\\log \\frac{N}{|\\xi(\\th)|} \\)\n  \\,d\\th.\n\\end{align*}\nIf $1<r<2$ then $U(N,u_0,u_1)<\\infty$ for any $N\\ge 2e$, hence Theorem \\ref{Thm2} (i) can be applied. \nIf $r=1$ then $U(N_0,u_0,u_1)<\\infty$ for $M_0 \\ge MN_0$, hence Theorem \\ref{Thm2} (ii) can be applied if $M_0$ is large enough. \n\\end{example}\n\n\\begin{example}\\label{Ex2Thm2}\nLet $a(t)$ be defined in Example \\ref{Ex1} with $m \\ge 1$, $0\\le p<q<1$ and $r=0$. \nThen we have \n\\begin{equation*}\n  \\Th(t)\\simeq (1+t)^{1-p},\\;\\;\n  \\Xi(t) \\simeq (1+t)^{\\frac{p}{m}-q+1}, \n\\end{equation*}\nand (H4*) dose not hold. \nLet us define $\\La(t)$ by \n\\begin{equation*}\n  \\La(t):=(1+t)^{1-q}\n =\\(\\frac{(1+t)^{1-p}}{(1+t)^{-p-m(1-q)+1}}\\)^{\\frac{1}{m}}\n \\simeq \\(\\frac{\\Th(t)}{\\int^\\infty_t \\Xi(s)^{-m}\\,ds}\\)^{\\frac{1}{m}}. \n\\end{equation*}\nThen \\eqref{La-infty}, (H5*) are (H6*) are valid. \nNoting that $\\Th(\\La^{-1}(\\tau)) \\simeq \\tau^{(1-p)\/(1-q)}$ for $\\tau\\ge 1$, there exists a positive constant $M_0$ such that\n\\begin{align*}\n  |\\xi(\\th)| \\Th\\(\\La^{-1}\\(\\frac{N}{|\\xi(\\th)|}\\)\\)\n  \\le\n  M_0 N^{\\frac{1-p}{1-q}}|\\xi(\\th)|^{-\\frac{q-p}{1-q}}\n\\end{align*}\nfor any $N\\ge 2$ on $\\T\\setminus\\{0\\}$. \nFor $\\rho>0$ and $\\ka \\ge (q-p)\/(1-q)$ we define $u_1=\\{u_1[k]\\}_{k\\in \\Z}$ by \n\\begin{equation*}\n  u_1[k]:=\\frac{1}{2\\pi}\\int_{\\T}\n  \\exp\\(-\\rho\\(\\sin^2\\frac{\\th}{2}\\)^{-\\frac{\\ka}{2}}\\)\n  e^{\\i k \\th}\\,d\\th,\n\\end{equation*}\nit follows that\n\\begin{align*}\n  \\cE(0,\\th)\n =\\exp\\(-2 \\rho \\(\\sin^2 \\frac{\\th}{2}\\)^{-\\frac{\\ka}{2}}\\)\n =\\exp\\(-2^{\\ka+1}\\rho|\\xi(\\th)|^{-\\ka}\\). \n\\end{align*}\nThen we have \n\\begin{align*}\n  U(N,u_0,u_1)\n  \\le\n  \\int_\\T \\exp\\(\n  -2|\\xi(\\th)|^{-\\ka}\\(\n  2^{\\ka}\\rho-M_0 N^{\\frac{1-p}{1-q}}|\\xi(\\th)|^{\\ka-\\frac{q-p}{1-q}}\n  \\)\\)\\,d\\th.\n\\end{align*}\nIf $\\ka > (q-p)\/(1-q)$, then $U(N,u_0,u_1)<\\infty$ for any large $N$, hence Theorem \\ref{Thm2} (i) can be applied. \nIf $\\ka = (q-p)\/(1-q)$ then $U(N_0,u_0,u_1)<\\infty$ for \n$\\rho \\ge 2^{-\\ka}M_0 N_0^{(1-p)\/(1-q)}$, hence Theorem \\ref{Thm2} (ii) can be applied if $\\rho$ is large enough. \n\\end{example}\n\n\n\nBy Theorem \\ref{Thm2}, Example \\ref{Ex2-Thm3} and Lemma \\ref{Lemma1-Thm3} we have the following corollary: \n\\begin{corollary}\nFor $\\nu>1$ and $\\rho>0$ we define the Gevrey classes $\\ga_\\rho^\\nu$ and $\\ga_\\infty^\\nu$ by \n\\begin{equation*}\n  \\ga^\\nu_\\rho:=\\left\\{f\\in C^\\infty\\(\\T^d\\)\\;;\\;\n\\sup_{\\th\\in\\T^d}\\left\\{\\left|\\pa_\\th^\\al f(\\th)\\right|\n  \\frac{\\rho^{|\\al|}}{\\al!^\\nu}\\right\\}<\\infty\\right\\},\n\\;\\;\n  \\ga_\\infty^\\nu:=\\bigcup_{\\rho>0} \\ga_\\rho^\\nu,\n\\end{equation*}\nrespectively. \nLet {\\rm (H1*)} and {\\rm (H2*)} hold for \n$\\Th(t)\\lesssim (1+t)^{1-p}$ and $\\Xi(t)=(1+t)^{\\frac{p}{m}-q+1}$ \nwith $m\\ge 1$ and $0\\le p < q<1$. \n\\begin{itemize}\n\\item[{\\rm (i)}]\nIf $\\xi_j \\hat{u}_0,\\, \\hat{u}_1 \\in \\ga_\\infty^\\nu$ \n$(j=1,\\ldots,d)$ with \n$\\nu<(1-p)\/(q-p)$ and \n\\begin{equation}\\label{eq1-cor}\n  \\(\\pa_\\th^\\al \\xi_j \\hat{u}_0\\)(0) = 0\\;\\;(j=1,\\ldots,d),\\;\\;\n  \\(\\pa_\\th^\\al \\hat{u}_1\\)(0) = 0,\\;\\;\n  \\al\\in \\N_0^d,\n\\end{equation}\nthen there exists a positive constant $N_0$ such that $U(N_0,u_0,u_1)<\\infty$ and \n\\eqref{Ebdd} is established. \n\\item[{\\rm (ii)}] \nThere exist positive constants $\\rho$ and $N_0$ such that \nfor any $\\xi_j \\hat{u}_0,\\, \\hat{u}_1 \\in \\ga_\\rho^\\nu$ \n$(j=1,\\ldots,d)$ with $\\nu=(1-p)\/(q-p)$ satisfying \\eqref{eq1-cor}, \n$U(N_0,u_0,u_1)<\\infty$ and \\eqref{Ebdd} is established.  \n\\end{itemize}\n\\end{corollary}\n\n\n\nLet us consider the conditions for \\eqref{est_Thm2} to the initial data $u_0$ and $u_1$ themselves instead of $\\hat{u}_0$ and $\\hat{u}_1$. \nIn order to describe the conditions, we introduce the logarithmic convexity and the associated functions for sequences.\nThe sequence of positive real numbers $\\{M_j\\}=\\{M_j\\}_{j=0}^\\infty$ is called logarithmically convex if the following estimate holds: \n\\begin{equation*}\n  \\frac{M_{j}}{M_{j-1}} \\le \\frac{M_{j+1}}{M_{j}},\\;\\;\n  j\\in \\N.\n\\end{equation*}\nFor a logarithmically convex sequence $\\{M_j\\}$, we define the associated function $T[\\{M_j\\}](\\tau)$ on $\\R_+$ by \n\\begin{equation*}\n  T[\\{M_j\\}](\\tau):=\\sup_{j\\in \\N}\\left\\{\\frac{\\tau^j}{M_j}\\right\\}. \n\\end{equation*}\nThen our third theorem is given as follows: \n\n\\begin{theorem}\\label{Thm3}\nLet $\\Th(t)$, $\\Xi(t)$ and $\\La(t)$ satisfy the same assumptions in Theorem \\ref{Thm3}, and $\\{M_j\\}$ be a logarithmically convex sequence \nsatisfying \n\\begin{equation}\\label{eq1-Thm3}\n  L\\(N,\\{M_j\\}\\):=\n  \\inf_{\\tau \\ge 1}\\left\\{\n  \\frac{T\\left[\\left\\{\\frac{M_j}{j!}\\right\\}\\right](\\tau)}\n       {\\exp\\(\\tau^{-1}\\Th\\(\\La^{-1}(N \\tau\\)\\)}\n  \\right\\}>0\n\\end{equation}\nfor a positive real number $N$. \n\\begin{itemize}\n\\item[{\\rm (i)}] \nIf $L(N,\\{M_j\\})>0$ for any $N>0$, and $(u_0,u_1)$ satisfies \n\\begin{equation}\\label{eq2-Thm3}\n  \\sum_{k\\in \\Z^d} k^{\\al} D_j^+ u_0[k]=0\n  \\;\\;(j=1,\\ldots,d), \\;\\;\n  \\sum_{k\\in \\Z^d} k^\\al u_1[k]=0, \\;\\;\n  \\al\\in \\N_0^d:=(\\N\\cup\\{0\\})^d\n\\end{equation}\nand \n\\begin{equation}\\label{eq3-Thm3}\n  \\sup_{k\\in \\N^d}\\left\\{\n  \\(\\sum_{j=1}^d|D_j^+ u_0[k]|+ |u_1[k]|\\)|k|^{d+1} T[\\{M_j\\}]\\(\\rho^{-1}|k|\\)\n  \\right\\} <\\infty \n\\end{equation}\nfor a positive constant $\\rho$, \nthen there exists a positive constant $N_0$ such that \n$U(N_0,u_0,u_1)<\\infty$ and the energy estimate \\eqref{Ebdd} is established. \n\\item[{\\rm (ii)}]\nIf the following estimate holds: \n\\begin{equation}\\label{eq4-Thm3}\n  \\lim_{N\\to\\infty} \\inf_{\\tau\\ge \\frac{1}{2\\sqrt{d}}}\n  \\left\\{\\frac{\\Th\\(\\La^{-1}(N\\tau)\\)}{N\\Th\\(\\La^{-1}(\\tau)\\)}\\right\\}\n  =\\infty,\n\\end{equation}\nthen there exist positive constants $\\rho$ and $N_0$ such that \nfor any $(u_0,u_1)$ satisfying \\eqref{eq2-Thm3} and \\eqref{eq3-Thm3}, \n$U(N_0,u_0,u_1)<\\infty$ and \\eqref{Ebdd} is established.  \n\\end{itemize}\n\\end{theorem}\n\nLet us introduce some examples of the choice of $\\{M_j\\}$ in Theorem \\ref{Thm3}. \n\\begin{example\nLet $a(t)$ be define in Example \\ref{Ex1} and Example \\ref{Ex1Thm2}. \nIf $\\{M_j\\} = \\{j! \\exp(b j^{\\si})\\}$ for $b>0$ and $\\si>1$, then there exists a positive constant $b^\\ast$ such that \n\\begin{align*}\n  T\\left[\\left\\{\\frac{M_j}{j!}\\right\\}\\right](\\tau)\n  \\gtrsim \\exp\\(b^\\ast \\(\\log\\tau\\)^{\\frac{\\si}{\\si-1}}\\)\n\\end{align*}\nfor any $\\tau\\ge 1$ (see \\cite{M}). \nBy the consideration of Example \\ref{Ex1Thm2}, there exists a positive constant $M$ such that\n\\begin{align*}\n  \\frac{T\\left[\\left\\{\\frac{M_j}{j!}\\right\\}\\right](\\tau)}\n       {\\exp\\(\\tau^{-1}\\Th\\(\\La^{-1}(N \\tau\\)\\)}\n  \\gtrsim \n  \\exp\\(b^\\ast \\(\\log\\tau\\)^{\\frac{\\si}{\\si-1}}\n       -M\\log(N\\tau)^{r-1}\\). \n\\end{align*}\nTherefore, for any $N>0$, \n\\eqref{eq1-Thm3} is valid for $\\si>1$ if $r\\le 2$, and for $(r-1)\/(r-2)>\\si>1$ if $r>2$. \nHere we note that \\eqref{eq4-Thm3} does not hold. \n\\end{example}\n\n\n\\begin{example}\\label{Ex2-Thm3}\nLet $a(t)$ be define in Example \\ref{Ex1} and Example \\ref{Ex2Thm2}. \nIf $\\{M_j\\} = \\{j!^{\\nu}\\}$ for $\\nu>1$, then there exists a positive constant $q$ such that \n\\begin{align*}\n  T\\left[\\left\\{\\frac{M_j}{j!}\\right\\}\\right](\\tau)\n  \\gtrsim \\tau^{-\\frac12}\\exp\\((\\nu-1) \\tau^{\\frac{1}{\\nu-1}}\\)\n\\end{align*}\nfor any $\\tau\\ge 1$ (see \\cite{M}). \nBy the consideration of Example \\ref{Ex1Thm2}, there exists a positive constant $M$ such that\n\\begin{align*}\n  \\frac{T\\left[\\left\\{\\frac{M_j}{j!}\\right\\}\\right](\\tau)}\n       {\\exp\\(\\tau^{-1}\\Th\\(\\La^{-1}(N \\tau)\\)\\)}\n  \\gtrsim \n  \\tau^{-\\frac12}\\exp\\((\\nu-1) \\tau^{\\frac{1}{\\nu-1}}\n       -M N^{\\frac{1-p}{1-q}}\\tau^{\\frac{q-p}{1-q}}\\). \n\\end{align*}\nTherefore, \\eqref{eq1-Thm3} is valid for any $N>0$ with $\\nu<(1-p)\/(q-p)$, \nthat is, $1\/(\\nu-1)>(q-p)\/(1-q)$, and thus Theorem \\ref{Thm3} (i) can be applied. \nIf $\\nu=(1-p)\/(q-p)$ then \\eqref{eq1-Thm3} is valid for \n$N \\le ((\\nu-1)\/M)^{(1-q)\/(1-p)}$. \nNoting that $\\Th(\\La^{-1}(N\\tau))\/(N\\Th(\\La^{-1}(\\tau)))\\simeq N^{(q-p)\/(1-q)}\\to\\infty$ as $N\\to\\infty$, Theorem \\ref{Thm3} (ii) can be applied. \n\\end{example}\n\n\n\\section{Proof of Theorem \\ref{Thm1}}\n\\subsection{Proof of Theorem \\ref{Thm1} (i)}\nLet $\\xi=\\xi(\\th)$ be defined by \\eqref{xith}. \nDenoting \n\\begin{equation*}\n  v=v(t,\\xi)=\\hat{u}(t,\\th),\n\\end{equation*}\nthe equation of \\eqref{hu} is represented as follows: \n\\begin{equation}\\label{v}\n\\partial_t^2 v(t,\\xi) + a(t)^2 |\\xi|^2 v(t,\\xi) =0, \\;\\;\n  (t,\\xi)\\in \\R_+ \\times [-2,2]^d. \n\\end{equation}\nHere we denote $\\cE(t,\\th)$ by $\\cE(t,\\xi)$: \n\\begin{align*}\n  \\cE(t,\\xi) = |\\pa_t v(t,\\xi)|^2 + a(t)^2 |\\xi|^2 |v(t,\\xi)|^2 \n\\end{align*}\nwithout any confusion. \nWe define $\\cE_\\infty(t,\\xi)$ by \n\\begin{equation}\n  \\cE_\\infty(t,\\xi):=|\\pa_t v(t,\\xi)|^2 + a_\\infty^2 |\\xi|^2 |v(t,\\xi)|^2. \n\\end{equation}\nThen we have \n\\begin{align*}\n  \\pa_t \\cE_\\infty(t,\\xi)\n=\\: &2\\(a_\\infty^2-a(t)^2\\)|\\xi|^2 \\Re\\(\\pa_t v\\, \\ol{v}\\) \n\\\\\n\\le\\: &\n  \\frac{\\left|a_\\infty^2-a(t)^2\\right||\\xi|}{a_\\infty} \n  \\cE_\\infty(t,\\xi)\n\\le\n  \\frac{2a_1}{a_0} \\left|a(t)-a_\\infty\\right||\\xi|\n  \\cE_\\infty(t,\\xi).\n\\end{align*}\nBy Gronwall's inequality and \\eqref{stb}, we have \n\\begin{align*}\n  \\cE_\\infty(t,\\xi)\n  \\le\\:& \\exp\\(\\frac{2a_1}{a_0}\\Th(t)|\\xi|\\) \\cE_\\infty(0,\\xi)\n\\\\\n  \\le\\:& \\exp\\(\\frac{4\\sqrt{d} \\, a_1}{a_0}\\sup_{t\\ge 0}\\{\\Th(t)\\}\\) \n  \\cE_\\infty(0,\\xi) \n  \\simeq \\cE_\\infty(0,\\xi) \n\\end{align*}\nfor any $(t,\\xi)\\in \\R_+ \\times [-2,2]^d$. \nAnalogously, we have \n\\begin{equation*}\n  \\cE_\\infty(t,\\xi) \\ge \n  \\exp\\(-\\frac{4\\sqrt{d} \\, a_1}{a_0}\\sup_{t\\ge 0}\\{\\Th(t)\\}\\) \n  \\cE_\\infty(0,\\xi)\n  \\simeq \\cE_\\infty(0,\\xi). \n\\end{equation*}\nTherefore, by Lemma \\ref{lemm-E-cE} and noting the estimate: \n\\begin{equation}\\label{cEinfcE}\n  \\cE_\\infty(t,\\xi) \\simeq \\cE(t,\\xi)\n\\end{equation}\ndue to \\eqref{a0a1}, we have \n\\begin{equation*}\n  E(t) \\simeq \\int_{\\T^d} \\cE_\\infty(t,\\xi(\\th))\\,d\\th\n  \\simeq  \\int_{\\T^d} \\cE_\\infty(0,\\xi(\\th))\\,d\\th\n  \\simeq E(0).\n\\end{equation*}\n\\qed\n\n\\begin{remark}\nIt is crucial for the proof of Theorem \\ref{Thm1} (i) that $\\sup_{\\th\\in\\T}\\{|\\xi(\\th)|\\}<\\infty$ which comes from the characteristics of the discrete model. \n\\end{remark}\n\n\\subsection{Proof of Theorem \\ref{Thm1} (ii)}\n\nBy \\eqref{ak} we have \n\\begin{align*}\n  \\pa_t \\cE(t,\\xi)\n = 2a'(t)a(t)|\\xi|^2 |v(t,\\xi)|^2\n \\le \\frac{2C_1}{a_0}\\Xi(t)^{-1} \\cE(t,\\xi). \n\\end{align*}\nTherefore, by Gronwall's inequality we have \n\\begin{equation*}\n  \\cE(t,\\xi)\n  \\le \\exp\\(\\frac{2C_1}{a_0}\\left\\|\\Xi(\\cd)^{-1}\\right\\|_{L^1(\\R_+)}\\) \n  \\cE(0,\\xi)\n  \\simeq \\cE(0,\\xi). \n\\end{equation*}\nAnalogously, we have \n\\begin{equation*}\n  \\cE(t,\\xi) \\ge \n  \\exp\\(-\\frac{2C_1}{a_0}\\left\\|\\Xi(\\cd)^{-1}\\right\\|_{L^2(\\R_+)}\\) \n  \\cE(0,\\xi)\n  \\simeq \\cE(0,\\xi).  \n\\end{equation*}\nThus the proof is concluded by using Lemma \\ref{lemm-E-cE}. \n\\qed\n\n\\subsection{Proof of Theorem \\ref{Thm1} (iii)}\n\\subsubsection{Zones}\nWe can suppose that $\\lim_{t\\to\\infty}\\Th(t)=\\infty$; otherwise we have GEC by Theorem \\ref{Thm1} (i). \nLet $N$ be a large constant depends on $a_0$, $a_1$ and $C_k$ ($k=0,\\cdots,m$), to be chosen later. \nWe define $T_0$ and $t_\\xi$ for $\\xi \\in [-2,2]^d$ by \n\\[\n  T_0:=\\max\\left\\{t\\ge 0\\;;\\; \\Th(t) = \\frac{N}{2\\sqrt{d}}\\right\\}\n\\]\nand\n\\[\n  t_\\xi:=\\max\\left\\{t\\ge T_0\\;;\\; \\Th(t)|\\xi| = N\\right\\},\n\\]\nrespectively. \nThen we divide the region $\\R_+\\times [-2,2]^d$ by $(t_\\xi,\\xi)$ into the pseudo-differential zone $Z_\\Psi$ and the hyperbolic zone $Z_H$ as follows: \n\\begin{equation*\n  Z_\\Psi:=\\left\\{(t,\\xi)\\in \\R_+\\times [-2,2]^d\\;;\\; t \\le t_\\xi \\right\\}\n\\end{equation*}\nand\n\\begin{equation*\n  Z_H:=\\left\\{(t,\\xi)\\in \\R_+\\times [-2,2]^d \\;;\\; t \\ge t_\\xi \\right\\},\n\\end{equation*}\nrespectively. \nWe shall estimate $\\cE(t,\\xi)$ in each zones by different methods. \n\n\\subsubsection{Estimate in $Z_\\Psi$}\nLet $(t,\\xi) \\in Z_\\Psi$, that is, $\\Th(t)|\\xi| \\le N$. \nBy the same way as the method for the proof of Theorem \\ref{Thm1} (i), we have\n\\begin{align}\\label{estcE0-ZP}\n  \\cE_\\infty(t,\\xi)\n  \\le \\exp\\(\\frac{2a_1}{a_0}\\Th(t)|\\xi|\\) \\cE_\\infty(0,\\xi)\n  \\le \\exp\\(\\frac{2a_1 N}{a_0}\\) \\cE_\\infty(0,\\xi) \n\\end{align}\nand\n\\begin{align*}\n  \\cE_\\infty(t,\\xi)\n  \\ge \\exp\\(-\\frac{2a_1 N}{a_0}\\) \\cE_\\infty(0,\\xi). \n\\end{align*}\nTherefore, by \\eqref{cEinfcE} we have \n\\begin{equation}\\label{estcE-ZP}\n  \\cE(t,\\xi)\\simeq \\cE(0,\\xi), \\;\\; t \\le t_\\xi.\n\\end{equation}\n\n\\subsubsection{Estimate in $Z_H$}\n\nLet us consider the following first order system:\n\\begin{equation}\\label{V}\n\\pa_t V=A V,\\;\\;\nV=\\begin{pmatrix}v_1 \\\\ v_2 \\end{pmatrix},\\;\\;\nA=\\begin{pmatrix} \\phi(t,\\xi) & \\ol{r(t,\\xi)} \\\\ \n  r(t,\\xi) & \\ol{\\phi(t,\\xi)} \\end{pmatrix}. \n\\end{equation}\nFor a complex valued function $\\phi$, we denote the real and the imaginary parts of $\\phi$ by $\\phi_\\Re$ and $\\phi_\\Im$, respectively. \nThen we have the following lemma: \n\\begin{lemma}\\label{lemm-estV}\nFor any $t,\\tilde{t} \\in \\R$ satisfying $\\tilde{t} < t$ \nthe solution $V=V(t,\\xi)$ of \\eqref{V} satisfies the following estimates: \n\\begin{equation}\\label{estV}\n  \\|V(t,\\xi)\\|_{\\C^2}^2\n  \\begin{cases}\n  \\ds{\\le  \\exp\\(2\\int^t_{\\tilde{t}} \\phi_\\Re(s,\\xi)\\,ds\n        + 2\\int^t_{\\tilde{t}} |r(s,\\xi)|\\,ds\\)\n  \\|V(t_0,\\xi)\\|_{\\C^2}^2},\n\\\\[3ex]\n  \\ds{\\ge \\exp\\(2\\int^t_{\\tilde{t}} \\phi_\\Re(s,\\xi)\\,ds\n        - 2\\int^t_{\\tilde{t}} |r(s,\\xi)|\\,ds\\) \n  \\|V(t_0,\\xi)\\|_{\\C^2}^2}.\n  \\end{cases}\n\\end{equation}\n\n\\end{lemma}\n\\begin{proof}\nBy Cauchy-Schwarz inequality, we have \n\\begin{align*}\n  \\pa_t \\|V\\|_{\\C^2}^2\n&=2\\Re\\(\\pa_t V,V\\)_{\\C^2}\n =2\\Re\\(\\begin{pmatrix} \\phi & 0 \\\\ 0 & \\ol{\\phi} \\end{pmatrix}V, V\\)_{\\C^2}\n +2\\Re\\(\\begin{pmatrix} 0 & \\ol{r} \\\\ r & 0 \\end{pmatrix}V, V\\)_{\\C^2}\n\\\\\n&=2\\phi_\\Re\\|V\\|_{\\C^2}^2 + 4\\Re\\(rv_1\\ol{v_2}\\)\n\\begin{cases}\n\\le 2\\(\\phi_\\Re + |r|\\)\\|V\\|_{\\C^2}^2,\n\\\\\n\\ge 2\\(\\phi_\\Re - |r|\\)\\|V\\|_{\\C^2}^2.\n\\end{cases}\n\\end{align*}\nThus we have \\eqref{estV} by Gronwall's inequality. \n\\end{proof}\n\nSince the equation of \\eqref{v} is reduced to the following first order system:\n\\begin{equation}\\label{V1}\n  \\pa_t V_1=A_1 V_1,\\;\\;\n  V_1:=\\begin{pmatrix}\n    \\pa_t v + \\i a(t)|\\xi|v \\\\ \\pa_t v -\\i a(t)|\\xi|v\n  \\end{pmatrix},\n  \\;\\;\n  A_1:=\\begin{pmatrix} \\phi_1 & \\ol{r_1} \\\\ r_1 & \\ol{\\phi_1} \\end{pmatrix},\n\\end{equation}\nwhere \n\\begin{align*}\n  r_1:=-\\frac{a'(t)}{2a(t)}=\\ol{r_1}\n  \\;\\text{ and }\\;\n  \\phi_1=\\frac{a'(t)}{2a(t)}+\\i a(t)|\\xi|,\n\\end{align*}\nby Lemma \\ref{lemm-estV} and noting the equality \n\\begin{equation}\\label{cEV1}\n  \\|V_1(t,\\xi)\\|_{\\C^2}^2=2\\cE(t,\\xi),\n\\end{equation}\nwe have \n\\begin{equation}\\label{estV1}\n  \\cE(t,\\xi)\n\\begin{cases}\n\\ds{\\le\n\\frac{a(t)}{a(t_0)}\n  \\exp\\(\\int^t_{\\tilde{t}} \\frac{|a'(s)|}{a(s)}\\,ds\\) \n  \\cE(t_0,\\xi)}, \n\\\\[3ex] \n\\ds{\\ge\n\\frac{a(t)}{a(t_0)}\n  \\exp\\(-\\int^t_{\\tilde{t}} \\frac{|a'(s)|}{a(s)}\\,ds\\) \n  \\cE(t_0,\\xi)}. \n\\end{cases}\n\\end{equation}\nIndeed, \\eqref{GEC} is concluded from the estimate \\eqref{estV1} \nif $\\Xi(t)^{-1}\\in L^1(\\R_+)$, but not for $\\Xi(t)\\not\\in L^1(\\R_+)$. \nTherefore, we introduce the following idea, which is called refined diagonalization procedure taking account the properties \\eqref{stb} and \\eqref{ak} with $m\\ge 2$. \nA basic idea of this method was introduced in \\cite{Yg}, and improved in \\cite{H07} to take the benefit of the property \\eqref{stb}. \nThe refined diagonalization procedure can be understood to construct a $2 \\times 2$ regular matrix $M=M(t,\\xi)$ in $Z_H$ satisfies the following properties: \n\\begin{itemize}\n\\item\n$M$ is defined in $Z_H$. \n\\item\n$V:=M V_1$ satisfies $\\|V(t,\\xi)\\|_{\\C^2}^2 \\simeq \\cE(t,\\xi)$. \n\\item\n$V$ is a solution of \\eqref{V}. \n\\item\n$\\int^t_{\\tilde{t}}\\phi_\\Re(s,\\xi)\\,ds$ is bounded in $Z_H$. \n\\item\n$\\int^t_{\\tilde{t}}|r(s,\\xi)|\\,ds$ is bounded in $Z_H$. \n\\end{itemize}\nIndeed, if there exists such a matrix $M$, then we immediately have \\eqref{GEC} from \\eqref{estcE-ZP} and \\eqref{estV} with $\\tilde{t}=t_\\xi$. \n\n\nFor $p\\in \\N_0$ and $q, r \\in \\Z$ we introduce the symbol-like class \n$S^{(p)}\\{q,r\\}$ as the set of functions $f(t,\\xi)$ satisfy the following estimates in $Z_H$: \n\\begin{equation*\n  \\left|\\pa_t^k f(t,\\xi)\\right|\\lesssim \n  |\\xi|^q \\Xi(t)^{-r-k}\n  \\;\\; (k=0,\\ldots,p).\n\\end{equation*}\nThen we have the following usual algebraic properties and a hierarchy of the classes in $Z_H$:\n\\begin{lemma}\\label{symbol}\nThe following properties are established:\n\n\\begin{itemize}\n\\item[\\rm (i)]\nIf $f\\in S^{(p)}\\{q,r\\}$ with $p\\ge1$, then $f\\in S^{(p-1)}\\{q,r\\}$ and $\\pa_t f\\in S^{(p-1)}\\{q,r+1\\}$.\n\n\\item[\\rm (ii)]\nIf $f_1\\in S^{(p)}\\{q_1,r_1\\}$ and $f_2\\in S^{(p)}\\{q_2,r_2\\}$, then\n$f_1f_2\\in S^{(p)}\\{q_1+q_2,r_1+r_2\\}$.\n\\item[\\rm (iii)]\nIf $f\\in S^{(p)}\\{q,r\\}$, then $f \\in S^{(p)}\\{q+1,r-1\\}$.\n\\item[\\rm (iv)]\nIf $f\\in S^{(p)}\\{-q,q\\}$ with $q\\ge 1$, then for any $\\ve>0$ there exists a positive constant $N_0$ such that $|f(t,\\xi)|\\le \\ve$ for any $N \\ge N_0$. \n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\n(i) and (ii) are trivial from the definition of $S^{(p)}\\{q,r\\}$. \nLet $f(t,\\xi)\\in S^{(p)}\\{q,r\\}$.\nBy (\\ref{ThXi}) we have\n\\begin{align*}\n  \\left|\\pa_t^k f(t,\\xi)\\right|\n  \\lesssim\\: & |\\xi|^q \\Xi(t)^{-r-k}\n  \\le N^{-1} |\\xi|^{q+1} \\frac{\\Th(t)}{\\Xi(t)} \\Xi(t)^{-(r-1)-k}\n\\\\\n  \\le\\:& N^{-1} C_0 |\\xi|^{q+1} \\Xi(t)^{-(r-1)-k}\n\\end{align*}\nfor $k=0,\\ldots,p$. \nIt follows that $f(t,\\xi)\\in S^{(p)}\\{q+1,r-1\\}$; thus (iii) is proved. \nIf $f(t,\\xi)\\in S^{(p)}\\{-q,q\\}$, then by \\eqref{ThXi} we have \n\\begin{align*}\n  |f(t,\\xi)|\n  \\le C|\\xi|^{-q}\\Xi(t)^{-q}\n  \\le C N^{-q}\\(\\frac{\\Th(t)}{\\Xi(t)}\\)^{q}\n  \\le C C_0^q N^{-q} \\le \\ve\n\\end{align*}\nby putting $N_0=C_0(C\\ve^{-1})^{1\/q}$. \n\\end{proof}\n\nBy Lemma \\ref{symbol} we have the following lemma: \n\\begin{lemma}\\label{symbol2}\nThere exists a positive constant $N_0$ such that the following properties are established for any $N \\ge N_0$:\n\\begin{itemize}\n\\item[\\rm (i)] \nIf $f(t,\\xi)\\in S^{(p)}\\{0,0\\}$ and $f(t,\\xi)\\gtrsim 1$, \nthen $1\/f(t,\\xi)\\in S^{(p)}\\{0,0\\}$. \n\\item[\\rm (ii)] \nIf $f(t,\\xi)\\in S^{(p)}\\{1,0\\}$ and $f(t,\\xi)\\gtrsim |\\xi|$, \nthen $1\/f(t,\\xi)\\in S^{(p)}\\{-1,0\\}$. \n\\item[\\rm (iii)] \nIf $f(t,\\xi)\\in S^{(p)}\\{-q,q\\}$ with $q \\ge 1$, \nthen $\\sqrt{1-|f(t,\\xi)|^2}\\in S^{(p)}\\{0,0\\}$. \n\\end{itemize} \n\\end{lemma}\n\\begin{proof}\nSince (i) is proved by the same way as the proof of (ii), we prove (ii). \nBy applying Fa\\`a di Bruno's formula:  \n\\begin{equation*\n  \\frac{d^k}{dt^k}F(G(x))\n =k!\\sum_{h=1}^k \n  F^{(h)}(G(t))\n  \\sum_{\\substack{h_1 + 2h_2 + \\cdots + n h_n=k \\\\ \n                  h_1+h_2+\\cdots + h_k = h}}\n  \\,\n  \\prod_{l=1}^k \\frac{1}{h_l!\\, l!^{h_l}}\\(G^{(l)}(t)\\)^{h_l} \n\\end{equation*}\nwith $F(G)=1\/G$ and $G(t)=f(t,\\xi)$, and noting \n$|F^{(h)}(G)|\\lesssim 1\/|G|^{h+1}$, we have \n\\begin{align*}\n  \\left|\\pa_t^k \\frac{1}{f(t,\\xi)}\\right|\n\\lesssim\\:&\n  \\sum_{h=1}^k \\frac{1}{f(t,\\xi)^{h+1}}\n  \\sum_{\\substack{h_1 + 2h_2 + \\cdots + n h_n=k \\\\ \n                  h_1+h_2+\\cdots + h_k = h}}\n  \\,\n  \\prod_{l=1}^k \\left|\\pa_t^l f(t,\\xi)\\right|^{h_l}\n\\\\\n\\lesssim\\:&\n  \\sum_{h=1}^k |\\xi|^{-h-1}\n  \\sum_{\\substack{h_1 + 2h_2 + \\cdots + n h_n=k \\\\ \n                  h_1+h_2+\\cdots + h_k = h}} \\,\n  \\prod_{l=1}^k \\(|\\xi| \\Xi(t)^{-l}\\)^{h_l}\n\\\\\n\\simeq\\:&\n  |\\xi|^{-1} \\Xi(t)^{-k}\n\\end{align*}\nfor $k=0,\\ldots,p$; thus (ii) is proved.  \nBy Lemma \\ref{symbol} (iv) we can suppose that \n$\\sqrt{1-|f(t,\\xi)|^2} \\ge 1\/2$. \nBy applying Fa\\`a di Bruno's formula with \n$F(G)=\\sqrt{1-G}$ and $G(t)=|f(t,\\xi)|^2 \\in S^{(p)}\\{-2q,2q\\}$, \nand noting the estimates \n\\begin{align*}\n  |F^{(h)}(G)| \\lesssim \\frac{F(G)}{(1-G)^h} \\le 4^h\n\\end{align*}\nfor $h=0,1,\\ldots$, we have \n\\begin{align*}\n  \\left|\\pa_t^k \\sqrt{1-|f(t,\\xi)|^2}\\right|\n\\lesssim\\:&\n  \\sum_{h=1}^k 4^h \n  \\sum_{\\substack{h_1 + 2h_2 + \\cdots + n h_n=k \\\\ \n                  h_1+h_2+\\cdots + h_k = h}}\n  \\,\n  \\prod_{l=1}^k \\left|\\pa_t^l |f(t,\\xi)|^2\\right|^{h_l}\n\\\\\n\\lesssim\\:&\n  \\sum_{h=1}^k \n  \\sum_{\\substack{h_1 + 2h_2 + \\cdots + n h_n=k \\\\ \n                  h_1+h_2+\\cdots + h_k = h}}\n  \\,\n  \\prod_{l=1}^k \\(|\\xi|^{-2q}\\Xi(t)^{-2q-l}\\)^{h_l}\n\\\\\n\\lesssim\\:&\n  \\sum_{h=1}^k |\\xi|^{-2qh}\\Xi(t)^{-2qh-k}\n\\le \\Xi(t)^{-k} \\sum_{h=1}^k \\(\\frac{C_0}{N}\\)^{2qh}\n\\\\\n\\simeq\\:& \\Xi(t)^{-k} \n\\end{align*}\nfor $k=1,\\ldots,p$; thus (iii) is prove. \n\\end{proof}\n\n\nAn eigenvalue of $A_1$ is represented by \n\\begin{equation*}\n  \\la_1=\\phi_{1\\Re} + \\i \\phi_{1\\Im}\\sqrt{1 -\\(\\frac{|r_1|}{\\phi_{1\\Im}}\\)^2}. \n\\end{equation*}\nBy \\eqref{ak}, \\eqref{ThXi} and choosing $N>C_0 C_1 a_0^{-2}\/2$, we have \n\\begin{align*}\n  \\(\\frac{|r_1|}{\\phi_{1\\Im}}\\)^2\n\\le \\(\\frac{C_1}{2a_0^2|\\xi|\\Xi(t)}\\)^2\n\\le \\(\\frac{C_1}{2a_0^2N}\\frac{\\Th(t)}{\\Xi(t)}\\)^2\n\\le \\(\\frac{C_0 C_1}{2a_0^2N}\\)^2<1.\n\\end{align*}\nIt follows that the other eigenvalue of $A_1$ is given by $\\ol{\\la_1}$. \nTherefore, a diagonalizer $M_1$ for $A_1$ is formally given by \n\\begin{equation*}\n  M_1=\\begin{pmatrix} 1 & \\ol{\\de_1} \\\\ \\de_1 & 1 \\end{pmatrix},\n  \\quad\n  \\de_{1}=\\frac{\\la_{1}-\\phi_1}{\\ol{r_1}}. \n\\end{equation*}\nHere we note that the following lemma is established which ensures the invertibility of $M_1$: \n\\begin{lemma}\\label{symbol_de1}\nThere exists a positive constant $N_0$ such that \n$r_1\\in S^{(m-1)}\\{0,1\\}$, $1\/\\phi_{1\\Im}\\in S^{(m)}\\{-1,0\\}$\nand $\\de_1\\in S^{(m-1)}\\{-1,1\\}$ for any $N \\ge N_0$. \n\\end{lemma}\n\\begin{proof}\nThe first two properties are trivial. \nLet $N_0$ be large enough so that Lemma \\ref{symbol} and Lemma \\ref{symbol2} can be applied. \nBy Lemma \\ref{symbol} (ii) and (iv), we have \n$(|r_1|\/\\phi_{1\\Im})^2 \\in S^{(m-1)}\\{-2,2\\}$ and \n$(|r_1|\/\\phi_{1\\Im})^2 \\le 1\/2$. \nBy Lemma \\ref{symbol2} (iii) we have \n\\begin{align*}\n  \\sqrt{1-\\(\\frac{|r_1|}{\\phi_{1\\Im}}\\)^2}+1\\in S^{m-1}\\{0,0\\},\n\\end{align*}\nit follows from Lemma \\ref{symbol2} (i) that \n\\begin{align*}\n  \\frac{1}{\\sqrt{1-\\(\\frac{|r_1|}{\\phi_{1\\Im}}\\)^2}+1}\n  \\in S^{(m-1)}\\{0,0\\}. \n\\end{align*}\nTherefore, by Lemma \\ref{symbol} (ii) and the first two properties of Lemma \\ref{symbol_de1}, we have \n\\begin{align*}\n  \\de_1\n=\\frac{\\i \\phi_{1\\Im}}{\\ol{r_1}}\n  \\(\\sqrt{1-\\(\\frac{|r_1|}{\\phi_{1\\Im}}\\)^2}-1\\)\n=-\\frac{\\i r_1}{\\phi_{1\\Im}}\n  \\frac{1}{\\sqrt{1-\\(\\frac{|r_1|}{\\phi_{1\\Im}}\\)^2}+1}\n\\in S^{(m-1)}\\{-1,1\\}. \n\\end{align*}\n\\end{proof}\n\nWe define $A_2=A_2(t,\\xi)$ by\n\\begin{align*}\n  A_2:=\\begin{pmatrix} \\phi_2 & \\ol{r_2} \\\\ r_2 & \\ol{\\phi_2} \\end{pmatrix},\n  \\;\\;\n  r_2:=-\\frac{(\\de_1)_t}{1-|\\de_1|^2},\\;\\;\n  \\phi_2:=\\la_1+\\frac{\\ol{\\de_1}(\\de_1)_t}{1-|\\de_1|^2}. \n\\end{align*}\nThen we see that $r_2 \\in S^{(m-2)}\\{-1,2\\}$ by the lemmas for the symbol classes above. \nNoting the equalities \n\\begin{align*}\n  M_1^{-1} \\(A_1 - \\pa_t I\\)M_1\n=\\begin{pmatrix} \\la_{1} & 0 \\\\ 0 & \\ol{\\la_{1}} \\end{pmatrix}\n -M_1^{-1} \\(\\pa_t M_1\\) -\\pa_t I\n=A_2 - \\pa_t I,\n\\end{align*}\n\\eqref{V1} is reduced to the following system: \n\\begin{equation*\n  \\pa_t V_2 = A_2 V_2,\\quad\n  V_2:=M_1^{-1}V_1.\n\\end{equation*}\nHere we remark the followings: \n\\begin{itemize}\n\\item \n$M_1$ is a diagonalizer for $A_1$ but not for $A_1-\\pa_t I$, \nthat is, $A_1 - M_1^{-1}(\\pa_t M_1)$ is not diagonal. \n\\item \nSince $r_1 \\in S^{(m-2)}\\{0,1\\}$ and $r_2 \\in S^{(m-2)}\\{-1,2\\}$, \n$M_1$ is a diagonalizer for $A_1-\\pa_t I$ modulo $S^{(m-2)}\\{0,1\\}$. \n\\item\nThe structure in which both the diagonal and the off-diagonal entries are complex conjugate are conserved by the diagonalization procedure due to $M_1$. \n\\end{itemize}\nTherefore, one can carry out the same diagonalization procedure for $A_2-\\pa_t I$ if $m\\ge 3$. \nGenerally, we have the following lemma, which is the essential of the refined diagonalization procedure. \n\\begin{lemma}\\label{ref_diag}\nLet $k$ be a positive integer satisfying $k<m$, $A_k$ be given by\n\\begin{equation*}\n  A_k=\\begin{pmatrix} \\phi_k & \\ol{r_k} \\\\ r_k & \\ol{\\phi_k} \\end{pmatrix},\n\\end{equation*}\nwhere $r_k$ and $\\phi_{k\\Im}$ satisfy \n\\begin{equation}\\label{rphik}\n  r_k\\in S^{(m-k)}\\{-k+1,k\\},\\;\\;\n  \\phi_{k\\Im} \\in S^{(m-k)}\\{1,0\\}\n  \\;\\text{ and }\\;\n  |\\phi_{k\\Im}| \\gtrsim |\\xi|. \n\\end{equation}\nThen there exists a positive constant $N_k$ such that the following properties are established for any $N \\ge N_k$: \n\\begin{itemize}\n\\item[\\rm (i)]\n$A_k$ has complex conjugate eigenvalues $\\la_k$ and $\\ol{\\la_k}$. \n\\item[\\rm (ii)]\nThe following matrix $M_k$ is invertible: \n\\begin{equation*}\n  M_k:=\\begin{pmatrix} 1 & \\ol{\\de_k} \\\\ \\de_k & 1 \\end{pmatrix},\n  \\quad\n  \\de_k:=\\frac{\\la_k-\\phi_k}{\\ol{r_k}}, \n\\end{equation*}\nmoreover, we have \n\\begin{equation}\\label{dek}\n  \\de_k \\in S^{(m-k)}\\{-k,k\\}. \n\\end{equation}\n\\item[\\rm (iii)]\n$A_{k+1}:=M_k^{-1}A_kM_k - M_k^{-1}(\\pa_t M_k)$ is represented as follows: \n\\begin{equation*}\n  A_{k+1}=\\begin{pmatrix}\n  \\phi_{k+1} & \\ol{r_{k+1}} \\\\ r_{k+1} & \\ol{\\phi_{k+1}} \\end{pmatrix},\n\\end{equation*}\nwhere $r_{k+1}$ and $\\phi_{k+1}=\\phi_{(k+1)\\Re}+\\i\\phi_{(k+1)\\Im}$ are given by \n\\begin{equation}\\label{rk+1}\n  r_{k+1}=-\\frac{(\\de_k)_t}{1-|\\de_k|^2},\n\\end{equation}\n\\begin{equation*\n  \\phi_{(k+1)\\Re} =\\phi_{k\\Re}-\\frac{\\pa_t \\log\\(1-|\\de_{k}|^2\\)}{2} \n\\end{equation*}\nand\n\\begin{equation*\n  \\phi_{(k+1)\\Im}\n =\\phi_{k\\Im}\\sqrt{1-\\(\\frac{|r_k|}{\\phi_{k\\Im}}\\)^2}\n   -\\Im\\(\\ol{\\de_k}r_{k+1}\\).\n\\end{equation*}\n\\item[\\rm (iv)]\n$r_{k+1}$ and $\\phi_{(k+1)\\Im}$ satisfy the followings: \n\\begin{equation*\n  r_{k+1}\\in S^{(m-k-1)}\\{-k,k+1\\},\n  \\quad\n  \\phi_{(k+1)\\Im} \\in S^{(m-k-1)}\\{1,0\\},\n  \\;\\text{ and }\\;\n  \\phi_{(k+1)\\Im} \\gtrsim |\\xi|.\n\\end{equation*}\n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\nLet $N \\ge N_k$ with a large constant $N_k$ so that Lemma \\ref{symbol} and Lemma \\ref{symbol2} can be applied. \nAn eigenvalue of $A_k$ is represented by \n\\begin{equation*}\n  \\la_k=\\phi_{k\\Re}+\\i\\phi_{k\\Im}\\sqrt{1-\\(\\frac{|r_k|}{\\phi_{k\\Im}}\\)^2}. \n\\end{equation*}\nBy \\eqref{rphik}, Lemma \\ref{symbol} (ii) and Lemma \\ref{symbol2} (ii), \nwe have \n\\begin{equation}\\label{rkphik}\n  \\(\\frac{|r_{k}|}{\\phi_{k\\Im}}\\)^2 \\in S^{(m-k)}\\{-2k,2k\\}. \n\\end{equation}\nIt follows from Lemma \\ref{symbol} (iv) that the other eigenvalue of \n$A_k$ is given by $\\ol{\\la_k}$. \nBy \\eqref{rphik}, \\eqref{rkphik} and using the same way as the proof of Lemma \\ref{symbol_de1}, we have \\eqref{dek}. \nMoreover, $M_k$ is invertible since $|\\de_k|^2 \\le 1\/2$ by Lemma \\ref{symbol} (iv). \n(iii) is prove by direct computations. \nBy applying Lemma \\ref{symbol2} (i) with $f(t,\\xi)=1-|\\de_k|^2\\ge 1\/2$, \nwe have $(1-|\\de_k|^2)^{-1}\\in S^{(m-k)}\\{0,0\\}$. \nTherefore, by Lemma \\ref{symbol} (i) and \\eqref{rk+1}, we have \n$r_{k+1}\\in S^{(m-k-1)}\\{-k,k+1\\}$, \nhence we have $\\phi_{(k+1)\\Im} \\in S^{m-k-1}\\{1,0\\}$ \nby Lemma \\ref{symbol} (ii), (iii), Lemma \\ref{symbol2} (iii), \n\\eqref{dek} and \\eqref{rkphik}. \nNoting \\eqref{rkphik}, $\\Im(\\ol{\\de_k}r_{k+1})\/\\phi_{k\\Im} \\in S^{(m-k-1)}\\{-2k-1,2k+1\\}$ and Lemma \\ref{symbol} (iv), we can suppose that \n\\begin{align*}\n  \\(\\frac{|r_{k}|}{\\phi_{k\\Im}}\\)^2 \\le \\frac14\n  \\;\\text{ and }\\;\n  \\frac{\\Im\\(\\ol{\\de_k}r_{k+1}\\)}{\\phi_{k\\Im}}\\le \\frac14. \n\\end{align*}\nIt follows that \n\\begin{align*}\n  \\phi_{(k+1)\\Im}\n =\\phi_{k\\Im}\\(\n  \\sqrt{1-\\(\\frac{|r_k|}{\\phi_{k\\Im}}\\)^2}\n   -\\frac{\\Im\\(\\ol{\\de_k}r_{k+1}\\)}{\\phi_{k\\Im}}\\)\n \\ge \\frac{\\sqrt{3}-1}{2}\\phi_{k\\Im}\n  \\gtrsim |\\xi|. \n\\end{align*}\n\\end{proof}\n\nBy Lemma \\ref{ref_diag} we have the following proposition: \n\\begin{proposition}\nThere exists a positive constant $N_m$ such that the following estimate is established for any $N \\ge N_m$ in $Z_H$: \n\\begin{equation}\n  \\cE(t,\\xi) \\simeq \\cE(t_\\xi,\\xi). \n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nLet $M_k$ ($k=1,\\ldots,m-1$) be defined by Lemma \\ref{ref_diag}. \nThen we can suppose that $|\\de_k(t,\\xi)|^2 \\le 1\/2$ for $k=1,\\ldots,m-1$. \nDenoting $\\cM_{m-1} = M_1\\cdots M_{m-1}$ and $V_m=\\cM_{m-1}^{-1}V_1$, we have \n\\begin{equation*}\n  \\(\\pa_t - A_m\\) V_m\n =\\cM_{m-1}^{-1} \\(\\pa_t - A_1\\)\\cM_{m-1} V_m\n =\\cM_{m-1}^{-1} \\(\\pa_t - A_1\\) V_1 = 0\n\\end{equation*}\nby \\eqref{V1}. \nTherefore, by \\eqref{estV} we have \n\\begin{equation*}\n  \\|V_m(t,\\xi)\\|_{\\C^2}^2\n\\begin{cases}\n  \\ds{\\le \\exp\\(2\\int^t_{t_\\xi} \\phi_{m\\Re}(s,\\xi)\\,ds\n        + 2\\int^t_{t_\\xi} |r_m(s,\\xi)|\\,ds\\) \n  \\|V_m(t_\\xi,\\xi)\\|_{\\C^2}^2}, \n\\\\[3ex]\n  \\ds{\\ge \\exp\\(2\\int^t_{t_\\xi} \\phi_{m\\Re}(s,\\xi)\\,ds\n        - 2\\int^t_{t_\\xi} |r_m(s,\\xi)|\\,ds\\) \n  \\|V_m(t_\\xi,\\xi)\\|_{\\C^2}^2}. \n\\end{cases}\n\\end{equation*}\nBy Lemma \\ref{ref_diag} (iii) and (iv) we have \n\\begin{align*}\n  \\exp\\(2\\int^t_{t_\\xi} \\phi_{m\\Re}(s,\\xi)\\,ds\\)\n&=\\exp\\(2\\int^t_{t_\\xi} \\phi_{1\\Re}(s,\\xi)\\,ds \n  + \\sum_{k=1}^{m-1} \\log\\frac{1-|\\de_k(t,\\xi)|^2}{1-|\\de_k(t_\\xi,\\xi)|^2}\\)\n\\\\\n&= \\frac{a(t)}{a(t_\\xi)}\n   \\prod_{k=1}^{m-1}\\frac{1-|\\de_k(t,\\xi)|^2}{1-|\\de_k(t_\\xi,\\xi)|^2}\n\\\\\n&\\begin{cases}\n\\le\n  \\(2^{m-1}a_0^{-1} a_1\\),\n  \\\\\n\\ge  \\(2^{m-1}a_0^{-1} a_1\\)^{-1}\n\\end{cases}\n\\end{align*}\nand\n\\begin{align*}\n  \\exp\\(2\\int^t_{t_\\xi} |r_m(s,\\xi)|\\,ds\\)\n\\le\\:&\\exp\\(C|\\xi|^{-m+1} \\int^t_{t_\\xi} \\Xi(s)^{-m}\\,ds \\)\n\\\\\n\\le\\:&\\exp\\(CN^{-m+1}\\Th(t_\\xi)^{m-1} \\int^\\infty_{t_\\xi} \\Xi(s)^{-m}\\,ds \\)\n\\simeq 1\n\\end{align*}\nby \\eqref{int_Cm}. \nIt follows that\n\\begin{equation}\\label{estVm}\n  \\|V_m(t,\\xi)\\|_{\\C^2}^2 \\simeq \\|V_m(t_\\xi,\\xi)\\|_{\\C^2}^2\n\\end{equation}\nin $Z_H$. \nTherefore, noting the estimates: \n\\begin{align*}\n  \\|V_k\\|_{\\C^2}^2\n =\\|M_k V_{k+1}\\|_{\\C^2}^2\n  \\le \\(1+|\\de_k|\\)^2 \\|V_{k+1}\\|_{\\C^2}^2 \n  \\le \\frac{3+2\\sqrt{2}}{2}\\|V_{k+1}\\|_{\\C^2}^2\n\\end{align*}\nand\n\\begin{align*}\n  \\|V_k\\|_{\\C^2}^2\n\\ge \\(1-|\\de_k|\\)^2 \\|V_{k+1}\\|_{\\C^2}^2 \n\\ge \\frac{3-2\\sqrt{2}}{2}\\|V_{k+1}\\|_{\\C^2}^2\n\\end{align*}\nfor $k=1,\\ldots,m-1$, it follows that \n\\begin{equation*\n  \\(\\frac{3 - 2\\sqrt{2}}{2}\\)^{m-1}\\|V_m\\|_{\\C^2}^2\n  \\le \\|V_1\\|_{\\C^2}^2\n  \\le \\(\\frac{3 + 2\\sqrt{2}}{2}\\)^{m-1}\\|V_m\\|_{\\C^2}^2,\n\\end{equation*}\nand \\eqref{cEV1}, we have \n\\begin{equation}\\label{estcE-ZH}\n  \\cE(t,\\xi) \\simeq \\|V_m(t,\\xi)\\|_{\\C^2}^2 \n  \\simeq \\|V_m(t_\\xi,\\xi)\\|_{\\C^2}^2\n  \\simeq \\cE(t_\\xi,\\xi).\n\\end{equation}\n\\end{proof}\n\nCombining the estimates \\eqref{estcE-ZP}, \\eqref{estcE-ZH} and \nusing Lemma \\ref{lemm-Parseval}, the proof of Theorem \\ref{Thm1} (iii) is concluded. \n\n\n\\section{Proof of Theorem \\ref{Thm2}}\n\nFor a large constant $\\tN$, we define $\\tilde{T}_0$ and $\\tt_\\xi$ for $\\xi \\in [-2,2]^d$ by \n\\[\n  \\tilde{T}_0:=\\max\\left\\{t\\ge 0\\;;\\; \\La(t) = \\frac{\\tN}{2\\sqrt{d}}\\right\\}\n  \\;\\text{ and }\\;\n  \\tt_\\xi:=\\max\\left\\{t\\ge \\tilde{T}\\;;\\; \\La(t)|\\xi| = \\tN\\right\\},\n\\]\nrespectively. \nThen we define the zones $\\tZ_H$ and $\\tZ_\\Psi$ by \n\\begin{equation*\n  \\tZ_\\Psi:=\\left\\{(t,\\xi)\\in  \\R_+\\times [-2,2]^d \\;;\\; t \\ge \\tt_\\xi \\right\\}\n\\end{equation*}\nand\n\\begin{equation*\n  \\tZ_H:=\\left\\{(t,\\xi)\\in  \\R_+\\times [-2,2]^d \\;;\\; t\\le \\tt_\\xi \\right\\},\n\\end{equation*}\nrespectively. \n\nLet $t\\le \\ttx$. \nBy the same way to derive the estimate \\eqref{estcE0-ZP} and using \\eqref{La-infty2}, we have \n\\begin{align*}\n  \\cE(t,\\xi)\n  \\lesssim\\:& \\exp\\(\\frac{2a_1}{a_0}\\Th\\(\\ttx\\)|\\xi|\\)\\cE(0,\\xi)\n\\\\\n  =\\: &\\exp\\(\\frac{2a_1 \\tN}{a_0}\\frac{|\\xi|}{\\tN}\n  \\Th\\(\\La^{-1}\\(\\frac{\\tN}{|\\xi|}\\)\\)\\)\n  \\cE(0,\\xi) \n\\\\\n  \\le\\: &\\exp\\(\\frac{2a_1 \\tN}{a_0} \\frac{|\\xi|}{N}\n  \\Th\\(\\La^{-1}\\(\\frac{N}{|\\xi|}\\)\\)\\)\n  \\cE(0,\\xi)\n\\\\\n  \\le\\: &\\exp\\(|\\xi|\\Th\\(\\La^{-1}\\(\\frac{N}{|\\xi|}\\)\\)\\)\n  \\cE(0,\\xi) \n\\end{align*}\nfor any $N \\ge 2a_1 \\tN \/ a_0$. \n\nLet $t\\ge \\ttx$. \nNoting that $\\ttx \\ge \\tx$ with $\\tN=N$, Lemma \\ref{ref_diag} is valid in $\\tZ_H$ for any large $\\tN$. \nTherefore, by (H6*) there exist positive constants $C$ and $N$ such that\n\\begin{align*}\n  \\int^t_{\\ttx} |r_m(s,\\xi)|\\,ds\n  \\le \\: &\n  C|\\xi|^{-m+1} \\int^\\infty_{\\ttx} \\Xi(s)^{-m}\\,ds\n\\\\\n = \\: & CN^{-m}  \n  \\La\\(\\ttx\\)^{m} \\Th(\\ttx)^{-1} \\int^\\infty_{\\ttx} \\Xi(s)^{-m}\\,ds\n  \\: |\\xi|\\Th(\\ttx) \n\\\\\n \\le \\: & C\n  N^{-m} \n  \\sup_{t\\ge 0}\\left\\{\n    \\La(t)^{m} \\Th(t)^{-1} \\int^\\infty_{t} \\Xi(s)^{-m}\\,ds\n  \\right\\}\n  |\\xi|\\Th\\(\\La^{-1}\\(\\frac{\\tN}{|\\xi|}\\)\\) \n\\\\\n \\le \\: & \n  |\\xi|\\Th\\(\\La^{-1}\\(\\frac{N}{|\\xi|}\\)\\). \n\\end{align*}\nBy \\eqref{cEV1} and the same way to derive the estimate \n\\eqref{estVm}, we have the following estimate:\n\\begin{equation*\n  \\cE(t,\\xi)\n  \\lesssim \\exp\\(|\\xi|\\Th\\(\\La^{-1}\\(\\frac{N}{|\\xi|}\\)\\)\\) \\cE(\\ttx,\\xi)\n\\end{equation*}\nfor any $t \\ge \\ttx$. \nCombining the estimates of $\\cE(t,\\xi)$ in $\\tZ_\\Psi$ and $\\tZ_H$, we have \n\\begin{align*}\n  \\cE(t,\\xi) \\lesssim \\exp\\(2|\\xi|\\Th\\(\\La^{-1}\\(\\frac{N}{|\\xi|}\\)\\)\\). \n\\end{align*}\n\\qed\n\n\\section{Proof of Theorem \\ref{Thm3}}\n\nThe following proposition is essential for the proof of Theorem \\ref{Thm3}. \n\n\\begin{proposition}\\label{Prop-Thm3}\nLet $\\{M_j\\}$ be a logarithmically convex sequence. \nIf $v=\\{v[k]\\}_{k\\in \\Z^d}$ satisfies \n\\begin{equation}\\label{eq1-Prop-Thm3}\n  \\sum_{k\\in \\Z^d} k^\\al v[k] = 0 \n\\end{equation}\nfor any $\\al\\in \\N_0^d$ and there exists a positive constant $\\rho$ such that \n\\begin{equation}\\label{eq2-Prop-Thm3}\n  \\sup_{k\\in \\N^d}\\left\\{\n   |v[k]| |k|^{d+1} T[\\{M_j\\}]\\(\\rho^{-1}|k|\\) \n  \\right\\} <\\infty, \n\\end{equation}\nthen the following estimate is established: \n\\begin{equation*}\n  \\sup_{\\th\\in \\T^d\\setminus\\{0\\}}\n  \\left\\{\n  \\left|\\hat{v}(\\th)\\right|T\\left[\\left\\{\\frac{M_j}{j!}\\right\\}\\right]\n  \\(\\frac{1}{d\\rho|\\th|}\\)\n  \\right\\}<\\infty.\n\\end{equation*}\n\\end{proposition}\n\nIn order to prove Proposition \\ref{Prop-Thm3}, we introduce the following lemma: \n\\begin{lemma}\\label{Lemma1-Thm3}\nIf $f\\in C^\\infty(\\T^d)$ satisfies \n$(\\pa_\\th^\\al f)(0)=0$ and \n\\begin{equation}\\label{eq-Lemma1-Thm3}\n  \\left|\\pa_\\th^\\al f(\\th)\\right| \\le C M_{|\\al|} \\rho^{|\\al|}\n\\end{equation}\nfor any $\\al \\in \\N_0^d$ with positive constants $C$, $\\rho$ and a logarithmically convex sequence $\\{M_j\\}$, then the following estimate is established: \n\\begin{equation*}\n  \\sup_{\\th\\in \\T^d\\setminus\\{0\\}}\\left\\{\n  T\\left[\\left\\{\\frac{M_j}{j!}\\right\\}\\right]\\(\\frac{1}{d \\rho|\\th|}\\)\n  |f(\\th)|\n  \\right\\}<\\infty.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nFor any $\\al\\in \\N_0^\\al$, there exists $\\ze \\in \\T^d$ such that \n$f(\\th)=(\\pa_\\th^\\al f)(\\ze) \\th^\\al\/\\al!$ by Taylor's theorem. \nBy \\eqref{eq-Lemma1-Thm3} and the inequalities $|\\th^\\al|\\le|\\th|^{|\\al|}$ and \n$|\\al|! \\le d^{|\\al|}\\al!$ we have\n\\begin{align*}\n  |f(\\th)|\n  \\le C \\frac{M_{|\\al|}\\rho^{|\\al|}}{\\al!} \\left|\\th^\\al\\right|\n  \\le C \\frac{M_{|\\al|}}{|\\al|!} \\(d\\rho|\\th|\\)^{|\\al|}.\n\\end{align*}\nTherefore, we have \n\\begin{align*}\n  |f(\\th)| \\le C\\inf_{j\\in \\N_0}\\left\\{\n  \\frac{M_j}{j!}\\(d\\rho |\\th|\\)^j\\right\\}\n =\\frac{C}{T\\left[\\left\\{\\frac{M_j}{j!}\\right\\}\\right]\n  \\(\\frac{1}{d \\rho|\\th|}\\)}.\n\\end{align*}\n\\end{proof}\n\n\\noindent\n{\\it Proof of Proposition \\ref{Prop-Thm3}.}\\; \nLet $\\al\\in \\N_0^d$. \nBy \\eqref{eq1-Prop-Thm3}, we have \n\\begin{align*}\n  \\pa_\\th^\\al\\hat{v}(\\th)|_{\\th=0}\n=\\sum_{k} (-ik)^\\al e^{-ik\\cd\\th} v[k]\\Bigr|_{\\th=0}\n=(-i)^{|\\al|}\\sum_{k} k^\\al v[k] = 0.\n\\end{align*}\nLet $\\al\\neq 0$. By \\eqref{eq2-Prop-Thm3}, there exists a positive constant $C$ such that\n\\begin{align*}\n  \\left|\\pa_\\th^\\al \\hat{v}(\\th)\\right|\n  \\le \\: & \\sum_{k} \\left|(-ik)^\\al e^{-ik\\cd\\th} v[k]\\right|\n  \\le C \\sum_{k\\in \\Z^d\\setminus\\{0\\}}\n    |k|^{-d-1+|\\al|}\\(T[\\{M_j\\}]\\(\\rho^{-1}|k|\\)\\)^{-1}\n\\\\\n  \\le \\: & C\\sum_{k\\in \\Z^d\\setminus\\{0\\}}\n  |k|^{-d-1+|\\al|}\\(\\frac{\\(\\rho^{-1}|k|\\)^{|\\al|}}{M_{|\\al|}}\\)^{-1}\n  \\simeq M_{|\\al|} \\rho^{|\\al|}.\n\\end{align*}\nTherefore, by Lemma \\ref{Lemma1-Thm3} we have\n\\begin{align*}\n  \\left|\\hat{v}(\\th)\\right|\n  \\lesssim \n  \\frac{1}{T\\left[\\left\\{\\frac{M_j}{j!}\\right\\}\\right]\n  \\(\\frac{1}{d \\rho|\\th|}\\)}. \n\\end{align*}\n\\qed\n\n\\noindent\n{\\it Proof of Theorem \\ref{Thm3} {\\rm (i)}.}\\; \nBy \\eqref{La-infty2}, \\eqref{eq1-Thm3}, $|\\th| \\le \\pi|\\xi(\\th)|\/2$, \nand applying Proposition \\ref{Prop-Thm3} with \n$v=D_j^+ u_0$ $(j=1,\\ldots,d)$ and $v=u_1$, we have \n\\begin{align*}\n  \\cE(0,\\th)\n\\simeq \\: &\n  \\left|\\hat{u}_1(\\th)\\right|^2\n +\\sum_{j=1}^d\\left|\\widehat{D_j^+ u_0}(\\th)\\right|^2\n\\lesssim\n  T\\left[\\left\\{\\frac{M_j}{j!}\\right\\}\\right]\n  \\(\\frac{1}{d \\rho|\\th|}\\)^{-2}\n\\\\\n\\le \\: &\n  L(N,\\{M_j\\})^{-2}\n  \\exp\\(-2d\\rho|\\th|\\Th\\(\\La^{-1}\\(\\frac{N}{d\\rho|\\th|}\\)\\)\\)\n\\\\\n = \\: &L(N,\\{M_j\\})^{-2}\\exp\\(-2N\\mu\\(\\frac{d\\rho|\\th|}{N}\\)\\)\n\\\\\n\\le \\: &\n  L(N,\\{M_j\\})^{-2}\\exp\\(-2N\\mu\\(\\frac{\\pi d\\rho|\\xi(\\th)|}{2N}\\)\\). \n\\end{align*}\nBy Theorem \\ref{Thm2} there exists a positive constant $\\tN_0 \\ge 1$ such that \n\\begin{equation}\\label{tcEtcE0_tN0}\n  \\cE(t,\\th) \\lesssim \n  \\exp\\(2\\tN_0\\mu\\(\\frac{|\\xi(\\th)|}{\\tN_0}\\)\\) \\cE(0,\\th). \n\\end{equation}\nIf \\eqref{eq1-Thm3} holds for any $N\\ge 1$, \nthen putting $N_0=\\max\\{\\tN_0,\\pi d\\rho \\tN_0\/2\\}$ and noting \n\\begin{align*}\n\\tN_0 \\mu\\(\\frac{|\\xi|}{\\tN_0}\\)-N_0\\mu\\(\\frac{\\pi d\\rho|\\xi|}{2N_0}\\)\n\\begin{cases}\n\\ds{=\\tN_0\\(1 -\\frac{\\pi d \\rho}{2}\\)\\mu\\(\\frac{|\\xi|}{\\tN_0}\\)} \n  \\le 0 & \\text{ for }\\; \\dfrac{\\pi d\\rho}{2} \\ge 1,\n\\\\[3ex]\n\\ds{  \\le\n  \\tN_0 \\mu\\(\\frac{|\\xi|}{\\tN_0}\\)-N_0\\mu\\(\\frac{|\\xi|}{N_0}\\)\n  =0}  &  \\text{ for }\\; \\dfrac{\\pi d\\rho}{2} < 1,\n\\end{cases}\n\\end{align*}\nwe have $\\cE(t,\\th) \\lesssim L(N_0,\\{M_j\\})^{-2}$. \nIt follows that \n\\[\n  U(N_0,u_0,u_1)\\lesssim (2\\pi)^d L(N_0,\\{M_j\\})^{-2} <\\infty, \n\\]\nand thus \\eqref{Ebdd} is established. \n\\qed\n\n\\noindent\n{\\it Proof of Theorem \\ref{Thm3} {\\rm (ii)}.}\\; \nBy the same way for the proof of (i), there exist positive constants $N_0\\ge 1$ and $N_1$ such that $L(N_1,\\{M_j\\})>0$ and \n\\begin{align*}\n  U(N_0,u_0,u_1)\n  \\lesssim \\: &\n  L(N_1,\\{M_j\\})^{-2}\n\\\\\n  & \\: \\times  \\int_{\\T^d}\n  \\exp\\(\n   2N_0\\mu\\(\\frac{|\\xi(\\th)|}{N_0}\\)\n  -2N_1\\mu\\(\\frac{\\pi d\\rho|\\xi(\\th)|}{2N_1}\\)\\)\\,d\\th. \n\\end{align*}\nBy \\eqref{eq4-Thm3} there exists a positive constant $N$ such that \n\\begin{equation*}\n  \\frac{\\mu\\(\\frac{1}{N}\\frac{|\\xi(\\th)|}{N_0}\\)}\n       {\\mu\\(\\frac{|\\xi(\\th)|}{N_0}\\)}\n =\\frac{\\Th\\(\\La^{-1}\\(N \\frac{N_0}{|\\xi(\\th)|}\\)\\)}\n       {N\\Th\\(\\La^{-1}\\(\\frac{N_0}{|\\xi(\\th)|}\\)\\)}\n \\ge \\frac{N_0}{N_1}. \n\\end{equation*}\nTherefore, putting $\\rho=2N_1\/(\\pi d N_0 N)$ we have \n\\begin{equation*}\n  N_0 \\mu\\(\\frac{|\\xi(\\th)|}{N_0}\\)\n  \\le\n  N_1 \\mu\\(\\frac{\\pi d \\rho|\\xi(\\th)|}{2N_1}\\),\n\\end{equation*}\nit follows that $U(N_0,u_0,u_1)<\\infty$, and thus \\eqref{Ebdd} is established. \n\\qed\n\n\n\\section{Appendix}\n\\subsection{Proof of Lemma \\ref{TDFT}}\nSince $f \\in l^1(\\Z^d)$, we have\n\\begin{align*}\n  \\sum_k e^{-\\i k\\cd \\th} f[k \\pm e_j]\n=e^{\\pm \\i e_j \\cd \\th} \\sum_k e^{-\\i(k \\pm e_j)\\cd \\th} f[k \\pm e_j]\n   = e^{\\pm \\i\\th_j} \\sum_k e^{-\\i k\\cd \\th}f[k]. \n\\end{align*}\nHence we have \n\\begin{align*}\n  \\cF_{\\Z^d}[D_j^+ f](\\th)\n =\\sum_{k} e^{-\\i k\\cd\\th}\\(f[k+e_j]-f[k]\\)\n =\\(e^{\\i\\th_j} - 1\\) \\cF_{\\Z^d}[f](\\th),\n\\end{align*}\nthat is, \\eqref{TDFT2} is valid. \nSimilarly, we have \n\\begin{align*}\n  \\cF_{\\Z^d}[D_j^- f](\\th)\n =\\(1-e^{-\\i\\th_j}\\) \\cF_{\\Z^d}[f](\\th).\n\\end{align*}\nTherefore, we have \\eqref{TDFT3} as follows:\n\\begin{align*}\n \\cF_{\\Z^d}[D_j^+ D_j^- f](\\th)\n= \\: &\\(e^{\\i\\th_j} - 1\\) \\cF_{\\Z^d}[D_j^- f](\\th) \n=\\(e^{\\i\\th_j} - 1\\)\\(1-e^{-\\i\\th_j}\\) \\cF_{\\Z^d}[f](\\th) \n\\\\\n= \\: &\\(e^{\\i\\frac{\\th_j}{2}}-e^{-\\i\\frac{\\th_j}{2}}\\)^2 \\hat{f}(\\th)\n =-4\\(\\sin\\frac{\\th_j}{2}\\)^2 \\hat{f}(\\th). \n\\end{align*}\n\\qed\n\n\\subsection{Proof of Lemma \\ref{lemm-Parseval}}\nFrom the definition of $\\hat{f}$ we have \n\\begin{align*}\n  \\int_{\\T^d}|\\hat{f}(\\th)|^2\\,d\\th\n= \\: &\\int_{\\T^d}\n  \\(\\sum_{k} e^{-\\i k\\cd\\th} f[k]\\)\n  \\ol{\\(\\sum_{l} e^{-\\i l\\cd\\th} f[l]\\)}\\,d\\th\n\\\\\n= \\: & \\sum_{k,l} f[k]\\ol{f[l]} \\int_{\\T}  e^{-\\i(k-l)\\cd\\th} \\,d\\th \n\\\\\n= \\: &(2\\pi)^d \\sum_{k,l} \\de_{kl} f[k] \\ol{f[l]}\n=(2\\pi)^d \\sum_{k} |f[k]|^2, \n\\end{align*}\nwhere $\\de_{kl}$ denotes the Kronecker delta. \n\\qed\n\n\\subsection{Proof of Lemma \\ref{lemm-E-cE}}\nBy Lemma \\ref{TDFT} and Lemma \\ref{lemm-Parseval}, we have \n\\[\n  (2\\pi)^d\\sum_{k\\in\\Z^d}|u'(t)[k]|^2\n =\\int_{\\T^d}\\left|\\cF_{\\Z^d}[u'(t)](\\th)\\right|^2\\,d\\th\n =\\int_{\\T^d}\\left|\\pa_t \\hat{u}(t,\\th)\\right|^2\\,d\\th,\n\\]\nand\n\\begin{align*}\n  (2\\pi)^d\\sum_{j=1}^d \\sum_{k\\in\\Z^d}|D_j^+ u(t)[k]|^2\n= \\: &\\sum_{j=1}^d \n  \\int_{\\T^d} \\left| \\cF_{\\Z^d}[D_j^+ u(t)](\\th)\\right|^2\\,d\\th\n\\\\\n= \\: &\\sum_{j=1}^d \n  \\int_{\\T^d} \\left| \\(e^{\\i\\th_j}-1\\)\\hat{u}(t,\\th)\\right|^2\\,d\\th\n\\\\\n= \\: &\\int_{\\T^d} \\sum_{j=1}^d \\xi_j(\\th)^2|\\hat{u}(t,\\th)|^2\\,d\\th.\n\\end{align*}\nThus we have \\eqref{EcE}. \n\\qed\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nUnderstanding the structure of wave functions in Fock space is a hot topic in many-body physics, intimately related to the concepts of ergodicity and thermalization in complex interacting systems.\nThe natural way to think of the interaction is as a cause of transitions between single-particle states, which are no longer eigenstates of a quantum system in the presence of interaction.\nSinge-particle excitations decay into three-particle states (two electrons and one hole), which then further decay into five-particle states, etc. The simplest way to characterize the process of quantum-mechanical spreading of an excitation in Fock space is to study its energy relaxation rate (inverse lifetime).\n\n\nThe study of the inelastic relaxation in a confined geometry has been pioneered by Sivan, Imry and Aronov (SIA) \\cite{SIA}, who calculated the quasiparticle lifetime in a diffusive quantum dot with chaotic electron dynamics.\nWorking within the conventional Fermi golden rule (FGR) picture, they calculated the relaxation rate of an excitation with the energy $\\varepsilon$, induced by the screened Coulomb interaction with the small momentum transfer:\n\\begin{equation}\n  \\gamma_0(\\varepsilon)\n  \\sim\n  \\lambda^2 \\Delta (\\varepsilon\/E_{\\text{Th}})^2 ,\n\\label{SIA}\n\\end{equation}\nwhere $\\Delta$ is the mean single-particle level spacing in the dot, and $E_{\\text{Th}}= D\/L^2$ is the Thouless energy determined by the inverse time of electron diffusion across the system ($D$ is the diffusion coefficient, and $L$ is the typical size of the quantum dot).\nFor generality, in Eq.~(\\ref{SIA}) we introduced the dimensionless interaction strength $\\lambda$, taking its maximal value, $\\lambda=1$, for the screened Coulomb potential.\nThe result (\\ref{SIA}) was derived in the hot-electron regime (negligible temperature, $T\\ll\\varepsilon$) under the assumption of the zero-dimensional geometry, $\\varepsilon \\ll E_{\\text{Th}}$.\nIt provides the diffusive contribution to the relaxation rate originating from the processes with momentum transfer of the order of the inverse system size, $1\/L$. The total relaxation rate is then the sum of $\\gamma_0(\\varepsilon)$ given by Eq.~(\\ref{SIA}) and the Fermi-liquid contribution due to the processes with large-momentum transfer, $\\gamma_*(\\varepsilon)\\sim \\varepsilon^2\/E_F$ ($E_F$ is the Fermi energy) \\cite{FL}. The latter can be neglected for sufficiently large quantum dots, $L\\gg k_Fl^2$ ($k_F$ is the Fermi momentum, and $l$ is the mean free path) \\cite{SIA,Blanter}, that will be assumed thereafter.\n\nRemarkably, Eq.~(\\ref{SIA}) derived in the zero-dimensional limit, $\\varepsilon \\ll E_{\\text{Th}}$, shows that in this regime the single-particle spectrum is well resolved, $\\gamma_0(\\varepsilon)\\ll\\Delta$. Though this conclusion is in a good agreement with the experimental results on tunneling spectroscopy of a disordered quantum dot~\\cite{SMM}, it raises the question of consistency of the derivation.\nIndeed, the FGR can be safely applied only in the case of continuous spectrum, while the result (\\ref{SIA}) implies that it is not. This subtle point was recognized already by SIA, who argued that their approach might be correct as summation over many final states is performed.\nThis idea was elaborated in a seminal paper by Altshuler, Gefen, Kamenev, and Levitov (AGKL) \\cite{AGKL}, who emphasized that it is the spectrum of final states that should be continuous for the FGR picture to be applicable. In the problem of the hot-electron decay, final states are three-particle states with two electrons and one hole, and the corresponding mean level spacing can be estimated as\n\\begin{equation}\n  \\Delta_3(\\varepsilon) \\sim \\Delta^3\/\\varepsilon^2\n  .\n\\label{Delta3}\n\\end{equation}\nComparing $\\gamma_0(\\varepsilon)$ and $\\Delta_3(\\varepsilon)$, AGKL came to the conclusion that the FGR description should be valid for sufficiently large energies, $\\varepsilon>\\varepsilon_\\text{FGR}$, where\n\\begin{equation}\n\\varepsilon_\\text{FGR}=\\Delta\\sqrt g,\n\\label{eps*}\n\\end{equation}\nand $g=E_{\\text{Th}}\/\\lambda\\Delta\\gg1$ sets the scale, $\\Delta\/g$, of the interaction matrix elements (for the screened Coulomb interaction with $\\lambda=1$, $g$ coincides with the dimensionless conductance of the dot).\n\nIn order to study the spreading of a single-particle excitation over the space of many-particle states beyond the FGR approximation,\nAGKL proposed a remarkable mapping of the initial problem to a tight-binding Anderson model on a hierarchical lattice, treating each site as a basis vector in the many-particle Fock space. Motivated by the observation that the structure of this lattice for the quantum dot problem locally looks like a tree, AGKL approximated it by the Bethe lattice with a large branching number. Then using the known solution for the Anderson model on the Bethe lattice \\cite{Cayley}, AGKL predicted the localization transition in Fock space of an interacting quantum dot at the energy $\\varepsilon_\\text{MBL}\\sim\\Delta\\sqrt{g\/\\ln g}$, which appeared to be parametrically smaller than the FGR breaking energy scale $\\varepsilon_\\text{FGR}$. Thus, in the AGKL picture of many-body localization one should distinguish between three regimes \\cite{AGKL,MF97}.\nIn the localized regime realized at $\\varepsilon<\\varepsilon_\\text{MBL}$, the Slater determinants of one-particle states are very close to the exact many-body states, and the width of quasiparticle states is exactly zero.\nIn the intermediate region, $\\varepsilon_\\text{MBL}<\\varepsilon<\\varepsilon_\\text{FGR}$, quasiparticle states are delocalized, but they are strongly fractal and non-ergodic, with the spectral weight given by a number of slightly broadened lines.\nThe spectral weight acquires a Lorentzian form with a well defined width $\\gamma$ (corresponding to a simple exponential decay $e^{-\\gamma t}$ in the time domain) only in the regime $\\varepsilon\\gg\\varepsilon_\\text{FGR}$, where the FGR description finally sets in.\n\nThe AGKL paper has triggered a boost of activity in the field of many-body localization. The suggested mapping onto the lattice model has been proved to be extremely fruitful: instead of studying an interacting problem one now can deal with a non-interacting quantum mechanics, but on a very complicated lattice with an exponentially large number of sites.\n(Though all states in a finite system are localized by definition, even an extremely weak coupling to the external reservoir providing a level width larger than the exponentially small distance between the many-body states, which can be estimated as $\\exp(-\\alpha\\sqrt{\\varepsilon\/\\Delta})$ with $\\alpha\\sim1$ \\cite{Silv,Bethe1936,BohrMottelson},\nrenders the spectrum of delocalized states effectively continuous.)\nSince the complexity of the problem is encoded in the lattice topology, the crucial point is\nto identify the relevant features responsible for many-body localization.\n\nQuite soon it was recognized that the Bethe lattice with a constant branching number is an oversimplified model of Fock space.\nFirstly, the coordination number of the lattice decreases with the number of generations \\cite{Silv,Jacquod97}. Secondly, the actual lattice is not a tree, and the presence of loops essentially modifies combinatorics of the perturbative expansion in the localized region \\cite{Silv}, increasing the AGKL estimate for $\\varepsilon_\\text{MBL}$.\nNumerical studies \\cite{LTB98,Kamenev2002} demonstrated gradual delocalization with the growth of the quasiparticle energy.\nAnyway, despite the lack of a rigorous theory of many-body localization in a quantum dot, the general understanding achieved in the beginning of 2000s was that in finite systems one should expect a localization\/delocalization crossover, though its position was not firmly established.\n\n\nLater, the concept of many-body localization developed in the quantum dot problem\nwas applied to extended systems of interacting electrons with spatially localized single-particle states \\cite{BAA,GMP}, where the transition between the localized and delocalized many-body phases occurs at a finite temperature.\nThe related issues of ergodicity and thermalization are currently being actively investigated (for a review, see Ref.~\\cite{Polkovnikov}).\n\nVery recently, many-body localization in a quantum dot was reconsidered by Gornyi, Mirlin, and Polyakov \\cite{Mirlin2015}. Working in the framework of the lattice model, they found an additional factorial contribution in the perturbative series in the number of involved generations, which renders coupling to distant generations less efficient, thus acting in favor of localization.\nFor the hot-electron decay problem, this leads to the estimate for the threshold energy $\\varepsilon_\\text{MBL}\\sim g\\Delta\/\\ln g$, which is much larger than the original AGKL estimate.\nFor a thermal many-body state with the temperature $T$ and the total energy ${\\cal E}\\sim T^2\/\\Delta$, the FGR temperature $T_\\text{FGR}\\sim \\Delta\\sqrt{g}$ was shown to be logarithmically larger than the localization transition point, $T_\\text{MBL}\\sim\\Delta\\sqrt{g\/\\ln g}$. This result formally  coincides with the AGKL \\emph{energy}\\ threshold and is in full agreement with Ref.~\\cite{BAA}.\n\nSince, contrary to AGKL, Refs.~\\cite{Silv,Mirlin2015} place the many-body localization threshold $\\varepsilon_\\text{MBL}$ for the hot-electron problem above the FGR-breaking scale $\\varepsilon_\\text{FGR}$,\none can ask how to reconcile the FGR description with many-body localization. This question was answered by Silvestrov \\cite{Silvestrov2001} who studied the temporal decay of a quasiparticle with $\\varepsilon\\gg\\varepsilon_\\text{FGR}$. He showed that the FGR relaxation rate $\\gamma_0(\\varepsilon)$ describes the initial stage of exponential relaxation, that slows down at larger times due to smaller relaxation rate of descendant states, and eventually due to many-body localization.\n\nIn this paper, we address the initial temporal stage of energy relaxation in a quantum dot, when quantum many-body localization effects are not yet visible. Assuming the FGR approach is applicable, we calculate mesoscopic fluctuations of the energy relaxation rate. The smallness of fluctuations compared to the average relaxation rate provides an \\emph{a posteriori}\\ condition for the FGR applicability. In our analysis, we do not use the lattice model of Fock space and work in terms of Keldysh diagram technique in real space, where non-perturbative disorder averaging is performed by means of the non-linear supersymmetric sigma model. Such an approach allows us to derive the expression for the variance of the relaxation rate without any simplifications concerning the nature of the electron-electron interaction.\n\nThe paper is organized as follows. In Sec.~\\ref{S:Model} we introduce the model and summarize the results. Section~\\ref{S:General} describes the Keldysh kinetic approach\nalong with its modification simplifying the study of weak nonequilibrium.\nIn Sec.~\\ref{S:Average} we rederive the SIA result and generalize it to the case of an arbitrary temperature and interaction radius. Section~\\ref{S:MF} is devoted to the discussion of the general strategy for the calculation of mesoscopic fluctuations of the energy relaxation rate in terms of the exact (disorder-dependent) electron Green functions. The product of several Green functions is averaged non-perturbatively in Sec.~\\ref{S:nonpert}. The final step of the calculation of mesoscopic fluctuations is performed, for not very short-range interaction, in Sec.~\\ref{S:final} and, in the general case, in Sec.~\\ref{S:mesofluct-rs}. Our results are summarized in Sec.~\\ref{Discussion}. Numerous technical details are relegated to several Appendices.\n\nWe use the system of units with $\\hbar=k_B=1$.\n\n\n\n\n\n\n\n\\section{Model description and results}\n\\label{S:Model}\n\n\\subsection{Model}\n\nWe consider an isolated chaotic quantum dot with the broken time-reversal symmetry (unitary symmetry class).\nImpurity scattering in the dot is supposed to be strong enough to completely\nrandomize the electron trajectories establishing the diffusive regime with\n$l\\ll L$, where $l$ is the elastic mean free path,\nand $L$ is the characteristic size of the dot.\nFor generality, we consider the case of an arbitrary spin degeneracy, $N_s$, which plays the role of the number of independent fermionic flavors (the case $N_s=1$ corresponds to completely spin-polarized electrons).\n\nWe will discuss two models of electron-electron interactions: (i) the long-range Coulomb interaction with a small gas parameter $r_s \\equiv e^2\/\\epsilon v_F \\ll 1$ (where $\\epsilon$ is the dielectric constant of the medium, and $v_F$ is the Fermi velocity), and (ii) an arbitrary weak interaction. In the case of the Coulomb interaction, its screening should be taken into account, whereas for a weak interaction this procedure is unnecessary.\nIn both cases, the statically screened interaction in the real space can be written in the form\n\\begin{equation}\n\\label{V(r)}\n  V({\\bf r})\n  =\n  \\frac{\\lambda}{N_s\\nu} \\delta_\\kappa({\\bf r}) ,\n\\end{equation}\nwhere $\\lambda$ is the dimensionless interaction strength,\nand $\\delta_\\kappa({\\bf r})$ is the delta function smeared on the scale of the interaction radius $1\/\\kappa$\n[the latter is assumed to be much smaller than the size of the quantum dot,\njustifying the notion of the smeared delta function in Eq.~(\\ref{V(r)})].\nFor the long-range Coulomb interaction, $\\lambda=1$ and $1\/\\kappa$ is the Thomas-Fermi screening length,\nsee Eq.~(\\ref{kappa}).\nBesides the dimensionless strength $\\lambda$, the only characteristic of the potential (\\ref{V(r)}) that will be important in the following is the ratio, $F$, of the Hartree and Fock diagrams, which is a measure of the interaction range \\cite{Altshulerbook,Akkermans}. In the three- and two-dimensional cases it is given by (see \\ref{S:Non-RPA} for details)\n\\begin{equation}\n\\label{f-gen}\n  F\n  =\n  \\int_0^{1} v_\\kappa(2p_Fx) \\varphi(x) \\, dx ,\n\\qquad\n  \\varphi(x)\n  =\n  \\begin{cases}\n    2 x, & \\text{in 3D}, \\\\\n    (2\/\\pi) (1-x^2)^{-1\/2}, & \\text{in 2D}, \\\\\n  \\end{cases}\n\\end{equation}\nwhere $v_\\kappa({\\bf q})$ is the Fourier transform of $\\delta_\\kappa({\\bf r})$ in Eq.~(\\ref{V(r)}).\nThe limiting cases of $F=0$ and $F=1$ correspond to long-range ($\\kappa\\ll k_F$) and point-like screened interactions, respectively.\n\nWe will assume that interaction can be taken into account perturbatively, that can be justified in the two partially overlapping limits:\n\n\\begin{equation}\n\\label{lambda-f}\n  \\text{$\\lambda\\ll1$ or $F\\ll1$} ,\n\\end{equation}\ni.e., in the limit of weak interaction or for sufficiently large interaction radius ($\\kappa\\ll k_F$).\nThe two models of electron-electron interaction discussed above conform to the condition (\\ref{lambda-f}):\nFor the Coulomb interaction with $\\lambda=1$, its applicability is provided by the smallness of the gas parameter $r_s$ since $F\\sim r_s\\ln 1\/r_s$ [see Eq.~(\\ref{f-Coulomb})]; for weak interaction with an arbitrary radius $1\/\\kappa$, it is justified as long as $\\lambda\\ll1$.\n\n\n\n\\subsection{Average energy relaxation rate}\n\nWe start with generalizing the SIA result (\\ref{SIA}) to the case of a finite temperature $T$, arbitrary spin degeneracy $N_s$ and arbitrary interaction radius measured by the parameter $F$ [assuming that the restriction (\\ref{lambda-f}) holds].\nThe temperature and excitation energy are supposed to be smaller than the Thouless energy, $\\max \\{ \\varepsilon, T \\}\\ll E_{\\text{Th}}$.\nThe average energy relaxation rate calculated in the second order in the statically screened interaction (\\ref{V(r)}) is given by\n\\begin{equation}\n  \\gamma_0(\\varepsilon,T)\n  =\n  \\frac{\\lambda^2c(N_s,F)}{\\pi}\n  \\frac{\\Delta}{E_2^2} (\\varepsilon^2+\\pi^2 T^2) ,\n\\label{SIAT}\n\\end{equation}\nwhere $\\Delta=(\\nu V)^{-1}$ is the mean level spacing in the dot\n($\\nu$ is the single-particle density of states at the Fermi level per one spin projection, and $V$ is volume of the dot), and the energy scale $E_2$ is determined by the diffusion inside the dot:\n\\begin{equation}\n\\label{En}\n  E_n\\equiv\\biggl[\\mathop{{\\sum}'}_m \\frac{1}{(Dq^2_m)^n}\\biggr]^{-1\/n} \\sim E_{\\text{Th}}\n  .\n\\end{equation}\nHere $D$ is the diffusion coefficient, and $q^2_m$ are non-zero eigenvalues\nof the Laplace operator in the dot, $- \\nabla^2 \\psi_m({\\bf r}) = q_m^2 \\psi_m({\\bf r})$,\nwith the von Neumann boundary conditions.\nThe magnitude of $E_n$ defined by Eq.~(\\ref{En}) is set by the Thouless energy,\n$E_{\\text{Th}}=D\/L^2$, where $L$ is the typical size of the quantum dot.\n\nThe factor $c(N_s,F)$ in Eq.~(\\ref{SIAT}) takes into account spin degeneracy and the spatial structure of the interaction potential (\\ref{V(r)}):\n\\begin{equation}\n\\label{c-res}\n  c(N_s,F)\n  =\n  \\frac{1}{N_s^2} \\left(N_s - 2F + N_s F^2 \\right) .\n\\end{equation}\nThe function $c(N_s,F)$ has an important property $c(1,1)=0$, which ensures vanishing of interaction effects for polarized ($N_s=1$) fermions with a contact interaction,\nas a consequence of the Pauli exclusion principle. It should be noted\nthat the anticipated cancellation of $c(1,1)$ takes place only if the Hartree-type diagrams yielding the terms with $F$ in Eq.~(\\ref{c-res}) are taken into account (see Sec.~\\ref{SS:rs}). This issue is often overlooked in literature, leading to a variety of claimed prefactors in Eq.~(\\ref{SIA}).\n\n\n\n\\subsection{Mesoscopic fluctuations of the energy relaxation rate}\n\nIn the FGR description, the relaxation rate $\\gamma_i$ of a given single-particle state $|i\\rangle$ is a function of its energy only, implying that the states close in energy should have nearly the same inelastic width.\nThe validity of this approximation can be verified \\emph{a posteriori}\\\nby studying mesoscopic, i.~e., level-to-level fluctuations of $\\gamma_i$.\nWe calculate them under the same assumptions of the zero-dimensional geometry,\n$\\max \\{ \\varepsilon, T \\}\\ll E_{\\text{Th}}$, used for the determination of the average $\\gamma_0(\\varepsilon,T)$.\n\nWe describe the relaxation rate in the framework of the quantum kinetic equation for electrons in dirty metals~\\cite{RS}. This approach differs significantly from the lattice model proposed by AGKL \\cite{AGKL} and extensively used in subsequent publications \\cite{MF97,Silv,Jacquod97,LTB98,Mirlin2015}.\nThe method of kinetic equation is not suitable for studying many-body localization, but is sufficiently simple to unveil the limits of applicability of the FGR description.\nNevertheless, the kinetic approach, working well for interacting systems with continuous spectrum, meets significant difficulties in the considered region $\\max\\{\\varepsilon,T\\}\\llE_{\\text{Th}}$, where individual energy levels are well-resolved, $\\gamma_{0}(\\varepsilon,T)\\ll \\Delta$.\nDiscreteness of energy spectrum requires non-perturbative averaging over disorder, which is performed by means of the non-linear supersymmetric sigma model \\cite{Efetov-book}.\n\nThe main idea behind our calculations is the following. In the zeroth approximation (which is equivalent to the FGR), energy levels acquire an inelastic width $\\gamma_0(\\varepsilon,T)$. The inverse of this quantity yields the temporal scale\nat which single-particle coherence is maintained.\nThis corresponds to the appearance of the `mass' of the order $\\gamma_{0}(\\varepsilon,T)$ for the zero-dimensional diffusons (diffusons with zero momentum).\nAt the next stage, when loop corrections to the FGR result are considered, this `mass' will regularize the otherwise divergent contributions, producing $\\gamma_0(\\varepsilon,T)$ in the denominator, precisely in the way it occurs in the case of the non-interacting quantum dot in an external field \\cite{dyn-loc,Kubo,Kubo4} and in the high-temperature phase in the problem of many-body localization \\cite{BAA}. Thus, with decreasing the energy of excitations and\/or temperature, the loop corrections to the quasiclassical rate increase, and at some energy scale they become comparable, indicating the breakdown of the FGR description.\n\n\nFluctuations of the energy relaxation rate $\\gamma(\\varepsilon,T)$ around its FGR average value $\\gamma_0(\\varepsilon,T)$ given by Eq.~(\\ref{SIAT}) are characterized by the irreducible average:\n\\begin{equation}\n\\label{corr-irr-def}\n  \\corr{\\gamma^2(\\varepsilon,T)}=\\gamma_0^2(\\varepsilon,T) + \\ccorr{\\gamma^2(\\varepsilon,T)} .\n\\end{equation}\nWe calculate the leading contribution to $\\ccorr{\\gamma^2(\\varepsilon, T)}$ in the delocalized regime and obtain\n\\begin{equation}\n  \\ccorr{\\gamma^2(\\varepsilon,T)}\n  =\n  \\frac{\\lambda^4\\Delta^5}{4\\pi^3}\n  \\left[\\frac{c_2(N_s,F)}{E_2^4}+\\frac{c_4(N_s,F)}{E_4^4}\\right]\n  \\int d\\varepsilon_1 d\\omega_1\n  \\frac{({\\cal F}_{\\varepsilon_1}-{\\cal F}_{\\varepsilon_1-\\omega_1})^2({\\cal B}_{\\omega_1}+{\\cal F}_{\\varepsilon-\\omega_1})^2}\n  {\\gamma_0(\\varepsilon-\\omega_1,T)+\\gamma_0(\\varepsilon_1,T)+\\gamma_0(\\varepsilon_1-\\omega_1,T)}\n  ,\n\\label{subfinal}\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{FandB}\n  {\\cal F}_{\\varepsilon}=\\tanh \\frac{\\varepsilon}{2T},\n  \\qquad\n  {\\cal B}_{\\omega}=\\coth\\frac{\\omega}{2T}\n\\end{equation}\nare equilibrium fermionic and bosonic distribution functions, respectively, while the energies $E_2$ and $E_4$ (both of the order of $E_{\\text{Th}}$) are defined in Eq.~(\\ref{En}). The functions $c_2(N_s,F)$ and $c_4(N_s,F)$ in Eq.~(\\ref{subfinal}) are given by\n\\begin{subequations}\n\\label{c2c4}\n\\begin{gather}\n\\label{c2}\n  c_2(N_s,F)\n  =\n  \\frac{1}{N_s^4}\n  \\left[\n    (3N_s^2+1) - 16N_s F + 2(5N_s^2+7) F^2\n  - 16N_s F^3 + (3N_s^2+1) F^4\n  \\right]\n,\n\\\\\n\\label{c4}\n  c_4(N_s,F)\n  =\n  \\frac{1}{N_s^4}\n  \\left[\n    2N_s^2 - 8N_s F + 4(N_s^2+2) F^2\n  - 8N_s F^3 + 2N_s^2 F^4\n  \\right] .\n\\end{gather}\n\\end{subequations}\nThey obey an important relation $c_2(1,1)=c_4(1,1)=0$, which guarantees vanishing of mesoscopic fluctuations [as well as the inelastic width itself, see Eq.~(\\ref{c-res})] for spin-polarized fermions with a contact interaction.\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=.55\\textwidth]{upsilon.pdf}}\n\\caption{The function $\\Upsilon(x)$ [Eq.~(\\ref{Upsilon-def})], which determines the energy and temperature dependence of $\\ccorr{\\gamma^2(\\varepsilon,T)}$, see Eq.~(\\ref{gamma-Upsilon}).}\n\\label{F:Upsilon}\n\\end{figure}\n\nTo single out the energy and temperature dependence of Eq.~(\\ref{subfinal}), we rewrite it in the form\n\\begin{equation}\n\\label{gamma-Upsilon}\n  \\ccorr{\\gamma^2(\\varepsilon,T)}\n  =\n  \\frac{\\lambda^2 \\Delta^4}{4\\pi^2} \\frac{E_2^2}{c(N_s,F)} \\left[\\frac{c_2(N_s,F)}{E_2^4}+\\frac{c_4(N_s,F)}{E_4^4}\\right]\n  \\Upsilon \\left( \\frac{\\varepsilon}{2T} \\right),\n\\end{equation}\nwhere the dimensionless function $\\Upsilon(x)$ plotted in Fig.~\\ref{F:Upsilon} is defined as\n\\begin{equation}\n\\label{Upsilon-def}\n  \\Upsilon(x)\n  =\n  \\int dy \\, dz \\,\n  \\frac{[\\tanh(y)-\\tanh (y-z)]^2[\\coth(z)-\\tanh(z-x)]^2}{(x-z)^2+y^2+(y-z)^2 + 3\\pi^2\/4}\n  =\n  \\begin{cases}\n    0.248 , & x = 0 ; \\\\\n    16.95 , & x = \\infty .\n  \\end{cases}\n\\end{equation}\nHence, the magnitude of mesoscopic fluctuations of the relaxation rate can be roughly estimated as\n\\begin{equation}\n  \\ccorr{\\gamma^2(\\varepsilon,T)}\n  \\sim\n  \\lambda^2\n  \\frac {\\Delta^4}{E_{\\text{Th}}^2} .\n\\label{qualresult1}\n\\end{equation}\nNote however that, due to a huge variation of the function $\\Upsilon(x)$, fluctuations at the Fermi energy ($\\varepsilon\\ll T$) are nearly 100 times smaller than the naive estimate (\\ref{qualresult1}).\n\n\nExpression (\\ref{subfinal}) for mesoscopic fluctuations of the energy relaxation rate in the FGR regime is the main result of our work.\nThe relative strength of mesoscopic fluctuations can be estimated as\n\\begin{equation}\n  \\frac{\\ccorr{ \\gamma^2(\\varepsilon,T)}}{\\gamma_{0}^2(\\varepsilon,T)}\n  \\sim\n  \\frac{\\Delta_3(\\max\\{\\varepsilon,T\\})}{\\gamma_{0}(\\varepsilon,T)}\n  \\sim\n  \\left( \\frac{\\varepsilon_\\text{FGR}}{\\max\\{ \\varepsilon, T\\}} \\right)^4 ,\n\\label{qualresult2}\n\\end{equation}\nwhere $\\Delta_3(\\varepsilon)$ is the mean three-particle level spacing defined in Eq.~(\\ref{Delta3}). The ratio (\\ref{qualresult2}) becomes of the order of unity as $\\max\\{\\varepsilon,T\\}$ approaches the FGR-breaking scale $\\varepsilon_\\text{FGR} = \\Delta\\sqrt{g}$. Hence for the validity of the FGR description of the initial stage of quasiparticle disintegration, the temperature $T$ of electrons in the dot and the excitation energy $\\varepsilon$ at which the decay rate is studied play nearly the same role. They will act differently at larger time scales, when many-body localization may have enough time to show up \\cite{Mirlin2015,Silvestrov2001}.\nPhysically, at $\\max\\{\\varepsilon,T\\}\\gg\\varepsilon_\\text{FGR}$ the state has many available routes to decay into three-particle excitations. Fluctuations of the number of such routes are small and are determined by Eq.~(\\ref{qualresult2}).\n\n\n\n\n\n\\section{General expression for the energy relaxation rate}\n\\label{S:General}\n\n\\subsection{Keldysh technique}\n\\label{SS:Keldysh}\n\nIn order to express the inelastic energy relaxation rate, we use the Keldysh technique \\cite{Keldysh} in the representation of the functional integral \\cite{KA99,KL09}. For an interacting system, the Keldysh action\nis a functional of the fermionic Grassmann fields $\\psi_\\sigma({\\bf r},t)$\n($\\sigma=1,\\dots,N_s$ counts spin projections)\nand the bosonic plasmon field $\\phi({\\bf r},t)$:\n\\begin{equation}\n\\label{action1}\n  S =\n  \\int dt\\left\\{\\sum_{\\sigma=1}^{N_s} \\int d{\\bf r} \\, \\psi^+_{\\sigma}({\\bf r},t) \\left( \\hat G_0^{-1} + \\phi_a({\\bf r},t) \\gamma^a \\right) \\psi_{\\sigma}({\\bf r},t)\n+ \\int(d{\\bf q}) V_0^{-1}({\\bf q}) \\, \\phi^T({\\bf q},t)\\gamma_2\\phi(-{\\bf q},t)\\right\\}.\n\\end{equation}\nThe fields $\\psi_\\sigma({\\bf r},t)$ and $\\phi({\\bf r},t)$ are two-component vectors in the Keldysh space.\nIn the Keldysh-rotated basis \\cite{KA99,KL09},\n\\begin{equation}\n\\hat G_0^{-1}=\\left(i\\frac{\\partial}{\\partial t}+\\frac{\\nabla^2}{2m}-U_{\\text{dis}}({\\bf r})\\right)\\gamma_1\\equiv G_0^{-1}\\gamma_1,\n\\qquad\n\\gamma^1=\\begin{pmatrix} 1&0\\\\0&1 \\end{pmatrix},\n\\qquad\n\\gamma^2=\\begin{pmatrix} 0&1\\\\1&0 \\end{pmatrix},\n\\end{equation}\nwhere $U_{\\text{dis}}({\\bf r})$ is the disorder potential\nwith the correlation function $\\corr{U_{\\text{dis}}({\\bf r})U_{\\text{dis}}({\\bf r}')} = \\delta({\\bf r}-{\\bf r}')\/(2\\pi\\nu\\tau)$\n($\\tau$~is the elastic mean free time, and $\\nu$ is the density of states per one spin projection\nat the Fermi energy). We assume no spin-dependent interactions so that all matrices\nact as a unit matrix in the spin space.\nThe second term in Eq.~(\\ref{action1}) is the action of the plasmon field,\nwith $V_0({\\bf q})$ being the bare interaction potential,\nand $(d{\\bf q})\\equiv d^dq\/(2\\pi)^d$ is the momentum integration measure in $d$ dimensions.\n\nThe Green functions are defined as ($\\xi_i$ denotes the pair ${\\bf r}_i,t_i$):\n\\begin{gather}\n\\hat G(\\xi_1,\\xi_2)=-i\\langle\\psi(\\xi_1)\\psi^+(\\xi_2)\\rangle\n=-i\\int D\\psi^{*}D\\psi D\\phi \\, e^{iS} \\, \\psi(\\xi_1)\\psi^+(\\xi_2),\n\\\\\n\\hat V(\\xi_1,\\xi_2)=-2i\\langle\\phi(\\xi_1)\\phi^T(\\xi_2)\\rangle\n=-2i\\int D\\psi^{*}D\\psi D\\phi \\, e^{iS} \\, \\phi(\\xi_1)\\phi^T(\\xi_2) .\n\\label{propV}\n\\end{gather}\nThe electron Green function has the triangular structure in the Keldysh space:\n\\begin{equation}\n\\hat G = \\begin{pmatrix} G^\\text{R} & G^\\text{K}\\\\ 0&G^\\text{A} \\end{pmatrix},\n\\qquad\nG^\\text{K} = G^\\text{R}\\circ {\\cal F}-{\\cal F}\\circ G^\\text{A},\n\\end{equation}\nwhere ${\\cal F}$ is the fermion distribution function, and the symbol ``$\\circ$''\ndenotes the convolution over intermediate spatial and time indices.\nAt the equilibrium with the temperature $T$, ${\\cal F}_\\varepsilon=\\tanh(\\varepsilon\/2T)$.\n\n\n\\subsection{Interaction propagator}\n\nIn the derivation below we will mainly assume that electrons in the dot interact via the Coulomb potential $V_0({\\bf r})=e^2\/\\epsilon r$, where $\\epsilon$ is the dielectric constant of the medium.\nDue to the long-range nature of the Coulomb interaction, $V_0^{-1}({\\bf q}\\to 0)=0$,\nthe propagator $\\hat V$ should be calculated taking screening into account\n(this procedure is not required for an arbitrary weak and not long-range potential).\nAs usual, we do this in the random phase approximation (RPA) \\cite{Mahan} justified in the case of a small gas parameter:\n\\begin{equation}\n\\label{rs}\n  r_s = \\frac{e^2}{\\epsilon v_F} \\ll 1 .\n\\end{equation}\nUnder this condition the ratio of the Hartree and Fock diagrams, $F$, introduced in Eq.~(\\ref{f-gen}) is small too, see Eq.~(\\ref{f-Coulomb}). Then one should sum only the bubble contributions to the effective propagator, with independent averaging over disorder in each bubble.\n\nThe RPA-screened\ninteraction propagator has the standard structure:\n\\begin{equation}\n  \\hat V = \\begin{pmatrix} V^\\text{K} & V^\\text{R} \\\\ V^\\text{A} & 0 \\end{pmatrix},\n\\qquad\n  V^\\text{K}=V^\\text{R}\\circ {\\cal B}-{\\cal B}\\circ V^\\text{A},\n\\label{CProp}\n\\end{equation}\nwhere ${\\cal B}$ is the boson distribution function defined as \\cite{KA99}\n\\begin{equation}\n{\\cal B}_{\\omega}=\\frac 1{2\\omega}\\int_{-\\infty}^{\\infty}(1-{\\cal F}_{\\varepsilon'}{\\cal F}_{\\varepsilon'-\\omega})d\\varepsilon' .\n\\end{equation}\nAt the thermal equilibrium, ${\\cal B}_\\omega=\\coth\\left(\\omega\/2T\\right)$.\n\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=120mm]{35textcropped.pdf}}\n\\caption{\nThe dynamically screened interaction $V(\\omega,{\\bf q})$ (wavy line) expressed in terms\nof the statically screened interaction (SSI) $V({\\bf q})$ (zigzag line)\nand the dynamic polarization operator $G^\\text{R}G^\\text{A}$.\nFor a quantum-dot at $(\\varepsilon,T)\\llE_{\\text{Th}}$, each dynamic polarization bubble contains a small factor of $\\omega\/E_{\\text{Th}}$.}\n\\label{F:SSI}\n\\end{figure}\n\nThe dynamically screened interaction can be written in the form (see Fig.~\\ref{F:SSI})\n\\begin{equation}\n  V^{\\text{R}(\\text{A})}(\\omega,{\\bf q})\n  =\n  \\bigl[\n    V^{-1}({\\bf q}) + \\Pi^{\\text{R}(\\text{A})}_\\text{dyn}(\\omega,{\\bf q})\n  \\bigr]^{-1} ,\n\\label{CProp10}\n\\end{equation}\nwhere the real function $V({\\bf q})$ is the statically screened interaction (SSI):\n\\begin{equation}\n  V({\\bf q})\n  =\n  \\bigl[\n    V_0^{-1}({\\bf q}) + N_s\\nu\n  \\bigr]^{-1} ,\n\\label{V-SSI}\n\\end{equation}\nand $\\Pi_\\text{dyn}$ is the dynamic part of the polarization operator originating from the bubble $G^\\text{R}G^\\text{A}$:\n\\begin{equation}\n  \\Pi^{\\text{R}(\\text{A})}_\\text{dyn}(\\omega,{\\bf q})\n  =\n  N_s\\nu \\frac{\\pm i\\omega}{Dq^2\\mp i\\omega}\n  .\n\\label{Pi10}\n\\end{equation}\nThe simple form of the polarization operator $\\Pi({\\bf q},\\omega)$ in Eqs.~(\\ref{V-SSI}) and (\\ref{Pi10}) corresponds to the limit $q\\ll p_F$ (and $\\omega\\ll E_F$). In the case of the Coulomb interaction with a small $r_s$, the spatial dispersion of $\\Pi({\\bf q},0)$ is irrelevant as the interaction in Eq.~(\\ref{f-gen}) is determined by $q\\sim\\kappa\\ll p_F$. In the case of a weak short-scale interaction, $\\lambda\\ll1$, static screening can be neglected and the form of $\\Pi({\\bf q},0)$ is not important.\n\nThe SSI defined in Eq.~(\\ref{V-SSI}) can be conveniently rewritten in the form [Fourier transform of Eq.~(\\ref{V(r)})]\n\\begin{equation}\n\\label{V-static}\n  V({\\bf q})\n  = \\frac{\\lambda}{N_s\\nu} v_\\kappa({\\bf q})\n ,\n\\end{equation}\nwhere\n$v_\\kappa({\\bf q})$\nis the momentum\nrepresentation of the $\\delta$ function smeared over the interaction radius, $\\delta_\\kappa({\\bf r})$.\nBy definition, $v_\\kappa(0)=1$.\nIn the important case of the Coulomb interaction, $\\lambda^\\text{Coul}=1$ and\n\\begin{equation}\n\\label{delta-kappa-Coul}\n  v_\\kappa^\\text{Coul}({\\bf q})\n  =\n  \\begin{cases}\n    \\kappa_\\text{3D}^2\/(q^2+\\kappa_\\text{3D}^2), & \\text{in 3D,} \\\\\n    \\kappa_\\text{2D}\/(q+\\kappa_\\text{2D}), & \\text{in 2D,}\n  \\end{cases}\n\\end{equation}\nwhere $1\/\\kappa$ is the Thomas-Fermi screening length given by\n\\begin{equation}\n\\label{kappa}\n  \\kappa^2_\\text{3D} = 4\\pi N_s\\nu_3 e^2\/\\epsilon,\n\\qquad\n  \\kappa_\\text{2D} = 2\\pi N_s\\nu_2 e^2\/\\epsilon .\n\\end{equation}\n\n\n\n\\subsection{Kinetic equation}\n\nIn this Section we sketch the main steps in the derivation of the quantum kinetic equation\nin the Keldysh formalism (for a detailed discussion see, e.g, Refs.~\\cite{RS,Altshuler}).\nFrom the Dyson equation for the exact (disorder-dependent)\nelectron Green function $\\hat G$ we obtain\n\\begin{equation}\n  \\bigl[\\hat G_0^{-1},\\hat G\\bigr]=\\bigl[\\hat{\\Sigma},\\hat G\\bigr],\n\\label{comm}\n\\end{equation}\nwhere $[A,B]=A\\circ B-B\\circ A$,\nand $\\hat{\\Sigma}$ is the irreducible self-energy due to interaction,\nbearing the same triangular structure as the electron Green function:\n\\begin{equation}\n\\label{Sigma-RAK}\n  \\hat{\\Sigma}=\\begin{pmatrix} \\Sigma^\\text{R}& \\Sigma^\\text{K} \\\\ 0& \\Sigma^\\text{A} \\end{pmatrix}.\n\\end{equation}\n\nAs we are interested in the slow dynamics of the system,\nit is convenient to switch to the mixed energy-time (Wigner) representation,\n\\begin{equation}\n  f_{\\varepsilon}(t) = \\int d\\tau \\, f\\left(t+\\frac {\\tau}2, t-\\frac{\\tau}2\\right)e^{i\\varepsilon \\tau} ,\n\\end{equation}\nwhich allows us to get rid of the convolution on the right-hand side\nof Eq.~(\\ref{comm}).\nThe kinetic equation is obtained from the Keldysh component of Eq.~(\\ref{comm})\nby tracing over the space.\nWe also assume that the distribution function does not depend\non the coordinates. Therefore the left-hand side of Eq.~(\\ref{comm})\nproduces only the time derivatives of ${\\cal F}$, and we arrive at\nthe standard form of the kinetic equation,\n\\begin{equation}\n\\label{F-St}\n  \\partial_t {\\cal F}_\\varepsilon = \\text{St}[{\\cal F}_\\varepsilon] .\n\\end{equation}\nThe collision integral $\\text{St}[{\\cal F}_\\varepsilon]$ is given by\n\\begin{equation}\n  \\text{St}[{\\cal F}_\\varepsilon]\n  \\int d{\\bf r} \\, \\Delta G_{\\varepsilon}({\\bf r},{\\bf r})\n  =\n  -i\\int d{\\bf r} \\, d{\\bf r}'\\left\\{\n  \\Delta \\Sigma_{\\varepsilon}({\\bf r}, {\\bf r}')G^\\text{K}_{\\varepsilon}({\\bf r}',{\\bf r})\n- \\Sigma^\\text{K}_{\\varepsilon}({\\bf r},{\\bf r}')\\Delta G_{\\varepsilon}({\\bf r}',{\\bf r})\n  \\right\\},\n\\label{kinur0}\n\\end{equation}\nwhere all functions have an implicit central-time argument $t$, and we denote\n\\begin{equation}\n  \\Delta G= G^\\text{R}-G^\\text{A},\\qquad \\Delta \\Sigma=\\Sigma^\\text{R}-\\Sigma^\\text{A}.\n\\label{DeltaG}\n\\end{equation}\n\n\n\n\\subsection{Modification of the Keldysh technique}\n\\label{SS:Modified}\n\nAs our future analysis will involve diagrams with many interaction propagators,\nit is convenient to modify the standard Keldysh diagrammatic technique\ndiscussed above to make it suitable for routine calculations.\nWe find it appropriate to `eliminate' the Keldysh component\nof the electron Green function and to describe the system\nin terms of the retarded and advanced Green functions only.\nTo this end, we note that in the case of weak nonequilibrium\none has $G^\\text{K}_{\\varepsilon}(t)={\\cal F}_{\\varepsilon}(t)\\Delta G_{\\varepsilon}(t)$,\nwhich allows us to diagonalize the Green function as\n\\begin{equation}\n\\hat G=U^{-1}_{\\varepsilon} g_{\\varepsilon} U_{\\varepsilon},\n\\qquad\ng_{\\varepsilon}=\\begin{pmatrix} G^\\text{R}_{\\varepsilon}&0\\\\0&G^\\text{A}_{\\varepsilon} \\end{pmatrix},\n\\qquad\nU_{\\varepsilon}=\\begin{pmatrix} 1&{\\cal F}_{\\varepsilon}\\\\0&-1\\end{pmatrix} ,\n\\label{gtransform}\n\\end{equation}\nwhere the time argument is suppressed for brevity.\nIn order to construct the perturbation theory in terms of\nthe Green function $g$,\nit is convenient to include $U_{\\varepsilon}$ into the definition\nof the interaction vertex:\n\\begin{equation}\n\\Gamma^k(\\varepsilon_1,\\varepsilon_2)=U_{\\varepsilon_1}\\gamma^k U^{-1}_{\\varepsilon_2}.\n\\end{equation}\nThe resulting $\\Gamma^k(\\varepsilon_1,\\varepsilon_2)$ depend on two\nenergy indices owing to the energy dependence of the distribution function:\n\\begin{equation}\n\\Gamma^1(\\varepsilon_1,\\varepsilon_2)=\\begin{pmatrix} 1&{\\cal F}_{\\varepsilon_2}-{\\cal F}_{\\varepsilon_1}\\\\0&1\\end{pmatrix}, \\qquad \\Gamma^2(\\varepsilon_1,\\varepsilon_2)=\\begin{pmatrix} {\\cal F}_{\\varepsilon_1}&-1+{\\cal F}_{\\varepsilon_1}{\\cal F}_{\\varepsilon_2}\\\\-1&-{\\cal F}_{\\varepsilon_2}\\end{pmatrix}.\n\\end{equation}\n\nThis modification of the Keldysh technique that will be used below allows us\nto simplify calculations with many interaction lines significantly,\nsince now the electron Green function $g$ is diagonal in the Keldysh space\nand does not contain the distribution function. The interaction line is defined now as\n\\begin{equation}\n\\label{Y-def}\nY_{abcd}(\\varepsilon,\\varepsilon',\\omega,{\\bf q})=\n\\raisebox{-24pt}{\n  \\begin{picture}(88,53)(-10,0)\n    \\put(0,10){\\includegraphics[scale=0.15]{Yabcd1.jpg}}\n    \\put(2,2){$\\varepsilon$}\n    \\put(2,48){$\\varepsilon-\\omega$}\n    \\put(43,48){$\\varepsilon'-\\omega$}\n    \\put(58,2){$\\varepsilon'$}\n    \\put(-5,15){$a$}\n    \\put(-5,34){$b$}\n    \\put(68,34){$c$}\n    \\put(68,15){$d$}\n  \\end{picture}\n  }\n  =\n  \\Gamma^k_{ab}(\\varepsilon,\\varepsilon-\\omega)\\Gamma^l_{cd}\\left(\\varepsilon'-\\omega,\\varepsilon'\\right)V^{kl}_{\\omega}({\\bf q}),\n\\end{equation}\nwhere indices $a,b,c,d$ take the values $\\text{R}$ and $\\text{A}$,\nand we imply the correspondence $\\text{R}\\leftrightarrow 1$ and $\\text{A}\\leftrightarrow 2$.\nExplicit expressions for $Y_{abcd}$ are listed in \\ref{A:Y}.\nAccording to the general rules of the diagrammatic technique,\nthe element $Y_{abcd}(\\varepsilon,\\varepsilon',\\omega,{\\bf q})$ enters with the coefficient $i\/2$\n[the factor $2$ is inherited from Eq.~(\\ref{propV})].\n\nThe irreducible self-energy in the new basis, $\\Sigma^{ij}$,\nis related to the self-energy (\\ref{Sigma-RAK}) as\n\\begin{equation}\n  \\Sigma^\\text{R}=\\Sigma^{\\text{RR}},\n\\quad\n  \\Sigma^\\text{A}=\\Sigma^{\\text{AA}},\n\\quad\n  \\Sigma^\\text{K}=\\left(\\Sigma^{\\text{RR}}-\\Sigma^{\\text{AA}}\\right) {\\cal F}_{\\varepsilon}-\\Sigma^{\\text{RA}},\n\\quad\n  \\Sigma^{\\text{AR}}=0.\n\\end{equation}\nThus, in the modified Keldysh technique, Eq.~(\\ref{kinur0})\nfor the collision integral takes a compact form:\n\\begin{equation}\n  \\text{St}[{\\cal F}_\\varepsilon] \\int d{\\bf r}\\,\\Delta G_{\\varepsilon}({\\bf r},{\\bf r})\n  =\n  -i\\int d{\\bf r}\\, d{\\bf r}'\\,\\Sigma^{\\text{RA}}_{\\varepsilon}({\\bf r},{\\bf r}') \\Delta G_{\\varepsilon}({\\bf r}',{\\bf r}).\n\\label{kinur1}\n\\end{equation}\n\n\n\\subsection{Collision integral and the energy relaxation rate}\n\nThe inelastic energy relaxation rate, $\\gamma(\\varepsilon,T)$, can be obtained\nfrom the collision integral in the usual way~\\cite{Altshuler,Schmid-ep,Reizer}:\n\\begin{equation}\n  \\gamma(\\varepsilon,T) = -\\frac{\\delta \\, \\text{St}[{\\cal F}_{\\varepsilon}]}{\\delta {\\cal F}_{\\varepsilon}}.\n\\label{tau}\n\\end{equation}\nUsing $\\text{St}[{\\cal F}_{\\varepsilon}]$ from Eq.~(\\ref{kinur1}), we arrive at the following\nexpression for $\\gamma(\\varepsilon,T)$:\n\\begin{equation}\n  \\gamma(\\varepsilon,T) \\int d{\\bf r}\\,\\Delta G_{\\varepsilon}({\\bf r},{\\bf r})\n  =\n  i\\frac{\\delta}{\\delta {\\cal F}_{\\varepsilon}}\n  \\int d{\\bf r} \\, d{\\bf r}'\\,\\Sigma^{\\text{RA}}_{\\varepsilon}({\\bf r},{\\bf r}') \\Delta G_{\\varepsilon}({\\bf r}',{\\bf r}).\n\\label{gamma}\n\\end{equation}\n\nEquation (\\ref{gamma}) is the starting point for the calculation of the energy relaxation rate.\nThis equation contains exact disorder-dependent Green functions, and averaging its $n$'th power\nover the random potential generates the $n$'th moment of $\\gamma(\\varepsilon,T)$.\nThe simplest is the first moment, $\\corr{\\gamma(\\varepsilon,T)}$, related to the average\ncollision integral $\\corr{\\text{St}[{\\cal F}_{\\varepsilon}]}$. In this case, the left-hand side of Eq.~(\\ref{kinur1})\ngives the average density of states, $\\corr{\\Delta G_{\\varepsilon}({\\bf r},{\\bf r})} = -2\\pi i \\nu$,\nand one arrives at\n\\begin{equation}\n  \\corr{\\text{St}[{\\cal F}_{\\varepsilon}]}\n  =\n  \\frac{\\Delta}{2\\pi} \\int d{\\bf r}\\, d{\\bf r}'\n  \\langle\\Sigma^{\\text{RA}}_{\\varepsilon}({\\bf r},{\\bf r}') \\Delta G_{\\varepsilon}({\\bf r}',{\\bf r})\\rangle .\n\\label{kinur01}\n\\end{equation}\nIn the next Section we show how this equation reproduces Sivan, Imry and Aronov result (\\ref{SIAT}) for the average inelastic rate in the limit $F\\to0$, and generalize it to the case of an arbitatry parameter $F$.\nThe second moment of $\\gamma(\\varepsilon,T)$ will be considered in Secs.~\\ref{S:MF}--\\ref{S:mesofluct-rs}.\n\n\n\n\n\\section{Average energy relaxation rate}\n\\label{S:Average}\n\nIn this Section we rederive the result of Sivan, Imry and Aronov \\cite{SIA}\nand generalize it to the case\nof an arbitrary temperature $T$, spin degeneracy $N_s$ and interaction radius characterized by the parameter $F$.\nOur treatment closely follows the standard derivation\nof the kinetic equation in the RPA approximation \\cite{RS}, but within the modified\nKeldysh technique introduced in Sec.~\\ref{SS:Modified}.\nThe purpose of this Section is to illustrate the usage of this technique\nand to prepare the ingredients for the analysis of mesoscopic fluctuations\nof $\\gamma(\\varepsilon,T)$ in Sec.~\\ref{S:MF}.\n\n\n\n\\subsection{Derivation of the Sivan, Imry and Aronov result}\n\\label{SS:SIA}\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=163mm]{30textcropped.pdf}}\n\\caption{(a) The simplest\n(and the most important for small $F$) contribution to the self-energy $\\Sigma^{\\text{RA}}_{\\varepsilon}$.\n(b) The graphic representation of the product $\\Sigma^{\\text{RA}}_{\\varepsilon} \\Delta G_{\\varepsilon}$\nwhich determines the collision integral (\\ref{kinur01}). The thick line\nstanding for $\\Delta G_{\\varepsilon} = G^\\text{R}_{\\varepsilon}-G^\\text{A}_{\\varepsilon}$ corresponds to\nthe electron state whose decay is studied.\n(c) The diagram (b) averaged over disorder, with the shaded block representing the diffuson.}\n\\label{F:SIA2}\n\\end{figure}\n\nThe simplest diagram for the self-energy $\\Sigma^{\\text{RA}}_{\\varepsilon}$\nis shown in Fig.~\\ref{F:SIA2}(a).\nIn the limit $F\\ll1$ it gives the leading contribution\nto the relaxation rate (a more general situation is discussed\nin Sec.~\\ref{SS:rs}).\nThe corresponding analytic expression\ncan be easily written in terms of the elements $Y$ introduced in Eq.~(\\ref{Y-def}):\n\\begin{equation}\n\\Sigma^{\\text{RA}}_{1\\varepsilon}({\\bf r},{\\bf r}')=\\frac i2\\int (d\\omega)\\left\\{G^\\text{R}_{\\varepsilon-\\omega}({\\bf r},{\\bf r}')Y_{\\text{RR},\\text{RA}}(\\varepsilon,\\varepsilon,\\omega, {\\bf r}-{\\bf r}')+G^\\text{A}_{\\varepsilon-\\omega}({\\bf r},{\\bf r}')Y_{\\text{RA},\\text{AA}}(\\varepsilon,\\varepsilon,\\omega,{\\bf r}-{\\bf r}')\\right\\},\n\\end{equation}\nwith $(d\\omega)\\equiv d\\omega\/2\\pi$.\nThe interaction propagators are assumed to be RPA screened\nand averaged over disorder (see Sec.~\\ref{SS:Keldysh}),\nwhile the electron Green functions are still exact in a given realization of disorder.\nUsing Eqs.~(\\ref{RRRA}) and (\\ref{RAAA}), and employing the analyticity properties\nwe obtain\n\\begin{equation}\n\\Sigma^{\\text{RA}}_{1\\varepsilon}({\\bf r},{\\bf r}')\n=\n\\frac i2 \\int (d\\omega) \\Phi(\\varepsilon,\\omega)\n\\Delta G_{\\varepsilon-\\omega}({\\bf r},{\\bf r}')\n\\Delta V(\\omega, {\\bf r}-{\\bf r}'),\n\\label{Sigma1}\n\\end{equation}\nwhere $\\Delta V=V^\\text{R}-V^\\text{A}$, and\n\\begin{equation}\n  \\Phi(\\varepsilon,\\omega)=({\\cal F}_{\\varepsilon}-{\\cal F}_{\\varepsilon-\\omega}){\\cal B}_{\\omega}-(1-{\\cal F}_{\\varepsilon}{\\cal F}_{\\varepsilon-\\omega})\n\\end{equation}\nis a combination of the fermionic and bosonic distribution functions which vanishes at the equilibrium (detailed balance).\n\nThe collision integral (\\ref{kinur01}) involves the product $\\Sigma^{\\text{RA}}_{\\varepsilon} \\Delta G_{\\varepsilon}$,\nwhich is to be averaged over disorder. To make this calculation more intuitive,\nwe represent it diagrammatically on a separate graph in Fig.~\\ref{F:SIA2}(b),\nwhere the thick line stands for the Green function $\\Delta G_{\\varepsilon}$\nof the initial state. To avoid confusion we emphasize that this picture is just\na simple graphical notation for $\\Sigma^{\\text{RA}}_{\\varepsilon} \\Delta G_{\\varepsilon}$, since\nthe vertices with a thick line are not described by the rule of Eq.~(\\ref{Y-def}).\nSubstituting the self-energy (\\ref{Sigma1}) into Eq.~(\\ref{kinur01})\nwe obtain for the averaged collision integral:\n\\begin{equation}\n  \\corr{\\text{St}[{\\cal F}_{\\varepsilon}]}\n  =\n  \\frac{i\\Delta}{4\\pi}\\int d{\\bf r} \\,d{\\bf r}'(d\\omega) \\Phi(\\varepsilon,\\omega)\n  \\langle\\Delta G_{\\varepsilon}({\\bf r}',{\\bf r}) \\Delta G_{\\varepsilon-\\omega}({\\bf r},{\\bf r}')\\rangle\n  \\Delta V(\\omega,{\\bf r}-{\\bf r}') .\n\\end{equation}\nThe averaged product of two Green functions,\n\\begin{equation}\n\\label{<GG>}\n  \\corr{\\Delta G_{\\varepsilon}({\\bf r}',{\\bf r}) \\Delta G_{\\varepsilon-\\omega}({\\bf r},{\\bf r}')}\n  =\n  -4\\pi\\nu \\mathop{\\rm Re} D_0^\\text{R}(\\omega,{\\bf r}-{\\bf r}')\n\\end{equation}\nis expressed in terms\nof the particle-hole ladder (diffuson), $D_0^\\text{R}(\\omega,{\\bf q}) = 1\/(Dq^2-i\\omega)$,\nsee Fig.~\\ref{F:SIA2}(c). Then one arrives at the well-known result for the collision integral in dirty metals (see, for example, Refs.~\\cite{RS,AltshulerAronov,Schmid-ee}):\n\\begin{equation}\n  \\corr{\\text{St}[{\\cal F}_\\varepsilon]}\n  =\n  2\\int(d{\\bf q})(d\\omega)\\Phi(\\varepsilon,\\omega)\\mathop{\\rm Re} D_0^\\text{R}(\\omega,{\\bf q})\\mathop{\\rm Im} V^\\text{R}(\\omega,{\\bf q}) ,\n\\label{StF}\n\\end{equation}\nwhere $(d{\\bf q})\\equiv d^dq\/(2\\pi)^d$, and $d$ is the space dimensionality.\n\nAccording to Eq.~(\\ref{CProp10}), the imaginary part of the fluctuation propagator is determined by the dynamic polarization operator $\\Pi_\\text{dyn}$.\nFor a quantum dot in the zero-dimensional regime ($\\omega\\llE_{\\text{Th}}$),\n$\\Pi_\\text{dyn}$ is a small correction to $V^{-1}({\\bf q})$,\nand $\\mathop{\\rm Im} V^\\text{R}(\\omega,{\\bf q})$ can be written as\n\\begin{equation}\n\\label{ImV}\n  \\mathop{\\rm Im} V^\\text{R}(\\omega,{\\bf q})\n  \\approx\n  - V^2({\\bf q}) \\mathop{\\rm Im} \\Pi^{\\text{R}}_\\text{dyn}(\\omega,{\\bf q}) .\n\\end{equation}\nUsing Eq.~(\\ref{Pi10}) and assuming that the system size, $L$, is larger than the interaction radius, $1\/\\kappa$, we obtain in the zero-dimensional regime:\n\\begin{equation}\n  \\corr{\\text{St}[{\\cal F}_{\\varepsilon}]}\n  =\n  -\\frac{2\\lambda^2}{N_s\\nu}\\int (d\\omega) (d{\\bf q})\\frac{\\omega \\, \\Phi(\\varepsilon,\\omega)}{(Dq^2)^2}\n  =\n  - \\frac{2\\lambda^2\\Delta}{N_s} \\mathop{{\\sum}'}_m\n  \\int (d\\omega)\\frac {\\omega\\,\\Phi(\\varepsilon,\\omega)}{(Dq_m^2)^2} ,\n\\label{StF1}\n\\end{equation}\nwhere $q_m^2$ are the eigenvalues of the operator $-\\nabla^2$ in the dot\nwith von Neumann boundary conditions, with the zero mode being excluded\ndue to electroneutrality \\cite{SIA,Blanter,AGKL,AG98,BlanterMirlin1997}.\nPerforming summation over discrete momenta, we get\n\\begin{equation}\n  \\corr{\\text{St}[{\\cal F}_{\\varepsilon}]}\n  =\n  - \\frac{2\\lambda^2\\Delta}{N_s E_2^2}\n  \\int (d\\omega) \\, \\omega \\, \\Phi(\\varepsilon,\\omega) ,\n\\end{equation}\nwhere the energy $E_2\\simE_{\\text{Th}}$ is defined in Eq.~(\\ref{En}).\nThe inelastic energy relaxation rate at the equilibrium\ncan be extracted with the help of Eq.~(\\ref{tau}):\n\\begin{equation}\n\\label{gamma0-integral}\n\\gamma_0^\\text{RPA}(\\varepsilon,T)=\\frac{2\\lambda^2\\Delta}{N_s E_2^2} \\int (d\\omega)\\,\\omega \\left\\{ \\coth\\left(\\frac{\\omega}{2T}\\right)+\\tanh\\left(\\frac{\\varepsilon-\\omega}{2T}\\right)\\right\\}.\n\\end{equation}\nIntegrating over $\\omega$, we arrive at Eq.~(\\ref{SIAT})\nfor the average energy relaxation rate $\\gamma_0(\\varepsilon,T)$\nin the limit $F\\to0$.\n\n\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=150mm]{401textcropped.pdf}}\n\\vspace{0mm}\n\\caption{\nThe diagrams for the collision integral with two SSI lines\nprior to disorder averaging.\n(a) The RPA diagram\n\\ref{F:SIA2}(b)\nredrawn in terms of the SSI lines (\\ref{V-static})\nand the dynamic polarization bubble $G^\\text{R}G^\\text{A}$. The corresponding contribution to the relaxation rate is given by the $X_4$ term in Eq.~(\\ref{gamma1}).\n(b) The other, non-RPA diagram discarded in Sec.~\\ref{SS:SIA}, leading to the $Y_4$ term in Eq.~(\\ref{gamma1}). Its contribution to the mean relaxation rate can be neglected\nat small $F$ [see Eq.~(\\ref{gamma1full})].}\n\\label{F:2Coulomb}\n\\end{figure}\n\n\n\n\n\n\\subsection{Physical interpretation and the other diagram}\n\\label{SS:Physical}\n\nIt is instructive to rederive the result for $\\gamma_0^\\text{RPA}(\\varepsilon,T)$\nin a slightly different manner to clarify the physics of the decay\nprocess responsible for the width of an one-electron level.\nReal decay processes are described by $\\mathop{\\rm Im} V^\\text{R}(\\omega,{\\bf q})$ [see Eq.~(\\ref{StF})],\nwhich is determined by the dynamic screening of the bare interaction.\nIn the zero-dimensional limit, $(E,T,\\omega)\\llE_{\\text{Th}}$,\nthe dynamic part of the polarization bubble,\n$\\Pi_\\text{dyn}(\\omega,{\\bf q})$, contains\na small factor $\\omega\/E_{\\text{Th}}$ [see Eq.~(\\ref{Pi10})].\nThis means that with the accuracy of order $\\omega\/E_{\\text{Th}}$ we may\nconsider $\\Pi_\\text{dyn}(\\omega,{\\bf q})$ as a small perturbation\nand retain only\nthe first dynamic bubble in the expansion of $V(\\omega,{\\bf q})$\nnear the statically screened interaction (SSI)\n$V({\\bf q})$, see Eq.~(\\ref{ImV}).\n\nTherefore with the accuracy of order $\\omega\/E_{\\text{Th}}$ the diagram in Fig.~\\ref{F:SIA2}(a)\nfor the average collision integral with one dynamically screened interaction\ncan be redrawn as the diagram in Fig.~\\ref{F:2Coulomb}(a)\nwith two SSI lines.\nThe advantage of the diagrammatic representation of Fig.~\\ref{F:2Coulomb}(a)\nis that it elucidates the physics of the inelastic collision:\nThe cross-section of this diagram corresponds to the decay of an electron with\nenergy $\\varepsilon$ into an electron with energy $\\varepsilon-\\omega$ and an electron-hole\npair with energies $\\varepsilon_1$ and $\\varepsilon_1-\\omega$.\nThe corresponding contribution to the self-energy\n(before disorder averaging)\ncan be easily read off from Eq.~(\\ref{Y-def}):\n\\begin{equation}\n\\label{SigmaRAa}\n\\Sigma^{\\text{RA}}_{a}=-N_s\\left(\\frac i2\\right)^2\\sum_{abc}G^a_{\\varepsilon-\\omega}({\\bf r}_1,{\\bf r}_2)G^b_{\\varepsilon_1-\\omega}({\\bf r}_4,{\\bf r}_3)G^c_{\\varepsilon_1}({\\bf r}_3,{\\bf r}_4) Y_{\\text{R}abc}(\\varepsilon,\\varepsilon_1,\\omega,{\\bf r}_1,{\\bf r}_3)Y_{cba\\text{A}}(\\varepsilon_1,\\varepsilon,\\omega,{\\bf r}_2,{\\bf r}_4),\n\\end{equation}\nwhere here and in what follows the interaction line $Y_{abcd}$ corresponds to the SSI $V({\\bf q})$.\nThe factor $-N_s$ in Eq.~(\\ref{SigmaRAa}) comes from the upper closed electron loop [the bottom bubble containing a highlighted Green function at the external energy $\\Delta G_{\\varepsilon}$\nis not a closed loop in the diagrammatic sense (see Fig.~\\ref{F:SIA2}) and does not contribute an extra factor $-N_s$].\n\nThe diagram \\ref{F:2Coulomb}(a) is not the only one with two SSI lines\nthat describes quasiparticle decay process. The other diagram is shown in Fig.~\\ref{F:2Coulomb}(b). Though it is not as intuitive as the diagram \\ref{F:2Coulomb}(a), it also makes a contribution to the relaxation rate. That contribution is usually discarded\nsince it vanishes in the limit $F\\to0$ (see below).\nHowever as we show in Sec.~\\ref{Seconddiagram},\nthe diagram \\ref{F:2Coulomb}(b) should be taken into account\nin calculating mesoscopic fluctuations of the relaxation rate,\nsince its contribution is comparable to that of the diagram \\ref{F:2Coulomb}(a)\neven in the limit $F\\to0$.\nThe self-energy for the diagram \\ref{F:2Coulomb}(b) is given by\n\\begin{equation}\n\\Sigma^{\\text{RA}}_{b}=\\left(\\frac i2\\right)^2\\sum_{abc}G^a_{\\varepsilon_1}({\\bf r}_1,{\\bf r}_2)G^b_{\\varepsilon_1-\\omega}({\\bf r}_2,{\\bf r}_3)G^c_{\\varepsilon-\\omega}({\\bf r}_3,{\\bf r}_4) Y_{\\text{R}abc}(\\varepsilon,\\varepsilon-\\omega,\\varepsilon-\\varepsilon_1,{\\bf r}_1,{\\bf r}_3)Y_{abc\\text{A}}(\\varepsilon_1,\\varepsilon,\\omega,{\\bf r}_2,{\\bf r}_4).\n\\end{equation}\nDue to the absence of the closed electron loop in the diagram \\ref{F:2Coulomb}(b), $\\Sigma^{\\text{RA}}_{b}$ does not contain an extra factor $-N_s$ compared to $\\Sigma^{\\text{RA}}_{a}$.\n\nThe energy relaxation rate $\\gamma(\\varepsilon, T)$ (in a given realization of disorder) due to processes with two SSI\nlines can be obtained from Eq.~(\\ref{gamma}) with $\\Sigma^{\\text{RA}}=\\Sigma^{\\text{RA}}_{a}+\\Sigma^{\\text{RA}}_{b}$.\nAfter some algebra involving analyticity properties we obtain\n\\begin{equation}\n  \\gamma(\\varepsilon,T)\\int d{\\bf r} \\, \\Delta G_{\\varepsilon}({\\bf r},{\\bf r})\n  =\n  - \\frac{i\\lambda^2}{4N_s^2\\nu^2}\\int d{\\bf r}_1\\ldots d{\\bf r}_4\\int (d\\varepsilon_1)(d\\omega)({\\cal F}_{\\varepsilon_1}-{\\cal F}_{\\varepsilon_1-\\omega})({\\cal F}_{\\varepsilon-\\omega}+{\\cal B}_{\\omega})\n  \\left[ N_s X_4-Y_4 \\right],\n\\label{gamma1}\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{X4}\n  X_4\n  =\n  \\delta_\\kappa({\\bf r}_1-{\\bf r}_3)\\delta_\\kappa({\\bf r}_2-{\\bf r}_4) \\,\n  \\Delta G_{\\varepsilon}({\\bf r}_2,{\\bf r}_1)\\Delta G_{\\varepsilon-\\omega}({\\bf r}_1,{\\bf r}_2)\n  \\Delta G_{\\varepsilon_1-\\omega}({\\bf r}_4,{\\bf r}_3)\\Delta G_{\\varepsilon_1}({\\bf r}_3,{\\bf r}_4)\n\\end{equation}\nand\n\\begin{equation}\n\\label{Y4}\nY_4=\\delta_{\\kappa}({\\bf r}_1-{\\bf r}_3)\\delta_{\\kappa}({\\bf r}_2-{\\bf r}_4)\\Delta G_{\\varepsilon}({\\bf r}_4,{\\bf r}_1)\\Delta G_{\\varepsilon_1}({\\bf r}_1,{\\bf r}_2)\\Delta G_{\\varepsilon_1-\\omega}({\\bf r}_2,{\\bf r}_3)\\Delta G_{\\varepsilon-\\omega}({\\bf r}_3,{\\bf r}_4).\n\\end{equation}\nIn Eq.~(\\ref{gamma1}), the term $N_s X_4$ comes from $\\Sigma^{\\text{RA}}_{a}$ [diagram \\ref{F:2Coulomb}(a)], and the term $-Y_4$ comes from $\\Sigma^{\\text{RA}}_{b}$ [diagram \\ref{F:2Coulomb}(b)].\nAn expression similar to Eq.~(\\ref{gamma1}) has been derived in Ref.~\\cite{Basko}, where only the contribution from the\ndiagram \\ref{F:2Coulomb}(a) has been considered.\nIn the limit when the interaction radius $1\/\\kappa$ exceeds the Fermi wavelength, i.e. at $F\\ll1$, disorder averaged $\\langle \\gamma(\\varepsilon,T)\\rangle$\nreproduces the result (\\ref{gamma0-integral}) for $\\gamma_0^\\text{RPA}(\\varepsilon, T)$ [see Eq.~(\\ref{gamma1full})].\n\n\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=150mm]{505textcropped.pdf}}\n\\caption{Possible ways to average the diagrams for the relaxation rate shown in Fig.~\\ref{F:2Coulomb} over disorder (other types of averaging with two diffusons exist but are suppressed at least by the factor $l\/L\\ll1$, where $l$ is the mean free path).\n(a) RPA averaging of the diagram \\ref{F:2Coulomb}(a)\nreproducing the SIA result for $\\gamma_0^\\text{RPA}(\\varepsilon,T)$ [Eq.~(\\ref{gamma0-integral})].\n(b) Non-RPA averaging of the diagram \\ref{F:2Coulomb}(a)\nwith the contribution $F^2\\gamma_0^\\text{RPA}(\\varepsilon,T)$.\n(c) and (d) Two ways to average the diagram \\ref{F:2Coulomb}(b), each producing the contribution $-(F\/N_s)\\gamma_0^\\text{RPA}(\\varepsilon,T)$ to the relaxation rate.}\n\\label{F:2Coulomb-av}\n\\end{figure}\n\n\n\\subsection{Disorder averaging and the role of the parameter $F$}\n\\label{SS:rs}\n\nEquation (\\ref{gamma1}) is the starting point for evaluating moments of the relaxation rate. In order to find its mean\nvalue one has to average the combinations of four Green functions in Eqs.~(\\ref{X4}) and (\\ref{Y4}).\n\nThe RPA result of Sec.~\\ref{SS:SIA} is reproduced if one takes only the diagram \\ref{F:2Coulomb}(a) (the term $X_4$) into account and average each bubble independently [see Fig.~\\ref{F:2Coulomb-av}(a)]:\n\\begin{equation}\n\\label{X4RPA}\n  \\corr{X_4}_\\text{RPA}\n  =\n  \\delta_\\kappa({\\bf r}_1-{\\bf r}_3)\\delta_\\kappa({\\bf r}_2-{\\bf r}_4)\n  \\corr{\\Delta G_{\\varepsilon}({\\bf r}_2,{\\bf r}_1)\\Delta G_{\\varepsilon-\\omega}({\\bf r}_1,{\\bf r}_2)}\n  \\corr{\\Delta G_{\\varepsilon_1-\\omega}({\\bf r}_4,{\\bf r}_3)\\Delta G_{\\varepsilon_1}({\\bf r}_3,{\\bf r}_4)} .\n\\end{equation}\nAveraging with the help of Eq.~(\\ref{<GG>}) and integrating\nover $\\varepsilon_1$, one arrives at Eq.~(\\ref{gamma0-integral}) for $\\gamma_0^\\text{RPA}(\\varepsilon,T)$.\n\nAlong with the RPA averaging (\\ref{X4RPA}), there exists another way to average $X_4$ over disorder shown in Fig.~\\ref{F:2Coulomb-av}(b). It has the same structure of diffusons as the RPA diagram \\ref{F:2Coulomb-av}(a),\nbut with the interaction lines taken at fast momenta $q\\sim\\min(\\kappa,p_F)$.\nThe corresponding contribution to the relaxation rate, $F^2\\gamma_0^\\text{RPA}(\\varepsilon,T)$,\ndiffers from the RPA-contribution by a factor $F^2$, where $F$ is the standard notation for the ratio of the Hartree and Fock diagrams \\cite{Altshulerbook,Akkermans,AAL} discussed in \\ref{S:Non-RPA}.\n\nFinally, consider disorder averaging of the diagram \\ref{F:2Coulomb}(b) corresponding to the term $Y_4$ in Eq.~(\\ref{gamma1}). For this diagram, there are also two ways to draw two-diffuson configurations shown in Figs.~\\ref{F:2Coulomb-av}(c) and (d).\nTheir contributions are equal and can be calculated similar to that of the\ndiagram \\ref{F:2Coulomb-av}(b), but now only one of the two SSI lines are taken at fast momentum making the result proportional to the first power of $F$.\nThe overall correction of the diagram \\ref{F:2Coulomb}(b) to the average relaxation rate\nis given by $-(2F\/N_s)\\gamma_0^\\text{RPA}(\\varepsilon, T)$.\n\nThus we see that among four possible contributions to the\naverage relaxation rate shown in Fig.~\\ref{F:2Coulomb-av},\nall non-RPA diagrams [(b), (c) and (d)] are proportional to some\npower of $F$ and hence are suppressed\nif the interaction radius is larger than the Fermi wavelength.\nThe smallness of the non-RPA diagrams\nin this limit should be attributed to the Friedel oscillations\nwhich suppress the contribution of the corresponding process \\cite{Mahan}.\nSuch a situation was considered, e.g., in Refs.~\\cite{RS,AltshulerAronov,Schmid-ee}.\nOn the other hand, for the point-like interaction (corresponding to $F=1$),\nthe diagrams \\ref{F:2Coulomb-av}(a) and \\ref{F:2Coulomb-av}(b) give the same\ncontribution. This short-range limit was considered in\nRefs.~\\cite{Blanter,AGKL,AG98,ABG}.\n\nCollecting the contributions of all diagrams with two SSI lines\nand two diffusons (Fig.~\\ref{F:2Coulomb-av}), we obtain the final result for the average energy relaxation rate $\\gamma_0(\\varepsilon,T)=\\corr{\\gamma(\\varepsilon,T)}$:\n\\begin{equation}\n\\label{gamma1full}\n  \\gamma_0(\\varepsilon, T)\n  =\n  \\left( 1 - 2F\/N_s + F^2 \\right) \\gamma_0^\\text{RPA}(\\varepsilon,T) ,\n\\end{equation}\nleading to Eq.~(\\ref{SIAT}).\nNote that for spinless electrons with point-like interaction,\nthe inelastic relaxation rate vanishes, $\\gamma=0$,\nwhich is a consequence of the Pauli exclusion principle.\nIt is essential that such a cancellation takes place only\nif the diagram \\ref{F:2Coulomb}(b) is taken into account.\n\n\n\n\n\n\n\n\n\\section{Mesoscopic fluctuations of the energy relaxation rate: general consideration}\n\\label{S:MF}\n\nIn this Section we discuss the general approach to the calculation of mesoscopic fluctuations of the energy relaxation rate $\\gamma(\\varepsilon,T)$.\nThe starting point is the diagrams for the collision integral shown\nin Figs.~\\ref{F:2Coulomb}(a) and \\ref{F:2Coulomb}(b).\nThe corresponding expression for $\\gamma(\\varepsilon,T)$ is given by\nEq.~(\\ref{gamma1}) which is written for a particular realization of impurities. Mesoscopic fluctuations of the relaxation rate are determined by the square of $\\gamma(\\varepsilon,T)$ averaged over disorder:\n\\begin{multline}\n  \\langle \\gamma^2(\\varepsilon,T)\\rangle\n  \\int d{\\bf r}\\, d{\\bf r}' \\,\n  \\langle \\Delta G_{\\varepsilon}({\\bf r},{\\bf r})\\Delta G_{\\varepsilon}({\\bf r}',{\\bf r}')\\rangle\n  =\n- \\frac{\\lambda^4}{16N_s^4\\nu^4}\\int d{\\bf r}_1\\ldots d{\\bf r}_8\n  \\int (d\\varepsilon_1)(d\\varepsilon_2)(d\\omega_1)(d\\omega_2)\n\\\\{}\n  \\times\n  ({\\cal F}_{\\varepsilon_1}-{\\cal F}_{\\varepsilon_1-\\omega_1})({\\cal F}_{\\varepsilon-\\omega_1}+{\\cal B}_{\\omega_1})\n  ({\\cal F}_{\\varepsilon_2}-{\\cal F}_{\\varepsilon_2-\\omega_2})({\\cal F}_{\\varepsilon-\\omega_2}+{\\cal B}_{\\omega_2})\n  \\corr{(N_s X_4-Y_4)(N_s X_4'-Y_4')} ,\n\\label{corr}\n\\end{multline}\nwhere the objects $X_4$ and $Y_4$ containing different products of four Green functions and two SSI lines are defined in Eqs.~(\\ref{X4}) and (\\ref{Y4}), while\n$X_4'$ and $Y_4'$ are obtained from them through the replacement\n\\begin{equation}\n  \\{{\\bf r}_i\\} \\to \\{{\\bf r}_{i+4}\\} ,\n\\quad\n  \\varepsilon_1 \\to \\varepsilon_2 ,\n\\quad\n  \\omega_1 \\to \\omega_2 .\n\\end{equation}\nEquation (\\ref{corr}) contains three different products of eight Green functions, $X_4X_4'$, $X_4Y_4'$ and $Y_4Y_4'$, which need to be separately averaged over disorder (the term $Y_4X_4'$ reduces to $X_4Y_4'$ after an obvious change of variables). For example, the explicit form of $X_4X_4'$ is given by\n\\begin{multline}\n  X_4X_4'\n  =\n  \\delta_\\kappa({\\bf r}_1-{\\bf r}_3)\\delta_\\kappa({\\bf r}_2-{\\bf r}_4)\n  \\delta_\\kappa({\\bf r}_5-{\\bf r}_7)\\delta_\\kappa({\\bf r}_6-{\\bf r}_8)\n  \\,\n  \\Delta G_{\\varepsilon}({\\bf r}_2,{\\bf r}_1) \\Delta G_{\\varepsilon-\\omega_1}({\\bf r}_1,{\\bf r}_2)\n\\\\{}\n  \\times\n  \\Delta G_{\\varepsilon_1-\\omega_1}({\\bf r}_4,{\\bf r}_3) \\Delta G_{\\varepsilon_1}({\\bf r}_3,{\\bf r}_4)\n  \\Delta G_{\\varepsilon}({\\bf r}_6,{\\bf r}_5) \\Delta G_{\\varepsilon-\\omega_2}({\\bf r}_5,{\\bf r}_6)\n  \\Delta G_{\\varepsilon_2-\\omega_2}({\\bf r}_8,{\\bf r}_7)\\Delta G_{\\varepsilon_2}({\\bf r}_7,{\\bf r}_8)\n  .\n\\label{X8}\n\\end{multline}\nSince $\\Delta G = G^\\text{R}-G^\\text{A}$, each of the products $X_4X_4'$, $X_4Y_4'$ and $Y_4Y_4'$ contains $2^8=256$ different elementary contributions in terms of $G^\\text{R}$ and $G^\\text{A}$.\n\n\nOur task is to calculate the irreducible part $\\ccorr{\\gamma^2(\\varepsilon,T)}$ defined in Eq.~(\\ref{corr-irr-def}).\nIn a usual situation, mesoscopic fluctuations of a random quantity $x$ are determined by irreducible averaging over disorder, when two copies of $x$ are connected by at least one impurity line. In the present case the situation is different as the left-hand side of the basic Eq.~(\\ref{corr}) contains the pairwise correlator $\\corr{ \\Delta G_{\\varepsilon} \\Delta G_{\\varepsilon} }$, which deviates significantly from its reducible part $\\corr{\\Delta G_{\\varepsilon}}^2$ (see Sec.~\\ref{paircorr}).\nTherefore, even extracting the square of the average, $\\corr{\\gamma(\\varepsilon,T)}^2$, from Eq.~(\\ref{corr}) should be done with care as it involves irreducible disorder averaging of\n$\\corr{(N_s X_4-Y_4)(N_s X_4'-Y_4')}$\nneeded for non-perturbative account for correlations between\ntwo Green functions with the energy $\\varepsilon$. The details of this calculation are presented in \\ref{reducible part}.\nSuch a complication is the price one has to pay for extracting the property of a single discrete level with the technique well suited for describing continuous spectra.\nKeeping that in mind we proceed with evaluation of the irreducible part of $\\corr{\\gamma^2(\\varepsilon,T)}$.\n\n\nIn order to determine $\\corr{\\gamma^2(\\varepsilon,T)}$ one has to compute\n$\\corr{X_4X_4'}$, $\\corr{X_4Y_4'}$ and $\\corr{Y_4Y_4'}$,\nand evaluate four remaining energy integrals in Eq.~(\\ref{corr}).\nThis calculation is rather nontrivial and will be performed in\nSecs.~\\ref{S:nonpert}, \\ref{S:final} and \\ref{S:mesofluct-rs}.\nMeanwhile we discuss the main idea and outline the principal technical steps of the derivation.\n\nThe most complicated task of this procedure is to perform disorder averaging which becomes rather involved in the low-energy limit, $(\\varepsilon,T)\\llE_{\\text{Th}}$, we are interested in.\nIndeed, the single-particle spectrum is well resolved in this case, $\\corr{\\gamma(\\varepsilon,T)}\\ll\\Delta$, indicating that the averaging is to be performed non-perturbatively.\nThat can be achieved with the help of the Efetov's nonlinear supersymmetric sigma model technique \\cite{Efetov-book}, properly generalized to the case of several Green functions with different energies.\n\nThe averaging of two Green functions on the left-hand side of Eq.~(\\ref{corr}) is straightforward and can be done exactly in the zero-dimensional geometry, see Sec.~\\ref{paircorr}.\n\nThe calculation of\n$\\corr{X_4X_4'}$, $\\corr{X_4Y_4'}$ and $\\corr{Y_4Y_4'}$,\ncontaining eight Green functions is much more complicated and cannot be performed exactly in the general case.\nFortunately, the exact expression is not required for determination of the leading contribution to mesoscopic fluctuations of the relaxation rate.\nA significant simplification is suggested by analysing the physics of the decay process.\nAccording to the FGR, $\\gamma(\\varepsilon, T)$ can be considered as a sum of the decay processes of a single-particle excitation into all possible final three-particle states allowed by the energy conservation:\n$\\varepsilon\\to(\\varepsilon-\\omega,\\varepsilon',-\\varepsilon'+\\omega)$. Consequently, $\\gamma^2(\\varepsilon, T)$ given by Eq.~(\\ref{corr}) contains,\nin principle, six different final states: $(\\varepsilon-\\omega_1,\\varepsilon_1,-\\varepsilon_1+\\omega_1, \\varepsilon-\\omega_2,\\varepsilon_2,-\\varepsilon_2+\\omega_2)$.\n\nHowever, as it was first demonstrated in Ref.~\\cite{BAA}, the leading contribution to mesoscopic fluctuations of the relaxation rate comes from those configurations that describe \\emph{the square of the same decay process}, i.e., from the terms with the identical set of the final states.\nSuch a situation can be realized with two choices:\n\\begin{equation}\n\\label{energy-matching}\n  \\text{(a) \\: $\\varepsilon_1 \\approx \\varepsilon_2$ and $\\omega_1 \\approx \\omega_2$},\n\\qquad\n  \\text{(b) \\: $\\varepsilon_1 \\approx \\varepsilon-\\omega_2$ and $\\varepsilon_2 \\approx \\varepsilon-\\omega_1$} ,\n\\end{equation}\nwhere the energies should coincide with the accuracy of the single level width $\\gamma$.\nTechnically, the importance of such configurations is related to the appearance of the formally divergent delta function of zero frequency, $\\delta(0)$, if conditions (\\ref{energy-matching}) are considered as strict equalities, corresponding to the use of non-interacting Green functions in\n$\\corr{X_4X_4'}$, $\\corr{X_4Y_4'}$ and $\\corr{Y_4Y_4'}$.\nIn order to take into account the single-particle level broadening due to interaction one should add an imaginary part to the energy argument $E$ of the Green function, replacing it by $E_+$ (for $G^\\text{R}$) and $E_-$ (for $G^\\text{A}$):\n\\begin{equation}\n\\label{width}\n  E_\\pm = E \\pm i\\gamma(E)\/2 ,\n\\end{equation}\nwhere $\\gamma(E) \\equiv \\gamma_0(E,T)$ is the average value of the relaxation rate obtained in the FGR approximation [Eq.~(\\ref{SIAT})].\nThe substitution (\\ref{width}) is equivalent to including the elastic part of the electron-electron interaction in the zero-dimensional diffuson (see Sec.~\\ref{slowaveraging} and \\ref{eldiffuson}).\nA finite imaginary part cuts the singularity in $\\delta(0)$ which should be replaced roughly by $1\/\\gamma$, making fluctuations\nfinite but divergent as $\\varepsilon$ and $T$ are decreased.\nThis enhancement of fluctuations at low energies and temperatures is precisely the effect we are looking for.\n\nSuch a mechanism of enhancement of mesoscopic fluctuations of the inelastic width with decreasing temperature was suggested in Ref.~\\cite{BAA} for a model when the matrix elements of the interaction $V_{\\alpha\\beta\\gamma\\delta}$ in the basis of exact single-particle states were assumed to be independently distributed Gaussian variables.\nOur task is more complicated as we do not make any assumptions about the structure of the matrix elements in Fock space and calculate them for the real Coulomb interaction. Though we do not use the language of the matrix elements $V_{\\alpha\\beta\\gamma\\delta}$, they are effectively generated after proper averaging over fast diffusive modes\n\\cite{Blanter,AGKL}.\n\nHaving discussed the main idea, we now outline the crucial steps in the calculation of\n$\\corr{X_4X_4'}$, $\\corr{X_4Y_4'}$ and $\\corr{Y_4Y_4'}$:\n\\begin{itemize}\n\\item\nEach of $X_4X_4'$, $X_4Y_4'$ and $Y_4Y_4'$ contains in general $2^8=256$ elementary products of $G^\\text{R}$ and $G^\\text{A}$.\nAs we discuss in Secs.~\\ref{slowaveraging} and \\ref{qual},\nthe principal contribution to $\\ccorr{\\gamma^2(\\varepsilon,T)}$ originates from the terms with four $G^\\text{R}$ and four $G^\\text{A}$. Each of these  $C_8^4=70$ terms\nmay be presented as a functional integral over the 16-component superfields which is then averaged over disorder following the standard technique \\cite{Efetov-book}.\nThe resulting supersymmetric nonlinear sigma model is formulated in terms of the functional integral over the $16\\times16$ superfield $Q({\\bf r})$, see Sec.~\\ref{SS:SUSY}.\n\n\\item\nAs the SSI lines carry nonzero momenta (otherwise the diagram is zero due to electroneutrality), momentum conservation requires the presence of some number of fast ($q\\neq0$) diffusive modes.\nSince each fast diffuson contributes a small factor of $\\Delta\/E_{\\text{Th}}$, their number should be minimized.\nFor $\\corr{\\gamma(\\varepsilon,T)}$ the minimal number was two [see Fig.~\\ref{F:2Coulomb-av}], and for $\\corr{\\corr{\\gamma^2(\\varepsilon,T)}}$ it is four, producing the\nfactors $1\/E_2^4$ and $1\/E_4^4$.\nFollowing the method developed in Ref.~\\cite{KM}, we integrate over these fast modes perturbatively and derive an effective action for the zero-mode (spatially-uniform supermatrix $Q$). This procedure is described in details in Sec.~\\ref{fastaveraging}, with the total number of emergent contributions $7!!=105$\nfor every given elementary term from $X_4X_4'$, $X_4Y_4'$ and $Y_4Y_4'$ with four $G^\\text{R}$ and four $G^\\text{A}$.\n\n\\item\nThe resulting zero-dimensional sigma model is still too complicated\nbecause of the large size of the $Q$ matrix.\nHowever, as described above, the leading contribution comes from configurations\nwhere among energy arguments of eight Green functions, four are pairwise equal\n(with an uncertainty of $\\gamma$).\nSince the typical energy difference between pairs is $(\\varepsilon,T)\\gg\\Delta$,\nour extended sigma model splits into four blocks, each corresponding to the standard Efetov sigma model for $\\corr{G^\\text{R}G^\\text{A}}$.\nThe integrals over these sigma models are then evaluated non-perturbatively using the standard technique \\cite{Efetov-book}.\n\nThe detailed discussion of this step in connection with the choice of pairs is presented in Secs.~\\ref{slowaveraging} and \\ref{SS:further}.\n\n\\item\nAs a result of the described procedure, quite a few terms will be generated.\nFortunately, most of them will be either zero or less singular than expected.\nOnly a small number of terms will effectively contribute to fluctuations of\nthe relaxation rate.\nThis selection is discussed in Sec.~\\ref{S:final} in the simplest case of $F\\to0$.\nThe final evaluation of $\\ccorr{\\gamma^2(\\varepsilon,T)}$ in the general case of arbitrary parameter $F$ is performed in Sec.~\\ref{S:mesofluct-rs}.\n\n\\end{itemize}\nThe announced program will be realized step by step in Secs.~\\ref{S:nonpert}, \\ref{S:final} and \\ref{S:mesofluct-rs}.\n\n\n\n\n\n\\section{Nonperturbative averaging of eight Green functions}\n\\label{S:nonpert}\n\n\\subsection{Pairwise correlator $\\corr{ \\Delta G_{\\varepsilon} \\Delta G_{\\varepsilon}}$}\n\\label{paircorr}\n\nWe start with discussing the pair correlation function on the left-hand side of Eq.~(\\ref{corr}). It is instructive to introduce a small energy mismatch $\\omega$ between the energies of two Green functions and to consider a more general correlation function\n\\begin{equation}\n\\label{R-def}\n  \\int d{\\bf r} \\, d{\\bf r}' \\,\n  \\corr{ \\Delta G_{\\varepsilon}({\\bf r},{\\bf r}) \\Delta G_{\\varepsilon-\\omega}({\\bf r}',{\\bf r}')}\n  =\n  -4\\pi^2\\nu^2 R_\\gamma(\\omega) .\n\\end{equation}\nAs usual, the terms $G^\\text{R}G^\\text{R}$ and $G^\\text{A}G^\\text{A}$ are averaged trivially, each contributing 1\/4 to $R_\\gamma(\\omega)$. Averaging of the cross term $G^\\text{R}G^\\text{A}$ is performed with the help of Efetov's zero-dimensional supersymmetric sigma model \\cite{Efetov-book}, where in accordance with Eq.~(\\ref{width}) one has to introduce a complex frequency\n\\begin{equation}\n  \\Omega = \\varepsilon_+ - (\\varepsilon-\\omega)_-\n  =\n  \\omega + i[\\gamma(\\varepsilon)+\\gamma(\\varepsilon-\\omega)]\/2 .\n\\end{equation}\nEvaluating the standard integrals with the help of the machinery developed in \\ref{slowcontrrules}, we obtain\n\\begin{equation}\n\\label{R-general}\n  R_\\gamma(\\omega)\n  =\n  1-\\mathop{\\rm Re} X(-i\\pi\\Omega\/\\Delta) ,\n\\end{equation}\nwhere the function $X(a)$ is given by Eq.~(\\ref{X}):\n\\begin{equation}\n  X(a)=\\frac{1-e^{-2a}}{2a^2} .\n\\end{equation}\n\nIn the limit of vanishing width ($\\gamma\\to0$), $R_0(\\omega)$ reproduces the pair correlation\nfunction for the Gaussian unitary ensemble in the random matrix theory \\cite{Mehta}:\n$\n  R_0^\\text{RMT}(\\omega) = 1-(\\sin x\/x)^2+\\pi \\delta(x)\n$,\nwhere $x=\\pi\\omega\/\\Delta$.\nA finite but small width acts as a regularizer of the $\\delta$ function, accounting for the contribution of the same broadened level into the correlation function. In the relevant limit of a resolved discrete spectrum, $(\\omega,\\gamma)\\ll\\Delta$, $R_\\gamma(\\omega)$ can be written as\n\\begin{equation}\n\\label{R-delta}\n  R_\\gamma(\\omega)\n  \\approx\n  \\Delta \\delta_\\gamma(\\omega) ,\n\\end{equation}\nwhere $\\delta_\\gamma(\\omega)$ is a Lorentzian approximation of the delta function:\n\\begin{equation}\n\\label{delta_gamma}\n  \\delta_\\gamma(\\omega)\n  =\n  \\frac{1}{\\pi} \\frac{\\gamma(\\varepsilon)}{\\gamma^2(\\varepsilon)+\\omega^2} .\n\\end{equation}\nIt is worth noting that Eq.~(\\ref{R-delta}) is non-perturbative. Though it looks just like a one-diffuson contribution, $R_\\gamma(\\omega) \\approx -\\mathop{\\rm Im} 1\/(\\omega+i\\gamma)$, it is an asymptotic expansion of the exact non-perturbative expression (\\ref{R-general}).\n\nThe left-hand side of Eq.~(\\ref{corr}) can be expressed with the help of Eq.~(\\ref{R-def}), with $R_\\gamma(0)=\\Delta\/\\pi\\gamma(\\varepsilon)$.\n\n\n\n\n\\subsection{Supersymmetric sigma model for eight Green functions (the case $F=0$)}\n\\label{SS:SUSY}\n\nThe objects $X_4X_4'$, $X_4Y_4'$ and $Y_4Y_4'$\ncontain the products of eight $\\Delta G = G^\\text{R}-G^\\text{A}$ which should be averaged over disorder.\nTo introduce the method we start with\nthe simplest case of $F\\to0$. Generalization to the case of an arbitrary $F$ will be performed in Sec.~\\ref{S:mesofluct-rs}. To be specific, we focus on calculating $\\corr{X_4X_4'}$ (disorder averaging of other products is performed analogously and the result is presented in Sec.~\\ref{Seconddiagram}).\nAs we will see in Secs.~\\ref{slowaveraging} and \\ref{qual}, only terms with the equal number of retarded and advanced Green functions make the leading contribution to mesoscopic fluctuations.\nConsider, e.~g., a particular choice of $G^\\text{R}$ and $G^\\text{A}$ from $X_4X_4'$ and calculate\n\\begin{equation}\nK=\\langle G^\\text{R}_{\\varepsilon}({\\bf r}_2,{\\bf r}_1) G^\\text{A}_{\\varepsilon}({\\bf r}_6,{\\bf r}_5) G^\\text{R}_{\\varepsilon-\\omega_2}({\\bf r}_5,{\\bf r}_6) G^\\text{A}_{\\varepsilon-\\omega_1}({\\bf r}_1,{\\bf r}_2)\nG^\\text{R}_{\\varepsilon_1-\\omega_1}({\\bf r}_4,{\\bf r}_3)\nG^\\text{A}_{\\varepsilon_2-\\omega_2}({\\bf r}_8,{\\bf r}_7)  G^\\text{R}_{\\varepsilon_2}({\\bf r}_7,{\\bf r}_8) G^\\text{A}_{\\varepsilon_1}({\\bf r}_3,{\\bf r}_4) \\rangle .\n\\label{GGGG}\n\\end{equation}\nDisorder averaging of other relevant terms from $X_4X_4'$ [listed in Eq.~(\\ref{RA-4ways})] can be performed analogously (see Secs.~\\ref{SS:X8a} and \\ref{SS:X8b}).\n\nIn order to calculate $K$ we follow the standard line of Efetov's supersymmetric sigma model \\cite{Efetov-book}, generalizing it to the case of eight Green functions. We group the Green functions into four RA pairs in the sequence they appear in Eq.~(\\ref{GGGG}):\n\\begin{equation}\n\\label{1234}\n  \\begin{tabular}{|c|cccc|}\n  \\hline\n      & 1 & 2 & 3 & 4 \\\\\n  \\hline\n    R & $\\varepsilon$ & $\\varepsilon-\\omega_2$ & $\\varepsilon_1-\\omega_1$ & $\\varepsilon_2$ \\\\\n  \\hline\n    A & $\\varepsilon$ & $\\varepsilon-\\omega_1$ & $\\varepsilon_2-\\omega_2$ & $\\varepsilon_1$ \\\\\n  \\hline\n  \\end{tabular}\n\\end{equation}\nthereby introducing a new space of pairs, that will be referred to as 1234.\nThe way this space is introduced is consistent with the pairing (a) in Eq.~(\\ref{energy-matching}) shown in Fig.~\\ref{F:AGKL31}(a). The leading contribution to mesoscopic fluctuations from this pairing will come from configurations when the energies in each pair nearly coincide, corresponding to $Q$ matrices which are block-diagonal in the space 1234 [see Eq.~(\\ref{Q-blocks}) below].\nThe other relevant contribution originates from the pairing (b) in Eq.~(\\ref{energy-matching}) shown in Fig.~\\ref{F:AGKL31}(b).\nIn principle, it can be also handled using the 1234 structure of Eq.~(\\ref{1234}), however the corresponding $Q$ matrices will have a cumbersome structure in this basis.\nTherefore in the study of the pairing (b) in Sec.~\\ref{SS:X8b} we will introduce 1234 space in a different way consistent with that pairing.\n\nThe resulting sigma model is formulated in terms of the $16\\times 16$ supermatrix field $Q({\\bf r})$ acting in the tensor product of the spaces $\\text{FB}\\otimes\\text{RA}\\otimes 1234$, where FB stands for the superspace. The matrix $Q$ can be written as $Q=T^{-1}\\Lambda T$, where $T$ spans the supersymmetric coset\n$\\text{U}(8|4,4)\/\\text{U}(4|4)\\times\\text{U}(4|4)$, and $\\Lambda=\\sigma_3^\\text{RA}$. The sigma-model action is given by (we adopt the fermion-dominated notation of Ref.~\\cite{Efetov-book})\n\\begin{equation}\n  S[Q]\n  =\n  \\frac{\\pi\\nu}4\\int d{\\bf r}\n  \\mathop{\\rm str} \\bigl[ D\\left( \\nabla Q({\\bf r})\\right)^2+4i\\hat E Q ({\\bf r}) \\bigr],\n\\label{action}\n\\end{equation}\nwhere $\\hat E$ is the diagonal matrix made of the energy arguments [with widths, according to Eq.~(\\ref{width})] of the corresponding Green functions:\n\\begin{equation}\n\\label{E8-def}\n  \\hat E\n  =\n  \\mathop{\\rm diag}\n\\bigl\\{\n\\varepsilon_+,\n(\\varepsilon-\\omega_2)_+, (\\varepsilon_1-\\omega_1)_+, (\\varepsilon_2)_+,\n\\varepsilon_-,\n(\\varepsilon-\\omega_1)_-, (\\varepsilon_2-\\omega_2)_-,(\\varepsilon_1)_-\n\\bigr\\}\\otimes 1_{\\text{FB}}.\n\\end{equation}\n\nIn the sigma-model language, the average product of eight Green functions in Eq.~(\\ref{GGGG}) transforms to\n\\begin{equation}\n  K=(\\pi \\nu)^8\\int Q^{\\text{AR}}_{21}({\\bf r}_1)Q^{\\text{RA}}_{12}({\\bf r}_2)Q^{\\text{RA}}_{21}({\\bf r}_5)Q^{\\text{AR}}_{12}({\\bf r}_6) Q^{\\text{AR}}_{43}({\\bf r}_3)Q^{\\text{RA}}_{34}({\\bf r}_4)Q^{\\text{RA}}_{43}({\\bf r}_7)Q^{\\text{AR}}_{34}({\\bf r}_8) e^{-S[Q]}DQ,\n\\label{Qcorr}\n\\end{equation}\nwhere all the elements of $Q$ matrices in the preexponent are taken from the BB sector (omitted for brevity). Equation (\\ref{Qcorr}) holds only in the limit $F\\to0$, when the interaction range is much larger than the Fermi wave length, and additional terms in the preexponent are suppressed by Friedel oscillations. In the general case discussed in Sec.~\\ref{S:mesofluct-rs}, Eq.~(\\ref{Qcorr}) should be replaced by Eq.~(\\ref{Qcorr-rs}).\n\n\n\n\\subsection{Integration over fast modes}\n\\label{fastaveraging}\n\n\nBesides eight Green functions, the block $X_4X_4'$ given by Eq.~(\\ref{X8}) contains four SSI lines, each carrying a non-zero momentum. Therefore in calculating $K$ in Eq.~(\\ref{Qcorr}) we should (i) allow fast ($q\\neq0$) diffuson modes to make $\\corr{X_4X_4'}$ nonzero, (ii) minimize their number as every fast mode brings a small factor of $\\Delta\/E_{\\text{Th}}$, and (iii) be able to handle the resulting zero-dimensional integral non-perturbatively in order to resolve discrete levels. This task can be accomplished following the strategy of Ref.~\\cite{KM}, where non-universal corrections to the random-matrix level statistics were calculated beyond the zero-dimensional limit.\nFor this purpose we write\n\\begin{equation}\n  Q({\\bf r}) = T^{-1} Q'({\\bf r}) T,\n\\label{param}\n\\end{equation}\nwhere the matrix $T$ is spatially uniform, and $Q'({\\bf r})$ describes all fast modes with non-zero momenta.\nSince the latter are to be accounted perturbatively, we expand $Q'({\\bf r})$ near the origin:\n\\begin{equation}\n  Q'({\\bf r})=\\Lambda\\left[1+W({\\bf r})+W^2({\\bf r})\/2+\\ldots\\right], \\qquad \\{W({\\bf r}),\\Lambda\\}=0.\n\\label{Q'-W}\n\\end{equation}\n\nTo get the leading contribution to $\\corr{X_4X_4'}$ we extract one fast $W$ from each of the eight $Q$ matrices in the preexponent of Eq.~(\\ref{Qcorr}), and average it with the Gaussian action\n\\begin{equation}\n  S^{(2)}[W] = - \\frac{\\pi\\nu D}4 \\int d{\\bf r}\\, \\mathop{\\rm str}[\\nabla W({\\bf r})]^2 ,\n\\end{equation}\ncoming from the gradient term of Eq.~(\\ref{action}).\nUsing Wick's theorem, the correlator of eight $W$ fields can be expressed as the sum of all possible products of four pairwise correlators. The latter can be easily calculated with the help of the following contraction rule valid for arbitrary matrices $P$ and $R$:\n\\begin{equation}\n  \\langle \\mathop{\\rm str} PW({\\bf r}) \\mathop{\\rm str} RW({\\bf r}') \\rangle_W\n  =\n  D({\\bf r},{\\bf r}')\\mathop{\\rm str}(P\\Lambda R\\Lambda -PR),\n\\qquad\n  D({\\bf r},{\\bf r}')=\\frac{\\corr{{\\bf r}|(-\\nabla^2)^{-1}|{\\bf r}'}}{\\pi\\nu D} .\n\\label{contruction rules}\n\\end{equation}\nAccording to Wick's theorem, averaging over fast modes generates $7!!=105$ different terms in the expression for $K$.\nNot all of them are equally important. To pick up the relevant terms one should understand how to perform further integration over the zero mode. This procedure will be discussed in Sec.~\\ref{slowaveraging}, and in Sec.~\\ref{SS:further} we will proceed with the derivation based on Eq.~(\\ref{contruction rules}).\n\n\n\n\n\\subsection{Block structure of the zero-dimensional sigma model and energy pairs}\n\\label{slowaveraging}\n\nHaving integrated out fast diffusive modes, we end up with the effective zero-dimensional sigma model for the $16 \\times 16$ supermatrix $Q=T^{-1}\\Lambda T$ with the action\n\\begin{equation}\n\\label{action0}\n  S_0[Q]=\\frac{\\pi i }{\\Delta}\\mathop{\\rm str} \\hat E Q.\n\\end{equation}\nOwing to a large size of the matrix $Q$, this is still a complicated theory. Fortunately, we do not need its exact solution since, as discussed in Sec.~\\ref{S:MF}, the most singular contribution to $\\ccorr{\\gamma^2(\\varepsilon,T)}$ comes from pairwise coinciding energies, see Eq.~(\\ref{energy-matching}). Since we are interested in the limit $(\\varepsilon,T)\\gg\\Delta$, energy difference between pairs is typically large, $|\\varepsilon_i-\\varepsilon_j|\\gg\\Delta$, and hence correlations between different pairs can be neglected (configurations with $|\\varepsilon_i-\\varepsilon_j|\\lesssim\\Delta$ which require non-perturbative treatment exist but their weight is small). On the other hand, energies from the same pair match with the accuracy of $\\gamma$ and should be treated non-perturbatively like in Sec.~\\ref{paircorr}. In the language of the zero-dimensional sigma model, large energy difference between pairs suppresses degrees of freedom in the supermatrix $Q$ which are off-diagonal in the space of pairs. As a result, $Q$ becomes block-diagonal in the space of pairs, and the full theory splits into a product of four standard Efetov's supermatrix sigma models for $\\corr{G^\\text{R}G^\\text{A}}$, see Eq.~(\\ref{S-splitting}) below.\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=1.0\\textwidth]{65textcropped.pdf}}\n\\caption{\nTwo types of arranging eight Green functions in\nthe product\n$X_4X_4'$ [Eq.~(\\ref{X8})] into four pairs that produce the most singular contributions $\\corr{X_4X_4'}_\\text{sing}^{(a)}$ and $\\corr{X_4X_4'}_\\text{sing}^{(b)}$ in Eq.~(\\ref{X8-sing-reg}).\nGreen functions from the same pair are marked by a dotted line. The pairings (a) and (b) correspond to the two ways to match energies in Eq.~(\\ref{energy-matching}). All energy and coordinate indices in (b) are the same as in (a).}\n\\label{F:AGKL31}\n\\end{figure}\n\nThe next step is to understand what are the possible ways to group eight Green functions into four pairs to maximize their contribution to fluctuations of $\\gamma(\\varepsilon,T)$. First of all, it is clear that the Green functions with the energies $\\varepsilon$ always need to be in a pair, as they immediately produce the smeared delta function of zero argument, $\\delta_\\gamma(0)=1\/\\pi\\gamma$ [cf.\\ Eq.~(\\ref{delta_gamma})]. It is this large factor that compensates the analogous contribution from $\\corr{G^\\text{R}_\\varepsilon G^\\text{A}_\\varepsilon}$ on the left-hand side of Eq.~(\\ref{corr}). Our aim then is to identify those pairings that may introduce an additional large factor of $\\delta_\\gamma(0)$ after integration over all intermediate energies. Such a situation can be realized only if coincidence of the energies in two pairs automatically implies coincidence of the energies in the third pair. That can be achieved only in two physically relevant cases, (a) and (b), listed in Eq.~(\\ref{energy-matching}) and shown diagrammatically in Fig.~\\ref{F:AGKL31}\n(there exist three other unphysical pairings which should be discarded as demonstrated in \\ref{A:spurious}). We emphasize that possible choices of pairing are dictated by the energy arguments of electron Green functions and are not specific for the particular arrangement of $G^\\text{R}$ and $G^\\text{A}$. The only  natural requirement is that each pair contains one retarded and one advanced Green function (otherwise, there is no correlations within a pair at all).\n\n\n\nAccording to this general logic, each diagram for $\\corr{X_4X_4'}$, $\\corr{X_4Y_4'}$ and $\\corr{Y_4Y_4'}$ can be written in the form\n\\begin{equation}\n  \\corr{A} = \\corr{A}_\\text{sing}^{(a)} + \\corr{A}_\\text{sing}^{(b)} + \\corr{A}_\\text{reg} ,\n\\label{X8-sing-reg}\n\\end{equation}\nwhere $\\corr{\\dots}_\\text{sing}^{(a)}$ and $\\corr{\\dots}_\\text{sing}^{(b)}$ denote singular (in the limit $\\gamma\\to0$) contributions from the pairings (a) and (b).\nWe proceed with calculating these singular contributions for the term $K$ [a particular element of $\\corr{X_4X_4'}$ defined in Eq.~(\\ref{GGGG})] from the zero-dimensional sigma model (\\ref{action0}).\nThe introduction of the 1234 space in Eq.~(\\ref{1234}) is consistent with the pairing (a): in the 1234 basis the corresponding matrix $Q$ becomes diagonal. On the contrary, the pairing (b) is described by non-diagonal matrices in the 1234 space. To calculate $K_\\text{sing}^{(b)}$ we find it more convenient to change the basis and introduce a new $1234'$ space consistent with the pairing (b), in which the matrix $Q$ is diagonal.\nThen the calculation of $K_\\text{sing}^{(b)}$ becomes completely analogous to the calculation of $K_\\text{sing}^{(a)}$.\nTo illustrate the technique, we focus on the contribution of $K$ due to the pairing (a) [the contribution of the pairing (b) is discussed in Sec.~\\ref{SS:X8b}].\n\n\n\n\n\n\n\n\n\n\\subsection{Singular contribution of the pairing (a)}\n\\label{SS:further}\n\nNow we are ready to proceed with the calculation started in Sec.~\\ref{fastaveraging}.\nIn order to apply the contraction rules (\\ref{contruction rules}) for averaging the preexponent in Eq.~(\\ref{Qcorr}) over fast modes, it is convenient to introduce projectors onto different sectors of the $Q$ manifold.\nBy definition, we write $Q^{ab}_{ij}({\\bf r})=\\mathop{\\rm str} P^{ab}_{ij}Q({\\bf r})$, where $a$ and $b$ refer to the RA space, whereas $i$ and $j$ refer to the 1234 space (BB sector is also implied). For example, the projector $P^{\\text{AR}}_{21}$ is given by\n\\begin{equation}\n  P^{\\text{AR}}_{21} = \\begin{pmatrix} 0&P^{\\text{AR}}&0&0\\\\0&0&0&0\\\\0&0&0&0\\\\0&0&0&0 \\end{pmatrix}_{1234},\n\\quad\n  P^{\\text{AR}} = \\begin{pmatrix} 0&P_\\text{BB}\\\\0&0 \\end{pmatrix}_{\\text{RA}},\n\\quad\n  P_\\text{BB} = \\begin{pmatrix} 0&0\\\\0&1 \\end{pmatrix}_{\\text{FB}}.\n\\end{equation}\n\nAs we discussed in Sec.~\\ref{slowaveraging}, the singular contribution $K_\\text{sing}^{(a)}$ originates from the matrices $Q$ which are diagonal in $1234$ space. Such a structure of the $Q$ matrix guarantees that only four out of $7!!=105$ terms appearing after averaging over fast modes [see Eq.~(\\ref{param})] turn out to be non-zero:\n\\begin{align}\n  K_\\text{sing}^{(a)}\n  =\n  (\\pi\\nu)^8\n  \\bigl\\langle\\bigl\\{\n    D({\\bf r}_1,{\\bf r}_2) & D({\\bf r}_5,{\\bf r}_6)\\mathop{\\rm str} \\left(P^{\\text{AR}}_{21}QP^{\\text{RA}}_{12}Q-P^{\\text{AR}}_{21} P^{\\text{RA}}_{12}\\right) \\mathop{\\rm str} \\left(P^{\\text{RA}}_{21}QP^{\\text{AR}}_{12}Q-P^{\\text{RA}}_{21} P^{\\text{AR}}_{12}\\right)\n\\nonumber\n\\\\{}\n  + D({\\bf r}_1,{\\bf r}_6) & D({\\bf r}_2,{\\bf r}_5) \\mathop{\\rm str} \\left(P^{\\text{AR}}_{21}QP^{\\text{AR}}_{12}Q-P^{\\text{AR}}_{21} P^{\\text{AR}}_{12}\\right) \\mathop{\\rm str} \\left(P^{\\text{RA}}_{21}QP^{\\text{RA}}_{12}Q- P^{\\text{RA}}_{21} P^{\\text{RA}}_{12}\\right)\\bigr\\}\n\\nonumber\n\\\\{}\n\\times \\bigl\\{ D({\\bf r}_3,{\\bf r}_4) & D({\\bf r}_7,{\\bf r}_8)\\mathop{\\rm str} \\left(P^{\\text{AR}}_{43}QP^{\\text{RA}}_{34}Q-P^{\\text{AR}}_{43} P^{\\text{RA}}_{34}\\right) \\mathop{\\rm str} \\left(P^{\\text{RA}}_{43}QP^{\\text{AR}}_{34}Q-P^{\\text{RA}}_{43} P^{\\text{AR}}_{34}\\right)\n\\nonumber\n\\\\{}\n+ D({\\bf r}_3,{\\bf r}_8) & D({\\bf r}_4,{\\bf r}_7) \\mathop{\\rm str} \\left(P^{\\text{AR}}_{43}QP^{\\text{AR}}_{34}Q-P^{\\text{AR}}_{43} P^{\\text{AR}}_{34}\\right) \\mathop{\\rm str} \\left(P^{\\text{RA}}_{43}QP^{\\text{RA}}_{34}Q- P^{\\text{RA}}_{43} P^{\\text{RA}}_{34}\\right)\\bigr\\}\\bigr\\rangle\n,\n\\label{Ka}\n\\end{align}\nwhere $\\langle \\ldots \\rangle$ denotes averaging over the $16\\times16$ zero-dimensional sigma-model manifold with the action (\\ref{action0}).\n\nSince $Q$ is diagonal in the 1234 space, it can be represented as\n\\begin{equation}\n\\label{Q-blocks}\n  Q = \\mathop{\\rm diag} \\{Q_1, Q_2, Q_3, Q_4\\} ,\n\\end{equation}\nwhere $Q_j$ are independent $4\\times4$ supermatrices in the $\\text{RA}\\otimes\\text{FB}$ space spanning the standard supermanifold of the sigma model for $\\corr{G^\\text{R}G^\\text{A}}$.\nThe action (\\ref{action0}) then splits into a sum of four separate Efetov's sigma-model actions:\n\\begin{equation}\n\\label{S-splitting}\n  S_0[Q]\n  =\n  \\sum_{j=1}^4 S_{0j}[Q_j]\n  =\n  \\frac{\\pi i}{\\Delta} \\sum_{j=1}^4 \\mathop{\\rm str} \\hat E_j Q_j,\n\\end{equation}\nwhere the elements of the diagonal matrix $\\hat E_j$ are taken from the corresponding sector of $\\hat E$ defined in Eq.~(\\ref{E8-def}). Now averaging over all four $Q_j$ is performed independently and can be done exactly in the standard way \\cite{Efetov-book}. To handle the supertrace structure in Eq.~(\\ref{Ka}), it is convenient to use the zero-dimensional contraction rules (\\ref{contr-rules-0D}) derived in \\ref{slowcontrrules}. The result can be expressed in a compact form in terms of the quantities\n\\begin{equation}\n  X_j \\equiv X\\left(-i\\pi\\Omega_j\/\\Delta \\right) ,\n\\qquad\n  Z_j \\equiv Z\\left(-i\\pi\\Omega_j\/\\Delta \\right) , \\label{XjZj}\n\\end{equation}\nwhere the functions $X(a)$ and $Y(a)$ are given by Eqs.~(\\ref{X}) and (\\ref{Z}):\n\\begin{equation}\n\\label{XZ}\n  X(a)=\\frac{1-e^{-2a}}{2a^2} ,\n\\qquad\n  Z(a)=\\frac 1a ,\n\\end{equation}\nand $\\Omega_j$ is the energy difference between the arguments of $G^\\text{R}$ and $G^\\text{A}$ in the corresponding pair [cf.\\ Eq.~(\\ref{1234})]:\n\\begin{gather}\n\\label{Omega1234}\n  \\Omega_1 = \\varepsilon_+ - \\varepsilon_- ,\n  \\!\\!\\qquad\n  \\Omega_2 = (\\varepsilon-\\omega_2)_+ - (\\varepsilon-\\omega_1)_- ,\n  \\!\\!\\qquad\n  \\Omega_3 = (\\varepsilon_1-\\omega_1)_+ - (\\varepsilon_2-\\omega_2)_-  ,\n  \\!\\!\\qquad\n  \\Omega_4 = (\\varepsilon_2)_+ - (\\varepsilon_1)_- .\n\\end{gather}\n\nTo demonstrate the technique, consider for example one of the four terms in Eq.~(\\ref{Ka}):\n\\begin{equation}\n  K_\\text{sing}^{(a)}\n  =\n  (\\pi\\nu)^8 D({\\bf r}_1,{\\bf r}_2)D({\\bf r}_5,{\\bf r}_6) D({\\bf r}_3,{\\bf r}_4)D({\\bf r}_7,{\\bf r}_8) L_1 + \\dots\n\\label{Ka2}\n\\end{equation}\nSubstituting $Q$ in the form (\\ref{Q-blocks}) and tracing over the 1234 space we obtain\n\\begin{align}\n  L_1\n  =\n  \\bigl\\langle & \\mathop{\\rm str} (P^{\\text{AR}}Q_2P^{\\text{RA}}Q_1-P^{\\text{AR}}_{21} P^{\\text{RA}}_{12}) \\mathop{\\rm str} (P^{\\text{RA}}Q_2P^{\\text{AR}}Q_1-P^{\\text{RA}}_{21} P^{\\text{AR}}_{12})\n\\nonumber\n\\\\{} \\times\n  & \\mathop{\\rm str} (P^{\\text{AR}}Q_4P^{\\text{RA}}Q_3-P^{\\text{AR}}_{43} P^{\\text{RA}}_{34}) \\mathop{\\rm str} (P^{\\text{RA}}Q_4P^{\\text{AR}}Q_3-P^{\\text{RA}}_{43} P^{\\text{AR}}_{34})\\bigr\\rangle.\n\\label{K11}\n\\end{align}\nSequentially applying the contraction rules (\\ref{contr-rules-0D}) for averaging over all $Q_j$, we arrive at the exact non-perturbative expression\n\\begin{equation}\n  L_1=(4+2X_1+2X_2+4X_1X_2)(4+2X_3+2X_4+4X_3X_4) .\n\\end{equation}\nContributions of the three other terms in Eq.~(\\ref{Ka2}) are calculated analogously, and we obtain finally the singular part of $K$ due to pairing (a):\n\\begin{align}\n  K_\\text{sing}^{(a)}\n  =(\\pi\\nu)^8\n  & \\left\\{D({\\bf r}_1,{\\bf r}_2)D({\\bf r}_5,{\\bf r}_6)(4+2X_1+2X_2+4X_1X_2)+ 4D({\\bf r}_1,{\\bf r}_6)D({\\bf r}_2,{\\bf r}_5)Z_1Z_2\\right\\}\n\\nonumber\n\\\\{}\n  \\times\n  & \\left\\{ D({\\bf r}_3,{\\bf r}_4)D({\\bf r}_7,{\\bf r}_8)(4+2X_3+2X_4+4X_3X_4)+4D({\\bf r}_3,{\\bf r}_8)D({\\bf r}_4,{\\bf r}_7)Z_3Z_4\\right\\}.\n\\label{K1}\n\\end{align}\n\nThough the singular contribution due to the energy pairing (a) may originate only from Eq.~(\\ref{K1}), not all terms in this equation do actually produce the singular contribution. We analyze this expression and perform the final step of calculation in the next Section.\n\n\n\n\n\\section{Mesoscopic fluctuations at $F=0$}\n\\label{S:final}\n\n\\subsection{General recipe}\n\\label{qual}\n\nThe most singular contribution from Eq.~(\\ref{K1}) originates from the terms which contain the maximal number (four) of $X_j$ and $Z_j$. We have already explained that this choice is dictated by the necessity to obtain two smeared delta-functions (\\ref{delta_gamma}) with zero argument, contributing a large factor $\\Delta\/\\gamma$ each.\nThis is also the reason why we have to choose each pair in the $1234$ space to consist of one $G^R$ and one $G^\\text{A}$: large factors $X_j$ and $Z_j$ appear as a result of averaging $\\langle G^\\text{R} G^\\text{A} \\rangle$ over the zero-dimensional diffusive model.\n\nIn calculating the energy integrals in Eq.~(\\ref{corr}), the important contribution comes from the vicinity of the poles of $X_j$ and $Z_j$. This observation allows us to work in the limit $\\Omega_j \\sim \\gamma \\ll\\Delta$ [with $\\Omega_j$ introduced in Eq.~(\\ref{Omega1234}) being the energy mismatch within a pair] and use the leading-order asymptotics of Eqs.~(\\ref{XZ}):\n\\begin{equation}\n  X_j \\approx Z_j = \\frac{i\\Delta}{\\pi\\Omega_j} .\n\\end{equation}\nThe chosen sequence of $G^\\text{R}$ and $G^\\text{A}$ in the correlator $K$ [Eq.~(\\ref{GGGG})] also ensures that the product $1\/\\Omega_2\\Omega_3\\Omega_4$ has poles both in the upper and lower half-planes of the energy variables $\\varepsilon_1$, $\\varepsilon_2$, $\\omega_1$ and $\\omega_2$. Hence integration over intermediate energies does not vanish and indeed produces the desired delta-function-like contribution.\n\nNow the recipe for extracting the leading singular term from Eq.~(\\ref{K1}) and  similar expressions for other diagrams can be formulated as follows.\nOne should substitute all $X_j$ by $Z_j$ and take the term proportional to ${\\cal Z} = Z_1Z_2Z_3Z_4$.\nThen the coefficient ${\\cal Z}$ should be replaced by\n\\begin{equation}\n  {\\cal Z}\n  \\longrightarrow\n  \\frac{4\\Delta^4}{\\pi^2 \\gamma(\\varepsilon)}\n  \\frac{\\delta(\\varepsilon_1-\\varepsilon_2)\\delta(\\omega_1-\\omega_2)}{\\gamma(\\varepsilon-\\omega_1)+\\gamma(\\varepsilon_1)+\\gamma(\\varepsilon_1-\\omega_1)} ,\n\\label{XXX}\n\\end{equation}\nwhere the factor $\\Delta\/\\pi\\gamma(\\varepsilon)$ originates from $Z_1$ and will be canceled by a similar factor from the left-hand side of Eq.~(\\ref{corr}) (see Sec.~\\ref{paircorr}), while the coefficient in front of the delta functions can be easily obtained by calculating, e.g., $\\int Z_2Z_3Z_4\\,d\\varepsilon_2d\\omega_2$.\nThe last step is to integrate four diffuson propagators in Eq.~(\\ref{K1}) over ${\\bf r}_i$ [see Eq.~(\\ref{corr})].\nIn doing that, $\\delta_\\kappa({\\bf r}-{\\bf r}')$ in the expressions for\n$X_4X_4'$, $X_4Y_4'$ and $Y_4Y_4'$\ncan be considered as the usual zero-range delta function.\nThen depending on the term under consideration, diffusons combine either into $(\\mathop{\\rm tr} D^2)^2$ or  $\\mathop{\\rm tr} D^4$. These traces evaluated in the Fourier space produce either $1\/E_2^4$ or $1\/E_4^4$, where $E_n\\simE_{\\text{Th}}$ are defined in Eq.~(\\ref{En}).\nAs a result, the contribution to mesoscopic fluctuations of the inelastic rate can be written in the form\n\\begin{equation}\n  \\ccorr{\\gamma^2(\\varepsilon,T)}_{A}^{(i)}\n  =\n  \\frac{{\\lambda^4}\\Delta^5}{4\\pi^3}\n  c_{A}^{(i)}\n  \\int d\\varepsilon_1 d\\omega_1\n  \\frac{({\\cal F}_{\\varepsilon_1}-{\\cal F}_{\\varepsilon_1-\\omega_1})^2({\\cal B}_{\\omega_1}+{\\cal F}_{\\varepsilon-\\omega_1})^2}\n  {\\gamma_0(\\varepsilon-\\omega_1,T)+\\gamma_0(\\varepsilon_1,T)+\\gamma_0(\\varepsilon_1-\\omega_1,T)}\n  ,\n\\label{gamma-c}\n\\end{equation}\nwhere $A$ is a particular diagram or a set of diagrams considered, and $i$ labels the type of the energy pairing (a or b).\n\n\n\\subsection{Contribution from $X_4X_4'$ due to the energy pairing (a)}\n\\label{SS:X8a}\n\nApplying this general recipe to expression (\\ref{K1}) for $K_\\text{sing}^{(a)}$, we first reduce it to the form\n\\begin{equation}\n  K_\\text{sing}^{(a)}\n  =16(\\pi\\nu)^8 \\, {\\cal Z}\n  \\left\\{D({\\bf r}_1,{\\bf r}_2)D({\\bf r}_5,{\\bf r}_6) + D({\\bf r}_1,{\\bf r}_6)D({\\bf r}_2,{\\bf r}_5)\\right\\}\n  \\left\\{ D({\\bf r}_3,{\\bf r}_4)D({\\bf r}_7,{\\bf r}_8) + D({\\bf r}_3,{\\bf r}_8)D({\\bf r}_4,{\\bf r}_7)\\right\\} ,\n\\label{K1-Z}\n\\end{equation}\nand then, tracing the diffuson propagators, obtain the coefficient $c_{K}^{(a)}$ in Eq.~(\\ref{gamma-c}):\n\\begin{equation}\n  c_{K}^{(a)}\n  =\n  \\frac{1}{2N_s^2}\\left(\\frac 1{E_2^4}+\\frac 1{E_4^4}\\right) .\n\\label{gamma1'}\n\\end{equation}\n\nTo complete the analysis of the contribution from $X_4X_4'$ due type-(a) energy pairing, one has to consider other arrangements of $G^\\text{R}$ and $G^\\text{A}$. The requirement of having one $G^\\text{R}$ and one $G^\\text{A}$ in each pair limits the number of various possibilities to $2^4=16$. However, not all of them should be taken into account. Consider, for example, the correlator\n$$\nK'=\\langle G^\\text{R}_{\\varepsilon}({\\bf r}_2,{\\bf r}_1) G^\\text{A}_{\\varepsilon}({\\bf r}_6,{\\bf r}_5)  G^\\text{R}_{\\varepsilon-\\omega_2}({\\bf r}_5,{\\bf r}_6) G^\\text{A}_{\\varepsilon-\\omega_1}({\\bf r}_1,{\\bf r}_2)\nG^\\text{R}_{\\varepsilon_1-\\omega_1}({\\bf r}_4,{\\bf r}_3) G^\\text{A}_{\\varepsilon_2-\\omega_2}({\\bf r}_8,{\\bf r}_7) G^\\text{A}_{\\varepsilon_2}({\\bf r}_7,{\\bf r}_8) G^\\text{R}_{\\varepsilon_1}({\\bf r}_3,{\\bf r}_4)\\rangle ,\n$$\nwhich differs from $K$ [Eq.~(\\ref{GGGG})] by changing R${}\\leftrightarrow{}$A in the last two Green functions.\nThe expression analogous to Eq.~(\\ref{K1}) will contain  $X_4^*$ instead of $X_4$ and $Z_4^*$ instead of $Z_4$, which renders the poles of both $X_3X_4^*$ and $Z_3Z_4^*$  lying in the upper half-plane of $\\varepsilon_2$. Therefore, the contribution of this term is non-singular and should be disregarded.\nA simple analysis demonstrates that there are only four  possibilities to arrange $G^\\text{R}$ and $G^\\text{A}$\nin $X_4X_4'$:\n\\begin{equation}\n\\label{RA-4ways}\n  \\text{RARARARA},\n\\quad\n  \\text{ARRARARA},\n\\quad\n  \\text{RAARARAR},\n\\quad\n  \\text{ARARARAR},\n\\end{equation}\nwhere all energy and coordinate indices follow Eq.~(\\ref{GGGG}). Each choice gives the same contribution given by Eq.~(\\ref{gamma1'}). As a result, the total contribution of the term $X_4X_4'$ due to type-(a) energy pairing (shown in Fig.~\\ref{F:AGKL31}a) is described by the coefficient\n\\begin{equation}\n  c_{XX}^{(a)}\n  =\n  \\frac{2}{N_s^2}\\left(\\frac 1{E_2^4}+\\frac 1{E_4^4}\\right) .\n\\label{gamma11}\n\\end{equation}\n\n\n\\subsection{Contribution from $X_4X_4'$ due to the energy pairing (b)}\n\\label{SS:X8b}\n\nFollowing the same line we can analyze the singular part of $K$ due to pairing (b) shown in Fig.~\\ref{F:AGKL31}(b). Instead of Eq.~(\\ref{K1-Z}) we obtain:\n\\begin{equation}\n  K_\\text{sing}^{(b)}\n  =\n  16 (\\pi\\nu)^8 \\, {\\cal Z} \\,\n  D({\\bf r}_1,{\\bf r}_2) D({\\bf r}_3,{\\bf r}_4)\n  D({\\bf r}_5,{\\bf r}_6) D({\\bf r}_7,{\\bf r}_8) .\n\\label{K2-Z}\n\\end{equation}\nFollowing the procedure described in Sec.~\\ref{qual} and utilizing the same four possibilities (\\ref{RA-4ways}) to arrange $G^\\text{R}$ and $G^\\text{A}$, we arrive at\n\\begin{equation}\n  c_{XX}^{(b)}\n  =\n  \\frac{1}{N_s^2E_2^4} .\n\\label{gamma12}\n\\end{equation}\n\n\n\n\n\\subsection{Contributions from $X_4Y_4'$ and $Y_4Y_4'$}\n\\label{Seconddiagram}\n\n\nIt is left to discuss the other two terms, $X_4Y_4'$ and $Y_4Y_4'$, in Eq.~(\\ref{corr}). Following the same steps we obtain that the contributions of the cross term $\\corr{X_4Y_4'}$ from both the pairings (a) and (b) vanish after integration over fast diffusive modes due to the diagonal structure of the $Q$ matrix in the 1234 space, while in calculating $\\corr{Y_4Y_4'}$ only the pairing (b) should be taken into account for the same reason.\n\nConsider, e.~g., a particular realization of $G^\\text{R}$ and $G^\\text{A}$ from $\\corr{Y_4Y_4'}$ [compare with $K$ defined in Eq.~(\\ref{GGGG})]:%\n\\begin{equation}\n\\tilde K=\\langle G^\\text{R}_{\\varepsilon}({\\bf r}_4,{\\bf r}_1) G^\\text{A}_{\\varepsilon_1}({\\bf r}_1,{\\bf r}_2) G^\\text{R}_{\\varepsilon_1-\\omega_1}({\\bf r}_2,{\\bf r}_3) G^\\text{A}_{\\varepsilon-\\omega_1}({\\bf r}_3,{\\bf r}_4) G^\\text{A}_{\\varepsilon}({\\bf r}_8,{\\bf r}_5) G^\\text{R}_{\\varepsilon_2}({\\bf r}_5,{\\bf r}_6) G^\\text{A}_{\\varepsilon_2-\\omega_2}({\\bf r}_6,{\\bf r}_7) G^\\text{R}_{\\varepsilon-\\omega_2}({\\bf r}_7,{\\bf r}_8) \\rangle .\n\\end{equation}\nIts singular contribution due to the pairing (b) is given by\n\\begin{equation}\n  \\corr{\\tilde K}_\\text{sing}^{(b)}\n  =\n  16 (\\pi\\nu)^8 \\, {\\cal Z} \\,\n  D({\\bf r}_1,{\\bf r}_2) D({\\bf r}_3,{\\bf r}_4)\n  D({\\bf r}_5,{\\bf r}_6) D({\\bf r}_7,{\\bf r}_8) ,\n\\end{equation}\nTracing the diffusons and taking into account four possibilities (\\ref{RA-4ways}), we get\n\\begin{equation}\n  c_{YY}^{\\text{(b)}}\n  =\n  \\frac{1}{N_s^4E_2^4} .\n\\label{gamma2}\n\\end{equation}\n\nFinally, adding the contributions (\\ref{gamma11}), (\\ref{gamma12}) and (\\ref{gamma2}) we obtain the final expression for mesoscopic fluctuations of the inelastic rate in the limit $F\\to0$:\n\\begin{equation}\n  \\ccorr{\\gamma^2(\\varepsilon,T)}\n  =\n  \\frac{\\lambda^4\\Delta^5}{4\\pi^3N_s^4}\n  \\left(\\frac{3N_s^2+1}{E_2^4}+\\frac{2N_s^2}{E_4^4}\\right)\n  \\int d\\varepsilon_1 d\\omega_1\n  \\frac{({\\cal F}_{\\varepsilon_1}-{\\cal F}_{\\varepsilon_1-\\omega_1})^2({\\cal B}_{\\omega_1}+{\\cal F}_{\\varepsilon-\\omega_1})^2}\n  {\\gamma_0(\\varepsilon-\\omega_1,T)+\\gamma_0(\\varepsilon_1,T)+\\gamma_0(\\varepsilon_1-\\omega_1,T)}\n  .\n\\label{gamma-total-f0}\n\\end{equation}\n\n\n\n\\section{Mesoscopic fluctuations at arbitrary $F$}\n\\label{S:mesofluct-rs}\n\n\\subsection{Account for a finite $F$ in the sigma-model language}\n\nIn this Section we generalize the result (\\ref{gamma-total-f0}) to the case of an arbitrary interaction parameter $F$ and derive the general expression (\\ref{subfinal}).\nFirst, we explain the main idea with an example of the type-(a) energy pairing in the diagram for $X_4X_4'$, and then apply it to the other diagrams ($X_4Y_4'$ and $Y_4Y_4'$) and energy pairing (b).\n\nWhen the range of the SSI (\\ref{V(r)}) characterized by the function $\\delta_\\kappa({\\bf r})$ becomes comparable to the Fermi wave length (i.~e., $F$ becomes non-negligible), a simple sigma-model expression (\\ref{Qcorr}) for the correlator $K$ breaks down. To modify it one has to turn back to the intermediate step in the derivation of the sigma model, prior to the final integration over the 16-component supervector $\\psi$ used to represent Green functions in the functional form.\nAt this stage, the correlator $K$ [Eq.~(\\ref{GGGG})] is written as\n\\begin{align}\n  K\n  =\n  \\int\n  \\langle\n& \\psi_\\text{R1}({\\bf r}_2) \\psi^*_\\text{R1}({\\bf r}_1)\n  \\psi_\\text{A1}({\\bf r}_6) \\psi^*_\\text{A1}({\\bf r}_5)\n  \\psi_\\text{R2}({\\bf r}_5) \\psi^*_\\text{R2}({\\bf r}_6)\n  \\psi_\\text{A2}({\\bf r}_1) \\psi^*_\\text{A2}({\\bf r}_2)\n\\nonumber\n\\\\{} \\times{}\n& \\psi_\\text{R3}({\\bf r}_4) \\psi^*_\\text{R3}({\\bf r}_3)\n  \\psi_\\text{A3}({\\bf r}_8) \\psi^*_\\text{A4}({\\bf r}_7)\n  \\psi_\\text{R4}({\\bf r}_7) \\psi^*_\\text{R4}({\\bf r}_8)\n  \\psi_\\text{A4}({\\bf r}_3) \\psi^*_\\text{A4}({\\bf r}_4)\n  \\rangle_{\\psi}\n  \\,\n  e^{-S[Q]} DQ ,\n\\label{psi16}\n\\end{align}\nwhere the bosonic components of the superfield $\\psi$ are implied.\nAveraging over $\\psi$ should be performed with the help of Wick's theorem with the correlation function\n\\begin{equation}\n  \\corr{\\psi({\\bf r})\\psi^*({\\bf r}')} = -i g({\\bf r},{\\bf r}') \\Lambda ,\n\\end{equation}\nwhere $g$ is the supermatrix Green function defined as\n\\begin{equation}\n  g({\\bf r},{\\bf r}')\n  =\n  \\langle\n    {\\bf r} | ( \\hat E - H_0 + iQ\/2\\tau )^{-1} | {\\bf r}'\n  \\rangle .\n\\end{equation}\n\nThe result of the $\\psi$-averaging is sensitive to the distance between the points ${\\bf r}_i$ controlled by the spatial range of the SSI propagators through the factors $\\delta_\\kappa({\\bf r}_i-{\\bf r}_j)$ in Eq.~(\\ref{X8}).\nSince the interaction is assumed to be short-range on the scale of the system size $L$, only correlations between $\\psi$'s coupled to the same SSI propagator should be taken into account. Therefore the $\\psi$-averaging in Eq.~(\\ref{psi16}) factorizes into four contributions corresponding to four SSI propagators in Eq.~(\\ref{X8}):\n\\begin{equation}\n\\label{K-KKKK}\n  K\n  =\n  \\int K^{(1,3)} K^{(2,4)} K^{(5,7)} K^{(6,8)} e^{-S[Q]} DQ .\n\\end{equation}\nConsider for example the group of four $\\psi$'s coupled by the interaction with $\\delta_\\kappa({\\bf r}_1-{\\bf r}_3)$:\n\\begin{equation}\n  K^{(1,3)}\n  =\n  \\langle\n  \\psi^*_\\text{R1}({\\bf r}_1)\n  \\psi_\\text{A2}({\\bf r}_1)\n  \\psi^*_\\text{R3}({\\bf r}_3)\n  \\psi_\\text{A4}({\\bf r}_3)\n  \\rangle_{\\psi}\n.\n\\label{psi4}\n\\end{equation}\nApplication of Wick's theorem generates two terms:\n\\begin{equation}\n  K^{(1,3)}\n  =\n  -\n  g_\\text{A2,R1}({\\bf r}_1,{\\bf r}_1)\n  g_\\text{A4,R3}({\\bf r}_3,{\\bf r}_3)\n  -\n  g_\\text{A2,R3}({\\bf r}_1,{\\bf r}_3)\n  g_\\text{A4,R1}({\\bf r}_3,{\\bf r}_1)\n.\n\\label{K13-gg}\n\\end{equation}\n\nThe first (diagonal) term in Eq.~(\\ref{K13-gg}) reduces to the $Q$ matrix due to the self-consistency equation $g({\\bf r},{\\bf r})=-i\\pi\\nu Q({\\bf r})$ \\cite{Efetov-book}, and then one immediately recovers Eq.~(\\ref{Qcorr}) used previously in the analysis of the $F=0$ case. In contrast, the second (off-diagonal) term in Eq.~(\\ref{K13-gg}) cannot be so easily expressed in terms of the $Q$ matrix. But here we can use the knowledge that at the later stage, in the process of averaging over fast modes (see Sec.~\\ref{fastaveraging}), each $Q$ will be expanded to the first power of $W$ [see Eqs.~(\\ref{param}), (\\ref{Q'-W})]. The linear-in-$W$ contribution to $g$ is given by:\n\\begin{equation}\n\\label{dg1}\n  \\delta g({\\bf r}_1,{\\bf r}_3)\n  =\n  - \\frac{i}{2\\tau}\n  \\int d{\\bf r}'\n  \\langle\n    {\\bf r}_1 | ( - H_0 + iQ\/2\\tau )^{-1} | {\\bf r}'\n  \\rangle\n  \\,\n  T^{-1} \\Lambda W({\\bf r}') T\n  \\,\n  \\langle\n    {\\bf r}' | ( - H_0 + iQ\/2\\tau )^{-1} | {\\bf r}_3\n  \\rangle ,\n\\end{equation}\nwhere we write $Q=T^{-1}\\Lambda T$, and neglect the term $\\hat E$ as we are working in the diffusive limit $\\max\\{\\varepsilon, T\\} \\tau \\ll 1$.\nSince $W$ anticommutes with $\\Lambda$, we can rewrite Eq.~(\\ref{dg1}) in the form\n\\begin{equation}\n\\label{dg2}\n  \\delta g({\\bf r}_1,{\\bf r}_3)\n  =\n  - \\frac{i}{2\\tau}\n  \\int d{\\bf r}'\n  \\,\n  T^{-1}\n  \\langle\n    {\\bf r}_1 | ( - H_0 + i\\Lambda\/2\\tau )^{-1} | {\\bf r}'\n  \\rangle\n  \\langle\n    {\\bf r}' | ( - H_0 - i\\Lambda\/2\\tau )^{-1} | {\\bf r}_3\n  \\rangle\n  \\Lambda W\n  T\n  =\n  - \\frac{i \\delta Q}{2\\tau}\n  (G^\\text{R}G^\\text{A})({\\bf r}_1-{\\bf r}_3)\n  ,\n\\end{equation}\nwhere $\\delta Q=T^{-1} \\Lambda W T$,\nand $G^\\text{R}G^\\text{A}$ is a scalar function given by Eq.~(\\ref{GRGA}).\n\nMultiplying the second term of Eq.~(\\ref{K13-gg}) by $\\delta_\\kappa({\\bf r}_1-{\\bf r}_3)$ and integrating over the difference ${\\bf r}_1-{\\bf r}_3$, we immediately recover the factor $F$ introduced in Eq.~(\\ref{f-I}).\nHence, Eq.~(\\ref{K13-gg}) can be written as\n\\begin{equation}\n  K^{(1,3)}\n  =\n  (\\pi\\nu)^2\n  \\left[\n  Q^{\\text{AR}}_{21}({\\bf r}_1) Q^{\\text{AR}}_{43}({\\bf r}_3)\n  + F\\,\n  Q^{\\text{AR}}_{23}({\\bf r}_1) Q^{\\text{AR}}_{41}({\\bf r}_3)\n  \\right] ,\n\\label{K13-QQ}\n\\end{equation}\nwhere with our accuracy we do not distinguish between ${\\bf r}_1$ and ${\\bf r}_3$.\nGenerally speaking, Eq.~(\\ref{K13-QQ}) is incorrect.\nWe write it in such a form for brevity, assuming that it will be further subject to the procedure of fast mode extraction described in Sec.~\\ref{fastaveraging}.\n\nPerforming the same analysis for other $K^{(i,j)}$ from Eq.~(\\ref{K-KKKK}), one concludes that for the purpose of calculation of mesoscopic fluctuations of the relaxation rate at $F\\neq0$, Eq.~(\\ref{Qcorr}) should be modified as\n\\begin{multline}\n  K\n  =\n  (\\pi \\nu)^8\\int\n  \\left[\n  Q^{\\text{AR}}_{21}({\\bf r}_1) Q^{\\text{AR}}_{43}({\\bf r}_3)\n  + F\\,\n  Q^{\\text{AR}}_{23}({\\bf r}_1) Q^{\\text{AR}}_{41}({\\bf r}_3)\n  \\right]\n  \\left[\n  Q^{\\text{RA}}_{12}({\\bf r}_2) Q^{\\text{RA}}_{34}({\\bf r}_4)\n  + F\\,\n  Q^{\\text{RA}}_{14}({\\bf r}_2) Q^{\\text{RA}}_{32}({\\bf r}_4)\n  \\right]\n\\\\{}\n  \\times\n  \\left[\n  Q^{\\text{RA}}_{21}({\\bf r}_5) Q^{\\text{RA}}_{43}({\\bf r}_7)\n  + F\\,\n  Q^{\\text{RA}}_{23}({\\bf r}_5) Q^{\\text{RA}}_{41}({\\bf r}_7)\n  \\right]\n  \\left[\n  Q^{\\text{AR}}_{12}({\\bf r}_6) Q^{\\text{AR}}_{34}({\\bf r}_8)\n  + F\\,\n  Q^{\\text{AR}}_{14}({\\bf r}_6) Q^{\\text{AR}}_{32}({\\bf r}_8)\n  \\right]\n  e^{-S[Q]}DQ,\n\\label{Qcorr-rs}\n\\end{multline}\nwhere the difference between the coordinates within the same bracket can be neglected.\nSimilar expressions originate in the analysis of other contributions from\n$X_4X_4'$, $X_4Y_4'$ and $Y_4Y_4'$.\n\n\n\n\\subsection{Contributions of different diagrams and energy pairings}\n\nTo find mesoscopic fluctuations at a finite $F$, one should use Eq.~(\\ref{Qcorr-rs}) as a starting point and perform all the steps outlined in Secs.~\\ref{S:nonpert} and \\ref{S:final}.\nThis is a routine procedure leading to the following results.\nIn the case $F\\neq0$, there are finite contributions from all the terms $X_4X_4'$, $X_4Y_4'$, and $Y_4Y_4'$, and from both energy pairings (a) and (b).\nThese six contributions are characterized by six coefficients $c_A^{(i)}$ in Eq.~(\\ref{gamma-c}). After some algebra, we obtain (the coefficients $c_{XY}^{(i)}$ already contain the factor 2 accounting for the mirror term $Y_4X_4'$):\n\\begin{gather}\n  c_{XX}^{(a)} =\n  \\frac{2(1+F^2)^2}{N_s^2 E_2^4} + \\frac{2(1+F^4)}{N_s^2 E_4^4} ,\n\\qquad\n  c_{XX}^{(b)} =\n  \\frac{1+6F^2+F^4}{N_s^2 E_2^4} + \\frac{4F^2}{N_s^2 E_4^4} ,\n\\label{gX}\n\\\\\n  c_{XY}^{(a)} = c_{XY}^{(b)} =\n  - \\frac{8(F+F^3)}{N_s^3 E_2^4} - \\frac{4(F+F^3)}{N_s^3 E_4^4} ,\n\\label{gXY}\n\\\\\n  c_{YY}^{(a)} = \\frac{8 F^2}{N_s^4 E_2^4} + \\frac{4 F^2}{N_s^4 E_4^4} ,\n\\qquad\n  c_{YY}^{(b)} = \\frac{1+6F^2+F^4}{N_s^4 E_2^4} + \\frac{4F^2}{N_s^4 E_4^4}.\n\\label{gY}\n\\end{gather}\nSumming the contributions (\\ref{gX})--(\\ref{gY}), we come to the final result (\\ref{subfinal}) valid for an arbitrary $F$.\n\n\n\n\n\\section{Discussion and conclusion}\n\\label{Discussion}\n\nIn this work we have analyzed the applicability of the Fermi-golden-rule description of the initial stage of quasiparticle decay in diffusive quantum dots in the regime when the single-particle levels are already resolved. Approaching the problem from the high energy\/temperature side, where each energy level can be characterized by a Lorentzian width $\\gamma_0(\\varepsilon,T)$, we have calculated mesoscopic fluctuations of the energy relaxation rate.\nThe leading contribution to fluctuations comes from the diagrams which describe the square of the same decay process, i.e.\\ have the same set of final states.\nThe resulting expression is non-perturbative in $\\gamma_0$, which appears in the denominator of Eq.~(\\ref{subfinal}), ensuring the growth of fluctuations with the decrease of the excitation energy and\/or temperature.\n\nQuantum relaxation of the initial state $|i\\rangle$ can be described by the return probability\n\\begin{equation}\n  P(t) = \\overline{\\bigl| \\corr{i|e^{-iHt}|i} \\bigr|^2} ,\n\\end{equation}\nwhere $H$ is the Hamiltonian of the interacting quantum dot, and the bar stands for the thermal average. In the semiclassical FGR picture this is just a pure exponential decay:\n\\begin{equation}\n  P_\\text{FGR}(t) = e^{-\\gamma(\\varepsilon,T)t} ,\n\\end{equation}\nwhere the rate $\\gamma(\\varepsilon,T)$ depends on a particular disorder realization.\nIts average value is given by Eq.~(\\ref{SIAT}), and we focused on its mesoscopic fluctuations.\nWe found that the FGR description of the initial stage of quasiparicle decay is applicable as long as $\\max\\{\\varepsilon,T\\}\\gg\\varepsilon_\\text{FGR}$. In this limit, each level is characterized by a well-defined energy width $\\gamma(\\varepsilon,T)$, which weakly fluctuates near the FGR mean:\n\\begin{equation}\n\\label{qualresult3}\n  \\frac{\\ccorr{ \\gamma^2(\\varepsilon,T)}}{\\gamma_{0}^2(\\varepsilon,T)}\n  \\sim\n  \\left( \\frac{\\varepsilon_\\text{FGR}}{\\max\\{ \\varepsilon, T\\}} \\right)^4 .\n\\end{equation}\nThe temperature and the excitation energy enter the result in a similar way [note, however, the presence of the factor $\\Upsilon(\\varepsilon\/2T)$ in Eq.~(\\ref{gamma-Upsilon}), that can change by two orders of magnitude]. This fact is a consequence of the lowest-order approximation. In higher orders their role is expected to be different, in accordance with the difference between the relaxation dynamics in the hot-electron and thermal problems \\cite{Mirlin2015}.\n\nIt is important that in the range of applicability of the FGR description, $\\max\\{\\varepsilon,T\\}\\gg\\varepsilon_\\text{FGR}$, the hybridiation with distant generations and eventually many-body localization effects (if any) become relevant at sufficiently large time scales. The characteristic time $t_*$ when $P_\\text{FGR}(t)$ crosses over to a weaker dependence is determined by the initial state.\nIn the limit $\\max\\{\\varepsilon,T\\}\\gg\\varepsilon_\\text{FGR}$, the scale $t_*$ satisfies $\\gamma_0(\\varepsilon,T)t_*\\gg1$, indicating that almost all the quasiparticle weight is lost during the FGR exponential relaxation\n(for the hot-electron problem, $\\gamma_0(\\varepsilon,0)t_* \\sim \\ln(\\varepsilon\/\\varepsilon_\\text{FGR})$ \\cite{Silvestrov2001}).\n\nOur approach is conceptually similar to the one used by Basko, Aleiner and Altshuler (BAA)~\\cite{BAA}, and hence it is instructive to compare the two results.\nBAA considered inelastic relaxation in a chaotic quantum dot (in Ref.~\\cite{BAA}, referred to as the localization cell), working in the basis of exact one-particle states $|\\alpha\\rangle$ and treating electron-electron interaction phenomenologically. Interaction matrix elements, $\\corr{\\alpha\\beta|V|\\gamma\\delta}$, were assumed to be independent normally distributed random variables with the standard deviation $\\lambda_\\text{BAA}\\Delta$ for energy difference smaller than the ultraviolet cutoff $M\\Delta$, and zero otherwise.\nThe FGR relaxation rate is then given by $\\gamma_{\\text{BAA}}(T)\\sim \\lambda_\\text{BAA}^2 MT$, and BAA obtained the following expression for  mesoscopic fluctuations:\n\\begin{equation}\n\\label{mesofluct-BAA}\n  \\frac{\\ccorr{\\gamma^2(T)}_{\\text{BAA}}}{\\gamma^2_{\\text{BAA}}(T)}\n  \\sim\n  \\frac{\\lambda_\\text{BAA}^4M\\Delta^2T}{\\gamma_{\\text{BAA}}^3(T)}\n  \\sim\n  \\frac{\\Delta^2}{\\lambda_\\text{BAA}^2M^2T^2}.\n\\end{equation}\nIn order to apply these results to the case of a diffusive quantum dot, one should put $\\lambda_\\text{BAA} \\sim \\lambda \\Delta\/E_{\\text{Th}}$~\\cite{AGKL} and $M\\sim T\/\\Delta$.\nThen $\\gamma_{\\text{BAA}}(T)$ coincides with the Sivan-Imry-Aronov relaxation rate (\\ref{SIAT}), while the estimate (\\ref{mesofluct-BAA}) reproduces our result (\\ref{qualresult3}) for mesoscopic fluctuations.\n\nFinally, we emphasize that our result (\\ref{subfinal}) for the leading contribution to mesoscopic fluctuations provides an exact account for the (screened) electron-electron interaction in a diffusive quantum dot. It has an important implementation for the statistics of the interaction matrix elements $\\corr{\\alpha\\beta|V|\\gamma\\delta}$ in the lattice-model language. Since Ref.~\\cite{AGKL}, it is usually assumed for simplicity that this statistics is Gaussian. However, our result (\\ref{subfinal}) demonstrates that for a real diffusive quantum dot such an assumption is generally incorrect. Indeed, for the Gaussian statistics all correlators of the four matrix elements in $\\ccorr{\\gamma^2(\\varepsilon,T)}$ can be expressed through pairwise correlators (the Wick theorem), which are known to be determined by $E_2$ only \\cite{Blanter,AGKL}.\nTherefore the presence of the quantity $E_4$ in Eq.~(\\ref{subfinal}) indicates that the statistics of the interaction matrix elements in a quantum dot is essentially non-Gaussian. This fact should be taken into account in constructing the theory of many-body localization in quantum dots.\n\nWe thank\nI. Aleiner,\nYa.~M. Blanter,\nM. V. Feigel'man,\nI. V. Gornyi,\nV. E. Kravtsov,\nA. D. Mirlin,\nand A. Silva\nfor useful discussions.\nThis work was supported by the Russian Science Foundation under Grant\nNo.\\ 14-42-00044 (M.A.S.).\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $\\left(X,Y\\right)$ be a random couple with joint cumulative distribution function $H$ and marginal distribution functions $F$ and $G$. The Sklar's theorem (see \\cite{skl}) says that there exists a bivariate distribution function $C$ on $[0,1]^2$  with uniform margins such that\n$$H(x,y)=C(F(x),G(y)).$$\nThe function $C$ is called the copula associated with $(X,Y)$ and couples the joint distribution $H$ with its marginals. If the marginal distribution functions $F$ and $G$ are continuous, then the copula $C$ is unique and we have  for all $(u,v)\\in [0,1]^2$,\n$$C(u,v)=H(F^{-1}(u),G^{-1}(v)),$$ \nwhere $F^{-1}(u)=\\inf\\{x: F(x)\\geq u\\}$ and $G^{-1}(v)=\\inf\\{y: G(y)\\geq v\\}$ are the generalized inverse functions of $F$ and $G$, respectively. \\\\\n\nThere are three main approaches for copula estimation : parametric, semiparametric and nonparametric. The parametric approach assumes parametric models for both the copula and the marginals and then deals with maximum likelihood or moment method estimation Oakes \\cite{oak}(1982). Semiparametric estimation specifies a parametric copula while leaving the marginals nonparametric (see, e.g. Genest \\textit{et al.}\\cite{ggr}(1995)). The nonparametric approch offers the greatest generality and was initiated by Deheuvels \\cite{deh} (1979), who proposed an estimator based on a multivariate empirical distribution function and its marginals. Afterward, some kernel smoothed estimators have been proposed in the literature (see for instance \\cite{r4},\\cite{r8},\\cite{r3},\\cite{r2},\\cite{r7}).\\\\\n\nIn this paper we are interested with the kernel estimators proposed in Chen and Huang \\cite{r2}(2007), Gijbels and Mielniczuk \\cite{r4}(1990) and Fermanian \\textit{et al.} \\cite{r3}(2004), res pectively called the local linear, the mirror-reflection and the transformation estimators. We will establish for each of them a uniform in bandwidth law of the iterated logarithm for its deviation. These results allows to study the uniform consistency of kernel copula estimators over compact sets $[a,b]^2$, with $0<a<b<1.$\\\\\n\nLet $(X_1,Y_1),...,(X_n,Y_n)$ be an independent and identically distributed sample of the bivariate random vector $(X,Y)$ with joint cumulative distribution function $H$ and marginal distribution functions $F$ and $G$. Denote by $F_n$ and $G_n$  the marginal empirical cumulative distribution functions. Let $\\phi:[0,1]\\mapsto[0,1]$ be an increasing transformation and  $K(\\cdot,\\cdot)$ a bivariate kernel function. For all $0\\leq u,v\\leq 1$ and $0<h<1$, define the general estimator\n\\begin{equation}\\label{ge}\n\\hat{C}_{n,h}^{(\\cdot)}(u,v)=\\frac{1}{n}\\sum_{i=1}^{n}K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\hat{U}_i)}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\hat{V}_i)}{h} \\right)\n\\end{equation}\nIf $K(x,y)$ is a multiplicative kernel, i.e. $K(x,y)=K(x)K(y)$ and the pseudo-observations are $\\hat{U}_i=\\frac{n}{n+1}F_n(X_i)$ and $\\hat{V}_i=\\frac{n}{n+1}G_n(Y_i)$, \\eqref{ge} is exactly the transformation estimator which is defined by\n\\begin{equation}\\label{tra}\n\\hat{C}_{n,h}^{(T)}(u,v)=\\frac{1}{n}\\sum_{i=1}^{n}K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\hat{U}_i)}{h} \\right)K\\left(\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\hat{V}_i)}{h} \\right).\n\\end{equation}\nFor the local linear estimator,  suppose first that the marginals $F$ and $G$ are estimated by\n\\begin{center} \n$\\displaystyle \\hat{F}_n(x)=\\frac{1}{n}\\sum_{i=1}^n K\\left(\\frac{x-X_i}{b_{n1}}\\right),\\ \\ \\ \\ \\ \\hat{G}_n(y)=\\frac{1}{n}\\sum_{i=1}^n K\\left(\\frac{y-Y_i}{b_{n2}}\\right),$\n\\end{center}  where $b_{n1}$ and $b_{n2}$ are  bandwidths converging to 0, as $n\\rightarrow\\infty$ and $K$ is the integral of a symmetric bounded kernel function $k$ supported on $[-1,1]$. \nNext, consider the pseudo-observations $\\hat{U}_i = \\hat{F}_n(X_i)$ and $\\hat{V}_i =\\hat{G}_n(Y_i)$ and choose a multiplicative $K(x,y)=K_{u,h}(x)K_{v,h}(y)$, for $u,v\\in[0,1]$, where $K_{w,h}(x)=\\int_{-\\infty}^{x}k_{w,h}(t)dt,\\; w=u,v$\n with $k_{w,h}$ a local linear version of the kernel $k$ given by\n$$\n k_{w,h}(t)=\\frac{k(t)\\{a_2(w,h)-a_1(w,h)t\\}}{a_0(w,h)a_2(w,h)-a_1^2(w,h)}\\mathbb{I}\\left\\{\\frac{w-1}{h}<t<\\frac{w}{h}\\right\\},$$\nwhere $ a_j(w,h)=\\int_{(w-1)\/h}^{w\/h} t^j k(t)dt$ for $j=0,1,2$ ; $w\\in[0,1]$ and $0<h<1$ is a bandwidth. Finally taking $\\phi(s)=t$, the identity function in \\eqref{ge}, we obtain the local linear kernel estimator  defined as\n\\begin{equation}\\label{lli}\n\\hat{C}_{n,h}^{(LL)}(u,v)=\\frac{1}{n}\\sum_{i=1}^{n}K_{u,h}\\left(\\frac{u-\\hat{U_i}}{h} \\right)K_{v,h}\\left(\\frac{v-\\hat{V_i}}{h} \\right).\n\\end{equation}\n  According to Omelka \\textit{et al} (2009), the mirror-reflection estimator can be represented as\n \\begin{equation}\\label{mrf}\n\\hat{C}_{n}^{(MR)}(u,v)=\\sum_{l=1}^9\\left[Z_n(l,u,v)-Z_n(l,u,0)-Z_n(l,0,v)+Z_n(l,0,0)\\right],\n\\end{equation}\nwhere\n\\begin{equation}\\label{bigz}\nZ_n(l,u,v)=\\frac{1}{n}\\sum_{i=1}^nK\\left(\\frac{u-\\hat{U}_i^{(l)}}{h_n}\\right)K\\left(\\frac{v-\\hat{V}_i^{(l)}}{h_n}\\right)\n\\end{equation}\nand \n\\begin{multline}\n\\left\\{\\left(\\hat{U}_i^{(l)},\\hat{V}_i^{(l)}\\right),i=1,...,n,l=1,...,9\\right\\}\\\\\n=\\left\\{\\left(\\pm\\hat{U}_i,\\pm\\hat{V}_i\\right),\\left(\\pm\\hat{U}_i,2-\\hat{V}_i\\right),\\left(2-\\hat{U}_i,\\pm\\hat{V}_i\\right),\\left(2-\\hat{U}_i,2-\\hat{V}_i\\right),i=1,...,n\\right\\}\\nonumber\n\\end{multline}\nSetting $\\phi(t)=t$ and using a multiplicative kernel $K(x,y)=K(x)K(y)$, one can see that each quantity $Z_n(l,u,v)$ may be put in the form \\eqref{ge}.\\\\\n\nThe remainder of the paper is organized as follows. In Section 2, we state our main results which consist of laws of the iterated logarithm for the deviations of the estimators \\ref{tra},\\ref{lli} and \\ref{mrf} from their means.  Section 3 is devoted to the proofs of the results. Finally, in Section 4 we give an appendix, where we show a key result for the proofs inspired by a general theorem of Mason and Swanpoel (2010), concerning the uniform in bandwidth consistency of kernel-type function estimators. \n\n\\section{Main results }\\label{ssec2}\nWe state our theoretical results in Theorems \\ref{t1}, \\ref{t2} and \\ref{t3}, which give  uniform in bandwidth laws of the iterated logarithm (LIL) for the maximal deviation of the estimators \\eqref{tra},\\eqref{lli} and \\eqref{mrf} from their expectations.\\\\\n Let $R_n =\\left(\\frac{n}{2\\log\\log n}\\right)^{1\/2}$. We have\n \\begin{theorem}\\label{t1}\nFor any sequence of positive constants $(b_n)_{n\\geq 1}$ satisfying $0<b_n<1, b_n\\rightarrow 0$, $b_n\\geq (\\log n)^{-1}$, and for some $c>0$, we have almost surely\n\\begin{equation}\\label{e2}\n\\limsup_{n\\rightarrow\\infty}\\left\\{R_n\\sup_{\\frac{c\\log n}{n}\\le h\\le b_n}\\sup_{(u,v)\\in(0,1)^2}\\left|\\hat{C}_{n,h}^{(LL)}(u,v)-\\mathbb{E}\\hat{C}_{n,h}^{(LL)}(u,v)\\right|\\right\\}\\leq 3.\n\\end{equation}\n\\end{theorem} \n\n\\begin{theorem}\\label{t2}\nSuppose that the increasing transformation $\\phi$ admits a bounded derivative $\\phi'$. Then, for any sequence of positive constants $(b_n)_{n\\geq 1}$ satisfying $0<b_n<1, b_n\\rightarrow 0$, $b_n\\geq (\\log n)^{-1}$, and for some $c>0$, we have almost surely \n\\begin{equation}\\label{ee2}\n\\limsup_{n\\rightarrow\\infty}\\left\\{R_n\\sup_{\\frac{c\\log n}{n}\\le h\\le b_n}\\sup_{(u,v)\\in(0,1)^2}\\left|\\hat{C}_{n,h}^{(T)}(u,v)-\\mathbb{E}\\hat{C}_{n,h}^{(T)}(u,v)\\right|\\right\\}\\leq 3.\n\\end{equation}\n\\end{theorem}\n\\begin{theorem}\\label{t3}\nFor any sequence of positive constants $(b_n)_{n\\geq 1}$ satisfying $0<b_n<1, b_n\\rightarrow 0$, $b_n\\geq (\\log n)^{-1}$, and for some $c>0$, we have almost surely \n\\begin{equation}\\label{eeq2}\n\\limsup_{n\\rightarrow\\infty}\\left\\{R_n\\sup_{\\frac{c\\log n}{n}\\le h\\le b_n}\\sup_{(u,v)\\in(0,1)^2}\\left|\\hat{C}_{n,h}^{(MR)}(u,v)-\\mathbb{E}\\hat{C}_{n,h}^{(MR)}(u,v)\\right|\\right\\}\\leq 3.\n\\end{equation}\n\\end{theorem}\n\nThe proofs of Theorems \\ref{t1},\\ref{t2},\\ref{t3} are similar and are postponded until Section 3. They are obtained by combining a general theorem of Mason and Swanpoel \\cite{r5}(2010) for proving the uniform in bandwidth consistency of kernel-type function estimators, and a law of the iterated logarithm for Kiefer processes due to Wichura \\cite{r10}(1973).\\\\\n\n\\textbf{Remark}.\n{\\rm \\begin{itemize}\n\\item[1)] Under smoothness conditions on the copula $C$, namely the existence bounded second-order partial derivatives, ensuring the uniform almost sure convergence of the bias of the estimators  to zero, we obtain the strong uniform in bandwidth consistency of the kernel copula estimators $\\hat{C}_{n,h}^{(LL)},\\hat{C}_{n,h}^{(MR)}$ and $\\hat{C}_{n,h}^{(T)}$.\n\\item[2)] The uniformity on the bandwidth allows the use of a large bandwidth selection methods including the method of shrinkage proposed by Omelka \\textit{et al.}\\cite{r7}(2009).\n\\item[3)] As the boundary bias is present, the uniform consistency of these kernel estimators is not valid over the entire $[0,1]^2$, but it is true for all  $(u,v)\\in[h,1-h]^2,\\; 0<h<1$.\n\\item[4)] If we consider a data-driven bandwidth ; that is $h=\\hat{h}_n$, we have the probability versions of Theorems \\ref{t1},\\ref{t2} and \\ref{t3}, stated in  Corollary \\ref{cr1} below.\\\\\n\\end{itemize} }\n\nLet $\\hat{C}_{n,h}^{(\\cdot)}$ denote a kernel estimator of the copula $C$, where $(\\cdot)$ stands for $(LL), (MR)$ and $(T)$, respectively. One has\n\\begin{corollary}\\label{cr1}\nUnder the assumptions of Theorems \\ref{t1},\\ref{t2} or \\ref{t3}, we have for all $0<\\epsilon <1$ and for any sequence of data-driven bandwidth $\\hat{h}_n$ satisfying \\begin{equation} \\label{ee4'}\n \\mathbb{P}(\\frac{c\\log n}{n}\\le \\hat{h}_n\\le b_n)\\rightarrow 1,\\; n\\rightarrow\\infty,\n \\end{equation}\n\\begin{equation}\\label{ee2}\n\\mathbb{P}\\left\\{R_n\\sup_{(u,v)\\in(0,1)^2}\\left|\\hat{C}_{n,\\hat{h}_n}^{(\\cdot)}(u,v)-\\mathbb{E}\\hat{C}_{n,\\hat{h}_n}^{(\\cdot)}(u,v)\\right|>  3(1+\\epsilon)\\right\\}=o(1).\n\\end{equation}\n\\end{corollary}\n\\textbf{Remark}. This result enables us to construct simultaneous confidence bands for the copula curve $C(u,v), 0<u,v<1$, with  asymptotic confidence level $100\\%$, under the condition that the bias of the estimator $\\mathbb{E}\\hat{C}_{n,\\hat{h}_n}^{(\\cdot)}-C(u,v)$ converges uniformly to zero with the same rate $R_n$, as $n\\rightarrow\\infty$, i.e.  for all $0<\\epsilon <1$,\n\\begin{equation}\\label{e2}\n\\mathbb{P}\\left\\{R_n\\sup_{(u,v)\\in(0,1)^2}\\left|\\mathbb{E}\\hat{C}_{n,\\hat{h}_n}^{(\\cdot)}(u,v)-C(u,v)\\right|>  \\epsilon\\right\\}=o(1).\n\\end{equation}\nAn example of such confidence bands is provided in \\cite{r1} for the local linear estimator.  \n\\section{Proofs}\nThe proofs of the theorems are similar. To simplify we will establish the results for a generalized estimator $\\hat{C}_{n,h}^{(\\cdot)}$ defined as follows. Let $\\phi:[0,1]\\mapsto[0,1]$ be an increasing transformation. For a bivariate kernel $K(\\cdot,\\cdot)$ and $0<u,v<1$, define the estimator\n\\begin{equation}\\label{ge}\n\\hat{C}_{n,h}^{(\\cdot)}(u,v)=\\frac{1}{n}\\sum_{i=1}^{n}K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\hat{U}_i)}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\hat{V}_i)}{h} \\right)\n\\end{equation}\nand set\n$$\\hat{D}_{n,h}^{(\\cdot)}(u,v)=:\\hat{C}_{n,h}^{(\\cdot)}(u,v)-\\mathbb{E}\\hat{C}_{n,h}^{(\\cdot)}(u,v).$$\nWe call $\\hat{D}_{n,h}^{(\\cdot)}(u,v)$ the deviation of the estimator $\\hat{C}_{n,h}^{(\\cdot)}(u,v)$ from its expectation and we will study its behavior by using general empirical process theory.\\\\\n\\indent We remark that if the kernel $K(\\cdot,\\cdot)$ is multiplicative, i.e. $K(x,y)=K(x)K(y)$, we obtain directly the transformation estimator. If $\\phi(t)=t$, the identity function and the local linear kernels $K_{u,h}(\\cdot)$ and $K_{v,h}(\\cdot)$ are employed, then we obtain the local linear kernel estimator. Note that for the mirror-reflection estimator, we consider the following decomposition due to Omelka \\textit{et al} (2009),\n \\begin{equation}\\label{mirror-local}\n\\hat{C}_{n}^{(MR)}(u,v)=\\sum_{l=1}^9\\left[Z_n(l,u,v)-Z_n(l,u,0)-Z_n(l,0,v)+Z_n(l,0,0)\\right],\n\\end{equation}\nwhere\n\\begin{equation}\\label{bigz}\nZ_n(l,u,v)=\\frac{1}{n}\\sum_{i=1}^nK\\left(\\frac{u-\\hat{U}_i^{(l)}}{h_n}\\right)K\\left(\\frac{v-\\hat{V}_i^{(l)}}{h_n}\\right).\n\\end{equation}\nSetting $\\phi(t)=t$ and using a multiplicative kernel, one can see that each quantity $Z_n(l,u,v)$ may be put in the form \\eqref{ge}.\\\\\n\nLet $H_n$, $F_n$ and $G_n$ be the empirical cumulative distribution functions of $H$, $F$ and $G$, respectively. Then  the estimator based directly on Sklar's Theorem is given by\n$$ C_n(u,v)= H_n(F_n^{-1}(u),G_n^{-1}(v)),$$\nwith $F_n^{-1}(u)=\\inf\\{x:F_n(x)\\geq u\\}$ and $G_n^{-1}(v)=\\inf\\{x:F_n(x)\\geq v\\}$. The bivariate empirical copula process is defined as\n$$\\mathbb{C}_n(u,v)=\\sqrt{n}[C_n(u,v)- C(u,v)],\\qquad (u,v)\\in [0,1]^2.$$\nIntroduce the following quantity.\n$$ \\widetilde{C}_n(u,v)=\\frac{1}{n}\\sum_{i=1}^n\\mathbb{I}\\{U_i\\leq u,V_i\\leq v\\}$$\nwhich represents the uniform bivariate empirical distribution function based on a sample $(U_1,V_1),\\cdots,(U_n,V_n)$ of i.i.d random  variables uniformly distributed on $[0,1]^2$. \nDefine the following empirical process\n $$\\mathbb{\\widetilde{C}}_n(u,v)=\\sqrt{n}[\\widetilde{C}_n(u,v)- C(u,v)],\\quad (u,v)\\in[0,1]^2.$$\nThen, wecan easily prove that that \n\\begin{equation}\\label{c3}\n\\mathbb{\\widetilde{C}}_n(u,v)=\\mathbb{C}_n(u,v)+ \\frac{1}{\\sqrt{n}}.\n\\end{equation}\nFor any given increasing transformation $\\phi$ and $(u,v)\\in [0,1]^2$, define \n\\begin{eqnarray*}\n&g_{n,h}=\\hat{C}_{n,h}^{(\\cdot)}(u,v)-\\widetilde{C}_n(u,v)&\\\\\n&  = \\frac{1}{n}\\sum_{i=1}^{n}\\left [K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\hat{U}_i)}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\hat{V}_i)}{h} \\right)- \\mathbb{I}\\{U_i\\leq u,V_i\\leq v\\}\\right ]&\\\\\n& =\\frac{1}{n}\\sum_{i=1}^{n}\\left [K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\hat{F}_n oF^{-1}(U_i))}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\hat{G}_n oG^{-1}(V_i))}{h} \\right)-\\mathbb{I}\\{U_i\\leq u,V_i\\leq v\\}\\right]& \\\\\n& =: \\frac{1}{n}\\sum_{i=1}^{n}g(U_i,V_i,h),&\n\\end{eqnarray*}\nwhere $g$ belongs to the class of measurable functions $\\mathcal{G}$ defined as \n$$\n\\mathcal{G}=\\left\\{\\begin{array}{c} \ng:(s,t,h)\\mapsto g(s,t,h)=K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(s))}{h}, \\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(t))}{h}\\right)- \\mathbb{I}\\{s\\leq u,t\\leq v\\},\\\\\n u,v\\in[0,1], 0<h<1 \\,\\text{and }\\, \\zeta_{1,n}\\zeta_{2,n}:[0,1]\\mapsto [0,1] \\,\\text{nondecreasing.}\n\\end{array} \\right\\}\n$$\nSince $\\mathbb{E}\\widetilde{C}_n(u,v)=C(u,v)$, one can observe that \n$$\n\\sqrt{n}\\vert g_{n,h}- \\mathbb{E}g_{n,h}\\vert= \\vert\\sqrt{n}\\hat{D}_{n,h}^{(\\cdot)}(u,v)-\\mathbb{\\widetilde{C}}_n(u,v)\\vert.\n$$\n\nNow, we want to apply the main Theorem in \\cite{r5} due to Mason and Swanpoel. Towards this end, the class of functions $\\mathcal{G}$ must verify the following  four conditions: \n\\begin{itemize}\n\\item[(G.i)]There exists a finite constant $\\kappa>0$ such that\n $$ \\sup_{0\\leq h\\leq 1}\\sup_{g\\in \\mathcal{G}}\\left\\|g\\left(\\cdot,\\cdot,h\\right)\\right\\|_\\infty=\\kappa <\\infty. $$\n\\item[(G.ii)]\\ \\ \\ There exists a constant $C'>0$ such that for all $h\\in [0,1]$,\n$$ \\sup_{g\\in \\mathcal{G}}\\mathbb{E}\\left[g^2\\left(U,V,h\\right)\\right]\\leq C'h. $$\n\\item[(F.i)]\\ \\ \\ \n$\\mathcal{G}$ satisfies the uniform entropy condition, i.e., \n$$\\exists \\, C_0>0, \\nu_0>0\\ :\\ N\\left(\\epsilon,\\mathcal{G}\\right)\\leq C_0\\epsilon^{-\\nu_0}.$$\n\\item[(F.ii)]\\ \\ \\ $\\mathcal{G}$ is a pointwise measurable class, i.e there exists a countable sub-class $\\mathcal{G}_0$ of $\\mathcal{G}$ such that for all $g\\in \\mathcal{G}$, there exits $\\left(g_m\\right)_m\\subset \\mathcal{G}_0$ such that $g_m\\longrightarrow g.$\\\\\n\\end{itemize}\n\nThe checking of these conditions will be done in Appendix and constitutes the proof of the following proposition.\n\\begin{proposition}\\label{p1}\nSuppose that the copula function $C$ has bounded first order partial derivatives on $(0, 1)^2$ and the transformation $\\phi$ admits a bounded derivative $\\phi'$. Then assuming (G.i), (G.ii), (F.i) and (F.ii), we have for some $c > 0,\\ 0 < h_0 < 1,$ with probability one, \n$$\n\\limsup_{n\\rightarrow\\infty}\\sup_{\\frac{c\\log n}{n}\\leq h \\leq h_0}\\sup_{(u,v)\\in(0,1)^2}\n\\frac{|\\sqrt{n}\\hat{D}_{n,h}^{(\\cdot)}(u,v)-\\tilde{\\mathbb{C}}_n(u,v)|}{\\sqrt{h(|\\log h\\vert\\vee\\log\\log n)}}=A(c),\n$$\nwhere $A(c)$ is a positive constant.\n\\end{proposition}\n\n\\begin{corollary}\\label{crl1}\n Under the assumptions of Proposition \\ref{p1}, one has for any sequence of constants $0<b_n<1,$ satisfying $\\ b_n\\rightarrow 0,\\ b_n\\geq (\\log n)^{-1}$, with probability one, \n $$\n \\sup_{\\frac{c\\log n}{n}\\leq h \\leq b_n}\\sup_{(u,v)\\in(0,1)^2}\n\\frac{|\\sqrt{n}\\hat{D}_{n,h}^{(\\cdot)}(u,v)-\\tilde{\\mathbb{C}}_n(u,v)|}{\\sqrt{\\log\\log n}}=O(\\sqrt{b_n}).\n $$\n \\end{corollary}\n\\begin{proof}{( \\textbf{Corollary \\ref{crl1})}}\\\\\n{\\rm First, observe that the condition $b_n\\geq (\\log n)^{-1}$ implies\n\\begin{equation}\\label{dc}\n\\frac{\\vert \\log b_n\\vert}{\\log\\log n}\\leq 1.\n\\end{equation}\nNext, by the monotonicity of the function $x\\mapsto x\\vert\\log x \\vert$ on $[0,1\/e]$, one can write for $n$ large enough, $h\\vert\\log h \\vert\\leq b_n\\vert\\log b_n \\vert$ and hence,\n\\begin{equation}\nh(\\vert\\log h \\vert  \\vee\\log\\log n)\\leq b_n(\\vert\\log b_n \\vert \\vee\\log\\log n).\n\\end{equation}\nCombining this and Proposition \\ref{p1}, we obtain\n$$\n \\sup_{\\frac{c\\log n}{n}\\leq h \\leq b_n}\\sup_{(u,v)\\in(0,1)^2}\n\\frac{|\\sqrt{n}\\hat{D}_{n,h}(u,v)-\\tilde{\\mathbb{C}}_n(u,v)|}{\\sqrt{b_n\\log\\log n\\left(\\frac{\\vert \\log b_n\\vert}{\\log\\log n}\\vee 1\\right)}}=O(1).\n $$\nThus the Corollary \\ref{crl1} follows from  \\eqref{dc}.}\n\\end{proof}\nComing back to the proof of our theorems, we have to show that\n\\begin{equation}\\label{lil}\n\\limsup_{n\\rightarrow\\infty}\\sup_{\\frac{c\\log n}{n}\\leq h \\leq b_n}\\sup_{(u,v)\\in[0,1]^2}\\frac{\\left|\\sqrt{n}\\hat{D}_{n,h}^{(\\cdot)}(u,v)\\right|}{\\sqrt{2\\log\\log n}}\\leq 3.\n\\end{equation}\nTowards this end, we make use of an approximation of the empirical copula process $\\mathbb{C}_n$ by a Kiefer process (see e.g., Zari\\cite{r11}, page 100). Let $\\mathbb{W}(u,v,t)$ be a $3$-parameters  Wiener process defined on $[0,1]^2\\times[0,\\infty)$. Then the Gaussian process $\\mathbb{K}(u,v,t)=\\mathbb{W}(u,v,t)-\\mathbb{W}(1,1,t).uv$ is called a $3$-parameters  Kiefer process defined on $[0,1]^2\\times[0,\\infty)$.\\\\\n\\indent By Theorem 3.2 in Zari\\cite{r11}, for $d=2$, there exists a sequence of Gaussian processes $\\left\\{\\mathbb{K}_{C}(u,v,n), u,v\\in[0,1], n>0\\right\\}$ such that\n$$ \\sup_{(u,v)\\in[0,1]^2}\\left|\\sqrt{n}\\mathbb{C}_n(u,v)-\\mathbb{K}_{C}^\\ast(u,v,n)\\right|=O\\left(n^{3\/8}(\\log n)^{3\/2}\\right),$$ where \n$$\\mathbb{K}_{C}^\\ast(u,v,n)=\\mathbb{K}_{C}(u,v,n)-\\mathbb{K}_{C}(u,1,n)\\frac{\\partial C(u,v)}{\\partial u}-\\mathbb{K}_\\mathbb{C}(1,v,n)\\frac{\\partial C(u,v)}{\\partial v}.$$\nThis yields\n\\begin{equation}\\label{c1}\n\\limsup_{n\\rightarrow\\infty}\\sup_{(u,v)\\in[0,1]^2}\\frac{\\left|\\mathbb{C}_n(u,v)\\right|}{\\sqrt{2\\log\\log n}}=\\limsup_{n\\rightarrow\\infty}\\sup_{(u,v)\\in[0,1]^2}\\frac{\\left|\\mathbb{K}_{C}^\\ast(u,v,n)\\right|}{\\sqrt{2n\\log\\log n}}.\n\\end{equation}\nBy the works of Wichura\\cite{r10} on the  law of the iterated logarithm , for $d=2$, one has almost surely\n\\begin{equation}\\label{c2}\n\\limsup_{n\\rightarrow\\infty}\\sup_{(u,v)\\in[0,1]^2}\\frac{\\left|\\mathbb{K}_\\mathbb{C}^\\ast(u,v,n)\\right|}{\\sqrt{2n\\log\\log n}}\\leq 3,\n\\end{equation}\n\n\\noindent which entails  \n$$\\limsup_{n\\rightarrow\\infty}\\sup_{(u,v)\\in[0,1]^2}\\frac{\\left|\\mathbb{C}_n(u,v)\\right|}{\\sqrt{2\\log\\log n}}\\leq 3.$$\nSince ${\\mathbb{C}}_n(u,v)$ and $\\tilde{\\mathbb{C}}_n(u,v)$ are asymptotically equivalent in view of (\\ref{c3}), one obtains\n$$\\limsup_{n\\rightarrow\\infty}\\sup_{(u,v)\\in[0,1]^2}\\frac{\\left|\\tilde{\\mathbb{C}}_n(u,v)\\right|}{\\sqrt{2\\log\\log n}}\\leq 3.$$\nApplying Corollary \\ref{crl1} and the fact that $\\sqrt{b_n}\\rightarrow 0$, we obtain \\eqref{lil} which proves the theorems.\n\n\\section*{Appendix}\n\\begin{proof}(\\textbf{Proposition \\ref{p1}})\\\\\nAssume that the function $K(\\cdot,\\cdot)$ is the integral of a symmetric bounded kernel $k(\\cdot,\\cdot)$, supported on $[-1,1]^2$, i.e. $K(x,y)=\\int_0^x\\int_0^y k(s,t)dsdt$. We have to check (G.i), (G.ii), (F.i) and (F.ii).\\\\\n\n\\noindent \\textbf{Checking for (G.i):}\nRecall that $(U_i,V_i), i\\geq 1$ are iid random variables uniformly distributed on $[0,1]^2$, $\\zeta_{1,n}(U_i)=\\hat{F}_n o F^{-1}(X_i)$ and $\\zeta_{2,n}(U_i)=\\hat{G}_n o G^{-1}(Y_i)$.\nFor any function $g\\in\\mathcal{G}$ and $0<h<1$, we can write \n\\begin{eqnarray*}\ng\\left(U_i,V_i,h\\right) & =&K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(U_i))}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(V_i))}{h}\\right)- \\mathbb{I}\\{U_i\\leq u,V_i\\leq v\\}\\\\\n& =&\\int_{-\\infty}^{\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(U_i))}{h}} \\int_{-\\infty}^{\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(V_i))}{h}}k(s,t)dsdt-\\mathbb{I}\\{U_i\\leq u,V_i\\leq v\\}\\\\\n& =&\\int_{-1}^{1}\\int_{-1}^{1}\\mathbb{I}\\left\\{U_i\\leq \\zeta_{1,n}^{-1}\\circ\\phi(\\phi^{-1}(u)-th) ,V_i\\leq \\zeta_{2,n}^{-1}\\circ\\phi(\\phi^{-1}(v)-sh)\\right\\}k(s,t)dsdt \\\\\n & & -\\mathbb{I}\\{U_i\\leq u,V_i\\leq v\\}\\\\\n&\\leq &\\int_{-1}^{1}\\int_{-1}^{1}k(t)k(s,t)dsdt-\\mathbb{I}\\{U_i\\leq u,V_i\\leq v\\}\\ \\leq 4 \\|k\\|^2+1,\\\\\n\\end{eqnarray*}\nwhere $\\displaystyle \\|k\\|=\\sup_{(s,t)\\in[-1,1]^2}|k(s,t)|$ represents the supremum norm on $[-1,1]^2$. Thus (G.i) holds by taking $\\kappa:=4\\|k\\|^2+1.$\\\\\n\n\\noindent \\textbf{Checking for (G.ii).} We have to show that $\\displaystyle \\sup_{g\\in \\mathcal{G}}\\mathbb{E}g^2(U,V,h)\\leq C_0 h$, where $C_0$ is a positive  constant. One can write\n\\begin{eqnarray*}\\small\n&\\mathbb{E}g^2(U,V,h) =  \\mathbb{E}\\left[K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(U))}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(V))}{h}\\right)- \\mathbb{I}\\{U\\leq u,V\\leq v\\}\\right]^2 &\\\\\n&=  \\mathbb{E}\\left[K^2\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(U))}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(V))}{h}\\right)\\right] &\\\\\n&  - 2\\mathbb{E}\\left[K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(U))}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(V))}{h}\\right)\\mathbb{I}\\{U\\leq u,V\\leq v\\}\\right] + C(u,v)&\\\\\n& =:  A -2B + C(u,v)&.\n\\end{eqnarray*}\n Since the function $K(\\cdot,\\cdot)$ is a kernel of a distribution function, we may assume without loss of generality that it takes its values in $[0,1]$. Then, we can use the inequality $K^2(x,y)\\leq K(x,y)$ to bound up the term $A$ in the right hand side of the previous egality.\n\\begin{eqnarray*}\n&A  =   \\mathbb{E}\\left[K^2\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(U))}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(V))}{h}\\right)\\right]&\\\\\n&\\leq  \\mathbb{E}\\left[K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(U))}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(V))}{h}\\right)\\right]&\\\\\n& \\leq \\mathbb{E}\\left[ \\int_{-1}^{1}\\int_{-1}^{1}\\mathbb{I}\\left\\{U\\leq \\zeta_{1,n}^{-1}\\circ\\phi(\\phi^{-1}(u)-sh),V\\leq \\zeta_{2,n}^{-1}\\circ\\phi(\\phi^{-1}(v)-th)\\right\\}k(s,t)dsdt \\right].&\n\\end{eqnarray*}\nThe other term $B$ is written into\n\\begin{eqnarray*}\n&B  = \\mathbb{E}\\left[K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(U))}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(V))}{h}\\right)\\mathbb{I}\\{U\\leq u,V\\leq v\\}\\right]&\\\\\n&= \\mathbb{E}\\left[ \\int_{-1}^{1}\\int_{-1}^{1}\\mathbb{I}\\left\\{s\\leq \\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(U))}{h},t\\leq \\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(V))}{h}\\right\\} \\mathbb{I}\\{U\\leq u,V\\leq v\\}k(s,t)dsdt\\right]&\\\\\n&= \\mathbb{E}\\left[ \\int_{-1}^{1}\\int_{-1}^{1}\\mathbb{I}\\left\\{U\\leq u\\wedge \\zeta_1^{-1}\\circ\\phi(\\phi^{-1}(u)-sh),V\\leq v\\wedge \\zeta_2^{-1}\\circ\\phi(\\phi^{-1}(v)-th)\\right\\}k(s,t)dsdt \\right],&\n\\end{eqnarray*} where $x\\wedge y=\\min (x,y)$.\nNote that $$ C(u,v)=\\int_{-1}^{1}\\int_{-1}^{1}C(u,v)k(s,t)dsdt,$$\nas the kernel $k(\\cdot,\\cdot)$ satisfies  $\\int_{-1}^{1}\\int_{-1}^{1}k(s,t)dsdt=1$. Thus\n\\begin{eqnarray*}\n&\\mathbb{E}g^2(U,V,h)\\leq \\mathbb{E}\\left[ \\int_{-1}^{1}\\int_{-1}^{1}\\mathbb{I}\\left\\{U\\leq \\zeta_{1,n}^{-1}\\circ\\phi(\\phi^{-1}(u)-sh),V\\leq \\zeta_{2,n}^{-1}\\circ\\phi(\\phi^{-1}(v)-th)\\right\\}k(s,t)dsdt \\right]&\\\\\n &  -2\\mathbb{E}\\bigg[ \\int_{-1}^{1}\\int_{-1}^{1}\\mathbb{I}\\left\\{U\\leq u\\wedge \\zeta_{1,n}^{-1}\\circ\\phi(\\phi^{-1}(u)-sh),V\\leq v\\wedge \\zeta_{2,n}^{-1}\\circ\\phi(\\phi^{-1}(v)-th)\\right\\}k(s,t)dsdt \\bigg]& \\\\\n & + \\int_{-1}^{1}\\int_{-1}^{1}C(u,v)k(s,t)dsdt.&\n\n\\end{eqnarray*}\nWe shall suppose that the empirical kernel distributions $\\hat{F}_n$ and $\\hat{G}_n$ are asymptotically equivalent the classical empirical distribution function $F_n$ $G_n$. From the Chung's (1949) LIL, we can infer that for all $u\\in [0,1]$, as $n\\rightarrow\\infty$,\n$$\\zeta_{1,n}^{-1}(u)-u = F\\circ\\hat{F}_n^{-1}(u)-F\\circ F^{-1}(u)=O(n^{-1}\\log\\log n).$$\nThat is $\\zeta_{1,n}^{-1}(u)$ is asymptotically equivalent to $u$. Same for $\\zeta_{2,n}^{-1}(u)=G\\circ\\hat{G}_n^{-1}(u)$. Thus, for all large $n$, we can write\n\\begin{eqnarray*}\n&\\mathbb{E}g^2(U,V,h)\\leq \\mathbb{E}\\left[ \\int_{-1}^{1}\\int_{-1}^{1}\\mathbb{I}\\left\\{U\\leq \\phi(\\phi^{-1}(u)-sh),V\\leq \\phi(\\phi^{-1}(v)-th)\\right\\}k(s,t)dsdt \\right]&\\\\\n &  -2\\mathbb{E}\\left[ \\int_{-1}^{1}\\int_{-1}^{1}\\mathbb{I}\\left\\{U\\leq u\\wedge \\phi(\\phi^{-1}(u)-sh),V\\leq v\\wedge \\phi(\\phi^{-1}(v)-th)\\right\\}k(s,t)dsdt \\right]&\\\\\n &  +\\int_{-1}^{1}\\int_{-1}^{1}C(u,v)k(s,t)dsdt.&\n\\end{eqnarray*}\nThat is,\n\\begin{eqnarray}\\label{eqcas}\n\\mathbb{E}g^2(U,V,h)&\\leq &\\int_{-1}^{1}\\int_{-1}^{1}C\\left(\\phi(\\phi^{-1}(u)-sh),\\phi(\\phi^{-1}(v)-th)\\right)k(s,t)dsdt \\nonumber\\\\\n & & -2\\int_{-1}^{1}\\int_{-1}^{1}C\\left(u\\wedge\\phi(\\phi^{-1}(u)-sh),v\\wedge \\phi(\\phi^{-1}(v)-th)\\right)k(s,t)dsdt \\\\\n &  & +\\int_{-1}^{1}\\int_{-1}^{1}C(u,v)k(s,t)dsdt.\\nonumber\n\\end{eqnarray}\nNow, we have to discuss condition (G.ii) in the four following cases:\\\\\n\n\\textbf{Case 1.} $u\\wedge\\phi(\\phi^{-1}(u)-sh)=\\phi(\\phi^{-1}(u)-sh)\\ \\text{et}\\ v\\wedge \\phi(\\phi^{-1}(v)-th)=\\phi(\\phi^{-1}(v)-th)$.\\\\\nIn this case the second member of inequality \\eqref{eqcas} is reduced and we have \n\\begin{eqnarray*}\n\\mathbb{E}g^2(U,V,h)&\\leq &-\\int_{-1}^{1}\\int_{-1}^{1}C\\left(\\phi(\\phi^{-1}(u)-sh),\\phi(\\phi^{-1}(v)-th)\\right)k(s,t)dsdt \\\\\n  &  & +\\int_{-1}^{1}\\int_{-1}^{1}C(u,v)k(s,t)dsdt\\\\\n\t&\\leq &\\int_{-1}^{1}\\int_{-1}^{1}\\left[C(u,v)-C\\left(\\phi(\\phi^{-1}(u)-sh),\\phi(\\phi^{-1}(v)-th)\\right)\\right]k(s,t)dsdt.\n\\end{eqnarray*}\nBy a Taylor expansion for the copula function $C$, we have\n\\begin{eqnarray*}\nC(u,v)-C(\\phi(\\phi^{-1}(u)-sh),\\phi(\\phi^{-1}(v)-th))&=&[u-\\phi(\\phi^{-1}(u)-sh)]C_u(u,v)\\\\\n &+ & [v-\\phi(\\phi^{-1}(v)-th)]C_v(u,v)+ o(h). \n\\end{eqnarray*}\nApplying again a Taylor-Young expansion for the function $\\phi$, we obtain\n$$ u-\\phi(\\phi^{-1}(u)-sh)=\\phi(\\phi^{-1}(u)) -\\phi(\\phi^{-1}(u)-sh)=\\phi'(u)sh+o(h)$$\nand\n$$ v-\\phi(\\phi^{-1}(v)-sh)=\\phi(\\phi^{-1}(v)) -\\phi(\\phi^{-1}(v)-th)=\\phi'(u)th+o(h).$$\nThus\n\\begin{eqnarray*}\n\\mathbb{E}g^2(U,V,h)&\\leq &\\int_{-1}^{1}\\int_{-1}^{1}\\left[\\phi'(u)C_u(u,v)h+\\phi'(v)C_v(u,v)h\\right]k(s,t)dsdt \\\\\n\t&\\leq & 4h\\left[\\|C_u\\|+\\|C_v\\|\\right]\\sup_{x\\in[0,1]}|\\phi'(x)|\\|k\\|.\n\\end{eqnarray*}\nTaking $C_0=4\\left[\\left\\|C_u\\right\\|+\\left\\|C_v\\right\\|\\right]\\left\\|\\phi'\\right\\|\\left\\|k\\right\\|$ gives condition (G.ii).\\\\\n\n\\textbf{Case 2.} $u\\wedge\\phi(\\phi^{-1}(u)-sh)=u\\ \\text{et}\\ v\\wedge \\phi(\\phi^{-1}(v)-th)=v$. \\\\\nHere the inequality \\eqref{eqcas} is reduced to\n\\begin{eqnarray*}\n\\mathbb{E}g^2(U,V,h)\\leq \\int_{-1}^{1}\\int_{-1}^{1}\\left[C\\left(\\phi(\\phi^{-1}(u)-sh),\\phi(\\phi^{-1}(v)-th)\\right)-C(u,v)\\right]k(s,t)dsdt.\n\\end{eqnarray*}\nUsing the same arguments as in Case 1, we obtain condition (G.ii):\n$\\displaystyle \\sup_{g\\in \\mathcal{G}}\\mathbb{E}g^2(U,V,h)\\leq C_0h$, with $C_0=4\\left[\\left\\|C_u\\right\\|+\\left\\|C_v\\right\\|\\right]\\left\\|\\phi'\\right\\|\\left\\|k\\right\\|$.\\\\\n\n\\textbf{Case 3.} $u\\wedge\\phi(\\phi^{-1}(u)-sh)=\\phi(\\phi^{-1}(u)-sh)\\ \\text{et}\\ v\\wedge \\phi(\\phi^{-1}(v)-th)=v$.\\\\\nHere, inequality \\eqref{eqcas} is rewritten into\n\\begin{eqnarray*}\n&\\mathbb{E}g^2(U,V,h)\\leq \\int_{-1}^{1}\\int_{-1}^{1}C\\left(\\phi(\\phi^{-1}(u)-sh),\\phi(\\phi^{-1}(v)-th)\\right)k(s,t)dsdt & \\\\\n & -2\\int_{-1}^{1}\\int_{-1}^{1}C\\left(\\phi(\\phi^{-1}(u)-sh),v\\right)k(s,t)dsdt  +\\int_{-1}^{1}\\int_{-1}^{1}C(u,v)k(s,t)dsdt &\\\\\n&\\leq \\int_{-1}^{1}\\int_{-1}^{1}\\left[C\\left(\\phi(\\phi^{-1}(u)-sh),\\phi(\\phi^{-1}(v)-th)\\right)-C\\left(\\phi(\\phi^{-1}(u)-sh),v\\right)\\right]k(s,t)dsdt &\\\\\n& -\\int_{-1}^{1}\\int_{-1}^{1}\\left[C\\left(\\phi(\\phi^{-1}(u)-sh),v\\right)-C(u,v)\\right]k(s,t)dsdt. &\n\\end{eqnarray*}\nBy applying successively a Taylor expansion for $C$ and for $\\phi$, we get\n\\begin{eqnarray*}\n\\mathbb{E}g^2(U,V,h)&\\leq &\\int_{-1}^{1}\\int_{-1}^{1}C_v\\left(\\phi(\\phi^{-1}(u)-sh),\\theta_1\\right)\\left[\\phi(\\phi^{-1}(v)-th)-\\phi(\\phi^{-1}(v))\\right]k(s,t)dsdt\\\\\n& & -\\int_{-1}^{1}\\int_{-1}^{1}C_u\\left(\\theta_2,v\\right)\\left[\\phi(\\phi^{-1}(u)-sh)-\\phi(\\phi^{-1}(u))\\right]k(s,t)dsdt\\\\\n&\\leq & \\int_{-1}^{1}\\int_{-1}^{1}C_v\\left(\\phi(\\phi^{-1}(u)-sh),\\theta_1\\right)\\phi'(\\gamma_1).(-th)k(s,t)dsdt\\\\\n& & -\\int_{-1}^{1}\\int_{-1}^{1}C_u\\left(\\theta_2,v\\right)\\phi'(\\gamma_2).(-sh)k(s,t)dsdt,\n\\end{eqnarray*}\nwhere $\\theta_1\\in \\left(\\phi(\\phi^{-1}(v)-th),v\\right)\\ ;\\ \\theta_2\\in \\left(\\phi(\\phi^{-1}(u)-sh),u\\right)\\ ;\\ \\gamma_1\\in\\left(\\phi^{-1}(v)-th,\\phi^{-1}(v)\\right)\\ ;\\ \\gamma_2\\in\\left(\\phi^{-1}(u)-sh,\\phi^{-1}(u)\\right).$\\\\\nThis implies \n\\begin{eqnarray*}\n\\mathbb{E}g^2(U,V,h)&\\leq &4h\\left\\|C_v\\right\\|\\left\\|\\phi'\\right\\|\\left\\|k\\right\\|^2\\left|t\\right|+4h\\left\\|C_u\\right\\|\\left\\|\\phi'\\right\\|\\left\\|k\\right\\|\\left|s\\right|\\\\\n&\\leq & 4h\\left\\|\\phi'\\right\\|\\left\\|k\\right\\|\\left(\\left\\|C_v\\right\\|+\\left\\|C_u\\right\\|\\right).\n\\end{eqnarray*}\nThus condition (G.ii) holds, with $C_0=4\\left\\|\\phi'\\right\\|\\left\\|k\\right\\|\\left(\\left\\|C_v\\right\\|+\\left\\|C_u\\right\\|\\right).$\\\\\n\n\\textbf{Case 4.} $u\\wedge\\phi(\\phi^{-1}(u)-sh)=u\\ \\text{et}\\ v\\wedge \\phi(\\phi^{-1}(v)-th)=\\phi(\\phi^{-1}(v)-th)$.\\\\ \nThis case is analogous to Case 3, where the roles of $u$ and $v$ are interchanged. Hence, condition (G.ii) is fulfilled, with the same constant  $C_0=4\\left\\|\\phi'\\right\\|\\left\\|k\\right\\|\\left(\\left\\|C_v\\right\\|+\\left\\|C_u\\right\\|\\right).$\\\\\n\n\\textbf{Checking for (F.i)}. We have to check the uniform entropy condition for the class of functions\n$$\n\\mathcal{G}=\\left\\{\\begin{array}{c} \nK\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(s))}{h}, \\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(t))}{h}\\right)- \\mathbb{I}\\{s\\leq u,t\\leq v\\},\\\\\n u,v\\in[0,1], 0<h<1 \\,\\text{and }\\, \\zeta_{1,n}\\zeta_{2,n}:[0,1]\\mapsto [0,1] \\,\\text{nondecreasing.}\n\\end{array} \\right\\}\n$$\n To this end, we consider the following classes of functions, where $\\varphi$ is an increasing function :\\\\\n\\noindent\n$ \\mathbb{F}=\\left\\{(\\varphi(x)+m)\/\\lambda,\\lambda > 0,\\;m\\in\\mathbb{R}\\right\\}$\\\\\n$\\displaystyle \\mathbb{K}_0=\\left\\{K((\\varphi(x)+m)\/\\lambda),\\lambda>0, \\;m\\in\\mathbb{R}\\right\\}$\\\\\n$\\displaystyle \\mathbb{K}=\\left\\{K((\\phi(x)+m)\/\\lambda,(\\phi(y)+m)\/\\lambda), \\lambda>0,\\;m\\in\\mathbb{R}\\right\\}$\\\\\n$\\displaystyle \\mathbb{H}=\\left\\{K((\\phi(x)+m)\/\\lambda,(\\phi(y)+m)\/\\lambda)-\\mathbb{I}\\left\\{x\\leq u,y\\leq v\\right\\}\\ ;\\lambda> 0, \\;m\\in\\mathbb{R}, (u,v)\\in [0,1]^2\\right\\}$.\\\\\n\nIt is clear that by applying  lemmas 2.6.15 and 2.6.18 in van der Vaart and Wellner (see \\cite{r9}, p. 146-147), the sets $\\mathbb{F},\\;\\mathbb{K}_0,\\;\\mathbb{K},\\;\\mathbb{H}$ are all VC-subgraph classes. Thus, by choosing  the constant function ${\\rm G}(x,y)\\mapsto {\\rm G}(x,y)=\\left\\|k\\right\\|^2+1 $ as an envelope function for the class $\\mathbb{H}$ ( indeed ${\\rm G}(x,y)\\geq \\sup_{g\\in\\mathbb{H}}\\left|g(x,y)\\right|,\\ \\forall (x,y))$, we can infer from Theorem 2.6.7 in \\cite{r9} that  $\\mathbb{H}$ satisfies the uniform entropy condition. Since $\\mathbb{H}$ and $\\mathcal{G}$ have the same structure, we can conclude that $\\mathcal{G}$ satisfies this property too, i.e.\n$$\\exists \\; C_0>0, \\nu_0>0\\ :\\ N\\left(\\epsilon,\\mathcal{G}\\right)\\leq C_0\\epsilon^{-\\nu_0},\\quad 0<\\epsilon<1.$$\n\n\\noindent \\textbf{Checking for (F.ii).}\\\\\n Define the class of functions \n$$\n\\mathcal{G}_0=\\left\\{\\begin{array}{c} \nK\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1}(s))}{h}, \\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2}(t))}{h}\\right)- \\mathbb{I}\\{s\\leq u,t\\leq v\\},\\\\\n u,v\\in[0,1]\\cap\\mathbb{Q}, 0<h<1 \\,\\text{and }\\, \\zeta_{1}\\zeta_{2}:[0,1]\\mapsto [0,1] \\,\\text{nondecreasing.}\n\\end{array} \\right\\}\n$$\nIt's clear that  $\\mathcal{G}_0$ is countable and  $\\mathcal{G}_0\\subset\\mathcal{G}$. Let\n$$g(s,t)=K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_1(s))}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_2(t))}{h}\\right)- \\mathbb{I}\\{s\\leq u,s\\leq v\\}\\in\\mathcal{G}, (s,t)\\in [0,1]^2$$\nand for $m\\geq 1$, \n$$g_m(s,t)=K\\left(\\frac{\\phi^{-1}(u_m)-\\phi^{-1}(\\zeta_1(s))}{h} ,\\frac{\\phi^{-1}(v_m)-\\phi^{-1}(\\zeta_2(y))}{h}\\right)- \\mathbb{I}\\{s\\leq u_m,t\\leq v_m\\},$$\nwhere $ u_m=\\frac{1}{m^2}[m^2u]+\\frac{1}{m^2}$ and $ v_m=\\frac{1}{m^2}[m^2v]+\\frac{1}{m^2}$.\\\\\n\\indent Let $\\alpha_m=u_m-u,\\quad \\beta_m=v_m-v$. Then, we have  $\\displaystyle 0<\\alpha_m\\leq\\frac{1}{m^2}$ and $\\displaystyle 0<\\beta_m\\leq\\frac{1}{m^2}$. Hence $u_m\\searrow u$ and $v_m\\searrow v$. By continuity $\\phi^{-1}(u_m)\\searrow \\phi^{-1}(u)$ and $\\phi^{-1}(v_m)\\searrow \\phi^{-1}(v)$. Define\n$$\\delta_{m,u}=\\left(\\frac{\\phi^{-1}(u_m)-\\phi^{-1}(\\zeta_1(x))}{h} \\right)-\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_1(x))}{h} \\right)=\\frac{\\phi^{-1}(u_m)-\\phi^{-1}(u)}{h} $$ and\n$$\\delta_{m,v}=\\left(\\frac{\\phi^{-1}(v_m)-\\phi^{-1}(\\zeta_2(y))}{h} \\right)-\\left(\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_2(y))}{h} \\right)=\\frac{\\phi^{-1}(v_m)-\\phi^{-1}(v)}{h}.$$\nThen\n$\\delta_{m,u}\\searrow 0$ and $\\delta_{m,v}\\searrow 0$, \nwhich are equivalent to $$ \\left(\\frac{\\phi^{-1}(u_m)-\\phi^{-1}(\\zeta_1(x))}{h} \\right)\\searrow \\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_1(x))}{h} \\right)$$\n and $$\\left(\\frac{\\phi^{-1}(v_m)-\\phi^{-1}(\\zeta_2(y))}{h} \\right)\\searrow \\left(\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_2(y))}{h} \\right).$$\nBy right-continuity of the kernel $K(\\cdot,\\cdot)$, we obtain\n$$\\forall (x,y)\\in[0,1]^2, g_m(x,y)\\longrightarrow g(x,y), m\\rightarrow \\infty$$\nand conclude that $\\mathcal{G}$ is pointwise measurable class.\\\\\n\\end{proof}\n\n\\addcontentsline{toc}{section}{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{intro}Introduction}\nQuantum information science is growing rapidly not only in theory but also at the hardware level; this leads us to believe that quantum computers will soon be practically applied on a large scale. Once quantum computers are realized on a commercial scale, quantum technology (such as devices or communications) will be widespread. In such a future, an interesting question that arises is how quantum computers can change the world. For example, in quantum game theory, Meyer \\cite{PhysRevLett.82.1052} showed that for a two-player zero-sum strategic game, a player's expectation can increase when using quantum strategies. After Meyer's work, Eisert, Wilkens, and Lewenstein (EWL) \\cite{PhysRevLett.83.3077} showed that quantum entanglement changes the Nash equilibrium \\cite{Nash48} between the strategies of two persons in Prisoners' Dilemma (PD). The strategy of whether to cooperate or defect adopted by one person affects another person's strategy through quantum entanglement between the two individual strategies. Prisoners can choose quantum strategies, which are described by unitary operators; therefore, they have more strategies at hand compared to the classical game and can use them to increase their profit. There are many studies inspired by their quantization scheme (e.g. summarized in \\cite{doi:10.1142\/S0219477502000981,Khan2018,GUO2008318}), and not a few quantum games induce nontrivial results that are not expected classically.\n\nThese ideas of quantizing the strategies can also be applied to gambling theory. A fair quantum gambling game played by two players far from each other was proposed by \\cite{PhysRevLett.82.3356}. In the classical case, it is difficult to ensure fair play without introducing a third party to oversee the game. However, by providing a quantum superposition state and then observing it, it is possible to achieve fair gambling. In addition, procedures were devised to protect the security of such remote gambling from entanglement attacks \\cite{PhysRevA.64.064302,PhysRevA.66.052311}. Zhang {\\it et al.} devised and realized an optical quantum circuit that helps achieve fair gambling and experimentally showed that the error is practically under control \\cite{Zhang_2008}. In the sense of quantizing a game, some quantized games have been proposed \\cite{Schmidt_2012,SCHMIDT2013400,FRACKIEWICZ20148,qchess,Fuchs_2020,bleiler2009quantized}. Quantum duel was proposed \\cite{Schmidt_2012}, and a Russian roulette game using a quantum gun was proposed and studied \\cite{SCHMIDT2013400,FRACKIEWICZ20148}. Quantum chess was devised as a more casual game \\cite{qchess}. Quantized strategies in poker, one of the most popular card games, were also considered in \\cite{bleiler2009quantized} using Meyer's quantized procedure \\cite{PhysRevLett.82.1052}. Recently, a game similar to poker was presented on a quantum computer by \\cite{Fuchs_2020}. In their experiment, the researchers investigated the expected value under quantum noise on a practical quantum computer. They also showed that the expected value can be improved using quantum error-mitigation techniques. These studies suggest that quite a few games based on quantum mechanics have been examined.\n\nRecent studies have shown that quantum bits yield advantages to blackjack, which is one of the most popular casino games \\cite{PhysRevA.102.012425}; the researchers showed that the expectation values of two cooperating persons can increase by imparting information of each individual's card with entangled qubits. In the present paper, strategies and cards are represented as qubits and we devised a quantum circuit for reproducing classical blackjack. Classical blackjack has been studied statistically for more than half a century \\cite{doi:10.1080\/01621459.1956.10501334,BOND1974413,Thorp1966BeatTD}, and it is decided whether the player should do {\\it hit} or {\\it stand}. This is called {\\it Basic Strategy}. Moreover, it is widely known in various rules, such as number of decks, options, and refunds when the player has blackjack \\cite{Thorp1966BeatTD,wizard}. In blackjack, as the player can choose strategies (e.g., hit or stand), we can quantize not only the game itself but also the strategies involved using a method similar to \\cite{PhysRevLett.82.1052,PhysRevLett.83.3077}. This quantum circuit reproducing blackjack can be easily realized in the near future when quantum computers are commonplace. Furthermore, we showed that the player's expectation value increases substantially when quantum strategies are employed in the presence of entanglement.\n\nIn the next section \\ref{rule}, we explain the blackjack rule that we adopted (simplified for ease of calculation). In section \\ref{cbj}, we reformulate the blackjack in order to regard it as a object of the quantum game theory. In section \\ref{qbj}, we propose a quantum circuit that reproduces blackjack, (especially a toy model,) and express the possibility of entanglement. In section \\ref{res}, we show classical expectation value and basic strategy; we also show that using quantum strategies in the presence of entanglement, the player's expectation value increases compared to the classical game. Some conclusions are shown in section \\ref{conclusion}.\n\n\\section{\\label{rule}Blackjack rules and toy model}\nBlackjack is a one-on-one card game between a dealer (Alice) and a player (Bob). One deck comprises 52 cards excluding jokers. Aces are worth 1 (hard) or 11 (soft), face cards (JQK) are worth 10, and every other card has its number value.\nAfter the player has placed his bet, the dealer distributes two cards to the player. Similarly, she distributes two cards to herself. At this time the dealer has one card face-up and the other face-down. The objective is to get a hand with a total closer to 21. It is worth noting however, that the 21 hand which is made of one ace and one face card or 10 is the strongest (called natural). \n\nFirst, the player must choose to hit or stand. If he chooses to hit, he draws a card. If he exceeds 21 (bust), he loses his bet regardless of the dealer's hand. Next, the dealer opens the face-down card, and if her total is soft 16 or less, she must hit. She repeats this procedure until her total reaches soft 17 or more. If the dealer busts and the player does not, or if the player's hand is closer to 21 than the dealer's, the player wins the bet. If the player's total equals the dealer's total, the bet will be returned (push). In blackjack, the player can do his best by calculating the expected value of the profit that he will obtain in each initial state. This is widely known as a basic strategy and has been calculated in various rules \\cite{Thorp1966BeatTD,wizard}. Note that only the player can choose whether to hit or stand. Conversely, the dealer has no choice of a strategy since she just draws the card mechanically until her total reaches soft 17 or more. At first glance, the player who has many choices seems to have a competitive edge. However, the dealer has the advantage because the player loses if both hands exceed 21. The reason many casinos adopt option rules, such as double down, split, and surrender is to increase the player's expectation.\n\n\nFor simplicity, we quantized a toy model of blackjack proposed by Ethier \\cite{Ethier2010} and called {\\it snackjack} by Epstein \\cite{epstein}. As snackjack is highly simplified compared to blackjack which uses 52 cards, it is easy to calculate the basic strategy by hand. Recently, Ethier calculated the expectation and basic strategy under various options, with card counting, and with bet variation in this classical snackjack \\cite{ethier2019snackjack}. Snackjack uses eight cards AA223333, aiming toward a maximum value of 7, with an ace having a value of either 1 or 4, and 2 and 3 having their own values. In this paper, we ignored various options and simplified them for calculation. Tab. \\ref{tab1} shows the rules we followed.\n\n\\begin{table}[h]\n    \\centering\n    \\caption{Our rulesets}\n    \\begin{tabular}{|c||c|} \\hline\n     ace means & 1 or 4 \\\\ \\hline\n     aim to & 7 \\\\ \\hline\n     natural is & ace and 3 \\\\ \\hline\n     player chooses  & hit or stand (only one time) \\\\ \\hline\n     dealer stands  & soft 6 or more \\\\ \\hline\n     blackjack pays & 1 to 1  \\\\ \\hline\n    \\end{tabular}\n    \\label{tab1}\n\\end{table}\n\n\n\\section{\\label{cbj}Classical snackjack as a quantum game}\nReferring to \\cite{doi:10.1080\/09500340008232180,doi:10.1142\/S0219749904000092}, we formulate the blackjack played by a player and a dealer as a {\\it 2 player quantum game}.\nBlackjack is regarded as the game in which the pay-off is determined by the initial hand cards of the player and the dealer.\nMore precisely, blackjack is a mapping that assigns a {\\it 2 player quantum game} $\\Gamma_{p,d}$ to an element $(p,d)$ of $T_{1}\\times T_{2}$ where $T_{1}$ is a set of a pair of initial hand cards of the player, $T_{2}$ is a set of an up card of the dealer.\nThe quantum games $\\Gamma_{p,d}=(\\mathcal{H},\\rho,\\Sigma_1,\\Sigma_2,P_1,P_2)$ are specified by Hilbert space $\\mathcal{H}$ for the states, the initial state $\\rho\\in\\mathcal{H}$, the sets $\\Sigma_1$ and $\\Sigma_2$ of the strategies of the player and the dealer, and the player's utility $P_{1}$ and the dealer's utility $P_{2}$.\nEach player has a qubit and the state of the system is represented as the tensor product of their qubit.\nThe strategies $\\hat{S}_1(\\in \\Sigma_1)$ and $\\hat{S}_2(\\in \\Sigma_2)$ selected by players are regarded as $2\\times 2$ unitary operators acting on their own qubit.\nIn this formalism, classical blackjack is the quantum game specified by\n\\begin{gather}\n\t\\mathcal{H}=\\left\\{a_{00}\\ket{0}\\otimes\\ket{0}+a_{01}\\ket{0}\\otimes\\ket{1}+a_{10}\\ket{1}\\otimes\\ket{0}+a_{11}\\ket{1}\\otimes\\ket{1}|\\,a_{00},a_{01},a_{10},a_{11}\\in\\mathbb{C}\\right\\}, \\notag \\\\\n\t\\rho = \\ket{0}\\otimes\\ket{1}, ~~~\\Sigma_1 = \\{\\hat{I},\\hat{X}\\}, ~~~\\Sigma_2 = \\{\\hat{I}\\},\n\\end{gather}\nwhere $\\ket{0}$ ($\\ket{1}$) denotes stand (hit) and $\\hat{X}$ is the first Pauli matrix. The player has the first qubit and the dealer has the second. We will call each of them {\\it strategy bit}.\nNote that the hit $\\ket{1}$ for the dealer means that he draws a card if the score of his hand is less than 6.\nSince the utilities are determined by $(p,d)\\in T_{1}\\times T_{2}$ and player's strategy $\\hat{S}_{1}\\in\\Sigma_{1}$, we can write them as $P_1(p,d,\\hat{S}_1)$ and $P_2(p,d,\\hat{S}_1)$.\nConcretely,\n\t\\begin{align}\n\tP_1(p,d,\\hat{S}_1)\n\t=E_{\\mathrm{std}}\\bra{0}\\otimes\\bra{1}\\left(\\hat{S}_1\\otimes\\hat{I}\\right)\\rho+E_{\\mathrm{hit}}\\bra{1}\\otimes\\bra{1}\\left(\\hat{S}_1\\otimes\\hat{I}\\right)\\rho,\n\t\\label{utiladd1}\n\t\\end{align}\nwhere $E_{\\mathrm{std}}$ and $E_{\\mathrm{hit}}$ are pay-offs corresponding to stand and hit.\nIt is a point to notice that $E_{\\mathrm{std}}$ and $E_{\\mathrm{hit}}$ depend on the initial hand cards $(p,d)$.\nThe utilities do not have $\\hat{S}_{2}$ dependence since $\\Sigma_{2}=\\{\\hat{I}\\}$ in the classical blackjack. Especially in our snackjack (Tab. \\ref{tab1}), $P_2(p,d,\\hat{S}_1)=-P_1(p,d,\\hat{S}_1)$, thus it is a zero-sum game.\n\n\\section{\\label{qbj}Entanglement and quantum circuits}\n\\subsection{\\label{entangle}The case of the presence of entangle}\nIn this section, using EWL quantization protocol \\cite{PhysRevLett.83.3077}, we investigate the quantum blackjack including an entanglement between the player and the dealer. Eisert {\\it et al.} prepared the entangled player's strategies in 2 players symmetric game, especially in PD game (Fig. \\ref{add1}). They showed that the dilemma disappears by using restricted quantum strategies and an entanglement. After their work, in PD game, some studies about generalized strategy spaces \\cite{PhysRevLett.87.069801,PhysRevLett.88.137902,Du_2003,ICHIKAWA2007531,Ichikawa_2008} or repeated games \\cite{IQBAL2002541,Frackiewicz_2012,aoki2020repeated} have been worked. In generalized strategy spaces, the players are not able to escape from the dilemma \\cite{PhysRevLett.87.069801,Ichikawa_2008}, and van Enk {\\it et al.} \\cite{PhysRevA.66.024306} pointed out that EWL protocol does not preserve the non-cooperative aspects of PD. Our blackjack game, however, does not have any dilemmas due to the asymmetry of only the player seeking a win and trying to beat the dealer who has limited strategy space $\\Sigma_2=\\{\\hat{I}\\}$. Thus we straightforwardly adopt EWL protocol. In section \\ref{conclusion}, we will comment on the degree of freedom of the dealer's strategy in this quantized blackjack game.\n \\begin{figure}[t]\n     \\centering\n     \\includegraphics[scale=0.8]{circuit4.pdf}\n     \\caption{EWL quantization scheme}The gate $\\mathcal{J}$ entangles the two qubits and $\\mathcal{J}^\\dagger$ disentangles those. The players choose an unitary operator $U$ as a quantum strategy.\n     \\label{add1}\n \\end{figure}\n \nWe consider entanglement between the player's and dealer's strategies. When the player selects the strategy $\\hat{S_{p}}$, the state of the system just before the observation is\n\\begin{equation}\n    \\mathcal{J}^\\dagger (\\hat{S}_p \\otimes \\hat{I})\\mathcal{J} \\ket{0} \\otimes \\ket{1},\n\\end{equation}\nwhere $\\mathcal{J}$ is $4\\times4$ complex matrix acting on both qubits.\n\nClassically, all strategies the player can choose are $\\hat{X}$ or $\\hat{I}$. We assume \n\\begin{equation}\n\\mathcal{J} = \\mathrm{exp}(-i\\frac{\\gamma}{2} \\hat{X} \\otimes \\hat{U})\n\\end{equation}\nso that the game reproduces the classical game if the player selects the classical strategies.\nThe first part is the player's strategy sector and the second is the dealer's, and operator $\\hat{U}$ is unitary and hermitian. $\\gamma$ is the measure for the game's entanglement \\cite{PhysRevLett.83.3077} and we will call $\\gamma$ {\\it entangle intensity} below. When $\\hat{S}_1 = \\hat{X} \\mspace{5mu} \\mathrm{or} \\mspace{5mu} \\hat{I}$, it follows\n\\begin{equation}\n\\mathcal{J}^\\dagger ( \\hat{X} \\otimes \\hat{I} ) \\mathcal{J} = \\hat{X} \\otimes \\hat{I},\n\\end{equation}\n\\begin{equation}\n\\mathcal{J}^\\dagger ( \\hat{I} \\otimes \\hat{I} ) \\mathcal{J} = \\hat{I} \\otimes \\hat{I}.\n\\end{equation}\nIn this way, without breaking the classical game, we can insert the operator making entangle strategies.\n\nA unitary and hermitian operator is written by\n\\begin{equation}\n\\hat{U} = s_0 \\hat{I} + s_1 \\hat{X} + s_2 \\hat{Y} + s_3 \\hat{Z} \n\\label{Uop1}\n\\end{equation}\nwhere\n\\begin{equation}\n(s_0^2 + s_1^2 + s_2^2 + s_3^2)\\hat{I} +2s_0(s_1\\hat{X} + s_2\\hat{Y} + s_3\\hat{Z}) = \\hat{I} \\ \\  (s_\\mu \\in \\mathbb{R}).\n\\label{Uop2}\n\\end{equation}\nIn this $\\Sigma_2=\\{\\hat{I}\\}$ game, except for relative phases, $\\hat{I}$, $\\hat{Z}$ and $\\hat{X}$, $\\hat{Y}$ have the same effect on a strategy bit respectively, hence we can set $s_0 = s_2 = 0,s_1 = \\sin\\theta , s_3 = \\cos\\theta$ without losing generality. Here $\\theta$ is {\\it game parameter}.\n\nIn the case of general player's strategy $\\hat{S}_1$, it follows\n\\begin{equation}\n\\begin{split}\n\\left( \\mathcal{J^\\dagger}(\\hat{S_1} \\otimes \\hat{I}) \\mathcal{J} \\right) \\ket{0} \\otimes \\ket{1} = \\left\\{ \\cos^2{\\frac{\\gamma}{2}} \\hat{X}\\{\\hat{X},\\hat{S_1} \\} + (\\sin^2{\\frac{\\gamma}{2}} - \\cos^2{\\frac{\\gamma}{2}}) \\hat{X}\\hat{S_1}\\hat{X} \\right\\} \\ket{0} &\\otimes \\ket{1}  \\\\\n+ i\\cos{\\frac{\\gamma}{2}}\\sin{\\frac{\\gamma}{2}}\\sin{\\theta} [ \\hat{X} , \\hat{S_1} ] \\ket{0} &\\otimes \\ket{0} \\\\ -i\\cos{\\frac{\\gamma}{2}}\\sin{\\frac{\\gamma}{2}}\\cos{\\theta} [ \\hat{X} , \\hat{S_1} ] \\ket{0} &\\otimes \\ket{1}.\n\\end{split}\n\\end{equation}\nFrom here, we limit $\\Sigma_1$ to $\\{\\hat{I},\\hat{X},\\hat{Y},\\hat{Z}\\}$. Both strategies bits after through $\\mathcal{J}^\\dagger$ are\n\\begin{align}\n\\begin{aligned}\n\\label{iop}\n\\hat{S_1} &= \\hat{I} ~:~\\ket{0}\\otimes\\ket{1} \\\\\n\\hat{S_1} &= \\hat{X} :~\\ket{1}\\otimes\\ket{1},\n\\end{aligned}\n\\end{align}\n\\begin{align}\n\\begin{aligned}\n\\label{zop}\n\\hat{S_1} &= \\hat{Y} :~i\\cos\\gamma \\ket{1} \\otimes \\ket{1} + \\sin\\gamma\\cos\\theta \\ket{0} \\otimes \\ket{1} - \\sin\\gamma\\sin\\theta \\ket{0} \\otimes \\ket{0} \\\\\n\\hat{S_1} &= \\hat{Z} :~\\cos\\gamma \\ket{0} \\otimes \\ket{1} - i\\sin\\gamma\\cos\\theta \\ket{1} \\otimes \\ket{1} + i\\sin\\gamma\\sin\\theta \\ket{1} \\otimes \\ket{0}.\n\\end{aligned}\n\\end{align}\nThe state of 01 means stand, and 11 means hit. As it is clear from (\\ref{zop}), when the player chooses $\\hat{Y}$ or $\\hat{Z}$, he can do operations 00 and 10 that do not exist in the classical game.\n\nWe summarize the quantized blackjack, employing the formalism of the 2 player quantum game $\\Gamma_{p,d}=(\\mathcal{H},\\rho,\\Sigma_1,\\Sigma_2,P_1,P_2)$ introduced in the previous subsection.\nIn order to get the quantized blackjack, we modify two settings of the classical blackjack.\nThe modifications are expanding the set of player's strategy to $\\Sigma_{1}=\\{\\hat{I},\\hat{X},\\hat{Y},\\hat{Z}\\}$ and changing the utilities to\n\t\\begin{align}\n\tP_1(p,d,\\hat{S}_1)\n\t~=~~&E_{00}\\bra{0}\\otimes\\bra{0}\\mathcal{J}\\left(\\hat{S}_1\\otimes\\hat{I}\\right)\\mathcal{J}^{\\dagger}\\rho\n\t +E_{\\mathrm{std}}\\bra{0}\\otimes\\bra{1}\\mathcal{J}\\left(\\hat{S}_1\\otimes\\hat{I}\\right)\\mathcal{J}^{\\dagger}\\rho \\notag \\\\\n\t+~&E_{10}\\bra{1}\\otimes\\bra{0}\\mathcal{J}\\left(\\hat{S}_1\\otimes\\hat{I}\\right)\\mathcal{J}^{\\dagger}\\rho\n\t +E_{\\mathrm{hit}}\\bra{1}\\otimes\\bra{1}\\mathcal{J}\\left(\\hat{S}_1\\otimes\\hat{I}\\right)\\mathcal{J}^\\dagger \\rho\n\t \\label{qutility}\n\t\\end{align}\nwhere $E_{00}$ and $E_{10}$ are the expectation values corresponding to operations 00 and 10 respectively. From (\\ref{iop})-(\\ref{qutility}), $P_1$ corresponding to each strategy $\\{\\hat{I}, \\hat{X}, \\hat{Y}, \\hat{Z}\\}$ is \n\\begin{equation}\n\\label{exp}\n\\{E_\\mathrm{std},E_\\mathrm{hit},\\sin^2\\gamma(\\cos^2\\theta E_\\mathrm{std} +\\sin^2\\theta E_{00}) +\\cos^2\\gamma E_\\mathrm{hit}, \\cos^2\\gamma E_\\mathrm{std} +\\sin^2\\gamma(\\cos^2\\theta E_\\mathrm{hit}+\\sin^2\\theta E_{10}) \\}.\n\\end{equation}\n\n\\subsection{\\label{q-circuit}Quantum circuit reproducing blackjack}\nIn this section, we show the circuit which reproduces the games discussed above. First of all, we introduce the circuit which reproduces the classical blackjack defined in section \\ref{cbj}. Fig. \\ref{circuit1} and Fig. \\ref{circuit2} show the quantum circuit.\nLet $\\mathcal{H}_{\\rm whole}$ be Hilbert space, on which the circuit in Fig. \\ref{circuit1} works and consists of some spaces as follows:\n    \\begin{align}\\label{whole Hilbert}\n    \\mathcal{H}_{\\rm whole}=\\mathcal{H}_{\\rm deck}\\otimes\\mathcal{H}_{\\rm deck\\,copy}\\otimes\\mathcal{H}_{\\rm p\\mathchar`-hand}\\otimes\\mathcal{H}_{\\rm control}\\otimes\\mathcal{H}_{\\rm p\\mathchar`-strategy}\\otimes\\mathcal{H}_{\\rm d\\mathchar`-hand}\\otimes\\mathcal{H}_{\\rm d\\mathchar`-strategy}.\n    \\end{align}\nWe in turn explain the component spaces of $\\mathcal{H}_{\\rm whole}$ appearing in eq. \\eqref{whole Hilbert}.\nWe first define a deck state $\\ket{D}\\in\\mathcal{H}_{\\rm deck}$, a copied deck state $\\ket{D}_{\\rm copy}\\in\\mathcal{H}_{\\rm deck\\,copy}$, an initial player's hand $\\ket{p}\\in\\mathcal{H}_{\\rm p\\mathchar`-hand}$, and dealer's hand $\\ket{d}\\in\\mathcal{H}_{\\rm d\\mathchar`-hand}$ to regard $p\\in T_{1}$ and $d\\in T_{2}$ as qubits. These take this form\n\\begin{equation}\n\\ket{A^{(1)}} \\otimes \\ket{A^{(2)}} \\otimes \\ket{2^{(1)}} \\otimes \\ket{2^{(2)}} \\otimes \\ket{3^{(1)}} \\otimes \\ket{3^{(2)}} \\otimes \\ket{3^{(3)}} \\otimes \\ket{3^{(4)}},\n\\label{deck}\n\\end{equation}\nwhere,\n\\begin{gather}\n\\ket{A^{(n)}},\\ket{2^{(n)}},\\ket{3^{(m)}} = \\alpha \\ket{0} + \\beta \\ket{1} \\;\\;\\;\\;(n=1,2,~m=1,2,3,4,~ \\alpha,\\beta \\in \\mathbb{C}), \\\\\n|\\alpha|^2 + |\\beta|^2 = 1,\n\\end{gather}\nand then $\\ket{1}$ means an existing card corresponding to each card sector and $\\ket{0}$ means not. $\\mathcal{H}_{\\rm deck},\\mathcal{H}_{\\rm deck\\,copy},\\mathcal{H}_{\\rm p\\mathchar`-hand}$ and $\\mathcal{H}_{\\rm d\\mathchar`-hand}$ are spanned by these $2^8$ bases. Secondly, $\\mathcal{H}_{\\rm p\\mathchar`-strategy}\\otimes\\mathcal{H}_{\\rm d\\mathchar`-strategy}$ is the same as the space $\\mathcal{H}$ in section \\ref{cbj} and section \\ref{entangle}. Finally, $\\ket{\\Psi}$ consists of three q-bits and lies on $\\mathcal{H}_{\\rm control}$, which is a vector space of dimension $2^{3}$. Note that in the actual blackjack there is no object to which the element of $\\mathcal{H}_{\\rm deck\\,copy}$ or $\\mathcal{H}_{\\rm control}$ corresponds. We, however, need to incorporate these two spaces into our circuit due to some technical reasons. We refer to this in detail below.\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=15.5cm]{circuit1.pdf}\n\\caption{Quantum circuit reproducing the classical game}\n\\label{circuit1}\n\\end{figure*}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=11cm]{circuit2.pdf}\n\\caption{Detail of {\\it Hit} operator }\n\\label{circuit2}\n\\end{figure}\n\nWe explain the outline of Fig. \\ref{circuit1} and Fig. \\ref{circuit2} as follows. The player's strategy is expressed by a unitary operator $\\hat{S_1}$ acting on its strategy bit. In Fig. \\ref{circuit1}, first the dealer and player get the cards classically, then the player chooses the strategy $\\hat{S}_1 = \\hat{X}$ or $\\hat{I}$ which means hit or stand respectively. If the player chooses hit, he has $\\hat{X}$ operate to his strategy bit. In that case {\\it Hit} operator $\\hat{\\mathcal{H}}$ acts on the deck bit and his hand bit (in Fig. \\ref{circuit2}). This operation signifies drawing a card from the deck. Now, we can expands the player's strategy space $\\Sigma_1$ to the set of $2 \\times 2$ unitary matrices thus he can overlap his strategy bit. For example, when he chooses the Hadamard gate, the cases of hit and stand overlap with a 1\/2 probability. As is clear from (\\ref{utiladd1}), this does not affect the maximum value expected by the player. After the player's operation, the dealer turns over her face-down card by $\\hat{\\mathcal{H}}$ operating, and if the total of her hand ($\\equiv |d|$) is 5 or less, she must draw a card one more time. This operation is repeated until $|d| \\ge 6$.\n\nWe next explain the detail of {\\it Hit} operator, surrounded by the dotted line in Fig. \\ref{circuit1}. Control bit $\\ket{\\Psi} (\\in \\mathcal{H}_{\\rm control})$ is responsible for the probability of appearance of cards and it is comprised of three qubits and takes the following form\n\\begin{eqnarray}\n\\ket{\\Psi} &=& \\textstyle\\sum\\limits_{ijk=0}^1 \\frac{1}{\\sqrt{8}}\\ket{ijk} \\\\ \\nonumber\n&=& \\frac{1}{\\sqrt{8}}  ( \\ket{000} + \\ket{001}  + \\ket{010} + \\ket{100} + \\ket{110} + \\ket{101} + \\ket{011} + \\ket{111} ).\n\\end{eqnarray}\nEach state corresponds to a sector of the card, e.g., 000 to the first ace, 001 to the second ace, 010 to the first deuce , and so on. Therefore, the probability that every particular card is drawn is equal when the unitary {\\it Hit} operator acts on the deck and hand bit. In Fig. \\ref{circuit2}, $\\ket{\\cdot_i} (i=1,2,\\cdots,8)$ means the sector of the card in (\\ref{deck}), e.g., $\\ket{\\cdot_1}$ corresponds to the sector of the first ace $\\ket{A^{(1)}}$. When $\\ket{D_1} = \\ket{D_1}_{copy} = \\ket{1}$, (which indicates the existence of the first ace,) $\\ket{D_1}$ and $\\ket{p_1}$, corresponding to $\\ket{000}$ which is a component of $\\ket{\\Psi}$, are swapped. Note that, the deck states can be copied without violating the no-cloning theorem \\cite{Wootters1982,DIEKS1982271} because the deck state comprises only $\\ket{0}$ and $\\ket{1}$, which are orthogonal. After measuring the control bit, it is determined which card has been drawn.\n\n\nNote that, $\\ket{\\Psi}$ converges on a state that corresponds to the sector of the card that has already been drawn with a 3\/8 probability at the time of first hit even though $\\hat{\\mathcal{H}}$ operates. For example, when a control bit converges on $\\ket{000}$ despite $\\ket{D_1} = \\ket{D_1}_{copy} = \\ket{0}$, there is no exchange. To avoid this problem, we ensure that this procedure is repeated until $\\ket{p} \\neq \\ket{p^\\prime}$. Even if a strategy bit has $\\ket{0}$ component, we can employ this procedure as we know which sector of the card bit was exchanged, by measuring not the hand bit, but the control bit. This is explained below. For example, in the case of the initial cards states \n\\begin{align}\n&\\ket{p}=\\ket{A^{(1)},2^{(1)}} \\equiv \\ket{1}\\ket{0}\\ket{1}\\ket{0}\\ket{0}\\ket{0}\\ket{0}\\ket{0}, \\notag \\\\ \n&\\ket{d}=\\ket{3^{(1)}}\\equiv \\ket{0}\\ket{0}\\ket{0}\\ket{0}\\ket{1}\\ket{0}\\ket{0}\\ket{0}, \\notag\\\\\n&\\ket{D}=\\ket{A^{(2)},2^{(2)},3^{(2)},3^{(3)},3^{(4)}} \\equiv \\ket{0}\\ket{1}\\ket{0}\\ket{1}\\ket{0}\\ket{1}\\ket{1}\\ket{1},\n\\end{align}\nand the overlapped player's strategy bit with Hadamard operator $\\hat{H}$ acting on, the first state is  \n\\begin{equation}\n    \\ket{p}\\ket{D}\\ket{\\Psi}\\ket{d}\\hat{H}\\ket{0} = \\ket{p}\\ket{D}\\ket{\\Psi}\\ket{d}\\frac{1}{\\sqrt{2}}(\\ket{0}+\\ket{1}),\n\\end{equation}\nwhere the last state is the player's strategy bit.\nHere, we omitted the dealer's strategy bit, and the copy bits. After thorough $\\hat{\\mathcal{H}}$, the state is\n\\begin{equation}\n\\frac{1}{\\sqrt{2}}\\left\\{\\ket{p}\\ket{D}\\ket{\\Psi}\\ket{d}\\ket{0} + \\hat{\\mathcal{H}}\\left(\\ket{p}\\ket{D}\\ket{\\Psi}\\right)\\ket{d}\\ket{1}\\right\\},\n\\end{equation}\nand after measuring the control bit, it will be\n\\begin{eqnarray}\n    \\begin{cases} \\ket{p}\\ket{D}\\ket{d}\\ket{0} + \\ket{p}\\ket{D}\\ket{d}\\ket{1} &: \\mspace{6mu} \\ket{\\Psi} = \\ket{000},  \\ket{010} \\mathrm{or} \\ket{110} \\\\  \\ket{p}\\ket{D}\\ket{d}\\ket{0} + \\ket{A^{(1)},A^{(2)},2^{(1)}}\\ket{2^{(2)},3^{(2)},3^{(3)},3^{(4)}}\\ket{d}\\ket{1} &: \\mspace{6mu} \\ket{\\Psi} =  \\ket{001} \\\\ \\ket{p}\\ket{D}\\ket{d}\\ket{0} + \\ket{A^{(1)},2^{(1)},2^{(2)}}\\ket{A^{(2)},3^{(2)},3^{(3)},3^{(4)}}\\ket{d}\\ket{1} &: \\mspace{6mu} \\ket{\\Psi} = \\ket{100} \\\\ \\mspace{30mu} \\vdots \\\\ \n     \\ket{p}\\ket{D}\\ket{d}\\ket{0} + \\ket{A^{(1)},2^{(1)},3^{(4)}}\\ket{A^{(2)},2^{(2)},3^{(2)},3^{(3)}}\\ket{d}\\ket{1} &: \\mspace{6mu} \\ket{\\Psi} = \\ket{111}. \\end{cases}\n\\end{eqnarray}\n\nThe overall constant has been dropped. When the control bit converges $\\ket{000}$, $\\ket{010}$ or $\\ket{110}$ in the first line, we repeat this one more time. In this way, we can judge whether we must repeat this procedure without measuring the strategy bit or hand bit. In the following discussions, though we omit specifying this, we always perform this procedure.\n\\clearpage\n\n\nDealer's operation {\\it Hit} is the same. First, the dealer hits regardless of her strategy bit (corresponding to turn over her face-down card), and if her total $|d|$ is 5 or less and her strategy bit is $\\ket{1}$, she repeats the hit operation. Again, we can determine her hand total only by measuring the control bit; therefore, we can build the algorithm where she must hit until her total $|d|$ reaches 6 without measuring her hand bit. Note that when the strategy bit has $\\ket{0}$ component, the same algorithm is used. The summary of the algorithm is shown by Fig. \\ref{flowchart}.\n\nFinally, after measuring both strategy bits, all states converge. As explained in section \\ref{rule}, the player wins when his hand does not exceed 7 and is closer to 7 than the dealer's hand. This quantum circuit reproduces classical snackjack, and therefore the player's expectation and basic strategy must be the same as the classical ones. Such a circuit can be applied to blackjack in a same way by increasing qubits, although it is not considered after this.\n\nWe can also design the quantum circuit which reproduces the quantized blackjack formulated in section \\ref{entangle}, inserting EWL protocol in  our circuit. The modified quantum circuit is shown in Fig. \\ref{fig3}. There is no difference between the circuit in Fig. \\ref{fig3} and Fig. \\ref{circuit1} except the insertion of EWL protocol.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[scale=0.6]{flowchart.pdf}\n    \\caption{The Process of the game}\n    \\label{flowchart}\n\\end{figure}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=15.3cm]{circuit3.pdf}\n\\caption{Entangled strategy}\n\\label{fig3}\n\\end{figure}\n\\clearpage\n\n\n\\section{\\label{res}Results}\n\nClassically, Tab. \\ref{tab2} shows $E_{\\mathrm{std}},E_{\\mathrm{hit}}$ in all initial states and basic strategy. Here, we assume that the player bets unit per one game. Some calculations of $E_{\\mathrm{std}}$ and  $E_{\\mathrm{hit}}$ in the case of No. 14 game are shown in Fig. \\ref{stdandhit}.\n\n\\begin{table}[h]\n\\centering\n\\scalebox{0.8}[0.8]{\n\\begin{tabular}{ccccccccc}\nNo & initial state (As,2s,3s) & dealer up& $E_{\\mathrm{std}}$ & $E_{\\mathrm{hit}}$ & number of cases & CBS \\\\ \\hline \\hline\n1 & (2,0,0) & 2 & 1\/5 & 1\/5 & 2 & I,X \\\\\n2 & & 3 & -2\/5 & 1\/5 & 4 & X \\\\\n3 & (0,2,0) & 1 & -3\/5 & -3\/5 & 2 & I,X \\\\\n4 & & 3 & -1 & -2\/5 & 4 & X \\\\\n5 &(0,0,2) & 1 & -8\/15 & -4\/5 & 12 & I \\\\\n6 & & 2 & -1\/30 & -1\/3 & 12 & I \\\\\n7 & & 3 & -2\/5 & -7\/15 & 12 & I \\\\\n8 & (1,1,0) & 1 & -4\/5 & -4\/5 & 4 & I,X \\\\\n9 & & 2 & 3\/5 & 3\/5 & 4 & I,X \\\\\n10 & & 3 & -1\/20 & -1\/20 & 16 & I,X \\\\\n11 & (1,0,1) & 1 & 2\/5 & -3\/10 & 8 & I \\\\\n12 & & 2 & 1 & 1\/2 & 16 & I \\\\\n13 & & 3 & 4\/5 & 1\/30 & 24 & I \\\\\n14 & (0,1,1) & 1 & -4\/5 & -17\/20 & 16 & I \\\\\n15 & & 2 & -2\/5 & -2\/5 & 8 & I,X \\\\\n16 & & 3 & -4\/5 & -13\/30 & 24 & X \\\\ \\hline\ntotal &&&&&168& \\\\\n\\end{tabular}\n}\n\\caption{Classical basic strategy (CBS)}No. 1 to 16 are the types of initial states. Each initial state is determined by classical probability.\n\\label{tab2}\n\\end{table}\n\\begin{figure}[h]\n  \\begin{minipage}[b]{0.45\\linewidth}\n    \\centering\n    \\mbox{\\raisebox{7.5mm}{\\includegraphics[scale=0.6]{stand.pdf}}}\n  \\end{minipage}\n  \\begin{minipage}[b]{0.45\\linewidth}\n    \\centering\n    \\includegraphics[scale=0.6]{hit.pdf}\n  \\end{minipage}\n  \\caption{}The calculations of $E_{\\mathrm{std}}$ and $E_{\\mathrm{hit}}$ in No. 14. The hand states are denoted as (As,2s,3s;total).\n  \\label{stdandhit}\n\\end{figure}\n\nIn the classical game, when the player chooses the basic strategy, the expectation value of this game is -1.7\\%; thus, the dealer has the advantage in this game. Although the player can also choose $\\hat{Y}$ or $\\hat{Z}$ in the absence of entanglement, these effects on the strategy bit are the same as $\\hat{X}$ and $\\hat{I}$, respectively. Therefore, it does not affect the overall expectation value and basic strategy. Note here, that our classical results are different from \\cite{ethier2019snackjack}, because of differences in the rulesets (Tab. \\ref{tab1}) of snackjack.\n\nIn the presence of entanglement, the player has a meaningful quantum strategy, $\\hat{Y}$ or $\\hat{Z}$. By the previous section (\\ref{zop})-(\\ref{exp}), when parameter $\\gamma = \\theta = \\frac{\\pi}{2}$, expectation value $E_Y$ which player chooses strategy $\\hat{Y}$, equal to $E_{00}$. And also $E_Z = E_{10}$. Tab. \\ref{tab3} shows quantum basic strategy (QBS) which the player can choose in entangled strategy. Fig. \\ref{00and10} shows the calculations of $E_{00}$ and $E_{10}$ in No. 6 game.\n\\clearpage\n\\begin{table}[h]\n\\centering\n\\scalebox{0.7}[0.7]{\n\\begin{tabular}{ccccccccc}\nNo & initial state (As,2s,3s) & dealer up& $E_{\\mathrm{std}}$ & $E_{\\mathrm{hit}}$ & $E_{Y} = E_{00}$ & $E_{Z} = E_{10}$ & number of cases &QBS \\\\ \\hline \\hline\n1 & (2,0,0) & 2 & 1\/5 & 1\/5 & 1\/5 & 2\/5 & 2 & Z \\\\\n2 & & 3 & -2\/5 & 1\/5 & -3\/5 & 1\/10 & 4 & X \\\\\n3 & (0,2,0) & 1 & -3\/5 & -3\/5 & -1 & -3\/5 & 2 & I,X,Z \\\\\n4 & & 3 & -1 & -2\/5 & -1 & -2\/5 & 4 & X,Z \\\\\n5 &(0,0,2) & 1 & -8\/15 & -4\/5 & -1\/5 & -4\/5 & 12 & Y \\\\\n6 & & 2 & -1\/30 & -1\/3 & 3\/5 & -1\/5 & 12 & Y \\\\\n7 & & 3 & -2\/5 & -7\/15 & 0 & -2\/5 & 12 & Y \\\\\n8 & (1,1,0) & 1 & -4\/5 & -4\/5 & -4\/5 & -4\/5 & 4 & I,X,Y,Z \\\\\n9 & & 2 & 3\/5 & 3\/5 & 4\/5 & 4\/5 & 4 & Y,Z \\\\\n10 & & 3 & -1\/20 & -1\/20 & 0 & 0 & 16 & Y,Z \\\\\n11 & (1,0,1) & 1 & 2\/5 & -3\/10 & 2\/5 & -3\/10 & 8 & I,Y \\\\\n12 & & 2 & 1 & 1\/2 & 1 & 4\/5 & 16 & I,Y \\\\\n13 & & 3 & 4\/5 & 1\/30 & 4\/5 & 1\/10 & 24 & I,Y \\\\\n14 & (0,1,1) & 1 & -4\/5 & -17\/20 & -4\/5 & -17\/20 & 16 & I,Y \\\\\n15 & & 2 & -2\/5 & -2\/5 & -2\/5 & -3\/10 & 8 & Z \\\\\n16 & & 3 & -4\/5 & -13\/30 & -4\/5 & -2\/5 & 24 & Z \\\\ \\hline\ntotal &&&&&&&168& \\\\\n\\end{tabular}\n}\n\\caption{Quantum basic strategy ($\\gamma = \\theta = \\pi\/2$)}\n\\label{tab3}\n\\end{table}\n\\begin{figure}[h]\n  \\begin{minipage}[b]{0.45\\linewidth}\n    \\centering\n    \\mbox{\\raisebox{7.5mm}{\\includegraphics[scale=0.6]{00.pdf}}}\n  \\end{minipage}\n  \\begin{minipage}[b]{0.45\\linewidth}\n    \\centering\n    \\includegraphics[scale=0.6]{10.pdf}\n  \\end{minipage}\n  \\caption{}\n  \\label{00and10}\n\\end{figure}\n\nWhen the player chooses QBS in Tab. \\ref{tab3}, the overall expected value is +10.2\\%; therefore, the player has the advantage in this quantum game. In the $\\gamma = \\theta = \\frac{\\pi}{2}$ game, the player has other choices that let the dealer definitely stand (prohibiting the second draw) therefore expectation value increases compared to the classical game. We can set various values in not only entangle intensity $\\gamma$ but also game parameter $\\theta$. For example in $\\gamma = \\frac{\\pi}{4},\\; \\theta= \\frac{\\pi}{2}$, the player's utilities are $\\{E_\\mathrm{std}, E_\\mathrm{hit}, \\frac{1}{2}(E_\\mathrm{hit}+E_{00}), \\frac{1}{2}(E_\\mathrm{std}+E_{10})\\}$ by (\\ref{exp}), and the best choice for the player becomes QBS in each initial hands. The whole expectation for the player in the game is the sum of the largest utility times the appearance ratio of the initial state from No. 1 to No. 16. Fig. \\ref{fig4} shows numerical results regarding the player's expectation in various parameters.\nAs is clear from (\\ref{iop})-(\\ref{exp}), in $\\theta = 0$ game, one can see any $\\gamma$ do not change the expectation value from the classical one. Otherwise, when $\\gamma = \\frac{\\pi}{4},\\; \\theta = \\frac{\\pi}{2}$, half entangled game, the expectation is +1.8\\%. From these results, we conclude that the player has the advantage in the appropriate parameter game.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=9cm]{qbjgraph1.1.pdf}\n\\caption{Player's expectation in various parameters \\\\ \\centering ($0 \\le \\gamma , \\theta \\le \\frac{\\pi}{2}$)}\n\\label{fig4}\n\\end{figure}\n\\clearpage\n\n\\section{\\label{conclusion}Conclusion and Discussion}\nIn section \\ref{intro}, we introduced quantum effects in game theory \\cite{PhysRevLett.83.3077,PhysRevLett.82.1052,doi:10.1142\/S0219477502000981,Khan2018,GUO2008318} and the application to gambling \\cite{PhysRevLett.82.3356,PhysRevA.64.064302,PhysRevA.66.052311,Zhang_2008,Schmidt_2012,SCHMIDT2013400,FRACKIEWICZ20148,qchess,Fuchs_2020,bleiler2009quantized,PhysRevA.102.012425}. We then proposed a toy model of blackjack, called {\\it snackjack} \\cite{Ethier2010,ethier2019snackjack,epstein} as a quantum game in section \\ref{rule}, \\ref{cbj}. In section \\ref{qbj}, we found that we can possibly entangle strategies without breaking the classical game and proposed a quantum circuit which reproduced classical blackjack. In section \\ref{res}, we showed that the player's expectation increases and the player has an advantage compared to the classical blackjack with QBS.\n\nThere remain some interesting questions: (1) The dealer's strategy space $\\Sigma_2=\\{\\hat{I}\\}$ can be expanded into more general strategy space, e.g. $\\Sigma_2 = \\{\\hat{I},\\hat{Z}\\}$. In other words, the dealer inherently has infinite options changing the phase of the initial state $\\ket{1}$. This phase affects the player's strategy bit and perhaps cause a dilemma between them depending on the first hand state. In this case, QBS may not be the best choice for the player. (2) It is a nontrivial question that how this quantized blackjack on noisy intermediate scale quantum computers affects the player's expectation, as in their work \\cite{Fuchs_2020}. (3) The operation of drawing cards in the circuit we proposed can be applied to other card games. It is also an open question of what kind of quantized casino game will make a difference from the classical one. In the proposed quantum blackjack, we have demonstrated that entanglement between strategies could be used to bankrupt a casino. In the forthcoming quantum era, casinos will have to set house rules that did not exist in the era of classical physics to avoid bankruptcy.\n\n\\section*{Acknowledgments}\nThis study was inspired by a discussion with K. Ikeda and S. Aoki about quantum economics. We acknowledge K. Ikeda for supporting us by proofreading and offering counsel. We also thank T. Onogi for proofreading the manuscript and providing us insightful comments. The authors would like to thank Enago (www.enago.jp) for the English language review.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nWe consider problems of scattering of time-harmonic electromagnetic waves by a locally\nperturbed infinite plane (which is called a locally rough surface).\nSuch problems occurs in many applications such as radar, remote sensing, geophysics, medical\nimaging and nondestructive testing (see, e.g. \\cite{SS01,V99,WC01}).\n\nIn this paper, we are restricted to the two-dimensional transverse electric (TE) case for a perfectly reflecting,\nlocally rough surface by assuming that the local perturbation is invariant in the $x_3$ direction.\nDenote by $\\G:=\\{(x_1,x_2)\\;:\\;x_2=h_\\G(x_1),x_1\\in\\R\\}$ the locally rough surface with a smooth surface profile\nfunction $h_\\G\\in C^2(\\R)$ having a compact support in $\\R$, and by $D_+$ the unbounded domain above $\\G$.\nLet $\\Sp^1_\\pm:=\\{x=(x_1,x_2):|x|=1,x_1\\gtrless0\\}$ be the upper and lower parts of the unit circle\n$\\Sp^1:=\\{x=(x_1,x_2):|x|=1\\}$. Suppose a time-harmonic ($e^{-i\\omega t}$ time dependence) plane wave\n\\ben\nu^i=u^i(x;d,k):=\\exp({ikd\\cdot x})\n\\enn\nis incident on the locally rough surface from the upper unbounded domain $D_+$,\nwhere $d=(\\sin\\theta,-\\cos\\theta)^T\\in\\Sp^1_-$ is the incident direction with $\\theta\\in(-\\pi\/2,\\pi\/2)$\nthe angle of incidence, $k=\\omega\/c>0$ is the wave number, $\\omega$ and $c$ are the wave frequency and\nspeed in $D_+$, respectively.\nThen the total field $u(x)$ is given as the sum of the incident wave $u^i(x)$,\nthe reflected wave $u^r(x)$ and the unknown scattered wave $u^s(x)$. Here,\n$u^r$ is the reflected wave with respect to the infinite plane $x_2=0$ given by\n\\ben\nu^r=u^r(x;d,k):=-\\exp({ikd'\\cdot x})\n\\enn\nwhere $d'=(\\sin\\theta,\\cos\\theta)\\in\\Sp^1_+$ is the reflected direction.\nFurthermore, the scattered field $u^s$ is required to satisfy the Helmholtz equation in $D_+$,\nthe boundary condition on $\\Gamma$ and the so-called Sommerfeld radiation condition at infinity, respectively:\n\\begin{align}\\label{eq1_nr}\n\\Delta u^s+k^2 u^s=0&\\quad \\textrm{in}\\;\\;D_+\\\\\n\\label{eq2_nr}u^s=f&\\quad \\textrm{on}\\;\\;\\G \\\\\n\\label{eq3_nr}\\lim_{r\\to\\infty}r^\\half\\left(\\frac{\\pa u^s}{\\pa r}-ik u^s\\right)=0&\\quad r=|x|\n\\end{align}\nwhere $f=-(u^i+u^r)$ has a compact support on $\\Gamma$. Here, $u^{near}:=u^r+u^s$ is called the near field.\nIn addition, (\\ref{eq3_nr}) implies that $u^s$ has the following asymptotic behavior\n(see \\cite{Willers1987,ZhangZhang2013}):\n\\be\\label{eq4}\nu^s(x;d,k)=\\frac{e^{ik|x|}}{\\sqrt{|x|}}\\left(u^\\infty(\\hat{x};d,k)+O\\Big(\\frac{1}{|x|}\\Big)\\right),\\qquad |x|\\to\\infty,\n\\en\nuniformly for all observation directions $\\hat{x}=x\/|x|\\in\\Sp^1_+$, where $u^\\infty$ is\ncalled the far-field pattern of the scattered field $u^s$.\nThe geometry of the scattering problem is presented in Figure \\ref{fig1}.\n\n\\begin{figure}\\label{fig1}\n  \\centering\n    \\includegraphics[width=4in]{example\/rough.eps}\n    \\vspace{-0.4in}\n \\caption{The scattering problem from a locally rough surface}\n\\end{figure}\n\nThe existence and uniqueness of solutions to the scattering problem (\\ref{eq1_nr})-(\\ref{eq3_nr})\nhas been studied by using the integral equation method in \\cite{Willers1987} and\na variational method in \\cite{BaoLin11}. Recently in \\cite{ZhangZhang2013},\na novel integral equation formulation was proposed for this scattering problem,\nwhich leads to a fast numerical solution of the scattering problem including the case with a large wave number.\nIt should be noted that, in the past years, the mathematical and computational aspects of the\nscattering problem (\\ref{eq1_nr})-(\\ref{eq3_nr}) and other boundary conditions\nhave been studied extensively by using the integral equation method and variational approaches\nfor the case when the surface $\\G$ is a non-local (or global) perturbation of the infinite plane $x_2=0$\n(which is called the rough surface scattering in the engineering community)\n(see, e.g. \\cite{CZ98,CRZ99,CM05,CHP06,CE10,SS01,V99,WC01,ZC03}).\n\nOn the other hand, many reconstruction algorithms have been developed for the inverse problem of reconstructing\nlocally rough surfaces from the scattered near-field or far-field data, corresponding to incident\npoint sources or plane waves (see, e.g. \\cite{AKY,BaoLin11,BaoLin13,CGHIR,DPT06,KressTran2000,LSZ16}).\nFor example, a Newton method was proposed in \\cite{KressTran2000} to reconstruct a locally rough surface\nfrom the far-field pattern under the condition that the local perturbation is both star-like\nand {\\em above} the infinite plane.\nAn optimization method was introduced in \\cite{BaoLin13} to recover a mild, locally rough surface\nfrom the scattered field measured on a straight line within one wavelength above the locally\nrough surface, under the assumption that the local perturbation is {\\em above} the infinite plane.\nIn \\cite{BaoLin11}, a continuation approach over the wave frequency was developed for\nreconstructing a general, locally rough surface from the scattered field measured on an upper\nhalf-circle enclosing the local perturbation, based on the choice of the descent vector field.\nA regularized Newton method was proposed in \\cite{ZhangZhang2013} to reconstruct a general, locally rough surface\nfrom multi-frequency far-field data, where the novel integral equation introduced in \\cite{ZhangZhang2013}\nis used to solve the forward scattering problem in each iteration.\nThe reconstruction results obtained in \\cite{BaoLin11,ZhangZhang2013} are stable and accurate even for\nmulti-scale surface profiles in view of using multiple frequency data and considering multiple scattering.\nWe point out that many reconstruction algorithms have also been developed for reconstructing non-locally rough surfaces\nfrom the scattered near-field data (see, e.g. \\cite{BaoLi13,BaoLi14,BP2010,BP2009,CL05,DeSanto1,DeSanto2,LS15}).\n\nIn practical applications, it is much harder to obtain data with accurate phase information\ncompared with just measuring the intensity (or the modulus) of the data, and therefore it is often desirable to\nreconstruct the scattering surface profile from the phaseless near-field or far-field data.\nHowever, not many results are available for such problems both mathematically and numerically.\nKress and Rundell first studied such inverse problems in \\cite{KR97} and proved that\nfor the sound-soft bounded obstacle case with one incidence plane wave, the modulus of the far-field pattern\nis invariant under translations of the obstacle, and therefore it is impossible to reconstruct the location of the\nobstacle from the phaseless far-field data for one incident plane wave.\nIt was further proved in \\cite{KR97} that this ambiguity cannot be remedied by using the phaseless\nfar-field pattern for finitely many incident plane waves with different wave numbers or different incident directions.\nRegularized Newton and Landweber iteration methods have also been discussed in \\cite{KR97} for recovering the\nshape of the obstacle from the phaseless far-field data.\nIn \\cite{Ivanyshyn07,IvanyshynKress2010}, a nonlinear integral equation method was proposed\nto reconstruct the shape of the obstacle from the phaseless far-field data.\nFurther, in \\cite{IvanyshynKress2010} after the shape of the obstacle is reconstructed from the phaseless far-field data,\nan algorithm is proposed for the localization of the obstacle by utilizing the translation invariance property\ntogether with several full far-field measurements at the backscattering direction.\nIn \\cite{IvanyshynKress2011}, a nonlinear integral equation method was developed to reconstruct the real-valued surface\nimpedance function from the phaseless far-field data provided that the bounded obstacle is known in advance.\nRecently, a continuation algorithm was proposed in \\cite{BaoLiLv13} to reconstruct the shape of a perfectly reflecting\nperiodic surface from the phaseless near-field data, in \\cite{BLT11} to deal with the phaseless measurements for\nan inverse source problem, and in \\cite{BaoZhang16} to recover the shape of multi-scale sound-soft large rough surfaces\nfrom phaseless measurements of the scattered field generated by tapered waves with multiple frequencies.\nRecently, for inverse acoustic scattering with bounded obstacles it was proved in \\cite{ZZ16}\nthat the translation invariance property of the phaseless far-field pattern can be broken by using\nsuperpositions of two plane waves as the incident fields in conjunction with all wave numbers in a finite interval.\nFurther, a recursive Newton-type iteration algorithm in frequencies was developed in \\cite{ZZ16} to numerically\nreconstruct both the location and the shape of the obstacle simultaneously from multi-frequency phaseless far-field data.\n\nThe purpose of this paper is to develop an efficient imaging algorithm to reconstruct the locally rough surface $\\G$\nfrom phaseless data associated with incident plane waves. Two types of phaseless data will be considered:\nthe phaseless far-field data and the phaseless near-field data (see Figure \\ref{fig1}).\nSimilarly as in the bounded obstacle case, the phaseless far-field pattern, $|u^\\infty(\\hat{x};d,k)|$,\nis also invariant under translations of the local perturbation along the $x_1$ direction for one incident plane wave,\nthat is, $|u^\\infty_\\ell(\\hat{x};d,k)|=|u^\\infty(\\hat{x};d,k)|$, $\\hat{x}\\in\\Sp^1_+$ for all $\\ell\\in\\R,$\nwhere $u^\\infty_\\ell(\\hat{x};d,k)$ is the far-field pattern of the scattering solution with\nrespect to the shifted surface $\\G^{\\ell}:=\\{(x_1+\\ell,h_\\G(x_1)):x_1\\in\\R\\}$ of $\\G$ along the $x_1$ direction\n(see Theorem \\ref{th2} below).\nThus, it is impossible to recover the location of the locally rough surface $\\G$ from phaseless far-field data\ncorresponding to one incident plane wave. To overcome this difficulty, motivated by \\cite{ZZ16},\nwe will use the following superposition of two plane waves rather than one plane wave as the incident field:\n\\be\\label{IW}\nu^i=u^i(x;d_1,d_2,k):=\\exp({ikd_1\\cdot x})+\\exp({ikd_2\\cdot x})\n\\en\nwhere, for $l=1,2$, $\\theta_l\\in(-\\pi\/2,\\pi\/2)$ is the incidence angle and\n$d_l=(\\sin\\theta_l,-\\cos\\theta_l)^T\\in\\Sp^1_-$ is the incident direction.\nThen the reflected wave with respect to the infinite plane $x_2=0$ will be given by\n\\ben\nu^r=u^r(x;d_1,d_2,k):=-\\exp({ikd_1'\\cdot x})-\\exp({ikd_2'\\cdot x})\n\\enn\nwhere, for $l=1,2$, $d'_l=(\\sin\\theta_l,\\cos\\theta_l)\\in\\Sp^1_+$ is the reflected direction,\nand the scattered field $u^s$ will have the asymptotic behavior\n\\be\\label{asymp-2}\nu^s(x;d_1,d_2,k)=\\frac{e^{ik|x|}}{\\sqrt{|x|}}\\left(u^\\infty(\\hat{x};d_1,d_2,k)\n+O\\Big(\\frac{1}{|x|}\\Big)\\right),\\qquad |x|\\to\\infty,\n\\en\nuniformly for all observation directions $\\hat{x}=x\/|x|\\in\\Sp^1_+$.\n\nWe will prove that, if the incident field is taken as $u^i=u^i(x;d_1,d_2,k)$ with\n$d_1\\not=d_2$ and all wave numbers $k$ in a finite interval,\nthen the translation invariance property of the phaseless far-field pattern does not hold\nfor non-trivial locally rough surfaces (that is, $h_\\G\\not\\equiv0$) (see Theorem \\ref{th3} below).\nThus, both the location and the shape of the local perturbation of the surface $\\G$ can be reconstructed from\nthe phaseless far-field data, corresponding to such incident fields with multiple wave numbers\n(see the numerical experiments in Section \\ref{se4}).\nFurthermore, a recursive Newton iteration algorithm in frequencies is developed to reconstruct both\nthe location and the shape of the surface $\\G$ simultaneously from multi-frequency phaseless far-field data.\nA similar Newton iteration algorithm is also developed for reconstructing the location and shape of\nthe surface $\\G$ from multi-frequency phaseless near-field data.\nIn our reconstruction algorithms the fast integral equation solver developed in \\cite{ZhangZhang2013}\nis used to solve the forward scattering problem in each iteration.\n\nThis paper is organized as follows. Section \\ref{se1} gives a brief introduction to the integral equation\nformulation of the forward problem proposed in \\cite{ZhangZhang2013}, and\nthe inverse scattering problem is studied in Section \\ref{se2} with phaseless far-field and near-field data.\nIn Section \\ref{se3}, a recursive Newton-type iteration algorithm in frequencies is proposed to solve the\ninverse problems. Numerical examples are carried out in Section \\ref{se4} to illustrate the effectiveness of\nour inversion algorithm. Concluding remarks are presented in Section \\ref{se5}.\n\n\n\\section{The integral equation formulation for the scattering problem}\\label{se1}\n\\setcounter{equation}{0}\n\n\nIn this section we give a brief introduction to the integral equation formulation\nproposed in \\cite{ZhangZhang2013} for the scattering problem (\\ref{eq1_nr})-(\\ref{eq3_nr})\nwhich leads to a fast numerical solution of the problem and will be used in\nour inversion algorithm. To this end, we need the following notations.\n\nLet $B_R\\coloneqq\\{x=(x_1,x_2)\\;|\\;|x|<R\\}$ be a circle with $R>0$ large enough so that\nthe local perturbation $\\{(x_1,h_{\\G}(x_1))\\;|\\;x_1\\in\\textrm{supp}(h_{\\G})\\}\\subset B_R$.\nThen $\\G_R\\coloneqq\\G\\cap B_R$ represents the part of $\\G$ containing the local perturbation of the infinite plane.\nDenote by $x_A:=(-R,0),x_B:=(R,0)$ the endpoints of $\\G_R$.\nWrite $\\mathbb{R}^2_\\pm\\coloneqq\\{(x_1,x_2)\\in\\R^2\\;|\\;x_2\\gtrless0\\}$,\n$D^\\pm_R\\coloneqq B_R\\cap D_\\pm$ and $\\pa B^\\pm_R\\coloneqq\\pa B_R\\cap D_\\pm$,\nwhere $D_-:=\\{(x_1,x_2)\\;|\\;x_2<h_\\G(x_1),x_1\\in\\R\\}.$ See Figure \\ref{fig2} for the problem geometry.\n\\begin{figure}\\label{fig2}\n  \\centering\n    \\includegraphics[width=4in]{example\/rough2.eps}\n    \\vspace{-0.4in}\n \\caption{Problem geometry}\n\\end{figure}\n\nFor $\\varphi\\in C(\\pa D_R^-)$,\ndefine $S_k,S_k^{re},K_k,K_k^{re}$ to be the boundary integral operators of the following form:\n\\ben\n(S_k\\varphi)(x)&:=&\\int_{\\pa D^-_R}\\Phi_k(x,y)\\varphi(y)ds(y),\\quad x\\in\\pa D^-_R\\\\\n(S_k^{re}\\varphi)(x)&:=&\n     \\left\\{\\begin{array}{l}\n     \\ds\\int_{\\pa D^-_R}\\Phi_k(x,y)\\varphi(y)ds(y),\\quad x\\in\\Gamma_R\\\\\n     \\ds\\int_{\\pa D^-_R}\\Phi_k(x^{re},y)\\varphi(y)ds(y),\\quad x\\in\\pa B^-_R\\cup\\{x_A,x_B\\}\n     \\end{array}\\right.\\\\\n(K_k\\varphi)(x)&:=&\\int_{\\pa D^-_R}\\frac{\\pa\\Phi_k(x,y)}{\\pa\\nu(y)}\\varphi(y)ds(y),\n     \\quad x\\in\\pa D^-_R\\\\\n(K_k^{re}\\varphi)(x)&:=&\n\\left\\{\\begin{array}{l}\\ds\\int_{\\pa D^-_R}\\frac{\\pa\\Phi_k(x,y)}{\\pa\\nu(y)}\\varphi(y)ds(y),\n    \\quad x\\in\\Gamma_R\\\\\n    \\ds\\int_{\\pa D^-_R}\\frac{\\pa\\Phi_k(x^{re},y)}{\\pa\\nu(y)}\\varphi(y)ds(y),\n    \\quad x\\in\\pa B^-_R\\cup\\{x_A,x_B\\}\n    \\end{array}\\right.\n\\enn\nwhere $x^{re}=(x_1,-x_2)$ is the reflection of $x=(x_1,x_2)$ about the $x_1$-axis,\n$\\Phi_k(x,y)$ is the fundamental solution of the Helmholtz equation $\\Delta w+k^2w=0$\nwith the wavenumber $k$, and $\\nu$ is the unit outward normal on $\\pa D^-_R$.\n\nIt was proved in \\cite{ZhangZhang2013} that the scattering solution $u^s$ of the scattering\nproblem (\\ref{eq1_nr})-(\\ref{eq3_nr}) is given as follows\n\\be\\label{eq10_nr}\nu^s(x)=\\int_{\\pa D^-_R}\\left[\\frac{\\pa\\Phi_k(x,y)}{\\pa\\nu(y)}-i\\eta\\Phi_k(x,y)\\right]\\varphi(y)ds(y),\\quad x\\in\\pa D^-_R,\n\\en\nwith $\\eta\\neq0$ a real coupling parameter, if and only if $\\varphi\\in C(\\pa D_R^-)$ is the solution of\nthe integral equation $P\\varphi(x)=g(x)$. Here,\n\\be\\label{eq11_nr}\nP\\varphi\\coloneqq\\left\\{\\begin{array}{ll}\n\\ds\\varphi+\\left(K_k\\varphi-i\\eta S_k\\varphi\\right)\n   +\\left(K^{re}_k\\varphi-i\\eta S_k^{re}\\varphi\\right), &x\\in\\G_R\\\\\n\\ds\\half\\varphi+\\left(K_k\\varphi-i\\eta S_k\\varphi\\right)\n   +\\left(K^{re}_k\\varphi-i\\eta S^{re}_k\\varphi\\right), &x\\in\\pa B_R^-\\cup\\{x_A,x_B\\}\n\\end{array}\\right.\n\\en\nand\n\\be\\label{eq12_nr}\ng(x):=\\left\\{\\begin{array}{ll}\n-2\\big(u^i(x)+u^r(x)\\big), &x\\in\\G_R\\\\\n0, &x\\in\\pa B_R^-\\cup\\{x_A,x_B\\}\n\\end{array}\\right.\n\\en\n\nIn \\cite[theorem 2.4]{ZhangZhang2013}, it was shown that the integral equation $P\\varphi=g$ is uniquely solvable\nin $C(\\pa D_R^-)$, which is solved numerically by using the Nystr\\\"{o}m method with a graded mesh\nintroduced in \\cite[Section 3.5]{ColtonKress1998} (see \\cite{ZhangZhang2013} for details).\n\n\\begin{remark}\\label{re3}{\\rm\nBy (\\ref{eq10_nr}) and the asymptotic behavior of the fundamental solution $\\Phi_k$, it follows that the\nfar-field pattern of the scattered field $u^s$ is given by\n\\be\\label{eq14_nr}\nu^\\infty(\\hat{x};d_1,d_2,k)=\\frac{e^{-i\\pi\/4}}{\\sqrt{8\\pi k}}\\int_{\\pa D_R^-}\n[k\\nu(y)\\cdot\\hat{x}+\\eta]e^{-ik\\hat{x}\\cdot y}\\varphi(y)ds(y),\\qquad \\hat{x}\\in\\Sp^1_+,\n\\en\nwhich is an analytic function on the unit circle $\\Sp^1_+$, where $\\varphi$ is the solution of the\nintegral equation $P\\varphi=g$.\n}\n\\end{remark}\n\n\n\n\\section{The inverse problems}\\label{se2}\n\\setcounter{equation}{0}\n\n\nIn this paper we consider two types of scattered field measurement data without phase information:\nthe phaseless far-field data and the phaseless near-field data (see Figure \\ref{fig1}).\nThe far-field pattern $u^\\infty$ is already defined in (\\ref{eq4}), while the near-field $u^{near}$\nis defined as $u^{near}(x;d,k)=u^r(x;d,k)+u^s(x;d,k)$, $x\\in\\G_{H,L}$ for positive constants $H$ and $L$,\nwhere $\\G_{H,L}:=\\{(x_1,H),x_1\\in[-L,L]\\}$ is the measurement straight line segment above the surface $\\G$.\n\nWe first study properties of the phaseless far-field pattern and the phaseless near-field under\ntranslations of the locally rough surface. To this end, for $\\ell\\in\\R$ define\n$\\G^{\\ell}:=\\{(x_1+\\ell,h_\\G(x_1)):x_1\\in\\R\\}$ to be the shifted surface of $\\G$ along the $x_1$ direction.\nThen we have the following theorem.\n\n\\begin{theorem}\\label{th2}\nGiven the wavenumber $k>0$, let the incident wave be given by $u^i=u^i(x;d,k)$ with the incident direction\n$d=(\\sin\\theta,-\\cos\\theta)$ and the incident angle $\\theta\\in(-\\pi\/2,\\pi\/2)$ and let $u^s(x;d,k),u^s_\\ell(x;d,k)$\nbe the scattering solutions of the scattering problem $(\\ref{eq1_nr})-(\\ref{eq3_nr})$, corresponding to the locally\nrough surface $\\G$ and the shifted one $\\G^\\ell$, respectively.\nAssume that $u^\\infty(\\hat{x};d,k),u^\\infty_\\ell(\\hat{x};d,k)$ ($u^{near}(x;d,k),u^{near}_\\ell(x;d,k)$) are the\nfar-field pattern (the near-field) of the scattering solutions $u^s,u^s_\\ell$, respectively.\nThen we have $u^s_\\ell(x^{\\ell};d,k)=e^{ik\\ell\\sin\\theta}u^s(x;d,k)$,\n$u^{near}_\\ell(x^{\\ell};d,k)=e^{ik\\ell\\sin\\theta}u^{near}(x;d,k)$, $x\\in D_+$\nand $u^\\infty_\\ell(\\hat{x};d,k)=e^{ik\\ell(\\sin\\theta-\\hat{x}_1)}u^\\infty(\\hat{x};d,k)$, $\\hat{x}\\in\\Sp^1_+$\nwhere $x^{\\ell}:=x+(\\ell,0)^T$ for $x\\in\\R^2$ and $\\hat{x}_1$ is the first component of $\\hat{x}$.\n\\end{theorem}\n\n\\begin{proof}\nAssume that $D^\\ell_+$ is the unbounded domain above $\\G^\\ell$.\nLet $v(x):=e^{ik\\ell\\sin\\theta}u^s(x^{-\\ell};d,k)$ for $x\\in D^\\ell_+$.\nThen it is easily seen that $v$ is well defined in $D^\\ell_+$.\nThus, by the properties of the scattered field $u^s$ it follows that $v$ satisfies the Helmholtz equation (\\ref{eq1_nr})\nin $D^\\ell_+$ and the Sommerfeld radiation condition (\\ref{eq3_nr}).\nFurther, it can be seen that\n\\ben\nu^i(x,d,k)=e^{ik\\ell\\sin\\theta}u^i(x^{-\\ell},d,k),\\quad\nu^r(x,d,k)=e^{ik\\ell\\sin\\theta}u^r(x^{-\\ell},d,k)\n\\enn\nThen from the boundary condition (\\ref{eq2_nr}) on $u^s$, we have that $u^i(x,d,k)+u^r(x,d,k)+v(x)=0$ for $x\\in\\G^\\ell$.\nNow, the uniqueness result of the scattering problem (\\ref{eq1_nr})-(\\ref{eq3_nr}) implies that\n$v(x)=u^s_\\ell(x;d,k)$ for $x\\in D^\\ell_+$.\nTherefore, we obtain that $u^s_\\ell(x^{\\ell};d,k)=e^{ik\\ell\\sin\\theta}u^s(x;d,k)$,\n$u^{near}_\\ell(x^{\\ell};d,k)=e^{ik\\ell\\sin\\theta}u^{near}(x;d,k)$ for $x\\in D_+$.\n\nFinally, from the asymptotic behavior (\\ref{eq4}) of the scattered field and the fact that for $a\\in\\R^2$,\n\\ben\n|x-a|-|x|=-a\\cdot\\hat{x}+O\\left(\\frac{1}{|x|}\\right),\\quad |x|\\rightarrow\\infty\n\\enn\nuniformly for all directions $\\hat{x}=x\/|x|\\in\\Sp^1$, it is derived that\n$u^\\infty_\\ell(\\hat{x};d,k)=e^{ik\\ell(\\sin\\theta-\\hat{x}_1)}u^\\infty(\\hat{x};d,k)$ for $\\hat{x}\\in\\Sp^1_+$.\nThe proof is thus completed.\n\\end{proof}\n\nTheorem \\ref{th2} indicates that the modulus of the far field pattern (or the phaseless far-field pattern)\nfor one incident plane wave is invariant under translations along the $x_1$ direction of the boundary,\nthat is, $|u^\\infty_\\ell(\\hat{x};d,k)|=|u^\\infty(\\hat{x};d,k)|$, $\\hat{x}\\in\\Sp^1_+$ for all $\\ell\\in\\R.$\nThis means that the location of the local perturbation on the boundary $\\G$ can not be determined from\nthe phaseless far-field pattern for one incident plane wave, as shown in Example \\ref{fig4-1}.\nIn the next theorem we prove that, if the locally rough surface is non-trivial (that is, $x_2\\not=0$)\nthen the translation invariance property of the phaseless far-field pattern only holds for a countably infinite\nnumber of real numbers $\\ell$ in the case when the incident field is taken as a superposition of two plane waves\nwith different directions, that is, $u^i=u^i(x;d_1,d_2,k)$ with $d_1\\not=d_2$.\n\n\\begin{theorem}\\label{th3}\nLet the incident wave be given by $u^i=u^i(x;d_1,d_2,k)$ with $d_j=(\\sin\\theta_j,-\\cos\\theta_j),$\n$\\theta_j\\in(-\\pi\/2,\\pi\/2),\\;j=1,2$ and $\\theta_1\\neq\\theta_2$.\nAssume that $u^s(x;d_1,d_2,k),u^s_l(x;d_1,d_2,k)$ are the scattering solutions of the problem\n$(\\ref{eq1_nr})-(\\ref{eq3_nr})$, corresponding to the locally rough surface $\\G$ and the shifted one $\\G^\\ell$,\nrespectively, with $u^\\infty(\\hat{x};d_1,d_2,k),u^\\infty_\\ell(\\hat{x};d_1,d_2,k)$ the corresponding far-field\npatterns.\nAssume further that the locally rough surface $\\G$ is non-trivial (that is, $x_2\\not=0$ or $h_\\G\\not\\equiv0$).\nThen we have\n\\be\\label{TI}\n|u^\\infty(\\hat{x};d_1,d_2,k)|=|u^\\infty_\\ell(\\hat{x};d_1,d_2,k)|,\\;\\;\\hat{x}\\in\\Sp^1_+\n\\en\nfor all $\\ell=\\ell_n:=2\\pi n\/[k(\\sin\\theta_1-\\sin\\theta_2)]$ with any $n\\in\\Z$.\nFurther, except for $\\ell_n$, there may exist at most one real constant $\\tau$ with $0<\\tau<2\\pi$ such that\n(\\ref{TI}) holds for $\\ell=\\ell_{n\\tau}:=(2\\pi n+\\tau)\/[k(\\sin\\theta_1-\\sin\\theta_2)]$ with any $n\\in\\Z$.\n\\end{theorem}\n\n\\begin{proof}\nLet $v_j(\\hat{x}):=u^\\infty(\\hat{x};d_j,k)$, where $u^\\infty(\\hat{x};d_j,k)$ is the far-field pattern of the\nsolution $u^s(x;d_j,k)$ to the scattering problem (\\ref{eq1_nr})-(\\ref{eq3_nr}), corresponding to the incident field\n$u^i=u^i(x;d_j,k)$, $j=1,2.$ From the linearity of the scattering problem and the definition of $u^i({x};d_1,d_2,k)$,\nwe have that $u^\\infty(\\hat{x};d_1,d_2,k)=v_1(\\hat{x})+v_2(\\hat{x})$.\nFurther, by Theorem \\ref{th2} it follows that\n$$\nu^\\infty_\\ell(\\hat{x};d_1,d_2,k)=e^{ik\\ell(\\sin\\theta_1-\\hat{x}_1)}v_1(\\hat{x})\n+e^{ik\\ell(\\sin\\theta_2-\\hat{x}_1)}v_2(\\hat{x}).\n$$\nThen (\\ref{TI}) becomes\n\\ben\n|v_1(\\hat{x})+v_2(\\hat{x})|=|e^{ik\\ell(\\sin\\theta_1-\\hat{x}_1)}v_1(\\hat{x})\n+e^{ik\\ell(\\sin\\theta_2-\\hat{x}_1)}v_2(\\hat{x})|,\\quad\\hat{x}\\in\\Sp^1_+\n\\enn\nfor $\\ell\\in\\R$. By a direct calculation, the above equation is reduced to\n\\be\\label{c5eq6}\n\\Rt\\left(v_1(\\hat{x})\\ol{v_2(\\hat{x})}\\right)=\n\\Rt\\left(e^{ik\\ell(\\sin\\theta_1-\\sin\\theta_2)}v_1(\\hat{x})\\ol{v_2(\\hat{x})}\\right),\\quad\\hat{x}\\in\\Sp^1_+\n\\en\nfor $\\ell\\in\\R$.\n\nWe can claim that $v_1(\\hat{x})\\ol{v_2(\\hat{x})}\\not\\equiv 0,\\;\\;\\hat{x}\\in\\Sp^1_+.$\nIn fact, if this is not true, then we have\n\\be\\label{c5eq7}\nv_1(\\hat{x})\\ol{v_2(\\hat{x})}=0,\\quad\\hat{x}\\in\\Sp^1_+\n\\en\nWe now distinguish between the following two cases.\n\n{\\bf Case 1.} $v_1\\equiv0$ on $\\Sp^1_+$.\nSince $v_1(\\hat{x})$ is the far-field pattern of $u^s(x;d_1,k)$, then, by \\cite[Lemma 3.2]{Ma05}\nwe have that $u^s(x;d_1,k)\\equiv0$ in $D_+$. Further, from the boundary condition (\\ref{eq2_nr})\nfor $u^s(x;d_1,k)$ it follows that $u^i(x;d_1,k)+u^r(x;d_1,k)\\equiv0$ on $\\G$. This implies that\n$\\exp(2ikh_\\G(x_1)\\cos\\theta_1)\\equiv 1$ for all $x_1\\in\\R$. Thus, $kh_\\G(x_1)\\cos\\theta_1\\equiv n\\pi$\nfor all $x_1\\in\\R$, where $n=0,\\pm1,\\pm2,\\cdots$, so $h_\\G\\equiv 0$. This is a contradiction.\n\n{\\bf Case 2.} There exists an $\\hat{x}_0\\in\\Sp^1_+$ such that $v_1(\\hat{x}_0)\\neq0$.\nThen we have $v_1\\neq0$ in a neighborhood of $x_0$, which, together with (\\ref{c5eq7}), implies that\n$v_2=0$ in a neighborhood of $x_0$. Since $v_2$ is an analytic function on $\\Sp^1_+$ (see Remark \\ref{re3}),\nwe have $v_2\\equiv0$ on $\\Sp^1_+$. Arguing similarly as in Case 1 gives that $h_\\G\\equiv 0$,\ncontradicting to the assumption of the theorem.\n\nNow, by (\\ref{c5eq6}) it is easy to see that (\\ref{c5eq6}) or equivalently (\\ref{TI}) holds if $\\ell$\nsatisfies the condition\n\\be\\label{TI-C1}\nk\\ell(\\sin\\theta_1-\\sin\\theta_2)=2\\pi n,\\qquad n\\in\\Z,\n\\en\nor if $\\ell=2\\pi n\/[k(\\sin\\theta_1-\\sin\\theta_2)]$ with any $n\\in\\Z$.\nFurther, assume that $v_1(\\hat{x}_0)\\ov{v_2(\\hat{x}_0)}\\not=0$ for some $\\hat{x}_0\\in\\Sp^1_+$ and write $v_1(\\hat{x}_0)\\ov{v_2(\\hat{x}_0)}=|v_1(\\hat{x}_0)\\ov{v_2(\\hat{x}_0)}|\\exp(i\\theta(\\hat{x}_0))$\nwith $0<\\theta(\\hat{x}_0)<\\pi$. Then (\\ref{c5eq6}) is reduced to the equation\n\\be\\label{TI-C1a}\n\\cos[k\\ell(\\sin\\theta_1-\\sin\\theta_2)+\\theta(\\hat{x}_0)]-\\cos[\\theta(\\hat{x}_0)]=0.\n\\en\nBy (\\ref{TI-C1a}) we know that, except for the above $\\ell$ satisfying the condition (\\ref{TI-C1}),\n(\\ref{c5eq6}) or equivalently (\\ref{TI}) also holds for $\\ell$ satisfying the condition\n\\be\\label{TI-C1+}\nk\\ell(\\sin\\theta_1-\\sin\\theta_2)=2\\pi n-2\\theta(\\hat{x}_0),\\qquad n\\in\\Z,\n\\en\nor equivalently for\n\\be\\label{TI-C1b}\n\\ell=\\frac{2\\pi n-2\\theta(\\hat{x}_0)}{k(\\sin\\theta_1-\\sin\\theta_2)},\\;\\; n\\in\\Z.\n\\en\nThus, except for $\\ell=\\ell_n$, there may exist at most one real number $\\tau$ with $0<\\tau<2\\pi$ such that\n(\\ref{c5eq6}) or equivalently (\\ref{TI}) holds for $\\ell=(2\\pi n+\\tau)\/[k(\\sin\\theta_1-\\sin\\theta_2)]$\nwith any $n\\in\\Z$. In fact, by (\\ref{TI-C1b}) we have $\\tau=2\\pi-2\\theta(\\hat{x}_0)$.\nThe proof is thus complete.\n\\end{proof}\n\n\nTheorem \\ref{th3} indicates that the translation invariance property of the phaseless far-field pattern can be broken\nfor non-trivial locally rough surfaces if the incident field is taken as $u^i=u^i(x;d_1,d_2,k)$ with\n$d_1\\not=d_2$ and all wave numbers $k\\in[k_1,k_N]$ for some wave numbers $k_N>k_1>0.$\nThen it is expected that both the location and the shape of the local perturbation part of the boundary $\\G$ can be\nreconstructed simultaneously from the phaseless far-field data, corresponding to such incident fields with multiple\nwave numbers, as demonstrated in the numerical experiments.\n\nOn the other hand, from Theorem \\ref{th2} and Figure \\ref{fig4-7} it is expected that the translation invariance\nproperty of the phaseless near-field data measured on the line segment $\\G_{H,L}$ does not hold even for one incident\nplane wave though we can not prove this rigorously.\nThus, both the location and the shape of the local perturbation part of the boundary $\\G$ can also be\nreconstructed from the phaseless near-field data, corresponding to one incident plane wave (see the numerical examples).\n\nBased on the above discussions, we consider the following two inverse problems.\n\n{\\bf Inverse problem (IP1):} Given the incident fields $u^i=u^i(x;d_1,d_2,k)$ with multiple wave numbers\n$k=k_1,\\cdots,k_N$, where $d_1,d_2\\in\\Sp^1_-$ and $d_1\\neq d_2$, to reconstruct the locally rough surface $\\G$\nfrom the corresponding intensity-only far-field data $|u^\\infty(\\hat{x};d_1,d_2,k)|^2$, $\\hat{x}\\in\\Sp^1_+$,\n$k=k_1,\\cdots,k_N$.\n\n{\\bf Inverse problem (IP2):} Given the wave number $k>0$ and the incident field $u^i=u^i(x;d,k)$\nwith $d\\in\\Sp^1_-$, to reconstruct the locally rough surface $\\G$ from the corresponding intensity-only\nnear-field data $|u^{near}(x;d,k)|^2$, $x\\in\\G_{H,L}$.\n\nIn the next section we develop a recursive Newton-type iteration algorithm in frequencies for solving\nthe inverse problems (IP1) and (IP2).\nTo this end, given the incident wave $u^i=u^i(x;d_1,d_2,k)$ with $d_1,d_2\\in\\Sp^1_-$ and $d_1\\neq d_2$,\ndefine the far-field operator $\\mathcal{F}_{(d_1,d_2,k)}$ mapping the function $h_\\G$ which describes the locally\nrough surface $\\G$ to the intensity of the corresponding far-field pattern,\n$|u^\\infty(\\hat{x};d_1,d_2,k)|^2$ in $L^2(\\Sp^1_+)$ of the scattered wave $u^s(x;d_1,d_2,k)$ of the\nscattering problem (\\ref{eq1_nr})-(\\ref{eq3_nr}), that is,\n\\be\\label{FFO}\n\\big(\\mathcal{F}_{(d_1,d_2,k)}[h_\\G]\\big)(\\hat{x})=|u^\\infty(\\hat{x};d_1,d_2,k)|^2,\\quad\n\\hat{x}\\in\\Sp^1_+\n\\en\nSimilarly, given the incident wave $u^i=u^i(x;d,k)$ with $d\\in\\Sp^1_-$, define the near-field operator\n$\\mathcal{N}_{(d,k)}$ mapping the function $h_\\G$ to the intensity of the corresponding near-field,\n$|u^{near}(x;d,k)|^2$ in $L^2(\\G_{H,L})$, of the scattering problem (\\ref{eq1_nr})-(\\ref{eq3_nr}), that is,\n\\be\\label{eq5}\n\\big(\\mathcal{N}_{(d,k)}[h_\\G]\\big)(x)=|u^{near}(x;d,k)|^2,\\quad x\\in\\G_{H,L}\n\\en\n\nOur Newton-type iterative algorithm consists in solving the nonlinear and ill-posed equation (\\ref{FFO})\nor (\\ref{eq5}) for the unknown $h_\\G$. To this end, we need to investigate the Frechet differentiability of\n$\\mathcal{F}_{(d_1,d_2,k)}$ and $\\mathcal{N}_{(d,k)}$ at $h_\\G.$\nNow, let $\\triangle h\\in C^2_{0,R}(\\R)\\coloneqq\\{h\\in C^2(\\R):\\textrm{supp}(h)\\subset(-R,R)\\}$ be a small\nperturbation and let $\\G_{\\triangle h}\\coloneqq \\{(x_1,h_\\G(x)+\\triangle h(x)):x_1\\in\\R\\}$ denote the\ncorresponding boundary for $h_\\G(x)+\\triangle h(x)$.\nThen $\\mathcal{F}_{(d_1,d_2,k)}$ is said to be Frechet differentiable at $h_\\G$ if there exists a linear\nbounded operator $\\mathcal{F}'_{(d_1,d_2,k)}:C^2_{0,R}(\\R)\\rightarrow L^2(\\Sp^1_+)$ such that\n\\ben\n\\left\\|\\mathcal{F}_{(d_1,d_2,k)}[h_\\G+\\triangle h]-\\mathcal{F}_{(d_1,d_2,k)}[h_\\G]\n-\\mathcal{F}'_{(d_1,d_2,k)}[h_\\G;\\triangle h]\\right\\|_{L^2(\\Sp^1_+)}=o\\left(||\\triangle h||_{C^2(\\R)}\\right)\n\\enn\nThe Frechet differentiable at $h_\\G$ of $\\mathcal{N}_{(d,k)}$ is defined similarly.\nWe have the following theorem.\n\n\\begin{theorem}\\label{th1_nr}\n(i) Given the incident wave $u^i=u^i(x;d_1,d_2,k)$ with $d_1,d_2\\in\\Sp^1_-$ and $d_1\\neq d_2$,\nlet $u(x)=u^i(x)+u^r(x)+u^s(x)$, where $u^s$ solves the scattering problem $(\\ref{eq1_nr})-(\\ref{eq3_nr})$\nwith the boundary data $f=-(u^i+u^r)$. If $h_\\G\\in C^2(\\R)$, then $\\mathcal{F}_{(d_1,d_2,k)}$ is Frechet\ndifferentiable at $h_\\G$ with the derivative given by\n$\\mathcal{F}'_{(d_1,d_2,k)}[h_\\G;\\triangle h]=2\\Rt\\left[\\ov{u^\\infty}(u')^\\infty\\right]$ for\n$\\triangle h\\in C^2_{0,R}(\\R)$.\nHere, $(u')^\\infty$ is the far-field pattern of $u'$ which solves the scattering problem\n$(\\ref{eq1_nr})-(\\ref{eq3_nr})$ with the boundary data $f=-(\\triangle h\\cdot\\nu_2){\\pa u}\/{\\pa\\nu}$,\nwhere $\\nu_2$ is the second component of the unit normal $\\nu$ on $\\G$ directed into the infinite domain $D_+$.\n\n(ii) Given the incident wave $u^i=u^i(x;d,k)$ with $d\\in\\Sp^1_-$, let $u$ and $u^{near}$ be\nthe total and near-field, respectively, corresponding to the scattering problem $(\\ref{eq1_nr})-(\\ref{eq3_nr})$.\nIf $h_\\G\\in C^2(\\R)$, then $\\mathcal{N}_{(d,k)}$ is Frechet differentiable at $h_\\G$ with the derivative\ngiven by $\\mathcal{N}'_{(d,k)}[h_\\G;\\triangle h]=2\\Rt\\left[\\ov{u^{near}}u'\\right]$\nfor $\\triangle h\\in C^2_{0,R}(\\R)$, where $u'$ is the solution of the problem ($(\\ref{eq1_nr})-(\\ref{eq3_nr})$\nwith the boundary data $f=-(\\triangle h\\cdot\\nu_2){\\pa u}\/{\\pa\\nu}$.\n\\end{theorem}\n\n\\begin{proof}\nThe proof is similar to that of Theorems 2.1 and 2.2 in \\cite{BaoLiLv13} for\ninverse diffraction grating problems with appropriate modifications.\n\\end{proof}\n\n\n\n\\section{Reconstruction algorithm}\\label{se3}\n\\setcounter{equation}{0}\n\n\nIn this section, we describe the Newton-type iteration algorithm for the inverse problem (IP1).\nFor the inverse problem (IP2), the approach is similar, so we omit it.\n\nLet $d_{1l},d_{2l}\\in\\Sp^1_-,l=1,2,\\ldots,n_d,$ be the incident directions and let $k>0$\nbe the fixed wave number. Assume that $h^{app}$ is an approximation to the function $h_\\G$.\nWe replace (\\ref{FFO}) by the linearized equations:\n\\be\\label{LFFO}\n\\big(\\mathcal{F}_{(d_{1l},d_{2l},k)}[h^{app}]\\big)(\\hat{x})\n+\\big(\\mathcal{F}'_{(d_{1l},d_{2l},k)}[h^{app};\\triangle h]\\big)(\\hat{x})\n\\approx|u^\\infty(\\hat{x};d_{1l},d_{2l},k)|^2,\\quad l=1,2,\\ldots,n_d,\n\\en\nwhere $\\triangle h$ is the update function to be determined. Our Newton iterative algorithm consists in\niterating the equations (\\ref{LFFO}) by using the Levenberg-Marquardt algorithm (see, e.g. \\cite{Hohage1999}).\n\nIn the numerical examples, the noisy phaseless far-field pattern\n$|u^\\infty_{\\delta}(\\hat{x_j};d_{1l},d_{2l},k)|,j=1,2,\\ldots,n_f,l=1,2,\\ldots,n_d,$\nare considered as the measurement data which satisfies\n\\ben\n\\big\\||u^\\infty_{\\delta}(\\,\\cdot\\,;d_{1l},d_{2l},k)|^2-|u^\\infty(\\,\\cdot\\,;d_{1l},d_{2l},k)|^2\\big\\|_{L^2(\\Sp^1_+)}\n\\leq\\delta\\big\\||u^\\infty(\\,\\cdot\\,;d_{1l},d_{2l},k)|^2\\big\\|_{L^2(\\Sp^1_+)},\\quad l=1,2,\\ldots,n_d,\n\\enn\nwhere $\\delta>0$ is called the noise ratio and the observation directions $\\hat{x_j},j=1,2,\\ldots,n_f,$\nare the equidistant points on $\\Sp^1_+$.\nIn practical computation, $h^{app}$ has to be taken from a finite-dimensional subspace $R_M$.\nHere, $R_M=\\textrm{span}\\{\\phi_{1,M},\\phi_{2,M},\\cdots,\\phi_{M,M}\\}$ is a subspace of $C^2_{0,R}(\\R)$,\nwhere $\\phi_{j,M},j=1,2,\\ldots,M,$ are spline basis functions with support in $(-R,R)$ (see Remark \\ref{re1_nr}).\nThen, by the strategy in \\cite{Hohage1999}, we seek an updated function\n$\\triangle h=\\sum^M_{i=1}\\triangle a_i\\phi_{i,M}$ in $R_M$ so as to solve the minimization problem:\n\\be\\no\n&&\\min_{\\triangle a_i}\\left\\{\\sum_{l=1}^{n_d}\\sum_{j=1}^{n_f}\n\\Big|\\big(\\mathcal{F}_{(d_{1l},d_{2l},k)}[h^{app}]\\big)(\\hat{x}_j)\n+\\big(\\mathcal{F}_{(d_{1l},d_{2l},k)}'[h^{app};\\triangle h]\\big)(\\hat{x}_j)\\right.\\\\ \\label{eq15_nr}\n&&\\qquad\\qquad\\left. -|u^\\infty_{\\delta}(\\hat{x_j};d_{1l},d_{2l},k)|^2\\Big|^2+\\beta\\sum_{i=1}^M|\\triangle a_i|^2\\right\\}\n\\en\nwhere $\\beta>0$ is chosen such that\n\\be\\label{eq16_nr}\n\\left[\\sum_{l=1}^{n_d}\\sum_{j=1}^{n_f}\\Big|\\big(\\mathcal{F}_{(d_{1l},d_{2l},k)}[h^{app}]\\big)(\\hat{x}_j)\n+\\big(\\mathcal{F}_{(d_{1l},d_{2l},k)}'[h^{app};\\triangle h]\\big)(\\hat{x}_j)\n-|u^\\infty_{\\delta}(\\hat{x_j};d_{1l},d_{2l},k)|^2\\Big|^2\\right]^{\\half}\\no\\\\\n=\\rho\\left[\\sum_{l=1}^{n_d}\\sum_{j=1}^{n_f}\\Big|\\big(\\mathcal{F}_{(d_{1l},d_{2l},k)}[h^{app}]\\big)(\\hat{x}_j)\n-|u^\\infty_{\\delta}(\\hat{x_j};d_{1l},d_{2l},k)|^2\\Big|^2\\right]^{\\half}\n\\en\nfor a given constant $\\rho\\in(0,1)$. Then the approximation function $h^{app}$ is updated by $h^{app}+\\triangle h$.\nFurther, define the error function\n\\ben\nErr_k:=\\frac{1}{n_d}\\sum_{l=1}^{n_d}\n\\begin{array}{c}\n{\\left[\\sum\\limits_{j=1}^{n_f}\\Big|\\big(\\mathcal{F}_{(d_{1l},d_{2l},k)}[h^{app}]\\big)(\\hat{x}_j)\n-|u^\\infty_{\\delta}(\\hat{x_j};d_{1l},d_{2l},k)|^2\\Big|^2\\right]^{1\/2}} \\\\\n\\hline\n{\\left[\\sum\\limits_{j=1}^{n_f}|u^\\infty_{\\delta}(\\hat{x_j};d_{1l},d_{2l},k)|^4\\right]^{1\/2}}\n\\end{array}\n\\enn\nThen the iteration is stopped if $Err_k< \\tau\\delta$, where $\\tau>1$ is a fixed constant.\n\n\\begin{remark}\\label{re1_nr} {\\rm\nFor a positive integer $M\\in\\N^+$ let $h=2R\/(M+5)$ and $t_i=(i+2)h-R$.\nThen the spline basis functions of $R_M$ are defined by\n$\\phi_{i,M}(t)=\\phi((t-t_i)\/h),i=1,2,\\ldots,M,$ where\n\\ben\n\\phi(t):=\\sum^{k+1}_{j=0}\\frac{(-1)^j}{k!}\\left(\\begin{array}{c}k+1\\\\\n                                      j\\end{array}\\right)\n       \\left(t+\\frac{k+1}{2}-j\\right)^k_+\n\\enn\nwith $z^k_+=z^k$ for $z\\geq0$ and $=0$ for $z<0$.\nIn this paper, we choose $k=4$, that is, $\\phi$ is the quartic spline function.\nNote that $\\phi_{i,M}\\in C^3(\\R)$ with support in $(-R,R)$. See \\cite{DeBoor} for details.\n}\n\\end{remark}\n\n\\begin{remark}\\label{re2_nr} {\\rm\nThe integral equation method proposed in \\cite{ZhangZhang2013} (cf. Section \\ref{se1}) is used to solve\nthe direct scattering problem in each iteration.\n}\n\\end{remark}\n\nOur recursive Newton iteration algorithm in frequencies is given in Algorithm \\ref{alg1} for the inverse problem (IP1).\n\n\\begin{algorithm}\\label{alg1}\nGiven the phaseless far-field data $|u^\\infty_{\\delta}(\\hat{x_j};d_{1l},d_{2l},k_m)|$,\n$j=1,2,\\ldots,n_f$, $l=1,2,\\ldots,n_d$, $m=1,2,\\ldots,N$ with $d_{1l},d_{2l}\\in\\Sp^1_-$, $d_{1l}\\neq d_{2l}$\nand $k_1<k_2<\\cdots<k_N$.\n\n1) Let $h^{app}$ be the initial guess of $h_\\G$. Set $i=0$ and go to Step 2).\n\n2) Set $i=i+1$. If $i>N$, then stop the iteration; otherwise, set $k=k_i$ and go to Step 3).\n\n3) If $Err_k<\\tau\\delta$, go to Step 2); otherwise, go to Step 4).\n\n4) Solve (\\ref{eq15_nr}) with the strategy (\\ref{eq16_nr}) to get an updated function $\\triangle h$.\nLet $h^{app}$ be updated by $h^{app}+\\triangle h$ and go to Step 3).\n\\end{algorithm}\n\n\\begin{remark}\\label{re3_nr} {\\rm\nSince $\\mathcal{F}'_{(d_1,d_2,k)}[0;\\triangle h]=0$ for any $d_1,d_2\\in\\Sp^1_-$, $k>0$ and\n$\\triangle h\\in C^2_{0,R}(\\R^2)$, then the initial guess $h^{app}$ should not be zero for\nthe inverse problem (IP1). However, in all the numerical examples for the inverse problem (IP2),\nthe initial guess of $h_\\G$ is chosen to be $h^{app}=0$.\n}\n\\end{remark}\n\n\n\\section{Numerical experiments}\\label{se4}\n\n\nIn this section, several numerical experiments are carried out to illustrate the effectiveness of\nthe inversion algorithm. The following assumptions are made in all numerical experiments.\n\n1) For each example we use multi-frequency data with the wave numbers $k=1,3,\\ldots,2N-1,$ where $N$\nis the total number of frequencies.\n\n2) To generate the synthetic data, the integral equation method is used to the direct scattering problem.\nFor the inverse problem (IP1), the intensity of the far-field pattern is measured along the upper half-aperture\n(that is, the measurement angle is between $0$ and $\\pi$) with $200$ equidistant measurement points.\nFor the inverse problem (IP2), the intensity of the near-field is measured on the straight line segment\n$\\G_{1,1}:=\\{(x_1,1)\\;:\\;x_1\\in[-1,1]\\}$ also with $200$ equidistant measurement points.\nThe corresponding noisy data $|u^\\infty_{\\delta}|$ and $|u^{near}_{\\delta}|$ are simulated as\n$|u^\\infty_{\\delta}|^2=|u^\\infty|^2(1+\\delta\\zeta)$ and $|u^{near}_{\\delta}|^2=|u^{near}|^2(1+\\delta\\zeta)$,\nrespectively, where $\\zeta$ is a normally distributed random number in $[-1,1].$\nIn all the numerical examples, the noise level is taken as $\\delta=5\\%$.\n\n3) The parameters are taken as $\\rho=0.8$ and $\\tau=1.5$.\n\n4) In all the numerical examples, the local perturbation of the infinite plane is assumed to be restricted to\nthe range $-1<x_1<1$, that is, $\\textrm{supp}(h_\\G)\\in(-1,1)$.\n\n\\textbf{Example 1 (Shape reconstruction).} We first demonstrate numerically that the location of the local\nperturbation on the infinite plane can not be determined from the intensity far-field data if only one plane\nwave is used as the incident field, that is, $u^i(x)=u^i(x;d,k)$. The locally rough surface is given by\n\\ben\nh_\\G(x_1)=\\phi(({x_1+0.2})\/{0.3}),\n\\enn\nwhere $\\phi$ is defined as in Remark \\ref{re1_nr}, and the incident direction is taken as\n$d=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$. The number of the spline basis functions is chosen to be $M=10$,\nand the total number of frequencies used is $N=10$. Further, we choose two different initial guesses\nfor the iteration inversion algorithm: 1) $h^{app}=0.1\\sum_{i=2}^4\\phi_{i,M}$, and\n2) $h^{app}=0.1\\sum_{i=5}^7\\phi_{i,M}$, where the corresponding reconstructed curves\nare denoted as \"Reconstructed curve 1\" and \"Reconstructed curve 2\".\nFigure \\ref{fig4-1} presents the initial curves and the reconstructed curves at the wavenumbers $k=1,7,19$.\nFrom the reconstructions presented in Figure \\ref{fig4-1} it is seen that\nthe shape of the local perturbation can be well reconstructed numerically, but its reconstructed location\nis translated along the $x_1$ direction, depending on the choice of initial guess. This means that\nthe location of the local perturbation can not be recovered from the intensity far-field data\nif one plane wave is taken as the incident field.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfigure{\\includegraphics[width=3in]{example\/IP0\/case3\/initial_curve.eps}}\n  \\hspace{-0.35in}\n  \\vspace{-0.5in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP0\/case3\/rough3_k_1_error5.eps}}\n \n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP0\/case3\/rough3_k_7_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP0\/case3\/rough3_k_19_error5.eps}}\n  \\vspace{-0.5in}\n\\caption{Shape reconstruction of a locally rough surface from $5\\%$ noisy intensity far-field data\nwith one incident plane wave $u^i=u^i(x;d,k)$, where $d=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$.\nHere, the initial and reconstructed curves are presented at the wave numbers $k=1,7,19.$\n}\\label{fig4-1}\n\\end{figure}\n\n\\textbf{Example 2 (Shape and location reconstruction).} We consider the same locally rough surface as\nin Example 1. Here, we demonstrate that by using a superposition of two plane waves with different\ndirections as the incident field, that is, $u^i=u^i(x;d_1,d_2,k)$ with $d_1\\not=d_2$,\nboth the location and the shape of the local perturbation on the rough surface can be reconstructed from\nthe corresponding intensity far-field data.\nIn the numerical experiment, the two incident directions are taken as $d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and\n$d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$, the number of the spline basis functions is chosen to be $M=10$, and\nthe total number of frequencies used is $N=13$. Further, we use the same initial guesses and\nthe same terms for the corresponding reconstructed curves of $h_\\G$ as in Example 1.\nThe initial and reconstructed curves are presented in Figure \\ref{fig4-2}, at the wavenumbers $k=1,7,25,$\nrespectively. From Figures \\ref{fig4-1} and \\ref{fig4-2} it can be seen that both the location and the shape\nof the local perturbation are accurately reconstructed with two different initial guesses.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case3\/initial_curve.eps}}\n  \\hspace{-0.35in}\n  \\vspace{-0.5in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case3\/rough3_k_1_error5.eps}}\n \n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case3\/rough3_k_7_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case3\/rough3_k_25_error5.eps}}\n  \\vspace{-0.5in}\n\\caption{Location and shape reconstruction of a locally rough surface from $5\\%$ noisy intensity far-field data\nwith a superposition of two plane waves as the incident field $u^i=u^i(x;d_1,d_2,k)$,\nwhere $d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nHere, the initial and reconstructed curves are presented at the wave numbers $k=1,7,25$.\n}\\label{fig4-2}\n\\end{figure}\n\n\\textbf{Example 3 (Piecewise linear curve).} We now consider the inverse problem (IP1) with a piecewise linear local perturbation\n(the solid line in Figure \\ref{c5fig4-9}). The number of the spline basis functions is chosen as $M=40$, the total number of\nfrequencies used is taken to be $N=18$, and the initial guess for the rough surface profile $h_\\G$ is chosen as\n$h^{app}=0.05\\sum_{i=5}^{15}\\phi_{i,M}$. Figure \\ref{c5fig4-9} presents the initial and reconstructed curves\nat the wavenumbers $k=7,21,35$, respectively, obtained by using the intensity far-field data with $5\\%$ noise,\ncorresponding to the incident wave $u^i=u^i(x;d_1,d_2,k)$ with\n$d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nThe reconstruction results show that the piecewise linear surface profile is very accurately reconstructed even\nat the corners of the boundary.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfigure{\\includegraphics[width=2.8in]{example\/IP1\/case9\/initial_curve.eps}}\n  \\hspace{-0.35in}\n  \\vspace{-0.5in}\n  \\subfigure{\\includegraphics[width=2.8in]{example\/IP1\/case9\/rough9_k_7_error5.eps}}\n \n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=2.8in]{example\/IP1\/case9\/rough9_k_21_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=2.8in]{example\/IP1\/case9\/rough9_k_35_error5.eps}}\n  \\vspace{-0.5in}\n\\caption{Location and shape reconstruction of a piecewise linear local perturbation from $5\\%$ noisy intensity\nfar-field data with a superposition of two plane waves as the incident field $u^i=u^i(x;d_1,d_2,k)$,\nwhere $d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nHere, the initial and reconstructed curves are presented at the wave numbers $k=7,21,35$.\n}\\label{c5fig4-9}\n\\end{figure}\n\n\\textbf{Example 4 (multi-scale curve).} Consider the inverse problem (IP1) again with a multi-scale surface profile given by\n\\ben\nh_\\G(x_1)=\\left\\{\\begin{array}{ll}\n\\ds \\exp\\left[16\/(25x_1^2-16)\\right]\\left[0.5+0.1\\sin(16\\pi x_1)\\right], &|x_1|<4\/5\\\\\n\\ds 0, &|x_1|\\geq4\/5.\n\\end{array}\\right.\n\\enn\nThis function has two scales: the macro-scale represented by the function $0.5\\exp[16\/(25x_1^2-16)]$ and the micro-scale\nrepresented by the function $0.1\\exp[16\/(25x_1^2-16)]\\sin(16\\pi x_1)$.\nFor the inverse problem, the number of the spline basis functions is chosen to be $M=40$, the total number of\nfrequencies used is $N=30$, and the initial guess for $h_\\G$ is $0.05\\sum_{i=10}^{30}\\phi_{i,M}$.\nFigure \\ref{fig4-6} presents the initial curve and the reconstructed curves at $k=19,33,39,49,59$,\nobtained by using the intensity far-field data with $5\\%$ noise, with respect to the incident wave\n$u^i=u^i(x;d_1,d_2,k)$, where $d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nIt is observed from Figure \\ref{fig4-6} that the macro-scale of $h_\\G$ is recovered at low frequencies (e.g. $k=19$),\nbut the micro-scale of $h_\\G$ is not recovered completely even in the case when data with higher frequencies are used.\n\nIn order to get a better reconstruction, more measurement data are used.\nFigure \\ref{fig4-3} presents the initial and reconstructed curves at $k=19,33,39,49,59,$\nobtained by using more intensity far-field data generated with the incident waves $u^i(x)=u^i(x;d_{1l},d_{2l},k),$ $l=1,2$,\nwhere $d_{11}=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6)),d_{21}=(0,-1),d_{12}=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$ and $d_{22}=(0,-1)$.\nCompared with Figure \\ref{fig4-6}, the micro-scale of the surface profile $h_\\G$ is accurately recovered.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/initial_curve.eps}}\n  \\hspace{-0.35in}\n  \\vspace{0in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/two_wave\/rough6_k_19_error5}}\n \n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/two_wave\/rough6_k_33_error5}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/two_wave\/rough6_k_39_error5}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/two_wave\/rough6_k_49_error5}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/two_wave\/rough6_k_59_error5}}\n\\caption{Reconstruction of a two-scale surface profile from $5\\%$ noisy intensity far-field data\nwith a superposition of two plane waves as the incident field $u^i=u^i(x;d_1,d_2,k)$,\nwhere $d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nHere, the initial and reconstructed curves are presented at the wave numbers $k=19,33,39,49,59$.\n}\\label{fig4-6}\n\\end{figure}\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/initial_curve.eps}}\n  \\hspace{-0.35in}\n  \\vspace{0in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/four_wave\/rough6_k_19_error5}}\n \n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/four_wave\/rough6_k_33_error5}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/four_wave\/rough6_k_39_error5}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/four_wave\/rough6_k_49_error5}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/four_wave\/rough6_k_59_error5}}\n\\caption{Reconstruction of a two-scale surface profile from $5\\%$ noisy intensity far-field data\ngenerated with two superpositions of two plane waves as the incident fields $u^i=u^i(x;d_{1l},d_{2l},k),l=1,2$,\nwhere $d_{11}=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6)),d_{21}=(0,-1),d_{12}=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$ and $d_{22}=(0,-1)$.\nHere, the initial and reconstructed curves are presented at the wave numbers $k=19,33,39,49,59$.\n}\\label{fig4-3}\n\\end{figure}\n\n\\textbf{Example 5 (multi-scale curve).} We consider the inverse problem (IP1) again with the multi-scale surface profile given by\n\\ben\nh_\\G(x_1)=\\left\\{\\begin{array}{ll}\n\\ds \\exp\\left[16\/(25x_1^2-16)\\right]\\left[0.5+0.1\\sin(16\\pi x_1)\\right]\\sin(\\pi x_1), &|x_1|<4\/5\\\\\n\\ds 0, &|x_1|\\geq4\/5\n\\end{array}\\right.\n\\enn\nThis profile consists of a macro-scale represented by $0.5\\exp\\left[16\/(25x_1^2-16)\\right]\\sin(\\pi x_1)$\nand a micro-scale represented by $0.1\\exp\\left[16\/(25x_1^2-16)\\right]\\sin(16\\pi x_1)\\sin(\\pi x_1)$.\nFor the inverse problem, the number of the spline basis functions is chosen to be $M=40$, the total number of frequencies\nused is $N=35$, and the initial guess for $h_\\G$ is $0.05\\sum_{i=25}^{35}\\phi_{i,M}$.\nFigure \\ref{c5fig4-10} presents the initial and reconstructed curves at $k=19,39,49,59,69$, respectively,\nobtained by using $5\\%$ noisy intensity far-field data generated with the incident wave $u^i=u^i(x;d_1,d_2,k)$,\nwhere $d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nFrom Figure \\ref{c5fig4-10} it is observed that the macro-scale features are captured at $k=19$\nand the reconstruction improves as the measurement data with higher frequencies are used.\nFinally, the whole locally rough surface (including the micro-scale features of the boundary)\nis accurately recovered at $k=69$.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfigure{\\includegraphics[width=2.8in]{example\/IP1\/case7\/initial_curve.eps}}\n  \\hspace{-0.35in}\n  \\vspace{0in}\n  \\subfigure{\\includegraphics[width=2.8in]{example\/IP1\/case7\/rough7_k_19_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n    \\subfigure{\\includegraphics[width=2.8in]{example\/IP1\/case7\/rough7_k_39_error5.eps}}\n  \\hspace{-0.35in}\n  \\vspace{0in}\n  \\subfigure{\\includegraphics[width=2.8in]{example\/IP1\/case7\/rough7_k_49_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=2.8in]{example\/IP1\/case7\/rough7_k_59_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=2.8in]{example\/IP1\/case7\/rough7_k_69_error5.eps}}\n  \\vspace{0in}\n\\caption{Reconstruction of a two-scale surface profile from $5\\%$ noisy intensity far-field data\ngenerated with a superposition of two plane waves as the incident field $u^i=u^i(x;d_1,d_2,k)$,\nwhere $d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nHere, the initial and reconstructed curves are presented at the wave numbers $k=19,39,49,59,69$.\n}\\label{c5fig4-10}\n\\end{figure}\n\n\\textbf{Example 6 (piecewise linear curve).} We now consider the inverse problem (IP2) with the locally rough surface\ngiven as in Example 3. For the inverse problem, the number of the spline basis functions is chosen to be $M=40$,\nthe total number of frequencies is assumed to be $N=18$, and the initial guess for the reconstructed curve is taken as\nthe infinite plane $x_2=0$. In Figure \\ref{fig4-7}, we present the initial curve and the reconstructed curves at $k=7,15,31,$\nrespectively, which are obtained from the $5\\%$ intensity near-field data generated with the incident wave $u^i=u^i(x;d,k)$,\nwhere $d=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$. It can be seen that the reconstruction at the illuminated part of the boundary is\nbetter than that at the shadowed part. Thus, in order to improve the reconstruction, more measurement data are needed.\nFigure \\ref{fig4-4} presents the initial curve and the reconstructed curves at $k=7,15,35,$ respectively, obtained by using\nthe $5\\%$ intensity near-field data corresponding to two incident waves $u^i=u^i(x;d_l,k),l=1,2$,\nwhere $d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nIt is found that the piecewise linear surface profile is accurately reconstructed even at the corners of the surface by\nusing two incident plane waves with different directions.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case9\/initial_curve.eps}}\n  \\hspace{-0.35in}\n  \\vspace{-0.5in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case9\/one_wave\/rough9_k_7_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case9\/one_wave\/rough9_k_15_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case9\/one_wave\/rough9_k_31_error5.eps}}\n  \\vspace{-0.5in}\n\\caption{Reconstruction of a piecewise linear surface profile from $5\\%$ noisy intensity near-field data\ngenerated with one incident plane wave $u^i=u^i(x;d,k)$, where $d=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$.\nHere, the initial and reconstructed curves are presented at the wavenumbers $k=7,15,31,$ respectively.\n}\\label{fig4-7}\n\\end{figure}\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case9\/initial_curve.eps}}\n  \\hspace{-0.35in}\n  \\vspace{-0.5in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case9\/two_wave\/rough9_k_7_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case9\/two_wave\/rough9_k_15_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case9\/two_wave\/rough9_k_35_error5.eps}}\n  \\vspace{-0.5in}\n\\caption{Reconstruction of a piecewise linear surface profile from $5\\%$ noisy intensity near-field data\ngenerated with two incident plane waves $u^i=u^i(x;d_l,k),l=1,2$, where $d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$\nand $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nHere, the initial and reconstructed curves are presented at the wave numbers $k=7,15,35$.\n}\\label{fig4-4}\n\\end{figure}\n\n\\textbf{Example 7 (multi-scale curve).} We consider the inverse problem (IP2) again with the multi-scale surface profile\ngiven as in Example 5. For the inverse problem, the number of the spline basis functions is chosen to be $M=40$,\nthe total number of frequencies is set to be $N=30$, and the initial guess for the reconstructed curve is taken as\nthe infinite plane $x_2=0$. In Figure \\ref{fig4-8},\nwe present the initial curve and the reconstructed curves at $k=13,29,39,49,59$,\nobtained from the $5\\%$ noisy intensity near-field data generated with one incident plane wave $u^i=u^i(x;d,k)$,\nwhere $d=(0,-1)$. It is observed that the micro-scale of the boundary surface is not recovered accurately.\nFigure \\ref{fig4-5} presents the initial curve and the reconstructed curves at $k=13,29,39,49,59,$ respectively,\nobtained from the $5\\%$ noisy intensity near-field data corresponding to two incident plane waves $u^i=u^i(x;d_l,k),l=1,2$,\nwith two different directions $d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nCompared with Figure \\ref{fig4-8} it can be seen that the multi-scale surface profile can be accurately recovered\nby using two incident plane waves with different directions.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/initial_curve.eps}}\n  \\hspace{-0.35in}\n  \\vspace{0in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/one_wave\/rough7_k_13_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/one_wave\/rough7_k_29_error5.eps}}\n  \\hspace{-0.35in}\n  \\vspace{0in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/one_wave\/rough7_k_39_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/one_wave\/rough7_k_49_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/one_wave\/rough7_k_59_error5.eps}}\n  \\vspace{0in}\n\\caption{Reconstruction of a multi-scale surface profile from $5\\%$ noisy intensity near-field data\ngenerated by one incident plane wave $u^i=u^i(x;d,k)$ with a normal incidence ($d=(0,-1)$).\nHere, the initial and reconstructed curves are presented at $k=13,29,39,49,59$.\n}\\label{fig4-8}\n\\end{figure}\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/initial_curve.eps}}\n  \\hspace{-0.35in}\n  \\vspace{0in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/two_wave\/rough7_k_13_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/two_wave\/rough7_k_29_error5.eps}}\n  \\hspace{-0.35in}\n  \\vspace{0in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/two_wave\/rough7_k_39_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/two_wave\/rough7_k_49_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/two_wave\/rough7_k_59_error5.eps}}\n  \\vspace{0in}\n\\caption{Reconstruction of a multi-scale surface profile from $5\\%$ noisy intensity near-field data\ngenerated by two incident plane waves $u^i=u^i(x;d_l,k),l=1,2$ with different directions\n$d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nHere, the initial and reconstructed curves are presented at $k=13,29,39,49,59.$\n}\\label{fig4-5}\n\\end{figure}\n\n\n\n\\section{Conclusion}\\label{se5}\n\n\nIn this paper, it is proved that the translation invariance along the $x_1$-direction of the phaseless far-field pattern\ncan be broken down if superpositions of two plane waves with different directions are used as the incident fields.\nAn efficient multi-frequency Newton iterative algorithm is then proposed for reconstructing the locally rough surface profile\nfrom the intensity of the far-field patterns (called the intensity-only far-field data) generated by such incident fields.\nThe algorithm can also be applied to recover the locally rough surface profile from the intensity of the scattering waves\nmeasured at a straight line segment with a constant height above the surface (called the intensity-only near-field data),\ngenerated by one incident plane wave.\nAt each iteration, an efficient integral equation solver is used to solve the direct scattering problem which\nwas proposed previously in \\cite{ZhangZhang2013}, so multiple scattering is considered.\nSeveral numerical examples presented here indicate that our inversion algorithm with multi-frequency phaseless far-field and\nnear-field data gives a stable and accurate reconstruction of the local perturbation of the infinite plane,\neven in the multi-scale case.\nThe reconstruction results are similar to those obtained by the inversion algorithms in \\cite{BaoLin11,BaoLin13,ZhangZhang2013}\nwith using the full far-field and near-field data (including the phase information).\nThe main reason for this, we think, is that our inversion algorithm considers both multiple scattering (through the use of\nthe fast integral equation solver for the direct problem) and multiple frequency data.\nIt was indicated in \\cite{pre06,josa08} that multiple scattering plays a key role in subwavelength imaging from far-field data.\nOur reconstruction results here together with those given in \\cite{BaoLin11,BaoLin13,ZhangZhang2013}\nillustrate that multiple scattering in conjunction with using multiple frequency data plays an essential role in\nsubwavelength imaging from both far-field and near-field data.\n\n\n\\section*{Acknowledgements}\n\nThis work was partly supported by the NNSF of China grants 61379093, 91430102 and 11501558.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:intro}\n\t\n\tWhen handling real-world optimization problems by means of mathematical modeling, it is often necessary to treat inexact or uncertain input data due to various measurement errors or estimations. Throughout the years, several approaches for solving optimization problems with imprecise data have emerged, based on different sources of uncertainty and different requirements imposed on the solutions. This paper adopts the approach of interval linear programming, which can be considered a special case of multi-parametric optimization with no dependencies among the coefficients. Interval-valued coefficients have been used for modeling uncertainty in many practical applications, such as portfolio selection \\cite{Lai:2002, Kumar:2016}, environmental management \\cite{Tan:2010, Cheng:2015} or group decision making \\cite{Groselj:2017}.\n\t\n\tIn interval optimization, it is assumed that the coefficients may perturb independently within the given lower and upper bounds. However, rather than focusing on the worst case and trying to find a stable solution (as in robust optimization), we aim to cover the properties of all possible scenarios. This approach leads to questions such as describing the set of feasible solutions over all scenarios \\cite{Oettli:1964, Rohn:InexactLP:2006} and the set of all optimal solutions \\cite{Allahdadi:2013:opt}, computing the optimal value range \\cite{Mraz:1998, Rohn:InexactLP:2006b, Hladik:2009} or characterizing the duality gaps \\cite{Novotna:2017}. For an overview of other results in interval linear programming see the survey by \\cite{Hladik:2012}.\n\t\n\tThis paper discusses an ever-present issue in interval optimization and interval analysis in general: \\emph{the dependency problem}. Since we do not allow any dependencies among the coefficients of an interval program and assume that the values of the coefficients perturb independently, any transformation that changes the program and duplicates some of the interval coefficients will also create new possible scenarios. Due to the dependency problem, transforming an interval program into a desired form by applying the standard transformations used in classical linear programming may change its feasible or optimal solutions and other important properties. Therefore, it may be necessary to study different types of programs separately, since the results obtained for one type do not have to hold for other types.\n\t\n\tWe examine the effects of the transformations on the optimal solution set and the optimal value range in order to identify the transformations that are also applicable to interval linear programs while preserving the considered properties. This will allow us to directly generalize the results known for one type of programs to other forms, as well as highlight the cases in which the particular properties may differ. We will also study the problem on a special class of interval linear programs with interval coefficients only appearing in the objective function and the right-hand-side vector in order to derive stronger results. These types of programs arise in applications, where the coefficient matrix represents some known relations, such as transportation problems \\cite{Safi:2013, Cerulli:2017:inttransp} or network flow problems \\cite{Hashemi:2006}.\n\t\n\tThe paper is structured as follows: Section~\\ref{sec:ilp} introduces the basic notions and terminology of interval linear programming. In Section~\\ref{sec:transf}, we review the standard transformations used in classical linear programming and discuss their applicability on interval programs. Sections~\\ref{sec:optsol} and~\\ref{sec:optval} explore the effects of the transformations on the set of all optimal solutions and optimal values of an interval linear program. Section~\\ref{sec:concl} summarizes and concludes the paper. \n\t\n\t\\section{Interval Linear Programming}\\label{sec:ilp}\n\tGiven two real matrices $\\lb{A}, \\ub{A} \\in \\mathbb{R}^{m\\times n}$ satisfying $\\lb{A} \\le \\ub{A}$, we define an \\emph{interval matrix} as the set \\[\\bint{A} = [\\lb{A}, \\ub{A}] = \\{A \\in \\mathbb{R}^{m \\times n} : \\lb{A} \\le A \\le \\ub{A}\\}.\\]\n\tThe matrices $\\lb{A}, \\ub{A}$ are the \\emph{lower} and \\emph{upper bound} of $\\bint{A}$, respectively. Alternatively, we can also define an interval matrix by its \\emph{center} $A_c = \\frac{1}{2}(\\ub{A} + \\lb{A})$ and \\emph{radius} $A_\\Delta = \\frac{1}{2}(\\ub{A} - \\lb{A})$.\n\tThroughout the paper, we denote interval objects by bold lowercase (for one-dimensional intervals and interval vectors) and bold uppercase (for interval matrices) letters. The symbol $\\mathbb{IR}$ is used to denote the set of all (real) intervals.\n\t\n\tFurthermore, let us extend the classical definition of linear programs to the case of uncertain interval coefficients. We define an \\emph{interval linear program} (ILP) as a family of linear programs\n\t\\begin{equation}\\label{eq:ilp:def}\n\t\\{ \\text{minimize } c^T x \n\t\\text{ subject to} \\  x \\in \\mathcal{M}(A,b) : A \\in \\bint{A}, b \\in \\bint{b}, c \\in \\bint{c} \\}\n\t\\end{equation}\n\twhere $\\mathcal{M}(A,b)$ denotes the feasible set described by some linear constraints. For the sake of brevity, we usually write ILPs in the form of linear programs with interval coefficients (see \\eqref{eq:ilp:A}--\\eqref{eq:ilp:C} below for examples). A particular linear program in the family is called a~\\emph{scenario}. Given an interval linear program, we can define its \\emph{dual} ILP as the family of all dual linear programs.\n\t\n\tSimilarly, we can also generalize the notion of feasibility and optimality to the interval case. We say that a vector $x \\in \\mathbb{R}^n$ is a \\emph{(weakly) feasible\/optimal} solution to the interval linear program, if there exists a scenario determined by the triplet $A \\in \\bint{A}$, $b \\in \\bint{b}, c \\in \\bint{c}$ such that $x$ is a feasible\/optimal solution for the scenario (see \\cite{Rohn:InexactLP:2006} and \\cite{Hladik:2017} for an overview of the feasibility properties of interval linear systems). \n\tThe \\emph{(weakly) feasible\/optimal set} is the set of all weakly feasible\/optimal solutions of an ILP.\n\tWe often omit the word ``weakly'', since no confusion should arise in the paper.\n\t\n\tConsidering the optimal objective values, we may be interested in the best and the worst value that can be achieved as optimal for some scenario of the given ILP. For a minimization program, these bounds are defined as \n\t\\begin{align*}\n\t&\\lb{f}(\\bint{A}, \\bint{b}, \\bint{c}) = \\inf \\{f(A,b,c): A \\in \\bint{A}, b \\in \\bint{b}, c \\in \\bint{c}\\},\\\\\n\t&\\ub{f}(\\bint{A}, \\bint{b}, \\bint{c}) = \\sup \\{f(A,b,c): A \\in \\bint{A}, b \\in \\bint{b}, c \\in \\bint{c}\\},\n\t\\end{align*}\n\twhere $f(A,b,c)$ denotes the optimal value of the corresponding linear program, allowing the values $-\\infty$ and $\\infty$ for unbounded programs or infeasible programs.\n\tThe interval $[\\lb{f}(\\bint{A}, \\bint{b}, \\bint{c}), \\ub{f}(\\bint{A}, \\bint{b}, \\bint{c})]$ is called the \\emph{optimal value range}.\n\t\n\tDue to the dependency problem, we usually assume that an~ILP is given either in the general form\n\t\\begin{equation*}\n\t\\begin{array}{lrrrr@{\\ }l}\n\t\\text{minimize } & \\multicolumn{5}{l}{\\bint{c_1}^T x_1 + \\bint{c_2}^T x_2} \\\\\n\t\\text{subject to } & \\bint{A}x_1 &+ & \\bint{B}x_2 & = & \\bint{b_1},\\\\\n\t&  \\bint{C}x_1 & + & \\bint{D}x_2 & \\le & \\bint{b_2},\\\\\n\t& &  \\multicolumn{2}{r}{x_1, x_2} &\\ge & 0,\n\t\\end{array}\n\t\\end{equation*}\n\tor in one of the following forms often encountered in optimization problems:\n\t\\begin{align}\n\t&\\text{minimize } \\bint{c}^T x \\text{ subject to } \\bint{A}x = \\bint{b},\\;x \\ge 0, \\tag{I}\\label{eq:ilp:A}\\\\ \n\t&\\text{minimize } \\bint{c}^T x \\text{ subject to } \\bint{A}x \\le \\bint{b}, \\tag{II}\\label{eq:ilp:B}\\\\\n\t&\\text{minimize } \\bint{c}^T x \\text{ subject to } \\bint{A}x \\le \\bint{b},\\;x \\ge 0. \\tag{III}\\label{eq:ilp:C}\n\t\\end{align}\n\t\n\t\\section{Transformations of Interval Linear Programs}\\label{sec:transf}\n\tLet us now briefly review the basic transformations that are used in linear programming to convert a given program into the desired form. While some of them can also be applied to interval programs, other transformations may change the set of feasible or optimal solutions and affect the properties of the program. The effects of these transformations on the set of all optimal solutions and optimal values will be discussed in more detail in Sections~\\ref{sec:optsol} and~\\ref{sec:optval}.\n\t\n\t\\paragraph{Changing the objective.} A commonly used trick in linear programming is to switch the objective from maximization to minimization (or vice versa) by taking the opposite of the objective function. Since this transformation does not duplicate any coefficients, it can also be directly applied to interval programs: an interval objective $\\max \\bint{c}^T x$ can be equivalently rewritten as $- (\\min -\\bint{c}^T x)$.\n\t\n\t\\paragraph{Adding slack variables.} In order to convert inequality constraints to equations in a linear program, sign-restricted slack variables are added to the constraints. Analogously, we can transform an interval inequality system $\\bint{A}x \\le \\bint{b}$ into an interval system \\[ \\bint{A}x + Iy = \\bint{b},\\; y \\ge 0. \\]\n\tAgain, the transformation does not duplicate any of the existing coefficients and only introduces some new crisp coefficients. Applying this transformation to a set of constraints results in an equivalent interval linear program, since each new scenario is equivalent to a corresponding scenario in the original interval system.\n\t\n\t\\paragraph{Splitting equations into inequalities.} When changing a general linear program into an inequality-constrained form, we split each equation into two opposite inequalities. However, splitting an interval equation into interval inequalities while preserving all properties is not always possible, due to the dependency problem. Breaking dependencies among multiple occurrences of an interval coefficient leads to generating new scenarios that do not have a counterpart in the original problem. More precisely, while an interval equation $\\bint{a}^T x = b$ consists of the scenarios $a^T x = b$ with all possible choices of $a \\in \\bint{a}$, the system of two interval inequalities $\\bint{a}^T x \\le b$, $\\bint{a}^T x \\ge b$ corresponds to the (possibly larger) set of all scenarios in the form $a_1^T x \\le b, a_2^T x \\ge b$ with coefficients $a_1, a_2 \\in \\bint{a}$.\n\t\n\t\\paragraph{Imposing non-negativity.} Another transformation used in linear programming is substituting the difference of two non-negative variables for a free variable. Along with introducing two new variables for each original free variables, the operation also duplicates the corresponding coefficients. This, again, leads to breaking the dependency in case of interval constraints: By replacing a constraint $\\bint{a}^T x = \\bint{b}$ with $\\bint{a}^T x^+ - \\bint{a}^T x^- = \\bint{b}$ and $x^+, x^- \\ge 0$, we create two independent occurrences of the interval coefficient $\\bint{a}$, which can take on different values in a particular scenario.\n\t\n\t\\section{Optimal Solution Set under Transformations}\\label{sec:optsol}\n\t\\subsection{General Case}\\label{ssec:gen}\n\tIn this section, we discuss the two types of transformations that can possibly change the properties of an interval linear program: splitting equation constraints and imposing non-negativity on the variables. First, we present examples of ILPs showing that these transformations can change the optimal solution set. Note that in both cases, the transformed interval program contains all of the original scenarios, therefore, the original optimal set is always a subset of the transformed one.\n\t\n\tWe have seen in Section~\\ref{sec:transf} that splitting an interval equation into two inequalities can cause a dependency problem and possibly change the feasible or optimal solutions of an ILP. However, it was proved by \\cite{Li:2015} that this is not the case for the feasible set.\n\t\n\t\\begin{theorem}[\\cite{Li:2015}]\\label{thm:Li}\n\t\tA vector $x \\in \\mathbb{R}^n$ is a weakly feasible solution of an~interval linear system $\\bint{A}x = \\bint{b}$ if and only if it is a weakly feasible solution of the system $\\bint{A}x \\le \\bint{b}, \\bint{A}x \\ge \\bint{b}$.\n\t\\end{theorem}\n\t\n\tEven though the transformation preserves the weakly feasible solution set, the following example shows that creating new scenarios may, in fact, lead to expanding the set of optimal solutions.\n\t\n\t\\begin{example}\\label{ex:optsol}\n\t\tConsider the interval linear program\n\t\t\\begin{equation}\\label{eq:ex}\n\t\t\\begin{array}{lr@{\\ }l}\n\t\t\\text{minimize } & \\multicolumn{2}{l}{-x_1} \\\\\n\t\t\\text{subject to } & [0,1]x_1 - x_2 = 0,\\\\\n\t\t&  x_2 \\le 1,\\\\\n\t\t&  x_1, x_2 \\ge 0.\n\t\t\\end{array}\n\t\t\\end{equation}\n\t\tThe optimal solution set of~\\eqref{eq:ex} consists of all values with $x_1 \\in [1, \\infty)$ and $x_2 = 1$. By splitting the equation $[0,1]x_1 - x_2 = 0$ into two inequalities, we allow the value for each coefficient to be chosen independently. This results in introducing additional scenarios into the problem, such as the scenario \n\t\t\\begin{equation*}\n\t\t1x_1 - x_2 \\le 0,\\quad 0x_1 - x_2 \\ge 0\n\t\t\\end{equation*}\n\t\twith the optimal solution $(0,0)$. Therefore, the optimal solution sets of the two problems are not equivalent.\n\t\\end{example}\n\t\n\tThe second type of transformation affected by the dependency problem is restricting the sign of the variables. However, unlike all of the other transformations, replacing a free variable by a difference of two non-negative variables does not even preserve the feasible solution set of an interval linear program, as already noted by \\cite{Hladik:2012}. This implies that the set of optimal solutions may also change, in general.\n\t\n\t\\subsection{Special Case: Fixed Coefficient Matrix}\\label{ssec:spec}\n\tThis section is devoted to a special class of interval linear programs, in which uncertainty only affects the objective function and the right-hand-side vector. We show that, unlike general interval linear programs, programs with a fixed coefficient matrix allow for all types of transformations while preserving the optimal solution set. Theorem~\\ref{thm:spec:AC} presents the result for the transformation of splitting an equation into two opposite inequalities, i.e. converting a program of type~\\eqref{eq:ilp:A} with a fixed matrix into a program of type~\\eqref{eq:ilp:C}.\n\t\n\t\\begin{theorem}\\label{thm:spec:AC}\n\t\tLet $\\mathcal{S}(A,\\bint{b},\\bint{c})$ be the optimal solution set of an interval linear program of type~\\eqref{eq:ilp:A} with a fixed coefficient matrix given by the triplet $(A, \\bint{b}, \\bint{c})$. Then, $\\mathcal{S}(A,\\bint{b},\\bint{c})$ is equal to the optimal solution set of the problem\n\t\t\\begin{equation}\\label{eq:transfAC}\n\t\t\\begin{array}{l@{\\hskip 8pt}r@{\\ }l}\n\t\t\\textnormal{minimize} & \\multicolumn{2}{l}{\\bint{c}^T x}\\\\\n\t\t\\textnormal{subject to} & A x &\\le \\bint{b}_1,\\\\\n\t\t& -A x &\\le -\\bint{b}_2,\\\\\n\t\t& x &\\ge 0.\n\t\t\\end{array}\n\t\t\\end{equation}\n\t\twith $\\bint{b}_1 = \\bint{b}_2 = \\bint{b}$.\n\t\\end{theorem}\n\t\\begin{proof}\n\t\tLet $\\mathcal{S}(A',\\bint{b'},\\bint{c})$ denote the optimal solution set of program~\\eqref{eq:transfAC}. Clearly, the inclusion $\\mathcal{S}(A,\\bint{b},\\bint{c}) \\subseteq \\mathcal{S}(A',\\bint{b'},\\bint{c})$ holds, since \\eqref{eq:transfAC} contains all the scenarios of \\eqref{eq:ilp:A}.\n\t\t\n\t\tOn the other hand, let $x'$ be an optimal solution of ILP \\eqref{eq:transfAC} for a scenario determined by the coefficient vectors $c \\in \\bint{c}$ and $b_1, b_2 \\in \\bint{b}$ satisfying \n\t\t\\begin{equation}\\label{eq:transf:scenario}\n\t\tAx' \\le b_1,\\; -Ax' \\le -b_2.\n\t\t\\end{equation}\n\t\tSince we have, $b_2 \\le Ax' \\le b_1$, there exists $b_3 \\in [b_2, b_1] \\subseteq \\bint{b}$ with $Ax' = b_3$. We claim that $x'$ is also an optimal solution of program~\\eqref{eq:ilp:A} for the scenario\n\t\t\\begin{equation*}\n\t\t\\begin{array}{lrl@{\\ }l}\n\t\t\\text{minimize} & \\multicolumn{2}{l}{c^T x} \\\\\n\t\t\\text{subject to} & Ax &=b_3,\\\\\n\t\t& x & \\ge 0.\n\t\t\\end{array}\n\t\t\\end{equation*}\n\t\tSuppose, for the sake of contradiction, that there exists another feasible solution $x^*$ with $c^T x^* < c^T x'$. By the choice of $b_3$, the vector $x^*$ is also feasible for scenario~\\eqref{eq:transf:scenario}. However, since the objective function is the same for both problems, this contradicts the assumption that $x'$ is optimal in scenario~\\eqref{eq:transf:scenario}.\n\t\\end{proof}\n\t\n\tNote that while the transformation preserves the optimal solutions, it may still change other properties of the program, such as the existence of infeasible scenarios or the range of optimal values, which will be discussed in more detail in Section~\\ref{sec:optval}. Theorem~\\ref{thm:spec:nonneg} shows an analogous result for the transformation of free variables into non-negative variables.\n\t\n\t\\begin{theorem}\\label{thm:spec:nonneg}\n\t\tLet $\\mathcal{S}(A, \\bint{b}, \\bint{c})$ denote the optimal solution set of an interval linear program of type~\\eqref{eq:ilp:B} with a fixed coefficient matrix and let $\\mathcal{S}(A', \\bint{b}, \\bint{c}')$ be the optimal solution set of the program\n\t\t\\begin{equation}\\label{eq:transfBC}\n\t\t\\begin{array}{lr@{\\ }l}\n\t\t\\textnormal{minimize} & \\multicolumn{2}{l}{\\bint{c}_1^T x^+ - \\bint{c}_2^T x^-}\\\\\n\t\t\\textnormal{subject to} & A x^+ - A x^- &\\le \\bint{b},\\\\\n\t\t& x^+, x^- &\\ge 0\n\t\t\\end{array}\n\t\t\\end{equation}\n\t\twith $\\bint{c}_1 = \\bint{c}_2 = \\bint{c}$. Then, the following properties hold:\n\t\t\\begin{enumerate}\n\t\t\t\\item If $x \\in \\mathcal{S}(A,\\bint{b},\\bint{c})$, then there exists $(x^+, x^-) \\in \\mathcal{S}(A', \\bint{b}, \\bint{c}')$ with $x = x^+ - x^-$.\n\t\t\t\\item Conversely, if $(x^+, x^-) \\in \\mathcal{S}(A', \\bint{b}, \\bint{c}')$, then $x^+ - x^- \\in \\mathcal{S}(A,\\bint{b},\\bint{c})$.\n\t\t\\end{enumerate}\n\t\\end{theorem}\n\t\\begin{proof}\n\t\tLet $x \\in \\mathcal{S}(A,\\bint{b},\\bint{c})$ be an optimal solution for a scenario $c \\in \\bint{c}$, $b \\in \\bint{b}$. Decompose $x$ into the non-negative vectors $x^+ = \\max(0,x)$ and $x^- = -\\min(0,x)$, where the operations $\\max$ and $\\min$ are understood entry-wise. The vectors satisfy $x = x^+ - x^-$ and it is easy to see that the pair $(x^+, x^-)$ is optimal in ILP~\\eqref{eq:transfBC} for the scenario determined by the objective vector $(c, -c)$ and the right-hand side $b$.\n\t\t\n\t\tNow, let $(x^+, x^-) \\in \\mathcal{S}(A', \\bint{b}, \\bint{c}')$ be optimal for some $b \\in \\bint{b}$ and $c_1, c_2 \\in \\bint{c}$. By duality in linear programming, there exists a dual feasible vector $y$ satisfying\n\t\t\\begin{eqnarray*}\n\t\t\t&&c_1^T x^+ - c_2^T x^- = b^T y,\\\\\n\t\t\t&&A x^+ - Ax^- \\le b,\\; x^+ \\ge 0,\\; x^- \\ge 0,\\\\\n\t\t\t&&A^T y \\le c_1,\\; -A^T y \\le -c_2,\\; y \\le 0.\n\t\t\\end{eqnarray*}\n\t\tFrom the dual feasibility constraints, we have $A^T y = c_3$ for some $c_3 \\in [c_2, c_1] \\subseteq \\bint{c}$. We will prove that $x = x^+ - x^-$ is optimal for the scenario with objective vector $c_3$ and right-hand-side vector $b$ in ILP~\\eqref{eq:ilp:B}, by showing that it satisfies the system\n\t\t\\begin{equation*}\n\t\tc_3^T x = b^T y,\\; Ax \\le b,\\; A^T y = c_3,\\; y \\le 0.\n\t\t\\end{equation*}\n\t\tNamely, it remains to show that $c_3^T (x^+ - x^-) = b^T y$ holds. Complementary slackness implies that for each $i \\in \\{1, \\dots, n\\}$ we have\n\t\t\\begin{eqnarray*}\n\t\t\t&&(x^+)_i = 0\\;\\vee\\;(c_1-A^T y)_i = 0, \\text{ and}\\\\\n\t\t\t&&(x^-)_i = 0\\;\\vee\\;(A^T y-c_2)_i = 0.\n\t\t\\end{eqnarray*}\n\t\tIf $(x^+)_i > 0$ and $(x^-)_i > 0$ for some index $i$, then we have $(c_1)_i = (c_2)_i = (c_3)_i$ by the choice of $c_3$. For $(x^+)_i = 0$ and $(x^-)_i > 0$ we have $(c_2)_i = (c_3)_i$, and analogically, $(c_1)_i = (c_3)_i$ for the symmetric case. By substituting into the equation\n\t\t\\begin{equation*}\n\t\tc_1^T x^+ - c_2^T x^- = b^T y,\n\t\t\\end{equation*}\n\t\twe can see that the desired constraint is satisfied in all cases. Therefore, $x^+ - x^-$ is an optimal solution of program~\\eqref{eq:ilp:B}.\n\t\\end{proof}\n\t\n\tFigure~\\ref{fig:gen} provides an overview of the transformations preserving the feasible and optimal solution set of general interval linear programs, as presented in Section~\\ref{ssec:gen}. Note that ILPs of type~\\eqref{eq:ilp:C} are a special case of type~\\eqref{eq:ilp:B} and the transformation of type~\\eqref{eq:ilp:A} to type~\\eqref{eq:ilp:B} can be obtained by transitivity. As shown in Figure~\\ref{fig:spec}, all of the considered transformations preserve the optimal solution set for ILPs with a fixed coefficient matrix. Apart from the direct consequences of Theorem~\\ref{thm:spec:AC} and Theorem~\\ref{thm:spec:nonneg}, the remaining results follow again by transitivity.\n\t\n\t\\begin{figure}[b]\n\t\t\\begin{minipage}[t]{0.49\\textwidth}\n\t\t\t\\begin{tikzpicture}[scale=0.5]\n\t\t\t\\tikzstyle{sipka}=[-{>[length=7pt, width=5pt]}]\n\t\t\t\\tikzstyle{rect}=[draw,rectangle,minimum height=12ex,inner sep=3pt,text width=50pt, text height=2.25ex, align=center]\n\t\t\t\\node[rect] (a) at (0,4) {Type (I): \\small{$\\bint{A} x=\\bint{b}$, $x\\ge 0$}};\n\t\t\t\\node[rect] (b) at (-3.8,-1.2) {Type (II): \\small{$\\bint{A}x \\le \\bint{b}$}};\n\t\t\t\\node[rect] (c) at (3.8,-1.2) {Type (III): \\small{$\\bint{A}x \\le \\bint{b}$, $x \\ge 0$}};\n\t\t\t\n\t\t\t\\draw[sipka,dashed] (a.south west) -- (b.north);\n\t\t\n\t\t\t\\draw[sipka,dashed] (a.south east) -- (c.north);\n\t\t\t\\draw[sipka] ([xshift=-15pt]c.north) -- ([xshift=-15pt]a.south east);\n\t\t\n\t\t\t\\draw[sipka] ([yshift=-5pt]c.west) -- ([yshift=-5pt]b.east);\n\t\t\t\\node at (0,1) {$\\min\\,\\bint{c}^T x$};\n\t\t\t\\end{tikzpicture}\n\t\t\t\\caption{Transformations preserving the feasible (dashed arrows) and optimal (solid arrows) solution set of a general interval linear program.}\\label{fig:gen}\n\t\t\\end{minipage}\n\t\t\\hspace{0.02\\textwidth}\n\t\t\\begin{minipage}[t]{0.49\\textwidth}\n\t\t\t\\begin{tikzpicture}[scale=0.5]\n\t\t\t\\tikzstyle{sipka}=[-{>[length=7pt, width=5pt]}]\n\t\t\t\\tikzstyle{rect}=[draw,rectangle,minimum height=12ex,inner sep=3pt,text width=50pt, text height=2.25ex, align=center]\n\t\t\t\\node[rect] (a) at (0,4) {Type (I): \\small{$A x=\\bint{b}$, $x\\ge 0$}};\n\t\t\t\\node[rect] (b) at (-3.8,-1.2) {Type (II): \\small{$Ax \\le \\bint{b}$}};\n\t\t\t\\node[rect] (c) at (3.8,-1.2) {Type (III): \\small{$Ax \\le \\bint{b}$, $x \\ge 0$}};\n\t\t\t\n\t\t\t\\draw[sipka] (a.south west) -- (b.north);\n\t\t\t\\draw[sipka] ([xshift=15pt]b.north) -- ([xshift=15pt]a.south west);\n\t\t\t\\draw[sipka] (a.south east) -- node[above,sloped] {Thm.~\\ref{thm:spec:AC}} (c.north);\n\t\t\t\\draw[sipka] ([xshift=-15pt]c.north) --  ([xshift=-15pt]a.south east);\n\t\t\t\\draw[sipka] ([yshift=5pt]b.east) -- node[above] {Thm.~\\ref{thm:spec:nonneg}} ([yshift=5pt]c.west);\n\t\t\t\\draw[sipka] ([yshift=-5pt]c.west) -- ([yshift=-5pt]b.east);\n\t\t\t\\node at (0,1) {$\\min\\, \\bint{c}^T x$};\n\t\t\t\\end{tikzpicture}\n\t\t\t\\caption{Transformations preserving the optimal solution set of an interval linear program with a fixed coefficient matrix.}\\label{fig:spec}\n\t\t\\end{minipage}\n\t\\end{figure}\n\t\n\t\\section{Optimal Values under Transformations}\\label{sec:optval}\n\tLet us now discuss the effects of transformations on the set of all optimal values of an interval linear program and the optimal value range. Recall that the optimal value range refers to the interval $[\\lb{f}, \\ub{f}]$, which is the smallest interval that encloses all optimal values. This interval may differ from the set of optimal values, since not all values in the range have to be attained as optimal for a scenario. Again, we consider the two transformations that can possibly lead to a dependency problem: splitting equations into inequalities and substituting a difference of two non-negative variables for a free variable.\n\t\n\t\\subsection{General Case}\\label{ssec:gen:val}\n\tWe have already seen in Section~\\ref{ssec:gen} that splitting an equation into two opposite inequalities may change the optimal solution set of an interval linear program. This also holds for the optimal value range, namely, the transformation can change the worst-case bound ($\\ub{f}$ for a minimization program). Moreover, this is caused not only by creating an infeasible scenario resulting in $\\ub{f} = \\infty$, but can also happen due to an expansion of the set of finite optimal values.\n\t\\addtocounter{example}{-1}\n\t\\begin{example}[continued]\n\t\tThe optimal value range of the interval linear program\n\t\t\\begin{equation*}\n\t\t\\begin{array}{lr@{\\ }l}\n\t\t\\text{minimize } & \\multicolumn{2}{l}{-x_1} \\\\\n\t\t\\text{subject to } & [0,1]x_1 - x_2 = 0,\\\\\n\t\t&  x_2 \\le 1,\\\\\n\t\t&  x_1, x_2 \\ge 0.\n\t\t\\end{array}\n\t\t\\end{equation*}\n\t\tis the interval $(-\\infty, -1]$. However, by splitting the equation $[0,1]x_1 - x_2 = 0$ into two inequalities, the solution $(0,0)$ becomes optimal and, thus, the value~$0$ belongs to the set of optimal values (and the optimal value range) of the transformed program. This shows that $\\ub{f} = -1$ is no longer the worst-case optimal value.\n\t\\end{example}\n\t\n\tOn the other hand, we will now show that the transformation preserves the best-case optimal value~$\\lb{f}$. The proof is a consequence of a result on a unified approach for computing the optimal value range by~\\cite{Hladik:2009}. For the purposes of the following theorem, let us consider an interval linear program in the general form\n\t\\begin{equation*}\n\t\\begin{array}{lrrrr@{\\ }l}\n\t\\text{minimize } & \\multicolumn{5}{l}{\\bint{c_1}^T x_1 + \\bint{c_2}^T x_2} \\\\\n\t\\text{subject to } & \\bint{A}x_1 &+ & \\bint{B}x_2 & = & \\bint{b_1},\\\\\n\t&  \\bint{C}x_1 & + & \\bint{D}x_2 & \\le & \\bint{b_2},\\\\\n\t& &  \\multicolumn{2}{r}{x_1, x_2} &\\ge & 0.\n\t\\end{array}\n\t\\end{equation*}\n\tLet $\\bint{b} = (\\bint{b_1}, \\bint{b_2})$, $\\bint{c} = (\\bint{c_1}, \\bint{c_2})$ and let $\\mathcal{M}$ denote the set of all feasible solutions. Furthermore, let $\\mathcal{N}$ be the weakly feasible set of the dual ILP. Then, we can apply the formulas presented in Theorem~\\ref{thm:hladik:f} to compute the optimal value range.\n\t\n\t\\begin{theorem}[\\cite{Hladik:2009}]\\label{thm:hladik:f}\n\t\tWe have\n\t\t\\begin{equation}\\label{eq:lb}\n\t\t\\lb{f} = \\inf \\{c_c^T x - c_\\Delta^T \\lvert x \\rvert : x \\in \\mathcal{M}\\}.\n\t\t\\end{equation}\n\t\tIf $\\ub{f} < \\infty$, then\n\t\t\\begin{equation}\\label{eq:ub}\n\t\t\\ub{f} = \\sup \\{b_c^T y + b_\\Delta^T \\lvert y \\rvert : y \\in \\mathcal{N}\\}.\n\t\t\\end{equation}\n\t\\end{theorem}\n\t\n\tTheorem~\\ref{thm:optval:AC} proves that the transformation of splitting an equation into two opposite inequalities does not change the best-case optimal value of an interval linear program. The result implies that it is possible to convert an equation-constrained ILP of type~\\eqref{eq:ilp:A} into an inequality constrained ILP of type~\\eqref{eq:ilp:C}, while preserving the best optimal value.\n\t\n\t\\begin{theorem}\\label{thm:optval:AC} \n\t\tTransforming $\\bint{A}x = \\bint{b}$ into $\\bint{A}x \\le \\bint{b}, \\bint{A}x \\ge \\bint{b}$ does not change the best optimal value $\\lb{f}$ of a minimization ILP.\n\t\\end{theorem}\n\t\\begin{proof}\n\t\tBy Theorem~\\ref{thm:hladik:f}, the best optimal value $\\lb{f}$ of a general interval linear program can be found by minimizing the fixed objective function $c_c^T x - c_\\Delta^T \\lvert x \\rvert$ over the set of all weakly feasible solutions. Since the applied transformation of splitting an equation into two inequalities does not change the weakly feasible set of an interval system (see Theorem~\\ref{thm:Li}), the value of $\\lb{f}$ remains the same.\n\t\\end{proof}\n\t\n\tBy the property of strong duality in classical linear programming, we can also derive analogous results for the second considered transformation. For this case, let us first show that substituting the difference of two non-negative variables for a free variables can change the best-case bound $\\lb{f}$, as well as the set of optimal values.\n\t\n\t\\begin{example}\n\t\tConsider the dual interval linear program to~\\eqref{eq:ex}, which can be rewritten into a minimization form as\n\t\t\\begin{equation}\\label{eq:ex:2}\n\t\t\\begin{array}{lrrrr@{\\ }l}\n\t\t\\text{minimize } & \\multicolumn{5}{l}{-y_2} \\\\\n\t\t\\text{subject to } & [0,1]y_1 & & & \\le & -1,\\\\\n\t\t&  -y_1 & + & y_2 & \\le & 0,\\\\\n\t\t& &  \\multicolumn{2}{r}{y_2} &\\le & 0.\n\t\t\\end{array}\n\t\t\\end{equation}\n\t\tThe optimal value range of program~\\eqref{eq:ex:2} is the interval $[1, \\infty)$. Let us now substitute the term $y_1^+-y_1^-$ with $y_1^+, y_1^- \\ge 0$ for the free variable $y_1$. Analogously to the previous example, the set of optimal values changes and the best-case bound $\\lb{f} = 1$ is no longer valid (again, the value $0$ becomes optimal).\n\t\\end{example}\n\t\n\tHowever, we can show that the transformation preserves the worst-case bound $\\ub{f}$. The proof uses the notion of strong feasibility, which also provides a characterization of finiteness of the bound $\\ub{f}$, as stated in Theorem~\\ref{thm:strfeas}. An interval linear program is said to be \\emph{strongly feasible}, if each scenario of the program is feasible.\n\t\n\t\\begin{theorem}[\\cite{Hladik:2009}]\\label{thm:strfeas}\n\t\tAn interval linear program (in the general form) is strongly feasible if and only if $\\ub{f} < \\infty$.\n\t\\end{theorem}\n\t\n\t\\begin{theorem}\n\t\tSubstituting $x = x^+ - x^-$ with $x^+, x^- \\ge 0$ for a free variable $x$ does not change the worst optimal value $\\ub{f}$ of a minimization ILP.\n\t\\end{theorem}\n\t\\begin{proof}\n\t\tLet us denote by $\\ub{f}$ and $\\ub{f}^\\pm$ the worst-case optimal value of the original program and the transformed program created by the introducing the substitution, respectively. \n\t\t\n\t\tFirst, assume that $\\ub{f} = \\infty$. Since all of the original scenarios are also included in the transformed ILP, the latter is a relaxation of the original program and $\\ub{f} \\le \\ub{f}^\\pm$ holds. Therefore, we also have $\\ub{f}^\\pm = \\infty$.\n\t\t\n\t\tFurther, let $\\ub{f} < \\infty$ hold for the original interval program. Then, the program is strongly feasible. Since the conditions for strong feasibility of the original and the transformed program are equivalent for both equation and inequality constraints (see \\cite{Hladik:2017}), the latter is also strongly feasible and,  by Theorem~\\ref{thm:strfeas}, the property $\\ub{f}^\\pm < \\infty$ holds. By formula~\\eqref{eq:ub} of Theorem~\\ref{thm:hladik:f} we can calculate the worst-case optimal values by optimizing the objective function $b_c^T y + b_\\Delta^T \\lvert y \\rvert$ over the dual feasible set of the respective ILPs. Note that applying the substitution to a free variable in the primal ILP corresponds to splitting an equation into two opposite inequalities in the dual ILP. As this preserves the set of feasible solutions (Theorem~\\ref{thm:Li}), the dual feasible sets of the original and the transformed program are equal, and thus $\\ub{f} = \\ub{f}^\\pm$. \n\t\\end{proof}\n\t\n\t\\subsection{Special Case: Fixed Coefficient Matrix}\\label{ssec:spec:val}\n\tIn Section~\\ref{ssec:gen:val} we have seen that the optimal values and bounds of the optimal value range may change under some transformations. Therefore, it is natural to ask whether these properties are preserved at least for ILPs with a fixed coefficient matrix. Unfortunately, even for this special class of programs, the transformations may change the optimal value range, as shown by the following trivial examples.\n\t\n\t\\begin{example}\\label{ex:optval:fixed}\n\t\tConsider the following interval linear programs:\n\t\t\\vskip\\abovedisplayskip\n\t\t\\noindent\n\t\t\\begin{subequations}\n\t\t\t\\begin{minipage}{.5\\linewidth}\n\t\t\t\t\\begin{equation}\\label{eq:ex:3a}\n\t\t\t\t\\begin{array}{lrrrr@{\\ }l}\n\t\t\t\t\\text{minimize } & \\multicolumn{5}{l}{[0,1]x} \\\\\n\t\t\t\t\\text{subject to } & x & \\ge & 1,\\\\\n\t\t\t\t\\end{array}\n\t\t\t\t\\end{equation}\n\t\t\t\\end{minipage\n\t\t\t\\begin{minipage}{.5\\linewidth}\n\t\t\t\t\\begin{equation}\\label{eq:ex:3b}\n\t\t\t\t\\begin{array}{lrrrr@{\\ }l}\n\t\t\t\t\\text{minimize } & \\multicolumn{5}{l}{-y} \\\\\n\t\t\t\t\\text{subject to } & y & = & [0,1].\\\\\n\t\t\t\t\\end{array}\\qquad\n\t\t\t\t\\end{equation}\n\t\t\t\\end{minipage}\n\t\t\\end{subequations}\n\t\t\\vskip\\belowdisplayskip\\noindent\n\t\tThe optimal value range of ILP~\\eqref{eq:ex:3a} is the interval $[0,1]$, for ILP~\\eqref{eq:ex:3b} it is the opposite interval $[-1,0]$. By substituting $x^+-x^-$ with non-negative variables $x^+, x^-$ for the free variable $x$ in \\eqref{eq:ex:3a}, we introduce an unbounded scenario (setting the objective to $0x^+ - 1x^-$) and the best-case bound of the optimal value range changes to $\\lb{f} = -\\infty$.\n\t\tSimilarly, by splitting the equation in \\eqref{eq:ex:3b} into two opposite inequalities $y \\le [0,1], y \\ge [0,1]$, we create an infeasible scenario leading to $\\ub{f} = \\infty$.\n\t\\end{example}\n\t\n\tHowever, note that there is an important difference between Example~\\ref{ex:optval:fixed} and the previous examples with an interval coefficient matrix. While in the examples of Section~\\ref{ssec:gen:val} we have seen that a transformation may cause a change in the set of optimal values, in programs \\eqref{eq:ex:3a} and \\eqref{eq:ex:3b} the transformations only change one of the bounds in the optimal value range due to infeasibility or unboundedness of a newly introduced scenario.\n\t\n\tLet us now consider the set of all finite optimal values of an interval linear program with a fixed coefficient matrix. The following theorems prove that even though the optimal value range may still change when transforming a~program with a fixed matrix, this can only be caused by the infinite optimal values for infeasible or unbounded scenarios and the finite optimal values remain the same.\n\t\n\t\\begin{theorem}\\label{thm:spec:optval}\n\t\tTransforming $Ax = \\bint{b}$ into $Ax \\le \\bint{b}, Ax \\ge \\bint{b}$ does not change the set of all finite optimal values of an ILP with a fixed coefficient matrix.\n\t\\end{theorem}\n\t\\begin{proof}\n\t\tClearly, all optimal values of the original program remain optimal in the transformed program. Let a solution $x^*$ be optimal for a scenario given by $c \\in \\bint{c}$ and $b_1, b_2 \\in \\bint{b}$ in the transformed program. It is easy to see that $x^*$ is also optimal for the scenario determined by $c$ and $b_3 = Ax^*$ of the original program, since it has the same objective function and a restricted feasible set. Therefore, the optimal value $c^T x^*$ of the transformed ILP is also optimal for the original ILP.\n\t\\end{proof}\n\t\n\t\\begin{theorem}\n\t\tSubstituting $x = x^+ - x^-$ with $x^+, x^- \\ge 0$ for a free variable $x$ does not change the set of all finite optimal values of an ILP with a fixed coefficient matrix.\n\t\\end{theorem}\n\t\\begin{proof}\n\t\tBy strong duality of linear programming, the sets of finite optimal values of an interval linear program and its dual are the same. As introducing the substitution in the primal ILP corresponds to the transformation of splitting an equation into two inequalities in the dual ILP, applying Theorem~\\ref{thm:spec:optval} yields the result.\n\t\\end{proof}\n\t\n\t\\section{Conclusion}\\label{sec:concl}\n\tWe addressed the dependency problem in transforming interval linear programs using the techniques known from classical linear programming. We showed that while it is possible to switch the objective of an interval linear program or add slack variables to convert inequalities into equations, other transformations are not always applicable to interval programs without affecting some of their properties.\n\t\n\tNamely, we considered three commonly used forms of interval linear programs. It was shown that the set of all optimal solutions may change, in general, under the transformations among these forms. Therefore, we also studied a special class of interval programs with a fixed coefficient matrix, for which we proved that all of the transformations preserve the optimal set.\n\t\n\tFurthermore, we also studied the effect of the transformations on the set of optimal values and the optimal value range of an interval linear program. We proved that the best-case optimal value $\\lb{f}$ of a minimization program remains the same when an equation is split into two opposite inequalities, while the worst-case optimal value $\\ub{f}$ is preserved when substituting the difference of two non-negative variables for a free variable. The complementary results do not hold, even in the case of a fixed coefficient matrix. However, the set of all finite optimal values does not change for transformations on the special class of programs.\n\t\n\tThe results allow us to generalize the theory that was derived for a particular form of ILPs to all other types that can be obtained by a transformation respecting the studied properties. We believe that they also provide a better insight into the nature of the dependency problem in interval optimization.\n\t\n\t\n\n\n\n\n\t\n\n\n\n\n\\bibliographystyle{abbrv}     \n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe versatile technique of positron annihilation makes use of the fact that positrons ($e^+$) are trapped at  free \nvolume-type defects which allows their detection by a specific variation of the positron-electron annihilation characteristics \\cite{Hautojaervi79, Brossmann12, Krause-Rehberg99, Puska94}. \nWhereas the kinetics of $e^+$ trapping at vacancy-type point defects can be well described by \nrate theory (so-called simple trapping model), it is well known that for trapping at extended defects like\ngrain boundaries, interfaces, voids, clusters, or precipitates, diffusion limitation of the trapping process may be an issue. \nDiffusion-limited positron trapping at interfaces and grain boundaries has been quantatively modeled by several groups,\nranging from entirely diffusion-controlled trapping \\cite{Brandt72}, diffusion-reaction controlled trapping including detrapping\n\\cite{Dupasquier93, Wuerschum96, Koegel96, dryzek1999}, up to diffusion-reaction controlled trapping at grain boundaries and competetive transition-limited trapping at point defects in crystals \n\\cite{Cizek02, Oberdorfer09, dryzek1998, Koegel96}.\n\nCompared to grain boundaries, diffusion-limited $e^+$ trapping at voids and clusters has not been studied in such detail \ndespite the undoubted relevance of positron annihilation for studying this important class of defects \\cite{Nieminen79a, Bentzon90, Eldrup03}.\nOne approach to deal with diffusion-limited trapping is based on effective diffusion-trapping rates \nwhich then allow an implementation in standard rate theory  (e.g., \\cite{Bentzon90}).\nDiffusion-limited trapping at point-like defects was studied by Dryzek \\cite{dryzek1998diffusion}\nfor the one-dimensional case. A full treatment of $e^+$ trapping and annihilation in voids\nin the framework of diffusion-reaction theory was given by Nieminen et al. \\cite{Nieminen79}.\nThis treatment of Nieminen et al. \\cite{Nieminen79} is conceptionally analogous to the subsequent work of Dupasquier et al. \n\\cite{Dupasquier93} for diffusion-limited $e^+$ trapping at grain boundaries, both of which lead to solutions exclusively in terms of infinite series.\n\nAnother treatment  of the diffusion-reaction problem of $e^+$ trapping at grain boundaries\nwas given by W\\\"urschum and Seeger \\cite{Wuerschum96} which yields closed-form expressions\nfor the mean $e^+$ lifetime and the intensity of the annihilation component associated with \nthe trapped state. This approach is applied in the present work to the diffusion-reaction problem \nof $e^+$ trapping and annihilation in spherical extended  defects (voids, clusters, precipitates).\\footnote{For the sake of\nsimplicity, representatively for all kinds of  spherical extended defects (voids, clusters, or precipitates) \nthe term voids is used in the following.} \nFollowing our earlier further work on grain boundaries \n\\cite{Oberdorfer09}, now in addition competitive reaction rate-limiting trapping at point defects is taken into account.\nThe present  treatment yields closed-form expressions of the major $e^+$ annihilation parameters\nfor this application-relevant case of competitive\n$e^{+}$ trapping in voids and point defects.\nThese closed-form expressions allow deeper insight in the physical details of\n$e^+$ annihilation characteristics as well as an assessment of the so far often used  approach based on\neffective diffusion-trapping rates. Above all, the results can be conveniently applied for\nthe analysis of experimental data.\n\nIn a further part, the model presented here and the previous model on positron trapping at grain boundaries\nare merged in order to study precipitates embedded in matrix.  \nHere, diffusion- and reaction limited trapping is considered for both the trapping\nfrom the matrix into the  precipitate$-$matrix interface and for  \nthe trapping from inside the precipitates into the interfaces.\n\n\\section{The Model}\nThe model describes positron ($e^+$) trapping and annihilation in  voids in the general case that both the $e^+$ diffusion and the transition reaction has to be taken into account (so called diffusion-reaction controlled trapping process).\nIn order to cover more complex cases, competitive transition-limited trapping at vacancy-type points defects is also considered\n(see Fig. \\ref{fig:1}).\nThis procedure follows our earlier study \nwhere concomitant positron trapping at grain boundaries and at point defects in crystallites has been considered \\cite{Oberdorfer09}.\n\n\n\nThe behavior of the positrons is described by their bulk (free) lifetime $\\tau_f$,  \nby their lifetime ($\\tau_t)$ in the  voids, \nby their lifetime ($\\tau_v)$ in the vacancy-type point defects in the lattice (matrix),\nand by their bulk diffusivity $D$.\nTrapping at the point defects of the matrix is characterized by the specific $e^+$ trapping rate $\\sigma_v$ (unit s$^{-1}$), as usual.\nThe  voids are considered as spherical-shaped extended defects (radius $r_0$) with a specific trapping rate $\\alpha$ (unit m\\,s$^{-1}$) which is related to the surface area of the void. In units of s$^{-1}$ the specific trapping rate of voids reads\n\\begin{equation}\n\\label{eq:sigma_t}\n\\sigma_t= \\frac{\\alpha 4\\pi r_0^2}{\\Omega},\n\\end{equation}\nwhere $\\Omega$ denotes the atomic volume.\n \nThe temporal and spatial evolution of the density  $\\rho_l$ of free positrons in the lattice is governed by:\n\\begin{equation}\n\\label{differential_eq}\n\\frac{\\partial \\rho_l}{\\partial t}=\nD\\nabla^{2}\\rho_l-\\rho_l\\left(\\frac{1}{\\tau_f}+\n\\sigma_v C_v\\right)\n\\end{equation}\nwhere $C_v$ denotes the concentration of vacancy-type point defects in the matrix.\nThe positrons trapped in the voids are described in terms of their density $\\rho_t$ obeying the rate equation\n\\begin{equation}\n\\label{eq:rho_t}\n\\frac{\\mathrm{d}\\rho_t}{\\mathrm{d}t}=\n\\alpha \\rho_l(r_{0},t)-\\frac{1}{\\tau_t}\\rho_t.\n\\end{equation}\nThe temporal evolution of the number $N_v$ of $e^+$ trapped in the point defects in the lattice is given by\n\\begin{equation}\n\\frac{\\mathrm{d}N_v}{\\mathrm{d}t}=\n-\\frac{1}{\\tau_v} N_v+\\sigma_v C_v N_f \\, ,\n\\end{equation}\nwhere the number $N_f$ of positrons in the free state follows from integration of $\\rho_l$:\n\\begin{equation}\nN_f=\\int \\rho_l \\mathrm{d}V.\n\\end{equation}\n\n\nThe continuity of the ${\\rm e}^{+}$ flux at the boundary between the lattice and the void surface is expressed by\\footnote{Note the negative sign in contrast to the model of $e^+$ trapping at grain boundaries (e.g., \\cite{Oberdorfer09}) where the corresponding continuity equation refers to the outer boundary.}\n\\begin{equation}\nD\\nabla \\rho_l\\Big|_{r=r_{0}}-\\alpha \\rho_l(r_{0},t)=0.\n\\end{equation}\n\nThe outer radius $R$ of the diffusion sphere is related to the void concentration \n\\begin{equation}\n\\label{eq:C_t}\nC_t = \\frac{3 \\Omega}{4 \\pi R^3} \\, .\n\\end{equation}\nThe outer boundary condition\n\\begin{equation}\n\\frac{\\partial \\rho_l}{\\partial r}\\Big|_{r=R}=0\n\\end{equation}\nreflects the vanishing $e^+$ flux through the outer border ($r = R$) of the diffusion sphere. \nThis boundary condition is the same as applied earlier in a quite\ndifferent diffusion-reaction model of ortho-para conversion of positronium at reaction cerntres \\cite{Wuerschum95}.\n\nAs initial condition we adopt the picture that at $t=0$ all thermalized positrons are in the free state and homogeneously distributed in the lattice, i.e., initial density $\\rho_l = \\rho_l (0)$,  $\\rho_t(0)=0$, $N_v (0) = 0$.\nUnder this initial condition the solution of Eq. (\\ref{differential_eq}) exhibits spherical symmetry.\n\nUp to this point the above formulated diffusion-reaction problem is identical to that of Nieminen et al. \\cite{Nieminen79}\napart from the additional rate-limited trapping at vacancy-type point defects which is considered here.\nHowever, compared to \\cite{Nieminen79}, in the following part of the present work the time dependence is handled by means of Laplace transformation which will lead to the more convenient\nclosed-form solutions. Applying the Laplace transformation \n\\begin{eqnarray}\n\\nonumber\n\\tilde{\\rho}_{l,t}(p)=\\int\\limits _{0}^{\\infty}\\exp(-pt)\\rho_{l,t}(t)\\mathrm{d}t,\n\\end{eqnarray}\n\\begin{equation}\n\\tilde{N}_{v,f}(p)=\\intop_{0}^{\\infty}\\exp(-pt)N_{v,f}(t)\\mathrm{d}t\n\\end{equation}\nleads to the basic equations\n\\begin{equation}\n\\label{differential_eq_r}\n\\frac{\\mathrm{d^{2}}\\tilde{\\rho}_l}{\\mathrm{d}r^{2}}+\\frac{2}{r}\n\\frac{\\mathrm{d}\\tilde{\\rho}_l}{\\mathrm{d}r}-\\gamma^{2}\\tilde{\\rho}_l\n=-\\frac{\\rho_l(0)}{D}\n\\end{equation}\nwith\n\\begin{eqnarray}\n\\label{eq:gamma}\n\\gamma^{2}=\\gamma^{2}(p)=\\frac{\\tau_f^{-1}+\\sigma_v C_v+p}{D} \\, ,\n\\end{eqnarray}\nand\n\\begin{equation}\n\\label{tilde_rho_t}\n\\tilde{\\rho}_t=\n\\frac{\\alpha \\tilde{\\rho}_l(r_{0},p)}{\n\\tau_{t}^{-1}+p} \\: ,\n\\end{equation}\n\\begin{equation}\n\\label{N_v}\n\\tilde{N}_v=\n\\frac{\\sigma_v C_v}{\\tau_v^{-1}+p} \\times \\int\\limits^{R}_{r_0} 4 \\pi r^2 \\tilde{\\rho_l}(r,p) {\\rm d} r\\: ,\n\\end{equation}\nwith the boundary conditions\n\\begin{equation}\n\\label{boundary_condition1}\nD\\,\\frac{\\mathrm{d}\\tilde{\\rho}_l}{\\mathrm{d}r}\\Bigg|_{r=r_{0}}-\n\\alpha \\tilde{\\rho}_l(r_{0},p)=0\n\\end{equation}\nand\n\\begin{equation}\n\\label{boundary_condition2}\n\\frac{\\mathrm{d}\\tilde{\\rho}_l}{\\mathrm{d}r}\\Bigg|_{r=R}=0 \\, .\n\\end{equation}\n\nThe solution of the differential equation (\\ref{differential_eq_r}) satisfying equations\n[Eq. (\\ref{boundary_condition1})] and [Eq. (\\ref{boundary_condition2})] can be written as\n\\begin{equation}\n\\label{solution_rho_tilde}\n\\tilde{\\rho}_l(r,p)=A\\, i_{0}^{(1)}(\\gamma r)+ B\\, i_{0}^{(2)}(\\gamma r) +\n\\frac{\\rho_l(0)}{\\tau_f^{-1}+\\sigma_v C_v + p}\n\\end{equation}\nwith\n\\begin{eqnarray}\n\\nonumber\nA:=& \\displaystyle{\\alpha~\\frac{\\rho_l(0)}{\\tau_f^{-1}+\\sigma_v C_v+p}~\\times~\\frac{i_{1}^{(2)}(\\gamma R)}{-D \\gamma F_1 + \\alpha F_2}} \\, , \\\\\n\\label{AB}\nB:= & \\displaystyle{\\alpha~\\frac{\\rho_l(0)}{\\tau_f^{-1}+\\sigma_v C_v+p}~\\times~\\frac{i_{1}^{(1)}(\\gamma R)}{D \\gamma F_1 - \\alpha F_2}}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\nonumber\nF1= i_1^{(2)}(\\gamma r_0) i_1^{(1)}(\\gamma R) - i_1^{(1)}(\\gamma r_0)  i_1^{(2)}(\\gamma R) \\, , \\\\\nF2= i_0^{(2)}(\\gamma r_0)  i_1^{(1)}(\\gamma R) - i_0^{(1)}(\\gamma r_0)  i_1^{(2)}(\\gamma R) \\, .\n\\end{eqnarray}\n\n \n$i_{n}^{(1)}$ and $i_{n}^{(2)}$ ($n = 0,1$) denote the modified spherical Bessel functions of order $n$\n\\cite{olver2010nist}\n\\begin{eqnarray}\n\\nonumber\ni_{n}^{(1)}(z):=(\\frac{\\pi}{2z})^{1\/2}I_{n+1\/2}(z), \\,     \n\\end{eqnarray}\n\\begin{equation}\ni_{0}^{(1)}=\\frac{\\sinh z}{z},\\quad                        \ni_{1}^{(1)}=\\frac{\\cosh z}{z}-\\frac{\\sinh z}{z^2}\n\\end{equation}\n\\begin{eqnarray}\n\\nonumber\ni_{n}^{(2)}(z):=(\\frac{\\pi}{2z})^{1\/2}I_{-n-1\/2}(z), \\,    \n\\end{eqnarray}\n\\begin{equation}\ni_{0}^{(2)}=\\frac{\\cosh z}{z},\\quad                        \ni_{1}^{(2)}=\\frac{\\sinh z}{z}-\\frac{\\cosh z}{z^2}\n\\end{equation}\nwhere $I_{\\pm n\\pm 1\/2}(z)$ represents the Bessel function.\n\nBasis for analyzing positron annihilation experiments is the total probability $n(t)$ that a\n$e^+$ implanted at $t=0$ has not yet been annihilated at time $t$.\nHere $n(t)$ is given by the number density of $e^+$ per lattice sphere at time\n$t$:\n\\begin{equation}\nn(t)= \\frac{1}{\\frac{4}{3}\\pi (R^3 - r_{0}^{3})\\rho_l(0)} \\times\n\\left\\{\n\\int\\limits _{r_{0}}^{R}4\\pi r^{2}\\rho_l(r,t)\\mathrm{d}r\n+4\\pi r_{0}^{2}\\rho_t (t) + N_v (t)\n\\right\\}       \\:.\n\\end{equation}\nThe Laplace transform of $n(t)$ can be calculated taking into account the solution of  $\\tilde{N}_v$ [Eq. (\\ref{N_v})]\nand the solution of the differential equation (\\ref{solution_rho_tilde}) which yields\n\\begin{equation}\n\\tilde{n}(p)=\\frac{1}{\\frac{4}{3}\\pi (R^3-r_{0}^{3})\\rho_l(0)}\n\\times\n\\left\\{\n \\left(1+\\frac{\\sigma_v C_v}{\\tau_v^{-1}+p}\\right)\n \\int\\limits^{R}_{r_0} 4\\pi r^{2} \\tilde{\\rho}_l(r,p) {\\rm d} r\n+ 4\\pi r^{2}_{0} \\tilde{\\rho}_t(p)\n\\right\\}       \\:.\n\\end{equation}\n\nSolving the integral after substituting $\\tilde{\\rho_t}(p)$ by Eq. (\\ref{tilde_rho_t}),  \ninsertion of $A$ and $B$ [Eq. (\\ref{AB})],\nyields after some algebra\n\n\\begin{equation}\n\\label{Laplace_n}\n\\tilde{n}(p)=\\frac{1}{t_{fc}^2 t_{v}  t_{t}} \\Biggl\\{t_{vc} t_{fc}  t_{t} + \\frac{K (t_{fc}  t_{v} - t_{vc}  t_{t})\n\\Bigl(\\gamma \\hat{R} - \\tanh(\\gamma \\hat{R}) [1- \\gamma^2 r_0 R]\\Bigr)}\n{\\gamma \\hat{R}- \\tanh(\\gamma \\hat{R}) [1-\\gamma^2 r_0 R] + \\frac{\\alpha r_0}{D}[\\gamma R - \\tanh(\\gamma \\hat{R})]} \\Biggr\\} \n\\end{equation}\nwith \n\\begin{equation}\n\\label{eq:K}\nK=\\frac{3\\alpha r_0^2}{R^3-r_0^3} \\, ,\n\\end{equation}\n\\begin{equation}\n\\label{eq:R}\n\\hat{R} = R-r_0 \\, ,\n\\end{equation}\nand the abbreviations\n\\begin{eqnarray}\n\\nonumber\nt_{t}=\\tau_t^{-1}+p \\, ; & t_{v}=\\tau_v^{-1}+p \\,; \\\\\nt_{vc}=\\tau_v^{-1}+\\sigma C+p \\, ; & t_{fc}=\\tau_f^{-1}+\\sigma C+p \\, .\n\\end{eqnarray}\nThe Laplace transform $\\tilde{n}(p)$ [Eq. (\\ref{Laplace_n})] represents the\nsolution of the present diffusion and trapping model from which\nboth the mean positron lifetime and the positron lifetime spectrum\ncan be deduced.\nThe mean positron lifetime $\\overline{\\tau}$ is obtained  by\ntaking the Laplace  transform  at $p = 0$:\n\\begin{equation}\n\\label{tauq_n_tilde}\n\\overline{\\tau} = \\tilde{n}(p=0) =  \\int\\limits^{\\infty}_{0}  n(t)\ndt \\, .\n\\end{equation}\nThe positron lifetime spectrum follows from  $\\tilde{n}(p)$ by means of Laplace inversion. The single poles $p = - \\lambda_i$\nof  $\\tilde{n}(p)$ in the complex $p$ plane define the decay rates $\\lambda_i (i=0,1,2,\\dots)$ of the positron lifetime spectrum:\n\\begin{equation}\n\\label{spectrum}\nn(t) = \\sum_{i=0}^{\\infty}I_i \\exp (-\\lambda_i t)  \\, ,\n\\end{equation}\nwhere $I_i$ denote the relative intensities.\n\n\n\n\\section{Analysis}\nAt first, we consider the most important case that $e^+$ trapping  exclusively occurs at  voids, i.e., \nwe omit $e^+$ trapping at point defects in the lattice ($C_v=0$).\nFor this case, we present the solution of the general diffusion-reaction theory (Sect.~\\ref{sec:general}) and compare \nit with the limiting cases of entirely reaction-controlled trapping (Sect.~\\ref{sec:rate_limit})\nand entirely diffusion-controlled trapping (Sect.~\\ref{sec:diffusion_limit}). Finally, the case of competitive \nreaction-controlled trapping at lattice defects is considered (Sect.~\\ref{sec:vacancies})\nand an extension to larger precipitates is presented for describing precipitate$-$matrix composite structures\n(Sect.~\\ref{sec:extended}).\n\n\\subsection{\\label{sec:general}\nGeneral case with trapping at voids, exclusively ($C_v=0$)}\n\nFor negligible trapping at vacancies within the lattice ($C_v=0$), \nthe diffusion-reaction model according to Eq.~(\\ref{Laplace_n})  yields\nfor positron trapping in voids as the single type of trap:\n\\begin{equation}\n\\label{eq:n}\n\\tilde{n}(p) = \\frac{1}{\\tau_f^{-1}+p} \\Biggl\\{ 1+\\frac{K(\\tau_f^{-1}-\\tau_t^{-1})}{(\\tau_t^{-1}+p)(\\tau_f^{-1}+p)} \\times \n\\frac{\\gamma \\hat{R}-\\tanh(\\gamma \\hat{R}) [1-\\gamma^2 r_0 R]}{\\gamma \\hat{R}-\\tanh(\\gamma \\hat{R}) [1-\\gamma^2 r_0 R]+ \\frac{\\alpha r_0}{D}[\\gamma R- \\tanh(\\gamma \\hat{R})]}\n \\Biggr \\} \n\\end{equation}\nand, hence, for the mean positron lifetime \n\\begin{equation}\n\\label{eq:tauq}\n\\overline{\\tau}=\\tilde{n}(0)= \\tau_f \\Biggl\\{1+ K (\\tau_t-\\tau_f) \\times \n\\frac{\\gamma_0 \\hat{R}-\\tanh(\\gamma_0 \\hat{R}) [1-\\gamma_0^2 r_0 R]}{\\gamma_0 \\hat{R}-\\tanh(\\gamma_0 \\hat{R}) [1-\\gamma_0^2 r_0 R]+ \\frac{\\alpha r_0}{D}[\\gamma_0 R- \\tanh(\\gamma_0 \\hat{R})]}\n \\Biggr \\} \\, .\n\\end{equation}\nThe pole of Eq.~(\\ref{eq:n}) for $p = - \\tau_t^{-1}$ corresponds to the positron lifetime  component $\\tau_t$ \nof the void-trapped state for which the following intensity is obtained:\n\\begin{equation}\n\\label{eq:I_t}\nI_t= \\frac{K}{\\tau_f^{-1}-\\tau_t^{-1}}\\times \n\\frac{\\gamma_t \\hat{R}-\\tanh(\\gamma_t \\hat{R}) [1-\\gamma_t^2 r_0 R]}{\\gamma_t \\hat{R}-\\tanh(\\gamma_t \\hat{R}) [1-\\gamma_t^2 r_0 R]+ \\frac{\\alpha r_0}{D}[\\gamma_t R- \\tanh(\\gamma_t \\hat{R})]} \\,.\n\\end{equation}\nIn equations (\\ref{eq:n}), (\\ref{eq:tauq}), (\\ref{eq:I_t}):\n\\begin{equation}\n\\label{eq:gamma_0_t}\n\\gamma^2=\\frac{\\tau _f^{-1}+p}{D}; \\, \n \\gamma_0^2=\\frac{\\tau _f^{-1}}{D}; \\, \n \\gamma_{t}^2=\\frac{\\tau _f^{-1}-\\tau_{t}^{-1}}{D} \\, .\n\\end{equation}\n\nIn addition to the annihilation component $\\tau_t^{-1}$ of the void-trapped state, \n$\\tilde{n}(p)$ [Eq.~(\\ref{eq:n})] comprises a sequence of first-order poles $p = - \\lambda_{0,j}$ for \n$\\lambda_{0,j} > \\tau_f^{-1}$. These  components $\\lambda_{0,j}$,  \nwhich define the fast decay rates ($\\lambda_{0,j} > \\tau_f^{-1}$) of the $e^+$ lifetime spectrum, \nare given by the solutions of the transcendental equation \n\\begin{equation}\n\\tan(\\gamma^{\\star} \\hat{R}) = \\frac{\\gamma^{\\star} (\\alpha r_0 R + D \\hat{R})}{D(1+\\gamma^{\\star 2} r_0 R) + \\alpha r_0}\n\\label{eq:transcendent}\n\\end{equation}\nwith\n\\begin{equation}\n\\gamma^{\\star 2} = \\frac{\\lambda_{0,j}-\\tau_f^{-1}}{D} \n\\label{eq:gamma'}\n\\end{equation}\nin agreement with the aforementioned earlier work of Nieminen et al. \\cite{Nieminen79}.\\footnote{Eq. (\\ref{eq:transcendent}) \nis identical to the corresponding eq. (15) in the work of Nieminen et al. when $\\nu$ in \\cite{Nieminen79}\nis identified with $4\\pi r_0^2 \\alpha$.}$^,$\\footnote{We note that   \nthe same problem was treated in the framework of a more general theoretical approach by K\\\"ogel \\cite{Koegel96}. The \nquoted specific function  in dependence of $\\gamma \\hat{R}$ [eq. (75) in \\cite{Koegel96}], which\ndetermines the mean $e^+$ lifetime and the intensity of the trap component,\n however, is not readily applicable.\n}\nAs usual for this kind of diffusion-reaction problem (see, e.g. \\cite{Oberdorfer09}),  \nthe intensities of these decay rates rapidly decrease.\nExperimentally only a single fast decay rate can be resolved in addition to the decay rate $\\tau_t^{-1}$ of the trapped state. An experimental two-component $e^+$ lifetime spectrum is practically entirely  defined by \n$\\overline{\\tau}$ [Eq.~(\\ref{eq:tauq})] and by $\\tau_t$ with the corresponding intensity $I_t$ [Eq.~(\\ref{eq:I_t})].\n\nThe appearance of a second-order pole in Eq.~(\\ref{eq:n}) at $p = - \\tau_f^{-1}$ (i.e., $\\gamma = 0$) is spurious. Closer inspection by applying Taylor expansion shows that the intensity associated with this pole cancels.\n\n\nFollowing the consideration of Dryzek \\cite{dryzek2002}, in analogy to the mean $e^+$ lifetime [Eq. (\\ref{eq:tauq})]\na respective relation for the mean line shape parameter $\\overline{S}$ of \nDoppler broadening of the positron-electron annihilation\ncan be given:\n\\begin{equation}\n\\label{eq:Doppler}\n\\overline{S}=  S_f \\Biggl\\{1+ K (S_t-S_f) \\times \n\\frac{\\gamma_0 \\hat{R}-\\tanh(\\gamma_0 \\hat{R}) [1-\\gamma_0^2 r_0 R]}{\\gamma_0 \\hat{R}-\\tanh(\\gamma_0 \\hat{R}) [1-\\gamma_0^2 r_0 R]+ \\frac{\\alpha r_0}{D}[\\gamma_0 R- \\tanh(\\gamma_0 \\hat{R})]}\n \\Biggr \\} \\, ,\n\\end{equation}\nwhere $S_f$ and $S_t$ denote the line shape parameters of the free and trapped state, respectively.\n\nFor the sake of completeness, we quote $\\tilde{n}(p)$ without derivation for the case that \nat time zero positrons are homogeneously distributed in the voids and the lattice, \ni.e., for the initial condition $\\rho_t (0) = r_0 \\rho_l (0)\/3$: \n\\begin{eqnarray}\n\\label{eq:n_homogen}\n\\nonumber\n\\tilde{n}(p) =  \\displaystyle{\\frac{1}{\\tau_f^{-1}+p} \\Biggl\\{ 1+ \\frac{r_0^3}{R^3} \\times\n\\frac{\\tau_f^{-1}-\\tau_t^{-1}}{\\tau_t^{-1} + p} +}\n \\frac{3 \\alpha r_0^2}{R^3} \\times \\frac{\\tau_f^{-1}-\\tau_t^{-1}}{(\\tau_t^{-1}+p)(\\tau_f^{-1}+p)} \n\\times \\\\\n \\displaystyle{\\frac{\\gamma \\hat{R}-\\tanh(\\gamma \\hat{R}) [1-\\gamma^2 r_0 R]}{\\gamma \\hat{R}-\\tanh(\\gamma \\hat{R}) [1-\\gamma^2 r_0 R]+ \\frac{\\alpha r_0}{D}[\\gamma R- \\tanh(\\gamma \\hat{R})]}\n \\Biggr \\}} \\, .\n\\end{eqnarray}\nEq.~(\\ref{eq:n_homogen}) includes in the limiting case  of negligible trapping ($\\alpha = 0$)\nas  mean $e^+$ lifetime $\\overline{\\tau} = \\tilde{n}(0) = \n[(R^3-r_0^3) \\tau_f + r_0^3 \\tau_t]\/R^3$ the expected volume-averaged mean value of $\\tau_f$ and $\\tau_t$.  \n\n\n\\subsection{\\label{sec:rate_limit}\nLimiting case of entirely reaction limited trapping ($C_v=0$)}\n\nIf the e$^+$ diffusivity is high ($\\gamma \\hat{R} \\ll 1$), the hyperbolic tangent \nin Eq.~(\\ref{eq:n})  can be expanded.\nExpansion up to the third order \n\\begin{equation}\n\\tanh(z)\\approx z-\\frac{z^3}{3}\n\\end{equation}   \nyields the mean $e^+$ lifetime \n\\begin{equation}\n\\label{eq:tauq_rate}\n\\overline{\\tau}=\\displaystyle{\n\\tau_f \\frac{1+ K \\tau_t}{1+K \\tau_f}}\n\\end{equation}\nand for the $e^+$ lifetime component $\\tau_t$ the intensity \n\\begin{equation}\n\\label{eq:I_t_rate}\nI_t=\\displaystyle{\n\\frac{K }{\\tau_f^{-1}+ K -\\tau_t^{-1}}\n}\n\\end{equation}\nwith $K$ according to Eq.~(\\ref{eq:K}).\nEquations (\\ref{eq:tauq_rate}) and (\\ref{eq:I_t_rate}) are the well-known solutions of the simple trapping model\nwhen we identify $K$ for vanishing defect volume with the trapping rate $\\sigma_t C_t$ \n[equations (\\ref{eq:sigma_t}) and (\\ref{eq:C_t})].\nNote that the standard trapping model does not take into account the finite defect volume (here $4 \\pi r_0^3\/3$)\nand, therefore, does not contain the subtrahend $r_0^3$ as in Eq. (\\ref{eq:K}). With this subtrahend,\nequations (\\ref{eq:tauq_rate}) and (\\ref{eq:I_t_rate}) correctly contain \nthe exact values  $\\overline{\\tau} = \\tau_t$ and $I_t = 1$ \nas limiting case for $R=r_0$.\n\n\n\\subsection{\\label{sec:diffusion_limit}\nLimiting case of entirely diffusion limited trapping ($C_v=0$)}\n\nThe present solution includes in the limiting special case $\\alpha \\rightarrow \\infty$ the relationships\nfor an entirely diffusion-limited trapping, i.e., for Smoluchowski-type boundary condition \n\\begin{equation}\n\\label{eq:Smoluchowski}\n\\rho_l (r_0, t) = 0 \\, .\n\\end{equation}\nIn this limit one obtains from the Laplace transform [Eq. (\\ref{eq:n})]\nthe mean $e^+$ lifetime \n\\begin{equation}\n\\label{eq:tauq_diffusion}\n\\overline{\\tau}=\\tau_f \\Biggl \\{ 1+ \\frac{3r_{\\rm 0}D}{R^3-r_{\\rm 0}^3}(\\tau_t-\\tau_f) \n\\frac{\\gamma_0 \\hat{R}-\\tanh(\\gamma_0 \\hat{R})[1-\\gamma_0^2 r_0 R]}{\\gamma_0 R-\\tanh(\\gamma_0 \\hat{R})} \\Biggr\\}\n\\end{equation}\nand for the trap component $\\tau_t$ the intensity\n\\begin{equation}\n\\label{eq:I_t_diffusion}\nI_t=\\frac{3 r_0 D}{R^3-r_0^3} \\times \\frac{1}{\\tau_f^{-1}-\\tau_t^{-1}} \\times \\frac{\\gamma_t \\hat{R}-\\tanh(\\gamma_t \\hat{R})[1-\\gamma_t^{2}r_0 R]}{\\gamma_t R-\\tanh(\\gamma_t \\hat{R})}\n\\end{equation}\nwith $\\gamma_0$, $\\gamma_t$ according to eq. (\\ref{eq:gamma_0_t}).\n\n\n \\subsection{\\label{sec:vacancies}\nGeneral case with voids {\\em and} lattice vacancies}\n\nThe positron annihilation characteristics of diffusion-reaction\ncontrolled trapping at voids and concomitant transition-limited\ntrapping at point defects in the lattice is given by Eq.\n(\\ref{Laplace_n}) in combination with Eq. (\\ref{tauq_n_tilde}) and\nEq. (\\ref{spectrum}).\nThe mean positron lifetime [Eq. \\ref{tauq_n_tilde}], obtained from\nEq. (\\ref{Laplace_n}) for $p=0$,  reads in the general case:\n\\begin{eqnarray}\n\\label{eq:tau_q_general}\n\\nonumber\n\\overline{\\tau}=  \\displaystyle{\n\\frac{1}{(\\tau_f^{-1}+\\sigma_v C_v)^{2}} }\n\\Biggl\\{(\\tau_f^{-1}+\\sigma_v C_v)  (\\tau_v^{-1}+\\sigma_v C_v) \\tau_v+  \\\\\n\\displaystyle{\n\\frac{K \\Bigl((\\tau_f^{-1}+\\sigma_v C_v)  \\tau_{t} - (\\tau_v^{-1}+\\sigma_v C_v)  \\tau_{v} \\Bigr)\\Bigl(\\gamma_0 \\hat{R} - \\tanh(\\gamma_0 \\hat{R})[1- \\gamma^2 r_0 R]\\Bigr)}\n{\\gamma_0 \\hat{R}- \\tanh(\\gamma_0 \\hat{R})[1-\\gamma_0^2 r_0 R] + \\frac{\\alpha r_0}{D}[\\gamma_0 R - \\tanh(\\gamma_0 \\hat{R})]}\\Biggr\\}\n}\n\\end{eqnarray}\nwith\n\\begin{equation}\n\\label{eq:gamma_v}\n\\gamma_0^2=\\frac{\\tau_f^{-1}+\\sigma_v C_v}{D} \\, .\n\\end{equation}\n\nIn addition to the pole $p = - \\tau_t^{-1}$ which characterizes the void trapped state,\n$\\tilde{n}(p)$ [Eq. (\\ref{Laplace_n})] contains the further defect-related pole  \n$p = - \\tau_v^{-1}$ for the vacancy-type defect in the lattice.\nFrom the residues of  $\\tilde{n}(p)$ [Eq. (\\ref{Laplace_n})], \nthe corresponding relative intensities\n\\begin{equation}\n\\label{eq:I_t_general}\nI_t=\\frac{K}{\\tau_f^{-1}+\\sigma_v C_v-\\tau_t^{-1}}\\times\\frac{\\gamma_t \\hat{R} - \\tanh(\\gamma_t \\hat{R})[1- \\gamma_t^2 r_0 R]}{\\gamma_t \\hat{R}- \\tanh(\\gamma_t \\hat{R})[1-\\gamma_t^2 r_0 R] + \n\\frac{\\alpha r_0}{D}[\\gamma_t R - \\tanh(\\gamma_t \\hat{R})]}\n\\end{equation}\nand \n\\begin{eqnarray}\n\\nonumber\nI_v=\n\\frac{\\sigma_v C_v}{\\tau_f^{-1}+\\sigma_v C_v-\\tau_v^{-1}}  \\Biggl\\{1- \\frac{K}{\\tau_f^{-1}+\\sigma_v C_v-\\tau_v^{-1}} \\times \\\\\n\\label{eq:I_v} \n\\frac{\\gamma_v \\hat{R} - \\tanh(\\gamma_v \\hat{R})[1- \\gamma_v^2 r_0 R]}{\\gamma_v \\hat{R}- \\tanh(\\gamma_v \\hat{R})[1-\\gamma_v^2 r_0 R] + \\frac{\\alpha r_0}{D}(\\gamma_v R - \\tanh(\\gamma_v \\hat{R})]}\\Biggr\\}\n\\end{eqnarray}\nare deduced with \n\\begin{equation}  \n\\label{eq:gamma_t_v}\n\\gamma_{t,v}^2=\\frac{\\tau _f^{-1}+\\sigma_v C_v-\\tau_{t,v}^{-1}}{D} \\, .\n\\end{equation}\n\n\n\\subsection{\\label{sec:extended}\nExtended model for larger preciptates  with  $e^+$-trapping from both sides of precipitate$-$matrix interface}\nThe model presented above describes $e^+$ annihilation from a trapped state ($\\tau_t$)\nin spherical defects. Particularly, for larger precipitate sizes a situation may prevail where \n$e^+$ annihilation inside the precipitates occurs from a free state with a characteristic $e^+$ lifetime $\\tau_p$\nand where also from this free precipitate state positrons may  get trapped into the spherical interfacial shell between the precipitate and the surrounding matrix. This means that the precipitates are characterized by two compoments, one corresponding to the precipitate volume ($\\tau_p$) and one corresponding to the trapped state in the matrix$-$precipitate interface ($\\tau_t$).\n\n\nThe present model can be extended in a straight forward manner to this case under the reasonable assumption that the $e^+$ trapping from inside the precipitates is entirely reaction controlled. This is pretty well fulfilled as long as\nthe precipitate diameter is remarkably lower than the $e^+$ diffusion length in the precipitate.\\footnote{A further model extension avoiding this\nconstraint will be outlined below.} \nIn this case the extension can be described by an additional rate equation for the temporal evolution of the number $N_p$ of $e^+$ \ninside the precipitates \n\\begin{equation}\n\\label{eq:N_p}\n\\frac{\\mathrm{d}N_p}{\\mathrm{d}t}=\n-\\Bigl( \\frac{1}{\\tau_p} + \\frac{3\\beta}{r_0} \\Bigr) N_p \\, ,\n\\end{equation}\nwhere $\\beta$ denotes the specific trapping rate (in units of m\/s) at the spherical interfacial shell.\nThis trapping from inside the precipitates, which occurs in addition to the diffusion- and reaction-limited trapping\ninto the interfacial shell from the surrounding matrix, has to be taken into account in the rate equation for \n$\\rho_t$ (Equation \\ref{eq:rho_t})\nby the additional summand $\\beta \\rho_p(t)$ with the number density\n$\\rho_p = 3 N_p \/ (4 \\pi r_0^3)$ of $e^+$ in the precipitate.\n\nAssuming a homogeneous distribution of $e^+$ at time zero  in the matrix and the \nprecipitate ($\\rho_l (0) = \\rho_p (0)$) without  $e^+$ in the trapped state ($\\rho_t (0) = 0$) for $t=0$,\none obtains with the Laplace transform of eq.~(\\ref{eq:N_p})\n\\begin{equation}\n\\label{eq:Laplace_N_p}\n\\tilde{N_p} = \\displaystyle{\\frac{N_p(0)}{\\tau_p^{-1} + \\frac{3 \\beta}{r_0} + p}}\n\\end{equation}\nthe additional summand\n\\begin{equation}\n\\label{eq:n_extension}\n\\Bigl( \\frac{r_0}{R} \\Bigr)^3 \n\\displaystyle{\\Bigl(\\frac{\\frac{3 \\beta}{r_0}}{\\tau_t^{-1} +p} +1 \\Bigr)\n\\frac{1}{\\tau_p^{-1} + \\frac{3 \\beta}{r_0} + p}}\n\\end{equation}\nin  eq. (\\ref{Laplace_n}) of $\\tilde{n}(p)$. \nMoreover, in the bracket of eq. (\\ref{Laplace_n}) the first summand  is extended by the weighting factor \n$[ 1 - (r_0\/R)^3]$ and  the  trapping rate $K$ (Equation \\ref{eq:K}) in the second summand is replaced by \n$3 \\alpha r_0^2\/R^3$.\n\nFor \\underline{$C_v = 0$} this leads to the mean $e^+$ lifetime \n\\begin{eqnarray}\n\\label{eq:tauq_extended}\n&\\overline{\\tau}= \\tau_f \\Biggl\\{\\displaystyle{ \\Bigl[1 - \\Bigl(\\frac{r_0}{R} \\Bigr)^3 \\Bigr]} + \n\\nonumber \\\\\n& \\displaystyle{\\frac{3 \\alpha r_0^2}{R^3} \n\\times (\\tau_t-\\tau_f) \\times \n\\frac{\\gamma_0 \\hat{R}-\\tanh(\\gamma_0 \\hat{R}) [1-\\gamma_0^2 r_0 R]}{\\gamma_0 \\hat{R}-\\tanh(\\gamma_0 \\hat{R}) [1-\\gamma_0^2 r_0 R]+ \\frac{\\alpha r_0}{D}[\\gamma_0 R- \\tanh(\\gamma_0 \\hat{R})]} \\Biggr \\} } + \\nonumber \\\\\n& \\displaystyle{\\Bigl(\\frac{r_0}{R} \\Bigr)^3 \\times \\tau_t \\times \\frac{\\tau_t^{-1} + \\frac{3 \\beta}{r_0}}{\\tau_p^{-1} + \\frac{3 \\beta}{r_0}}}\n\\, , \n\\end{eqnarray}\nas compared to eq. (\\ref{eq:tauq}).\nEq.~(\\ref{eq:tauq_extended}) includes in the limiting case  of negligible trapping ($\\alpha = \\beta = 0$)\nas  mean $e^+$ lifetime $\\overline{\\tau} = \n[(R^3-r_0^3) \\tau_f + r_0^3 \\tau_p]\/R^3$ the expected volume-averaged mean value of $\\tau_f$ and $\\tau_p$.  \n\nThe additional pole for $p = - (\\tau_p^{-1} + 3 \\beta \/r_0)$ of  $\\tilde{n}(p)$ yields the intensity \nof the $e^+$ lifetime component $\\tau_p$ in the precipitate:\n\\begin{equation}\nI_p = \\Bigl(\\frac{r_0}{R} \\Bigr)^3 \\times \n\\Biggl(1 - \\displaystyle{\\frac{\\frac{3 \\beta}{r_0}}{\\tau_p^{-1} + \\frac{3 \\beta}{r_0} - \\tau_t^{-1}} \\Biggr)}\n\\, . \n\\label{eq:I_p}\n\\end{equation}\nApart from the weighting prefactor ($(r_0\/R)^3$), $I_p$ corresponds to the solution of the simple trapping model.\\footnote{\nNote that $I_p$ characterizes the free state in the precipitate.} Without trapping ($\\beta = 0$), $I_p$ simply takes  the form of the weighting prefactor $(r_0\/R)^3$.\n\nSince $e^+$ trapping into the precipitate$-$matrix interface occurs both from inside the precipitate and from the surrounding matrix, the intensity  \nof the trap component $\\tau_t$ is given by the sum\n\\begin{equation}\nI_t = I_t^{precip} + I_t^{matrix} \\text{ with } \nI_t^{precip} = \\Bigl(\\frac{r_0}{R} \\Bigr)^3 \\times \n\\displaystyle{\\frac{\\frac{3 \\beta}{r_0}}{\\tau_p^{-1} + \\frac{3 \\beta}{r_0} - \\tau_t^{-1}} }\n\\, ,\n\\label{eq:I_t_tot}\n\\end{equation}\nwhere $I_t^{matrix}$ corresponds to the intensity $I_t$ according to eq.~(\\ref{eq:I_t}) with  \n$K$   replaced by $3 \\alpha r_0^2\/R^3$.\\footnote{The identical equation for $I_t$ (equation~\\ref{eq:I_t_tot})  follows \nfrom the root $p = - \\tau_t$ of the Laplace transform $\\tilde{n}(p)$ in which the above mentioned extensions \nof eq. (\\ref{Laplace_n}) are taken into consideration.}\n \nWe note that the two $e^+$ trapping processes into the precipitate$-$matrix interface, namely that from inside the precipitate and that from the surrounding matrix, are completely decoupled. The trapping process from inside the precipitate can, therefore, be treated independently.\nThis also means that the process has not to be restricted to the case of entirely reaction-controlled trapping as given above, but \nthat $e^+$ trapping at the precipitate$-$matrix interface from inside the spherical precipitates can also be treated \nin the framework of diffusion-reaction theory. Hence, the available solutions for \ndiffusion- and reaction-limited trapping at grain boundaries (GBs) of spherical crystallites \\cite{Wuerschum96, Oberdorfer09} can \nbe directly applied. For this purpose the solutions for the GB-model have simply to be weighted by the factor $(r_0\/R)^3$ which denotes the volume fraction of the precipitates.\\footnote{Given the above initial condition $\\rho_l (0) = \\rho_p (0)$ and $\\rho_t (0) = 0$.}\n\nFor instance, for the mean $e^+$ lifetime, the last summand in Eq.~(\\ref{eq:tauq_extended}), i.e., the rate-equation solution has \nto be replaced by that calculated for diffusion- and reaction-limited trapping at GBs \\cite{Wuerschum96, Oberdorfer09},\nyielding:\n\\begin{eqnarray}\n\\label{eq:tauq_double_extended}\n&\\overline{\\tau}= \\tau_f \\Biggl\\{\\displaystyle{ \\Bigl[1 - \\Bigl(\\frac{r_0}{R} \\Bigr)^3 \\Bigr]} + \n\\nonumber \\\\\n& \\displaystyle{\\frac{3 \\alpha r_0^2}{R^3} \n\\times (\\tau_t-\\tau_f) \\times \n\\frac{\\gamma_0 \\hat{R}-\\tanh(\\gamma_0 \\hat{R}) [1-\\gamma_0^2 r_0 R]}{\\gamma_0 \\hat{R}-\\tanh(\\gamma_0 \\hat{R}) [1-\\gamma_0^2 r_0 R]+ \\frac{\\alpha r_0}{D}[\\gamma_0 R- \\tanh(\\gamma_0 \\hat{R})]} \\Biggr \\} } + \\nonumber \\\\\n& \\displaystyle{\\Bigl(\\frac{r_0}{R} \\Bigr)^3 \\Biggl\\{ \n\\tau_p +  (\\tau_t-\\tau_p) \\times \n\\frac{3 \\beta L(\\gamma'_0 r_0)}{r_0 \\gamma'_0\n\\Bigl( \\beta + \\gamma'_0 D L (\\gamma'_0 r_0) \\Bigr) }\\Biggr\\}}\n\\, , \n\\end{eqnarray}\nwith $\\gamma'_0 = (\\tau_p D)^{-1\/2}$,  $\\gamma_0 = (\\tau_f D)^{-1\/2}$\n and the Langevin function\n\\begin{equation}\nL (z) = \\coth z - \\frac{1}{z} \\, .\n\\label{eq:langevin}\n\\end{equation}\n\nLikewise the intensity component $I_t^{precip}$\n of the rate-equation solution in Eq.~(\\ref{eq:I_t_tot}) has to be replaced by \\cite{Wuerschum96, Oberdorfer09}:\n\\begin{equation}\n\\label{eq:I_p_diff}\nI_t^{precip} = \\Bigl(\\frac{r_0}{R} \\Bigr)^3 \\times \\frac{3 \\beta}{r_0(\\tau_p^{-1}-\\tau_t^{-1})} \\times\n\\Biggl\\{\n\\frac{\\gamma'_t D L(\\gamma'_t r_0)}{\n \\beta + \\gamma'_t D L(\\gamma'_t r_0)  }\n\\Biggr\\}\n\\, \n\\end{equation}\nwith \n\\begin{equation}\n\\gamma'^2_t = \\frac{\\tau_p^{-1} - \\tau_t^{-1}}{D} \\, .\n\\label{eq:gamma'_t}\n\\end{equation}\nFor the sake of completeness we quote the mean $e^+$ lifetime for reaction-controlled trapping from both in- and outside:\n\\begin{equation}\n\\label{eq:tauq_extended_rate}\n\\overline{\\tau}= \\displaystyle{\\tau_t \\Bigl(1 - \\Bigl[\\frac{r_0}{R} \\Bigr]^3 \\Bigr) \n\\frac{\\tau_t^{-1} + \\frac{3 \\alpha r_0^2}{R^3-r_0^3}}{\\tau_f^{-1} + \\frac{3 \\alpha r_0^2}{R^3-r_0^3}}\n+ \\tau_t \\Bigl(\\frac{r_0}{R} \\Bigr)^3  \\, \\, \\frac{\\tau_t^{-1} + \\frac{3 \\beta}{r_0}}{\\tau_p^{-1} + \\frac{3 \\beta}{r_0}}}\n\\, . \n\\end{equation}\nA further extension for taken into account additional $e^+$ trapping at point defects\ninside the matrix (Sect.~\\ref{sec:vacancies})\nand inside the precipitates (in analogy to the GB model \\cite{Oberdorfer09}) is straightforward, so that \nthe corresponding equations have not to be stated explicitly.\n\n\n\\section{\\label{discussion} Discussion}\n\\subsection{\\label{sec:voids}\nVoids, clusters, small precipitates}\n\nThe presented model with the exact solution of diffusion-reaction\ncontrolled trapping at voids (or other extended spherical defects like clusters and small precipitates)\nand competitive transition-limited\ntrapping at vacancy-type defects yields closed-form expressions\nfor the mean positron lifetime $\\overline{\\tau}$ [Eq. (\\ref{eq:tau_q_general})]\nand for the relative intensities $I_t$ [Eq. (\\ref{eq:I_t_general})]\nand $I_v$ [Eq. (\\ref{eq:I_v})] of the $e^+$ lifetime components\n$\\tau_t$ and $\\tau_v$ of the void and the vacancy\ntrapped states, respectively.\n\nWe start the discussion considering exclusively diffusion-reaction\ncontrolled trapping at voids  (Sect. \\ref{sec:general}).\nThe model contains as limiting cases both the solution of the simple trapping model \n(Sect. \\ref{sec:rate_limit}) and the one of the entirely diffusion-limited trapping (Sect. \\ref{sec:diffusion_limit}).\nThe mean $e^+$ lifetime $\\overline{\\tau}$ [Eq. (\\ref{eq:tauq})] and the intensity \n$I_t$ [Eq. (\\ref{eq:I_t})] in dependence of the radius $R$ of the diffusion sphere\nare compared in Fig. \\ref{fig:2} with the two limiting cases. Note, that $R$ is related to the the void concentration \n[Eq. (\\ref{eq:C_t})].\nFor illustration the following characteristic e$^+$ annihilation parameters are used:\na free $e^+$ lifetime  $\\tau_f=160$~ps as typical for aluminium, \na $e^+$ lifetime $\\tau_t=400$~ps as typical for voids \\cite{Eldrup03},\na $e^+$ diffusion coefficient $D=2 \\times 10^{-5}$~m$^2$s$^{-1}$, a void radius $r = 3$~nm, and  \na specific e$^+$ trapping rate  $\\alpha = 3 \\times 10^3$~ms$^{-1}$\nreported by Dupasquier et al. \\cite{Dupasquier93} for interfaces in Al. For surfaces of Al \na value $\\alpha = 7.6 \\times 10^3$~ms$^{-1}$ was calculated by Nieminen and Lakkonnen \\cite{Nieminen79a}.\nUsing an atomic volume $\\Omega$ for Al of $\\Omega^{-1} = 6 \\times 10^{28}$~m$^{-3}$,   \n$\\alpha = 3 \\times 10^3$~ms$^{-1}$ corresponds to  a trapping rate $\\sigma_t = 2\\times 10^{16}$ s$^{-1}$ \n[Eq. \\ref{eq:sigma_t}] which is similar to that deduced by Bentzon and Evans \\cite{Bentzon90} for voids in Mo.\\footnote{\nA value $\\sigma_t = 4\\times 10^{16}$ s$^{-1}$ is deduced from \nthe trapping rate of $3.2 \\times 10^9$~s$^{-1}$ at 300~K and \na void number density of $5.3 \\times 10^{21}$~m$^{-3}$\nquoted in \\cite{Bentzon90}.}\n\nBoth $\\overline{\\tau}$ (Fig. \\ref{fig:2}a) and $I_t$ (Fig. \\ref{fig:2}b) exhibit the characteristic sigmoidal increase\nfrom the free state to the saturation-trapped state with decreasing $R$, i.e., increasing void concentration $C_t$.\nCompared to the exact solution of the present model, the standard trapping model \nand the limiting case of entirely diffusion-limited trapping \nshow qualitatively the same trend for $\\overline{\\tau}$ and $I_t$.\nHowever, both special cases systematically overestimate $\\overline{\\tau}$ and $I_t$, i.e., predict \nstronger trapping since either the rate-limiting effect or the diffusion-limiting effect are neglected in these\napproximations. For instance, if one would determine  the void concentration from a typical, experimentally measured intensity $I_t$ of\n45\\,\\% \\cite{Nambissan1989}, a concentration 36\\,\\% too low would be deduced from the standard trapping model \ncompared to the exact theory for the parameter set according to Fig. \\ref{fig:2}b.\n\nThe deviations of the two limiting cases from the exact solution become even more clear when \nthe ratios of the trap component intensities of the limiting and exact solution is considered\nas shown in the upper part of Fig. \\ref{fig:3}.\nThe deviation from the exact solution substantially increases with decreasing intensity, i.e., with\ndecreasing void concentration. In this low concentration regime, the deviations \nattain a factor of ca. 1.5 (reaction limit) or larger than 3 (diffusion limit)\nfor the present set of parameters, i.e., the entirely diffusion-limiting case\ndeviates in this example more strongly than the reaction-limited case. \nDiffusion limitation gets even more pronounced when $e^+$ diffusivity is reduced, e.g., due to\nscattering at lattice imperfections.\nRegarding the opposite side of high defect concentrations,  \nFig. \\ref{fig:3} (upper part) nicely demonstrates that deviations from the exact theory  \nvanishes upon approaching $e^+$ saturation trapping since in this regime  \nkinetic effects tends to become irrelevant.\n\n\n\n\\subsubsection{\\label{sec:effective_rate} Comparison with effective rate approach}\nNext we compare the present model with approximations according to which\ndiffusion limitation is taking account in the standard trapping model by means of a diffusion-limited trapping rate \n\\cite{Seeger74,Bentzon90}:\n\\begin{equation}\nK_{diff} = \\frac{4\\pi r_0 D}{\\Omega} \\times C_t \\,.\n\\label{eq:K_diffusion}\n\\end{equation}\nThe case of both transition- and diffusion-limited trapping, is treated in this approximation by means of\nthe effective trapping rate \\cite{Seeger74,Bentzon90}\n\\begin{equation}\nK_{eff} = \\frac{K_{diff} \\sigma_t C_t}{K_{diff} + \\sigma_t C_t} \n\\label{eq:K_eff}\n\\end{equation}\nwith $\\sigma_t$ and $C_t$ according to equations (\\ref{eq:sigma_t}) and  (\\ref{eq:C_t}), respectively.\nWe note that the diffusion-limited trapping rate according to eq. (\\ref{eq:K_diffusion}) is also included in the present model; in fact $K_{diff}$ is identical to the pre-factor of $I_t$ for entirely diffusion limited trapping [Eq. (\\ref{eq:I_t_diffusion})] when the \nsubtrahend $r_0^3$ in the nominator, which is associated with the defect volume,  is omitted.\n\nIn figure \\ref{fig:4} the concentration dependence of the relative intensity \n$I_t$ of the $e^+$ lifetime component $\\tau_t$ \nin voids is shown for the exact models of diffusion-reaction  [Eq. (\\ref{eq:I_t})] or \nentire diffusion limitation [Eq. (\\ref{eq:I_t_diffusion})] in comparison with the corresponding approximations\nusing the above mentioned effective or  diffusion-trapping rates [Equations (\\ref{eq:K_diffusion}), (\\ref{eq:K_eff})] \nwith the simple trapping  model [Eq. (\\ref{eq:I_t_rate})].\nAlthough the effective-rate approximations of the diffusion limitation describe the sigmoidal curve fairly well,\ndeviations from the exact diffusion models are also apparent, e.g., for the example, $I_t = 45$\\,\\%, mentioned above \nthe deviation in concentration is ca. 7\\,\\% compared to the exact diffusion-reaction theory.\n\nThe deviations become clearer once more when we consider the intensity ratio of the effective-rate model and the exact\ntheory, as plotted in the lower part of Fig. \\ref{fig:3}.\nRemarkably, since the effectice trapping rate $K_{eff}$ is lower than both the reaction-trapping rate \n$\\sigma_t C_t$ and the diffusion-trapping rate $K_{diff}$,\nthe intensity $I_t$ deduced from the effective trapping model is smaller than the exact value.\nDeviations from the full model occur throughout the entire intensity regime, although these \ndeviations  are less pronounced compared to the two limiting cases \n(fully reaction- or diffusion limited, upper part of Fig. \\ref{fig:3}). \nFor applications in the analysis of experimental data, \nthe accuracy of the effective rate approach [equation~(\\ref{eq:K_eff})] can be assessed \nby plotting the intensity ratio (lower part of Fig. \\ref{fig:3}) for the respective parameter set.\nIrrespectively whether deviations of the effective-rate approach are strong or  minor only, \nthe present model founded on diffusion-reaction theory is that which covers the underlying physics\nmost accurately.\n  \n\n\\subsubsection{\\label{sec:competitive} Competitive trapping at point defects}\nNow, we discuss the general case that in addition to diffusion-reaction\ncontrolled trapping at voids also competitive transition-limited trapping\nat vacancy-type defects in the lattice occurs (Sect. \\ref{sec:vacancies}).\nThe relative intensities of the void component $I_t$  [Eq. (\\ref{eq:I_t_general})] and \nof the vacancy component $I_v$ [Eq. (\\ref{eq:I_v})] is plotted in figure \\ref{fig:5}\nin dependence of void concentration $C_t$ (a) and vacancy concentration $C_v$ (b), \nfor a given fixed $C_v$ or $C_t$, respectively. For the \nvacancy-type defect a $e^+$ lifetime component $\\tau_v=250$~ps and \na specific trapping rate $\\sigma_v=4\\times10^{14}$~s$^{-1}$ \\cite{Schaefer87} is assumed; the other parameters \nare the same as used above.\nThe competitive $e^+$ trapping at voids and  vacancy-type defects becomes evident.\nFor a given vacancy concentration the\nintensity $I_t$ of the void increases and the\nintensity $I_v$ of the vacancy  component decreases with increasing \nvoid concentration due to the increasing fraction of e$^+$ that reaches the\nvoids (Fig.~\\ref{fig:5}.a). Likewise, for a given void concentration,\n$I_v$  increases and  $I_t$ decreases with increasing vacancy concentration\n(Fig. \\ref{fig:5}.b).\n\n\n\n\\subsubsection{\\label{sec:gb} Comparison with $e^+$ trapping at grain boundaries}\nIn the end of this subsection (\\ref{sec:voids}), the results of the present model on diffusion-reaction limited $e^+$ trapping\nat extended spherical defects will briefly be compared with the corresponding model \nof $e^+$ trapping at grain boundaries of spherical crystallites with radius $R$ \\cite{Wuerschum96, Oberdorfer09}.\nWhereas in the latter case the surface of the diffusion sphere  \nwith area $4 \\pi R^2$ acts as $e^+$ trap, in the present case with voids of radius $r_0$, the trapping active area \n$4 \\pi r_0^2$ is much smaller. Moreover,  the trapping rate $3 \\alpha \/R$ for grain boundary trapping \\cite{Oberdorfer09}\ndecreases much more slowly with increasing $R$ compared to the trapping rate $3 \\alpha r_0^2 \/ (R^3-r_0^3)$ of \nspherical extended defects with radius $r_0$ [Eq. \\ref{eq:K}].\nThis is the reason why diffusion limitation \naffects the kinetics of $e^+$ trapping at grain boundaries more strongly than in the case of voids\nwhich is nicely  demonstrated in Fig. \\ref{fig:6} where the exact solutions are compared with those of infinite diffusivities.\nIn Fig. \\ref{fig:6} the mean $e^+$ lifetime according to the exact solutions  and those of the standard rate theory  \nfor the two types of extended traps are plotted.\nThe exact solution for $e^+$ trapping at grain boundaries of spherical crystallites with radius $R$ reads \n\\cite{Wuerschum96, Oberdorfer09}\n\\begin{equation}\n\\label{eq:tauq_GB}\n\\overline{\\tau} = \\tau_f +  (\\tau_t-\\tau_f) \\times \n\\frac{3 \\alpha L(\\gamma_0 R)}{R \\gamma_0\n\\Bigl\\{ \\alpha + \\gamma_0 D L(\\gamma_0 R) \\Bigl\\}} \\text{ with } L (z) = \\coth z - \\frac{1}{z}\\, .\n\\end{equation}\nThe more stronger deviation between the exact solution and the rate theory in the case of grain boundary trapping is obvious (Fig. \\ref{fig:6}).\n\n\\subsection{\\label{sec:composite}\nLarger precipitates: $e^+$-trapping from both sides of precipitate$-$matrix interface }\n\nIn Sect.~(\\ref{sec:extended}) we extended the model for applying it to larger precipitates taking into account free $e^+$ annihilation\nwithin the precipitate. The $e^+$ trapping from the precipitate into the precipitate$-$matrix interface is handled \neither by rate theory, for special cases where the precipitate radius is well below the $e^+$ diffusion length, or else \nby diffusion-reaction theory, for the more general case that the precipitate radius is in the range of or larger than \nthe $e^+$ diffusion length.\nWith this extension the present model is applicable to a wide variety of structurally complex scenarios, namely to all type\nof composite structures where spherical precipitates are embedded in a matrix irrespective of the size and the number density of \nthe precipitates.\n\nWhereas for extended defects with smaller size, which were discussed in Sect.~(\\ref{sec:voids}),\nthe deviations between the exact model and the rate theory may be of less relevance since the \n trapping active area $4 \\pi r_0^2$ is small,  \nfor larger precipitates the diffusion-limitation in any case gets relevant owing\nto the much larger trapping active area, similar as for $e^+$ trapping at GBs \n(see Fig.~\\ref{fig:6}).\nThis is demonstrated in Fig.~\\ref{fig:7}, where the variation of the mean $e^+$ lifetime with radius $R$\nis compared for four different solutions, namely diffusion-limitation of trapping into the \nprecipitate$-$matrix interface from both the matrix and the precipitate, from the matrix only, and for\nentirely reaction-limited trapping from both sides with standard-trapping rate or with effective diffusion-limited trapping rate.\nThe latter is obtained by replacing in equation~(\\ref{eq:tauq_extended_rate}) the standard-trapping rates by \nthe  effective diffusion-limited trapping rate according to equation~(\\ref{eq:K_eff}), i.e., \n$3 \\alpha r_0^2 (R^3-r_0^3)^{-1}$ by $3 \\alpha D r_0^2 R^{-3} (\\alpha r_0 +D)^{-1}$ and \n$3 \\beta r_0^{-1}$ by $3 \\beta D r_0^{-1} (\\beta r_0 +D)^{-1}$.\n\nIn contrast to the case of small extended defects (Fig.~\\ref{fig:6}), for larger precipitates \n(example $r_0=100$~nm) substantial deviations between the solutions occur for the entire concentration regime  \nif the diffusion-limitation is neglected (Fig.~\\ref{fig:7}). \nEven the rate approach with effective diffusion-limited trapping rate,\nwhich at least for small extended defects is a reasonable approximation (Sect.~\\ref{sec:effective_rate}, Fig.~\\ref{fig:4}),\nturns out to be completely inadequate for the larger precipitate size.\nThe deviations are much less if the diffusion-limitation is only neglected\nfor the trapping from the precipitate into the interface, since the precipitate size (in contrast to the precipitate distance) remains in the range of the $e^+$ diffusion length independent of the \nprecipitate concentration. Anyhow, for a precise description even for such small precipitate sizes, \nthe exact theory of diffusion- and reaction controlled trapping has to be applied for\nthe trapping from the interior of the precipitates.\n\nFinally, we compare this model with that presented by Dryzek \\cite{dryzek1999, dryzek2016}\nfor studying recrystallization in highly deformed metals. In that case recrystallized grains are embedded in a highly deformed matrix. Diffusion-limited $e^+$ trapping occurs from the grains into the matrix,\nwhereas within the matrix saturation trapping of $e^+$ prevails due to the high defect density.\nIn this sense, the model of Dryzek represents an extension of the diffusion-reaction theory for trapping at grain boundaries, where instead of GBs a surrounding deformed matrix is considered. The model presented here, represents a further extension where  diffusion- and reaction-controlled trapping also from the matrix into the interfaces is considered.  \n\n \n\n\\section{\\label{conclusion} Conclusion}\n\nThe present model with the exact solution\nof  the diffusion-reaction theory for the $e^+$ trapping at \n extended spherical defects  \nand competitive transition-limited trapping at  atomic defects \nyields a basis for the quantitative description of the\n $e^+$  behaviour in materials with complex defect structure.\nIt could be shown that the model includes  as special cases\nthe simple trapping model and the entirely diffusion-limited trapping,\nbut both of these limiting cases represent approximations, only.\nFor the full model, closed-form expressions were obtained for\nthe mean positron lifetime  $\\overline{\\tau}$ and for the intensities\nof the e$^+$ lifetime components associated with\ntrapping. This exact model allowed a quantitative assessment of  the usual approach, \nwhich takes  diffusion limitation for the trapping at voids into account \nby effective diffusion-trapping rates. The present closed-form solutions also  \nrenders this effective rate approach unnecessary.\n\nThe presented theory  goes even much  far beyond existing models,  \nsince it is not only applicable to small extended defects (such as voids or clusters), but also \nto larger precipitates where positron trapping from the precipitates into the\nprecipitate$-$matrix interface is taken into consideration.\nTherefore, the model presents the basis for studying all type\nof composite structures where spherical precipitates are embedded in a matrix irrespective of their size and  their number density.\n\n\n\\begin{acknowledgments}\nThe senior author (R.W.) dedicates this work to Alfred Seeger \nwhose numerous pioneering works also included modeling of positron annihilation.\nThis work was performed in the framework of the inter-university cooperation of TU Graz and Uni Graz on natural science (NAWI\nGraz).\n\\end{acknowledgments}\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nConformal field theories (CFTs) in two dimensions are of interest for various areas of physics, from condensed matter physics to string theory. In string theory they naturally arise on the worldsheet of the string. In the context of holographic duality, certain two-dimensional CFTs are also known to be \\emph{dual to} string theories on three-dimensional anti de Sitter spacetimes~\\cite{Maldacena:1997re,Giveon:1998ns,Kutasov:1999xu}. An important instance of the $AdS_3\/CFT_2$ duality is obtained by studying string theory on $AdS_3\\times S^3\\times T^4$. In the case of pure NSNS flux the string is described by a Wess-Zumino-Witten (WZW) model on the group $F=SL(2,\\mathbb R)\\times SU(2)$, see~\\cite{Maldacena:2000hw,Maldacena:2000kv,Maldacena:2001km} and references there. We will be interested mainly in this setup. The CFT description of the worldsheet theory allows to make precise statements about the AdS\/CFT duality in this case. A recent example is the duality between the symmetric product orbifold CFT and the string WZW model at level $k=1$~\\cite{Eberhardt:2018ouy}.\n\nGenerally speaking it is interesting to understand the conformal manifold of a CFT, i.e. the space of marginal deformations generated by adding a local perturbation to the Lagrangian. When applied to the WZW model under study, the marginal deformations correspond to deformations of the supergravity background. They give us (at least in principle) a way to go beyond the usual $AdS_3\/CFT_2$ duality and extend it to cases in which e.g. the supersymmetry or the conformal symmetry of the dual $CFT_2$ are broken. Local marginal deformations of WZW models were studied by Chaudhuri and Schwartz (CS)~\\cite{Chaudhuri:1988qb}. They found that {a necessary condition for} local operators constructed out of the chiral and antichiral currents as \n\\begin{equation}\nO(\\sigma,\\bar \\sigma)= c^{a b}J_a(\\sigma)\\bar J_{ b}(\\bar \\sigma)\\,,\n\\end{equation}\nwith $c^{ab}$ some constant coefficients, {to give a marginal deformation is that $c^{ab}$ satisfy}\n\\begin{equation}\\label{eq:CS-weak-intro}\nC\\cdot C+ \\bar C\\cdot \\bar C=0\\,,\n\\end{equation}\nwhere we have defined $C^{abc}\\equiv c^{da}c^{eb}f_{de}^{\\ c}$, and $\\bar C^{abc}\\equiv c^{ad}c^{be} f_{de}^{\\ c}$, and the product is obtained using the Killing metric $K_{ab}$, e.g. $C\\cdot C\\equiv C^{abc}C^{def}  K_{ad} K_{be}K_{cf}$.\nEquation~\\eqref{eq:CS-weak-intro} is quartic in $c^{ab}$, and it involves also the structure constants of the algebra of $F$, the Lie group of the WZW model. We will call it the \\emph{weak} CS condition. CS were interested in the case of CFTs where the group $F$ is compact. In that case~\\eqref{eq:CS-weak-intro} reduces to \n\\begin{equation}\\label{eq:CS-strong-intro}\nC^{abc}=0\\,,\\qquad\\text{and}\\qquad\n\\bar C^{abc}=0\\,,\n\\end{equation}\nwhich is an equation quadratic in $c^{ab}$ that we will call the \\emph{strong} CS condition. It imposes, in fact, a stronger constraint, since it is only in the case of compact groups that it is equivalent to~\\eqref{eq:CS-weak-intro}. CS also showed that solutions of the strong condition correspond to abelian subalgebras of $Lie(F)$, since when~\\eqref{eq:CS-strong-intro} holds it is always possible to identify linear combinations of $J_a,\\bar J_a$ such that their OPEs do not have the term involving the structure constants --- the so-called ``no simple-pole condition'', cf.~\\eqref{eq:JJ-OPE}. {In that case the correlation functions of $O$ are the same as for an $O$ constructed out of free bosons, which in turn implies that the deformation can be completed to all orders in conformal perturbation theory in the deformation parameter. For deformations satisfying only the weak CS condition on the other hand there is no guarantee that they remain marginal beyond lowest order in the deformation parameter.}\nIn the literature on marginal deformations of WZW models, see e.g.~\\cite{Chaudhuri:1992yca,Hassan:1992gi,Kiritsis:1993ju,Tseytlin:1993hm,Forste:2003km,Israel:2004vv,Israel:2004cd,Orlando:2006cc,Detournay:2005fz,Fredenhagen:2007rx}, we did not find examples that satisfy the weak CS condition but not the strong one. Here we will construct such examples by involving sufficient components of the chiral and antichiral currents of $SL(2,\\mathbb R)$. In this sense our results identify new directions to explore the conformal manifold.\n\n\nIn~\\cite{Hassan:1992gi,Henningson:1992rn,Kiritsis:1993ju} it was argued that $O(d,d)$ transformations provides the correct language to obtain the exact (in the deformation parameter) version of the CFTs deformed by the abelian current-current operators. Indeed, such  transformations do not break the isometries involved in the deformation and one can show that the derivative of the action with respect to the deformation parameter is given by\n\\begin{equation}\n\\frac{dS}{d\\eta}=-\\frac{T}{2}\\int d^2\\sigma\\,J^\\eta\\bar J^\\eta\\,,\n\\end{equation}\nwhere $J^\\eta$ and $\\bar J^\\eta$ are (anti)chiral currents of the \\emph{deformed} theory corresponding to the isometries involved in the deformation. This makes it clear that the infinitesimal deformation can be integrated to a finite one.\nThe relevant so-called $\\beta$-shifts of $O(d,d)$, corresponding to a simple shift of the $B$-field of the dual model obtained by performing T-duality on two $U(1)$ isometries, are also known as TsT (T-duality, shift, T-duality) transformations~\\cite{Lunin:2005jy,Frolov:2005ty,Frolov:2005dj,Alday:2005ww}. A TsT transformation exploits an abelian $U(1)^2$ global symmetry of the sigma model to construct a deformation parameterised by a continuous deformation parameter. Since the deformation is constructed by exploiting T-duality, on-shell the deformed model is equivalent to the undeformed one, and the deformation can be equivalently understood as a twist in the boundary conditions in the compact direction of the worldsheet.  \n\nTsT transformations are known to belong to a larger class of deformations of sigma models, usually called Yang-Baxter (YB) deformations. They first appeared in the context of integrable models, since the deformations do not break the classical integrability of the original model~\\cite{Klimcik:2008eq,Delduc:2013qra,Kawaguchi:2014qwa}. YB deformations are particularly interesting in the context of the $AdS\/CFT$ correspondence, since they can be used to generate string backgrounds deforming the standard ones appearing in the $AdS_{d+1}\/CFT_d$ dualities. Particularly important cases are those for which integrability techniques may be applied. In the case of $AdS_5\/CFT_4$ it was proposed that the deformations of the $AdS_5\\times S^5$ background should correspond to non-commutative deformations of $\\mathcal N=4$ super Yang-Mills~\\cite{Matsumoto:2014gwa,vanTongeren:2015uha,vanTongeren:2016eeb,Araujo:2017jkb}.\nThe name YB comes from the fact that the deformation is controlled by an object $R$ which is an element of $\\mathfrak g\\wedge \\mathfrak g$ (where $\\mathfrak g$ is the algebra of isometries of the starting background) and solves the classical Yang-Baxter equation\\footnote{There is also a version of these models where $R$ solves instead the \\emph{modified} CYBE~\\cite{Klimcik:2008eq,Delduc:2013qra}. The story in that case is quite different and we will not consider it here.} (CYBE) on $\\mathfrak g$. The simplest solutions to the CYBE are the so-called abelian $R$-matrices, e.g. $R=T_1\\wedge T_2$ with $T_i\\in \\mathfrak g$ and $[T_1,T_2]=0$. In this case the CYBE is trivially satisfied because the relevant structure constants vanish. Abelian YB deformations were shown to be equivalent to TsT transformations in~\\cite{Osten:2016dvf}. On compact algebras, the CYBE only admits abelian solutions. On non-compact algebras, instead, more interesting non-abelian solutions (i.e. $R$-matrices constructed out of generators of a non-abelian subalgebra) are possible. \n\nOriginally, in the construction of the YB-deformed sigma models, the CYBE was necessary in order to preserve the classical integrability. Later it was understood that YB deformations may be obtained from non-abelian T-duality (NATD)~\\cite{Hoare:2016wsk,Borsato:2016pas}. That interpretation revealed a consistent generalisation of what is known about TsT transformations, since it became clear that YB deformations correspond to a shift of the $B$-field of the dual (undeformed) sigma model; the deformed model is then obtained by applying NATD on the subalgebra of $\\mathfrak g$ where $R$ is non-degenerate. After restricting the domain, $R$ may be inverted and its inverse $R^{-1}$ is a Lie algebra 2-cocycle. The shift of the dual $B$-field is given by this 2-cocycle. In other words, it is possible to go beyond the construction related to integrable models, and understand the CYBE as being a constraint necessary to shift the dual $B$-field without modifying its field strength $H=dB$. Consistently with this interpretation, in~\\cite{Lust:2018jsx,Sakamoto:2018krs} it was proposed  to identify YB deformations with the $\\beta$-shifts of a larger group extending the known $O(d,d)$ group of abelian T-duality, that in~\\cite{Lust:2018jsx} was dubbed ``non-abelian T-duality group''.\nThe  logic of NATD\/$\\beta$-shifts may be used to construct YB deformations of generic sigma models~\\cite{Lust:2018jsx,Sakamoto:2018krs,Borsato:2018idb}, beyond those for which YB deformations were first introduced, the Principal Chiral Model and (super)cosets.\\footnote{Ref.~\\cite{Bakhmatov:2017joy} put forward a first proposal for a set of transformation rules to go beyond these cases.}   Here we will use the transformation rules of~\\cite{Borsato:2018idb}, which  were derived from the NATD construction and have the advantage of being applicable to a generic background with isometries  (even when the initial $G-B$ is not invertible, as is the case for the background metric and Kalb-Ramond field that we have to consider in this paper).\n\nBecause of their realization via NATD, YB models will be Weyl invariant at least to one loop in $\\sigma$--model perturbation theory (and exactly in the deformation parameter), provided that the Lie algebra on which the $R$-matrix is non-trivial is unimodular (i.e. the trace of its structure constants vanish). In this case the deformed background solves the standard supergravity equations of motion. When the algebra is not unimodular there is a potential Weyl anomaly \\cite{Alvarez:1994np,Elitzur:1994ri}. In that case the resulting background solves instead a generalization of the standard supergravity equations \\cite{Arutyunov:2015mqj,Wulff:2016tju} controlled by a Killing vector field $K$. Even in these non-unimodular cases, it can happen that there is actually no Weyl anomaly. This is reflected in the fact that the generalized supergravity equations can have ``trivial'' solutions, i.e. solutions with $K\\neq0$ but where nevertheless the other fields solve the standard supergravity equations \\cite{Wulff:2018aku}. In \\cite{Wulff:2018aku} it was shown that this can happen if $K$ is null. In appendix \\ref{app:trivial} we show that this condition can be weakened and $K$ does not have to be null if the one-form {$\\rm X$ appearing in the generalized supergravity equations takes the form $\\rm X=d\\phi+\\tilde K$ with $\\phi$ the dilaton and $\\tilde K$ another Killing vector}.\nWe will see that YB deformations of the $AdS_3\\times S^3$ WZW model give rise also to ``trivial'' solutions, both ones with $K^2=0$ and with $K^2\\neq0$. We will also find some examples with a genuine anomaly, corresponding to $K$ not being null (and $\\tilde K$ not defining a Killing vector).\n\nAs we will discuss in more detail in section~\\ref{sec:YB}, at leading order in the deformation parameter YB deformations correspond to current-current deformations. We are therefore led to study YB deformations of strings on backgrounds containing an $AdS_3$ subspace, expecting to find marginal deformations of the corresponding WZW model. Particularly  interesting for us are the deformations that do not solve the strong version of the CS condition, but only the weak one. We will construct explicitly such examples. Such possibilities are allowed because we exploit also the non-compact part of the current algebra to generate the deformations. \n\nThe paper is organised as follows. In section~\\ref{sec:WZW} we review some aspects of the $SL(2,\\mathbb R)\\times SU(2)$ WZW model and of marginal current-current deformations that are important for our discussion. In section~\\ref{sec:YB} we review the transformation rules of YB deformations and explain in which cases we can understand them as compositions of simpler YB transformations. We will also explain the connection to the marginal current-current deformations. Using the classification of $R$-matrices in appendix~\\ref{app:R-matrices}, we later study deformations of $AdS_3$ and $AdS_3\\times S^3$. We give our conclusions in section~\\ref{sec:disc}.\nAppendix~\\ref{app:field-red} collects some details on the field redefinition used in section~\\ref{sec:YB}, and  appendix~\\ref{app:onshell} discusses the on-shell equivalence of the YB models to the undeformed ones. In appendix~\\ref{app:sl3} we consider the case of the $\\mathfrak{sl}_3$ algebra, which is separate from the rest of the paper. In appendix~\\ref{app:trivial} we extend the triviality condition of~\\cite{Wulff:2018aku}.\n\n\n\n\\section{Wess-Zumino-Witten model and marginal current-current deformations}\\label{sec:WZW}\nIn this section we review certain aspects of WZW models and their marginal current-current deformations. Although the discussion can be made general, for concreteness we will take the example of the $SL(2,\\mathbb R)\\times SU(2)$ WZW model, since it is important for string theory applications and it already contains all the salient features.\n\n\\subsection{The $AdS_3\\times S^3$ sigma model}\\label{sec:sigmamodel}\nWe start with a sigma model describing the propagation of a string in $AdS_3\\times S^3$, that can be viewed (after adding four free bosons) as the bosonic sector of the superstring . The sigma model action is\\footnote{\nWe work with a Lorentzian worldsheet and we introduce worldsheet coordinates $\\sigma^\\pm=\\sigma^0\\pm\\sigma^1$, so that $\\eta^{+-}=\\eta^{-+}=-2$, $\\epsilon^{+-}=-\\epsilon^{-+}=-2$ and $d^2 \\sigma=\\frac12d\\sigma^+d\\sigma^-$. We also use the standard notation $\\sigma,\\bar \\sigma$ in place of $\\sigma^+,\\sigma^-$, as well as $\\partial =\\partial_\\sigma,\\bar \\partial=\\partial_{\\bar \\sigma}$.}\n\\begin{equation}\nS=\\frac{k}{2\\pi}\\int d^2\\sigma\\ \\left(\\frac{-\\partial x^-\\bar\\partial x^++\\partial z\\bar\\partial z}{z^2}+\\frac{1}{4} \\partial\\phi_i\\bar\\partial\\phi_i+\\frac{1}{2} \\partial\\phi_2\\bar \\partial\\phi_1\\sin \\phi_3\\right)\\,.\n\\end{equation}\nHere we are considering the pure NSNS background, and $k$ will be the level of the WZW model. The string tension is $T=k\/\\pi$, and the metric and $B$-field appearing in the sigma model action follow from $S=T\\int  d^2 \\sigma\\ L=\\frac{T}{2}\\int d^2 \\sigma\\ \\partial x^m (G_{mn}-B_{mn})\\bar \\partial x^n$ and are\\footnote{In our conventions $B=\\frac{1}{2} B_{nm}dx^m\\wedge dx^n$.}\n\\begin{equation}\n\\begin{aligned}\nds^2&=ds_{\\text{AdS}_3}^2+ds_{\\text{S}^3}^2=\n\\frac{-dx^+dx^-+dz^2}{z^2}\n+\\frac{1}{4}\\left[d\\phi_3^2+\\cos^2\\phi_3d\\phi_1^2+(d\\phi_2+\\sin \\phi_3 d\\phi_1)^2\\right]\\,,\\\\\nB&=\\frac{dx^+\\wedge dx^-}{2z^2}-\\frac{1}{4}\\sin \\phi_3 d\\phi_1\\wedge d\\phi_2\\,.\n\\end{aligned}\n\\end{equation}\n$AdS_3$ is parameterised by the boundary coordinates $x^\\pm$ and the radial coordinate $z$, while the angles $\\phi_i$ parameterise the sphere. The $AdS_3$ metric admits the following Killing vectors \n\\begin{equation}\n\\begin{aligned}\n&k_{0}^m\\partial_m=x^+\\partial_{x^+}+\\tfrac{1}{2}z\\partial_{z}\\,,\\qquad\n&&k_{+}^m\\partial_m=\\partial_{x^+}\\,,\\qquad\n&&&k_{-}^m\\partial_m=-(x^+)^2\\partial_{x^+}-z^2\\partial_{x^-}-x^+z\\partial_{z}\\,,\\\\\n&\\bar k_{0}^m\\partial_m=-x^-\\partial_{x^-}-\\tfrac{1}{2}z\\partial_{z}\\,,\\qquad\n&&\\bar k_{-}^m\\partial_m=\\partial_{x^-}\\,,\\qquad\n&&&\\bar k_{+}^m\\partial_m=-(x^-)^2\\partial_{x^-}-z^2\\partial_{x^+}-x^-z\\partial_{z}\\,.\n\\end{aligned}\n\\end{equation}\nThey satisfy $[k_a^m\\partial_m,k_b^n\\partial_n]=-f_{ab}^{\\ c}k_c^p\\partial_p$ (and similarly for $\\bar k_a$), where $f_{ab}^{\\ c}$ are the structure constants of the algebra of $SL(2,\\mathbb R)$\n\\begin{equation}\n[S_0,S_\\pm]=\\pm S_\\pm\\,,\\qquad[S_+,S_-]=2S_0\\,.\n\\label{eq:sl2}\n\\end{equation}\nIn these formulas and in the following we use a bar to distinguish the right copy of the algebra from the left copy.\\footnote{We will interchangeably place the bar on an object or on its index, in other words $\\bar k_a$ or $k_{\\bar a}$ have the same meaning. For readability sometimes we will prefer the former.}\nFor the sphere we have two copies of $SU(2)$, whose algebra is generated by $T_a$ ($a=1,2,3$) with commutation relations $[T_a,T_b]=-\\epsilon_{abc}T_c$.\nWe will not write explicitly all Killing vectors of $S^3$ since we will not need them. For our purposes it will be enough to use the two commuting Killing vectors\n\\begin{equation}\nk_1=-\\partial_{\\phi_1}\\qquad\\mbox{and}\\qquad\\bar k_2=\\partial_{\\phi_2}\\,.\n\\end{equation}\n\nThe sigma-model action is invariant under the transformations generated by the above Killing vectors, although in certain cases the $B$-field is not invariant but changes by a total derivative. Therefore in general the corresponding Noether currents are given by\n\\begin{equation}\n\\mathcal J_{A,\\pm}=k_A^{m}(G_{mn}\\pm B_{mn})\\partial_{\\pm}x^n+j_{A,\\pm}\\,,\n\\label{eq:Noether}\n\\end{equation}\nwhere $j_{A,\\pm}$ is defined by looking at the variation of the Lagrangian $\\delta_A L=\\varepsilon\\partial_ij_A^i$ under the infinitesimal global transformation.\nBecause of our choice of gauge, in the $AdS_3$ part only $j_-^i$ and $\\bar j_+^i$ are non-zero, and we also have $j_1^i=\\bar j_2^i=0$. In the following we will ignore the transformations generated by $S_-,\\bar S_+$, since for our discussion it will be enough to focus on the (maximal solvable) subalgebra generated by\n\\begin{equation}\\label{eq:subalgebra}\nS_0,\\quad S_+,\\quad \\bar S_0,\\quad \\bar S_-,\\quad T_1,\\quad \\bar T_2\\,.\n\\end{equation} \nAll Noether currents that we will need to consider will therefore have $j_{A,\\pm}=0$.\nLet us anticipate that these Noether currents are not always equal to the chiral (resp. antichiral) currents of the WZW description, which we shall denote by $J$ (resp. $\\bar J$) and write explicitly in the next subsection. They agree up to ``improvement terms'' that do not spoil the current conservation, of the type $\\epsilon^{ij}\\partial_jc$ for some $c$. Restricting to the generators in~\\eqref{eq:subalgebra}, for $AdS_3$ we have\n\\begin{equation}\\label{eq:noether-chiralAdS3}\n\\begin{aligned}\n&\\mathcal J_{0,+}=J_0-\\tfrac{1}{2}\\partial \\log z\\,,\\qquad\n&\\mathcal J_{0,-}=+\\tfrac{1}{2}\\bar\\partial \\log z\\,,\\qquad\n&\\mathcal J_{+,+}=J_+\\,,\\qquad\n&\\mathcal J_{+,-}=0\\,,\\\\\n&\\bar{\\mathcal J}_{0,-}=\\bar J_0+\\tfrac{1}{2}\\bar\\partial \\log z\\,,\\qquad\n&\\bar{\\mathcal J}_{0,+}=-\\tfrac{1}{2}\\partial \\log z\\,,\\qquad\n&\\bar{\\mathcal J}_{-,-}=\\bar J_-\\,,\\qquad\n&\\bar{\\mathcal J}_{-,+}=0\\,,\n\\end{aligned}\n\\end{equation}\nwhile for $S^3$\n\\begin{equation}\\label{eq:noether-chiralS3}\n\\begin{aligned}\n&\\mathcal J_{1,+}=J_1+\\tfrac{1}{4}\\partial\\phi_1\\,,\\qquad\n&\\mathcal J_{1,-}=-\\tfrac{1}{4}\\bar\\partial\\phi_1\\,,\\\\\n&\\bar{\\mathcal J}_{2,-}=\\bar J_2-\\tfrac{1}{4}\\bar\\partial\\phi_2\\,,\\qquad\n&\\bar{\\mathcal J}_{2,+}=\\tfrac{1}{4}\\partial\\phi_2\\,.\n\\end{aligned}\n\\end{equation}\nThis fact will later play an important role in our discussion.\n\n\n\\subsection{The $SL(2,\\mathbb R)\\times SU(2)$ WZW model}\nThe action of the WZW model is $S_{WZW}=S_1+k\\Gamma$ where $k$ is the level and\n\\begin{equation}\nS_1[g] = \\frac{k}{4 \\pi} \\int_{\\partial\\mathcal B} d^2\\sigma \\mathrm{Tr}(\\partial^ig^{-1}\\partial_i g )\\,,\\qquad\n\\Gamma[g] = -\\frac{1}{6\\pi}\\int_{\\mathcal B}d^3\\sigma \\epsilon^{ijk}\\mathrm{Tr}(g^{-1} \\partial_ig g^{-1} \\partial_jg g^{-1} \\partial_kg)\\,.\n\\end{equation}\nHere $g$ is an element of a group $G$, depending on coordinates on $\\mathcal B$, whose boundary $\\partial\\mathcal B$ is the worldsheet of the string.\nIn the following we will take the action\\footnote{The relative minus sign is needed to get the correct sign in front of the $S^3$ metric. In the supersymmetric case it is naturally accounted for by the supertrace.} $S_{WZW}[g_{\\rm a}]-S_{WZW}[g_{\\rm s}]$ with\n\\begin{equation}\ng_{\\rm a} = e^{x^+S_+}z^{2S_0}e^{-x^-S_-}\\in SL(2,\\mathbb R)\\,,\\qquad\ng_{\\rm s} = e^{\\phi_1T_1}e^{\\phi_3T_3}e^{\\phi_2T_2}\\in SU(2)\\,.\n\\end{equation}\nWe realise the generators of the algebra of $SL(2,\\mathbb R)$ in terms of the Pauli matrices as $S_0=\\sigma_3\/2$, $S_+=(\\sigma_1+i\\sigma_2)\/2$, $S_-=(\\sigma_1-i\\sigma_2)\/2$, and similarly for $SU(2)$ we take $T_a=\\tfrac{i}{2}\\sigma_a$.\nThe Killing form is related to the trace in this representation as $K_{ab}=f_{ac}^{\\ d}f_{bd}^{\\ c}=4\\mathrm{Tr}(S_aS_b)$, and similarly for $T_a$. We will use the bilinear form induced by the trace, rather than the Killing form, to raise and lower algebra indices.\n\nThe equations of motion for the action $S_{WZW}[g]$ imply chirality for the current $J=\\partial g g^{-1}$, and equivalently antichirality for the current $\\bar J=- g^{-1} \\bar \\partial g$, i.e. $\\bar \\partial J=0$, $\\partial \\bar J=0$.\nWe decompose the currents as $J=J_aS^a$ for $AdS_3$ and $J=J_aT^a$ for $S^3$, and similarly for $\\bar J$, where $S^0=2S_0, S^\\pm=S_\\mp$ and $T^a=-2T_a$. Thanks to these definitions the component $J_a$ of the chiral current corresponds to the action of the generator $S_a$ (or $T_a$ in the case of the sphere) from the left, while the component $\\bar J_a$ of the antichiral current corresponds to the action of the same generator from the right. The same holds for the corresponding Killing vectors $k_a$ and $\\bar k_a$. In particular we have $k_a^m\\partial_mg_{\\rm a}=+S_ag_{\\rm a}$, $\\bar k_{a}^m\\partial_mg_{\\rm a}=-g_{\\rm a}S_{a}$ for $AdS$ and $k_1^m\\partial_mg_{\\rm s}=-T_1g_{\\rm s}$, $\\bar k_{2}^m\\partial_mg_{\\rm s}=+g_{\\rm s}T_{2}$ for the sphere.\\footnote{The relative minus sign between $AdS_3$ and $S^3$ is again related to the fact that we are not using the supertrace in the action and in order to define the components of the currents.}  In our  parameterisation the components of the $SL(2)$ currents read \n\\begin{equation}\n\\begin{aligned}\n&J_0={\\phantom{-}}\\frac{z \\partial z-x^+\\partial x^-}{z^2}\\,,\n&& J_-=\\frac{x^+ (x^+\\partial x^--2 z \\partial z)}{z^2}+\\partial x^+\\,,\n&&& J_+=-\\frac{\\partial x^-}{z^2}\\,,\n\\\\\n&\\bar J_0=-\\frac{z \\bar\\partial z-x^-\\bar\\partial x^+}{z^2}\\,,\n&&\\bar J_+=\\frac{x^-(x^-\\bar\\partial x^+-2 z\\bar \\partial z)}{z^2}+\\bar\\partial x^-\\,,\n&&&\\bar J_-=-\\frac{\\bar\\partial x^+}{z^2}\\,,\n\\end{aligned}\n\\end{equation}\nwhile for the $SU(2)$ currents we have\n\\begin{equation}\n\\begin{aligned}\n&J_1=\\tfrac{1}{2} (-\\partial \\phi_1 - s_3\\partial\\phi_2 )\\,,\n&& J_2=\\tfrac{1}{2} (-c_1 c_3 \\partial \\phi_2 - s_1\\partial \\phi_3 )\\,,\n&&& J_3=\\tfrac{1}{2} (-c_1 \\partial\\phi_3 + c_3 s_1\\partial\\phi_2 )\\,,\n\\\\\n& \\bar J_2= \\tfrac{1}{2} (+\\bar\\partial \\phi_2 + s_3\\bar\\partial \\phi_1 )\\,, \n&&\\bar J_1=\\tfrac{1}{2} (+c_2 c_3 \\bar\\partial \\phi_1 + s_2\\bar\\partial \\phi_3 )\\,,\n&&&\\bar J_3=\\tfrac{1}{2} (+c_2 \\bar\\partial \\phi_3 - c_3 s_2\\bar\\partial \\phi_1 )\\,,\n\\end{aligned}\n\\end{equation}\nwhere we use the shorthand notation $s_i=\\sin\\phi_i$, $c_i=\\cos\\phi_i$.\nThe (anti)chiral currents appear also when computing the Noether currents from the action $S_{WZW}=S_1+k\\Gamma$. In fact, invariance of the WZW action under left transformations $g\\to (1+\\varepsilon_L+\\ldots)g$ implies the conservation of the Noether current\n$\\mathscr J^i=\\tfrac{1}{4} (\\eta^{ij} -\\epsilon^{ij})\\partial_j gg^{-1}$\nwhich is related to the chiral current as\n\\begin{equation}\\label{eq:noether-WZW}\n\\mathscr J=(\\mathscr J_+,\\mathscr J_{-})=(J,0)\\,. \n\\end{equation}\nSimilarly, from the right transformations $g\\to g(1+\\varepsilon_R+\\ldots)$ one finds the Noether current $\\bar{\\mathscr J}^i=-\\tfrac{1}{4} (\\eta^{ij} +\\epsilon^{ij})g^{-1}\\partial_j g$\nrelated to the antichiral current as\n\\begin{equation}\\label{eq:noetherbar-WZW}\n\\bar{\\mathscr J}=(\\bar{\\mathscr J}_{+},\\bar{\\mathscr J}_{-})=(0,\\bar J)\\,.\n\\end{equation}\nConservation of the Noether current $\\partial_i\\mathscr J^i=0$ (respectively $\\partial_i\\bar{\\mathscr J}^i=0$) implies chirality of $J$ (respectively antichirality of $\\bar J$).\nAs we have already pointed out, in general these Noether currents are not the same as those of the sigma model description, which we denoted by $\\mathcal J$.\n\n\\subsection{Marginal deformations}\\label{sec:marginal}\nIn~\\cite{Chaudhuri:1988qb} Chaudhuri and Schwartz considered two-dimensional CFTs with $J_a,\\bar J_{a}$ satisfying current algebra relations\\footnote{Since we are not normalising the currents with an explicit $k$ and we raise\/lower indices with the bilinear form induced by the trace, as opposed to the Killing form, certain factors differ from \\cite{Chaudhuri:1988qb}.}\n\\begin{equation}\\label{eq:JJ-OPE}\n\\begin{aligned}\nJ_a(\\sigma)J^b(\\sigma')\\sim \\frac{i\\ \\delta_a^b}{2k(\\sigma-\\sigma')^2}+ \\frac{if_{ac}^{\\ b}J^c}{2k(\\sigma-\\sigma')}\\,,\\\\\n\\bar J_{ a}(\\bar \\sigma)\\bar J^{ b}(\\bar \\sigma')\\sim \\frac{i\\ \\delta_{ a}^{ b}}{2k(\\bar \\sigma-\\bar \\sigma')^2}+ \\frac{if_{ a c}^{\\  b}\\bar J^{ c}}{2k(\\bar \\sigma-\\bar \\sigma')}\\,,\n\\end{aligned}\n\\end{equation}\nwhere we use $\\sim$ since we are omitting regular terms and $f_{ab}^{\\ c}$ are structure constants of a Lie algebra $\\mathfrak f$. \nThe authors of~\\cite{Chaudhuri:1988qb} were interested in exploring the space of marginal deformations induced by dimension $(1,1)$ operators of the type\n\\begin{equation}\ng\\mathcal O(\\sigma,\\bar \\sigma)= g c^{a b}J_a(\\sigma)\\bar J_{ b}(\\bar \\sigma)\\,,\n\\end{equation}\nwhere $c^{ab}$ are constant coefficients. The above operator is ``integrably'' or exactly marginal (i.e. can be completed to all orders in conformal perturbation theory in $g$) if it has no anomalous dimension, and they found that {a necessary condition for this to hold is that}\n\\begin{equation}\\label{eq:CS-weak}\nC^{abc}C^{def}  K_{ad} K_{be}K_{cf}\n+ \\bar C^{abc}\\bar C^{def}  K_{ad} K_{be}K_{cf}=0\\,,\n\\end{equation}\nwhere $K_{ab}$ is the Killing form and we have defined \n\\begin{equation}\\label{eq:def-C-Cbar}\nC^{abc}\\equiv c^{da}c^{eb}f_{de}^{\\ c}\\,,\\qquad\n\\bar C^{abc}\\equiv c^{ad}c^{be} f_{de}^{\\ c}\\,.\n\\end{equation}\nWe will call~\\eqref{eq:CS-weak} the \\emph{weak} Chaudhuri-Schwartz (CS) condition. Ref.~\\cite{Chaudhuri:1988qb} considered only the case of compact algebras, meaning that the Killing form $K_{ab}$ is negative definite and can be taken to be diagonal. In this case~\\eqref{eq:CS-weak} becomes a sum of squares of $C^{abc}$ and $\\bar C^{abc}$, and it holds if and only if\n\\begin{equation}\\label{eq:CS-strong}\nC^{abc}=0\\,,\\qquad\\text{and}\\qquad\n\\bar C^{abc}=0\\,.\n\\end{equation}\nWe will call~\\eqref{eq:CS-strong} the \\emph{strong} CS condition, because it is a stronger constraint in the case of non-compact algebras.\nIn~\\cite{Chaudhuri:1988qb} it was also shown that the strong condition is equivalent to being able to rewrite \n\\begin{equation}\n\\mathcal O(\\sigma,\\bar \\sigma)= \\tilde c^{a b}\\tilde J_a(\\sigma)\\tilde{\\bar{J}}_{ b}(\\bar \\sigma)\\,,\n\\end{equation}\nwhere $\\tilde J_a(\\sigma),\\tilde{\\bar{J}}_{ b}(\\bar \\sigma)$ are linear combinations of the original $J_a(\\sigma)$, ${\\bar{J}}_{ b}(\\bar \\sigma)$ such that \n\\begin{equation}\n\\begin{aligned}\n\\tilde J_a(\\sigma)\\tilde J^b(\\sigma')\\sim \\frac{i\\ \\delta_a^b}{2k(\\sigma-\\sigma')^2}\\,,\\qquad\n\\tilde{\\bar J}_{ a}(\\bar \\sigma)\\tilde{\\bar J}^{ b}(\\bar \\sigma')\\sim \\frac{i\\ \\delta_{ a}^{ b}}{2k(\\bar \\sigma-\\bar \\sigma')^2}\\,,\n\\end{aligned}\n\\end{equation}\ni.e. the structure constants for this particular set of currents vanish. {The absence of a simple pole in these OPEs means that they are the same as similar ones for free bosons, which in turn means that the $\\beta$-function for the deformation parameter $g$ vanishes and the deformation is exactly marginal. In other words, deformations corresponding to abelian subalgebras, which is the only possibility in the compact case, are exactly marginal.} When the Lie algebra $\\mathfrak f$ is non-compact {it is possible to find deformations that satisfy only the weak CS condition}, as we will see. {A priori they are not guaranteed to be marginal beyond lowest order, and indeed we will find both examples which are and those which are not.}\n\nIn fact a \\emph{sufficient} condition on $c^{ab}$ such that the weak CS~\\eqref{eq:CS-weak} holds is that the coefficients $c^{ab}$ identify two solvable subalgebras of $\\mathfrak f$ (one corresponding to $J_a$ and one to $\\bar J_b$). This follows directly from Cartan's criterion for a solvable Lie algebra $\\mathfrak h$\n\\begin{equation}\n\\mathfrak h \\text{ solvable } \\iff \\mathrm{Tr}(ab)=0, \\qquad \\forall a\\in\\mathfrak h, b\\in [\\mathfrak h,\\mathfrak h]\\,.\n\\end{equation}\nIf we are in such a situation then the two terms in~\\eqref{eq:CS-weak} separately vanish because\n\\begin{equation}\nC^{abc}C^{def}  K_{cf}=0\\,,\\qquad\\qquad\n \\bar C^{abc}\\bar C^{def}  K_{cf}=0\\,.\n\\end{equation}\nIn  the case of the $SL(2,\\mathbb R)\\times SU(2)$ WZW model, we may for example identify the two solvable subalgebras generated by $\\{S_0,S_+,T_1\\}$ and $\\{\\bar S_0,\\bar S_-,\\bar T_2\\}$. Then if we call $Y_a\\equiv\\{J_0,J_+,J_1\\}$ and $\\bar Y_a=\\{\\bar J_0,\\bar J_-,\\bar J_2\\}$ the list of the corresponding (anti)chiral currents, an operator\n\\begin{equation}\n\\mathcal O(\\sigma,\\bar \\sigma)=  c^{a b}Y_a(\\sigma)\\bar Y_{ b}(\\bar \\sigma)\\,,\n\\end{equation}\nwill be {marginal to lowest order} for \\emph{generic} coefficients $c^{ab}$. Notice that generically $c^{ab}$ will not solve the strong CS condition \\eqref{eq:CS-strong}.\n\nAll the solutions to the weak CS condition that we will generate from the CYBE on $\\mathfrak g=\\mathfrak f_{\\mbox{\\tiny L}}\\oplus \\mathfrak f_{\\mbox{\\tiny R}}=\\mathfrak{sl}(2,\\mathbb R)_{\\mbox{\\tiny L}}\\oplus \\mathfrak{su}(2)_{\\mbox{\\tiny L}}\\oplus \\mathfrak{sl}(2,\\mathbb R)_{\\mbox{\\tiny R}}\\oplus \\mathfrak{su}(2)_{\\mbox{\\tiny R}}$ will be of this type. Indeed to solve the CYBE it is enough to look at the subalgebra  generated by  $\\{S_0,S_+,T_1,\\bar S_0,\\bar S_-,\\bar T_2\\}$. When they come from the YB construction, the coefficients $c^{ab}$ will obviously not be generic, and we will relate them to certain components of the $R$-matrix, see the discussion at the end of section~\\ref{sec:YB-CS}. The YB construction has the advantage of giving a way to go beyond the infinitesimal deformation driven by $\\mathcal O(\\sigma,\\bar \\sigma)$, and gives a sigma-model action that is exact in the deformation parameter.\n\nAs we have argued, we expect the CYBE to give solutions to the weak CS condition in more generic situations. In appendix~\\ref{app:sl3} we discuss a solution of the CYBE that provides coefficients $c^{ab}$ that solve the weak CS condition without identifying solvable subalgebras.\n\n\\section{Yang-Baxter and current-current deformations}\\label{sec:YB}\n\n\\subsection{Yang-Baxter deformations}\nWe now review the transformation rules for the target space fields for YB deformations derived in~\\cite{Borsato:2018idb}.\nGiven an initial sigma model with metric and Kalb-Ramond fields $G_{mn}$, $B_{mn}$, the background of the YB deformed model is given by\n\\begin{equation}\n\\tilde G-\\tilde B = (G-B)[1+\\eta\\Theta(G-B)]^{-1}\\,,\n\\label{eq:GBtilde}\n\\end{equation}\nwhere for simplicity we are suppressing all spacetime indices. Here $\\Theta^{mn}=k^m_AR^{AB}k^n_B$ is a tensor constructed out of the Killing vectors $k^m_A$ and of $R^{AB}$, which is a solution to the CYBE on the Lie algebra $\\mathfrak g$\n\\begin{equation}\\label{eq:CYBE}\nR^{D[A}R^{|E|B}f_{DE}^{C]}=0\\,.\n\\end{equation}\nIn our case $\\mathfrak g= \\mathfrak f_{\\mbox{\\tiny L}}\\oplus \\mathfrak f_{\\mbox{\\tiny R}}$ is the sum of a left and a right copy of $\\mathfrak f=\\mathfrak{sl}(2)\\oplus \\mathfrak{su}(2)$, and $G,B$ were given in section~\\ref{sec:sigmamodel}. \nThe derivation of~\\cite{Borsato:2018idb} assumes that the $B$-field is invariant under the isometries used in the deformation, i.e. the ones appearing in $\\Theta$. This is ensured by picking the form of $B$ in section~\\ref{sec:sigmamodel} and using only the isometries generated by (\\ref{eq:subalgebra}), which is enough to generate any Yang-Baxter deformation (see appendix \\ref{app:R-matrices}).\n\nThe deformation produces also a shift of the dilaton calculated from the determinant\\footnote{In the supersymmetric case the determinant is replaced by the superdeterminant.}\n\\begin{equation}\ne^{-2\\tilde\\Phi}=e^{-2\\Phi}\\det[1+\\eta\\Theta(G-B)]\\,,\n\\label{eq:Phitilde}\n\\end{equation}\nwhere $\\Phi$ is the dilaton of the original background (in our case $\\Phi=0$).\nIn general YB backgrounds are solutions to the  equations of generalised supergravity~\\cite{Arutyunov:2015mqj,Wulff:2016tju}, so that in addition to the usual fields one may have also a vector $K$ computed as\\footnote{Eq.~\\eqref{eq:K} may be obtained from the formula derived in~\\cite{Borsato:2018idb} (i.e. $K^m=\\eta \\Theta^{mn}n_n=\\eta k^m_AR^{AB}n_B$ with $n_A=f_{AB}^{\\ B}$) after using the identity $R^{AB}f_{AB}^{\\ C}=-2f_{AB}^{\\ A}R^{BC}$, which is a consequence of the CYBE. It is also easy to check that~\\eqref{eq:K} agrees with $K^m=\\eta \\nabla_n^{(0)}\\Theta^{mn}$ proposed in~\\cite{Araujo:2017jkb}, where $\\nabla_n^{(0)}$ is the covariant derivative of the original undeformed background. Indeed, using first the Killing equation for $k_A^m$ and then the anti-symmetry of $R$, we have $\\nabla_m^{(0)}\\Theta^{mn}=k_A^mR^{AB}\\nabla_m^{(0)}k_B^n=R^{AB}k_A^m\\partial_mk_B^n$. Knowing that Killing vectors satisfy $[k_A^m\\partial_m,k_B^n\\partial_n]=-f_{AB}^{\\ C}k_C^p\\partial_p$ we obtain again~\\eqref{eq:K}.}\n\\begin{equation}\\label{eq:K}\nK^m = -\\frac{\\eta}{2}R^{AB}f_{AB}^{\\ C}k_C^m\\,,\n\\end{equation}\nwhich is a Killing vector of the YB background $\\nabla_{(m}K_{n)}=0$.\nFor such generalised supergravity solutions the role of (the derivative of) the dilaton is replaced by the vector\\footnote{This expression applies in a gauge where $B$ is invariant under the isometry generated by $K$, $\\mathcal L_KB=0$. Here we stick with the original notation of~\\cite{Wulff:2016tju}. In~\\cite{Borsato:2018idb} and~\\cite{Wulff:2018aku} $X_m$ was used instead, but there is a risk of confusing it with $X_m={\\rm X}_m+K_m$ of~\\cite{Wulff:2016tju}.}\n\\begin{equation}\\label{eq:X}\n{\\rm X}_m=\\partial_m\\tilde\\Phi -B_{mn}K^n\\,.\n\\end{equation}\nWhen $K^m$ vanishes one goes back to a standard supergravity solution. From~\\eqref{eq:K} the relation to the unimodularity condition of~\\cite{Borsato:2016ose} is manifest. There exist also so-called ``trivial solutions'' of generalised supergravity~\\cite{Wulff:2018aku}, i.e. when $K$ does not vanish but it decouples from the equations. A trivial solution is therefore both a solution of the generalised and the standard supergravity equations. Later we will encounter examples of this type.\n\n\n\\vspace{12pt}\n\n{Let us comment on the fact that the YB transformations constructed in~\\cite{Borsato:2018idb} were derived by assuming a group of left isometries for the sigma model. This is necessary in order to apply the NATD construction and twist the model with the corresponding Killing vectors $k_A$. The isometries that we will exploit here to deform $AdS_3\\times S^3$, corresponding to the generators in~\\eqref{eq:subalgebra}, belong both to the left and to the right copy of the symmetry group of the WZW model. The reason why we can apply the above rules of YB transformations is that the corresponding sigma model may be constructed as a coset on $SO(2,2)\/SO(1,2)\\times SO(4)\/SO(3)$ with a WZ term. For example, focusing on $AdS_3$, we may relate the generators of $\\mathfrak{sl}(2,\\mathbb R)$ to those of the conformal algebra as\n\\begin{equation}\n\\begin{aligned}\n&S_0=+\\tfrac{1}{2}(D-J_{01}),\\qquad\n&&S_+=p_+,\\qquad\n&&&S_-=k_-,\\\\\n&\\bar S_0=-\\tfrac{1}{2}(D+J_{01}),\\qquad\n&&\\bar S_+=k_+,\\qquad\n&&&\\bar S_-=p_-,\n\\end{aligned}\n\\end{equation}\nwhere e.g. $p_\\pm=\\tfrac12 (p_0\\pm p_1)$.\nThen we obtain the wanted sigma model action from $S=\\tfrac{k}{2\\pi}\\int d^2 \\sigma \\mathrm{Tr}[g^{-1}\\partial g(P+b)g^{-1}\\bar \\partial g]$ where $g=\\exp(x^+p_++x^-p_-)\\exp(D\\log z)$, $P$ projects on the generators of the coset $p_i-k_i$ and $D$, and finally  $b(p_\\pm-k_\\pm)=\\pm(p_\\pm-k_\\pm)$ produces the $B$-field. In this formulation the isometries that we want to exploit, generated by $S_0,S_+,\\bar S_0,\\bar S_-$, act from the left as $g\\to hg$ and leave also the $B$-field invariant.\n}\n \n\\vspace{12pt}\n\nFor later convenience, let us say at this point that it is easy to check that when an $R$-matrix is given by the sum of two $R$-matrices, the corresponding background can be understood as the composition of two successive YB transformations.\nThis is easily seen using the following identity valid when $\\Theta=\\Theta_1+\\Theta_2$\n\\begin{equation}\n[1+\\eta\\Theta_1(G-B)]^{-1}\\left[1+\\eta\\Theta_2(G-B)[1+\\eta\\Theta_1(G-B)]^{-1}\\right]^{-1}=[1+\\eta\\Theta(G-B)]^{-1}\\,,\n\\end{equation}\nwhich holds without assuming any property\\footnote{Obviously we need to assume invertibility of the above operators.} for $\\Theta_i$, neither antisymmetry nor CYBE.\nThanks to this formula it is straightforward to argue that the background metric and $B$-field of a YB deformation generated by $\\Theta=\\Theta_1+\\Theta_2$ are equivalent to those coming from the composition of two successive deformations, e.g. first one generated by $\\Theta_1$ and then one generated by $\\Theta_2$ (or vice versa). The same holds for the transformation rule of the dilaton, and of the vector $K$, which is linear in $\\Theta$ (or equivalently $R$). Obviously, the interpretation as YB deformations in the intermediate steps will be possible only if $\\Theta_1,\\,\\Theta_2$ separately solve the CYBE, and if the isometries needed to implement the second deformation are not broken by the first one. In this case we will say that $\\Theta$ is ``decomposable''. Apart from these subtleties, it will often prove useful to interpret a deformation generated by $\\Theta=\\Theta_1+\\Theta_2$ as a composition of two transformations.\n\nLater we will encounter examples in which $\\Theta_1$ generates the undeformed background \\emph{up to} a ($\\eta$-dependent) field redefinition. In this case one can say that $\\Theta=\\Theta_1+\\Theta_2$ is equivalent to the YB deformation generated by $\\Theta_2$ alone \\emph{only if} the field redefinition  $x^m\\to x^m(x')=x'{}^m+\\eta f^m(x')$ needed to trivialise $\\Theta_1$ is compatible with $\\Theta_2$. It is easy to convince oneself that the necessary compatibility condition is\n\\begin{equation}\\label{eq:compTheta2}\nA^{-1}\\Theta_2(x'{}^m+\\eta f^m(x'))A^{-T}=\\Theta_2(x'{}^m)\\,,\n\\end{equation}\nwhere $A^m_{\\ n}=\\frac{\\partial x^m}{\\partial x^{'n}}$, and we are writing the explicit dependence of $\\Theta_2$ on the coordinates. \n\n\n\\subsection{Relation to marginal current-current deformations}\\label{sec:YB-CS}\nBefore discussing YB deformations of $AdS_3\\times S^3$, let us make a simple observation: at leading order in the deformation parameter, the YB deformation is of the form $\\mathcal J\\mathcal J$, where $\\mathcal J$ are the Noether currents of the sigma model. This is straightforwardly checked by expanding the sigma model action $S=\\tfrac{T}{2}\\int d^2 \\sigma\\ \\partial x^m (\\tilde G_{mn}-\\tilde B_{mn})\\bar \\partial x^n$ to lowest order in the deformation parameter\\footnote{Recall that we are restricting to the case when the isometries used to construct the deformation leave not just the action but also the Lagrangian invariant, so that $j_{Ai}=0$. This was the assumption also in the derivation done in~\\cite{Borsato:2018idb}.}\n\\begin{equation}\\label{eq:expand-YB}\nS=S_0-\\eta\\frac{T}{2}\\int d^2\\sigma\\,R^{AB}\\mathcal J_{A+}\\mathcal J_{B-} +\\mathcal{O}(\\eta^2)\\,.\n\\end{equation}\nWhile this is true for a generic sigma model, this observation is particularly interesting when the original sigma model is related to a WZW model. If the Noether currents $\\mathcal J_{Ai}$ coincided with the chiral Noether currents of the WZW model $\\mathscr J_{Ai}=\\{\\mathscr J_{ai},\\bar{\\mathscr J}_{ai}\\}$, then we would automatically obtain a current-current deformation of the type $J\\bar J$. In fact, from~\\eqref{eq:noether-WZW} and ~\\eqref{eq:noetherbar-WZW} one immediately finds that\\footnote{Notice that the $R$-matrix may have also non-vanishing components with both indices in the left (or both in the right) copy of the algebra, but these will not contribute in the final expression, and the contributions coupling currents with the same chirality cancel out.} $\\int d^2\\sigma \\ \\epsilon^{ij}\\, R^{BA}\\mathscr J_{Ai} \\mathscr J_{Bj}=4\\int d^2\\sigma\\ R^{a\\bar b} J_a \\bar J_{\\bar b}$. \nAs we have seen, though, in general $\\mathcal J_{A}^i= \\mathscr J_{A}^i+\\epsilon^{ij}\\partial_jc_A$, and the  discussion is more subtle because~\\eqref{eq:expand-YB} will contain additional terms together with the wanted $J\\bar J$ ones. We are about to show that for YB deformations of the $AdS_3\\times S^3$ sigma model these additional terms can be removed by proper field redefinitions. We can therefore relate YB deformations to the deformations of the type $c^{a b}J_a\\bar J_{ b}$ considered by CS. From this discussion it is also clear that we should identify the coefficients $c^{a b}$ of CS with an ``off-diagonal'' block of the $R$-matrix. More explicitly, since $R\\in \\mathfrak g\\wedge \\mathfrak g$ we are solving the CYBE on an algebra which is the sum of a left and a right copy $\\mathfrak g=\\mathfrak f_{\\mbox{\\tiny L}}\\oplus \\mathfrak f_{\\mbox{\\tiny R}}$, the $R$-matrix can be decomposed as\n\\begin{equation}\nR=\\left(\n\\begin{array}{cc}\nR_{\\mbox{\\tiny L}\\sL} & R_{\\mbox{\\tiny L}\\mbox{\\tiny R}}\\\\\nR_{\\mbox{\\tiny R}\\mbox{\\tiny L}} & R_{\\mbox{\\tiny R}\\sR}\n\\end{array}\n\\right)\n\\end{equation}\nwith $R_{\\mbox{\\tiny L}\\sL}^{T}=-R_{\\mbox{\\tiny L}\\sL}, R_{\\mbox{\\tiny R}\\sR}^{T}=-R_{\\mbox{\\tiny R}\\sR}$ and $R_{\\mbox{\\tiny L}\\mbox{\\tiny R}}^T=-R_{\\mbox{\\tiny R}\\mbox{\\tiny L}}$. The relation to the coefficients of CS is therefore $c^{a b}=R_{\\mbox{\\tiny L}\\mbox{\\tiny R}}^{ab}$.\nWe can therefore generate solutions to the (weak) CS condition from solutions of the CYBE, and we will find several non-trivial examples in the following.\nObviously abelian $R$-matrices ($R=a\\wedge b, [a,b]=0$) will give solutions of the strong CS condition. When dealing with non-compact algebras the CYBE allows also for solutions that are not of the abelian type. Some of them will give coefficients $c^{ab}$ that do not solve the strong CS condition. They all solve the weak CS condition as already explained at the end of section~\\ref{sec:marginal}. \n\nIt would be very interesting to understand more deeply the relation between the space of solutions of the CYBE~\\eqref{eq:CYBE} and the weak CS condition~\\eqref{eq:CS-weak} in the case of a generic algebra $\\mathfrak g=\\mathfrak f_{\\mbox{\\tiny L}}\\oplus \\mathfrak f_{\\mbox{\\tiny R}}$. {It is interesting to notice that in order to solve the CYBE one may need also components of the diagonal blocks $R_{\\mbox{\\tiny L}\\sL}$ and $R_{\\mbox{\\tiny R}\\sR}$, while in the weak CS condition these will not enter. In fact, given $R_{\\mbox{\\tiny L}\\mbox{\\tiny R}}^{a\\bar b}=c^{a \\bar b}$ and taking the CYBE on mixed left\/right indices (where we use an explicit bar for indices of the right copy of the algebra), one gets for example $c^{d\\bar a}R^{eb}f_{de}^{\\ c}+c^{b\\bar d}c^{c\\bar e}f_{\\bar d\\bar e}^{\\ \\bar a}+R^{dc}c^{e\\bar a}f_{de}^b=0$. Depending on the coefficients $c^{a\\bar b}$ one may also need non-vanishing left-left $R^{ab}$ components in order to solve this equation,\\footnote{Such an example is given by the fourth $R$-matrix in table~\\ref{tab:non-ab-sl2}.} but the CS condition is not sensitive to them.}\n\nLet us also comment that, differently from what was claimed in~\\cite{Araujo:2018rho}, the strong CS condition~\\eqref{eq:CS-strong} is \\emph{not} the CYBE, not even when one further imposes the unimodularity condition (and in fact our $\\mathbf R_9$ in table~\\ref{tab:non-ab-sl2-su2-r4} is a counter example to that claim).\n\n\n\\subsection{Field redefinition}\nIn order to display the current-current structure of the deformed model it is convenient to write the Lagrangian in the form\n\\begin{equation}\nL=L_0-\\tfrac12\\eta\\mathcal J_{A+}[(1+\\eta RM)^{-1}R]^{AB}\\mathcal J_{B-}\\,,\n\\label{eq:Lprime}\n\\end{equation}\nwhere $L_0$ is the undeformed Lagrangian and $M_{AB}=k_A^mk_B^n(G-B)_{mn}$. This follows directly from the form of $\\tilde G-\\tilde B$ in (\\ref{eq:GBtilde}) and the definition of the Noether currents in (\\ref{eq:Noether}) upon recalling that we will pick $R$ so that the last term in $\\mathcal J$ does not contribute. To compare this to the discussion of current-current deformations of the WZW model we need to perform a field redefinition that replaces the Noether currents in the above expression with the chiral currents.\\footnote{It is actually enough to look at the lowest order in $\\eta$ if we are considering infinitesimal deformations. {The action written in terms of the Noether currents as in~\\eqref{eq:Lprime} was given also in~\\cite{Araujo:2018rho} but the rewriting in terms of the chiral currents is missing there.}} In appendix~\\ref{app:field-red} we find such a field redefinition for a general deformation specifying to $AdS_3\\times S^3$ for concreteness.\nHere we will just say that for all deformations that we consider we find that the Lagrangian can be written in the form\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta\\hat J_a[(1+\\eta RM')^{-1}R]^{a\\bar a}\\hat{\\bar J}_{\\bar a}\\,,\n\\label{eq:LdefAdS3}\n\\end{equation}\nwhere $M'$ is a shorthand for $M$ after the field redefinition. $\\hat J_a,\\hat{\\bar J}_{\\bar a}$ are modifications of the chiral currents of the undeformed WZW model. Their explicit form is given in~\\eqref{eq:JhatAdS3} for deformations of $AdS_3$, and in section~\\ref{sec:def-AdS3S3} for deformations of $AdS_3\\times S^3$.\n\nAt leading order in $\\eta$ the above Lagrangian becomes\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta J_aR^{a\\bar a}\\bar J_{\\bar a}+\\mathcal{O}(\\eta^2)\\,,\n\\end{equation}\nso that the comparison to the current-current deformations considered by CS is now manifest.\n\n\\subsection{YB deformations of $AdS_3$}\\label{sec:def-AdS3}\nLet us start by looking at YB deformations that deform only $AdS_3$. We will start with the simplest ones which are TsT-transformations. They come from the abelian $R$-matrices of $\\mathfrak{sl}(2,\\mathbbm R)_{\\mbox{\\tiny L}}\\oplus \\mathfrak{sl}(2,\\mathbbm R)_{\\mbox{\\tiny R}}$ which (up to automorphisms) are\\footnote{From now on we will use the components $R^{AB}$ to construct $R=R^{AB}\\mathbf T_A\\wedge \\mathbf T_B\\in \\mathfrak g\\wedge \\mathfrak g$.\n} \\footnote{We could consider also $R=S_0\\wedge \\bar S_-$, but it is related to $R=S_+\\wedge \\bar S_0$ if we also exchange the left and right copy of the algebra.}\n\\begin{equation}\nS_+\\wedge\\bar S_-\\,,\\qquad S_0\\wedge\\bar S_0\\,,\\qquad S_+\\wedge\\bar S_0\\,.\n\\end{equation}\nFor the first one we obtain, from (\\ref{eq:GBtilde}) and (\\ref{eq:Phitilde}), the supergravity background of a deformation of $AdS_3\\times S^3\\times T^4$\n\\begin{equation}\n\\begin{aligned}\nds^2&=\n-\\frac{dx^+dx^-}{z^2-\\eta}\n+\\frac{dz^2}{z^2}+ds_{S^3}^2+ds_{T^4}^2\\,,\\\\\nB&=\\frac{dx^+\\wedge dx^-}{2(z^2-\\eta)}\n-\\frac{1}{4}\\sin \\phi_3 d\\phi_1\\wedge d\\phi_2\\,,\\qquad\\quad\ne^{-2\\tilde\\Phi}=1-\\frac{\\eta}{z^2}\\,.\n\\end{aligned}\n\\end{equation}\nIn this case the isometries involved in the deformation procedure correspond to Noether currents that agree with the (anti)chiral currents, see~\\eqref{eq:noether-chiralAdS3}. We therefore automatically get that to leading order the deformation of the Lagrangian is given by the marginal operator $\\eta J_+\\bar J_-$. In~\\cite{Giveon:2017nie} the above  background was argued to be the dual of the ``single-trace'' $T\\bar T$ deformation of the symmetric product orbifold CFT$_2$. This is also is accordance with the fact this particular YB deformation is just a TsT transformation involving the two boundary coordinates.\\footnote{We can have a TsT interpretation because we can implement the sequence T-duality, shift, T-duality in terms of the coordinates $x^0,x^1$, instead of the null coordinates $x^\\pm$.}\nAt finite order in the deformation parameter the Lagrangian is given by (\\ref{eq:LdefAdS3}), (\\ref{eq:JhatAdS3}) and (\\ref{eq:M})\n\\begin{equation}\nL=L_0-\\tfrac12\\frac{\\eta}{1-\\frac{\\eta}{z^2}}J_+\\bar J_-\\,.\n\\label{eq:TsT1}\n\\end{equation}\nThe derivative of the action with respect to the deformation parameter is given by\n\\begin{equation}\n\\frac{dS}{d\\eta}=-\\frac{T}{2}\\int d^2\\sigma \\ J_+^\\eta\\bar J_-^\\eta\\,,\n\\end{equation}\nwhere $J_+^\\eta=(1-\\eta z^{-2})^{-1}J_+$ and $\\bar J_-^\\eta=(1-\\eta z^{-2})^{-1}\\bar J_-$ are (anti)chiral currents of the \\emph{deformed} model.{ This background was analysed in~\\cite{Forste:1994wp}.}\nLet us also note that in this case the deformation parameter can be absorbed by a rescaling of the coordinates. There are therefore only three cases: $\\eta>0$, $\\eta=0$ and $\\eta<0$. The first of these is not globally well behaved since the dilaton becomes imaginary when crossing $z=\\sqrt\\eta$. For $\\eta<0$ the solution interpolates between two CFTs: the $SL(2,\\mathbbm R)$ WZW model and a linear dilaton background (plus two decoupled bosons). It would be interesting to study further the implications of the existence of this interpolating solution but we will not do so here. See also~\\cite{Giveon:2017nie} for a discussion on the different interpretations depending on the sign of the deformation parameter.\n\nFor the remaining two abelian $R$-matrices we obtain by a similar calculation the Lagrangians\\footnote{{The $J_0\\bar J_0$-deformation was considered also in~\\cite{Israel:2003ry}.}}\n\\begin{equation}\n\\begin{aligned}\n&L=L_0-\\tfrac12\\eta\\frac{(4+\\eta)z^2}{4z^2+\\eta(4+\\eta)x^+x^-}J_0 \\bar J_0\\,,\\\\\n&L=L_0-\\tfrac12\\eta\\frac{z^2}{z^2+\\eta  x^-}J_+ \\bar J_0\\,.\n\\end{aligned}\n\\end{equation}\nFor all the abelian examples one finds, as already mentioned, that\n\\begin{equation}\n\\frac{dS}{d\\eta}=-\\frac{T}{2}\\int d^2\\sigma \\ R^{a\\bar a}J_a^\\eta\\bar J_{\\bar a}^\\eta\\,,\n\\end{equation}\nwhere $J_a^\\eta,\\bar J_{\\bar a}^\\eta$ are (anti)chiral currents of the deformed model. For the last two abelian examples they are\n\\begin{equation}\n\\begin{aligned}\n&R=S_0\\wedge \\bar S_0: && \nJ_0^\\eta=\\alpha J_0\\,, \\qquad \n\\bar J_0^\\eta=\\alpha \\bar J_0\\,,\\qquad\n\\alpha\\equiv\\frac{4 z^2\\sqrt{1+\\frac{\\eta }{2}}}{4z^2+\\eta(4+\\eta)x^+x^-}\\,,\\\\\n&R=S_+\\wedge \\bar S_0: && J_+^\\eta=\\alpha J_+\\,, \\qquad \n\\bar J_0^\\eta=\\alpha \\bar J_0\\,,\\qquad\n\\alpha\\equiv\\frac{z^2}{z^2+\\eta  x^-}\\,.\n\\end{aligned}\n\\end{equation}\nFollowing~\\cite{Forste:1994wp}, the above result is another way to see that the YB model provides a deformation that is marginal exactly in the deformation parameter.\n\n\\vspace{12pt}\n\nIn order to find deformations that at least potentially are not TsT, one should look at the class of non-abelian $R$-matrices. The full list of non-abelian $R$-matrices of $\\mathfrak{sl}(2,\\mathbb{R})_{\\mbox{\\tiny L}}\\oplus\\mathfrak{sl}(2,\\mathbb{R})_{\\mbox{\\tiny R}}$ (up to $SL(2,\\mathbb{R})_{\\mbox{\\tiny L}}\\times SL(2,\\mathbb{R})_{\\mbox{\\tiny R}}$ automorphisms) is given in table~\\ref{tab:non-ab-sl2}. They are special cases of the $R$-matrices for $\\mathfrak{sl}(2,\\mathbbm R)_{\\mbox{\\tiny L}}\\oplus \\mathfrak{sl}(2,\\mathbbm R)_{\\mbox{\\tiny R}}\\oplus\\mathfrak{su}(2)_{\\mbox{\\tiny L}}\\oplus \\mathfrak{su}(2)_{\\mbox{\\tiny R}}$ classified in appendix \\ref{app:R-matrices}.\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|}\n\\hline\n& & & \\\\\n$R=R^{AB}\\mathbf T_A\\wedge \\mathbf T_B$ & Deformation?& $c^{ab}J_a\\bar J_b$ & Strong CS?\\\\\n\\hline\n$R_1=S_0\\wedge S_+$ & Trivial deformation & 0& Yes\\\\\n$R_2=(S_0\\mp\\bar S_-)\\wedge S_+$ & TsT & $\\pm J_+\\bar J_-$& Yes\\\\\n$R_3=(S_0-a\\bar S_0)\\wedge S_+$ & TsT & $aJ_+\\bar J_0$& Yes\\\\\n$R_4=(S_0-\\bar S_0)\\wedge (S_+\\pm\\bar S_-)$ & Not SUGRA & $J_+\\bar J_0\\pm J_0\\bar J_-$& No\\\\\n$R_5=S_0\\wedge S_+\\pm\\bar S_0\\wedge \\bar S_-+\\lambda {S_+\\wedge \\bar S_-}$ & {TsT} & $\\lambda {J_+\\bar J_-}$& Yes\\\\\n\\hline\n\\end{tabular}\n\\caption{Non-abelian $R$-matrices of $\\mathfrak{sl}(2)_{\\mbox{\\tiny L}}\\oplus\\mathfrak{sl}(2)_{\\mbox{\\tiny R}}$ up to $SL(2,\\mathbb{R})_{\\mbox{\\tiny L}}\\times SL(2,\\mathbb{R})_{\\mbox{\\tiny R}}$ inner automorphisms and swaps of $\\mbox{\\tiny L}\\leftrightarrow \\mbox{\\tiny R}$. For convenience we also write the marginal operators that they give rise to, and whether they satisfy the strong CS condition.}\n\\label{tab:non-ab-sl2}\n\\end{center}\n\\end{table}\nAnalysing the first example $R_1=S_0\\wedge S_+$ one finds that, after the field redefinition $x^+\\to x^+-\\frac{\\eta}{2}\\log z$, not only the leading order in $\\eta$ vanishes in the action, but also all the higher orders. This is obvious from eq. (\\ref{eq:LdefAdS3}) and the fact that $R$ with an anti-chiral index vanishes. In other words the deformation is trivial, since its effect is to give the undeformed $AdS_3$ background in new ($\\eta$-dependent) coordinates. To be more precise, from $R_1$ we get back undeformed $AdS_3$ \\emph{up to} a non-vanishing $K=-\\eta\\partial_{x^+}$, from (\\ref{eq:K}). This is of course a ``trivial solution''  of generalised supergravity (i.e. one that solves also standard supergravity upon dropping $K$, notice  that $K$ is null).\\footnote{Obviously a similar discussion holds also for $R=\\bar S_0\\wedge \\bar S_-$.}\n\nThe above result for $R_1=S_0\\wedge S_+$ turns out to be useful to analyse some of the following examples in the table. \nIt is easy to see that $R_2$ and $R_3$ in table~\\ref{tab:non-ab-sl2} are of the form $R= S_0\\wedge S_++R'$ and that in both cases $R'$ is compatible with the field redefinition  that trivialises the effect of the piece $S_0\\wedge S_+$, see~\\eqref{eq:compTheta2}. Therefore these two $R$-matrices are equivalent to two of the TsT transformations already discussed,\\footnote{Also in this case the equivalence holds up to a non-vanishing $K=-\\eta\\partial_{x^+}$ that is null and decouples from the equations of generalised supergravity.} generated by $S_+\\wedge \\bar S_-$ and $S_+\\wedge \\bar S_0$. Alternatively it is easy to see this directly from the Lagrangian in (\\ref{eq:LdefAdS3}) and (\\ref{eq:JhatAdS3}).\n\n\nThe last two $R$-matrices in table~\\ref{tab:non-ab-sl2}, instead, give backgrounds that are not of this type. From (\\ref{eq:K}) we find that they have $K=-\\eta(\\partial_{x^+}\\pm\\partial_{x^-})$ and $K=-\\eta(\\partial_{x^+}\\mp\\partial_{x^-})$ respectively, neither of which is null. The only way they can give solutions of standard supergravity is then, {as shown in appendix \\ref{app:trivial}, if ${\\rm X}=d\\phi+\\tilde K$ with $\\phi$ the dilaton and $\\tilde K$ an independent Killing vector field. We can extract $\\tilde K$ from the equation $dK+i_{\\tilde K}H=0$ in (\\ref{eq:trivial}) and for $R_4$ one finds\n$\\tilde K=\\eta(\\partial_{x^+}\\mp\\partial_{x^-})\\pm\\eta^2z^{-1}\\partial_z+\\mathcal O(\\eta^3)$ which, at lowest order, differs from $K$ only in the sign of the $\\partial_{x^+}$-term. However from the lowest order correction to the action $J_+\\bar J_0\\pm J_0\\bar J_-$ we read off the deformed metric \n\\begin{equation}\nz^{-2}(-dx^-dx^++dzdz)-\\eta z^{-4}(zdz(dx^-\\mp dx^+)+(\\pm x^+-x^-)dx^+dx^-)+\\mathcal O(\\eta^2)\\,,\n\\end{equation}\nwhich is only invariant under $\\delta x^+=\\eta\\epsilon$, $\\delta x^-=\\pm\\eta\\epsilon$ which is generated by $K$, and not under $\\delta x^+=\\eta\\epsilon$, $\\delta x^-=\\mp\\eta\\epsilon$, $\\delta z=\\pm\\eta^2\\epsilon z^{-1}$ which is generated by $\\tilde K$. Therefore $\\tilde K$ is not Killing and the $R_4$-background is not a solution to standard supergravity. Note that it only fails to be a solution at order $\\eta^2$ which is consistent with the fact that the leading-order deformation  satisfies the weak CS condition.}\n We will not consider this background further here since our interest is mainly in string theory applications. For $R_5$, instead, one can show that $\\tilde K$ is a Killing vector of the deformed metric and it satisfies~\\eqref{eq:trivial}, meaning that we get a ``trivial solution'' of generalised supergravity for generic $\\lambda$. Actually, using (\\ref{eq:LdefAdS3}) and (\\ref{eq:JhatAdS3}), for $R_5$ we find the Lagrangian\\footnote{For concreteness we take $R_5=S_0\\wedge S_++\\bar S_0\\wedge \\bar S_-+\\lambda {S_+\\wedge \\bar S_-}$, but similar results apply also for $R_5=S_0\\wedge S_+-\\bar S_0\\wedge \\bar S_-+\\lambda {S_+\\wedge \\bar S_-}$.}\n\\begin{equation}\nL=L_0+{\\frac{\\eta  z^2 (\\eta -4 \\lambda )}{2 \\left(\\eta  (\\eta -4 \\lambda )+4 z^2\\right)}}J_+\\bar J_-\\,,\n\\label{eq:lambda0}\n\\end{equation}\nwhich is exactly that of the TsT example in~\\eqref{eq:TsT1} with {$\\eta\\rightarrow \\eta\\lambda-\\eta^2\/4$}. This background therefore provides an example of the more general kind of trivial solution of the generalized supergravity equations discussed in appendix \\ref{app:trivial} for which $K^2\\neq0$.\n\n{This example also shows how the identification of the deformation parameters can be non-trivial. In fact, although at leading order the marginal deformation is given only by $\\eta\\lambda J_+\\bar J_-$, the deformation exact in $\\eta$ shows that the deformation parameter of the TsT is instead $\\eta\\lambda-\\eta^2\/4$, and in particular it does not vanish even when $\\lambda=0$.}\n\nIt is interesting to look at the marginal operators of the type $c^{ab}J_a\\bar J_b$  that are generated by each $R$-matrix. We write them for convenience in table~\\ref{tab:non-ab-sl2}. While they all solve the weak CS condition, they all solve also the strong CS condition {(which guarantees that they are exactly marginal)} except for the fourth one {(which we have seen fails to be marginal beyond lowest order)}. \n\n\\subsection{YB deformations of $AdS_3\\times S^3$}\\label{sec:def-AdS3S3}\nLooking at deformations of $AdS_3\\times S^3$ gives a richer set of possibilities. In this case we want to solve the CYBE on the algebra $\\mathfrak{sl}(2)_{\\mbox{\\tiny L}}\\oplus\\mathfrak{sl}(2)_{\\mbox{\\tiny R}}\\oplus\\mathfrak{su}(2)_{\\mbox{\\tiny L}}\\oplus\\mathfrak{su}(2)_{\\mbox{\\tiny R}}$.\n\n\nThe simplest example to start with is $R=S_+\\wedge \\bar T_2$. This is an abelian $R$-matrix and therefore corresponds to a TsT mixing $AdS_3$ and $S^3$. In~\\cite{Chakraborty:2018vja,Apolo:2018qpq} it was argued that this background is dual to the single-trace $T\\bar J$ deformation of the CFT$_2$. From the YB procedure one explicitly finds\n\\begin{equation}\n\\begin{aligned}\nds^2&=ds_{\\text{AdS}_3}^2+ds_{\\text{S}^3}^2+\\frac{\\eta}{4z^2}dx^-(d\\phi_2+2\\sin \\phi_3d\\phi_1),\\\\\nB&=B_0\n+\\frac{\\eta  dx^-\\wedge(d\\phi_2 +2 \\sin \\phi_3 d\\phi_1)}{8 z^2},\n\\qquad e^{-2\\Phi}=1.\n\\end{aligned}\n\\end{equation}\nFrom~\\eqref{eq:noether-chiralS3} we see that the Noether current $\\bar{\\mathcal J}_2$ differs from the antichiral current $\\bar J_2$, and therefore field redefinitions are needed in order to put the action in the form that makes the chirality structure manifest. After the field redefinition $x^+\\to x^+-\\frac{\\eta}{4}\\phi_2$, or equivalently by looking directly at (\\ref{eq:Ldef}), (\\ref{eq:Jhat}) and (\\ref{eq:Jbarhat}) the Lagrangian becomes\n\\begin{equation}\nL=L_0-\\tfrac12\\eta J_+\\bar J_2\\,.\n\\end{equation}\nThis deformation is special since the leading linear order is exact. Obviously $dS\/d\\eta=- \\tfrac{T}{2} \\int J_+\\bar J_{2}$.\n\nWe will now focus on non-abelian $R$-matrices. These are classified for $\\mathfrak{sl}(2)_{\\mbox{\\tiny L}}\\oplus\\mathfrak{sl}(2)_{\\mbox{\\tiny R}}\\oplus\\mathfrak{su}(2)_{\\mbox{\\tiny L}}\\oplus\\mathfrak{su}(2)_{\\mbox{\\tiny R}}$ in appendix~\\ref{app:R-matrices}. Since the CYBE on the two copies of $\\mathfrak{su}(2)$ implies that the subset of generators from this part of the algebra should be abelian, our classification is useful also to study deformations of the full $AdS_3\\times S^3\\times T^4$ or the more generic $AdS_3\\times S^3\\times S^3\\times S^1$. Every time an $\\mathfrak{su}(2)$ generator appears, it may as well be replaced by another compact generator, as long as all compact generators involved form an abelian subalgebra. For concreteness we will only look at deformations of $AdS_3\\times S^3$. The rank-2 non-abelian $R$-matrices are collected in table~\\ref{tab:non-ab-sl2-su2-r2}. Since they are special cases of the rank-4 $R$-matrices, listed in table~\\ref{tab:non-ab-sl2-su2-r4}, we will not consider them separately.\n\n\n\\begin{table}[ht]\n\\begin{center}\n\\begin{tabular}{|l|c|}\n\\hline\n&   \\\\\n$R=R^{AB}\\mathbf T_A\\wedge \\mathbf T_B$ & Deformation?\\\\\n\\hline\n&   \\\\\n$\\mathbf R_1=(S_0-\\bar S_0+T)\\wedge (S_+\\pm\\bar S_-)$ & $R_4$+TsT$\\implies$ not SUGRA  \\\\\n$\\mathbf R_2=(S_0-a\\bar S_0+T)\\wedge S_+$ & $R_3$+TsT$\\implies$ TsT \\\\\n$\\mathbf R_3=(S_0\\mp\\bar S_-+T)\\wedge S_+$ & $R_2$+TsT$\\implies$ TsT  \\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|}\n\\hline\n& &  \\\\\n$R$& $c^{ab}J_a\\bar J_b$ & Strong CS?\\\\\n\\hline\n&  & \\\\\n$\\mathbf R_1$ &  $\\pm(J_0+aJ_1)\\bar J_-+J_+(\\bar J_0+b\\bar J_2)$ & No \\\\\n$\\mathbf R_2$ & $J_+(a\\bar J_0+b\\bar J_2)$& Yes \\\\\n$\\mathbf R_3$ & $J_+(\\pm\\bar J_-+b\\bar J_2)$& Yes \\\\\n\\hline\n\\end{tabular}\n\\caption{Rank-2 non-abelian $R$-matrices of $\\mathfrak{sl}(2)_{\\mbox{\\tiny L}}\\oplus\\mathfrak{sl}(2)_{\\mbox{\\tiny R}}\\oplus \\mathfrak{su}(2)_{\\mbox{\\tiny L}}\\oplus\\mathfrak{su}(2)_{\\mbox{\\tiny R}}$ up to $SL(2,\\mathbb{R})_{\\mbox{\\tiny L}}\\times SL(2,\\mathbb{R})_{\\mbox{\\tiny R}}\\times SU(2)_{\\mbox{\\tiny L}}\\times SU(2)_{\\mbox{\\tiny R}}$ inner automorphisms and swaps of $\\mbox{\\tiny L}\\leftrightarrow \\mbox{\\tiny R}$. With $T$ we denote a generic linear combination of $T_1, \\bar T_2$. }\n\\label{tab:non-ab-sl2-su2-r2}\n\\end{center}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|c|}\n\\hline\n& \\\\\n$R=R^{AB}\\mathbf T_A\\wedge \\mathbf T_B$ & Deformation?\\\\\n\\hline\n&  \\\\\n$\\mathbf R_4=\\mathbf R_1+a T_1\\wedge \\bar T_2$ & $\\mathbf R_1+$TsT$\\implies$not SUGRA \\\\\n$\\mathbf R_5=\\mathbf R_2+b T_1\\wedge \\bar T_2$ & $\\mathbf R_2+$TsT$\\implies$TsT  \\\\\n$\\mathbf R_6=(S_0+T)\\wedge S_++T' \\wedge (\\bar S_0+T'')$ & SUGRA  \\\\\n$\\mathbf R_7=(S_0+T)\\wedge S_++T' \\wedge (\\bar S_-+T'')$ & TsT  \\\\\n$\\mathbf R_8=\\mathbf R_3+a T_1\\wedge \\bar T_2$ & $\\mathbf R_3+$TsT$\\implies$TsT \\\\\n$\\mathbf R_9=(S_0+\\bar S_0+T)\\wedge T'+S_+\\wedge \\bar S_-$ & SUGRA, not TsT \\\\\n$\\mathbf R_{10}={(S_0+T)\\wedge S_+\\pm(\\bar S_0+T')\\wedge \\bar S_-+\\lambda S_+\\wedge \\bar S_-}$ & {TsT}  \\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|}\n\\hline\n&  & \\\\\n$R$& $c^{ab}J_a\\bar J_b$ & Strong CS?\\\\\n\\hline\n& &  \\\\\n$\\mathbf R_4$  & $(J_0+aJ_1)\\bar J_-+J_+(\\bar J_0+b\\bar J_2)+a J_1\\bar J_2$ & No \\\\\n$\\mathbf R_5$  & $J_+(a\\bar J_0+b\\bar J_2)+bJ_1\\bar J_2$ & Yes \\\\\n$\\mathbf R_6$ & $aJ_+\\bar J_2+bJ_1\\bar J_0+cJ_1\\bar J_2$ & Yes \\\\\n$\\mathbf R_7$  &  $aJ_+\\bar J_2+bJ_1\\bar J_-+cJ_1\\bar J_2$ & Yes \\\\\n$\\mathbf R_8$ & $J_+(\\bar J_-+b\\bar J_2)+aJ_1\\bar J_2$ & Yes \\\\\n$\\mathbf R_9$ & $aJ_0\\bar J_2+bJ_1\\bar J_0+cJ_1\\bar J_2+J_+\\bar J_-$ & No ($a\\neq 0$ or $b\\neq 0$)\\\\\n$\\mathbf R_{10}$ & ${cJ_+\\bar J_2+d J_1\\bar J_-+\\lambda J_+\\bar J_-}$ & {Yes} \\\\\n\\hline\n\\end{tabular}\n\\caption{Rank-4 non-abelian $R$-matrices of $\\mathfrak{sl}(2)_{\\mbox{\\tiny L}}\\oplus\\mathfrak{sl}(2)_{\\mbox{\\tiny R}}\\oplus \\mathfrak{su}(2)_{\\mbox{\\tiny L}}\\oplus\\mathfrak{su}(2)_{\\mbox{\\tiny R}}$ up to $SL(2,\\mathbb{R})_{\\mbox{\\tiny L}}\\times SL(2,\\mathbb{R})_{\\mbox{\\tiny R}}\\times SU(2)_{\\mbox{\\tiny L}}\\times SU(2)_{\\mbox{\\tiny R}}$ inner automorphisms and swaps of $\\mbox{\\tiny L}\\leftrightarrow \\mbox{\\tiny R}$. With $T,T',T''$ we denote  generic linear combinations of $T_1, \\bar T_2$.}\n\\label{tab:non-ab-sl2-su2-r4}\n\\end{center}\n\\end{table}\n\nTo compute the Killing vector $K$ that appears in the generalized supergravity equations we note that eq. (\\ref{eq:K}) implies that only the non-abelian generators matter and we can set $T=T'=T''=0$ (where these are   generic linear combinations of $T_1, \\bar T_2$). We find therefore the same answer as for the $SL(2,\\mathbbm R)$ case in the previous section and comparing to that analysis we see that only $\\mathbf R_4$ (and therefore $\\mathbf R_1$) does not give a supergravity solution. Since we are interested in string theory applications we will focus on the analysis of $\\mathbf R_5$ through $\\mathbf R_{10}$. Recall that in this way we automatically consider also the rank-2 $R$-matrices $\\mathbf R_2$ and $\\mathbf R_3$.  From (\\ref{eq:Ldef}), (\\ref{eq:Jhat}) and (\\ref{eq:Jbarhat}) it follows that (after the field redefinitions) the Lagrangian takes the form\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta\\hat J_a[(1+\\eta RM')^{-1}R]^{a\\bar a}\\hat{\\bar J}_{\\bar a}\\,,\n\\end{equation}\nwhere for $\\mathbf R_5$--$\\mathbf R_8$\n\\begin{equation}\n\\hat J_a=J_a+\\eta\\delta_a^0y_iY^i_AR^{A+}J_+\\,,\\qquad\n\\hat{\\bar J}_{\\bar a}=\\bar J_{\\bar a}-\\eta\\delta_{\\bar a}^{\\bar0}Y^i_AR^{A\\bar-}y_i\\bar J_-\\,,\n\\end{equation}\nwhile for $\\mathbf R_9$\n\\begin{equation}\n\\hat J_a=e^{\\delta_a}J_a\\,,\\qquad\n\\hat{\\bar J}_{\\bar a}=e^{\\bar\\delta_{\\bar a}}\\bar J_{\\bar a}\\,,\\qquad\n\\delta_+=-\\bar\\delta_{\\bar-}=-\\eta Y^i_AR^{A0}y_i\\,.\n\\end{equation}\nIt is not hard to show that $\\mathbf R_5$, $\\mathbf R_7$ and $\\mathbf R_8$ are in fact equivalent to TsT transformations. Consider the first one. The deformed Lagrangian is\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta[J_a+\\eta\\delta_a^0y_iY^i_AR^{A+}J_+][(1+\\eta RM')^{-1}R]^{a\\bar a}\\bar J_{\\bar a}\\,.\n\\end{equation}\nHowever, due to the form of the $R$-matrix and the fact that $M'_{+A}=0$ this is equal to\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta J_a[(1+\\eta RM')^{-1}R]^{a\\bar a}\\bar J_{\\bar a}\\,,\n\\end{equation}\nwhere furthermore we can replace $R$ by the abelian $R$-matrix obtained by dropping the $S_0\\wedge S_+$-term in $\\mathbf R_5$. This deformation therefore reduces to a sequence of commuting TsT transformations. The same conclusion applies to $\\mathbf R_8$ as is easily seen from the form of the $R$ matrix. For $\\mathbf R_7$ we have\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta[J_a+\\eta\\delta_a^0y_iY^i_AR^{A+}J_+][(1+\\eta RM')^{-1}R]^{a\\bar a}[\\bar J_{\\bar a}-\\eta\\delta_{\\bar a}^{\\bar0}Y^i_AR^{A\\bar-}y_i\\bar J_-]\\,,\n\\end{equation}\nbut again the terms with $a=0$ and $\\bar a=\\bar0$ drop out due to the form of the $R$-matrix and the fact that $M_{+A}=0$ and this reduces to\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta J_a[(1+\\eta RM')^{-1}R]^{a\\bar a}\\bar J_{\\bar a}\\,.\n\\end{equation}\nIt is also not hard to see, using again the form of the matrices $M$ and $R$, that one can again replace $R$ by the abelian $R$-matrix obtained by dropping the $S_0\\wedge S_+$-term in $\\mathbf R_7$. The resulting background is therefore also a TsT. \nThe fact that $\\mathbf R_5$, $\\mathbf R_7$ and $\\mathbf R_8$ are equivalent to TsT backgrounds may be argued also from the fact that the $R$-matrices are abelian up to the $S_0\\wedge S_+$-term, and that~\\eqref{eq:compTheta2} holds.\n\n\nFor $\\mathbf R_6$ the deformed Lagrangian is\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta[J_a+\\eta\\delta_a^0y_iY^i_AR^{A+}J_+][(1+\\eta RM')^{-1}R]^{a\\bar a}\\bar J_{\\bar a}\\,.\n\\end{equation}\nAgain, using the form of $R$ and the fact that $M_{+A}=0$, this simplifies to\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta J_a[(1+\\eta RM')^{-1}R]^{a\\bar a}\\bar J_{\\bar a}\\,.\n\\end{equation}\nHowever, in this case we cannot get an equivalent background simply by dropping the $S_0\\wedge S_+$-term in $\\mathbf R_6$.\\footnote{One way to see this is that $\\mathbf R_6-R_1$ is not compatible with the coordinate redefinition needed to undo the transformation with $R_1=S_0\\wedge S_+$ (see eq.~\\eqref{eq:compTheta2}), therefore one does not expect to be able to undo the effect of the $R_1$ piece of the $R$-matrix.} Let us work out the background for the simplest case, $T=T''=0$ and $T'=aT_1+b\\bar T_2$. One finds\n\\begin{equation}\nL\n=\nL_0\n-\\eta f^{-1}\n\\Big(\n2b\\eta J_+\\bar J_2\n+2ab\\eta(1-\\eta\\frac{x^-}{z^2})J_1\\bar J_2\n-\\tfrac12\\eta^2(a^2+b^2-2ab\\sin\\phi_3)J_+\\bar J_0\n+8aJ_1\\bar J_0\n\\Big)\n\\end{equation}\nwhere $f=16+\\eta^2(a^2+b^2-2ab\\sin\\phi_3)[1-\\eta x^-\/z^2]$. One can show that in fact the (anti)chiral currents entering this action $J_+$, $J_1$, $\\bar J_0$ and $\\bar J_2$ extend to chiral currents to all orders in $\\eta$, i.e. the corresponding isometries are not broken by the deformation. Since this is a characteristic feature of TsT backgrounds it is natural to guess that this background can be generated in that way. It is not hard to show this explicitly in the special cases $a=1$, $b=0$ and $a=0$, $b=1$ in which cases it is equivalent to the backgrounds generated by\n\\begin{equation}\nR=T_1\\wedge\\bar S_0-\\frac{\\eta^2}{16}S_+\\wedge\\bar S_0\\qquad\\mbox{and}\\qquad\nR={\\frac{4\\eta}{16+\\eta^2}S_+\\wedge\\bar T_2}-\\frac{\\eta^2}{16+\\eta^2}S_+\\wedge\\bar S_0\\,,\n\\end{equation}\nrespectively.\n\n$\\mathbf R_9$ is the only unimodular example, i.e. the only one with $K=0$. The deformed Lagrangian is\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta\\hat J_a[(1+\\eta RM')^{-1}R]^{a\\bar a}\\hat{\\bar J}_{\\bar a}\\,,\n\\end{equation}\nwith\n\\begin{equation}\n\\hat J_a=e^{\\delta_a}J_a\\,,\\qquad\n\\hat{\\bar J}_{\\bar a}=e^{\\bar\\delta_{\\bar a}}\\bar J_{\\bar a}\\,,\\qquad\n\\delta_+=-\\bar\\delta_{\\bar-}=-\\eta Y^i_AR^{A0}y_i\n\\end{equation}\nFor simplicity we will set $T=0$ and $T'=T_1$. We can argue that this example is not equivalent to a TsT as follows. To order $\\eta^2$ the Lagrangian is\n\\begin{equation}\nL=\nL_0\n-\\tfrac12\\eta(1+\\frac{\\eta}{z^2})J_+\\bar J_-\n+\\tfrac12\\eta J_1\\bar J_0\n-\\tfrac18\\eta^2J_0\\bar J_0\n-\\tfrac12\\eta^2\\frac{x^-}{z^2}J_1\\bar J_-\n+\\mathcal O(\\eta^3)\\,.\n\\end{equation}\nThe action is clearly not invariant under the isometry corresponding to constant shifts of $x^-$ and therefore the corresponding chiral current $J_+$ does not extend to the deformed theory. Instead the equations of motion lead to chiral currents\n\\begin{equation}\nJ_+^\\eta=(1+\\frac{\\eta}{z^2})J_+-\\eta\\frac{x^-}{z^2}J_1+\\mathcal O(\\eta^2)\n\\,,\\qquad\nJ_1^\\eta=J_1+\\mathcal O(\\eta^2)\\,,\n\\end{equation}\nwhile the remaining equations of motion read\n\\begin{equation}\n\\partial[(1+\\frac{\\eta}{z^2})\\bar J_-]-\\eta J_1\\bar J_-=\\mathcal O(\\eta^2)\\,,\\qquad\n\\partial[\\bar J_0+\\eta\\frac{x^-}{z^2}\\bar J_-]-\\eta J_+\\bar J_-=\\mathcal O(\\eta^2)\\,.\n\\end{equation}\nAt the same time we have\n\\begin{equation}\n\\frac{dS}{d\\eta}=\n-\\frac{T}{2}\\int\\,\\left(J_+^\\eta(1+\\frac{\\eta}{z^2})\\bar J_--J_1\\bar J_0+3\\eta\\frac{x^-}{z^2}J_1\\bar J_-+\\tfrac12\\eta J_0\\bar J_0+\\mathcal O(\\eta^2)\\right)\\,,\n\\end{equation}\nwhich clearly cannot be written as a bilinear in deformed chiral currents.\nExplicitly, we obtain the background\n\\footnote{Here we are writing the background that is obtained \\emph{before}  doing the field redefinition.}\n\\begin{equation}\n\\begin{aligned}\nds^2=&\n-\\frac{\\eta  d\\phi_1 (x^+ dx^-+x^-  dx^+)+4 dx^- dx^++\\left(\\eta -z^2\\right) d\\phi_1^2}{\\eta  (\\eta  x^- x^+-4)+4  z^2}\n+\\frac{dz^2}{z^2}\\\\\n&+\\frac{d\\phi_2^2+d\\phi_3^2}{4}\n-\\frac{2 d\\phi_2 \\sin \\phi_3 \\left(\\eta  x^- dx^++\\left(\\eta -z^2\\right) d\\phi_1\\right)}{\\eta  (\\eta  x^- x^+-4)+4 z^2},\\\\\nB=&\\frac{-4 dx^-\\wedge dx^+-2 \\sin \\phi_3 \\left( d\\phi_2\\wedge(\\eta  x^- dx^++\\left(\\eta -z^2\\right) d\\phi_1)\\right)+\\eta d\\phi_1\\wedge(  x^+  dx^--  x^-  dx^+)}{2 \\eta  (\\eta  x^- x^+-4)+8 z^2},\\\\\ne^{-2\\Phi}=&1+\\frac{\\eta(-4+\\eta x^+x^-)}{4z^2}.\n\\end{aligned}\n\\end{equation}\n{Finally, $\\mathbf R_{10}=R_5+T\\wedge S_++T'\\wedge \\bar S_-$. Since the $\\mathfrak{sl}(2,\\mathbb R)$ $R$-matrix $R_5$ does not break the isometries generated by $S_+,\\bar S_-$, we conclude that the additional terms in $\\mathbf R_{10}$ have the only effect of adding further TsT transformations on top of the background generated by $R_5$, which is itself a TsT background.}\n\n\nInterestingly, the matrices $\\mathbf R_4$ (which may be decomposed in terms of $\\mathbf R_1$) and $\\mathbf R_9$  give rise (at leading order in the deformation parameter) to marginal deformations of the WZW model that obviously satisfy the weak CS condition, but not the strong one.\n\n\n\\section{Discussion}\\label{sec:disc}\nIn this paper we have constructed YB deformations of strings on the pure NSNS $AdS_3\\times S^3\\times T^4$ background. Together with abelian YB deformations, which are known to reproduce TsT transformations, we also  constructed non-abelian YB deformations. While some non-abelian $R$-matrices give rise to backgrounds that cannot be obtained simply from TsT transformations, we found that  others generate again TsT backgrounds, or even no deformation at all.\\footnote{Except for the introduction of a (decoupled) non-vanishing vector $K$ in the equations of generalised supergravity.} We expect this to be related to the fact that the initial $G-B$ is degenerate.\n\nFor example, the Jordanian $R$-matrix $R_1=S_0\\wedge S_+$ gives back the undeformed $AdS_3$ background up to an $\\eta$-dependent field redefinition (and up to a non-vanishing $K=-\\eta \\partial_{x^+}$). Recalling that the YB deformation is equivalent to a shift of the $B$-field plus NATD, this observation suggests that $AdS_3$ with NSNS flux has a certain property of self T-duality, when we dualise the non-abelian algebra of isometries generated by $S_0,S_+$ and we also regularise the action by performing a $B$-field gauge transformation.\n\nAlthough we used our classification of $R$-matrices to deform, for concreteness, only the $AdS_3\\times S^3$ part of the background, our results may also be used to obtain deformations involving the $T^4$ of $AdS_3\\times S^3\\times T^4$, or even deformations of the more general $AdS_3\\times S^3\\times S^3\\times S^1$ background. Indeed, in all our expressions of the $R$-matrices the generators $T_1,\\bar T_2$ may be substituted with any other two commuting generators of the compact part of the isometry algebra.\\footnote{That is because the CYBE implies that the restriction of an $R$-matrix to a compact algebra must be abelian. We are therefore free to choose which abelian subalgebra we wish to consider.}\nLet us also mention that the string on these $AdS_3$ backgrounds is integrable~\\cite{Cagnazzo:2012se,Sundin:2012gc} and that our deformations preserve the classical integrability.\n\n\nTo leading order in the deformation parameter, all our YB deformations reduce to the marginal current-current deformations of the type considered by Chaudhuri and Schwartz in~\\cite{Chaudhuri:1988qb}. While they are all {marginal to lowest order}, since  they satisfy what we called the ``weak CS condition'', some of them do not satisfy the ``strong CS condition'' and the celebrated ``no simple-pole condition'' {which guarantee exact marginality. Indeed we found examples which failed to be marginal beyond lowest order (and at one-loop in $1\/k$), all involving the $R$-matrix $R_4$ in Table \\ref{tab:non-ab-sl2}, and one example, $\\mathbf R_9$ in Table \\ref{tab:non-ab-sl2-su2-r4}, which remains marginal to all orders in $\\eta$ at least up to one loop in $1\/k$.} The relation between the space of solutions of the CYBE and that of the weak CS condition is an interesting question. The former is a quadratic equation for $R$, while the latter is a quartic equation for the coefficients $c^{ab}$ of the current-current deformation, related to the left-right block of the $R$-matrix simply as $c^{ab}=R_{\\mbox{\\tiny L}\\mbox{\\tiny R}}^{ab}$. While we expect all solutions of the CYBE to generate solutions of the weak CS condition (including the trivial ones) it seems hard to prove this statement for a generic Lie algebra. In appendix~\\ref{app:sl3} we took a digression from the setup of the paper and we considered the CYBE on the $\\mathfrak{sl}_3$ algebra, finding again that it generates non-trivial solutions to the weak CS condition (that do not solve the strong one). We do not rule out the possibility of having solutions to the weak CS condition that cannot be ``completed'' to  a solution of the CYBE equation. \nIn section~\\ref{sec:YB} we actually discussed a more generic criterion (related to the solvability of the subalgebras involved) not requiring CYBE, to construct solutions of the weak CS condition.\n\nIn~\\cite{Azeyanagi:2012zd} a marginal deformation constructed out of abelian currents (a TsT transformation) was interpreted in terms of spectral flow. It would be interesting to understand if this can be generalised to the non-abelian set-up.\n\nWe would like to stress that we worked out the YB deformations in the sigma model description. It would be very interesting to understand how to formulate the YB deformation directly at the level of the WZW action. Such a construction was performed in~\\cite{Delduc:2014uaa,Delduc:2017fib,Delduc:2018xug} for $R$ a solution of the \\emph{modified} CYBE.\\footnote{There the special propriety $R^3=-R$ was used, so that we do not expect their results to be immediately applicable to the case of the homogeneous CYBE. Moreover, here we want $R$ to be a solution of the CYBE on $\\mathfrak f_{\\mbox{\\tiny L}}\\oplus \\mathfrak f_{\\mbox{\\tiny R}}$ that also couples the left and right copy of the algebra.} The formulation of the deformation of the WZW action may be obtained from the construction in terms of NATD, in the spirit of~\\cite{Hoare:2016wsk} and~\\cite{Borsato:2016pas}. An alternative may be to use the language of $\\mathcal E$ models~\\cite{Klimcik:1995dy,Stern:1998my,Klimcik:2015gba}, see~\\cite{Demulder:2018lmj} for a recent application in similar contexts.\n\nOne motivation to carry out this work came from the recent developments on the $T\\bar T$ deformation~\\cite{Smirnov:2016lqw,Cavaglia:2016oda} and its generalisations. The components $T,\\bar T$ of the stress-energy tensor of a (quite generic) two-dimensional relativistic field theory may be used to construct a ``double-trace'' operator generating an irrelevant perturbation of the theory. The deformation is solvable in the sense that the spectrum of the deformed theory may be computed \\emph{exactly} in the deformation parameter  as a function of the spectrum of the original undeformed theory. In~\\cite{Giveon:2017nie} a ``single-trace'' version of the $T\\bar T$ deformation of the symmetric product orbifold CFT was considered. It was argued that the irrelevant deformation of the ``spacetime'' CFT governed by\\footnote{We refer to~\\cite{Giveon:2017nie} for the connection to Little String Theories. The above operator should be compared to the original double-trace version studied in~\\cite{Smirnov:2016lqw,Cavaglia:2016oda} and given by $T(x)\\bar T(\\bar x)$, where $T(x)=\\sum_{i=1}^NT_i(x)$ and $\\bar T(\\bar x)=\\sum_{i=1}^N\\bar T_i(\\bar x)$.} $\\mathcal O(x)\\propto \\sum_{i=1}^NT_i(x)\\bar T_i(\\bar x)$, where $i$ labels each copy in the symmetric product,  corresponds to a marginal deformation of the dual WZW model that infinitesimally is just the current-current deformation  $J_+(\\sigma)\\bar J_-(\\bar \\sigma)$, where $J_+,\\bar J_-$ are the left and right $SL(2,\\mathbb R)$ currents generating shifts of the boundary coordinates $x^+,x^-$. \nAnother deformation, similar in spirit to the above one, was studied in~\\cite{Chakraborty:2018vja} and~\\cite{Apolo:2018qpq} after replacing the $\\bar T$ with an antichiral $U(1)$ current of the compact factor.\\footnote{This   deformation is in fact the single-trace version of the one first constructed in~\\cite{Guica:2017lia}.}\nIt was argued in~\\cite{Chakraborty:2018vja,Apolo:2018qpq} that the deformation of the dual WZW model is governed again by a marginal deformation bilinear in the currents (where now the antichiral current belongs to the compact part of the algebra). Such marginal deformations of the WZW model may be completed to finite values of the deformation parameter in terms of TsT (or equivalently certain $O(d,d)$) transformations. \nSince TsT deformations are a subclass of YB ones, it would be interesting to understand if it is possible to provide a holographic interpretation also for the YB deformations of $AdS_3\\times S^3\\times T^4$ considered here. (The connection to YB models was also pointed out the recent paper~\\cite{Araujo:2018rho}.) We expect our marginal deformations of the WZW model to correspond to deformations of the dual CFT$_2$ which generalise the (single trace version of the) $T\\bar T$ construction. \nIt would be very interesting to understand for example the case of the non-abelian $R$-matrix $\\mathbf R_9$, which gives rise to the marginal deformation $aJ_0\\bar J_2+bJ_1\\bar J_0+cJ_1\\bar J_2+J_+\\bar J_-$. The non-abelianity of the generators involved forbids the usual iteration of the infinitesimal deformation in order to obtain the exact one. The YB deformation, despite the non-abelianity, provides the realisation of the finite deformation on the worldsheet of the string.\n\n\n\\section*{Acknowledgements}\nWe thank  R. Conti, B. Hoare and D. Thompson for useful discussions. LW thanks the participants of the workshop ``A fresh look at $AdS_3\/CFT_2$'' in Villa Garbald, Castasegna, for stimulating discussions. The work of RB is supported by the Maria de Maeztu Unit of Excellence MDM-2016-0692, by FPA2017-84436-P, by Xunta de Galicia (ED431C 2017\/07), and by FEDER. His work was supported by the ERC advanced grant No 341222 while at Nordita.\n\n\\vspace{2cm}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA widely studied class of disordered systems in statistical physics\nconsist in adding random impurities to a field coupled with the order\nparameter. A textbook example of such a system is the random field Ising model (RFIM), introduced by \\cite{Larkin},  that has been a very useful playground for theoretical ideas. \nThe Hamiltonian of the standard RFIM reads\n\\begin{equation}\n{\\cal H} = - \\sum_{i<j} J_{ij} S_i S_j + \\sum_i h_i S_i \\; ,\n\\label{Hamiltonian}\n\\end{equation} \nwhere all the non-zero interactions are ferromagnetic, i.e. $J_{ij} \\ge 0$. The $N$ Ising spins $S_i=\\pm1$, $i=1,\\dots,N$, are placed at the\nvertices of a graph (lattice), and $h_i$ are quenched random\nmagnetic fields. The fact that all the interactions $J_{ij}$ are non-negative is fundamental, it means that in the absence of the fields there is no explicit frustration in the problem.\n\nThe case where the graph of interactions is a finite-dimensional lattice and where the fields are taken from a Gaussian distribution with zero mean and a\nvariance $H_R$ has received a lot of attention. Of particular interest\nis the phase diagram in the $T$-$H_R$ plane, where $T$ is the\ntemperature. Several authors have suggested, based on non-rigorous field\ntheoretic arguments, that there exists an equilibrium spin glass phase\nin the three-dimensional RFIM, that is a phase with a random frozen ordering\n\\cite{intermediate,intermediate2,CyranoBrezin,CyranoBrezin2,CyranoBrezin3}.\nThese suggestions were disproved rigorously in \\cite{KrzakalaRicci10}\nfor the RFIM defined by Hamiltonian (\\ref{Hamiltonian}). In particular\n\\cite{KrzakalaRicci10} showed that for the RFIM (\\ref{Hamiltonian}) a special case of the Fortuin-Kasteleyn-Ginibre (FKG) inequality \\cite{FKG} implies that the spin glass susceptibility is upper-bounded by the ferromagnetic\nsusceptibility. Since the spin glass susceptibility diverges in the\nwhole spin glass phase, a spin glass phase can not exist away from the\nferromagnetic critical point\/line in the RFIM.\n\nThe field theoretic approach of\n\\cite{intermediate,intermediate2,CyranoBrezin,CyranoBrezin2,CyranoBrezin3},\nhowever, was not formulated with the Ising spin Hamiltonian\n(\\ref{Hamiltonian}) but instead with the soft-spin\ndescription of the random field model. This is the well-known\nGinzburg-Landau model (or the so-called\n$\\phi^4$-theory) which is defined by the following Hamiltonian\n\\begin{equation}\n{\\cal H}_N = - \\sum_{ij} J_{ij} \\phi_i \\phi_j - \\sum_i h_i \\phi_i\n+ \\sum_i r_i \\phi_i^2 + \\sum_i u_i \\phi_i^4 \\; ,\n\\label{Ham_LG}\n\\end{equation}\nwhere $\\phi_i$ are now real numbers, $\\phi_i \\in \\mathbb{R}$, and the interactions are ferromagnetic, $J_{ij} \\ge 0$ (this will be the case in the whole article).\n\nThe generalized model (\\ref{Ham_LG}) includes several special cases. \nThe Ising model is recovered in the limit where $r_i=-2u_i$ and $u_i\\to \\infty$.\nThe most common {\\it random field model} is obtained when $h_i$ are random variables while $r_i=r$ and $u_i=u$ are fixed, and $J_{ij}=0$ or $J_{ij}=1$ depending if the spins $ij$ interact or not. Another version that was considered in the literature, the {\\it random temperature model}, is when $r_i$ are random while $h_i=0$, $u_i=u$ and $J_{ij}\\in \\{0,1\\}$. The\nexistence of a spin-glass phase was also\npredicted in the random temperature model \\cite{MaRudnick,Tarjus02}, based again on some non-rigorous arguments using\nperturbation theory; this result was, however, questioned in \\cite{Sherrington90}.\n\nOur results work even for the slightly more general Hamiltonian\n\\begin{equation}\n{\\cal H}_N = - \\sum_{ij} J_{ij} \\phi_i \\phi_j + \\sum_i f_i(\\phi_i) \\; ,\\label{Ham_gen}\n\\end{equation}\nwhere $J_{ij} \\ge 0$ and the local constraining potentials $f_i()$ are arbitrary analytic functions, but for the requirement that the partition function\n\\begin{equation}\nZ_{N} = \\int_{-\\infty}^\\infty \\prod_{i=1}^N {\\rm d}\\phi_i\\;\ne^{-\\beta {\\cal H}_{N}(\\{\\phi_i\\}) } \\label{Z_N} \\; .\n\\end{equation}\nmust exists for any $N \\in \\mathbb{N}$.\nThis is the most minimalist requirement, since the non-convergence of the integral in (\\ref{Z_N}) would make the Gibbs-Boltzmann measure ill defined and the model would not be a physical one.\n\nThe Gibbs-Boltzmann average at temperature $T=\\beta^{-1}$ is defined by\n\\begin{equation}\n\\langle A\\rangle^{(N)} = \\frac{1}{Z_N} \\int_{-\\infty}^\\infty \\prod_{i=1}^N {\\rm d}\\phi_i A\\, e^{-\\beta {\\cal H}_{N}(\\{\\phi_i\\}) } \\; ,\n\\end{equation}\nThe superscript $(N)$ on the angular brackets will be written explicitly only when the size dependence is crucial, while the temperature dependence is always made implicit.\nConnected correlation functions are defined as\n\\begin{equation}\n\\langle A\\,B\\rangle_c = \\langle A\\,B\\rangle - \\langle A\\rangle\n\\langle B\\rangle \\; .\n\\end{equation}\n\nIt is worth noticing that the convergence of the integral in (\\ref{Z_N}) ensures that the partition function $Z_N$ is an analytic function of the coupling constants $J_{ij}$ for any finite value of $N$. Then the derivative\n\\begin{equation}\n\\beta^{-1} \\frac{\\partial \\ln Z_N}{\\partial J_{ij}} =\n\\langle \\phi_i \\phi_j \\rangle\n\\end{equation}\nexists as well for any pair of indices $i,j$, and this implies that single-variable marginal probability distributions have the first and the second moment, $\\langle \\phi_i \\rangle$ and $\\langle \\phi_i^2 \\rangle$.\nActually in soft-spin models used in the literature, such as the spherical model and the $\\phi^4$ model, single-variable marginal probabilities decay exponentially fast for large values of $\\phi_i$, and so all the moments $\\langle \\phi_i^k \\rangle$ exist. However, our proof only requires the first two moments to exist.\n\nNote also that any type of lattice can be encoded in model (\\ref{Ham_gen}) by setting $J_{ij}=0$ if spins $i$ and $j$ do not interact. \n\nThe main contribution of this paper is a rigorous proof that the soft-spin random-field random-temperature model defined by (\\ref{Ham_gen}) does not have a spin glass phase as long as the interactions are ferromagnetic (non-negative). This generalizes the result of \\cite{KrzakalaRicci10}. \n\n\n\\section{Definitions of susceptibilities}\n\nWe define the ferromagnetic and the spin glass phases using the\nproperties of the ferromagnetic and spin glass susceptibilities. \n\nThe order parameter that characterizes a ferromagnetic transition is\nthe magnetization $m= \\sum_i \\langle \\phi_i\\rangle\/N $. However, a non-zero\nmagnetization does not imply a ferromagnetic phase. Indeed, $m>0$ even at\nlarge temperatures when a uniform positive external magnetic field is applied. A convenient way to characterize the ferromagnetic phase is to\ndefine the ferromagnetic susceptibility as\n\\begin{equation}\n\\chi^0_F(N) = \\frac{1}{N} \\sum_{ij} \\langle \\delta\\phi_i\\, \\delta\\phi_j\\rangle \\; ,\n\\end{equation}\nwhere\n\\begin{equation}\n\\delta\\phi_i =\n\t\\frac{\\phi_i - \\langle \\phi_i \\rangle}\n\t{\\sqrt{\\langle \\phi_i^2 \\rangle - \\langle \\phi_i \\rangle^2}}\\;,\n\\label{fluct}\n\\end{equation}\nare the fluctuations with respect to the average values, normalized by the variances.\n\nIn the thermodynamic limit ($N \\to \\infty$), $\\chi^0_F(\\infty)$ is finite in the high temperature ($T > T_c$) paramagnetic phase and it diverges approaching the ferromagnetic critical point from above ($T \\searrow T_c$).\nRight at the critical point ($T=T_c$), $\\chi^0_F(N)$ diverges with $N\\to \\infty$ signaling that the system is critical, i.e.\\ has long range correlations between fluctuations of its variables. Unfortunately, the ferromagnetic susceptibility $\\chi^0_F(N)$ diverges with $N$ also in the whole low temperature ($T<T_c$) ferromagnetic phase: however this divergence is not due to criticality (i.e.\\ long range correlation of fluctuations), but only because below $T_c$ two ferromagnetic states coexist\\footnote{In the presence of two or more equivalent states, an appropriately chosen perturbation, although of infinitesimal strength, may induce a macroscopic change of state, thus leading to an infinite susceptibility.}.\n\nGiven that we are interested in finding critical points and critical lines where a phase transition takes place, we would like to measure an observable that diverges only at criticality, and so we consider the following {\\em ferromagnetic susceptibility}\n\\begin{equation}\n\\chi_F = \\lim_{h \\searrow 0} \\lim_{N \\to \\infty} \\chi_F(h,N)\n\t  = \\lim_{h \\searrow 0} \\lim_{N \\to \\infty} \\frac{1}{N}\n\t  \\sum_{ij} \\langle \\delta\\phi_i\\, \\delta\\phi_j \\rangle \\, ,\n\\label{chi_ferro}\n\\end{equation}\nwhere $h$ is an auxiliary uniform magnetic field (in practice one needs to add a term $-h \\sum_i \\phi_i$ in the Hamiltonian).\nDue to the order of the limits in (\\ref{chi_ferro}), below $T_c$, the infinitesimal external field $h$ makes the two ferromagnetic states no longer equivalent, and consequently $\\chi_F$ is finite everywhere, but at the critical point $T_c$ (which is indeed defined as the point where $\\chi_F$ diverges).\n\nIn general to define a susceptibility that diverges only when a critical state is present one should explicitly break (by adding infinitesimal perturbations) all the symmetries of the Hamiltonian. In our case, the Hamiltonian (\\ref{Ham_gen}) is very general, but the first term is invariant under the transformation $\\phi_i \\to -\\phi_i\\; \\forall i$. In case the potentials too are invariant under such a transformation, $f_i(\\phi)=f_i(-\\phi)$, then the infinitesimal auxiliary uniform field in (\\ref{chi_ferro}) is strictly required.\n\nThe spin glass phase is characterized by a freezing of spins in\nrandom directions \\cite{EA}, hence the {\\em spin glass susceptibility} is defined as\n\\begin{equation}\n\\chi_{SG} = \\lim_{h \\searrow 0} \\lim_{N \\to \\infty} \\chi_{SG}(h,N)\n\t   = \\lim_{h \\searrow 0} \\lim_{N \\to \\infty} \\frac{1}{N}\n\t   \\sum_{ij} \\langle \\delta\\phi_i\\, \\delta\\phi_j \\rangle^2 \\,.\n\\label{chi_SG}\n\\end{equation}\nAgain we use the infinitesimal auxiliary external field to break the $\\phi \\to -\\phi$ symmetry, if present.\nThe susceptibility $\\chi_{SG}$ is related closely to what is measured in simulations and experiments \\cite{HERTZ}, and it is predicted to diverge at the critical point in spin glass theories such as replica symmetry breaking \\cite{MF}, or the droplet description \\cite{DROPLET}.\nIn a spin glass phase $\\chi_{SG}$ is infinite, because of the presence of at least two states\\footnote{The number of states depends on the model and for some models, like the 3D Edwards-Anderson model, it is still a matter of debate.} related by symmetries, which are not broken by the auxiliary field.\nFor this reason we can define that a system is in a spin glass phase {\\it if and only if} the ferromagnetic susceptibility (\\ref{chi_ferro}) is finite, while the spin glass susceptibility (\\ref{chi_SG}) is infinite.\n\nMore precisely the computation of these two susceptibilities must proceed by first taking the thermodynamic limit in the presence of the external field, $\\chi_F(h,\\infty)$ and $\\chi_{SG}(h,\\infty)$, and then studying the limit $h \\searrow 0$ of these two functions. If such a limit exists, then we say that the susceptibility is finite and we are away from the critical point, while if a divergence is found while decreasing $h$, then we say that the susceptibility is infinite.\n\nIn the next Section we prove that $\\chi_{SG}(h,N) \\le \\chi_F(h,N)$, for any value of $h$ and $N$, thus excluding the possibility of a spin glass phase (defined by $\\chi_F<\\infty$ and $\\chi_{SG} \\to \\infty$) in the model (\\ref{Ham_gen}) in the absence of explicit frustration in the couplings.\n\n\n\\section{Results}\n\nWe start by proving a generalization of the 2nd Griffith's inequality \\cite{Griffiths67b}. The following Lemma is also a consequence of much more general FKG inequalities \\cite{FKG,FKG_gen}, we, however, find useful to present an independent and more elementary proof. \n\n\\begin{lemma}\nIn the model defined by the Hamiltonian (\\ref{Ham_gen}) with non-negative couplings, $J_{ij} \\ge 0 \\; \\forall i,j$, under the Gibbs-Boltzmann measure $e^{-\\beta {\\cal H}_N}\/Z_N$ the correlation between fluctuations of any two variables is non-negative and bounded by 1,\n\\begin{equation}\n0 \\le \\langle \\delta\\phi_i\\, \\delta\\phi_j \\rangle \\le 1 \\qquad \\forall i,j\\;. \n\\label{eq_lemma_FKG}\n\\end{equation}\n\\label{lemma_FKG}\n\\end{lemma}\n\n\\begin{proof}[Lemma \\ref{lemma_FKG}]\nLet us prove first the second inequality in (\\ref{eq_lemma_FKG}).\nFrom the definition (\\ref{fluct}) of the relative fluctuations we have that $\\langle \\delta\\phi_i^2 \\rangle = 1$ for any $i$.\nMoreover for any pair of indices $i,j$ we have that\n\\begin{equation}\n0 \\le \\langle (\\delta\\phi_i - \\delta\\phi_j)^2 \\rangle =\n\\langle \\delta\\phi_i^2 \\rangle + \\langle \\delta\\phi_j^2 \\rangle\n- 2 \\langle \\delta\\phi_i\\, \\delta\\phi_j \\rangle =\n2 (1 - \\langle \\delta\\phi_i\\, \\delta\\phi_j \\rangle)\n\\end{equation}\nfrom which $\\langle \\delta\\phi_i\\, \\delta\\phi_j \\rangle \\le 1$ follows.\n\nIn order to prove the first inequality in (\\ref{eq_lemma_FKG}) we notice that it is equivalent to the inequality\n\\begin{equation}\n\\langle \\phi_i \\phi_j \\rangle_c \\ge 0\\;,\n\\label{simpler}\n\\end{equation}\nthanks to the fact that all denominators in the definition (\\ref{fluct}) of $\\delta\\phi_i$ are positive and can be canceled without changing the sign of the correlation.\n\nThen we prove (\\ref{simpler}), by induction in the system size. In a system of $N=1$ spin \n\\begin{equation}\n\\langle \\phi_1^2 \\rangle^{(1)}_c \\ge 0 \\, ,\n\\end{equation} \nsince the variance is always non-negative. \nThen we assume the property to hold in a system of $N$ spins and we consider a system with $N+1$ spins. The Hamiltonian of that system is related to the $N$-spin system as\n\\begin{equation}\n      {\\cal H}_{N+1}   = {\\cal H}_{N} - \\sum_{i=1}^N J_{N+1,i} \\phi_{N+1} \\phi_i +f_{N+1}(\\phi_{N+1})  \\, .\n\\end{equation}\nWe denote \n\\begin{eqnarray}\n     P(x) = \\frac{1}{Z_{N+1}}   \\int_{-\\infty}^\\infty \\prod_{i=1}^N\n     {\\rm d}\\phi_i  \\hspace{7cm} \\\\ \\exp{\\left[-\\beta\n         {\\cal H}_{N}(\\{\\phi_i\\}) + \\beta \\sum_{i=1}^N J_{N+1,i} x \\phi_i\n         - \\beta f_{N+1}(x) \\right]}\\, . \\nonumber\n\\end{eqnarray}\nLet us denote the thermodynamic average in a modified external magnetic field as\n\\begin{equation}\n\\langle A \\rangle_x^{(N)}    =  \\frac{\\int \\prod_{i=1}^N {\\rm d}\\phi_i   A \\, e^{-\\beta {\\cal H}_{N}(\\{\\phi_i\\}) + \\beta \\sum_{i=1}^N J_{N+1,i} x \\phi_i   }}{\\int \\prod_{i=1}^N {\\rm d}\\phi_i \\,   e^{-\\beta {\\cal H}_{N}(\\{\\phi_i\\}) + \\beta \\sum_{i=1}^N J_{N+1,i} x \\phi_i  }}\\, .\n\\end{equation}\nThe connected correlation between spins $\\phi_{N+1}$ and $\\phi_i$ in the $N+1$-spin system can then be rewritten as\n\\begin{eqnarray}\n     \\langle \\phi_{N+1} \\phi_i \\rangle_c^{(N+1)} &=& \\int_{-\\infty}^\\infty  {\\rm d}x  \\,  x \\, P(x)   \\langle \\phi_i \\rangle_x^{(N)}   -   \\int_{-\\infty}^\\infty  {\\rm d}y \\,   y \\, P(y)  \\int_{-\\infty}^\\infty  {\\rm d}x\\,  P(x)   \\langle \\phi_i \\rangle_x^{(N)} \\nonumber \\\\  & =&   \\int_{-\\infty}^\\infty  {\\rm d}x  \\,  \\left[ x -  \\int_{-\\infty}^\\infty  {\\rm d}y \\,   y \\, P(y)   \\right]\\, P(x)   \\langle \\phi_i \\rangle_x^{(N)} \\, .\n\\end{eqnarray}\nWe can then use the following inequality: For any real non-decreasing function $g(x)$ such that \n\\begin{equation}\n     \\int_{-\\infty}^\\infty  {\\rm d}x  \\, g(x) = 0\\, ,\n\\end{equation} \nand any non-decreasing function $f(x)$ one has\n\\begin{equation}\n     \\int_{-\\infty}^\\infty  {\\rm d}x \\,  g(x) \\, f(x) \\ge 0 \\, . \\label{lemma2}\n\\end{equation}\nProof of this statement is elementary, function $g(x)$ has to be non-positive for some $x \\le x_0$ and non-negative for $x \\ge x_0$. Since $f(x)$ is non-decreasing one has \n\\begin{equation}\n    \\int_{-\\infty}^{x_0}  {\\rm d}x \\,  |g(x)| \\, f(x) \\le   \\int_{x_0}^\\infty  {\\rm d}x \\,  |g(x)| \\, f(x) \\, ,\n\\end{equation}\nfrom which (\\ref{lemma2}) follows.\nWe observe that $\\langle \\phi_i \\rangle_x^{(N)}$ is a non-decreasing function of $x$. Indeed\n\\begin{equation}\n        \\frac{{\\rm d} \\langle \\phi_i \\rangle_x^{(N)}}{{\\rm d}x} = \\beta \\sum_{j=1}^N J_{N+1,j}  \\langle \\phi_j \\phi_i \\rangle_c^{(N)} \\ge 0\\, ,\n\\end{equation}\nwhere the last inequality follows from the inductive assumption and since $J_{N+1,j}\\ge 0$.\nAnd that\n\\begin{equation}\n     \\int_{-\\infty}^\\infty  {\\rm d}x  \\,  \\left[ x -  \\int_{-\\infty}^\\infty  {\\rm d}y \\,   y \\, P(y)   \\right]\\, P(x)  =0\\, .\n\\end{equation}\nHence (\\ref{lemma2}) implies\n\\begin{equation}\n   \\langle \\phi_{N+1} \\phi_i \\rangle_c^{(N+1)} \\ge 0\\, .\n\\end{equation} \n\nWe proceed similarly for the correlation function between two spins that were already present in the $N$-spin system\n\\begin{eqnarray}\n    \\langle \\phi_i \\phi_j \\rangle_c^{(N+1)}   &=&\n    \\int_{-\\infty}^\\infty  {\\rm d}x  \\, P(x)   \\langle \\phi_i\n    \\phi_j \\rangle_x^{(N)} -  \\int_{-\\infty}^\\infty  {\\rm d}x  \\,\n    P(x)   \\langle \\phi_i \\rangle_x^{(N)} \\int_{-\\infty}^\\infty\n    {\\rm d}y  \\, P(y)   \\langle \\phi_j \\rangle_y^{(N)} \\nonumber \\\\ &=& \n\\int_{-\\infty}^\\infty  {\\rm d}x  \\, P(x)   \\langle \\phi_i \\phi_j \\rangle_{x,c}^{(N)}  + \\nonumber \\\\ && \\int_{-\\infty}^\\infty  {\\rm d}x  \\, P(x)   \\langle \\phi_i \\rangle_x^{(N)}\\left[  \\langle \\phi_j \\rangle_x^{(N)}   -  \\int_{-\\infty}^\\infty  {\\rm d}y  \\, P(y)   \\langle \\phi_j \\rangle_y^{(N)} \\right]\n\\end{eqnarray}\nwhere the first term is non-negative by the inductive assumption, and the second term is non-negative according to (\\ref{lemma2}), because $\\langle \\phi_i \\rangle_x^{(N)}$ and $\\langle \\phi_j \\rangle_x^{(N)}$ are non-decreasing functions of $x$ and \n\\begin{equation}\n    \\int_{-\\infty}^\\infty  {\\rm d}x  \\, P(x)  \\left[  \\langle \\phi_j \\rangle_x^{(N)}   -  \\int_{-\\infty}^\\infty  {\\rm d}y  \\, P(y)   \\langle \\phi_j \\rangle_y^{(N)} \\right] = 0\\, .\n\\end{equation}\nHence\n\\begin{equation}\n   \\langle \\phi_i \\phi_j \\rangle_c^{(N+1)}   \\ge 0.\n\\end{equation} \nThis concludes the proof of Lemma \\ref{lemma_FKG}. \\qed\n\\end{proof}\n\nBased on the previous Lemma we can now easily state the main result of the present paper.\n\n\\begin{theorem}\nIn the model defined by the Hamiltonian (\\ref{Ham_gen}) with non-negative couplings, $J_{ij} \\ge 0 \\; \\forall i,j$, under the Gibbs-Boltzmann measure $e^{-\\beta {\\cal H}_N}\/Z_N$, the spin glass susceptibility $\\chi_{SG}(h,N)$ is always upper-bounded by the ferromagnetic susceptibility $\\chi_F(h,N)$. Consequently the model does not posses a thermodynamic spin glass phase.\n\\label{main_lemma}\n\\end{theorem}\n\\begin{proof}[Theorem \\ref{main_lemma}]\nThe hypothesis of the present Theorem are the same as those of Lemma \\ref{lemma_FKG}, with the only difference that to properly define the susceptibilities we need to add the external auxiliary field term to the original model Hamiltonian.\nThen Lemma \\ref{lemma_FKG} can be used only if\n\\begin{equation}\nZ_N(h) = \\int_{-\\infty}^\\infty \\prod_{i=1}^N {\\rm d}\\phi_i\\;\ne^{-\\beta {\\cal H}_{N}(\\{\\phi_i\\}) + \\beta h \\sum_i \\phi_i}\\;\n\\end{equation}\nexists also for $h > 0$, and this is easy to prove. Indeed $Z_N(0)$ exists (otherwise the Gibbs-Boltzmann measure would be ill-defined) and also the first two derivatives of $Z_N(h)$ with respect to $h$ exist (because $\\langle \\phi_i \\rangle$ and $\\langle \\phi_i^2 \\rangle$ exist): so $Z_N(h)$ can be continued in a neighborhood on $h=0$, that we call $S_0$, and this is enough to take the limit $h \\searrow 0$ that is required to define properly the susceptibility. Please note that the region $S_0$ coincide with $\\mathbb{R}$ for all the models used in the literature, such as the spherical model and the $\\phi^4$ model.\n\nGiven that the hypothesis of Lemma \\ref{lemma_FKG} are satisfied in $S_0$, we can make use of inequalities in (\\ref{eq_lemma_FKG}) and find that\n\\begin{equation}\n\\langle \\delta\\phi_i \\delta\\phi_j \\rangle^2 \\le\n\\langle \\delta\\phi_i \\delta\\phi_j \\rangle \\quad \\Longrightarrow \\quad\n\\chi_{SG}(h,N) \\le \\chi_F(h,N)\n\\label{main}\n\\end{equation}\nfor any value of $N$ and $h \\in S_0$.\nEven in the thermodynamic limit the inequality holds\n\\begin{equation}\n\\chi_{SG}(h,\\infty) \\le \\chi_F(h,\\infty)\n\\end{equation}\nand so the spin glass susceptibility can not diverge if the ferromagnetic one stays finite.\n\nIn other words, from the definitions given in the previous Section it is clear that if a thermodynamic spin glass phase exists, then for a sufficiently large value of $N$ and a sufficiently small value of $h$ the spin glass susceptibility must be larger than the ferromagnetic one and this would violate the inequality in (\\ref{main}). Then we conclude that a thermodynamic spin glass phase does not exists in the model defined in the hypothesis. \\qed\n\\end{proof}\n \n \n\\section{Discussion}\n\nWe have shown rigorously that there is no spin glass phase in the scalar soft-spin random-field random-temperature Ginzburg-Landau model with ferromagnetic interactions defined by (\\ref{Ham_gen}).\nThis shows that with two-body interactions and a scalar order\nparameter one cannot obtain a genuine spin glass phase at\nequilibrium without {\\it explicit frustration} in the couplings\n(another possibility to frustrate the system is to impose a non-equilibrium value of magnetization, see \\cite{KrzakalaRicci10}).\n\nOur proof contradicts the conclusions of some works that used field theoretic arguments\n\\cite{intermediate,intermediate2,CyranoBrezin,CyranoBrezin2,CyranoBrezin3,MaRudnick,Tarjus02}.\nIt is yet to be discovered where the problem lies in those approaches.  \nOne possibility to consider is that the spin glass instability could be an artifact of some truncation in the perturbative expansion. For some of\nthese works the discrepancy may stem from the use of vectorial soft-spin models instead of scalar ones.\nAnother possibility, that is related to what was suggested recently in \\cite{TarjusRecent}, is that the observed \"replica symmetry\nbreaking\" instabilities arise only in disorder averaged quantities and never in the thermodynamic limit of a single instance quantities. These instabilities would then not be equivalent to the divergence of the spin glass susceptibility (which we prove impossible out of the ferromagnetic critical point), but they could instead be connected to some subtle non-self-averaging effects between different realizations of the system. Indeed all the above-mentioned works considered a \"replicated\" field theory, that is, a field theory averaged over\nmany realizations of the disorder. The divergences that they observed could hence be coming from strong sample to sample fluctuation. \nThe fact that some non-self-averaging is present in the RFIM has been suggested by Parisi and Sourlas \\cite{P-S}. They argue that the correlation function, or equivalently the ferromagnetic susceptibility, of the RFIM is non self-averaging in the critical region, and they argue that this was the source of the problems with perturbative expansions. Note, however, that such simple non-self-averaging effects can {\\it only}\ntake place {\\it at the ferromagnetic critical point} in any finite\ndimensional system. This is a straightforward\nconsequence of a theorem by Wehr and Aizenman \\cite{AW} who proved\nthat any extensive quantities (such as the ferromagnetic\nsusceptibility away from the critical point) is self-averaging in finite dimensional systems. In other words, if this effect was the one observed in the field theoretic approaches, it has to be limited to the ferromagnetic critical point\nitself.\n\nFinally, it would be very interesting to see if our proof can be generalized further. There are two interesting\ncounter-examples that seem to put strong limits to such\ngeneralizations. Matsuda and Nishimori (private communication) showed that a random field Ising model\nwith 3-spins interactions on the\nBethe lattice can have a spin glass phase. And so moving beyond pairwise interacting models seems impossible in full generality.\nMoreover Parisi (private communication) provided an interesting example of a\npairwise interacting $n=2$ component vector spin system where the two point connected correlation can be negative even if all couplings are positive.\nIt is a chain of spins with an external field that smoothly rotates by 180 degrees along the chain, such that the field on the last spin is opposite to field on the first spin.\nIf the field strength is strong enough, each spin will be mostly aligned along the local field and will thermally fluctuate around this position. \nHowever, given that the extremal spins are in opposite directions, their thermal fluctuations will be negatively correlated.\nThis is a very specific configuration which may not happen in typical samples, but its existence implies that the proof strategy presented in this paper cannot be straightforwardly generalized to vector spin models.\n\n\n\\vspace{5mm}\n\n\\noindent {\\bf Acknowledgment:} We thank G. Parisi, H. Nishimori, F. Toninelli and  for very useful comments and enlightening discussions. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\begin{figure*}[ht!]\n\\centering\n\\begin{tikzpicture}\n    \\node[rectangle, inner sep=10pt, draw, text width=1cm] at (-1, 1) (GD Teacher){GD Teacher};\n    \\node[rectangle, inner sep=5pt, draw] at (-1, -2) (GD Baseline){GD Baseline}; \n    \\node[rectangle, inner sep=5pt, draw] at (-1, -3) (GD Student){GD Student}; \n    \n    \\node[rectangle, inner sep=10pt, draw, text width=1cm] at (3.5, 1) (Baseline Teacher) {Baseline Teacher};\n    \\node[rectangle, inner sep=10pt, draw, text width=1cm] at (5.5, 1) (Adapted Teacher) {Adapted Teacher};\n    \n    \\node[rectangle, inner sep=5pt, draw] at (1.5, -1) (1){1}; \n    \\node[rectangle, inner sep=5pt, draw] at (3.5, -1) (2){2}; \n    \\node[rectangle, inner sep=5pt, draw] at (5.5, -1) (3){3}; \n    \\node[rectangle, inner sep=5pt, draw] at (1.5, -2) (4){4}; \n    \\node[rectangle, inner sep=5pt, draw] at (3.5, -2) (5){5}; \n    \\node[rectangle, inner sep=5pt, draw] at (5.5, -2) (6){6}; \n    \\node[rectangle, inner sep=5pt, draw] at (1.5, -3) (7){7}; \n    \\node[rectangle, inner sep=5pt, draw] at (3.5, -3) (8){8}; \n    \\node[rectangle, inner sep=5pt, draw] at (5.5, -3) (9){9}; \n    \n    \\draw[dotted] (0.5, 1.7) -- (0.5, -3.4);\n    \\node at (1.5, 1) {In-Domain};\n\n    \\draw[->] (GD Teacher) .. controls (1.5, 2) and (5.5, 2) .. (Adapted Teacher.north);\n    \\draw[->, dashed] (GD Teacher.west) .. controls (-3, -1.5) and (-3,-3) .. (GD Student.west);\n    \n    \\draw[->, dashed] (Baseline Teacher) -- (2.north);\n    \\draw[->, dashed] (Baseline Teacher.south) .. controls (4.5, -1) and (4.5, -2) .. (5.east);\n    \\draw[->, dashed] (Baseline Teacher.south) .. controls (4.5, -1) and (4.5, -2) .. (8.east);\n    \n    \\draw[->, dashed] (Adapted Teacher) -- (3.north);\n    \\draw[->, dashed] (Adapted Teacher.south) .. controls (6.5, -1) and (6.5, -2) .. (6.east);\n    \\draw[->, dashed] (Adapted Teacher.south) .. controls (6.5, -1) and (6.5, -2) .. (9.east);\n\n    \\draw[->] (GD Baseline) -- (4.west);\n    \\draw[->] (3.5, -2.5) -- (5.south);\n    \\draw[->] (GD Baseline.east) .. controls (1.5, -2.5) .. (3.5, -2.5) -- (5.5, -2.5) -- (6.south);\n    \n    \\draw[->] (GD Student) -- (7.west);\n    \\draw[->] (3.5, -3.5) -- (8.south);\n    \\draw[->] (GD Student.east) .. controls (1.5, -3.5) .. (3.5, -3.5) -- (5.5, -3.5) -- (9.south);\n    \n\\end{tikzpicture}\n\\caption{There are 9 possible configurations for training small, in-domain models with knowledge distillation and domain adaptation. Models trained on general-domain data are shown on the left, and in-domain models are shown on the right. Solid arrows represent domain adaptation via continued training. Dashed arrows represent improved optimization via sequence-level knowledge distillation. Configuration 1 is the model which is trained  on in-domain data with random initializations and without the assistance of a teacher. }\n\\label{fig:configs}\n\\end{figure*}\n\n\nMachine translation systems rely on large amounts of data to deduce the rules underlying translation from one language to another. This presents challenges in some important niche domains, such as patent and medical literature translation, due to the high cost of hiring experts to generate suitable training data. A cost-effective alternative is \\emph{domain adaptation}, which leverages large amounts of parallel documents from less difficult and readily-available domains, such as movie subtitles and news articles.\n\nDomain adaptation works well in practice. However, these large datasets, which we call \\emph{general domain} datasets, introduce some scalability problems. Large datasets require large models; neural machine translation systems can take days or weeks to train. Some models require gigabytes of disk space, making deployment to edge computing devices challenging. They can also require excessive compute during inference, making them slow and costly to scale up in production environments.\\cite{gordon_2019}\n\nTo alleviate these issues, \\emph{knowledge distillation} (aka Teacher-Student) \\cite{Hinton2015-on} is used to compress models into a manageable form. But although knowledge distillation is the most commonly used form of model compression in practice, it is also one of the least understood.\n\nIn this work, we perform a \\textbf{large-scale empirical analysis} to attempt to discover best practices when using knowledge distillation in combination with domain adaptation. Out of several common-sense configurations, we find that two stages of knowledge distillation give the best performance: one using general-domain data and another using in-domain data with an adapted teacher. We perform experiments on multiple language pairs (Russian-English, German-English, Chinese-English), domains (patents, subtitles, news, TED talks), and student sizes.\n\n\\section{Background}\n\n\\paragraph{Domain Adaptation} helps overcome a lack of quality training data in niche domains by leveraging large amounts of data in a more accessible general-domain. Domain adaptation is usually accomplished by \\emph{continued training} \\cite{Luong2015-ym,Zoph2016-pt}, which involves two steps:\n\n\\begin{enumerate}\n    \\item A model is randomly initialized and trained until convergence on the general-domain data.\n    \\item A new model is initialized with the parameters resulting from Step 1 and trained until convergence on the in-domain dataset.\n\\end{enumerate}\n\nWe can consider domain adaptation as extracting a useful \\emph{inductive-bias} from the general-domain dataset, which is encoded and passed along to the in-domain model as a favorable weight initialization. While there are other methods of extracting inductive bias from general-domain datasets (including mixed fine-tuning \\cite{Chu2017-db} and cost weighting \\cite{Chen2017-vw}), continued training is most common and the focus of this paper. \n\n\\paragraph{Knowledge Distillation} is a method for improving the performance of under-parameterized ``Student\" models by exploiting the probability distribution of a more computationally complex ``Teacher\" network. Kim and Rush \\shortcite{Kim2016-st} presented an extension of knowledge distillation to machine translation in two flavors: word-level and sequence-level knowledge distillation.\n\n \\emph{Sequence-level knowledge distillation}, which is more general, involves three steps:\n \n \\begin{enumerate}\n     \\item A large Teacher network is randomly initialized and trained until convergence on the data.\n     \\item The source-side of the training data is decoded using the Teacher to produce ``distilled\" target data.\n     \\item A smaller Student model is randomly initialized and trained until convergence on the distilled source-target pairs (discarding the original target sequences in the data).\n \\end{enumerate}\n\nThe goal of knowledge distillation is to train the student model to mimic the teacher's probability distribution over translations. Since the teacher and the student are trained on the same dataset, they should be capable of learning the same distribution in theory. In practice, however, using the teacher as an additional training signal improves student test performance.\\footnote{Interestingly, this can be true even when the student has the same computational resources as the teacher \\cite{Furlanello2018-lp}} Explanations for this phenomenon include dark knowledge \\cite{Furlanello2018-lp}, mode reduction \\cite{Zhou2019-wp}, and regularization \\cite{Gordon2019-qs,Dong2019-ve}, but no definitive evidence has been given.\n\n Sequence-level knowledge distillation is widely used in both industry \\cite{Xia2019-iq} and research and is the second focus of this paper. \\footnote{Sequence-level knowledge distillation is also commonly used to train non-autoregressive machine translation models \\cite{Zhou2019-wp}.}\n\n\\section{Distilling and Adapting}\n\nHow domain adaptation and knowledge distillation would interact when applied in combination was not previously clear. Specifically, our research questions are:\n\n\\begin{itemize}\n    \\item Is a distilled model easier or harder to adapt to new domains?\n    \\item Should knowledge distillation be used on in-domain data? If so, how should the teacher be trained?\n\\end{itemize}\n\nTo answer these questions, we performed experiments on 9 possible configurations which are assigned configuration numbers in Figure \\ref{fig:configs}. For ease of reference, we will primarily refer to small, in-domain models by their configuration number and encourage readers to consult Figure \\ref{fig:configs}. Each configuration has two attributes of interest. \n\n\\paragraph{Distilling In-Domain Data} How is in-domain data pre-processed using knowledge distillation? Some models are trained with no pre-processing (configurations 1, 4, and 7), while others use a teacher to pre-process the in-domain training data. This teacher might be a baseline trained on in-domain data only (configurations 2, 5, and 8) or it can be trained on general-domain data and then adapted to in-domain via continued training (configurations 3, 6, and 9).\n\n\\paragraph{Initialization} How are models initialized? A model might be randomly initialized (configurations 1, 2, and 3), or it might be adapted from a model trained on general-domain data. This general-domain model might be a baseline trained directly on the general-domain data (configurations 4, 5, and 6) or it might be a student model trained on the output of a general-domain teacher (configurations 7, 8, 9).\n\n\n\\section{Experiments}\n\\subsection{Data}\n\\paragraph{General-Domain Data}\nWe train models in multiple settings: 3 language pairs (German-English, Russian-English, and Chinese-English) each with 1 general-domain dataset and 2 different in-domain datasets. \nThe general-domain datasets for each language are a concatenation of data from OpenSubtitles2018 \n\\cite{Tiedemann2016-ob,Lison2016-kr} (which contains translated movie subtitles) and the WMT 2017 datasets \\cite{Ondrej2017-dw} (which includes a variety of sources, including news commentary, parliamentary proceedings, and web-crawled data).\n\n\\paragraph{In-Domain Data}\nWe use the World International Property Organization (WIPO) COPPA-V2 dataset \\cite{Junczys-Dowmunt2018-oh} and the TED Talks dataset \\cite{Qi2018-le} as our two in-domain datasets. The WIPO data contains parallel sentences from international patent abstracts, while the TED Talks dataset consists of translated transcripts of public speeches.\n\n\\paragraph{Data Statistics} The size of each training dataset is presented in Table \\ref{tab:data-sizes}. General-domain datasets contain tens of millions of sentences, while in-domain datasets contain much less. German-English WIPO has an exceptional amount of training data (4.5 times more than the next biggest in-domain dataset) and helps qualify how our results might change when more in-domain data is available.\n\n\\begin{table}[]\n    \\centering\n    \\begin{tabular}{lrrr}\n        Language & General-Domain & WIPO & TED\\\\\n        \\hline\n        De-En & 28.3 M & 821 k & 15 k\\\\\n        Ru-En & 51.1 M & 29 k & 180 k\\\\\n        Zh-En & 35.9 M & 154 k & 169 k\\\\\n        \n    \\end{tabular}\n    \\caption{The number of training sentences in each dataset.}\n    \\label{tab:data-sizes}\n\\end{table}\n\n\\paragraph{Pre-processing} All datasets are tokenized using the Moses\\footnote{\\href{http:\/\/statmt.org\/moses}{statmt.org\/moses}} tokenizer. A BPE vocabulary \\cite{Sennrich2016-an} of 30,000 tokens is constructed for each language using the training set of the general-domain data. This BPE vocabulary is then applied to both in-domain and general-domain datasets. This mimics the typical scenario of a single, general-domain model being trained and then adapted to new domains as they are encountered. Note that re-training BPE on in-domain data to produce a different vocabulary would force us to re-build the model, making adaptation impossible.\n\n\\paragraph{Evaluation} The general-domain development set for each language contains newstest2016 concatenated with the last 2500 lines of OpenSubtitles2018. We reserve 3000 lines of WIPO to use as the in-domain development set. TED talks development sets are provided by the authors and contain around 2000 lines each. Evaluations of each model are performed by decoding the appropriate development set with a beam-search size of 10 and comparing to the reference using multi-bleu.perl from the Moses toolkit.\n\n\\subsection{Architectures and Training}\n A list of architecture sizes is provided in Table \\ref{tab:sizes}. Teachers are trained using the Large hyper-parameter settings, while we experiment with Medium, Small, and Tiny students for each configuration and language\/domain setting.\n \n All models are Transformers \\cite{Vaswani2017-vu}. We use the same hyper-parameters (which are based on a template from \\cite{Duh_undated-uo}\\footnote{\\href{https:\/\/git.io\/JvL85}{https:\/\/git.io\/JvL85}}) for every model, except those that affect the size of the model (Table \\ref{tab:sizes}). Models are trained either for 300,000 updates, 100 epochs, or until the model does not improve for 10 checkpoints (early-stopping), whichever comes first.\n \n \\begin{table}[]\n     \\centering\n     \\begin{tabular}{lccc}\n          Size & Layers & FF Size & Hidden Size \\\\\n          \\hline\n          Large & 12 & 2048 & 512 \\\\\n          Medium & 6 & 2048 & 512 \\\\\n          Small & 6 & 1024 & 256 \\\\\n          Tiny & 2 & 1024 & 256 \\\\\n     \\end{tabular}\n     \\caption{Hyper-parameters of various model sizes used in this work. For example, the Large Transformer model architecture uses 6 encoder and 6 decoder layers, a feed-forward hidden dimension of 2048 at each layer, and a word-embedding \/ hidden dimension of 512.}\n     \\label{tab:sizes}\n \\end{table}\n \n \n\n\\paragraph{Continued Training} Work by \\cite{Gordon2019-qs} suggests that students may benefit from training on some combination of the distilled and un-distilled reference dataset. We experimented with this by continuing to train each in-domain student model on the original, un-distilled dataset, using similar stopping criterion to the first round of training. This improved some models by up to 1 BLEU. Because of this, we recommend that any distilled model continue training on the original dataset as long as development accuracy improves. When continued training improves performance of a student, we show that score instead of the score without continued training.\n\n\\section{Recommendations}\n\\subsection{Adapt Teachers}\n    \nIn this section, we compare training in-domain models with no teacher (config 1), a teacher trained on in-domain data only (config 2), and a teacher adapted from the general domain (config 3). The performance of the two teachers in each language-pair and domain is listed in Table \\ref{tab:id-teachers}. It shows that adaptation greatly improves the performance of every in-domain teacher except German-English WIPO.\\footnote{German WIPO is also the largest in-domain dataset we test, which might make adaptation unnecessary. Another explanation might be that the German-English general-domain is not similar enough to the patent domain in this case to improve performance.}\n\nTable \\ref{tab:id-students} shows the results of using these teachers to distill the in-domain data before training student models in various settings. \\textbf{We see that in almost every case, using an adapted teacher gives the best or close to the best results.} This is somewhat expected since models with better development scores tend to make better teachers \\cite{Zhang-2018ak}. Although knowledge distillation is typically seen as ``simplifying\" data for students, in this case we suspect that the adapted teacher's knowledge about the general-domain is making its way to students via the distilled in-domain data.\n    \\begin{table}\n    \\centering\n    \\begin{tabular}{lllccc}\n    Domain & Size & Adapted From & de-en & ru-en & zh-en\\\\\n    \n\\hline\nted & Large & None                       & 29.25 & 19.38 & 14.79\\\\\n &  & Large                      & \\color{ForestGreen}37.64 & \\color{ForestGreen}26.57 & \\color{ForestGreen}20.45\\\\\n\\hline\nwipo & Large & None                      & 48.31 & 21.36 & 31.02\\\\\n &  & Large                     & \\color{ForestGreen} 48.56 & \\color{ForestGreen}37.08 & \\color{ForestGreen}36.80\\\\\n\n    \\end{tabular}\n    \\caption{BLEU development score of in-domain teachers. Adaptation drastically improves performance on every language pair and domain, except de-en WIPO.}\n    \\label{tab:id-teachers}\n    \\end{table}\n\n    \\begin{table}\n    \\centering\n    \\begin{tabular}{llcccc}\n    Domain & Size & Config \\# & de-en & ru-en & zh-en\\\\\n    \n &  & 1                                  & 27.73 & 19.34 & 15.17\\\\\nted & medium & 2                         & 29.11 & 20.31 & 15.71\\\\\n &  & 3                                  & \\textbf{29.54} & \\textbf{20.56} & \\textbf{15.90}\\\\\n\\hline\n &  & 1                                  & 27.89 & 18.42 & 14.87\\\\\n & small & 2                             & 28.93 & 19.65 & 14.95\\\\\n &  & 3                                  & \\textbf{29.52} & \\textbf{19.88} & \\textbf{15.79}\\\\\n\\hline\n &  & 1                                  & 25.78 & 17.48 & 13.03\\\\\n & tiny & 2                              & 27.20 & 17.87 & 13.39\\\\\n &  & 3                                  & \\textbf{27.58} & \\textbf{19.27} & \\textbf{13.74}\\\\\n\\hline\n &  & 1                                  & 48.89 & 24.45 & 30.13\\\\\nwipo & medium & 2                        & \\textbf{50.66} & \\textbf{24.62} & 32.13\\\\\n &  & 3                                  & 50.23 & 24.60 & \\textbf{33.16}\\\\\n\\hline\n &  & 1                                  & 47.94 & 21.91 & 30.66\\\\\n & small & 2                             & 49.46 & \\textbf{23.70} & 32.19\\\\\n &  & 3                                  & \\textbf{49.72} & 23.50 & \\textbf{32.61}\\\\\n\\hline\n &  & 1                                  & 44.15 & 21.39 & 27.67\\\\\n & tiny & 2                              & 48.03 & \\textbf{22.24} & 28.18\\\\\n &  & 3                                  & \\textbf{48.51} & 22.03 & \\textbf{29.88}\\\\\n\n\n    \\end{tabular}\n    \\caption{BLEU development scores for in-domain students with no teacher (config 1), an in-domain only teacher (config 2), or an adapted teacher continued from the general-domain (config 3). In almost every case, using an adapted teacher gives the best or close to the best results. }\n    \\label{tab:id-students}\n    \\end{table}\n\n\n\n\\subsection{Adapt the Best Student}\n\nWe also train small models directly on the general-domain data and adapt them to in-domain data. The possible configurations are random initialization (config 1), initializing from a baseline model trained on general-domain data (config 4), or initializing from a student model distilled from a general-domain teacher (config 7). Table \\ref{tab:gd} shows the performance of these models on the general-domain datasets, and Table \\ref{tab:id-adapted} shows their performance on in-domain datasets.\n\nWhile adapting teachers gives modest gains on in-domain datasets (0-2 BLEU), training small models directly on the general-domain data gives much more substantial gains (5-10 BLEU). We believe this is because a large amount of data is required to fully reveal the teacher's probability distribution over translations \\cite{Fang2019-lw}. While an adapted teacher might contain much information from the general-domain, it is unable to express that knowledge to students just by translating the smaller in-domain dataset. \\textbf{To get the full benefit of general-domain data, the small models must be directly trained on general-domain data.}\\footnote{A reasonable alternative to this might include data-free KD \\cite{Yin2019-bm}, which explores the teacher's probability distribution without any dependence on data.}\n\nWe also observe that Medium-sized models are not small enough to benefit from knowledge distillation in the general-domain, and so their general-domain scores do not improve with distillation. These distilled Medium-sized models (config 7) also tend to do slightly worse than their baseline counter-parts (config 4) on in-domain data. Indeed, Figure \\ref{fig:gd-id} shows that in-domain performance is roughly linearly related to general-domain performance regardless of whether distillation is applied before adaptation.\n\nThis implies that \\textbf{distillation does not interfere with the adaptability of a model}, so the model with the best general-domain performance should be adapted, regardless of whether distillation was applied. Adapting a distilled model can improve performance by up to 1 BLEU over adapting the baseline model without distillation.\n\n    \\begin{table}\n    \\centering\n    \\begin{tabular}{lccc}\n    Model & de-en & ru-en & zh-en\\\\\n    \n\\hline\nTeacher                                  & 41.08 & 32.25 & 47.17\\\\\n\\hline\nMedium Baseline                          & 39.86 & 30.81 & 45.40\\\\\nMedium Student                           &\\color{red} 39.40 & \\color{red} 30.65 & \\color{red} 45.11\\\\\n\\hline\nSmall Baseline                           & 36.78 & 27.54 & 42.09\\\\\nSmall Student                            & \\color{ForestGreen} 38.51 & \\color{ForestGreen}28.88 & \\color{ForestGreen}42.73\\\\\n\\hline\nTiny Baseline                            & 31.27 & 23.63 & 34.71\\\\\nTiny Student                             & \\color{ForestGreen}34.58 & \\color{ForestGreen}25.86 & \\color{ForestGreen}36.09\\\\\n\n    \\end{tabular}\n    \\caption{General-domain models, teachers and students. While knowledge distillation improves small and tiny models, it appears medium-sized models are not under-parameterized enough for knowledge distillation to improve performance.}\n    \\label{tab:gd}\n    \\end{table}\n\n    \\begin{table}\n    \\centering\n    \\begin{tabular}{llcccc}\n    Domain & Size & Config \\# & de-en & ru-en & zh-en\\\\\n    \n\n\\hline\n &  & 1                                  & 27.73 & 19.34 & 15.17\\\\\nted & medium & 4                         & \\textbf{36.94} & \\textbf{25.82} & 20.13\\\\\n &  & 7                                  & 35.93 & 25.43 & \\textbf{20.18}\\\\\n\\hline\n &  & 1                                  & 27.89 & 18.42 & 14.87\\\\\n & small & 4                          & 34.78 & 24.10 & 18.84\\\\\n &  & 7                                  & \\textbf{35.33} & \\textbf{24.30} & \\textbf{19.32}\\\\\n\\hline\n &  & 1                                  & 25.78 & 17.48 & 13.03\\\\\n & tiny & 4                           & 31.52 & 21.30 & 16.51\\\\\n &  & 7                                  & \\textbf{32.30} & \\textbf{21.65} & \\textbf{17.06}\\\\\n\\hline\n &  & 1                                  & \\textbf{48.89} & 24.45 & 30.13\\\\\nwipo & medium & 4                        & 48.58 & \\textbf{35.98} & \\textbf{35.33}\\\\\n &  & 7                                  & 48.53 & 35.55 & 35.27\\\\\n\\hline\n &  & 1                                  & 47.94 & 21.91 & 30.66\\\\\n & small & 4                         & 48.13 & \\textbf{35.30} & \\textbf{34.90}\\\\\n &  & 7                                  & \\textbf{48.31} & 35.18 & 34.52\\\\\n\\hline\n &  & 1                                  & 44.15 & 21.39 & 27.67\\\\\n & tiny & 4                          & 46.06 & 31.13 & 28.45\\\\\n &  & 7                                  & \\textbf{46.54} & \\textbf{31.74} & \\textbf{29.07}\\\\\n\n    \\end{tabular}\n    \\caption{In-domain models that are initialized randomly (config 1), initialized from a baseline trained on general-domain data directly (config 4), or initialized from a  general-domain student trained using a general-domain teacher (config 7). }\n    \\label{tab:id-adapted}\n    \\end{table}\n    \n    \\begin{figure}\n        \\centering\n        \\includegraphics[width=9cm]{images\/gd-vs-id.png}\n        \\caption{The BLEU of general-domain models vs. their corresponding in-domain scores when adapted to a different domain. We see that in-domain performance is roughly linearly related to general-domain performance regardless of whether distillation is applied before adaptation.}\n        \\label{fig:gd-id}\n    \\end{figure}\n\n\\subsection{Distill, Adapt, Distill}\nFinally, we test whether these two ways of improving small, in-domain models are orthogonal. We might hypothesize that training small models directly on general-domain data eliminates the need to adapt teachers or use an in-domain teacher at all. To test this, we also train adapted student models using a baseline teacher (config 8) and an adapted teacher (config 9).\n\n   \\begin{table}\n    \\centering\n    \\begin{tabular}{llcccc}\n    Domain & Size & Config \\# & de-en & ru-en & zh-en\\\\\n    \n\\hline\n &  & 7                                  & 35.93 & 25.43 & \\textbf{20.18}\\\\\nted & medium & 8                         & 35.23 & 25.18 & 19.96\\\\\n &  & 9                                  & \\textbf{36.65} & \\textbf{25.91} & 20.13\\\\\n\\hline\n &  & 7                                  & 35.33 & 24.30 & 19.32\\\\\n & small & 8                             & 35.11 & 23.97 & 19.17\\\\\n &  & 9                                  & \\textbf{35.57} & \\textbf{24.95} & \\textbf{19.48}\\\\\n\\hline\n &  & 7                                  & 32.30 & 21.65 & 17.06\\\\\n & tiny & 8                              & 32.21 & 21.45 & 16.72\\\\\n &  & 9                                  & \\textbf{33.12} &\\textbf{22.49} & \\textbf{17.54}\\\\\n\\hline\n &  & 7                                  & 48.53 & 35.55 & 35.27\\\\\nwipo & medium & 8                        & 49.07 & 34.71 & 35.09\\\\\n &  & 9                                  & \\textbf{49.82} & \\textbf{35.83} & \\textbf{36.48}\\\\\n\\hline\n &  & 7                                  & 48.31 & \\textbf{35.18} & 34.52\\\\\n & small & 8                             & \\textbf{48.79} & 34.27 & 34.89\\\\\n &  & 9                                  & 48.35 & 35.10 & \\textbf{35.55}\\\\\n\\hline\n &  & 7                                  & 46.54 & 31.74 & 29.07\\\\\n & tiny & 8                              & \\textbf{49.90} & 31.12 & 30.05\\\\\n &  & 9                                  & 49.70 & \\textbf{31.75} & \\textbf{31.82}\\\\\n\n    \\end{tabular}\n    \\caption{In-domain models which are initialized from a general-domain student and trained on in-domain data which is pre-processed either with no teacher (config 7), an in-domain only teacher (config 8), or an adapted teacher continued from general-domain data (config 9).}\n    \\label{tab:id-adapt-both}\n    \\end{table}\n\nTable \\ref{tab:id-adapt-both} shows that \\textbf{distilling a second time using in-domain data with an adapted teacher can further boost performance of an already distilled model}, while using an un-adapted in-domain teacher can sometimes hurt performance. \n\nThese results lead us to a general recipe for training small, in-domain models using knowledge distillation and domain adaptation in combination:\n\n\\begin{enumerate}\n    \\item Distill general-domain data to improve general-domain student performance.\n    \\item Adapt the best model from Step 1 to in-domain data.\\\\\n    (2-10 BLEU better than no adaptation)\n    \\item Adapt the teacher and distill again in-domain. \\\\\n    (0-2 BLEU better than no or non-adapted teacher)\n\\end{enumerate}\n\nFollowing this procedure will result in either configuration 6 or 9 as described in Figure \\ref{fig:configs}. And indeed, configuration 9 performs the best or near best (within 0.1 BLEU) in almost every case, as shown in Table \\ref{tab:best-configs}. For those Medium sized models which were not improved by distillation in the general-domain, configuration 6 performs the best. \n\n\n    \\begin{table}\n    \\centering\n    \\begin{tabular}{llcccc}\n    Domain & Size & Config \\# & de-en & ru-en & zh-en\\\\\n    \n\\hline\n & medium & 6                            & 36.80 & 26.26 & 20.13\\\\\nted & small & 6                          & 35.50 & 24.68 & 19.31\\\\\n & tiny & 6                              & 32.09 & 22.20 & 17.25\\\\\n\\hline\n & medium & 6                            & 48.31 & 35.82 & 36.58\\\\\nwipo & small & 6                         & 49.04 & 35.30 & 35.40\\\\\n & tiny & 6                              & 48.02 & 31.57 & 30.53\\\\\n\n    \\end{tabular}\n    \\caption{Development scores for models initialized from a model trained on general-domain data. The in-domain data is pre-processed with a teacher adapted from the general-domain (config 6).}\n    \\label{tab:config-6}\n    \\end{table}\n    \n    \nModels trained on German-English WIPO are an exception, with adaptation from the general-domain not improving performance. This is in line with the results from Table \\ref{tab:id-teachers} which shows adaptation does not improve teachers, either. We suspect this is because the German-English WIPO dataset is the biggest out of any in-domain dataset, making adaptation unnecessary. Future work might also benefit from a quantification of domain similarity between datasets \\cite{Britz2017-qg}, which would guide the use of domain adaptation in cases like these.\n    \n    \\begin{table}\n    \\centering\n    \\begin{tabular}{llccc}\n    Domain & Size & de-en & ru-en & zh-en\\\\\n    \n\\hline\n & medium & 4\/6 & 6 & 4\/6\/7\/9\\\\\nted & small & 6\/9 & 9 & 9\\\\\n & tiny & 9 & 9 & 9\\\\\n\\hline\n & medium & 2 & 4\/6\/9 & 6\\\\\nwipo & small & 3 & 4\/6 & 9\\\\\n & tiny & 8 & 7\/9 & 9\\\\\n\n    \\end{tabular}\n    \\caption{Best configurations for each setting. Scores within 0.1 BLEU of the best are also listed. Configuration 9 generally performs best, while configuration 6 is best for those medium-sized models which were not improved by distillation in the general-domain.}\n    \\label{tab:best-configs}\n    \\end{table}\n       \n\n\\subsection{Training Times}\nThe models trained in this work collectively required 10 months of single-GPU compute time. Table \\ref{tab:times} breaks this down by model size and dataset.\n\nWhile distilling twice might give the best performance, it also increases the amount of computation time required. Rather than training a single in-domain model, configuration 9 requires training a general-domain teacher, a general-domain student, and then adapting both. This can increase compute required to train models by 2-4x.\n\nA huge portion of computation was also spent on decoding the general-domain data using a teacher model for sequence-level knowledge distillation, which could take up to 24 days of GPU time (using a beam size of 10 and a batch size of 10). This can be arbitrarily sped up using multiple GPUs in parallel, but future work might explore how to distill teachers in a less expensive way.\n\n\\begin{table}[]\n    \\centering\n    \\begin{tabular}{lccc}\n        Model & General-Domain & In-Domain & Adapting \\\\\n        \\hline\n        Large & 2-4 days & 2-4 days & 7-48 hours \\\\ \n        Medium & 2-4 days & 2-4 days & 1-48 hours \\\\ \n        Small & 1-2 days & 1-2 days & 2-14 hours \\\\ \n        Tiny & 1 days & 1-24 hours & 2-24 hours \\\\\n        \\hline\n        Distilling & 10-24 days & 1-2 days &\n    \\end{tabular}\n    \\caption{Estimates of the computation time required for training randomly initialized models on just general-domain data or just in-domain data. We also show the time required for adapting general-domain models and distilling data using teachers.}\n    \\label{tab:times}\n\\end{table}\n\n\\section{Related Work}\nOur work is one the few that focuses specifically on training small, under-parameterized in-domain models. There is, however, similar work which is \\emph{not directly comparable} but uses knowledge distillation to adapt to new domains.\n\n\\paragraph{Knowledge Adaptation} uses knowledge distillation to transfer knowledge from multiple, labeled source domains to un-labeled target domains. This is in contrast to our setting, which has labels for both general-domain and in-domain data. Ruder et al. \\shortcite{Ruder2017-zt} introduced this idea as ``Knowledge Adaptation,\" using multi-layer perceptrons to provide sentiment analysis labels for unlabeled in-domain data. Similar work includes Iterative Dual Domain Adaptation \\cite{Zeng2019-eg} and Domain Transformation Networks \\cite{Wang2019-hs}. These ideas are not limited to machine translation; recent work by Meng et al. \\shortcite{Meng2020-yu} trains in-domain speech recognition systems with knowledge distillation, while Orbes-Arteaga et al. \\shortcite{Orbes-Arteaga2019-de} does similar work on segmentation of magnetic resonance imaging scans. \n\n\\paragraph{Compressing Pre-trained Language Models} Domain adaptation via continued training in NMT is closely related to the idea of pre-training a language model and fine-tuning to different tasks, which might come from different data distributions than the pre-training data. Because language models tend to be extremely cumbersome to train and evaluate, more focus is given to the compression aspect of knowledge distillation. Sanh et al. \\shortcite{Sanh2019-gl}, Sun et al. \\shortcite{Sun2019-io}, and Liu et al. \\shortcite{Liu2019-ho} independently showed that knowledge distillation could be used to compress pre-trained models without affecting downstream tasks. Tang et al. \\shortcite{Tang2019-qc} showed that task-specific information could be distilled from a large Transformer into a much smaller Bi-directional RNN. These methods might reasonably be extended to domain adaptation for NMT.\n\n\\section{Conclusion}\nIn this work, we conducted a large-scale empirical investigation to determine best practices when using sequence-level knowledge distillation and domain adaptation in combination. We found that adapting models from the general-domain makes them better teachers and that distilling using general-domain data does not impact a model's adaptability. This leads us to recommend distilling twice for best results: once in the general-domain to possibly improve student performance, and again using an adapted in-domain teacher. The results are robust among multiple language pairs, student sizes, in-domain settings.\n\\bibliographystyle{named}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:introduction}\nIn this paper, we consider a one-dimensional chain of coupled harmonic oscillators with masses $m_x$ and Hamiltonian \n$$\n\tH \\; = \\; \\sum_{x\\in \\mathbb{Z}} \\left( \\frac{p_x^2}{2 m_x} + g\\frac{(q_{x+1} - q_{x})^2}{2} \\right) \n$$\nBy changing units, one can assume that the stiffness coefficient $g$ is equal to $1$.\nThe dynamics is governed by Hamilton's equations: \n$$\nm_x \\dot q_x =p_x, \\qquad \\dot p_x = (\\Delta q)_x,\n$$ \nwhere we have used the notation $\\Delta = \\nabla_- \\nabla_+ = \\nabla_+ \\nabla_-$ for the discrete Laplacian, \nwith $(\\nabla_+ f)_x = f_{x+1} - f_{x}$ and $(\\nabla_- f)_x = f_x - f_{x-1}$. \n{For the sake of simplicity, we consider for now the system on the infinite lattice $\\mathbb{Z}$. \nThis is by no means necessary, and starting from the next section, we will restrict ourselves to a finite box and take the thermodynamic limit properly.}\n\nThis system was first analyzed in finite volume when all masses $m_x$ are equal. \nPutting the chain in a non-equilibrium stationary state (NESS) between two heat reservoirs at different temperatures, \nit was found in \\cite{rieder} that the energy current does not decay with the size of the system, indicating that energy propagates ballistically.\nThe situation changes if the masses are taken to be i.i.d.\\@ random variables. \nThis case was first investigated in \\cite{rubin_greer,casher_lebowitz} and subsequently studied in \\cite{verheggen,dhar,ajanki_huveneers}. \nAs it turns out, the disordered harmonic chain is an Anderson insulator in disguise \\cite{anderson}. \nHowever, as a consequence of the conservation of momentum, \nthe ground state of the operator $M^{-1}\\Delta$, featuring in Newton's equation $\\ddot q = M^{-1}\\Delta q$ with $M$ the diagonal matrix of the masses,\nis a ``symmetry protected mode'' \\cite{halperin}, implying a divergent localization length in the lower edge of the spectrum. \nThis leads to a rich and unexpected phenomenology. \nIn particular, if the chain is again in a NESS, the scaling of the energy current with the system size \nhappens to depend on boundary conditions and spectral factors of the reservoirs \\cite{dhar}.\nThis finding reveals also the complete lack of local thermal equilibrium, \nthat results eventually from integrability (see Section \\ref{subsec: invariant quantities}).\n\nThe harmonic chain has three ``obvious'' conserved quantities: \nthe total energy $H$, the total momentum $P = \\sum_x p_x$ and the total stretch or elongation $R = \\sum_x r_x$ with $r_x = (\\nabla_+ q)_x$.   \nThis gives rise to the following microscopic conservation laws: \n$$\n\t\\dot r_x = \\frac{p_{x+1}}{m_{x+1}} - \\frac{p_x}{m_x}, \n\t\\qquad\n\t\\dot p_x = r_x - r_{x-1}, \n\t\\qquad \n\t\\dot e_x = \\frac{r_x p_{x+1}}{m_{x+1}} - \\frac{r_{x-1}p_x}{m_x}\n$$\nwith $e_x = \\frac{1}{2} \\left( \\frac{p_x^2 }{ m_x } + r_x^2 \\right)$. \nAfter a hyperbolic rescaling of space and time, we ask in this paper whether the empirical densities of these conserved quantities converge to the densities $\\mathbf r, \\mathbf p$ and $\\mathbf e$ governed by the macroscopic laws\n$$\n\t\\partial_t \\mathbf r (y,t) = \\frac{1}{\\overline{m}} \\partial_y \\mathbf p (y,t), \n\t\\quad \n\t\\partial_t \\mathbf p (y,t) = \\partial_y \\mathbf r (y,t), \n\t\\quad \n\t\\partial_t \\mathbf e (y,t) = \\frac{1}{\\overline{m}}  \\partial_y \\big(\\mathbf r (y,t) \\mathbf p (y,t)\\big) ,\n$$\ncorresponding to Euler equations in Lagrangian coordinates, with $\\overline{m}$ the average mass. \n\nInstances of rigorous derivation of Euler equations in the smooth regime rest on the ergodicity of the microscopic dynamics.\nIn \\cite{eo,ovy,komorowski_olla}, the Hamiltonian dynamics is perturbed by some stochastic noise \nacting in such a way that conserved quantities are not destroyed but that the ergodicity of the dynamics can be established rigorously.  \n\n{One of the main motivations of this work is to show that ergodicity \nis not in general a necessary assumption for Euler equations to hold. Indeed,} \nthe dynamics considered here is purely Hamiltonian, non-ergodic, \nand possesses actually a full set of invariant quantities (see Section \\ref{subsec: invariant quantities}). \nIn the clean case, i.e.\\@ when the masses are all equal, we show in Section \\ref{subsec: clean} that Euler equations hold if and only if the temperature profile is constant.\nInstead, in Section \\ref{subsec: disordered}, we argue that Euler equations hold even out of thermal equilibrium if there is disorder on the masses. \nWe briefly discuss the fate of other conserved quantities in Section \\ref{subsec: other conserved}. \nTheorem \\ref{the: main result} in Section \\ref{sec: model and results} constitutes the main result of our paper: \nWe show the convergence to Euler equations for the disordered harmonic chain, almost surely with respect to the masses and on average with respect to an initial local Gibbs state. \nThe rest of the paper is devoted to the proof of this theorem.  \n\n\n\n\\subsection{Clean harmonic chain}\\label{subsec: clean}\nLet us assume that all masses $m_x$ are equal, say $m_x = 1$ for simplicity. \nIn this case, the equations of motion read\n\\begin{equation}\\label{eq:1}\n\t\\dot q_x = p_x , \\qquad \\dot p_x = \\Delta q_x .\n\\end{equation}\nLet us first consider the thermal equilibrium case: \nAssume that the initial configuration of the chain is random and distributed according to a Gaussian law $\\mu_0$ with covariance matrix\n\\begin{equation}\\label{eq:2}\n\t\\lgs{(\\nabla_+ q)_x ; (\\nabla_+ q)_y} =  \\lgs{ p_x ; p_y} = \\beta^{-1} \\delta_{x,y}, \\qquad \n  \t\\lgs{q_x; p_y} = 0,\n\\end{equation}\nfor some inverse temperature $\\beta$. \nIt is easy to prove that at time $t>0$, the distribution $\\mu_t$ in the phase space is still given by a Gaussian law with the same covariance matrix.  \nTo see this, just use Fourier transforms to diagonalize the dynamics:\n\\begin{equation}\\label{eq:3}\n  \t\\hat q(k,t) = \\sum_{x\\in \\ZZ} e^{i 2\\pi k x} q_x(t), \\qquad  \\hat p(k,t) = \\sum_{x\\in \\ZZ} e^{i 2\\pi k x} p_x(t),\n\\end{equation}\nand define the wave function \n\\begin{equation}\\label{eq:4}\n  \t\\hat \\phi (k,t) = \\omega(k)  \\hat q(k,t) + i  \\hat p(k,t) \n\\end{equation}\nwhere $\\omega(k) = |2\\sin(\\pi k)|$ is the dispersion relation. Then, the explicit solution of \\eqref{eq:1} is given by\n\\begin{equation}\n  \\label{eq:5}\n   \\hat \\phi (k,t) = e^{-i\\omega(k) t} \\,  \\hat \\phi (k,0).\n\\end{equation}\nThe correlations \\eqref{eq:2} imply that \n\\begin{equation}\\label{eq:5-0}\n\t\\lgs{ \\hat \\phi (k,0)^* ;  \\hat \\phi (k',0)}  = 2 \\beta^{-1} \\delta(k-k'), \\qquad  \\lgs{ \\hat \\phi (k,0) ; \\hat \\phi (k',0)}=0.\n\\end{equation}\nConsequently\n\\begin{equation}\n\\label{eq:6}\n\\begin{split}\n& \\lgs{ \\hat \\phi (k,t)^* ;  \\hat \\phi (k',t)} =   e^{i(\\omega(k)- \\omega(k')) t}\\,  \\lgs{ \\hat \\phi (k,0)^* ;  \\hat \\phi (k',0)}\n= 2\\beta^{-1} \\delta(k-k'),\\\\\n&\\lgs{ \\hat \\phi (k,t) ;  \\hat \\phi (k',t)} =   e^{-i(\\omega(k)+\\omega(k')) t} \\, \\lgs{ \\hat \\phi (k,0) ;  \\hat \\phi (k',0)}\n=0. \n\\end{split} \n\\end{equation}\nFrom (\\ref{eq:5-0}) and (\\ref{eq:6}), we deduce that the covariances  \\eqref{eq:2} are the same at any time $t$:\n\\begin{equation}\n \\label{eq:2222}\n  \\lgs{(\\nabla_+ q)_x (t) ; (\\nabla_+ q)_y (t) } =  \\lgs{ p_x (t)  ; p_y (t) } = \\beta^{-1} \\delta_{x,y}, \\qquad \n  \\lgs{q_x (t) ; p_y (t) } = 0,\n\\end{equation}\nwhich implies that the Gaussian distribution $\\mu_t$ differs from $\\mu_0$ only by the averages $\\bar r_x (t) = \\lgs{r_x (t)} = \\lgs{(\\nabla_+ q)_x(t)}$ and $\\bar p_x (t) = \\lgs{p_x(t)}$ that, \nby linearity of the dynamics, evolve following the same equation \\eqref{eq:1}. \n\nAssume now that the initial averages of the momentum $p_x$ and stretch $r_x = (\\nabla_+ q)_x$ are slowly varying on a macroscopic scale. \n{More precisely, let $N$ be an integer representing a macroscopic number of sites in the chain, \nand let $\\mathsf p, \\mathsf r : \\mathbb{R} \\to \\mathbb{R}$ be smooth and fast decaying initial macroscopic profiles. \nWe let}\n\\begin{equation}\n  \\label{eq:7}\n  {\\bar r_{[Ny]}(0) = \\mathsf r (y),\\qquad\n  \\bar p_{[Ny]}(0) = \\mathsf p (y)} . \n\\end{equation}\nLet $\\widehat {\\mathsf p} (\\xi)$ and $\\widehat {\\mathsf r} (\\xi)$ be the Fourier transforms (in $\\RR$) of $\\mathsf p(y)$ and $\\mathsf r(y)$. \nThen, as $N\\to\\infty$, \n$\\frac 1N \\widehat{\\bar p}(\\tfrac \\xi N) \\longrightarrow \\widehat {\\mathsf p}(\\xi)$ and \n$\\frac 1N \\widehat{\\bar r}(\\tfrac \\xi N) \\longrightarrow \\widehat {\\mathsf r} (\\xi)$. After a straightforward analysis \nwe have that\n\\begin{equation}\n  \\label{eq:8}\n  \\begin{split}\n    \\frac 1N \\widehat{\\bar p}\\Big(\\tfrac \\xi N, Nt\\Big) \\longrightarrow \\widehat {\\mathbf p} \\, (\\xi,t), \\qquad \n    \\frac 1N \\widehat{\\bar r} \\Big(\\tfrac \\xi N, Nt \\Big)\n    \\longrightarrow \\widehat {\\mathbf r}\\, (\\xi,t )\n  \\end{split}\n\\end{equation}\nwhere \n\\begin{equation}\n  \\label{eq:9}\n \\partial_t  \\widehat {\\mathbf r} \\, (\\xi,t) = -i {2\\pi} \\xi  \\,  \\widehat {\\mathbf p} \\, (\\xi,t), \\qquad \n \\partial_t  \\widehat {\\mathbf p} \\, (\\xi,t) = -i {2\\pi} \\xi \\,  \\widehat {\\mathbf r} \\, (\\xi,t).\n\\end{equation}\nConsequently {$\\bar r_{[Ny]}(Nt)$ and $\\bar p_{[Ny]}(Nt)$} converge {(as distributions)} to the solution of the linear wave equation\n\\begin{equation}\n  \\label{eq:10}\n  \\partial_t  \\mathbf r \\,(y,t) = \\partial_y \\mathbf p \\, (y,t), \\qquad \n \\partial_t \\mathbf p \\,(y,t) = \\partial_y   \\mathbf r \\, (y,t).\n\\end{equation}\nLet us now consider the energy per particle $e_x = \\frac 12\\left(p_x^2 + r_x^2\\right)$. \nIts average under the distribution $\\mu_t$ is \n$\\lgs{ e_x (t)} = \\beta^{-1} + \\frac 12\\left(\\bar p_x^2(t) + \\bar r_x^2(t)\\right)$ since by (\\ref{eq:2222}), the variance of $p_x$ and $r_x$, i.e. the temperature, remains constant in time. In the limit $N\\to \\infty$ we have\n$$\\lgs{ e_{[Ny]}(Nt)} \\longrightarrow {\\bf e} \\, (y,t) = \\beta^{-1} + \\frac 12\\left( {\\bf p}^2 (y,t) + {\\bf r}^2(y,t)\\right),$$\ni.e. it solves the equation\n\\begin{equation}\n  \\label{eq:11}\n   \\partial_t {\\bf e}\\, (y,t) = \\partial_y \\left( {\\bf p}\\, (y,t)   \\, {\\bf r}\\, (y,t)\\right).\n\\end{equation}\nWe recognize that (\\ref{eq:10} - \\ref{eq:11}) are the Euler equations. \nThe above is the simplest example of propagation of local equilibrium and hydrodynamic \nlimit in hyperbolic scaling: in a harmonic chain in thermal equilibrium at temperature $\\beta^{-1}$, \nand the mechanical modes not in equilibrium, we  will have a persistence of \nthe thermal equilibrium {at any time $t$}, \nwhile the mechanical modes evolve independently from the thermal mode  following the linear wave equation. \n\nNotice that the argument above does not require the distribution $\\mu_0$ to be the thermal equilibrium measure defined by \\eqref{eq:2}, \nand that it holds for any measure $\\mu_0$ with translation invariant covariance given by\n\\begin{equation}\n  \\label{eq:2-c}\n  \\lgs{\\nabla q_x ; \\nabla q_y} =  \\lgs{ p_x ; p_y} = C(x-y), \\qquad \n  \\lgs{q_x; p_y} = 0\n\\end{equation}\nfor a positive definite function $C(x)$. \nThe only difference is then that in (\\ref{eq:5-0}- \\ref{eq:6}) the term $\\beta^{-1}$ has to be replaced by the Fourier transform ${\\widehat C} (k)$. \nActually, the measure $\\mu_t$ is not even a local equilibrium state \\cite{BO_book}, underlining that the validity of Euler equations in this example does not require the propagation of local equilibrium.\n\nThe above argument rests on the translation invariance of the distribution of the thermal modes and fails if it is space inhomogeneous, \nfor example if the starting distribution is given by a local Gibbs state with a slowly varying temperature $\\beta_{N,x}^{-1} = \\beta^{-1}(x\/N)$, \ni.e.\\@ a Gaussian measure with covariances\n\\begin{equation}\n  \\label{eq:12}\n  \\lgs{\\nabla q_x ; \\nabla q_y} =  \\lgs{ p_x ; p_y} = \\beta_{N,x}^{-1}\\delta_{x,y}, \\qquad \n  \\lgs{q_x; p_y} = 0.\n\\end{equation}\nIn this case, even though the wave equation \\eqref{eq:10} still holds, \ngenerally the energy \nequation \\eqref{eq:11} is not valid. \nIn fact the energies of each mode $k$ evolves autonomously, \nas we can see studying the limit evolution of the Wigner distribution defined by\n\\begin{equation}\\label{eq:wigner1}\n\\begin{split}\n    \t\\widehat W_N(\\xi, k,t) &:= \\frac 2N \n    \t\\Blgs{\\widehat\\phi^*\\left(k - \\tfrac{\\xi}{2N}, Nt\\right)\n       \t\\widehat\\phi\\left(k + \\tfrac{\\xi}{2N}, Nt\\right)}\\\\\n   \t W_N(y, k,t) &:= \\int e^{-i 2\\pi \\xi y}   \\widehat W_N (\\xi, k,t) \\; d\\xi,\n\\end{split}\n\\end{equation}\n(the above definitions should be understood as distributions on $\\mathbb{R}\\times \\Pi$ with $\\Pi = \\mathbb{R}\\backslash \\mathbb{Z}$).\n\nIn the limit as $N\\to\\infty$ the Wigner distribution converge to a positive distribution\nwith an absolutely continuous part, the local distribution of the thermal modes, and \na singular part concentrated on $k=0$, the mechanical modes:\n\\begin{equation}\n  \\label{eq:18}\n  \\lim_{N\\to\\infty} \\widehat W_N(\\xi, k, t) = \\widehat W_{th}(\\xi, k,t) + \\widehat W_m(\\xi, t)\\;  \\delta_0(dk) \n\\end{equation}\nThe mechanical part $\\widehat W_m(\\xi, t)$ is the Fourier transform of \n$\\frac 12\\left( {\\bf p}^2 (y,t) + {\\bf r}^2(y,t)\\right)$.\n\nA straightforward calculation gives for the thermal part (see \\cite{dobru} or \\cite{bos} for a rigorous argument):\n\\begin{equation}\n  \\label{eq:14}\n  \\widehat W_{th}(\\xi, k,t) = e^{-i\\omega'(k)\\xi t} \\; \\widehat W_{th}(\\xi, k,0).\n\\end{equation}\nThis implies that the inverse Fourier transform  $W_{th}(y, k,t)$ satisfies the transport equation\n\\begin{equation}\n  \\label{eq:13}\n  \\partial_t W_{th}(y, k,t) + \\frac{\\omega'(k)}{2\\pi} \\partial_y W_{th}(y, k,t) = 0.  \n\\end{equation}\nIt also follow that \n\\begin{equation}\n  \\label{eq:15}\n  \\int  W_{th}(y, k,t) \\; dk \\ = \\ \\tilde {\\bf e}(y,t)\n\\end{equation}\nwhere $\\tilde {\\bf e}(y,t)$ is the limit profile of thermal energy (or temperature) defined as\n\\begin{equation}\n  \\label{eq:16}\n  \\frac 12 \\left(\\lgs{ r_{[Ny]}(Nt); r_{[Ny]}(Nt)} + \\lgs{ p_{[Ny]}(Nt); p_{[Ny]}(Nt)}\\right) \\rightharpoonup \\tilde {\\bf e}(y,t).\n\\end{equation}\nConsequently the thermal energy  $\\tilde {\\bf e}(y,t)$ evolves non autonomously following the equation\n\\begin{equation}\n  \\label{eq:17}\n  \\partial_t \\tilde {\\bf e}(y,t) + \\partial_y J(y,t) = 0, \\qquad J(y,t) = \\int \\omega'(k)  W_{th}(y, k,t) \\; dk.\n\\end{equation}\nWe say that the system is in {\\it local equilibrium} if $W_{th}(y,k) = \\beta^{-1}(y)$ constant in $k$. \nThis correspond to the fact that Gibbs measure gives uniform distribution on the modes.\nStarting in thermal equilibrium means $W_{th}(y,k,0) = \\beta^{-1}$ and trivially $W_{th}(y,k,t) = \\beta^{-1}$ \nfor any $t>0$.\nBut starting with local equilibrium, i.e. $W(y,k,0) = \\beta^{-1}(y)$ constant in $k$, \nwe have a non autonomous evolution of $\\tilde e(y,t)$. \n\n\n\n\n\n\n\n\n\n\\subsection{Disordered harmonic chain}\\label{subsec: disordered}\nThe situation so far can be summarized as follows. \nBy linearity, the variables $r_x$ and $p_x$ admit a macroscopic limit described by \\eqref{eq:10} independently of the initial temperature profile. \nThe macroscopic equation \\eqref{eq:11} predicts that the evolution of the energy is purely mechanical and that the temperature does not evolve with time. \nAs it turns out, the evolution of the mechanical energy is correctly described by Euler equation (see the term $\\mathcal A_N (t)$ in our decomposition \\eqref{eq: average and fluctuation} below), \nbut thermal fluctuations do in general evolve with time as well, except if the temperature profile is initially flat. \n\nThis picture gets strongly modified if the masses are taken to be random. \nOn the one hand, deriving the macroscopic evolution of the fields $r_x$ and $p_x$ becomes less obvious because some homogenization over the masses is required. \nThis difficulty can be solved by the elegant method of the \n``corrected empirical measure'', see \\cite{goncalves_jara,jara_landim,bernardin} \n(though we will actually solve it another way). \nOn the other hand, and this is the main point in considering random masses, \nthe evolution of the energy $e_x$ is now much better approximated by Euler equation.\nIndeed, at a microscopic level, all thermal fluctuations are frozen thanks \nto Anderson localization and the evolution of the energy becomes purely mechanical. \n\nTo understand this a little bit better, it is good to realize how the disorder modifies the nature \nof the eigenmodes $(\\psi^k)_{1 \\le k \\le N}$ of the operator $M^{-1}\\Delta$ for a finite chain of size $N$. \nAs a consequence of Anderson localization \\cite{anderson}, all modes at positive energy are spatially localized. \nHowever the localization length $\\zeta_k$ diverges as one approaches the ground state: \n$$\n\t\\zeta_k^{-1} \\sim \\omega^2_k \\sim \\Big(\\frac{k}{N}\\Big)^2,\n$$\nso that only the modes with $k\\gtrsim \\sqrt N$ are actually localized, while the modes $k \\lesssim \\sqrt N$ remain comparable to the modes of the clean chain \\cite{verheggen,ajanki_huveneers}.\nBy imposing a smooth initial profile $\\mathsf r, \\mathsf p$, the initial local Gibbs state attributes a weight of order 1 to a few first modes above the ground state, and a weight of order 1 to all other modes together. \nThe first ones are responsible for the transport of mechanical energy; \nall modes with $k \\gg \\sqrt N$ are localized and do not transport any thermal energy;  \nall modes with $1  \\ll k \\le o (N)$ have a vanishing weight in the thermodynamic limit and can be neglected in the analysis. \n\nFinally, we would like to mention that, while the disorder considered here \nand the stochastic velocity exchange noise considered in \\cite{eo,komorowski_olla} \nact in an obviously very different way, \ne.g.\\@ the disorder preserves integrability while the stochastic noise makes the dynamic ergodic, \nthey do produce the same effects in some respect. \n{Indeed the noise has only a very slow (negligeable) effect on the macroscopic modes.}\n{ This bares some similarity with the fact that the disorder has very little influence on the low modes of the disordered chain,\nwhile the dynamical noise provides an active hopping mechanism among the high modes. \nConsequently the dynamical noise produces a superdiffusive sub-ballistic spreading of the thermal energy \\cite{jara_kom_olla,komorowski_olla}, \nthat is not visible in the hyperbolic scaling.\nThus, the dynamical noise plays here as well a role analogous to the disorder only in the hyperbolic scaling by freezing the temperature profile.\n\nIt is important to notice that in the disordered case the temperature profile remains frozen at any time scale, \nincluding the diffusive time scale and further, see remark \\ref{frozenforever}.\nIn particular this implies a vanishing thermal diffusivity for the disordered unpinned harmonic chain. This is not in contradiction with the divergence of the thermal conductivity observed in the NESS of the same unpinned system when connected to Langevin thermostats at different temperatures with free boundary conditions, see \\cite{casher_lebowitz,ajanki_huveneers}, and our result sheds actually some light on the various behaviors for the conductivity found in \\cite{dhar} depending on the boundary conditions. In fact the thermal conductivity divergence in the NESS is due to the fluctuations of the low mechanical modes, and in our analysis there is a clear separation of the behavior of the mechanical modes (responsable for the ballistic motion) and the high thermal modes that give the temperature profile.}\n\n\n\n\n\n\n\\subsection{Other conserved quantities}\\label{subsec: other conserved}\nBefore moving on, let us briefly comment on the issue of the other conserved quantities of the system. \nThese can also be written as a sum of local terms and lead thus to additional conservation laws. \nFor example, \n$$\n\tI \\; = \\; \\sum_{x} d_x \\; = \\; \\frac{1}{2}\\sum_x\\left( \\frac{(r_x - r_{x-1})^2}{m_x} + \\left( \\frac{p_{x+1}}{m_{x+1}} - \\frac{p_x}{m_x} \\right)^2 \\right)\n$$\nis conserved (see Sections \\ref{subsec: a priori} and \\ref{subsec: invariant quantities}) and leads to the microscopic conservation law\n$$\n\t\\dot d_x = \\left( \\frac{p_{x+1}}{m_{x+1}} - \\frac{p_{x}}{m_{x}} \\right) \\frac{r_{x+1} - r_x}{m_{x+1}} -  \\left( \\frac{p_{x}}{m_{x}} - \\frac{p_{x-1}}{m_{x-1}} \\right) \\frac{r_{x} - r_{x-1}}{m_x} .\n$$\n\nIt is thus natural to ask whether this relation generates also some macroscopic law. \nIn the cases where we can derive the macroscopic evolution equation \\eqref{eq:11} for the energy, it is easy to argue that the corresponding macroscopic density $\\mathbf d (y,t)$ does not evolve with time in the hyperbolic scaling.\nIndeed, we can decompose $d_x$ as the sum of a mechanical and a thermal contribution, as we do in \\eqref{eq: average and fluctuation} below for the energy.\nIn this case, contrary to what happens for the energy, the mechanical contribution vanishes in the thermodynamic limit since $d_x$ depends on $r$ and $p$ only through their gradients, \nwhile the contribution from the thermal modes does not evolve with time, for the same reasons as it does not for the energy.  \n\nAll the other conserved quantities in this model that can be written as a sum of local terms are obtained by taking further gradients in the variables $r$ and $p$ (see Section \\ref{subsec: invariant quantities}), \nand have thus no evolution either in the hyperbolic scaling.   \n\n\n\n\n\n\n\\section{Model and results}\\label{sec: model and results}\n\nWe define the model studied in this paper and we state our main result. \nFor technical reasons, it is easier to work on a finite system of size $N$ and then let $N\\to \\infty$.\n\n\\subsection{Hamiltonian model}\nThe Hamiltonian $H$ on $\\mathbb{R}^{2N}$ is defined by \n$$\n\tH (q,p) = \\frac{1}{2} \\sum_{x=1}^N \\left( \\frac{p_x^2}{m_x} + ((\\nabla_+q)_{x})^2 \\right).\n$$\n{For concreteness, we assume} free boundary conditions, i.e.\\@ $q_0 = q_1$ and $q_{N+1} = q_N$; \n{other boundary conditions such as fixed or periodic could be considered just as well.} \nThe masses $(m_x)_{1 \\le x \\le N}$ are i.i.d.\\@ random variables. \nIn order to avoid any technical difficulty in exploiting known results from the Anderson localization literature, we assume that the law of $m_x$ admits a smooth density compactly supported in $[m_-,m_+]$ with $m_->0$. \n\nThe equations of motion read $M \\dot q = p$ and $\\dot p = \\Delta q$ where $M$ is the square diagonal matrix of size $N$ with entries defined by $M_{x,y} = \\delta (x-y) m_x$ ($\\delta(z)$ is defined by $1$ for $z=0$ and $0$ otherwise).\nIt is more convenient to express the equations of motion in terms of the displacement variables \n\\begin{equation*}\n\tr_x  =  (\\nabla_+ q)_x \\qquad (1 \\le x \\le N-1) .  \n\\end{equation*}\nThe equations of motion become\n\\begin{equation}\\label{eq: equations of motion}\n\t\\dot r_x = \\big( \\nabla_+ M^{-1}p \\big)_x \\quad (1 \\le x \\le N-1), \n\t\\qquad \\dot p_x = (\\nabla_- r)_x \\quad (1 \\le x \\le N)\n\\end{equation}\nwhere we use fixed boundary conditions for $r$ in the second equation: $r_0 = r_N = 0$.\n\n\\subsection{Gibbs and locally Gibbs states}\nWe consider three locally conserved quantities in the bulk: \n$$ H = \\sum_{x=1}^N e_x =  \\sum_{x=1}^N \\left( \\frac{p^2_x}{2m_x} + \\frac{r_x^2}{2} \\right), \\qquad P = \\sum_{x=1}^N p_x, \\qquad R = \\sum_{x=1}^{N-1} r_x. $$\nThe energy $H$ and the momentum $P$ are actually truly conserved, but the conservation of $R$ is broken at the boundary: $\\dot R = m_N^{-1}p_N - m_1^{-1}p_1$. \n\nThe Gibbs states are characterized by three parameters: \n$\\beta>0$ and $\\mathsf p, \\mathsf r \\in \\mathbb{R}$. Its probability density writes\n$$ \n\\rho_{\\text{G}} (r,p) = \\frac{1}{Z_{\\text{G}}} \\exp \\Big\\{- \\frac{\\beta}{2} \\sum_{x=1}^{N} m_x\\Big(\\frac{p_x}{m_x} - \\frac{\\mathsf p}{\\overline m}\\Big)^2 - \\frac{\\beta}{2} \\sum_{x=1}^{N-1} (r_x - \\mathsf r)^2  \\Big\\} .\n$$\nwhere $\\overline m$ denotes the mean mass and $Z_{\\text{G}} :=Z_{\\text{G}} (\\beta, \\mathsf p, \\mathsf r)$ is a normalizing constant.\nLocal Gibbs states are obtained by replacing the constant parameters $\\beta, \\mathsf p, \\mathsf r$ by functions \n$$\\beta,\\mathsf p, \\mathsf r : [0,1] \\to \\mathbb{R},$$  \nwith $\\beta(x)>0$ for all $x \\in [0,1]$, and by considering the measure with density \n\\begin{equation}\\label{eq: local Gibbs}\n\t\\rho_{\\text{loc}} (r,p) = \\frac{1}{Z_{\\text{loc}}} \\exp \\Big\\{- \\frac{1}{2} \\sum_{x=1}^{N} \\beta(x\/N) m_x\\Big(\\frac{p_x}{m_x} - \\frac{\\mathsf p(x\/N)}{\\overline{m}}\\Big)^2 - \\frac{1}{2} \\sum_{x=1}^{N-1} \\beta(x\/N) (r_x - \\mathsf r(x\/N))^2  \\Big\\}\n\\end{equation}\nwhere $Z_{\\text{loc}}:=Z_{\\text{loc}} (\\beta, \\mathsf p, \\mathsf r)$ is a normalizing constant.\nWe impose the following regularity conditions on $\\beta, \\mathsf p, \\mathsf r$: \n\\begin{equation}\\label{eq: regularity beta r p}\n\t\\beta \\in \\mathcal C^0([0,1]), \n\t\\quad \\mathsf r \\in \\mathcal C^1([0,1]) \\text{ with }\\mathsf r(0) = \\mathsf r(1) = 0, \n\t\\quad \\mathsf p \\in \\mathcal C^1([0,1]).\n\\end{equation}\nWe take such a local Gibbs state as initial state.\nBelow, we denote the expectation with respect to it by $\\lgs{\\cdot}$:\n$$\n\t\\lgs{F} = \\int F(r,p) \\rho_{\\text{loc}} (r, p) \\, \\mathrm{d} r \\mathrm{d} p.\n$$ \nInstead, expectation (resp.\\@ probability) with respect to the masses is denoted by $\\mathsf E$ (resp.\\@ $\\mathsf P$). \n\n\n\\subsection{Evolution of the locally conserved quantities} \nLet us fix some maximal time $T>0$. \nLet us define the fields $\\mathcal R$, $\\mathcal P$ and $\\mathcal E$ acting on functions $f \\in \\mathcal C^0([0,1])$ as \n\\begin{equation}\\label{eq:rpe}\n\\begin{split}\n&\\mathcal R(f,t) = \\int_0^1 \\mathbf r (y,t)\\,  f(y) \\, \\mathrm{d} y, \\quad  \\mathcal P(f,t) = \\int_0^1 \\mathbf p (y,t) \\, f(y)\\,  \\mathrm{d} y,\\\\\n&\\mathcal E(f,t) = \\int_0^1 \\mathbf e (y,t) \\, f(y) \\, \\mathrm{d} y. \n\\end{split}\n\\end{equation}\nfor all $t \\in [0,T]$. \nThe kernels $\\mathbf r$, $\\mathbf p$ and $\\mathbf e$ are defined as follows. \nFirst, at $t=0$, we impose\n$$\n\t\\mathbf r(y,0) = \\mathsf r(y), \\qquad \n\t\\mathbf p(y,0) = \\mathsf p(y), \\qquad\n\t\\mathbf e(y,0) =  \\frac1{\\beta(y)} +  \\frac{\\mathsf p^2 (y)}{2\\overline m} + \\frac{\\mathsf r^2 (y)}{2}. \n$$\nNext, the evolution at all further time is governed by the following system of conservation laws: \n\\begin{align}\n\t&\\partial_t \\mathbf r(y,t) = \\frac1{\\overline m} \\partial_y \\mathbf p(y,t), \\qquad \\mathbf r(0,t) = \\mathbf r(1,t) = 0, \n\t\\label{eq:equa diff r}\\\\\n\t&\\partial_t \\mathbf p(y,t) = \\partial_y \\mathbf r (y,t),\n\t\\label{eq:equa diff v}\\\\\n\t&\\partial_t \\mathbf e(y,t) = \\frac1{\\overline m} \\partial_y (\\mathbf r(y,t) \\mathbf p(y,t)).\n\t\\label{eq:equa diff e}\n\\end{align}\n\nThanks to the regularity conditions on $\\mathsf r, \\mathsf p$ in \\eqref{eq: regularity beta r p}, the solutions of these equations are classical.\nSince $(\\mathbf r, \\mathbf p)$ are solution of wave equations with suitable boundary conditions, they can be obtained explicitly by expanding them in Fourier series.  \nThen, by a time integration, $\\mathbf e$ may be expressed as a function of $(\\mathbf r, \\mathbf p)$, see (\\ref{eq: evolution energy limit}). \nLater we will use that a classical solution for the system governing $(\\mathbf r, \\mathbf p)$ coincides with the (unique) weak solution of this system. \nBecause of the boundary conditions, test functions will have to be chosen appropriately (see (\\ref{eq: R characterization}-\\ref{eq: P characterization})). \n\n\\begin{Theorem}\\label{the: main result}\nLet $t\\in [0,T]$ and $f \\in \\mathcal C^0([0,1])$. \nLet us assume that the system is initially prepared in a locally Gibbs state such that $\\beta$, $\\mathsf r$ and $\\mathsf p$ satisfy \\eqref{eq: regularity beta r p}. \nThen, as $N\\to \\infty$, almost surely (w.r.t.\\@ $\\mathsf P$), \n\\begin{align}\n\t&\\mathcal R_N (f,t) \\; = \\;  \\frac{1}{N} \\sum_{x=1}^N f(x\/N) \\; \\lgs{r_x (Nt)} \\qquad \\to \\qquad \\mathcal R(f,t), \\label{R limit}\\\\\n\t&\\mathcal P_N (f,t) \\; = \\; \\frac{1}{N} \\sum_{x=1}^N f(x\/N) \\; \\lgs{p_x (Nt)} \\qquad \\to \\qquad \\mathcal P(f,t), \\label{P limit}\\\\\n\t&\\mathcal E_N (f,t) \\; = \\; \\frac{1}{N} \\sum_{x=1}^N f(x\/N) \\; \\lgs{e_x (Nt)} \\qquad \\to \\qquad \\mathcal E(f,t) \\label{E limit}.\n\\end{align}\n\\end{Theorem}\n\\begin{Remark}\nAs pointed out in the introduction, the situation is much simpler at thermal equilibrium, i.e.\\@ for $\\beta$ constant, and these limits hold even for the non-disordered chain.\nSee Section \\ref{subsec: thermal equilibrium} for a derivation along the lines used to derive Theorem \\ref{the: main result}. \n\\end{Remark}\n\n\n\n\n\n\\section{Evolution of $\\mathcal R_N$ and $\\mathcal P_N$}\n\nIn this section, we show the limits (\\ref{R limit}-\\ref{P limit}). \nMoreover, in order to later deal with the field $\\mathcal E_N$, we show more: \n\nThe functions $\\lgs{r_{[Ny]} (Nt)}$ and $m_{[Ny]}^{-1}\\lgs{ p_{[Ny]} (Nt)}$ are uniformly (in $N$) H\\\"older regular in $y\\in [0,1]$, with exponent at least $1\/2$. \nHence they converge pointwise to $\\mathbf r(y,t)$ and $\\mathbf p (y,t)$ respectively. \n\n\\subsection{A priori estimates}\\label{subsec: a priori}\nGiven $d\\in \\mathbb{N}$, we denote the standard scalar product on $\\mathbb{R}^d$ by $\\langle \\cdot , \\cdot \\rangle_d$ (we will drop the subscript $d$ when no confusion seems possible).\nLet us consider the two following conserved quantities: \n\\begin{align}\n\tH(r,p) &= \\frac12 \\big(  \\langle p ,M^{-1 } p \\rangle_N + \\langle r ,r \\rangle_{N-1} \\big), \\\\\n\tI(r,p) &= \\frac12 \\big( \\langle \\nabla_- r, M^{-1} \\nabla_- r \\rangle_N + \\langle \\nabla_+ M^{-1} p , \\nabla_+ M^{-1}p \\rangle_{N-1} \\big).\\label{eq: I conserved}\n\\end{align}\nThe conservation of $I$ follows from the fact that, if $(r,p)$ solve \\eqref{eq: equations of motion}, then $(\\nabla_+ M^{-1}p, \\nabla_- r)$ solve the same equation, the corresponding Hamiltonian being $I$ \n(since we have that $H (\\nabla_+ M^{-1} p, \\nabla_- r) =I (r,p)$).\nNotice also that a full set of conserved quantities can be generated by further taking gradients, see Section \\ref{subsec: invariant quantities}. \n\nThanks to these two conservation laws, and to the smoothness assumptions on $\\mathsf r$ and $\\mathsf p$, we deduce\n\\begin{Lemma}\\label{lem-bounds}\nThere exists {a deterministic} $\\mathrm{C}$ such that, for any $t \\ge 0$ and any $N \\in \\mathbb{N}$, \n\\begin{align} \n\t&\\sum_{x=1}^{N-1} \\lgs{ r_x(Nt) }^{2} \\le \\mathrm{C} N, \\qquad \n\t\\sum_{x=1}^N \\lgs{ p_x(Nt) }^2 \\le \\mathrm{C} N, \n\t\\label{eq: L2 estimate}\\\\\n\t&\\sum_{x=1}^N \\lgs{ (\\nabla_- r)_x (Nt)}^2 \\le \\frac\\mathrm{C}{N}, \\qquad \n\t\\sum_{x=1}^{N-1} \\lgs{ (\\nabla_+ M^{-1} p)_x (Nt) }^2 \\le \\frac\\mathrm{C}{N}.\n\t\\label{eq: H1 estimate}\n\\end{align}\n\\end{Lemma}\n\\begin{proof}\nBy linearity of the equations of motion \\eqref{eq: equations of motion}, $(\\lgs{ r }, \\lgs{ p } )$ solve the same equations as $(r,p)$. \nTherefore, the conservation of $H(r,p)$ and $I(r,p)$ implies the conservation of $H(\\lgs{ r }, \\lgs{ p })$ and $I(\\lgs{ r }, \\lgs{ p })$. \nSince the quantities to be estimated in \\eqref{eq: L2 estimate} are bounded by $H(\\lgs{ r }, \\lgs{ p })$ \nand the quantities to be estimated in \\eqref{eq: H1 estimate} are bounded by $I(\\lgs{ r }, \\lgs{ p })$, \nwe conclude that is it enough to establish them respectively for $H(\\lgs{r}, \\lgs{p})$ and  $I(\\lgs{r}, \\lgs{p})$ at $t=0$. \nThis follows from a direct computation, thanks to the product structure of the local Gibbs state \\eqref{eq: local Gibbs} and to the hypotheses on $\\mathsf r$ and $\\mathsf p$ in \\eqref{eq: regularity beta r p} \n(in particular, this is the place where the boundary condition on $\\mathsf r$ plays a role). \n\\end{proof}\n\\noindent\n\n{\n\\begin{Remark}\nNotice that the bounds in Lemma \\ref{lem-bounds} are actually valid for any time scale $N^\\alpha t$, for any $\\alpha>0$.\n\\end{Remark}\n}\n\nAs a corollary, we deduce the existence of a constant $\\mathrm{C}\\in \\mathbb{R}$ such that, for any $x,y \\in \\mathbb{Z} \\cap [1,N]$,\n\\begin{equation}\n\\label{eq: Holder continuity}\n\\begin{split}\n&\\big| \\lgs{r_{x'}(Nt) } - \\lgs{r_x(Nt) } \\big| \\le  \\mathrm{C} \\left|\\tfrac{x'-x}N \\right|^{1\/2},\\\\\n&\\big| m_{x'}^{-1} \\lgs{ p_{x'}(Nt) } -  m_x^{-1} \\lgs{ p_x(Nt) } \\big|  \\le  \\mathrm{C} \\Big|\\tfrac{x'-x}N \\Big|^{1\/2},\n\\end{split}\n\\end{equation}\nand therefore also such that\n\\begin{equation}\\label{eq: bounded r and p}\n\t|\\lgs{ r_x(Nt) }| \\le \\mathrm{C}, \\quad |\\lgs{ p_x(Nt) }| \\le \\mathrm{C} .\n\\end{equation}\nIndeed, to get e.g.\\@ \\eqref{eq: Holder continuity} for $r$, we deduce from \\eqref{eq: H1 estimate} that \n\\begin{equation*}\n\\begin{split}\n\\big|\\lgs{ r_{x'}(Nt) } - \\lgs{ r_x(Nt) } \\big|&=\\left|\\sum_{z=x+1}^{x'} \\lgs{ (\\nabla_- r)_z(Nt)} \\right| \\le \\left( \\sum_{z=1}^N \\lgs{ (\\nabla_- r)_z (Nt) }^2 \\right)^{1\/2} |x-x'|^{1\/2} \\\\\n&\\le \\mathrm{C} \\Big|\\tfrac{x'-x}N \\Big|^{1\/2}.\n\\end{split}\n\\end{equation*}\n{Next \\eqref{eq: bounded r and p} follows from \\eqref{eq: Holder continuity} if, given $N,t$, there exists at least some $x_0$ such that the inequalities hold.\nThis in turn follows from \\eqref{eq: L2 estimate}.}\n\n\n\\subsection{Averaging lemma for the field $\\mathcal P_N$}\nThe method of the corrected empirical measure is an elegant method to deal with the randomness on the masses in deriving the hydrodynamic limit for $\\mathcal R_N$ and $\\mathcal P_N$ \\cite{goncalves_jara,jara_landim,bernardin}.\nHowever, in our case, it seems more convenient to use the following lemma: \n\n\\begin{Lemma}\\label{lem: replacement}\nLet $f \\in \\mathcal C^0([0,1])$ and $t\\ge 0$. \nAlmost surely (w.r.t.\\@ the masses), for $N\\to\\infty$, \n\\begin{align}\n\t &\\frac1N \\sum_{x=1}^N f (x\/N) \\frac{\\lgs{ p_x (Nt) }}{m_{x}} (m_x - \\overline m) \\quad \\to \\quad 0,\n\t \\label{eq: replacement}\\\\\n\t &\\frac1N \\sum_{x=1}^N f (x\/N) \\left(\\frac{\\lgs{ p_x (Nt) }}{m_{x}}\\right)^2 (m_x - \\overline m) \\quad \\to \\quad 0.\n\t \\label{eq: replacement bis}\n\\end{align}\n\\end{Lemma}\n\n\\begin{proof}\nLet us start with \\eqref{eq: replacement}.\nLet $A_N$ be the quantity in the left hand side of \\eqref{eq: replacement},\nlet $\\widetilde m_x = m_x - \\overline m$, and let \n\\begin{equation}\\label{eq: some def of varphi}\n\t\\varphi(x) =  f(x\/N) \\frac{\\lgs{ p_x (Nt) }}{m_{x}}\n\\end{equation}\n{where, for simplicity, we do not write explicitly the dependence of $\\varphi$ on $N$ and $t$}.\nLet $0 < \\tau < 1$. \n{Let}\n$$\n\t{\\Gamma_N^\\tau = \\{\u00a01 + k \\lfloor N^\\tau \\rfloor : k \\in \\mathbb{Z} \\} \\cap [1,N] } \n$$ \n{and, given $x\\in \\Gamma_N^\\tau$, let }\n$$\n\t{\u00a0x' = \\min \\{ y \\in \\Gamma_N^\\tau \\cup \\{\u00a0N+1\\}\u00a0: y > x\u00a0\\} .} \n$$\nWe decompose $A_N$ as\n\\begin{align*}\n\tA_N\n\t=& {\\frac{1}{N^{1 - \\tau}} \\sum_{x \\in \\Gamma_N^\\tau} \\frac{1}{N^\\tau} \\sum_{x \\le z < x'} \\varphi (z) \\tilde m_z }\\\\\n\t=&{ \\frac{1}{N^{1 - \\tau}} \\sum_{x \\in \\Gamma_N^\\tau} \\frac{\\varphi (x)}{N^\\tau} \\sum_{x \\le z < x'} \\tilde m_z  \n\t + \\frac{1}{N^{1 - \\tau}} \\sum_{x \\in \\Gamma_N^\\tau} \\frac{1}{N^\\tau} \\sum_{x \\le z < x'} (\\varphi (z) - \\varphi (x)) \\tilde m_z }\\\\\n\t=: & A_N^{(1)} + A_N^{(2)}.\n\\end{align*}\nTo deal with $A_N^{(1)}$, we observe that $\\varphi$ is bounded, see \\eqref{eq: bounded r and p}, so that by Jensen's inequality, \n$$\n\t\\mathsf E((A_N^{(1)})^4) \n\t\\le {\\frac{\\mathrm{C}}{N^{1-\\tau}} \\sum_{x \\in \\Gamma_N^\\tau} \\mathsf E \\bigg(\\bigg( \\frac{1}{N^\\tau} \\sum_{x\\le z <x'} \\widetilde m _{z}  \\bigg)^4\\bigg) }\n\t\\le \\frac{\\mathrm{C}}{N^{2 \\tau}}\n$$\n{(since the masses $m_x$ are i.i.d.).} \nTaking $\\tau > 1\/2$, this shows that $A_N^{(1)} \\to 0$ almost surely by Borel-Cantelli's lemma.\n{To deal with $A_N^{(2)}$, we start from the definition \\eqref{eq: some def of varphi} of $\\varphi$ and we bound}\n\\begin{equation}\\label{eq: bound varphi in proof lemma 2}\n\t{\n\t|\\varphi (z) - \\varphi(x)| \n\t\\le \n\t\\mathrm{C} \\left| m_z^{-1}\\lgs{ p_z (Nt) } -  m_x^{-1}\\lgs{ p_x (Nt) } \\right|\n\t+ \\mathrm{C} |f(z\/N) - f(x\/N)|.\n\t}\n\\end{equation}\n{The fact that $A_N^{(2)}\\to 0$ (deterministically) follows then from the bound \\eqref{eq: Holder continuity} and from the uniform continuity of $f$. }\n\n{The proof of \\eqref{eq: replacement bis} is entirely analogous, with now \n$\\varphi (x) = f(x\/N) (\\lgs{ p_x (Nt) } \/ m_{x} )^2$ instead of \\eqref{eq: some def of varphi}. \nBy \\eqref{eq: bounded r and p}, this function is bounded and \\eqref{eq: bound varphi in proof lemma 2} is still satisfied since\n\\begin{align*}\n\t&\\left| (m_z^{-1}\\lgs{ p_z (Nt) })^2 -  (m_x^{-1}\\lgs{ p_x (Nt) })^2 \\right|\\\\\n\t&\\le \n\t\\left| m_z^{-1}\\lgs{ p_z (Nt) } +  m_x^{-1}\\lgs{ p_x (Nt) } \\right| \\times \\left| m_z^{-1}\\lgs{ p_z (Nt) } -  m_x^{-1}\\lgs{ p_x (Nt) } \\right|\\\\\n\t&\\le \n\t\\mathrm{C} \\left| m_z^{-1}\\lgs{ p_z (Nt) } -  m_x^{-1}\\lgs{ p_x (Nt) } \\right|.\n\\end{align*}\n}\n\\end{proof}\n\n\n\n\n\\subsection{Proof of  the convergence to the linear wave equation (\\ref{R limit}-\\ref{P limit})}\\label{sec:proof-}\nFor any smooth functions $f, g:[0,1] \\in \\mathbb{R}$ such that $f(0)= f(1) = 0$, \nthe limiting fields $\\mathcal R$ and $\\mathcal P$ defined in (\\ref{eq:rpe}) \ncan be equivalently characterized as follows: \n\\begin{align}\n\t\\mathcal R (f,t) &= \\mathcal R (f,0) - \\frac{1}{\\overline m} \\int_0^t \\mathcal P (f',s) \\mathrm{d} s, \n\t\\label{eq: R characterization}\\\\\n\t\\mathcal P (g,t) &= \\mathcal P (g,0) - \\int_0^t \\mathcal R (g',s) \\mathrm{d} s,\n\t\\label{eq: P characterization}\n\\end{align}\nand \n\\begin{equation}\\label{eq: limiting fields t=0}\n\t\\mathcal R (f,0) = \\int_0^1 f(x) \\mathsf r(x) \\mathrm{d} x, \\qquad \\mathcal P(g,0) = \\int_0^1 g(x) \\mathsf p (x) \\mathrm{d} x.\n\\end{equation}\nLet us use this characterization to show that $\\mathcal R_N (f,t) \\to \\mathcal R(f,t)$ \nand  $\\mathcal P_N (g,t) \\to \\mathcal P(g,t)$.\n\nThe convergence at $t=0$ follows from the strong law of large numbers: $\\mathcal R_N (f,0)$ and $\\mathcal P_N (g,0)$ converge almost surely to $\\mathcal R (f,0)$ and $ \\mathcal P(g,0)$ given by \\eqref{eq: limiting fields t=0}. \n\nLet us next consider $t\\ge 0$, and let us first deal with $\\mathcal R_N$. \nIntegrating the equations of motion yields \n\\begin{align*}\n\t\\mathcal R_N(f,t) \n\t&= \\mathcal R_N (f ,0) + \\int_0^{Nt}   \\frac1{N}\\sum_{x=1}^N f(x\/N)\\, \\blgs{ \\big( \\nabla_+ M^{-1} p \\big)_x (s)} \\,\\mathrm{d} s \\\\\n\t&= \\mathcal R_N (f, 0) - \\int_0^{Nt}   \\frac1{N}\\sum_{x=1}^N \\nabla_- f(x\/N) m_x^{-1}  \\lgs{ p_x (s) } \\,\\mathrm{d} s\n\\end{align*}\nwhere we used the boundary condition $f(0) = f (1) = 0$ to perform the integration by part. \nSince $\\nabla_- f(x\/N) = N^{-1} f'(x\/N) + \\mathcal O (N^{-2})$, we obtain\n$$\n\t\\mathcal R_N (f,t) =  \\mathcal R_N (f, 0) - \\int_0^{t}   \\frac1{N}\\sum_{x=1}^N f'(x\/N) m_x^{-1}  \\lgs{ p_x (Ns) } \\mathrm{d} s + \\mathcal O \\Big(\\frac1N \\Big).\n$$\n{Using \\eqref{eq: replacement} in Lemma \\ref{lem: replacement}, as well as the dominated convergence theorem to deal with the time integral, \nwe may replace} $m_x^{-1}$ by $(\\overline m)^{-1}$ up to an error that vanishes almost surely in the limit $N\\to \\infty$.\nThus \n\\begin{equation}\\label{eq: RN relation}\n\t\\mathcal R_N (f,t) =  \\mathcal R_N (f, 0) -  \\frac{1}{\\overline{m}}\\int_0^{t} \\mathcal P_N (f',s) \\mathrm{d} s + \\varepsilon_N,\n\\end{equation}\nwhere $\\varepsilon_N \\to 0$ almost surely as $N \\to \\infty$. \nLet us next deal with $\\mathcal P_N$. This case is simpler since no homogenization over the masses is needed. Proceeding similarly, we find\n\\begin{align}\n\t\\mathcal P_N(g,t)\n\t&= \\mathcal P_N (g,0) + \\int_0^{Nt} \\frac1N \\sum_{x=1}^N g(x\/N)\\, \\blgs{\\big(\\nabla_- r\\big)_x(s)}\\, \\mathrm{d} s \n\t\\nonumber\\\\\n\t&= \\mathcal P_N (g,0) - \\int_0^{Nt} \\frac1N \\sum_{x=1}^N \\nabla_+ g(x\/N) \\lgs{ r_x (s)}\\, \\mathrm{d} s \n\t\\nonumber\\\\\n\t& = \\mathcal P_N (g,0) - \\int_0^t \\mathcal R_N (g',s) \\mathrm{d} s + \\tilde\\varepsilon_N\n\t\\label{eq: PN relation}\n\\end{align}\nwhere we used the boundary condition $r_0(s) = r_N(s) = 0$ for all time $s\\ge 0$ to perform the integration by part, and where $\\tilde\\varepsilon_N \\to 0$ deterministically as $N\\to \\infty$. \n\nThe families $\\big( \\mathcal R_N (f,\\cdot)\\big)_N$ and $\\big( \\mathcal P_N (g,\\cdot)\\big)_N$ are equicontinuous since a uniform bound on the time derivative of $\\mathcal R_N(f,\\cdot)$ and $ \\mathcal P_N (g,\\cdot)$ holds. \nHence, the relations \\eqref{eq: RN relation} and \\eqref{eq: PN relation} implies that any limiting point must satisfy (\\ref{eq: R characterization}-\\ref{eq: P characterization}).\n\n\n\n\\subsection{Pointwise convergence}\\label{sec:poit}\nThanks to the H\\\"older regularity\nof both $\\lgs{ r_{x}(Nt) }$ and $m_x^{-1}\\lgs{ p_{x}(Nt) }$ expressed by \\eqref{eq: Holder continuity}, \nwe deduce a stronger result: \n\\begin{Proposition}\\label{prop: pointwise convergence}\nLet $y\\in ]0,1[$ and let $t\\in ]0,1[$. \nAs $N\\to \\infty$, almost surely (w.r.t.\\@ $\\mathsf P$),  \n$$\n\t\\lgs{ r_{[Ny]}(Nt) } \\quad\\to\\quad \\mathbf r(y,t), \n\t\\qquad \n\t\\frac{\\lgs{ p_{[Ny]} (Nt) }}{m_{[Ny]}} \\quad\\to\\quad \\frac{\\mathbf p(y,t)}{\\overline{m}}.\n$$\n\\end{Proposition}\n\\begin{proof}\nLet us first deal with $\\lgs{ r_{[Ny]}(Nt) }$.\nLet $(\\rho_\\epsilon)_{\\epsilon > 0}$ be a regularizing family: \n$\\rho_\\epsilon \\in \\mathcal C^\\infty (\\mathbb{R})$, \n$\\mathrm{supp} (\\rho_\\epsilon) = [-\\epsilon, \\epsilon]$, $\\rho_\\epsilon \\ge 0$ and $\\int \\rho_\\epsilon (y) \\, \\mathrm{d} y = 1$.\nFor $y\\in ]\\epsilon, 1-\\epsilon[$, we decompose\n\\begin{equation*}\n\t\\begin{split}\n    \t&\\lgs{ r_{[Ny]} (Nt) } = \\int \\rho_\\epsilon \\Big( y-y' \\Big) \\lgs{ r_{[Ny]} (Nt) } \\mathrm{d} y'\\\\\n\t&=  \\int \\rho_\\epsilon \\Big( y-y' \\Big) \\lgs{ r_{[Ny']} (Nt) } \\mathrm{d} y' + \\int \\rho_\\epsilon \\Big( y-y' \\Big) \\left( \\lgs{r_{[Ny]} (Nt)} -\\lgs{ r_{[Ny']} (Nt) } \\right)\\mathrm{d} y'.\n  \t\\end{split}\n\\end{equation*}\nBy \\eqref{eq: Holder continuity}, the second term is bounded in absolute value by\n\\begin{equation}\t\\label{eq: Holder rest}\n  \\int \\rho_\\epsilon \\Big( y-y' \\Big) \\left| \\lgs{r_{[Ny]} (Nt)} -\\lgs{ r_{[Ny']} (Nt) } \\right|\\mathrm{d} y' \\ \\le\\\n  \\mathrm{C} \\sqrt\\epsilon,\n\\end{equation}\nwhile the first term is approximated uniformly in $N$ by\n\\begin{equation*}\n \\frac{1}N \\sum_{x=1}^N \\rho_\\epsilon \\Big( \\frac{x}N - y \\Big) \\lgs{ r_x (Nt) }.\n\\end{equation*}\nThus, by the result shown {in Section~\\ref{sec:proof-}}, this term converges to \n\\begin{equation}\\label{eq: regularized integral}\n\t\\int_0^1 \\rho_\\epsilon (y - y') \\mathbf r(y',t) \\mathrm{d} y'\n\\end{equation}\nas $N\\to \\infty$. \nLetting next $\\epsilon\\to 0$,  the continuity of $\\mathbf r(\\cdot , t)$ \nimplies that \\eqref{eq: regularized integral} converges to $\\mathbf r(y,t)$ \nwhile \\eqref{eq: Holder rest} converges to $0$ as $N\\to \\infty$. \nTo deal with $m_{[Ny]}^{-1}\\lgs{ p_{[Ny]} (Nt) }$, we proceed similarly, \n{using \\eqref{eq: replacement} in} Lemma \\ref{lem: replacement}, to get the analog of \\eqref{eq: regularized integral}. \n\\end{proof}\nFinally, thanks to the bound \\eqref{eq: bounded r and p} and the pointwise convergence result in Proposition \\ref{prop: pointwise convergence}, \nand thanks to {using \\eqref{eq: replacement} in} Lemma \\ref{lem: replacement} for the field $\\mathcal P_N$, \nwe derive  (\\ref{R limit}-\\ref{P limit}) by applying the dominated convergence theorem. \n\n\n\n\n\n\n\\section{Evolution of the energy $\\mathcal E_N$}\\label{sec: energy}\n\nIn this section we show the limit \\eqref{E limit}. We will assume that $f \\in \\mathcal C^1([0,1])$. \nWe can then recover the result \\eqref{E limit} for $f \\in \\mathcal C^0([0,1])$ by density, and using the a priori estimate $\\sum_{x} \\lgs{e_x (t)} \\le \\mathrm{C} N$ at all time $t \\ge 0$. \n\n\\subsection{Main decomposition of the energy}\\label{sec:textbfm-decomp-energ}\nIn order to derive the limit of $\\mathcal E_N$, we separate the contribution to the total energy from the temperature (that does not evolve with time) \nand from mechanical energy, i.e.\\@ the average kinetic and potential energy (that does evolve due to the transport of momentum and displacement). \n\nAt the macroscopic level, we deduce from (\\ref{eq:equa diff r}-\\ref{eq:equa diff e}) that \n\\begin{align}\n\t\\mathbf e(y,t) \n\t&= \\mathbf e (y,0) + \\frac{1}{\\overline m} \\int_0^t \\partial_y  (\\mathbf r(y,s) \\mathbf p(y,s)) \\mathrm{d} s \\nonumber\\\\\n\t&= \\frac1{\\beta(y)} + \\frac{\\mathbf p{^2} (y,0)}{2 \\overline m} + \\frac{\\mathbf r{^2} (y,0)}{2} + \\frac{1}{\\overline m} \\int_0^t \\partial_s \\Big( \\mathbf p^2(y,s)\/2 + \\overline m \\mathbf r{^2}(y,s)\/2 \\Big) \\mathrm{d} s \\nonumber\\\\\n\t&= \\frac{\\mathbf p^2 (y,t)}{2 \\overline m} + \\frac{\\mathbf r^2(y,t)}{2} + \\frac1{\\beta(y)}.\\label{eq: evolution energy limit}\n\\end{align}\nAt the microscopic level, we decompose\n\\begin{align}\n\t\\mathcal E_N(f,t)\n\t=& \\frac1N \\sum_{x=1}^N f(x\/N) \\left(  \\frac{\\lgs{ p_x^2 } }{2m_x} + \\frac{\\lgs{ r_x^2 }}{2}  \\right) (Nt) \n\t\\nonumber\\\\\n\t=& \\frac1N \\sum_{x=1}^N  f(x\/N) \\left(  \\frac{\\lgs{ p_x}^2}{2m_x} + \\frac{\\lgs{ r_x }^2}{2}  \\right) (Nt)\n\t+\\frac1N \\sum_{x=1}^N  f(x\/N) \\left(  \\frac{\\lgs{ \\widetilde p_x^2 }}{2m_x} + \\frac{\\lgs{ \\widetilde r_x^2 }}{2}  \\right) (Nt) \n\t\\nonumber\\\\ \n\t=:& \\mathcal A_N(t) + \\mathcal F_N(t), \n\t\\label{eq: average and fluctuation}\n\\end{align}\nwith \n$$ \n\t\\widetilde p_x = p_x - \\lgs{p_x}, \\qquad \\widetilde r_x = r_x - \\lgs{r_x},\n$$\nand where $\\mathcal A$ and $\\mathcal F$ stands respectively for ``average'' and ``fluctuations''. \nComparing \\eqref{eq: evolution energy limit} and \\eqref{eq: average and fluctuation}, we conclude that it is enough to show that, $\\mathsf P$ almost surely, as $N\\to \\infty$, \n\\begin{align} \n\t& \\mathcal A_N(t) \\quad \\to \\quad  \\int_0^1 f (y) \\left(  \\frac{\\mathbf p^2 (y,t)}{2 \\overline m} + \\frac{\\mathbf r^2(y,t)}{2} \\right) \\mathrm{d} y, \n\t\\label{eq: 1st limit energy}\\\\\n\t& \\mathcal F_N (t) - \\mathcal F_N (0) \\quad \\to \\quad  0, \\qquad \\mathcal F_N (0) \\quad \\to \\quad \\int_0^1 \\frac{f(y)}{\\beta (y)} \\mathrm{d} y. \n\t\\label{eq: 2d limit energy}\n\\end{align}\n\nThe limit \\eqref{eq: 1st limit energy} is deduced in the same ways as (\\ref{R limit}-\\ref{P limit}): \n{\nWe first use \\eqref{eq: replacement bis} in Lemma \\ref{lem: replacement} to replace $\\lgs{p_x}^2\/2m_x$ by $\\frac{\\overline m}{2}(\\lgs{p_x}\/m_x)^2$\nup to an error that vanishes as $N\\to \\infty$. }\nNext, thanks to the bound \\eqref{eq: bounded r and p} and the pointwise convergence result in Proposition \\ref{prop: pointwise convergence}, we derive \\eqref{eq: 1st limit energy}  by applying the dominated convergence theorem. \n\n{\nThe limit \\eqref{eq: 2d limit energy} express the fact that the profile of thermal energy (temperature) remains frozen in time. }\nIt will be established thanks to the localization of the high modes of the chain; \nthis is the only place where localization is used. \nMoreover, we will show in Section \\ref{subsec: thermal equilibrium} that in thermal equilibrium, \nthe equality $\\mathcal F_N (t) = \\mathcal F_N (0)$ holds without any assumption on the distribution of the masses (besides positivity).\nThis shows thus that Theorem \\ref{the: main result} holds actually also for a clean chain if $\\beta$ is constant.  \n\n{\n  \\begin{Remark}\\label{frozenforever}\n    The proof of \\eqref{eq: 2d limit energy} holds for any larger time scale $N^\\alpha t$, $\\alpha \\ge 1$, i.e. \n    \\begin{align} \n\t& \\mathcal F_N (N^{\\alpha -1} t) - \\mathcal F_N (0) \\quad \\to \\quad  0.\n\t\\label{eq: 2d limit energy-s}\n\\end{align}\n  \\end{Remark}\n}\n\n\\subsection{Convergence of $\\mathcal F_N (t)$}\nTo deal with the limit \\eqref{eq: 2d limit energy}, we will use the fact that any mode of the chain at positive energy is spatially localized in the thermodynamic limit. \nHence, we will expand the solutions to the equations of motion into the eigenmodes of the chain. \nIn Section \\ref{sec: eigenmodes expansion} below, we carry this expansion in details and we deduce the needed localization estimates. \nFor our present purposes, it suffices to know the following: \nThere exists a basis $\\{\\psi^k \\}_{0\\le k \\le N-1}$ of $\\mathbb{R}^N$, the basis of the eigenmodes of the chain, so that the solutions to the equations of motion read\n\\begin{align}\n&\\widetilde r_x (t)   = \\sum_{k=1}^{N-1} \\Big(  \\frac{\\langle  \\nabla_+ \\psi^k, \\widetilde r(0) \\rangle}{\\omega_k} \\cos \\omega_k t + \\langle \\psi^k , \\widetilde p(0) \\rangle \\sin \\omega_k t \\Big)  \\frac{(\\nabla_+ \\psi^k)_x}{\\omega_k}, \n\t\\label{eq:r solution bis}\\\\\n\t& \\widetilde p_x(t) = \\sum_{k = 0}^{N-1} \\Big( \\langle \\psi^k, \\widetilde p(0) \\rangle \\cos \\omega_k t  -  \\frac{\\langle \\nabla_+ \\psi^k ,\\widetilde r(0) \\rangle}{\\omega_k} \\sin \\omega_k t \\Big) (M \\psi^k)_x,\n\t\\label{eq:p solution bis}\n\\end{align}\nwhere $\\omega_0 = 0$  and $\\omega_k >0$ for $1 \\le k \\le N$ are the corresponding eigenfrequencies of the chain\n{(we assume that the $\\omega_k$ are sorted by increasing order)}.\nObserve that the first term starts from $k=1$ while the second one starts from $k=0$.\nMoreover, the orthogonality relation $\\langle \\psi^k , M \\psi^j \\rangle = \\delta (k - j)$ holds and $\\{ \\omega_k^{-1}\\nabla_+ \\psi^k  \\}_{1 \\le k \\le N-1}$ forms an orthonormal basis of $(\\mathbb{R}^{N-1}, \\langle \\cdot, \\cdot \\rangle_{N-1})$. \nSee Section  \\ref{subsec: eigenmodes solution} for more details. \nThis representation is useful to exploit localization: all modes with $k\\gtrsim \\sqrt N$ are spatially localized. \nSee Section \\ref{subsec: localization} for more quantitative estimates. \nHowever, low modes with $k\\lesssim \\sqrt N$ remain extended, and we will have to show that the contribution of these modes vanish since their proportion $N^{1\/2}\/N \\to 0$ in the thermodynamic limit. \nBelow, for technical reasons, we will replace $1\/2$ by $1-\\alpha$, for some $\\alpha > 0 $ that we will need to choose small enough.  \n\nLet $0 < \\alpha \\ll 1$, let \n$$ \n\tF_1 = \\mathbb{Z} \\cap [0,N^{1-\\alpha}], \\qquad F_{2} = \\mathbb{Z} \\cap ]N^{1-\\alpha}, N-1],\n$$\nand let us decompose $\\widetilde r(t) = \\widetilde r^{(1)}(t) + \\widetilde r^{(2)}(t)$ and $\\widetilde p (t) = \\widetilde p^{(1)} (t) + \\widetilde p^{(2)}(t)$ with \n$$\n\t\\widetilde r^{(i)}(t) = \\sum_{k\\in F_{i} \\backslash \\{0\\}} (\\dots), \\qquad \n\t\\widetilde p^{(i)}(t) = \\sum_{k\\in F_{i}} (\\dots),\n$$\nfor $i=1,2$ and with $(...)$ the summand featuring in \\eqref{eq:r solution bis} or \\eqref{eq:p solution bis}.\nWe insert this decomposition in $\\mathcal F_N$:\n$$ \n\t\\mathcal F_N (t) =  \\frac1N \\sum_{x=1}^N  f(x\/N) \\left(  \n\t\\frac{\\blgs{ \\big( \\widetilde p_x^{(1)} + \\widetilde p_x^{(2)} \\big)^2 }}{2m_x} \n\t+ \\frac{\\blgs{ \\big(\\widetilde r_x^{(1)} +  \\widetilde r_x^{(2)} \\big)^2 } }{2}  \\right) (Nt). \n$$\nLet us show the two following limits: \n\\begin{align}\n\t&\\mathcal F^{(1)}_N (t) = \\frac1N \\sum_{x=1}^N  f (x\/N) \\left(  \n\t\\frac{\\blgs{ \\big( \\widetilde p_x^{(1)} \\big)^2} }{2m_x} \n\t+ \\frac{\\blgs{ \\big(\\widetilde r_x^{(1)} \\big)^2 } }{2}  \\right) (Nt)  \\quad \\to \\quad  0, \n\t\\label{eq:F1}\\\\\n\t&\\mathcal F^{(2)}_N (t) = \\frac1N \\sum_{x=1}^N  f(x\/N) \\left(  \n\t\\frac{\\blgs{ \\big( \\widetilde p_x^{(2)}  \\big)^2} }{2m_x} \n\t+ \\frac{\\blgs{ \\big(\\widetilde r_x^{(2)}  \\big)^2 } }{2}  \\right) (Nt)  \\quad \\to \\quad  \\int\\frac{f(y)}{\\beta (y)} \\mathrm{d} y,\n\t\\label{eq:F2}\n\\end{align}\nwhich, by Cauchy-Schwarz inequality, implies \\eqref{eq: 2d limit energy}. \n\nLet us show \\eqref{eq:F1}.\nLet us bound $|f (x\/N)| \\leq \\mathrm{C}$, and use the explicit solution (\\ref{eq:r solution bis}-\\ref{eq:p solution bis}): \n\\begin{align*}\n\t&\\quad|\\mathcal F^{(1)}_N (t)| \\\\\n\t&\\le \\frac\\mathrm{C}{2N}\\sum_{x=1}^N \\frac{1}{m_x}\n\t\\BBlgs{\\bigg( \n\t\\sum_{k\\in F_1} \\Big(   \\langle \\psi^k ,\\widetilde p (0) \\rangle  \\cos (\\omega_k Nt) - \\tfrac{\\langle \\nabla_+ \\psi^k,\\widetilde r(0)\\rangle}{\\omega_k} \\sin (\\omega_k Nt ) \\Big) \n\tm_x \\psi^k_x \\bigg)^2} \\\\\n\t&+\\frac\\mathrm{C}{2N}\\sum_{x=1}^N\n\t\\BBlgs{\\bigg( \n\t\\sum_{k\\in F_1\\backslash \\{0\\}} \\Big( \\tfrac{\\langle \\nabla_+ \\psi^k, \\widetilde r(0)\\rangle}{\\omega_k} \\cos (\\omega_k Nt) + \\langle \\psi^k, \\widetilde p (0) \\rangle \\sin (\\omega_k Nt) \\Big) \n\t\\frac{(\\nabla_+ \\psi^k)_x}{\\omega_k}\n\t\\bigg)^2}.\n\\end{align*}\nIn both terms, one may expand the square so as to get a double sum over $k,j\\in F_1$ or $k,j\\in F_1 \\backslash \\{0\\}$, and insert the sum over $x$ inside the sum over $k,j$. \nThis yields\n$$ \n\t\\sum_{x=1}^N \\frac{m_x^2}{m_x} \\psi^j_x \\psi^k_x  = \\langle \\psi^j, M\\psi^k \\rangle = \\delta(k-j), \\qquad \n\t\\sum_{x=1}^N \\frac{(\\nabla_+ \\psi^j)_x (\\nabla_+ \\psi^k)_x}{\\omega_j \\omega_k} = \\delta (k-j).\n$$\nThus \n\\begin{align*}\n\t|\\mathcal F^{(1)}_N (t)| \\le\n\t&\\frac\\mathrm{C}{2N} \\sum_{k \\in F_1} \\Blgs{ \\Big(  \\langle \\psi^k ,\\widetilde p (0) \\rangle  \\cos (\\omega_k Nt) - \\frac{\\langle \\nabla_+ \\psi^k ,\\widetilde r(0)\\rangle}{\\omega_k} \\sin (\\omega_k Nt)  \\Big)^2 } \\\\\n\t&+\\frac\\mathrm{C}{2N} \\sum_{k \\in F_1 \\backslash \\{0\\}} \\Blgs{ \\Big(  \\frac{\\langle \\nabla_+ \\psi^k ,\\widetilde r(0)\\rangle}{\\omega_k} \\cos (\\omega_k Nt) + \\langle \\psi^k, \\widetilde p(0) \\rangle \\sin (\\omega_k Nt)  \\Big)^2}.\n\\end{align*}\nAt this point, it suffices to show that there exists a constant $\\mathrm{C}$ such that, for all $k \\in F_1$, \n\\begin{equation}\n\\label{eq:avant30}\n\\lgs{ \\langle \\psi^k,\\widetilde p(0) \\rangle^2 } \\le \\mathrm{C}, \\qquad \\frac{\\lgs{ \\langle \\nabla_+ \\psi^k, \\widetilde r(0) \\rangle^2 }}{\\omega^2_k} \\le \\mathrm{C}, \n\\end{equation}\nsince, bounding $\\sin$ and $\\cos$ by 1, and using Cauchy-Schwarz, we obtain \n$$\n\t|\\mathcal F^{(1)}_N (t)| \\le \\frac{\\mathrm{C}}{N} \\sum_{k \\in F_1} 1 = \\frac{\\mathrm{C} N^{1 - \\alpha}}{N} \\to 0.\n$$\nLet us deal with $\\lgs{ \\langle \\psi^k,\\tilde p(0) \\rangle^2 }$ (the other case is analogous): \n\\begin{equation*}\n\\begin{split}\n\\lgs{ \\langle \\psi^k , \\tilde p (0) \\rangle^2 }\n\t&= \\Blgs{\\Big( \\sum_x \\psi^k_x  \\widetilde p_x (0)  \\Big)^2}\n\t= \\Blgs{ \\sum_{x,y}   \\psi^k_x  \\psi^k_y \\widetilde p_x (0) \\widetilde p_y (0)  }\\\\\n\t&= \\sum_x (\\psi^k_x)^2  \\lgs{ (\\widetilde p_x (0))^2 }\n\\end{split}\n\\end{equation*}\nwhere we have used the fact that $\\lgs{\\cdot}$ is a product measure and that $\\lgs{\\widetilde p_x(0)} = 0$\n for all $x\\in \\{1, \\dots , N\\}$. \nWe compute $ \\lgs{ (\\widetilde p_x (0))^2 } = \\frac{m_x}{\\beta(x\/N)}$. \nSince $\\beta$ is positive and continuous on $[0,1]$, there exists $\\beta_->0$\n such that $\\beta (x\/N) \\ge \\beta_-$ for all $x\\in \\{1, \\dots , N\\}$. \nHence\n\\begin{equation}\\label{eq: beta minus equation}\n\t\\lgs{ \\langle \\psi^k, \\widetilde p (0) \\rangle^2 } \\le \\frac{1}{\\beta_-} \\sum_{x=1}^N m_x (\\psi^k_x)^2 = \\frac{1}{\\beta_-} \\langle \\psi^k , M \\psi^k \\rangle=\\frac{1}{\\beta_-}. \n\\end{equation}\nLet us now show \\eqref{eq:F2}. \nA computation using the initial measure shows that \n$\\mathcal F(0) \\to \\int \\frac{f(y)}{\\beta (y)} \\mathrm{d} y$ as $N\\to \\infty$. \nHence, thanks to \\eqref{eq:F1}, it holds that $\\mathcal F^{(2)}_N (0)\n \\to  \\int \\frac{f(y)}{\\beta (y)} \\mathrm{d} y$ as $N\\to \\infty$. \nThus it suffices to show that $\\mathcal F^{(2)}_N (t) - \\mathcal F^{(2)}_N (0) \\to 0$ as $N\\to \\infty$. \nLet us write $\\mathcal F^{(2)}_N(t)$ as a scalar product and expand it in the eigenmodes basis: \n\\begin{align*}\n\t\\mathcal F^{(2)}_N(t) \n\t=&  \\frac{1}{2N}\\blgs{ \\langle (f \\cdot \\widetilde p^{(2)})(Nt), M^{-1} \\widetilde p^{(2)} (Nt)\\rangle  + \\langle ( f \\cdot \\widetilde r^{(2)} )(Nt), \\widetilde r^{(2)}(Nt) \\rangle  }\\\\\n\t=& \\frac{1}{2N} \\sum_{k\\in F_2}\\Blgs{ \\langle ( f \\cdot \\widetilde p^{(2)})(Nt), \\psi^k \\rangle \\langle \\psi^k, \\widetilde p (Nt)\\rangle \\\\\n\t&\\phantom{ \\frac{1}{2N} \\sum_{k\\in F_2}\\langle\\langle} +\\frac{1}{\\omega_k^2}  \\langle (f \\cdot \\widetilde r^{(2)}) (Nt), \\nabla_+ \\psi^k \\rangle \\langle \\nabla_+\\psi^k ,  \\widetilde r(Nt) \\rangle} .\n\\end{align*}\nHere $g\\cdot h$ denotes a function on $\\mathbb{Z}\\cap [1,N]$ obtained by the usual multiplication in real space between a function $g$ on $[0,1]$ and $h$ on $\\mathbb{Z}\\cap [1,N]$, i.e.\\@ $(g\\cdot h)_x = g(x\/N) h_x$.\nBy Lemma \\ref{lem: localization} stated in Section \\ref{sec: eigenmodes expansion} below, one may associate a localization center $x_0(k)$ to each mode $\\psi^k$ with $k\\in F_2$: \n$x_0(k)$ is the center of the interval $J(k)$ featuring there \n(assuming that $\\alpha$ is small enough so that the hypotheses of Lemma \\ref{lem: localization} are satisfied).\nFor each $k\\in F_2$, let us decompose $f$ as \n$$\n\tf = f_0 (k) + \\widetilde f_k \\quad \\text{with} \\quad f_0 (k) = f (x_0(k)\/N) \n$$\n(thus $f_0(k)$ is a constant and $\\widetilde f_k$ vanishes at $x_0 (k)\/N$).\nWe insert this decomposition in the above expression for $\\mathcal F_N^{(2)}(t)$: \n\\begin{align}\n\t\\mathcal F^{(2)}_N(t) \n\t= &\\frac{1}{2N} \\sum_{k\\in F_2} f_0 (k) \\,\\BBlgs{ \\langle \\widetilde p(Nt) , \\psi^k \\rangle^2 + \\frac{\\langle \\widetilde r (Nt),\\nabla_+ \\psi^k \\rangle^2}{\\omega_k^2}   } \\label{eq: F2 part 1} \\\\\n\t&+ \\frac1{2N}\\sum_{k\\in F_2} \\BBlgs{ \\langle (\\widetilde f_k \\cdot \\widetilde p^{(2)})(Nt), \\psi^k \\rangle \\langle \\psi^k, \\widetilde p (Nt)\\rangle \\nonumber\\\\\n\t&\\phantom{\\frac1{2N}\\sum_{k\\in F_2} \\Blgs{.}} +\\frac{1}{\\omega_k^2}  \\langle (\\widetilde f_k \\cdot \\widetilde r^{(2)}) (Nt), \\nabla_+ \\psi^k \\rangle \\langle \\nabla_+\\psi^k ,  \\widetilde r(Nt) \\rangle}. \\label{eq: F2 part 2}\n\\end{align}\nEach expression between $\\lgs{\\dots}$ in the sum in \\eqref{eq: F2 part 1} represents the energy of the mode $\\psi^k$ and does not evolve with time, \nsee \\eqref{eq: energy of the modes} in Section \\ref{sec: eigenmodes expansion} below. \nHence, to show $\\mathcal F^{(2)}_N (t) - {\\mathcal F}^{(2)}_N (0) \\to 0$, we only need to show that the sum in \\eqref{eq: F2 part 2} converges to 0 as $N\\to \\infty$.\n\nLet us consider a single term in the sum \\eqref{eq: F2 part 2}, and let us focus on the term involving $\\widetilde p$ (the one involving $\\widetilde r$ is treated the same way). \nFirst, by Cauchy-Schwarz, \n\\begin{equation}\\label{eq: to show for localization}\n\\blgs{\\langle (\\widetilde f_k \\cdot\\widetilde p^{(2)})(Nt), \\psi^k \\rangle \\langle \\psi^k, \\widetilde p (Nt)\\rangle }\n\\le \\lgs{\\langle (\\widetilde f_k \\cdot \\widetilde p^{(2)})(Nt), \\psi^k \\rangle^{2}}^{1\/2} \\lgs{ \\langle \\psi^k, \\widetilde p (Nt)\\rangle^{2}}^{1\/2}.\n\\end{equation}\nThe second factor in \\eqref{eq: to show for localization} is bounded by a constant: \n\\begin{align}\n\t \\lgs{ \\langle \\psi^k, \\widetilde p (Nt)\\rangle^{2}}\n\t &= \\BBlgs{ \\Big( \\langle \\psi^k , \\widetilde p(0) \\rangle \\cos \\omega_k Nt - \\frac{\\langle \\nabla_+ \\psi^k , \\widetilde r(0) \\rangle}{\\omega_k} \\sin \\omega_k Nt  \\Big)^2 }  \\nonumber\\\\\n\t &\\le 2 \\,\\BBlgs{  \\langle \\psi^k , \\widetilde p(0) \\rangle^2 + \\frac{\\langle \\nabla_+ \\psi^k , \\widetilde r(0) \\rangle^2}{\\omega_k^2} }\\, \\le \\mathrm{C},\\label{eq: to show for localization next 2}\n\\end{align}\nsee \\eqref{eq:avant30}.\nFor the first factor in \\eqref{eq: to show for localization}, we use again Cauchy-Schwarz to get\n\\begin{equation}\n\\label{eq: to show for localization next}\n\\begin{split}\n\t\\lgs{\\langle (\\widetilde f_k \\cdot \\widetilde p^{(2)})(Nt), \\psi^k \\rangle^{2}}\n\t&= \\Blgs{\\Big( \\sum_x \\widetilde f_k(x\/N) \\widetilde p^{(2)}_x (Nt) \\psi^k_x \\Big)^2}\\\\\n\t&\\le \\Big( \\sum_x \\widetilde f_k^2 (x\/N) (\\psi^k_x)^2 \\Big) \\,\\Blgs{\\sum_x \\big( \\widetilde p^{(2)}_x\\big)^2 (Nt)}\\, .\n\t\\end{split}\n\\end{equation}\nFor $\\widetilde f_k$, we have the bound \n$$\n\t|\\widetilde f_k (x\/N)| = |f_k (x\/N) - f_k (x_0(k)\/N) |\\le \\mathrm{C} \\,\\frac{|x-x_0 (k) |}{N}\n$$\n(this is the only place where we use $f \\in \\mathcal C^1([0,1])$).\nHence, form Lemma \\ref{lem: localization} below, we deduce that for any $\\epsilon > 0$, the first factor in the right hand side of \\eqref{eq: to show for localization next} can be bounded by $1\/N^{2-\\epsilon}$ by taking $\\alpha > 0$ small enough.\nFrom the conservation of energy (see (\\ref{eq: energy of the modes}) and the bound (\\ref{eq:avant30})) , the second factor in \\eqref{eq: to show for localization next} is $\\mathcal O (N)$.\nHence, for $\\alpha>0$ small enough, \\eqref{eq: to show for localization next} goes to zero as $N\\to \\infty$. \nCombining this with \\eqref{eq: to show for localization next 2}, we find that  \\eqref{eq: to show for localization} goes to zero as $N \\to \\infty$, and hence that \\eqref{eq: F2 part 2} converges to 0 as $N\\to \\infty$.\n\n\n\n\n\n\\subsection{Thermal equilibrium case}\\label{subsec: thermal equilibrium}\n\nAssume here that\n there exists some $\\overline{\\beta} >0$ such that $\\beta (y) = \\overline{\\beta}$ for all $y \\in [0,1]$. \nThen, we may relax the assumptions on the masses: requiring only that they are all strictly positive, let us show that $\\mathcal F_N (t) = F_N (0)$ for all $t \\ge 0$. \nThis results from an exact computation. \n\nSince $f$ is arbitrary, it is necessary and sufficient to prove that, for any $x$, \n$$ \n\t\\frac{\\mathrm{d} }{ \\mathrm{d} t} \\left(  \\frac{\\lgs{\\tilde p_x^2(t)}}{2 m_x} + \\frac{\\lgs{\\tilde r_x^2(t)}}{2} \\right) = 0. \n$$\nWe compute \n\\begin{multline*}\n\t\\frac{\\tilde p_x^2(t)}{2 m_x} \n\t=\\frac12 \\sum_{j,k} \\Big( \\langle \\psi^j , \\tilde p(0) \\rangle \\cos \\omega_j t - \\frac{\\langle \\nabla_+ \\psi^j , \\tilde r(0)\\rangle}{\\omega_j} \\sin \\omega_j t  \\Big)\n\t {\\,\\times} \\\\\n\t \\Big( \\langle \\psi^k, \\tilde p(0) \\rangle \\cos \\omega_k t - \\frac{\\langle \\nabla_+ \\psi^k , \\tilde r(0)\\rangle}{\\omega_j} \\sin \\omega_k t  \\Big)  m_x \\psi^j_x \\psi^k_x.\n\\end{multline*}\nA similar expression holds for $\\tilde r^{2}(t)\/2$. \nIn order to obtain the expectation with respect to $\\lgs{\\cdot}$, we compute\n\\begin{eqnarray}\n\t\t\\blgs{\\langle \\psi^k , \\tilde p (0) \\rangle \\langle \\psi^j , \\tilde p (0) \\rangle} & = & \\frac{\\delta (k-j)}{\\overline{\\beta}}, \\label{cancellation beta const 1} \\\\\n\t\t\\blgs{\\langle \\psi^k , \\tilde p (0) \\rangle \\langle \\nabla_+ \\psi^j , \\tilde r (0) \\rangle} &  = & 0, \\label{cancellation beta const 2} \\\\\n\t\t\\blgs{\\langle \\nabla_+ \\psi^k , \\tilde r (0) \\rangle \\langle \\nabla_+ \\psi^j , \\tilde r (0) \\rangle} &  = & \\frac{\\omega_k^2 \\delta (k-j)}{\\overline\\beta}. \\label{cancellation beta const 3}\n\\end{eqnarray}\nThese three properties result from the fact the product structure of $\\rho_{\\mathrm{loc}}$, from the fact that the variables $\\tilde p$ and $\\tilde r$ are centered, \nand from the the fact that $\\beta$ is constant for  \\eqref{cancellation beta const 1} and \\eqref{cancellation beta const 3}.\nE.g.\\@ \\eqref{cancellation beta const 1} is obtained by \n\\begin{equation*}\n\\begin{split}\n\\blgs{\\langle \\psi^k , \\tilde p (0) \\rangle \\langle \\psi^j , \\tilde p (0) \\rangle} \n\t&= \\sum_{x,y}\\psi^k_x  \\psi^j_y \\lgs{ \\tilde p_x (0) \\tilde p_y (0)} = \\frac{1}{\\beta} \\sum_x m_x \\psi^k_x \\psi^j_x \\\\\n\t&= \\frac{\\delta (k-j)}{\\overline \\beta}.\n\\end{split}\n\\end{equation*}\nHence we have that\n$$\n\t \\frac{\\lgs{\\tilde p_x^2(t)}}{2 m_x} = \\frac{1}{2\\beta} \\sum_k( \\cos^2 \\omega_kt + \\sin^2 \\omega_k t ) m_x (\\psi^k_x)^2 = \\frac{1}{2\\beta}\n$$\nand similarly $\\lgs{\\tilde r_x^2(t)}\/2 = \\frac1{2\\beta}$. \n\n\n\n\n\n\n\\section{Eigenmodes expansion: integrability, localization}\\label{sec: eigenmodes expansion}\n\nWe describe an explicit solution to the equations of motion \\eqref{eq: equations of motion} in terms of the eigenmodes of the system. \nThis representation is useful to establish the integrability of the system and to exploit the localization at all energies above the ground states (in the thermodynamic limit).  \n\n\\subsection{Solution to the equations of motion}\\label{subsec: eigenmodes solution} \nFrom \\eqref{eq: equations of motion}, one can deduce second order equations for $r$ and $p$ separately:\n$$ \n\t \\ddot r_x = \\big(\\nabla_+ M^{-1} \\nabla_- r\\big)_x \\quad (1 \\le x \\le N-1), \\qquad \\ddot p_x = \\big( \\Delta M^{-1}p\\big)_x \\quad (1 \\le x \\le N),\n$$ \nwhere, besides the boundary conditions $r_0 = r_N = 0$, we have assumed free boundary conditions for $M^{-1}p$, i.e. $m_0^{-1}p_0 = m_1^{-1}p_1$ and $m_{N+1}^{-1}p_{N+1} = m_N^{-1}p_{N}$. \nNotice that there are two different vector spaces: a $(N-1)$-dimensional space for $r$ with fixed boundary conditions, and a $N$-dimensional space for $M^{-1}p$ with free boundary conditions.\nMoreover, we observe that $\\nabla_+ = - (\\nabla_-)^\\dagger$ with fixed boundary conditions, and that $\\Delta = \\Delta^\\dagger$ with free boundary conditions. \n\nIn order to solve the equations of motion, we need to diagonalize two matrices: \n$\\big(\\nabla_+ M^{-1} \\nabla_-\\big)^\\dagger=\\nabla_+ M^{-1} \\nabla_-$ (of size $N-1$) \nand $\\big(\\Delta M^{-1} \\big)^\\dagger=M^{-1}\\Delta$ (of size $N$). \n{Let us start with the latter:}\nThis matrix is not symmetric but the matrix $ M^{-1\/2} (-\\Delta) M^{-1\/2} $ is symmetric and non-negative.\nIt admits thus an orthonormal set of eigenvectors, $\\{\\varphi^k\\}_{0 \\le k \\le N-1}$ and corresponding eigenvalues $\\omega^2_k$\n{that we assume to be sorted by increasing order}. \nMoreover, the spectrum is $\\mathsf P$-almost surely non-degenerate \n(see e.g.\\@ Proposition II.1 in \\cite{kunz_souillard}, considering here a perturbation around the non-degenerate equal masses case). \nTherefore the vectors $\\psi^k = M^{-1\/2} \\varphi^k$ are such that \n\\begin{equation}\\label{eq: eigenmodes equation}\n\tM^{-1} (-\\Delta) \\psi^k = \\omega^2_k  \\psi^k, \\qquad \\langle \\psi^j, M \\psi^k \\rangle = \\delta(j-k).\n\\end{equation}\nBecause of free boundary conditions, $\\omega^2_0 = 0$, and {one may chose $\\psi_0$ to be given by}\n$$\n\t\\psi_0 = \\Big(\\sum_{x}m_x\\Big)^{-1\/2} (1, \\dots , 1)^\\dagger .   \n$$\n{Next}, the matrix ${-} \\nabla_+ M^{-1} \\nabla_- $ is symmetric and non-negative, \nand we denote the eigenvectors by $|\\widetilde\\psi^k \\rangle$ for $1 \\le k \\le N-1$.\nIt is readily checked that {they may be chosen to be given by}\n$$\n\t\\widetilde \\psi^k = \\frac{1}{\\omega_k} \\nabla_+ \\psi^k\n$$\nwith the corresponding eigenvalue given by $\\omega_k$ for $1 \\le k \\le N-1$. \nWe observe that, by the free boundary conditions on $\\psi^k$,  $\\widetilde \\psi^k (0) = \\widetilde \\psi^k (N) = 0$. \n \nGiven initial conditions $r(0),p(0)$, we can write an explicit solution for $r(t),p(t)$: \n$$ \n\t\\langle \\widetilde \\psi^k , \\ddot{r} \\rangle = -\\omega^2_k \\langle \\widetilde \\psi^k, r\\rangle \\quad (1 \\le k \\le N-1), \n\t\\qquad \n\t\\langle \\psi^k, \\ddot p \\rangle = -\\omega^2_k \\langle \\psi^k, p \\rangle \\quad (0 \\le k \\le N-1).\n$$\nThus\n\\begin{align*}\n\t&\\langle \\widetilde\\psi^k, r(t) \\rangle = \\langle \\widetilde \\psi^k, r(0) \\rangle \\cos \\omega_k t + \\frac{\\langle \\widetilde\\psi^k, \\nabla_+ M^{-1}p(0) \\rangle}{\\omega_k} \\sin \\omega_k t \\quad (1 \\le k \\le N- 1),\\\\\n\t&\\langle \\psi^k ,p(t) \\rangle = \\langle \\psi^k ,p(0) \\rangle \\cos \\omega_k t + \\frac{\\langle \\nabla_- r(0),\\psi^k \\rangle}{\\omega_k} \\sin \\omega_k t \\quad (0 \\le k \\le N-1)\n\\end{align*}\nwith the convention $\\frac{\\sin 0}{0} = 1$ in the second expression at $k=0$ (notice that $\\langle \\nabla_- r(0) ,\\psi^k \\rangle = - \\langle r(0) , \\nabla_+\\psi^k \\rangle = 0$ for $k=0$). This yields therefore\n\\begin{align}\n\t&r(t) = \\sum_{k=1}^{N-1} \\Big(  \\frac{\\langle  \\nabla_+ \\psi^k ,r(0) \\rangle}{\\omega_k} \\cos \\omega_k t + \\langle \\psi^k , p(0) \\rangle \\sin \\omega_k t \\Big)  \\frac{\\nabla_+ \\psi^k}{\\omega_k}, \n\t\\label{eq:r solution}\\\\\n\t&p(t) = \\sum_{k = 0}^{N-1} \\Big( \\langle \\psi^k ,p(0) \\rangle \\cos \\omega_k t  -  \\frac{\\langle \\nabla_+ \\psi^k ,r(0) \\rangle}{\\omega_k} \\sin \\omega_k t \\Big) M \\psi^k.\n\t\\label{eq:p solution}\n\\end{align}\n\n\n\\subsection{Full set of invariant quantities}\\label{subsec: invariant quantities} \nWe observe that the dynamics has $N$ invariant quantities, corresponding to the energy of each mode. \nIt is thus an integrable system. \nIndeed, let us write the full energy as \n\\begin{align*}\n\tH &= \\frac12 \\big( \\langle p, M^{-1}p \\rangle + \\langle r,r \\rangle \\big)\n\t= \\frac12 \\sum_{k=0}^{N-1} \\langle p,\\psi^k \\rangle \\langle M\\psi^k, M^{-1}p \\rangle + \\frac12 \\sum_{k=1}^{N-1} \\langle r, \\widetilde \\psi^k \\rangle \\langle \\widetilde \\psi^k, r \\rangle \\\\\n\t&= \\frac12 \\sum_{k=0}^{N-1} \\Big(  \\langle p,\\psi^k \\rangle^2 + \\frac{\\langle r ,\\nabla_+ \\psi^k \\rangle^2}{\\omega_k^2} \\Big)\n\\end{align*}\nwith the convention that the second term in the last sum is $0$ at $k=0$.\nFrom the evolution equation of the dynamics, one gets that actually \n\\begin{equation}\n\\label{eq: energy of the modes}\n\t\\frac{\\mathrm{d} }{ \\mathrm{d} t} \\left(  \\langle p ,\\psi^k \\rangle^2 + \\frac{\\langle r, \\nabla_+ \\psi^k \\rangle^2}{\\omega_k^2} \\right) = 0 \\quad\\text{for all} \\quad 0 \\le k \\le N-1. \n\\end{equation}\n\nMoreover, by taking specific linear combinations of these conserved quantities, one can obtain conserved quantities that can be written as a sum of local terms. \nThis is for instance the case of the quantity $I$ defined in \\eqref{eq: I conserved}, that reads also \n$$\n\tI(r,p) = \\frac12 \\sum_{k=1}^{N-1} \\omega_k^2 \\, \\left(  \\langle p,\\psi^k \\rangle^2 + \\frac{\\langle r,\\nabla_+ \\psi^k \\rangle^2}{\\omega_k^2} \\right).\n$$\n\n\n\n\n\n\\subsection{Localization}\\label{subsec: localization} \nLocalization can be expressed mathematically in the following strong sense, \nsee \\cite{kunz_souillard,aizenman_graff,aizenman_molchanov} for the general theory and \\cite{verheggen,ajanki_huveneers} for precise estimates on the localization length as one approaches the ground state. \nLet $0 < \\alpha < \\frac12$ and let $I(\\alpha) = ]N^{(1- \\alpha)},N-1]\\cap \\mathbb{Z}$. \nThere exist constants $\\mathrm{C}, c >0$ such that \n$$\n\t\\mathsf E \\Big( \\sum_{k \\in I(\\alpha)} |\\psi^k_x \\psi^k_y | \\Big) \\le \\mathrm{C} \\mathrm{e}^{-c|x-y|\/\\zeta(\\alpha)} \\quad \\text{with} \\quad \\zeta(\\gamma) = N^{2\\alpha}.\n$$\nWe will use this estimate to show that every mode in $k \\in I(\\alpha)$ is supported in a small subset of $[1,N] \\cap \\mathbb{Z}$ up to a small error: \n\n\\begin{Lemma}\\label{lem: localization}\nLet $\\alpha, \\gamma>0$ be such that $2 \\alpha < \\gamma < 1$.\nThere exists almost surely $N_0 \\in\\mathbb{N}$ so that for all $N \\ge N_0$, and for all $k \\in I (\\alpha)$, \nthere exists an interval $J(k)$ with $|J(k)| \\le 2N^\\gamma$ such that $|\\psi^k_x | \\le N^{-1\/\\gamma}$ for all $x \\notin J(k)\\cap \\mathbb{Z}$. \n\\end{Lemma}\n\n\n\\begin{proof}\nLet us first show that \n\\begin{align}\n\tP &:= \\mathsf P\\big( \\exists k \\in I(\\alpha), \\exists x,y\\in [1,N]\\cap \\mathbb{Z} : |x-y| \\ge N^\\gamma, |\\psi^k_x | \\ge N^{-1\/\\gamma},  |\\psi^k_y | \\ge N^{-1\/\\gamma}   \\big)  \n\t\\nonumber\\\\\n\t&\\le\\; \\mathrm{C}(\\alpha, \\gamma) \\mathrm{e}^{- N^{(\\gamma - 2\\alpha)\/2}}\n\t\\label{eq: proba 0 event localization}\n\\end{align}\nIndeed we compute\n\\begin{align*}\n\tP &\\le \\sum_{k\\in I(\\alpha)} \\sum_{x,y : |x-y| \\ge N^\\gamma} \\mathsf P \\big(  |\\psi^k_x | \\ge N^{-1\/\\gamma},  |\\psi^k_y | \\ge N^{-1\/\\gamma} \\big) \\\\\n\t&\\le \\sum_{k\\in I(\\alpha)}  \\sum_{x,y : |x-y| \\ge N^\\gamma} \\mathsf P \\big(  |\\psi^k_x\\psi^k_y | \\ge N^{-2\/\\gamma} \\big) \\\\\n\t&\\le \\sum_{k\\in I(\\alpha)}  \\sum_{x,y : |x-y| \\ge N^\\gamma} N^{2\/\\gamma} \\mathsf E ( |\\psi^k_x\\psi^k_y |) \\\\\n\t&\\le \\mathrm{C} N^{2\/\\gamma} \\sum_{x,y : |x-y| \\ge N^\\gamma} \\mathrm{e}^{-c|x-y|\/\\zeta(\\alpha)} \n\t\\le \\mathrm{C} N^{2\/\\gamma} N^{2} \\mathrm{e}^{- N^{\\gamma - 2 \\alpha}} \n\t\\le  \\mathrm{C}(\\alpha, \\gamma) \\mathrm{e}^{- N^{(\\gamma - 2\\alpha)\/2}}.\n\n\\end{align*}\nTherefore, there exists almost surely $N_0$ so that for for all $N\\ge N_0$, the event featuring in \\eqref{eq: proba 0 event localization} does not occur. \nHence in this case, for all $k \\in I(\\alpha)$ and all $|x - y|>N^{\\gamma}$, we must have either $|\\psi^k_x | \\le 1\/N^{1\/\\gamma}$ or $|\\psi^k_y | \\le 1\/N^{1\/\\gamma}$.\nThis means thus that for any $k\\in I(\\alpha)$  there exists an interval $J(k)$ with $|J(k)| \\le 2N^\\gamma$ such that $|\\psi^k_x | \\le N^{-1\/\\gamma}$ for all $x \\notin J(k)\\cap \\mathbb{Z}$. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{section::introduction}\n\nThe magnetotelluric (MT) method uses electromagnetic passive sources\nin the magnetosphere and ionosphere to estimate electrical resistivity\nin Earth's interior \\cite{chave1991electrical}. This method was\nintroduced almost 70 years ago by Cagniard\n\\cite{cagniard1953basic}. The passive sources excite a certain\nspectrum, that is, interval of frequencies. There is a long history of\nstudies based on some form of data fitting including a Bayesian\napproach \\cite{grandis1999bayesian}. It has been understood that high\nfrequency data enable determining the resistivity in the ``near''\nsubsurface, while low frequency ones enable obtaining an estimate of\nthe resistivity in the ``deep'' interior. The MT method has been used\nto study melt and hydration in Earth's crust and mantle\n\\cite{rokityansky2012geoelectromagnetic, feucht2017magnetotelluric,\n  evans1994electrical, rippe2013magnetotelluric}, and general\nelectrical structure in the lithosphere\n\\cite{malleswari2019magnetotelluric, thiel2008modelling} and below the\nocean floor \\cite{cox1971electromagnetic}. Moreover, the MT method has\nbeen applied to earthquake prediction \\cite{chouliaras1988application}.\n\n\\smallskip\n\nIn this paper, we analyze the inverse problem for the MT method. We\nlet $\\Omega \\subset \\R^3$ be an open bounded domain with $C^{1,1}$\nboundary and let $\\varepsilon, \\sigma, \\mu \\in C^2(\\overline\\Omega)$\nbe non-negative. We consider the time-harmonic Maxwell equations for\nelectromagnetic fields, $E$ and $H$,\n\\begin{equation}\\label{eqn::Maxwell}\n   \\nabla \\times E = i \\omega \\mu H\\quad\\text{and}\\quad\n   \\nabla \\times H = -i \\omega\n          \\Big(\\varepsilon + \\frac{i \\sigma}{\\omega}\\Big) E\n   \\quad\\text{in}\\quad\\Omega ,\n\\end{equation}\nwhere $\\omega > 0$ is a fixed frequency. The functions $\\varepsilon$,\n$\\sigma$ and $\\mu$ represent the material parameters, namely, electrical\npermittivity, electrical conductivity and magnetic permeability,\nrespectively. In the case of MT, one invokes the following\n\n\\begin{Assumption} \\label{MT assumption}\nThe electrical permittivity vanishes, $\\varepsilon = 0$, and the\nelectrical conductivity and magnetic permeability satisfy $\\sigma \\ge\n\\sigma_0$, $\\mu \\ge \\mu_0$ on $\\overline\\Omega$ for some constants\n$\\sigma_0, \\mu_0 > 0$.\n\\end{Assumption}\n\n\\noindent\nThis assumption has its origin in the low-frequency reduction of\nMaxwell's equations.\n\nWe let $\\nu$ be the outer unit normal to the boundary $\\p \\Omega$, and\ndefine the trace operator $\\bt: C^\\infty(\\overline\\Omega;\\C^3) \\to\nC^\\infty(\\p\\Omega;\\C^3)$ as\n$$\n   \\bt(u) := \\nu \\times u|_{\\p\\Omega}\\quad\\text{for}\\quad\n   u \\in C^\\infty(\\overline\\Omega;\\C^3) .\n$$\nThe trace $\\bt$ can be extended to a bounded linear operator from\n$H_{\\Div}^1(\\Omega)$ into $TH^{1\/2}_{\\Div}(\\p\\Omega)$, where\n$$\n   H^1_{\\Div}(\\Omega) := \\left\\{\n   u \\in H^1(\\Omega;\\C^3) : \\bt(u) \\in TH^{1\/2}_{\\Div}(\\p\\Omega)\\right\\}\n$$\nand\n$$\n   TH^{1\/2}_{\\Div}(\\p\\Omega) := \\left\\{f \\in H^{1\/2}(\\p \\Omega;\\C^3) :\n   \\nu \\cdot f = 0\\quad\\text{and}\\quad\n   \\Div(f)\\in H^{1\/2}(\\p \\Omega;\\C^3)\\right\\} ,\n$$\nand $\\Div$ denotes the surface divergence on $\\p\\Omega$. We refer the\nreader to \\cite{ola1993inverse} for more details. Under\nAssumption~\\ref{MT assumption}, when $\\omega > 0$ does not belong to a\ndiscrete set of magnetic resonant frequencies, the equation\n\\eqref{eqn::Maxwell} with the boundary condition $\\bt(H) = f \\in\nTH^{1\/2}_{\\Div}(\\p\\Omega)$ has a unique solution $(H,E) \\in\nH^1_{\\Div}(\\Omega)\\times H^1_{\\Div}(\\Omega)$; see\nSection~\\ref{section::well posedness}. Then the impedance map\n$Z^\\omega_{\\sigma,\\mu}: TH^{1\/2}_{\\Div}(\\p\\Omega)\\to\nTH^{1\/2}_{\\Div}(\\p\\Omega)$ is defined as\n$$\n   Z^\\omega_{\\sigma,\\mu}(f)\n           := \\bt(E),\\quad f\\in TH^{1\/2}_{\\Div}(\\p\\Omega) .\n$$\nThe inverse MT problem (IMTP) is to determine $\\sigma$ and $\\mu$ from\nthe knowledge $Z^\\omega_{\\sigma,\\mu}$. In the geophysics literature,\none commonly considers the resistivity, that is, the reciprocal of\nconductivity \\cite{weckmann2003images}.\n\n\\smallskip\n\n\\begin{Remark}\nIn practice, the data are represented as the graph of\n$Z^\\omega_{\\sigma,\\mu} : TH^{1\/2}_{\\Div}(\\p\\Omega) \\to\nTH^{1\/2}_{\\Div}(\\p\\Omega)$.\n\\end{Remark}\n\nOur first main result is\n\n\\begin{Theorem}\\label{main thm}\nLet $\\Omega\\subset \\R^3$ be a bounded domain with $C^{1,1}$ boundary\nand let $\\sigma_j,\\mu_j\\in C^2(\\overline\\Omega)$, $j=1,2$, be such\nthat $\\sigma_j \\ge \\sigma_0$ and $\\mu_j \\ge \\mu_0$ for some constants\n$\\sigma_0, \\mu_0 > 0$. Suppose that $\\omega > 0$ is not a resonant\nfrequency for $(\\sigma_1,\\mu_1)$ and $(\\sigma_2,\\mu_2)$ and that\n\\begin{equation}\\label{boundary assumption}\n   \\p^\\alpha \\sigma_1|_{\\p\\Omega} = \\p^\\alpha \\sigma_2|_{\\p\\Omega}\n   \\quad\\text{and}\\quad\n   \\p^\\alpha \\mu_1|_{\\p\\Omega} = \\p^\\alpha \\mu_2|_{\\p\\Omega}\n   \\quad\\text{for}\\quad|\\alpha| \\le 2 .\n\\end{equation}\nThen $Z_{\\sigma_1,\\mu_1}^\\omega = Z_{\\sigma_2,\\mu_2}^\\omega$ implies\nthat $\\sigma_1 = \\sigma_2$ and $\\mu_1 = \\mu_2$.\n\\end{Theorem}\n\n\\noindent\nCondition \\eqref{boundary assumption} is not important. We expect that\na suitable boundary determination result would allow to remove it as\nin \\cite{caro2019boundary, joshi2000total, mcdowall1997boundary}. See\nalso Remark~\\ref{remark in Appendix B}.\n\n\\smallskip\n\nThe inverse problem considered in the present paper formally looks\nlike a standard inverse electromagnetic problem (IEMP) proposed in\n\\cite{somersalo1992linearized}: Determine $\\varepsilon$, $\\mu$ and\n$\\sigma$, from the knowledge of the admittance map\n$\\Lambda^\\omega_{\\varepsilon,\\sigma,\\mu} : \\bt(E) \\mapsto \\bt(H)$ for\nall $(E,H) \\in H_{\\Div}^1(\\Omega) \\times H_{\\Div}^1(\\Omega)$ solving\n\\eqref{eqn::Maxwell}. However, the conditions in the IEMP do not allow\nthe vanishing of $\\varepsilon$ as in our Assumption~\\ref{MT\n  assumption}. To be precise, IEMP invokes\n\n\\begin{Assumption} \\label{EM assumption}\nThe electrical permittivity, electrical conductivity and magnetic\npermeability satisfy $\\varepsilon \\ge \\varepsilon_0$, $\\mu \\ge \\mu_0$\nand $\\sigma \\ge 0$ on $\\overline\\Omega$ for some constants\n$\\varepsilon_0, \\mu_0 > 0$.\n\\end{Assumption}\n\n\\noindent\nAs a consequence, the IMTP and IEMP problems are in fact\ndifferent. For comparison, we also mention the EIT, or inverse\nconductivity problem, also known as electrical resistivity tomography\n(ERT) in the geophysics literature, proposed in\n\\cite{calderon2006inverse}: Determine $\\sigma$, satisfying $\\sigma \\ge\n\\sigma_0$ for some constant $\\sigma_0 > 0$, from the\nDirichlet-to-Neumann map $u|_{\\p\\Omega} \\mapsto \\sigma\\p_\\nu\nu|_{\\p\\Omega}$ for all $u \\in H^1(\\Omega)$ solving the conductivity\nequation $\\nabla \\cdot (\\sigma \\nabla u) = 0$ in $\\p\\Omega$. A\nlow-frequency limit of IEMP as in \\cite{lassas1997impedance} will not\nconverge to EIT. However, one expects that a low-frequency limit of\nIMTP meaningfully relates to EIT.\n\n\\smallskip\n\nWe now give a brief overview of results pertaining to the IEMP and its\nhistory. The standard approach to solve this problem is to\nconstruct a family of exponentially growing solutions, also known as\n\\emph{complex geometric optics solutions}, following the celebrated\npaper~\\cite{sylvester1987global} on the inverse conductivity\nproblem. One of the main challenges in adopting the method of\n\\cite{sylvester1987global} is the fact that \\eqref{eqn::Maxwell} with\nAssumption~\\ref{EM assumption} is not elliptic. The linearized problem\nat constant material parameters was studied in\n\\cite{somersalo1992linearized}. For the nonlinear problem, a\nuniqueness result was given in \\cite{sun1992inverse} when the\nelectromagnetic parameters are close to constants. In this paper, to\nget ellipticity, equation \\eqref{eqn::Maxwell} with Assumption~\\ref{EM\n  assumption} was reduced to a system whose principal part is the\nLaplacian. However, this reduction gives some first order terms. For\nmaterial parameters that are nearly constant, the authors were able to\nmanage the first-order terms and introduce complex geometrical solutions\nfor \\eqref{eqn::Maxwell}. The first global uniqueness result was\nproven in~\\cite{ola1993inverse}. This proof was later simplified\nin~\\cite{ola1996electromagnetic}. The important point in the\nsimplified proof is to augment \\eqref{eqn::Maxwell} with\nAssumption~\\ref{EM assumption} to a certain $8 \\times 8$ Dirac\nequation and connect it via some other Dirac operator to an $8 \\times\n8$ system whose principal part is the Laplacian while its remainder\ninvolves only zeroth-order terms. This allowed the authors to\nconstruct complex geometric optics solutions for the latter system and\nconnect them to \\eqref{eqn::Maxwell} with Assumption~\\ref{EM\n  assumption} by applying the Dirac operator that was initially\nintroduced. This technique became popular in the subsequent works on\nvarious aspects of IEMP~\\cite{assylbekov2017kerrinverse,\n  caro2014global, kenig2011inverse}.\n\n\\smallskip\n\nIn the setting of the IMTP, however, one cannot simply employ the\ncomplex geometric optics solutions constructed in\n\\cite{ola1993inverse, ola1996electromagnetic} for the IEMP. Moreover,\nthe elliptization argument of \\cite{ola1996electromagnetic}, applied\nto \\eqref{eqn::Maxwell} with Assumption~\\ref{MT assumption}, does not\nhelp avoiding first-order terms. Instead of that, we\nfollow~\\cite{sun1992inverse} and reduce \\eqref{eqn::Maxwell} with\nAssumption~\\ref{MT assumption} to a system whose principal part is the\nLaplacian. We then introduce novel complex geometrical optics\nsolutions for the reduced system that are essentially solutions for\n\\eqref{eqn::Maxwell} with Assumption~\\ref{MT assumption}. Moreover,\nusing this reduction gives an integral identity with a clear relation\nto \\eqref{eqn::Maxwell}. To deal with the first-order terms, we use\nthe ideas from \\cite{colton1992uniqueness} with substantial\nmodifications since the latter paper assumes that $\\mu$ is constant.\n\n\\smallskip\n\nIn the MT method, performing measurements on the entire boundary (that\nis, the surface of the earth) is impossible. Therefore, the analysis\nof the inverse problem with local measurements is important. We can\nassume that the measurements are performed on a nonempty open subset\n$\\Gamma$ of $\\p\\Omega$ only and that the inaccessible part of the\nboundary $\\Gamma_0 = \\overline{\\p\\Omega \\setminus \\Gamma}$ is a part\nof a sphere (our planet's surface). Our second main result is the\nfollowing\n\n\\begin{Theorem}\\label{main thm 2}\nLet $\\Omega \\subset B_0$ be a bounded domain with $C^{1,1}$ boundary\nincluded in an open ball $B_0 \\subset \\R^3$ and let $\\Gamma_0 =\n\\p\\Omega \\cap \\p B_0$, $\\Gamma_0 \\neq \\p B_0$ and $\\Gamma =\n\\overline{\\p\\Omega \\setminus \\Gamma_0}$. Suppose that $\\sigma_j, \\mu_j\n\\in C^2(\\overline\\Omega)$, $j=1,2$, satisfy $\\sigma_j \\ge \\sigma_0$\nand $\\mu_j \\ge \\mu_0$, for some constants $\\sigma_0, \\mu_0 > 0$, and\n\\begin{equation}\\label{boundary assumption on Gamma}\n   \\p^\\alpha \\sigma_1|_{\\Gamma} = \\p^\\alpha \\sigma_2|_{\\Gamma}\n   \\quad\\text{and}\\quad\n   \\p^\\alpha \\mu_1|_{\\Gamma} = \\p^\\alpha \\mu_2|_{\\Gamma}\n   \\quad\\text{for}\\quad|\\alpha| \\le 2 .\n\\end{equation}\nIn addition, assume that $\\sigma_j$ and $\\mu_j$, $j=1,2$, can be\nextended to $\\R^3$ as $C^2$ functions which are invariant under\nreflection across $\\p B_0$. Suppose that $\\omega > 0$ is not a\nresonant frequency for $(\\sigma_1,\\mu_1)$ and $(\\sigma_2,\\mu_2)$. If\n$$\n   Z_{\\sigma_1,\\mu_1}^\\omega(f)|_{\\Gamma}\n             = Z_{\\sigma_2,\\mu_2}^\\omega(f)|_{\\Gamma}\n   \\quad\\text{for all}\\quad f \\in TH^{1\/2}_{\\Div}(\\p\\Omega)\n   \\quad\\text{with}\\quad \\supp(f)\\subset\\Gamma ,\n$$\nthen $\\sigma_1 = \\sigma_2$ and $\\mu_1 = \\mu_2$.\n\\end{Theorem}\n\n\\noindent\nFor the proof of Theorem~\\ref{main thm}, we follow Isakov's reflection\napproach \\cite{isakov2007uniqueness} which was originally proposed for\nthe inverse conductivity problem. An analogous result for IEMP was\nobtained in \\cite{caro2009inverse}.\n\n\\smallskip\n\nWe briefly describe a connection of our results to the land-based CSEM\n(controlled source electromagnetic) method in geophysical exploration\n\\cite{streich2016controlled}. Contrary to the MT method, the CSEM\nmethod employs active sources. In recent work by Schaller \\textit{et\n  al.}~\\cite{schaller2018land}, a land-based CSEM survey was designed\nand performed at the Schoonebeek oil field. The application of\nland-based CSEM for low-cost ${\\rm CO}_2$ monitoring was studied in\n\\cite{mcaliley2019analysis}. The marine CSEM method, which was\nintroduced by Cox \\textit{et al.}~\\cite{cox1971electromagnetic}, would\nrequire a careful incorporation of the ocean layer in the analysis,\nwhich we do not pursue in this paper. It has successfull applications\nin direct identification of hydrocarbons \\cite{chave1991electrical,\n  eidesmo2002sea}, and the study of the oceanic lithosphere and active\nspreading centers \\cite{chave1990some, constable1996marine,\n  cox1986controlled, evans1994electrical, macgregor2002use,\n  young1981electromagnetic}. For a more detailed exposition of\nprogress made on the marine CSEM, we refer to a review paper by\nConstable \\cite{constable2010ten}. Various basic data fitting\napproaches have been developed for CSEM \\cite{abubakar20082,\n  gribenko2011joint, li20072dp1, li20072dp2}. From a mathematical\npoint of view, the data for the land-based CSEM is modeled by point\nsource measurements. More precisely, for an arbitrary unit vector\n$\\alpha$ and $y \\in \\p\\Omega$, consider the equation\n$$\n   \\nabla_x \\times E_\\alpha(x,y) = i \\omega \\mu(x) H_\\alpha(x,y)\n   + \\delta(x-y) \\nu(y) \\times \\alpha\n   \\quad\\text{and}\\quad\n   \\nabla_x \\times H_\\alpha(x,y) = \\sigma(x) E_\\alpha(x,y)\n   \\quad\\text{in}\\quad\\R^3 ,\n$$\nwith the outgoing radiation condition. The equation governs the\nelectromagnetic field of a magnetic dipole (active source) tangential\nto the boundary $\\p\\Omega$. Then the inverse problem for the\nland-based CSEM is to determine $\\sigma$ and $\\mu$ from\n$$\n   \\mathcal A_{\\sigma,\\mu} := \\left\\{\\big(\\nu(x) \\times H_{e_j}(x,y) ,\\\n   \\nu(x) \\times E_{e_j}(x,y)\\big) : x, y \\in\\p\\Omega ,\\quad\n   x \\neq y ,\\quad j=1,2,3\\right\\} ,\n$$\nwhere $e_j$, $j=1,2,3$, denote the Cartesian coordinate vectors. We\nexpect that following the arguments similar to\n\\cite{ola1996electromagnetic}, one can show that to the knowledge of\n$\\mathcal A_{\\sigma,\\mu}$ is equivalent the knowledge of the graph of\n$Z^\\omega_{\\sigma,\\mu}$ via layer potentials. Then the land-based CSEM\nand MT would concern the same inverse problem with boundary\ndata.\n\n\\smallskip\n\nThe paper is organized as follows. In Section~\\ref{section::well\n  posedness}, we prove the well-posedness of the direct problem using\nstandard arguments. In Section~\\ref{section::CGOs}, we first rewrite\n\\eqref{eqn::Maxwell} as the curl-curl equation and then construct\ncomplex geometric optics solutions for it. We use these solutions to\nprove Theorem~\\ref{main thm} in Section~\\ref{section::proof of main\n  thm}. Next, in Section~\\ref{section::reflection approach} we perform\nthe reflection approach of Isakov \\cite{isakov2007uniqueness} and\nprove an analog of Theorem~\\ref{main thm 2} but in the case when the\npart of the boundary inaccessible for measurements is a subset of the\nplane $\\{x\\in\\R^3 : x_3=0\\}$. Theorem~\\ref{main thm 2} is then proved\nin Section~\\ref{section::proof of thm 2} by analyzing the behavior of\n\\eqref{eqn::Maxwell} under the Kelvin\ntransform. Appendix~\\ref{section::pullbacks} contains properties of\npullbacks used in the main text. Finally, in\nAppendix~\\ref{section::Appendix B} we show that the impedance map\n$Z^\\omega_{\\sigma,\\mu}$ is a pseudodifferential operator of order $1$\nif $\\sigma, \\mu \\in C^\\infty(\\overline\\Omega)$. Using this fact, we\ngain insight in the notion of apparent resistivity used in geophysics\nfrom a mathematical point of view.\n\n\\section{Well-posedness of the direct problem}\n\\label{section::well posedness}\n\nLet $\\Omega\\subset\\R^3$ be a bounded domain with $C^{1,1}$ boundary as\nbefore, and let $\\sigma,\\mu\\in C^1(\\overline\\Omega)$ be such that\n$\\sigma \\ge \\sigma_0$ and $\\mu \\ge \\mu_0$ for some constants\n$\\sigma_0, \\mu_0 > 0$. Consider the following system of equations for\nelectromagnetic fields $E$ and $H$:\n\\begin{equation}\\label{eqn::Maxwell homogenous in appendix}\n\\nabla\\times E=i\\omega\\mu H\\quad\\text{and}\\quad \\nabla\\times H=\\sigma E \\quad\\text{in}\\quad\\Omega,\n\\end{equation}\nwith the tangential boundary condition $\\mathbf{t}(H)=f$, where\n$\\omega$ is a complex number. The main result of the present section is\n\n\\begin{Theorem}\\label{thm::well posedness new version homogeneous}\nLet $\\Omega\\subset\\R^3$ be a bounded domain with $C^{1,1}$ boundary and let $\\sigma,\\mu\\in C^1(\\overline\\Omega)$ be such that $\\sigma \\ge \\sigma_0$ and $\\mu \\ge \\mu_0$ for some constants $\\sigma_0, \\mu_0 > 0$. There is a discrete subset $\\Sigma$ of $\\C$ such that for all $\\omega\\notin \\Sigma$ and for a given $f\\in TH_{\\Div}^{1\/2}(\\p \\Omega)$ the system \\eqref{eqn::Maxwell homogenous in appendix} with $\\mathbf t(H)=f$ has a unique solution $(E,H)\\in H_{\\Div}^1(\\Omega)\\times H_{\\Div}^1(\\Omega)$ satisfying\n$$\n\\|E\\|_{H_{\\Div}^1(\\Omega)}+\\|H\\|_{H_{\\Div}^1(\\Omega)}\\le C\\|f\\|_{TH_{\\Div}^{1\/2}(\\p \\Omega)}\n$$\nfor some constant $C>0$ independent of $f$.\n\\end{Theorem}\n\nFor $\\omega>0$ with $\\omega\\notin \\Sigma$, we define the \\emph{impedance map} $Z_{\\sigma,\\mu}^\\omega$ as\n$$\nZ_{\\sigma,\\mu}^\\omega(f) := \\mathbf t(E),\\quad f\\in TH_{\\Div}^{1\/2}(\\p \\Omega),\n$$\nwhere $(E,H)\\in H_{\\Div}^1(\\Omega)\\times H_{\\Div}^1(\\Omega)$ is the unique solution of the system \\eqref{eqn::Maxwell homogenous in appendix} with $\\mathbf t(H)=f$, guaranteed by Theorem~\\ref{thm::well posedness new version homogeneous}. Moreover, the estimate provided in Theorem~\\ref{thm::well posedness new version homogeneous} implies that the impedance map is a well-defined and bounded operator $Z_{\\sigma,\\mu}^\\omega: TH_{\\Div}^{1\/2}(\\p \\Omega)\\to TH_{\\Div}^{1\/2}(\\p \\Omega)$.\\smallskip\n\nTo prove Theorem~\\ref{thm::well posedness new version homogeneous}, we\nconsider the following non-homogeneous problem. Let $J_e$ and $J_m$ be\nvector fields defined in $\\Omega$ representing current sources. We\nconsider the non-homogenous time-harmonic Maxwell equations,\n\\begin{equation}\\label{eqn::Maxwell in appendix}\n\\nabla\\times E=i\\omega\\mu H + J_m\\quad\\text{and}\\quad \\nabla\\times H=\\sigma E + J_e \\quad\\text{in}\\quad\\Omega.\n\\end{equation}\nWe have\n\n\\begin{Theorem}\\label{thm::well posedness new version}\nLet $\\Omega\\subset\\R^3$ be a bounded domain with $C^{1,1}$ boundary and let $\\sigma,\\mu\\in C^1(\\overline\\Omega)$ be such that $\\sigma \\ge \\sigma_0$ and $\\mu \\ge \\mu_0$ for some constants $\\sigma_0, \\mu_0 > 0$. Suppose that $J_e,J_m\\in L^2(\\Omega;\\C^3)$ such that $\\nabla\\cdot J_e,\\nabla\\cdot J_m\\in L^2(\\Omega;\\C^3)$ and $\\nu\\cdot J_e|_{\\p\\Omega}, \\nu\\cdot J_m|_{\\p\\Omega}\\in H^{1\/2}(\\p\\Omega)$. Then there is a discrete subset $\\Sigma$ of $\\C$ such that for all $\\omega\\notin \\Sigma$ the boundary value problem \\eqref{eqn::Maxwell in appendix} with $\\mathbf t(H) = 0$ has a unique solution $(E,H)\\in H^1_{\\Div}(\\Omega)\\times H^1_{\\Div}(\\Omega)$ satisfying\n$$\n\\|E\\|_{H_{\\Div}^1(\\Omega)}+\\|H\\|_{H_{\\Div}^1(\\Omega)}\\le C\\big(\\|J_e\\|_{L^2(\\Omega;\\C^3)}+\\|J_m\\|_{L^2(\\Omega;\\C^3)} + \\|\\nabla\\cdot J_e\\|_{L^2(\\Omega)}+\\|\\nabla\\cdot J_m\\|_{L^2(\\Omega)} + \\|\\nu\\cdot J_e\\|_{H^{1\/2}(\\p\\Omega)}+\\|\\nu\\cdot J_m\\|_{H^{1\/2}(\\p\\Omega)}\\big)\n$$\nfor some constant $C>0$ independent of $J_e$ and $J_m$\n\\end{Theorem}\n\n\\noindent\nWe first prove Theorem~\\ref{thm::well posedness new version} and then\nshow that it can be used to prove Theorem~\\ref{thm::well posedness new\n  version homogeneous}.\n\n\\subsection{Proof of Theorem~\\ref{thm::well posedness new version}}\n\nWe introduce some notion that will be used for the proof. We work with the following Hilbert space which is the largest domain of $\\nabla\\times$:\n$$\nH(\\curl;\\Omega):=\\{w\\in L^2(\\Omega;\\C^3):\\nabla\\times w\\in L^2(\\Omega;\\C^3)\\}\n$$\nendowed with the norm $\\|w\\|_{H(\\curl;\\Omega)}:=\\|w\\|_{L^2(\\Omega;\\C^3)} + \\|\\nabla\\times w\\|_{L^2(\\Omega;\\C^3)}$. Then the tangential trace operator has its extensions to bounded operators $\\mathbf{t}:H(\\curl; \\Omega)\\to H^{-1\/2}(\\p \\Omega;\\C^3)$. We also work with the space of vector fields in $H(\\curl;\\Omega)$ having zero tangential trace\n$$\nH(\\curl, 0; \\Omega):=\\{w\\in H(\\curl;\\Omega):\\mathbf{t}(w)=0\\}.\n$$\nFor the short proof, we follow the standard variational methods used in \\cite{kirsch2016mathematical, monk2003finite}. Substituting the first equation of \\eqref{eqn::Maxwell in appendix} into the second one, we obtain\n\\begin{equation}\\label{eqn::second order equation}\n\\nabla\\times(\\sigma^{-1}\\nabla\\times H) - i\\omega\\mu H = J_m +\\nabla\\times(\\sigma^{-1} J_e)\\quad\\text{in}\\quad\\Omega.\n\\end{equation}\nOur first step is to find a unique solution $H\\in H(\\curl, 0;\\Omega)$ of this equation satisfying\n\\begin{equation}\\label{eqn::estimate for H}\n\\|H\\|_{H(\\curl;\\Omega)}\\le C(\\|J_e\\|_{L^2(\\Omega;\\C^3)}+\\|J_m\\|_{L^2(\\Omega;\\C^3)}).\n\\end{equation}\nBy Helmholtz type decompositions in \\cite[Section~4.1.3]{kirsch2016mathematical} or \\cite[Section~3.7]{monk2003finite}, we can uniquely decompose\n\\begin{align*}\nH &= H_0+\\nabla h,\\quad H_0\\in H(\\curl, 0;\\Omega)_{0,\\mu} := \\{w\\in H(\\curl, 0; \\Omega): \\nabla\\cdot(\\mu w) = 0\\},\\quad h\\in H^1_0(\\Omega;\\C),\\\\\n\\mu^{-1}J_m &= J_{m,0}+\\nabla j_m,\\quad J_{m,0}\\in L^2(\\Omega;\\C^3)_{0,\\mu} := \\{w\\in L^2(\\Omega;\\C^3): \\nabla\\cdot(\\mu w) = 0\\},\\quad j_m\\in H^1_0(\\Omega;\\C).\n\\end{align*}\nWe note here that\n\\begin{equation}\\label{eqn::estimate for j_e in H^1 norm}\n\\|j_m\\|_{H^1(\\Omega;\\C)}\\le C\\|J_m\\|_{L^2(\\Omega;\\C^3)}.\n\\end{equation}\nUsing these decompositions, \\eqref{eqn::second order equation} can be rewritten as\n\\begin{equation}\\label{eqn::second order equation in a weak form -- rewritten}\n\\nabla\\times(\\sigma^{-1}\\nabla\\times H_0) - i\\omega\\mu H_0 - i\\omega\\mu \\nabla h = \\mu J_{m,0} + \\mu\\nabla j_m +\\nabla\\times(\\sigma^{-1} J_e)\\quad\\text{in}\\quad\\Omega.\n\\end{equation}\nTo extract $h$ from \\eqref{eqn::second order equation in a weak form -- rewritten}, we simply set $h=-(i\\omega)^{-1}j_m$. Thus, we need to find a unique $H_0\\in H(\\curl, 0;\\Omega)$ with $\\nabla\\cdot(\\mu H_0) = 0$ satisfying\n\\begin{equation}\\label{eqn::second order equation -- rewritten}\n\\nabla\\times(\\sigma^{-1}\\nabla\\times H_0) - i\\omega\\mu H_0 = \\mu J_{m,0} + \\nabla\\times(\\sigma^{-1} J_e)\\quad\\text{in}\\quad\\Omega.\n\\end{equation}\nTo solve this equation, we need the following result on existence of a solution operator\n\\begin{Proposition}\\label{prop::resonances}\nThere exist a constant $\\lambda>0$ and a bounded linear map $T_\\lambda : H(\\curl, 0;\\Omega)'\\to H(\\curl, 0;\\Omega)$ such that\n\\begin{equation}\\label{eqn::T_lambda is an inverse of a certain PDoperator}\n\\nabla \\times (\\sigma^{-1} \\nabla \\times T_\\lambda u) + \\lambda \\mu T_\\lambda u = u,\\quad u\\in H(\\curl, 0;\\Omega)'\n\\end{equation}\nand\n$$\nT_\\lambda(\\nabla\\times(\\sigma^{-1}\\nabla\\times e)+\\lambda\\mu e)=e,\\quad e\\in H(\\curl, 0;\\Omega).\n$$\nFurthermore, if $\\nabla\\cdot u = 0$, then $T_\\lambda u\\in H(\\curl, 0;\\Omega)_{0,\\mu}$.\n\\end{Proposition}\n\nHere and in what follows, $\\langle\\cdot,\\cdot\\rangle_\\Omega$ is the duality between $H(\\curl, 0;\\Omega)'$ and $H(\\curl, 0;\\Omega)$ naturally extending the inner product of $L^2(\\Omega;\\C^3)$.\n\n\\begin{proof}\nThe proof is similar to that of \\cite[Proposition~5.1]{assylbekov2016note} using Lax-Milgram's lemma.\n\\end{proof}\n\n\n\n\n\nThen $H_0\\in H(\\curl, 0;\\Omega)$ with $\\nabla\\cdot(\\mu H_0) = 0$ solves \\eqref{eqn::second order equation -- rewritten} if and only if\n\\begin{equation}\\label{eqn::second order equation -- rewritten in terms of T_lambda operators}\nH_0-(i\\omega+\\lambda)\\widetilde T_{\\lambda}H_0=T_\\lambda\\left(\\mu J_{m,0}+\\nabla\\times(\\sigma^{-1}J_e)\\right)\n\\end{equation}\nwhere $\\widetilde T_{\\lambda}=T_\\lambda \\circ m_\\mu\\circ P_\\mu$, $m_\\mu$ is multiplication by $\\mu$, and $P_\\mu$ is the bounded orthogonal projection of $L^2(\\Omega;\\C^3)$ onto $L^2(\\Omega;\\C^3)_{0,\\mu}$ constructed in \\cite[Section~4.1.3]{kirsch2016mathematical}. Since $J_{m,0}\\in L^2(\\Omega;\\C^3)_{0,\\mu}$ and $\\nabla\\cdot\\nabla\\times = 0$, we then have $\\nabla\\cdot\\left(\\mu J_{m,0}+\\nabla\\times(\\sigma^{-1}J_e)\\right) = 0$. Therefore, by the second part of Proposition~\\ref{prop::resonances}, this implies that $T_\\lambda\\left(\\mu J_{m,0}+\\nabla\\times(\\sigma^{-1}J_e)\\right)$ belongs to $H(\\curl, 0;\\Omega)_{0,\\mu}$. The second part of Proposition~\\ref{prop::resonances} implies also that $\\widetilde T_{\\lambda}$ can be considered as a bounded linear operator\n$$\n\\widetilde T_{\\lambda} : L^2(\\Omega;\\C^3)_{0,\\mu}\\overset{m_\\mu}{\\longrightarrow} L^2(\\Omega;\\C^3)_{0,1}\\overset{T_\\lambda}{\\longrightarrow} H(\\curl, 0;\\Omega)_{0,\\mu}\\hookrightarrow L^2(\\Omega;\\C^3)\\overset{P_\\mu}{\\longrightarrow} L^2(\\Omega;\\C^3)_{0,\\mu}.\n$$\nUsing the compactness of the inclusion $H(\\curl, 0;\\Omega)_{0,\\mu}\\hookrightarrow L^2(\\Omega;\\C^3)$ \\cite{weber1980local} and following similar reasoning as at the end of \\cite[Section~5]{assylbekov2016note}, one can show that  for any $\\omega\\notin \\Sigma$, where $\\Sigma:=\\{\\omega\\in \\C\\setminus\\{\\pm i\\lambda\\}:(i\\omega+\\lambda)^{-1}\\in \\Spec(\\widetilde T_{\\lambda})\\}$ which is discrete, \\eqref{eqn::second order equation -- rewritten in terms of T_lambda operators} has a unique solution $H_0\\in H(\\curl, 0;\\Omega)_{0,\\mu}$ satisfying\n$$\n\\|H_0\\|_{H(\\curl;\\Omega)}\\le C(\\|J_e\\|_{L^2(\\Omega;\\C^3)}+\\|J_{m,0}\\|_{L^2(\\Omega;\\C^3)}).\n$$\nNext, setting $H = H_0-(i\\omega)^{-1}j_m$, we obtain a unique $H(\\curl, 0;\\Omega)$ solution for \\eqref{eqn::second order equation} satisfying \\eqref{eqn::estimate for H} thanks to \\eqref{eqn::estimate for j_e in H^1 norm}. Defining $E := \\sigma^{-1}(\\nabla\\times H - J_e)$ we obtain a unique $(E,H)\\in H(\\curl, 0;\\Omega)\\times H(\\curl;\\Omega)$ solving \\eqref{eqn::Maxwell in appendix} and satisfying\n$$\n\\|E\\|_{H(\\curl;\\Omega)} + \\|H\\|_{H(\\curl;\\Omega)}\\le C(\\|J_e\\|_{L^2(\\Omega;\\C^3)}+\\|J_m\\|_{L^2(\\Omega;\\C^3)}).\n$$\nTo prove that $(E,H)\\in H^1_{\\Div}(\\Omega)\\times H^1_{\\Div}(\\Omega)$, apply $\\nabla\\cdot$ to \\eqref{eqn::Maxwell in appendix} and get $\\nabla\\cdot(i\\omega\\mu H) =-\\nabla\\cdot J_m$ and $\\nabla\\cdot(\\sigma E) = -\\nabla\\cdot J_e$. Hence $\\nabla\\cdot E, \\nabla\\cdot H\\in L^2(\\Omega)$, since $\\nabla\\cdot J_e, \\nabla\\cdot J_m\\in L^2(\\Omega)$ by assumption. Then $\\bt(H)=0$ and the results in \\cite{costabel1990remark} imply that $H\\in H^1_{\\Div}(\\Omega)$ and\n$$\n\\|H\\|_{H^1_{\\Div}(\\Omega)} \\le C\\big(\\|H\\|_{H(\\curl;\\Omega)} + \\|J_e\\|_{L^2(\\Omega;\\C^3)} + \\|J_m\\|_{L^2(\\Omega;\\C^3)} + \\|\\nabla\\cdot J_e\\|_{L^2(\\Omega)} + \\|\\nabla\\cdot J_m\\|_{L^2(\\Omega)}\\big).\n$$\nTo show that $E\\in H^1(\\Omega;\\C^3)$, observe that $\\nu\\cdot (\\nabla\\times H)|_{\\p\\Omega} = - \\Div(\\bt(H))=0$ by \\cite[Corollary~A.20]{kirsch2016mathematical}. Then by \\eqref{eqn::Maxwell in appendix}, $\\nu\\cdot E|_{\\p\\Omega} = \\sigma^{-1} \\nu\\cdot (\\nabla\\times H)|_{\\p\\Omega} - \\sigma^{-1} \\nu\\cdot J_e|_{\\p\\Omega} = - \\sigma^{-1} \\nu\\cdot J_e|_{\\p\\Omega} \\in H^{1\/2}(\\p\\Omega)$. According to the results in \\cite{costabel1990remark}, this implies that $E\\in H^1(\\Omega;\\C^3)$ and\n$$\n\\|E\\|_{H^1_{\\Div}(\\Omega)} \\le C\\big(\\|E\\|_{H(\\curl;\\Omega)} + \\|J_e\\|_{L^2(\\Omega;\\C^3)} + \\|\\nabla\\cdot J_e\\|_{L^2(\\Omega)} + \\|\\nu\\cdot J_e\\|_{H^{1\/2}(\\p\\Omega)}\\big).\n$$\nNext, using \\cite[Corollary~A.20]{kirsch2016mathematical} and \\eqref{eqn::Maxwell in appendix}, we can show $\\Div(\\bt(E)) = - \\nu\\cdot (\\nabla\\times E)|_{\\p\\Omega} = - i\\omega\\mu \\nu\\cdot H|_{\\p\\Omega} + \\nu\\cdot J_m|_{\\p\\Omega} \\in H^{1\/2}(\\p\\Omega)$. Thus, $E\\in H^1_{\\Div}(\\Omega)$. Finally, the estimate in the statement of the theorem follows by combining all the above estimates. The proof of Theorem~\\ref{thm::well posedness new version} is thus complete.\n\n\\subsection{Proof of Theorem~\\ref{thm::well posedness new version homogeneous}}\n\nFirst prove the uniqueness of the solution. For a fixed $\\omega\\in\n\\C$, suppose that $(E_j,H_j)\\in H^1_{\\Div}(\\Omega)\\times\nH^1_{\\Div}(\\Omega)$, $j=1,2$, solve \\eqref{eqn::Maxwell homogenous in\n  appendix} and satisfy $\\mathbf{t}(H_1)=\\mathbf{t}(H_2)$. Then\n$(E,H)\\in H^1_{\\Div}(\\Omega)\\times H^1_{\\Div}(\\Omega)$ also solve\n\\eqref{eqn::Maxwell homogenous in appendix} and satisfy\n$\\mathbf{t}(H)=0$, where $E:=E_1-E_2$ and $H:=H_1-H_2$. The uniqueness\npart of Theorem~\\ref{thm::well posedness new version} (with\n$J_e=J_m=0$) gives that $E=0$ and $H=0$.\\smallskip\n\nNext, we prove existence of a solution. For a given $f\\in TH^{1\/2}_{\\Div}(\\p\\Omega)$, there is $H'\\in H^1_{\\Div}(\\Omega)$ such that $\\mathbf{t}(H')=f$ and $\\|H'\\|_{H^1_{\\Div}(\\Omega)}\\le C\\|f\\|_{TH^{1\/2}_{\\Div}(\\p\\Omega)}$. Applying Theorem~\\ref{thm::well posedness new version} with $J_e=-\\nabla\\times H'$ and $J_m=i\\omega\\mu H'$, we obtain a unique $(E_0,H_0)\\in H^1_{\\Div}(\\Omega)\\times H^1_{\\Div}(\\Omega)$ solving\n$$\n\\nabla\\times E_0=i\\omega\\mu H_0+i\\omega\\mu H',\\quad \\nabla\\times H_0=\\sigma E_0 - \\nabla\\times H',\\quad \\bt(H_0)=0\n$$\nand satisfying $\\|E_0\\|_{H_{\\Div}^1(\\Omega)}+\\|H_0\\|_{H_{\\Div}^1(\\Omega)}\\le C \\|f\\|_{TH^{1\/2}_{\\Div}(\\p\\Omega)}$. Here, we used the fact that $\\nu\\cdot (\\nabla\\times H')|_{\\p\\Omega} = - \\Div(\\bt(H'))\\in H^{1\/2}(\\p\\Omega)$ by \\cite[Corollary~A.20]{kirsch2016mathematical}. Then $(E,H)\\in H^1_{\\Div}(\\Omega)\\times H^1_{\\Div}(\\Omega)$ solves \\eqref{eqn::Maxwell homogenous in appendix} with $\\mathbf{t}(E)=f$, where $E:=E_0+E'$ and $H:=H_0$. The proof is complete.\n\n\n\n\\section{Construction of complex geometric optics solutions}\\label{section::CGOs}\n\nThroughout this section, we assume that $\\sigma$ and $\\mu$ can be extended to the whole $\\R^3$ so that $\\sigma\\ge \\sigma_0$, $\\mu\\ge \\mu_0$ and\n\\begin{equation}\\label{sigma and mu are constants outside of a compact set}\n\\sigma - \\sigma_0,\\quad \\mu - \\mu_0 \\in C^2_0(\\R^3)\n\\end{equation}\nfor some constants $\\sigma_0, \\mu_0 > 0$. We also let $R>0$ be large enough (but fixed) so that $B_R(0)$ contains both $\\supp(\\sigma - \\sigma_0)$ and $\\supp(\\mu - \\mu_0)$.\\smallskip\n\nSubstituting the first equation of \\eqref{eqn::Maxwell} into the second one, we obtain the following second-order equation\n\\begin{equation}\\label{eqn::curl-curl equation}\n\\nabla\\times(\\sigma^{-1}\\nabla\\times H) - i\\omega \\mu H = 0\\quad\\text{in}\\quad \\Omega\n\\end{equation}\nThe aim of the present section is to construct a complex geometric optics solution in $H\\in H_{\\Div}^1(\\Omega)$ for the above equation. Instead of working in $\\Omega$, we conduct our analysis in the whole $\\R^3$. Therefore, we consider\n\\begin{equation}\\label{eqn::curl-curl equation R^3}\n\\nabla\\times(\\sigma^{-1}\\nabla\\times H) - i\\omega \\mu H = 0\\quad\\text{in}\\quad \\R^3\n\\end{equation}\n\nTaking the divergence of \\eqref{eqn::curl-curl equation R^3}, it straightforwardedly follows that $\\nabla\\cdot(\\mu H) = 0$ in $\\R^3$. Therefore, we obtain\n$$\n\\nabla\\times\\nabla\\times H =  - \\Delta H - \\nabla(\\nabla\\beta \\cdot H)\\quad\\text{in}\\quad \\R^3,\n$$\nwhere $\\beta := \\log\\mu$. Then, we use the latter identity in \\eqref{eqn::curl-curl equation R^3} to show that this equation is equivalent to the system\n\\begin{align}\nL_{\\sigma, \\mu} H := -\\Delta H - \\nabla(\\nabla\\beta \\cdot H) - \\nabla\\alpha \\times \\nabla\\times H - i\\omega \\sigma \\mu H = 0\\quad\\text{in}\\quad \\R^3, \\label{eqn3-1}\\\\\n\\nabla\\cdot(\\mu H) = 0\\quad\\text{in}\\quad \\R^3, \\label{eqn3-2}\n\\end{align}\nwhere $\\alpha := \\log\\sigma$. We note that the derivatives $\\p^\\kappa \\alpha$ and $\\p^\\kappa \\beta$ are uniformly continuous on $\\R^3$ for $|\\kappa|=0, 1, 2$.\\smallskip\n\n\n\nThe complex geometric optics solutions we aim to construct are of the form\n$$\nH(x; \\zeta) = e^{i \\zeta\\cdot x}(a(x; \\zeta) + r(x; \\zeta)),\n$$\nwhere $\\zeta\\in \\C^3\\setminus\\{0\\}$ such that $\\zeta\\cdot\\zeta = i\\omega \\sigma_0 \\mu_0$, $a$ is a specific complex-valued smooth vector field on $\\R^3$ and $r$ is the correction term. Then \\eqref{eqn3-1} is equivalent to\n\\begin{equation} \\label{eq:Lsm}\ne^{-i \\zeta\\cdot x}L_{\\sigma,\\mu}(e^{i \\zeta\\cdot x}r) = - f,\\quad f := e^{-i \\zeta\\cdot x} L_{\\sigma,\\mu}(e^{i \\zeta\\cdot x}a).\n\\end{equation}\n\n\\subsection{Solution operator}\n\nFor $\\zeta\\in \\C^3\\setminus\\{0\\}$ such that $\\zeta\\cdot\\zeta = i\\omega \\sigma_0 \\mu_0$, we define the operators\n$$\n\\nabla_\\zeta := \\nabla + i\\zeta,\\quad \\Delta_\\zeta := \\Delta + 2i \\zeta\\cdot \\nabla.\n$$\nThen\n\\begin{equation}\\label{eqn:: conj grad and laplacian}\ne^{-i \\zeta\\cdot x}\\circ \\nabla\\circ e^{i \\zeta\\cdot x} = \\nabla_\\zeta\\quad\\text{and}\\quad e^{-i \\zeta\\cdot x}\\circ \\Delta\\circ e^{i \\zeta\\cdot x} = \\Delta_\\zeta - i\\omega \\sigma_0 \\mu_0.\n\\end{equation}\nFor $\\delta\\in\\R$, we define the $L^2$-based weighted space on $\\R^3$\n$$\nL^2_\\delta := \\left\\{f: \\R^3 \\to \\C^3: \\|f\\|_{L^2_\\delta}:=\\Big(\\int_{\\R^3} (1+|x|^2)^{\\delta}|f(x)|^2\\,dx\\Big)^{1\/2} < \\infty\\right\\}\n$$\nand\n$$\nH_\\delta^1 := \\left\\{f \\in L^2_\\delta: \\|f\\|_{H^1_\\delta}:=\\|f\\|_{L^2_\\delta} + \\sum_{j=1}^3\\|\\p_j f\\|_{L^2_\\delta} < \\infty\\right\\}.\n$$\n\n\\begin{Proposition}\\label{prop unique solvability}\nFor $k\\in \\C$, suppose $\\zeta\\in \\C^3$ with $\\zeta \\cdot \\zeta = k$, $-1 < \\delta < 0$. Assume that $\\gamma\\in C^2(\\R^3)$ is positive. Then for $f\\in L^2_{\\delta+1}$ there is a unique  $u\\in L^2_\\delta$ solving\n\\begin{equation}\\label{eqn elliptic}\n(-\\Delta_\\zeta - \\nabla\\log\\gamma \\cdot \\nabla_\\zeta) u = f\\quad\\text{in}\\quad\\R^3\n\\end{equation}\nsuch that\n$$\n\\|u\\|_{L^2_\\delta} \\le \\frac{C}{|\\zeta|}\\|f\\|_{L^2_{\\delta+1}}\n$$\nfor some constant $C>0$. Furthermore, $u$ belongs to $H^1_\\delta$.\n\\end{Proposition}\n\n\\begin{proof}\nIt follows from the identity\n$$\n(-\\Delta_\\zeta - \\nabla\\log\\gamma \\cdot \\nabla_\\zeta) u = \\gamma^{-1\/2}(-\\Delta_\\zeta + q) (\\gamma^{1\/2}u),\\quad q:=\\gamma^{-1\/2}\\Delta\\gamma^{1\/2}\\in C_0(\\R^3),\n$$\nthat solving \\eqref{eqn elliptic} is equivalent to solving\n$$\n(-\\Delta_\\zeta + q) \\tilde u = \\gamma^{1\/2} f\\quad\\text{in}\\quad\\R^3,\n$$\nwhere $\\tilde u := \\gamma^{1\/2} u$. By \\cite[Theorem~1.6]{sylvester1987global} there is a unique $\\tilde u\\in L^2_\\delta$  solving the above equation and satisfying\n$$\n\\|\\tilde u\\|_{L^2_\\delta} \\le \\frac{C}{|\\zeta|}\\|\\gamma^{1\/2}f\\|_{L^2_{\\delta+1}}.\n$$\nNext, \\cite[Lemma~1.15]{sun1992inverse} implies that $\\tilde u \\in H^1_\\delta$.\nThe result now follows immediately by setting $u = \\gamma^{-1\/2} \\tilde u$.\n\\end{proof}\n\nAccording to Proposition~\\ref{prop unique solvability}, for sufficiently large $|\\zeta|$, there is a bounded inverse $G_{\\zeta, \\gamma}: L^2_{\\delta + 1} \\to L^2_\\delta$ of $-\\Delta_\\zeta - \\nabla\\log\\gamma \\cdot \\nabla_\\zeta$ such that\n$$\n\\|G_{\\zeta,\\gamma}\\|_{L^2_{\\delta + 1} ; L^2_\\delta} = \\mathcal O\\Big(\\frac{1}{|\\zeta|}\\Big)\\quad\\text{as}\\quad |\\zeta|\\to\\infty\n$$\nMoreover, $G_{\\zeta, \\gamma}$ maps $L^2_{\\delta + 1}$ into $H^1_\\delta$.\n\n\\subsection{Mollified $\\sigma$ and $\\mu$}\nLet $\\Phi\\in C^\\infty_0(\\R^3)$ with $0 \\le \\Phi \\le 1$ and $\\int_{\\R^3} \\Phi(x)\\,dx=1$. For a fixed $\\epsilon$ with $0 < \\epsilon < 1\/8$, we consider\n$$\n\\Phi_\\tau(x) := \\Big(\\frac{1}{\\tau^{-\\epsilon}}\\Big)^3 \\Phi\\Big(\\frac{x}{\\tau^{-\\epsilon}}\\Big)\\quad\\text{for large } \\tau>0.\n$$\nWe define \n$$\n\\alpha^\\sharp(x;\\tau) := \\alpha * \\Phi_\\tau (x),\\quad \\beta^\\sharp(x;\\tau) := \\beta * \\Phi_\\tau(x)\\quad\\text{for}\\quad x\\in \\R^3.\n$$\nThen $\\alpha^\\sharp(\\cdot;\\tau), \\beta^\\sharp(\\cdot;\\tau) \\in C^\\infty(\\R^3)$. From\n$\\p^\\kappa \\alpha^\\sharp = (\\p^\\kappa \\alpha) * \\Phi_\\tau$ and $\\p^\\kappa \\beta^\\sharp = (\\p^\\kappa \\beta) * \\Phi_\\tau$,\nit follows that\n\\begin{equation}\\label{supports of derivatives of alpha and beta}\n\\supp(\\p^\\kappa \\alpha^\\sharp) ,\\ \\supp(\\p^\\kappa \\beta^\\sharp) \\subset B_{R+2\\tau^{-\\epsilon}}(0),\\quad |\\kappa| = 1, 2.\n\\end{equation}\nWe also have\n\\begin{equation}\\label{L^infty norms of derivatives of alpha and beta errors}\n\\|\\alpha -  \\alpha^\\sharp\\|_{W^{2,\\infty}(\\R^3)} = o(1), \\quad \\| \\beta - \\beta^\\sharp\\|_{W^{2,\\infty}(\\R^3)} = o(1)\\quad\\text{as}\\quad \\tau\\to\\infty.\n\\end{equation}\nIndeed,\n$$\n\\p^\\kappa \\alpha(x) - \\p^\\kappa \\alpha^\\sharp(x;\\tau) = \\int_{\\R^3} \\Phi(y) [\\p^\\kappa\\alpha(x) - \\p^\\kappa\\alpha(x - \\tau^{-\\epsilon} y)]\\,dy,\\quad |\\kappa|=0, 1, 2,\n$$\nand uniform continuity of $\\p^\\kappa\\alpha$ on $\\R^3$ gives the desired estimate using \\cite[Theorem~0.13]{folland1995introduction}. A similar argument can be used for $\\beta$.\\smallskip\n\nFinally, using that $\\p^\\kappa \\alpha^\\sharp = \\alpha * (\\p^\\kappa\\Phi_\\tau)$ and $\\p^\\kappa \\beta^\\sharp = \\beta * (\\p^\\kappa\\Phi_\\tau)$, a direct calculation shows that\n$$\n\\|\\p^\\kappa\\alpha^\\sharp\\|_{L^{\\infty}(\\R^3)},\\ \\|\\p^\\kappa\\beta^\\sharp\\|_{L^{\\infty}(\\R^3)} = \\mathcal O_\\kappa(\\tau^{|\\kappa|\\epsilon})\\quad\\text{for}\\quad |\\kappa|\\ge 0\\quad\\text{as}\\quad \\tau\\to \\infty,\n$$\nwhich implies that\n\\begin{equation}\\label{eqn::L^infty norms of derivatives of alpha & beta}\n\\|\\alpha^\\sharp\\|_{W^{k,\\infty}(\\R^3)},\\ \\|\\beta^\\sharp\\|_{W^{k,\\infty}(\\R^3)} = \\mathcal O_k(\\tau^{k\\epsilon})\\quad\\text{for}\\quad k = 0,1,2,\\ldots\\quad\\text{as}\\quad \\tau\\to \\infty.\n\\end{equation}\nWe note that stronger estimates follow from \\eqref{L^infty norms of derivatives of alpha and beta errors}\n\\begin{equation}\\label{eqn::W^2,infty norms of alpha & beta sharps}\n\\|\\alpha^\\sharp\\|_{W^{2,\\infty}(\\R^3)},\\ \\|\\beta^\\sharp\\|_{W^{2,\\infty}(\\R^3)} = \\mathcal O(1)\\quad\\text{as}\\quad \\tau\\to \\infty.\n\\end{equation}\n\n\\subsection{Transport equation}\n\nSince $\\zeta \\cdot \\zeta = i\\omega \\sigma_0 \\mu_0$, using \\eqref{eqn:: conj grad and laplacian}, we obtain (cf.~(\\ref{eq:Lsm}))\n\\begin{align*}\nf &= -\\Delta_\\zeta a - \\nabla_\\zeta(\\nabla\\beta\\cdot a) - \\nabla\\alpha\\times \\nabla_\\zeta\\times a  - i\\omega\\sigma\\mu a + i\\omega \\sigma_0 \\mu_0 a \\\\\n& = -\\Delta a - \\nabla(\\nabla\\beta\\cdot a) - \\nabla\\alpha\\times \\nabla\\times a  - i\\omega (\\sigma\\mu - \\sigma_0 \\mu_0) a\n- 2i \\zeta\\cdot \\nabla a - (\\nabla\\beta\\cdot a) i\\zeta - \\nabla\\alpha \\times (i\\zeta \\times a).\n\\end{align*}\nWe shall consider $\\zeta$ of the form $\\zeta = \\tau \\rho + \\zeta_1$ where $\\tau > 0$ is a large parameter, $\\rho \\in \\C^3$  is independent of $\\tau$ and satisfies $\\Re\\rho \\cdot \\Im\\rho = 0$ and $|\\Re\\rho| = |\\Im\\rho| = 1$, and $\\zeta_1 = \\mathcal O(1)$ as $\\tau \\to \\infty$. Then\n\\begin{align*}\nf &= -\\Delta a - \\nabla(\\nabla\\beta\\cdot a) - \\nabla\\alpha\\times \\nabla\\times a  - i\\omega (\\sigma\\mu - \\sigma_0 \\mu_0) a\n- 2i \\zeta_1\\cdot \\nabla a - (\\nabla\\beta\\cdot a) i\\zeta_1 - \\nabla\\alpha \\times (i\\zeta_1 \\times a)\\\\\n& \\quad - i\\tau ((\\nabla(\\beta-\\beta^\\sharp)\\cdot a) \\rho + \\nabla(\\alpha-\\alpha^\\sharp) \\times (\\rho \\times a))\n - i\\tau (2\\rho\\cdot \\nabla a + (\\nabla\\beta^\\sharp\\cdot a) \\rho + \\nabla\\alpha^\\sharp \\times (\\rho \\times a)).\n\\end{align*}\nIn order to get $\\|f \\|_{L^2_{\\delta+1}} = o(\\tau)$ as $\\tau \\to\n\\infty$, for $-1 < \\delta < 0$, we should construct $a$ satisfying the\ntransport equation, that is,\n\\begin{equation}\\label{transport eqn}\n2\\rho\\cdot \\nabla a + (\\nabla\\beta^\\sharp\\cdot a) \\rho + \\nabla\\alpha^\\sharp \\times (\\rho \\times a) = o(1)\\quad\\text{in}\\quad L^2_{\\delta+1}\\quad\\text{as}\\quad\\tau\\to\\infty.\n\\end{equation}\nWe\nget\n\\begin{equation}\\label{transport eqn rewritten}\n2\\rho \\cdot \\nabla a + (\\nabla\\beta^\\sharp \\cdot a) \\rho + (\\nabla\\alpha^\\sharp \\cdot a) \\rho - (\\rho \\cdot \\nabla\\alpha^\\sharp) a = o(1)\\quad\\text{in}\\quad L^2_{\\delta+1}\\quad\\text{as}\\quad\\tau\\to\\infty.\n\\end{equation}\nFor arbitrary $s_0\\in \\R$, we seek solutions of \\eqref{transport eqn rewritten} in the form\n$$\na = e^{-\\alpha^\\sharp \/ 2} \\rho + s_0 e^{\\alpha^\\sharp \/ 2}e^{\\chi_\\tau \\Psi^\\sharp} \\overline{\\rho},\n$$\nwhere $\\chi_\\tau(x) := \\chi(\\tau^{-\\theta}x)$, $0< \\theta < 1\/2$ with $\\chi\\in C^\\infty_0(\\R^3)$ such that $\\chi(x) \\equiv 1$ for $|x| < 1\/2$ and $\\chi(x) \\equiv 0$ for $|x| \\ge 1$. Then\n\\begin{equation}\\label{nabla a}\n\\p_j a = - \\frac{1}{2} \\p_j \\alpha^\\sharp e^{-\\alpha^\\sharp \/ 2} \\rho + \\frac{s_0}{2}\\p_j \\alpha^\\sharp e^{\\alpha^\\sharp \/ 2}e^{\\chi_\\tau \\Psi^\\sharp} \\overline{\\rho}\n + \\frac{s_0}{\\tau^\\theta} (\\p_j\\chi)(\\theta^{-\\theta} x)\\Psi^\\sharp e^{\\alpha^\\sharp \/ 2}e^{\\chi_\\tau \\Psi^\\sharp} \\overline{\\rho} + s_0 \\chi_\\tau \\p_j \\Psi^\\sharp e^{\\alpha^\\sharp \/ 2}e^{\\chi_\\tau \\Psi^\\sharp} \\overline{\\rho}.\n\\end{equation}\nSubstituting these into \\eqref{transport eqn rewritten}, we come to\n$$\ns_0 \\Big\\{ \\tau^{-\\theta}\\Psi^\\sharp 2\\rho\\cdot(\\nabla \\chi)(\\tau^{-\\theta}\\,\\cdot\\,) + \\chi_\\tau 2\\rho \\cdot \\nabla \\Psi^\\sharp + \\overline{\\rho} \\cdot \\nabla(\\alpha^\\sharp + \\beta^\\sharp) \\Big\\} e^{\\alpha^\\sharp \/ 2} e^{\\chi_\\tau \\Psi^\\sharp} \\overline{\\rho} = o(1)\\quad\\text{in}\\quad L^2_{\\delta+1}\\quad\n\\text{as $\\tau\\to\\infty$}. \n$$\nWe observe that by \\eqref{supports of derivatives of alpha and beta}, $\\nabla(\\alpha^\\sharp + \\beta^\\sharp) = \\chi_\\tau \\nabla(\\alpha^\\sharp + \\beta^\\sharp)$ for large enough $\\tau>0$. Therefore, it is enough to find $\\Psi^\\sharp$ in $C^\\infty(\\R^3)$ with a nice control on $\\|\\Psi^\\sharp\\|_{L^\\infty(\\R^3)}$ and satisfying\n\\begin{equation}\\label{eqns for Phi and Psi}\n2\\rho \\cdot \\nabla \\Psi^\\sharp + \\overline{\\rho} \\cdot \\nabla(\\alpha^\\sharp + \\beta^\\sharp) = 0 \\quad\\text{in}\\quad \\R^3.\n\\end{equation}\nSince $\\rho\\cdot \\rho = 0$ and $\\Re\\rho \\cdot \\Im\\rho = 0$, the operator $N_\\rho := \\rho \\cdot \\nabla$ is just the $\\overline\\p$-operator in certain linear coordinates. Its inverse is defined as\n$$\nN_\\rho^{-1} f(x) := \\frac{1}{2\\pi} \\int_{\\R^2} \\frac{f(x - y_1 \\Re\\rho - y_2 \\Im\\rho)}{y_1 + i y_2}\\,dy,\\quad f\\in C_0(\\R^3).\n$$\nThen by \\cite[Lemma~4.6]{salo2004inverse},\n$$\n\\Psi^\\sharp(x,\\rho;\\tau) := - \\frac{1}{2} N_\\rho^{-1} \\{\\overline{\\rho} \\cdot \\nabla(\\alpha^\\sharp + \\beta^\\sharp)\\} \\in C^{\\infty}(\\R^3)\n$$\nsatisfies equation \\eqref{eqns for Phi and Psi}. It also follows from \\cite[Lemma~4.6]{salo2004inverse} and \\eqref{L^infty norms of derivatives of alpha and beta errors} that\n\\begin{equation}\\label{estimate W^1,infty of Psi^sharp}\n\\|\\Psi^\\sharp\\|_{W^{1,\\infty}(\\R^3)} \\le \\mathcal O(1) \\big\\{\\|\\alpha^\\sharp\\|_{W^{2,\\infty}(\\R^3)} + \\|\\beta^\\sharp\\|_{W^{2,\\infty}(\\R^3)} \\big\\} = \\mathcal O(1)\\quad\\text{as}\\quad \\tau\\to\\infty.\n\\end{equation}\nFurthermore, \\cite[Lemma~4.6]{salo2004inverse} and \\eqref{eqn::L^infty norms of derivatives of alpha & beta} imply that\n\\begin{equation}\\label{estimate W^k,infty of Psi^sharp}\n\\begin{aligned}\n\\|\\Psi^\\sharp\\|_{W^{k,\\infty}(\\R^3)} &\\le \\mathcal O_k(1) \\big\\{\\|\\alpha^\\sharp\\|_{W^{k+1,\\infty}(\\R^3)} + \\|\\beta^\\sharp\\|_{W^{k+1,\\infty}(\\R^3)} \\big\\}\\\\\n&= \\mathcal O_k(\\tau^{k\\epsilon+\\epsilon})\\quad\\text{for}\\quad k = 0,1,\\ldots \\quad\\text{as}\\quad \\tau\\to\\infty.\n\\end{aligned}\n\\end{equation}\nWe set\n$$\n\\Psi(x,\\rho) := -\\frac{1}{2} N_\\rho^{-1} \\{\\overline{\\rho} \\cdot \\nabla(\\alpha + \\beta)\\} \\in L^\\infty(\\R^3).\n$$\nThen \\cite[Lemma~3.1]{sylvester1987global} together with \\eqref{supports of derivatives of alpha and beta} and \\eqref{L^infty norms of derivatives of alpha and beta errors} implies that\n$$\n\\|\\Psi^\\sharp(\\cdot,\\rho;\\tau) - \\Psi(\\cdot,\\rho)\\|_{L^2_{\\rm loc}(\\R^3)} = o(1)\\quad\\text{as}\\quad \\tau\\to \\infty.\n$$\nFinally, by \\cite[Lemma~3.1]{salo2006semiclassical} together with \\eqref{eqn::L^infty norms of derivatives of alpha & beta} and \\eqref{eqn::W^2,infty norms of alpha & beta sharps},\n\\begin{equation}\\label{estimate pointwise derivatives of Psi^sharp}\n\\begin{aligned}\n|\\p^\\kappa \\Psi^\\sharp(x,\\rho;\\tau)| &\\le \\mathcal O_\\kappa(1) \\begin{cases}\n(1 + |x_T|^2)^{-1\/2}\\chi_{B(0,R)}(x_\\perp),&\\text{if } |\\kappa| = 0, 1,\\\\\n\\tau^{|\\kappa|\\epsilon + \\epsilon}(1 + |x_T|^2)^{-1\/2}\\chi_{B(0,R)}(x_\\perp),&\\text{otherwise}\n\\end{cases}\\\\\n&\\quad\\,\\,\\text{as}\\quad \\tau\\to\\infty,\n\\end{aligned}\n\\end{equation}\nwhere $x_T$ is the projection of $x$ onto $\\Span\\{\\rho_1,\\rho_2\\}$ and $x_\\perp = x - x_T$, and $R>0$ is such that $B(0,R)$ contains $\\supp(\\sigma - \\sigma_0)$ and $\\supp(\\mu - \\mu_0)$.\\smallskip\n\nIt follows that\n\\begin{equation}\\label{integral Japanese symbol estimate}\n\\int_{|x_T| \\le \\tau^\\theta,\\, |x_\\perp| \\le R} (1 + |x_T|^2)^{-1} (1 + |x|^2)^{\\delta+1}\\, dx\n\\le \\mathcal O(1) \\int_{|x_T| \\le \\tau^\\theta} (1 + |x_T|^2)^{\\delta}\\, dx_T = \\mathcal O(\\tau^{2(\\delta+1)\\theta})\\quad\\text{as}\\quad\\tau \\to \\infty.\n\\end{equation}\nSimilarly,\n\\begin{equation}\\label{integral Japanese symbol estimate -j}\n\\int_{|x_T| \\le \\tau^\\theta,\\, |x_\\perp| \\le R} (1 + |x_T|^2)^{-j} (1 + |x|^2)^{\\delta+1}\\, dx\n\\le \\mathcal O(1) \\Big(1-\\frac{1}{(1+\\tau^{2\\theta})^{j-2-\\delta}}\\Big) = \\mathcal O(1)\\quad\\text{as}\\quad\\tau \\to \\infty\\quad\\text{for}\\quad j\\ge 2.\n\\end{equation}\nWe have\n\n\\begin{Lemma}\nLet $\\chi_\\tau$ and $\\Psi^\\sharp$ be as above. Then\n\\begin{align}\n\\|\\p_j(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} &= \\mathcal O(\\tau^{(\\delta+1)\\theta}),\\label{lemma tech 1}\\\\\n\\|\\p_j(\\chi_\\tau \\Psi^\\sharp) \\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} &= \\mathcal O(1),\\label{lemma tech 2}\\\\\n\\|\\p_j(\\chi_\\tau \\Psi^\\sharp) \\p_k(\\chi_\\tau \\Psi^\\sharp) \\p_l(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} &= \\mathcal O(1),\\label{lemma tech 3}\\\\\n\\|\\p_j\\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} &= o(\\tau),\\label{lemma tech 4}\\\\\n\\|\\p_j\\p_k(\\chi_\\tau \\Psi^\\sharp) \\p_l(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} &= \\mathcal O(\\tau^{3\\epsilon}),\\\\\n\\|\\p_l\\p_j\\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} &= o(\\tau^{1+\\epsilon})\\label{lemma tech 5}\n\\end{align}\nas $\\tau \\to \\infty$.\n\\end{Lemma}\n\n\\begin{proof}\nUsing \\eqref{estimate pointwise derivatives of Psi^sharp} and \\eqref{integral Japanese symbol estimate}, we obtain\n\\begin{align*}\n\\|\\p_j(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} &\\le \\frac{1}{\\tau^\\theta}\\|\\p_j\\chi(\\tau^{-\\theta}\\,\\cdot\\,) \\Psi^\\sharp\\|_{L^2_{\\delta+1}} + \\|\\chi_\\tau \\p_j\\Psi^\\sharp\\|_{L^2_{\\delta+1}} \\\\\n&\\le \\mathcal O\\Big(\\frac{1}{\\tau^\\theta} + 1\\Big) \\int_{|x_T| \\le \\tau^\\theta,\\, |x_\\perp| \\le R} (1 + |x_T|^2)^{-1} (1 + |x|^2)^{\\delta+1}\\, dx\\\\\n& = \\mathcal O(\\tau^{(\\delta+1)\\theta}) \\quad\\text{as}\\quad\\tau \\to \\infty.\n\\end{align*}\nThe other estimates follow readily.\n\\end{proof}\n\n\\subsection{Estimating $\\|f\\|_{L^2_{\\delta+1}}$}\\label{section on L^2 norm of f}\n\nWith our choice of $a$, we have\n\\begin{equation}\\label{transport equation part}\n2\\rho\\cdot \\nabla a + (\\nabla\\beta^\\sharp\\cdot a) \\rho + \\nabla\\alpha^\\sharp \\times (\\rho \\times a) = s_0 \\tau^{-\\theta} 2\\rho\\cdot(\\nabla \\chi)(\\tau^{-\\theta}\\,\\cdot\\,) \\Psi^\\sharp e^{\\alpha^\\sharp \/ 2} e^{\\chi_\\tau \\Psi^\\sharp} \\overline{\\rho}.\n\\end{equation}\nThen, as in the proof of \\eqref{lemma tech 1}, we use \\eqref{eqn::W^2,infty norms of alpha & beta sharps}, \\eqref{estimate W^1,infty of Psi^sharp}, \\eqref{estimate pointwise derivatives of Psi^sharp} and \\eqref{integral Japanese symbol estimate} to obtain\n\\begin{multline*}\n\\| i\\tau (2\\rho\\cdot \\nabla a + (\\nabla\\beta^\\sharp\\cdot a) \\rho + \\nabla\\alpha^\\sharp \\times (\\rho \\times a)) \\|_{L^2_{\\delta+1}} \\\\\n\\le \\mathcal O(\\tau^{1-\\theta}) \\Big(\\int_{|x_T| \\le \\tau^\\theta,\\, |x_\\perp| \\le R} (1 + |x_T|^2)^{-1} (1 + |x|^2)^{\\delta+1}\\, dx \\Big)^{1\/2} = \\mathcal O(\\tau^{1+\\delta\\theta})\\quad\\text{as}\\quad\\tau \\to \\infty.\n\\end{multline*}\nUsing \\eqref{sigma and mu are constants outside of a compact set},\n\\eqref{supports of derivatives of alpha and beta},\n\\eqref{L^infty norms of derivatives of alpha and beta errors},\n\\eqref{eqn::W^2,infty norms of alpha & beta sharps} and\n\\eqref{estimate W^1,infty of Psi^sharp}, it is then straightforward to show\nthat\n\\begin{align*}\n\\| i\\tau ((\\nabla(\\beta-\\beta^\\sharp)\\cdot a) \\rho + \\nabla(\\alpha-\\alpha^\\sharp) \\times (\\rho \\times a)) \\|_{L^2_{\\delta+1}} &= o(1)\\ \\quad\\text{as}\\quad\\tau \\to \\infty,\\\\[0.05cm]\n\\| (\\nabla\\beta\\cdot a) i\\zeta_1 + \\nabla\\alpha \\times (i\\zeta_1 \\times a) \\|_{L^2_{\\delta+1}} &= \\mathcal O(1)\\quad\\text{as}\\quad\\tau \\to \\infty,\\\\\n\\|i\\omega (\\sigma\\mu - \\sigma_0 \\mu_0) a\\|_{L^2_{\\delta+1}} &= \\mathcal O(1)\\quad\\text{as}\\quad\\tau \\to \\infty.\n\\end{align*}\nNow, using expressions \\eqref{eqn::W^2,infty norms of alpha & beta sharps} and \\eqref{estimate W^1,infty of Psi^sharp}, we find that\n$$\n\\|\\p_j a \\|_{L^2_{\\delta+1}} \\le \\mathcal O(1)\\|\\p_j\\alpha^\\sharp\\|_{L^2_{\\delta+1}} + \\mathcal O(1)\\|\\p_j(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\n$$\nThus with \\eqref{supports of derivatives of alpha and beta}, \\eqref{eqn::W^2,infty norms of alpha & beta sharps} and \\eqref{lemma tech 1}, we obtain\n\\begin{equation}\\label{L2 norm of der a}\n   \\|\\p_j a \\|_{L^2_{\\delta+1}}\n   = o(\\tau^{\\theta})\\quad\\text{as}\\quad\\tau \\to \\infty,\n\\end{equation}\nand therefore,\n$$\n\\| 2i \\zeta_1\\cdot \\nabla a \\|_{L^2_{\\delta+1}} = o(\\tau^{\\theta})\\quad\\text{as}\\quad\\tau \\to \\infty,\n$$\n$$\n\\|\\nabla\\alpha\\times \\nabla\\times a\\|_{L^2_{\\delta+1}} \\le \\mathcal O(1) \\|\\nabla\\times a\\|_{L^2_{\\delta+1}} \\le \\mathcal O(1)\\sum_{j=1}^3\\|\\p_j a \\|_{L^2_{\\delta+1}} = o(\\tau^{\\theta})\\quad\\text{as}\\quad\\tau \\to \\infty\n$$\nand\n$$\n\\|\\nabla(\\nabla\\beta\\cdot a)\\|_{L^2_{\\delta+1}} \\le \\|(\\nabla\\nabla\\beta) a\\|_{L^2_{\\delta+1}} + \\mathcal O(1) \\sum_{j=1}^3 \\|\\p_j a\\|_{L^2_{\\delta+1}} = o(\\tau^\\theta)\\quad\\text{as}\\quad\\tau \\to \\infty.\n$$\nIn the last step, we used \\eqref{sigma and mu are constants outside of a compact set} and that $\\supp(\\nabla\\nabla\\beta)$ is compact.\nFinally, by \\eqref{eqn::W^2,infty norms of alpha & beta sharps} and\n\\eqref{estimate W^1,infty of Psi^sharp}\n\\begin{multline*}\n\\|\\p_j\\p_k a\\|_{L^2_{\\delta+1}} \\le \\mathcal O(1) \\|\\p_j\\p_k \\alpha^\\sharp\\|_{L^2_{\\delta+1}} + \\mathcal O(1) \\|\\p_j\\alpha^\\sharp \\p_j\\alpha^\\sharp\\|_{L^2_{\\delta+1}}\\\\\n+ \\mathcal O(1)\\|\\p_j\\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} + \\mathcal O(1)\\|\\p_j(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\\\\\n+ \\mathcal O(1)\\|\\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} + \\mathcal O(1)\\|\\p_j(\\chi_\\tau \\Psi^\\sharp)\\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\n\\end{multline*}\nwhence, by \\eqref{supports of derivatives of alpha and beta}, \\eqref{eqn::W^2,infty norms of alpha & beta sharps}, \\eqref{lemma tech 1}, \\eqref{lemma tech 2} and \\eqref{lemma tech 4},\n\\begin{equation}\\label{L2 norm of second der of a}\n   \\|\\p_j\\p_k a\\|_{L^2_{\\delta+1}}\n   = o(\\tau)\\quad\\text{as}\\quad\\tau \\to \\infty.\n\\end{equation}\nThus,\n$$\n\\|\\Delta a\\|_{L^2_{\\delta+1}} \\le \\sum_{j=1}^3 \\|\\p_j^2 a\\|_{L^2_{\\delta+1}} = o(\\tau)\\quad\\text{as}\\quad\\tau \\to \\infty.\n$$\nCombining all of the above estimates, we come to\n$$\n\\|f\\|_{L^2_{\\delta+1}} = o(\\tau)\\quad\\text{as}\\quad \\tau\\to\\infty.\n$$\n\n\\subsection{Estimating $\\|\\nabla_\\zeta \\times f\\|_{L^2_{\\delta+1}}$}\n\\label{section on L^2 norm of weighted curl of f}\n\nSince\n$$\n\\nabla_\\zeta\\times f = \\nabla\\times f + i\\zeta\\times f\\quad\\text{and}\\quad \\|i\\zeta\\times f\\|_{L^2_{\\delta+1}} = o(\\tau^2),\n$$\nwe need to estimate $\\|\\nabla\\times f\\|_{L^2_{\\delta+1}}$. By straightforward calculations,\n\\begin{align*}\n  \\nabla\\times f &= -\\nabla\\times \\Delta a - \\nabla\\times(\\nabla\\alpha\\times \\nabla\\times a)  - i\\omega \\nabla\\times\\big((\\sigma\\mu - \\sigma_0 \\mu_0) a \\big)\n  - 2i \\nabla\\times(\\zeta_1\\cdot \\nabla a) - \\nabla (\\nabla\\beta\\cdot a)\\times i\\zeta_1 - \\nabla\\times\\big(\\nabla\\alpha \\times (i\\zeta_1 \\times a)\\big)\\\\\n& \\quad - i\\tau \\nabla(\\nabla(\\beta-\\beta^\\sharp)\\cdot a)\\times \\rho - i\\tau\\nabla\\times\\big(\\nabla(\\alpha-\\alpha^\\sharp) \\times (\\rho \\times a)\\big)\n - i\\tau \\nabla\\times \\big(2\\rho\\cdot \\nabla a + (\\nabla\\beta^\\sharp\\cdot a) \\rho + \\nabla\\alpha^\\sharp \\times (\\rho \\times a)\\big).\n\\end{align*}\nBy \\eqref{transport equation part},\n$$\n\\nabla\\times \\big(2\\rho\\cdot \\nabla a + (\\nabla\\beta^\\sharp\\cdot a) \\rho + \\nabla\\alpha^\\sharp \\times (\\rho \\times a)\\big) = \\frac{s_0}{\\tau^{\\theta}} \\nabla\\Big(2\\rho\\cdot(\\nabla \\chi)(\\tau^{-\\theta}\\,\\cdot\\,) \\Psi^\\sharp e^{\\alpha^\\sharp \/ 2} e^{\\chi_\\tau \\Psi^\\sharp}\\Big) \\times \\overline{\\rho}.\n$$\nThen, as in the proof of \\eqref{lemma tech 4}, we use\n\\eqref{eqn::W^2,infty norms of alpha & beta sharps} and\n\\eqref{estimate W^1,infty of Psi^sharp},\n\\begin{align*}\n\\|i\\tau \\nabla &\\times \\big(2\\rho\\cdot \\nabla a + (\\nabla\\beta^\\sharp\\cdot a) \\rho + \\nabla\\alpha^\\sharp \\times (\\rho \\times a)\\big)\\|_{L^2_{\\delta+1}}\n\\le \\mathcal O(\\tau^{1-\\theta})\\sum_{j,k=1}^3\\|\\p_j(\\p_k\\chi(\\tau^{-\\theta}\\,\\cdot\\,)\\Psi^\\sharp e^{\\alpha^\\sharp \/ 2} e^{\\chi_\\tau \\Psi^\\sharp})\\|_{L^2_{\\delta+1}}\\\\\n&\\le \\mathcal O(\\tau^{1-2\\theta})\\sum_{j,k=1}^3\\|\\p_j\\p_k\\chi(\\tau^{-\\theta}\\,\\cdot\\,)\\Psi^\\sharp\\|_{L^2_{\\delta+1}} + \\mathcal O(\\tau^{1-\\theta})\\sum_{j,k=1}^3\\|\\p_k\\chi(\\tau^{-\\theta}\\,\\cdot\\,)\\p_j\\Psi^\\sharp\\|_{L^2_{\\delta+1}}\n+ \\mathcal O(\\tau^{1-\\theta})\\sum_{j,k=1}^3\\|\\p_k\\chi(\\tau^{-\\theta}\\,\\cdot\\,)\\p_j\\alpha^\\sharp \\Psi^\\sharp\\|_{L^2_{\\delta+1}} \\\\\n&\\quad+ \\mathcal O(\\tau^{1-2\\theta})\\sum_{j,k=1}^3\\|\\p_k\\chi(\\tau^{-\\theta}\\,\\cdot\\,)\\p_j\\chi(\\tau^{-\\theta}\\,\\cdot\\,) \\Psi^\\sharp{}^2\\|_{L^2_{\\delta+1}}\n+ \\mathcal O(\\tau^{1-\\theta})\\sum_{j,k=1}^3\\|\\p_k\\chi(\\tau^{-\\theta}\\,\\cdot\\,)\\p_j\\Psi^\\sharp \\chi_\\tau \\Psi^\\sharp\\|_{L^2_{\\delta+1}}\n= o(\\tau)\n   \\quad\\text{as}\\quad \\tau\\to\\infty,\n\\end{align*}\nwhere in the last step, we also used \\eqref{supports of derivatives of\n  alpha and beta}, \\eqref{estimate pointwise derivatives of\n  Psi^sharp}, \\eqref{integral Japanese symbol estimate} and\n\\eqref{integral Japanese symbol estimate -j}. Next, using \\eqref{sigma\n  and mu are constants outside of a compact set}, \\eqref{supports of\n  derivatives of alpha and beta}, \\eqref{L^infty norms of derivatives\n  of alpha and beta errors}, \\eqref{eqn::W^2,infty norms of alpha &\n  beta sharps}, \\eqref{estimate W^1,infty of Psi^sharp} and \\eqref{L2\n  norm of der a}, we obtain\n\\begin{equation*}\n\\| i\\tau \\nabla(\\nabla(\\beta-\\beta^\\sharp)\\cdot a)\\times \\rho\\|_{L^2_{\\delta+1}}\\le \\mathcal O(\\tau)\\sum_{j,k} \\|\\p_j\\p_k (\\beta - \\beta^\\sharp)a\\|_{L^2_{\\delta+1}} + \\mathcal O(\\tau)\\sum_{j,k}\\|\\p_k (\\beta - \\beta^\\sharp) \\p_j a\\|_{L^2_{\\delta+1}}\n= o(\\tau^{1+\\theta})\\quad\\text{as}\\quad \\tau\\to\\infty.\n\\end{equation*}\nSimilarly,\n\\begin{equation*}\n\\| i\\tau \\nabla\\times\\big(\\nabla(\\alpha-\\alpha^\\sharp) \\times (\\rho \\times a)\\big) \\|_{L^2_{\\delta+1}}\\le \\mathcal O(\\tau)\\sum_{j,k} \\|\\p_j\\p_k (\\alpha-\\alpha^\\sharp)a\\|_{L^2_{\\delta+1}} + \\mathcal O(\\tau)\\sum_{j,k}\\|\\p_k (\\alpha-\\alpha^\\sharp) \\p_j a\\|_{L^2_{\\delta+1}} = o(\\tau^{1+\\theta}),\n\\end{equation*}\n\\begin{equation*}\n\\|\\nabla (\\nabla\\beta\\cdot a) \\times i\\zeta_1 \\|_{L^2_{\\delta+1}}\\le \\mathcal O(1)\\sum_{j,k} \\|\\p_j\\p_k \\beta\\, a\\|_{L^2_{\\delta+1}} + \\mathcal O(1)\\sum_{j,k}\\|\\p_k \\beta\\, \\p_j a\\|_{L^2_{\\delta+1}} = o(\\tau^{\\theta}),\n\\end{equation*}\n\\begin{equation*}\n\\|\\nabla\\times\\big(\\nabla\\alpha \\times (i\\zeta_1 \\times a)\\big) \\|_{L^2_{\\delta+1}}\\le \\mathcal O(1)\\sum_{j,k} \\|\\p_j\\p_k \\alpha\\, a\\|_{L^2_{\\delta+1}} + \\mathcal O(1)\\sum_{j,k}\\|\\p_k \\alpha\\, \\p_j a\\|_{L^2_{\\delta+1}} = o(\\tau^{\\theta}),\n\\end{equation*}\nand\n\\begin{equation*}\n\\|i\\omega\\nabla\\times\\big((\\sigma\\mu - \\sigma_0 \\mu_0) a\\big)\\|_{L^2_{\\delta+1}}\\le \\mathcal O(1)\\sum_{j} \\|\\p_j(\\sigma\\mu) a\\|_{L^2_{\\delta+1}} + \\mathcal O(1)\\sum_{j,k}\\|(\\sigma\\mu - \\sigma_0\\mu_0) \\p_j a\\|_{L^2_{\\delta+1}} = o(\\tau^\\theta)\n\\end{equation*}\nas $\\tau \\to \\infty$. Using \\eqref{L2 norm of second der of a} we find that\n$$\n\\|2i\\nabla\\times(\\zeta_1\\cdot \\nabla a)\\|_{L^2_{\\delta+1}} \\le \\mathcal O(1) \\sum_{j,k} \\|\\p_j\\p_k a\\|_{L^2_{\\delta+1}} = o(\\tau)\n$$\nand\n$$\n\\|\\nabla\\times(\\nabla\\alpha\\times \\nabla\\times a)\\|_{L^2_{\\delta+1}}\\le \\mathcal O(1)\\sum_{j,k,l} \\|\\p_j\\p_k \\alpha\\, \\p_l a\\|_{L^2_{\\delta+1}} + \\mathcal O(1)\\sum_{j,k,l}\\|\\p_k \\alpha\\, \\p_j\\p_l a\\|_{L^2_{\\delta+1}} = o(\\tau)\n$$\nas $\\tau \\to \\infty$.\n\nUsing \\eqref{eqn::W^2,infty norms of alpha & beta sharps} and \\eqref{estimate W^1,infty of Psi^sharp}, we estimate\n\\begin{align*}\n\\|\\p_l\\p_j\\p_k a\\|_{L^2_{\\delta+1}} &\\le \\mathcal O(1)\\|\\p_l\\p_j\\p_k \\alpha^\\sharp\\|_{L^2_{\\delta+1}} + \\mathcal O(1)\\|\\p_l\\p_j \\alpha^\\sharp \\p_k \\alpha^\\sharp\\|_{L^2_{\\delta+1}}\n+ \\mathcal O(1)\\|\\p_j \\alpha^\\sharp \\p_l\\p_k \\alpha^\\sharp\\|_{L^2_{\\delta+1}}\n \\\\ &\\\n + \\mathcal O(1)\\|\\p_j\\p_k \\alpha^\\sharp \\p_l \\alpha^\\sharp\\|_{L^2_{\\delta+1}}\n + \\mathcal O(1)\\|\\p_l \\alpha^\\sharp \\p_j \\alpha^\\sharp \\p_k \\alpha^\\sharp\\|_{L^2_{\\delta+1}}\n + \\mathcal O(1) \\|\\p_l\\p_j\\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\n \\\\ &\\\n + \\mathcal O(1) \\|\\p_l\\p_j\\alpha^\\sharp \\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\n + \\mathcal O(1) \\|\\p_l\\p_j(\\chi_\\tau \\Psi^\\sharp) \\p_k\\alpha^\\sharp\\|_{L^2_{\\delta+1}} + \\mathcal O(1) \\|\\p_l\\p_j(\\chi_\\tau \\Psi^\\sharp) \\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\n \\\\ &\\\n + \\mathcal O(1) \\|\\p_j\\alpha^\\sharp \\p_l\\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} + \\mathcal O(1) \\|\\p_j(\\chi_\\tau \\Psi^\\sharp) \\p_l\\p_k\\alpha^\\sharp\\|_{L^2_{\\delta+1}} + \\mathcal O(1) \\|\\p_j(\\chi_\\tau \\Psi^\\sharp) \\p_l\\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\n \\\\ &\\\n + \\mathcal O(1) \\|\\p_j\\p_k\\alpha^\\sharp \\p_l(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} + \\mathcal O(1) \\|\\p_j \\p_k(\\chi_\\tau \\Psi^\\sharp) \\p_l\\alpha^\\sharp\\|_{L^2_{\\delta+1}}+ \\mathcal O(1) \\|\\p_j \\p_k(\\chi_\\tau \\Psi^\\sharp) \\p_l(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\n \\\\ &\\\n + \\mathcal O(1) \\|\\p_j\\alpha^\\sharp \\p_k\\alpha^\\sharp \\p_l(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\n + \\mathcal O(1) \\|\\p_j\\alpha^\\sharp \\p_k(\\chi_\\tau \\Psi^\\sharp) \\p_l\\alpha^\\sharp\\|_{L^2_{\\delta+1}}\n + \\mathcal O(1) \\|\\p_j\\alpha^\\sharp \\p_k(\\chi_\\tau \\Psi^\\sharp) \\p_l(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\n \\\\ &\\\n + \\mathcal O(1) \\|\\p_j(\\chi_\\tau \\Psi^\\sharp) \\p_k\\alpha^\\sharp \\p_l\\alpha^\\sharp\\|_{L^2_{\\delta+1}}\n + \\mathcal O(1) \\|\\p_j(\\chi_\\tau \\Psi^\\sharp) \\p_k\\alpha^\\sharp \\p_l(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\n + \\mathcal O(1) \\|\\p_j(\\chi_\\tau \\Psi^\\sharp) \\p_k(\\chi_\\tau \\Psi^\\sharp) \\p_l\\alpha^\\sharp\\|_{L^2_{\\delta+1}}\n \\\\ &\\hspace*{4.55cm}\n + \\mathcal O(1) \\|\\p_j(\\chi_\\tau \\Psi^\\sharp) \\p_k(\\chi_\\tau \\Psi^\\sharp) \\p_l(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\\quad\\text{as}\\quad\\tau \\to \\infty\n\\end{align*}\nand conclude that\n\\[\n   \\|\\p_l\\p_j\\p_k a\\|_{L^2_{\\delta+1}}\n       = o(\\tau^{1+\\epsilon})\\quad\\text{as}\\quad\\tau \\to \\infty.\\ \\\n\\]\nThis implies that\n$$\n\\|\\nabla\\times \\Delta a\\|_{L^2_{\\delta+1}} = o(\\tau^{1+\\epsilon})\\quad\\text{as}\\quad\\tau \\to \\infty.\n$$\nCombining all of these, we finally come to $\\|\\nabla\\times\nf\\|_{L^2_{\\delta+1}} = o(\\tau^{1+\\epsilon})$ and, hence,\n$\\|\\nabla_\\zeta\\times f\\|_{L^2_{\\delta+1}} =\no(\\tau^2)\\quad\\text{as}\\quad\\tau \\to \\infty$.\n\n\\subsection{Construction of complex geometric optics solutions} Now, we are ready to construct complex geometric optics solutions for the system \\eqref{eqn3-1} and \\eqref{eqn3-2} which is equivalent to\n\\begin{align}\ne^{-i \\zeta\\cdot x}L_{\\sigma,\\mu}(e^{i \\zeta\\cdot x}r) & = - f, \\label{eqn LE=0 rewritten}\\\\\n\\nabla_\\zeta \\cdot r + \\nabla\\log\\mu \\cdot r & = - \\nabla_\\zeta \\cdot a + \\nabla\\log\\mu \\cdot a. \\label{eqn div(sigma E)=0 rewritten}\n\\end{align}\nAccording to the discussion in Section~\\ref{section on L^2 norm of f}, we have $\\|f\\|_{L^2_{\\delta+1}} = o(\\tau)$ as $\\tau \\to \\infty$.\\smallskip\n\nFirst, we need to show that for sufficiently large $|\\zeta|$ there is $r\\in L^2_\\delta$ solving \\eqref{eqn LE=0 rewritten}. Using \\eqref{eqn:: conj grad and laplacian},\n\\begin{equation}\\label{another pde for r}\ne^{-i \\zeta\\cdot x} L_{\\sigma,\\mu}(e^{i \\zeta\\cdot x}r) = -\\Delta_\\zeta r - \\nabla_\\zeta(\\nabla\\log\\mu\\cdot r) - \\nabla\\log\\sigma\\times \\nabla_\\zeta\\times r  - i\\omega (\\sigma\\mu - \\sigma_0 \\mu_0) r.\n\\end{equation}\nWe have\n$$\n\\nabla_\\zeta(\\nabla\\log\\mu\\cdot r) = \\nabla\\log\\mu \\times\n\\nabla_\\zeta\\times r + \\nabla\\log\\mu \\cdot \\nabla_\\zeta r +\n\\nabla\\nabla(\\log\\mu)r\n$$\nTherefore, \\eqref{eqn LE=0 rewritten} can be written as\n\\begin{equation}\\label{pre final pde for r}\n-\\Delta_\\zeta r - \\nabla\\log\\mu \\cdot \\nabla_\\zeta r - \\nabla\\log(\\sigma\\mu) \\times \\nabla_\\zeta\\times r - V_1 r = - f,\n\\end{equation}\nwhere\n$$\nV_1 := \\nabla\\nabla(\\log\\mu) + i\\omega (\\sigma\\mu - \\sigma_0 \\mu_0).\n$$\nWe need to deal with the third term on the left hand-side of\n\\eqref{pre final pde for r}. We define $Q := \\nabla_\\zeta\\times r$, and\nfind that\n$$\n\\Delta_\\zeta r = \\nabla_\\zeta \\nabla_\\zeta \\cdot r - \\nabla_\\zeta \\times \\nabla_\\zeta \\times r.\n$$\nAlso, it follows from \\eqref{eqn div(sigma E)=0 rewritten} that\n$$\n\\nabla_\\zeta \\nabla_\\zeta \\cdot r = - \\nabla_\\zeta \\nabla_\\zeta \\cdot a - \\nabla_\\zeta(\\nabla\\log\\mu \\cdot a) - \\nabla_\\zeta(\\nabla\\log\\mu \\cdot r).\n$$\nSubstituting these into \\eqref{another pde for r}, we come to\n$$\ne^{-i \\zeta\\cdot x} L_{\\sigma,\\mu}(e^{i \\zeta\\cdot x}r) = \\nabla_\\zeta \\times \\nabla_\\zeta \\times r + \\nabla_\\zeta \\nabla_\\zeta \\cdot a + \\nabla_\\zeta(\\nabla\\log\\mu \\cdot a) - \\nabla\\log\\sigma\\times \\nabla_\\zeta\\times r  - i\\omega (\\sigma\\mu - \\sigma_0 \\mu_0) r.\n$$\nHence, \\eqref{eqn LE=0 rewritten} implies\n$$\n\\nabla_\\zeta \\times Q + \\nabla_\\zeta \\nabla_\\zeta \\cdot a + \\nabla_\\zeta(\\nabla\\log\\mu \\cdot a) - \\nabla\\log\\sigma\\times Q  - i\\omega (\\sigma\\mu - \\sigma_0 \\mu_0) r = - f.\n$$\nApplying $\\nabla_\\zeta\\times$ to it, we get\n$$\n\\nabla_\\zeta \\times \\nabla_\\zeta \\times Q - \\nabla_\\zeta\\times (\\nabla\\log\\sigma\\times Q) - i\\omega \\nabla(\\sigma \\mu) \\times r - i\\omega (\\sigma \\mu - \\sigma_0 \\mu_0) Q = - \\nabla_\\zeta\\times f\n$$\nas\n$$\n\\nabla_\\zeta\\times (\\nabla_\\zeta \\nabla_\\zeta \\cdot a) = 0\\quad\\text{and}\\quad \\nabla_\\zeta\\times (\\nabla_\\zeta(\\nabla\\log\\mu \\cdot a) )= 0.\n$$\nUsing\nthe fact that $\\nabla_\\zeta \\cdot Q = 0$, we can write\n$$\n\\nabla_\\zeta\\times (\\nabla\\log\\sigma\\times Q) = \\nabla\\nabla(\\log\\sigma) Q - (\\Delta\\log\\sigma)Q - \\nabla\\log\\sigma \\cdot \\nabla_\\zeta Q.\n$$\nThus, we come to\n\\begin{equation}\\label{final eqn for Q}\n-\\Delta_\\zeta Q - \\nabla\\log\\sigma \\cdot \\nabla_\\zeta Q - V_2 Q \\\\\n= i\\omega \\nabla(\\sigma\\mu)\\times r - \\nabla_\\zeta \\times f,\n\\end{equation}\nwhere\n$$\nV_2:= \\nabla\\nabla(\\log\\sigma) + i\\omega(\\sigma\\mu - \\sigma_0 \\mu_0) - \\Delta\\log\\sigma.\n$$\nAccording to the discussion in Section~\\ref{section on L^2 norm of weighted curl of f}, for sufficiently large $\\tau$, there are bounded inverses\n$$\nG_{\\zeta, \\mu}: L^2_{\\delta + 1} \\to L^2_\\delta \\quad\\text{and}\\quad G_{\\zeta, \\sigma}: L^2_{\\delta + 1} \\to L^2_\\delta\n$$\nof $-\\Delta_\\zeta - \\nabla\\log\\mu \\cdot \\nabla_\\zeta$ and $-\\Delta_\\zeta - \\nabla\\log\\sigma \\cdot \\nabla_\\zeta$, respectively, satisfying\n\\begin{equation}\\label{ineq::operator norms of G sigma}\n\\|G_{\\zeta,\\mu}\\|_{L^2_{\\delta + 1} ; L^2_\\delta} \\le \\mathcal O\\Big(\\frac{1}{\\tau}\\Big)\\quad\\text{as}\\quad\\tau \\to \\infty\\end{equation}\nand\n\\begin{equation}\\label{ineq::operator norms of G mu}\n\\|G_{\\zeta,\\sigma}\\|_{L^2_{\\delta + 1} ; L^2_\\delta} \\le \\mathcal O\\Big(\\frac{1}{\\tau}\\Big)\\quad\\text{as}\\quad\\tau \\to \\infty.\\end{equation}\nFurthermore, $G_{\\zeta,\\mu}$ and $G_{\\zeta,\\sigma}$ map the space $L^2_{\\delta + 1}$ into $H^1_\\delta$.\\smallskip\n\nIf $\\tau$ is large enough, we apply $G_{\\zeta, \\sigma}$ to \\eqref{final eqn for Q} and obtain the following identity\n\\begin{equation}\\label{int identity for Q}\n(\\id - G_{\\zeta, \\sigma}V_2) Q = G_{\\zeta, \\sigma}\\big\\{i\\omega\\nabla(\\sigma\\mu)\\times r - \\nabla_\\zeta \\times f \\big\\}.\n\\end{equation}\nSince the support of $V_2$ is compact in $\\R^3$, the operator $\\id - G_{\\zeta, \\sigma}V_2$ is invertible in $L^2_\\delta$ for large enough $\\tau$. Also, if $r\\in L^2_\n\\delta$, then $\\nabla(\\sigma\\mu)\\times r\\in L^2_{\\delta+1}$ by \\eqref{sigma and mu are constants outside of a compact set}. Thus, one can solve the above identity for $Q\\in L^2_\\delta$,\n\\begin{equation}\\label{solution for Q}\nQ = (\\id - G_{\\zeta, \\sigma} V_2)^{-1} G_{\\zeta, \\sigma}\\big\\{i\\omega\\nabla(\\sigma\\mu)\\times r - \\nabla_\\zeta \\times f \\big\\}.\n\\end{equation}\nSubstituting this solution into \\eqref{pre final pde for r}, we obtain\n\\begin{equation}\\label{final integro pde for r}\n-\\Delta_\\zeta r - \\nabla\\log\\mu \\cdot \\nabla_\\zeta r - i\\omega W r - V_1 r = - F\n\\end{equation}\nwhere\n\\begin{align*}\nW &:=  \\nabla\\log(\\sigma\\mu) \\times (\\id - G_{\\zeta, \\sigma} V_1)^{-1}\\circ G_{\\zeta, \\sigma}\\circ \\nabla(\\sigma\\mu)\\times\\ ,\\\\\nF &:= f + \\nabla\\log(\\sigma\\mu) \\times (\\id - G_{\\zeta, \\sigma} V_1)^{-1}\\circ G_{\\zeta, \\sigma}\\circ \\nabla_\\zeta \\times f.\n\\end{align*}\nIt follows from \\eqref{ineq::operator norms of G mu} and \\eqref{sigma and mu are constants outside of a compact set} that\n\\begin{equation}\\label{operator norm of W}\n\\|W\\|_{L^2_{\\delta} ; L^2_{\\delta}} = \\mathcal O\\Big(\\frac{1}{\\tau}\\Big)\\quad\\text{as}\\quad\\tau \\to \\infty.\n\\end{equation}\nand\n$$\n \\|F\\|_{L^2_{\\delta+1}} \\le  \\|f\\|_{L^2_{\\delta+1}} + \\mathcal O\\Big(\\frac{1}{\\tau}\\Big)\\| \\nabla_\\zeta \\times f\\|_{L^2_{\\delta+1}} = o(\\tau)\\quad\\text{as}\\quad\\tau \\to \\infty.\n$$\nApplying $G_{\\zeta, \\mu}$ to \\eqref{final integro pde for r}, we get\n\\begin{equation}\\label{int identity for r}\n(\\id - i\\omega G_{\\zeta, \\mu} W - G_{\\zeta, \\mu} V_1) r = - G_{\\zeta, \\mu} F.\n\\end{equation}\nSince $V_1$ is compactly supported and $W$ satisfies \\eqref{operator norm of W}, the operator $\\id - i\\omega G_{\\zeta, \\mu} W - G_{\\zeta, \\mu} V_1$ is invertible in $L^2_\\delta$ for $\\tau$ sufficiently large. Therefore, one can solve the above identity for $r\\in L^2_\\delta$ by\n$$\nr = - (\\id - i\\omega G_{\\zeta, \\mu} W - G_{\\zeta, \\mu} V_1)^{-1} G_{\\zeta, \\mu}F.\n$$\nFinally, by \\eqref{ineq::operator norms of G mu} and \\eqref{solution for Q}, we can show that\n$$\n\\|r\\|_{L^2_{\\delta}} \\le \\mathcal O\\Big(\\frac{1}{\\tau}\\Big)\\|F\\|_{L^2_{\\delta+1}} = o(1)\\quad\\text{as}\\quad\\tau \\to \\infty\n$$\nand\n$$\n\\|\\nabla_\\zeta\\times r\\|_{L^2_{\\delta}} \\le \\mathcal O\\Big(\\frac{1}{\\tau}\\Big) \\Big\\{\\|r\\|_{L^2_{\\delta}} + \\|\\nabla_\\zeta\\times f\\|_{L^2_{\\delta+1}}\\Big\\} = o(\\tau)\\quad\\text{as}\\quad\\tau \\to \\infty.\n$$\nIt follows from \\eqref{int identity for Q} and \\eqref{int identity for r} that\n\\begin{align*}\nr &= i\\omega G_{\\zeta, \\mu} (Wr) + G_{\\zeta, \\mu} (V_1 r) - G_{\\zeta, \\mu} F,\\\\\nQ &= G_{\\zeta, \\sigma}(V_2 Q) + G_{\\zeta, \\sigma}\\big\\{i\\omega\\nabla(\\sigma\\mu)\\times r - \\nabla_\\zeta \\times f \\big\\}.\n\\end{align*}\nSince $V_1$, $V_2$ and $W$ are compactly supported and $G_{\\zeta,\\mu}$ and $G_{\\zeta,\\sigma}$ map the space $L^2_{\\delta + 1}$ into $H^1_\\delta$, this implies that $r,\\nabla_\\zeta\\times r \\in H^1_\\delta$. Thus, we have constructed the following complex geometric optics solution for \\eqref{eqn3-1}\n$$\nH(x;\\zeta) = e^{i\\zeta\\cdot x}\\big\\{e^{-\\alpha^\\sharp(x;\\tau)\/2}\\rho + s_0 b e^{\\alpha^\\sharp(x;\\tau)\/2} e^{\\Psi^\\sharp(x;\\rho)}\\overline\\rho + r(x;\\zeta)\\big\\}.\n$$\nOur next step is to show that $\\nabla\\cdot(\\mu H)=0$.\nWe observe that \\eqref{eqn3-1} is equivalent to\n$$\n\\nabla\\times(\\sigma^{-1}\\nabla\\times H) - \\sigma^{-1}\\nabla(\\mu^{-1}\\nabla\\cdot(\\mu H)) - i\\omega\\mu H = 0.\n$$\nApplying the divergence to this identity and setting $v = \\mu^{-1}\\nabla\\cdot(\\mu H)$, we get\n$$\n- \\nabla\\cdot(\\sigma^{-1}\\nabla v) - i\\omega\\mu v = 0\n$$\nwhich can be written as\n$$\n- \\Delta \\tilde v + \\tilde q \\tilde v = 0,\\quad \\tilde v := \\sigma^{-1\/2} v,\\quad \\tilde q := \\sigma^{1\/2} \\Delta \\sigma^{-1\/2} - i\\omega \\sigma \\mu.\n$$\nStraightforward calculations give\n$$\n\\tilde v = e^{i\\zeta\\cdot x} u,\\quad u:= \\sigma^{-1\/2}\\mu^{-1}\\nabla_\\zeta\\cdot\\big(\\sigma(a+r)\\big).\n$$\nThen, by \\eqref{eqn:: conj grad and laplacian}, $u$ satisfies\n$$\n-\\Delta_\\zeta u= q u,\\quad q := - \\sigma^{1\/2} \\Delta \\sigma^{-1\/2} + i\\omega ( \\sigma \\mu - \\sigma_0 \\mu_0 ).\n$$\nBy \\cite[Theorem~1.6]{sylvester1987global}, again, there is a bounded inverse, $G_\\zeta : L^2_{\\delta+1} \\to L^2_{\\delta}$ of $-\\Delta_\\zeta$ such that $\\|G_\\zeta\\|_{L^2_{\\delta+1} ; L^2_{\\delta}} \\le \\mathcal O(|\\zeta|^{-1})$ as $|\\zeta|\\to\\infty$. Since $u = G_\\zeta (qu)$ for large enough $|\\zeta|$ and $q$ is compactly supported,\n$$\n\\|u\\|_{L^2_{\\delta}} \\le \\|G_\\zeta(qu)\\|_{L^2_{\\delta}} \\le \\mathcal O\\Big(\\frac{1}{|\\zeta|}\\Big)\\|qu\\|_{L^2_{\\delta+1}} \\le \\mathcal O\\Big(\\frac{1}{|\\zeta|}\\Big)\\|u\\|_{L^2_{\\delta}}\\quad\\text{as}\\quad |\\zeta|\\to\\infty.\n$$\nThis implies that $u=0$ and hence $\\nabla\\cdot(\\mu H)=0$ for sufficiently large $|\\zeta|$.\nThus, restricting $H$ onto $\\Omega$, we get $H\\in H^1(\\Omega;\\C^3)$ solving \\eqref{eqn::curl-curl equation} and satisfying $\\nabla\\times H\\in H^1(\\Omega;\\C^3)$. Clearly, $\\nu\\times H|_{\\p\\Omega}\\in H^{1\/2}(\\p\\Omega;\\C^3)$. Then by \\cite[Corollary~A.20.]{kirsch2016mathematical},\n$$\n\\Div(\\nu\\times H|_{\\p\\Omega}) = - \\nu\\cdot (\\nabla\\times H)|_{\\p\\Omega} \\in H^{1\/2}(\\p\\Omega;\\C^3).\n$$\nTherefore, $\\nu\\times H|_{\\p\\Omega}\\in TH^{1\/2}_{\\Div}(\\p\\Omega)$ and hence $H\\in H^1_{\\Div}(\\Omega)$. In a similar way, also using the fact that $H$ is a solution of \\eqref{eqn::curl-curl equation}, one can easily show that $\\nu\\times (\\nabla\\times H)|_{\\p\\Omega} \\in TH^{1\/2}_{\\Div}(\\p\\Omega)$ and hence $\\nabla \\times H\\in H^1_{\\Div}(\\Omega)$.\nThus, we proved\n\n\\begin{Proposition}\\label{prop CGO}\nLet $\\Omega\\subset \\R^3$ be an open bounded set with $C^{1,1}$ boundary and $\\sigma,\\mu\\in C^2(\\overline\\Omega)$ with $\\sigma \\ge \\sigma_0$, $\\mu \\ge \\mu_0$ for some $\\sigma_0,\\mu_0>0$. Assume that $\\sigma$ and $\\mu$ can be extended positively to $\\R^3$ so that $\\sigma - \\sigma_0, \\mu - \\mu_0 \\in C^2_0(\\R^3)$. Let $\\zeta\\in \\C^3$ be such that $\\zeta\\cdot\\zeta=i\\omega\\sigma_0\\mu_0$, $\\zeta = \\tau \\rho + \\zeta_1$ where $\\tau > 0$ is a large parameter, $\\rho \\in \\C^3$ is independent of $\\tau$ and satisfies $\\Re\\rho \\cdot \\Im\\rho = 0$ and $|\\Re\\rho| = |\\Im\\rho| = 1$, and $\\zeta_1 = \\mathcal O(1)$ as $\\tau \\to \\infty$. Then, for any $s_0 \\in \\R$, there is a solution $H\\in H^1_{\\Div}(\\Omega)$ for \\eqref{eqn::curl-curl equation} of the form\n$$\nH(x;\\zeta) = e^{i\\zeta\\cdot x}\\big\\{e^{-\\alpha^\\sharp(x;\\tau)\/2}\\rho + s_0 b e^{\\alpha^\\sharp(x;\\tau)\/2} e^{\\Psi^\\sharp(x;\\rho)}\\overline\\rho + r(x;\\zeta)\\big\\}.\n$$\nFurthermore, $\\nabla\\times H\\in H^1_{\\Div}(\\Omega)$. The function\n$\\Psi^\\sharp(\\cdot, \\rho; \\tau)\\in C^\\infty(\\R^3)$ satisfies\n$\\|\\Psi^\\sharp\\|_{W^{1,\\infty}(\\R^3)} = \\mathcal O(1)$ as\n$\\tau\\to\\infty$ and converges to $\\Psi(\\cdot, \\rho) := - N_\\rho^{-1}\n\\{\\overline{\\rho} \\cdot \\nabla\\log(\\sigma\\mu)^{1\/2}\\} \\in\nL^\\infty(\\R^3)$ in $L^2_{\\rm loc}(\\R^3)$ as $\\tau\\to\\infty$. The\nfunction $\\alpha^\\sharp(\\,\\cdot\\,; \\tau)\\in C^\\infty(\\R^3)$ satisfies\n$\\| \\alpha^\\sharp \\|_{W^{2,\\infty}(\\R^3)} = \\mathcal O(1)$ as\n$\\tau\\to\\infty$ and $\\| \\alpha^\\sharp - \\log\\sigma\n\\|_{W^{2,\\infty}(\\R^3)} = o(1)$ as $\\tau\\to\\infty$. The correction\nterm, $r$, satisfies $\\|r\\|_{L^2(\\Omega;\\C^3)} = o(1)$ and\n$\\|\\nabla_\\zeta\\times r\\|_{L^2(\\Omega;\\C^3)} = o(\\tau)$ as\n$\\tau\\to\\infty$.\n\\end{Proposition}\n\n\\section{Proof of Theorem~\\ref{main thm}}\\label{section::proof of main thm}\n\nSince we assume that $\\p^\\alpha \\sigma_1|_{\\p\\Omega} = \\p^\\alpha\n\\sigma_2|_{\\p\\Omega}$ and $\\p^\\alpha \\mu_1|_{\\p\\Omega} = \\p^\\alpha\n\\mu_2|_{\\p\\Omega}$ for $|\\alpha| \\le 2$, we can extend $\\sigma_j$ and\n$\\mu_j$, $j=1,2$, to $C^2$ functions defined on $\\R^3$, still denoted\nby $\\sigma_j$ and $\\mu_j$, such that $\\sigma_j \\ge \\sigma_0$, $\\mu_j\n\\ge \\mu_0$ on $\\R^3$, $\\sigma_j - \\sigma_0, \\mu_j - \\mu_0 \\in\nC^2_0(\\R^3)$ and $\\sigma_1=\\sigma_2$ and $\\mu_1=\\mu_2$ on\n$\\R^3\\setminus\\overline\\Omega$. These kind of extensions (of Whitney\ntype) hold for all functions defined on any closed subset of $\\R^3$\nthat can be approximated by certain polynomials. The argument to prove\nthe existence of such polynomials is similar to the one in\n\\cite[Section~2]{caro2013stability} for $C^{1,\\varepsilon}$ functions\non $\\overline\\Omega$. The only difference, here, is that the authors\nof \\cite{caro2013stability} refer to \\cite[Section~2 of\n  Chapter~VI]{stein1970singular}, while we refer to \\cite[Section~4.7\n  of Chapter~VI]{stein1970singular}; see also\n\\cite[Section~3]{caro2014global}.\n\n\\begin{Proposition}\nLet $\\Omega\\subset \\R^3$ be a bounded domain with $C^{1,1}$ boundary and $\\sigma_j,\\mu_j\\in C^2(\\overline\\Omega)$, $j=1,2$, with $\\sigma_j \\ge \\sigma_0$, $\\mu_j \\ge \\mu_0$ for some $\\sigma_0,\\mu_0>0$. Suppose that $Z_{\\sigma_1,\\mu_1}^\\omega = Z_{\\sigma_2,\\mu_2}^\\omega$; then\n\\begin{equation}\\label{main integral identity}\n\\int_\\Omega (\\mu_1 - \\mu_2) H_1\\cdot H_2\\,dx + \\frac{1}{i\\omega} \\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\nabla\\times H_1\\cdot \\nabla\\times H_2\\,dx = 0\n\\end{equation}\nfor all $H_j\\in H_{\\Div}^1(\\Omega)$ with $\\nabla\\times H_j\\in H_{\\Div}^1(\\Omega)$ solving\n$$\n\\nabla\\times(\\sigma_j^{-1}\\nabla\\times H_j) - i\\omega \\mu_j H_j = 0\\quad\\text{in}\\quad\\Omega,\\quad j=1,2.\n$$\n\\end{Proposition}\n\\begin{proof}\nDefine\n$$\nE_j := \\sigma_j^{-1} \\nabla\\times H_j,\\quad j=1,2.\n$$\nThen $E_j\\in H_{\\Div}^1(\\Omega)$ and $\\nabla\\times E_j = i\\omega\\mu_j H_j$. Hence $(H_j, E_j) \\in H^1_{\\Div}(\\Omega) \\times H^1_{\\Div}(\\Omega)$, $j=1,2$, solve\n$$\n\\nabla\\times E_j=i\\omega\\mu_j H_j\\quad\\text{and}\\quad \\nabla\\times H_j=\\sigma_j E_j \\quad\\text{in}\\quad\\Omega,\\quad j=1,2.\n$$\nThen the assumption $Z_{\\sigma_1,\\mu_1}^\\omega = Z_{\\sigma_2,\\mu_2}^\\omega$ implies existence of $(H', E')\\in H^1_{\\Div}(\\Omega) \\times H^1_{\\Div}(\\Omega)$ satisfying\n$$\n\\nabla\\times E'=i\\omega\\mu_2 H'\\quad\\text{and}\\quad \\nabla\\times H'=\\sigma_2 E'\\quad\\text{in}\\quad\\Omega\n$$\nand\n$$\n\\bt(H') = \\bt(H_1)\\quad\\text{and}\\quad \\bt(E') = \\bt(E_1)\\quad\\text{on}\\quad\\p \\Omega.\n$$\nIntegrating by parts,\n\\begin{align*}\n\\int_\\Omega \\nabla\\times(H'-H_1)\\cdot E_2\\,dx - \\int_\\Omega i\\omega\\mu_2 (H'-H_1)\\cdot H_2\\,dx&= \\int_\\Omega \\nabla\\times(H'-H_1)\\cdot E_2\\,dx - \\int_\\Omega (H'-H_1)\\cdot \\nabla\\times E_2\\,dx\\\\\n&= \\int_{\\p\\Omega} \\bt (H'-H_1)\\cdot E_2\\,dS(x) = 0,\n\\end{align*}\nwhere $dS$ is the surface measure on $\\p \\Omega$. Similarly,\n$$\n\\int_\\Omega \\nabla\\times(E'-E_1)\\cdot H_2\\,dx - \\int_\\Omega \\sigma_2 (E'-E_1)\\cdot E_2\\,dx = 0.\n$$\nAdding these two identities, we obtain\n$$\n\\int_\\Omega [\\nabla\\times(H'-H_1) - \\sigma_2 (E'-E_1)]\\cdot E_2\\,dx + \\int_\\Omega [\\nabla\\times(E'-E_1) - i\\omega\\mu_2 (H'-H_1)]\\cdot H_2\\,dx = 0.\n$$\nIt is easy to show that\n$$\n\\nabla\\times(H'-H_1) - \\sigma_2 (E'-E_1) = (\\sigma_2 - \\sigma_1) E_1,\\quad \\nabla\\times(E'-E_1) - i\\omega\\mu_2 (H'-H_1) = i\\omega(\\mu_2 - \\mu_1) H_1.\n$$\nSubstituting these into the latter integral identity, we come to\n$$\n\\int_\\Omega (\\sigma_2 - \\sigma_1) E_1\\cdot E_2\\,dx + \\int_\\Omega i\\omega(\\mu_2 - \\mu_1) H_1\\cdot H_2\\,dx = 0.\n$$\nThis implies \\eqref{main integral identity}.\n\\end{proof}\n\nLet $\\xi,\\rho_1,\\rho_2\\in \\R^3$ be such that $|\\rho_1| = |\\rho_2| = 1$ and $\\rho_1\\cdot\\rho_2 = \\rho_1 \\cdot \\xi = \\rho_2 \\cdot \\xi =0$. Consider\n\\begin{align*}\n\\zeta^1 &= \\phantom{-}\\frac{\\xi}{2} + i \\tau \\rho_2 + \\tau \\sqrt{1 - \\frac{|\\xi|^2}{4\\tau^2} + \\frac{i\\omega\\sigma_0\\mu_0}{\\tau^2}} \\rho_1,\\\\\n\\zeta^2 &= -\\frac{\\xi}{2} + i \\tau \\rho_2 + \\tau \\sqrt{1 - \\frac{|\\xi|^2}{4\\tau^2} + \\frac{i\\omega\\sigma_0\\mu_0}{\\tau^2}} \\rho_1.\n\\end{align*}\nHere, by $\\sqrt{\\, \\cdot \\,}$ we mean its principal branch. Then\n$$\n  \\zeta^j = \\tau \\rho + \\zeta^j_1\\quad\\text{with}\\quad \\zeta^j_1 = \\mathcal O(1)\\quad\\text{as}\\ \\tau \\to \\infty\\quad\\text{and}\\quad\n  \\zeta^1 - \\zeta^2 = \\xi,\\quad \\zeta^j\\cdot\\zeta^j = i\\omega\\sigma_0\\mu_0,\n$$\nwhere $\\rho:=\\rho_1 + i\\rho_2$. By Proposition~\\ref{prop CGO}, there are complex geometric optics solutions $H_1,H_2\\in H^1_{\\Div}(\\Omega)$, with $\\nabla\\times H_1, \\nabla\\times H_2 \\in H^1_{\\Div}(\\Omega)$ satisfying\n$$\n\\nabla\\times(\\sigma_1^{-1}\\nabla\\times H_1) - i\\omega \\mu_1 H_1 = 0\\quad\\text{and}\\quad \\nabla\\times(\\sigma_2^{-1}\\nabla\\times H_2) - i\\omega \\mu_2 H_2 = 0 \\quad\\text{in}\\quad\\Omega,\n$$\nrespectively, which have the following forms\n$$\nH_1(x;\\zeta^1) = e^{i\\zeta^1\\cdot x} \\Big(a_1 \\rho + \\frac{1}{2} b_1 \\overline\\rho + r_1\\Big),\\quad H_2(x;\\zeta^2) = e^{-i\\zeta^2\\cdot x} \\Big(-a_2 \\rho - \\frac{1}{2} b_2 \\overline\\rho + r_2\\Big),\n$$\nwhere\n\\begin{align*}\na_1 &= e^{-\\alpha_1^\\sharp(x;\\tau)\/2},\\quad b_1 = e^{\\alpha_1^\\sharp(x;\\tau)\/2} e^{\\Psi^\\sharp_1(x, \\rho; \\tau)},\\\\\na_2 &= e^{-\\alpha_2^\\sharp(x;\\tau)\/2},\\quad b_2 = e^{\\alpha_2^\\sharp(x;\\tau)\/2} e^{\\Psi^\\sharp_2(x, \\rho; \\tau)}.\n\\end{align*}\nThe functions $\\Psi_1^\\sharp(\\cdot, \\rho; \\tau),\\Psi_2^\\sharp(\\cdot, \\rho; \\tau)\\in C^\\infty(\\R^3)$ satisfy\n\\begin{align}\n\\|\\Psi_1^\\sharp\\|_{W^{1,\\infty}(\\R^3)}&,\\ \\|\\Psi_2^\\sharp\\|_{W^{1,\\infty}(\\R^3)} = \\mathcal O(1) \\quad \\text{as}\\quad\\tau\\to\\infty,\\label{eqn 4.2}\\\\\n\\|\\Psi_1^\\sharp - \\Psi_1\\|_{L^2_{\\rm loc}(\\R^3)}&,\\ \\|\\Psi_2^\\sharp - \\Psi_2\\|_{L^2_{\\rm loc}(\\R^3)} = o(1) \\quad\\text{as}\\quad\\tau\\to\\infty,\\label{eqn 4.3}\n\\end{align}\nwhere\n$$\n\\Psi_j(\\cdot, \\rho) := - N_\\rho^{-1} \\{\\overline{\\rho} \\cdot \\nabla\\log(\\sigma_j\\mu_j)^{1\/2}\\} \\in L^\\infty(\\R^3),\\quad j=1,2.\n$$\nFurthermore, the functions $\\alpha_1^\\sharp(\\,\\cdot\\,;\\tau),\\ \\alpha_2^\\sharp(\\,\\cdot\\,;\\tau) \\in C^\\infty(\\R^3)$ satisfy\n\\begin{align}\n\\| \\alpha_1^\\sharp \\|_{W^{2,\\infty}(\\R^3)}&,\\ \\| \\alpha_2^\\sharp \\|_{W^{2,\\infty}(\\R^3)} = \\mathcal O(1) \\quad\\text{as}\\quad\\tau\\to\\infty,\\label{eqn 4.4}\\\\\n\\| \\alpha_1^\\sharp - \\log\\sigma_1 \\|_{W^{2,\\infty}(\\R^3)}&,\\ \\| \\alpha_2^\\sharp - \\log\\sigma_2 \\|_{W^{2,\\infty}(\\R^3)} = o(1) \\quad\\text{as}\\quad\\tau\\to\\infty.\\label{eqn 4.5}\n\\end{align}\nThe correction terms, $r_1$, $r_2 \\in H_{\\Div}^1(\\Omega)$, satisfy\n\\begin{equation}\\label{eqn 4.6}\n\\|r_j\\|_{L^2(\\Omega)} = o(1)\\quad\\text{and}\\quad\\|\\nabla_{\\zeta^j}\\times r_j\\|_{L^2(\\Omega)} = o(\\tau) \\quad \\text{as}\\quad \\tau\\to\\infty,\\quad j=1,2.\n\\end{equation}\nThen, using that $\\zeta^j = \\tau\\rho + \\zeta^j_1$, we find that\n\\begin{align*}\n\\nabla\\times E_1 &= e^{i\\zeta^1\\cdot x}\\,\\nabla_{\\zeta^1}\\times\\Big(a_1 \\rho + \\frac{1}{2} b_1 \\overline\\rho + r_1\\Big\n= e^{i\\zeta^1\\cdot x} \\Big(\\nabla_{\\zeta^1_1}a_1\\times \\rho+\\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho + b_1 \\tau \\rho_1\\times \\rho_2 + \\nabla_{\\zeta^1}\\times r_1\\Big),\\\\%\\Big\\{e^{-\\beta^\\sharp\/2}\\Big(-\\frac{1}{2}\\nabla\\beta^\\sharp + i\\zeta^1_1\\Big)\\times \\rho + \\frac{1}{2}e^{\\beta^\\sharp\/2}e^{\\Psi^\\sharp_1}\\Big(\\frac{1}{2}\\nabla\\beta^\\sharp + \\nabla\\Psi^\\sharp_1 + i\\zeta^1_1\\Big)\\times \\overline{\\rho}\\\\\n\\nabla\\times E_2 &= e^{-i\\zeta^2\\cdot x} \\,\\nabla_{-\\zeta^2}\\times\\Big(- a_2 \\rho - \\frac{1}{2} b_2 \\overline\\rho + r_2\\Big\n= e^{-i\\zeta^2\\cdot x} \\Big(- \\nabla_{-\\zeta^2_1}a_2\\times \\rho - \\frac{1}{2}\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho + b_2 \\tau \\rho_1\\times \\rho_2 + \\nabla_{-\\zeta^2}\\times r_2\\Big)\n\\end{align*}\nSubstituting $H_1$, $H_2$, $\\nabla\\times H_1$ and $\\nabla\\times H_2$ into \\eqref{main integral identity} and dividing the whole identity by $\\tau^2$, we obtain\n\\begin{equation}\\label{integral identity to find mu}\n\\begin{aligned}\n\\frac{1}{\\tau^2}\\int_\\Omega &(\\mu_1 - \\mu_2) e^{i\\xi\\cdot x}\\Big(a_1\\rho + \\frac{1}{2} b_1 \\overline\\rho + r_1\\Big)\\cdot \\Big(- a_2\\rho - \\frac{1}{2} b_2 \\overline\\rho + r_2\\Big)\\,dx\\\\\n&- \\frac{1}{\\tau^2}\\int_\\Omega \\frac{1}{i\\omega}\\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x}\\Big(\\nabla_{\\zeta^1_1}a_1\\times \\rho+\\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho\\Big)\\cdot \\Big(\\nabla_{-\\zeta^2_1}a_2\\times \\rho+\\frac{1}{2}\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho\\Big)\\,dx\\\\\n&+ \\frac{1}{\\tau}\\int_\\Omega \\frac{1}{i\\omega}\\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x}\\Big(\\nabla_{\\zeta^1_1}a_1\\times \\rho+\\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho\\Big)\\cdot (b_2 \\rho_1\\times\\rho_2)\\,dx\\\\\n&- \\frac{1}{\\tau}\\int_\\Omega \\frac{1}{i\\omega}\\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x} (b_1 \\rho_1\\times \\rho_2) \\cdot \\Big(\\nabla_{-\\zeta^2_1}a_2\\times \\rho+\\frac{1}{2}\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho\\Big)\\,dx\\\\\n&+ \\frac{1}{\\tau^2}\\int_\\Omega \\frac{1}{i\\omega}\\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x}\\Big(\\nabla_{\\zeta^1_1}a_1\\times \\rho+\\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho\\Big)\\cdot (\\nabla_{-\\zeta^2}\\times r_2)\\,dx\\\\\n&- \\frac{1}{\\tau^2}\\int_\\Omega \\frac{1}{i\\omega}\\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x} (\\nabla_{\\zeta^1}\\times r_1) \\cdot \\Big(\\nabla_{-\\zeta^2_1}a_2\\times \\rho+\\frac{1}{2}\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho\\Big)\\,dx\\\\\n&+ \\frac{1}{\\tau}\\int_\\Omega \\frac{1}{i\\omega}\\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x} (b_1 \\rho_1\\times \\rho_2) \\cdot (\\nabla_{-\\zeta^2}\\times r_2)\\,dx\\\\\n&+ \\frac{1}{\\tau}\\int_\\Omega \\frac{1}{i\\omega}\\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x} (\\nabla_{\\zeta^1}\\times r_1) \\cdot (b_2 \\rho_1\\times\\rho_2)\\,dx + \\int_\\Omega \\frac{1}{i\\omega}\\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x} b_1 b_2 \\,dx = 0.\n\\end{aligned}\n\\end{equation}\nFor the last term on the left-hand side, we also used the property $|\\rho_1\\times\\rho_2| = 1$ as $\\rho_1 \\cdot \\rho_2 = 0$ and $|\\rho_1| = |\\rho_2| = 1$. By the Cauchy-Schwartz inequality,\n\\begin{multline*}\n\\Big|\\int_\\Omega (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x}\\Big(a_1\\rho + \\frac{1}{2} b_1 \\overline\\rho + r_1\\Big)\\cdot \\Big(- a_2\\rho - \\frac{1}{2} b_2 \\overline\\rho + r_2\\Big)\\,dx\\Big|\\\\\n\\le \\mathcal O(1) \\Big\\|a_1\\rho + \\frac{1}{2} b_1 \\overline\\rho + r_1\\Big\\|_{L^2(\\Omega)} \\Big\\|- a_2\\rho - \\frac{1}{2} b_2 \\overline\\rho + r_2\\Big\\|_{L^2(\\Omega)}\\quad\\text{as}\\quad \\tau\\to\\infty.\n\\end{multline*}\nThen, by \\eqref{eqn 4.2}, \\eqref{eqn 4.4} and \\eqref{eqn 4.6},\n$$\n\\Big|\\int_\\Omega (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x}\\Big(a_1\\rho + \\frac{1}{2} b_1 \\overline\\rho + r_1\\Big)\\cdot \\Big(- a_2\\rho - \\frac{1}{2} b_2 \\overline\\rho + r_2\\Big)\\,dx\\Big| = \\mathcal O(1)\\quad\\text{as $\\tau\\to\\infty$}.\n$$\nIn a similar way, we obtain\n\\begin{align*}\n\\Big|\\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x}\\Big(\\nabla_{\\zeta^1_1}a_1\\times \\rho+\\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho\\Big)\\cdot \\Big(\\nabla_{-\\zeta^2_1}a_2\\times \\rho+\\frac{1}{2}\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho\\Big)\\,dx\\Big| &= \\mathcal O(1),\\\\\n\\Big|\\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x}\\Big(\\nabla_{\\zeta^1_1}a_1\\times \\rho+\\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho\\Big)\\cdot (b_2 \\rho_1\\times\\rho_2)\\,dx\\Big| &= \\mathcal O(1),\\\\\n\\Big|\\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x} (b_1 \\rho_1\\times \\rho_2) \\cdot \\Big(\\nabla_{-\\zeta^2_1}a_2\\times \\rho+\\frac{1}{2}\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho\\Big)\\,dx\\Big| &= \\mathcal O(1),\\\\\n\\Big|\\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x}\\Big(\\nabla_{\\zeta^1_1}a_1\\times \\rho+\\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho\\Big)\\cdot (\\nabla_{-\\zeta^2}\\times r_2)\\,dx\\Big| &= o(\\tau),\\\\\n\\Big|\\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x} (\\nabla_{\\zeta^1}\\times r_1) \\cdot \\Big(\\nabla_{-\\zeta^2_1}a_2\\times \\rho+\\frac{1}{2}\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho\\Big)\\,dx\\Big| &= o(\\tau),\\\\\n\\Big|\\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x} (b_1 \\rho_1\\times \\rho_2) \\cdot (\\nabla_{-\\zeta^2}\\times r_2)\\,dx\\Big| &= o(\\tau),\\\\\n\\Big|\\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x} (\\nabla_{\\zeta^1}\\times r_1) \\cdot (b_2 \\rho_1\\times\\rho_2)\\,dx\\Big| &= o(\\tau)\n\\end{align*}\nas $\\tau\\to\\infty$. Here, we used again that $\\zeta^j_1 = \\mathcal O(1)$ as $\\tau\\to\\infty$, $j=1,2$. Finally, we use \\eqref{eqn 4.2}-\\eqref{eqn 4.5}, to show that\n\\begin{align*}\n\\Big|\\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} & e^{i\\xi\\cdot x} b_1 b_2 \\, dx - \\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x} \\sigma_1^{1\/2}\\sigma_2^{1\/2} e^{\\Psi_1+\\Psi_2}\\,dx\\Big| \\\\\n&\\le \\mathcal O(1) \\big\\| e^{\\alpha_1^\\sharp\/2 + \\alpha_2^\\sharp\/2} e^{\\Psi^\\sharp_1+\\Psi^\\sharp_2} - \\sigma_1^{1\/2}\\sigma_2^{1\/2} e^{\\Psi_1+\\Psi_2} \\big\\|_{L^2(\\Omega)}\\\\\n&\\le \\mathcal O(1) \\big\\| e^{\\alpha_1^\\sharp\/2 + \\alpha_2^\\sharp\/2} e^{\\Psi^\\sharp_1+\\Psi^\\sharp_2} - \\sigma_1^{1\/2}\\sigma_2^{1\/2} e^{\\Psi^\\sharp_1+\\Psi^\\sharp_2} \\big\\|_{L^2(\\Omega)}\n+ \\mathcal O(1) \\big\\| \\sigma_1^{1\/2}\\sigma_2^{1\/2} e^{\\Psi^\\sharp_1+\\Psi^\\sharp_2} - \\sigma_1^{1\/2}\\sigma_2^{1\/2} e^{\\Psi_1+\\Psi_2} \\big\\|_{L^2(\\Omega)}\\\\\n&\\le \\mathcal O(1) \\big\\| e^{\\alpha_1^\\sharp\/2 + \\alpha_2^\\sharp\/2} - e^{(\\log \\sigma_1)\/2 +(\\log\\sigma_2)\/2} \\big\\|_{L^2(\\Omega)}\n+ \\mathcal O(1) \\big\\| e^{\\Psi^\\sharp_1+\\Psi^\\sharp_2} - e^{\\Psi_1+\\Psi_2} \\big\\|_{L^2(\\Omega)}\\\\\n&\\le \\mathcal O(1) \\big\\| \\alpha_1^\\sharp- \\log \\sigma_1\\|_{L^2(\\Omega)} + \\mathcal O(1) \\|\\alpha_2^\\sharp  - \\log\\sigma_2 \\big\\|_{L^2(\\Omega)}\n + \\mathcal O(1) \\big\\| \\Psi^\\sharp_1 - \\Psi_1 \\|_{L^2(\\Omega)} + \\mathcal O(1) \\big\\| \\Psi^\\sharp_2 - \\Psi_2 \\big\\|_{L^2(\\Omega)}\\\\[0.15cm]\n&= o(1)\\quad\\text{as}\\quad \\tau\\to\\infty,\n\\end{align*}\nemploying the basic inequality, \n\\begin{equation}\\label{technical ineq from complex analysis}\n|e^z - e^w| \\le |z - w| e^{\\max(\\Re z,\\Re w)},\\quad z,w\\in \\C\n\\end{equation}\nfrom \\cite{krupchyk2014uniqueness}. According to these estimates, taking the limit as $\\tau \\to \\infty$ in \\eqref{integral identity to find mu}, we come to\n$$\n\\int_{\\R^3} e^{i\\xi\\cdot x} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1^{1\/2} \\sigma_2^{1\/2}} e^{\\Psi_1 + \\Psi_2}\\,dx = 0.\n$$\nNote that the integration is extended to all of $\\R^3$ since $\\sigma_1 - \\sigma_2 = 0$ on $\\R^3\\setminus\\overline\\Omega$. This implies that $\\sigma_1=\\sigma_2$.\\smallskip\n\nNext, we set $\\sigma = \\sigma_1=\\sigma_2$. By Proposition~\\ref{prop CGO}, there are complex geometric optics solutions $H_1,H_2\\in H^1_{\\Div}(\\Omega)$, with $\\nabla\\times H_1, \\nabla\\times H_2 \\in H^1_{\\Div}(\\Omega)$ satisfying\n$$\n\\nabla\\times(\\sigma^{-1}\\nabla\\times H_1) - i\\omega \\mu_1 H_1 = 0\\quad\\text{and}\\quad \\nabla\\times(\\sigma^{-1}\\nabla\\times H_2) - i\\omega \\mu_2 H_2 = 0 \\quad\\text{in}\\quad\\Omega,\n$$\nrespectively, which have the following forms\n$$\nH_1(x;\\zeta^1) = e^{i\\zeta^1\\cdot x} \\Big(a_1 \\rho + r_1\\Big),\\quad H_2(x;\\zeta^2) = e^{-i\\zeta^2\\cdot x} \\Big(-a_2 \\rho - \\frac{1}{2} b_2 \\overline\\rho + r_2\\Big),\n$$\nwhere\n$$\na_1 = e^{-\\alpha^\\sharp(x;\\tau)\/2},\\quad a_2 = e^{-\\alpha^\\sharp(x;\\tau)\/2},\\quad b_2 = e^{\\alpha^\\sharp(x;\\tau)\/2} e^{\\Psi^\\sharp(x, \\rho; \\tau)}.\n$$\nThe function $\\Psi^\\sharp(\\cdot, \\rho; \\tau)\\in C^\\infty(\\R^3)$ satisfies\n\\begin{equation}\\label{eqn 4.7}\n\\|\\Psi^\\sharp\\|_{W^{1,\\infty}(\\R^3)} = \\mathcal O(1) \\quad \\text{and}\\quad  \\|\\Psi^\\sharp - \\Psi\\|_{L^2_{\\rm loc}(\\R^3)} = o(1) \\quad\\text{as}\\quad\\tau\\to\\infty,\n\\end{equation}\nwhere\n$$\n\\Psi(\\cdot, \\rho) := - N_\\rho^{-1} \\{\\overline{\\rho} \\cdot \\nabla\\log(\\sigma\\mu_2)^{1\/2}\\} \\in L^\\infty(\\R^3),\\quad j=1,2.\n$$\nFurthermore, the function $\\alpha^\\sharp(\\,\\cdot\\,;\\tau) \\in C^\\infty(\\R^3)$ satisfies\n\\begin{equation}\\label{eqn 4.8}\n\\| \\alpha^\\sharp \\|_{W^{2,\\infty}(\\R^3)} = \\mathcal O(1) \\quad\\text{and}\\quad  \\| \\alpha^\\sharp - \\log\\sigma \\|_{W^{2,\\infty}(\\R^3)} = o(1) \\quad\\text{as}\\quad\\tau\\to\\infty.\n\\end{equation}\nThe correction terms $r_1$, $r_2 \\in H_{\\Div}^1(\\Omega)$ satisfy\n\\begin{equation}\\label{eqn 4.9}\n\\|r_j\\|_{L^2(\\Omega)} = o(1)\\quad\\text{and}\\quad\\|\\nabla_{\\zeta^j}\\times r_j\\|_{L^2(\\Omega)} = o(\\tau) \\quad \\text{as}\\quad \\tau\\to\\infty,\\quad j=1,2.\n\\end{equation}\nSubstituting $H_1$, $H_2$ and $\\sigma = \\sigma_1=\\sigma_2$ into \\eqref{main integral identity}, we come to\n\\begin{align*}\n-\\int_\\Omega (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x} a_1 b_2\\,dx &+ \\int_\\Omega (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x} a_1 \\rho\\cdot r_2\\,dx\\\\\n-&\\int_\\Omega (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x} r_1\\cdot \\Big(a_2\\rho+\\frac{1}{2}b_2\\overline\\rho\\Big)\\,dx + \\int_\\Omega (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x} r_1\\cdot r_2\\,dx= 0.\n\\end{align*}\nBy the Cauchy-Schwartz inequality together with \\eqref{eqn 4.8} and \\eqref{eqn 4.9},\n$$\n\\Big|\\int_\\Omega (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x} a_1 \\rho\\cdot r_2\\,dx\\Big| \\le \\mathcal O(1) \\int_\\Omega | a_1 \\rho\\cdot r_2 |\\,dx \\le \\mathcal O(1) \\|a_1\\|_{L^2(\\Omega)} \\|r_2\\|_{L^2(\\Omega)} = o(1)\n$$\nas $\\tau\\to\\infty$. In a similar way, and also using \\eqref{eqn 4.7}, one can show that\n$$\n\\Big|\\int_\\Omega (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x} r_1\\cdot \\Big(a_2\\rho+\\frac{1}{2}b_2\\overline\\rho\\Big)\\,dx\\Big| = o(1)\\quad\\text{and}\\quad \\Big| \\int_\\Omega (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x} r_1\\cdot r_2\\,dx \\Big| = o(1)\\quad\\text{as}\\quad \\tau \\to\\infty.\n$$\nFinally, using \\eqref{eqn 4.7} and \\eqref{eqn 4.8},\n$$\n\\Big| \\int_\\Omega (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x} a_1 b_2\\,dx - \\int_{\\R^3} (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x} e^{\\Psi}\\,dx \\Big| \\le \\mathcal O(1) \\| e^{\\Psi^\\sharp} - e^{\\Psi}\\|_{L^2(\\Omega)} \\le \\mathcal O(1) \\|\\Psi^\\sharp - \\Psi\\|_{L^2(\\Omega)} = o(1)\\quad\\text{as}\\quad \\tau \\to\\infty.\n$$\nHere, we have again employed inequality \\eqref{technical ineq from complex analysis}. Thus, letting $\\tau\\to\\infty$, we obtain\n$$\n\\int_{\\R^3} e^{i\\xi\\cdot x} (\\mu_1 - \\mu_2) e^{\\Psi}\\,dx = 0.\n$$\nThe integration is extended to all of $\\R^3$ since $\\mu_1 - \\mu_2 = 0$ on $\\R^3\\setminus\\overline\\Omega$. This implies that $\\mu_1 = \\mu_2$ completing the proof of Theorem~\\ref{main thm}.\n\n\\section{Reflection approach} \\label{section::reflection approach}\n\nIn this section, we use Isakov's reflection approach \\cite{isakov2007uniqueness} to prove the following local uniqueness result where the region of the boundary that is inaccessible for measurements is a part of a plane. For a closed $\\Gamma\\subset\\p\\Omega$, define\n$$\nC_{\\Gamma}(\\sigma, \\mu; \\omega) := \\{(\\bt(H) |_{\\Gamma}, \\bt(E)|_{\\Gamma}): (H, E)\\in H^1_{\\Div}(\\Omega)\\times H^1_{\\Div}(\\Omega)\\text{ is a solution to \\eqref{eqn::Maxwell} with }\\supp(\\bt(H))\\subseteq\\Gamma\\}.\n$$\n\n\\begin{Theorem}\\label{main thm flat}\nLet $\\Omega\\subset \\{x\\in \\R^3: x_3 < 0 \\}$ be a bounded domain with $C^{1,1}$ boundary and let $\\Gamma_0 = \\p\\Omega\\cap \\{x\\in \\R^3: x_3 = 0\\}$ and $\\Gamma = \\overline{\\p\\Omega\\setminus \\Gamma_0}$. Suppose that $\\sigma_j,\\mu_j\\in C^2(\\overline\\Omega)$, $j=1,2$, satisfy $\\sigma_j \\ge \\sigma_0$ and $\\mu_j \\ge \\mu_0$, for some constants $\\sigma_0, \\mu_0 > 0$, and\n\\begin{equation}\\label{boundary assumption on Gamma flat}\n\\p^\\alpha \\sigma_1|_{\\Gamma} = \\p^\\alpha \\sigma_2|_{\\Gamma}\\quad\\text{and}\\quad \\p^\\alpha \\mu_1|_{\\Gamma} = \\p^\\alpha \\mu_2|_{\\Gamma}\\quad\\text{for}\\quad|\\alpha| \\le 2.\n\\end{equation}\nIn addition, assume that $\\sigma_j$ and $\\mu_j$, $j=1,2$, can be extended into $\\R^3$ as $C^2$ functions which are invariant under reflection across the plane $\\{x\\in \\R^3: x_3 = 0\\}$. Then $C_\\Gamma(\\sigma_1, \\mu_1; \\omega) = C_\\Gamma(\\sigma_2, \\mu_2; \\omega)$ implies $\\sigma_1=\\sigma_2$ and $\\mu_1=\\mu_2$.\n\\end{Theorem}\n\n\\noindent\nSimilar results were obtained for the inverse conductivity problem in\n\\cite{isakov2007uniqueness} and for the IEMP in\n\\cite{caro2009inverse}. Consider the reflected domain\n$$\n\\Omega^* := \\{(x_1, x_2, -x_3)\\in\\R^3 : (x_1, x_2, x_3)\\in\\Omega\\}\n$$\nand define\n$$\n\\mathcal U := \\Omega \\cap \\Gamma_0^{\\rm int} \\cap \\Omega^*.\n$$\nBy the assumptions in Theorem~\\ref{main thm flat}, we can extend the coefficients $\\sigma_j$ and $\\mu_j$ into $\\mathcal U$ as $C^2$ functions which are even with respect to $x_3$ for $j = 1,2$.  Next, by the assumption \\eqref{boundary assumption on Gamma flat}, we can extend $\\sigma_j$ and $\\mu_j$, $j=1,2$, to $C^2$ functions defined on $\\R^3$, still denoted by $\\sigma_j$ and $\\mu_j$, such that $\\sigma_j \\ge \\sigma_0$, $\\mu_j \\ge \\mu_0$ on $\\R^3$, $\\sigma_j - \\sigma_0, \\mu_j - \\mu_0 \\in C^2_0(\\R^3)$ and $\\sigma_1=\\sigma_2$ and $\\mu_1=\\mu_2$ on $\\R^3\\setminus\\overline{\\mathcal U}$.\n\n\\begin{Proposition}\\label{prop main integral identity local}\nLet $\\Omega\\subset \\{x\\in \\R^3: x_3 < 0 \\}$ be a bounded domain with $C^{1,1}$ boundary . Let $\\Gamma_0 := \\p\\Omega\\cap \\{x\\in\\R^3: x_3=0\\}$ and $\\Gamma:=\\overline{\\p\\Omega\\setminus \\Gamma_0}$. Suppose that\n\\begin{equation}\\label{local data assumption}\nZ_{\\sigma_1,\\mu_1}^\\omega(f)|_{\\Gamma} = Z_{\\sigma_2,\\mu_2}^\\omega(f)|_{\\Gamma}\\quad\\text{for all}\\quad f\\in TH^{1\/2}_{\\Div}(\\p\\Omega)\\quad\\text{with}\\quad \\supp(f)\\subset\\Gamma;\n\\end{equation}\nthen\n\\begin{equation}\\label{main integral identity local}\n\\int_\\Omega (\\mu_1 - \\mu_2) H_1\\cdot H_2\\,dx + \\frac{1}{i\\omega}\\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} (\\nabla\\times H_1)\\cdot (\\nabla\\times H_2)\\,dx = 0\n\\end{equation}\nfor all $H_j\\in H_{\\Div}^1(\\Omega)$ with $\\nabla\\times H_j\\in H_{\\Div}^1(\\Omega)$ solving\n$$\n\\nabla\\times(\\sigma_j^{-1}\\nabla\\times H_j) - i\\omega \\mu_j H_j = 0\\quad\\text{in}\\quad\\Omega,\\quad j=1,2.\n$$\nand satisfying $\\supp(\\bt(H_j))\\subseteq \\Gamma$.\n\\end{Proposition}\n\\begin{proof}\nSimilarly as in the proof of  \\eqref{main integral identity}, define\n$$\nE_j := \\sigma_j^{-1} \\nabla\\times H_j,\\quad j=1,2.\n$$\nThen $E_j\\in H_{\\Div}^1(\\Omega)$ and $\\nabla\\times E_j = i\\omega\\mu_j H_j$. Hence $(H_j, E_j) \\in H^1_{\\Div}(\\Omega) \\times H^1_{\\Div}(\\Omega)$, $j=1,2$, solve\n$$\n\\nabla\\times E_j=i\\omega\\mu_j H_j\\quad\\text{and}\\quad \\nabla\\times H_j=\\sigma_j E_j \\quad\\text{in}\\quad\\Omega,\\quad j=1,2.\n$$\nThen by the assumption \\eqref{local data assumption}, there is $(H', E')\\in H^1_{\\Div}(\\Omega) \\times H^1_{\\Div}(\\Omega)$ with $\\supp(\\bt(H'))\\subseteq\\Gamma$ satisfying\n$$\n\\nabla\\times E'=i\\omega\\mu_2 H'\\quad\\text{and}\\quad \\nabla\\times H'=\\sigma_2 E'\\quad\\text{in}\\quad\\Omega\n$$\nand\n$$\n\\bt(H')|_{\\Gamma} = \\bt(H_1)|_{\\Gamma}\\quad\\text{and}\\quad\\bt(E')|_{\\Gamma} = \\bt(E_1)|_{\\Gamma}.\n$$\nIntegrating by parts, leads to\n\\begin{align*}\n\\int_\\Omega \\nabla\\times(H'-H_1)\\cdot E_2\\,dx - \\int_\\Omega i\\omega\\mu_2 (H'-H_1)\\cdot H_2\\,dx&= \\int_\\Omega \\nabla\\times(H'-H_1)\\cdot E_2\\,dx - \\int_\\Omega (H'-H_1)\\cdot \\nabla\\times E_2\\,dx\\\\\n& = \\int_{\\p\\Omega} \\bt (H'-H_1)\\cdot E_2\\,dS(x) = \\int_{\\Gamma_0^{\\rm int}} \\bt (H'-H_1)\\cdot \\bt(E_2)\\,dS(x) = 0,\n\\end{align*}\nsince both $\\bt(H')$ and $\\bt(H_1)$ are supported on $\\Gamma$. Similarly,\n$$\n\\int_\\Omega \\nabla\\times(E'-E_1)\\cdot H_2\\,dx - \\int_\\Omega \\sigma_2 (E'-E_1)\\cdot E_2\\,dx = \\int_{\\Gamma_0^{\\rm int}} \\bt (E'-E_1)\\cdot \\bt(H_2)\\,dS(x) = 0,\n$$\nsince $\\bt(H_2)$ is supported on $\\Gamma$. The remainder of the proof of \\eqref{main integral identity local} is similar to that of \\eqref{main integral identity}.\n\\end{proof}\n\nFor $\\beta:\\R^3\\to \\C$ and $X:\\R^3\\to \\C^3$, we define the reflections as\n$$\n\\beta^*(x) := \\beta(x_1, x_2, - x_3),\\quad X^*(x) := \\big(X_1(x_1, x_2, - x_3), X_2(x_1, x_2, - x_3), - X_3(x_1, x_2, - x_3)\\big)\n$$\nwith the properties,\n\\begin{equation}\\label{eqn::reflection identities}\n\\nabla \\beta^* = (\\nabla \\beta)^*,\\quad (\\beta X)^* = \\beta^* X^*,\\quad \\nabla\\times X^* = - (\\nabla\\times X)^*\n\\end{equation}\n\n\\subsection* {Proof of Uniqueness}\n\nConsider $\\zeta^1$ and $\\zeta^2$ defined as in the proof of Theorem~\\ref{main thm}.\nThen, by Proposition~\\ref{prop CGO}, there are complex geometric optics solutions $\\tilde H_1, \\tilde H_2\\in H^1_{\\Div}(\\mathcal U)$, with $\\nabla\\times \\tilde H_1, \\nabla\\times \\tilde H_2 \\in H^1_{\\Div}(\\mathcal U)$, for\n$$\n\\nabla\\times(\\sigma_1^{-1}\\nabla\\times\\tilde H_1) - i\\omega \\mu_1\\tilde H_1 = 0\\quad\\text{and}\\quad \\nabla\\times(\\sigma_2^{-1}\\nabla\\times\\tilde H_2) - i\\omega \\mu_2\\tilde H_2 = 0 \\quad\\text{in}\\quad\\mathcal U,\n$$\nrespectively, which have the following forms\n$$\n\\tilde H_1(x;\\zeta^1) = e^{i\\zeta^1\\cdot x} \\Big(a_1 \\rho + \\frac{1}{2} b_1 \\overline\\rho + r_1\\Big),\\quad \\tilde H_2(x;\\zeta^2) = e^{-i\\zeta^2\\cdot x} \\Big(-a_2 \\rho - \\frac{1}{2} b_2 \\overline\\rho + r_2\\Big),\n$$\nwhere\n\\begin{align*}\na_1 &= e^{-\\alpha_1^\\sharp(x;\\tau)\/2},\\quad b_1 = e^{\\alpha_1^\\sharp(x;\\tau)\/2} e^{\\Psi^\\sharp_1(x, \\rho; \\tau)},\\\\\na_2 &= e^{-\\alpha_2^\\sharp(x;\\tau)\/2},\\quad b_2 = e^{\\alpha_2^\\sharp(x;\\tau)\/2} e^{\\Psi^\\sharp_2(x, \\rho; \\tau)}.\n\\end{align*}\nThe functions $\\Psi_1^\\sharp(\\cdot, \\rho; \\tau),\\Psi_2^\\sharp(\\cdot, \\rho; \\tau)\\in C^\\infty(\\R^3)$, $\\alpha_1^\\sharp(\\,\\cdot\\,;\\tau), \\alpha_2^\\sharp(\\,\\cdot\\,;\\tau) \\in C^\\infty(\\R^3)$ and the correction terms $r_1$, $r_2 \\in H^1_{\\Div}(\\mathcal U)$ satisfy \\eqref{eqn 4.3}-\\eqref{eqn 4.6}. Using \\eqref{eqn::reflection identities}, it follows that\n$$\n\\nabla\\times (\\sigma_j^{-1}\\nabla\\times \\tilde H_j^*) - i\\omega \\mu_j \\tilde H_j^* = - \\nabla\\times (\\sigma_j^{-1}\\nabla\\times\\tilde H_j)^* - (i\\omega \\mu_j \\tilde H_j)^* = \\big(\\nabla\\times (\\sigma_j^{-1}\\nabla\\times\\tilde H_j) - i\\omega \\mu_j\\tilde H_j\\big)^* = 0\\quad\\text{in}\\quad \\mathcal U.\n$$\nTherefore,\n$$\nH_1(x;\\rho) := \\tilde H_1(x;\\rho) - \\tilde H_1^*(x;\\rho),\\quad H_2(x;\\rho) := \\tilde H_2(x;\\rho) - \\tilde H_2^*(x;\\rho).\n$$\nalso satisfy\n$$\n\\nabla\\times(\\sigma_1^{-1}\\nabla\\times\\tilde H_1) - i\\omega \\mu_1\\tilde H_1 = 0\\quad\\text{and}\\quad \\nabla\\times(\\sigma_2^{-1}\\nabla\\times\\tilde H_2) - i\\omega \\mu_2\\tilde H_2 = 0 \\quad\\text{in}\\quad\\mathcal U.\n$$\nAs in the proof of Proposition~\\ref{prop CGO}, it is not difficult to see that $H_1|_\\Omega$ and $H_2|_\\Omega$, still denoted by $H_1$ and $H_2$, respectively, belong to $H^1_{\\Div}(\\Omega)$. Moreover, $\\nabla\\times H_1, \\nabla\\times H_2 \\in H^1_{\\Div}(\\Omega)$. Using the fact that $\\nu = (0, 0, 1)$ on $\\Gamma$, we have  $\\nu\\times H_1|_{\\Gamma}=\\nu\\times H_2|_{\\Gamma}=0$. Thus, $H_1$ and $H_2$ satisfy the hypotheses of Proposition~\\ref{prop main integral identity local}, and hence we can substitute $H_1$ and $H_2$ into \\eqref{main integral identity local}. To that end, we first compute $\\nabla\\times H_1$ and $\\nabla\\times H_2$. Using \\eqref{eqn::reflection identities},\n\\begin{align*}\n\\nabla\\times H_1 &= \\nabla\\times \\tilde H_1 + (\\nabla\\times \\tilde H_1)^* \\\\\n&= e^{i\\zeta^1\\cdot x} \\Big(\\nabla_{\\zeta^1_1}a_1\\times \\rho+\\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho + b_1 \\tau \\rho_1\\times \\rho_2 + \\nabla_{\\zeta^1}\\times r_1\\Big)\\\\\n&\\quad + e^{i{\\zeta^1}^*\\cdot x} \\Big((\\nabla_{\\zeta^1_1}a_1\\times \\rho)^*+\\frac{1}{2}(\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho)^* + b_1^* \\tau (\\rho_1\\times \\rho_2)^* + (\\nabla_{\\zeta^1}\\times r_1)^*\\Big).\n\\end{align*}\nSimilarly,\n\\begin{align*}\n\\nabla\\times H_2 &= e^{-i\\zeta^2\\cdot x} \\Big(- \\nabla_{-\\zeta^2_1}a_2\\times \\rho - \\frac{1}{2}\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho + b_2 \\tau \\rho_1\\times \\rho_2 + \\nabla_{-\\zeta^2}\\times r_2\\Big)\\\\\n&\\quad + e^{-i{\\zeta^2}^*\\cdot x} \\Big(- (\\nabla_{-\\zeta^2_1}a_2\\times \\rho)^* - \\frac{1}{2}(\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho)^* + b_2^* \\tau (\\rho_1\\times \\rho_2)^* + (\\nabla_{-\\zeta^2}\\times r_2)^*\\Big).\n\\end{align*}\nWe take a closer look at the phases of products of these vector fields. We have\n$$\ni(\\zeta^1 - \\zeta^2)\\cdot x = i\\xi\\cdot x,\\quad i({\\zeta^1}^* - {\\zeta^2}^*)\\cdot x = i\\xi^*\\cdot x,\n$$\n$$\ni(\\zeta^1 - {\\zeta^2}^*)\\cdot x = i\\tilde\\xi_{+}\\cdot x - 2\\tau\\rho_{2,3} x_3 - \\eta_{+},\\quad i({\\zeta^1}^* - \\zeta^2)\\cdot x = i\\tilde\\xi_{-}\\cdot x + 2\\tau\\rho_{2,3} x_3 - \\eta_{-},\n$$\nwhere\n$$\n\\tilde\\xi_{\\pm} = \\Bigg(\\xi', \\pm 2\\tau\\sqrt{1 - \\frac{|\\xi|^2}{4\\tau^2}}\\rho_{1,3}\\Bigg),\\quad |\\tilde\\xi_{\\pm}|\\to\\infty\\quad\\text{as}\\quad \\tau\\to \\infty\n$$\nand\n$$\n\\eta_{\\pm}:= \\pm\\frac{2\\omega \\sigma_{0} \\mu_{0} \\rho_{1,3} x_3}{\\tau\\Big(\\sqrt{1 - \\frac{|\\xi|^2}{4\\tau^2} + \\frac{i\\omega\\sigma_0\\mu_0}{\\tau^2}} + \\sqrt{1 - \\frac{|\\xi|^2}{4\\tau^2}}\\Big)} ,\\quad |\\eta_{\\pm}| = \\mathcal O(\\tau^{-1})\\quad\\text{as}\\quad \\tau\\to \\infty.\n$$\nFurthermore, we assume that $\\rho_{1,3} \\neq 0$ and $\\rho_{2,3} = 0$. Then\n\\begin{align*}\n\\int_\\Omega & e^{i\\tilde\\xi_{+}\\cdot x - \\eta_{+}} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big(\\nabla_{\\zeta^1_1} a_1\\times \\rho + \\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho\\Big) \\cdot \\Big((\\nabla_{-\\zeta^2_1}a_2\\times \\rho)^* + \\frac{1}{2}(\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho)^*\\Big)\\,dx\\\\\n& = \\int_\\Omega e^{i\\tilde\\xi_{+}\\cdot x} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big\\{e^{- \\eta_{+}}\\Big(\\nabla_{\\zeta^1_1} a_1\\times \\rho + \\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho\\Big) \\cdot \\Big((\\nabla_{-\\zeta^2_1}a_2\\times \\rho)^* + \\frac{1}{2}(\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho)^*\\Big)\\\\\n&\\qquad\\qquad\\qquad\\qquad - \\Big(\\nabla_{\\zeta^1_1} \\mu_1^{1\/2}\\times \\rho + \\frac{1}{2}\\nabla_{\\zeta^1_1}(\\mu^{-1\/2}e^{\\Psi_1})\\times \\overline\\rho\\Big) \\cdot \\Big((\\nabla_{-\\zeta^2_1}\\mu_2^{1\/2}\\times \\rho)^* + \\frac{1}{2}(\\nabla_{-\\zeta^2_1}(\\mu_2^{-1\/2}e^{\\Psi_2})\\times \\overline\\rho)^*\\Big)\\Big\\}\\,dx\\\\\n&\\quad + \\int_\\Omega e^{i\\tilde\\xi_{+}\\cdot x} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big(\\nabla_{\\zeta^1_1} \\mu_1^{1\/2}\\times \\rho + \\frac{1}{2}\\nabla_{\\zeta^1_1}(\\mu^{-1\/2}e^{\\Psi_1})\\times \\overline\\rho\\Big) \\cdot \\Big((\\nabla_{-\\zeta^2_1}\\mu_2^{1\/2}\\times \\rho)^* + \\frac{1}{2}(\\nabla_{-\\zeta^2_1}(\\mu_2^{-1\/2}e^{\\Psi_2})\\times \\overline\\rho)^*\\Big)\\,dx\n\\end{align*}\nThe first integral on the right-side goes to zero as $\\tau\\to\\infty$ according to estimates \\eqref{eqn 4.2}-\\eqref{eqn 4.5} and the fact that $|\\eta_{\\pm}| = \\mathcal O(\\tau^{-1})$ as $\\tau\\to \\infty$. The second integral goes to zero as $\\tau\\to\\infty$ by the Riemann-Lebesgue lemma. Therefore,\n$$\n\\Big|\\int_\\Omega  e^{i\\tilde\\xi_{+}\\cdot x - \\eta_{+}} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big(\\nabla_{\\zeta^1_1} a_1\\times \\rho + \\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho\\Big) \\cdot \\Big((\\nabla_{-\\zeta^2_1}a_2\\times \\rho)^* + \\frac{1}{2}(\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho)^*\\Big)\\,dx\\Big| = o(1)\\quad\\text{as}\\quad\\tau\\to\\infty.\n$$\nIn a similar way, one can show that\n\\begin{align*}\n\\Big|\\int_\\Omega e^{i\\tilde\\xi_{-}\\cdot x - \\eta_{-}} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big((\\nabla_{\\zeta^1_1}a_1\\times \\rho)^*+\\frac{1}{2}(\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho)^*\\Big) \\cdot \\Big(\\nabla_{-\\zeta^2_1}a_2\\times \\rho + \\frac{1}{2}\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho\\Big)\\,dx\\Big| &= o(1)\\\\\n\\Big|\\int_\\Omega  e^{i\\tilde\\xi_{+}\\cdot x - \\eta_{+}} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big(\\nabla_{\\zeta^1_1} a_1\\times \\rho + \\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho\\Big) \\cdot \\Big(b_2^* \\tau (\\rho_1\\times \\rho_2)^*\\Big)\\,dx\\Big| &= o(\\tau)\\\\\n\\Big|\\int_\\Omega e^{i\\tilde\\xi_{-}\\cdot x - \\eta_{-}} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big(b_1^* \\tau (\\rho_1\\times \\rho_2)^*\\Big) \\cdot \\Big(\\nabla_{-\\zeta^2_1}a_2\\times \\rho + \\frac{1}{2}\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho\\Big)\\,dx\\Big| &= o(\\tau)\\\\\n\\Big|\\int_\\Omega  e^{i\\tilde\\xi_{+}\\cdot x - \\eta_{+}} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big(b_1 \\tau \\rho_1\\times \\rho_2\\Big) \\cdot \\Big((\\nabla_{-\\zeta^2_1}a_2\\times \\rho)^* + \\frac{1}{2}(\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho)^*\\Big)\\,dx\\Big| &= o(\\tau)\\\\\n\\Big|\\int_\\Omega e^{i\\tilde\\xi_{-}\\cdot x - \\eta_{-}} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big((\\nabla_{\\zeta^1_1}a_1\\times \\rho)^*+\\frac{1}{2}(\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho)^*\\Big) \\cdot \\Big(b_2\\tau \\rho_1\\times\\rho_2\\Big)\\,dx\\Big| &= o(\\tau)\\\\\n\\Big|\\int_\\Omega  e^{i\\tilde\\xi_{+}\\cdot x - \\eta_{+}} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big(b_1 \\tau \\rho_1\\times \\rho_2\\Big) \\cdot \\Big(b_2^*\\tau (\\rho_1\\times\\rho_2)^*\\Big)\\,dx\\Big| &= o(\\tau^2)\\\\\n\\Big|\\int_\\Omega  e^{i\\tilde\\xi_{-}\\cdot x - \\eta_{-}} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big(b_1^* \\tau (\\rho_1\\times \\rho_2)^*\\Big) \\cdot \\Big(b_2\\tau \\rho_1\\times\\rho_2\\Big)\\,dx\\Big| &= o(\\tau^2)\\\\\n\\Big|\\int_\\Omega(\\mu_1 - \\mu_2) e^{i\\tilde\\xi_{+}\\cdot x - \\eta_{+}} \\Big(a_1 \\rho + \\frac{1}{2} b_1 \\overline\\rho\\Big)\\cdot \\Big(a_2^* \\rho^* + \\frac{1}{2} b_2^* \\overline\\rho^*\\Big)\\,dx\\Big| &= o(1)\\\\\n\\Big|\\int_\\Omega(\\mu_1 - \\mu_2) e^{i\\tilde\\xi_{-}\\cdot x - \\eta_{-}} \\Big(a_1^* \\rho^* + \\frac{1}{2} b_1^* \\overline\\rho^*\\Big)\\cdot \\Big(a_2 \\rho + \\frac{1}{2} b_2 \\overline\\rho\\Big)\\,dx\\Big| &= o(1)\n\\end{align*}\nas $\\tau\\to\\infty$. Now we substitute $H_1$, $H_2$, $\\nabla\\times H_1$ and $\\nabla\\times H_2$ into \\eqref{main integral identity local} and divide the whole identity by $\\tau^2$. According to the estimates obtained above, the terms with phases $i\\tilde\\xi_{\\pm}\\cdot x - \\eta_{\\pm}$ that do not involve the correction terms $r_1$ and $r_2$ go to zero as $\\tau\\to\\infty$. The terms with phases $i\\tilde\\xi_{\\pm}\\cdot x - \\eta_{\\pm}$ that involve the correction terms $r_1$ and $r_2$ go to zero as $\\tau\\to\\infty$, because of the correction terms. Finally, the terms with phases $i\\xi\\cdot x$ or $i\\xi^*\\cdot x$ can be controlled exactly as in the proof of Theorem~\\ref{main thm} using \\eqref{eqn 4.2} - \\eqref{eqn 4.6} and \\eqref{technical ineq from complex analysis}. Thus, letting $\\tau\\to\\infty$, we obtain\n$$\n\\int_{\\Omega} e^{i\\xi\\cdot x} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1^{1\/2} \\sigma_2^{1\/2}}e^{\\Psi_1 + \\Psi_2}\\,dx + \\int_{\\Omega} e^{i\\xi^*\\cdot x} \\Bigg(\\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1^{1\/2} \\sigma_2^{1\/2}}e^{\\Psi_1 + \\Psi_2}\\Bigg)^*\\,dx = 0.\n$$\nMaking the change of the variables $(x_1, x_2, x_3)\\mapsto (x_1, x_2, - x_3)$ in the second integral, we come to\n$$\n\\int_{\\mathcal D} e^{i\\xi\\cdot x} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1^{1\/2} \\sigma_2^{1\/2}} e^{\\Psi_1 + \\Psi_2}\\,dx = 0.\n$$\nThis integral can be extended to all of $\\R^3$ since $\\sigma_1 - \\sigma_2 = 0$ on $\\R^3\\setminus\\overline{\\mathcal D}$. Then this will imply that $\\sigma_1=\\sigma_2$ in $\\R^3$.\\smallskip\n\nNext, we set $\\sigma = \\sigma_1 = \\sigma_2$. By Proposition~\\ref{prop CGO}, there are complex geometric optics solutions $\\tilde H_1, \\tilde H_2\\in H^1_{\\Div}(\\mathcal U)$, with $\\nabla\\times \\tilde H_1, \\nabla\\times\\tilde H_2 \\in H^1_{\\Div}(\\mathcal U)$, for\n$$\n\\nabla\\times(\\sigma^{-1}\\nabla\\times\\tilde H_1) - i\\omega \\mu_1\\tilde H_1 = 0\\quad\\text{and}\\quad \\nabla\\times(\\sigma^{-1}\\nabla\\times\\tilde H_2) - i\\omega \\mu_2\\tilde H_2 = 0 \\quad\\text{in}\\quad\\mathcal U,\n$$\nrespectively, which have the following forms\n$$\n\\tilde H_1(x;\\zeta^1) = e^{i\\zeta^1\\cdot x} (a_1 \\rho + r_1),\\quad \\tilde H_2(x;\\zeta^2) = e^{-i\\zeta^2\\cdot x} \\Big(-a_2 \\rho - \\frac{1}{2} b_2 \\overline\\rho + r_2\\Big),\n$$\nwhere\n$$\na_1 = e^{-\\alpha^\\sharp(x;\\tau)\/2},\\quad a_2 = e^{-\\alpha^\\sharp(x;\\tau)\/2},\\quad b_2 = e^{\\alpha^\\sharp(x;\\tau)\/2} e^{\\Psi^\\sharp(x, \\rho; \\tau)}.\n$$\nThe functions $\\Psi^\\sharp(\\cdot, \\rho; \\tau), \\alpha^\\sharp(\\,\\cdot\\,;\\tau)\\in C^\\infty(\\R^3)$ and the correction terms $r_1$, $r_2 \\in H_{\\Div}^1(\\mathcal U)$ satisfy \\eqref{eqn 4.7} - \\eqref{eqn 4.9}. In a similar way as before, one can show that\n$$\nH_1(x;\\rho) := \\tilde H_1(x;\\rho) - \\tilde H_1^*(x;\\rho),\\quad H_2(x;\\rho) := \\tilde H_2(x;\\rho) - \\tilde H_2^*(x;\\rho).\n$$\nsatisfy\n$$\n\\nabla\\times(\\sigma_1^{-1}\\nabla\\times\\tilde H_1) - i\\omega \\mu_1\\tilde H_1 = 0\\quad\\text{and}\\quad \\nabla\\times(\\sigma_2^{-1}\\nabla\\times\\tilde H_2) - i\\omega \\mu_2\\tilde H_2 = 0 \\quad\\text{in}\\quad\\mathcal U.\n$$\nAlso, the restrictions of $H_1$ and $H_2$ onto $\\Omega$, still denoted by $H_1$ and $H_2$, respectively, belong to $H^1_{\\Div}(\\Omega)$ and satisfy $\\nu\\times H_1|_{\\Gamma}=\\nu\\times H_2|_{\\Gamma}=0$. Thus, $H_1$ and $H_2$ satisfy the hypotheses of Proposition~\\ref{prop main integral identity local}. We substitude $H_1$, $H_2$ and $\\sigma = \\sigma_1=\\sigma_2$ into \\eqref{main integral identity local}. As before, we assume that $\\rho_{1,3} \\neq 0$ and $\\rho_{2,3} = 0$. Therefor, we obtain\n\\begin{align*}\n0 &= \\int_\\Omega (\\mu_1-\\mu_2) e^{i\\xi\\cdot x}(a_1\\rho + r_1)\\cdot \\Big(-a_2\\rho-\\frac{1}{2}b_2\\overline{\\rho}+r_2\\Big)\\,dx\\\\\n&\\quad+\\int_\\Omega (\\mu_1-\\mu_2) e^{i\\xi^*\\cdot x}(a_1^*\\rho^*+r_1^*)\\cdot \\Big(-a_2^*\\rho^*-\\frac{1}{2}b_2^*\\overline{\\rho}^*+r_2^*\\Big)\\,dx\\\\\n&\\quad-\\int_\\Omega (\\mu_1-\\mu_2) e^{i\\tilde\\xi_+\\cdot x-\\eta_+}(a_1\\rho+r_1)\\cdot \\Big(-a_2^*\\rho^*-\\frac{1}{2}b_2^*\\overline{\\rho}^*+r_2^*\\Big)\\,dx\\\\\n&\\quad-\\int_\\Omega (\\mu_1-\\mu_2) e^{i\\tilde\\xi_-\\cdot x-\\eta_-}(a_1^*\\rho^*+r_1^*)\\cdot \\Big(-a_2\\rho-\\frac{1}{2}b_2\\overline{\\rho}+r_2\\Big)\\,dx.\n\\end{align*}\nAs before, we use \\eqref{eqn 4.7} - \\eqref{eqn 4.8}, the fact that $|\\eta_{\\pm}| = \\mathcal O(\\tau^{-1})$ as $\\tau\\to \\infty$ and the Riemann-Lebesgue lemma, to show that\n\\begin{align*}\n\\Big|\\int_\\Omega (\\mu_1-\\mu_2) e^{i\\tilde\\xi_+\\cdot x-\\eta_+}a_1\\rho\\cdot \\Big(-a_2^*\\rho^*-\\frac{1}{2}b_2^*\\overline{\\rho}^*\\Big)\\,dx\\Big| &= o(1),\\\\\n\\Big|\\int_\\Omega (\\mu_1-\\mu_2) e^{i\\tilde\\xi_-\\cdot x-\\eta_-}a_1^*\\rho^*\\cdot \\Big(-a_2\\rho-\\frac{1}{2}b_2\\overline{\\rho}\\Big)\\,dx\\Big| &= o(1)\n\\end{align*}\nas $\\tau\\to\\infty$. These estimates guarantee that the terms with phases $i\\tilde\\xi_{\\pm}\\cdot x - \\eta_{\\pm}$ that do not involve the correction terms $r_1$ and $r_2$ go to zero as $\\tau\\to\\infty$. The terms with phases $i\\tilde\\xi_{\\pm}\\cdot x - \\eta_{\\pm}$ that involve the correction terms $r_1$ and $r_2$ also go to zero as $\\tau\\to\\infty$ by \\eqref{eqn 4.9}. Finally, the terms with phases $i\\xi\\cdot x$ or $i\\xi^*\\cdot x$ can be controlled exactly as in the proof of Theorem~\\ref{main thm} using \\eqref{eqn 4.7} - \\eqref{eqn 4.9} and \\eqref{technical ineq from complex analysis}. Thus, letting $\\tau\\to\\infty$, we obtain\n$$\n\\int_{\\Omega} e^{i\\xi\\cdot x} (\\mu_1 - \\mu_2) e^{\\Psi}\\,dx + \\int_{\\Omega} e^{i\\xi^*\\cdot x} \\Big((\\mu_1 - \\mu_2) e^{\\Psi}\\Big)^*\\,dx = 0.\n$$\nMaking the change of the variables $(x_1, x_2, x_3)\\mapsto (x_1, x_2, - x_3)$ in the second integral, we come to\n$$\n\\int_{\\mathcal U} e^{i\\xi\\cdot x} (\\mu_1 - \\mu_2) e^{\\Psi}\\,dx = 0\n$$\nThis integral can be extended to all of $\\R^3$ since $\\mu_1 - \\mu_2 = 0$ on $\\R^3\\setminus\\overline\\Omega$. This implies that $\\mu_1 = \\mu_2$ completing the proof of Theorem~\\ref{main thm flat}.\n\n\\section{Proof of Theorem~\\ref{main thm 2}}\\label{section::proof of thm 2}\n\nWithout loss of generality, we can assume that $B_0$ is the open ball of radius $1\/2$ centered at $x_0 = (0,0,1\/2)$ and that $0\\notin \\overline\\Omega$. We recall that\n$$\n\\Gamma_0 = \\p\\Omega\\cap \\p B_0,\\quad \\Gamma_0 \\neq \\p B_0\\quad\\text{and}\\quad\\Gamma = \\overline{\\p\\Omega\\setminus \\Gamma_0}.\n$$\nWe define the map\n$$\nK:\\Omega\\to \\R^3\\setminus\\{0\\},\\quad K(x) := |x|^{-2}x\n$$\nwhich is known as the Kelvin transform. One can easily verify that $K^{-1}(y) = |y|^{-2}y$ for $y\\in K(\\Omega)$. We let $DK$ and $DK^{-1}$ denote the Jacobi matrices of $K$ and $K^{-1}$, respectively.\n\n\\smallski\n\nNext, we define $\\widetilde\\Omega := \\{- y + x_0 : y\\in K(\\Omega)\\}$ and\n$$\n F : \\widetilde\\Omega\\to \\Omega,\\quad F(y) := - K^{-1}(y) + x_0,\\quad y\\in\\widetilde\\Omega.\n$$\nThen\n$$\nF^{-1}(x) = - K(x-x_0),\\quad x\\in \\Omega.\n$$\nIt is not difficult to verify that $\\widetilde\\Omega\\subset \\{x\\in\n\\R^3: x_3 < 0 \\}$ and $\\widetilde\\Gamma_0:=F^{-1}(\\Gamma_0)$ is a\nsubset of the plane $\\{x\\in \\R^3: x_3 = 0\\}$. We also write\n$\\widetilde\\Gamma:=F^{-1}(\\Gamma)$. Thus, we are in a situation when\nthe inaccessible part of the boundary is part of a plane. A direct\ncalculation gives\n$$\nDF^{-1} = - |x-x_0|^{-2} I + 2 |x-x_0|^{-4} (x-x_0)(x-x_0)^T ,\\quad DF = - |y|^{-2} I + 2 |y|^{-4} yy^T,\n$$\nwhere $x-x_0$ and $y$ are considered as column vectors and $I$ is the $3\\times 3$ identity matrix. These identities can be used to show that\n\\begin{equation}\\label{eqn::DFDF^T}\nDF=(DF)^T\\quad\\text{and}\\quad DF(DF)^T=|y|^{-4}I\n\\end{equation}\nand\n\\begin{equation}\\label{eqn::key identities for the Kelvin transform}\nDF^{-1}\\circ F = |y|^{4} DF,\\quad DF = (DF)^T \\quad\\text{and}\\quad \\det(DF) = |y|^{-6}.\n\\end{equation}\n\n\\begin{Lemma}\\label{invariance of maxwell equation under pullback}\nLet $(H_j, E_j)\\in H^1_{\\Div}(\\Omega)\\times H^1_{\\Div}(\\Omega)$, $j=1,2$. Consider their pullbacks onto $\\Omega$,\n$$\n\\widetilde H_j := F^* H_j,\\quad \\widetilde E_j := F^* E_j,\\quad\\widetilde\\sigma_j = \\sigma_j\\circ F,\\quad \\widetilde\\mu_j = \\mu_j\\circ F,\\quad j=1,2.\n$$\nThen\n$$\n\\nabla\\times E_j=i\\omega\\mu_j H_j\\quad\\text{and}\\quad \\nabla\\times H_j=\\sigma_j E_j \\quad\\text{in}\\quad\\Omega\n$$\nif and only if\n$$\n\\widetilde\\nabla\\times \\widetilde E_j=i\\omega|y|^{-2}\\widetilde\\mu_j \\widetilde H_j\\quad\\text{and}\\quad \\widetilde\\nabla\\times \\widetilde H_j=|y|^{-2}\\widetilde\\sigma_j\\widetilde E_j \\quad\\text{in}\\quad\\widetilde\\Omega.\n$$\n\\end{Lemma}\n\n\\noindent\nHere and in what follows, $\\widetilde\\nabla\\times$ denotes the curl operator with respect to the coordinates in $\\widetilde\\Omega$.\n\\begin{proof}\nThe claim of the lemma is easy to prove by straightforward calculations using \\eqref{eqn::key identities for the Kelvin transform} and the facts from Appendix~\\ref{section::pullbacks}.\n\\end{proof}\n\n\\begin{Lemma}\\label{impedance map under pullback}\nThe following holds true,\n\\[\n   Z_{|y|^{-2}\\widetilde\\sigma_j ,|y|^{-2}\\widetilde\\mu_j}^\\omega(F^*f)\n        = F^*\\big(Z_{\\sigma_j ,\\mu_j}^\\omega(f)\\big)\n\\]\nfor all $f \\in TH^{1\/2}_{\\Div}(\\p\\Omega)$, $j=1,2$.\n\\end{Lemma}\n\n\\begin{proof}\nConsider a $C^{1,1}$ boundary defining function $\\rho$ for $\\p\\Omega$. Then using \\eqref{eqn::DFDF^T},\n$$\n\\widetilde\\nu = - \\frac{\\nabla(\\rho\\circ F)}{|\\nabla(\\rho\\circ F)|}\\Bigg|_{\\p\\widetilde\\Omega} = - \\frac{(DF)^T (\\nabla \\rho)\\circ F}{|(DF)^T (\\nabla \\rho)\\circ F|}\\Bigg|_{\\p\\widetilde\\Omega} = - \\frac{|y|^2 F^*(\\nabla \\rho)}{|(\\nabla\\rho)\\circ F|}\\Bigg|_{\\p\\widetilde\\Omega} = |y|^2 F^*\\Bigg(-\\frac{\\nabla \\rho}{|\\nabla\\rho|}\\Bigg)\\Bigg|_{\\p\\widetilde\\Omega} = |y|^2 F^*\\nu.\n$$\nUsing Lemma~\\ref{lemma::pullback of the cross product} and \\eqref{eqn::key identities for the Kelvin transform},\n$$\n\\tilde\\nu\\times\\tilde H_j = |y|^2 F^*\\nu\\times F^*H_j = |y|^{-4} \\big((DF)^{-1}(\\nu\\times H_j)\\big)\\circ F = DF\\,\\big((\\nu\\times H_j)\\circ F\\big) = (DF)^T\\big((\\nu\\times H_j)\\circ F\\big) = F^*(\\nu\\times H_j).\n$$\nSimilarly, $\\tilde\\nu\\times\\tilde E_j =  F^*(\\nu\\times E_j)$.\n\\end{proof}\n\nWith $Z_{\\sigma_1 ,\\mu_1}^\\omega = Z_{\\sigma_2 ,\\mu_2}^\\omega$, it follows from these lemmas that\n$$\nZ_{|y|^{-2}\\widetilde\\sigma_1 ,|y|^{-2}\\widetilde\\mu_1}^\\omega = Z_{|y|^{-2}\\widetilde\\sigma_2 ,|y|^{-2}\\widetilde\\mu_2}^\\omega.\n$$\nFinally, by hypothesis, $\\sigma_j$ and $\\mu_j$, $j=1,2$, can be extended to $\\R^3$ as $C^2$ functions which are invariant under reflection across $\\p B_0$. This is equivalent to the invariance of such extensions of $\\sigma_j$ and $\\mu_j$, $j=1,2$, under the map $x\\mapsto F\\circ R\\circ F^{-1}(x)$, where $R(y_1,y_2,y_3)=(y_1,y_2,-y_3)$. Therefore, $\\tilde\\sigma_j$ and $\\tilde\\mu_j$ can be extended into $\\R^3$ as $C^2$ functions which are invariant under reflection across the plane $\\{x\\in\\R^3: x_3=0\\}$. Thus, by Theorem~\\ref{main thm flat}, we get $|y|^{-2}\\tilde\\sigma_1=|y|^{-2}\\tilde\\sigma_2$ and $|y|^{-2}\\tilde\\mu_1=|y|^{-2}\\tilde\\mu_2$ in $\\widetilde\\Omega$. Hence, $\\sigma_1=\\sigma_2$ and $\\mu_1=\\mu_2$ in $\\Omega$ as desired.\n\n\n\\section*{Ackkowledgments}\n\nYA would like to thank Total E \\& P Research \\& Technology USA, for\nfinancial support. MVdH was supported by the Simons Foundation under\nthe MATH + X program, the National Science Foundation under grant\nDMS-1815143, and the corporate members of the Geo-Mathematical Imaging\nGroup at Rice University.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe potential effect of technical networks on territories, and more particularly of transportation networks, have fed scientific debates which remain relatively open in the current state-of-the-art, such as the issue of identifying of structuring effects of infrastructures~\\citep{offner1993effets}. These can be observed on long time periods, but several works recall that caution is key regarding the contingency of each situation and the dangers of a political application of the concept~\\citep{espacegeo2014effets}.\n\n\nA relevant approach to this question, explored by \\cite{raimbault2018caracterisation} of which we do a short synthesis here, is to understand territories and transportation networks as co-evolving, i.e. exhibiting strongly coupled dynamics which are difficult to isolate~\\citep{bretagnolle:tel-00459720}. The construction of the concept of territory by \\cite{raimbault2018caracterisation} at the intersection of Raffestin's approach and Pumain's approach, i.e. by combining the concept of human territory~\\citep{raffestin1988reperes} with the one of territory as the space in which systems of cities are embedded~\\citep{pumain2018evolutionary}, shows that it implies potential relations between geographical objects, and by induction the emergence of technical networks through the realization of transactional projects~\\citep{dupuy1987vers}. Therefore, the interaction between networks and territories can be intrinsically understood as endogenous to the concept of territory itself. We go further by postulating the potential existence of a \\emph{co-evolution} between transportation networks and territories, that we will define below from an interdisciplinary point of view in a dedicated section. We study more particularly in our work the \\emph{transportation networks}, in particular because these are typically representative of potentialities to shape territories that are attributed to technical networks, more specifically through the effective transportation of flows~\\citep{bavoux2005geographie}.\n\n\nThe research work we summarize proposes to explore this perspective of a co-evolution by studying it through the lens of modeling and simulation, considering the model as an instrument of knowledge in itself~\\citep{banos2013pour} complementary to the theoretical and empirical aspects~\\citep{raimbault2017applied}, and which impact is amplified by the use of methods and tools for model exploration and intensive computation~\\citep{pumain2017urban}. The use of generative simulation models for the considered systems allows to construct an indirect knowledge on implied processes and for example to directly test and compare hypothesis in this virtual laboratory~\\citep{epstein1996growing}.\n\n\nFollowing \\cite{raimbault2017invisible} which establishes a map of scientific approaches to the question, domains which studied the modeling of interactions between transportation networks and territories are much varied, from geography to planning of urban economics, and more recently physics, but these seem to be highly isolated~\\citep{raimbault2017models}. According to a systematic litterature review and a modelography done by~\\cite{raimbault2018caracterisation}, and a typology of interaction processes, two scales appear as relevant to unveil the possible existence of co-evolution processes : the mesoscopic scale, which will typically be a metropolitan spatial scale, and the macroscopic scale, which corresponds to the scale of the system of cities.\n\n\nTwo complementary modeling directions corresponding to these two scales have thus been developed, extending the previous works modeling this co-evolution at the macroscopic~\\citep{baptistemodeling,schmitt2014modelisation} and mesoscopic~\\citep{raimbault2014hybrid} scale. These two axes correspond to different theoretical frameworks, namely the evolutive urban theory for the macroscopic \\citep{pumain2018evolutionary} and urban morphogenesis for the mesoscopic, which we understand as the emergent relation between form and function~\\citep{doursat2012morphogenetic}.\n\n\nThis contribution aims at giving a synthesis of these research works, and is organized the following way : we firstly define the concept of co-evolution from a multi-disciplinary viewpoint. We then detail the results obtained for the modeling at the macroscopic scale, and then the ones for the mesoscopic scale. We finally discuss the perspectives opened and the future developments in the context of modeling co-evolution between transportation networks and territories.\n\n\n\\section{Defining co-evolution}\n\n\nTransfer of concepts between disciplines is always difficult, and as co-evolution has been used by several disciplines, we propose here a definition inspired by the various ones in which it was developed. A multi-disciplinary review and a more general definition not specific to territorial systems is developed in \\cite{raimbault2018co}. The detailed description of the different approaches is given in this paper, of which we give here only a broad summary.\n\n\nOriginating in biology, the concept of evolution requires typical features for a system to exhibit it~\\citep{durham1991coevolution}, namely (i) transmission processes between agents; (ii) transformation processes; and (iii) isolation of sub-system such that differentiations emerge from the previous processes. Co-evolution then corresponds to entangled evolutionary changes in two species~\\citep{janzen1980coevolution}. It was generalized to diffuse co-evolution by taking into account the broader context of numerous species in the ecological interaction network and of the environment~\\citep{strauss2005toward}.\n\nThe concept was transferred to disciplines closer to social and human sciences, including the emerging field of cultural evolution \\cite{Mesoudi25072017} for which building bricks of culture are transmitted and mutated, possibly with an interplay with biological evolution itself~\\citep{bull2000meme}, but also sociology to interpret for example the interactions between social organizations as entities themselves~\\citep{volberda2003co}. In the frame of evolutionary economics~\\citep{nelson2009evolutionary}, the concept was also largely applied in economic geography~\\citep{schamp201020}, to investigate for example the link between economic clusters and knowledge spillovers~\\citep{doi:10.1080\/00343400802662658}, the link between territories and technological innovation~\\citep{colletis2010co}, or environmental economics issues~\\citep{kallis2007coevolution}. The concept was particularly developed in geography in the frame of the evolutive urban theory \\citep{pumain1997pour}. Its operational definition in that context taken for example by \\cite{paulus2004coevolution} or \\cite{schmitt2014modelisation} relies on systems of cities constituted by strongly coupled subsystems and entangled interactions.\n\n\n\n\n\nWe can observe that these various definitions of co-evolution mostly correspond to the theoretical framework introduced by \\cite{holland2012signals}, in which hierarchically imbricated subsystems correspond to ecological niches and are therefore containing co-evolutive dynamics.\n\n\n\nThe definition of co-evolution we construct for the particular case of transportation networks and territories is the following, similar to the definition of \\cite{raimbault2018co}:\n\n\\begin{enumerate}\n    \\item the evolutive processes are carried by the transformation of territorial components at different scales;\n   \n\t\\item a co-evolution may occur in the strong coupling of such evolutive processes, with different levels of strength, namely as circular causal relation (i) between individual entities; (ii) within populations of individual, i.e. with a certain statistical sense within a given territory; (iii) between most elements of the system with a difficulty to disentangle these relations.\n\n\t\\item these different level and the spatial isolation typically enhancing evolutionary drift imply the existence of \\emph{territorial niches}, i.e. niches of co-evolution, imbricated at different scales.\n\n\\end{enumerate}\n\n\nOur definition is strongly anchored within a dynamical vision of processes, within the initial spirit of the introduction of the concept in biology. The opening on the different levels to which the co-evolution can occur yields a generality but also a precision and the construction of empirical characterization methods, such as the one introduced for the second level (population co-evolution) by \\cite{raimbault2017identification}. Furthermore, the implicit integration of the concept of niche implies a suitability with territoriality and its declination at different scales within territorial subsystems both independent and interdependent. Our approach is more general than the notion of congruence proposed by \\cite{offner1993effets}, which remains fuzzy in the interdependency relations between the entities concerned, and could be similar to the third level of systemic interdependencies.\n\n\n\n\n\n\n\n\\section{Macroscopic scale}\n\n\nThe first modeling axis relates to the macroscopic scale and is based on the principles of the evolutionary urban theory~\\citep{pumain1997pour}. The family of Simpop models is mainly situated in corresponding ontologies and scales, i.e. elementary entities constituted by cities themselves, at the spatial scale of the system of cities (regional to continental) and on relatively long time scales (longer than half a century)~\\citep{pumain2012multi}.\n\n\n\\subsection{Network effects}\n\nA first model that can be interpreted as a control, in which the network is static but has a retroaction on cities, indirectly suggests network effects. This preliminary work is detailed by~\\cite{raimbault2018indirect} which details the model and applies it to the French system of cities on a long time scale (1830-1999). The model is based on expected populations and captures complexity through non-linear interactions between cities, which are carried by the network. Three processes are added to determine the growth rate of cities: (i) an endogenous growth fixed by a parameter, corresponding to the Gibrat model ; (ii) direct interaction processes described as a gravity potential which influences the growth rate ; (iii) a retroaction of flows circulating in the network on the cities traversed. The model is initialized with real populations at the start of a period, and then evaluated by comparison to simulated populations on the full period, on two objectives which allow to take into account the adjustment of the total population or of their logarithm. The production of Pareto fronts shows that it is not possible to uniformly adjust the model for the full spectrum of city sizes. We show that the addition of the network component provides an effective increase in the adjustment, what suggests network effects.\n\n\n\\subsection{Co-evolution model}\n\n\nThis model is then extended to a co-evolutive model by~\\cite{raimbault2018modeling}, in which cities and links of the transportation network are both dynamic and within a reciprocal dependency. This model is close to~\\cite{schmitt2014modelisation} for the rules of population evolution, and to~\\cite{baptistemodeling} for the evolution of the network. More precisely, it operates iteratively with the following steps : (i) population of cities evolve according the the specification of the static model described above ; (ii) the network evolves, following an abstract implementation such that distances between cities are updated with a self-reinforcement function depending on flows between each city, with a threshold parameter. This version of the model is strictly macroscopic and does not include the spatial form of the network since it updates the distance matrix only.\n\n\nThe systematic exploration of this model using the OpenMOLE software~\\citep{reuillon2013openmole} and the application of an empirical method to characterize co-evolution~\\citep{raimbault2017identification} allow us to show that it captures a large variety of coupled dynamics, including indeed co-evolutive dynamics: among the broad range of interaction regimes between network variables and population variables (in the sense of \\cite{raimbault2017identification}, by classifying lagged correlation patterns), more than half of the regimes are effectively co-evolutive, i.e. exhibit circular causalities. This aspect could appear as trivial for a model conceived to integrate a co-evolution, but one has to realize that processes integrated in models are at the microscopic scale whereas the co-evolution is quantified at the macroscopic scale: it is indeed a property of the model to make co-evolution emerge. For example in the case of the SimpopNet model~\\citep{schmitt2014modelisation}, \\cite{raimbault2018unveiling} shows that co-evolution regimes are much more rare and that this other model produces more often configurations of the type structuring effects or without any relation. Furthermore, the exploration unveils the existence of an optimal value for an interaction range parameter, at which the system exhibits a maximal complexity of city trajectories. This range correspond to the appearance of territorial niches, within which populations are co-evolving, corresponding to the third point of our definition.\n\n\nThe calibration on the French system of cities on the same time period than the static model, with population data and dynamical railway network data taken into account with dynamical distance matrices constructed from the database of \\cite{thevenin2013mapping}, unveils Pareto fronts between the objective on the distance between cities and population objective, suggesting an impossibility to simultaneously calibrate this type of models for the network component and for the territorial component. Moreover, the adjustment on populations is improved by this model for a certain number of periods, compared to the model with a static network, suggesting the relevance of taking into account co-evolutive dynamics. The evolution of the calibrated threshold parameter suggests that the model captures a ``High Speed Rail (TGV) effect'', i.e. an increase in the effective accessibility for the metropolitan areas concerned but a loss of speed for the territories left behind.\n\n\n\\section{Mesoscopic scale}\n\n\nThe second axis, at the mesoscopic scale, considers the approach through urban morphogenesis, understood as the simultaneous emergence of the form and the function of a system \\citep{doursat2012morphogenetic}. It allows to consider a more precise description of territories, at the scale of fine population grids (500m resolution) and of a vectorial representation of the network at the same scale.\n\n\n\\subsection{Morphogenesis by aggregation-diffusion}\n\nThe territorial systems produced are quantified with morphological indicators for the population \\citep{le2015forme}. A first preliminary model, integrating only the population, shows that aggregation and diffusion processes are sufficient to explain a large majority of urban forms existing in Europe \\citep{raimbault2018calibration}. This result suggests that taking into account the form only can be achieved in an autonomous way, but that functional processes will then not be taken into account in the core of dynamics. In order to grasp morphogenesis processes, i.e. the strong link between form and function during the emergence of these, we follow the idea of using the transportation network as a proxy of functional properties of territories, in particular through centrality properties. This leads us to consider a morphogenesis model by co-evolution at the mesoscopic scale.\n\n\n\\subsection{Morphogenesis by co-evolution}\n\n\nThe urban form indicators computed on moving windows of size 50km for the whole Europe are completed with structural network indicators for the road network, computed from OpenStreetMap data after application of a specific simplification algorithm which conserves topological properties \\citep{raimbault2018urban}. These indicators and their static spatial correlations are thus computed on windows of a similar size covering the whole Europe. We show through the study of the spatial study of these correlations the non-stationarity of interaction processes at the second order, confirming the relevance of the notion of niche as a consistent territorial sub-system.\n\nWe then introduce in \\cite{raimbault2018urban} a morphogenesis model capturing the co-evolution of the spatial distribution of population and of the road network. This model combines the logic of \\cite{raimbault2018calibration} for the complementarity of aggregation and diffusion processes to the one of \\cite{raimbault2014hybrid} for the hybrid grid and vectorial network structure and also the influence of local explicative variables on the territorial evolution. This model works the following way: (i) morphological and functional local properties, integrated as local normalized explicative variables (including population, distance to the network, betweenness centrality, closeness centrality, accessibility), determine the value of a utility function from which new population is added through preferential attachment, and a diffusion of population is achieved; (ii) the road network evolves following rules depending on different heuristics (multi-modeling approach), which include, after the addition of nodes preferentially to the new population and their direct connection to the existing network, an addition of links with an heuristic among random links, random potential breakdown \\citep{schmitt2014modelisation}, deterministic potential breakdown \\citep{raimbault2016generation}, biological self-organized network \\citep{tero2010rules}, cost-benefit compromises \\citep{louf2013emergence}.\n\n\nThe computation of topological indicators for the road network allows to calibrate the model, and \\cite{raimbault2018multi} shows that the different network growth processes which have been included following the multi-modeling procedure are complementary to reach a maximum a real network configurations. Furthermore, the model is simultaneously calibrated on morphological indicators, topological indicators, and their correlation matrices, and the relatively low distances to data for a non negligible number of points suggest that the model is able to reproduce outcomes of processes at the first order (indicators) but also at the second order (interactions between indicators). Regarding the causality regimes produced by the model, we obtain configurations corresponding to a co-evolution, but a much lower diversity than for the more simple model of \\cite{raimbault2014hybrid} which was used as a toy model to show the relevance of the method of causality regimes in \\cite{raimbault2017identification}, suggesting a tension between static performance (reproduction of outcomes of processes) and dynamical performance (reproduction of processes themselves) for this kind of models.\n \n \n\n\\subsection{Transportation governance}\n \nFinally, a last metropolitan model (Lutecia model) is described in \\cite{raimbault2018caracterisation}, extending the one proposed by \\citep{lenechet:halshs-01272236}. It explores the role of governance processes in the growth of the transportation network, within a co-evolution model. Here, the metropolitan scale indeed corresponds to the mesoscopic scale. The collaboration between local actors for the construction of transportation infrastructures is included using game theory paradigms. This allows to simulate the emergence of the transportation network and its interaction with the urban form quantified by spatial accessibility patterns. This model allows for example to show that co-evolutive dynamics can in some case lead to the inversion of the behavior of accessibility gains in comparison to a situation without evolution of land-use, i.e. to qualitatively change the regime of the metropolitan system. The calibration of this model on the stylized case of the mega-urban region of Pearl River Delta in China allows to extrapolate on governance processes, and suggests that the shape of the current network is more likely to be the consequence either of fully regional decisions or of fully local decisions, but never an intermediate configuration, contradicting the view of the Chinese political context as a multi-level decision system~\\citep{liao2017ouverture}.\n\n\n\n\\section{Perspectives}\n\nThis research therefore developed complementary approaches at different scales of interactions between transportation networks and territories by modeling their co-evolution. We finally detail some immediate development perspectives opened by this work.\n\n\n\\subsection{Developments at the macroscopic scale}\n\n\nOur macroscopic models have not been tested yet on other urban systems and other time spans, and future developments will have to study which conclusions obtained here are specific to the French urban system on these periods, and which are more general are could be more generic in systems of cities. The application of the model to other systems of cities also recalls the difficulty to define urban systems. In our case, a strong bias must be induced by the fact of considering France only, since the insertion of its urban system within an European system is a reality we had to neglect. The span and scale of such models is always a difficult subject. We rely here on the administrative consistence and on the consistence of the database \\citep{pumain1986fichier}, but the sensitivity to the definition of the system and to its spatial extent must still be tested.\n\n\nFurthermore, the calibration used only the railway network for the distances between cities. Considering a single transportation mode is naturally a reduction, and an immediate direction for developments is testing the model with real distance matrices for other types of networks, such as the freeway network which has undergone a significant growth in France in the second half of the 20th century. This application would necessitate the construction of a dynamical database for the freeway network spanning 1950-2015, since classical bases (IGN or OpenStreetMap) do not integrate the opening date of segments. A natural extension of the model would then consist in the implementation of a multilayer network, what is a typical approach to represent multi-modal transportation networks~\\citep{gallotti2014anatomy}. Each layer of the transportation network should have a co-evolutive dynamic with populations, possibly with the existence of inter-layer dynamics.\n\n\n\n\\subsection{Developments at the mesoscopic scale}\n\n\nThe issue of the generic character of the co-evolution morphogenesis model is also opened, i.e. if it would work similarly to reproduce urban forms on very different systems such as the United States or China. A first interesting development would be to test it on these systems and at slightly different scales (1km cell size for example).\n\n\nFurthermore, the Lutecia model is also a fundamental contribution towards the inclusion of more complex processes implied in co-evolution, such as the governance of the transportation system. This model paves the way to a new generation of models, that could potentially become operational in the case of regional systems with a very high evolution speed such as in the Chinese case.\n\n\n\\subsection{Towards multi-scale models}\n\n\nOur work finally opens perspectives for integrated approaches, towards multi-scale models of these interactions, which appear to be more and more necessary for the construction of operational models that can be applied to the design of sustainable planning policies \\citep{rozenblat2018conclusion}.\n\n\nA first contribution towards multi-scalar models would be to take into account in a finer way the physical network in macroscopic models, what is for example the object of~\\cite{mimeur:tel-01451164}, which produces interesting results regarding the influence of the centralization of network investment decision-making on final forms, but keeps static populations and does not produce a co-evolution model. Similarly, the choice of indicators to quantify the distance of the simulated network to a real network is a difficult question in that context: indicators such as the number of intersections taken by~\\cite{mimeur:tel-01451164} is associated to procedural modeling and does not reflect structural indicators. It is probably for the same reason that~\\cite{schmitt2014modelisation} is only interested in population trajectories and not in network indicators: the superposition and adjustment of population and network dynamics at different scales remains an open problem.\n\n\nA second entry consists in the integration of the mesoscopic morphogenesis model into population of models in interaction. This approach would allow to take into account the non-stationarity of territorial systems that we moreover showed empirically. Here at the mesoscopic scale, total population and growth rate are fixed by exogenous conditions due to processes at the macroscopic scale. It is in particular the aim of spatial growth models such as the macroscopic model we already introduced to determine such parameters through the relations between cities as agents. It would then be possible to condition the morphological development of each area to the values of parameters determined at the upper level. In that context, one must remain careful on the role of emergence: should the emergent urban form influence the macroscopic behavior in its turn ? Such complex multi-scalar models are promising but must be considered with caution for the level of complexity required and the way to couple scales.\n\n\n\\section{Conclusion}\n\nWe did here a synthesis of main results of \\cite{raimbault2018caracterisation}, confirming the relevance of the co-evolution approach to focus on interactions between transportation networks and territories, in particular to model these interactions. We developed a precise definition of co-evolution, moreover associated to an empirical characterization method. The models at different scales, and the perspective of multi-scale models, can become precious tools for territorial prospective in the context of sustainable transitions on long time, allowing to quantify the possible territorial systems and the one desirables regarding sustainability, taking into account scales and processes still relatively poorly tackled in the literature.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{The shift property}\nWe consider in this paper a property of Schauder bases that has come up on several\noccasions since the first construction of a truly non-classical Banach space\nby B. S. Tsirelson in 1974 \\cite{tsirelson}. It is a weakening of the property of  perfect homogeneity, which\nreplaces the condition \n\\begin{quote}\n{\\em all normalised block bases are equivalent}\n\\end{quote}\nwith the weaker \n\\begin{quote}\n{\\em all normalised block bases with the same growth rate are equivalent,}\n\\end{quote}\nand is satisfied by bases constructed along the lines of the Tsirelson basis,\nincluding the standard bases for the Tsirelson space and its dual.\n\n\nTo motivate our study and in order to fix ideas,  in the following result\nwe sum up a number of conditions that have been studied at various occasions in the literature and that can all be seen to be reformulations of the aforementioned property. Though I know of no single reference for the proof of the equivalence, parts of it are implicit in J. Lindenstrauss and L. Tzafriri's paper \\cite{lindenstrauss} and the paper by P. G.  Casazza, W. B. Johnson and L. Tzafriri \\cite{casazza}. Moreover, any idea needed for the proof can be found in, e.g., the book by F. Albiac and N. J. Kalton \\cite{albiac} (see Lemma 9.4.1, Theorem 9.4.2. and Problem 9.1) and the statement should probably be considered folklore knowledge.\n\\begin{thm}\\label{equivalences}\nLet $(e_n)_{n=1}^\\infty$ be a normalised unconditional Schauder basis for a Banach space $X$. Then the following conditions are equivalent.\n\\begin{enumerate}\n\\item Any block subspace is complemented.\n\\item Any block subspace $[x_n]_{n=1}^\\infty$ is complemented by a projection $P$ such that \n$$\nPz=\\sum_{n=1}^\\infty x^*_n(z)x_n,\n$$\nwhere $x_n^*\\in X^*$ satisfy ${\\rm supp}\\; x^*_n\\subseteq {\\rm supp}\\; x_n$.\n\\item  If $(x_n)_{n=1}^\\infty$ and $(y_n)_{n=1}^\\infty$ are normalised block sequences of $(e_n)_{n=1}^\\infty$ with\n$$\nx_1<y_1<x_2<y_2<\\ldots,\n$$\nthen $(x_n)_{n=1}^\\infty\\sim (y_n)_{n=1}^\\infty$.\n\\item If $(x_n)_{n=1}^\\infty$ is a normalised block basis, then $(x_{n})_{n=1}^\\infty\\sim(x_{n+1})_{n=1}^\\infty$.\n\\item If $(x_i)_{i=1}^\\infty$ and $(y_i)_{i=1}^\\infty$ are normalised block sequences such that \n$$\n\\max({\\rm supp}\\; x_i\\cup {\\rm supp}\\; y_i)<\\min({\\rm supp}\\; x_{i+1}\\cup {\\rm supp}\\; y_{i+1})\n$$\nfor all $i$, then $(x_i)_{i=1}^\\infty\\sim(y_i)_{i=1}^\\infty$.\n\\item For all normalised block bases $(x_n)_{n=1}^\\infty$, if $k_n\\in {\\rm supp}\\; x_n$ for all $n$,\nthen $(e_{k_n})_{n=1}^\\infty\\sim (x_n)_{n=1}^\\infty$.\n\\end{enumerate}\nMoreover, if any of the above properties hold, then they do so uniformly, e.g.,\nin (4) there is a constant $C$ such that for all normalised block bases\n$(x_n)_{n=1}^\\infty$, we have $(x_n)_{n=1}^\\infty\\sim_C(x_{n+1})_{n=1}^\\infty$. \n\\end{thm}\n\n\nAn unconditional basis satisfying the above equivalent conditions will be said to have the {\\em shift property}. This is a natural weakening of {\\em perfect homogeneity}, i.e., that all normalised block bases are equivalent, which was shown to be just a reformulation of being equivalent to the standard unit vector basis of $c_0$ or $\\ell_p$, $1\\leqslant p<\\infty$, by M. Zippin \\cite{zippin}.  Let us also note that the shift property is stronger than what it is called the {\\em block property} in \\cite{kutzarova}, which is the requirement that every block sequence is equivalent with some subsequence of the basis. Finally, we remark that the shift property is obviously hereditary, that is, any normalised block basis of an unconditional basis with the shift property will itself have the shift property. \n\n\nMoreover, while the canonical bases of both Tsirelson's space and its dual have the shift property, only one of them contains a {\\em minimal} subspace, i.e., an infinite-dimensional subspace that embeds into all of its further infinite-dimensional subspaces. On the other hand, recall that a space $E$ is {\\em locally minimal} \\cite{minimal} if there is a constant $K$ such that for all finite-dimensional\n$F\\subseteq E$ and infinite-dimensional $X\\subseteq E$, $F\\sqsubseteq_KX$,\ni.e., $F$ embeds with constant $K$ into $X$. As was pointed out in \\cite{minimal} (Proposition 6.7), the proof of \nTheorem 14 in  \\cite{casazza} essentially shows that any locally minimal space with a basis having the shift property is minimal.\n\n\n\nThe goal of the present paper is not to study the shift property per se, but rather to characterise the separable reflexive spaces that embed into a Banach space having a Schauder basis with the shift property. This will require some rather sophisticated techniques developed by E. Odell and T. Schlumprecht in a series of papers (see, e.g., \\cite{haydon, odell}) and that we shall  summarise and slightly develop here. As a first application of their techniques, they characterised in \\cite{odell} the separable reflexive Banach spaces embedding into an $\\ell_p$-sum of finite-dimensional spaces for $1<p<\\infty$ and their result was further improved in \\cite{IAG} to the following statement.\n\n\\begin{thm}[see \\cite{odell, IAG}]\\label{OS}\nLet $E$ be a separable reflexive Banach space such that any normalised weakly null tree $T$ in $E$ has a  branch $(x_i)_{i=1}^\\infty\\in [T]$ equivalent with all its subsequences. Then $E$ embeds into an $\\ell_p$-sum, $1<p<\\infty$, of finite-dimensional spaces.\n\\end{thm} \n\n\nThe result we shall obtain here has a weaker, though similar sounding hypothesis, but its conclusion is perhaps more satisfactory, since it provides a  basis rather than a finite-dimensional decomposition.\n\n\\begin{thm}\\label{ros}\nLet $E$ be a separable reflexive Banach space such that any normalised weakly null tree $T$ in $E$ has a  branch $(x_i)_{i=1}^\\infty\\in [T]$ satisfying $(x_{2i-1})_{i=1}^\\infty\\sim(x_{2i})_{i=1}^\\infty$. Then $E$ embeds into a reflexive space having an unconditional basis with the shift property.\n\\end{thm} \n\nIf the reader is not familiar with the techniques of Odell and Schlumprecht, this  should not be a hindrance to understanding the present construction, as we shall take certain of their technical results as black boxes that are directly applicable in our situation.\n\n\nWithout further introduction, let us commence the technical part of the paper by proving Theorem \\ref{equivalences} for the record and the convenience of the reader.\n\n\\begin{proof}\nThe implication (1)$\\Rightarrow$(2) follows directly from Lemma 9.4.1 in \\cite{albiac}, so we shall not repeat the proof here.\n\n(2)$\\Rightarrow$(3): Suppose (2) holds and $(x_n)_{n=1}^\\infty$ and $(y_n)_{n=1}^\\infty$ are normalised block sequences satisfying\n$$\nx_1<y_1<x_2<y_2<\\ldots.\n$$\nAssume that $a_n$ are scalars such that $\\sum_{n=1}^\\infty a_nx_n$ converges and choose $s_n>0$ converging to $0$ such that also \n$\\sum_{n=1}^\\infty  \\frac{a_n}{s_n} x_n$ converges. Put $w_n=x_n+s_ny_n$ and find $w_n^*\\in X^*$ such that ${\\rm supp}\\;w_n^*\\subseteq {\\rm supp}\\;w_n$ and \n$$\nPz=\\sum_{n=1}^\\infty w_n^*(z)w_n\n$$\ndefines a bounded projection onto $[w_n]_{n=1}^\\infty$, whence $\\sup\\norm{w_n^*}<\\infty$. Then\n$$\nP\\big(\\sum_{n=1}^\\infty  \\frac{a_n}{s_n}x_n\\big)=\\sum_{n=1}^\\infty  \\frac{a_n}{s_n}P(x_n)=\\sum_{n=1}^\\infty  \\frac{a_n}{s_n}w_n^*(x_n)w_n=\\sum_{n=1}^\\infty  \\frac{a_n}{s_n}w_n^*(x_n)(x_n+s_ny_n)\n$$\nand so the last series is norm convergent. By unconditionality, it follows that the series $\\sum_{n=1}^\\infty a_nw_n^*(x_n)y_n$ is norm convergent too. Thus, as \n$$\nw_n^*(x_n)=w_n^*(w_n)-w^*_n(s_ny_n)=1-s_nw_n^*(y_n)\\Lim{n\\rightarrow \\infty}1,\n$$\nusing unconditionality again, we find that also $\\sum_{n=1}^\\infty a_ny_n$ is norm convergent. A symmetric argument shows that if  $\\sum_{n=1}^\\infty a_ny_n$ converges, then so does  $\\sum_{n=1}^\\infty a_nx_n$, whence $(x_n)_{n=1}^\\infty$ and $(y_n)_{n=1}^\\infty$ are equivalent.\n\n\n(3)$\\Rightarrow$(4): Assume that (3) holds and that $(x_n)_{n=1}^\\infty$ is a normalised block sequence. Then using (3) \n$$\n(x_{2n-1})_{n=1}^\\infty\\sim (x_{2n})_{n=1}^\\infty\\sim (x_{2n+1})_{n=1}^\\infty.\n$$\nBy unconditionality, it follows that the sequence $(x_n)_{n=1}^\\infty$, which is the disjoint union of the sequences $(x_{2n-1})_{n=1}^\\infty$ and  $(x_{2n})_{n=1}^\\infty$, is equivalent to the sequence $(x_{n+1})_{n=1}^\\infty$, which itself is the disjoint union of the sequences $(x_{2n})_{n=1}^\\infty$ and  $(x_{2n+1})_{n=1}^\\infty$.\n\n(4)$\\Rightarrow$(5): If $(x_i)_{i=1}^\\infty$ and $(y_i)_{i=1}^\\infty$ are normalised block sequences such that \n$$\n\\max({\\rm supp}\\; x_i\\cup {\\rm supp}\\; y_i)<\\min({\\rm supp}\\; x_{i+1}\\cup {\\rm supp}\\; y_{i+1}),\n$$\nthen both $x_1,y_2,x_3,y_4,\\ldots$ and $x_2,y_3,x_4,y_5,\\ldots$ are normalised block sequences, whence $(x_{2i-1})_{i=1}^\\infty\\sim(y_{2i})_{i=1}^\\infty$ and $(x_{2i})_{i=1}^\\infty\\sim(y_{2i+1})_{i=1}^\\infty$. By unconditionality, it follows that $(x_i)_{i=1}^\\infty\\sim(y_{i+1})_{i=1}^\\infty\\sim (y_i)_{i=1}^\\infty$.\n\n\n\n\n\n\n(5)$\\Rightarrow$(6): Trivial.\n\n(6)$\\Rightarrow$(1): If (6) holds, then it does so uniformly, that is, there is a constant $C$ such that $(x_n)_{n=1}^\\infty\\sim_C(e_{k_{n}})_{n=1}^\\infty$ whenever $(x_n)_{n=1}^\\infty$ is a normalised block basis and $k_n\\in {\\rm supp}\\;x_n$.\nThis can easily be seen, as otherwise one would be able to piece together finite bits of sequences  with worse and worse constants of equivalence to get a counter-example to (6). Let also $K_u$ be the constant of unconditionality of $(e_n)_{n=1}^\\infty$.\n\nSuppose $(x_n)_{n=1}^\\infty$ is a normalised block sequence and let $I_1<I_2<I_3<\\ldots$ be a partition of $\\N$ into successive finite intervals such that ${\\rm supp}\\;x_n\\subseteq I_n$. Find also norm $1$ functionals $x_n^*\\in X^*$ such that ${\\rm supp}\\; x_n^*\\subseteq {\\rm supp}\\; x_n$ and $x_n^*(x_n)=1$. We claim that\n$$\nP(z)=\\sum_{n=1}^\\infty x_n^*(z)x_n\n$$\ndefines a projection of norm $\\leqslant K_uC^2$ from $X$ onto $[x_n]_{n=1}^\\infty$. To see this, suppose $z\\in X$ and write $z=\\sum_{n=1}^\\infty a_nz_n$, where the $z_n$ are normalised block vectors such that ${\\rm supp}\\;z_n\\subseteq I_n$. Modulo, perturbing $x_n$ and $z_n$ ever so slightly to get ${\\rm supp}\\; x_n=I_n={\\rm supp}\\; z_n$ and picking $k_n\\in I_n$, we see that $(x_n)_{n=1}^\\infty\\sim_C(e_{k_n})_{n=1}^\\infty\\sim_C(z_n)_{n=1}^\\infty$. So $\\sum_{n=1}^\\infty a_n x_n$ converges and, by unconditionality, so does $\\sum_{n=1}^\\infty x_n^*(z_n)a_n x_n=\\sum_{n=1}^\\infty x_n^*(z) x_n$. Therefore, $P$ is defined and satisfies\n\\[\\begin{split}\n\\norm{P(z)}&=\\norm{\\sum_{n=1}^\\infty x_n^*(z) x_n}=\\norm{\\sum_{n=1}^\\infty x_n^*(z_n) a_nx_n}\\\\\n&\\leqslant K_u\\norm{\\sum_{n=1}^\\infty a_nx_n}\\leqslant K_uC^2\\norm{\\sum_{n=1}^\\infty a_nz_n}=K_uC^2\\norm{z},\n\\end{split}\\]\nproving the estimate on the norm.\n\\end{proof}\n\n\n\n\n\n\n\nFinally, let us also remark that unconditionality is already implied by conditions (4), (5) and (6) of Theorem \\ref{equivalences}. E.g., if a normalised basis  $(e_n)_{n=1}^\\infty$ satisfies (4) and\n$(\\theta_n)_{n=1}^\\infty\\in\\{-1,1\\}^\\infty$, then \n$$\n(e_1,\\theta_1\ne_2,\\theta_1\\theta_2e_3,\\theta_1\\theta_2\\theta_3e_4,\\ldots)\\sim (\\theta_1\ne_2,\\theta_1\\theta_2e_3,\\theta_1\\theta_2\\theta_3e_4,\\theta_1\\theta_2\\theta_3\\theta_4e_5,\\ldots),\n$$\nand therefore, multiplying both sides with $(\\theta_1,\\theta_1\\theta_2,\\theta_1\\theta_2\\theta_3,\\theta_1\\theta_2\\theta_3\\theta_4,\\ldots)$, we have\n$$\n(\\theta_1 e_1,\\theta_2e_2,\\theta_3e_3,\\theta_4e_4,\\ldots)\\sim (e_2,e_3,e_4,e_5,\\ldots)\\sim(e_1,e_2,e_3,e_4,\\ldots).\n$$\nSince $(\\theta_n)_{n=1}^\\infty\\in\\{-1,1\\}^\\infty$ is arbitrary, this shows that $(e_n)_{n=1}^\\infty$ is unconditional.\n\n\n\n\n\n\nBefore continuing with the proof of Theorem \\ref{ros}, let us note that, while Theorem \\ref{ros} characterises reflexive spaces embeddable into a space with a basis having the shift property, we do not know of any significant characterisation of the spaces containing a basic sequence with the shift property. Using W. T. Gowers' block Ramsey theorem from \\cite{gowers} and Lemma 6.4 of \\cite{minimal}, we can conclude that if $X$ is a Banach space with a Schauder basis $(e_n)_{n=1}^\\infty$, then $X$ contains a normalised block sequence $(y_n)_{n=1}^\\infty$ that either is unconditional and has the shift property or such that there is  a non-empty tree $T$ consisting of finite normalised block sequences of $(y_n)_{n=1}^\\infty$ with the following property:\n\\begin{itemize}\n\\item[(a)] if $(z_1,\\ldots,z_m)\\in T$ and $Z$ is a block subspace of $[y_n]_{n=1}^\\infty$, then there is $z\\in Z$ such that $(z_1,\\ldots,z_m, z)\\in T$, and\n\\item[(b)] if $(z_1,z_2,z_3,\\ldots)$ is an infinite branch of $T$, then $(z_{2n-1})_{n=1}^\\infty\\not \\sim(z_{2n})_{n=1}^\\infty$.\n\\end{itemize}\nHowever, it is not clear what can be concluded from the existence of such a tree $T$ and one would like to draw stronger or more informative consequences from this.\n\\begin{prob}\nFormulate and prove a dichotomy that characterises the Banach spaces containing a unconditional basis sequence with the shift property.\n\\end{prob}\n\n\n\n\n\n\n\\section{Subspaces of spaces with an F.D.D.}\\label{subspaces of\nspaces with FDD's}\n\nWe fix in the following Banach spaces $E\\subseteq F$ and an F.D.D. $(F_i)_{i=1}^\\infty$ of $F$. For each interval $I\\subseteq \\N$, we let $I(x)$  denote the canonical projection of  $x\\in F$\nonto the subspace $\\sum_{i\\in I}F_i$ and shall also sometimes write\n$[\\sum_{i\\in I}F_i](x)$ for $I(x)$ if there is any chance of confusion. So, if $K$ denotes the constant of the decomposition $(F_i)_{i=1}^\\infty$, then $\\norm{I}\\leqslant 2K$ for any interval $I\\subseteq \\N$.\n\nFixing notation, if $A$ is a set, we let $A^\\infty$ denote the set of all infinite sequences $(a_i)_{i=1}^\\infty$ of elements of $A$ and let $A^{<\\infty}$ denote the set of all finite sequences $(a_1,\\ldots,a_n)$ of elements of $A$, including the empty sequence $\\tom$. A {\\em tree} on $A$ is a subset $T\\subseteq A^{<\\infty}$ closed under initial segments, i.e., such that  $(a_1,\\ldots,a_n)\\in T$ implies that $(a_1,\\ldots,a_m)\\in T$ for all $m\\leqslant n$. When $T$ is a tree on $A$, we let $[T]$ denote the set of all {\\em infinite branches} of $T$, i.e., the set of all sequences $(a_i)_{i=1}^\\infty$ such that $(a_1,\\ldots,a_{n})\\in T$ for all $n$.\n\nTo simplify notation, if $\\Delta=(\\delta_i)_{i=1}^\\infty$  \nis a decreasing sequence of real numbers $\\delta_i>0$ tending to $0$, we will denote this simply by $\\Delta\\searrow0$.\nSimilarly, if $M=(m_i)_{i=1}^\\infty$ is a strictly increasing sequence of natural numbers, we shall denote this by $M\\nearrow\\infty$.\n\n\n\nIf $\\B\\subseteq S_E^\\infty$ is a set of normalised sequences in $E$, we let \n$$\n\\B_\\Delta=\\big\\{(x_i)_{i=1}^\\infty\\in S_E^\\infty\\del \\e (y_i)_{i=1}^\\infty\\in \\B\\; \\a i\\; \\norm{x_i-y_i}<\\delta_i\\big\\}\n$$\nand\n$$\n{\\rm Int}_\\Delta(\\B)=\\big\\{(x_i)_{i=1}^\\infty\\in S_E^\\infty\\del \\a (y_i)_{i=1}^\\infty\\in S_E^\\infty\\; \\big(\\a i\\; \\norm{x_i-y_i}<\\delta_i\\rightarrow (y_i)_{i=1}^\\infty\\in  \\B\\big)\\big\\},\n$$\nand note that ${\\rm Int}_\\Delta(\\B)=\\;\\sim\\!(\\sim \\B)_\\Delta$, where the complement is taken with respect to $S_E^\\infty$.\n\n\n\n\n\\begin{defi}\\label{delta-block}\nGiven $\\Delta\\searrow0$, a normalised sequence $(x_i)_{i=1}^\\infty\\in S_E^\\infty$ is said to be\na $\\Delta$-{\\em block sequence} if there are intervals $I_i\\subseteq \\N$ such that\n$$\nI_1<I_2<I_3<\\ldots\n$$\nand for every $i$,\n$$\n\\|{I_i}(x_i)-x_i\\|<\\delta_i.\n$$\nMoreover, if $M\\nearrow \\infty$, we say that $(x_i)_{i=1}^\\infty$ is {\\em $M$-separated} if the witnesses $I_i\\subseteq \\N$ can be chosen such that \n$$\nm_1<I_1\\quad \\&\\quad \\a i\\;\\e j\\;\\;\\; I_i<m_j<m_{j+1}<I_{i+1}.\n$$\n\nWe let ${\\rm bb}_{E,\\Delta}(F_i)$ denote the set of  $\\Delta$-block sequences in $E$ and let ${\\rm bb}_{E,\\Delta, M}(F_i)$ denote the set of $M$-separated $\\Delta$-block sequences in $E$.\n\\end{defi}\n\nWe notice that if $K$ is the constant of the decomposition $(F_i)_{i=1}^\\infty$ and $(x_i)_{i=1}^\\infty$ and $(y_i)_{i=1}^\\infty$ are normalised sequences such that $ \\|x_i-y_i\\|<\\delta_i$ for all $i$, then if $(x_i)_{i=1}^\\infty$\nis a $\\Delta$-block, $(y_i)_{i=1}^\\infty$ is a $4K\\Delta$-block (with the same sequence of witnesses $I_1<I_2<\\ldots$).\n\nAlso, since $\\Delta\\searrow0$ is a decreasing sequence, the sets  ${\\rm bb}_{E,\\Delta}(F_i)$ and  ${\\rm bb}_{E,\\Delta, M}(F_i)$ are closed under taking subsequences, that is, if $(x_i)_{i=1}^\\infty\\in {\\rm bb}_{E,\\Delta, M}(F_i)$, as witnessed by a sequence $(I_i)_{i=1}^\\infty$, and $A\\subseteq \\N$, then $(I_i)_{i\\in A}$ witnesses that $(x_i)_{i\\in A}\\in {\\rm bb}_{E,\\Delta, M}(F_i)$. Lemma \\ref{all blocks} below essentially improves this to closure under taking normalised block sequences.\n\n\n\n\n\n\\begin{lemme}\\label{interior}\nSuppose $E$ is a subspace of a space $F$ with an F.D.D. $(F_i)_{i=1}^\\infty$. Let\n$\\B\\subseteq  S_E^\\infty$ be a set of sequences invariant under equivalence.\nThen there is a $\\Delta\\searrow0$ such that\n$$\n\\B_{\\Delta}\\cap {\\rm bb}_{E,\\Delta}(F_i)\\subseteq {\\rm Int}_{\\Delta}(\\B).\n$$\n\\end{lemme}\n\n\n\n\\begin{proof}Pick a $\\Delta\\searrow0$ depending on the constant of the decomposition $(F_i)_{i=1}^\\infty$ such that if $(y_i)_{i=1}^\\infty$ is a normalised block sequence in $F$ and $(v_i)_{i=1}^\\infty$ is a sequence in $F$ satisfying $\\norm{v_i-y_i}<5\\delta_i$ for all $i$, then $(v_i)_{i=1}^\\infty\\sim (y_i)_{i=1}^\\infty$. Assume also that $\\delta_i>2\\delta_i^2$ for every $i$.\n\n\nNow, suppose $(x_i)_{i=1}^\\infty\\in\\B_{\\Delta}\\cap {\\rm bb}_{E,\\Delta}(F_i)$ and let $(u_i)_{i=1}^\\infty$  be a normalised sequence in $E$ such that $\\norm{x_i-u_i}<\\delta_i$ for all $i$. We must show that $(u_i)_{i=1}^\\infty\\in \\B$, which will imply that $(x_i)_{i=1}^\\infty\\in {\\rm Int}_\\Delta(\\B)$.\n\n\nBy assumption on $(x_i)_{i=1}^\\infty$, we can find $(z_i)_{i=1}^\\infty\\in \\B$ and intervals $I_1<I_1<\\ldots$ such that $\\norm{x_i-z_i}<\\delta_i$  and $\\norm {{I_i}(x_i)-x_i}<\\delta_i$ for all $i$. Letting $y_i=\\frac{{I_i}(x_i)}{\\norm{{I_i}(x_i)}}$, we see that $(y_i)_{i=1}^\\infty$ is a normalised block sequence in $F$ and a simple calculation using $\\delta_i>2\\delta_i^2$ gives $\\norm{x_i-y_i}<4\\delta_i$, whence $\\norm{u_i-y_i}<5\\delta_i$ and $\\norm{z_i-y_i}<5\\delta_i$. It follows that \n$(u_i)_{i=1}^\\infty\\sim (y_i)_{i=1}^\\infty\\sim (z_i)_{i=1}^\\infty\\in \\B$ and so also $(u_i)_{i=1}^\\infty\\in \\B$.\n\\end{proof}\n\n\n\n\n\n\\begin{lemme}\\label{all blocks}\nSuppose $E$ is a subspace of a space $F$ with an F.D.D. $(F_i)_{i=1}^\\infty$ and $\\Theta\\searrow0$. Then there is $\\Gamma\\searrow0$ such that for any $M\\nearrow0$ and $(x_i)_{i=1}^\\infty\\in  {\\rm bb}_{E,\\Gamma, M}(F_i)$,\n\\begin{enumerate}\n\\item $(x_i)_{i=1}^\\infty$ is a normalised basic sequence, and\n\\item any normalised block sequence $(z_i)_{i=1}^\\infty$ of $(x_i)_{i=1}^\\infty$ belongs to ${\\rm bb}_{E,\\Theta,M}(F_i)$.\n\\end{enumerate}\n\\end{lemme}\n\n\n\\begin{proof}Let $K$ be the constant of the decomposition $(F_i)_{i=1}^\\infty$. As in the proof of Lemma \\ref{interior}, there is some $\\Lambda\\searrow0$ such that if $(x_i)_{i=1}^\\infty\\in {\\rm bb}_{E,\\Lambda}(F_i)$, as witnessed by a sequence of intervals $(I_i)_{i=1}^\\infty$, then\n$$\n(x_i)_{i=1}^\\infty\\sim_2\\Big(\\frac{I_ix_i}{\\norm{I_ix_i}}\\Big)_{i=1}^\\infty.\n$$\nLet now $\\Gamma\\searrow0$ be chosen such  that\n$12K^2\\sum_{i=m}^\\infty\\gamma_i<\\theta_m$ and $\\gamma_m<\\lambda_m$ for all $m$.\n\nNow suppose $(x_i)_{i=1}^\\infty\\in  {\\rm bb}_{E,\\Gamma, M}(F_i)$ for some $M\\nearrow \\infty$, as witnessed by a sequence of intervals\n$(I_i)_{i=1}^\\infty$. Then $(x_i)_{i=1}^\\infty\\in  {\\rm bb}_{E,\\Lambda}(F_i)$ and hence is $2$-equivalent to the normalised block basis $\\Big(\\frac{I_ix_i}{\\norm{I_ix_i}}\\Big)_{i=1}^\\infty$, whence $(x_i)_{i=1}^\\infty$ is itself a basic sequence.\n\n\nSuppose also that $z=\\sum_{i=n}^ma_ix_i$ is a block vector. We claim\nthat if we let $J=[\\min I_n,\\max I_m]$, then\n$$\n\\norm{Jz-z}<\\theta_n\\norm{z},\n$$\nwhich is enough to obtain condition (2). To see this, notice first that for $i=n,\\ldots m$,\n\\begin{displaymath}\\begin{split}\n\\norm{Jx_i-x_i}\n=&\\Norm{[1,\\min I_n-1](x_i)+[\\max I_m+1,\\infty[(x_i)}\\\\\n=&\\Norm{[1,\\min I_n-1](x_i-I_ix_i)+[\\max I_m+1,\\infty[(x_i-I_ix_i)}\\\\\n\\leqslant&\\Norm{[1,\\min I_n-1](x_i-I_ix_i)}+\\Norm{[\\max I_m+1,\\infty[(x_i-I_ix_i)}\\\\\n\\leqslant& K\\norm{x_i-I_ix_i}+2K\\norm{x_i-I_ix_i}\\\\\n<&3K\\gamma_i.\n\\end{split}\\end{displaymath}\nSince $\\Norm{P_{I_i}}\\leqslant 2K$ and $(x_i)_{i=1}^\\infty$ is $2$-equivalent to $\\Big(\\frac{I_ix_i}{\\norm{I_ix_i}}\\Big)_{i=1}^\\infty$, we have\n$$\n\\sup_{n\\leqslant i\\leqslant m}|a_i|\n=\\sup_{n\\leqslant i\\leqslant m}\\Norm{a_i\\frac{I_ix_i}{\\norm{I_ix_i}}}\n\\leqslant 2K\\Norm{\\sum_{i=n}^ma_i\\frac{I_ix_i}{\\norm{I_ix_i}}}\n\\leqslant 4K\\Norm{\\sum_{i=n}^ma_ix_i},\n$$\nand therefore\n\\begin{displaymath}\\begin{split}\n\\Norm{J(\\sum_{i=n}^ma_ix_i)-(\\sum_{i=n}^ma_ix_i)}\n=&\\Norm{\\sum_{i=n}^ma_i(Jx_i-x_i)}\\\\\n\\leqslant&\\sum_{i=n}^m|a_i|\\;\\norm{Jx_i-x_i}\\\\\n<& \\sup_{n\\leqslant i\\leqslant m}|a_i|\\cdot \\sum_{i=n}^m3K\\gamma_i\\\\\n\\leqslant& 12K^2\\Norm{\\sum_{i=n}^ma_ix_i}\\sum_{i=n}^m\\gamma_i\\\\\n\\leqslant&\\theta_n\\|\\sum_{i=n}^ma_ix_i\\|,\n\\end{split}\\end{displaymath}\nwhich shows that $\\norm{Jz-z}<\\theta_n\\norm z$.\n\\end{proof}\n\n\n\n\n\n\\begin{defi}\nGiven $\\Delta\\searrow0$, a $\\Delta$-{\\em block tree} $T$ is a non-empty tree on $S_E$ such that for all\n$(x_1,\\ldots,x_{n})\\in T$ the set\n$$\n\\{y\\in S_E\\del (x_1,\\ldots,x_{n},y)\\in T\\}\n$$\ncan be written as $\\{y_i\\}_{i=0}^\\infty$, where for each $i$ there is an interval $I_i\\subseteq \\N$ satisfying\n\\begin{itemize}\n\\item $\\norm{I_iy_i-y_i}<\\delta_{n+1}$,\n\\item $\\min I_i\\Lim{i\\rightarrow \\infty}\\infty$.\n\\end{itemize}\n\\end{defi}\n\nNow, an easy inductive construction shows that any $\\Delta$-block tree $T$ contains a subtree $T'\\subseteq T$ such that any infinite branch in $T'$ is a $\\Delta$-block sequence, i.e., $[T']\\subseteq {\\rm bb}_{E,\\Delta}(F_i)$.\nSo, without loss of generality, we can always assume that any $\\Delta$-block tree satisfies this additional hypothesis.\n\n\n\n\n\nWe recall the following result from \\cite{IAG}, which is proved using infinite-dimensional Ramsey theory. A similar statement for closed sets was proved earlier by Odell and Schlumprecht in \\cite{odell}.\n\n\n\\begin{thm}\\label{strategy for I}\nLet $\\B\\subseteq S_E^\\infty$ be a  coanalytic set. Then the following\nare equivalent.\n\\begin{enumerate}\n\\item $\\e \\Delta\\searrow0\\;\\;\\; \\e M\\nearrow\\infty\\quad  {\\rm bb}_{E,\\Delta,M}(F_i)\\subseteq   {\\rm Int}_\\Delta(\\B)$,\n\\item $\\e \\Delta\\searrow0$  such that any $\\Delta$-block tree has a branch in ${\\rm Int}_\\Delta(\\B)$.\n\\end{enumerate}\n\\end{thm}\n\n\n\n\n\n\\begin{defi}\nA {\\em weakly null tree} is a tree $T$ on $S_E$ such that, for any $(x_1,\\ldots,x_{n})\\in T$, the set\n$$\n\\{y\\in S_E\\del (x_1,\\ldots,x_{n},y)\\in T\\}\n$$\ncan be written as $\\{y_i\\}_{i=1}^\\infty$ for some weakly null sequence $(y_i)_{i=1}^\\infty$.\n\\end{defi}\n\n\n\n\n\nWe recall also a statement from \\cite{IAG} that sums up some of the elements of the construction of Odell and Schluprecht from \\cite{odell}\nthat we shall use in the following.\n\n\n\n\n\\begin{prop}\\label{wbjohnson}\nLet $E$ be a separable reflexive Banach space. Then there is a reflexive Banach space\n$F\\supseteq E$ having an F.D.D. $(F_i)_{i=1}^\\infty$  and a constant $c>1$ such\nthat whenever $\\Delta\\searrow0$ and $T$ is a $\\Delta$-block\ntree in $S_E$ with respect to $(F_i)_{i=1}^\\infty$, there is a weakly null tree $S$ in $S_E$ such that\n$$\n[S]\\subseteq [T]_{\\Delta c}\\quad\\&\\quad[T]\\subseteq [S]_{\\Delta c}.\n$$\n\\end{prop}\n\n\n\nWe can now assemble the above results into the following general lemma.\n\n\\begin{lemme}\\label{basic}\nSuppose $E$ is a separable reflexive Banach space and $\\B\\subseteq S_E^\\infty$ is a coanalytic set, invariant under equivalence, such that any weakly null tree on $S_E$ has a branch in $\\B$. \nThen there are $\\Gamma\\searrow0$, $M\\nearrow \\infty$ and  a reflexive space  $F\\supseteq E$ with an F.D.D. $(F_i)_{i=1}^\\infty$ such that  any element of ${\\rm bb}_{E,\\Gamma, M}(F_i)$ is a  basic sequence all of whose normalised block sequences belong to  $\\B$.\n\\end{lemme}\n\n\n\n\\begin{proof}\nPick first, by Proposition \\ref{wbjohnson}, a space $F$\ncontaining $E$ with a shrinking F.D.D. $(F_i)_{i=1}^\\infty$ and a constant $c>1$ such that,\nfor any $\\Delta\\searrow0$ and $\\Delta$-block tree $T$ in $E$, \nthere is a weakly null tree $S$ in $E$ with\n\\begin{equation}\\label{close}\n[S]\\subseteq [T]_{\\Delta c}\\quad\\&\\quad[T]\\subseteq [S]_{\\Delta c}.\n\\end{equation}\nChoose also, by Lemma \\ref{interior}, some  $\\Delta\\searrow0$ such that\n$$\n\\B_{\\Delta c}\\cap {\\rm bb}_{E,\\Delta c}(F_i)\\subseteq {\\rm Int}_{\\Delta c}(\\B).\n$$\n\nWe claim that any $\\Delta$-block tree has a branch in ${\\rm Int}_\\Delta(\\B)$. To see this, suppose\n$T$ is a $\\Delta$-block tree and assume without loss of generality that  $[T]\\subseteq {\\rm bb}_{E,\\Delta}(F_i)\\subseteq {\\rm bb}_{E,\\Delta c}(F_i)$.\nPick also a  weakly null tree $S$ satisfying (\\ref{close}). Then, as $[S]\\cap \\B\\neq \\tom$, also \n$$\n\\tom\\neq [T]\\cap \\B_{\\Delta c}\\subseteq[T]\\cap  {\\rm bb}_{E,\\Delta c}(F_i)\\cap \\B_{\\Delta c}\\subseteq [T]\\cap {\\rm Int}_{\\Delta c}(\\B)\\subseteq [T]\\cap {\\rm Int}_{\\Delta}(\\B),\n$$\nshowing that $T$ has a branch in ${\\rm Int}_\\Delta(\\B)$.\n\n\nApplying Theorem \\ref{strategy for I}, we find some $\\Theta\\searrow0$ and $M\\nearrow\\infty$ such that ${\\rm bb}_{E,\\Theta,M}(F_i)\\subseteq   {\\rm Int}_\\Theta(\\B)\\subseteq \\B$ and, applying Lemma \\ref{all blocks}, the statement follows.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Killing the overlap}\n\n\n\nThe next proposition is Corollary 4.4 in \\cite{odell}, except that condition (5) is not listed in the statement of the corollary. However, it can easily be gotten from the proof, provided that one chooses, in the notation of the paper, $\\epsilon_i<\\delta_i$.\n\n\\begin{prop}\\label{overlap}\nSuppose $F$ is a reflexive space with an F.D.D. $(H_i)_{i=1}^\\infty$, $E\\subseteq F$ is a subspace and $\\Sigma\\searrow0$. Then there are integers $0=a_0<a_1<\\ldots$ such that for all $x\\in S_E$ there are a sequence $(x_i)_{i=1}^\\infty$ in $E$, a subset\n$D\\subseteq \\N$ and numbers $a_{i-1}<b_i\\leqslant a_i$, $b_0=0$, satisfying the following five conditions.\n\\begin{enumerate}\n  \\item $x=\\sum_{i=1}^\\infty x_i$,\n  \\item $\\a i\\notin D\\; \\norm{x_i}<\\sigma_i$,\n  \\item $\\a i\\in D\\;\\Norm{[H_{b_{i-1}+1}\\oplus\\ldots\\oplus H_{b_i-1}]x_i-x_i}<\\sigma_i\\|x_i\\|$,\n  \\item $\\a i\\;\\Norm{[H_{b_{i-1}+1}\\oplus\\ldots\\oplus H_{b_i-1}]x-x_i}<\\sigma_i$,\n  \\item $\\a i\\; \\norm{H_{b_i}x}<\\sigma_i$.\n\\end{enumerate}\n\\end{prop}\n\n\n\n\n\nCombining Lemma \\ref{basic} and Proposition \\ref{overlap}, we are now in a position to prove our main result, Theorem \\ref{ros}.\n\n\n\n\n\n\\begin{thm}\nSuppose that $E$ is a separable reflexive Banach space such that any weakly\nnull tree in $E$ has a branch $(x_i)_{i=1}^\\infty$ satisfying $(x_{2i-1})_{i=1}^\\infty\\sim(x_{2i})_{i=1}^\\infty$. Then $E$\nembeds into a reflexive space with an unconditional Schauder basis having the shift property.\n\\end{thm}\n\n\n\\begin{proof}Applying Lemma \\ref{basic} to the set \n$$\n\\B=\\big\\{(x_i)_{i=1}^\\infty\\in S_E^\\infty\\del (x_{2i-1})_{i=1}^\\infty\\sim(x_{2i})_{i=1}^\\infty\\big\\},\n$$\nwe find $\\Gamma\\searrow0$, $M\\nearrow \\infty$ and a reflexive space  $F\\supseteq E$ with an F.D.D. $(F_i)_{i=1}^\\infty$ such that  any element of ${\\rm bb}_{E,\\Gamma, M}(F_i)$ is a  basic sequence all of whose normalised block sequences $(y_i)_{i=1}^\\infty$ satisfy $(y_{2i-1})_{i=1}^\\infty\\sim(y_{2i})_{i=1}^\\infty$.\n\nWe claim that there is a constant $C\\geqslant 1$ such that $(y_{2i-1})_{i=1}^\\infty\\sim_C(y_{2i})_{i=1}^\\infty$ for any such normalised block basis $(y_i)_{i=1}^\\infty$. For if not, then,  by concatenating finite bits of sequences, we would be able to produce  some $(u_i)_{i=1}^\\infty \\in {\\rm bb}_{E,\\Gamma, M}(F_i)$ and a normalised block sequence $(y_i)_{i=1}^\\infty$  of $(u_i)_{i=1}^\\infty$ failing \n$(y_{2i-1})_{i=1}^\\infty\\sim(y_{2i})_{i=1}^\\infty$, which is impossible.\n\n\nSince it suffices to prove the conclusion of the theorem for a cofinite-dimensional subspace of $E$, by considering the cofinite-dimensional subspaces $F_{m_1+1}\\oplus F_{m_1+2}\\oplus F_{m_1+3}\\oplus\\ldots $ and $E\\cap \\big(F_{m_1+1}\\oplus F_{m_1+2}\\oplus F_{m_1+3}\\oplus\\ldots\\big)$ of respectively $F$ and $E$, we can, without loss of generality, assume that $m_1=0$ and thus not worry about the initial offset by $m_1$ in the definition of $M$-separation (cf. Definition \\ref{delta-block}).\n\n\n\nPick $(u_i)_{i=1}^\\infty\\in {\\rm bb}_{E,\\Gamma,M}(F_i)$. Then, for any choice of signs $\\epsilon_i\\in\\{-1,1\\}$, also $(\\epsilon_iu_i)_{i=1}^\\infty\\in {\\rm bb}_{E,\\Gamma,M}(F_i)$ and hence $(\\epsilon_{2i-1}u_{2i-1})_{i=1}^\\infty\\sim(\\epsilon_{2i}u_{2i})_{i=1}^\\infty$. It follows that $(u_{2i-1})_{i=1}^\\infty$ is a basic sequence, equivalent to $(\\epsilon_{2i}u_{2i})_{i=1}^\\infty$ for any choice of signs $\\epsilon_i\\in \\{-1,1\\}$, and thus must be unconditional. \n\nNow $[u_{2i-1}]_{i=1}^\\infty$ can be equivalently renormed so that $(u_{2i-1})_{i=1}^\\infty$ is $1$-unconditional and, by a result of A. Pe\\l czy\\'nski \\cite{pelczynski}, this renorming extends to an equivalent renorming of $F$. So, without loss of generality, we shall assume that $(u_{2i-1})_{i=1}^\\infty$ is $1$-uncondi\\-tio\\-nal and has the shift property with some constant $C$.\nMoreover, as $E$ is reflexive, it follows by a theorem of R. C. James (Theorem 3.2.13 in \\cite{albiac}) that $(u_{2i-1})_{i=1}^\\infty$ is both shrinking and boundedly complete.\n\nWe let $v_i=u_{2i+1}$, whence $(v_i)_{i=1}^\\infty$ is the subsequence of $(u_{2i-1})_{i=1}^\\infty$ omitting the first term.\nChoose also $\\sigma_i<\\gamma_{2i-1}$ such that $\\sum_{i=1}^\\infty\\sigma_i<\\frac 1{24KC^2}$, , where $K$ denotes the constant of the decomposition $(F_i)_{i=1}^\\infty$.\n\n\n\nSince $(u_i)_{i=1}^\\infty$ is an $M$-separated $\\Gamma$-block sequence,  $\\N$ can be partitioned into successive finite intervals\n$$\nL_1<I_1<R_1\\;\\;\\;<\\;\\;\\;L_2<I_2<R_2\\;\\;\\;<\\;\\;\\;L_3<I_3<R_3\\;\\;\\;<\\ldots\n$$\nsuch that\n\\begin{itemize}\n\\item[(a)] $\\norm {I_i(v_i)-v_i}<\\gamma_{2i+1}$,\n\\item[(b)] for every $i>1$ there is a $j$ such that $[m_j,m_{j+1}]\\subseteq L_i$\n\\item[(c)] and for every $i$ there is a $j$ such that $[m_j,m_{j+1}]\\subseteq R_i$.\n\\end{itemize}\nMoreover, for \n$$\nH_i=\\sum_{j\\in L_i\\cup I_i\\cup R_i}F_j,\n$$\nlet $(a_i)_{i=0}^\\infty$ be given as in Proposition \\ref{overlap} and set\n$$\nA_i=H_{a_{i-1}+1}\\oplus\\ldots\\oplus H_{a_i}.\n$$\nWe define a new norm $\\triple\\cdot$ on ${\\rm span}(\\bigcup_{i=1}^\\infty A_i)$ by setting\n$$\n\\triple y=\\Big\\|\\sum_{i=1}^\\infty \\|A_iy\\|v_{a_i}\\Big\\|.\n$$\nSince $(v_i)_{i=1}^\\infty$ is  $1$-unconditional,  $\\triple\\cdot$ is\nindeed a norm and we can therefore consider the completion $V=\\overline{\\rm span}^{\\triple\\cdot}\\big(\\bigcup_{i=1}^\\infty A_i\\big)$.\nMoreover, we claim that the mapping \n$$\nT\\colon x\\in E\\mapsto \\sum_{i=1}^\\infty A_ix\\in V\n$$\nis a well-defined isomorphic embedding of $E$ into $V$.\n\n\n\n\nTo see this, suppose $x\\in S_E$ is fixed and let $(x_i)_{i=1}^\\infty$, $(b_i)_{i=0}^\\infty$ and $D\\subseteq \\N$\nbe given as in Proposition \\ref{overlap}. Let also\n$$\nB_i=H_{b_{i-1}+1}\\oplus\\ldots\\oplus H_{b_i-1}.\n$$\nThen the decomposition $F=F_1\\oplus F_2\\oplus F_3\\oplus\\ldots$ blocks as\n\\[\\begin{split}\nF=&A_1\\oplus A_2\\oplus A_3\\oplus\\ldots\\\\\n=&B_1\\oplus H_{b_1}\\oplus B_2\\oplus H_{b_2}\\oplus B_3\\oplus H_{b_3}\\oplus\\ldots,\n\\end{split}\\]\nwhere, moreover, \n$$\nA_i\\;\\subseteq\\; B_i\\oplus H_{b_i}\\oplus B_{i+1}\n$$\nand, letting $A_0$ be the trivial space $\\{0\\}$,\n$$\nB_{i}\\;\\subseteq\\; A_{i-1}\\oplus A_{i}.\n$$\nIt follows that with respect to the ordering of the original decomposition $(F_i)_{i=1}^\\infty$, we have\n\\begin{equation}\nB_1<L_{b_1}<I_{b_1}<R_{b_1}<B_2<L_{b_2}<I_{b_2}<R_{b_2}<B_3<\\ldots.\n\\end{equation}\nNow, by condition (4) of Proposition \\ref{overlap},\n$$\n\\big | \\|B_ix\\|-\\|x_i\\|\\big|\\leqslant \\|B_ix-x_i\\|<\\sigma_i,\n$$\nand so, using condition (5) of Proposition \\ref{overlap}, we have\n\\begin{displaymath}\\begin{split}\n\\|A_ix\\|\n\\leqslant&2K\\Norm{[B_i\\oplus H_{b_i}\\oplus B_{i+1}]x}\\\\\n\\leqslant& 2K \\big(\\|B_ix\\|+\\norm{H_{b_i}x}+\\|B_{i+1}x\\|\\big)\\\\\n<& 2K \\big(\\|x_i\\|+\\|x_{i+1}\\|+3\\sigma_i\\big).\n\\end{split}\\end{displaymath}\nNote also that\n$$\n\\|x_i\\|\\leqslant \\|B_ix\\|+\\sigma_i\\leqslant 2K\\|A_{i-1}x\\|+2K\\|A_ix\\|+\\sigma_i,\n$$\nand, by condition (3) of Proposition \\ref{overlap}, for any $i\\in D$, we have\n$$\n\\Norm{B_ix_i-x_i}<\\sigma_i\\norm {x_i}<\\gamma_{2i-1}\\norm {x_i}.\n$$\nList now $D$ increasingly as $D=\\{d_1,d_2,d_3,\\ldots\\}$ and note that, as $2i< 2b_{d_i}+1$,\n$$\n\\norm{I_{b_{d_i}}(v_{b_{d_i}})-v_{b_{d_i}}}<\\gamma_{2b_{d_i}+1}\\leqslant  \\gamma_{2i}.\n$$\nTherefore, by the ordering (2) above, we see that\n$$\n\\Big(\\frac{x_{d_1}}{\\norm{x_{d_1}}},v_{b_{d_1}},\\frac{x_{d_2}}{\\norm{x_{d_2}}},v_{b_{d_2}},\\frac{x_{d_3}}{\\norm{x_{d_3}}},v_{b_{d_3}},\\frac{x_{d_4}}{\\norm{x_{d_4}}},\\ldots\\Big)\n$$\nis an $M$-separated $\\Gamma$-block sequence, as witnessed by the sequence of interval projections\n$$\nB_{d_1},I_{b_{d_1}},B_{d_2},I_{b_{d_2}},B_3,I_{b_{d_3}},B_4,\\ldots,\n$$\nand hence $\\big(\\frac{x_{i}}{\\norm{x_{i}}}\\big)_{i\\in D}\\sim_C\\big(v_{b_i}\\big)_{i\\in D}$.\nFurthermore, as $(v_i)_{i=1}^\\infty$ has the shift property with constant $C$ and $b_1\\leqslant a_1<b_2\\leqslant a_2<\\ldots$, we have \n\\begin{equation}\\label{equiv}\n(v_{b_{i+1}})_{i=1}^\\infty\\sim_C(v_{b_i})_{i=1}^\\infty\\sim_C(v_{a_i})_{i=1}^\\infty\\sim_C(v_{a_{i+1}})_{i=1}^\\infty\n\\end{equation}\nand therefore $\\big(\\frac{x_{i}}{\\norm{x_{i}}}\\big)_{i\\in D}\\sim_{C^2}\\big(v_{a_i}\\big)_{i\\in D}$. Since now $\\sum_{i=1}^\\infty x_i$ converges and $\\sum_{i\\notin D}\\norm{x_i}<\\sum_{i\\notin D}\\sigma_i<\\infty$, it follows that also $\\sum_{i=1}^\\infty \\norm{x_i}v_{a_i}$ and  $\\sum_{i=1}^\\infty \\norm{x_{i+1}}v_{a_i}$ converge. Since $\\norm{A_ix}\\leqslant2K \\big(\\|x_i\\|+\\|x_{i+1}\\|+3\\sigma_i\\big)$ and $(v_i)_{i=1}^\\infty$ is unconditional,  we finally see that the sum $\\sum_{i=1}^\\infty \\norm{A_ix}v_{a_i}$ converges and hence that $Tx=\\sum_{i=1}^\\infty A_ix\\in V$ is well-defined.\n\nBy the same mode of reasoning, one verifies the following sequence of inequalities.\n\\begin{displaymath}\\begin{split}\n\\|x\\|\n=&\\Norm{\\sum_{i=1}^\\infty x_i}\\\\\n\\leqslant&\\Big\\|\\sum_{i\\in D}\\|x_i\\|\\frac{x_i}{\\|x_i\\|}\\Big\\|+\\Norm{\\sum_{i\\notin D} x_i}\\\\\n\\leqslant&C\\Big\\|\\sum_{i\\in D}\\|x_i\\|v_{b_i}\\Big\\|+\\sum_{i\\notin D}\\|x_i\\|\\\\\n\\leqslant&C^2\\Big\\|\\sum_{i=1}^\\infty \\|x_i\\|v_{a_i}\\Big\\|+\\sum_{i\\notin D}\\sigma_i\\\\\n\\leqslant&C^2\\Big\\|\\sum_{i=1}^\\infty \\big(2K\\|A_{i-1}x\\|+2K\\|A_ix\\|+\\sigma_i\\big)v_{a_i}\\Big\\|+\\frac14\\\\\n\\leqslant&2KC^2\\big\\|\\sum_{i=1}^\\infty \\|A_{i-1}x\\|v_{a_{i}}\\big\\|+2KC^2\\big\\|\\sum_{i=1}^\\infty \\|A_ix\\|v_{a_i}\\big\\|+C^2\\sum_{i=1}^\\infty \\sigma_i+\\frac14\\\\\n\\leqslant&2KC^2(C+1)\\big\\|\\sum_{i=1}^\\infty \\|A_ix\\|v_{a_i}\\big\\|+\\frac12\\\\\n\\leqslant&4KC^3\\triple{Tx}+\\frac12.\n\\end{split}\\end{displaymath}\nThus, as $\\|x\\|-\\frac 12=\\frac12\\|x\\|$, we have $\\|x\\|\\leqslant 8KC^3\\triple{Tx}$. \nA similar argument shows that $\\triple{Tx}\\leqslant 5KC^3\\norm{x}$, whereby $T$ is an isomorphic embedding of $E$ into $V$.\n\n\n\nWe shall now show how to embed $V$ into a space with a basis having the block property, which will finish the proof of the theorem.\nFirst, to simplify notation, we let $w_i=v_{a_i}$. Fix also  $k_i\\geqslant 1$ such that $A_i$ embeds\nwith constant $2$ into $Z_i=\\ell_\\infty^{k_i}$. Then $V$ clearly embeds with\nconstant $2$ into $Z=\\sum_{i=1}^\\infty Z_i$ equipped with the norm\n$$\n\\triple{y}'=\\Big\\|\\sum_{i=1}^\\infty\\|Z_iy\\|w_i\\Big\\|.\n$$\nMoreover, since $(w_i)_{i=1}^\\infty=(v_{a_i})_{i=1}^\\infty$ is both shrinking and boundedly complete, $Z$ is\nreflexive.\n\nFor each $i$, we let $(e_1^i,e_2^i,\\ldots,e_{k_i}^i)$ be the standard unit\nvector basis for $\\ell_\\infty^{k_i}$. Then\n$$\n(f_i)_{i=1}^\\infty=(e_1^1,e_2^1,\\ldots,e_{k_1}^1,e_1^2,e_2^2,\\ldots,e_{k_2}^2,\\ldots)\n$$\nis a $1$-unconditional basis for $Z$, which we claim has the block property.\nTo see this, suppose $(y_i)_{i=1}^\\infty$ is a normalised\nblock sequence of $(f_i)_{i=1}^\\infty$ and set $r_i=\\min {\\rm supp}\\; y_i$ and\n$$\ni\\in A\\Leftrightarrow \\e j\\;\\; y_i\\in Z_j.\n$$\nNotice that for all $j$ there are at most two distinct $i\\notin A$ such that\n$Z_jy_i\\neq 0$. We can therefore split $\\sim A$ into two sets $B$ and $D$ such that\nfor all $j$ there is at most one $i$ from each of $B$ and $D$ such that\n$Z_jy_i\\neq 0$. By unconditionality, it is enough to show that $(y_i)_{i\\in\nA}\\sim(f_{r_i})_{i\\in A}$, $(y_i)_{i\\in B}\\sim(f_{r_i})_{i\\in B}$ and\n$(y_i)_{i\\in D}\\sim(f_{r_i})_{i\\in D}$. Since the cases $B$ and $D$ are\nsimilar, let us just do $A$ and $B$.\n\n\nFor each $i\\in B$, let $n_i$ and $m_i$ be respectively the minimal and maximal $j$ such that $Z_jy_i\\neq0$, whence\n$y_i=Z_{n_i}y_i+\\ldots+Z_{m_i}y_i$ and $n_i<m_i<n_j<m_j$ for $i<j$ in $B$. In particular, this means that if\n$$\nz_i=\\sum_{j=n_i}^{m_i}\\|Z_jy_i\\|w_{j},\n$$\nthen $(z_i)_{i\\in B}$ is a block sequence of $(w_i)_{i=1}^\\infty$ and\n$$\n\\norm{z_i}=\\Norm{\\sum_{j=n_i}^{m_i}\\|Z_jy_i\\|w_{j}}=\\triple{y_i}'=1.\n$$ \nAs $(w_i)_{i=1}^\\infty$ has the shift property, this means that $(z_i)_{i\\in B}\\sim(w_{n_i})_{i\\in B}\\\n\\sim(f_{r_i})_{i\\in B}$.\nOn the other hand, if $(\\lambda_i)_{i=1}^\\infty\\in c_{00}$, then\n\\begin{displaymath}\\begin{split}\n\\Triple{\\sum_{i\\in B}\\lambda_iy_i}' \n=& \\Big\\|\\sum_{i\\in B}\\sum_{j=n_i}^{m_i}\\|Z_j\\lambda_iy_i\\|w_{j}\\Big\\|\\\\\n=& \\Big\\|\\sum_{i\\in B}|\\lambda_i|\\sum_{j=n_i}^{m_i}\\|Z_jy_i\\|w_{j}\\Big\\|\\\\\n=& \\Big\\|\\sum_{i\\in\nB}|\\lambda_i|z_i\\Big\\|.\n\\end{split}\\end{displaymath}\nSince $(z_i)_{i\\in B}$ is unconditional, it follows that $(y_i)_{i\\in\nB}\\sim(z_i)_{i\\in B}\\sim(f_{r_i})_{i\\in B}$.\n\nWe now partition $A$ into finite sets $a_j$ by setting\n$$\ni\\in a_j\\Leftrightarrow y_i\\in Z_j.\n$$\nThen for all $(\\lambda_i)_{i=1}^\\infty\\in c_{00}$\n\\begin{displaymath}\\begin{split}\n\\Triple{\\sum_{i\\in A}\\lambda_iy_i}' \n=& \\Big\\|\\sum_{j=1}^\\infty\\Norm{\\sum_{i\\in a_j}\\lambda_iy_i}w_{j}\\Big\\|\\\\\n=& \\Big\\|\\sum_{j=1}^\\infty\\;\\big(\\sup_{i\\in a_j}|\\lambda_i|\\big)w_{j}\\Big\\|\\\\\n=& \\Big\\|\\sum_{j=1}^\\infty\\Norm{\\sum_{i\\in a_j}\\lambda_if_{r_i}}w_{j}\\Big\\|\\\\\n=&\\Triple{\\sum_{i\\in A}\\lambda_if_{r_i}}'.\n\\end{split}\\end{displaymath}\nSo $(y_i)_{i\\in A}\\sim(f_{r_i})_{i\\in A}$, which finishes the proof.\n\\end{proof}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLearned image compression (LIC) has outperformed traditional methods like JPEG~\\cite{jpeg} and BPG~\\cite{bpg} in terms of PSNR and MS-SSIM. Its outstanding rate-distortion performance mostly derives from jointly training all parameters instead of handcrafted and independent module design. Hyperprior and context frameworks~\\cite{imagecnn5, imagecnn7, minnen2020channel, elic} dramatically improve compression gain across all ranges of bit rates. However, traditional methods have their own merits for industrial applications including smaller memory footprint, less computational complexity and hardware-friendly characteristics. LIC has an extremely high demand for hardware resources. Large model size and intensive computation hinder real-world deployment of LIC. Furthermore, correct decoding across platforms is not guaranteed due to the non-determinism of model inference and error can propagate catastrophically in entropy decoding~\\cite{balle2018integer}. All these obstacles restrict commercial deployment of LIC even though it obtains a better trade-off between bit rate and PSNR or MS-SSIM. In this paper, we report the performance of our optimized LIC implementation, which outperforms the fastest industrial-level codec implementation, i.e., jpeg-turbo~\\cite{jpegturbo} regarding rate-distortion performance and visual quality while achieving comparable speed. Compared with the most competitive next-generation image codec JPEG XL~\\cite{alakuijala2019jpeg}, our model shows superior compression performance with several times of advantage regarding inference speed.\n\nTo obtain a practical LIC model, we apply neural architecture search (NAS) and quantization to make LIC lighter and faster. NAS has been proved useful in finding efficient models for high-level vision tasks, such as object detection~\\cite{chen2019detnas},  recognition~\\cite{ding2020autospeech} and segmentation~\\cite{cubuk2019autoaugment}, while researches on NAS for low-level tasks such as denoising, super-resolution and image compression are still in early stages. In this paper, we aim to design a more efficient network and hardware-friendly architecture for LIC to enable its industrial deployment. Hardware-aware NAS is dedicated to designing promising networks for various devices.\n\nBesides, quantization has exhibited promising results for decreasing model size as well as accelerating inference. Methods for neural network quantization can be divided into two categories of quantization-aware training (QAT) and post-training quantization (PTQ). We exploit quantization not only to accelerate the inference of LIC but also to ensure the correctness of cross-platform decoding~\\cite{balle2018integer}. \n\nThe paper is organized as follows. Section \\uppercase\\expandafter{\\romannumeral2} introduces related works. Section \\uppercase\\expandafter{\\romannumeral3} gives an overview of our architecture and pipeline. Section \\uppercase\\expandafter{\\romannumeral4} describes the details of our method. Experimental results and discussion are given in section \\uppercase\\expandafter{\\romannumeral5}, and conclusion is drawn in section \\uppercase\\expandafter{\\romannumeral6}.\n\n\\section{Related Work}\n\\subsection{NAS}\nSearch space, optimization methods and search strategy are the major components of NAS. In this paper, we follow the framework of BigNAS~\\cite{yu2020bignas}, which only trains one big model and searches efficient sub-networks for direct deployment without additional training. For optimization methods, Yu \\emph{et al.}~\\cite{yu2018slimmable, yu2019universally} propose a slimmable neural network with parameter sharing which can adaptively prune output channels with little performance degradation. HyperNet~\\cite{ha2016hypernetworks} trains an auxiliary network to generate weight parameters for the trainable model. For search strategy,  an effective evolution method~\\cite{real2019regularized}  is proposed to obtain architecture with higher accuracy, and HyperBand~\\cite{li2017hyperband} can assess candidate neural architectures to avoid redundant training which results in worse performance.  \n\n\\begin{figure}[tb]\n    \\centering\n    \\includegraphics[width=0.99\\linewidth]{imgs\/framework.pdf}\n    \\caption{Framework of our architecture. $x, \\hat{x}$ are the original image and the reconstructed one. $y, \\hat{y}, z, \\hat{z}$ are latent variables. $\\mu, \\sigma$ are mean and variance values of Gaussian distribution. $Q$ represents quantization. \\textit{AE} and \\textit{AD} are the arithmetic encoding and decoding. The blue blocks are (transposed) convolutional layers controlled by \\textit{supernet} in Fig.~\\ref{fig:supernet}, and $k5s2$ means $5 \\times 5$ kernel with stride $2$. \\textit{ReLU, ReLU6} are the activation functions.}\n    \\label{fig:framework}\n\\end{figure}\n\n\\subsection{Quantization}\n\nQuantization-Aware Training (QAT)~\\cite{choi2018pact, gong2019differentiable, esser2019learned} and Post-Training Quantization (PTQ)~\\cite{nagel2020up, li2021brecq, wei2022qdrop} are two major categories of model quantization. QAT has attracted much attention because it compensates for the quantization error in an end-to-end optimization manner and achieves relatively better performance even in extremely low bit-width scenarios. Besides speeding up the model inference, the quantization techniques are also used in LIC for another purpose: eliminating the cross-platform inconsistency. Ball{\\'e} \\emph{et al.}~\\cite{balle2018integer} first introduces a QAT-like integer-only arithmetic technique to determinize the hyperprior model. Sun \\emph{et al.}~\\cite{sun2021learned} instead adopts PTQ-based approaches and He \\emph{et al.}\\cite{he2022post} extends the deterministic inference to more elaborate joint autoregressive models.\n\n\n\\section{Architecture}\nOur framework is shown in Fig.~\\ref{fig:framework}. We follow the pipeline in ~\\cite{imagecnn5} but use less complex activation functions and supernet to search the parameters of convolution layers, whose weight and bias are generated by supernet in Fig.~\\ref{fig:supernet}. \n\n\\begin{figure}[tb]\n    \\centering\n    \\includegraphics[width=0.95\\linewidth]{imgs\/supernet.pdf}\n    \\caption{Supernet of our NAS architecture. $w, b$ are the weight and bias parameters. The notation $kmsn$ means $m \\times m$ kernel with stride $n$. \\textit{Search Space} is the combination of parameters for channels. \\textit{Linear} is the linear convolutional layer. \\textit{GeLU, Softmax} are the activation functions where $GeLU(x)=0.5 x(1+\\tanh{[\\sqrt{\\frac{\\pi}{2}} (x+0.004715 x^3)]})$, and $Softmax(x_i)=\\frac{e^{x_i}}{\\sum^l_{i=0} e^{x_i}}$, $l$ is the length of input vector $\\mathrm{x}$. The blue block is the convolution adopted in Fig.~\\ref{fig:framework}. }\n    \\label{fig:supernet}\n\\end{figure}\n\n\\section{Methodology}\n\n\\subsection{Conditional and Hardware-aware NAS}\nFirstly we define the search space. The minimal and maximal numbers of parameters to be searched are given empirically, and the search stride is predefined for each convolutional layer. BigNAS can train several models at the same time, and the sandwich rule with combinations of ${min, random, random, max}$ is embedded to guarantee the effectiveness of sub-models, where ${min, random}$ and ${max}$ denote the sub-models with minimal, random and maximal number of parameters. After training, we can directly deploy sub-models for validation. \n\nFor each convolutional layer in Fig.~\\ref{fig:framework}, weight and bias parameters are generated by supernet in Fig.~\\ref{fig:supernet}. A group of weights from the initial transposed convolution or convolution layers will be combined by \\textit{concat} operation. Then the dot product of concatenated tensor and latent variables extracted from search space parameters by multiple layer perception (MLP), which consists of a single \\textit{Linear} layer and a \\textit{Softmax} layer, will be calculated to generate weight. Meanwhile, bias will be directly generated by latent variables from MLP, which consists of two \\textit{Linear} layers and a \\textit{GeLU} activation layer. Then the original parameters are replaced with generated weight and bias for training and testing. \n\nDuring the searching process, we balance the rate-distortion performance and computational complexity by taking FLOPS and inference latency into consideration. FLOPS is calculated to restrict the model size and a latency lookup table obtained from the statistical characteristics of various operations on specific hardware is used to constrain the inference latency. \n\n\\subsection{Other Optimization}\\label{otheropt}\nEfforts have been made in engineering, resulting in a highly optimized software implementation. We leverage multi-threading and instruction-level parallelism to accelerate our implementation, reaching throughput higher than the reciprocal of latency. For entropy coding, multi-threading and AVX-512 are also used for acceleration.\n\n\\section{Experiments}\n\\subsection{Setup}\\label{setup}\nWe train our model with 8000 images picked from ImageNet~\\cite{deng2009imagenet} dataset and randomly crop each image to $256\\times256$. For evaluation, we use 24 Kodak~\\cite{1993kodak} images.\n\nFor training settings, we set learning rate to $1e-4$ and train the network for 2000 epochs. We follow~\\cite{minnen2020channel} and use the mixed quantization trick to train our network. Channel numbers in main and hyper networks are obtained with NAS. In terms of quantization experiments, we apply LSQ~\\cite{esser2019learned} to quantize all layers in our network following the general quantization rules of NVIDIA's TensorRT Framework. We use the full precision model to initialize weights. Then we train the network for 400 epochs with learning rate set to $1e-4$ and another 100 epochs with learning rate decreased to its $\\frac{1}{10}$. \n\nFor traditional codecs, we measure the latency for executing the whole pipeline including disk i\/o and png format conversion for a fair comparison.\nFor JPEG XL~\\cite{alakuijala2019jpeg}, we tune the distance parameter for rate-distortion trade-off and leave other parameters as default. For AVIF, we test with AOM codec by adjusting its quantization step tuned by PSNR unless otherwise specified. \n\nTo test the average latency of our LIC model and traditional codecs, we use 100 1080p images captured from a video and measure its average processing time. For a fair comparison, when comparing latency between LIC model and jpeg-turbo~\\cite{jpegturbo}, we test these images saved in \\emph{bmp} file format while we use images saved in \\emph{png} format when comparing LIC model, JPEG XL and AVIF. Unless otherwise mentioned, \nwe test on a Intel(R) Core(TM) i9-10900X CPU @ 3.70GHz and a Tesla T4 GPU for 1080p dataset with png format.   \n\\subsection{Rate Distortion Performance}\n\\begin{figure}[tb]\n\\centering\n    \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/bpp_psnr_avif_ssim.pdf}%\n        \\caption{BPP-PSNR curves of codecs}\n        \\label{fig:bpp-psnr}\n    \\end{subfigure}\n    \\\\\n    \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/bpp_msssim_avif_ssim.pdf}%\n        \\caption{BPP-MS-SSIM curves of codecs}\n        \\label{fig:bpp-msssim}\n    \\end{subfigure}\n    \\caption{Performance of our LIC model compared with traditional codecs with respect to bpp v.s. psnr and bpp v.s. ms-ssim. ``avif-0.10.0 (ssim)\" denotes AVIF tuned by ssim.}\n\\end{figure}\nAs shown in Fig.~\\ref{fig:bpp-psnr}, our LIC model achieves superior performance with a large margin regarding BPP-PSNR compared with jpeg-turbo, especially in low bit rates. It also defeats JPEG XL at low bit rates. Its performance is still comparable to JPEG XL when bpp exceeds 1.0. For AVIF, our LIC model is comparable at low bit rates but lags as bit rate increases.\n\nOur model surpasses traditional codecs in terms of BPP-MS-SSIM, as shown in Fig.~\\ref{fig:bpp-msssim}. MS-SSIM has a closer relationship with perceptual quality~\\cite{cheng2019perceptual} and for deep networks, they tend to obtain a larger advantage over traditional codecs in terms of BPP-MS-SSIM, which can be verified from the fact that both LIC (Learned Image Compression) and DVC (Deep Video Compression) surpass traditional codecs in terms of BPP-MS-SSIM before BPP-PSNR~\\cite{imagecnn7, lu2019dvc}. We follow the same rationale by using a hybrid loss of PSNR and MS-SSIM~\\cite{sun2021hlic} to optimize perceptual quality, which leads to our model performing better than traditional codecs including AVIF tuned by ssim by a large margin in terms of BPP-MS-SSIM but worse than AVIF in terms of BPP-PSNR.\n\\subsection{Speed Analysis}\nWe compare the inference latency of our LIC model with traditional codecs and the results are shown in Fig.~\\ref{fig:latency}. Our model achieves \\textbf{18\/12ms} for encoding\/decoding latency, which is at most 20\\% higher in encoding latency and at least 35\\% lower in decoding latency than jpeg-turbo with \\emph{bmp} format input. Compared with JPEG XL and AVIF, our LIC model is several times faster, reaching \\textbf{50\/19ms} for encoding and decoding processes respectively. To further demonstrate the practicality of our implementation of the LIC model, we also test its latency on a less expensive GPU (NVIDIA GeForce GTX 1660 SUPER) and a less powerful CPU (Intel(R) Core(TM) i7-8700 CPU @ 3.20GHz). Encoding\/decoding latency is \\textbf{58\/25ms}. In addition, we have also tested the latency of our LIC model on CPU, reaching encoding\/decoding latency of \\textbf{199\/284ms, comparable to JPEG XL} as shown in Fig.\\ref{fig:latency-xl-avif}. We leave the optimization of CPU deployment as future work.\n\nIn terms of throughput, with techniques mentioned in Section \\ref{otheropt}, when processing images in parallel, our implementation reaches stunning \\textbf{145\/208fps} for \\emph{bmp} format input and \\textbf{126\/199fps} when input is saved in \\emph{png} format for encoding and decoding processes. On a less expensive GPU (NVIDIA GeForce GTX 1660 SUPER), our model still reaches throughput of \\textbf{92\/82fps}.\n\n\n\\begin{figure}[tb]\n\\centering\n    \\begin{subfigure}[b]{0.4\\textwidth}\n        \\centering\n        \\captionsetup{width=80mm}\n        \\includegraphics[width=\\linewidth]{imgs\/comparison_latency_jpeg_turbo.pdf}%\n        \\caption{Comparison of latency between our LIC model and jpeg-turbo for 1080p \\emph{bmp} format images.}\n        \\label{fig:latency-turbo}\n    \\end{subfigure}\n    \\\\\n    \\begin{subfigure}[b]{0.4\\textwidth}\n        \\centering\n        \\captionsetup{width=80mm}\n        \\includegraphics[width=\\linewidth]{imgs\/comparison_latency_jpeg_xl_avif.pdf}%\n        \\caption{Comparison of latency between our LIC model, AVIF and JPEG XL for 1080p \\emph{png} format images.}\n        \\label{fig:latency-xl-avif}\n    \\end{subfigure}\n\\caption{Latency comparison of our LIC model and traditional codecs. \n``Quality\" stands for jpeg qp while ``Reference Bpp\" stands for bit rates on 1080p dataset for JPEG XL and AVIF.\n}\n\\label{fig:latency}\n\\end{figure}\n\nWe also report latency of each subnetwork in our model in Table.~\\ref{table:latency} which is obtained using TensorRT's trtexec benchmarking tool with an input of resolution $1088\\times1920$.\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{ c |c c c c}\n\\hline \\hline\n \\textbf{GPU Type}& $g_a$ & $g_s$ &$h_a$&$h_s$ \\\\\n \\hline\n \\textbf{Tesla T4} & 7.25&\t5.48&\t1.15&\t1.17 \\\\  \n \\textbf{GeForce GTX 1660 SUPER} & 10.08 & 10.75 & 1.68 & 2.80   \\\\\n \\hline\n\\end{tabular}\n\\caption{Latency of each subnetwork in our model. $g_a$, $g_s$, $h_a$ and $h_s$ above refer to analysis and synthesis transforms of main and hyperprior autoencoders.}\n\\label{table:latency}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Memory Analysis}\nWe show the memory footprint of our model during encoding and decoding processes of 100 1080p images in Fig.~\\ref{fig:mem}, peaking at 1.8GB and 1.5GB for CPU and GPU respectively. \n\\begin{figure}[tb]\n\\centering\n    \\begin{subfigure}[t]{0.23\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/cpu_mem.pdf}%\n        \\label{fig:cpu_mem}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.23\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/gpu_mem.pdf}%\n        \\label{fig:gpu_mem}\n    \\end{subfigure}\n    \\caption{CPU and GPU memory usage during encoding and decoding.}\n    \\label{fig:mem}\n\\end{figure}\n\n\\subsection{Visual Quality}\nIn this subsection, we compare the visual quality of our LIC model with that of traditional codecs on Kodak~\\cite{1993kodak} dataset. As can be seen in Fig.~\\ref{fig:kodim24}, the image compressed by JPEG~\\cite{jpeg} displays blocking artifacts while the image compressed by AVIF is oversmoothed, losing details. Image compressed by JPEG XL~\\cite{alakuijala2019jpeg} suffers from blocking artifacts, similar to JPEG, but to a lesser extent. The image compressed by our LIC model, on the other hand, shows better perceptual quality.\n\n\n\\begin{figure}[tb]\n\\centering\n    \\begin{subfigure}[t]{0.15\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/display\/kodim24_box.png}%\n        \\caption{Original image}\n    \\end{subfigure}\n    \\\\\n    \\begin{subfigure}[t]{0.11\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/output_decoded_kodim24_30_512x768_box.png}%\n        \\caption{JPEG \\\\44.4KB}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.11\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/output_decoded_kodim24_35_512x768.png}%\n        \\caption{AVIF \\\\43.75KB}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.11\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/output_decoded_kodim24_3.35_512x768.png}%\n        \\caption{JPEG XL \\\\44.35KB}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.11\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/kodim24.png}%\n        \\caption{our LIC \\\\44.44KB}\n    \\end{subfigure}\n    \\\\\n    \\begin{subfigure}[t]{0.11\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/output_decoded_kodim24_30_512x768_region.png}%\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.11\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/output_decoded_kodim24_35_512x768_region.png}%\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.11\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/output_decoded_kodim24_3.35_512x768_region.png}%\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.11\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/kodim24_region.png}%\n    \\end{subfigure}\n    \\caption{Visual quality comparison of kodim24.png~\\cite{1993kodak}. Images on the third row are obtained by zooming in the blue square box region of images above.}\n    \\label{fig:kodim24}\n\\end{figure}\n\\subsection{NAS}\n For NAS, we reduce about $10\\%$ inference time while improving rate-distortion performance.\n We get lower bpp ($0.7281 \\rightarrow 0.7278$), with higher PSNR ($33.5550 \\rightarrow 34.0997$) and higher MS-SSIM ($0.9876 \\rightarrow 0.9880$), but with lower latency ($15.2059 \\rightarrow 13.9574$) as shown in Table.~\\ref{tab:Bignas}.\n\n\\begin{table}[]\n\\setlength{\\tabcolsep}{3pt}\n\\begin{tabular}{ccccc}\n\\hline \\hline\nStructure   & $g_a$          & $h_a$       & $h_s$       & $g_s$     \\\\ \\hline\nOrigin      &48,96,112,176& 176,246,176 & 246,176,176 & 112,96,3  \\\\ \nNAS      & 32,120,104,220 & 248,224,256 & 236,200,220 & 112,112,3 \\\\ \\hline \\hline\nPerformance & BPP            & PSNR        & MS-SSIM     & Lantency(ms)  \\\\ \\hline\nOrigin      & 0.7281         & 33.5550     & 0.9876      & 15.2059   \\\\ \nNAS      & 0.7278         & 34.0997     & 0.9880      & 13.9547   \\\\ \\hline\n\\end{tabular}\n\\caption{Output channels of each (de)convolution block in $g_a$, $g_s$, $h_a$ and $h_s$  and rate-distortion performance with latency are listed above.}\\label{tab:Bignas}\n\\end{table}\n\n\\subsection{Quantization}\n\\begin{table}[]\n\\centering\n\\setlength{\\tabcolsep}{6pt}\n\\begin{tabular}{c|ccc}\n\\hline \\hline\nmodel   & bpp         & psnr     &ms-ssim       \\\\ \\hline\nFP32      &0.728 & 33.55 & 0.9876 \\\\ \nQuantized      & 0.704     & 32.91    & 0.9839 \\\\\n$g_a$-$h_a$-Quantized      &0.716\t&33.49&\t0.9873 \\\\\\hline\n\n\\end{tabular}\n\\caption{BPP-PSNR-MS-SSIM of quantized models. ``Quantized\" stands for the model with all layers quantized, evaluated on a NVIDIA GeForce GTX 1660 SUPER GPU. ``$g_a$-$h_a$-Quantized\" denotes the model with its main and hyper encoders quantized, evaluated in pytorch's fake quantize mode.} \n\\label{table:quant}\n\\end{table}\nWe show the rate-distortion performance of the quantized models in Table.~\\ref{table:quant}. We notice that quantizing main and hyper encoders barely hurt performance while quantizing all layers leads to a minor loss in rate-distortion performance. As for speedup on a real device, quantization brings about approximately 20\\% to 35\\% increase in throughput (images processed in parallel) on a NVIDIA GeForce GTX 1660 SUPER GPU, from 92\/82 fps to 114\/112 fps.\n\n\n\\section{Conclusion}\nWe evaluate the practicality of learned image compression by optimizing and assessing rate-distortion performance, latency, throughput, memory footprint and visual quality. We compare these results with those of traditional codecs. LIC has comparable or even faster inference speed compared with the fastest industrial codec, i.e., jpeg-turbo, while being superior in terms of rate-distortion performance and perceptual quality. Besides, LIC achieves astonishing throughput of 145\/208 fps for 1080p images, making it applicable for industrial scenarios. In addition, NAS and quantization show promising results for obtaining more efficient and more effective LIC models. All these results show that learned image compression is becoming ready for industrial applications. Future works will be devoted to optimizing LIC for mobile devices.\n\n\n\\balance\n{\\small\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Conclusion and Limitation}\n\\label{sec:discussion}\n\n\\textbf{Limitations}\nIn our experiments, we only train and test our method on two tasks, which limits the scope of the proposed method. More dexterous manipulation tasks will be our future research direction. Another potential improvement of our method is to use Recurrent Neural Network and temporal information for policy networks. It will enable us to do long-horizon tasks.\n\n\\textbf{Conclusion}\nTo the best of our knowledge, our approach is the first work to train a dexterous manipulation reinforcement learning policy with point cloud inputs that can transfer to the real world. We justified that direct sim-to-real transfer is possible for two manipulation tasks with point cloud representation. \n\n\\acknowledgments{This work was supported, in part, by grants from NSF CCF-2112665 (TILOS), NSF 1730158 CI-New: Cognitive Hardware and Software Ecosystem Community Infrastructure (CHASE-CI), NSF ACI-1541349 CC*DNI Pacific Research Platform, the Industrial Technology Innovation Program (20018112, Development of autonomous manipulation and gripping technology using imitation learning based on visualtactile sensing) funded by the Ministry of Trade, Industry and Energy of the Republic of Korea, and gifts from Meta, Google, Qualcomm. We also thanks Fanbo Xiang for suggestion on shader setting in SAPIEN, Ruihan Yang for helpful discussion on RL training.}\n\n\\section{Experiments}\n\\label{sec:exp}\n\n\\subsection{Experimental Setup}\n\\label{sec:exp_setup}\n\n\\begin{figure*}[t]\n    \\begin{minipage}{0.20\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figs\/bottle_exp.pdf} \\\\\n    \\vspace{-1em}\n    \\quad {(a) }\n\t\\end{minipage}\n\t\\begin{minipage}[h]{0.20\\linewidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figs\/can_exp.pdf} \\\\\n    \\vspace{-1em}\n    \\quad {(b)}\n    \\end{minipage}\n    \\begin{minipage}{0.20\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figs\/abl_bottle.pdf} \\\\\n    \\vspace{-1em}\n    \\quad {(c)}\n\t\\end{minipage}\n\t\\begin{minipage}[h]{0.20\\linewidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figs\/abl_can.pdf}\\\\\n    \\vspace{-1em}\n    \\quad  {(d)}\n    \\end{minipage}%\n    \\begin{minipage}{0.20\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figs\/abl_door.pdf} \\\\\n    \\vspace{-1em}\n    \\quad {(e)}\n\t\\end{minipage}\n    \n    \\caption{\\small{\\textbf{Training Curves.} The left two plots show the single-object and multi-object training curve of (a) bottle category and (b) can category. The right three plots show the ablation results on the (c) grasping bottle (d) grasping can and (e) door opening. The x-axis is the training iterations and y-axis is the normalized episodic return. The shaded area indicates standard error and the performance is evaluated on five random seeds.}}\n    \\label{fig:main_exp}\n\\end{figure*}\n\n\n\n\\begin{table}[t]\n\\renewcommand{\\arraystretch}{1.2}\n\\centering\n\\begin{tabular}{c|c|c|c|c|c}\n         \\multirow{2}{*}{\\textbf{Settings}} & \\multicolumn{2}{c}{\\textbf{Bottle}} \\vline  & \\multicolumn{2}{c}{\\textbf{Can}} \\\\ \n        \n         & Known Obj. & Novel Obj. & Known Obj. & Novel Obj. \\\\\\shline\n         \n         Single Obj. Training & $0.81\\pm0.09$ & $0.60\\pm0.06$ & $0.96\\pm0.04$ & $0.63\\pm0.18$ \\\\\n         \n         Multi Obj. Training & $\\textbf{0.83}\\pm\\textbf{0.16}$ & $\\textbf{0.81}\\pm\\textbf{0.15}$ & $\\textbf{0.93}\\pm\\textbf{0.07}$ & $\\textbf{0.68}\\pm\\textbf{0.09}$\\\\\n\\end{tabular}\n\\vspace{0.5em}\n\\caption{\\small{\\textbf{Experiment on Multi-object Training.} We evaluate the policy trained with single and multiple objects on bottle (upper two rows) and can (bottom two rows) categories with point cloud input. We test them on both known or novel objects. The success rate are reported on 5 seeds.}}\n\\vspace{-2em}\n\\label{tab:single_multi}\n\\end{table} \n\n\\textbf{Point Cloud Pre-processing:}\nTo enable smooth transfer from simulation to real-world, we apply the same data preprocessing procedure to the point cloud captured by the camera. It involves four steps: (i) Crop the point cloud to the work region with a manually-defined bounding-box; (ii) Down-sample the point cloud uniformly to $512$ points; (iii) Add a distance-dependent Gaussian noise to the simulated point cloud to improve the sim2real robustness; (iv) Transform point cloud from the camera frame to the robot base frame using camera pose. In simulation, we use the ground-truth camera pose with multiplicative noise for frame transformation. In the real-world, we perform hand-eye calibration to get the camera extrinsic parameters. \n\n\\textbf{Evaluation Criterion:}\nWe evaluate the performance of a policy by its success rate. For grasping tasks, a task is considered a success if $d_{ot} < 0.05m$ in simulation, where $d$ is the distance between object position and goal position. In real world, the task is considered as success if the XY position of the object is within 5cm from the target position and the height of the object is at least 15cm from table top. For the door opening, the task is considered as success if the door is opened to at least around 45 degrees.\n\n\\textbf{EigenGrasp Baseline:}\nWe choose the EigenGrasp~\\cite{ciocarlie2007dexterous} as the grasp representation. Given an object mesh model, we use the GraspIt~\\cite{miller2004graspit} to search valid grasp for Allegro Hand. Then, we use the RRTConnect~\\cite{kuffner2000rrt} motion planner implemented in OMPL~\\cite{sucan2012open} to plan a joint trajectory to the pre-grasp pose and then plan a screw motion from pre-grasp pose to the grasp pose. Finally, we close all fingers based on searched grasp pose and lift the object to the target. Note that different from our approach, the baseline method \\textbf{requires complete object model to search for grasps and ground-truth object pose} to align the grasp pose in the robot frame. To evaluate the performance of baseline on novel objects, we first build a grasp database on ShapeNet bottle and can categories using GraspIt. Given the sensory data of a new object, we search for the most similar objects in the dataset and use the query grasps for the novel object. Here we compare the performance of our method with baselines in the real-world. \n\n\\textbf{Training:}\nWe train RL in two settings for grasping: (i) training on a single object (ii) training on multiple objects jointly. For the single object grasping, we perform experiments on both ``tomato soup can'' and ``mustard bottle'' from YCB. For the multi-object training, we choose 10 objects from the can or bottle categories of ShapetNet. We randomly choose one object to train for each episode in the multi-object training. Here, we use can and bottle as experimental subjects since they represent two different basic grasping patterns~\\cite{napier1956prehensile} for anthropomorphic hand: precision grasp and power grasp. For door opening, we only train the policy on the door with fixed lever geometry.\n\n\\subsection{Comparison of Single-object and Multi-object Training}\nWe plot the training curve of RL in Figure~\\ref{fig:main_exp} (a) and (b). In general, our method can learn to grasp and move the object to the goal pose within 600 iterations, where each iteration contains 20K environment steps. Then we evaluate the policy trained on both known and novel objects, and the results are shown in Table~\\ref{tab:single_multi}. We run 100 trials to compute the average success rate. \nOn grasping known objects, we find that agent trained on single-object outperforms the agent trained on multi-object by a small margin, for both learning efficiency and final success rate. However, agent trained on multi-objects does much better at grasping novel objects. Our results suggest that using multiple object during training is important, and is of great importance for novel object generalization.\n\n\\subsection{Ablation Results in Simulation}\n\\vspace{-0.5em}\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=1\\linewidth]{figs\/real_expriment.pdf}\n    \\caption{\\small{\\textbf{Real-experiment:} We evaluate our point cloud policy on various unseen objects.}}\n    \\label{fig:real_expriment}\n    \\vspace{-0.1in}\n\\end{figure}\n\n\\begin{table}[t]\n\t\\centering\n        \\resizebox{\\textwidth}{!}{\n\t\\begin{tabular}{l|c|c|c|c|c|c}\n        \\multirow{2}{*}{\\textbf{Settings}} & \\multicolumn{2}{c}{\\textbf{Bottle}}  \\vline  & \\multicolumn{2}{c}{\\textbf{Can}} \\vline& \\multicolumn{2}{c}{\\textbf{Door}}\\\\\n        \n        & Known Obj. & Novel Obj. & Known Obj. & Novel Obj. & Known Obj. & Novel Obj \\\\\\shline\n        w\/o Imagined PC. & $0.60\\pm0.46$ & $0.56\\pm0.51$ & $0.91\\pm0.17$ & $0.63\\pm0.07$& $0.14\\pm0.28$ & $0.11\\pm0.27$ \\\\\n        \n        w\/o  Contact Rew. & $0.00\\pm0.00$ & $0.00\\pm0.00$ & $0.06\\pm0.08$ & $0.03\\pm0.06$& $0.21\\pm0.26$ & $0.20\\pm0.25$\\\\\n        \n        w\/o  Both & $0.00\\pm0.00$ & $0.00\\pm0.00$ & $0.00\\pm0.00$ & $0.00\\pm0.00$& $0.00\\pm0.00$ & $0.00\\pm0.00$\\\\\n        \n        Ours & $\\textbf{0.83}\\pm\\textbf{0.16}$ & $\\textbf{0.81}\\pm\\textbf{0.15}$ & $\\textbf{0.93}\\pm\\textbf{0.07}$ & $\\textbf{0.68}\\pm\\textbf{0.09}$&$\\textbf{0.92}\\pm\\textbf{0.06}$&$\\textbf{0.79}\\pm\\textbf{0.11}$\\\\\n        \\end{tabular}\n        }\n    \\vspace{0.5em}\n    \\caption{\\textbf{Ablation Study:} we investigate the influence of contact-based reward design and imaged point cloud. We evaluate the success rate on both known and novel objects under four settings: (i) without imaged point cloud; (ii) without contact reward; (iii) without both; (iv) with both. }\n    \\label{tab:ablation}\n   \n\\end{table}\\hfill\n\n\nWe ablate two key innovations of the work: the reward design with oracle contact and the imaged hand point cloud. We perform experiments on four different variants: (i) without imagined point cloud; (ii) without contact-based reward design; (iii) without both imagined point cloud and contact based reward design; (iv) our standard approach with both techniques. Note that the variant (iii) is an approximation of~\\citep{corl21-inhand} in our environments and tasks. We compare both the learning curve and the evaluation success rate of these four variants. For grasping task, Figure~\\ref{fig:main_exp} (c) and (d) show the results on bottle and can categories, and Table~\\ref{tab:ablation} shows the success rate. Our findings can be summarized as follows. \n\nFirst, we find that contact reward information is of vital importance for training the point cloud RL policy on the multi-finger robot hand. Without using contact reward, the agent can hardly learn anything (red and green curve in the figure) and get nearly zero success rate during evaluation for both bottle and can categories. \nBy encouraging the contact between fingers and object, the RL agent can avoid getting stuck in local minimums and learn meaningful manipulation behavior. \n\nSecond, the imagined point cloud can also improve the training and test performance for both categories, though it is not so important as the contact reward. As is shown in the learning curves of Figure~\\ref{fig:main_exp} (c) and (d), the policy utilizing the imagined point cloud as input can learn faster in the early stage of the training and show much smaller variances. An interesting fact is that imagined point cloud is more beneficial for the bottle category than the can category. One possible reason is that grasping a bottle requires multi-finger coordination to perform a power grasp. Such coordination is highly dependent on the detailed finger information provided by the imagined point cloud. The imagined point cloud can help the agent to better see the fingers even if they are occluded by the object, e.g., fingers behind the object. \n\nThe experiments on the door opening task also support these findings. As shown in Figure~\\ref{fig:main_exp} (e), the policy trained with contact based reward and imaged hand point cloud outperform other ablation methods, which also demonstrate the effectiveness of our two key design. Compared with the grasping experiments, we can observe larger variations during policy training. One possible reason is that door opening suffers from heavier occlusion than grasping, when the door lever is grasped by the robot hand, which influence the temporal consistence of PointNet feature.\n\n\\subsection{Real-World Evaluation}\n\n\\begin{table}[h]\n\\begin{minipage}[h]{0.6\\linewidth}\n\\vspace{0.4em}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{l|c|c|c}\n    Method & Bottle & Can  & Mixed Category\\\\ \\shline\n    EigenGrasp Oracle & $0.66$ & $0.45$ & $N\/A$ \\\\\n    EigenGrasp & $0.50$& $0.41$ & $N\/A$  \\\\\n    Single Obj. Train & $0.75\\pm0.06$ & $0.75\\pm0.08$& $N\/A$ \\\\ \n    Multi Obj. Train & $\\textbf{0.87}\\pm\\textbf{0.03}$ & $\\textbf{0.83}\\pm\\textbf{0.13}$ & $\\textbf{0.73}\\pm\\textbf{0.12}$\n\\end{tabular}\n}\n\\vspace{0.1em}\n\n\\caption{\\small{\\textbf{Real-World Grasping Experiment}: This experiment consist of 3 categories, which incorporate 26 objects: 10 bottles, 6 cans, and 10 other objects in multiple mixed categories.}}\n\\label{tab:real_grasping}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{l|c|c|c}\n     Settings & Original Door & Novel Door 1 & Novel Door 2  \\\\ \\shline\n     Single Door Train & $0.72\\pm0.07$ & $0.60\\pm0.03$ &$0.67\\pm0.01$\\\\\n\\end{tabular}\n}\n\\caption{\\small{\\textbf{Real-World Door Opening Experiment}: This experiment consist of 3 different doors, the first one on the left is used for training and the other two are for testing.}}\n\\label{tab:real_door_opening}\n\\end{minipage}\n\\begin{minipage}[h]{0.4\\linewidth}\n\\centering\n\\includegraphics[width=\\textwidth]{figs\/objects_door.pdf}\n\\end{minipage}\\\\\n\\vspace{1em}\n\\vspace{-2em}\n\\end{table}\n\n\n\nWe perform sim2real experiments to evaluate the performance of our method in the real world. As is shown in Figure~\\ref{fig:robot_setup}, we attached an Allegro hand onto a XArm-6 robot arm to grasp the object on the front table. We apply the same data-preprocessing steps for both simulated environment and real-world as mentioned in Sec.\\ref{sec:exp_setup}. \n\nThe task execution sequence is visualized on the bottom row of Figure~\\ref{fig:teaser}. Both Single Obj. Training and Multi Obj. Training will be evaluated in the corresponding category, while policies training with multiple objects even evaluated in the Mixed category with unseen objects. We run 10 independent trials seeds for each object-policy pair. \n\nThe real-world evaluation results are shown in Table~\\ref{tab:real_grasping} and Table~\\ref{tab:real_door_opening}. The policy trained in the simulator with point cloud input can directly transfer to the real-world without fine-tuning.\nFor both two tasks, our policy can even deploy on objects that have never been seen during the training. Moreover, We find that for grasping, training on multiple objects can ensure better performance than single-object training. Compared with the EigenGrasp baseline shown in Table~\\ref{tab:real_grasping}, our policies trained on both single-object and multi-object performs better, even if the EigenGrasp baseline use the object model information. Since EigenGrasp + motion planning is a open-loop manipulation policy, small error in object and scene modeling, e.g. initial object pose or object geometry, can lead to a failure in final grasp results. In contrast, our methods works in a closed-loop fashion with point cloud observation, which does not require privilege knowledge about the object.\n\n\\section{Introduction}\n\\label{sec:intro}\n\nDexterous manipulation has remained to be one of the most challenging problems in robotics~\\cite{rl-openai}. While multi-finger hands create ample opportunities for robots to flexibly manipulate objects in our daily life, the nature of the high degree of freedom and high-dimensional action space creates significant optimization challenges for both search-based planning algorithms and policy learning algorithms. Recent efforts using model-free Reinforcement Learning have achieved encouraging results on complex manipulation tasks~\\cite{rl-openai, dex-rl-valve}. However, it still faces many challenges in generalizing to diverse objects and being deployed on multi-finger hand in the real-world. \n\nFor example, the dexterous manipulation framework proposed by OpenAI et al.~\\cite{rl-openai} can solve in-hand manipulation of Rubik's Cube with RL and transfer to the real robot hand. However, the policy is only trained with one particular object and it is not able to \\emph{generalize to diverse objects}. \nTo achieve cross-object generalization, recent efforts proposed to learn robust 3D point cloud representations~\\cite{corl21-inhand, ilad, rl-pc-inhand-generalization, mu2021maniskill} with diverse objects using RL in simulation. While point cloud input has also been shown easier for Sim2Real transfer~\\cite{zhang2022close} given its focus on geometry instead of texture, the assumption on the access of complete object point clouds and ground truth states limit the transferability of above methods to the \\emph{real robot deployment}. Among these works, Chen et al.~\\cite{corl21-inhand} showed that cross-object generalization is achievable in simulation without knowing the shape of the object to grasp, but its requirement of real-time access to the object states is itself a very challenging robot perception problem, especially under large occlusions during hand object interaction.  \n\nIn this paper, we provide a sim-to-real reinforcement learning framework for generalizable dexterous manipulation, using two tasks with the Allegro Hand\\cite{allegro}: (i) object grasping where the test objects has not been seen during training; (ii) door opening where the test doors have levers of novel shape that has not been used in training.\nThe tasks are visualized in Figure~\\ref{fig:teaser}.\n\nthe robot needs to first rotate the lever to unlock the door latch and then pull the lever in a circular motion to open the door. \n\nWe perform our studies by training a point cloud based reinforcement learning policy in the grasping and door opening task. With this approach, we list the three key discoveries of our framework for learning generalizable point cloud policy below:\n\n(i) We justified that it is possible to achieve \\emph{direct sim-to-real transfer} for a dexterous manipulation policy with category-level generalizability when we use point cloud as the data representation.\n\n(ii) Raw point clouds captured by sensors often come with heavy occlusions and noise: only a very small portion of the points from the observation are representing the robot fingers. We propose to \\emph{imagine} the complete robot finger point clouds according to the robot kinematic model and use them to augment the occluded real point cloud observations. We find that explicitly augmenting the input by \\emph{imagined} points can help achieve better robustness and sample efficiency for reinforcement learning.\n\n(iii) Different from existing works that add contact information to the input of RL, we design a novel reward using contact pair information without adding contact to the observation. This practice remarkably improves sample efficiency as well as learning stability and avoids the dependency on contact sensor that is often unavailable for real robot models. \n\\section{Approach}\n\\label{sec:approach}\n\n\\begin{wrapfigure}{r}{7cm}\n\\vspace{-0.45cm}\n    \\centering\n    \\includegraphics[width=0.97\\linewidth]{figs\/real-experiment_setting.pdf}\n    \\caption{\\small{\\textbf{Real-experiment Setup:} we use an Allegro Hand attached on an XArm6 and a RealSense D435 camera facing forward the robot.}}\n    \\label{fig:robot_setup}\n    \\vspace{-0.2in}\n\\end{wrapfigure}\n\nOur objective is to train a generalizable point cloud policy on a dexterous robot hand-arm system that is able to grasp a wide range of objects or open an closed door with RL. We aims at Sim-to-Real transfer without any real-world training or data. During testing, the robot can only access the single-viewed point cloud and the robot proprioception data. As is discussed before, training such a policy comes with numerous technical challenges, including reward design and imperfect point cloud information. In this work, we propose a novel reward design technique based on contact and imagined point cloud model to deal with these challenges.\n\n\\textbf{Preliminaries:}\nWe model the dexterous manipulation problem as a Partially Observable Markov Decision Process~(POMDP) $\\mathcal{M} = (\\mathcal{O}, \\mathcal{S}, \\mathcal{A}, \\mathcal{R}, \\mathcal{T}, \\mathcal{U}).$ Here, $\\mathcal{O}$ is the observation space, $\\mathcal{S}$ is the underlying state space, $\\mathcal{A}$ is the action space, $\\mathcal{R}$ is the reward function, $\\mathcal{T}$ is the transition dynamics, and $\\mathcal{U}$ generates agent's observation. At timestep $t$, the environment is at the state $s_t\\in\\mathcal{S}$. The agent observes $o_t\\sim\\mathcal{U}(\\cdot|s_t)\\in\\mathcal{O}$. The agent takes action $a_t$ and receives reward $r_t = \\mathcal{R}(s_t, a_t)$. The environment state at timestep $t+1$ then transit to $s_{t+1}\\sim\\mathcal{T}(s_t, a_t)$. The objective of the agent is to maximize the return $\\sum_{t=0}^T\\gamma^tr_t$, where $\\gamma$ is a discount factor. \n\n\\textbf{System Setup:} \nMany previous works on learning-based dexterous manipulation attach the hand to a fixed platform to simplify the experiment environment in the real world. In this work, we create a more flexible and powerful dexterous manipulation system which includes both the robotic hand and the arm~(Figure~\\ref{fig:robot_setup}). Concretely, we attach the Allegro Hand to an XArm6 robot. Allegro Hand is a 16-DoF anthropomorphic hand with four fingers and XArm6 is a 6-DoF robot arm. We place a RealSense D435 camera at the right front of the robot to capture the point cloud. This setup brings additional challenges to RL exploration and Sim2Real deployment. We use SAPIEN~\\cite{sapien} platform with uses a full-physics simulator to build the environment of the whole system. The simulation time step is $0.005s$ to ensure stable contact simulation. Each control step lasts for $0.05s$.\n\n\\textbf{Tasks and Objects:}\nIn this paper, we expect our robot to perform the grasping task over a diverse set of objects and to open a locked door by rotating the lever. In the grasping task, we first select a random object from an object dataset and place it on the table. The robot is then required to move it to a target pose. Moreover, the robot should be able to generalize to different initial states, so we randomize both the initial pose and the goal pose for each trial. In simulation experiments, we use bottles and cans from both ShapeNet~\\cite{shapenet} dataset and YCB~\\cite{ycb} dataset. \nIn real-world experiments, we use novel unseen objects to test the policy. \nIn the door opening task, the robot hand is required first to rotate the lever to unlock the door latch and then pull the lever in a circular motion. We use three doors to test the policies both in the simulation and the real world, only one is used for training and the other two are unseen doors. We also randomize the initial pose for each trial. \n\n\\textbf{Observation Space:} \nThe observation contains both visual and proprioceptive information with four modalities: (1) Observed point cloud provided by the camera; (2) Proprioception signals of the robot including joint positions and end-effector position; (3) Imagined hand point cloud proposed in Sec.~\\ref{sec:imagination}; (4) Object goal position provided in each trial. All the information is accessible on the real robot. The dimension of each observation modality is shown in Figure~\\ref{fig:pipeline}.\n\n\\textbf{Action Space:}\nThe action is responsible for controlling both the $6$-DOF robot arm and the $16$-DOF hand. It has $6+16=22$ dimension in total. The robot arm is parametrized by the 6D translation and rotation of the end-effector relative to a reference pose. We use the damped least square inverse kinematics solver with a damping constant $\\lambda = 0.05$ to compute the joint motion. Each finger joint of the Allegro hand is controlled by a position controller. Both robot arm and hand are controlled by PD controllers. \n\n\\textbf{Network Architecture:}\nThe network architecture is visualized in Figure~\\ref{fig:pipeline}, \n\n\\subsection{Reward Design with Oracle Contact}\n\\label{sec:reward}\nSince we aim to solve the dexterous manipulation problem with pure RL, the reward design is central to the method. We need a good reward function to ensure proper interaction between the robotic hand and the object. The whole interaction process consists of two phases. The first phase is to simply reach the object. The second phase is to grasp the object and move it to the target, which is more challenging. For the first phase, we encourage reaching with the following reaching reward:\n\\begin{equation}\n    r_{\\rm reach} = \\sum_{\\rm finger}\\frac{1}{\\epsilon_r + d(\\textbf{x}_{\\rm finger}, \\textbf{x}_{\\rm obj})}.\n    \\label{reach_reward}\n\\end{equation}\nHere, $\\textbf{x}_{\\rm finger}$ and $\\textbf{x}_{\\rm obj}$ are the Cartesian position of each fingertip and the target object. Note that $\\textbf{x}_{\\rm obj}$ is available when we perform training in simulation. However, using this reward alone cannot ensure proper contact for grasping. For example, the robot can touch the object with the back of the hand rather than the palm and then get stuck in this local minimum. Therefore, we introduce a novel contact reward to guarantee meaningful contact behavior:\n\\begin{equation}\n    r_{\\rm contact} = {\\rm \\textbf{IsContact}}({\\rm thumb}, {\\rm object})\\ {\\rm \\textbf{AND}}\\  \\left(\\sum_{\\rm finger} \\textbf{IsContact}({\\rm finger}, {\\rm object}) \\geq 2 \\right).\n\\end{equation}\nThis contact reward function outputs a boolean value in $\\{0, 1\\}$. It outputs $1$ only if the thumb is in contact with the object and there are more than one finger in contact with the object. Intuitively, it encourages the robot to cage the object within fingers. In this case, the robot can quickly find out stable grasping and lift the object to the target location. The lifting behavior is encouraged by\n\\begin{equation}\n    r_{\\rm lift} = r_{\\rm contact}{\\rm \\textbf{Lift}}(\\textbf{x}_{\\rm obj}, \\textbf{x}_{\\rm target}).\n\\end{equation}\nThe \\textbf{Lift} function is basically in the form of Equation~\\ref{reach_reward} and the main difference is that it will return a large reward value upon task completion. The overall reward function is a weighted combination of the terms above plus a control penalty:\n\\begin{equation}\n    \\mathcal{R} = w_{\\rm reach} r_{\\rm reach} + w_{\\rm contact} r_{\\rm contact} + w_{\\rm lift} r_{\\rm lift} + w_{\\rm penalty} r_{\\rm penalty}.\n\\end{equation}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/CoRL_sim_pipeline_camera_ready.pdf}\n    \\caption{\\small{\\textbf{Architecture:} our feature extractor takes the observed point cloud, imagined point cloud, robot proprioception, and goal pose as input to output a feature embedding. Both actor and critic take the same feature to predict action and value. The red point represented the imaged point cloud of robot hand. Note that our network does not require RGB information.}}\n    \\label{fig:pipeline}\n    \\vspace{-0.1in}\n\\end{figure}\n\n\n\\subsection{Imagined Hand Point Cloud}\n\\label{sec:imagination}\nThe usage of point cloud comes with two challenges. The first challenge is occlusion, which may occur to both the object under manipulation and the hand itself. When the robot hand is interacting with an object, the fingers may be occluded by the object. Since we do not assume tactile sensors in this work, this occlusion problem can be serious. The second challenge is the low point cloud resolution during RL training, where we can only use a limited number of points due to the memory limit. In this case, the number of points from hand finger may not be adequate to precisely capture the spatial relationship between the robot and the object. We propose a simple yet effective method to handle both issues in a unified manner. Our idea is to use an imagined hand point cloud in the observation to help the robot to \\textit{see the interaction}. \n\nWe provide one example in Figure~\\ref{fig:pipeline}. Black points indicate the point cloud captured by the camera, in which some important details of fingers are missing. These missing details provide crucial information of the interaction. Though such interaction information can also be inferred by combining the information from both the proprioception and visual input, we find that the best way is to synthesize these missing details. Concretely, we can compute the pose for each finger link via forward kinematics given the joint position from the robot joint encoder and the robot kinematics model. Then, we synthesize the imagined point cloud~(blue points in Figure~\\ref{fig:pipeline}) by sampling the points from the mesh of each finger link. This process is possible in both simulation and the real world. \n\n\\subsection{Training}\n\\label{sec:impl}\nWe adopt Proximal Policy Optimization~(PPO)~\\cite{schulman2017proximal} to train the agent in simulation, and then deployed to real without real-world fine-tuning. The network architecture is illustrated in Figure~\\ref{fig:pipeline}. Both value and policy networks share the same visual feature extraction backbone. We concatenate the observed point cloud with the imagined hand point cloud together as the input to the feature extractor. We also attach a one-hot encoding to each point which indicates whether it is observed or an imagined point. \n\\section{Related Work}\n\n\\textbf{Dexterous Manipulation.} \nDexterous manipulation aims to enable robotic hands to achieve human-level dexterity for grasping and manipulating objects. Analytical methods have been proposed to adopt planning to solve this problem~\\citep{salisbury1982articulated,analytical-dex,rus1999hand, grasping-book, vkumar-survey, bohg-survey,Dogar2010}. However, they rely on detailed object models, which are not accessible when testing on unseen objects. To mitigate this issue, researchers propose a learning-based method for dexterous manipulation~\\citep{learning-based-review}. \nSome methods~\\citep{gps-manipulation, rl-openai, dex-rl-valve, corl21-inhand, rl-pc-inhand-generalization} take a reinforcement learning~(RL) approach, while another line of work~\\citep{il-dex, il-vr, soil, coarse-to-fine-il, chen2022learning} also proposes to learn from demonstration using imitation learning~(IL) to acquire the control policy. Recently, the combination of RL and IL~\\citep{dapg, tactile-guided, hand-teleop, corl20-suboptimal, dexmv, ilad} has also shown encouraging dexterous manipulation results in simulation. However, most methods are either learning policy for one single object or assume access to object states produced by perfect detector which increases challenges to Sim2Real transfer. In this paper,  we surpass these limitations by introducing training on multiple objects and applying point cloud inputs for control. \n\n\\textbf{Point Cloud in Robotic Manipulation.} \nPoint cloud representation has been widely applied in the robotics community. Researchers have studied matching the observed point clouds to an object in a grasping dataset~\\citep{pointcloud-registration} and executing the corresponding grasping action~\\citep{dexnet-2017rss, bohg-survey}. For learning-based approaches, one line of research has focused on first estimating grasp proposals or affordance given the point cloud input and then planning accordingly for manipulation~\\citep{grasp-detection-pc, pointnet-gpd, 6dof-grasp, s4g, sparse-pc-grasp, wu2020grasp, wei2021gpr, ral2022-grasp-pc-dvgg, wang2022goal}. While these methods are designed for parallel-jaw grippers, recent advancements have also been made for grasping with dexterous hands with similar approaches~\\cite{Andrews2013, varley2015generating, brahmbhatt2019contactgrasp,lu2020planning}. However, this line of methods requires feedback from motion planning to estimate whether a grasp is plausible. Oftentimes, a stable grasp pose is proposed but it is not achievable by planning. To achieve efficient and flexible manipulation, a Reinforcement Learning policy with point cloud inputs is proposed~\\cite{corl21-inhand, rl-pc-inhand-generalization,ilad}, which allows the robot hand to flexibly adjust its pose while interacting with the object. However, these approaches still face challenges in transferring to the real robot, given the noisy, occluded point clouds in the real world, and training RL policy with noisy point clouds increases the optimization difficulty. In this paper, we propose to use imagined point clouds, contact information, and multi-object training to combat these problems and achieve Sim2Real generalization.\n\n\\textbf{Using Contact Information for Manipulation.}\nHumans can manipulate objects purely from tactile sensing without seeing them. This biological fact inspires researchers to integrate contact and tactile information into the learning pipeline~\\cite{tactile-feedback,yuan2017gelsight,calandra2018more,murali2018learning,lambeta2020digit,lee2020making,bhirangi2021reskin}. For example, both tactile and visual information inputs are combined together in~\\citep{lee2020making} for decision making. The tactile information is also utilized as inputs for RL-based manipulation policies~\\cite{tactile-features, tactile-intrinsic, tactile1, tactile-guided,xu2021towards}. For example, visual-tactile sensor is used with an Allegro hand in simulator for playing piano~\\citep{xu2021towards}. Instead of using contacts as inputs, some methods also use contact and tactile information as reward to encourage exploration and boost policy learning~\\cite{tactile-intrinsic, tactile-guided}. Motivated by these works, we provide a novel design of reward based on each contact link of the robot hand, which encourages more reasonable grasping behavior. Our design does not require a real tactile sensor when training in simulation or deploying in the real robot. \n\\section{Environment Details}\n\\vspace{-0.08in}\n\n\\textbf{Object Set.} \nFor single-object experiments, we use the single object mesh from the YCB dataset for training and testing. For multi-object experiments, we chooses 10 objects from the ShapeNet dataset~\\cite{shapenet} for training, and another 40 objects for testing in simulation. The test objects of real-world experiments are described in the main paper.\n\n\\textbf{Object Pre-processing.} \nTo use the above object models in the simulation experiments, we apply the object preprocessing. The preprocessing includes two steps: scaling and convex decomposition.\nThe scaling step is to ensure that the object size is within a proper ranger for manipulation. The YCB objects~\\cite{ycb} are captured by the real scan, so the scale of object mesh from YCB dataset aligns with the real counterpart and can be used for robot manipulation directly. Different from the YCB dataset, the ShapeNet dataset does not contain any scale information, e.g. the mug in ShapeNet can be larger than the robot. Therefore,  we scale each object based on the diagonal length of its bounding box. For each category, we manually select a diagonal length so that all object instances from the category will have the same bounding-box diagonal length after scaling. Besides, we will not use objects with non-manifold geometry to avoid instability in physical simulation. The next step is convex decomposition. We use V-HACD~\\cite{mamou2009simple} to decompose both YCB object and ShapeNet object into convex parts with default parameters. We will not use the object with more than 40 parts after convex decomposition for both stability and efficiency of the physical simulation.\n\n\\textbf{Reward.}\nAs is mentioned in Section 3.2 of the main paper, the overall reward for our task is composed of four parts: reach, contact, lift, and action penalty.\n\n\\begin{equation}\n    \\mathcal{R} = w_{\\rm reach} r_{\\rm reach} + w_{\\rm contact} r_{\\rm contact} + w_{\\rm lift} r_{\\rm lift} + w_{\\rm penalty} r_{\\rm penalty}.\n\\end{equation}\nIn the implementation, we set the lift reward as the the difference between object current height and object initial height $r_{\\rm lift} = h_{\\rm  current} - h_{\\rm init}$. The action penalty reward $r_{\\rm penalty} = - ||a||_2^2$. For reaching reward, it consists of the distance between object and target, the distance between finger tip and the target. \nThe weight for the four reward terms are: $w_{\\rm reach}=1$, $w_{\\rm contact}=0.5$, $w_{\\rm lift}=10$, $w_{\\rm penalty}=0.01$. Recall that the lift reward $r_{\\rm lift}$ is set to 0 if the contact reward $r_{\\rm contact}$ is $0$. Intuitively, it ensures that the robot can only receive the reward for lifting the object up if the robot hand is in good contact with the object. This design prevents the agent from using a large force to knock the object into the air or using unstable contact to lift the object up.\n\n\\section{Learning Details}\n\n\\textbf{RL Training.}\nWe use on-policy RL training for setting except the distilling experiments. We select PPO as the RL algorithm to train the point cloud based manipulation policy. The hyper-parameters of the PPO are shown in Table~\\ref{tab:ppo}.\n\n\\textbf{Network Architecture}\nThe RL agent use the PointNet based architecture as the visual backbone. As shown in Figure 3 of the main paper, we first concatenate the visual features from PointNet and the propriception feature from the MLP. This concatenated feature is then shared by both value network and policy network to predict value and action. The details for the network architecture are shown in Table~\\ref{tab:arch}.\n\n\\begin{minipage}{.25\\textwidth}\n\\resizebox{0.95\\columnwidth}{!}{%\n\\begin{tabular}{l|c} \\shline\nParameter & Value   \\\\ \\shline\nMini-Batch Size & 500  \\\\ \\hline\nLearning Rate & 3e-4 \\\\ \\hline\nClip Range & 0.8 \\\\ \\hline\nHorizon & 200 \\\\ \\hline\nEpoch & 10 \\\\ \\hline\nSteps per Iteration & 10 \\\\ \\hline\n\\end{tabular}\n}\n\\captionof{table}{PPO Parameters.}\n\\label{tab:ppo}\n\\end{minipage}\n\\hspace{0.5em}\n\\begin{minipage}{.73\\textwidth}\n\\resizebox{0.95\\columnwidth}{!}{%\n\\begin{tabular}{ccc}\\shline\nModule & Architecture & Output Dim\\\\\\shline\n\n\\multirow{2}*{Visual Feature Extractor}  & PointNet Local Channel: (64, 128, 256) & 256 \\\\ \n& PointNet Global Channel: (256, ) & 256 \\\\ \\hline\n\n\\multirow{1}*{State Feature Extractor}  & MLP: (64, 64) & 64 \\\\ \\hline\n\n\\multirow{1}*{Actor}  & MLP: (64, 64) & 64 \\\\ \\hline\n\n\\multirow{1}*{Critic}  & MLP: (64, 64) & 64 \\\\ \\hline\n \n\\end{tabular}\n}\n\n\\captionof{table}{Network Architecture.}\n\\label{tab:arch}\n\\end{minipage}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Background and Related Work}\\label{sec:related-work}\nThe original author of grammar-based visualization concept is \\cite{Wilkinson}, changing the way\nscientists and developers think, as well as inspiring them. Stanford's team in \\cite{Polaris} has\nplanted the seed of the grammar of graphics' software, after which gobs of systems developed ---such\nas Tableau in \\cite{tableau}, ggplot2 in \\cite{ggplot2}; although their user preferences'\ncustomization is limited, they brought in abstraction of data models, graphical geometries, visual\nencoding channels, scales, and guides (e.g., axes and legends), yielding more expressive design\nspace. JSOL's architecture heavily influenced by these works; inheriting from Wilkinson's\ngrammar and components; rendering basic graphical scenes using these components and the grammar that\nmakes these components generate a full scene. The component instance could be a visual channel such\nas \\textit{position}, \\textit{color}, \\textit{shape}, and \\textit{size}, may also include common\ndata transformations, as a sample, binning, aggregation, sorting, and filtering. The grammar is the\nvalidation step that is responsible for making these components work in flow and mapping data\nattributes to its component in the scene.\n\n\\subsection{Specification}\nVisualization libraries are of a distinct number of architectures' types: \n\\begin{itemize}\n  \\item \\textbf{Hierarchical:} Layers and components are implemented in a hierarchical view, so that\n  when a new scene graph introduced, authors build it in layers employing the components. In case\n  that the graphic depends on the same component they will be linked.\n  \\item \\textbf{Parallel:} Building layers and components in two parallel independent levels, so\n  when a scene graph proposed users use the component given by the system to implement it or to\n  customize their graphics.\n  \\item \\textbf{Hybrid:} It is a combination of the previous\n  couple of types. The compiler and layers are hierarchical; however, layers' design is parallel\n  (JSOL\\space adopts this type).\n\\end{itemize}\n\nThe compiler added on the top layer makes JSOL\\space a declarative \\textit{domain specific\n  language (DSL)} for visualization design; by decoupling specification from execution details,\ndeclarative systems allow users to focus on specifying their application domain without limiting\ntheir abilities to customize.\n\n\\cite{vega-lite} and \\cite{Bostock2009Protovis} followed the same approach of the DSL compilation criterion;\nhowever, they use a declarative framework for mapping data to visual elements. Nevertheless,\nJSOL\\space does not strictly impose a toolkit-specific lexicon of graphical marks; instead,\nJSOL\\space directly maps data attributes to the HTML5 canvas element.\n\n\\subsection{Other Libraries}\\label{sec:comparative-study}\n\\subsubsection{GGPlot2}\\label{sec:ggplot2}\nIn \\cite{ggplot2}, there is excessive concentration on low-level details which makes plotting a\nhassle (e.g., drawing guides). However, it provides a dominant model for graphics that makes it easy\nto produce complex multi-layered scenes. You can hardly find demerits in \\textit{ggplot2}, but one\nof them is the language it uses, \\textit{R}, as discussed earlier, is a programming language that\nuses a specific interpreter, making it laborious for users to install for developing some simple\nvisualization scenes, yet it is normal for R audience to use. Furthermore, R does not support\nvariables' or methods' labeling which makes users struggle when using it. Besides, it only works on\nCPUs, making it slow against other libraries.\n\n\\subsubsection{D3}\\label{sec:d3}\nMany scientists, researchers, developers, and even programs make use of \\textit{D3}, it is nearly\nsuitable for each user, but there are few crucial drawbacks; it is entirely low-level visualization\ngrammar which is hard for novice users to manage, one extra drawback appears when working on an SVG\nelement in big data, which is generating a ton of elements that may break browsers as a result of\nthe load added on the browser's bag.\n\n\\subsubsection{Protovis}\\label{sec:protovis}\nWe conducted a comparison between \\cite{Bostock2009Protovis} and JSOL\\space in Table \\ref{table:1},  \\cite{Bostock2009Protovis} composes custom views of data with simple marks such as pies (\\includegraphics*[height=\\fontcharht\\font`\\B]{basic_pie}) and dots. Unlike low-level graphics libraries that quickly become tedious for visualization, Protovis defines marks through dynamic properties that encode data, allowing inheritance, scales and layouts to simplify construction. However it is no longer under active development and that make it unbearable for different types of users.\n\n\\subsubsection{Vega}\\label{sec:vega}\nIn a previous section we saw that Vega specification is simply a JSON object that describes an interactive visualization, and that may appear akin to JSOL. However vega uses \\cite{2011-d3} as a backend engine to produce and provide a SVG visulization components given a user-grammars. On the other hand, JSOL\\space specification may be cross-compiled to provide a reusable visualization component, of course given the user input grammars.\n\n\n\\subsubsection{Vega-Lite}\\label{sec:vega-lite}\nWe conducted a comparative study between \\cite{vega-lite} and JSOL\\space in Table \\ref{table:1},\nthese two libraries are chosen for a particular purpose which is that the users may confound that\nthey are equal, true that they are similar in attributes, but not in drawbacks. JSOL\\space\nspecifically developed for making a complete detailed scene, that is why it collects all\nspecification that developers or scientists seek to configure their scene.\n\\begin{table}[tbh]\n  \\centering\n  \\caption{Comparative Study}\\label{table:1}\n  \\renewcommand{\\arraystretch}{1.0}\n  \\begin{tabular}{|l|l|l|l|l|l|}\n    \\hline\n    \\multicolumn{3}{|l|}{Grammar of Graphics layers} & Protovis & Vega & JSOL  \\\\ \\hline\n    \\multicolumn{3}{|l|}{Transformation layer} & Done & Done & Done \\\\ \\hline\n    \\multicolumn{3}{|l|}{Data layer} & Done & Done & Done \\\\ \\hline\n    \\multirow{17}{*}{\\begin{tabular}[c]{@{}l@{}}Geometry \\\\ layer\\end{tabular}} & \\multicolumn{2}{l|}{Point} & Done & Done & Done \\\\ \\cline{2-6} \n    & \\multirow{4}{*}{Bar} & bar chart & Done & Done & Done \\\\ \\cline{3-6} &  & \\begin{tabular}[c]{@{}l@{}}Stacked bar\\\\ chart\\end{tabular} & Done & Done & Done \\\\ \\cline{3-6}\n    &  & Histogram & Done & Done & Done \\\\ \\cline{3-6} &  & \n    \\begin{tabular}[c]{@{}l@{}}Vertical bar\\\\ chart\\end{tabular}  & Done & Done & Done \\\\ \\cline{2-6}\n\t& \\multirow{2}{*}{Area} & Area chart & Done & Done & Done \\\\ \\cline{3-6}\n    &  & \\begin{tabular}[c]{@{}l@{}}Stacked Area \\\\ chart\\end{tabular} & Done & Done & - \\\\ \\cline{2-6}\n\t& \\multicolumn{2}{l|}{Text} & Done & Done & Done \\\\ \\cline{2-6}\n\t& \\multicolumn{2}{l|}{Line} & Done & Done & Done \\\\ \\cline{2-6}\n\t& \\multicolumn{2}{l|}{Marks} & Done & Done & Done \\\\ \\cline{2-6}\n\t& \\multicolumn{2}{l|}{HLine} & Done & Done & Done \\\\ \\cline{2-6}\n\t& \\multicolumn{2}{l|}{VLine} & Done & Done & Done \\\\ \\cline{2-6}\n\t& \\multicolumn{2}{l|}{Pie chart} & Done & Done & Done \\\\ \\cline{2-6}\n\t& \\multicolumn{2}{l|}{Arc chart} & Done & Done & Done \\\\ \\cline{2-6}\n\t& \\multicolumn{2}{l|}{Picture} & Done & Done & Done \\\\ \\cline{2-6}\n\t& \\multirow{2}{*}{Tick} & Dot plot & Done & -& - \\\\ \\cline{3-6}\n\t& & Strip plot & Done & - & - \\\\ \\hline\n    \\multicolumn{3}{|l|}{Scale layer} & Done & Done & Done \\\\ \\hline\n    \\multirow{6}{*}{Axes Layer} & \\multicolumn{2}{l|}{Cartesian coordinate} & Done & Done & Done \\\\ \\cline{2-6}\n    & \\multicolumn{2}{l|}{Coordinate equal} & Done & - & Done \\\\ \\cline{2-6}\n    & \\multicolumn{2}{l|}{Coordinate flip} & Done & - & Done \\\\ \\cline{2-6}\n    & \\multicolumn{2}{l|}{Coordinate polar} & Done & - & Done \\\\ \\cline{2-6}\n    & \\multicolumn{2}{l|}{Parallel coordinate} & Done & Done & Done \\\\ \\cline{2-6}\n    & \\multicolumn{2}{l|}{Polar parallel coordinate} & - & - & Done \\\\ \\hline\n    \\multicolumn{3}{|l|}{Aesthetics layer} & Done & Done & Done \\\\ \\hline\n  \\end{tabular}\n\\end{table}\n\n\n\\section{Conclusion}\\label{sec:conclusion}\nWe have presented the JavaScript Open-source Library (JSOL), the grammar of graphics JavaScript library which has the power to\ngenerate the most captivating plots with no limitations. JSOL\\space is implemented using\nJavaScript, which enabled it to be integrable to many other fields. We also demonstrated how it\nworks internally by describing its layers' specification, role, and the architectural design. We\ndiscussed how JSOL\\space accompanies the Human-Computer Interaction (HCI) principles. Manipulating data is a crucial part of JSOL, the way data loaded from various sources, and how transformation (e.g., filtering and grouping) and statistical (e.g., mean and std)\noperations are applied, the way of generating scales for mapping data fed to the geometries layer\nproffered JSOL\\space an edge while comparing to other libraries.  Another contribution of the\nlibrary is the way of combining low-level visualization layer with the compilation process. And we\nconsidered a comparative study between JSOL\\space and great libraries such \\cite{2011-d3} and\n\\cite{ggplot2}, and a comprehensive comparison with \\cite{vega-lite}. Finally, we explained how to\nuse JSOL\\space through numerous examples such as the parallel plot in Cartesian and Polar\nscales.\n\n\n\\section{The JSOL}\\label{sec:brief-insight-libn}\nJSOL\\space amalgamates graphics' grammar with a state-of-art compilation process. Throughout\nthis section, we will cover how a simple scene is generated and constructed, together with how data\nprocessed; besides, the layers' design.\n\n\\subsection{Unit Specification}\nDoubtlessly, a scene must have a data variable. After all, that is what going to be a graphic. There\nare also some customization parameters ---such as transformation, geometries, properties, and a set\nof encodings. Transformation layer is responsible for applying filters and aggregation. After which,\nthe geometry layer visually encodes the incoming input.\n\n\\lstinline[style=customJS]{scene := (data, tansformations, geometries, properties)}\n\nDefining \\textit{properties} is optional; however, it is significant when it comes to details; as a\nway of illustration, imagine the case where a user wants to declare points' color in the scene or\nthe type of the data variable (e.g., CSV). \\textit{Scale} is also necessary as it determines how\ndata attributes mapped to traits of geometries. \\textit{Axes} enhances the readability of scales.\n\n\\lstinline[style=customJS]{properties := (geometries, data, functions, scale, axes, guide)}\n\n\\subsection{Layers of~JSOL}\nAs the unit specification complexity issue is evolving daily, layers built with simplicity in mind. Several parts of the kernel is the same as \\cite{2011-d3}'s, the other parts modified to flatter the interaction between the user and the library. We will discuss each layer's structure and role.\n\n\\subsubsection{Data Layer}\nAs the layer's name states, its role is to read data from various sources, for example, flat files (e.g., CSV and TSV), and open-standard files (e.g., JSON). Data could be an array of arbitrary values, numbers, strings,\nor objects. After reading the data source, the layer stores it in a pre-defined data structure for easily referencing through other layers; the data structure also supports CRUD operations. As shown in the example below, each dataset has a \\textit{name} for the consistency of multiple dataset loading, the \\textit{values} parameter is the source that contains the dataset, the \\textit{format} defines the source type.\n\n\n\\begin{lstlisting}\n\t\"data\": [{\n\t\t\"name\": \"troops\",\n\t\t\"values\": \"troops.csv\",\n\t\t\"format\": { \"type\": \"csv\" }\n\t\t}, {\n\t\t\"name\": \"cities\",\n\t\t\"values\": \"cities.csv\",\n\t\t\"format\": { \"type\": \"csv\" }\n\t}]\n\\end{lstlisting}\n\n\n\\subsubsection{Transformation Layer}\nThis layer's responsibility is to perform analytical operations on data\ngathered by the data layer, and it helps the user to perform many\ntransformations including filtering and grouping. Executing these\noperations is necessary to optimize the processing time. The layer\nexpects that its input comes from the data layer to do a valid procedure on the dataset. It consists of two sub-layers taken from \\cite{Wilkinson} which are \\textit{variables} and \\textit{algebra}.\n\n\\begin{lstlisting}\n\"transform\":[{\n  \"lang\": \"R\",\n  \"function\": \"fibonnaci\",\n  \"properties\": {\n    \"data\" : \"static\",\n    \"length\" : 20,\n    \"field\" : \"x\",\n    \"name\": \"fibonnaci_x\"\n  }\n}]\n\\end{lstlisting}\n\n\\paragraph{High-dimensional Spaces}\nLiving in a 3-D world restricts us from visualizing structures in high-dimensional spaces. The curse of dimensionality, as called by \\cite{bellman}, has been an impediment for so long with various solutions; however, we will only recall \\textit{nesting} as embracing all solutions is currently out of our scope.  Nesting is a way to circumvent the challenge, in which we represent any two dimensions of the data on the X-axis and Y-axis. Nevertheless, points designed by these axes are separate charts (e.g., pie, bar, image and so on); in JSOL, we use a 9-block greyscale image in which each block's greyscale level is equal to the dimension's value \\textit{(see fig. \\ref{fig:nesting})}.\n\n\n\\begin{figure}[tbh]\n\t\\centering\n\t\\includegraphics[width=0.5\\textwidth]{nesting_property_figure}\n\t\\caption{nesting caption}\n\t\\label{fig:nesting}\n\\end{figure}\n\\subsubsection{Scales Layer}\nA visual encoding is called \\textit{scale}. As to draw data in a scene, we need to map data values\nto their corresponding geometries. This layer is taken from \\cite{2011-d3}, supports both ordinal\nand quantitative (linear, logarithmic, exponential, quantile) values.\n\n\n\\begin{lstlisting}\n\"scales\": [{\n  \"name\": \"yscale\",\n  \"type\": \"linear\",\n  \"range\": {\n    \"type\": \"range\",\n    \"value\": \"height\"\n  },\n  \"domain\": {\n    \"data\": \"crimea\",\n    \"field\": \"economy (mpg)\"\n  }\n}]\n\\end{lstlisting}\n\n\\subsubsection{Statistics Layer}\nThe use of the statistics layer is optional since applying statistical functions is not the central\nobjective of each user. It is managed using the R programming language which gives us a handicap\nwhile comparing to other libraries as R is a user-friendly language built specially for statistical\nmodeling and inference, making it light to execute any statistical function on data in a few lines\nof codes. Moreover, it has a knowledgeable community and rich documentation.\n\n\\subsubsection{Geometry Layer}\nJSOL~implements the same \\texttt{d3.svg.shape} element provided by \\cite{2011-d3} in an HTML5 canvas element that is suitable for charting, providing the power of computational speed supported from \\textbf{WebGL}; the arc, for example, builds elliptical arcs such as pie (\\includegraphics*[height=\\fontcharht\\font`\\B]{basic_pie}), donut and cox-comb (\\includegraphics*[height=\\fontcharht\\font`\\B]{cox-comb}) charts via formulating arbitrary data to paths. Typically, this function bounded to the arc attribute, note that the radius and angles of the arc can be specified both as constants or callback functions. Additional shapes provided for areas, lines, mark symbols, etc.\n\n\\begin{lstlisting}\n\"geom\" : [{\n  \"type\" : \"Point\",\n  \"data\" : \"static\",\n  \"properties\" : {\n    \"x\" : \"xscale\",\n    \"y\" : \"yscale\",\n    \"fillColor\" : \"zscale\"\n  }\n}]\n\\end{lstlisting}\n\n\\subsubsection{Axes Layer}\nThe layer is a crucial step in the graphical scene since it maps the scale to a meaningful form that is comprehensible by human eyes. Axes visualize spatial scale mappings using \\textit{ticks}, \\textit{grid lines}, and \\textit{labels}. JSOL\\space supports lot of axis based on a given scale, and currently supports axes for Cartesian (rectangular) and Polar coordinates.\n\n\\begin{lstlisting}\n\"axes\": [{\n  \"type\": \"x\",\n  \"data\": \"static\"\n  \"field\": \"x\"\n  \"orient\" : \"bottom\"\n  \"grid\" : true\n}, \n{\n  \"type\": \"y\",\n  \"scale\": \"yscale\"\n}]\n\\end{lstlisting}\n\n\\subsubsection{Guide Layer}\nBoth \\textit{guides} and \\textit{axes} visualize scales; but guides aid interpretation of scales\nwith ranges such as colors, shapes, and sizes, whereas axes aid interpretation of scales with\nspatial ranges. Similar to scales and axes, guides can be defined either as a top-level or low-level\nvisualization.\n\n\\begin{lstlisting}\n\"guides\": [{\n  \"type\": \"legend\",\n  \"domain\": { ... },\n  \"properties\": {\n    \"title\": { \n      \"name\": \"key map\"\n    },\n    \"position\": { ... }\n  }\n}]\n\\end{lstlisting}\n\n\\subsection{Compiler of JSOL} \nThe JSOL's compiler is made purposefully to transform the library from low-level visualization library to a high-level one, making it effortless for users to interpret their parameters to build imaginative scenes, the compiler also has predefined value for each parameter; consequently, the headache of passing values uprooted.\n\nThe user's specifications pass through several phases or stages\nto become a beautiful chart, the first one is \\textit{scanning} user's\nspecifications and divide them into sections basing on their's role; \\textit{parse}, which is subject to prepare the low-level representation and fill the missing parameters; \\textit{linking}, where layers' objects attached to each other, and finally \\textit{assemble} is where the full chart comes alive.\\\\\n\nIn JSOL~compilation process the first stage is scanning, here where the users' specification been parsed and validated, JSOL~ provides tons of rules, for example each type of scales like linear scale is considered a rule, can be applied by users and developers each of these rules follow some validation steps so we should certain that the JSOL~ users will follow it ensure that the scene will be generated. For example some geometry components must have a color pallets, so if the user forgot to set it the compiler must set it to a default color pallet and so on.\n\nSecond things second is connecting phase, after the user's specifications are validated, this phase is responsible for generating these specifications, by generating we means that should transform the user's specifications to layers executable classes, function and api's. Also these transformations require searching for a huge combinations tree of components, and might be some specification exists that are not required so these phase is responsible to check that each specification transformation is required to build the scene, for example the user might put for the same data source a file path and file url, and one of them are necessary.\\\\\n\nAfter builing phase connecting is required, and here's where linking phase responsibility comes. Linking main responsibility is to connect layers to each other, for example axis and geometry layers requires scale layer, so the JSOL~ requires to traverse for all user's specification and search for each node connection and connect\/link it for it's requirements. Another responsibility is to check for linking or connecting acceptability, not all layers accept to connect to each other, so JSOL~ require to check if each user's connection is accepted or not.\\\\\n\nLast thing last is assemble phase, where the user will see the result of his\/her hand. assemble phase take all transformed specification that turned to layers functions\/api and execute it to the selected HTML5 Canvas, the phase is similar to code generating and optimizing where it takes each layer's functions and put them in a queue to be executed in the same order, for example, we have to execute scales before axis, and run color palettes before geometries, and so on.\n\n\n\\subsection{JSOL\\space from \\textbf{H}uman \\textbf{C}omputer \\textbf{I}nteraction perspective}\n\\subsubsection{Discoverability}\nFrom (Norman, 2002)'s point of view, \\textit{discoverability} is to figure out possible actions and\nhow to do them. Meanwhile, (Nielsen, 1994) suggests that \\textit{discoverability} is minimizing the\nuser's memory load by making objects, actions, and options visible as possible. Instructions for the\nuse of the system should be visible or easily retrievable. In JSOL, layers are understood by\ntheir name (i.e., geometries layer clearly read that it is for generating geometrical objects in a\nscene).\n\n\\subsubsection{Mapping}\n\\cite{Norman} believes that \\textit{mapping} is a technical term which means the relationship\nbetween two instances of things (data and its visual representation in our case). On the other hand,\n\\cite{Nielsen} says that a system would exhibit \\textit{mapping} if it speaks the users' language,\nwith words, phrases, and concepts familiar to the user, rather than system-oriented terms. Follow\nreal-world conventions, making information appear in a natural and logical order. As JSOL\\space\nis a visualization library, its foremost concern is mapping data to graphics, using same words and\nphrases the other libraries apply, making understanding it or switching from any library a soft\ntouch.\n\n\\subsubsection{Affordance}\nOne way to make the interface both manageable and usable is to design interfaces that by their very\ndesign inform users how to. \\cite{Norman} defined \\textit{affordance} to be a relationship between\nthe properties of an object and the capabilities of the agent that determine how the object could be\npossibly used. As illustrated earlier, JSOL\\space uses keywords and functions which tell the\nuser how it operates.\n\n\\subsubsection{Structure}\nAs \\cite{Constantine} proposes, a software would employ the \\textit{structure} principle if its\nuser's interface design is organized purposefully, in meaningful and useful ways based on precise,\nconsistent modes that are apparent and recognizable to users, putting related things together and\nseparating unrelated things, differentiating different things and making similar things resemble one\nanother. JSOL's user grammar based components which come from internal layers provide a\nstructure by itself that helps to manage the overall graphical scene generation.\n\n\\subsubsection{Ease\/Comfort}\nEase and Comfort are two similar ideas come from the principles of universal design,\n\\cite{uni_design} defined \\textit{ease} as using software efficiently, comfortably and with a minimum\nof fatigue. While defining \\textit{comfort} as presenting appropriate size and space for approach,\nreach, manipulation, and use regardless of user's body size, posture, or mobility. It is quite\napparent that these two concepts achieved in JSOL\\space because of the compiler as illustrated\npreviously.\n\n\\subsubsection{Flexibility}\nDefined by \\cite{Nielsen} as speeding up the interaction for the expert user such that the system\ncan cater to both inexperienced and experienced users. Allowing users to tailor frequent\nactions. \\cite{uni_design} represents it as either the design accommodated a wide range of individual\npreferences and abilities or not. JSOL\\space obeys both definitions since it allows its users to\ngenerate the same plot in many ways, hence making proper for both novice and expert users from\nvarious domains.\n\n\n\\section{Introduction}\\label{sec:introduction}\nThe most popular visualization steps, generally speaking, and particularly for data visualization is as easy as to get your dataset into a visualization library (or system), defining your charting information, and then you have the chart you need. These steps could be implemented in a multitude of ways; while the technology is rising, most of the software is attempting to cope up and contain the top used graphs and visualization scenes and also modifying the way to generate these graphs by using a user interface to make it milder to construct graphs with few clicks. This methodology has the advantage when it comes to quick and easy use, but the disadvantage is that this software is restrictive and does not allow internal customizations and modification. The other alternative is to use a charting system such as vector drawing which permits the user to customize every single piece of the graph, the interest here is that the user can build the chart freely without any restrictions. However, it is troublesome to use for most of the users and consumes a considerable amount of time to build a full scene; thus, the need arises for a visualization library that combines all of the merits.\n\n\\subsection{Grammar of Graphics and Other Systems} \nGrammar of Graphics (GoG) is a set of rules put together to create a scene that expresses the data; divided into two categories, \\textit{Low-Level} grammar, that is used to customize each piece of the visualization scenes; mainly used for exploratory data analysis as their primitives offer fine-grained control ---like scenes used in analysis tools. By way of illustration, \\cite{2011-d3} and \\cite{Bostock2009Protovis}. On the other hand, \\textit{High-Level} grammar is used to make traditional visual plots expeditiously. It is useful to users and analysts for rapid development. Examples include \\cite{ggplot2}, \\cite{vega-lite}, and grammar-based systems such as \\cite{tableau}. High-level grammars are more prevalent as users prefer conciseness over expressiveness. Furthermore, In contrast to the \\textit{low-level} grammars, the \\textit{high-level} grammars use default values to resolve ambiguities of visualization, hence development is comfortable for analysts and developers, At last there is some libraries that are in the middle level such \\textbf{protovis}, those libraries tries to combines the ease of use and the non-restrictive methodology to generate a middle-ware level that combine the advantages of each of low-level and high-level.\\\\\nChoosing between the low-level or high-level is not easy, user should take in consideration some concepts, such as the time-consumption or in other words how long it takes to build the scene? and thats called \"\\textit{efficiency}\", another concept is do the user have the knowledge to build the scene or is the knowledge exists to make the user learn how to use the system\/library and that's called \"\\textit{accessibility}\", at last can the user build the scene or not? and this is the \"\\textit{expressiveness}\", every user should take these concepts in consideration when it comes to generate a beautiful plots, sometimes it confuses the users when it comes to choose between those levels and that's the main reason for inventing the middle-level and the hybrid-libraries.\n\n\n\\subsection{Why Another Library?}\\label{sec:why-another-library}\nJSOL~belongs to high-level grammars because of the compiler it presents, which has built-in default values passed to low-level layers when not specified by the user. As a result, rapid development accomplished. Moreover, JSOL's layers are a real representation of low-level grammars: representing customization for each visualization component.\n\n\\bigskip\n\nJSOL's layers are built using \\textit{JavaScript} scripting language with its compiler which uses \\textit{JSON} (JavaScript Object Notation) to interpret the user's definition of specifications. Consequently, it entirely works and runs in web browsers that support JavaScript. Specifically, JSOL\\space runs in \\textit{HTML5 canvas} elements that allow for high\nperformance.\n\nUsers may think that the debugging will be problematic; however, that is not the case because of JSOL's compiler having an inherent error handling layer that provides the user with a proper debugger and informative error\/warning messages.\n\nThere are some objectives we cannot miss in any visualization library,\nwhich are:\n\\begin{itemize}\n  \\item\\textbf{Performance}: high-level abstraction may limit the user's ability to generate fast\n  visualization scenes. However, building charts on top of HTML5 Canvas elements allow\n  JSOL\\space to use \\textit{GPU} acceleration to speed up the process, even though we will be\n  shifting the responsibility of data representation and transformation to the user and no longer\n  treat it as our concern.\n  \\item\\textbf{Debugging}: trial and error is a fundamental part of the development and learning\n  processes; accessible tools must be designed to support debugging when errors occur. As\n  JSOL\\space was built using JavaScript, it allows users to use various types of debugging\n  tools, on account of the built-in test\/debug layer that enhances the package's efficiency.\n\\end{itemize}\n\n\\bigskip\n\nJSOL~comprises low-level layers, called \\textit{modules} or \\textit{kernels}, that are made specially to carry all the burden of a given task. Visualization primitives alternatively named \\textit{marks} in \\cite{vega-lite} or shape in \\cite{2011-d3}, provide geometries or shapes for chartings, such as a bar, point, and an arc. Another layer is the data encoding layer, also called \\textit{scale}, it allows mapping data points to encoded pixels to bring data to life, for using a particular scale layer you need to generate a reference via the \\textit{Axes} layer which allows creating \\textit{Cartesian} or \\textit{Polar} coordinated charts and plots. There is also a layer for data munging, called \\textit{algebra}, made for practicing operations on data such as \\textit{join} (left and right), \\textit{cross}, and \\textit{nesting} which allows visualizing in higher dimensions as we will later see in the examples section. Over and above, there are statistics for making summary statistical operations on data, such as \\textit{mean}, \\textit{std}, and so on. An expansive and comprehensive description of the library illustrated in Sec \\ref{sec:brief-insight-libn}.\n\n\\bigskip\n\nThe JSOL~compiler synthesizes and combines all low-level rules and specification gained from the other layers within respect for a given data and validates the users' rules through all layers in JSOL\\space using the handler which manages default values for visualization primitives. In a wide range of examples, we will confirm how the compiler takes a tremendous advantage of the lower-level encoding and visualization layer of the library to bring a high-level specification to visualizations, and later will demonstrate how the compiler works and what are the minimum values must be given to make visual plots.\n\n\\bigskip\n\nOn the one hand, visualization systems considered a subordinate of graphical ones. Moreover, they are the responsible entities to produce and process representation of data graphically and its interaction to gain insight into the data. On the other hand, graphical systems used to generate drawings in general, offering the utmost flexibility. They also have different types (discussed in Sec. \\ref{sec:related-work}). Nevertheless, primarily, they were not tailored for visualization purposes.\n\n\n\\section{Examples}\\label{sec:exampl-results-disc}\n\\subsection{Parallel Coordinates Example}\n\\begin{figure}[tbh]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{parallel_coordinates}\n  \\caption{Caption place holder}\\label{fig:parallel_cooridnates}\n\\end{figure}\nFigure~\\ref{fig:parallel_cooridnates} presents the benchmarking results, as illustrated in the previous\nsection, for all JSOL\\space visualization, the time it takes starting from page loading to\nrendering scene graphs typically faster than most of the libraries; the main reason is the\n\\textit{HTML5 Canvas} built on top of WebGL that uses graphics card as a computational booster.\n\n\\bigskip\n\nFirst and foremost, we have to load the data variable, CSV type in this situation (For JSON, specify\nthe type to be \\textit{\"json\"}). We can import from multiple data sources and execute some\noperation on them using \\textit{transformation} and \\textit{statistics} layers.\n\\begin{lstlisting}\n\"data\": [{\n  \"name\": \"crimea\",\n  \"values\": \"crimea-parallel.csv\",\n  \"format\": {\n    \"type\": \"csv\"\n  }\n}]\n\\end{lstlisting}\nAfter finishing data loading, the \\textit{transformation} phase begins, since we do not need it in the situation, we specify it to be empty.\n\\begin{lstlisting}\n\"transform\": []\n\\end{lstlisting}\n\nWe then move to set \\textit{scales} used for the axes, in this example, we will use the \\textit{linear} scale as the data does not need scaling. We can use as many scales as we want.\n\\begin{minipage}[t]{0.25\\textwidth}\n\\begin{lstlisting}\n\"scales\": {\n  \"name\": \"xscale\",\n  \"type\": \"linear\",\n  \"range\": {\n    \"type\": \"range\",\n    \"value\": [0, 330]\n  },\n  \"domain\": {\n    \"data\": \"crimea\",\n    \"field\": \"name\"\n  }\n}\n\\end{lstlisting}\n\\end{minipage}%\n\\begin{minipage}[t]{0.25\\textwidth}\n\\begin{lstlisting}\n\"encoding\": {\n  \"x\": {\n    \"field\": \"wavelength\",\n    \"type\": \"quantitative\",\n    \"scale\": {\n      \"domain\": [300,450]\n    }\n  },\n  \"y\": {\n    \"field\": \"power\",\n    \"type\": \"quantitative\"\n  }\n}\n\\end{lstlisting}\n\\end{minipage}\nDefining \\textit{axes} follows placing scales. An axes must have some characteristics such as:\n\\begin{itemize}\n  \\item \\texttt{Type:} the type in the snippet below implemented uniquely for polar parallel coordinates. There are tons of axes types.\n  \\item \\texttt{Properties:} define aesthetics of the axes; annotation sets axes' title text, position, and color.\n  \\item \\texttt{Transform:} sets the transformation of axes, in our case, the axes uses the \\textit{power function} transformation.\n\\end{itemize}\nAnd as you see in comparison code snippets, the difference between JSOL~ and \\textbf{Vega-Lite}, you can see that vega-lite for each variable you define it's scale or uses the default scale for the type(quantitative in example), but if you are using JSOL~ you only need to define the scale and use it's name to invoke it on as many variables\/axes as you want, and here you can sense the power of dynamicity.\n\n\\begin{minipage}{0.3\\textwidth}\n  \\lstset{framexrightmargin=-2pt, basicstyle=\\footnotesize}\n\\begin{lstlisting}{}\n\"axes\": {\n  \"type\": \"coord_polar_parallel\",\n  \"properties\": [{\n     \"type\": \"y\",\n     \"data\": \"crimea\",\n     \"field\": \"economy-mpg\",\n     \"orient\": \"left\",\n     \"grid\": false,\n     \"text\": {\n       \"font\": \"10px tohma\",\n       \"colour\": \"blue\"\n     },\n     \"annotation\": {\n       \"title\": \"economy-mpg-\",\n       \"position\":\n          \"edge\",\n       \"font\": \"10px Arial\",\n       \"colour\": \" blue\"\n     },\n     \"transform\": {\n       \"function\": \"pow\",\n       \"properties\": {\n         \"power\": 2,\n         \"name\": \"power\"\n\t}\n  }]\n}\n\\end{lstlisting}\n\\end{minipage}%\n\\begin{minipage}{0.20\\textwidth}\n  \\begin{figure}[H]\n    \\includegraphics[width=\\textwidth]{enhanced_parallel_coord}\n  \\end{figure}\n  \\begin{figure}[H]\n    \\includegraphics[width=\\textwidth]{arc_polar_coord}\n  \\end{figure}\n\\end{minipage}\nThe\nThe figures above reveal the power of using JSOL, both are \\cite{parallel_coord}'s parallel coordinates, but the difference as shown is that one of them is polar which is a compelling point in the library, that you can easily convert any visualization\nscene from cartesian to polar coordinate in a single command.\n\n\n\n\n\\section*{References}\n    \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:level1}INTRODUCTION}\nElectrons in materials with open d- or f-shells possess an electronic correlation that would significantly affect the electronic structure in materials and give rise to a variety of exotic properties such as electronic topology\\cite{PhysRevB.104.235108,PhysRevB.91.125139,PhysRevLett.121.066402,PhysRevLett.122.016803,PhysRevResearch.3.013265}, magnetism\\cite{PhysRevB.97.184404, PhysRevB.95.075124,2021Electron}, and metal-insulator transitions \\cite{PhysRevLett.116.116403,PhysRevB.98.075117,PhysRevB.104.035108}. Generally, electron correlation effects are more pronounced in two-dimensional (2D) systems than in three-dimensional (3D) systems with the same chemical composition. This is because Coulomb screening, which suppresses the long-range Coulomb interaction between electrons, is inhibited by dimensionality reduction in 2D systems. Therefore, it is easier to tune the electronic correlation strength of 2D systems experimentally, e.g., through the use of\nstructured substrate \\cite{2017Coulomb,2019A,PhysRevLett.123.206403} or the growth of multilayer films with different thicknesses\\cite{2021Anomalous}. On the other hand, the properties developed by Coulomb engineering have been theoretically revelled in 2D materials\\cite{PhysRevB.105.115115,PhysRevB.104.085149,PhysRevB.98.075117,PhysRevB.104.035108}. For example, a coexisting quantum anomalous Hall insulation state in the VSi$_2$P$_4$ monolayer has been proposed with tuning of the Hubbard \\textit{U}-constant in first-principles calculations \\cite{PhysRevB.104.085149}; By varying the Coulomb repulsion \\textit{U}, the location of the metal-insulator transition and the magnetism as well as the superconductivity of some 2D systems were revealed \\cite{PhysRevB.98.075117}. Thus, 2D materials provide a good playground to investigate numerous electronic correlation effects.\n\nRecently, a new class of 2D van der Waals systems, single-layer (SL) H-FeX$_2$ (X=Cl, Br, I) \\cite{2020The,hu2020concepts,ZHAO202256}, has attracted much attention. It is reported that the SL H-FeCl$_2$ belongs to ferrovalley (FV) materials \\cite{2016Concepts} with spontaneously valley polarization coupled with their ferromagnetism.\n Theoretical calculations have shown that  the orientation of easy magnetization axis of SL H-FeClF would rotate from out-of-plane to in-plane  \\cite{PhysRevB.105.104416} with increasing the strength of the electronic correlation (\\textit{U$_{\\textup{eff}}$}). In addition, the electronic energy bands of SL H-FeClF exhibit a noticeable evolution with \\textit{U$_{\\textup{eff}}$}, giving rise to topological phase transitions  \\cite{PhysRevB.105.104416}.  Intuitively, varying the strength of electronic correlations alters the local distribution of electronic valence charges on the open d-shell of Fe ions and changes the local magnetism and the strength of spin-orbital coupling (SOC). This, of course, drives the evolution of the atomic-scale magnetic anisotropy (MA)  of the system. As we know, the atomic-scale MA is not only tightly related to the performance of the upper limit of the system's magnetic memory, but also in connection with the polarization of valleys in FV materials due to the magneto-valley coupling \\cite{PhysRevB.102.035412, PhysRevB.104.085149}. So, it is necessary to study the MA of  SL H-FeX$_2$ (X=Cl, Br, I). Unfortunately, the evolutionary behaviour of MA with \\textit{U$_{\\textup{eff}}$} of SL H-FeX$_2$ (X=Cl, Br, I) members is unknown. In addition, since the modulation of both the electronic structure and the magnetic properties in each member of the SL H-FeX$_2$ (X=Cl, Br, I) family is all driven by the Coulomb correlation on Fe, the performance in MA of each member of the family would likely share a common trait. On the other hand, different member of SL H-FeX$_2$ (X=Cl, Br, I) has different halogen element, and these different halogen elements should, to some extent, alter the modulation of the electronic and magnetic properties of the system in different manner. The study of these concerns is clearly of great importance in both the scientific interest and the technological importance of spintronics.\n \nIn this work, we investigated the effect of electronic correlation strength on electronic structures as well as the magnetic anisotropy energy (MAE), represented by SL H-FeBr$_2$. Based on the first-principles calculations with DFT+\\textit{U} approach, We found that the trend of MAE with increasing value of \\textit{U$_{\\textup{eff}}$} imposed on Fe shows a non-monotonic evolution behavior corresponding to a flip between in-plane and out-of-plane magnetization. We suggested that this non-monotonic evolutionary behavior of MAE is  attributed to the competition between the element resolved MAE  contributed by Fe and Br atoms. The above-mentioned MAE behavior with \\textit{U$_{\\textup{eff}}$} and its underlying mechanism is universal for the SL H-FeX$_2$ (X=Cl, Br, I) family. In addition, the band inversion occurs in SL H-FeBr$_2$ during the rise of \\textit{U$_{\\textup{eff}}$}, leading to occurrence of topological phase transitions and quantum anomalous valley Hall (QAVH) states. Our work highlights the correlation effects on the MAE and the topological phase transition in the H-FeX$_2$ (X=Cl, Br, I) family.\n\n\n\\section{\\label{sec:level1}METHODS}\nWe performed first-principles density functional theory (DFT) calculations implemented in the Vienna \\textit{ab initio} Simulations Package (VASP) \\cite{PhysRevB.54.11169} to study all our concerns in the present work. In our theoretical treatment, the plane-wave cut-off energy was set to be 600 eV and the Brillouin zone was sampled with a $12\\times 12\\times 1$ $\\Gamma$-centered $k$-point mesh. The generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) realization was used for the exchange correlation functional \\cite{PhysRevLett.77.3865}.    \nTo treat the effect of correlation between electrons in the 3d orbitals of Fe atoms, the Dudarev's approach of the Coulomb correction implemented in  DFT+\\textit{U} scheme was applied to Fe ions in the system, where only the \\textit{U$_{\\textup{eff}}$} =$\\textit{U}-\\textit{J}$ is meaningful (Dudarev approach) \\cite{PhysRevB.57.1505}. In addition, the vdW corrections included in the DFT-D3 method was considered \\cite{2010A}.\nDuring the structural optimization, both the atomic positions and the lattice constants of each system were fully relaxed until the Hellmann-Feynman forces acting on each atom were less than $10^{-3}$ eV; The electronic convergence criteria was set to be $10^{-6}$eV. Since our concerned system is a two-dimensional nanosheet, a 20\\AA vacuum region was added along the direction perpendicular to the surface of the nanosheet was added to avoid the interaction between the periodic images. \nTo calculated the Berry curvature, the maximally localized Wannier functions (MLWFs) were constructed using the WANNIER90 package \\cite{MOSTOFI20142309}. The edge states were calculated using the iterative Green function method, which is implemented in the WannierTools package \\cite{2017WannierTools}.\n\nThe MAE is defined as the total energy difference between the in-plane ferromagnetic (FM) configuration (\\textit{E$_{\\textup{x}}$}) and the out-of-plane FM configuration (\\textit{E$_{\\textup{z}}$}), namely \n\\begin{equation}\nMAE = \\textit{E$_{\\textup{x}}$}-\\textit{E$_{\\textup{z}}$}. \n\\end{equation}\n\nHere, the positive or negative value of MAE indicates that the orientation of easy magnetization axis is along the out-of-plane or in-plane direction. Moreover, the element- and orbital-resolved MAE were calculated from the difference of SOC energies between in-plane and out-of-plane ferromagnetic configurations, i.e., \\cite{PhysRevB.101.214436}\n\\begin{equation}\n\\Delta{E}_{\\textup{SOC}} = E^{x}_{\\textup{SOC}}- E^{z}_{\\textup{SOC}},\n\\end{equation}\nwith \n\\begin{equation}\nE_{\\textup{SOC}} = \\langle \\frac{h^{2}}{2m^{2}c^{2}}  \\frac{1}{r} \\frac{dV}{dr} \\hat{L} \\cdot \\hat{S}  \\rangle,\n\\end{equation}\nwhere $V(r)$ is the spherical  part of the effective potential inside the PAW sphere, while $\\hat{L}$ and $\\hat{S}$ are the orbital and spin angular momentum respectively.  According to the second-order perturbation theory, only about 50\\% of the SOC energy difference contributes to the MAE, i.e., $\\textup{MAE} \\approx \\frac{1}{2} \\Delta{E}_{\\textup{SOC}}$ \\cite{ANTROPOV201435,PhysRevB.96.014435}, and the remaining SOC energies might be translated into both crystal-field energy and unquenched orbital moments \\cite{2011Is}.  \n\n\\section{\\label{sec:level1}RESULTS AND DISCUSSION}\n\\subsection{\\label{sec:level2}Basic structural and magnetic properties}\n\n\\begin{figure}[ht]\n\\includegraphics[scale=0.27]{1.png}\n\\caption{\n(a) Top and (b) side view of the atomic structure of SL H-FeBr$_2$.(c) The splitting of 3d orbitals of Fe atom under the trigonal prismatic crystal field. The trigonal prismatic crystal structure is also shown. (d)The Brillouin zone with high-symmetry points labeled.\n }\n\\label{fig1}\n\\end{figure}\n\nThe single layer (SL) H-FeBr$_2$ is a hexagonal structure with the space group of \\textit{P$\\overline{6}$m}2 (No.187). A monolayer of Fe atoms is sandwiched by two adjacent Br monolayers and each Fe atom is surrounded by six Br atoms, forming a local FeBr$_6$ trigonal prism, as shown in Figs.\\ref{fig1}(a) and (b). Our calculations showed that the lattice constant of this system is \\textit{a}= 3.57\\AA and the bond angle of Fe-Br-Fe is $\\theta=85.3^{\\circ}$, which are in excellent agreement with the literature \\cite{2020The}. In particular, the six Br ions in each FeBr$_6$ trigonal prism provide a local crystal field on each Fe atom (seen in Fig. \\ref{fig1}(c)). This causes the 3d orbitals of the Fe atom to split into three groups in the energy landscape: the $d_{z^{2}}$ orbital (denoted as A$_1$), the degenerate $(d_{xy}, d_{x^{2}-y^{2}})$ orbitals (denoted as E$_1$), and the  $(d_{xz}, d_{yz})$ orbitals (denoted as E$_2$), \n which are schematically displayed in Fig. \\ref{fig1}(c). Since the electronic configuration of the Fe$^{2+}$ ion is 3d$^{6}$, the spin-up channels of the 3d orbitals are fully occupied. For the spin-down channel, only the $d_{z^{2}}$ orbital is occupied by an electron, and the other d orbitals are empty. This gives rise to a spin magnetic moment of 4$\\mu_{B}$ at each Fe$^{2+}$ ion. \n \n \\begin{figure}[ht]\n\\includegraphics[scale=0.4]{2.png}\n\\caption{\n(a) The schematic top views of the FM, Stripy-AFM, and Zigzag-AFM magnetic configurations. The solid and open circles represent spin-up and spin-down states, respectively. \n(b) The evolution of energy difference between FM and Stripy-AFM (Zigzag-AFM) states with \\textit{U$_{\\textup{eff}}$}. The energy difference is defined as $\\Delta E = E_{\\textup{Stripy\/Zigzag}}-E_{\\textup{FM}}$.\n }\n\\label{fig2}\n\\end{figure}\n\nTo investigate the magnetic ground state of the SL H-FeBr$_2$, three types of magnetic configurations, i.e., a FM configuration, a stripy antiferromagnetic (AFM) configuration and a zigzag AFM configuration (seen in Fig.\\ref{fig2} (a)), were considered. Performing calculations of the nanosheet with each concerned magnetic configuration when\\textit{U$_{\\textup{eff}}$} imposed on Fe and varied from 0.0 to 1.8 eV, we found that the energy of the system with either the stripy AFM or the zigzag AFM is significantly higher than that with the FM configuration (Fig.\\ref{fig2}(b)), which agrees well with the previous study \\cite{2020The}.\n\n\nThe origin of the above FM state can be understood in terms of the super-exchange interaction.\nNote that the bond angle (85.3\u00b0) of Fe-Br-Fe in SL H-FeBr$_2$ is close to 90\u00b0. Combining this structural feature with the Goodenough-Kanamori-Anderson (GKA) rules \\cite{PhysRev.100.564,KANAMORI195987}, we know that the super-exchange interaction between the two nearest neighbouring Fe atoms predominately characterizes the feature of FM interaction. In addition to this FM super-exchange interaction, there is also a weak direct AFM exchange between the two nearest neighbouring Fe ions. However, its strength is usually weaker than that of the FM super-exchange interaction, due to the localization of d-orbitals on each magnetic atom.\nHence, the FM super-exchange interaction dominates the interaction between two nearest neighbouring Fe ions in this nanosheet when \\textit{U$_{\\textup{eff}}$} is small. \n\n\nAnother aspect shown in Fig. \\ref{fig2}(b) is that the energy difference between the stripy AFM and the FM configurations or between the zigzag AFM and the FM configurations (defined as $\\Delta E = E_{\\textup{Stripy\/Zigzag}}-E_{\\textup{FM}}$) decreases with increasing the value of \\textit{U$_{\\textup{eff}}$}, showing a weakening of FM coupling. When \\textit{U$_{\\textup{eff}}$} reaches 2.0eV, the Zigzag AFM configuration has a lower energy than that of the FM configuration, leading to an AFM ground state. \n\n\nIn fact, this FM-AFM transition behaviour can be understood as a consequence of the competition between the indirect FM superexchange and the direct AFM exchange. As we know, the indirect FM superexchange strength is proportional to \n $-\\frac{ t^{4}_{pd} J^{p}_{H} }{(\\Delta_{pd}+U_{d}  )^{4}} $, \n with $t_{pd}$, $J^{p}_{H}, \\Delta_{pd}$ and $U_{d}$ \nrepresenting the hybridization strength between Fe-d orbitals and Br-p orbitals, the Hund's coupling strength of Br-p orbitals, the energy interval between Fe-d orbitals and Br-p orbitals, and the spin-splitting energy of Fe-d orbitals, respectively\\cite{PhysRevB.105.085129}. Meanwhile, the strength of direct AFM exchange is proportional to \n$\\frac{t_{dd}}{U_{d}}$, \nwith $t_{dd}$ being the strength of direct hybridization between d orbitals of nearest neighbouring Fe ions. \n\nWith the increase of \\textit{U$_{\\textup{eff}}$}, the spin splitting $U_{d}$ increases significantly. From the above formulas, it can be seen that the indirect FM superexchange strength is inversely proportional to the fourth power of the $U_{d}$ value, while the direct AFM exchange strength is only inversely proportional to the first power of the $U_{d}$ value. Obviously, increasing the $U_{d}$ value leads to a more pronounced decrease in the strength of the indirect FM superexchange than that of the direct AFM exchange. As a result, the energy difference between AFM and FM decreases with \\textit{U$_{\\textup{eff}}$}, giving rise to a transition between AFM and FM magnetic configurations. \n\nWe emphasize that although the energy of stripy AFM state is slightly lower than that of the FM state (for about 0.007eV\/Fe) when \\textit{U$_{\\textup{eff}}$}=2eV, the FM is the most energetically favourable state in most of the considered range (from 0.0eV to 2.0eV) of \\textit{U$_{\\textup{eff}}$}. Thus, in our following calculations, only the FM state of the system will be investigated.\n\n\\subsection{\\label{sec:level2}Electronic correlation effects on MAE}\n\n\n\\begin{figure}[t]\n\\includegraphics[scale=0.27]{3.png}\n\\caption{\nThe evolution of (a) total MAE and (b) element-resolved MAE in a unit cell with \\textit{U$_{\\textup{eff}}$}.\n }\n\\label{fig4}\n\\end{figure}\n Now we turn to investigate the evolution of MAE with different correlation strength represented by \\textit{U$_{\\textup{eff}}$} in Fe. \nFigure \\ref{fig4} displays our calculated MAE as a function of \\textit{U$_{\\textup{eff}}$}. As shown in Fig \\ref{fig4}(a), the MAE value increases with increasing the value of \\textit{U$_{\\textup{eff}}$} until the \\textit{U$_{\\textup{eff}}$} value reaches 0.8 eV, followed by a rapid decrease in MAE. Therefore, by increasing the value of \\textit{U$_{\\textup{eff}}$}, the MAE varies from negative values to positive values and then to negative values again. This corresponds to the change of the magnetic state from the out-of-plane FM state to the in-plane FM state before returning to the out-of-plane FM state. The system prefers an in-plane FM state if $\\textit{U$_{\\textup{eff}}$}$\\textless0.1eV or $\\textit{U$_{\\textup{eff}}$}$\\textgreater1.1 eV, otherwise, the systems shows an out-of-plane FM state. So, there exists a perpendicular magnetic anisotropy (PMA) in the range of 0.1 eV\\textless\\textit{U$_{\\textup{eff}}$}\\textless1.1 eV.  \n   \nBasically, the entire MAE of a system is contributed from each atom. Thus, the element-resolved MAE as a function of \\textit{U$_{\\textup{eff}}$} is calculated, which is shown in Fig. \\ref{fig4}(b). It is remarkable that the MAE from Fe and Br elements show a completely different behavior with increasing \\textit{U$_{\\textup{eff}}$}: the MAE of Br element (denoted as Br-MAE) increases monotonically with increasing \\textit{U$_{\\textup{eff}}$}; However, the MAE of Fe (denoted as Fe-MAE) remains almost constant before \\textit{U$_{\\textup{eff}}$} reaches 0.8 eV, and when \\textit{U$_{\\textup{eff}}$}\\textgreater0.8 eV Fe-MAE decreases sharply and varies from positive values to negative values. \nApparently, the non-monotonic behaviour of the whole MAE in a unit cell could be the result of the competition of the element-resolved MAE between Fe and Br. For \\textit{U$_{\\textup{eff}}$}\\textless0.8 eV, the enhancement of Br-MAE dominates, which is responsible for the increase in the total MAE. For \\textit{U$_{\\textup{eff}}$}\\textgreater0.8 eV, the faster change of the Fe-MAE dominates and gives rise to a decrease in the total MAE. \n\n\\begin{figure*}[ht]\n\\includegraphics[scale=0.5]{4.pdf}\n\\caption{\n(a) The difference of orbital-resolved MAE of each Br atom between \\textit{U$_{\\textup{eff}}$}=0.0 eV and \\textit{U$_{\\textup{eff}}$}=2.0 eV, which is defined as $\\Delta\\textup{MAE} = {\\textup{MAE}}_{\\textit{U$_{\\textup{eff}}$}= 2 eV}-{\\textup{MAE}}_{\\textit{U$_{\\textup{eff}}$}= 0 eV}$. The spin-polarized DOS of Br-$p_{x}$ and Br-$p_{y}$ orbitals at (b) \\textit{U$_{\\textup{eff}}$}=0.0eV and (c) \\textit{U$_{\\textup{eff}}$}=2.0 eV is given. \n(d) The difference of orbital resolved MAE of each Fe atom between \\textit{U$_{\\textup{eff}}$}=0.8eV and \\textit{U$_{\\textup{eff}}$}=2.0 eV, which is defined as $\\Delta\\textup{MAE} = {\\textup{MAE}}_{\\textit{U$_{\\textup{eff}}$}=2 eV}-{\\textup{MAE}}_{\\textit{U$_{\\textup{eff}}$}=0.8 eV}$. The spin-polarized DOS of Fe-d$_{xy}$ and Fe -$d_{x^{2}-y^{2}}$ orbitals at (e) \\textit{U$_{\\textup{eff}}$}=0.8 eV and (f) \\textit{U$_{\\textup{eff}}$}=2.0 eV is given. \n}\n\\label{fig5}\n\\end{figure*}\n    \nWhy does the evolution of Br-MAE behave differently than that of Fe-MAE with changing \\textit{U$_{\\textup{eff}}$}?  To uncover the nature underlying this concern, we recall that based on the second-order perturbation theory, the MAE essentially correlates with the SOC interaction between the occupied and unoccupied states around the Fermi level, which is expressed as \\cite{PhysRevB.47.14932,PhysRevB.103.195402}\n\\begin{equation}\n\\begin{aligned}\n& \\textup{MAE} \\\\\n  & =  \\sum_{\\sigma,\\sigma'}(2\\delta_{\\sigma,\\sigma'}-1)\\xi^{2}\\sum_{o^{\\sigma},u^{\\sigma'}} \\frac{|\\langle o^{\\sigma} |\\hat{L}_{z} |u^{\\sigma'} \\rangle |^{2} - |\\langle o^{\\sigma} |\\hat{L}_{x} |u^{\\sigma'} \\rangle |^{2}}{E^{\\sigma'}_{u}-E^{\\sigma}_{o}}    \n\\end{aligned}\n\\end{equation}\n\nHere $\\xi$ is the strength of SOC, $\\sigma$ and $\\sigma'$ are spin indexs. $\\hat{L}_{z}$ and $\\hat{L}_{x}$ are angular momentum operators.   $|o^{\\sigma} \\rangle $ ($|u^{\\sigma'} \\rangle $) is the occupied (unoccupied) state with spin $\\sigma$ ($\\sigma'$), whose energy is $E^{\\sigma}_{o}$ ($E^{\\sigma'}_{u}$).  According to the expression above,\nthe MAE is sensitive to the energy interval between occupied and unoccupied states, $ E^{\\sigma'}_{u}-E^{\\sigma}_{o}$. Thus, by tuning the value of \\textit{U$_{\\textup{eff}}$}, the value of $ E^{\\sigma'}_{u}-E^{\\sigma}_{o}$ could change, giving rise to the variation of MAE.    \n\n\nFurthermore, we computed the orbital-resolved MAE for each Br atom with \\textit{U$_{\\textup{eff}}$}= 0 eV and 2 eV  and for each Fe atom with \\textit{U$_{\\textup{eff}}$}=0.8 eV and 2 eV respectively (as shown in Fig. S1 in the Supplemental Material \\cite{SM}), followed by making difference $\\Delta \\textup{MAE} = {\\textup{MAE}}_{ \\textit{U$_{\\textup{eff}}$}= 2 eV}-{\\textup{MAE}}_{ \\textit{U$_{\\textup{eff}}$}= 0 eV}$ for Br and $\\Delta \\textup{MAE} = {\\textup{MAE}}_{\\textit{U$_{\\textup{eff}}$}=2 eV}-{\\textup{MAE}}_{\\textit{U$_{\\textup{eff}}$}=0.8 eV}$ for Fe, as displayed in Fig.\\ref{fig5}(a) and \\ref{fig5}(d). \n\nIn the case of Br, it is observed in Fig. \\ref{fig5}(a) that as the value of  \\textit{U$_{\\textup{eff}}$} increases from 0.0 eV to 2.0 eV, the $p_{x}-p_{y}$ orbital pairs make more positive contributions to the Br-MAE, while those from other orbital pairs have only minor contributions, thus only the Br $p_{x}$ and $p_{y}$ orbitals are considered. To understand the phenomenon observed above, we calculated the density of states (DOS) of spin-polarized Br-$p_{x}$ and Br-$p_{y}$ orbitals in SL H-FeBr$_{2}$ when \\textit{U$_{\\textup{eff}}$} = 0.0 eV and \\textit{U$_{\\textup{eff}}$} = 2.0 eV, which are shown in Fig. \\ref{fig5}(b) and \\ref{fig5}(c). It can be observed that the $p_{x}$ and $p_{y}$ orbitals of Br are all degenerate. Furthermore, the unoccupied states of Br near the Fermi level are the spin-down $p_{x\/y}$ states and the occupied states of Br near the Fermi level are the spin-up $p_{x\/y}$ states.\n\nWe first investigate the MAE contributed by the SOC interaction between the unoccupied spin-down Br-$p_{y}$ states (denoted as $|p^{\\downarrow}_{y,u} \\rangle$) and the occupied spin-up Br-$p_{x}$ states (denoted as $|p^{\\uparrow}_{x,o}\\rangle$), which can be expressed as:\n\n\\begin{equation}\n\\begin{aligned}\n    \\textup{MAE}_{\\textup{Br-$p_{x\/y}$}} & =  -\\xi ^{2} \\sum_{o,u} \\frac{|\\langle p^{\\uparrow}_{x,o}|\\hat{L_{z}}| p^{\\downarrow}_{y,u} \\rangle |^{2} -|\\langle p^{\\uparrow}_{x,o}|\\hat{L_{x}}| p^{\\downarrow}_{y,u} \\rangle |^{2}}{E^{\\downarrow}_{p_{y,u}}  - E^{\\uparrow}_{p_{x,o}} }.\\\\\n\\end{aligned}\n\\end{equation}\n\nNoting that the matrix elements are \\cite{doi:10.1021\/acs.inorgchem.9b00687}\n\\begin{equation}\n    \\langle p_{x} |\\hat{L_{z}} | p_{y} \\rangle = i, \n\\end{equation}\nand\n\\begin{equation}\n    \\langle p_{x} |\\hat{L_{x}} | p_{y} \\rangle = 0.\n\\end{equation}\n Thus, the $\\textup{MAE}_{\\textup{Br-$p_{x\/y}$}}$  can be simplified as :\n \\begin{equation}\n \\begin{aligned}\n     \\textup{MAE}_{\\textup{Br-$p_{x\/y}$}} = -\\xi ^{2} \\sum_{o,u} \\frac{1}{E^{\\downarrow}_{p_{y,u}}  - E^{\\uparrow}_{p_{x,o}}}.\n \\end{aligned}\n \\end{equation}\n \nApparently, the $\\textup{MAE}_{\\textup{Br-$p_{x\/y}$}}$ contributes to the negative value of Br-MAE. Meanwhile, the amplitude of the $\\textup{MAE}_{\\textup{Br-$p_{x\/y}$}}$ is inversely proportional to the energy gap between the occupied spin-up $p_{x}$ states and the unoccupied spin-down $p_{y}$ states near the Fermi level. Here, the energy gap is approximately described as the energy difference between the two major peaks in DOS of occupied spin-up $p_{x}$ states and unoccupied spin-down $p_{y}$ states. Adjusting the value of \\textit{U$_{\\textup{eff}}$} from 0 eV to 2 eV, the aforementioned energy gap  was changed from 1.51 eV to 2.09 eV, as shown in Fig. \\ref{fig5}(b) and (c). This increase in the energy gap gives rise to a smaller magnitude of negative $\\textup{MAE}_{\\textup{Br-$p_{x\/y}$}}$, which is responsible for decreasing the value of negative Br-MAE. \n\nRecall that the $p_{x}$ and $p_{y}$ orbitals of Br are degenerate, by doing the same analysis on the unoccupied spin-down $p_{x}$ states and the occupied spin-up $p_{y}$ states of Br near the Fermi level, we found that the gap width between the main peaks in the DOS corresponding to these two orbitals increase as well, resulting in a decreasing magnitude in the negative Br-MAE. Therefore, the negative Br-MAE contributes less with an increase in \\textit{U$_{\\textup{eff}}$}.\n\nFor Fe, it is shown in Fig. \\ref{fig5}(d) that as the value of  \\textit{U$_{\\textup{eff}}$} increases from 0.8 eV to 2.0 eV, the $d_{xy}-d_{x^{2}-y^{2}}$ orbital pairs make more negative contributions to the Fe-MAE, and those from other orbital pairs have minor contributions, thus only $d_{x^{2}-y^{2}}$ and $d_{xy}$ orbitals of Fe are considered. To explain this phenomenon, the spin-polarized DOS of Fe-$d_{x^{2}-y^{2}}$ orbital and Fe-$d_{xy}$ orbital in SL H-FeBr$_2$ are calculated when \\textit{U$_{\\textup{eff}}$} =0.8 eV and 2.0 eV, respectively, which are shown in Figs. \\ref{fig5}(e) and (f)). \nIt is easy to find that the $d_{xy}$ and $d_{x^{2}-y^{2}}$ orbitals of Fe are all degenerate.\n\nIn the case of \\textit{U$_{\\textup{eff}}$} =0.8 eV, the occupied spin-down $d_{x^{2}-y^{2}}$ states (denoted as $|d^{\\downarrow}_{x^{2}-y^{2},o}\\rangle $) interact strongly with the unoccupied spin-down $d_{xy}$ states (denoted as $|d^{\\downarrow}_{xy,u} \\rangle$) near the Fermi level under SOC (as circled in Fig.\\ref{fig5} (e)),\nIt is noted that the orientations of the spins in these states are same. The MAE of Fe contributed by these two states is called as spin-conserved MAE, denoted as $\\textup{MAE}_{\\textup{Fe},\\textup{spin-conserved}}$. \nHere, we have\n\\begin{equation}\n\\begin{aligned}\n    &\\textup{MAE}_{\\textup{Fe},\\textup{spin-conserved}} \\\\\n    & =  \\xi ^{2} \\sum_{o,u} \\frac{|\\langle d^{\\downarrow}_{xy,u}|\\hat{L_{z}}| d^{\\downarrow}_{x^{2}-y^{2},o} \\rangle |^{2} -|\\langle d^{\\downarrow}_{xy,u}|\\hat{L_{x}}| d^{\\downarrow}_{x^{2}-y^{2},o} \\rangle |^{2}}{E^{\\downarrow}_{d_{xy,u}}  - E^{\\downarrow}_{d_{x^{2}-y^{2},o}} }\\\\\n\\end{aligned}\n\\end{equation}\nwith using the matrix elements  \\cite{doi:10.1021\/acs.inorgchem.9b00687} \n\\begin{equation}\n    \\langle d_{xy} |\\hat{L_{z}} | d_{x^{2}-y^{2}} \\rangle = 2i, \n\\end{equation}\nand\n\\begin{equation}\n    \\langle d_{xy} |\\hat{L_{x}} | d_{x^{2}-y^{2}} \\rangle=0,\n\\end{equation}\n\nthe $\\textup{MAE}_{\\textup{Fe},\\textup{spin-conserved}}$ can be simplified as\n\\begin{equation}\n\\begin{aligned}\n    \\textup{MAE}_{\\textup{Fe},\\textup{spin-conserved}} = \\xi ^{2} \\sum_{o,u} \\frac{4}{E^{\\downarrow}_{d_{xy,u}}  - E^{\\downarrow}_{d_{x^{2}-y^{2},o}}}\n\\end{aligned}    \n\\end{equation}\n\nClearly, the value of $\\textup{MAE}_{\\textup{Fe},\\textup{spin-conserved}}$ is positive, which contributes to the positive Fe-MAE when \\textit{U$_{\\textup{eff}}$} is about 0.8 eV. \nHowever, when \\textit{U$_{\\textup{eff}}$} reaches 2 eV, the occupied spin-down $d_{x^{2}-y^{2}}$ orbitals almost vanish on the VBM, as seen in Fig. \\ref{fig5}(f). This means that the SOC interaction between unoccupied spin-down $d_{xy}$ states and occupied spin-down $d_{x^{2}-y^{2}}$ states  almost disappears at this value of \\textit{U$_{\\textup{eff}}$}. \nInstead, the unoccupied spin-down $d_{xy}$ states near the Fermi level mainly interact with the occupied spin-up $d_{x^{2}-y^{2}}$ states (denoted as $|d^{\\uparrow}_{x^{2}-y^{2},o}\\rangle  $) lying deeply in the valance bands, as circled in Fig. \\ref{fig5}(f). This SOC interaction between these two states with anti-paralleled spins contributes to the Fe-MAE as well, and we denoted this MAE contribution as $\\textup{MAE}_{\\textup{Fe},\\textup{spin-flip}}$. It can be written as:\n\\begin{equation}\n\\begin{aligned}\n        &\\textup{MAE}_{\\textup{Fe},\\textup{spin-flip}} \\\\\n    & =  -\\xi ^{2} \\sum_{o,u} \\frac{|\\langle d^{\\downarrow}_{xy,u}|\\hat{L_{z}}| d^{\\uparrow}_{x^{2}-y^{2},o} \\rangle |^{2} -|\\langle d^{\\downarrow}_{xy,u}|\\hat{L_{x}}| d^{\\uparrow}_{x^{2}-y^{2},o} \\rangle |^{2}}{E^{\\downarrow}_{d_{xy,u}}  - E^{\\uparrow}_{d_{x^{2}-y^{2},o}} }\\\\\n    & = -\\xi ^{2} \\sum_{o,u} \\frac{4}{E^{\\downarrow}_{d_{xy,u}}  - E^{\\uparrow}_{d_{x^{2}-y^{2},o}}}\n\\end{aligned}\n\\end{equation}\n\n\\begin{figure*}[ht]\n\\includegraphics[scale=0.42]{5.png}\n\\caption{\nThe band structure of ferromagnetic SL H-FeBr$_2$ calculated with (a) spin-polarized case without including SOC, (b) out-of-plane magnetism with SOC, and (c) in-plane magnetization magnetism with SOC. In (b) and (c), the red, blue and green dots represent Fe-A$_{1}$, Fe-E$_{1}$, and Fe-E$_{2}$ orbitals, respectively. \n }\n\\label{fig3}\n\\end{figure*}\n\n\nThe value of $\\textup{MAE}_{\\textup{Fe},\\textup{spin-flip}}$ is negative, which is now responsible for the negative Fe-MAE. \nNote that the $d_{xy}$ and $d_{x^{2}-y^{2}}$ orbitals of Fe are degenerate in this case, we can do the same analysis on the occupied spin-up $d_{xy}$ states and unoccupied spin-down $d_{x^{2}-y^{2}}$ states of Fe near the Fermi level. After carefully examining the features of the DOS corresponding to these two orbitals, we found that the gap between the major peaks in the DOS becomes narrow with increasing \\textit{U$_{\\textup{eff}}$}. This directly contributes to the negative Fe-MAE value. In total, with increasing \\textit{U$_{\\textup{eff}}$}, the competition between $\\textup{MAE}_{\\textup{Fe},\\textup{spin-conserved}}$ and $\\textup{MAE}_{\\textup{Fe},\\textup{spin-flip}}$ causes the Fe-MAE to switch from positive values to negative values .\n\n\n\n\\subsection{\\label{sec:level2}Evolution of band structures with electronic correlation strength}\n\n\\begin{figure*}[ht]\n\\includegraphics[scale=0.25]{6.png}\n\\caption{\nThe band structure of SL H-FeBr$_2$ calculated at (a) \\textit{U$_{\\textup{eff}}$}=0.0 eV, (b) \\textit{U$_{\\textup{eff}}$}=0.6 eV, (c) \\textit{U$_{\\textup{eff}}$}=0.7 eV, (d) \\textit{U$_{\\textup{eff}}$}=0.8 eV, (e) \n\\textit{U$_{\\textup{eff}}$}=1.0 eV, (f)\n\\textit{U$_{\\textup{eff}}$}=1.2 eV. The red, blue and green dots represent Fe-A$_{1}$, Fe-E$_{1}$, and Fe-E$_{2}$ orbitals, respectively. \n}\n\\label{fig6}\n\\end{figure*}\n\n\n\\begin{figure*}[ht]\n\\includegraphics[scale=0.65]{7.png}\n\\caption{\nThe Berry curvature of SL H-FeBr$_2$ in the 2D Brillouin zone calculated at (a) \\textit{U$_{\\textup{eff}}$}=0eV and (b) \\textit{U$_{\\textup{eff}}$}=0.8 eV;\nBoth units of Berry curvature in (a) and (b) are Bohr$^{2}$.\n(c) The topological edge state of SL H-FeBr$_2$ along the (100) direction calculated at \\textit{U$_{\\textup{eff}}$}=0.8 eV;\n(d) The Topological phase diagram of SL H-FeBr$_2$ with varied \\textit{U$_{\\textup{eff}}$}. \n}\n\\label{fig7}\n\\end{figure*}\n\nLet us again pay our attention to Figs. \\ref{fig5}(e) and (f). It can be observed that the electronic structures associated with the Fe-3d states vary greatly with \\textit{U$_{\\textup{eff}}$}, which could be reflected in the variation in the energy bands. Therefore, we carefully calculated the band structures with different \\textit{U$_{\\textup{eff}}$} values.\n\n\n\nFor comparison, the band structure of FM SL H-FeBr$_{2}$ at \\textit{U$_{\\textup{eff}}$} = 0 eV was first examined. In this case, the spin-polarized band structure without considering the SOC is plotted in Fig. \\ref{fig3}(a). It can be clearly seen that the spin-up and spin-down channels are split, where the spin-down component dominates near the Fermi level.\nMeanwhile, the electronic band structure with spin-down has a band gap of about 0.253 eV. Therefore, this system characterizes the FM half-semiconductor. In addition, the energy values of the valence band maximum (VBM) at K+ and K- are equal. This scenario also occurs at the conduction band minimum (CBM) at the two k-points. Thus, there are degenerate valleys in this energy band. This feature indicates that SL H-FeBr$_{2}$ belongs to the 2D valleytronic materials \\textbf{\\cite{2016Valleytronics}}.\n\n\n\n\nIn fact, the SOC cannot be ignored in this system, so the SOC is taken into account in the following treatment. As mentioned before, the easy axis of magnetization of this system can be in the \\textit{ab} plane or the out-of-plane. When the easy axis of magnetization is in the \\textit{ab} plane, the calculated energy bands show that the energy gap at K+ and K- are equal (Fig. \\ref{fig3}(c)), and since the CBM (VBM) at K+ and K- are degenerate, there is no spontaneous polarization of valleys. When we switch the easy axis to out-of-plane, the band gap width at the K+ point is 0.314 eV and that at the K- point is 0.191 eV (Figure \\ref{fig3}(b)), showing the spontaneous polarization of the valley. In this case the system is in the FV state. Strikingly, the difference of the band gap between K+ and K- is 123 meV, being larger than other typical FV materials such as SL H-FeCl$_{2}$ (106 meV) \\cite{hu2020concepts}, Nb$_{3}$I$_{8}$ (107 meV) \\cite{PhysRevB.102.035412}, LaBr$_{2}$ (33 meV) \\cite{2019Single} and MnPX$_{3}$ (43 meV) \\cite{doi:10.1073\/pnas.1219420110}. Apparently, the valley state strongly couples with the magnetization direction in SL H-FeBr$_{2}$, which is explained theoretically in the Appendix \\ref{app}. According to the previous reports \\cite{PhysRevB.105.104416,2020The}, such FV materials with out-of-plane magnetization potentially possess topological nontrivial states. Therefore, in our following calculations, the magnetization orientation was set to be out-of-plane.\n\nThe evolution of electronic band structures with \\textit{U$_{\\textup{eff}}$} is investigated, and the representative band structures with different \\textit{U$_{\\textup{eff}}$} values are shown in Fig. \\ref{fig6}. When \\textit{U$_{\\textup{eff}}$} increases to 0.6 eV, both the conduction bands and valence bands move towards the Fermi level, thereby reducing the band gap at the K+ and K- points. When increasing the value of \\textit{U$_{\\textup{eff}}$} to 0.7 eV, the band gap at the K+ point is still non-zero, but the band gap at the K- point is closed (as shown in Fig. \\ref{fig6}(c)). This shows that the system has Dirac-like band crossing characteristic at K- \\cite{RevModPhys.90.015001}. In this case, the system becomes the so-called half valley metal (HVM) \\cite{hu2020concepts}, which can provide a massless elementary excitation that potentially contributes to well-behaved charge transport. \n\nThe band gap closed at the K- point reopens when \\textit{U$_{\\textup{eff}}$} is slightly larger than 0.7 eV . \nInterestingly, at the K- point, the low-lying E$_{1}$ orbitals on the VBM shift to the CBM, while the high-lying A$_{1}$ orbital on the CBM shifts down to the VBM (as shown in Fig. \\ref{fig6}(d)) . Despite of this, the orbital composition of the energy band at the K+ point remains intact, forming a single-valley band-inverted state. When the value of \\textit{U$_{\\textup{eff}}$} is adjusted to about 1.0 eV, the bandgap at the K- point reopens, and the bandgap at the K+ point closes (as shown in Fig. \\ref{fig6}(e)). Compared to the case of \\textit{U$_{\\textup{eff}}$}=0.8 eV, the orbital composition of CBM and VBM at the K+ point are reversed here. Meanwhile, we found that when \\textit{U$_{\\textup{eff}}$} is larger than 1.0 eV, the conduction band (valance band) near the Fermi level is contributed only by the orbitals of Fe-E$_{1}$ (A$_{1}$) (as shown in Fig. \\ref{fig6}(f)). If the value of \\textit{U$_{\\textup{eff}}$} is further increased, Fe-A$_{1}$ orbitals and Fe-E$_{1}$ orbitals are no longer entangled with each other on the conduction band (valance band) near the Fermi level. During this disentanglement process with increasing \\textit{U$_{\\textup{eff}}$}, the SOC interaction between the occupied spin-down Fe-d$_{xy}$ (d$_{x^{2}-y^{2}}$) on the valance bands near the Fermi level and the unoccupied spin-down Fe-d$_{x^{2}-y^{2}}$ (d$_{xy}$) orbitals on the conduction bands near the Fermi level weakens, which is the origin of the rapid  switch in Fe-MAE. \n\n\n\n\nTo further characterize the effect of \\textit{U$_{\\textup{eff}}$} on valley-contrasting physics, the Berry curvature along $\\textit{z}$ direction was calculated based on the Kubo formula \\cite{PhysRevLett.49.405}:\n\\begin{equation}\n    \\Omega_{z} (\\textbf{k}) = -\\sum_{n}\\sum_{n\\neq n'} f_{n} \\frac{2 \\textup{Im} \\langle \\psi_{n\\textbf{k}} | v_{x} | \\psi_{n'\\textbf{k}} \\rangle \\langle \\psi_{n'\\textbf{k}} | v_{y} | \\psi_{n\\textbf{k}} \\rangle }{(E_{n}-E_{n'})^{2}}\n\\end{equation}\nHere, $f_{n}$ is the Fermi-Dirac distribution function, $E_{n}$ is the eigenvalue of Bloch state $| \\psi_{n\\textbf{k}} \\rangle $, and $v_{x\/y}$ is the vector operator.  \n\n\n\nThe k-resolved Berry curvatures for \\textit{U$_{\\textup{eff}}$} = 0 eV and \\textit{U$_{\\textup{eff}}$} = 0.8 eV are plotted in Fig. \\ref{fig7}(a) and Fig. \\ref{fig7}(b), respectively. In the case of \\textit{U$_{\\textup{eff}}$} = 0 eV, the Berry curvatures at K+ and K- have peaks of opposite signs and different absolute values, which originate from the spontaneous breaking of time-reversal symmetry and space-reversal symmetry. For such a system with a single valley band inversion, the Berry curvature peaks at K+ and K- have the same sign when \\textit{U$_{\\textup{eff}}$} = 0.8 eV. In this case, the full-space integral over the Berry curvature yields a nonzero Chern number, which shows a topologically nontrivial character. To uncover the nature of this topologically nontrivial feature, we computed the edge states along the (100) direction in the system at \\textit{U$_{\\textup{eff}}$} = 0.8 eV. As shown in Fig. \\ref{fig7}(c), there is a gapless chiral edge state, which simultaneously connecting the valence band at K+ and the conduction band at K-. Such gapless chiral edge states are the fingerprints of quantum anomalous Hall (QAH) states in topologically nontrivial systems with ferromagnetism \\cite{doi:10.1146\/annurev-conmatphys-031115-011417,doi:10.1146\/annurev-conmatphys-033117-054144}. Furthermore, the anomalous Hall conductivity (AHC) is calculated by the following formula \\cite{PhysRevLett.88.207208,PhysRevLett.92.037204}:\n\\begin{equation}\n\\sigma_{xy} = \\frac{e^{2}}{h} \\int_{BZ} \\frac{d\\textbf{k} }{(2\\pi)^{2}} \\Omega_{z} (\\textbf{k})   \n\\end{equation}\n As shown in Fig. S2 in the Supplemental Material \\cite{SM}), the AHC in the energy gap is  $-\\frac{e^{2}}{h}$ , confirming a QAH state with a Chern number of \\textit{C} = -1 in this system. Note that this single-valley band-inverted state with QAH effect is the so-called QAVH state \\cite{hu2020concepts}. Therefore, when the electron correlation strength is adequate, SL H-FeBr$_{2}$ can possess a QAVH state, which has great potential applications for the development of spintronic devices. \n\nIn principle, the presence of the QAVH state in the system is relevant to the value of \\textit{U$_{\\textup{eff}}$}. We therefore carefully look for the range of \\textit{U$_{\\textup{eff}}$} in which the QAVH effect appears in the system. Fig. \\ref{fig7}(d) displays the topological phase diagram in which the QAVH state is predicted to exist in the range of \\textit{U$_{\\textup{eff}}$} varying from 0.7 eV to 1.0 eV. As we know, the QAVH state can only survive when the system hosts PMA. The value of \\textit{U$_{\\textup{eff}}$} corresponding to the QAVH state ranges from 0.7 eV to 1.0 eV, which belongs to the \\textit{U$_{\\textup{eff}}$} value interval of PMA (from 0.1eV to 1.1eV). Thus, the QAVH state could naturally exist in this system without external magnetic fields. \n\n\\subsection{\\label{sec:level2}Discussions}\n \\begin{figure*}[ht]\n\\includegraphics[scale=0.25]{8.png}\n\\caption{\nThe evolution of total MAE of (a) SL H-FeCl$_2$, (b) SL H-FeBr$_2$, (c) SL H-FeI$_2$. The evolution of element-resolved MAE of (d) SL H-FeCl$_2$, (e) SL H-FeBr$_2$, (f) SL H-FeI$_2$ are also given.  \n}\n\\label{fig8}\n\\end{figure*}\n\nWe are now curious if the behavior of \\textit{U$_{\\textup{eff}}$}-dependent MAE in SL H-FeBr$_2$ is present in SL H-FeX$_2$ (X = Cl, I), which are the counterparts of SL H-FeBr$_2$. To this end, the total MAEs of SL H-FeX$_2$ (X = Cl, I) in FM state were computed, which are all plotted in Figs. \\ref{fig8}(a)-(c). Obviously, as the value of \\textit{U$_{\\textup{eff}}$} increases in the considered range, all of the total MAEs of SL H-FeX$_2$ (X = Cl, I) experience sensitive change. \n\nCommonly, the total MAE of SL H-FeX$_2$ (X = Cl, I) shows a rapid decrease after \\textit{U$_{\\textup{eff}}$} reaches a critical magnitude.\nTo understand this common feature, the element resolved MAE of SL H-FeX$_2$ (X = Cl, Br, I) under different values of \\textit{U$_{\\textup{eff}}$} is plotted in  Figs. \\ref{fig8}(d)-(f). It can be observed that when \\textit{U$_{\\textup{eff}}$} is small, the Fe-MAE of SL H-FeX$_2$ (X = Cl, Br, I) remains almost constant. However, after \\textit{U$_{\\textup{eff}}$} reaches a critical value, the Fe-MAE would suddenly decrease with \\textit{U$_{\\textup{eff}}$}.   \nSimilar to the case in SL H-FeBr$_2$, these common features in Fe-MAE of SL H-FeX$_2$ (X = Cl, Br, I) are all originated from the disentanglement between Fe-A$_1$ and E$_1$ orbitals, which are all reflected in the evolution of their band structures with \\textit{U$_{\\textup{eff}}$} (as shown in Fig. S3 and S5 in the Supplemental Material \\cite{SM}). This disentanglement weakens the SOC interaction between occupied Fe- $d_{xy}$ ($d_{x^{2}-y^{2}}$) orbitals at VBM and unoccupied $d_{x^{2}-y^{2}}$ ($d_{xy}$ ) orbitals at CBM, giving rise to a strong decrease of the Fe-MAE. It is worth noting that if the magnetization directions of all SL H-FeX$_2$ (X = Cl, Br, I) compounds are forced to be out-of-plane, the topological phase transitions might commonly happen during the continuous tuning of \\textit{U$_{\\textup{eff}}$}, giving rise to HVM and QAVH states.\n\n\n\nIn addition to the common features mentioned above, the MAEs of different members in SL H-FeX$_{2}$ (X = Cl, Br, I) also show significant differences with \\textit{U$_{\\textup{eff}}$}. First, the MAEs of SL H-FeCl$_{2}$ and H-FeBr$_{2}$ exhibit transitions between positive and negative MAEs with increasing \\textit{U$_{\\textup{eff}}$}, which corresponds to the reversal of their magnetization behaviors between out-of-plane and in-plane, but the MAE of SL H-FeI$_{2}$ does not behave in this way. Second, when the value of \\textit{U$_{\\textup{eff}}$} is small, the MAE from X (X=Br,I) elements (denoted as X-MAE) in both SL H-FeBr$_{2}$ and H-FeI$_{2}$ increase significantly with the increase of \\textit{U$_{\\textup{eff}}$} ; as the \\textit{U$_{\\textup{eff}}$} value continues to rise, the X-MAEs (X=Br, I) of both systems increase slowly. However, the Cl-MAE of SL H-FeCl$_{2}$ is not sensitive to \\textit{U$_{\\textup{eff}}$} in the above-mentioned process of increasing \\textit{U$_{\\textup{eff}}$}, as shown in Figs. \\ref{fig8} (d)-(f). Apparently, the halogen atoms of Br and I tend to provide negative values of MAEs. Among them, the amplitude of the negative I-MAE in H-FeI$_{2}$ is larger than that of the positive Fe-MAE, so the total MAE of H-FeI$_{2}$ is always negative. However, in SL H-FeCl$_{2}$, the total MAE of H-FeCl$_{2}$ is almost contributed by Fe-MAE since Cl atoms hardly contribute to MAE. Therefore, the evolution behavior of total MAE in SL H-FeCl$_{2}$ is dominated by Fe-MAE. Based on these calculations, we found that the heavier the halogen atom, the greater the contribution of the halogen atom to the negative MAE.\n\n Finally, similar with the case of SL H-FeBr$_{2}$, in SL H-FeX$_{2}$ (X = Cl, I) with out-of-plane magnetization, we also observed the chiral edge states connecting the conduction bands and valance bands, as well as an AHC of $-\\frac{e^{2}}{h}$ lying in the energy gap (see Fig. S4 and S6 in the Supplemental Material \\cite{SM}) in specific intervals of \\textit{U$_{\\textup{eff}}$}. These characteristics confirm the QAH states with Chern numbers being \\textit{C} = -1. We emphasize that these common features of correlation-driven electronic topology stem from the Fe-3d orbitals, which dominates the low-energy states of   SL H-FeX$_{2}$ (X = Cl, Br, I). \n\n\n\n\n\n\n\n\\section{\\label{sec:level1}CONCLUSION}\nWe investigated the evolution of MAE as well as electronic structures of SL H-FeBr$_2$ under varying correlation strength, quantified as \\textit{U$_{\\textup{eff}}$} imposed on Fe ions. It is found that the MAE would decrease after its increase with \\textit{U$_{\\textup{eff}}$}, and the transition between negative and positive MAE reflects the switching between out-of-plane magnetization and in-plane magnetization. This non-monotonic evolution behaviour of MAE stems from the competition of element-resolved MAE between Fe and Br. The evolution of element-resolved MAE was found to arise from the variation of spin-orbital coupling (SOC) interaction between different orbitals. Further investigation revealed that as  \\textit{U$_{\\textup{eff}}$} increases, the energy bands of SL H-FeBr$_2$ at K+ and K- invert in turn, giving rise to topological phase transition, and a QAVH state with chiral edge states was predicted. By comparing the MAE evolution behaviours of different members in the SL H-FeX$_2$ (X = Cl, Br, I) family, the underlying mechanism on the universality and specificity of MAE evolution behaviour is provided. Our study has deepened the understanding of correlation-induced electronic structural transition of SL H-FeX$_2$ (X = Cl, Br, I) family , which would open new perspectives of possible spintronics and valleytronics  applications on nanoelectronic devices based on these materials.\n\n\\section{\\label{sec:level1}ACKNOWLEDGEMENTS}\nThe author sincerely thanks Wenhui Duan, Yong Xu, Haowei Chen, and Zhiming Xu for helpful discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Role of Random Occlusion}\n\nThe random occlusion (RO) we designed for data augmentation is similar to the random erasing (RE) \\cite{zhong2020random} and cutout \\cite{devries2017improved} methods. In the RE implementation, the target erasing area is sampled from a combination of random area and aspect ratio, which could exceed the original image height or width. Therefore, it needs to try multiple times (100 by default) to generate a reasonable region for erasing. In contrast, in our implementation of the random occlusion, a square area is used, with the size randomly sampled at most $0.8\\times width$ of the image, and randomly put in a valid location. Then the square area is filled with white pixels. Note that with a simple square area, there is no need to sample multiple times of areas and aspect ratios and check the validity, and hence the generation process is more efficient. As for the cutout method, it uses multiple square regions in fixed sizes specified by hyperparameters, but not in random. The fixed-size regions may make the cut either too small or too large, and so it is not very convenient to set.\n\nTo show their differences, in the training of QAConv, we compare these data augmentation methods as well as a baseline without any random occlusion. From the results shown in Table \\ref{tab:occlusion}, it can be observed that the three data augmentation methods generally improve the baseline which does not apply any random occlusion. Intuitively, they are useful for QAConv because random occlusion forces QAConv to learn various local correspondences, instead of only salient but easy ones. Besides, the three data augmentation methods perform comparable, with the RO implementation being slightly better. Therefore, considering also the efficiency of the RO implementation, it is adopted in the training of the proposed QAConv algorithm.\n\n\\begin{table}\n    \\centering\n    \\caption{Role of random occlusion.}\\label{tab:occlusion}\n    \\begin{tabular}{|l|c|c|c|c|}\n      \\hline\n      \\multirow{2}{*}{\\tabincell{c}{Method}}  & \\multicolumn{2}{|c|}{Market$\\rightarrow$Duke} &  \\multicolumn{2}{|c|}{Duke$\\rightarrow$Market} \\\\\n      \\cline{2-5}\n        & Rank-1 & mAP  & Rank-1 & mAP \\\\\n      \\hline\n      QAConv without occlusion & 50.5 & 29.5 & 61.6 & 28.4\\\\\n      QAConv with RE~\\cite{zhong2020random} & 51.6 & 30.6 & 62.0 & 29.8\\\\\n      QAConv with cutout~\\cite{devries2017improved} & 51.6 & 30.8 & 62.6 & 30.3\\\\\n      QAConv with RO & \\textbf{54.4} & \\textbf{33.6} &\\textbf{62.8} &\\textbf{31.6}\\\\\n      \\hline\n    \\end{tabular}\n\\end{table}\n\n\n\n\\section{Complete Comparisons of Backbone Networks}\nTables \\ref{tab:duke} and \\ref{tab:market} show complete comparisons between the QAConv results with the ResNet-50 as backbone (denoted as QAConv$_{50}$) and with the ResNet-152 as backbone (denoted as QAConv$_{152}$), with DukeMTMC-reID and Market-1501 as the target datasets, respectively. Results of applying re-ranking alone are not shown in the main paper.\n\\begin{table}\n    \\centering\n    \\caption{Comparison (\\%) of backbone networks with DukeMTMC-reID as the target dataset.}\\label{tab:duke}\n    \\begin{tabular}{|l|c|c|c|c|}\n      \\hline\n      \\multirow{2}{*}{\\tabincell{c}{Method}}  & \\multicolumn{2}{|c|}{Training}  & \\multicolumn{2}{|c|}{Test: Duke}\\\\\n      \\cline{2-5}\n       & Source & Target & R1 & mAP \\\\\n      \\hline\n      \\hline\n      QAConv$_{50}$ & Market & & 48.8 & 28.7\\\\\n      QAConv$_{152}$ & Market & & 54.4 & 33.6 \\\\\n      QAConv$_{50}$ + RR & Market & & 56.9 & 47.8 \\\\\n      QAConv$_{152}$ + RR & Market & & 61.8 & 52.4 \\\\\n      QAConv$_{50}$ + RR + TLift & Market & & 64.5 & 55.1 \\\\\n      QAConv$_{152}$ + RR + TLift & Market & & 70.0 & 61.2 \\\\\n      \\hline\n      \\hline\n      QAConv$_{50}$ & MSMT & & 69.4 & 52.6\\\\\n      QAConv$_{152}$ & MSMT & & 72.2 & 53.4\\\\\n      QAConv$_{50}$ + RR & MSMT & & 76.7 & 71.2 \\\\\n      QAConv$_{152}$ + RR & MSMT & & 78.1 & 72.4 \\\\\n      QAConv$_{50}$ + RR + TLift & MSMT & & 80.3 & 77.2 \\\\\n      QAConv$_{152}$ + RR + TLift & MSMT & & 82.2 & 78.4\\\\\n      \\hline\n    \\end{tabular}\n\\end{table}\n\\begin{table}\n    \\centering\n    \\caption{Comparison (\\%) of backbone networks with Market-1501 as the target dataset.}\\label{tab:market}\n    \\begin{tabular}{|l|c|c|c|c|}\n      \\hline\n      \\multirow{2}{*}{\\tabincell{c}{Method}}  & \\multicolumn{2}{|c|}{Training}  & \\multicolumn{2}{|c|}{Test: Market}\\\\\n      \\cline{2-5}\n       & Source & Target & R1 & mAP \\\\\n      \\hline\n      \\hline\n      QAConv$_{50}$ & Duke & & 58.6 & 27.2 \\\\\n      QAConv$_{152}$  & Duke & & 62.8 & 31.6 \\\\\n      QAConv$_{50}$ + RR & Duke & & 65.7 & 45.8 \\\\\n      QAConv$_{152}$ + RR & Duke & & 68.5 & 51.2 \\\\\n      QAConv$_{50}$ + RR + TLift & Duke & & 74.6 & 51.5  \\\\\n      QAConv$_{152}$ + RR + TLift & Duke & & 78.7 & 58.2  \\\\\n      \\hline\n      \\hline\n      QAConv$_{50}$  & MSMT & & 72.6 & 43.1  \\\\\n      QAConv$_{152}$  & MSMT & & 73.9 & 46.6  \\\\\n      QAConv$_{50}$ + RR   & MSMT & & 77.4 & 65.6 \\\\\n      QAConv$_{152}$ + RR  & MSMT & & 79.2 & 69.1 \\\\\n      QAConv$_{50}$ + RR + TLift   & MSMT & & 86.5 & 72.2\\\\\n      QAConv$_{152}$ + RR + TLift  & MSMT & & 88.4 & 76.0\\\\\n      \\hline\n    \\end{tabular}\n\\end{table}\n\n\n\\section{Comparisons to Other Losses}\n\nSince the loss of hard triplet mining \\cite{hermans2017defense} is popular in person re-identification, we further include it in the loss comparisons. Besides, we provide a further analysis on different loss configurations of the QAConv. The results are shown in Table \\ref{tab:loss} under Market$\\rightarrow$Duke, where triplet results are each with its best margin. While the mini-batch hard triplet loss does improve the softmax cross-entropy loss, it seems that it is not efficient in learning the QAConv, possibly because local matching requires large pairs to learn, as done with the proposed class memory and focal loss, but not in mini-batches. Note that focal loss is a bit aggressive in learning, but softly. However, the hard triplet loss is in fact more aggressive.\n\n\\begin{table}\n    \\centering\n    \\caption{Role of loss functions under Market$\\rightarrow$Duke (\\%).}\\label{tab:loss}\n    \\begin{tabular}{|c|c|c|c|}\n      \\hline\n      \\multicolumn{2}{|c|}{Method} & Rank-1 & mAP \\\\\n      \\hline\n      \\multirow{5}{*}{\\tabincell{c}{ResNet-152}} & Softmax cross-entropy & 34.9 & 18.4\\\\\n      & Softmax cross-entropy + triplet & 39.6 & 23.0\\\\\n      & Arc loss~\\cite{Deng-CVPR19-ArcFace} & 35.3 & 17.1\\\\\n      & Center loss~\\cite{Wen-ECCV16-CenterLoss,Jin-IJCB17-CenterLossReID} & 38.9 & 22.1\\\\\n      & Class memory loss & 40.7 & 21.8\\\\\n      \\hline\n      \\multirow{7}{*}{\\tabincell{c}{QAConv$_{50}$}} & Mini-batch triplet (w\/o class memory) & 42.2 & 23.7\\\\\n      & Softmax cross-entropy & 43.4 & 24.9\\\\\n      & Binary cross-entropy & 46.1 & 27.3\\\\\n      & Softmax cross-entropy + triplet & 44.3 & 24.2 \\\\\n      & Binary cross-entropy + triplet & 44.7 & 23.6\\\\\n      & Focal loss + triplet & 43.3 & 23.2\\\\\n      & Focal loss (default) & \\textbf{48.8} & \\textbf{28.7}\\\\\n      \\hline\n    \\end{tabular}\n\\end{table}\n\n\n\\section{Fusion of Global Similarity}\n\nTo see whether fusing a global similarity branch helps improving the performance, we tried an extra global feature learning branch by performing a global average pooling on the final feature maps, and a softmax cross-entropy loss for classification. During testing, the cosine similarity computed from this global feature branch is fused to the QAConv similarity. However, after trying different weights of the two losses, the best mAP we can get is 28.4\\% under Market$\\rightarrow$Duke, with the weight 0.001 of the global branch. It is a bit worse than the default QAConv (28.7\\%). This may be because the vanilla global feature branch cannot handle misalignments and occlusions, and so more advanced techniques are needed here. This deserves a further study.\n\n\n\\section{TLift for Other Methods}\n\nNote that TLift can also be generally applied to other methods for improvements. To demonstrate this, Tables \\ref{tab:tlift-market} and \\ref{tab:tlift-duke} show results of applying TLift to all baseline methods under Market$\\rightarrow$Duke and Duke$\\rightarrow$Market, respectively. It can be observed that, beyond the improvements made by re-ranking, TLift can further improve all baseline methods. The improvements are consistently large, with Rank-1 improved by 10.1\\%-14.1\\%, and mAP improved by 3.6\\%-11.1\\%.\n\n\\begin{table}\n    \\centering\n    \\caption{Role of TLift under Market$\\rightarrow$Duke (\\%).}\\label{tab:tlift-market}\n    \\begin{tabular}{|c|c|c|c|c|c|c|}\n      \\hline\n      \\multirow{2}{*}{\\tabincell{c}{Method}}  & \\multicolumn{2}{|c|}{Original} &  \\multicolumn{2}{|c|}{+ RR} & \\multicolumn{2}{|c|}{+ RR + TLift}\\\\\n      \\cline{2-7}\n        & Rank-1 & mAP  & Rank-1 & mAP & Rank-1 & mAP\\\\\n      \\hline\n      Softmax cross-entropy & 34.9 & 18.4 & 41.5 & 30.5 & 51.7 & 39.7\\\\\n      Arc loss~\\cite{Deng-CVPR19-ArcFace} & 35.3 & 17.1 & 39.8 & 26.3 & 51.0 & 34.8\\\\\n      Center loss~\\cite{Wen-ECCV16-CenterLoss,Jin-IJCB17-CenterLossReID} & 38.9 & 22.1 & 42.5 & 31.5 & 56.6 & 42.6\\\\\n      Class memory loss & 40.7 & 21.8 & 47.8 & 36.1 & 59.6 & 46.2\\\\\n      QAConv & \\textbf{54.4} & \\textbf{33.6} & \\textbf{61.8} & \\textbf{52.4} & \\textbf{70.0} & \\textbf{61.2}\\\\\n      \\hline\n    \\end{tabular}\n\\end{table}\n\n\\begin{table}\n    \\centering\n    \\caption{Role of TLift under Duke$\\rightarrow$Market (\\%).}\\label{tab:tlift-duke}\n    \\begin{tabular}{|c|c|c|c|c|c|c|}\n      \\hline\n      \\multirow{2}{*}{\\tabincell{c}{Method}}  & \\multicolumn{2}{|c|}{Original} &  \\multicolumn{2}{|c|}{+ RR} & \\multicolumn{2}{|c|}{+ RR + TLift}\\\\\n      \\cline{2-7}\n        & Rank-1 & mAP  & Rank-1 & mAP & Rank-1 & mAP\\\\\n      \\hline\n      Softmax cross-entropy & 48.5 & 21.4 & 53.2 & 33.7 & 63.3 & 38.0\\\\\n      Arc loss~\\cite{Deng-CVPR19-ArcFace} & 48.9 & 21.4 & 54.5 & 34.8 & 64.8 & 39.3\\\\\n      Center loss~\\cite{Wen-ECCV16-CenterLoss,Jin-IJCB17-CenterLossReID} & 48.8 & 22.0 & 52.5 & 33.3 & 63.0 & 36.9\\\\\n      Class memory loss & 47.8 & 20.5 & 52.9 & 33.1 & 63.4 & 37.5\\\\\n      QAConv & \\textbf{62.8} & \\textbf{31.6} & \\textbf{68.5} & \\textbf{51.2} & \\textbf{78.7} & \\textbf{58.2}\\\\\n      \\hline\n    \\end{tabular}\n\\end{table}\n\n\\begin{table}\n    \\centering\n    \\caption{Influence of TLift parameters under Market$\\rightarrow$Duke (\\%). Bold numbers are with the default parameters.}\\label{tab:tlift-para-market}\n    \\linespread{1.2}\\selectfont\n    \\setlength{\\tabcolsep}{2mm}{\n    \\begin{tabular}{|c|c c c c c c c c c c|}\n      \\hline\n      $\\tau$ & 50 & \\textbf{100}\t& 150 &\t200\t& 250 &\t300\t& 350 &\t400\t& 450 &\t500\\\\\n      \\hline\n      Rank-1 & 69.3 & \\textbf{70.0} & 69.7 & 69.8 & 69.1 & 68.3 & 66.8 & 65.5 & 64.4 & 63.9 \\\\\n      mAP & 60.7 & \\textbf{61.2} & 60.7 & 59.9 & 58.8 & 57.3 & 55.7 & 54.0 & 52.4 & 51.2\\\\\n      \\hline\n    \\end{tabular}\n    \\\\[5mm]\n    \\begin{tabular}{|c|c c c c c c c c c c|}\n      \\hline\n      $\\sigma$ & 50 & 100 & 150 & \\textbf{200}\t& 250 &\t300\t& 350 &\t400\t& 450 &\t500\\\\\n      \\hline\n      Rank-1 & 67.4 & 69.5 & 70.4 & \\textbf{70.0} & 69.4 & 69.2 & 68.9 & 68.4 & 68.0 & 67.7\\\\\n      mAP & 55.4 & 59.6 & 60.9 & \\textbf{61.2} & 61.0 & 60.8 & 60.5 & 60.1 & 59.8 & 59.5 \\\\\n      \\hline\n    \\end{tabular}\n    \\\\[5mm]\n    \\begin{tabular}{|c|c c c c c c c c c c|}\n      \\hline\n      $K$ & 5 & \\textbf{10} & 15 & 20 & 30 & 40 & 50 & 100 & 150 & 200\\\\\n      \\hline\n      Rank-1 & 69.7 & \\textbf{70.0} & 70.2 & 70.0 & 69.4 & 68.9 & 68.2 & 67.0 & 65.5 & 64.8\\\\\n      mAP & 60.8 & \\textbf{61.2} & 61.2 & 61.0 & 60.3 & 59.6 & 58.8 & 56.8 & 55.7 & 55.2\\\\\n      \\hline\n    \\end{tabular}\n    \\\\[5mm]\n    \\begin{tabular}{|c|c c c c c c c c c c|}\n      \\hline\n      $\\alpha$ & 0.01 & 0.02 & 0.05 & 0.1 & \\textbf{0.2} & 0.3 & 0.4 & 0.5 & 0.7 & 1\\\\\n      \\hline\n      Rank-1 & 70.4 & 70.4 & 70.2 & 70.2 & \\textbf{70.0} & 69.4 & 69.1 & 68.6 & 68.3 & 67.5 \\\\\n      mAP & 60.8 & 60.8 & 60.8 & 61.0 & \\textbf{61.2} & 61.1 & 61.0 & 60.9 & 60.4 & 59.7\\\\\n      \\hline\n    \\end{tabular}}\n\\end{table}\n\n\n\\section{Parameter Analysis}\n\nConsidering the memory consumption and the efficiency, the kernel size of QAConv is set to $s=1$. Parameters for TLift are $\\tau=100$, $\\sigma=200$, $K=10$, and $\\alpha=0.2$. They were fixed in all experiments after some initial tries. To understand their influence, we vary them one by one, with corresponding results shown in Tables \\ref{tab:tlift-para-market} and \\ref{tab:tlift-para-duke}. It can be observed that, the parameters are not sensitive in a broad range, so that they are easy to select. Besides, some better results can be obtained by varying parameters other than the defaults.\n\n\\begin{table}\n    \\centering\n    \\caption{Influence of TLift parameters under Duke$\\rightarrow$Market (\\%). Bold numbers are with the default parameters.}\\label{tab:tlift-para-duke}\n    \\linespread{1.2}\\selectfont\n    \\setlength{\\tabcolsep}{2mm}{\n    \\begin{tabular}{|c|c c c c c c c c c c|}\n      \\hline\n      $\\tau$ & 50 & \\textbf{100}\t& 150 &\t200\t& 250 &\t300\t& 350 &\t400\t& 450 &\t500\\\\\n      \\hline\n      Rank-1 & 76.2 & \\textbf{78.7} & 79.8 & 79.7 & 79.9 & 79.0 & 78.6 & 78.2 & 77.6 & 77.2 \\\\\n      mAP & 57.2 & \\textbf{58.2} & 58.6 & 58.4 & 58.2 & 57.7 & 57.2 & 56.6 & 56.0 & 55.4\\\\\n      \\hline\n    \\end{tabular}\n    \\\\[5mm]\n    \\begin{tabular}{|c|c c c c c c c c c c|}\n      \\hline\n      $\\sigma$ & 50 & 100\t& 150 &\t\\textbf{200}\t& 250 &\t300\t& 350 &\t400\t& 450 &\t500\\\\\n      \\hline\n      Rank-1 & 76.1 & 78.5 & 78.6 & \\textbf{78.7} & 78.6 & 78.1 & 78.0 & 77.9 & 77.9 & 77.6 \\\\\n      mAP & 55.6 & 57.6 & 58.1 & \\textbf{58.2} & 58.5 & 58.7 & 58.8 & 59.0 & 59.1 & 59.2\\\\\n      \\hline\n    \\end{tabular}\n    \\\\[5mm]\n    \\begin{tabular}{|c|c c c c c c c c c c|}\n      \\hline\n      $K$ & 5 & \\textbf{10} & 15 & 20 & 30 & 40 & 50 & 100 & 150 & 200\\\\\n      \\hline\n      Rank-1 & 79.6 & \\textbf{78.7} & 78.1 & 77.6 & 76.6 & 76.2 & 75.8 & 74.4 & 73.4 & 72.7\\\\\n      mAP & 56.9 & \\textbf{58.2} & 58.4 & 58.3 & 58.0 & 57.9 & 57.8 & 57.3 & 56.6 & 55.9\\\\\n      \\hline\n    \\end{tabular}\n    \\\\[5mm]\n    \\begin{tabular}{|c|c c c c c c c c c c|}\n      \\hline\n      $\\alpha$ & 0.01 & 0.02 & 0.05 & 0.1 & \\textbf{0.2} & 0.3 & 0.4 & 0.5 & 0.7 & 1\\\\\n      \\hline\n      Rank-1 & 78.4 & 78.5 & 78.7 & 78.8 & \\textbf{78.7} & 78.5 & 78.0 & 77.6 & 76.5 & 75.4\\\\\n      mAP &  53.8 & 54.1 & 55.0 & 56.3 & \\textbf{58.2} & 59.4 & 59.9 & 60.0 & 59.5 & 58.5\\\\\n      \\hline\n    \\end{tabular}}\n\\end{table}\n\n\n\\section{Memory Usage}\n\nOne drawback of QAConv is that it requires more memory to run than other methods, and it needs to store feature maps of images, rather than features, where feature maps are generally larger in size than representation features. For training on the DukeMTMC-reID, the GPU memory consumption for the QAConv is about 2.83GB, while that for the softmax baseline is about 2.78GB. They are comparable because though QAConv spends some more on class memory, it uses three layers of the ResNet-50, while the softmax baseline uses four layers. For inference, the peak GPU memory for the QAConv is about 2.3GB, while that for the softmax baseline is about 1.7GB.\n\n\n\\section*{Biography}\n\nShengcai Liao is a Lead Scientist in the Inception Institute of Artificial Intelligence (IIAI), Abu Dhabi, UAE. He is a Senior Member of IEEE. Previously, he was an Associate Professor in the Institute of Automation, Chinese Academy of Sciences (CASIA). He received the B.S. degree in mathematics from the Sun Yat-sen University in 2005 and the Ph.D. degree from CASIA in 2010. He was a Postdoc in the Michigan State University during 2010-2012. His research interests include object detection, face recognition, and person re-identification. He has published over 100 papers, with over 11,000 citations according to Google Scholar. He was awarded the Best Student Paper in ICB 2006, ICB 2015, and CCBR 2016, and the Best Paper in ICB 2007. He was also awarded the IJCB 2014 Best Reviewer and CVPR 2019 Outstanding Reviewer. He was an Assistant Editor for the book \"Encyclopedia of Biometrics (2nd Ed.)\". He also served as Area Chairs for ICPR 2016, ICB 2016, and ICB 2018, and reviewers for ICCV, CVPR, ECCV, TPAMI, IJCV, TIP, TIFS, etc. He was the Winner of the CVPR 2017 Detection in Crowded Scenes Challenge and the ICCV 2019 NightOwls Pedestrian Detection Challenge. Homepage: \\url{https:\/\/liaosc.wordpress.com\/}\n\n~\\\\\n\nLing Shao (Senior Member, IEEE) is currently the Executive Vice President and a Provost of the Mohamed bin Zayed University of Artificial Intelligence. He is also the CEO and the Chief Scientist of the Inception Institute of Artificial Intelligence (IIAI), Abu Dhabi, United Arab Emirates. His research interests include computer vision, machine learning, and medical imaging. He is a fellow of IAPR, IET, and BCS.\n\n\n\\bibliographystyle{splncs04}\n\n\\section{Introduction}\nPerson re-identification is an active research topic in computer vision. It aims at finding the same person as the query image from a large volume of gallery images. With the progress in deep learning, person re-identification has been largely advanced in recent years. However, when generalization ability becomes an important concern, required by practical applications, existing methods usually lack satisfactory performance, evidenced by direct cross-dataset evaluation \\cite{yi2014deep,Hu2014Cross}. To address this, many transfer learning, domain adaptation, and unsupervised learning methods, performed on the target domain, have been proposed. However, these methods require heavy computations in deployment, limiting their application in practical scenarios where the deployment machine may have limited resources to support deep learning and users may cannot wait for a time-consuming adaptation stage. Therefore, improving the baseline model's generalization ability to support ready usage is still of urgent importance.\n\nMost existing person re-identification methods compute a fixed representation vector, also known as a feature vector, for each image, and employ a typical distance or similarity metric (e.g. Euclidean distance or cosine similarity) for image matching. Without domain adaptation or transfer learning, the learned model is fixed as is, which is not adaptable for handling various unseen scenarios. Therefore, when generalization ability is a concern, it is expected to have an adaptive ability for the given model architecture.\n\nIn this paper, we focus on generalizable and ready-to-use person re-identification, through direct cross-dataset evaluation. Beyond representation learning, we consider how to formulate query-adaptive image matching directly in deep feature maps.\nSpecifically, we treat image matching as finding local correspondences in feature maps, and construct query-adaptive convolution kernels on the fly to achieve local matching (see Fig. \\ref{fig:qaconv}). In this way, the learned model benefits from adaptive convolution kernels in the final layer, specific to each image, and the matching process and result are interpretable (see Fig. \\ref{fig:top-examples}), similar to traditional feature correspondence approaches \\cite{SIFT,Bay-CVIU-08}. Probably because finding local correspondences through query-adaptive convolution is a common process among different domains, this explicit matching is more generalizable than representation features to unseen scenarios, such as unknown misalignments, pose or viewpoint changes. We call this Query-Adaptive Convolution QAConv. To facilitate end-to-end training of this architecture, we further build a class memory module to cache feature maps of the most recent samples of each class, so as to compute image matching losses for metric learning.\n\nThrough direct cross-dataset evaluation without further transfer learning, the proposed method achieves comparable results to many transfer learning methods for person re-identification. Besides, to explore the prior spatial-temporal structure of a camera network, a model-free temporal cooccurrence based score weighting method is proposed, named Temporal Lifting (TLift).\nThis is also computed on the fly for each query image, without statistical learning of a transition time model in advance. As a result, TLift improves person re-identification to a further extent, resulting in state-of-the-art results in cross-dataset evaluations.\n\nTo summarize, the novelty of this work include (i) a new deep image matching approach with query-adaptive convolutions, along with a class memory module for end-to-end training, and (ii) a model-free temporal cooccurrence based score weighting method. The advantages of this work are also two-fold. First, the proposed image matching method is interpretable, it is well-suited in handling misalignments, pose or viewpoint changes, and it also generalizes well in unseen domains. Second, both QAConv and TLift can be computed on the fly, and they are complementary to many other methods. For example, QAConv can serve as a better pre-trained model for transfer learning, and TLift can be readily applied by most person re-identification algorithms as a post-processing step.\n\n\\section{Related Works}\nDeep learning approaches have largely advanced person re-identification in recent years~\\cite{Ye2020Survey}. However, due to limited labeled data and a big diversity in real-world surveillance,\nthese methods usually have poor generalization ability in unseen scenarios. To address this, many unsupervised domain adaption (UDA) methods have been proposed~\\cite{peng2016unsupervised,chang2019disjoint,wang2018transferable,li2018unsupervised,fan2018unsupervised,li2019unsupervised,Zhong-CVPR19-ECN,Yu-CVPR19-MAR,Yang-CVPR19-PAUL}, which show improved cross-dataset results than traditional methods, though requiring further training on the target domain. QAConv is orthogonal to transfer learning methods as it can provide a better baseline model for them (see Section \\ref{subsec:sota} and Table \\ref{tab:duke+market}).\n\nThere are many representation learning methods proposed to deal with viewpoint changes and misalignments in person re-identification, such as part-aligned feature representations~\\cite{sun2017pcb,wang2018learning,suh2018part,zhao2017deeply}, pose-adapted feature representations~\\cite{zhao2017spindle,saquib2018pose}, human parsing based representations~\\cite{kalayeh2018human}, local neighborhood matching~\\cite{ahmed2015improved,Li-CVPR-2014-DeepReID}, and attentional networks \\cite{liu2017end,qian2017multi,liu2017hydraplus,xu2017jointly,si2018dual,li2018harmonious,xu2018attention}. While these methods present high accuracy when trained and tested on the same dataset, their generalization ability to other datasets is mostly unknown. Besides, beyond representation learning, QAConv focuses on image matching via local correspondences.\n\nGeneralizable person re-identification was first studied in our previous works \\cite{yi2014deep,Hu2014Cross}, where direct cross-dataset evaluation was proposed. More recently, Song et al. \\cite{song2019generalizable} proposed a domain-invariant mapping network by meta-learning, and Jia et al. \\cite{jia2019frustratingly} applied the IBN-Net \\cite{pan2018two} to improve generalizability, while QAConv is preliminarily reported in \\cite{Liao-arXiv2019-QAConv}. QAConv is orthogonal to methods of network design, for example, it can also be applied on the IBN-Net for improvements.\n\nFor deep feature matching, Kronecker-Product Matching (KPM) \\cite{shen2018end} computes a cosine similarity map by outer product for softly aligned element-wise subtraction. Besides, Bilinear Pooling \\cite{Lin2015Bilinear,Ustinova2017bilinear,suh2018part-aligned} and Non-local Neural Networks \\cite{wang2018non} also apply the outer product for part-aligned or self-attended representation learning. Different to the above methods, QAConv is a convolutional matching method but not simply outer product especially when its kernel size $s>1$. It is explicitly designed for local correspondence matching, interpretation, and generalization, in a straightforward way without other branches.\n\nFor post-processing, re-ranking is a technique of refining matching scores, which further improves person re-identification~\\cite{liu2013pop,yu2017divide,zhong2017re,saquib2018pose}. Besides, temporal information is also a useful cue to facilitate cross-camera person re-identification~\\cite{lv2018unsupervised,wang2019spatial-temporal}. While existing methods model transition times across different cameras but encounter difficulties in complex transition time distributions, the proposed TLift method applies cooccurrence constraint within each camera to avoid estimating transition times, and it is model-free and can be computed on the fly.\n\nFor memory based loss, ECN \\cite{Zhong-CVPR19-ECN} proposed an exemplar memory which caches feature vectors of every instance for UDA. This makes the instance-level label inference convenient but limits its scalability. In contrast, class memory is independently designed \\cite{Liao-arXiv2019-QAConv}, which is more efficient working in class level.\n\n\\section{Query-adaptive Convolution}\n\\subsection{Query-adaptive Convolutional Matching}\n\nFor face recognition and person re-identification, most existing methods do not explicitly consider the relationship between two input images under matching, but instead, like classification, they treat each image independently and apply the learned model to extract a fixed feature representation. Then, image matching is simply a distance measure between two representation vectors, regardless of the direct relationship between the actual contents of the two images.\n\nIn this paper, we consider the relationship between two images, and try to formulate adaptive image matching directly in deep feature maps. Specifically, we treat image matching as finding local correspondences in feature maps, and construct query-adaptive convolution kernels on the fly to achieve local matching. As shown in Fig. \\ref{fig:qaconv} and Fig. \\ref{fig:arch}, to match two images, each image is firstly fed forward into a backbone CNN, resulting in a final feature map of size $[1, d, h, w]$, where $d$ is the number of output channels, and $h$ and $w$ are the height and width of the feature map, respectively. Then, the channel dimension of both feature maps is normalized by the $\\ell2$-norm. After that, local patches of size $[s, s]$  at every location of the query feature map are extracted, and then reorganized into $[hw, d, s, s]$ as a convolution kernel, with input channels $d$, output channels $hw$, and kernel size $[s, s]$. This acts as a query-adaptive convolution kernel, with parameters constructed on the fly from the input, in contrast to fixed convolution kernels in the learned model. Upon this, the adaptive kernel can be used to perform a convolution on another feature map, resulting in $[1, hw, h, w]$ similarities.\n\nSince feature channels are $\\ell2$-normalized, when $s=1$, the convolution in fact measures the cosine similarity at every location of the two feature maps.\nBesides, since the convolution kernel is adaptively constructed from the image content, these similarity values exactly reflect the local matching results between the two input images. Therefore, an additional global max pooling (GMP) operation will output the best local matches, and the maximum indices found by GMP indicate the best locations of local correspondences, which can be further used to interpret the matching result, as shown in Fig. \\ref{fig:top-examples}.\nNote that GMP can also be done along the $hw$ axis of the $[1, hw, h, w]$ similarity map. That is, seeking the best matches can be carried out from both sides of the images. Concatenating the output will result in a similarity vector of size $2hw$ for each pair of images.\n\n\\subsection{Network Architecture}\n\\begin{figure}\n\\begin{minipage}{58mm}\n\\centering\n\\includegraphics[width=58mm]{arch}\n\\caption{Architecture of the QAConv. GMP: global max pooling. BN: batch normalization. FC: fully connection.}\n\\label{fig:arch}\n\\end{minipage}\n\\hspace{4mm}\n\\begin{minipage}{58mm}\n\\centering\n\\includegraphics[width=58mm]{TLift}\n\\caption{Illustration of the proposed TLift approach.\n\\label{fig:TLift}\n\\end{minipage}\n\\end{figure}\nThe architecture of the proposed query-adaptive convolution method is shown in Fig. \\ref{fig:arch}, which consists of a backbone CNN, the QAConv layer for local matching, a class memory layer for training, a global max pooling layer, a BN-FC-BN block, and, finally, a similarity output by a sigmoid function for evaluation in the test phase or loss computation in the training phase.\nThe output size of the FC layer is $1$, which acts as a binary classifier or a similarity metric, indicating whether or not one pair of images belongs to the same class. The two BN (batch normalization \\cite{Ioffe-BatchNorm-ICML15}) layers are all one-dimensional. They are used to normalize the similarity output and stabilize the gradient during training.\n\n\\subsection{Class Memory and Update}\nWe propose a class memory module to facilitate the end-to-end training of  the QAConv network. Specifically, a $[c, d, h, w]$ tensor buffer is registered, where $c$ is the number of classes. For each mini batch of size $b$,\nthe $[b, d, h, w]$ feature map tensor of the mini batch will be updated into the memory buffer. We use a direct assignment update strategy, that is, each $[1, d, h, w]$ sample of class $i$ from the mini batch will be assigned into location $i$ of the $[c, d, h, w]$ memory buffer.\n\nAn exponential moving average update can also be used here. However, in our experience this is inferior to the direct replacement update. There might be two reasons for this. First, the replacement update caches feature maps of the most recent samples of each class, so as to reflect the most up-to-date state of the current model for loss computation. Second, since our task is to carry out image matching with local details in feature maps for correspondences, exponential moving average may smooth the local details of samples from the same class\n\n\\subsection{Loss Function}\nWith a mini batch of size $[b, d, h, w]$ and class memory of size $[c, d, h, w]$, $b\\times c$ pairs of similarity values will be computed by QAConv after the BN-FC-BN block. We use a sigmoid function to map the similarity values into $[0,1]$, and compute the binary cross entropy loss. Since there are far more negative than positive pairs, to balance them and enable online hard example mining, we apply the focal loss \\cite{lin2017focal} to weight the binary cross entropy. That is,\n\\begin{equation}\\label{eq:loss}\n\\ell(\\theta) = -\\frac{1}{b}\\sum_{i=1}^{b}\\sum_{j=1}^{c}(1-\\hat{p}_{ij}(\\theta))^{\\gamma}log(\\hat{p}_{ij}(\\theta)),\\\\\n\\end{equation}\nwhere $\\theta$ is the network parameter,\n$\\gamma=2$ is the focusing parameter \\cite{lin2017focal}, and\n\\begin{equation}\n\\begin{aligned}\n&\\hat{p}_{ij}=\n\\begin{cases}\np_{ij}  & \\mbox{if $y_{ij}=1$,}\\\\\n1-p_{ij}& \\mbox{otherwise,}\n\\end{cases}\n\\end{aligned}\n\\end{equation}\nwhere $y_{ij}=1$ indicates a positive pair, while a negative pair otherwise, and $p_{ij}\\in[0,1]$ is the sigmoid probability.\n\n\\section{Temporal Lifting}\nFor person re-identification, to explore the prior spatial-temporal structure of a camera network, usually a transition time model is learned to measure the transition probability. However, for a complex camera network and various person transition patterns, it is not easy to learn a robust transition time distribution. In contrast, in this paper a model-free temporal cooccurrence based score weighting method is proposed, which is called Temporal Lifting (TLift). TLift does not model cross-camera transition times which could be variable and complex. Instead, TLift makes use of a group of nearby persons in each single camera, and find similarities between them.\n\nFig. \\ref{fig:TLift} illustrates the idea. A basic assumption is that people nearby in one camera are likely still nearby in another camera. Therefore, their corresponding matches in other cameras can serve as pivots to enhance the weights of other nearby persons. In Fig. \\ref{fig:TLift}, $A$ is the query person. $E$ is more similar than $A'$ to $A$ in another camera. With nearby persons $B$ and $C$, and their top retrievals $B'$ and $C'$ acting as pivots, the matching score of $A'$ can be temporally lifted since it is a nearby person of $B'$ and $C'$, while the matching score of $E$ will be reduced since there is no such pivot.\n\nFormally, suppose $A$ is the query person in camera $Q$, then, the set of nearby persons to $A$ in camera $Q$ is defined as $R = \\{B | \\Delta T_{AB} < \\tau, \\forall B \\in Q\\}$,\nwhere $\\Delta T_{AB}$ is the within-camera time difference between persons $A$ and $B$, and $\\tau$ is a threshold on $\\Delta T$ to define nearby persons. Then, for each person in $R$, cross-camera person retrieval will be performed on a gallery camera $G$ by QAConv or other methods, and the overall top K retrievals for $R$ are defined as the pivot set $P$. Next, each person in $P$ acts as an ensemble point for 1D kernel density estimation on within-camera time differences in $G$, and the temporal matching probability between $A$ and any person $X$ in camera $G$ will be computed as\n\\begin{equation}\np_t(A,X) = \\frac{1}{|P|}\\sum_{B\\in P}e^{-\\frac{\\Delta T_{BX}^2}{\\sigma^2}},\n\\end{equation}\nwhere $\\sigma$ is the sensitivity parameter of the time difference. Then, this temporal probability is used to weight the similarity score of appearance models using a multiplication fusion as $p(A,X) = (p_t(A,X) + \\alpha)p_a(A,X)$,\nwhere $p_a(A,X)$ is the appearance based matching probability (e.g. by QAConv), and $\\alpha$ is a regularizer.\n\nThis way, true positives near pivots will be lifted, while hard negatives far from pivots will be suppressed. Note that this is also computed on the fly for each query image, without learning a transition time model in advance. Therefore, it does not require training data, and can be readily applied by many other person re-identification methods.\n\n\\section{Experiments}\n\\subsection{Implementation Details}\nThe proposed method is implemented in PyTorch, based upon an adapted version \\cite{zhong2018camstyle} of the open source person re-identification library (open-reid)\\footnote{https:\/\/cysu.github.io\/open-reid\/}. Person images are resized to $384\\times128$. The backbone network is the ResNet-152 \\cite{he2016resnet} pre-trained on ImageNet, unless otherwise stated. The layer3 feature map of the backbone network is used, since the size of the layer4 feature map is too small. A $1\\times1$ convolution with 128 channels is further appended to reduce the final feature map size. The batch size of samples for training is 32. The SGD optimizer is applied, with a learning rate of 0.001 for the backbone network, and 0.01 for newly added layers. They are decayed by 0.1 after 40 epochs, and the training stops at 60 epochs. The whole QAConv is end-to-end jointly trained, while class memory is updated only after the loss computation. Considering the memory consumption and the efficiency, the kernel size of QAConv is set to $s=1$. Parameters for TLift are $\\tau=100$, $\\sigma=200$, $K=10$, and $\\alpha=0.2$. They are not sensitive in a broad range, as analyzed in the Appendix.\n\n\nA random occlusion module is implemented for data augmentation, which is similar to the random erasing \\cite{zhong2020random} and cutout \\cite{devries2017improved} methods (see Appendix for comparisons). Specifically, a square area is generated with the size randomly sampled at most $0.8\\times width$ of the image. Then this square area is filled with white pixels. It is useful for QAConv because random occlusion forces QAConv to learn various local correspondences, instead of only saliency but easy ones. Beyond this, only a random horizontal flipping is used for data augmentation.\n\n\\subsection{Datasets}\nExperiments were conducted on four large person re-identification datasets, Market-1501 \\cite{zheng2015smarket}, DukeMTMC-reID \\cite{ristani2016duke,zheng2017unlabeled}, CUHK03 \\cite{Li-CVPR-2014-DeepReID}, and MSMT17~\\cite{Wei-CVPR18-PTGAN}. The Market-1501 dataset contains 32,668 images of 1501 identities captured from 6 cameras. There are 12,936 images from 751 identities for training, and 19,732 images from 750 identities for testin\n. The DukeMTMC-reID is a subset of the multi-target and multi-camera pedestrian tracking dataset DukeMTMC~\\cite{ristani2016duke}. It includes 1,812 identities and 36,411 images, where 16,522 images of 702 identities are used for training, and the remainings for test. The CUHK03 dataset includes 13,164 images of 1,360 pedestrians. We adopted the CUHK03-NP protocol provided in \\cite{zhong2017re}, where images of 767 identities were used for training, and other images of 700 identities were used for test. Besides, we used the detected subset for evaluation, which is more challenging. The MSMT17 dataset is the largest person re-identification dataset to date, which contains 4,101 identities and 126,441 images captured from 15 cameras. It is divided into a training set of 32,621 images from 1,041 identities, and a test set with the remaining images from 3,010 identities.\n\nCross-dataset evaluation was performed in these datasets, by training on the training subset of one dataset (except that in MSMT17 we used all images for training following \\cite{Yu-CVPR19-MAR,Yang-CVPR19-PAUL}), and evaluating on the test subset of another dataset. The cumulative matching characteristic (CMC) and mean Average Precision (mAP) were used as the performance evaluation metrics. All evaluations followed the single-query evaluation protocol.\n\nThe Market-1501 and DukeMTMC-reID datasets are with frame numbers available, so that it is able to evaluate the proposed TLift method. The DukeMTMC-reID dataset has a good global and continuous record of frame numbers, and it is synchronized by providing offset times. In contrast, the Market-1501 dataset has only independent frame numbers for each session of videos from each camera.\nAccordingly we simply made a cumulative frame record by assuming continuous video sessions. After that, frame numbers were converted to seconds in time by dividing the Frames Per Second (FPS) in video records, where FPS=59.94 for the DukeMTMC-reID dataset and FPS=25 for the Market-1501 dataset.\n\n\\subsection{Ablation Study}\nSome ablation studies have been conducted to understand the proposed method, in the context of direct cross-dataset evaluation between the Market-1501 and DukeMTMC-reID datasets.\nFirst, to understand the QAConv loss, several other loss functions, including the classical softmax based cross entropy loss, the center loss~\\cite{Wen-ECCV16-CenterLoss,Jin-IJCB17-CenterLossReID}, the Arc loss (derived from the ArcFace method~\\cite{Deng-CVPR19-ArcFace} which is effective for face recognition), and the proposed class memory based loss, are evaluated for comparison. For these compared loss functions, the global average pooling of layer4 (better than layer3) of the ResNet-152 is used for feature representation, and the cosine similarity measure is adopted instead of the QAConv similarity. For the class memory loss, feature vectors are cached in memory instead of learnable parameters, and the same BN layer and Eq. (\\ref{eq:loss}) are applied after calculating the cosine similarity values between mini-batch features and memory features.\n\nFrom results shown in Table \\ref{tab:loss}, it is obvious that QAConv improves existing loss functions by a large margin, with 13.7\\%-19.5\\% improvements in Rank-1, and 9.6\\%-11.1\\% in mAP. Interestingly, large margin classifiers improves the softmax cross-entropy baseline when trained on the Market-1501 dataset, but do not have such improvements when trained on DukeMTMC-reID. This is probably due to many ambiguously labeled or closely walking persons in DukeMTMC-reID (see Section \\ref{sec:discuss}), which may confuse the strict large margin training. Note that the class memory based loss only performs comparable to other existing losses, indicating that the large improvement of QAConv is mainly due to the new matching mechanism, rather than the class memory based loss function. Besides, the Arc loss published recently is one of the best face recognition method, but it does not seem to be powerful when applied in person re-identification\\footnote{We have tried different hyper parameters and reported the best results. The best margin values were found to be 0.5 on Market-1501 and 0.2 on DukeMTMC-reID.}. In our experience, the choice of loss functions does not largely influence person re-identification performance. Similar as in face recognition, existing studies \\cite{Wen-ECCV16-CenterLoss,Deng-CVPR19-ArcFace} show that new loss functions do have improvements, but cannot be regarded as significant ones over the softmax cross entropy baseline. Therefore, we may conclude that the large improvement observed here is due to the new matching scheme, instead of different loss configurations (see Appendix for more analyses).\n\\begin{table}\n    \\centering\n    \\caption{Role of loss functions (\\%).}\\label{tab:loss}\n    \\begin{tabular}{|c|c|c|c|c|}\n      \\hline\n      \\multirow{2}{*}{\\tabincell{c}{Method}}  & \\multicolumn{2}{|c|}{Market$\\rightarrow$Duke} &  \\multicolumn{2}{|c|}{Duke$\\rightarrow$Market} \\\\\n      \\cline{2-5}\n        & Rank-1 & mAP  & Rank-1 & mAP \\\\\n      \\hline\n      Softmax cross-entropy & 34.9 & 18.4 & 48.5 & 21.4\\\\\n      Arc loss~\\cite{Deng-CVPR19-ArcFace} & 35.3 & 17.1 & 48.9 & 21.4\\\\\n      Center loss~\\cite{Wen-ECCV16-CenterLoss,Jin-IJCB17-CenterLossReID} & 38.9 & 22.1 & 48.8 & 22.0\\\\\n      Class memory loss & 40.7 & 21.8 & 47.8 & 20.5\\\\\n      QAConv & \\textbf{54.4} & \\textbf{33.6} & \\textbf{62.8} & \\textbf{31.6}\\\\\n      \\hline\n    \\end{tabular}\n\\end{table}\n\nNext, to understand the role of re-ranking (RR), the k-reciprocal encoding method \\cite{zhong2017re} is applied upon QAConv. From results shown in Table \\ref{tab:tlift}, it can be seen that enabling re-ranking do improve the performance a lot, especially with mAP, which is increased by 18.8\\% under Market$\\rightarrow$Duke, and 19.6\\% under Duke$\\rightarrow$Market. This improvement is much more significant based on QAConv than that based on other methods as reported in \\cite{zhong2017re}. This is probably because the new QAConv matching scheme better measures the similarity between images, which benefits the reverse neighbor based re-ranking method\n\nFurthermore, based on QAConv and re-ranking, the contribution of TLift is evaluated, compared to a recent method called TFusion (TF) \\cite{Lv18-TFusion}, which is originally designed to iteratively improve transfer learning. From results shown in Table \\ref{tab:tlift}, it can be observed that employing TLift to explore temporal information further improves the results, with Rank-1 improved by 8.2\\%-10.2\\%, and mAP by 7.0\\%-8.8\\%. This improvement is complementary to re-ranking, so they can be combined. As for the existing method TFusion, it appears to be not stable, as a large improvement can be observed under Market$\\rightarrow$Duke, but little improvement can be obtained under Duke$\\rightarrow$Market, or even the mAP is clearly decreased\\footnote{Note that TFusion parameters were optimized on each dataset to get the best results, but for TLift we used fixed parameters for all datasets (see Appendix for analysis).}. This may be because TFusion is based on learning transition time distributions across cameras, which is not easy to deal with complex camera networks and person transitions as in the Market-1501 (various repeated presences per person in one camera). In contrast, the TLift method only depends on single-camera temporal information which is relatively more easy to handle. Note that TLift can also be generally applied to other methods for improvements, as shown in the Appendix. Besides, as shown in Table \\ref{tab:tlift}, directly applying TLift to QAConv without re-ranking also improves the performance a lot.\n\\begin{table}\n    \\centering\n    \\caption{Performance (\\%) of different post-processing methods.}\\label{tab:tlift}\n    \\begin{tabular}{|l|c|c|c|c|}\n      \\hline\n      \\multirow{2}{*}{\\tabincell{c}{Method}}  & \\multicolumn{2}{|c|}{Market$\\rightarrow$Duke} &  \\multicolumn{2}{|c|}{Duke$\\rightarrow$Market} \\\\\n      \\cline{2-5}\n        & Rank-1 & mAP  & Rank-1 & mAP \\\\\n      \\hline\n      QAConv & 54.4 & 33.6 & 62.8 & 31.6 \\\\\n      QAConv + TLift & 62.7 & 45.3 & 61.5 & 40.6\\\\\n      QAConv + RR~\\cite{zhong2017re} & 61.8 & 52.4 & 68.5 & 51.2 \\\\\n      QAConv + RR + TF~\\cite{Lv18-TFusion} & \\textbf{70.7} & \\textbf{61.9} & 68.6 & 47.2\\\\\n      QAConv + RR + TLift & 70.0 & 61.2 & \\textbf{78.7} & \\textbf{58.2} \\\\\n      \\hline\n    \\end{tabular}\n\\end{table}\n\nFinally, to understand the effect of the backbone network, the QAConv results with the ResNet-50 as backbone are also reported in Tables \\ref{tab:duke+market} and \\ref{tab:cuhk+msmt}, compared to the default ResNet-152 (denoted as QAConv$_{50}$ and QAConv$_{152}$, respectively). As can be observed, a larger network ResNet-152 does have a better performance due to its larger learning capability. It can improve the Rank-1 accuracy over the QAConv$_{50}$ by 1.3\\%-7.3\\%, and the mAP by 0.8\\%-5.5\\%. Besides, there are also consistent improvements in case of combining re-ranking and TLift. Hence, it seems that this larger network, which contains more learnable parameters, does not have the overfitting problem when equipped with QAConv. Note that, though ResNet-152 is a very large network requiring heavy computation, in practice, it can be efficiently reduced by knowledge distillation \\cite{hinton2015distilling}.\n\n\\subsection{Comparison to the State of the Arts}\\label{subsec:sota}\nThere are a great number of person re-identification methods since this is a very active research area. Here we only list recent results for comparison due to limited space. The cross-dataset evaluation results on the four datasets are listed in Tables \\ref{tab:duke+market} and \\ref{tab:cuhk+msmt}. Considering that many person re-identification methods employ the ResNet-50 network, for a fair comparison, the following analysis is based on the QAConv$_{50}$ results. Note that this paper mainly focuses on cross-dataset evaluation. Therefore, some recent methods performing unsupervised learning on the target dataset are not compared here, such as the TAUDL~\\cite{li2018unsupervised}, UTAL~\\cite{li2019unsupervised}, and UGA~\\cite{Wu-CVPR19-UGA}, and also partially due to the fact that they use single-camera target identity labels for training. There are mainly two groups of methods listed in Table \\ref{tab:duke+market}, namely unsupervised transfer learning based methods, and direct cross-dataset evaluation based methods. The first group of methods require images from the target dataset for unsupervised learning, which are not directly comparable to the second one that directly evaluates on the target dataset in consideration of real applications. The proposed QAConv method belongs to the second group. There are very few existing results of the same setting for the second group, except some baselines of other recent methods and the PN-GAN \\cite{qian2018pose} which aims at augmenting source training data by GAN. For the comparison to the transfer learning methods, we consider that QAConv can serve as a better pre-trained model for them, and computing the RR+TLift on the fly is also more efficient than training on target dataset.\n\\begin{table}\n    \\centering\n    \\caption{Comparison of the state-of-the-art cross-dataset evaluation results (\\%) with DukeMTMC-reID and Market-1501 as the target datasets.}\\label{tab:duke+market}\n    \\begin{tabular}{|l||c|c|c|c||c|c|c|c|}\n      \\hline\n      \\multirow{2}{*}{\\tabincell{c}{Method}}  & \\multicolumn{2}{|c|}{Training}  & \\multicolumn{2}{|c||}{Test: Duke}  & \\multicolumn{2}{|c|}{Training}  & \\multicolumn{2}{|c|}{Test: Market}\\\\\n      \\cline{2-9}\n       & Source & Target & R1 & mAP & Source & Target & R1 & mAP \\\\\n      \\hline\n      \\hline\n      PUL, TOMM18~\\cite{fan2018unsupervised} & Market & Duke & 30.4 & 16.4& Duke & Market & 44.7 & 20.1\\\\\t\t\t\t\t\n      TJ-AIDL, CVPR18~\\cite{wang2018transferable} & Market & Duke & 44.3 & 23.0 & Duke & Market & 58.2 & 26.5\\\\\n      MMFA, BMVC18~\\cite{lin2018multi} & Market & Duke & 45.3 & 24.7 & Duke &\tMarket & 56.7 & 27.4\\\\\n      CFSM, AAAI19~\\cite{chang2019disjoint} & Market & Duke & 49.8 & 27.3 & Duke & Market & 61.2 & 28.3\\\\\n      DECAMEL, TPAMI19~\\cite{Yu-TPAMI19-DECAMEL} &-&-&-&-& Multi & Market & 60.2 & 32.4\\\\\n      PAUL, CVPR19~\\cite{Yang-CVPR19-PAUL} & Market & Duke & 56.1 & 35.7 & Duke & Market & 66.7 & 36.8\\\\\n      ECN, CVPR19~\\cite{Zhong-CVPR19-ECN} & Market & Duke & 63.3 & 40.4 & Duke & Market & 75.1 & 43.0\\\\\n      CDS, ICME19~\\cite{Wu-ICME19-CDS} & Market & Duke & 67.2 & 42.7 & Duke & Market & 71.6 & 39.9\\\\\n     \n     \n      \\hline\n      ECN baseline, CVPR19~\\cite{Zhong-CVPR19-ECN} & Market &  & 28.9 & 14.8 & Duke &  & 43.1 & 17.7\\\\\n      PN-GAN, ECCV18~\\cite{qian2018pose} & Market & & 29.9 & 15.8&-&&-&-\\\\\n      QAConv$_{50}$ & Market & & 48.8 & 28.7  & Duke & & 58.6 & 27.2 \\\\\n      QAConv$_{152}$ & Market & & 54.4 & 33.6  & Duke & & 62.8 & 31.6 \\\\\n      QAConv$_{50}$ + RR + TLift & Market & & 64.5 & 55.1 & Duke & & 74.6 & 51.5  \\\\\n      QAConv$_{152}$ + RR + TLift & Market & & 70.0 & 61.2 & Duke & & 78.7 & 58.2  \\\\\n      \\hline\n      \\hline\n      MAR, CVPR19~\\cite{Yu-CVPR19-MAR} & MSMT & Duke & 67.1 & 48.0 & MSMT & Market & 67.7 & 40.0\\\\\n      PAUL, CVPR19~\\cite{Yang-CVPR19-PAUL} & MSMT & Duke & 72.0 & 53.2 & MSMT & Market & 68.5 & 40.1\\\\\n     \n      \\hline\n      MAR baseline, CVPR19~\\cite{Yu-CVPR19-MAR} & MSMT & & 43.1 & 28.8 & MSMT & & 46.2 & 24.6\\\\\n      PAUL baseline, CVPR19~\\cite{Yang-CVPR19-PAUL} & MSMT & & 65.7 & 45.6 & MSMT & & 59.3 & 31.0\\\\\n      QAConv$_{50}$ & MSMT & & 69.4 & 52.6 & MSMT & & 72.6 & 43.1  \\\\\n      QAConv$_{152}$ & MSMT & & 72.2 & 53.4 & MSMT & & 73.9 & 46.6  \\\\\n      QAConv$_{50}$ + RR + TLift & MSMT & & 80.3 & 77.2  & MSMT & & 86.5 & 72.2\\\\\n      QAConv$_{152}$ + RR + TLift & MSMT & & 82.2 & 78.4  & MSMT & & 88.4 & 76.0\\\\\n      \\hline\n    \\end{tabular}\n\\end{table}\n\n\\textbf{DukeMTMC-reID dataset.} As can be observed from Table \\ref{tab:duke+market}, when trained on the Market-1501 dataset, QAConv achieves the best performance in the direct evaluation group with a large margin. When compared to transfer learning methods, QAConv also outperforms many of them except some very recent methods, indicating that QAConv enables the network to learn how to match two person images, and the learned model generalizes well in unseen domains without transfer learning. Besides, by enabling re-ranking and TLift, the proposed method achieves the best result among all except Rank-1 of CDS. Note that the re-ranking and TLift methods can also be incorporated into other methods, though. Therefore, we list their results separately. However, both of these are calculated on the fly without learning in advance, so together with QAConv, it appears that a ready-to-use method with good generalization ability can also be achieved even without further UDA, which is a nice solution considering that UDA requires heavy computation for deep learning in deployment phase.\n\nWhen trained on MSMT17, QAConv itself beats all other methods except the transfer learning method PAUL. This is also the second best result among all existing methods taking DukeMTMC-reID as the target dataset, regardless of the training source. This clearly indicates QAConv's superiority in learning from large-scale data. It is preferred in practice in the sense that, when trained with large-scale data, there may be no need to adapt the learned model in deployment.\n\n\\textbf{Market-1501 dataset.} With Market-1501 as the target dataset as shown in Table \\ref{tab:duke+market}, similarly, when trained with MSMT17, QAConv itself also achieves the best performance among others except Rank-1 of ECN. This can be considered a large advancement in cross-dataset evaluation, which is a better evaluation strategy for understanding the generalization ability of algorithms. Besides, when equipped with RR+TLift, the proposed method achieves the state of the art, with Rank-1 accuracy of 86.5\\% and mAP of 72.2\\%. Note that this comparison is not in a sense of \\textit{fair}. We would like to share that beyond many recent efforts in UDA, enlarging the training data and exploiting on-the-fly computations in re-ranking and temporal fusion may also lead to good performance in unknown domain, with the advantage of no cost in training deep models everywhere.\n\n\\textbf{CUHK03 dataset.} The CUHK03 and MSMT17 datasets present large domain gaps to others. For CUHK03, it can be observed from Table \\ref{tab:cuhk+msmt} that, with either Market-1501 or DukeMTMC-reID dataset as training set, QAConv without UDA performs better than a UDA method PUL~\\cite{fan2018unsupervised}, and fairly comparable to another recent transfer learning method CDS \\cite{Wu-ICME19-CDS}. However, all methods perform not well on the CUHK03 dataset. Only with the large MSMT17 data set as the source training data, the proposed method performs relatively better.\n\n\\textbf{MSMT17 dataset.} With the MSMT17 as target, only QAConv does not require adaptation in Table \\ref{tab:cuhk+msmt}. However, it performs better than PTGAN \\cite{Wei-CVPR18-PTGAN} and in part comparable to ECN \\cite{Zhong-CVPR19-ECN}. This further confirms the generalizability of QAConv under large domain gaps, since without UDA it is already in part comparable to the state-of-the-art UDA methods. Note that TLift is not applicable on CUHK03 and MSMT17 due to no temporal information provided.\n\\begin{table}\n   \n    \\caption{Comparison of the state-of-the-art cross-dataset evaluation results (\\%) with CUHK03-NP (detected) and MSMT17 as the target datasets.}\\label{tab:cuhk+msmt}\n    \\hspace{-4mm}\n    \\begin{tabular}{|l||c|c|c|c||c|c|c|c|}\n      \\hline\n      \\multirow{2}{*}{\\tabincell{c}{Method}}  & \\multicolumn{2}{|c|}{Training}  & \\multicolumn{2}{|c||}{Test: CUHK03}  & \\multicolumn{2}{|c|}{Training}  & \\multicolumn{2}{|c|}{Test: MSMT}\\\\\n      \\cline{2-9}\n       & Source & Target & R1 & mAP & Source & Target & R1 & mAP \\\\\n      \\hline\n      \\hline\n      PUL, TOMM18~\\cite{fan2018unsupervised} & Market & CUHK03 & 7.6 & 7.3 & - & -  & - & - \\\\\n      CDS, ICME19~\\cite{Wu-ICME19-CDS} & Market & CUHK03 & 9.1 & 8.7 & - & -  & - & - \\\\\n      PTGAN~\\cite{Wei-CVPR18-PTGAN}, CVPR18 & - & -  & - & -  & Market & MSMT & 10.2 & 2.9\\\\\n      ECN, CVPR19~\\cite{Zhong-CVPR19-ECN} & - & -  & - & -  & Market & MSMT & 25.3 & 8.5\\\\\n      \\hline\n      QAConv$_{50}$ & Market & & 9.9 & 8.6 & Market & & 22.6 & 7.0\\\\\n      QAConv$_{152}$ & Market & & 14.1 & 11.8  & Market & & 25.6 & 8.2\\\\\n      \\hline\n      \\hline\n      PUL, TOMM18~\\cite{fan2018unsupervised} & Duke & CUHK03 & 5.6 & 5.2 & - & -  & - & - \\\\\n      CDS, ICME19~\\cite{Wu-ICME19-CDS} & Duke & CUHK03 & 8.1 & 7.1 & - & -  & - & - \\\\\n      PTGAN~\\cite{Wei-CVPR18-PTGAN}, CVPR18 & - & -  & - & -  & Duke & MSMT & 11.8 & 3.3\\\\\n      ECN, CVPR19~\\cite{Zhong-CVPR19-ECN} & - & -  & - & -  & Duke & MSMT & 30.2 & 10.2\\\\\n      \\hline\n      QAConv$_{50}$ & Duke & & 7.9 & 6.8 & Duke & & 29.0 & 8.9\\\\\n      QAConv$_{152}$ & Duke & & 11.0 & 9.4 & Duke & & 32.7 & 10.4\\\\\n      \\hline\n      \\hline\n      QAConv$_{50}$ & MSMT & & 25.3 & 22.6 & -& -& -& -\\\\\n      QAConv$_{152}$ & MSMT & & 32.6 & 28.1 &- & -& -& -\\\\\n      \\hline\n    \\end{tabular}\n\\end{table}\n\n\\subsection{Qualitative Analysis and Discussion}\n\\label{sec:discuss}\nA unique characteristic of the proposed QAConv method is its interpretation ability of the matching. Therefore, we show some qualitative matching results in Fig. \\ref{fig:samples} for a better understanding of the proposed method. The model used here is trained on the MSMT17 dataset, and the evaluations are done on the query subsets of the Market-1501 and DukeMTMC-reID datasets. Results of both positive pairs and hard negative pairs are shown. Note that only reliable correspondences with matching scores over 0.5 are shown, and the local positions are coarse due to the $24\\times8$ size of the feature map. As can be observed from Fig. \\ref{fig:samples}, the proposed method is able to find correct local correspondences for positive pairs of images, even if there are notable misalignments in both scale and position, pose\/viewpoint changes, occlusions, and mix up of other persons, thanks to the local matching mechanism of QAConv instead of global feature representations. Besides, for hard negative pairs, the matching of QAConv still appears to be mostly reasonable, by linking visually similar parts or even the same person (may be ambiguously labeled or walking closely to other persons). Note that the QAConv method gains the matching capability by automatic learning, from supervision of only class labels but not local correspondence labels\n\\begin{figure*}\n\\centering\n\\includegraphics[width=13mm]{samples\/market\/pos\/13_0,7121_0720_c5s2_061977_00-0720_c3s2_061603_00.jpg} \\hspace{1mm}\n\\includegraphics[width=13mm]{samples\/market\/pos\/11_0,6259_1299_c4s6_008360_00-1299_c2s3_025582_00.jpg}\\hspace{1mm}\n\\includegraphics[width=13mm]{samples\/market\/pos\/8_0,6189_1255_c4s5_051210_00-1255_c2s3_015432_00.jpg}\\hspace{1mm}\n\\includegraphics[width=13mm]{samples\/market\/pos\/10_0,6574_0776_c4s4_028210_00-0776_c1s4_018156_00.jpg}\n\\hspace{3mm}\n\\includegraphics[width=13mm]{samples\/market\/neg\/7_0,7262_1074_c1s5_007861_00-1247_c3s3_020678_00.jpg}\\hspace{1mm}\n\\includegraphics[width=13mm]{samples\/market\/neg\/8_0,6691_0060_c1s1_008501_00-0182_c5s1_034601_00.jpg}\\hspace{1mm}\n\\includegraphics[width=13mm]{samples\/market\/neg\/8_0,5749_1438_c3s3_055978_00-1485_c2s3_089827_00.jpg}\\hspace{1mm}\n\\includegraphics[width=13mm]{samples\/market\/neg\/11_0,5823_0514_c4s2_068373_00-0964_c1s4_061111_00.jpg}\\\\\n0.71 \\hspace{7mm} 0.63 \\hspace{7mm} 0.62 \\hspace{7mm} 0.66\n\\hspace{10mm}\n0.73 \\hspace{7mm} 0.67 \\hspace{7mm} 0.57 \\hspace{7mm} 0.58\\\\\n(a) Positive pairs on Market-1501\n\\hspace{12mm}\n(b) Negative pairs on Market-1501\\\\\n\\vspace{3mm}\n\n\\includegraphics[width=13mm]{samples\/duke\/pos\/11_0,6435_2988_c5_f0089268-2988_c8_f0044540.jpg}\\hspace{1mm}\n\\includegraphics[width=13mm]{samples\/duke\/pos\/9_0,6157_4405_c6_f0090766-4405_c7_f0092580.jpg}\\hspace{1mm}\n\\includegraphics[width=13mm]{samples\/duke\/pos\/9_0,6628_4508_c8_f0083757-4508_c6_f0110908.jpg}\\hspace{1mm}\n\\includegraphics[width=13mm]{samples\/duke\/pos\/11_0,6045_0295_c2_f0099605-0295_c1_f0099259.jpg}\n\\hspace{3mm}\n\\includegraphics[width=13mm]{samples\/duke\/neg\/5_0,5375_0360_c1_f0108265-0361_c5_f0110785.jpg}\\hspace{1mm}\n\\includegraphics[width=13mm]{samples\/duke\/neg\/6_0,5773_4572_c8_f0105003-4573_c7_f0134670.jpg}\\hspace{1mm}\n\\includegraphics[width=13mm]{samples\/duke\/neg\/9_0,5446_0612_c2_f0163295-4759_c7_f0184320.jpg}\\hspace{1mm}\n\\includegraphics[width=13mm]{samples\/duke\/neg\/6_0,4970_4519_c6_f0115726-4719_c7_f0171037.jpg}\\\\\n0.64 \\hspace{7mm} 0.62 \\hspace{7mm} 0.66 \\hspace{7mm} 0.60\n\\hspace{10mm}\n0.54 \\hspace{7mm} 0.58 \\hspace{7mm} 0.54 \\hspace{7mm} 0.50\\\\\n(c) Positive pairs on DukeMTMC-reID\n\\hspace{5mm}\n(d) Negative pairs on DukeMTMC-reID\\\\\n\\caption{Examples of qualitative matching results by the proposed QAConv method using the model trained on the MSMT17 dataset. Numbers represent similarity scores.}\\label{fig:samples}\n\\end{figure*}\n\nThe QAConv network was trained on an NVIDIA DGX-1 server, with two V100 GPU cards. With the backbone network ResNet-50, the training time of QAConv on the DukeMTMC-reID dataset was 1.22 hours. In contrast, the most efficient softmax baseline took 0.72 hour for training. For deployment, the ECN~\\cite{Zhong-CVPR19-ECN} reported 1 hour of transfer learning time with DukeMTMC-reID as target, while MAR~\\cite{Yu-CVPR19-MAR} and DECAMEL~\\cite{Yu-TPAMI19-DECAMEL} reported 10 and 35.2 hours of total learning time, respectively, compared to the ready-to-use QAConv. For inference, with the DukeMTMC-reID dataset as target, QAConv took 26 seconds for feature extraction and 26 seconds for similarity computation. In contrast, the softmax baseline took 26 seconds for feature extraction and 0.2 seconds for similarity computation. Besides, the proposed method took 303 seconds for reranking, and 67 seconds for TLift. This is still efficient, especially for RR+TLift, compared to transfer learning for deployment. Therefore, the overall solution of QAConv+RR+TLift is promising in practical applications.\n\n\nFor further analysis on memory usage, please see the Appendix. As for the TLift, it can only be applied on datasets with good time records. Though this information is easy to obtain in real surveillance, most existing person re-identification datasets do not contain it. Another drawback of TLift is that it cannot be applied to arbitrary query images beyond a camera network, though once an initial match is found, it can be used to refine the search. Besides, it cannot help when there is no nearby person with the query.\n\n\\section{Conclusion}\nIn this paper, through extensive experiments we show that the proposed QAConv method is quite promising for person matching without further transfer learning, and it has a much better generalization ability than existing baselines. Though QAConv can also be plugged into other transfer learning methods as a better pre-trained model, in practice, according to the experimental results of this paper, we suggest a ready-to-use solution which works in the following principles. First, a large-scale and diverse training data (e.g. MSMT17) is required to learn a generalizable model. Second, a larger network (e.g. ResNet-152) benefits a better overall performance, which could be further distilled into smaller ones for efficiency. Finally, score re-ranking and temporal fusion model such as TLift can be computed on the fly in deployment, which can largely improve performance and they are more efficient to use than transfer learning.\n\n\\section*{Acknowledgements}\nThis work was partly supported by the NSFC Project \\#61672521. The authors would like to thank Yanan Wang who helped producing several illustration figures in this paper, Jinchuan Xiao who optimized the TLift code, and Anna Hennig who helped proofreading the paper.\n\n\n\\clearpage\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Illustrative Examples}\n\n\\subsection{Use of Kronecker Products}\n\nThe example below illustrates the use of Kronecker products to establish the two stated properties of $P,$ to demonstrate \\eqref{miss1}, and to provide insights into the two observations.\n\\begin{example}\t\\label{ex1}\nConsider $X^{\\prime}$ of \\eqref{Xprimedef} with $N=\\{1,2,3\\}$ in the $n=3$ variables $x_1, x_2,$ and $x_3.$ The $2^3=8$ inequalities of \\eqref{RLTstep} are expressed in \\eqref{prodkron2} as \n\\begin{eqnarray} \\scriptsize\n\\left[\\begin{array}{rr|rr|rr|rr}   \nU_1U_2U_3 & -U_1U_2 & -U_1U_3 & U_1 & -U_2U_3 & U_2 & U_3 & -1 \\\\  \n-U_1U_2L_3 & U_1U_2 & U_1L_3 & -U_1 & U_2L_3 & -U_2 & -L_3 & 1 \\\\ \\cline {1-8}\n-U_1L_2U_3 & U_1L_2 & U_1U_3 & -U_1 & L_2U_3 & -L_2 & -U_3 & 1 \\\\  \nU_1L_2L_3 & -U_1L_2 & -U_1L_3 & U_1 & -L_2L_3 & L_2 & L_3 & -1 \\\\ \\cline {1-8}\n-L_1U_2U_3 & L_1U_2 & L_1U_3 & -L_1 & U_2U_3 & -U_2 & -U_3 & 1 \\\\  \nL_1U_2L_3 & -L_1U_2 & -L_1L_3 & L_1 & -U_2L_3 & U_2 & L_3 & -1 \\\\ \\cline {1-8}\nL_1L_2U_3 & -L_1L_2 & -L_1U_3 & L_1 & -L_2U_3 & L_2 & U_3 & -1 \\\\  \n-L_1L_2L_3 & L_1L_2 & L_1L_3 & -L_1 & L_2L_3 & -L_2 & -L_3 & 1 \\\\\n\\end{array}\\right]\\left(\\begin{array}{cccccccc} 1 \\\\ x_3 \\\\ \\cline {1-1} x_2 \\\\ x_2x_3 \\\\ \\cline {1-1} x_1 \\\\ x_1x_3 \\\\ \\cline {1-1} x_1x_2 \\\\ x_1x_2x_3\\\\ \\end{array} \\right) \\geq \\left(\\begin{array}{cccccccc} 0 \\\\ 0 \\\\  \\cline {1-1} 0 \\\\ 0 \\\\  \\cline {1-1} 0 \\\\ 0 \\\\  \\cline {1-1} 0 \\\\ 0\\\\ \\end{array} \\right). \\label{larger}\n\\end{eqnarray} \nThe equations of \\eqref{ca6} in nonnegative variables $\\boldsymbol{\\lambda}$ take the form\n\\begin{eqnarray} \\tiny\n\\left(\\begin{array}{cccccccc} 1 \\\\ x_3 \\\\ \\cline {1-1} x_2 \\\\ w_{23} \\\\ \\cline {1-1} x_1 \\\\ w_{13} \\\\ \\cline {1-1} w_{12} \\\\ w_{123}\\\\ \\end{array} \\right) =  \\frac{1}{d_1d_2d_3} \\left[\\begin{array}{rr|rr|rr|rr}   \n1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\  \nL_3 & U_3 & L_3 & U_3 & L_3 & U_3 & L_3 & U_3 \\\\ \\cline {1-8}\nL_2 & L_2 & U_2 & U_2 & L_2 & L_2 & U_2 & U_2 \\\\  \nL_2L_3 & L_2U_3 & U_2L_3 & U_2U_3 & L_2L_3 & L_2U_3 & U_2L_3 & U_2U_3 \\\\ \\cline {1-8}\nL_1 & L_1 & L_1 & L_1 & U_1 & U_1 & U_1 & U_1 \\\\  \nL_1L_3 & L_1U_3 & L_1L_3 & L_1U_3 & U_1L_3 & U_1U_3 & U_1L_3 & U_1U_3 \\\\ \\cline {1-8}\nL_1L_2 & L_1L_2 & L_1U_2 & L_1U_2 & U_1L_2 & U_1L_2 & U_1U_2 & U_1U_2 \\\\  \nL_1L_2L_3 & L_1L_2U_3 & L_1U_2L_3 & L_1U_2U_3 & U_1L_2L_3 & U_1L_2U_3 & U_1U_2L_3 & U_1U_2U_3 \\\\\n\\end{array}\\right]\\left(\\begin{array}{cccccccc} \\lambda_1 \\\\ \\lambda_2 \\\\ \\cline {1-1} \\lambda_3 \\\\ \\lambda_4 \\\\ \\cline {1-1} \\lambda_5 \\\\ \\lambda_6 \\\\ \\cline {1-1} \\lambda_7 \\\\ \\lambda_8\\\\ \\end{array} \\right), \\label{sume}\n\\end{eqnarray}\nwhere the RLT linearization step sets $w_{12}=x_1x_2,$ $w_{13}=x_1x_3,$ $w_{23}=x_2x_3,$ and $w_{123}=x_1x_2x_3.$ \n\nThere are eight extreme points to this system, with extreme point $j$ having $\\lambda_j=d_1d_2d_3$ and $\\lambda_i=0$ for $i \\neq j.$ Then the eight extreme points to $P$ of \\eqref{RLTstep2} and \\eqref{ca4} are given by the eight columns of the above $8 \\times 8$ matrix, less the first row. Consequently, we have that $P=conv(T)$ with \n\\begin{equation}\nT=\\tiny\\left\\{ \\left(\\begin{array}{c} x_1 \\\\  x_2 \\\\ x_3 \\\\  w_{12} \\\\ w_{13} \\\\ w_{23} \\\\ w_{123}\\\\ \\end{array} \\right):  L_j \\leq x_j \\leq U_j \\; \\forall \\; j=1,2,3, \\; \\left(\\begin{array}{c} x_1x_2 \\\\ x_1x_3 \\\\ x_2x_3 \\\\ x_1x_2x_3 \\\\ \\end{array} \\right)=\\left(\\begin{array}{c} w_{12} \\\\ w_{13} \\\\ w_{23} \\\\ w_{123} \\\\ \\end{array} \\right)\\right\\}, \\nonumber\n\\end{equation}\nas in \\eqref{miss1}.\n\nNow consider any multilinear polynomial $\\sum_{J \\subseteq N}\\alpha_J\\prod_{j \\in J}x_j$ as found in \\mythref{obs1,obs2}, and the corresponding system in variables $\\boldsymbol{\\pi}\\in \\real^{2^n}.$ Using obvious notation, denote \\eqref{larger} by $A\\boldsymbol{v} \\geq \\boldsymbol{0}$ so that the (scaled) $8 \\times 8$ matrix of \\eqref{sume} is $A^{-1}.$ Observe that the eight inequalities of \\eqref{larger} correspond, in order, to the functions $F(K)$ of \\eqref{Fun} having $K= \\emptyset, \\{3\\}, \\{2\\}, \\{2,3\\}, \\{1\\}, \\{1,3\\}, \\{1,2\\}, \\{1,2,3\\}.$ (This order coincides with the variable indices of the products within the vector $\\boldsymbol{v}.$) Accordingly define $\\boldsymbol{\\alpha}^T=(\\alpha_{\\emptyset},\\alpha_3,\\alpha_2,\\alpha_{23},\\alpha_1,\\alpha_{13},\\alpha_{12},\\alpha_{123}),$ and $\\boldsymbol{\\pi} \\in \\real^{8}$ by $\\boldsymbol{\\pi}^T=(\\pi_{\\emptyset},\\pi_3,\\pi_2,\\pi_{23},\\pi_1,\\pi_{13},\\pi_{12},\\pi_{123})$ so that $\\boldsymbol{\\pi}^T=\\boldsymbol{\\alpha}^TA^{-1}.$ Then we have\n\\begin{equation}\n\\sum_{J \\subseteq N} \\alpha_J\\prod_{j \\in J}x_j=\\boldsymbol{\\alpha}^T\\boldsymbol{v}=(\\boldsymbol{\\alpha}^TA^{-1})(A\\boldsymbol{v})=\\boldsymbol{\\pi}^T(A\\boldsymbol{v}) =\\sum_{K \\subseteq N}\\pi_KF(K), \\label{includee}\n\\end{equation}\nwhere the four equalities follow, from left to right, from the definitions of $\\boldsymbol{\\alpha}$ and $\\boldsymbol{v},$ the multiplicative inverse of $A,$ the definition of $\\boldsymbol{\\pi},$ and the stated equivalence between the vector $A\\boldsymbol{v}$ and the functions $F(K)$ of \\eqref{Fun}. The computing of the multipliers $\\pi_K$ by \\mythref{obs2} follows from \\eqref{includee} since each entry of the vector $\\boldsymbol{\\alpha}^TA^{-1}$ corresponds to a distinct $K \\subseteq N,$ and realizes value $\\frac{1}{D_3}(\\sum_{J \\subseteq N} \\alpha_J\\prod_{j \\in J}\\hat{x}_j),$ where $\\hat{x}_j=U_j$ for all $j \\in K$ and $\\hat{x}_j=L_j$ for all $j \\notin K.$ For each such $K,$ this value is the multiplier $\\pi_K$ on the associated function $F(K).$ \\mythref{obs2} gives us, provided it is nonnegative over the extreme points of $X^{\\prime},$ that the polynomial $\\sum_{J \\subseteq N}\\alpha_J\\prod_{j \\in J}x_j$ vanishes at a point $\\hat{\\x} \\in X^{\\prime}$ if and only if $\\pi_KF(K)=0$ for all $K \\subseteq N,$ where each $F(K)$ is evaluated at $\\hat{\\x}.$ \\hfill$\\diamond$\n\\end{example}\n\n\\subsection{Convex Hull via Projection}\n\nThe example below illustrates the computation of the set $conv\\left(G^{\\prime}\\right)$ via the projection from the extended variable space $(\\x,\\boldsymbol{w},y)$ of $P_y$ onto the space of the variables $(\\x,y),$ as stated in \\mythref{equalities}. For simplicity, the sets $X^{\\prime}$ and $G^{\\prime}$ of \\eqref{Xprimedef} and \\eqref{marker} are reduced to the sets $X$ and $G$, respectively, by setting $L_j=\\ell$ and $U_j =u$ for all $j \\in N.$ The example also demonstrates the result of \\mythref{obs2} that allows for the identification of those points within $X$ at which each facet is satisfied exactly. Example~\\ref{ex2} builds upon Example~\\ref{ex1}, and will be later referenced.\n\n\\begin{example}\t\\label{ex2}\nConsider the set $G$ with $N=\\{1,2,3\\}$ in the $n=3$ variables $x_1, x_2,x_3,$ and the variable $y,$ with $y=m(\\x)=x_1x_2x_3,$. The set $P_y$ of \\eqref{handy}, whose projection onto the $(\\x,y)$ variable space gives $\\conv{G}$ as stated in \\mythref{equalities}, is expressed in matrix form below. The matrix partitioning is used to emphasize that the first eight restrictions are inequalities and the last restriction is equality.\n\\begin{align}\n&\\scriptsize\\left[\\begin{array}{rr|rr|rr|rr|r}   \nu^3 & -u^2 & -u^2 & u & -u^2 & u & u & -1 & 0 \\\\  \n-\\ell u^2 & u^2 & \\ell u & -u & \\ell u & -u & -\\ell & 1 & 0 \\\\ \\cline {1-9}\n-\\ell u^2 & \\ell u & u^2 & -u & \\ell u & -\\ell & -u & 1 & 0 \\\\  \n\\ell^2 u & -\\ell u & -\\ell u & u & -\\ell^2 & \\ell & \\ell & -1 & 0 \\\\ \\cline {1-9}\n-\\ell u^2 & \\ell u & \\ell u & -\\ell & u^2 & -u & -u & 1 & 0 \\\\  \n\\ell^2 u & -\\ell u & -\\ell^2 & \\ell & -\\ell u & u & \\ell & -1 & 0 \\\\ \\cline {1-9}\n\\ell^2 u & -\\ell^2 & -\\ell u & \\ell & -\\ell u & \\ell & u & -1 & 0 \\\\  \n-\\ell^3 & \\ell^2 & \\ell^2 & -\\ell & \\ell^2 & -\\ell & -\\ell & 1 & 0 \\\\ \n\\end{array}\\right]\\left(\\begin{array}{cccccccc} 1 \\\\ x_3 \\\\ \\cline {1-1} x_2 \\\\ w_{23} \\\\ \\cline {1-1} x_1 \\\\ w_{13} \\\\ \\cline {1-1} w_{12} \\\\ w_{123} \\\\  \\end{array} \\right) \\geq \\left(\\begin{array}{cccccccc} 0 \\\\ 0 \\\\  \\cline {1-1} 0 \\\\ 0 \\\\  \\cline {1-1} 0 \\\\ 0 \\\\  \\cline {1-1} 0 \\\\ 0 \\\\  \\end{array} \\right)\\nonumber  \\\\\n&\\scriptsize\\phantom{|}\\left[\\begin{array}{rr|rr|rr|rr|r}   \n\\phantom{\\ell u^2|}0 & \\phantom{-\\ell |}0 & \\phantom{\\ell u}0 & \\phantom{-|}0\\phantom{l} & \\phantom{\\ell u}0 & \\phantom{-\\ell}0 & \\phantom{a|}0 & -1 &  1\\\\ \n\\end{array}\\right]\\phantom{|}\\left(\\begin{array}{cccccccc}  \\phantom{aa}y\\phantom{aa} \\\\ \\end{array} \\right) \\phantom{|}=\\phantom{|} \\left(\\begin{array}{cccccccc} \\phantom{|}0\\phantom{|} \\\\  \\end{array} \\right) \\label{largeee}\n\\end{align}\n\n\\noindent The eight inequalities are the linearized form of \\eqref{larger} when $L_1,$ $L_2,$ and $L_3$ are set to $\\ell,$ and when $U_1,$ $U_2,$ and $U_3$ are set to $u$, while the equation is $-\\{m(\\x)\\}_L+y=0.$ Since we desire to project \\eqref{largeee} onto the space of the variables $(x_1,x_2,x_3,y),$ the projection cone takes the form\n\\begin{eqnarray}  \n\\scriptsize\n\\left[\\begin{array}{rrrrrrrrr}   \nu & -u & -u & u & -\\ell & \\ell & \\ell & -\\ell & 0\\\\  \nu & -u & -\\ell & \\ell & -u & u & \\ell & -\\ell & 0\\\\\nu & -\\ell & -u & \\ell & -u & \\ell & u & -\\ell & 0\\\\\n-1 & 1 & 1 & -1 & 1 & -1 & -1 & 1 & -1\\\\\n\\end{array}\\right]\\left(\\begin{array}{cccccccc} \\pi_{\\emptyset} \\\\ \\pi_3 \\\\ \\pi_2 \\\\ \\pi_{23} \\\\ \\pi_1 \\\\ \\pi_{13} \\\\ \\pi_{12} \\\\ \\pi_{123} \\\\ \\beta^{\\prime} \\\\ \\end{array} \\right) = \\left(\\begin{array}{ccc} 0 \\\\ 0 \\\\  0\\\\ 0\\\\ \\end{array} \\right), \\label{largend}\n\\end{eqnarray}\nwhere $\\boldsymbol{\\pi}$ are the nonnegative multipliers on the eight inequality restrictions of \\eqref{largeee}, and where $\\beta^{\\prime}$ is the multiplier on the equation. When $\\ell > 0,$ there are fifteen extreme directions to this cone, with six ``trivial directions\" resulting in the six facets $x_j \\geq \\ell$ and $-x_j \\geq -u$ for $j \\in \\{1,2,3\\}.$ Specifically, setting $\\beta^{\\prime}=0$ for each $j,$ the ``trivial direction\" having $\\pi_K=1$ if $ j \\in K$ and $\\pi_K=0$ otherwise gives $x_j- \\ell \\geq 0,$ and the ``trivial direction\" having $\\pi_K=1$ if $j \\in (N-K)$ and $\\pi_K=0$ otherwise gives $u-x_j \\geq 0,$ all inequalities scaled by $\\frac{1}{(u-\\ell)^2}.$ (Clearly, for each $j \\in \\{1,2,3\\},$ the inequality $x_j - \\ell \\geq 0$ is satisfied exactly at all points $(x_1,x_2,x_3,y) \\in G$ having $x_j = \\ell,$ and the inequality $u-x_j \\geq 0$ is satisfied exactly at all points $(x_1,x_2,x_3,y) \\in G$ having $x_j = u.$) Each of the remaining nine directions is depicted as a column of the below matrix.\n\\begin{eqnarray} \n\\scriptsize\n\\left[\\begin{array}{ccccccccc}   \n0 & 0 & 0 & 0 & 0 & 0 & 0 & \\ell & \\ell+2u\\\\  \n\\ell+u & \\ell & \\ell+u & \\ell & 0 & 0 & 0 & 0 & u\\\\\n\\ell & \\ell+u & 0 & 0 & \\ell+u & \\ell & 0 & 0 & u\\\\\n\\ell+u & \\ell+u & u & 0 & u & 0 & \\ell & 0 & 0\\\\\n0 & 0 & \\ell & \\ell+u & \\ell & \\ell+u & 0 & 0 & u\\\\\nu & 0 & \\ell+u & \\ell+u & 0 & u & \\ell & 0 & 0\\\\\n0 & u & 0 & u & \\ell+u & \\ell+u & \\ell & 0 & 0\\\\\n0 & 0 & 0 & 0 & 0 & 0 & 2\\ell+u & u & 0\\\\\n\\ell-u & \\ell-u & \\ell-u & \\ell-u & \\ell-u & \\ell-u & -\\ell+u & -\\ell+u & -\\ell+u\\\\\n\\end{array}\\right] \\nonumber\n\\end{eqnarray}\nThe set $\\conv{G}$ is then defined in terms of the $x_j - \\ell \\geq 0$ and $u-x_j \\geq 0$ restrictions for $j \\in \\{1,2,3\\},$ together with the nine facets that are listed in the first column of the below table, upon dividing each inequality by $(u-\\ell).$\n\n\\begin{table\n\\begin{center}\n\\caption{Facet and Points in $G$ where Satisfied Exactly when $\\ell >0$}\n\\label{tab1}\n\\begin{tabular}{|c | c c c |}\n\\hline\n\\vspace{-.1 in}\n& & & \\\\\n\\vspace{-.1 in}\nFacet &  \\multicolumn{3}{c} {Points in $G$ where Satisfied Exactly}  \\\\ \n& & & \\\\\n\\hline\n$(\\ell^2)x_1+(\\ell u)x_2+(u^2)x_3-y \\geq \\ell u(\\ell+u)$     &$(x_1,\\ell,\\ell)$   &$(u,x_2,\\ell)$    &$(u,u,x_3)$        \\\\ \n$(\\ell^2)x_1+(u^2)x_2+(\\ell u)x_3-y \\geq \\ell u(\\ell+u)$     &$(x_1,\\ell,\\ell)$   &$(u,x_2,u)$    &$(u,\\ell,x_3)$       \\\\\n$(\\ell u)x_1+(\\ell^2)x_2+(u^2)x_3-y \\geq \\ell u(\\ell+u)$     &$(x_1,u,\\ell)$   &$(\\ell,x_2,\\ell)$    & $(u,u,x_3)$      \\\\\n$(u^2)x_1+(\\ell^2)x_2+(\\ell u)x_3-y \\geq \\ell u(\\ell+u)$     &$(x_1,u,u)$   &$(\\ell,x_2,\\ell)$    & $(\\ell,u,x_3)$     \\\\\n$(\\ell u)x_1+(u^2)x_2+(\\ell^2)x_3-y \\geq \\ell u(\\ell+u)$     &$(x_1,\\ell,u)$   &$(u,x_2,u)$    & $(\\ell,\\ell,x_3)$       \\\\\n$(u^2)x_1+(\\ell u)x_2+(\\ell^2)x_3-y \\geq \\ell u(\\ell+u)$     &$(x_1,u,u)$   &$(\\ell,x_2,u)$    & $(\\ell,\\ell,x_3)$      \\\\\n$-(\\ell^2)x_1-(\\ell^2)x_2-(\\ell^2)x_3+y \\geq -2\\ell^3$     &$(x_1,\\ell,\\ell)$   &$(\\ell,x_2,\\ell)$    & $(\\ell,\\ell,x_3)$      \\\\\n$-(\\ell u)x_1-(\\ell u)x_2-(\\ell u)x_3+y \\geq -\\ell u(\\ell+u)$     & \\multicolumn{3}{c|} {Any $x_i=\\ell,$ any $x_j=u, \\; i \\neq j.$}  \\\\\n$-(u^2)x_1-(u^2)x_2-(u^2)x_3+y \\geq -2u^3$   &$(x_1,u,u)$   &$(u,x_2,u)$    & $(u,u,x_3)$    \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nBy letting $y=m(\\x)$ within each facet of Table~\\ref{tab1}, \\mythref{obs2} allows us to identify, in terms of the positive multipliers $\\boldsymbol{\\pi}$ of \\eqref{largend}, the points in $G$ where each such inequality is satisfied exactly. Definition \\eqref{Fun} gives us, for this example,  that a function $F(K)$ vanishes at a point $\\hat{\\x} \\in X$ if and only if either $\\hat{x}_j=\\ell$ for some $j \\in K$ or $\\hat{x}_j=u$ for some $j \\notin K.$ The second column of Table~\\ref{tab1}, which then logically follows, lists the set of all points $(x_1,x_2,x_3,y) \\in G$ where each facet is satisfied exactly, with the value $y=x_1x_2x_3$ suppressed for simplicity. The notation $``x_j\"$ found within this table indicates that the variable $x_j$ can have $\\ell \\leq x_j \\leq u.$ Observe that, given any realization of $(x_1,x_2,x_3)$ wherein two of the variables have values at either their lower or upper bounds, the first six facets enforce $y \\leq x_1x_2x_3$ while the last three facets enforce $y \\geq x_1x_2x_3,$ ensuring that $y=x_1x_2x_3.$\n\nFor the case when $\\ell=0,$ each of the first eight facets of Table~\\ref{tab1} is twice listed by repetition, and the inequalities $x_j \\geq 0$ for $j \\in \\{1,2,3\\}$ are not facets. A statement of the five facets resulting from Table~\\ref{tab1} and those points $(x_1,x_2,x_3,y) \\in G,$ with $y=x_1x_2x_3$ suppressed for simplicity, where each facet is satisfied exactly is given in Table~\\ref{tab2}. Here, the notation $``x_j\"$ indicates that the variable $x_j$ can have $0 \\leq x_j \\leq u.$ \\hfill$\\diamond$\n\n\\begin{table\n\\begin{center}\n\\caption{Facet and Points in $G$ where Satisfied Exactly when $\\ell =0$}\n\\label{tab2}\n\\begin{tabular}{|c | c c c |}\n\\hline\n\\vspace{-.1 in}\n& & & \\\\\n\\vspace{-.1 in}\nFacet &  \\multicolumn{3}{c} {Points in $G$ where Satisfied Exactly}  \\\\ \n& & & \\\\\n\\hline\n$(u^2)x_3-y \\geq 0$     &$(x_1,x_2,0)$   & $(u,u,x_3)$   &     \\\\ \n$(u^2)x_2-y \\geq 0$     &$(x_1,0,x_3)$   &  $(u,x_2,u)$ &      \\\\\n$(u^2)x_1-y \\geq 0$     &$(0,x_2,x_3)$   &   $(x_1,u,u)$ &     \\\\\n$y \\geq 0$     & \\multicolumn{3}{c|} {Any $x_i=0$}   \\\\\n$-(u^2)x_1-(u^2)x_2-(u^2)x_3+y \\geq -2u^3$   &$(x_1,u,u)$   &$(u,x_2,u)$    & $(u,u,x_3)$    \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\end{example}\n\n\n\\subsection{Convex Hull for Supermodular Function}\n\n\\begin{example} \\label{ex3}\nConsider $G$ in the $n=3$ variables $x_1, x_2,x_3,$ and the variable $y,$ with $m(\\x)=x_1x_2x_3$ as in Example~\\ref{ex2}, and with $\\ell \\geq 0$. Inequalities \\eqref{rest0} hold true because $(m(\\x^k)-m(\\x^{k-1}))$ takes values $\\ell^2(u-\\ell),$ $(\\ell u)(u-\\ell),$ and $u^2(u-\\ell)$ when $k=1,$ $2,$ and $3,$ respectively. Then \\mythref{result1} is applicable, and core facet \\eqref{res1} is\n\\begin{equation}\n-\\ell u(\\ell +u)+(\\ell^2)x_1+(\\ell u)x_2+(u^2)x_3-y \\geq 0, \\label{testing1}\n\\end{equation}\nand core facets \\eqref{res2} are\n\\begin{eqnarray}\n-\\ell^2u-\\ell^2\\left(x_1+x_2+x_3-[u+2\\ell]\\right)+y &\\geq& 0,\\label{testing2}\\\\\n-\\ell u^2-\\ell u\\left(x_1+x_2+x_3-[2u+\\ell]\\right)+y &\\geq& 0,\\label{testing3}\\\\\n-u^3-u^2\\left(x_1+x_2+x_3-[3u]\\right)+y &\\geq& 0. \\label{testing4}\n\\end{eqnarray}\nFor $\\ell>0,$ $m(\\x)$ is strictly supermodular and facet \\eqref{testing1}, together with the additional five facets obtained by permuting the coefficients $\\x[\\beta],$ are the first six facets of Table~\\ref{tab1}. Facets \\eqref{testing2}--\\eqref{testing4} are the last three facets of Table~\\ref{tab1}. In addition, the six inequalities $\\ell \\leq x_j \\leq u$ for $j\\in \\{1,2,3\\}$ comprise the remaining facets of $\\conv{G}.$ For $\\ell=0,$ $m(\\x)$ is supermodular (but not strictly supermodular because $(m(\\x^k)-m(\\x^{k-1}))$ takes values $0,$ $0,$ and $u^3$ when $k=1,$ $2,$ and $3,$ respectively), and three of the six facets obtained by permuting $\\x[\\beta]$ in \\eqref{testing1} are repetitive. The three resulting facets are the first three inequalities of Table~\\ref{tab2}. Also, and consistent with Remark 1, facets \\eqref{testing2} and \\eqref{testing3} are the same. Facets \\eqref{testing2} and \\eqref{testing4} are the last two inequalities of Table~\\ref{tab2}. In addition for $\\ell=0,$  $x_j \\geq 0$ for $j \\in \\{1,2,3\\}$ are not facets.\n\nSuppose that $G$ in the $n=3$ variables $x_1, x_2,x_3,$ and the variable $y,$ is changed to have $y=m(\\x)=(2u-\\ell)(x_1x_2+x_1x_3+x_2x_3)-x_1x_2x_3,$ with arbitrary $\\ell$ and $u.$ Inequalities \\eqref{rest0} hold true because the difference between the two sides takes values $2(u-\\ell)^3$ and $(u-\\ell)^3$ when $k=1$ and $k=2,$ respectively, so that \\mythref{result1} is applicable with $m(\\x)$ strictly supermodular. Core facet \\eqref{res1} is\n\\begin{equation}\n(\\ell u)(4\\ell -5u)+(4\\ell u-3\\ell^2)x_1+\\left(2u^2-\\ell^2\\right)x_2+\\left(3u^2-2\\ell u\\right)x_3-y \\geq 0, \\nonumber\n\\end{equation}\nand core facets \\eqref{res2} are\n\\begin{eqnarray*}\n-(4\\ell u^2-\\ell^3 -\\ell^2 u)-(4\\ell u - 3\\ell^2)\\left(x_1+x_2+x_3-[u+2\\ell]\\right)+y &\\geq& 0,\\\\\n-(2\\ell u^2-2\\ell^2 u+2u^3)-(2u^2-\\ell^2)\\left(x_1+x_2+x_3-[2u+\\ell]\\right)+y &\\geq& 0,\\\\\n-(5u^3-3\\ell u^2)-(3u^2-2\\ell u)\\left(x_1+x_2+x_3-[3u]\\right)+y &\\geq& 0.\\\\\n\\end{eqnarray*}\nThere are exactly nine facets \\eqref{sum24} with $\\beta^{\\prime}= \\pm 1$ for $\\conv{G},$ including the four listed above and the five additional obtained by permuting the coefficients $\\x[\\beta]$ in the first inequality. The six inequalities $\\ell \\leq x_j \\leq u$ for $j\\in \\{1,2,3\\}$ comprise the remaining facets of $\\conv{G},$ as $\\ell <u.$ \\hfill$\\diamond$\n\\end{example}\n\n\n\\subsection{Convex Hull for Monomial}\n\n\\begin{example} \\label{ex4}\nConsider $G$ in the $n=3$ variables $x_1,x_2,x_3,$ and the variable $y,$ with $m(\\x)=5x_1x_2x_3,$ and with $-\\ell=u=2$. Since $n \\geq 3,$ the convex hull representation is given by the (scaled) six inequalities $-2 \\leq x_j \\leq 2$ for $j \\in \\{1,2,3\\}$ found in \\eqref{con258}, together with the remaining (scaled) two inequalities of \\eqref{con258}, $-40 \\leq x_4 \\leq 40,$ and the eight inequalities of \\eqref{con269}, with the last ten such inequalities summarized in Table~\\ref{tab3} below. Here, $y$ is used in lieu of $x_4$ for clarity. Each inequality $\\beta_0 +\\sum_{j=1}^n\\beta_jx_j+\\beta^{\\prime}y \\geq 0$ has been scaled to $\\beta_0 +\\sum_{j=1}^n\\left(\\frac{\\beta_j}{u}\\right)x_j+\\left(\\frac{\\beta^{\\prime}}{c_nu^n}\\right)y \\geq 0$ as discussed at the beginning of this section to handle the coefficient $c_n=5$ found in $m(\\x)$ and the variable bounds $-\\ell=u=2,$ and has then been rescaled to have integer coefficients. The first column of the table gives the facet and the second column, explained later in Example~\\ref{ex6}, gives the set of all points $(x_1,x_2,x_3,y) \\in G$ where each facet is satisfied exactly. Consistent with Tables 1 and 2, the notation $``x_j\"$ indicates that the variable $x_j$ can have $-2 \\leq x_j \\leq 2,$ and the value $y=x_1x_2x_3$ is suppressed for simplicity.\\hfill$\\diamond$\n\n\\begin{table\n\\begin{center}\n\\caption{Facet and Points in $G$ where Satisfied Exactly}\n\\label{tab3}\n\\begin{tabular}{|c | c c c|}\n\\hline\n\\vspace{-.1 in}\n& & & \\\\\n\\vspace{-.1 in}\nFacet &  \\multicolumn{3}{c} {Points in $G$ where Satisfied Exactly}  \\\\ \n& & &\\\\\n\\hline\n$40-y \\geq 0$     & \\multicolumn{3}{c|} {$(2,2,2)\\;\\;\\;\\;(2,-2,-2)\\;\\;\\;\\;(-2,2,-2)\\;\\;\\;\\;(-2,-2,2)$}     \\\\\n$40+y \\geq 0$     & \\multicolumn{3}{c|} {$(-2,-2,-2)\\;\\;\\;\\;(-2,2,2)\\;\\;\\;\\;(2,-2,2)\\;\\;\\;\\;(2,2,-2)$}     \\\\\n\\hline\n$80+20x_1+20x_2+20x_3-y\\geq 0$     &$(x_1,-2,-2)$   &$(-2,x_2,-2)$ &$(-2,-2,x_3)$\\\\\n\\hline\n$80-20x_1-20x_2-20x_3+y\\geq 0$     &$(x_1,2,2)$   &$(2,x_2,2)$ &$(2,2,x_3)$\\\\\n\\hline\n$80-20x_1+20x_2+20x_3+y\\geq 0$     &$(x_1,-2,-2)$   &$(2,x_2,-2)$ &$(2,-2,x_3)$\\\\\n$80+20x_1-20x_2+20x_3+y\\geq 0$     &$(x_1,2,-2)$   &$(-2,x_2,-2)$ &$(-2,2,x_3)$\\\\\n$80+20x_1+20x_2-20x_3+y\\geq 0$     &$(x_1,-2,2)$   &$(-2,x_2,2)$ &$(-2,-2,x_3)$\\\\\n\\hline\n$80-20x_1-20x_2+20x_3-y\\geq 0$     &$(x_1,2,-2)$   &$(2,x_2,-2)$ &$(2,2,x_3)$\\\\\n$80-20x_1+20x_2-20x_3-y\\geq 0$     &$(x_1,-2,2)$   &$(2,x_2,2)$ &$(2,-2,x_3)$\\\\\n$80+20x_1-20x_2-20x_3-y\\geq 0$     &$(x_1,2,2)$   &$(-2,x_2,2)$ &$(-2,2,x_3)$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\end{example}\n\n\\subsection{Exactness for Supermodular Functions}\n\n\\begin{example} \\label{ex5}\nConsider $G$ in the $n=3$ variables $x_1,x_2,x_3,$ and the variable $y,$ with $m(\\x)=x_1x_2x_3$ as introduced in Example~\\ref{ex2} and revisited in Example 3 with $\\ell \\geq 0.$ As noted in Example~\\ref{ex3}, \\mythref{result1} is applicable, and \\eqref{res1} takes the form \\eqref{testing1}, while \\eqref{res2} takes the forms \\eqref{testing2}, \\eqref{testing3}, and \\eqref{testing4}. For $\\ell >0,$ $m(\\x)$ was noted to be strictly supermodular. Then \\mythref{exactstrictsupermod} gives us inequality \\eqref{testing1} is satisfied exactly at only those points $(\\x,y) \\in G$ where $\\x$ is of the form $(x_1,\\ell,\\ell),$ $(u,x_2,\\ell),$ or $(u,u,x_3),$ matching the first inequality of Table~\\ref{tab1}. The next five  inequalities of Table~\\ref{tab1}, which follow from permutations of the coefficients of \\eqref{testing1}, have the set of points at which each is satisfied exactly obtained via the same permutations. \\mythref{exactstrictsupermod} gives us that inequalities \\eqref{testing2}, \\eqref{testing3}, and \\eqref{testing4} are satisfied exactly at only those points $(\\x,y) \\in G$ where $\\x$ has, respectively, at least two entries of value $\\ell,$ at least one entry of value $u$ and at least one entry of value $\\ell,$ and at least two entries of value $u.$ These results of \\mythref{exactstrictsupermod} match the last three rows of Table~\\ref{tab1}. Also as noted in Example~\\ref{ex3}, the six inequalities $\\ell \\leq x_j \\leq u$ for $j\\in \\{1,2,3\\}$ are all facets of $\\conv{G}.$ For each of these last six facets, the set of points $(\\x,y) \\in G$ which satisfy it exactly is obvious.  For $\\ell =0,$ $m(\\x)$ was noted to be supermodular, but not strictly supermodular. In this case, \\mythref{newsuper100} gives us inequality \\eqref{testing1} is satisfied exactly at only those points $(\\x,y) \\in G$ having either $x_3=0$ or $x_1=x_2=u,$ matching the first inequality of Table~\\ref{tab2}. The next two inequalities of Table~\\ref{tab2}, which follow from permutations of the coefficients of \\eqref{testing1}, have the set of points at which each is satisfied exactly obtained via the same permutations. Also as shown in Example~\\ref{ex3}, inequalities \\eqref{res2} are the last two inequalities of Table~\\ref{tab2}. Then \\mythref{newsuper100} gives us that the $y \\geq 0$ inequality of \\eqref{testing2} and the $-(u^2)x_1-(u^2)x_2-(u^2)x_3+y \\geq -2u^3$ inequality of \\eqref{testing4} are satisfied exactly at only those points $(\\x,y) \\in G$ where $\\x$ has, respectively, at least one entry of value $0,$ and at least two entries of value $u,$ matching Table~\\ref{tab2}.\\hfill$\\diamond$\n\\end{example}\n\n\n\\subsection{Exactness for Monomial}\n\n\\begin{example} \\label{ex6}\nReconsider $G$ in the $n=3$ variables $x_1,x_2,x_3,$ with $m(\\x)=5x_1x_2x_3,$ and with $-\\ell=u=2$, as found in Example~\\ref{ex4} and Table~\\ref{tab3}. Recall that each facet within Table~\\ref{tab3} has been suitably scaled to handle the coefficient $c_n=5$ found in $m(\\x)$ and the variable bounds $-\\ell=u=2.$ Therefore, consistent with the discussion at the beginning of this subsection, every point $(\\tilde{\\x},\\tilde{y})$ identified in \\mythref{finalet} as satisfying an inequality of the form \\eqref{trivial}, \\eqref{pair1}, \\eqref{pair2}, or \\eqref{dominate2} exactly for the case in which $c_n=1$ and $-\\ell=u=1$ must be scaled to $(u\\tilde{\\x},c_nu^n\\tilde{y})=(2\\tilde{\\x},40\\tilde{y}).$ The second column of Table~\\ref{tab3} gives these scaled points for each such inequality, with the value $y=5x_1x_2x_3$ suppressed for simplicity. Here, the two inequalities in the first block of constraints within Table~\\ref{tab3} are scaled \\eqref{trivial} for $\\beta^{\\prime}=-1$ and $\\beta^{\\prime}=1,$ respectively, the inequality in the second block is scaled \\eqref{pair1}, the inequality in the third block is scaled \\eqref{pair2}, and the inequalities in the fourth and fifth blocks are the three inequalities resulting from coefficient permutations of the scaled \\eqref{dominate2}, with $t=0$ and $t=1,$ respectively. \\hfill$\\diamond$\n\\end{example}\n\\section{Introduction}\n\nA polynomial is multilinear if every monomial is square-free in the sense that it is a product of a subset of variables raised to the power one. Multilinear polynomials have degree $n$ and they become linear functions when $n-1$ variables are fixed (hence the name). A multilinear polynomial can be expressed as $\\sum_{J\\subseteq N}c_{J}\\prod_{j\\in J}x_{j}$ for some $c\\in\\real^{2^{n}}$. We are interested in symmetric multilinear polynomials (SMPs) in this paper, by which we mean the polynomial\n\\begin{subequations}\n\\begin{equation}\nm(\\x) \\eq \\sum_{i=2}^n c_i \\sum_{\\substack{J \\subseteq N\\\\|J| =i}}\\prod_{j \\in J}x_j, \\label{functionm}\n\\end{equation}\nwhere we have dropped the constant and linear terms since they are inconsequential to us from the point of view of convexification. The symmetry of $m$ refers to the fact that for every $x\\in\\real^{n}$ and $\\bar{x}$ equal to a permutation of $x$, we have $m(x) = m(\\bar{x})$. Our goal is to study the convex hull of the graph of an SMP over a symmetric box in $\\real^{n}$. Namely, we study $\\conv{G}$ which is the convex hull of the set\n\\begin{equation}\nG = \\left\\{(\\x,y)\\in \\real^n \\times \\real: \\x \\in X, \\; y= m(\\x)\\right\\}, \\label{Sdef}\n\\end{equation}\nwhere $X$ is a box imposing the same lower and upper bounds (finite $\\ell < u$) on all variables,\n\\begin{equation}\t\nX = \\left\\{\\x\\in \\real^n: \\ell \\leq x_j \\leq u, \\ j \\in N\\right\\}.\t\\label{Xdef}\n\\end{equation}\n\\end{subequations}\nThroughout this paper, we use $N:= \\{1, \\ldots, n\\}$. \\akshay{Although coordinatewise scaling and translation does not break the symmetry of $m(x)$ and can reduce the box $X$ to be the unit hypercube $[0,1]^{n}$, we work with arbitrary $\\ell$ and $u$ to avoid this affine transformation which can be cumbersome to perform.\n\nThe graph $G_{p} = \\{(\\x,y)\\in B\\times\\real \\colon y = p(x)\\}$ of a general multilinear polynomial $p(x) = \\sum_{J\\subseteq N}c_{J}\\prod_{j\\in J}x_{j}$ over an arbitrary box $B\\subset\\real^{n}$ appears not only as a substructure in some important applications but also when optimizing a polynomial over binary variables \\citep{del2016polyhedral}, which is equivalent to pseudo-Boolean optimization \\citep{boros2002pseudo}, and as an intermediate set when performing factorable reformulations of general polynomial optimization problems \\citep{bao2015global,del2019impact,buchheim2008efficient}. Hence, convexification of $G_{p}$ has been the subject of many studies in the literature. It is known that this convex hull is a polytope and exponential-sized extended formulations are available \\cite{rikun1997convex,sherali1997convex,ballerstein2014extended} but an explicit description in the $(\\x,y)$-space is not known in general. Earlier studies focused on  convexifying multilinear monomials \\cite{cafieri2010convex,meyer2004trilinear,ryoo2001analysis,benson2004concave}, motivated by the classical linearization of a monomial $x_{1}x_{2}\\dots x_{n}$ over $[0,1]^{n}$ \\citep{glover74lin} which defines the convex hull for the graph of the monomial \\cite{crama1993concave} and leads to the standard linearization for $G_{p}$. Separation over $G_{p}$ is NP-hard, and so different classes of valid inequalities and cutting plane procedures have been developed for use in global optimization algorithms  \\citep{bao2015global,del2019impact,fomeni2015cutting,crama2017class}. There have also been many recent studies on describing $G_{p}$ or generalizations of it in the monomial space, which is obtained by adding a new variable for each monomial, under different assumptions on the structure of the polynomial \\cite{del2018multilinear,del2018running,del2016polyhedral,gupte2020bilinear,chen2020multilinear,fischer2018matroid,fischer2020matroid,luedtke2012strength,buchheim2019berge}.\n\nSymmetry has not been exploited in the rich body of literature on convexifying multilinear polynomials. The main objective of this paper is to initiate a systematic polyhedral analysis of $\\conv{G}$ by exploiting symmetry in this set. Our focus is on the minimal inequality description of this full-dimensional polytope in the original $(x,y)$-space, as opposed to the many studies in the literature about convexifying general multilinear polynomials in the monomial space. Note that there will be exponentially many monomials in $m(x)$ when $c_{i}\\neq 0$ where $i=n\/k$ for some constant $k\\ge 1$, and so a reformulation to the monomial space will not always be tractable. The symmetric structure we assume is interesting not only because it enables a thorough analysis of the convex hull and hence adds to the convexification literature, but also because it arises in many applications of combinatorial optimization \\cite{anthony2016quadratization,boros2019compact,dey2020spca,kim2019convexification} and with regards to chromatic number of graphs and other areas of combinatorics \\citep[cf.][]{eagles2020h,stanley1995symmetric}.\n} \n\n \n\\subsection{Our Contributions}\n\nThere are exponentially many facets, but symmetry of the function $m$ and of the set $X$ means that we only need to focus on a certain subset of inequalities, which we call core inequalities, and all other inequalities are generated as permutations of coefficients in core inequalities. In theory, all these inequalities can be obtained by projecting an extended formulation having  $O(n^{2})$ many variables and $O(n)$ many constraints, which is much smaller in size than the exponential-sized extensions for general multilinear polynomials. However, projecting this extension is a tedious task due to the combinatorial explosion that generally occurs with the projection operation. Instead, we give several necessary conditions and some sufficient conditions for a core valid inequality to be facet-defining, and these are more tractable to verify than the conditions that come from the use of polarity since the latter require enumeration of extreme points of a polyhedron. They can be applied to certify whether a given description of $\\conv{G}$ is minimal or not. In that regard, we use our conditions to obtain explicit listing of all the facet-defining inequalities for two families of SMPs. The first family is that of supermodular SMPs and the second family is that of monomials whose lower and upper bounds are reflections of each other ($-\\ell = u > 0$).\n\nThe Reformulation-Linearization Technique (RLT) is known to convexify the graph of a general multilinear polynomial over an arbitrary box. We show that RLT also implies that such a polynomial is nonnegative on a box if and only if it is nonnegative at every vertex of the box. This consequence enables us to characterize the set of points on $G$ that lie on a facet of the convex hull of $G$. The sets, by derivation, turn out to be different unions of $d $-dimensional ($0 \\le d\\le n-1$) faces of $\\conv{G}.$\n\n\nThe questions of optimization and separation are also answered for $\\conv{G}$ for any SMP without using the quadratic-sized extension. A linear function can be optimized over $\\conv{G}$ in $O(n\\log{n})$ time (assuming the value of function $m$ at a vertex of $X$ can be computed in $O(1)$ time) without using the extended formulation. Thus, a point can be separated from $\\conv{G}$ in polynomial time via the ellipsoid method. There is also a direct separation algorithm that runs in $O(nt + n\\log{n})$ time where $t$ is the number of core inequalities, and hence has polynomial time complexity when $t$ is bounded by a polynomial in input size. \n\n\n\\subsubsection{Related Results in Literature}\n\n\\akshay{Although we recognize that our convex hull descriptions for the two special families have been established before in literature, our necessary conditions for core facets of $\\conv{G}$ yield alternate proofs for them. In this context, the following results are known. A set function is supermodular if and only if a certain extension of it from the vertices of a box to the entire box is a concave function \\cite[Proposition 4.1]{lovasz1983submod}; see \\cite[Theorem 4]{iwata2008submodular} for a simple proof. The projection of this extension generates the concave envelope of a supermodular function, and this envelope is described in the $x$-space by so-called polymatroid inequalities; see \\cite[Theorem 1]{atamturk2008polymatroids}, \\cite[Lemma 4]{vondrak2010lecture}, \\cite[Theorem 3.3]{tawarmalani2013explicit} for some proofs of this well-known fact. These polymatroid inequalities can be derived from polarity and the results on optimizing over the polymatroid polyhedron \\cite{edmonds1970submodular}. There is one inequality for each $n$-permutation, allowing for repetitions, and this immediately relates to there being a single core facet of the envelope and all other facets being permutations of it.\nThe convex envelope of a supermodular function is not known in general and is NP-hard to separate over. For the symmetric structure that is considered in this paper, this envelope was first described by \\cite[Theorem 4.6]{tawarmalani2013explicit} using their approach of strategically computing an exponential number of subdivisions of $X$ and by using the extreme points of these subdivisions to calculate the facets.\nA similarity between this proof and ours is that they both rely on the Kuhn's triangulation \\cite{kuhn1960lemma} of a box. The convex hull of a monomial with $-\\ell = u$, was first established by the authors in \\citep[Theorem 4.1]{monomerr}. However, it was done so without using the symmetry in the monomial to recognize the core facets. \n}\n\n\nSince the original submission of this paper, convexification of permutation-invariant sets, such as the graph $G$, has been studied by \\cite{kim2019convexification} who give a general framework for obtaining an extended formulation with $O(n^{2})$ many variables and inequalities for the convex hulls of such sets. This framework is based on first convexifying a strategically-defined subset of the region of interest, and then obtaining the convex hull for all permutations of each point within the convexified set. In the realm of convex hulls for SMPs, \\cite{kim2019convexification} and this paper both utilize information relative to a specific simplex that generates the Kuhn's triangulation of a box, and this similarity is not entirely surprising because of symmetry in the problem.\n\n\n\\subsection{Organisation of the Paper}\nSection~2 presents an extended formulation for $\\conv{G}$, gives a polar description of this convex hull, introduces the concept of core inequalities from which all valid linear inequalities can be derived upto permutations the coefficients, and answers the questions of optimization and separation. Section~3 provides various conditions for a core valid equality to be a core facet. Section~4 gives the RLT theory for general multilinear polynomials and derives consequences of it on nonnegativity of the polynomial over a box. This RLT machinery enables us to characterize the set of points in the graph at which a core facet is satisfied exactly. Section~5 considers the case of $m(\\x)$ being a supermodular function and Section~6 considers the case of $m(\\x)$ being a monomial with the variable domain being a box that allows reflections across the origin. Finally, Section~7 provides a summary of the paper and highlights some outstanding open questions for future research. The Appendix gives alternate arguments for deriving basic properties of $\\conv{G}$ using the RLT, and also has illustrative examples for our main results.\n\n\n\\section{Preliminaries}\n\n\\akshay{The convex hull of $G$ is a full-dimensional set because $G$ is the surface of a nonlinear function taken over an $n$-dimensional box, and this set is a polytope whose vertices are in bijection to the vertices of the box $X$ since this is known for general multilinear functions \\citep{rikun1997convex,sherali1997convex}.} Throughout this paper, we will study inequalities of the form\n\\begin{equation}\n\\beta_0+\\sum_{j=1}^n\\beta_jx_j+\\beta^{\\prime}y \\geq 0. \\label{sum24}\n\\end{equation}\n\\akshay{Trivial facets of $\\conv{G}$ (also referred to as vertical facets in the literature) are the facets generated by valid $\\x[\\beta]$ with $\\beta^{\\prime}=0$. For $n\\ge 3$, trivial facets are precisely the bounds on the variables, and for $n=2$ there are no trivial facets \\citep[Theorem 2.4 and Remark 2.1]{bao2009multiterm}. Wlog and upto scaling, we can assume that every nontrivial valid inequality has $\\beta^{\\prime}=\\pm{1}$. Since $\\conv{G}$ is the intersection of the epigraph of the convex envelope of $m$ and hypograph of the concave envelope of $m$, a facet-defining inequality (facet) with $\\beta^{\\prime}=1$ (resp. $\\beta^{\\prime}=-1$) represents a nontrivial facet of the epigraph (resp. hypograph). Since $\\conv{G}$ is full-dimensional, for every nontrivial facet of $\\conv{G}$ there exists a unique nontrivial valid inequality. We will characterize nontrivial facets using the concept of core inequalities.\n\nCertain notation will be useful throughout our study. Our polyhedral analysis will rely on the simplex\n\\begin{equation}\n\\mathcal{S} \\equiv \\left\\{\\x\\in \\real^n: u \\geq x_1\\geq\\ldots\\geq x_n\\geq \\ell \\right\\}, \\label{simpleex}\n\\end{equation}\nwhose $n+1$ extreme points are\n\\begin{equation}\t\\label{def:xk}\n\\x^{k} \\equiv \\left(\\underbrace{u, \\ldots, u}_{k},\\, \\underbrace{\\ell,\\ldots,\\ell}_{n-k} \\right), \\quad k= 0,\\dots,n.\n\\end{equation}\n\\akshay{The simplex $\\mathcal{S}$ is the one that has been used in combinatorial geometry to yield the Kuhn's triangulation of a box  \\cite{kuhn1960lemma}. It is also known to be useful for convexifying general submodular\/supermodular functions \\cite{lovasz1983submod,tawarmalani2013explicit} and we will see this also in \\textsection\\ref{sec:supermod}. The value of the multilinear function at each $\\x^{k}$ can be computed by substituting it into equation~\\eqref{functionm}. In the special case of $\\ell=0$ and $u=1$, this becomes the combinatorial formula $m(\\x^{k}) = c_{k} + \\sum_{j=2}^{k-1}\\binom{k}{j}\\,c_{j}$.} For notational convenience, we let ${\\cal L}_k(\\beta_0,\\x[\\beta])= \\beta_0+\\sum_{j=1}^n\\beta_jx^k_j$ be that value obtained by inserting extreme point $\\x^k$ of $\\mathcal{S}$ into the expression $\\beta_0+\\sum_{j=1}^n\\beta_jx_j$ of \\eqref{sum24}, where $x^k_j$ denotes entry $j$ of $\\x^k,$ so that \n\\begin{equation}\n{\\cal L}_k(\\beta_0,\\x[\\beta])=\\beta_0+u\\left(\\sum_{j=1}^k\\beta_j\\right)+\\ell \\left(\\sum_{j=k+1}^n\\beta_j\\right), \\qquad \\forall \\; k \\in \\{0, \\ldots, n\\}, \\label{anoto25}\n\\end{equation} \nwith $\\sum_{j=1}^0\\beta_j=0$ and $\\sum_{j=n+1}^n\\beta_j=0$ in ${\\cal L}_0(\\beta_0,\\x[\\beta])$ and ${\\cal L}_n(\\beta_0,\\x[\\beta]),$ respectively. \n\nWe begin by noting two implicit descriptions of $\\conv{G}$, one is an extended formulation that projects onto this convex hull and another is a polarity result that gives a characterization of all the facets. Then we introduce core inequalities as those inequalities having a nondecreasing order on the coefficients and permutations of which generate all the valid inequalities. Lastly, we give algorithms for optimizing and separating over $\\conv{G}$.\n}\n\n\n\\subsection{Implicit Descriptions of the Convex Hull}\n\n\\akshay{A straightforward application of disjunctive programming along with using the well-known result that the envelopes of a multilinear function are generated by the extreme points of the box gives us a $O(n^{2})$-sized extended formulation for $\\conv{G}$.\n\n\\begin{proposition}\t\\thlabel{extform}\nThe following polyhedron projects onto $\\conv{G}$,\n\\begin{align*}\n\\Big\\{(x,\\{w^{k}\\}_{k=0}^{n}, y , v, \\lambda) \\sep & x_{j} = \\sum_{k=0}^{n}w^{k}_{j}, \\; \\forall j\\in N, \\ y = \\sum_{k=0}^{n}v_{k}, \\ \\sum_{k=0}^{n}\\lambda_{k}=1, \\\\ \n& \\sum_{j=1}^{n}w^{k}_{j} = (ku + (n-k)\\ell)\\lambda_{k}, \\ v_{k} = \\mk\\lambda_{k}, \\ k=0,\\dots,n \\\\\n& \\ell \\lambda_{k} \\le w^{k}_{j} \\le u\\lambda_{k} \\; \\forall j,k, \\ v \\in \\real^{n+1}, \\ \\lambda \\in \\real^{n+1}_{+} \\Big\\}\n\\end{align*}\n\\end{proposition}\n\\begin{proof}\nIt is well-known \\cite{sherali1997convex,rikun1997convex} that envelopes of a multilinear function over a box have the vertex extendability property meaning that for any multilinear function $p(x)$ and box $B\\subset\\real^{n}$, the convex hull of $\\{(x,y)\\in B\\times\\real\\colon y = p(x) \\}$ is equal to the convex hull of $\\{(x,y)\\in \\vertex{B}\\times\\real\\colon y = p(x) \\}$. Therefore, we have $\\conv{G}$ being equal to $\\conv{\\{(x,y)\\in\\{\\ell,u\\}^{n}\\times\\real\\colon y = m(x)\\}}$. The vertex set $\\{\\ell,u\\}^{n}$ can be partitioned into $n+1$ sets with each set for $k\\in\\{0,\\dots,n\\}$ corresponding to points having $k$ entries of $u$, thereby giving us\n\\begin{equation}\t\\label{convunion}\n\\vertex{(\\conv{G})} \\eq \\bigcup_{k=0}^{n}\\bigcup_{\\sigma\\in\\Sigma_{n}} \\left\\{(\\sigma\\cdot\\x^{k},m(\\sigma\\cdot\\x^{k})) \\right\\}\n\\eq \\bigcup_{k=0}^{n} \\left\\{ Q_{k}  \\times \\{\\mk\\} \\right\\},\n\\end{equation}\nwhere $\\Sigma_{n}$ is the symmetric group of $n$ elements, $\\sigma\\cdot v = (v_{\\sigma(1)}, v_{\\sigma(2)}, \\dots, v_{\\sigma(n)})$ for a vector $v\\in\\real^{n}$, $Q_{k} := \\cup_{\\sigma\\in\\Sigma_{n}}\\{\\sigma\\cdot\\x^{k}\\}$,  and the second equality is due to symmetry of $m(x)$. Since $\\conv{G}$ is the convex hull of its vertices, we get $\\conv{G} = \\conv{\\left(\\bigcup_{k=0}^{n} \\left\\{ \\conv{Q_{k}}  \\times \\{\\mk\\} \\right\\}\\right)}$. Since $Q_{k}$ is the set of all points that have $k$ entries of $u$ and $n-k$ entries of $\\ell$, we have $\\conv{Q_{k}} = \\{x\\in[\\ell,u]^{n}\\colon \\sum_{j=1}^{n}x_{j} = ku + (n-k)\\ell \\}$. Set $P_{k} := \\conv{Q_{k}}\\times\\{\\mk\\} = \\{(x,y)\\in[\\ell,u]^{n}\\times\\real\\colon \\sum_{j=1}^{n}x_{j} = ku + (n-k)\\ell, \\, y = \\mk\\}$, which is a polyhedron. Since $\\conv{G} = \\conv(\\cup_{k=0}^{n}P_{k})$, applying disjunctive programming \\cite{balas1979disjunctive} yields the desired extended formulation.\n\\end{proof}\n\nAnother way of implicitly describing the convex hull is to characterise all its facets using the extreme points of the polar of the convex hull. This is known from \\cite[Theorem 2]{sherali1997convex} and \\cite[Theorem 2.4]{bao2009multiterm} for general multilinear functions over arbitrary boxes, and hence we state the below result for an SMP without proof.\n\n\\begin{proposition}\nAn inequality~\\eqref{sum24} with $\\beta' = 1$ (resp. $\\beta' = -1$) is a facet of $\\conv{G}$ if and only if $(\\beta_{0},\\beta)$ is an extreme point of the polyhedron $E := \\{(\\beta_{0},\\beta)\\colon \\beta_{0}+\\beta^{\\top}\\bar{\\x} \\ge - m(\\bar{\\x}), \\ \\bar{x}\\in\\{\\ell,u\\}^{n} \\}$ (resp. $H := \\{(\\beta_{0},\\beta)\\colon \\beta_{0}+\\beta^{\\top}\\bar{\\x} \\ge m(\\bar{\\x}), \\ \\bar{x}\\in\\{\\ell,u\\}^{n} \\}$).\n\\end{proposition}\n\nThe challenge with using this polarity result is that it requires enumeration of extreme points of the set $E$ and $H$, which are exponentially many in general. The next section observes that it suffices to focus attention on only a subset of inequalities since all other inequalities are obtained via permutations.\n}\n\n\\subsection{Core Inequalities}\n\nWe define \\emph{core inequalities} as being those inequalities~\\eqref{sum24} that have $\\beta^{\\prime}=\\pm{1}$ and $\\beta_1 \\leq \\ldots \\leq \\beta_n.$ The restriction that $\\beta^{\\prime}=\\pm{1}$ for nonzero $\\beta^{\\prime}$ is nonrestrictive, as it follows from scaling. The restriction that $\\beta_1 \\leq \\ldots \\leq \\beta_n$ follows from the problem symmetry, as an inequality will be valid (a facet) for $\\conv{G}$ if and only if every symmetric copy obtained by permuting the entries in $\\x[\\beta]^{\\top}$ is also valid (a facet). A \\emph{valid core inequality} is a core inequality that is valid for $\\conv{G}.$ A valid core inequality that is also a facet for $\\conv{G}$ is a \\emph{core facet}. \n\nSymmetry implies that the validity of a core inequality \\eqref{sum24} can be checked in terms of only the $(n+1)$ extreme points of the simplex $\\mathcal{S}.$ \n\n\\begin{lemma}\t\\thlabel{foundation}\nA core inequality \\eqref{sum24} is a valid core inequality if and only if \n\\begin{equation}\n{\\cal L}_k(\\beta_0,\\x[\\beta]) + \\beta^{\\prime} m(\\x^k) \\geq 0, \\qquad \\forall \\; k \\in \\{0, \\ldots, n\\}. \\label{anoto}\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nThe only if direction is trivial and so we consider the if direction. It is sufficient to show that the $(n+1)$ inequalities of \\eqref{anoto} imply that \\eqref{sum24} is nonnegative for all $2^n$ extreme points of $\\conv{G}.$ Toward this end, consider any $k \\in\\{0, \\ldots, n\\},$ and note that each of the $n \\choose k$ extreme points $(\\x,y)$ of $\\conv{G}$ with $\\x$ having $k$ entries of value $u$ and $(n-k)$ entries of value $\\ell$ also has $y = m(\\x^k).$ In addition, each such extreme point yields a value for $\\left(\\beta_0+\\sum_{j=1}^n\\beta_jx_j\\right)$ that is at least as large as ${\\cal L}_k(\\beta_0,\\x[\\beta]),$ so that \\eqref{sum24} is nonnegative for all $n \\choose k$ such extreme points. As the result holds true for every such $k,$ it holds true for all the extreme points of $\\conv{G}.$ This completes the proof.\n\\end{proof}\n\n\\subsection{Optimization and Separation}\n\n\\akshay{\nThe question of optimizing a linear function over $\\conv{G}$ can be solved via a sorting algorithm  without using the quadratic-sized extended formulation of \\mythref{extform}.\n\n\\begin{proposition}\t\\thlabel{optcompl}\nFor any $(\\alpha,\\alpha')\\in\\real^{n}\\times\\real$,\n\\[\n\\max\\left\\{\\alpha^{\\top}x + \\alpha^{\\prime}y \\sep (x,y)\\in\\conv{G} \\right\\} \\eq \\max_{k=0,\\dots,n}\\left\\{u\\sum_{j=1}^{k}\\alpha_{\\sigma(j)} + \\ell\\sum_{j=k+1}^{n}\\alpha_{\\sigma(j)} + \\alpha^{\\prime}\\mk \\right\\},\n\\]\nwhere the permutation $\\sigma$ is such that $\\alpha_{\\sigma(1)}\\ge \\alpha_{\\sigma(2)} \\ge \\cdots \\ge \\alpha_{\\sigma(n)}$.\n\\end{proposition}\n\\begin{proof}\nThe linear function has an optimum at a vertex of $\\conv{G}$, and so the left-hand side is equivalent to maximising $\\alpha^{\\top}x + \\alpha^{\\prime}y$ over $\\vertex{(\\conv{G})}$. Our claim follows after using $\\vertex{(\\conv{G})} = \\cup_{k=0}^{n}\\{Q_{k}\\times\\{\\mk\\} \\}$ from \\eqref{convunion} and observing that $\\max\\{\\alpha^{\\top}x \\colon x\\in Q_{k}\\} = u\\sum_{j=1}^{k}\\alpha_{\\sigma(j)} + \\ell\\sum_{j=k+1}^{n}\\alpha_{\\sigma(j)}$.\n\\end{proof}\n\nHence, assuming computation of each $\\mk$ takes $O(1)$ time, linear optimization over $\\conv{G}$ can be solved in $O(n\\log{n})$ time by sorting the vector $\\alpha$.\n\nDue to the well-known equivalence of complexity of optimization and separation over a polyhedron, it follows that a given point can be separated from the convex hull of $G$ in polynomial time. However, this connection invokes the ellipsoid algorithm whose complexity is a high degree polynomial in the input encoding. The question of separation can be answered directly if there are polynomially many core facets. The approach is similar to that used for \\mythref{optcompl} but here the point $(\\bar{x},\\bar{y})$ to be separated has $\\bar{x}$ sorted in nonincreasing order. \n\n\\begin{proposition}\t\\thlabel{sepcompl}\nIf $\\conv{G}$ has $t$ core facets, then a point can be separated from $\\conv{G}$ in $O(nt+n\\log{n})$ time.\n\\end{proposition}\n\\begin{proof}\nLet $(\\bar{x},\\bar{y})\\in X\\times\\real$ be a given point. Take $\\sigma\\in\\Sigma_{n}$ such that $\\bar{x}_{\\sigma(1)} \\le \\bar{x}_{\\sigma(2)} \\le \\cdots \\le \\bar{x}_{\\sigma(n)}$. Choose any core facet~\\eqref{sum24}. We have $\\beta_{1}\\le \\cdots\\le\\beta_{n}$ and  need to check if this facet or any permuted facet is violated, i.e., whether there exists a permutation $\\sigma\\in\\Sigma_{n}$ such that $\\beta_0+\\sum_{j=1}^n\\beta_{\\sigma(j)} \\bar{x}_j+\\beta^{\\prime}\\bar{y} < 0$. This is equivalent to checking whether $- \\beta_{0} - \\beta^{\\prime}\\bar{y} > \\min\\{\\bar{x}^{\\top}v\\colon v\\in P_{n}(\\beta) \\}$, where $P_{n}(\\beta) = \\conv{\\{(\\beta_{\\sigma(1)},\\beta_{\\sigma(2)},\\dots,\\beta_{\\sigma(n)}) \\colon \\sigma\\in\\Sigma_{n} \\}}$ is the generalized permutahedron \\cite{bowman1972permutation} with respect to $\\beta$. Linear optimization over the permutahedron can be done via the sorting algorithm since the permutahedron is the base polymatroid polyhedron corresponding to a certain submodular function \\cite{rado52ineq} and \\citet{edmonds1970submodular} established that the sorting algorithm works for optimizing over any base polymatroid. In particular, we have that our minimum is attained at $v^{*} = \\sigma^{-1}\\cdot\\hat{\\beta}$, where $\\hat{\\beta} := (\\beta_{n},\\beta_{n-1},\\dots,\\beta_{1})$ is the reverse sorting of $\\beta$, so that $v^{*}_{j} = \\beta_{n+1-\\sigma^{-1}(j)}$ for all $j\\in N$. Thus, checking separation of a single core facet and its permuted copies is equivalent to checking whether $-\\beta_{0} - \\beta^{\\prime}\\bar{y} > \\sum_{j=1}^{n}\\beta_{n+1-\\sigma^{-1}(j)}\\bar{x}_{j}$, which runs in $O(n)$ time. The overall complexity then becomes $O(nt+n\\log{n})$ due to the sorting of $\\bar{x}$ initially and enumeration over $t$ core facets.\n\\end{proof}\n\nAs seen in the above proof, the overall complexity is composed of $nt$ additions and multiplications and a sorting step which requires $O(n\\log{n})$ comparisons.\n}\n\n\\section{Conditions for Core Facets}\n\nA challenge in determining whether a valid core inequality is a core facet is the identification of a maximum number of affinely independent points within $\\conv{G}$ that satisfies the inequality exactly. Of course, we can restrict attention to only those points that are extreme to $\\conv{G}.$ The following proposition shows that the set of extreme points of $\\conv{G}$ which satisfy a core  inequality exactly can be completely characterized in terms of the extreme points of the simplex $\\mathcal{S}$.\n\n\\begin{lemma}\t\\thlabel{foundation45}\nGiven an extreme point $(\\tilde{\\x},m(\\tilde{\\x}))$ of $\\conv{G}$ with $\\tilde{\\x}$ not an extreme point of $\\mathcal{S}$, let $r$ be the smallest index $j$ such that $x_j =\\ell,$ and $s$ be the largest index $j$ such that $x_j =u.$ Further let $p$ be the number of entries of $\\tilde{\\x}$ having value $u$ (so that $r \\leq p \\leq s-1$). Then $(\\x,y)=(\\tilde{\\x},m(\\tilde{\\x}))$ satisfies a valid core inequality \\eqref{sum24} exactly if and only if $(\\x,y)=(\\x^p,m(\\x^p))$ satisfies the inequality exactly, and $\\beta_r = \\ldots = \\beta_s.$\n\\end{lemma}\n\\begin{proof}\nWe have $\\beta_0+\\sum_{j=1}^n\\beta_j\\tilde{x}_j+\\beta^{\\prime}m(\\tilde{\\x}) \\geq {\\cal L}_p(\\beta_0,\\x[\\beta])+\\beta^{\\prime}m(\\x^p) \\geq 0$, where the first inequality is due to the nondecreasing values of $\\beta_j$ and the symmetry of $m(\\x),$ and the second inequality is due to the validity of \\eqref{sum24}. The first inequality is satisfied exactly if and only if $\\beta_r = \\ldots = \\beta_s,$ while the second inequality is satisfied exactly if and only if \\eqref{sum24} is satisfied exactly at $(\\x^p,m(\\x^p)).$\n\\end{proof}\n\nA consequence is that given a valid core inequality, we can identify the subset of $2^n$ extreme points of $\\conv{G}$ that satisfy the inequality exactly by considering only the $(n+1)$ extreme points of $\\mathcal{S}.$ The next few results build upon this consequence to provide characteristics of core facets in terms of the extreme points of $\\mathcal{S}.$\n\n\\begin{proposition}\t\\thlabel{foundation22}\nA valid core inequality \\eqref{sum24} is a core facet only if it is satisfied exactly at $(\\x^k,m(\\x^k))$ for at least two extreme points $\\x^k$ of the simplex $\\mathcal{S},$ with at least one extreme  point not being $\\x^0$ or $\\x^n.$\n\\end{proposition}\n\\begin{proof}\nSuppose that a valid core inequality \\eqref{sum24} is satisfied exactly at fewer than two such points $(\\x^k,m(\\x^k)),$ or at only the points $(\\x^0,m(\\x^0))$ and $(\\x^n,m(\\x^n)).$ Since the dimension of $\\conv{G}$ is $n+1,$ it is sufficient to show that the inequality is satisfied exactly at no more than $n$ affinely independent extreme points of $\\conv{G}.$ Three cases arise. First, if the inequality is satisfied exactly at no such $(\\x^k,m(\\x^k)),$ then \\eqref{sum24} is not satisfied exactly at any of the $2^n$ extreme points of $\\conv{G}$ by \\mythref{foundation45}. Second, if the inequality is satisfied exactly at precisely one such $(\\x^{\\tilde{k}},m(\\x^{\\tilde{k}}))$ then, by \\mythref{foundation45}, the only extreme points $(\\x,m(\\x))$ of $\\conv{G}$ that can possibly satisfy \\eqref{sum24} exactly are the $n \\choose \\tilde{k}$ points with $\\x$ having $\\tilde{k}$ entries of value $u$ and $(n-\\tilde{k})$ entries of value $\\ell.$ However, as the two linearly independent hyperplanes $\\sum_{j=1}^nx_j=\\tilde{k}u+(n-\\tilde{k})\\ell$ and $y=m(\\x^{\\tilde{k}})$ both pass through these $n \\choose \\tilde{k}$ points, there exist at most $n$ such affinely independent points. Finally, if the inequality is satisfied exactly at only $(\\x^0,m(\\x^0))$ and $(\\x^n,m(\\x^n)),$ then \\mythref{foundation45} gives us that these are the only extreme points of $\\conv{G}$ that satisfy \\eqref{sum24} exactly.\n\\end{proof}\n\nThe necessary condition in  \\mythref{foundation22} to be a core facet is also sufficient when all the coefficients are equal.\n\n\\begin{proposition}\t\\thlabel{equalcoeff}\nA valid core inequality \\eqref{sum24} with $\\beta_1 = \\ldots = \\beta_n$ is a core facet if and only if it is satisfied exactly at $(\\x^k,m(\\x^k))$ for at least two extreme points $\\x^k$ of the simplex $\\mathcal{S},$ with at least one extreme point not being $\\x^0$ or $\\x^n.$\n\\end{proposition}\n\\begin{proof}\nThe only if direction follows directly from \\mythref{foundation22}, and so we consider the if direction. Suppose the inequality is satisfied exactly at two such points $(\\x^p,m(\\x^p))$ and $(\\x^q,m(\\x^q)),$ with $\\x^p$ not being $\\x^0$ or $\\x^n.$ Then \\mythref{foundation45} gives us that the $n \\choose p$ extreme points $(\\x,m(\\x))$ of $\\conv{G}$ with $\\x$ having $p$ entries of value $u$ and $(n-p)$ entries of value $\\ell$ satisfy the inequality exactly. The proof is to show that there exist $n$ affinely independent points from amongst this set of $n \\choose p$ points. Then these $n$ points, together with $(\\x^q,m(\\x^q)),$ will form an affinely independent set of $(n+1)$ points because each of the first $n$ points satisfies the equation $\\sum_{j=1}^nx_j=pu+(n-p)\\ell,$ but $(\\x^q,m(\\x^q))$ does not. In fact, it is sufficient to show that the extreme points of $X$ associated with these $n \\choose p$ points are affinely independent.\n\nSince the affine independence of a collection of points is unaffected when the same value is subtracted from every entry of each point, and when each point is multiplied by a nonzero scalar, the affine independence of the associated $n \\choose p$ extreme points of $X$ remains unchanged when every $u$ is replaced with 1 and every $\\ell$ is replaced with $0.$ Consider the $n \\times {{n} \\choose {p}}$ matrix $A$ defined so that each column corresponds to one such point, upon application of these operations. If $p=1,$ then $A$ is a permutation matrix, and the $n \\choose p$ points are affinely independent. Otherwise, $A$ has the two properties that: each row contains $\\kappa_1 \\equiv  {{n-1} \\choose {p-1}}$ entries of value 1, and every pair of two distinct rows contains $\\kappa_2 \\equiv  {{n-2} \\choose {p-2}}$ common entries of value $1.$ Hence, $AA^T$ is the $n \\times n$ matrix having $\\kappa_1$ along the main diagonal and $\\kappa_2$ elsewhere. Since $\\mbox{rank}(AA^T)=n,$ because the common row sum allows us to subtract $\\kappa_2$ from every entry of $AA^T$ to obtain a lower-triangular matrix with $(\\kappa_1-\\kappa_2) \\neq 0$ along the diagonal, then $\\mbox{rank}(A)=n,$ and the proof is complete.\n\\end{proof}\n\nThe below proposition gives further conditions, in terms of the extreme points $\\x^k$ of $\\mathcal{S},$ for a valid core inequality to be a core facet.\n\n\\begin{proposition}\t\\thlabel{foundation24}\nGiven any two valid core inequalities of the form \n\\begin{equation}\n\\bar{\\beta}_0+\\sum_{j=1}^n\\bar{\\beta}_jx_j+\\beta^{\\prime}y \\geq 0 \\mbox{ and } \\hat{\\beta}_0+\\sum_{j=1}^n\\hat{\\beta}_jx_j+\\beta^{\\prime}y \\geq 0, \\label{firsti}\n\\end{equation}\nif \n\\begin{equation}\n{\\cal L}_k(\\bar{\\beta}_0,\\bar{\\x[\\beta]}) \\leq {\\cal L}_k(\\hat{\\beta}_0,\\hat{\\x[\\beta]}) \\; \\forall \\; k \\in \\{0, \\ldots, n\\}, \\label{secondi} \n\\end{equation}\nwith strict inequality holding for at least one $k$ in \\eqref{secondi}, then the right inequality of \\eqref{firsti} is not a facet.\n\\end{proposition}\n\\begin{proof}\nIt is sufficient to show that every extreme point $(\\x,m(\\x))$ of $\\conv{G}$ that satisfies the right inequality of \\eqref{firsti} exactly also satisfies the left inequality of \\eqref{firsti} exactly. This statement holds for the $(n+1)$ extreme points $(\\x^k,m(\\x^k)), k \\in \\{0, \\ldots, n\\},$ by \\eqref{secondi}, and\nso we arbitrarily select any one of the remaining $2^n-(n+1)$ extreme points, say $(\\tilde{\\x},m(\\tilde{\\x})),$ and suppose that the right inequality holds exactly at this point. Some $p \\in \\{1, \\ldots, n-1\\}$ entries of $\\tilde{\\x}$ have value $u$ and $(n-p)$ entries have value $\\ell,$ with the first $p$ entries not all equal to $u.$ \nThe proof reduces to showing that \n\\begin{equation}\n0={\\cal L}_p(\\hat{\\beta}_0,\\hat{\\x[\\beta]})+\\beta^{\\prime}m(\\x^p)={\\cal L}_p(\\bar{\\beta}_0,\\bar{\\x[\\beta]})+\\beta^{\\prime}m(\\x^p)=\\bar{\\beta}_0+\\sum_{j=1}^n\\bar{\\beta}_j\\tilde{x}_j+\\beta^{\\prime}m(\\tilde{\\x}).\n\\label{summary}\n\\end{equation}\nEach equality of \\eqref{summary} is considered separately.\n\\begin{itemize}\n\\item Since, by assumption, $(\\tilde{\\x},m(\\tilde{\\x}))$ satisfies the right inequality of \\eqref{firsti} exactly, \\mythref{foundation45} gives us that the first equality of \\eqref{summary} holds true, and also that $\\hat{\\beta}^*=\\hat{\\beta}_r = \\ldots =\\hat{\\beta}_s$ for some scalar $\\hat{\\beta}^*,$ where $r$ and $s$ are, respectively, the indices of the first and last entries of $\\tilde{\\x}$ which differ from $\\x^p.$\n\\item The second equality of \\eqref{summary} holds true by \\eqref{secondi} with $k=p,$ the first equality of \\eqref{summary}, and the validity of the left inequality of \\eqref{firsti} at $(\\x^p,m(\\x^p)).$ \n\\item\n Since as noted above, $\\hat{\\beta}^*=\\hat{\\beta}_r = \\ldots =\\hat{\\beta}_s$ for some scalar $\\hat{\\beta}^*,$ we have that\n\\begin{equation}\n{\\cal L}_p(\\hat{\\beta}_0,\\hat{\\x[\\beta]})+\\hat{\\beta}^*(u-\\ell)(s-p)={\\cal L}_s(\\hat{\\beta}_0,\\hat{\\x[\\beta]}) \\geq {\\cal L}_s(\\bar{\\beta}_0,\\bar{\\x[\\beta]})={\\cal L}_p(\\bar{\\beta}_0,\\bar{\\x[\\beta]})+(u-\\ell)(\\sum_{j=p+1}^s\\bar{\\beta}_j) \\label{balance3}\n\\end{equation}\nand\n\\begin{equation}\n{\\cal L}_p(\\hat{\\beta}_0,\\hat{\\x[\\beta]})-\\hat{\\beta}^*(u-\\ell)(p-r+1)={\\cal L}_{r-1}(\\hat{\\beta}_0,\\hat{\\x[\\beta]}) \\geq {\\cal L}_{r-1}(\\bar{\\beta}_0,\\bar{\\x[\\beta]})={\\cal L}_p(\\bar{\\beta}_0,\\bar{\\x[\\beta]})-(u-\\ell)(\\sum_{j=r}^p\\bar{\\beta}_j), \\label{balance4}\n\\end{equation}\nwhere, within \\eqref{balance3} and \\eqref{balance4}, the equalities follow from \\eqref{anoto25}, and the inequalities are due to \\eqref{secondi}. Combine these expressions and invoke ${\\cal L}_p(\\hat{\\beta}_0,\\hat{\\x[\\beta]})={\\cal L}_p(\\bar{\\beta}_0,\\bar{\\x[\\beta]})$ from the second equality of \\eqref{summary} to obtain \n\\begin{equation}\n\\hat{\\beta}^* \\leq \\left(\\frac{1}{p-r+1}\\right)\\left(\\sum_{j=r}^p\\bar{\\beta}_j\\right) \\leq \\left(\\frac{1}{s-p}\\right)\\left(\\sum_{j=p+1}^s\\bar{\\beta}_j\\right) \\leq \\hat{\\beta}^*, \\nonumber\n\\end{equation}\nwhere the three inequalities follow from \\eqref{balance4}, the nondecreasing property of $\\bar{\\x[\\beta]},$ and \\eqref{balance3}, respectively. Again by the nondecreasing property of $\\bar{\\x[\\beta]},$ we have that $\\hat{\\beta}^*=\\bar{\\beta}_r = \\ldots =\\bar{\\beta}_s,$ giving the third equality of \\eqref{summary} by \\mythref{foundation45}.\\qedhere\n\\end{itemize}\n\\end{proof}\n\nThis leads us to the following necessary and sufficient condition for a valid inequality with distinct coefficients to be a core facet.\n\n\\begin{corollary}\\label{sfirst}\nA valid inequality $\\bar{\\beta}_0+\\sum_{j=1}^n\\bar{\\beta}_jx_j+\\beta^{\\prime}y \\geq 0$ with $\\bar{\\beta}_1 < \\ldots < \\bar{\\beta}_n$ is a core facet if and only if it is satisfied exactly at $(\\x^k,m(\\x^k))$ for $k \\in\\{0, \\ldots, n\\}$. In this case, no other core facet can exist with the given $\\beta^{\\prime}.$\n\\end{corollary}\n\\begin{proof}\nThe if direction follows from the $n+1$ points $(\\x^k,m(\\x^k))$ for $k \\in\\{0, \\ldots, n\\}$ being affinely independent, and so we consider the only if direction. \\mythref{foundation45} gives us that the only extreme points to $\\conv{G}$ that can possibly satisfy the given inequality exactly are $(\\x^k,m(\\x^k))$ for $k \\in\\{0, \\ldots, n\\}.$ Since the inequality is a facet, these $(n+1)$ affinely independent points must satisfy the inequality exactly, giving ${\\cal L}_k(\\bar{\\beta}_0,\\bar{\\x[\\beta]})+\\beta^{\\prime}m(\\x^k)=0$ for $k \\in\\{0, \\ldots, n\\}.$ No other valid inequality $\\hat{\\beta}_0+\\sum_{j=1}^n\\hat{\\beta}_jx_j+\\beta^{\\prime}y \\geq 0$ for $\\conv{G}$ with the given $\\beta^{\\prime}$ can then be a facet by \\mythref{foundation24}.\n\\end{proof}\n\nAnother set of necessary conditions for being a core facet is obtained below.\n\n\\begin{proposition}\t\\thlabel{equalcoeff7}\nGiven any two valid core inequalities of the form \\eqref{firsti}, suppose that the following conditions hold.\n\\begin{enumerate}\n\\item $\\bar{\\beta}_u=\\bar{\\beta}_v$ for each $(u,v),u<v,$ having $\\hat{\\beta}_u=\\hat{\\beta}_v.$  \n\\item $\\{k:{\\cal L}_k(\\hat{\\beta}_0,\\hat{\\x[\\beta]})+\\beta^{\\prime}m(\\x^k)=0\\} \\subset  \\{k:{\\cal L}_k(\\bar{\\beta}_0,\\bar{\\x[\\beta]})+\\beta^{\\prime}m(\\x^k)=0\\}.$\n\\end{enumerate}\nThen the right inequality of \\eqref{firsti} is not a facet.\n\\end{proposition}\n\\begin{proof}\nAs with the proof of \\mythref{foundation24}, it is sufficient to show that every extreme point of $\\conv{G}$ that satisfies the right inequality of \\eqref{firsti} exactly also satisfies the left inequality of \\eqref{firsti} exactly. This statement holds for the $(n+1)$ extreme points $(\\x^k,m(\\x^k)), k \\in\\{0, \\ldots, n\\},$ by Condition 2 above, and\nso we arbitrarily select any one of the remaining $2^n-(n+1)$ extreme points, say $(\\tilde{\\x},m(\\tilde{\\x})),$ and suppose that the right inequality holds exactly at this point. Some $p \\in \\{1, \\ldots, n-1\\}$ entries of $\\tilde{\\x}$ have value $u$ and $(n-p)$ entries have value $\\ell,$ with the first $p$ entries not all equal to $u.$ By \\mythref{foundation45}, we have that $\\hat{\\beta}_r = \\ldots = \\hat{\\beta}_s$ where $r$ and $s$ are, respectively, the indices of the first and last entries of $\\tilde{\\x}$ which differ from $\\x^p.$ The first condition above then gives $\\bar{\\beta}_r = \\ldots = \\bar{\\beta}_s,$ so that by again invoking \\mythref{foundation45}, we have the left inequality of \\eqref{firsti} is satisfied exactly at $(\\tilde{\\x},m(\\tilde{\\x})).$\n\\end{proof}\n\nThe below result identifies instances in which select coefficients of a core facet must have equal values. We do not need to consider $r=n$ or $s=0$ because such cases are not possible by \\mythref{foundation22}. \n\n\\begin{proposition} \\thlabel{equalcoeff8}\nGiven a core facet \\eqref{sum24}, let \n\\[\n\\begin{split}\nr&=\\min\\left\\{k \\in \\{0, \\ldots, n-1\\}\\colon {\\cal L}_k(\\beta_0,\\x[\\beta]) +\\beta^{\\prime} m(\\x^k)=0\\right\\}, \\\\ \ns&=\\max\\left\\{k \\in \\{1, \\ldots, n\\}\\colon{\\cal L}_k(\\beta_0,\\x[\\beta]) +\\beta^{\\prime} m(\\x^k)=0\\right\\}. \n\\end{split}\n\\]\nThen $\\beta_1 = \\ldots = \\beta_{r+1}$ and $\\beta_{s} = \\ldots = \\beta_n.$\n\\end{proposition}\n\\begin{proof}\nWe consider separately the conclusions that $\\beta_1 = \\ldots = \\beta_{r+1}$ and $\\beta_{s} = \\ldots = \\beta_n.$\n\\begin{itemize}\n\\item If $r=0,$ the first conclusion follows trivially. Otherwise, $r \\geq 1$ and, by contradiction, define $p \\leq r$ so that $\\beta_p < \\beta_{p+1} = \\beta_{r+1}.$ It is sufficient to show that there exists an $\\epsilon>0$  so that the inequality\n\\begin{equation}\n(\\beta_0-pu\\epsilon)+\\sum_{j=1}^{p}(\\beta_j+\\epsilon)x_j+\\sum_{j=p+1}^n\\beta_jx_j+\\beta^{\\prime}y \\geq 0, \\label{lastly}\n\\end{equation}\nwith $(\\beta_p+\\epsilon) \\leq \\beta_{p+1},$ is valid for $\\conv{G}.$ Then every extreme point $(\\x,m(\\x))$ to $\\conv{G}$ with $x_j=u$ for $j \\in \\{1, \\ldots, p\\}$ will have the left side of \\eqref{lastly} equal to the left side of \\eqref{sum24}, while every extreme point $(\\x,m(\\x))$ to $\\conv{G}$ with $x_j=\\ell$ for at least one $j \\in \\{1, \\ldots, p\\}$ will have the left side of \\eqref{lastly} strictly less than the left side of \\eqref{sum24}. Relative to the extreme points of the simplex $\\mathcal{S}$, the left sides of \\eqref{lastly} and \\eqref{sum24} will both take value ${\\cal L}_k(\\beta_0,\\x[\\beta]) +\\beta^{\\prime} m(\\x^k)$ at $(\\x^k,m(\\x^k))$ for each $k \\in \\{p, \\ldots, n\\},$ but the left side of \\eqref{lastly} will be ${\\cal L}_k(\\beta_0,\\x[\\beta]) +\\beta^{\\prime} m(\\x^k)-(p-k)(u-\\ell)\\epsilon$ at $(\\x^k,m(\\x^k))$ for each $\\x^k, k \\in \\{0, \\ldots, p-1\\},$ which is $(p-k)(u-\\ell)\\epsilon>0$ less than the left side of \\eqref{sum24}. Define $\\epsilon=\\min\\{\\epsilon^{\\prime},(\\beta_{p+1}-\\beta_{p})\\},$ where $\\epsilon^{\\prime}=\\mbox{min}_{k \\in \\{0, \\ldots, p-1\\}}\\left\\{\\frac{{\\cal L}_k(\\beta_0,\\x[\\beta]) +\\beta^{\\prime} m(\\x^k)}{(p-k)(u-\\ell)}\\right\\}.$  Then $\\epsilon >0$ and inequality \\eqref{lastly} is valid for $\\conv{G}$ by \\mythref{foundation}, as it is valid at all $(\\x^k,m(\\x^k)),k \\in \\{0, \\ldots, n\\}.$ \n\\item If $s=n,$ the second conclusion follows trivially. Otherwise $s \\leq (n-1)$ and, by contradiction, define $p \\geq s$ so that $\\beta_{s} = \\beta_p < \\beta_{p+1}.$  It is sufficient to show that there exists an $\\epsilon>0$ so that the inequality\n\\begin{equation}\n\\beta_0+(n-p)\\ell \\epsilon+\\sum_{j=1}^{p}\\beta_jx_j+\\sum_{j=p+1}^n(\\beta_j-\\epsilon)x_j+\\beta^{\\prime}y \\geq 0, \\label{lastly2}\n\\end{equation}\nwith $\\beta_p \\leq (\\beta_{p+1}-\\epsilon),$ is valid for $\\conv{G}.$ Then every extreme point $(\\x,m(\\x))$ to $\\conv{G}$ with $x_j=\\ell$ for $j \\in \\{p+1, \\ldots, n\\}$ will have the left side of \\eqref{lastly2} equal to the left side of \\eqref{sum24}, while every extreme point $(\\x,m(\\x))$ to $\\conv{G}$ with $x_j=u$ for at least one $j \\in \\{p+1, \\ldots, n\\}$ will have the left side of \\eqref{lastly2} strictly less than the left side of \\eqref{sum24}. Relative to the extreme points of the simplex $\\mathcal{S}$, the left sides of \\eqref{lastly2} and \\eqref{sum24} will both take value ${\\cal L}_k(\\beta_0,\\x[\\beta]) +\\beta^{\\prime} m(\\x^k)$ at $(\\x^k,m(\\x^k))$ for each $k \\in \\{1, \\ldots, p\\},$ but the left side of \\eqref{lastly2} will be ${\\cal L}_k(\\beta_0,\\x[\\beta]) +\\beta^{\\prime} m(\\x^k)-(k-p)(u-\\ell)\\epsilon$ at $(\\x^k,m(\\x^k))$ for each $\\x^k, k \\in \\{p+1, \\ldots, n\\},$ which is $(k-p)(u-\\ell)\\epsilon>0$ less than the left side of \\eqref{sum24}. Define $\\epsilon=\\min\\{\\epsilon^{\\prime},(\\beta_{p+1}-\\beta_{p})\\},$ where $\\epsilon^{\\prime}=\\mbox{min}_{k \\in \\{p+1, \\ldots, n\\}}\\left\\{\\frac{{\\cal L}_k(\\beta_0,\\x[\\beta]) +\\beta^{\\prime} m(\\x^k)}{(k-p)(u-\\ell)}\\right\\}.$  Then $\\epsilon >0$ and inequality \\eqref{lastly2} is valid for $\\conv{G}$ by \\mythref{foundation}, as it is valid at all $(\\x^k,m(\\x^k)),k \\in \\{0, \\ldots, n\\}.$ \\qedhere\n\\end{itemize}\n\\end{proof}\n\n\\section{RLT for Multilinear Functions}\nIn this section, we present the Reformulation-Linearization Technique (RLT) as it pertains to the set $G$. We provide a brief description of select aspects of the general RLT process that are relevant to this study, emphasizing some key properties. Then we   review the mathematical details of the RLT in terms of Kronecker products of matrices. This RLT machinery enables us to characterize exactness of the core facets for $\\conv{G}$.\n\n\\subsection{Main Ideas}\nThe RLT is a general methodology for reformulating mixed-integer linear and polynomial programs for the purpose of obtaining tight linear programming relaxations. While there is a rich body of literature on the topic \\citep{sherali1994hierarchy,sherali1990hierarchy,sherali99rlt} we focus attention here on a box-constrained region of $n$ variables $x_j,$ where each $x_j$ is restricted to lie between variable bounds $L_j$ and $U_j.$  The RLT gives a hierarchy of successively tighter polyhedral relaxations, but we consider only the highest level $n$ which affords the convex hull representations.\n\nSpecifically, consider the set\n\\begin{equation}\nX^{\\prime} \\equiv \\left\\{\\x\\in \\real^n: L_j \\leq x_j \\leq U_j \\; \\forall \\; j \\in N\\right\\} \\label{Xprimedef}\n\\end{equation}\nhaving each $L_j <U_j,$ which is a generalization of the set $X$ of \\eqref{Xdef} that reduces to \\eqref{Xdef} when $L_j=\\ell$ and $U_j=u$ for all $j \\in N.$ The RLT process that we apply to \\eqref{Xprimedef} consists of the two steps of \\emph{reformulation} and \\emph{linearization}. The \\emph{reformulation} step computes products of the expressions $(x_j-L_j)$ and $(U_j-x_j),$ taken $n$ at a time, such that one such expression is chosen for each $j.$ In this manner, $2^n$ multilinear polynomial functions of degree $n$ emerge. To elaborate, define the $2^n$ functions $F_{K}(x)$ so that \n\\begin{equation}\nF_{K}(x)= \\prod_{j\\in K}(x_j-L_j)\\prod_{j\\notin K}(U_j-x_j), \\quad K \\subseteq N. \\label{Fun}\n\\end{equation}\nThen we have $2^n$ multilinear polynomial functions of the form\n\\begin{equation}\nF_{K}(x), \\quad K \\subseteq N. \\label{functions}\n\\end{equation}\nEach of these functions is nonnegative for all $\\x \\in X^{\\prime},$ and the RLT enforces this nonnegativity to obtain the $2^n$ multilinear polynomial inequalities\n\\begin{equation}\nF_{K}(x) \\geq 0, \\quad K \\subseteq N, \\label{RLTstep}\n\\end{equation}\nthat are satisfied for all $\\x \\in X^{\\prime}.$ \n\nThe \\emph{linearization} step then substitutes a continuous variable $w_J$ for each of the $2^n-(n+1)$ distinct product terms $\\prod_{j \\in J}x_j$ with $J\\subseteq N$ and $|J| \\geq 2$ that are found in \\eqref{RLTstep}. Denote the linearized form of each function $F_{K}(x)$ that is obtained via such substitutions as $F_{K}(x).$ The RLT gives the following \\emph{polyhedral} set:\n\\begin{equation}\nP\\equiv\\left\\{(\\x,\\x[w]) \\in \\real^n\\times\\real^{2^n-(n+1)}:F_{K}(x) \\geq 0, \\ K \\subseteq N\\right\\}.  \\label{RLTstep2}\n\\end{equation}\nWe adopt the notation that $\\{\\bullet\\}_L$ is the linearized form of the vector $\\bullet$ that is obtained by substituting $w_J=\\prod_{j \\in J}x_j$ for all $J \\subseteq N$ with $|J| \\geq 2$ throughout $\\bullet.$ In this manner, $F_{K}(x)=\\{F_{K}(x)\\}_L$ for all $K \\subseteq N.$ Two properties of the set $P$ are as follows.\n\n\\begin{proposition}\t\\thlabel{propP}\nThe set $P$ is a polytope having exactly $2^n$ extreme points and at each such extreme point $(\\x,\\x[w])$ we have\n\\begin{enumerate}\n\\item $x_j\\in\\{L_j,U_j\\}$ for all $j \\in N,$\n\\item $w_J=\\prod_{j \\in J}x_j$ for all $J \\subseteq N$ with $|J| \\geq 2.$\n\\end{enumerate}\nConsequently, we have $P = \\conv{T}$ where \n\\begin{equation}\nT\\equiv\\left\\{(\\x,\\x[w]) \\in \\prod_{j\\in N}[L_{j},U_{j}] \\, \\times\\real^{2^n-(n+1)}: w_{J} = \\prod_{j \\in J}x_j, \\ J \\subseteq N \\mbox{ with }|J| \\geq 2\\right\\}. \\label{miss1}\n\\end{equation}\n\\end{proposition}\n\nThese properties of $P$ were originally established by \\citet{sherali1990hierarchy} when $P$ has $L_j=0$ and $U_j=1$ for all $j \\in N$; see also \\cite[Theorem 1]{fomeni2015cutting}. They were more generally proven in \\cite{adams2005hierarchy,henry} using Kronecker products of matrices for richer families of polytopes that subsume $P.$ These same type Kronecker products are also found in the generalization of the RLT constructs to handle general integer variables \\cite{adams2011,adams2005hierarchy}. For completeness, we present a tailored version of the arguments in the next subsection. In addition to establishing the two stated properties of $P,$ this Kronecker product representation of \\eqref{functions} through \\eqref{RLTstep2} motivates two consequences that will be used later to characterize exactness of facet-defining inequalities. These apply to any multilinear polynomial and therefore encompass the functions $m(\\x)$.\n\n\\begin{corollary} \\thlabel{obs1}\nFor every $p(x) = \\sum_{J \\subseteq N}\\alpha_j\\prod_{j \\in J}x_j,$ there exists a unique $\\x[\\pi] \\in \\real^{2^n}$ so that $p(x) = \\sum_{K \\subseteq N}\\pi_{K}F_{K}(x)$. In particular, $\\pi_K=\\frac{p(\\hat{\\x})}{F_{K}(\\hat{\\x})},$ where $\\hat{x}_j=U_j$ for all $j \\in K$ and $\\hat{x}_j=L_j$ for all $j \\notin K.$\n\\end{corollary}\n\n\n\n\\begin{corollary} \\thlabel{obs2}\nFor every $p(x) = \\sum_{J \\subseteq N}\\alpha_j\\prod_{j \\in J}x_j,$ we have $p(x) \\ge 0$ for all $x\\in X^{\\prime}$ if and only if $p(x) \\ge 0$ for every extreme point $x\\in X^{\\prime}$. \n\\end{corollary}\n\n\\mythref{propP} can be extended to include the variable $y$ and restriction $y= m(\\x)$ of \\eqref{Sdef}, so that the projection operation from the resulting higher-variable space onto the original $(\\x,y)$ space can, in theory, be used to compute $\\conv{G}.$ \\akshay{These properties were already noted by \\citet{sherali1997convex} using a different approach,  \nwe give our arguments in Appendix~\\ref{sec:propG} for sake of completeness.}\n\n\n\\subsection{Use of Kronecker Products}\n\n\\begin{proof}[\\textbf{Proof of \\mythref{propP}}]\nWe prove \\mythref{propP} by expressing \\eqref{functions} and \\eqref{RLTstep} as Kronecker products of matrices, and then using a known result of Kronecker products. The consequence $P = \\conv{T}$ follows since every extreme point of the polytope $P$ is in $T,$ and since $T \\subseteq P$ by construction.\n\nRecall that the Kronecker product of an $m_1 \\times n_1$ matrix $A$ with an $m_2 \\times n_2$ matrix $B$, denoted by $A \\otimes B,$ is that $m_1m_2 \\times n_1n_2$ matrix  $A \\otimes B = \\scriptsize \\left[\\begin{array}{ccc} a_{11}B & \\hdots & a_{1n_1}B \\\\ \\vdots & \\ddots & \\vdots \\\\ a_{m_11}B & \\hdots & a_{m_1n_1}B \\end{array}\\right], \\normalsize$\n\\noindent where $a_{ij}$ is the $(i,j)^{th}$ entry of $A.$ A result of Kronecker products is that, given any collection of $r$ pairs of matrices $A_i$ and $B_i$ for $i=1,...,r$ such that the products $A_iB_i$ are defined, we have that \n\\begin{equation}\nA_1B_1 \\otimes A_2B_2 \\otimes \\ldots \\otimes A_rB_r= (A_1 \\otimes A_2 \\otimes \\ldots \\otimes A_r)(B_1 \\otimes B_2 \\otimes \\ldots \\otimes B_r). \\label{kroneck}\n\\end{equation}\nAs a consequence, if each of the matrices $A_i$ is invertible with inverse $A_i^{-1},$ then by setting $B_i=A_i^{-1}$ for all $i$ within \\eqref{kroneck}, we obtain that\n\\begin{equation}\n(A_1 \\otimes A_2 \\otimes \\ldots \\otimes A_{r})^{-1} = A_1^{-1} \\otimes A_2^{-1} \\otimes \\ldots A_r^{-1}, \\label{kroneck2}\n\\end{equation}\nsince the left side of \\eqref{kroneck} becomes the suitably-sized identity matrix. \n\nRelative to the functions $F_{K}(x)$ of \\eqref{functions} and \\eqref{RLTstep}, and the functions $F_{K}(x)$ and set $P$ of \\eqref{RLTstep2}, define for each variable $x_j,$ the two matrices $\\scriptsize \\left[\\begin{array}{cc} U_j &  -1 \\\\ -L_j & 1 \\end{array}\\right] \\normalsize$ and $\\scriptsize \\left[\\begin{array}{c} 1 \\\\ x_j \\end{array}\\right] \\normalsize$ so that the functions $(U_j-x_j)$ and $(x_j-L_j)$ can be represented in matrix form as the two entries of the vector $\\scriptsize \\left[\\begin{array}{cc} U_j &  -1 \\\\ -L_j & 1 \\end{array}\\right] \\left[\\begin{array}{c} 1 \\\\ x_j \\end{array}\\right] \\normalsize.$ Then the $2^n$ functions of \\eqref{functions} can be collectively expressed using Kronecker products as \n\\begin{equation}\n\\scriptsize \\left(\\left[\\begin{array}{cc} U_1 &  -1 \\\\ -L_1 & 1 \\end{array}\\right] \\left[\\begin{array}{c} 1 \\\\ x_1 \\end{array}\\right]\\right)\\normalsize \\otimes \\ldots \\otimes \\scriptsize \\left(\\left[\\begin{array}{cc} U_n &  -1 \\\\ -L_n & 1 \\end{array}\\right] \\left[\\begin{array}{c} 1 \\\\ x_n \\end{array}\\right]\\right)\\normalsize, \\nonumber\n\\end{equation}\nand the $2^n$ inequalities of \\eqref{RLTstep} can be similarly expressed as \n\\begin{equation}\n\\scriptsize \\left(\\left[\\begin{array}{cc} U_1 &  -1 \\\\ -L_1 & 1 \\end{array}\\right] \\left[\\begin{array}{c} 1 \\\\ x_1 \\end{array}\\right]\\right)\\normalsize \\otimes \\ldots \\otimes \\scriptsize \\left(\\left[\\begin{array}{cc} U_n &  -1 \\\\ -L_n & 1 \\end{array}\\right] \\left[\\begin{array}{c} 1 \\\\ x_n \\end{array}\\right]\\right)\\normalsize  \\geq\t \\boldsymbol{0}, \\label {prodkron}\n\\end{equation}\nwhere $\\boldsymbol{0}$ is the $2^n$-dimensional column vector of all zeroes. Identity \\eqref{kroneck} allows us to rewrite \\eqref{prodkron} as\n\\begin{equation}\n\\scriptsize \\left(\\left[\\begin{array}{cc} U_1 &  -1 \\\\ -L_1 & 1 \\end{array}\\right] \\normalsize \\otimes \\ldots \\otimes \\scriptsize \\left[\\begin{array}{cc} U_n &  -1 \\\\ -L_n & 1 \\end{array}\\right]\\right)\\left(\\left[\\begin{array}{c} 1 \\\\ x_1 \\end{array}\\right]\\normalsize \\otimes \\ldots \\otimes \\scriptsize \\left[\\begin{array}{c} 1 \\\\ x_n \\end{array}\\right]\\right) \\normalsize \\geq \\boldsymbol{0}. \\label {prodkron2}\n\\end{equation}\n\nThe set $P$ of \\eqref{RLTstep2} can then be expressed in terms of \\eqref{prodkron2} as \n\\begin{alignat}{2} \nP=\\mbox{\\Huge \\{} \\normalsize &(\\x,\\x[w]) \\in \\real^n\\times\\real^{2^n-(n+1)}:   \\nonumber \\\\ \n&\\scriptsize \\left(\\left[\\begin{array}{cc} U_1 &  -1 \\\\ -L_1 & 1 \\end{array}\\right] \\normalsize \\otimes \\ldots \\otimes \\scriptsize \\left[\\begin{array}{cc} U_n &  -1 \\\\ -L_n & 1 \\end{array}\\right]\\right)\\left\\{\\left[\\begin{array}{c} 1 \\\\ x_1 \\end{array}\\right]\\normalsize \\otimes \\ldots \\otimes \\scriptsize \\left[\\begin{array}{c} 1 \\\\ x_n \\end{array}\\right]\\right\\}_L \\normalsize \\geq \\boldsymbol{0}\\mbox{\\Huge \\}} \\normalsize, && \\label{ca4} \n\\end{alignat}\nwhich is equivalent to \n\\begin{alignat}{2} \nP=\\mbox{\\Huge \\{} \\normalsize &(\\x,\\x[w]) \\in \\real^n\\times\\real^{2^n-(n+1)}:   \\nonumber \\\\ \n&\\scriptsize \\left(\\left[\\begin{array}{cc} U_1 &  -1 \\\\ -L_1 & 1 \\end{array}\\right] \\normalsize \\otimes \\ldots \\otimes \\scriptsize \\left[\\begin{array}{cc} U_n &  -1 \\\\ -L_n & 1 \\end{array}\\right]\\right)\\left\\{\\left[\\begin{array}{c} 1 \\\\ x_1 \\end{array}\\right]\\normalsize \\otimes \\ldots \\otimes \\scriptsize \\left[\\begin{array}{c} 1 \\\\ x_n \\end{array}\\right]\\right\\}_L \\normalsize = \\x[\\lambda] \\nonumber \\\\ & \\mbox{for some nonnegative } \\x[\\lambda] \\in \\real^{2^n}\\mbox{\\Huge \\}} \\normalsize. && \\label{ca5} \n\\end{alignat}\n\nFor each $j,$ the inverse of the matrix $\\scriptsize \\left[\\begin{array}{cc} U_j &  -1 \\\\ -L_j & 1 \\end{array}\\right]$ \\normalsize is $\\scriptsize \\frac{1}{d_j}\\left[\\begin{array}{cc} 1 &1 \\\\ L_j & U_j \\end{array}\\right],$ \\normalsize where $d_j=(U_j-L_j)$ denotes the difference between $U_j$ and $L_j.$ Then we apply \\eqref{kroneck2} to \\eqref{ca5} to rewrite the set $P$ as\n\\begin{alignat}{2} \nP=\\mbox{\\Huge \\{} \\normalsize &(\\x,\\x[w]) \\in \\real^n\\times\\real^{2^n-(n+1)}:  \\nonumber \\\\ \n&\\left\\{\\scriptsize\\left[\\begin{array}{c} 1 \\\\ x_1 \\end{array}\\right]\\normalsize \\otimes \\ldots \\otimes \\scriptsize \\left[\\begin{array}{c} 1 \\\\ x_n \\end{array}\\right]\\right\\}_L \\normalsize =   \\left(\\scriptsize \\frac{1}{d_1}\\left[\\begin{array}{cc} 1 &  1 \\\\ L_1 & U_1 \\end{array}\\right] \\normalsize \\otimes \\ldots \\otimes \\scriptsize \\frac{1}{d_n}\\left[\\begin{array}{cc} 1 &  1 \\\\ L_n & U_n \\end{array}\\right]\\right)\\normalsize\\x[\\lambda] \\nonumber \\\\ & \\mbox{for some nonnegative } \\x[\\lambda] \\in \\real^{2^n}\\mbox{\\Huge \\}} . && \\label{ca6} \n\\end{alignat}\n\nThe two claimed properties of $P$ become apparent in light of \\eqref{ca6}. The first equation of \\eqref{ca6} is equivalent via scaling to $D_n=\\sum_{j=1}^{2^n}\\lambda_j,$ with $D_n \\equiv \\prod_{j=1}^nd_j,$ thereby establishing $P$ as a polytope having exactly $2^n$ extreme points. These points can be numbered so that extreme point $j$ has $\\x[\\lambda]$ given by $\\lambda_j = D_n$ and $\\lambda_i=0$ for $i \\neq j,$ and $(\\x,\\x[w])$ given by column $j$ of the matrix $\\scriptsize \\left[\\begin{array}{cc} 1 &  1 \\\\ L_1 & U_1 \\end{array}\\right] \\normalsize \\otimes \\ldots \\otimes  \\scriptsize \\left[\\begin{array}{cc} 1 &  1 \\\\ L_n & U_n \\end{array}\\right], \\normalsize$ less the first row. Each of the $2^n$ columns of this last matrix consists of a distinct realization of the vector $\\scriptsize\\left[\\begin{array}{c} 1 \\\\ x_1 \\end{array}\\right]\\normalsize \\otimes \\ldots \\otimes \\scriptsize \\left[\\begin{array}{c} 1 \\\\ x_n \\end{array}\\right] \\normalsize,$ less the first entry, evaluated at some $\\x$ having each $x_j$ fixed at either its lower bound $L_j$ or its upper bound $U_j.$ \n\\end{proof}\n\n\\begin{proof}[\\textbf{Proof of \\mythref{obs1}}]\nInvertibility of the matrix $\\left(\\scriptsize\\left[\\begin{array}{cc} U_1 &  -1 \\\\ -L_1 & 1 \\end{array}\\right] \\normalsize \\otimes \\ldots \\otimes \\scriptsize \\left[\\begin{array}{cc} U_n &  -1 \\\\ -L_n & 1 \\end{array}\\right]\\right)$ from \\eqref{prodkron2} establishes the $2^n$ functions $F_{K}(x)$ of \\eqref{functions} as a basis for the vector space consisting of all multilinear polynomials having degree at most $n.$ This space clearly has dimension $2^n,$ and the invertibility demonstrates the functions to be linearly independent. As a result, associated with every multilinear polynomial $\\sum_{J \\subseteq N}\\alpha_j\\prod_{j \\in J}x_j,$ there exists a unique $\\x[\\pi] \\in \\real^{2^n}$ so that $p(x) = \\sum_{K \\subseteq N}\\pi_{K}F_{K}(x)$ holds. This identity can be readily solved in terms of $\\pi_K$ due to the structure of the functions $F_{K}(x)$ to obtain the claimed expression for $\\pi$.\n\\end{proof}\n\n\\begin{proof}[\\textbf{Proof of \\mythref{obs2}}]\nThe multipliers $\\pi_K$ of \\mythref{obs1} can be used to establish, in terms of the extreme points of the set $X^{\\prime}$, the nonnegativity of $\\sum_{J \\subseteq N}\\alpha_J\\prod_{j \\in J}x_j$ over $X^{\\prime},$ and the collection of points at which the polynomial vanishes. Clearly, such a polynomial is nonnegative over $X^{\\prime}$ if and only if $\\pi_K \\ge 0$ for all $K$. \\mythref{obs1} uniquely defines each such multiplier to be the positive scalar $\\frac{p(\\hat{\\x})}{F_{K}(\\hat{\\x})}$ evaluated at a distinct extreme point $\\hat{x}$. \n\\end{proof}\n\n\n\\subsection{Exactness of Core Inequalities}\n\nIn this section, we exploit the RLT properties to characterise the set of points $(\\x,y) \\in G$ at which each valid core inequality is satisfied exactly. We consider only core inequalities since other inequalities can be handled via permutations of the variables $x_j$ to coincide with the permutations of the coefficients $\\beta_j.$ We begin with three propositions and a theorem. The first proposition provides a necessary and sufficient condition for a point $\\tilde{\\x} \\in X$ to satisfy a function $F_{K}(x)$ of \\eqref{Fun} exactly, and the second gives three equivalent conditions for exactness to hold. The third proposition uses the RLT results of Observations 1 and 2 as a theoretical bridge to translate the conditions of the second proposition to that of satisfying a valid inequality for $\\conv{G}$ exactly. The theorem uses this third proposition to give a necessary and sufficient condition for a point $(\\tilde{\\x},\\tilde{y}) \\in G$ to satisfy a valid core inequality exactly. We subsequently present two special cases of the theorem as corollaries. Consistent with our earlier work, we then separately address the core facets when the SMP $m(\\x)$ is supermodular over the extreme points of $X$ of \\eqref{Xdef}, and when $m(\\x)$ is a monomial having $-\\ell=u>0.$ The two corollaries serve to simplify this analysis. Throughout, the inequalities $F_{K}(x) \\geq 0$ of \\eqref{RLTstep} are assumed to have $L_j=\\ell$ and $U_j=u$ for all $j \\in N$ within \\eqref{Fun} as in $X$ of \\eqref{Xdef}.   \n\n\\begin{lemma}\\thlabel{fite000}\nGiven any $K \\subseteq N$ and any $\\tilde{\\x} \\in X$ of \\eqref{Xdef}, partition $N$  into $N_1,$ $N_2,$ and $N_3$ so that $N_1 \\equiv\\{j:\\tilde{x}_j=\\ell\\},$ $N_2 \\equiv\\{j:\\tilde{x}_j=u\\},$ and $N_3 \\equiv\\{j:\\ell<\\tilde{x}_j<u\\}.$ Then $\\tilde{\\x}$ satisfies $F_{K}(x)=0$ if and only if $\\left([K \\cap N_1]  \\cup [(N\\setminus K) \\cap N_2]\\right) \\neq \\emptyset.$\n\\end{lemma}\n\\begin{proof}\nTrivial by the definition of the polynomial $F_{K}(x)$ in \\eqref{Fun}.\n\\end{proof}\n\n\\begin{proposition}\\thlabel{fite00}\nGiven any $K \\subseteq N$ and any $\\tilde{\\x} \\in X$ of \\eqref{Xdef}, partition $N$  into $N_1,$ $N_2,$ and $N_3$ so that $N_1 \\equiv\\{j:\\tilde{x}_j=\\ell\\},$ $N_2 \\equiv\\{j:\\tilde{x}_j=u\\},$ and $N_3 \\equiv\\{j:\\ell<\\tilde{x}_j<u\\}.$ The following statements are equivalent:\n\\begin{enumerate}\n\\item $\\tilde{\\x}$ satisfies $F_{K}(x)=0,$ \n\\item $\\hat{\\x}$ satisfies $F_{K}(x)=0$ at every $\\hat{\\x} \\in X$ having $\\hat{x}_j=\\ell \\; \\forall \\; j \\in N_1$ and $\\hat{x}_j=u\\; \\forall \\; j \\in N_2,$\n\\item $\\hat{\\x}$ satisfies $F_{K}(x)=0$ at every extreme point $\\hat{\\x}$ of $X$ having $\\hat{x}_j=\\ell \\; \\forall \\; j \\in N_1$ and $\\hat{x}_j=u\\; \\forall \\; j \\in N_2.$\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nGiven any $K \\subseteq N$ and any $\\tilde{\\x} \\in X$ with $N$ partitioned into $N_1,$ $N_2,$ and $N_3$ as described, the proof is to show that $1 \\rightarrow 2 \\rightarrow 3 \\rightarrow 1.$ The case $2 \\rightarrow 3$ is trivial and so we consider the remaining two cases. For convenience, we define the set $\\mathcal{N}_K \\subseteq N$ as $\\mathcal{N}_K\\equiv \\left([K \\cap N_1]  \\cup [(N\\setminus K) \\cap N_2]\\right).$ \\\\\n$\\left(1 \\rightarrow 2\\right)$ \n\\mythref{fite000} gives us that $\\tilde{\\x}$ satisfies $F_{K}(x)=0$ only if $\\mathcal{N}_K \\neq \\emptyset.$ But $\\mathcal{N}_K$ remains nonempty for every $\\hat{\\x} \\in X$ having $\\hat{x}_j=\\ell \\; \\forall \\; j \\in N_1$ and $\\hat{x}_j=u \\; \\forall \\; j \\in N_2$ so that the ``if\" condition of \\mythref{fite000} establishes the result. \\\\\n$\\left(3 \\rightarrow 1\\right)$ \nConsider that extreme point $\\hat{\\x}$ of $X$ having $\\hat{x}_j=\\ell \\; \\forall \\; j \\in N_1,$ $\\hat{x}_j=u \\; \\forall \\; j \\in N_2,$  $\\hat{x}_j=\\ell \\; \\forall \\; j \\in (N_3-K),$ and $\\hat{x}_j=u \\; \\forall \\; j \\in (K \\cap N_3).$ Then \n\\begin{equation}\nF_{K}(x)=0=(u-\\ell)^{(n-|\\mathcal{N}_K|)}\\left(\\prod_{j \\in (K \\cap N_1)}(\\hat{x}_j-\\ell)\\prod_{j \\in (N\\setminus K) \\cap N_2}(u-\\hat{x}_j)\\right)\\nonumber\n\\end{equation} \nwhen evaluated at $\\hat{\\x},$ implying that $\\mathcal{N}_K \\neq \\emptyset.$ Then the ``if\" condition of \\mythref{fite000} establishes the result.\n\\end{proof}\n\nThe result below mirrors that of \\mythref{fite00}, but invokes Observations 1 and 2 to extend the results from the functions $F_{K}(x)$ to the inequalities \\eqref{sum24}.\n\\begin{proposition}\\thlabel{fitt}\nGiven any inequality \\eqref{sum24} that is valid for $\\conv{G}$ and any $\\tilde{\\x} \\in X$ of \\eqref{Xdef}, partition $N$ into $N_1,$ $N_2,$ and $N_3$ so that $N_1 \\equiv\\{j:\\tilde{x}_j=\\ell\\},$ $N_2 \\equiv\\{j:\\tilde{x}_j=u\\},$ and $N_3 \\equiv\\{j:\\ell<\\tilde{x}_j<u\\}.$ The following statements are equivalent:\n\\begin{enumerate}\n\\item $(\\x,y)=(\\tilde{\\x},m(\\tilde{\\x}))$ satisfies \\eqref{sum24} exactly,\n\\item $(\\x,y)=(\\hat{\\x},m(\\hat{\\x}))$ satisfies \\eqref{sum24} exactly at every $\\hat{\\x} \\in X$ having $\\hat{x}_j=\\ell \\; \\forall \\; j \\in N_1$ and $\\hat{x}_j=u\\; \\forall \\; j \\in N_2,$\n\\item $(\\x,y)=(\\hat{\\x},m(\\hat{\\x}))$ satisfies \\eqref{sum24} exactly at every extreme point $\\hat{\\x}$ of $X$ having $\\hat{x}_j=\\ell \\; \\forall \\; j \\in N_1$ and $\\hat{x}_j=u\\; \\forall \\; j \\in N_2.$\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nGiven any $K \\subseteq N$ and any $\\tilde{\\x} \\in X$ with $N$ partitioned into $N_1,$ $N_2,$ and $N_3$ as described, the proof is to show that $1 \\rightarrow 2 \\rightarrow 3 \\rightarrow 1.$ The case $2 \\rightarrow 3$ is trivial and so we consider the remaining two cases. To begin, note that Observations 1 and 2 combine to show, given any inequality \\eqref{sum24} that is valid for $\\conv{G},$ there exists a unique, nonnegative $\\x[\\pi] \\in \\real^{2^n}$ satisfying $\\beta_0+\\sum_{j=1}^n\\beta_jx_j+\\beta^{\\prime}m(\\x)= \\prod_{K \\subseteq N}\\pi_KF_{K}(x)$ so that, given any $\\bar{\\x} \\in X,$\n\\begin{equation}\n\\beta_0+\\sum_{j=1}^n\\beta_j\\bar{x}_j+\\beta^{\\prime}m(\\bar{\\x})= 0 \\mbox{ if and only if every } \\pi_K>0 \\mbox{ has } F_{K}(x)=0 \\mbox{ at } \\bar{\\x}. \\label{newbeed}\n\\end{equation}\n$\\left(1 \\rightarrow 2\\right)$ Given $(\\x,y)=(\\tilde{\\x},m(\\tilde{\\x}))$ satisfies \\eqref{sum24} exactly, the ``only if\" direction of \\eqref{newbeed} with $\\bar{\\x}=\\tilde{\\x}$ gives us that every $\\pi_K>0$ has $F_{K}(x)=0$ at $\\tilde{\\x}.$ Implication $1 \\rightarrow 2$ of \\mythref{fite00} then gives us that every $\\pi_K>0$ has $F_{K}(x)=0$ at every $\\hat{\\x} \\in X$ having $\\hat{x}_j=\\ell \\; \\forall \\; j \\in N_1$ and $\\hat{x}_j=u\\; \\forall \\; j \\in N_2.$ Then the ``if\" direction of \\eqref{newbeed} with each such $\\hat{\\x}$ substituted for $\\bar{\\x}$ gives the result.\\\\\n$\\left(3 \\rightarrow 1\\right)$ Given $(\\x,y)=(\\hat{\\x},m(\\hat{\\x}))$ satisfies \\eqref{sum24} exactly at every extreme point $\\hat{\\x}$ of $X$ having $\\hat{x}_j=\\ell \\; \\forall \\; j \\in N_1$ and $\\hat{x}_j=u\\; \\forall \\; j \\in N_2,$ the ``only if\" direction of \\eqref{newbeed} with each such $\\hat{\\x}$ substituted for $\\bar{\\x}$ gives us that every $\\pi_K>0$ has $F_{K}(x)=0$ at every such $\\hat{\\x}.$ Implication $3 \\rightarrow 1$ of \\mythref{fite00} then gives us that every $\\pi_K>0$ has $F_{K}(x)=0$ at $\\tilde{\\x}.$ Then the ``if\" direction of \\eqref{newbeed} with $\\bar{\\x}=\\tilde{\\x}$ gives the result.\n\\end{proof}\n\nWe invoke \\mythref{foundation45} and \\mythref{fitt} to establish a theorem and two corollaries. The theorem gives, in terms of the extreme points of the simplex $\\mathcal{S}$ of \\eqref{simpleex}, a necessary and sufficient condition for a valid core inequality \\eqref{sum24} to be satisfied exactly at a point $(\\tilde{\\x},\\tilde{y})\\in G.$ This theorem is a generalization of \\mythref{foundation45} in that, by restricting $n=(p+b)$ within the theorem, the stated point $(\\tilde{\\x},\\tilde{y})\\in G$ must be the extreme point $(\\tilde{\\x},m(\\tilde{\\x}))\\in G$ of $\\conv{G}$ found within \\mythref{foundation45}. The corollaries are special cases of the theorem when the valid core inequality \\eqref{sum24}: is satisfied exactly at $(\\x^j,m(\\x^j))$ for all $j \\in \\{0, \\ldots, n\\},$ and has all coefficients $\\beta_j$ equal to the same scalar, say $\\bar{\\beta},$ respectively.\n\n\\begin{theorem}\\thlabel{exact1111}\nGiven a point $(\\tilde{\\x},\\tilde{y})\\in G$ with $\\tilde{\\x}$ not an extreme point of the simplex $\\mathcal{S}$ of \\eqref{simpleex}, let $r$ be the smallest index $j$ such that $\\tilde{x}_j<u$ and $s$ be the largest index $j$ such that $\\tilde{x}_j> \\ell.$ Further let $p$ be the number of entries of $\\tilde{\\x}$ having value $u$ and $b$ be the number of entries of $\\tilde{\\x}$ having value $\\ell.$ Then $(\\tilde{\\x},\\tilde{y})$ satisfies a valid core inequality \\eqref{sum24} exactly if and only if $(\\x,y)=(\\x^j,m(\\x^j))$ satisfies \\eqref{sum24} exactly for all $j \\in\\{p,\\ldots,n-b\\},$ and $\\beta_r=\\ldots=\\beta_s.$ \n\\end{theorem}\n\\begin{proof}\nIf $n=(p+b)$ so that all entries of $\\tilde{\\x}$ take either value $\\ell$ or $u,$ then $(\\tilde{\\x},\\tilde{y})\\in G$ is the extreme point $(\\tilde{\\x},m(\\tilde{\\x}))$ of $\\conv{G},$ so the result is \\mythref{foundation45}. Otherwise, $n \\geq (p+b+1),$ and we adopt the notation of \\mythref{fitt} that $N_1 \\equiv\\{j:\\tilde{x}_j=\\ell\\},$ $N_2 \\equiv\\{j:\\tilde{x}_j=u\\},$ and $N_3 \\equiv\\{j:\\ell<\\tilde{x}_j<u\\},$ so that $|N_1|=b,$ $|N_2|=p,$ and $|N_3|=n-(p+b).$ For any chosen $k \\in\\{0,\\ldots,n-(p+b)\\},$ define a new point $\\hat{\\x}$ in terms of $\\tilde{\\x}$ by setting $k$ values of $\\hat{x}_j$ with $j \\in N_3$ to the value $u$ and the remaining $(n-(p+b)-k)$ values of $\\hat{x}_j$ with $j \\in N_3$ to the value $\\ell,$ and by setting $\\hat{x}_j=\\tilde{x}_j$ for all $j \\in (N_1 \\cup N_2).$ Then, if $\\hat{\\x} \\neq \\x^{k+p},$ \\mythref{foundation45} gives us that $(\\x,y)=(\\hat{\\x},m(\\hat{\\x}))$ satisfies \\eqref{sum24} exactly if and only if $(\\x^{k+p},m(\\x^{k+p}))$ satisfies \\eqref{sum24} exactly, and $\\beta_{\\hat{r}}=\\ldots=\\beta_{\\hat{s}},$ where $\\hat{r}$ is the smallest index $j$ such that $\\hat{x}_j = \\ell$ and $\\hat{s}$ is the largest index $j$ such that $\\hat{x}_j=u.$ Consequently, by considering all $k \\in\\{0,\\ldots,n-(p+b)\\}$ and all corresponding ${n-(p+b)} \\choose {k}$ possible fixings of $\\hat{x}_j$ for $j \\in N_3,$ $(\\x,y)=(\\hat{\\x},m(\\hat{\\x}))$ will satisfy \\eqref{sum24} exactly at every extreme point $\\hat{\\x}$ of $X$ having $\\hat{x}_j= \\ell$ for all $j \\in N_1$ and $\\hat{x}_j= u$ for all $j \\in N_2$ if and only if $(\\x,y)=(\\x^j,m(\\x^j))$ satisfies \\eqref{sum24} exactly for all $j \\in\\{p,\\ldots,n-b\\},$ and $\\beta_r=\\ldots=\\beta_s.$ The ``only if\" consequence that $\\beta_r=\\ldots=\\beta_s$ is due to the following: if $r=s,$ the result is trivial while if $r<s,$ then any $\\hat{\\x}$ defined as above so that $\\hat{x}_r=\\ell$ and $\\hat{x}_s=u,$ regardless of $k,$ will yield $r=\\hat{r}$ and $s=\\hat{s}.$ Then implication $3 \\rightarrow 1$ of \\mythref{fitt} gives us that $(\\tilde{\\x},\\tilde{y})$ satisfies \\eqref{sum24} exactly, as desired.\n\\end{proof}\n\n\\mythref{exact1111} simplifies when the valid core inequality \\eqref{sum24} is satisfied exactly at $(\\x^j,m(\\x^j))$ for all $j \\in \\{0, \\ldots, n\\}.$ In this case, the parameters $p$ and $b$ within the theorem are no longer needed, as they serve only to restrict the points $(\\x^j,m(\\x^j))$ that must satisfy the inequality exactly. The simplification is below.\n\n\\begin{corollary}\\thlabel{exactly1110}\nGiven a point $(\\tilde{\\x},\\tilde{y})\\in G$ with $\\tilde{\\x}$ not an extreme point of the simplex $\\mathcal{S}$ of \\eqref{simpleex}, let $r$ be the smallest index $j$ such that $\\tilde{x}_j<u$ and $s$ be the largest index $j$ such that $\\tilde{x}_j> \\ell.$ Given a valid core inequality \\eqref{sum24} that is satisfied exactly at $(\\x^j,m(\\x^j))$ for all $j \\in \\{0, \\ldots, n\\},$ the point $(\\tilde{\\x},\\tilde{y})$ satisfies this inequality exactly if and only if $\\beta_r=\\ldots=\\beta_s.$ \n\\end{corollary}\n\n\\mythref{exact1111} also simplifies when the valid core inequality \\eqref{sum24} has, for some scalar $\\bar{\\beta},$ $\\beta_j=\\bar{\\beta}$ for all $j \\in N.$ In this case, the parameters $r$ and $s$ within the theorem are no longer needed, as they serve only to restrict a subset of the coefficients $\\beta_j$ to equal. The simplification is stated formally below.\n\n\\begin{corollary}\\thlabel{exactly1111}\nGiven a point $(\\tilde{\\x},\\tilde{y})\\in G$ with $\\tilde{\\x}$ not an extreme point of the simplex $\\mathcal{S}$ of \\eqref{simpleex}, let $p$ be the number of entries of $\\tilde{\\x}$ having value $u$ and $b$ be the number of entries of $\\tilde{\\x}$ having value $\\ell.$ Then $(\\tilde{\\x},\\tilde{y})$ satisfies a valid core inequality \\eqref{sum24} having $\\beta_j=\\bar{\\beta}$ for all $j \\in N$ exactly if and only if $(\\x,y)=(\\x^j,m(\\x^j))$ satisfies \\eqref{sum24} exactly for all $j \\in\\{p,\\ldots,n-b\\}.$ \n\\end{corollary}\n\n\\section{Supermodular Functions}\t\\label{sec:supermod}\n\nIn this subsection, we describe $\\conv{G}$ for SMPs $m(\\x)$ that are \\emph{supermodular} over the extreme points of $X$. Such a function $m(\\x)$ over these $2^n$ points can be expressed as a set function $f(A)$ defined over $A \\subseteq N$ so that $f(A)=m(\\x)$ when evaluated at that extreme point $\\x$ having $x_j=u$ for $j \\in A$ and $x_j=\\ell$ for $j \\notin A.$  Recall that a set function $f$ is defined to be supermodular over $N$ if and only if\n\\begin{equation}\nf(S \\cup \\{r\\})-f(S) \\leq f(S \\cup \\{r,t\\})-f(S \\cup \\{t\\}) \\; \\mbox{ for } r,t \\in N, r \\neq t, \\mbox{ and }S\\subseteq N \\backslash \\{r,t\\}, \\label{rest0777}\n\\end{equation}\nwe have that the function $m(\\x)$ is supermodular over the $2^n$ extreme points of $X$ if and only if \n\\begin{equation}\nm(\\x^{k})-m(\\x^{k-1}) \\leq m(\\x^{k+1})-m(\\x^{k}), \\quad  \\forall \\; k \\in \\{1, \\ldots, n-1\\}. \\label{rest0}\n\\end{equation}\nThis equivalence follows by considering, for each $k \\in \\{1, \\ldots, n-1\\},$ all sets $S \\subseteq N$ within \\eqref{rest0777} having $|S|=(k-1),$ and all $\\{r,t\\} \\in N \\backslash S,$ and by invoking the symmetry of $m(\\x)$ to obtain that $f(T)=m(\\x^k)$ for all $T \\subseteq N$ with $|T|=k.$ The function $m(\\x)$ is \\emph{strictly supermodular} over the $2^n$ extreme points of $X$ if and only if the $(n-1)$ inequalities of \\eqref{rest0} are satisfied strictly. Henceforth, for brevity, we will refer to functions $m(\\x)$ that are (strictly) supermodular over the $2^n$ extreme points of $X$ as being (strictly) supermodular.\n\nThe below theorem explicitly states all core facets for supermodular $m(\\x).$\n\n\\begin{theorem} \\thlabel{result1}\nWhen $m(\\x)$ is supermodular, there exist at most $(n+1)$ core facets for $\\conv{G},$ and all such facets, subject to repetition, are \n\\begin{equation}\n\\frac{um(\\x^0)-\\ell m(\\x^n)}{u-\\ell}+\\sum_{j=1}^n\\left(\\frac{m(\\x^j)-m(\\x^{j-1})}{u-\\ell}\\right)x_j-y \\geq 0, \\label{res1}\n\\end{equation}\nand\n\\begin{equation}\n-m(\\x^k) - \\left(\\frac{m(\\x^{k})-m(\\x^{k-1})}{u-\\ell}\\right)\\left(\\sum_{j=1}^nx_j-[ku+(n-k)\\ell]\\right)+y \\geq 0, \\quad \\forall \\; k \\in N. \\label{res2}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe $(n+1)$ inequalities of \\eqref{res1} and \\eqref{res2} are valid for $\\conv{G}$ by \\mythref{foundation}, as they are readily verified to hold for all $(\\x^k,m(\\x^k)), k \\in\\{0, \\ldots, n\\}.$ Inequality~\\eqref{res1} is a facet because these same $(n+1)$ affinely independent points satisfy it exactly, and it is a core facet because the $\\beta_j$ are nondecreasing by \\eqref{rest0}. In addition, no other core facet can exist with $\\beta^{\\prime}=-1$ by \\mythref{foundation24} because the functions ${\\cal L}_k(\\beta_0,\\x[\\beta])$ from \\eqref{anoto25}, with $(\\beta_0,\\x[\\beta])$ defined in terms of \\eqref{res1}, have ${\\cal L}_k(\\beta_0,\\x[\\beta])=m(\\x^k)$ for all $k \\in \\{0, \\ldots, n\\}.$ Relative to \\eqref{res2}, for each $k \\in N,$ the corresponding inequality is a core facet by \\mythref{equalcoeff}, as it is satisfied exactly at the two points $(\\x^{k-1},m(\\x^{k-1}))$ and $(\\x^k,m(\\x^k)).$ To show that no other core facet can exist with $\\beta^{\\prime}=1$ and complete the proof, it is sufficient to show that every core facet $\\beta_0+\\sum_{j=1}^n\\beta_jx_j+y \\geq 0$ which is satisfied exactly at some $(\\x^{r},m(\\x^{r}))$ and $(\\x^{s},m(\\x^{s}))$ with $r<s,$ is also satisfied exactly at $(\\x^{r+1},m(\\x^{r+1})).$ Then induction has the core facet being satisfied exactly at $(\\x^{p},m(\\x^{p}))$ for all $p \\in \\{r, \\ldots, s\\}$ so that, for each $p \\in \\{r+1, \\ldots, s\\},$ by inserting $(\\x^{p-1},m(\\x^{p-1}))$ and $(\\x^{p},m(\\x^{p}))$ into the facet and subtracting the first expression from the second, we obtain $\\beta_{p}(u-\\ell)=-\\left(m(\\x^{p})-m(\\x^{p-1})\\right).$ As the $\\beta_p$ are nondecreasing and the values $-\\left(m(\\x^{p})-m(\\x^{p-1})\\right)$ are nonincreasing, it will then follow that $\\beta_{p}(u-\\ell)=-\\left(m(\\x^{r+1})-m(\\x^r)\\right)$ for all $p \\in \\{r+1, \\ldots, s\\}.$ Hence, by selecting $r=\\mbox{argmin}_{k \\in \\{0, \\ldots, n-1\\}}\\left\\{{\\cal L}_k(\\beta_0,\\x[\\beta]) +m(\\x^k)=0\\right\\}$ and $s=\\mbox{argmax}_{k \\in \\{1, \\ldots, n\\}}\\left\\{{\\cal L}_k(\\beta_0,\\x[\\beta]) + m(\\x^k)=0\\right\\}$, \\mythref{equalcoeff8} will give us that $\\beta_j=-\\left(\\frac{m(\\x^{r+1})-m(\\x^r)}{u-\\ell}\\right)$ for all $j \\in N.$ Then the chosen core facet must be inequality \\eqref{res2} with $k=(r+1).$ \n\nTo show that the core facet is satisfied exactly at $(\\x^{r+1},m(\\x^{r+1})),$ we restrict attention to $s \\geq (r+2)$ since the result is trivial for $s=(r+1).$ We have that\n\n\\begin{eqnarray*}\nm(\\x^{r+1}) &\\le& m(\\x^{r})+\\frac{m(\\x^s)-m(\\x^r)}{s-r} \\\\\n&=&-\\beta_0-u\\left(\\sum_{j=1}^{r}\\beta_j\\right)-\\ell\\left(\\sum_{j=r+1}^{n}\\beta_j\\right)-\\left(\\frac{u-\\ell}{s-r}\\right)\\left(\\sum_{j=r+1}^{s}\\beta_j\\right) \\\\\n&=&-\\beta_0-u\\left(\\sum_{j=1}^{r+1}\\beta_j\\right)-\\ell\\left(\\sum_{j=r+2}^{n}\\beta_j\\right)-\\left(\\frac{u-\\ell}{s-r}\\right)\\left(\\sum_{j=r+1}^{s}(\\beta_j-\\beta_{r+1})\\right) \\\\\n&\\leq& -\\beta_0-u\\left(\\sum_{j=1}^{r+1}\\beta_j\\right)-\\ell\\left(\\sum_{j=r+2}^{n}\\beta_j\\right) \\\\\n&\\leq& m(\\x^{r+1}),\n\\end{eqnarray*}\nwhere the first inequality follows from summing the $(s-r)$ inequalities $m(\\x^{r+1})-m(\\x^r) \\leq m(\\x^{k+1})-m(\\x^k)$ for $k \\in \\{r, \\ldots, s-1\\}$ that are implied by \\eqref{rest0} for $k>r,$ the first equality follows from the facet holding exactly at $(\\x^r,m(\\x^r))$ and $(\\x^{s},m(\\x^{s})),$ the second equality is algebra, the second inequality follows from the nondecreasing values of $\\beta_j,$ and the final inequality follows from the feasibility of $(\\x^{r+1},m(\\x^{r+1}))$ to $\\conv{G}.$ Then $m(\\x^{r+1})=-\\beta_0-u\\left(\\sum_{j=1}^{r+1}\\beta_j\\right)-\\ell\\left(\\sum_{j=r+2}^{n}\\beta_j\\right)$ so that the core facet is satisfied exactly at $(\\x^{r+1},m(\\x^{r+1})).$ The proof is complete.\n\\end{proof}\n\n\\mythref{result1} states that there exist \\emph{at most} $(n+1)$ core facets because the inequalities of \\eqref{res2} can repeat. Repetition will occur whenever $\\left(m(\\x^{p})-m(\\x^{p-1})\\right)=\\left(m(\\x^{q})-m(\\x^{q-1})\\right)$ for distinct $p,q \\in N.$ Notably, if $m(\\x)$ is strictly supermodular, then  repetitions will not occur so that \\eqref{res2} will contain $n$ distinct facets. \n\nSubject to permutations of $\\x[\\beta],$ inequalities \\eqref{res1} define the concave envelope of $m(\\x).$ \\akshay{These inequalities are a special case of the polymatroid inequalities that are known for general supermodular functions \\cite{lovasz1983submod,tawarmalani2013explicit}. Polymatroid inequalities are known to be separated easily in $O(n\\log{n})$ time using a sorting algorithm since \\citet{edmonds1970submodular} showed how to optimize a linear function over the polymatroid polyhedron corresponding to a submodular function.} Inequalities~\\eqref{res2} have equal coefficients on the variables and hence they do not need to be permuted and define the convex envelope of $m(\\x)$. \\akshay{Our proof for these inequalities is an alternative proof to that of \\citet[Theorem 4.6]{tawarmalani2013explicit}.} For a submodular function $m(\\x)$ (meaning that the inequality in \\eqref{rest0} is switched to $\\ge$), inequalities~\\eqref{res1}, subject to permutations of $\\x[\\beta],$ define the convex envelope, and inequalities~\\eqref{res2} define the concave envelope subject to the following modifications:  all occurrences of $m(\\x^j),$ $j \\in \\{0, \\ldots, n\\},$ and $y$ are negated.\n\n\nNow we use \\mythref{exactly1110,exactly1111} to identify, for each of the $(n+1)$ core facets \\eqref{res1} and \\eqref{res2} of \\mythref{result1}, the set of all points $(\\x,y) \\in G$ that satisfies it exactly. A key ingredient  of \\mythref{result1} that is useful in our upcoming proof is that every core facet $\\beta_0+\\sum_{j=1}^n\\beta_jx_j+y \\geq 0$ of the form \\eqref{res2} which is satisfied exactly at some $(\\x^{r},m(\\x^{r}))$ and $(\\x^{s},m(\\x^{s}))$ with $r<s,$ is also satisfied exactly at $(\\x^{p},m(\\x^{p}))$ for all $p \\in \\{r, \\ldots, s\\}.$ We then have that the set of points $(\\x^j,m(\\x^j))$ which satisfies the facet exactly must be consecutive. \n\n\\begin{theorem}\t\\thlabel{newsuper100}\nLet $m(\\x)$ be supermodular and $(\\x,y)\\in G$.\n\\begin{enumerate}\n\\item $(\\x,y)$ satisfies \\eqref{res1} exactly if and only if $\\x=\\x^{k}$ for some $k$, or $\\beta_{r}=\\cdots=\\beta_{s}$ where $r=\\min\\{j\\colon x_{j}<u\\}$ and $s = \\max\\{j\\colon x_{j} > \\ell\\}$.\n\\item $(\\x,y)$ satisfies \\eqref{res2} exactly if and only if $(\\x^{j},m(\\x^{j}))$ satisfies \\eqref{res2} exactly for $j=p,\\dots,n-q$, where $p=|\\{j\\colon x_{j}=u \\} |$ and $q=|\\{j\\colon x_{j}=\\ell \\} |$.\n\\item $(\\x,y)$ satisfies \\eqref{res2} exactly if and only if $|\\{j\\colon x_{j}=u \\}| \\ge v$ and $|\\{j\\colon x_{j}=\\ell \\}| \\ge n-w$, where $v$ and $w$ are the smallest and largest, respectively, values of $j$ such that the point $(\\x^{j},m(\\x^{j}))$ satisfies \\eqref{res2} exactly.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n(1) As noted in the proof of \\mythref{result1}, the points $(\\x^k,m(\\x^k))$ satisfy \\eqref{res1} exactly for all $k \\in \\{0, \\ldots, n\\}.$ Then the result trivially holds true when ${\\x}$ is an extreme point of $\\mathcal{S},$ and it holds true when ${\\x}$ is not an extreme point of $\\mathcal{S}$ by \\mythref{exactly1110}.\n\n(2) The result trivially holds true when ${\\x}$ is an extreme point of $\\mathcal{S},$ and it holds true when ${\\x}$ is not an extreme point of $\\mathcal{S}$ by \\mythref{exactly1111} with $\\bar{\\beta}=-\\left(\\frac{m(\\x^k)-m(\\x^{k-1}}{u-\\ell}\\right).$\n\n(3) This is a restatement of above using Remark 6 of \\mythref{result1} which states that the set of extreme points of $\\mathcal{S}$ satisfying the facet exactly must be consecutive.\n\\end{proof}\n\n\nMore refined characterizations for exactness hold when the function is strictly supermodular.\n\n\\begin{corollary} \\thlabel{exactstrictsupermod}\nLet $m(\\x)$ be strictly supermodular and $(\\x,y)\\in G$.\n\\begin{enumerate}\n\\item $(\\x,y)$ satisfies \\eqref{res1} exactly if and only if $\\x$ is a convex combination of two consecutive extreme points $\\x^j$ and $\\x^{j+1}$ of $\\mathcal{S}.$\n\\item $(\\x,y)$ satisfies \\eqref{res2} exactly if and only if $|\\{j\\colon x_{j}=u_{j} \\} | \\ge k-1$ and $| \\{j\\colon x_{j} = \\ell \\}| \\ge n-k$.\n\\end{enumerate} \n\\end{corollary}\n\\begin{proof}\n(1) Follows directly from \\mythref{newsuper100} since the coefficients $\\beta_j= \\left(\\frac{m(\\x^j)-m(\\x^{j-1})}{u-\\ell}\\right)$ for all $j \\in N$ of \\eqref{res1} are distinct, as every inequality within \\eqref{rest0} is satisfied strictly for strictly supermodular $m(\\x).$\n\n(2) Consider any $k \\in N.$ As noted in the proof of \\mythref{result1} and readily verified, $(\\x,y)=(\\x^{k-1},m(\\x^{k-1}))$ and $(\\x,y)=(\\x^k,m(\\x^k))$ each satisfy the core facet exactly. Thus, it is sufficient to show that supermodular $m(\\x)$ enforces $r=(k-1)$ and $s=k$ in \\mythref{newsuper100}. By contradiction, suppose that $s \\geq (r+2)$ so that each of $(\\x^{r},m(\\x^{r})),$ $(\\x^{r+1},m(\\x^{r+1})),$ and $(\\x^{r+2},m(\\x^{r+2}))$ satisfy the facet exactly. As in the proof of \\mythref{result1}, for each $p \\in \\{r+1,r+2\\},$ by inserting $(\\x^{p-1},m(\\x^{p-1}))$ and $(\\x^p,m(\\x^p))$ into the facet and subtracting the second expression from the first, we obtain\n\\begin{equation}\n \\frac{-\\left(m(\\x^{r+1})-m(\\x^r)\\right)}{u-\\ell}=\\beta_{r+1}=\\beta_{r+2}=\\frac{-\\left(m(\\x^{r+2})-m(\\x^{r+1})\\right)}{u-\\ell},\\nonumber\n\\end{equation}\nwhere $\\beta_{r+1}=\\beta_{r+2}$ is due to the form of \\eqref{res2}. The strict inequality of \\eqref{rest0} for supermodular $m(\\x)$ with $k=(r+1)$ yields the contradiction that $\\beta_{r+1} > \\beta_{r+2}.$\n\\end{proof}\n\nWe finish this section by remarking on supermodularity of an SMP, giving us some important families of functions $m(\\x)$ so that \\mythref{result1} characterizes their convex hull. First we provide an alternate characterization to~\\eqref{rest0} by expressing these requirements in terms of the coefficients $c_{j}$ of the function $m$.\n\n\\begin{proposition}\t\\thlabel{supermodcond}\n$m(\\x)$ is supermodular over $X$ if and only if $\\sum_{d=2}^{n}c_{d}(u-\\ell)^{2}\\vartheta_{k,d} \\ge 0$ for $k=1,\\dots,n-1$, where $\\vartheta_{2,2} = 1$ and\n\\[\n\\vartheta_{k,d} = \\sum_{\\substack{J \\subseteq N\\setminus\\{k,k+1\\} \\\\ |J|=d-2}}\\,\\prod_{j \\in J}x^{k-1}_j, \\qquad d = 3,\\dots,n,\\ k =1,\\dots,n-1,\n\\]\n\\end{proposition}\n\\begin{proof}\nFor each $k \\in \\{1, \\ldots, n-1\\},$ the difference $m(\\x^{k+1})-m(\\x^{k})-\\left(m(\\x^{k})-m(\\x^{k-1})\\right)$ is computable in terms of only those expressions $\\prod_{j \\in J}x_j$ within $m(\\x)$ that contain both $k \\in J$ and $\\{k+1\\} \\in J.$ For $d=2,$ this difference for the $n \\choose d$ expressions in $m(\\x)$ of degree $d$ is given by $c_2(u-\\ell)^2,$ while for each $d \\in \\{3, \\ldots, n\\},$ it is given by\n\\begin{equation} \nc_d(u-\\ell)^2\\left(\\sum_{\\substack{J \\subseteq N-\\{k,k+1\\} \\\\ |J|=d-2}}\\left(\\prod_{j \\in J}x^{k-1}_j\\right)\\right). \\label{newbee}\n\\end{equation}\nThen \\eqref{rest0} is equivalent to having the sum of these expressions from 2 to $n$ being nonnegative for all $k \\in \\{1, \\ldots, n-1\\}.$ \n\\end{proof}\n\n\\akshay{\nA multilinear monomial $x_{1}x_{2}\\cdots x_{n}$ is supermodular over the nonnegative orthant \\citep[cf.][]{tawarmalani2013explicit}. Since supermodularity is preserved under taking nonnegative combinations of functions, it follows that an SMP with $c_{j}\\ge 0$ for all $j$ is supermodular over $\\real^{n}_{+}$. This fact for nonnegative valued SMPs also follows from our characterization.}\n\n\\begin{corollary}\t\\thlabel{nonnegsupermod}\n$m(\\x)$ is supermodular over $X$ if $\\ell \\ge 0$ and $c_{j} \\ge 0$ for all $j=2,\\dots,n$.\n\\end{corollary}\n\\begin{proof}\n$\\ell\\ge 0$ implies that $\\x^{k}_{j} \\ge 0$ for all $j,k$, which implies $\\vartheta_{k,d}\\ge 0$ for all $k,d$. The summation in \\mythref{supermodcond} is nonnegative because $c_{j}\\ge 0$ for all $j$.\n\\end{proof}\n\nAnother consequence is that symmetric quadratic polynomials are always either submodular or supermodular, regardless of the box in $\\real^{n}$.\n\n\\begin{corollary}\nThe symmetric quadratic polynomial $\\tau\\sum_{i\\neq j}x_{i}x_{j}$ is either submodular or supermodular over $X$ for any $\\tau\\neq 0$.\n\\end{corollary}\n\\begin{proof}\nA symmetric quadratic polynomial is $m(\\x)$ with $c_{3} = \\cdots = c_{n} = 0$, and $c_{2}=\\tau$. Since all monomials are of the same degree 2, we can assume wlog that $\\ell\\ge 0$ because otherwise we can negate all the variables and consider the reflected box $X' = \\{x'\\sep -u_{j}\\le x'_{j} \\le -\\ell_{j} \\}$. Applying \\mythref{nonnegsupermod} to $m(\\x)$ if $\\tau > 0$ or to $-m(\\x)$ if $\\tau < 0$ yields the desired claim. \n\\end{proof}\n\nWhen considering the unit hypercube in $\\real^{n}$, supermodularity is attained through nonnegativity of a partial sum. Denote ${0 \\choose 0}=1$.\n\n\\begin{corollary}\n$m(\\x)$ is supermodular over $[0,1]^{n}$ if and only if\n\\begin{equation}\n\\sum_{d=2}^{k+1}{{k-1} \\choose {d-2}}c_d \\geq 0, \\quad \\forall \\; k \\in \\{1, \\ldots, n-1\\}. \\label{newbee2}\n\\end{equation} \n\\end{corollary}\n\\begin{proof}\nFollows immediately by simplifying the sum in~\\eqref{newbee}.\n\\end{proof}\n\n\n\nMore general families of SMPs are encompassed by \\eqref{newbee2} including, for example, those having $c_d \\geq 0$ for $d\\in\\{2, \\ldots, n-1\\}$ and $c_n \\geq -\\sum_{d=2}^{n-1}{{n-2} \\choose {d-2}}c_d,$ which reduces to $c_2 \\geq \\mbox{max}\\{0,(2-n)c_3\\}$ for cubic functions.\n\n\n\\section{Monomials with Reflection Symmetry\n\n\\akshay{We consider $m(\\x)$ to be a monomial $m(\\x)=c_n\\prod_{j=1}^n x_j$ over the box $X$. The coefficient $c_{n}$ can be taken to be 1 since we can scale $y$ to $y\/c_{n}$. Three cases arise depending on the location of the box in $\\real^{n}$: (1) $\\ell u = 0$, (2) $\\ell u > 0$, and (3) $\\ell u < 0$. After performing appropriate scalings, the first two cases are equivalent to that of $\\ell = 0, u =1$ and $\\ell = 1, u = r$ for some fixed $r > 1$, respectively. These two cases fall within the class of supermodular SMPs (cf.~\\mythref{nonnegsupermod}), and so \\mythref{result1} gives their convex hull. Indeed, it is easily verified that the resulting description of the convex hull matches the known envelopes for $\\prod_{j=1}^{n} x_{j}$ over $[0,1]^{n}$ \\cite{crama1993concave} and over $[1,r]^{n}$ for some fixed $r > 1$ (see \\cite[Proposition 4.1]{monomerr} which is a direct consequence of results from \\cite[Theorem 1]{benson2004concave} and \\cite[Theorem 4.6]{tawarmalani2013explicit}). The third case $\\ell u < 0$ is equivalent (upto scaling) to taking $\\ell = -1$ and $u = r$ for some $r > 0$ so that $X = [-1,r]^{n}$, and it is readily verified that a monomial is neither submodular nor supermodular over $[-1,r]^{n}$. Note that since we are considering monomials, scaling means that if each variable $x_{j}$ has different lower and upper bounds $\\ell_{j}$ and $u_{j}$, then as long as $u_{j} = -\\ell_{j}$ we can reduce to the case $X = [-1,r]^{n}$. The convex hull for the specific subcase having $r=1$ was first established by the authors in \\citep[Theorem 4.1]{monomerr}.  However, it was done so without using the symmetry of the monomial and hence did not recognize the core facets. Our main goal in this section is to independently describe this convex hull by identifying the core facets and their properties. \n}\n\n\n\n\\begin{theorem}\t\\thlabel{resulting3}\nFor $m(\\x)=\\prod_{j=1}^nx_j$ and $-\\ell=u=1,$ there exist precisely $n+3$ core facets, and these are \n\\begin{align}\n&1-y \\geq 0 \\mbox{ and } 1+y \\geq 0 \\label{trivial} \\\\\n&(n-1)+\\sum_{j=1}^nx_j+(-1)^ny \\geq 0, \\label{pair1} \\\\\n&(n-1)-\\sum_{j=1}^nx_j+y \\geq 0. \\label{pair2}\\\\\n&(n-1)-\\left(\\sum_{j=1}^{t+1}x_j-\\sum_{j=t+2}^nx_j\\right) +(-1)^{n-(t+1)}y \\geq 0, \\quad \\forall \\; t \\in \\{0, \\ldots, n-2\\}.  \\label{dominate2}\n\\end{align}\n\\end{theorem}\n\n\\akshay{The general case of convexifying $\\prod_{j=1}^{n}x_{j}$ over $[-1,r]^{n}$ for $r > 0, r \\neq 1$ remains an open question. Before giving our proof for the above theorem, let us note the complexity of separation and comment on the structure of the proposed inequalities which includes identifying exactness of the core facets. }\n\n\\begin{corollary}\nFor $m(\\x)=\\prod_{j=1}^nx_j$ and $-\\ell=u=1,$ a point can be separated from $\\conv{G}$ in $O(n\\log{n})$ time.\n\\end{corollary}\n\\begin{proof}\n\\mythref{resulting3} tells us that there are $t=n+3$ core facets. \\mythref{sepcompl} then implies that the complexity of separation is $O(n^{2} + n\\log{n})$. In the proof of this result, the complexity $O(nt)$ came from having to take a summation of $n$ terms while separating each of the $t$ core facets. Since the summation in both \\eqref{pair1} and \\eqref{pair2} is the same for any permutation of variables and therefore can be stored and reused, we get that the overall complexity is $O(t+n\\log{n}) = O(n\\log{n})$.\n\\end{proof}\n\n\n\\subsection{Structure of the inequalities}\n\nAll facets for $\\conv{G}$ are computable in terms of the core facets enumerated in \\mythref{resulting3} via permutations of $\\x[\\beta].$ Since each of the four core facets of \\eqref{trivial}, \\eqref{pair1}, and \\eqref{pair2} have all $\\beta_j$ equal, no permutations of $\\x[\\beta]$ will produce additional facets. However, for each $t \\in \\{0, \\ldots, n-2\\},$ the core facet \\eqref{dominate2} admits ${n} \\choose {t+1}$ facets. Then \\eqref{pair1}, \\eqref{pair2}, and \\eqref{dominate2} motivate the $2^n$ inequalities\n\\begin{equation}\n(n-1)-\\left(\\sum_{j \\in J}x_j-\\sum_{j \\in N\\setminus J}x_j\\right)+(-1)^{n-|J|}y \\geq 0, \\quad \\forall \\; J \\subseteq N,  \\label{dominate3}\n\\end{equation}\nwhere \\eqref{dominate3} with $J = \\emptyset$ is \\eqref{pair1}, where \\eqref{dominate3} with $J=N$ is \\eqref{pair2}, and where \\eqref{dominate3} for each $J \\subseteq N$ with $|J| \\in \\{1, \\ldots, n-1\\}$ are the ${n \\choose |J|}$ facets obtained by permutations of $\\x[\\beta]$ in \\eqref{dominate2} when $(t+1)=|J|.$ The desired representation of $\\conv{G}$ is then the two inequalities $-1 \\leq y \\leq 1$ of \\eqref{trivial}, the $2^n$ inequalities of \\eqref{dominate3} and, for the case in which $n \\geq 3,$ the $2n$ inequalities $-1 \\leq x_j \\leq 1 \\; \\forall \\; j \\in N.$ We can combine and more succinctly write the $(2+2^n+2n)$ inequalities of \\eqref{trivial}, \\eqref{dominate3}, and the restrictions $-1 \\leq x_j \\leq 1$ for $j \\in N$ for the cases having $n \\geq 3,$ by letting the variable $x_{n+1}$ denote $y$ and by letting $N^{\\prime} = \\{1, \\ldots, n+1\\}.$ Using this new definition of variables, \\eqref{dominate3} can be partitioned in terms of the exponents $(n-|J|)$ as\n\\begin{equation}\n(n-1)-\\left(\\sum_{j \\in J}x_j-\\sum_{j \\in (N\\setminus J) \\cup \\{n+1\\}}x_j\\right) \\geq 0,\\quad \\forall \\; J \\subseteq N \\mbox{ with }n-|J| \\mbox{ even},  \\nonumber\n\\end{equation}\nand\n\\begin{equation}\n(n-1)-\\left(\\sum_{j \\in J\\cup \\{n+1\\}}x_j-\\sum_{j \\in N\\setminus J}x_j\\right) \\geq 0,\\quad \\forall \\; J \\subseteq N \\mbox{ with }n-|J| \\mbox{ odd}.  \\nonumber\n\\end{equation}\nConsequently, $-1 \\leq x_j \\leq 1$ for $j \\in N,$ \\eqref{trivial}, and \\eqref{dominate3} can be expressed as \\eqref{con258} and \\eqref{con269} below, where \\eqref{con258} encompasses $-1 \\leq x_j \\leq 1$ for $j \\in N$ and \\eqref{trivial}, and \\eqref{con269} encompasses the above two families of inequalities.\n\\begin{eqnarray}\n-1 \\leq x_j \\leq 1 \\; &\\forall \\; j \\in N^{\\prime} \\label{con258}\\\\\n(n-1)-\\left(\\sum_{j \\in J}x_j-\\sum_{j \\in N^{\\prime}\\setminus J}x_j\\right) \\geq 0, &\\quad \\forall \\;J \\subseteq N^{\\prime} \\mbox{ with } n+1-|J| \\mbox{ odd} \\label{con269}\n\\end{eqnarray}\nOf course, if $n=2,$ then \\eqref{con258} simplifies to $-1 \\leq x_{n+1} \\leq 1.$\n\nNow we use \\mythref{exact1111} and \\mythref{exactly1111} to identify, for each of the $(n+3)$ core facets \\eqref{trivial}, \\eqref{pair1}, \\eqref{pair2}, and \\eqref{dominate2} of \\mythref{resulting3}, the set of all points $(\\x,y) \\in G$ that satisfies it exactly. \n\n\\begin{theorem}\\thlabel{finalet}\nFor $m(\\x)=\\prod_{j=1}^nx_j$ and $-\\ell=u=1,$ $(\\tilde{\\x},\\tilde{y}) \\in G$ satisfies the core facet\n\\begin{enumerate}\n\t\\item \\eqref{trivial} exactly if and only if $\\tilde{\\x}$ contains\n\t\\begin{enumerate}\n\t\\item even number of entries of value $-1$ and the remaining entries of value 1 when $\\beta^{\\prime}=-1,$ \n\t\\item odd number of entries of value $-1$ and the remaining entries of value 1 when $\\beta^{\\prime}=1,$  \n\t\\end{enumerate}\n\t\\item \\eqref{pair1} exactly if and only if $\\tilde{\\x}$ contains at least $(n-1)$ entries of value $-1,$ \n\t\\item \\eqref{pair2} exactly if and only if $\\tilde{\\x}$ contains at least $(n-1)$ entries of value $1,$\n\t\\item \\eqref{dominate2} for $t \\in \\{0, \\ldots, n-2\\}$ if and only if $\\tilde{\\x}$ differs from $\\x^{t+1}$ in at most one entry.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nEach statement is considered separately. \n\\begin{enumerate}\n\\item Follows directly from the definition of $m(\\x)$ when $-\\ell =u=1.$ \n\\item \\mythref{foundation900} and its proof give us that \\eqref{pair1} is satisfied exactly at $(\\tilde{\\x},\\tilde{y})=(\\x^{k},m(\\x^{k}))$ for $k \\in \\{0,1\\},$ but at no other $k.$ Then \\mythref{exactly1111} with $\\bar{\\beta}=1$ and $\\tilde{\\x}$ having $p=0$ and $b=(n-1)$ gives the result.\n\\item \\mythref{foundation900} and its proof give us that \\eqref{pair2} is satisfied exactly at $(\\tilde{\\x},\\tilde{y})=(\\x^{k},m(\\x^{k}))$ for $k \\in \\{n-1,n\\},$ but at no other $k.$ Then \\mythref{exactly1111} with $\\bar{\\beta}=-1$ and $\\tilde{\\x}$ having $p=(n-1)$ and $b=0$ gives the result.\n\\item For each $t \\in \\{0, \\ldots, n-2\\},$ \\mythref{resulting} and its proof give us that the associated inequality of \\eqref{dominate2} is satisfied exactly at $(\\tilde{\\x},\\tilde{y})=(\\x^{k},m(\\x^{k}))$ for $k \\in \\{t,t+1,t+2\\},$ but at no other $k.$ Then \\mythref{exact1111} with $\\tilde{\\x}$ having $p=t$ and $b=(n-t-1)$ allows $\\tilde{\\x},$ with suitable $r$ and $s,$ to differ from $x^{t+1}$ in only a single entry $j$ less than or equal to $(t+1).$ Similarly, \\mythref{exact1111} with $\\tilde{\\x}$ having $p=(t+1)$ and $b=(n-t-1)$ allows $\\tilde{\\x},$ with suitable $r$ and $s,$ to differ from $x^{t+1}$ in only a single entry $j$ greater than or equal to $(t+1).$ Combining these outcomes gives the result.\\qedhere\n\\end{enumerate}\n\\end{proof}\n\n\n\\subsection{Proof of the Convex Hull}\n\nWe prove \\mythref{resulting3} using three main lemmas. Let us begin with two simple observations. First, there exist no facets \\eqref{sum24} with $\\beta^{\\prime}=0$ if $n=2,$ and that there exist $2n$ such facets of the form $-1 \\leq x_j \\leq 1$ for $j \\in N$ if $n \\geq 3.$ Second, the two inequalities of \\eqref{trivial} are core facets. They are trivially valid, and they are core facets by \\mythref{equalcoeff} because each has $\\beta_j=0$ for all $j \\in N,$ and each is satisfied exactly at two extreme points $\\x^k$ of $\\mathcal{S},$ with one point not being $\\x^0$ or $\\x^n;$ equality occurs at $\\x^{n-2}$ and $\\x^n$ for the left inequality and occurs at $\\x^{n-3}$ and $\\x^{n-1}$ for the right inequality. \n\nNow, to facilitate the derivation of the remaining core facets and thereby characterize $\\conv{G},$ we define, for each $(p,q),p<q,$ the value \n\\begin{equation}\nD(p,q)=2\\sum_{j=p+1}^q\\beta_j+\\beta^{\\prime}\\left(m(\\x^q)-m(\\x^p)\\right)\n\\nonumber \n\\end{equation} \nthat is computed by subtracting the left side of \\eqref{sum24} evaluated at $(\\x,y)=(\\x^{p},m(\\x^{p}))$ from the left side evaluated at $(\\x^{q},m(\\x^{q})),$ upon recalling that $-\\ell=u=1.$\n\nThe following result identifies two core facets in terms of the extreme points $\\x^k$ of $\\mathcal{S},$ and also eliminates, in terms of these same points, other inequalities from consideration. \n\n\\begin{lemma}\t\\thlabel{foundation900}\nThe only core facets that are satisfied exactly at $(\\x^k,\\mk)$ for two consecutive extreme points $\\x^k$ of the simplex $\\mathcal{S},$ and for no other such extreme points, are~\\eqref{pair1} and \\eqref{pair2}.\n\\end{lemma} \n\\begin{proof}\nSuppose that a core facet is satisfied exactly at $(\\x^t,m(\\x^t))$ and $(\\x^{t+1},m(\\x^{t+1}))$ for extreme points $\\x^t$ and $\\x^{t+1}$ of the simplex $\\mathcal{S},$ and for no other such extreme points. Then \\mythref{equalcoeff8} with $r=t$ and $s=(t+1)$ states that there exists a $\\bar{\\beta}$ so that $\\bar{\\beta}=\\beta_j$ for all $j \\in N.$ Three cases exist.\n\n\\begin{itemize}\n\\item $t \\in\\{1, \\ldots, n-2\\}.$ We have $D(t-1,t)=2\\bar{\\beta}+\\beta^{\\prime}\\left(m(\\x^{t+2})-m(\\x^{t+1})\\right) < 0$ because $m(\\x^{t})=m(\\x^{t+2})$ and $m(\\x^{t-1})=m(\\x^{t+1}),$ and we have $D(t+1,t+2)=2\\bar{\\beta}+\\beta^{\\prime}\\left(m(\\x^{t+2})-m(\\x^{t+1})\\right) > 0.$ Combining, $0 < 2\\bar{\\beta}+\\beta^{\\prime}\\left(m(\\x^{t+2})-m(\\x^{t+1})\\right)< 0,$ which is not possible.\n\\item\n$t=0.$ We have $D(0,1)=2\\bar{\\beta}+\\beta^{\\prime}\\left(2(-1)^{n+1}\\right)=0,$ with the first equality holding since $\\mk[0]=-m(\\x^{1})=(-1)^{n}.$ Then $\\bar{\\beta}=\\beta^{\\prime}(-1)^n$ and \\eqref{sum24} evaluated at $(\\x,y)=(\\x^0,\\mk[0])$ gives $\\beta_0=\\beta^{\\prime}\\left((-1)^n(n-1)\\right)$ because $\\mk[0]=(-1)^{n}.$ Inequality \\eqref{sum24} becomes \n\\begin{equation}\n\\beta^{\\prime}\\left((-1)^n(n-1)+(-1)^n\\sum_{j=1}^nx_j+y\\right) \\geq 0. \\nonumber\n\\end{equation} \nThis inequality is valid when $\\beta^{\\prime} = (-1)^n$ by the ``if\" direction of \\mythref{foundation} since it holds true at $(\\x,y)=(\\x^k,\\mk)$ for $k \\in \\{0, \\ldots, n\\},$ but it is invalid when $\\beta^{\\prime}=(-1)^{n+1}$ since it is then violated at $(\\x,y)=(\\x^k,\\mk)$ for $k \\in \\{2, \\ldots, n\\}.$ The inequality with $\\beta^{\\prime} = (-1)^n$ is \\eqref{pair1}, and is a facet by the ``if\" direction of \\mythref{equalcoeff} since it holds exactly at $(\\x,y)=(\\x^k, \\mk)$ for $k \\in \\{0,1\\}.$\n\\item\n$t=(n-1).$ We have $D(n-1,n)=2\\bar{\\beta}+2\\beta^{\\prime}=0,$ with the first equality holding since $-m(\\x^{n-1})=\\mk[n]=1.$  Then $\\bar{\\beta}=-\\beta^{\\prime}$ and \\eqref{sum24} evaluated at $(\\x,y)=(\\x^{n-1},m(\\x^{n-1}))$ gives $\\beta_0=\\beta^{\\prime}(n-1)$ because $m(\\x^{n-1})=-1.$ Inequality \\eqref{sum24} becomes \n\\begin{equation}\n\\beta^{\\prime}\\left((n-1)-\\sum_{j=1}^nx_j+y\\right) \\geq 0. \\nonumber\n\\end{equation} \nThis inequality is valid when $\\beta^{\\prime}=1$ by the ``if\" direction of \\mythref{foundation} since it holds true at $(\\x,y)=(\\x^k,\\mk)$ for $k \\in \\{0, \\ldots, n\\},$ but it is invalid when $\\beta^{\\prime}=-1$ since it is then violated at $(\\x,y)=(\\x^k,\\mk)$ for $k \\in \\{0, \\ldots, n-2\\}.$ The inequality with $\\beta^{\\prime} = 1$ is \\eqref{pair2}, and is a facet by the ``if\" direction of \\mythref{equalcoeff} since it holds exactly at $(\\x,y)=(\\x^{k}, \\mk)$ for $k \\in \\{n-1,n\\}.$ \\qedhere\n\\end{itemize}\n\\end{proof} \n\nThe below result builds further by providing, in terms of the extreme points $\\x^k$ of $\\mathcal{S},$ necessary conditions for valid core inequalities that are not found in \\eqref{trivial}, \\eqref{pair1}, or \\eqref{pair2} to be facets. \n\n\\begin{lemma}\t\\thlabel{foundation15}\nEvery core facet for $\\conv{G}$ which is not found in \\eqref{trivial}, \\eqref{pair1}, or \\eqref{pair2} is satisfied exactly at $(\\x^k,\\mk)$ for three consecutive extreme points $\\x^k$ of the simplex $\\mathcal{S},$ and for no other such extreme points.\n\\end{lemma}\n\\begin{proof}\n\\mythref{foundation22} gives us that a core facet is satisfied exactly at $(\\x^k,\\mk)$ for at least two extreme points of the simplex $\\mathcal{S}.$ Let $r$ and $s$ be \nas in \\mythref{equalcoeff8}. Let $d=(s-r) \\geq 1$ denote the difference between $s$ and $r.$ \\mythref{foundation900} exhausts the cases having $d=1,$ and so it is sufficient to show three results: a core facet having $d=2$ that is not of the form \\eqref{trivial} must be satisfied exactly at $(\\x,y)=(\\x^{r+1},\\mk[r+1]),$ there exists no valid inequality having $d \\geq 3$ and $d$ odd, and a core facet having $d \\geq 4$ and $d$ even must be of the form \\eqref{trivial}. \n\\begin{itemize}\n\\item Suppose that a core facet $\\beta_0+\\sum_{j=1}^n\\beta_jx_j +\\beta^{\\prime}y \\geq 0$ has $d=2$ and is not of the form \\eqref{trivial}. We have $\\beta_1 = \\ldots = \\beta_{r+1}$ and $\\beta_{r+2}=\\ldots=\\beta_n$ by \\mythref{equalcoeff8}, so that we cannot have $\\beta_{r+1}=\\beta_{r+2}=0$ since the facet would be of the form \\eqref{trivial}. As a result, $D(r,r+2)=0$ with $\\mk[r]=m(\\x^{r+2})$ gives $0<-\\beta_{r+1} = \\beta_{r+2}=\\bar{\\beta}$ for some $\\bar{\\beta}>0.$ Then $D(r+1,r+2) \\leq 0$ gives $\\bar{\\beta} \\leq \\beta^{\\prime}\\mk[r+1]$ because $m(\\x^{r+2}) = -\\mk[r+1]$ and, since $0 < \\bar{\\beta}$ and $\\mk[r+1]=(-1)^{n-(r+1)},$ we have $0 < \\bar{\\beta}\\leq \\beta^{\\prime}\\mk[r+1]=\\beta^{\\prime}(-1)^{n-(r+1)}$ so that $\\beta^{\\prime}=(-1)^{n-(r+1)}.$ Consequently, the core facet takes the form \n\\begin{equation} \n\\beta_0-\\bar{\\beta}\\left(\\sum_{j=1}^{r+1}x_j-\\sum_{j=r+2}^{n}x_j\\right)+(-1)^{n-(r+1)}y \\geq 0. \\label{trivia0}\n\\end{equation} \nWe now use \\mythref{equalcoeff7} to show that the core facet must be satisfied exactly at $(\\x,y)=(\\x^{r+1},\\mk[r+1])$ by setting $\\beta_0 = (n-1)$ and $\\bar{\\beta}=1$ within \\eqref{trivia0}. The resulting inequality is valid for $\\conv{G}$ and is satisfied exactly at $(\\x,y)=(\\x^k,\\mk)$ for $k \\in \\{r,r+1,r+2\\}.$ The validity follows from \\mythref{foundation} since, for any $r \\in \\{0, \\ldots, n-2\\},$ the left side is $2|k-(r+1)|-1+(-1)^{k-(r+1)}$ at $(\\x,y)=\\left(\\x^k,\\mk\\right)$ for each $k \\in \\{0, \\ldots, n\\}$ because $\\mk=(-1)^{n-k}.$ \n\n\\item By contradiction, suppose there exists a valid inequality $\\beta_0+\\sum_{j=1}^n\\beta_jx_j +\\beta^{\\prime}y \\geq 0$ having $d \\geq 3$ and $d$ odd. Then $D(r,s)=0$ gives $\\sum_{j=r+1}^s\\beta_j=\\beta^{\\prime}\\mk[r]$ since $\\mk[r]=-\\mk[s].$ If $\\beta^{\\prime}\\mk[r]=1$ so that $\\beta^{\\prime}\\left(\\mk[r+1]-\\mk[r]\\right)=-2,$ then $D(r,r+1) \\geq 0$ gives $\\beta_{r+1} \\geq 1,$ and nondecreasing $\\beta_j$ gives $\\beta_{r+1} \\leq \\frac{1}{d}.$ Thus, a contradiction.  Similarly, if $\\beta^{\\prime}\\mk[r]=-1$ so that $\\beta^{\\prime}\\left(\\mk[s]-m(\\x^{s-1})\\right)=2,$ then $D(s-1,s) \\leq 0$ gives $\\beta_{s} \\leq -1,$ and nondecreasing $\\beta_j$ gives $\\beta_{s} \\geq \\frac{-1}{d}.$ Again a contradiction.\n\\item Suppose that a core facet $\\beta_0+\\sum_{j=1}^n\\beta_jx_j +\\beta^{\\prime}y \\geq 0$ has $d \\geq 4$ and $d$ even. The value $D(r,s)=0$ gives $\\sum_{j=r+1}^s \\beta_j=0$ since $\\mk[r]=\\mk[s].$ But $D(r,r+2) \\geq 0$ gives $\\left(\\beta_{r+1}+\\beta_{r+2}\\right) \\geq 0$ since $\\mk[r]=m(\\x^{r+2}),$ and the nondecreasing $\\beta_j$ gives $\\beta_j=0$ for $j \\in \\{r+1, \\ldots, s\\}.$ Then $\\beta_j=0$ for $j \\in N$ by \\mythref{equalcoeff8}, and the facet is of the form \\eqref{trivial}. \\qedhere\n\\end{itemize}\n\\end{proof}\n\nA key component of the proof of \\mythref{foundation15} is the family of $(n-1)$ valid core inequalities \\eqref{trivia0} with $\\beta_0 = (n-1)$ and $\\bar{\\beta}=1$ that has the property that, for each $r \\in \\{0, \\ldots, n-2\\},$ the corresponding inequality is satisfied exactly at $(\\x,y)=(\\x^k,\\mk)$ for $k \\in \\{r,r+1,r+2\\}.$ It turns out that each such inequality is a core facet, and that there exist no other core facets that are satisfied exactly at $(\\x^k,\\mk)$ for three consecutive extreme points $\\x^k$ of the simplex $\\mathcal{S}.$ This result is established below. Here, we find it convenient to substitute the index $t$ in \\eqref{dominate2} for $r$ in \\eqref{trivia0}.\n\n\\begin{lemma}\t\\thlabel{resulting}\nThere exist precisely $(n-1)$ core facets that are satisfied exactly at $(\\x^k,\\mk)$ for three consecutive extreme points $\\x^k$ of the simplex $\\mathcal{S},$ and these inequalities are~\\eqref{dominate2}.\n\\end{lemma}\n\\begin{proof}\nThe proof consists of two parts: the first part shows that inequalities \\eqref{dominate2} are the only candidate core facets that are satisfied exactly at three consecutive extreme points of the simplex $\\mathcal{S},$ and the second part shows that each such inequality is a core facet.\n\nSuppose, for some $t \\in \\{0,\\ldots, n-2\\},$ that $\\beta_0+\\sum_{j=1}^n\\beta_jx_j +\\beta^{\\prime}y \\geq 0$ is a core facet that is satisfied exactly at $(\\x,y)=(\\x^k,\\mk)$ for $k \\in \\{t,t+1,t+2\\}.$ Then $d=2$ in the proof of \\mythref{foundation15} since $d \\ge 3,$ $d$ odd, was shown not possible, and $d \\geq 4,$ $d$ even, was shown to yield a core facet of the form \\eqref{trivial}. Clearly, the facet is not of the form \\eqref{trivial} since $D(t,t+1) = 0$ gives $\\beta_{t+1}=\\beta^{\\prime}m(\\x^t) \\neq 0.$ As a result, the first part of the proof of \\mythref{foundation15} gives us that the facet must be of the form \\eqref{trivia0}. The two equations in two unknowns $\\beta_0$ and $\\bar{\\beta}$ that are obtained by evaluating $(\\x,y)=(\\x^k,\\mk)$ for $k \\in \\{t,t+1\\}$ within \\eqref{trivia0} give $\\beta_0=(n-1)$ and $\\bar{\\beta}=1$ as in \\eqref{dominate2}, since $-m(\\x^t)=m(\\x^{t+1})=(-1)^{n-(t+1)}.$\n\nThe proof of \\mythref{foundation15} showed inequalities \\eqref{dominate2} to be valid for $\\conv{G}$ and, for each $t \\in \\{0,\\ldots, n-2\\},$ the associated inequality to be satisfied exactly at $(\\x,y)=(\\x^k,\\mk)$ for $k \\in \\{t, t+1, t+2\\}.$ Thus, given such a $t,$ it is sufficient to identify $(n+1)$ affinely independent points in $\\conv{G}$ that satisfy the inequality exactly. Consider the $n$ extreme points of $\\conv{G},$ denoted by $(\\x[p]^1,m(\\x[p]^1)), \\ldots, (\\x[p]^n,m(\\x[p]^n)),$ so that $\\x[p]^i$ differs from $\\x^{t+1}$ in only position $i.$ Since $\\beta_1= \\ldots = \\beta_{t+1}$ and $(\\x,y)=(\\x^t,m(\\x^t))$ satisfies the inequality exactly, \\mythref{foundation45} gives us that $(\\x,y)=(\\x[p]^i,m(\\x[p]^i))$ for $i \\in \\{1,\\ldots, t+1\\}$ satisfies the inequality exactly. Similarly, since $\\beta_{t+2}= \\ldots = \\beta_n$ and $(\\x,y)=(\\x^{t+1},m(\\x^{t+1}))$ satisfies the inequality exactly, \\mythref{foundation45} gives us that $(\\x,y)=(\\x[p]^i,m(\\x[p]^i))$ for $i \\in \\{t+2,\\ldots, n\\}$ satisfies the inequality exactly. Subtract $\\x^{t+1}$ from every such point to reduce $\\x[p]^i$ to $-2\\boldsymbol{e}^i$ for $i \\leq t+1$ and to $2\\boldsymbol{e}^i$ for $i \\geq t+2,$ where $\\boldsymbol{e}^i$ is the unit vector in $\\real^n$ having a 1 in position $i$ and $0$ elsewhere. Hence, $\\x^{t+1},$ together with $\\x[p]^1, \\ldots, \\x[p]^n,$ is an affinely independent set of points, so that $(\\x^{t+1},m(\\x^{t+1})),$ together with $(\\x[p]^1,m(\\x[p]^1)), \\ldots, (\\x[p]^n,m(\\x[p]^n)),$  is an affinely independent set of points.\n\\end{proof}\n\n\\begin{proof}[\\textbf{Proof of \\mythref{resulting3}}]\nFollows from \\mythref{foundation900,foundation15,resulting}.\n\\end{proof}\n\n\n\\section{Summary and Open Questions}\n\nThis paper derives polyhedral results for the convex hull of symmetric multilinear polynomials (SMPs) taken over a box domain. \\akshay{Exponential-sized extended formulations of general multilinear polynomials are available via the reformulation-linearization-technique (RLT), but symmetry and disjunctive programming enable a quadratic-sized extended formulation.} The goal of this paper is to obtain the convex hulls in the original variable spaces. Instead of adopting the tedious method of projecting the extended formulations, our approach is more elegant in the sense that we directly exploit the problem structure to define special core facets by which all facets can be characterized, and to then devise necessary and\/or sufficient conditions on the coefficients of these facets. Whereas much of the theory is applicable to general SMPs over box constraints, we focus attention on two special problem classes: general supermodular (submodular) functions, and monomials having the variable lower bound equal to the negative of the upper bound. For each class, we use the necessary conditions to motivate families of core facets, and then prove that no other such facets can exist. \\akshay{Our derivations of these convex hulls provides alternate proofs to those in literature.} For both classes, we use RLT results to characterize for each facet the set of all points at which the inequality is satisfied exactly.\n\nA direction of future research is the identification of convex hull forms for more general families of SMPs than the two types within this paper, and likewise for the identification of all points within the resulting graphs that satisfy each facet exactly. \\akshay{One open question in this regard is to generalize \\mythref{resulting3} to monomials taken over $[-1,r]^{n}$ for arbitrary $r > 0$. Even more broadly, we do not know an explicit minimal description for convex hulls of general SMPs in the original variable space. Projecting the extended formulation of \\mythref{extform} is an option, but this is likely to result in a combinatorial explosion as is usually the case when projecting from a higher-dimensional space. Another question that is open is whether similar to the two families of SMPs analysed in this paper, every SMP has a linear (or even polynomial) number of core facets. A positive answer to this question would imply a straightforward separation algorithm for the convex hull due to \\mythref{sepcompl} without invoking the ellipsoid method and the optimization algorithm in \\mythref{optcompl}.}\n\n\\paragraph{Acknowledgement.}\nThis research was initiated when the authors were in the School of Mathematical and Statistical Sciences at Clemson University, USA, during which the first two authors (YX and WA) were supported by ONR grant N00014-16-1-2168 and the third author (AG) was supported by ONR grant N00014-16-1-2725.\n\n{\n\\newrefcontext[sorting=nyt]\n\\printbibliography[heading=bibliography]\n}\n\n\\newpage\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n\\setcounter{equation}{0}\n\n\\begin{appendices}\n\n\\section{Properties of $\\conv{G}$ from RLT}\t\\label{sec:propG}\n\nThe key observation is that the equivalence of \\eqref{miss1} between the polyhedral sets $P$ of \\eqref{RLTstep2} and $\\conv{T}$ of \\eqref{miss1} continues to hold true with the inclusion of the restriction $y= m(\\x)$  within these two sets. Consider a generalization of the set $G$ given by \n\\begin{equation}\nG^{\\prime} \\equiv\\left\\{(\\x,y)\\in \\real^n \\times \\real: \\x \\in X^{\\prime}, \\; y= m(\\x)\\right\\}, \\label{marker}\n\\end{equation}\nthat is obtained by replacing $\\x \\in X$ of \\eqref{Xdef} with $\\x \\in X^{\\prime}$ of \\eqref{Xprimedef}. Let\n\\begin{equation}\nP_y\\equiv\\left\\{(\\x,\\x[w],y) \\in \\real^n\\times\\real^{2^n-(n+1)}\\times \\real:(\\x,\\x[w]) \\in P, \\; y=\\left\\{m(\\x)\\right\\}_L\\right\\}  \\label{handy}\n\\end{equation}\nbe the set $P$ of \\eqref{RLTstep2} that is modified to include the additional variable $y$ and additional restriction $y=\\left\\{m(\\x)\\right\\}_L,$ and let \n\\begin{equation}\nT_y\\equiv\\left\\{(\\x,\\x[w],y) \\in \\real^n\\times\\real^{2^n-(n+1)}\\times \\real:(\\x,\\x[w]) \\in T, \\; y=m(\\x)\\right\\}  \\nonumber\n\\end{equation}\nbe the set $T$ of \\eqref{miss1} that is modified to include the additional variable $y$ and additional restriction $y=m(\\x).$ Then we have that\n\\begin{equation}\nP_y=conv\\left(T_y \\right). \\label{pete}\n\\end{equation}\nEquality \\eqref{pete} holds true from \\eqref{miss1} because the extreme points of $P$ and $P_y$ correspond in a one-to-one fashion in such a manner that $(\\x,\\x[w])$ is an extreme point of $P$ if and only if $(\\x,\\x[w],y)$ is an extreme point of $P_y$ having $y=\\left\\{m(\\x)\\right\\}_L=m(\\x).$ Therefore, every extreme point of the polytope $P_y$ is in $T_y$ and $T_y \\subseteq P_y$ by construction. \n\nThe three propositions below are consequences of \\eqref{pete}, where $\\mbox{Proj}_{(\\x,y)}(\\bullet)$ denotes the projection of the set $\\bullet$ onto the space of the variables $(\\x,y).$ \n\n\\begin{proposition}\t\\thlabel{equalities}\n$\\conv{G^{\\prime}} = \\conv{\\left(\\mbox{Proj}_{(\\x,y)}(T_y)\\right)} = \\mbox{Proj}_{(\\x,y)}\\left(\\conv{T_y}\\right) = \\mbox{Proj}_{(\\x,y)}\\left(P_y\\right)$. \n\\end{proposition}\n\\begin{proof}\nThe first equality follows because $G^{\\prime}=\\mbox{Proj}_{(\\x,y)}(T_y),$ the second equality follows from interchanging the projection and convex hull operators, and the third equality follows from \\eqref{pete}.\n\\end{proof} \n\n\\begin{proposition}\t\\thlabel{equalities21}\n$conv\\left(G^{\\prime}\\right)$ is a polytope with $2^n$ extreme points that have one-to-one correspondence with the extreme points of $X^{\\prime}$ in such a manner that $y=m(\\x)$ at each such point $\\x.$ \n\\end{proposition}\n\\begin{proof}\n$conv\\left(G^{\\prime}\\right)$ is a polytope with no more than $2^n$ extreme points since it is the projection of the polytope $P_y$ having $2^n$ extreme points onto the $(\\x,y)$ space, as stated in \\mythref{equalities}. However, each of the $2^n$ points of $conv\\left(G^{\\prime}\\right)$ satisfying the property that every entry of $\\x$ has either $x_j=L_j$ or $x_j=U_j$ for all $j \\in N,$ and that $y=m(\\x),$ is trivially an extreme point of $conv\\left(G^{\\prime}\\right).$\n\\end{proof} \n\n\\mythref{equalities21} also follows from results in \\citet{rikun1997convex}. As opposed to using the projection from a higher-dimensional RLT space as in the above proof, \\citeauthor{rikun1997convex} shows that every $(\\x,y) \\in G^{\\prime}$ which has some $x_j \\in (L_j,U_j)$ can be expressed as a strict convex combination of two distinct points in $G^{\\prime}.$\n\nThe following result addresses the validity of a linear inequality for $conv\\left(G^{\\prime}\\right)$ in terms of the restrictions of the set $P_y.$\n\n\\begin{corollary}\t\\thlabel{equalities24}\nA linear inequality $\\beta_0+\\sum_{j=1}^n\\beta_jx_j+\\beta^{\\prime}y \\geq 0$ is valid for $conv\\left(G^{\\prime}\\right)$ if and only if it can be uniquely expressed as a linear combination of the restrictions of $P_y$ using nonnegative multipliers $\\x[\\pi] \\in \\real^{2^n}$ and the scalar $\\beta^{\\prime},$ so that\n\\begin{equation}\n\\beta_0+\\sum_{j=1}^n\\beta_jx_j+\\beta^{\\prime}y=\\sum_{K \\subseteq N}\\pi_{K}F_{K}(x)+\\beta^{\\prime}\\left(y-\\left\\{m(\\x)\\right\\}_L\\right). \\label{ruby}\n\\end{equation}\n\\end{corollary}\n\\begin{proof}\nThe existence of nonnegative multipliers $\\x[\\pi]$ satisfying \\eqref{ruby} follows from the result of \\mythref{equalities} that $\\conv{G^{\\prime}}\\mbox{Proj}_{(\\x,y)}\\left(P_y\\right),$ as then a linear inequality will be valid for $\\conv{G^{\\prime}}$ if and only if it is valid for $P_y.$ The uniqueness follows from the invertibility of the matrix $\\scriptsize\\left[\\begin{array}{cc} U_1 &  -1 \\\\ -L_1 & 1 \\end{array}\\right] \\normalsize \\otimes \\ldots \\otimes \\scriptsize \\left[\\begin{array}{cc} U_n &  -1 \\\\ -L_n & 1 \\end{array}\\right]$ of \\eqref{ca4}.\n\\end{proof}\n\n\\input{Examples}\n\\end{appendices}\n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:intro}\n\nThe \\emph{$s$-independence number} of a graph $G$, denoted $\\alpha_s(G)$, is the maximum number of vertices in a $K_s$-free induced subgraph of $G$ (so the standard independence number is the same as the $2$-independence number). For a given graph $H$, the \\emph{Ramsey-Tur\\'an number} $\\textup{\\textbf{RT}}_s(n, H, f)$ is the maximum number of edges in any $H$-free graph $G$ on $n$  vertices with $\\alpha_s(G) < f$. If there does not exist any $H$-free graph $G$ on $n$ vertices with $\\alpha_s(G) < f$ then we put $\\textup{\\textbf{RT}}_s(n,H,f)=0$.  Observe that the lower bound $k \\le \\textup{\\textbf{RT}}_s(n,H,f)$ means that there exists an $H$-free graph~$G$ of order~$n$ with $\\alpha_s(G) < f$ and at least $k$ edges. The upper bound $\\textup{\\textbf{RT}}_s(n,H,f) < \\ell$ says that there is no $H$-free graph~$G$ of order $n$, $\\alpha_s(G) < f$, and at least $\\ell$ edges.\n\nIn general it is far from trivial to even determine the existence of any $H$-free graph~$G$ on $n$ vertices with $\\alpha_s(G) <f$, let alone to maximize the number of edges in such graphs if they do exist. The \\emph{Erd\\H{o}s-Rogers number} $\\textup{\\textbf{f}}_{s, t}(n)$ is the minimum possible $s$-independence number taken over all $K_t$-free graphs $G$ of order~$n$. Note that if $f \\le \\textup{\\textbf{f}}_{s,t}(n)$ then $\\textup{\\textbf{RT}}_s(n, K_t, f)=0$. Erd\\H{o}s and Rogers \\cite{ERog} were the first who studied $\\textup{\\textbf{f}}_{s,t}(n)$ for fixed $s$ and $t=s+1$ and $n$ going to infinity. They  proved that for every $s$ there is a positive $\\varepsilon(s)$ such that $\\textup{\\textbf{f}}_{s, s+1}(n) \\le n^{1-\\varepsilon(s)}$, where $\\lim_{s\\to\\infty}\\varepsilon(s)=0$. This question was subsequently addressed by Bollob\\'as and Hind~\\cite{BH}, Krivelevich~\\cite{KR2, KR}, Alon and Krivelevich \\cite{AK}, Dudek and R\\\"odl \\cite{DR}, Wolfovitz~\\cite{Wo}, and most recently by Dudek and Mubayi~\\cite{DM}, and Dudek, R\\\"odl and Retter~\\cite{DRR}. Due to~\\cite{DM} and~\\cite{DRR} it is known that for any $s\\ge 3$,\n\\begin{equation}\\label{f:1}\n\\Omega\\left(\\sqrt{\\frac{n \\log n }{\\log\\log n}} \\right) = \\textup{\\textbf{f}}_{s, s+1}(n) = O\\left( (\\log n)^{4s^2}\\sqrt{n} \\right).\n\\end{equation}\nFor $s=2$ very sharp bounds are known: \n\\begin{equation}\\label{f:1b}\n(1\/\\sqrt{2} - o(1))\\sqrt{n\\log n} \\le \\textup{\\textbf{f}}_{2,3}(n) \\le (\\sqrt{2} + o(1))\\sqrt{n\\log n},\n\\end{equation}\nwhere the upper bound was obtained by Shearer~\\cite{Shearer} and the lower bound, independently, by Bohman and Keevash~\\cite{BK2}, and Pontiveros, Griffiths, and Morris~\\cite{PGM}.\nFurthermore, a result of Sudakov~\\cite{SU} together with~\\cite{DRR} imply that for any $\\varepsilon > 0$ and an integer $r\\ge 2$\n\\begin{equation}\\label{f:r}\n\\Omega\\big{(}n^{\\frac{1}{2} - \\varepsilon}\\big{)} = \\textup{\\textbf{f}}_{s,s+r}(n) = O(\\sqrt{n})\n\\end{equation}\nfor all sufficiently large~$s$.\n\nThe above bounds are quite recent and therefore the Ramsey-Tur\\'an number\\linebreak $\\textup{\\textbf{RT}}_s(n, H, f)$ was not extensively studied for small $f$. \nPreviously, most researchers investigated the Ramsey-Tur\\'an number for $f$ not too much smaller than $n$. Perhaps the first paper written about this problem was by S\\'os~\\cite{VS}. In particular there are many results on $\\theta_r(H)$, where\n\\[\n\\theta_r(H) = \\lim_{\\varepsilon \\rightarrow 0} \\lim_{n \\rightarrow \\infty} \\frac{1}{n^2}\\textup{\\textbf{RT}}(n, H, \\varepsilon n).\n\\] \nIt is perhaps surprising that $\\theta_r(H)$ would ever be positive, but it is known for example that $\\theta_2(K_4)= \\frac{1}{8}$ (upper bound by Szemer\\'edi in \\cite{Sz}, lower bound by Bollob\\'as and Erd\\H{o}s in \\cite{BE}). Several other exact values for $\\theta_r(H)$ are known, and even more bounds are known where exact values are not, see, for example, the papers by Balogh and Lenz~{\\cite{BL,BL2}}; Erd\\H{o}s, Hajnal, Simonovits, S\\'os, and Szemer\\'edi~\\cite{EHSSS}; \nErd\\H{o}s, Hajnal, S\\'os, and Szemer\\'edi~\\cite{EHSS}, and Simonovits and S\\'os~\\cite{SS}.  Until recently only a very few results for $f \\le n^{\\delta}$ with $\\delta<1$ were given. For example, Sudakov~\\cite{SU} gave upper and lower bounds on $\\textup{\\textbf{RT}}_2(n, K_{4}, n^{\\delta})$ (see also \\cite{SU3} and~\\cite{FLZ}). Balogh, Hu, and Simonovits~\\cite{BHS} studied $\\textup{\\textbf{RT}}_2(n,K_t,f)$ for several pairs of $t$ and $f$.\n\n\n\nIn this paper we study $\\textup{\\textbf{RT}}_s(n, K_{s+r}, f)$ for $\\textup{\\textbf{f}}_{s,s+r}(n) +1 \\le f\\le n^{\\delta}$ with $\\delta<1$. In view of \\eqref{f:1} and \\eqref{f:r}, it is of interest to ask about $\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^\\delta)$ for $r\\ge 1$ and $1\/2 < \\delta < 1$. Moreover, for $r\\ge 2$ it also makes sense to ask for $\\textup{\\textbf{RT}}_s(n, K_{s+r}, Cn^{1\/2})$ for a sufficiently large constant $C$ (which may depend on $s$). We will show (Theorem~\\ref{thm:main1}, \\ref{thm:main2}, and \\ref{thm:ub_delta}) that for all $r\\ge 1$, $\\varepsilon>0$ and $1\/2 < \\delta < 1$, and all sufficiently large $s$,\n\\[\n\\Omega\\of{n^{2 - (1-\\delta)\/r  - \\varepsilon}} = \\textup{\\textbf{RT}}_s(n, K_{s+r}, n^\\delta) <\n\\begin{cases}\nn^{2-\\frac{(1-\\delta)^2}{r-\\delta}}, & \\text{ if $\\frac{r-\\delta}{1-\\delta}$ is an integer},\\\\ \nn^{2-\\frac{(1-\\delta)^2}{r+1-2\\delta}}, & \\text{ otherwise}. \n\\end{cases}\n\\]\nIn particular, this implies for $r=1$ and $1\/2 < \\delta < 1$ nearly optimal bounds,\n\\[\n\\Omega\\of{n^{1+\\delta  - \\varepsilon}} = \\textup{\\textbf{RT}}_s(n, K_{s+1}, n^\\delta) = O\\of{n^{1+\\delta}}.\n\\]\nAs a matter of fact the upper bound is trivial since as it was already observed in~\\cite{EHSSS} if $G$ is $K_{s+1}$-free, then $\\Delta(G) < \\alpha_s(G)$.\nWe will also show that for specific values of $\\delta$ one can remove $\\varepsilon$ from the exponent (see Theorem~\\ref{thm:quad} and Corollary~\\ref{cor:quad}) and we conjecture that this should be true for all $1\/2 < \\delta < 1$.\n\nVery recently Balogh, Hu, and Simonovits~\\cite{BHS} introduced an interesting concept of a (Ramsey-Tur\\'an) phase transition. Following this direction we will study behavior of $\\textup{\\textbf{RT}}_s(n, K_{2s+1}, f)$. In particular, for $1\/2 < \\delta < 1$ we show that $\\textup{\\textbf{RT}}_s(n, K_{2s}, n^\\delta)$ is subquadratic, while $\\textup{\\textbf{RT}}_s(n, K_{2s+1}, n^\\delta)$ is quadratic (and we actually find its value asymptotically exactly). But for $\\delta = 1\/2$, $\\textup{\\textbf{RT}}_s(n, K_{2s+1}, n^\\delta)$ is subquadratic again, yielding a ``jump''  in the critical window.\n\nOur proofs of the lower bounds are built upon the ideas in~\\cite{DR,DRR,Wo} and are based on certain models of  random graphs that are constructed using some finite geometries. Roughly speaking, finite geometries provide us with a structure that allows us to bound the number of vertices that interact in certain ways, which helps us show that in the random graph we construct, we do not expect to see the forbidden subgraph (say if we are doing $\\textup{\\textbf{RT}}_s(n, K_{s+r}, f)$ then the forbidden subgraph is $K_{s+r}$) while we do expect to see many copies of $K_s$. The proofs use the probabilistic method, but the use of probability is relatively elementary. The proofs of the upper bounds use the dependent random choice technique (see, e.g., \\cite{FS}).\n\nThe rest of the paper is structured as follows. In Section~\\ref{sec:prelim} we state the probabilistic tools we will use. In Sections~\\ref{sec:main1} and~\\ref{sec:main2} we prove lower bounds on $\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^{\\delta})$ for $\\delta$ close to $1\/2$ (in Section~\\ref{sec:main1}) and then for larger values $\\delta <1$ (in Section~\\ref{sec:main2}). In Section~\\ref{sec:ub} we prove upper bounds on $\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^{\\delta})$. In Section~\\ref{sec:r_eq_1} we further discuss the case $r=1$ and show that for a few values $\\delta$ we can prove upper and lower bounds matching up to a constant factor. In Section~\\ref{sec:big_r} we discuss a phase transition of $\\textup{\\textbf{RT}}_s(n, K_{2s+1}, f)$.\n\n\\section{Preliminaries}\\label{sec:prelim}\n\nThis paper uses the probabilistic method in the most classical sense: if we define a random structure and show that with some positive probability the random structure has a certain property, then there must exist a structure with that property. The probabilistic aspect of this paper is elementary. We use only standard bounds on the probability of certain events, which we state here. \n\nWe state basic forms of the Chernoff and Markov bounds (see, e.g., \\cite{AS,JLR}).\n\\begin{c2}\nIf X is any nonnegative random variable and $\\zeta > 0$, then\n\\[\n\\Pr\\left(X \\geq \\zeta\\cdot \\mathbb{E}(X)\\right) \\leq \\frac{1}{\\zeta}.\n\\]\n\\end{c2}\n\nLet $\\textrm{Bin}(n,p)$ denotes the random variable with binomial distribution with number of trials~$n$ and probability of success~$p$.\n\\begin{c1}\nIf $X \\sim \\textrm{Bin}(n,p)$ and $0 < \\varepsilon \\leq \\frac{3}{2}$, then\n\\[\n\\Pr \\left( |X - \\mathbb{E}(X)| \\geq \\varepsilon \\cdot \\mathbb{E}(X) \\right) \\leq 2 \\exp \\left\\{ -\\frac{\\mathbb{E}(X) \\varepsilon^2 }{3} \\right\\} .\n\\]\n\\end{c1}\n\nWe will also use the union bound.\n\\begin{ub}\nIf $E_i$ are events, then \n\\[\n\\Pr \\Big( \\bigcup_{i=1}^k E_i \\Big) \\leq k \\cdot \\max \\{\\Pr(E_i): i \\in [k]\\}.\n\\]\n\\end{ub}\n\nFinally, we say that an event $E_n$ occurs \\emph{with high probability}, or {\\textit{w.h.p. }} for brevity, if $\\lim_{n\\rightarrow\\infty}\\Pr(E_n)=1$.\n\nAll logarithms in this paper are natural (base $e$). Asymptotic notation can be viewed as either in the variable $n$ (the number of vertices in the graphs we are interested in) or $q$ (another parameter that will go to infinity along with $n$). When we say that a statement \\emph{holds for $s$ sufficiently large}, we mean that there exists some $s_0$ (which may dependent on some other parameters) such that the statement holds for any $s \\ge s_0$. For simplicity, in the asymptotic notion we do not round numbers that are supposed to be integers either up or down. This is justified since these rounding errors are negligible.\n\n\n\\section{Lower bound on $\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^\\delta)$ for small values of $\\delta$}\\label{sec:main1}\n\nIn this section we will construct (randomly) a graph $G_1$ that gives our lower bound for $\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^\\delta)$ for relatively small $\\delta$ (not much bigger than $1\/2$). The construction uses several ideas from~\\cite{DR,DRR,Wo}.\n\nWe start with the affine plane, which is a  hypergraph with certain desirable properties. For this range of $\\delta$ we want a somewhat sparser hypergraph, so we randomly remove some edges from the affine plane to form a new hypergraph $\\mathcal{H}_1$. We then construct $G_1$ by taking the vertices in each edge of $\\mathcal{H}_1$ and putting a complete $s$-partite graph (with a random $s$-partition) on them together with a large independent set. Consequently, $G_1$ will have many copies of $K_s$. On the other hand, since each edge of $\\mathcal{H}_1$ contains only copies of $K_s$ (and no larger complete subgraph), any possible copy of $K_{s+r}$ in $G_1$ must not be entirely contained in one edge of $\\mathcal{H}_1$. We will exploit the properties  $\\mathcal{H}_1$ and how its edges interact to show that this is unlikely, and therefore we do not expect to see any $K_{s+r}$ in $G_1$.\n\n\\subsection{The hypergraph $\\mathcal{H}_1$} \\label{sec:H1}\n\nThe {\\em affine plane} of order $q$ is an incidence structure on a set of $q^2$ points and a set of $q^2+q$ lines such that: any two points lie on a unique line; every line contains~$q$ points; and every point lies on~$q+1$ lines. It is well known that affine planes exist for all prime power orders. (For more details see, e.g., \\cite{CA}.)\nClearly, an incidence structure can be viewed as a hypergraph with points corresponding to vertices and lines corresponding to hyperedges; we will use this terminology interchangeably.\n\nIn the affine plane, call lines $L$ and $L'$ \\emph{parallel} if $L \\cap L' = \\emptyset$. In the affine plane there exist $q+1$ sets of $q$ pairwise parallel lines. Let $\\mathcal{H}=(V, \\mathcal{L})$ be the hypergraph  obtained by removing a parallel class of $q$ lines from the affine plane or order~$q$. Thus, $\\mathcal{H}$ is $q$-regular hypergraph of order~$q^2$.\n\nThe objective of this section is to establish the existence of a certain hypergraph $\\mathcal{H}_1 = (V_1,\\mathcal{L}_1) \\subseteq \\mathcal{H}$ by considering a random sub-hypergraph of $\\mathcal{H}$. Preceding this, we introduce some terminology. \nCall $S \\subseteq V_1$ \\emph{complete} if every pair of points in $S$ is contained in some common line in~$\\mathcal{L}_1$. \nWe distinguish 2 types of complete \\emph{dangerous} subsets $S\\subseteq V$. \\emph{Type~1} dangerous set consists of $|S|$ points in general position. \\emph{Type~2} dangerous set consists of $|S|-r$ points that lie on some line $L\\in\\mathcal{L}_1$ and a set $R$ of $r$ many other points that do not belong to $L$. \n\n\n\n\n\\begin{lemma} \\label{lem:1}\nLet $q$ be a sufficiently large prime and $\\log q \\ll \\lambda \\le q$. \nThen, there exists a $q$-uniform hypergraph $\\mathcal{H}_1=(V_1,\\mathcal{L}_1)$ of order $q^2$ such that:\n\\begin{enumerate}[label=$({\\textup{H}}_{1}\\textup{\\alph*})$]\n\\item\\label{H1:a} Any two vertices are contained in at most one hyperedge;\n\\item\\label{H1:b} For every $v \\in V_1$, $\\frac{\\lambda}{2} \\le \\deg_{\\mathcal{H}_1}(v) \\leq \\frac{3\\lambda}{2}$; \n\\item\\label{H1:c} $|\\mathcal{D}_a|  \\leq  \\frac{4\\lambda^{a^2\/2} }{ q^{(a^2-5a)\/2}}$ and $| \\mathcal{D}_b| \\leq \\frac{4\\lambda^{br } }{ q^{b(r-1) - 5r^3}} $, where $\\mathcal{D}_a$ is the set of all dangerous sets of Type 1 and size~$a$, and $\\mathcal{D}_b$ is the set of all dangerous sets of Type 2 and size~$b$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof} Let $V_1 = V$ be the same vertex set as $\\mathcal{H}$, and let $\\mathbb{H}_1=(V_1,\\mathcal{L}_1)$ be a random sub-hypergraph of $\\mathcal{H}$ where every line in $\\mathcal{L}$ is taken independently with probability \n${\\lambda}\/{q}$.\n\nSince $\\mathbb{H}_1$ is a subgraph of $\\mathcal{H}$, any two vertices are in at most one line, so $\\mathbb{H}_1$ always satisfies \\ref{H1:a}. \nWe will show that $\\mathbb{H}_1$ satisfies {\\textit{w.h.p. }} \\ref{H1:b} and satisfies \\ref{H1:c} with probability at least $1\/2$. \nTogether this implies that $\\mathbb{H}_1$ satisfies \\ref{H1:a}-\\ref{H1:c} with probability at least $1-\\frac{1}{2}-o(1)$, establishing the existence of a hypergraph $\\mathcal{H}_1$ that satisfies \\ref{H1:a}-\\ref{H1:c}. \n\n\\medskip\n\n\\textbf{\\ref{H1:b}:}  Observe for fixed $v \\in V_1$, $\\deg_{\\mathbb{H}_1}(v) \\sim \\textrm{Bin}(q, \\frac{\\lambda}{q})$ and the expected value $\\mathbb{E}(\\deg_{\\mathbb{H}_1}(v))= \\lambda.$ So by the Chernoff bound with $\\varepsilon = \\frac{1}{2}$, \n\\[\n \\Pr \\Big( |\\deg_{\\mathbb{H}_1}(v)-\\lambda| \\geq \\frac{\\lambda}{2} \\Big) \\leq 2  \\exp \\left\\{ -\\frac{\\lambda}{12} \\right\\}. \n\\]\nThus by the union bound the probability that there exists some $v \\in V_1$ with $\\deg_{\\mathbb{H}_1}(v) \\notin \\sqbs{\\frac{\\lambda}{2} , \\frac{3\\lambda}{2}}$ is at most\n\\[\nq^2 \\cdot 2  \\exp \\left\\{-\\frac{\\lambda}{12}\\right\\} = 2 \\exp \\left\\{2 \\log q-\\frac{\\lambda}{12}\\right\\} = o(1),\n\\]\nsince $\\lambda\\gg\\log q$.\n\n\\medskip\n\n\\textbf{\\ref{H1:c}:} In order to show  we have both $|\\mathcal{D}_a|  \\leq  \\frac{4\\lambda^{a^2\/2} }{ q^{(a^2-5a)\/2}}$ and $| \\mathcal{D}_b| \\leq \\frac{4\\lambda^{br } }{ q^{b(r-1) - 5r^3}} $ with probability at least $\\frac{1}{2}$, we begin by counting the number of dangerous subsets of each type. Clearly the number of Type 1 dangerous subsets is at most ${q^2 \\choose a}$ and each of them contains $\\binom{a}{2}$ lines. To count the number of Type 2 dangerous subsets first we choose a line $L$ out of $q^2$ lines in~$\\mathcal{L}$. Next we choose $b-r$ points in $L$ having $\\binom{q}{b-r}$ choices, and finally, we choose the remaining $r$ points not in $L$. Thus, the number of configurations of Type~2 is at most $\\binom{q^2}{1} \\cdot \\binom{q}{b-r} \\cdot q^{2r}$. Now we bound the number of lines in a dangerous set of Type 2. First there is the line $L$ containing $|S|-r$ points. Then, every pair of points $u,v$, where $u \\in L \\cap S$ and $v \\in R$, must be contained in some line $L_{u,v}$ in $\\mathcal{L}$. No line $L_{u,v}$ can contain more than one vertex in $L$, but it is possible for some line $L_{u,v}$ to contain multiple vertices in $R$. However for $v, v' \\in R$, if $L_{u,v}$ contains $v'$ then no other line can contain both $v, v'$. Thus, the number of lines of the form $L_{u,v}$ such that $|L_{u,v} \\cap R|\\ge 2$ is at most $\\binom{r}{2}$. Now since each of the $r(b-r)$ many pairs $\\{u,v\\}$ must be covered by some $L_{u,v}$, and the number of lines covering multiple pairs is at most $\\binom{r}{2}$, and no line covers more than $r$ many pairs, the total number of lines $L_{u,v}$ is at least $r(b-r) - r\\binom{r}{2} \\ge br - 2r^3$. \n\nBy the linearity of expectation, we now compute \n\\[\n\\mathbb{E}(| \\mathcal{D}_a|) \n\\leq \\binom{q^2}{a}  \\cdot \\left( \\frac{\\lambda}{q} \\right)^{\\binom{a}{2}} \\leq q^{2a} \\cdot \\left( \\frac{\\lambda}{q} \\right)^{\\binom{a}{2}} \\le  \\frac{\\lambda^{a^2 \/2 }}{ q^{(a^2-5a)\/2}}\n\\]\nand\n\\begin{align*}\n\\mathbb{E}(|\\mathcal{D}_b|) \n&\\leq  \\binom{q^2}{1} \\cdot \\binom{q}{b-r} \\cdot q^{2r} \\cdot \\left( \\frac{\\lambda}{q} \\right)^{br - 2r^3}\\\\\n&\\leq   q^{2} \\cdot q^{b-r} \\cdot q^{2r} \\cdot \\left( \\frac{\\lambda}{q} \\right)^{br - 2r^3} \\le \\frac{\\lambda^{br } }{ q^{b(r-1) - 5r^3}}.\n\\end{align*}\nThus, the Markov bound yields \n\\[\n\\Pr \\left( |\\mathcal{D}_a| \\ge   \\frac{4\\lambda^{a^2 \/2 }}{ q^{(a^2-5a)\/2}} \\right)\n\\le \\Pr \\left( |\\mathcal{D}_a| \\ge 4 \\mathbb{E} ( |\\mathcal{D}_a|) \\right) \\le \\frac{1}{4}\n\\]\nand \n\\[\n\\Pr \\left( | \\mathcal{D}_b| \\ge    \\frac{4\\lambda^{br } }{ q^{b(r-1) - 5r^3}}\\right)\n\\le \\Pr \\left( |\\mathcal{D}_b| \\ge 4 \\mathbb{E} ( | \\mathcal{D}_b|) \\right) \\le \\frac{1}{4},\n\\]\nand finally\n\\[\n\\Pr \\left( |\\mathcal{D}_a| \\le   \\frac{4\\lambda^{a^2 \/2 }}{ q^{(a^2-5a)\/2}} \\mbox{ and } | \\mathcal{D}_b| \\le    \\frac{4\\lambda^{br } }{ q^{b(r-1) - 5r^3}}\\right) \\ge 1 - \\frac 14 - \\frac 14 = \\frac 12,\n\\]\nas required.\n\\end{proof}\n\n\n\\subsection{The graph $G_1$}\\label{sec:G1}\nBased upon the hypergraph $\\mathcal{H}_1$ established in the previous section, we will construct a graph $G_1$ with the following properties. \n\\begin{lemma} \\label{lem:3}\nLet $r\\ge 1$ and $s$ be sufficiently large constant.\nLet  $q$ be a sufficiently large prime, $q \\ge \\lambda \\gg \\log q$, $ 1 \\ge p \\gg (\\log q) \/ \\lambda$, and $\\alpha \\ge (10 s \\log s)q\/p$. Furthermore, let $a$ be a positive constant and $b=\\left\\lceil(s+r) \/{ \\binom{a-1}{2} }\\right\\rceil +r$. Then, there exists a graph $G_1$ of order $q^2$ such that:\n\\begin{enumerate}[label=$({\\textup{G}}_{1}\\textup{\\alph*})$]\n\\item\\label{G1:a} $\\alpha_s(G_1) < \\alpha$;\n\\item\\label{G1:b} For every vertex $v$ in $G_1$, $ \\deg_{G_1}(v)= \\Theta( \\lambda q p^2)$;\n\\item\\label{G1:c} $G_1$ has at most  $8\\left(\\frac{\\lambda^{a^2 \/2} p^{a^2 -a}}{ q^{(a^2-5a)\/2}} + \\frac{\\lambda^{br } p^{(2r+1)b - 4r^3}}{ q^{b(r-1) -5r^3}}\\right)$ copies of  $K_{s+r}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}  Fix a hypergraph $\\mathcal{H}_1=(V_1,\\mathcal{L}_1)$ as established by Lemma~\\ref{lem:1}. Construct the random graph $\\mathbb{G}_1=(V_1,E)$ as follows. For every $L\\in \\mathcal{L}_1$, let $\\chi_L : L \\to  {[s+1]}$ be a random partition of the vertices of $L$ into $ {s+1}$ classes, where for every $v \\in L$, \n\\[\n\\Pr(\\chi(v) = i) = \n\\begin{cases}\n{p}\/{s} & \\text{for } 1\\le i\\le s,\\\\\n1-p & \\text{for } i=s+1,\n\\end{cases}\n\\]\nand $\\chi(v)$ is assigned independently from other vertices.\n If $\\chi_L(v) =s+1$, then we say that $v$ is {\\em $L$-isolated}. Let $\\{x,y\\} \\in E$ if $\\{x,y\\} \\subseteq L$ for some $L \\in \\mathcal{L}_1$ and $\\chi_L(x),  \\chi_L(y)$ are distinct and neither $x$ nor $y$ is $L$-isolated. Thus for every $L \\in \\mathcal{L}_1$, $\\mathbb{G}_1[L]$ consists of a set of isolated vertices (the $L$-isolated vertices) together with a complete $s$-partite graph with vertex partition $L = \\chi^{-1}_L (1) \\cup \\chi^{-1}_L (2) \\cup \\dots \\cup \\chi^{-1}_L (s)$ (where the classes need not have the same size and the unlikely event that a class is empty is permitted). Observe that not only are $\\mathbb{G}_1[L]$ and $\\mathbb{G}_1[L']$ edge disjoint for distinct $L, L' \\in \\mathcal{L}_1$, but also that the partitions for $L$ and $L'$ were determined independently.\n\n\nWe will show that $\\mathbb{G}_1$ satisfies {\\textit{w.h.p. }} \\ref{G1:a} and \\ref{G1:b} and satisfies \\ref{G1:c} with probability at least $1\/2$. \n\n\\medskip\n\n\\textbf{\\ref{G1:a}:} First fix $C \\in {V_1 \\choose \\alpha}$. We will bound the probability that $\\mathbb{G}_1[C] \\not \\supseteq K_s$. For a fixed $L$, the probability that one of the classes $\\chi_L^{-1}(1), \\dots , \\chi_L^{-1}(s)$ contains no element of $C$ is at most $s\\of{1-\\frac ps}^{|L \\cap C|} \\le s e^{-\\frac {p}{s} |L \\cap C|}$. Note that \n\\[\n\\sum_{L \\in \\mathcal{L}_1} |L \\cap C| \\ge \\frac 12 \\lambda |C| =  \\frac 12 \\lambda \\alpha,\n\\] \nsince each point is in at least $\\frac 12 \\lambda$ lines due to condition~\\ref{H1:b}. Now since $\\chi_L$ and $\\chi_{L'}$ are chosen independently for $L \\neq L'$ we get, \n\\[\n\\Pr\\Big( K_s \\not \\subseteq \\mathbb{G}_1[C] \\Big) \\le \\prod_{L \\in \\mathcal{L}_1}\\Pr \\Big( K_s \\not \\subseteq \\mathbb{G}_1[L \\cap C] \\Big) \\le s^{|\\mathcal{L}_1|} \\exp\\left\\{ -\\frac ps \\sum_{L \\in \\mathcal{L}_1} |L \\cap C| \\right\\},\n\\]\nand since $|\\mathcal{L}_1| \\le 3\\lambda q$ by~\\ref{H1:b}, \n\\[\n\\Pr\\Big( K_s \\not \\subseteq \\mathbb{G}_1[C] \\Big) \\le \\exp \\left\\{(3\\log s) \\lambda q  - \\frac{1}{2s} \\lambda \\alpha p \\right\\}.\n\\]\n\nSo by the union bound, the probability that there exists a set $C$ of $\\alpha$ vertices in $\\mathbb{G}_1$ that contains no $K_s$ is at most\n\\begin{align*}\n&q^{2 \\alpha} \\exp \\left\\{(3\\log s) \\lambda q  - \\frac{1}{2s} \\lambda \\alpha p  \\right\\}\\le   \\exp \\left\\{2\\alpha  \\log q +(3\\log s) \\lambda q  - \\frac{1}{2s} \\lambda \\alpha p \\right\\}=o(1),\n\\end{align*}\nsince $ p \\gg  (\\log q) \/ \\lambda$ and $\\alpha \\ge (10 s \\log s)q\/p$.\n\nThus, {\\textit{w.h.p. }} $\\alpha_s(\\mathbb{G}_1) < \\alpha$.\n\n\\medskip\n\n\\textbf{\\ref{G1:b}:} Observe that for a fixed $L\\in \\mathcal{L}_1$, the number of non-$L$-isolated vertices $|\\chi_L^{-1}([s])|$ is distributed as $\\textrm{Bin}(q, p)$ which has expectation $pq $ so by the Chernoff bound with $\\varepsilon = 1\/2$ get that\n\\[ \n\\Pr\\of{ \\left| |\\chi_L^{-1}([s])| -pq \\right| > \\frac 12 pq} \\le 2\\exp \\left\\{-\\frac{pq}{12} \\right\\}\n\\]\nand so by the union bound, the probability that there exists some $L$ such that\\linebreak $\\left| |\\chi_L^{-1}([s])| -pq \\right| > \\frac 12 pq$ is at most \n\\[\nq^2 \\cdot  2\\exp \\left\\{-\\frac{pq}{12} \\right\\} = 2\\exp \\left\\{2\\log q-\\frac{pq}{12} \\right\\} = o(1),\n\\] \nsince $p\\gg (\\log q) \/ \\lambda$ and $\\lambda \\le q$.\nThus, {\\textit{w.h.p. }} every line has between $\\frac 12 pq$ and $\\frac 32 pq$ many non-$L$-isolated vertices. \n\nNow for fixed $v$, let $X_v$ be the number of lines $L$ in which $v$ is non-$L$-isolated. $X_v$ is distributed as $\\textrm{Bin}(\\deg_{\\mathcal{H}_1} (v), p)$ which has expectation $\\deg_{\\mathcal{H}_1} (v)p  \\ge \\lambda p\/2$. Now by the Chernoff bound with $\\varepsilon = 1\/2$ we get \n\\[\n\\Pr\\of{|X_v - \\deg_{\\mathcal{H}_1} (v)p| > \\frac 12 \\deg_{\\mathcal{H}_1} (v)p} \\le 2 \\exp \\left\\{-\\frac{\\deg_{\\mathcal{H}_1} (v)p}{12} \\right\\} \\le 2 \\exp \\left\\{-\\frac{\\lambda p}{24} \\right\\}\n\\]\nand so by the union bound, the probability that there exists some point $v$ with $|X_v - \\deg_{\\mathcal{H}_1} (v)p| > \\frac 12 \\deg_{\\mathcal{H}_1} (v)p$ is at most \n\\[\nq^2 \\cdot 2 \\exp \\left\\{-\\frac{\\lambda p}{24} \\right\\} = 2 \\exp \\left\\{2\\log q-\\frac{\\lambda p}{24} \\right\\} = o(1),\n\\]\nsince $p \\gg (\\log q) \/ \\lambda$.\nSo {\\textit{w.h.p. }} for every $v$, $v$ is non-$L$-isolated for some number of lines $L$ that is at least  $\\frac 12 \\deg_{\\mathcal{H}_1} (v)p \\ge \\frac 14 \\lambda p$ and at most $\\frac 32 \\deg_{\\mathcal{H}_1} (v)p \\le \\frac 94 \\lambda p$.\n\nNow assume for each $L$ and each $v \\in L$ we have revealed whether $v$ is $L$-isolated, but we have not revealed $\\chi_L(v)$ when $v$ is non-$L$-isolated. When we do reveal values $\\chi_L(v)$ to form the graph $\\mathbb{G}_1$, we have for each non-$L$-isolated vertex $v$ that\n$\\deg_{\\mathbb{G}_1[L]}(v) \\sim \\textrm{Bin}(|\\chi_L^{-1}([s])|-1, \\frac{s-1}{s})$. Thus,\n$\\mathbb{E}(\\deg_{\\mathbb{G}_1[L]}(v)) = (|\\chi_L^{-1}([s])|-1)(s-1)\/s \\ge (\\frac 12 pq-1)(s-1)\/s$  and the Chernoff bound with $\\varepsilon = 1\/2$ tells us that \n\\begin{align*}\n\\Pr\\of{|\\deg_{\\mathbb{G}_1[L]}(v) - \\mathbb{E}(\\deg_{\\mathbb{G}_1[L]}(v))| \\ge \\frac 12 \\mathbb{E}(\\deg_{\\mathbb{G}_1[L]}(v))} &\\le 2\\exp\\left\\{- \\frac{\\mathbb{E}(\\deg_{\\mathbb{G}_1[L]}(v))}{12} \\right\\}\\\\\n& \\le 2\\exp\\left\\{- \\frac{(\\frac 12 pq-1)(s-1)\/s}{12} \\right\\}\n\\end{align*}\nso by the union bound, the probability that there is a vertex $v$ with $|\\deg_{\\mathbb{G}_1[L]}(v) - \\mathbb{E}(\\deg_{\\mathbb{G}_1[L]}(v))| \\ge \\frac 12 \\mathbb{E}(\\deg_{\\mathbb{G}_1[L]}(v))$ is at most \n\\[\nq^2 \\cdot 2\\exp\\left\\{- \\frac{(\\frac 12 pq-1)(s-1)\/s}{12} \\right\\} = o(1),\n\\] \nsince $p \\gg (\\log q) \/ \\lambda$ and $\\lambda \\le q$. Thus {\\textit{w.h.p. }} for every vertex $v$ and line $L$ for which $v$ is non-$L$-isolated we have that $\\deg_{\\mathbb{G}_1[L]}(v)$ is at least $\\frac 12 (|\\chi_L^{-1}([s])|-1)(s-1)\/s \\ge \\frac 18 pq$ and at most $\\frac 32 (|\\chi_L^{-1}([s])|-1)(s-1)\/s \\le \\frac 94 pq$. \n\nThus, for each vertex $v$, {\\textit{w.h.p. }} its degree in $\\mathbb{G}_1$ is $\\Theta( \\lambda q p^2)$.\n\n\\medskip\n\n\\textbf{\\ref{G1:c}:} Recall that $a$ is some positive integer and $b=\\left\\lceil(s+r)\/{ \\binom{a-1}{2} }\\right\\rceil +r.$\nFirst we show that every copy of $K_{s+r}$ in $\\mathbb{G}_1$ contains a dangerous subset in $\\mathcal{D}_a \\cup \\mathcal{D}_b$.\nLet $K$ be any copy of $K_{s+r}$ in $\\mathbb{G}_1$. Clearly if $K$ contains a subset of $a$ points in general position then such subset is in $\\mathcal{D}_a$. Assume that $K$ contains at most $a-1$ points in general position. These points can be covered by at most $\\binom{a-1}{2}$ lines. In fact, all of the  $s+r$ points must belong to those lines. Thus, there is a line $L$ with at least $\\lceil (s+r)\/\\binom{a-1}{2}\\rceil =b-r$ points from $K$. Moreover, since each line can contain at most $s$ points from $K$ there is a set $R$ of $r$ additional points in~$K$  that do not belong to~$L$. Hence, set $L\\cup R$ is in $\\mathcal{D}_b$. \n\nBy \\ref{H1:c},  $|\\mathcal{D}_a|  \\leq  \\frac{4\\lambda^{a^2 \/ 2} }{ q^{(a^2-5a)\/2}}$, and $| \\mathcal{D}_b| \\leq \\frac{4\\lambda^{br } }{ q^{b(r-1) - 5r^3}} $. We will show that {\\textit{w.h.p. }} none of these dangerous sets gives rise to a $K_{s+r}$ in $\\mathbb{G}_1$. In order for such dangerous set to give a $K_{s+r}$ in $\\mathbb{G}_1$, none of the vertices can be $L$-isolated in any of the lines in the dangerous set. For a Type 1 dangerous set, there are $\\binom{a}{2}$ lines, each containing $2$ vertices, so the probability that no vertex $v$ is $L$-isolated for $L$ containing $v$ is $p^{2 \\binom{a}{2}} = p^{a^2 - a}$. Thus, the expected number of copies of $K_{s+r}$ that arise from Type 1 dangerous sets  is at most\n\\[\n\\frac{4\\lambda^{a^2\/2}}{ q^{(a^2-5a)\/2}}\\cdot p^{a^2-a}.\n\\]\nNow for dangerous sets of Type 2, the probability that a dangerous set in $\\mathcal{D}_b$ gives rise to a $K_{s+r}$ is $p^x$, where $x$ is the number of point-line incidences there are within the dangerous set. We observed before that each Type~2 dangerous set consists of a line~$L$ with $b-r$ points and a set $R$ of $r$~points and at least $r(b-r) - r\\binom{r}{2}$ lines containing one point from $L$ and at least one point from $R$. Thus, the number of point-line incidences is at least\n\\[\n(b-r) + 2 \\left(r(b-r) - r\\binom{r}{2} \\right) \\ge (2r+1)b - 4r^3.\n\\]\nTherefore, the expected number of copies of $K_{s+r}$ arising from Type 2 dangerous sets is at most\n\\[\n\\frac{4\\lambda^{br}}{ q^{b(r-1) - 5r^3}} \\cdot p^{(2r+1)b - 4r^3}.\n\\]\n\nThus by linearity of expectation the total number of copies of $K_{s+r}$ has expectation at most\n\\[\n\\frac{4\\lambda^{a^2\/2} p^{a^2-a}}{ q^{(a^2-5a)\/2}} + \\frac{4\\lambda^{br} p^{(2r+1)b - 4r^3}}{ q^{b(r-1) - 5r^3}}\n\\]\nand so we are done by the Markov bound applied with $\\zeta= 2$. \n\\end{proof}\n\n\n\n\n\\subsection{Deriving the lower bound for small values of $\\delta$} \\label{sec:LB1}\n\n\\begin{theorem}\\label{thm:main1}\nLet $r\\ge 1$, $\\varepsilon>0$, and $\\frac 12 < \\delta \\le \\frac 12 + \\frac{1}{2(2r+1)}$. Then for all sufficiently large $s$, we have \n\\begin{equation}\\label{thm:main1:eq1}\n\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^\\delta) = \\Omega\\of{n^{2 - (1-\\delta)\/r  - \\varepsilon}}.\n\\end{equation}\nFurthermore, if $r\\ge 2$, then there exists a positive constant $C=C(s)$ such that for all sufficiently large $s$\n\\begin{equation}\\label{thm:main1:eq2}\n\\textup{\\textbf{RT}}_s(n, K_{s+r}, C\\sqrt{n}) = \\Omega\\of{n^{2 - 1\/(2r)  - \\varepsilon}}.\n\\end{equation}\n\\end{theorem}\n\n\n\\begin{proof}\nWe will apply Lemma~\\ref{lem:3} after discussing how to set parameters. First we prove~\\eqref{thm:main1:eq1} assuming that $\\frac 12 < \\delta \\le \\frac 12 + \\frac{1}{2(2r+1)}$.\n\nFix $r\\ge 1$ and $\\varepsilon>0$. It is known that for large $x$ there exists a prime number between $x$ and $x(1+o(1))$ (see, e.g.,~\\cite{BHP}). Hence, for large $n$ there is a prime number $q$ such that $\\sqrt{n} \\le q \\le (1+o(1))\\sqrt{n}$. Set $\\alpha = n^\\delta$ and $p = \\kappa q^{-(2 \\delta -1)}$, where $\\kappa = 20s\\log s$. \nWe will show that the above parameters satisfy all assumptions of Lemma~\\ref{lem:3} implying the existence of a graph $G_1$ of order $q^2$ satisfying \\ref{G1:a}-\\ref{G1:c}.\n\nFirst observe that $(10 s \\log s)q\/p = q^{2\\delta} \/ 2 \\le n^{\\delta} = \\alpha$, as required by Lemma~\\ref{lem:3}. Thus, due to~\\ref{G1:a} the $s$-independence number of $G_1$ is less than $\\alpha$. \n\nSet $a =20r$, and $b=\\left\\lceil(s+r)\/ \\binom{a-1}{2} \\right\\rceil +r$. We will assume that $s$, and consequently $b $, is large enough such that for example\n\\begin{equation}\\label{thm:main1:eps}\n\\varepsilon > 5r^2\/b.\n\\end{equation}\nFurthermore, let\n\\[\n\\lambda = q^{1-1\/r + (2\\delta -1)(2+1\/r) - 10r^2\/b}.\n\\]\nObserve that \n\\[\n\\lambda p = \\kappa q^{1-1\/r + (2\\delta -1)(1+1\/r) - 10r^2\/b} \\gg \\log q,\n\\]\nsince $\\delta>1\/2$ and $b$ is sufficiently large. This implies that $p \\gg (\\log q) \/ \\lambda$, as required in Lemma~\\ref{lem:3}. Clearly this also implies that $\\lambda \\gg \\log q$. Also note that since $\\delta \\le \\frac 12 + \\frac{1}{2(2r+1)}$,\n\\[\n1- \\frac 1r + (2\\delta -1)\\of{2+ \\frac 1r} - \\frac{10r^2}{b} \\le 1 - \\frac 1r + \\frac{1}{2r+1}\\of{2+ \\frac 1r} - \\frac{10r^2}{b}=1 - \\frac{10r^2}{b}\n\\] \nso $\\lambda < q$ and hence all assumptions of Lemma~\\ref{lem:3} are satisfied.\n\nNow we show that $G_1$ is $K_{s+r}$-free. By~\\ref{G1:c} The number of copies of $K_{s+r}$ is at most \n\\begin{equation} \\label{nums+r}\n8 \\left(\\frac{\\lambda^{a^2 \/2} p^{a^2 -a}}{ q^{(a^2-5a)\/2}} + \\frac{\\lambda^{br } p^{(2r+1)b - 4r^3}}{ q^{b(r-1) -5r^3}}\\right).\n\\end{equation}\nThe order of magnitude of the first term in~\\eqref{nums+r} is \n\\begin{align*}\n\\frac{q^{\\left(1-1\/r + (2\\delta -1)(2+1\/r) - 10r^2\/b\\right)  a^2\/2} \\cdot q^{(1-2\\delta)(a^2-a)}}{q^{(a^2-5a)\/2}}\n&= q^{(a^2\/2) \\cdot \\left( -(2-2\\delta)\/r - 10r^2\/b \\right) + (a\/2) \\cdot \\left( 2(2\\delta-1) +5\\right)} \\\\\n& \\le q^{-a^2\/(2r+1) + 7a\/2} = o(1),\n\\end{align*}\nwhere in the last line we used $\\delta \\le \\frac 12 + \\frac{1}{2(2r+1)}$ and $- 10r^2\/b < 0$, and the fact that $a=20r$. Now the second term of \\eqref{nums+r} has order of magnitude at most\n\\[\n\\frac{q^{\\left( 1-1\/r + (2\\delta -1)(2+1\/r) - 10r^2\/b\\right) \\cdot br} q^{(1-2\\delta)\\of{(2r+1)b-4r^3}} }{ q^{b(r-1) -5r^3}}\n=q^{-5r^3 + 4r^3(2\\delta-1)} \\le q^{-r^3} = o(1).\n\\]\n\nNow let $G$ be any induced subgraph of $G_1$ of order $n$. Clearly, $G$ is $K_{s+r}$-free with $\\alpha_s(G) < n^{\\delta}$. Furthermore, \n\\begin{align*}\n|E(G)| &\\ge |E(G_1)| - |V(G_1) - V(G)| \\cdot \\Delta(G_1)\\\\ \n& \\ge |V(G_1)|\\cdot \\delta(G_1) - o(1) \\cdot \\Delta(G_1)  = \\Omega\\of{ q^2 \\cdot \\lambda q p^2},\n\\end{align*}\nby~\\ref{G1:b}. Finally observe that \n\\begin{align*}\nq^2 \\cdot \\lambda q p^2 = \\lambda q^3 p^2 &\\ge q^{1-1\/r + (2\\delta -1)(2+1\/r) - 10r^2\/b} \\cdot q^3 \\cdot \\of{ q^{1-2\\delta}}^2\\\\\n& \\ge q^{4-2(1-\\delta)\/r - 10r^2\/b} = n^{2-(1-\\delta)\/r - 5r^2\/b } >  n^{2-(1-\\delta)\/r - \\varepsilon},\n\\end{align*}\nbecause of~\\eqref{thm:main1:eps}. Thus, $|E(G)| = \\Omega(n^{2-(1-\\delta)\/r - \\varepsilon})$ yielding the lower bound in~\\eqref{thm:main1:eq1}.\n\n\nThe proof of~\\eqref{thm:main1:eq2} is very similar. Assume that $r\\ge 2$ and let $\\delta=1\/2$, $p=1$, and  $\\alpha = C \\sqrt{n}$ with $C=20s(\\log s)$. Other parameters are the same. Then, as in the previous case the assumptions of Lemma~\\ref{lem:3} hold.\n\\end{proof}\n\n\n\\section{Lower bound on $\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^\\delta)$ for intermediate values of  $\\delta$}\\label{sec:main2}\n\nRecall that in Section~\\ref{sec:main1} we started with the affine plane and made it sparser by taking edges with probability $\\lambda \/ q$. In Section~\\ref{sec:LB1}, to optimize our result we set \n\\[\n\\lambda = q^{1-1\/r + (2\\delta -1)(2+1\/r) - 10r^2\/b},\n\\]\nso when $\\delta = \\frac 12 + \\frac{1}{2(2r+1)}$, we are setting $\\lambda$ to be nearly $q$ (since $r^2 \/ b$ is small), which is about as large as $\\lambda$ can possibly be (and then we are making our hypergraph $\\mathcal{H}_1$ nearly as dense as the original affine plane). Thus it makes sense that if $\\delta$ is bigger than $\\frac 12 + \\frac{1}{2(2r+1)}$ then we no longer want a sparser version of the affine plane, but a denser version. Therefore in this section we will construct a denser hypergraph $\\mathcal{H}_2$ by keeping all of the edges and ``eliminating\" some vertices. That is the key difference between Sections~\\ref{sec:main1} and~\\ref{sec:main2}. Otherwise the proofs are quite similar.\n\n\\subsection{The hypergraph $\\mathcal{H}_2$} \\label{sec:H2}\n\nIn this section we establish the existence of a hypergraph $\\mathcal{H}_2$ with certain properties.\n\n\\begin{lemma} \\label{lem:4}\nLet $q$ be a sufficiently large prime and $\\log q \\ll \\lambda \\le q$. \nThen, there exists a $q$-regular hypergraph $\\mathcal{H}_2 = (V_2, \\mathcal{L}_2)$ of order $\\lambda q (1+o(1))$ such that:\n\\begin{enumerate}[label=$({\\textup{H}}_{2}\\textup{\\alph*})$]\n\\item\\label{H2:a} Any two vertices are contained in at most one hyperedge;\n\\item\\label{H2:b} For every $L \\in \\mathcal{L}_2$, $\\frac{\\lambda}{2} \\le |L| \\leq \\frac{3\\lambda}{2}$; \n\\item\\label{H2:c} $|\\mathcal{D}_a|  \\leq  4\\lambda^a q^a$ and $| \\mathcal{D}_b| \\leq 4\\lambda^b q^{r+2}$, where $\\mathcal{D}_a$ is the set of all dangerous sets of Type~1 and size~$a$ and $\\mathcal{D}_b$ is the set of all dangerous sets of Type~2 and size~$b$.\n\\end{enumerate}\n\\end{lemma}\n\n\n\n\\begin{proof}\nStarting with the hypergraph $\\mathcal{H}$, we randomly ``eliminate\" some points. For each $v\\in V$ we randomly (and independently from other vertices) choose to eliminate $v$  with probability $1 -  \\lambda \/ q$. Say $X$ is the set of vertices chosen for elimination. By ``elimination\", we mean that we will form a new hypergraph $\\mathbb{H}_2$ with vertex set $V_2 = V \\setminus X$, and edge set \n\\[\n\\mathcal{L}_2 = \\{L \\setminus X: L \\in \\mathcal{L} \\}.\n\\]\n\nFirst we will prove that {\\textit{w.h.p. }} $\\mathbb{H}_2$ has $\\lambda q (1+o(1))$ vertices. The number of vertices~$V$ is distributed as $\\textrm{Bin}(q^2, \\lambda \/ q)$ which has expectation $\\lambda q$. By the Chernoff bound with $\\varepsilon = (\\lambda q)^{-1\/4}$,\n\\[\n\\Pr\\of{|V - \\lambda q| > (\\lambda q)^{3\/4}} \\le 2 \\exp \\left\\{ - \\frac{(\\lambda q )^{1\/2}}{3}\\right\\} = o(1).\n\\]\n\nNow by the construction of $\\mathbb{H}_2$ and the properties of $\\mathcal{H}$, two vertices are in at most one line, so $\\mathbb{H}_2$ always satisfies \\ref{H2:a}. \nWe will show that $\\mathbb{H}_2$ satisfies {\\textit{w.h.p. }} \\ref{H2:b} and satisfies \\ref{H2:c} with probability at least $1\/2$. \nTogether this implies $\\mathbb{H}_2$ satisfies \\ref{H2:a}-\\ref{H2:c} with probability at least $1-\\frac{1}{2}-o(1)$, establishing the existence of a hypergraph $\\mathcal{H}_2$ that satisfies \\ref{H2:a}-\\ref{H2:c}. \n\n\\medskip\n\n\\textbf{\\ref{H2:b}:}  Observe that for fixed $L \\in \\mathcal{L}_2$, $|L| \\sim \\textrm{Bin}(q, \\frac{\\lambda}{q})$ and $\\mathbb{E}(|L|)= \\lambda.$ So by the Chernoff bound with $\\varepsilon = \\frac{1}{2}$, \n\\[\n \\Pr \\Big( \\left||L|-\\lambda\\right| \\geq \\frac{\\lambda}{2} \\Big) \\leq 2  \\exp \\left\\{ -\\frac{\\lambda}{12} \\right\\}. \n\\]\nThus by the union bound the probability that there exists some $L \\in \\mathcal{L}_2$ with $|L| \\notin \\sqbs{\\frac{\\lambda}{2} , \\frac{3\\lambda}{2}}$ is at most\n\\[\nq^2 \\cdot 2  \\exp \\left\\{-\\frac{\\lambda}{12}\\right\\} = 2 \\exp \\left\\{2 \\log q-\\frac{\\lambda}{12}\\right\\} = o(1).\n\\]\n\n\\medskip\n\n\\textbf{\\ref{H2:c}:} In order to show that both $|\\mathcal{D}_a|  \\leq  4\\lambda^{a} q^a$ and $|\\mathcal{D}_b| \\leq 4\\lambda^b q^{r+2}$ with probability at least $\\frac{1}{2}$, we recall from Section~\\ref{sec:H1} the number of dangerous subsets of each type. The number of Type~1 dangerous subsets is at most ${q^2 \\choose a}$, and the number of Type~2 dangerous subsets is at most $\\binom{q^2}{1} \\cdot \\binom{q}{b-r} \\cdot q^{2r}$. In order for $\\mathbb{H}_2$ to inherit a dangerous set, none of its vertices can be eliminated. \nBy the linearity of expectation, we now compute \n\\[\n\\mathbb{E}(| \\mathcal{D}_a|) \\leq \\binom{q^2}{a}  \\cdot \\left( \\frac{\\lambda}{q} \\right)^{a} \\leq \\lambda^a q^a\n\\]\nand\n\\[\n\\mathbb{E}(|\\mathcal{D}_b|) \n\\leq  \\binom{q^2}{1} \\cdot \\binom{q}{b-r} \\cdot q^{2r} \\cdot \\left( \\frac{\\lambda}{q} \\right)^{b}\n\\leq   q^{2} \\cdot q^{b-r} \\cdot q^{2r} \\cdot \\left( \\frac{\\lambda}{q} \\right)^{b} = \\lambda^b q^{r+2},\n\\]\nand we are done by the Markov bound applied twice with $\\zeta=4$.\n\\end{proof}\n\n\\subsection{The graph $G_2$}\\label{sec:G2}\n\nBased upon the hypergraph $\\mathcal{H}_2$ established in the previous section, we will construct a graph $G_2$ with the following properties. \n\n\\begin{lemma} \\label{lem:5}\nLet $r\\ge 1$ and $s$ be sufficiently large constant.\nLet  $q$ be a sufficiently large prime, $q \\ge \\lambda \\gg \\log q $, $1 \\ge p \\gg (\\log q) \/ \\lambda$, and $\\alpha \\ge (10 s \\log s)q\/p$. Furthermore, let $a$ be a positive constant and $b=\\left\\lceil(s+r)\/\\binom{a-1}{2}\\right\\rceil +r$. Then, there exists a graph $G_2$ with $\\lambda q (1+o(1))$ vertices such that:\n\\begin{enumerate}[label=$({\\textup{G}}_{2}\\textup{\\alph*})$]\n\\item\\label{G2:a} $\\alpha_s(G_2) < \\alpha$;\n\\item\\label{G2:b} For every vertex $v$ in $G_2$, $ \\deg_{G_2}(v)= \\Theta( \\lambda q p^2)$;\n\\item\\label{G2:c} $G_2$ has at most  $8\\left(\\lambda^a q^a p^{a^2 -a} +  \\lambda^b q^{r+2} p^{(2r+1)b - 4r^3}\\right)$ copies of  $K_{s+r}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nStarting with the hypergraph $\\mathcal{H}_2=(V_2,\\mathcal{L}_2)$, we form the random graph $\\mathbb{G}_2=(V_2,E)$ as follows. For every $L\\in \\mathcal{L}_2$, let $\\chi_L : L \\to  {[s+1]}$ be a random partition of the vertices of $L$ into $ {s+1}$ classes, where for every $v \\in L$,\n\\[\n\\Pr(\\chi(v) = i) = \n\\begin{cases}\n{p}\/{s} & \\text{for } 1\\le i\\le s,\\\\\n1-p & \\text{for } i=s+1,\n\\end{cases}\n\\]\nand $\\chi(v)$ is assigned independently from other vertices.\nIf $\\chi_L(v) =s+1$, then we say that $v$ is {\\em $L$-isolated}. Let $\\{x,y\\} \\in E$ if $\\{x,y\\} \\subseteq L$ for some $L \\in \\mathcal{L}_2$ and $\\chi_L(x),  \\chi_L(y)$ are distinct and neither $x$ nor $y$ is $L$-isolated. Thus for every $L \\in \\mathcal{L}_2$, $\\mathbb{G}_2[L]$ consists of a set of isolated vertices (the $L$-isolated vertices) together with a complete $s$-partite graph with vertex partition $L = \\chi^{-1}_L (1) \\cup \\chi^{-1}_L (2) \\cup \\dots \\cup \\chi^{-1}_L (s)$.\n\nWe will show that $\\mathbb{G}_2$ satisfies {\\textit{w.h.p. }} \\ref{G2:a} and \\ref{G2:b} and satisfies \\ref{G2:c} with probability at least $1\/2$. \n\n\\medskip\n\n\\textbf{\\ref{G2:a}:} First fix $C \\in {V_2 \\choose \\alpha}$. We will bound the probability that $\\mathbb{G}_2[C] \\not \\supseteq K_s$. For a fixed $L$, the probability that one of the classes $\\chi_L^{-1}(1), \\dots, \\chi_L^{-1}(s)$ contains no element of $C$ is at most $s\\of{1-\\frac ps}^{|L \\cap C|} \\le s e^{-\\frac {p}{s} |L \\cap C|}$. Note that \n\\[\n\\sum_{L \\in \\mathcal{L}_2} |L \\cap C| = q |C| =   \\alpha q,\n\\] \nsince each point is in  $q$ lines. Now since $\\chi_L$ and $\\chi_{L'}$ are chosen independently for $L \\neq L'$ we get, \n\\begin{align*}\n\\Pr\\Big( K_s \\not \\subseteq \\mathbb{G}_2[C] \\Big) &\\le \\prod_{L \\in \\mathcal{L}_2}\\Pr \\Big( K_s \\not \\subseteq \\mathbb{G}_2[L \\cap C] \\Big)\\\\\n& \\le s^{q^2} \\exp\\left\\{ -\\frac ps \\sum_{L \\in \\mathcal{L}_2} |L \\cap C| \\right\\} \\le \\exp \\left\\{(\\log s)  q^2  - \\frac{1}{s} \\alpha q p \\right\\}\n\\end{align*} \n\nSo by the union bound, the probability that there exists a set $C$ of $\\alpha$ vertices in $\\mathbb{G}_2$ that contains no $K_s$ is at most\n\\[\nq^{2 \\alpha} \\cdot \\exp \\left\\{(\\log s)  q^2  - \\frac{1}{s} \\alpha q p \\right\\} \\le   \\exp \\left\\{2\\alpha  \\log q +(\\log s)  q^2  - \\frac{1}{s} \\alpha q p \\right\\}=o(1),\n\\]\nsince $p \\gg (\\log q) \/ \\lambda$, $\\lambda \\le q$, and  $\\alpha \\ge (10 s \\log s) q\/p$. \n\nThus, {\\textit{w.h.p. }} $\\alpha_s(\\mathbb{G}_2) < \\alpha$.\n\n\\medskip\n\n\n\n\\textbf{\\ref{G2:b}:} Observe that for a fixed $L\\in \\mathcal{L}_2$, the number of non-$L$-isolated vertices $|\\chi_L^{-1}([s])|$ is distributed as $\\textrm{Bin}(|L|, p)$ which has expectation $p|L| $ so by the Chernoff bound with $\\varepsilon = 1\/2$ we get that\n\\[\n\\Pr\\of{ \\left| |\\chi_L^{-1}([s])| -p|L| \\right| > \\frac 12 p|L|} \\le 2\\exp \\left\\{-\\frac{p|L|}{12} \\right\\}\\le 2\\exp \\left\\{-\\frac{p\\lambda}{24} \\right\\}\n\\]\nand so by the union bound, the probability that there exists some $L$ such that\\linebreak \n$\\left| |\\chi_L^{-1}([s])| -p|L| \\right| >  \\frac 12 p |L|$ is at most \n\\[\nq^2 \\cdot  2\\exp \\left\\{-\\frac{p \\lambda}{24} \\right\\} = o(1),\n\\]\nsince $p\\gg (\\log q) \/ \\lambda$.\nThus, {\\textit{w.h.p. }} every line $L$ has at least $\\frac 12 p|L| \\ge \\frac 14 p \\lambda$ and at most $\\frac 32 p|L| \\le \\frac 94 p \\lambda$ many non-$L$-isolated vertices. \n\nNow for fixed $v$, let $X_v$ be the number of lines $L$ in which $v$ is non-$L$-isolated. $X_v$ is distributed as $\\textrm{Bin}(q , p)$ which has expectation $qp $. Now by the Chernoff bound with $\\varepsilon = 1\/2$ we get \n\\[\n\\Pr\\of{|X_v - qp| > \\frac 12 qp} \\le 2 \\exp \\left\\{-\\frac{qp}{12} \\right\\}\n\\] \nand so by the union bound, the probability that there exists some point $v$ with $|X_v - qp| > \\frac 12 qp$ is at most \n\\[\nq^2 \\cdot 2 \\exp \\left\\{-\\frac{q p}{12} \\right\\} = o(1),\n\\]\nsince $ p \\gg (\\log q) \/ \\lambda$ and $\\lambda \\le q$.\nSo {\\textit{w.h.p. }} for every $v$, $v$ is non-$L$-isolated for some number of lines $L$ that is between  $\\frac 12 qp $ and at most $\\frac 32qp$.\n\nNow assume for each $L$ and each $v \\in L$ we have revealed whether $v$ is $L$-isolated, but we have not revealed $\\chi_L(v)$ when $v$ is non-$L$-isolated. When we do reveal values $\\chi_L(v)$ to form the graph $\\mathbb{G}_2$, we have for each non-$L$-isolated vertex $v$ that\n$\\deg_{\\mathbb{G}_2[L]}(v) \\sim \\textrm{Bin}(|\\chi_L^{-1}([s])|-1, \\frac{s-1}{s})$. Thus,\n$\\mathbb{E}(\\deg_{\\mathbb{G}_2[L]}(v)) = (|\\chi_L^{-1}([s])|-1)(s-1)\/s \\ge (\\frac 14 p\\lambda-1)(s-1)\/s$  and the Chernoff bound with $\\varepsilon = 1\/2$ tells us that \n\\begin{align*}\n\\Pr\\of{|\\deg_{\\mathbb{G}_2[L]}(v) - \\mathbb{E}(\\deg_{\\mathbb{G}_2[L]}(v))| \\ge \\frac 12 \\mathbb{E}(\\deg_{\\mathbb{G}_2[L]}(v))} &\\le 2\\exp\\left\\{- \\frac{\\mathbb{E}(\\deg_{\\mathbb{G}_2[L]}(v))}{12} \\right\\}\\\\ \n&\\le 2\\exp\\left\\{- \\frac{(\\frac 14 p\\lambda-1)(s-1)\/s}{12} \\right\\}\n\\end{align*}\nso by the union bound, the probability that there is a vertex $v$ with $|\\deg_{\\mathbb{G}_2[L]}(v) - \\mathbb{E}(\\deg_{\\mathbb{G}_2[L]}(v))| \\ge \\frac 12 \\mathbb{E}(\\deg_{\\mathbb{G}_2[L]}(v))$ is at most \n\\[\nq^2 \\cdot 2\\exp\\left\\{- \\frac{(\\frac 14 p\\lambda-1)(s-1)\/s}{12} \\right\\} = o(1),\n\\]\nsince $p \\gg (\\log q) \/ \\lambda$. Thus {\\textit{w.h.p. }} for every vertex $v$ and line $L$ for which $v$ is non-$L$-isolated we have that $\\deg_{\\mathbb{G}_2[L]}(v)=\\Theta(p \\lambda)$. \n\nThus, for each vertex $v$, {\\textit{w.h.p. }} its degree in $\\mathbb{G}_2$ is $\\Theta( \\lambda q p^2)$.\n\n\\medskip\n\n\\textbf{\\ref{G2:c}:} Recall that $a$ is some positive integer and $b=\\left \\lceil(s+r)\/ { \\binom{a-1}{2} }\\right \\rceil +r$. \nFirst we show that every $K_{s+r}$ in $\\mathbb{G}_2$ contains a dangerous subset in $\\mathcal{D}_a \\cup \\mathcal{D}_b$.\nLet $K$ be any copy of $K_{s+r}$ in $\\mathbb{G}_2$. Similarly to the graph $\\mathbb{G}_1$, we can conclude that the vertices of $K$ must contain a dangerous set in $\\mathcal{D}_a$ or in $\\mathcal{D}_b$.\n\nIn order for such a dangerous set to give a $K_{s+r}$ in $\\mathbb{G}_2$  none of the vertices can be $L$-isolated in any of the lines in the dangerous set. For a Type 1 dangerous set, there are $\\binom{a}{2}$ lines, each containing $2$ vertices, so the probability that no vertex $v$ is $L$-isolated for $L$ containing $v$ is $p^{2 \\binom{a}{2}} = p^{a^2 - a}$. Thus, the expected number of copies of $K_{s+r}$ that arise from Type 1 dangerous sets  is at most $4\\lambda^{a}q^a p^{a^2-a}.$ \nNow for dangerous sets of Type 2, the probability that a dangerous set in $\\mathcal{D}_b$ gives rise to a $K_{s+r}$ is $p^x$, where $x$ is the number of point-line incidences there are within the dangerous set. As in Section~\\ref{sec:G1} the number of point-line incidences is at least $  (2r+1)b - 4r^3$. Therefore the expected number of copies of $K_{s+r}$ arising from Type~2 dangerous sets is at most $4\\lambda^b q^{r+2} p^{(2r+1)b - 4r^3}$.\n\nThus by linearity of expectation the total number of copies of $K_{s+r}$ has expectation at most \n\\[\n4\\lambda^{a}q^a p^{a^2-a} + 4\\lambda^b q^{r+2} p^{(2r+1)b - 4r^3}\n\\] \nand so we are done by the Markov bound applied with $\\zeta=2$. \n\n\n\\end{proof}\n\n\\subsection{Deriving the lower bound for intermediate values of  $\\delta$}\n\n\\begin{theorem}\\label{thm:main2}\nLet $r\\ge 1$, $\\varepsilon>0$, and $\\frac 12 + \\frac{1}{2(2r+1)} < \\delta < 1$. Then for all sufficiently large $s$, we have \n\\[\n\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^\\delta) = \\Omega\\of{n^{2 - (1-\\delta)\/r  - \\varepsilon}}.\n\\]\n\\end{theorem}\n\nFirst we state a proposition that we will use to estimate fractions that have ``error\" in the numerator and denominator. \n\n\\begin{proposition} \\label{estlem}\n\nFor any real numbers $x, y, \\epsilon_x, \\epsilon_y$, if $x,y \\neq 0$ and $|\\frac{\\epsilon_x}{x}|, |\\frac{\\epsilon_y}{y}| \n\\le \\frac{1}{2}$, then \n\\[\n\\left|\\frac{x+ \\epsilon_x}{y+\\epsilon_y} - \\frac{x}{y} \\right| \\le \\frac{|\\epsilon_x y| + 3|\\epsilon_y x|}{y^2}. \n\\]\n\\end {proposition}\n\n\n\\begin{proof}\nObserve that \n\\begin{align*}\n\\left|\\frac{x+ \\epsilon_x}{y+\\epsilon_y} - \\frac{x}{y}\\right| &= \\left|\\frac{x}{y} \\left[ \\left(1+\\frac{\\epsilon_x}{x} \\right) \\cdot \\frac{1}{1+\\frac{\\epsilon_y}{y}} -1 \\right] \\right|\\\\\n&= \\left|\\frac{x}{y} \\left[ \\left(1+\\frac{\\epsilon_x}{x} \\right) \\cdot \\left(1 + \\sum_{n=1}^{\\infty} \\of{-\\frac{\\epsilon_y}{y}}^n \\right) -1 \\right]\\right|\\\\\n&= \\left|\\frac{x}{y} \\left[ \\frac{\\epsilon_x}{x}  +\\of{1+\\frac{\\epsilon_x}{x}}\\sum_{n=1}^{\\infty} \\of{-\\frac{\\epsilon_y}{y}}^n \\right]\\right|\\\\\n&\\le\\left| \\frac{ \\epsilon_x }{y}\\right| + \\left|  \\frac{x}{y} \\right| \\cdot \\left|1+\\frac{\\epsilon_x}{x}\\right| \\cdot \\sum_{n=1}^{\\infty} \\left|\\frac{\\epsilon_y}{y}\\right|^n\\\\\n&\\le\\left| \\frac{ \\epsilon_x }{y}\\right| + \\left|  \\frac{x}{y} \\right| \\cdot \\frac{3}{2} \\cdot 2 \\left|\\frac{\\epsilon_y}{y}\\right|.\n\\end{align*}\nIn the last line we have used $|\\epsilon_x \/ x| \\le 1\/2$ and\n\\[\n\\sum_{n=1}^{\\infty} \\left|\\frac{\\epsilon_y}{y}\\right|^n = \\left|\\frac{\\epsilon_y}{y}\\right| \\sum_{n=0}^{\\infty} \\left|\\frac{\\epsilon_y}{y}\\right|^n \\le \\left|\\frac{\\epsilon_y}{y}\\right| \\cdot 2\n\\] \nwhich follows from $|\\epsilon_y \/ y| \\le 1\/2$.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:main2}]\nWe will apply Lemma~\\ref{lem:5} after discussing how to set parameters.\n\nFix $r\\ge 1$, $\\varepsilon>0$ and $\\frac 12 + \\frac{1}{2(2r+1)} < \\delta < 1$. Set \n\\[\na = 2 + \\max\\left\\{ \\left\\lceil \\frac{1}{\\delta}  \\right\\rceil , \\;\\;\\left\\lceil  \\of{\\frac{(1-\\delta)(2r+1)}{\\delta(2r+1)-1} + 1} \\cdot \\frac{\\delta(2r+1)-1}{1-\\delta} \\right\\rceil \\right\\}\n\\]\nand\n\\[\nb=\\left \\lceil(s+r) \/ \\binom{a-1}{2} \\right \\rceil +r.\n\\] \nWe will assume that $s$, and consequently $b$, is large enough which satisfies for instance the following\n\\begin{equation}\\label{thm:main2:eps}\n\\varepsilon > 104r^2\/b.\n\\end{equation}\n\nSimilarly as in the proof of Theorem~\\ref{thm:main1} for large $n$ there is a prime number $q$ such that \n\\[\nn^{1\/\\of{\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3}+1}} \\le q\n\\le (1+o(1)) n^{1\/\\of{\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3}+1}}.\n\\]\n(It will be shown soon that the exponents are positive.)\nLet \n\\[\n\\lambda = q^\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3} \\quad \\text{ and }\\quad \np = \\kappa q^{1 - \\delta \\of{\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3}+1}},\n\\]\nwhere $\\kappa = 20 s \\log s$. Finally set $\\alpha = n^\\delta $. \n\nWe will show that the above parameters satisfy all assumptions of Lemma~\\ref{lem:5} implying the existence of a graph $G_2$ of order $\\lambda q (1+o(1))$ satisfying \\ref{G2:a}-\\ref{G2:c}.\n\nFirst observe that \n\\[\n\\frac{(10s\\log s) q}{p} = \\frac{q^{\\delta\\of{\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3}+1}}}{2}\n=\\frac{(1+o(1))n^{\\delta}}{2} \\le \\alpha.\n\\]\nyielding by~\\ref{G2:a} that the $s$-independence number of $G_2$ is less than $\\alpha$. \n\nNow we examine the exponent of~$\\lambda$. Clearly, \n\\[\n\\lim_{b\\to\\infty} \\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3} = \\frac{(1-\\delta)(2r+1) }{\\delta(2r+1)-1}.\n\\]\nWe will show that for $b \\ge 20r^2$ the exponent is very close to the above limit. Indeed, since $r \\ge 1$ and $1\/2  < \\delta < 1$, we have \n\\[\n\\left|\\frac{  -4(1-\\delta)r^3 +r+3}{(1-\\delta)(2r+1)b} \\right| \\le \\frac{8r^3}{rb} \\le \\frac 12 \\quad \\text{ and }  \\quad  \\left|\\frac{-4\\delta r^3}{(\\delta(2r+1)-1)b} \\right| \\le \\frac{8r^3}{rb} \\le \\frac 12\n\\]\nand so by Proposition~\\ref{estlem} we obtain\n\\begin{align*}\n&\\left|\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{(\\delta(2r+1)-1)b - 4\\delta r^3} - \\frac{(1-\\delta)(2r+1) }{\\delta(2r+1)-1} \\right| \\\\ \n&\\qquad\\qquad\\qquad\\le \\frac{\\of{4(1-\\delta)r^3 +r+3}(\\delta(2r+1)-1)b + 3\\cdot 4\\delta r^3(1-\\delta)(2r+1)b  }{\\of{(\\delta(2r+1)-1)b}^2}\\\\\n&\\qquad\\qquad\\qquad\\le \\frac{\\of{8r^3}(2r)b + 12r^3(3r)b  }{\\of{rb}^2} = \\frac{52r^2}{b},\n\\end{align*}\nwhere the last inequality follows from $r \\ge 1$ and $1\/2 < \\delta < 1$.  Also note that  \n\\[\n\\frac{(1-\\delta)(2r+1) }{\\delta(2r+1)-1} \\ge 1-\\delta\n\\] \nand\n\\[\n\\frac{(1-\\delta)(2r+1) }{\\delta(2r+1)-1} = 1- \\frac{(2\\delta-1)(2r+1)-1}{\\delta(2r+1)-1} \\le  1- \\frac{(2\\delta-1)(2r+1)-1}{2r}.\n\\]\nObserve that since $\\delta > \\frac 12 + \\frac{1}{2(2r+1)}$, we get $\\frac{(2\\delta-1)(2r+1)-1}{2r} > 0$. Thus, if \n\\[\n\\frac{52r^2}{b} < \\min\\left\\{1-\\delta, \\frac{(2\\delta-1)(2r+1)-1}{2r} \\right\\},\n\\] \nthen\n\\begin{equation}\\label{eq:main2:1}\n\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{(\\delta(2r+1)-1)b - 4\\delta r^3} \\le \\frac{(1-\\delta)(2r+1) }{\\delta(2r+1)-1} + \\frac{52r^2}{b} < 1\n\\end{equation}\nand\n\\begin{equation}\\label{eq:main2:2}\n\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{(\\delta(2r+1)-1)b - 4\\delta r^3} \\ge \\frac{(1-\\delta)(2r+1) }{\\delta(2r+1)-1} - \\frac{52r^2}{b} > 0.\n\\end{equation}\nConsequently, $\\lambda$ is at least $q^{\\Omega(1)}$ and less than $q$, so $\\log q \\ll \\lambda \\le q$, as required in Lemma~\\ref{lem:5}.\n\nFurthermore, \n\\[\n\\lambda p = \\kappa q^{1+(1-\\delta)\\of{ \\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3} } - \\delta}\n\\ge \\kappa q^{1-\\delta} \\gg \\log q\n\\]\nand\n\\begin{align*}\np &= \\kappa q^{1 - \\delta \\of{\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3}+1}}\n\\overset{\\eqref{eq:main2:2}}{\\le} \\kappa q^{ 1-\\delta\\of{ \\frac{(1-\\delta)(2r+1) }{\\delta(2r+1)-1} - \\frac{52r^2}{b} +1  }  } \\\\\n&= \\kappa q^{ 1-\\delta\\of{ \\frac{1-\\delta}{\\delta} + \\frac{1-\\delta}{\\delta\\of{\\delta(2r+1)-1}} - \\frac{52r^2}{b} +1  }  } \n= \\kappa q^{-\\frac{1-\\delta}{\\delta(2r+1)-1} + \\frac{52r^2\\delta}{b}} = o(1)\n\\end{align*}\nfor $b$ sufficiently large. Consequently, $1 \\ge p \\gg (\\log q) \/ \\lambda$ and all assumptions of Lemma~\\ref{lem:5} are satisfied.\n\n\nNow we will see that our choice of parameters makes $G_2$ a $K_{s+r}$-free graph. By~\\ref{G2:c}, the number of copies of $K_{s+r}$ is at most \n\\begin{equation} \\label{nums+r2}\n8\\left(\\lambda^a q^a p^{a^2 -a} +  \\lambda^b q^{r+2} p^{(2r+1)b - 4r^3}\\right).\n\\end{equation}\nThe first term is on the order of $\\lambda^a q^a p^{a^2 -a} = O\\of{\\left(\\lambda q p^{a-1}\\right)^a}$.\nWe show that $\\lambda q p^{a-1} = o(1)$. The order of magnitude of the latter is\n\\begin{align*}\nq^\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3} &\\cdot q \\cdot q^{\\of{1 - \\delta \\of{\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3}+1}}(a-1)}\\\\\n&=q^{\\of{\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3} }\\cdot (1-\\delta(a-1)) + 1+(1-\\delta)(a-1)}  \\\\\n&\\overset{\\eqref{eq:main2:1}}{\\le} q^{\\of{ \\frac{(1-\\delta)(2r+1)}{\\delta(2r+1)-1} + \\frac{52r^2}{b} }\\cdot (1-\\delta(a-1)) + 1+(1-\\delta)(a-1)}  \\\\\n&=q^{ \\frac{(1-\\delta)(2r+1)}{\\delta(2r+1)-1} + \\frac{52r^2}{b}(1- \\delta(a-1))  + 1 + \\of{1 - \\delta - \\frac{(1-\\delta)(2r+1)}{\\delta(2r+1)-1} \\delta }(a-1)} \\\\\n&\\le q^{ \\frac{(1-\\delta)(2r+1)}{\\delta(2r+1)-1}   + 1 + \\of{1 - \\frac{(2r+1)\\delta}{\\delta(2r+1)-1} }(1-\\delta)(a-1)} \\\\\n&= q^{ \\frac{(1-\\delta)(2r+1)}{\\delta(2r+1)-1} + 1 -  \\frac{1}{\\delta(2r+1)-1} (1-\\delta)(a-1)} = o(1),\n\\end{align*}\nwhere the second to last line follows from $a > 1+ 1\/\\delta$ and the last line follows from\n\\[\na > 1 + \\of{\\frac{(1-\\delta)(2r+1)}{\\delta(2r+1)-1} + 1} \\cdot \\frac{\\delta(2r+1)-1}{1-\\delta}.\n\\] \nThus, $\\lambda^a q^a p^{a^2 -a}=o(1)$.\nNow we bound the order of the magnitude of the second term in~\\eqref{nums+r2}. Observe that \n\\begin{align*}\nq^{\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3} b} &\\cdot q^{r+2} \\cdot q^{\\of{1 - \\delta \\of{\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3}+1}}\\of{(2r+1)b - 4r^3}} = \\frac{1}{q} = o(1).\n\\end{align*}\nThus, $G_2$ is $K_{s+r}$-free.\n\nNow let $G$ be any induced subgraph of $G_2$ of order $n$. Clearly, $G$ is $K_{s+r}$-free with $\\alpha_s(G) < n^{\\delta}$. Furthermore, since $n = (1+o(1))\\lambda q$,\n\\begin{align*}\n|E(G)| &\\ge |E(G_2)| - |V(G_2) - V(G)| \\cdot \\Delta(G_2)\\\\ \n& \\ge |V(G_2)|\\cdot \\delta(G_2) - o(1) \\cdot \\Delta(G_2)  = \\Omega\\of{ \\lambda q \\cdot \\lambda q p^2} = \\Omega(np^2),\n\\end{align*}\nby~\\ref{G2:b}. Since \n\\[\np = (1+o(1))\\kappa n^{1\/\\of{\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3}+1}-\\delta}\n\\overset{\\eqref{eq:main2:1}}{\\ge} (1+o(1))\\kappa n^{1\/\\of{\\frac{(1-\\delta)(2r+1) }{\\delta(2r+1)-1} + \\frac{52r^2}{b}+1}-\\delta},\n\\]\nwe get\n\\[\n\\Omega\\of{n^2 p^2} \n= \\Omega \\of{n^{2-2\\delta + 2\/\\of{1+ \\frac{(1-\\delta)(2r+1) }{\\delta(2r+1)-1} + \\frac{52r^2}{b}}} }\n= \\Omega \\of{n^{2-2\\delta + 2\/\\of{\\frac{2r }{\\delta(2r+1)-1} + \\frac{52r^2}{b}}} }.\n\\]\nSince for any positive real numbers $x, y, z$ with $y > z$,\n\\[\n\\frac{x}{y+z} = \\frac{x}{y} \\cdot \\frac{1}{1+\\frac{z}{y}} \\ge \\frac xy \\cdot \\of{1- \\frac zy} = \\frac{x}{y} - \\frac{xz}{y^2},\n\\]\nwe get\n\\[\n2\\left\/\\right.\\!\\of{\\frac{2r }{\\delta(2r+1)-1} + \\frac{52r^2}{b}} \\ge \\frac{\\delta(2r+1)-1}{r} - \\frac{104r^2}{b} \\cdot \\of{\\frac{\\delta(2r+1)-1}{2r}}^2\n\\ge \\frac{\\delta(2r+1)-1}{r} - \\varepsilon,\n\\]\nthe latter is due to~\\eqref{thm:main2:eps}.\nThus,\n\\[\n\\Omega\\of{n^2 p^2} = \\Omega\\of{ n^{2-2\\delta + \\frac{\\delta(2r+1)-1}{r} - \\varepsilon} } = \\Omega\\of{n^{2-\\frac{1-\\delta}{r}-\\varepsilon}}\n\\]\ncompleting the proof.\n\\end{proof}\n\n\n\\section{Upper bound on $\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^\\delta)$}\\label{sec:ub}\n\nFirst we state the well-known dependent random choice lemma from a survey paper of Fox and Sudakov~\\cite{FS}. Early versions of this lemma were proved and applied by various researchers,\nstarting with Gowers~\\cite{Gowers}, R\\\"odl and Kostochka~\\cite{KRodl}, and Sudakov~\\cite{SU3,SU,SU2}.\n\n\\begin{lemma}\\label{lem:dependent}\nLet $a, d, m, n, r$ be positive integers. Let $G = (V,E)$ be a graph with $|V| = n$ vertices and\naverage degree $d = 2|E(G)|\/n$. If there is a positive integer $t$ such that\n\\[\n\\frac{d^t}{n^{t-1}} - n^r \\left( \\frac{m}{n} \\right)^{t} \\ge a,\n\\]\nthen $G$ contains a subset $U$ of at least $a$ vertices such that every $r$ vertices in $U$ have at least $m$\ncommon neighbors.\n\\end{lemma}\n\n\\begin{corollary}\\label{cor:sudakov}\nLet $r\\ge 1$ be an integer and $0<\\delta<1$.\nLet $G$ be a graph on $n$ vertices with at least $n^{2-(1-\\delta)\/\\lceil \\frac{r-\\delta}{1-\\delta} \\rceil}$ edges. \nThen $G$ contains a subset $U$ of at least $n^{\\delta}$ vertices such that every $r$ vertices in $U$ have at least $n^{\\delta}$\ncommon neighbors.\n\\end{corollary}\n\n\\begin{proof}\nLet $t=\\lceil \\frac{r-\\delta}{1-\\delta} \\rceil$, $a=m=n^{\\delta}$, and $d=2n^{1-(1-\\delta)\/\\lceil \\frac{r-\\delta}{1-\\delta} \\rceil}$. Then,\n\\[\n\\frac{d^t}{n^{t-1}} - {n^r} \\left( \\frac{m}{n} \\right)^{t} \n= 2^{\\lceil \\frac{r-\\delta}{1-\\delta} \\rceil} n^{\\delta}  - n^r n^{-(1-\\delta)\\cdot \\lceil \\frac{r-\\delta}{1-\\delta} \\rceil}\n\\ge 2^{\\lceil \\frac{r-\\delta}{1-\\delta} \\rceil} n^{\\delta} - n^{\\delta}\n\\ge a.\n\\]\nNow the corollary follows from the above lemma.\n\\end{proof}\n\nThe next theorem is an easy generalization of a result of Sudakov from~\\cite{SU}.\n\\begin{theorem}\\label{thm:ub_delta}\nLet $s \\ge r\\ge 1$ and $0<\\delta<1$. Then,\n\\[\n\\textup{\\textbf{RT}}_s(n,K_{s+r}, n^{\\delta}) < \n\\begin{cases}\nn^{2-\\frac{(1-\\delta)^2}{r-\\delta}}, & \\text{ if $\\frac{r-\\delta}{1-\\delta}$ is an integer},\\\\ \nn^{2-\\frac{(1-\\delta)^2}{r+1-2\\delta}}, & \\text{ otherwise}. \n\\end{cases}\n\\]\n\\end{theorem}\n\n\\begin{proof}\nClearly if $\\frac{r-\\delta}{1-\\delta}$ is an integer, then $n^{2-\\frac{(1-\\delta)^2}{r-\\delta}} = n^{2-(1-\\delta)\/\\lceil \\frac{r-\\delta}{1-\\delta} \\rceil}$. Otherwise, \n\\[\nn^{2-\\frac{(1-\\delta)^2}{r+1-2\\delta}} = n^{2-(1-\\delta)\/ \\frac{r+1-2\\delta}{1-\\delta} }\n= n^{2-(1-\\delta)\/ \\left(\\frac{r-\\delta}{1-\\delta} +1\\right)}\n> n^{2-(1-\\delta)\/\\lceil \\frac{r-\\delta}{1-\\delta} \\rceil}.\n\\]\n\nLet $G$ be a graph on $n$ vertices with at least $n^{2-(1-\\delta)\/\\lceil \\frac{r-\\delta}{1-\\delta} \\rceil}$ edges which contains no copy of $K_{s+r}$. We show that $\\alpha_s(G) \\ge n^{\\delta}$. By Corollary~\\ref{cor:sudakov} graph $G$ contains a subset of vertices $U$ of size~$n^\\delta$ such that any $W\\subseteq U$ of size $r$ has $|N(W)|\\ge n^{\\delta}$. If $G[U]$ contains no $K_s$, then $U$ is an $s$-independent set and we are done. So suppose it contains a copy of $K_s$ and denote by $W$ any $r$ vertices of such copy (recall that $r\\le s$). Clearly, $|N(W)|\\ge n^{\\delta}$. If $N(W)$ contains a copy of~$K_s$, then together with\nthe vertices in $W$ we obtain a complete subgraph of $G$ on $s+r$ vertices, a contradiction. Thus $N(W)$ is an $s$-independent set of size at least $n^{\\delta}$.\n\\end{proof}\n\n\n\n\\section{Better lower bounds on $\\textup{\\textbf{RT}}_s(n,K_{s+1}, n^{\\delta})$ for certain values of $\\delta$}\\label{sec:r_eq_1}\n\nObserve that Theorems \\ref{thm:main1}, \\ref{thm:main2} and \\ref{thm:ub_delta} (applied with $r=1$) immediately yield the following statement.\n\\begin{theorem}\\label{thm:r_eq_1}\nLet $\\varepsilon>0$ and $\\frac 12 < \\delta < 1$. Then for all sufficiently large $s$, we have \n\\[\n\\Omega\\of{n^{1+\\delta  - \\varepsilon}} = \\textup{\\textbf{RT}}_s(n, K_{s+1}, n^\\delta) = O\\of{n^{1+\\delta}}.\n\\]\n\\end{theorem}\nAs it was already observed this is also optimal with respect to $\\delta$, since for $\\delta\\le 1\/2$ $\\textup{\\textbf{RT}}_s(n, K_{s+1}, n^\\delta)=0$. We will show now that for specific values of $\\delta$ one can basically remove $\\varepsilon$ from the exponent. \n\nFirst we recall some basic properties of generalized quadrangles. A \\textit{generalized quadrangle} of order $(p,q)$ is an incidence structure on a set~${P}$ of points and a set~$\\mathcal{L}$ of lines such that:\n\\begin{enumerate}[label=$({\\textup{Q}}\\textup{\\arabic*})$]\n\\item any two points lie in at most one line,\n\\item\\label{Q:2} if $u$ is a point not on a line~$ L$, then there is a unique point $w\\in L$ collinear with~$u$, and hence, no three lines form a triangle,\n\\item every line contains~$p+1$ points, and every point lies on~$q+1$ lines,\n\\item\\label{Q:4} $|P| = (pq+1)(p+1)$ and $|\\mathcal{L}| = (pq+1)(q+1)$,\n\\item\\label{Q:5} $p\\le q^2$ and $q\\le p^2$.\n\\end{enumerate}\n\n\\begin{theorem}\\label{thm:quad}\nLet $s\\ge 2$. If a generalized quadrangle of order $(p,q)$ exists, then\n\\begin{equation}\\label{thm:quad:1}\n\\textup{\\textbf{RT}}_s((pq+1)(p+1), K_{s+1}, \\Theta(pq)) = \\Theta(p^3q^2),\n\\end{equation}\nwhere the hidden constants depend only on $s$.\n\\end{theorem}\n\n\\begin{proof}\nFor a given generalized quadrangle $(P,\\mathcal{L})$ we construct the random graph $\\mathbb{G}=(V,E)$ with $V=P$ as follows. For every $L\\in \\mathcal{L}$, let $\\chi_L : L \\to  {[s]}$ be a random partition of the vertices of $L$ into $s$ classes chosen uniformly at random, where the classes need not have the same size and the unlikely event that a class is empty is permitted. Next we embed the $s$-partite complete graph on $\\chi^{-1}_L (1) \\cup \\chi^{-1}_L (2) \\cup \\dots \\cup \\chi^{-1}_L (s)$. Observe that not only are $\\mathbb{G}[L]$ and $\\mathbb{G}[L']$ edge disjoint for distinct $L, L' \\in \\mathcal{L}$, but also that the partitions for $L$ and $L'$ were determined independently.\n\nObserve that by~\\ref{Q:2} $\\mathbb{G}$ is $K_{s+1}$-free and by the Chernoff bound {\\textit{w.h.p. }} $|E| = |\\mathcal{L} |\\cdot \\Omega(p^2) = \\Omega(p^3q^2)$.\n\nNow we will show that $\\alpha_s(\\mathbb{G}) \\le s^2 p q$. Consider any $C \\in {V \\choose s^2 pq}$. We will bound the probability that $\\mathbb{G}[C] \\not \\supseteq K_s$. \nFor each $L \\in \\mathcal{L}$, let $X_L$ be the event that $K_s \\not \\subseteq \\mathbb{G}[L \\cap C]$. Clearly, \n\\[\n\\Pr(X_L) \\le s \\left( 1-\\frac{1}{s}\\right)^{|L\\cap C|} \\le s e^{-|L\\cap C|\/s}\n\\]\nand by independence, \n\\[\n\\Pr \\Big( K_s \\not \\in \\mathbb{G}[C] \\Big) \n\\leq \\Pr \\Big( \\bigcap_{L \\in \\mathcal{L}} X_L \\Big) \n\\leq \\prod_{L\\in \\mathcal{L}} s e^{-|L\\cap C|\/s}\n\\leq s^{|\\mathcal{L}|} e^{-\\sum_{L\\in \\mathcal{L}} |L\\cap C|\/s }.\n\\]\nSince $\\sum_{L\\in \\mathcal{L}} |L\\cap C| = |C|(q+1)$, we get\n\\[\n\\Pr \\Big( K_s \\not \\in \\mathbb{G}[C] \\Big) \n\\leq s^{|\\mathcal{L}|} e^{-|C|(q+1)\/s}\n= e^{|\\mathcal{L}| \\log s -|C|(q+1)\/s}.\n\\]\nSo by the union bound, the probability that there exists a subset of $s^2 p q$ vertices in $V$ that contains no $K_s$ is at most\n\\[\n\\binom{|V|}{s^2 pq} e^{|\\mathcal{L}| \\log s -|C|(q+1)\/s}\n\\le  |V|^{s^2 pq} e^{|\\mathcal{L}| \\log s -|C|(q+1)\/s}\n= e^{s^2 pq \\log |V| + |\\mathcal{L}| \\log s -|C|(q+1)\/s} = o(1),\n\\]\nbecause of~\\ref{Q:4} and $|C| = s^2pq$. Thus, {\\textit{w.h.p. }} $\\alpha_s(\\mathbb{G}) \\le s^2 pq$ and consequently\n\\[\n\\textup{\\textbf{RT}}_s((pq+1)(p+1), K_{s+1}, \\Theta(pq)) = \\Omega(p^3q^2).\n\\]\n\nFinally observe that the upper bound in~\\eqref{thm:quad:1} is trivial, since if $G$ is a $K_{s+1}$-free graph of order $(pq+1)(p+1)$, then $\\Delta(G) < \\alpha_s(G) = O(pq)$ yielding $|E(G)| = O(p^3q^2)$.\n\\end{proof}\n\nIt is known that when $(p,q)\\in \\{(r,r), (r,r^2),(r^2,r),(r^2,r^3),(r^3,r^2)\\} $ for any arbitrary prime power $r$, then the generalized quadrangle exists (see, e.g., ~\\cite{GO,TH}) yielding the following:\n\n\\begin{center}\n\\begin{tabular}{c | c | c | c | c }\n$p$ & $q$ & $(pq+1)(p+1)$  & $pq$ & $p^3 q^2$ \\\\\n\\hline\\hline\n$r$ & $r$ & $ \\Theta(r^3)$ & $r^2$ & $r^5$\\\\ \\hline\n$r$ & $r^2$ & $ \\Theta(r^4)$ & $r^3$ & $r^7$\\\\ \\hline\n$r^2$ & $r$ & $ \\Theta(r^5)$ & $r^3$ & $r^8$\\\\ \\hline\n$r^2$ & $r^3$ & $ \\Theta(r^7)$ & $r^5$ & $r^{12}$\\\\ \\hline\n$r^3$ & $r^2$ & $ \\Theta(r^8)$ & $r^5$ & $r^{13}$\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\n\\smallskip\n\nThus, by letting $n$ to be $\\Theta(r^3)$, $\\Theta(r^4)$, $\\Theta(r^5)$, $\\Theta(r^7)$, and $\\Theta(r^8)$, respectively,  we get the following corollary.\n\\begin{corollary}\\label{cor:quad}\nLet $\\delta \\in \\{3\/5, 5\/8, 2\/3, 5\/7, 3\/4\\}$. Then\n\\[\n\\textup{\\textbf{RT}}_s(n,K_{s+1}, \\Theta(n^{\\delta})) = \\Theta(n^{1+\\delta}).\n\\]\n\\end{corollary}\n\n\n\nIn view of Theorems~\\ref{thm:r_eq_1} and \\ref{thm:quad} and the above corollary we conjecture that actually\n\\[\n\\textup{\\textbf{RT}}_s(n,K_{s+1}, n^{\\delta}) = \\Theta(n^{1+\\delta})\n\\] \nfor any $1\/2 < \\delta < 1$.\n\n\n\n\n\n\n\n\\section{Phase transition of $\\textup{\\textbf{RT}}_s(n, K_{2s+1}, f)$}\\label{sec:big_r}\n\nThroughout this section $\\omega = \\omega(n)$ is a function which goes to infinity arbitrarily slowly together with $n$.\n\nHere we briefly discuss an extension  of a recent result of Balogh, Hu, and Simonovits~\\cite{BHS}, who showed (answering a question of Erd\\H{o}s and S\\'os~\\cite{ES}) that\n\\begin{equation}\\label{eq:BHS:1}\n\\textup{\\textbf{RT}}_2(n,K_5, \\sqrt{n\\log n} \/ \\omega) = o(n^2)\n\\end{equation}\nand\n\\begin{equation}\\label{eq:BHS:2}\n\\textup{\\textbf{RT}}_2(n,K_5, c \\sqrt{n\\log n}) = n^2\/4 + o(n^2)\n\\end{equation}\nfor any $c>1$. Thus, one can say that $K_5$ has a (Ramsey-Tur\\'an) phase transition at $c\\sqrt{n\\log n}$ for every $c>1$. One can easily show that for any $s\\ge 2$, $K_{2s+1}$ also has a phase transition (with respect to the $s$-independence number). This will follow from the following two theorems.\n\n\\begin{theorem}\\label{thm:ub_ks_r}\nLet $s\\ge 2$. Then \n\\[\n\\textup{\\textbf{RT}}_s(n,K_{2s+1}, \\textup{\\textbf{f}}_{s,s+1}(n\/\\omega)) = o(n^2).\n\\]\n\\end{theorem}\n\nFirst we derive another corollary from Lemma~\\ref{lem:dependent}.\n\n\\begin{corollary}\\label{cor:dependent2}\nLet $\\varepsilon>0$ be fixed and $s\\ge 1$ be an integer. Let $G$ be a graph on $n$ vertices with at least $\\varepsilon n^2$ edges. \nThen for sufficiently large $n$, $G$ contains a subset $U$ of at least $n\/\\omega$ vertices such that every $s+1$ vertices in $U$ have at least $\\textup{\\textbf{f}}_{s,s+1}(n\/\\omega)$ common neighbors.\n\\end{corollary}\n\n\\begin{proof}\nLet $r=s+1$, $t=2s+1$, $a = n\/\\omega$, $m = \\textup{\\textbf{f}}_{s,s+1}(n\/\\omega)$, and $d=2\\varepsilon n$. Observe that by~\\eqref{f:1}, $\\textup{\\textbf{f}}_{s,s+1}(n\/\\omega) = o(n^{(s+1)\/(2s+1)})$.\nThus,\n\\[\n\\frac{d^t}{n^{t-1}} - {n^r} \\left( \\frac{m}{n} \\right)^{t} \n\\ge (2\\varepsilon)^{2s+1} n - \\frac{(\\textup{\\textbf{f}}_{s,s+1}(n\/\\omega))^{2s+1}}{n^s} \\ge n\/\\omega = a,\n\\]\nfor sufficiently large~$n$.\nNow the corollary follows from Lemma~\\ref{lem:dependent}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:ub_ks_r}]\nLet $\\varepsilon>0$ be an arbitrarily small constant and let $G$ be a $K_{2s+1}$-free graph of order $n$ with $\\varepsilon n^2$ edges and $\\alpha_s(G)< \\textup{\\textbf{f}}_{s,s+1}(n\/\\omega)$. By Corollary~\\ref{cor:dependent2} there is a set $U$ of size $n\/\\omega$ such that every $s+1$ vertices in $U$ have at least $\\textup{\\textbf{f}}_{s,s+1}(n\/\\omega)$ common neighbors. First observe that $G[U]$ contains a copy of $K_{s+1}$. Otherwise, $G[U]$ is a $K_{s+1}$-free graph of order $n\/\\omega$ with $\\alpha_s(G[U])<\\textup{\\textbf{f}}_{s,s+1}(n\/\\omega)$. But clearly this cannot happen.\nNow let $W$ be the set of vertices of a copy of $K_{s+1}$ in $U$. If $N(W)$ contains a copy of $K_s$, then such a copy together with $G[W]$ gives a copy of $K_{2s+1}$. Thus, $G[N(W)]$ is $K_s$-free. But this yields that $\\alpha_s(G)\\ge |N(W)|\\ge \\textup{\\textbf{f}}_{s,s+1}(n\/\\omega)$, a contradiction.\n\\end{proof}\n\nThe following extends a result of Erd\\H{o}s, Hajnal, Simonovits, S\\'os, and Szemer\\'edi~\\cite{EHSSS} for small $s$-independence numbers.\n\n\\begin{theorem}\\label{thm:lb_ks_r}\nLet $k\\ge 2$, $s\\ge 2$, $1\\le r\\le k-1$. Let $c$ be a constant satisfying\n\\[\nc > \n\\begin{cases}\n1 \\text{ if } s=2,\\\\\nk \\text{ if } s\\ge 3.\\\\\n\\end{cases}\n\\]\nThen,\n\\[\n\\textup{\\textbf{RT}}_s(n,K_{ks+r}, c\\;\\!\\textup{\\textbf{f}}_{s,s+1}(n\/k)) \\ge \\frac{1}{2} \\left(1-\\frac{1}{k} \\right) n^2 + o(n^2). \n\\]\nFurthermore, if $r=1$, then \n\\[\n\\textup{\\textbf{RT}}_s(n,K_{ks+1}, c\\;\\!\\textup{\\textbf{f}}_{s,s+1}(n\/k)) = \\frac{1}{2} \\left(1-\\frac{1}{k} \\right) n^2 + o(n^2).\n\\]\n\\end{theorem}\n\n\\begin{proof}\nFirst we assume that $s\\ge 3$ and $c > k$.\nLet $H$ be a $K_{s+1}$-free graph of order $n\/k  + o(n)$ with $\\alpha_s(H) < c\/k\\cdot\\textup{\\textbf{f}}_{s,s+1}(n\/k)$. The existence of such graph follows from the definition of the Erd\\H{o}s-Rogers number. Let $G=(V,E)$ be a graph of order $n$ such that $V=V_1\\cup\\dots\\cup V_{k}$, $|V_1| = \\dots = |V_k| = n\/k+o(n)$, $G[V_i]$ is isomorphic to $H$ for each $1\\le i\\le k$, and for each $v\\in V_i$ and $w\\in V_j$ we have $\\{v,w\\}\\in E$ for any $1\\le i < j\\le k$. Observe that $G$ is $K_{ks+r}$-free and $\\alpha_{s}(G) < c\\;\\!\\textup{\\textbf{f}}_{s,s+1}(n\/k)$. Thus,\n\\[\n\\textup{\\textbf{RT}}_s(n,K_{ks+r}, c\\;\\!\\textup{\\textbf{f}}_{s,s+1}(n\/k)) \\ge \\binom{k}{2} \\left( \\frac{n}{k} \\right)^2 + o(n^2) = \\frac{1}{2} \\left(1-\\frac{1}{k} \\right) n^2 + o(n^2).\n\\]\n\nNow observe that if $s=2$ and $c>1$, then it suffices to assume that $\\alpha_s(H) < c\\;\\!\\textup{\\textbf{f}}_{s,s+1}(n\/k)$, since any independent set in $G$ must lie entirely in some $V_i$.\n\nFinally, if $r=1$, then\n\\[\n\\textup{\\textbf{RT}}_s(n,K_{ks+1}, c\\;\\!\\textup{\\textbf{f}}_{s,s+1}(n\/k)) \\le \\textup{\\textbf{RT}}_s(n,K_{ks+1}, o(n)) \\le \\frac{1}{2} \\left(1-\\frac{1}{k} \\right) n^2 + o(n^2),\n\\]\nwhere the latter follows from Theorem 2.6\\,(a) in~\\cite{EHSSS}.\n\\end{proof}\n\nSince by~\\eqref{f:1b} we have  $ \\textup{\\textbf{f}}_{2,3}(n\/2) \\le (1+o(1))\\sqrt{n\\log n}$ and $  \\textup{\\textbf{f}}_{2,3}(n\/\\omega) \\ge \\sqrt{n\\log n}\/\\omega$, the above theorems can be viewed as generalizations of \\eqref{eq:BHS:1} and \\eqref{eq:BHS:2}, which are the case $k=s=2$ and $r=1$.\n\nIn particular, when $k=2$ and $r=1$, we get that $\\textup{\\textbf{RT}}_s(n,K_{2s+1}, n^{\\delta}) = n^2\/4 + o(n^2)$  for any $1\/2<\\delta<1$ and $\\textup{\\textbf{RT}}_s(n,K_{2s+1}, n^{\\delta})=o(n^2)$ for $\\delta \\le 1\/2$. Also observe that if we replace $K_{2s+1}$ by $K_{2s}$, then $\\textup{\\textbf{RT}}_s(n,K_{2s}, n^{\\delta}) = o(n^2)$ for any $\\delta<1$ (by Theorem \\ref{thm:ub_delta}).\nHere we bound $\\textup{\\textbf{RT}}_s(n,K_{2s}, n^{\\delta})$ from below using the Zarankiewicz function.\nRecall that the \\emph{Zarankiewicz function} $\\textrm{\\textbf{z}}(n,s)$ denotes the maximum possible number of edges in a $K_{s,s}$-free bipartite graph $G = (U\\cup V, E)$ with $|U| = |V| = n$.\n\n\\begin{theorem}\\label{thm:2s}\nLet $s\\ge 2$ and $c>2$. Then\n\\[\n\\textup{\\textbf{RT}}_s(n,K_{2s}, c\\;\\!\\textup{\\textbf{f}}_{s,s+1}(n\/2)) \\ge \\textrm{\\textbf{z}}(n\/2,s).\n\\]\n\\end{theorem}\n\n\\begin{proof}\nLet $H$ be a $K_{s+1}$-free graph of order $n\/2$ with $\\alpha_s(H) < c\/2 \\cdot\\textup{\\textbf{f}}_{s,s+1}(n\/2)$. Let $G=(V,E)$ be a graph of order $n$ such that $V=V_1\\cup V_2$, $|V_1| = |V_2| = n\/2$, and $G[V_1]$ and $G[V_2]$ are isomorphic to $H$. Between $V_1$ and $V_2$ we embed a $K_{s,s}$-free bipartite graph with $z(n\/2,s)$ edges. Clearly $G$ is $K_{2s}$-free and $\\alpha_{s}(G) < c\\;\\!\\textup{\\textbf{f}}_{s,s+1}(n\/2)$.  Thus,\n\\[\n\\textup{\\textbf{RT}}_s(n,K_{2s}, c\\;\\!\\textup{\\textbf{f}}_{s,s+1}(n\/2)) \\ge |E| \\ge \\textrm{\\textbf{z}}(n\/2, s).\n\\]\n\\end{proof}\n\nIt is known due to a result of Bohman and Keevash~\\cite{BK} that for sufficiently large $n$ there exists a $K_{s,s}$-free graph of order~$n$ with \n$\\Omega\\left(n^{2-2\/(s+1)} \\cdot (\\log n)^{1\/(s^2-1)}\\right)$ edges. Since each graph has a bipartite subgraph with at least half of the edges, we get that \n\\[\n\\textrm{\\textbf{z}}(n, s) \\ge \\frac{1}{2}\\textrm{\\textbf{ex}}(n,K_{s,s}) = \\Omega\\left(n^{2-2\/(s+1)} \\cdot (\\log n)^{1\/(s^2-1)}\\right).\n\\]\nThis together with Theorem~\\ref{thm:2s} implies that \n\\begin{equation}\\label{eq:K2s}\n\\textup{\\textbf{RT}}_s(n,K_{2s}, f) = \\Omega\\left(n^{2-2\/(s+1)} \\cdot (\\log n)^{1\/(s^2-1)}\\right)\n\\end{equation}\nfor any $s\\ge 2$ and $f \\ge c\\;\\!\\textup{\\textbf{f}}_{s,s+1}(n\/2)$ with $c>2$.\n\n\n\\section{Concluding remarks}\n\nThe study of the Ramsey-Tur\\'an number for small $s$-independence number brings several new problems and challenges. We believe that now the most interesting question is to decide whether the lower bounds given by Theorems~\\ref{thm:main1} and~\\ref{thm:main2} are nearly optimal for any $r\\ge 2$. We believe that the upper bound in Theorem~\\ref{thm:ub_delta} can be improved.\n\nIt is also  not hard to verify that the proofs of Theorems~\\ref{thm:main1} and~\\ref{thm:main2} can allow $r$ to grow with $s$. For fixed $\\varepsilon$  and $\\delta$, these theorems still hold for $r = r(s) \\le c s^{1\/5}$ for some small constant $c$ which may depend on $\\varepsilon$ and $\\delta$. Of course, since $\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^\\delta)$ is nondecreasing in $r$, we get some lower bound for larger $r$ as well, but this fact is not particularly satisfying. While we did not put much effort into allowing $r$ to grow large with $s$, new ideas would be needed to prove lower bounds on $\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^\\delta)$ that get better as $\\delta$ increases and which apply to say, all $1 \\le r \\le s$. (Recall that the upper bound in Theorem~\\ref{thm:ub_delta} applies to all $1 \\le r \\le s$, and that we have the lower bound~\\eqref{eq:K2s} but this bound stays the same even for $\\delta$ quite close to $1$.)\n\n\n\n\n\n\\section{Acknowledgment}\nWe are grateful to an anonymous referee for bringing to our attention reference~\\cite{BHS}.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSince the successful exfoliation of single layer graphene, 2D materials have been attracting significant interest due to their highly tunable physical properties and immense potential in scalable device applications \\cite{novoselov2004electric,novoselov2005two,zhang2005experimental,geim2007rise,RevModPhys.81.109}. However, pristine graphene exhibits no band gap and its inherent inversion symmetry suppresses the spin-orbit coupling (SOC) \\cite{zibouche2014transition, PhysRevB.92.144404, RevModPhys.76.323, macdonald2005ferromagnetic, dietl2010ten, deng2011li}. The weak SOC and zero band gap eliminate graphene as a potential candidate for being applied to spintronic devices, which require one to search for alternative 2D materials that extend beyond graphene to other layered materials with van der Waals gaps \\cite{PhysRevB.92.144404, RevModPhys.76.323, macdonald2005ferromagnetic, dietl2010ten, deng2011li}. For example, in single-layer MoS$_2$, the large SOC leads to a unique spin-valley coupling which may be useful for spintronic applications \\cite{PhysRevB.84.153402,PhysRevLett.108.196802,PhysRevB.86.165108,PhysRevB.88.125301,xu2014spin}. Whereas spintronic devices using 2D materials are still in their infancy \\cite{han2014graphene,ohno2000electric,chang2013experimental,PhysRevLett.113.137201}, which is due to the lack of long-range ferromagnetic order that is crucial for macroscopic magnetic effects \\cite{PhysRevLett.114.016603,gonzalez2016atomic}. The emergence of ferromagnetism in 2D materials in combination with their rich electrical and optical properties could open up ample opportunities for 2D magnetic, magneto-electric, and magneto-optic applications \\cite{ohno2000electric,chang2013experimental,gong2017discovery}.\n\nRecently, Chromium Tellurides Cr$X$Te$_3$ ($X$ = Si, Ge, and Sn) with the centrosymmetric have arisen significant attention because they belong to a rare category of ferromagnetic semiconductors possessing a 2D layered structure\\cite{PhysRevB.92.144404,gong2017discovery,PhysRevB.91.235425,carteaux1995crystallographic,ji2013ferromagnetic,tian2016magneto,PhysRevB.92.035407,siberchicot1996band,PhysRevX.3.031002,li2014crxte,PhysRevB.92.165418,PhysRevLett.117.257201,liu2016critical,carteaux19952d}. Extensive theoretical and experiment efforts have been extended towards understanding the properties of these 2D magnets. On the theoretical side, recent studies on Cr$X$Te$_3$ have been focusing on their electronic structure and magnetic properties, particularly predictions of the single-layer properties \\cite{PhysRevB.91.235425,PhysRevB.92.035407,siberchicot1996band,PhysRevX.3.031002,li2014crxte,PhysRevB.92.165418,PhysRevLett.117.257201}. On the experimental side, CrSiTe$_3$ and CrGeTe$_3$ have been prepared and characterized \\cite{PhysRevB.92.144404,gong2017discovery,carteaux1995crystallographic,ji2013ferromagnetic,tian2016magneto,liu2016critical,carteaux19952d}. Comparing with  CrSiTe$_3$, showing characteristics of a 2D-Ising behavior\\cite{PhysRevB.92.144404,liu2016critical,carteaux19952d}, the smaller van der Waals gap and the larger in-plane nearest-neighbor Cr-Cr distance in CrGeTe$_3$  enhance the Curie temperature from 32 K for the  CrSiTe$_3$ to 61 K for the CrGeTe$_3$ \\cite{PhysRevB.92.144404,carteaux1995crystallographic,PhysRevB.92.035407,li2014crxte}. In addition, theoretical investigations have suggested that the single-layer CrGeTe$_3$ presents characteristics of 2D-Ising behavior similar to CrSiTe$_3$\\cite{li2014crxte,PhysRevLett.117.257201}. By contrast, a scanning magneto-optic Kerr microscopy experiment, single-layer CrGeTe$_3$ represents a close-to-ideal 2D Heisenberg ferromagnetic system using the rigorous renormalized spin wave theory analysis and calculations\\cite{gong2017discovery}. It is known that, with the increase of the $X$ atom radius, Cr$X$Te$_3$ presents the smaller van der Waals gap and the larger cleavage energy \\cite{PhysRevB.92.144404,carteaux1995crystallographic,PhysRevB.92.035407,li2014crxte}. We suppose that Cr$X$Te$_3$ system may undergo a three dimensional (3D) magnetic phase transition from 2D with the increase of the $X$ atom radius. Therefore, a method to rapidly characterize the critical behavior of single-crystalline CrGeTe$_3$ is crucial. For this purpose, we present a detailed investigation of the critical phenomena of CrGeTe$_3$ using the initial isothermal $M(H)$ curves around the Curie temperature $T_\\mathrm{C}$. We find that the critical exponents of CrGeTe$_3$ satisfy the universality class of the tricritical mean-field theory. This indicates that the magnetic phase transition of CrGeTe$_3$ should be close to a tricritical point from 2D to 3D.\n\n\\section{Methods}\n\\label{sec:methods}\nSamples of single-crystalline CrGeTe$_3$ were prepared by the self-flux technique \\cite{ji2013ferromagnetic}. The XRD data indicated that the powders are single phase with the rhombohedral structure (see Supporting Information). We measured the heat capacity using the Quantum Design physical properties measurement system (PPMS-9T) and characterized the magnetic properties by the magnetic property measurement system (MPMS-XL5). Density functional theory (DFT) calculations were performed using Vienna {\\it Ab-initio} Simulation Package \\cite{Ref36}. We used the local density approximation \\cite{Ref37, Ref38} to treat the electron-electron exchange-correlation interactions. The electron-ion interactions are described by the potentials based on the projector augmented wave method \\cite{Ref39, Ref40}.\n\\section{Results}\nFigure~\\ref{fig:1}(a) and (b) show the temperature-dependent inverse susceptibility 1\/$\\chi(T)$ of CrGeTe$_3$ under field cooled cooling with applied magnetic field $H$ = 100 Oe, parallel to the $ab$ plane and $c$ axis, respectively. We observe a paramagnetic-ferromagnetic (PM-FM) transition that occurs at a critical temperature of 67.3 K, as determined by the derivative of the susceptibility. This temperature is consistent with the values of 61 K or 70 K reported previously \\cite{carteaux1995crystallographic,ji2013ferromagnetic,tian2016magneto}. For a FM system, the 1\/$\\chi(T)$ above can be described by the Curie-Weiss law resulting from the mean-field theory \\cite{Ref41}. The red curves showing the Curie-Weiss law are obeyed only at high temperatures. A close observation of Fig.\\ref{fig:1}(a) and (b) reveals that the curves deviate from straight lines at around 150 K, which is much higher than $T_C^\\mathrm{mag}$, indicating strong short-range FM spin interactions in CrGeTe$_3$ above $T_C^\\mathrm{mag}$. The effective magnetic moment $\\mu_\\mathrm{eff}$ is determined to be around 4.22$\\mu_\\mathrm{B}$  (parallel the ab plane) and 4.35$\\mu_\\mathrm{B}$ (parallel the c axis), which are close to the theoretical value expected for Cr$^{3+}$ of 3.87$\\mu_\\mathrm{B}$ \\cite{carteaux1995crystallographic}. The insets of Fig.\\ref{fig:1}(a) and (b) show the isothermal magnetization $M(H)$ at 5 K exhibiting a typical FM behavior with the saturation field $H_S$ of about 5 kOe (parallel to the ab plane) and 2.5 kOe (parallel to the c axis). In addition, the $M(H)$ curves show almost no coercive force for CrGeTe$_3$.\n\n\\begin{figure}[t]\n  \\includegraphics[width=8cm]{Fig1.eps}\n  \\caption{(a) and (b) Temperature-dependent inverse susceptibility 1\/$\\chi(T)$ of CrGeTe$_3$ under field cooled cooling with an applied magnetic field $H$ of 100 Oe, parallel to the $ab$ plane and $c$ axis, respectively. The red solid lines are the fitted results according to the Curie-Weiss law. The insets show the isothermal magnetization curves $M$($H$) at 5 K. (c) Specific heat $C_p$ as a function of $T$ for CrGeTe$_3$ and the fitted $C_V^\\mathrm{Debye}(T)$ using Eqs.\\ref{eq:1} and 2; Temperature-dependent magnetic (d) specific heat $C_\\mathrm{mag}(T)$ and (e) entropy $S_\\mathrm{mag}(T\\rightarrow\\infty)$. The blue dashed line refers to $S_\\mathrm{mag}(T)$ calculated with the magnetic moment S of Cr$^{3+}$ being 3\/2.}\n  \\label{fig:1}\n\\end{figure}\n\nFigure~\\ref{fig:1}(c) shows the variation of the zero-field specific heat (SH) $C_p(T)$ with temperature. The sharp anomaly in $C_p(T)$ at 64.8 K corresponds to the Curie temperature $T_C^\\mathrm{SH}$. Since CrGeTe$_3$ is a seminconductor \\cite{carteaux1995crystallographic}, the electronic contribution to the heat capacity is not considered. The $C_\\mathrm{mag}$ can be calculated by the following equations \\cite{Ref41}:\n\\begin{equation}\n  \\label{eq:1}\nC_\\mathrm{mag}(T) =C_p(T)-NC_V^\\mathrm{Debye}(T)\n\\end{equation}\nand\n\\begin{equation}\n  \\label{eq:2}\nC_V^\\mathrm{Debye}(T) = 9R\\bigg(\\frac{T}{\\Theta_\\mathrm{D}}\\bigg)^3\\int_{0}^{\\Theta_\\mathrm{D}\/T} \\frac{x^4e^x}{(e^x-1)^2} dx,\n\\end{equation}\nwhere $R$ is the molar gas constant, $\\Theta_\\mathrm{D}$ is the Debye temperature, and $N$ = 5 is the number of atoms per formula unit. The sum of Debye functions accounts for the lattice contribution to the specific heat. We can extract the magnetic contribution $C_\\mathrm{mag}(T)$ from the measured specific heat of CrGeTe$_3$. The fitted $C_p(T)$  for CrGeTe$_3$ by Eqs.\\ref{eq:1} and 2 over the temperature range from about 7 to 250 K is shown by the red curve in Fig.\\ref{fig:1}(c) using the Debye temperature  $\\Theta_\\mathrm{D}$ = 476.5 K. We observe a sharp peak at $T_C^\\mathrm{SH}$ of 64.8 K and there is strong dynamic short-range FM spin interactions above $T_C^\\mathrm{SH}$  (see Fig.\\ref{fig:1}(d)). The magnetic entropy $S_\\mathrm{mag}(T)$ is calculated by\n\\begin{equation}\n  \\label{eq:3}\nS_\\mathrm{mag}(T) =\\int_{0}^{T}\\frac{C_\\mathrm{mag}(T)}{T}dT.\n\\end{equation}\nFig.\\ref{fig:1}(e)  shows the temperature dependence of $S_\\mathrm{mag}(T)$. The entropy of CrGeTe$_3$ per mole with completely disordered spins S is\n\\begin{equation}\n  \\label{eq:4}\nS_\\mathrm{mag}(T\\rightarrow\\infty)=6R\\mathrm{ln}(2\\mathrm{S}+1).\n\\end{equation}\nUsing S = 3\/2 for Cr$^{3+}$, we obtain $S_\\mathrm{mag}(T\\rightarrow\\infty)$ of 69.2 J\/(mol$\\cdot$K). However, we observe the $S_\\mathrm{mag}$ is 64.7 J\/(mol$\\cdot$K) at 150 K in Fig.\\ref{fig:1}(e), which is smaller than $S_\\mathrm{mag}(T\\rightarrow\\infty)$.  Note that there is an error of about 10\\% \\cite{PhysRevB.72.174404} in our measurement due to the fitting of the optical phonon contributions at high temperatures. In spite of this small error, our result indicates the strong short-range FM spin interactions above $T_C^\\mathrm{SH}$.\n\n\\begin{figure}[t]\n  \\includegraphics[width=8.5cm]{Fig2.eps}\n  \\caption{(a) Initial magnetization of CrGeTe$_3$ around $T_\\mathrm{C}$; (b) Arrott plots of $M^2$ versus $H\/M$ (the $M(H)$ curves are measured at temperature intervals of 1 K and 0.5 K approaching $T_\\mathrm{C}$); (c) Normalized slopes as a function of temperature; (d) Modified Arrott plot ($M^{1\/\\beta}$ versus $(H\/M)^{1\/\\gamma}$) of isotherms with $\\beta$ = 0.24 and $\\gamma$ = 1 for CrGeTe$_3$. The red dashed line is the linear fit of isotherm at 67.9 K; (e) Temperature dependence of $M_S$ and $\\chi_0^{-1}$. The $T_\\mathrm{C}$ and critical exponents are obtained from the fitting of Eqs.S1 and S2; (f) The Kouvel-Fisher plot. The  $T_\\mathrm{C}$ and critical exponents are obtained from the linear fit.}\n  \\label{fig:2}\n\\end{figure}\n\nAs mentioned above, with the $X$ atom radius increases, the Cr$X$Te$_3$ compounds present the smaller van der Waals gap and the larger cleavage energy \\cite{PhysRevB.92.144404,carteaux1995crystallographic,PhysRevB.92.035407,li2014crxte}, which may induce a 3D magnetic phase transition. For the purpose of confirmation, we performed a detailed characterization of the critical phenomena using the initial isothermal $M(H)$ curves around $T_\\mathrm{C}$ for the CrGeTe$_3$, which are shown in Fig.\\ref{fig:2}(a). In the mean-field theory, the critical exponents and critical temperature can be determined from the Arrott plot with $\\beta$ of 0.5 and $\\gamma$ of 1.0 \\cite{Ref49, Ref50}. According to this method, the $M^2$ versus $H\/M$ (shown in Fig.\\ref{fig:2}(b)) should be a series of parallel straight lines in the higher field range around $T_\\mathrm{C}$ and the line at $T$ = $T_\\mathrm{C}$ should pass through the origin. Note that the lower-field data mainly represent the arrangement of magnetic domains, which should be excluded from the fitting process \\cite{Ref51}. However, all the curves in Fig.\\ref{fig:2}(b) show nonlinear behaviors having downward curvature even at high fields, which indicates an non-mean-field-like behavior. Moreover, the positive slope reveals a second-order phase transition according to the criterion proposed by Banerjee \\cite{Ref52}. As such, a modified Arrott plot should be employed to obtain the critical exponents.\n\n\nTo determine an accurate model, we obtain a modified Arrott plot following Eq.S5 for single-crystalline CrGeTe$_3$ at different temperatures. Three groups of possible exponents belonging to the 3D Heisenberg model ($\\beta$ = 0.365, $\\gamma$ = 1.386), 3D-Ising model ($\\beta$ = 0.325, $\\gamma$ = 1.24) and tricritical mean-field model ($\\beta$ = 0.25, $\\gamma$ = 1.0) exhibit nearly straight lines in the high field region \\cite{Ref45, Ref53}. We calculate their normalized slopes (NS) defined as NS =$ S(T)\/S(T_C^\\mathrm{mag}$ = 67.3 K). By comparing NS with the ideal value of unity, one can identify the most suitable model \\cite{Ref45, Ref53}. Fig.\\ref{fig:2}(c) shows the plots of NS versus $T$ employing the three different models, revealing that the tricritical mean-field model is the most appropriate to describe the critical behavior of CrGeTe$_3$.\n\nBy proper selections of $\\beta$ and $\\gamma$, one can clearly show the isotherms are a set of parallel straight lines at high fields as displayed in the Fig.\\ref{fig:2}(d). The linear extrapolation from the high-field region gives the spontaneous magnetization $M_S(T,0)$ and the initial  inverse susceptibility $\\chi_0^{-1}(T,0)$ (see Fig.\\ref{fig:2}(e)) corresponding to the intercepts on the $M^{1\/\\beta}$ and $(H\/M)^{1\/\\gamma}$ axes, respectively. By fitting the data of $M_S(T,0)$ and to Eqs.S1 and S2, one obtains two new values of $\\beta$ = 0.242$\\pm$0.006 with $T_\\mathrm{C}$ = 67.95$\\pm$0.01 and $\\gamma$ = 0.985$\\pm$0.009 with $T_\\mathrm{C}$  = 67.90$\\pm$0.09. These results are again very close to the critical exponents of tricritical mean-field model. In addition, these critical exponents and $T_\\mathrm{C}$ can be obtained more accurately from the Kouvel-Fisher (KF) method \\cite{Ref54}. Hence, one can find that the temperature dependence of $M_S(dM_S\/dT)^{-1}$  and $\\chi_0^{-1}(d\\chi_0^{-1}\/dT)^{-1}$  should be straight lines with slopes 1\/$\\beta$ and 1\/$\\gamma$, respectively. As seen in Fig.\\ref{fig:2}(f), the linear fit yields the $\\beta$ of 0.240$\\pm$0.006 with $T_\\mathrm{C}$ of 67.91$\\pm$0.07 and $\\gamma$ of 1.000$\\pm$0.005 with $T_\\mathrm{C}$ of 67.88$\\pm$0.05, respectively. Remarkably, the obtained values of the critical exponents and $T_\\mathrm{C}$ using the KF method are in excellent agreement with those using the modified Arrott plot based on the tricritical mean-field model. This suggests that the estimated values are self-consistent and unambiguous.\n\nTo further validate the above critical exponents $\\beta$ and $\\gamma$, we study the relation among these exponents. According to Eq.S3, $\\delta$ can be directly estimated from the critical isotherm at $T_\\mathrm{C}$. Figure \\ref{fig:3}(a) shows the isothermal magnetization $M(H)$ at $T_\\mathrm{C}$ = 67.9 K. The inset of the same plot has been demonstrated on a log-log scale. The solid straight line with a slope 1\/$\\delta$ is the fitted result using Eq.S3. From the linear fit we obtained the third critical exponent $\\delta$ of 5.032$\\pm$0.005. Moreover, the exponent $\\delta$ can be calculated by the Widom scaling relation \\cite{Ref55,Ref56}\n\n\\begin{equation}\n  \\label{eq:5}\n\\delta =1+\\gamma\/\\beta.\n\\end{equation}\nBased on the $\\beta$ and $\\gamma$ values calculated in  Fig.\\ref{fig:2}(e) and (f), Eq.\\ref{eq:5} yields $\\delta$ of 5.070 $\\pm$ 0.006 and 5.167 $\\pm$ 0.006, respectively. We emphasize that these values are very close to the results from the experimental critical isothermal analysis. Therefore, the critical exponents obtained in this study basically obey the Widom scaling relation, showing that the obtained $\\beta$, $\\gamma$ and $\\delta$ are reliable.\n\nFinally, these critical exponents should follow the scaling equation (Eq.S6) in the critical region. The scaling equation indicates that $m$ versus $h$ forms two universal curves for $T > T_\\mathrm{C}$ and  $T < T_\\mathrm{C}$, respectively. Based on Eq.S7, the isothermal magnetization around the critical temperatures for CrGeTe$_3$ has been plotted in Fig.\\ref{fig:3}(b). All experimental data in the higher-field region collapse onto two universal curves, in agreement with the scaling theory. The inset of Fig.\\ref{fig:3}(b) shows the corresponding log-log plot. Similarly, all the points  collapse into two different curves in the higher-field region. This result shows again that the obtained results of the critical exponents and $T_\\mathrm{C}$ are valid.\n\n\\begin{figure}\n  \\includegraphics[width = 8cm]{Fig3.eps}\n  \\caption{(a) Isothermal $M(H)$ at $T_\\mathrm{C}$. The inset shows the alternative plot on a log-log scale and the straight line is the linear fit following Eq.S3; (b) Renormalized magnetization $m$ versus renormalized field $h$ at several typical temperatures around the $T_\\mathrm{C}$. The inset shows an alternative plot on a log-log scale; the effective exponents (c) below $T_\\mathrm{C}$ and (d)  above $T_\\mathrm{C}$ as a function of the reduced temperature $\\varepsilon$.}\n  \\label{fig:3}\n\\end{figure}\n\n\nTo further examine the convergence of the critical exponents, the effective exponents $\\beta_\\mathrm{eff}$ and $\\gamma_\\mathrm{eff}$ can be are obtained by Eqs.S8 and S9 for CrGeTe$_3$. As shown in Fig.\\ref{fig:3}(c) and (d), both $\\beta_\\mathrm{eff}$ and $\\gamma_\\mathrm{eff}$ show a non-monotonic variation with $\\varepsilon$ (see Eq.S4). The lowest $\\varepsilon$ ($\\varepsilon_\\mathrm{min}$) are 5.89$\\times$10$^{-3}$ and 1.47$\\times$10$^{-3}$ for $\\beta_\\mathrm{eff}$ and $\\gamma_\\mathrm{eff}$, respectively. We obtain the effective exponents $\\beta_\\mathrm{eff}$ of 0.242 and $\\gamma_\\mathrm{eff}$ of 1.069, indicating that both $\\beta_\\mathrm{eff}$ and $\\gamma_\\mathrm{eff}$ are converged when the temperature approaches $T_\\mathrm{C}$.\n\nThe experimental critical exponents of CrGeTe$_3$, as well as the theoretical values of CrSiTe$_3$, MnSi and some other manganites based on various models, are summarized in Table~\\ref{table:1}. It is seen that the critical exponents for MnSi and doped manganites are consistent with those of tricritical mean-field theory \\cite{Ref45,Ref53,Ref60,Ref61}. These compounds have the same characteristics,{\\it i.e.}, a tricritical point separating the first-order from the second-order ferromagnetic phase transitions. This phenomenon shows that the element substitution \\cite{Ref53,Ref61}, hole or electric doping \\cite{Ref60}, and external magnetic field\\cite{Ref45} can induce the tricritical behavior. However, CrGeTe$_3$ presents a second-order ferromagnetic phase transitions\\cite{PhysRevB.92.144404,carteaux1995crystallographic,ji2013ferromagnetic,tian2016magneto,PhysRevB.92.035407} and our results indicate that the critical behavior of CrGeTe$_3$ is close to the theoretical value of tricritical mean-field model. Comparing with CrSiTe$_3$, showing characteristic of 2D-Ising model \\cite{PhysRevB.92.144404,liu2016critical,carteaux19952d}, the smaller van der Waals gap and the larger planar nearest-neighbor Cr-Cr distance of CrGeTe$_3$ enhances the Curie temperature from 32 K for the CrSiTe$_3$ to 61 K for the CrGeTe$_3$ \\cite{PhysRevB.92.144404,carteaux1995crystallographic,ji2013ferromagnetic,tian2016magneto,PhysRevB.92.035407}. In addition, the neutron scattering and isothermal magnetization experiments yield a critical exponent $\\beta$ of around 0.151 or 0.17 for CrSiTe$_3$ \\cite{PhysRevB.92.144404,liu2016critical,carteaux19952d}, which is close to the value expected for a 2D transition ($\\beta_\\mathrm{2D}^\\mathrm{Ising}$= 0.125) and well below the values expected for a 3D transition ($\\beta_\\mathrm{2D}^\\mathrm{Ising}$ = 0.326). Our results yield a critical exponent $\\beta$ of 0.24 for CrGeTe$_3$ that is close to the critical exponent of the tricritical mean-field model. Hence, the increase of the $X$ atom radius, facilitating super exchange coupling between the Cr atoms via the Te atom and leading to the smaller van der Waals gap in Cr$X$Te$_3$ system \\cite{PhysRevB.92.144404,carteaux1995crystallographic,ji2013ferromagnetic,tian2016magneto,PhysRevB.92.035407}, could induce a tricritical magnetic phase transition in the CrGeTe$_3$ single crystal.\n\n\n\\begin{table*}[tb]\n  \\caption{Critical exponents of CrGeTe$_3$ with various theoretical models, CrSiTe$_3$ and other related materials with tricritical mean-field model (SC = single crystal; PC = polycrystalline; cal = calculated from Eq.\\ref{eq:5}).}\n  \\begin{ruledtabular}\n    \\begin{center}\n      \\begin{tabular}{ccccccc}\n       Composition & Referecne & $T_\\mathrm{C}(\\mathrm{K})$ & Technique & $\\beta$ & $\\gamma$ & $\\delta$\\\\\n        \\hline\n       CrGeTe$_3^\\mathrm{SC}$ & This work & 67.9 & Modified Arrott plot &0.242$\\pm$0.006 &0.985$\\pm$0.003&5.070$\\pm$0.006$^\\mathrm{cal}$\\\\\n                                                   &                 &         & Kouvel-Fisher method &0.240$\\pm$0.006 &1.000$\\pm$0.005&5.167$\\pm$0.006\\\\\n                                                   &                 &         & Critical isotherm & & &5.032$\\pm$0.005$^\\mathrm{cal}$\\\\\n         Tricritical mean-field                                         &  [\\onlinecite{Ref52}]      &         & Theory &0.25 & 1 &5\\\\\nMean-field                                         &  [\\onlinecite{Ref49}][\\onlinecite{Ref50}]      &         & Theory &0.5 & 1 &3\\\\\n3D-Heisenberg theory                                         &  [\\onlinecite{Ref49}][\\onlinecite{Ref50}]      &         & Theory &0.365 & 1.386 &4.8\\\\\n3D-Ising                                        &  [\\onlinecite{Ref49}][\\onlinecite{Ref50}]      &         & Theory &0.325 & 1.24 &4.82\\\\\n       CrSiTe$_3^\\mathrm{SC}$ & [\\onlinecite{liu2016critical}]  & 31 & Modified Arrott plot &0.170$\\pm$0.008 &1.532$\\pm$0.001&10.012$\\pm$0.047$^\\mathrm{cal}$\\\\\n       MnSi$^\\mathrm{SC}$ & [\\onlinecite{Ref45}]  & 30.5 & Modified Arrott plot &0.242$\\pm$0.006 &0.915$\\pm$0.003&4.734$\\pm$0.006\\\\\nLa$_{0.1}$Nd$_{0.6}$Sr$_{0.3}$MnO$_3$$^\\mathrm{PC}$ & [\\onlinecite{Ref53}]  & 249.3 & Modified Arrott plot &0.257$\\pm$0.005 &1.12$\\pm$0.03&5.17$\\pm$0.02\\\\\nLa$_{0.9}$Te$_{0.1}$MnO$_3$$^\\mathrm{PC}$ & [\\onlinecite{Ref60}]  & 239.5 & Modified Arrott plot &0.201$\\pm$0.003 &1.27$\\pm$0.04&7.14$\\pm$0.04\\\\\nLa$_{0.6}$Ca$_{0.4}$MnO$_3$$^\\mathrm{PC}$ & [\\onlinecite{Ref61}]  & 265.5 & Modified Arrott plot &0.25$\\pm$0.03 &1.03$\\pm$0.05&5.0$\\pm$0.8\\\\\n\n      \\end{tabular}\n    \\end{center}\n  \\end{ruledtabular}\n  \\label{table:1}\n\\end{table*}\n\n\nAlthough single-crystalline CrSnTe$_3$ has not yet been synthesized, we speculate that the magnetism of CrSnTe$_3$ should be closer to the 3D-Ising model. To support this assumption, we perform DFT calculations with the same calculation parameters that were used in Ref.[\\onlinecite{Ref62}]. Figure~\\ref{fig:4}(a) shows the calculated formation energy $E_\\mathrm{f}$, which is defined as the energy cost of extracting a sheet of single-layer Cr$X$Te$_3$ from their bulk counterparts. As can be seen, $E_\\mathrm{f}$ increases as the species vary from Si to Ge. This is consistent with the larger theoretical cleavage energy of single-layer CrGeTe$_3$ than that of CrSiTe$_3$, which indicates that the layers are coupled more strongly in CrGeTe$_3$\\cite{li2014crxte}. The formation energy of CrSnTe$_3$ is even higher than the other two compounds, revealing that it presents the strongest interlayer coupling, which leads to its 3D characteristics.  Figure~\\ref{fig:4}(b), (c), and (d) illustrate the charge density of the three compounds. Consistent with the trend of the $E_\\mathrm{f}$ results, the electron density around the Sn-Sn pair is the least among the three materials. Namely, more electrons in CrSnTe$_3$ participate the interlayer coupling.\n\\begin{figure}[t]\n\\center\n  \\includegraphics[width=8.5cm]{Fig4.eps}\n  \\caption{(a) Formation energy of single-layer Cr$X$Te$_3$; The formation energy of single-layer CrSiTe$_3$ is adopted from Ref.[\\onlinecite{Ref62}] (b), (c), and (d) charge density of bulk Cr$X$Te$_3$ with an isosurface value of 0.05 $e$\/$r_\\mathrm{Bohr}^3$.}\n  \\label{fig:4}\n\\end{figure}\n\n\\section{Conclusions}\nIn conclusion, we have performed a comprehensive experimental study on the critical properties of single-crystalline CrGeTe$_3$ using isothermal magnetization around the Curie temperature $T_\\mathrm{C}$. Based on various experimental techniques including the modified Arrott plot, Kouvel-Fisher method and critical isotherm analysis, we obtained the critical exponents $\\beta$, $\\gamma$, and $\\delta$ of 0.240 $\\pm$ 0.006, 1.000 $\\pm$ 0.005, and 5.070 $\\pm$ 0.006, respectively, at the Curie temperature of 67.9K. These numerical results are similar to the theoretical values in the tricritical mean-field model, which is therefore capable of describing the critical magnetic behavior of 2D CrGeTe$_3$. DFT calculations show that the formation energy of CrGeTe$_3$ lies between those of CrSiTe$_3$ and CrSnTe$_3$, which is in line with a crossover of the magnetic phase transition from 2D to 3D. Overall, our findings provide a fundamental understanding of the anomalous PM-FM transition in a novel 2D ferromagnetic semiconductor.\n\n\\begin{acknowledgments}\nThis work was supported by the National Key Research and Development Program under contracts 2016YFA0300404 and 2016YFA0401003, the Joint Funds of the National Natural Science Foundation of China and the Chinese Academy of Sciences' Large-Scale Scientific Facility under contract U1432139 and U1532153, the National Natural Science Foundation of China under contract 11674326, Key Research Program of Frontier Sciences, CAS (QYZDB-SSW-SLH015), and the Nature Science Foundation of Anhui Province under contract 1508085ME103. J. Z. is supported by the Nature Science Foundation of China (Grant No. 51602079) and the Fundamental Research Funds for the Central Universities of China (Grant No. 372 AUGA5710013115). This research also used computational resources of the National Supercomputing Center of China in Shenzhen (Shenzhen Cloud Computing Center).\n\\end{acknowledgments}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro}Introduction}\n\nThere is at least one Higgs boson~\\cite{LHC}; maybe there are more.\nMulti-Higgs doublet models contain new sources of CP violation, which is one\nof the required ingredients~\\cite{Sakharov} for baryogenesis. It is therefore\ninteresting to consider whether CP violation from the Higgs sector could be\nused to generate the baryon asymmetry of the Universe~\\cite{BAU,BAU2}. This\ncan occur in electroweak baryogenesis scenarios \\cite{BAU@EPT}; here we are\ninterested in asymmetries produced before the electroweak phase transition\n(EWPT).\n\nIn this paper we consider two-Higgs doublet models (2HDM)~\\cite{hhg,2HDM}. If\ninteractions which exchange Higgs flavour are sufficiently weak, then the two\npopulations of Higgs fields could  contain independent asymmetries in the\nearly Universe. Since at least one of the Higgs must couple to Standard Model\n(SM) fermions, its asymmetry is redistributed among other SM particles by\nYukawa interactions,  prior to the electroweak phase transition.\nHowever, hypercharge neutrality of the Universe relates the asymmetries among\nall charged particles. This implies that a relative asymmetry among the Higgs\nscalars, generated by out-of-equilibrium CP-violating processes in the Higgs\nsector, could be transformed into a baryon asymmetry in the presence of\n($B+L$)-violating sphalerons~\\cite{sphalerons}. The interest of such\nbaryogenesis scenarios is that they require no $B$ or $L$-violating\ninteractions beyond the non-perturbative sphalerons of the SM, relying only on\nCP violation in an extended Higgs sector.\n\nThe issue of ``basis-independence'' is of particular\nimportance~\\cite{Lavoura:1994fv,DH}. The point is that physical observables\ncannot depend on a basis choice in the Lagrangian ---one may ask, for\ninstance, what $\\phi_1$ and $\\phi_2$ are in the 2HDM. Clearly, the survival of\na relative asymmetry between the $\\phi_1$'s and $\\phi_2$'s in early Universe\nwill depend on the speed of interactions that exchange $\\phi_1$ with $\\phi_2$.\nBut the pertinent coupling constants naively appear to depend on the choice of\n$\\phi_1$ and $\\phi_2$. We show that such washout interactions are controlled\nby the misalignment among different couplings, and can be parameterised in a\nbasis-independent way.\n\nThe paper is organized as follows. A compendium of relevant results for the\n2HDM is given in section~\\ref{sec:2HDM}, followed by some estimates for\ninteraction rates in the early Universe. Section~\\ref{sec:TE} constrains\nparameters of the Higgs potential by requiring Higgs flavour exchange to be\nout of equilibrium. In the second part of this section, we discuss the\nbasis-independence of these bounds. In Section~\\ref{sec:chemeq} we derive the\nequations of chemical equilibrium~\\cite{KS},  which relate the asymmetries\namong SM particles and Higgs fields, due to the interactions which are in\nequilibrium. As a result, a nonvanishing equilibrium baryon asymmetry is\nobtained in the presence of a relative Higgs asymmetry, even with $B-L$\nconservation. Simple scenarios based on the out-of-equilibrium decay of\nsinglet scalar fields into Higgs doublets are presented in\nSection~\\ref{sec:scenarios}. Finally, our conclusions are summarized in\nSection~\\ref{sec:summary}.\n\n\\section{The 2HDM at finite temperature}\n\\label{sec:notn}\n\n\\subsection{Notation and review}\n\\label{sec:2HDM}\n\nThe interaction Lagrangian for  the  general 2HDM~\\cite{hhg,2HDM} consists of\na scalar potential plus Yukawa coupling terms. The most general gauge\ninvariant scalar potential can be written as\n\\begin{eqnarray}\n\\label{pot}\nV&=& m_{11}^2\\phi_1^\\dagger\\phi_1+m_{22}^2\\phi_2^\\dagger\\phi_2\n-[m_{12}^2\\phi_1^\\dagger\\phi_2+{\\rm h.c.}]\\nonumber\\\\[6pt]\n&&\\quad +\\frac{1}{2}\\lambda_1(\\phi_1^\\dagger\\phi_1)^2\n+\\frac{1}{2}\\lambda_2(\\phi_2^\\dagger\\phi_2)^2\n+\\lambda_3(\\phi_1^\\dagger\\phi_1)(\\phi_2^\\dagger\\phi_2)\n+\\lambda_4(\\phi_1^\\dagger\\phi_2)(\\phi_2^\\dagger\\phi_1)\n\\nonumber\\\\[6pt]\n&&\\quad +\\left[\\frac{1}{2}\\lambda_5(\\phi_1^\\dagger\\phi_2)^2\n+\\lambda_6(\\phi_1^\\dagger\\phi_1)(\\phi_1^\\dagger\\phi_2)\n+\\lambda_7(\\phi_2^\\dagger\\phi_2)(\\phi_1^\\dagger\\phi_2)+{\\rm H.c.}\\right]\\,,\n\\end{eqnarray}\nwhere $\\phi_1$ and $\\phi_2$ are two complex SU(2)$_{L}$ doublet scalar fields\nof unit hypercharge; $m_{11}^2$, $m_{22}^2$, and $\\lambda_1\\ldots\\lambda_4$\nare real parameters, while $m_{12}^2$ and $\\lambda_5,\\lambda_6,\\lambda_7$ can\nbe complex. In  general, both $\\phi_1$ and $\\phi_2$ can have Yukawa couplings\nto all the SM fermions. The Yukawa interactions are\n\\begin{eqnarray}\n- {\\cal L}_Y &=&\n\\overline{Q_L}\\, (\\mathbf{\\Gamma}_1 \\phi_1 + \\mathbf{\\Gamma}_2 \\phi_2)\\, d_R\n+ \\overline{Q_L}\\, (\\mathbf{\\Delta}_1 \\tilde{\\phi}_1 + \\mathbf{\\Delta}_2\n\\tilde{\\phi}_2)\\, u_R +\n\\nonumber\\\\\n&& +\\, \\overline{L_L}\\, (\\mathbf{\\Pi}_1 \\tilde{\\phi}_1 + \\mathbf{\\Pi}_2\n\\tilde{\\phi}_2)\\, \\ell_R + \\textrm{H.c.}, \\label{Yuk_Z2}\n\\end{eqnarray}\nwhere $Q_L = (u_L, d_L)^T$ ($u_R$ and $d_R$) is a vector in the 3-dimensional\ngeneration space of left-handed quark doublets (right-handed charge $+2\/3$ and\n$-1\/3$ quarks). Accordingly, $\\mathbf{\\Gamma}_1$, $\\mathbf{\\Gamma}_2$,\n$\\mathbf{\\Delta}_1$, and $\\mathbf{\\Delta}_2$ are $3 \\times 3$ matrices in the\nrespective quark generation spaces. Similarly, $L_L = (\\nu_L, \\ell_L)^T$ and\n$\\ell_R$ are vectors in the 3-dimensional generation space of left-handed\nlepton doublets and right-handed charged leptons, respectively, while\n$\\mathbf{\\Pi}_1$ and $\\mathbf{\\Pi}_2$ are $3 \\times 3$ matrices. For\nsimplicity, we assume that there are no right-handed neutrino fields.\n\nUnder global SU(2) transformations in $(\\phi_1, \\phi_2)$ space, the kinetic\nterms of the Higgs doublets are invariant, whereas the parameters of the\nscalar potential (and the Yukawa couplings) will be modified.  Such basis\ntransformations in the Lagrangian cannot affect observables,  so the numerical\nvalue of the parameters in Eq.~(\\ref{pot}) is only meaningful when the basis\nis specified. Three obvious Higgs basis choices can be envisaged:\n\\begin{itemize}\n\\item   $m_{12}^2 = 0$ basis, where we put a tilde on the parameters\n    ($\\tilde{\\lambda}_i, \\tilde{m}_{ii}^2$, $\\tilde{y}^f_i$),\n\\item symmetry basis, where the parameters are lower case with a prime\n    ($\\lambda'_i, m_{ij}^{'2}$, ${y'}^f_i$),\n\\item (thermal) mass eigenstate basis, where the parameters are uppercase\n    ($\\Lambda_i, M_{ij}$, ${Y}^f_i$).\n\\end{itemize}\nHere, $\\tilde{y}^f_i$, ${y'}^f_i$ and ${Y}^f_i$ denote the Yukawa matrices of\nthe SM fermions $f$ interacting with the Higgs $i$, in the corresponding Higgs\nbasis (so $y^u_i = (\\Delta_1,\\Delta_2)$, and so on).\n\nSince our goal is to store an asymmetry between the Higgs populations prior to\nthe EWPT, interactions which exchange $\\phi_1 \\leftrightarrow \\phi_2$ must be\nsmall (see next section). We refer to such interactions as (Higgs)\nflavour-exchanging processes. For instance, in the $m_{12}^2 = 0$ basis, the\noffending parameters from the potential are $\\tilde{\\lambda}_5$,\n$\\tilde{\\lambda}_6$ and $\\tilde{\\lambda}_7$. In the Yukawa sector,\ninteractions  of both Higgs doublets to either quarks or leptons will be\nstrongly constrained. This is because a relative asymmetry in the two Higgs\npopulations should be preserved, so the two Higgs fields cannot both share\ntheir asymmetry with the same fermions.\n\nSome of the undesirable couplings can be suppressed by imposing a  discrete\n$Z_2$  symmetry\n\\begin{equation} \\phi_1 \\to -\\phi_1\\,,\\quad \\phi_2 \\to \\phi_2\\,.\\label{Z2}\n\\end{equation}\nIn the basis where the symmetry has the above form, it implies $m'_{12}\n=\\lambda'_6 =   \\lambda'_7 = 0$, so the Higgs sector contains no explicit CP\nviolation, because the phase of $\\lambda'_5$ can be rotated away by a phase\nchoice of the Higgs fields. If both scalar fields couple to fermions of the\nsame charge, then there will be flavour changing neutral scalar interactions,\nwhich are strongly constrained by experiment. The undesirable Yukawa\ninteractions can be removed by extending the $Z_2$ symmetry of Eq.~(\\ref{Z2})\nto the fermion sector, so that each fermion charge sector only couples to one\nof the Higgs scalars. The four ways to implement this symmetry are shown in\nTable~\\ref{table1}.\n\\begin{table*}[t!]\n\\centering\n\n\\begin{tabular}{|c|cc|}\n\\hline\nModel type & $\\phi_1$ & $\\phi_2$\\\\\n\\hline\nType I &   & $u$, $d$, $\\ell$\\\\\nType II & $d$, $\\ell$  & $u$ \\\\\nType X  &  $\\ell$ & $u$, $d$ \\\\\nType Y &  $d$  & $u$, $\\ell$\\\\\n\\hline\n\\end{tabular}\n\\caption{The four types of $Z_2$ models and the corresponding Higgs couplings\nto fermions. Type X is also known as ``lepton specific'', and type Y as\n``flipped''. In the usual $\\phi_{u,d}$ notation, $\\phi_u = \\phi_2$\nalways.}\n\\label{table1}\n\\end{table*}\n\n\nThe discovery of a 125 GeV scalar at LHC places constraints on the 2HDM\nparameter space, studied so far in the context of a $Z_2$ symmetry\n($\\lambda'_6 = \\lambda'_7 = 0$), occasionally exact ($m'_{12} =\n0$)~\\cite{reviews}. After electroweak symmetry breaking, the neutral\ncomponents of the scalar fields acquire the vacuum expectation values (VEVs)\n$\\langle \\phi_1^0 \\rangle = v_1$ and $\\langle \\phi_2^0 \\rangle = v_2$, where\n$v = (v_1^2 + v_2^2)^{1\/2} \\simeq 174$~GeV. Of the eight components in the\ntwo Higgs doublets, three provide longitudinal components to $W^\\pm$ and $Z$,\ntwo create a $H^\\pm$ pair, two yield the Higgs scalar ($m_h = 125$ GeV) and\nanother neutral scalar ($H$), and the last gives a pseudoscalar ($A$). These\nmasses and the VEVs are related to the parameters of the potential. In\nparticular, \\textit{if both $v_1 \\neq 0$ and $v_2 \\neq 0$}, then the\nstationarity conditions can be used to write the pseudoscalar mass\n\\begin{equation}\nm_A^2 = \\frac{v^2}{v_1 v_2}\\, m^{'2}_{12} - 2\\lambda'_5\\, v^2.\n\\end{equation}\nThis shows that requiring small $\\phi_1 \\leftrightarrow \\phi_2$ exchanges\nthrough $m'_{12} \\sim 0$ and $\\lambda'_5 \\sim 0$, leads to $m_A \\sim 0$,\nunless $v_1 v_2 \\sim 0$. This occurs because, in the $m'_{12} = \\lambda'_5 =\n\\lambda'_6 = \\lambda'_7 = 0$ limit, the potential in Eq.~\\eqref{pot} has a\nglobal $U(1)$ symmetry, which is broken by $v_1 v_2 \\neq 0$, with the\nconsequent appearance of a massless Goldstone boson ($m_A=0$).\n\nOne solution is to consider the inert model~\\cite{inert}, which is a Type I\n2HDM with exact $Z_2$ and $v_1 = 0$\\footnote{In the usual notation for the\ninert doublet model, only $\\phi_1$ couples to fermions, while $v_2=0$. In the\nnotation used here, the role of $\\phi_1$ and $\\phi_2$ are reversed, implying\nthe changes $m'_{11} \\leftrightarrow m'_{22}$ and $\\lambda'_1 \\leftrightarrow\n\\lambda'_2$.}. In that case,\n\\begin{equation}\nm_A^2 = m^{'2}_{22} + (\\lambda'_3 + \\lambda'_4- \\lambda'_5)\\, v^2.\n\\label{mA_inert}\n\\end{equation}\nThis mass can be kept nonzero, even if $\\lambda'_5 = 0$, because the vacuum\nwith $v_1=0$ does not break the global $U(1)$. The only consequence of\n$\\lambda'_5 = 0$ is $m_H=m_A$. Because the inert $\\phi_1$ does not couple to\nfermions, the lightest particle is a candidate for dark matter. A series of\nvery clear analyses of this model, including constraints from both LHC and\nWMAP, have been performed by the Warsaw group~\\cite{warsaw}. They find large\nregions of parameter space consistent with all known data, especially if the\n$h \\rightarrow \\gamma \\gamma$ signal is consistent with the SM\n($R_{\\gamma\\gamma} \\sim 1$). This is within the 2$\\sigma$ ranges of current\nATLAS~\\cite{Aad:2013wqa} and CMS~\\cite{CMS_gg} data\n\\begin{eqnarray}\n\\textrm{ATLAS:}\n&\\ \\ \\ &\nR_{\\gamma\\gamma} = 1.55^{+0.33}_{-0.28}\\,,\n\\nonumber\\\\\n\\textrm{CMS:}\n&\\ \\ \\ &\nR_{\\gamma\\gamma} = {0.78}^{+0.28}_{-0.26}\\,.\n\\end{eqnarray}\nValues of $R_{\\gamma\\gamma}$ larger than one restrict considerably the\nparameter space.\n\nAn additional constraint imposed on the Higgs spectrum by our baryogenesis\nscenario is that $\\phi_1$ and $\\phi_2$ should be present in the thermal bath\nuntil the EWPT. If one of the $\\phi_i$ is sufficiently heavy that its\npopulation decays away prior to the EWPT, then the relative Higgs asymmetry\nis lost.\n\n\\subsection{Thermal masses and interaction rates}\n\nIn this section we review interaction rates in a thermal bath.  The relevant\neigenbasis for the external leg particles should be the thermal mass\neigenstate basis, so we start by estimating the thermal mass matrix of the\nHiggs scalars, at temperatures $T \\gg |m_{ij}|$. At finite temperature, the\nlowest order contribution to the mass-squared matrix is~\\cite{CE}\n\\begin{equation}\nm^2_{ij}(T) \\simeq\n\\frac{\\partial^2 V_{eff}(T, \\phi_k)}{\\partial\\phi_i \\partial\\phi^*_j},\n\\label{defnmT}\n\\end{equation}\nwhere  $V_{eff}(T, \\phi_k)= V + V_T$ is the effective potential,\nwith $V$ given in Eq.~(\\ref{pot}).\nIn the high temperature limit ($T \\gg |m_{ij}|$),\n\\begin{equation}\nV_T = \\frac{T^2}{12} {\\rm Tr} \\big[ \\mathbf{m}^2 \\big] =\n\\frac{T^2}{24}\\sum_{i=1,2} g_i M^2(\\phi_i)\\,,\n\\label{VT}\n\\end{equation}\nwhere  $g_i = 4$  for a complex doublet field, and the trace is calculated\nover the  $T=0$ scalar mass-squared matrix in background field, {\\it  i.e.}\nallowing $\\phi_1$ and $\\phi_2$ to have non-zero values so that $M(\\phi_i)$\nare field-dependent masses.\n\nNeglecting zero-temperature loop contributions and finite-temperature fermion\nand gauge contributions, we find\n\\begin{equation}\nV_T =\\frac{T^2}{12}\\left\\{\n 3\\lambda_1|\\phi_1|^2  +(2 \\lambda_{3} + \\lambda_{4})(|\\phi_2|^2+|\\phi_1|^2)\n+  3\\lambda_2|\\phi_2|^2\n-\n[3(\\lambda_6 +\n \\lambda_7)\\phi^{\\dagger}_1 \\phi_2\n+{\\rm H.c.}]\\right\\},\n\\end{equation}\nwhere the $m_{ij}^2$ terms are dropped because they give no contribution to\nEq.~(\\ref{defnmT}). For an arbitrary basis in the Higgs doublet space~\\cite{Ivanov},\nthis gives the thermal mass-squared matrix\n\\begin{eqnarray}\n\\mathbf{m}^2(T)& \\simeq&\n\\frac{T^2}{12}\\left(\n\\begin{array}{cc}\n3\\lambda_1 + 2 \\lambda_{3}   + \\lambda_{4}&\n-3\\lambda_{6}   - 3\\lambda_{7}\\\\\n -3\\lambda^*_{6}   - 3\\lambda^*_{7}\n & 3\\lambda_2 + 2 \\lambda_{3}   + \\lambda_{4}\\\\\n\\end{array}\n\\right)\n+\n\\left(\n\\begin{array}{cc}\n m_{11}^2&m_{12}^2 \\\\\nm_{12}^{*2} &m_{22}^2\n\\end{array}\n\\right).\n\\label{mtherm}\n\\end{eqnarray}\nDiagonalising this matrix gives the thermal mass eigenstate basis. In the\npresence of a $Z_2$ symmetry, the thermal masses are simply given by\n\\begin{eqnarray}\nm_{11}^2(T) &=& \\frac{T^2}{12} (3 \\lambda_1 + 2 \\lambda_3 + \\lambda_4) + m_{11}^2,\n\\nonumber\\\\\nm_{22}^2(T) &=& \\frac{T^2}{12} (3 \\lambda_2 + 2 \\lambda_3 + \\lambda_4) + m_{22}^2,\n\\end{eqnarray}\nso that no term of the type $m_{12}^2(T)\\, \\phi_1^\\dagger \\phi_2$ is\ngenerated. Therefore, in the latter case, the only link between $\\phi_1$ and\n$\\phi_2$ in the Higgs potential comes from the $\\lambda_5$ term, both at zero\nand at finite temperature.\n\n\nWe now review the assumptions and approximations involved in our estimates for\nthe interaction rates. We take ``thermal equilibrium\" to describe a particle\nspecies distributed following a Maxwell-Boltzmann distribution.  At\ntemperatures $T \\ll m_{\\rm GUT} \\simeq 10^{16}$~GeV, this will be the case for\nparticles with SM gauge interactions. We define an interaction to be in\n``chemical equilibrium'' if it is fast enough to impose relations among the\nasymmetries in the participating particles. This will be the case if its\ntimescale, $1\/\\Gamma$, is much shorter than the age of the Universe $\\sim\n1\/H$, i.e. $\\Gamma \\gg H$, where\n\\begin{equation}\nH = \\sqrt{\\frac{4\\pi^3 g_\\ast}{45}} \\frac{T^2}{m_P} \\simeq \\frac{\n17\\, T^2}{m_P}, \\label{hubble}\n\\end{equation}\nis the Hubble expansion parameter, $g_\\ast$ is the number of\nrelativistic degrees of freedom ($g_\\ast=107.75$ in the 2HDM) and $m_P = 1.22\n\\times 10^{19}$~GeV is the Planck mass.\n\nWe estimate the interaction rate $\\Gamma = \\gamma\/n$, where $n_i   \\simeq\ng_iT^3\/\\pi^2$ is the equilibrium density of an incident (massless) particle,\nand $\\gamma$ is the interaction density.  For a process $ij \\to mn$,  where\nall the participating particles are in thermal equilibrium, $\\gamma$  is the\nthermally averaged scattering rate,\n\\begin{eqnarray} \\label{gammarate}\n\\gamma(ij &\\to& mn)\n= \\langle n_i n_j \\sigma(i+j \\to..) \\rangle \\nonumber\\\\\n && = \\int d\\Pi_i  d\\Pi_j f^{eq}_i f^{eq}_j\n\\int |{\\cal M}(i+j \\rightarrow m+n )|^2\n~(2\\pi)^4{\\delta}^4( p_i+p_j  -  p_m - p_n)~\nd\\Pi_m d\\Pi_n \\nonumber \\\\\n&&= \\frac{g_1 g_2 |\\Lambda|^2\\, T^4}{ 32\\pi^5},\n\\label{gamlam}\n\\end{eqnarray}\nwhere $g_i$ is the number of degrees of freedom of the particle in the  bath\n(2 for a doublet), $d\\Pi = \\dfrac{d^3p}{2E(2 \\pi)^3}$ is the relativistic\nphase space, and $f^{eq}$ is the Maxwell-Boltzmann equilibrium distribution.\nThe last equality in Eq.~\\eqref{gammarate} is the result for $|{\\cal M}(i+j\n\\rightarrow m+n )|^2 = |\\Lambda|^2$.\n\n\\section{Keeping Higgs flavour-exchanging interactions out of equilibrium}\n\\label{sec:TE}\n\nWe suppose that particle-antiparticle asymmetries in $\\phi_1$ and $\\phi_2$\nwere generated at some earlier epoch of the Universe. In\nsection~\\ref{sec:scenarios}, we shall illustrate this in a simple framework.\nWe focus on the relative asymmetry between the two Higgs doublets:\n\\begin{equation}\n\\mathcal{Y}_{\\Delta \\phi_1} - \\mathcal{Y}_{\\Delta \\phi_2} \\equiv\n\\frac{n_{\\phi_1}- n_{\\overline{\\phi}_1}}{s}- \\frac{n_{\\phi_2}-\nn_{\\overline{\\phi}_2}}{s}, \\label{asym}\n\\end{equation}\nwhere $s$ is the entropy density of the Universe. We use the notation\n$\\mathcal{Y}_{\\Delta X}$ for the asymmetry $\\mathcal{Y}_X -\n\\mathcal{Y}_{\\overline{X}}$, where $\\mathcal{Y}_X = n_X\/s$ is the comoving number\ndensity.\nThis asymmetry will be conserved as long as Higgs flavour-exchanging\ninteractions are out of equilibrium. In this section, we identify these\ninteractions,  estimate the constraints on the couplings, and express these\nbounds in some useful bases.\n\nIn the thermal mass eigenstate basis, the flavour-exchanging Higgs\ninteractions that must be out of equilibrium are mediated by the quartic\ncouplings $\\Lambda_5$, $\\Lambda_6$, and $\\Lambda_7$. Requiring $ \\Gamma \\ll H$\nat $T \\simeq 100$~GeV, and using Eqs.~(\\ref{hubble}) and (\\ref{gamlam}),\nimplies\n\\begin{equation}\n|\\Lambda_n|\n\\lesssim {\\rm few} \\times 10^{-7},\\quad n =5,6,7,\n\\label{bdL}\n\\end{equation}\nto keep the Higgs asymmetries separate for temperatures down to the EWPT. This\ncondition applies in the thermal mass eigenstate basis; we translate it below\nto other bases.\n\nHiggs flavour could also be exchanged via Yukawa couplings, if both Higgs\ndoublets interact with the same fermions.  For simplicity, we only consider\nthe third generation of fermions. The $t$, $b$ and $\\tau$ have Yukawa\ninteractions to both Higgs fields, so their Yukawa couplings are vectors in\nHiggs doublet space, which we represent capitalised in the thermal mass\neigenstate basis. For instance, the top Yukawa coupling is $(Y^t_1, Y^t_2)$,\nwith $m_t = Y^t_1\\, v_{1}^{T} +  Y^t_2\\,v_{2}^{T}$, where the $v_{i}^{T}$ are\nthe zero-temperature Higgs VEVs in the thermal basis. The survival of the\nrelative Higgs asymmetry requires that the Yukawa interactions, between a\nfermion species $f = t,b,\\tau$ and one of the Higgs doublets, be out of\nequilibrium:\n\\begin{equation}\n\\min_i \\gamma(f+g\/\\gamma \\to f+\\phi_i) \\ll n_{f} H.\n\\end{equation}\nFor instance, in the case of the top quark, this gives\n\\begin{equation}\n \\theta_t^2 \\frac{\\alpha_s}{16\\pi^2} \\frac{m_t^2}{v^2}   \\ll 17 \\frac{T}{m_P},\n\\label{topbd}\n\\end{equation}\nwhere $\\theta_t$ is the rotation angle between the thermal mass eigenstate\nHiggs basis and the eigenvector of the top Yukawa coupling (cf.\nAppendix~\\ref{sec:appendix}):\n\\begin{equation} \\label{thetatapprox}\n\\theta_t \\simeq \\frac{|Y^t_1\\,Y^t_2|}{\\sqrt{|Y^t_1|^2 + |Y^t_2|^2}}\\,\n\\simeq\n\\frac{\\left|m_{12}^2(T)\\right|}{\\left|m_{11}^2(T) - m_{22}^2(T)\\right|}\n\\,,\n\\end{equation}\nwhere the last expression is in the Yukawa eigenbasis.\n\nAt $T \\simeq 100$~GeV, Eq.~\\eqref{topbd} requires\n\\begin{equation}\\label{thetatbound}\n\\theta_t\\lesssim {\\rm few}\\times 10^{-7}.\n\\end{equation}\nFurthermore, using $|\\lambda_6|, |\\lambda_7| \\lesssim$ few $\\times 10^{-7}$ to\nsatisfy Eq.~(\\ref{bdL}), and assuming $m_{11}^2$ and $m_{22}^2$ of the order\nof the lightest Higgs mass, it follows from Eqs.~\\eqref{mtherm}\nand~\\eqref{thetatapprox} that\n\\begin{equation}\\label{m12bound}\n|m_{12}^2| \\lesssim (100\\,\\textrm{MeV})^2,\n\\end{equation}\nin the Yukawa eigenbasis.\n\nWe remark that, in obtaining the bound (\\ref{topbd}), we approximate\n$m_t=(|Y^t_1|^2 + |Y^t_2|^2)^{1\/2}\\, v $,  that is, we neglect the\nmisalignment between the top Yukawa coupling vector and the  zero-temperature\nVEVs. This could underestimate the magnitude of the Yukawa coupling (as arises\nfor the $b$ and $\\tau$ in the large $\\tan \\beta$ limit of the supersymmetric\nSM). Therefore, the interaction rates we obtain will be lower bounds. Similar\nbounds apply to other fermions $f$, with the replacement $m_t \\to m_f$ (and\n$\\alpha_s \\to \\alpha_{\\rm QED}\/4 $ for leptons). This leads to the bounds\n\\begin{eqnarray}\\label{thetabtau}\n\\theta_b &\\lesssim& 10^{-5} \\to  10^{-7}\\quad \\text{for the b quark}, \\nonumber\\\\\n\\theta_\\tau &\\lesssim& 10^{-4} \\to  10^{-6}\\quad \\text{for the $\\tau$-charged lepton}.\n\\end{eqnarray}\nThe weaker limit corresponds to $|Y^f_1|^2 +|Y^f_2|^2=(m_f\/v)^2$ and the\nstronger one to $|Y^f_1|^2 +|Y^f_2|^2 \\sim 1$.\n\n\\subsection{Basis-independent conditions}\n\\label{sec:basis}\n\nIn this section, the conditions given in Eqs.~(\\ref{bdL}) and (\\ref{topbd}),\nwhich ensure the survival of a relative Higgs asymmetry, are expressed in a\nway which is independent of the (Higgs) basis transformation\n\\begin{equation}\n\\Phi \\rightarrow \\mathbf{U}\\, \\Phi,\n\\label{HBT}\n\\end{equation}\nwhere $\\Phi = ( \\phi_1,\\ \\phi_2)^T$ and $\\mathbf{U}$ is a $2 \\times 2$ unitary\nmatrix in Higgs flavour space. In Refs.~\\cite{Lavoura:1994fv,DH},\nbasis-independent combinations of potential parameters were constructed by\ncontracting the parameters with the Higgs VEVs (a vector in Higgs doublet\nspace). We will construct similar invariants here, but replacing the Higgs VEV\nwith the top Yukawa coupling, which is more relevant for our scenario, and is\nalso a vector in Higgs space (in the one generation approximation). Indeed,\none can combine the top Yukawa couplings in\n\\begin{equation}\n- {\\cal L}_Y\n=\n\\overline{t_L} \\left( y_1^t,\\ y_2^t\\right)\\,\n\\left(\n\\begin{array}{cc}\n\\phi_1\\\\\n\\phi_2\n\\end{array}\n\\right)\\, t_R\n+ \\textrm{h.c.},\n\\end{equation}\ninto a vector\n\\begin{equation}\n\\hat{y}^t\n=\n\\frac{1}{\\sqrt{|y^t_1|^2 +|y^t_2|^2}}\n\\left(\n\\begin{array}{c}\ny^t_1\\\\\ny^t_2\n\\end{array}\n\\right),\n\\label{hat_y}\n\\end{equation}\ntransforming as $\\hat{y}^t \\rightarrow \\mathbf{U}\\, \\hat{y}^t$,\nand its orthogonal\n\\begin{equation}\n\\hat{\\epsilon}^t\n=\n\\frac{1}{\\sqrt{|y^t_1|^2 +|y^t_2|^2}}\n\\left(\n\\begin{array}{c}\n- y^{t\\,\\ast}_2\\\\\ny^{t\\,\\ast}_1\n\\end{array}\n\\right),\n\\label{hat_eps}\n\\end{equation}\ntransforming as $\\hat{\\epsilon}^t \\rightarrow\n[\\textrm{det}\\,\\mathbf{U}]^{-1}\\, \\mathbf{U}\\, \\hat{\\epsilon}^t$. In the top\nbasis (see Appendix~\\ref{sec:appendix}), these vectors become\n\\begin{equation}\n\\hat{y}^t\n=\n\\left(\n\\begin{array}{c}\n1\\\\\n0\n\\end{array}\n\\right),\n\\ \\ \\ \\\n\\hat{\\epsilon}^t\n=\n\\left(\n\\begin{array}{c}\n0\\\\\n1\n\\end{array}\n\\right).\n\\label{topbasis}\n\\end{equation}\nFrom Eq.~(\\ref{topbd}), it is clear that the direction in Higgs space of the\ntop Yukawa coupling $\\hat{y}^t$ should approximately correspond to $\\phi_1$\nor $\\phi_2$ of the thermal mass eigenstate basis. We then simply impose the\nbounds of Eqs.~(\\ref{bdL}) and (\\ref{topbd}) in the basis of\nEq.~(\\ref{topbasis}).\n\nIt is convenient to introduce some notation patterned on Ref.~\\cite{DH}. The\nquartic Higgs interactions can be represented via a four-index tensor which\nappears in the Lagrangian as $\\frac{1}{2} Z_{a \\bar{b} c \\bar{d}}\n\\Phi_{\\bar{a}}^\\dagger \\Phi_b \\Phi_{\\bar{c}}^\\dagger \\Phi_d$ where\n$\\Phi^\\dagger = (\\phi_1^\\dagger, \\phi_2^\\dagger)$, and $a,b,c,d=1,2$. The\nbarred (unbarred) notation keeps track of which indices transform as\n$\\mathbf{U}^\\dagger$ ($\\mathbf{U}$), under the basis\ntransformation~\\eqref{HBT}. The elements of $Z_{a \\bar{b} c \\bar{d}}$ are\n\\begin{eqnarray}\n&& Z_{1\\bar{1}1\\bar{1}}=\\lambda_1\\,,\\qquad\\qquad \\,\\,\\phantom{Z_{2\\bar{2}2\\bar{2}}=}\nZ_{2\\bar{2}2\\bar{2}}=\\lambda_2\\,,\\nonumber\\\\\n&& Z_{1\\bar{1}2\\bar{2}}=Z_{2\\bar{2}1\\bar{1}}=\\lambda_3\\,,\\qquad\\qquad\nZ_{1\\bar{2}2\\bar{1}}=Z_{2\\bar{1}1\\bar{2}}=\\lambda_4\\,,\\nonumber \\\\\n&& Z_{1\\bar{2}1\\bar{2}}=\\lambda_5\\,,\\qquad\\qquad \\,\\,\\phantom{Z_{2\\bar{2}2\\bar{2}}=}\nZ_{2\\bar{1}2\\bar{1}}=\\lambda_5^*\\,,\\\\\n&& Z_{1\\bar{1}1\\bar{2}}=Z_{1\\bar{2}1\\bar{1}}=\\lambda_6\\,,\\qquad\\qquad\nZ_{1\\bar{1}2\\bar{1}}=Z_{2\\bar{1}1\\bar{1}}=\\lambda_6^*\\,,\\nonumber \\\\\n&& Z_{2\\bar{2}1\\bar{2}}=Z_{1\\bar{2}2\\bar{2}}=\\lambda_7\\,,\\qquad\\qquad\nZ_{2\\bar{2}2\\bar{1}}=Z_{2\\bar{1}2\\bar{2}}=\\lambda_7^*\\,.\\nonumber\n\\label{znum}\n\\end{eqnarray}\nBy analogy with the invariants $|Z_5|$, $|Z_6|$, and $|Z_7|$ presented in\nRef.~\\cite{DH}, the following basis invariant quantities can be constructed:\n\\begin{eqnarray}\n|S_5|\n& \\equiv &\n| Z_{a \\bar{b} c \\bar{d}}\n\\, \\hat{y}_{\\bar{a}}^{t \\ast}\n\\, \\hat{\\epsilon}_{b}^{t}\n\\, \\hat{y}_{\\bar{c}}^{t \\ast}\n\\, \\hat{\\epsilon}_{d}^{t} |,\n\\label{szvv5}\n\\nonumber\\\\\n|S_6|\n& \\equiv &\n| Z_{a \\bar{b} c \\bar{d}}\n\\, \\hat{y}_{\\bar{a}}^{t \\ast}\n\\, \\hat{y}_{b}^{t}\n\\, \\hat{y}_{\\bar{c}}^{t \\ast}\n\\, \\hat{\\epsilon}_{d}^{t} |,\n\\label{szvv6}\\\\\n|S_7|\n& \\equiv &\n| Z_{a \\bar{b} c \\bar{d}}\n\\, \\hat{y}_{\\bar{a}}^{t \\ast}\n\\, \\hat{\\epsilon}_{b}^{t}\n\\, \\hat{\\epsilon}_{\\bar{c}}^{t \\ast}\n\\, \\hat{\\epsilon}_{d}^{t} |.\\nonumber\n\\label{szvv7}\n\\end{eqnarray}\nThese correspond to $|\\lambda_5|$, $|\\lambda_6|$,  and $|\\lambda_7|$ in the\nbasis of Eq. (\\ref{topbasis}) and, consequently, $|S_n| \\lesssim {\\rm few}\n\\times 10^{-7}$ to satisfy Eq.~(\\ref{bdL}).\n\nAs seen in Appendix~\\ref{sec:appendix}, the rotation angle between the thermal\nmass eigenstate basis and the top Yukawa eigenbasis of Eq.~(\\ref{topbasis})\ncan be written in the top basis as\n\\begin{equation}\n|\\tan{(2 \\theta_{t})}|\n=\n\\frac{2 |m_{12}^{t 2}|}{|m_{11}^{t 2} -  m_{22}^{t 2}|}\n=\n\\frac{2\n\\left|m_{a \\bar{b}}^{t 2}\\ \\hat{y}_{\\bar{a}}^{t \\ast}\\ \\hat{\\epsilon}_b^t\n\\right| }{ \\left|\nm_{a \\bar{b}}^{t 2}\\ \\hat{y}_{\\bar{a}}^{t \\ast}\\ \\hat{y}_b^t -\nm_{a \\bar{b}}^{t 2}\\ \\hat{\\epsilon}_{\\bar{a}}^{t \\ast}\\ \\hat{y}_b^t\n\\right|},\n\\end{equation}\nwhere the last expression is manifestly basis invariant and, according to\nEq.~\\eqref{thetatbound}, $\\theta_t \\lesssim  {\\rm few} \\times 10^{-7}$ to\nsatisfy Eq.~(\\ref{topbd}).\n\nFinally, the misalignment angles between the top, bottom and tau eigenbases\ncan be formulated in basis-independent notation as\n\\begin{eqnarray}\n{\\rm min} {\\Big\\{} \\hat{y}^b\\cdot \\hat{y}^t,\n \\hat{y}^b\\cdot \\hat{\\epsilon}^t  {\\Big\\}}  &\\simeq & \\theta_b \\lesssim 10^{-5}\n\\to 10^{-7},\n\\nonumber\\\\\n{\\rm min} {\\Big\\{} \\hat{y}^\\tau\\cdot \\hat{y}^t,\n \\hat{y}^\\tau\\cdot \\hat{\\epsilon}^t  {\\Big\\}}  &\\simeq & \\theta_\\tau\n \\lesssim 10^{-4}\\to 10^{-6},\n\\end{eqnarray}\nwhere $\\hat{y}^b$ and  $\\hat{y}^\\tau$ are defined analogously to $\\hat{y}^t$\nin Eq.~\\eqref{hat_y} and the upper bounds on the right-hand sides follow from\nEq.~\\eqref{thetabtau}.\n\nIn summary, in a 2HDM prior to the EWPT, a relative Higgs asymmetry can\nsurvive provided the Yukawa interactions and the Higgs potential have a\ncertain form. The Higgs potential parameters should satisfy the constraints\n$|\\lambda_n| \\lesssim$ few $\\times 10^{-7}\\ (n=5,6,7)$, and $|m_{12}^2|\n\\lesssim $ (100 MeV)$^2$. In the same basis,  SM singlet  fermions of a given\ncharge (up-type quarks, down-type quarks, charged leptons) should interact\nwith approximately only one Higgs field, that is, the model should be of type\nI, II, X, or Y.\n\n\\section{Chemical equilibrium relations}\n\\label{sec:chemeq}\n\nLet us now study the redistribution of asymmetries in conserved quantum\nnumbers due to interactions in equilibrium. We neglect lepton flavour\nasymmetries, so that the (exactly and effectively) conserved quantum numbers\nare the hypercharge, $B-L$, and the relative Higgs asymmetry. Assuming that\nthe asymmetries in all species are small, they are related to the chemical\npotential $\\mu$ as\n\\begin{equation}\nn_i - n_{\\overline{i}} =\n\\frac{g_i T^2}{6} \\, \\mu_i \\times\n\\left\\{\n\\begin{array}{cl}\n2 & \\textrm{for bosons}\\\\\n1 & \\textrm{for fermions}\n\\end{array}\n\\right.,\n\\end{equation}\nwhere $g_i$ is the number of degrees of freedom of the particle.\n\nWe take the thermal bath to contain the SM fermions and gauge bosons, and two\nHiggs doublets.  We consider temperatures just prior to the EWPT, when all the\nYukawa interactions are in equilibrium, but gauge symmetries are unbroken, so\nthat the gauge bosons have zero chemical potential. Our aim is to investigate\nwhether a Higgs asymmetry, as given in Eq.~\\eqref{asym}, can be used to\ngenerate a baryon asymmetry.\n\nIf an interaction is in chemical equilibrium, then the sum of the chemical\npotentials of the participating particles should vanish. The SM Yukawa\ninteractions impose the relations\n\\begin{eqnarray}\n- \\mu_q + \\mu_{\\phi_d} + \\mu_{d_R} &=& 0,\n\\label{yuk_d}\n\\\\\n- \\mu_q - \\mu_{\\phi_u} + \\mu_{u_R} &=& 0,\n\\label{yuk_u}\n\\\\\n- \\mu_\\ell + \\mu_{\\phi_e} + \\mu_{e_R} &=& 0,\n\\label{yuk_e}\n\\end{eqnarray}\nwhere $\\phi_d$, $\\phi_u$ and $\\phi_e$ denote the scalar that couples to the\ndown-type quarks, up-type quarks and charged leptons, respectively. Since in\nthe usual notation $\\phi_u = \\phi_2$, the various models in Table~\\ref{table1}\ndiffer by whether $\\phi_d$ and\/or $\\phi_e$ coincide with $\\phi_2$.\n\nThe electroweak sphalerons  impose\n\\begin{equation}\n3 \\mu_q  + \\mu_\\ell = 0,\n\\label{Ewk_sphal}\n\\end{equation}\nwhile the QCD sphalerons lead to the chemical equilibrium condition\n\\begin{equation}\n2 \\mu_q - \\mu_{u_R}  - \\mu_{d_R} = 0.\n\\label{QCD_sphal}\n\\end{equation}\nAdding Eqs.~\\eqref{yuk_d}, \\eqref{yuk_u}, and \\eqref{QCD_sphal}, we find\n\\begin{equation}\n\\mu_{\\phi_d} - \\mu_{\\phi_u} = 0.\n\\label{crucial}\n\\end{equation}\nIn type II and type Y models, where $\\phi_2$ couples  to up-type quarks, and\n$\\phi_1$  to down-type quarks, this forces $\\mu_{\\phi_1} - \\mu_{\\phi_2}$ to\nvanish. Therefore, in 2HDM of type II and type Y, a relative Higgs asymmetry\nwould be washed out. In contrast, in type I and type X models, $\\phi_d =\n\\phi_2 = \\phi_u$, Eq.~\\eqref{crucial} is trivially satisfied, and thus\none can have $\\mu_{\\phi_1} - \\mu_{\\phi_2} \\neq 0$. Next we show that, provided\na Higgs asymmetry was created in early Universe, it can be used to generate a\nbaryon asymmetry at later times.\n\nThe baryon and lepton number comoving asymmetries are given by\n\\begin{eqnarray}\n\\mathcal{Y}_{\\Delta B} &=&N_g (\n2 \\mu_q + \\mu_{u_R} + \\mu_{d_R})\\,\\frac{T^2}{3s}= 4N_g \\mu_q \\frac{T^2}{3s},\n\\nonumber\\\\\n\\mathcal{Y}_{\\Delta L} &=& N_g (2 \\mu_\\ell + \\mu_{e_R})\\,\\frac{T^2}{3s} =\n- N_g (9 \\mu_q + \\mu_{\\phi_e})\\,\\frac{T^2}{3s},\n\\end{eqnarray}\nwhere $N_g = 3$ is the number of generations, and\nEqs.~\\eqref{yuk_e}-\\eqref{QCD_sphal} have been used to rewrite the right-hand\nsides of these expressions. As a result,\n\\begin{equation}\n\\mathcal{Y}_{\\Delta B} - \\mathcal{Y}_{\\Delta L} =\nN_g \\left( 13\\, \\mu_q + \\mu_{\\phi_e} \\right)\\,\\frac{T^2}{3s}.\n\\label{BmL}\n\\end{equation}\n\nFinally, hypercharge (or, equivalently, electric) neutrality of the plasma\ngives\n\\begin{eqnarray}\nN_g &&\\left(\n- \\mu_{e_R} - \\mu_\\ell + \\mu_q + 2 \\mu_{u_R} - \\mu_{d_R}\n\\right)\n+ 2 ( \\mu_{\\phi_1} + \\mu_{\\phi_2} )\n\\nonumber\\\\\n=&& N_g \\left(\n8 \\mu_q + \\mu_{\\phi_e} + 3 \\mu_{\\phi_2}\n\\right)\n+ 2 ( \\mu_{\\phi_1} + \\mu_{\\phi_2} )=0.\n\\label{Qem}\n\\end{eqnarray}\nThe equilibrium  baryon asymmetry can then be written as a function of $B-L$\nand the Higgs asymmetry:\n\\begin{eqnarray}\n\\mathcal{Y}_{\\Delta B}\n&=&\n\\frac{8}{23} (\\mathcal{Y}_{\\Delta B} - \\mathcal{Y}_{\\Delta L})\n+ \\frac{3}{46}\n\\left( \\mathcal{Y}_{\\Delta \\phi_1} - \\mathcal{Y}_{\\Delta \\phi_2} \\right)\n\\hspace{5mm}\n\\textrm{for type I},\n\\label{BI_TypeI}\n\\\\\n\\mathcal{Y}_{\\Delta B}\n&=&\n\\frac{8}{23} (\\mathcal{Y}_{\\Delta B} - \\mathcal{Y}_{\\Delta L})\n- \\frac{33}{92}\n\\left( \\mathcal{Y}_{\\Delta \\phi_1} - \\mathcal{Y}_{\\Delta \\phi_2} \\right)\n\\hspace{5mm}\n\\textrm{for type X}.\n\\label{BI_TypeX}\n\\end{eqnarray}\n\nEqs.~\\eqref{BI_TypeI} and \\eqref{BI_TypeX} give a baryon asymmetry in the\npresence of a relative Higgs asymmetry, even if $\\mathcal{Y}_{\\Delta B}\n-\\mathcal{Y}_{\\Delta L}=0$. Thus, in these cases, the baryon asymmetry is due\nexclusively to the initial imbalance between the asymmetry in $\\phi_1$ and\nthe asymmetry in $\\phi_2$. We dub this scenario as split Higgsogenesis. As\nfar as we know, this is a novel mechanism for baryogenesis. This is\nreminiscent of certain asymmetric dark matter (DM) models~\\cite{ADMrev},\nwhere an asymmetry is generated in a new dark sector (which contains the DM\ncandidate), and then is shared with the SM fermions. The role that the Higgs\ncan play in transferring asymmetries between the SM fermions and the dark\nsector has been recently emphasized in Ref.~\\cite{ST}. However, $\\phi_1$ does\nnot seem to be a successful asymmetric DM candidate in the simple model\ndiscussed here.\\footnote{Symmetric dark matter and electroweak baryogenesis\nhave been recently discussed in the inert doublet model in\nRefs.~\\cite{BAUIDM}.}\n\nLet us assume that, indeed, $\\mathcal{Y}_{\\Delta B} =\\mathcal{Y}_{\\Delta L}$. Using\nEqs.~\\eqref{BmL}-\\eqref{BI_TypeX}, we find\n\\begin{eqnarray}\n\\mathcal{Y}_{\\Delta B}\n&=&\n\\frac{3}{46}\n\\left( \\mathcal{Y}_{\\Delta \\phi_1} - \\mathcal{Y}_{\\Delta \\phi_2} \\right)\n=\n-\\frac{6}{13}\\, \\mathcal{Y}_{\\Delta \\phi_2} = \\frac{6}{79}\\,  \\mathcal{Y}_{\\Delta \\phi_1}\n\\hspace{6mm}\n\\textrm{for type I},\n\\label{B_TypeI}\n\\\\*[2mm]\n\\mathcal{Y}_{\\Delta B}\n&=&\n- \\frac{33}{92}\n\\left( \\mathcal{Y}_{\\Delta \\phi_1} - \\mathcal{Y}_{\\Delta \\phi_2} \\right)\n=\n-\\frac{6}{13}\\,  \\mathcal{Y}_{\\Delta \\phi_1} = \\frac{66}{41}\\,  \\mathcal{Y}_{\\Delta \\phi_2}\n\\hspace{5mm}\n\\textrm{for type X}.\n\\label{B_TypeX}\n\\end{eqnarray}\nThe type I model is particularly interesting because $\\phi_1$ does not couple\nto any fermion and could act as dark matter. In that case,\n$\\mathcal{Y}_{\\Delta \\phi_1}\/\\mathcal{Y}_{\\Delta B} = 79\/6$, so the DM scalar\nshould be lighter than the proton\\footnote{ Alternatively, one could assume a\nprimordial $B-L$ asymmetry, no relative Higgs asymmetry, and let the\n$\\lambda_{5}$ coupling (allowed by the $Z_2$ symmetry which ensures DM\nstability) to equilibrate the asymmetry between the two Higgs fields. Then,\nEq.~\\eqref{Qem} yields a Higgs asymmetry smaller than the baryon asymmetry: $\n\\mathcal{Y}_{\\Delta \\phi_1} = \\mathcal{Y}_{\\Delta \\phi_2} = -\n\\mathcal{Y}_{\\Delta B}\/8$. This corresponds to a scalar DM mass $\\sim 20$\nGeV, which is ruled out by the width of the $Z$ boson. Furthermore,\n$\\lambda_5$ would mediate DM-anti-DM oscillations which would wash out the\nasymmetry.} to obtain $\\Omega_{\\textrm{DM}} \\sim 5\\, \\Omega_{B}$. We recall\nthat the mass density of baryons in the Universe today, as inferred from WMAP\nin the context of $\\Lambda$CDM cosmology, is~\\cite{Komatsu:2010fb}\n\\begin{equation}\\label{nBobs}\n \\frac{m_p}{\\rho_c}( n_B - n_{\\overline B})=\n \\Omega_{B} h^2 = 0.02255 \\pm 0.0054,\n\\end{equation}\nor equivalently,\n\\begin{equation}\\label{YBobs}\n  \\mathcal{Y}_{\\Delta B} = (8.79\\pm0.44)\\times 10^{-11},\n\\end{equation}\nwhere $m_p$ is the proton mass, $h \\equiv H_0\/(100\\, $km s$^{-1}\\,$Mpc$^{-1})\n= 0.742 \\pm 0.036$ is the present Hubble parameter, and $\\rho_c = 3H_0^2\/(8\n\\pi G)$ is the critical density of a spatially flat Universe. On the other\nhand, for the $\\Lambda$CDM cosmology with three light neutrinos, the cold dark\nmatter relic abundance is $\\Omega_{\\textrm{DM}} h^2 = 0.1126\\pm\n0.0036$~\\cite{Komatsu:2010fb}, so that the ratio of dark matter particles to\nbaryons is $\\mathcal{Y}_{\\textrm{DM}}\/\\mathcal{Y}_{\\Delta B} \\sim 5\\,\nm_p\/m_{\\textrm{DM}}$.\n\n\\section{Simple split-Higgsogenesis scenarios}\n\\label{sec:scenarios}\n\nOur goal in this section is to provide a few simple scenarios for\nbaryogenesis through split-Higgsogenesis, where the cosmological baryon\nasymmetry could in principle be generated via the out-of-equilibrium decay of\nheavy singlet scalars into Higgs doublets.\n\n\\subsection{One extra singlet scalar}\n\nWe consider an inert 2HDM extended by one real scalar singlet. The two Higgs\ndoublets, $\\phi_1$ and $\\phi_2$, and the singlet scalar $S$ transform under\n$Z_2$ as\n\\begin{equation}\n\\phi_1 \\rightarrow - \\phi_1,\\quad\n\\phi_2 \\rightarrow \\phi_2,\\quad\nS \\rightarrow -S.\n\\label{Z2_2d1s}\n\\end{equation}\nThe $Z_2$-invariant Higgs potential can be written as\n\\begin{equation}\nV = V_\\phi + V_S + V_{S \\phi},\n\\label{VPPS}\n\\end{equation}\nwhere\n\\begin{eqnarray}\\label{V_vp_Z2}\nV_\\phi\n&=&\nm_{11}^2\\, |\\phi_1|^2 + m_{22}^2\\, |\\phi_2|^2\n+\\, \\tfrac{1}{2}\\, \\lambda_1\\, |\\phi_1|^4 + \\tfrac{1}{2}\\, \\lambda_2\\,\n|\\phi_2|^4 + \\lambda_3\\, |\\phi_1|^2 |\\phi_2|^2 + \\lambda_4\\, (\\phi_1^\\dagger\n\\phi_2)(\\phi_2^\\dagger \\phi_1)\n\\nonumber\\\\\n&&\n+\\, \\tfrac{1}{2}\\, \\lambda_5 \\left[ (\\phi_1^\\dagger \\phi_2)^2 +\n\\textrm{h.c.} \\right],\\nonumber\\\\\nV_S &=& M^2\\, S^2 + \\lambda_S\\, S^4,\\\\\nV_{S \\phi}\n&=&\nz_1\\,M (\\phi_1^\\dagger \\phi_2)\\, S + z_1^\\ast\\,M (\\phi_2^\\dagger \\phi_1)\\, S\n+\n\\beta_1\\, |\\phi_1|^2\\, S^2 + \\beta_2\\, |\\phi_2|^2\\, S^2.\\nonumber\n\\end{eqnarray}\nAll parameters are real but $z_1$, so that CP violation in the scalar sector\nis related with a complex $z_1$.\n\nWe address now the question whether this model can be used to generate a CP\nasymmetry in the Higgs sector that could be converted into a baryon\nasymmetry. The basic idea is analogous to that of the standard leptogenesis\nscenario. A population of $S$'s is produced through scattering processes at\ntemperatures $T \\sim M \\gg m_{\\phi_1,\\phi_2}$. This population decays away at\n$T < M$, when the singlet scalar equilibrium density is Boltzmann suppressed.\nIf the interactions of the heavy singlet $S$ are CP-violating, and provided\nthat the relevant interactions are out of equilibrium, a net Higgs asymmetry\ncan be generated. The latter is then converted into a baryon asymmetry by the\nsphalerons. To illustrate our mechanism, let us consider the tree-level and\none-loop diagrams\\footnote{There is also a one-loop vertex correction to the\ntree-level diagram, but since it carries the same phase $z_1$, it does not\ncontribute to the Higgs CP asymmetry.} depicted in Fig.~\\ref{fig1}.\n\n\\begin{figure}[t]\n\\includegraphics*[height=3.5cm]{fig1.eps}\n\\caption{\\label{fig1}Diagrams contributing to the Higgs CP asymmetry. The\nnotation $1$ and $\\bar{2}$ refers to $\\phi_1$ and $\\bar{\\phi}_2$,\nrespectively.}\n\\end{figure}\nWe have two contributions with different CP-odd phases; $z_1$ and $z_1^\\ast$.\nBecause the second diagram is a loop diagram, a cut on this diagram leads to\nan absorptive part that contains the CP-even phase needed for CP violation in\ndecays. As a result, the interference of the tree-level and one-loop\namplitudes leads to a nonvanishing CP asymmetry in the final Higgs states.\nDefining this asymmetry as\n\\begin{equation} \\label{CPasym}\n\\epsilon = \\frac{\\Gamma(S\\rightarrow \\phi_1\\,\\bar{\\phi}_2)-\\Gamma(S\\rightarrow\n\\bar{\\phi}_1\\,\\phi_2)}{ \\Gamma(S\\rightarrow\n\\phi_1\\,\\bar{\\phi}_2)+\\Gamma(S\\rightarrow \\bar{\\phi}_1\\,\\phi_2)},\n\\end{equation}\nwe find\n\\begin{equation}\n\\epsilon \\simeq\\frac{\\overline{\\left|c_0 \\mathcal{A}_0+c_1\n\\mathcal{A}_1\\right|^2} -\\overline{\\left|c^\\ast_0 \\mathcal{A}_0+c^\\ast_1\n\\mathcal{A}_1\\right|^2}}{ \\overline{\\left|c_0 \\mathcal{A}_0+c_1\n\\mathcal{A}_1\\right|^2}+ \\overline{\\left|c^\\ast_0 \\mathcal{A}_0+c^\\ast_1\n\\mathcal{A}_1\\right|^2}} \\simeq -4\\frac{\\text{Im}\\left[c_0^\\ast c_1\\right]\n\\text{Im}\\left[\\overline{\\mathcal{A}_0^\\ast \\mathcal{A}_1}\\right]}{\n2|c_0|^2\\overline{|\\mathcal{A}_0|^2}},\n\\end{equation}\nwhere $\\mathcal{A}_0$ and $\\mathcal{A}_1$ are the tree-level and one-loop\namplitudes, respectively. For the decay of Fig.~\\ref{fig1}, one has $c_0 = -\nz_1$, $c_1= 3\\,\\lambda_5\\, z_1^\\ast$ and $\\mathcal{A}_0=1$. Thus, the weak\nphase gives\n\\begin{equation}\n\\text{Im}\\left[c_0^\\ast\\, c_1\\right]=- 3\\, \\text{Im}\\left[z_1^{\\ast}\\,\n\\lambda_5\\, z_1^{\\ast} \\right],\n\\end{equation}\nwhile the strong phase comes from\n\\begin{equation}\n\\text{Im}\\left[\\overline{\\mathcal{A}_0^\\ast \\mathcal{A}_1}\\right]\n=-\\frac{1}{16\\pi}.\n\\end{equation}\nWe then get\n\\begin{equation} \\label{eps1}\n\\epsilon \\simeq - \\frac{3}{8\\pi} \\frac{\\text{Im}\\left[z_1^{\\ast}\\, \\lambda_5\\, z_1^{\\ast} \\right]}{2 |z_1|^2} =\n\\frac{3}{16\\pi} \\lambda_5 \\sin[2\\,\\text{arg}(z_1)].\n\\end{equation}\nThus, in this simple scenario, the Higgs asymmetry is controlled by the\nstrength of the quartic parameter $\\lambda_5$ and the phase of the coupling\n$z_1$.\n\nThe final baryon asymmetry (baryon-to-entropy ratio) can be approximated as\n\\begin{equation}\\label{YB}\n  \\mathcal{Y}_{\\Delta B} \\simeq \\mathcal{Y}^{\\rm eq}_S\\times C\\times\\epsilon\\,\n  \\eta,\n\\end{equation}\nwhere the first factor is the equilibrium $S$ number density divided by the\nentropy density, the second factor is the fraction of the Higgs asymmetry\nconverted into a baryon asymmetry by the sphalerons ($C=3\/46$ in the present\ncase), and the efficiency factor $\\eta\\, (0 \\leq \\eta \\leq 1)$ measures how\nefficient the out-of-equilibrium $S$-decays are in producing the asymmetry.\n\nAlthough a precise computation of $\\eta$ requires the solution of a full set\nof Boltzmann equations, simple analytical estimates can be given. It is useful\nto introduce the decay parameter\n\\begin{equation}\\label{Kappa}\n  K=\\frac{\\Gamma_D(T=0)}{H(T=M)},\n\\end{equation}\nwhere $\\Gamma_D=|z_1|^2 M\/(8\\pi)$ is the tree-level decay rate of the singlet\n$S$ into the two Higgs doublets. In the so-called weak washout regime ($K \\ll\n1$), i.e. when the scalar singlet decays strongly out of equilibrium, the\nefficiency factor is $\\eta\\simeq 1$. In the strong washout regime ($K\\gg 1$),\nthe efficiency does not depend on the initial conditions, and is mildly\nsuppressed as $\\eta \\simeq 1\/K$. For intermediate values of $K$ ($K \\lesssim\n1$ or $K \\gtrsim 1$) the efficiency depends on the assumed initial conditions.\nWe can roughly approximate it as $\\eta \\sim \\min(1, 1\/K)$, if $S$ has thermal\ninitial abundance, or $\\eta \\sim \\min(K, 1\/K)$, if $S$ has zero initial\nabundance.\n\nAn estimate for $\\mathcal{Y}_{\\Delta B}$ can be obtained from\nEq.~\\eqref{YB} in the form:\n\\begin{equation}\\label{YBapp}\n  \\mathcal{Y}_{\\Delta B} \\simeq \\frac{135 \\zeta(3)}{92\\pi^4 g_\\ast}\n  \\epsilon\\, \\eta \\simeq 2\\times10^{-4}  \\epsilon\\, \\eta .\n\\end{equation}\nThis is to be compared with the WMAP inferred value given in\nEq.~\\eqref{YBobs}. Since Eq.~\\eqref{eps1} leads to the upper bound $|\\epsilon|\n\\lesssim 6\\times 10^{-2} \\lambda_5$, then Eq.~\\eqref{YBapp}, when combined with\nEq.~\\eqref{YBobs}, requires\n\\begin{equation}\n\\lambda_5 \\, \\eta \\gtrsim 7\\times 10^{-6}.\n\\end{equation}\n\nNotice that between the mass scale $M$ and the EW scale an effective\nquartic coupling $\\lambda_5^\\text{eff}=\\lambda_5 +\\tfrac{1}{2} z_1^2$ is\ngenerated by $S$-exchange. Recalling that, for the relative asymmetry between\n$\\phi_1$ and $\\phi_2$ to survive, interactions which exchange $\\phi_1\n\\leftrightarrow \\phi_2$ must be out of equilibrium until the electroweak\nscale, then Eq.~\\eqref{bdL} imposes $|\\lambda_5^\\text{eff}| \\lesssim {\\rm\nfew}\\times 10^{-7}$. Thus, in this simple setup, unless there is a fine-tuned\ncancellation between $\\lambda_5$ and $z_1^2$ to satisfy this bound, we cannot\naccommodate the observed baryon asymmetry~\\eqref{YBobs}, even with a maximal\nefficiency $\\eta\\simeq 1$.\n\n\\subsection{One extra singlet  and a third doublet}\n\nWe consider now a model with three doublet scalars and one real scalar singlet.\nThe Higgs doublets, $\\phi_1$, $\\phi_2$ and $\\phi_3$, and the singlet scalar\n$S$ transform under $Z_2$ as\n\\begin{equation}\n\\phi_1 \\rightarrow - \\phi_1,\\quad\n\\phi_3 \\rightarrow - \\phi_3,\\quad\n\\phi_2 \\rightarrow \\phi_2,\\quad\nS \\rightarrow -S.\n\\label{Z2_3d1s}\n\\end{equation}\nThe singlet and the third doublet $\\phi_3$  will be significantly heavier\nthan the EW scale; $\\phi_1$ and $\\phi_3$ are defined as the $Z_2$-odd mass\neigenstates. The $Z_2$-invariant Higgs potential can be written as\n\\begin{equation}\nV_3 = V + \\Delta V_\\phi + \\Delta V_{S \\phi},\n\\label{V3}\n\\end{equation}\nwhere $V$ is given in Eqs.~\\eqref{VPPS}-\\eqref{V_vp_Z2} and\n\\begin{eqnarray}\\label{V3_vp_Z2}\n\\Delta V_\\phi\n&=&\nm_{33}^2\\, |\\phi_3|^2\n+ \\, \\tfrac{1}{2}\\, \\lambda_{3333}\\, |\\phi_3|^4\n+ \\lambda_{1133}\\, |\\phi_1|^2 |\\phi_3|^2\n+ \\lambda_{2233}\\, |\\phi_2|^2 |\\phi_3|^2\n\\nonumber\\\\\n&&\n + \\lambda_{1331}\\, (\\phi_1^\\dagger \\phi_3)(\\phi_3^\\dagger \\phi_1)\n+ \\lambda_{2332}\\, (\\phi_2^\\dagger \\phi_3)(\\phi_3^\\dagger \\phi_2)\n\\nonumber\\\\\n&&\n+  \\left[\\tfrac{1}{2}\\lambda_{1313}(\\phi_1^\\dagger \\phi_3)^2\n+  \\tfrac{1}{2}\\lambda_{3232}(\\phi_3^\\dagger \\phi_2)^2\n+  \\lambda_{1223}(\\phi_1^\\dagger \\phi_2) (\\phi_2^\\dagger \\phi_3)\n+  \\lambda_{1232}(\\phi_1^\\dagger \\phi_2) (\\phi_3^\\dagger \\phi_2) \\right.\\nonumber\\\\\n&&\n+  \\left. \\lambda_{1311}(\\phi_1^\\dagger \\phi_3) (\\phi_1^\\dagger \\phi_1)\n+  \\lambda_{1322}(\\phi_1^\\dagger \\phi_3) (\\phi_2^\\dagger \\phi_2)\n+  \\lambda_{1333}(\\phi_1^\\dagger \\phi_3) (\\phi_3^\\dagger \\phi_3) + \\textrm{H.c.} \\right]\n ,\\nonumber\\\\\n\\Delta V_{S \\phi}\n&=&\nz_3\\,M (\\phi_3^\\dagger \\phi_2)\\, S + z_3^\\ast\\,M (\\phi_2^\\dagger \\phi_3)\\, S\n+\n\\beta_3\\, |\\phi_3|^2\\, S^2 + \\left[\\beta_{13}\\, \\phi_1^\\dagger \\phi_3 \\, S^2 +\n \\textrm{H.c.} \\right].\n\\end{eqnarray}\n\nThe basic idea is similar to the 2HDM with an extra singlet. A population of\n$S$'s is produced through scattering processes at temperatures $T \\sim M >\nm_{33} \\gg m_{\\phi_1,\\phi_2}$. This population decays away at $T < M$, when\nthe singlet scalar equilibrium density is Boltzmann suppressed. If the\ninteractions of the heavy singlet $S$ are CP-violating, asymmetries among the\nthree Higgs doublets can be generated. When washout interactions are out of\nequilibrium, the asymmetries can survive. The $\\phi_3$ later decay to\n$\\phi_1$, leaving an asymmetry between $\\phi_2$ and $\\phi_1$. Assuming the\nasymmetry from $S\\to \\phi_1 \\phi_2^*$ to be negligible, due to the bounds on\n$\\lambda_5$ and $z_1^2$,  we neglect it and focus on a possible asymmetry\nfrom $S\\to \\phi_3 \\phi_2^*$.\n\nWe consider tree-level and one-loop diagrams analogous to those depicted in\nFig.~\\ref{fig1}, with $\\phi_1 \\rightarrow \\phi_3$, $z_1 \\rightarrow z_3$, and\n$\\lambda_5 \\rightarrow \\lambda_{3232}$. The interference of the\ntree-level and one-loop amplitudes leads to a nonvanishing CP asymmetry in\nthe final Higgs states:\n\\begin{equation} \\label{CPasym3}\n\\epsilon = \\frac{\\Gamma(S\\rightarrow \\phi_3\\,\\bar{\\phi}_2)-\\Gamma(S\\rightarrow\n\\bar{\\phi}_3\\,\\phi_2)}{ \\Gamma(S\\rightarrow {\\rm all})}\n\\simeq - \\frac{3}{8\\pi} \\frac{\\text{Im}\\left[z_3^{\\ast}\\,\n\\lambda_{3232}\\, z_3^{\\ast} \\right]}{2 |z_3|^2} = \\frac{3}{16\\pi} \\lambda_{3232}\n\\sin[2\\,\\text{arg}(z_3)],\n\\end{equation}\nwhere the contribution of $z_1$ to the total decay rate has been neglected.\n\nFor the relative asymmetry between $\\phi_3 + \\phi_1$ and $\\phi_2$ to survive,\ninteractions which exchange $\\phi_3 $ with $\\phi_2$ must be out of\nequilibrium until the $\\phi_3$ decay. (Recall that we have already imposed\nthat interactions exchanging $\\phi_1 \\leftrightarrow \\phi_2$ are out of\nthermal equilibrium). By extrapolating Eq.~\\eqref{bdL} to higher\ntemperatures, this imposes\n\\begin{equation}\n|z_3|^2 , |\\lambda_{3232}| <  {\\rm few}\\times\n10^{-7} \\sqrt{\\frac {m_{33}}{\\text{TeV}}}\\,,\n\\end{equation}\nassuming that $S$-exchange generates an effective\n$\\lambda_{3232}^\\text{eff}=\\lambda_{3232}+\\tfrac{1}{2} z_3^2$ between the\ntemperature scales $M$ and $m_{33}$, and that $\\phi_3$ decay at $T \\simeq\n10\\, m_{33}$. From Eqs.~\\eqref{YBapp} and \\eqref{YBobs}, a large enough\nasymmetry is obtained for $m_{33} \\gtrsim 10^8$~GeV, provided the washout\neffects are not too strong, $\\eta \\simeq \\mathcal{O}(0.1)$. Since the\nefficiency of Higgsogenesis is controlled by the decay parameter $K$ defined\nin Eq.~\\eqref{Kappa},\n\\begin{equation}\nK= \\frac{|z_3|^2}{136\\pi}\\frac{m_P}{M} \\simeq 3 \\times\n\\left(\\frac{|z_3|^2}{10^{-4}}\\right)\\times\\left(\\frac{10^{12}\\,\\text{GeV}}{M}\\right),\n\\end{equation}\nthis implies $M \\gtrsim 10^{12}$~GeV.\n\n\\subsection{Two extra singlet scalars}\n\nLet us now consider a model with two real scalar singlets $S_i\\ (i=1,2)$, both\ntransforming under $Z_2$ as $S_i \\rightarrow - S_i$. The Higgs potential can\nbe written as in Eqs.~\\eqref{VPPS}-\\eqref{V_vp_Z2}, but in this case $V_S$\nand $V_{S \\phi}$ contain additional terms. In particular, $V_{S \\phi}$\ncontains the cubic terms $z_i\\,M_i\\, (\\phi_1^\\dagger \\phi_2)\\, S_i+\nz_i^\\ast\\,M_i\\, (\\phi_2^\\dagger \\phi_1)\\, S_i\\,$, in addition to six quartic\nterms. Similarly, $V_S$ has several quartic terms and, without loss of\ngenerality, we choose a $\\left\\{ S_1, S_2\\right\\}$ basis where the quadratic\nterms are already diagonalized, and assume $M_1 < M_2$.\n\nIn what follows, we make the following simplifying assumptions: The heavy\nsinglet spectrum is hierarchical, $M_1 \\ll M_2$; there is a thermal production\nof $S_1$ and negligible production of $S_2$. With these assumptions, the\nHiggsogenesis mechanism will proceed via the out-of-equilibrium decays of\n$S_1$. The decay $S_1 \\rightarrow \\phi_1 \\bar{\\phi}_2$ is still mediated by\nthe diagrams in Fig~\\ref{fig1}, but we also have the additional (vertex and\nself-energy) diagrams depicted in Fig.~\\ref{fig2}.\n\\begin{figure}[t]\n\\includegraphics*[height=3.5cm]{fig2.eps}\n\\caption{\\label{fig2}New diagrams contributing to the Higgs CP asymmetry in\nthe presence of two singlet scalars. The notation $1$ and $\\bar{2}$ refers to\n$\\phi_1$ and $\\bar{\\phi}_2$, respectively.}\n\\end{figure}\n\nThese diagrams carry the phase of $z_1^\\ast z_2^2$, which will be beaten\nagainst the $z_1$ phase from the tree level diagram in Fig.~\\ref{fig1}. As a\nresult, in this model, even if $\\lambda_5$ vanishes, there is a CP-violating\ncontribution proportional to $\\textrm{Im}(z_1^{\\ast 2} z_2^2)$. The resulting\nCP asymmetry, as given in Eq.~\\eqref{CPasym}, can be evaluated following\nthe standard procedure. Neglecting the $\\lambda_5$ contribution coming from\nthe one-loop diagram of Fig.~\\ref{fig1}, we obtain\n\\begin{equation}\n\\epsilon_1 \\simeq -\\frac{1}{4\\pi}\n\\frac{\\text{Im}\\left[(z^\\ast_1z_2)^2\\right]}{|z_1|^2}\\left[\\frac{1}{2}f(x_2)+g(x_2)\\right],\n\\end{equation}\nwhere $x_2 = M_2^2\/M_1^2$, and\n\\begin{equation}\nf(x)=x\\ln \\Big(\\frac{x}{1+x}\\Big),\\quad g(x)=\\frac{x}{1-x},\n\\end{equation}\nare the vertex and self-energy one-loop functions, respectively.\n\nIn the hierarchical limit, $x_2\\gg 1$, $f(x_2)\\simeq -1$, $g(x_2) \\simeq -1$,\nand the CP asymmetry is approximately given by\n\\begin{equation} \\label{eps2}\n\\epsilon_1 \\simeq \\frac{3}{8\\pi}\n\\frac{\\text{Im}\\left[(z^\\ast_1z_2)^2\\right]}{|z_1|^2}=\\frac{3}{8\\pi} |z_2|^2\n\\sin[2\\arg (z_1^\\ast z_2)].\n\\end{equation}\n\nFrom Eqs.~\\eqref{YBapp}, \\eqref{YBobs} and \\eqref{eps2} we then conclude that\na successful generation of the baryon asymmetry within the present model\nrequires\n\\begin{equation}\\label{z2constraint}\n  |z_2|^2\\, \\eta \\gtrsim {\\rm few} \\times 10^{-6},\n\\end{equation}\nwhich in turn implies that $|z_2|^2 \\gtrsim {\\rm few} \\times 10^{-6}$. Yet,\nas in the case with one extra singlet, an effective quartic coupling\n$\\lambda_5^{\\rm eff}=\\lambda_5 + \\tfrac{1}{2} z_1^2 +\\tfrac{1}{2} z_2^2$ is\ngenerated by $S$-exchange between the $M$ and EW temperature scales. The\nlatter should satisfy the bound $|\\lambda_5^\\text{eff}| \\lesssim {\\rm\nfew}\\times 10^{-7}$. So, in this case, to accommodate the observed baryon\nasymmetry~\\eqref{YBobs} we should have some relation between $\\lambda_5$,\n$z_1^2$ and\/or $z_2^2$; for example, $z_1 \\simeq i z_2$ to avoid the\nrestrictive bounds on these couplings.\n\n\\section{Summary}\n\\label{sec:summary}\n\nIn this work, we have studied the possibility of generating the cosmological\nbaryon asymmetry in the context of 2HDM extensions of the SM, prior to the\nelectroweak phase transition. We have shown that if the Higgs-flavour\nexchanging interactions are sufficiently slow in the early Universe, then a\nrelative asymmetry among the Higgs doublets corresponds to an effectively\nconserved quantum number. Such a relative Higgs asymmetry can be transformed\ninto a baryon asymmetry by the sphalerons, without the need for $B-L$\nviolation.\n\nAmong the four possible types of $Z_2$ models considered, we have\ndemonstrated that this  ``split Higgsogenesis'' mechanism is only possible in\nthe framework of a type-I or type-X 2HDM. We then  presented simple scenarios\nto generate a Higgs asymmetry,  based on inert type-I 2HDMs extended by heavy\nsinglet scalar fields and\/or one extra Higgs doublet. In the presence of\nCP-violating interactions, the out-of-equilibrium decays of the heavy\nsinglets into the Higgs doublets can produce a net Higgs asymmetry and the\nmechanism of baryogenesis through (split) Higgsogenesis can be viable. Since\na successful implementation of our mechanism requires the scalar potential\nparameters to satisfy definite bounds, we have also paid particular attention\nto their basis-independent formulation.\n\n\\acknowledgments\n We are grateful to A. Barroso, M.~Krawczyk and R. Santos for\nuseful discussions. S.D. acknowledges partial support from the EU FP7 ITN\nINVISIBLES (MC actions, PITN-GA-2011-289-442) and the Lyon Institute of\nOrigins. The work of R.G.F. and J.P.S. was partially supported by FCT -\n\\textit{Funda\\c{c}\\~{a}o para a Ci\\^{e}ncia e a Tecnologia}, under the Projects\nPEst-OE\/FIS\/UI0777\/2011 and CERN\/FP\/123580\/2011, and by the EU RTN Marie\nCurie Project PITN-GA-2009-237920. The work of H.S. is funded by the European\nFEDER and Spanish MINECO, under the Grant FPA2011-23596.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{S:1}\n\nThis paper is concerned with scattering of time-harmonic electromagnetic plane waves by periodic structures, which is known as gratings in optics.\nThe goal is to investigate stability properties of the Helmholtz equation in both deterministic and random periodic structures.\n\nFor the scattering by a periodic structure, considerable progress has been made mathematically in literatures.\nBao, Dobson and Cox \\cite{Bao1995} reduced the scattering problem into a bounded domain problem by introducing a transparent boundary condition and proved that there exists a unique solution at all but a sequence of countable frequencies.\nLord and Mulholland \\cite{Lord2013} used the variational formula  to derive a priori estimate  for periodic structures, which can be viewed as an extension of  \\cite{Bao1996}.\nMore results on the Helmholtz and Maxwell equations in periodic structures can be found in \\cite{Petit1980, Bao2001, Bao2021book}.\n\nRecently, there is an increasing interest in the study of wavenumber-explicit bounds for scattering problems.\nChandler-Wilde and Monk \\cite{Chandler2005} obtained a priori bounds explicit with the wavenumber for the scattering problem by unbounded rough surfaces.\nHetmaniuk \\cite{Hetmaniuk2007} established stability estimates for the Helmholtz equation with mixed boundary conditions. Esterhazy and Melenk \\cite{Esterhazy2012} further established an estimate for bounded Lipschitz domains with a Robin boundary condition.\nThe stability for the scattering problem by a large rectangular cavity was obtained by Bao, Yun and Zhou \\cite{Bao2012} in transverse electric polarization and was improved and extended to transverse magnetic polarization in  \\cite{Bao2016}.\nDu, Li and Sun \\cite{LiBuyang2015} presented a numerical study of the stability estimate of the scattering by a rectangular cavity.\nWavenumber-explicit stability on the scattering problem by an obstacle in homogeneous media case \\cite{Spence2014} and heterogeneous media case \\cite{Pembery2019} were obtained\nunder the assumption that the scatterer $D$ is a star-shaped domain satisfying nontrapping conditions.\nHowever, for periodic structures, little is known about stability analysis with explicit dependence on the wavenumber, which will be derived in this paper.\nAs noted in \\cite{Bao2012, Bao2016, Chandler2005}, both the geometry and the type of boundary condition strongly affect the wavenumber-dependent stability.\nThe transparent boundary conditions (TBC) of periodic structures  are different from those of open cavities and unbounded rough surfaces, which leads to additional difficulties on resonances.\nTo overcome the difficulties, a uniform distance $\\varepsilon$ is introduced in this paper and a $k$-explicit stability for periodic structures is established by a combination of the variational method and Fourier analysis.\n\n\nWe also analyze the stability for scattering by random periodic structures in this paper.\nProgress has been made recently in the exterior scattering problem by an obstacle with randomness\\cite{Spence2014, Hiptmair2018} and scattering problems in random media \\cite{Pembery2019, Pembery2020}.\nHowever, no stability result is available for scattering by random periodic structures, which has many applications in diffractive optics \\cite{Rico-Garcia2009}.\nNumerical methods for the scattering by random periodic structures have been developed recently in \\cite{Feng2018, BaoLinSINUM2020}.\nIn this work, we focus on the stability for gratings with uncertainty.\nOne difficulty is the lack of compactness. In fact, since both the deterministic and the stochastic Helmholtz equations are not coercive and the necessary compactness results are not valid any more in Bochner spaces, Fredholm theory cannot be used to the stochastic Helmholtz equation to compensate for the lack of coercivity as for the deterministic Helmholtz equation. Consequently, it is difficult to obtain the well-posedness for the random case directly as that for the deterministics case.\nIn this work, we employ Pettis measurability theorem and Bochner's theorem as in \\cite{Pembery2020} to obtain the stability estimate explicit on the wavenumber for the random case based on the stability result for the deterministic case.\nHowever, the randomness of the integral domain for the scattering by random periodic structures prevents a direct application of the general framework in  \\cite{Pembery2020}, which leads to the other difficulty.\nTo overcome this difficulty, a variable transform is introduced here so that the transformed random variational form is defined on a deterministic domain with random coefficients. Similar transformation idea has also been used for other types of random differential equations \\cite{Xiu2006}.\nTherefore, for the stochastic case, by integrating the deterministic result over the probability space and introducing a transform to change the stochastic integral area into a definite one,  regularity and stochastic regularity of the scattering surface, Pettis measurability theorem and Bochner's theorem yield the well-posedness and a stability estimate of our model problem.\n\nThe rest of the paper is as follows. In Section \\ref{sec:results}, the model problem is introduced, the variational formulation is described, and the results (i.e., Theorems \\ref{thm:1}-\\ref{thm:3}) are presented.\nTheorem \\ref{thm:1} concerns the stability for the Helmholtz equation in a deterministic periodic structure.\nTheorem \\ref{thm:3} gives the stability estimate for large wavenumber for the Helmholtz equation in a random periodic structure.\nThe following sections are devoted to the proofs of the two results.\nSection \\ref{sec:thm1} and Section \\ref{sec:thm23} give the detailed proofs of Theorem \\ref{thm:1} and Theorem \\ref{thm:3} respectively, followed by\nconclusion given in Section \\ref{sec:conslusion}.\n\n\n\n\n\\section{Main results}\n\\label{sec:results}\n\n\n\\subsection{Model problem}\nConsider a plane wave incident on a random periodic structure\n$$S := \\{x\\in \\mathbb{R}^2: x_2=f(\\omega;x_1),\\omega \\in \\Omega,  x_1 \\in [0,\\Lambda]\\},$$\nwhich is characterized by the wavenumber $k$ ruled on a perfect conductor.\nHere, $\\omega \\in \\Omega$ denotes the random sample in a complete probability space $(\\Omega, \\mathcal{F}, \\mu)$,\n$x=(x_1,x_2)\\in \\mathbb{R}^2$ are the spatial variables, and the random surface $f: \\Omega \\times \\mathcal{X}   \\rightarrow \\mathbb{R}$ is a stationary Gaussian process, each of which is a Lipschitz function, i.e. $f\\in L^2(\\Omega;Lip)$, where $Lip$ is the space of all Lipschitz functions.\nThe medium and material are assumed to be invariant in the $x_3$ direction and  $\\Lambda$-periodic in the $x_1$ direction.\nThere are two fundamental polarizations for the electromagnetic fields: transverse-electric (TE) and transverse-magnetic (TM) polarization. Here, we consider the random periodic perfectly conducting grating problem for TE polarization.\n\n\\begin{figure}[h]\n\t\\centering\n\n\t\\includegraphics[width=0.9\\textwidth]{geometry.eps}\\\\\n\t\\caption{Problem geometry}\\label{fig:geometry}\n\\end{figure}\n\n\nAs shown in Figure \\ref{fig:geometry}, the grating is illuminated from above by a time-harmonic plane wave\n$\tu^{i}=e^{i\\alpha x_1-i\\beta x_2},$\nwhere\n$ \\alpha = k \\sin\\theta,~\\beta=k \\cos\\theta$,\n$\\theta\\in(-\\pi\/2,\\pi\/2)$ is the incident angle with respect to the positive $x_2$-axis.\nDenote $$D^+= \\{x\\in\\mathbb{R}^2: x_2>f(\\omega; x_1),\\ \\omega \\in \\Omega,  x_1 \\in [0,\\Lambda]\\}.$$ Since the medium below $S$ is perfectly electric conducting, the scattering problem for TE polarization can be modeled by the following two-dimensional Helmholtz equation with the homogeneous boundary condition:\n\\begin{eqnarray}\\label{eq:u0}\n\t\\left\\{\\begin{array}{ll}\n\t\t\\Delta u(\\omega ; \\cdot)+k^{2} u(\\omega ; \\cdot)=0 & \\text { in } \\Omega \\times D^+, \\\\\n\t\tu(\\omega;\\cdot)=0 & \\text { on } \\Omega \\times S ,\n\t\\end{array}\\right.\n\\end{eqnarray}\nwhere $u(\\omega ; \\cdot)$ is the total field.\n\n\\subsection{Variational form}\n\\label{sec:variational}\nIn order to obtain a stability estimate, an important step is to reduce the infinite scattering problem into a bounded domain problem by introducing a transparent boundary condition \\cite{Bao1995SINUM}.\n\nSince the total field can be decomposed into\n$\nu(\\omega;\\cdot) = u^{i} + u^s(\\omega;\\cdot)\n$\nwhere\nthe incident wave satisfies\n\\begin{equation*}\n\t\\Delta u^{i}+k^{2} u^{i}=0 \\quad  \\text { in } \\Omega \\times D^+,\n\\end{equation*}\nthe scattered field $u^s(\\omega;\\cdot)$ satisfies\n\\begin{eqnarray}\\label{eq:u}\n\t\\left\\{\\begin{array}{ll}\n\t\t\\Delta u^s(\\omega ; \\cdot)+k^{2} u^s(\\omega ; \\cdot)=0 & \\text { in } \\Omega \\times D^+, \\\\\n\t\tu^s(\\omega ; \\cdot)+u^{i}=0 & \\text { on } \\Omega \\times S .\n\t\\end{array}\\right.\n\\end{eqnarray}\n\nDenote by $\\Gamma = \\{ (x_1,x_2)\\in\\mathbb{R} ^2 : x_2=b, x_1 \\in [0,\\Lambda] \\}  $\nwith\n$b > \\max_{\\omega\\in \\Omega, x_1\\in(0,\\Lambda)}  f(\\omega;x_1). $\nThe scattered field $u^s$ above $\\Gamma$ admits the Rayleigh expansion\n\\begin{equation}\\label{Rayleigh}\n\tu^s(\\omega;x_1,x_2)=\\sum_{n \\in Z}  A_{n}  e^{i\\left(\\alpha_{n}  x_1+\\beta_{n}  x_2\\right)} ,\n\\end{equation}\nwhere\n\\begin{equation}\\label{alpha_n}\n\n\t\\alpha_{n}=\\alpha+\\dfrac{2 \\pi n}{\\Lambda}, \\beta_{n}^{2}=k^{2}-\\alpha_{n}^{2} \\text { with } \\Im \\beta_{n} \\geq 0.\n\\end{equation}\n\n\\begin{assumption}\\label{assump:k}\n\t(Exclusion of resonances). Assume there exists $\\varepsilon>0$, such that $\\vert k-\\alpha_n \\vert\\geq \\varepsilon, \\ \\forall n \\in \\mathbb{Z} $.\n\\end{assumption}\n\n\\begin{remark}\n\tA constant  $\\varepsilon$ is introduced here to exclude\tall possible resonances ($\\kappa = \\alpha_n$) uniformly and thus a stability explicit with the wavenumber $k$ is obtained by a combination of the variational method and Fourier analysis.\n\\end{remark}\n\n\n\nDefine a boundary operator $T:H^{\\frac{1}{2}}(\\Gamma) \\rightarrow H^{-\\frac{1}{2}}(\\Gamma)$ by\n\\begin{equation*}\n\t(T v)\\left(x_{1} \\right)=\\sum_{n \\in \\mathbb{Z} }  \\mathrm{i}  \\beta_{n}  v_{n}  e^{\\mathrm{i}  \\alpha_{n}  x_{1} } ,\n\\end{equation*}\nwhere\n$v_{n} (\\omega)=\\frac{1} {\\Lambda}  \\int_{0} ^{\\Lambda}  v\\left(\\omega;x_{1} ,b\\right) e^{-\\mathrm{i}  \\alpha_{n}  x_{1} }  \\mathrm{d} x_{1} , $\nwhich yields a transparent boundary condition on $\\Gamma$:\n\\begin{equation*}\n\t\\partial_{\\nu} u = T u +g(x_1) \\quad \\text{ on } \\Gamma,\n\\end{equation*}\nand $g(x_1)= \\partial_{\\nu} u^{i} - T u^{i} = -2 i \\beta e^{i \\alpha x_1 - i \\beta b}$.\n\nDenote by $D= \\{x\\in\\mathbb{R}^2: f(\\omega; x_1)<x_2<b,\\ \\omega \\in \\Omega,  x_1 \\in [0,\\Lambda]\\}$. The scattering problem \\eqref{eq:u0} in an unbounded domain may be reduced to the following problem in the bounded domain:\n\\begin{equation}\\label{eq:u_total}\n\t\\left\\{\\begin{array}{ll}\n\t\t\\Delta u(\\omega;\\cdot)+k^{2} u(\\omega;\\cdot)=0 & \\text { in } \\Omega \\times D, \\\\\n\t\tu(\\omega;\\cdot)=0 & \\text { on } \\Omega \\times S, \\\\\n\t\t\\partial_{\\nu}u(\\omega;\\cdot) = Tu(\\omega;\\cdot)+g(x) & \\text{ on } \\Omega\\times \\Gamma.\n\t\\end{array}\\right.\n\\end{equation}\n\nMotivated by uniqueness, we seek for quasi-periodic solutions of $u(\\omega;\\cdot)$, that is, $u(\\omega;x_1+\\Lambda,x_2) = e^{i \\alpha \\Lambda} u(\\omega;x_1,x_2)$.\nLet $\\tau$ denote a trace operator from $D$ or $S$ to $\\Gamma$.\nDefine the function spaces inside $D$ by\n\\begin{equation*}\n\t\\begin{aligned}\n\t\tH^1(D)& = \\{ v\\in L^2(D): \\nabla v \\in L^2(D)^2 \\},\\\\\n\t\tH^1_S(D) &= \\{v\\in H^1(D): \\tau v=0 \\text{ on } S\\}, \\\\\n\t\tH^1_{S,qp}(D)&=\\{v\\in H^1_S(D): v(0,y) = v(\\Lambda,y)e^{-i\\alpha \\Lambda}\\}.\n\t\\end{aligned}\n\\end{equation*}\n\n\nLet $c:\\Omega \\rightarrow \\mathcal{C}$ be defined by\n\\begin{equation}\\label{def:c}\n\tc(\\omega)=f(\\omega),\n\\end{equation}\nwhere $\\mathcal{C}:= Lip$ is the space of all Lipschitz functions.\n\n\n\\textit{Variational form.}\nDefine the sesquilinear forms $\\mathcal{A}$ on $L^2(\\Omega;H^1_{S,qp}(D)) \\times L^2(\\Omega;H^1_{S,qp}(D))$ and  $\\tilde{a}_{c(\\omega)}$   on $H^1_{S,qp}(D) \\times H^1_{S,qp}(D)$ by\n\\begin{equation}\\label{def:AA}\n\t\\mathcal{A}(u,v) := \\int_{\\Omega}\\tilde{a}_{c(\\omega)} (u,v) \\mathrm{d} \\mathbb{P}(\\omega),\n\\end{equation}\nand\n\\begin{equation}\\label{aw}\n\t\\tilde{a}_{c(\\omega)} (u,v):= \\int_D \\nabla u \\cdot \\nabla \\bar{v} - k^2 \\int_D u \\bar{v} - \\int_\\Gamma T \\tau u \\overline{ \\tau v}.\n\\end{equation}\nDefine the antilinear functionals $\\mathcal{G}$ on $L^2(\\Omega;H^1_{S,qp}(D))$ and  $\\tilde{G}_{c(\\omega)}$  on  $H^1_{S,qp}(D)$ by\n\\begin{equation}\\label{def:GG}\n\t\\mathcal{G}(v) := \\int_{\\Omega} \\tilde{G}_{c(\\omega)}(v) \\mathrm{d} \\mathbb{P}(\\omega),\n\\end{equation}\nand\n\\begin{equation}\\label{Gw}\n\t\\tilde{G}_{c(\\omega)}(v):= \\int_\\Gamma g \\overline{ \\tau v}.\n\\end{equation}\nConsider the following variational form of the Helmholtz equation in a random periodic structure:\n\\begin{problem}\\label{P3}\n\tFind $u\\in L^2(\\Omega;H^1_{S,qp}(D))$ such that\n\t\\begin{equation}\\label{variational}\n\t\t\\mathcal{A} (u,v) = \\mathcal{G} (v), \\  \\forall   v \\in L^2(\\Omega;H^1_{S,qp} (D)).\n\t\\end{equation}\n\\end{problem}\n\n\n\n\\subsection{Stability results}\n\nWe first present a stability result explicit with respect to the wavenumber $k$ for the scattering problem by a deterministic periodic structure is given in Theorem \\ref{thm:1}.\n\n\n\\begin{theorem}\\label{thm:1}\n\t(Stability for the deterministic case). \tFix one sample $\\omega_0$. Let $\\tilde{u}=u(\\omega_0) \\in H^1_{S,qp}(D)$ be the solution of the deterministic scattering problem. Under Assumption \\ref{assump:k}, there exists a constant $C$ independent of  $b>1$ and large $k$, such that\n\t\\begin{equation}\\label{Destimate}\n\t\t\\|\\nabla \\tilde{u} \\|_{L^2(D)} + k \\|\\tilde{u}\\|_{L^2(D)} \\leq C \\max \\left\\{ \\dfrac{ b^2 {k} ^{2} } {\\sqrt{\\varepsilon} } ,b^3 {k} ^{\\frac{5} {2} }   \\right\\}  \\|g\\|_{L^2(\\Gamma)}.\n\t\\end{equation}\n\\end{theorem}\n\n\n\\begin{remark} \\label{re:1}\n\tFor the estimate, $\\varepsilon$ is independent of the wavenumber $k$ and the measured height $b$. Note that for small \t$\\varepsilon$, wavenumber is close to resonances and then the corresponding bound is very large.\n\n\tIf $\\varepsilon\\geq \\frac{1} { b^2 k } $, then the estimate becomes\n\t\\begin{equation}\\label{reduced}\n\t\t\\|\\nabla  \\tilde{u}  \\|_{L^2(D)}  + k \\| \\tilde{u} \\|_{L^2(D)}  \\leq C b^3 k ^{\\frac{5} {2} }     \\|g\\|_{L^2(\\Gamma)}.\n\t\\end{equation}\n\\end{remark}\n\nNext, the first stability result in a random periodic structure is shown in Theorem \\ref{thm:3} under Assumption \\ref{assump:k} and Assumption \\ref{cond:f}.\n\n\\begin{assumption}\\label{cond:f}\n\t(Regularity and stochastic regularity of $f$). Assume the random structure  $ f\\in L^2(\\Omega;Lip)$.\n\\end{assumption}\n\n\\begin{remark}\\label{remark:f0}\n\t(Random surfaces satisfying Assumption \\ref{cond:f}). Motivated by the uncertainty quantification such as Karhunen-Lo\\`eve expansion and other similar expansions,  it makes sense to consider $f$ as series expansions around the known deterministic surface $f_0$ satisfying the Lipschitz condition.\n\\end{remark}\n\n\\begin{theorem}\\label{thm:3}\n\t(Stability for the random case).\n\tUnder Assumption \\ref{assump:k} and Assumption \\ref{cond:f},\n\tthere exists a unique solution $u\\in L^2(\\Omega; H_{S,qp}^1(D))$ to Problem \\ref{P3}.\n\tMoreover, there exists a constant $C$ independent of  $b>1$ and large $k$,  such that\n\t\\begin{equation}\\label{Sestimate}\n\t\t\\|\\nabla u \\|_{L^2(\\Omega;L^2(D))} + k \\|u\\|_{L^2(\\Omega;L^2(D))} \\leq C \\max \\left\\{ \\dfrac{b^2 {k} ^{2} } {\\sqrt{\\varepsilon} }\n\t\t,b^3 {k} ^{\\frac{5} {2} }   \\right\\}   \\|g\\|_{L^2(\\Omega;L^2(\\Gamma))}.\n\t\\end{equation}\n\\end{theorem}\n\n\n\\section{Stability for the deterministic case}\n\\label{sec:thm1}\n\nThis section is devoted to the proof of Theorem \\ref{thm:1}. Our approach based on the variational method and Fourier analysis consists of two steps: the first step is to estimate the norms of $u$ in the domain $D$ by the norms of $u$ on the boundary $\\Gamma$ and the second step is to estimate the norms of $u$ on $\\Gamma$ by the norm of $g$.\n\nAlthough our approach is related to \\cite{Bao2012,Bao2016} for the scattering by a large rectangular cavity, additional difficulties arise: (1) Because of the different model geometry, quasiperiodic solutions have to be considered for our model here. (2) Instead of the Dirichlet or Neumann  boundary condition for the cavity problem, the (nonlocal) transparent boundary conditions must be dealt with. (3) Unlike the cavity problem, our model problem may not have unique solutions due to the resonances. Therefore, new techniques must be developed to prove the stability result.\n\nBefore proceeding to the detailed proof,  some useful notations and preliminary results are introduced first.\n\n\n\n\\subsection{Notations and preliminary results}\n\\label{S:preliminary}\n\nLet $\\tilde{u}(x_1,x_2)=u(\\omega_0;x_1,x_2)$ be the solution to the deterministic problem for a given sample $\\omega_0$. Then the total field $\\tilde{u}$ satisfies\n\\begin{equation}\\label{eq:u_deter}\n\t\\left\\{\\begin{array}{ll}\n\t\t\\Delta \\tilde{u}+k^{2} \\tilde{u}=0 & \\text { in } D, \\\\\n\t\t\\tilde{u}=0 & \\text { on } S, \\\\\n\t\t\\partial_{\\nu}\\tilde{u} = T\\tilde{u}+g & \\text{ on }  \\Gamma.\n\t\\end{array}\\right.\n\\end{equation}\n\nLet $C$ denote positive constants which are independent of large $k$ and $b>1$. Denote by $ \\alpha \\simeq \\beta$ if there exists a positive constant $C$ independent of large $k$ and $b>1$ such that $\\frac{1}{C} \\alpha \\leq \\beta\\leq C \\alpha$.\nWithout loss of generality, assume the period $\\Lambda = 2\\pi$ and denote $\\tilde{u}$ as $u$ for simplicity.\n\n\nThe scattered field $u$ can be written in the  single-layer potential representation:\n\\begin{equation}\\label{u1}\n\tu(x_1,x_2)=\\int_{0}^{\\Lambda} \\phi(s) G(x_1, x_2 ; s, 0) \\mathrm{d} s,\n\\end{equation}\nwhere\n$\\phi \\in L^2(0,\\Lambda)$ is an unknown periodic density function and $G$ is a quasi-periodic Green function given explicitly as\n$$\nG(x_1, x_2 ; s, t)=\\frac{i}{2 \\pi} \\sum_{n \\in \\mathbb{Z}} \\frac{1}{\\beta_{n}} e^{i \\alpha_{n}(x_1-s)+i \\beta_{n}\\vert x_2-t\\vert }, \\quad(x_1,x_2) \\neq(s, t).\n$$\nWithout loss of generality, it is assumed that the normal incidence is with $2 \\pi$-periodicity, i.e.\n$\\theta=0, \\alpha=0, \\beta=k, \\Lambda = 2 \\pi.$\nTherefore\n$\\alpha_n=n$ and $\\beta_n=\\sqrt{k^2-n^2}$,\nand thus\n\\begin{equation}\\label{Gn}\n\tG(x_1,x_2; s, t)=\\frac{i}{2 \\pi} \\sum_{n \\in \\mathbb{Z}} \\frac{1}{\\sqrt{k^2-n^2}} e^{i n(x-s)+i \\sqrt{k^2-n^2}\\vert y-t\\vert }\n\n\t\\text{~~and~~}\n\n\t\\phi(s)=\\sum_{n \\in \\mathbb{Z}} \\phi_{n} e^{i n s}.\n\\end{equation}\nSubstituting \\eqref{Gn} into \\eqref{u1} yields\n$$\nu^s (x_1,x_2)=\\sum_{n \\in \\mathbb{Z}} \\psi_n e^{i nx_1+i \\sqrt{k^2-n^2}  x_2}.\n$$\nSince $u=u^{inc} +u^s$, the solution $u$ can be expressed as\n\\begin{equation*}\n\t\\begin{aligned}\n\t\tu(x_1,x_2)=&\n\t\t\\displaystyle\\sum_{n=0} ^{\\infty}  a_{n}  \\sin \\left(\\sqrt{{k} ^{2} -n^{2} }  x_2\\right) \\sin (n x_1) + \\sum_{n=0} ^{\\infty}  b_{n}  \\cos \\left(\\sqrt{{k} ^{2} -n^{2} }  x_2\\right) \\cos (n x_1)\\\\\n\t\t& + i \\left[ \\displaystyle\\sum_{n=0} ^{\\infty}  c_n \\cos \\left(\\sqrt{{k} ^{2} -n^{2} }  x_2\\right) \\sin (n x_1) + \\sum_{n=0} ^{\\infty}  d_{n}  \\sin \\left(\\sqrt{{k} ^{2} -n^{2} }  x_2\\right) \\cos (n x_1)\\right]\\\\\n\t\t\\triangleq& \\mu+i \\nu.\n\t\\end{aligned}\n\\end{equation*}\nwhere $a_n=-\\psi_n+\\psi_{-n} $, $b_n=\\psi_n+\\psi_{-n} $, $n\\in \\mathbb{N} _+$, $a_0=-\\psi_0+1$, $b_0=\\psi_0+1$;  $c_n=-a_n$, $d_n=b_n$, $n\\in \\mathbb{N} _+$, $c_0=\\psi_0-1$, $d_0=\\psi_0+1$.\nFor the case of non-normal cases, we only need to replace $n$ with $\\alpha_n$ and $\\sqrt{k^2-n^2} $ with $\\sqrt{k^2-{\\alpha_n} ^2} $.\nSince $\\mu$ and $\\nu$ can be studied similarly, only detailed study on $\\mu$ is provided here. Also, for convenience, in the following proof, the notation $\\mu$ is denoted as $u$.\n\nLet $u_n$, $(\\partial_{\\nu}u)_n$, $(Tu)_n$ be the sine coefficients, and $v_n$, $(\\partial_{\\nu}v)_n$, $(Tv)_n$ be the cosine coefficients of the expansions of $u$, $\\partial_{\\nu}u$, $Tu$, on $\\Gamma$, respectively. That is,\n\\begin{equation}\\label{unvn}\n\tu_{n}:=\n\ta_{n} \\sin \\left(\\sqrt{k^{2}-n^{2}} b\\right),\n\tv_{n}:=\n\tb_{n} \\cos \\left(\\sqrt{k^{2}-n^{2}} b\\right),\n\\end{equation}\n\\begin{equation}\\label{partialun}\n\t\\left(\\partial_{\\nu} u\\right)_{n}:=\n\ta_{n} \\sqrt{k^{2}-n^{2}} \\cos \\left(\\sqrt{k^{2}-n^{2}} b\\right),\n\\end{equation}\n\\begin{equation}\\label{partialvn}\n\t\\left(\\partial_{\\nu} v\\right)_{n}:=\n\t-b_{n} \\sqrt{k^{2}-n^{2}} \\sin \\left(\\sqrt{k^{2}-n^{2}} b\\right).\n\\end{equation}\nHence\n\\begin{equation*}\n\tu\\vert_{\\Gamma}(x_1) = \\sum_{n=0}^{\\infty} u_n \\sin(n x_1) + \\sum_{n=0}^{\\infty} v_n \\cos(n x_1),\n\\end{equation*}\n\\begin{equation*}\n\t\\partial_{\\nu} u\\vert_{\\Gamma}(x_1) = \\sum_{n=0}^{\\infty} (\\partial_{\\nu} u)_n \\sin(n x_1) + \\sum_{n=0}^{\\infty} (\\partial_{\\nu} v)_n \\cos(n x_1),\n\\end{equation*}\nand\n\\begin{equation*}\n\tTu\\vert_{\\Gamma}(x_1) = \\sum_{n=0}^{\\infty} (Tu)_n \\sin(n x_1) + \\sum_{n=0}^{\\infty} (Tv)_n \\cos(n x_1).\n\\end{equation*}\n\n\n\\begin{definition}\\label{def:AB}\n\tFor $u =\\displaystyle\\sum_{n=0}^{\\infty} u_n \\sin(n x_1) + \\sum_{n=0}^{\\infty} v_n \\cos(n x_1)$, define\n\t$$\\|u\\|_{A}=\\sqrt{\\sum_{n=0}^{\\infty}\\left( \\lvert u_{n}\\rvert^{2} +\\lvert v_{n}\\rvert^{2} \\right) \\frac{1}{\\sqrt{\\lvert k^{2}-n^{2}\\rvert}+\\frac{1}{b}}},$$\n\tand\n\t$$\\|u\\|_{B}=\\sqrt{\\sum_{n=0}^{\\infty}\\left( \\lvert u_{n}\\rvert^{2} +\\lvert v_{n}\\rvert^{2}\\right) \\left(\\sqrt{\\lvert k^{2}-n^{2}\\rvert}+ \\frac{1}{b}\\right)}.$$\nHere, the norms $A$ and $B$ are formerly dual to each other and behave like $H^{-1\/2}$, $H^{1\/2}$, respectively. Moreover, a simple calculation gives that\n\t\\begin{equation}\\label{norm}\n\t\t\\int_{\\Gamma} u_1(x_1) \\overline{u_2(x_1)} \\mathrm{d} x_1 \\leq \\pi \\|u_1\\|_{A}\\|u_2\\|_{B}.\n\t\\end{equation}\n\\end{definition}\n\n\n\\begin{definition}\n\tDefine the lower frequency set $\\mathbf{L}$ and the higher frequency set $\\mathbf{H}$ by\n\t$$\\mathbf{L}=\\left\\{n \\in \\mathbb{N} \\mid n < k \\right\\}\n\t\\text{~and~~}\n\t\\mathbf{H}=\\left\\{n \\in \\mathbb{N} \\mid n > k \\right\\}.$$\n\\end{definition}\n\nThen, $u$, $\\partial_{\\nu}u$, $T(u)$ on $\\Gamma$ defined above can be separated into the lower frequency part and the higher frequency part:\n$$\nu(x_1) =\\mathcal{V}_\\mathbf{L} +\\mathcal{V}_\\mathbf{H}, \\quad\n\\partial_{\\nu} u(x_1) =\\mathcal{P}_\\mathbf{L} +\\mathcal{P}_\\mathbf{H},\\quad\n(T (u))(x_1) = \\mathcal{T}_\\mathbf{L} +\\mathcal{T}_\\mathbf{H},$$\nwith\n\\begin{equation*}\n\t\\begin{aligned}\n\t\t&\\mathcal{V}_\\mathbf{L} =\\sum_{n \\in \\mathbf{L}} u_{n} \\sin (n x_1)\n\t\t+\\sum_{n \\in \\mathbf{L}} v_{n} \\cos (n x_1),\\\\\n\t\t&\\mathcal{V}_\\mathbf{H}=\\sum_{n \\in \\mathbf{H}} u_{n} \\sin (n x_1)\n\t\t+\\sum_{n \\in \\mathbf{H}} v_{n} \\cos (n x_1), \\\\\n\t\t&\\mathcal{P}_\\mathbf{L}=\\sum_{n \\in \\mathbf{L}}\\left(\\partial_{\\nu} u\\right)_{n} \\sin (n x_1)\n\t\t+\\sum_{n \\in \\mathbf{L}}\\left(\\partial_{\\nu} v\\right)_{n} \\cos (n x_1),\\\\\n\t\t&\\mathcal{P}_\\mathbf{H}=\\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} u\\right)_{n} \\sin (n x_1)\n\t\t+\\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} v\\right)_{n} \\cos (n x_1),\\\\\n\t\t&\\mathcal{T}_\\mathbf{L} =\\sum_{n \\in \\mathbf{L}}(T u)_{n} \\sin (n x_1)\n\t\t+\\sum_{n \\in \\mathbf{L}}(T v)_{n} \\cos (n x_1),\\\\\n\t\t&\\mathcal{T}_\\mathbf{H}=\\sum_{n \\in \\mathbf{H}}(T u)_{n} \\sin (n x_1)\n\t\t+\\sum_{n \\in \\mathbf{H}}(T v)_{n} \\cos (n x_1).\n\t\\end{aligned}\n\\end{equation*}\n\n\nThe higher frequency terms have useful properties presented in Lemma\n\\ref{Lem:H}.\n\\begin{lemma}\\label{Lem:H}\n\tIf $u(x_1,x_2)$ is the solution to the scattering problem \\eqref{eq:u_deter}, then\n\t$$ \\left\\|\\mathcal{P}_\\mathbf{H}\\right\\|_{A}^{2}\n\t\\simeq \\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} u\\right)_{n} \\overline{u_{n}} + \\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} v\\right)_{n}\\overline{v_{n}}   \\simeq\\left\\|\\mathcal{V}_\\mathbf{H}\\right\\|_{B}^{2} .$$\n\\end{lemma}\n\\begin{proof}\n\tIf $n\\in \\mathbf{H}$,\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\t&\\left( \\lvert a_{n} \\sqrt{k^{2}-n^{2}} \\cos \\left(\\sqrt{k^{2}-n^{2}} b\\right) \\rvert^2 + \\lvert -b_{n} \\sqrt{k^{2}-n^{2}} \\sin \\left(\\sqrt{k^{2}-n^{2}} b\\right) \\rvert^2 \\right) \\frac{1}{\\sqrt{\\lvert k^{2}-n^{2}\\rvert}+\\frac{1}{b}}\\\\\n\t\t\t\\simeq & \\left( \\lvert a_{n} \\sin \\left(\\sqrt{k^{2}-n^{2}} b\\right) \\rvert^2 + \\lvert  b_{n} \\cos \\left(\\sqrt{k^{2}-n^{2}} b\\right) \\rvert^2 \\right) \\left(\\sqrt{\\lvert k^{2}-n^{2}\\rvert}+ \\frac{1}{b}\\right).\n\t\t\\end{aligned}\n\t\\end{equation*}\n\tHence\n\t\\begin{equation*}\n\t\t\\left( \\lvert  (\\partial_{\\nu}u)_n \\rvert^2 + \\lvert  (\\partial_{\\nu}v)_n \\rvert^2 \\right) \\frac{1}{\\sqrt{\\lvert k^{2}-n^{2}\\rvert}+\\frac{1}{b}}\n\t\t\\simeq \\left( \\lvert  u_n \\rvert^2 + \\lvert  v_n \\rvert^2 \\right) \\left(\\sqrt{\\lvert k^{2}-n^{2}\\rvert}+ \\frac{1}{b}\\right).\n\t\\end{equation*}\n\tFrom \\eqref{unvn}-\\eqref{partialvn}, one can show that for any $n \\in \\mathbf{H}$,\n\t$\\left(\\partial_{\\nu} u\\right)_{n} \\overline{u_{n}} \\geq 0$, $\\left(\\partial_{\\nu} v\\right)_{n} \\overline{v_{n}} \\geq 0$.\n\tTherefore,\n\t$$ \\left\\|\\mathcal{P}_\\mathbf{H} \\right\\|_{A}^{2}\n\t\\simeq \\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} u\\right)_{n} \\overline{u_{n}} + \\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} v\\right)_{n}\\overline{v_{n}} .$$\n\\end{proof}\n\nThe next result provides a quantitative estimate of the function $u$ and its normal derivative.\n\n\\begin{lemma}\\label{Lem:sqrtk}\n\tIf $u(x_1,x_2)$ is the solution to the scattering problem \\eqref{eq:u_deter}, then\n\t\\begin{equation}\\label{Lemineq}\n\t\t\\Lambda \\sum_{n\\in \\mathbb{Z} }  \\sqrt{\\vert {k} ^{2} -\\alpha_n^2\\vert}  \\vert\\hat{u} _n\\vert^2 \\geq \\sqrt{\\varepsilon}  \\sqrt{k}  \\|u\\|_{L^2(\\Gamma)} ^2.\n\t\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n\tSince $\\Lambda \\sum_{n\\in \\mathbb{Z} }  \\vert\\hat{u} _n\\vert^2 = \\|u\\|_{L^2(\\Gamma)} ^2$ and\n\t$\\sqrt{\\vert {k} ^{2} -\\alpha_n^2 \\vert}  \\geq \\sqrt{\\varepsilon}  \\sqrt{k} $ for $\\vert k-\\alpha_n\\vert\\geq \\varepsilon, \\ \\forall n \\in \\mathbb{Z} $,\n\t$$\\Lambda \\sum_{n\\in \\mathbb{Z} }  \\sqrt{\\vert{k} ^{2} -\\alpha_n^2\\vert}  \\vert\\hat{u} _n\\vert^2 \\geq \\sqrt{\\varepsilon}  \\sqrt{k}  \\|u\\|_{L^2(\\Gamma)} ^2,$$\n\twhich completes the proof.\n\\end{proof}\n\n\\begin{remark}\n\tDifferent from Lemma 3.5 in \\cite{Bao2012} for the scattering by a cavity, TBC of gratings yield additional difficulties on resonances and  stability.\n\tHence, in this paper, a uniform distance $\\varepsilon$ is introduced to exclude resonances and thus obtain the estimate \\eqref{Lemineq} in Lemma \\ref{Lem:sqrtk}, which is essential for the proof procedure of the $k$-explicit stability for periodic structures.\n\\end{remark}\n\nNext, the relation between the norm of $A$ (or $B$) and the $L^2$ norm is established in Remark \\ref{rem:AL}.\n\n\\begin{remark}\\label{rem:AL}\n\tFor the norm $A$, if there exists a natural number $n^*$ such that $\\sqrt{\\vert k^2-{n^*}^2 \\vert}\\leq \\frac{1}{b}$, then $n^*$ is very close to $k$ so that $\\vert k-{n^*} \\vert\\leq \\frac{1}{ b^2 k}$. Since $\\sqrt{\\vert k^2-n^2 \\vert}\\geq \\sqrt{\\frac{k}{2}}$,\n\tthe norm $A$ can be simplified as\n\t$$\\|u\\|_{A} \\simeq \\sqrt{\\left(\\sum_{n \\in \\mathbb{N} \\backslash\\left\\{n^{*}\\right\\}}\\left(\\lvert u_{n}\\rvert^{2}+\\lvert v_{n}\\rvert^{2}\\right) \\frac{1}{\\sqrt{\\lvert k^{2}-n^{2}\\rvert}}\\right)+\\left(\\lvert u_{n^*}\\rvert^{2}+\\lvert v_{n^*}\\rvert^{2}\\right) b}.$$\t\n\tIf such $n^*$ does not exist, then the norm $A$ has no relation to $b$ since\n\t$$\\|u\\|_A \\simeq \\sqrt{\\sum_{n \\in \\mathbb{N}}\\left(\\lvert u_{n}\\rvert^{2}+\\lvert u_{n}\\rvert^{2}\\right) \\frac{1}{\\sqrt{\\lvert k^{2}-n^{2}\\rvert}}}.$$\n\tHence, whether $n^*$ exists or not, there always exists a constant $C$ independent of $k$ and $b$ such that\n\t$$\\|u\\|_{A}^{2} \\leq C\\left(\\frac{1}{\\sqrt{k}}\\|u\\|_{L^{2}(\\Gamma)}^{2}+b \\left( \\lvert u_{k^{*}}\\rvert^{2}+\\lvert v_{k^{*}}\\rvert^{2}\\right)\\right),$$\n\twhere $k^*=\\arg\\min_{n\\in\\mathbb{N}} \\vert n-k\\vert $, $u_{k^*} = \\frac{1}{\\pi} \\int_{\\Gamma} u(x_1) \\sin \\left(k^{*} x_1\\right) \\mathrm{d} x_1 $ and $v_{k^*} = \\frac{1}{\\pi} \\int_{\\Gamma} u(x_1) \\cos \\left(k^{*} x_1\\right) \\mathrm{d} x_1$.\n\tIt follows immediately that\n\t\\begin{equation}\n\t\t\\|u\\|_{A}^{2} \\leq C b\\|u\\|_{L^{2}(\\Gamma)}^{2}.\n\t\\end{equation}\n\tThe similar approximation can be deduced for the norm $B$:\n\t\\begin{equation}\\label{ineq:normB}\n\t\tb\\|u\\|_{B}^{2} \\leq C \\|u\\|_{L^{2}(\\Gamma)}^{2}.\n\t\\end{equation}\n\\end{remark}\n\nAs for the proof of Theorem \\ref{thm:1},\nthe first step given in Lemma \\ref{lem:step1} is to estimate the norm of $u$ in $D$ by the norms of $u$ on $\\Gamma$, i.e., the norms $A$ and $B$, and the second step given in Lemma \\ref{lem:step2} is to estimate $\\|\\partial_{\\nu}u\\|_A$ and $\\|u\\|_B$ by  $\\|g\\|_A$. Finally, with the correlation between the norm $A$ and the $L^2$ norm discussed in Remark \\ref{rem:AL}, the stability estimate of Theorem \\ref{thm:1} is proved.\n\n\\begin{lemma}\\label{lem:step1}\n\tLet $u$ be a solution to the scattering problem \\eqref{eq:u_deter}. Under Assumption \\ref{assump:k}, there exists a constant $C$ independent of $b>1$ and large $k$,  such that\n\t\\begin{equation}\n\t\t\\|\\nabla u \\|_{L^2(D)} + k \\|u\\|_{L^2(D)} \\leq C  b k \\left( \\| \\partial_{\\nu}u \\|_A + \\|u\\|_B \\right),\n\t\\end{equation}\n\twhere the norms $A$ and $B$ are defined in Definition \\ref{def:AB}.\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:step2}\n\tLet $u$ be a solution to the scattering problem \\eqref{eq:u_deter}. Under Assumption \\ref{assump:k}, there exists a constant $C$ independent of $b>1$ and large $k$, such that\n\t\\begin{equation}\n\t\t\\| \\partial_{\\nu}u \\|_A + \\|u\\|_B \\leq C \\max \\left\\{ \\dfrac{  b^{\\frac{1} {2} } {k}  }  {\\sqrt{\\varepsilon} }\n\t\t,  b^{\\frac{3} {2} }  {k} ^{\\frac{3} {2} }  \\right\\}   \\|g\\|_A,\n\t\\end{equation}\n\twhere the norms $A$ and $B$ are defined in Definition \\ref{def:AB}.\n\\end{lemma}\n\n\\subsection{Proof of Lemma \\ref{lem:step1} and Lemma \\ref{lem:step2}}\n\n\nThe proof of the bound in Lemma \\ref{lem:step1} is mainly based on the definition of the norms and the representation of the series, which means that a stability estimate for the solution $u$ in $D$ can be given in terms of its Dirichlet and Neumann data on $\\Gamma$.\n\n\\begin{proof}[Proof of Lemma~{\\upshape\\ref{lem:step1}}]\n\tFrom the equation $\\Delta u + k^2 u = 0 $ in $D$ and its boundary condition on $S$\n\n\tas well as the quasi-periodicity of the solution $u$, one has\n\t$$ \\int_{\\Gamma} u \\overline{\\partial_{\\nu} u} \\mathrm{d} x_1=\\int_{D}\\left(\\vert\\nabla u\\vert^{2}-k^{2}\\vert u\\vert^{2}\\right) \\mathrm{d} x_1 \\mathrm{d} x_2. $$\n\t\n\tThe left-hand side of the equality above is obviously bounded by $ \\left\\|\\partial_{\\nu} u\\right\\|_{A}^{2}+\\|u\\|_{B}^{2}$.\n\tHowever, the right-hand side has a negative sign for the term $\\vert u\\vert^2$. In order to prove\n\tLemma \\ref{lem:step1}, it suffices to show that\n\t\\begin{equation}\\label{6.1}\n\t\t\\|u\\|_{L^{2}(D)}^{2} \\leq C b^{2}\\left(\\left\\|\\partial_{\\nu} u\\right\\|_{A}^{2}+\\|u\\|_{B}^{2}\\right).\n\t\\end{equation}\n\t\n\tNote that $\\partial_{\\nu}u$, $u$ on $\\Gamma$, and $u$ in $D$ can be expressed in terms of sine and cosine functions in the direction of $x_1$. Because of the orthogonality of sine and cosine functions, one may assume\n\tthat\n\t$$ u(x_1,x_2)=\\eta \\sin \\left(\\sqrt{k^{2}-n^{2}} x_2\\right) \\sin (n x_1) + \\zeta   \\cos \\left(\\sqrt{k^{2}-n^{2}} x_2\\right) \\cos (n x_1)\\text { in } D, \\ n\\neq k. $$\n\n\tFor $n \\neq k$, consider three cases:\n\t(i) $n<k$ and $b \\sqrt{k^2-n^2}\\geq 1$;\n\t(ii) $b \\sqrt{\\vert k^2-n^2 \\vert}<1$; and\n\t(iii) $n>k$ and $b \\sqrt{\\vert k^2-n^2 \\vert}\\geq 1$.\n\t\n\tFor Case (i), it is easy to see that $\\|u\\|_{L^{2}(D)}^{2} \\simeq b$. On the boundary $\\Gamma$,\n\t$$\n\t\\begin{aligned}\n\t\t\\partial_{\\nu} u &=\\eta \\sqrt{k^{2}-n^{2}} \\cos \\left(\\sqrt{k^{2}-n^{2}} b\\right) \\sin (n x_1) - \\zeta \\sqrt{k^{2}-n^{2}} \\sin \\left(\\sqrt{k^{2}-n^{2}} b\\right) \\cos (n x_1), \\\\\n\t\tu &=\\eta \\sin \\left(\\sqrt{k^{2}-n^{2}} b\\right) \\sin (n x_1) + \\zeta   \\cos \\left(\\sqrt{k^{2}-n^{2}} b\\right) \\cos (n x_1).\n\t\\end{aligned}\n\t$$\n\tFrom the definition of the norms A and B, one has\n\t$$\n\t\\begin{array}{l}\n\t\t\\left\\|\\partial_{\\nu} u\\right\\|_{A}^{2}+\\|u\\|_{B}^{2} \\\\\n\t\t\\qquad \\begin{aligned}\n\t\t\t=& \\vert\\eta\\vert^2 \\left(\\frac{k^{2}-n^{2}}{\\sqrt{k^{2}-n^{2}}+\\frac{1}{b}}\\lvert \\cos \\left(\\sqrt{k^{2}-n^{2}} b\\right)\\rvert^{2}\n\t\t\t+\\left(\\sqrt{k^{2}-n^{2}}+\\frac{1}{b}\\right)\\lvert \\sin \\left(\\sqrt{k^{2}-n^{2}} b\\right)\\rvert^{2} \\right)\\\\\n\t\t\t& + \\vert\\zeta\\vert^2 \\left(\\frac{k^{2}-n^{2}}{\\sqrt{k^{2}-n^{2}}+\\frac{1}{b}}\\lvert \\sin \\left(\\sqrt{k^{2}-n^{2}} b\\right)\\rvert^{2}\n\t\t\t+\\left(\\sqrt{k^{2}-n^{2}}+\\frac{1}{b}\\right)\\lvert \\cos \\left(\\sqrt{k^{2}-n^{2}} b\\right)\\rvert^{2} \\right)\\\\\n\t\t\t\\geq & \\frac{1}{2} (\\vert\\eta\\vert^2+\\vert\\zeta\\vert^2)\\sqrt{k^{2}-n^{2}} \\geq \\frac{(\\vert\\eta\\vert^2+\\vert\\zeta\\vert^2)}{2 b} \\geq C \\frac{1}{b^{2}}\\|u\\|_{L^{2}(D)}^{2}.\n\t\t\\end{aligned}\n\t\\end{array}\n\t$$\n\t\n\tFor Case (ii), one has\n\t$$ \\lvert \\sin \\left(\\sqrt{k^{2}-n^{2}} x_2\\right)\\rvert \\simeq \\sqrt{\\lvert k^{2}-n^{2}\\rvert} x_2 \\text{  and }\n\t\\lvert \\cos \\left(\\sqrt{{k} ^{2} -n^{2} }  x_2\\right)\\rvert \\simeq 1- \\dfrac{1} {2}  \\lvert {k} ^{2} -n^{2} \\rvert x_2^2$$\n\tfrom $ \\sqrt{\\lvert k^{2}-n^{2}\\rvert} b<1 $.\n\tTherefore, $\\|u\\|_{L^{2}(D)}^{2} \\simeq\\lvert k^{2}-n^{2}\\rvert b^{3}$ and\n\t$$ \\left\\|u\\left(x, b\\right)\\right\\|_{B}^{2} \\simeq\\left(\\sqrt{\\lvert k^{2}-n^{2}\\rvert}+\\frac{1}{b}\\right)\\lvert k^{2}-n^{2}\\rvert b^{2} \\simeq b\\lvert k^{2}-n^{2}\\rvert, $$\n\twhich implies the inequality \\eqref{6.1}.\n\t\n\tFinally, for Case (iii),\n\t$$\n\t\\begin{aligned}\n\t\tu&=\\eta \\sin \\left(\\sqrt{k^{2}-n^{2}} x_2\\right) \\sin (n x_1) + \\zeta   \\cos \\left(\\sqrt{k^{2}-n^{2}} x_2\\right) \\cos (n x_1) \\\\\n\t\t&= \\eta i \\sinh \\left(\\sqrt{n^{2}-k^{2}} x_2\\right) \\sin (n x_1) + \\zeta  \\cosh \\left(\\sqrt{n^{2}-k^{2}} x_2\\right) \\cos (n x_1) \\text { in } D.\n\t\\end{aligned}\n\t$$\n\tThe representation of $u$ yields\n\t$$ \\|u\\|_{L^{2}(D)}^{2}  \\leq C \\frac{1}{\\sqrt{n^{2}-k^{2}}} e^{2 \\sqrt{n^{2}-k^{2}} b} \\leq C b^{2}\\left(\\sqrt{n^{2}-k^{2}}+\\frac{1}{b}\\right) e^{2 \\sqrt{n^{2}-k^{2}} b} \\leq C b^{2}\\left\\|u\\left(\\cdot, b\\right)\\right\\|_{B}^{2}. $$\n\\end{proof}\n\n\nFor the proof of Lemma \\ref{lem:step2}, the main difficulty is due to the fact that  the boundary data $\\partial_{\\nu} u = T u +g$ does not yield direct estimates on $\\| \\partial_{\\nu}u \\|_A$ and  $\\|u\\|_B$. To be specific, let us consider the key term $-\\mathfrak{Im}\\int_{\\Gamma} g \\bar{u} \\mathrm{d} x_1$. Obviously,\n\\begin{equation}\\label{diff}\n\t- \\mathfrak{Im}\\int_{\\Gamma} g \\bar{u} \\mathrm{d} x_1 =  \\Lambda \\sum_{\\vert \\alpha_n \\vert<k}  \\sqrt{{k} ^{2} -{\\alpha_n} ^2}   \\vert\\hat{u} _n\\vert^2 \\geq 0.\n\\end{equation}\nHowever, the integral in \\eqref{diff} does not contain information on $\\hat{u}$ for $\\vert \\alpha_n \\vert > k $.\nIn order to obtain the stability, we have to analyze $\\hat{u}$ for $\\vert \\alpha_n \\vert > k $, and thus to generate a bound on $u$. Next, we give the proof of Lemma \\ref{lem:step2}.\n\n\\begin{proof}[Proof of Lemma~{\\upshape\\ref{lem:step2}}]\n\tFrom the definition,\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\t\\int_{\\Gamma} g \\bar{u} \\mathrm{d} x_1 & = \\int_{\\Gamma} (\\partial_{\\nu} u - T(u)) \\bar{u} \\mathrm{d} x_1\\\\\n\t\t\t& = \\int_{\\Gamma}  \\partial_{\\textbf{n} }  u \\bar{u}  \\mathrm{d} x_1- i  \\Lambda \\sum_{\\vert n \\vert<k}  \\sqrt{{k} ^{2} -n^2}   \\vert\\hat{u} _n\\vert^2  +\n\t\t\t\\Lambda \\sum_{\\vert n\\vert>k}  \\sqrt{n^2-{k} ^{2} }  \\vert \\hat{u} _n \\vert^2 \\\\\n\t\t\t& \\triangleq \\mathfrak{Re} + i \\mathfrak{Im},\n\t\t\\end{aligned}\n\t\\end{equation*}\n\twith\n\t$\\mathfrak{Re}  = \\int_{\\Gamma}  \\partial_{\\textbf{n} }  u \\bar{u}  \\mathrm{d} x_1 + \\Lambda \\sum_{\\vert n \\vert>k}  \\sqrt{n^2-{k} ^{2} }  \\vert \\hat{u} _n \\vert^2$ and\n\t$ \\mathfrak{Im}   = - \\Lambda \\sum_{\\vert n \\vert<k}  \\sqrt{{k} ^{2} -n^2}   \\vert \\hat{u} _n \\vert^2  $.\n\n\tObviously,\n\t\\begin{equation}\\label{case2Im}\n\t\t\\Lambda \\sum_{\\vert n \\vert<k}  \\sqrt{{k} ^{2} -n^2}   \\vert \\hat{u} _n \\vert^2 = \\lvert \\mathfrak{Im}\\rvert \\leq \\lvert  \\int_{\\Gamma} g \\bar{u}  \\mathrm{d} x_1 \\rvert.\n\t\\end{equation}\n\tAs for the real part, since\n\t$$\n\t\\lvert\\mathfrak{Re}\\rvert \\leq \\lvert  \\int_{\\Gamma} g \\bar{u} \\mathrm{d} x_1 \\rvert \\leq \\pi\\|g\\|_A \\|u\\|_B \\leq \\gamma \\pi \\left\\| \\partial_{\\nu}u  \\right\\|_A \\|u\\|_B\n\t$$\n\tprovieded that \n\t\\begin{equation}\\label{condcase2}\n\t\t\\| g \\|_A \\leq \\gamma \\| \\partial_{\\nu}u  \\|_A,\n\t\\end{equation}\n\tand\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\\mathfrak{Re} = &\\sum_{n \\in \\mathbf{L}} \\left( \\partial_{\\nu}u \\right)_n \\bar{u}_n + \\sum_{n \\in \\mathbf{L}} \\left( \\partial_{\\nu}v \\right)_n \\bar{v}_n + \\sum_{n \\in \\mathbf{H}} \\left( \\partial_{\\nu}u \\right)_n  \\bar{u}_n + \\sum_{n \\in \\mathbf{H}} \\left( \\partial_{\\nu}v \\right)_n \\bar{v}_n \\\\\n\t\t&+\\Lambda \\sum_{\\vert n \\vert>k}  \\sqrt{n^2-{k} ^{2} }  \\vert \\hat{u} _n \\vert^2,\n\t\t\\end{aligned}\n\t\\end{equation*}\n\tone obtains\n\t\\begin{equation}\\label{case2Re}\n\t\t\\begin{aligned}\n\t\t\t&\\gamma \\pi \\left\\|  \\partial_{\\nu}u  \\right\\|_A \\|u\\|_B +  \\lvert   \\sum_{n \\in \\mathbf{L}} \\left( \\partial_{\\nu}u \\right)_n \\bar{u}_n + \\sum_{n \\in \\mathbf{L}} \\left( \\partial_{\\nu}v \\right)_n \\bar{v}_n \\rvert \\\\\n\t\t\t& \\geq \\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} u\\right)_{n} \\overline{u_{n}} + \\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} v\\right)_{n}\\overline{v_{n}}  \n\t\t\t + \\Lambda \\sum_{\\vert n \\vert>k}  \\sqrt{n^2-{k} ^{2} }  \\vert \\hat{u} _n \\vert^2,\n\t\t\\end{aligned}\n\t\\end{equation}\n\twith\n\t$$ \\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} u\\right)_{n} \\overline{u_{n}} + \\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} v\\right)_{n}\\overline{v_{n}} \n\t \\simeq \\left\\| \\mathcal{P}_\\mathbf{H} \\right\\|_A^2 \\geq 0$$\n\tand\n\t$$\\Lambda \\sum_{\\vert n \\vert>k}  \\sqrt{n^2-{k} ^{2} }  \\vert \\hat{u} _n \\vert^2 \\geq 0.\n\t$$\n\n\tThe estimate \\eqref{case2Re} implies that  $\\displaystyle\\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} u\\right)_{n} \\overline{u_{n}} + \\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} v\\right)_{n}\\overline{v_{n}}  $ and $ \\Lambda \\displaystyle\\sum_{\\vert n \\vert>k}  \\sqrt{n^2-{k} ^{2} }  \\vert \\hat{u} _n \\vert^2$ can be bounded by $\\lvert   \\displaystyle\\sum_{n \\in \\mathbf{L}} \\left( \\partial_{\\nu}u \\right)_n \\bar{u}_n + \\sum_{n \\in \\mathbf{L}} \\left( \\partial_{\\nu}v \\right)_n \\bar{v}_n \\rvert$ \nsince $\\gamma \\pi \\left\\| \\left( \\partial_{\\nu}u \\right)_n  \\right\\|_A \\|u\\|_B$ may be absorbed by choosing an appropriate $\\gamma$.\n\nEvidently,  in order to estimate  $ \\Lambda \\displaystyle\\sum_{\\vert n \\vert>k}  \\sqrt{n^2-{k} ^{2} }  \\vert \\hat{u} _n \\vert^2$, it is sufficient to estimate $\n\t\\lvert   \\displaystyle\\sum_{n \\in \\mathbf{L}} \\left( \\partial_{\\nu}u \\right)_n \\bar{u}_n + \\sum_{n \\in \\mathbf{L}} \\left( \\partial_{\\nu}v \\right)_n \\bar{v}_n \\rvert\\;.\n\t$\n\tSince the estimate \\eqref{case2Re} is based on the condition \\eqref{condcase2}, we will consider $\\| g \\|_A \\leq \\gamma \\| \\partial_{\\nu}u  \\|_A$ and $\\| g \\|_A \\geq \\gamma \\| \\partial_{\\nu}u  \\|_A$ separately. For the case $\\| g \\|_A \\leq \\gamma \\| \\partial_{\\nu}u  \\|_A$, since the term $\n\t\\lvert   \\displaystyle\\sum_{n \\in \\mathbf{L}} \\left( \\partial_{\\nu}u \\right)_n \\bar{u}_n + \\sum_{n \\in \\mathbf{L}} \\left( \\partial_{\\nu}v \\right)_n \\bar{v}_n \\rvert\n\t$ concerns the low frequency part for ${u}_n$ and $\\left( \\partial_{\\nu}u \\right)_n$, to characterize which term is bigger, $\\delta \\left\\| \\mathcal{P}_\\mathbf{L}  \\right\\|_A \\geq  \\left\\|  \\mathcal{V}_\\mathbf{L}   \\right\\|_B$ and $\\delta \\left\\| \\mathcal{P}_\\mathbf{L}  \\right\\|_A \\leq  \\left\\|  \\mathcal{V}_\\mathbf{L}   \\right\\|_B$ are considered separately. \t\n\tTo sum up, we consider the following three cases:\n\t\\begin{itemize}\n\t\t\\item Case (i)\n\t\t\\begin{equation}\\label{case2.1}\n\t\t\t\\| g \\|_A \\leq \\gamma \\| \\partial_{\\nu}u  \\|_A, \\quad \\delta \\left\\| \\mathcal{P}_\\mathbf{L}  \\right\\|_A \\geq  \\left\\|  \\mathcal{V}_\\mathbf{L}   \\right\\|_B;\n\t\t\\end{equation}\n\t\t\\item Case (ii)\n\t\t\\begin{equation}\\label{case2.2}\n\t\t\t\\| g \\|_A \\leq \\gamma \\| \\partial_{\\nu}u  \\|_A, \\quad \\delta \\left\\| \\mathcal{P}_\\mathbf{L}  \\right\\|_A \\leq  \\left\\|  \\mathcal{V}_\\mathbf{L}   \\right\\|_B.\n\t\t\\end{equation}\n\t    \\item Case (iii)\n\t    \\begin{equation}\\label{case1.1}\n\t    \t\\| g \\|_A \\geq \\gamma \\| \\partial_{\\nu}u  \\|_A;\n\t    \\end{equation}\n\t\\end{itemize}\n\tThe constant $\\gamma$ and $\\delta$ are small numbers which will be chosen later.\n\t\n\t\n\tFor Case (i),\n\tit yields from \\eqref{case2Re} that\n\t\\begin{equation}\\label{case2.1.1}\n\t\t\\begin{aligned}\n\t\t&\\delta \\left\\| \\mathcal{P}_\\mathbf{L}  \\right\\|_A^2  + \\gamma \\pi \\left\\| \\left(  \\partial_{\\nu}u \\right)_n  \\right\\|_A \\|u\\|_B  \\\\\n\t\t&\\geq  \\left(\\sum_{n \\in \\mathbf{H}} \\left( \\partial_{\\nu}u \\right)_n  \\bar{u}_n + \\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} v\\right)_{n}\\overline{v_{n}}  \\right)\n\t\t+\\Lambda \\sum_{\\vert n \\vert>k}  \\sqrt{n^2-{k} ^{2} }  \\vert \\hat{u} _n \\vert^2.\n\t\t\\end{aligned}\n\t\\end{equation}\n\tFor the second term in \\eqref{case2.1.1}, Case (i) and Lemma \\ref{Lem:H} yield\n\t\\begin{equation}\\label{case2.1.2}\n\t\t\\begin{aligned}\n\t\t\t\\gamma \\pi \\left\\| \\left( \\partial_{\\nu}u \\right)_n  \\right\\|_A \\|u\\|_B\n\t\t\t\\leq & \\gamma \\pi \\left( \\left\\| \\mathcal{P}_\\mathbf{L}  \\right\\|_A + C \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B \\right) \\times \\left( \\delta \\left\\| \\mathcal{P}_\\mathbf{L}  \\right\\|_A + C \\left\\| \\mathcal{V}_\\mathbf{H}  \\right\\|_B \\right)\\\\\n\t\t\t\\leq & C' \\left( \\gamma \\delta \\left\\| \\mathcal{P}_\\mathbf{L}  \\right\\|_A^2 +  \\gamma^2 \\left\\| \\mathcal{P}_\\mathbf{L}  \\right\\|_A^2 + \\frac{1}{2} \\left\\| \\mathcal{V}_\\mathbf{H}  \\right\\|_B^2 \\right).\n\t\t\\end{aligned}\n\t\\end{equation}\n\tSince\n\t\\begin{equation}\\label{case2.1frac12}\n\t\t\\gamma^2 \\| \\partial_{\\nu}u  \\|_A^2\n\t\t\\geq  \\| \\partial_{\\nu}u -T(u) \\|_A^2\n\t\t\\geq  \\left\\| \\mathcal{P}_\\mathbf{L}  - \\mathcal{T}_\\mathbf{L}   \\right\\|_A^2\n\t\t\\geq   \\frac{1}{2} \\left\\| \\mathcal{P}_\\mathbf{L} \\right\\|_A^2 - \\left\\| \\mathcal{T}_\\mathbf{L} \\right\\|_A^2,\n\t\\end{equation}\n\tone has\n\t\\begin{equation}\\label{case2.1.3}\n\t\t\\frac{1}{2} \\left\\| \\mathcal{P}_\\mathbf{L} \\right\\|_A^2  \\leq  \\left\\| \\mathcal{T}_\\mathbf{L} \\right\\|_A^2 + \\gamma^2 \\| \\partial_{\\nu}u  \\|_A^2\n\t\t\\leq \\left\\| \\mathcal{T}_\\mathbf{L} \\right\\|_A^2  + \\gamma^2 \\left( \\left\\| \\mathcal{P}_\\mathbf{L} \\right\\|_A^2 + C \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B^2 \\right).\n\t\\end{equation}\n\tBy \\eqref{case2.1.1}, \\eqref{case2.1.2} and \\eqref{case2.1.3},\n\tchoosing\n\t\\begin{equation}\\label{gamma}\n\t\t\\gamma=C_{(i)}\\sqrt{\\delta},\n\t\\end{equation}\n\twhere $C_{(i)}$ is an  arbitrarily sufficiently small constant independent of $k$ and $b$, it can be deduced  that\n\t\\begin{equation}\\label{Tu_L}\n\t\t\\left\\| \\mathcal{T}_\\mathbf{L} \\right\\|_A^2 \\geq \\dfrac{C_1}{\\delta} \\left( \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B^2 + \\Lambda \\sum_{\\vert n \\vert>k}  \\sqrt{n^2-{k} ^{2} }  \\vert \\hat{u} _n \\vert^2 \\right).\n\t\\end{equation}\n\t\n\n\tIn order to get a bound for $\\left\\| \\mathcal{T}_\\mathbf{L} \\right\\|_A^2$, it is sufficient to estimate $(Tu)_n$ and $(Tv)_n$ for $n\\leq k$ due to the definition of $\\mathbf{L}$.\n\tBy the Cauchy-Schwarz inequality, one has\n\t\\begin{equation}\n\t\t\\begin{aligned}\n\t\t\t&  \\sum_{\\vert n \\vert<k}  \\sqrt{{k} ^{2} -n^2}   \\vert \\hat{u} _n \\vert^2  +   \\sum_{\\vert n \\vert>k}  \\sqrt{n^2-{k} ^{2} }  \\vert \\hat{u} _n \\vert^2 \\\\\n\t\t\t\\geq & \\lvert  \\sum_{n\\in \\mathbb{Z} }  \\sqrt{{k} ^{2} -n^2}   \\hat{u} _n \\overline{\\widehat{\\sin(n x_1)} } \\rvert^2 \\left( \\sum_{n\\in \\mathbb{Z} }  \\sqrt{\\vert{k} ^{2} -n^2\\vert}   \\lvert \\widehat{\\sin(n x_1)} \\rvert^2   \\right)^{-1}  \\\\\n\t\t\t\\geq & \\lvert (Tu)_n \\rvert^2 \\left( \\sum_{n\\in \\mathbb{Z} }  \\sqrt{\\vert{k} ^{2} -n^2\\vert}   \\lvert \\widehat{\\sin(n x_1)} \\rvert^2 \\right)^{-1}  \\\\\n\t\t\t\\geq & \\frac{1} {2k}  \\lvert (Tu)_n \\rvert^2 \\quad n < k,\n\t\t\\end{aligned}\n\t\\end{equation}\n\tand\n\t\\begin{equation}  \\label{CSv}\n\t\t\\begin{aligned}\n\t\t\t& \\sum_{\\vert n \\vert<k}  \\sqrt{{k} ^{2} -n^2}   \\vert \\hat{u} _n \\vert^2  +   \\sum_{\\vert n \\vert>k}  \\sqrt{n^2-{k} ^{2} }  \\vert \\hat{u} _n \\vert^2 \\\\\n\t\t\t\\geq & \\lvert  \\sum_{n\\in \\mathbb{Z} }  \\sqrt{{k} ^{2} -n^2} \\hat{u} _n \\overline{\\widehat{\\cos(n x_1)} }   \\rvert^2 \\left( \\sum_{n\\in \\mathbb{Z} }  \\sqrt{\\vert{k} ^{2} -n^2\\vert}  \\lvert \\widehat{\\cos(n x_1)} \\rvert^2   \\right)^{-1}  \\\\\n\t\t\t\\geq & \\lvert (Tv)_n \\rvert^2 \\left( \\sum_{n\\in \\mathbb{Z} }  \\sqrt{\\vert{k} ^{2} -n^2\\vert}  \\lvert \\widehat{\\cos(n x_1)} \\rvert^2 \\right)^{-1}  \\\\\n\t\t\t\\geq & \\frac{1} {2k}  \\lvert (Tv)_n \\rvert^2 \\quad n < k.\n\t\t\\end{aligned}\n\t\\end{equation}\n\t\n\tNote that\n\t\\begin{equation}\\label{LCy0}\n\t\\begin{aligned}\n\t\t\\sum_{n\\in\\mathbf{L}} \\frac{1}{\\sqrt{\\lvert k^{2}-n^{2}\\rvert}+\\frac{1}{b}}  &= \\sum_{n=0}^{\\lfloor k \\rfloor-1}\\frac{1}{\\sqrt{\\lvert k^{2}-n^{2}\\rvert}+\\frac{1}{b}} +  \\frac{1}{\\sqrt{\\lvert k^{2}-\\lfloor k \\rfloor^2  \\rvert}+\\frac{1}{b}} \\\\\n\t\t&\\leq \\displaystyle\\int_0^{\\lfloor k \\rfloor} \\frac{1}{\\sqrt{\\lvert k^{2}-x^{2}\\rvert}+\\frac{1}{b}} \\mbox{d}x + b \\\\\n\t\t&\\leq C + b \\leq {C_2}b\n\t\\end{aligned}\t\n\t\\end{equation}\n\tsince $b>1$.\n\tA combination of the above estimate \\eqref{LCy0} with \\eqref{Tu_L}-\\eqref{CSv} yields\n\t\\begin{equation} \\label{case2.1leqgeq}\n\t\t\\begin{aligned}\n\t\t\t& C_2  b k \\left(  \\Lambda \\sum_{\\vert n \\vert<k}  \\sqrt{{k} ^{2} -n^2}   \\vert \\hat{u} _n \\vert^2  +  \\Lambda \\sum_{\\vert n \\vert>k}  \\sqrt{n^2-{k} ^{2} }  \\vert \\hat{u} _n \\vert^2 \\right) \\\\\n\t\t\t\\geq & \\sum_{n\\in\\mathbf{L} }  \\frac{1} {\\sqrt{\\lvert {k} ^{2} -n^{2} \\rvert} +\\frac{1} {b} }  \\left( \\lvert (Tu)_n \\rvert^2 + \\lvert (Tv)_n \\rvert^2  \\right) \\\\\n\t\t\t= & \\left\\| \\mathcal{T}_\\mathbf{L}  \\right\\|_A^2 \\\\\n\t\t\t\\geq & \\frac{C_1} {\\delta}  \\left( \\left\\|\\mathcal{V}_\\mathbf{H}\\right\\|_B^2 +  \\Lambda \\sum_{\\vert n \\vert>k}  \\sqrt{n^2-{k} ^{2} }  \\vert \\hat{u} _n \\vert^2 \\right) \\\\\n\t\t\t\\geq & \\frac{C_1} {\\delta}  \\Lambda \\sum_{\\vert n \\vert>k}  \\sqrt{n^2-{k} ^{2} }  \\vert \\hat{u} _n \\vert^2.\n\t\t\\end{aligned}\n\t\\end{equation}\n\tBy choosing\n\t\\begin{equation}\\label{delta}\n\t\t\\delta = \\frac{C_{(ii)}}{ b k},\n\t\\end{equation}\n\twith $C_{(ii)} = \\frac{2 C_1}{1 + 2 C_2}>0$, which is independent of $k$ and $\\varepsilon$, one has\n\t\\begin{equation}\\label{ineq:Cnk}\n\t\tC  \\sum_{\\vert n \\vert<k}  \\sqrt{{k} ^{2} -n^2}   \\vert \\hat{u} _n \\vert^2  \\geq \\sum_{\\vert n \\vert>k}  \\sqrt{n^2-{k} ^{2} }  \\vert \\hat{u} _n \\vert^2.\n\t\\end{equation}\n\tApplying the above estimate \\eqref{ineq:Cnk}  and the choice of $\\delta$ \\eqref{delta} to \\eqref{case2.1leqgeq}, one has\n\t\\begin{equation}\\label{ineq:bk}\n\t\tC  \\sum_{\\vert n \\vert<k}  \\sqrt{{k} ^{2} -n^2}   \\vert \\hat{u} _n \\vert^2 \\geq  \\left\\|\\mathcal{V}_\\mathbf{H}\\right\\|_B^2 +  \\Lambda \\sum_{\\vert n \\vert>k}  \\sqrt{n^2-{k} ^{2} }  \\vert \\hat{u} _n \\vert^2.\n\t\\end{equation}\n\tBy combining \\eqref{case2Im}, \\eqref{ineq:bk}, Lemma \\ref{Lem:sqrtk} and the proposition \\eqref{ineq:normB} of norm $B$, one gets\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\tC  k \\lvert  \\int_{\\Gamma}  g \\bar{u}  \\mathrm{d} x_1\\rvert\n\t\t\n\t\t\t\\geq &  k \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B^2 + C'' \\sqrt{\\varepsilon}   k \\sqrt{k}  \\|u\\|_{L^2(\\Gamma)} ^2\n\t\t\t\\geq  k \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B^2 + \\sqrt{\\varepsilon}   \\sqrt{k}   \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B^2\n\t\t\t\\geq  \\sqrt{\\varepsilon}   \\sqrt{k}   \\|u\\|_B^2.\n\t\t\\end{aligned}\n\t\\end{equation*}\n\tHence,\n\t\\begin{equation}\\label{case21uB}\n\t\t\\dfrac{ C \\sqrt{k} } {\\sqrt{\\varepsilon} }  \\|g\\|_A \\geq \\|u\\|_B.\n\t\\end{equation}\n\t\n\tMoreover, since Lemma \\ref{Lem:H} indicates\n\t\\begin{equation}\\label{case2.1.H}\n\t\t\\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B^2 \\geq C \\left\\| \\mathcal{P}_\\mathbf{H} \\right\\|_A^2,\n\t\\end{equation}\n\tand \\eqref{case2.1frac12} implies\n\t\\begin{equation}\\label{case2.1frac12.2}\n\t\t\\left\\| \\mathcal{T}_\\mathbf{L} \\right\\|_A^2 \\geq \\frac{1}{2} \\left\\| \\mathcal{P}_\\mathbf{L} \\right\\|_A^2 - \\gamma^2 \\| \\partial_{\\nu}u  \\|_A^2,\n\t\\end{equation}\n\ta combination of the above \\eqref{case2.1.H}-\\eqref{case2.1frac12.2} with \\eqref{case2Im},  \\eqref{case2.1leqgeq} and Lemma \\ref{Lem:H} yields\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\tC b k \\pi \\|g\\|_A \\|u\\|_B\n\t\t\t\\geq  C b k \\lvert \\int_{\\Gamma} g \\bar{u} \\mathrm{d} x_1\\rvert\n\t\t\t\\geq  C b k \\Lambda \\sum_{\\vert n \\vert<k}  \\sqrt{{k} ^{2} -n^2}   \\vert \\hat{u} _n \\vert^2\n\t\t\n\t\t\t\\geq   \\sqrt{\\varepsilon} \\left\\| \\partial_{\\nu}u \\right\\|_A^2.\n\t\t\\end{aligned}\n\t\\end{equation*}\n\tTherefore, using \\eqref{case21uB}, one has\n\t\\begin{equation*}\n\t\t\\dfrac{C b {k} ^{\\frac{3} {2} }     } {\\varepsilon } \\|g\\|_A^2 \\geq \\left\\| \\partial_{\\nu}u \\right\\|_A^2.\n\t\\end{equation*}\n\tHence,\n\t\\begin{equation}\\label{case21puA}\n\t\t\\dfrac{C b^{\\frac{1} {2} } {k} ^{\\frac{3} {4} }     } {\\sqrt{\\varepsilon} } \\|g\\|_A \\geq \\left\\| \\partial_{\\nu}u \\right\\|_A.\n\t\\end{equation}\n\tThe estimate \\eqref{case21uB} and \\eqref{case21puA} indicate that\n\t\\begin{equation}\\label{case21result}\n\t\t\\dfrac{C b^{\\frac{1} {2} } {k} ^{\\frac{3} {4} }     } {\\sqrt{\\varepsilon} } \\|g\\|_A \\geq \\left\\| \\partial_{\\nu}u \\right\\|_A + \\|u\\|_B.\n\t\\end{equation}\n\t\n\tFor Case (ii),\n\tApplying Lemma \\ref{Lem:H} to \\eqref{case2Re}, one has\n\t\\begin{equation}\\label{case2.2.0.1}\n\t\t\\frac{1}{\\delta} \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B^2 + \\gamma\\pi \\| \\partial_{\\nu}u \\|_A \\|u\\|_B  \\geq C \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B^2 + \\Lambda \\sum_{\\vert n \\vert>k}  \\sqrt{n^2-{k} ^{2} }  \\vert \\hat{u} _n \\vert^2.\n\t\\end{equation}\n\tFor the second term in \\eqref{case2.2.0.1}, Case (ii) and Lemma \\ref{Lem:H} yield\n\t\\begin{equation}\\label{case2.2.0.2}\n\t\t\\begin{aligned}\n\t\t\t\\gamma\\pi \\|\\partial_{\\nu}u \\|_A \\|u\\|_B\n\t\t\t\\leq & \\gamma\\pi \\left( \\frac{1}{\\delta} \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B + C \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B \\right)  \\times \\left( \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B + \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B \\right) \\\\\n\t\t\t\\leq & C \\frac{\\gamma}{\\delta} \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B^2 + C \\gamma^2 \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B^2 + \\frac{1}{2}\\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B^2.\n\t\t\\end{aligned}\n\t\\end{equation}\n\tSince $\\gamma=C_{(i)}\\sqrt{\\delta}$ is sufficiently small, the estimate \\eqref{case2.2.0.1} and \\eqref{case2.2.0.2} indicate that\n\t\\begin{equation}\\label{case2.2.0}\n\t\t\\frac{C}{\\delta} \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B^2\n\t\t\\geq \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B^2 + \\Lambda \\sum_{\\vert n \\vert>k}  \\sqrt{n^2-{k} ^{2} }  \\vert \\hat{u} _n \\vert^2  \\geq \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B^2.\n\t\\end{equation}\n\tOn the other hand, for the lower frequency, it is obvious that\n\t\\begin{equation*}\n\t\tC   \\Lambda \\sum_{\\vert n \\vert<k}  \\sqrt{{k} ^{2} -n^2}   \\vert \\hat{u} _n \\vert^2  \\geq \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B^2.\n\t\\end{equation*}\n\tHence, it follows from \\eqref{case2Im} that\n\t\\begin{equation*}\n\t\tC  \\lvert \\int_{\\Gamma}  g \\bar{u}  \\mathrm{d} x_1\\rvert  \\geq C   \\Lambda \\sum_{\\vert n \\vert<k}  \\sqrt{{k} ^{2} -n^2}   \\vert \\hat{u} _n \\vert^2   \\geq \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B^2.\n\t\\end{equation*}\n\tThus\n\t\\begin{equation}\\label{case2.2.1}\n\t\tC   \\|g\\|_A \\|u\\|_B \\geq \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B^2.\n\t\\end{equation}\n\tBy \\eqref{case2.2.0} and $\\delta=\\dfrac{C_{(ii)}}{b k}$, one has\n\t\\begin{equation*}\n\t\tC  b k \\|g\\|_A \\|u\\|_B \\geq \\left\\| u \\right\\|_B^2.\n\t\\end{equation*}\n\tThus\n\t\\begin{equation}\\label{case2.2.2}\n\t\tC  b k \\|g\\|_A \\geq \\left\\| u \\right\\|_B.\n\t\\end{equation}\n\t\n\tNote that\n\t\\begin{equation}\\label{case2.2.3}\n\t\t\\|u\\|_B \\geq \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B \\geq \\dfrac{C_{(ii)}}{ bk} \\left\\| \\mathcal{P}_\\mathbf{L}  \\right\\|_A,\n\t\\end{equation}\n\tdue to the assumption of Case (ii).\n\tApplying \\eqref{case2.2.3} to \\eqref{case2.2.1}, one gets\n\t\\begin{equation*}\n\t\tC  b^2 {k} ^2\\|g\\|_A \\|u\\|_B \\geq \\left\\| \\mathcal{P}_\\mathbf{L}  \\right\\|_A^2.\n\t\\end{equation*}\n\tUsing \\eqref{case2.2.0} and Lemma \\ref{Lem:H}, this implies that\n\t\\begin{equation}\\label{case2.2.4}\n\t\tC b^2  {k} ^2 \\|g\\|_A \\|u\\|_B \\geq \\|\\partial_{\\nu}u \\|_A^2.\n\t\\end{equation}\n\tCombining the above estimate \\eqref{case2.2.4} with \\eqref{case2.2.2} yields\n\t\\begin{equation}\\label{case2.2.5}\n\t\tC   b^{\\frac{3} {2} } {k} ^{\\frac{3} {2} }  \\|g\\|_A \\geq \\|\\partial_{\\nu}u \\|_A.\n\t\\end{equation}\n\tThe estimate \\eqref{case2.2.2} and \\eqref{case2.2.5} indicate that\n\t\\begin{equation}\\label{case22result}\n\t\tC b^{\\frac{3} {2} } {k} ^{\\frac{3} {2} }    \\|g\\|_A \\geq \\|\\partial_{\\nu}u \\|_A + \\|u\\|_B.\n\t\\end{equation}\n\n\tFor Case (iii), it suffices to estimate $\\|u\\|_B$.\n\tUsing the definition of $T(u)$, the property of the norm defined above and Lemma \\ref{Lem:sqrtk},\n\tone obtains\n\t\\begin{equation} \\label{ineq1}\n\t\t\\begin{aligned}\n\t\t\t\\left( 1+\\frac{1} {\\gamma}  \\right)  \\sqrt{k}  \\pi \\|g\\|_A \\|u\\|_B\n\t\t\t& \\geq \\sqrt{k}  \\pi \\|T(u)\\|_A \\|u\\|_B \\\\\n\t\t\n\t\t\t& \\geq  \\sqrt{k}  \\lvert  \t\t\\Lambda \\sum_{n\\in \\mathbb{Z} }  \\sqrt{{k} ^{2} -n^2}  \\vert \\hat{u} _n \\vert^2 \t\t\\rvert\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\t\\left( \\geq \\sqrt{k}  \\lvert\n\t\t\t\\frac{\\Lambda} {2}  \\sum_{n\\in \\mathbb{Z} }  \\sqrt{\\vert {k} ^{2} -n^2\\vert}  \\vert \\hat{u} _n \\vert^2\n\t\t\t\\rvert \\right) \\\\\n\t\t\t& \\geq \\dfrac{\\sqrt{\\varepsilon} } {2}  k\\|u\\|_{L^2(\\Gamma)} ^2.\n\t\t\\end{aligned}\n\t\\end{equation}\n\tSince $k \\geq \\sqrt{k^2-n^2}$  for $n\\in \\mathbf{L}$, it is obvious that there exists a constant $C$ such that\n\t\\begin{equation}\\label{kinL}\n\t\tCk \\geq \\sqrt{\\lvert k^{2}-n^{2}\\rvert}+ \\frac{1}{b}.\n\t\\end{equation}\n\tApplying  \\eqref{kinL} to \\eqref{ineq1}, the bound for $n\\in\\mathbf{L}$ can be deduced:\n\t\\begin{equation}\\label{uL}\n\t\tC \\dfrac{\\sqrt{k}  } {\\sqrt{\\varepsilon}  \\gamma}  \\|g\\|_A \\|u\\|_B\n\t\t\\geq \\sum_{n \\in \\mathbf{L}} \\left(\\sqrt{\\lvert k^{2}-n^{2}\\rvert}+ \\frac{1}{b}\\right) \\left( \\vert u_n \\vert^2 + \\vert v_n \\vert^2 \\right) = \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B^2.\n\t\\end{equation}\n\tUsing Lemma \\ref{Lem:H}, the bound for $n\\in \\mathbf{H}$ can be deduced:\n\t\\begin{equation}\\label{uH}\n\t\t\\frac{1}{\\gamma^2} \\|g\\|_A^2 \\geq \\| \\partial_{\\nu}u  \\|_A^2 \\geq \\left\\| \\mathcal{P}_\\mathbf{H} \\right\\|_A^2 \\simeq \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B^2.\n\t\\end{equation}\n\tCombining \\eqref{uL} and \\eqref{uH} yields\n\t\\begin{equation*}\n\t\tC \\left( \\frac{1}{\\gamma^2} \\|g\\|_A^2+\\dfrac{\\sqrt{k}  } {\\sqrt{\\varepsilon}  \\gamma}\\|g\\|_A \\|u\\|_B \\right) \\geq \\|u\\|_B^2.\n\t\\end{equation*}\n\tSolving the second order polynomial above, one has\n\t\\begin{equation}\\label{case1.2}\n\t\tC  \\dfrac{\\sqrt{k}  } {\\sqrt{\\varepsilon}  \\gamma}  \\|g\\|_A \\geq \\|u\\|_B.\n\t\\end{equation}\n\tTherefore, for Case (i), one obtains from \\eqref{case1.1} and \\eqref{case1.2} that\n\t\\begin{equation}\\label{case1result}\n\t\tC  \\dfrac{\\sqrt{k}  } {\\sqrt{\\varepsilon} \\gamma} \\|g\\|_A \\geq \\| \\partial_{\\nu}u  \\|_A + \\|u\\|_B.\n\t\\end{equation}\t\n\tApplying choice of $\\gamma$ \\eqref{gamma} and $\\delta$ \\eqref{delta} to the estimate \\eqref{case1result} immediately yields that\n\t\\begin{equation} \\label{case1resultnew}\n\t\tC  \\dfrac{ b^{\\frac{1} {2} } k } {\\sqrt{\\varepsilon} }  \\|g\\|_A \\geq \\| \\partial_{\\textbf{n} } u  \\|_A + \\|u\\|_B.\n\t\\end{equation}\n\t\n\t\n\tFinally, we complete the proof of Lemma \\ref{lem:step2} based on the above proofs for Case (i)-(iii).\n\tCombining the bounds  \\eqref{case21result} , \\eqref{case22result} and \\eqref{case1resultnew} for Case (i)-(iii) respectively, one arrives\n\t\\begin{equation*}\n\t\t\\| \\partial_{\\textbf{n} } u \\|_A + \\|u\\|_B \\leq C \\max \\left\\{ \\dfrac{ b^{\\frac{1} {2} } k  } {\\sqrt{\\varepsilon} }\n\t\t, b^{\\frac{3} {2} } k ^{\\frac{3} {2} }   \\right\\}   \\|g\\|_A,\n\t\\end{equation*}\n\twhich completes the proof.\n\\end{proof}\n\n\n\\subsection{Proof of Theorem \\ref{thm:1}}\n\\label{sec:proofthm1}\n\nNow we are ready to prove the bounds in Theorem \\ref{thm:1}.\n\n\\begin{proof}\n\tLemma \\ref{lem:step2} yields that: Under Assumption \\ref{assump:k},\n\tthere exists a constant $C$ independent of $b>1$ and large $k$, such that\n\t\\begin{equation*}\n\t\t\\| \\partial_{\\nu}u \\|_A + \\|u\\|_B \\leq C \\max \\left\\{ \\dfrac{  b^{\\frac{1} {2} } k } {\\sqrt{\\varepsilon} }\n\t\t,  b^{\\frac{3} {2} } {k} ^{\\frac{3} {2} }    \\right\\}  \\|g\\|_A .\n\t\\end{equation*}\n\tFurthermore, Lemma \\ref{lem:step1} gives\n\t\\begin{equation*}\n\t\t\\|\\nabla u \\|_{L^2(D)} + k \\|u\\|_{L^2(D)}  \\leq C  b k \\left( \\| \\partial_{\\nu}u \\|_A + \\|u\\|_B \\right) .\n\t\\end{equation*}\n\tTherefore,\n\t\\begin{equation}\\label{eq:2.1}\n\t\t\\| \\partial_{\\nu}u \\|_A + \\|u\\|_B \\leq C  \\max \\left\\{ \\dfrac{ b^{\\frac{3} {2} } {k} ^2  } {\\sqrt{\\varepsilon} }\n\t\t,  b^{\\frac{5} {2} }  {k} ^{\\frac{5} {2} }   \\right\\} \\|g\\|_A .\n\t\\end{equation}\n\tFrom the definition and the property of the norm $A$ mentioned in Definition \\ref{def:AB} and Remark \\ref{rem:AL},\n\t\\begin{equation}\\label{eq:2.2}\n\t\t\\|g\\|_A \\leq C \\sqrt{b} \\|g\\|_{L^2(\\Gamma)} .\n\t\\end{equation}\n\t\n\tApplying \\eqref{eq:2.2} to \\eqref{eq:2.1}, one derives the bounds in Theorem \\ref{thm:1}, which completes the proof.\n\\end{proof}\n\n\n\\section{Stability for the random case}\n\\label{sec:thm23}\n\nIn this section, Theorem \\ref{thm:3} is proved by  utilizing the general framework developed in  \\cite{Pembery2020} based upon the stability in Theorem \\ref{thm:1} for the deterministic case.\nNote that the randomness of integral domain prevents direct application of general framework in  \\cite{Pembery2020}.\nThus the difficulty lies in that the scattering surface in model problem is\nwith randomness, which implies the stochastic domain for the scattering problem.\nFor this, a variable transform, which changes the random variational form with stochastic domains into a transformed one with a definite domain and random medium, is introduced to prove all necessary propositions, including the continuity of  sesquilinear form $\\tilde{a}$ and antilinear functional $\\tilde{G}$, and stability and uniqueness which hold almost surely, which implies the stability for the random case.\n\n\n\\subsection{Prerequisites}\n\\label{sec:4.preliminary}\n\nDenote $\\tilde{a}_{c(\\omega)}$ and $\\tilde{G}_{c(\\omega)}$ (defined by \\eqref{aw} and \\eqref{Gw} in Sec \\ref{sec:variational}) by $(\\tilde{a} \\circ c)(\\omega)$ and $(\\tilde{G} \\circ c)(\\omega)$, respectively.\nDefine the norm $\\|v\\|^2_{1,k} := \\|\\nabla v \\|_{L^2(D)}^2 + k^2 \\|v\\|_{L^2(D)}^2$ and $\\|v\\|_{1,\\infty}:=\\|v\\|_{\\infty}+\\|v'\\|_{\\infty}$ on $H_{S,qp}^1(D)$.\nNecessary propositions required in the general framework \\cite{Pembery2020} include the continuity of $\\tilde{a}$ and $\\tilde{G}$, regularity of $\\tilde{a}\\circ c$ and $\\tilde{G}\\circ c$, measurability and $\\mu$-essentially separability of $c$, stability a.s. and uniqueness a.s., described in Proposition \\ref{prop:1} and Proposition \\ref{prop:2}.\n\n\\begin{proposition}\n\t\\label{prop:1}\n\t$\\tilde{a}$, $\\tilde{G}$ and $c$ in the variational form \\eqref{variational} of the scattering problem in a random periodic structure satisfies the following properties:\n\t\\begin{itemize}\n\t\t\\item[(i)] The function $c: \\Omega \\rightarrow \\mathcal{C}$ defined by \\eqref{def:c} is measurable and $\\mu$-essentially separably valued.\n\t\t\\item[(ii)]\n\t\tLet $B(H_{S,qp}^1(D), H_{S,qp}^1(D))$ denote the space of bounded linear maps $H_{S,qp}^1(D) \\rightarrow H_{S,qp}^1(D)$. The functions $\\tilde{a}: \\mathcal{C}\\rightarrow B(H_{S,qp}^1(D),H_{S,qp}^1(D))$ and $\\tilde{G}:\\mathcal{C}\\rightarrow H_{S,qp}^1(D)$ defined by \\eqref{aw} and \\eqref{Gw} are continuous,\n\t\tthe maps $\\tilde{a}\\circ c\\in L^{\\infty}(\\Omega;B(H_{S,qp}^1(D),H_{S,qp}^1(D)))$, $\\tilde{G}\\circ c\\in L^2(\\Omega;H_{S,qp}^1(D))$.\n\t\n\t\\end{itemize}\n\\end{proposition}\n\n\n\\begin{definition}\n\t(The solution operator $\\mathcal{U}$)\n\tDefine $\\mathcal{U}:\\mathcal{C}\\rightarrow H_{S,qp}^1(D)$ by letting $\\mathcal{U}(f_0) \\in H_{S,qp}^1(D)$ be the solution of the deterministic Helmholtz problem with sampling $f_0\\in\\mathcal{C}$.\n\\end{definition}\n\n\n\\begin{proposition} \\label{prop:2}\n\tFor $f_0\\in Lip$,  the solution of the variational form \\eqref{variational} exists and is unique.\n\tLet $v_0=\\mathcal{U}(f_0)$ be the solution. Under Assumption \\ref{assump:k}, there exists a constant $C$ independent of  $b>1$ and large $k$, such that\n\t\\begin{equation}\\label{Destimatelem}\n\t\t\\|\\nabla v_0 \\|_{L^2(D)} + k \\|v_0\\|_{L^2(D)} \\leq C \\max \\left\\{ \\dfrac{b^2 {k} ^{2} } {\\sqrt{\\varepsilon} } ,b^3 {k} ^{\\frac{5} {2} }   \\right\\}  \\|g\\|_{L^2(\\Gamma)}.\n\t\\end{equation}\n\tMoreover, the scattering problem \\eqref{eq:u_total} in a random periodic structure satisfies the following stability and uniqueness properties:\n\t\\begin{itemize}\n\t\t\\item[(i)] (Uniqueness almost surely) $ker(\\tilde{a}_{c(\\omega)})=\\{0\\}$ $\\mu$-almost surely.\n\t\t\\item[(ii)] (Stability almost surely) There exist $C_1, h_1:\\Omega\\rightarrow\\mathbb{R}$ such that $C_1 h_1\\in L^1(\\Omega)$ and the bound\n\t\t\\begin{equation}\n\t\t\t\\|u(\\omega)\\|_{1,k}^2\\leq C_1(\\omega) h_1(\\omega)\n\t\t\\end{equation}\n\t\tholds almost surely.\n\t\\end{itemize}\n\\end{proposition}\n\n\\begin{remark}\n\tFrom the general framework in  \\cite{Pembery2020}, one can conclude that (i) with properties of $\\tilde{a}$, $\\tilde{G}$ and $c$ stated in Proposition \\ref{prop:1}, the maps $\\mathcal{A}$ and $\\mathcal{G}$ defined by \\eqref{def:AA} and \\eqref{def:GG} are well-defined;\n\t(ii) with the almost surely uniqueness property in Proposition \\ref{prop:2}, the solution to the Helmholtz equation in the stochastic case is unique in $L^2(\\Omega;H_{S,qp}^1(D))$;\n\t(iii) with the almost surely stability property in Proposition \\ref{prop:2}, integrability and measurability implies the stochastic a priori bound.\n\\end{remark}\n\nIn order to verify Proposition \\ref{prop:1} and Proposition \\ref{prop:2}, we give the following four lemmas. Lemma \\ref{lem:Kirsch} means that the solution to the scattering problem depends continuously on the scattering surface.\nLemma \\ref{lem:Pettis} shows the equivalence between strong measurability and measurability as well as $\\mu$-essentially separability.\nLemma \\ref{lem:Bochner} shows the necessary and sufficient condition for Bochner integrability.\nLemma \\ref{u_welldefined} shows that the solution operator $\\mathcal{U}$ is well defined.\n\n\\begin{lemma}\\label{lem:Kirsch}\n\t(continuous dependence on the boundary curve $f$   \\cite{Kirsch1993}) Let $f\\in C^2(\\mathbb{R})$ be $2 \\pi$-periodic and $u\\in H_{S,qp}^1$ be the unique solution of the scattering problem \\eqref{eq:u_deter}. Let $K \\subset D$ be compact. Then there exists $\\gamma>0$ and $C>0$ both depending on $k, u^i, f$ and $K$, such that for all $2\\pi$-periodic $r\\in C^1(\\mathbb{R})$ with $\\|r-f\\|_{1,\\infty}\\leq\\gamma$, the unique solution $u_r$ of the scattering problem corresponding to $r$ satisfies\n\t$$ \\|u_r-u\\|_{H^1(K)} \\leq C \\|f-r\\|_{1,\\infty}.$$\n\tHere, $\\|q\\|_{1,\\infty}:=\\|q\\|_{\\infty}+\\|q'\\|_{\\infty}$ denotes the norm in $C^1[0,\\Lambda]$, where $\\Lambda$ is the period of the scattered structures.\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:Pettis}\n\t(Pettis measurability theorem\n\t(Proposition 2.15 in  \\cite{Ryan2002})). Let  $(\\Omega, \\mathcal{F}, \\mu)$ be a complete $\\sigma$-finite measure space. The following are equivalent for a function $f: \\Omega \\rightarrow X$ (i) $f$ is strongly measurable, (ii) $f$ is measurable and $\\mu$-essentially separably valued.\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:Bochner}\n\t(Bochner's Theorem (Proposition 2.16 in \\cite{Ryan2002})). If $f: \\Omega \\rightarrow X$ is a strongly measurable function, then $f$ is Bochner integrable if and only if the scalar function $\\|f\\|$ is integrable.\n\\end{lemma}\n\n\n\\begin{lemma}\\label{u_welldefined}\n\t( $\\mathcal{U}$ is well defined)\n\tFor $f_0\\in\\mathcal{C}$, the solution  $\\mathcal{U}$ of the scattering problem exists, is unique, and depends continuously on $f_0$.\n\\end{lemma}\n\n\\begin{proof}\t\n\tFor $f_0\\in\\mathcal{C}$, the scattering problem is a deterministic case satisfying the condition in Theorem \\ref{thm:1}. Based on the previous work in   \\cite{Bao1995} on the existence and uniqueness, it follows from Theorem \\ref{thm:1} and Lemma \\ref{lem:Kirsch} that the solution $\\mathcal{U}$ is  well defined.\n\\end{proof}\n\nHere the basic propositions on measure theory and Bochner spaces needed for the proof are omitted. See  \\cite{Bogachev2007} and  \\cite{Diestel1977} for more details.\n\n\n\\subsection{Proof of Proposition \\ref{prop:1} and Proposition \\ref{prop:2}}\\label{sec:4.verification}\n\nIn this section, we verify that the random periodic structure scattering problem has all necessary propositions required as in the general framework \\cite{Pembery2020}.\nProposition \\ref{prop:1} includes\nmeasurability  and\n$\\mu$-essentially separability of $c$, continuity of $\\tilde{a}  $ and $\\tilde{G}  $ and\nregularity of $\\tilde{a}  \\circ c$ and  $\\tilde{G}  \\circ c$.\nProposition \\ref{prop:2} includes the necessary\nstability and uniqueness which hold almost surely. To be specific, under the assumpiton of resonance exclusion and stochastic regularity of the scattering surface, a variable transform is introduced and theory of Bochner spaces such as Pettis measurability theorm and Bochner's theorem is used to complete the proof.\n\nFirst, we give the proof of Proposition \\ref{prop:1}.\n\n\\begin{proof}[Proof of Proposition~{\\upshape\\ref{prop:1}}]\n\n\tSince each of $f$ is a Lipschitz function, $f$ is strongly measurable. By Pettis measurability theorem (Lemma \\ref{lem:Pettis}), it follows that $c$ defined by Definition \\ref{def:c} is measurable and $\\mu$-essentially separably valued, so properties of $c$ in Proposition \\ref{prop:1} are satisfied.\n\t\n\n\tIn order to prove the continuity of $\\tilde{a}$ and $\\tilde{G}$, we need to show that if $(f_m)\\rightarrow(f_0)$ in $\\mathcal{C}$ then $\\tilde{a}(f_m)\\rightarrow \\tilde{a}(f_0)$ in $B(H_{S,qp}^1(D),H_{S,qp}^1(D))$, and similarly for $\\tilde{G}$. Since\n\t\\begin{equation}\\label{varia:f_m}\n\t\t\\tilde{a}_{f_m}(v_m,\\phi)=\\tilde{G}_{f_m}(\\phi),\n\t\\end{equation}\n\twhere\n\t\\begin{equation}\n\t\t\\tilde{a}_{f_m} =\\int_{D_{f_m}}  \\nabla v_m \\cdot \\nabla \\bar{\\phi}\n\t\t- k^2 \\int_{D_{f_m}} v_m \\bar{\\phi}\n\t\t- \\int_{\\Gamma} T \\tau v_m \\overline{\\tau \\phi}  , \\quad  \\tilde{G}_{f_m} = \\int_{\\Gamma} g \\overline{\\tau \\phi} ,\n\t\\end{equation}\n\tand\n\t\\begin{equation}\\label{varia:f_0}\n\t\t\\tilde{a}_{f_0}(v_0,\\phi)=\\tilde{G}_{f_0}(\\phi),\n\t\\end{equation}\n\twhere\n\t\\begin{equation}\\label{varia:f_0aG}\n\t\t\\tilde{a}_{f_0} =\\int_{D_{f_0}}  \\nabla v_0 \\cdot \\nabla \\bar{\\phi}\n\t\t- k^2 \\int_{D_{f_0}} v_0 \\bar{\\phi}\n\t\t- \\int_{\\Gamma} T \\tau v_0 \\overline{\\tau \\phi} , \\quad  \\tilde{G}_{f_0} = \\int_{\\Gamma} g \\overline{\\tau \\phi}.\n\t\\end{equation}\n\tOur goal is to  transform the volume integral in $D_{f_m}$ into an integral over $D_{f_0}$ by a suitable change of the variable $x$.\n\t\n\n\tChoose $\\gamma_0>0$ such that\n\t\\begin{equation}\\label{def:gamma}\n\t\t\\gamma_0<\\min\\{\\min\\{x_2:(x_1,x_2)\\in D_{f_0}\\}-f_0(x_1):x_1\\in\\mathbb{R}\\},\n\t\\end{equation} and a function $\\alpha\\in C^1(\\mathbb{R})$ with\n\t\\begin{equation}\\label{def:alpha}\n\t\t\\alpha(t) =\\left\\{ \\begin{array}{ll}\n\t\t\t1, & t \\leq \\gamma_0\/2 \\\\\n\t\t\t0, & t\\geq \\gamma_0.\n\t\t\\end{array} \\right..\n\t\\end{equation}\t\n\tFor $f_m$ such that $\\|f_m-f_0\\|_{1,\\infty} \\leq \\gamma_0$, define $\\mathcal{F}: D_{f_0} \\rightarrow D_{f_m}$ by\n\t$$\n\t\\mathcal{F}(y) = y_2 + \\alpha(y_2-f_0(y_1)) [f_m(y_1)-f_0(y_1)] \\hat{e}_2,\\quad y\\in D_{f_0},\n\t$$\n\twhere $\\hat{e}_2=(0,1)^T$.\t\n\tFor $\\|f_m-f_0\\|_{1,\\infty} \\leq \\gamma$ with sufficiently small $\\gamma\\leq \\gamma_0$, $\\mathcal{F}$ is a diffeomorphism and $\\mathcal{F}=\\mathcal{I}$ on $K$. The Jacobian $J_{\\mathcal{F}}$ of $\\mathcal{F}$ satisfies $J_{\\mathcal{F}}(y) = \\mathcal{I} + \\mathcal{O}(\\|f_m-f_0\\|_{1,\\infty})$ uniformly in $y\\in D_{f_0}$. For the inverse $\\mathcal{Q}$ of $\\mathcal{F}$, one also has $J_{\\mathcal{Q}}(y) = \\mathcal{I} + \\mathcal{O}(\\|f_m-f_0\\|_{1,\\infty})$. The change of variable $x=\\mathcal{F}(y)$ then transforms \\eqref{varia:f_m} into\n\t\\begin{equation}\\label{eq:varia_new}\n\t\t\\int_{D_{f_0}}\\left[\\sum_{i, j=1}^{2} b_{i j} \\frac{\\partial \\hat{v}_{m}}{\\partial y_{i}} \\frac{\\partial \\overline{\\hat{\\phi}}}{\\partial y_{j}}-k^{2} \\hat{v}_{m} \\overline{\\hat{\\phi}}\\right] \\operatorname{det} (J_{\\mathcal{F}}) \\mathrm{d} y-\\int_{\\Gamma} \\overline{\\tau \\hat{\\phi}} T \\tau \\hat{v}_{m} \\mathrm{d} s=\\int_{\\Gamma} g \\overline{\\tau \\hat{\\phi}} \\mathrm{d} s,\n\t\\end{equation}\n\twhere $\\hat{v}_{m}=v_{m} \\circ \\mathcal{F}, \\hat{\\phi}=\\phi \\circ \\mathcal{F}$ both in $H_{S,qp}^1(D_{f_0})$, and\n\t\\begin{equation}\\label{bij}\n\t\tb_{i j}(y)=\\sum_{l=1}^{2} \\frac{\\partial \\mathcal{Q}_{i}(x)}{\\partial x_{l}} \\frac{\\partial \\mathcal{Q}_{j}(x)}{\\partial x_{l}}\\rvert_{x=\\mathcal{F}(y)}, \\quad i, j = 1, 2.\n\t\\end{equation}\n\tThe left hand side of \\eqref{eq:varia_new} defines a sesquilinear form $\\tilde{a}_{f_m}(v,\\phi)$ on $H_{S,qp}^1(D_{f_0})$.\n\tSince $ \\operatorname{det} (J_{\\mathcal{F}}) = 1 + \\mathcal{O}(\\|f_m-f_0\\|_{1,\\infty}) $ and $ b_{ij}=\\delta_{ij}+\\mathcal{O}(\\|f_m-f_0\\|_{1,\\infty}) $, one concludes that\n\t$$\\lvert \\tilde{a}_{f_m}(v, \\phi)-\\tilde{a}_{f_0}(v, \\phi)\\rvert \\leq C\\|f_m-f_0\\|_{1, \\infty}\\|v\\|_{H^{1}\\left(D_{f_0}\\right)}\\|\\phi\\|_{H^{1}\\left(D_{f_0}\\right)} \\text { for all } v, \\phi \\in H_{S,qp}^1(D_{f_0}).$$\n\tHence if $(f_m)\\rightarrow(f_0)$ in $\\mathcal{C}$, then $\\tilde{a}_{f_m}\\rightarrow \\tilde{a}_{f_0}$ in $B(H_{S,qp}^1(D),H_{S,qp}^1(D))$. For $\\tilde{G}$, it follows from the definition \\eqref{varia:f_0}-\\eqref{varia:f_0aG}, the right hand side of \\eqref{eq:varia_new},  $\\hat{\\phi}=\\phi \\circ \\mathcal{F} $ and $\\phi=\\phi \\circ \\mathcal{I}$ that\n\tif $(f_m)\\rightarrow(f_0)$ in $\\mathcal{C}$, then $\\tilde{G}_{f_m}\\rightarrow \\tilde{G}_{f_0}$ in $H_{S,qp}^1(D)$.\n\t\n\n\tSince $c$ is strongly measurable and the map $\\tilde{a}$ is continuous,  $\\tilde{a}\\circ c $ is strongly measurable. Recall that the operator $T$ is continuous from $H^{1\/2}(\\Gamma)$ to $H^{-1\/2}(\\Gamma)$   \\cite{Nedelec2001}. For $v\\in H_{S,qp}^1(D)$, $\\phi\\in H_{S,qp}^1(D)$, one has\n\t\\begin{equation}\\label{eq:afomega}\n\t\t\\tilde{a}_{f(\\omega)}(v, \\phi) =\\int_{D_{f(\\omega)}}  \\nabla v \\cdot \\nabla \\bar{\\phi}\n\t\t- k^2 \\int_{D_{f(\\omega)}} v \\bar{\\phi}\n\t\t- \\int_{\\Gamma} T \\tau v \\overline {\\tau \\phi}.\n\t\\end{equation}\n\tSame as above, transform the volume integral in $D_{f(\\omega)}$ into an integral over $D_{f_0}$.\n\tFor $f_1=f(\\omega)$ such that $\\|f_1-f_0\\|_{1,\\infty} \\leq \\gamma_0$ (defined same as \\eqref{def:gamma}), define $\\mathcal{F}: D_{f_0} \\rightarrow D_{f_1}$ by\n\t$$\n\t\\mathcal{F}(y) = y_2 + \\alpha(y_2-f_0(y_1)) [f_1(y_1)-f_0(y_1)] \\hat{e}_2,\\quad y\\in D_{f_0},\n\t$$\n\twhere $\\hat{e}_2=(0,1)^T$ and $\\alpha$ is defined by \\eqref{def:alpha}.\t\n\tFor $\\|f_1-f_0\\|_{1,\\infty} \\leq \\gamma$ with sufficiently small $\\gamma \\leq \\gamma_0$, $\\mathcal{F}$ is a diffeomorphism and $\\mathcal{F}=\\mathcal{I}$ on $K$.  The change of variable $x=\\mathcal{F}(y)$ then transforms \\eqref{eq:afomega} into\n\t\\begin{equation}\n\t\t\\tilde{a}_{f(\\omega)}(\\hat{v}, \\phi)=\\int_{D}\\left[\\nabla \\hat{v}\\operatorname{det} (J_{\\mathcal{F}}) B \\nabla \\bar{\\phi}-k^{2}(\\operatorname{det} (J_{\\mathcal{F}}) \\hat{v} \\bar{\\phi}\\right] d y,\n\t\\end{equation}\n\twhere $\\hat{v}=v \\circ \\mathcal{F}, \\hat{\\phi}=\\phi \\circ \\mathcal{F}$ both in $H_{S,qp}^1(D_{f_0})$, and $B=(b_{ij})_{i,j=1,2}$ from \\eqref{bij}. \t\t\n\tSince $ \\operatorname{det} (J_{\\mathcal{F}}) = 1 + \\mathcal{O}(\\|f_m-f_0\\|_{1,\\infty}) $ and $ b_{ij}=\\delta_{ij}+\\mathcal{O}(\\|f_m-f_0\\|_{1,\\infty}) $, observe that the Cauchy-Schwarz inequality and properties of $T$ imply that there exists $C>0$ such that\n\t$$\n\t\\lvert \\tilde{a}_{f(\\omega)}(\\hat{v}, \\phi)\\rvert \\leq\n\tC \\|f(\\omega)-f_0\\|_{1,\\infty}^2 \\|v\\|_{H^1(D)} \\|\\phi\\|_{H^1(D)} \\text { for all } v, \\phi \\in H_{S,qp}^1(D_{f_0}),\n\t$$\n\tand hence $\\tilde{a}\\circ c \\in L^{\\infty}(\\Omega;B(H_{S,qp}^1(D),H_{S,qp}^1(D))) $.\n\t\n\n\tSince $c$ is strongly measurable and the map $\\tilde{G}$ is continuous,  $\\tilde{G}\\circ c $ is strongly measurable. It is clear that $\\| (\\tilde{G}\\circ c)(\\omega) \\|_{H_{S,qp}^1(D)} \\leq \\|g\\|_{L^2(D)}$, and thus $\\tilde{G}\\circ c \\in L^2(\\Omega;H_{S,qp}^1(D))$ since $g \\in L^2(\\Omega;L^2(D))$.\n\\end{proof}\n\nNow prove Proposition \\ref{prop:2}.\n\n\\begin{proof}[Proof of Proposition~{\\upshape\\ref{prop:2}}]\n\tFor $f_0\\in \\mathcal{C}$, it follows from Theorem \\ref{thm:1} and Lemma \\ref{lem:Kirsch} that the solution $\\mathcal{U}(f_0)$ of the variational problem exists, is unique, and has the $k$-explicit stability estimate \\eqref{Destimatelem}. \t\n\tUniqueness almost surely holds immediately from Lemma \\ref{u_welldefined}.\n\t\n\tFor stability almost surely, choose\n\t$C_1 =C'  \\max \\left\\{ \\dfrac{b^2 {k} ^{2} } {\\sqrt{\\varepsilon} } ,b^3 {k} ^{\\frac{5} {2} }   \\right\\} $ with $C'=\\sup_\\Omega {C}$, $C$ being the previous constant in \\eqref{Destimatelem}, and $h_1 = \\|g\\|_{L^2(\\Gamma)}$.\n\tIt remains to show that $C_1 h_1 \\in L^1(\\Omega)$.\n\tWe first show that $C_1 h_1$ is measurable and then show that it lies in $ L^1(\\Omega)$.\n\t\n\tTo conclude $C_1$ is measurable, use the fact that the product, sum and maximum of two measurable functions are measurable. Since $g$ is measurable and the map $g\\mapsto \\|g\\|_{L^2(\\Gamma)}^2$ is clearly continuous, $h_1$ is measurable. As the product of two measurable functions is measurable, it follows that $C_1 h_1$ is measurable.\n\t\n\tNow show that $C_1 h_1 \\in L^1(\\Omega)$. Using the Cauchy-Schwarz inequality yields\n\t\\begin{eqnarray} \\label{C1h1L1}\n\t\t\\left\\|C_{1}h_1\\right\\|_{L^{1}(\\Omega)}=\\int_{\\Omega} C_{1}(\\omega)h_1 (\\omega) \\mathrm{~d} \\mathbb{P}(\\omega)\\leq \\left\\|C_{1}\\right\\|_{L^{1}(\\Omega)}\\left\\|\\left\\|g\\right\\|_{L^{2}\\left(\\Gamma\\right)}^{2}\\right\\|_{L^{1}(\\Omega)}<\\infty.\n\t\\end{eqnarray}\n\tTherefore $C_1 h_1 \\in L^1(\\Omega)$ as required. Integrating \\eqref{Destimatelem} in the probability space and using \\eqref{C1h1L1} obviously yield \\eqref{Sestimate}.\n\\end{proof}\n\n\n\\subsection{Proof of Theorem \\ref{thm:3}}\n\\label{sec:4.proof}\n\nBefore the proof, the continuity of the solution operator $\\mathcal{U}$ is given in the following lemma.\n\n\\begin{lemma}\\label{u_continuity}\n\t(Continuity of $\\mathcal{U}$)\n\tFor the scattering problem by a random periodic structure, the solution operator $\\mathcal{U}:\\mathcal{C}\\rightarrow H_{S,qp}^1(D)$ is continuous.\n\\end{lemma}\n\\begin{proof}\n\tLet $f_0, f_1 \\in \\mathcal{C}$, with $\\mathcal{U}(f_0)=u_0$ and $\\mathcal{U}(f_1)=u_1$. Then for any $v\\in H_{S,qp}^1(D)$,\n\t$$\\tilde{a}_{f_j}(u_j,v)=\\tilde{G}_{f_j}(v),\\ j=0,1.$$\n\t\n\tSince one has\n\t$$\\lvert \\tilde{a}_{f_1}(u, \\phi)-\\tilde{a}_{f_0}(u, \\phi)\\rvert \\leq C\\|f_1-f_0\\|_{1, \\infty}\\|u\\|_{H^{1}\\left(D_{f_0}\\right)}\\|\\phi\\|_{H^{1}\\left(D_{f_0}\\right)} \\text { for all } u, \\phi \\in H_{S,qp}^1(D_{f_0})$$\n\tfrom the proof of Proposition \\ref{prop:1},\n\twhere\n\tboth $\\tilde{a}_{f_1}(u, \\phi)$\n\tand $\\tilde{a}_{f_0}(u, \\phi)$ are sesquilinear forms,\n\tone can applies the general perturbation theory of variational equation \\cite{Kato1976} which yields\n\t$$\\|u_1\\circ\\mathcal{F} - u_0\\|_{H^1(D_{f_0})} = \\|\\hat{u}_1-u_0\\|_{H^1(D_{f_0})}\\leq C \\|f_m-f_0\\|_{1,\\infty},$$\n\tand thus, since $\\mathcal{F}=\\mathcal{I}$ on $D_{f_0}$,\n\t$$\\|u_1-u_0\\|_{H^1(K)} \\leq C \\|f_1-f_0\\|_{1,\\infty}.$$\n\tLet $u_d:=u_0-u_1$. It can be deduced that  $u_d\\rightarrow 0$ in $H_{S,qp}^1(D_{f_0})$ as $f_1 \\rightarrow f_0$ in $\\mathcal{C}$. This ends the proof of this lemma.\n\\end{proof}\n\nNow we are ready to prove Theorem \\ref{thm:3}.\n\n\\begin{proof}\n\tLet $u = \\mathcal{U} \\circ c$ (which\n\tis well-defined by Lemma \\ref{u_welldefined}). By construction, $a_{c(\\omega)}(u(\\omega), v)=G_{c(\\omega)}(v)$ for all $v \\in H_{S,qp}^1(D)$ almost surely.\n\tSince it follows from Assumption \\ref{cond:f} and Lemma \\ref{u_continuity} that $u$ is measurable, $u$ solves the variational problem.\n\t\n\tUnder Assumption \\ref{cond:f}, continuity of $\\tilde{a}  $ and $\\tilde{G}  $,\n\tregularity of $\\tilde{a}  \\circ c$ and  $\\tilde{G}  \\circ c$, and\n\tmeasurability  and\n\t$\\mu$-essentially separability of $c$ hold by Proposition \\ref{prop:1};\n\tstability a.s. and uniqueness a.s. hold by Proposition \\ref{prop:2}.\n\tMoreover, since there is no trapping cases for the scattering problem by a Lipschitz perfectly conductor, the nontrapping condition required in   \\cite{Pembery2020} is naturally satisfied.\n\tTherefore\n\tthe maps $\\mathcal{A}$ and $\\mathcal{G}$ (defined by \\eqref{def:AA} and \\eqref{def:GG}) are well-defined;\n\tthere exists $u\\in L^2(\\Omega;H^1_{S,qp}(D))$ being the solution to Problem \\ref{P3}; the solution to Problem \\ref{P3} is unique in $L^2(\\Omega;H^1_{S,qp}(D))$.\n\t\n\n\tCombining the stability for the scattering problem by deterministic periodic structures in Theorem \\ref{thm:1} with integrability and measurability properties on stochastic quantities verified by Proposition \\ref{prop:1} and stability and uniqueness which hold almost surely given by Proposition \\ref{prop:2} yields the well-posedness and the $k$-explicit stability for the scattering problem by random periodic structures, which completes the proof of Theorem \\ref{thm:3}.\n\\end{proof}\n\n\\begin{remark}\n\n\tIf the random structure is (uncertainty) quantified using the Karhunen-Lo\\`eve (KL) expansion as in   \\cite{BaoLinSINUM2020}, then the random structure can be represented by the following  Karhunen-Lo\\`eve expansion\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\tf(\\omega;x_1)&=\\tilde{f}(x_1) + \\displaystyle \\sum_{j=0}^{\\infty}\\sqrt{\\lambda_j} \\xi_{j}(\\omega)\\varphi_{j}(x_1),\n\t\t\\end{aligned}\n\t\\end{equation*}\n\twhere $\\tilde{f}(x_1)$ is a $\\Lambda$-periodic deterministic function, eigenvalues $\\{ \\lambda_j \\}_{j=0}^{\\infty}$ arranged in a descending order are corresponding to the orthonormal eigenfunctions $\\{ \\varphi_{j} \\}_{j=0}^{\\infty}$ of the covariance operator $C_f$\n\tand $\\{\\xi_j \\}_{j=0}^{\\infty}$ is a random variable with zero mean and unit covariance. The covariance function   \\cite{bookRandomSurface, bookcovariance} takes the following form\n\t\\begin{equation*}\n\t\tc(x_1-y_1)=\\sigma^2 \\exp(-\\dfrac{\\vert x_1-y_1\\vert^2}{l^2}), 0<l\\ll \\Lambda,\n\t\\end{equation*}\n\twhere $\\sigma$ is the root mean square of the surface and $l$ is the correlation length.\n\tIt follows that the KL expansion of the random process $f(\\omega;x_1)$ may be written as\n\t\\begin{equation*}\\label{eq:fKL}\n\t\t\\begin{aligned}\n\t\t\tf(\\omega;x_1)&=\\tilde{f}(x_1)+\\sqrt{\\lambda_0}\\xi_0(\\omega) \\sqrt{\\dfrac{1}{\\Lambda}} \\\\\n\t\t\t& + \\displaystyle \\sum_{j=1}^{\\infty}\\sqrt{\\lambda_j}\\left( \\xi_{j,s}(\\omega)\\sqrt{\\dfrac{2}{\\Lambda}} \\sin\\left(\\dfrac{2j\\pi x_1}{\\Lambda}\\right)\n\t\t\t+ \\xi_{j,c}(\\omega)\\sqrt{\\dfrac{2}{\\Lambda}} \\cos\\left(\\dfrac{2j\\pi x_1}{\\Lambda}\\right) \\right),\n\t\t\\end{aligned}\n\t\\end{equation*}\n\twhere $\\xi_0$, $\\xi_{j,s}$ and $\\xi_{j,c}$ are independent and identically distributed random variables with zero mean and unit covariance.\n\t\n\tSince the eigenvalues $\\{ \\lambda_j \\}_{j=0}^{\\infty}$ decrease exponentially, it is easy to show that the derivative of the random surface $f(\\omega;x_1)$ respect to $x_1$ is bounded and there exists a positive constant $C$ such that $\\vert f(\\omega;x_1)-f(\\omega;x_2)\\vert \\leq C \\vert x_1-x_2\\vert$. Thus the random surface $f(\\omega;x_1)$ in KL expansion is Lipschitz. Therefore, we have the results on the well-posedness of variational formulations of the Helmholtz equation on a random periodic structure proved in this paper, which verifies the well-posedness of the direct problem in   \\cite{BaoLinSINUM2020}.\n\\end{remark}\n\n\n\\section{Conclusions}\n\\label{sec:conslusion}\n\nIn this work, we have established stability results for the Helmholtz equation on deterministic and random periodic structures, respectively.  Both estimates are explicit with respect to the wavenumber. To the authors' best knowledge, these are the first stability results explicit with respect to the wavenumber for the scattering problem in periodic structures.\nAn interesting future direction is to apply the stability results for conducting convergence analysis of numerical methods for solving the scattering problems. \nOur techniques may be applicable to other (stochastic) scattering problems. In particular, another future direction is to\nconduct stability analysis for the electromagnetic scattering problems.\n\n\n\n\\bmhead{Acknowledgments}\n\nThis work was supported in part by National Natural Science\nFoundation of China (11621101, U21A20425, 12071430, 12201404), a Key Laboratory of Zhejiang Province, and Postdoctoral Science Foundation of China (2021TQ0203).\n\n\\section*{Declarations}\n\n\n\\begin{itemize}\n\t\\item[] \\textbf{Conflict of interest} The authors declare that they have no conflict of interest.\n\t\\item[] \\textbf{Data availability} Data sharing is not applicable to this article as obviously no datasets were generated or analyzed during the current study.\n\t\\item[] \\textbf{Publisher's Note} Springer Nature remains neutral with regard to jurisdictional claims in\n\tpublished maps and institutional affiliations.\n\\end{itemize}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSince 1998 when a deviation on the luminosity distance of Supernovae Ia (Sne Ia) was observed by two independent groups \\cite{SN1}, and later on by other proofs, the expansion of the universe is thought to be accelerating, a phenomena that has been widely accepted by the scientific community since then. In order to explain such behaviour of the universe expansion, plenty of theoretical models  have been proposed to sort out this challenge, under the name of dark energy. The list of theoretical models includes canonical\/phantom scalar fields, vector fields, modifications of General Relativity (GR), or a cosmological constant, among others\\cite{Nojiri:2010wj}. The latter, known as $\\Lambda$CDM model , has been the one that serves as a reference to test the others.\\\\\nHowever, plenty of models of dark energy are able to fit the data as good as $\\Lambda$CDM model, what increases the difficulty to sort out this problem. This degeneracy among models does not seem to break at the background level at least with the present data, but other tests are required, as the analysis of the perturbations. Then, a useful tool can be to develop an approach able to provide a particular dynamical behaviour of dark energy without providing explicitly the underlying theoretical framework. Roughly speaking, an independent-model approach that serves just for phenomenological purposes. In this sense, some interesting approaches have been analysed over the last years. In this sense, some parametrizations of the dark energy equation of state (EoS) may provide  information about the dynamical evolution of dark energy, while other model-independent methods as cosmography may lead to information about the background evolution and the possibility of testing $\\Lambda$CDM model. Here some issues regarding these model-independent methods are analysed and their usefulness for testing the concordance model of cosmology.\n\\section{Parametrizations of the equation of state for dark energy}\n\\label{reconstruction}\nIn order to test the dynamics of dark energy, some parametrizations of the EoS for dark energy have been proposed in the literature, in such a way that may show up how the evolution of dark energy is, at least at low redshifts, regardless of the underlying theory. In this sense, we may highlight the ones proposed  by Huterer and Turner in 2001\\cite{Huterer:2000mj}, and the Chevallier-Polarski-Linder parametrization\\cite{Huterer:2000mj}, which are given by\n\\begin{equation}\nw_{HT}(z) = w_0 + w_1 z, \\quad w_{CPL}=w_0+w_1\\frac{z}{1+z}\\ .\n\\label{ST11}\n\\end{equation}\nThe first parametrization, also called Linear Redshift Parametrization,  is assumed to describe the behaviour of the dark energy fluid at low redshift ($z < 1$), while the second one tends to a constant for large redshifts. Fitting both parametrizations with Supernovae Ia data, the best fit gives $w_0=-1.4$ and $w_1=1.67$ (HT parametrization), and $w_0=-0.82$ and $w_1=0.58$ (CPL parametrization), leading to a very similar goodness of fit in both cases. Note that both parametrizations include $\\Lambda$CDM as a particular case which does not coincide with the best fit although can not be discarded. In addition, both phenomenological descriptions predict a very different EoS at $z=0$, where $w_{HT}=-1.4$ (phantom-like fluid) and $w_{CPL}=-0.82$, a degeneracy problem that commonly arise when comparing theoretical models directly.\\\\\nSimilarly, other authors have investigated other parametrizations of the EoS where a fast transition -to a phantom epoch- may occur\\cite{Leanizbarrutia:2014xta}:\n\\begin{equation}\nw_{1}(z)=- 1+ w_{0}\\left[ \\tanh\\left(z-z_0\\right)-1\\right]\\ ,\\quad w_2(z)=-1+ w_0 \\tanh\\left(z-z_0\\right)\\ .  \n\\label{2.1}\n\\end{equation}\nHere $w_{0}$ and $z_0$ are free parameters, being $z_0$ the turning point of both functions along the cosmological evolution and $w_0$ the value of the EoS parameter when $z\\leq z_0$. Note also that for $w_{0}=0$, $\\Lambda$CDM is recovered in both cases, while the phantom transition may never occur for some particular ranges of the free parameters. Consequently, in the first parametrisation in (\\ref{2.1}), a future singularity will occur in case that $w_0>0$ whereas  the expansion would be smooth when $w_0<0$. In the second one, the value of $\\tilde{w}_2$ is above -1 for negative (positive) values of $w_0$ and $z_0 > -1$ ($z_0 < -1$), so there is no singularity. For positive (negative) values of $w_0$ and $z_0 > -1$ ($z_0 < -1$), the value of $\\tilde{w}_2$ is below $-1$ and a singularity occurs.  Thus, depending on the free parameters, the above models may lead to some kind of future singularity. Then, by using Sne Ia data, the best fits are obtained for both models in (\\ref{2.1}) and compared with $\\Lambda$CDM. The results are summarised in the following table, where the best $\\chi^2$ is included as well as the reduced $\\chi^2_{red}$ in order to compare the goodness of the fit for every model. As shown, every model gives a very similar fit, but different cosmological evolutions. Similarly, by using other data sets as BAO, the degeneracy of the results remains, such that parameterisations of the EoS for dark energy are not very useful to find out how the dynamics of dark energy are, at least with the available data sets.\n\\begin{table}[h!] \n\\begin{center}\n\\begin{tabular}{ccccccc}\n\\hline\n\\hline\n\\bf{Model} & $\\bf{\\chi_{min}^2}$ &  $\\bf{w_0}$ & $\\bf{z_0}$ & $\\bf{\\Omega_{0m}}$ & $\\bf{\\chi_{red}^2}$\\\\\n\\hline \\vspace{-5pt}\\\\\n$\\Lambda$CDM  & $542.685$ & - & - & $0.27 \\pm 0.02$ & $0.978 $\\\\\n\\\\\n$ w_1(z)$ & $542.683$ & $0.0045 \\pm 0.1$ & $-25 \\pm 30$ & $0.27$ & $0.981 $ \\\\\n\\\\\n$ w_2(z)$ & $541.583$ & $-0.03 \\pm 0.07$ & $22 \\pm 45$ & $0.27$ & $0.979 $\\\\\\\\\n \\hline \\hline\n\\end{tabular}\n\\caption{Best fit for the models (\\ref{2.1}) with $\\Omega_{0m}=0.27$ by using the Sne Ia dataset \\cite{union2.1}. The result for the $\\Lambda$CDM model is also shown. \\label{table1}}\n\\end{center}\n\\end{table}\n\\section{Cosmography}\n\\label{cosmography}\nAnother model-independent approach to test cosmology is the so-called cosmography, which is based solely on the cosmological principle regardless of the underlying theoretical model. To do so, the Hubble parameter is expanded in terms of an auxiliary variable\\cite{Weinberg-Harrison}:\n\\begin{equation}\nH(z)=\\frac{\\dot{a}}{a}=H_0+H_{z0}z+\\frac{H_{zz0}}{2}z^2+\\frac{H_{zzz0}}{6}z^3+...\\ ,\n\\label{cosmo1}\n\\end{equation}\nHere we have used the redshift $1+z=\\frac{1}{a}$ as the auxiliary variable and the subscript $0$ refers to quantities evaluated today. Then, the cosmographic parameters are defined in terms of the derivatives of the Hubble parameter, or equivalently in terms of the scale factor as\n\\begin{equation}\nH_0=\\frac{\\dot{a}_0}{a_0}\\ ,\\, q_0=-\\frac{\\ddot{a}_0}{a_0H^2_0}\\ , \\, j_0=\\frac{a_0^{(3)}}{a_0H_0^3}\\ , \\,  s_0=\\frac{a_0^{(4)}}{a_0H_0^4}\\ ,  ...\\ \n\\label{cosmo2}\n\\end{equation}\nwhere the dots are cosmic time derivatives. Then, by inserting (\\ref{cosmo2}) in (\\ref{cosmo1}), the Hubble parameter is written in terms of the cosmographic parameters which should be set with  observational data. However, note that the series (\\ref{cosmo1}) converges for $|z|<1$. Hence, alternatively the expansion (\\ref{cosmo1})  may be expressed in terms of another auxiliary variable $y = \\frac{z}{1+z}$ which successes to describe the entire universe history by ensuring the convergence of the series\\cite{Visser}. By generating mock data from a fiducial spatially flat $\\Lambda$CDM model, where we have assumed the same redshifts as the Union 2.1 catalogue \\cite{union2.1}, with errors of magnitude $\\sigma_{\\mu}=0.15$, we can show which variable and order of the series behaves better. We have run 100 simulations and fit the cosmographic parameters by using two different sets of parameters for each variable: $\\bm{\\theta_1} = \\{H_0,q_0,j_0,s_0\\}$ and $\\bm{\\theta_2} = \\{H_0,q_0,j_0,s_0,l_0\\}$, where $H_0$ is marginalised. Then, by using Monte Carlo Markov Chain (MCMC), the corresponding constraints for each set and each variable are obtained and shown in the following table. The table contains the number of times the true parameters were inside the confidence region bounds. It is expected the true value to lie within $1\\sigma$, \n68\\% of the times and $2\\sigma$, 95\\% of the times. The variable $z$ gives well-behaved coverage \nresults for the set $\\theta_1$, overestimates the errors while considering a higher order in the expansion $\\theta_2$. On the other hand, the $y$-parametrisation gives completely biased \nestimators for the set $\\theta_1$, and overestimates the errors for $\\theta_2$. These results show that the variable $z$ is preferable in comparison with $y$ for testing models\\cite{Busti:2015xqa}.\n\\begin{table}[htbp]\n\\caption{Coverage test for $\\bm{\\theta_1}$ and $\\bm{\\theta_2}$. Refer to the bulk of the text for further details.}\n\\label{tables1}\n\\begin{center}\n\\begin{tabular}{@{}cccccccccccccc@{}}\n\\hline\n&     &     &     &  $\\bm{\\theta_1}$   &      &     &     &      &     &  $\\bm{\\theta_2}$   &      &     \\\\ \\hline \n& \\vline    &   $y$   &   \\vline  & \\vline    &  $z$   &   \\vline &   & \\vline    &   $y$   &   \\vline  & \\vline    &   $z$   &   \\vline  \\\\\n\\hline  & $1\\sigma$ & $2\\sigma$ &\n 3$\\sigma$ & $1\\sigma$ & $2\\sigma$ &\n 3$\\sigma$ &  & $1\\sigma$ & $2\\sigma$ &\n 3$\\sigma$ & $1\\sigma$ & $2\\sigma$ &\n 3$\\sigma$\n\\\\ \\hline\n$q_0$ & 26 & 32 & 42 & 67   & 27 & 6 &  & 82 & 12 & 6 & 82   & 18 & 0  \\\\ \n$j_0$ & 10  &  45 & 45 & 64  & 29 & 7 &  & 93  &  5 & 2 & 88  & 12 & 0  \\\\ \n$s_0$ & 10  & 67 & 23 & 83  & 15 & 2 &  & 92  & 7 & 1 & 93  & 6 & 1 \\\\\n$l_0$ & - & - & - & - & - & - &  & 100 & 0 & 0 & 100 & 0 & 0 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\subsection{Testing $\\Lambda$CDM with cosmography}\nSince flat $\\Lambda$CDM model gives unequivocally $j_0=1$ regardless of the matter -and dark energy- content, cosmography can be used as a test for the $\\Lambda$CDM model. In order to show the usefulness of the approach, we have generated Sne Ia data by assuming a XCDM model, different from $\\Lambda$CDM, given by: $H^2\/H_0^2=\\Omega_m (1+z)^3+(1-\\Omega_m)(1+z)^{3(1+w)}$ with $\\Omega_m=0.3$ and $w=-1.3$, which gives $j_0=1.945$. Then, by assuming the variable $z$ and fitting the two parameters sets $\\bm{\\theta_1}$ (fourth order) and $\\bm{\\theta_2}$ (fifth order), we have tested whether the cosmographic approach can rule out $\\Lambda$CDM by obtaining the posterior probability for $j_0$. As depicted in Fig.~\\ref{fig2}, there is some evidence of $j_0 \\neq 1$ when considering $\\bm{\\theta_1}$ but such evidence disappears when $\\bm{\\theta_2}$ is assumed. In addition, we have also included the posterior probability for $j_0$ while considering directly the expression of the Hubble parameter for the XCDM model, which leads to a much better fit.  Consequently, the constraints obtained by using cosmography present clear limits while comparing the concordance model with close-enough competitors. \n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{j0_posterior_3cases.eps}\n\\end{center}\n\\caption{Posterior probability for $j_0$ considering 4 parameters $(\\bm{\\theta_1})$, 5 parameters $(\\bm{\\theta_1})$ and XCDM model\\cite{Busti:2015xqa}.}\n\\label{fig2}\n\\end{figure*}\n\\subsection{Reconstructing dark energy models}\nCosmography have been also used for reconstructing some particular models for dark energy, since the underlying action can be expanded around $z=0$ and a correspondence with the cosmographic parameters is obtained. For instance, by considering the usual quintessence\/phantom scalar field model for dark energy, $\\mathcal{S}=\\int {\\rm d}^4x^{}\\sqrt{-g}\\left[-\\frac{1}{2} \\omega (\\phi)\\partial_{\\mu} \\phi \\partial^{\\mu }\\phi -V(\\phi )\\right]\\ ,$ where $\\omega(\\phi)$ is the factor that renormalises the scalar field $\\phi$ and $V(\\phi)$ \nits potential. Then, the derivatives of the potential evaluated today, i.e., at redshift zero,  can be expressed in terms of the cosmographic parameters as follows\n\\begin{eqnarray}\n\\label{DE3}\n&\\frac{V_0}{H_0^2}&=2-q_0-\\frac{3\\Omega_m}{2}\\ , \\quad \\frac{V_{z0}}{H_0^2}=4+3q_0-j_0-\\frac{9\\Omega_m}{2}\\ , \\nonumber \\\\\n&\\frac{V_{2z0}}{H_0^2}&=4+8q_0+j_0(4+q_0)+s_0-9\\Omega_m\\ ,\\nonumber \\\\\n\\end{eqnarray}\nwhere we have used the FLRW equations for this model. We can assume  $\\Omega_m \\approx 2\/3(1+q_0)$ by considering the universe to be close enough to $\\Lambda$CDM today, what yields a one-to-one correspondence between the derivatives of the potential and the cosmographic parameters. Then, fitting the cosmographic parameters leads to constraints over the derivatives of the scalar potential. However, as shown by Dunsby et al (2015)\\cite{Busti:2015xqa}, the constraints obtained by means of the cosmographic approach provides larger errors than other model-independent methods.\\\\\nSimilarly, higher-order derivatives models as Galileons or $f(R)$ gravities can be evaluated at $z=0$ in terms of the cosmographic parameters. However, in this case the higher number of degrees of freedom does not allow to get a one-to-one correspondence as in (\\ref{DE3}). In order to show this, let us consider $f(R)$ gravity, whose derivatives evaluated today lead to\n\\begin{eqnarray}\n\\label{DE4}\n&&\\frac{f_0}{6H_0^2}\\,=\\,-\\alpha q_0 +\\Omega_m + 6\\beta\\left( 2+q_0-j_0\\right)\\ , \\quad \\frac{f_{z0}}{6H_0^2}\\,=\\,\\alpha\\left(2+q_0-j_0\\right)\\ , \\label{f2z0} \\\\\n&&\\frac{f_{2z0}}{6H_0^2}\\,=\\,6\\beta\\left(2+q_0-j_0\\right)^2+\\alpha\\left[2+4q_0+(2+q_0)j_0+s_0\\right]\\,.\\nonumber \n\\end{eqnarray}\nIn this case, there are two extra free parameters, $\\frac{{\\rm d}f}{{\\rm d}R}|_{R=R_0}=\\alpha$ and $\\frac{{\\rm d}^2f}{{\\rm d}R^2}|_{R=R_0}=\\frac{\\beta}{H_0^2}$.\nThis means we need either theoretical priors or additional tests since data does not provide any constraints over $\\alpha$ and $\\beta$. Some previous works assumed the values of $\\alpha=1$ and $\\beta=0$ a priori, such that the model coincides with General Relativity at $z=0$, but this may lead to instabilities. Let us illustrate the difficulties in getting good constraints for $f(R)$ gravities by generating mock data and assuming some sensible priors over the aforementioned parameters for the following toy-model: $f(R)=R+aR^2+bR^3$ where $\\alpha=2.81$ and $\\beta=0.06$. In Fig.~\\ref{fig3}, the probability for $\\{f_0, f_{z0}, f_{zz0}\\}$  are depicted. Here three different hypotheses have been assumed: the true values of $\\{\\alpha, \\beta\\}$, $\\{\\alpha=1, \\beta=0\\}$ and a ``broad'' marginalisation ($\\alpha \\sim N(1,0.05)$ and $\\beta \\sim N(0.07,0.05)$). The probability of $f_0$ is highly dependent on the choice of $\\{\\alpha, \\beta\\}$ which may even lead to ruling out the true values of $f_0$, whether are not known in advance - as is the case when dealing with real data. There are not large differences for the values of $f_{z0}$ and $f_{zz0}$, but the errors are so large than almost any $f(R)$ may be valid, leading to a completely degenerated result. Consequently cosmography is extremely weak when reconstructing $f(R)$ gravities, since it does not allow to distinguish among different Lagrangians.\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.3\\textwidth]{f0_posterior.eps}\n\\includegraphics[width=0.3\\textwidth]{fp0_posterior.eps}\n\\includegraphics[width=0.3\\textwidth]{fpp0_posterior.eps}\n\\end{center}\n\\caption{Probabilities for $f(R)$ and its derivatives evaluated today and the effects of the different choices of the free parameters $\\alpha$ and $\\beta$\\cite{Busti:2015xqa}.}\n\\label{fig3}\n\\end{figure*}\n\\section*{Acknowledgments}\nD.S.-G. acknowledges support from a postdoctoral fellowship Ref.~SFRH\/BPD\/95939\/2013 by Funda\\c{c}\\~ao para a Ci\\^encia e a Tecnologia (FCT, Portugal) and the support through the research grant UID\/FIS\/04434\/2013 (FCT, Portugal).\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nWhy do electrically charged particles exist but magneti-cally charged particles, i.e., magnetic monopoles, appa-rently do not~\\cite{Rajantie:2012xh,Rajantie:2016paj}? If they did, Maxwell's equations would have a perfect duality symmetry between electricity and magnetism.\nIn 1931, Dirac showed that (static) magnetic monopoles are compatible with quantum mechanics~\\cite{Dirac:1931kp}, provided that their magnetic charge $g$ and the electric charges $e$ of all electrically charged particle satisfy the Dirac quantisation condition,\n\\begin{equation}\n\\frac{eg}{2\\pi}\\in{\\mathbb Z}.\n\\label{equ:Diraccondition}\n\\end{equation}\nThis implies that both electric and magnetic charges have to be quantised, and if the elementary electric charge is the charge of the positron, then the elementary magnetic charge is the Dirac charge,\n\\begin{equation}\ng_D=\\frac{2\\pi}{e}\\approx 20.7.\n\\end{equation}\n\n\n\n\n\n\nBecause $g_D\\gg 1$, quantum field theory of magnetic monopoles cannot be studied using standard perturbation theory techniques.\nOne consequence of this failure of perturbation theory is that it is not possible to compute the production cross section of magnetic monopoles in particle collisions. Collider searches of magnetic monopoles can therefore only report upper bounds on the cross section, rather than actual constraints on the theory parameters such as the monopole mass. In practice, it is customary for the experiments to quote mass bounds obtained by assuming the tree-level Drell-Yan cross section, but these bounds cannot be taken literally and serve mainly as a way of comparing the performance of different experiments.\n\nFurthermore, for solitonic monopoles such as the 't~Hooft-Polyakov monopole~\\cite{tHooft:1974kcl,Polyakov:1974ek}, there are semiclassical arguments that their production cross section in elementary particle collisions would be exponentially suppressed~\\cite{Witten:1979kh,Drukier:1981fq}. The argument does not apply, at least in the same form, to elementary magnetic monopoles, but it still raises the question of whether their production cross section may also be very different from the Drell-Yan estimates.\n\nIn this paper, I will discuss a different monopole production process, namely Schwinger pair production in strong magnetic fields~\\cite{Affleck:1981ag,Affleck:1981bma,Gould:2017zwi}. Its rate can be computed using semiclassical instanton techniques, without having to assume a weak coupling. Therefore it does not suffer from the same strong-coupling issue as the perturbative calculations. It is also largely independent of the microscopic details of the monopoles, and within limits, the same results will therefore apply to both elementary and solitonic monopoles. This means that it is possible to obtain actual, largely model-indepenent lower bounds on the magnetic monopole mass.\n\nThe strongest known magnetic field in the Universe are in certain neutron stars called magnetars and in heavy ion collision experiments. I discuss the Schwinger process in both cases, and review the monopole mass bounds obtained from them.\n\n\\section{Elementary and Solitonic Monopoles}\nIn principle, magnetic monopoles can appear in quantum field theory as either elementary or composite particles. In the former case, there is a separate quantum field associated with the monopole, whereas in the latter case, they are states made up of other fields. At weak coupling, this state is usually a semiclassical soliton solution, and therefore I will refer to these as solitonic monopoles. \n\nThe best known example of a solitonic monopole is the 't~Hooft-Polyakov monopole solution~\\cite{tHooft:1974kcl,Polyakov:1974ek} in the Georgi-Glashow model~\\cite{Georgi:1972cj}, an ${\\rm SU}(2)$ gauge theory with an adjoint scalar field $\\Phi$. When the scalar field has a non-zero vacuum expectation value, it breaks the gauge symmetry to ${\\rm U}(1)$, and therefore the low energy effective theory corresponds to electrodynamics.\n\nIn the classical theory, the 't~Hooft-Polyakov monopole is a smooth solution of the field equations, of the form\n\\begin{equation}\n\\Phi^a(\\vec{x})\\propto x^a,\\quad A_i^a \\propto \\epsilon_{iaj}x_j,\n\\end{equation}\nand it has the magnetic charge $g=4\\pi\/e=2g_D$, where $e$ is the ${\\rm SU}(2)$ gauge coupling and also corresponds to the elementary electric charge. The monopole mass is given by the energy of the solution,\nand is approximately\n\\begin{equation}\nM\\approx \\frac{4\\pi m}{e^2}\\approx 137m,\n\\label{equ:tPmass}\n\\end{equation}\nwhere $m$ is the mass of the massive gauge bosons. The monopole also has a finite size,\n\\begin{equation}\nR\\approx \\frac{1}{m} \\approx \\frac{4\\pi}{e^2 M}\\approx 137M^{-1}.\n\\label{equ:tPradius}\n\\end{equation}\n\nThe same monopole exists also in the quantum theory as a non-perturbative state, and at weak coupling it is well approximated by the classical solution. Quantum corrections to it can be calculated perturbatively~\\cite{Kiselev:1988gf}, because the coupling constant of the theory is $e\\ll 1$, not $g=4\\pi\/e\\gg 1$. One can also go beyond perturbation theory by using lattice Monte Carlo methods~\\cite{Rajantie:2005hi,Rajantie:2011nq}.\n\n\nThe 't~Hooft-Polyakov solution can be found in any Grand Unified Theory (GUT)~\\cite{Preskill:1979zi}, and therefore the existence of magnetic monopoles is an unavoidable consequence of grand unification.\nThe mass of these GUT monopoles (\\ref{equ:tPmass}) would typically be around\n$10^{16}~{\\rm GeV}$, which is well above the energies of any foreseeable particle collider experiments. However, there are theories that have lighter solitonic monopole solutions~\\cite{Cho:2013vba,Ellis:2016glu}, possibly even within the reach of the Large Hadron Collider.\n\nAlthough most theoretical work has focused on solitonic monopoles, it is also important to consider the case of elementary monopoles.\nIn practice, theoretical calculations with them are difficult, not only because of their strong charge, which makes perturbation theory invalid, but also because the existing quantum theory formulations are cumbersome~\\cite{Schwinger:1966nj,Zwanziger:1970hk,Milton:2006cp}. However, that is not an argument against their existence.\nIf the magnetic monopole is an elementary particle, its mass is a free parameter. It is therefore perfectly possible that it is at the TeV scale or even lower, as long as it is compatible with the current experimental bounds.\n\n\nBecause of quantum effects, even elementary monopoles would have to have a non-zero effective size~\\cite{goebel1970spatial,goldhaber1982monopoles},\n\\begin{equation}\nR\\gtrsim R_{\\rm QM}=\\frac{g^2}{8\\pi M},\n\\label{equ:quantumradius}\n\\end{equation}\nwhere $M$ is the monopole mass. This is significantly larger than their Compton radius, and therefore even elementary monopoles would not actually appear fully pointlike. However, because of the calculational difficulties due to the strong magnetic charge, there is no detailed understanding of how this finite size arises. It is, nevertheless, interesting to note that it agrees with the size (\\ref{equ:tPradius}) of the 't~Hooft-Polyakov monopole. Indeed, the distinction between elementary and solitonic monopoles is not necessarily clear-cut, and there are examples of theories which have two dual descriptions, in one of which the monopoles are solitonic and in the other one elementary~\\cite{Montonen:1977sn}.\n\n\n\n\n\n\n\n\n\\section{Production Cross Section}\n\nThere have been several searches for magnetic monopoles in particle colliders~\\cite{Patrizii:2015uea}, including LEP, Tevatron and the LHC. The more recent results are from the ATLAS and MoEDAL experiments at the LHC~\\cite{Aad:2019pfm,Acharya:2019vtb}. \nBecause there has been no positive discovery, these experiments place upper bounds on the monopole production cross section.\nIn order to constrain actual theory parameters such as the monopole mass, one would need a reliable theoretical prediction for the production cross section. Sadly such a prediction does not exist for monopole production in collisions of elementary particles.\n\nThe main obstacle to the calculation of the production cross section of magnetic monopoles is their strong magnetic charge $g=ng_D$, $n\\in\\mathbb{Z}$, which the Dirac quantisation condition (\\ref{equ:Diraccondition}) requires to be much greater than one. This means that the calculation cannot be carried out using perturbation theory. \nThere are also strong arguments that the production cross section of solitonic monopoles, such as GUT monopoles, is suppressed by an exponential factor $\\exp(-4\/\\alpha)\\sim 10^{-238}$~\\cite{Witten:1979kh,Drukier:1981fq}. This is because they are highly ordered coherent lumps of field consisting of $O(1\/\\alpha)$ quanta. Even if there is enough energy available to produce the monopoles, it is much more likely that the same energy gets distributed to a large number of particles in a less ordered fashion. It is not known if the production cross section of elementary monopoles is suppressed by a similar factor, because we currently do not have the tools to carry out the calculation.\n\nBecause of this theoretical uncertainty, experiments tend to quote nominal mass bounds based on the tree-level Drell Yan cross section.\nFor monopoles with a single Dirac charge, $g=g_D$, this gives~\\cite{Aad:2019pfm} $M\\gtrsim 1850~{\\rm GeV}$ or $M\\gtrsim 2370~{\\rm GeV}$, depending on whether they have spin $0$ or $1\/2$, respectively.\nHowever, these nominal bounds are mainly useful for comparing different experiments and should not be interpreted as actual lower bounds on monopole masses. Much lighter monopoles can exist if their production cross sections is low. In order to obtain an actual mass bound, one therefore has to consider other processes than monopole pair production from elementary particle collisions.\n\n\n\\section{Schwinger Pair Production}\nIn addition to elementary particle collisions, magnetic monopoles can also be produced in a strong external magnetic field by the Schwinger process. \n\nIt was shown by Sauter~\\cite{Sauter:1931zz} and Schwinger~\\cite{Schwinger:1951nm} that electrically charged particles are\npair produced in a sufficiently strong electric field. This can be understood as tunneling through the Coulomb potential barrier. The rate of this process can be calculated using semiclassical instanton techniques even at strong coupling~\\cite{Affleck:1981ag,Affleck:1981bma}.\n\nIf state $|\\Omega\\rangle$ is unstable, then its decay probability is given by\n\\begin{eqnarray}\n\tP&=&1-\\left|\\langle\\Omega|\\hat{S}|\\Omega\\rangle\\right|^2\n\t=1-\\exp\\left[2\\,{\\rm Im}\\left(i\\log \\langle\\Omega|\\hat{S}|\\Omega\\rangle\\right)\\right]\n\t\\nonumber\\\\\n\t\t&=&1-\\exp\\left[2\\,{\\rm Im}\\left(\\log \\langle\\Omega|\\hat{S}_E|\\Omega\\rangle\\right)\\right],\n\\end{eqnarray}\nwhere the S-matrix $\\hat{S}$ is the time evolution operator from past infinity to future infinity in Minkowski space and $\\hat{S}_E$ the corresponding operator in Euclidean space. In the semiclassical approximation, the exponent is given by\n\\begin{equation}\n{\\rm Im}\\left(\\log \\langle\\Omega|\\hat{S}_E|\\Omega\\rangle\\right)\\sim e^{-S_{\\rm inst}},\n\\end{equation}\nwhere $S_{\\rm inst}$ is the action of the instanton solution, which is a classical solution of the Euclidean equations of motion with one negative mode, i.e. a saddle point solution. One negative mode is needed so that the solution gives an imaginary contribution to the path integral.\n\nFor an electrically charged point particle with mass $m$ and electric charge $e$ in a background gauge field $A^{\\rm ext}_\\mu$, the Euclidean action is\n\\begin{eqnarray}\nS_E[x^\\mu]&=&m\\int_0^1  d\\tau\\, \\left(\\dot{x}_\\mu \\dot{x}^\\mu\\right)^{1\/2}-ie\\oint dx^\\mu A^{\\rm ext}_\\mu\n\\nonumber\\\\\n&&\n-\\frac{e^2}{8\\pi^2}\\int d\\tau d\\tau'\n \\frac{\\dot{x}^\\mu(\\tau) \\dot{x}_\\mu(\\tau')}{|x(\\tau)-x(\\tau')|^2}\n,\n\\end{eqnarray}\nwhere $\\tau\\in [0,1)$ is a parameter along the worldline and $A^{\\rm ext}_\\mu$ is the background gauge field. The last term corresponds to the self-interactions of the particle.\n\nIn a constant background electric field $\\vec{E}$, rotation symmetry implies that the solution is a circle in the plane defined by the field $\\vec{E}$ and the time direction. Denoting the radius of the circle by $r$, its action is\n\\begin{equation}\nS_E(r)=2\\pi m r - e|\\vec{E}| \\pi r^2 - \\frac{e^2}{4},\n\\end{equation}\nwhere the unstable direction corresponds to change of $r$. \nTherefore the saddle point corresponds to the radius\n\\begin{equation}\nr=r_{\\rm inst}=\\frac{m}{e|\\vec{E}|},\n\\label{equ:instantonradius}\n\\end{equation}\nwhich maximises the action, and gives\n\\begin{equation}\nS_{\\rm inst}=S_E\\left(r_{\\rm inst}\\right)=\\frac{\\pi m^2}{e|\\vec{E}|} - \\frac{e^2}{4}.\n\\end{equation}\nThe solution has zero modes corresponding to translations in the four Euclidean directions, which contribute a spacetime volume factor, which means that there is a non-zero, finite rate per unit spacetime volume~\\cite{Affleck:1981bma}\n\\begin{equation}\n\\Gamma= D e^{-S_E}=\n\\frac{e^2|\\vec{E}|^2}{8\\pi^3}\n\\exp\\left( - \\frac{\\pi m^2}{e|\\vec{E}|} + \\frac{e^2}{4} \\right),\n\\label{equ:GammaE}\n\\end{equation}\nwhere the prefactor $D$ is given by a functional determinant of the second derivatives of the action.\n\nThe result (\\ref{equ:GammaE}) means that if the field is sufficiently strong,\n\\begin{equation}\n|\\vec{E}|\\gtrsim \\frac{\\pi m_{\\rm e}^2}{e}\\approx 10^{18}~{\\rm V\/m},\n\\end{equation}\nwhere $m_{\\rm e}$ is the electron mass,\nelectron-positron pairs are produced at an unsuppressed rate. This field is a few orders of magnitude stronger than what can be  currently reached with the most powerful lasers, and therefore Schwinger pair production has not yet been confirmed directly in experiments~\\cite{Turcu:2016dxm,Gould:2018efv}.\n\nBy the electromagnetic duality, if magnetic monopoles exist, they would be pair produced by the same mechanism in sufficiently strong external magnetic field. The rate can be obtained from Eq.~(\\ref{equ:GammaE}) by replacing $e\\rightarrow g$ and $\\vec{E}\\rightarrow \\vec{B}$~\n\\cite{Affleck:1981ag,Affleck:1981bma},\n\\begin{equation}\n\\Gamma=\n\\frac{g^2|\\vec{B}|^2}{8\\pi^3}\n\\exp\\left( - \\frac{\\pi M^2}{g|\\vec{B}|} + \\frac{g^2}{4} \\right).\n\\label{equ:GammaB}\n\\end{equation}\nBecause $g\\gg 1$, the second term in the exponent is important, and therefore the field strength needed to produce monopoles of mass $M$ is\n\\begin{equation}\n|\\vec{B}|\\gtrsim \\frac{4\\pi M^2}{g^3}.\n\\end{equation}\nCorrespondingly, if monopole production is not observed in field $\\vec{B}$, it implies a lower mass bound\n\\begin{equation}\nM\\gtrsim \\sqrt{\\frac{g^3|\\vec{B}|}{4\\pi}}.\n\\label{equ:vacuumbound}\n\\end{equation}\n\nThis calculation assumes that the monopoles are pointlike.\nThe radius of the instanton is given by the electromagnetic dual of Eq.~(\\ref{equ:instantonradius}),\n\\begin{equation}\nr_{\\rm inst}=\\frac{M}{g|\\vec{B}|},\n\\label{equ:instantonradius2}\n\\end{equation}\nso this assumption is justified if the monopole size (\\ref{equ:quantumradius}) is less than this,\n$R_{\\rm QM}\\ll r_{\\rm inst}$. It is easy to check that this is true if the monopole mass satisfies Eq.~(\\ref{equ:vacuumbound}).\n\nAs a simple application, one can consider Schwinger pair production of monopoles by the LHC magnets,\nwhich have field strength $|\\vec{B}|\\approx 8.3~{\\rm T}\\approx 1.6\\times  10^{-15}~{\\rm GeV}^2$. Even before any particle collisions were carried out, the fact that this field did not produce magnetic monopoles when the magnets were first switched on, implies the lower mass bound\n\\begin{equation}\nM\\gtrsim 1.5\\left(\\frac{g}{g_D}\\right)^{3\/2}~{\\rm keV}\n\\label{equ:LHCbound0}\n\\end{equation}\nfor the mass of monopoles.\n\n\\section{Neutron Stars}\nTo improve the bound (\\ref{equ:LHCbound0}), one needs to find stronger magnetic fields. The strongest known magnetic fields currently existing in the Universe and in neutron stars known as magnetars, 23 of which have been found~\\cite{Olausen:2013bpa}. \nThe one with the strongest field is SGR 1806-20, with $|\\vec{B}|\\approx 2\\times 10^{11}~{\\rm T}\\approx 4\\times 10^{-5}~{\\rm GeV}^2$.\nIts temperature is low in comparison, $T\\approx 0.55~{\\rm keV}$, and therefore the Schwinger pair production rate is given by\nthe zero-temperature expression~(\\ref{equ:GammaB}).\n\nAt the surface of SGR 1806-20, the ratio of the gravitational and magnetic forces acting on a monopole\nis\n\\begin{equation}\n\\frac{F_G}{F_B}=\\frac{G_N M_{\\rm NS} M}{g|\\vec{B}|R_{\\rm NS}^2}\n\\approx\n\\left(\n\\frac{g}{g_D}\\right)^{-1}\n\\frac{M}{1.8\\times 10^{17}~{\\rm GeV}},\n\\end{equation}\nwhere $R_{\\rm NS}\\sim 10~{\\rm km} \\approx  5\\times 10^{19}~{\\rm GeV}^{-1}$ is the radius of the magnetar and $M_{\\rm NS}\\sim 1.5M_{\\odot}\\approx 1.6\\times 10^{57}~{\\rm GeV} $ is its mass. \nIf a pair of magnetic monopoles with mass $M\\ll 1.8\\times 10^{17}~{\\rm GeV}$ are produced near the surface of a magnetar, the magnetic field would therefore pull one of them to the surface of the star and expel the  other one into space. This would reduce the strength of the magnetic field, in contradiction with observations~\\cite{Gould:2017zwi}. Using Eq.~(\\ref{equ:vacuumbound}), one therefore obtains a bound\n\\begin{equation}\nM\\gtrsim 0.17\\left(\\frac{g}{g_D}\\right)^{3\/2}~{\\rm GeV}.\n\\label{equ:NSbound}\n\\end{equation}\nA more detailed calculation, which takes into account the long time scale over which the field has to survive and grow, gives a somewhat stronger bound $M\\gtrsim 0.31~{\\rm GeV}$ for $g=g_D$~\\cite{Gould:2017zwi}.\n\n\nThe bound (\\ref{equ:NSbound}) was obtained by considering the magnetic field outside the magnetar, but the magnetic fields in the interior are even stronger.\nThere are also many other open questions about magnetars,\nand with a better understanding of them, one may well be able to improve the bound further.\n\n\n\n\\section{SPS Heavy Ion Collisions}\nEven stronger magnetic fields than those around magnetars are present in relativistic heavy ion collisions. The heavy ions are nuclei of heavy elements such as Au or Pb, and therefore they have a high electric charge $Q=Ze\\sim 100e$.  In these experiments, these nuclei are collided at relativistic speeds. When the collision is not head-on, one therefore has two very high electric currents moving in opposite directions past each other at the time of the collision. This induces a very strong magnetic field for a short period of time.\n\nThe most recent published monopole search in heavy ion collisions was carried out at CERN Super Proton Synchrotron (SPS) in 1997~\\cite{He:1997pj}.\nIt was a fixed-target Pb collision with beam energy $160A~{\\rm GeV}$, corresponding to a centre-of-mass energy\n$\\sqrt{s_{NN}}\\approx 17~{\\rm GeV}$ per nucleon. This produces a magnetic field \n$|\\vec{B}|\\approx 0.01~{\\rm GeV}^2$~\\cite{Skokov:2009qp} and a fireball with a high temperature~$T\\approx 0.185~{\\rm GeV}$~\\cite{Schlagheck:1999aq}. \n\nThe Schwinger process at non-zero temperature was studied in Ref.~\\cite{Gould:2017fve,Gould:2018ovk}. \nTo calculate the rate in the semiclassical approximation, one needs to find the instanton in Euclidean space with a compact imaginary time direction of length $\\beta=1\/T$. This breaks the Euclidean rotation symmetry and therefore the instanton is no longer a circle. Because of the strong coupling, the solutions have to be found numerically, and this was done in Ref.~\\cite{Gould:2017fve}. The result for the SPS case is, however, simple. Because the temperature is sufficiently high, the relevant instanton is a time-dependent sphaleron solution, which corresponds to a static monopole-antimonopole pair. \n\nThe energy of a static monopole-antimonopole pair separated by the distance $\\vec{r}$ in a constant magnetic field $\\vec{B}$ is\n\\begin{equation}\nE(\\vec{r})=2M-\\frac{g^2}{4\\pi |\\vec{r}|}-g\\vec{B}\\cdot\\vec{r}.\n\\end{equation}\nThe sphaleron configuration corresponds to the distance\n\\begin{equation}\n|\\vec{r}|=r_{\\rm sph}=\\sqrt{\\frac{g}{4\\pi |\\vec{B}|}},\n\\end{equation}\nwhich maximises the energy, which gives the sphaleron energy\n\\begin{equation}\nE_{\\rm sph}\n=E(r_{\\rm sph})=2M-2\\left(\\frac{g^3|\\vec{B}|}{4\\pi}\\right)^{1\/2}.\n\\end{equation}\nIncluding the prefactor,\nthe pair production rate is~\\cite{Gould:2018ovk}\n\\begin{equation}\n\\Gamma\\approx\n\\left(\\frac{M^5 T^9}{64\\pi^7 g B^3}\\right)^{1\/2}\\exp\\left[-\\frac{2M}{T}\\left(1-\\sqrt{\\frac{g^3|\\vec{B}|}{4\\pi M^2}}\\right)\\right],\n\\end{equation}\nand the predicted monopole pair production cross section is\n\\begin{equation}\n\\sigma_{M\\overline{M}}\\approx \\sigma_{\\rm tot}{\\cal V}\\Gamma,\n\\end{equation}\nwhere $\\sigma_{\\rm tot}\\approx 6.3~{\\rm b}$ is the total inelastic cross section, and ${\\cal V}$ is the spacetime volume of the collision.\nThe failure of SPS heavy ion collisions to produce magnetic monopoles implies an upper bound \n$\\sigma_{M\\overline{M}}\\lesssim 1.9~{\\rm nb}$ on the monopole pair production cross section~\\cite{He:1997pj}.\nThis translates to a bound on the monopole mass~\\cite{Gould:2017zwi},\n\\begin{equation}\nM\\gtrsim\n\\left(2.0+2.6\\left(\\frac{g}{g_D}\\right)^{3\/2}\\right)~{\\rm GeV}.\n\\label{equ:SPSbound}\n\\end{equation}\n\n\n\\section{LHC Heavy Ion Collisions}\nThe magnetic field produced by a heavy ion collision increases with the collision energy, and therefore the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) should be able to provide much stronger bounds on monopole pair production and, therefore, on the monopole mass. There is no published data on monopole searches at RHIC, so I will focus on the LHC, which carried out a month-long heavy ion run in November 2018, with collision energy per nucleon\n$\\sqrt{s_{NN}}=5.02~{\\rm TeV}$.\n\nThe evolution of the electromagnetic fields during the collision was studied in Ref.~\\cite{Deng:2012pc}. The field strength grows linearly with collision energy, and is highest for collisions with impact parameter $b\\approx 13~{\\rm fm}$, which corresponds to the diameter of the nucleus. Therefore we are mainly interested in peripheral collisions, and can ignore thermal effects.\n\nFor the LHC energy, the peak magnetic field strength is\n$B\\approx 7.3~{\\rm GeV}^2$, reached at the time of collision. The field is highly time-dependent, and can be approximately described by the analytic fit~\\cite{Gould:2019myj}\n\\begin{eqnarray}\n\\vec{B}&=&\\frac{B\\hat{y}}{2}\\left[\n\\left(1+\\omega^2\\left(t-\\frac{z}{v}\\right)^2\\right)^{-3\/2}\n\\right.\n\\nonumber\\\\&&\n\\left.\n+\n\\left(1+\\omega^2\\left(t+\\frac{z}{v}\\right)^2\\right)^{-3\/2}\n\\right],\n\\label{equ:timedepB}\n\\end{eqnarray}\nwhere\n$\\hat{y}$ is the direction of the impact parameter, and\n$\\omega\\approx 73~{\\rm GeV}$ parameterises the rate of the field evolution in time.\nThe time-dependence is important if the\nthe ratio of this to the inverse radius (\\ref{equ:instantonradius2})) of the constant-field instanton,\n\\begin{equation}\n\\xi=\\omega r_{\\rm inst}=\\frac{\\omega M}{gB}\n\\approx \\left(\\frac{g}{g_D}\\right)^{-1}\\frac{M}{2~{\\rm GeV}},\n\\end{equation}\nis large. Therefore\nwe see that time dependence cannot be ignored for monopoles heavier than a few ${\\rm GeV}$.\n\nIn Ref.~\\cite{Gould:2019myj}, the pair production rate was computed\nfor the time-dependent field (\\ref{equ:timedepB}), by analytically continuing it to Euclidean time $\\tau$, \n\\begin{eqnarray}\n\\vec{B}\\longrightarrow \\vec{B}_E&=&\\frac{B\\hat{y}}{2i}\\left[\n\\left(1+\\omega^2\\left(i\\tau-\\frac{z}{v}\\right)^2\\right)^{-3\/2}\n\\right.\\nonumber\\\\&&\\left.\n+\n\\left(1+\\omega^2\\left(i\\tau+\\frac{z}{v}\\right)^2\\right)^{-3\/2}\n\\right],\n\\label{equ:timedepBE}\n\\end{eqnarray}\nand then finding the instanton solution in this Euclidean background field.\n\nIn the zeroth-order approximation, in which the monopole worldline self-interactions are ignored completely,\nthe instanton solution can be found analytically~\\cite{Dunne:2005sx,Dunne:2006st,Gould:2019myj} to be an ellipse with semi-major axis $a_\\tau=M\/gB\\sqrt{1+\\xi^2}$ in the Euclidean time direction and semi-minor axis\n$a_y=M\/gB(1+\\xi^2)$ in the $y$ direction.\nIts action is~\\cite{Gould:2019myj}\n\\begin{equation}\nS_E^{(0)}=\\frac{4M^2}{gB\\xi^2}\n\\left[\n{\\bf E}(-\\xi^2)\n-{\\bf K}(-\\xi^2)\n\\right]\\sim \\frac{4M^2}{gB\\xi}\n,\n\\label{equ:SEzero}\n\\end{equation}\nwhere ${\\bf E}$ and ${\\bf K}$ are elliptic integrals, and the last expression is valid at $\\xi\\gg 1$, which corresponds to $M\\gg 1~{\\rm GeV}$. At the zeroth order, the prefactor in the rate $\\Gamma$ can also be computed analytically~\\cite{Dunne:2005sx,Dunne:2006st}. \nImportantly, the result (\\ref{equ:SEzero}) shows that the rate decreases when $\\xi$ increases.\n\nOne can also go beyond the zeroth-order approximation analytically, and compute the next-to-leading order self-interaction correction to \nthe instanton action~\\cite{Gould:2019myj},\n\\begin{equation}\nS_E^{(1)}=-\\frac{g^2}{8}\\left(\n\\sqrt{1+\\xi^2}\n+\\frac{1}{\\sqrt{1+\\xi^2}}\n\\right)\n\\sim -\\frac{g^2}{8}\\xi\n.\n\\label{equ:SEone}\n\\end{equation}\nComparing Eq.~(\\ref{equ:SEzero}) and (\\ref{equ:SEone}), we see that\nthe total next-to-leading order action $S_E^{(0)}+S_E^{(1)}$ is negative if\n$\\xi^2>32M^2\/g^3B$, which is always satisfied for LHC collisions, irrespective of the monopole mass $M$.\nAt the face value this would imply that the pair production is unsuppressed, but in practice it is more likely to be a signal that the approximations have broken down.\n\nIt is possible to find the instanton to all orders in self-interactions by solving the full non-linear equations numerically~\\cite{Gould:2019myj}. Because the equations are non-local, this is a computationally heavy task, and therefore it has currently only been done for relatively small $\\xi$, where the NLO approximation should be valid, and indeed, the results agree with it.\n\nAgain, this calculation is based on the assumption of a pointlike monopole. \nFor it to be valid, the monopole size (\\ref{equ:quantumradius}) needs to be smaller than the semi-minor axis $a_y$ of the instanton, which gives the condition $\\xi^2\\lesssim 8\\pi M^2\/g^3 B$. This coincides approximately with the value where the NLO action in the pointlike approximation becomes negative, and is not satisfied for any monopole masses in LHC heavy ion collisions. For an accurate and reliable estimate of the production rate, it will therefore be necessary to include the non-zero monopole size.\nFor 't~Hooft-Polyakov monopoles this can be done, at least in principle, by finding the instanton solution in the full field theory.\n\nAs a rough estimate of the production cross section, one can use the locally constant field approximation, in which the constant-field rate (\\ref{equ:GammaB}) is integrated over the spacetime volume were the fields are strongest. \nWithin this approximation,\nif the LHC searches do not find monopoles,  Eq.~(\\ref{equ:vacuumbound}) would give a very rough bound\n\\begin{equation}\nM\\gtrsim 70~{\\rm GeV}.\n\\label{equ:LHCbound}\n\\end{equation}\nBecause the effect of the time-dependence appears to enhance the production, this can be expected to be a conservative estimate. A complete calculation may therefore make the bound significantly stronger. \n\n\n\n\n\n\n\\section{Conclusion}\nThe electromagnetic dual of\nSchwinger pair production provides a new way of searching for magnetic monopoles.\nIf monopoles exist, they would be produced in a sufficiently strong magnetic field. Conversely, if this does not happen, monopoles either do not exist or their mass is too high. By considering physical instances of strong magnetic fields, we can therefore derive lower bounds on magnetic monopole masses. This requires a theoretical calculation of the pair production rate, which can be carried out using the semiclassical instanton method, which does not require perturbation theory or the assumption of a weak coupling. Therefore it can be applied to magnetic monopoles, whose coupling to the electromagnetic field is necessarily strong. \n\nIn a constant field at zero temperature, the rate can be computed analytically and the result is independent of the microscopic details of the monopoles and whether they are elementary and solitonic. The resulting mass bounds are therefore universal. In a time-dependent field, the full result requires a numerical calculation, and the finite monopole size needs to be taken into account. Therefore the precise result will also depend on the microscopic nature of the monopoles. For solitonic 't~Hooft-Polyakov monopoles, the numerical calculation is straightforward in principle, although computationally very demanding.\n\nUsing this approach, the magnetic fields of magnetars imply a bound~(\\ref{equ:NSbound}) of slightly below one ${\\rm GeV}$, and heavy ion collisions at SPS an order of magnitude stronger~(\\ref{equ:SPSbound}). The LHC should be able to improve significantly on these. The results of the one-month heavy-ion run in November 2018 have not yet been published, but the estimate (\\ref{equ:LHCbound}) suggests that if monopoles with mass $M\\lesssim 70~{\\rm GeV}$ exist, they would have been produced then. If data shows no monopoles, it will therefore imply a lower mass bound of the same order. However, obtaining the precise mass bound will need further theoretical work in order to take the time dependence of the collision and the finite monopole size into account.\n\nAt the face value, this appears to be $20-30$ times lower than the current bounds from ATLAS and MoEDAL. However, it is important to remember that all existing mass bounds from proton-proton collisions are based on perturbation theory, which is not actually valid for magnetic monopoles because of their strong magnetic charge. Therefore one cannot currently rule out the existence of monopoles with masses of even a few ${\\rm GeV}$, and the only way to do that is to carry out these calculations and experiments.\n\n \n\\acknowledgments\nThe author would like to thank O.~Gould, D.L.-J.~Ho, S.~Mangles, S.~Rose, D.J.~Weir and C.~Xie and the whole MoEDAL collaboration for useful discussions and collaboration on this topic. This work was supported by the UK Science and Technology Facilities Council grant ST\/P000762\/1.\n\n\n\\input RSTA_Rajantie_arxiv.bbl\n\n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{}\n\n\n\n\\section{\\(CP\\)-violating weak phase \\(\\phis\\) and \\(\\DGs\\) from flavor-tagged\n         time-dependent angular analysis of \\(\\Bst\\) by ATLAS}\n\\label{delgamma}\n  New phenomena beyond the predictions of the Standard Model (SM) may\n  alter \\(CP\\) violation in \\(B\\)-decays. A channel that is expected to\n  be sensitive to new physics contributions is the decay\n  \\(\\overline{\\ensuremath{B_{s}^{0}}\\xspace}\\ensuremath{\\rightarrow}\\xspace\\ensuremath{J\/\\psi}\\phi\\) (quark content\n  \\(\\aBs\\equiv\\ensuremath{b}\\xspace\\bar{\\ensuremath{s}\\xspace}\\)~\\cite{Beringer:1900zz}).\n  \\(CP\\) violation in the \\(\\overline{\\ensuremath{B_{s}^{0}}\\xspace}\\ensuremath{\\rightarrow}\\xspace\\ensuremath{J\/\\psi}\\phi\\) decay\n  occurs due to interference between direct decays and decays occurring\n  through \\BsaBs mixing.  The oscillation frequency of \\ensuremath{B_{s}^{0}}\\xspace\\ meson mixing\n  is characterized by the mass difference \\dms\\ of the heavy (\\BH) and\n  light (\\BL) mass eigenstates. The \\(CP\\)-violating phase \\phis\\ is\n  defined as the weak phase difference between the \\( \\BsaBs\\) mixing\n  amplitude and the \\(b\\ensuremath{\\rightarrow}\\xspace{c}\\ensuremath{\\overline c}\\xspace{s}\\) decay amplitude. In the SM the\n  phase \\phis\\ is small and can be related to CKM quark mixing matrix\n  elements; a value of\n  \\(\\phis\\simeq-2\\beta_{s}=-0.0368\\pm0.0018\\)~rad\n  is predicted in the SM~\\cite{Bona:2006sa}.\n  Many models describing a new physics predict\n  larger \\(\\phis\\) values whilst satisfying all existing constraints,\n  including the precisely measured value of\n  \\dms\\ \\cite{Abulencia:2006ze, Aaij:2011qx}.  Another physical quantity\n  involved in \\(\\BsaBs\\) mixing is the width difference\n  \\(\\DGs=\\GL-\\GH\\) of \\(\\BL\\) and \\(\\BH\\).  Physics beyond the SM is not\n  expected to affect \\DGs\\ as significantly as \\phis~\\cite{Lenz:2011ti}.\n  The decay of the pseudoscalar \\(\\aBs\\) to the vector-vector\n  final-state \\(J\/\\psi\\phi\\) results in an admixture of \\(CP\\)-odd and\n  \\(CP\\)-even states, with orbital angular momentum \\(L = 0\\), \\(1\\) or\n   \\(2\\). The \\(CP\\) states are separated statistically through the\n  time-dependence of the decay and angular correlations among the\n  final-state particles.\n\\par\n  The analysis is based on a data sample of an integrated luminosity\n  \\(4.9\\,\\ifb \\) collected in 2011 by the ATLAS detector in \\(\\ensuremath{pp}\\xspace\\)\n  collisions at \\(\\sqrt{s} = 7\\TeV\\) with di-muon triggers selecting\n  \\(\\ensuremath{J\/\\psi}\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}\\) candidates~\\cite{atlas-conf-phis}.  The triggers\n  select di-muon events requiring both muons to have\n  \\(\\mbox{$p_{\\rm T}$}\\xspace(\\ensuremath{\\mu^\\pm}\\xspace)>4\\gevc\\), or with asymmetric requirements of\n  \\(\\mbox{$p_{\\rm T}$}\\xspace(\\mu_{1})>6\\gevc\\), and \\(\\mbox{$p_{\\rm T}$}\\xspace(\\mu_{1})>4\\gevc\\) with a rapidity\n  range of \\(\\left|\\eta(\\ensuremath{\\mu^\\pm}\\xspace)\\right|<2.4\\) for both cases.  The\n  candidates for \\(\\phi\\ensuremath{\\rightarrow}\\xspace\\ensuremath{K^-}\\xspace\\ensuremath{K^+}\\xspace \\) are reconstructed from all pairs of\n  oppositely charged particles with \\(\\mbox{$p_{\\rm T}$}\\xspace(h)>1\\gevc\\) and\n  \\(\\left|\\eta(h)\\right|<2.5\\), that are not identified as muons.\n  The \\(\\aBs\\) candidates are reconstructed using measurements provided\n  by the inner tracking detectors and the muon\n  spectrometers~\\cite{Aad:2008zzm}.\n  Candidates for\n  \\(\\aBs\\ensuremath{\\rightarrow}\\xspace\\ensuremath{J\/\\psi}(\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\mathrm{\\mu^+ \\mu^-}})\\phi(\\ensuremath{\\rightarrow}\\xspace\\ensuremath{K^-}\\xspace\\ensuremath{K^+}\\xspace)\\)~\\footnote{Unless otherwise\n    stated all references to the specific charge combination imply the\n    charge conjugate combination as well.}\n  are sought by fitting the tracks for each combination of\n  \\(\\ensuremath{J\/\\psi}\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\mathrm{\\mu^+ \\mu^-}} \\) and \\(\\phi\\ensuremath{\\rightarrow}\\xspace\\ensuremath{K^-}\\xspace\\ensuremath{K^+}\\xspace\\) to a common vertex with\n  \\(m(\\ensuremath{\\mathrm{\\mu^+ \\mu^-}})\\) constrained to the world average \n  \\(m(\\ensuremath{J\/\\psi})\\)~\\cite{Beringer:1900zz}.\n  In total \\(N(\\ensuremath{B_{s}^{0}}\\xspace)=131,000\\) candidates are collected within a mass\n  range of \\(m(\\ensuremath{B_{s}^{0}}\\xspace)\\in(5.15,\\,5.65)\\gevcc\\) to be used in the likelihood\n  fit.\n  To infer the initial flavor of the \\(\\ensuremath{B}\\xspace\\)-meson (\\(\\aBs\\) or \\(\\ensuremath{B_{s}^{0}}\\xspace\\))\n  the flavor of the \\(\\ensuremath{B}\\xspace\\)-hadron originating from the other \\(b\\)-quark\n  (OS tagging) is determined.  The flavor tagging probabilities are\n  evaluated using the calibration mode \\(\\ensuremath{B^-}\\xspace\\ensuremath{\\rightarrow}\\xspace{J}\/{\\psi{K}^{-}}\\)\n  reconstructed in the same data sample.\n  A maximum likelihood fit is performed over an unbinned set of the\n  reconstructed mass \\(m\\), the measured proper decay time, \n  \\(t\\equiv\\mbox{$L_{xy}$}\\xspace\\cdot{m}(\\ensuremath{B_{s}^{0}}\\xspace)\/({c}\\cdot\\mbox{$p_{\\rm T}$}\\xspace(\\ensuremath{B_{s}^{0}}\\xspace))\\), the\n  measured mass and proper decay time uncertainties \\(\\sigma_{m}\\) and\n  \\(\\sigma_{t}\\), and the transversity angles \\(\\Omega\\) of each \n  \\(\\overline{\\ensuremath{B_{s}^{0}}\\xspace}\\ensuremath{\\rightarrow}\\xspace\\ensuremath{J\/\\psi}\\phi\\) decay\n  candidate satisfying the selection criteria~\\cite{atlas-conf-phis}.\n  The fit finds the following values for the physics parameters of the interest:\n\\begin{align*}\n   \\phis                & = 0.12\\pm0.25\\,\\rm{(stat.)}\\pm0.11\\,\\rm{(syst.)}\\,rad\\\\\n   \\DGs                 & = 0.053\\pm0.021\\,\\rm{(stat.)}\\pm0.009\\,\\rm{(syst.)}\\,{ps}^{-1}\\\\ \n   \\ensuremath{ \\Gamma_{s}     }                  & = 0.677\\pm0.007\\,\\rm{(stat.)}\\pm0.003\\,\\rm{(syst.)}\\,{ps}^{-1}\\\\\n   |A_{0}(0)|^2         & = 0.529\\pm0.006\\,\\rm{(stat.)}\\pm0.011\\,\\rm{(syst.)}\\\\\n   |A_{\\parallel}(0)|^2 & = 0.220\\pm0.008\\,\\rm{(stat.)}\\pm0.009\\,\\rm{(syst.)}\\\\\n   {\\delta_{\\perp}}     & = 3.89\\pm0.46\\,\\rm{(stat.)}\\pm0.13\\,\\rm{(syst.)}\\,{rad}\n\\end{align*}\n  The resulting contours for the several confidence intervals are\n  produced using a profile likelihood method and are shown in\n  Fig.~\\ref{fig:Contour}.\n  \\begin{figure}[hbt]\n  \\begin{center}\n    \\includegraphics[width=0.34\\textwidth]{.\/figures\/atl-dgamma.png}\n    \\caption{Likelihood contours in the \\(\\phis{-}\\DGs\\) plane.  Three\n             contours show the \\(68\\%,\\,90\\%\\) and \\(95\\%\\) confidence intervals\n             (statistical errors only). }\n  \\end{center}\n  \\label{fig:Contour}\n  \\end{figure}\n  The values are consistent with those obtained in the previous untagged\n  analysis~\\cite{Aad:2012kba}, and as expected improving significantly\n  on the overall uncertainty on \\(\\phis\\). These results are also\n  consistent with theoretical expectations, in particular \\(\\phis\\) and\n  \\DGs\\ are in good agreement with the\n  values predicted in the Standard Model.\n\\section{Angular analysis of the decay \\(\\ensuremath{B^0}\\xspace\\ensuremath{\\rightarrow}\\xspace{K^{*0}}\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}\\)}\n\\label{kstarmumu}\n   Another productive area for indirect searches of new phenomena, in\n   flavor physics, is the study of flavor-changing neutral current\n   decays of \\(b \\) hadrons such as the semileptonic decay mode\n   \\(\\ensuremath{\\Bbar^0}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Kbar^{*0}}\\xspace}\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}\\).\n   This decay is forbidden at tree level in the SM, resulting in small\n   SM rates. From the theoretical side, robust calculations are now\n   possible for much of the phase space of this decay and the\n   calculations also indicate that new physics could give rise to\n   readily observable effects. Finally, this decay mode is relatively\n   easy to select and reconstruct at hadron colliders.  Two important\n   observables\n   in the \\(\\ensuremath{\\Bbar^0}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Kbar^{*0}}\\xspace{(892)}}(\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{K^-}\\xspace\\ensuremath{\\pi^+}\\xspace})\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}\\) \n   decay are the forward-backward asymmetry of the muons, \\({A_{FB}}\\),\n   and the longitudinal polarization fraction of the\n   \\(\\ensuremath{\\Kbar^{*0}}\\xspace{(892)}\\), \\({F_{L}}\\).\n   The relevant angular variables are the angle \\(\\theta_{l}\\) defined\n   as the angle between the positive (negative) muon momentum and the\n   direction opposite to the \\(\\ensuremath{\\Bbar^0}\\xspace\\,(\\ensuremath{B^0}\\xspace)\\) in the dimuon reference\n   frame and the angle \\(\\theta_{K}\\) defined as the angle between the\n   kaon momentum and the direction opposite to the \\(\\ensuremath{\\Bbar^0}\\xspace\\,(\\ensuremath{B^0}\\xspace)\\) in\n   the \\(\\ensuremath{\\Kbar^{*0}}\\xspace\\,(\\ensuremath{K^{*0}}\\xspace)\\) rest frame.  The decay rate distribution\n   of \\(\\ensuremath{\\Bbar^0}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Kbar^{*0}}\\xspace}\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}\\) is described as a function of\n   \\(\\theta_{l}\\) and \\(\\theta_{K}\\) and is measured in several\n   \\({q^{2}}\\) (\\(\\equiv{m^{2}(\\ensuremath{\\mathrm{\\mu^+ \\mu^-}})}\\)) bins. The main results of the\n   analysis, \\({F_{L}}\\) and \\({A_{FB}}\\) are extracted from unbinned\n   extended maximum likelihood fits to three variables: \\(m(\\ensuremath{\\Bbar^0}\\xspace)\\)\n   and the two angular variables.  The results yielded by fits for every\n   \\({q^{2}}\\) bin are compared to SM\n   predictions~\\cite{Kruger:1999xa}. Deviations from the SM predictions\n   may indicate new phenomena.\n\\par  \n   The CMS analysis~\\cite{cms-conf-afb} uses the data sample of\n   \\({\\ensuremath{\\int\\!{\\cal{L}}\\,dt}}\\xspace\\approx5.2\\,\\ifb \\) collected by several flavors of CMS dimuon\n   trigger.  The CMS trigger acceptance criteria are\n   \\(\\left|\\eta(\\ensuremath{\\mu^\\pm}\\xspace)\\right|<2.2\\), \\(\\mbox{$p_{\\rm T}$}\\xspace(\\ensuremath{\\mu^\\pm}\\xspace)>3,\\,4,\\,4.5,\\,5\\gevc\\)\n   (depending on trigger flavor) and \\(\\mbox{$p_{\\rm T}$}\\xspace(\\ensuremath{\\mathrm{\\mu^+ \\mu^-}})>6.9\\gevc\\). The CMS\n   trigger fits \\(\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}\\) pairs to a common point required to be\n   displaced from a vertex of the origin of \\(\\ensuremath{pp}\\xspace\\) interaction. The\n   \\(\\ensuremath{\\Kbar^{*0}}\\xspace\\) candidates are reconstructed through their decay mode\n   \\(\\ensuremath{\\Kbar^{*0}}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{K^-}\\xspace\\ensuremath{\\pi^+}\\xspace\\)\n   and the \\(\\ensuremath{\\Bbar^0}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Kbar^{*0}}\\xspace{(892)}}(\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{K^-}\\xspace\\ensuremath{\\pi^+}\\xspace})\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}\\)\n   is reconstructed by fitting the identified \\(\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}\\) pair and the two\n   hadron tracks each with \\(\\mbox{$p_{\\rm T}$}\\xspace(h)>0.75\\gevc\\) to a common vertex.\n\\par  \n   The ATLAS analysis~\\cite{atlas-conf-afb} is based on\n   \\({\\ensuremath{\\int\\!{\\cal{L}}\\,dt}}\\xspace\\approx4.9\\,\\ifb\\) of data collected by ATLAS single muon and\n   dimuon triggers.  The main ATLAS triggers select di-muon events\n   requiring both muons to have \\(\\mbox{$p_{\\rm T}$}\\xspace(\\ensuremath{\\mu^\\pm}\\xspace)>4\\gevc\\) or alternatively\n   \\(\\mbox{$p_{\\rm T}$}\\xspace(\\mu_{1})>6\\gevc\\) and \\(\\mbox{$p_{\\rm T}$}\\xspace(\\mu_{2})>4\\gevc\\) with a rapidity\n   range of \\(\\left|\\eta(\\ensuremath{\\mu^\\pm}\\xspace)\\right|<\\)2.4. In most ATLAS triggers no constraint\n   on the di-muon invariant mass is applied. The \\ensuremath{\\Kbar^{*0}}\\xspace candidates are\n   reconstructed from hadron tracks with \\(\\left|\\eta(h)\\right|<\\)2.5 and\n   \\(\\mbox{$p_{\\rm T}$}\\xspace(h)>0.5\\gevc\\).\n\\par  \n   The results are shown in Fig.~\\ref{fig:fl-afb}.  No deviations from\n   the SM predictions~\\cite{Kruger:1999xa} are found by both\n   experiments. There is a slight tension between ATLAS data and SM\n   expectations for low \\({q^{2}}\\) bins.\n  \\begin{figure}[hbt]\n  \\begin{center}\n       \\subfigure{%\n           \n         \\includegraphics[width=0.25\\textwidth]{.\/figures\/cms-fl-data.png}\n       }%\n       \\subfigure{%\n           \n         \\includegraphics[width=0.25\\textwidth]{.\/figures\/atl-fl-data.png}\n       }\\\\%\n       \\subfigure{%\n           \n         \\includegraphics[width=0.25\\textwidth]{.\/figures\/cms-afb-data.png}\n        }%\n       \\subfigure{%\n           \n          \\includegraphics[width=0.25\\textwidth]{.\/figures\/atl-afb-data.png}\n       }%\n  \\end{center}\n  \\caption{ Results of the measurement of \\({F_{L}}\\) (upper plots) and\n    \\({A_{FB}}\\) (bottom plots) versus the dimuon \\({q^{2}}\\).  The\n    \\({q^{2}}\\) bins corresponding to \\(\\ensuremath{J\/\\psi} \\) and \\(\\ensuremath{\\psi{(2S)}}\\xspace\\)\n    resonances are not filled.  Good agreement with SM predictions~\\cite{Kruger:1999xa}\n    is found. }\n  \\label{fig:fl-afb}\n  \\end{figure}\n\\section{\\(\\Y1S,\\,\\Y2S,\\,\\Y3S\\) cross section measurements by CMS}\n\\label{y123s}\n  In this section we present the production of the lowest\n  \\(\\Upsilon(nS)\\) states in \\(\\ensuremath{pp}\\xspace \\) collisions studied with the CMS\n  detector. These bottomonium (\\(\\ensuremath{b\\overline b}\\xspace\\)) states are produced promptly,\n  contrary to charmonium (\\(\\ensuremath{c\\overline c}\\xspace\\)) states originating partially from\n  weak \\ensuremath{b}\\xspace-decays.  Their dominant production mechanism is through the\n  fragmentation of partons, \\(\\exegrat\\), gluons, \\(\\ensuremath{g}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{b\\overline b}\\xspace\\) though some\n  fraction of \\(S\\)-wave states result from the strong or radiative\n  decays of higher, \\(\\exegrat\\), \\({P}\\)-wave \\(\\ensuremath{b\\overline b}\\xspace\\) states.  There are\n  several bottomonium production models which predict different shapes\n  of \\(\\mbox{$p_{\\rm T}$}\\xspace\\) production spectra at their high range in \\(\\ensuremath{pp}\\xspace \\)\n  collisions~\\cite{Cho:1995vh}.  The experimental measurements from LHC\n  provide access to high-\\mbox{$p_{\\rm T}$}\\xspace range of \\(\\Upsilon(nS)\\) production\n  spectra to make a useful comparison with the predictions.\n\\par  \n  Earlier, the ATLAS Collaboration has published the production\n  cross section measurements of all three \\(\\Y1S,\\,\\Y2S,\\,\\Y3S\\)\n  states~\\cite{Aad:2012yna}. The analysis was based on\n  \\({\\ensuremath{\\int\\!{\\cal{L}}\\,dt}}\\xspace\\approx1.8\\,\\ifb\\) collected by ATLAS dimuon triggers. The total\n  production cross-section over \\(p_{\\rm T}^{\\Upsilon}<70\\gevc \\) and\n  \\(\\left|y^{\\Upsilon}\\right|<2.25\\) and the \\(\\mbox{$p_{\\rm T}$}\\xspace \\) spectra in central\n  rapidity \\(\\left|y^{\\Upsilon}\\right|<1.2\\) and forward\n  \\(1.2<|y^{\\Upsilon}|<2.25\\) rapidity intervals have been\n  measured~\\cite{Aad:2012yna}.\n\\par  \n  The latest analysis by the CMS Collaboration presented here is based on an\n  unprescaled dimuon trigger involving the tracker and muon\n  systems~\\cite{cms-conf-yns}. The larger data sample of\n  \\({\\ensuremath{\\int\\!{\\cal{L}}\\,dt}}\\xspace\\approx4.9\\,\\ifb\\) contains \\(\\Upsilon(nS)\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}\\) candidates\n  with vertex fitted and produced in a central area,\n  \\(\\left|y^{\\Upsilon}\\right|<0.6\\) with a broad \\(\\mbox{$p_{\\rm T}$}\\xspace\\) range of\n  \\(10-100\\gevc\\). The experimental spectra are found from fits of the\n  invariant mass \\(m(\\ensuremath{\\mathrm{\\mu^+ \\mu^-}})\\) distributions of the candidates\n  reconstructed in several \\mbox{$p_{\\rm T}$}\\xspace bins of \\(\\left|y^{\\Upsilon}\\right|<0.6\\)\n  rapidity range.  The raw spectra are corrected by the trigger\n  efficiency, by the acceptance of analysis criteria and normalized to\n  the luminosity and \\({\\ensuremath{\\cal B}\\xspace}(\\Upsilon(nS)\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}) \\).\n  The results are shown in Fig.~\\ref{fig:prod-yns} for \\(\\Y1S\\) and\n  \\(\\Y2S\\) states~\\cite{cms-conf-yns}.  The \\mbox{$p_{\\rm T}$}\\xspace-spectra reveal at\n  \\(\\mbox{$p_{\\rm T}$}\\xspace\\gsim{20}\\gevc\\) the change of an exponential shape to a\n  power-law behavior.\n  To emphasize the transition area the production\n  ratio\n  \\({\\frac{d\\sigma}{d{p_{T}}}}\\cdot{\\ensuremath{\\cal B}\\xspace}(\\Y2S)\/{\\frac{d\\sigma}{d{p_{T}}}}\\cdot{\\ensuremath{\\cal B}\\xspace}(\\Y1S)\\)\n  is presented at the bottom right plots both for CMS and\n  ATLAS. Interestingly enough, the recently published ATLAS\n  spectra~\\cite{Aad:2012yna} do show the similar break-down of the shape\n  around \\(\\mbox{$p_{\\rm T}$}\\xspace\\sim20\\gevc \\) as the right bottom plot of\n  Fig.~\\ref{fig:prod-yns} demonstrates. In a summary, new measurements\n  of\n  \\({\\frac{d\\sigma}{d{p_{T}}}}\\Big|_{|y|<0.6}\\times{{\\ensuremath{\\cal B}\\xspace}\\big(\\Upsilon(nS)\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\mathrm{\\mu^+ \\mu^-}}\\big)}\\)\n  in a wide range of \\(\\mbox{$p_{\\rm T}$}\\xspace\\in(10-100)\\gevc\\) are made with CMS detector.\n  The transition from nearly exponential cross section decrease with\n  \\(\\mbox{$p_{\\rm T}$}\\xspace \\) to power-law behavior for all three \\(\\Upsilon(nS)\\) is\n  observed presenting a challenge to theoretical models.\n\\begin{figure}[hbt]\n  \\begin{center}\n       \\subfigure{%\n           \n         \\includegraphics[width=0.25\\textwidth]{.\/figures\/cms-xs_fit_1S.pdf}\n       }%\n       \\subfigure{%\n           \n         \\includegraphics[width=0.25\\textwidth]{.\/figures\/cms-xs_fit_2S.pdf}\n       }\\\\%\n       \\subfigure{%\n           \n         \\includegraphics[width=0.25\\textwidth]{.\/figures\/cms-Ratio2S-1S.pdf}\n        }%\n       \\subfigure{%\n           \n          \\includegraphics[width=0.25\\textwidth]{.\/figures\/atlas-ratio2S-1S.png}\n       }%\n  \\end{center}\n  \\caption{ CMS results: \\(\\Y1S \\) production cross section spectrum\n            (upper left plot), \\(\\Y2S \\) production spectrum cross section\n            (upper right plot) and the ratio \n            \\({\\frac{d\\sigma}{d{p_{T}}}}\\cdot{\\ensuremath{\\cal B}\\xspace}(\\Y2S)\/{\\frac{d\\sigma}{d{p_{T}}}}\\cdot{\\ensuremath{\\cal B}\\xspace}(\\Y1S)\\)\n            of production spectra (bottom left plot).\n            ATLAS result to compare: ATLAS\n            \\((\\Y2S)\/(\\Y1S)\\) \n            production ratio (bottom right plot). The same change in\n            shape of the production ratio at high \\(\\mbox{$p_{\\rm T}$}\\xspace \\) is observed.}\n  \\label{fig:prod-yns}\n\\end{figure}\n\\section{\\(\\ensuremath{B^+}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{J\/\\psi}\\ensuremath{K^+}\\xspace\\) cross section measurements by ATLAS}\n\\label{bpxsec}\n  Measurements of the \\ensuremath{b}\\xspace-hadron production cross section in \\(\\ensuremath{pp}\\xspace\\)\n  collisions at LHC provide further tests of QCD calculations for\n  heavy-quark production at higher center-of-mass energies and in wider\n  transverse momentum (\\mbox{$p_{\\rm T}$}\\xspace) and rapidity (\\(y\\)) ranges, thanks to the\n  extended coverage and excellent performance of the LHC\n  detectors. ATLAS and CMS are sensitive in the central rapidity region,\n  so their measurements are complementary to open beauty measurements\n  with LHCb detector.\n\\par  \n  In this section we present the production cross section measurement\n  for \\(\\ensuremath{B^+}\\xspace \\) reconstructed in its fully exclusive decay mode to\n  \\(\\ensuremath{J\/\\psi}(\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\mathrm{\\mu^+ \\mu^-}})\\ensuremath{K^+}\\xspace \\) using the ATLAS detector. The data for this\n  analysis correspond to an integrated luminosity\n  \\({\\ensuremath{\\int\\!{\\cal{L}}\\,dt}}\\xspace\\approx2.4\\,\\ifb\\) collected at \\(\\mbox{$\\sqrt{s}$}\\xspace=7\\tev \\) by a dimuon\n  trigger, which requires the presence of at least two muon candidates\n  of \\(\\mbox{$p_{\\rm T}$}\\xspace(\\ensuremath{\\mu^\\pm}\\xspace)>4.0\\gevc\\) and \\(\\left|\\eta(\\ensuremath{\\mu^\\pm}\\xspace)\\right|<2.4\\) each.  Offline,\n  the events are required to contain at least one pair of reconstructed\n  muons, and each pair is fitted using a vertexing algorithm. The\n  corresponding \\(\\ensuremath{J\/\\psi}\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\mathrm{\\mu^+ \\mu^-}} \\) candidate with fitted common vertex\n  is selected with the invariant mass, \\(m(\\ensuremath{\\mathrm{\\mu^+ \\mu^-}})\\in(2.7,\\,3.5)\\gevcc \\).\n  The muon tracks of the selected \\(\\ensuremath{J\/\\psi}\\) are again fitted to a common\n  vertex with an additional hadron track of \\(\\mbox{$p_{\\rm T}$}\\xspace(h)>1\\gevc\\) and with\n  the \\(\\ensuremath{K^\\pm}\\xspace\\) mass assigned.  The three-track vertex fit is performed\n  by constraining the muon tracks to the \\(\\ensuremath{J\/\\psi} \\) world average\n  mass~\\cite{Beringer:1900zz}.  The \\(\\ensuremath{B^+}\\xspace\\) candidates with\n  \\(\\mbox{$p_{\\rm T}$}\\xspace(\\ensuremath{B^+}\\xspace)>9\\gevc\\) and \\(\\left|{y(\\ensuremath{B^+}\\xspace)}\\right|< 2.3\\) in the mass range\n  \\(m(\\ensuremath{B^+}\\xspace)\\in(5.040,\\,5.800)\\gevcc\\) are kept for further analysis.\n  The number of reconstructed \\(\\ensuremath{B^+}\\xspace\\) mesons is obtained using a binned\n  maximum likelihood fit to the invariant mass of the selected\n  candidates per every \\((\\Delta{\\mbox{$p_{\\rm T}$}\\xspace},\\,\\Delta{y})\\) bin.\n  The cross section \\(d^{2}\\sigma(\\ensuremath{pp}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{B^+}\\xspace\\,+X)\/d{\\mbox{$p_{\\rm T}$}\\xspace}d{y}\\) for\n  four \\(\\Delta{y} \\) and eight \\(\\Delta{\\mbox{$p_{\\rm T}$}\\xspace} \\) intervals, covering the\n  range of \\(\\left|{y}\\right|< 2.25\\) and \\(\\mbox{$p_{\\rm T}$}\\xspace\\in(9,\\,120)\\gevc \\) are\n  presented in Fig.~\\ref{fig:prod-bp}~\\cite{ATLAS:2013cia}.\n  The measured differential cross section is compared with the\n  {\\sc QCD NLO} calculations.  The predictions are obtained using\n  {\\sc Powheg+Pythia} and {\\sc MC$@$NLO+Herwig} and are quoted with an\n  uncertainty from renormalization and factorization scales and \\ensuremath{b}\\xspace-quark\n  mass of the order of \\((20-40)\\% \\). Within these uncertainties,\n  {\\sc Powheg+Pythia} predictions are in agreement both in absolute scale\n  and in the shape with the measured \\(\\mbox{$p_{\\rm T}$}\\xspace\\) and \\(y\\) double\n  differential distributions. At low \\(\\left|{y}\\right|\\), \n  {\\sc MC$@$NLO+Herwig} predicts lower production cross section and a\n  softer \\(\\mbox{$p_{\\rm T}$}\\xspace \\) spectrum than the one observed in data, which becomes\n  harder for \\(\\left|{y}\\right|>1.0\\).  An {\\sc FONLL} calculation\n  with \\({f_b}\\rightarrow\\ensuremath{B^+}\\xspace=(40.1\\pm1.3)\\%\\) to fix the overall scale,\n \n \n  is in a good agreement with the measured spectra, in particular at \n  \\(\\mbox{$p_{\\rm T}$}\\xspace<30\\gevc\\) range.\n\\begin{figure}[hbt]\n  \\begin{center}\n       \\subfigure{%\n           \n            \\includegraphics[width=0.25\\textwidth]\n            {.\/figures\/atl-bp-xsec-pt.pdf}\n       }%\n       \\subfigure{%\n          \n           \\includegraphics[width=0.25\\textwidth]\n           {.\/figures\/atl-cms-bp-xsec-pt.pdf}\n       }%\n  \\end{center}\n  \\caption{ Double-differential cross section of \\ensuremath{B^+}\\xspace production as a\n            function of \\mbox{$p_{\\rm T}$}\\xspace for different rapidity ranges: (left plot) The data\n            points are compared to {\\sc NLO} predictions from {\\sc Powheg} and\n            {\\sc MC$@$NLO}; \n            (right plot) the data points are compared with \n            {\\sc  FONLL} predictions (see also inset) and \n            the CMS results as well.\n  }\n  \\label{fig:prod-bp}\n\\end{figure}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and main results}\n\nWe begin by introducing some notation. By $H$ we denote the Hilbert space\nwith inner product $\\langle \\cdot,\\cdot \\rangle$ and norm $\\|\\cdot\\|$,\nand by $A$ we denote an unbounded operator from its domain $D(A)\n\\subset H$ to $H$.\n\nIf $A$ is self-adjoint and has a compact resolvent operator, then it has an orthonormal basis of eigenvectors. Unfortunately, even a slight perturbation of $A$ can destroy the self-adjointness of $A$ and so also the orthonormal basis property of the eigenvectors. However, in general the (normalized) eigenvectors, $\\{\\phi_n\\}_{n \\in {\\mathbb N}}$  will still form a Riesz basis, i.e., their span is dense in $H$ and there exist (positive) constants $m$ and $M$ such that\n\\begin{equation}\n  \\label{eq:1.1}\n    m \\sum_{n=1}^N |\\alpha_n|^2 \\leq \\left\\|\\sum_{n=1}^N \\alpha_n \\phi_n \\right\\|^2 \\leq  m \\sum_{n=1}^N |\\alpha_n|^2\n\\end{equation}\nfor every sequence $\\{\\alpha_n\\}_{n=1}^N$.\nIf $A$ processes a Riesz basis of eigenvectors, then many system theoretic properties like stability, controllability, etc.\\ are easily checkable, see e.g.\\ \\cite{CuZw95}. \n\nSince this Riesz-basis property is so important there is an extensive\nliterature on this problem. We refer to the book of Dunford and\nSchwartz \\cite{DuSc71}, where this problem is treated for discrete\noperators, i.e., the inverse of $A$ is compact. They apply these\nresults to differential operators. Riesz spectral properties for\ndifferential operators is also the subject of Mennicken and Moller\n\\cite{MeMo03} and Tretter \\cite{Tret00}. In these references no use is\nmade of the fact that for many differential operators the abstract\ndifferential equation\n\\begin{equation}\n  \\label{eq:1.1a}\n  \\dot{x}(t) = A x(t), \\qquad x(0)=x_0\n\\end{equation}\non the Hilbert space $H$ has a unique solution for every initial\ncondition $x_0$, i.e., $A$ is the infinitesimal generator of a\n$C_0$-semigroup. In \\cite{XuYu05} this property is used. The property\nwill also be essential in our paper. In many applications the differential\noperator $A$ arises from a partial differential equation, for which it is\nknown that (\\ref{eq:1.1a}) has a solution. Hence the assumption that $A$ generates a\n$C_0$-semigroup is not strong. However, for our result the semigroup\nproperty does not suffice, we need that $A$ generates a group, i.e.,\n(\\ref{eq:1.1a}) possesses a unique solution forward and backward in\ntime. Since we also assume that the eigenvalues lie in a strip\nparallel to the imaginary axis, the group condition is not very\nrestrictive. For more information on groups and semigroups, we refer\nthe reader to \\cite{CuZw95,EnNa00}.\n\nThe approach which we take to prove our result is different to the one\ntaken in \\cite{DuSc71, MeMo03, Tret00}, and \\cite{XuYu05}. We use\nthe fact that every generator of a group has a bounded ${\\mathcal\n  H}_{\\infty}$-calculus on a strip. This means that to every complex\nvalued function $f$ bounded and analytic in a strip parallel to the\nimaginary axis, there exists a bounded operator $f(A)$. For more detail\nwe refer to \\cite{Haas06}. Note that the ${\\mathcal\n  H}_{\\infty}$-calculus extends the functional calculus of Von~Neumann\n\\cite{Neum96} for self-adjoint operators.\n\nWe formulate our main results.\n\\begin{theorem}\n  \\label{T1.1}\n  Let $A$ be the infinitesimal generator of the $C_0$-group $(T(t))_{t\n    \\in {\\mathbb R}}$ on the Hilbert space $H$. We denote the\n  eigenvalues of $A$ by $\\lambda_n$ (counting with multiplicity), and\n  the corresponding (normalized) eigenvectors by $\\{\\phi_n\\}$. If the\n  following two conditions hold,\n  \\begin{enumerate}\n  \\item The span of the eigenvectors form a dense set in $H$;\n  \\item The point spectrum has a {\\em uniform gap}, i.e.,\n    \\begin{equation}\n    \\label{eq:1.2}\n    \\inf_{n \\neq m} |\\lambda_n - \\lambda_m| > 0.\n    \\end{equation}\n  \\end{enumerate}\n  Then the eigenvectors form a Riesz basis on $H$.\n\\end{theorem} \n\n\\mbox{}From this result, we have some easy consequences.\n\\begin{remark}%\n\\label{R1.2}\n  \\begin{itemize}\n  \\item Since we are counting the eigenvalues with multiplicity, we see\n    from (\\ref{eq:1.2}) that all eigenvalues are simple.\n  \\item If the operator possesses a Riesz basis of eigenvectors, then\n    the spectrum equals the closure of the point spectrum. Using\n    (\\ref{eq:1.2}), we see that the spectrum of $A$, $\\sigma(A)$,\n    is pure point spectrum, and $\\sigma(A) =\\{\\lambda_n\\}_{n\\in\n      {\\mathbb N}}$.\n  \\item It is easy to see that for $\\lambda \\in \\rho(A)$, the\n    finite-range operator $\\sum_{n=1}^N \\frac{1}{\\lambda-\\lambda_n}\n    \\phi_n$ converges uniformly to $(\\lambda I - A)^{-1}$. Thus under\n    the conditions of Theorem \\ref{T1.1}, the resolvent operator of\n    $A$ is compact, i.e., $A$ is discrete.\n  \\end{itemize}\n\\end{remark}\n\nIf the eigenvalues do not satisfy (\\ref{eq:1.2}), then Theorem \\ref{T1.1} need not to hold. A simple counter example is given next.\n\\begin{example}\n  \\label{E1.3}  Let $H$ be a Hilbert space with orthonormal basis $\\{e_n\\}_{n \\in {\\mathbb N}}$. Define $A$ as\n  \\begin{equation}\n    \\label{eq:6}\n    A \\sum_{n \\in {\\mathbb N}} \\alpha_n e_n = \\sum_{k \\in {\\mathbb N}} (ki \\alpha_{2k-1} + \\alpha_{2k} ) e_{2k-1} + (k+\\frac{1}{k})i \\alpha_{2k} e_{2k}\n  \\end{equation}\n  with domain\n  \\begin{equation}\n    \\label{eq:5}\n    D(A) =\\left\\{ x= \\sum_{n=1}^{\\infty} \\alpha_n e_n \\in H \\mid \\sum_{n=1}^{\\infty} |n\\alpha_n|^2 < \\infty \\right\\}.\n  \\end{equation}\n  Hence the operator $A$ is block diagonal, i.e.,\n  \\begin{equation}\n    \\label{eq:7}\n    A = \\mathrm{diag}\\, \\left[ \\begin{array}{cc} ki & 1 \\\\ 0 & (k +\\frac{1}{k})i \\end{array} \\right].\n  \\end{equation}\n  Using this structure, it is easy to see that $A$ generates a strongly continuous group on $H$, and that its eigenvalues are given by $\\{ ki, (k+\\frac{1}{k})i; k \\in {\\mathbb N}\\}$, with the normalized eigenvectors $\\phi_{2k-1} = e_{2k-1}$ and $\\phi_{2k} = \\frac{1}{\\sqrt{k^2+1}} \\left( ki e_{2k-1} + e_{2k} \\right)$, $k \\in {\\mathbb N}$. \n\n  Since\n  \\[\n     \\inf_k \\| i \\phi_{2k-1} - \\phi_{2k} \\| =0\n  \\]\n  we have that the eigenvectors do not from a Riesz basis.\n\\end{example}\n  \n  It is trivial to see that the eigenvalues in the above example can be written as the union of two sets with every subset satisfying (\\ref{eq:1.2}). Furthermore, if we would group the eigenvectors as $\\Phi_n = \\{ \\phi_{2n-1},\\phi_{2n} \\}$, then the (spectral) projections, $P_n$, on the span of $\\Phi_n$ satisfy\n  \\begin{equation}\n    \\label{eq:8}\n    \\sum_{n} \\|P_n x \\|^2 = \\|x\\|^2.\n  \\end{equation}\nThis is equivalent to the fact that the projections are orthonormal. The concept of a Riesz basis is an extension of the concept of an orthonormal basis. Similarly, we can extend the concept of orthonormal projections.\n\\begin{definition}\n  \\label{D1.4}\n  The family of projections $\\{P_n, n \\in {\\mathbb N}\\}$ is a Riesz family if there exists constants $m_1$ and $M_1$ such that\n  \\begin{equation}\n    \\label{eq:9}\n    m_1 \\|x\\|^2 \\leq \\sum_{n} \\|P_n x\\|^2 \\leq M_1 \\|x\\|^2\n  \\end{equation}\n  for all $x \\in H$.\n\\end{definition}\n\nPlease note that in \\cite[section I.1.4]{AvIv95} the range of the projections is called a (Riesz) basis. We found it confusing with the standard definition of a Riesz basis, and therefor we use Riesz family. This concept is equivalent to Riesz basis in parenthesis, see \\cite{Shka86}.\nWermer \\cite{Werm54} proved the following characterization.\n\\begin{lemma}\n\\label{L1.5}\n  A family of projections is a Riesz family if and only if there exists a $M_2$ such that for every subset ${\\mathbb J}$ of ${\\mathbb N}$ there holds\n  \\begin{equation}\n    \\label{eq:10}\n    \\|\\sum_{n \\in {\\mathbb J}} P_n \\| \\leq M_2.\n  \\end{equation}\n\\end{lemma}\n\nIn the example we saw that we had a Riesz family of spectral projections. The following theorem states that this always hold when the eigenvalues can be decomposed in  a finite number of sets with every set satisfying (\\ref{eq:1.2}).\n\\begin{theorem}\n\\label{T1.6}\n  Let $A$ be the infinitesimal generator of the $C_0$-group $(T(t))_{t\n    \\in {\\mathbb R}}$ on the Hilbert space $H$. We denote the\n  eigenvalues of $A$ by $\\lambda_n$ (counting with multiplicity). If the\n  following two conditions hold,\n  \\begin{enumerate}\n  \\item The span of the (generalized) eigenvectors form a dense set in $H$;\n  \\item The eigenvalues $\\{\\lambda_n\\}$ can be decomposed into $K$ sets, with every set having a uniform gap. \n  \\end{enumerate}\n  Then there are spectral projections $P_n$, $n \\in {\\mathbb N}$ such that\n  \\begin{enumerate}  \n  \\item $\\sum_n P_n = I$, i.e., for every $x\\in H$ there holds $\\lim_{N\\rightarrow \\infty} \\sum_{n=1}^N P_n x = x$;\n  \\item The dimension of the range of $P_n$ is at most $K$;\n  \\item The family of projections $\\{P_n, n \\in {\\mathbb N}\\}$ is a Riesz family.\n  \\end{enumerate}\n\\end{theorem}\n\nSo the difference with Theorem \\ref{T1.1} is that we allow for non-simple eigenvalues, and that the eigenvalues may cluster. Apart from these differences, the other remarks of Remark \\ref{R1.2} still hold. An operator satisfying the conditions of Theorem \\ref{T1.6} has pure point spectrum and has compact resolvent.\n\n\\section{Functional calculus for groups.}\n\\label{sec:2}\n\nWe begin by introducing some notation. \nSince $\\left(T(t)\\right)_{t \\in {\\mathbb R}}$ is a group, there exists\na $\\omega_0$ and $M_0$ such that $\\|T(t)\\| \\leq M_0 e ^{\\omega_0 |t|}$.\n\nFor $\\alpha>0$ we define a strip parallel to the imaginary axis by $S_{\\alpha}:=\\{ s\n\\in {\\mathbb C} \\mid - \\alpha < \\mathrm{Re}(s) < \\alpha \\}$.\n\nBy ${\\mathcal H}^{\\infty}(S_{\\alpha})$ we denote the linear space of\nall functions from $S_{\\alpha}$ to ${\\mathbb C}$ which are analytic\nand (uniformly) bounded on $S_{\\alpha}$. The norm of a function in\n${\\mathcal H}^{\\infty}(S_{\\alpha})$ is given by\n\\begin{equation}\n  \\label{eq:2.1}\n  \\|f\\|_{\\infty}  = \\sup_{s \\in S_{\\alpha}} |f(s)|.\n\\end{equation}\nIn Haase \\cite{Haas04,Haas06} it is shown that the generator of the\ngroup, $A$, has a ${\\mathcal H}^{\\infty}(S_{\\alpha})$-calculus for\n$\\alpha > \\omega_0$. This we explain in a little bit more detail.\n\nChoose $\\omega_1$ and $\\omega$ such that $\\omega_0 < \\omega_1 <\\alpha < \\omega$. Furthermore, let $\\Gamma = \\gamma_1 \\oplus \\gamma_2$ with $\\gamma_1 = -\\omega_1- ir$, $\\gamma_2=\\omega_1 + i r$, $r \\in {\\mathbb R}$. For $f \\in {\\mathcal H}^{\\infty}(S_{\\alpha})$ and $x \\in D(A^2)$ we define\n\\begin{equation}\n\\label{eq:2.2}\n  f(A)x = \\frac{1}{2\\pi i} \\int_{\\Gamma} \\frac{f(z)}{z^2- \\omega^2} (z I - A)^{-1} d z \\cdot (A^2 - \\omega^2)x .\n\\end{equation}\nFor $\\lambda \\not\\in \\overline{S_{\\alpha}}$ we have that \n\\[\n  \\left( \\frac{1}{\\lambda - \\cdot} \\right)(A)x = (\\lambda I - A)^{-1} x\n\\]\nFurthermore, the operator defined in (\\ref{eq:2.2}) extends to a bounded operator on $H$, and\n\\begin{equation}\n  \\label{eq:2.3}\n  \\|f(A)\\| \\leq c \\|f\\|_{\\infty},\n\\end{equation}\nwith $c$ independent of $f$.\n\nIn the following lemma we show that this functional calculus behaves\nlike one would expect from the functional calculus of von~Neumann and\nDunford.\n\\begin{lemma}\n  \\label{L2.1}\n  Let $A$ be the infinitesimal generator of a group and let $\\phi_n$\n  be an eigenvector for the eigenvalue $\\lambda_n$. Then for every $f\n  \\in {\\mathcal H}^{\\infty}(S_{\\alpha})$ there holds that\n  \\begin{equation}\n    \\label{eq:2.4}\n      f(A) \\phi_n = f(\\lambda_n) \\phi_n, \\qquad n \\in {\\mathbb N}.\n  \\end{equation}\n  Furthermore, if $\\phi_{n,j}$ is the $j$-th generalized eigenvector for the eigenvalue $\\lambda_n$, i.e., $(A-\\lambda_n) \\phi_{n,j} =\\phi_{n,j-1}$, $j\\geq 1$, with $\\phi_{n,0}=\\phi_n$, then\n  \\begin{equation}\n    \\label{eq:1}\n     f(A) \\phi_{n,j} = \\sum_{m=0}^j \\frac{f^{(j-m)}(\\lambda_n)}{(j-m)!} \\phi_{n,m}, \\qquad n \\in {\\mathbb N},\n \\end{equation}\n  where $f^{(\\ell)}$ denotes the $\\ell$-th derivative of $f$.\n\\end{lemma}\n  {\\bf Proof}:\\\/ Since $\\phi_n$ is an eigenvector, it is an element of\n  $D(A^2)$ and so we may use equation (\\ref{eq:2.2}).\n  Hence\n  \\begin{eqnarray*}\n    f(A)\\phi_n &=&  \\frac{1}{2\\pi i} \\int_{\\Gamma} \\frac{f(z)}{z^2- \\omega^2} (z I - A)^{-1} d z \\cdot (A^2 - \\omega^2) \\phi_n\\\\\n    &=& \\frac{1}{2\\pi i} \\int_{\\Gamma} \\frac{f(z)}{z^2- \\omega^2} (z I - A)^{-1} d z\\, (\\lambda_n^2-\\omega^2)\\phi_n\\\\\n    &=& (\\lambda_n^2-\\omega^2) \\frac{1}{2\\pi i} \\int_{\\Gamma} \\frac{f(z)}{z^2- \\omega^2} (z I - \\lambda_n)^{-1}\\phi_n d z.\n  \\end{eqnarray*}\n  Since $f$ is bounded on $S_{\\alpha}$, we see that the integrand\n  converges quickly to zero for $|z|$ large. Hence we may apply Cauchy\n  residue theorem. The only pole within the contour is $\\lambda_n$,\n  and so we obtain\n  \\[\n    f(A) \\phi_n = (\\lambda_n^2-\\omega^2) \\frac{f(\\lambda_n)}{\\lambda_n^2- \\omega^2} \\phi_n = f(\\lambda_n) \\phi_n.\n  \\]\n  This shows equation (\\ref{eq:2.4}). We now prove the assertion for the generalized eigenvectors. \n\n  By induction it is easy to show that\n  \\begin{equation}\n    \\label{eq:2}\n    (zI-A)^{-1} \\phi_{n,j} = \\sum_{m=0}^j \\frac{1}{(z-\\lambda_n)^{j+1-m}} \\phi_{n,m}.\n  \\end{equation}\n  Using (\\ref{eq:2.2}) we have that\n  \\begin{align*}\n    f(A) \\phi_{n,j}  &=  \\frac{1}{2\\pi i} \\int_{\\Gamma} \\frac{f(z)}{z^2- \\omega^2} (z I - A)^{-1} d z \\cdot (A^2 - \\omega^2) \\phi_{n,j}\\\\\n      &= \\frac{1}{2\\pi i} \\int_{\\Gamma} \\frac{f(z)}{z^2- \\omega^2} (z I - A)^{-1} d z ((\\lambda_n^2 - \\omega^2)\\phi_{n,j} + 2\\lambda_n \\phi_{n,j-1} + \\phi_{n,j-2})\\\\\n      &=  (\\lambda_n^2-\\omega^2) \\frac{1}{2\\pi i} \\int_{\\Gamma} \\frac{f(z)}{z^2- \\omega^2} \\sum_{m=0}^j (z I - \\lambda_n)^{-(j-m+1)}\\phi_{n,m} d z + \\\\\n   &\\qquad 2\\lambda_n \\frac{1}{2\\pi i} \\int_{\\Gamma} \\frac{f(z)}{z^2- \\omega^2} \\sum_{m=0}^{j-1} (z I - \\lambda_n)^{-(j-m)}\\phi_{n,m} d z + \\\\\n   &\\qquad\n  \\frac{1}{2\\pi i} \\int_{\\Gamma} \\frac{f(z)}{z^2- \\omega^2} \\sum_{m=0}^{j-2} (z I - \\lambda_n)^{-(j-m-1)}\\phi_{n,m} d z \\\\\n   &= (\\lambda_n^2-\\omega^2) \\sum_{m=0}^{j} \\frac{g^{(j-m)}(\\lambda_n)}{(j-m)!} \\phi_{n,m} + \\\\\n   &\\qquad 2\\lambda_n \\sum_{m=0}^{j-1} \\frac{g^{(j-1-m)}(\\lambda_n)}{(j-m-1)!} \\phi_{n,m} +\n   \\sum_{m=0}^{j-2} \\frac{g^{(j-2-m)}(\\lambda_n)}{(j-m-2)!} \\phi_{n,m},\n  \\end{align*}\n  where we have introduced $g(z)=f(z)\/(z^2-\\omega^2)$. Using the fact that $f(z) = (z^2-\\omega^2) g(z)$, we see that for $\\ell \\geq 2$\n  \\[\n     f^{(\\ell)}(z) = (z^2-\\omega^2) g^{(\\ell)}(z) + 2 z\\ell g^{(\\ell-1)}(z) + \\ell(\\ell-1) g^{(\\ell-2)}(z) \n  \\]\n  and\n  \\[\n     f^{(1)}(z) = (z^2-\\omega^2) g^{(1)}(z) + 2 z g(z).\n  \\]\n  This implies that\n  \\begin{eqnarray*}\n   f(A) \\phi_{n,j} &=& \\sum_{m=0}^{j-2} \\left[ (\\lambda_n^2-\\omega^2) \\frac{g^{(j-m)}(\\lambda_n)}{(j-m)!} +\\right. \\\\\n   && \\hphantom{\\sum_{m=0}^{j-2} \\left[ \\right]}\n   \\left. 2\\lambda_n \\frac{g^{(j-1-m)}(\\lambda_n)}{(j-m-1)!} +\n   \\frac{g^{(j-2-m)}(\\lambda_n)}{(j-m-2)!} \\right] \\phi_{n,m} + \\\\\n   &&\n  \\left[ (\\lambda_n^2 - \\omega^2) g^{(1)}(\\lambda_n) + 2 \\lambda_n g(\\lambda_n) \\right] \\phi_{n,j-1}+ (\\lambda_n^2 - \\omega^2) g(\\lambda_n) \\phi_{n,j}\\\\\n  &=&  \\sum_{m=0}^{j-2} \\frac{1}{(j-m)!}f^{(j-m)}(\\lambda_{n}) \\phi_{n,m} + f^{(1)}(\\lambda_n)  \\phi_{n,j-1}  + f(\\lambda_n) \\phi_{n,j}.\n  \\end{eqnarray*}\n  Hence we have proved the assertion. \\hfill$\\Box$\n\\medskip\n\n\\section{Interpolation sequences}\n\nFor the proof of Theorem \\ref{T1.1} we need the following\ninterpolation result, see \\cite[Theorem VII.1.1]{Garn07}.\n\\begin{theorem}\n  \\label{T2.2}\n  Consider the sequence ${\\mu_n}$ which satisfies $\\beta_1 >\n  \\mathrm{Re}(\\mu_n) > \\beta_2 >0$ and $\\inf_{n\\neq m} |\\mu_n - \\mu_m|\n  >0$. Then for every bounded sequence $\\{\\alpha_n\\}_{n\\in {\\mathbb\n      N}}$ of complex numbers there exists a function $g$ holomorphic\n  and bounded in the right-half plane ${\\mathbb C}_+:=\\{ s \\in\n  {\\mathbb C} \\mid \\mathrm{Re}(s) >0\\}$ such that\n  \\begin{equation}\n    \\label{eq:2.5}\n      g(\\mu_n) = \\alpha_n.\n    \\end{equation}\n  Furthermore, there exists an $M$ independent of $g$ and $\\{\\alpha_n\\}_{n \\in {\\mathbb N}}$ such that\n  \\begin{equation}\n    \\label{eq:2.6}\n      \\sup_{\\mathrm{Re}(s) >0} |g(s)| \\leq M \\sup_{n \\in {\\mathbb N}} |\\alpha_n|.\n  \\end{equation}\n\\end{theorem}\n\nA sequence $\\{\\mu_n\\}_{n \\in {\\mathbb N}}$ satisfying the conditions of the above theorem is called an {\\em interpolation sequence}. It is well known that for any interpolation sequence, we can find a Blaschke product with exactly this sequence as its zero set. Using Schwartz lemma, and Lemma VII.5.3 of \\cite{Garn07} the following is easy to show.\n  \\begin{lemma}\n  \\label{L3.2}\n    Let $\\{\\mu_n\\}_{n \\in {\\mathbb N}}$ be an interpolating sequence, and let $0< \\delta := \\inf_{n\\neq m} |\\mu_n - \\mu_m|$. Furthermore, let $B(a, r)$ denote the ball in the complex plane with center $a$ and radius $r$. \n\n  Let $f$ be a function defined on $B(\\mu_n,\\delta\/2)$ and which is analytic and bounded on this set. Furthermore, $f$ is zero at $\\mu_n$, i.e., $f(\\mu_n)=0$. Then there exists a constant $m_1$ independent of $n$ and $f$ such that\n  \\begin{equation}\n    \\label{eq:11}\n      \\sup_{s \\in B(\\mu_n,\\delta\/2)} \\left| \\frac{f(s)}{Bl(s)} \\right| \\leq m_1 \\sup_{s \\in B(\\mu_n,\\delta\/2)} |f(s)|,\n  \\end{equation}\n  where $Bl(s)$ is Blaschke product with zeros $\\{\\mu_n\\}_{n \\in {\\mathbb N}}$.\n\n  Similarly,  let $f$ be a function defined on $B(\\mu_n,\\delta\/2)$ and which is analytic and bounded on this set. Furthermore, $f$ has $\\ell$ zeros at $\\mu_n$, i.e., $f(\\mu_n)=\\cdots = f^{(\\ell)}(\\mu_n)=0$. Then there exists a constant $m_\\ell$ independent of $n$ and $f$ such that\n  \\begin{equation}\n    \\label{eq:11a}\n      \\sup_{s \\in B(\\mu_n,\\delta\/2)} \\left| \\frac{f(s)}{Bl(s)^\\ell} \\right| \\leq m_\\ell \\sup_{s \\in B(\\mu_n,\\delta\/2)} |f(s)|.\n  \\end{equation}\n  \\end{lemma}\n\\begin{theorem}\n  \\label{T3.3} \n  Let $\\{\\mu_n\\}_{n \\in {\\mathbb N}}$ be a sequence of numbers in the right-half plane, and let $\\{\\zeta_n\\}_{n \\in {\\mathbb N}}$ be an interpolation sequence. Furthermore, let $K$ be a positive natural number. Suppose that we can find positive real numbers $r_n$ such that \n  \\begin{itemize}\n  \\item $0< \\inf_n r_n \\leq \\sup_n r_n < \\infty$;\n  \\item $\\{\\mu_n\\}_{n \\in {\\mathbb N}} \\subset \\cup_{n\\in {\\mathbb N}} B(\\zeta_n,r_n)$;\n  \\item The number of $\\mu_n$'s in $B(\\zeta_k,r_k)$ is bounded by $K$, and \n  \\item The distance between every two balls $B(\\zeta_n,r_n)$ and $B(\\zeta_m,r_m)$, $n \\neq m$, is bounded away from zero.\n  \\end{itemize}\n  Let $\\nu$ be an element of ${\\mathbb N} \\cup \\{0\\}$. Under these assumptions we have that for every bounded sequence $\\{\\alpha_n\\}_{n \\in {\\mathbb N}}$ there exists a function $g$ holomorphic and bounded in the right-half plane ${\\mathbb C}_+$ such that\n  \\begin{equation}\n    \\label{eq:4}\n     g(\\mu_k) = \\alpha_n \\quad \\mbox{if}\\quad  \\mu_k \\in B(\\zeta_n,r_n)\n  \\end{equation}\n  and \n  \\begin{equation}\n    \\label{eq:4a}\n     g^{(q)}(\\mu_k) = 0 \\quad \\mbox{for}\\quad  1\\leq q \\leq \\nu,\\mbox{ and } \\mu_k \\in B(\\zeta_n,r_n)\n  \\end{equation}\n  Furthermore, there exists an $M$ independent of $g$ and $\\{\\alpha_n\\}_{n \\in {\\mathbb N}}$ such that\n  \\begin{equation}\n    \\label{eq:16}\n    \\sup_{\\mathrm{Re}(s) >0} |g(s)| \\leq M \\sup_{n \\in {\\mathbb N}} |\\alpha_n|.\n  \\end{equation}\n  Hence the function $g$ interpolates the points $\\mu_k$, but points close to each other are given the same value.\n\\end{theorem}\n  {\\bf Proof}:\\\/ We present the proof for the case that $\\kappa =2$ and $\\nu=1$. For higher values of $\\kappa$ and $\\nu$ the proof goes similarly. We denote the points $\\mu_n \\in B(\\zeta_n,r_n)$ by $\\xi_n$ and $\\gamma_n$, and we assume that $\\gamma_n \\neq \\xi_n$.  By the assumptions, we know that these are interpolation sequences.\n\nLet $Bl_1(s)$ and $BL_2(s)$ be the Blaschke products with zeros $\\{\\xi_n\\}_{n \\in {\\mathbb N}}$ and $\\{\\gamma_n\\}_{n \\in {\\mathbb N}}$, respectively.\n\nWe write $g$ in the following form\n\\begin{equation}\n  \\label{eq:12}\n  g(s) = g_1(s) + Bl_1(s)g_2(s) + Bl_1(s)^2 g_3(s) + Bl_1(s)^2Bl_2(s) g_4(s), \n\\end{equation}\nand we construct bounded analytic functions $g_j$ such that (\\ref{eq:4})--(\\ref{eq:16}) are satisfied. For our situation, the interpolation conditions become $g(\\xi_n)=g(\\gamma_n)=\\alpha_n$ and $g^{(1)}(\\xi_n)=g^{(1)}(\\gamma_n)=0$. Using the form (\\ref{eq:12}), these conditions are equivalent to\n\\begin{align}\n  \\label{eq:3}\n   g_1(\\xi_n) = & \\, \\alpha_n\\\\ \n   \\label{eq:13}\n   g_2(\\xi_n) = & -\\frac{g_1^{(1)}(\\xi_n)}{Bl_1^{(1)}(\\xi_n)}\\\\\n   \\label{eq:14}\n   g_3(\\gamma_n) =& \\,\\frac{\\alpha_n - g_1(\\gamma_n)- Bl_1(\\gamma_n) g_2(\\gamma_n)}{Bl_1(\\gamma_n)^2} \\\\\n   \\label{eq:15}\n   g_4(\\gamma_n) = &  \\left[ g_1^{(1)}(\\gamma_n) + Bl_1(\\gamma_n)g_1^{(1)}(\\gamma_n) + Bl_1^{(1)}(\\gamma_n) g_2(\\gamma_n) +\\right.\\\\\n\\nonumber\n    &\\left. 2 Bl_1(\\gamma_n) Bl_1^{(1)}(\\gamma_n)g_3(\\gamma_n) + Bl_1(\\gamma_n)^2g_3^{(1)}(\\gamma_n)\\right]\\left[Bl_1(\\gamma_n)^2Bl_2^{(1)}(\\gamma_n)\\right]^{-1}.\n\\end{align}\nSince $\\{\\xi_n\\}$ is an interpolation sequences, and since $\\{\\alpha_n\\}$ is a bounded sequence, we have by Theorem \\ref{T2.2} the existence of a bounded $g_1$ for which (\\ref{eq:2.6}) holds.\n\n\\mbox{} Define the function $f(s)=-g_1(s)+\\alpha_n$. This is clearly bounded and analytic on $B(\\zeta_n,r_n)$ and zero at $s=\\xi_n$. By Lemma \\ref{L3.2} the function $f(s)\/Bl_1(s)$ is bounded (uniformly in $n$) in this ball. l'Hopital gives that the value at $s=\\xi_n$ equals the right-hand side of (\\ref{eq:13}), and so the right hand-side is a bounded sequence. Furthermore, since $\\{\\xi_n\\}_{n \\in {\\mathbb N}}$ is an interpolation sequence we can apply Theorem \\ref{T2.2} and find a bounded analytic function $g_2$ satisfying (\\ref{eq:13}).\n\nConsider the function\n\\begin{equation}\n  \\label{eq:17}\n  q_1(s) = \\frac{\\alpha_n - g_1(s)- Bl_1(s) g_2(s)}{Bl_1(s)^2} \n\\end{equation}\nBy (\\ref{eq:3}) and (\\ref{eq:13}), we have that the denominator has two zero's in $\\xi_n$. Furthermore, it is analytic on $B(\\zeta_n,r_n)$ and bounded independently of $n$. Lemma \\ref{L3.2} implies that $q_1(s)$ is uniformly bounded. In particular, the sequence $q_1(\\gamma_n)$ is uniformly bounded. By Theorem \\ref{T2.2} we construct a bounded $g_3$ satisfying (\\ref{eq:14}).\n\nIt remains to show that the right hand-side of (\\ref{eq:15}) is a uniformly bounded sequence. The sequence $-\\frac{g_3^{(1)}(\\gamma_n)}{Bl_2^{(1)}(\\gamma_n)}$ is uniformly bounded for the same reason why the sequence in (\\ref{eq:13}) was bounded. So we may disregard that term in (\\ref{eq:15}). Using the value of $g_3(\\gamma_n)$ as found in (\\ref{eq:14}), we have that the denominator of (\\ref{eq:15}) becomes\n\\begin{align}\n  \\label{eq:18}\n    g_1^{(1)}(\\gamma_n) + Bl_1(\\gamma_n)g_1^{(1)}(\\gamma_n) +& Bl_1^{(1)}(\\gamma_n) g_2(\\gamma_n) + \\\\\n\\nonumber\n   &2 Bl^{(1)}_1(\\gamma_n) \\frac{\\alpha_n - g_1(\\gamma_n)- Bl_1(\\gamma_n) g_2(\\gamma_n)}{Bl_1(\\gamma_n)}.\n\\end{align}\nBased on this and using equation (\\ref{eq:17}) we define in $B(\\zeta_n,r_n)$ the bounded analytic function\n\\begin{equation}\n  \\label{eq:20}\n  f(s)= g_1^{(1)}(s) + Bl_1(s)g_1^{(1)}(s) + Bl_1^{(1)}(s) g_2(s) + 2 Bl_1(s)Bl^{(1)}_1(s) q_1(s).\n\\end{equation}\nUsing (\\ref{eq:13}), we have that $f(\\xi_n)=0$, and using that \n\\[\n   q_1(\\xi_n) = \\frac{g_1^{(2)}(\\xi_n) - Bl_1^{(2)}(\\xi_n) g_2(\\xi_n) - 2 Bl_1^{(1)}(\\xi_n) g_2^{(1)}(\\xi_n)}{2 \\left( Bl_1^{(1)}(\\xi_n)\\right)^2}\n\\]\nwe find that $f^{(1)}(\\xi_n)$ is zero as well. By Lemma \\ref{L3.2}, we conclude that \n  \\begin{align*}\n    \\left[ g_1^{(1)}(\\gamma_n) + Bl_1(\\gamma_n) \\right.  & g_1^{(1)}(\\gamma_n) + Bl_1^{(1)}(\\gamma_n) g_2(\\gamma_n) +\\\\\n     & \\left. 2 Bl_1(\\gamma_n) Bl_1^{(1)}(\\gamma_n)g_3(\\gamma_n) + Bl_1(\\gamma_n)^2g_3^{(1)}(\\gamma_n)\\right]\\left[Bl_1(\\gamma_n)^2\\right]^{-1}\n  \\end{align*}\n  is uniformly bounded. Since $Bl_2(s)$ is a Blaschke product with zeros $\\{\\gamma_n\\}$ we have that $Bl_2^{(1)}(\\gamma_n)$ is bounded away from zero. Hence the right hand-side of (\\ref{eq:15}) is a bounded sequence, and so by Theorem \\ref{T2.2} we can find the interpolating $g_4$. This concludes the construction.\n \\hfill$\\Box$\n\\medskip\n\nNow we have all the ingredients for the proof of Theorem \\ref{T1.1} and \\ref{T1.6}.\n\n\n\\section{Proof of Theorem \\ref{T1.1} and \\ref{T1.6} }\n\nAs may be clear from the formulation of the Theorems \\ref{T1.1} and \\ref{T1.6}, Theorem \\ref{T1.1} is a special case of Theorem \\ref{T1.1}. However, since the Riesz basis property is for applications more important than Riesz family, we we decided to formulate them separately. The proof of Theorem \\ref{T1.1} is more simple than that of the general theorem, but the underlying ideas are the same. \n\\medskip\n\n\\noindent{\\bf Proof of Theorem \\ref{T1.1}}: \nLet $\\alpha$ be the positive number defined at the beginning of Section \\ref{sec:2}. We define the complex numbers $\\mu_n$ as\n\\begin{equation}\n  \\label{eq:2.7}\n  \\mu_n = \\lambda_n + \\alpha \\qquad n \\in {\\mathbb N}.\n\\end{equation}\nBy the conditions on $\\alpha$ and $\\lambda_n$, we see that $\\{\\mu_n\\}_{n \\in {\\mathbb N}}$ satisfies the conditions of Theorem \\ref{T2.2}.\n\nLet ${\\mathbb J}$ be a subset of ${\\mathbb N}$. Since the eigenvalues\nsatisfy (\\ref{eq:1.2}), we conclude by Theorem \\ref{T2.2} there exists\na function $g_{{\\mathbb J}}$ bounded and analytic in ${\\mathbb C}_+$\nsuch that\n\\begin{equation}\n  \\label{eq:2.8}\n  g_{\\mathbb J}(\\mu_n)= \n  \\begin{cases} 1, & \\quad \\mbox{if } n \\in {\\mathbb J}\\\\\n                0, & \\quad \\mbox{if } n \\not\\in {\\mathbb J}. \n  \\end{cases} \n\\end{equation}\nFurthermore, see (\\ref{eq:2.6})\n\\begin{equation}\n  \\label{eq:2.10}\n  \\sup_{s \\in {\\mathbb C}_+} |g_{{\\mathbb J}}(s)| \\leq M.\n\\end{equation}\n\nGiven this $g_{\\mathbb J}$ we define $f_{\\mathbb J}$ as\n\\begin{equation}\n  \\label{eq:2.11}\n  f_{\\mathbb J}(s) = g_{\\mathbb J}(s+ \\alpha), \\qquad s \\in S_{\\alpha}.\n\\end{equation}\nThen using the properties of $g_{\\mathbb J}$ we have that $f_{\\mathbb J} \\in {\\mathcal H}_{\\infty}(S_{\\alpha})$, \n\\begin{equation}\n  \\label{eq:2.12}\n  f_{\\mathbb J}(\\lambda_n)= \n  \\begin{cases}  1, & \\quad \\mbox{if } n \\in {\\mathbb J}\\\\\n                 0, & \\quad \\mbox{if } n \\not\\in {\\mathbb J},\n  \\end{cases} \n\\end{equation}\nand there exists a $M>0$ independent of $f_{\\mathbb J}$ such that \n\\begin{equation}\n  \\label{eq:2.13}\n  \\|f_{{\\mathbb J}}\\|_{\\infty} \\leq M.\n\\end{equation}\n\nNext we identify the operator $f_{\\mathbb J}(A)$. Combining\n(\\ref{eq:2.4}) with (\\ref{eq:2.12}) gives\n\\[\n  f_{\\mathbb J}(A)\\phi_n = \n  \\begin{cases}  \\phi_n, & \\quad \\mbox{if } n \\in {\\mathbb J}\\\\\n                 0, & \\quad \\mbox{if } n \\not\\in {\\mathbb J}.\n  \\end{cases}\n\\]\nSince $f_{\\mathbb J}(A)$ is a linear operator, we obtain that\n\\begin{equation}\n  \\label{eq:2.14}\n  f_{\\mathbb J}(A) \\left(\\sum_{n=1}^N \\alpha_n \\phi_n\\right) = \\sum_{n\\in {\\mathbb J} \\cap \\{1,\\cdots,N\\}} \\alpha_n \\phi_n.\n\\end{equation}\nBy assumption the span of $\\{\\phi_n\\}_{n \\in {\\mathbb N}}$ is dense in\n$H$. Furthermore, $f_{\\mathbb J}(A)$ is a bounded operator. So we\nconclude that $f_{\\mathbb J}$ is the spectral projection associated to\nthe spectral set $\\{ \\lambda_n \\mid n \\in {\\mathbb J}\\}$.\n\nCombining (\\ref{eq:2.3}) with (\\ref{eq:2.13}) we have that these\nspectral projections are uniformly bounded. Since the eigenvalues are\nsimple, this implies that the (normalized) eigenvectors form a Riesz\nbasis, see Lemma \\ref{L1.5}. \\hfill$\\Box$\n\\medskip\n\n\\noindent{\\bf Proof of Theorem \\ref{T1.6}}:\nAs in the previous proof we can shift the eigenvalues by $\\alpha$ such that they all lies in the right half plane, and they are bounded away from the imaginary axis. We denote these shifted eigenvalues by $\\mu$. \nSince the $\\lambda_n$'s can be decomposed into $K$ interpolation sequences, the same holds for $\\mu_n$. Hence we can group the $\\mu_n$'s as in Theorem \\ref{T3.3}. Note that we are counting the eigenvalues $\\lambda_n$'s, and thus $\\mu_n$, with their multiplicity, and so in every ball there can at most be $K$ different values, and the multiplicity of every value is also bounded by $K$. Let us renumber the $\\mu_n$'s such that the values in the $n$'th ball are given by $\\mu_{n,k}$, $k=1,\\cdots,n_k$.  The eigenvectors corresponding to $\\lambda_{n,k}=\\mu_{n,k}-\\alpha$ are denoted by $\\phi_{n,k,0}$, and the generalized eigenvectors by $\\phi_{n,k,j}$, $j =1,\\cdots, j_{n,k}$. By construction we have that \n\\begin{equation}\n  \\label{eq:21}\n  \\sum_{k=1}^{n_k} \\left[ j_{n,k}+1\\right] \\leq K.\n\\end{equation}\n\nLet ${\\mathbb J}$ be a subset of ${\\mathbb N}$. By the above, we conclude from Theorem \\ref{T3.3} that there exists a function $g_{\\mathbb J}$ bounded and analytic in ${\\mathbb C}_+$ such that\n\\begin{equation}\n  \\label{eq:2.15}\n  g_{\\mathbb J}(\\mu_{n,k})= \n  \\begin{cases} 1, & \\quad \\mbox{if } n \\in {\\mathbb J}\\\\\n                0, & \\quad \\mbox{if } n \\not\\in {\\mathbb J}. \n  \\end{cases} \n\\end{equation}\nand\n\\begin{equation}\n  \\label{eq:2.16}\n   g_{\\mathbb J}^{(j)}(\\mu_{n,k})=0, \\qquad j=1,\\cdots, K.\n\\end{equation}\nFurthermore, see (\\ref{eq:16})\n\\begin{equation}\n  \\label{eq:2.17}\n  \\sup_{s \\in {\\mathbb C}_+} |g_{{\\mathbb J}}(s)| \\leq M.\n\\end{equation}\n\nGiven this $g_{\\mathbb J}$ we define $f_{\\mathbb J}$ as\n\\begin{equation}\n  \\label{eq:2.18}\n  f_{\\mathbb J}(s) = g_{\\mathbb J}(s+ \\alpha), \\qquad s \\in S_{\\alpha}.\n\\end{equation}\nThen using the properties of $g_{\\mathbb J}$ we have that $f_{\\mathbb J} \\in {\\mathcal H}_{\\infty}(S_{\\alpha})$, \n\\begin{equation}\n  \\label{eq:2.19}\n  f_{\\mathbb J}(\\lambda_{n,k})= \n  \\begin{cases}  1, & \\quad \\mbox{if } n \\in {\\mathbb J}\\\\\n                 0, & \\quad \\mbox{if } n \\not\\in {\\mathbb J},\n  \\end{cases} \n\\end{equation}\n\\begin{equation}\n  \\label{eq:2.20}\n   f_{\\mathbb J}^{(j)}(\\lambda_{n,k})=0, \\qquad j=1,\\cdots, K.\n\\end{equation}\nand there exists a $M>0$ independent of ${\\mathbb J}$ such that \n\\begin{equation}\n  \\label{eq:2.21}\n  \\|f_{{\\mathbb J}}\\|_{\\infty} \\leq M.\n\\end{equation}\n\nNext we identify the operator $f_{\\mathbb J}(A)$. Combining\n(\\ref{eq:1}) with (\\ref{eq:2.19}) and (\\ref{eq:2.20}), gives\n\\[\n  f_{\\mathbb J}(A)\\phi_{n,k,j} = \n  \\begin{cases}  \\phi_{n,k,j}, & \\quad \\mbox{if } n \\in {\\mathbb J}\\\\\n                 0, & \\quad \\mbox{if } n \\not\\in {\\mathbb J}.\n  \\end{cases}\n\\]\nSince $f_{\\mathbb J}(A)$ is a linear operator, we obtain that\n\\begin{equation}\n  \\label{eq:2.22}\n  f_{\\mathbb J}(A) \\left(\\sum_{n=1}^N \\sum_{k=1}^{n_k} \\sum_{j=0}^{j_{n,k}} \\alpha_{n,k,j} \\phi_{n,k,j}\\right) = \\sum_{n\\in {\\mathbb J} \\cap \\{1,\\cdots,N\\}}\\sum_{k=1}^{n_k} \\sum_{j=0}^{j_{n,k}}  \\alpha_{n,k,j} \\phi_{n,k,j}.\n\\end{equation}\nBy assumption the span of $\\{\\phi_{n,k,j}\\}_{n \\in {\\mathbb N}, k=1,\\cdots,n_k,,j=0,\\cdots,j_{n,k}}$ is dense in\n$H$. Furthermore, $f_{\\mathbb J}(A)$ is a bounded operator. So we\nconclude that $f_{\\mathbb J}$ is the spectral projection associated to\nthe spectral set $\\{ \\lambda_{n,k} \\mid n \\in {\\mathbb J}\\}$.\n\nCombining (\\ref{eq:2.21}) with (\\ref{eq:2.22}) we have that these\nspectral projections are uniformly bounded. By Lemma \\ref{L1.5} we conclude that the spectral projections $\\{P_n\\}_{n \\in {\\mathbb N}}$, where $P_n$ is the spectral projection associated to the eigenvalues in the $n$-th ball, are a spectral family. From (\\ref{eq:21}) we see that the dimension of the range of $P_n$ is bounded by $K$. \\hfill$\\Box$\n\\medskip\n \n\n\n\n\n\n\n\n\\section{Closing remarks}\n\\label{sec:3}\n\nA natural question is the following: {\\em If $A$ is the infinitesimal generator of a group and $A$ has only point spectrum with multiplicity one, is the span over all eigenvectors dense in $H$}? \n\nIn general the answer to this question is negative. On page 665 of Hille and Phillips \\cite{HiPh57} one may find an example of a generator of a group without any spectrum. \n\nHowever, there are some interesting cases for which the answer is positive. If $A$ generates a bounded group, i.e., $\\sup_{t \\in {\\mathbb R}}\\|T(t)\\| < \\infty$, then $A$ is similar to a skew-adjoint operator, and so there is a complete spectral measure, see \\cite{Cast83,Zwar01}. Another interesting situation is the following. Since $A$ generates a group, it can be written as $A=A_0+Q$, where $A_0$ generates a bounded group, and $Q$ is a bounded linear operator, see \\cite{Haas04}. If $A_0$ has only point spectrum which satisfies (\\ref{eq:1.2}), then by Theorem XIX.5.7 of \\cite{DuSc71} we know that condition 1.\\ holds for $A$.\n\nWhen calculating the eigenvalues of a differential operator, one normally finds that these eigenvalues are the zeros of an entire function. If this function has its zeros in a strip parallel to the imaginary axis, and on the boundary of this strip the function is bounded and bounded away from zero, then its zeros can be decomposed into finitely many interpolation sequences, see Proposition II.1.28 of \\cite{AvIv95}. They name this class of entire function {\\em sine type functions}, but in Levin \\cite{Levi96} this name is restricted to a smaller class of functions.\n\n\n\\subsection*{Acknowledgment}\n\nThe author wants to thank Markus Haase and Jonathan Partington for their help .\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nEarthquake catalogs are a key seismological tool for problems involving location, precursors, and earthquake source parameters \\citep{Gutenberg56,Scholz68,AkiRichards,Meier2017}. Catalogs are fundamental to determining where, when, and how a fault has ruptured \\citep{AkiRichards,Scholz2002} and they are central to studies of earthquake precursors and seismic hazard based on both field observations \\citep{Marsan2014,Bouchon2008,Bouchon2013,Huang2017,Meng2018} and theory \\citep{AkiRichards,Ferdowsi2013,Kazemian}. Laboratory studies of earthquake precursors also rely on event catalogs \\citep{Lei2014,JohnsonPrecursor,riviere2018}. Such studies include the temporal evolution of the Gutenberg-Richter b-value, which is a fundamental parameter relating laboratory and field studies of earthquake physics \\citep{Smith1980,Wang2016,Scholz2015,Gulia,riviere2018,Kwiatek2014}. The seismic b-value has emerged as a key parameter for estimating seismic hazard and for connecting laboratory and field studies of seismicity \\citep{Scholz2002}.\n\nLaboratory-based studies of seismic hazard and earthquake forecasting have traditionally relied on earthquake catalogs. However, recent work shows that lab earthquakes can be predicted based on continuous acoustic data \\citep{rouet17ML,BRLFriction,Hulbert2017SlowSlip}. These works show that statistical characteristics of the continuous seismic signal emanating from lab fault zones can predict the timing of future failure events as well as the frictional state of the fault zone. Moreover, recent work has extended this approach to field observation by showing that statistical characteristics of continuous seismicity recorded in Cascadia contain a fingerprint of the megathrust displacement rate that can be used to predict the timing of episodic tremor and slip events \\citep{RouetCascadia18}. The methods are based on machine learning (ML) and rely exclusively on continuous measurements of acoustic emission (AE). Thus, a key question involves whether this approach can be extended to catalog based measurements of seismic activity.\n\nApplications of ML to seismology and geosciences are becoming increasingly common \\citep{lary2016,khawaja18,kenta18,zefeng18,Holtzman18}. Here, we apply the ML Random Forest method \\citep{Breiman2001} to study laboratory earthquake catalogs.  We find that as the catalog degrades, so does our ability to infer fault physics, as quantified the ability to predict shear stress and the time to and since failure.  We begin by describing the biaxial shearing experiment, our methods to construct a catalog from the continuous waveform data, the statistical descriptors we extract from the catalog, and the random forest method used to make inferences from these descriptors. This results in successful models for estimating the fault physics variables using local-in-time catalog data.  Then, from the original catalog we construct a series of progressively more degraded catalogs, each of which contains only the events that exceed a variable magnitude of completeness.  We examine the performance of this method as a function of the magnitude of completeness, and observe that the learned signatures of fault physics degrade dramatically if small enough events are not detected. However, the performance plateaus for low magnitudes of completeness; the smallest 80\\% events in the catalog can be dropped without sacrificing performance. Our results show that ML methods based on event catalogs can successfully model laboratory faults, with the caveat that predictive power requires a sufficiently complete catalog.\n\n\\section{Data Collection}\n\n\\subsection{Biaxial Shear Experiment}\n\nWe used data from laboratory friction experiments conducted with a biaxial shear apparatus \\citep{marone98,JohnsonPrecursor} pictured in Fig.~\\ref{figstress}a. Experiments were conducted in the double direct shear configuration in which two fault zones are sheared between three rigid forcing blocks. Our samples consisted of two 5 mm-thick layers of simulated fault gouge with a nominal contact area of 10 x 10 cm$^2$. Gouge material consisted of soda-lime glass beads with initial particle size between 105-149 {\\textmu}m. Additional details about the apparatus and sample construction can be found in \\citet{Anthony05}, \\citet{riviere2018}, and Text S1.  Prior to shearing, we impose a constant fault normal stress of 2 MPa using a servo-controlled load-feedback mechanism and allow the sample to compact. Once the sample has reached a constant layer thickness, the central block is driven down at constant rate of 10~\\textmu m\/s. In tandem, we collect an AE signal continuously at 4~MHz (Fig.~\\ref{figstress}b, yellow curve) from a piezoceramic sensor embedded in a steel forcing block $\\approx 22$ mm from the gouge layer. \n\nFigure~\\ref{figstress}c shows the shear stress measured on the fault interface for a full experiment.  Experiments begin with a run-in stage where the shear stress increases and macroscopic shearing of the fault zone transitions from stable sliding to stick-slip failure. The repetitive cycles of loading and failure represent laboratory seismic cycles and transition from periodic to aperiodic as a function of load-point displacement in our experiments \\citet{Anthony05,JohnsonPrecursor}. Here, we focus on aperiodic slip cycles and measure the slip history for a set of 23 failure events that occur over 268~seconds~\\ref{figstress}c. We define large failure events as times for which stress drop exceeds 0.05 MPa within 1 ms.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=32pc]{fig1.pdf}\n\n \\caption{{\\bf a)} Diagram of biaxial shear apparatus. The blue arrow indicates the direction of shear, and the red arrow locates the piezoceramic sensor that records AE. {\\bf b)} Example cataloged events (grey circles) derived from continuously recorded AE (yellow) using the statistics of the smoothed signal envelope (black). {\\bf c)} Observed shear stress for an experiment at fixed strain rate. Sharp drops in stress correspond to failure events. Inset: The data segment analyzed in this paper. An event catalog is constructed from AE data sampled at 4 MHz. Our models are constructed using event data from the training segment (blue) and evaluated on data from the test segment (green). \n }\n \\label{figstress}\n \\end{figure}\n\n\n\\subsection{Event Catalog}\n\\label{sec:events}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=32pc]{fig2.pdf}\n \\caption{ {\\bf a)} Normalized count of training events with magnitude exceeding $M$. The dashed line represents ideal Gutenberg-Richter scaling with a $b$-value of 1.66. {\\bf b)} Event rates vs. TTF stacked over all 23 failure cycles. {\\bf c)} Stacked event statistics for median event magnitude vs. time to failure.}\n \\label{figstack}\n \\end{figure}\n\nOur event catalog is built from the acoustic signal (example in Fig.~\\ref{figstress}b, yellow) during the time interval noted in Fig.~\\ref{figstress}c, inset. The catalog is composed of event amplitudes  $A(t_k)$ and times $t_k$, where $k$ is an event index, built following methods described in \\citet{riviere2018} and Text S2. We define laboratory magnitudes $m_k = \\log_{10}{A(t_k)}$. The catalog contains  $N_\\textrm{tot} \\approx 3.8 \\times 10^5$ slip events with laboratory magnitudes that range from about 1.0 to 4.2. Our lab magnitudes are based only on acoustic signal amplitude, and although they do not connect directly to earthquake magnitude, previous works have demonstrated the connection between seismic b-value and lab frequency magnitude statistics \\citep{Scholz2002}.\n\nTo visualize the distribution of magnitudes, we construct the cumulative event distribution\n\\begin{equation}\nN(M) = \\sum_{m_k \\geq M} 1. \\label{eq:n_M}\n\\end{equation}That is, $N(M)$ is the count of all events with magnitude at least $M$. We plot $\\log N(M) \/ N_\\textrm{tot}$ in Fig.~\\ref{figstack}a and observe reasonable Gutenberg-Richter scaling with a $b$-value of 1.66. \n\nFigure~\\ref{figstack}b shows the rate of events as a function of time to failure (TTF), averaged over all slip cycles.  The rate of events increases significantly as failure is approached, similar to what has been observed in prior works \\citep{JohnsonPrecursor,weeks1978,Ponomarev1997,riviere2018,Goebel2013}.  Figure~\\ref{figstack}c shows that the median event magnitude changes steadily but slowly from approximately $M_{\\textrm{median}}=1.4$ to $M_{\\textrm{median}}=1.5$  as a function of time during the lab seismic cycle. The trends of the catalog event count and size reflected in Fig.~\\ref{figstack}b and Fig.~\\ref{figstack}c motivate the creation of machine learning features that describe the evolution event count and size in more detail.\n\n\\section{Machine Learning Methods}\n\n\\subsection{Regression with Random Forests}\n\nAbstractly, a regression model maps input points $\\mathbf{X}_i$ to continuous output labels  $\\hat y_i$. Input points are represented by a feature vectors $\\mathbf X_i = (X_{i,1}, X_{i,2},\\dots X_{i, N_\\textrm{features}})$ that are designed to capture the relevant information about $\\hat y_i$. The training dataset $\\{(\\mathbf X_i, y_i)\\}$ consists of many points $\\mathbf X_i$ associated with true labels $y_i$. From this training set, one aims to learn a model that is successful at predicting labels $\\hat y_i$ for {\\em new} data points that were unseen within the training process. In this work, the features will be statistics extracted from the catalog, the labels will describe the macroscopic state of the fault (such as shear stress); both are indexed by time window $i$. \n\nWe  use the random forest (RF) algorithm \\citep{Breiman2001} as it is implemented in the scikit-learn package \\citep{pedregosa2011scikit}. A single RF model contains many decision trees, each of which is a simple regression model constructed stochastically from the training data. To make a prediction, the decision tree begins at its root node and asks a series of yes\/no questions. Each question has the form: {\\em Is some input feature above or below some threshold value?} Eventually a unique leaf node is reached, from which the decision tree makes its prediction. The RF makes a prediction by averaging over many decision trees, and tends to be more robust than any individual tree. Additional details about our training procedure are available in Text S3. We investigated other regression methods \\citep{Friedman2001,hui2005,tibshirani96,hoerl1970} besides the RF, and although all were successful to some extent, none offered definitive improvement over the RF (see Table~S1). \n\nA regression model is successful if its predicted labels $\\hat y_i$ agree with the true labels $y_i$ for data points in the {\\em testing dataset}, which is separate from the training dataset. We use the dimensionless $R^2$ value to quantify the performance of the model,\n\n\\begin{equation}\nR^2 = 1 - \\frac{\\sum_i (\\hat{y_i}-y_i)^2}{\\sum_i (\\bar{y}-y_i)^2},\n\\end{equation}\nwhere $\\bar{y}$ is the mean output label for the testing dataset. Let us review the significance of the $R^2$ metric, which is maximized at $R^2=1$, and can take on arbitrarily low values. Using the mean label as a constant predictor for all data points, $\\hat y_i = \\bar y$, yields $R^2=0$. Thus, $R^2<0$ indicates that a model performs worse than a constant learned only from the labels, without utilizing the features; if $R^2 \\leq  0$, a model has failed to discover useful correlations between the features and the labels. The condition $R^2=1$ would indicate that a model makes perfect predictions for all labels in the test set.\n\n\\subsection{Feature and Label Creation}\n\nOur task is to build a regression model that uses the event distribution within a {\\em local} time window to understand and make predictions about the stress state and times to and from large failure events. Each time window is labeled, and predictions of the labels will depend only on features from within the window. Thus, each test prediction is independent of prior predictions, and the model cannot exploit the quasi-periodicity of the fault state.\n\nThere is significant interest in predicting the TTF for an upcoming earthquake event. \\citet{rouet17ML} and \\citet{BRLFriction} showed that the continuous AE from a fault is remarkably predictive of the TTF and instantaneous friction. In both previous works, $\\mathbf X_i$ is a collection of statistical features extracted from the {\\em continuous} AE signal for the time window. Here, we build features for each time window using only the event catalog, rather than the continuous acoustic signal. \n\nWe divide our full event catalog (cf. Sec.~\\ref{sec:events}) into sub-catalogs that contain only the events occurring in non-overlapping time windows of length $\\Delta T = 1$ second. The feature vector contains information about events in the sub-catalog for this time window. Our analysis region consists of 270 seconds, which we divide into 1 second windows. We train on the first $60\\%$ (159 windows) and test on the remaining $40\\%$ (111 windows).\n\nWe construct regression models for three labels: (a) the shear stress, (b) the time to the next failure event (TTF), and (c) the time since the last failure event (TSF). The shear stress label is assigned using the mean shear stress over that window. TTF is measured following the end of time window, and TSF is measured preceding the beginning of the time window. For windows including failures, both TTF and TSF are zero.\n\nAbstractly, our the features for each time window describe the cumulative statistics of event counts and amplitudes from the sub-catalog evaluated at magnitude thresholds extracted from the full catalog. More precisely, the feature vector $\\mathbf X_i$ for time window $i$ will have components $X_{i,1}, X_{i,2}, \\dots X_{i,N_\\textrm{features}}$. Each feature component $X_{i,j}$  is associated with a characteristic magnitude $M_j$. For each $j$, we define $M_j$ to be the largest observed magnitude that satisfies\n\\begin{equation}\n\\frac{N(M_j)}{N_\\textrm{tot}} \\geq \\alpha^j,\n\\end{equation}\nwhere the $j$ in right hand side is applied as an exponent; $\\alpha$ is a parameter of range $ 0 < \\alpha <1$ that controls the fineness of the magnitude bins and the total number of bins, $N_\\textrm{features}$. Recall from Eq.~\\eqref{eq:n_M} that $N(M)$ is the cumulative event count of our {\\em entire} training catalog, and that $N_\\textrm{tot} \\approx 3.8\\times 10^5$. In this work, we select $\\alpha=0.7$, and this leads to $N_\\textrm{feature} = 35$ non-empty bins. We emphasize that the characteristic magnitudes $M_j$ are derived from the entire training catalog, independent of window index $i$.\n\nWe now define the feature vector $\\mathbf{X}_i$ for each window $i$ from the sub-catalog of event magnitudes $m^{(i)}_k$, where $k$ indexes the events within the window. We test two simple schemes for $\\mathbf X_i$, sensitive to the counts and amplitudes in the catalog. Mathematically, the $j$th components of these two feature vectors are\n\\begin{align}\nX^{\\textrm{count}}_{i,j} &=\\sum_{m^{(i)}_k \\geq M_j} 1 \\\\\nX^{\\textrm{ampl}}_{i,j} &= \\sum_{m^{(i)}_k \\geq M_j} 10^{m^{(i)}_k}  \\label{eq:ampl_features}\n\\end{align}\nwhere the sums run over all sub-catalog events $k$ with magnitude $m^{(i)}_k$ at least $M_j$. In words,  $X_{i,j}^\\textrm{count}$ is the count of events in the $i$th window with magnitude exceeding $M_j$ (analogous to $N(M_j)$, but restricted to sub-catalog $i$). The features $X_{i,j}^\\textrm{ampl}$ measure the total amplitude of all events in window $i$ above the magnitude $M_j$.\n\n\n\\subsection{Catalog Ablation Test}\n\nTo understand how our catalog-RF model performs with decreasing information, we vary the magnitude of completeness of our catalog by discarding all events with magnitude below a cutoff,  $M_\\textrm{cut}$. In practice, we implement this ablation by removing features  $X_{i,j}$ for indices  $j$ that correspond to magnitudes $M_j$ below the cutoff $M_{\\textrm{cut}}$. We can then quantify the performance as a function of the ablation cutoff $M_\\textrm{cut}$.\n\n\\section{Results}\n\\subsection{Performance on Full Catalogs}\n\nTable \\ref{tabregression} shows the $R^2$ performance of our catalog-RF models for shear stress, TSF, and TTF, trained using the full catalogs.  The TTF models ($R^2 = 0.551, R^2=0.612$) do not capture fine details. The TTF prediction is worse than the TSF and shear stress models ($R^2 > 0.8$)---in other words, it is harder to predict the future than to estimate aspects of the past or present. Table \\ref{tabregression} also indicates that the cumulative counts  $\\mathbf{X}^\\textrm{count}$ are better for predicting the shear stress, and the cumulative amplitudes  $\\mathbf{X}^\\textrm{ampl}$ are slightly better for TTF and TSF. The count-based catalog features predict shear stress with an accuracy of $R^2 = 0.898$, nearly equal to the continuous acoustic approach of \\citet{BRLFriction}, who report $R^2 = 0.922$ using the same data (that is, same train and test segments on the same experiment). We note that although the same data are analyzed, the methods cannot be compared exactly; for example, the time window in \\citet{BRLFriction} is $1.33$ seconds, and each time window has $90\\%$ overlap with the previous window. (Recall that in our work the time window length is $1$ second, and windows do not overlap.)  Figure~\\ref{figregression} shows the predictions compared to the true labels over time for the amplitude-based features $\\mathbf{X}^\\textrm{ampl}$, as well as scatter-plots showing a comparison of the true and predicted labels. Figure~S1 shows that models perform nearly as well for shear stress and TSF when analyzing a larger region of 573 s of data (44 failure cycles), but with reduced performance for TTF. This fact can be accounted for by drift in cycle length over time; TTF performance is restored by randomly assigning cycles to testing data from throughout the entire data analysis region (Fig.~S2).\n\n\\begin{table}[h]\n \\centering\n \\begin{tabular}{l  r  r  r }\n \\toprule\n   & Shear Stress & TSF  & TTF \\\\\n \\midrule\n  $\\mathbf{X}^\\textrm{count}$ & 0.898 & 0.842 & 0.551 \\\\\n  $\\mathbf{X}^\\textrm{ampl}$ & 0.848 & 0.882 & 0.612 \\\\\n  \\bottomrule\n \\end{tabular}\n \\caption{$R^2$  performance for regression of shear stress, TSF, and TTF for catalog-RF models on complete catalogs. The choice of count or amplitude based features ( $\\mathbf{X}^\\textrm{count}$ or  $\\mathbf{X}^\\textrm{ampl}$) affects performance slightly.}\n \\label{tabregression}\n\\end{table}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=32pc]{fig3.pdf}\n\n \\caption{Regression performance for the catalog-ML model using amplitude-based features. The rows show performance for shear stress, TSF, and TTF, respectively. The left panel of each row shows predictions on training and testing data over time. True values are shown as black squares, and predicted values as colored circles. Time windows for features are contiguous and non-overlapping. Each point is plotted at the end of the time window represented.  The right panel of each row plots the prediction vs. true values for each label. Testing data is shown with dark markers, and training data is shown with faint markers.}\n \\label{figregression}\n \\end{figure}\n\n\\subsection{Performance on Ablated Catalogs}\n\n\n \\begin{figure}[h]\n \\centering\n \\includegraphics[width=32pc]{fig4.pdf}\n\n \\caption{The accuracy of the catalog-RF model decreases as training sub-catalogs are artificially limited to varying magnitudes of completeness, $M_\\textrm{cut}$. Left: $R^2$ performance vs. $M_{\\textrm{cut}}$. The drop of $R^2$ for $M_\\textrm{cut}$ between 2.0 and 3.0 indicates that this range of event magnitudes plays a key role in our RF models. The dashed vertical line at $M_{\\textrm{cut}} = 3.67$ denotes the smallest event magnitude associated with a large stress drop. Right: $R^2$ is plotted against $N(M_\\textrm{cut})\/N_\\textrm{tot}$, i.e. the fraction of events remaining in the full catalog after ablation. One observes that most events contribute very little to the catalog-RF performance.}\n \\label{figablation}\n \\end{figure}\n\nNext we build our catalog-RF models on ablated catalogs that include events only above an imposed magnitude of completeness, $M_\\textrm{cut}$. For this study, we use the amplitude features $\\mathbf{X}^\\textrm{ampl}$ of Eq.~\\eqref{eq:ampl_features}.  Figure~\\ref{figablation}, left side, shows the $R^2$ performance as a function of the imposed magnitude cutoff.  As one might expect, catalog-RF models generally perform worse when provided less information. However, it is interesting to note that $R^2$ decays relatively slowly until $M_\\textrm{cut}$ reaches about 2.0, after which the performance drops precipitously, plateauing near magnitude 3.0. In this regard we say that $M_{\\textrm{cut}} = 2.0$ provides a {\\em sufficient magnitude of completeness} for the catalog-RF model with the given window size.  In Fig.~\\ref{figablation}, right side, we plot $R^2$ as a function of the remaining event fraction $N(M_\\textrm{cut})\/N_\\textrm{tot}$. This highlights that the performance drop at $M_{\\textrm{cut}} \\approx 2.0$ corresponds to ablating $\\approx 80\\%$ of the events from the catalog. Likewise, the sharp decay in performance indicates that events $M \\lesssim 2.5$ are {\\em required} to make an accurate determination of the fault state. The performance of the method for $M_{\\textrm{cut}} \\gtrsim 3 $ plateaus at a non-zero value of $R^2$ because the algorithm is still able to differentiate between windows that contain a large failure and those that do not, but is otherwise unable to discern useful differences between sub-catalogs. \n\nA study of the ablation curve as a function of window size (Fig.~S3) shows that the sharp decay in performance shifts to smaller cutoff magnitudes as the window size is decreased; while regression can be performed with small window sizes, the sufficient magnitude of completeness is lower. This leads us to hypothesize that the sharp decay in performance may be related to the number of events within a window; to keep the number of events in a window fixed, one must lower the magnitude of completeness along with the window size. \n\n\\section{Conclusions}\n\nWe analyze a dense acoustic event catalog obtained from biaxial shearing friction experiment.  The device produces dozens of aperiodic stick-slip events during an experiment. Stacking catalog statistics over many cycles shows that as failure approaches, events are more common and typically larger (Fig.~\\ref{figstack}).  We then use a machine learning workflow to show that physical characteristics of the fault (shear stress, TSF, and TTF) can be obtained from catalog statistics over a short time window using a Random Forest regression model. Previous work employed the continuous AE, and showed remarkable ability to forecast failure \\citep{rouet17ML}, as well as to infer the instantaneous shear stress and friction \\citep{BRLFriction}.  Here, we demonstrate that the continuous acoustic waveform is not needed if a catalog {\\em with sufficient completeness} is available. Our models achieve similar accuracy to the continuous approach.\n\nWe note, however, that our catalogs are extraordinarily well resolved. To create these catalogs, we recorded AE at very high frequency (4 MHz) using a sensor very near the fault (distance $\\approx 22$ mm). One would not expect such fidelity in real earthquake catalogs, except perhaps if the sensors happened to be located very close to the slip patch.  We find that the smallest events in our catalog contribute little to the prediction accuracy; increasing $M_\\textrm{cut}$ from 1.0 to 2.0 only slightly diminishes the catalog-RF model performance (visible in Fig.~\\ref{figablation}).\n\nThe fact that the smallest events are not {\\em required} offers hope that a catalog-based ML approach could eventually contribute to the understanding of natural seismicity. We note a few potential differences between laboratory conditions and natural seismicity that may pose challenges. Laboratory data corresponds to a single isolated fault. Tectonic fault zones are typically comprised of many sub-faults, and predictions may be vastly more difficult. While precursors are frequently observed in laboratory studies, they are not as reliably observed in natural faults. Moreover, large seismic cycles in the earth are far slower \\citep{nelson2006,Ellsworth2013}, and thus fewer cycles are available to train ML models. \n\nWe note that \\citet{RouetCascadia18} have applied the continuous approach developed in the laboratory to slow slip in Cascadia with surprisingly good results.  Thus we are hopeful that the catalog-based approach described here may prove fruitful to the study of real earthquakes, given sufficient catalog quality. It may have particular importance in regions where continuous data are not yet available, where the continuous data is quite noisy, or for examining archival catalogs.\n\n\\acknowledgments\nWork supported by institutional support (LDRD) at the Los Alamos National Laboratory.  We gratefully acknowledge the support of the Center for Nonlinear Studies. CM was supported by NSF-EAR1520760 and DE-EE0006762. We thank Andrew Delorey, Robert A. Guyer, Bertrand Rouet-Leduc, Claudia Hulbert, and James Theiler for productive discussions. The data used here are freely available from the Penn State Rock Mechanics lab at \\url{http:\/\/www3.geosc.psu.edu\/~cjm38\/}.\n\n\n\\section*{Contents}\n\n\\begin{enumerate}\n\\item Text S1: Experimental details\n\\item Text S2: Catalog generation from acoustic emissions\n\\item Text S3: Random forest details\n\\item Table S1: Performance of various regression algorithms\n\\item Figure S1: Regression for long time segment with contiguous train\/test split\n\\item Figure S2: Regression for long time segment with random cycle train\/test split\n\\item Figure S3: Ablation analysis for varying time-window sizes\n\\end{enumerate}\n\n\\section*{Text S1: Experimental details}\n\nWe constructed samples for the double direct shear (DDS) assembly following the procedures detailed in previous works \\citep{Anthony05,riviere2018}. Steel guide plates are mounted on the side blocks of the DDS to prevent gouge loss on the front and back edges. The faces of the forcing blocks in contact with the gouge layers are grooved to ensure that shear occurs within the layer; grooves are 0.8 mm deep and spaced every 1mm perpendicular to the direction of shear. In addition, a small rubber jacket covers the bottom half of the sample to minimize material loss throughout the experiment. Forces on faults are measured with strain-gauge load cells placed in series with the horizontal and vertical pistons. Displacements on the fault are measured using direct current displacement transducers that are coupled to the horizontal and vertical pistons. Stress and displacement data are measured continuously at 1kHz throughout the experiment from a 24-bit recording system. All measurements of shear stress and displacement for the DDS configuration are relative to one fault. In addition, we have instrumented 36 broad-band piezoceramic sensors (~0.02-2 MHz) into steel 10 x 10 cm\\textsuperscript{2} blocks. The sensors are in blind holes 2-mm from the block face and the block is placed adjacent to the fault zone to monitor acoustic emission activity \\citep{riviere2018}. The acoustic emission data analyzed here are recorded continuously at 4 MHz from a single piezoceramic sensor using a 14-bit Verasonics data acquisition system.\n\n\\section*{Text S2: Catalog generation from acoustic emissions}\n\n\nWe use a simple event detection algorithm to construct a catalog from the acoustic emission time series signal.  The algorithm has been designed to account for both the response of the sensor and the background noise level. The first step is to compute the envelope of the signal and use a moving average to smooth it over the time scale $\\tau_\\textrm{smooth}$. From this smoothed envelope we examine peaks and label them as events if they satisfy the following criteria: 1) The peak is above a minimum amplitude threshold, $A_\\textrm{min}$, related to the average background noise. 2) The peak is separated from previous peaks by a minimum inter-event time threshold, $\\tau_\\textrm{min}$. 3) The peak does not occur during in the coda of an event -- this is defined using an exponential ring-down function computed over the ten previous events. Mathematically, event $k$ is not part of a coda if the peak amplitude $A(t_k)$ satisfies\n \\begin{equation}\nA(t_k) > A(t_{k-i})\\exp{\\frac{-{(t_k-t_{k-i})}}{\\tau_\\textrm{ring-down}}}\n\\label{eq:A_t0}\n\\end{equation}\nfor $i = 1...10$.  The parameters of this algorithm are selected by evaluating the catalog quality on random 5~$\\mu s$ snapshots of the acoustic signal using trial parameters and visual evaluation of catalog quality. A high-quality catalog maximizes the number of distinct visual phenomena selected as events, while at the same time minimizes the assignment of multiple events to a single phenomenon. In the end, we selected $A_\\textrm{min} = 15$, $\\tau_\\textrm{smooth} = 7.75$ {\\textmu}s,  $\\tau_\\textrm{min} = 150$ {\\textmu}s, and $\\tau_\\textrm{ring-down} = 156.7$ {\\textmu}s.\n\n\\section*{Text S3: Random forest details}\nWe employed 100 decision trees per random forest, each of which sees a bootstrap resampling of the training set, with a mean-squared-error cost function. Before training a random forest model (i.e., designing each tree's yes\/no questions from the data), one may specify the maximum depth of the decision trees, which controls the allowable complexity of the random forest model. (If the maximum depth is larger, the random forest can fit more complicated datasets, but also becomes more prone to overfitting irrelevant details of the data.) For each regression task, we used 10-fold cross-validation of the training set to select the maximum depth; the possible depths are 1, 2, 4, 6, 8, and 12. The folds for cross-validation are selected by randomly partitioning the training set. Cross-validation uses the $R^2$ score as the criterion for the best model.  After identifying the tree depth which cross-validates most successfully, we retrain a forest with that maximal depth to the entire training set. Other regularization hyperparameters such as the minimal leaf size and random feature sub-selection were not utilized.\n\n\\begin{table}[h]\n \\centering\n\\begin{tabular}{llrrrrr}\n\\toprule\n           &       &    RFR &    GBR &  ElasticNet &  Lasso &  Ridge \\\\\nRegression task & Feature vector &        &        &               &          &          \\\\\n\\midrule\nShear stress & $\\mathbf X^\\textrm{count}$ &  {\\bf 0.901} &  0.893 &         0.715 &    0.715 &    0.800 \\\\\n           & $\\mathbf X^\\textrm{ampl}$ &  {\\bf 0.860} &  0.827 &         0.817 &    0.817 &    0.796 \\\\\nTSF & $\\mathbf X^\\textrm{count}$ &  0.843 &  \\bf{0.886} &         0.630 &    0.634 &    0.743 \\\\\n           & $\\mathbf X^\\textrm{ampl}$ &  0.888 &  \\bf{0.891} &         0.779 &    0.779 &    0.771 \\\\\nTTF & $\\mathbf X^\\textrm{count}$ &  0.547 &  \\bf{0.579} &         0.533 &    0.533 &    0.571 \\\\\n           & $\\mathbf X^\\textrm{ampl}$ &  \\bf{0.618} &  0.578 &         0.588 &    0.588 &    0.471 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{$R^2$ performance for various regression models. We compare random forest regression (RFR) \\citep{Breiman2001}, gradient boosting regression (GBR) \\citep{Friedman2001}, and linear regression with ElasticNet \\citep{hui2005}, lasso \\citep{tibshirani96}, and ridge \\citep{hoerl1970} regularization penalties as implemented in scikit-learn \\citep{pedregosa2011scikit}. The best performing model for each regression task and feature vector type is shown in bold. RFR and GBR produce models of similar quality and both outperform linear regression schemes.}\n \\label{methodchoices}\n\\end{table}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=32pc]{figS1.pdf}\n\n \\caption{Regression performance for the catalog-ML model using amplitude-based features using 573 seconds of data. The rows show performance for shear stress ($R^2 =0.869$) , TSF ($R^2 =0.813$), and TTF ($R^2 =0.284$), respectively. The left panel of each row shows predictions on training (region shaded blue) and testing (region shaded green) data over time. True values are shown as black squares, and predicted values as colored circles. The right panel of each row plots the prediction vs. true values for each label. Testing data is shown with bold markers, and training data is shown with faint markers.}\n \\label{figregression}\n \\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=32pc]{figS2.pdf}\n\n \\caption{Regression performance for the catalog-ML model using amplitude-based features using 573 seconds of data, where each failure cycle is individually assigned to either training or testing data. The rows show performance for shear stress ($R^2 =0.881$), TSF ($R^2 =0.869$), and TTF ($R^2 =0.511$), respectively. The left panel of each row shows predictions on training (region shaded blue) and testing (region shaded green) data over time. True values are shown as black squares, and predicted values as colored circles. The right panel of each row plots the prediction vs. true values for each label. Testing data is shown with bold markers, and training data is shown with faint markers.}\n \\label{figregression}\n \\end{figure}\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=20pc]{figS3.pdf}\n\n \\caption{Ablation analysis for time since failure (TSF) showing $R^2$ performance vs. artificial magnitude of completeness for different window sizes. As the window size decreases, performance decreases. Furthermore, the curve shifts left; the magnitude of completeness necessary to achieve a given $R^2$ performance lowers along with window size. This figure demonstrates that while regression can be performed with small window sizes, the process relies on having information about progressively smaller events.}\n \\label{fig:windowsizeabalation}\n \\end{figure}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nNon-Hermitian physics has stimulated significant interest in recent years~\\cite{PTreview1,PTreview2,PTreview3,Uedareview}, where particular attention has been devoted to its unconventional dynamics, peculiar critical behavior, and exotic band topology. A major driving force behind the booming field is the experimental implementation or simulation of these intriguing phenomena, particularly in open dissipative quantum systems~\\cite{dalibard,daley,openbook}.  Therein, the system undergoes particle or energy loss to its environment, and a non-Hermitian effective Hamiltonian becomes relevant by imposing postselection~\\cite{dalibard,daley}.\nSo far, a wide spectrum of non-Hermitian phenomena, ranging from parity-time (PT)-symmetry breaking and non-Hermitian criticality~\\cite{benderreview,ptcrit,kznc}, to non-Hermitian skin effects and non-Bloch topology~\\cite{WZ1,Budich,alvarez,mcdonald,ThomalePRB,Lee,WZ2,murakami}, have been experimentally implemented and explored in quantum mechanical systems such as the single-photon interferometry network~\\cite{xuept,crit1,photonskin}, cold atoms~\\cite{luole,bryce,yan,NHSOCexp,yanzeno}, nitrogen-vacancy centers~\\cite{crit2,epencircle}, superconducting qubits~\\cite{scpt}, and trapped ions~\\cite{chenion,zhangion}. While most of these experiments investigate the single-particle aspects of the non-Hermitian physics, the interplay of non-Hermiticity and interaction is a fast-growing frontier with many open questions and fresh challenges~\\cite{sf1,sf2,nonHtwobody,Yu,Cui}.\n\nOne of the key issues here is the relevance of non-Hermitian many-body Hamiltonians to open dissipative quantum systems~\\cite{fu,michishita}.\nWhile the latter is naturally characterized by trace-preserving density-matrix dynamics, the former requires a biorthogonal construction to ensure othornormality and recover the bosonic\/fermionic statistics~\\cite{DCB}.\nFurther, the postselection framework in a many-body setting requires an unchanged particle number~\\cite{michishita}, thus exacting a stringent limit on the timescale within which the non-Hermitian description can be applied.\nBy contrast, this is not an issue with a non-interacting system, as is the case with the recent observations of PT transition and exceptional-point encircling in cold atoms~\\cite{luole,NHSOCexp}. Therein,\na non-interacting atomic gas undergoes particle loss to the environment through optical pumping---a single-particle process. The dynamics is driven by a non-Hermitian effective Hamiltonian derived from the Lindblad master equation by dropping the quantum jump term $c\\rho c^\\dag$ (here $c$ is the atomic annihilation operator and $\\rho$ the full density matrix of the non-interacting system).\nThis is equivalent to focusing only on atoms that are not lost to the environment, in the spirit of postselection.\nSpecifically, under the quantum-trajectory description~\\cite{daley}, since dynamics of individual atoms are decoupled, they constitute an ensemble of independent trajectories with no quantum jumps, all driven by the non-Hermitian effective Hamiltonian.\nTherefore, given a large number of atoms, the corresponding non-unitary dynamics can be probed by making measurements on the remaining atoms. Should interactions exist, however, dynamics of atoms in general would not decouple. Applying postselection would then amount to requiring the complete absence of quantum jumps for any single atom within the ensemble, which becomes exponentially unlikely with an increasing atom number.\n\n\n\nNevertheless, we demonstrate in this work that, key non-Hermitian physics can still be probed in the full density-matrix dynamics over fairly long times, provided the open many-body system be dominated by few-body correlations.\nUsing a minimal model of either two or three interacting fermions in a one-dimensional lattice, we first show that, under the non-Hermitian effective Hamiltonian of the corresponding Lindblad master equation, two-body scattering states of the system feature state- and interaction-dependent PT transitions. We then evolve the three-fermion dissipative system using the quantum trajectory scheme, taking into account the quantum jump processes. Remarkably, the decay of two-body correlations in this three-fermion open system follows the imaginary components of the complex eigenenergies of the two-body non-Hermitian scattering states. This enables us to extract the global exceptional point of the underlying two-body non-Hermitian system in the three-atom dissipative dynamics, from which the impact of interaction on the exceptional point is identified.\nOur results suggest that, non-Hermitian many-body physic can in principle be probed in the context of an open dissipative many-body setting,\nat least when both are dominated by few-body correlations.\n\nOur paper is organized as follows. In Sec.~II, we present the model configuration, the corresponding Lindblad master equation, and the non-Hermitian effective Hamiltonian. We solve the two-body eigen problem of the non-Hermitian Hamiltonian using exact diagonalization in Sec.~III, where we reveal the existence of PT transitions and exceptional points in the scattering states. In Sec.~IV, we solve the Lindblad master equation for a three-fermion system using the quantum trajectory approach, and show the relevance between the two-body correlations therein and the complex eigenenergies of the non-Hermitian scattering states. We summarize in Sec.~V.\n\n\\section{Dissipative many-body system and non-Hermitian Hamiltonian}\n\nAs illustrated in Fig.~\\ref{fig:model}, we consider fermionic atoms with two hyperfine states in a one-dimensional optical lattice potential. The corrsponding Hamiltonian is given by\n\\begin{align}\nH= & \\sum_{k, \\sigma} \\epsilon_{k} c_{k, \\sigma}^{\\dagger} c_{k, \\sigma}+J\\sum_{k}\\left(c_{k, \\uparrow}^{\\dagger} c_{k, \\downarrow} + c_{k, \\downarrow}^{\\dagger} c_{k, \\uparrow}\\right)\\nonumber \\\\\n&+\\frac{U_s}{\\mathcal{L}} \\sum_{k, k^{\\prime}, q} c_{k+q, \\uparrow}^{\\dagger}\nc_{k^{\\prime}-q, \\downarrow}^{\\dagger} c_{k^{\\prime}, \\downarrow}\nc_{k, \\uparrow}\\nonumber\\\\\n&+\\frac{U_p}{\\mathcal{L}} \\sum_{k, k^{\\prime}, q, \\sigma} \\sin(\\frac{k-k^\\prime+2q}{2})\\sin(\\frac{k-k^\\prime}{2})\\nonumber\\\\\n&\\quad \\quad\\quad\\quad\\quad \\times c_{k+q, \\sigma}^{\\dagger}\nc_{k^{\\prime}-q, \\sigma}^{\\dagger} c_{k^{\\prime}, \\sigma}\nc_{k, \\sigma}.\n\\label{eq:H}\n\\end{align}\nHere $c_{k, \\sigma} $ ($c^\\dag_{k, \\sigma} $) annihilates (creates) a fermionic atom with quasimomentum $k$ ($k\\in [-\\pi,\\pi)$) in the hyperfine state $|\\sigma\\rangle$ ($\\sigma=\\uparrow,\\downarrow$), $\\epsilon_k=-t [\\cos(ka_0)-1]$ with a hopping rate $t$ under the tight-binding approximation, $J$ is the radio-frequency (r.f.) coupling rate between different hyperfine spins, and $\\mathcal{L}$ is the quantization length. We consider both $s$-wave and $p$-wave interactions, characterized by $U_s$ and $U_p$~\\cite{pinteraction}, respectively, with $U_s,U_p<0$.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=0.4\\textwidth]{figure1}\n\\caption{Schematics of the dissipative lattice gas driven by the master equation (\\ref{eq:lind}). The single-particle loss (with rate $\\Gamma$) is induced by optical pumping, via an electronically excited state, to a third state (not drawn) that is not trapped by the lattice potential. The r.f. coupling rate $J$ and hopping rate $t$ are defined in the main text.\n}\n\\label{fig:model}\n\\end{figure}\n\nWe further consider the case where one of the hyperfine spin states ($\\left|\\downarrow\\right\\rangle$) is subject to optical pumping, via an electronically excited state, out of the lattice potential. Under the Markovian approximation, the dynamics of the system is captured by the Lindblad master equation\n\\begin{align}\n\\frac{d \\rho}{dt}=-i\\Big(H_{\\text{eff}}\\rho-\\rho H^\\dag_{\\text{eff}}\\Big)+\\Gamma\\sum_{k}c_{k,\\downarrow} \\rho c_{k,\\downarrow}^{\\dagger},\n\\label{eq:lind}\n\\end{align}\nwhere $\\rho$ is the density matrix, the non-Hermitian effective Hamiltonian of the Lindblad equation is given by $H_{\\text{eff}}=H-i\\frac{\\Gamma}{2}\\sum_k c^\\dag_{k,\\downarrow}c_{k,\\downarrow}$, and $\\Gamma$ is the single-particle loss rate.\n\nWhile the full dynamics of the open system is governed by Eq.~(\\ref{eq:lind}), under the quantum trajectory framework, the dynamics is understood as a non-unitary time evolution driven by $H_{\\text{eff}}$, which is further interrupted by quantum jumps $\\{c_{k,\\downarrow}\\}$ with relative probabilities $\\{\\Gamma\\delta t|c_{k,\\downarrow}|\\psi(t)\\rangle|^2\\}$. Here $\\delta t$ is the coarse-grained time step, and $|\\psi(t)\\rangle$ is the instantaneous state of the system. It is often argued that, when the effects of quantum jumps are negligible, the open system would evolve under the non-Hermitian Hamiltonian $H_{\\text{eff}}$. Such a postselection argument plays a key role in connecting realistic open quantum systems to the rich and exotic non-Hermitian physics that has attracted much attention of late~\\cite{Uedareview}.\n\nFor a non-interacting atomic gas with $U_s=U_p=0$, imposing postselection is conveniently equivalent to focusing only on the dynamics of atoms that are not lost to the environment.\nSince the full density matrix is just a direct product of single-particle density matrices, dynamics of each individual atom is driven by the same\nmaster equation. Further, as atoms that remain necessarily have not undergone the quantum jump process, the trajectories of remaining atoms are\nnon-unitary evolutions driven by the same non-Hermitian effective Hamiltonian.\nThis is indeed the case with the recent experimental demonstrations of PT symmetry and exceptional-point encircling in cold atoms~\\cite{luole,NHSOCexp}.\n\nHowever, in an interacting system, the full density matrix can no longer be decomposed into single-particle ones.\nPostselection thus corresponds to a complete absence of quantum jumps, i.e., it requires an unchanged total particle number.\nThe non-Hermitian effective Hamiltonian is then applicable at short times,\nwhen the impact of quantum jump terms are negligibly small. This is equivalent to the practice in Ref.~\\cite{Yu}, which projects the time evolution of the Lindblad equation onto the maximum-atom-number subspace. However, the corresponding time scale should become exponentially short with increasing particle number.\nGenerally, consider an $N$-particle system undergoing single-particle loss with the rate $\\Gamma$.\nThe probability that not a single quantum jump occurs scales as $\\sim e^{-N\\Gamma \\tau}$, where $\\tau$ is the evolution time.\nTherefore, the probability of all $N$ particles still remain at the time $1\/\\Gamma$ (for an evolution starting at $t=0$) is of the order $e^{-N}$, and the time scale at which the non-Hermitian effective Hamiltonian dominates should be $t< 1\/N\\Gamma$.\nNevertheless, we show in the following that, two-body physics under the non-Hermitian effective Hamiltonian {\\it can} be probed in the full density-matrix dynamics of the corresponding Lindblad equation, on time scales that exceed $1\/\\Gamma$.\n\n\n\n\n\\section{Non-Hermitian two-body scattering state}\n\nWe first characterize the two-body problem under the non-Hermitian effective  Hamiltonian $H_{\\text{eff}}$. To connect with previous cold-atom experiments on PT symmetry~\\cite{luole,NHSOCexp}, we define the PT symmetric Hamiltonian $H_{\\text{PT}}=H_{\\text{eff}}+i\\Gamma$. While the addition of the pure imaginary energy shift $i\\Gamma$ does not change key physics such as the emergence and location of exceptional points, it renders Hamiltonian $H_{\\text{PT}}$ PT symmetric in the non-interacting limit, with purely real (imaginary) eigenspectrum for $J>\\Gamma\/4$ ($J<\\Gamma\/4$).\n\n\\begin{figure}[tbp]\n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth] {figure2.pdf}\n\t\\caption{(Color online) Complex energy spectra of two fermions under the PT symmetric Hamiltonian $H_{\\text{PT}}$ for (a) $ U_s=0$, $U_p=0$, $J\/t=0.04 $; (b) $ U_s=0$, $U_p\/t=-0.2$, $J\/t=0.04$; and (c) $ U_s\/t=-2$, $U_p\/t=-0.2$, $J\/t=0.04 $. The left (right) panel shows the real (imaginary) components of the eigenspectra, with $K_c$ the center-of-mass momentum of the two-body state. For all numerical calculates, we take a one-dimensional lattice with $N=16$ sites.}\n\\label{fig:fig1}\n\\end{figure}\n\nIn Fig.~\\ref{fig:fig1}, we show the numerically evaluated eigenspectra for two fermions along a lattice of $N=16$ sites, with the parameters $J\/t=0.04$ and $\\Gamma\/t=0.1$. Here $K_c$ is the center of mass of the two-body state, which is a good quantum number of the system. In the non-interacting case [see Fig.~\\ref{fig:fig1}(a)(b)], $H_{\\text{PT}}$ is in the PT-unbroken regime, with purely real eigenspectra. The PT symmetry is broken under a sufficiently large $p$-wave interaction, as the eigenspectra acquire imaginary components under a finite $U_p$ [see Fig.~\\ref{fig:fig1}(c)(d)]. This in contrast to the $s$-wave interaction, which does not affect the imaginary components of the eigenspectra [see Fig.~\\ref{fig:fig1}(e)(f)]. Note that while discrete two-body bound states can be identified, for instance in Fig.~\\ref{fig:fig1}(e), it is the two-body scattering states within the continuum that acquire imaginary components.\nImportantly, we expect the PT transition point (or the exceptional points) of the non-interacting Hamiltonian be shifted by the $p$-wave interaction.\n\n\\begin{figure}[tbp]\n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth] {figure3.pdf}\n\t\\caption{(Color online) Complex energy spectra of $ H_{\\text{PT}} $ as functions of $ J\/t$ for (a) $ K_c=0$ and (b) $ K_c=\\pi $. As $ J\/t$ decreases, the scattering states coalesce in pairs at state-dependent exceptional points, through which their real components merge and imaginary components bifurcate. The full eigenspectra become completely real above a critical $J_c\/t$, which is identified as the global exceptional point (or the global PT transition point).\n}\n\\label{fig:fig2}\n\\end{figure}\n\nThis is confirmed in Figs.~\\ref{fig:fig2} and \\ref{fig:fig3}. Specifically, in Fig.~\\ref{fig:fig2}, we show the splitting of exceptional points in different $K_c$ sectors under a finite $U_p$. With increasing $J$, scattering states sequentially coalesce in pairs at an array of second-order exceptional points. We identify the exceptional point with the largest $J$ as the global PT transition point under the $p$-wave interaction,denoted by $J_c$. The resulting PT phase diagram is shown in Fig.~\\ref{fig:fig3}. Apparently, $p$-wave interactions shift the global exceptional point toward larger $J$, consistent with the results in Fig.~\\ref{fig:fig1}.\n\n\n\\begin{figure}[tbp]\n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth] {figure4.pdf}\n\t\\caption{(Color online) PT phase diagram for $H_{\\text{PT}}$ with $U_s=0$ and $U_p<0$. The global exceptional point $J_c\/t$ increases with larger $p$-wave interaction.\nThe parameter used in Fig.~\\ref{fig:fig2} is indicated by the vertical dashed line.}\n\\label{fig:fig3}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\t\\centering\n\\includegraphics[width=0.45\\textwidth] {figure5.pdf}\n\\caption{(Color online) Comparison of the total particle-number evolution $\\sum_{k,\\sigma}\\text{Tr}\\Big[ \\rho(\\tau)c^\\dag_{k,\\sigma}c_{k,\\sigma}\\Big]$ under the Lindblad master equation for systems initialized with\nthree atoms (blue solid), two atoms (red solid), and a single atom (yellow dashed), respectively, in the states $|\\uparrow,k=-\\pi\\rangle\\otimes |\\uparrow,k=0\\rangle\\otimes |\\uparrow, k=\\pi-2\\pi\/N$ ($N=8$), $|\\uparrow,k=\\pi\\rangle\\otimes |\\uparrow,k=0\\rangle$, and $|\\uparrow,k=0\\rangle$. The interaction parameters are (a) $U_p\/t=-0.5$, (b) $U_p=0$, with $U_s=0$, $J\/t=0.08$, and $\\gamma\/t=0.1$ for both cases. To facilitate comparison, the particle numbers in all cases are normalized to one.\n}\n\\label{fig:N}\n\\end{figure}\n\n\\section{Probing non-Hermitian physics in open system}\n\nWe now show that the global exceptional point $J_c$ of the two-body scattering states in Fig.~\\ref{fig:fig3} can be probed from the density-matrix dynamics under the full Lindblad equation. Under interactions, information of the two-body exceptional point is difficult to extract from the particle-number dynamics of a many-body system. This is illustrated in Fig.~\\ref{fig:N}, where we compare the particle-number evolution under the Lindblad master equation for systems initialized with different particle numbers, either with (Fig.~\\ref{fig:N}(a)) or without (Fig.~\\ref{fig:N}(b)) interactions.\nWhile the dynamics for different initial particle numbers appear to be the same without interactions, they generally differ under a finite $U_p$.\nThis shows that non-Hermitian two-body physics cannot be directly probed using particle-number dynamics in a three-body dissipative system. Note that non-Hermitian two-body physics can be fully captured by a Lindblad equation initialized in the two-body sector, since quantum jump terms in a two-body sector only couple to three-body states~\\cite{Yu}.\n\nInstead, other observables should be adopted. For a minimal demonstration, we solve the Lindblad equation in a three-fermion system, initialized in the state $|\\psi^{(S)}(K_c=-\\pi)\\rangle\\otimes |\\uparrow,k=\\pi-2\\pi\/N\\rangle$. Here $|\\uparrow,k\\rangle$ is a single-particle state with hyperfine spin $|\\uparrow\\rangle$ and momentum $k$, $|\\psi^{(S)}(K_c)\\rangle$ is the two-body scattering state of $H_{\\text{eff}}$ with a center-of-mass momentum $K_c$, and $|\\psi^{(S)}(K_c=\\pi)\\rangle$ is the eigenstate with the largest imaginary eigenenergy component, denoted as $\\text{Im}(E^{(S)})$. We take the system size $N=8$ for numerical calculations.\nFurther, we define a normalized two-body correlation function\n\\begin{align}\nG_{\\alpha\\beta}(k_1,k_2)=\\frac{\\text{Tr} \\Big[\\rho(\\tau) c_{k_1,\\alpha}^{\\dagger} c_{k_2,\\beta}^{\\dagger} c_{k_2,\\beta} c_{k_1,\\alpha}\\Big]}{\\text{Tr} \\Big[\\rho(0) c_{k_1,\\alpha}^{\\dagger} c_{k_2,\\beta}^{\\dagger} c_{k_2,\\beta} c_{k_1,\\alpha}\\Big]}. \\label{eq:G}\n\\end{align}\n\nWe evolve the Lindblad equation using the quantum trajectory approach, and plot the correlation function $ G_{\\uparrow\\uparrow}(k_1=-\\pi,k_2=0) $ in Fig.~\\ref{fig:fig4}.\nDue to the particular choice of initial state and the two-body correlation function, the decay of the correlation function follows the imaginary component of the two-body scattering state with the largest critical $J$.\nThus, by fitting the exponents of decay in $G$, we are able to map out the global exceptional points in the phase diagram Fig.~\\ref{fig:fig3}, and retrieve key properties of a two-body non-Hermitian Hamiltonian from the full dynamics of a interacting three-body open system.\nWe expect that this would hold true for many-body open systems, as long as the dominant correlations remain few-body in nature.\n\nFinally, we note that, should we choose a different initial state and two-body correlation function, we would be able to extract information of other two-body scattering states.\n\n\n\n\n\n\\begin{figure}[tbp]\n    \\centering\n   \\includegraphics[width=0.45\\textwidth] {figure6.pdf}\n   \\caption{(Color online) (a)(b)(c) Time evolution of the normalized two-body correlation function $ G_{\\uparrow\\uparrow}(k_1=-\\pi,k_2=0)$ (blue lines) under the quantum trajectory approach for (a) $ J\/t=0.04 $, (b) $ J\/t=0.08 $, and (c) $ J\/t=0.1 $. We take $ U_p\/t=-0.2$, $U_s=0$, and $\\Gamma\/t=0.1 $ for numerical calculations.\n  \n   The red lines show the time evolution of the norm of the corresponding two-body eigenstate of the non-Hermitian Hamiltonian $ H_{\\text{PT}} $.\n   (d) Comparison between the imaginary component of the two-body eigenenergy under $ H_{\\text{PT}} $ (red line), with the numerically fitted exponent of the correlation-function decay (blue line and symbol). For the exponent, we numerically fit the time-dependent correlation function up to the time $\\tau t=5$. For all our numerical calculations here, we take the lattice size $N=8$. We average over $2000$ trajectories for the quantum-trajectory calculations.}\n      \\label{fig:fig4}\n\\end{figure}\n\n\n\\section{Summary and Discussion}\\label{sec:sum}\n\nAdopting a minimal model of a few dissipative fermions on a one-dimensional lattice, we show that PT transitions exist in the scattering states of the non-Hermitian Hamiltonian, and are shifted by the $p$-wave inter-atomic interactions. The interaction-shifted global exceptional point can be probed by measuring two-body correlations in the trace-preserving density-matrix dynamics driven by the Lindblad master equation. We therefore explicitly demonstrate a minimal scenario where key properties of a non-Hermitian interacting Hamiltonian can be probed in the context of an open system.\n\nIn particular, while one expects the non-Hermitian many-body Hamiltonian to be of relevance on a short time scale of $1\/N\\Gamma$ (see discussions in Sec.~II), it is remarkable that the two-body correlation captures key features of the two-body non-Hermitian scattering states at time scales even longer than $1\/\\Gamma$ (see Fig.~\\ref{fig:fig4}). We attribute such a phenomenon to the dominance of few-body correlations in the open dissipative system. Our result is complementary to previous attempts at connecting non-Hermitian many-body Hamiltonians with open dissipative systems~\\cite{fu,michishita}, and is relevant to cold atomic gases where few-body correlations dominate.\n\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\nWe thank Xiaoling Cui for helpful discussions.\nThis work has been supported by the Natural Science Foundation of China (Grant No. 11974331) and the National Key R\\&D Program (Grant Nos. 2016YFA0301700, 2017YFA0304100).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=7cm]{MilieuPerio.png}\n\\qquad\\begin{tikzpicture}[scale=0.7]\n\\pgfmathsetmacro{\\cubexa}{8}\n\\pgfmathsetmacro{\\cubeya}{1}\n\\pgfmathsetmacro{\\cubeza}{1}\n\n\\draw[black,dotted] (-0.5,-\\cubexa\/2,-0.5) -- ++(0,-0.8,0);\n\\draw[black,dotted] (-\\cubexa\/2,-0.5,-0.5) -- ++(-0.8,0,0);\n\n\\draw[black,fill=gray!30] (0.5,0.5,-0.5) -- ++(0,0,-\\cubexa\/2+0.5) -- ++(-\\cubeya,0,0) -- ++(0,0,\\cubexa\/2-0.5) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,-0.5) -- ++(0,-\\cubeza,0) -- ++(-\\cubeya,0,0) -- ++(0,\\cubeza,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,-0.5) -- ++(0,0,-\\cubexa\/2+0.5) -- ++(0,-\\cubeza,0) -- ++(0,0,\\cubexa\/2-0.5) -- cycle;\n\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2,0.5) -- ++(0,-\\cubexa,0) -- ++(0,0,-\\cubeya) -- ++(0,\\cubexa,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2,0.5) -- ++(-\\cubeza,0,0) -- ++(0,0,-\\cubeya) -- ++(\\cubeza,0,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2,0.5) -- ++(0,-\\cubexa,0) -- ++(-\\cubeza,0,0) -- ++(0,\\cubexa,0) -- cycle;\n\n\\draw[black,fill=gray!30] (\\cubexa\/2,0.5,0.5) -- ++(-\\cubexa,0,0) -- ++(0,-\\cubeya,0) -- ++(\\cubexa,0,0) -- cycle;\n\\draw[black,fill=gray!30] (\\cubexa\/2,0.5,0.5) -- ++(0,0,-\\cubeza) -- ++(0,-\\cubeya,0) -- ++(0,0,\\cubeza) -- cycle;\n\\draw[black,fill=gray!30] (\\cubexa\/2,0.5,0.5) -- ++(-\\cubexa,0,0) -- ++(0,0,-\\cubeza) -- ++(\\cubexa,0,0) -- cycle;\n\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2,0.5) -- ++(0,-\\cubexa\/2+0.5,0) -- ++(0,0,-\\cubeya) -- ++(0,\\cubexa\/2-0.5,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2,0.5) -- ++(-\\cubeza,0,0) -- ++(0,0,-\\cubeya) -- ++(\\cubeza,0,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2,0.5) -- ++(0,-\\cubexa,0) -- ++(-\\cubeza,0,0) -- ++(0,\\cubexa,0) -- cycle;\n\n\\draw[black,fill=gray!30] (0.5,0.5,\\cubexa\/2) -- ++(0,0,-\\cubexa\/2+0.5) -- ++(-\\cubeya,0,0) -- ++(0,0,\\cubexa\/2-0.5) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,\\cubexa\/2) -- ++(0,-\\cubeza,0) -- ++(-\\cubeya,0,0) -- ++(0,\\cubeza,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,\\cubexa\/2) -- ++(0,0,-\\cubexa\/2+0.5) -- ++(0,-\\cubeza,0) -- ++(0,0,\\cubexa\/2-0.5) -- cycle;\n\\draw[black,dotted] (\\cubexa\/2,0.5,0.5) -- ++(0.8,0,0);\n\\draw[black,dotted] (\\cubexa\/2,-0.5,0.5) -- ++(0.8,0,0);\n\\draw[black,dotted] (\\cubexa\/2,0.5,-0.5) -- ++(0.8,0,0);\n\\draw[black,dotted] (\\cubexa\/2,-0.5,-0.5) -- ++(0.8,0,0);\n\\draw[black,dotted] (-\\cubexa\/2,0.5,0.5) -- ++(-0.8,0,0);\n\\draw[black,dotted] (-\\cubexa\/2,-0.5,0.5) -- ++(-0.8,0,0);\n\\draw[black,dotted] (-\\cubexa\/2,0.5,-0.5) -- ++(-0.8,0,0);\n\n\\draw[black,dotted] (0.5,\\cubexa\/2,0.5) -- ++(0,0.8,0);\n\\draw[black,dotted] (0.5,\\cubexa\/2,-0.5) -- ++(0,0.8,0);\n\\draw[black,dotted] (-0.5,\\cubexa\/2,0.5) -- ++(0,0.8,0);\n\\draw[black,dotted] (-0.5,\\cubexa\/2,-0.5) -- ++(0,0.8,0);\n\\draw[black,dotted] (0.5,-\\cubexa\/2,0.5) -- ++(0,-0.8,0);\n\\draw[black,dotted] (0.5,-\\cubexa\/2,-0.5) -- ++(0,-0.8,0);\n\\draw[black,dotted] (-0.5,-\\cubexa\/2,0.5) -- ++(0,-0.8,0);\n\n\\draw[black,dotted] (0.5,0.5,\\cubexa\/2) -- ++(0,0,0.8);\n\\draw[black,dotted] (-0.5,0.5,\\cubexa\/2) -- ++(0,0,0.8);\n\\draw[black,dotted] (0.5,-0.5,\\cubexa\/2) -- ++(0,0,0.8);\n\\draw[black,dotted] (-0.5,-0.5,\\cubexa\/2) -- ++(0,0,0.8);\n\\draw[black,dotted] (0.5,0.5,-\\cubexa\/2) -- ++(0,0,-0.8);\n\\draw[black,dotted] (-0.5,0.5,-\\cubexa\/2) -- ++(0,0,-0.8);\n\\draw[black,dotted] (0.5,-0.5,-\\cubexa\/2) -- ++(0,0,-0.8);\n\\draw[black,dotted] (-0.5,-0.5,-\\cubexa\/2) -- ++(0,0,-0.8);\n\\end{tikzpicture}\n\\caption{Left: unbounded periodic cubic lattice made of thin bars of width $\\varepsilon$ with square cross-sections. Right: geometry $\\Omega$ of the near field, aka boundary layer, problem at the junction regions. \\label{PeriodicLattice}}\n\\end{figure}\n\nWith the profusion of works related to graphene in physics, an important effort has been made in the mathematical community to understand the asymptotic behaviour of the spectrum of the Dirichlet operator in quantum waveguides made of thin ligaments, of characteristic width $\\varepsilon\\ll1$, forming unbounded periodic lattices. Various geometries have been considered and we refer the reader to \\cite{Kuch02} for a review article. \nTo address such problems, the general approach can be summarized as follows. Using the Floquet-Bloch-Gelfand theory \\cite{Gelf50,Kuch82,Skri87,Kuch93}, one shows that the spectrum of the operator in the periodic domain has a band-gap structure, the bands being generated by the eigenvalues of a spectral problem set on the periodicity cell with quasi-periodic boundary conditions involving the Floquet-Bloch parameter. The first step consists in applying techniques of dimension reduction to derive a 1D model for this spectral problem on the periodicity cell. Then one studies precisely this 1D model depending on the Floquet-Bloch parameter to get information on the spectral bands.\\\\\n\\newline\nLet us mention that in the past, slapdash and casual conclusions have been made concerning the 1D model problem. This model consists of ordinary differential equations on the ligaments obtained when taking $\\varepsilon\\to0$ supplemented by transmission conditions at the nodes of the graph. Certain authors have inappropriately applied L. Pauling's model \\cite{Paul36} and imposed Kirchoff transmission conditions at the nodes. These Kirchoff conditions boil down to impose continuity of the field and zero outgoing flux (the sum of the derivatives of the field along the outgoing directions at the node vanishes). This is correct for the Laplacian with Neumann boundary conditions (BC) and has been rigorously justified in \\cite{KuZe03,ExPo05,Post12}. However it has been shown by D. Grieser in \\cite{Grie08} (see also \\cite{MoVa07}) that for the Dirichlet problem, in general the right conditions to impose at the nodes are Dirichlet ones. More precisely, it has been proved in \\cite{Grie08} that the transmission conditions to impose depend on the existence or absence of so-called threshold resonances for the near field operator defined as the Laplacian in the geometry obtained when zooming at the junction regions (denoted $\\Omega$ in the sequel, see Figure \\ref{PeriodicLattice} right). We say that there is a threshold resonance if there is a non zero bounded function which solves the homogeneous problem at the frequency coinciding with the bottom of the essential spectrum (the threshold) of the Laplace operator. For the Neumann problem, the threshold is $\\Lambda_\\dagger=0$ and  there is a threshold resonance because the constants solve $-\\Delta u=0$ in $\\Omega$ + Neumann BC. Due to this property, one must impose Kirchoff conditions at the nodes. For the Dirichlet problem, the continuous spectrum starts at a positive threshold $\\Lambda_\\dagger>0$ and in general the only solution to the problem \n\\begin{equation}\\label{PbThreshold}\n\\begin{array}{|rcll}\n-\\Delta u&=&\\Lambda_\\dagger u &\\mbox{ in } \\Omega\\\\[3pt]\nu&=&0&\\mbox{ on } \\partial\\Omega\n\\end{array}\n\\end{equation}\nwhich remains bounded at infinity is zero, i.e. there is no threshold resonance. Because of this feature, one should impose Dirichlet condition at the nodes of the 1D model (see (\\ref{farfield_BC}) for the precise moment where this pops up in the analysis below).\n\n\n\n\\begin{figure}[!ht]\n\\centering\n\\begin{tikzpicture}[scale=1]\n\\draw[fill=gray!30,draw=none](-2,-1\/2) rectangle (2,1\/2);\n\\draw[fill=gray!30,draw=none](-1\/2,-1\/2) rectangle (1\/2,2);\n\\draw[black] (-2,-1\/2)--(2,-1\/2);\n\\draw[black] (-2,1\/2)--(-1\/2,1\/2)--(-1\/2,2);\n\\draw[black] (2,1\/2)--(1\/2,1\/2)--(1\/2,2);\n\\draw[black,dotted] (-2.5,-1\/2)--(2.5,-1\/2);\n\\draw[black,dotted] (-2.5,1\/2)--(-2,1\/2);\n\\draw[black,dotted] (2.5,1\/2)--(2,1\/2);\n\\draw[black,dotted] (-1\/2,2.5)--(-1\/2,2);\n\\draw[black,dotted] (1\/2,2.5)--(1\/2,2);\n\\end{tikzpicture}\\qquad\\begin{tikzpicture}[scale=1]\n\\begin{scope}[rotate=20]\n\\draw[fill=gray!30,draw=none](-1\/2,0) rectangle (1\/2,2);\n\\draw[black] (-1\/2,1\/2)--(-1\/2,2);\n\\draw[black] (1\/2,0)--(1\/2,2);\n\\draw[black,dotted] (-1\/2,2.5)--(-1\/2,2);\n\\draw[black,dotted] (1\/2,2.5)--(1\/2,2);\n\\end{scope}\n\\draw[fill=gray!30,draw=none](-2,-1\/2) rectangle (2,1\/2);\n\\begin{scope}[rotate=20]\n\\draw[black,dashed] (0,0)--(0,2.5);\n\\end{scope}\n\\draw[black,dashed] (-2.5,0)--(2.5,0);\n\\draw[black] (-2,-1\/2)--(2,-1\/2);\n\\draw[black] (-2,1\/2)--(-0.7,1\/2);\n\\draw[black] (2,1\/2)--(0.35,1\/2);\n\\draw[black,dotted] (-2.5,-1\/2)--(2.5,-1\/2);\n\\draw[black,dotted] (-2.5,1\/2)--(-2,1\/2);\n\\draw[black,dotted] (2.5,1\/2)--(2,1\/2);\n\\pgfmathparse{.5*cos(110)}\\let\\x\\pgfmathresult\n\\pgfmathparse{.5*sin(110)}\\let\\y\\pgfmathresult\n\\draw[blue,->] (\\x,\\y) arc (110:180:.5cm);\n\\node[left,blue] at (145:.5) {\\small $\\alpha$};\n\\end{tikzpicture}\\qquad\\raisebox{-0.6cm}{\\begin{tikzpicture}[scale=0.75]\n\\draw[fill=gray!30,draw=none](-2,-1\/2) rectangle (2,1\/2);\n\\draw[fill=gray!30,draw=none](-1\/2,-2) rectangle (1\/2,2);\n\\draw[black] (-2,-1\/2)--(-1\/2,-1\/2)--(-1\/2,-2);\n\\draw[black] (2,-1\/2)--(1\/2,-1\/2)--(1\/2,-2);\n\\draw[black] (-2,1\/2)--(-1\/2,1\/2)--(-1\/2,2);\n\\draw[black] (2,1\/2)--(1\/2,1\/2)--(1\/2,2);\n\\draw[black,dotted] (2.5,-1\/2)--(2,-1\/2);\n\\draw[black,dotted] (-2.5,-1\/2)--(-2,-1\/2);\n\\draw[black,dotted] (-2.5,1\/2)--(-2,1\/2);\n\\draw[black,dotted] (2.5,1\/2)--(2,1\/2);\n\\draw[black,dotted] (-1\/2,2.5)--(-1\/2,2);\n\\draw[black,dotted] (1\/2,2.5)--(1\/2,2);\n\\draw[black,dotted] (-1\/2,-2.5)--(-1\/2,-2);\n\\draw[black,dotted] (1\/2,-2.5)--(1\/2,-2);\n\\begin{scope}[rotate=45]\n\\draw[fill=gray!60,draw=none](-0.707,-0.707) rectangle (0.707,0.707);\n\\end{scope}\n\\end{tikzpicture}}\n\\caption{Examples of T- and X-shaped 2D geometries.\\label{2Dexamples}} \n\\end{figure}\n\n\\noindent From there, authors have worked to establish rigorous results showing the absence of non zero bounded solution to (\\ref{PbThreshold}). First, different planar\nquantum waveguides made of T-, X- and Y-shaped junctions of thin ligaments have been considered in \\cite{Naza10a,Naza14,Naza14a,NaRU14,NaRU15,Naza17,Pank17}. In these articles, additionally it has been proved that the near field operator in $\\Omega$, depending on the considered geometry, may have discrete spectrum (one or several eigenvalue below the continuous spectrum). When the latter exists, the low-frequency range of the spectrum in the periodic domain is not described by the above mentioned 1D model with Dirichlet conditions at the nodes. Instead, the first spectral bands, their number being equal to the multiplicity of the discrete spectrum, are generated by functions which are localized at the junctions regions. Let us mention that for certain exceptional geometries, for example for a sequence of angles $\\alpha$ in the central domain of Figure \\ref{2Dexamples}, one may have non zero solutions to (\\ref{PbThreshold}) which remain bounded at infinity, i.e. existence of threshold resonance. In these situations, at least if the dimension of the space of bounded solutions to (\\ref{PbThreshold}) is one, the good 1D model describing the spectral bands which are not associated with the discrete spectrum in $\\Omega$, has certain Kirchoff transmission conditions at the nodes which depend on the geometry. We emphasize that this leads to very different spectra for the operator in the periodic medium. More precisely, when there is no threshold resonance, the bands of the spectrum in the periodic material become very small as $\\varepsilon\\to0$ and the spectral gaps enlarge. In other words, the spectrum becomes rather sparse, for most of the spectral parameters waves cannot propagate and the limit 1D ligaments are somehow disconnected. This is what we will obtain below in our configuration. On the other hand, when Problem (\\ref{PbThreshold}) admits a space of dimension one of bounded solutions, the spectral bands in the periodic material are much larger.\\\\\n\\newline\nAfterwards, 3D geometries were considered in \\cite{BaMaNa17} (see also the corresponding note \\cite{BaMaNa15}). In these articles, the authors consider quantum waveguides for which the near field domain $\\Omega$ is a cruciform junction of two cylinders whose cross section coincides with the unit disc. The passage from the planar case to the spatial case requires the non obvious adaptation of the methods. In particular the characterization of the discrete spectrum of the near field operator and the proof of absence of threshold resonances are much more involved. \nIn the present work, we study an even more intricate 3D geometry for which the near field geometry is the union of three waveguides. Rather precise Friedrichs estimates are required to prove that the discrete spectrum of the near field operator contains exactly one eigenvalue and to show that at the threshold, zero is the only bounded solution. This work also complements the study of \\cite{BaMaNa17} thanks to the numerical experiments. Let us mention that the cruciform junction of two 3D cylinders with square cross section reduces to a 2D problem in a X-shaped geometry (see again Figure \\ref{2Dexamples} right) and due to factoring out, is of no interest.\\\\\n\\newline\nThe outline is as follows. First we describe the problem, introduce the notation and present the main results. Then we study the discrete spectrum of the near field operator. In section \\ref{sectionAbsenceofBoundedSol}, we demonstrate the absence of threshold resonance for the near field operator. Section \\ref{SectionModels} is dedicated to the analysis of the main theorem of the article (Theorem \\ref{MainThmPerio}) with the derivation of asymptotic models for the spectral bands in the original periodic domain. Finally we show some numerics to complement the results and conclude with some appendix containing the proof of two lemmas needed in sections \\ref{Section_DiscreteSpectrum}, \\ref{sectionAbsenceofBoundedSol}. \n\n\\section{Notation and main results}\n\nFor $j=1,2,3$, introduce the cylinder with square cross section\n\\begin{equation}\\label{defBars}\nL_j  :=  \\{x:=(x_1,x_2,x_2)\\in\\mathbb{R}^3\\,|\\,|x_k|<1\/2\\mbox{ for }k\\in\\{1,2,3\\}\\setminus\\{j\\}\\}\n\\end{equation}\nand for $\\varepsilon>0$ small, $m,n\\in\\mathbb{Z}:=\\{0,\\pm1,\\pm2,\\dots\\}$, set\n\\[\n\\begin{array}{rcl}\nL^{mn\\varepsilon}_1 & := & \\{x\\in\\mathbb{R}^3\\,|\\,|x_2-m|<\\varepsilon\/2,\\,|x_3-n|<\\varepsilon\/2\\}\\\\[2pt]\nL^{mn\\varepsilon}_2 & := & \\{x\\in\\mathbb{R}^3\\,|\\,|x_1-m|<\\varepsilon\/2,\\,|x_3-n|<\\varepsilon\/2\\}\\\\[2pt]\nL^{mn\\varepsilon}_3 & := & \\{x\\in\\mathbb{R}^3\\,|\\,|x_1-m|<\\varepsilon\/2,\\,|x_2-n|<\\varepsilon\/2\\}.\n\\end{array}\n\\]\nFinally define the unbounded periodic domain \n\\[\n\\Pi^\\varepsilon:=\\bigcup_{m,n\\in\\mathbb{Z}}L^{mn\\varepsilon}_1\\cup L^{mn\\varepsilon}_2\\cup L^{mn\\varepsilon}_3\n\\]\n(see Figure \\ref{PeriodicLattice} left). Consider the Dirichlet spectral problem for the Laplace operator \n\\begin{equation}\\label{PbSpectral}\n\\begin{array}{|rcll}\n-\\Delta u^\\varepsilon&=&\\lambda^\\varepsilon\\,u^\\varepsilon &\\mbox{ in }\\Pi^\\varepsilon\\\\[3pt]\nu^\\varepsilon&=&0&\\mbox{ on } \\partial\\Pi^\\varepsilon.\n\\end{array}\n\\end{equation}\nThe variational form associated with this problem writes\n\\begin{equation}\\label{variationPb}\n\\int_{\\Pi^\\varepsilon}\\nabla u^\\varepsilon\\cdot\\nabla v^\\varepsilon\\,dx=\\lambda^\\varepsilon\\int_{\\Pi^\\varepsilon} u^\\varepsilon v^\\varepsilon\\,dx,\\qquad\\forall v^\\varepsilon\\in\\mrm{H}^1_0(\\Pi^\\varepsilon).\n\\end{equation}\nHere $\\mrm{H}^1_0(\\Pi^\\varepsilon)$ stands for the Sobolev space of functions of $\\mrm{H}^1(\\Pi^\\varepsilon)$ which vanish on the boundary $\\partial\\Pi^\\varepsilon$. Classically (see e.g \\cite[\\S10.1]{BiSo87}), the variational problem (\\ref{variationPb}) gives rise to an unbounded, positive definite, selfadjoint operator $A^\\varepsilon$ in the Hilbert space $\\mrm{L}^2(\\Pi^\\varepsilon)$, with domain $\\mathcal{D}(A^\\varepsilon)\\subset\\mrm{H}^1_0(\\Pi^\\varepsilon)$. Note that this operator is sometimes called the quantum graph Laplacian \\cite{BeKu13,Post12}. Since $\\Pi^\\varepsilon$ is unbounded, the embedding $\\mrm{H}^1_0(\\Pi^\\varepsilon)\\subset \\mrm{L}^2(\\Pi^\\varepsilon)$ is not compact and $A^\\varepsilon$ has a non empty essential component $\\sigma_e(A^\\varepsilon)$ (\\cite[Thm. 10.1.5]{BiSo87}). Actually, due to the periodicity, we have $\\sigma_e(A^\\varepsilon)=\\sigma(A^\\varepsilon)$. The Gelfand transform (see the surveys \\cite{Kuch82,Naza99a} and books \\cite{Skri87,Kuch93}) \n\\[\nu^{\\varepsilon}(x)\\mapsto U^\\varepsilon(x,\\eta)=\\cfrac{1}{(2\\pi)^{3\/2}}\\sum_{\\iota\\in\\mathbb{Z}^3} e^{i\\eta\\cdot \\iota}u^{\\varepsilon}(x+\\iota),\\quad\\eta=(\\eta_1,\\eta_2,\\eta_3),\n\\]\nchanges Problem (\\ref{PbSpectral}) into the following spectral problem with quasi-periodic at the faces located at $x_1=\\pm1\/2$, $x_2=\\pm1\/2$, $x_3=\\pm1\/2$,\n\\begin{equation}\\label{PbSpectralCell}\n\\begin{array}{|rcll}\n-\\Delta U^\\varepsilon(x,\\eta)&=&\\Lambda^\\varepsilon\\,U^\\varepsilon(x,\\eta) & x\\in\\omega^\\varepsilon\\\\[3pt]\nU^\\varepsilon(x,\\eta)&=&0&x\\in\\partial\\omega^\\varepsilon\\cap\\partial\\Pi^\\varepsilon\\\\[3pt]\nU^\\varepsilon(-1\/2,x_2,x_3,\\eta) & = & e^{i\\eta_1}U^\\varepsilon(+1\/2,x_2,x_3,\\eta) & (x_2,x_3)\\in(-\\varepsilon\/2;\\varepsilon\/2)^2\\\\[3pt]\n\\partial_{x_1}U^\\varepsilon(-1\/2,x_2,x_3,\\eta) & = & e^{i\\eta_1}\\partial_{x_1}U^\\varepsilon(+1\/2,x_2,x_3,\\eta) & (x_2,x_3)\\in(-\\varepsilon\/2;\\varepsilon\/2)^2\\\\[3pt]\nU^\\varepsilon(x_1,-1\/2,x_3,\\eta) & = & e^{i\\eta_2}U^\\varepsilon(x_1,+1\/2,x_3,\\eta) & (x_1,x_3)\\in(-\\varepsilon\/2;\\varepsilon\/2)^2\\\\[3pt]\n\\partial_{x_2}U^\\varepsilon(x_1,-1\/2,x_3,\\eta) & = & e^{i\\eta_2}\\partial_{x_2}U^\\varepsilon(x_1,+1\/2,x_3,\\eta) & (x_1,x_3)\\in(-\\varepsilon\/2;\\varepsilon\/2)^2\\\\[3pt]\nU^\\varepsilon(x_1,x_2,-1\/2,\\eta) & = & e^{i\\eta_3}U^\\varepsilon(x_1,x_2,+1\/2,\\eta) & (x_1,x_2)\\in(-\\varepsilon\/2;\\varepsilon\/2)^2\\\\[3pt]\n\\partial_{x_3}U^\\varepsilon(x_1,x_2,-1\/2,\\eta) & = & e^{i\\eta_3}\\partial_{x_3}U^\\varepsilon(x_1,x_2,+1\/2,\\eta) & (x_1,x_2)\\in(-\\varepsilon\/2;\\varepsilon\/2)^2,\n\\end{array}\n\\end{equation}\nset in the periodicity cell \n\\[\n\\omega^\\varepsilon:=\\omega^\\varepsilon_1\\cup\\omega^\\varepsilon_2\\cup\\omega^\\varepsilon_3\\quad\\mbox{ with }\\quad\\begin{array}{|l}\n\\omega^\\varepsilon_1:=(-1\/2;1\/2)\\times(-\\varepsilon\/2;\\varepsilon\/2)^2\\\\[2pt]\n\\omega^\\varepsilon_2:=(-\\varepsilon\/2;\\varepsilon\/2)\\times(-1\/2;1\/2)\\times(-\\varepsilon\/2;\\varepsilon\/2)\\\\[2pt]\n\\omega^\\varepsilon_3:=(-\\varepsilon\/2;\\varepsilon\/2)^2\\times(-1\/2;1\/2).\n\\end{array}\n\\]\nProblem (\\ref{PbSpectralCell}) is formally selfadjoint for any value of the dual variable $\\eta\\in\\mathbb{R}^3$. Additionally it is $2\\pi$-periodic with respect to each of the $\\eta_j$ because the transformation $\\eta_j\\mapsto\\eta_j+2\\pi$ leaves invariant the quasiperiodicity conditions. For any $\\eta\\in[0;2\\pi)^3$, the spectrum of (\\ref{PbSpectralCell}) is discrete, made of a monotone increasing positive sequence of eigenvalues\n\\[\n0<\\Lambda_1^\\varepsilon(\\eta)\\le \\Lambda_2^\\varepsilon(\\eta)\\le \\dots \\le \\Lambda_p^\\varepsilon(\\eta)\\le \\dots\n\\]\nwhere the $\\Lambda_p^\\varepsilon(\\eta)$ are counted according to their multiplicity. The functions \n\\[\n\\eta\\mapsto \\Lambda_p^\\varepsilon(\\eta)\n\\]\nare continuous (\\cite[Chap. 9]{Kato95}) so that the sets \n\\begin{equation}\\label{SpectralBands}\n\\Upsilon^\\varepsilon_p=\\{\\Lambda_p^\\varepsilon(\\eta),\\,\\eta\\in[0;2\\pi)^3\\}\n\\end{equation}\nare connected compact segments. Finally, according to the theory (see again \\cite{Gelf50,Kuch82,Skri87,Naza99a,Kuch93}), the spectrum of the operator $A^{\\varepsilon}$ has the form\n\\[\n\\sigma(A^\\varepsilon)=\\bigcup_{p\\in\\mathbb{N}^\\ast}\\Upsilon^\\varepsilon_p\n\\]\nwhere $\\mathbb{N}^\\ast:=\\{1,2,\\dots\\}$. At this stage, we see that to clarify the behaviour of the spectrum of $A^\\varepsilon$ with respect to $\\varepsilon\\to0^+$, we need to study the dependence of the $\\Upsilon^\\varepsilon_p$ with respect to $\\varepsilon$.\\\\\n\\newline\nAs already mentioned in the introduction, the analysis developed for example in \\cite{Grie08,Naza05,Naza14} shows that the asymptotic behaviour of the $\\Upsilon^\\varepsilon_p$ with respect to $\\varepsilon$ depends on the features of the Dirichlet Laplacian in the geometry obtained when zooming at the junction region of the periodicity cell $\\omega^\\varepsilon$. More precisely, introduce the unbounded domain\n\\begin{equation}\\label{DefOmega}\n\\Omega:=L_1\\cup L_2\\cup L_3\n\\end{equation}\n(see Figure \\ref{PeriodicLattice} right) where the $L_j$ are the cylinders with unit square cross section appearing in (\\ref{defBars}). In $\\Omega$, consider the Dirichlet spectral problem for the Laplace operator \n\\begin{equation}\\label{PbSpectralZoom}\n\\begin{array}{|rcll}\n-\\Delta v&=&\\mu\\,v &\\mbox{ in }\\Omega\\\\[3pt]\nv&=&0&\\mbox{ on } \\partial\\Omega\n\\end{array}\n\\end{equation}\nwhich is now independent of $\\varepsilon$. We denote by $A^\\Omega$ the unbounded, positive definite, selfadjoint operator naturally associated with this problem defined in $\\mrm{L}^2(\\Omega)$ and of domain $\\mathcal{D}(A^\\Omega)\\subset\\mrm{H}^1_0(\\Omega)$. Its continuous spectrum $\\sigma_c(A^\\Omega)$ occupies the ray $[2\\pi^2;+\\infty)$ and the threshold point $\\Lambda_\\dagger:=2\\pi^2$ is the first eigenvalue of the Dirichlet problem in the cross sections of the branches of $\\Omega$ (which are unit squares). The main goal of this article is to show the following results.\n\\begin{theorem}\\label{ThmDiscreteSpectrum}\nThe discrete spectrum of the operator $A^{\\Omega}$ contains exactly one eigenvalue $\\mu_1\\in(0;2\\pi^2)$.\n\\end{theorem}\n\\begin{remark}\nNumerically, in Section \\ref{SectionNumerics}, we find $\\mu_1\\approx 12.9\\approx1.3\\pi^2$. Note that the eigenfunctions associated with $\\mu_1$ decay at infinity as $O(-\\sqrt{2\\pi^2-\\mu_1}\\,|x_j|)$ in $L_j$, $j=1,2,3$.\n\\end{remark}\n\\noindent As classical in literature, we shall say that there is a threshold resonance for Problem (\\ref{PbSpectralZoom}) if there is non trivial function which solves (\\ref{PbSpectralZoom}) with the threshold value $\\mu=2\\pi^2$ of the spectral parameter.\n\\begin{theorem}\\label{ThmNoBoundedSol}\nThere is no threshold resonance for Problem (\\ref{PbSpectralZoom}). \n\\end{theorem}\n\\noindent From Theorems \\ref{ThmDiscreteSpectrum} and \\ref{ThmNoBoundedSol}, we will derive the final result for the spectrum of $A^\\varepsilon$ in the initial unbounded periodic lattice:\n\\begin{theorem}\\label{MainThmPerio}\nThere are positive values $\\varepsilon_p>0$ and $c_p>0$ such that for the spectral bands introduced in (\\ref{SpectralBands}), we have the estimates\n\\[\n\\begin{array}{ll}\n\\Upsilon^\\varepsilon_1\\subset \\varepsilon^{-2}\\mu_1+\\varepsilon^{-2}e^{-\\sqrt{2\\pi^2-\\mu_1}\/\\varepsilon}(-c_1;c_1),&\\  \\varepsilon\\in(0;\\varepsilon_1]\\\\[8pt]\n\\Upsilon^\\varepsilon_{1+q+3(p-1)}\\subset \\varepsilon^{-2}2\\pi^2+p^2\\pi^2+\\varepsilon\\,(-c_p;c_p),&\\  \\varepsilon\\in(0;\\varepsilon_p],\\  q=1,2,3,\\  p\\in\\mathbb{N}^\\ast.\\\\\n\\end{array}\n\\]\n\\end{theorem}\n\\noindent Let us comment this result. First, as already mentioned in the introduction, the spectrum of the operator $A^\\varepsilon$ goes to $+\\infty$ as $\\varepsilon\\to0$. Additionally this spectrum becomes very sparse. Indeed Theorem \\ref{MainThmPerio} implies the following results. The length of the spectral bands are infinitesimal as $\\varepsilon\\to0$.  Moreover, between the bands $\\Upsilon^\\varepsilon_1$ and $\\Upsilon^\\varepsilon_2$, there is a gap, that is a segment of spectral parameters $\\Lambda$ such that waves cannot propagate, of size $O(\\varepsilon^{-2}(2\\pi^2-\\mu_1))$ while between $\\Upsilon^\\varepsilon_{1+q+3(p-1)}$ and $\\Upsilon^\\varepsilon_{1+q+3p}$, the gap is of width $O(\\pi^2(2p-1))$. Therefore the main message here is that in the thin lattice $\\Pi^\\varepsilon$ the propagation of waves is hampered and occurs for very narrow intervals of frequencies.\n\n\n\\section{Properties of the discrete spectrum of $A^{\\Omega}$}\\label{Section_DiscreteSpectrum}\nThe goal of this section is to prove Theorem \\ref{ThmDiscreteSpectrum}. We start by showing that the discrete spectrum of $A^{\\Omega}$ is non empty (for related multidimensional problems, see \\cite{Naza12a}). \n\\begin{proposition}\\label{PropoAtLeastOne}\nThe discrete spectrum of $A^{\\Omega}$ has at least one eigenvalue.\n\\end{proposition}\n\\begin{proof}\nDefine the 2D X-shaped geometry $\\Omega^{2D}:=\\Omega^{2D}_1\\cup\\Omega^{2D}_2$ with $\\Omega^{2D}_1:=\\mathbb{R}\\times(-1\/2;1\/2)$ and $\\Omega^{2D}_2:=(-1\/2;1\/2)\\times\\mathbb{R}$ (see Figure \\ref{2Dexamples} right). According to \\cite{ScRW89,ABGM91} (see also \\cite[Thm. 2.1]{Naza14b} as well as the discussion at the end of this proof), we know that the Dirichlet Laplacian in $\\Omega^{2D}$ admits exactly one eigenvalue $\\mu^{2D}$ below the continuous spectrum which coincides with $[\\pi^2;+\\infty)$. Let $\\varphi\\in\\mrm{H}^1_0(\\Omega^{2D})$ be a corresponding eigenfunction. In the domain $\\Omega\\subset\\mathbb{R}^3$ introduced in (\\ref{DefOmega}), consider the function $v$ such that\n\\[\n\\begin{array}{ll}\nv(x_1,x_2,x_3) = \\varphi(x_1,x_2)\\cos(\\pi x_3)\\ \\mbox{ in }L_1\\cup L_2,\\qquad\\qquad &v(x_1,x_2,x_3)=0\\ \\mbox{ in }\\Omega\\setminus\\overline{L_1\\cup L_2}.\n\\end{array}\n\\]\nWe have \n\\[\n\\int_{\\Omega}|\\nabla v|^2\\,dx=\\int_{L_1\\cup L_2}|\\nabla v|^2\\,dx=-\\int_{L_1\\cup L_2}\\Delta v v\\,dx=(\\mu^{2D}+\\pi^2)\\int_{\\Omega} v^2\\,dx.\n\\]\nBut according to the max-min principle (cf. \\cite[Thm. 10.2.2]{BiSo87}), we know that\n\\begin{equation}\\label{CaracInfSup}\n\\inf \\sigma(A^{\\Omega})=\\inf_{w\\in\\mrm{H}^1_0(\\Omega)\\setminus\\{0\\}}\\cfrac{\\displaystyle\\int_{\\Omega}|\\nabla w|^2\\,dx}{\\displaystyle\\int_{\\Omega}w^2\\,dx}\\,.\n\\end{equation}\nInserting the above $v$ in the right hand side of (\\ref{CaracInfSup}), we deduce that the discrete spectrum of $A^{\\Omega}$ contains an eigenvalue below $\\mu^{2D}+\\pi^2$.\\\\\n\\newline\nWe end this proof by describing the elegant trick proposed in \\cite{ScRW89,ABGM91} to show the existence of an eigenvalue in the discrete spectrum of the Dirichlet Laplacian in $\\Omega^{2D}$. Consider the square coloured in dark gray of Figure \\ref{2Dexamples} right. It has side $\\sqrt{2}$. Therefore the first eigenvalue of the Dirichlet Laplacian in this geometry is equal to $\\pi^2$. Hence, in any domain containing strictly this square and included in $\\Omega^{2D}$, the first eigenvalue of the Dirichlet Laplacian is strictly less than $\\pi^2$. Extending a corresponding eigenfunction by zero to $\\Omega^{2D}$ and using the max-min principle, we infer that the discrete spectrum of the Dirichlet Laplacian in $\\Omega^{2D}$ is not empty.\n\\end{proof}\n\n\n\\noindent It remains to show that the discrete spectrum of $A^{\\Omega}$ has at most one eigenvalue. To proceed, first we establish a result of symmetry. \n\\begin{lemma}\\label{LemmaSym}\nLet $v\\in\\mrm{H}^1_0(\\Omega)$ be an eigenfunction of the operator $A^{\\Omega}$ associated with an eigenvalue $\\mu<2\\pi^2$. Then $v$ is symmetric with respect to the three planes $(Ox_2x_3)$, $(Ox_3x_1)$, $(Ox_1x_2)$.\n\\end{lemma}\n\\begin{proof}\nWe present the proof of symmetry with respect to the plane $(Ox_2x_3)$. The two other symmetries can be established similarly. Introduce the function $\\varphi$ such that\n\\[\n\\varphi(x_1,x_2,x_3)=v(x_1,x_2,x_3)-v(-x_1,x_2,x_3).\n\\]\nOur goal is to prove that $\\varphi\\equiv0$. Set $\\Omega^+_1:=\\{(x_1,x_2,x_3)\\in\\Omega\\,|\\,x_1>0\\}$. The function $\\varphi$ satisfies  $-\\Delta\\varphi=\\mu\\varphi$ in $\\Omega^+_1$ and $\\varphi=0$ on $\\partial\\Omega^+_1$. Therefore we have \n\\begin{equation}\\label{IdentityEigen}\n\\int_{\\Omega^+_1}|\\nabla\\varphi|^2\\,dx=\\mu\\int_{\\Omega^+_1}\\varphi^2\\,dx.\n\\end{equation}\n\n\n\\begin{figure}[!ht]\n\\centering\\begin{tikzpicture}[scale=0.75]\n\\pgfmathsetmacro{\\cubexa}{8}\n\\pgfmathsetmacro{\\cubeya}{1}\n\\pgfmathsetmacro{\\cubeza}{1}\n\\begin{scope}[rotate=90]\n\\draw[black,dotted] (0,-\\cubexa\/2,-0.5) -- ++(0,-0.8,0);\n\n\\draw[black,fill=gray!30] (0.5,0.5,-1) -- ++(0,0,-\\cubexa\/2) -- ++(-\\cubeya\/2,0,0) -- ++(0,0,\\cubexa\/2) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,-1) -- ++(0,-\\cubeza,0) -- ++(-\\cubeya\/2,0,0) -- ++(0,\\cubeza,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,-1) -- ++(0,0,-\\cubexa\/2) -- ++(0,-\\cubeza,0) -- ++(0,0,\\cubexa\/2) -- cycle;\n\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2+1,0.5) -- ++(0,-\\cubexa\/2,0) -- ++(0,0,-\\cubeya) -- ++(0,\\cubexa\/2,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2+1,0.5) -- ++(-\\cubeza\/2,0,0) -- ++(0,0,-\\cubeya) -- ++(\\cubeza\/2,0,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2+1,0.5) -- ++(0,-\\cubexa\/2,0) -- ++(-\\cubeza\/2,0,0) -- ++(0,\\cubexa\/2,0) -- cycle;\n\n\\draw[black,fill=gray!30] (\\cubexa\/2+1,0.5,0.5) -- ++(-\\cubexa\/2,0,0) -- ++(0,-\\cubeya,0) -- ++(\\cubexa\/2,0,0) -- cycle;\n\\draw[black,fill=gray!30] (\\cubexa\/2+1,0.5,0.5) -- ++(0,0,-\\cubeza) -- ++(0,-\\cubeya,0) -- ++(0,0,\\cubeza) -- cycle;\n\\draw[black,fill=gray!30] (\\cubexa\/2+1,0.5,0.5) -- ++(-\\cubexa\/2,0,0) -- ++(0,0,-\\cubeza) -- ++(\\cubexa\/2,0,0) -- cycle;\n\n\\draw[black,fill=gray!30] (0.5,0.5,0.5) -- ++(-0.5,0,0) -- ++(0,-\\cubeya,0) -- ++(0.5,0,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,0.5) -- ++(0,0,-\\cubeza) -- ++(0,-\\cubeya,0) -- ++(0,0,\\cubeza) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,0.5) -- ++(-0.5,0,0) -- ++(0,0,-\\cubeza) -- ++(0.5,0,0) -- cycle;\n\n\\draw[black,fill=gray!30] (0.5,-1,0.5) -- ++(0,-\\cubexa\/2,0) -- ++(0,0,-\\cubeya) -- ++(0,\\cubexa\/2,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,-1,0.5) -- ++(-\\cubeza\/2,0,0) -- ++(0,0,-\\cubeya) -- ++(\\cubeza\/2,0,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,-1,0.5) -- ++(0,-\\cubexa\/2,0) -- ++(-\\cubeza\/2,0,0) -- ++(0,\\cubexa\/2,0) -- cycle;\n\n\\draw[black,fill=gray!30] (0.5,0.5,\\cubexa\/2+1) -- ++(0,0,-\\cubexa\/2) -- ++(-\\cubeya\/2,0,0) -- ++(0,0,\\cubexa\/2) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,\\cubexa\/2+1) -- ++(0,-\\cubeza,0) -- ++(-\\cubeya\/2,0,0) -- ++(0,\\cubeza,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,\\cubexa\/2+1) -- ++(0,0,-\\cubexa\/2) -- ++(0,-\\cubeza,0) -- ++(0,0,\\cubexa\/2) -- cycle;\n\n\\draw[black,dotted] (\\cubexa\/2+1,0.5,0.5) -- ++(0.8,0,0);\n\\draw[black,dotted] (\\cubexa\/2+1,-0.5,0.5) -- ++(0.8,0,0);\n\\draw[black,dotted] (\\cubexa\/2+1,0.5,-0.5) -- ++(0.8,0,0);\n\\draw[black,dotted] (\\cubexa\/2+1,-0.5,-0.5) -- ++(0.8,0,0);\n\n\\draw[black,dotted] (0.5,\\cubexa\/2+1,0.5) -- ++(0,0.8,0);\n\\draw[black,dotted] (0.5,\\cubexa\/2+1,-0.5) -- ++(0,0.8,0);\n\\draw[black,dotted] (0,\\cubexa\/2+1,0.5) -- ++(0,0.8,0);\n\\draw[black,dotted] (0,\\cubexa\/2+1,-0.5) -- ++(0,0.8,0);\n\\draw[black,dotted] (0.5,-\\cubexa\/2-1,0.5) -- ++(0,-0.8,0);\n\\draw[black,dotted] (0.5,-\\cubexa\/2-1,-0.5) -- ++(0,-0.8,0);\n\\draw[black,dotted] (0,-\\cubexa\/2-1,0.5) -- ++(0,-0.8,0);\n\n\\draw[black,dotted] (0.5,0.5,\\cubexa\/2+1) -- ++(0,0,0.8);\n\\draw[black,dotted] (0,0.5,\\cubexa\/2+1) -- ++(0,0,0.8);\n\\draw[black,dotted] (0.5,-0.5,\\cubexa\/2+1) -- ++(0,0,0.8);\n\\draw[black,dotted] (0,-0.5,\\cubexa\/2+1) -- ++(0,0,0.8);\n\\draw[black,dotted] (0.5,0.5,-\\cubexa\/2-1) -- ++(0,0,-0.8);\n\\draw[black,dotted] (0,0.5,-\\cubexa\/2-1) -- ++(0,0,-0.8);\n\\draw[black,dotted] (0.5,-0.5,-\\cubexa\/2-1) -- ++(0,0,-0.8);\n\n\\node at (3,0,0.5){$L_1^+$};\n\\node at (0,2.6,-1){$S_2^+$};\n\\node at (0,-3.6,-1){$S_2^-$};\n\\node at (0,-0.5,2){$S_3^+$};\n\\node at (0,-0.5,-4.5){$S_3^-$};\n\\draw[black,->] (1.6,-1.6,0) -- ++(-1,1,0);\n\\node at (1.8,-2,0){$Q^+_1$};\n\\end{scope}\n\\end{tikzpicture}\\qquad \\begin{tikzpicture}[scale=0.75]\n\\pgfmathsetmacro{\\cubexa}{8}\n\\pgfmathsetmacro{\\cubeya}{1}\n\\pgfmathsetmacro{\\cubeza}{1}\n\\begin{scope}[rotate=90]\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2+1,0.5) -- ++(0,-\\cubexa\/2,0) -- ++(0,0,-\\cubeya\/2) -- ++(0,\\cubexa\/2,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2+1,0.5) -- ++(-\\cubeza\/2,0,0) -- ++(0,0,-\\cubeya\/2) -- ++(\\cubeza\/2,0,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2+1,0.5) -- ++(0,-\\cubexa\/2,0) -- ++(-\\cubeza\/2,0,0) -- ++(0,\\cubexa\/2,0) -- cycle;\n\n\\draw[black,fill=gray!30] (\\cubexa\/2+1,0.5,0.5) -- ++(-\\cubexa\/2,0,0) -- ++(0,-\\cubeya\/2,0) -- ++(\\cubexa\/2,0,0) -- cycle;\n\\draw[black,fill=gray!30] (\\cubexa\/2+1,0.5,0.5) -- ++(0,0,-\\cubeza\/2) -- ++(0,-\\cubeya\/2,0) -- ++(0,0,\\cubeza\/2) -- cycle;\n\\draw[black,fill=gray!30] (\\cubexa\/2+1,0.5,0.5) -- ++(-\\cubexa\/2,0,0) -- ++(0,0,-\\cubeza\/2) -- ++(\\cubexa\/2,0,0) -- cycle;\n\n\\draw[black,fill=gray!30] (0.5,0.5,0.5) -- ++(-0.5,0,0) -- ++(0,-\\cubeya\/2,0) -- ++(0.5,0,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,0.5) -- ++(0,0,-\\cubeza\/2) -- ++(0,-\\cubeya\/2,0) -- ++(0,0,\\cubeza\/2) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,0.5) -- ++(-0.5,0,0) -- ++(0,0,-\\cubeza\/2) -- ++(0.5,0,0) -- cycle;\n\n\\draw[black,fill=gray!30] (0.5,0.5,\\cubexa\/2+1) -- ++(0,0,-\\cubexa\/2) -- ++(-\\cubeya\/2,0,0) -- ++(0,0,\\cubexa\/2) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,\\cubexa\/2+1) -- ++(0,-\\cubeza\/2,0) -- ++(-\\cubeya\/2,0,0) -- ++(0,\\cubeza\/2,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,\\cubexa\/2+1) -- ++(0,0,-\\cubexa\/2) -- ++(0,-\\cubeza\/2,0) -- ++(0,0,\\cubexa\/2) -- cycle;\n\n\\draw[black,dotted] (\\cubexa\/2+1,0.5,0.5) -- ++(0.8,0,0);\n\\draw[black,dotted] (\\cubexa\/2+1,0,0.5) -- ++(0.8,0,0);\n\\draw[black,dotted] (\\cubexa\/2+1,0.5,0) -- ++(0.8,0,0);\n\\draw[black,dotted] (\\cubexa\/2+1,0,0) -- ++(0.8,0,0);\n\n\\draw[black,dotted] (0.5,\\cubexa\/2+1,0.5) -- ++(0,0.8,0);\n\\draw[black,dotted] (0.5,\\cubexa\/2+1,0) -- ++(0,0.8,0);\n\\draw[black,dotted] (0,\\cubexa\/2+1,0.5) -- ++(0,0.8,0);\n\\draw[black,dotted] (0,\\cubexa\/2+1,0) -- ++(0,0.8,0);\n\n\\draw[black,dotted] (0.5,0.5,\\cubexa\/2+1) -- ++(0,0,0.8);\n\\draw[black,dotted] (0,0.5,\\cubexa\/2+1) -- ++(0,0,0.8);\n\\draw[black,dotted] (0.5,0,\\cubexa\/2+1) -- ++(0,0,0.8);\n\\draw[black,dotted] (0,0,\\cubexa\/2+1) -- ++(0,0,0.8);\n\\draw[black,->] (1.6,1.6,0) -- ++(-1,-1,0);\n\\node at (1.8,1.8,0){$Q^+$};\n\\end{scope}\n\\end{tikzpicture}\n\\caption{Left: exploded decomposition of $\\Omega^+_1$. Right: exploded decomposition of $\\Omega^+$. \\label{ExplodedDomain}}\n\\end{figure}\n\n\\noindent Define the domains \n\\[\n\\begin{array}{rcl}\nL^+_1 & := & \\{x\\in L_1\\,|\\,x_1>1\/2\\}\\\\[2pt]\nS_2 & := & \\{x\\in L_2\\,|\\,x_1>0\\}=\\Omega^+_1\\cap L_2\\\\[2pt]\nS_3 & := & \\{x\\in L_3\\,|\\,x_1>0\\}=\\Omega^+_1\\cap L_3\\\\[2pt]\nS^{\\pm}_2 & := & \\{x\\in S_2\\,|\\,\\pm x_2>1\/2\\}\\\\[2pt]\nS^{\\pm}_3 & := & \\{x\\in S_3\\,|\\,\\pm x_3>1\/2\\}\\\\[2pt]\nQ^+_1& := & \\{x\\in\\mathbb{R}^3\\,|\\,0<x_1<1\/2,\\,|x_2|<1\/2,\\,|x_3|<1\/2\\}=S_2\\cap S_3\n\\end{array}\n\\]\n(see the exploded view of Figure \\ref{ExplodedDomain} left). Using the Poincar\\'e inequality in a section of $L^+_1$, which is a square of unit side, and then integrating with respect to the $x_1$ variable, first we find\n\\begin{equation}\\label{PoincareTranche0}\n2\\pi^2 \\int_{L^+_1}\\varphi^2\\,dx \\le \\int_{L^+_1}|\\nabla\\varphi|^2\\,dx.\n\\end{equation}\nIn $S^\\pm_2$, $S^\\pm_3$, since the transverse section is smaller than in $L^+_1$, we have better estimates, namely\n\\begin{equation}\\label{PoincareTranche}\n5\\pi^2\\int_{S_2^\\pm}\\varphi^2\\,dx \\le \\int_{S_2^\\pm}(\\partial_{x_1}\\varphi)^2+(\\partial_{x_3}\\varphi)^2\\,dx,\\qquad 5\\pi^2\\int_{S_3^\\pm}\\varphi^2\\,dx \\le \\int_{S_3^\\pm}(\\partial_{x_1}\\varphi)^2+(\\partial_{x_2}\\varphi)^2\\,dx.\n\\end{equation}\nThe difficulty to complete the proof however is that in $Q^+_1$, using a 1D  Poincar\\'e inequality with respect to the $x_1$ variable, we can simply get \n\\begin{equation}\\label{PoincareTranche0bis}\n\\pi^2 \\int_{Q^+_1}\\varphi^2\\,dx \\le \\int_{Q^+_1}(\\partial_{x_1}\\varphi)^2\\,dx.\n\\end{equation}\nThis is enough to conclude directly that $\\varphi\\equiv0$ when $\\mu\\in(0;\\pi^2)$ but not in the general situation $\\mu\\in(0;2\\pi^2)$ that we wish to deal with. Therefore we have to refine the strategy. What we will do is to exploit the ``extra'' terms coming from the efficiency of estimates (\\ref{PoincareTranche}) to control the $\\mrm{L}^2$-norm of $\\varphi$ in $Q^+_1$. More precisely, the Lemma 5.1 of \\cite{BaMaNa17} that we recall in Appendix (Lemma \\ref{LemmaPoincareFriedrichs}) guarantees that for a given $a>0$, we have the Friedrichs inequality \n\\begin{equation}\\label{PoincareModified}\n\\kappa(a)\\int_{0}^{1\/2}\\phi^2\\,dt \\le \\int_{0}^{+\\infty}(\\partial_t\\phi)^2\\,dt+a^2\\int_{1\/2}^{+\\infty}\\phi^2\\,dt,\\qquad \\forall \\phi\\in\\mrm{H}^1(0;+\\infty),\n\\end{equation}\nwhere $\\kappa(a)$ is the smallest positive root of the transcendental equation\n\\begin{equation}\\label{TransEqn}\n\\sqrt{\\kappa}\\tan\\bigg(\\cfrac{\\sqrt{\\kappa}}{2}\\bigg)=a.\n\\end{equation}\nIn particular, solving (\\ref{TransEqn}) with $a=\\pi\\sqrt{5\/2}$, we find $\\kappa(a)>\\pi^2\/2$. Therefore, using (\\ref{PoincareModified}), we can write\n\\begin{equation}\\label{PoincareTranche01}\n\\cfrac{\\pi^2}{2}\\int_{Q^+_1}\\varphi^2\\,dx \\le \\int_{S_2}(\\partial_{x_2}\\varphi)^2\\,dx+\\cfrac{5\\pi^2}{2}\\int_{S_2^+\\cup S_2^-}\\varphi^2\\,dx\n\\end{equation}\nas well as \n\\begin{equation}\\label{PoincareTranche02}\n\\cfrac{\\pi^2}{2}\\int_{Q^+_1}\\varphi^2\\,dx \\le \\int_{S_3}(\\partial_{x_3}\\varphi)^2\\,dx+\\cfrac{5\\pi^2}{2}\\int_{S_3^+\\cup S_3^-}\\varphi^2\\,dx.\n\\end{equation}\nThen inserting (\\ref{PoincareTranche}) in (\\ref{PoincareTranche01}), (\\ref{PoincareTranche02}) and summing up the resulting estimates, we obtain\n\\begin{equation}\\label{PoincareTranche1}\n\\begin{array}{rcl}\n\\pi^2\\displaystyle\\int_{Q^+_1}\\varphi^2\\,dx &\\le&\\displaystyle \\int_{S_2}(\\partial_{x_2}\\varphi)^2\\,dx+\\cfrac{1}{2}\\int_{S_2^+\\cup S_2^-}(\\partial_{x_3}\\varphi)^2\\,dx+\\displaystyle \\int_{S_3}(\\partial_{x_3}\\varphi)^2\\,dx+\\cfrac{1}{2}\\int_{S_3^+\\cup S_3^-}(\\partial_{x_2}\\varphi)^2\\,dx\\\\[10pt]\n & & +\\displaystyle\\cfrac{1}{2}\\int_{S_2^+\\cup S_2^-\\cup S_3^+\\cup S_3^-}(\\partial_{x_1}\\varphi)^2\\,dx.\n\\end{array}\n\\end{equation}\nOn the other hand, (\\ref{PoincareTranche}) also yields\n\\begin{equation}\\label{PoincareTranche2}\n2\\pi^2\\int_{S^+_2\\cup S^-_2\\cup S^+_3\\cup S^-_3}\\varphi^2\\,dx \\le \\cfrac{1}{2} \\int_{S^+_2\\cup S^-_2\\cup S^+_3\\cup S^-_3}(\\partial_{x_1}\\varphi)^2\\,dx+\\cfrac{1}{2}\\int_{S_2^+\\cup S_2^-}(\\partial_{x_3}\\varphi)^2\\,dx+\\cfrac{1}{2}\\int_{S_3^+\\cup S_3^-}(\\partial_{x_2}\\varphi)^2\\,dx.\n\\end{equation}\nFinally, summing up (\\ref{PoincareTranche0}), (\\ref{PoincareTranche0bis}), (\\ref{PoincareTranche1}) and (\\ref{PoincareTranche2}), we get \n\\[\n2\\pi^2 \\int_{\\Omega^+_1}\\varphi^2\\,dx \\le \\int_{\\Omega^+_1}|\\nabla\\varphi|^2\\,dx.\n\\]\nThis estimate together with the identity (\\ref{IdentityEigen}) imply $\\varphi\\equiv0$.\n\\end{proof}\n\\noindent We can now establish the main result of this section.\\\\[10pt]\n\\textit{Proof of Theorem \\ref{ThmDiscreteSpectrum}.} From Proposition \\ref{PropoAtLeastOne}, we know that there is at least one eigenvalue $\\mu_1$ of $A^{\\Omega}$ below $2\\pi^2$. Assume that $A^{\\Omega}$ has a second eigenvalue $\\mu_2$ such that $\\mu_2<2\\pi^2$. Set $\\Omega^+:=\\{x\\in\\Omega\\,|\\,x_1>0,\\,x_2>0,\\,x_3>0\\}$ (see Figure \\ref{ExplodedDomain} right). Then according to the result of symmetry of Lemma \\ref{LemmaSym}, the problem \n\\[\n\\begin{array}{|rcll}\n-\\Delta v &=&\\mu v  &\\mbox{in }\\Omega^+\\\\[3pt]\nv&=&0 &\\mbox{on }\\Sigma_0:=\\partial\\Omega\\cap \\partial \\Omega^+\\\\[3pt]\n\\partial_nv&=&0 &\\mbox{on }\\partial \\Omega^+\\setminus\\Sigma_0\n\\end{array}\n\\]\nadmits the two eigenvalues $\\mu_1$, $\\mu_2$. Besides, from the max-min principle (\\cite[Thm. 10.2.2]{BiSo87}), we have\n\\[\n\\mu_2=\\max_{E\\subset\\mathscr{E}_1}\\inf_{w\\in E\\setminus\\{0\\}}\\cfrac{\\displaystyle\\int_{\\Omega^+}|\\nabla w|^2\\,dx}{\\displaystyle\\int_{\\Omega^+} w^2\\,dx}\\,\n\\]\nwhere $\\mathscr{E}_1$ denotes the set of subspaces of $\\mrm{H}^1_{0}(\\Omega^+;\\Sigma_0):=\\{\\varphi\\in\\mrm{H}^1(\\Omega^+)\\,|\\,\\varphi=0\\mbox{ on }\\Sigma_0\\}$ of codimension one. In particular, we have \n\\begin{equation}\\label{Contra1}\n\\mu_2 \\ge \\inf_{w\\in E\\setminus\\{0\\}}\\cfrac{\\displaystyle\\int_{\\Omega^+}|\\nabla w|^2\\,dx}{\\displaystyle\\int_{\\Omega^+} w^2\\,dx}\n\\end{equation}\nwith $E=\\{\\varphi\\in\\mrm{H}^1_{0}(\\Omega^+;\\Sigma_0)\\,|\\,\\int_{Q^+}\\varphi\\,dx=0\\}$, $Q^+:=(0;1\/2)^3$. However from the Poincar\\'e inequality, we can write for $w\\in E\\setminus\\{0\\}$, \n\\begin{equation}\\label{IdentityEigenD}\n2\\pi^2\\int_{\\Omega^+\\setminus\\overline{Q^+}}w^2\\,dx \\le \\int_{\\Omega^+\\setminus\\overline{Q^+}} |\\nabla w|^2\\,dx\n\\end{equation}\nand there holds, according to the max-min principle, \n\\begin{equation}\\label{Contra2}\n\\inf_{w\\in E\\setminus\\{0\\}}\\cfrac{\\displaystyle\\int_{Q^+}|\\nabla w|^2\\,dx}{\\displaystyle\\int_{Q^+} w^2\\,dx}=4\\pi^2\n\\end{equation}\nbecause the first positive eigenvalue of the Neumann Laplacian in $Q^+$ is equal to $4\\pi^2$. Using (\\ref{IdentityEigenD}) and (\\ref{Contra2}) in (\\ref{Contra1}) leads to $\\mu_2\\ge2\\pi$ which contradicts the initial assumption. Therefore $A^{\\Omega}$ cannot have two eigenvalues below the continuous spectrum.\\hfill $\\square$\n\n\n\\section{Absence of threshold resonance}\\label{sectionAbsenceofBoundedSol}\n\nIn this section, we establish Theorem \\ref{ThmNoBoundedSol}. To proceed, we apply the tools of \\cite{Pank17,BaNa21} that we recall now. For $R\\ge1\/2$ and $j=1,2,3$, define the truncated cylinder\n\\[\nL^R_j  :=  \\{x\\in L_j\\,|\\,|x_j|<R\\}\n\\]\nand set $\\Omega^R:=L^R_1\\cup L^R_2 \\cup L^R_3$. In $\\Omega^R$, consider the spectral problem with mixed boundary conditions\n\\begin{equation}\\label{SpectralRMixed}\n\\begin{array}{|rcll}\n-\\Delta v & = & \\mu v &\\mbox { in }\\Omega^R\\\\[2pt]\n v & = & 0 &\\mbox{ on }\\partial\\Omega^R\\cap\\partial\\Omega\\\\[2pt]\n\\partial_n v & = & 0 &\\mbox{ on }\\partial\\Omega^R\\setminus\\partial\\Omega\n\\end{array}\n\\end{equation}\nwhere $n$ stand for the unit normal vector to $\\partial\\Omega^R$ directed to the exterior of $\\Omega^R$. Set $\\mrm{H}^1_{0}(\\Omega^R;\\partial\\Omega^R\\cap\\partial\\Omega):=\\{\\varphi\\in\\mrm{H}^1(\\Omega^R)\\,|\\,\\varphi=0\\mbox{ on }\\partial\\Omega^R\\cap\\partial\\Omega\\}$ and finally denote by $B^R$ the unbounded operator associated with (\\ref{SpectralRMixed}) of domain $\\mathcal{D}(B^R)\\subset\\mrm{H}^1_{0}(\\Omega^R;\\partial\\Omega^R\\cap\\partial\\Omega)$.\\\\\n\\newline\nThe Theorem 3 of \\cite{Pank17} (see also the necessary and sufficient condition of \\cite[Thm. 1]{BaNa21}) guarantees that there is no threshold resonant for Problem (\\ref{PbSpectralZoom}) if there exists for some $R\\ge0$ such that the second eigenvalue $\\mu^R_2$ of $B_R$ satisfies $\\mu^R_2>2\\pi^2$.\\\\\n\\newline\nTherefore from now our goal is to show that the second eigenvalue of $B_R$ is larger than $2\\pi^2$ (Proposition \\ref{PropoBoundedSecond} below). We start with a result of symmetry similar to Lemma \\ref{LemmaSym}. \n\n\n\\begin{lemma}\\label{LemmaSymR}\nFor $R$ large enough, if $v\\in\\mrm{H}^1_{0}(\\Omega^R;\\partial\\Omega^R\\cap\\partial\\Omega)$ is an eigenfunction of the operator $B^R$ associated with an eigenvalue $\\mu\\le2\\pi^2$, then $v$ is symmetric with respect to the planes $(Ox_2x_3)$, $(Ox_3x_1)$, $(Ox_1x_2)$.\n\\end{lemma}\n\\begin{proof}\nThe demonstration follows the lines of the one of Lemma \\ref{LemmaSym}. However we write the details for the sake of clarity. We focus our attention on the symmetry with respect to the plane $(Ox_2x_3)$, the two other ones being similar. Introduce the function $\\varphi$ such that\n\\[\n\\varphi(x_1,x_2,x_3)=v(x_1,x_2,x_3)-v(-x_1,x_2,x_3).\n\\]\nWe wish to show that $\\varphi\\equiv0$. Set $\\Omega^{R+}_1:=\\{x\\in\\Omega^R\\,|\\,x_1>0\\}$. We have \n\\begin{equation}\\label{IdentityEigenR}\n\\int_{\\Omega^{R+}_1}|\\nabla\\varphi|^2\\,dx=\\mu\\int_{\\Omega^{R+}_1}\\varphi^2\\,dx.\n\\end{equation}\nDefine the domains \n\\[\n\\begin{array}{rcl}\nL^{R+}_1 & := & \\{x\\in L^R_1\\,|\\,x_1>1\/2\\}\\\\[2pt]\nS^R_2 & := & \\{x\\in L^R_2\\,|\\,x_1>0\\}=\\Omega^{R+}_1\\cap L^R_2\\\\[2pt]\nS^R_3 & := & \\{x\\in L^R_3\\,|\\,x_1>0\\}=\\Omega^{R+}_1\\cap L^R_3\\\\[2pt]\nS^{R\\pm}_2 & := & \\{x\\in S^R_2\\,|\\,\\pm x_2>1\/2\\}\\\\[2pt]\nS^{R\\pm}_3 & := & \\{x\\in S^R_3\\,|\\,\\pm x_3>1\/2\\}.\n\\end{array}\n\\]\nIn $L^{R+}_1$ the Poincar\\'e inequality gives\n\\begin{equation}\\label{PoincareTranche0R}\n2\\pi^2 \\int_{L^{R+}_1}\\varphi^2\\,dx \\le \\int_{L^{R+}_1}|\\nabla\\varphi|^2\\,dx.\n\\end{equation}\nOn the other hand, in $Q^+_1=(0;1\/2)\\times(-1\/2;1\/2)^2$ there holds\n\\begin{equation}\\label{PoincareTranche0bisR}\n\\pi^2 \\int_{Q^+_1}\\varphi^2\\,dx \\le \\int_{Q^+_1}(\\partial_{x_1}\\varphi)^2\\,dx.\n\\end{equation}\nThe Lemma \\ref{LemmaPoincareFriedrichsR} in Appendix (see also Lemma 5.1 of \\cite{BaMaNa17}) guarantees that for $R>1\/2$ and $a>0$, we have\n\\begin{equation}\\label{PoincareModifiedR}\n \\kappa(a,R)\\int_{0}^{1\/2}\\phi^2\\,dt \\le \\int_{0}^{R}(\\partial_t\\phi)^2\\,dt+a^2\\int_{1\/2}^{R}\\phi^2\\,dt,\\qquad \\forall\\phi\\in\\mrm{H}^1(0;R),\n\\end{equation}\nwhere $\\kappa(a,R)>0$ converges to the constant $\\kappa(a)$ appearing in (\\ref{TransEqn}) as $R\\to+\\infty$. For $a=\\pi\\sqrt{5\/2}$, as already said, one finds $\\kappa(a)>\\pi^2\/2$. Introduce $\\delta>0$ such that $\\kappa(\\pi\\sqrt{5\/2})>\\pi^2\/2+\\delta$. We know that there is $R_0$ large enough such that we have $\\kappa(\\pi\\sqrt{5\/2},R)>\\pi^2\/2+\\delta\/2$ for all $R\\ge R_0$. We infer that we have\n\\begin{equation}\\label{PoincareTranche01R}\n\\cfrac{\\pi^2+\\delta}{2}\\int_{Q^+_1}\\varphi^2\\,dx \\le \\int_{S^R_2}(\\partial_{x_2}\\varphi)^2\\,dx+\\cfrac{5\\pi^2}{2}\\int_{S_2^{R+}\\cup S_2^{R-}}\\varphi^2\\,dx\n\\end{equation}\nand\n\\begin{equation}\\label{PoincareTranche02R}\n\\cfrac{\\pi^2+\\delta}{2}\\int_{Q^+_1}\\varphi^2\\,dx \\le \\int_{S^R_3}(\\partial_{x_3}\\varphi)^2\\,dx+\\cfrac{5\\pi^2}{2}\\int_{S_3^{R+}\\cup S_3^{R-}}\\varphi^2\\,dx.\n\\end{equation}\nBut from the Poincar\\'e inequality in the transverse section of $S_2^{R\\pm}$, $S_3^{R\\pm}$, we know that \n\\begin{equation}\\label{PoincareTrancheR}\n5\\pi^2\\int_{S_2^{R\\pm}}\\varphi^2\\,dx \\le \\int_{S_2^{R\\pm}}(\\partial_{x_1}\\varphi)^2+(\\partial_{x_3}\\varphi)^2\\,dx,\\qquad 5\\pi^2\\int_{S_3^{R\\pm}}\\varphi^2\\,dx \\le \\int_{S_3^{R\\pm}}(\\partial_{x_1}\\varphi)^2+(\\partial_{x_2}\\varphi)^2\\,dx.\n\\end{equation}\nInserting (\\ref{PoincareTrancheR}) in (\\ref{PoincareTranche01R}), (\\ref{PoincareTranche02R}) and summing up the resulting estimates, we obtain\n\\begin{equation}\\label{PoincareTranche1R}\n\\begin{array}{rcl}\n(\\pi^2+\\delta)\\displaystyle\\int_{Q^+_1}\\varphi^2\\,dx &\\le&\\displaystyle \\int_{S^R_2}(\\partial_{x_2}\\varphi)^2\\,dx+\\displaystyle \\int_{S_3^R}(\\partial_{x_3}\\varphi)^2\\,dx\\\\[10pt]\n & &+\\displaystyle\\cfrac{1}{2}\\int_{S_2^{R+}\\cup S_2^{R-}}(\\partial_{x_3}\\varphi)^2\\,dx+\\cfrac{1}{2}\\int_{S_3^{R+}\\cup S_3^{R-}}(\\partial_{x_2}\\varphi)^2\\,dx\\\\[10pt]\n & & +\\displaystyle\\cfrac{1}{2}\\int_{S_2^{R+}\\cup S_2^{R-}\\cup S_3^{R+}\\cup S_3^{R-}}(\\partial_{x_1}\\varphi)^2\\,dx.\n\\end{array}\n\\end{equation}\nOn the other hand, (\\ref{PoincareTrancheR}) also yields\n\\begin{equation}\\label{PoincareTranche2R}\n\\begin{array}{l}\n\\dsp2\\pi^2\\int_{S^{R+}_2\\cup S^{R-}_2\\cup S^{R+}_3\\cup S^{R-}_3}\\varphi^2\\,dx \\\\[10pt]\n\\displaystyle\\le \\cfrac{1}{2} \\int_{S^{R+}_2\\cup S^{R-}_2\\cup S^{R+}_3\\cup S^{R-}_3}(\\partial_{x_1}\\varphi)^2\\,dx+\\cfrac{1}{2}\\int_{S_2^{R+}\\cup S_2^{R-}}(\\partial_{x_3}\\varphi)^2\\,dx+\\cfrac{1}{2}\\int_{S_3^{R+}\\cup S_3^{R-}}(\\partial_{x_2}\\varphi)^2\\,dx.\n\\end{array}\n\\end{equation}\nFinally, summing up (\\ref{PoincareTranche0R}), (\\ref{PoincareTranche0bisR}), (\\ref{PoincareTranche1R}) and (\\ref{PoincareTranche2R}), we get \n\\[\n\\delta\\displaystyle\\int_{Q^+_1}\\varphi^2\\,dx+2\\pi^2 \\int_{\\Omega^{R+}_1}\\varphi^2\\,dx \\le \\int_{\\Omega^{R+}_1}|\\nabla\\varphi|^2\\,dx.\n\\]\nThis estimate together with the identity (\\ref{IdentityEigenR}) imply $\\int_{Q^+_1}\\varphi^2\\,dx=0$ and so $\\varphi\\equiv0$ in $Q^+_1$. From the unique continuation principle, this gives $\\varphi\\equiv0$ in $\\Omega^{R+}_1$.\n\\end{proof}\n\n\\begin{proposition}\\label{PropoBoundedSecond}\nFor $R$ large enough, the second eigenvalue $\\mu_2^R$ of $B^R$ satisfies $\\mu_2^R>2\\pi^2$. \n\\end{proposition}\n\\begin{proof}\nOne applies Lemma \\ref{LemmaSymR} to reduce the analysis to $\\Omega^{R+}:=\\{x\\in\\Omega^R\\,|\\,x_1>0,\\,x_2>0,\\,x_3>0\\}$. Here in particular we use the assumption that $R$ is large enough. The rest of the proof is completely similar to the one of Theorem \\ref{ThmDiscreteSpectrum}. \n\\end{proof}\n\n\\section{Model problems for the spectral bands}\\label{SectionModels}\n\nIn this section, we establish Theorem \\ref{MainThmPerio} and obtain models for the spectral bands $\\Upsilon^\\varepsilon_p$ appearing in (\\ref{SpectralBands}). We recall that the spectrum of $A^\\varepsilon$ is such that $\\sigma(A^\\varepsilon)=\\bigcup_{p\\in\\mathbb{N}^\\ast}\\Upsilon^\\varepsilon_p$.\n\n\\subsection{Study of $\\Upsilon^\\varepsilon_1$}\nBy definition, we have \n\\begin{equation}\\label{Def_FirstBande}\n\\Upsilon^\\varepsilon_1=\\{\\Lambda_1^\\varepsilon(\\eta),\\,\\eta\\in[0;2\\pi)^3\\}\n\\end{equation}\nwhere $\\Lambda_1^\\varepsilon(\\eta)$ is the first eigenvalue of (\\ref{PbSpectralCell}). Therefore our goal is to obtain an asymptotic expansion of $\\Lambda_1^\\varepsilon(\\eta)$ with respect to $\\varepsilon$ as $\\varepsilon\\to0^+$.\\\\\n\\newline\nPick $\\eta\\in[0;2\\pi)^3$. Let $u^\\varepsilon(\\cdot,\\eta)$ be an eigenfunction associated with $\\Lambda_1^\\varepsilon(\\eta)$. As a first approximation when $\\varepsilon\\to0^+$, it is natural to consider the expansion\n\\begin{equation}\\label{ExpansionTR0}\n\\Lambda_1^\\varepsilon(\\eta)=\\varepsilon^{-2}\\mu_1+\\dots,\\qquad u^\\varepsilon(x,\\eta)=v_1(x_1\/\\varepsilon)+\\dots\n\\end{equation}\nwhere $\\mu_1\\in(0;2\\pi^2)$ stands for the eigenvalue of the discrete spectrum of the operator $A^{\\Omega}$ introduced in Theorem \\ref{ThmDiscreteSpectrum} and $v_1\\in\\mrm{H}^1_0(\\Omega)$ is a corresponding eigenfunction that we choose such that $\\|v_1\\|_{\\mrm{L}^2(\\Omega)}=1$. Indeed, inserting $(\\varepsilon^{-2}\\mu_1,v_1(\\cdot\/\\varepsilon))$ in Problem (\\ref{PbSpectralCell}) only leaves a small discrepancy on the square faces of $\\partial\\omega^\\varepsilon$ because $v_1$ is exponentially decaying at infinity. Let us write more precisely the decomposition of $v_1$ at infinity because this will be useful in the sequel. To proceed and keep short notation, we shall work with the coordinates $(z_j,y_j)$, $j=1,\\dots,6$, such that\n\\begin{equation}\\label{Def_coor_local}\n\\begin{array}{|l}\nz_1=x_1,\\quad y_1=(x_2,x_3);\\\\[3pt]\nz_2=x_2,\\quad y_2=(x_3,x_1);\\\\[3pt]\nz_3=x_3,\\quad y_3=(x_1,x_2);\n\\end{array}\\qquad\\qquad \\begin{array}{|l}\nz_4=-x_1,\\quad y_4=(x_2,-x_3);\\\\[3pt]\nz_5=-x_2,\\quad y_5=(x_3,-x_1);\\\\[3pt]\nz_6=-x_3,\\quad y_6=(x_1,-x_2).\n\\end{array}\n\\end{equation}\nWe also define for $j=1,\\dots,6$, the branch\n\\[\n\\mathcal{L}_j:=\\{x\\in\\Omega\\,|\\,z_j>1\/2\\}.\n\\]\nThen Fourier decomposition together with the result of symmetry of Lemma \\ref{LemmaSym} and the fact that $\\mu_1$ is a simple eigenvalue\\footnote{This is needed to show that $K$ is the same in the branches $\\mathcal{L}_j$, $j=1,\\dots,6$.} guarantee that for $j=1,\\dots,6$, we have \n\\[\nv_1(x)=K\\,e^{-\\beta_1 z_j}\\,U_{\\dagger}(y_j)+O(e^{-\\beta_2 z_j})\\quad\\mbox{ in }\\mathcal{L}_j\\quad\\mbox{ as } z_j\\to+\\infty.\n\\]\nHere $K\\in\\mathbb{R}^*$ is independent of $j$, $\\beta_1:=\\sqrt{2\\pi^2-\\mu_1}$, $\\beta_2:=\\sqrt{5\\pi^2-\\mu_1}$ and \n\\begin{equation}\\label{defUdagger}\nU_{\\dagger}(s_1,s_2)=2\\cos(\\pi s_1)\\cos(\\pi s_2).\n\\end{equation}\nThe first model (\\ref{ExpansionTR0}) is interesting but does not comprise the dependence with respect $\\eta$. To improve it, consider the more refined ans\\\"atze\n\\begin{equation}\\label{ExpansionTR1}\n\\Lambda_1^\\varepsilon(\\eta)=\\varepsilon^{-2}\\mu_1+\\varepsilon^{-2}e^{-\\beta_1\/\\varepsilon}M(\\eta)+\\dots,\\qquad u^\\varepsilon(x,\\eta)=v_1(x_1\/\\varepsilon)+e^{-\\beta_1\/\\varepsilon}V(x_1\/\\varepsilon,\\eta)+\\dots\n\\end{equation}\nwhere the quantities $M(\\eta)$, $V(\\cdot,\\eta)$ are to determine. Inserting (\\ref{ExpansionTR1}) into (\\ref{PbSpectralCell}), first we obtain that $V(\\cdot,\\eta)$ must satisfy\n\\begin{equation}\\label{Pb_corrector_TR}\n\\begin{array}{|rccl}\n-\\Delta V(\\cdot,\\eta)-\\mu_1V(\\cdot,\\eta)&=&M(\\eta)v_1 & \\mbox{ in }\\Omega\\\\[3pt]\n V(\\cdot,\\eta) &=&0 &\\mbox{ on }\\partial\\Omega.\n\\end{array}\n\\end{equation}\nIn order to have a non zero solution to (\\ref{Pb_corrector_TR}), we must look for a $V(\\cdot,\\eta)$ which is growing at infinity. Due to (\\ref{Pb_corrector_TR}), the simplest growth that we can allow is\n\\[\nV(x,\\eta)=B_j\\,e^{\\beta_1 z_j}\\,U_{\\dagger}(y_j)+O(e^{-\\beta_1 z_j})\\quad\\mbox{ in }\\mathcal{L}_j\\quad\\mbox{ as } z_j\\to+\\infty\n\\]\nwhere the $B_j$ are some constants. Then the quasi-periodic conditions satisfied by $u^\\varepsilon(\\cdot,\\eta)$ at the faces located at $x_1=\\pm1\/2$, $x_2=\\pm1\/2$, $x_3=\\pm1\/2$ (see (\\ref{PbSpectralCell})) lead us to choose the $B_j$ such that\n\\[\n\\begin{array}{|l}\nK+B_4=e^{i\\eta_1}(K+B_1)\\\\[2pt]\nK-B_4=e^{i\\eta_1}(-K+B_1)\n\\end{array}\\qquad \\begin{array}{|l}\nK+B_5=e^{i\\eta_2}(K+B_2)\\\\[2pt]\nK-B_5=e^{i\\eta_2}(-K+B_2)\n\\end{array}\\qquad\\begin{array}{|l}\nK+B_6=e^{i\\eta_3}(K+B_3)\\\\[2pt]\nK-B_6=e^{i\\eta_3}(-K+B_3).\n\\end{array}\n\\]\nSolving these systems, we obtain $B_j=K\\,e^{-i\\eta_j}$, $B_{j+3}=K\\,e^{+i\\eta_j}$ for $j=1,2,3$. Now since $\\mu_1$ is a simple eigenvalue of $A^\\Omega$, multiplying (\\ref{Pb_corrector_TR}) by $v_1$, integrating by part in $\\Omega^R$ and taking the limit $R\\to+\\infty$, we find that there is a solution if and only if the following compatibility condition \n\\[\nM(\\eta)\\|v_1\\|_{\\mrm{L}^2(\\Omega)}=-2\\beta_1K^2\\sum_{j=1}^3e^{i\\eta_j}+e^{-i\\eta_j}\\qquad\\Leftrightarrow\\qquad M(\\eta)=-4\\beta_1K^2\\sum_{j=1}^3\\cos(\\eta_j)\n\\]\nis satisfied. This defines the value of $M(\\eta)$ in the expansion (\\ref{ExpansionTR1}). From (\\ref{Def_FirstBande}), this gives the inclusion $\\Upsilon^\\varepsilon_1\\subset \\varepsilon^{-2}\\mu_1+\\varepsilon^{-2}e^{-\\sqrt{2\\pi^2-\\mu_1}\/\\varepsilon}(-c_1;c_1)$ for $\\varepsilon$ small enough of Theorem \\ref{MainThmPerio}. For $c_1$, we can take any constant larger than $12\\beta_1K^2$. Note that we decided to focus on a rather formal presentation above for the sake of conciseness. We emphasize that all these results can be completely justified by proving rigorous error. This has been realized in detail in \\cite{Naza17,Naza18} for similar problems and can be repeated with obvious modifications.\n\n\n\n\n\n\n\n\n\\subsection{Study $\\Upsilon^\\varepsilon_k$, $k\\ge 2$}\n\nWe turn our attention to the asymptotic of the spectral bands of higher frequency\n\\begin{equation}\\label{Def_FirstBande_2}\n\\Upsilon^\\varepsilon_k=\\{\\Lambda_k^\\varepsilon(\\eta),\\,\\eta\\in[0;2\\pi)^3\\},\\qquad k\\ge2,\n\\end{equation}\nas $\\varepsilon\\to0^+$. Pick $\\eta\\in[0;2\\pi)^3$ and introduce $u^\\varepsilon(\\cdot,\\eta)$ an eigenfunction associated with $\\Lambda_k^\\varepsilon(\\eta)$. In the sequel, to simplify, we remove the subscript ${}_k$ and do not indicate the dependence on $\\eta$. As a first approximation when $\\varepsilon\\to0^+$, we consider the expansion\n\\begin{equation}\\label{ExpansionTR0_2}\n\\Lambda^\\varepsilon=\\varepsilon^{-2}2\\pi^2+\\nu+\\dots,\\qquad u^\\varepsilon(x)=v^\\varepsilon(x)+\\dots\n\\end{equation}\nwith $v^\\varepsilon$ of the form\n\\[\nv^\\varepsilon(x)=\\begin{array}{|rl}\n\\gamma^\\pm_{1}(x_1)\\,U_{\\dagger}(x_2\/\\varepsilon,x_3\/\\varepsilon)&\\mbox{ in }\\omega^{\\varepsilon\\pm}_1:=\\{x\\in\\omega^{\\varepsilon}_1\\,|\\,\\pm x_1>\\varepsilon\/2\\},\\\\[3pt]\n\\gamma^\\pm_{2}(x_2)\\,U_{\\dagger}(x_3\/\\varepsilon,x_1\/\\varepsilon)&\\mbox{ in }\\omega^{\\varepsilon\\pm}_2:=\\{x\\in\\omega^{\\varepsilon}_2\\,|\\,\\pm x_2>\\varepsilon\/2\\},\\\\[3pt]\n\\gamma^\\pm_{3}(x_3)\\,U_{\\dagger}(x_1\/\\varepsilon,x_2\/\\varepsilon)&\\mbox{ in }\\omega^{\\varepsilon\\pm}_3:=\\{x\\in\\omega^{\\varepsilon}_3\\,|\\,\\pm x_3>\\varepsilon\/2\\},\\\\[3pt]\n\\end{array}\n\\]\nwhere the functions $\\gamma^\\pm_{j}$, $j=1,2,3$, are to determine ($U_{\\dagger}$ is defined in (\\ref{defUdagger})). Inserting (\\ref{ExpansionTR0_2}) into Problem (\\ref{PbSpectralCell}), we obtain for $j=1,2,3$,\n\\begin{equation}\\label{farfield}\n\\begin{array}{|rcl}\n\\partial^2_s\\gamma^+_{j}+\\nu\\gamma^+_{j}&=&0\\quad\\mbox{ in }(0;1\/2)\\\\[3pt]\n\\partial^2_s\\gamma^-_{j}+\\nu\\gamma^-_{j}&=&0\\quad\\mbox{ in }(-1\/2;0)\\\\[3pt]\n\\gamma^-_{j}(-1\/2)&=&e^{i\\eta_j}\\gamma^+_{j}(+1\/2)\\\\[3pt]\n\\partial_s\\gamma^-_{j}(-1\/2)&=&e^{i\\eta_j}\\partial_s\\gamma^+_{j}(+1\/2).\n\\end{array}\n\\end{equation}\nTo uniquely define the $\\gamma^\\pm_{j}$, we need to complement (\\ref{farfield}) with some conditions at the origin. To proceed, we match the behaviour of the $\\gamma^\\pm_{j}$ with the one of some inner field expansion of $u^\\varepsilon$. More precisely, in a neighbourhood of the origin we look for an expansion of $u^\\varepsilon$ of the form \n\\begin{equation}\\label{NearField}\nu^\\varepsilon(x)=W(x_1\/\\varepsilon)+\\dots\n\\end{equation}\nwith $W$ to determine. Inserting (\\ref{NearField}) and (\\ref{ExpansionTR0_2}) in (\\ref{PbSpectralCell}), we find that $W$ must satisfy \n\\begin{equation}\\label{NearFieldPb}\n\\begin{array}{|rcll}\n\\Delta W+2\\pi^2W&=&0&\\mbox{ in }\\Omega\\\\\nW&=&0 &\\mbox{ on }\\partial\\Omega.\n\\end{array}\n\\end{equation}\nNow we come to the point where Theorem \\ref{ThmNoBoundedSol} appears in the analysis. Indeed it guarantees that the only solution of (\\ref{NearFieldPb}) which is bounded at infinity is the null function. Therefore we take $W\\equiv0$ and impose \n\\begin{equation}\\label{farfield_BC}\n\\gamma^\\pm_{j}(0)=0.\n\\end{equation}\nThen solving the spectral problem (\\ref{farfield}), (\\ref{farfield_BC}), we obtain \n\\begin{equation}\\label{DefModel1D}\n\\nu=p^2\\pi^2 \\mbox{ for }p\\in\\mathbb{N}^\\ast,\\qquad  \\begin{array}{|l}\n\\gamma^+_{j}(s)=\\sin(p\\pi s)\\\\[2pt]\n\\gamma^-_{j}(s)=-e^{i\\eta_j}\\sin(p\\pi s).\n\\end{array}\n\\end{equation}\nNote that $\\nu$ is a triple eigenvalue (geometric multiplicity equal to three) because the problems (\\ref{farfield}), (\\ref{farfield_BC}) for $j=1,2,3$ are uncoupled. Additionally $\\nu$ is independent of $\\eta\\in[0;2\\pi)^3$. This latter fact is not completely satisfactory and in the sequel we wish to improve the model obtained above. Let us refine the expansion proposed in (\\ref{ExpansionTR0_2}) and work with\n\\begin{equation}\\label{ExpansionTR0_3}\n\\Lambda^\\varepsilon=\\cfrac{2\\pi^2}{\\varepsilon^2}+p^2\\pi^2+\\varepsilon\\tilde{\\nu}+\\dots,\\quad u^\\varepsilon(x)=\\begin{array}{|rl}\n(a_1\\gamma^\\pm_{1}(x_1)+\\varepsilon\\tilde{\\gamma}^{\\pm}_{1}(x_1))\\,U_{\\dagger}(x_2\/\\varepsilon,x_3\/\\varepsilon)&\\mbox{ in }\\omega^{\\varepsilon\\pm}_1\\\\[3pt]\n(a_2\\gamma^\\pm_{2}(x_2)+\\varepsilon\\tilde{\\gamma}^{\\pm}_{2}(x_2))\\,U_{\\dagger}(x_3\/\\varepsilon,x_1\/\\varepsilon)&\\mbox{ in }\\omega^{\\varepsilon\\pm}_2\\\\[3pt]\n(a_3\\gamma^\\pm_{3}(x_3)+\\varepsilon\\tilde{\\gamma}^{\\pm}_{3}(x_3))\\,U_{\\dagger}(x_1\/\\varepsilon,x_2\/\\varepsilon)&\\mbox{ in }\\omega^{\\varepsilon\\pm}_3\\\\[3pt]\n\\end{array}+\\dots .\n\\end{equation}\nHere $a:=(a_1,a_2,a_3)\\in\\mathbb{R}^3$, $\\tilde{\\nu}$ as well as the $\\tilde{\\gamma}^{\\pm}_{j}$, $j=1,2,3$, are to determine. Since we are working with eigenfunctions, we can impose the normalization condition $a_1^2+a_2^2+a_3^2=1$. Inserting (\\ref{ExpansionTR0_3}) into Problem (\\ref{PbSpectralCell}) and extracting the terms in $\\varepsilon$, we get for $j=1,2,3$,\n\\begin{equation}\\label{farfield_3}\n\\begin{array}{|rcl}\n\\partial^2_s\\tilde{\\gamma}^+_{j}+p^2\\pi^2\\tilde{\\gamma}^+_{j}&=&-\\tilde{\\nu}a_j\\gamma^+_j\\quad\\mbox{ in }(0;1\/2)\\\\[3pt]\n\\partial^2_s\\tilde{\\gamma}^-_{j}+p^2\\pi^2\\tilde{\\gamma}^-_{j}&=&-\\tilde{\\nu}a_j\\gamma^-_j\\quad\\mbox{ in }(-1\/2;0)\\\\[3pt]\n\\tilde{\\gamma}^-_{j}(-1\/2)&=&e^{i\\eta_j}\\tilde{\\gamma}^+_{j}(+1\/2)\\\\[3pt]\n\\partial_s\\tilde{\\gamma}^-_{j}(-1\/2)&=&e^{i\\eta_j}\\partial_s\\tilde{\\gamma}^+_{j}(+1\/2).\n\\end{array}\n\\end{equation}\nTo define properly the $\\tilde{\\gamma}^\\pm_{j}$, we need to add to (\\ref{farfield_3}) conditions at the origin. To identify them, again we match with the behaviour of some inner field representation of $u^\\varepsilon$. In a neighbourhood of the origin we look for an expansion of $u^\\varepsilon$ of the form \n\\begin{equation}\\label{NearField_3}\nu^\\varepsilon(x)=\\varepsilon \\tilde{W}(x_1\/\\varepsilon)+\\dots.\n\\end{equation}\nInserting (\\ref{NearField_3}) and (\\ref{ExpansionTR0_3}) in (\\ref{PbSpectralCell}), we find that $\\tilde{W}$ must satisfy \n\\begin{equation}\\label{NearFieldPb_3}\n\\begin{array}{|rcll}\n\\Delta \\tilde{W}+2\\pi^2\\tilde{W}&=&0&\\mbox{ in }\\Omega\\\\\n\\tilde{W}&=&0 &\\mbox{ on }\\partial\\Omega.\n\\end{array}\n\\end{equation}\nThis problem admits solutions $\\tilde{W}_{j}$, $j=1,\\dots,6$, with the expansions\n\\begin{equation}\\label{DefPolMat1}\n\\tilde{W}_j(x)=\n\\begin{array}{|rl}\n(z_j+M_{jj})\\,U_{\\dagger}(y_j)+\\dots&\\mbox{ in }\\mathcal{L}_j\\mbox{ as }z_j\\to+\\infty \\\\[2pt]\nM_{jk}\\,U_{\\dagger}(y_k)+\\dots&\\mbox{ in }\\mathcal{L}_k\\mbox{ as }z_k\\to+\\infty,\\quad k\\ne j.\n\\end{array}\n\\end{equation}\nHere we use the coordinates $(z_j,y_j)$ introduced in (\\ref{Def_coor_local}). Note that at infinity $\\tilde{W}_j$ is growing only in the branch $\\mathcal{L}_j$. Let us explain how to show the existence of these functions (see \\cite[Chap. 4, Prop. 4.13]{NaPl94} for more details). Introduce some $\\psi\\in\\mathscr{C}^{\\infty}(\\mathbb{R})$ such that $\\psi(s)=1$ for $s>1$ and $\\psi(s)=0$ for $s\\le 1\/2$. Then for $j=1,\\dots,6$, define $f_j$ such that $f_j(x)=(\\Delta+2\\pi^2)(\\psi(z_j) \\,z_jU_{\\dagger}(y_j))$. Observe that $f_j$ is compactly supported. Then the theory of \\cite[Chap. 5]{NaPl94} together with Theorem \\ref{ThmNoBoundedSol} above ensure that there is a solution to the problem\n\\[\n\\begin{array}{|rcll}\n\\Delta \\hat{W}_j+2\\pi^2\\hat{W}_j&=&-f_j&\\mbox{ in }\\Omega\\\\\n\\hat{W}_j&=&0 &\\mbox{ on }\\partial\\Omega\n\\end{array}\n\\]\nwhich is bounded at infinity. Finally we take $\\tilde{W}_j$ such that $\\tilde{W}_j(x)=\\psi(z_j) \\,z_jU_{\\dagger}(y_j)+\\hat{W}_j(x)$. The coefficients $M_{jk}$ form the so-called polarization matrix\n\\begin{equation}\\label{def_Pola_matrix}\n\\mathbb{M}:=\\left(M_{jk}\\right)_{1\\le j,k\\le 6}\n\\end{equation}\nwhich is real and symmetric even in non symmetric geometries (see \\cite[Chap. 5, Prop. 4.13]{NaPl94}).\\\\\n\\newline\nFor the functions $\\gamma^{\\pm}_{j}$ in (\\ref{DefModel1D}), we have the Taylor expansion, as $s\\to0$,\n\\begin{equation}\\label{ExpanFarField}\n\\gamma^+_{j}(s)=0+p\\pi s+\\dots=\\varepsilon p\\pi\\,\\cfrac{s}{\\varepsilon}+\\dots,\\qquad \\gamma^-_{j}(s)=0-e^{i\\eta_j}p\\pi s+\\dots=-\\varepsilon e^{i\\eta_j} p\\pi \\,\\cfrac{s}{\\varepsilon}+\\dots.\n\\end{equation}\nComparing (\\ref{ExpanFarField}) with (\\ref{DefPolMat1}) leads us to choose $\\tilde{W}$ in the expansion $u^\\varepsilon(x)=\\varepsilon \\tilde{W}(x_1\/\\varepsilon)+\\dots$ (see (\\ref{NearField_3})) such that\n\\[\n\\tilde{W}=p\\pi \\sum_{j=1}^3a_j\\,(\\tilde{W}_j+e^{i\\eta_j}\\tilde{W}_{3+j}).\n\\]\nThis sets the constant behaviour of $\\tilde{W}$ at infinity and we now match the later with the behaviour of the $\\tilde{\\gamma}^\\pm_{j}$ at the origin to close system (\\ref{farfield_3}). This step leads us to impose \n\\begin{equation}\\label{ConditionAtZero}\n\\begin{array}{ll}\n\\displaystyle\\tilde{\\gamma}^+_1(0)=p\\pi \\sum_{j=1}^3 a_j\\,(M_{j1}+e^{i\\eta_j}M_{3+j1}); &\\quad\\displaystyle \\tilde{\\gamma}^-_1(0)=p\\pi \\sum_{j=1}^3 a_j\\,M_{j4}+e^{i\\eta_j}M_{3+j4});\\\\[14pt]\n\\displaystyle\\tilde{\\gamma}^+_2(0)=p\\pi \\sum_{j=1}^3 a_j\\,(M_{j2}+e^{i\\eta_j}M_{3+j2}); &\\quad\\displaystyle \\tilde{\\gamma}^-_2(0)=p\\pi \\sum_{j=1}^3 a_j\\,(M_{j5}+e^{i\\eta_j}M_{3+j5});\\\\[14pt]\n\\displaystyle\\tilde{\\gamma}^+_3(0)=p\\pi \\sum_{j=1}^3 a_j\\,(M_{j3}+e^{i\\eta_j}M_{3+j3}); &\\quad\\displaystyle \\tilde{\\gamma}^-_3(0)=p\\pi \\sum_{j=1}^3 a_j\\,(M_{j6}+e^{i\\eta_j}M_{3+j6}).\n\\end{array}\n\\end{equation}\nEquations (\\ref{farfield_3}), (\\ref{ConditionAtZero}) form a boundary value problem for a system of ordinary differential equations. For this problem, there is a kernel and co-cokernel. In order to have a solution, the following compatibility conditions, obtained by multiplying (\\ref{farfield_3}) by $\\overline{\\gamma^\\pm_j}$ and integrating by parts,\n\\[\n\\overline{\\partial_s\\gamma^+_j(0)}\\,\\tilde{\\gamma}^+_j(0)-\\overline{\\partial_s\\gamma^-_j(0)}\\,\\tilde{\\gamma}^-_j(0)=-\\tilde{\\nu}a_j\\left(\\int_{-1\/2}^{0}|\\gamma^-_j|^2\\,ds+\\int_{0}^{1\/2}|\\gamma^+_j|^2\\,ds\\right)\n\\]\nmust be satisfied for $j=1,2,3$ (note that we used that $\\gamma^{\\pm}_j(0)=0$ according to (\\ref{farfield_BC})). Since $\\overline{\\partial_s\\gamma^+_j(0)}=p\\pi$ and $\\overline{\\partial_s\\gamma^-_j(0)}=-p\\pi e^{-i\\eta_j}$, this gives\n\\[\n2(p\\pi)^2\\bigg(\\sum_{k=1}^3 a_k\\,(M_{kj}+e^{i\\eta_k}M_{3+kj})+e^{-i\\eta_j}\\sum_{k=1}^3 a_k\\,(M_{k3+j}+e^{i\\eta_k}M_{3+k3+j})\\bigg)=-\\tilde{\\nu}a_j.\n\\]\nIn a more compact form and by rewriting the dependence with respect to $\\eta$, we obtain\n\\begin{equation}\\label{CompactForm}\n\\mathbb{A}(\\eta) a^{\\top}=-\\tilde{\\nu}(\\eta)a^{\\top},\n\\end{equation}\nwith \n\\[\n\\mathbb{A}(\\eta):=\\Theta(\\eta)\\mathbb{M}\\Theta^\\ast(\\eta)\\in\\mathbb{C}^{3\\times 3},\\quad\n\\Theta(\\eta):=\\left(\\begin{array}{cccccc}\n1 & 0 & 0 & e^{-i\\eta_1} & 0 & 0\\\\[2pt]\n0& 1 & 0 & 0 & e^{-i\\eta_2} & 0\\\\[2pt]\n0 & 0 & 1 & 0 & 0 & e^{-i\\eta_3} \n\\end{array}\\right),\\quad\\Theta^{\\ast}(\\eta)=\\overline{\\Theta(\\eta)}^{\\top}.\n\\]\nAbove we used that the polarization matrix $\\mathbb{M}$ defined in (\\ref{def_Pola_matrix}) is symmetric. By solving the spectral problem (\\ref{CompactForm}), we get the values for $\\tilde{\\nu}(\\eta)$ and $a$. Once $\\tilde{\\nu}(\\eta)$ and $a$ are known, one can compute the solution to the system (\\ref{farfield_3}), (\\ref{ConditionAtZero}) to obtain the expressions of the $\\tilde{\\gamma}_j(\\eta)$. This ends the definition of the terms appearing in the expansions (\\ref{ExpansionTR0_3}). Let us exploit and comment these results.\\\\ \n\\newline\n$\\star$ First, observe that the $\\tilde{\\nu}(\\eta)$ are real. Indeed, since $\\mathbb{M}$ is real and symmetric, we infer that $\\mathbb{A}(\\eta)$ is hermitian. \\\\\n\\newline\n$\\star$ We have obtained \n\\begin{equation}\\label{ExpanConclu}\n\\Lambda^\\varepsilon(\\eta)=\\varepsilon^{-2}2\\pi^2+p^2\\pi^2+\\varepsilon\\tilde{\\nu}(\\eta)+\\dots\\,.\n\\end{equation}\nNote that in this expansion, the third term, contrary to the first two ones, depends on $\\eta$. For $\\eta\\in[0;2\\pi)^3$, denote by $\\tilde{\\nu}_1(\\eta)$, $\\tilde{\\nu}_2(\\eta)$, $\\tilde{\\nu}_3(\\eta)$ the three eigenvalues of \n(\\ref{CompactForm}) numbered such that\n\\[\n\\tilde{\\nu}_1(\\eta) \\le \\tilde{\\nu}_2(\\eta) \\le \\tilde{\\nu}_3(\\eta).\n\\]\nDefine the quantity \n\\begin{equation}\\label{defAleph}\n\\aleph:=\\bigcup_{j=1}^3\\{\\tilde{\\nu}_j(\\eta),\\,\\eta\\in[0;2\\pi)^3\\}.\n\\end{equation}\nSince $\\Upsilon^\\varepsilon_{1+q+3(p-1)}=\\{\\Lambda_{1+q+3(p-1)}^\\varepsilon(\\eta),\\,\\eta\\in[0;2\\pi)^3\\}$, this analysis shows that we have the inclusion\n\\begin{equation}\\label{MainInclusion}\n\\Upsilon^\\varepsilon_{1+q+3(p-1)}\\subset \\varepsilon^{-2}2\\pi^2+p^2\\pi^2+\\varepsilon\\,(-c_p;c_p)\n\\end{equation}\nfor $\\varepsilon$ small enough as stated in Theorem \\ref{MainThmPerio}. For $c_p$, we can take any value such that $\\overline{\\aleph}\\subset (-c_p;c_p)$. \\\\\n\\newline\n$\\star$ Due to the symmetries of $\\Omega$, $\\mathbb{M}$ is of the form\n\\begin{equation}\\label{MatPolSimple}\n\\left(\\begin{array}{cccccc}\nr_m & t^{\\perp}_m & t^{\\perp}_m & t_m & t^{\\perp}_m & t^{\\perp}_m \\\\[2pt]\nt^{\\perp}_m & r_m & t^{\\perp}_m & t^{\\perp}_m & t_m & t^{\\perp}_m \\\\[2pt]\nt^{\\perp}_m & t^{\\perp}_m & r_m & t^{\\perp}_m & t^{\\perp}_m & t_m \\\\[2pt]\nt_m & t^{\\perp}_m & t^{\\perp}_m & r_m & t^{\\perp}_m & t^{\\perp}_m  \\\\[2pt]\nt^{\\perp}_m & t_m & t^{\\perp}_m & t^{\\perp}_m & r_m & t^{\\perp}_m   \\\\[2pt]\nt^{\\perp}_m & t^{\\perp}_m & t_m & t^{\\perp}_m & t^{\\perp}_m & r_m  \n\\end{array}\\right)\n\\end{equation}\nwhere $r_m$, $t_m$, $t_m^\\perp$ are real coefficients. For $\\eta=(0,0,0)$, we find that the eigenvalues of (\\ref{CompactForm}) are \n\\[\n2r_m+2t_m+8t^{\\perp}_m,\\qquad 2r_m+2t_m-4t^{\\perp}_m,\\qquad 2r_m+2t_m-4t^{\\perp}_m.\n\\] \nFor $\\eta=(\\pi,\\pi,\\pi)$, we obtain \n\\[\n2r_m-2t_m,\\qquad 2r_m-2t_m,\\qquad  2r_m-2t_m.\n\\]\nIn the numerics of \\S\\ref{Section_PolMat}, we compute $\\mathbb{M}$. From the values of $r_m$, $t_m$, $t^{\\perp}_m$ obtained in (\\ref{ValuePolMat}), we find for example\n\\[\n2r_m+2t_m+8t^{\\perp}_m \\ne 2r_m-2t_m.\n\\]\nThis shows that the set $\\aleph$ appearing in (\\ref{defAleph}) has a non empty interior. Additionally, since $\\nu_1(\\pi)=\\nu_2(\\pi)=\\nu_3(\\pi)$ and the $\\nu_j$ depend continuously on $\\eta$, we deduce that $\\aleph$ is a connected segment. This is obtained in the numerics of \\S\\ref{Section_PolMat} (see (\\ref{NumAleph})) and due to the symmetries of $\\omega^\\varepsilon$, this was somehow expected.\\\\\n\\newline\nFinally, let us mention again that all the formal presentation above can be justified rigorously by a direct adaptation of the proofs of error estimates presented in \\cite{Naza17,Naza18}.\n\n\n\n\\section{Numerics}\\label{SectionNumerics}\n\n\\subsection{Spectrum of $A^{\\Omega}$}\n\nWe start the numerics by computing the spectrum of the operator $A^{\\Omega}$ defined after (\\ref{PbSpectralZoom}). More precisely, in order to approximate also the eigenvalues which are embedded in the continuous spectrum of $A^{\\Omega}$ and to reveal complex resonances which would be located close to the real axis, we work with Perfectly Matched Layers \\cite{Bera94,KiPa10,BoCP18} (see also the techniques of analytic dilatations \\cite{aguilar1971class,balslev1971spectral,moiseyev1998quantum}). For $\\theta\\in(0;\\pi\/2)$ and $L>1\/2$, define the complex valued parameters \n\\[\n\\alpha^\\theta_1=\\begin {array}{|ll}\n1 & \\mbox{for }|x_1|\\le L\\\\[2pt]\ne^{-i\\theta} & \\mbox{for }|x_1|>L\n\\end{array},\\qquad \\alpha^\\theta_2=\\begin {array}{|ll}\n1 & \\mbox{for }|x_2|\\le L\\\\[2pt]\ne^{-i\\theta} & \\mbox{for }|x_2|>L\n\\end{array},\\qquad \\alpha^\\theta_3=\\begin {array}{|ll}\n1 & \\mbox{for }|x_3|\\le L\\\\[2pt]\ne^{-i\\theta} & \\mbox{for }|x_3|>L.\n\\end{array}\n\\]\nHere the coefficient $\\theta$ will drive the rotation of the continuous spectrum while $L$ marks the beginning of the PML region. Then consider the spectral problem\n\\begin{equation}\\label{ProblemPMLs}\n\\begin{array}{|rcll}\n-\\alpha^\\theta_1\\cfrac{\\partial}{\\partial x_1}\\Big(\\alpha^\\theta_1\\cfrac{\\partial u}{\\partial x_1}\\,\\Big)+\\alpha^\\theta_2\\cfrac{\\partial}{\\partial x_2}\\Big(\\alpha^\\theta_2\\cfrac{\\partial u}{\\partial x_2}\\,\\Big)+\\alpha^\\theta_3\\cfrac{\\partial}{\\partial x_3}\\Big(\\alpha^\\theta_3\\cfrac{\\partial u}{\\partial x_3}\\,\\Big)&=&\\Lambda u &\\mbox{ in } \\Omega\\\\[3pt]\nu&=&0&\\mbox{ on } \\partial\\Omega.\n\\end{array}\n\\end{equation}\nDenote by $A^\\Omega_\\theta$ the unbounded operator associated with (\\ref{ProblemPMLs}). Observe that $A^\\Omega_\\theta$ is not selfadjoint due to the complex parameters $\\alpha^\\theta_j$. However the theory guarantees that the real eigenvalues of $A^\\Omega_\\theta$ coincide exactly with the real eigenvalues of $A^{\\Omega}$. What is interesting is that one can show that the essential spectrum of $A^\\Omega_\\theta$, that is the values of $\\Lambda$ such that $A^\\Omega_\\theta-\\Lambda\\,\\mrm{Id}:\\mathcal{D}(A^\\Omega_\\theta)\\to\\mrm{L}^2(\\Omega)$ is not Fredholm, corresponds to the set\n\\[\n\\bigcup_{m,n\\in\\mathbb{N}^\\ast}\\{\\pi^2(m^2+n^2)+t\\,e^{-2i\\theta},\\,t\\ge0\\},\n\\]\nso that the real eigenvalues of $A^\\Omega_\\theta$ are isolated in the spectrum. As a consequence, we can compute them by truncating the branches of $\\Omega$ at a certain distance without producing spectral pollution. Then we approximate the spectrum in this bounded geometry by using a classical P1 finite element method. We construct the matrices with the library \\texttt{Freefem++} \\cite{Hech12} and compute the spectrum with \\texttt{Matlab}\\footnote{\\textit{Matlab}, \\url{http:\/\/www.mathworks.com\/}.}.\\\\\n\\newline\nIn Figure \\ref{FigSpectrum}, we display in the complex plane the approximation of the spectrum of $A^\\Omega_\\theta$ obtained with this approach. We observe that the branches of essential spectrum of $A^\\Omega_\\theta$ are somehow discretized. This is due to the fact that the approximated problem is set in finite dimension. Moreover, we note that $A^\\Omega_\\theta$, and so $A^\\Omega$, have eigenvalues on the real line. In accordance with Theorem \\ref{ThmDiscreteSpectrum}, we find exactly one eigenvalue $\\mu_1\\approx 12.9\\approx1.3\\pi^2$ on the segment $(0;\\pi^2)$. We also note the presence of eigenvalues embedded in the continuous spectrum for the operator $A^{\\Omega}$. For the first one, we get $\\mu_2\\approx 46.7\\approx4.7\\pi^2$. \n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=10cm]{Spectrum.pdf}\n\\caption{Approximation of the spectrum of $A^\\Omega_\\theta$ for $\\theta=\\pi\/4$ ($L=1\/2$).}\n\\label{FigSpectrum}\n\\end{figure}\n\n\\noindent In Figure \\ref{FigEigenfunctions}, we represent an eigenfunction (trapped mode) associated with the eigenvalue $\\mu_1$ of the discrete spectrum of $A^{\\Omega}$. As guaranteed by Lemma \\ref{LemmaSym}, we observe that it is indeed symmetric with respect to the planes $(Ox_2x_3)$, $(Ox_3x_1)$, $(Ox_1x_2)$.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=5.8cm]{TMx0}\\qquad\\quad\\includegraphics[width=5.8cm]{TMy0}\n\\caption{Eigenfunction associated with the eigenvalue $\\mu_1$: cuts $x_1=0$ (left) and $x_2=0$ (right).}\n\\label{FigEigenfunctions}\n\\end{figure}\n\n\\subsection{Polarization matrix and threshold scattering matrix for the operator $A^\\Omega$}\\label{Section_PolMat}\nIn this section, we explain how to compute the polarization matrix $\\mathbb{M}$ introduced after (\\ref{CompactForm}) and whose properties allow one to assess the second corrector term in the expansion $\\Lambda^\\varepsilon=\\varepsilon^{-2}2\\pi^2+p^2\\pi^2+\\varepsilon\\tilde{\\nu}+\\dots$ of the eigenvalues generating the spectral bands $\\Upsilon^\\varepsilon_k$, $k\\ge 2$. We work in the geometry $\\Omega$ defined in (\\ref{DefOmega}) and study the problem (\\ref{PbSpectralZoom}) at the threshold, namely \n\\begin{equation}\\label{PbSpiderThreshold}\n\\begin{array}{|rcll}\n\\Delta v +2\\pi^2v&=& 0 &\\mbox{on }\\Omega\\\\\nv&=&0 &\\mbox{on }\\partial\\Omega.\n\\end{array}\n\\end{equation}\nTo obtain $\\mathbb{M}$, we will first compute the so-called threshold scattering matrix that we define now. In $\\mathcal{L}_j$, $j=1,\\dots,6$, set \n\\[\nw_{j}^{\\pm}(x)=\\cfrac{z_j \\mp i}{\\sqrt{2}}\\,U_{\\dagger}(y_j).\n\\]\nLet us work again with the function $\\psi\\in\\mathscr{C}^{\\infty}(\\mathbb{R})$ introduced after (\\ref{DefPolMat1}) such that $\\psi(s)=1$ for $s>1$ and $\\psi(s)=0$ for $s\\le 1\/2$. For $j=1,\\dots,6$, define $\\psi_j$ such that $\\psi_j(x)=\\psi(z_j,y_j)$ (observe that $\\psi_j$ is non zero only in the branch $\\mathcal{L}_j$). For $j=1,\\dots,6$, the theory of \\cite[Chap. 5]{NaPl94} guarantees that problem (\\ref{PbSpiderThreshold}) admits a solution with the decomposition\n\\begin{equation}\\label{decompo_vj}\nv_j=\\psi_j\\,w_j^{-}+\\sum_{k=1}^6\\psi_k\\,s_{jk}\\,w_k^++\\tilde{u}_j,\n\\end{equation}\nwhere the $s_{jk}$ are complex numbers and the $\\tilde{u}_j$ decay exponentially at infinity. The matrix \n\\[\n\\mathbb{S}:=\\left(s_{jk}\\right)_{1\\le j,k\\le 6}\n\\]\nis called the threshold scattering matrix. It is symmetric ($\\mathbb{S}=\\mathbb{S}^{\\top}$) but not necessarily hermitian and unitary ($\\mathbb{S}\\,\\overline{\\mathbb{S}}^{\\top}=\\mrm{Id}$). It is known (see relation (7.9) in \\cite{Naza16}) that $\\mathbb{M}$ coincides with the Cayley transform of $\\mathbb{S}$, i.e. we have\n\\begin{equation}\\label{DefMatPol}\n\\mathbb{M}=i(\\mrm{Id}+\\mathbb{S})^{-1}(\\mrm{Id}-\\mathbb{S}).\n\\end{equation}\nNote that one can show that we have $\\dim\\,\\ker\\,(\\mrm{Id}+\\mathbb{S})=\\dim\\,(\\mathscr{B}\/\\mathscr{B}_{\\mrm{tr}})$ (the quotient space) where $\\mathscr{B}$ denotes the space of bounded solutions of (\\ref{PbSpiderThreshold}) and $\\mathscr{B}_{\\mrm{tr}}$ the space of trapped modes of (\\ref{PbSpiderThreshold}). For the proof, we refer the reader for example to Theorem 1 in \\cite{Naza14}. Since Theorem \\ref{ThmNoBoundedSol} ensures that $\\mathscr{B}$ reduces to the null function, we infer that $\\mrm{Id}+\\mathbb{S}$ is invertible which guarantees that $\\mathbb{M}$ is well defined via formula (\\ref{DefMatPol}). Additionally, due to the symmetries of $\\Omega$, $\\mathbb{S}$ is of the form\n\\begin{equation}\\label{DefMatScaSimplifiee}\n\\left(\\begin{array}{cccccc}\nr & t^{\\perp} & t^{\\perp} & t & t^{\\perp} & t^{\\perp} \\\\[2pt]\nt^{\\perp} & r & t^{\\perp} & t^{\\perp} & t & t^{\\perp} \\\\[2pt]\nt^{\\perp} & t^{\\perp} & r & t^{\\perp} & t^{\\perp} & t \\\\[2pt]\nt & t^{\\perp} & t^{\\perp} & r & t^{\\perp} & t^{\\perp}  \\\\[2pt]\nt^{\\perp} & t & t^{\\perp} & t^{\\perp} & r & t^{\\perp}   \\\\[2pt]\nt^{\\perp} & t^{\\perp} & t & t^{\\perp} & t^{\\perp} & r  \n\\end{array}\\right)\n\\end{equation}\nwhere $r$, $t$, $t^{\\perp}$ are complex reflection and transmission coefficients. Therefore it is sufficient to compute $v_1$. To proceed, we shall work in the bounded domain $\\Omega^R$ (see before (\\ref{SpectralRMixed})) and impose approximated radiation conditions on the artificial cuts. Denote by $n$ the unit normal to $\\partial\\Omega^R$ directed to the exterior of $\\Omega^R$, set \n\\[\n\\begin{array}{l}\n\\Gamma_1:=\\{R\\}\\times(-1\/2;1\/2)^2,\\quad  \\Gamma_2:=(-1\/2;1\/2)\\times\\{R\\}\\times(-1\/2;1\/2),\\quad  \\Gamma_3:=(-1\/2;1\/2)^2\\times\\{R\\}\\\\[4pt]\n\\Gamma_4:=\\{-R\\}\\times(-1\/2;1\/2)^2,\\  \\Gamma_5:=(-1\/2;1\/2)\\times\\{-R\\}\\times(-1\/2;1\/2),\\  \\Gamma_6:=(-1\/2;1\/2)^2\\times\\{-R\\}\n\\end{array}\n\\]\nand $\\Gamma:=\\cup_{j=1}^6\\Gamma_j$. On $\\Gamma_1$, according to (\\ref{decompo_vj}), we have\n\\[\n\\partial_n(v_1-\\psi_1w_{1}^{-})=\\partial_{z_1}(v_1-w_{1}^{-})=2^{-1\/2}r\\,U_{\\dagger}+\\dots\n\\]\nwhere the dots stand for terms which are small for large values of $R$. On the other hand on $\\Gamma_1$, there holds\n\\[\nv_1-w_{1}^{-}=2^{-1\/2}r\\,(R-i)\\,U_{\\dagger}+\\dots .\n\\]\nTherefore, this gives, still on $\\Gamma_1$, \n\\begin{equation}\\label{Approx_RC1}\n\\partial_n v_1=\\cfrac{v_1-w_{1}^{-}}{R-i}+\\partial_{z_1} w_{1}^{-}+\\dots=\\cfrac{v_1}{R-i}+w_{1}^{-}\\left(\\cfrac{1}{R+i}-\\cfrac{1}{R-i}\\right)+\\dots=\\cfrac{v_1}{R-i}-\\cfrac{2i}{R^2+1}\\,w_{1}^{-}+\\dots.\n\\end{equation}\nOn $\\Gamma_j$, $j\\ne1$, the situation is simpler because $v_1$ is outgoing in the corresponding branches and we have\n\\begin{equation}\\label{Approx_RC2}\n\\partial_n v_1=\\cfrac{v_1}{R-i}+\\dots .\n\\end{equation}\nFinally, using the Robin conditions (\\ref{Approx_RC1}), (\\ref{Approx_RC2}) as approximated radiation conditions, we consider the variational formulation\n\\begin{equation}\\label{Pb_Approximation}\n\\begin{array}{|l}\n\\mbox{Find }\\hat{v}_1\\in\\mrm{H}^1_{0}(\\Omega^R;\\partial\\Omega^R\\cap\\partial\\Omega)\\mbox{ such that for all }v\\in\\mrm{H}^1_{0}(\\Omega^R;\\partial\\Omega^R\\cap\\partial\\Omega) \\\\[6pt]\n\\displaystyle\\int_{\\Omega^R}\\nabla \\hat{v}_1\\cdot\\nabla v\\,dx-\\cfrac{1}{R-i}\\int_{\\Gamma} \\hat{v}_1 v\\,d\\sigma-(2\\pi)^2\\displaystyle\\int_{\\Omega^R}\\hat{v}_1 v\\,dx=-\\cfrac{2i}{R^2+1}\\int_{\\Gamma_1} w_{1}^{-} v\\,d\\sigma.\n\\end{array}\n\\end{equation}\nOne can prove that $\\hat{v}_1$ yields a good approximation of $v_1$ with an error which is exponentially decaying with $R$. In practice, we solve the problem (\\ref{Pb_Approximation}) with a P1 finite element method thanks to \\texttt{Freefem++}. Then replacing $v_1$ by $\\hat{v}_1$ in the exact formulas\n\\begin{equation}\\label{FormulaApproxCoef}\nr=\\cfrac{2}{R^2+1}\\,\\int_{\\Gamma_1} (v_1-w_{1}^{-})\\,w_{1}^{-}\\,d\\sigma,\\quad t=\\cfrac{2}{R^2+1}\\,\\int_{\\Gamma_4} v_1\\,w_4^{-}\\,d\\sigma,\\quad t^\\perp=\\cfrac{2}{R^2+1}\\,\\int_{\\Gamma_2} v_1\\,w_2^{-}\\,d\\sigma,\n\\end{equation}\nwe get an approximation of the threshold scattering matrix $\\mathbb{S}$ given by (\\ref{DefMatScaSimplifiee}). Finally with (\\ref{DefMatPol}), we obtain an approximation of the polarization matrix $\\mathbb{M}$ which appears in the $3\\times3$ spectral problems (\\ref{CompactForm}).\\\\\n\\newline\nOur computations give\n\\[\nr\\approx 0.66+0.11i,\\qquad t\\approx 0.08-0.70i,\\qquad t^{\\perp}\\approx -0.08-0.08i.\n\\]\nThe eigenvalues of $\\mathbb{S}\\in\\mathbb{C}^{6\\times6}$ are approximately equal to \n\\[\n0.44 - 0.9i\\mbox{ (simple)},\\quad 0.9 - 0.44i\\mbox{ (double)}\\quad\\mbox{ and }\\quad 0.58 + 0.81i\\mbox{ (triple).}\n\\]\nThey have modulus one which is consistent with the fact that $\\mathbb{S}$ is unitary. Moreover, we indeed observe that they are different from $-1$ which is coherent with the discussion following (\\ref{DefMatPol}) (absence of threshold resonance). On the other hand, for the coefficients of $\\mathbb{M}$ (see (\\ref{MatPolSimple})), we get\n\\begin{equation}\\label{ValuePolMat}\nr_m\\approx 0.08,\\qquad t_m\\approx -0.44,\\qquad t^{\\perp}_m\\approx -0.06.\n\\end{equation}\nWith these values, solving the $3\\times3$ eigenvalue problem (\\ref{CompactForm}) for $\\eta\\in[0;2\\pi)^3$, we find that the segment $\\aleph$ defined in (\\ref{defAleph}) satisfies\n\\begin{equation}\\label{NumAleph}\n\\aleph\\approx (-1.24;1.04).\n\\end{equation}\n\n\\noindent Let us mention that the Robin conditions (\\ref{Approx_RC1}), (\\ref{Approx_RC2}) are rather crude approximations of the exact radiation conditions. To get good errors estimates, we should take rather large values of $R$. However in practice, large $R$ are no so simple to handle and can create important numerical errors. Therefore a compromise must be found and we take $R=2.5$. Admittedly, this point should more investigated. \n\n\n\\section*{Apprendix}\n\n\\subsection{Friedrichs inequality}\nWe reproduce here the Lemma 5.1 of \\cite{BaMaNa17}.\n\\begin{lemma}\\label{LemmaPoincareFriedrichs}\nAssume that $a>0$. Then we have the Friedrichs inequality \n\\begin{equation}\\label{Fried_ineq}\n \\kappa(a)\\int_{0}^{1\/2}\\phi^2\\,dt \\le \\int_{0}^{+\\infty}(\\partial_t\\phi)^2\\,dt+a^2\\int_{1\/2}^{+\\infty}\\phi^2\\,dt,\\qquad \\forall \\phi\\in\\mrm{H}^1(0;+\\infty),\n\\end{equation}\nwhere $\\kappa(a)$ is the smallest positive root of the transcendental equation\n\\begin{equation}\\label{TransEqnLemma}\n\\sqrt{\\kappa}\\tan\\bigg(\\cfrac{\\sqrt{\\kappa}}{2}\\bigg)=a.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFor $a>0$, consider the spectral problem \n\\begin{equation}\\label{Eigen1D}\n\\begin{array}{|rcll}\n-\\partial^2_t \\phi+a^2\\mathbbm{1}_{(1\/2;+\\infty)} \\phi &=& \\lambda(a)\\,\\mathbbm{1}_{(0;1\/2)}\\phi&\\mbox{ in }(0;+\\infty)\\\\[4pt]\n\\partial_t\\phi(0)&=&0\n\\end{array}\n\\end{equation}\nwhere $\\mathbbm{1}_{(1\/2;+\\infty)}$, $\\mathbbm{1}_{(0;1\/2)}$ stand for the indicator functions of the sets $(1\/2;+\\infty)$, $(0;1\/2)$ respectively. Let us equip $\\mrm{H}^1(0;+\\infty)$ with the inner product\n\\[\n(\\phi,\\phi')_a=\\int_{0}^{+\\infty}\\partial_t\\phi\\,\\partial_t\\phi'\\,dt+a^2\\phi\\,\\phi'\\,dt.\n\\]\nWith the Riesz representation theorem, define the linear and continuous operator $T:\\mrm{H}^1(0;+\\infty)\\to \\mrm{H}^1(0;+\\infty)$ such that\n\\[\n(T\\phi,\\phi')_a=\\int_{0}^{1\/2}\\phi\\,\\phi'\\,dt.\n\\]\nWith this definition, we find that $(\\lambda(a),\\phi)$ is an eigenpair of (\\ref{Eigen1D}) if and only if we have \n\\[\nT\\phi=(\\lambda(a)+a^2)^{-1}\\phi.\n\\]\nSince $T$ is bounded and symmetric, it is self-adjoint. Additionally the Rellich theorem ensures that $T$ is compact. This guarantees that the spectrum of (\\ref{Eigen1D}) coincides with a sequence of positive eigenvalues whose only accumulation point is $+\\infty$. Let us denote by $\\kappa(a)$ the smallest eigenvalue of (\\ref{Eigen1D}). From classical results concerning compact self-adjoint (see e.g. \\cite[Thm. 2.7.2]{BiSo87}), we know that\n\\begin{equation}\\label{Def_quotient}\n(\\kappa(a)+a^2)^{-1}=\\underset{\\phi\\in\\mrm{H}^1(0;+\\infty),\\,(\\phi,\\phi)_a=1}{\\sup} (T\\phi,\\phi)_a.\n\\end{equation}\nRearranging the terms, we find that (\\ref{Def_quotient}) provides the desired estimates (\\ref{Fried_ineq}). Now we compute $\\kappa(a)$. Solving the ordinary differential equation (\\ref{Eigen1D}) with $\\lambda(a)=\\kappa(a)$, we obtain, up to a multiplicative constant,\n\\[\n\\phi(t)=\\begin{array}{|ll}\n\\cos(\\sqrt{\\kappa(a)}t) & \\mbox{ for }t\\in(0;1\/2)\\\\[3pt]\nc\\,e^{-at} & \\mbox{ for }t\\ge 1\/2\n\\end{array}\n\\]\nwhere $c$ is a constant to determine. Writing the transmission conditions at $t=1\/2$, we find that a non zero solution exists if and only if $\\kappa(a)>0$ satisfies the relation (\\ref{TransEqnLemma}).\n\\end{proof}\n\\begin{lemma}\\label{LemmaPoincareFriedrichsR}\nAssume that $a>0$ and $R>1\/2$. Then we have the Friedrichs inequality \n\\begin{equation}\\label{Fried_ineq_R}\n \\kappa(a,R)\\int_{0}^{1\/2}\\phi^2\\,dt \\le \\int_{0}^{R}(\\partial_t\\phi)^2\\,dt+a^2\\int_{1\/2}^{R}\\phi^2\\,dt,\\qquad \\forall \\phi\\in\\mrm{H}^1(0;R),\n\\end{equation}\nwhere $\\kappa(a,R)$ is the smallest positive root of the transcendental equation\n\\begin{equation}\\label{TransEqnLemmaR}\n\\sqrt{\\kappa}\\tan\\bigg(\\cfrac{\\sqrt{\\kappa}}{2}\\bigg)=a\\,\\tanh(a(R-1\/2)).\n\\end{equation}\nTherefore, we have $\\lim_{R\\to+\\infty}\\kappa(a,R)=\\kappa(a)$ where $\\kappa(a)$ is the constant appearing in Lemma \\ref{LemmaPoincareFriedrichs}.\n\\end{lemma}\n\\begin{proof}\nThe demonstration is completely similar to the one of Lemma \\ref{LemmaPoincareFriedrichs} above. We find that the largest constant $ \\kappa(a,R)$ such that (\\ref{Fried_ineq_R}) holds coincides with the smallest eigenvalue of the problem\n\\[\n\\begin{array}{|rcll}\n-\\partial^2_t \\phi+a^2\\mathbbm{1}_{(1\/2;R)} \\phi &=& \\kappa(a,R)\\mathbbm{1}_{(0;1\/2)}\\phi&\\mbox{ in }(0;R)\\\\[4pt]\n\\partial_t\\phi(0)=\\partial_t\\phi(R)&=&0.\n\\end{array}\n\\]\nSolving it, we find that if $\\phi$ is a corresponding eigenfunction, up to a multiplicative constant, we have\n\\[\n\\phi(t)=\\begin{array}{|ll}\n\\cos(\\sqrt{\\kappa(a,R)}t) & \\mbox{ for }t\\in(0;1\/2)\\\\[3pt]\nc\\,\\cosh(a(t-R)) & \\mbox{ for }t\\in(1\/2;R)\n\\end{array}\n\\]\nfor some constant $c$. This times, writing the transmission conditions at $t=1\/2$, we find that a non zero solution exists when $\\kappa(a,R)$ satisfies (\\ref{TransEqnLemmaR}). Finally we obtain that $\\kappa(a,R)\\to\\kappa(a)$ because $\\tanh(a(R-1\/2))$ in (\\ref{TransEqnLemmaR}) tends to $1$ when $R\\to+\\infty$.\n\\end{proof}\n\n\\section{Acknowledgements}\nThe work of the second author was supported by the Russian Science Foundation, project 22-11-00046.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOver the past decade, advances in computing devices and navigation sensors accelerated the development and the range of missions USVs can undertake. USV are extensively utilised for applications such as seabed mapping, marine animals tracking, and maintenance of offshore facilities such as wind farms or oil pipelines. Compared to manned vehicles, a USV could be deployed for dangerous and boring tasks continuously in shallow and restricted waters \\cite{ma2018multi}. As a USV could carry heavy payload, it can install multiple energy resources to achieve long endurance performance \\cite{niu2018energy}. However, USV can only perceive dynamic sea condition in a short range. By contrast, UAV is capable of detecting large-scale sea condition at a high altitude and has better maneuverability than USV. However, UAV has limited flight endurance due to its limited battery capacity. By developing a system that can allow for a UAV to autonomously land on the USV, the flight range of UAVs will be considerably increased. UAV could also detect the USV surrounding environment at a high altitude \\cite{xiao2017uav}, which improves the detection range of USV. Moreover, UAV can be used as a communication relay between USV and ground station, which enlarges the communication range of USV. This research proposes an affordable system that allows the UAV and USV work collaboratively and autonomously for making uses of the strengths and diminishing the weaknesses of both UAVs and USVs.\n\n\n\\begin{figure} [!t]\n  \\begin{center}\n    \\includegraphics[width=3.4in]{UAVUSV.PNG}\n    \\caption{3D printed UAV and USV.}\n    \\label{fig1}\n  \\end{center}\n\\end{figure}\n\n\n\\section{UAV and USV Design}\n\n\\subsection{UAV Design}\nTo protect the electronic components from sea water, we designed a waterproof UAV, which was 3D printed using Selective Laser Sintering (SLS) printer and PA2200 printing material, as shown in Fig.~\\ref{fig21}. The volume of the shell is 0.004006 cubic meters and its buoyancy can support a mass of $4.11kg$, which enables the UAV to float on water. To enable the UAV to localize the USV position precisely under various lighting conditions, we used an IR beacon detector on the bottom of UAV. The IR beacon detector was designed to filter most light noises and only detect the light within certain frequency spectrum emitted by the IR beacon, which was installed on the USV landing platform. The GPS, autopilot, battery and IR beacon detector were installed on the component tray, which is inside of the shell, as shown in Fig.~\\ref{fig21}.\n\n\n\\begin{figure}[h]\n  \\begin{center}\n    \\includegraphics[width=3.25in,height=1.6in]{MainstructualofUAV.PNG}\n    \\caption{CAD design of waterproof UAV.}\n    \\label{fig21}\n  \\end{center}\n\\end{figure}\n\n\n\\subsection{Design of the USV and its Landing Platform}\n\nTo improve the stability of the landing platform, a catamaran USV was designed. The CAD design of hull and wooden keel are shown in Fig.~\\ref{fig24}. It is calculated to carry a payload of up to $32.5kg$ per hull when fully submerged. Polyethylene Terephthalate Glycol (PETG) filament was used to print hulls since it can produce strong parts with slight flexibility and has a relatively low shrinkage ratio compared to other filament types such as Acrylonitrile Butadiene Styrene (ABS) and Polylactic Acid (PLA). The final hull dimensions was $1.655m \\times 0.301m \\times 0.155m$ and the total mass was 8.1kg per hull. MotorGuide R3-30 trolling motors were selected and fitted via modified mounts that allowed the motors position adjustable, as shown in Fig.~\\ref{fig25}. The maximum thrust of motors can be up to $133.8N$. The weight of each hull with fully assembled motors and deck was $17.4kg$.\nAll control units of the USV, including raspberry pi, speed controller RoboClaw 2x30A, Battery, Telemetry Radio, Ublox Neo M8 compass and Emlid reach GPS, were installed in a waterproof box, as shown in Fig.~\\ref{fig26}, which was located in the centre of the landing platform with aluminium frame supports, as shown in Fig.~\\ref{fig27}.\n\n\\begin{figure}[hbt!]\n\\centering\n\\subfigure[]{\\label{fig24}\n\\includegraphics[width=0.23\\textwidth,height=0.16\\textwidth]{Woodenkeelandhull.PNG}}\n\\subfigure[]{\\label{fig25}\n\\includegraphics[width=0.23\\textwidth,height=0.16\\textwidth]{propulsion.jpg}}\n\\subfigure[]{\\label{fig26}\n\\includegraphics[width=0.23\\textwidth,height=0.16\\textwidth]{USVcontrolbox.PNG}}\n\\subfigure[]{\\label{fig27}\n\\includegraphics[width=0.23\\textwidth,height=0.16\\textwidth]{finalusvdesign.jpg}}\n\\caption{Catamaran USV: (a) Wooden keel and 3D printed hull (b) Propulsion system of USV (c) USV control box (d) Catamaran USV and landing platform}\n\\end{figure}\n\n\n\\section{Experiments and Discussion}\n\\subsection{Two-Phase Precise Landing Test}\nA two-phase landing method was proposed to land the UAV on the USV precisely using standard GPS and IR beacon. At first, the UAV navigates to the USV location using USV GPS data. Standard GPS data are not accurate enough to allow accurate landing on the platform due to the limited dimension. When the UAV is above the USV and localizes the IR beacon, precise landing using IR beacon will be activated. The accuracy of precise landing is evaluated using twenty tests. As shown in Fig.~\\ref{fig41}, the relative position between the IR beacon detector on the UAV and the IR beacon in the centre of the USV landing platform satisfies: $\\Delta X < 20cm$, $\\Delta Y < 20cm$. Note that the offset along y axis was caused by the wind during the tests. As the landing platform dimension is $1.2m \\times 1m$ that is larger than the landing derivation, the UAV could land on the USV safely. \n\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[]{\\label{fig41}\n\\includegraphics[width=3.0in]{Landingprecision.PNG}}\n\\subfigure[]{\\label{fig42}\n\\includegraphics[width=2.8in]{carrotchasing.PNG}}\n\n\\caption{UAV and USV experiments: (a) Accuracy results of twenty landing tests (b) Aerial view of USV control algorithm and path following algorithm test}\n\\end{figure}\n\n\\subsection{Control and Path Following Algorithm of USV}\nPI controller and PD controller were implemented for speed control and heading control \\cite{niu2016efficient}, respectively. To navigate the USV to follow the pre-defined path precisely, carrot chasing path following algorithm \\cite{niu2016efficient} was implemented on USV for generating required heading and speed command. As shown in Fig.~\\ref{fig42}, two waypoints were pre-defined by ground station. The USV was able to follow the pre-defined path, represented by the line between point1 and point2.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and main results}\n\nThe geometric properties of random interfaces are a vast arena of study in rigorous statistical mechanics.\nTwo important classes of interface models are {\\em phase separation} models that idealize the boundary between a droplet of one substance suspended in another, and {\\em last passage percolation} models, where a directed path in independent local randomness maximizes a random weight determined by the environment.\n\nOne of the best known mathematical examples of phase separation is the two dimensional supercritical Ising model in a large box with negative boundary condition. Conditioned to have an atypically high number of positive spins in the box, the vertices in the plus phase tend to form a droplet surrounded by a sea of negative spins. The random phase boundary of such a droplet has been the object of intense study.  \nWulff proposed that the profile of such constrained circuits would macroscopically resemble\na dilation of an isoperimetrically optimal curve. This was established rigorously in \\cite{DKS} and \\cite{IS}.\nVia the FK representation of the Ising model, \n one observes a similar situation in the setting of subcritical two dimensional percolation, where the analogous object is the boundary of the cluster containing the origin after this cluster is conditioned to be atypically large.  The Wulff shape captures the macroscopic profile of the circuit in such models, but what of fluctuations? Several definitions may be considered that seek to capture fluctuation behaviour on the part of the circuit, including  the deviation in the Hausdorff metric of the convex hull of the circuit from an appropriately scaled Wulff shape.\nAlternative definitions, more local in nature, serve better to capture the transition in circuit geometry from local Brownian randomness to a smoother profile dictated by the constraints of global curvature.\nIn fact, a pair of definitions is natural, one to capture the longitudinal distance at which the transition takes place, and the second to treat the orthogonal inward deviation of the interface at this transition scale. \nTo specify the characteristic longitudinal distance, we may note that the convex hull of the circuit is a polygonal path that is composed of planar line segments or {\\em facets}; we may treat the typical or maximum facet length as a barometer of the transition from the shorter scale of local randomness to the longer scale of curvature. Latitudinally, we may note that any point in the circuit has a {\\em local roughness}, given by its distance from the convex hull boundary. The typical or maximum local roughness along the circuit is a latitudinal counterpart to facet length. \n Alexander \\cite{Alexander}, and Hammond \\cite{AH1,AH2,AH3} analysed such conditioned circuit models and determined that when the area contained in the circuit is of order $n^2$, so that the circuit has diameter of order $n$, facet length and local roughness scale as $n^{2\/3}$ and $n^{1\/3}$. A similar situation is witnessed when a parabola $x \\to t^{-1}x^2$ is subtracted from a two-sided Brownian motion $B:\\mathbb{R} \\to \\mathbb{R}$. When $t > 0$ is large, facets of the motion's convex hull have length $\\Theta(t^{2\/3})$ and inward deviation $\\Theta(t^{1\/3})$.\nThis phenomenon is expected to be universal when the local structure of the interface is Brownian and other examples include \na Brownian bridge pinned at $-T$ and $T$ and conditioned to remain above the semi-circle of radius $T$ centred at the origin which was analysed by Ferrari and Spohn \\cite{FS}. \n\nThe second class of random geometric paths we have mentioned are last passage percolation models. These models form part of a huge, Kardar-Parisi-Zhang, class of statistical mechanical models in which a path through randomness is selected to be extremal for a natural weight determined by that randomness. The maximizing paths are often called {\\em polymers}. \nThe fluctuation behaviour of a length $n$ polymer with given endpoints may be gauged either in terms of the scale of deviation of its weight from the mean value, or by the scale of deviation of say the polymer's midpoint from the planar line segment that interpolates the polymer's endpoints. The two deviations are given by scales $n^{1\/3}$ and $n^{2\/3}$. \nThese were first proved for planar Poissonian\ndirected last passage percolation, in the seminal work of Baik, Deift and Johansson \\cite{BDJ99}. Since then, this and other integrable models in the same KPZ universality class have been extensively analysed and detailed information about this model, both geometric and algebraic, has been obtained \\cite{J00, LM01, LMS02}. \n\nIt is of much interest to study models that combine phase separation and path minimization (or maximization) in random environment. Consider an environment with local independent randomness, and the circuit through that randomness whose weight determined by that randomness is extremal among those circuits that trap a given area. Studying the random geometry of such a circuit is a problem of extremal isoperimetry. Taking the randomness to be supercritical percolation, Itai Benjamini conjectured the first-order, Wulff-like behaviour of the boundary of the set in the supercritical percolation cluster attaining the so called anchored expansion constant. This was proved by~\\cite{BLPR} by showing that the curve in the limit solves a natural isoperimetric variational problem; this has recently been extended to higher dimensions by Gold \\cite{G16}. Given the role of facet length and local roughness in capturing the local random to global curvature transition in circuit geometry, the natural next problem is to understand the scaling exponents of such objects, which is the pursuit we undertake in this paper.  \nOur choice of model preserves the qualitative features of the problem of interface fluctuation in an isoperimetrically  extremal droplet in supercritical percolation and  at the same time ensures that key  algebraic aspects of KPZ theory can be harnessed to yield sharp fluctuation estimates. \n\n\nThe particular setting we consider in this paper is planar Poissonian \ndirected last passage percolation. \nTo impose the required properties, we study this model under a quadratic curvature constraint: that is, we force the best path to move away from the straight line and have a quadratic curvature on the average. This is done naturally by considering the longest upright path in a Poissonian environment joining $(0,0)$ and $(n,n)$ which has the additional \\textit{area trap} property that the area under the curve is at least $(\\frac{1}{2}+\\alpha)n^2$ for some  $\\alpha\\in (0,\\frac{1}{2})$. Note that from the discussion above it follows that the unconstrained longest path stays close to the diagonal and hence encloses area $(\\frac{1}{2}+o(1))n^2$. \nPostponing precise statements until later (see Section \\ref{scale-exp}), our main results quantify the competition between the global parabolic nature  and local behaviour guided by KPZ relations of the contour.\nThey show that the maximum facet length of the contour's least concave majorant scales as  $n^{3\/4+o(1)}$, while the inward deviation from the concave majorant (local roughness) \nscales as $n^{1\/2+o(1)}$.\n\nWe also establish a law of large numbers for the length of the optimal path. The proof  proceeds by setting up a variational problem, as is natural in such contexts (see \\cite{DZ2}, \\cite{BLPR}). Perhaps  surprisingly, however, the problem turns out to have a very explicit solution. The geometric information about the limit shape of the constrained polymer is also used as input in some of the arguments about fluctuations. \nThe question of fluctuation in the context of  isoperimetrically extremal circuits in supercritical percolation can be formulated as a first passage percolation analogue of our setting of last passage percolation. Although the first passage percolation model is not exactly solvable, it, too, is believed to be in the KPZ universality class, and the geodesics there are believed to have the same $n^{2\/3}$ scaling of transversal fluctuation as in our case \n(see \\cite{FPPsurvey} and the references therein). \nThus our results suggest that a certain universality is being witnessed and, according to this belief, we formulate a conjecture concerning percolation. \nWe elaborate on this and a number of other interesting questions in Section \\ref{oq}.\n\nFinally, we discuss briefly the key inputs used in our paper and how it contrasts with the other examples of fluctuation results already mentioned.\nThe results in \\cite{AH1,AH2,AH3} crucially used the refined understanding and geometric estimates for percolation clusters while a study of area trapping planar Brownian loop \\cite{HP} used well known estimates for Brownian motion.  Exact expressions involving Brownian motion conditioned to stay over a parabola was also the key ingredient in the proofs in \\cite{FS}. \nHowever, in our setting, even though the unconstrained model has integrable properties, the lack of general geometric understanding as one deviates slightly from integrable models causes a big challenge for us.  \nNonetheless, using certain known facts about the unconstrained model and their robust variants established recently in \\cite{BSS14} as blackbox estimates, we rigorously establish local roughness exponents for our model which we henceforth call the \\emph{Area Trapping Polymer} model (see next section for precise definitions). \nThus, this work is an example  where one can use inputs from the integrable literature to make geometric conclusions about settings which are beyond the exactly solvable world (see also \\cite{BSS14}).\nWe believe a general program of developing such geometric arguments would lead to more robust proofs which could work for settings that are non-integrable but are variants of solvable models. \n\n\\subsection{Model Definitions}\\label{def1012}\nWe now recall the planar Poissonian directed last passage percolation model. \n\nLet $\\Pi$ be a homogeneous rate one Poisson Point Process (PPP) on the plane. A partial order on $\\mathbb{R}^2$ is given by $(x_1,y_1)\\preceq (x_2,y_2)$ if $x_1\\leq x_2$ and $y_1\\leq y_2$. \nFor $u\\preceq v$, a directed path $\\gamma$ from $u$ to $v$ is a piecewise linear path that joins points $u=\\gamma_{0}\\preceq \\gamma_1 \\preceq \\cdots \\preceq \\gamma_{k}= v$ where each $\\gamma_i$  for $i\\in [k-1]$ (throughout the article we will adopt the standard notation $[n]=\\{1,2,\\ldots,n\\}$)\nis a point of $\\Pi$. \nDefine the length of $\\gamma$, denoted $|\\gamma|$, to be the number of $\\Pi$-points on $\\gamma$. \n\\begin{definition}\\label{geodesic1} \n Define the last passage time from $u$ to $v$, denoted by $L(u,v)$, to be the maximum  of $|\\gamma|$ as $\\gamma$ varies over all directed paths from $u$ to $v$. There may be several maximizing paths between $u$ and $v$, and throughout the paper we will refer to the top most path (it is easy to see that the top most path is well defined here) among those, as the \\emph{geodesic} between $u$ and $v$ and denote it by  $\\gamma(u,v)$. \n \\end{definition}\n\nWe will often call $\\gamma(u,v)$ as the polymer between $u$ and $v$ and $|\\gamma(u,v)|$ as the polymer length\/weight respectively.\nNext we introduce a constraint in this classical model. \n\n\\subsubsection{Area Trapped by a path}\\label{def10}\nConsider a path $\\gamma$ between the origin $(0,0)$ and a point $(x,y)$ in the positive quadrant. The area trapped by the path $\\gamma$, denoted by $A(\\gamma)$, is defined to be the area of the closed polygon determined by the $x$-axis, the vertical line segment joining $(x,0)$ to $(x,y)$ together with the line segments of the path $\\gamma$. Let $\\gamma_n$ denote the geodesic  between $(0,0)$ and $(n,n)$. Well-known facts about Poissonian LPP readily imply that $A(\\gamma_{n})= (\\frac{1}{2}+o(1))n^2$ asymptotically almost surely. We constrain the model and consider maximizing paths subject to trapping a much larger area. To this end fix $\\alpha\\in (0,\\frac{1}{2})$, and let \n\\begin{equation}\n\\label{e:cop}\nL_{\\alpha}(n):=\\max\\left\\{|\\gamma|: \\gamma~\\text{path from}~(0,0)~\\text{to}~(n,n)~\\text{and}~ A(\\gamma)\\geq \\big(\\tfrac{1}{2}+\\alpha\\big)n^2\\right\\}.\n\\end{equation}\nIt is easily seen that, among the paths that attain this maximum, there is almost surely exactly one that traps the least area. This path will be called $\\Gamma_{\\alpha,n}$. We will write   $\\Gamma_{n}$ provided that the context clarifies the value of $\\alpha$ in question. We shall call $\\Gamma_n$ the constrained (or $\\alpha$-constrained) geodesic.\nIn analogy with the phase separation example in percolation mentioned the introduction, it might seem more natural to consider the family of  down right paths joining $(n,0)$ and $(0,n)$ since all of these curves enclose the origin. However because of the obvious underlying symmetry, and to take advantage of standard notational conventions, throughout the sequel  we will consider the contour joining the origin to the point $(n,n)$.\n\n\nOur main objects of interest are two quantities that measure local regularity of the constrained geodesic $\\Gamma_n$. The following definitions are illustrated by Figure \\ref{def}.\n\n\\begin{definition}\n\\label{d:mlrmfl}\nLet ${\\rm conv}(\\Gamma_{\\alpha, n})$ denote the convex hull of the polygon determined by the path $\\Gamma_{\\alpha, n}$ and the coordinate axes. Let $\\Gamma^{*}_{\\alpha, n}$ denote the closure of the polygonal part of the boundary of ${\\rm conv}(\\Gamma_n)$ between $(0,0)$ and $(n,n)$ above the $x$-axis. Thus $\\Gamma^{*}_{\\alpha, n}$ is the least concave majorant of $\\Gamma_{\\alpha,n}$ and is an union of finitely many line segments. These  segments will be called {\\rm facets}.\nDefine {\\rm{maximum facet length}} of $\\Gamma_n$, denoted $\\mathrm{MFL}(\\Gamma_{\\alpha, n})$ to be the maximum Euclidean length of the facets. For $x\\in \\Gamma_{\\alpha, n}$, let ${\\rm d}(x,\\Gamma^{*}_{\\alpha, n})$ denote the distance from $x$ to $\\Gamma^*_{\\alpha, n}$. This is a natural notion of  the \\emph{local roughness at $x$}. \nDefine the {\\rm{maximum local roughness}} of $\\Gamma_n$, \ndenoted $\\mathrm{MLR}(\\Gamma_n)$ by\n$$\\mathrm{MLR}(\\Gamma_{\\alpha, n}):= \\sup\\{x\\in \\Gamma_{\\alpha, n}: {\\rm d}(x,\\Gamma^{*}_{\\alpha, n})\\}.$$\n\\end{definition} \n\nIn the following subsections we present our main results. We first state our results regarding the fluctuation exponents of $\\Gamma_{\\alpha, n}$.\n\\subsection{Scaling exponents for geodesic geometry}\\label{scale-exp}\nThe sense in which we capture the exponents is stronger and easier to state for the lower bounds, and so we begin with them. \n\n\\begin{theorem}\n\\label{t:mfllb}\nFix $\\alpha \\in (0,\\frac{1}{2})$ and $\\varepsilon>0$. Then   there exists $c=c(\\alpha,\\varepsilon)>0$ such that for all large enough $n,$\n$$\\mathbb{P}(\\mathrm{MFL}(\\Gamma_n) \\geq n^{3\/4-\\varepsilon})\\ge 1-e^{-n^c}.$$ \n\\end{theorem}\n\nAs we have mentioned, the scaling exponent for transversal fluctuation of point-to-point geodesic in unconstrained Poissonian last passage percolation is known to be $2\/3$ \nThis fact, put together with the above theorem yields the following lower bound on maximum local roughness.\n\n\\begin{theorem}\n\\label{t:mlrlb}\nFix $\\alpha \\in (0,\\frac{1}{2})$ and $\\varepsilon>0$. Then   there exists $c=c(\\alpha, \\varepsilon)>0$ such that for all large enough $n,$\n$$\\mathbb{P}(\\mathrm{MLR}(\\Gamma_n) \\geq n^{1\/2-\\varepsilon})\\ge 1-e^{-n^c}.$$ \n\\end{theorem}\n\nRegarding the matching upper bound, \nwe prove that, with high probability, there exists a dense set of $\\alpha \\in (0,2^{-1})$ for which the maximum length of the facets away from the boundary is bounded above by $n^{3\/4+o(1)}$. To make this precise, fix $\\delta \\in (0,\\pi\/4)$, and consider a facet in $\\Gamma_{\\alpha, n}$ with endpoints $A$ and $B$\nrecorded in clockwise order. Setting $O = (n,0)$,  let $\\theta_{A}$ denote the acute angle that $OA$ makes with the $y$-axis and $\\theta_{B}$, the acute angle  that $OB$ makes with the $x$-axis; see Figure~\\ref{def}. \n\n\\begin{definition}\n\\label{interior1}\nThe facet $AB$ is called $\\delta$-interior if $\\min (\\theta_{A}, \\theta_{B})\\geq \\delta$.\n\\end{definition} \n\nNote that the union of the $\\delta$-interior facets forms a polygonal path.\n(The union could be empty, but we will infer later from Theorem~\\ref{lln2}\nthat this event has an exponentially small probability.)\n\\footnote{Note that at this point, it is not a priori clear if the set of $\\delta-$interior facets is non-empty. However, this will be a consequence of Theorem \\ref{lln2}, stated later, which implies for any $\\delta$, the length of all the facets \nwill be less than $O(\\delta n)$ with exponentially small failure probability.} Let $A_0$ and $B_0$ denote the extremities of this union path, and let $\\Gamma_{\\delta, \\alpha, n}$ denote the subpath of  $\\Gamma_{\\alpha,n}$ between $A_0$ and $B_0$. Define the maximum $\\delta$-interior facet length of $\\Gamma_{\\alpha, n}$, denoted by $\\mathrm{MFL}(\\Gamma_{\\delta, \\alpha,n})$, to be the maximum length of the $\\delta$-interior facets.  Define the $\\delta$-interior maximum local roughness, denoted by $\\mathrm{MLR}(\\Gamma_{\\delta, \\alpha, n})$ by altering  Definition \\ref{d:mlrmfl} so that now the supremum is taken over all $x \\in \\Gamma_{\\delta, \\alpha, n}$. \n\\begin{definition}\\label{good10}\nFix $\\varepsilon>0$. We say $\\alpha\\in (0,\\frac{1}{2})$ is  $(n,\\varepsilon,\\delta)$-good if $\\mathrm{MFL}(\\Gamma_{\\delta, \\alpha, n}) \\leq n^{3\/4+\\varepsilon}$. \n\\end{definition}\nHere is our upper bound concerning the maximum length of facets.\n\n\\begin{theorem}\n\\label{t:mflub}\nFix $\\varepsilon>0$, $\\delta\\in (0,\\pi\/4)$ and an interval $[\\alpha_1, \\alpha_2] \\subset (0,\\frac{1}{2})$. Let $I_{n, \\alpha_1,\\alpha_2}$ denote the set of  $(n,\\varepsilon, \\delta)$-good $\\alpha \\in [\\alpha_1, \\alpha_2]$. Then there exists $c=c(\\varepsilon,\\delta,\\alpha_1,\\alpha_2)>0$ such that the  probability that  $I_{n, \\alpha_1,\\alpha_2} \\not= \\emptyset$ is at least $1-e^{-n^c}$ for all large enough $n.$\n\\end{theorem} \n\nOur final principal result concerning exponents asserts that, for $\\alpha\\in I_{n,\\alpha_1,\\alpha_2}$, it is highly likely that  $\\mathrm{MLR}(\\Gamma_{\\delta, \\alpha, n})\\geq n^{1\/2+\\varepsilon}$. Thus we obtain,  with high probability,  an $n^{1\/2+\\varepsilon}$ upper bound for interior maximum local roughness for a dense, though possibly $n$-dependent and random, set of~$\\alpha$. \n\n\\begin{theorem}\n\\label{t:mlrub}\nIn the setting of Theorem \\ref{t:mflub}, there exists $c>0$, such that the event that there exists $\\alpha\\in I_{n, \\alpha_1,\\alpha_2}$ such that \n$\\mathrm{MLR}(\\Gamma_{\\delta, \\alpha, n})> n^{1\/2+\\varepsilon},$ occurs with probability at most  $e^{-n^c} $ for all large enough $n$.\n\\end{theorem}\n\n\\begin{figure}[h] \n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=0.38\\textwidth]{flatfacet.pdf} &\\quad\\includegraphics[width=0.4\\textwidth]{fluccurv.pdf} \\\\\n(a) & (b)\n\\end{tabular}\n\\caption{ (a) The three different curves correspond to $\\psi_{\\alpha,n},\\Gamma_{\\alpha,n}, \\Gamma^*_{\\alpha,n}$.  The  dashed lines indicate the $\\delta-$ interior and hence the facets inside the sector bounded by them, are the $\\delta-$interior facets.\n(b)The scale at which the competition from two sources are equal to each other.}\n\\label{def}\n\\end{figure}\n\nWe now state our results about law of large numbers  for $L_{\\alpha}(n)$ and the constrained geodesic. \nAlthough the route taken here of setting up an appropriate variational problem is by now classical \\cite{DZ2}, we point out that it was not at all obvious that the solution can be explicitly described. Moreover some of the consequences of the results in the following section are used as geometric inputs for the fluctuation results stated before.  \n\\subsection{Law of Large Numbers}\nFor the unconstrained model, a straightforward subadditivity argument yields that $\\frac{\\mathbb{E} L_{n}}{n}$ converges to a limit. The evaluation of the limiting constant is classical: \\cite{VerKer77, LogShep77} showed that the limit equals $2$ by using Young tableaux combinatorics and the RSK correspondence (see also \\cite{AD95}). However, for the constrained model, the subadditive structure is lost and it is not clear a priori that a law of large numbers for $L_{\\alpha}(n)$ exists. Our first result here is to establish the law of large numbers for the area trapping polymer model; we are also able to evaluate the limiting constant implicitly as a function of $\\alpha$. \n\nGiven $\\alpha \\in (0,1\/2)$, let $c_{\\alpha}$ be given implicitly by the following equation:\n\\begin{equation}\\label{implicit1}\n\\frac{1+c_{\\alpha}}{c_{\\alpha}}\\left(1-\\frac{\\log(1+c_{\\alpha})}{c_{\\alpha}}\\right)=\\frac{1}{2}+\\alpha.\n\\end{equation}\nOne can see that the function $f(c)=\\frac{1+c}{c}[1-\\frac{\\log(1+c)}{c}]$ is strictly increasing \\footnote{$f'(c)=\\frac{(2+c)\\log(1+c)-2c}{c^3}$ and $\\frac{d}{dc}[(2+c)\\log(1+c)-2c]=\\log(1+c)-\\frac{c}{1+c} >0$.} in $c$, and converges to $1\/2$ and $1$ at $0$ and $\\infty$ respectively. \nLet ${\\mathrm{w}}_{\\alpha}= \\sqrt{1+c_{\\alpha}}\\frac{\\log(1+c_{\\alpha})}{c_{\\alpha}}$.\n\\begin{theorem} \n\\label{t:lln} \nFor any $\\alpha\\in (0,1\/2)$ \n$$\\mathbb{E} L_{\\alpha}(n)=2{\\mathrm{w}}_{\\alpha} n+o(n)$$\nas $n\\to \\infty$.\n\\end{theorem}\n\nNotice that ${\\mathrm{w}}_{\\alpha}\\to 1$ as $\\alpha\\to 0$; hence the above theorem  \nis consistent with the result in the classical unconstrained case. In the unconstrained model, one also has a law of large numbers for the geodesic, i.e., it is known that the geodesic is concentrated around the straight line joining $(0,0)$ and $(n,n)$. More precisely, under the rescaling that takes the $n\\times n$ square to a unit square, the geodesic converges almost surely in Hausdorff distance to the diagonal of the unit square \\cite{DZ2} (as mentioned before, the precise order of the transversal fluctuations are known to be $n^{2\/3}$ as a consequence of integrability). Even though we do not have any exactly solvable structure in the constrained model, we can establish a similar law of large numbers here too asserting that the constrained geodesic concentrates around a deterministic curve. Moreover, we can identify the limiting curve in a fairly explicit manner. \n\nLet $\\psi_{\\alpha}:[0,1]\\to [0,1]$ be defined as \n$\\psi_{\\alpha}(x)=\\frac{(1+c_{\\alpha})x}{1+c_{\\alpha}x}$ where $c_{\\alpha}$ is as above. Also let $\\psi_{\\alpha,n}: [0,n]\\to [0,n]$ be the $n$ blow up of $\\psi_{\\alpha}$ i.e. $\\psi_{\\alpha,n}(x)=n \\psi_{\\alpha}(x\/n)$. We denote by ${\\rm dist}(\\cdot,\\cdot)$ the Hausdorff distance. \nThe following theorem is our law of large numbers for the constrained geodesic.\n\n\\begin{theorem}\\label{lln2} For any $\\alpha \\in (0,1\/2)$ and $\\Delta>0$, there exists $c=c(\\alpha,\\Delta)$ such that, for all large enough $n$, it is  with probability at least $1-e^{-cn}$ that $${\\rm dist}(\\Gamma_{\\alpha,n},\\psi_{\\alpha,n})\\le \\Delta n.$$\n\\end{theorem}\n\nIn particular, this theorem states that to first order, the constrained geodesics behave like a given smooth curve. This result will be useful to us while studying the scaling exponents for local roughness of the constrained geodesic, in particular, when we see to  rule out long and flat facets (see Theorem \\ref{flat0}).\n\n\\subsection{Open questions and future directions}\\label{oq}\n\n\nBelow we list  below several interesting questions for future research:\n\\begin{enumerate}\n\\item The upper bound Theorem \\ref{t:mflub} is weaker than the lower bound  Theorem \\ref{t:mfllb}. Strengthening the former is a natural open problem. \n\\item By definition, $A(\\Gamma_{\\alpha,n})\\ge (\\frac{1}{2}+\\alpha)n^2$. The typical order of $A(\\Gamma_{\\alpha,n})- (\\frac{1}{2}+\\alpha)n^2$ remains unknown. \n\n\\item For $\\alpha \\in (0,1\/2)$,\nwhat is the typical deviation of $\\Gamma_{\\alpha,n}$ from the curve $\\psi_{\\alpha,n}$? What is the order of fluctuations of $L_{\\alpha}(n)$?\n\\item In a phase separation problem, \\cite{AH1,AH2,AH3} determines the polylogarithmic corrections to both elements of the $(2\/3,1\/3)$ (facet length,local roughness) exponent pair. The powers of the logarithm are $(1\/3,2\/3)$. Finding such corrections for the \\emph{Area Trapping Polymer} model would refine  the identification of the exponent pair $(3\/4,1\/2)$ made in this article and suggested by the first point.\nThe first two points will be discussed further in \nSection~\\ref{areafluc}.\n\\end{enumerate}\n\\noindent\n\\textbf{Supercritical Percolation}: We end this section with a conjecture regarding fluctuation exponents in the context of supercritical percolation on the nearest neighbor graph on $\\mathbb{Z}^2$. As already mentioned before, \\cite{BLPR} settles a conjecture of Benjamini regarding the limit shape of  isoperimetrically extremal  sets. Formally, for supercritical percolation, with the origin $\\mathbf{0}$ conditioned to be in the infinite cluster $\\mathcal{C}_{\\infty}({\\bf{0}})$, the authors in \\cite{BLPR} consider  the `anchored isoperimetric profile', i.e. for any $r>0$ they look at the  set $\\mathcal{B}_{r}$ which solves the following isoperimetric problem:\n$$\\inf \\left\\{\\frac{|\\partial B|}{|B|}: {\\bf{0}}\\in B, B\\subset \\mathcal{C}_{\\infty}({\\bf{0}}) \\text{ is connected}, |B|\\le r \\right\\}$$ where $\\partial B$ denotes the edge boundary of $B,$  restricted to $\\mathcal{C}_{\\infty}(\\bf{0})$ (for more details see \\cite{BLPR}).\nThe main result in \\cite{BLPR} shows the convergence of the set $\\mathcal{B}_r$  as $r\\to \\infty$ after suitable renormalization, to a deterministic Wulff crystal, in the Hausdorff sense.  To go beyond first order behaviour one has to understand local  geometric properties of the boundary of  $\\mathcal{B}_r$. \nThe  extremal circuit broadly has to satisfy the:\n\\begin{enumerate}\n\\item Volume condition\n\\item Extremal isoperimetry condition\n\\end{enumerate}\nThe heuristic now is that the former is a global constraint while the latter is only local. \nNamely, each local part of the boundary does not feel the volume  constraint $|\\mathcal{B}_r|\\le r,$ and thereby just tries to move through the `open' edges while trying to minimize the number of open edges that it cuts across, since these are precisely the edges that contribute to $|\\partial \\mathcal{B}_r|$.\n This brings us within the realm of first passage percolation  predicted to be in the KPZ universality class as well(see \\cite{FPPsurvey} for more on first passage percolation).\n Thus Theorems, \\ref{t:mfllb},  \\ref{t:mlrlb},  \\ref{t:mflub}  and \\ref{t:mlrub}  can be thought of as rigorous   counterparts to the above discussion in the integrable last passage percolation setting and combined with the above discussion allows us to leads us to conjecture fluctuation exponents for the above model. Below we formulate a precise statement in a slightly simpler  setting: \n \n Recall the standard definition of the dual graph of  the nearest neighbor lattice on $\\mathbb{Z}^2$. \nGiven a supercritical bond percolation environment on the edge set of $\\mathbb{Z}^2$ with density $p>p_c(\\mathbb{Z}^2)=1\/2$, for every positive integer $n,$ consider the set $\\mathcal{D}_{n^2},$ of dual  circuits (simple loop consisting of dual edges) enclosing a connected subset  of $\\mathbb{Z}^2$ of size  $n^2$ and containing the origin such that the number of primal open edges in the percolation environment that the circuit cuts through is minimized. Note that such a circuit is not necessarily unique due to the discrete nature of the problem. \nThus the above model is an exact analogue of the model considered in this paper, in the context of  first passage percolation in a bond percolation environment.\nWe now state precisely our conjecture:  \n\n\\noindent\n\\textbf{Conjecture}:\nConsider  bond percolation on the nearest neighbor lattice on $\\mathbb{Z}^2$ with any supercritical parameter value \n  $p>p_c(\\mathbb{Z}^2)$.  The random variables \n$$\\max_{D\\in \\mathcal{D}_{n^2}}\\left|\\frac{\\log(\\mathrm{MFL}(D))}{\\log n}-3\/4\\right| \\, \\, \\text{ and } \\, \\,  \\max_{D\\in \\mathcal{D}_{n^2}}\\left|\\frac{\\log(\\mathrm{MLR}({D}))}{\\log n}-1\/2\\right|$$ converge to zero  in probability   as $n$ grows to infinity\nwhere for any $D\\in \\mathcal{D}_{n^2}$, the maximum facet length $\\mathrm{MFL}(D)$ and maximum local roughness $\\mathrm{MLR}(D)$ are defined in the same way as in this paper by considering the convex hull of the points in $D$.\n\nWe end with the remark that  the problem considered in \\cite{BLPR}\ncorresponds to a similar first passage percolation problem where the environment has bounded dependence range and hence should have the same fluctuation behaviour as the simpler model\njust described. \n \n\\subsection{Organization of the rest of the article}\n In Section \\ref{s:aux2} we collect some preliminary probabilistic results: some for the constrained model and a few from the unconstrained model. The results for the unconstrained model follow from the sharp moderate deviation estimates in \\cite{LM01, LMS02} and the consequences established in \\cite{BSS14}. In Section~\\ref{s:var}, we set up and solve the variational problem for the law of large numbers and the following Section \\ref{s:lln} is devoted to the proofs of Theorem \\ref{t:lln} and Theorem \\ref{lln2}: the law of large numbers for the length of constrained geodesic and the path itself.\n We next turn to the proofs of the results on the scaling exponents. In Section \\ref{s:lb} we provide proofs of Theorems \\ref{t:mfllb} and Theorem \\ref{t:mlrlb}. The proofs of Theorem \\ref{t:mflub} and Theorem \\ref{t:mlrub} are completed in Section \\ref{s:ub}. Proofs of some of the auxiliary results stated and used throughout the article are postponed to Section \\ref{es}. \n\n\n\\subsection*{Acknowledgments}\nThe authors thank Marek Biskup, Craig Evans and Ofer Zeitouni for useful discussions. S.G.'s research is supported by a Miller Research Fellowship at UC Berkeley. A.H. is supported by NSF grant DMS-1512908. \n\n\\section{Important probability estimates}\n\\label{s:aux2}\nIn this section we gather the  probabilistic inputs needed for our proof. First we shall record a few useful facts about the length and geometry of the constrained geodesics that will be used repeatedly later. The bulk of this section will then recall results about the unconstrained model. These results are all consequences of the exactly solvable nature of the Poissonian LPP model and can be derived starting with the basic integrable ingredients of the exactly solvable model obtained by Lowe, Merkl and Rolles \\cite{LM01,LMS02}. Some of these consequences were established and used recently in \\cite{BSS14} and we quote the relevant results here.\n\nWe start with some basic facts about the constrained model.\n\n\\subsection{Some Basic Results for the Area Trapping Polymer Model}\nWe first start with concentration for $L_{\\alpha}(n)$. In absence of integrability we use the standard Poinc\\'are inequality techniques and obtain a concentration at $n^{1\/2+o(1)}$ scale. \n\n\\begin{theorem}\\label{concentration1} Fix any $\\alpha \\in (0,1\/2)$. Then there exists a constant $C>0$ such that for all $t>0$ $$\\mathbb{P}(|L_{\\alpha}(n)-\\mathbb{E}(L_{\\alpha}(n))| \\ge t) \\le Ce^{-t^{2}\/Cn \\log^{2}(n)}.$$\n\\end{theorem} \nThe proof is standard and is postponed until the end of the paper in Section \\ref{es}. \n\nOur next result  rules out flat facets.\nThis is a consequence of Theorem \\ref{lln2}, which asserts that it is extremely unlikely that any interior facets (which are those in the bulk) have  very shallow or steep gradient.\n\\begin{theorem}\\label{flat0} For any small enough $\\delta>0$ there exists $\\gamma>0$ such that,   with probability $1-e^{-cn}$, all $\\delta-$interior facets make an angle with the $x-$axis which lies in the interval $(\\omega,\\pi\/2-\\omega)$.\n\\end{theorem}\n\n \\begin{proof} \nWe start by recalling a  simple fact: \n for two facets of $\\Gamma^*_{\\alpha,n}$ with starting points $u_1,u_3$ and ending points $u_2, u_4$ respectively where  $u_1 \\preceq u_2 \\preceq u_3 \\preceq u_4,$ by convexity of $\\Gamma^*_{\\alpha,n}$, the angle made with the $x-$axis by the  facet $(u_1,u_2)$ is larger than the angle made by $(u_3,u_4)$. \n\n\nConsider any $\\delta-$interior facet $(u_1,u_2),$ and let $u_{1}=(x,y)$.\nAlso let $L_1,L_2$ be the straight lines joining the origin to $u_1$ and $(n,n)$ to $u_2$ respectively. \nLet $v_1$ and $v_2$ be the points of intersection of $L_1$ and $L_2$ with $\\psi_{\\alpha,n}$. Theorem \\ref{lln2} implies that, for any $\\varepsilon > 0$, with  probability at least $1-e^{-cn},$  the bound  \n$$\\max (|u_1-v_1|,|u_2-v_2|)\\le \\varepsilon n$$\nholds for all $\\delta$-interior facets, simultaneously. \nNow, by the convexity of $\\Gamma^{*}_{\\alpha, n}$, the gradient of the facet $(u_1,u_2)$ is between the gradient of the lines joining $(0,0)$ to $u_1$ and $(n,n)$ to $u_2$.  Choosing $\\varepsilon$ to be much smaller than $\\delta,$ we see that the gradient of the facet $(u_1,u_2)$ plus an error of $O(\\varepsilon)$ lies between the gradients of the lines joining  the origin and $v_1$ and $(n,n)$ and $v_2$ respectively. \nThus we are done by choosing $\\varepsilon$ to be small enough and using the strict convexity of $\\psi_{\\alpha,n}$; (see Figure \\ref{def}(a) for illustration). \n\\end{proof}\n\nWe will later need Theorem \\ref{flat1}, a strengthening \nof the above result, which is uniform \nin $\\alpha$.  \nWe next state useful results that concern the unconstrained, exactly solvable, model. All of these are corollaries of the following moderate deviation estimates.\n\n\\subsection{Moderate deviation estimates}\nRecall the polymer length \n$L(\\cdot,\\cdot)$ from Definition \\ref{geodesic1}.\nLet $u=(u_1,u_2)\\preceq v=(v_1,v_2)$ be such that $|u_1-v_1||u_2-v_2|=t$; i.e., $t$ is the area of the rectangle whose opposite corners are $u$ and $v$. Notice that,  by scale invariance of the Poisson point process, the distribution of $L(u,v)$ is a function merely of $t$.\n\n\\begin{theorem}[\\cite{LM01,LMS02}]\n\\label{t:moddev}\nFix $\\kappa>1$. Let $u$, $v$ as above be points such that the straight line joining $u$ and $v$ has gradient $m$, where \n$m\\in ( \\kappa^{-1} , \\kappa)$. There exist positive constants $s_0$, $t_0$, $C$ and $c$ depending only on $\\kappa$ such that, for all $t>t_0$ and $s>s_0$,\n$$\\mathbb{P}[|L(u,v)-2\\sqrt{t}| \\geq st^{1\/6}]\\leq Ce^{-cs^{3\/2}}.$$\n\\end{theorem}\n\nObserve that the above theorem implies that $|\\mathbb{E} L(u,v) - 2\\sqrt{t}|=O(t^{1\/6})$; and hence similar tail bounds are true for the quantity $|L(u,v)-\\mathbb{E} L(u,v)|$. When we make use of Theorem \\ref{t:moddev}, it will often be in order to obtain tail bounds for the quantities $|L(u,v)-\\mathbb{E} L(u,v)|$. Also the results of \\cite{BDJ99} establish that $t^{-1\/6} (L(u,v)-2\\sqrt{t})$ converges weakly to the  GUE Tracy-Widom distribution (which is defined for example in \\cite{BDJ99}). This law has negative mean, leading to the next bound.  \n\n\\begin{equation}\n\\label{e:mean}\n\\mathbb{E} L(u,v)\\leq 2\\sqrt{t}-Ct^{1\/6}\n\\end{equation}\nfor some $C>0$.\n\n\n\\subsection{Transversal fluctuations of point-to-point geodesics}\nLet $u=(u_1,u_2)\\preceq v=(v_1,v_2)$. Let $y=p_{u,v}(x)$ denote the equation of the line segment joining $u$ and $v$.\nFor any path $\\gamma$ from $u$ to~$v$, define the maximum transversal fluctuation of the path $\\gamma$ by \n$${\\rm TF}(\\gamma):= \\sup_{x\\in [u_1,v_1]} \\{\\sup |p_{u,v}(x)-y|: (x,y)\\in \\gamma\\}.$$\nFor points $u=(u_1,u_2)$ and $v=(v_1,v_2)$, we  call $|v_1-u_1|$ the $x$-coordinate distance between $u$ and $v$ and similarly define the $y$-coordinate distance. \n\nTransversal fluctuations for paths between $(0,0)$ and $(n,n)$ were shown to be $n^{2\/3+o(1)}$ with high probability in \\cite{J00}. The following more precise estimate was established in \\cite{BSS14}.\n\n\\begin{theorem}[Tails for Transversal Fluctuations, \\cite{BSS14}]\n\\label{t:tftail}\nFix $\\kappa>1$. Let $u\\preceq v$ be points such that the $x$-coordinate distance between $u$ and $v$ is $r$ and the gradient of the line joining $u$ and~$v$ is $m$, where $m\\in (\\kappa^{-1}, \\kappa)$. Then, for the geodesic $\\Gamma$ from $u$ to $v$, \n$$\\mathbb{P}[{\\rm TF}(\\Gamma) > sr^{2\/3}]\\leq Ce^{-cs}$$\nfor $\\kappa$-dependent constants  $C,c$, and $r,s$ sufficiently large. \n\\end{theorem}\n\nThe proof of Theorem \\ref{t:tftail} \nfrom \\cite{BSS14} in fact shows something more. Any path that has  transversal fluctuations much bigger than $r^{2\/3}$\nis not only unlikely to be a geodesic; it is typically much shorter than a geodesic. In particular, the proof of Theorem 11.1 in \\cite{BSS14} implies the next result (which  can also be derived more straightforwardly).\n\n\\begin{theorem}[Paths with off-scale ${\\rm TF}$ are uncompetitive]\n\\label{t:tftail1}\nFix $\\kappa>1$ and $\\varepsilon>0$. Let $u\\preceq v$ be points such that the $x$-coordinate distance between $u$ and $v$ is $r$ and the gradient of the straight line joining $u$ and $v$ is $m$ where $m\\in ( \\kappa^{-1}, \\kappa)$. \nLet $\\mathcal{E}=\\mathcal{E}_{u,v}$ denote the event that there exists a path $\\gamma$ from $u$ to $v$ with ${\\rm TF}(\\gamma) \\geq r^{2\/3+\\varepsilon}$ and $|\\gamma|> \\mathbb{E}[L(u,v)]-r^{1\/3+\\varepsilon\/10}$ (we suppress the dependence of $\\varepsilon$ in $\\mathcal{E}$ for brevity). Then there exists $c=c(\\varepsilon)>0$ such that\n$$\\mathbb{P}(\\mathcal{E}) \\leq e^{-r^{c}}$$ \nfor $r$ sufficiently large. \n\\end{theorem}\n\nThe stretched exponential nature of these estimates allow us to prove  uniform properties of the noise space which will be convenient for some of the arguments later. This point is a theme in this article: points to which we seek apply the above results may be special, so that estimates that are uniform over all points will be needed. \nFor the next result, recall that our noise space is nothing other than a rate one Poisson point process in the box  $[0,n]^2$. \nFor $u,v\\in [0,n]^2$, let $\\mathds{L}(u,v)$ denote  the line joining $u$ and $v$. Now, for any $u\\preceq v$, the pair $(u,v)$ is called $\\kappa$-steep if the gradient \nof $\\mathds{L}(u,v)$ lies in the interval $[\\kappa^{-1},\\kappa]$. Lastly,\nfor $\\tau>0$,  \n\\begin{equation}\\label{notation540}\nS(u,v)= S_{\\tau}(u,v):=\\left\\{ \\begin{array}{cc}\n|u-v|^{1\/3+\\varepsilon\/10} & \\text{ if } |u-v|>n^{\\tau},\\\\  \nn^{2\\tau\/3} & \\text{ otherwise }.\n\\end{array}\\right. \n\\end{equation}\n\n\\begin{corollary}\n\\label{uniform1}\nFix  $\\varepsilon>0$, $\\kappa>1$, and $\\tau$ sufficiently small. Then there exists $c=c(\\varepsilon,\\tau)$ such that\n$$\\mathbb{P}\\left(\\bigcup_{u,v: (u,v) \\text{is } \\kappa-\\text{steep}}\n|L(u,v)-\\mathbb{E}(L(u,v))|\\ge S(u,v)\\right) \\le e^{-n^c}.$$\n\\end{corollary}\n\\begin{proof}\nThis follows by using Theorem \\ref{t:moddev} and a standard coarse graining argument. Since we are not attempting to prove optimal bounds, we simply partition the box into squares of area one. Thus the corner points are the lattice points, i.e., elements of the set $\\{0,1,\\ldots,n\\}\\times\\{0,1,\\ldots,n\\}$. The proof now follows by first proving the desired statement for lattice points $u,v$. This just follows by Theorem \\ref{t:moddev} and a union bound over all pairs of lattice points $u,v$ (of which there are $n^2$). The proof is then completed by  showing that, since any box of area one is unlikely (with stretched exponentially small failure probability) to contain more than $n^{\\varepsilon\/100}$ points, for any $u$ and $v$, one can approximate $L(u,v)$ by $L(u_*,v_*)$, where $u_*,v_*$ are the nearest lattice points.\n\\end{proof}\n\nWe now specify a modification of the event $\\mathcal{E}_{u,v}$ from Theorem \\ref{t:tftail1}.  Let $\\mathcal{E}^{*}_{u,v}$ denote the event $\\mathcal{E}_{u,v}$ where the permitted error term $r^{1\/3+\\varepsilon\/10}$ is instead taken to be the  quantity~$S(u,v)$ defined a few moments ago. The next corollary is proved along the same lines as Theorem~\\ref{t:tftail1}, in combination with the arguments in the proof of Corollary \\ref{uniform1},\nand we omit the proof. \n\\begin{corollary}\n\\label{uniform0}\nFix  $\\varepsilon>0$, $\\kappa>1$, and $\\tau$ small enough. Then there exists $c=c(\\varepsilon,\\tau)$ such that\n$$\\mathbb{P}\\left(\\bigcup_{u,v:|(u,v) \\text{is } \\kappa-\\text{steep}}\\mathcal{E}^*_{u,v}\\right) \\le e^{-n^c}.$$\n\\end{corollary}\n\n\\subsection{Paths between points in an on scale parallelogram}\nNext we control the deviation of lengths between pairs of points within certain parallelograms. We first need some definitions. Let  $U(r,m,h)$ denote a parallelogram with the  properties that:\n\\begin{enumerate}\n\\item[(a)] One pair of parallel sides are vertical.\n\\item[(b)] The other pair of parallel sides have gradient $m$.\n\\item[(c)] The (horizontal) distance between the vertical sides is $r$.\n\\item[(d)] The height of the vertical sides is $h$.\n\\end{enumerate}\nNote that this defines a unique parallelogram up to translation. This will be enough for our purposes because the underlying noise field is translation invariant. \n\n\\begin{theorem}[\\cite{BSS14}]\n\\label{t:par1}\nLet $m\\in (\\kappa^{-1},\\kappa)$ and $h=W r^{2\/3}$ for some $W>0$. Then there exist $(\\kappa,W)$-dependent constants $C,c>0$,  such that, for all sufficiently large $r$ and $s$, \n$$ \\mathbb{P}\\left( \\sup_{u\\in A, v\\in B} (L(u,v)- \\mathbb{E} L(u,v)) \\geq  sr^{1\/3} \\right) \\leq Ce^{-cs},$$\nand\n$$ \\mathbb{P}\\left( \\inf_{u\\in A, v\\in B} (L(u,v)- \\mathbb{E} L(u,v)) \\leq  -sr^{1\/3} \\right) \\leq Ce^{-cs},$$\nwhere $A$ and $B$ respectively denote the right third and the left third of $U(r,m,h)$.\n\\end{theorem}\n\nThe next theorem states that, even if the paths are restricted to stay inside the parallelogram, the fluctuation remains on scale. \nLet $L(u,v;U)$ denote the length of the longest path from $u$ to $v$ that does not exit $U$. \n\n\\begin{theorem}[\\cite{BSS14}]\n\\label{t:par2}\nUnder the assumptions of Theorem \\ref{t:par1}, there exist $(\\kappa,W)$-dependent constants $C,c>0$  such that, for all sufficiently large $r$ and $s$, \n$$ \\mathbb{P}\\left( \\inf_{u\\in A, v\\in B} (L(u,v;U)- \\mathbb{E} L(u,v)) \\leq  -sr^{1\/3} \\right) \\leq Ce^{-cs^{1\/2}}$$\nwhere $A$ and $B$ respectively denote the right third and the left third of $U(r,m,h)$.\n\\end{theorem}\n\n\\subsection{Paths constrained in a thin parallelogram}\nThe next set of results shows that, if a path is constrained to have much smaller than typical transversal fluctuation, then it must have much smaller length than a typical geodesic's.\n\\begin{theorem}[Paths with small ${\\rm TF}$ are uncompetitive]\n\\label{t:constrained}\nFix $\\varepsilon>0$. Consider the parallelogram $U=U(r,m,h)$, where $m\\in (\\kappa^{-1},\\kappa)$ and $h=r^{2\/3-\\varepsilon}$. Let $u_0$ and $v_0$ denote the midpoints of the vertical sides $A_0$ and $B_0$ of $U$. Then, for some $c=c(\\varepsilon)>0$,\n$$\\mathbb{P}\\left(L(u_0,v_0; U) \\geq \\mathbb{E} L(u_0,v_0)- r^{1\/3+\\varepsilon\/3}\\right) \\leq e^{-r^{c}}$$\nfor $r$ sufficiently large.\n\\end{theorem}\n\nThe proof of Theorem \\ref{t:constrained} follows a strategy, by now well known, that has been used to show Gaussian fluctuation of paths constrained to stay in a thin rectangle in \\cite{CD13} in the context of first passage percolation. In the context of LPP, the Gaussian fluctuation has recently been shown in \\cite{DPM17}, and tail bounds are proved in a more general context in the preprint \\cite{BH17}. We now  provide a sketch of Theorem~\\ref{t:constrained}'s proof. \n\n\\begin{proof}[Sketch of Proof]\nWithout loss of generality let us assume $m=1$. Also, we can replace the parallelogram $U$ by the rectangle $U',$ one of whose pairs of sides is parallel to the line segment $u_0v_0$, with the other pair having midpoints $u_0$ and $v_0$ \nand  length $r^{2\/3-\\varepsilon}$. Divide the rectangle $U'$ into $K$ equal parts (each of width $\\frac{r}{K}$) by parallel line segments perpendicular to $u_0v_0$. Let $L_{i}$ denote the left side of the $i$-th rectangle and let $u_{i}$ be its midpoint. Now let $\\gamma_{i}$ be the best \npath (i.e. with maximal value of $|\\gamma|$) between $L_{i}$ and $L_{i+1}$ that stays within $U'$. It follows from Theorem \\ref{t:par1} that it is extremely likely that $|\\gamma_{i}|-L(u_i,u_{i+1})\\ll (K^{-1}r)^{1\/3}$. It follows that up to an error of order much smaller than $K^{2\/3}r^{1\/3}$ we can approximate $L(u_0,v_0; U')$ by $\\sum_{i} L(u_i,u_{i+1})$. Now observe that $L(u_i,u_{i+1})$ are independent and identically distributed with mean $2r\/K-C(K^{-1}r)^{1\/3}$ (here we use~\\eqref{e:mean}) and standard deviation of the order of $ (K^{-1}r)^{1\/3}$. By standard concentration inequalities, one then shows that $\\sum_{i} L(u_i,u_{i+1})$ concentrates around $2r-K^{2\/3}r^{1\/3}$ at scale $K^{1\/6}r^{1\/3}$. By choosing $K\\gg r^{\\varepsilon\/2}$ properly one gets the result. \n\\end{proof}\n\nIn the same vein, we have the following corollary.\n\\begin{corollary}\n\\label{uniform} \nFix  $\\varepsilon>0$, $\\kappa>1$, and $\\tau$ small enough. Then there exists $c=c(\\varepsilon,\\tau)$ such that, with probability at least $1-e^{-n^c}$,\n$$\n\\displaystyle{\\sup_{u,v:|u-v|\\ge n^{\\tau}, (u,v) \\text{is } \\kappa-\\text{steep}} L(u,v,U)-\\mathbb{E}(L(u,v)) \\le  -|u-v|^{\\frac{1}{3}+\\varepsilon}}.\n$$\n\\end{corollary} \n\n\n\\subsection{Estimates for One-Sided Geodesics}\nWe end this section by considering geodesics in a different constrained model. Baik and Rains \\cite{br01,br} considered increasing\npaths from $(0,0)$ to $(n,n)$ that lie above the diagonal line joining the points. Recall that $L(u,v)$ denotes the length of the longest increasing path between points $u \\preceq v$. Let $L^{\\boxslash}(u,v)$ denote the length of the  longest path between $u$ and $v$ restricted to lie \nabove the line joining $u$ and $v$ and, similarly to Definition \\ref{geodesic1}, let $\\gamma^{\\boxslash}(u,v)$ denote the corresponding uppermost one-sided geodesic between $u$ and $v$. Moreover, let $L^{\\boxslash}(n)$ denote $L^{\\boxslash}(u,v)$ in the special case that $u=(0,0)$ and $v=(n,n)$. Baik and Rains \\cite{br01,br} proved that $\\mathbb{E} L^{\\boxslash}(n)=2n+o(n)$ and that the fluctuations are again of order $n^{1\/3}$ (although in this case the scaling limit is different; it is the GSE Tracy Widom distribution instead of the GUE Tracy Widom distribution). We shall need the corresponding moderate deviation estimates, consequences of  Theorem \\ref{t:par2}. \n\n\\begin{theorem}\n\\label{baikrains} \nLet $u,v$ be as in the hypothesis of Theorem \\ref{t:moddev}. There exist positive constants $s_0, t_0, C$ and $c$ such that, for all $t>t_0$ and $s>s_0$,\n$$\\mathbb{P}[|L^{\\boxslash}(u,v)-2\\sqrt{t}| \\geq st^{1\/6}]\\leq Ce^{-cs^{1\/2}}.$$\n\\end{theorem}\n\n\\begin{proof}\nThe upper tail result follows from Theorem \\ref{t:moddev} due to $L^{\\boxslash}(u,v)\\leq L(u,v)$. For the lower tail, we use Theorem \\ref{t:par2} with the straight line joining $u$ and $v$ being the bottom side of the parallelogram and use the fact that $\\mathbb{E} L(u,v)=2\\sqrt{t}- \\Theta(t^{1\/6})$.\n\\end{proof}\n\nWe need a uniform version of this result. Recall $S(u,v)=S_{\\tau}(u,v)$ from \\eqref{notation540}.\n\\begin{corollary}\n\\label{uniform2}\nFix  $\\varepsilon>0$, $\\kappa>1$, and $\\tau > 0$ small enough. Then there exists $c=c(\\varepsilon,\\tau)$ such that, with probability at least $1-e^{-n^c}$,\n$$\n\\displaystyle{\\sup_{u,v:(u,v) \\text{is } \\eta-\\text{steep}} L^{\\boxslash}(u,v)-\\mathbb{E}(L(u,v)) \\ge  -S(u,v)} .\n$$\n\\end{corollary}\n\nProofs of Corollaries \\ref{uniform0} \\ref{uniform}, \\ref{uniform2} follow from the corresponding theorems for fixed points, in the same way as  Corollary \\ref{uniform1} follows from Theorem \\ref{t:moddev}.  We omit the details. \n\n\\section{A Variational Approach to the Constrained Geodesic}\n\\label{s:var}\nWe now move towards proving Theorem \\ref{t:lln} and Theorem \\ref{lln2}.\nDeuschel and Zeitouni \\cite{DZ1,DZ2} studied in detail the hydrodynamic limit of the unconstrained geodesic for inhomogeneous point processes. We follow their strategy broadly and study the limit of the constrained curve by means of appropriate variational problems. \nWe first  explain the idea. Fix $\\alpha\\in (0,\\frac{1}{2})$ for the rest of this section. First let us assume that a limiting continuous curve $\\phi$ of the constrained geodesics exists after scaling. This curve $\\phi:[0,1] \\to [0,1]$ will be  continuous, non-decreasing and surjective.  By the area constraint, $\\int_0^1\\phi(s)ds \\ge \\tfrac{1}{2}+\\alpha$. Now heuristically, since in practice there is always a little bit of area excess,  one can approximate $\\phi$ by a piecewise affine function at a scale local enough that each piece is exempt from the area constraint; and we can then use the law of large numbers for the unconstrained geodesic (see Theorem \\ref{t:moddev}) at every local scale and  sum over them. This argument suggests that the approximating path will have length $2J({\\phi})n +o(n)$ where  \n$$J({\\phi})=\\int_{0}^1\\sqrt {\\dot{\\phi}(s)}ds.$$\nIt thus seems that we should maximize $J({\\phi})$ among all curves $\\phi$ satisfying the area constraint, and the maximum will be the constant appearing in the required law of large numbers. We now proceed to make this precise. \nLet $\\mathcal{B}$ be the collection of all right-continuous non-decreasing functions from $[0,1]$ to $[0,1]$. \nThus $\\mathcal{B}$ is in bijection with the set of all sub-probability measures on $[0,1]$. Now, for any $\\phi\\in \\mathcal{B},$ by the Lebesgue decomposition theorem we can write   \n \\begin{equation}\\label{decomposition23}\n \\phi=\\phi_{ac}+\\phi_{s}\n \\end{equation}\n  in a unique way as a sum of a pair of sub-probability measures, with $\\phi_{ac}$ being the absolutely continuous part and $\\phi_s$ the  singular part (with respect to Lebesgue measure). \nNote that this implies $\\phi_{ac}$ has a derivative \n$\\dot \\phi_{ac}$ almost everywhere \\footnote{Throughout the article, for any function $f$ on $[0,1],$ which is differentiable almost everywhere, we will denote its derivative by $\\dot f$. Recall, that this does not necessarily imply $\\int_{0}^x \\dot f(s)ds=f(x)$ for all $x$.}  \nsuch that, for any $0\\le x\\le 1$, we have $$\\int_{0}^x \\dot \\phi_{ac}(s)ds =\\phi_{ac}(x)$$ while $\\phi_s$ is almost surely flat and hence has derivative $0$ almost surely.  Thus $\\dot \\phi=\\dot \\phi_{ac}$ almost surely.\nAlso define \n\\begin{equation}\\label{variational1}\nJ(\\phi)= \\int_{0}^1\\sqrt {\\dot{\\phi}(s)}{\\rm d}s .\n\\end{equation} \nLet \n\\begin{equation}\\label{restr304}\n\\mathcal{B}_{\\alpha}=\\left\\{\\phi\\in \\mathcal{B}: \\int_0^1\\phi(s)ds \\ge \\tfrac{1}{2}+\\alpha\\right\\}.\n\\end{equation}\nWe shall often omit the subscript $\\alpha$. Finally let \n\\begin{equation}\\label{functional}\nJ_{\\alpha}=\\sup_{\\phi \\in \\mathcal{B}_{\\alpha}} J(\\phi).\n\\end{equation}\n\nThe first step is to show existence and uniqueness of the solution. \n\n\\begin{proposition}\n\\label{opt1} \nThere exists a unique element $\\psi=\\psi_{\\alpha} \\in \\mathcal{B}$ that attains the supremum  in \\eqref{functional}.\n\\end{proposition}\n\nNote that the function $\\psi=\\psi_{\\alpha}$ in Proposition \\ref{opt1} will be the same in the statement of Theorem~\\ref{lln2}. The existence and uniqueness parts of Proposition \\ref{opt1} have separate proofs. We first state the existence result.\n\n\\begin{lemma}\n\\label{l:exist}\nThere exists $\\psi\\in \\mathcal{B}$ that achieves the supremum  in \\eqref{functional}.\n\\end{lemma}\n\nThe proof of Lemma \\ref{l:exist} is technical: it uses a compactness argument on the space of probability measures following a similar argument from \\cite{DZ2}. We postpone the proof until Section \\ref{es}. \n\nIn the next part we show uniqueness. \n\n\\begin{lemma}\n\\label{l:uni}\nSuppose $\\psi_1$ and $\\psi_2$ are in $\\mathcal{B}_{\\alpha}$ and satisfy  $J(\\psi_i)= J_{\\alpha}$ for $i=1,2$. Then $\\psi_1=\\psi_2$. \n\\end{lemma}\n\nThe proof of Lemma \\ref{l:uni} is rather straightforward once we establish the following technical lemma that rules out the possibility that any of the optimizing functions have a nontrivial singular part. \n\n\\begin{lemma}\n\\label{l:ac}\n Let $\\psi \\in \\mathcal{B}_{\\alpha}$ be such that $J(\\psi)= J_{\\alpha}$. Then $\\psi$ corresponds to a probability measure which is absolutely continuous with respect to Lebesgue measure.\n\\end{lemma}\n\nHere is the basic idea of the proof of this lemma. From the results of \\cite{DZ2} it follows that without the area constraint the optimizing curve is the diagonal line, i.e., the preferred gradient for the graph of the function is one. If the singular part is non-trivial,  then the graph of the function $\\psi$ may be expected to have a flat piece. Because the gradient of the graph has to be one on average due to boundary conditions, it follows that there must be a piece of the graph having gradient away from one. We shall show that one can modify the flat and the steep parts of the curve locally by pieces with more moderate gradient in a way that increases not only the value of the functional $J$; and  thereby optimality will be contradicted. Also, clearly in the case that $\\psi$ does not have total mass one, one can add to it another function to make it (or, more precisely of course, to bring it  into correspondence with)  a probability measure, while also increasing the value of $J(\\psi),$ and thus contradict maximality in this case also. Thus $\\psi$ is also a probability measure. The details of the proof are postponed to Section \\ref{es}.\n\n\n\nWe can now prove the uniqueness result using Lemma \\ref{l:ac}. \n\n\\begin{proof}[Proof of Lemma \\ref{l:uni}]\nWe use the fact that the square-root  function is strictly concave. Given any $\\psi_1$ and $\\psi_2$ as in the statement of the lemma, we consider $\\psi=\\frac{\\psi_1+\\psi_2}{2}$.  Clearly, $\\psi$ satisfies the area constraint and $\\dot{(\\frac{\\psi_1+\\psi_2}{2})}=\\frac{\\dot {\\psi_1}+\\dot{\\psi_2}}{2}$. Thus using Jensen's inequality $J(\\psi)$ is necessarily larger than $J(\\psi_1)=J(\\psi_2)$ unless $\\dot \\psi_1=\\dot \\psi _2$ almost surely. Since by the previous result the absolutely continuous parts contain mass one, $\\psi_1(0)=\\psi_2(0)=0$. Hence for all $0\\le x\\le1 $ \n$$\\psi_1(x)=\\int_0^{x}\\dot \\psi_1(s)ds=\\int_0^{x}\\dot \\psi_2(s)ds=\\psi_2(x).$$ Thus we are done. \n\\end{proof}\n\nIt still remains to identify the optimizer $\\psi$ in \\eqref{functional}; in particular we need to show this is the same $\\psi$ as defined in Theorem \\ref{lln2}. To this end, we shall now record some properties of the unique optimizer $\\psi$ that will be useful later. We show that the derivative is almost surely positive and decreasing. The proofs are easy and will be postponed until Section \\ref{es}.\n\\begin{lemma}\\label{decreasing1} \nLet $\\psi=\\psi_{\\alpha}$ denote the unique optimizer in Proposition \\ref{opt1}. Then\n\\begin{enumerate}\n\\item[(i)] The derivative $\\dot \\psi$ is almost surely decreasing i.e. there is a set of full measure on which it is decreasing. \n\\item[(ii)] $\\dot \\psi$ is almost surely positive.\n\\end{enumerate}\n\\end{lemma}\n\n\n\\subsection{Identifying the curve $\\psi_{\\alpha}$}\nIn this subsection we determine the curve $\\psi_{\\alpha}$ and show that it is the same as the curve in Theorem \\ref{lln2}. We first start with the following proposition.\n\n\\begin{proposition}\n\\label{solution1} \nGiven $\\alpha \\in (0, 1\/2)$ there exists $c=c_\\alpha>0,$ such that\n$\\psi_\\alpha(x)=\\frac{(1+c)x}{1+cx}$.\n\\end{proposition}\n\n\\begin{proof}The proof proceeds by showing that, given any $0<a<b<1$, $$\\frac{1}{\\sqrt{\\dot \\psi(x)}}=c_1x+c_2 \\text{ a.s. on } [a,b].$$  \nfor some constants $c_1$ and $c_2$. Fix $0<a<b<1$ and let $h_*$ be a polynomial such that \n\\begin{align}\n\\label{int20}\n\\int_a^b h_*(x){\\rm{d}}x  &=0\\\\\n\\label{int21}\n\\int_a^b x h_*(x){\\rm{d}}x &=0\n\\end{align}\nDefine $h_*(x)$ on $[0,1]$ by defining it to be $0$ outside $[a,b]$. \nLet $g=\\dot \\psi+\\varepsilon h_*$. Now since $\\dot \\psi$ is monotonically decreasing and positive almost surely (see Lemma \\ref{decreasing1}), $\\dot \\psi$ is bounded above and below by constants on $[a,b]$ and hence for all $\\varepsilon$ (with sufficiently small absolute\nvalue depending on $[a,b]$, we will need $\\varepsilon$ to take both positive and negative values in the argument later), we conclude  $g$ is strictly positive. Moreover by \\eqref{int20}, \n$\\int_0^1 g(x){\\rm{d}}x =1$. \nWe now verify the area constraint. It is easy to see by Fubini's theorem (see \\eqref{fubini} in the proof of Lemma \\ref{decreasing1}) that $\\int_0^1 (1-x)\\dot \\psi(x)\\ge 1\/2+ \\alpha$. Thus $$\\int_0^1 (1-x)g(x){\\rm{d}}x  =\\int_0^1 (1-x)\\dot \\psi(x)+\\varepsilon \\int_0^1 (1-x)h_*(x){\\rm{d}}x \\ge \\frac{1}{2}+\\alpha$$\nby \\eqref{int20} and \\eqref{int21}. Now, by Taylor expansion, for all small enough $\\varepsilon$ (depending on $a$, $b$, $h$ and $\\psi$),\n\\begin{equation}\\label{variational20}\n\\int_0^1 \\left[\\sqrt{g(x)}-\\sqrt{\\dot \\psi(x)} \\right]{\\rm{d}}x  = \\int_{a}^b \\left(\\sqrt{\\dot \\psi(x)}+\\varepsilon \\frac{h_*(x)}{2\\sqrt{\\dot \\psi(x)}}-  \\varepsilon^2 O\\biggl(\\frac{h_*(x)^2}{{\\dot \\psi(x)}^{3\/2}}\\biggr)\\right){\\rm{d}}x  , \n\\end{equation}\nwhere the constants in the $O$- notation just depend on $[a,b],\\psi$ (since $\\dot \\psi$ is in $[c_3,c_4]$ almost surely for some $c_3<c_4$ on $[a,b]$ by the preceding discussion). Since $J(\\psi)= J_{\\alpha}$ (and $\\varepsilon$ can be both positive and negative\nit follows that, \n\\begin{equation}\\label{ortho}\n\\int_{a}^b\\frac{h_*(x)}{\\sqrt{\\dot \\psi(x)}}{\\rm{d}}x =0 .\n\\end{equation}\nSince $\\frac{1}{\\sqrt{\\dot \\psi}}$ is bounded above and below on $[a,b]$, one can find a linear function $L(x):=a_1x +b_1$ such that \n\\begin{align*}\n\\int_{a}^{b} \\left[\\frac{1}{\\sqrt{\\dot \\psi(x)}}-L(x)\\right]{\\rm{d}}x =0 \\,\\,\\text{and }\\,\\,  \\int_{a}^b \\left[\\frac{1}{\\sqrt{\\dot \\psi(x)}}-L(x)\\right]x{\\rm{d}}x =0. \n\\end{align*}\nIndeed, such $L$ can be found by solving the two linear equations in $a_1$ and $b_1$ given by the above two equations. To see that, for any $a<b$, the equations are non-degenerate, first observe that one can by a change of variable assume that $a=0$ and $b=1$. Then the matrix corresponding to the linear equations is nothing but $\\left[\\begin{array}{cc}\n1\/2 &1\\\\\n1\/3 &1\/2\n\\end{array}\n\\right]$  and hence admits a unique solution. Together with \\eqref{ortho} this now implies that \n\\begin{align*}\\int_{a}^b \\left[\\frac{1}{\\sqrt{\\dot \\psi(x)}}-L(x)\\right]P(x){\\rm{d}}x =0\n\\end{align*}\nfor all polynomials $P(x)$. For this, observe that by Gram-Schmidt orthogonalization any polynomial $P$ can be decomposed as  $P(x)=L'(x)+h_*(x)$ where $L'$ is a linear function and $h_*$ is a polynomial satisfying \\eqref{int20} and \\eqref{int21}. Since polynomials are dense in $\\mathbb{L}_2[a,b]$ by the Stone-Weierstrass theorem, this implies \n$$\\frac{1}{\\sqrt{\\dot \\psi(x)}}-L(x)=0~\\text{a.e. on}~[a,b].$$ \n\n\nSince  $\\frac{1}{\\sqrt{\\dot \\psi}}$ is a linear function on all intervals $[a,b]$, it follows  that it has to be the same linear function on all such intervals and hence $\\frac{1}{\\sqrt{\\dot \\psi(x)}}=L(x)$ for some linear function $L(x)$ almost everywhere on the entire interval $[0,1]$.\nThus $\\dot \\psi(x)=\\frac{d}{(1+cx)^2}$ for some constants $c,d$. Integrating and using that $\\psi(0)=0$ and $\\psi(1)=1$ (see Lemma \\ref{l:ac}) we are done. \n\\end{proof}\n\nThe next proposition completes the identification of $\\psi_{\\alpha}$ by showing that the constant $c_{\\alpha}$ is given by \\eqref{implicit1}.\n\n\\begin{proposition}\n\\label{p:calpha}\nThe constant $c_{\\alpha}$ is given implicitly by the following equation:\n$$\\frac{1+c_{\\alpha}}{c_{\\alpha}}\\left[1-\\frac{\\log(1+c_{\\alpha})}{c_{\\alpha}}\\right]=\\frac{1}{2}+\\alpha$$\nwhere $c_{\\alpha}$ is the constant in Theorem \\ref{solution1} corresponding to $\\alpha$.\n\\end{proposition}\n\n\n\\begin{proof}\nClearly we have\n\\begin{align*}\n\\int_{0}^1 \\psi(x){\\rm{d}}x =\\int_{0}^1 \\frac{(1+c)x}{1+cx}=\\frac{1+c}{c}\\left[1-\\frac{\\log(1+c)}{c}\\right]=\\frac{1}{2}+\\beta\n\\end{align*}\nfor some $\\beta=\\beta(\\alpha)\\geq \\alpha$. We shall show that $\\beta=\\alpha$ which will establish the proposition. Suppose not. As we have pointed out,  the left-hand side of this equation is increasing in $c$. Let $c_{*}$ be the unique solution to the equation \n\\begin{align*}\n\\int_{0}^1 \\psi(x){\\rm{d}}x =\\int_{0}^1 \\frac{(1+c)x}{1+cx}=\\frac{1+c}{c}\\left[1-\\frac{\\log(1+c)}{c}\\right]=\\frac{1}{2}+\\alpha.\n\\end{align*}\nIt is easy to check that $\\beta> \\alpha$ implies $c_{*}<c_{\\alpha}$. Define $\\psi^{*}$ by \n$\\psi^*(x)=\\frac{(1+c_*)x}{1+c_*x}$.\nClearly $\\psi_{*}\\in \\mathcal{B}_{\\alpha}$. A straightforward computation shows that \n\\begin{align}\n\\label{weight}\nJ(\\psi^*)=\\int_{0}^1 \\sqrt{\\dot \\psi(x)}{\\rm{d}}x = \\sqrt{(1+c_*)}\\int_{0}^1\\frac{1}{1+c_*x} = \\sqrt{1+c_*}\\frac{\\log(1+c_*)}{c_*}.\n\\end{align}\nThe right-hand side here is readily seen to be decreasing in $c_*$ \nand hence $J(\\psi^*)>J(\\psi)$. This contradicts the optimality of $\\psi$, and  completes the proof. \n\\end{proof}\nObserve that, $J_{\\alpha}=\\sqrt{1+c_{\\alpha}}\\frac{\\log(1+c_{\\alpha})}{c_{\\alpha}}$.\nThis expression is the same as the constant $\\mathrm{w}_{\\alpha}$ in Theorem \\ref{t:lln} which is consistent with our heuristic explanation at the beginning of this section. We prove this theorem next. \n\n\\section{Law of large numbers}\n\\label{s:lln}\nWith the preparation from the previous section we now turn to the proof of Theorem \\ref{t:lln}. Fix $\\alpha\\in (0,\\frac{1}{2})$ as before. The idea of the proof is as follows. We first show that if a path follows approximately the blow up $\\psi_{\\alpha, n}$ of the deterministic curve $\\psi$ (and also satisfies the area constraint) then it is overwhelmingly likely that the path has length at least $2(\\mathrm{w}_{\\alpha}-\\varepsilon)n$ for $\\varepsilon$ arbitrarily small. We further show that any path satisfying the area constraint is extremely likely to have length less than $2(\\mathrm{w}_{\\alpha}+\\varepsilon)n$. We begin with the following trivial but useful lemma that is an immediate consequence of continuity and monotonicity of ${\\rm{w}}_{\\alpha}$.\n\n\n\\begin{lemma} Given any $\\alpha \\in (0,\\frac{1}{2}),$ for all small enough $\\varepsilon>0,$ there exist $\\delta_1,\\delta_2 > 0$ such that $|{\\rm{w}}_{\\alpha}-{\\rm{w}}_{\\alpha_1}| < \\varepsilon$ if $|\\alpha-\\alpha_1|\\le \\delta_1 $ and $|{\\rm{w}}_{\\alpha}-{\\rm{w}}_{\\alpha_1}| > \\varepsilon$ if $|\\alpha-\\alpha_1| \\ge \\delta_2$.\n\\end{lemma}\nNext we need a couple of preparatory lemmas about appropriate discretisations. Let $I_{\\delta}=\\{0,\\delta,2\\delta,\\ldots,1\\}$ be the discretisation of the unit interval; (we will take $\\delta$ to be the reciprocal of a positive integer to avoid rounding errors). Consider any non-decreasing function $L: [0,1] \\to [0,1]$ that corresponds to an absolutely continuous measure on $[0,1]$: by identifying $L$ with its graph, we shall interpret $L$ as an increasing path on the unit square directed from $(0,0)$ to $(1,1)$. \nAlso let $L_{\\delta}$ be  the piecewise affine function that agrees on $I_\\delta$ with $L$.\n\n\\begin{lemma} \n\\label{l:approx3}\nWe have $\\left | \\int_{0}^1[L(x)-L_{\\delta}(x)]{\\rm{d}}x \\right | \\le 2\\delta$.\n\\end{lemma}\n\n\\begin{proof}First notice that by integration by parts or by Fubini's theorem, for any absolutely continuous function $f$ on $[0,1],$  $\\int_0^1 f(x){\\rm{d}}x =\\int_0^1 \\dot f(x) (1-x){\\rm{d}}x $. The lemma now follows from the observation that $\\left |\\int_{0}^1 x \\dot L(x){\\rm{d}}x-\\int_{0}^1 x \\dot L_{\\delta}(x){\\rm{d}}x\\right|\\le   2\\delta$.\n{Let $x_{i}$ be the midpoint of the interval $I_\\delta^{(i)}=[i\\delta,(i+1)\\delta]$. \nNotice that by definition $\\dot L_{\\delta}$ is constant on $I_\\delta^{(i)}$ and is equal to $\\frac{1}{\\delta}\\int_{I_\\delta^{(i)}}\\dot L(x){\\rm{d}}x $. \nThus \n$$\n\\sum_{i=0}^{\\frac{1}{\\delta}-1}\\int_{I_{\\delta}^{(i)}} x_i \\dot L(x) {\\rm{d}}x=\\sum_{i=0}^{\\frac{1}{\\delta}-1}x_i \\int_{I_{\\delta}^{(i)}}  \\dot L(x){\\rm{d}}x=\\sum_{i=0}^{\\frac{1}{\\delta}-1}x_i \\int_{I_{\\delta}^{(i)}}  \\dot L^{\\delta}(x){\\rm{d}}x \n=\\sum_{i=0}^{\\frac{1}{\\delta}-1}\\int_{I_{\\delta}^{(i)}} x \\dot L^{\\delta}(x)=\\int_0^1 x \\dot L^{\\delta}(x)\n$$\nThe proof now follows by observing $|\\int_{0}^1 x \\dot L(x){\\rm{d}}x -\\int_{0}^1 x \\dot L^{\\delta}(x){\\rm{d}}x | \\le \\sum_{i=0}^{\\frac{1}{\\delta}-1}\\int_{I_{\\delta}^{(i)}} |x - x_i| \\dot L(x){\\rm{d}}x\\le \\delta$.\n}\n\\end{proof}\nWe now discretise the vertical direction as well. \nLet us choose $\\eta\\ll \\delta$ (to be specified exactly later),  and  let $B_{\\delta,\\eta}= I_{\\delta} \\times I_{\\eta}$. Given $L$ as before let us now set, \n$L_{\\delta,\\eta}$ to be the piecewise linear curve determined by the points  $(i\\delta, \\eta \\lfloor \\frac{L(i\\delta)}{\\eta} \\rfloor)$.\nClearly for all $x,$ $|\\dot L_{\\delta}-\\dot L_{\\delta,\\eta}| \\le \\frac{2\\eta}{\\delta},$ and hence for $\\eta\\le \\delta^2,$\n\\begin{equation}\\label{area14}\n\\bigg\\vert \\,  \\int_{0}^1[L(x)-L_{\\delta,\\eta}(x)]{\\rm{d}}x \\, \\bigg\\vert \\le 4\\delta. \\end{equation}\n\n\n\nBefore proceeding further we make a few comments about notation. Recall that till now,  our underlying noise space has been a point process $\\Pi$ on $\\mathbb{R}^2$ and we have been concerned about geodesics in $[0,n]^2$. However in the last section while solving variational problems we switched to a normalized picture where every path lives in $[0,1]^2$. We will continue with this convention throughout this section. Equivalently the noise space can be thought of as a Poisson point process of intensity $n^{2}$ in $[0,1]^2$. In an abuse of notation,  we will define $L(u,v)$ for two points $u,v \\in [0,1]^2$ to be the length of the geodesic $\\gamma(u,v)$ between $u$ and $v$ in the same noise space. In light of  properties of Poisson process under scaling, \n{Moreover, all the estimates regarding $\\gamma(u,v)$ stated in the previous sections continue to hold after the appropriate variable change. We omit further elaboration.}  \nNext, we state a uniform version of  Theorem \\ref{t:moddev} in the large deviation regime. \n \n \n \n\\begin{lemma}\n\\label{uni100} Fix $\\delta,\\eta$ as above.\nThere exists $c=c(\\delta,\\eta)$ such that, simultaneously for all $x\\in I_{\\delta}$ and $y_2> y_1 \\in I_{\\eta}$ for all large $n$:\n\\begin{enumerate}\n\\item With probability at least $1-e^{-cn},$ $$L((x,y_1),((x+\\delta),y_2))- 2 n \\sqrt{\\delta(y_2-y_1)}\\le \\eta \\delta n,$$ \n\\item With probability at least $1-e^{-cn^2},$ $$L((x,y_1),((x+\\delta),y_2))- 2 n \\sqrt{\\delta(y_2-y_1)}\\ge - \\eta \\delta n.$$  \n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof} The proof follows from the large deviation probabilities for the  length of unconstrained geodesics in \\cite{sepLDP}.\nIn particular the theorem states that for any $x,y_1,y_2$ as in the statement of the theorem,  \n\\begin{align*}\n\\mathbb{P}(L((x,y_1),((x+\\delta),y_2))- 2 n \\sqrt{\\delta(y_2-y_1)}\\ge \\eta \\delta n)&\\le e^{-cn},\\\\\n\\mathbb{P}(L((x,y_1),((x+\\delta),y_2))- 2 n \\sqrt{\\delta(y_2-y_1)} -\\eta \\delta n)&\\le e^{-cn^2},\n\\end{align*}\nfor some constant $c=c(\\delta,\\eta)$.\nA union bound over points $x,y_1,y_2$, the total number of which is $\\frac{1}{\\delta \\eta^2}$, completes the proof.\n\\end{proof}\n\nWe are now ready to prove Theorem \\ref{t:lln}. We start by showing the upper bound. Recall that $L_{\\alpha}(n)$ denotes the length of the constrained geodesic.\n\n\\begin{proposition}\n\\label{p:llnub}\nFix $\\varepsilon>0$. There exists $c=c(\\alpha,\\varepsilon)>0$ such that with probability at least $1-e^{-cn}$ we have \n$$L_{\\alpha}(n) \\leq 2(\\mathrm{w}_{\\alpha}+\\varepsilon)n.$$ \n\\end{proposition}\n\n\\begin{proof}\nRecall that for any increasing path $\\gamma,$ we denote its length or the number of points of the Poisson process it passes through by $|\\gamma|$.\nLet us now notice that $$|\\gamma|\\le \\sum_{i\\in I_{\\delta}}L\\biggl(\\bigl(i\\delta, \\gamma_{\\delta,\\eta}(i\\delta)\\bigr), \\bigl((i+1)\\delta, \\gamma_{\\delta,\\eta}((i+1)\\delta)+\\eta\\bigr)\\biggr).$$ \nThus by the previous lemma, with probability at least  $1-e^{-cn},$ we have for all increasing paths, $\\gamma$ (from $(0,0)$ to $(1,1)$ in the rescaled space)\n\\begin{align*}\n|\\gamma|&\\le \\sum_{i\\in I_{\\delta}} 2\\delta \\sqrt{\\frac{\\gamma_{\\delta,\\eta}((i+1)\\delta)+\\eta)-\\gamma_{\\delta,\\eta}(i\\delta)}{\\delta}}n +2\\eta n, \\\\\n&= \\sum_{i\\in I_{\\delta}} 2n \\delta \\sqrt{\\dot \\gamma_{\\delta,\\eta}+\\frac{\\eta}{\\delta}} +2\\eta n.\n\\end{align*}\nAn argument involving truncating at $\\dot \\gamma_{\\delta,\\eta}< \\sqrt{\\frac{\\eta}{\\delta}}$ shows that, with probability at least  $1-e^{-cn}$, for all increasing paths $\\gamma,$\n$$\\sum_{i\\in I_{\\delta}} n\\delta \\sqrt{\\dot \\gamma_{\\delta,\\eta}+\\frac{\\eta}{\\delta}}\\le n\\sum_{i} \\delta \\sqrt{\\dot \\gamma_{\\delta,\\eta}}  +{\\left(\\frac{\\eta}{\\delta}\\right)}^{1\/4}O(n).\n$$\n\nThus, for $\\eta < \\delta^5$, with probability at least $1-e^{-cn},$ for all $\\gamma,$ \n\\begin{equation}\\label{approx34}\n|\\gamma|\\le n \\int_{0}^1 2\\sqrt{\\dot \\gamma_{\\delta,\\eta}}(x){\\rm{d}}x +O(\\delta n).\n\\end{equation} \n\nNow let $\\gamma$ be an \nincreasing path that traps area at least $(1\/2+\\alpha)$ (which corresponds to area $(1\/2+\\alpha)n^2$ in the unscaled model).  By \\eqref{area14},  it follows that $\\int_0^1 \\gamma_{\\delta,\\eta}(x){\\rm{d}}x \\ge \\frac{1}{2}+\\alpha- O(\\delta)$ and hence, by continuity of $\\mathrm{w}_{\\alpha}$, we have $\\int_{0}^1 \\sqrt{\\dot \\gamma_{\\delta,\\eta}}(x){\\rm{d}}x\\le ({\\rm w}_{\\alpha}+\\varepsilon\/2)$ by choosing $\\delta$ sufficiently small. By choosing $\\delta$ suitably small, this implies that, with probability at least $1-e^{-cn}$, the bound $|\\gamma|\\le 2( w_{\\alpha}+\\varepsilon)n$ for such increasing paths $\\gamma$. This completes the proof of the proposition.\n\\end{proof}\n\nWe now show the corresponding lower bound. \n\n\\begin{proposition}\n\\label{p:llnlb}\nFix $\\varepsilon>0$. There exists $c=c(\\alpha,\\varepsilon)>0$ such that, with probability at least $1-e^{-cn^2}$, we have \n$$L_{\\alpha}(n) \\geq 2(\\mathrm{w}_{\\alpha}-\\varepsilon)n.$$ \n\\end{proposition}\n\n \\begin{proof}\nThe derivation of this lower bound is straightforward:  we will take a discretisation of $\\psi_{\\alpha}$ and then use the continuity of $\\dot \\psi$.\nFix $\\varepsilon>0$. Choose $\\theta>0$ and consider $\\psi_*=\\psi_{\\alpha+\\theta}$. Then, by definition, \n$\n\\int_{0}^1 \\psi_*(x){\\rm{d}}x\\ge \\frac{1}{2}+\\alpha+\\theta .\n$\nChoose $\\theta$ small enough that \n$\n{\\rm w}_{\\alpha+\\theta}=\\int_{0}^1 \\sqrt{\\dot \\psi_*(x)}{\\rm{d}}x\\ge {\\rm w}_{\\alpha}-\\frac{\\varepsilon}{2}.\n$\n\nBy using the continuity of $\\psi$ and $\\dot \\psi$, we see that, for given $\\varepsilon$, there exist sufficiently small choices of  $\\delta,\\eta,$ such that the piecewise affine function $\\psi_{*,\\delta, \\eta}$ interpolating between the points $u_i:=(i\\delta,\\eta\\lfloor\\frac{\\psi_*(i\\delta)}{\\eta}\\rfloor)$\nsatisfies \n \\begin{align*}\n \\int_{0}^1 \\psi_{*,\\delta,\\eta}(x){\\rm{d}}x\\ge \\frac{1}{2}+\\alpha+\\frac{\\theta}{2},&\n \\int_{0}^1 \\sqrt{\\dot \\psi_{*,\\delta,\\eta}}(x){\\rm{d}}x\\ge {\\rm w}_\\alpha-\\frac{3\\varepsilon}{4}.\n \\end{align*}\nNow let $\\gamma_{(i,i+1)}$ \nbe the geodesic between the points $u_i$ and $u_{i+1}$.  Let $\\gamma$ be the path obtained by concatenating the paths $\\gamma_{(i,i+1)}$ for $i=0,\\ldots,\\frac{1}{\\delta}-1$ \nUsing Lemma \\ref{uni100}, it follows that, with probability at least $1-e^{-cn^2}$, \n$$\n|\\gamma|\\ge \\sum_{i\\in I_{\\delta}} 2n\\delta \\sqrt{\\dot \\psi_{*,\\delta,\\eta}} -2n\\eta \\geq 2(\\mathrm{w}_{\\alpha}-\\varepsilon)n ,\n$$\nwhere we take $\\eta$ and $\\delta$ sufficiently small. Now  by\nLemma \\ref{l:approx3}, we find that \n$$\\left |\\int_{0}^{1} \\gamma(x){\\rm{d}}x - \\int_{0}^1 \\psi_{*,\\delta,\\eta}(x){\\rm{d}}x \\right|\\le 2\\delta,$$ since $\\gamma$ and $\\psi_{*,\\delta,\\eta}$ agree  on $I_{\\delta}$.\nHence, for $\\delta$ small enough,\n$\n\\int_{0}^1 \\gamma(x){\\rm{d}}x \\ge \\frac{1}{2}+\\alpha\n$.\nThus, $L_{\\alpha,n}$ is at least as large as $|\\gamma|$. Hence, we are done.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{t:lln}]\nCombining Proposition \\ref{p:llnub} and Proposition \\ref{p:llnlb}, we complete the proof of Theorem \\ref{t:lln} by noting that an upper tail bound of the form $\\mathbb{P}(L_{\\alpha}(n)\\geq k)\\leq e^{-k}$ for all $k\\geq 5n^2$ is easily obtained by bounding the upper tail of the Poisson-distributed  total number of points. \n\\end{proof}\n\n\\subsection{Law of large numbers for the geodesic}\n\nWe now prove  Theorem \\ref{lln2}. The proof is by contradiction:  if ${\\rm dist}(\\Gamma_{\\alpha},\\psi_{\\alpha})\\ge \\Delta$, and the noise space is such that the events listed in Lemma \\ref{uni100}\noccur, then we would be able to construct an absolutely continuous function $h$ that traps area at least  $1\/2+\\alpha$ and for which   $\\int_0^1 \\sqrt{\\dot h}(x){\\rm{d}}x>\\int_0^1\\sqrt{\\dot \\psi_{\\alpha}}(x){\\rm{d}}x $. This would contradict extremality  of $\\psi_{\\alpha}$.\nTo show the above inequality, we will employ a concavity argument. \nFor notational brevity, let $\\gamma=\\Gamma_{\\alpha}$ and $\\psi=\\psi_{\\alpha}$.  For $\\delta,\\eta>0$, recall the definition of $\\gamma_{\\delta,\\eta}$ from the previous section. \nWe will consider the function $h=\\frac{\\gamma_{\\delta,\\eta}+\\psi}{2}$.\nFixing parameters $\\varepsilon,\\delta$ and $\\eta\\le \\delta^5,$\nwe will restrict attention to the event $\\mathcal{A}$,\non which:\n\\begin{enumerate}\n\\item $|\\gamma|\\le 2n \\int_{0}^1 \\sqrt{\\dot \\gamma_{\\delta,\\eta}}(x){\\rm{d}}x +O(\\delta n),$ \\\\\n\\item $2({\\rm w}_\\alpha -\\varepsilon)n \\le |\\gamma|\\le 2({\\rm w}_\\alpha +\\varepsilon)n$.\n\\end{enumerate}\nBy Propositions \\ref{p:llnub} and \\ref{p:llnlb}, and \\eqref{approx34}, $\\mathcal{A}$ occurs with probability at least $1-e^{-cn}$. \nObserve that Proposition \\ref{solution1} implies that $\\dot \\psi$ is bounded away from zero and infinity on $[0,1]$.  Thus, $\\dot \\psi$ lies in  $[c_1,C_2]$ where $0<c_1<C_2 < \\infty$.\nSuppose that $\\displaystyle{\\sup_x|\\gamma(x)-\\psi(x)|\\ge \\Delta}$.  We now claim that  if $\\delta,\\eta$ are small enough compared to $\\Delta$,  then  there exists $y\\in[0,1]$ such that    \n\\begin{equation}\\label{conseq203}\n|\\gamma_{\\delta,\\eta}(y)-\\psi(y)|\\ge \\frac{\\Delta}{2}.\n\\end{equation}\nTo see this, we choose $y\\in[0,1]$ such that $|\\gamma(y)-\\psi(y)|\\ge \\Delta$. Let $y_1\\in I_{\\delta}$  be such that $y_1\\le y< y_1+\\delta$.\nNow we consider two cases: \n\\begin{enumerate}\n\\item \nIf $\\gamma(y)\\ge \\psi(y)+\\Delta$, then\n \\begin{align*}\n\\gamma_{\\delta,\\eta}(y_1+\\delta) \\ge \\gamma(y)-\\eta\\ge \\psi(y)+\\Delta-\\eta &\\ge \\psi(y_1+\\delta)+\\Delta-O(\\delta), \\text{ and hence,}\\\\\n\\gamma_{\\delta,\\eta}(y_1+\\delta)& \\ge  \\psi(y_1+\\delta)+\\Delta-O(\\delta).\n\\end{align*}\n These implications, excepting the last, follow by definition. The last implication is due to  $\\psi$ having Lipschitz constant at most $C_2$.\n\\item If $\\psi(y)\\ge \\gamma(y)+\\Delta$, then, similarly,\n \\begin{align*}\n \\psi(y_1)\\ge \\psi(y)-O(\\delta)&\\ge \\gamma(y)+\\Delta-O(\\delta)\\ge \\gamma_{\\delta,\\eta}(y_1)+\\Delta-O(\\delta) \\text{ and hence,}\\\\\n  \\psi(y_1)&\\ge \\gamma_{\\delta,\\eta}(y_1)+\\Delta-O(\\delta).\n\\end{align*}\n \\end{enumerate}\nThus, by \\eqref{conseq203}, \n$$\\int_0^1|\\dot \\gamma_{\\delta,\\eta}(x)-\\dot \\psi(x)|{\\rm{d}}x\\ge  \\left|\\int_{0}^y[\\dot \\gamma_{\\delta,\\eta}(x)-\\dot \\psi(x)]{\\rm{d}}x \\right| \\ge \\frac{\\Delta}{2}.$$\nAlso, note that, since both $\\psi$ and $\\gamma_{\\delta,\\eta}$ start at $(0,0)$ and end at $(1,1)$,  \n$\\int_0^1\\dot \\gamma_{\\delta,\\eta}(x){\\rm{d}}x =\\int_0^1\\dot \\psi(x){\\rm{d}}x$.\nHence, \n\\begin{equation}\\label{algebra102}\n\\int_0^1 [\\dot \\psi(x)- \\dot \\gamma_{\\delta,\\eta}(x)]\\mathbf{1}(S){\\rm{d}}x \\ge \\Delta\/4,\n\\end{equation}\nwhere $S$ denotes $\\{x\\in [0,1]: \\dot \\psi(x)\\ge \\dot \\gamma_{\\delta,\\eta}\\}$.   Moreover, on  $S$, \n\\begin{equation}\\label{bound303}\n\\dot \\gamma_{\\delta,\\eta}\\le \\dot \\psi \\le C_2\n\\end{equation} \nfor some $C_2 >1$.\nObserving that $\\dot h=\\frac{\\dot \\gamma_{\\delta,\\eta}+{\\dot \\psi}}{2}$, \n we see that the  concavity of  $x\\to \\sqrt x$ implies that $\\sqrt{\\dot h}-\\frac{\\sqrt{\\dot \\gamma_{\\delta,\\eta}}+\\sqrt{\\dot \\psi}}{2},$ is  non-negative.  \nFurthermore, simple algebra shows that, for all $x\\in S$,\n\\begin{align}\\label{algebra101}\n\\sqrt{\\dot h}(x)-\\frac{[\\sqrt{\\dot \\psi}(x)+\\sqrt{\\dot \\gamma_{\\delta,\\eta}}(x)]}{2} &= \\frac{(\\sqrt{\\dot \\psi}(x)-\\sqrt{\\dot \\gamma_{\\delta,\\eta}}(x))^2}{4[\\sqrt{\\dot h}(x)+\\frac{[\\sqrt{\\dot \\psi}(x)+\\sqrt{\\dot \\gamma_{\\delta,\\eta}}(x)]}{2}]}\\stackrel{\\eqref{bound303}}{\\ge}  \\frac{(\\sqrt{\\dot \\psi}(x)-\\sqrt{\\dot \\gamma_{\\delta,\\eta}}(x))^2}{8 C_2},\\\\\n\\nonumber\n&\\ge \\frac{({\\dot \\psi}(x)-{\\dot \\gamma_{\\delta,\\eta}}(x))^2}{8C_2[\\sqrt {\\dot \\psi}(x)+\\sqrt{\\dot \\gamma_{\\delta,\\eta}}(x)]^2}\\stackrel{\\eqref{bound303}}{\\ge} \\frac{({\\dot \\psi}(x)-{\\dot \\gamma_{\\delta,\\eta}}(x))^2}{100 C_2^2}.\n\\end{align} \n\n\nThe proof is now completed by observing that a length gain has been realized:\\begin{align*}\n\\int_0^1 \\left[\\sqrt{\\dot h}(x)-\\frac{[\\sqrt{\\dot \\psi}(x)+\\sqrt{\\dot \\gamma_{\\delta,\\eta}}(x)]}{2}\\right]{\\rm{d}}x& \\ge \\int_0^1 \\left[\\sqrt{\\dot h}(x)-\\frac{[\\sqrt{\\dot \\psi}(x)+\\sqrt{\\dot \\gamma_{\\delta,\\eta}}(x)]}{2}\\right] \\mathbf{1}(S){\\rm{d}}x \\\\\n&\\stackrel{\\eqref{algebra101}}{\\ge} \\int_0^1 \\frac{({\\dot \\psi}(x)-{\\dot \\gamma_{\\delta,\\eta}}(x))^2}{100 C_2^2}\\mathbf{1}(S){\\rm{d}}x \\\\\n&\\ge  \\frac{1}{100C_2^2} \\left(\\int |({\\dot \\psi}(x)-{\\dot \\gamma_{\\delta,\\eta}}(x)|\\mathbf{1}(S){\\rm{d}}x \\right)^2\\\\\n &\\stackrel{\\eqref{algebra102}}{\\ge} \\frac{1}{2000C_2^2} \\Delta^2. \n\\end{align*}\nOn the event $\\mathcal{A},$ we see, by means of this inequality and the definition of  $\\mathcal{A}$, that if  $\\displaystyle{\\sup_x|\\gamma(x)-\\psi(x)|\\ge \\Delta} $ holds, \nthen $$\\int_0^1\\sqrt{\\dot h}(x){\\rm{d}}x \\ge [({\\rm w}_\\alpha-\\varepsilon)+\\frac{1}{C^2} \\Delta^2]$$ for some constant $C$ that  depends only on $\\alpha$.\nAlso, both  $\\int_{0}^1\\psi(x){\\rm{d}}x $ and $\\int_{0}^1\\gamma(x){\\rm{d}}x $ are at least $\\frac{1}{2}+\\alpha$ by definition,  and hence  by \\eqref{area14}, we have \n$$\n\\int_0^1 h(x){\\rm{d}}x =\\frac{\\int_0^1 \\psi(x){\\rm{d}}x +\\int_0^1 \\gamma_{\\delta,\\eta}(x){\\rm{d}}x }{2}\\ge \\frac{1}{2}+\\alpha-\\varepsilon\/2,\n$$ where $\\varepsilon$ can be made arbitrarily small by choosing $\\delta$ and hence $\\eta$ small enough. This inference contradicts the continuity of ${\\rm w}_{\\alpha}$ in $\\alpha$.\nHence we are done.\n\\qed\n\n\nWe now use similar arguments as those employed to prove the law of large numbers in order to establish a variant of Theorem \\ref{flat0} that is also uniform in $\\alpha$. This particular variant will be crucial in the proof of Theorem \\ref{t:mflub}.\n\n\n\\begin{theorem}\n\\label{flat1} \nFix $0< \\alpha_1<\\alpha_2 <\\frac{1}{2}$. For any small enough $\\delta>0$, there exist $\\gamma>0$ and $c>0$ such that, with probability at least $1-e^{-cn}$, all $\\delta$-interior facets of $\\Gamma_{\\alpha,n}$ for which $\\alpha\\in[\\alpha_1,\\alpha_2]$ make an angle with the $x$-axis that lies $(\\omega,\\pi\/2-\\omega)$.\n\\end{theorem}\n\n\\begin{proof} \nRecall that the proof of  Theorem \\ref{flat0}  \nused Theorem \\ref{lln2} and the strict convexity of the function $\\psi_\\alpha$. Since $\\psi_{\\alpha}$ is uniformly convex for all $\\alpha\\in [\\alpha_1,\\alpha_2]$, this proof will  be complete, using the same arguments as in the proof of Theorem \\ref{flat0}, once we prove the following uniform version of Theorem \\ref{lln2}: for any $\\Delta>0$, with probability at least $1-e^{-cn}$,  $$\\sup_{\\alpha\\in[\\alpha_1,\\alpha_2]}\\sup_x|\\Gamma_{\\alpha}(x)-\\psi_{\\alpha}(x)|\\le \\Delta,$$ where $c$ depends on $\\Delta$ and the interval $[\\alpha_1,\\alpha_2]$.\nWe fix an $\\varepsilon$ to be specified later and discretise the interval $[\\alpha_1,\\alpha_2]$ to obtain the set $\\mathcal{B}=\\{\\alpha_1, \\alpha_1+\\varepsilon, \\alpha_1+2\\varepsilon,\\ldots,\\alpha_2\\}$.\nFixing parameters $\\varepsilon,\\delta$ and $\\eta\\le \\delta^5$,\nagain as before we restrict attention to the event $\\mathcal{A}$ on which, for all $\\alpha \\in \\mathcal{B}$,\n\\begin{enumerate}\n\\item $|\\Gamma_{\\alpha}|\\le 2 n \\int_{0}^1 \\sqrt{\\dot{ \\Gamma}_{\\alpha,\\delta,\\eta}}(x){\\rm{d}}x +O(\\delta n)$; \n\\item and $2({\\rm w}_\\alpha -\\varepsilon)n \\le |\\Gamma_{\\alpha}|\\le 2({\\rm w}_\\alpha +\\varepsilon)n$.\n\\end{enumerate}\nSimilarly to a previous argument, by Propositions \\ref{p:llnub} and \\ref{p:llnlb} and \\eqref{approx34}, followed by a union bound, $P(\\mathcal{A})\\ge 1-e^{-cn}$. \nNow if $\\delta$ and $\\eta$ are chosen to be sufficiently small depending on $\\Delta,$  by the previous result, Theorem \\ref{lln2}, and  a simple union bound,  we obtain $$\\sup_{\\alpha\\in \\mathcal{B}}\\sup_x|\\Gamma_{\\alpha}(x)-\\psi_{\\alpha}(x)|\\le \\Delta.$$\nThe proof will proceed along the same lines as the proof of Theorem \\ref{lln2} did.  With the aim of  arriving at a contradiction, let $\\alpha\\in [\\alpha_1+i\\varepsilon,\\alpha_1+(i+1)\\varepsilon]$ be such that \n\\begin{equation}\\label{violate}\n\\sup_x|\\Gamma_{\\alpha}(x)-\\psi_{\\alpha}(x)|> \\Delta.\n\\end{equation}\nClearly, $|\\Gamma_{\\alpha_1+(i+1)\\varepsilon}|\\le |\\Gamma_{\\alpha}|\\le |\\Gamma_{\\alpha_1+i\\varepsilon}|$ holds by definition.\nSince  $\\psi_{\\beta}$ is a continuous function of  $\\beta$ in the supremum norm, we see that, for small enough $\\varepsilon$, $$\\displaystyle{\\sup_x|\\Gamma_{\\alpha}(x)-\\psi_{\\alpha_1+i\\varepsilon}(x)|\\ge \\Delta\/2}.$$\nNow,  as we argued in the proof of  Theorem \\ref{lln2}, this implies that \n\\begin{align}\\label{dev100}\n\\sup_x|\\Gamma_{\\alpha,\\delta,\\eta}(x)-\\psi_{\\alpha_1+i\\varepsilon}(x)|\\ge \\Delta\/2.\n\\end{align}\nNote then that\n$$\n|\\Gamma_{\\alpha}|\\ge |\\Gamma_{\\alpha_1+(i+1)\\varepsilon}|\\ge  2n({\\rm{w}}_{\\alpha_1+(i+1)\\varepsilon}-\\varepsilon) \\ge 2n({\\rm{w}}_{\\alpha}-\\varepsilon_1),\n$$ \nwhere where $\\varepsilon_1$ can be made small enough by choosing $\\varepsilon$ small enough. The first inequality follows by definition, and the second by the occurrence of $\\mathcal{A}$.\n\nThus, using \\eqref{approx34}, we find that  \n$$\\int_{0}^1 \\sqrt{\\dot \\Gamma_{\\alpha,\\delta,\\eta}}(x){\\rm{d}}x\\ge {\\rm{w}}_{\\alpha}-\\varepsilon_1.$$ \nThis along with \\eqref{dev100} allows us to  apply the concavity argument that appears in the proof of Theorem \\ref{lln2}:  by choosing $\\varepsilon,\\delta,\\eta$ much smaller than $\\Delta$, we thus contradict the continuity of ${\\rm w }_{\\beta}$ at $\\beta=\\alpha$.\n\\end{proof}\n\n\n\\section{Lower Bound for Scaling Exponents}\n\\label{s:lb}\nIn this section we provide proofs of the lower bounds on ${\\rm MFL}(\\Gamma_n)$ and ${\\rm MLR}(\\Gamma_n)$, i.e., we prove Theorem \\ref{t:mfllb} and Theorem \\ref{t:mlrlb}. Most of the work goes into proving the ${\\rm MFL}$ lower bound Theorem~\\ref{t:mfllb}, since the lower bound for local roughness is a reasonably easy corollary of Theorem \\ref{t:tftail}. Let $\\alpha\\in (0,\\frac{1}{2})$ and $\\varepsilon>0$ be fixed for the rest of the section. Let $\\mathcal{A}_{\\varepsilon}$ denote the event that ${\\rm MFL}(\\Gamma_n)\\leq n^{3\/4-\\varepsilon}$. We shall show that the event $\\mathcal{A}_{\\varepsilon}$ is extremely unlikely. We start with an overview of the proof. We shall need a geometric definition. \n\n\n\\begin{definition}\n\\label{d:reg}\nLet $C>1,\\kappa>1$ be given constants. A sequence of points $u_0\\preceq u_1 \\preceq \\cdots \\preceq u_k$ is called a $(C,\\kappa)$-\\textbf{regular} sequence if the following conditions hold.\n\\begin{enumerate}\n\\item[(i)] The union of line segments joining $u_{i}$ to $u_{i+1}$ for $i=0,1,\\ldots,k-1,$ is convex.\n\\item[(ii)] The gradient of the all the line segments joining $u_i$ to $u_{i+1}$ is $\\in (\\frac{1}{\\kappa}, \\kappa)$.\n\\item[(iii)]The distance between the first and last point in the sequence i.e., $|u_k-u_0|\\in (\\frac{1}{C}n^{3\/4-\\varepsilon\/2}, Cn^{3\/4-\\varepsilon\/2})$.\n\\item[(iv)] The distance between the consecutive points in the sequence is small: $|u_i-u_{i+1}|\\leq n^{3\/4-\\varepsilon}$.\n\\item[(v)] Let $\\theta_1$ and $\\theta_2$ be the angles that the line segments $(u_0,u_1)$ and $(u_{k-1},u_{k})$ make with the positive $x$-axis (clearly $\\theta_1\\geq \\theta_2$ by hypothesis). Then $\\theta_1-\\theta_2 \\leq 100C^2 n^{-1\/4-\\varepsilon\/2}$.\n\\end{enumerate}\n\\end{definition}\n\nSee Figure \\ref{f:corner} for an illustration of this definition. \n\n\\begin{figure}[h] \n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=0.4\\textwidth]{regcorner.pdf} &\\includegraphics[width=0.3\\textwidth]{corner2.pdf} \\\\\n(a) & (b)\n\\end{tabular}\n\\caption{ (a) A sequence of regular corners $u_0,u_1,\\ldots, u_k$. The triangle $T$ is an isosceles triangle formed by vertices $u_0, u_k$ and $u_*$ such that the adjacent sides lie above the piecewise linear path passing through the points $u_0,u_1,\\ldots, u_k$.  By definition of regularity, the triangle $T$ is contained in the parallelogram $R$ formed by the vertices $u_0, v_0,v_k$ and $u_k$ which has vertical height $n^{1\/2-\\varepsilon\/2}$. Observe that the height of this parallelogram is much smaller than the transversal fluctuation of paths between $u_0$ and $u_k$. \n(b) On $\\mathcal{A}_{\\varepsilon}$, there is a regular sequence of corners $u_0, u_1,\\ldots, u_{k}$ and $\\Gamma$ is the best constrained path. We show that there is an alternative path $\\gamma$ between $u_0$ and $u_1$ above the triangle $T$ that is extremely likely to be strictly larger in length than the restriction of $\\Gamma$ between $u_0$ and $u_{k}$. Then the path obtained by replacing $\\Gamma$ with $\\gamma$ between $u_0$ and $u_k$ has a larger length and traps a larger area, thus leading to a contradiction.\n}\n\\label{f:corner}\n\\end{figure}\n\n\nBefore proceeding let us try to motivate the item (v) of the above definition. Consider the least concave majorant of an increasing path from $(0,0)$ to $(n,n)$ as described in Definition \\ref{d:mlrmfl}  \nThe total change of angle made by the  line segments constituting the concave majorant with the $x$-axis is roughly around $\\pi\/2$ over a length of around $n$. So over a distance of around $n^{3\/4-\\varepsilon\/2}$, the change of angle should be around $n^{1\/4-\\varepsilon\/2}$ on average. The item (v) of the above definition asserts that on a regular sequence the change of angle is not much more than this average. \n\nThe following basic geometric consequence will be useful.\n\n\\begin{lemma}\n\\label{l:reggeo}\nLet $S=\\{u_0,u_1,\\ldots, u_k\\}$ be a $(C,\\kappa)$-regular sequence for some $C,\\kappa>1$. Let $L_{S}$ denote the union of line segments joining the consecutive points of $S$. Let $R$ denote the parallelogram with two vertical sides of length \n$n^{1\/2-\\varepsilon\/2}$ whose bottom side is the line segment joining $u_0$ and $u_k$; see Figure \\ref{f:corner}(a). Then there exists a point $u_*$ in $R$ such that the triangle $T$ formed by the points $u_0,u_k, u_*$ is an isosceles triangle (with the sides adjacent to $u_*$ being of equal length) contained in $R$ such that the two sides of $T$ adjacent to $u_*$ lie above the piecewise affine curve obtained by joining the consecutive points of $S$. \n\\end{lemma} \n\\begin{proof}\nConsider the angles $\\omega_1$ and $\\omega_2$ made by the line segments $(u_0,u_1)$ and $(u_{k-1},u_k)$ respectively with the line segment $(u_0,u_k)$.  Also, let $v$ be the point of intersection obtained by extending the line segments  $(u_0,u_1)$ and $(u_{k-1},u_k)$. By convexity, the piecewise affine curve obtained by joining the points of $S$ lies inside the triangle $(u_0,u_k,v)$. Moreover, it follows from elementary geometric arguments that $\\theta_1-\\theta_2=\\omega_1+\\omega_2$. Now let us consider the isosceles triangle $(u_0,u_k,u_*)$ where $(u_0,u_k)$ forms the base and $\\theta_1-\\theta_2$ is the value of the two equal angles. Since $\\theta_1-\\theta_2$ is at least as big as $\\omega_1$ and $\\omega_2$, it follows that the triangle $(u_0,u_k,v)$ is contained inside the triangle $(u_0,u_k,u_*)$.  Moreover, the latter is clearly contained in a parallelogram of height $O(n^{3\/4-\\varepsilon\/2}n^{-1\/4-\\varepsilon\/2})$ where the constant in the $O(\\cdot)$ notation depends on $\\kappa$. Since for all large enough $n$,  $n^{3\/4-\\varepsilon\/2}n^{-1\/4-\\varepsilon\/2}\\ll n^{1\/2-\\varepsilon\/2}$, we are done. \n\\end{proof}\n\nLet us now record the main steps of the proof of Theorem \\ref{t:mfllb}. Let $\\Gamma$ denote the constrained geodesic. Recall that $\\Gamma^{*}_{\\alpha,n}$ denotes least concave majorant of $\\Gamma$ and is an union of the facets as described in Definition \\ref{d:mlrmfl}.  \n\n\\textbf{Step 1:} We shall fix a $(C,\\kappa)$-regular sequence $S=\\{u_0,u_1,\\ldots, u_k\\}$, and consider the best path~$\\gamma_{S}$ from $u_0$ to $u_{k}$ that passes through all the points in the sequence. We shall show that with very high probability the length of this path is much smaller than the length of the best path between $u_0$ and $u_{k}$. The reason that this event is likely is the following: because the lengths of the segments $\\{u_{i}, u_{i+1}\\}$ are much smaller than that of the segment $\\{u_{0}, u_{k}\\}$, it will turn out that the path $\\gamma$ will have a much smaller transversal fluctuation compared to the typical geodesic between $u_0$ and $u_{k}$. Thus, by Theorem~\\ref{t:constrained},  it is extremely likely that this path has a much smaller length. \n\n\\textbf{Step 2:} Next we will show that there exists a path between $u_0$ and $u_{k}$ that lies above the line segments joining the consecutive points of $S$ and whose length is comparable to that of the geodesic between $u_0$ and $u_{k}$. This will follow from the definition of regular sequence together with an application of Lemma \\ref{l:reggeo} and Theorem \\ref{baikrains}.  \n\n\\textbf{Step 3:} Finally, we will show that, on the event $\\mathcal{A}_{\\varepsilon}$, it is overwhelmingly likely that there exists a $(C,\\kappa)$-regular sequence made out of consecutive corners of $\\Gamma^{*}_{\\alpha,n}$. Once we establish this, we will arrive at a contradiction: using the first two steps, we will construct a path that traps more area than does $\\Gamma$ and that also has a greater length.\n\n\nThe next proposition treats the first step.\n\n\\begin{proposition}\n\\label{p:step1}\nLet $S=\\{u_0,u_1,\\ldots, u_k\\}$ be a $(C,\\kappa)$-regular sequence. For $i=0,1,\\ldots,k-1,$ let $\\gamma_{i}$ be the geodesic\nbetween $u_{i}$ and $u_{i+1}$. Let $\\gamma_{S}$ denote the concatenation of the paths $\\gamma_{i}$. Then there exists $c>0$ such that, with probability at least $1-e^{-n^{c}}$, we have \n$$|\\gamma_{S}|\\leq \\mathbb{E} L(u_0,u_k)-n^{1\/4-\\varepsilon\/12}.$$\n\\end{proposition}\n\n\\begin{proof}\nLet $R'$ be the parallelogram with sides parallel to the sides of $R$ such that $u_0$ and $u_{k}$ are the midpoints of the vertical sides of $R'$; and the height of the vertical sides is $4n^{1\/2-\\varepsilon\/2}$.  For each $i$, let $R_i$ denote the parallelogram with the following properties:\n\\begin{enumerate}\n\\item $R_i$ has one pair of vertical sides and the other pair of sides is parallel to the line segment joining $u_{i}$ and $u_{i+1}$. \n\\item The vertical sides have midpoints $u_{i}$ and $u_{i+1}$. \n\\item The height of the vertical sides is $n^{1\/2-3\\varepsilon\/5}$. \n\\end{enumerate} \nIt is straightforward to check from Definition \\ref{d:reg} that all the rectangles $R_{i}$ are contained in the rectangle $R'$. Now it follows from Theorem \\ref{t:tftail1} that, with probability at least $1-e^{-n^{c}}$, each geodesic $\\gamma_i$ is contained in the parallelogram $R_i$. In particular, on this event, \n$$ |\\gamma_{S}|\\leq L(u_0,u_{k}; R').$$\nThe result now follows using Theorem \\ref{t:constrained} and the assumed lower bound on $|u_0-u_{k}|$.\n\\end{proof}\n\n\n\n\\begin{figure}[h] \n\\centering\n\\begin{tabular}{ccc}\n\\includegraphics[width=0.4\\textwidth]{corner3.pdf} &\\includegraphics[width=0.4\\textwidth]{mlr.pdf} \\\\\n(a) & (b) \n\\end{tabular}\n\n\\caption{ (a) Illustrates the situation when all the facets are small. Using a priori bounds on the transversal fluctuations of the unconstrained geodesic, it follows that the segment of $\\Gamma_{\\alpha,n}$ between $u_0$ and $u_k$ lies in a thin parallelogram with width much smaller than the characteristic fluctuation scale. This then implies that the length of this segment is atypically low (see Theorem \\ref{t:constrained}).\n(b) Illustrates how in this situation one can replace the segment of $\\Gamma_{\\alpha,n}$ between $u_0$ and $u_k$ by a path that lies strictly above the parallelogram and fluctuates on a characteristic scale, and hence captures more area along with more length, which contradicts extremality of $\\Gamma_{\\alpha,n}$. This is used in the proof of Theorem \\ref{t:mlrub} } \n\\label{f:alternatepath20}\n\\end{figure}\nNext we move onto Step 2 where we show the existence of a ``good\" path between $u_0$ and $u_{k}$. We have the following proposition. \n\n\\begin{proposition}\n\\label{p:step2}\nLet $S=\\{u_0,u_1,\\ldots, u_k\\}$ be a $(C,\\kappa)$-regular sequence. Let $\\mathbb{L}_{S}$ denote the union of line segments joining $u_{i}$ to $u_{i+1}$.Then there exists $c>0$ such that, with probability at least $1-e^{-n^{c}}$, there exists a path $\\gamma_{*}$\nfrom $u_0$ to $u_{k}$ lying above $\\mathbb{L}_{S}$ such that\n$$|\\gamma_{*}| > \\mathbb{E} L(u_0,u_k)-n^{1\/4-\\varepsilon\/12}.$$\n\\end{proposition} \n\n\\begin{proof}\nWe will use the notation from Lemma \\ref{l:reggeo}. Let $u_*$ be the point in $R$ that satisfies the conclusion of that lemma.\nLet $\\gamma_{1}$ denote a path from $u_0$ to $u_{*}$ lying above the line segment joining $u_0$ and $u_{*}$ that achieves the length $L^{\\boxslash}(u_0,u_{*})$. Similarly, let $\\gamma_{2}$ denote a path from $u_{*}$ to $u_{1}$ lying above the line segment joining $u_*$ and $u_{1}$ that achieves the length $L^{\\boxslash}(u_*,u_{1})$. See Figure \\ref{f:corner} (b).\n\n\nLet $\\gamma_{*}$ denote the concatenation of the paths $\\gamma_1$ and $\\gamma_2$. Clearly $\\gamma_{*}$ lies above $\\mathbb{L}_{S}$. We shall show that, with overwhelming probability, $\\gamma_*$ also satisfies the other condition in the statement of the proposition. It follows by definition and some elementary geometry that both of the equal sides of the isosceles triangle $T$ have gradient in $(\\frac{1}{2\\kappa}, 2\\kappa)$, and also $|u_0-u_*|=|u_*-u_k|=\\Theta (n^{3\/4-\\varepsilon\/2})$. It follows using Theorem \\ref{baikrains} that with probability at least $1-e^{-n^{c}}$ one has \n$$ L^{\\boxslash}(u_0,u_*)\\geq \\mathbb{E} L(u_0,u_*)- n^{1\/4-\\varepsilon\/8}; \\qquad L^{\\boxslash}(u_*,u_k)\\geq \\mathbb{E} L(u_*,u_k)- n^{1\/4-\\varepsilon\/8}. $$\nAlso observe that $u_*$ is equidistant from $u_0$ and $u_k$ and that this point lies within the parallelogram $R$ whose height is $o(|u_0-u_k|^{2\/3})$. Thus, by a standard calculation, \n$$ \\mathbb{E} L(u_0,u_*)+ \\mathbb{E} L(u_*,u_k) -\\mathbb{E} L(u_0,u_k)= o(n^{1\/4-\\varepsilon\/6}).$$\nCombining these inferences, we find that, with probability at least  $1-e^{-n^{c}}$,  \n$$|\\gamma_{*}| > \\mathbb{E} L(u_0,u_k)-n^{1\/4-\\varepsilon\/8}.$$\nThis completes the proof of the proposition.\n\\end{proof}\n\nIn the final step, we establish that, on $\\mathcal{A}_{\\varepsilon}$, that is if ${\\rm MFL}(\\Gamma_{n})$ is smaller than $n^{3\/4-\\varepsilon}$, it is extremely likely that there exists a regular sequence whose points are consecutive corners of  $\\Gamma^*_{\\alpha, n}$ (the least concave majorant of $\\Gamma_{\\alpha, n}$). \nLet ${\\rm Reg}(C,\\kappa)$ denote the event that there exist consecutive corners $u_0,u_1,\\ldots, u_{k}$ of $\\Gamma^*_{\\alpha, n}$ such that $\\{u_0,u_1,\\ldots,u_k\\}$ is a $(C,\\kappa)$-regular sequence. \n\n\\begin{proposition}\n\\label{l:hullreg} There exist $C,\\kappa$ such that, except for a sub-event of exponentially small probability, \nthe event $\\mathcal{A}_{\\varepsilon}$ is contained in the event ${\\rm Reg}(C,\\kappa)$ i.e. $$\\mathbb{P}(\\mathcal{A}_{\\varepsilon}\\setminus  {\\rm Reg}(C,\\kappa)) \\le e^{-cn},$$ for some universal constant $c>0$.\n\\end{proposition}\n\n\n\nOn $\\mathcal{A}_{\\varepsilon}$, we shall find a regular sequence among interior facets.  \nBefore proceeding we state an easy geometric lemma  that will be used. The proof is provided in Section \\ref{es}.\n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[scale=.4]{fig1.pdf}\n\\caption{Crude estimates on length of facets which subtend an angle $\\theta$ at the origin. }\n\\label{fig1}\n\\end{figure}\n\nFor any two points $p_1,p_2 \\in [0,n]^2$, denote by $\\theta(p_1,p_2)$ the acute angle between the lines joining the points $p_1$ and $p_2$ to the point $(n,0)$, i.e., the bottom right corner of the square (see Figure \\ref{fig1}).\n\\begin{lemma}\\label{crude1}  There exists $\\theta_0$ such that with probability at least $1-e^{-cn}$  simultaneously for all $u,v$ on $\\Gamma^*_{\\alpha,n}$ with $\\theta(u,v)\\le \\theta_0$, then  $|u-v|= \\Theta (\\theta(u,v) n)$. \n\\end{lemma} \n\n\\begin{proof}[Proof of Proposition \\ref{l:hullreg}]\nObserve that by definition, on $\\mathcal{A}_{\\varepsilon}$, any sequence $u_0\\preceq u_1 \\preceq \\cdots \\preceq u_k$ of consecutive corners of $\\Gamma^*_{\\alpha, n}$ satisfies condition (i) and (iv) of Definition \\ref{d:reg}. Fix $\\delta>0$.  By Theorem \\ref{flat1}, with failure probability at most $e^{-cn}$ all the $\\delta-$interior facets have gradient $s$ in the interval $(1\/\\kappa, \\kappa)$ for some $\\kappa(\\delta)>1$.  For the remainder of the proof, we shall assume that this event occurs. \nNow we  partition all the $\\delta$-interior vertices of $\\Gamma^*_{\\alpha, n}$ (it is easy to see they are non-empty on $\\mathcal{A}_{\\varepsilon}$) into consecutive segments \n\\begin{equation}\\label{partition1}\n(v_{1},v_{2},\\ldots, v_{i_1}), (v_{i_1+1},v_{i_1+2}, \\ldots, v_{i_2}),\\ldots,\n\\end{equation} (where $v_i$'s are the consecutive corners on $\\Gamma^*_{\\alpha,n}$)\n such that \n\\begin{align*}\n\\theta(v_{i_j},v_{i_{j+1}-1}) &< n^{-1\/4-\\varepsilon\/2} \\le \\theta(v_{i_j+1},v_{i_{j+1}})\n\\end{align*}\nfor all $j=0,1,\\ldots,$ where $i_0=1$.\n\nWe denote the segment $(v_{i_j},v_{i_j+1}, \\ldots, v_{i_{j+1}})$ by $S_j$ \nThus if $S_1,S_2,\\ldots, S_{m}$ are all such segments, then $m=\\Theta(n^{1\/4+\\varepsilon\/2})$. It follows from convexity and  the boundedness of the gradients (see Lemma \\ref{crude1} and Figure \\ref{fig1}) that  $|v_{i_j}-v_{i_{j+1}}|=\\Theta (n^{3\/4-\\varepsilon\/2})$. Let $\\omega_{k}$ denote the angle made by the line segment $(v_{k}, v_{k+1})$ with the positive $x$-axis. For the sequence $S_j$, let $\\theta_{j}=\\omega_{i_j}-\\omega_{i_{j+1}-1}$. Notice that, by convexity, $\\theta_j$ is nonnegative for all $j$. Also observe that \n$$\\sum_{j}  \\theta_{j}\\leq \\frac{\\pi}{2}.$$\nThis together with Markov's inequality and the fact that the number of $S_j$'s is $\\Theta(n^{1\/4+\\varepsilon\/2})$ implies that, for some constant $C$, there exists at least one $S_j$ such that $\\theta_j \\leq Cn^{-1\/4-\\varepsilon\/2}$. \nSetting $S=(u_0,u_1,\\ldots , u_{k})$ to be equal to such a sequence, we see that $S$ satisfies all conditions in the Definition \\ref{d:reg} for some $C,\\kappa>1$. This completes the proof of the proposition.\n\\end{proof}\n\n\n\nWith all this preparation, finally we are ready to prove Theorem \\ref{t:mfllb}.\n\n\\begin{proof}[Proof of Theorem \\ref{t:mfllb}] \nLet $\\alpha\\in (0,\\frac{1}{2})$ and $\\varepsilon>0$ be fixed as before. Let $(C,\\kappa)$ be such that the conclusion of Proposition \\ref{l:hullreg} holds. Let $\\mathcal{S}$ denote the event that there exists a $(C,\\kappa)$-regular sequence for which at least one of the conclusions in Proposition \\ref{p:step1} and Proposition \\ref{p:step2} does not hold.\nUsing the proofs of the said propositions, along with  taking an union bound over all pairs of grid points obtained by coarse graining $[0,n]^2$ (exactly as Corollary \\ref{uniform1} followed from Theorem \\ref{t:moddev}), that $\\mathbb{P}(\\mathcal{S})\\leq e^{-n^{c}}$ for some $c>0$.\nBy Proposition \\ref{l:hullreg},  it suffices to show that  \n$$\\mathcal{A}_{\\varepsilon}\\cap \\mathcal{S}^c \\cap {\\rm Reg}(C,\\kappa)=\\emptyset$$\nBy way of contradiction, let us assume that the set  is non-empty. Fix a configuration in the set $\\mathcal{A}_{\\varepsilon}\\cap \\mathcal{S}^c \\cap {\\rm Reg}(C,\\kappa)$, and consider the least concave majorant $\\Gamma^*_{\\alpha, n}$ of the constrained geodesic. On $\\mathcal{A}_{\\varepsilon}\\cap  {\\rm Reg}(C,\\kappa)$, there exists a $(C,\\kappa)$-regular sequence $S=\\{u_0,u_1,\\ldots , u_{k}\\}$ consisting of consecutive corners of $\\Gamma^{*}_{\\alpha,n}$. On $\\mathcal{S}^c$, there exists a path $\\gamma_*$ from $u_0$ to $u_{k}$ satisfying the conclusion of Proposition \\ref{p:step2}. Let $\\gamma_{0}$ denote the restriction of $\\Gamma$ between $u_0$ and $u_{k}$. Clearly, \n$$ |\\gamma_{0}|\\leq \\sum_{i} L(u_i,u_{i+1}).$$ It follows from Propositions \\ref{p:step1} and \\ref{p:step2} that, on $\\mathcal{S}^c$,  $|\\gamma_{0}|< |\\gamma_{*}|$. Also observe that $\\gamma_{0}$ lies below the line segments joining $u_{i}$ to $u_{i+1}$ whereas $\\gamma_{*}$ lies above these segments. Hence the increasing path $\\Gamma'$ from $(0,0)$ to $(n,n)$ that coincides with $\\Gamma$ between $(0,0)$ and $u_0$ and between $u_{k}$ and $(n,n)$ and coincides with $\\gamma_{*}$ between $u_0$ and $u_{k}$ traps at least as much area as does $\\Gamma$. However, by the above discussion, $|\\Gamma'|> |\\Gamma|$. This contradicts the extremality of $\\Gamma$ and completes the proof of the theorem. \n\\end{proof}\n\n\\subsection{Proof of lower bound of ${\\rm MLR}$}\nIn this subsection we prove Theorem \\ref{t:mlrlb}. That is, we show that the maximum local roughness of the facets has scaling exponent at least $\\frac{1}{2}$. The basic idea of the proof is to  use KPZ scaling to argue that, if there is a facet of length roughly $n^{3\/4}$, then along that facet, the local roughness is likely to be at least $(n^{3\/4})^{2\/3}=n^{1\/2}$. We now make this more precise.\n\nFix $\\varepsilon>0$. We have just proved that, with high probability, there exists a facet of length at least $n^{3\/4-\\varepsilon}$. In fact, by the proof of Theorem \\ref{t:mfllb}, more is true. Fix $\\kappa\\gg 1$ a large constant. Let $\\mathcal{B}_{\\varepsilon}=\\mathcal{B}_{n,\\varepsilon,\\kappa}$ denote the event that $\\Gamma^{*}_{\\alpha,n}$ (see Definition \\ref{d:mlrmfl}) has a facet of length at least $n^{3\/4-\\varepsilon}$ with endpoints $v_0$ and $v_{1}$ such that the straight line joining $v_0$ and $v_1$ has gradient  $\\in (1\/\\kappa, \\kappa)$. The following lemma is a consequence of the proof of Theorem \\ref{t:mfllb}. \n\n\\begin{lemma}\n\\label{l:be}\nFor all large enough $\\kappa$ (depending on $\\alpha$),\n$\\mathbb{P}(\\mathcal{B}_{\\varepsilon})\\ge 1-e^{-n^c}$ for all $n$ sufficiently large, \nwhere $c$ is a constant depending only on $\\varepsilon$ and $\\alpha$.\n\\end{lemma}\n\n\\begin{proof}\nNote that the proof of Theorem \\ref{t:mfllb} in fact showed that there must be a $\\delta$-interior facet of length at least $n^{3\/4-\\varepsilon}$ with sufficiently large probability. Since all the $\\delta$-interior facets satisfy the gradient requirement (see Theorem \\ref{flat1}), we are done. \n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{t:mlrlb}]\nLet $\\mathcal{C}_{\\varepsilon}$ denote the event that ${\\rm MLR}(\\Gamma_n) \\leq n^{1\/2-\\varepsilon}$. Thus, to prove Theorem~\\ref{t:mlrlb}, it suffices to show that  $$\\mathbb{P}(\\mathcal{C}_{\\varepsilon}\\cap \\mathcal{B}_{\\varepsilon})\\le e^{-n^c}.$$ \n\nLet $\\Gamma$ denote the constrained geodesic. On $\\mathcal{C}_{\\varepsilon}\\cap \\mathcal{B}_{\\varepsilon}$, let $v_0$ and $v_1$ be corners of $\\Gamma^*_{\\alpha, n}$ satisfying the conditions in the definition of $\\mathcal{B}_{\\varepsilon}$. Consider the restriction $\\Gamma(v_0,v_1)$ of $\\Gamma$ between $v_0$ and $v_1$. As ${\\rm MLR}(\\Gamma) \\leq n^{1\/2-\\varepsilon}$; it follows that there exists $C=C(\\kappa)>0$ such that the restriction of $\\Gamma$ (call it $\\gamma$) between $v_0$ and $v_1$ is contained in the parallelogram $U$ that has a pair of vertical sides of height $Cn^{1\/2-\\varepsilon}$ and whose top side is the line segment joining $v_0$ and $v_1$. Now observe that the height of the parallelogram is much smaller than the on scale transversal fluctuation of geodesics between $v_0$ and $v_1$, these fluctuations being at least $$(n^{3\/4-\\varepsilon})^{2\/3-\\varepsilon\/100}\\ge n^{1\/2-{3\\varepsilon}\/{4}} . $$ \nThus, Corollary \\ref{uniform0} implies that the bound \n$$\\mathbb{E}(L(v_0, v_1))-|\\gamma| \\ge |v_0-v_1|^{1\/3- 5\\varepsilon\/9}$$\nfails with  probability at most $e^{-n^{c}}$. By Corollary \\ref{uniform2}, \n$$\n L^{\\boxslash}(v_0,v_1)-\\mathbb{E}(L(v_0,v_1)) \\ge  -|v_0-v_1|^{\\frac{1}{3}+\\varepsilon\/100}\n $$\nexcept, again, on a set of probability at most $e^{-n^{c}}$.\nConsider now the path $\\Gamma'$ that agrees with $\\Gamma$ outside the segment $(v_0,v_1)$ and that between $v_0$ and $v_1$ equals $\\gamma^{\\boxslash}(v_0,v_1)$, which recall is a path from $v_0$ to $v_1$ lying above the line segment that joins these two points and achieves length $L^{\\boxslash}(v_0,v_1)$.\nWe see then that $\\Gamma'$  has greater length, and traps as much area as,~$\\Gamma$. \nThis contradicts the assumption of length maximality for $\\Gamma$; (similarly to the proof of Theorem \\ref{t:mfllb}. See Figure \\ref{f:alternatepath20}(b).)\n\\end{proof}\n\n\\section{Upper Bound for Scaling Exponents}\n\\label{s:ub}\nThis section is devoted to the proof of Theorem \\ref{t:mflub}. That is,  it will be shown here that, for a dense set of $\\alpha$ that may depend on $n$, the maximum facet length of the constrained geodesic is very likely to be at most \n$n^{3\/4+o(1)}$. Theorem \\ref{t:mlrub} will then follow by the KPZ scaling enjoyed by transversal fluctuations.  \n\n\nWe explain the main ideas before giving proofs. Recall Definition \\ref{good10}. The basic idea is to show, using Corollary \\ref{uniform2}, \nthat if a certain value of $\\alpha$ is not `good' (recall Definition \\ref{good10}),\nwe may  perform  a ``landgrab\" operation. More formally let us assume that there is a facet $(v_0,v_1)$ which has length $n^{3\/4+\\varepsilon}$ and has a reasonable gradient bounded away from zero and infinity. Theorem \\ref{baikrains} allows us to find a path $\\gamma$ from $v_0$ to $v_1$ which lies above the facet $(v_0,v_1)$ at a characteristic height of $(n^{3\/4+\\varepsilon})^{2\/3}$ and has characteristic fluctuations (we have room to allow some error) i.e. by Corollary \\ref{uniform2} we can assume \n$$|{\\gamma}|\\ge \\mathbb{E}(L(v_0,v_1)) -(n^{3\/4+\\varepsilon})^{1\/3+\\varepsilon\/1000}\\ge \\mathbb{E}(L(v_0,v_1))-n^{1\/4+2\\varepsilon\/5}.$$\nSimilarly by an Corollary \\ref{uniform1} we can assume that $$|\\Gamma(v_0,v_1)| \\le \\mathbb{E}(L(v_0,v_1))+ n^{1\/4+2\\varepsilon\/5}$$ \nwhere $\\Gamma(v_0,v_1)$ is the restriction of the constrained geodesic $\\Gamma$ between  $v_0$ and $v_1$. \nThus the path $\\Gamma'$ which is obtained from $\\Gamma$ by replacing $\\Gamma(v_0,v_1)$ by $\\gamma$ traps at least  $n^{5\/4+{5\\varepsilon}\/{3}}$ more area than  does $\\Gamma$ and loses at most $n^{1\/4+2\\varepsilon\/5}$ in length. Repeating this operation about $n^{3\/4-5\\varepsilon\/3}$ times (we can do that provided there are no good $\\alpha$ in this interval), one gains an area that is $\\Theta(1)$ while losing at most $O(n^{1-\\varepsilon})$ in length with very high probability. This contradicts the fact that $\\mathbb{E} L_{\\alpha_1}(n) -\\mathbb{E} L_{\\alpha_2}(n)=\\Theta(n)$ for $\\alpha_1<\\alpha_2$ (an easy consequence of Theorem \\ref{t:lln} using ${\\rm w}_{\\alpha}$ is strictly decreasing) using Theorem \\ref{concentration1}. Next we provide  the details needed to make the above argument precise. \n\nFix $0<\\alpha_1 <\\alpha_2 <\\frac{1}{2}$ and also $\\varepsilon>0$ and $\\delta\\in (0,\\pi\/4)$. Let $\\mathcal{G}_{\\alpha_1,\\alpha_2}$ denote the event that no $\\alpha\\in [\\alpha_1,\\alpha_2]$ is  $(n,\\varepsilon, \\delta)$-good. Let $B_{n}=B_{n,\\varepsilon,\\kappa}$ be the set of all pairs $(x,y)\\in [0,n]^2$ satisfying the following conditions. \n\\begin{enumerate}\n\\item $x\\preceq y$.\n\\item The straight line joining $x$ and $y$ has gradient $\\in (\\frac{1}{\\kappa}, \\kappa)$.\n\\item $|x-y|\\geq n^{3\/4}$. \n\\end{enumerate} \nFor $(x,y)\\in B_n$, let $A_{x,y}$ denote the event that $|L^{\\boxslash}(x,y)-\\mathbb{E} L(x,y)|\\leq |x-y|^{1\/3+\\varepsilon\/1000}$ and $|L^{*}(x,y)-\\mathbb{E} L(x,y)|\\leq |x-y|^{1\/3+\\varepsilon\/1000}$ where $L^*(x,y)$ denotes the best path between $x$ and $y$ that is constrained to stay below the straight line joining $x$ and $y$. Let $\\mathcal{A}_{\\varepsilon}$ denote the event that $A_{x,y}$ holds for all $(x,y)\\in B_n$. Further let $S_{\\delta}$ denote the event in the statement of Theorem \\ref{flat1}, i.e., all the $\\delta$ interior facets of  $\\Gamma^*_{\\alpha, n}$ has moderate gradients for all $\\alpha\\in [\\alpha_1, \\alpha_2]$. We have the following proposition.\n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.3\\textwidth]{landgrab.pdf}\n\\caption{The landgrab operation. By replacing the path $\\Gamma$ by the path $\\gamma$ one gains at least the amount of the area in the shaded region, whereas the event $A_{\\varepsilon'}$ ensures that the length loss is not too much.}\n\\label{f:landgrab}\n\\end{figure}\n\n\n\\begin{proposition}\n\\label{p:landgrab}\nThere exists $\\kappa=\\kappa(\\delta,\\varepsilon)$ sufficiently small such that, on $\\mathcal{A}_{\\varepsilon}\\cap S_{\\delta}\\cap \\mathcal{G}_{\\alpha_1,\\alpha_2}$, there exists $\\alpha_* \\geq \\alpha_2$ for which \n$$L_{\\alpha_1}(n)-L_{\\alpha_*}(n)\\leq (\\alpha_{*}-\\alpha_1) n^{1-6\\varepsilon\/5}.$$\n\\end{proposition}\n\n\\begin{proof}\nWe perform the following recursive construction. Let $\\beta_0=\\alpha_1$. For $\\beta_{i}\\in [\\alpha_1,\\alpha_2]$ construct $\\beta_{i+1}$ recursively as follows: find the longest $\\delta$-interior facet in $\\Gamma_{\\beta_i,n}$. On $\\mathcal{G}_{\\alpha_1,\\alpha_2}$, the longest $\\delta$-interior facet $(x_i,y_i)$ has length at least $n^{3\/4+\\varepsilon}$.  Note that on $S_{\\delta}$, the endpoints $x_{i}$ and $y_{i}$ of this facet satisfy  $(x_{i},y_{i})\\in B_n$ for some $\\kappa=\\kappa(\\delta)$. Now pick a point $z_{i}$  at orthogonal distance $|x_{i}-y_{i}|^{2\/3}$ from  the midpoint of the  line segment joining $x_i$ and $y_{i}$; (see Figure \\ref{f:landgrab}). Thus the area of the triangle $(x_i,y_i,z_i)$ is at least $c|x_{i}-y_{i}|^{5\/3}$ for some constant $c$ that does not depend on $i$. Set $\\beta_{i+1}=\\beta_{i}+ cn^{-2}|x_{i}-y_{i}|^{5\/3}$. Now consider the path $\\gamma$ that coincides with $\\Gamma_{\\beta_{i},n}$ outside the facet $(x_i,y_i)$, and is formed by concatenating $\\gamma^{\\boxslash}(x_i,z_i)$ and $\\gamma^{\\boxslash}(z_i,y_i)$ between $x_i$ and $y_{i}$. Clearly the area under the curve $\\gamma$ is at least $(\\frac{1}{2}+\\beta_{i+1})n^2$. Also, on $A_{\\varepsilon}$, \n$$|\\Gamma_{\\beta_{i},n}|-|\\gamma|\\leq |x_{i}-y_{i}|^{1\/3+\\varepsilon\/500}.$$\nIt follows that  \n$$L_{\\beta_i}(n)-L_{\\beta_{i+1}}(n) \\leq |x_{i}-y_{i}|^{1\/3+\\varepsilon\/500}.$$\n\n\n\nDenote $\\alpha_{*}=\\beta_{i_0}$, where $i_0$ is the smallest index $i$ for which $\\beta_{i}\\geq \\alpha_2$. It follows that \n$$L_{\\alpha_1}(n)-L_{\\alpha_*}(n)\\leq \\sum_{i=0}^{i_0-1} |x_{i}-y_{i}|^{1\/3+\\varepsilon\/500}\\leq c^{-1}(\\alpha_*-\\alpha_1)n^{2} \\left(\\max_{i\\leq i_0} |x_i-y_i|^{-4\/3+\\varepsilon\/500}\\right).$$\nSince $|x_i-y_{i}|\\geq n^{3\/4+\\varepsilon}$ for all $i$, it follows that the final term is at most $n^{-1-4\\varepsilon\/3}$   and that completes the proof of the proposition.\n\\end{proof}\n\nWe may now  complete the proof of Theorem \\ref{t:mflub}. \n\n\\begin{proof}[Proof of Theorem \\ref{t:mflub}]\nLet $\\kappa$ be  large enough that the conclusion of Proposition \\ref{p:landgrab} holds. Using Theorem \\ref{flat1} and Corollary \\ref{uniform}, it follows that, for this choice of $\\kappa$ (and for given  $\\alpha_1,\\alpha_2, \\delta$ and $\\varepsilon$), one has that, for some constant $c>0$,\n$$\\mathbb{P}[\\mathcal{A}_{\\varepsilon}^{c}\\cup S_{\\delta}^{c}] \\leq e^{-n^c}.$$\nHence, using Proposition \\ref{p:landgrab}, it suffices to show that \n\\begin{equation}\n\\label{e:suffice}\n\\mathbb{P}\\left (L_{\\alpha_1}(n)-L_{\\alpha_2}(n)\\leq (1\/2-\\alpha_1)n^{1-6\\varepsilon\/5}\\right )\\leq e^{-n^c}\n\\end{equation}\nfor some $c>0$. Notice that in the above step we have used the trivial inequality $L_{\\alpha_2}(n)\\geq L_{\\alpha_*}(n)$ for all $\\alpha_{*}\\in [\\alpha_2,\\frac{1}{2})$.\nIt follows now from Theorem \\ref{t:lln} that $\\mathbb{E} L_{\\alpha_1}(n)-\\mathbb{E} L_{\\alpha_2}(n)\\geq \\frac{{\\rm w}_{\\alpha_1}-{\\rm w}_{\\alpha_2}}{2}n$ for $n$ sufficiently large. The claimed bound \\eqref{e:suffice} now follows from Theorem \\ref{concentration1} and the fact that ${\\rm w}_{\\alpha_1}-{\\rm w}_{\\alpha_2} >0$. This completes the proof.\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{t:mlrub}}\nThis derivation is similar to that by which Theorem \\ref{t:mlrlb} follows from Theorem \\ref{t:mfllb}. \nBy Theorem \\ref{flat1}, for all $\\alpha \\in [\\alpha_1,\\alpha_2]$ and all small enough $\\delta$, there is a probability at least $1-e^{-cn}$ that all the $\\delta$-interior facets of $\\Gamma^*_{\\alpha,n}$ are $\\kappa-$ steep for some value of $1\\le \\kappa<\\infty$.\nNow for any good $\\alpha\\in [\\alpha_1, \\alpha_2]$ (note that the set of good $\\alpha'$s is a function of the underlying point process and hence is random), \nby definition, all the interior facets of $\\Gamma^*_{\\alpha,n}$ have length at most~$n^{3\/4+\\varepsilon}$. Moreover, by the uniform bounds  discussed above we may restrict to the case where all the facets are $\\kappa$-steep. Suppose that the maximum local roughness restricted to the $\\delta$-interior facets  is  at least $n^{1\/2+\\varepsilon}$.  Let $u$ and $v$ be the endpoints of the facet  that attains this maximum local roughness. Let $\\gamma(u,v)$ be the segment of $\\Gamma_{\\alpha,n}$ between $u$ and $v$. \nBy using Corollary \\ref{uniform0}, we see that, with probability at least $1-e^{-n^c}$, \n$$|\\gamma(u,v)|< \\mathbb{E}(L(u,v))- S(u,v).$$ On the other hand,\n by Corollary \\ref{uniform2} again, we find that, except on an event of probability at most $e^{-n^c}$,\n$$L^{\\boxslash}(u,v)\\ge \\mathbb{E}(L(u,v)))- S(u,v).$$\nThus, the path that agrees with $\\Gamma$ from $(0,0)$ to $u$ and from $v$ to $(n,n)$ \nand equals the path $\\gamma^{\\boxslash}(u,v)$ between the points $u$ and $v$ clearly captures at least as much area as does $\\Gamma_{\\alpha,n}$ and also contains more points  than does $\\Gamma_{\\alpha,n}$. This  contradicts the extremality of the latter path. \n \\qed\n\n\\subsection{Possible extensions and difficulties}\\label{areafluc} \n We end this section with a few remarks related to the first two open problems mentioned in Section \\ref{oq}. A natural approach to  quantifying the  proof of Theorem \\ref{t:mflub} is via Proposition \\ref{p:landgrab}. Namely, one can hope to bound below the density of those $\\alpha\\in [\\alpha_1,\\alpha_2]$ that are $(n,\\varepsilon)$ good (where here we ignore the parameter $\\delta$ for the purpose of illustration.) This is because using similar arguments to those used in deriving these two results, one can hope to prove the following uniform bound: with high probability the noise space is such that for all $\\alpha\\in [\\alpha_1,\\alpha_2]$ we have $L_{\\alpha}(n)-L_{\\alpha_*}(n)\\le \\tilde O(\\ell^{1\/3})$ for $\\alpha_*=\\alpha+ \\ell^{5\/3}n^{-2},$ where $\\ell=\\max({\\rm{MFL}}(\\Gamma_{\\alpha,n}),n^{3\/4}),$ and where the $\\tilde O$ notation hides poly-logarithmic multiplicative factors (which arise due to the exceptional nature of the facet endpoints that contributes a polynomial entropy factor).  Thus, the above implies that the length-to-area gradient in scaled coordinates (with length measured in units of $n$ and area measured in units of $n^2$) is $\\tilde O(\\frac{n}{\\ell^{4\/3}})=\\tilde O(1)$ as by definition $\\ell \\ge n^{3\/4}$. Also, we know that, for all $\\alpha$ which are\n  $(n,\\varepsilon)$ bad, the gradient is polynomially small (since by the land-grab argument (see Figure \\ref{f:landgrab}), the length loss is much smaller than the area gain). This coupled with the fact that $L_{\\alpha_1}-L_{\\alpha_2}=\\Theta(n),$ should then allow to conclude that the density of good $\\alpha$ is at least $\\frac{1}{\\tilde O(1)}$.\n\nNotice that our approach for proving upper bounds relies on the joint coupling of the process for various values of $\\alpha$. This is quite different in spirit from the known proofs of similar statements in other contexts such as phase separation \\cite{AH1} where the upper bound was proven for a fixed value of $\\alpha$. The key ingredient there was an understanding of the excess area fluctuation (see Open question (2) in Section \\ref{oq}) which then allowed a resampling argument to work. However, since we do not have such bounds, we have to work across various values of $\\alpha$ simultaneously. \n\n A heuristic argument in our setting proving an upper bound of $n^{5\/4+o(1)}$ on the excess area trapped by $\\Gamma=\\Gamma_{\\alpha}$, assuming that $\\alpha$ is $(n,\\varepsilon\/10)$ good (recall Definition \\ref{good10}) can be made as follows: Suppose that the excess area is more than $n^{5\/4+\\varepsilon}$. Since, by hypothesis the longest facet length is at most $n^{3\/4+\\varepsilon\/10}$ , we may find two points $x$ and $y$ on $\\Gamma^*_{\\alpha,n}$  such that $|x-y|\\approx n^{3\/4+\\varepsilon\/5}$ and segment of  $\\Gamma^*_{\\alpha,n}$ between $x$ and $y$ has average curvature (the average distance of the curve from the line segment joining $x$ and $y$ is what it should be on for a circle; see Figure \\ref{def} (b)). Now consider shortcutting $\\Gamma$ between $x$ and $y$ and hence replacing the subpath of $\\Gamma_{\\alpha,n}$ between $x$ and $y$ by the unconstrained geodesic between $x$ and $y$. This operation will create an area loss (approximately $n^{5\/4+3\\varepsilon\/5}$) which is not enough to violate the area constraint, because the excess area before the shortcut was at least $n^{5\/4+\\varepsilon}$. Moreover, the shortcut increases the overall weight and hence contradicts extremality of $\\Gamma_{\\alpha,n}$.\n\n\\section{Proofs of some of the earlier statements}\\label{es}\nIt remains to provide proofs of Lemmas \\ref{l:exist},\\ref{l:ac}, \\ref{decreasing1}, \\ref{crude1} and Theorem \\ref{concentration1} that were postponed.\n\n\n\\subsection{Proof of Lemma \\ref{l:exist}} This subsection follows closely the arguments presented in \\cite[Section 3]{DZ2}. Let $\\phi_1,\\phi_2,\\ldots$ be a sequence of elements in $\\mathcal{B}_{\\alpha}$ such that $$\\lim_{n}J(\\phi_n)=J_{\\alpha}.$$ \nWe equip the space of bounded signed measures on $[0,1]$ with the weak topology generated by $\\mathcal{C}=C[0,1]$ (the space of continuous functions on $[0,1]$).  \nSince for all $n\\ge1,$ $\\phi_n$ correspond to sub-probability measures, using compactness, by passing to a subsequence (denoted again by $\\{n\\}$ for convenience) let $\\phi_n$ converge to $\\psi\\in \\mathcal{M},$  in the weak topology.  \nThis in particular implies $\\phi_n(x)$ converges to $\\psi(x)$ almost everywhere (since the convergence happens at all continuity points of $\\psi$, see e.g., \\cite[Section 3.2]{Dur} ). Since $\\phi_n$'s are all bounded by $1$ and belong to $\\mathcal{B}_{\\alpha},$ by the bounded convergence theorem, it follows that $\\psi \\in \\mathcal{B}_{\\alpha}$. \nNext we will show that \n\\begin{equation}\\label{claim203}\n\\lim_{n}J(\\phi_n)\\le J(\\psi).\n\\end{equation}\nClearly this implies $J(\\psi)=J_{\\alpha}$ and hence we would be done.\nThe claim in \\eqref{claim203} follows from the upper semicontinuity of $J(\\cdot)$ or equivalently the lower semicontinuity of $-J(\\cdot)$.\nTo show the latter one represents $-J(\\cdot)$ as an appropriate Legendre transform. \nTo proceed we need some definitions: Clearly \\eqref{variational1} extends naturally to all non-negative measures. We extend $J$ to all of $\\mathcal{M}$ by setting it to be $\\infty$ for all measures which are not non-negative. \nMoreover, for any $f\\in \\mathcal{C},$ define\n $$\\Lambda(f)=\\left \\{ \\begin{array}{cc}\n\\infty & \\text {if } \\int_0^1\\frac{1}{|f(s)|}ds =\\infty  \\text{ or if } f\\ge 0 \\text{ on a set of positive Lebesgue measure}. \\\\\n-\\frac{1}{4} \\int_{0}^1 \\frac{1}{f(s)}ds  & \\text{ otherwise}\n\\end{array}\n\\right.\n$$\nNow for any $\\phi \\in \\mathcal{B},$ define $\\Lambda^*(\\phi)=\\displaystyle{\\sup_{f\\in C[0,1]}  \\left[\\int_{0}^1 f d(\\phi) -\\Lambda(f)\\right]}$. \nThis function, by definition, is lower semicontinuous, since for any $f\\in C[0,1],$ and a sequence $\\phi_n$ converging to $\\phi,$ by definition of weak convergence, $$\\lim_{n\\to \\infty}\\int_0^1 f d(\\phi_n) = \\int_0^1 f d(\\phi).$$ Thus the proof is complete by \\cite[Lemma 5]{DZ2} which says $\\Lambda^*(\\phi)=-J(\\phi)$. \n\\qed\n\n\n\\subsection{Proof of Lemma \\ref{l:ac}}\nAs already hinted right after the statement of Lemma \\ref{l:ac}, the proof is by contradiction. Assuming that the singular part of $\\psi,$ which we denote by $\\psi_s$ (see \\eqref{decomposition23}). has positive mass, we create a modification $\\psi_1$ such that $J(\\psi_1)> J(\\psi)$ (see \\eqref{variational1}). However in the process, it is possible that we violate the area constraint, i.e. $\\int_0^1\\psi_1(x){\\rm{d}}x < \\frac{1}{2}+\\alpha$. Thus we make a second modification to construct a function $\\psi_2$  with the property $J(\\psi_2)> J(\\psi),$ and moreover \nit satisfies the area constraint as well. This contradicts the extremality of $\\psi$ which was a part of the hypothesis. The details follow.\n\nLet us assume that $\\psi_s$ has total mass $k>0$. Then clearly, without loss of generality, we can assume $\\psi_s=k\\delta_0$, i.e., there is an atom of mass $k$ at zero, since \n\\begin{align*}\n\\int_{0}^1 (\\psi_{ac}(x)+k) {\\rm{d}}x \\ge \\int_{0}^1 (\\psi_{ac}(x)+\\psi_{s}(x)) {\\rm{d}}x\\ge \\frac{1}{2}+\\alpha. \n \\end{align*}\nAlso, because the absolutely continuous part stays the same,  $J(\\psi_{ac}+k\\delta_0)=J(\\psi)$. \n  We will now make some local modifications to contradict extremality of $\\psi$. \n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[scale=.7]{fig4.pdf}\n\\caption{Local improvement}\n\\label{fig9}\n\\end{figure}\n\nLet $r>0$ be a small number to be chosen later.  Consider a new function $\\psi_1$, as illustrated in Figure \\ref{fig9}, which is a linear function interpolating $(0,0)$ and $(r,\\psi(r))$  and which agrees with $\\psi$ on $[r,1]$. Since $\\psi(r)=\\psi_{ac}(r)+k$,\nit follows that\n\\begin{equation}\\label{arealoss}\n\\int_{0}^1 \\psi_1(x){\\rm{d}}x \\ge \\int_{0}^1 \\psi(x){\\rm{d}}x -\\frac{1}{2}kr-O(\\psi_{ac}(r)r).\n\\end{equation}\nOn the other hand, for all small enough $r$,\n\\begin{align}\\label{lengthgain}\n\\int_{0}^1 \\sqrt{\\dot \\psi_1(x)} {\\rm{d}}x &\\ge \\int_{0}^1 \\sqrt{\\dot\\psi(x)} {\\rm{d}}x  -\\int_{0}^r \\sqrt{\\dot\\psi(x)} {\\rm{d}}x + \\sqrt{kr}\\\\\n\\nonumber\n&\\ge  \\int_{0}^1 \\sqrt{\\dot\\psi(x)} {\\rm{d}}x -\\sqrt{\\psi_{ac}(r) r}+ \\sqrt{kr},\\\\\n\\nonumber\n& \\ge  \\int_{0}^1 \\sqrt{\\dot\\psi(x)} {\\rm{d}}x + \\frac{\\sqrt{kr}}{2}.\n\\end{align}\nThe second last inequality uses the bound $\\int_0^r \\sqrt{\\dot \\psi(x)} {\\rm{d}}x \\le \\sqrt{r \\psi_{ac}(r)},$ which follows from the Cauchy-Schwarz inequality.  To see the last inequality,  note that $\\psi_{ac}(r)$ goes to zero as $r$ goes to zero. \nHowever, it is easy to see that  $$\\int_{0}^1  \\psi_1(x){\\rm{d}}x < \\int_0^1 \\psi(x){\\rm{d}}x .$$ Thus a priori $\\psi_{1}$ need not be an element of $\\mathcal{B}_{\\alpha}$; (see \\eqref{restr304}). To ensure that indeed $\\int_0^1\\psi_{1}(x){\\rm{d}}x \\ge \\frac{1}{2}+\\alpha$,  we have to make another modification.\nNote that already in the proof of Lemma \\ref{l:uni} we argued that a priori even if $\\psi$ is not unique, $\\dot \\psi$ is unique. Thus there exist $c_{\\alpha}>0$ and $0<c_1<c_2$  depending only on  $\\alpha$ such that the set $(\\dot \\psi)^{-1}[c_1,c_2]$ has Lebesgue measure at least $c_{\\alpha}$. (Note that even though the proof of Lemma \\ref{l:uni} relied on Lemma \\ref{l:ac}, the uniqueness of the density was argued independently and hence this is not a circular argument.) \n\nFor brevity, let us call the set $(\\dot \\psi)^{-1}[c_1,c_2]$ by the name~$I$. We now choose arbitrary disjoint subsets of $I\\cap [\\varepsilon,1]$ for some $\\varepsilon>0$, denoted by $I_1$ and $I_2$, both of measure $k$; (we take the intersection with  $[\\varepsilon,1]$ to ensure that both the sets are away from $0$).  Note that this can always be done if $\\mu(I)> 2k$. In case $\\mu(I)\\le 2k$, we can modify $\\psi_{1}$ by taking it to be  a linear function interpolating $(0,k-k_*)$ and $(r,\\psi(r))$ for some $k_*$ small enough so that $\\mu(I)>2k_*$; (see Figure \\ref{fig9} ii.)\nNothing changes in any of the arguments that follow as well as the conclusions and hence we will pretend that $k=k_*$ throughout the rest of the argument.\nLet us also choose $I_1$ and $I_2$ \nsuch that $\\sup(I_1)< \\inf(I_2)$, \nand let $\\psi_2(x):=\\int_0^x \\dot \\psi_2(y){\\rm{d}}y$ where \n\\begin{equation}\\label{modi24}\n\\dot \\psi_2:=\\dot \\psi_1 + \\frac{r}{k} \\mathbf{1}(I_1) - \\frac{r}{k}\\mathbf{1}(I_2).\n\\end{equation}\nWe will also choose $r<\\varepsilon$.\nNote that since $\\dot \\psi_1=\\dot \\psi \\ge c_1$ on $I_2$, for all small enough $r,$   $\\dot \\psi_2$ is non-negative. Also, since the measures of $I_1$ and $I_2$ are the same, $$\\int_0^1 \\dot \\psi_2(x){\\rm{d}}x =\\int_0^1  \\dot \\psi_1(x){\\rm{d}}x .$$\n\nWe now compute the area under the curve $\\psi_2(\\cdot)$ \nand see how it differs from that of $\\psi_1(\\cdot)$.\nFor every $y\\in [0,1],$ we have $$\\psi_2(y)-\\psi_1(y)=\\frac{r}{k}[ \\mu(I_1\\cap [0,y])- \\mu(I_2\\cap [0,y])].$$\nThus, \n\\begin{align}\n\\label{area123}\n\\int_0^1 [\\psi_2(y)-\\psi_1(y)]{\\rm{d}}y &=\\frac{r}{k} \\int_0^1  [\\mu(I_1\\cap [0,y])-\\mu(I_2\\cap [0,y])]{\\rm{d}}y\\\\\n\\nonumber\n&= \\frac{r}{k} \\int_{0}^1 (1-y) [\\mathbf{1}(I_1)-\\mathbf{1}(I_2)]{\\rm{d}}y\\\\\n\\nonumber\n& \\ge rk.\n\\end{align}\nThe last inequality follows since, as $\\mu(I_1)=\\mu(I_2)= k$ and $\\sup(I_1)< \\inf(I_2)$, the integral above is at least $k^2$.\nThus, the new function $\\psi_2(x)$ traps at least as much area as does $\\psi(x)$ since \n\\begin{align*}\n\\int_{0}^{1}\\psi_2(x){\\rm{d}}x -\\int_{0}^{1}\\psi(x){\\rm{d}}x &=[\\int_{0}^{1}\\psi_2(x){\\rm{d}}x -\\int_{0}^{1}\\psi_1(x){\\rm{d}}x ]+[\\int_{0}^{1}\\psi_1(x){\\rm{d}}x -\\int_{0}^{1}\\psi(x){\\rm{d}}x ],\\\\\n&\\ge rk-\\frac{rk}{2}-O(\\psi_{ac}(r)r)>0,\n\\end{align*}\n for all small enough $r$; (the last inequality uses \\eqref{arealoss}).\n\nMoreover,  by Taylor expansion (using \\eqref{modi24} and  $\\dot \\psi_1\\ge c_1$ on $I_1 \\cup I_2$), \n\\begin{align*}\n\\int_0^1 \\sqrt{\\dot \\psi_2(x)}{\\rm{d}}x -\\sqrt{\\dot \\psi_1(x)}{\\rm{d}}x  &\\ge \\int_{I_1} \\sqrt{\\dot \\psi_1(x)}(\\frac{r}{2k\\dot \\psi_1(x)}- C(\\frac{r}{k\\dot \\psi_1(x)})^2){\\rm{d}}x  \\\\\n& +\\int_{I_2} \\sqrt{\\dot \\psi_1(x)}(-\\frac{r}{2k\\dot \\psi_1(x)}- C(\\frac{r}{k\\dot \\psi_1(x)})^2){\\rm{d}}x \\\\\n\\nonumber\n&\\ge -O(\\frac{r}{k})k=-O(r).\n\\end{align*}\nWe use the bound  $\\dot \\psi_1> c_1$  and that $\\mu(I_1)=\\mu(I_2)=k$ crucially in the last inequality, and all the constants  depend only on $c_1$ through the Taylor expansion.\nThus,  by choosing $r\\ll k$ we have $\\sqrt{rk}\\gg r$, and hence using \\eqref{lengthgain}, \n\\begin{align*}\n\\int_0^1 \\sqrt{\\dot \\psi_2(x)}{\\rm{d}}x -\\int_0^1 \\sqrt{\\dot \\psi(x)}{\\rm{d}}x &=\\int_0^1 \\sqrt{\\dot \\psi_2(x)}{\\rm{d}}x -\\int_0^1 \\sqrt{\\dot \\psi_1(x)}{\\rm{d}}x + \\int_0^1 \\sqrt{\\dot \\psi_2(x)}{\\rm{d}}x -\\int_0^1 \\sqrt{\\dot \\psi_1(x)}{\\rm{d}}x ,\\\\\n&\\ge -O(r) + \\sqrt{kr} -O(\\sqrt{\\psi_{ac}(r) r}),\\\\\n&\\ge \\sqrt{kr}\/2,\n\\end{align*}\nfor all small enough $r$. Hence, we obtain a contradiction to the extremality of $\\psi$.\n\\qed\n\n\n\\subsection{Proof of Lemma \\ref{decreasing1}}\n(i) The proof follows by taking the monotone rearrangement and using uniqueness. Let $\\dot \\psi_{mon}$  be the monotone rearrangement of $\\dot \\psi$. \nNotice that, by Fubini's theorem, \n\\begin{align}\n\\label{fubini}\n\\frac{1}{2}+\\alpha\\le \\int_0^1 \\psi(x){\\rm{d}}x =\\int_0^1 \\dot \\psi(x) (1-x){\\rm{d}}x  \\le \\int_0^1 (\\dot \\psi)_{mon}(x) (1-x){\\rm{d}}x  = \\int_0^1 \\psi_1(x){\\rm{d}}x  \n\\end{align} \nwhere $\\dot \\psi_1=(\\dot \\psi)_{mon}$. Only the second inequality above needs justification and it is a consequence of a standard rearrangement inequality. Thus $\\psi_1$ satisfies the area constraint. \nAlso clearly  $ \\int_0^1 \\sqrt {\\dot\\psi}(x){\\rm{d}}x = \\int_0^1 \\sqrt {\\dot\\psi_1}(x){\\rm{d}}x $ as rearrangement keeps integrals unchanged. \nThis contradicts the uniqueness of $\\psi$ unless $\\psi=\\psi_{1}$\n\n(ii) Let $\\dot\\psi$ be zero on a set of positive measure. By the first part of the lemma, this implies that there exists $a$ such that $\\psi(y)$ is a constant on the interval $[a,1]$. Note that it is easy to contradict extremality of $\\psi$ if $\\psi$ is not one on this interval.  In case it is one on this interval, consider the function $\\psi_1(x)=1-\\psi^{-1}(1-x)$ (where $\\psi^{-1}(1)=a$).   We claim that \n\\begin{align*}\n\\int_0^1 \\psi_1(y){\\rm{d}}y&=\\int_0^1 \\psi(x){\\rm{d}}x ,\\,\\,\\text{ and }\n\\int_0^1 \\sqrt{\\dot{\\psi_1}(y)}{\\rm{d}}y=\\int_0^1 \\sqrt{\\dot\\psi(x)}{\\rm{d}}x .\n\\end{align*}\nThe first equality follows by Fubini's theorem. The second becomes clear after the change of variable $y=1-\\phi(x)$ is made.\nNow, by uniqueness, $ \\psi_1=\\psi$, and by Lemma \\ref{l:ac}  $\\psi$ \nhas no singular part. However, $\\psi_1$ has an atom of mass $1-a$ at $0$ which implies $a=1$.\n\\qed\n\n\n\n\\subsection{Proof of Lemma \\ref{crude1}} The reader may find it  useful to refer to Figure~\\ref{fig1}. Let $\\omega_1$ and $\\omega_2$ be the acute angles that the line segments $((n,0),u)$ and $((n,0),v)$ make with the \n$x$-axis and the $y$-axis. Now, without loss of generality, we can assume $0\\le \\omega_1 \\le \\pi\/4$; otherwise, we could work with $\\omega_2$, since clearly one of the two quantities is at most $\\pi\/4$.\nNow, as a simple consequence of Theorem \\ref{lln2}, we see that, with probability at least $1-e^{-cn},$ $(n,0)$ is at a $\\Theta(n)$ distance from $\\Gamma^*_{\\alpha,n}$. Also, since $\\psi_{\\alpha}$  is strictly concave, and $\\omega_1 \\le \\pi\/4$, the line segment $(u,v)$  makes at least an angle $c(\\alpha)>0$ (depending only on $\\alpha$ and not on $u$ and $v$) with the $x$-axis. \nThe proof is now completed by  considering the triangle $((n,0),u,v)$   and simple geometric arguments. We omit the details. \n\\qed\n\\subsection{Proof of Theorem \\ref{concentration1}} The proof will follow by constructing a suitable martingale and appealing to well known concentration results for martingales. However things are slightly complicated by the possibility that the  martingale may not have bounded increments. A simple truncation argument will allow us to take the increments to be bounded.  \nOur proof strategy follows closely arguments in \\cite{BB}.\nThe proof relies on some coarse graining. We start by introducing some notation. \nLet  $$A_{i}:=\\{(x,y):i\\le x+y < {i+1}\\}.$$ \nAlso let $B_{i,j}= [{i-1}, {i})\\times [{j-1}, {j})$. Fix a number $k$ to  be specified soon.  Recall  the point process $\\Pi$ and the path $\\Gamma=\\Gamma_{\\alpha,n}$ from Section \\ref{def10}. Let $\\Gamma_{k}=\\Gamma_{k,\\alpha,n}$ be the  increasing path between $(0,0)$ and $(n,n)$ with the same constraints as $\\Gamma$ along with the additional  constraint that the intersection with $A_i$ is less than $k$ for all $i \\in [0,\\ldots, 2n-1]$.  Let us denote  by $|\\Gamma_{k}|$ the weight of $\\Gamma_{k}$. Consider the Doob Martingale $\\{\\mathbb{E}\\left[|\\Gamma_k|\\mid \\mathcal{F}_i\\right]\\}_{0\\le i\\le 2n-1}$, \nalong  the filtration $\\mathcal{F}_i=\\{\\Pi \\cap A_j : j\\le i\\}$.   Note that the martingale increments are deterministically bounded by $k$.\nThe following is a standard consequence of the Azuma-Hoeffding inequality. \n\\begin{lemma} $\\mathbb{P}(\\bigl|  |\\Gamma_{k}|-\\mathbb{E}(|\\Gamma_{k}|)|\\ge t)\\le e^{-\\frac{t^{2}}{4k^2n}}$.\n\\end{lemma}\n\nIn the remainder of the section, we obtain a bound on $\\bigl| |\\Gamma|-|\\Gamma_{k}|\\bigr|$ and thus also on $\\bigl|\\mathbb{E}|\\Gamma|-\\mathbb{E}|\\Gamma_{k}|\\bigr|$.\nLet $\\Pi_*$ be the point process obtained from $\\Pi$ by removing points arbitrarily if necessary from  $\\Pi \\cap B_{i,j}$ to make sure that  $|\\Pi_{*}\\cap B_{ij}| \\le k\/3$ for all $1\\le i,j\\le n$.\nLet $\\Gamma_{*}:=\\Gamma_{\\alpha,*}$ be the longest path with the same constraints as $\\Gamma_{\\alpha}$ in the environment $\\Pi_{*}$.\n\n\\begin{lemma} Deterministically as a consequence of the definitions it follows that $|\\Gamma_{k}| \\ge |\\Gamma_{*}|$.\n\\end{lemma}\n\n\\begin{proof} Clearly it suffices to show  that $\\Gamma_{*} $ intersects none of the $A_{i}$'s at more than $k$ points. The proof is by contradiction: assume that there exists $i$ such that \n$|\\Gamma_*\\cap A_i|\\ge k+1$. Since the points on $\\Gamma_*$ are totally ordered (recall the ordering introduced in Section \\ref{def1012}), let $(x_1,y_1)$ and $(x_2,y_2)$ be the smallest and largest\npoints in the set $\\Gamma_* \\cap A_i$. By definition of $A_i$, \n\\begin{equation}\\label{last}\n(x_2-x_1)+(y_2-y_1) \\le 1,\n\\end{equation}  \nso that $\\Gamma_*\\cap A_i$ intersects at most three $B_{i,j}'s$,  since the number of different indices it can cross in each direction  is at most one if \\eqref{last} is to be satisfied.\nThus $\\Gamma_*\\cap A_i$ \nhas to intersect at least one of the boxes at more than $\\frac{k+1}{3}$ points. This contradicts the definition of $\\Gamma_*$.\n\\end{proof}\nThus, it follows that $|\\Gamma|-|\\Gamma_{k}|\\le |\\Gamma|-|\\Gamma_*| \\le C$, where $C=\\sum_{i,j} \\max(|\\Pi_{i,j}|-k\/3,0)$ and $\\Pi_{i,j}=\\Pi \\cap B_{i,j}$. Taking $k=6\\frac{\\log n}{\\log\\log n}$, the proof is now complete in view of the next result. \n\\begin{lemma}The random variable $C$ satisfies the following:\n\\begin{enumerate} \n\\item\n$\\mathbb{P}(C\\ge \\lambda n^{1\/2}\\frac{\\log n}{\\log \\log n}) \\le 2 \\lambda ^2 e^{-\\lambda^2}$\n\\item $\\mathbb{E}(C)\\le 1$.\n\\end{enumerate}\n\\end{lemma}\n\nThis is a simple consequence of the observation that $|\\Pi_{i,j}|$ are independent Poisson variables with mean one and the  following tail bound of a standard Poisson variable $X$:\n$P(X>r)\\le e^{-r\\log r +r}$.\n\\qed\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Real-Time Maude and Sockets}\n\\label{sec:maude-background}\n\nIn this section we briefly cover the important constructs we used from \nReal-Time Maude and Maude Sockets. We assume the reader is familiar with basic \nMaude constructs including modules ({\\tt mod}), sorts ({\\tt sort}), operators \n({\\tt op}), unconditional and conditional equations ({\\tt eq} and {\\tt ceq}) \nand unconditional and conditional rules ({\\tt rl} and {\\tt crl}).\n\n\\subsection{Full Maude and Real-Time Maude}\n\nFull Maude \\cite{fullmaude} is a Maude interpreter written in Maude, which in\naddition to the Core Maude constructs provides syntactic constructs such as\nobject oriented modules. Object oriented (OO) modules implicitly add sorts\n{\\tt Object} and {\\tt Msg}. Furthermore, OO-modules add a sort called {\\tt\nConfiguration} which consists of a multiset of terms of sort {\\tt Object} or {\\tt\nMsg}.\n\nObjects are represented as records:\n\n\\begin{footnotesize}\n\\begin{alltt}\n< {\\em objectID} : {\\em classID} | {\\em AttributeName} : {\\em Attribute}, ... >\n\\end{alltt}\n\\end{footnotesize}\n\nRewrite rules are then used to describe state transitions of objects\nbased on consumption of messages. For example, the following rule expresses\nthe fact that a pacemaker object consumes a message to set the pacing\nperiod to T:\n\n\\begin{footnotesize}\n\\begin{alltt}\nrl setPeriod(pm, T)\n   < pm : Pacing-Module | pacing-period : PERIOD >\n   => < pm : Pacing-Module | pacing-period : T > .\n\\end{alltt}\n\\end{footnotesize}\n\nReal-Time Maude \\cite{rt-maude} is a real-time extension of \nMaude in Full Maude. It adds syntactic constructs for defining timed \nmodules. Timed modules automatically import the {\\tt TIME} module, which defines \nthe sort {\\tt Time} (which can be chosen to be discrete or continuous) along\nwith various arithmetic and comparison operations on {\\tt Time}. Timed modules\nalso provide a sort {\\tt System} which encapsulates a {\\tt Configuration} and\nimplicitly associates with it a time stamp of sort {\\tt Time}. After defining a\ntime-advancing strategy, Real-Time Maude provides timed execution ({\\tt\ntrew}), timed search ({\\tt tsearch}), which performs search on\na term of sort {\\tt System} based on the time advancement strategy, and\ntimed and untimed LTL model checking commands.\n\n\\subsection{Deterministic Timed Rewriting in Real-Time Maude}\n\nWe are interested in emulations of real-time systems specified in Real-Time \nMaude. For useful real execution, a self-evident condition is that the \ntime-advancing rewrite rules in the specification should be deterministic. This \ncan be achieved by defining only one time-advancing rewrite rule on the system \nwith two auxiliary operators {\\em tick} and {\\em mte} \\cite{rtmaude-manual}. In \nour specification this is captured in {\\tt TIME-ADV-SEMANTICS}, which is \nincluded by all other timed modules:\n\n\\begin{footnotesize}\n\\begin{verbatim} \n(mod TIME-ADV-SEMANTICS is ...\n  op mte : Configuration ~> TimeInf . ...\n   eq mte(none) = INF .\n   eq mte(C C') = minimum(mte(C), mte(C')) .\n  op tick : Configuration Time ~> Configuration . ...\n   eq tick(none, T) = none .\n   eq tick(C C', T) = tick(C, T) tick(C', T) .\n  op def-te : -> Time .\n  op max-te : -> TimeInf .\n\n   crl {GC} => {tick(GC, T)} in time T if T <= mte(GC) [nonexec] .\nendm)\n\\end{verbatim}\n\\end{footnotesize}\n\nThe rewrite rule at the end is assumed to be the only timed rewrite rule (a \nrewrite rule that advances the time stamp of the system) in the system \nspecification. The {\\tt mte} operator defines the maximum time elapse before \nany 0-time rewrite rule can be applied. The {\\tt tick} operator defines how \nthe system state changes due to time advancement between applications of \n0-time rewrite rules. We also define a default time elapse, {\\tt def-te}, \nand maximum time elapse, {\\tt max-te}, inside the module to be used as \nparameters during real execution.\n\n\\subsection{Socket Programming in Maude}\n\nMaude supports the Berkley sockets API for TCP communication. This is done by \nhaving a special gateway object, denoted {\\tt <>}, to consume all the messages \nresponsible for setting up sockets and communicating to an external environment \n(e.g. {\\tt createClientTcpSocket}, {\\tt send}, {\\tt receive}). The gateway \nobject will also generate messages upon status updates from the socket (e.g. \n{\\tt sent}, {\\tt received}, {\\tt closedSocket}). Consuming and generating \nmessages from the gateway object is captured by external rewrite rules which \ncan be executed using the {\\tt erew} command in Core Maude. An important thing \nworth pointing out about external rewrite rules is that {\\em external rewrite \nrules are only applied when no internal rewrite rules can be applied}. Also, \nusing external rewrite rules with Real-Time Maude specifications (built on top \nof Full Maude) requires reflecting the specification down to a Core Maude \nmodule before executing.\n\n\\section{Case Studies}\n\n\\label{sec:case}\n\n\\subsection{A Pattern for Medical Device Execution}\n\nWe briefly described in the introduction at a high level that the model \nexecution framework is to support rapid prototyping of instantiated medical \ndevice safety patterns. In \\cite{sun-meseguer-sha-wrla10} and \\cite{tech-rep}, \nwe have described in detail the command shaper pattern for medical device \nsafety. In essence, the command shaper pattern can modify commands to an \nexisting medical device to guarantee specific safety properties in terms of \nlimiting durations of stressful states and limiting the rate of change (Figure \n\\ref{fig:commandshaper}).\n\n\\begin{figure}\n\\centering \\includegraphics[width=10cm, angle=0]{figures\/pacemaker_wrapper}\n\\caption{Command Shaper Pattern for a Pacemaker}\n\\label{fig:commandshaper}\n\\end{figure}\n\n\n\\subsection{Pacemaker Simulation Case Study}\n\nOne of the applications for the command shaper pattern is a pacemaker system \n\\cite{sun-meseguer-sha-wrla10}. At a high level the safety properties \nguaranteed by the command shaper pattern is that the pacemaker will not pace at \nfast heart rates too frequently or for too long, and the pacing rate will \nchange gradually. We omit the details of instantiating the medical device \npattern, but the final wrapper object provided by the pattern is:\n\n\\begin{footnotesize}\n\\begin{verbatim}\n(tomod PARAM-PACEMAKER is pr EPR-WRAPPER-EXEC{Safe-Pacer} ...\n   eq wrapper-init =\n      < pacing-module : EPR-Wrapper{Safe-Pacer} |\n         inside :\n            < pacing-module : Pacing-Module |\n               nextPace : t(0),\n               period : safe-dur >,\n         val : safe-dur,\n         next-val : safe-dur,\n         disp : t(period),\n         stress-intervals : (nil).Event-Log{Stress-Relax} > .\n... endtom)\n\\end{verbatim}\n\\end{footnotesize} \n\nThis says that a wrapper is placed around a pacing module, and the initial \npacing rate is set as the default safe-duration ({\\tt safe-dur} is 750 ms or 80 \nheart beats per minute). Verifying this instantiation (with a simple pacemaker \nlead model \\cite{tech-rep}) indicates that the safety properties are met by the \npattern. However, with the power of the model emulation framework, we can \nimmediately use this specification to run with an actual pacemaker. In this \npaper we demonstrate this emulation capability not on an actual pacemaker but \non a pacemaker simulator (a Java widget that receives messages about when to \npace and draws a simple line graph resembling an ECG trace). Before the system \ncan be emulated with the pacemaker simulator, some interface information must \nbe provided. The entire module providing all the necessary interface \ninformation is shown below:\n\n\\begin{footnotesize}\n\\begin{verbatim}\n(mod CREATE-TICKER is\n   inc PARAM-PACEMAKER .\n   inc TIME-CLIENT .\n   inc SEND-RECEIVE-CLIENT .\n\n   eq addr = \"localhost\" .\n   eq port = 4444 .\n\n   eq def-te = 1 .\n   eq max-te = INF .\n   eq time-grain = 10 . --- milliseconds\n\n   op pacer-client : -> Oid .\n   eq internal = wrapper-init .\n   \n   eq out-adapter(shock)\n      = createSendReceiveClient(pacer-client, \"localhost\", 4451, \"shock\") .\n   eq in-adapter(msg-received(pacer-client, \"shocked\\n\"))\n      = set-period(pacing-module, 50) .\nendm)\n\\end{verbatim}\n\\end{footnotesize} \n\nThe module first indicates that the TCP\nsocket interface to the pacemaker simulator is {\\em localhost} on {\\em port}\n4444. The default time elapse for one tick is 1 time unit. The maximum time elapse\nfor one tick step is infinity (i.e. there is no maximum). The duration of one time\nunit is 10 milliseconds. The time units are in terms of milliseconds since the\nminimum time granularity provided by the Java time interfaces is 1 millisecond.\n\nThe equation for {\\tt internal} specifies that the internal configuration to be\nexecuted is the configuration defined by {\\tt wrapper-init} (as defined in {\\tt\nPARAM-PACEMAKER}). Also, the last two equations specify that the output message\nshock should be mapped to a string ``shock'' sent over the socket, and upon\nreceiving the acknowledgment message ``shocked'' set the pacing period to 500\nms (120 bpm - a really fast heart rate). The last equation creates the scenario\nwhere a stressful heart rate is always being sent to the pacing module. Since\nthe command shaper pattern should prevent this unsafe behavior, we should see\nthe pacing automatically slow down from 120 bpm after some time interval.\n\nThe module is executed by first reflecting the {\\tt CREATE-TICKER} module down \nto Core-Maude (with the command {\\tt show all CREATE-TICKER}), and executing \nwith the {\\tt erew} command. A snapshot of the ``ECG'' trace of the pacemaker \nsimulator is show in Figure \\ref{fig:pacing}. For validation, we measured the \njitter for executing such a system -- the physical time required to completely \nexecute 0-time rules and finish communication (Figure \\ref{fig:jitter}). The \nresults were obtained from a 1.67 GHz Dual-Core Intel Centrino with Maude \nrunning in Windows through Cygwin (tracing was turned off). The main thing to \nnotice is that the jitter is mostly below 0.1 seconds and almost never exceeds \n0.2 seconds. This amount of jitter is tolerable since most medical devices need \nto respond in the order of seconds. The pacemaker is a bit more strict in terms \nof its timing requirements. To evaluate suitability for the pacemaker, we \nplotted the recorded the physical time duration between pacing events (Figure \n\\ref{fig:pacingintervals}). Notice in this example the heart rate increases \n(duration decreases) up to a limit and then the heart rate starts to decrease \n(duration increases) and the cycle repeats. It is clear that the jitter in \ncontrol seems tolerable since there are no sharp spikes in the graph of the \npacing durations.\n\n\n\\begin{figure}\n\\centering \\includegraphics[width=6cm, angle=0]{figures\/pacing}\n\\caption{Trace from Pacemaker Simulator}\n\\label{fig:pacing}\n\\end{figure}\n\n\\begin{figure}\n\\centering \\includegraphics[width=12cm, angle=0]{figures\/execution_jitter}\n\\caption{Model Execution Jitter Distribution}\n\\label{fig:jitter}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering \\includegraphics[width=12cm, angle=0]{figures\/pacing_intervals}\n\\caption{Pacing periods recorded by the pacemaker simulator (jitter effects\nare reflected by noise on the curve)}\n\\label{fig:pacingintervals}\n\\end{figure}\n\n\\subsection{Syringe Pump Case Study}\n\nThe pacemaker emulation example was demonstrated through a simulated\npacemaker mostly because current pacemakers do not have external interfaces for\nsetting when to pace (and rightly so). However, for devices such as electronic\nsyringe pumps these interfaces are available. Syringe pumps and infusion pumps\nin general deliver intravenous injections into a patient. For this scenario, we\nassume that the syringe pump is delivering an analgesic (e.g.\\ morphine) to the\npatient, and we would like to prevent overdose. We assume that for a normal\npatient overdoses do not occur at the base rate of infusion and can only result\nif a bolus dose is administered too often for the patient. We again use the\ncommand shaper pattern to limit the frequency and duration of bolus doses. The\ninstantiated pump is as follows:\n\n\\begin{footnotesize}\n\\begin{verbatim}\n(tomod PARAM-PUMP is\n   pr EPR-WRAPPER-EXEC{Safe-Pump} .\n   pr DELAY-MSG .\n...\n   eq msgs-init =\n      delay(set-mode(pump-module, bolus), t(9))\n      delay(set-mode(pump-module, bolus), t(11))\n      delay(set-mode(pump-module, bolus), t(12)) ... .\n   eq wrapper-init =\n      < pump-module : EPR-Wrapper{Safe-Pump} |\n         inside :\n            < pump-module : Pump-Module |\n               mode : base\n            > base,\n         val : base,\n         next-val : base,\n         disp : t(period),\n         stress-intervals : (nil).Stress-Relax-Log > .\n... endtom)\n\\end{verbatim}\n\\end{footnotesize}\n\nThis module shows the initialized wrapper object for the pump, with the initial\nstate being the base rate of infusion. Furthermore, there is also a set of\ndelayed messages that will be sent to the pump. In the term {\\tt msgs-init}, the\nmodel will send bolus requests at 9 time units, 11 time units, 12 time units,\n\\ldots after the start of execution for the system. Again, creating a simulated\npatient model, we can verify the safety of the instantiated pattern\n\\cite{tech-rep}. Instantiating the pump is similar to instantiating the\npacemaker, except that there are a few more types of output messages.\n\n\\begin{footnotesize}\n\\begin{verbatim}\n(mod CREATE-TICKER is\n   inc PARAM-PUMP .\n   inc TIME-CLIENT .\n   inc SEND-RECEIVE-CLIENT .\n\n   eq addr = \"localhost\" .\n   eq port = 4444 .\n\n   eq def-te = 1 .\n   eq max-te = INF .\n   eq time-grain = 1000 . --- milliseconds\n\n   ops pump-client pump-client' : -> Oid .\n   eq internal = wrapper-init msgs-init .\n   \n   eq out-adapter(stop)\n      = createSendReceiveClient(pump-client, \"localhost\", 1234, \"STP\") .\n   eq out-adapter(base)\n      = createSendReceiveClient(pump-client, \"localhost\", 1234, \"RAT1\")\n      createSendReceiveClient(pump-client', \"localhost\", 1234, \"RUN\") .\n   eq out-adapter(bolus)\n      = createSendReceiveClient(pump-client, \"localhost\", 1234, \"RAT2\")\n      createSendReceiveClient(pump-client', \"localhost\", 1234, \"RUN\") .\n   var S : String .\n   eq in-adapter(msg-received(pump-client, S))\n      = none .\n   eq in-adapter(msg-received(pump-client', S))\n      = none .\nendm)\n\\end{verbatim}\n\\end{footnotesize}\n\nThe model is communicating with {\\em localhost} on {\\em port} 4444. The time \ngranularity is 1 second. The internal configuration being executed is the \nwrapped pump as well as the set of messages that will deliver bolus requests. \nThe output requests are handled by a Java thread listening on port 1234 and \nforwarding the request string to the actual {\\em Multi-Phaser NE-500} Syringe \nPump (Figure \\ref{fig:pump}). A few important requests to the pump are: {\\tt \nSTP} stop the pump, {\\tt RAT <n>} set infusion rate to {\\tt n} ml\/hr, {\\tt RUN} \nstart the infusion. Reflecting down the {\\tt CREATE-TICKER} module and \nexecuting with {\\tt erew} will now control the physical pump motor!\n\n\\begin{figure}\n\\centering \\includegraphics[width=8cm, angle=0]{figures\/pump}\n\\caption{Multi-Phaser NE-500 Syringe Pump}\n\\label{fig:pump}\n\\end{figure}\n\nAs a validation for correct pump control, we used a Salter Brecknell 7010SB \nscale to weigh the amount of liquid infused from the syringe pump over time \n(Figure \\ref{fig:pumpdata}). The data granularity is a bit rough since the \nscale can only measure within a precision of 0.1 oz. For this example, to \nclearly distinguish between two pump states, we let the base rate of infusion \nbe zero (horizontal parts of the graph) and the bolus rate be the maximum \ninfusion rate provided by the pump (positive sloped parts of the graph). Bolus \nrequests are continuously sent to the pump. The safety properties require that \nbolus doses last no longer than 30 seconds, and there must be 10 seconds \nbetween bolus doses, and at most 3 bolus doses for a window size of 3 minutes. \nThe graph validates that these properties are indeed satisfied for this \nparticular execution of the pump.\n\n\\begin{figure}\n\\centering \\includegraphics[width=8cm, angle=0]{figures\/pump_data}\n\\caption{Infusion Volume over Time}\n\\label{fig:pumpdata}\n\\end{figure}\n\n\n\n\\section{Conclusion}\n\\label{sec:conc}\n\nSafety of medical devices and of their interoperation is an unresolved issue \ncausing severe and sometimes deadly accidents for patients.  Formal methods, \nparticularly in support of highly generic and reusable formal patterns whose \nsafety properties have been verified can help in ensuring the safety of \nspecific components, but this still leaves several open problems including: (i) \nhow to pass from specifications to code and from logical time to physical time \nin a correctness-preserving ways; and (ii) how to experimentally validate \nmedical safety architectures in realistic scenarios in which actual devices and \nmodels of patients and doctors can interact with formally specified and \nprovably safe designs of device components.\n\nBy developing virtual emulation environments in which highly generic and \nreusable formally verified patterns in Real-Time Maude can be easily \ntransformed into emulations in physical time which can interact with other real \ndevices and with simulations of patient and\/or doctor behaviors, we have taken \nsome first steps towards a seamless integration of formal specification and \nverification with emulation and testing, and ultimately with deployment of \nmedical DES systems that offer much stronger safety guarantees than what is \ncurrently available.  Much work remains ahead.  As we explain in \n\\cite{sun-meseguer-sha-wrla10}, the provably safe formal pattern used in the \nexperiments of this paper is just one such pattern: it covers a useful class, \nbut does not cover other kinds of safety needed in other medical devices. Also, \nother safety concerns, such as so-called open-loop safety, ensuring that \nmedical devices will always be in states safe for the patient even under key \ninfrastructure failures, such as network disconnection, have not been addressed\nin this work.  However, we believe that the general methodology presented here \nto pass from formal specifications to virtual emulation environments and \neventually to deployed systems should also be applicable to those new formally \nverified patterns that have yet to be developed.\n\n\n\\section{Introduction}\n\n\\label{sec:intro}\n\nEach year, just in the US hospitals, a shocking and almost unacceptable number \nof medical accidents occur. In a 2009 study, reports estimate 40,000 instances \nof medical harm occur daily, and from the 2005 through 2007 period, at least \n92,882 deaths were potentially preventable \\cite{HealthGrades2009}. Many of \nthese accidents happen due to mistakes and failures in the \n\\emph{interoperation} of medical devices. A modern hospital's operating room is \nin fact a quite complex distributed embedded system (DES) with many devices \ninvolved in either passively monitoring the patient state or actively \nperforming different parts of a procedure. Both the safety of the individual \ndevices and the safety of interoperation between devices (and between the \npatients and doctors) are of paramount importance. Presently, this safety is \nnot adequately guaranteed.\n\nWhen the reported accidents are analyzed, it becomes clear that many of them \ncould and should have been avoided if the DES formed by the devices, the \npatient, and the doctors had been properly designed and analyzed, so that many \nunsafe interactions become \\emph{impossible} by design.  The use of formal \nmethods can clearly help in this respect, and promising research advances have \nalready been made in this direction (see, e.g., \n\\cite{alur04,arney07,ray04,jetley06}).\n\nIn our recent work \\cite{tech-rep}\\cite{sun-meseguer-sha-wrla10}, we have \ndeveloped a scalable and highly reusable approach to the safety of medical \ndevices by means of \\emph{formally verified patterns} that: (i) are formally \nspecified as real-time rewrite theories in Real-Time Maude; (ii) are generic, \nso that they apply not to a single device but to a wide range of devices, and \nare therefore specified as \\emph{parameterized} modules; (iii) come with \nexplicit \\emph{formal requirements} (specified in their parameter theories) \nthat must be met by any pattern instantiation to be correct; and (iv) come with \n\\emph{formal safety guarantees} that will be satisfied by any correct \ninstantiation of the pattern. For example, in \\cite{tech-rep} we present one \nsuch pattern and show how it can be instantiated to obtain safe controllers for \nquite different devices, such as a pacemaker, an infusion pump for analgesia, \nand the interoperation of a ventilator with an X-ray machine.\n\nHowever, there is still a substantial gap between the verified safety of \ndesigns in formal specifications and the actual safety of real medical devices, \npatients and doctors in an operating room for at least two reasons.  First, the \nformal specifications are somewhat idealized abstractions in which, for \nexample, time is not the actual physical time that devices need to operate in, \nbut logical time, and the code of the actual devices and controllers is not \nused but instead some formal specifications are used.  Second, it is important \nto consider not just the safety of a single device or small collection of \ndevices, but also that of their \\emph{interoperation} with other devices and \nwith the patient and the doctors.  This work takes some first steps towards the \ngoal of bridging the gap between formal specifications and actual devices in a \nhospital to help ensure that safety properties are preserved in the passage \nfrom specification to actual code and physical devices.  To achieve this goal\nwe propose the use of \\emph{virtual emulation environments} in which:\n\\begin{enumerate}\n\\item formally verified patterns \\cite{sun-meseguer-sha-wrla10}  can be \ninstantiated to obtain various concrete specifications of desired devices and \ncontrollers;\n\n\\item the so-obtained formal executable specifications of devices and \ncontrollers are used \\emph{directly} to generate emulators that perform the \nsame specified behavior in physical time;\n\n\\item actual devices, as well as actual executable models of patient and doctor \nbehavior, can be seamlessly integrated with specification-based emulators to \nvalidate the safety not just of individual devices but also of various \nDESs that are needed in practice in actual operating room conditions and \nscenarios.\n\\end{enumerate}\nThe advantage of point (1) is a great degree of reusability, and amortizing the \nformal verification effort across potentially many devices.  The advantage of \n(2) is that, since each specification-based emulator executes the exact same \nformal specification that has been proved safe for the given device, the safety \nof such an emulator is automatically guaranteed, and a path remains open to \ncorrectly generate actual code for it preserving such safety in an actual \nimplementation. The advantage of point (3) is that system experimentation with \nphysical time and actual devices becomes available from very early in the \ndesign process and are available afterwards along the entire development \nprocess: initially, only specifications may be emulated; at intermediate \nstages, both specifications and actual devices form the virtual emulation \nenvironment; and in the end the emulating environment seamlessly becomes an \nactual implementation.\n\nTechnically, the way such virtual emulation environments are obtained from \nformal specifications is by using a key idea first demonstrated by Musab \nAl-Turki to semi-automatically pass from a Real-Time Maude formal executable \nspecification operating in \\emph{logical time} to a corresponding \n\\emph{physical emulation} of the same specification operating in \\emph{physical \ntime} and possibly interacting with other devices in a distributed way.  The \nkey observations are: (i) in Real-Time Maude rewrite rules are either 0-time \nrules requiring no time, or time-advancing rules moving the entire system \nforward in logical time; (ii) time advancing rules (typically a single such \nrule) can be physically implemented by an external object that sends time ticks \naccording to physical time;  (iii) although so-called 0-time rules do take some \nphysical time to be executed, if this time is small enough in comparison with \nthe time granularity of the physical time period chosen, for all practical \npurposes they can be considered to take 0 time units to execute; and (iv) the \nMaude infrastructure for Maude computations to interact with external objects \nvia sockets can be used to interface the Maude objects in the formal \nspecification with the external ticker object and also to other external \ndevices.\n\n\\subsection{Our Contribution.}  This is the first work we are aware of in which \nformal specifications of real-time components are directly used in the area of \ndistributed embedded systems for medical applications to obtain a virtual \nemulation environment in which specifications, patients, doctors, and actual \ndevices can be emulated in physical time, and such that the correctness of \nverified specifications is preserved, provided adequate timing \nconstraints are obeyed (see Section \\ref{sec:issues}).  This work is a first \nstep towards a seamless integration of formal specification, verification, and \nsystem development and testing for safe medical systems.  The associated notion \nof a virtual emulation environment plays a crucial role in passing from \nspecifications to code and devices, and from logical time to physical time.  We \nhave demonstrated both the feasibility and the usefulness of these methods in \ntwo concrete scenarios: one in which a pacemaker interacts adaptively in \nphysical time with a simulated model of a patient heart and keeps heart rates \nwithin a safe envelope; and another in which a safety controller for \npatient-controlled analgesia interacts in physical time with an actual drug \ninfusion device and with a simulation of patient behavior.\n\nIn the passage from formal real-time specifications to their corresponding \nemulators there are additional novel contributions that were required for this \nwork, including: (i) advancing time by the maximum time elapsable as opposed to \nby a fixed ticking period; (ii) handling asynchronous interrupts in addition to \nsynchronous communication; (iii) emulating the interaction of real medical \ndevices, patient models, and formal specifications; and (iv) generating a timed \nwrapper for each component specification almost for free with deterministic \nReal-Time Maude specifications using both a time ticker and the computation of \nthe maximum time elapsable for each time advance.\n\nThe paper is organized as follows: Section \\ref{sec:overview} provides the high \nlevel ideas of the framework from formal patterns to real-time execution. \nSection \\ref{sec:maude-background} covers the basics of Real-Time Maude and \nMaude's support for socket programming. Section \\ref{sec:io} and \n\\ref{sec:emulation} describes the core of the execution framework which allows \nseamless passage from Real-Time Maude specifications to execution with physical \ntime and physical devices. Section \\ref{sec:case} covers case studies for a \npacemaker and a syringe pump to evaluate the feasibility of using formal models \nto execute medical devices. Finally, we describe some fundamental assumptions \nrequired for formal model execution to work for medical devices in Section \n\\ref{sec:issues}, and we conclude in Section \\ref{sec:conc}.\n\\section{Mapping Internal Messages to External I\/O}\n\\label{sec:io}\n\nValidating the design of a device in an execution environment requires handling \nits outputs. After all, the end validation of a system's behavior is based on \nits outputs. Thus, it seems reasonable to talk about how internal messages in \nthe model can be converted into messages for communicating with the external \nworld. This section also serves as an explanation for unfamiliar readers of how \nMaude sockets are used.\n\nIn order to talk about external communication, we must first define in the \nmodel what is external. The model will have an internal distributed actor \nconfiguration with internal messages as well as messages to be output to the \nexternal world. Thus, the first definition is {\\tt EXTERNAL-CONFIGURATION} \nwhich defines external messages {\\tt ExtMsg} as subsort of {\\tt Msg}. \nFurthermore, external messages are classified in terms of incoming external \nmessages {\\tt InExtMsg} and outgoing external messages {\\tt OutExtMsg}. A \nconfiguration is called {\\em open} if there are external messages present in \nthe configuration: either an incoming external message has not been delivered, \nor an outgoing external message has not been sent. The predicate {\\tt open?} is \ndefined accordingly.\n\n\\begin{footnotesize}\n\\begin{verbatim}   \nsubsorts InExtMsg OutExtMsg < ExtMsg < Msg .\nop open? : Configuration -> Bool .\n eq open?(C C') = open?(C) or open?(C') .\n eq open?(O) = false .\n eq open?(M) = M :: ExtMsg .\n\\end{verbatim}\n\\end{footnotesize}\n\nActually sending an external message may be more complex than just forwarding \nthe message through the gateway object. External messages may not be the same \nin the internal configuration and in the external configuration. For example, a \nsimple output message in the internal configuration may need to be mapped to a \nclient object that initiates the communication to deliver the message. \nOperators {\\tt in-adapter} and {\\tt out-adapter} are defined to perform these \nmappings from external message client configurations to internal messages.\n\nAn example of an output adapter for a pacemaker message to beat the heart\nmay be:\n\n\\begin{footnotesize}\n\\begin{verbatim}\neq out-adapter(shock)\n    = createSendReceiveClient(pacer-client, \"localhost\", 4451, \"SetLeadVoltage 5V\")\n\\end{verbatim}\n\\end{footnotesize}\n\nIn this example, the message {\\tt shock} is transformed into a client object \nwhich sends a message on port 4451 with the string {\\tt \"SetLeadVoltage 5V\"} \nindicating that the proxy server will then proceed to set a 5V voltage on the \npacemaker lead.\n\n\\subsection{One-Round Communication Clients}\n\nOnce the external message is mapped into a client configuration, we must define\nthe rewrite rules to specify how the communication protocol works with the\nexternal device. Here we describe a simple {\\tt SEND-RECEIVE-CLIENT} which is\nresponsible for establishing communication, sending a message, receiving a\nreply, and then closing the communication. Although simple, this type of\nprotocol is sufficient for most of the communication for medical devices we have\nused in our case studies.\n\n\\begin{footnotesize}\n\\begin{verbatim}\n(mod SEND-RECEIVE-CLIENT is ...\n  op createSendReceiveClient : Oid String Nat String -> Configuration .\n   eq createSendReceiveClient(CLIENT, ADDRESS, PORT, SEND-CONTENTS)\n      = < CLIENT : SendReceiveClient | ... >\n      createClientTcpSocket(socketManager, CLIENT, ADDRESS, PORT) .\n  op msg-received : Oid String -> InExtMsg .\n...endm)\n\\end{verbatim}\n\\end{footnotesize}\n\nAfter creating the client and establishing communication, the client goes into \none round of send and receive before the socket is closed. Once the socket is \nclosed, the entire client object is converted into one reply message to be \ndelivered to the internal configuration using the operator {\\tt msg-received}.\n\n\\begin{footnotesize}\n\\begin{verbatim}\n--- send contents\nrl createdSocket(CLIENT, socketManager, SOCKET-DST)\n   < CLIENT : SendReceiveClient | ... send-contents : SEND-CONTENTS >\n   => < CLIENT : SendReceiveClient | ... > send(SOCKET-DST, CLIENT, SEND-CONTENTS) .\n--- receive contents\nrl sent(CLIENT, SOCKET-DST) < CLIENT : SendReceiveClient | ... >\n   => < CLIENT : SendReceiveClient | ... > receive(SOCKET-DST, CLIENT) .\n--- close socket\nrl received(CLIENT, SOCKET-DST, RECEIVE-CONTENTS) < CLIENT : SendReceiveClient | ... >\n   => < CLIENT : SendReceiveClient | ... recv-contents : RECEIVE-CONTENTS >\n      closeSocket(SOCKET-DST, CLIENT) .\n--- done\nrl closedSocket(CLIENT, SOCKET-DST, \"\")\n   < CLIENT : SendReceiveClient | ... recv-contents : RECEIVE-CONTENTS >\n   => msg-received(CLIENT, RECEIVE-CONTENTS) .\n\\end{verbatim}\n\\end{footnotesize}\n\n\\section{Assumptions and Issues}\n\n\\label{sec:issues}\n\nIn this section we discuss the timing assumptions that need to be taken into\naccount to ensure that the emulation of a Real-Time Maude specification\ncorrectly implements the logical time behavior.\n\nFigure \\ref{fig:timing} shows one round of communication between the time server\nand the formal model.\n$t_{comm,i}$ denotes the delays due to each stage of communication.\n$t_{rew,i}$ denotes the delays incurred by each stage of rewriting. $t_{proc}$ \ndenotes the time needed at the physical device interface to process\nthe commands. Thus, the entire time to finish a round is $t_{round} =\nt_{comm1} + t_{rew1} + t_{comm2} + t_{proc} + t_{comm3} +\nt_{rew2} + t_{comm4}$.\n\n\\begin{figure}\n\\centering \\includegraphics[width=10cm, angle=0]{figures\/timing}\n\\caption{Timing Considerations}\n\\label{fig:timing}\n\\end{figure}\n\nIn logical time, no time advancement should actually take place in a \ncommunication round. All computation and message communication is assumed to \ntake zero time. Of course, for proper timed operation we can relax these \nconstraints to first allow the model to have a non-zero (but bounded) delay for \nthese computations and communications. The maximum bound on these \ncommunications is $t_{round} \\leq mte(C_{next})$. Otherwise, by the time the \nround has completed, the execution is already delayed passed the time for the \nmaximum time elapse for the next 0-timed rewrite rule; i.e., the maximum speed \nof execution of the formal model and communication is slower than the real time \nrequirements. Now, assuming that the constraint $t_{round} \\leq mte(C_{next})$ \nis satisfied, there is still another problem we must deal with. The actual \ncommands sent to the device are not received until $t_{jitter} = t_{comm1} + \nt_{rew1} + t_{comm2}$ time after they are actually supposed to be executed. This \ncould be very problematic. Even if the model can keep up with real-time, the \ntime in which it issues commands will be delayed. For example, the shocks from \na pacemaker may be issued at the correct time by the model, but the real shock \nis not delivered until 0.1 seconds later. To meet this requirement, we need to \nlook at the finer requirements of medical devices and patient parameters. How \nmuch jitter in control can a patient tolerate? As we have seen in the Section \n\\ref{sec:case}, the jitter seems to be tolerable for the applications we \nconsidered, and furthermore, the end-to-end round communication timing \nconstraints are also satisfied by our case studies.\n\n\\section{Overview of the Model Execution Framework}\n\n\\label{sec:overview}\n\nAn envisioned design framework from generic design patterns to executable \nspecifications is shown in Figure \\ref{fig:pattern-to-execution}. We start with \na safety pattern, which is a parameteric module with well-defined parameters \nwith formal requirements provided by an input theory. The next stage is to \ninstantiate the pattern to a concrete instance, which will of course still \nsatisfy all the safety properties ensured by the pattern. Finally, in the last \nstage (the focus of this paper), the entire executable specification of a \nsystem can really be executed in the real world by encapsulating the \nspecification in an external wrapper for model execution. Figure \n\\ref{fig:pattern-to-execution} illustrates these various stages for the design \nof a cardiac pacemaker system.\n\n\\begin{figure}\n\\centering \\includegraphics[width=9cm, angle=0]{figures\/mde_process}\n\\caption{From Formal Patterns to Real-Time Execution}\n\\label{fig:pattern-to-execution}\n\\end{figure}\n\nThe first step formally defines a safety pattern as a parameterized module. In \nour pacemaker example, the pattern is a generic safety wrapper for medical \ndevices. We briefly describe the pattern in this paper, summarizing the details \npresented in \\cite{tech-rep}\\cite{sun-meseguer-sha-wrla10}. The safety wrapper \nfilters the input commands, so that state changes in a medical device fall \nwithin safe physiological ranges and constraints. The white boxes in the \ndiagram denote pattern parameters that must be instantiated. For the device \nsafety wrapper these parameters include the type of device that is being \nconsidered, the period for updating device states, states of the device \nconsidered stressful for the patient, etc. Aside from the parameters, there are \nalso formal constraints that these parameters must conform to in order for a \nparameter instance to be acceptable. For example, stress states and relaxed \nstates must be disjoint for a device.\n\nThe next stage is to instantiate the parameters of the pattern. In our \nexample, the medical device safety wrapper is instantiated to filter the input \ncommands to a pacemaker. The state of the pacemaker is assumed to be its pacing \nrate. The necessary parameters are then filled in. The period for updating the \npacing rate is every 10 ms; pacing rates between 90 bpm to 120 bpm are \nconsidered stressful for the patient; etc. This instantiated model will satisfy \nthe safety properties guaranteed by the safety pattern, provided all the \nparameters satisfy the necessary constraints and formal parameter requirements. \nFurthermore, once instantiated, the wrapped pacemaker is just another \nexecutable Real-Time Maude model. Thus, we can use this model for simulation and\nmodel checking purposes in Real-Time Maude.\n\nIn the final stage, which is the main focus of this paper, we transform the \nmodel to execute in {\\em real world} time with physical devices in a medical \ndevice emulation environment. For this purpose, we take the model of the \nwrapped pacemaker and wrap it again in an external execution wrapper (Figure \n\\ref{fig:mdewrapper}). The execution wrapper is responsible for conveying to \nthe model the notion of real world time as well as providing a communication \ninterface to the external world. A dedicated timer thread is responsible for \n``ticking'' the model by sending a minimal number of messages to advance the \nmodel's logical time. The timer thread also intercepts all asynchronous \n(interrupt) messages and relays them to the model. Another aspect of the \nexecution wrapper is the ability to map external I\/O messages to communicate \nwith the external devices. For example, in the pacemaker specification, an \ninternal message called {\\em paceVentricle} may be mapped into an entire client \nconfiguration to send a message for setting the final voltage on a pacing lead.\n\n\\begin{figure}\n\\centering \\includegraphics[width=6cm, angle=-90]{figures\/mde_wrapper}\n\\caption{Real-Time Model Execution Wrapper}\n\\label{fig:mdewrapper}\n\\end{figure}\n\nFor the design of a medical device or, more generally, of any safety-critical \nsystem, all the stages can work together to achieve a modular component design, \nand to support experimentation and testing in the context of other real devices \nthat the safe component being designed has to interact with. In the first \nstages of safety pattern specification, we create a parameteric formal \nspecification that essentially isolates important safety properties of a device \nfrom the rest of the system. We then can use theorem proving techniques to \nprovide provable properties of the safety pattern. Although theorem proving \nmay be time-consuming, as we show with our case studies one safety pattern can \nbe applied to many different applications, so the time spent proving properties \nof the pattern is well worth the effort. The second stage with pattern \ninstantiation is necessary to obtain a fully specified and executable model. \nThe last stage of course takes the existing models, with minimal auxiliary \ninformation for external interfaces, and provides an executable prototype \nessentially for free. In this way, it becomes possible to emulate the behavior\nof safe medical components in an experimental environment involving interactions\nwith real medical devices.\n\n\\section{Distributed Emulation of Safe Medical Devices}\n\\label{sec:emulation}\n\nThe external execution wrapper is an object that encapsulates the original \nformal model. It is primarily responsible for interfacing constructs between \nthe physical world (the real interfaces to devices) and the logical world (the \nworld as seen by the formal model). In particular, the execution wrapper is \nresponsible for conveying the measurement of real time elapsed to the model and \nalso for mapping logical communication messages to communication configurations \nthat can deliver the message to real devices. The most important feature of the \nexternal execution wrapper is its modularity. Aside from adding the minimal \ninformation about how to map external I\/O to messages in the model, no further \nspecifications are required to execute the logical model within an external \nenvironment.\n\n\\subsection{Mapping Logical Time to Physical Time}\n\nAs mentioned earlier, time advancement of the system is achieved by defining \nthe {\\em tick} and {\\em mte} operators. Ideally the system continuously evolves \nover time (possibly nondeterministically). Of course, we cannot capture the \nnotion of continuous time without abstractions in the model, so to advance time \ndiscretely, an {\\em mte} (maximum time elapsable) operator is introduced. A \ncorrectly defined {\\em mte} operator ensures that if a system is in state $S$, \nthen for any time $T < mte(S)$, no 0-time rewrite rules (state transitions)\n can apply to $tick(S, T)$. That is, if a system is in state $S$, and $T \\leq \nmte(S)$, then $tick(S, T)$ will be equivalent to the state $S$ advancing in \ncontinuous time for $T$ time units. This ideal semantics of time is shown on the \nleft side of Figure \\ref{fig:time}. The figure shows that 0-time rewrite \nrules are assumed to take zero time, and ideally, the system continuously \nevolves over time between the 0-time rewrite rules.\n\n\\begin{figure}\n\\centering \\includegraphics[width=14cm, angle=0]{figures\/logical_time_to_physical_time}\n\\caption{From Ideal Time Advancement Semantics to Physical Time Advancement}\n\\label{fig:time}\n\\end{figure}\n\nOf course, in a real execution of the model, the ideal notion of time with \n0-time rewrite rules and time-advancing rules is only an idealized \nabstraction. Performing rewrites cannot take zero time, and we cannot \ncontinuously rewrite states of the system over time. We could of course create \na model in discrete time with very fine time granularity and drive it by a high \nfrequency clock like in hardware. However, this would introduce a lot of \nunnecessary overhead in terms of communication of timing messages and \nperforming rewrite rules to change the model for every clock tick. We resolve \nthis problem by observing that the actual internal state of the model is not \nimportant at most instants in time unless it is communicating with the external \nworld. The model states only generate output messages with 0-time rewrite \nrules, so we can essentially let the model in state $S$ remain unaffected by \nthe passage of time until the next time instant in which a 0-time rewrite \nrule can be applied; this is exactly $mte(S)$ time units later. This method of \ndriving execution is shown in the right part of Figure \\ref{fig:time}. We have \ncreated a dedicated timer server thread (in Java) that has access to the system \ntime. When the execution of the wrapped model starts, it will send a start \nrequest which includes the time units of the model or the minimum granularity \nof time for model execution in milliseconds. Once the timer thread processes \nall the initial information, it will send a {\\em Go!} message to signal the \nmodel to start executing. The model then calculates the maximum time elapsable \n(which is 10 seconds in the example) and sends this information to the timer \nthread. The model then proceeds to sleep until the timer thread wakes it in \ntime for the next 0-time rewrite rule. The process then continues. There are \ntwo key points to notice about this example. The input and output messages from \nthe model may be delayed by an amount of time equal to the communication jitter \nplus the time to complete rewriting. Normally this delay is on the order of 10 \nms, but this is still suitable for medical devices which normally receive \ncommands on the order of seconds or more. Also, the timer thread sets the \ntimeout from the last time it sent a time advancement message to the model and \nnot from the time it receives the $mte$ message from the model. This ensures \nthat clock skew and jitter are bounded over time.\n\n\\subsection{Synchronous Timed Execution}\n\nThe {\\em communication wrapper} ({\\tt commwrap}) is represented as an object \nwith the attributes for the communication state, the socket information for \ncommunication, and the internal wrapped (Real-Time Maude) system model being \nexecuted. The top level system is of sort {\\tt CommWrapConfiguration} for any \ncommunicating model.\n\n\\begin{footnotesize}\n\\begin{verbatim}\nop commwrap : Configuration -> CommWrapConfiguration .\nop wrap-client : Configuration -> Configuration .\n eq wrap-client(C) = < client : TickClient |\n                       state : start, internal : [ {C} in time 0 ], socket-name : no-oid > .\nop init-client : -> CommWrapConfiguration .\n eq init-client = commwrap( <> wrap-client(internal)\n                            createClientTcpSocket(socketManager, client, addr, port) ) .\n\\end{verbatim}\n\\end{footnotesize}\n\nThe communication wrapper initializes a wrapped communication client that \nreceives messages from the tick server (a Java thread executing in real-time \nthat sends it messages for time advancement). After creating the TCP socket, \nthe first message sent from the client to the tick server is the \ntime-granularity ({\\tt time-grain}), which is a rational number specifying the \nnumber of milliseconds in one time unit. Then, the actual execution starts when \nthe communication wrapper receives a {\\tt GO} message from the tick server. The \ntime when the tick server sends the {\\tt GO} message is the starting point from \nwhich time elapses are being measured. Upon receiving the {\\tt GO} message, the \nformal model will immediately start to execute ({\\tt state : run}).\n\n\n\\begin{footnotesize}\n\\begin{verbatim}   \nrl [send-init] :\n commwrap( <> createdSocket(...) < client : TickClient | ... > )\n  => commwrap( <> < client : TickClient | ... > send(..., string(time-grain)) ) .\nrl [wait-for-go] :\n commwrap( <> sent(...) < client : TickClient | ... > )\n  => commwrap( <> < client : TickClient | ... > receive(...) ) .\nrl [start-running] :\n commwrap( <> received(..., \"GO\\r\\n\") < client : TickClient | ... > )\n  => commwrap( <> < client : TickClient | state : run, ... > .\n\\end{verbatim}\n\\end{footnotesize}\n\nThe formal model executes until {\\tt mte} becomes non-zero (no other 0-time \nrewrite rules can be applied), and the model sends a message to request the \nnext time advancement message after the maximum time elapse and blocks. After \nsending this waiting duration, the tick server will sleep for this time \nduration and then send a time advancement message when the time has expired. \nThe model will then advance time (tick) the model for the time duration expired\nand perform 0-time rewrite rules. The model now blocks again for the next {\\tt\nmte}, and the cycle repeats.\n\n\\begin{footnotesize}\n\\begin{verbatim}   \ncrl [request-wait-timer] :\n commwrap( <>\n   < client : TickClient |\n      state : run,\n      internal : [ {C} in time T ], ... > )\n => commwrap( <>\n   < client : TickClient |\n      state : request, ... >\n   send(..., string(mte(C, T))) )\n if mte(C,T) :: TimeInf \/\\ mte(C,T) > 0 \/\\ not open?(C) . ...\n\nrl [block] :\n commwrap( <> sent(...)\n   < client : TickClient |\n      state : request, ... > )\n => commwrap( <>\n   < client : TickClient |\n     state : wait, ... >\n     receive(SOCKET-NAME, client) ) .\n\nrl [wake-up] :\n commwrap( <> received(..., ADV-STR)\n   < client : TickClient |\n      state : wait, ... > )\n => commwrap( <>\n   < client : TickClient |\n      state : run,\n      internal : [ {tick(C, rat(ADV-STR))} in time rat(ADV-STR) in time T ], ... > ) .\n\\end{verbatim}\n\\end{footnotesize}\n\n\n\\subsection{Handling Asynchronous External Events}\n\nSo far, the model can only handle synchronous events (polling and blocking \ncommunication). However, in general a useful design must be able to react to \nexternal events from the environment. For example, an EKG sensor detects a QRS \nwaveform, and sends this information to the pacemaker. This points to the fact \nthat our model needs to be able to handle external events asynchronously.\n\n\\begin{figure}\n\\centering \\includegraphics[width=8cm, angle=0]{figures\/interrupts}\n\\caption{Handling Interrupts and Asynchronous Communication Semantics}\n\\label{fig:async}\n\\end{figure}\n\nAn external message would trigger a 0-time rewrite rule to receive the message \nby some object and process it. More precisely, if we have $C_M$ and $C_{Ext}$ \nas the model configuration and the external (environment) configuration \nrespectively, the maximal time elapse for the system $C_M C_{Ext}$ should be \n$min(mte(C_M), mte(C_{Ext}))$, where $mte(C_{Ext})$ denotes time duration \nbefore the next interrupt message. This semantics is captured by having \ninterrupt messages forwarded by the timer thread, as shown in Figure \n\\ref{fig:async}. The timer thread will only check for interrupts when it is \nwaiting for the next timeout, so when the interrupt message arrives, it will \nwake up and immediately forward the interrupt message to the model with the \namount of time that has elapsed. Any future timeouts are canceled. Introducing \nthe notion of interrupts requires us to modify the wake-up rule for the model \nto not only advance time, but also check for potential interrupt messages as \nwell.\n\n\\begin{footnotesize}\n\\begin{verbatim}\nrl [wake-up] :\ncommwrap( <> received(client, SOCKET-NAME, INTR-STR)\n  < client : TickClient |\n  state : wait,\n  internal : [ {C} in time T ],\n  socket-name : SOCKET-NAME > )\n=>\ncommwrap( <> < client : TickClient |\n  state : run,\n  internal : [\n    {tick(C, recv->rat(INTR-STR)) recv->conf(INTR-STR)}\n      in time recv->rat(INTR-STR) in time T\n  ],\n  socket-name : SOCKET-NAME > ) .\n\\end{verbatim}\n\\end{footnotesize}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Omitted Proof from \\Cref{app:universal}}\nHere we provide the proof for \\Cref{lem:bcard}.\n\\begin{proof}[Proof for \\Cref{lem:bcard}] Fix an ordering on the elements of $\\mathrm{F}$. Let $\\mathrm{F}(i)$ denote the $i$'th frequency element of $\\mathrm{F}$. For all $\\phi_{S} \\in B$, the set of distinct frequencies in $\\phi_{S}$ is a subset of $\\mathrm{F}$ and the length of $\\phi_{S}$ is equal to $n$. Therefore, any element $\\phi_{S} \\in B$ can be encoded as a unique vector $v_{\\phi_{S}} \\in [0,n]^{\\mathrm{F}}$, where $v_{\\phi_{S}}(i) \\eqdef \\phi(\\mathrm{F}(i))$ denotes the number of elements in $\\phi_{S}$ that have frequency $F(i)$. Using the previous discussion, we have $|B| \\leq |[0,n]^{\\mathrm{F}}| \\leq (n+1)^{|\\mathrm{F}|}$.\n\\end{proof}\n\n\n\\section{Omitted Proofs from \\Cref{sec:appl}}\\label{sec:omitted}\nHere, we present and prove results related to the existence of an estimator for entropy and distance to uniformity on a fixed subset $S\\subseteq \\mathcal{D}$. Note the estimator we provide here is exactly same to the one presented in~\\cite{ADOS16} but defined only on subset $S\\subseteq \\mathcal{D}$. All the results and proofs presented here are similar to the ones in~\\cite{ADOS16} and for completeness and verification purposes we reprove (with slight modifications) these results. As in~\\cite{ADOS16}, we first provide a general definition of an estimator that works both for entropy and distance to uniformity. In \\Cref{lem:general_est}, we prove a result that captures the maximum change of this general estimator by changing one sample. In section \\ref{subsec:entr} and \\ref{subsec:dtu}, we provide proofs for entropy and distance to uniformity respectively.\n\nGiven $2n$ samples $x^{2n}=(x_1^n,x_2^n)$ from distribution $\\textbf{p}$. Let\n$n_{\\smb}^{'}\\eqdef \\textbf{f}(x_1^n,y)$, and $n_{\\smb}\\eqdef \\textbf{f}(x_2^n,y)$ be the number of appearances of symbol $y$ in the first and second half respectively. We define the following estimator which is exactly the same as \\cite{ADOS16} but defined only on subset $S$. For all $x \\in S$,\n\\[\n\\hat{g}_{S}(x^{2n})\n=\\max\\left\\{\\min\\left\\{\\sum_{y \\in S}{g_{y}},f_{S,\\max}\\right\\}, 0\\right\\}.\n\\]\nwhere $f_{S,\\max}$ is the maximum value of the property $f$ on subset $S$ and for all $y \\in S$,\n\\[\ng_{y}=\n\\begin{cases}\nG_{L,g}(n_{\\smb}), & \\text{ for } n_{\\smb}^{'}<c_2\\log N, \\text{ and }\nn_{\\smb}< c_1\\log N,\\\\ 0, & \\text{ for } n_{\\smb}^{'}<c_2\\log k, \\text{\n  and } n_{\\smb}\\ge c_1\\log N,\\\\ g \\left(\\frac{n_{\\smb}}{n}\\right)+ g_n\n, & \\text{ for } n_{\\smb}^{'}\\ge c_2\\log N,\n\\end{cases}\n\\]\nwhere $g_n$ is the first order bias correction term for $g$,\n$G_{L,g}(n_{\\smb}) = \\sum^L_{i=1} b_i {(\\frac{n_{\\smb}}{n})}^{i}$ is\nthe unbiased estimator for $P_{L,g}(\\textbf{p}_{y})$, the optimal uniform approximation of function $g$ by degree-$L$ polynomials on $[0; c_{1} \\frac{\\log n}{n}]$. \n\\begin{lemma}\n  \\label{lem:general_est}\n  For any estimator $\\hat{g}$ defined as above, changing any one of\n  the sample changes the estimator by at most\n  \\[\n  9\\max\\left(e^{L^2\/n}\\max\n  |b_i|, \\frac{L_g}{n}, g\\left(\\frac{c_1 \\log (n)}n\\right), g_n\\right),\n  \\]\n  where $L_g = n \\max_{i\\in \\mathbb{N}} |g(i\/n)- g((i-1)\/n)|$.\n\\end{lemma}\n\\begin{proof}\nGiven $2n$ samples $x^{2n}=(x_1^n,x_2^n)$ from distribution $\\textbf{p}$. Recall the estimator for entropy and distance to uniform from \\cite{ADOS16},\n\\[\n\\hat{g}(x^{2n})\n=\\max\\left\\{\\min\\left\\{\\sum_{y \\in S}{g_{y}},f_{\\max}\\right\\}, 0\\right\\}.\n\\]\nwhere $f_{\\max}$ is the maximum value of the property $f$ and for all $y \\in \\mathcal{D}$,\n\\[\ng_{y}=\n\\begin{cases}\nG_{L,g}(n_{\\smb}), & \\text{ for } n_{\\smb}^{'}<c_2\\log N, \\text{ and }\nn_{\\smb}< c_1\\log N,\\\\ 0, & \\text{ for } n_{\\smb}^{'}<c_2\\log k, \\text{\n\tand } n_{\\smb}\\ge c_1\\log N,\\\\ g \\left(\\frac{n_{\\smb}}{n}\\right)+ g_n\n, & \\text{ for } n_{\\smb}^{'}\\ge c_2\\log N,\n\\end{cases}\n\\]\nNow construct a new sequence from $x^n$ as follows: replace all symbols in $\\bar{S}$ (appearing in $x^n$) by a unique symbol $y' \\in \\bar{S}$ and call this new sequence $z^n$. Now note our estimator is unaffected by this change, because it only depends on the occurrences of elements in $S$. The change in the value of estimator in~\\cite{ADOS16} by changing one sample in $z^n$ is upper bounded by:\n\\begin{equation}\n8\\max\\left(e^{L^2\/n}\\max\n|b_i|, \\frac{L_g}{n}, g\\left(\\frac{c_1 \\log (n)}n\\right), g_n\\right),\n\\end{equation}\nThe above result follows by Lemma 5 in~\\cite{ADOS16}. We next study the change in the value of our estimator by changing one sample in $z^n$. Note this is equivalent to the change in the value of our estimator by changing one sample in $x^n$. The worst case change in the value of our estimator is when we take a symbol in $S$ (or $\\bar{S} \\backslash \\{y'\\}$) and replace it by a symbol in $\\bar{S}\\backslash \\{y'\\}$ (or $S$). In this case, by triangle inequality change in our estimator is upper bounded by $8\\max\\left(e^{L^2\/n}\\max|b_i|, \\frac{L_g}{n}, g\\left(\\frac{c_1 \\log (n)}n\\right), g_n\\right) + G_{L,g}(1)$ that is further upper bounded by $9\\max\\left(e^{L^2\/n}\\max|b_i|, \\frac{L_g}{n}, g\\left(\\frac{c_1 \\log (n)}n\\right), g_n\\right)$ and the result follows.\n\n\t\\end{proof}\n\n\\subsection{Entropy}\\label{subsec:entr}\nHere we present proof sketch for the following: for entropy the estimator defined above has low bias and the value of the estimator does not change too much by change in one sample. This result is analogous to Lemma 6 in~\\cite{ADOS16} and our proof for this lemma is very similar to~\\cite{ADOS16} and for completeness sketch for the proof.\n\\begin{lemma}\n  Let $g_n = 1\/(2n)$ and $\\alpha > 0$.  Suppose $c_1=2c_2$, and $c_2>35$,\n  Further suppose that\n  $n^3\\left(\\frac{16c_1}{\\alpha^2}+\\frac1{c_2}\\right)>\\log k\\cdot\\log\n  n$. Then for all subset $S\\subseteq \\mathcal{D}$, there exists a polynomial approximation of $- y \\log y$ {with degree $L = 0.25 \\alpha \\log n$}, over\n  $[0, c_1\\frac{\\log N}{n}]$ such that $\\max_{i} |b_i| \\leq\n  n^{\\alpha}\/n$ and the bias of the entropy estimator on subset $S$ ($\\sum_{y \\in S}\\textbf{p}_{y} \\log \\frac{1}{\\textbf{p}_{y}}$) is at most\n  $O\\left(\\left(1+\\frac{1}{\\alpha^2}\\right)\\frac{N}{n\\log N} + \\frac{\\log N}{N^4}\\right)$.\n\\end{lemma}\n\\begin{proof}\n\t\\newcommand{{E}_{1}}{{E}_{1}}\n\t\\newcommand{{E}_{2}}{{E}_{2}}\n\t\\newcommand{{E}_{3}}{{E}_{3}}\n\t\\newcommand{{E}}{{E}}\n\t\\newcommand{n_{y}}{n_{y}}\n\t\\newcommand{n'_{y}}{n'_{y}}\n\t\\newcommand{\\expt}[1]{\\mathbb{E}\\left[ #1 \\right]}\n\t\nWe first upper bound the bias of our estimator. We consider three events, \n$${E}_{1}\\eqdef \\cap _{y \\in S} \\left\\{ n'_{y} \\leq  c_{2} \\log N, n'_{y} \\leq  c_{1} \\log N \\implies \\textbf{p}_{y} \\leq \\frac{c_1 \\log N}{n}\\right\\},$$ \n$${E}_{2}\\eqdef\\cap _{y \\in S} \\left\\{ n'_{y} > c_{2} \\log N \\implies \\textbf{p}_{y} > \\frac{c_3 \\log N}{n}\\right\\},$$ $${E}_{3}\\eqdef \\cap _{y \\in S}\\left\\{ n'_{y} \\leq  c_{2} \\log N \\text{ and } n'_{y} > c_{1} \\log N \\right\\}~.$$ \nBy proof of Lemma 6 in~\\cite{ADOS16} we have,\n\\begin{equation}\\label{prob:e3}\n\\Prob{{E}_{3}^c} \\leq \\frac{1}{n^{4.9}}\n\\end{equation}\nBy equations 48 and 49 in~\\cite{WY16} combined with \\Cref{prob:e3}, we get,\n$$\\Prob{{E}_{1}^c} \\leq \\frac{2}{N^4} \\text{ and } \\Prob{{E}_{2}^c} \\leq \\frac{1}{N^4} $$\nDefine ${E}\\eqdef {E}_{1} \\cap {E}_{2}$, then \n\\begin{equation}\\label{prob:eall}\n\\Prob{{E}^c} \\leq \\Prob{{E}_{1}^c}+\\Prob{{E}_{2}^c} \\leq \\frac{2}{N^4}~. \n\\end{equation}\n\n\\newcommand{I_1}{I_1}\n\\newcommand{I_2}{I_2}\nFurther we define random sets\n$I_1\\eqdef \\left\\{y \\in S |  n'_{y} < c_{2} \\log N, n_{y} < c_{1} \\log N  \\text{ and } \\textbf{p}_{y} \\leq \\frac{c_1 \\log N}{n}\\right\\}$ and $I_2 \\eqdef \\left\\{ y \\in S ~|~n'_{y} > c_{2} \\log N \\text{ and } \\textbf{p}_{y} > \\frac{c_3 \\log N}{n}\\right\\}$.\nWe first bound the conditional bias and we later use it to bound the bias of our estimator. Our next statement follows from uniform approximation error~\\cite{Tim63} and is explicitly written in to Equation 53 of~\\cite{WY16}.\n\\begin{equation}\\label{eq:2}\n|\\expt{\\textbf{f}_{I_1}(\\textbf{p})-\\hat{g}_{I_1} | I_1}=|\\sum_{y \\in I_1} \\textbf{p}_{y} \\log \\frac{1}{\\textbf{p}_y}-P_{L,g}(\\textbf{p}_{y})| \\leq \\frac{N}{\\alpha^2 n \\log N}~.\n\\end{equation}\nLet $\\hat{\\textbf{p}}$ be the empirical distribution on $x_2^n$. Similarly by analysis of Case 2 and Equation 58 in~\\cite{WY16} we have,\n\\begin{equation}\\label{eq:3}\n|\\expt{\\textbf{f}_{I_2}(\\textbf{p})-\\hat{g}_{I_2} | I_2}=|\\expt{\\sum_{y \\in I_2} (\\textbf{p}_{y} \\log \\frac{1}{\\textbf{p}_y}-\\hat{\\textbf{p}}_{y} \\log \\frac{1}{\\hat{\\textbf{p}}_y})|I_2}| \\leq \\frac{N}{n \\log N}~.\n\\end{equation}\nCombining equations \\ref{eq:2}, \\ref{eq:3}, \\ref{prob:eall} and \\ref{prob:e3} we can upper bound the bias of our estimator by $\\frac{2N}{n \\log N}+\\frac{4 \\log N}{N^4}$. Note here we use the fact that in the case of bad event (${E}^c$ or ${E}_{3}^c$) the bias of our estimator is upper bounded by $\\log N$.\n\nOur analysis for largest change in the value of estimator by changing one sample is exactly the same as~\\cite{ADOS16} and for completeness we describe it next. The largest coefficient of the optimal uniform polynomial approximation of degree $L$ for function $x \\log x$ in the interval $[0,1]$ is upper bounded by $2^{3L}$. This result follows from the proof of Lemma 2 in~\\cite{CL11} and is also explicitly mentioned in the proof of Proposition 4 in~\\cite{WY16}. Therefore, the largest change (after appropriately normalizing) is the largest value of $b_i$ (co-efficient of the optimal uniform polynomial approximation) which is\n\\[\n\\frac{2^{3L}e^{L^2\/n}}{n}.\n\\]\nFor $L = 0.25\\alpha\\log n$, this is at most $\\frac {n^\\alpha}{n}$. \n\\end{proof}\n\nThe proof of Lemma~\\ref{thm:entrochange} for entropy follows from the above lemma and Lemma~\\ref{lem:general_est} by substituting $n= O\\left(\\frac{N}{\\log N} \\frac{1}{\\epsilon}\\right)$ and $\\epsilon>\\Omega\\left(\\frac{\\log N}{N^{1-\\alpha}}\\right)$.\n\n\\subsection{Distance to uniformity}\\label{subsec:dtu}\nHere we provide proof sketch for the existence of an estimator with desired properties. This result is analogous to Lemma 7 in~\\cite{ADOS16} and proof for this lemma is very similar to that of \\cite{ADOS16} and for completeness we sketch the proof for this result. \n\\begin{lemma}\n\n  Let $c_1>2c_2$, $c_2=35$. Then for all subset $S\\subseteq \\mathcal{D}$, there is an estimator for distance to uniformity on subset $S$ ($\\sum_{y \\in S}|\\textbf{p}_{y}-\\frac{1}{N}|$) that changes by at most $n^{\\alpha}\/n$ when a sample is changed, and the bias of the estimator is at most $O(\\frac{1}{\\alpha}\\sqrt{\\frac{c_1\\log N}{N\\cdot n}})$.\n\\end{lemma}\n\\begin{proof}\nWe divide estimation of distance to uniformity into two cases based on $n$. Note the proof for this lemma follows along the lines of \\cite{ADOS16}.\n\\paragraph{Case 1: $\\frac1N < c_2\\log N\/n$.} In this case, we use the estimator defined in the last section for $g(x) = |x-1\/k|$.\n\n\\paragraph{Case 2: $\\frac1N > c_2\\log N\/n$.} The estimator is as follows for all $y \\in S$:\n\\[\ng_{y}=\n\\begin{cases}\nG_{L,g}(n_{\\smb}), & \\text{ for }\n\\absv{\\frac{n_{\\smb}^{'}}{n}-\\frac1{N}}<\\sqrt{\\frac{c_2\\log N}{Nn}}, \\text{\n  and } \\absv{\\frac{n_{\\smb}}{n}-\\frac1{N}}<\\sqrt{\\frac{c_1\\log N}{Nn}},\\\\ 0,\n& \\text{ for }\n\\absv{\\frac{n_{\\smb}^{'}}{n}-\\frac1{N}}<\\sqrt{\\frac{c_2\\log N}{Nn}}, \\text{\n  and } \\absv{\\frac{n_{\\smb}}{n}-\\frac1{N}}\\ge\\sqrt{\\frac{c_1\\log N}{Nn}},\\\\ g\n\\left(\\frac{n_{\\smb}}{n}\\right) , & \\text{ for }\n\\absv{\\frac{n_{\\smb}^{'}}{n}-\\frac1{N}}\\ge\\sqrt{\\frac{c_2\\log N}{Nn}}.\n\\end{cases}\n\\]\n\nThe estimator proposed in~\\cite{ADOS16} is exactly the same as ours, but we define our estimator only for the domain elements in $S\\subseteq \\mathcal{D}$. Note the estimator defined in~\\cite{JHW17} is slightly different, assigning $G_{L,g}(n_{\\smb})$ for the first two cases. As in~\\cite{ADOS16}, this second case is designed to bound the change in value of the estimator by changing one sample. Using~\\cite{Tim63}(Equation 7.2.2),~\\cite{JHW17} (for their estimator) show that, contribution towards bias (conditioned on \"good\" event \\footnote{\\label{event}Refer~\\cite{JHW17} for definitions of \"good\" and \"bad\" events.}) by any domain element $y \\in \\mathcal{D}$ (note we only need this result to hold for $y\\in S$) satisfying $n_{\\smb}^{'}<c_2\\log N, \\text{ and } n_{\\smb}< c_1\\log N$ for case 1 and $\\absv{\\frac{n_{\\smb}^{'}}{n}-\\frac1{N}}<\\sqrt{\\frac{c_2\\log N}{Nn}}, \\text{ and } \\absv{\\frac{n_{\\smb}}{n}-\\frac1{N}}<\\sqrt{\\frac{c_1\\log N}{Nn}}$ for case 2, using polynomial approximation (Lemma 27 in~\\cite{JHW17}) is upper bounded by $O\\left(\\frac1{L}\\sqrt{\\frac{\\log N}{N\\cdot n\\log N}}\\right)$, where $L$ is the degree of optimal uniform approximation for function $|x-\\frac{1}{N}|$ in the interval $[0,2c_1 \\frac{\\log N}{N}]$ for case 1 (Equation (351) in~\\cite{JHW17}) and $[\\frac{1}{N}-\\sqrt{\\frac{c_1\\log N}{Nn}},\\frac{1}{N}-\\sqrt{\\frac{c_1\\log N}{Nn}}]$ for case 2 (Equation (367) in~\\cite{JHW17}). Further, the bias (conditioned on the \"good\" event) by empirical estimate for domain element $y \\in \\mathcal{D}$ (as before, we need this result to hold only for $y\\in S$) satisfying $n_{\\smb}^{'}\\ge c_2\\log N$ for case 1 and $\\absv{\\frac{n_{\\smb}^{'}}{n}-\\frac1{N}}\\ge\\sqrt{\\frac{c_2\\log N}{Nn}}$ for case 2, is zero (Refer proof of Theorem 2~\\cite{JHW17}).~\\cite{JHW17} also bound the probability of \"bad\" event\\footnoteref{event} (Refer proof of Lemma 2 in~\\cite{JHW17}), thus bounding the bias with respect to these domain elements. Further similar to~\\cite{ADOS16}, by our choice of $c_1, c_2$, the contribution to bias by domain element $y \\in S$ satisfying $n_{\\smb}^{'}<c_2\\log k, \\text{\n\tand } n_{\\smb}\\ge c_1\\log N$ for case 1 and $\\absv{\\frac{n_{\\smb}^{'}}{n}-\\frac1{N}}<\\sqrt{\\frac{c_2\\log N}{Nn}}, \\text{\n\tand } \\absv{\\frac{n_{\\smb}}{n}-\\frac1{N}}\\ge\\sqrt{\\frac{c_1\\log N}{Nn}}$ for case 2, is upper bounded by $1\/n^4<\\epsilon^2$. Combining analysis of all these cases together, we have our result for bias.\n\nThe proof for largest change in the estimator value by changing one sample is exactly same as~\\cite{ADOS16}. Similar to~\\cite{ADOS16}, here we use the fact~\\cite{CL11} (Lemma 2) that the largest coefficient of the optimal uniform polynomial approximation of degree $L$ for function $|x|$ in the interval $[-1,1]$ is upper bounded by $2^{3L}$. $2^{3L}$. Similar to entropy (after appropriate normalization), the largest difference in estimation will be at most $n^{\\alpha}\/n$.\n\\end{proof}\n\nThe proof of \\Cref{thm:dtuchange} for distance to uniformity follows from the above lemma and \\Cref{lem:general_est} and by substituting $n= O\\left(\\frac{N}{\\log N} \\frac{1}{\\epsilon^2}\\right)$.\n\n\\section{Support Estimation}\\label{sec:support}\nHere we study the PML based plug-in estimator for support estimation. \\cite{ADOS16} showed that PML based plug-in estimator is sample complexity optimal for estimating support within additive accuracy $\\epsilon k$ for all $\\epsilon>\\frac{1}{k^{0.2499}}$. Further for any $\\epsilon<\\frac{1}{k^{\\delta}}$ for some constant $\\delta>0$, the empirical distribution based plug-in estimator is exact with high probability. Here we provide proofs for two main results described in \\Cref{sec:results} for support. In \\Cref{thm:supp} and \\Cref{thm:approxsupp}, we show that PML and approximate PML distributions (under the constraint that all its probability values are $\\geq \\frac{1}{k}$) based plug-in estimators are sample complexity optimal for all parameter regimes, thus providing a better analysis for \\cite{ADOS16}.\n\nWe next define a function that outputs the number of distinct frequencies in the profile. Later in \\Cref{lem:pmlprob}, we show that the support of PML and approximate PML distribution is at least the number of distinct elements in the sequence.\n\\begin{defn}\n\tFor any $S \\subseteq \\mathcal{D}$, the function $\\mathrm{Distinct}:\\Phi^{n}\\rightarrow \\mathbb{Z}_{+}$, takes input $\\phi$ and returns $\\sum_{j\\in[n]}\\phi_j$. For any sequence $x^n$, we overload notation and use $\\mathrm{Distinct}(x^n)$ to denote $\\mathrm{Distinct}(\\Phi(x^n))$. Note $\\mathrm{Distinct}(\\phi)$ and $\\mathrm{Distinct}(x^n)$ denote the number of distinct domain elements observed in profile $\\phi$ or sequence $x^{n}$ respectively.\n\t\\end{defn}\n\\begin{lemma}\\label{lem:pmlprob}\n\tFor any distribution $\\textbf{p} \\in \\Delta^{\\mathcal{D}}$ such that $\\textbf{p}_x \\in \\{0\\} \\cup [\\frac{1}{k},1]$ and a profile $\\phi \\in \\Phi^{n}$, if $S(\\textbf{p})<\\mathrm{Distinct}(\\phi)$ then $\\textbf{p}(\\phi)=0$.\n\\end{lemma}\n\\begin{proof}\nConsider sequences $x^{n}$ with $\\Phi(x^n)=\\phi$. All such sequences have $\\mathrm{Distinct}(\\phi)$ number of distinct observed elements that is strictly greater than $S(\\textbf{p})$ and distribution $\\textbf{p}$ assigns probability zero for all these sequences.\n\\end{proof}\n\\newcommand{\\mathrm{Support}}{\\mathrm{Support}}\n\\begin{proof}[Proof for \\Cref{thm:supp}]\n\tGiven $\\phi$, let $\\textbf{p}_{\\phi} \\in \\Delta^{\\mathcal{D}}$, be the distribution with $\\textbf{p}_{\\phi}(x) \\in \\{0\\}\\cup[\\frac{1}{k},1]$. If $\\epsilon>\\frac{1}{k^{0.2499}}$, we know that plug-in approach on $\\textbf{p}_{\\phi}$ is sample complexity optimal~\\cite{ADOS16}. We consider the regime where $\\epsilon \\leq \\frac{1}{k^{0.2499}}$ and here the number of samples $n=c \\cdot k\\log k$ for some constant $c \\geq 2$. If $S(\\textbf{p}_{\\phi}) < \\mathrm{Distinct}(\\phi)$, then by \\Cref{lem:pmlprob} we have $\\textbf{p}_{\\phi}(\\phi)=0$ a contradiction because the empirical distribution assigns a non-zero probability value for observing $\\phi$. Therefore, without loss of generality we assume $S(\\textbf{p}_{\\phi}) \\geq \\mathrm{Distinct}(\\phi)$. We next argue that $S(\\textbf{p}_{\\phi}) = \\mathrm{Distinct}(\\phi)$. We prove this statement by contradiction. Suppose $S(\\textbf{p}_{\\phi}) > \\mathrm{Distinct}(\\phi)$, then define $\\textbf{p}_{\\phi}' \\in \\Delta^{\\mathcal{D}}$ to be the PML distribution under constraints $S(\\textbf{p}_{\\phi}')=\\mathrm{Distinct}(\\phi)$ and $\\textbf{p}_{\\phi}'(x) \\in \\{0\\}\\cup[\\frac{1}{k},1]$. Let $\\mathrm{Support}:\\Delta^{\\mathcal{D}} \\rightarrow 2^{\\mathcal{D}}$ be a function that takes distribution $\\textbf{p}$ as input and returns index set for the support of $\\textbf{p}$. Now consider $\\Prob{\\textbf{p}_{\\phi},\\phi}$, and recall $\\Prob{\\textbf{p}_{\\phi},\\phi}=\\sum_{\\{x^n\\in \\mathcal{D}^n~|~\\Phi(x^n)=\\phi\\}}\\Prob{\\textbf{p}_{\\phi},x^n}$. Further note that $\\sum_{\\{x^n\\in \\mathcal{D}^n~|~\\Phi(x^n)=\\phi\\}}\\Prob{\\textbf{p}_{\\phi},x^n}=\\sum_{\\{S\\subseteq \\mathrm{Support}(\\textbf{p}_{\\phi})~|~|S|=\\mathrm{Distinct}(\\phi) \\}} \\sum_{\\{x^n\\in S^n~|~\\Phi(x^n)=\\phi\\}}\\Prob{\\textbf{p}_{\\phi},x^n}$, therefore,\n\\begin{equation}\\label{eq:a1}\n\\begin{split}\n\\Prob{\\textbf{p}_{\\phi},\\phi}&=\\sum_{\\{S\\subseteq \\mathrm{Support}(\\textbf{p}_{\\phi})~|~|S|=\\mathrm{Distinct}(\\phi) \\}} \\sum_{\\{x^n\\in S^n~|~\\Phi(x^n)=\\phi\\}}\\Prob{\\textbf{p}_{\\phi},x^n}\\\\\n& \\leq \\sum_{\\{S\\subseteq \\mathrm{Support}(\\textbf{p}_{\\phi})~|~|S|=\\mathrm{Distinct}(\\phi) \\}} \\left(1-\\frac{S(\\textbf{p}_{\\phi})-|S|}{k} \\right)^n \\Prob{\\textbf{p}_{\\phi}',\\phi}~.\n\\end{split}\n\\end{equation}\n\\newcommand{\\textbf{p}_{\\phi,S}}{\\textbf{p}_{\\phi,S}}\nIn the second inequality, we use for all $x \\in \\mathcal{D}$, $\\textbf{p}_{\\phi}(x)\\in \\{0\\}\\cup [\\frac{1}{k},1]$ and we have $\\sum_{x\\in S}\\textbf{p}_{\\phi}(x) \\leq \\left(1-\\frac{S(\\textbf{p}_{\\phi})-|S|}{k} \\right)$ and the inequality follows. \n\nWe next upper bound the term $\\sum_{\\{S\\subseteq \\mathrm{Support}(\\textbf{p}_{\\phi})~|~|S|=\\mathrm{Distinct}(\\phi) \\}} \\left(1-\\frac{S(\\textbf{p}_{\\phi})-|S|}{k} \\right)^n$. Note that, \n\\begin{align*}\n\\sum_{\\{S\\subseteq \\mathrm{Support}(\\textbf{p}_{\\phi})~|~|S|=\\mathrm{Distinct}(\\phi) \\}} & \\left(1-\\frac{S(\\textbf{p}_{\\phi})-|S|}{k} \\right)^n  \\leq \\expo{-n\\frac{S(\\textbf{p}_{\\phi})-\\mathrm{Distinct}(\\phi)}{k}} \\binom{S(\\textbf{p}_{\\phi})}{\\mathrm{Distinct}(\\phi)} \\\\\n& \\leq  \\expo{-n\\frac{S(\\textbf{p}_{\\phi})-\\mathrm{Distinct}(\\phi)}{k} + (S(\\textbf{p}_{\\phi})-\\mathrm{Distinct}(\\phi))\\log S(\\textbf{p}_{\\phi}) }\\\\\n& \\leq   \\expo{ -\\log k }~.\n\\end{align*}\nIn the second inequality, we use a weak upper bound on the quantity $\\binom{S(\\textbf{p}_{\\phi})}{\\mathrm{Distinct}(\\phi)}$. In the third and fourth inequality, we use $n=ck\\log k$, $c\\geq2$ and $k \\geq  S(\\textbf{p}_{\\phi}) > \\mathrm{Distinct}(\\phi)$. Combining everything together we get, $\\Prob{\\textbf{p}_{\\phi},\\phi} \\leq \\expo{ -\\log k }\\Prob{\\textbf{p}_{\\phi}',\\phi}$. A contradiction because $\\textbf{p}_{\\phi}$ is the PML distribution.\n\n\n\nTherefore if $n>2k\\log k$, then the previous derivation implies, \n$$\\Prob{S(\\textbf{p}_{\\phi})=\\mathrm{Distinct}(\\phi)}=1~.$$ \nFurther if $n>2k\\log k$, then \n$$\\Prob{S(\\textbf{p}) =\\mathrm{Distinct}(\\phi)} \\geq 1-k\\expo{\\frac{-n}{k}}~.$$ \nCombining previous two inequalities and substituting $n>2k\\log k$ we get, $\\Prob{S(\\textbf{p}) =S(\\textbf{p}_{\\phi})} \\geq 1-\\exps{-\\log k}$, thus concluding the proof.\n\\end{proof}\n\n\\renewcommand{\\bp^{\\beta}_{\\phi_{S}}}{\\textbf{p}^{\\beta}_{\\phi}}\n\\begin{proof}[Proof for \\Cref{thm:approxsupp}]\n\t\tThe proof for this result is similar to \\Cref{thm:supp} and for completeness we reprove it. Given $\\phi$, let $\\textbf{p}_{\\phi},\\bp^{\\beta}_{\\phi_{S}} \\in \\Delta^{\\mathcal{D}}$, be PML and $\\beta$-approximate PML distributions respectively under the constraint $\\textbf{p}_{\\phi}(x),\\bp^{\\beta}_{\\phi_{S}}(x) \\in \\{0\\}\\cup[\\frac{1}{k},1]$. If $\\epsilon>\\frac{1}{k^{0.2499}}$, by \\cite{ADOS16} we already know that plug-in approach on $\\bp^{\\beta}_{\\phi_{S}}$ for $\\beta=\\exps{-\\epsilon^2 n^{1-\\alpha}}$ is sample complexity optimal with high probability. Here we consider the regime $\\epsilon \\leq \\frac{1}{k^{0.2499}}$ and in this case the number of samples $n=c \\cdot k\\log k$ for some large constant $c \\geq 2$. If $S(\\bp^{\\beta}_{\\phi_{S}}) < \\mathrm{Distinct}(\\phi)$, then by \\Cref{lem:pmlprob} we have $\\bp^{\\beta}_{\\phi_{S}}(\\phi)=0$ which is a contradiction, because the empirical distribution clearly returns a non-zero probability value. Therefore, without loss of generality we assume $S(\\bp^{\\beta}_{\\phi_{S}}) \\geq \\mathrm{Distinct}(\\phi)$. We next argue that $S(\\bp^{\\beta}_{\\phi_{S}}) \\leq \\mathrm{Distinct}(\\phi)+\\epsilon k$. We prove this statement by contradiction. Suppose $S(\\bp^{\\beta}_{\\phi_{S}}) > \\mathrm{Distinct}(\\phi)+\\epsilon k$, then consider the $\\Prob{\\bp^{\\beta}_{\\phi_{S}},\\phi}$, and recall $\\Prob{\\bp^{\\beta}_{\\phi_{S}},\\phi}=\\sum_{\\{x^n\\in \\mathcal{D}^n|\\Phi(x^n)=\\phi\\}}\\Prob{\\bp^{\\beta}_{\\phi_{S}},x^n}$. Further note that $\\sum_{\\{x^n\\in \\mathcal{D}^n|\\Phi(x^n)=\\phi\\}}\\Prob{\\bp^{\\beta}_{\\phi_{S}},x^n}=\\sum_{\\{S\\subseteq \\mathrm{Support}(\\bp^{\\beta}_{\\phi_{S}})||S|=\\mathrm{Distinct}(\\phi) \\}} \\sum_{\\{x^n\\in S^n|\\Phi(x^n)=\\phi\\}}\\Prob{\\bp^{\\beta}_{\\phi_{S}},x^n}$. Therefore,\n\t\\begin{equation}\n\t\\begin{split}\n\t\\Prob{\\bp^{\\beta}_{\\phi_{S}},\\phi}&=\\sum_{\\{S\\subseteq \\mathrm{Support}(\\bp^{\\beta}_{\\phi_{S}})||S|=\\mathrm{Distinct}(\\phi) \\}} \\sum_{\\{x^n\\in S^n|\\Phi(x^n)=\\phi\\}}\\Prob{\\bp^{\\beta}_{\\phi_{S}},x^n}\\\\\n\t& \\leq \\sum_{\\{S\\subseteq \\mathrm{Support}(\\bp^{\\beta}_{\\phi_{S}})~|~|S|=\\mathrm{Distinct}(\\phi) \\}} \\left(1-\\frac{S(\\bp^{\\beta}_{\\phi_{S}})-|S|}{k} \\right)^n \\Prob{\\textbf{p}_{\\phi},\\phi}~.\n\t\\end{split}\n\t\\end{equation}\n\tIn the final inequality we used for all $x \\in \\mathcal{D}$, $\\bp^{\\beta}_{\\phi_{S}}(x)\\in \\{0\\}\\cup [\\frac{1}{k},1]$ and we have $\\sum_{x\\in S}\\bp^{\\beta}_{\\phi_{S}}(x) \\leq \\left(1-\\frac{S(\\bp^{\\beta}_{\\phi_{S}})-|S|}{k} \\right)$ and using the definition of $\\textbf{p}_{\\phi}$ the inequality follows. \n\t\n\tWe next upper bound the term $\\sum_{\\{S\\subseteq \\mathrm{Support}(\\bp^{\\beta}_{\\phi_{S}})~|~|S|=\\mathrm{Distinct}(\\phi) \\}} \\left(1-\\frac{S(\\bp^{\\beta}_{\\phi_{S}})-|S|}{k} \\right)^n$. Note that, \n\t\\begin{align*}\n\t\\sum_{\\{S\\subseteq \\mathrm{Support}(\\bp^{\\beta}_{\\phi_{S}})~|~|S|=\\mathrm{Distinct}(\\phi) \\}} & (1-\\frac{S(\\bp^{\\beta}_{\\phi_{S}})-|S|}{k} )^n \\leq \\exps{-n\\frac{S(\\bp^{\\beta}_{\\phi_{S}})-\\mathrm{Distinct}(\\phi)}{k}} \\binom{S(\\bp^{\\beta}_{\\phi_{S}})}{\\mathrm{Distinct}(\\phi)} \\\\\n\t& \\leq \\exps{-n\\frac{S(\\bp^{\\beta}_{\\phi_{S}})-\\mathrm{Distinct}(\\phi)}{k} + (S(\\bp^{\\beta}_{\\phi_{S}})-\\mathrm{Distinct}(\\phi))\\log S(\\bp^{\\beta}_{\\phi_{S}}) } \\\\\n\t& \\leq  \\exps{ (S(\\bp^{\\beta}_{\\phi_{S}})-\\mathrm{Distinct}(\\phi))(\\log k- c \\log k } \\\\\n\t& \\leq  \\exps{ -\\epsilon k \\log k } < \\exps{ -\\epsilon^2 n^{1-4\\alpha} }~.\n\t\\end{align*}\n\t\\newcommand{\\textbf{p}_{\\phi,S}}{\\textbf{p}_{\\phi,S}}\n\t In the second inequality, we use a weak upper bound for the quantity $\\binom{S(\\textbf{p}_{\\phi})}{\\mathrm{Distinct}(\\phi)}$. In the third and fourth inequality, we use $n=ck\\log k$, $c\\geq2$ and $S(\\bp^{\\beta}_{\\phi_{S}}) > \\mathrm{Distinct}(\\phi)+\\epsilon k$. In the final inequality, we use $n^{1-\\alpha}\\leq k\\log k$ for constant $\\alpha>0$. Combining everything together we get $\\Prob{\\bp^{\\beta}_{\\phi_{S}},\\phi} < \\exps{ -\\epsilon^2 n^{1-4\\alpha} } \\Prob{\\textbf{p}_{\\phi},\\phi}$, a contradiction on the definition of $\\bp^{\\beta}_{\\phi_{S}}$.\n\t\n\tTherefore if $n>2k\\log k$, then the previous derivation implies $\\Prob{|S(\\bp^{\\beta}_{\\phi_{S}})-\\mathrm{Distinct}(\\phi)|\\geq \\epsilon k}=1$. Further if $n>2k\\log k$, then $\\Prob{S(\\textbf{p}) =\\mathrm{Distinct}(\\phi)} \\geq 1-k\\exps{\\frac{-n}{k}}$. Combining the previous two inequalities and substituting $n>2k\\log k$ we get, $\\Prob{|S(\\bp^{\\beta}_{\\phi_{S}})-S(\\textbf{p})|\\geq \\epsilon k} \\geq 1-\\exps{-\\log k}$, thus concluding the proof.\n\t\\end{proof}\n\\section{Experiments}\\label{sec:exp}\nWe performed two different sets of experiments for entropy estimation -- one to compare performance guarantees and the other to compare running times. In our pseudo PML approach, we divide the samples into two parts. We run the empirical estimate on one (this is easy) and the PML estimate on the other. For the PML estimate, any algorithm to compute an approximate PML distribution can be used in a black box fashion. \nAn advantage of the pseudo PML approach is that it can use any algorithm to estimate the PML distribution as a black box, providing both competitive performance and running time efficiency. \nIn our experiments, we use the heuristic algorithm in \\cite{PJW17} to compute an approximate PML distribution. In the first set of experiments detailed below, we compare the performance of the pseudo PML approach with raw \\cite{PJW17} and other state-of-the-art estimators for estimating entropy. Our code is available at \\url{https:\/\/github.com\/shiragur\/CodeForPseudoPML.git}\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{plotJuly30cite.pdf}\n\t\\label{fig:entr}\\end{figure}\n\\vspace{-24pt}\n\nEach plot depicts the performance of various algorithms for estimating entropy of different distributions with domain size $N=10^5$. \nEach data point represents 50 random trials. ``Mix 2 Uniforms'' is a mixture of two uniform distributions, with half the probability mass on\nthe first $N\/10$ symbols, and $\\mathrm{Zipf}(\\alpha) \\sim 1\/i^{\\alpha}$ with $i \\in [N]$. \nMLE is the naive approach of using the empirical distribution with correction bias;\nall the remaining algorithms are denoted using bibliographic citations.\nIn our algorithm we pick $threshold=18$ (same as \\cite{WY16}) and our set $\\mathrm{F}=[0,18]$ (input of \\Cref{algpml}), i.e. we use the PML estimate on frequencies $\\leq 18$ and empirical estimate on the rest. Unlike \\Cref{algpml}, we do not perform sample splitting in the experiments -- we believe this requirement is an artifact of our analysis. For estimating entropy, the error achieved by our estimator is competitive with \\cite{PJW17} and other state-of-the-art entropy estimators. Note that our results match \\cite{PJW17} for small sample sizes because not many domain elements cross the threshold and for a large fraction of the samples, we simply run the \\cite{PJW17} algorithm.\n\n\nIn the second set of experiments we demonstrate the running time efficiency of our approach. In these experiments, we compare the running time of our algorithm using \\cite{PJW17} as a subroutine to the raw \\cite{PJW17} algorithm on the $\\mathrm{Zipf(1)}$ distribution. The second row is the fraction of samples on which our algorithm uses the empirical estimate (plus correction bias). The third row is the ratio of the running time of \\cite{PJW17} to our algorithm. \nFor large sample sizes, the entries in the EmpFrac row have high value, i.e. our algorithm applies the simple empirical estimate on large fraction of samples; therefore, enabling $10$x speedup in the running times. \n\\vspace*{-0.1in}\n\\begin{center}\n\t\\begin{tabular}{|c||c|c|c|c|c|c|c|c|} \n\t\t\\hline\n\t\tSamples size & $10^3$ & $5*10^3$ & $10^4$ & $5*10^4$ & $10^5$ & $5*10^5$ &$10^6$ & $5*10^6$ \\\\ [0.5ex]   \n\t\t\\hline\n\t\tEmpFrac & 0.184 & 0.317 & 0.372 & 0.505 & 0.562 & 0.695 & 0.752 & 0.886 \\\\  \n\t\t\\hline\n\t\tSpeedup & 0.824 & 1.205 & 1.669 & 3.561 & 4.852 & 9.552 & 13.337 & 12.196 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{center}\n\\section{Introduction}\nSymmetric property estimation is a fundamental and well studied problem in machine learning and statistics. In this problem, we are given $n$ i.i.d samples from an unknown distribution\\footnote{Throughout the paper, distribution refers to discrete distribution.} $\\textbf{p}$ and asked to estimate $\\textbf{f}(\\textbf{p})$, where $\\textbf{f}$ is a symmetric property (i.e. it does not depend on the labels of the symbols).\nOver the past few years, the computational and sample complexities for estimating many symmetric properties have been extensively studied.\nEstimators with optimal sample complexities have been obtained for several properties including entropy~\\cite{VV11a, WY16, JVHW15}, distance to uniformity~\\cite{VV11b, JHW16},  and support~\\cite{VV11a, WY15}. \n\nAll aforementioned estimators were property specific and therefore, a natural question is to design a universal estimator. \nIn \\cite{ADOS16}, the authors showed that the distribution that maximizes the profile likelihood, i.e. the likelihood of the multiset of frequencies of elements in the sample, referred to as \\emph{profile maximum likelihood (PML) distribution}, can be used as a universal plug-in estimator. \\cite{ADOS16} showed that computing the symmetric property on the PML distribution is sample complexity optimal in estimating support, support coverage, entropy and distance to uniformity within accuracy $\\epsilon > \\frac{1}{n^{0.2499}}$. Further, this also holds for distributions that approximately optimize the PML objective, where the approximation factor affects the desired accuracy.\n\nAcharya et al. \\cite{ADOS16} posed two important and natural open questions. The first was to give an efficient algorithm for finding an approximate PML distribution, which was recently resolved in \\cite{CSS19}. \nThe second open question is whether PML is sample competitive in all regimes of the accuracy parameter $\\epsilon$? In this work, we make progress towards resolving this open question.\n\nFirst, we show that the PML distribution based plug-in estimator achieves optimal sample complexity for all $\\epsilon$ for the problem of estimating support size.\nNext, we introduce a variation of the PML distribution that we call the \\emph{pseudo PML distribution}.\nUsing this, we give a general framework for estimating a symmetric property.\nFor entropy and distance to uniformity, this pseudo PML based framework achieves optimal sample complexity for a broader regime of the accuracy parameter than was known for the vanilla PML distribution.\n\nWe provide a general framework that could, in principle be applied to estimate any separable symmetric property $\\textbf{f}$, meaning $\\textbf{f}(\\textbf{p})$ can be written in the form of $\\sum_{x \\in \\mathcal{D}}\\textbf{f}(\\textbf{p}_x)$. This motivation behind this framework is that for any symmetric property $\\textbf{f}$ that is separable, the estimate for $\\textbf{f}(\\textbf{p})$ can be split into two parts: $\\textbf{f}(\\textbf{p})=\\sum_{x \\in B}\\textbf{f}(\\textbf{p}_x)+\\sum_{x \\in G}\\textbf{f}(\\textbf{p}_x)$, where $B$ and $G$ are a (property dependent) disjoint partition of the domain $\\mathcal{D}$. \nWe refer to $G$ as the good set and $B$ as the bad set. Intuitively, $G$ is the subset of domain elements whose contribution to $\\textbf{f}(\\textbf{p})$ is easy to estimate, i.e a simple estimator such as empirical estimate (with correction bias) works. \nFor many symmetric properties, finding an appropriate partition of the domain is often easy.\nMany estimators in the literature~\\cite{JVHW15, JHW16, WY16} make such a distinction between domain elements.\nThe more interesting and difficult case is estimating the contribution of the bad set: $\\sum_{x \\in B}\\textbf{f}(\\textbf{p}_x)$.\nMuch of the work in these estimators is dedicated towards estimating this contribution\nusing sophisticated techniques such as polynomial approximation.\nOur work gives a unified approach to estimating the contribution of the bad set. We propose a PML based estimator for estimating $\\sum_{x \\in B}\\textbf{f}(\\textbf{p}_x)$. \nWe show that computing the PML distribution only on the set $B$ is sample competitive for entropy and distance to uniformity for almost all interesting parameter regimes thus (partially) handling the open problem proposed in \\cite{ADOS16}. \nAdditionally, requiring that the PML distribution be computed on a subset $B \\subseteq \\mathcal{D}$ reduces the input size for the PML subroutine and results in practical algorithms (See \\Cref{sec:exp}).\n\nTo summarize, the main contributions of our work are:\n\\vspace*{-0.1in}\n\\begin{itemize}\n\\item We make progress on an open problem of \\cite{ADOS16} on broadening the range of error parameter $\\epsilon$ that one can obtain for universal symmetric property estimation via PML.\n\\item We give a general framework for applying PML to new symmetric properties.\n\\item As a byproduct of our framework, we obtain more practical algorithms that invoke PML on smaller inputs (See \\Cref{sec:exp}).\n\\end{itemize}\n\n\n\\subsection{Related Work}\nFor many natural properties, there has been extensive work on designing efficient estimators both with respect to computational time and sample complexity \\cite{HJW17, HJM17, AOST14, RVZ17, ZVVKCSLSDM16, WY16a, RRSS07, WY15, OSW16, VV11a, WY16, JVHW15, JHW16, VV11b}.\nWe define and state the optimal sample complexity for estimating support, entropy and distance to uniformity. For entropy, we also discuss the regime in which the empirical distribution is sample optimal.\n\n\\noindent{\\bf Entropy:} For any distribution $\\textbf{p} \\in \\Delta^{\\mathcal{D}}$, the entropy $H(\\textbf{p})\\eqdef -\\sum_{x\\in \\mathcal{D}}\\textbf{p}_{x} \\log \\textbf{p}_{x}$. For $\\epsilon \\geq  \\frac{\\log N}{N}$ (the interesting regime), where $N\\eqdef |\\mathcal{D}|$, the optimal sample complexity for estimating $H(\\textbf{p})$ within additive accuracy $\\epsilon$ is $O(\\frac{N}{\\log N}\\frac{1}{\\epsilon})$~\\cite{WY16}. Further if $\\epsilon <  \\frac{\\log N}{N}$, then \\cite{WY16} showed that empirical distribution is optimal.\n\n\\noindent{\\bf Distance to uniformity:} For any distribution $\\textbf{p} \\in \\Delta^{\\mathcal{D}}$, the distance to uniformity $\\|\\textbf{p}-u\\|_1\\eqdef \\sum_{x\\in \\mathcal{D}}|\\textbf{p}_{x}-\\frac{1}{N}|$, where $u$ is the uniform distribution over $\\mathcal{D}$. The optimal sample complexity for estimating $\\|\\textbf{p}-u\\|_1$ within additive accuracy $\\epsilon$ is $O(\\frac{N}{\\log N}\\frac{1}{\\epsilon^2})$~\\cite{VV11b, JHW16}.\n\n\\newcommand{k}{k}\n\\noindent{\\bf Support:} For any distribution $\\textbf{p} \\in \\Delta^{\\mathcal{D}}$, the support of distribution $S(\\textbf{p})\\eqdef |\\{x\\in \\mathcal{D}~|~\\textbf{p}_{x}>0 \\}|$. Estimating support is difficult in general because we need sufficiently large number of samples to observe elements with small probability values. Suppose for all $x \\in \\mathcal{D}$, if $\\textbf{p}_{x} \\in \\{0\\} \\cup [\\frac{1}{k},1]$, then \\cite{WY15} showed that the optimal sample complexity for estimating support within additive accuracy $\\epsilon k$ is $O(\\frac{k}{\\log k}\\log^2 \\frac{1}{\\epsilon})$.\n\nPML was introduced by Orlitsky et al.~\\cite{OSSVZ04} in 2004. \nThe connection between PML and universal estimators was first studied in~\\cite{ADOS16}. \nAs discussed in the introduction, PML based plug-in estimator applies to a restricted regime of error parameter $\\epsilon$.\nThere have been several other approaches for designing universal estimators for symmetric properties. Valiant and Valiant~\\cite{VV11a} adopted and rigorously analyzed a linear programming based approach for universal estimators proposed by \\cite{ET76} and showed that it is sample complexity optimal in the constant error regime for estimating certain symmetric properties (namely, entropy and support size). Recent work of Han et al.~\\cite{HJW18} applied a local moment matching based approach in designing efficient universal symmetric property estimators for a single distribution. \\cite{HJW18} achieves the optimal sample complexity in restricted error regimes for estimating the power sum function, support and entropy.\n\nRecently, \\cite{YOSW18} gave a different unified approach to property estimation.\nThey devised an estimator that uses $n$ samples and achieves the performance attained by the empirical estimator with $n \\sqrt{\\log n}$ samples for a wide class of properties and for all underlying distributions. \nThis result is further strengthened to $n \\log n$ samples for Shannon entropy and a broad class of other properties including $\\ell_1$-distance in \\cite{HO19b}.\n\nIndependently of our work, authors in \\cite{HO19a} propose \\emph{truncated PML} that is slightly different but similar in the spirit to our idea of pseudo PML. They use the approach of truncated PML and study its application to symmetric properties such as: entropy, support and coverage; refer \\cite{HO19a} for further details.\n\\subsection{Organization of the Paper}\nIn \\Cref{sec:prelims} we provide basic notation and definitions. \nWe present our general framework in \\Cref{sec:results} and state all our main results. \nIn \\Cref{app:universal}, we provide proofs of the main results of our general framework. \nIn \\Cref{sec:appl}, we use these results to establish the sample complexity of our estimator in the case of entropy (See \\Cref{sec:entr}) and distance to uniformity (See \\Cref{sec:dtu}). Due to space constraints, many proofs are deferred to the appendix. In  \\Cref{sec:exp}, we provide experimental results for estimating entropy using pseudo PML and other state-of-the-art estimators. Here we also demonstrate the practicality of our approach.\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection*{Acknowledgments}\nWe thank the reviewers for the helpful comments, great suggestions, and positive feedback.\nMoses Charikar was supported by a Simons Investigator Award, a Google Faculty Research Award and an Amazon Research Award.\nAaron Sidford was partially supported by NSF CAREER Award CCF-1844855.\n\\bibliographystyle{alpha}\n\n\\section{Preliminaries}\\label{sec:prelims}\nLet $[a]$ denote all integers in the interval $[1,a]$. Let $\\Delta^{\\mathcal{D}} \\subset [0,1]_{\\mathbb{R}}^{\\mathcal{D}}$ be the set of all distributions supported on domain $\\mathcal{D}$ and let $N$ be the size of the domain. Throughout this paper we restrict our attention to discrete distributions and assume that we receive a sequence of $n$ independent samples from an underlying distribution $\\textbf{p} \\in \\Delta^{\\mathcal{D}}$. Let $\\mathcal{D} ^n$ be the set of all length $n$ sequences and $y^n \\in \\mathcal{D}^n$ be one such sequence with $y^n_{i}$ denoting its $i$th element. The probability of observing sequence $y^n$ is:\n$$\\mathbb{P}(\\textbf{p},y^n) \\eqdef \\prod_{x \\in \\mathcal{D}}\\textbf{p}_x^{\\textbf{f}(y^n,x)}$$\nwhere $\\textbf{f}(y^n,x)= |\\{i\\in [n] ~ | ~ y^n_i = x\\}|$ is the frequency\/multiplicity of symbol $x$ in sequence $y^n$ and $\\textbf{p}_x$ is the probability of domain element $x\\in \\mathcal{D}$. We next formally define profile, PML distribution and approximate PML distribution.\n\n\\begin{defn}[Profile]\nFor a sequence $y^n \\in \\mathcal{D}^n$, its \\emph{profile} denoted $\\phi=\\Phi(y^n) \\in \\mathbb{Z}_{+}^{n}$ is $\\phi\\eqdef(\\phi(j))_{j \\in [n]} $ where $\\phi(j)\\eqdef|\\{x\\in \\mathcal{D} |\\textbf{f}(y^n,x)=j \\}|$ is the number of domain elements with frequency ${j}$ in $y^{n}$. We call $n$ the length of profile $\\phi$ and use $\\Phi^n$ denote the set of all profiles of length $n$.\n\\footnote{The profile does not contain $\\phi(0)$, the number of unseen domain elements.}\n\\end{defn}\n\nFor any distribution $\\textbf{p} \\in \\Delta^{\\mathcal{D}}$, the probability of a profile $\\phi \\in \\Phi^n$ is defined as:\n\\begin{equation}\\label{eqpml1}\n\\bbP(\\textbf{p},\\phi)\\eqdef\\sum_{\\{y^n \\in \\mathcal{D}^n~|~ \\Phi (y^n)=\\phi \\}} \\mathbb{P}(\\textbf{p},y^n)\\\\\n\\end{equation}\n\nThe distribution that maximizes the probability of a profile $\\phi$ is the profile maximum likelihood distribution and we formally define it next.\n\\begin{defn}[Profile maximum likelihood distribution] For any profile $\\phi \\in \\Phi^{n}$, a \\emph{Profile Maximum Likelihood} (PML) distribution $\\textbf{p}_{pml,\\phi} \\in \\Delta^{\\mathcal{D}}$ is:\n $\\textbf{p}_{pml,\\phi} \\in \\argmax_{\\textbf{p} \\in \\Delta^{\\mathcal{D}}} \\bbP(\\textbf{p},\\phi)$ and $\\bbP(\\textbf{p}_{pml,\\phi},\\phi)$ is the maximum PML objective value. Further, a distribution $\\textbf{p}^{\\beta}_{pml,\\phi} \\in \\Delta^{\\mathcal{D}}$ is a $\\beta$-\\emph{approximate PML} distribution if $\\bbP(\\textbf{p}^{\\beta}_{pml,\\phi},\\phi)\\geq \\beta \\cdot \\bbP(\\textbf{p}_{pml,\\phi},\\phi)$.\n\\end{defn}\n\nWe next provide formal definitions for separable symmetric property and an estimator.\n\\begin{defn}[Separable Symmetric Property] A symmetric property $\\textbf{f}:\\Delta^{\\mathcal{D}} \\rightarrow \\mathbb{R}$ is separable if for any $\\textbf{p} \\in \\Delta^{\\mathcal{D}}$, $f(\\textbf{p})\\eqdef\\sum_{x\\in \\mathcal{D}}\\textbf{g}(\\textbf{p}_{x})$, for some function $\\textbf{g}:\\mathbb{R} \\rightarrow \\mathbb{R}$. Further for any subset $S\\subset \\mathcal{D}$, we define $f_{S}(\\textbf{p})\\eqdef\\sum_{x\\in S}\\textbf{g}(\\textbf{p}_{x})$.\n\t\\end{defn}\n\n\\begin{defn}\n\tA property estimator is a function $\\hat{\\textbf{f}}:\\mathcal{D}^n \\rightarrow \\mathbb{R}$, that takes as input $n$ samples and returns the estimated property value. The sample complexity of $\\hat{\\textbf{f}}$ for estimating a symmetric property $\\textbf{f}(\\textbf{p})$ is the number of samples needed to estimate $\\textbf{f}$ up to accuracy $\\epsilon$ and with constant probability. The optimal sample complexity of a property $\\textbf{f}$ is the minimum number of samples of any estimator.\n\t\\end{defn} \n\\section{Main Results}\\label{sec:results}\nAs discussed in the introduction, one of our motivations was to provide a better analysis for the PML distribution based plug-in estimator. In this direction, we first show that the PML distribution is sample complexity optimal in estimating support in all parameter regimes. Estimating support is difficult in general and all previous works make the assumption that the minimum non-zero probability value of the distribution is at least $\\frac{1}{k}$. In our next result, we show that the PML distribution under this constraint is sample complexity optimal for estimating support.\n\\begin{thm}\\label{thm:supp}\n\tThe PML distribution \\footnote{\\label{suppfoot} Under the constraint that its minimum non-zero probability value is at least $\\frac{1}{k}$. This assumption is also necessary for the results in \\cite{ADOS16} to hold.} based plug-in estimator is sample complexity optimal in estimating support for all regimes of error parameter $\\epsilon$.\n\\end{thm}\nFor support, we show that an approximate PML distribution is sample complexity optimal as well.\n\\begin{thm}\\label{thm:approxsupp}\n\tFor any constant $\\alpha>0$, an $\\exps{-\\epsilon^2 n^{1-\\alpha}}$-approximate PML distribution \\footnoteref{suppfoot} based plug-in estimator is sample complexity optimal in estimating support for all regimes of error $\\epsilon$.\n\\end{thm}\nWe defer the proof of both these theorems to \\Cref{sec:support}.\n\nFor entropy and distance to uniformity, we study a variation of the PML distribution we call the pseudo PML distribution and present a general framework for symmetric property estimation based on this. We show that this pseudo PML based general approach gives an estimator that is sample complexity optimal for estimating entropy and distance to uniformity in broader parameter regimes. To motivate and understand this general framework we first define new generalizations of the profile, PML and approximate PML distributions.\n\\newcommand{\\phis(j)}{\\phi_{S}(j)}\n\\begin{defn}[$S$-pseudo Profile]\n\tFor any sequence $y^{n} \\in \\mathcal{D}^{n}$ and $S \\subseteq \\mathcal{D}$, its $S$-\\emph{pseudo} profile denoted $\\phi_{S}=\\Phi_{S}(y^n)$ is $\\phi\\eqdef(\\phis(j))_{j\\in[n]} $ where $\\phis(j)\\eqdef|\\{x\\in S ~|~\\textbf{f}(y^n,x)=j \\}|$ is the number of domain elements in $S$ with frequency ${j}$ in $y^{n}$. We call $n$ the length of $\\phi_{S}$ as it represents the length of the sequence $y^n$ from which this pseudo profile was constructed. Let $\\Phi_{S}^{n}$ denote the set of all $S$-pseudo profiles of length $n$.\n\\end{defn}\n\nFor any distribution $\\textbf{p} \\in \\Delta^{\\mathcal{D}}$, the probability of a $S$-pseudo profile $\\phi_{S} \\in \\Phi_{S}^{n}$ is defined as:\n\\begin{equation}\\label{eqpml1}\n\\bbP(\\textbf{p},\\phi_{S})\\eqdef\\sum_{\\{y^n \\in \\mathcal{D}^n~|~ \\Phi_{S}(y^n)=\\phi_{S} \\}} \\mathbb{P}(\\textbf{p},y^n)\\\\\n\\end{equation}\nWe next define the $S$-pseudo PML and $(\\beta,S)$-approximate pseudo PML distributions that are analogous to the PML and approximate PML distributions.\n\\begin{defn}[$S$-pseudo PML distribution]\n\tFor any $S$-pseudo profile $\\phi_{S} \\in \\Phi_{S}^{n}$, a distribution $\\textbf{p}_{\\phi_{S}} \\in \\Delta^{\\mathcal{D}}$ is a $S$-\\emph{pseudo PML} distribution if $\\textbf{p}_{\\phi_{S}} \\in \\argmax_{\\textbf{p} \\in \\Delta^{\\mathcal{D}}} \\bbP(\\textbf{p},\\phi_{S})$.\n\\end{defn}\n\n\\begin{defn}[$(\\beta,S)$-approximate pseudo PML distribution]\n\tFor any profile $\\phi_{S} \\in \\Phi_{S}^{n}$, a distribution $\\bp^{\\beta}_{\\phi_{S}} \\in \\Delta^{\\mathcal{D}}$ is a $(\\beta,S)$-\\emph{approximate pseudo PML} distribution if $\\bbP(\\bp^{\\beta}_{\\phi_{S}},\\phi_{S})\\geq \\beta \\cdot \\bbP(\\bp^{\\beta}_{\\phi_{S}},\\phi_{S})$.\n\\end{defn}\nFor notational convenience, we also define the following function.\n\\begin{defn}\n\tFor any subset $S \\subseteq \\mathcal{D}$, the function $\\mathrm{Freq}:\\Phi_{S}^{n} \\rightarrow 2^{\\mathbb{Z}_{+}}$ takes input a $S$-psuedo profile and returns the set with all distinct frequencies in $\\phi_{S}$.\n\\end{defn}\n\nUsing the definitions above, we next give an interesting generalization of Theorem 3 in \\cite{ADOS16}.\n\\begin{thm}\\label{thmbeta}\n\tFor a symmetric property $\\textbf{f}$ and $S \\subseteq \\mathcal{D}$, suppose there is an estimator $\\hat{\\textbf{f}}:\\Phi_{S}^{n} \\rightarrow \\mathbb{R}$, such that for any $\\textbf{p}$ and $\\phi_{S} \\sim \\textbf{p}$ the following holds, $$\\Prob{|\\textbf{f}_{{S}}(\\textbf{p})-\\hat{\\textbf{f}}(\\phi_{S})|\\geq \\epsilon} \\leq \\delta~,$$ then for any $\\mathrm{F} \\in 2^{\\mathbb{Z}_{+}}$, a $(\\beta,S)$-approximate pseudo PML distribution $\\bp^{\\beta}_{\\phi_{S}}$ satisfies: $$\\Prob{|\\textbf{f}_{{S}}(\\textbf{p})-\\textbf{f}_{{S}}(\\bp^{\\beta}_{\\phi_{S}})| \\geq 2 \\epsilon} \\leq \\frac{\\delta n^{|\\mathrm{F}|}}{\\beta} \\Prob{\\dfreq{\\phi_{S}} \\subseteq \\mathrm{F}} +\\Prob{\\dfreq{\\phi_{S}} \\not\\subseteq \\mathrm{F}}~.$$\n\\end{thm}\n\nNote that in the theorem above, the error probability with respect to a pseudo PML distribution based estimator has dependency on$\\frac{\\delta n^{|\\mathrm{F}|}}{\\beta}$ and $\\Prob{\\dfreq{\\phi_{S}} \\not\\subseteq \\mathrm{F}}$. However Theorem 3 in \\cite{ADOS16} has error probability $\\frac{\\delta e^{\\sqrt{n}}}{\\beta}$. This is the bottleneck in showing that PML works for all parameter regimes and the place where pseudo PML wins over the vanilla PML based estimator, getting non-trivial results for entropy and distance to uniformity. We next state our general framework for estimating symmetric properties.\nWe use the idea of sample splitting which is now standard in the literature \\cite{WY16,JVHW15,JHW16,CL11,Nem03}.\n\\begin{algorithm}[H]\n\t\\caption{General Framework for Symmetric Property Estimation}\\label{algpml}\n\t\\begin{algorithmic}[1]\n\t\t\\Procedure{Property estimation}{$x^{2n},\\textbf{f}, \\mathrm{F}$}\n\t\t\\State Let $x^{2n}=(x_1^n,x_2^n)$, where $x_1^n$ and $x_2^n$ represent first and last $n$ samples of $x^{2n}$ respectively.\n\t\t\\State Define $S\\eqdef \\{y \\in \\mathcal{D}~|~f(x_1^n,y) \\in \\mathrm{F} \\}$.\n\t\t\\State Construct profile $\\phi_{S}$, where $\\phis(j)\\eqdef|\\{y\\in S ~|~\\textbf{f}(x_2^n,y)=j \\}|$.\n\t\t\\State Find a $(\\beta,S)$-approximate pseudo PML distribution $\\bp^{\\beta}_{\\phi_{S}}$ and empirical distribution $\\hat{\\textbf{p}}$ on $x_2^n$.\n\t\t\\State \\textbf{return} $\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})+\\textbf{f}_{\\bar{S}}(\\hat{\\textbf{p}}) + \\text{correction bias with respect to }\\textbf{f}_{\\bar{S}}(\\hat{\\textbf{p}})$.\n\t\t\\EndProcedure\n\t\\end{algorithmic}\n\\end{algorithm}\nIn the above general framework, the choice of $\\mathrm{F}$ depends on the symmetric property of interest. Later, in the case of entropy and distance to uniformity, we will choose $\\mathrm{F}$ to be the region where the empirical estimate fails; it is also the region that is difficult to estimate. One of the important properties of the above general framework is that $\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})$ (recall $\\bp^{\\beta}_{\\phi_{S}}$ is a $(\\beta,S)$-approximate pseudo PML distribution and $\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})$ is the property value of distribution $\\bp^{\\beta}_{\\phi_{S}}$ on subset of domain elements $S \\subseteq \\mathcal{D}$) is close to $\\textbf{f}_{S}(\\textbf{p})$ with high probability. Below we state this result formally.\n\\begin{thm}\\label{thm:main}\n\tFor any symmetric property $\\textbf{f}$, let $\\mathrm{G} \\subseteq {\\mathcal{D}}$ and $\\mathrm{F},\\mathrm{\\bar{F}'} \\in 2^{\\mathbb{Z}_{+}}$. If for all $S' \\in 2^\\mathrm{G}$, there exists an estimator $\\hat{\\textbf{f}}:\\Phi_{S'}^{n} \\rightarrow \\mathbb{R}$, such that for any $\\textbf{p}$ and $\\phi_{S'} \\sim \\textbf{p}$ satisfies, \n\t\\begin{equation}\\label{eq:condio}\n\t\\Prob{|\\textbf{f}_{{S'}}(\\textbf{p})-\\hat{\\textbf{f}}(\\phi_{S'})|\\geq2 \\epsilon} \\leq \\delta \\text{ and }\\Prob{\\dfreq{\\phi_{S'}} \\not\\subseteq \\mathrm{F}'} \\leq \\gamma~.\n\t\\end{equation}\n\tThen for any sequence $x^{2n}=(x_1^n,x_2^n)$, $$\\Prob{|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})|>4\\epsilon}\\leq \\frac{\\delta n^{|F'|}}{\\beta}+\\gamma + \\Prob{S \\notin 2^\\mathrm{G}}~,$$ where $S$ is a random set $\\textbf{S}\\eqdef \\{y \\in \\mathcal{D}~|~f(x_1^n,y) \\in \\mathrm{F} \\}$ and $\\phi_{S}\\eqdef \\Phi_{S}(x_2^n)$.\n\\end{thm}\n\nUsing the theorem above, we already have a good estimate for $\\textbf{f}_{S}(\\textbf{p})$ for appropriately chosen frequency subsets $\\mathrm{F}, \\mathrm{\\bar{F}'}$ and $\\mathrm{G} \\subseteq \\mathcal{D}$. Further, we choose these subsets $\\mathrm{F}, \\mathrm{\\bar{F}'}$ and $\\mathrm{G}$ carefully so that the empirical estimate $\\textbf{f}_{\\bar{S}}(\\hat{p})$ plus the correction bias with respect to $\\textbf{f}_{\\bar{S}}$ is close to $\\textbf{f}_{\\bar{S}}(\\textbf{p})$. Combining these together, we get the following results for entropy and distance to uniformity. \n\\begin{thm}\\label{thm:entr}\n\tIf error parameter $\\epsilon>\\Omega\\left(\\frac{\\log N}{N^{1-\\alpha}}\\right)$ for any constant $\\alpha>0$,\n\tthen for estimating entropy, the estimator \\ref{algpml} for $\\beta=n^{-\\log n}$ is sample complexity optimal.\n\\end{thm}\nFor entropy, we already know from \\cite{WY16} that the empirical distribution is sample complexity optimal if $\\epsilon <c \\frac{\\log N}{N}$ for some constant $c>0$. Therefore the interesting regime for entropy estimation is when $\\epsilon>\\Omega\\left(\\frac{\\log N}{N}\\right)$ and our estimator works for almost all such $\\epsilon$.\n\\begin{thm}\\label{thm:dtu}\n\tLet $\\alpha>0$ and error parameter  $\\epsilon>\\Omega\\left(\\frac{1}{N^{1-8\\alpha}}\\right)$, then for estimating distance from uniformity, the estimator \\ref{algpml} for $\\beta=n^{-\\sqrt{\\frac{n \\log n}{N}}}$ is sample complexity optimal.\n\\end{thm}\nNote that the estimator in \\cite{JHW17} also requires that the error parameter $\\epsilon \\geq \\frac{1}{N^C}$, where $C>0$ is some constant.\n\\section{Analysis of General Framework for Symmetric Property Estimation}\\label{app:universal}\nHere we provide proofs of the main results for our general framework (Theorem`\\ref{thmbeta} and \\ref{thm:main}). These results weakly depend on the property and generalize results in \\cite{ADOS16}. The PML based estimator in \\cite{ADOS16} is sample competitive only for a restricted error parameter regime and this stems from the large number of possible profiles of length $n$. Our next lemma will be useful to address this issue and later we show how to use this result to prove Theorems \\ref{thmbeta} and \\ref{thm:main}.\n\n\\begin{lemma}\\label{lem:bcard}\n\t For any subset $S \\subseteq \\mathcal{D}$ and $\\mathrm{F} \\in 2^{\\mathbb{Z}_{+}}$, if set $B$ is defined as $B\\eqdef \\{\\phi_{S} \\in \\Phi_{S}^{n}~|~\\dfreq{\\phi_{S}} \\subseteq \\mathrm{F} \\}$, then the cardinality of set $B$ is upper bounded by $(n+1)^{|\\mathrm{F}|}$.\n\t\\end{lemma}\n\\begin{proof}[Proof of \\Cref{thmbeta}]\n\tUsing the law of total probability we have,\n\t\\begin{align*}\n\t\\begin{split}\n\t\\Prob{|\\textbf{f}_{{S}}(\\textbf{p})-\\textbf{f}_{{S}}(\\bp^{\\beta}_{\\phi_{S}})|\\geq 2\\epsilon} &=\\Prob{|\\textbf{f}_{{S}}(\\textbf{p})-\\textbf{f}_{{S}}(\\bp^{\\beta}_{\\phi_{S}})|\\geq 2\\epsilon~,~\\dfreq{\\phi_{S}} \\subseteq \\mathrm{F}}\\\\\n\t&\\quad + \\Prob{ |\\textbf{f}_{{S}}(\\textbf{p})-\\textbf{f}_{{S}}(\\bp^{\\beta}_{\\phi_{S}})|\\geq 2\\epsilon~,~\\dfreq{\\phi_{S}} \\not\\subseteq \\mathrm{F}},\\\\\n\t& \\leq \\Prob{|\\textbf{f}_{{S}}(\\textbf{p})-\\textbf{f}_{{S}}(\\bp^{\\beta}_{\\phi_{S}})|\\geq 2\\epsilon~,~\\dfreq{\\phi_{S}} \\subseteq \\mathrm{F}}  + \\Prob{\\dfreq{\\phi_{S}} \\not\\subseteq \\mathrm{F}}~.\n\t\\end{split}\n\t\\end{align*}\n\t\n\tConsider any $\\phi_{S} \\sim \\textbf{p}$. If $\\probbp{\\phi_{S}}>\\delta\/\\beta$, then we know that $\\probbpml{\\phi_{S}}>\\delta$. For $\\beta \\leq 1$, we have $\\probbp{\\phi_{S}}>\\delta$ that implies $|\\textbf{f}_{S}(\\textbf{p})-\\estiS{\\phi_{S}}| \\leq \\epsilon$. Further $\\probbpml{\\phi_{S}}>\\delta$ implies $|\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})-\\estiS{\\phi_{S}}| \\leq \\epsilon$. Using triangle inequality we get,\n\t$|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})| \\leq |\\textbf{f}_{S}(\\textbf{p})-\\estiS{\\phi_{S}}|+|\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})-\\estiS{\\phi_{S}}| \\leq 2\\epsilon$.\n\tNote we wish to upper bound the probability of set: $\\textbf{B}_{\\mathrm{F},S,\\hat{\\bff}}\\eqdef \\{\\phi_{S} \\in \\Phi_{S}^{n}~|~\\dfreq{\\phi_{S}} \\subseteq \\mathrm{F} \\text{ and }|\\textbf{f}_{{S}}(\\textbf{p})-\\textbf{f}_{{S}}(\\bp^{\\beta}_{\\phi_{S}})|\\geq 2 \\epsilon\\}$. From the previous discussion, we get $\\probbp{\\phi_{S}} \\leq \\delta\/\\beta$ for all $\\phi_{S} \\in \\textbf{B}_{\\mathrm{F},S,\\hat{\\bff}}$. Therefore,\n\t\\begin{align*}\n\t\\Prob{|\\textbf{f}_{{S}}(\\textbf{p})-\\textbf{f}_{{S}}(\\bp^{\\beta}_{\\phi_{S}})|\\geq 2\\epsilon~,~\\dfreq{\\phi_{S}} \\subseteq \\mathrm{F}}=\\sum_{\\phi_{S} \\in \\textbf{B}_{\\mathrm{F},S,\\hat{\\bff}} }\\probbp{\\phi_{S}} \\leq \\frac{\\delta}{\\beta}|\\textbf{B}_{\\mathrm{F},S,\\hat{\\bff}}| \\leq \\frac{\\delta}{\\beta} (n+1)^{|\\mathrm{F}|}~.\n\t\\end{align*}\n\tIn the final inequality, we use $\\textbf{B}_{\\mathrm{F},S,\\hat{\\bff}} \\subseteq \\{\\phi_{S} \\in \\Phi_{S}^{n}~|~\\dfreq{\\phi_{S}} \\subseteq \\mathrm{F}\\}$ and invoke \\Cref{lem:bcard}.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof for \\Cref{thm:main}]\nUsing Bayes rule we have:\n\t\\begin{equation}\n\t\\begin{split}\n\t\\Prob{|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})|>2\\epsilon}&=\\sum_{S'\\subseteq \\mathcal{D}} \\Prob{|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})|>2\\epsilon~|~S=S'}\\Prob{S=S'}\\\\\n\t&\\leq \\sum_{S' \\in 2^\\mathrm{G}} \\Prob{|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})|>2\\epsilon~|~S=S'}\\Prob{S=S'}+ \\Prob{S \\notin 2^\\mathrm{G}}~.\n\t\\end{split}\n\t\\end{equation}\n\tIn the second inequality, we use $\\sum_{S' \\notin 2^\\mathrm{G}} \\Prob{|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})|>2\\epsilon ~,~ S=S'} \\leq \\Prob{S \\notin 2^\\mathrm{G}}$. Consider the first term on the right side of the above expression and note that it is upper bounded by, $\\sum_{S' \\in 2^\\mathrm{G}} \\Prob{|\\textbf{f}_{S'}(\\textbf{p})-\\textbf{f}_{S'}(\\bp_{\\phi_{S'}}^{\\beta})|>2\\epsilon}\\Prob{S=S'}\n\\leq \\sum_{S' \\in 2^\\mathrm{G}} \\left[\\frac{\\delta n^{|F'|}}{\\beta}+\\Prob{\\dfreq{\\phi_{S'}} \\not\\subseteq \\mathrm{\\bar{F}'}} \\right] \\Prob{S=S'}\n\\leq \\frac{\\delta n^{|F'|}}{\\beta}+\\gamma$.\n\tIn the first upper bound, we removed randomness associated with the random set $S$ and used $\\Prob{|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})|>2\\epsilon~|~S=S'}= \\Prob{|\\textbf{f}_{S'}(\\textbf{p})-\\textbf{f}_{S'}(\\bp_{\\phi_{S'}}^{\\beta})|>2\\epsilon}$.\n\tIn the first inequality above, we invoke \\Cref{thmbeta} using conditions from \\Cref{eq:condio}. In the second inequality, we use $\\sum_{S' \\in 2^\\mathrm{G}} \\Prob{S=S'} \\leq 1$ and $\\Prob{\\dfreq{\\phi_{S}} \\not\\subseteq \\mathrm{\\bar{F}'}, S=S'} \\leq \\gamma$. The theorem follows by combining all the analysis together.\n\\end{proof}\n \n \n \\section{Applications of the General Framework}\\label{sec:appl}\nHere we provide applications of our general framework (defined in \\Cref{sec:results}) using results from the previous section. We apply our general framework to estimate entropy and distance to uniformity. In \\Cref{sec:entr} and \\Cref{sec:dtu} we analyze the performance of our estimator for entropy and distance to uniformity estimation respectively.\n\\subsection{Entropy estimation}\\label{sec:entr}\nIn order to prove our main result for entropy (\\Cref{thm:entr}), we first need the existence of an estimator for entropy with some desired properties. The existence of such an estimator will be crucial to bound the failure probability of our estimator. A result analogous to this is already known in \\cite{ADOS16} (Lemma 2) and the proof of our result follows from a careful observation of \\cite{ADOS16, WY16}. We state this result here but defer the proof to appendix.\n\\begin{lemma}\\label{thm:entrochange}\n\tLet $\\alpha>0$, $\\epsilon>\\Omega\\left(\\frac{\\log N}{N^{1-\\alpha}}\\right)$ and $S \\subseteq \\mathcal{D}$, then for entropy on subset $S$ ($\\sum_{y\\in S}\\textbf{p}_{y}\\log \\frac{1}{\\textbf{p}_{y}}$) there exists an $S$-pseudo profile based estimator that use the optimal number of samples, has bias less than $\\epsilon$ and if we change any sample, changes by at most $c\\cdot \\frac{n^{\\alpha}}{n}$, where $c$ is a constant.\n\\end{lemma}\nCombining the above lemma with \\Cref{thm:main}, we next prove that our estimator defined in \\Cref{algpml} is sample complexity optimal for estimating entropy in a broader regime of error $\\epsilon$.\n\\begin{proof}[Proof for \\Cref{thm:entr}]\n\tLet $\\textbf{f}(\\textbf{p})$ represent the entropy of distribution $\\textbf{p}$ and $\\hat{\\textbf{f}}$ be the estimator in \\Cref{thm:entrochange}.\n\tDefine $\\mathrm{F} \\eqdef [0, c_{1} \\log n]$ for constant $c_{1}\\geq 40$. Given the sequence $x^{2n}$, the random set $S$ is defined as $S\\eqdef \\{y \\in \\mathcal{D}~|~f(x_1^n,y) \\leq c_{1} \\log n \\}$.\n\tLet $\\mathrm{\\bar{F}'}\\eqdef[0,8c_{1} \\log n]$, then by derivation in Lemma 6~\\cite{ADOS16} (or by simple application of Chernoff \\footnote{Note probability of many events in this proof can be easily bounded by application of Chernoff. These bounds on probabilities are also shown in \\cite{ADOS16, WY16} and we use these inequalities by omitting details.}) we have,\n\t$$ \\Prob{\\dfreq{\\phi_{S}} \\not\\subseteq \\mathrm{\\bar{F}'}} =\\Prob{\\exists y\\in \\mathcal{D} \\text{ such that }\\textbf{f}(x_1^n,y)\\leq c_{1} \\log n \\text{ and }\\textbf{f}(x_2^n,y)> 8c_{1} \\log n}\\leq \\frac{1}{n^{5}}~.$$\n\tFurther let $\\mathrm{G}\\eqdef \\{x \\in \\mathcal{D} ~|~\\textbf{p}_{x}\\leq \\frac{2c_{1} \\log N}{n}\\}$, then by Equation 48 in \\cite{WY16} we have, $\\Prob{S \\notin 2^\\mathrm{G}} \\leq \\frac{1}{n^4}$.\n\tFurther for all $S' \\in 2^{\\mathrm{G}}$ we have,\n\t$$\\Prob{\\dfreq{\\phi_{S'}} \\not\\subseteq \\mathrm{\\bar{F}'}}=\\Prob{\\exists y\\in S' \\text{ such that }\\textbf{f}(x_2^n,y)> 8c_{1} \\log n}\\leq \\gamma \\text{ for } \\gamma=\\frac{1}{n^{5}}~.$$\n\tNote for all $x \\in S'$, $\\textbf{p}_{x} \\leq \\frac{2c_{1} \\log N}{n}$ and the above inequality also follows from Chernoff. All that remains now is to upper bound $\\delta$. Using the estimator constructed in \\Cref{thm:entrochange} and further combined with McDiarmid's inequality, we have,\n\t$$\\Prob{|\\textbf{f}_{{S'}}(\\textbf{p})-\\hat{\\textbf{f}}(\\phi_{S'})|\\geq 2 \\epsilon} \\leq 2\\expo{\\frac{-2\\epsilon^2}{n(c\\frac{n^\\alpha}{n})^2}} \\leq \\delta \\text{ for }\\delta= \\expo{-2\\epsilon^2n^{1-2\\alpha}}~.$$\nSubstituting all these parameters together in \\Cref{thm:main} we have,\n\t\\begin{equation}\n\t\\begin{split}\n\t\\Prob{|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})|>2\\epsilon}&\\leq \\frac{\\delta n^{|F'|}}{\\beta}+\\Prob{\\dfreq{\\phi_{S}} \\not\\subseteq \\mathrm{\\bar{F}'}} + \\Prob{S \\notin 2^\\mathrm{G}}\\\\\n\t&\\leq  \\expo{-2\\epsilon^2n^{1-2\\alpha}} n^{9 c_{1} \\log n}+\\frac{1}{n^4} \\leq \\frac{2}{n^4}~.\n\t\\end{split}\n\t\\end{equation}\n\tIn the first inequality, we use \\Cref{thm:main}. In the second inequality, we substituted the values for $\\delta, \\gamma, \\beta$ and $\\Prob{S \\notin 2^\\mathrm{G}}$. In the final inequality we used $n=\\Theta(\\frac{N}{\\log N}\\frac{1}{\\epsilon})$ and $\\epsilon>\\Omega\\left(\\frac{\\log^3 N}{N^{1-4\\alpha}}\\right)$.\n\t\n\t\tOur final goal is to estimate $\\textbf{f}(\\textbf{p})$, and to complete the proof we need to argue that $\\textbf{f}_{\\bar{S}}(\\hat{\\textbf{p}})$ + the correction bias with respect to $\\textbf{f}_{\\bar{S}}$ is close to $\\textbf{f}_{\\bar{S}}(\\textbf{p})$, where recall $\\hat{\\textbf{p}}$ is the empirical distribution on sequence $x_2^n$. The proof for this follows immediately from \\cite{WY16} (Case 2 in the proof of Proposition 4). \\cite{WY16} bound the bias and variance of the empirical estimator with a correction bias and applying Markov inequality on their result we get $\\Prob{|\\textbf{f}_{\\bar{S}}(\\textbf{p})-(\\textbf{f}_{\\bar{S}}(\\hat{\\textbf{p}})+\\frac{|\\bar{S}|}{n})|>2\\epsilon} \\leq \\frac{1}{3}$, where $\\frac{|\\bar{S}|}{n}$ is the correction bias in \\cite{WY16}. Using triangle inequality, our estimator fails if either $|\\textbf{f}_{\\bar{S}}(\\textbf{p})-(\\textbf{f}_{\\bar{S}}(\\hat{\\textbf{p}})+\\frac{|\\bar{S}|}{n})|>2\\epsilon$ or $|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})|>2\\epsilon$. Further by union bound the failure probability is at most $\\frac{1}{3}+\\frac{2}{n^4}$, which is a constant.\n\\end{proof}\n\\subsection{Distance to Uniformity estimation}\\label{sec:dtu}\nHere we prove our main result for distance to uniformity estimation (\\Cref{thm:dtu}). First, we show existence of an estimator for distance to uniformity with certain desired properties. Similar to entropy, a result analogous to this is shown in \\cite{ADOS16} (Lemma 2) and the proof of our result follows from the careful observation of \\cite{ADOS16, JHW17}. We state this result here but defer the proof to \\Cref{sec:omitted}.\n\\begin{lemma}\\label{thm:dtuchange}\n\tLet $\\alpha>0$ and $S \\subseteq \\mathcal{D}$, then for distance to uniformity on $S$ ($\\sum_{y \\in S}|\\textbf{p}_{y}-\\frac{1}{N}|$) there exists an $S$-pseudo profile based estimator that use the optimal number of samples, has bias at most $ \\epsilon$ and if we change any sample, changes by at most $c\\cdot \\frac{n^{\\alpha}}{n}$, where $c$ is a constant.\n\\end{lemma}\nCombining the above lemma with \\Cref{thm:main} we provide the proof for \\Cref{thm:dtu}.\n\\begin{proof}[Proof for \\Cref{thm:dtu}]\n\tLet $\\textbf{f}(\\textbf{p})$ represent the distance to uniformity for distribution $\\textbf{p}$ and $\\hat{\\textbf{f}}$ be the estimator in \\Cref{thm:dtuchange}.\n\tDefine $\\mathrm{F} = [\\frac{n}{N}-\\sqrt{\\frac{c_{1}n \\log n}{N}}, \\frac{n}{N}+\\sqrt{\\frac{c_{1}n \\log n}{N}}]$ for some constant $c_1 \\geq 40$. Given the sequence $x^{2n}$, the random set $S$ is defined as $S\\eqdef \\{y \\in \\mathcal{D}~|~f(x_1^n,y) \\in \\mathrm{F} \\}$. Let $\\mathrm{\\bar{F}'}=[\\frac{n}{N}-\\sqrt{\\frac{8c_{1}n \\log n}{N}}, \\frac{n}{N}+\\sqrt{\\frac{8c_{1} n\\log n}{N}}]$, then by derivation in Lemma 7 of \\cite{ADOS16} (also shown in \\cite{JHW17} \\footnote{Similar to entropy, for many events their probabilities can be bounded by simple application of Chernoff and have already been shown in \\cite{ADOS16, JHW17}. We omit details for these inequalities.}) we have,\n\t$$ \\Prob{\\dfreq{\\phi_{S}} \\not\\subseteq \\mathrm{\\bar{F}'}} =\\Prob{\\exists y\\in \\mathcal{D} \\text{ such that }\\textbf{f}(x_1^n,y)\\in \\mathrm{F} \\text{ and }\\textbf{f}(x_2^n,y) \\notin \\mathrm{\\bar{F}'}}\\leq \\frac{1}{n^{4}}~.$$\n\t Further let $\\mathrm{G}\\eqdef \\{x \\in \\mathcal{D} ~|~\\textbf{p}_{x}\\in [\\frac{1}{N}-\\sqrt{\\frac{2c_{1} \\log n}{nN}}, \\frac{1}{N}+\\sqrt{\\frac{2c_{1} \\log n}{nN}}]\\}$, then using Lemma 2 in \\cite{JHW17} we get,\n\t$$\\Prob{S \\notin 2^\\mathrm{G}}=\\Prob{\\exists y \\in \\mathcal{D} \\text{ such that }\\textbf{f}(x_1^n,y)\\in \\mathrm{F} \\text{ and }\\textbf{p}_{x} \\notin G} \\leq \\frac{\\log n}{n^{1-\\epsilon}}~.$$\n\tFurther for all $S' \\in 2^{\\mathrm{G}}$ we have,\n\t$$\\Prob{\\dfreq{\\phi_{S'}} \\not\\subseteq \\mathrm{\\bar{F}'}}=\\Prob{\\exists y\\in S' \\text{ such that }\\textbf{f}(x_2^n,y)> 8c_{1} \\log n}\\leq \\gamma \\text{ for } \\gamma=\\frac{1}{n}~.$$\n\tNote for all $x \\in S'$, $\\textbf{p}_{x} \\in G$ and the above result follows from \\cite{JHW17} (Lemma 1). All that remains now is to upper bound $\\delta$. Using the estimator constructed in \\Cref{thm:dtuchange} and further combined with McDiarmid's inequality, we have,\n\t$$\\Prob{|\\textbf{f}_{{S'}}(\\textbf{p})-\\hat{\\textbf{f}}(\\phi_{S'})|\\geq 2 \\epsilon} \\leq 2\\expo{\\frac{-2\\epsilon^2}{n(c\\frac{n^\\alpha}{n})^2}} \\leq \\delta \\text{ for }\\delta= \\expo{-2\\epsilon^2n^{1-2\\alpha}}~.$$\n\tSubstituting all these parameters in \\Cref{thm:main} we get,\n\t\\begin{equation}\\label{eq:entropml}\n\t\\begin{split}\n\t\\Prob{|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})|>2\\epsilon}&\\leq \\frac{\\delta n^{|F'|}}{\\beta}+\\Prob{\\dfreq{\\phi_{S}} \\not\\subseteq \\mathrm{\\bar{F}'}} + \\Prob{S \\notin 2^\\mathrm{G}}\\\\\n\t&\\leq  \\expo{-2\\epsilon^2n^{1-2\\alpha}} n^{2\\sqrt{\\frac{8c_{1}n \\log n}{N}}}+\\frac{\\log n}{n^{1-\\epsilon}}+\\frac{1}{n} \\leq o(1)~.\n\t\\end{split}\n\t\\end{equation}\n\tIn the first inequality, we use \\Cref{thm:main}. In the second inequality, we substituted values for $\\delta, \\gamma, \\beta$ and $\\Prob{S \\notin 2^\\mathrm{G}}$. In the final inequality we used $n=\\Theta(\\frac{N}{\\log N}\\frac{1}{\\epsilon^2})$ and $\\epsilon>\\Omega\\left(\\frac{1}{N^{1-8\\alpha}}\\right)$.\n\t\n\tOur final goal is to estimate $\\textbf{f}(\\textbf{p})$, and to complete the proof we argue that $\\textbf{f}_{\\bar{S}}(\\hat{\\textbf{p}})$ + correction bias with respect to $\\textbf{f}_{\\bar{S}}$ is close to $\\textbf{f}_{\\bar{S}}(\\textbf{p})$, where recall $\\hat{\\textbf{p}}$ is the empirical distribution on sequence $x_2^n$. The proof for this case follows immediately from \\cite{JHW17} (proof of Theorem 2). \\cite{JHW17} define three kinds of events $\\mathcal{E}_{1},\\mathcal{E}_{2}$ and $\\mathcal{E}_{3}$, the proof for our empirical case follows from the analysis of bias and variance of events $\\mathcal{E}_{1}$ and $\\mathcal{E}_{2}$. Further combining results in \\cite{JHW17} with Markov inequality we get $\\Prob{|\\textbf{f}_{\\bar{S}}(\\textbf{p})-\\textbf{f}_{\\bar{S}}(\\hat{\\textbf{p}})|>2\\epsilon} \\leq \\frac{1}{3}$, and the correction bias here is zero. Using triangle inequality, our estimator fails if either $|\\textbf{f}_{\\bar{S}}(\\textbf{p})-(\\textbf{f}_{\\bar{S}}(\\hat{\\textbf{p}})+\\frac{|\\bar{S}|}{n})|>2\\epsilon$ or $|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})|>2\\epsilon$. Further by union bound the failure probability is upper bounded by $\\frac{1}{3}+o(1)$, which is a constant.\n\\end{proof}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nPhotonic crystals are periodically structured\ndielectric media possessing photonic band gaps:\nranges of frequency in which strong reflection\noccurs for all angles of incidence and all\npolarizations. They then behave as\nomnidirectional reflectors, free of\ndissipative losses. Since the initial predictions\nof Yablonovitch~\\cite{Yablo87} and John~\\cite{John87},\nphotonic crystals have been attracting a lot\nof attention and a wide variety of applications\nhave been suggested~\\cite{Dowling}.\n\nIn the one-dimensional case, a photonic crystal\nis nothing more than a periodic dielectric\nstructure. Bragg mirrors consisting of\nalternating low- and high-index layers\nconstitute, perhaps, the archetypical example~\\cite{Yeh88}.\nIn particular, quarter-wave stacks (at normal\nincidence) are the most thoroughly studied\nin connection with omnidirectional reflection\n(ODR)~\\cite{Fink98,Dowl98,Yablo98,Chig99,South99,Lekner00}.\n\nThe introduction of Fibonacci multilayers by\nKohmoto and coworkers~\\cite{KSI87} spurred the\ninterest for both possible optical\napplications~\\cite{Schw88} and theoretical\naspects of light transmission in aperiodic\nmedia~\\cite{Dulea90,Latge92,Liu97,Vasco98,Macia01}.\nIn fact, the possibility of obtaining ODR in\nquasiperiodic Fibonacci multilayers has been\nput forward recently~\\cite{Macia98,Cojo01,Lusk01,Peng02,Dong03}.\n\nUnderlying all these efforts a crucial\nquestion remains concerning whether\nquasiperiodic Fibonacci multilayers would\nachieve better performance than usual\nperiodic ones. To answer such a fundamental\nquestion one first needs to quantify the\nidea of ODR performance in a unique manner\nthat permits unambiguous comparison between\ndifferent structures. Only quite recently\na suitable figure of merit has been introduced:\nthe area under the transmittance curve as a\nfunction of the incidence angle~\\cite{Yonte04}.\nIn this paper, we resort to this concept of\narea to rank in a consistent way the ODR\ncharacteristics of these quasiperiodic systems.\n\n\\section{Quasiperiodic Fibonacci multilayers}\n\nA Fibonacci system is based on the recursive\nrelation $S_0 = \\{H \\}$, $S_1 = \\{ L \\} $  and\n$S_j = S_{j-1} S_{j-2}$ for $j \\ge 2$. Here $H$\nand $L$ are defined as being two dielectric layers\nwith refractive indices $(n_H, n_L)$ and thicknesses\n$(d_H, d_L)$, respectively. The material $H$ has a\nhigh refractive index while $L$ is of low refractive\nindex. The number of layers is given by $F_j$,\nwhere $F_j$ is a Fibonacci number obtained from\nthe recursive law $F_j = F_{j-1} + F_{j-2}$, with\n$F_0 = F_1 = 1$.\n\nIn order to properly compare the optical response\nof these systems we will rely on the transfer-matrix\ntechnique.  The transfer matrix $\\mathsf{M}_j$ for the\nFibonacci system $S_j$ can be computed as~\\cite{KSI87}\n\\begin{eqnarray}\n& \\mathsf{M}_0  = \\mathsf{M}_H ,\n\\qquad\n\\mathsf{M}_1  = \\mathsf{M}_L , &  \\nonumber \\\\\n& & \\\\\n& \\mathsf{M}_j  =  \\mathsf{M}_{j-1}\n\\mathsf{M}_{j-2}  , \\qquad j \\ge 2 . &\n\\nonumber\n\\end{eqnarray}\nThe transfer matrix for the single layer $H$ is\n\\begin{equation}\n\\mathsf{M}_H  =\n\\left (\n\\begin{array}{cc}\n\\cos \\beta_H &  q_H \\sin \\beta_H \\\\\n\\displaystyle\n\\frac{1}{q_H} \\sin \\beta_H  & \\cos \\beta_H\n\\end{array}\n\\right ) ,\n\\end{equation}\nand a analogous expression for $L$. Here $\\beta_H =\n(2 \\pi \/ \\lambda) n_H d_H \\cos \\theta_H$ is the layer\nphase thickness, $\\theta_H$ being the angle of\nrefraction, which is determined by Snell law.\nThe wavelength in vacuum of the incident\nradiation  is $\\lambda$. The parameter $q_H$\ncan be written for each basic polarization\n($p$ or $s$) as\n\\begin{equation}\nq_H (p) = \\frac{n_H \\cos \\theta}{n \\cos \\theta_H} ,\n\\qquad\nq_H (s) =  \\frac{n \\cos \\theta}{n_H \\cos \\theta_H} ,\n\\end{equation}\nwhere we have assumed that the layer is imbedded in a\nmedium of refractive index $n$. Henceforth $\\theta$\nwill denote the angle of incidence and, for simplicity,\nthe surrounding medium we will supposed to be air\n($n = 1$).\n\nLet us consider a $N$-period finite structure whose\nbasic cell is precisely the Fibonacci multilayer $S_j$.\nWe denote this system as $[S_j]^N$ and its overall transfer\nmatrix is\n\\begin{equation}\n\\mathsf{M}_j^{(N)} = (\\mathsf{M}_j)^N .\n\\end{equation}\nWhen the unit cell is $S_2$, the resulting multilayer\n$[S_2]^N$ is periodic. For $j > 2$, $[S_j]^N$ are\nquasiperiodic.\n\nThe transmittance $\\mathcal{T}_j^{(N)}$ is given\nin terms of the matrix $\\mathsf{M}_j^{(N)}$\nas~\\cite{KSI87}\n\\begin{equation}\n\\mathcal{T}_j^{(N)} =\n\\frac{4}{|| \\mathsf{M}_j^{(N)} ||^2 + 2} ,\n\\end{equation}\nwhere $|| \\mathsf{M}_j^{(N)} ||^2$ denotes the\nsum of the squares of the matrix elements.\n\nIn the theory of periodic systems it is well\nestablished that band gaps appear whenever\nthe trace of the basic period satisfies~\\cite{Yeh88}\n\\begin{equation}\n\\label{ODR}\n| \\mathop{\\mathrm{Tr}}\\nolimits ( \\mathsf{M}_j) | \\ge 2  .\n\\end{equation}\nThis should be worked out for both basic\npolarizations. The trace map is a powerful\ntool to investigate this condition, especially\nwhen the index $j$ is high~\\cite{KKT83}. In\nour context, it reads as\n\\begin{equation}\n\\mathop{\\mathrm{Tr}}\\nolimits ( \\mathsf{M}_{j+1}) = \\mathop{\\mathrm{Tr}}\\nolimits ( \\mathsf{M}_j)\n\\mathop{\\mathrm{Tr}}\\nolimits ( \\mathsf{M}_{j-1}) - \\mathop{\\mathrm{Tr}}\\nolimits ( \\mathsf{M}_{j-2})  .\n\\end{equation}\nThis simple recurrence relation allows us\nto compute easily the band gaps. We quote here\nthe first nontrivial cases, namely, the pure periodic\nsystem $S_2 = \\{ LH \\}$ and the first quasiperiodic\none $S_3 = \\{ LHL \\}$, respectively:\n\\begin{eqnarray}\n\\label{bands}\n& |\\cos \\beta_L \\cos \\beta_H -\n\\Lambda_{LH} \\\n\\sin \\beta_L \\sin \\beta_H | \\ge 1 , &\n\\nonumber \\\\\n& & \\\\\n& |\\cos ( 2 \\beta_L) \\cos \\beta_H -\n\\Lambda_{LH} \\\n\\sin (2 \\beta_L) \\sin \\beta_H | \\ge 1 . &\n\\nonumber\n\\end {eqnarray}\nThe function $\\Lambda_{LH}$ is\n\\begin{equation}\n\\Lambda_{LH} = \\frac{1}{2}\n\\left (\n\\frac{q_L}{q_H} +\n\\frac{q_H}{q_L}\n\\right ) ,\n\\end {equation}\nwhich is frequency independent but takes\ndifferent values for $p$ and $s$ polarizations.\nHowever, one can check that, irrespective of the\nangle of incidence, the following relation for both\nbasic polarizations holds:\n\\begin{equation}\n\\label{qps}\n\\frac{\\Lambda_{LH}(p)}{\\Lambda_{LH}(s)} \\le  1 .\n\\end{equation}\nDue to the restriction (\\ref{qps}),\nwhenever Eqs.~(\\ref{bands}) are fulfilled\nfor $p$ polarization, they are always\ntrue also for $s$ polarization. In\nconsequence, the $p$-polarization bands\nare more stringent than the corresponding\n$s$-polarization ones~\\cite{Dong03}.\n\n\\section{Assessing ODR from quasiperiodic multilayers}\n\nWe first investigate the range of layer thicknesses\nfor which ODR exists; that is, when condition~(\\ref{ODR})\nholds true for all the incidence angles. Although\nfor the simple periodic system $S_2$ analytic\napproximations are at hand, the general problem\nseems to be very involved and we content\nourselves with a numerical exploration.\n\nFor definiteness, we fix the refractive indices to\nthe values $n_L = 1.75$ and $n_H = 3.35$ at\n$\\lambda = 10 \\ \\mu$m. In Fig.~1 we have plotted\nthe zones of ODR for the basic periods $S_j$ (with\n$j=2, 3, 4, 5$) in terms of the adimensional thicknesses\n$n_L d_L\/\\lambda$ and $n_H d_H\/\\lambda$. Note that\nthe use of these adimensional variables not only\nsimplifies the presentation of the results, but,\nas dispersion  can be neglected, the\nresults apply to more general situations.\n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[height=6cm]{Figure1.eps}}\n\\caption{Regions where ODR (for $p$ polarization) occurs\nfor the Fibonacci systems $S_2 = \\{LH\\}$, $S_3 = \\{LHL\\}$,\n$S_4 =\\{LHLLH\\}$, and  $S_5 = \\{LHLLHLHL\\}$. We have taken\n$n_L = 1.75$ and $n_H = 3.35$ at $\\lambda = 10 \\ \\mu$m.\nThe inset identifies the filled ellipses. The marked\npoints correspond to the minimum area for each one\nof the systems.}\n\\end{figure}\n\nThe contours of all these regions are approximately\nelliptical. For every allowed value of $n_L\nd_L\/\\lambda$ there are two values of $n_H d_H\/\n\\lambda$. This can be traced back to the\nexplicit form of Eqs.~(\\ref{bands}) for the band\ngaps. The ellipses for $S_2$ are the biggest, which\nconfirms that this simple system has the best range of\nODR in terms of $n d\/\\lambda$ variables. Note also\nthat the usual Bragg solution with layers of a\nquarter-wavelength thick at normal incidence,\nnamely\n\\begin{equation}\n\\label{lambda4}\nn_L d_L\/\\lambda = n_H d_H\/\\lambda = 1\/4 ,\n\\end{equation}\nworks for $S_2$, but not for the others.\n\nIt is worth stressing the fact that the regions of\nODR for $S_2$ and $S_3$ are disjoint. This increases\nthe difficulty of comparison between these systems.\nOn the contrary, all the quasiperiodic multilayers\nhave a significant region of common parameters.\nIn fact, from the system $S_6$ onwards, all the\nelliptic contours are essentially the same as for\nthe $S_5$.\n\nThese regions of ODR are not enough to fully\nquantify the performance of the multilayer. In\nRef.~\\cite{Yonte04} we have proposed that,\nonce the materials and the wavelength are fixed,\nthe area under the transmittance as a function\nof the angle of incidence $\\theta$\n\\begin{equation}\n\\label{area}\n\\mathcal{A}^{(N)}_j = \\int_0^{\\pi\/2}\n\\mathcal{T}^{(N)}_j (\\theta) \\ d \\theta ,\n\\end{equation}\nis an appropriate figure of merit for the\nstructure: the smaller this area, the better\nthe performance as ODR. In Fig.~2 we have\nplotted this area as a function of\n$n_L d_L\/\\lambda$ and $n_H d_H\/\\lambda$\nfor $S_2$. The area has been computed solely\nfor the points fulfilling the ODR condition,\nso the abrupt steps give the boundaries of ODR\nplotted in Fig.~1.  However, this function\nvaries significantly in the ODR region.\n\n\\begin{figure}\n\\centering\n\\centerline{\\includegraphics[height=6cm]{Figure2.eps}}\n\\caption{Area under the transmittance curve,\ndefined in Eq.~(\\ref{area}), as a function of\n$n_L d_L \/\\lambda$ and $ n_H d_H\/ \\lambda$ for\nthe system $S_2$, with the same data as in Fig.~1.}\n\\end{figure}\n\nIn fact, for the present case the minimum of this\narea is reached at the point\n\\begin{equation}\nn_L d_L\/\\lambda = 0.34305  ,\n\\qquad\nn_H d_H\/\\lambda = 0.25416 .\n\\end{equation}\nWhile the value of $n_H d_H\/ \\lambda $ essentially\ncoincide with the quarter wavelength solution\n(\\ref{lambda4}),  $n_L d_L\/ \\lambda $ differs\nmore than 30 \\% of that solution.\n\n\\begin{figure}[t]\n\\centering\n\\centerline{\\includegraphics[height=6cm]{Figure3.eps}}\n\\caption{Regions of ODR for the same Fibonacci\nmultilayers as in Fig.~1 in the plane $(n_L, n_H)$\nof refractive indices. The curves show the limit\nof  ODR for each stack with the optimum thicknesses\nmarked in Fig.~1.}\n\\end{figure}\n\nIn Fig.~1 we have marked the points of minimum\narea for each one of the Fibonacci systems $S_j$.\nWe see the strong difference for the periodic and\nthe quasiperiodic systems. In fact, for the latter\n($S_j$ with $j \\ge 3$) we can summarize the results\nsaying that the optimum  area is reached approximately\nat the values of the parameters\n\\begin{equation}\nn_L d_L\/\\lambda = 1\/8  ,\n\\qquad\nn_H d_H\/\\lambda = 1\/4  .\n\\end{equation}\nIn our view, this is a  remarkable result: from the\nprinciple of minimum area, we have consistently derived\noptimum parameters for ODR, which differ a lot from\nthe usual solutions found in the literature.\n\n\n\nFor the thicknesses giving minimum area, we have\ncalculated the region in the $(n_L, n_H)$\nplane for which ODR occurs. In Fig.~3 we have\nplotted the boundary of such a region for\nthe same Fibonacci multilayers as before: above\nsuch curves we have the ODR region. It is again\nthe periodic system the first in fulfilling ODR:\nthe onset of the ODR curve is at $n_H \\simeq\n2.5$, in agreement with previous estimations~\\cite{Lekner00}.\n\n\nOf course, the optimum parameters for the system\n$S_j$ do not need to be optimum for $[S_j]^N$. To\nelucidate this question, we have computed numerically\nthe values of $n_L d_L \/\\lambda$ and $n_H d_H \/\\lambda$\nfor different systems containing up to 42 layers and\nfor the same indices as before. In Table 1 we have\nsummarized the corresponding data. We have included\nonly results for the five first periods $N=1, \\ldots, 5$,\nsince from $[S_j]^5$ onwards, all the thicknesses are\nfairly stable, while the area tends rapidly to 0,\nas one would expect from a band gap. We can conclude\nthat the optimum parameters do not depend strongly\non the number of layers.\n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[height=6cm]{Figure4.EPS}}\n\\caption{Logarithm of the area computed for the systems\n$[S_j]^N$ as a function of the number of layers.}\n\\end{figure}\n\n\nFrom previous results for the case of Bragg mirrors,\nit is reasonable to assume that the transmittance of\n$[S_j]^N$ tends to zero exponentially with the number\nof layers. To test such an \\textit{ansatz}, we have plotted\nthe area (in a logarithmic scale) for all these systems.\nThe results are presented in Fig.~4. We think that a\nsimple glance at this figure is enough to decide on the\nperformance of the quasiperiodic systems as omnidirectional\nreflectors.\n\n\n\nIt is quite clear that all the quasiperiodic systems,\nirrespective of the index $j$, behave essentially in\nthe same way as far as ODR is concerned. All the points\nfor these systems fit into a straight line. On the\nother hand, the periodic system $S_2$ lies on another\nstraight line, but with a better slope. That is,\nfor a given number of layers of the system (and under\nthe hypothesis of optimum thicknesses), the system\n$[LH]^N$ offers better performance than any other.\n\n\n\\begin{table}\n\\caption{Optimum parameters for the systems $[S_j]^N$\nfor different basic periods $S_j$ and the number of\nperiods $N$ ranging from 1 to 5.}\n\\begin{center}\n\\begin{tabular}{cccccc}\n\\textbf{Basic} & \\textbf{Bandwidth} & $\\# $ \\textbf{Periods} &\n$\\mathbf{n_L d_L\/\\lambda}$ & $\\mathbf{n_H d_H\/\\lambda}$ &\n\\textbf{Area}  \\\\\n\\hline\n\\ & \\            & 1 & 0.34305 & 0.25416 & 1.01660 \\\\\n\\ & \\            & 2 & 0.33807 & 0.22187 & 0.47147 \\\\\n$S_2$ & 0.217    & 3 & 0.30817 & 0.23429 & 0.17894 \\\\\n\\ & \\            & 4 & 0.29821 & 0.23926 & 0.06294 \\\\\n\\ & \\            & 5 & 0.29572 & 0.24422 & 0.02171 \\\\\n\\hline\n\\ & \\            & 1 & 0.11978 & 0.26906 &   1.07725 \\\\\n\\ & \\            & 2 & 0.12917 & 0.26657 &   0.47955 \\\\\n$S_3$ & 0.233    & 3 & 0.13319 & 0.26409 &   0.17693 \\\\\n\\ & \\            & 4 & 0.13587 & 0.26409 &   0.06092  \\\\\n\\ & \\            & 5 & 0.13722 & 0.26409 &   0.02055  \\\\\n\\hline\n\\ & \\            & 1 & 0.14795 & 0.25912 & 0.48632 \\\\\n\\ & \\            & 2 & 0.16538 & 0.25912 & 0.10423 \\\\\n$S_4$ & 0.195    & 3 & 0.16002 & 0.29389 & 0.01647 \\\\\n\\ & \\            & 4 & 0.16136 & 0.29389 & 0.00251 \\\\\n\\ & \\            & 5 & 0.16270 & 0.29389 & 0.00038 \\\\\n\\hline\n\\ & \\            & 1 & 0.14929 & 0.28396 &  0.24841 \\\\\n\\ & \\            & 2 & 0.15063 & 0.28396 &  0.01365 \\\\\n$S_5$ & 0.198    & 3 & 0.15331 & 0.28396 &  0.00073 \\\\\n\\ & \\            & 4 & 0.15197 & 0.28644 &  0.00004 \\\\\n\\ & \\            & 5 & 0.15331 & 0.28644 &  0.00002 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nOf course, one may think that the bandwidth of these\nsystems is different. Sometimes the bandwidth is\ndefined at normal incidence, and then it has been\nargued that quasiperiodic systems offer fundamental\nadvantages~\\cite{Lusk01}.  If we denote by\n$\\lambda_{\\mathrm{short}}$ and $\\lambda_{\\mathrm{long}}$\nthe longer- and shorter-wavelength edges for given\nODR bands (of the basic period), it seems more\nappropriate to define the ODR bandwidth as~\\cite{South99}\n\\begin{equation}\nB = \\frac{\\lambda_{\\mathrm{long}} -\n\\lambda_{\\mathrm{short}}}{\\frac{1}{2}\n(\\lambda_{\\mathrm{long}} +\n\\lambda_{\\mathrm{short}})} .\n\\end{equation}\nNote that this is the appropriate definition in our case.\nObviously, the parameters chosen for the purpose of\ncomparison must be the ones giving minimum area; i. e.,\noptimum ODR behavior. In fact, we have numerically\nchecked that the parameters giving optimum area offer\nalso a good bandwidth. In Table 1 we have indicated\nthis parameter, confirming again that with the proper\ndefinition the periodic system is the best.\n\n\\section{Concluding remarks}\n\nIn summary, we have exploited the idea of minimum area\nto fully assess in a systematic way the performance of\nomnidirectional reflectors. Although quasiperiodic\nsystems has attracted a lot of interest due to their\nunusual physical properties, Bragg reflectors offer\nthe best performance, although not at a quarter-wavelength\nthick at normal incidence. We believe that the best\nfeature of our approach is that it provides  a very\nclear thread to deal with omnidirectional reflection\nproperties in a systematic way. Our method is general\nand can be applied to any spectral region.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIt is generally believed that metric type II solar radio bursts\nare excited by energetic electrons accelerated at coronal shocks\n(e.g., Wild, 1950; Dulk, 1985; Pick \\& Vilmer, 2008). Although\nextensive studies exist in the literature investigating the\nphysics of type IIs, the exact emission site and the associated\nshock properties remain controversial mainly due to a lack of\ndirect high-resolution imagings of the bursts. The focus of the\ndebate lies in whether type II shocks are driven by flare heating\nor coronal mass ejection (CME), (e.g., Wagner {\\&}\nMcQueen, 1983; Gosling, 1993; Gopalswamy et al., 1998; Cliver et\nal., 1999, 2005; Mancuso {\\&} Raymond, 2004; Cane\n{\\&} Erickson, 2005; Vr\\v snak {\\&} Cliver, 2008;\nMagdaleni\\' c et al., 2008; Pohjolainen et al., 2008a), and whether the radio\n\\textbf{bursts are generated at the shock nose or at the shock flank}  (e.g. Reiner et al.,\n2003; Mancuso {\\&} Raymond, 2004; Cho et al., 2007, 2008, 2011;\nShen et al., 2013).\n\nRecently we have proposed that source properties including the\nemission site as well as the shock geometry can be inferred by\ncombining radio spectral data and solar imaging observations (Feng\net al., 2012; Kong et al., 2012). The main idea is to establish the physical\nconnection between certain morphological features of the type II spectrum (e.g., bump or\nbreak) and the specific eruptive processes as recorded by coronal\nimaging instruments (e.g., shocks crossing dense streamers).\nTheoretical basis of this idea stems from the plasma emission hypothesis of type II\ngeneration at coronal shocks, in which scenario the emission\nfrequencies and therefore the spectral shape is decided by the densities of\ncoronal structures along the shock path (Ginzburg \\& Zheleznyakov,\n1958).\n\nStreamers are the most prominent bright large-scale structures in\nthe corona. They are expected to have effects on the type II radio\nspectrum. Feng et al. (2012) and Kong et al. (2012) investigated\ntwo CME-streamer interaction events with accompanying type II\nradio bursts which occurred on November 1 2003 and March 27 2011. They\ndefined two special type II morphological features: spectral break\nand spectral bump. According to their studies, a spectral break is\nthe result of density decrease when a type II emission\nsource propagates from inside of a streamer to outside,\nand a spectral bump is produced when a type II source\npropagates across a streamer structure from one\nside to the other. Based on these  analyses, Feng et al.\n(2012) and Kong et al. (2012) remarked that the emission site can\nbe pinpointed. Note that in some recent studies it has\nbeen pointed out that when a shock crosses other localized\ncoronal and solar wind structures, such as dense coronal loops\n(e.g., Pohjolainen et al., 2008b), pre-existing dense CME materials\nand corotating interaction regions\n(e.g., Knock {\\&} Cairns, 2005; Schmidt {\\&} Cairns, 2012;\nHillan et al., 2012), it may result in similar spectral shape changes.\n\nTo apply the above type II radio source diagnostic method to more\nevents, in this paper we examine another CME event which occurred\non December 31 2007. This event is associated with a multiple\ntype II radio burst (e.g., Robinson \\&\nSheridan 1982; Shanmugaraju et al., 2005), well observed by the\nimaging instruments on board\nthe Solar TErrestrial RElations Observatory (STEREO) A and B\n(SA and SB for short) spacecraft (Kaiser et al., 2008) as well as\nthe Solar and Heliospheric Observatory (SOHO) spacecraft. The\nevent was associated with a radio burst\ncontaining clear and rich signals including type III and type IV\nbursts, and several episodes of type IIs. Many authors have\ninvestigated different aspects of this event (Autunes et al.,\n2009; Gopalswamy et al., 2009; de Koning et al., 2009; Liu et al.,\n2009a, 2009b; Odstrcil \\& Pizzo, 2009; Dai et al., 2010; Lin et\nal., 2010; Moran et al., 2010; Cho et al., 2011; Rigozo et al.,\n2011). In particular, Liu et al. (2009a), Gopalswamy et al. (2009)\nand Cho et al. (2011) have discussed the origin of the type II\nbursts. Inspecting the radio dynamic spectrum of this event, we\nrecognize that spectral bumps, similar to that studied by Feng et\nal. (2012), are present on the type II spectrum. Considering the\nextensive interests in this event and the potential important role\nof spectral bump in revealing the type II origin, in this paper we\nre-examine this event. Different from all previous studies, we\nfocus on the type II spectral bump and discuss how this feature\ncan be used to shed new lights on the type II origin.\n\n\n\\section{General properties of the event and previous studies}\n\nIn Figures 1a-1c we present three sets of EUV and white light\nobservations of the CME from SA, SOHO, and SB. The angle between SA and SB was 44$^\\circ$ at the\ntime of this event. The superposed Inner Coronagraph (COR1) at $\\sim$\n01:00 UT and Extreme UltraViolet Imager (EUVI) at $\\sim$ 00:55 UT\nimages from SA and SB and the difference Large Angle and Spectrometric\nCoronagraph (LASCO) image together with the  Extreme-ultraViolet Imaging\nTelescope (EIT) data from SOHO are shown. Black arrows denote the\neruption source active region (AR)10980, which is located at about E102S08,\nE58S08, and E81S08 as viewed from SA, SB, and SOHO\n(\\url{http:\/\/www.lmsal.com\/solarsoft\/}). So, the CME is\nseen  as a near-limb event by all three spacecraft. This\ngreatly constrains the projection effect on the measurements of\nthe CME dynamics. The CME is first observed by COR1A and COR1B at\n00:55 UT and by LASCO C2 at 01:31 UT. The corresponding CME\nleading edges are located at 1.55, 1.65, and 4.80 R$_\\odot$,\nrespectively. A C8.3 flare is associated with the CME. According\nto the GOES observation, the X-ray flux starts to increase\nrapidly at 00:30 UT, and reaches its peak at 00:50 UT.\n\nAs clearly seen from the SA and SOHO observations,\nstreamers are present on both sides of the CME source.\n Figures 1d and 1e present the coronal magnetic field\nconfigurations from the potential-field source-surface model\n(PFSS; Schatten et al., 1969; Schrijver \\& Derosa 2003) based on\nthe magnetic field measurements with Michelson Doppler\nImager (MDI; Scherrer et al., 1995) for the Carrington Rotation\n2065. The magnetic configurations have been rotated to the view\nangles of SA and SB, respectively. Large-scale closed field lines,\ncorresponding to the white light streamers, can be seen on both\nsides of the active regions. The northern streamer is narrower and\nweaker than the southern one. This is consistent with the fact\nthat the northern streamer is a pseudo streamer (PS: Wang et al., 2007;\n Liu et al., 2009b) while the southern streamer is a typical helmet streamer (HS).\nFor a PS, the polarities of the open field enclosing\nthe streamer-like structure are the same, as seen from the PFSS\nresults. In Figure 1a, we delineate the\nestimated overall magnetic topologies of both streamers with cyan\nlines.\n\n\nSeveral additional features in Figures 1a-1c worth to be noted.\nFirst, as seen from the LASCO difference image, there is a diffuse\nstructure ahead of the bright CME front. Such a structure\nis usually regarded  as the signature of shock sheath (Vourlidas\net al., 2003). Small white arrows point to different locations of\nthe sheath front. As a result of the asymmetric expansion of the\nCME, the stand-off distance varies significantly along the front.\nSecond, the PS is deflected\nsignificantly as shown from the obvious white-black difference structure,\neven in the absence of direct contact with\nthe bright ejecta. Streamer deflection without a CME\ncontact has been attributed to the CME-driven shock (e.g., Sheeley\net al., 2000). Finally, there is a concave-outward structure at\nthe CME front along the direction of the stalk of the southern\nstreamer, which is usually interpreted as a result of a CME\npropagating into denser and slower plasma sheet structure (see,\ne.g., Riley {\\&} Crokker, 2004; Odstrcil et al., 2004). This\nconcave-outward structure therefore indicates a very strong\ninteraction of the CME with this streamer.\nAs to be shown below, this observation is consistent with the SA\ndata.\n\nNow we turn our attention to the dynamic spectrum of the\nassociated radio burst, which was recorded by Learmonth (Kennewell \\&\nSteward 2003) and BIRS (Bruny Island Radio Spectrometer; Erickson\n1997) and shown in the left panel of Figure 2. An enlargement of\nthe spectrum in the square region is given in the right panel with\nthe y-axis on a linear scale. Several type II episodes are\nobserved. The first episode shows fundamental (F) and\nharmonic (H) branches with clear signatures of band\nsplitting on the H branch. It starts at $\\sim$ 00:53 UT and ends\nat $\\sim$ 01:20 UT lasting for about 30 minutes. The H branch\nspans from $>$ 100 MHz to 14 MHz.\n\nThe other two episodes are denoted by ``a'' (01:04 UT -\n01:10 UT, 85 MHz - 35 MHz) and ``b'' (01:11 UT - 01:14 UT, 57 MHz\n- 40 MHz) in Figure 2. These two emissions were regarded as\nthe fundamental and harmonic branches of a type II burst by\nGopalswamy et al. (2009) and Cho et al. (2011). However, with a\ncareful examination we conclude that this may not be the case. Note\nthat there are no temporally-overlapping emissions between ``a'' and ``b''.\nTherefore, to determine their frequency ratio we need to fit\nthe dynamic spectra to allow a\ndirect comparison of their frequencies. The fitting curve to ``a''\nfrom 01:08 UT - 01:11 UT \\textbf{is drawn by} the white solid line in\nFigure 2, using the 0.4$\\times$ Saito density model\n(Saito et al., 1977). \\textbf{The fitting gives a shock speed of\n550 km s$^{-1}$ (the reason of employing such a density model is\nexplained in the Appendix). Two white dashed lines with frequencies\n1.8 and 2 times larger than the fitted curve are also plotted.} It can be seen that the\nfrequency ratio between the emission denoted by ``b'' and the fitting to\n``a'' is less than $1.8$.\nThis excludes the possibility of ``b'' being the harmonic\ncounterpart of ``a'' (Nelson \\& Melrose, 1985; Mann et al., 1995, 1996).\nTherefore, we conclude that this multiple\ntype II event consists of three separate episodes of emissions.\n\\textbf{Since at metric wavelengths the harmonic emission is usually\nbrighter than the fundamental one (e.g., Cairns \\& Robinson, 1987), we\nregard ``a'' to be a harmonic-band emission.}\n\n\nAt $\\sim$ 01:00 UT the lower band of the H branch of the first episode\n($\\sim$ 48 MHz) become discontinuous. Shortly after this, the band rises up to\na higher frequency of $\\sim$ 52 MHz at 01:02 UT, then its frequency\ndecreases rapidly to $\\sim 37$ MHz at 01:05 UT. Such a\nnon-monotonic variation of frequencies has been referred to as type\nII spectral bump by Feng et al. (2012). Another bumping feature,\nsimilar in shape, but weaker in amplitude, is present on ``a''\nspanning from $\\sim$01:06:10 UT - $\\sim$01:07:40 UT around 60 MHz. The two\nbumps are indicated by white arrows in the figure. Their origin\nand physical implication are the focus of this study.\n\nBefore further discussion, we summarize relevant results from\nprevious studies on this type II event by Liu et al. (2009a),\nGopalswamy et al. (2009), and Cho et al. (2011).\n\nLiu et al. (2009a) focused on the driver of the first type II\nepisode. They used the 1.3$\\times$ Saito density model (Saito et al., 1977) with a shock\nspeed of 616 km s$^{-1}$ to fit the dynamic spectrum and deduce\nthe shock heights. The deduced heights were then compared to the\ndistance measurements of the CME front and the propagating streamer\nkink induced by the CME shock-streamer interaction. This\nestablished the physical connection between the metric and the\ndecametric-hectometric bursts. They concluded that this episode\nwas driven by the CME, rather than by the associated flare.\nGopalswamy et al. (2009) measured the height of the CME leading\nedge at the time of the onset of the first episode and concluded\nthat the type II is emitted at a few tenths of a solar radius\nabove the solar surface. Cho et al. (2011) examined the type II\nbursts as a multi-band event (e.g., Robinson \\& Sheridan 1982; Shanmugaraju et al., 2005).\nThey used a Newkirk density model (Newkirk, 1961) to convert the\nfrequencies of the two episodes into shock heights. They found\nthat the obtained two sets of shock heights were consistent with\nthe measured CME-nose heights and the interaction heights of the\nshock with the northern streamer, respectively. They then\nconcluded that the first episode was excited at the CME shock\nnose, and the second episode ``a'' excited at the interaction\nregion of the shock with the northern streamer.\n\nAll of the above authors agreed that the type II burst was excited\nby the CME-driven shock. None of them, however, discussed\nthe bumping features of the spectrum. As noted, these features\nprovide significant amount of information in diagnosing the origin\nof the type II radio burst. In particular, one may infer the location\nof the source and physical conditions for the generation of\ntype II radio bursts. We point out that features like these in\nradio spectra should be carefully examined in relevant CME\nstudies. We note that the back-extrapolation of the second\nepisode ``a'' maps to the onset of a cluster of type III bursts, which\nindicates a possible connection to the flare impulsive phase. Therefore,\na flare origin of this episode can not be ruled out completely. Nevertheless,\nwe only consider in this study the possibility that the type II emitting\nshock is driven by the CME.\n\n\\section{CME-shock profiles and origin of the type II spectral\nbumps}\n\nIn this section, we first introduce how we delineate the CME-shock\nprofiles from the EUVI and COR1 data so as to present a clear\npicture of how the shock evolves and interacts with the streamers.\nThen, we relate the deduced shock-streamer interactions to the\nobserved type II spectral bumps.\n\nCoronal shock profiles can be determined directly using EUV and white light\nimaging data. Observational signatures of a coronal shock include\ndiffuse sheath structure ahead of the bright CME ejecta (Vourlidas\net al., 2003), deflection and kink of streamer stalks and coronal\nrays as swept by the CME shock (e.g., Sheeley et al., 2000), and\nthe EUV propagating disturbance revealing the expanding shock front.\n\nIn our event, both the diffuse sheath structure and streamer deflection\/kink are\nobserved in the LASCO difference image at 01:32 UT. Since the streamers are\nbest seen by SA (see Figure 1), in Figure 3 we plot the eruption\nsequence between 00:55 UT-01:15 UT as observed by COR1 and EUVI of\nSA.  The upper and middle panels are the COR1 running difference and\ndirect images, respectively. The EUVI 195\n{\\AA} difference and direct data are included when available.\nCorresponding animations can be viewed online.\n\nIn the upper panels of Figure 3, we use arrows of different color\nto point out various shock signatures. The resultant continuous\nshock profiles are plotted in the lower panels. The solid part of\nthe shock profiles is determined by recognizable shock signatures,\nand the dotted part represents the profile without a clear shock\nsignature and is derived by assuming a smooth shock\npropagation.\n\nAt 00:55 UT, a bright loop is observed by both EUVI\nand COR1. This is the first appearance of the CME in the COR1\nfield of view. None of the above shock signatures are present at\nthis moment mostly because the shock is still rather close to the\nbright ejecta at this very-early stage of the eruption. Note that\nthe shock is already formed since the corresponding\ntype II burst started earlier at 00:53 UT. We thus take the\nposition of the bright loop structure as the shock front location.\nFive minutes later (01:00 UT), the loop propagates outward with a\nfast lateral expansion. A diffuse sheath structure appears at the\nnortheastern quadrant as pointed by the upper arrows. No observable\nsheath structure is present in the southeastern quadrant partially\ndue to the asymmetric propagation of the CME. The obtained shock\nprofile for 01:00 UT is shown in Figure 3g.\n\nAt 01:05 UT, the diffuse sheath structure is well observed ahead\nof the ejecta in the northeastern quadrant. In addition, from the\nEUVI 195 {\\AA} data we can recognize the EUV fronts associated\nwith the eruption as indicated by the black arrows, which are\nconsistent with the deflection fronts of the EUV rays. We use these\nfeatures to determine the lower extension of the shock.\nThis allows us to plot a more complete shock envelope than at 01:00\nUT. In the southeastern quadrant of the difference image, a weak\nconcave-outward structure along the direction of the stalk of the\nsouthern streamer is discernible. The structure is more\nprominent in the subsequent COR1 data. It was also seen\nin LASCO, indicating a direct interaction between the CME and\nthe streamer, as mentioned.\n\nAt 01:10 UT and 01:15 UT, besides the diffuse sheath structure\n(see white arrows) a new shock signature appears along the\nnorthern streamer as indicated by the blue arrow. The streamer is\nstrongly deflected without a direct contact with the bright\nejecta, indicating a shock front propagating along this direction\n(e.g., Sheeley et al., 2000).\n\nAt each time step, we connect the front of the diffuse structure,\nthe front of the streamer deflection, and the concave\noutward-structure to obtain the full envelope of the CME-driven\nshock (see Figures 3i and 3j). The obtained shock envelopes are\nshown all-together in Figure 3k, where the circle represents the\nsolar limb on which the projected eruption source is plotted as\nthe plus sign. The magnetic configurations of the two streamers\nare copied from Figure 1a. Also included are the shock profiles as\ndetermined from the COR1A running difference images at 01:20 UT\nand 01:25 UT, as well as that given by LASCO C2 at 01:32 UT. Since\nthe angular separation between SOHO and SA is about 22 degrees,\nthe shock profiles seen from SOHO and SA should be similar.\n\\textbf{This allows us to estimate} the shock speeds along\ndifferent directions. For example, the average speeds in the time\nrange of 00:55 UT - 01:10 UT along four directions pointing from the\neruption source \\textbf{ (see arrows in Figure 3k) are estimated to 560, 620,\n760, and 1000 km s$^{-1}$, respectively. These values reveal} a very asymmetric\npropagation of the CME front. The blue dashed shock envelopes at\n01:04 UT, 01:06:10 UT, and 01:07:40 UT are given by interpolations\nbetween nearby shock profiles or extrapolations using the obtained\naverage shock speeds along relevant directions. Thus, Figure 3k presents a\nrelatively-complete description of the shock propagation from the\ninner to the outer corona and provides important clues to\nunderstanding the origin of type II spectral bumps.\n\nNote that the presence of a spectral bump requires a high\ndensity structure along the shock path according to the plasma\nhypothesis of type II radio bursts (Ginzburg \\& Zheleznyakov,\n1958; Feng et al., 2012). From the coronagraph data alone, both the\nnorthern and the southern streamers can be the candidate for the\ntwo spectral bumps. From Figures 2 and 3 it can be seen that the\nfirst bump ends at 01:04 UT, before the shock contacts the\nnorthern PS ($\\sim$01:05 UT). This rules out the possibility of\nthe PS accounting for this bump. \\textbf{Further evidence supporting\nthe hypothesis that this spectral bump is caused by the shock transit\nacross the southern streamer can be found by examining the radio-source\nheight. Assuming a 1$\\times$Saito density model (see the Appendix), we infer\nthat the radio-source at the onset ($\\sim$48 MHz) and the end (37 MHz) of the\ntype II bursts was located at 1.69  R$_{\\odot}$ and 1.83  R$_{\\odot}$, respectively.}\n The two heights are presented in Figure 3k with two black dotted arcs. \\textbf{The intersections\n of these arcs with the shock envelopes are indicated by small red ellipses},\n which are presumably the source locations of the corresponding type II episode at the onset and\nthe end of the bump.\n\n\\textbf{It can be seen that the two intersections are located at the opposite\nsides of the pre-disturbed southern streamer.} This strongly indicates that the shock\ntransit across this streamer causes the first spectral bump.\nIn addition, from the bump duration ($ \\tau \\sim$ 4 minutes) and\nthe estimated shock speed in this region ($ v \\sim$ 600 km s$^{-1}$,\nsee Figure 3k), one infers the width $ D \\sim \\tau v$ of the dense\nstructure to be 0.2 R$_\\odot$, in agreement with the\nwhite light data of the southern streamer. Based on these arguments,\nwe conclude that the first bump is caused by the shock transit\nacross the southern streamer.\n\nThe second bump starts at 01:06:10 UT and lasts for $\\sim$1.5\nminutes. Its beginning is coincident with the first interaction of the\nshock with the northern PS. As mentioned earlier, the\ncorresponding type II emission is regarded as the harmonic branch. To\nexamine the role of the PS in the formation of the\nspectral bump, we convert the frequencies at the beginning and end of\nthis bump to shock heights using the 0.4$\\times$ Saito model\n(which is appropriate to describe the density distribution near the\nPS, see the Appendix). \\textbf{The radio-source heights are found to\nbe 1.32 R$_\\odot$ and 1.38 R$_\\odot$ and we depict these height levels\nby two dashed arcs in Figure 3k. Their intersections with the shock envelopes\nat 1:06:10 UT and 1:07:40 UT, marked by small red ellipses, are found to be\nlocated at the two sides of the pre-disturbed PS. This is similar to the situation\nfor the first bump. In addition, the product }of the bump duration ($\\sim$1.5\nminutes) and the estimated shock speed ($\\sim$1000 km s$^{-1}$, see\nFigure 3k) agrees with the observed streamer width, and the\nfact that the PS is narrower in width and weaker in brightness (smaller in density) is also\nconsistent with the corresponding spectral bump being weaker in\namplitude and shorter in duration. We therefore conclude that the\nshock transit across the PS causes the second bump.\n\nIn summary, by combining the dynamic spectral and imaging data we\nare able to determine the physical origin of the\ntwo spectral bumps. It suggests that the two type II episodes are\nproduced separately at the two flanks of the same CME shock. From\nthe shock envelopes and the estimated coronal magnetic field\ntopologies from the PFSS model, one finds that the shock geometry\nat the two flanks are more perpendicular or oblique than\nquasi-parallel.\n\nLast, in Figure 2 we have presented the fitting curves\nwith the 1$\\times$ and 0.4$\\times$ Saito density models to the pre-bump\nspectra of the first and the second episodes. From the\nfittings we deduce the radio source speeds to be 530 km s$^{-1}$ for the first\nepisode and 650 km s$^{-1}$ for the second episode. Note\nthat in our event the \\textbf{radio source propagates} along a highly non-radial direction,\n so these speeds are representative of the radial components.\nThese values agree with the speed measurements deduced\nfrom the white light images as shown in Figure 3k, therefore providing a\nself-consistency check of the validity of the density model used\nfor the fittings.\n\n\\section{Discussion on the inferred source size of type II bursts}\n\nWe point out that observations of spectral bumps of type II radio burst\ncan be employed as a unique method to infer the\nsource size of the type II radio bursts. From the dynamic spectrum\nwe see that the type II emission lane is raised up as a whole\nwithin a relatively short interval. This implies that the size of\nthe radio source is smaller than the associated dense streamer\nstructure ($\\sim$ 0.1 - 0.3 R$_\\odot$). The short duration of the rising\npart also suggests that the source is compact in\nspatial extension. Due to the intermittency of the radio signals\nat the onset of the first type II bump, we are unable to determine\nthe exact time duration for the spectral elevation. However, one\ncan infer an upper limit of $\\sim 1$ minute. Then assuming the\nshock crosses the streamer with a speed of $\\sim$ 600-1000 km\ns$^{-1}$, we deduce that the spatial dimension of the radio source\nneeds to be smaller than 0.05-0.1 R$_\\odot$  at a fundamental frequency\nlevel of 20-30 MHz. Comparing to the very broad extension\nof the shock surface (easily $>$ 1 R$_\\odot$), the\ntype II source can be practically regarded as ``point-like''.\n\nEarlier data analyses of radioheliographs revealed that type II\nsources were restricted to discrete sectors of an arc around the\nflare center (Wild 1970; Wild \\& Smerd, 1972), and sometimes of\nlarge dimension ($\\sim$ 0.5 R$_\\odot$) (Kai \\& McLean 1968).\nIt is well known that the radio source size obtained by radio-heliographs tends to\nbe larger than the real value due to propagation and scattering of\nradiation from the source to the observer. Also, earlier radio\nimaging instruments such as the Culgoora and Clark Lake\nRadio-heliographs have an angular resolution of several\narcminutes at frequencies $<$ 100 MHz, which corresponds to a\nspatial resolution of a few tenths of a solar radius near the Sun.\nTherefore these instruments were not able to resolve the type II\nsource as small as that deduced from our study, although the\ndiscrete feature of the imaging observations agrees with our\nresult.\n\nThe suggestion that the type II source is compact implies that\nthe presence of shocks is only a necessary condition for the\ngeneration of type II bursts. Other strict physical conditions\nmust also be satisfied at the shock in order to create a\nnon-thermal distribution of electrons which is unstable to plasma\ninstabilities. For example, the radio-emitting shock front may be\nquasi-perpendicular, as revealed by our study. Early theoretical\nstudies have proposed that quasi-perpendicular shocks can\naccelerate electrons by the shock drift acceleration mechanism\n(Holman \\& Pesses 1983, Wu 1984). When the shock speed is\nsufficiently large or $\\theta_{Bn}$ (the angle between shock\nnormal and upstream magnetic field) is close to 90 degrees, some\nelectrons can be accelerated to non-thermal energy and excite\nplasma waves. However it is known that in this process both the\nfraction and achievable energy of the accelerated particles are\nlimited (e.g., Ball \\& Melrose 2001). Other effects, such as MHD\nturbulence, shock ripples and\/or magnetic\ncollapsing trap geometries (e.g., Decker, 1990; Zlobec et al., 1993;\nMagdaleni\\'{c} et al., 2002; Guo {\\&} Giacalone, 2010; Schmidt {\\&}\nCarins, 2012; Hillan et al., 2012) may be required to yield more\nefficient electron acceleration and radio emission. If\nindeed local structures, such as ripples or magnetic traps, play\nan important role in accelerating electrons, then it is understandable\nthat the radio-emitting region at the shock front is spatially confined.\n\nIn space weather studies, the frequency drift of type II burst is\noften employed as an important input to predict the shock arrival\ntime at Earth. However, if the type II burst is generated from a\nspecial discrete part of the shock flank, as inferred from our\nstudy, then the speeds obtained from the curve fitting to the\ndynamic spectrum may not be accurate. This should be taken into\nconsideration when using type II spectrum as inputs to drive space\nweather forecastings.\n\n\\section{Conclusions}\n\nIn this paper we investigated the physical origin of two bump\nfeatures on the dynamic spectrum of a multiple type II radio\nburst, which were associated with a CME event occurred on December\n31 2007. Combining radio spectral data and EUV\/white light imaging\nobservations, we found that the type II bumps are caused by the\nsource density variation when the CME shock propagates through\nnearby dense streamers. It is suggested that the two type II\nepisodes are generated separately at the two flanks of the\nCME-driven shock. It is further inferred that the type II signals\nare emitted from discrete spatially-confined sources at the CME-shock\nflank with the source spatial extension smaller than 0.05-0.1 R$_\\odot$\nat a fundamental frequency level of 20-30 MHz\nand the large-scale shock geometry is close to quasi-perpendicular and\/or\n oblique.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLabeled transition systems constitute a widely used model of\nconcurrent computation. They model processes by explicitly\ndescribing their states and their transitions from state to state,\ntogether with the actions that produce these transitions. Several\nnotions of behavioral semantics have been proposed, with the aim\nto identify those states that afford the same observations\n\\cite{Gla01,Gla93}. For equational reasoning about processes, one\nneeds to find an axiomatization that is sound and\n\\emph{ground-complete} modulo the semantics under consideration,\nmeaning that all equivalent closed terms can be equated. Ideally,\nsuch an axiomatization is also \\emph{$\\omega$-complete}, meaning\nthat all equivalent \\emph{open} terms can be equated. If such a\nfinite axiomatization exists, it is said that there is a\n\\emph{finite basis} for the equational theory.\n\nFor concrete semantics, so in the absence of the silent action\n$\\tau$, the existence of finite bases is well-studied\n\\cite{Gro90,Gla01,CFLN07}, in the context of the process algebra\nBCCSP, containing the basic process algebraic operators from CCS\nand CSP\\@. However, for weak semantics, that take into account the\n$\\tau$, hardly anything is known on finite bases. In \\cite{Gla93},\nVan Glabbeek presented a spectrum of weak semantics. For several\nof the semantics in this spectrum, a sound and ground-complete\naxiomatization has been given, in the setting of the process\nalgebra BCCS (BCCSP extended by $\\tau$), see, e.g., \\cite{Gla97}.\nBut a finite basis has been given only for \\emph{weak}, \\emph{delay},\n$\\eta$- and \\emph{branching bisimulation} semantics \\cite{Mi89a,vG93a},\nand in case of an infinite alphabet of actions also for \\emph{weak\nimpossible futures} semantics \\cite{VM01}. The reason for this lack of\nresults on finite bases, apart from the inherent difficulties arising\nwith weak semantics, may be that it is usually not so straightforward\nto define a notion of unique normal form for \\emph{open} terms in a\n\\emph{weak} semantics. Here we will employ a saturation technique,\nin which normal forms are saturated with subterms.\n\nIn this paper, we focus on two closely related weak semantics,\nbased on failures and impossible futures. A \\emph{weak failure}\nconsists of a trace $a_1\\cdots a_n$ and a set $A$, both of\nconcrete actions. A state exhibits this weak failure pair if it\ncan perform the trace $a_1\\cdots a_n$ (possibly intertwined with\n$\\tau$'s) to a state that cannot perform any action in $A$ (even\nafter performing $\\tau$'s). In a \\emph{weak impossible future},\n$A$ can be a set of traces. Weak failures semantics plays an\nessential role for the process algebra CSP \\cite{BHR84}. For\nconvergent processes, it coincides with testing semantics\n\\cite{DeNHen84,RenVog07}, and thus is the coarsest congruence for\nthe CCS parallel composition that respects deadlock behavior.\nWeak impossible futures semantics \\cite{Vog92} is a natural\nvariant of possible futures semantics \\cite{RB81}. In \\cite{GV06}\nit is shown that weak impossible futures semantics, with an\nadditional root condition, is the coarsest congruence containing\nweak bisimilarity with explicit divergence that respects\ndeadlock\/livelock traces (or fair testing, or any liveness\nproperty under a global fairness assumption) and assigns unique\nsolutions to recursive equations.\n\nThe heart of our paper is a finite basis for the inequational theory\nof BCCS modulo the weak failures \\emph{preorder}. The axiomatization\nconsists of the standard axioms A1-4 for bisimulation, three extra\naxioms WF1-3 for failures semantics, and in case of a finite alphabet\n$A$, an extra axiom WF$_A$. The proof that A1-4 and WF1-3 are a finite\nbasis in case of an infinite alphabet is a sub-proof of the proof that\nA1-4, \\mbox{WF1-3} and WF$_A$ are a finite basis in case of a finite\nalphabet. Our proof has the same general structure as the beautiful\nproof for testing equivalences given in \\cite{DeNHen84} and further\ndeveloped in \\cite{Hen88}. Pivotal to this is the construction of\n``saturated'' sets of actions within a term \\cite{DeNHen84}.  Since\nhere we want to obtain an $\\omega$-completeness result, we extend this\nnotion to variables.  Moreover, to deal with $\\omega$-completeness, we\nadopt the same general proof structure as in the strong case\n\\cite{FN05}. In this sense, our proof strategy can be viewed as a\ncombination of the strategies proposed in \\cite{DeNHen84} and\n\\cite{FN05}.  Furthermore, we apply an algorithm from\n\\cite{AFI07,FG07,CFG08b} to obtain a finite basis for BCCS modulo weak\nfailures \\emph{equivalence} for free.\n\nAt the end, we investigate the equational theory of BCCS modulo\nweak impossible futures semantics. This shows a remarkable\ndifference with weak failures semantics, in spite of the strong\nsimilarity between the definitions of these semantics (and between\ntheir ground-complete axiomatizations). As said, in case of an\ninfinite alphabet, BCCS modulo the weak impossible futures\npreorder has a finite basis \\cite{VM01}. However, we show that in\ncase of a finite alphabet, such a finite basis does not exist.\nMoreover, in case of weak impossible futures \\emph{equivalence},\nthere is no finite ground-complete axiomatization, regardless of the\ncardinality of the alphabet.\n\nA finite basis for the equational theory of BCCSP modulo\n(concrete) failures semantics was given in \\cite{FN05}. The\nequational theory of BCCSP modulo (concrete) impossible futures\nsemantics is studied in \\cite{CF08}. It is interesting to see that\nour results for weak semantics agree with their concrete\ncounterparts, with very similar proofs. This raises a challenging open\nquestion: can one establish a general theorem to link the\naxiomatizability (or nonaxiomatizability) of concrete and weak semantics?\n\nAn extended abstract of this paper appears as \\cite{CFG08}.\n\\newpage\n\n\\section{Preliminaries} \\label{sec2}\n\n${\\rm BCCS}(A)$ is a basic process algebra for expressing finite\nprocess behavior. Its signature consists of the constant $\\mathbf{0}$,\nthe binary operator $\\_+\\_$\\,, and unary prefix operators $\\tau\\_$\nand $a\\_$\\,, where $a$ is taken from a nonempty set $A$ of visible\nactions, called the \\emph{alphabet}, ranged over by $a,b,c$. We\nassume that $\\tau\\notin A$ and write $A_\\tau$ for $A\\cup\\{\\tau\\}$,\nranged over by $\\alpha,\\beta$.\n\\[ t::= 0 \\mid at \\mid \\tau t \\mid t+t \\mid x\\]\nClosed ${\\rm BCCS}(A)$ terms, ranged over by $p,q$, represent\nfinite process behaviors, where $\\mathbf{0}$ does not exhibit any\nbehavior, $p+q$ offers a choice between the behaviors of $p$ and\n$q$, and $\\alpha p$ executes action $\\alpha$ to transform into\n$p$.  This intuition is captured by the transition rules below.\nThey give rise to $A_\\tau$-labeled transitions between closed BCCS terms.\n\\[\n  \\frac{~}{\\alpha x\\mv{\\alpha}x}\n\\qquad\n  \\frac{x\\mv{\\alpha}x'}{x+y\\mv \\alpha x'}\n\\qquad\n  \\frac{y\\mv{\\alpha}y'}{x+y\\mv \\alpha y'}\n\\]\nWe assume a countably infinite set $V$ of variables; $x,y,z$\ndenote elements of $V$. Open BCCS terms, denoted by $t,u,v,w$, may\ncontain variables from $V$. Write $\\var(t)$ for the set of variables\noccurring in $t$.\nThe operational semantics is extended verbatim to open terms;\nvariables generate no transition.\nWe write $t\\Rightarrow u$ if there is a sequence of\n$\\tau$-transitions $t\\mv{\\tau}\\cdots\\mv{\\tau}u$; furthermore\n$t\\mv{\\alpha}$ denotes that there is a term $u$ with\n$t\\mv{\\alpha}u$, and likewise $t\\Rightarrow\\mv{\\alpha}$ denotes that\nthere are a terms $u,v$ with $t\\Rightarrow u \\mv{\\alpha} v$.\n\nThe \\emph{depth} of a term $t$, denoted by $|t|$, is the length of\nthe \\emph{longest} trace of $t$, not counting $\\tau$-transitions.\nIt is defined inductively as follows: $|\\mathbf{0}|=|x|=0$;\n$|at|=1+|t|$; $|\\tau t| = |t|$; $|t+u|=\\max\\{|t|, |u|\\}$.\n\nA (closed) substitution, ranged over by $\\sigma,\\rho$, maps variables\nin $V$ to (closed) terms. For open terms $t$ and $u$, and a\npreorder $\\sqsubseteq$ (or equivalence $\\equiv$) on closed terms, we\ndefine $t\\sqsubseteq  u$ (or $t\\equiv u$) if $\\sigma(t)\\sqsubseteq\\sigma(u)$\n(resp.\\ $\\sigma(t)\\equiv\\sigma(u)$) for all closed substitutions\n$\\sigma$. Clearly, $t\\mv{a}t'$ implies that\n$\\sigma(t)\\mv{a}\\sigma(t')$ for all substitutions $\\sigma$.\n\nAn \\emph{axiomatization} is a collection of equations $t \\approx\nu$ or of inequations $t \\preccurlyeq u$. The (in)equations in an\naxiomatization $E$ are referred to as \\emph{axioms}.  If $E$ is an\nequational axiomatization, we write $E\\vdash t\\approx u$ if the\nequation $t\\approx u$ is derivable from the axioms in $E$ using\nthe rules of equational logic (reflexivity, symmetry,\ntransitivity, substitution, and closure under BCCS contexts). For\nthe derivation of an inequation $t\\preccurlyeq u$ from an\ninequational axiomatization $E$, denoted by $E\\vdash t\\preccurlyeq\nu$, the rule for symmetry is omitted. We will also allow equations\n$t\\approx u$ in inequational axiomatizations, as an abbreviation\nof $t\\preccurlyeq u\\land u\\preccurlyeq t$.\n\nAn axiomatization $E$ is \\emph{sound} modulo a preorder $\\sqsubseteq$\n(or equivalence $\\equiv$) if for all terms $t,u$, from $E\\vdash\nt\\preccurlyeq u$ (or $E\\vdash t\\approx u$) it follows that\n$t\\sqsubseteq u$ (or $t\\equiv u$).  $E$ is \\emph{ground-complete} for\n$\\sqsubseteq$ (or $\\equiv$) if $p\\sqsubseteq q$ (or $p\\equiv q$)\nimplies $E\\vdash p\\preccurlyeq q$ (or $E\\vdash p\\approx q$) for all\nclosed terms $p,q$. Moreover, $E$ is \\emph{$\\omega$-complete} if for\nall terms $t,u$ with $E\\vdash\\sigma(t)\\preccurlyeq\\sigma(u)$ (or\n$E\\vdash\\sigma(t)\\approx\\sigma(u)$) for all closed substitutions\n$\\sigma$, we have $E\\vdash t\\preccurlyeq u$ (or $E\\vdash t\\approx u$).\nWhen $E$ is $\\omega$-complete as well as ground-complete, it is\n\\emph{complete} for $\\sqsubseteq$ (or $\\equiv$) in the sense that\n$t\\sqsubseteq u$ (or $t\\equiv u$) implies $E\\vdash t\\preccurlyeq u$\n(or $E\\vdash t\\approx u$) for all terms $t,u$.  The equational\ntheory of BCCS modulo a preorder $\\sqsubseteq$ (or equivalence\n$\\equiv$) is said to be \\emph{finitely based} if there exists a\nfinite, $\\omega$-complete axiomatization that is sound and\nground-complete for BCCS modulo $\\sqsubseteq$ (or $\\equiv$).\n\nA1-4 below are the core axioms for BCCS modulo bisimulation\nsemantics. We write $t=u$ if $\\mbox{A1-4}\\vdash t\\approx u$.\n{\\small\\[\n\\begin{array}{l@{\\qquad}rcl}\n{\\rm A}1&x+y &~\\approx~& y+x\\\\\n{\\rm A}2&(x+y)+z &~\\approx~& x+(y+z)\\\\\n{\\rm A}3&x+x &~\\approx~& x\\\\\n{\\rm A}4&x+\\mathbf{0} &~\\approx~& x\\\\\n\\end{array}\n\\]}%\nSummation $\\sum_{i\\in\\{1,\\ldots,n\\}}t_i$ denotes $t_1+\\cdots+t_n$,\nwhere summation over the empty set denotes $\\mathbf{0}$. As binding\nconvention, $\\_+\\_$ and summation bind weaker than $\\alpha\\_$\\,.\nFor every term $t$ there exists a finite set $\\{\\alpha_i t_i\\mid\ni\\in I\\}$ of terms and a finite set $Y$ of variables such that\n$t=\\sum_{i\\in I} \\alpha_i t_i + \\sum_{y\\in Y}y$. The $\\alpha_i\nt_i$ for $i\\in I$ and the $y\\in Y$ are called the \\emph{summands}\nof $t$.  For a set of variables $Y$, we will often denote the term\n$\\sum_{y\\in Y}y$ by $Y$.\n\n\\begin{definition}[Initial actions]\\rm\nFor any term $t$, the set $\\mc{I}(t)$ of initial actions is\ndefined as $\\mc{I}(t)=\\{a\\in A\\mid t\\Rightarrow \\mv{a}\\}$.\n\\end{definition}\n\n\\begin{definition}[Weak failures]\\rm \\label{def:weak-failures}\n\\vspace{-1ex}\n\\begin{itemize}\n  \\item A pair $(a_1\\cdots a_k,B)$, with $k\\geq 0$ and $B \\subseteq A$,\n  is a \\emph{weak failure pair} of a process $p_0$ if there is a path\n  $p_0\\Rightarrow \\mv{a_1} \\Rightarrow \\cdots \\Rightarrow \\mv{a_k} \\Rightarrow p_k$\n  with $\\mc{I}(p_k) \\cap B = \\emptyset$.\n\n  \\item Write $p\\leq_{\\rm WF} q$ if the weak failure pairs of $p$ are\n  also weak failure pairs of $q$.\n\n  \\item The \\emph{weak failures preorder} $\\sqsubseteq_{\\rm WF}$ is\n  given by\\\\ $p\\sqsubseteq_{\\rm WF} q$ iff (1) $p\\leq_{\\rm WF} q$ and\n        (2) $p\\mv{\\tau}$ implies that $q\\mv{\\tau}$.\n\n  \\item \\emph{Weak failures equivalence} $\\equiv_{\\rm WF}$ is defined\n  as $\\sqsubseteq_{\\rm WF} \\cap \\sqsubseteq_{\\rm WF}^{-1}$.\n\\end{itemize}\n\\end{definition}\nIt is well-known that $p\\leq_{\\rm WF} q$ is \\emph{not} a\n\\emph{precongruence} for BCCS: e.g., $\\tau\\mathbf{0}\\leq_{\\rm WF}\\mathbf{0}$\nbut $\\tau\\mathbf{0}+a\\mathbf{0} \\not\\leq_{\\rm WF} \\mathbf{0}+a\\mathbf{0}$. However,\n$\\sqsubseteq_{\\rm WF}$ is, meaning that $p_1\\sqsubseteq_{\\rm WF} q_1$\nand $p_2\\sqsubseteq_{\\rm WF} q_2$ implies $p_1+p_2\\sqsubseteq_{\\rm WF}\nq_1+q_2$ and $\\alpha p_1\\sqsubseteq_{\\rm WF} \\alpha q_1$ for\n$\\alpha\\in A_\\tau$. In fact, $\\sqsubseteq_{\\rm WF}$ is the coarsest\nprecongruence contained in $\\leq_{\\rm WF}$.\nLikewise, $\\equiv_{\\rm WF}$ is a \\emph{congruence} for BCCS.\n\n\\section{A Finite Basis for Weak Failures Semantics} \\label{sec3}\n\n\\subsection{Axioms for the Weak Failures Preorder}\n\nOn BCCS processes, the weak failures preorder as defined above\ncoincides with the inverse of the must-testing preorder of \\cite{DeNHen84}.\nA sound and ground-complete axiomatization of the must-testing preorder\npreorder has been given in \\cite{DeNHen84}, in terms of a language\nricher than BCCS\\@. After restriction to BCCS processes, and reversing\nthe axioms, it consists of A1-4 together with the axioms: {\\small\n$$\\begin{array}{l@{\\qquad}rcl}\n\\mbox{N1} & \\alpha x + \\alpha y &\\approx& \\alpha (\\tau x + \\tau y)\\\\\n\\mbox{N2} & \\tau(x+y) &\\preccurlyeq& x + \\tau y\\\\\n\\mbox{N3} & \\alpha x + \\tau(\\alpha y + z) &\\approx& \\tau(\\alpha x + \\alpha y + z)\\\\\n\\mbox{E1} & x &\\preccurlyeq& \\tau x + \\tau y\n\\end{array}$$}%\nHere we simplify this axiomatization to A1-4 and WF1-3 from Tab.~\\ref{tab1}.\nIn fact it is an easy exercise to derive WF1-3 from N1, N2 and E1,\nand N1, N2 and E1 from WF1-3. It is a little harder to check that N3\nis derivable from the other three axioms (cf.~Lem.~\\ref{lemma5}).\n\n\\begin{table}[t] \\center\n \\begin{tabular}{l@{\\qquad}rcl}\n WF1   & $a x+ay$  & $\\approx$ & $a(\\tau x+\\tau y)$\\\\\n WF2   & $\\tau(x+y)$  & $\\preccurlyeq$ & $\\tau x+ y$\\\\\n WF3   & $x$  & $\\preccurlyeq $ & $ \\tau x+  y$\\medskip\\\\\n\\end{tabular}\\caption{Axiomatization for the weak failures preorder} \\label{tab1}\n\\vspace{-5mm}\n\\end{table}\n\n\\begin{theorem} \\label{theo:ground-complete}\n\\mbox{\\rm A1-4+WF1-3} is sound and ground-complete for\n$\\mathrm{BCCS}(A)$ modulo $\\sqsubseteq_{\\rm WF}$.\n\\end{theorem}\nIn this section, we extend this ground-completeness result with\ntwo $\\omega$-completeness results. The first one says, in\ncombination with Theo.~\\ref{theo:ground-complete}, that as long\nas our alphabet of actions is infinite, the axioms A1-4+WF1-3\nconstitute a finite basis for the inequational theory of BCCS$(A)$\nmodulo $\\sqsubseteq_{\\rm WF}$.\n\n\\begin{theorem} \\label{theo:A-infinite}\nIf $|A|\\mathbin=\\infty$, then \\mbox{\\rm A1-4+WF1-3} is $\\omega$-complete for\n$\\mathrm{BCCS}(A)$ modulo $\\sqsubseteq_{\\rm WF}$.\n\\end{theorem}\nTo get a finite basis for the inequational theory of BCCS modulo\n$\\sqsubseteq_{\\rm WF}$ in case $|A|<\\infty$, we need to add the\nfollowing axiom:\n\\[\\mathrm{WF}_A\\qquad \\sum_{a\\in A} ax_a \\preccurlyeq \\sum_{a\\in A}a x_a +y\\]\nwhere the $x_a$ for $a\\in A$ and $y$ are distinct variables.\n\n\\begin{theorem} \\label{theo:A-finite}\nIf $|A|<\\infty$, then \\mbox{\\rm A1-4+WF1-3+WF}$_A$ is $\\omega$-complete for\n $\\mathrm{BCCS}(A)$ modulo $\\sqsubseteq_{\\rm WF}$.\n\\end{theorem}\nThe rest of this section up to Sec.~\\ref{sec:wfe} is devoted to the\nproofs of Theorems~\\ref{theo:ground-complete}--\\ref{theo:A-finite}.\nFor a start, the inequations in Tab.~\\ref{tab:D-eqs} can be derived\nfrom A1-4+WF1-3:\n\n\\vspace{-3mm}\n\\begin{table}\n\\begin{center}\n \\begin{tabular}{l@{\\qquad}rcl}\n D1 & $ \\tau (x+y)+x$  & $\\approx$ & $\\tau (x+y)$\\\\\n D2 & $\\tau(\\tau x+y)$  & $\\approx$ & $\\tau x+ y$\\\\\n D3 & $ ax + \\tau(ay+z)$  & $\\approx$ & $\\tau(a x+ a y+z)$\\\\\n D4 & $ \\tau x $  & $\\preccurlyeq$ & $\\tau x +y$\\\\\n D5 & $ \\sum_{i\\in I} a x_i $  & $\\approx$ & $a (\\sum_{i\\in I}\\tau\n x_i)\\mbox{ for finite nonempty index sets }I $\\\\\n D6 & $ \\tau x + y$  & $\\approx$ & $\\tau x + \\tau (x+y) $\\\\\n D7 & $ \\tau x + \\tau y$  & $\\approx$ & $\\tau x + \\tau (x+y) + \\tau y $\\\\\n D8 & $ \\tau x + \\tau (x+y+z)$  & $\\approx$ & $\\tau x + \\tau (x+y) + \\tau(x+y+z) $\\\\\n D9 & $ \\sum_{i\\in I}\\tau (at_i+y_i)$ & $\\approx$ & $\\sum_{i\\in I}\\tau (at + y_i)$\n      for finite $I$, where $t = \\sum_{i\\in I} \\tau t_i$.\n\\end{tabular}\n\\end{center}\n\\caption{Derived inequations} \\label{tab:D-eqs}\n\\end{table}\n\\vspace{-7mm}\n\\begin{lemma} \\label{lemma5}\n  \\mbox{\\rm D1-9} are derivable from \\mbox{\\rm A1-4+WF1-3}.\n\\end{lemma}\n\n\\begin{proof}\nWe shorten ``$\\mbox{A1-4+WF1-3} \\vdash$'' to ``$\\vdash$''.\\vspace{-1ex}\n\\begin{enumerate}\n  \\item\n By WF3, $\\vdash x \\preccurlyeq \\tau x$, and thus $\\vdash\\tau x+x\\preccurlyeq\n \\tau x$. Moreover, by WF2,\\\\ $\\vdash\\tau (x+x) \\preccurlyeq \\tau x+x$,\n hence $\\vdash\\tau x\\preccurlyeq \\tau x+x$. In summary, $\\vdash\\tau x\\approx\n \\tau x+x$.\n\n So $\\vdash \\tau(x+y)\\approx \\tau(x+y)+x+y+x \\approx \\tau(x+y)+x$.\n\n \\medskip\n\n  \\item\n By WF2, $\\vdash \\tau(x+\\tau x)\\preccurlyeq \\tau x+\\tau x = \\tau\n x$, so by D1, $\\vdash \\tau\\tau x\\preccurlyeq \\tau x$.\n Hence, by WF2, $\\vdash\\tau (\\tau x+y) \\preccurlyeq \\tau \\tau x+y \\preccurlyeq \\tau x+y$.\n\n Moreover, by WF3, $\\vdash\\tau  x+y \\preccurlyeq \\tau(\\tau x+y)$.\n\n \\medskip\n\n \\item By WF3, $\\vdash y \\preccurlyeq \\tau y+\\tau x$. So by WF1, $\\vdash ay\\preccurlyeq a(\\tau x+\\tau y)\\approx ax+ay$. This implies $\\vdash \\tau(ay+z)\\preccurlyeq \\tau(ax+ay+z)$. Hence, by D1, $\\vdash ax+\\tau(ay+z)\\preccurlyeq ax+ \\tau(ax+ay+z) \\approx \\tau(ax+ay+z)$.\n\nMoreover, by WF2, $ \\vdash \\tau(a x+ a y+z) \\preccurlyeq ax +\n\\tau(ay+z)$.\n\n\\medskip\n  \\item By WF3 and D2, $\\vdash \\tau x\\preccurlyeq \\tau\\tau x+y \\approx \\tau x+y$.\n  \\medskip\n\n  \\item By induction on $|I|$, using WF1 and D2.\n  \\medskip\n\n  \\item By D4 and D1, $\\vdash \\tau x+y \\preccurlyeq \\tau x+\\tau(x+y)+y \\approx \\tau x+\\tau(x+y)$.\n\n        Moreover, by WF2, $\\vdash \\tau x+\\tau(x+y)\\preccurlyeq \\tau x+\\tau x+y=\\tau x+y$.\n  \\medskip\n\n  \\item By D4 in one direction; by D6 and D1 in the other.\n  \\medskip\n\n  \\item By D4 in one direction; by D6 and D1 in the other.\n  \\medskip\n\n  \\item By D1, $\\vdash \\sum_{i\\in I}\\tau (at_i+y_i) \\approx\n        \\sum_{i\\in I}\\tau (at_i+y_i) +u$, where $u=\\sum_{i\\in I}at_i$.\n    Thus, by repeated application of D3, $\\vdash \\sum_{i\\in I}\\tau (at_i+y_i) \\approx\n        \\sum_{i\\in I}\\tau (at_i+u+y_i) = \\sum_{i\\in I}\\tau (u+y_i)$.\n    By D5 we have $u=at$.\n\\qed\n\\end{enumerate}\n\\end{proof}\n\n\\subsection{Normal Forms}\n\nThe notion of a normal form, which is formulated in the following\ntwo definitions, will play a key role in the forthcoming proofs.\nFor any set $L\\subseteq A\\cup V$ of actions and variables let $A_L\n= L \\cap A$, the set of actions in $L$, and $V_L = L \\cap V$, the\nset of variables in $L$.\n\n\\begin{definition}[Saturated family]\\rm Suppose $\\mc{L}$ is a finite family of\nfinite sets of {actions} and {variables}. We say $\\mc{L}$ is\n\\emph{saturated} if it is nonempty and\n\\begin{itemize}\n  \\item $L_1, L_2\\in \\mc{L}$ implies that $L_1\\cup L_2 \\in\n  \\mc{L}$; and\n\n  \\item $L_1, L_2\\in \\mc{L}$ and $L_1\\subseteq L_3 \\subseteq L_2$\n  imply that $L_3\\in \\mc{L}$.\n\\end{itemize}\n\\end{definition}\n\n\\begin{definition}[Normal form]\\rm\n\\begin{list}{$\\bullet$}{\\leftmargin 18pt\n                        \\labelwidth\\leftmargini\\advance\\labelwidth-\\labelsep}\n  \\item[(i)] A term $t$ is in $\\tau$ normal form if\\vspace{-6pt}\n  \\[t = \\sum_{L\\in \\mc{L}}\\tau \\left(\\sum_{a \\in A_L} at_a+ V_L\\right)\\vspace{-3pt}\\]\nwhere the $t_a$ are in normal form and $\\mc{L}$ is a saturated\nfamily of sets of actions and variables. We write $L(t)$ for\n$\\bigcup_{L\\in\\mc{L}}L$; note that $L(t) \\in \\mc{L}$.\n\\medskip\n  \\item[(ii)] $t$ is in action normal form if\n\\[t= \\sum_{a\\in A_L} at_a + V_L \\]\nwhere the $t_a$ are in normal form and $L\\subseteq A \\cup V$. We\nwrite $L(t)$ for $L$.\n\\medskip\n\\item[(iii)] $t$ is in normal form if it is either in $\\tau$\nnormal form or in action normal form.\\pagebreak[3]\n\\end{list}\n\\end{definition}\nNote that the definition of a normal form requires that for any $a\\in A$, if\n$t\\Rightarrow\\mv{a} t_1$ and $t\\Rightarrow\\mv{a} t_2$, then $t_1$ and $t_2$ are\nsyntactically identical.\n\nWe prove that every term can be equated to a normal form. We start\nwith an example.\n\n\\begin{example}\n Suppose $t= \\tau(at_1+\\tau(bt_2+ct_3)+x) +\\tau(at_4+\\tau x+\\tau y)+z$.\n Then $t$ can be equated to a $\\tau$ normal form with\n $\\mc{L}=\\{${\\small$\\{a,b,c, x\\},\\{a, x,y\\}, \\{a,b,c, x,y,z\\}$,\n $\\{a,b,c, x,y\\},\\hspace{-.6pt}\\{a,b,c, x,z\\},\\hspace{-.6pt}\n \\{a,b, x,y\\},\\hspace{-.5pt} \\{a,c, x,y\\},\\hspace{-.5pt}\n \\{a, x,y,z\\},\\hspace{-.6pt} \\{a,b, x,y,z\\},\\hspace{-.6pt}\n \\{a,c, x,y,z\\}$}$\\}$.\n We give a detailed derivation. By D2,\n \\[\n   \\vdash t \\approx \\tau(at_1+bt_2+ct_3+x) +\\tau(at_4+ x+ y)+z\n \\]\n By D6,\n \\[\n   \\vdash t \\approx \\tau(at_1+bt_2+ct_3+x) +\\tau(at_4+x+y)+\\tau(at_4+x+y+z)\n \\]\n Let $u_a =  \\tau t_1+\\tau t_4$, $u_b=t_2$ and $u_c=t_3$. By D9,\n \\[\n   \\vdash t \\approx \\tau(au_a+bu_b+cu_c+x)+\\tau(au_a+x+y)+\\tau(au_a+x+y+z)\n \\]\nBy induction, $u_a$ can be brought into a normal form $t_a$, and\nlikewise for $u_b$ and $u_v$. So\n \\[\n   \\vdash t \\approx \\tau(tu_a+bt_b+ct_c+x)+\\tau(at_a+x+y)+\\tau(at_a+x+y+z)\n \\]\nBy D7,\n \\[\n   \\vdash t \\approx \\begin{array}[t]{l}\n                    \\tau(at_a+bt_b+ct_c+x) +\\tau(at_a+x+y)\\\\\n            +\\tau(at_a+x+y+z) + \\tau(at_a+bt_b+ct_c+x+y+z)\n            \\end{array}\n \\]\nFinally, by D8,\n \\[\n   \\vdash t \\begin{array}[t]{l@{}}\n  \\approx \\begin{array}[t]{@{}l@{}} \\tau(at_a+bt_b+ct_c+x)\n          +\\tau(at_a+x+y)   +\\tau(at_a+x+y+z)\\\\\n          + \\tau(at_a+bt_b+ct_c+x+y+z)\n          + \\tau(at_a+bt_b+ct_c+x+y)\\\\\n          + \\tau(at_a+bt_b+ct_c+x+z)\n          + \\tau(at_a+bt_b+x+y)\n          + \\tau(at_a+ct_c+x+y)\\\\\n          + \\tau(at_a+bt_b+x+y+z)\n          + \\tau(at_a+ct_c+x+y+z)\n             \\end{array}\\\\\n   = \\sum_{L\\in \\mc{L}}\\tau(\\sum_{a\\in A_L} a t_a + V_L)\n  \\end{array}\n \\]\n\\end{example}\n\n\\begin{lemma} \\label{nf}\n  For any term $t$, $\\mbox{\\rm A1-4+WF1-3} \\vdash t\\approx t'$ for some normal form $t'$.\n\\end{lemma}\n\n\\begin{proof}\nBy induction on $|t|$. We distinguish two cases.\n\\medskip\n\\begin{itemize}\n \\item $t\\,\\not\\!\\mv{\\tau}$. Let $t= \\sum_{i\\in I}a_i t_i+Y$. By D5,\n\\[\\vdash t\\approx \\sum_{a\\in \\mc{I}(t)} a(\\!\\!\\!\\!\\sum_{i\\in I, a_i=a}\\!\\!\\!\\!\\tau t_i)+Y~.\\]\nBy induction, for each $a\\in\\mc{I}(t)$,\n\\[\\vdash \\!\\!\\!\\!\\sum_{i\\in I,a_i=a}\\!\\!\\!\\! \\tau t_i\\approx t_a\\]\nfor some normal form $t_a$. So we are done.\n\\medskip\n \\item $t \\mv{\\tau}$. By D6, $t$ can be brought in the form\n $\\sum_{i\\in I}\\tau t_i$ with $I\\neq\\emptyset$, and using D2 one can\n even make sure that $t_i\\,\\not\\!\\mv{\\tau}$ for $i\\in I$.\n Using the first case in this proof, we obtain,\n for each $i\\in I$,\n\\[\n \\vdash t_i \\approx \\sum_{a\\in A_{L(i)}}a t_{a,i} + V_{L(i)}\n\\]\nfor some $L(i) \\subseteq A \\cup V$.\nThus\\vspace{-3pt}\n\\[\n\\vdash t \\approx \\sum_{i\\in I}\n\\tau \\left(\\sum_{a\\in A_{L(i)}}a t_{a,i} + V_{L(i)}\\right) ~.\n\\vspace{-3pt}\\]\nFor each $a\\in \\mc{I}(t)$, we define \\qquad\n$\\displaystyle u_a=\\!\\!\\!\\!\\sum_{i\\in I,~ a \\in A_{L(i)}}\\!\\!\\!\\! \\tau t_{a,i}~.$\\\\[1ex]\nThen $|u_a| < |t|$.\nBy induction, $\\vdash u_a \\approx t_a$ for some normal form $t_a$.\\\\\nDefine $\\mc{L} = \\{L(i) \\mid i \\in I\\}$.\nBy repeated application of D9 we obtain\\vspace{-3pt}\n\\[\n\\vdash t \\approx \\sum_{i\\in I} \\tau \\left( \\sum_{a\\in\n  A_{L(i)}}\\!\\!a u_{a} + V_{L(i)}\\right)\n\\approx \\sum_{L\\in \\mc{L}}\\tau\\left( \\sum_{a\\in A_L}\\ at_a +V_L\\right)\n~.\\vspace{-3pt}\\]\nThe latter term has the required form, except that the family $\\mc{L}$\nneed not be saturated. However, it is straightforward to saturate\n$\\mc{L}$ by application of D7 and D8.\n\\qed\n\\end{itemize}\n\\end{proof}\n\n\\begin{lemma} \\label{essential}\n  Suppose $t$ and $u$ are both in normal forms and $t\\sqsubseteq_{\\rm WF} u$.\n  If $t\\Rightarrow \\mv{a}t_a$, then there exists a term $u_a$ such that $u\\Rightarrow\\mv{a} u_a$ and $t_a\\leq_{\\rm WF} u_a$.\n\\end{lemma}\n\n\\begin{proof}\n Suppose $t\\sqsubseteq_{\\rm WF} u$ and $t\\Rightarrow \\mv{a} t_a$.\n Let $\\sigma$ be the closed substitution given by $\\sigma(x)=\\mathbf{0}$ for\n all $x\\in V$.  As $(a,\\emptyset)$ is a weak failure\n pair of $\\sigma(t)$ and $\\sigma(t)\\sqsubseteq_{\\rm WF} \\sigma(u)$, it\n is also a weak failure pair of $u$. Thus there\n exists a term $u_a$ such that\n \\raisebox{0pt}[0pt][0pt]{$u\\Rightarrow\\mv{a} u_a$}.\n By the definition of a normal form, this term is unique.\\hfill(*)\n\n We now show that $t_a \\leq_{\\rm WF} u_a$.\n Let $\\rho$ be a closed substitution.\n Consider a weak failure pair $(a_1\\cdots a_k,B)$ of $\\rho(t_a)$.\n Then $(aa_1\\cdots a_k,B)$ is a weak failure pair of\n $\\rho(t)$, and hence also of $\\rho(u)$. It suffices to conclude that\n $(a_1\\cdots a_k,B)$ is a weak failure pair of $\\rho(u_a)$.\n However, we can \\emph{not} conclude this directly, as\n possibly $u\\Rightarrow x+u'$ where\n $(a a_1\\cdots a_k,B)$ is a weak failure pair of $\\rho(x)$.\n To ascertain that nevertheless $(a_1\\cdots a_k,B)$ is a weak\n failure pair of $\\rho(u_a)$,\n we define a modification $\\rho'$ of $\\rho$ such that\n for all $\\ell\\leq k$ and for all terms $v$, $\\rho(v)$ and $\\rho'(v)$\n have the same weak failure pairs $(c_1\\cdots c_\\ell,B)$, while for all\n $x\\in V$, $(a a_1\\cdots a_k,B)$ is not a weak failure pair of $\\rho'(x)$.\n\n We obtain $\\rho'(x)$ from $\\rho(x)$ by replacing subterms $bp$ at\n depth $k$ by $\\mathbf{0}$ if $b\\not\\in B$ and by $bb\\mathbf{0}$ if $b\\in B$.\n That is,\n \\newcommand{\\chop}[1][]{\\mathit{chop}_{#1}}\n \\[\n   \\rho'(x)=\\chop[k](\\rho(x))\n \\]\n with ${\\it chop}_m$ for all $m\\geq 0$ inductively defined by\n \\[ \\begin{array}{lcl}\n        \\chop[m](\\mathbf{0}) &=& \\mathbf{0} \\\\\n        \\chop[m](p+q)  &=& \\chop[m](p)+\\chop[m](q) \\\\\n        \\chop[m](\\tau p) &=& \\tau\\,\\chop[m](p)\\\\\n        \\chop[0](bp)   &=& \\left \\{ \\begin{array}{ll}\n                              \\mathbf{0}   & \\mbox{if $b\\not\\in B$} \\\\\n                              bb\\mathbf{0} & \\mbox{if $b\\in B$}\n                                    \\end{array} \\right. \\\\\n        \\chop[m+1](bp) &=& b\\,\\chop[m](p)\n      \\end{array}\n \\]\n We proceed to prove that $\\rho'$ has the desired properties\n mentioned above.\n\n \\begin{enumerate}\n \\renewcommand{\\theenumi}{\\Alph{enumi}}\n \\item \\label{rhomodA}\n   For all $\\ell\\leq k$ and $c_1,\\ldots,c_\\ell\\in A$ and for all terms\n   $v$, $\\rho(v)$ and\n   $\\rho'(v)$ have the same weak failure pairs $(c_1\\cdots c_\\ell,B)$,\n\n   The difference between $\\rho(v)$ and $\\rho'(v)$ only appears within\n   subterms of depth $k$, that is for terms $p$ such that\n   $\\rho(v)\\Rightarrow \\mv{c_1} \\Rightarrow \\cdots \\Rightarrow\n   \\mv{c_k} \\Rightarrow p$ for certain $c_1,\\ldots,c_k \\in A$. Such a\n   subterm $p$ of $\\rho(v)$ corresponds to a subterm $p'$ of\n   $\\rho'(v)$---still satisfying $\\rho'(v)\\Rightarrow \\mv{c_1}\n   \\Rightarrow \\cdots \\Rightarrow \\mv{c_k} \\Rightarrow p'$---in which\n   certain subterms $bq$ are replaced by $\\mathbf{0}$ if $b\\not\\in B$ and by\n   $bb\\mathbf{0}$ if $b\\in B$.  For such corresponding subterms $p$ and $p'$\n   we have $\\mc{I}(p)\\cap B=\\emptyset$ if and only if $\\mc{I}(p')\\cap\n   B=\\emptyset$.  From this the claim follows immediately.\n\n   \\medskip\n\n \\item \\label{rhomodB}\n   For all $x\\in V$,\n   $(a a_1\\cdots a_k,B)$ is not a weak failure pair of $\\rho'(x)$.\n\n   \\medskip\n\n   To this end we show that for all closed terms $p$, $\\chop[m](p)$\n   does not have any weak failure pair $(c_0\\cdots c_{m},B)$\n   with $c_0,\\ldots,c_m\\in A$.\n   We apply induction on $m$.\n\n   \\smallskip\n\n   \\noindent\n   \\textit{Base case:} Since the summands of $\\chop[0](p)$, when\n   skipping over initial $\\tau$-steps, are\n   $bb\\mathbf{0}$ with $b\\in \\mc{I}(p)\\cap B$, $\\chop[0](p)$ does not\n   have a weak failure pair $(c_0,B)$.\n\n   \\smallskip\n\n   \\noindent\n   \\textit{Induction step:} Let $m>0$. By induction, for closed terms $q$,\n   $\\chop[m-1](q)$ does not have weak failure pairs $(c_1\\cdots c_{m},B)$.\n   Since the transitions of $\\chop[m](p)$ are\n   $\\chop[m](p)\\mv{c}\\chop[m-1](q)$ for $p\\mv{c}q$, it\n   follows that $\\chop[m](p)$ does not have weak failure pairs\n   $(c_0\\cdots c_{m},B)$.\n \\end{enumerate}\n\n \\noindent\n Now, since $(a_1\\cdots a_k,B)$ is a weak failure pair of $\\rho(t_a)$,\n by property (\\ref{rhomodA}) it is also a weak failure pair of $\\rho'(t_a)$,\n Therefore $(a a_1\\cdots a_k,B)$ is a weak failure pair of $\\rho'(t)$,\n and hence also of $\\rho'(u)$.\n Since according to property (\\ref{rhomodB}) it is \\emph{not} the case that\n $u\\Rightarrow x+u'$ with $(a  a_1\\cdots a_k,B)$ a weak failure pair\n of $\\rho'(x)$, it must be the case that $u\\Rightarrow \\mv{a}u''$ such\n that $(a_1\\cdots a_k,B)$ is a weak failure pair of $\\rho'(u'')$.\n By (*), $u''=u_a$.\n Again by property (\\ref{rhomodA}), $(a_1\\cdots a_k,B)$ is a\n weak failure pair of $\\rho(u_a)$.\n\\qed\n\\end{proof}\n\n\n\\subsection{$\\omega$-Completeness Proof}\n\nWe are now in a position to prove Theo.~\\ref{theo:A-infinite}\n($\\omega$-completeness in case of an infinite alphabet) and\nTheo.~\\ref{theo:A-finite} ($\\omega$-completeness in case of a\nfinite alphabet), along with Theo.~\\ref{theo:ground-complete}\n(ground completeness). We will prove these three theorems in one\ngo. Namely, in the proof, two cases are distinguished; only in the\nsecond case ($\\mc{I}(t)=A$), in which the $A$ is guaranteed to be\nfinite, will the axiom WF$_A$ play a role.\n\n\\begin{proof}\nLet $t\\sqsubseteq_{\\rm WF} u$. We need to show that $\\vdash\nt\\preccurlyeq u$. We apply induction on $|t|+|u|$. By\nLem.~\\ref{nf}, we can write $t$ and $u$ in normal form.\n\nWe first prove that $L(t)\\subseteq L(u)$. Suppose this is not the\ncase. Then there exists some $a\\in A_{L(t)}\\setminus A_{L(u)}$ or\nsome $x\\in V_{L(t)}\\setminus V_{L(u)}$. In the first case, let\n$\\sigma$ be the closed substitution with $\\sigma(z)=\\mathbf{0}$ for all\n$z\\in V$; we find that $(a,\\emptyset)$ is a weak failure pair of\n$\\sigma(t)$ but not of $\\sigma(u)$, which contradicts the fact that\n$\\sigma(t)\\sqsubseteq_{\\rm WF} \\sigma(u)$. In the second case, pick some\n$d>\\max\\{|t|,|u|\\}$, and consider the closed substitution\n$\\sigma(x)=a^d\\mathbf{0}$ and $\\sigma(z)=\\mathbf{0}$ for $z\\neq x$. Then\n$(a^d,\\emptyset)$ is weak failure pair of $\\sigma(t)$. However, it\ncan \\emph{not} be a weak failure pair of $\\sigma(u)$, again\ncontradicting $\\sigma(t)\\sqsubseteq_{\\rm WF} \\sigma(u)$.\n\nWe distinguish two cases, depending on whether $\\mc{I}(t)=A$ or\nnot.\n\n  \\begin{enumerate}\n\n  \\item $\\mc{I}(t) \\neq A$.  We distinguish three cases.\nDue to the condition that $t\\mv{\\tau}$ implies  $u\\mv{\\tau}$, it\ncannot be the case that $t$ is an action normal form and $u$ a\n$\\tau$ normal form.\n\n\\medskip\n  \\begin{enumerate}\n\\item $t$ and $u$ are both action normal forms.\nSo $t= \\sum_{a\\in A_L} at_a + V_L$ and $u= \\sum_{a\\in A_M}\nau_a+V_M$. We show that $L(t)=L(u)$. Namely, pick $b\\in A\\setminus\nA_L$, and let $\\sigma$ be the closed substitution with\n$\\sigma(z)=\\mathbf{0}$ for any $z\\in V_L$, and $\\sigma(z)=b\\mathbf{0}$ for\n$z\\not\\in V_L$. As $(\\varepsilon,A\\setminus\\mc{I}(t))$ is a weak\nfailure pair of $t$, and\n  hence of $u$, it must be that $L(u)\\subseteq L(t)$.\nTogether with $L(t)\\subseteq L(u)$ this gives $L(t)=L(u)$. By\nLem.~\\ref{essential}, for each $a\\in \\mc{I}(t)$, $t_a \\leq_{\\rm WF}\nu_a$, and thus clearly  $t_a\\sqsubseteq_{\\rm WF} \\tau u_a$. By\ninduction, $\\vdash t_a\\preccurlyeq \\tau u_a$ and hence $\\vdash\nat_a\\preccurlyeq au_a$. It follows that\n\\[\\vdash t = \\sum_{a\\in A_L} at_a +V_L \\preccurlyeq \\sum_{a\\in A_L}\nau_a +V_L  =  \\sum_{a\\in A_M} au_a +V_M = u\\]\n\n\\item Both $t$ and $u$ are $\\tau$ normal forms:\n\\[ t= \\sum_{L\\in \\mc{L}}\\tau (\\sum_{a \\in A_L} at_a+ V_L)\\]\nand\n\\[ u = \\sum_{M\\in \\mc{M}}\\tau (\\sum_{a \\in A_M} au_a+ V_M)\\]\n\nBy Lem.~\\ref{essential}, for each $a\\in  \\mc{I}(t)$, $t_a \\leq_{\\rm WF}\nu_a$, and thus clearly $t_a\\sqsubseteq_{\\rm WF} \\tau u_a$. By\ninduction, $\\vdash t_a\\preccurlyeq \\tau u_a$. By these\ninequalities, together with D4,\n\\begin{equation} \\label{eqnX}\n\\vdash t \\preccurlyeq \\sum_{L\\in \\mc{L}} \\tau(\\sum_{a\\in A_L} au_a\n+ V_L) + u\n\\end{equation}\n\n\\medskip\n\nWe now show that $\\mc{L}\\subseteq \\mc{M}$. Take any $L\\in \\mc{L}$,\npick $b\\in A \\setminus A_L$, and consider the closed substitution\n$\\sigma(z)=\\mathbf{0}$ for any $z\\in V_L$, and $\\sigma(z)=b\\mathbf{0}$ for\n$z\\not\\in V_L$. Since $\\sigma(t)\\mv{\\tau} \\sigma(\\sum_{a\\in L}at_a)$\nand $\\sigma(t)\\sqsubseteq_{\\rm WF} \\sigma(u)$, there exists an $M\\in\n\\mc{M}$ with $A_M\\subseteq A_L$ and $V_M \\subseteq V_L$. Since\nalso $L\\subseteq L(t) \\subseteq L(u)$, and $\\mc{M}$ is saturated,\nit follows that $L\\in\\mc{M}$. Hence, $\\mc{L} \\subseteq \\mc{M}$.\n\n\\medskip\n\nSince $\\mc{L} \\subseteq \\mc{M}$,\n\\begin{equation} \\label{eqnY}\n\\sum_{L\\in \\mc{L}}\\tau(\\sum_{a\\in A_L} au_a + V_L) + u = u\n\\end{equation}\nBy (\\ref{eqnX}) and (\\ref{eqnY}), $ \\vdash t\\preccurlyeq u$.\n\n\\medskip\n\n\\item $t$ is an action normal form and $u$ is a $\\tau$ normal\nform. Then $\\tau t \\sqsubseteq_{\\rm WF} u$. Note that $\\tau t$ is a\n$\\tau$ normal form, so according to the previous case,\n\\[\\vdash \\tau t \\preccurlyeq u\\]\nBy WF3,\n\\[\\vdash t\\preccurlyeq \\tau t \\preccurlyeq u \\]\n\n\\end{enumerate}\n\n   \\medskip\n\n   \\item $\\mc{I}(t)=A$. Note that in this case, $|A|<\\infty$. So, according to\nTheo.~\\ref{theo:A-finite}, axiom WF$_A$ is at our disposal. As\nbefore, we distinguish three cases.\n\\medskip\n   \\begin{enumerate}\n    \\item Both $t$ and $u$ are action normal forms. Since\n    $L(t)\\subseteq L(u)$ we have\n   $t= \\sum_{a\\in A} at_a +W$ and $u= \\sum_{a\\in A} au_a+X$ with $W\n   \\subseteq X$. By WF$_A$,\n   \\[ \\vdash \\sum_{a\\in A}a t_a \\preccurlyeq \\sum_{a\\in A}a t_a + u\\]\n   By Lem.~\\ref{essential}, for each $a\\in A$, $t_a \\leq_{\\rm WF} u_a$, and\n   thus clearly $t_a\\sqsubseteq_{\\rm WF} \\tau u_a$. By induction,\n   $\\vdash t_a\\preccurlyeq \\tau u_a$. It follows, using $W\\subseteq X$, that\n  \\[ \\vdash t=\\sum_{a\\in A}a t_a +W \\preccurlyeq \\sum_{a\\in A}a u_a + u + W=u\\]\n\n    \\medskip\n\n   \\item Both $t$ and $u$ are $\\tau$ normal forms.\n          \\[t= \\sum_{L\\in \\mc{L}}\\tau (\\sum_{a \\in A_L} at_a+V_L)\\]\n    and\n    \\[     u = \\sum_{M\\in \\mc{M}}\\tau (\\sum_{a \\in A_M} au_a+V_M)\\]\n    By D1 and WF$_A$ (clearly, in this case $A_{L(t)}=A$),\n   \\begin{equation} \\label{eqPP}\n   \\vdash t\\approx t+\\sum_{a\\in A}a t_a \\preccurlyeq t+\\sum_{a\\in A}a t_a+u\n   \\end{equation}\n   By Lem.~\\ref{essential}, for each $a\\in A$, $t_a \\leq_{\\rm WF} u_a$, and thus clearly\n   $t_a\\sqsubseteq_{\\rm WF} \\tau u_a$. By induction, $\\vdash\n   t_a\\preccurlyeq \\tau u_a$. By these inequalities, together with (\\ref{eqPP}),\n   \\[\n   \\vdash t \\preccurlyeq \\sum_{L\\in \\mc{L}}\\tau (\\sum_{a \\in A_L} au_a+V_L)\n   + \\sum_{a\\in A}a u_a + u\n   \\]\n   So by D1,\n   \\begin{equation} \\label{eqnQQ} \\vdash t \\preccurlyeq \\sum_{L\\in \\mc{L}}\n   \\tau (\\sum_{a \\in A_L} au_a+V_L) + u\\end{equation}\n    Now for $L\\in \\mc{L}$ with $A_L\\neq A$ we have $L \\in \\mc{M}$ using\n    the same reasoning as in 1(b). For $L\\in \\mc{L}$ with $A_L=A$ we\n    have $V_L \\subseteq V_{L(t)} \\subseteq V_{L(u)}$. By WF$_A$ we have\n    \\begin{equation} \\label{eqnSS}\n    \\vdash \\tau(\\sum_{a \\in A_L} au_a+V_L) \\preccurlyeq\n    \\tau(\\sum_{a \\in A} au_a+V_{L(u)})\n    \\end{equation}\n    As the latter is a summand of $u$ we obtain $t\\preccurlyeq u$.\n\n\\medskip\n\n  \\item $t$ is an action normal form and $u$ is a $\\tau$ normal\n  form. This can be dealt with as in case 1(c).\n   \\end{enumerate}\n  \\end{enumerate}\n  This completes the proof. \\qed\n\\end{proof}\n\n\\subsection{Weak Failures Equivalence} \\label{sec:wfe}\n\nIn \\cite{AFI07,FG07} an algorithm is presented which takes as\ninput a sound and ground-complete inequational axiomatization $E$\nfor BCCSP modulo a preorder $\\sqsubseteq$ which \\emph{includes the\nready simulation preorder} and is \\emph{initials\npreserving},\\footnote{meaning that $p\\sqsubseteq q$ implies that\n$I(p)\\subseteq I(q)$, where the set $I(p)$ of \\emph{strongly}\ninitial actions is $I(p) = \\{\\alpha \\in A_{\\tau} \\mid\np\\mv{\\alpha}\\}$} and generates as output an equational\naxiomatization $\\mc{A}(E)$ which is sound and ground-complete for\nBCCSP modulo the corresponding equivalence---its kernel: $\\sqsubseteq\n\\cap \\sqsubseteq^{-1}$. Moreover, if the\noriginal axiomatization $E$ is $\\omega$-complete, so is the\nresulting axiomatization. The axiomatization $\\mc{A}(E)$ generated\nby the algorithm from $E$ contains the axioms A1-4 for\nbisimulation equivalence and the axioms $\\beta(\\alpha x + z) +\n\\beta (\\alpha x + \\alpha y + z) \\approx\n \\beta (\\alpha x + \\alpha y + z)$ for $\\alpha,\\beta\\in A_\\tau$\nthat are valid in ready simulation semantics, together with the\nfollowing equations, for each inequational axiom $t \\preccurlyeq\nu$ in $E$:\n\\begin{itemize}\n\n\\item $ t + u \\approx u$; and\n\n\\item  $\\alpha(t + x) + \\alpha(u + x) \\approx \\alpha(u + x)$ (for\neach $\\alpha\\in A_\\tau$, and some variable $x$ that does not occur\nin $t + u$).\n\\end{itemize}\nMoreover, if $E$ contains an equation (formally abbreviating two\ninequations), this equation is logically equivalent to the four\nequations in $\\mc{A}(E)$ that are derived from it, and hence can be\nincorporated in the equational axiomatization unmodified.\n\nRecently, we lifted this result to weak semantics\n\\cite{CFG08b}, which makes the aforementioned algorithm applicable\nto all 87 preorders surveyed in \\cite{Gla93} that are at least as\ncoarse as the ready simulation preorder. Namely, among others, we\nshow that\n\n\\begin{theorem}\\label{alg}\nLet $\\sqsubseteq$ be a weak initials preserving precongruence%\n\\footnote{meaning that $p\\sqsubseteq q$ implies that\n$\\mc{I}_{\\tau}(p)\\subseteq \\mc{I}_\\tau(q)$, where the set\n$\\mc{I}_\\tau(p)$ of \\emph{weak} initial actions is $\\mc{I}_\\tau(p)\n= \\{\\alpha \\in A_{\\tau} \\mid p\\Rightarrow\\mv{\\alpha}\\}$} that\ncontains the strong ready simulation preorder $\\sqsubseteq_{\\rm RS}$\nand satisfies T2 (the second $\\tau$-law of CCS: $\\tau x \\approx\n\\tau x + x$), and let $E$ be a sound and ground-complete\naxiomatization of $\\sqsubseteq$. Then $\\mc{A}(E)$ is a sound and\nground-complete axiomatization of the kernel of $\\sqsubseteq$.\nMoreover, if $E$ is $\\omega$-complete, then so is $\\mc{A}(E)$.\n\\end{theorem}\nIt is straightforward to check that weak failures meets the\nprerequisites of Theo.~\\ref{alg}, and thus we can run the\nalgorithm  and obtain the axiomatization in Tab.~\\ref{tab-aux} for\nweak failures equivalence.\n\\begin{table}\n\\vspace{-5pt}\n\\begin{center}\n \\begin{tabular}{l@{\\qquad}rcl}\n WF1   & $ax+ay$  & $\\approx$ & $a(\\tau x+\\tau y)$\\\\\n WF2$^a$ & $\\tau(x+y)+\\tau x +y$  & $\\approx$ & $\\tau x+ y$\\\\\n WF2$^b$ & $\\alpha(\\tau(x+y)+z) + \\alpha (\\tau x+y +z) $  & $\\approx$ & $\\alpha(\\tau x+ y+z)$\\\\\n WF3$^a$ & $x+\\tau x+y$  & $\\approx $ & $ \\tau x+  y$\\\\\n WF3$^b$ & $\\alpha(x+z)+\\alpha(\\tau x+y+z)$  & $\\approx $ & $ \\alpha(\\tau x+  y+z)$\\\\\n RS   & $\\beta(\\alpha x + z) + \\beta(\\alpha x + \\alpha y + z)$  & $\\approx $ &\n $\\beta(\\alpha x + \\alpha y + z)$\\\\\n WF$_A^{~~a}$  & $\\sum_{a\\in A} ax_a + \\sum_{a\\in A} ax_a + y$  & $\\approx $ &\n $\\sum_{a\\in A} ax_a + y$\\\\\n WF$_A^{~~b}$  & $\\beta(\\sum_{a\\in A} ax_a + z) + \\beta(\\sum_{a\\in A} ax_a + y+z)$  & $\\approx $ & $\\beta(\\sum_{a\\in A} ax_a + y+z)$\n\\end{tabular}\n\\end{center}\\caption{Axiomatization generated from the algorithm}\\label{tab-aux}\n\\vspace{-5mm}\n\\end{table}\n\\noindent After simplification and omission of redundant axioms,\nwe obtain the axiomatization in Tab.~\\ref{tab5}.\n\\begin{table}\n\\vspace{-5pt}\n\\begin{center}\n \\begin{tabular}{l@{\\qquad}rcl}\n WF1  & $ax+ay$  & $\\approx$ & $a(\\tau x+\\tau y)$\\\\\n WFE2 & $\\tau(x+y)+\\tau x$  & $\\approx$ & $\\tau x+ y$\\\\\n WFE3 & $ax+\\tau(ay+z)$  & $\\approx$ & $\\tau(a x+ a y+z)$\\\\\n WFE$_A$ & $\\tau(\\sum_{a\\in A} ax_a + z) + \\tau(\\sum_{a\\in A} ax_a + y+z)$\n  & $\\approx $ & $\\tau(\\sum_{a\\in A} ax_a + y+z)$\n\\end{tabular}\n\\end{center}\n\\caption{Axiomatization for weak failures equivalence} \\label{tab5}\n\\vspace{-5mm}\n\\end{table}\n\n  \\begin{lemma} \\label{lem:four}\n     The axioms in \\emph{Tab.~\\ref{tab-aux}} are\n     derivable from the axioms in\n     \\emph{Tab.~\\ref{tab5}} together with A1-4.\n  \\end{lemma}\n  \\begin{proof}\n \n   WF1 is unmodified. WF2$^a$ and WF3$^a$ can be trivially\n  derived from WFE2. WF$_A^{~~a}$ is derivable using A3.\n\n  To proceed, we have that WFE2$\\vdash \\tau\\tau x \\approx \\tau x$ (namely by\n  substituting $\\tau x$ for $y$ and invoking D1) and hence also WFE2$\\vdash$D2\n  (namely by substituting $\\tau x $ for $x$ in WFE2 and invoking D1);\n  using D2, the instances of WF2$^b$ and WF3$^b$ with $\\alpha=\\tau$, as well as the\n  instance of RS with $\\beta=\\alpha=\\tau$,\n  are derivable from WFE2.\n\n  The instances of WF2$^b$ and WF3$^b$ with $\\alpha\\neq\\tau$,\n  are derivable from WF1 and the instances with $\\alpha=\\tau$;\n  the same holds for the instances of RS and WF$_A^{~~b}$ with\n  $\\beta\\neq\\tau$.\n\n  Finally in the remaining instances of RS\n  (with $\\beta=\\tau$ and $\\alpha=a\\in A$), we have WFE2$\\vdash \\tau(ax+z)+\\tau(ax+ay+z)\\approx\n  \\tau(ax+z)+ay$,\n \n  and thus it can be derived from WFE3.  The instance of\n  WF$_A^{~~b}$ with $\\beta=\\tau$ is exactly WFE$_A$. \\qed\n\n \n  \\end{proof}\n\nThe axioms WF1, \\mbox{WFE2-3} already appeared in \\cite{Gla97}.\nA1-4+WF1+WFE2-3 is sound and ground-complete for BCCS modulo\n$\\equiv_{\\rm WF}$ (see also \\cite{Gla97,CFG08b}). By\nTheo.~\\ref{theo:A-infinite} and Theo.~\\ref{theo:A-finite}\n(together with Lem.~\\ref{lem:four}), we have:\n\n\\begin{corollary}\n  If $|A|=\\infty$, then the axiomatization \\mbox{\\rm A1-4+WF1+WFE2-3} is\n  $\\omega$-complete for $\\mathrm{BCCS}(A)$ modulo $\\equiv_{\\rm WF}$.\n\\end{corollary}\n\n\\begin{corollary}\n  If $|A| <\\infty$, then the axiomatization \\mbox{\\rm A1-4+WF1+WFE2-3+WFE$_A$} is\n  $\\omega$-complete for\n   $\\mathrm{BCCS}(A)$ modulo $\\equiv_{\\rm WF}$.\n\\end{corollary}\n\n\\section{Weak Impossible Futures Semantics} \\label{sec6}\n\n\\emph{Weak impossible futures} semantics is closely related to\nweak failures semantics. Only, instead of the set of actions in\nthe second argument of a weak failure pair (see\nDef.~\\ref{def:weak-failures}), an impossible future pair contains\na set of \\emph{traces}.\n\n\\begin{definition}[Weak impossible futures]\\rm\\label{wif}\n\\vspace{-1ex}\n\\begin{itemize}\n  \\item A sequence $a_1\\cdots a_k \\in A^*$, with $k\\geq 0$, is a\n  \\emph{trace} of a process $p_0$ if there is a path\n  $p_0\\Rightarrow \\mv{a_1} \\Rightarrow \\cdots \\Rightarrow \\mv{a_k} \\Rightarrow p_k$;\n  it is a \\emph{completed trace} of $p_0$ if moreover $\\mathcal{I}(p_k)=\\emptyset$.\n  Let $\\mc{T}(p)$ denote the set of traces of process $p$, and\n  $\\mc{CT}(p)$ its set of completed traces.\n\n  \\item A pair $(a_1\\cdots a_k,B)$, with $k\\geq 0$ and $B \\subseteq A^*$,\n  is a \\emph{weak impossible future} of a process $p_0$ if there is a\n  path $p_0\\Rightarrow \\mv{a_1} \\Rightarrow \\cdots \\Rightarrow \\mv{a_k} \\Rightarrow p_k$\n  with $\\mc{T}(p_k) \\cap B = \\emptyset$.\n\n  \\item The \\emph{weak impossible futures preorder} $\\sqsubseteq_{\\rm WIF}$ is\n  given by $p\\sqsubseteq_{\\rm WIF} q$ iff (1) the weak impossible\n  futures of $p$ are also weak impossible futures of $q$,\n  (2) $\\mc{T}(p)=\\mc{T}(q)$ and\n  (3) $p\\mv{\\tau}$ implies that $q\\mv{\\tau}$.\n\n  \\item \\emph{Weak impossible futures equivalence} $\\equiv_{\\rm WIF}$ is defined\n  as $\\sqsubseteq_{\\rm WIF} \\cap \\sqsubseteq_{\\rm WIF}^{-1}$.\n\\end{itemize}\n\\end{definition}\n$\\sqsubseteq_{\\rm WIF}$ is a precongruence, and $\\equiv_{\\rm WF}$ a\ncongruence, for BCCS \\cite{VM01}.\n   The requirement (2) $\\mc{T}(p)=\\mc{T}(q)$ is necessary for this\n   precongruence property. Without it we would have $\\tau a\\mathbf{0} \\sqsubseteq\n   \\tau a\\mathbf{0} + b\\mathbf{0}$ but $c(\\tau a\\mathbf{0}) \\not\\sqsubseteq c(\\tau a\\mathbf{0} + b\\mathbf{0})$.\n\nA sound and ground-complete axiomatization for $\\sqsubseteq_{\\rm WIF}$\nis obtained by replacing axiom WF3 in Tab.~\\ref{tab1} by the\nfollowing axiom (cf.\\ \\cite{VM01}, where a slightly more\ncomplicated, but equivalent, axiomatization is given):\n\\[\n \\mbox{WIF3}~~~~  x  ~\\preccurlyeq~ \\tau x\n\\]\nHowever, surprisingly, there is no finite sound and\nground-complete axiomatization for $\\equiv_{\\rm WIF}$.\nWe will show this in Sec.~\\ref{nonax}. A similar\ndifference between the impossible futures preorder and equivalence\nin the concrete case (so in the absence of $\\tau$) was found\nearlier in \\cite{CF08}. We note that, since weak impossible\nfutures semantics is not coarser than ready simulation semantics,\nthe algorithm from \\cite{AFI07,FG07,CFG08b} to generate an\naxiomatization for the equivalence from the one for the preorder,\ndoes not work in this case.\n\nIn Sec.~\\ref{sec:42} we establish that the sound and ground-complete\naxiomatization for BCCS modulo $\\sqsubseteq_{\\rm WIF}$ is\n$\\omega$-complete in case $|A|=\\infty$, and in Sec.~\\ref{sec:43} that\nthere is no such finite basis for the inequational theory of BCCS\nmodulo $\\sqsubseteq_{\\rm WIF}$ in case $|A|<\\infty$. Again, these\nresults correspond to (in)axiomatizability results for the impossible\nfutures preorder in the concrete case \\cite{CF08}, with very similar\nproofs.\n\n\\subsection{Nonexistence of an Axiomatization for Equivalence}\\label{nonax}\nWe now prove that for any (nonempty) $A$ there does \\emph{not} exist\nany finite, sound, ground-complete axiomatization for\n$\\mathrm{BCCS}(A)$ modulo $\\equiv_{\\rm WIF}$. The cornerstone for this\nnegative result is the following infinite family of closed equations,\nfor $m\\geq 0$:\n\\[ \\tau a^{2m}\\mathbf{0}+\\tau(a^m\\mathbf{0}+a^{2m}\\mathbf{0}) ~\\approx~ \\tau(a^m\\mathbf{0}+a^{2m}\\mathbf{0}) \\]\nIt is not hard to see that they are sound modulo $\\equiv_{\\rm WIF}$.\nWe start with a few lemmas.\n\n\\begin{lemma}\\label{res}\nIf $p\\sqsubseteq_{\\rm WIF} q$ then $\\mc{CT}(p) \\subseteq \\mc{CT}(q)$.\n\\end{lemma}\n\n\\begin{proof}\nA process $p$ has a completed trace $a_1 \\cdots a_k$ iff it has a weak\nimpossible future $(a_1 \\cdots a_k,A)$.\n\\qed\n\\end{proof}\n\n\\begin{lemma} \\label{newvariable}\nSuppose $t\\sqsubseteq_{\\rm WIF} u$. Then for any $t'$ with\n$t\\Rightarrow\\mv{\\tau} t'$ there is some $u'$ with\n$u\\Rightarrow\\mv{\\tau} u'$ such that $\\var(u')\\subseteq \\var(t')$.\n\\end{lemma}\n\n\\begin{proof}\nLet $t\\Rightarrow\\mv{\\tau} t'$. Fix some $m>|t|$, and consider\nthe closed substitution $\\rho$ defined by $\\rho(x)=\\mathbf{0}$ if\n$x\\in\\var(t')$ and $\\rho(x)=a^m\\mathbf{0}$ if $x\\not\\in\\var(t')$. Since\n$\\rho(t)\\Rightarrow\\rho(t')$ with $|\\rho(t')|=|t'|<m$, and\n$\\rho(t)\\sqsubseteq_{\\rm WIF}\\rho(u)$, clearly $\\rho(u)\\Rightarrow q$\nfor some $q$ with $|q|<m$. From the definition of $\\rho$ it then\nfollows that there must exist $u\\Rightarrow u'$ with\n$\\var(u')\\subseteq \\var(t')$. In case $u \\Rightarrow\\mv{\\tau} u'$ we\nare done, so assume $u'=u$. Let $\\sigma$ be the substitution with\n$\\sigma(x)=\\mathbf{0}$ for all $x\\in V$. Since $\\sigma(t) \\mv{\\tau}$ and $t\n\\sqsubseteq_{\\rm WIF} u$ we have $\\sigma(u)\\mv{\\tau}$, so $u \\mv{\\tau}\nu''$ for some $u''$. Now $\\var(u'') \\subseteq \\var(u) = \\var(u')\n\\subseteq \\var(t')$.\\qed\n\\end{proof}\n\\pagebreak[3]\n\n\\begin{lemma} \\label{key2}\nAssume that, for terms $t,u$, closed substitution $\\sigma$, action $a$\nand integer $m$:\n\n\\begin{enumerate}\\vspace{-1ex}\n  \\item $t \\equiv_{\\rm WIF} u$;\n\n  \\item $m > |u|$;\n\n  \\item $\\mc{CT}(\\sigma(u)) \\subseteq \\{a^{m}, a^{2m}\\}$; and\n\n  \\item there is a closed term $p'$ such that\n  $\\sigma(t)\\Rightarrow\\mv{\\tau} p'$ and $\\mc{CT}(p') = \\{a^{2m}\\}$.\n\\vspace{-1ex}\n\\end{enumerate}\nThen there is a closed term $q'$ such that $\\sigma(u)\n\\Rightarrow\\mv{\\tau} q'$ and $\\mc{CT}(q') = \\{a^{2m}\\}$.\n\\end{lemma}\n\n\\begin{proof}\nAccording to proviso (4) of the lemma, we can distinguish two\ncases.\n\n\\begin{itemize}\n  \\item There exists some $x\\in V$ such that $t \\Rightarrow t'$ with\n  $t'=t''+x$ and $\\sigma(x)\\Rightarrow\\mv{\\tau} p'$ where $\\mc{CT}(p')=\\{a^{2m}\\}$.\n  Consider the closed substitution $\\rho$ defined by $\\rho(x)=a^m\\mathbf{0}$\n  and $\\rho(y)=\\mathbf{0}$ for any $y\\neq x$. Then $a^m \\in \\mc{CT}(\\rho(t))\n  = \\mc{CT}(\\rho(u))$, using Lem.~\\ref{res}, and this is only\n  possible if $u \\Rightarrow u'$ for some $u'=u''+x$. Hence\n  $\\sigma(u)\\Rightarrow\\mv{\\tau} p'$.\n\n  \\medskip\n\n  \\item $t\\Rightarrow\\mv{\\tau} t'$ with $\\mc{CT}(\\sigma(t'))=\\{a^{2m}\\}$.\n  Since $|t'|\\leq|t|=|u|<m$, clearly, for any \\mbox{$x\\mathbin\\in \\var(t')$},\n  either $|\\sigma(x)|\\mathbin=0$ or $\\ensuremath{\\mathit{norm}}(\\sigma(x))\\mathbin>m$,\n  where $\\ensuremath{\\mathit{norm}}(p)$ denotes the length of the shortest completed trace of $p$.\n  Since $t\\equiv_{\\rm WIF} u$, by Lem.~\\ref{newvariable},\n  $u\\mathbin\\Rightarrow\\mv{u} u'$ with $\\var(u')\\subseteq \\var(t')$. Hence,\n  for any $x\\mathbin\\in \\var(u')$, either $|\\sigma(x)|\\mathbin=0$\n  or $\\ensuremath{\\mathit{norm}}(\\sigma(x))\\mathbin>m$.  Since $|u'|\\mathbin<m$,\n  $a^m\\mathbin{\\notin} \\mc{CT}(\\sigma(u'))$.  It follows from\n  $\\mc{CT}(\\sigma(u))\\subseteq\\{a^{m}, a^{2m}\\}$ that\n  $\\mc{CT}(\\sigma(u'))= \\{a^{2m}\\}$. And $u\\Rightarrow\\mv{\\tau} u'$\n  implies $\\sigma(u)\\Rightarrow\\mv{\\tau} \\sigma(u')$.\\hfill$\\Box$\n\\end{itemize}\n\\end{proof}\n\n\\begin{lemma} \\label{key2cont}\nAssume that, for $E$ an axiomatization sound for\n$\\sqsubseteq_{\\rm WIF}$, closed terms $p,q$, closed\nsubstitution $\\sigma$, action $a$ and integer $m$:\n\\begin{enumerate}\\vspace{-1ex}\n  \\item $E\\vdash  p \\approx q$;\n\n  \\item $m > \\max \\{ |u|\\mid t \\approx u \\in  E\\}$;\n\n  \\item $\\mc{CT}(q) \\subseteq \\{a^{m},a^{2m}\\}$; and\n\n  \\item there is a closed term $p'$ such that $p\\Rightarrow\\mv{\\tau} p'$ and\n  $\\mc{CT}(p') = \\{a^{2m}\\}$.\n\\vspace{-1ex}\n\\end{enumerate}\nThen there is a closed term $q'$ such that $q \\Rightarrow\\mv{\\tau} q'$ and\n$\\mc{CT}(q') = \\{a^{2m}\\}$.\n\\end{lemma}\n\n\\begin{proof}\n\nBy induction on the derivation of $E\\vdash  p \\approx q$.\n\n\\begin{itemize}\n\n\\item Suppose $E\\vdash p\\approx q$ because $\\sigma(t)=p$ and\n$\\sigma(u)=q$ for some $t\\approx u\\in E$ or $u\\approx t\\in E$ and\nclosed substitution $\\sigma$. The claim then follows by Lem.~\\ref{key2}.\n\n\\medskip\n\n\\item Suppose $E\\vdash p\\approx q$ because $E\\vdash p\\approx r$ and\n$E\\vdash r\\approx q$ for some $r$. Since $r\\equiv_{\\rm WIF} q$, by\nproviso (3) of the lemma and Lem.~\\ref{res},\n$\\mc{CT}(r)\\subseteq \\{a^{m},a^{2m}\\}$. Since there is a $p'$\nsuch that $p\\Rightarrow\\mv{\\tau} p'$ with $\\mc{CT}(p') = \\{a^{2m}\\}$,  by\ninduction, there is an $r'$ such that $r\\Rightarrow\\mv{\\tau} r'$ and\n$\\mc{CT}(r') = \\{a^{2m}\\}$. Hence, again by induction, there is a\n$q'$ such that $q\\Rightarrow\\mv{\\tau} q'$ and $\\mc{CT}(q') = \\{a^{2m}\\}$.\n\n\\medskip\n\n\\item Suppose $E\\vdash p\\approx q$ because $p=p_1+p_2$ and\n$q=q_1+q_2$ with $E\\vdash p_1\\approx q_1$ and $E\\vdash p_2\\approx\nq_2$. Since there is a $p'$ such that $p\\Rightarrow\\mv{\\tau} p'$ and\n$\\mc{CT}(p') = \\{a^{2m}\\}$, either $p_1\\Rightarrow \\mv{\\tau} p'$ or\n$p_2\\Rightarrow\\mv{\\tau} p'$. Assume, without loss of generality, that\n$p_1\\Rightarrow\\mv{\\tau} p'$. By induction, there is a $q'$ such that\n$q_1\\Rightarrow\\mv{\\tau} q'$ and $\\mc{CT}(q') = \\{a^{2m}\\}$.\nNow $q\\Rightarrow\\mv{\\tau} q'$.\n\n\\medskip\n\n\\item Suppose $E\\vdash p \\approx q$ because $p=cp_1$ and $q=cq_1$\nwith $c\\in A$ and $E\\vdash p_1\\approx q_1$. In this case, proviso (4)\nof the lemma can not be met.\n\n\\medskip\n\n\\item Suppose $E\\vdash p \\approx q$ because $p=\\tau p_1$ and $q=\\tau\nq_1$ with $E\\vdash p_1\\approx q_1$. By proviso (4) of the lemma,\neither $\\mc{CT}(p_1)=\\{a^{2m}\\}$ or there is a $p'$ such that\n$p_1\\Rightarrow\\mv{\\tau} p'$ and $\\mc{CT}(p') = \\{a^{2m}\\}$.\nIn the first case, $q \\Rightarrow\\mv{\\tau} q_1$ and $\\mc{CT}(q_1) = \\{a^{2m}\\}$\nby Lem.~\\ref{res}. In the second, by induction, there is a $q'$\nsuch that $q_1\\Rightarrow\\mv{\\tau} q'$ and $\\mc{CT}(q') = \\{a^{2m}\\}$.\nAgain $q\\Rightarrow\\mv{\\tau} q'$.\n\\qed\n\\end{itemize}\n\\end{proof}\n\n\\begin{theorem}\nThere is no finite, sound, ground-complete axiomatization for\n$\\mathrm{BCCS}(A)$ modulo $\\equiv_{\\rm WIF}$.\n\\end{theorem}\n\n\\begin{proof}\nLet $E$ be a finite axiomatization over $\\mathrm{BCCS}(A)$ that is\nsound modulo $\\equiv_{\\rm WIF}$. Let $m$ be greater than the depth\nof any term in $E$. Clearly, there is no term $r$ such that\n$\\tau(a^m\\mathbf{0}+a^{2m}\\mathbf{0}) \\Rightarrow\\mv{\\tau} r$ and\n$\\mc{CT}(r)=\\{a^{2m}\\}$. So according to Lem.~\\ref{key2cont}, the\nclosed equation $\\tau a^{2m}\\mathbf{0}+\\tau(a^m\\mathbf{0}+a^{2m}\\mathbf{0}) \\approx\n\\tau(a^m\\mathbf{0}+a^{2m}\\mathbf{0})$ cannot be derived from $E$. Nevertheless, it is\nvalid modulo $\\equiv_{\\rm WIF}$. \\qed\n\\end{proof}\n\nIn the same way as above, one can establish the nonderivability of\nthe equations $a^{2m+1}\\mathbf{0}+a(a^m\\mathbf{0}+a^{2m}\\mathbf{0}) ~\\approx~\na(a^m\\mathbf{0}+a^{2m}\\mathbf{0})$ from any given finite equational\naxiomatization sound for $\\equiv_{\\rm WIF}$. As these equations\nare valid modulo (strong) 2-nested simulation equivalence, this\nnegative result applies to all BCCS-congruences that are at least\nas fine as weak impossible futures equivalence and at least as\ncoarse as strong 2-nested simulation equivalence. Note that\nthe corresponding result of \\cite{AFGI04} can be inferred.\n\n\\subsection{A Finite Basis for Preorder if $|A|=\\infty$} \\label{sec:42}\n\n\nIn this section, we show that A1-4+WF1-2+WIF3 is $\\omega$-complete\nin case $|A|=\\infty$. Note that this result was originally\nobtained in \\cite{VM01}. However, our proof is much simpler.\nFirst, let us note that A1-4+WF1-2+WIF3 $\\vdash$ D1, D2, D5.\n\\begin{lemma}\n  For any closed terms $p, q$, if $p\\sqsubseteq_{\\rm WIF} q$, then\n  \\mbox{\\rm A1-4+WF1-2+WIF3} $\\vdash p\\preccurlyeq q$.\n\\end{lemma}\n\n\\begin{proof} Let $p\\sqsubseteq_{\\rm WIF} q$. We prove $\\vdash\n  p\\preccurlyeq q$ by induction on $|p|+|q|$.  We distinguish two\n  cases:\n  \\begin{itemize}\n     \\item $q\\not\\!\\mv{\\tau}$. Then $p\\not\\!\\mv{\\tau}$ since $p\\sqsubseteq_{\\rm WIF} q$. Suppose\n      $p=\\sum_{i\\in I}a_ip_i$ and $q=\\sum_{j\\in J}b_j q_j$. Clearly,\n      we have $\\mc{I}(p)=\\mc{I}(q)$.\n    By D5, we have\n    \\[ \\vdash p\\approx \\sum_{a\\in \\mc{I}(p)} a(\\sum_{a_i=a, i\\in I}\\tau p_i)\\]\n    and\n    \\[ \\vdash q\\approx \\sum_{a\\in \\mc{I}(p)} a(\\sum_{b_j=a, j\\in J}\\tau q_j)\\]\n    Since $p\\sqsubseteq_{\\rm WIF} q$, for each $a\\in \\mc{I}(p)$, the following relation holds:\n    \\[  \\sum_{a_i=a, i\\in I}\\tau p_i \\sqsubseteq \\sum_{b_j=a, j\\in J} \\tau q_j\\]\n    By induction,\n    \\[ \\vdash \\sum_{a_i=a, i\\in I}\\tau p_i \\preccurlyeq \\sum_{b_j=a, j\\in J}\\tau q_j\\]\n    and thus\n    \\[ \\vdash a(\\sum_{a_i=a, i\\in I}\\tau p_i) \\preccurlyeq a(\\sum_{b_j=a, j\\in J}\\tau q_j)\\]\n    Summing these up for $a\\in \\mc{I}(p)$, we obtain that\n    \\[ \\vdash p\\preccurlyeq q\\]\n\n\\item $q\\mv{\\tau}$. By D2, we can write $ p\\approx\n\\sum_{i\\in I}\\alpha_ip_i$ and $q\\approx \\sum_{j\\in J} \\beta_jq_j$ such that\nfor each $\\alpha_i=\\tau$ (resp. $\\beta_j=\\tau$), $p_i\\not\\!\\mv{\\tau}$\n(resp. $q_j\\not\\!\\mv{\\tau}$).\nApplying D1, for each $i\\in I$ with $\\alpha_i=\\tau$, the summands of\n$p_i$ are also made summands of $p$, and likewise for $q$.\\hfill (*)\n\n\\medskip\n\nFor each $i\\in I$ with $\\alpha_i=\\tau$ we have $p\\mv{\\tau} p_i$. Since\n$p\\sqsubseteq_{\\rm WIF}q$ and no $q_j$ with $\\beta_j=\\tau$ contains a\n$\\tau$-summand, either $\\mc{T}(q)\\subseteq\\mc{T}(p_i)$ or there exists\n\\raisebox{0pt}[0pt][0pt]{$q\\mv{\\tau}q_j$} such that $\\mc{T}(q_j)\\subseteq \\mc{T}(p_i)$.\nSince \\raisebox{0pt}[0pt][0pt]{$q\\mv{\\tau}$}, in either case there exists some $j_i\\in J$ such\nthat $b_{j_i}=\\tau$ and  $\\mc{T}(q_{j_i})\\subseteq \\mc{T}(p_i)$. It follows that\n      \\[ p_i \\sqsubseteq_{\\rm WIF} p_i + q_{j_i}\\]\nSince $p_i \\not\\!\\mv{\\tau}$ and $q_{j_i} \\not\\!\\mv{\\tau}$, by the previous\ncase,\n      \\[ \\vdash p_i \\preccurlyeq  p_i + q_{j_i}\\]\nHence by WF2,\n\\[ \\vdash \\tau p_i \\preccurlyeq  \\tau (p_i + q_{j_i}) \\preccurlyeq p_i\n+\\tau q_{j_i} \\]\nand thus\n\\[ \\vdash p= \\sum_{\\alpha_i=\\tau} \\tau p_i + \\sum_{a\\in \\mc{I}(p)}\\sum_{\\alpha_i=a, i\\in I} ap_i\n\\preccurlyeq  \\sum_{\\alpha_i=\\tau} (p_i +\\tau q_{j_i}) + \\sum_{a\\in\n\\mc{I}(p)}\\sum_{\\alpha_i=a, i\\in I} ap_i\\]\nBy (*),\n\\[\\vdash \\sum_{\\alpha_i=\\tau} p_i + \\sum_{a\\in\n\\mc{I}(p)}\\sum_{\\alpha_i=a, i\\in I} ap_i \\approx \\sum_{a\\in\n\\mc{I}(p)}\\sum_{\\alpha_i=a, i\\in I} ap_i \\]\nSince $p\\sqsubseteq_{\\rm WIF} q$, $\\mc{I}(p)=\\mc{I}(q)$.\nUsing (*), it is easy to see that for each $a\\in \\mc{I}(p)$,\n\\[ \\sum_{\\alpha_i=a, i\\in I} ap_i \\sqsubseteq_{\\rm WIF}  \\sum_{\\beta_j=a,j\\in J} aq_j\\]\nSo by the previous case,\n\\[ \\vdash \\sum_{\\alpha_i=a, i\\in I} ap_i \\preccurlyeq  \\sum_{\\beta_j=a,j\\in J} aq_j\\]\nIt follows that\n\\[ \\vdash p \\preccurlyeq \\sum_{\\alpha_i=\\tau} \\tau q_{j_i} + \\sum_{a\\in\n\\mc{I}(p)}\\sum_{\\alpha_i=a, i\\in I} ap_i \\preccurlyeq \\sum_{\\alpha_i=\\tau}\n\\tau q_j + \\sum_{a\\in \\mc{I}(p)}\\sum_{\\beta_j=a,j\\in J} aq_j\\]\nBy WIF3,\n\\[\\vdash \\sum_{\\alpha_i=\\tau} \\tau q_j + \\sum_{a\\in\n\\mc{I}(p)}\\sum_{\\beta_j=a,j\\in J} aq_j \\preccurlyeq q \\]\nHence \\hfill $\\vdash p \\preccurlyeq q$\\hfill\\mbox{} \\qed\n  \\end{itemize}\n\\end{proof}\nWith this ground-completeness result at hand, it is\nstraightforward to apply the \\emph{inverted substitution}\ntechnique of Groote \\cite{Gro90} to derive (see also \\cite{CF08}):\n\\begin{theorem}\nIf $|A|=\\infty$, then \\mbox{\\rm A1-4+WF1-2+WIF3} is $\\omega$-complete\nfor $\\mathrm{BCCS}(A)$ modulo $\\sqsubseteq_{\\rm WIF}$.\n\\end{theorem}\n\\begin{proof}\nGiven an inequational axiomatization $E$ and open terms $t,u$ such\nthat $E \\vdash \\sigma(t) \\preccurlyeq \\sigma(u)$ for all closed\nsubstitutions $\\sigma$, the technique of inverted substitutions is a\nmethod to prove $E \\vdash t \\preccurlyeq u$. It does so by means of a\nclosed substitution $\\rho$ encoding open terms into closed terms, and\nan decoding operation $R$ that turns closed terms back into\nopen terms. By assumption we have $E \\vdash \\rho(t) \\preccurlyeq\n\\rho(u)$. The pair $(\\rho,R)$ should be chosen in such a way that, in\nessence, applying $R$ to all terms occurring in a proof of $\\rho(t)\n\\preccurlyeq \\rho(u)$ yields a proof of $t \\preccurlyeq u$.  As\nobserved in \\cite{Gro90}, this technique is applicable when three\nconditions are met, one of which being that $R(\\rho(t))=t$ and\n$R(\\rho(u))=u$. In fact, \\cite{Gro90} dealt with equational logic\nonly, but the very same reasoning applies to inequational logic.\n\nHere we use the same pair $(\\rho,R)$ that was used by Groote to obtain\nmost of the applications of the technique in \\cite{Gro90}---it could\nbe called the \\emph{default} (inverted) substitution. It is obtained\nby selecting for each variable $x\\in V$ an action $a_x\\in A$, not\noccurring in $t$ or $u$. This is possible because $|A|=\\infty$. Now\nthe default substitution $\\rho$ is given by $\\rho(x)=a_x\\mathbf{0}$ and the\ndefault inverted substitution $R$ replaces any maximal subterm of the\nform $a_x p$ into the variable $x$. Groote showed that with this\nparticular (inverted) substitution, 2 out of his 3 conditions are\nalways met, and the third one simply says that for each axiom $t\n\\preccurlyeq u$ in $E$ we should have that $E \\vdash R(t) \\preccurlyeq\nR(u)$.  This condition is clearly met for the axioms A1-4+WF1-2+WIF3,\nand hence this axiomatization is $\\omega$-complete.\\qed\n\\end{proof}\nNote that we could have used the same method to obtain\nTheo.~\\ref{theo:A-infinite}, but not Theo.~\\ref{theo:A-finite}.\n\n\\subsection{Nonexistence of a Finite Basis for Preorder if\n$|A|<\\infty$} \\label{sec:43}\n\n\\subsubsection{$1< |A| <\\infty$.}\nWe prove that the inequational theory of $\\mathrm{BCCS}(A)$ modulo\n$\\sqsubseteq_{\\rm WIF}$ does \\emph{not} have a finite basis in case of\na finite alphabet with at least two elements.\nThe cornerstone for this negative result is the following\ninfinite family of inequations, for $m\\geq 0$:\n\\[\\tau(a^mx) +  \\Phi_m ~\\preccurlyeq~ \\Phi_m \\]\nwith\n\\[\\Phi_m ~=~ \\tau(a^mx+x) + \\sum_{b\\in A} \\tau(a^mx+a^mb\\mathbf{0})\\]\nIt is not hard to see that these inequations are sound modulo\n$\\sqsubseteq_{\\rm WIF}$. Namely, given a closed substitution $\\rho$,\nwe have $\\mc{T}(\\rho(\\tau(a^mx))) \\subseteq \\mc{T}(\\rho(\\Phi_m))$ and\n$\\rho(\\Phi_m)\\mv{\\tau}$.\nTo argue that $\\rho(\\tau(a^mx) + \\Phi_m)$ and $\\rho(\\Phi_m)$ have the\nsame impossible futures, we only need to consider the\ntransition $\\rho(\\tau(a^mx)+\\Phi_m)\\mv{\\tau}a^m\\rho(x)$ (all other\ncases being trivial). If\n$\\rho(x)=\\mathbf{0}$, then $\\rho(\\Phi_m)\\mv{\\tau}a^m\\mathbf{0} + \\mathbf{0}$ generates\nthe same impossible futures $(\\varepsilon,B)$.  If, on the other hand,\n$b\\mathbin\\in\\mc{I}(\\rho(x))$ for some $b\\mathbin\\in A$, then\n$\\mc{T}(a^m\\rho(x)+a^mb\\mathbf{0})=\\mc{T}(a^m\\rho(x))$, so\n$\\rho(\\Phi_m)\\mv{\\tau} a^m\\rho(x)+a^mb\\mathbf{0}$ generates the same\nimpossible futures $(\\varepsilon,B)$.\n\nWe have already defined the traces and completed traces of closed\nterms. Now we extend these definitions to open terms by allowing\n(completed) traces of the form $a_1 \\cdots a_k x \\in A^\\ast V$.\nWe do this by treating each variable occurrence $x$ in a term as if it\nwere a subterm $x\\mathbf{0}$ with $x$ a visible action, and then apply\nDef.~\\ref{wif}. Under this convention,\n$\\mc{CT}(\\Phi_m) =\\{a^{m}x, x, a^{m}b \\mid b\\mathbin\\in A\\}$.\nWe write $\\mc{T}_V(t)$ for the set of traces of $t$ that end in a\nvariable, and $\\mc{T}_A(t)$ for ones that end in an action.\n\n\\begin{observation}\\label{traces-substitution}\nLet $m>|t|$ or $a_m \\in V$.\nThen $a_1\\cdots a_m \\in \\mc{T}(\\sigma(t))$ iff there is a $k < m$\nand $y\\mathbin\\in V$ such that $a_1\\cdots a_k y \\in \\mc{T}_V(t)$ and\n$a_{k+1} \\cdots a_m \\in \\mc{T}(\\sigma(y))$.\n\\end{observation}\n\n\\begin{lemma}\\label{trace-preservation}\nIf $|A|>1$ and $t \\sqsubseteq_{\\rm WIF} u$ then\n$\\mc{T}_A(t)= \\mc{T}_A(u)$ and $\\mc{T}_V(t)= \\mc{T}_V(u)$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\sigma$ be the closed substitution defined by\n$\\sigma(x)\\mathbin=\\mathbf{0}$ for all $x\\mathbin\\in V$.  Then\n$t\\sqsubseteq_{\\rm WIF} u$ implies $\\sigma(t)\\sqsubseteq_{\\rm WIF}\n\\sigma(u)$ and hence $\\mc{T}_A(t)=\\mc{T}(\\sigma(t))=\n\\mc{T}(\\sigma(u))=\\mc{T}_A(u)$ by Def.~\\ref{wif}.\n\nFor the second statement fix distinct actions $a,b\\mathbin\\in A$ and\nan injection $\\ulcorner \\cdot \\urcorner: V \\rightarrow\n\\mathbb{Z}_{>0}$ (which exists because $V$ is countable). Let\n$m=|u|+1=|t|+1$. Define the closed substitution $\\rho$ by\n$\\rho(z)=a^{\\ulcorner\\! z\\!\\urcorner{\\cdot}m}b\\mathbf{0}$ for all $z\\in V$.\nAgain, by Def.~\\ref{wif}, $t\\sqsubseteq_{\\rm WIF} u$ implies\n$\\mc{T}(\\rho(t))= \\mc{T}(\\rho(u))$. By Obs.~\\ref{traces-substitution},\nfor all terms $v$ we have $a_1\\cdots a_k y \\in \\mc{T}_V(v)$ iff $a_1\\cdots a_k\na^{\\ulcorner\\!  y\\!\\urcorner{\\cdot}m}b \\in \\mc{T}(\\rho(v))$ with\n$k\\mathbin<m$.  Hence $\\mc{T}_V(v)$ is completely determined by\n$\\mc{T}(\\rho(v))$ and thus $\\mc{T}_V(t) \\mathbin= \\mc{T}_V(u)$.\\qed\n\\end{proof}\n\n\\begin{lemma} \\label{newvariable3}\nLet $|A|>1$. Suppose $t\\sqsubseteq_{\\rm WIF} u$ and $t\\Rightarrow\\mv{\\tau} t'$.\nThen there is a term $u'$ such that $u\\Rightarrow\\mv{\\tau} u'$ and\n$\\mc{T}_V(u')\\subseteq \\mc{T}_V(t')$.\n\\end{lemma}\n\n\\begin{proof}\nDefine $\\rho$ exactly as in the previous proof. Since\n$\\rho(t)\\Rightarrow\\rho(t')$ and $t\\sqsubseteq_{\\rm WIF} u$ there must\nbe a $u'$ with $\\rho(u) \\Rightarrow q$ and $\\mc{T}(q)\\subseteq\n\\mc{T}(\\rho(t'))$. Since $\\rho(x)$ is $\\tau$-free for $x\\in V$ it must\nbe that $q=\\rho(u')$ for some term $u'$ with $u \\Rightarrow u'$.\nGiven the relationship between $\\mc{T}_V(v)$ and $\\mc{T}(\\rho(v))$ for\nterms $v$ observed in the previous proof, it follows that\n$\\mc{T}_V(u')\\subseteq\\mc{T}_V(t')$. In case $u \\Rightarrow\\mv{\\tau}\nu'$ we are done, so assume $u'=u$. Let $\\sigma$ be the substitution\nwith $\\sigma(x)=\\mathbf{0}$ for all $x\\in V$. Since $\\sigma(t) \\mv{\\tau}$\nand $t \\sqsubseteq_{\\rm WIF} u$ we have $\\sigma(u)\\mv{\\tau}$, so $u\n\\mv{\\tau} u''$ for some $u''$. Now $\\mc{T}_V(u'') \\subseteq \\mc{T}_V(u) =\n\\mc{T}_V(u') \\subseteq \\mc{T}_V(t')$.\n\\qed\n\\end{proof}\n\n\\begin{lemma} \\label{no-traces1}\n  Let $|A|>1$. Assume that, for some terms $t,u$, substitution\n  $\\sigma$, action $a$ and integer $m$:\n  \\begin{enumerate}\\vspace{-1ex}\n  \\item $t \\sqsubseteq_{\\rm WIF} u$;\n\n  \\item $m \\geq |u|$; and\n\n  \\item $\\sigma(t)\\Rightarrow\\mv{\\tau} \\hat t$ for a term $\\hat t$ without\n  traces $ax$ for $x\\mathbin\\in V$ or $a^mb$ for $b\\mathbin\\in A$.\n\\vspace{-1ex}\\end{enumerate}\nThen $\\sigma(u)\\Rightarrow\\mv{\\tau} \\hat u$ for a term $\\hat u$ without\n  traces $ax$ for $x\\mathbin\\in V$ or $a^mb$ for $b\\mathbin\\in A$.\n\\end{lemma}\n\\begin{proof}\nBased on proviso (3) there are two cases to consider.\\vspace{-1ex}\n\\begin{itemize}\n\\item $y \\in \\mc{T}_V(t)$ for some $y\\mathbin\\in V$ and\n  $\\sigma(y)\\Rightarrow\\mv{\\tau} \\hat t$.\n  In that case $y \\mathbin\\in \\mc{T}_V(u)$ by\n  Lem.~\\ref{trace-preservation}, so $\\sigma(u)\\Rightarrow\\mv{\\tau} \\hat t$.\n\\item $t \\Rightarrow\\mv{\\tau} t'$ for some term $t'$ such that $\\hat t\n  = \\sigma(t)$.  By Lem.~\\ref{newvariable3} there is a term $u'$ with\n  $u\\Rightarrow\\mv{\\tau} u'$ and $\\mc{T}_V(u')\\subseteq \\mc{T}_V(t')$.\n  Take $\\hat u = \\sigma(u')$. Clearly $\\sigma(u)\\Rightarrow\\mv{\\tau}\n  \\sigma(u')$.  Suppose $\\sigma(u')$ would have a trace $a^mb$. Then,\n  by Obs.~\\ref{traces-substitution}, there is a $k \\leq m$ and\n  $y\\mathbin\\in V$ such that $a^k y \\in \\mc{T}_V(u')$ and\n  $a^{m-k}b \\in \\mc{T}(\\sigma(y))$. Since $\\mc{T}_V(u')\\subseteq\n  \\mc{T}_V(t')$ we have $a^mb \\in \\mc{T}(\\sigma(t'))$, which is a\n  contradiction. The case $ax \\in \\mc{T}(\\sigma(u))$ is dealt with in\n  the same way.\\qed\n\\end{itemize}\n\\end{proof}\n\n\\begin{lemma} \\label{no-traces2}\n  Let $|A|>1$ and let $E$ be an axiomatization sound for\n  $\\sqsubseteq_{\\rm WIF}$. Assume that, for some terms $v,w$, action\n  $a$ and integer $m$:\n  \\begin{enumerate}\\vspace{-1ex}\n  \\item $E\\vdash  v \\preccurlyeq w$;\n\n  \\item $m \\geq \\max \\{|u| \\mid t \\preccurlyeq u \\in  E\\}$; and\n\n  \\item $v\\Rightarrow\\mv{\\tau} \\hat v$ for a term $\\hat v$ without\n  traces $ax$ for $x\\mathbin\\in V$ or $a^mb$ for $b\\mathbin\\in A$.\n\\vspace{-1ex}\\end{enumerate}\nThen $w\\Rightarrow\\mv{\\tau}  \\hat w$ for a term $\\hat w$ without\n  traces $ax$ for $x\\mathbin\\in V$ or $a^mb$ for $b\\mathbin\\in A$.\n\\end{lemma}\n\n\\begin{proof}\nBy induction on the derivation of $E\\vdash v \\preccurlyeq w$.\n\\begin{itemize}\n\n\\item Suppose $E\\vdash v\\preccurlyeq w$ because $\\sigma(t)=v$ and\n$\\sigma(u)=w$ for some $t\\preccurlyeq u\\in E$ and substitution\n$\\sigma$. The claim then follows by Lem.~\\ref{no-traces1}.\n\n\\medskip\n\n\\item Suppose $E\\vdash v\\preccurlyeq w$ because $E\\vdash v\\preccurlyeq u$\nand $E\\vdash u\\preccurlyeq w$ for some $u$. By induction, $u\n\\Rightarrow\\mv{\\tau} \\hat u$ for a term $\\hat u$ without traces $ax$ or\n$a^mb$. Hence, again by induction, \\mbox{$w\\Rightarrow\\mv{\\tau} \\hat w$}\nfor a term $\\hat w$ without traces $ax$ or $a^mb$.\n\n\\medskip\n\n\\item Suppose $E\\vdash v\\preccurlyeq w$ because $v=v_1+v_2$ and\n$w=w_1+w_2$ with $E\\vdash v_1 \\preccurlyeq w_1$ and $E\\vdash\nv_2\\preccurlyeq w_2$. Since $v \\Rightarrow\\mv{\\tau} \\hat v$, either\n$v_1 \\Rightarrow\\mv{\\tau} \\hat v$ or $v_2 \\Rightarrow\\mv{\\tau} \\hat v$.\nAssume, without loss of generality, that $v_1 \\Rightarrow\\mv{\\tau} \\hat v$.\nBy induction, $w_1 \\Rightarrow\\mv{\\tau} \\hat w$ for a term $\\hat w$ without\ntraces $ax$ or $a^mb$. Now $w \\Rightarrow\\mv{\\tau} \\hat w$.\n\n\\medskip\n\n\\item Suppose $E\\vdash v \\preccurlyeq w$ because $v=cv_1$ and $w=cw_1$ with\n$c\\in A$ and $E\\vdash v_1\\approx w_1$. In this case, proviso (3) of\nthe lemma can not be met.\n\n\\medskip\n\n\\item Suppose $E\\vdash v \\preccurlyeq w$ because $v=\\tau v_1$ and\n$w=\\tau w_1$ with $E\\vdash v_1\\approx w_1$. Then either $v_1=\\hat\nv$ or $v_1 \\Rightarrow\\mv{\\tau} \\hat v$. In the first case, $w_1$\nhas no traces $ax$ or $a^{m}b$ by Lem.~\\ref{trace-preservation}\nand proviso (3) of the lemma; hence $w$ has no such traces either.\nIn the second case, by induction, $w_1\\Rightarrow\\mv{\\tau} \\hat w$\nfor a term $\\hat w$ without traces $ax$ or $a^mb$. Again\n$w\\Rightarrow\\mv{\\tau} \\hat w$. \\qed\n\\end{itemize}\n\\end{proof}\n\n\\begin{theorem} \\label{thm:alphabetn}\nIf $1<|A|<\\infty$, then the inequational theory of $\\mathrm{BCCS}(A)$\nmodulo $\\sqsubseteq_{\\rm WIF}$ does not have a finite basis.\n\\end{theorem}\n\n\\begin{proof}\nLet $E$ be a finite axiomatization over $\\mathrm{BCCS}(A)$ that is\nsound modulo $\\sqsubseteq_{\\rm WIF}$. Let $m$ be greater than the\ndepth of any term in $E$.  According to Lem.~\\ref{no-traces2}, the\ninequation $\\tau(a^mx) + \\Phi_m \\preccurlyeq \\Phi_m$ cannot be derived\nfrom $E$.  Yet it is sound modulo $\\sqsubseteq_{\\rm WIF}$.\n\\qed\n\\end{proof}\n\n\n\\subsubsection{$|A|=1$.} We prove that the inequational theory of\n$\\mathrm{BCCS}(A)$ modulo $\\sqsubseteq_{\\rm WIF}$ does \\emph{not} have\na finite basis in case of a singleton alphabet. The cornerstone\nfor this negative result is the following infinite family of\ninequations, for $m\\geq 0$:\n\\[ a^m x ~\\preccurlyeq~  a^m x + x\\]\nIf $|A|=1$, then these inequations are clearly sound modulo\n$\\sqsubseteq_{\\rm WIF}$. Note that given a closed substitution\n$\\rho$, $\\mc{T}(\\rho(x))\\subseteq \\mc{T}(\\rho(a^m x))$.\n\n\\begin{lemma}\\label{trace-preservation2}\nIf $t \\sqsubseteq_{\\rm WIF} u$ then\n$\\mc{T}_V(t) \\subseteq \\mc{T}_V(u)$.\n\\end{lemma}\n\n\\begin{proof}\nFix $a\\in A$ and an injection $\\ulcorner \\cdot \\urcorner: V\n\\rightarrow \\mathbb{Z}_{>0}$. Let $m=|u|+1$. Define the closed\nsubstitution $\\rho$ by $\\rho(z)=a^{\\ulcorner\\!\nz\\!\\urcorner{\\cdot}m}\\mathbf{0}$ for all $z\\in V$.  By Lem.~\\ref{res},\n$\\mc{CT}(\\rho(t)) \\subseteq \\mc{CT}(\\rho(u))$.  Now suppose\n$a_1\\cdots a_k y \\in \\mc{T}_V(t)$. Then $a_1\\cdots a_k a^{\\ulcorner\\!\ny\\!\\urcorner{\\cdot}m} \\in \\mc{CT}(\\rho(t)) \\subseteq\n\\mc{CT}(\\rho(u))$ and $k<m$. This is only possible if $a_1\\cdots a_k y\n\\in \\mc{T}_V(u)$. \\qed\n\\end{proof}\n\n\\begin{lemma} \\label{1}\n Assume that, for terms $t,u$, substitution $\\sigma$, action $a$,\n variable $x$, integer $m$:\n  \\begin{enumerate}\\vspace{-1ex}\n  \\item $t \\sqsubseteq_{\\rm WIF} u$;\n\n  \\item $m>|u|$; and\n\n  \\item $x \\in \\mc{T}_V(\\sigma(u))$ and\n  $a^k x \\not\\in \\mc{T}_V(\\sigma(u))$ for $1\\leq k <m$.\n\n\\vspace{-1ex}\\end{enumerate}\nThen $x \\in \\mc{T}_V(\\sigma(t))$ and\n$a^k x \\not\\in \\mc{T}_V(\\sigma(t))$ for $1\\leq k <m$.\n\\end{lemma}\n\n\\begin{proof}\n\nSince $x\\in \\mc{T}_V(\\sigma(u))$, by\nObs.~\\ref{traces-substitution} there is a variable $y$ with $y\\in\n\\mc{T}_V(u)$ and $x \\in \\mc{T}_V(\\sigma(y))$. Consider the closed\nsubstitution $\\rho$ given by $\\rho(y)=a^m\\mathbf{0}$ and $\\rho(z)=\\mathbf{0}$\nfor $z\\neq y$. Then $m>|u|=|t|$, and $y\\in \\mc{T}_V(u)$ implies\n$a^m \\in \\mc{T}(\\rho(u)) = \\mc{T}(\\rho(t))$, so by\nObs.~\\ref{traces-substitution} there is some $k < m$ and\n$z\\mathbin\\in V$ such that $a^k z \\in \\mc{T}_V(t)$ and $a^{m-k}\n\\in \\mc{T}(\\rho(z))$. As $k<m$ it must be that $z=y$. Since\n$a^ky\\in \\mc{T}_V(t)$ and $x\\in \\mc{T}_V(\\sigma(y))$,\nObs.~\\ref{traces-substitution} implies that $a^k x \\in\n\\mc{T}_V(\\sigma(t))$. By Lem.~\\ref{trace-preservation2}, $a^k x\n\\not\\in \\mc{T}_V(\\sigma(t))$ for $1\\leq k < m$. Hence we obtain\n$k=0$.\\qed\n\\end{proof}\n\n\\begin{lemma} \\label{lemmaalphabet1}\nAssume that, for $E$ an axiomatization sound for $\\sqsubseteq_{\\rm\nWIF}$ and for terms $v,w$, action $a$, variable $x$ and integer $m$:\n \\begin{enumerate}\\vspace{-1ex}\n  \\item $E\\vdash  v \\preccurlyeq w$;\n\n  \\item $m > \\max \\{ |u| \\mid t \\preccurlyeq u \\in  E\\}$; and\n\n  \\item $x \\in \\mc{T}_V(w)$ and\n  $a^k x \\not\\in \\mc{T}_V(w)$ for $1\\leq k < m$.\n\\vspace{-1ex}\\end{enumerate}\nThen $x \\in \\mc{T}_V(v)$ and\n$a^k x \\not\\in \\mc{T}_V(v)$ for $1\\leq k < m$.\n\\end{lemma}\n\n\\begin{proof}\nBy induction on the derivation of $E\\vdash v \\preccurlyeq w$.\n\\begin{itemize}\n\n\\item Suppose $E\\vdash v\\preccurlyeq w$ because $\\sigma(t)=v$ and\n$\\sigma(u)=w$ for some $t\\preccurlyeq u\\in E$ and substitution\n$\\sigma$. The claim then follows by Lem.~\\ref{1}.\n\n\\medskip\n\n\\item Suppose $E\\vdash v\\preccurlyeq w$ because $E\\vdash v\\preccurlyeq\nu$ and $E\\vdash u\\preccurlyeq w$ for some $u$. By induction, $x \\in\n\\mc{T}_V(u)$ and $a^k x \\not\\in \\mc{T}_V(u)$ for $1\\leq k < m$.\nHence, again by induction, $x \\in \\mc{T}_V(v)$ and\n$a^k x \\not\\in \\mc{T}_V(v)$ for $1\\leq k < m$.\n\n\\medskip\n\n\\item Suppose $E\\vdash v\\preccurlyeq w$ because $v=v_1+v_2$ and\n$w=w_1+w_2$ with $E\\vdash v_1 \\preccurlyeq w_1$ and $E\\vdash\nv_2\\preccurlyeq w_2$. Since $x \\in \\mc{T}_V(w)$, either $x \\in\n\\mc{T}_V(w_1)$ or $x \\in \\mc{T}_V(w_2)$.  Assume, without loss of\ngenerality, that $x \\in \\mc{T}_V(w_1)$.  Since $a^k x \\not\\in\n\\mc{T}_V(w)$ for $1\\leq k < m$, surely $a^k x \\not\\in \\mc{T}_V(w_1)$\nfor $1\\leq k < m$.  By induction, $x \\in \\mc{T}_V(v_1)$, and hence $x\n\\in \\mc{T}_V(v)$.  For $1\\leq k < m$ we have $a^k x \\not\\in\n\\mc{T}_V(w)$ and hence $a^k x \\not\\in \\mc{T}_V(v)$, by\nLem.~\\ref{trace-preservation2}.\n\n\\medskip\n\n\\item Suppose $E\\vdash v \\preccurlyeq w$ because $v=cv_1$ and $w=cw_1$ with\n$c\\in A$ and $E\\vdash v_1\\approx w_1$. In this case, proviso (3) of\nthe lemma can not be met.\n\n\\medskip\n\n\\item Suppose $E\\vdash v \\preccurlyeq w$ because $v=\\tau v_1$ and\n$w=\\tau w_1$ with $E\\vdash v_1\\approx w_1$. Then, by proviso (3) of\nthe lemma, $x \\in \\mc{T}_V(w_1)$ and $a^k x \\not\\in \\mc{T}_V(w_1)$ for\n$1\\leq k < m$. By induction, $x \\in \\mc{T}_V(v_1)$ and $a^k x \\not\\in\n\\mc{T}_V(v_1)$ for $1\\leq k < m$.  Hence $x \\in \\mc{T}_V(v)$ and $a^k\nx \\not\\in \\mc{T}_V(v)$ for $1\\leq k < m$.  \\qed\n\\end{itemize}\n\n\\end{proof}\n\n\\begin{theorem}\\label{thm:alphabet1}\nIf $|A|=1$, then the inequational theory of $\\mathrm{BCCS}(A)$\nmodulo $\\sqsubseteq_{\\rm WIF}$ does not have a finite basis.\n\\end{theorem}\n\n\\begin{proof}\nLet $E$ be a finite axiomatization over $\\mathrm{BCCS}(A)$ that is\nsound modulo $\\sqsubseteq_{\\rm WIF}$. Let $m$ be greater than the\ndepth of any term in $E$.  According to Lem.~\\ref{lemmaalphabet1}, the\ninequation $a^mx \\preccurlyeq a^m x + x$ cannot be derived\nfrom $E$.  Yet, since $|A|=1$, it is sound modulo $\\sqsubseteq_{\\rm WIF}$.\n\\qed\n\\end{proof}\nTo conclude this subsection, we have\n\n\\begin{theorem}\nIf $|A|<\\infty$, then the inequational theory of $\\mathrm{BCCS}(A)$\nmodulo $\\sqsubseteq_{\\rm WIF}$ does not have a finite basis.\n\\end{theorem}\nConcluding, in spite of the close resemblance between weak\nfailures and weak impossible futures semantics, there is a\nstriking difference between their axiomatizability properties.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\subsubsection*{Classical backround}\nThe quantization of systems with constraints is of considerable importance in a variety of applications. \nLet $\\{p_j,q^j\\}$, $1\\leq j\\leq J$, denote a set of dynamical variables, $\\{\\lambda^a\\}$, $1\\leq a\\leq A\\leq 2J$, a set of Lagrange multipliers, and $\\{\\phi_a(p,q)\\}$ a set of constraints. Then the dynamics of a constrained system may be summarized in the form of an action principle by means of the classical action (summation implied)\n  \\bn I={\\textstyle{\\int}}[p_j{\\dot q}^j-H(p,q)-\\lambda^a\\phi_a(p,q)]\\,dt\\;.  \\en\n The resultant equations that arise from the action read\n  \\bn  &&{\\dot q}^j=\\frac{\\partial H(p,q)}{\\partial p_j}+\\lambda^a\\frac{\\partial\\phi_a(p,q)}{\\partial p_j}\\equiv\\{q^j,H\\}+\\lambda^a\\{q^j,\\phi_a\\}\\;,\\nonumber\\\\\n  &&{\\dot p}_j=-\\frac{\\partial H(p,q)}{\\partial q^j}-\\lambda^a\\frac{\\partial\\phi_a(p,q)}{\\partial q^j}\\equiv\\{p_j,H\\}+\\lambda^a\\{p_j,\\phi_a\\}\\;,\\nonumber\\\\\n  &&\\phi_a(p,q)=0\\;,  \\en\nwhere $\\{\\cdot,\\cdot\\}$ denotes the Poisson bracket. The set of conditions $\\{\\phi_a(p,q)=0\\}$ define the {\\it constraint hypersurface}. If the constraints satisfy \n\\bn &&\\{\\phi_a(p,q),\\phi_b(p,q)\\}=c_{ab}^{\\;\\;\\;\\;c}\\,\\phi_c(p,q)\\;,\\\\\n  &&\\{\\phi_a(p,q),H(p,q)\\}=h_a^{\\;\\;b}\\,\\phi_b(p,q)\\;,  \\en\nthen we are dealing with a system of first-class constraints. If the coefficients $c_{ab}^{\\;\\;\\;\\;c}$ and $h_a^{\\;\\;b}$ are constants, then it is a closed system of first-class constraints; if they are suitable functions of the variables $p,q$, then it is called an open first-class constraint system. If one or both of the conditions in (3) or (4) fails, then the constraints are said to be second class. \n\nFor first-class constraints it is sufficient to impose the constraints at the initial time inasmuch as the equations of motion will ensure that the constraints are fulfilled at all future times. Such an initial imposition of the constraints is called an {\\it initial value equation}. Furthermore, the Lagrange multipliers are not determined by the equations of motion; rather they must be specified (a choice of ``gauge'') in order for a solution of the dynamical equations to be given. For second-class constraints, on the other hand, the Lagrange multipliers are determined by the equations of motion in such a way that the constraints are satisfied for all time. \n\nIn the remainder of this section we review standard quantization procedures for systems with closed first-class constraints, both of the operator and path integral variety, pointing out some problems in each approach. In the following two sections we develop our coherent state approach, first for closed first-class constraints, and second for general constraints, i.e., open first-class constraints as well as second-class constraints \\cite{kla}.\n\nDue to space limitations, we are only able to offer here a few examples; the reader may consult \\cite{kla} for a discussion of additional examples.\n\\subsubsection*{Standard operator quantization}\nFor a system of closed first-class constraints we assume (with $\\hbar=1$) that \n\\bn &&[\\Phi_a(P,Q),\\Phi_b(P,Q)]=ic_{ab}^{\\;\\;\\;\\;c}\\,\\Phi_c(P,Q)\\;,\\\\\n  &&[\\Phi_a(P,Q),{\\cal H}(P,Q)]=ih_a^{\\;\\;b}\\,\\Phi_b(P,Q)\\;,  \\en\nwhere $\\Phi_a$ and ${\\cal H}$ denote self-adjoint constraint and Hamiltonian operators, respectively. Following Dirac \\cite{dir}, we adopt the quantization prescription given by\n  \\bn i{\\dot W}(P,Q)=[W(P,Q),{\\cal H}(P,Q)]  \\en\nwhere $W$ denotes any function of the kinematical operators $\\{Q^j\\}$ and $\\{P_j\\}$ which are taken as a self-adjoint, irreducible representation of the commutation rules $[Q^j,P_k]=i\\delta^j_k\\one$, with all other commutators vanishing. The equations of motion hold for all time $t$, say $0<t<T$. On the other hand, the conditions\n  \\bn  \\Phi_a(P,Q)|\\psi\\>_{\\rm phys}=0  \\en\nto select the physical Hilbert space are imposed only at time $t=0$ as the analog of the initial value equation; the quantum equations of motion ensure that the constraint conditions are fulfilled for all time.\n\nThe procedure of Dirac has potential difficulties if zero lies in the continuous spectrum of the constraint operators for in that case there are no normalizable solutions of the constraint condition. We face the same problem, of course, and our resolution is discussed in detail in Ref.~\\cite{kla}.\n\\subsubsection*{Standard path integral quantization}\nFaddeev \\cite{fad} has given a path integral formulation in the case of closed first-class constraint systems as follows. The formal path integral\n \\bn &&\\int\\exp\\{i{\\textstyle{\\int}}_0^T[p_j{\\dot q}^j-H(p,q)-\\lambda^a\\phi_a(p,q)]\\,dt\\}\\,{\\cal D} p\\,{\\cal D} q\\,{\\cal D}\\lambda \\nonumber\\\\\n  &&\\hskip1.5cm=\\int\\exp\\{i{\\textstyle{\\int}}_0^T[p_j{\\dot q}^j-H(p,q)]\\,dt\\}\\,\\delta\\{\\phi(p,q)\\}\\,{\\cal D} p\\,{\\cal D} q \\en\nmay well encounter divergences in the remaining integrals. Therefore, subsidiary conditions in the form $\\chi^a(p,q)=0$, $1\\leq a\\leq A$, are imposed picking out (ideally) one gauge equivalent point per gauge orbit, and in addition a  factor (in the form of a determinant) is introduced to formally preserve canonical covariance. The result is the path integral \n \\bn  \\int\\exp\\{i{\\textstyle{\\int}}_0^T[p_j{\\dot q}^j-H(p,q)]\\,dt\\}\\,\\delta\\{\\chi(p,q)\\} \\det(\\{\\chi^a,\\phi_b\\})\\delta\\{\\phi(p,q)\\}\\,{\\cal D} p\\,{\\cal D} q\\,.  \\en\nThis result may also be expressed as\n \\bn \\int\\exp\\{i{\\textstyle{\\int}}_0^T[p^*_j{\\dot q}^{*j}-H^*(p^*,q^*)]\\,dt\\}\\,{\\cal D} p^*\\,{\\cal D} q^*\\;,  \\en\nnamely, as a path integral\nover a reduced phase space in which the $\\delta$-functionals have been used to eliminate $2A$ integration variables. \n\nThe final expression generally involves curvilinear phase-space coordinates for which the definition of the path integral is typically ill defined. Additionally, in the form (10), the Faddeev-Popov determinant often suffers from ambiguities connected with inadmissible gauge fixing conditions \\cite{gri}. Thus this widely used prescription is not without its difficulties.\n\\section{Coherent State Path Integral}\nCanonical coherent states may be defined by the relation\n \\bn  |p,q\\>\\equiv e^{-iq^jP_j}\\,e^{ip_jQ^j}\\,|0\\>\\;,  \\en\nwhere $|0\\>$ traditionally denotes a normalized, unit frequency, harmonic oscillator ground state. Here $\\{Q^j\\}$ and $\\{P_j\\}$, $1\\leq j\\leq J$, denote an irreducible set of self-adjoint operators satisfying the Heisenberg commutation relations. The coherent states admit a resolution of unity in the form\n \\bn  \\one={\\textstyle{\\int}}\\,|p,q\\>\\<p,q|\\,d\\mu(p,q)\\;,\\hskip1.5cm d\\mu(p,q)\\equiv d^J\\!p\\,d^J\\!q\/(2\\pi)^J\\;,  \\en\nwhere the integration is over ${\\rm I}\\hskip-.2em{\\rm R}^{2J}$ and this integration domain and the form of the measure are unique. For a general operator ${\\cal H}(P,Q)$ we introduce the upper symbol\n   \\bn  H(p,q)\\equiv\\<p,q|{\\cal H}(P,Q)|p,q\\>=\\<p,q|:H(P,Q):|p,q\\> \\;  \\en\nwhich is related to the normal-ordered form as shown. If $\\cal H$ denotes the quantum Hamiltonian, then we shall adopt $H(p,q)$ as the classical Hamiltonian. We also note that an important one-form is given by\n$i\\<p,q|d|p,q\\>=p_j\\,dq^j$. \n\nUsing these quantities, the coherent state path integral for the  time-dependent Hamiltonian ${\\cal H}(P,Q)+\\lambda^a(t)\\Phi_a(P,Q)$ is readily given by\n\\bn  &&\\<p'',q''|{\\sf T}e^{-i{\\textstyle{\\int}}_0^T[{\\cal H}(P,Q)+\\lambda^a(t)\\Phi_a(P,Q)]\\,dt}|p',q'\\>\\nonumber\\\\\n&&\\hskip.6cm=\\lim_{\\epsilon\\ra0}\\int\\prod_{l=0}^N\\<p_{l+1},q_{l+1}|e^{-i\\epsilon({\\cal H}+\\lambda^a_l\\Phi_a)}\\,|p_l,q_l\\>\\prod_{l=1}^N\\,d\\mu(p_l,q_l)\\nonumber\\\\\n&&\\hskip.6cm=\n \\int\\exp\\{i{\\textstyle{\\int}}[i\\<p,q|(d\/dt)|p,q\\>-\\<p,q|{\\cal H}+\\lambda^a\\Phi_a|p,q\\>]\\,dt\\}\\,{\\cal D}\\mu(p,q)\\nonumber\\\\\n&&\\hskip.6cm= {\\cal M}\\int\\exp\\{i{\\textstyle{\\int}}[p_j{\\dot q}^j-H(p,q)-\\lambda^a\\phi_a(p,q)]\\,dt\\}\\,{\\cal D} p\\,{\\cal D} q \\;.  \\en\nIn the second line we have set $p_{N+1},q_{N+1}=p'',q''$ and $p_0,q_0=p',q'$, and repeatedly inserted the resolution of unity; in the third and fourth lines we have formally interchanged the continuum limit and the integrations, and written for the integrand the form it assumes for continuous and differential paths ($\\cal M$ denotes a formal normalization constant). The result evidently depends on the chosen form of the functions $\\{\\lambda^a(t)\\}$. \n\\subsubsection*{Enforcing the quantum constraints}\nLet us next introduce the quantum analog of the initial value equation. For simplicity we assume that the constraint operators form a compact group; the case of a noncompact group is dealt with in \\cite{kla}. In that case \n  \\bn {\\rm E}\\hskip-.55em{\\rm I}\\,\\equiv{\\textstyle{\\int}} e^{-i\\xi^a\\Phi_a(P,Q)}\\,\\delta\\xi  \\en\ndefines a {\\it projection operator} onto the subspace for which $\\Phi_a=0$ provided that $\\delta\\xi$ denotes the normalized, ${\\textstyle{\\int}}\\delta\\xi=1$, group invariant measure. Based on (5) and (6) it follows that\n  \\bn && \\hskip1.5cm e^{-i\\tau^a\\Phi_a}{\\rm E}\\hskip-.55em{\\rm I}\\,={\\rm E}\\hskip-.55em{\\rm I}\\,\\;,  \\\\\n      && e^{-i{\\cal H}T}{\\rm E}\\hskip-.55em{\\rm I}\\,={\\rm E}\\hskip-.55em{\\rm I}\\,e^{-i{\\cal H}T}{\\rm E}\\hskip-.55em{\\rm I}\\,={\\rm E}\\hskip-.55em{\\rm I}\\,e^{-i({\\rm E}\\hskip-.55em{\\rm I}\\,{\\cal H}{\\rm E}\\hskip-.55em{\\rm I}\\,)T}{\\rm E}\\hskip-.55em{\\rm I}\\,\\;.  \\en\nWe now project the propagator (15) onto the quantum constraint subspace \nwhich leads to the following set of relations\n \\bn &&\\hskip-1cm\\int\\<p'',q''|{\\sf T}e^{-i{\\textstyle{\\int}}[{\\cal H}+\\lambda^a(t)\\Phi_a]\\,dt}\\,|{\\overline p}',{\\overline q}'\\>\\<{\\overline p}',{\\overline q}'|{\\rm E}\\hskip-.55em{\\rm I}\\,|p',q'\\>\\,d\\mu({\\overline p}',{\\overline q}')\\nonumber\\\\\n&&=\\<p'',q''|{\\sf T}e^{-i{\\textstyle{\\int}}[{\\cal H}+\\lambda^a(t)\\Phi_a]\\,dt}\\,{\\rm E}\\hskip-.55em{\\rm I}\\,|p',q'\\>\\nonumber\\\\\n&&=\\lim\\,\\<p'',q''|[\\prod^{\\leftarrow}_l(e^{-i\\epsilon{\\cal H}}e^{-i\\epsilon\\lambda^a_l\\Phi_a})]\\,{\\rm E}\\hskip-.55em{\\rm I}\\,|p',q'\\>\\nonumber\\\\\n&&=\\<p'',q''|e^{-iT{\\cal H}}e^{-i\\tau^a\\Phi_a}\\,{\\rm E}\\hskip-.55em{\\rm I}\\,|p',q'\\>\\nonumber\\\\\n&&=\\<p'',q''|e^{-iT{\\cal H}}\\,{\\rm E}\\hskip-.55em{\\rm I}\\,|p',q'\\>\\;,  \\en\nwhere $\\tau^a$ incorporates the functions $\\lambda^a$ as well as the structure parameters $c_{ab}^{\\;\\;\\;\\;c}$ and $h_a^{\\;\\;b}$.\nAlternatively, this expression has the formal path integral representation\n \\bn \\int\\exp\\{i{\\textstyle{\\int}}[p_j{\\dot q}^j-H(p,q)-\\lambda^a\\phi_a(p,q)]\\,dt-i\\xi^a\\phi_a(p',q')\\}\\,{\\cal D}\\mu(p,q)\\,\\delta\\xi\\;.  \\en\nOn comparing (19) and (20) we observe that {\\it after projection onto the quantum constraint subspace the propagator is entirely independent of the choice of the Lagrange mutiplier functions. In other words, the projected propagator is gauge invariant.} \n\nWe may also express the physical (projected) propagator in a more general form, namely,\n\\bn &&\\hskip-1cm\\int\\exp\\{i{\\textstyle{\\int}}[p_j{\\dot q}^j-H(p,q)-\\lambda^a\\phi_a(p,q)]\\,dt\\}\\,{\\cal D}\\mu(p,q)\\,{\\cal D} C(\\lambda)\\nonumber\\\\\n&&\\hskip.1cm=\\<p'',q''|e^{-iT{\\cal H}}\\,{\\rm E}\\hskip-.55em{\\rm I}\\,|p',q'\\>  \\en \nprovided that ${\\textstyle{\\int}}{\\cal D} C(\\lambda)=1$ and that such an average over the functions $\\{\\lambda^a\\}$ introduces (at least) one factor ${\\rm E}\\hskip-.55em{\\rm I}\\,$. \n\n\\section{Application to General Constraints}\n\\subsubsection*{Classical considerations}\nWhen dealing with a general constraint situation it will typically happen that the self-consistency of the equations of motion will determine some or all of the Lagrange multipliers in order for the system to remain on the classical constraint hypersurface. For example, if the Hamiltonian attempts to force points initially lying on the constraint hypersurface to leave that hypersurface, then the Lagrange multipliers must supply the necessary forces for the system to remain on the constraint hypersurface. \n\\subsubsection*{Quantum considerations}\nAs in the previous section we let ${\\rm E}\\hskip-.55em{\\rm I}\\,$ denote the projection operator onto the quantum constraint subspace. Motivated by the classical comments given above we consider the quantity\n\\bn \\lim\\,\\<p'',q''|{\\rm E}\\hskip-.55em{\\rm I}\\,e^{-i\\epsilon{\\cal H}}{\\rm E}\\hskip-.55em{\\rm I}\\,e^{-i\\epsilon{\\cal H}}\\cdots{\\rm E}\\hskip-.55em{\\rm I}\\,e^{-i\\epsilon{\\cal H}}{\\rm E}\\hskip-.55em{\\rm I}\\,|p',q'\\> \\en\nwhere the limit, as usual, is for $\\epsilon\\ra0$. The physics behind this expression is as follows. Reading from right to left we first impose the quantum initial value equation, and then propagate for a small amount of time ($\\epsilon$). Next we recognize that the system may have left the quantum constraint subspace, and so we project it back onto that subspace, and so on over and over. In the limit that $\\epsilon\\ra0$ the system remains within the quantum constraint subspace and (22) actually leads to\n \\bn \\<p'',q''|{\\rm E}\\hskip-.55em{\\rm I}\\,e^{-iT({\\rm E}\\hskip-.55em{\\rm I}\\;{\\cal H}{\\rm E}\\hskip-.55em{\\rm I}\\;)}{\\rm E}\\hskip-.55em{\\rm I}\\,|p',q'\\>\\;, \\en\nwhich clearly illustrates temporal evolution entirely within the quantum constraint subspace. If we assume that ${\\rm E}\\hskip-.55em{\\rm I}\\,{\\cal H}{\\rm E}\\hskip-.55em{\\rm I}\\,$ is a self-adjoint operator, then we conclude that (23) describes a unitary time evolution within the quantum constraint subspace. \n\nThe expression (22) may be developed in two additional ways. First, we repeatedly insert the resolution of unity in such a way that (22) becomes\n \\bn \\lim\\,\\int\\prod_{l=0}^N\\<p_{l+1},q_{l+1}|{\\rm E}\\hskip-.55em{\\rm I}\\,e^{-i\\epsilon{\\cal H}}{\\rm E}\\hskip-.55em{\\rm I}\\,|p_l,q_l\\>\\prod_{l=1}^Nd\\mu(p_l,q_l)\\;. \\en\nWe wish to turn this expression into a formal path integral, but the procedure used previously relied on the use of unit vectors, and the vectors ${\\rm E}\\hskip-.55em{\\rm I}\\,|p,q\\>$ are generally not unit vectors. Thus let us rescale the factors in the integrand introducing \\bn  |p,q\\>\\!\\>\\equiv{\\rm E}\\hskip-.55em{\\rm I}\\,|p,q\\>\/\\|{\\rm E}\\hskip-.55em{\\rm I}\\,|p,q\\>\\| \n \\en which are unit vectors. If we let $M''=\\|{\\rm E}\\hskip-.55em{\\rm I}\\,|p'',q''\\>\\|$, $M'=\\|{\\rm E}\\hskip-.55em{\\rm I}\\,|p',q'\\>\\|$, and observe that $\\|{\\rm E}\\hskip-.55em{\\rm I}\\,|p,q\\>\\|^2=\\<p,q|{\\rm E}\\hskip-.55em{\\rm I}\\,|p,q\\>$, it follows that (24) may be rewritten as\n \\bn M''M'\\lim\\,\\int\\prod_{l=0}^N\\<\\!\\<p_{l+1},q_{l+1}|e^{-i\\epsilon{\\cal H}}|p_l,q_l\\>\\!\\>\\prod_{l=1}^N\\<p_l,q_l|{\\rm E}\\hskip-.55em{\\rm I}\\,|p_l,q_l\\>\\,d\\mu(p_l,q_l)\\;. \\en\nThis expression is represented by the formal path integral\n\\bn M''M'\\int\\exp\\{i{\\textstyle{\\int}}[i\\<\\!\\<p,q|(d\/dt)|p,q\\>\\!\\>-\\<\\!\\<p,q|{\\cal H}|p,q\\>\\!\\>]\\,dt\\}\\,{\\cal D}_E\\mu(p,q)\\;,  \\en\nwhere the new formal measure for the path integral is defined in an evident fashion from its lattice prescription. We can also reexpress this formal path integral in terms of the original bra and ket vectors in the form \n \\bn &&\\hskip-1cm M''M'\\int\\exp\\{i{\\textstyle{\\int}}[i\\<p,q|{\\rm E}\\hskip-.55em{\\rm I}\\,(d\/dt){\\rm E}\\hskip-.55em{\\rm I}\\,|p,q\\>\/\\<p,q|{\\rm E}\\hskip-.55em{\\rm I}\\,|p,q\\>\\nonumber\\\\\n    &&\\hskip2.3cm-\\<p,q|{\\rm E}\\hskip-.55em{\\rm I}\\,{\\cal H}{\\rm E}\\hskip-.55em{\\rm I}\\,|p,q\\>\/\\<p,q|{\\rm E}\\hskip-.55em{\\rm I}\\,|p,q\\>]\\,dt\\}\\,{\\cal D}_E\\mu(p,q)\\;.\\en\nThis last relation concludes our second route of calculation beginning with (22).\n\nThe third relation we wish to derive uses an integral representation for the projection operator ${\\rm E}\\hskip-.55em{\\rm I}\\,$ generally given by\n  \\bn {\\rm E}\\hskip-.55em{\\rm I}\\,={\\textstyle{\\int}} e^{-i\\xi^a\\Phi_a(P,Q)}\\,f(\\xi)\\,\\delta\\xi  \\en\nfor a suitable function $f$. Thus we rewrite (22) in the form\n\\bn &&\\hskip-1.5cm\\lim\\int\\<p'',q''|e^{-i\\epsilon\\lambda^a_{N}\\Phi_a}e^{-i\\epsilon{\\cal H}}e^{-i\\epsilon\\lambda^a_{N-1}\\Phi_a}e^{-i\\epsilon{\\cal H}}\\cdots e^{-i\\epsilon\\lambda^a_1\\Phi_a}e^{-i\\epsilon{\\cal H}}e^{-i\\epsilon\\lambda^a_0\\Phi_a}|p',q'\\>\\nonumber\\\\\n&&\\hskip2cm \\times\\,f(\\epsilon\\lambda_{N})\\cdots f(\\epsilon\\lambda_0)\\,\\delta\\epsilon\\lambda_{N}\\cdots\\delta\\epsilon\\lambda_0\\;.  \\en\nNext we insert the coherent-state resolution of unity at appropriate places to find that (30) may also be given by\n \\bn &&\\hskip-1.5cm\\lim\\int\\<p_{N+1},q_{N+1}|e^{-i\\epsilon\\lambda^a_{N}\\Phi_a}|p_N,q_N\\>\\prod_{l=0}^{N-1}\\<p_{l+1},q_{l+1}|e^{-i\\epsilon{\\cal H}}e^{-i\\epsilon\\lambda^a_{l}\\Phi_a}|p_l,q_l\\>\\nonumber\\\\   &&\\times [\\prod_{l=1}^{N}d\\mu(p_l,q_l)\\,f(\\epsilon\\lambda_l)\\,\\delta\\epsilon\\lambda_l]\\,f(\\epsilon\\lambda_0)\\,\\delta\\epsilon\\lambda_0\\;.  \\en\nFollowing the normal pattern, this last expression may readily be turned into a formal coherent-state path integral given by\n\\bn \\int\\exp\\{i{\\textstyle{\\int}}[p_j{\\dot q}^j-H(p,q)-\\lambda^a\\phi_a(p,q)]\\,dt\\}\\,{\\cal D}\\mu(p,q){\\cal D} E(\\lambda)\\;,  \\en\nwhere $E(\\lambda)$ is a measure designed so as to insert the projection operator ${\\rm E}\\hskip-.55em{\\rm I}\\,$ at every time slice. Unlike the case of the first-class constraints, we observe that the measure on the Lagrange multipliers is fixed. This usage of the Lagrange multipliers to ensure that the quantum system remains within the quantum constraint subspace is similar to their usage in the classical theory to ensure that the system remains on the classical constraint hypersurface. Thus it is not surprising that a fixed integration measure emerges for the Lagrange multipliers. On the other hand, it is also possible to use the measure $E(\\lambda)$ in the case of closed {\\it first}-class constraints as well; this would be just one of the acceptable choices for the measure $C(\\lambda)$ designed to put at least one projection operator ${\\rm E}\\hskip-.55em{\\rm I}\\,$ into the propagator.\n\nIn summary, we have established the equality of the three expressions\n\\bn &&\\hskip-1cm\\<p'',q''|{\\rm E}\\hskip-.55em{\\rm I}\\;e^{-iT({\\rm E}\\hskip-.55em{\\rm I}\\;{\\cal H}{\\rm E}\\hskip-.55em{\\rm I}\\;)}{\\rm E}\\hskip-.55em{\\rm I}\\,|p',q'\\>\\nonumber\\\\\n&&=M''M'\\int\\exp\\{i{\\textstyle{\\int}}[i\\<p,q|{\\rm E}\\hskip-.55em{\\rm I}\\,(d\/dt){\\rm E}\\hskip-.55em{\\rm I}\\,|p,q\\>\/\\<p,q|{\\rm E}\\hskip-.55em{\\rm I}\\,|p,q\\>\\nonumber\\\\\n    &&\\hskip2.3cm-\\<p,q|{\\rm E}\\hskip-.55em{\\rm I}\\,{\\cal H}{\\rm E}\\hskip-.55em{\\rm I}\\,|p,q\\>\/\\<p,q|{\\rm E}\\hskip-.55em{\\rm I}\\,|p,q\\>]\\,dt\\}\\,{\\cal D}_E\\mu(p,q)\\nonumber\\\\\n&&= \\int\\exp\\{i{\\textstyle{\\int}}[p_j{\\dot q}^j-H(p,q)-\\lambda^a\\phi_a(p,q)]\\,dt\\}\\,{\\cal D}\\mu(p,q){\\cal D} E(\\lambda)\\;.  \\en\nThis concludes our derivation of path integral formulas for general constraints. Observe that we have not introduced any $\\delta$-functionals, nor, in the middle expression, reduced the number of integration variables or the limits of integration in any way even though in that expression the integral over the Lagrange multipliers has been effected. \n\\section{Examples}\n\\subsubsection*{First-class constraint}\nConsider the system with two degrees of freedom, a single constraint, and a vanishing Hamiltonian characterized by the action\n \\bn I={\\textstyle{\\int}}[\\textstyle{\\frac{1}{2}}(p_1{\\dot q}_1-q_1{\\dot p}_1+p_2{\\dot q}_2-q_2{\\dot p}_2)-\\lambda(q_2p_1-p_2q_1)]\\,dt\\;,  \\en\nwhere for notational convenience we have lowered the index on the $q$ variables.\nNote that we have chosen a different form for the kinematic part of the action which amounts to a change of phase for the coherent states. It follows in this case that\n \\bn &&{\\cal M}\\int\\exp\\{i{\\textstyle{\\int}}[\\textstyle{\\frac{1}{2}}(p_1{\\dot q}_1-q_1{\\dot p}_1+p_2{\\dot q}_2-q_2{\\dot p}_2)-\\lambda(q_2p_1-p_2q_1)]\\,dt\\}\\,{\\cal D} p\\,{\\cal D} q\\,{\\cal D} C(\\lambda)\\nonumber\\\\\n&&\\hskip2cm=\\<p'',q''|{\\rm E}\\hskip-.55em{\\rm I}\\,|p',q'\\>\\;,  \\en\nwhere\n  \\bn  {\\rm E}\\hskip-.55em{\\rm I}\\,=(2\\pi)^{-1}\\int_0^{2\\pi}e^{-i\\xi(Q_2P_1-P_2Q_1)}\\,d\\xi={\\rm E}\\hskip-.55em{\\rm I}\\,(L_3=0)\\;.  \\en\nBased on the fact that\n \\bn \\<p'',q''|p',q'\\>=\\exp(-\\textstyle{\\frac{1}{2}}|z''_1|^2-\\textstyle{\\frac{1}{2}}|z''_2|^2+z''^*_1z'_1+z''^*_2z'_2-\\textstyle{\\frac{1}{2}}|z'_1|^2-\\textstyle{\\frac{1}{2}}|z'_2|^2)\\;,  \\en\nwhere $z'_1\\equiv(q'_1+ip'_1)\/\\sqrt{2}$, etc., it is straightforward to show that\n \\bn  &&\\<p'',q''|{\\rm E}\\hskip-.55em{\\rm I}\\,|p',q'\\>  =\\exp(-\\textstyle{\\frac{1}{2}}|z''_1|^2-\\textstyle{\\frac{1}{2}}|z''_2|^2-\\textstyle{\\frac{1}{2}}|z'_1|^2-\\textstyle{\\frac{1}{2}}|z'_2|^2)\\nonumber\\\\\n &&\\hskip3.3cm\\times I_0(\\sqrt{(z''^{*2}_1+z''^{*2}_2)(z'^2_1+z'^2_2)}\\,)\\;,\\en\nwith $I_0$ a standard Bessel function. We emphasize again that although the Hilbert space has been reduced by the introduction of ${\\rm E}\\hskip-.55em{\\rm I}\\,$, the reproducing kernel (38) leads to a reproducing kernel Hilbert space with an inner product having the same number of integration variables and domain of integration as in the unconstrained case.\n\\subsubsection*{Second-class constraint}\nConsider the two degree of freedom system determined by\n\\bn  I={\\textstyle{\\int}}[p{\\dot q}+r{\\dot s}-H(p,q,r,s)-\\lambda_1r-\\lambda_2s]\\,dt\\;, \\en\nwhere we have called the variables of the second degree of freedom $r,s$, and $H$ is not specified further. The coherent states satisfy $|p,q,r,s\\>=|p,q\\>\\otimes|r,s\\>$, which will be useful. We adopt (28) as our formal path integral in the present case, and choose \\cite{kla}\n  \\bn   &&{\\rm E}\\hskip-.55em{\\rm I}\\,={\\textstyle{\\int}} e^{-i(\\xi_1R+\\xi_2S)}\\,e^{-(\\xi_1^2+\\xi_2^2)\/4}\\,d\\xi_1d\\xi_2\/2\\pi\\nonumber\\\\  &&\\hskip.4cm=|r=0,s=0\\>\\<r=0,s=0|\\equiv|0_2\\>\\<0_2|  \\en\nwhich is a projection operator onto a coherent state for the second (constrained) degree of freedom only. With this choice it follows that\n \\bn &&i\\<p,q,r,s|{\\rm E}\\hskip-.55em{\\rm I}\\,(d\/dt){\\rm E}\\hskip-.55em{\\rm I}\\,|p,q,r,s\\>\/\\<p,q,r,s|{\\rm E}\\hskip-.55em{\\rm I}\\,|p,q,r,s\\>\\nonumber\\\\\n&&\\hskip2cm=i\\<p,q|(d\/dt)|p,q\\>-\\Im(d\/dt)\\ln[\\<0_2|r,s\\>]\\nonumber\\\\\n&&\\hskip2cm=p{\\dot q}-\\Im(d\/dt)\\ln[\\<0_2|r,s\\>]\\;,  \\en\nand \n\\bn &&\\hskip-1cm\\<p,q,r,s|{\\rm E}\\hskip-.55em{\\rm I}\\,{\\cal H}(P,Q,R,S){\\rm E}\\hskip-.55em{\\rm I}\\,|p,q,r,s\\>\/\\<p,q,r,s|{\\rm E}\\hskip-.55em{\\rm I}\\,|p,q,r,s\\>\\nonumber\\\\  &&\\hskip1cm=\\<p,q,0,0|{\\cal H}(P,Q,R,S)|p,q,0,0\\>=H(p,q,0,0)\\;. \\en\nConsequently, for this example, (28) becomes\n \\bn {\\cal M}\\int\\exp\\{i{\\textstyle{\\int}}[p{\\dot q}-H(p,q,0,0)]\\,dt\\}\\,{\\cal D} p\\,{\\cal D} q\n\\;\\times\\;\\<r'',s''|0_2\\>\\<0_2|r',s'\\>\\;, \\en\nwhere we have used the fact that at every time slice\n \\bn {\\textstyle{\\int}}\\<r,s|{\\rm E}\\hskip-.55em{\\rm I}\\,|r,s\\>\\,dr\\,ds\/(2\\pi)={\\textstyle{\\int}}|\\<0_2|r,s\\>|^2\\,dr\\,ds\/(2\\pi)=1\\;. \\en\n\nObserve that in this path integral quantization no variables have been eliminated nor has any domain of integration been reduced; moreover, the operators $R$ and $S$ have remained unchanged. The result in (43) is clearly a product of two distinct factors. The first factor describes the true dynamics as if we had solved for the classical constraints and substituted $r=0$ and $s=0$ in the classical action from the very beginning, while the second factor characterizes a one-dimensional Hilbert space for the second degree of freedom. Thus we can also drop the second factor completely as well as all the integrations over $r$ and $s$ and still retain the same physics. In this manner we recover the standard result without the use of Dirac brackets or having to eliminate the second-class constraints from the theory initially.\n\n\\section*{Acknowledgements}\nJ. Govaerts is thanked for his continued interest in this work.\n Some aspects of a coherent state quantization procedure that emphasizes projection operators for systems with closed first-class constraints have been anticipated by Shabanov \\cite{sha}. Thanks are expressed to S. Shabanov for bringing this work to the author's attention. Projection operators for closed first-class constraints also appear in the text of Henneaux and Teitelboim \\cite{hen}.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{section:introduction}\n\n\\IEEEPARstart{F}{uture} wireless systems require fundamental and crisp understanding of design principles and control mechanisms to efficiently manage network resources. Resource allocation policies lie at the heart of wireless communication networks, since they aim at guaranteeing the required Quality of Service (QoS) at the user level, while ensuring efficient and optimized operation at the network level to maximize operators' revenue.\nResource allocation management in wireless communications may include a wide spectrum of network functionalities, such as scheduling, transmission rate control, power control, bandwidth reservation, call admission control, transmitter assignment, and handover \\cite{Stanczak2009,Lee2014b,Ahmed2005}.  \nIn this survey, a resource allocation policy is defined by the following components: \\textit{i}) a multiple access technique and a scheduling component that distributes resources among users subject to individual QoS requirements; \\textit{ii}) a signaling strategy that allows simultaneous transmission of independent data streams to the scheduled users; and \\textit{iii}) rate allocation and power control that guarantee QoS and harness potential interference. \nFig.~\\ref{fig:mu-mimo-layered-tasks} illustrates these components and the interconnection between them. The figure highlights the fact that each function of the resource allocation strategy can be performed in either optimal or suboptimal way, which is elaborated upon below.\n\n\nThe multiple access schemes can be classified as orthogonal or non-orthogonal. The former is a conventional scheme that assigns radio resources, e.g., code, sub-carrier, or time slot, to one user per transmission interval. The main characteristic of orthogonal multiple access schemes is their reliability, since there is no need to deal with co-resource interference. The resource allocation policy can optimize with reasonable complexity several performance metrics, such as throughput, fairness, and QoS \\cite{Asadi2013}. The multiplexing gain, i.e., the number of scheduled users, is limited by the number of available radio resources in the system.  \nIn non-orthogonal multiple access,  a set of users concurrently superimpose their transmissions over the same radio resource, and potentially interfere with each other. \nIn this scheme the co-resource interference can be mitigated by signal processing and transmission techniques implemented at the transmitter and\/or receiver sides. Such techniques exploit different resource domains, e.g., power, code, or spatial domain, and a combination of them are envisaged to cope with the high data rate demands and system efficiency expected in the next generation of wireless networks \\cite{Li2014a,Dai2015}.\n\n\nHereinafter, we focus on multiple access schemes based on multi-antenna transceivers operating at the spatial domain, i.e., multiple-input multiple-output (MIMO). MIMO communication, where a multi-antenna base station (BS) or access point (AP) transmits one or many data streams to one or multiple user equipments simultaneously, is a key technology to provide high throughput in broadband wireless communication systems. MIMO systems have evolved from a fundamental research concept to real-world deployment, and they have been integrated in state-of-the-art wireless network standards \\cite{Li2010,Jones2015,Kim2015}, e.g., IEEE 802.11n, 802.11ac WLAN, 802.16e (Mobile WiMAX), 802.16m (WiMAX), 802.20 (MBWA), 802.22 (WRAN), 3GPP long-term evolution (LTE) and LTE-Advanced (E-UTRA). \nResource allocation is particularly challenging in wireless communication systems mainly due to the wireless medium variability and channel randomness, which renders the overall performance location-dependent and time-varying \\cite{Wang2007}. \nNevertheless, high spectral efficiency and multiplexing gains can be attained in MIMO systems since multiple data streams can be conveyed to independent users. By exploiting the spatial degrees of freedom (DoF) offered by multiple antennas we can avoid system resource wastage \\cite{Ajib2005}. Multiuser (MU) MIMO systems have been extensively investigated over the last years from both theoretical and practical perspective. In a recent evolution of MU-MIMO technology, known as massive MIMO or large-scale MIMO \\cite{Lu2014,Zheng2015}, few hundreds of antennas are employed at the BS to send simultaneously different data streams to tens of users. Massive MIMO has been identified as one of the promising air interface technologies to address the massive capacity requirement required demanded by 5G networks \\cite{Andrews2014,Boccardi2014}. \n\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{figure1.pdf}\\\\\n\t\\caption{Components of the resource allocation policy in MU-MIMO systems} \n\t\\label{fig:mu-mimo-layered-tasks}\n\\end{figure}\n\n\nThe downlink transmission is particularly challenging in MU-MIMO scenarios because the geographic location of the receivers is random and joint detection cannot be performed. The main goal is to convey independent data streams to a set of properly selected users, attaining spatial multiplexing gain offered by MU-MIMO. However, determining such a users set is a very challenging task, which depends on all elements of the resource allocation strategy, e.g., individual QoS requirements, signaling schemes, rate allocation and power control strategies implemented at the transmitter. \nMIMO systems allow for a plethora of mighty signal processing techniques that enhance the system performance by  exploiting a multi-dimensional pool of resources. This pool is composed of resources with different nature, e.g., signal spaces, transmission powers, time slots, sub-carriers, codes, and users. Efficient allocation designs over such a large set of resources implies that a tradeoff between optimality and feasibility. On the one hand, optimality can be reached by solving optimization problems over a set of integer and continuous variables, which may be a thoroughly complex task. Feasibility, on the other hand, implies that suboptimal resource allocation takes place by relaxing and reformulating optimization problems whose solutions can be found by practical and reliable algorithms. \n\n\n \n\n\n\n\n{\\color{black}{ \n\n\\subsection{Contributions of the Survey} \n\\label{section:introduction:contributions}\n\n\t\t\nThere exists a very rich literature on MIMO communications, and this paper complements it by providing a classification of different aspects of MU-MIMO systems and resource allocation schemes. Users with independent channels provide a new sort of diversity to enhance the overall performance. However, in contrast to systems where each user accesses a dedicated (orthogonal) resource \\cite{Wang2007,Asadi2013,Capozzi2013,Li2014}, in MU-MIMO systems the additional diversity is realized when several users access the same resource simultaneously. \n\nAccounting for multiple antennas at both ends of the radio link allows spatial steering of independent signals using precoding schemes, which results in the coexistence of many data streams conveyed to the concurrent users. Some of the main contributions of this survey are the description and classification of linear and non-linear precoding schemes, considering the amount of channel information available at the transmitter, the network scenario (e.g. single-cell or multi-cell), and the antenna settings. Each precoding scheme relies on different characteristics of the MU-MIMO channels to fully exploit the spatial domain. The paper provides a comprehensive classification of metrics that quantify the spatial compatibility, which can be used to select users and improve the precoding performance.\n\nThe spectral efficiency, error rates, fairness, and QoS are common criteria to assess performance in the MU-MIMO literature. Optimizing each one of these metrics requires specific problem and constrains formulations. The type of precoding, antenna configuration, and upper layer demands can be taken into account to design robust resource allocation algorithms, i.e. cross layer designs. Other contributions of this paper are the description and classification of different optimization criteria and general constraints used to characterize MU-MIMO system. The proposed classification  incorporates the antenna configuration, the amount of channel information available at the transmitter, and upper layer requirements. \n\nEarly surveys on MU-MIMO have pointed out that resource allocation can be opportunistically enhanced by tracking the instantaneous channel fluctuations for scenarios with a single transmitter \\cite{Ajib2005,Gesbert2007a}. However, in recent years, a large number of techniques have been developed for very diverse and heterogeneous MIMO scenarios. The paper presents a classification of state-of-the-art scheduling algorithms for MU-MIMO scenarios for single and multiple transmitter scenarios. We consider the channel state information, the objective functions optimized by the scheduler, the degree of cooperation\/coordination between transmitters, and the power allocation techniques. \n\nThe goal of this survey is not to describe in detail the theory behind precoding design, rate allocation, power control or user scheduling, but rather to use their fundamental principles to get insight on the interplay among them. Our aim is to describe state-of-the-art processing techniques for MU-MIMO, point out practical challenges, and present general guidelines to design efficient resource allocation algorithms. The material favors broad intuition over detailed mathematical formulations, which are left to the references. Although the list of references is certainly not intended to be exhaustive, the cited works and the references therein may serve as a starting point for readers aiming to go beyond a tutorial.\n\n}}\n\n\n\n\n\n\\subsection{Organization} \n\\label{section:introduction:organization}\n\n\nThe paper is organized as follows.\nIn Section~\\ref{section:preliminaries} we present the basic ideas behind MIMO wireless communications, introduce MU-MIMO systems, and discuss the main challenges.\nIn Section~\\ref{section:system-set-ups} we introduce the most commonly studied MU-MIMO channel and system models, their characteristics and conventional assumptions.\nSection~\\ref{section:precoding} is devoted to signal design and precoding schemes under different conditions of channel information. \nIn Section~\\ref{section:metrics-spatial-compatibility} we introduce the most common metrics of spatial compatibility, which are used to categorize users and reduce the scheduling complexity.\nSection~\\ref{section:performance-uf}  presents a classification of optimization criteria  and describes the usual constraints considered in MU-MIMO systems. \nIn Section~\\ref{section:algorithms} we propose a classification of the several techniques to address the user scheduling problem. The specific characteristics, limitations and use cases for each technique are discussed.    \nSection~\\ref{section:algorithms:selection-partial-csit} is focused on scheduling algorithms with partial channel information at the transmitter. We categorize the existing approaches and present guidelines to minimize complexity and improve efficiency.\nIn Section~\\ref{section:power-allocation} we present the most common power allocation schemes and discuss their role in MU-MIMO systems.\nFinally, we conclude the paper in Section~\\ref{section:remarks}.\nThe reader can find a list of technical terms and abbreviations summarized in Table~\\ref{table:abbreviations}.\n\n\nWe adopt the following notation: matrices and vectors are set in upper and lower boldface, respectively. $(\\cdot)^{T}$, $(\\cdot)^{H}$, $|\\cdot|$, $\\|\\cdot\\|_{p}$ denote the transpose, the Hermitian transpose, the absolute value, and the $p$-norm, respectively. $rank(\\mathbf{A})$, $null(\\mathbf{A})$ denote the rank and null space of matrix $\\mathbf{A}$. $Span(\\mathbf{A})$ and $Span(\\mathbf{A})^{\\perp}$ denote the subspace and orthogonal subspace spanned by the columns of matrix $\\mathbf{A}$. Calligraphic letters, e.g. $\\mathcal{G}$, denote sets, and $|\\mathcal{G}|$ denotes cardinality.  $\\mathbb{R}_{+}$ is the set of nonnegative real numbers and $\\mathbb{C}^{N \\times M}$ is the space of $N \\times M$ matrices. $\\mathcal{CN}(\\mathbf{a},\\mathbf{A})$ is the complex Gaussian distribution with mean $\\mathbf{a}$ and covariance matrix $\\mathbf{A}$. $\\mathbb{E}[\\cdot]$ denotes expectation. \n\n\n\n\n\n\n\n\\section{Preliminaries}\n\\label{section:preliminaries}\n\n\n\n\\subsection{Multiple Antenna Systems}\n\\label{section:introduction:mimo-systems}\n\nA MIMO system employs multiple antennas at the transmitter ($M$) and receiver ($N$) sides to improve communication performance.\nThe seminal works \\cite{Foschini1998,Telatar1999} provide a mathematical motivation behind multiple antenna processing and communications. Theoretical analysis has shown that the spectral efficiency, i.e., the amount of error-free bits per second per Hertz (bps\/Hz), follows the scaling low $\\min(M,N)$, without increasing the power or bandwidth requirements.\nThe signal processing techniques in multi-antenna systems can be classified as \\textit{spatial diversity techniques} and \\textit{spatial multiplexing techniques} \\cite{Mietzner2009}. \n\n\nSpatial diversity techniques (see \\cite{Mietzner2009} and references therein), provide transmission reliability and minimize error rates. This is attained by transforming a fading wireless channel into an additive white Gaussian noise (AWGN)-like channel, i.e., one can mitigate signal degradation due to fading \\cite{Ajib2005}. The probability that multiple statistically independent channels experience simultaneously deep fading gets very low as the number of independent paths increases. The spatial diversity  techniques can be applied at both transmission and reception sides of the link. Transmit diversity schemes include space diversity, polarization diversity, time diversity, frequency diversity, and angle diversity. Examples of receive diversity schemes are selection combining, maximum ratio combining (MRC), and equal gain combining \\cite{Tse2005,Goldsmith2005}.\n\n\n\n\\subsection{Multiuser MIMO}\n\\label{section:introduction:mu-mimo}\nThis paper focuses on spatial multiplexing techniques, which exploit the DoF provided by MIMO. Spatial multiplexing is tightly related to multiuser communications and smart antennas processing \\cite{Tse2005}. In multiuser systems, spatial multiplexing gains can be attained by steering signals toward specific receivers, such that the power to intended users is boosted. Simultaneously, co-channel interference to unintended users can be partially or completely suppressed. \n\n\nIn MU-MIMO systems, the available resources (power, bandwidth, antennas, codes, or time slots) must be assigned among $K$ active users.\nThere are two kinds of multiuser channels: the downlink channel, also known as \\textit{broadcast channel} ({BC}), where a single transmitter sends different messages to many receivers; and the uplink channel, also called \\textit{multiple access channel} ({MAC}), where many transmitters communicate with a single receiver.\nThere are several \\textit{explicit} differences between {BC} and {MAC}. In the former, the transmitted signal is a combination of the signals intended for all co-scheduled users, subject to total transmit power, $P$, constraints. In contrast, in the {MAC} channel, the signal from the $k$-th user is affected by other co-scheduled users, subject to individual power constraints, i.e. $P_k$, \\cite{Tse2005}. \nThere exists an \\textit{implicit} connection between BC and MAC, known as \\textit{duality}, which establishes the relationship between the capacity regions of both access channels \\cite{Tse2005,Goldsmith2005}. The BC-MAC duality has been fundamental to define optimal policies for power allocation, signaling, and QoS guaranteeing in MU-MIMO systems, see \\cite{Schubert2004,Schubert2006,Yu2006a,Chiang2008}.\nThe capacity regions include  operative point where transmission to multiple users do not interfere with each other. Every transmission is performed over orthogonal signaling dimensions, which is a signal separation called duplexing \\cite{Goldsmith2005}. This operation is performed by allocating communications across different time slots, known as time-division-duplex ({TDD}), or across separated frequency bands, known as frequency-division-duplex ({FDD}).\n\n\n\nIn the literature of {MU-MIMO}, two types of diversity are studied: \\textit{spatial multiplexing diversity} and \\textit{multiuser diversity} ({MUDiv}). The former is a consequence of the independent fading across MIMO links of different users. This means that independent data streams can be transmitted over parallel spatial channels, increasing the system capacity \\cite{Zheng2003}. The latter arises when users that are geographically far apart have channels that fade independently at any point in time. Such independent fading processes can be exploited so that users with specific channel conditions are simultaneously scheduled \\cite{Viswanath2002}. \nThere are two modes of transmission in MIMO systems, see Fig.~\\ref{fig:su-mu-mimo}: single user (SU) and multiuser (MU) mode. The SU-MIMO mode improves the performance of a single user, allocating one or many data streams in the same radio resource. In the MU-MIMO mode, different data streams are sent to different users such that a  performance metric is optimized, e.g., the average sum rate.\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=0.98\\linewidth]{figure2.pdf}\\\\\n\t\\caption{(a) Single-user (SU) and (b) Multi-user (MU) MIMO modes}\n\t\\label{fig:su-mu-mimo}\n\\end{figure}\n\nSelecting between SU- or MU-MIMO transmission modes depends on the accuracy of the channel state information at the transmitter (CSIT), the amount of allowed interference, the target rate per user, the number of user, the signal-to-noise ratio (SNR) regime, and the achievable capacity in each mode \\cite{Tang2010}.\nNonetheless, by assuming sufficient CSIT knowledge, MU-MIMO processing techniques provide several performance gains \\cite{Gesbert2007a}: multiple antennas attain diversity gain, which improves bit error rates ({BER}); directivity gains realized by {MUDiv}, since the spatial signatures of the users are uncorrelated, which mitigates inter-user interference (IUI); immunity to propagation limitations in SU-MIMO, such as rank loss or antenna correlation; and multiplexing gains that scale, at most, with the minimum number of deployed antennas. \n\n\n\\begin{figure*}[th!]\n\t\\centering\n\t\\includegraphics[width=0.98\\linewidth]{figure3.pdf}\\\\\n\t\\caption{Processing blocks and input signals in a MU-MIMO scenario. A transmitter with $M$ antennas serves $K$ multiple antenna receivers.} \n\t\\label{fig:system-processing-blocks}\n\\end{figure*} \n\n\\subsection{The need of User Scheduling}\n\\label{section:introduction:user-selection}\n\n\nIn {MU-MIMO} {BC} systems, the overall performance depends on how efficiently the resource allocation algorithms manage the hyper-dimensional pool of resources (carriers, time slots, codes, power, antennas, users, etc.). \nConsider a system with a transmitter equipped with $M$ antennas, and let $\\bar{\\mathcal{K}}=\\{1,2,3, \\ldots, K\\}$ be the set of all active users, illustrated in Fig.~\\ref{fig:system-processing-blocks}. To qualitatively determine the objectives of scheduling, we provide the following definitions:     \n\n\\begin{definition}\\label{defn:quality-of-service}\n\t\\textit{Quality of Service}. We say that QoS defines a set of prescribed network-\/user-based performance targets (e.g., peak rates, error rates, average delays, or queue stability), that can be measured, improved, and guaranteed for a specific upper layer application. \n\\end{definition}\n\n\\begin{definition}\\label{defn:user_scheduling}\n\t\\textit{User scheduling}. We say that a set of radio resources (e.g. time slot, codes, sub-channels, powers, etc.), has been assigned to a group of scheduled users, $\\mathcal{K} \\subseteq \\bar{\\mathcal{K}}$, so that a global performance metric is optimized subject to power and QoS constraints. Moreover, each user $k \\in \\mathcal{K}$, achieves non-zero rate with successful information reception.\n\\end{definition}\n\nConsider that each user $k \\in \\bar{\\mathcal{K}}$, is equipped with $N_k$ antennas. By having more receive than transmit antennas ($M< \\sum_{k} N_k$), one can solve a selection problem to achieve MUDiv gains in fading fluctuating channels \\cite{Ajib2005,Sharif2005,Sharif2007,Gesbert2007a,Yang2011}. A fundamental task in resource management is to select a subset of users $\\mathcal{K} \\subseteq \\bar{\\mathcal{K}}$, and assign resource to it, so that a given performance metric is optimized.\nFor the sake of illustration, consider a single-transmitter scenario, and let us formulate a general user scheduling problem for a single resource (sub-carrier, time slot, or code) as follows:\n\n\\begin{IEEEeqnarray}{rrCl}\n\t& {\\text{maximize}} & \\quad &  \\sum_{k =1 }^{K} \\xi_{\\pi(k)} U_{\\pi(k)}  \\label{eq:general-combinatorial-problem:cost-function}   \\\\\n\t& \\text{subject to} && \\sum_{k =1}^{K} \\xi_{k} p_{k} \\leq P \\IEEEyessubnumber \\label{eq:general-combinatorial-problem:total-power}\\\\\n\t&&& 0 \\leq \\xi_{k} p_{k} \\leq P_{k} \\ \\ \\ \\forall k \\in \\bar{\\mathcal{K}} \\IEEEyessubnumber \\label{eq:general-combinatorial-problem:individual-power} \\\\\n\t&  & & \\sum_{k =1}^{K} \\xi_{k} \\leq c_{p} \\IEEEyessubnumber \\label{eq:general-combinatorial-problem:set-cardinality} \\\\\n\t&&& \\xi_{k} \\in \\{0, 1\\} \\ \\ \\ \\forall k \\in \\bar{\\mathcal{K}} \\IEEEyessubnumber \\label{eq:general-combinatorial-problem:binary-vars}\n\\end{IEEEeqnarray}\n\nOur goal is to maximize the sum of the utility functions, $U_{\\pi(k)}$, $\\forall k$, which depends on several parameters: the multiuser MIMO channel, $\\mathbf{H}_{k}$, the allocated power, $p_{k}$, the individual data queues, $q_{k}$, and the encoding order, $\\pi(\\cdot)$. The QoS requirements can be included in the definition of $U_{\\pi(k)}$, as individual weights [cf. Section~\\ref{section:performance-uf}]. Equations (\\ref{eq:general-combinatorial-problem:total-power}) and (\\ref{eq:general-combinatorial-problem:individual-power}) define total and individual power constraints, which are set according to the scenario [cf. Section~\\ref{section:system-set-ups}]. The term $\\xi_{\\pi(k)}$ is a binary variable with value equal to 1 if the $\\pi(k)$-th user is scheduled and 0 otherwise. The set of selected user is given by $\\mathcal{K} = \\{ k \\in \\bar{\\mathcal{K}}: \\xi_{k} = 1 \\}$. \nThe system operates in MU-MIMO mode if $1 < |\\mathcal{K}| \\leq c_{p}$, where $c_{p}$ in (\\ref{eq:general-combinatorial-problem:set-cardinality}) denotes the maximum number of users or transmitted data streams, that can be sent over $M$ antennas. If the solution of (\\ref{eq:general-combinatorial-problem:cost-function}) only exists for $|\\mathcal{K}|=1$, the system operates in SU-MIMO mode. In such a case, the optimization problem can be formulated to attain MUDiv, multiplexing (high rate), and diversity (high reliability) gains, see \\cite{Tse2005,Heath2005,Chen2006}.\nDepending on the CSIT, the type of signaling design and coding applied to the data, theoretical analysis show that the number of users with optimal nonzero power is upper bounded\\footnote{The upper bound is tight for small values of $M$ and it becomes loose as the number of transmit antennas grow large. Numerical results comparing the upper bound of $|\\mathcal{K}|$ for several coding techniques can be found in \\cite{Boccardi2006}. } as $|\\mathcal{K}| \\leq c_p \\leq M^2$, \\cite{Yu2006a}. \nIn practical systems, multiplexing gain can be scaled up to $|\\mathcal{K}| \\leq M$, by means of linear signal processing   [cf. Section~\\ref{section:precoding:linear}]. \n\n\n\nThe mathematical formulation in (\\ref{eq:general-combinatorial-problem:cost-function}) resembles a knapsack or subset-sum problem \\cite{Rondeau2009,Williamson2010}, which is known to be non-polynomial time complete (NP-C). Although the users are fixed items that must be chosen to construct $\\mathcal{K}$, their associated utility functions change according to the channel conditions and the resource allocation of the co-selected users. This implies that the optimization variables are, in general, globally coupled.\nFinding the optimal set $\\mathcal{K}$, is a combinatorial problem due to the binary variables $\\xi_{k}$, and the encoding order $\\pi(\\cdot)$. Moreover, depending on $U_{k}$, problem (\\ref{eq:general-combinatorial-problem:cost-function}) might deal with non-convex functions on the multiple parameters, e.g. $K$, $M$, $N_k$, etc. The feasibility of (\\ref{eq:general-combinatorial-problem:cost-function}) relies on the constraints and processing, e.g. the precoding schemes, the power allocation, the CSIT accuracy, \\cite{Gesbert2007a}, [cf. Section~\\ref{section:performance-uf}]. The scheduling problem can be solved optimally by exhaustively searching (ExS) over all possible set sizes and user permutations. However, the computational complexity of ExS is prohibitively high, even for small values of $K$ \\cite{Ajib2005}. Furthermore, problem (\\ref{eq:general-combinatorial-problem:cost-function}) can be modified to include additional dimensions, such as multiple carriers for OFDMA systems (e.g. \\cite{Chan2007,Moretti2013}) or codes for CDMA systems (e.g. \\cite{Driouch2012}).\n\n\n\n\\subsection{The need of CSI availability} \n\\label{section:introduction:needof-csit}\n\nChannel knowledge at the transmitter can be modeled taking into account instantaneous or statistical information, e.g. variance, covariance, angles of arrival\/departure, and dominant path in line-of-sight \\cite{Vu2007}. \nThere are two main strategies used to obtain CSI, reciprocity and feedback, which provide different feedback requirements and robustness to CSI errors. The former, known as open-loop feedback, uses uplink channel information to define the downlink channel in the next transmission interval. It is suitable for TDD since transmit directions are in identical frequencies, and the channel can be reversed. The latter, known as closed-loop feedback, requires sending the downlink channel to the transmitter using dedicated pilots, and is commonly used in FDD.  \nThe majority of the papers reviewed in Section~\\ref{section:algorithms} assume closed-loop with full or limited channel feedback, and we refer the reader to \\cite{Love2008,Vu2007,Kobayashi2011} and references therein for further discussion on CSI acquisition and its impact on system performance.\n\n\n\nTo perform multiple antenna processing, interference mitigation, user scheduling, efficient power allocation, and to profit from MUDiv, knowledge of CSIT is compulsory.  The complete lack of CSIT reduces the multiplexing gain to one, and cannot use MUDiv for boosting the achievable capacity \\cite{Jafar2005, Sharif2005,Hassibi2007}. In such scenarios, the optimum resource allocation and transmission schemes are performed over orthogonal dimensions \\cite{Koutsopoulos2008}.    \nIn the literature of MU-MIMO, a large number of works assume full CSI (error-free),  at both the receiver (CSIR) and transmitter (CSIT) sides. \nIn practical systems, a strong downlink pilot channel, provided by the transmitter, is available to the users, hence the CSIR estimation error is negligible relative to that of the CSIT \\cite{Lau2009}. For simplicity, it is widely assumed that CSIR is perfectly known at the mobile terminals. \n{\\color{black}{\nIn cellular MU-MIMO systems, channel estimation relies on having orthogonal pilots allocated to different users. The orthogonality is guaranteed for the users within the same cell, but not for those scattered across different cells. The number of BS antennas and bandwidth constraints may not allow orthogonal pilots for each user in the system, resulting in pilot contamination \\cite{Marzetta2010}. Under universal frequency reuse, the pilots can be drastically polluted by users at adjacent cells, when the serving BS performs channel estimation \\cite{Elijah2016}.\n}}\n\n\n\nAchieving full CSIT (ideal noiseless and delay-free feedback) is highly challenging in practice. Feeding back the CSI requires rates that grow rapidly with the transmit power and the number of antennas \\cite{Jindal2006}. However, by assuming full CSIT, it is possible to derive upper bounds on the performance of different signal processing techniques and scheduling algorithms. The information-theoretic and numerical results using full CSIT provide useful insights regarding the system performance bounds (e.g., \\cite{Caire2003,Weingarten2006,Sharif2007}). Resource allocation strategies that optimize spectral efficiency, fairness, power consumption, and error probability can be designed to characterize optimal operating points \\cite{Schubert2006,Bjornson2013,Chiang2008}. Analytical results for full CSIT reveal the role of each parameter in the system, e.g., number of deployed transmit and receive antennas, number of active users, SNR regime, etc. \n\nIf channel knowledge is obtained via partial (rate-limited) feedback, the information available at the transmitter has finite resolution, resulting in quantization errors. \\textit{Partial CSIT} is comprised of two quantities: channel quality information (CQI) and channel direction information (CDI) \\cite{Gesbert2007a}. The CQI measures the achievable SINR, the channel magnitude, or any other function of the link quality. The CDI is the quantized version of the original channel direction, which is determined using codebooks [cf. Section~\\ref{section:precoding:partial-csit}]. The transmitter uses both indicators for scheduling [cf. Section~\\ref{section:algorithms:selection-partial-csit}], and the CQI is particularly used for power control, link adaptation, and interference management \\cite{Huang2012a}. The CSI feedback interval highly depends on the users' mobility,\\footnote{The authors in \\cite{Kobayashi2007a} showed that mobility defines the best reliable transmission strategy for capacity maximization, i.e., space-time coding or space division multiplexing. The results in \\cite{Zhang2011} defined acceptable mobility ranges for MU-MIMO scenarios.} and even for short-range communications (e.g., WiFi), immediate feedback is needed to achieve and maintain good performance \\cite{Jones2015}.\nBy considering high mobility and limited feedback rates, one cannot rely on instantaneous or full CSI. In such cases, the transmitters perform resource allocation based on \\textit{statistical} CSI, which vary over larger time scales than the instantaneous CSI. The statistics for the downlink and uplink are reciprocal in both FDD and TDD, which can be used to perform resource allocation, see \\cite{Gao2009} and references therein.\nTable~\\ref{tab:channel-information} summarizes the different types of CSIT in MU-MIMO systems, and the antenna configuration at the receivers. Partial CSIT refers to quantized channel information, which will be elaborated upon Sections \\ref{section:precoding:partial-csit} and \\ref{section:algorithms:selection-partial-csit}.\n\n\n\\begin{table}[t!] \\scriptsize\n\t\\renewcommand{\\arraystretch}{1.3}\t\n\t\\caption{Summary of the Type of CSI at the Transmitter for MISO ($N=1$) and MIMO ($N>1$) Configurations}\n\t\\label{tab:channel-information\n\t\\centering\n\t\\begin{tabularx}{0.48\\textwidth}{l *{2}{>{\\centering\\arraybackslash}X} }\n\t\t\\toprule\n\t\t&  \\textbf{MISO}  &  \\textbf{MIMO} \\\\\n\t\t\\midrule\n\t\t\\textbf{Full CSIT} & \\cite{Dimic2005, Huang2013, Tu2003, Wang2008a, Yoo2006,  Jiang2006, Dai2009, Lee2014, Huang2012b, Razi2010, Mao2012, Bjornson2014, Shi2012, Maciel2010, Chung2010, Driouch2012, Yoo2005a, Christopoulos2013, Xia2009, Matskani2008, Lau2005a, Cottatellucci2006, Wang2015, Yang2011, Koutsopoulos2008, Kobayashi2007, Boccardi2006, Destounis2015, Zhang2007b, Swannack2004, Tsai2008, Shirani-Mehr2011, Hammarwall2007, Seifi2011, Castaneda2015, Huh2012, Jang2011, Park2014, Ku2015, Song2008, Tu2003, Maciel2010,Bogale2015,Liu2015a} &  \\cite{Bayesteh2008, Shen2006, Wang2010, Bayesteh2008, Ko2012, Elliott2009, Chan2007, Tran2010, Wang2010a, Nam2014a, Torabzadeh2010, Souihli2010, Moretti2013, Zhang2009, Cui2011a, Yu2013, Park2010, Lossow2013,  Jagannathan2007, Chen2007a, Fuchs2007, Tejera2006, Zhang2005a, Wang2006, Chen2008, Sigdel2009a, Yi2011, Sun2010, Tran2012, Elliott2012, Hei2009, Lim2009, Cheng2014, Aniba2007} \\\\\n\t\t\\midrule\n\t\t\\textbf{Statistical} &  \\cite{Liu2014a, Nam2014, Lee2014, Adhikary2013, Adhikary2013a, Stridh2006a,Kountouris2006,Kountouris2006b, Hammarwall2008,Hammarwall2008a, Dartmann2013,YiXu2014,Lee2014c,Liu2015a}  & \\cite{Raghavan2007,Fuchs2007, Wang2006, Rico-Alvarino2014, Fan2014,Gao2009} \\\\\n\t\t\\midrule\n\t\t\\textbf{Outdated} & \\cite{Liu2014a,Driouch2012, Lau2009, Kobayashi2007,Kobayashi2007a, Tang2010, Shirani-Mehr2010, Zhang2011,Liu2015a} & \\cite{Wang2006} \\\\\n\t\t\\midrule\n\t\t\\textbf{Correlated} & \\cite{Kountouris2006b,Xu2009a,Weich2006,Gao2009, Raghavan2015,Liu2014a,Liu2015a} & \\cite{Schellmann2010,Chen2007a,Raghavan2007,Weich2006,Tran2012,Wang2006,Chen2008,Xu2009a,Bjornson2009a,Bjornson2010a,Godana2013a,Rao2013} \\\\\n\t\t\\midrule\n\t\t\\textbf{Partial CSIT} & \\cite{ Moon2013, Xia2009, Wagner2008,Yang2011,Jindal2006, Zorba2008, Conte2010, Tang2010, Huang2009a, Vicario2008, Yoo2007, Kountouris2007, Kountouris2008, Dai2008, Kountouris2006, Zhang2011, Ravindran2012, Kountouris2006a, Wang2007a, Choi2007c, Sohn2010, Kountouris2008a, Kountouris2005a, Kountouris2005, Xu2010, Khoshnevis2013, Simon2011, Zakhour2007, Huang2012a, Huang2007} & \\cite{Raghavan2007,Chae2008, Chen2013, Trivellato2008, Jindal2008, Nam2014a, VanRensburg2009, Hosein2009, Schellmann2010,Min2013, Sohn2012,  Zhang2007a, Fan2014, Wang2012d} \\\\\n\t\t\\bottomrule\n\t\\end{tabularx}%\n\\end{table}%\n \n\n\n\n{\\color{black}{\n\\section{MU-MIMO Channel and System Models}\\label{section:system-set-ups}\n}}\n\nThe signal processing and scheduling algorithms described in the following sections have been developed and studied for single-hop MU-MIMO scenarios. We have classified the scenarios in two groups, see Fig.~\\ref{fig:generic-deployments}: single transmitter and multiple transmitters scenarios.\n\n\n\nThe implemented resource allocation strategies, user scheduling, and signal processing techniques depend on the number of coordinated transmitters, the number of antennas ($M$ and $N$), the number of users ($K$), the SNR regime, and the CSIT accuracy. The system optimization relies on close-loop (e.g. \\cite{Vu2007, Mietzner2009}) or open-loop (e.g. \\cite{Destounis2015}) feedback,  to achieve spatial multiplexing gains, multiuser diversity gains, and to combat interference.\nIn cellular systems, there are two main sources of interference \\cite{Andrews2005}: other active devices in the same co-channel and same cell, i.e., intra-cell or IUI; and from transmissions in other cells, i.e., inter-cell interference (ICI). The techniques to mitigate IUI and ICI depend on the type of scenario and the optimization criterion. There exist a number of scenarios where the interference cannot be reduced, see \\cite{Jindal2006,Lozano2013}, whose characteristics are described in the following definition: \n\n\n\\begin{definition}\\label{defn:interference-limited-system}\n\t\\textit{Interference-limited system}. An MU-MIMO system is said to be interference limited if the performance metric saturates (ceiling effect) with the transmit SNR. This might occur due to CSIT inaccuracy, highly correlated multiuser channels (IUI), and irreducible ICI.\n\\end{definition}\n\n\\begin{figure}[b!]\n\t\\centering\n\t\\includegraphics[width=0.98\\linewidth]{figure4.pdf}\\\\\n\t\\caption{Classification of MU-MIMO Scenarios and two examples of commercial technologies} \\label{fig:generic-deployments}\n\\end{figure}\t\n\n\n\n\n\n\\subsection{Scenarios with a single transmitter}\n\\label{section:system-set-ups:single-cell-scenarios} \n\nThe objective of MU-MIMO processing is to accommodate many users per resource. Therefore, resource allocation strategies are commonly analyzed at the basic resource unit, e.g., code, single-carrier, time-slot, or frequency-time resource block. This can be done regardless the global system model (single-carrier, OFDM, or CDMA), since the same resource allocation strategy is applied over all resources, e.g., \\cite{Lau2006,Fuchs2007,Koutsopoulos2008,Vicario2008,Tsai2008,Conte2010, Driouch2012, Huang2013, Cheng2014, Fan2014,Bogale2016a}. We adopt a signal model using the most general approach in the reviewed references.  \nConsider a scenario where the transmitter is equipped with $M$ antennas, and $K$ active users are equipped with $N$ antennas. Let $\\mathbf{H}_{k} \\in \\mathbb{C}^{N \\times M}$, be the discrete-time complex baseband MIMO channel  of the $k$-th user for a given carrier. The received signal can be expressed as:\n\n\\begin{equation}\\label{eq:received_signal}\n\\mathbf{y}_{k} = \\mathbf{H}_{k}\\mathbf{x} + \\mathbf{n}_{k}  \n\\end{equation}\nwhere $\\mathbf{x} \\in \\mathbb{C}^{M \\times 1}$, is the joint transmitted signal for all users.\nThe MIMO channel is usually assumed to be ergodic, i.e., it evolves over time and frequency in an independent and identically distributed (i.i.d.) manner. The channel is commonly modeled as \\textit{Rayleigh fading}, which is suitable for non line-of-sight communications. The complete spatial statistics can be described by the second-order moments of the channel \\cite{Raghavan2015}. Define the channel covariance matrix as $\\mathbf{\\Sigma}_{k} = \\mathbb{E}[\\mathbf{H}_{k}^{H} \\mathbf{H}_{k}]$, which depends on the antenna configuration, propagation environment, scattering conditions and mobility. The channel can be decomposed as $\\mathbf{H}_{k} = \\sqrt{\\mathbf{\\Sigma}_{k}} \\mathbf{H}_{iid}$, where $\\mathbf{H}_{iid}$ has i.i.d. entries with distribution $\\mathcal{CN}(0,1)$.  \t\n\n\nAssuming spatially uncorrelated Rayleigh fading channels, i.e.,  $\\mathbf{\\Sigma}_{k} = \\gamma \\mathbf{I}$, $\\forall k$ and some $\\gamma > 0$, is the most common practice in the literature \\cite{Biglieri1998}. Physically, this implies rich scattering environments with sufficient antenna spacing at both ends of the radio link \\cite{Foschini1998}. Under these conditions, the fading paths between the multi-antenna transmitter and receiver become independent. The MU-MIMO channels are modeled as narrow band, experiencing frequency-flat (constant) fading, where there is no inter-symbol interference \\cite{Mietzner2009}. A common simplification used in OFDM broadband systems, is to assume multiple flat-fading sub-channels \\cite{Lau2006,Wang2007}. The received signal per sub-channel can be modeled as  (\\ref{eq:received_signal}), avoiding frequency selectivity \\cite{Ho2009}. More realistic channel models for broadband MIMO systems and their performance analysis were proposed in \\cite{Couillet2011}.\nThe Rayleigh model can be thought as a particular case of the asymmetric \\textit{Ricean fading} channel model. In this case, the entries of $\\mathbf{H}_{iid}$ have non-zero mean and there exists a dominant line-of-sight (LoS) component that increases the average SNR \\cite{Goldsmith2005}. \n{\\color{black}{Such a model is less common in MU-MIMO systems operating at microwave (i.e., sub-6 GHz bands), since the channels become more static and the benefits of MUDiv vanish with the magnitude of the LoS paths \\cite{Hammarwall2008a}.\nIn recent years, wireless communication over the millimeter wave (mmWave) frequency range (30-300 GHz) has proved to be a feasible and reliable technology with a central role to play in 5G \\cite{Rappaport2014,Heath2016,Bogale2016}. The mmWave technology rely on directional antennas to overcome propagation loss, penetration loss, and rain fading. High directivity implies that Ricean fading channels can be used to characterize both LoS and non-LoS components present in the mmWave channels \\cite{Bai2015,Shokri-Ghadikolaei2015,Kutty2016,Bogale2016}. \t\t\n}}\t\t\n\n\n\n\nSeveral authors model the MIMO channel such that the  correlation at transmit and receive antennas is distinguishable, and $\\mathbf{\\Sigma}_{k} \\neq \\gamma \\mathbf{I}$, $\\forall k$, see references in Table~\\ref{tab:channel-information}. There are two approaches to model and analyze performance under MIMO correlation, the \\textit{jointly correlated model} \\cite{Weich2006,Gao2009} and the simplified \\textit{Kronecker model} \\cite{Jorswieck2006,Hanlen2012}. The former assumes separability between transmit and receive eigen-directions, and characterizes their mutual dependence. The latter assumes complete correlation separability between the transmitter and receiver arrays, see \\cite{Shiu2000,Raghavan2010,Bjornson2010a,Hanlen2012} and references therein.\\footnote{Experimental results in \\cite{Czink2009} and theoretical analysis in \\cite{Raghavan2010} have shown that the conventional Kronecker model may not be well suited for MU-MIMO scenarios resulting in misleading estimates for the capacity of realistic scattering environments. This occurs due to the sparsity of correlated channel matrices, and the fact that the parameters of the Kronecker model change with time and position. The model can be used in scenarios with particular conditions on the local scattering at the transmitter and receiver sides \\cite{Jorswieck2006}.}  We refer to \\cite{Couillet2011} for a comprehensive analysis of Rayleigh and Rician correlated MU-MIMO channels. Note that in some MIMO propagation scenarios with uncorrelated antennas, the MIMO capacity can be low as compared to the SISO one due to the \\textit{keyhole} or \\textit{pinhole} effect. This is related to environments where rich scattering around the transmitter and receiver leads to low correlation of the signals, while other\npropagation effects, like diffraction or waveguiding, lead to a rank reduction of the transfer function matrix \\cite{Keyhole2002,Keyhole2006}.\n\n\nThe MIMO channels $\\mathbf{H}_{k} \\ \\forall k$, may also include large-scale fading effects due to shadowing and path loss \\cite{Goldsmith2005,Tse2005}. Depending on the type of access technique (OFDMA, CDMA, or TDMA), the channel model can take into account multi-path components, correlation, Doppler spread, and angular properties \\cite{Koutsopoulos2008}.  Another common assumption to avoid frequency dependency (particularly in low mobility scenarios), is to account for \\textit{block-fading channels} \\cite{Biglieri1998}. This means that the CSI is constant (within a coherence time duration) for a block of consecutive channel uses before changing independently for the next block.\n\nThe noise $\\mathbf{n}_{k} \\sim \\mathcal{CN}(\\mathbf{0}, \\mathbf{I}_{N})$, is usually modeled by i.i.d. normalized entries according to a circular normal (complex) Gaussian distribution with zero mean and unit variance \\cite{Love2008}. \nThe transmitted signal, $\\mathbf{x}$, can be defined according to the encoding applied over the user data, the number of spatial streams per user, and the power allocation. If linear precoding is used [cf. Section~\\ref{section:precoding}], the transmitted signal is defined as\n\n\\begin{equation}\\label{eq:transmitted_signal}\n\\mathbf{x} = \\sum_{k=1}^{K} \\mathbf{W}_{k}\\mathbf{d}_{k} \n\\end{equation}\nwhere $\\mathbf{W}_{k} \\in \\mathbb{C}^{M \\times d_k}$, is the precoding matrix, $\\mathbf{d}_{k} \\sim \\mathcal{CN}(\\mathbf{0}, \\mathbf{I}_{d_k})$ is the data signal, and $d_k$ is the number of multiplexed data streams of user $k$.\nIn single-transmitter  MU-MIMO scenarios with signal models defined by (\\ref{eq:received_signal}) and (\\ref{eq:transmitted_signal}), the ICI is negligible or assumed to be part of the additive background noise. Therefore, IUI is the main performance limiting factor, which can be addressed by precoding the user data, [cf. Section~\\ref{section:precoding}]. \n\n\nThe active users might experience similar average long-term channel gains (large-scale fading or path loss) and SNR regimes. For some practical cellular systems, assuming homogeneous users is valid if open-loop power control is used to compensate for cell-interior and cell-edge path losses. Therefore, the resultant effective multiuser channels have quasi-identical variances  \\cite{Huang2012a,Destounis2015}. The user distribution affects the fading statistics of $\\mathbf{H}_k$ $\\forall k$, and the general design of resource allocation algorithms \\cite{Jorswieck2006}.\n\n\nA possible single-transmitter MU-MIMO scenario arises in satellite communications. However, due to the characteristics of the satellite channels, marginal MIMO gains can be realized. The absence of scatterers in the satellite vicinity yields a Rician-type channel with a strong line-of-sight component, turning off the capabilities of MIMO processing. Due to the large coverage area in satellite communications, the users have heterogeneous long-term channel gains, which directly affects the resource allocation decisions. Regardless of the limited literature about MU-MIMO satellite communications, recent works show promising results and discussions about how intensive frequency reuse, user scheduling, and multibeam signal processing will be implemented in next generation broadband satellite systems \\cite{Arapoglou2011,Zorba2008,Cottatellucci2006}.\n\n\n\n\n\n\n\\subsection{Scenarios with multiple transmitters}\n\\label{section:system-set-ups:multi-cell-scenarios}\n\nIn this scenarios, the channel models and assumptions aforementioned are applied. Yet, some additional considerations are made so that signaling and connections between network entities can be modeled.   \nDeploying several transmitters across a geographic area can provide reliable communication for heterogeneous mobile terminals (different path losses or SNR regimes relative to each transmitter). This kind of infra-structure based systems include cellular and wireless local area networks (WLAN). \nThe resource allocation and access control can be performed based on CSI and knowledge of the interference structure \\cite{Osseiran2011}. If the transmitters are allowed to cooperate, e.g., through a central processing unit (CU), IUI can be mitigated using CSIT for signal design [cf. Sections~\\ref{section:introduction:needof-csit} and \\ref{section:precoding}]. Global knowledge or estimation of the interference can be used to avoid poor spectral efficiency or inaccurate assignment of radio resources. \n\n\nThis paper focuses on scenarios where roaming (mobility) and  reuse of resources are central management tasks. The premise behind cellular communications, is to exploit the power falloff with distance of signal propagation to reuse the same channel at spatially-separated locations. This means that the serving area is divided in non overlapping cells. Any cell site within a neighborhood cannot use the same frequency channel, which makes the same reused frequency channels sufficiently far apart \\cite{Gesbert2010,Nguyen2014}. \nIn traditional cellular systems, a given user belongs to only one cell at a time and resource allocation is performed unilaterally by its serving {BS} (non-cooperative approach in Fig.~\\ref{fig:generic-deployments}). \nEach transmitter serves its own set of users, transmission parameters are adjusted in a selfish manner by measuring ICI (simple interference-awareness), and there is no information exchange between BSs \\cite{Nguyen2014}.\nIf frequency reuse is employed, the {BS} can make autonomous resource allocation decisions and be sure that no uncoordinated {ICI} appears within the cell \\cite{Bjornson2013}. However, in many practical systems, universal frequency reuse is applied, which means that neighboring cells can access the same frequencies and time-slots simultaneously. This might increase the ICI and potentially degrade performance \\cite{Dahlman2013}. \nThe mitigation of {ICI} is a fundamental problem since the transmit strategy chosen by one {BS} will affect the reception quality of the users served by adjacent {BS}s. \n\n\nA cluster of BSs can coordinate the resource allocation, scheduling decisions, and ICI mitigation techniques  (cooperative approach in Fig.~\\ref{fig:generic-deployments}). Dynamic clustering is an ongoing research topic (see \\cite{Gesbert2010,Bjornson2013,Nguyen2014,Li2014} and references therein), which promises to meet the requirements established in the third generation partnership project ({3GPP}) standards \\cite{Dahlman2013}.\nDifferent forms of {ICI} control have been proposed over the last years. Extensions of space multiple access techniques for multi-cell systems have received several names, \\textit{coordinated multi-point} ({CoMP}) \\cite{Marsch2011,Lee2012a,Dahlman2013}, \\textit{network {MIMO}} \\cite{Zhang2009}, or  \\textit{joint signal transmission\/processing} ({JT}) \\cite{Nguyen2014}. These techniques exploit the spatial dimensions, serving multiple users (specially cell edge users \\cite{Bjornson2013}), while mitigating {ICI} of clustered {BS}s.\\footnote{Theoretical analysis in \\cite{Lozano2013} shows that in the high {SNR} regime, the achievable system capacity is fundamentally interference-limited due to the out-of-cluster interference. This occurs regardless the level of coordination and cooperation between clustered BSs. However, coordinated scheduling and user clustering can provide means to improve spectral efficiency and mitigate interference at high SNR.} \nFor these approaches, a cluster can be treated as a super cell, for which mathematical models from the single-cell scenario can be applied straightforwardly, e.g. \\cite{Liu2011,Khoshnevis2013,Marsch2011}. \n\n\nIf user data is shared among {BS}s, the use of \\textit{proactive} interference mitigation within a cluster can take place. This implies that coordinated BSs do not separately design their physical (PHY) and media access control layer parameters. Instead, the BSs coordinate their coding and decoding, exploiting knowledge of global data and {CSI} \\cite{Gesbert2010}. However, to guarantee large performance gains for these systems, several conditions must be met \\cite{Bjornson2013,Nguyen2014}: global {CSI} and data sharing availability, which scales up requirements for  channel estimation, backhaul capacity, and cooperation; coherent joint transmission and accurate synchronization; and  centralized  resource allocation algorithms, which may be infeasible in terms of computation load and scalability.\n\n\nThere is another approach of multi-cell cooperation, coined as \\textit{coordinated scheduling} (CS) with \\textit{coordinated beamforming} ({CBF}), which is a form of coordinated transmission for interference mitigation \\cite{Dahlman2013,Nguyen2014}.  CS\/CBF refers to the partial or total sharing of {CSI} between {BS}s to estimate spatial signaling, power allocation, and scheduling without sharing data or performing signal-level synchronization \\cite{Gesbert2010}.  {CBF} implies that each {BS} has a disjoint set of users to serve, but selects transmit strategies jointly with all other {BS}s to reduce {ICI}. In this approach, exchange of user data is not necessary, but control information and {CSI} can be exchanged to simultaneously transmit to a particular set of users \\cite{Lee2012a}.\nCBF is more suitable than JT for practical implementations, since it requires less information exchange. Nevertheless, CSI acquisition, control signaling, and coordinated scheduling are challenging tasks due to limited feedback bandwidth and finite capacity backhaul \\cite{Li2014}.\n\n\n\n\\subsection{Commercial Deployments}\n\\label{section:system-set-ups:practical-deployments}\n\nThis paper covers two wireless technologies, whose specifications already support MU-MIMO communications:  LTE-Advanced for cellular networks and IEEE 802.11ac for wireless local area networks (WLAN). \n\n\n$\\bullet$ \\textit{LTE-Advanced}: This is the 3GPP cellular system standard for 4G and beyond communications \\cite{Dahlman2013}. Several capabilities have been added to the LTE standard to increase capacity demands and integrated a large number of features in the access network. Among these attributes, the ones related to MIMO processing are the most relevant in the context of this paper: enhanced downlink MIMO, multi-point and coordinated transmission schemes, and multi-antenna enhancements. Due to the fact that LTE is a cellular technology, most of the studied deployments in the literature lie in the category of multiple transmitters scenarios, see Fig.~\\ref{fig:generic-deployments}. \n\n\nMU-MIMO communication has been incorporated in  LTE  with the following maximum values: four users in MU-MIMO configuration, two layers (spatial streams) per user, four simultaneous layers, and robust CSI tracking. Practical antenna deployments at the transmitter use dual-polarized arrays, and the expected number of co-scheduled users for such configurations will be two for most cases \\cite{Lim2013,Fan2014}. LTE provides a mechanism to improve performance by switching between MU or SU-MIMO mode on a per sub-frame basis, based on CSI, traffic type, and data loads. The goal of dynamic mode switching is to balance the spectral efficiency of cell edge and average cell users. This can be achieved, for instance, by using transmit diversity for users at the cell edge, or implementing spatial multiplexing for cell center users \\cite{Liu2012}. \n\n\t\n$\\bullet$ \\textit{IEEE 802.11ac}: This WLAN standard supports multiuser downlink transmission, and the number of simultaneous data streams is limited by the number of antennas at the transmitter. In MU-MIMO mode, it is possible to simultaneously transmit up to 8 independent data streams and up to 4 users \\cite{Kim2015,Jones2015}. Spatial multiplexing is achieved using different modulation and coding schemes per stream. MU-MIMO transmission prevents the user equipment with less antennas to limit the achievable capacity of other multiple antenna users, which generates rate gains for all receivers. A unique \\textit{compressed explicit feedback} protocol, based on channel sounding sequences, guarantees interoperability and is used to estimate CSI and define the steering matrices (beamforming). Other methods for channel estimation are described in \\cite{Liao2016} and references therein. \nCompared to cellular networks, WLANs usually have fewer users moving at lower speeds, the APs are less powerful that the BSs and the network topology is ad hoc. Although multiple APs could be deployed, most of the works in the literature focus on MU-MIMO systems with a single transmitter.\n{\\color{black}{Nonetheless, joint transmission from several APs to different mobile users is feasible in WLANs, which requires coordinated power control, distributed CSI tracking, as well as synchronization in time, frequency, and phase. The authors in \\cite{Hamed2016} have shown that distributed MIMO can be achieved by enhancing the physical layer for coordinated transmission, and by implementing time-critical functions for the media access control layer.\nIt is likely that in the next generation of 802.11 standard, coordination schemes between APs will be adopted to enable MU-MIMO communications \\cite{Liao2016}.\n}}\n\n\n\\section{MU-MIMO Precoding}\n\\label{section:precoding}\n\nThe spatial dimension provided by the multi-antenna transceivers can be used to create independent channelization schemes. In this way, the transmitter serves different users simultaneously over the same time slot and frequency band, which is known as space-division multiple access (SDMA) \\cite{Godara2002Ch18, Tse2005, Gesbert2007a, Mietzner2009}.\nThe spatial steering of independent signals consists of manipulating their amplitude and phases (the concept of \\textit{beamforming} in classic array signal processing), in order to add them up constructively in desired directions and destructively in the undesired ones \\cite{Mietzner2009,Bjornson2013}. By jointly encoding all (co-resource) signals using channel information, it is possible to increase the signal-to-interference-plus-noise ratio (SINR) at the intended receiver and mitigate interference for non-intended receivers.\n\nIn the literature of MU-MIMO systems, the term beamforming refers to the signal steering by means of beams to achieve SDMA. The term \\textit{precoding} is used to denote the scaling and rotation of the set of beams, so that, their power and spatial properties are modified according to a specific goal. Hereinafter, we use the term precoding\\footnote{Some authors denote as precoding all processing techniques over the transmitted signals, which achieve multiplexing or diversity gains, i.e., both space-time coding and beamforming \\cite{Shirani-Mehr2010}.} to describe the signal processing (i.e., beam vector\/matrix computation, scaling, rotation, and projection), applied to the independent signals prior to transmission. \nIn this section we describe the most common precoding techniques used in MU-MIMO scenarios, and their characteristics according to CSIT. Table~\\ref{tab:precoding} summarizes the precoding schemes used in the surveyed literature, as well as their associated methods for user selection, which will be elaborated upon in the following sections. \nAn important performance metric determined by the precoders $\\mathbf{W}_{k}$, $\\forall k$, related to the delivered energy through the MU-MIMO channels, is provided in the following definition.  \n\n\n\n\\begin{definition}\\label{defn:effective_channel_gains}\n\t\\textit{Effective channel gain}. It is the magnitude of the channel projected onto its associated precoding weight. Let $\\mathbf{H}_{k}^{(eff)} = \\mathbf{H}_{k} \\mathbf{W}_{k}$, be the \\textit{effective channel} after spatial steering, thus, the effective channel gain is given by: \\textit{i}) $|\\mathbf{H}_{k}^{(eff)}|^2$ for MISO scenarios; \\textit{ii}) for MIMO scenarios, it is given by a function of the eigenvalues of $\\mathbf{H}_{k}^{(eff)}$:  $\\det \\left(  \\mathbf{H}_{k}^{(eff)}( \\mathbf{H}_{k}^{(eff)} )^ {H} \\right)$ or $\\| \\mathbf{H}_{k}^{(eff)}\\|_{F}^{2}$.\n\\end{definition}\n \n\n\n\\begin{table*}[t] \\scriptsize\n\t\\renewcommand{\\arraystretch}{1.3}\t\n\t\\caption{Precoding Schemes and their associated Scheduling Methods}\n\t\\label{tab:precoding\n\t\\centering\t\t\n\t\\begin{tabularx}{\\textwidth}{ *{6}{>{\\centering\\arraybackslash}X} }  \n\t\t\\toprule\t\n\t\t\\textbf{Method} &   \\textbf{Utility-based}    &   \\textbf{CSI-Mapping}    & \\textbf{Metaheuristic}\\newline \\textbf{(stochastic)} & \\textbf{Classic}\\newline \\textbf{Optimization}& \\textbf{Exhaustive} \\textbf{Search} \\\\\n\t\t\\midrule\n\t\t\\textbf{DPC}   & \\cite{Boccardi2006}  &   -    & \\cite{Elliott2009}  &   -    & - \\\\\t\t\n\t\t\\midrule\n\t\t\\textbf{THP}   & \\cite{Boccardi2006}  &   -    &   -    &   -    &  - \\\\\n\t\t\\midrule\n\t\t\\textbf{VP} &  \\cite{Boccardi2006}  & \\cite{Razi2010}  &    -   &   -    & - \\\\\n\t\t\\midrule\n\t\t\\textbf{MRT}   & \\cite{Moon2013, Zhang2011}  & \\cite{Wang2015}  & \\cite{Bjornson2014}  &   -    & - \\\\\n\t\t\\midrule\n\t\t\\textbf{ZFBF}  & \\cite{Dimic2005, Moon2013, Huang2013, Wang2008a, Huang2012b, Zhang2005a, Lau2005a, Chan2007, Conte2010, Chae2008, Kountouris2007, Shirani-Mehr2010, Zhang2011, Ravindran2012, Jindal2008, Boccardi2006, Tsai2008, Huh2012, Park2010, Lossow2013,Hammarwall2008,YiXu2014,Bogale2015}  & \\cite{Wang2008a, Yoo2006, Nam2014, Bayesteh2008, Adhikary2013, Adhikary2013a, Wang2010, Mao2012, Shi2012, Maciel2010, Zhang2005a, Chung2010, Driouch2012, Yoo2005a, Wang2006, Sun2010, Wang2015, Yang2011, Conte2010, Huang2012a, Chae2008, Yoo2007, Kountouris2007, Dai2008, Kountouris2006a, Wang2007a, Trivellato2008, Min2013, Sohn2010, Souihli2010, Swannack2004, Shirani-Mehr2011, Xu2010, Sohn2012, Khoshnevis2013, Castaneda2015, Jang2011,Hammarwall2008,Lee2014c,Liu2015a}  & \\cite{Bjornson2014, Lau2005a, Cottatellucci2006}   & \\cite{Chan2007}   & \\cite{Chan2007, Zakhour2007, Destounis2015, Swannack2004,Bogale2015}   \\\\\n\t\t\\midrule\n\t\t\\textbf{ZFDP} & \\cite{Dimic2005, Tran2010}   & \\cite{Tu2003, Jiang2006, Dai2009, Tejera2006, Sigdel2009a, Sun2010}   &    -   &   -    &  - \\\\\n\t\t\\midrule\n\t\t\\textbf{SZF}   &    -   & \\cite{Zorba2008}  & \\cite{Elliott2012}   &   -    &  -  \\\\\n\t\t\\midrule\n\t\t\\textbf{CIZF}  &   -    & \\cite{Christopoulos2013}  &    -   &   -    &  - \\\\\n\t\t\\midrule\n\t\t\\textbf{BD}    & \\cite{Shen2006, Chen2007a, Chen2008, Ko2012, Tran2012, Lim2009, Cheng2014, Rico-Alvarino2014, Zhang2009,YiXu2014}   &  \\cite{Nam2014, Shen2006, Lee2014, Adhikary2013, Adhikary2013a, Chen2007a, Fuchs2007, Wang2006, Sigdel2009a, Yi2011, Lim2009, Wang2010a, Nam2014a}  & \\cite{Hei2009}   & \\cite{Chan2007, Moretti2013}  & - \\\\\n\t\t\\midrule\n\t\t\\textbf{MMSE\/SLNR}  &  \\cite{Zhang2011, Torabzadeh2010, Cui2011a, Seifi2011,Liu2014a}  & \\cite{Zorba2008, Lau2009, Kountouris2005a,Kountouris2006b}, \\cite{Xia2009, Castaneda2015}, \\cite{Adhikary2013}  & \\cite{Cottatellucci2006}   &   -    &  - \\\\\n\t\t\\midrule\n\t\t\\textbf{Adaptive}  & \\cite{Koutsopoulos2008, Kobayashi2007, Boccardi2006, Hammarwall2007, Dartmann2013, Yu2013}   & \\cite{Zhang2007b}  &   -    &  \\cite{Stridh2006a, Matskani2008, Ku2015, Song2008}  &  - \\\\\n\t\t\\midrule\n\t\t\\textbf{Codebook-based} & \\cite{Fan2014, Wang2012d, Tang2010, Vicario2008, Kountouris2006, Ravindran2012}  & \\cite{Kountouris2006b,Lee2014, Yang2011, Huang2009a, Kountouris2008, Chen2013, Zhang2007a, Choi2007c, Kountouris2008a, Kountouris2005a, Kountouris2005, Huang2007, VanRensburg2009, Hosein2009, Schellmann2010, Simon2011,Xu2009a}&  -     &   -    &  - \\\\\n\t\t\\bottomrule\n\t\\end{tabularx}%\n\\end{table*}%\n\n\n\n\n\n\\subsection{Non-linear Precoding with full CSIT}\n\\label{section:precoding:non-linear}\n\nFrom an information-theoretic perspective, the optimal transmit strategy for the MU-MIMO BC is \\textit{dirty paper coding} ({DPC}) \\cite{Costa1983}, and theoretical results showed that such strategy  achieves the entire BC capacity region \\cite{Caire2003,Weingarten2006}. The principle behind this optimum coding technique is that the transmitter knows the interference for each user. Therefore, interference can be pre-subtracted (from the information theoretical standpoint) before transmission, which yields the capacity of an interference free channel. DPC is a non-linear process that requires successive encoding and decoding, whose performance depends on the particular sequential order $\\pi(\\cdot)$ assigned to the co-scheduled users \\cite{Caire2006}. Although the implementation complexity of DPC for practical systems is prohibitively high, it establishes the fundamental capacity limits for MU-MIMO broadcast channels \\cite{Weingarten2006,Hassibi2007}. \n\nSuboptimal yet more practical non-linear precoding schemes have been proposed as an alternative to DPC \\cite{Boccardi2006}. The error rate and interference can be minimized at the symbol level by the Tomlinson-Harashima precoding (THP),\\footnote{The application of THP in MU-MIMO has been of particular interest in recent research on multibeam satellite communications \\cite{Arapoglou2011}.} which is not limited by the number of transmit or receive antennas \\cite{Tse2005}.  By modifying or perturbing the characteristics of the transmitted signal, the power consumption can be minimized (compared to traditional channel inversion filtering), using the non-linear vector perturbation (VP) scheme \\cite{Hochwald2005, Kurve2009}. This technique requires a multidimensional integer-lattice least squares optimization, whose solution can be found by several approaches.\n\n\n\\subsection{Linear Precoding with full CSIT}\n\\label{section:precoding:linear}\n\nLinear precoding is a generalization of traditional SDMA \\cite{Gesbert2007a}, which matches the signal on both ends of the radio link. This is attained by decoupling the input data into orthogonal spatial beams and allocating power according to CSIT \\cite{Vu2007}. The precoders weights are  vectors or matrices jointly designed at the transmitter, according to several parameters: the type of CSIT, the coding order, the performance metric (e.g., mean-square error (MSE), error probability, power consumption, or achievable SNR), and the system constraints (e.g., power and QoS). \nThe optimal precoder technique (linear filtering in the spatial domain), would be able to balance between signal power maximization and interference power minimization \\cite{Larsson2008,Jorswieck2008}. Precoding design subject to general constraints can be performed using standard optimization techniques (see \\cite{Hong2012} for a comprehensive survey) or by heuristic approaches, e.g., maximizing the signal-to-leakage-plus-noise ration (SLNR\\footnote{SLNR is also known in the literature as the transmit Wiener filter, transmit MMSE beamforming, or virtual SINR beamforming \\cite{Bjornson2014b}. }) \\cite{Sadek2007, Bjornson2013}.\nHowever, determining the optimal precoders is an NP-hard problem for many performance metrics \\cite{Liu2011,Weeraddana2012}, whose evaluation is performed by computationally demanding algorithms \\cite{Schubert2004,Christensen2008,Godara2002Ch18}. Therefore, many works in the literature focus on more suboptimal, yet practical, schemes that can achieve spatial multiplexing gains with low computational complexity.  \nA large number of linear precoding techniques have been developed, for which having more transmit than receive antennas, i.e., $M\\geq N$, is a condition required in most cases. There are some particular precoding schemes (e.g. SLNR) that can be implemented if $M < N$, but the system becomes interference limited in the moderate and high signal-to-noise (SNR) regimes, [cf. Definition~\\ref{defn:interference-limited-system}]. Therefore, power control and recursive adaptation of the precoders are mandatory to operate in those SNR regimes \\cite{Sadek2007}. \n\n\nIn a MISO antenna configuration, the matched filter or maximum ratio transmission (MRT) precoding maximizes the signal power at the intended users. This is performed by projecting the data symbol onto the beamforming vector given by the spatial direction of the intended channel \\cite{Bjornson2013}. A similar precoding scheme for MIMO scenarios is the singular value decomposition (SVD) beamforming \\cite{Tse2005}, which uses the eigenvectors of the channel as beamforming weights.\nThe zero-forcing beamforming\\footnote{Precoding based on ZFBF obtains a multiplexing gain of $M$ and can asymptotically achieve the DPC performance when $K \\rightarrow \\infty$ \\cite{Yoo2006}.} (ZFBF) \\cite{Caire2003}, also known as channel inversion precoding \\cite{Peel2005}, completely suppresses IUI in MISO scenarios. This technique is based on prefiltering the transmit signal vector by means of the Moore-Penrose inverse \\cite{Gentle2007}. An extension of ZFBF for MIMO scenarios is the block diagonalization (BD) precoding, where multiple data streams can be transmitted per user \\cite{Spencer2004a}. \n\n\nIn MISO scenarios, the regularized channel inversion (often coined as MMSE precoding) enhances ZF, taking into account the noise variance to improve performance in the low SNR regime \\cite{Peel2005}. An extension of MMSE for the general MIMO scenario is the regularized BD \\cite{Stankovic2008}.\nZero-forcing dirty paper (ZFDP) coding \\cite{Caire2003}, is a technique designed for MISO settings. For a given user $k$, ZFDP suppresses the interference coming from the next encoded users $\\{k+1,\\ldots,K\\}$, combining QR decomposition \\cite{Gentle2007} and DPC. Extensions of ZFDP for the MIMO scenarios were defined in \\cite{Dabbagh2007,Spencer2004a} (an iterative SVD method), and  \\cite{Sun2010} (combining ZF, DPC, and eigen-beamforming). Successive zero-forcing (SZF) was proposed in \\cite{Dabbagh2007}, for MIMO settings. SZF partially suppresses IUI, by encoding users similar to ZFDP, but DPC is not applied in the encoding process.\nThe generalization of precoding schemes based on ZF for multiple antenna receivers is not trivial. This is because applying MISO decompositions methods to the MIMO channel is equivalent to treating each receive antenna as an independent user. This process does not completely exploit the multiplexing and diversity gains of MIMO systems \\cite{Dabbagh2007}.\n\n\nThe aforementioned precoding schemes can be classified as user-level precoders, i.e., independent codewords intended to different users are transmitted simultaneously. There is another class of precoding schemes, where simultaneous transmitted symbols are addressed to different users \\cite{Alodeh2015}. This class of symbol-level precoding, e.g. constructive interference zero forcing (CIZF) precoding \\cite{Masouros2009,Alodeh2015}, has been developed for MISO settings. CIZF constructively correlates the interference among the spatial streams, rather than decorrelating them completely as in the case of user-level precoding schemes. \n\n\n\n\\subsection{Precoding with partial CSIT}\n\\label{section:precoding:partial-csit}\n\nIn the literature of limited feedback systems, CSIT acquisition relies on {\\color{black}{collections of predefined codewords (vector or matrix weights) or codebooks,}} that are known a priori at the transmitter and receiver sides. The codewords can be deterministic or randomly constructed, which defines the type of signal processing applied to achieve spatial multiplexing and interference mitigation \\cite{Love2008}:\n\n\n$\\bullet$ \\textit{Channel quantization and precoding}: The codebook $\\mathcal{C} = \\{ \\mathbf{c}_{1}, \\ldots, \\mathbf{c}_{b}, \\ldots, \\mathbf{c}_{2^B} \\}$, is used by user $k$ to quantize its channel direction with $B$ bits. This means that each user feeds back the index $b$, of the most co-linear codeword to its channel, [cf. Section~\\ref{section:metrics-spatial-compatibility:spatial-clustering}]. \nThis is illustrated in Fig.~\\ref{fig:hyperslab_cones}, where the user $k$ would report the index $b$, related to the cone where $\\mathbf{H}_{k}$ has been clustered. The precoding weights are usually computed using ZFBF over the quantized channels. Due to quantization errors, the signals cannot be perfectly orthogonalized, and the sum-rate reaches a ceiling as the SNR regime increases \\cite{Jindal2006}. In other words, resource allocation is performed over non-orthogonal spatial directions.\n\nThe optimal codebook design has not been fully solved in the literature. However, if the channel is assumed to have spatially i.i.d. entries, ($\\mathbf{\\Sigma}_{k} =  \\mathbf{I}$, and homogeneous long-term channel gains, $\\forall k$), off-line designs of $\\mathcal{C}$ can be realized using different approaches, e.g., the Grassmannian design \\cite{Love2003}, random vector quantization (VQ) \\cite{Jindal2006}, quantization cell approximation (QCA) \\cite{Mukkavilli2003},  and other techniques described in \\cite{Love2008,Trivellato2008}. This sort of isotropically distributed codebooks achieve acceptable performance, since they mirror the statistical properties of the eigen-directions of $\\mathbf{H}_{k}$, $\\forall k$. For correlated channels ($\\mathbf{\\Sigma}_{k} \\neq  \\mathbf{I}$), non-uniform or skewed codebooks must be constructed, taking into account the statistical characteristics of the dominant eigen-directions of $\\mathbf{\\Sigma}_{k}$, $\\forall k$, see \\cite{Raghavan2007,Rao2013,Godana2013a,Raghavan2015}.\n\n\n\n$\\bullet$ \\textit{Random beamforming (RBF)} \\cite{Sharif2005}: The available precoding vectors,  $\\mathcal{F} = \\{ \\mathbf{f}_{1}, \\ldots, \\mathbf{f}_{b}, \\ldots, \\mathbf{f}_{2^B} \\}$, are constructed at the transmitter according to a known distribution,\\footnote{The goal is to generate $2^B$ codewords that are i.i.d., according to the stationary distribution of the unquantized beamforming vector \\cite{Love2008}.}  or by methods that yield random orthonormal basis in $\\mathbb{C}^{M \\times 1}$, \\cite{Choi2007c}.  The transmitter sends pilot symbols to different user through the beams in $\\mathcal{F}$, and each user feeds back the index $b$, of its best beam [cf. Section~\\ref{section:algorithms:selection-partial-csit}]. RBF is an extension of the opportunistic beamforming scheme in \\cite{Viswanath2002}, and attempts to sustain multiuser diversity over fading channels with partial CSIT. Random changes of amplitude and phase at each transmit antenna result in channels experiencing accelerated fluctuations, which enhances the MIMO processing gains. This approach is only effective when the number of users is large,\\footnote{RBF asymptotically achieves the performance of downlink MIMO systems with full CSIT as $K \\rightarrow \\infty$. However, it does not achieve the full multiplexing gain at high SNR \\cite{Hassibi2007}, and suffers from large quantization error as $M$ grows \\cite{Kim2005a}.} $K \\gg M$, and the number of antennas at the transmitter $M$ is moderate \\cite{Choi2007c}. In some scenarios the performance can be enhanced by power control, or employing statistical channel information to compensate poor MUDiv.  \n\n\n\nWe hasten to say that defining the optimum $B$, either for $\\mathcal{C}$ or $\\mathcal{F}$, is not a trivial optimization problem. In practice, only a tradeoff between feed back load and performance should be sought. Under mild conditions, $B$ can be minimized for certain codebook designs while guaranteeing diversity gains \\cite{Castanheira2014a}. Large codebooks provide accurate CSIT at a price of factors, large feedback signaling overhead and memory requirements at the receiver, which increase exponentially with $B$ \\cite{Wang2007a}.  \n\n\n\n\n\n\n\n\\subsection{Precoding in Multiple Transmitters Scenarios}\n\\label{section:precoding:multiple-transmitters}\n\nRecent research work have focused on transmission cooperation and coordination in multi-cellular and heterogeneous networks. The signal processing techniques performed at the transmitters depend on several aspects  \\cite{Gesbert2010,Nguyen2014}: shared or non-shared user data; global or local CSIT; full or partial CSI; level of synchronization; per-transmitter power, backhaul, and delay  constraints; full or limited coordination; heterogeneous SNR regime across the receivers; and the number of clustered transmitters.\n\nIn multi-cell scenarios, assuming full CSIT and shared user data across the transmitter, the precoders can be computed by techniques described in Section~\\ref{section:precoding:linear}. However, the power should be constrained on a per-transmitter basis, instead of global power allocation as in single-transmitter scenarios \\cite{Silva2013}.\nThe precoding design can be oriented to optimize different objective functions, e.g. power minimization or sum-rate maximization. Optimizing such functions deals, in general, with non-convex problems, whose solutions are non-linear precoders \\cite{Nguyen2014}. Precoding weights designed for common utility functions, e.g. weighted-sum-rate maximization or mean-square-error (MSE) minimization, can be realized via standard optimization techniques and sophisticated high-performance algorithms, see \\cite{Hong2012,Kaviani2012} and references therein. \n\nHowever, a more pragmatic approach for MU-MIMO, is given by linear precoding schemes (e.g. ZFBF, SLNR, or BD), which can be extended for multi-cell systems and providing reliable and low-complexity solutions \\cite{Zhang2009,Bjornson2010}. In \\textit{network MIMO} scenarios, multi-cell BD precoding can remove ICI for the clustered cells using centralized processing at a central controller (full coordination) \\cite{Zhang2009}.\nIn MISO multi-cell scenarios, linear precoding can be performed in a distributed fashion or with partial coordination. There are two fundamental schemes \\cite{Bjornson2013}: MRT (a competitive or egoistic scheme) and inter-cell interference cancellation (ICIC is a cooperative or altruistic scheme). For the MRT scheme, the transmitters ignore the ICI that they cause to unintended receivers in their vicinity. The goal is to maximize the received signal power of local users (effective in non-cooperative systems). The ICIC is a ZF-based scheme, which means that the transmitters design their precoders so that no interference is cause to non-intended receivers (effective in cooperative systems). The design of ICIC\/ZF precoding is subject to constraints in the number of transmit and receive antennas, and its effectiveness highly depends on the CSIT across the transmitters. A close-to-optimal\\footnote{The optimal precoding design maximizes the received signal powers at low SNRs, minimizes the interference leakage at high SNRs, and balances between these conflicting goals at moderate SNRs.} distributed precoding scheme (for an arbitrary SNR), is attained by balancing MRT and ICIC, whose mathematical formulation is discussed in \\cite{Bjornson2013,Bjornson2014b}.\nIf the system operates under a limited feedback constraint, the CSIT is acquired using the principles described in Section~\\ref{section:precoding:partial-csit}. Precoding design extensions from single to multi-cell scenarios are not straightforward. For instance, in multi-cell cooperative systems, the codebooks sizes may significantly increase \\cite{Gesbert2010}, and depending on the delay tolerance and codebook granularity, the clustered transmitters should switch between MRT and ICIC schemes \\cite{Godana2013}.\n\n\nSeveral multi-cell CBF designs attempt to suppress IUI and ICI simultaneously with minimum coordination or control signaling between transmitters. These schemes rely on instantaneous or statistical CSI, and can be classified as \\cite{Zheng2015}: \\textit{hierarchical} and \\textit{coupled} precoders. The hierarchical approach is implemented in systems where each transmitter suppresses ICI using ZF. This is attained through the sequential construction of outer and inner precoders that reduce ICI and IUI, respectively. The ICI cancellation is achieved by aligning interference subspaces at the receivers, whilst the IUI is suppressed at the transmitters using linear precoding and local CSI, e.g., \\cite{Adhikary2013,Suh2011,Liu2014a,YiXu2014,Ferrand2014,Chen2014}.\nThe coupled or nested-structure approach, is implemented in interference limited system, [cf. Definition~\\ref{defn:interference-limited-system}]. The precoding design (beamforming weights and power allocation) optimizes an objective function subject to a set of QoS constraints, [cf. Section~\\ref{section:performance-uf}]. This kind of optimization problems has been extensively studied in the literature, covering power allocation \\cite{Chiang2008} and beamforming design \\cite{Hong2012}. \nUnder certain conditions, two specific problem formulations admit global optimal solutions via standard  optimization in multi-cellular scenarios \\cite{Boyd2004,Stanczak2009,Tan2012}: \n\\textit{i) power minimization subject to individual SINR constraints}. The CBF approach in \\cite{Zakhour2013} mitigates ICI by iteratively adjusting beamforming weights and transmit powers according to the experienced interference per user.\n\\textit{ii) the maximization of minimum SINR subject to power constraints}. The authors in \\cite{Huang2013b} tackled this problem using the Perron-Frobenius theory \\cite{Stanczak2009,Chiang2008}. The joint beamforming and power allocation design is reformulated as an eigenvalue-eigenvector optimization problem, whose optimal solution can be found by geometrically-fast convergent algorithms. \n\n \n\n\n\n\\subsection{Precoding in LTE-Advanced}\n\\label{section:precoding:lte-advanced}\n\nIn the LTE specification, CSI acquisition is called \\textit{implicit feedback} and relies on the following parameters \\cite{Li2010,Liu2012}: rank indicator (RI), which defines the number of data streams recommended for SU-MIMO transmission; precoding matrix index (PMI), which is the index of the best precoding matrix in the codebook; and channel quality indicator (CQI), which contains information of the channel quality corresponding to the reported RI and PMI \\cite{Dahlman2013}. The RI and PMI indexes provide the CDI of the MIMO channels, while the CQI indicates the strength of the corresponding spatial direction. \nThe precoding matrices are defined at the BSs using different approaches \\cite{Li2014,Lim2013}: codebook-based precoding or non-codebook-based precoding (arbitrary precoder selection based on RI). The standard also supports a dual codebook structure in MU-MIMO mode. This increases codebook granularity, suppresses IUI more efficiently, and enhances the overall performance. The idea is that one codebook tracks the wideband and long-term channel characteristics, while the other tracks the frequency-selective and short-term channel variations \\cite{Lim2013}. In this way, the transmitter has more flexibility and accuracy when designing the precoding matrix. \nAnother supported scheme is the \\textit{Per Unitary basis stream User and Rate Control} (PU2RC), which allows multiple orthonormal bases per codebook, increasing quantization granularity \\cite{Huang2009a, Tang2010}. Although PU2RC uses deterministic codebooks, random codebooks can be used to simplify theoretical analysis, e.g. \\cite{Sharif2005,Jindal2006,Huang2009a,Tang2010}. The optimum number of bases in the codebook depends on the number of active users $K$, and should be optimized to maximize MUDiv and multiplexing gains \\cite{Huang2009a}.\n\n\nLTE standard supports interference mitigation based on linear precoding schemes, such as SLNR and ZFBF with quantized channels \\cite{Lee2012a}. The precoders can be dynamically recalculated after each CSI update, i.e., tracking channel variations. Linear precoding based on quantized channels outperforms codebook-based precoding (RBF or PU2RC), when then number of active users is small, i.e., \\textit{sparse networks}, $K \\approx  M$ \\cite{Gesbert2007a}. The reverse holds as MUDiv increases, $K \\gg  M$, \\cite{Huang2009a}. \n \n\n\n\\subsection{Precoding in 802.11ac}\n\\label{section:precoding:208_11ac} \n\n{\\color{black}{\nSounding frames (null data packet) for MU-MIMO beamforming were introduced in 802.11n for channel estimation purposes. The transmitter sends known symbol patters from each antenna, allowing the receiver to sequentially construct the channel matrix, which is compressed and sent back to the transmitter \\cite{Rico-Alvarino2014}. The method employed to calculate the steering matrix is implementation and vendor specific relying on explicit CSI feedback, and is not defined by the 802.11ac standard \\cite{Bejarano2013,Arubanetworks2014,Liao2016}.}} \nOne popular technique to construct the steering or precoding matrix is through SVD precoding \\cite{Rico-Alvarino2014,Kim2015}. Since knowledge of CSIT is mandatory and feedback rates are limited, the receivers represent their estimated precoding matrix with orthogonal columns using Givens rotations \\cite{Gentle2007}. Then, the set of calculated parameters (angles) at the receivers are adjusted, quantized, and fed back to the transmitter. In general, the final precoding matrix calculated by the transmitter will be different from the weights reported by the users due to the orthogonalization process \\cite{Kim2015}.\n\nAnother practical approach is to compute the steering matrix using linear precoding schemes, e.g. ZFBF or MMSE  \\cite{Jones2015}. In \\cite{Cheng2014}, the authors used BD and regularized BD with geometric-mean decomposition for a MU-MIMO scenario. The designs' goal is to balance the achievable SNR across the users, meeting the requirements of the standard. If the number of transmit antennas is larger than the total number of receive antennas, the additional DoF can be used to efficiently nullify inter-stream interference \\cite{Jones2015}. \n\n\n\n{\\color{black}{\n\t\t\n\\subsection{Discussion and Future Directions}\n\\label{section:precoding:massive-mimo} \n\nThe objective of precoding is to achieve spatial multiplexing, enhance link reliability and improve coverage in MIMO systems. Every physically realizable precoding design depends on the objective function, the CSIT accuracy, the number of transmitters involved, and more recently, the hardware characteristics \\cite{Bjornson2014c,Kutty2016,Heath2016}. The schemes described in previous subsections can be classified as \\textit{full digital precoders}, where the signal processing happens in the baseband at sub-6 GHz bands. However, these schemes cannot be directly implemented in state-of-the-art transceiver architectures with many antenna elements, neither upon higher frequency bands, i.e., mmWave \\cite{Heath2016}.\n\n\nMassive MIMO \\cite{Lu2014} differentiates itself from classical MU-MIMO by the fact that the number of antennas at the transmitter is larger compared to the number of served users. In conventional digital precoding, [cf. Section~\\ref{section:precoding:linear}], each antenna element requires a radio frequency (RF) chain, i.e., signal mixer and analog-to-digital converter (ADC). In massive MIMO, although the number of antennas and RF chains is much larger than in conventional MU-MIMO, hundreds of low-cost amplifiers with low output power are used to replace the high power amplifier used in the latter. In order to keep the hardware cost and the circuit power consumption low, cost-effective and power-efficient hardware components are employed, e.g. low resolution ADC. And yet, this results in hardware impairments, especially low resolution quantization, that may affect the system performance. However,  recent results and implementations show that the effect of hardware impairments, similarly to the effect of noise and interference, is averaged out due to the excess number of antennas \\cite{Bjornson2014c}. Furthermore, the effect of quantization and AD conversion with very low number of bits can be taken into account in the precoding and signal design, providing schemes with promising spectral efficiency performance. In TDD massive MIMO, the major limiting factors were considered to be pilot contamination and channel reciprocity\/calibration. However, both issues are now well studied and understood, and efficient transceiver designs compensating for pilot contamination and imperfect calibration are available. The major bottleneck for practical implementations of TDD-based massive MIMO remains the amount of training required. High peak rates have been demonstrated in downlink massive MIMO (in TDD and sub-6 GHz bands) with linear and non-linear precoding by several companies. Nevertheless, the amount of uplink training required can reduce the net throughput by at least half. A major challenge for massive MIMO is its successful deployment in FDD systems, in which CSI should be obtained by feedback. Efficient approaches for channel representation and CSI quantization and compression are necessary for viable implementations. Codebook-based approaches would require a relatively high number of bits for channel quantization and feedback, which not only will reduce the net throughput but also is not supported by current standards. Various new approaches for precoding and feedback in FDD massive MIMO are expected in the near future. \n\nThe current trend of using frequencies above 6 GHz for broadband wireless communications put an extra stress to massive MIMO systems. Current MIMO transceiver architectures may not be cost-effective and realistic in mmWave frequencies due to extremely high cost and power consumption. Different transceiver architectures have been recently proposed to address the hardware limitations. These new schemes require the joint optimization of precoding weights in the digital and analog domains, the so called \\textit{hybrid precoding} \\cite{Ayach2014,Gao2015,Kutty2016,Heath2016}. In the digital domain, the low-dimensional precoding weights are computed using microprocessors. In the analog domain the RF precoders are implemented by phase shifters and variable gain amplifiers \\cite{Bogale2015}. \nThe main goal of hybrid precoding schemes is to achieve the performance of full digital precoders, but with a reduced number of RF chains \\cite{Sohrabi2016}. The performance gap between the full digital and hybrid precoders depends on the spatial load at the transmitter, i.e., the ratio between the number of active data streams over the number of RF chains, $M_{RF}$, \\cite{Bogale2016a}. Notice that the number of co-scheduled users per resource is limited by $M_{RF}$ in hybrid transceiver architectures, [cf. Section~\\ref{section:algorithms:aggregated-utility:massive-mimo}]. \nThe optimal design of hybrid precoders has not been fully understood, and due to the power and amplitude constraints the sum-rate optimization problem becomes non-convex \\cite{Gao2016}. Therefore, ongoing research is focused on designing sub-optimal, yet efficient and practical, architectures that improve the joint performance of digital and analog precoders \\cite{Sohrabi2016,Yu2016,Kutty2016,Bogale2016a,Heath2016}. Although hybrid precoding provides a compromise between system performance and hardware complexity, it still remains challenging to implement reliable and cost-effect analog beamforming schemes at mmWave. Despite the complexity reduction using hybrid precoding, the performance gains using fully digital beamforming remain attractive. Hardware complexity may not be the major issue with digital beamforming at mmWave; significant challenges will arise in signal compression and CPRI protocols \\cite{DelaOliva2016}, which will require innovative solutions.\n}}\n\n\n\n\n\\section{Spatial Compatibility Metrics}\n\\label{section:metrics-spatial-compatibility}\n\nLet $\\mathcal{K} \\subseteq \\bar{\\mathcal{K}}$ and $\\mathcal{K}' \\subseteq \\bar{\\mathcal{K}}$ be two sets of user with at least one non-common user, where $\\mathbf{H}(\\mathcal{K})= \\{\\mathbf{H}_{i}\\}_{i \\in \\mathcal{K}}$ and $\\mathbf{H}(\\mathcal{K}')=\\{\\mathbf{H}_{j}\\}_{j \\in \\mathcal{K}'}$ are their associated channels. A metric for spatial compatibility is a function of the CSIT that maps the spatial properties of the multiuser MIMO channels to a positive scalar value quantifying how efficiently such channels can be separated in space \\cite{Maciel2010}, i.e., $f(\\mathbf{H}(\\mathcal{K})):\\mathbb{C}^{ |\\mathcal{K}|N \\times M} \\mapsto \\mathbb{R}_{+}$. \nConsider the two subsets $\\mathcal{K}$ and $\\mathcal{K}'$, a metric of spatial compatibility can be used to estimate the achievable performance of their associated multiuser channels, e.g., having $f(\\mathbf{H}(\\mathcal{K})) > f(\\mathbf{H}(\\mathcal{K}'))$ may imply that $\\mathcal{K}$ is the set that achieves the maximum capacity. The mapping function $f(\\cdot)$ can be used for scheduling purposes, depending on its definition and other system parameters (e.g. SNR regime and $M$), which will be discussed in Section~\\ref{section:algorithms:aggregated-utility:csi-mapping}. \nBelow we provide two definitions related to the design of user scheduling policies. \n\n\n\\begin{definition}\\label{defn:compatible_set_of_users}\n\t\\textit{Spatially compatible users}. A feasible set of users $\\mathcal{K}$, is spatially compatible if the multiuser MIMO channels, $\\mathbf{H}(\\mathcal{K})$, can be separated in the spatial domain by means of beamforming\/precoding.\n\\end{definition}\n\n\\begin{definition}\\label{defn:user_grouping}\n\t\\textit{User grouping}. It is the task of forming a subset of users $\\mathcal{K}$, according a compatibility criterion, e.g. spatial separability in Definition~\\ref{defn:compatible_set_of_users}, to maximize the resource allocation and scheduling efficiency.  \n\\end{definition}\n\n{\\color{black}{\n\t\t\nUser grouping can be the initial step in a MU-MIMO scheduling algorithm since the characteristics of the joint channels dictate the transmission reliability and the resource allocation feasibility [cf. Definition~\\ref{defn:resource_feasibility}].  \nIn MU-MIMO scenarios, there exists a correspondence between the precoding capability to reduce IUI and the user grouping technique. The performance achieved by a precoder scheme is determined by the characteristics of the selected multiuser channels, i.e., providing \\textit{spatially compatible users} to the precoding processing block (see Fig.~\\ref{fig:system-processing-blocks}), is fundamental to guarantee high attainable SINRs at the receivers. \nIt is worth mentioning that the vast majority of scheduling algorithms in the literature focuses on constructing sets of users with orthogonal or semi-orthogonal MIMO channels. Yet, for some particular signal designs, the best scheduling strategy is to group users whose channels are parallel or semi-parallel, see \\cite{Jang2011,Christopoulos2013}. This can also be the case in scheduling for non-orthogonal multiple access (NOMA)-MIMO schemes \\cite{Liu2015,Dai2015}, where the notion of spatial compatibility may be revised.\nThe spatial compatibility metrics are used in the MU-MIMO literature to pair users and optimize the performance of a particular utility function. They are also used to quantize the channels in systems with limited feedback rates. \n\n}}\n\n\\subsection{Null Space Projection}\n\\label{section:metrics-spatial-compatibility:projection-spaces}\n\nOne of the objectives of MU-MIMO technology is to multiplex independent data streams to different users, which implies that only a subset of the transmitted data symbols are useful for each co-scheduled user. In such a scenario, a fundamental problem is to mitigate IUI, i.e., suppress the information intended to other receivers. There exist several signal processing techniques that can achieve such a goal, e.g., linear precoding or interference alignment. The effectiveness of such techniques rely upon the characteristics of the subspaces spanned by the MIMO channels, i.e, IUI is a function of the overlapped interference subspaces \\cite{Jafar2011}.\n\nConsider a set of user $\\mathcal{K}$ with $K$ users, let $\\tilde{\\mathbf{H}}_{k} = [\\mathbf{H}_{1}^{T}, \\ldots, \\mathbf{H}_{k-1}^{T}, \\mathbf{H}_{k+1}^{T},\\ldots, \\mathbf{H}_{K}^{T}]^{T}$, be the aggregated interference matrix of the user $k$ such that $M>\\max_{k} \\ \\text{rank}(\\tilde{\\mathbf{H}}_{k})$, which is a necessary condition to suppress IUI \\cite{Wang2006}. Define $\\mathcal{V}_{k} = \\text{Span}(\\tilde{\\mathbf{H}}_{k})$, as the subspace spanned by the channels of the subset of users $\\mathcal{K} \\setminus \\{k\\}$, and let $\\mathcal{V}_{k}^{\\perp}  = \\text{Span}(\\tilde{\\mathbf{H}}_{k})^{\\perp}$, be its orthogonal complement subspace. \nIn other words, $\\mathcal{V}_{k}^{\\perp}$ spans the null space of $\\tilde{\\mathbf{H}}_{k}$, i.e., $null(\\tilde{\\mathbf{H}}_{k}) = \\{\\mathbf{x} \\in \\mathbb{C}^{M \\times 1} : \\tilde{\\mathbf{H}}_{k} \\mathbf{x} = \\mathbf{0} \\}$.\nThe channel of the $k$-th user can be expressed as the sum of two vectors $\\mathbf{H}_{k} = \\mathbf{H}_{k}^{(\\parallel)} + \\mathbf{H}_{k}^{(\\perp)}$, each one representing the projection of $\\mathbf{H}_{k}$ onto the subspaces $\\mathcal{V}_{k}$ and $\\mathcal{V}_{k}^{\\perp}$ respectively, as illustrated in Fig.~\\ref{fig:space-projections}. \n\n\nFor all user-level ZF-based precoding schemes described in Section~\\ref{section:precoding:linear}, $\\mathcal{V}_{k}^{\\perp} = \\bigcup_{i=1, i \\neq k}^{K}  \\text{Span}(\\mathbf{H}_{i})^{\\perp}$, i.e., $\\mathcal{V}_{k}^{\\perp}$ contains all overlapped interference subspaces of channel $\\mathbf{H}_{k}$. The component $\\mathbf{H}_{k}^{(\\parallel)}$ is related to the signal degradation of the user $k$ due to channel correlation, whereas $\\mathbf{H}_{k}^{(\\perp)}$ defines the \\textit{zero-forcing direction}, i.e. the spatial direction that is free of IUI. The squared magnitude of $\\mathbf{H}_{k}^{(\\perp)}$ is known as the null space projection (NSP), and directly computes the effective channel gain obtained by ZF precoding \\cite{Caire2003}. \n\n\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{figure5.pdf}\\\\\n\t\\caption{Decomposition of the channel $\\mathbf{H}_{k}$ given by its projection onto the subspaces $\\mathcal{V}_{k}$ and $\\mathcal{V}_{k}^{\\perp}$.}\n\t\\label{fig:space-projections}\n\\end{figure}\n\n\nA common approach in the literature is to define the mapping $f(\\mathbf{H}(\\mathcal{K}))$ as a function of the exact or approximated \\textit{effective channel gains} [cf. Definition~\\ref{defn:effective_channel_gains}]. In other words, the spatial compatibility metric has to consider the precoding scheme, the SNR regime, and the available spatial DoF. Analytical results in \\cite{Lau2006,Chan2007,Shi2012} have shown that the set of users $\\mathcal{K}$ that maximizes the product of their effective channel gains\\footnote{Some works in the literature (e.g. \\cite{Swannack2004,Zakhour2007,Castaneda2014b}) optimize the sum of effective channel gains instead of their product, which yields similar performance for the high SNR and large number of users, i.e. $K \\gg M$.} is also the set that maximizes the sum-rate in the high SNR regime. The NSP has been extensively used for user grouping and scheduling purposes whenever ZF-based precoding is implemented [cf. Section~\\ref{section:algorithms:aggregated-utility}]. Note that a set of spatially compatible users would have large NSP components, which means that their associated channels are semi-orthogonal. \n\nThe computation of NSP for the general MU-MIMO scenario where $\\mathbf{H}_{k} \\in \\mathbb{C}^{N_k \\times M}$ $\\forall k$, can be performed by several approaches, namely SVD \\cite{Golub1996,Gentle2007,Yanai2011,Wang2010,Tran2010}, orthogonal projection matrix \\cite{Golub1996,Yanai2011,Fuchs2007,Caire2003}, Gram-Schmidt orthogonalization (GSO) \\cite{Golub1996,Gentle2007,Yoo2006,Sigdel2009a,Shen2006}, QR decomposition \\cite{Tran2012}, products of the partial correlation coefficients \\cite{Yanai2011,Castaneda2014b}, and ratio of determinants \\cite{Castaneda2014b,Razi2010}. \nSeveral works find sets of spatially compatible users by computing an approximation of the NSP (e.g. \\cite{Fuchs2007,Bjornson2013,Cheng2014,Castaneda2015,Xu2010,Bayesteh2008,Hammarwall2008}), which yields efficient scheduling designs that require low computational complexity.\nMoreover, recent works (\\cite{Manolakos2015} and references therein), have proposed efficient methods to compute and track null spaces for a set of co-scheduled users, which is fundamental to mitigate IUI for ZF-based precoding in single and multiple transmitter scenarios.\n\n\n\\subsection{Spatial Clustering}\n\\label{section:metrics-spatial-compatibility:spatial-clustering}\n\n{\\color{black}{\nSpatial clustering refers to the association of a channel matrix to a spatial subspace. This technique has been used in the literature to perform user\/channel grouping or scheduling in compliance with the precoding scheme and CSIT availability. In feedback limited scenarios, spatial clustering is also used to quantize the channel or precoder weights [cf. Section~\\ref{section:precoding:partial-csit}].  \nLet $\\measuredangle(\\mathbf{H}_{i},\\mathbf{H}_{j})$ denote the angle between i.i.d. channels $\\mathbf{H}_{i},\\mathbf{H}_{j} \\in \\mathbb{C}^{1 \\times M}$, and define the normalized inner product, also known as \\textit{coefficient of correlation}, as \\cite{Gentle2007}:\n}}\n\n\\begin{equation}\\label{eq:coefficient_correlation}\n\\cos (\\measuredangle(\\mathbf{H}_{i},\\mathbf{H}_{j})) =  \\frac{|\\mathbf{H}_{i}\\mathbf{H}_{j}^{H}|}{ \\| \\mathbf{H}_{i} \\| \\| \\mathbf{H}_{j} \\|},\n\\end{equation}\nwhere $\\cos(\\measuredangle(\\mathbf{H}_{i},\\mathbf{H}_{j})) \\in [0, 1]$, and $\\measuredangle(\\mathbf{H}_{i},\\mathbf{H}_{j}) = \\frac{\\pi}{2}$, implies that the channels are spatially uncorrelated or orthogonal.\nThe coefficient of correlation indicates how efficiently the transmitter can serve user $i$ without affecting user $j$, and vice versa. For particular MU-MISO scenarios, in which the set of scheduled users is subject to the cardinality constraint $|\\mathcal{K}|=2$, the NSP is a function of $\\sin^2 (\\measuredangle(\\mathbf{H}_{i},\\mathbf{H}_{j}))$, which can be evaluated from  (\\ref{eq:coefficient_correlation})  simplifying the user pairing and scheduling design \\cite{Yang2011}. Several works focused on sum-rate analysis (e.g., \\cite{Lu2009,Yang2011,Wang2015}) limit the cardinality to $|\\mathcal{K}|=2$ for simplicity and tractability. Other works analyzed scenarios with more practical constraints and considered two or four users, which are common values of $K$ in standardized systems, see \\cite{Kountouris2008,Lim2013,Jones2015}. \nThe metric (\\ref{eq:coefficient_correlation}) has been extensively used to define policies for user grouping. In MU-MIMO systems a set of user $\\mathcal{K}$ is called $\\epsilon$-orthogonal if $\\cos (\\measuredangle(\\mathbf{H}_{i},\\mathbf{H}_{j})) < \\epsilon$ for every $i \\neq j$ with $i,j \\in \\mathcal{K}$ \\cite{Swannack2004,Yoo2005a,Bayesteh2008}. The users can be grouped into disjoint sets according to a desired threshold $\\epsilon$, and the semi-orthogonal sets can be scheduled over independent carriers (e.g. \\cite{Zhang2005a,Chung2010,Driouch2012}), codes (e.g. \\cite{Driouch2012}), or time slots (e.g. \\cite{Yoo2006}). \nThe optimum value of $\\epsilon$ depends on the deployment parameters ($K$ and $M$) and is usually calculated through simulations, since for $M>2$ it is very hard or impossible to find the optimal $\\epsilon$ in closed-form. Nevertheless, there exist closed-form expressions to compute the ergodic capacity as a function of $\\epsilon$ for MU-MISO scenarios where the transmitter has two antennas $M=2$ and the users have homogeneous \\cite[Ch. 7]{Yang2011} or heterogeneous \\cite{Wang2015} large-scale fading gains.\n\n\n\n{\\color{black}{\nConsider that a codebook, $\\mathcal{F}$, is used to define the CDI of the MIMO channels, i.e., assume partial CSIT, [cf. Section~\\ref{section:precoding:partial-csit}]. In such a scenario, spatial clustering can be performed for a given  parameter $\\theta$ and the codeword $\\mathbf{f}_{i} \\in \\mathbb{C}^{1 \\times M}$. Define the hyperslab $\\mathfrak{F}_{i}(\\theta)$ as \\cite{Lee2014}:\n\\begin{equation}\\nonumber\n\t\\mathfrak{F}_{i}(\\theta) =\\left\\{ \\mathbf{H}_{k} \\in \\mathbb{C}^{1 \\times M}, k \\in \\mathcal{K}: \\cos (\\measuredangle(\\mathbf{H}_{k},\\mathbf{f}_{i})) \\leq \\cos(\\theta) \\right\\}.\t\\label{eq:hyperslab}\n\\end{equation}\t\t\nThe hyperslab defines a vector subspace whose elements attain a spatial correlation not greater than $\\cos(\\theta)$, w.r.t. the codeword $\\mathbf{f}_{i}$, as illustrated in Fig.~\\ref{fig:hyperslab_cones}. The parameter $\\cos(\\theta)$ is set to guarantee a target $\\epsilon$-orthogonality. \n}}\nThe generalization of the hyperslab clustering\\footnote{Some authors (e.g., \\cite{Swannack2004,Lau2009}) define $\\mathfrak{F}_{i}(\\theta)$ as a function of two parameters, $\\theta$ and the minimum acceptable channel magnitude. In this way only sets of spatially compatible and strong channels can be constructed.} for MIMO settings is straightforward, i.e. $\\mathbf{H}_{k} \\in \\mathbb{C}^{N_{k} \\times M}$, $\\mathbf{f}_{i} \\in \\mathbb{C}^{M \\times M}$, and the coefficient of correlation between matrices can be defined as in \\cite{Sigdel2009a,Tran2012}. \nThe parameter $\\phi$ shown in Fig.~\\ref{fig:hyperslab_cones} can be adjusted to fix the maximum co-linearity between cones \\cite{Wang2012d}. \n\nSeveral scheduling algorithms based on spatial clustering (e.g., \\cite{Wang2006,Razi2010,Yoo2006,Kountouris2007,Wang2008a,Lau2009,Lu2009,Sun2010,Sohn2010,Mao2012,Tran2012,Min2013,Lee2014}), can achieve MUDiv gains and improve the overall performance by adjusting the parameter $\\theta$ (or threshold $\\epsilon$) according to the number of competing users \\cite{Yoo2006}, the SNR regime, the large-scale fading gain, and the precoding scheme \\cite{Wang2015}. \nIn \\cite{Lau2009}, it was shown that the optimal cardinality of the user set grouped based on spatial clustering is a function of $K$, $\\theta$, and $M$.\nThe statistical properties of (\\ref{eq:coefficient_correlation}) have been extensively studied in the literature cf. \\cite{Mukkavilli2003,Swannack2004,Yoo2006,Jindal2006,Au-Yeung2007,Zhang2007b,Jagannathan2007}, and such properties depend on the type of CSIT (full or partial), the MIMO channel distribution, and the system parameters (e.g. $M$ and $B$). \n\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=0.8\\linewidth]{figure6.pdf}\\\\\n\t\\caption{The channels $\\{\\mathbf{H}_{j}\\}_{j=1}^{K}$ can be clustered within cones given a set $\\mathcal{F} = \\{\\mathbf{f}_{1},\\mathbf{f}_{2},\\mathbf{f}_{3}\\}$ formed either by orthonormal basis or codebook words. }\n\t\\label{fig:hyperslab_cones}\n\\end{figure}\n\n\n\n\\subsection{Compatibility between Subspaces}\n\\label{section:metrics-spatial-compatibility:other-metrics}\n\n{\\color{black}{\nIn heterogeneous MU-MIMO scenarios where each user is equipped with $N_{k}$ antennas $\\forall k$, the signal spaces span several dimension and mutual interference between user can be measured either in the angular or in the subspace domains \\cite{Wang2006,Yi2011}. Moreover, metric (\\ref{eq:coefficient_correlation}) cannot be used directly in heterogeneous MU-MIMO scenarios to measure spatial correlation between channels of different dimensions, thus, several alternative metrics have been proposed in the literature.}}\nConsider a set of users $\\mathcal{K}$ and its MU-MIMO channel $\\mathbf{H}(\\mathcal{K})$; the spatial compatibility can be measured as a function of the corresponding eigenvalues:\n\\begin{enumerate\n\t\n\t\\item  The \\textit{orthogonality defect} \\cite{Bayesteh2008} is a metric derived from Hadamard's inequality \\cite{Cover2006} that measures how close a basis is to orthogonal. It quantifies the energy degradation of the channel matrix due to the correlation between all its column vectors. User grouping algorithms based on this metric have been developed for homogeneous MIMO systems, e.g. \\cite{Tran2010,Castaneda2014b}. \n\t\n\t\\item The determinant geometrically represents the volume of the parallelepiped defined by the column vectors of the channel matrix. Larger determinant values implies that the column vectors of a matrix are more orthogonal \\cite{Gentle2007}. The \\textit{matrix volume ratio} used in \\cite{Wang2010a,Castaneda2015} measures the volume reduction of $\\mathbf{H}(\\mathcal{K})$ w.r.t. $\\mathbf{H}(\\mathcal{K} + \\{k'\\})$, where $k' \\notin \\mathcal{K}$ is a candidate user attempting to be grouped. Different approximations of such a metric can be done using the arithmetic-geometric mean inequality over the squared singular values of $\\mathbf{H}(\\mathcal{K})$, e.g., \\cite{Xu2009a,Castaneda2014b}.\n\t\n\t\\item The \\textit{geometrical angle} (see \\cite{Yi2011,Nam2014a} and references therein), is a metric similar to the matrix volume ratio that measures the spatial compatibility as a function of the all possible correlation coefficients [cf. (\\ref{eq:coefficient_correlation})] between the basis of two subspaces. It can be computed from the eigenvalues of two MIMO matrices  $\\mathbf{H}_{i}  \\in \\mathbb{C}^{N_{i} \\times M}$, $\\mathbf{H}_{j}  \\in \\mathbb{C}^{N_{j} \\times M}$, whose subspaces have different dimensions, i.e., $N_{i} \\neq N_{j}$.\n\t\n\t\\item If the elements of a multiuser channel matrix are highly correlated, the matrix is said to be ill-conditioned, i.e., it is close to singular and cannot be inverted. In numerical analysis, the \\textit{condition number} \\cite{Gentle2007} quantifies whether a matrix is well- or ill-conditioned, and is computed as the ratio between the maximum and minimum eigenvalues. This metric is used to measure how the eigenvalues of $\\mathbf{H}(\\mathcal{K})$ spread out due to spatial correlation. The ratio between the condition numbers of two MIMO channels can be used to quantify their spatial distance and compatibility \\cite{Czink2009}. \n\\end{enumerate}\t\n\nFor a given user $k \\in \\mathcal{K}$, the spatial compatibility between $\\mathbf{H}_{k}$  and $\\tilde{\\mathbf{H}}_{k}$, can be measured by a function of $\\theta_{\\mathcal{V}_{k}\\mathbf{H}_{k}}$, which is the angle between $\\mathcal{V}_{k}$ and $\\text{Span}(\\mathbf{H}_{k})$, see Fig.~\\ref{fig:space-projections}. The following metrics assess the spatial compatibility based on geometrical properties of the multiuser channels:\n\\begin{enumerate\n\t%\n\t\\item The \\textit{principal angle between subspaces} \\cite{Wang2006,Yi2011,Nam2014a}  measures the relative orientation of the basis of $\\mathcal{V}_{\\mathbf{H}_{k}} = \\text{Span}(\\mathbf{H}_{k})$ regarding the basis of $\\mathcal{V}_{\\tilde{\\mathbf{H}}_{k}} = \\text{Span}(\\tilde{\\mathbf{H}}_{k})$. Given the bases of $\\mathcal{V}_{\\mathbf{H}_{k}}$ and $\\mathcal{V}_{\\tilde{\\mathbf{H}}_{k}}$, the principal angle is associated with the largest coefficient of correlation [cf. (\\ref{eq:coefficient_correlation})] between the bases of both subspaces.\n\n\t\\item The \\textit{chordal distance} is extensively used in limited feedback systems for codebook design \\cite{Barg2002,Love2003}, user grouping and scheduling \\cite{Chae2008,Yi2011,Ko2012,YiXu2014}. This metric can be computed either from the principal angles between subspaces or from the projection matrices of $\\mathcal{V}_{\\mathbf{H}_{k}}$ and $\\mathcal{V}_{\\tilde{\\mathbf{H}}_{k}}$ in heterogeneous MU-MIMO scenarios, i.e., $N_{i} \\neq N_{j}$ $\\forall i, j \\in \\mathcal{K}$ with $i \\neq j$. \n\t%\n\t\\item The \\textit{subspace collinearity} \\cite{Czink2009,Yi2011} quantifies how similar the subspaces spanned by two channel matrices are, following the rationale behind (\\ref{eq:coefficient_correlation}). Given $\\mathbf{H}_{i}, \\mathbf{H}_{j} \\in \\mathbb{C}^{N \\times M}$ the subspace collinearity of the matrices compares the singular values and the spatial alignment of their associated singular vectors. \n\t%\n\t\\item Other metrics measuring the distance between subspaces are the \\textit{weighted likelihood similarity measure}, the \\textit{subspace projection measure}, and the \\textit{Fubini-Study similarity metric}. These metrics have been recently propounded in \\cite{YiXu2014} and used for user grouping based on statistical CSI, i.e., $\\mathbf{\\Sigma}_{k} \\ \\forall k$.\t\n\t%\n\\end{enumerate}\n\n\n{\\color{black}{\t\t\n\\subsection{Discussion}\n\\label{section:metrics-spatial-compatibility:discussion}\n\nIt is worth mentioning that neither the principal angles, nor the chordal distance can fully measure spatial compatibility in heterogeneous MU-MIMO scenarios. This is due to the fact that these metrics take into account the smallest dimension between two subspaces, potentially neglecting useful spatial correlation information \\cite{Yi2011}. Moreover, metrics that only evaluate the spatial separation between hyperplanes or the eigenvalue dispersion of $\\mathbf{H}(\\mathcal{K})$ neglect the degradation of the MIMO channel magnitude due to the interference subspace. Such metrics fail at maximizing the capacity since they do not evaluate or approximate the effective channel gain \\cite{Ko2012}. As the number of active users grows, the set of users that maximizes a given  spatial compatibility metric may diverge from the set that maximizes the capacity, either in their elements, cardinalities, or both \\cite{Yi2011,Castaneda2014b}.\n\n \nAuthors in \\cite{Wang2010a} pointed out that user grouping should be a function of the spatial correlation between $\\mathbf{H}_{k}$ and $\\tilde{\\mathbf{H}}_{k}$, and also consider the inner correlation of each multi-antenna user, i.e., the magnitudes of the eigenvectors of $\\mathbf{H}_{k}$ $\\forall k$.\nAccording to \\cite{Vu2007,Jorswieck2006,Godana2013a}, the precoding performance is primarily affected by the correlation between transmit antennas, whereas receive antenna correlation has marginal or no impact on the precoding design. Nonetheless, the effects of receive antenna correlation has not been fully studied in the user selection literature and conclusions are usually drawn based on specific correlation models. \nThe knowledge of statistical CSI, i.e., $\\mathbb{E}[\\mathbf{H}_{k}^{H} \\mathbf{H}_{k}]$, may be assumed in scenarios with practical constraints such as limited feedback rates [cf. Section~\\ref{section:precoding:partial-csit}].  If the transmitter has knowledge of statistical CSI, the dominant eigen-directions of the channel covariance matrix ($\\mathbf{\\Sigma}_{k}$, $\\forall k$) can be used as metrics to identify compatible users, e.g., \\cite{Kountouris2006b,Hammarwall2008,Gao2009,Li2016,YiXu2014,Adhikary2013a,Nam2014}. Spatial clustering based on $\\mathbf{\\Sigma}_{k}$, $\\forall k$, has been recently proposed to identify users with similar channel statistics in massive MIMO settings, see \\cite{YiXu2014,Nam2014,Zheng2015}. \n\n}}\n\n\n\n\n\\section{System Optimization Criteria}\n\\label{section:performance-uf}\n\n\n\n\\begin{table*}[t!] \\scriptsize\n\t\\renewcommand{\\arraystretch}{1.3}\t\n\t\\caption{Summary of System Optimization Criteria for Scheduling in MISO ($N=1$) and MIMO ($N>1$) configurations with full ($B=\\infty$) and partial ($B<\\infty$) CSIT. QoS refers to SINR, BER, or Queue Stability requirements}\n\t\\label{tab:optimization-criteria\n\t\\centering\t\t\n\t\\begin{tabularx}{\\textwidth}{l *{5}{>{\\centering\\arraybackslash}X} } \n\t\t\\toprule\t\t\n\t\t& \\textbf{Sum Rate} & \\textbf{Fairness} & \\textbf{WSR}   & \\textbf{QoS} & \\textbf{Round Robin} \\\\\n\t\t\\midrule\n\t\t\\textbf{MISO}, $B=\\infty$ & \\cite{Dimic2005, Huang2013, Tu2003, Wang2008a, Yoo2006, Jiang2006, Dai2009, Lee2014, Adhikary2013, Huang2012b, Razi2010, Mao2012, Bjornson2014, Shi2012, Maciel2010, Driouch2012, Yoo2005a, Christopoulos2013, Xia2009, Lau2005a, Cottatellucci2006, Wang2015, Yang2011, Boccardi2006, Zhang2007b, Dartmann2013, Castaneda2015, Jang2011, Park2014,  Ku2015,YiXu2014,Bogale2015,Lee2014c} & \\cite{Moon2013, Yoo2006, Maciel2010, Driouch2012, Christopoulos2013, Lau2005a, Lee2014, Dartmann2013, Seifi2011,YiXu2014,Liu2015a} & \\cite{Jagannathan2007, Hammarwall2007, Huh2012,Hammarwall2008,Liu2014a} & \\cite{Jagannathan2007, Maciel2010, Chung2010, Stridh2006a, Matskani2008, Koutsopoulos2008, Destounis2015, Swannack2004, Tsai2008, Shirani-Mehr2011, Song2008,  Chung2010, Koutsopoulos2008,Christopoulos2013,Kountouris2006b} & \\cite{Yoo2006, Cottatellucci2006, Lee2014, Dartmann2013, Jang2011,Liu2015a} \\\\\n\t\t\\midrule\n\t\t\\textbf{MISO}, $B<\\infty$ & \\cite{Jindal2006, Wagner2008, Nam2014, Lee2014, Adhikary2013, Adhikary2013a, Maciel2010, Xia2009, Zorba2008, Yang2011, Zakhour2007, Tang2010, Huang2009a, Vicario2008, Yoo2007, Kountouris2007, Kountouris2008, Dai2008, Kountouris2006, Zhang2011, Ravindran2012, Kountouris2006a, Wang2007a, Choi2007c, Min2013, Sohn2010, Kountouris2005a, Huang2007, Sohn2012, Khoshnevis2013,Xu2009a} & \\cite{Moon2013, Adhikary2013a, Wagner2008, Zorba2008, Lee2014, Huang2012a, Shirani-Mehr2010, Kountouris2008a, Kountouris2005, Xu2010, Khoshnevis2013,Xu2009a} & \\cite{Conte2010, Shirani-Mehr2010, Kobayashi2007} & \\cite{Huang2012a, Shirani-Mehr2010, Simon2011,Tang2010,Lau2009,Kountouris2006b} & \\cite{Lee2014, Zhang2011,Simon2011} \\\\\n\t\t\\midrule\n\t\t\\textbf{MIMO}, $B=\\infty$ & \\cite{Bayesteh2008, Shen2006, Wang2010, Bayesteh2008, Ko2012, Elliott2009, Chan2007, Tran2010, Wang2010a, Nam2014a, Zhang2009, Park2010, Lossow2013,Fuchs2007, Tejera2006, Wang2006, Chen2008, Sigdel2009a, Yi2011, Sun2010, Tran2012, Elliott2012, Hei2009, Lim2009, Cheng2014} & \\cite{Elliott2009, Nam2014a, Torabzadeh2010, Cui2011a, Yu2013,Fuchs2007, Sigdel2009a, Cheng2014, Aniba2007} & \\cite{Tran2012} & \\cite{Souihli2010, Moretti2013,Zhang2005a, Aniba2007,Chen2007a} & \\cite{Wang2010, Bayesteh2008, Chan2007,Fuchs2007} \\\\\n\t\t\\midrule\n\t\t\\textbf{MIMO}, $B<\\infty$ & \\cite{Sharif2005, Chae2008, Trivellato2008, Jindal2008, Nam2014a,Fuchs2007, Wang2006, Rico-Alvarino2014, Wang2012d, Zhang2007a} & \\cite{Fan2014, Nam2014a, VanRensburg2009, Hosein2009,Fuchs2007, Rico-Alvarino2014} & \\cite{Schellmann2010} & \\cite{Chen2013,Zhang2005a,Rico-Alvarino2014} & \\cite{Fuchs2007,Jindal2008} \\\\\n\t\t\\bottomrule\n\t\\end{tabularx}%\n\\end{table*}%\n\n\nThe optimization criteria determines the optimal resource allocation strategy \\cite{Jorswieck2006}, and can be  classified in two groups according to the objective function and constraints \\cite{Lau2006}. \\textit{i}) \\textit{PHY layer} based criteria, where a objective function, $U(\\cdot)$, must be optimized and channel information is the only input to the resource allocation algorithms. \\textit{ii}) \\textit{Cross layer} based criteria, where optimization of $U(\\cdot)$, takes into account QoS requirements (defined by upper layers) and channel information, see Fig.~\\ref{fig:system-processing-blocks}.\nThis section presents an overview and classification of the objective functions (criteria), and their associated constraints in the MU-MIMO literature. A summary of the content and general organization of this section are presented in Fig.~\\ref{fig:utility-functions-and-constraints} and Table~\\ref{tab:optimization-criteria}.\nTwo relevant concepts in multiuser system optimization are defined below.\n\n{\\color{black}{\n\\begin{definition}\\label{defn:resource_feasibility}\n\t\\textit{Resource allocation feasibility}. For given a set of users $\\mathcal{K}$, a resource allocation strategy is called feasible if it fulfills all individual and global constraints (e.g. power and QoS), implementing precoding, power control, or a combination of both.\n\\end{definition}\n\n\\begin{definition}\\label{defn:feasible_set_of_users}\n\t\\textit{Feasible set of users}. Given the set of all competing users $\\bar{\\mathcal{K}}$, the subset   $\\mathcal{K} \\subseteq \\bar{\\mathcal{K}}$, is called feasible if there exist precoding weights and powers, $\\forall k \\in \\mathcal{K}$, such that $U(\\cdot)$ has a solution meeting individual QoS and power constraints.\n\\end{definition} \n\n}}\n\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=0.98\\linewidth]{figure7.pdf}\\\\\n\t\\caption{Utility Functions and constraints for PHY layer and cross layer based optimization in MU-MIMO systems} \n\t\\label{fig:utility-functions-and-constraints}\n\\end{figure}\n\n\n\\subsection{Weighted Sum Rate (WSR) Maximization}\n\\label{section:performance-uf:weighted-sum-rate}\n\n\n\nThe system optimization is achieved by maximizing the sum of individual utility functions, $U_{k}(\\cdot)$, $\\forall k \\in \\mathcal{K}$, subject to a set of power constraints and set cardinality ($|\\mathcal{K}|$). \nGlobal optimization based on concave and differentiable utility functions is desirable, since efficient numerical methods can be applied to guarantee optimality. A common family of functions studied in the literature is defined by the $\\alpha$\\textit{-fair} utility function \\cite{Kelly1997,Mo2000,Hong2012}, which can be used to characterize the objective of the resource allocation strategies.\n\nThe $\\alpha$\\textit{-fair} function is mathematically expressed as \\cite{Mo2000}:\n\n\\begin{subnumcases}{ U\\left( \\{{R}_{k}\\}_{k=1}^{K} \\right) =}\n\\sum_{k=1}^{K} \\log({R}_k), & \\text{if $\\alpha=1$} \\label{eq:alpha-fair-utility-function-1}\n\\\\\n\\sum_{k=1}^{K} \\frac{ ({R}_k)^{1-\\alpha}  }{  1-\\alpha  }, & \\text{otherwise} \\label{eq:alpha-fair-utility-function-2} \n\\end{subnumcases}\nwhere ${R}_k$ denotes the achievable transmission rate of the $k$-th user. \nChanging the values of $\\alpha$ yields different priorities to the users, and can be used to define performance tradeoffs (e.g. throughput-fairness). For instance, $\\alpha = 1$, yields maximum fairness, $\\alpha = 0$, generates the sum-rate utility, and $\\alpha \\rightarrow \\infty$ defines the minimum-rate function. These functions are, in general, non-concave,\\footnote{For instance, non-concave utility functions may arise in the context of real-time applications with hard QoS constraints \\cite{Utschick2012}.} and several methods have been developed to optimize them subject to power and QoS constraints, see \\cite{Hong2012} for an in-depth review. \n\n\nA more common formulation of the weighted sum rate in MU-MIMO scenarios is given by the following expression: \n\n\\begin{equation}\\label{eq:weighted-sum-rate-function}\nU\\left( \\{{R}_{k}\\}_{k=1}^{K} \\right) = \\sum_{k=1}^{K} \\omega_{k} {R}_{k}\n\\end{equation}\nwhere $\\omega_{k}$ is a non-negative time-varying weight, which defines the priority of the $k$-th user. This expression is related to the $\\alpha$-fair function, e.g. if $\\omega_{k}=1$, then  (\\ref{eq:weighted-sum-rate-function}) converges to (\\ref{eq:alpha-fair-utility-function-2}) with $\\alpha = 0$. The dynamic adaptation of $\\omega_{k}$ can provide different levels of fairness (temporal, utilitarian, rate-proportional, fair-queuing, etc.) over the resource allocation at the \\textit{user level} \\cite{Lau2005}. \n\n\nAnother approach to provide fairness is defining $U\\left( \\{{R}_{k}\\}_{k=1}^{K} \\right)$ as a function of the rates' statistics. For instance, one might use the cumulative distribution function (CDF) of ${R}_{k}$ $\\forall k$, which can be empirically estimated at the transmitter \\cite{Park2005,Kountouris2008a}. \nFairness at the \\textit{system level} is another optimization criterion, measured over the average rates, transmit powers, time-slots, or any other allocated resource.  Different metrics to quantify fairness can be found in the literature of resource allocation \\cite{HuaizhouShi2014}. For a single resource, one can use the Gini index \\cite{Bartolome2006},  Jain's index \\cite{Jain1984}, or other fairness measures in \\cite{Lan2010} and reference therein. Generalized fairness metrics over multiple resources allocated simultaneously were defined in \\cite{Joe-Wong2013,HuaizhouShi2014} and references therein.\nThe weights in (\\ref{eq:weighted-sum-rate-function}) might be determined by upper layer requirements, such as buffer sizes, traffic\/service priority, packet loads, or any other QoS-related metric. The individual weights affect the scheduling decisions, and can be used to improve precoding design \\cite{Park2012a}.\n\n\n\nA special case of (\\ref{eq:weighted-sum-rate-function}) is given by setting $\\omega_{k}=1$ $\\forall k$, which represents the sum rate. This criterion defines the maximum amount of error-free information successfully received by a set of co-scheduled users, regardless of fairness \\cite{Goldsmith2005}. \nThe resource allocation strategies are usually assessed in terms of sum rate, since it quantifies the effectiveness of an algorithm and simplifies scheduling rules. However, maximizing the sum rate in scenarios where the users have heterogeneous SNRs might be very inefficient, since users with poor channel conditions would experience starvation \\cite{Yoo2006,Jagannathan2007}. A classical solution to such a problem is assigning the weights $\\omega_{k} \\ \\forall k$, according to a fairness criterion.\nThe works listed under the \\textit{fairness} category in Table~\\ref{tab:optimization-criteria}, optimize the long-term proportional fairness, extending the original idea from \\cite{Chaponniere2002,Viswanath2002} to MU-MIMO scenarios. \n\n\nRound robin is a channel-unaware scheduling method, that allocates the same amount of time to all competing users. It is the simplest form of fair resource-access, but neglects MUDiv, the amount of occupied resources, and QoS requirements \\cite{Capozzi2013,Lee2014b}. Equal opportunity for resource sharing may not mean equal distribution of resources, which results in wastage, redundant allocation, or resource starvation \\cite{HuaizhouShi2014}. Nonetheless, round robin might be useful in ultra dense heterogeneous networks, where transitions from non-LoS to strong LoS channels alter the performance of conventional resource allocation strategies \\cite{Lopez-Perez2015}. The works listed under the round robin category in Table~\\ref{tab:optimization-criteria}, use the method as a benchmark to assess MUDiv gains in MU-MIMO systems.\nIt is worth mentioning that in the reviewed literature, the system performance is assessed over a large number of channel realizations. For example, using the average sum rate per channel use, also called spectral efficiency, measured in bit\/s\/Hz. Other common performance metrics are \\cite{Feng2013}: the \\textit{throughput} or average rates achieved for packet transmission with practical modulations and realistic coding schemes; and  the \\textit{goodput}, which is the total average rate successfully delivered to the scheduled users, including layer 2 overheads and packet retransmission due to physical errors \\cite{Capozzi2013}.\n\n\n\\subsection{WSR with QoS Optimization}\n\\label{section:performance-uf:wsr-with-qos}\n\nThe WSR optimization problem can incorporate different types of QoS requirements. The QoS defines  a network- or user-based objective [cf. Definition~\\ref{defn:quality-of-service}], and the weights $\\omega_{k} \\ \\forall k$, establish priorities according to the service provided to each user.\nTaking into account QoS requirements reduces the flexibility of the resource allocation strategies. For instance, in the case of opportunistic transmission, achieving a target QoS reduces the value of CSIT, i.e., the channel conditions have to be weighted by requirements coming from the upper layers \\cite{Gesbert2007a}. System optimization considering QoS has a broader scope and once information from upper layers is considered, a more complex cross-layer optimization problem arises.\n\nThe QoS is defined according to the system model, which might include individual traffic characteristics or SINR requirements. Satisfying QoS requirements can be realized at different time granularities: each user must achieve a prescribed performance metric, e.g. instantaneous SINR or data rate \\cite{Schubert2006}; or a network performance metric must reach a stable behavior over time, e.g. attaining finite buffer sizes \\cite{Kobayashi2007}. Below, we provide a classification of the optimization criteria subject to QoS constraints in MU-MIMO scenarios.\n\n\n\n\\subsubsection{Target SINR and Error Rates}\n\\label{section:performance-uf:qos:target-rates}\n\nThe QoS can be guaranteed as long as individual SINR requirements are met, which also satisfies target error and peak rates. The QoS can be defined by a monotonic and bijective function of the SINR: one example is the bit error rate (BER), given a particular modulation and coding scheme (MCS) \\cite{Tsai2008,Chung2010}; another example is the Shannon capacity \\cite{Schubert2006}. \nDifferent tradeoffs between global and individual performance are achieved under SINR constraints: maximizing the WSR \\cite{Weeraddana2012}, max-min weighted SINR \\cite{Schubert2004,Bartolome2006,Cai2011,Tan2012,Stanczak2009}, sum power minimization \\cite{Schubert2004,Schubert2006,Zhang2007b,Hong2012}, and other hybrid formulations, see \\cite{Hong2012} and references therein. \n\nIt is worth noticing that in systems with SINR requirements, it is usually assumed that for each channel realization, a set of  compatible users $\\mathcal{K}$ has been previously selected. The system optimization is realized by power allocation and precoder design \\cite{Weeraddana2012}. This means that $\\mathcal{K}$ must be a feasible set of users [cf. Definition~\\ref{defn:feasible_set_of_users}], and user scheduling is required only if the constraints associated with $\\mathcal{K}$ are infeasible \\cite{Stanczak2009,Hong2012}. In that case, it is necessary to relax the initial conditions taking one of the following actions: \n\n\\begin{enumerate\n\t\\item Reducing the number of selected users applying \\textit{admission control} \\cite{Ahmed2005,Stridh2006a,Bartolome2006} or \\textit{user removal} schemes \\cite{Koutsopoulos2008,Mahdavi2010, Maciel2010,Castaneda2013}. These procedures are implemented in systems where the achievable SINRs take any positive real value, e.g. \\cite{Schubert2004,Caire2006,Zhang2007b,Cai2011,Tan2012}. The removal process can be thought as a form of scheduling, used to generate a feasible set of users [cf. Definition~\\ref{defn:feasible_set_of_users}]. By identifying users with unfeasible constraints, one can drop them temporarily (according to priorities), or re-schedule them over orthogonal resources (e.g. other carriers or time slots) \\cite{Wang2007}.\n\t\\item By relaxing the individual QoS requirements $\\forall k \\in \\mathcal{K}$, one can achieve resource allocation feasibility [cf. Definition~\\ref{defn:resource_feasibility}]. For instance, consider homogeneous target rates, then all scheduled users can be balanced to a common SINR \\cite{Koutsopoulos2008}. For heterogeneous QoS, one can iterate over different SINR levels per user (e.g. provided a set of MCSs), so that link adaptation can take place, see \\cite{Weeraddana2012,Castaneda2013,Destounis2015,Bartolome2006}.  \n\\end{enumerate}\n\nCross-layer algorithms subject to rate constraints can be designed so that $\\mathcal{K}$ and its associated resources are jointly optimized. Analytical results in \\cite{Lau2009} show that under mild conditions, there exists, with probability one, a feasible set of users $\\mathcal{K}$ that can be found though low-complexity algorithms [cf. Section~\\ref{section:algorithms:aggregated-utility}]. \n\nThe transmission errors (bit, frame, symbol, packet) are usually due to noisy channels and inaccurate CSIT. For the former factor, the errors occur due to the effect of non-ideal channel coding and finite blocklength channel coding. The error rates can be reduced by implementing stronger channel codes and longer blocklengths.\\footnote{In practice, for reasonable block length (e.g. 8 kbits) and strong coding (e.g. LDPC), the Shannon capacity can be approached to within 0.05 dB for a target FER of 10$^{\u22123}$ \\cite{Lau2005a,Tse2005,Lau2009}.} The error rates also occur due to CSI quantization (limited feedback), estimation errors, delays (channel outage), and distortion during the feedback \\cite{Shirani-Mehr2010,Zhang2011,Kobayashi2011}. Therefore, IUI  cannot be fully suppressed by precoding techniques [cf. Definition~\\ref{defn:interference-limited-system}], which degrades the achievable SINRs and BER figures. Nonetheless, multi-antenna receivers can implement efficient processing techniques to increase their SINRs and combat quantization errors, e.g., quantization-based combining (QBC) \\cite{Jindal2008}. Under partial CSIT conditions, any practical rate adaptation scheme achieves performance between the worst-case outage rate and the ideal-case, where the achievable rate equals the mutual information \\cite{Shirani-Mehr2010}.\n\n\n\n\\subsubsection{Queue Stability in MU-MIMO Scenarios}\n\\label{section:performance-uf:qos:queue-delay-stability}\n\nA network-based optimization requires cross-layer algorithms taking as inputs the queue state information (QSI), CSIT, and their inter-dependence \\cite{Swannack2004}. There exist a rich literature on optimization subject to queue or packet delay constraints, specially for systems with orthogonal resource allocation (e.g. one user per time-slot or sub-carrier),  see \\cite{Wang2007,Capozzi2013,Asadi2013} for an in-depth overview.  \nAccording to the QSI relevance, MU-MIMO systems fall in two categories  \\cite{Lau2006,Kobayashi2007}:\n\n\\begin{enumerate\n\t\\item \\textit{Systems with infinite backlogs}: \n\t%\n\tIn this scenario, the transmitter is assumed to always have data to send and infinite buffer capacity. In other words, the transmitter fully knows the intended data for each user. The objective of the resource allocation strategies is, in general, to maximize the WSR [cf. Section~\\ref{section:performance-uf:weighted-sum-rate}]. The weight $\\omega_{k}$ of the $k$-th user is used to reach a desired throughput-fairness tradeoff, instead of expressing the urgency of data flows. Thus, it can be assumed that the queues are balanced and users cannot be discriminated based on them \\cite{Swannack2004}. The service provided is delay-insensitive, and the traffic is managed so that the scheduled users attain non-zero rates \\cite{Souihli2010}. MU-MIMO systems without QSI information fall into this category.\n\t%\n\t\\item \\textit{Systems with bursty traffic and limited buffer size}: \n\t%\n\tIn these systems, packet data models with stochastic traffic arrivals are considered. The performance assessment is analyzed from the delaying and queuing perspectives. The WSR maximization attempts to balance between opportunistic channel access and urgency of data flows. The resource allocation algorithm must guarantee finite average buffer occupancy, i.e., \\textit{stability of the queues lengths} for all users \\cite{Caire2006,Chen2013}. It is worth mentioning that by Little's theorem \\cite{Lau2006,Wang2007,Chen2013}, achieving stability in the average queues is equivalent to minimize the average packet delay in the steady state.\\footnote{For systems with queue or delay requirements, a channel\/QoS-aware scheduling algorithm must be supported by admission control mechanisms in order to guarantee feasibility \\cite{Capozzi2013}.} The system model might consider different sources of delay, e.g., buffer congestion or destination unavailability (outage delay), which define the type of scheduling policy to be implemented \\cite{Souihli2010}. The choice of the system model and problem formulation depend on the desired tradeoff between tractability and accuracy in each particular scenario \\cite{Lau2006}.\n\\end{enumerate}\n\n\nIn MU-MIMO systems where queue stability is optimized, the scheduling algorithm has to guarantee that the average queue lengths of all users are bounded. Simultaneously, the available CSIT should be opportunistically exploited for throughput maximization \\cite{Caire2006}. \nThis goal can be achieved by establishing queue-based WSR as the optimization criterion. The weights $\\omega_{k}$ in (\\ref{eq:weighted-sum-rate-function}) can be defined according to the system requirements: considering the packet queue length or packet departure rates at each scheduling interval \\cite{Swannack2004,Shirani-Mehr2010,Huang2012a,Chen2013,Destounis2015}; considering fairness with heterogeneous traffic rates \\cite{Torabzadeh2010,Tan2012}; or considering service-oriented requirements, e.g., BER, delay tolerance, and packet dropping ratio for real- or non-real-time services \\cite{Chung2010,Souihli2010,Tsai2008}. \nThe queue lengths not only define the priority or urgency of the traffic, but they can also define the encoding order, $\\pi(\\cdot)$ in (\\ref{eq:general-combinatorial-problem:cost-function}), see \\cite{Swannack2004,Caire2006,Destounis2015}. In general, queue lengths are not symmetric or balanced, which means that the WSR optimization is dominated by the QSI, rather than by the CSIT \\cite{Swannack2004,Huang2012a}. \n\n\n\nThe characteristics of the MU-MIMO system model and the set of constraints define the resource allocation strategy that achieves stability. Several factors may affect the performance under QoS constraints:  CSIT accuracy \\cite{Huang2012a,Chen2013}, CSIT statistics \\cite{Shirani-Mehr2010}, and the number of resources used for channel estimation \\cite{Destounis2015}; the spatial compatibility between co-scheduled user and the SNR regime \\cite{Swannack2004}; the user priority based on the type of service \\cite{Tsai2008,Chung2010}; or the number of simultaneous spatial streams \\cite{Souihli2010}.\nThe overall optimization can also consider delays due to packet losses and retransmissions. This is supported by ACK\/NAK (handshake) signaling exchange in the upper layers. Such a protocol is used for automatic repeat request (ARQ), to convey an error-free logical channel to the application layers \\cite{Love2008}. Works in \\cite{Shirani-Mehr2010,Shirani-Mehr2011} and references therein, discuss algorithms to solve the WSR with queue constraints taking into account ARQ protocols. We refer the reader to \\cite{Lau2006,Georgiadis2006} for a comprehensive texts on cross-layer design under queue and delay constraints.      \n\n\n\n{\\color{black}{\n\n\\subsection{Discussion and Future Directions}\n\\label{section:performance-uf:discussion}\n\nThe sum-rate maximization is the main criterion to assess conventional cellular system, specially in scenarios with scarce and expensive radio resources, e.g., crowded sub-6 GHz bands. Conventional cellular planning usually considers few high-power transmitters that provide high spectrum efficiency, at expense of other performance metrics, such as energy efficiency (EE) \\cite{Tombaz2011}. 5G mobile communications will include dense deployments, operating at higher frequencies, i.e., mmWave, and with very heterogeneous radio resources per base station \\cite{Andrews2014,Baldemair2015,Nguyen2015a}.\nThe criteria presented in Sections~\\ref{section:performance-uf:weighted-sum-rate} and \\ref{section:performance-uf:wsr-with-qos} are used to define single objective functions subject to a set of constraints. However, 5G networks will require the simultaneous optimization of multiple criteria: peak data rates, traffic and user load across the network, fairness, quality of service and experience, EE, etc. These multiple objectives are usually coupled in a conflicting manner, such that optimization of one objective degrades the other objectives. \n\n\nOne approach to find a tradeoff between objectives is by deriving an explicit expression that can be used as a single objective function. For instance, EE has been jointly optimized with peak rates, WSR, or load balance in heterogeneous networks, see \\cite{Li2015,Yang2015a,Liu2016,Nguyen2015a} and references therein. Fundamental tradeoffs among EE and delay, sum rate, bandwidth and deployment cost have been derived in \\cite{Chen2011d}. Balancing  fairness and spectral efficiency is a well known problem strived for wireless communications \\cite{HuaizhouShi2014}, and several works have formulated its joint optimization as a single objective function, e.g., \\cite{HoTingCheng2008,Sediq2013,BinSediq2014}. To illustrate several conflicting objectives, consider that allocating resources to users with strong channels can satisfy QoS requirements and improve EE. However, it might also incur in unfair resource distribution across users and unbalanced load among BSs. Now assume that all users have equally good channels, and transmitting low traffic loads to all users increases the coverage area, but this might be very energy inefficient. Unconstrained EE maximization may result in operating points with low spectral efficiency per user \\cite{Bjornson2016a}. \n\n\nAnother approach for the joint optimization of multiple objectives, is to sample the solution space and chose the operative point that satisfies a predefined tradeoff. A mathematical framework to address multi-objective optimization problems for wireless communications has been proposed in \\cite{Bjornson2013,Bjornson2014a}. The solution space of such kind of mathematical problems, generally, does not have a unique point that can optimally satisfy all objectives. The main challenges are to characterize and understand efficient operating points within the solution space, so that the objectives are balanced.\nAuthors in \\cite{Bjornson2014a} provide an example of this approach by jointly optimizing individual peak rates, average area rates, and EE for a massive MIMO setting. Numerical results illustrate the conflicting nature of the objectives: the average area rates increases with the number of served users; the individual peak rates can be increased when the power is split among few users; and high EE is attained if the rate per user is small. \n\n\nTechniques such as multi-objective optimization \\cite{Bjornson2014a} and metaheuristic optimization [cf. Section~\\ref{section:algorithms:bio-inspired}], sample the solution space to find close-to-optimal solutions. However, the former is constructed from a mathematical framework, which characterizes the solution space, in particular, the desirable operative points.\nWe foresee that multi-objective optimization techniques will play a primal role towards 5G, as the overall network optimization become more complex. Those techniques can provide guidelines for network design, simplify parametrization, and assess compounded optimization criteria for heterogeneous networks. The network designer must establish resource allocation strategies to conciliate conflicted objectives, keeping in mind that the ultimate goals are user experience, satisfaction, and operators' interest \\cite{Sharifian2016}. \n}}\n\n\n\n\n\n\\section{MU-MIMO Scheduling}\n\\label{section:algorithms}\n\n\\begin{table*}[t] \\scriptsize\n\t\\renewcommand{\\arraystretch}{1.3}\t\n\t\\caption{User Scheduling Methods for MISO ($N=1$) and MIMO ($N>1$) configurations, and full ($B=\\infty$) and partial ($B<\\infty$) CSIT}\n\t\\label{tab:algorithms\n\t\\centering\t\t\n\t\\begin{tabularx}{\\textwidth}{ *{6}{>{\\centering\\arraybackslash}X} }  \n\t\t\\toprule\t\n\t\t\\textbf{Method} &   \\textbf{Utility-based}    &   \\textbf{CSI-Mapping}    & \\textbf{Metaheuristic}\\newline \\textbf{(stochastic)}  & \\textbf{Classic}\\newline \\textbf{Optimization}& \\textbf{Exhaustive}\\newline \\textbf{Search} \\\\\n\t\t\\midrule\n\t\t\\textbf{MISO}, $B=\\infty$ & \\cite{Dimic2005, Moon2013, Huang2013, Wang2008a, Huang2012b, Maciel2010, Driouch2012, Lau2005a, Koutsopoulos2008, Kobayashi2007,Liu2014a,YiXu2014,Bogale2015} & \\cite{Tu2003, Wang2008a, Yoo2006, Jiang2006, Dai2009, Lee2014, Adhikary2013, Adhikary2013a, Razi2010, Mao2012, Shi2012, Maciel2010, Chung2010, Driouch2012, Yoo2005a, Christopoulos2013, Wang2015, Yang2011, Zhang2007b, Swannack2004, Shirani-Mehr2011, Castaneda2015, Jang2011,Kountouris2006b,Lee2014c,Liu2015a} & \\cite{Bjornson2014, Stridh2006a, Lau2005a, Cottatellucci2006, Wang2015, Zhang2007b, Park2014} & \\cite{Maciel2010, Stridh2006a, Matskani2008, Ku2015, Song2008} & \\cite{Lau2005a, Destounis2015, Swannack2004} \\\\\n\t\t\\midrule\n\t\t\\textbf{MISO}, $B<\\infty$ & \\cite{Xia2009, Conte2010, Tang2010, Vicario2008, Kountouris2007, Kountouris2006, Shirani-Mehr2010, Zhang2011, Ravindran2012,Hammarwall2008} & \\cite{Nam2014, Lee2014, Adhikary2013, Adhikary2013a, Xia2009, Zorba2008, Yang2011, Conte2010, Zakhour2007, Huang2009a, Lau2009, Huang2012a, Yoo2007, Kountouris2007, Kountouris2008, Dai2008, Ravindran2012, Kountouris2006a, Wang2007a, Choi2007c, Min2013, Sohn2010, Kountouris2008a, Kountouris2005a,Kountouris2006b, Kountouris2005, Huang2007, Xu2010, Sohn2012, Khoshnevis2013, Simon2011, Wagner2008,Xu2009a,Hammarwall2008} & \\cite{Nam2014, Zhang2011} & -   & \\cite{Zakhour2007} \\\\\n\t\t\\midrule\n\t\t\\textbf{MIMO}, $B=\\infty$ & \\cite{Shen2006, Ko2012, Chan2007, Tran2010, Torabzadeh2010, Zhang2009, Cui2011a, Yu2013, Park2010, Lossow2013,Chen2007a, Fuchs2007, Zhang2005a, Chen2008, Tran2012, Lim2009, Cheng2014} & \\cite{Bayesteh2008, Shen2006, Wang2010, Ko2012, Tran2010, Wang2010a, Nam2014a, Souihli2010,  Jagannathan2007, Chen2007a, Fuchs2007, Tejera2006, Zhang2005a, Wang2006, Sigdel2009a, Yi2011, Sun2010, Tran2012, Lim2009, Cheng2014, Aniba2007} & \\cite{Elliott2009,Elliott2012, Hei2009} & \\cite{Chan2007, Moretti2013} & \\cite{Chan2007, Gesbert2007a} \\\\\n\t\t\\midrule\n\t\t\\textbf{MIMO}, $B<\\infty$ & \\cite{Chae2008, Jindal2008,Rico-Alvarino2014, Fan2014, Wang2012d} & \\cite{Chae2008, Chen2013, Trivellato2008, Nam2014a, VanRensburg2009, Hosein2009, Schellmann2010, Sharif2005,Zhang2007a} & -  & \\cite{Fan2014} & - \\\\\n\t\t\\bottomrule\n\t\\end{tabularx}%\n\\end{table*}%\n\n\nResource allocation can be performed over orthogonal radio resources \\cite{Capozzi2013,Asadi2013},  where the allocation decisions are made using individual estimates of  BER, SINR, or rate. Scheduling the best user per resource is performed by low-complexity algorithms that do not depend on full CSIT. For instance, a sorting-based algorithm at the transmitter can schedule users based on the CQI. In contrast, the scheduling algorithms for MU-MIMO systems attempt to allocate resources in a non-orthogonal fashion. The users cannot compute their achievable BER or rates since those metrics depend on the CSIT, powers, and precoders assigned to other co-scheduled users. \n\n\nTo illustrate the scheduling in problem (\\ref{eq:general-combinatorial-problem:cost-function}), consider a MU-MIMO system with a single transmitter equipped with $M$ antennas, and $K$ competing users, each equipped with $N$ antennas. Assuming that linear precoding is implemented [cf. Section~\\ref{section:precoding:linear}], the maximum number of co-scheduled users per resource is given by $K_{\\max} =\\lfloor M\/N \\rfloor$. \nDefine $\\hat{\\mathcal{K}}^{(ss)} = \\{ \\mathcal{K} \\subseteq \\{1,2, \\ldots, K\\}: 1 \\leq |\\mathcal{K}| \\leq K_{\\max} \\}$ as the \\textit{search space} set containing all possible user groups with limited cardinality.\\footnote{$\\hat{\\mathcal{K}}^{(ss)}$ contains the sets of cardinality one, i.e., SU-MIMO configuration. Efficient scheduling rules must dynamically switch between SU- and MU-MIMO configurations  \\cite{Chen2008,Ho2009,Sigdel2009a,Schellmann2010,Liu2012,Fan2014}.} The optimal set, $\\mathcal{K}^{\\star}$, that solves (\\ref{eq:general-combinatorial-problem:cost-function}) lies in a search space of size $|\\hat{\\mathcal{K}}^{(ss)}| = \\sum_{m=1}^{K_{\\max}} {K \\choose m}$. The set $\\mathcal{K}^{\\star}$ can be found by brute-force exhaustive search over $\\hat{\\mathcal{K}}^{(ss)}$, which is computationally prohibitive if $K \\gg K_{\\max}$. \nThe search space, and in turn the complexity of the problem, varies according to the system model and constraints:  the values of $M$ and $N$, the number of streams per user, individual QoS or power constraints, the set of MCSs, etc. \n\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=0.98\\linewidth]{figure8.pdf}\\\\\n\t\\caption{Classification of the User Scheduling Algorithms for MU-MIMO}\n\t\\label{fig:algorithm_calssification}\n\\end{figure}\n\nBy plugging the utility function $U(\\cdot)$, [cf. Section~\\ref{section:performance-uf}], into (\\ref{eq:general-combinatorial-problem:cost-function}), we obtain a mathematical formulation of the MU-MIMO resource allocation problem. \nThe optimal scheduling rule that solves (\\ref{eq:general-combinatorial-problem:cost-function}) in a feasible and efficient way is still an open problem. A number of algorithms have been proposed in the literature to circumvent the high complexity of finding $\\mathcal{K}^{\\star}$, and its associated resource allocation. \nA sub-discipline of optimization theory, known as heuristic search \\cite{Williamson2010}, provides solutions (approximation algorithms) to a category of discrete problems. Such problems cannot be solved any other way or whose solutions take a very long time to be computed (e.g. NP-hard problems). User scheduling falls in such a problem category, and the common approach in the literature is to design suboptimal heuristic algorithms that  balance complexity and performance. The optimal solution, $\\mathcal{K}^{\\star}$, is usually computed for benchmarking in simple MU-MIMO scenarios. \n\nIn this section, we provide a classification of the most common scheduling algorithms in the literature of MU-MIMO. Table~\\ref{tab:algorithms} and Fig.~\\ref{fig:algorithm_calssification} show the scheduling algorithms classification, the scenarios in which they are applied, and the methodology they follow. Most algorithms operate at the single resource level, i.e., considering optimization over one sub-carrier or time-slot. Extensions to scenarios with multiple carriers or codes are straightforward, e.g., \\cite{Chung2010, Driouch2012, Tsai2008, Park2014,Conte2010, Khoshnevis2013,Elliott2009, Chan2007, Yu2013,Schellmann2010, Lossow2013,Fuchs2007, Tejera2006, Zhang2005a, Cheng2014}.   \n\n\n\n\n\n\\subsection{Aggregated Utility Based Selection}\n\\label{section:algorithms:aggregated-utility}\n\nLet us reformulate (\\ref{eq:general-combinatorial-problem:cost-function}) as a multiuser WSR maximization problem as follows\n\n\\begin{equation} \\label{eq:inner-outer-problem}\n\t\\underbrace{ \\underset{ \\mathcal{K} \\subseteq \\bar{\\mathcal{K}}: |\\mathcal{K}| \\leq c_p }{\\max} \\ \\\n\t\\underbrace{ \\underset{ \\mathbf{W}, \\mathbf{P}, \\pi }{\\max} \\ \\  \\sum_{k \\in \\mathcal{K}} \\omega_{k} \\log_{2} \\left( 1+ \\text{SINR}_{k} \\right)  }_\\text{Inner problem: $R(\\mathcal{K})$} }_\\text{Outer problem}\n\\end{equation}\nwhere $\\mathbf{W}$ and $\\mathbf{P}$ summarize the precoding weights and the powers assigned to the users in $\\mathcal{K}$, and $\\pi$ defines an encoding order. \nThe formulation in (\\ref{eq:inner-outer-problem}) illustrates the fact that the resource allocation problem can be decomposed into different subproblems, which are related to different network layers. Furthermore, under specific system settings and constraints the subproblems can be decoupled. \n\nThe weights $\\omega_{k}$, $\\forall k \\in \\mathcal{K}$, can be defined according to the service or application delivered to the users, [cf. Section~\\ref{section:performance-uf:wsr-with-qos}]. The inner problem in (\\ref{eq:inner-outer-problem}) attempts to maximize the WSR for a given set $\\mathcal{K}$. Finding a solution requires joint admission control, precoding design, and power control, e.g. \\cite{Stridh2006a, Matskani2008, Liu2011, Hong2012}. A solution to the inner problem exists when the multiuser channel associated with $\\mathcal{K}$, has specific spatial dimension defined by the precoding scheme. The channel dimensions must be guaranteed by the solution of the outer problem. If linear precoding schemes are implemented, [cf. Section~\\ref{section:precoding:linear}], the original problem can be relaxed: the cardinality of $\\mathcal{K}$ is upper bounded by $K_{\\max}$; the encoding order $\\pi$ can be omitted since it does not affect the precoding performance;  and power allocation can be computed via convex optimization or by heuristic algorithms [cf. Section~\\ref{section:power-allocation}]. \nFinding a solution to the combinatorial outer problem implies that the scheduler accomplishes several goals: it exploits MUDiv to maximize multiplexing gains; and it finds the multiuser channel with the best set of spatially compatible users [cf. Definition~\\ref{defn:compatible_set_of_users}].\nProblem (\\ref{eq:inner-outer-problem}) might include constraints over several discrete parameter sets, such as $\\mathcal{K}$, the encoding orders, MCSs, spatial streams per user, etc. The complexity of the combinatorial problem grows exponentially with each discrete set size \\cite{Koutsopoulos2008}.\n\n\n\nDue to the hardness of problem (\\ref{eq:inner-outer-problem}), one common approach is to find suboptimal solutions  through heuristic \\textit{greedy opportunistic algorithms}. Given a fixed precoding structure and CSIT, the set $\\mathcal{K}$ is constructed by a sequence of decisions. Every newly selected user finds a locally maximum for an objective function. The optimized utility $U(\\cdot)$ can be defined by the WSR (direct approach) or by a metric of spatial compatibility (indirect approach). The greedy scheduling rule requires low computational complexity and is easy to implement. However, it does not guarantee neither performance nor convergence to the optimal solution \\cite{Williamson2010}.\n\n\n\\subsubsection{Direct Approach}\n\\label{section:algorithms:aggregated-utility:objective-function}\n\nThe seminal works \\cite{Dimic2005,Shen2006} proposed a simple construction of $\\mathcal{K}$, by iteratively selecting users as long as the aggregated objective function improves its value. \nIn \\cite{Dimic2005}, a greedy user selection (GUS) based on sum-rate maximization was proposed. First, the user with the highest rate is selected, and the following selected user will maximize the total sum-rate. This method was improved in \\cite{Wang2008a},  using sequential water-filling to eliminate users with zero transmit power after selection and power allocation. The algorithms in \\cite{Dimic2005,Wang2008a} might be computational demanding because calculation of the precoders and power allocation is required each iteration. Furthermore, in the case of imperfect CSI, the above algorithms fail to determine the optimal number of selected users. \nA generic structure of this kind of user selection is presented in Algorithm~\\ref{alg:algorithms:direct-approach}. In this approach, the transmitter must have channel information so that every iteration the precoders $\\mathbf{W}_k$, $\\forall k \\in \\mathcal{K}$, are recalculated to solve the inner problem in (\\ref{eq:inner-outer-problem}). If linear precoder schemes are implemented, the maximum number of iterations is at most $K_{\\max}$. \nSince the transmitter knows the precoders and the transmit powers, the achievable rates, SINR, or BERs are known each iteration, allowing the scheduler to identify when it is worth adding more users to $\\mathcal{K}$.\n\n\nThis approach is particularly effective if there are no constraints on the cardinality of $\\mathcal{K}$, or if the network is sparse, i.e., $|\\batchmode\\mathcal{K}|\\approx M$ \\cite{Gesbert2007a}. For instance, if ZF-based precoding is used, the multiplexing gain is not maximized at the low SNR regime, i.e., the total number of scheduled users must be strictly less than $K_{\\max}$. In the high SNR regime, multiplexing gains can be maximized if full CSIT is available \\cite{Caire2003}, otherwise the system may become interference limited [cf. Definition~\\ref{defn:interference-limited-system}].\nSeveral works have modified the structure of Algorithm~\\ref{alg:algorithms:direct-approach} to reduce complexity and improve performance, see \\cite{Lau2005a,Jiang2006,Hammarwall2007,Chen2008,Wang2008a,Lim2009,Tran2010,Yi2011,Sigdel2009a,Tran2012,Adhikary2013a}. The works \\cite{Dimic2005,Hammarwall2007,Wang2008a,Huang2013} showed that the solution space can be shrunk at each iteration, reducing the computational complexity. Based on the characteristics of the powers that solve the inner problem in (\\ref{eq:inner-outer-problem}), the scheduler defines a reduced set of candidate users, $\\mathcal{K}^{temp}$, for the next iteration. \n\n\nNonetheless, such methods also exhibit the flaw of not being able to identify the optimal cardinality, $|\\mathcal{K}|$. As such, there are redundant users in the selected user subset, i.e., users that can be deleted from the selected user subset to yield a performance increase. This is an inherent flaw of any greedy incremental algorithm, due to the non-iterative cumulative user selection procedure. In \\cite{Huang2012b,Huang2013}, it is proposed to add the \\textit{delete} and \\textit{swap} operations as means to tackle the redundant user issue. However, this approach increases the complexity, even as compared to GUS, as it involves matrix inversions, projections, and an iterative procedure. Moreover, its performance is sensitive to CSI inaccuracies due to the increased error in the projection operation using imperfect CSI, and it does not necessarily find the best user subset in the imperfect CSI case.\n\n\n\\begin{algorithm}[t!] \\small\n\t\\begin{algorithmic}[1]\n\t\t\\STATE Set $\\mathcal{K}^{temp} = \\bar{\\mathcal{K}}$, select the user $k$ with the strongest channel, update $\\mathcal{K} = \\{k\\}$, calculate precoders and powers, compute the achievable rate $R(\\mathcal{K})$ in (\\ref{eq:inner-outer-problem}).\n\t\t\\STATE Find the user $i \\in \\mathcal{K}^{temp} \\setminus \\mathcal{K}$ that maximizes $R(\\mathcal{K} + \\{i\\})$.\n\t\t\\STATE\tIf $|\\mathcal{K} + \\{i\\}| \\leq K_{\\max}$ and $R(\\mathcal{K} + \\{i\\}) > R(\\mathcal{K})$, then update $\\mathcal{K} = \\mathcal{K} + \\{i\\}$, modify $\\mathcal{K}^{temp}$, and go back to step 2.\\\\ Otherwise go to step 4.\t\n\t\t\\STATE Compute final precoders, powers, and WSR for $\\mathcal{K}$.\n\t\\end{algorithmic}\n\t\\caption{Utility-Based Scheduling (Direct)}\n\t\\label{alg:algorithms:direct-approach}\n\\end{algorithm}\n\n\n\\subsubsection{Indirect Approach}\n\\label{section:algorithms:aggregated-utility:csi-mapping}\n\nIn this approach, the inner and outer problems in (\\ref{eq:inner-outer-problem}) are decoupled. To illustrate the structure of the resource allocation strategy, let us reformulate (\\ref{eq:inner-outer-problem}) as two problems that must be solved sequentially:\n\n\\begin{subnumcases}{}  \n\t\\mathcal{K}^{map} =  \\underset{ \\mathcal{K} \\subseteq \\bar{\\mathcal{K}}: |\\mathcal{K}| \\leq K_{\\max} }{ \\arg \\max} \\ f(\\mathbf{H}(\\mathcal{K})) &   \\label{eq:csi-mapping-problem-1} \n\t\\\\\n\t\\underset{ \\mathbf{W}, \\mathbf{P}, \\pi }{\\max} \\ \\  \\sum_{k \\in \\mathcal{K}^{map}} \\omega_{k} \\log_{2} \\left( 1+ \\text{SINR}_{k} \\right) & \\label{eq:csi-mapping-problem-2}\n\\end{subnumcases}\n\nEquation (\\ref{eq:csi-mapping-problem-1}) describes a combinatorial user grouping problem, where $f(\\mathbf{H}(\\mathcal{K}))$ is a metric of spatial compatibility, [cf. Section~\\ref{section:metrics-spatial-compatibility}]. The user grouping is an NP-C problem, whose solution is found via ExS \\cite{Maciel2010}. Observe that finding a solution to problem (\\ref{eq:csi-mapping-problem-1}) does not require the computation of the precoders and powers. Instead, it depends on CSIT, algebraic operations defined by a given mapping function $f(\\mathbf{H}(\\mathcal{K}))$, and an iterative procedure. It is worth noting that (\\ref{eq:csi-mapping-problem-1}) may include the weights $\\omega_{k}$ to cope with QoS or fairness requirements, see \\cite{Yoo2006}. \nIn contrast, provided the set $\\mathcal{K}^{map}$, problem (\\ref{eq:csi-mapping-problem-2}) is tractable and can be solved efficiently via convex optimization \\cite{Lau2005a,Caire2003,Godara2002Ch18,Schubert2004}.   \n\n\nEarly works on MU-MIMO systems (e.g. \\cite{Caire2003,Tu2003,Swannack2004}) pointed out that under certain conditions of the weights, $\\omega_{k}$, and SNR regime, the objective function of the inner problem in (\\ref{eq:inner-outer-problem}), is dominated by the geometry of the multiuser channel $\\mathbf{H}(\\mathcal{K})$.\nThe rational behind the reformulation in (\\ref{eq:csi-mapping-problem-1})-(\\ref{eq:csi-mapping-problem-2}), is that scheduling a set of spatially compatible users [cf. Definition~\\ref{defn:compatible_set_of_users}], is crucial to suppress IUI. Refining the power allocation and precoding weights in (\\ref{eq:csi-mapping-problem-2}) requires less computational effort than solving the combinatorial problem (\\ref{eq:csi-mapping-problem-1}).\nA generic structure of this scheduling algorithms is presented in Algorithm~\\ref{alg:algorithms:indirect-approach}.\n\n\n\nThe complexity of finding a solution to problem (\\ref{eq:csi-mapping-problem-1}) can be simplified under certain conditions. Assuming that ZF-based precoding schemes are used, results in \\cite{Caire2003,Lau2006} establish that the optimal admitted set of users achieves full multiplexing gains, i.e., $|\\mathcal{K}|=K_{\\max}$, if the system operates in the high SNR regime. Therefore, the cardinality constraint in (\\ref{eq:csi-mapping-problem-1}) is given by $|\\mathcal{K}| = K_{\\max}$, which shrinks the search space size. A common approach in the literature is to assume that $K_{\\max}$ users can be co-scheduled when solving problem (\\ref{eq:csi-mapping-problem-1}), and refining the set $\\mathcal{K}$ when solving problem (\\ref{eq:csi-mapping-problem-2}).\n\n \n\nThe majority of the works that have proposed algorithms to solve problem (\\ref{eq:csi-mapping-problem-1}), e.g. \\cite{Tu2003, Jiang2006, Dai2009, Tejera2006,Yoo2006, Nam2014, Bayesteh2008,Mao2012,Maciel2010}, use ZF precoding for the second problem (\\ref{eq:csi-mapping-problem-2}). Consequently, the NSP is the metric of spatial compatibility that maximizes the sum-rate for the set of selected users $\\mathcal{K}$, [cf. Section~\\ref{section:metrics-spatial-compatibility:projection-spaces}].\nA considerable amount of research has been focused on reducing the complexity of the NSP computation, e.g., by reusing previous NSP calculations at each iteration, implementing efficient multiuser channel decompositions, or computing approximations of the NSP. The methods highly depend on $M$, $N$, the CSIT accuracy, and the system model. \n\n\nDifferent works have proposed a search space reduction per iteration, recalculating and reducing the number of competing users so that spatial compatibility is preserved. A common approach is to preselect a group of candidate users, $\\mathcal{K}^{temp}$ in Algorithm~\\ref{alg:algorithms:indirect-approach}, based on spatial clustering [cf. Section~\\ref{section:metrics-spatial-compatibility:spatial-clustering}]. The candidate users at the next iteration will exhibit channel directions fulfilling an $\\epsilon$-orthogonality criterion. This approach, coined as semi-orthogonal user selection (SUS), was originally proposed by Yoo and Goldsmith \\cite{Yoo2006,Yoo2005a}. Several variations of this technique can be found in the literature, see \\cite{Wang2008a,Lau2009,Sun2010,Razi2010,Sohn2010,Mao2012,Driouch2012,Tran2012,Min2013,Lee2014c}.\nOnce that $\\mathcal{K}$ has been constructed, it may be necessary to modify it, so that problem (\\ref{eq:csi-mapping-problem-2}) yields the maximum WSR. A common approach to refine $\\mathcal{K}$, is to apply user removal techniques [cf. Section~\\ref{section:performance-uf:wsr-with-qos}]. This might be necessary if: the selected channels lacks spatial compatibility; the system operates in extreme SNR regimes; or if the constraints related to $\\mathcal{K}$ turn problem (\\ref{eq:csi-mapping-problem-2}) infeasible \\cite{Lim2009,Maciel2010}. If linear precoding and water-filling are used to solve (\\ref{eq:csi-mapping-problem-2}), the set $\\mathcal{K}$ can be refined by dropping users that do not achieve a target rate or whose allocated power is zero. \n\n\nAuthors in \\cite{Fuchs2007} pointed out that regardless of the number of deployed antennas, it is more efficient to schedule users so that $|\\mathcal{K}| < K_{\\max}$. This means that the scheduler must seek a tradeoff between maximizing spatial multiplexing gains, and optimizing the WSR for a small set of spatially compatible users. To solve this cardinality problem, some algorithms, e.g. \\cite{Fuchs2007,Maciel2010,Ko2012,Cheng2014}, maximize the WSR by comparing the solution of (\\ref{eq:csi-mapping-problem-2}) for multiple sets that solve (\\ref{eq:csi-mapping-problem-1}). This is performed by sequentially constructing groups of spatially compatible users with different cardinalities, and then selecting the best group based on the achievable WSR. \n\n\nFor the general heterogeneous MU-MIMO scenario, $M\\geq N_k > 1$, $\\forall k$, the function $f(\\mathbf{H}(\\mathcal{K}))$ in problem (\\ref{eq:csi-mapping-problem-1}) can be defined by a metric of subspace compatibility [cf. Section~\\ref{section:metrics-spatial-compatibility:other-metrics}]. The user grouping procedure using such a metric can operate in a greedy fashion, as in Algorithm~\\ref{alg:algorithms:indirect-approach}, but every scheduler design depends on the system model.\nIt is worth pointing out that in heterogeneous systems, maximizing spatial compatibility is not enough to guarantee WSR maximization. Multidimensional metrics measure the potentially achievable spatial separability but not the effective channel gain per spatial dimension, [cf. Definition~\\ref{defn:effective_channel_gains}]. Efficient user selection algorithms in scenarios with heterogeneous users should not only consider spatial compatibility, but also an estimation of the achievable WSR, SINRs, or effective channel gains \\cite{Yi2011}.\nMoreover, dynamic allocation of the number of data streams per user must be included in the scheduler, to take into account CSIT accuracy and fulfill QoS constraints. Users with small signal spaces experience less IUI, and results in \\cite{Bjornson2013a} show that under certain conditions, transforming a MIMO channel into an equivalent MISO channel can improve spatial separability at the expense of multiplexing gain.\n\n\n\nThe reliability of scheduling based on metrics of spatial compatibility is highly sensitive to CSIT accuracy. Most of the works in the literature (e.g. \\cite{Bayesteh2008, Shen2006, Wang2010, Ko2012, Tran2010}), perform scheduling by evaluating a form of spatial correlation between MIMO channels of different user. However, the correlation between antennas at each receiver is usually not taken into account. Such a correlation may affect the achievable SINR, depending on the signal processing at the receiver, e.g., receive ZF processing \\cite{Wang2010}, or receive combining \\cite{Jindal2008,Bjornson2013a}.\n\nThe inner correlation of the channel $\\mathbf{H}_{k}$ of the $k$-th user can be measured by its rank or by the magnitude (dispersion) of its eigenvalues \\cite{Tejera2006}. Transmitting multiple spatial streams per user cannot be reliable, specially for high inner correlation. A common approach to simplify the scheduling designs is by limiting the number of streams per user, e.g., using only the strongest eigenvector per selected user  \\cite{Bayesteh2008,Bjornson2013a, Wang2010}. For each channel $\\mathbf{H}_{k}$, there exists a dominant spatial direction that can be selected for transmission without creating severe IUI, or performance degradation after precoding. In this way, problem (\\ref{eq:csi-mapping-problem-1}) can be solved based on antenna or eigen-direction selection, e.g., \\cite{Chen2007a,Chen2008,Bayesteh2008,Lim2009,Sun2010,Ko2012,Wang2010}.\n \n\n\\begin{algorithm}[t!] \\small\n\t\\begin{algorithmic}[1]\n\t\t\\STATE Set $\\mathcal{K}^{temp} = \\bar{\\mathcal{K}}$, select the user $k$ with the strongest channel, update $\\mathcal{K} = \\{k\\}$.\n\t\t\\STATE Find the user $i \\in \\mathcal{K}^{temp} \\setminus \\mathcal{K}$ that maximizes $f(\\mathbf{H}(\\mathcal{K}+\\{i\\}))$.\n\t\t\\STATE\tIf $|\\mathcal{K} + \\{i\\}| \\leq K_{\\max}$, then update $\\mathcal{K} = \\mathcal{K} + \\{i\\}$, preselect users in $\\mathcal{K}^{temp}$, and go back to step 2.\\\\ Otherwise go to step 4.\t\n\t\t\\STATE Perform operations to refine $\\mathcal{K}$.\n\t\t\\STATE Compute precoders and powers to solve (\\ref{eq:csi-mapping-problem-2}) for $\\mathcal{K}$.\n\t\\end{algorithmic}\n\t\\caption{CSI-Mapping-Based Scheduling (Indirect)}\n\t\\label{alg:algorithms:indirect-approach}\n\\end{algorithm}\n\n\n\n\\subsection{Metaheuristic Algorithms}\n\\label{section:algorithms:bio-inspired}\n\nSeveral works in the literature address problem (\\ref{eq:inner-outer-problem}) using stochastic optimization methods, which find close to optimal solutions to complex and non-convex mixed problems. These non-traditional or metaheuristic methods are an alternative to classical programming techniques \\cite{Rao2009,Williamson2010}. The mathematical proof of convergence to the optimal solution cannot be demonstrated, and analytical results or performance analysis cannot be derived. However, these methods may find a close to optimal solution with high probability, without the need of computing derivatives or satisfying convexity requirements (as in standard optimization techniques).    \nThe principle behind metaheuristic methods is the following: a set of feasible solutions is iteratively combined and modified until some convergence criterion is met. The term stochastic comes from the fact that initial conditions are generally chosen randomly. For each iteration, the internal parameters change according to dynamic probability functions or even randomly. \nThe algorithms designed under this approach are based on certain characteristics and behavior of natural phenomena, e.g., swarm of insects, genetics, biological or molecular systems \\cite{Rao2009,Zhang2014b}.  \nIn the context of MU-MIMO systems, two techniques have been used to solve the user scheduling problem: genetic algorithms (GA) and particle swarm optimization (PSO) algorithms.\n\n\nGA are bio-inspired or evolutionary algorithms, that can solve multiple objective optimization problems with many mixed continuous-discrete variables, and poorly behaved non-convex solution spaces. Unlikely standard optimization methods that rely on a single starting point (e.g. algorithms sensitive to initial conditions \\cite{Hammarwall2007}), GA start with a set of points (population). These points increase the argument domain of the optimized function and avoid local optimal problems due to the evaluation of the solution space \\cite{Rao2009}.\nGA encode potential solutions to the optimization problem in data structures called chromosomes. Due to the binary nature of the variables $\\xi_{k}$ in problem (\\ref{eq:general-combinatorial-problem:cost-function}), the chromosomes can be represented as binary strings containing valid configurations of the variables $\\xi_{k}$, [cf. (\\ref{eq:general-combinatorial-problem:set-cardinality})]. A set of chromosomes is referred to as a population, and at each iteration, the chromosomes combine, exchange critical information, mutate, and eventually evolve toward the optimal solution \\cite{Lau2005a}.\nPractical application of GA to solve the WSR maximization problem (\\ref{eq:inner-outer-problem}) have been proposed for different precoding schemes: ZFBF \\cite{Lau2005a,Lau2006}, ZFDP \\cite{Elliott2012}, and DPC \\cite{Elliott2009}. \nRecall that given a set $\\mathcal{K}$, the performance of DPC is sensitive to a given user order $\\pi$ selected out of $|\\mathcal{K}|!$ valid orders. The works \\cite{Tu2003,Tejera2006} have proposed heuristic algorithms to define $\\pi$. Since the chromosome structure contains different optimization variables, it can include not only the set of selected users but can also the order in which the users are encoded for ZFDP \\cite{Elliott2012} and DPC \\cite{Elliott2009}. \n\n\n\nPSO are behavior-inspired algorithms that mimic the distributed behavior of social organisms (particles). The particles use their own intelligence (based on an individual utility function) and the swarm intelligence (global utility function), to discover good directions that can be followed and verified by the swarm. \nPSO initially defines the swarm as a set of particles randomly sampled from the search space. Each particle has two associated parameters that define how fast it can move toward the optimal solution: position and velocity. In this approach, each particle wanders around in the search space to collect information about the achievable utility function at different locations. This means that the particle looks for the configuration that steers toward the global maximum or the average direction of the swarm. The particles exchange information regarding the best directions, and adjust their positions and velocities accordingly until the swarm converge to the same solution \\cite{Rao2009}.  \nAuthors in \\cite{Hei2009} designed PSO algorithms to solve problem (\\ref{eq:inner-outer-problem}) using BD precoding. Such an optimization uses the utility function, $U(\\cdot)$, of the direct and indirect approaches described in Section~\\ref{section:algorithms:aggregated-utility}. The PSO approach can define the swarm direction using the function $R(\\mathcal{K})$ in the inner problem (\\ref{eq:inner-outer-problem}), or the mapping function $f(\\mathbf{H}(\\mathcal{K}))$ in (\\ref{eq:csi-mapping-problem-1}). This illustrates the fact that the computational complexity of metaheuristic algorithms depends on the optimized objective function. \n\n\n\nAlgorithm~\\ref{alg:algorithms:metaheuristic} shows a sketch of the steps used to solve problem (\\ref{eq:inner-outer-problem}) following the GA or PSO approaches. The constants $K_{smp}$ and $K_{best}$ are arbitrarily defined to limit the overall computational complexity and to speed up convergence. The convergence criterion can be defined by the maximum number of iterations or a performance threshold. The details of the operations required to generate new chromosomes or particles, [Step 3 in Algorithm~\\ref{alg:algorithms:metaheuristic}], can be found in the reference aforementioned. The robustness of GA and PSO lies in the fact that the best solutions are systematically combined, and the algorithms have reasonable immunity from getting stuck in local minimums. \n\nMetaheuristic algorithms are efficient techniques for problems where the desired solution does not need to be computed in short time scales.  Although bio-inspired algorithms have been used to optimize resource allocation in wireless networks, e.g. \\cite{Rondeau2009,Sampaio2013,Lee2014b,Zhang2014b}, they seem to have limited application in practical MU-MIMO systems. This is because the channel conditions, rate demands, and even the number of competing users may change very rapidly over time. Metaheuristic algorithms require centralized processing and the convergence time remains a critical issue \\cite{Lee2014b}. They might suffer from a high computational cost because the objective function is evaluated for each member of the population (swarm). Nevertheless, these methods can provide performance benchmarks to assess other suboptimal, yet practical, resource allocation algorithms.\n\n\n\\begin{algorithm}[t!] \\small\n\t\\begin{algorithmic}[1]\n\t\t\\STATE For the $i=1$ iteration, extract $K_{smp}$ elements (chromosomes or particles) from the search space  $\\hat{\\mathcal{K}}^{(smp)}(i) =  \\{\\mathcal{K}_{i,1}, \\ldots, \\mathcal{K}_{i,K_{smp}}\\} \\subset \\hat{\\mathcal{K}}^{(ss)}$. \n\t\t\\STATE For the $i$-th iteration, evaluate the objective function for each element in $\\hat{\\mathcal{K}}^{(smp)}(i)$, select the $K_{best}$ elements with the largest objective function, and discard the other bad elements.\n\t\t\\STATE Perform operations over the remaining elements to generate an improved population\/swarm, $\\hat{\\mathcal{K}}^{(smp)}(i+1)$, for iteration $i+1$. \n\t\t\\STATE If convergence criterion is met, then compute precoders and powers for the solution set.\\\\ Otherwise go to step 2.\n\t\\end{algorithmic}\n\t\\caption{Metaheuristic Optimization}\n\t\\label{alg:algorithms:metaheuristic}\n\\end{algorithm}\n\n\n\n\\subsection{Classical Optimization}\n\\label{section:algorithms:classical-optimization}\n\n\nClassical optimization techniques \\cite{Rao2009,Boyd2004},  have been used for several resource allocation problems in MU-MIMO systems, e.g., WSR maximization, sum power minimization, max-min SINR, and other objective functions, see \\cite{Bjornson2013,Hong2012,Utschick2012,Stanczak2009,Athanasiou2015,Bethanabhotla2016} and references therein. Numerous works have included user scheduling for the optimization of the WSR \\cite{Chan2007}, sum power minimization \\cite{Matskani2008,Ho2009,Moretti2013}, or some metrics of spatial compatibility \\cite{Maciel2010,Castaneda2014}. A large number of optimization problems in MU-MIMO systems are non-convex or non-polynomial time solvable, depending on the system models and constraints \\cite{Hong2012}. Due to the high complexity of the resource allocation problem, the conventional optimization solutions must relax the original problem, approximate non-convex constraints in an iterative fashion, or change the domain of the optimization, sacrificing optimality for the sake of tractability.  Nevertheless, by employing a fixed structure for the precoders, the original problem can be simplified, and efficient resource allocation can be performed.\n\n\n\nAuthors in \\cite{Chan2007} presented different approaches to solve (\\ref{eq:general-combinatorial-problem:cost-function}) in a OFDMA MU-MIMO scenario. The mathematical formulation renders the combinatorial problem into a convex problem. By analyzing the entire search space, the proposed algorithms perform channel assignment and signal space selection, i.e., the algorithm defines the scheduled users per carrier and the number of spatial streams per user. Such a reformulation of (\\ref{eq:general-combinatorial-problem:cost-function}) yields algorithms that decouple the channel assignment in the OFDMA system, i.e., the global optimization is attained by solving problem (\\ref{eq:inner-outer-problem}) per sub-carrier. Furthermore, the objective function of the inner problem in (\\ref{eq:inner-outer-problem}) is defined in terms of rates and powers, which optimizes the total capacity and power consumption.  \nFor classical programming techniques, the set of constraints might turn the problem infeasible. The resource allocation algorithms proposed in \\cite{Chan2007} identify the assignment sets for which time sharing is the best resource allocation strategy. As discussed in Section~\\ref{section:performance-uf:wsr-with-qos}, admission control or user removal techniques can be implemented as a form of scheduling to guarantee feasibility.\n\n\n\nAuthors in \\cite{Moretti2013} have reformulated the user scheduling problem in multi-carrier scenarios as an extension of problems (\\ref{eq:csi-mapping-problem-1}) and (\\ref{eq:csi-mapping-problem-2}). A close to optimal solution can be found by a sequence of convex and linear programming optimizations. To reduce complexity of the combinatorial problem (\\ref{eq:csi-mapping-problem-1}), the mapping function $f(\\mathbf{H}(\\mathcal{K}))$ is defined as the magnitude of the MIMO channels, and a close to optimal scheduling sequence is designed based on that metric. According to \\cite{Ho2009,Moretti2013}, the generalization of (\\ref{eq:csi-mapping-problem-2}) for OFDMA systems yields a non-convex problem, but quasi-optimal solutions can be attained by applying the classical dual decomposition method. \nThe works \\cite{Moretti2013,Maciel2010} have modeled the user scheduling problem in OFDMA systems as a channel assignment problem, which can be solved by efficient algorithms that run in strong  polynomial time \\cite{Burkard2009}.  \n\n\n\nA novel reformulation of the WSR maximization problem, (\\ref{eq:inner-outer-problem}), as a semi-definite program  in \\cite{Ku2015}, sheds some light on non-explicit convex properties of the joint precoding design, power allocation, and users selection problem. The reformulation requires semi-define relaxation, and  the solution is found by combining convex optimization and sub-gradient projection methods. \nAnother approach in \\cite{Song2008} extends the max-min fair rate allocation problem in \\cite{Schubert2004,Tan2012,Cai2011,Schubert2006} so that joint optimization of precoding, power allocation, and user selection was attained. The problem was formulated as the difference of convex functions and solved by the branch and bound (BB) technique, which is a method used to solve mixed-integer programming problems \\cite{Rao2009}. \nThe approach in \\cite{Song2008} yields an interference limited systems [cf. Definition~\\ref{defn:interference-limited-system}], i.e., it is allowed that $|\\mathcal{K}|\\geq M$, which highly increases the search space and computational complexity. Notice that even if the search space is constrained so that $|\\mathcal{K}|\\leq M$, there are $2^K -1$ possible user schedules in a MISO configuration that will be enumerated by the BB technique. \n\n\nAs discussed in Section~\\ref{section:performance-uf}, admission control is a form of user selection that may be required to guarantee feasibility. In \\cite{Matskani2008}, the joint sum power minimization and maximization  of $|\\mathcal{K}|$ was formulated as a second-order cone program (SOCP). The set $\\mathcal{K}$ is refined iteratively by dropping the user with the largest gap to its target SINR,\\footnote{See \\cite{Mahdavi2010} for optimal and suboptimal user removal algorithms for QoS and power constrained systems.} (i.e. the most infeasible user), and optimal precoders and powers are found by the SOCP optimization.  \nAlthough the approaches in \\cite{Song2008,Matskani2008,Ku2015} provide suboptimal solutions due to the relaxation of non-convex problems, they show that joint optimization of the three variable sets $\\mathcal{K}$, $\\mathbf{P}$, and $\\mathbf{W}$ in (\\ref{eq:inner-outer-problem}), is feasible through convex optimization. \n\n\nOther classical optimization methods have been used in the single-carrier MISO scenario with ZF precoding. \nQuadratic optimization was used in \\cite{Maciel2010} to solve problem (\\ref{eq:csi-mapping-problem-1}), by optimizing a heuristic function $f(\\mathbf{H}(\\mathcal{K}))$ that linearly combines channel magnitudes and spatial correlation of the MIMO channels.\nIn \\cite{Castaneda2014}, a heuristic objective function that approximates the NSP has been proposed, and the set $\\mathcal{K}$ was found via integer programming. \nThe authors in \\cite{Chen2013} jointly addressed scheduling and WSR maximization subject to QoS constraints using geometric programming. However, similar to the approach in \\cite{Chan2007}, the optimization in \\cite{Chen2013} iterates over all solutions of the search space, which limits its application for practical scenarios.\n\n\n\nOther works model the resource allocation problem (\\ref{eq:csi-mapping-problem-1}), as mathematical problems that have been extensively studied \\cite{Korte2006,Burkard2009}: the minimum set cover problem from graph theory (e.g., \\cite{Yoo2005a,Driouch2012}); and the sum assignment problem (e.g., \\cite{Aniba2007,Castaneda2013b,Dartmann2013,Wang2011b}). The modeling used in these approaches requires the calculation of weights or costs values associated with every possible subset in the solution space, i.e., $\\{ \\forall \\mathcal{K}  \\subseteq \\bar{\\mathcal{K}} : |\\mathcal{K}| \\leq c_p \\}$, where $c_p$ depends on the system parameters. The solutions of such general problems can be found by heuristic algorithms.\nDue to the exponentially increasing complexity on $M$, $N$, and $K$, the algorithms based on classical optimization are suitable for offline implementation, benchmarking, and to assess other suboptimal heuristic resource allocation strategies for relatively small values of $K$ \\cite{Utschick2012,Hong2012}. \n\n\n\\subsection{Scheduling in Multi-Cell Scenarios}\n\\label{section:algorithms:multi-cell-selection-classification}\n\n\nIn Section~\\ref{section:system-set-ups:multi-cell-scenarios}, we have described the types of transmission schemes in multi-cellular scenarios. Different levels of coordination are required to improve and maintain the performance of users geographically spread within the coverage area. For users located at the cell edge, the simultaneous transmission from different clustered BSs can boost their throughputs (e.g. CoMP \\cite{Marsch2011}), whereas users close to their BS do not require interaction with adjacent BSs.\nIn full signal-level coordinated systems (e.g., Network MIMO or JT), the global user data and CSI knowledge enables a CU to perform CS using the approaches described in Section~\\ref{section:algorithms:aggregated-utility} and Section~\\ref{section:algorithms:selection-partial-csit}. \nAssuming partial coordination, the CS\/CBF can offer flexibility, scalability, and viability, since centralized or distributed algorithms can be implemented with limited signaling overhead and using local CSIT.\nThe minimum level of cooperation between transmitters can be attained through orthogonal resource allocation. For instance, in LTE systems a technique known as \\textit{dynamic point blanking},  prevents transmission at a certain time-frequency resource to reduce interference over such a resource used at a neighboring transmission point \\cite{Dahlman2013,Lopez-Perez2015}.\n\n\n\nThe optimization of CS and CBF can be done based on the methodologies described in Section~\\ref{section:algorithms:aggregated-utility}, i.e., the two problems can be addressed either independently or jointly. If CS and CBF are decoupled, CS can be tackled by processing shared information related to statistical CSI or performance metrics extracted from local CSI [cf. Section~\\ref{section:metrics-spatial-compatibility}]. The CBF optimization can be solved by optimal or heuristic techniques for precoding design and power allocation, see \\cite{Bjornson2013} for a comprehensive review.  \nA joint optimization of CS and CBF requires iterative calculation of the precoders and selected users, since both optimization spaces are coupled \\cite{Li2014,Ku2015}. Efficient scheduling rules must determine the set of users that maximizes the performance, and generates less interference to neighboring cells.   \nThe following classification presents some existing methods to implement CS and CBF in multi-cell networks \\cite{Pateromichelakis2013,Li2014}.\n\n\n\\subsubsection{Joint CS-CBF}\n\\label{section:algorithms:multi-cell-selection-classification:join-cs-cbf}\n\n\nIn full coordinated multi-cellular systems, a cluster of BSs defines a super-cell or virtual distributed antenna system with per-BS power constraints. Full coordination allows different transmission strategies: \\textit{i}) all clustered BSs send data to all the users in $\\mathcal{K}$; \\textit{ii}) some BSs jointly serve a subset of $\\mathcal{K}$; \\textit{iii}) each BS serves a set of users associated to it; or \\textit{iv}) each BS performs single transmission (ST) and serves only one user per scheduling interval, which is known as the interference channel model (IFC), see \\cite{Cover2006,Jorswieck2008,Larsson2008,Bjornson2013}. The fist and second strategies are exclusive of fully coordinated systems since payload data is shared among BSs. The other strategies are also implemented for distributed CS\/CBF. Recent works have extended these transmission schemes to heterogeneous networks, e.g., \\cite{Lossow2013,Park2014,Ku2015}. \n\nAccounting for global CSIT provides flexibility to the CS\/CBF design, and several ICI cancellation techniques can be applied. Using classical optimization, the authors in \\cite{Ku2015} solved the joint CS\/CBF optimization, formulated as an extension of problem (\\ref{eq:inner-outer-problem}) for heterogeneous networks.  \nThe authors in \\cite{Zhang2009,Huh2012} addressed the CS by iteratively evaluating the precoders and powers, following the methodology described in Section~\\ref{section:algorithms:aggregated-utility:objective-function}. \nThe approach in \\cite{Park2010} tackled the CBF problem using interference alignment \\cite{Jafar2011}, whereas the CS was sequentially solved in a greedy fashion. A dynamic transmission switching between ST and JT was presented in \\cite{Park2014}, where a centralized CS exploited individual rate statistics (CDF scheduling). \nIn \\cite{Marsch2011}, problem (\\ref{eq:csi-mapping-problem-1}) was solved using metrics of spatial compatibility (NSP-based user selection), and a centralized resource allocation solved problem (\\ref{eq:csi-mapping-problem-2}), [cf. Section~\\ref{section:algorithms:aggregated-utility:csi-mapping}]. \nPractical CS schemes were proposed in \\cite{Lossow2013} for heterogeneous cellular systems, where macro and small cells coordinate scheduling decisions using an the approach described in Section~\\ref{section:algorithms:aggregated-utility:objective-function}. The CBF was optimized using linear precoding and dynamic SU\/MU-MIMO switching \\cite{Schellmann2010}.   \n\n\t\n\\subsubsection{CS with cyclic CBF}\n\\label{section:algorithms:multi-cell-selection-classification:cs-cyclic-cbf}\t\t\n\n\nGiven a pool of precoding weights, each BS picks a different subset for transmission every scheduling interval, and switches them periodically, in a temporal beam-reuse fashion \\cite{VanRensburg2009,Hosein2009}. This method requires minimum coordination and uses information regarding the allocated beams per BS and the interference environment. This allows practical scheduling and mitigates the flashlight effect (changes of the active beams at neighboring transmitters). Another approach in \\cite{Dartmann2013}, performs user scheduling by solving a series of assignment problems heuristically, and the precoding weights are optimized every iteration. \n\n\n\\subsubsection{Sequential CS-CBF}\n\\label{section:algorithms:multi-cell-selection-classification:sequential-cs-cbf}\n\t \t\n\nThe clustered BSs jointly select users and assign resources in a sequential fashion. The first BS selects its users and broadcast its decision, then the second BS selects its user based on the decision made by the first BS and so on, see \\cite{Cui2011a,Wang2011b,Hossain2011,Dartmann2013,Moon2013}. The user selection per BS can be done using the approaches presented in Section~\\ref{section:algorithms:aggregated-utility}, and the scheduling order of the cells can be assigned according to interference levels or using the round robin approach. \nThe CS can be performed based on interference constraints as in \\cite{Seifi2011}. The BSs schedule their users and  perform resource allocation sequentially, so that the ICI generated to previously selected users is below a threshold. If the interference cannot be mitigated as desired, the interfering BSs remain silent during the scheduling interval, which is a low-complexity dynamic clustering strategy. \n\t\t\n\n\\subsubsection{Decoupled CS-CBF}\n\\label{section:algorithms:multi-cell-selection-classification:decoupled-cs-cbf}\n\t\nA distributed optimization policy interconnects clustered BSs through a CU or master BS, limiting the message exchange to local CSI and scheduling control signaling \\cite{Bjornson2013}. \nProblem (\\ref{eq:inner-outer-problem}) has three optimization variable sets, namely, $\\mathcal{K}$, $\\mathbf{W}$, and $\\mathbf{P}$, and their joint optimization is difficult to solve optimally and distributively. Still, authors in \\cite{Chiang2008} proposed an iterative approach that decouples and optimizes some optimization variables, while the others remain fixed. \nYu \\textit{et.al} in \\cite{Yu2013} have followed such an approach and developed a strategy to tackle (\\ref{eq:inner-outer-problem}) in a semi-distributed manner, demanding limited communication between BSs and the CU: fix the first two set of variables and optimize the third set, then fix the second and third sets and optimize the first one, and so on until convergence. The BSs jointly update parameters associated with all variable sets in an semi-distributed fashion. By fixing the set of users and precoder weights, the power allocation can be performed based on the information shared between BSs, see \\cite{Stridh2006a,Chiang2008,Stanczak2009}. By fixing the precoders and powers, the user scheduling can be performed using the approaches described in Sections~\\ref{section:algorithms:aggregated-utility}. By fixing the set of users and power allocation, the precoder weights can be computed using distributed algorithms, see \\cite{Bjornson2013}. \t\nA semi-distributed CS was proposed in \\cite{Castaneda2015}, where the users are selected using an approximation of the NSP [cf. Section~\\ref{section:metrics-spatial-compatibility:projection-spaces}], and the CBF is computed with local CSI. A similar approach was proposed in \\cite{Jang2011}, where the BSs dedicate spatial DoF to cancel ICI for some selected users at neighboring cells. This approach performs semi-distributed spatial clustering for CS [cf. Section~\\ref{section:metrics-spatial-compatibility:spatial-clustering}], and distributed linear precoding for CBF. \n\t\n\n\n\n\n\n  \n\n\n{\\color{black}{\n\n\n\\begin{figure*}[t!]\n\t\\centering\n\t\\includegraphics[width=0.98\\linewidth]{figure9.pdf}\\\\\n\t\\caption{Average sum rate vs SNR vs the number of users ($K$), for $M=8$ (left plot) and $M=64$ (right plot).} \n\t\\label{fig:k-vs-snr}\n\\end{figure*} \n\n\\subsection{Scheduling for Massive MIMO}\n\\label{section:algorithms:aggregated-utility:massive-mimo}\n\n\nMassive or large-scale MIMO has been widely envisaged as a key transmission technology for next generation of  wireless communication, 5G \\cite{Andrews2014,Zheng2015}. In massive MIMO systems, the BS is equipped with a large number of antennas (few hundreds) and serve multiple users (normally few tens). The excess amount of antennas enables focusing the transmission and reception of signal energy into smaller regions of space. Joint optimization over the spatial and multi-user domains can provide significant gains in throughput and EE  \\cite{Marzetta2010, Yang2013b, Rusek2013, Zheng2015}.\n\n\n\nIn conventional MU-MIMO systems, the number of co-scheduled users $K$ is usually larger than the number of BS antennas $M$. In contrast, in massive MIMO systems the  transmit antennas outnumber the active users ($M\\gg K$), which reduces the signal processing complexity and achieves large peak rates. However, the promising performance improvements come at the expense of hardware complexity. Due to cost and power consumption, practical transceiver architectures have a different number of RF chains ($M_{RF}$) compared to the number of antennas $M$, [cf. Section~\\ref{section:precoding:massive-mimo}]. If every antenna has its own RF chain, i.e. $M_{RF} = M$, digital beamforming, [cf. Section~\\ref{section:precoding:linear}], can allocate the whole spectrum to each active user \\cite{Bjornson2016}. In practice, the total number of simultaneous users is constrained by the number of RF chains at the base stations \\cite{Bogale2015,Shokri-Ghadikolaei2015,Bogale2016}. When the number of antennas is larger than the number of RF chains and users, i.e., $M\\gg K \\geq M_{RF}$, the system performance can be boosted from the diverse path losses and shadow fading conditions of different users. \n\n\nThe channel hardening effect in massive MIMO removes frequency selectivity, and avoids complex scheduling and power control designs \\cite{Bjornson2016}. As $M \\rightarrow \\infty$, the MIMO channels become spatially uncorrelated, user separability (in the sense of Definition~\\ref{defn:compatible_set_of_users}) plays a minor role in the scheduling decisions and the performance optimization relies on the channel magnitudes, see \\cite{Zhang2007b,Bjornson2009a,Liu2015a,Hong2015}. If sum-rate is the ultimate metric to be optimized, a highly efficient scheduler only needs to assign resources to the users with the largest channel magnitudes. \n\n\nMost of the scheduling designs for massive MIMO are based on the greedy approaches presented in Section~\\ref{section:algorithms:aggregated-utility}, e.g. \\cite{Huh2012,Adhikary2013,Nam2014,YiXu2014,Lee2014c,Bogale2015,Bogale2016a}. In these works is common to assume clusters of users sharing similar slow fading characteristics. Approaches such as the \\textit{Joint Spatial Division and Multiplexing} (JSDM) \\cite{Adhikary2013,Nam2014} first partition cell users into groups with distinguishable linear subspace spanned by the dominant eigenvectors of the group's channel covariance matrix.  The transmit beamforming design is performed in two stages: a pre-beamforming that separates groups by filtering the dominant eigenvectors of each group's channel covariance matrix, followed by precoding for separating the users within a group based on the effective channel. \n\n\nThe user clustering in \\cite{Adhikary2013,Nam2014} can be implemented using  spatial subspace compatibility metrics [cf. Section~\\ref{section:metrics-spatial-compatibility:other-metrics}]. It is worth noticing that such a grouping  process might be necessary to eliminate pilot contamination, by allocating identical pilot sequences to groups of users with similar channel characteristics \\cite{Elijah2016}. The overlap between user clusters and their sizes are features dictated by two factors, the number of RF chains, $M_{RF}$, and the number of geographically co-located uses \\cite{Shokri-Ghadikolaei2015}. The principle of the two-stage beamforming technique used by JSDM can be applied to multi-cell systems, where each  cluster of users is associated to one BS, and the outer precoders are used to cancel ICI \\cite{Liu2014a}. Note that widely used scheduling algorithms in small-scale MU-MIMO, e.g. SUS \\cite{Yoo2006}, result in extremely high complexity (of the order of $O(M^3K))$ in the large antenna regime \\cite{Lee2014c,Hong2015}. In contrast, low-complexity scheduling algorithms may attain acceptable performance with full digital beamforming in certain scenarios \\cite{Hong2015,Bai2014,Liu2015a}. However, further research must be done to design efficient scheduling rules for heterogeneous networks using hybrid precoding [cf. Section~\\ref{section:precoding:massive-mimo}].\n\n\nA process closely related to scheduling is user association, whose mathematical formulation resembles problem (\\ref{eq:general-combinatorial-problem:cost-function}). Matching a user to a particular transmitter, so that the association optimizes an objective function [cf. Section~\\ref{section:performance-uf}], is a complex combinatorial assignment problem \\cite{Liu2016}. Ongoing research on this topic have covered cellular (e.g. \\cite{Bethanabhotla2016}) and WLAN (e.g. \\cite{Athanasiou2015}) systems, but literature on heterogeneous networks is still limited \\cite{Liu2016}. \nHeuristic and simple association rules employed in current cellular networks, e.g. the max-RSS (received signal strength) or the biased-received-power based criteria, cannot include all system constraints neither fully exploit massive MIMO. In heterogeneous networks,  one must consider  per-base-station load and power constraints, per-user QoS constraints, and different number of antennas per transmitter. Construction of efficient rules for user association for future heterogeneous 5G networks is an emerging research topic, which must consider channel conditions, load balancing, and EE \\cite{Liu2016}.   \nRecent results in \\cite{Li2015} show that user association to a single BS is optimal most of the times, which can simplify the association rules and algorithms. However, in dense scenarios where the coverage range per transmitter is limited (e.g., due to power constraints), multiple user-base station associations can reduce handover and waste of resources in the backhaul network \\cite{Shokri-Ghadikolaei2015,Bogale2016}. \n\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=.98\\linewidth]{figure10.pdf}\\\\\n\t\\caption{Average sum rate vs number of antennas ($M$) vs SNR, and fixed number of active users, $K=128$.} \n\t\\label{fig:m-vs-snr}\n\\end{figure}\n\n \n\n}}\n\n\n\n\n\n{\\color{black}{\n\n\n\\subsection{Discussion and Future Directions}\n\\label{section:algorithms:discussion}\t\n\n\nTo analyze the performance of the scheduling schemes described in Section~\\ref{section:algorithms:aggregated-utility}, we consider uniformly distributed users deployed within a single cell with radius 250 m. The users have heterogeneous channel conditions, the path loss exponent is 3.5, and the log-normal shadow-fading has 8 dB in standard deviation.  For the direct selection [cf. Algorithm~\\ref{alg:algorithms:direct-approach}], the utility function is the sum rate, in the indirect selection [cf. Algorithm~\\ref{alg:algorithms:indirect-approach}], the mapping function is the NSP, and the Max-RSS selects the users with larger channel gains. It is assumed full CSIT at the BS, the precoding scheme is full digital ZFBF (i.e. $M_{RF}=M$), and water-filling is used for power allocation. \n\n\nThe average achievable sum rate is shown in Fig.~\\ref{fig:k-vs-snr}, for a MISO configuration with $N=1$, $M \\in \\{8, 64\\}$, a cell edge SNR in the range $[0,20]$ dB, and several number of users, $K \\in [8, 128]$.\nFor $M=8$ (left-plot), we have a conventional MU-MIMO scenario with $K \\geq M$, $|\\mathcal{K}| \\leq M$, and the performance gap between the direct and indirect selection vanishes as $K$ increases. The Max-RSS selection partially exploits the MUDiv by only considering the channel magnitudes, which results in a considerable performance gap as $K$ grows.\nFor $M=64$ (right plot), we can divide the users ($K$) axis in two regions. For $K\\leq 32$, the system operates in the massive MIMO region, where there are enough spatial resources to allocate one stream per active user. Notice that as $K$ approaches $M$, e.g. $K=32$, the direct selection overcomes the other methods specially in the low SNR regime. For $K=64$, the indirect and Max-RSS methods attempt to allocate resources to $M$ users, whereas the direct selection optimizes its objective function regardless the attainable multiplexing gain. This result shows that even if $M$ is large, the performance metric is optimized when $|\\mathcal{K}| < M$, see \\cite{Bjornson2016}. It is worth noticing that for $M=64$ and $K=128$, the performance gap between the direct and indirect selection decreases, but the latter approach might require less computational processing. This illustrates the fact that each selection method provides different performance-complexity tradeoffs, depending on the parameters $M$, $N$, $K$, and the SNR regime. \nFig.~\\ref{fig:m-vs-snr} compares the performance of the selection methods with fixed $K=128$, and different values of $M$. The performance gap between the direct and indirect selection is small for $M\\leq 32$, where the MUDiv is rich. The Max-RSS selection might be an efficient alternative, if $M$ is large and the number of co-scheduled users is small, i.e., $M \\gg |\\mathcal{K}|$. \n\n\nScheduling in massive MIMO systems has not attracted the same attention as in conventional MU-MIMO systems because the excess number of antennas, additional DoF in the scheduling procedure, and the channel hardening effect result in marginal scheduling gains. However, in real-world massive MIMO systems with imperfect CSI, errors in the channel estimation and calibration, there will be some remaining interference among users, especially using linear precoding. The system will inevitably become interference-limited and the sum-rate may saturate (ceiling effect) at high SNR if the CSI and estimation accuracies do not improve accordingly. In this case, it is crucial to select the right number of users to serve, independently of spatial compatibility and channel correlation metrics. Conventional user selection algorithms [cf. Section~\\ref{section:algorithms:aggregated-utility}], may fail to serve the optimal number of users, even if the selected users have good spatial characteristics. \nIn other words, in massive MIMO systems, it is more important to identify the optimal number of users to serve rather than a set of users with certain channel characteristics \\cite{Bjornson2016}. Even random or max-RSS user selection will perform well if they select the right amount of users.\n\n\nThe overhead due to channel estimation is proportional to the number of transmit antennas $M$. Thus, massive MIMO is more adequate for TDD operation, and its implementation in FDD mode is still an open problem \\cite{YiXu2014,Bjornson2016}. Although uplink and downlink scheduling are independent tasks with different traffic loads and access techniques \\cite{Abu-Ali2013}, the CSI necessary to optimize the downlink transmission is estimated via uplink pilots (channel reciprocity).\\footnote{Experimental work for distributed MIMO \\cite{Hamed2016}, presented a channel reciprocity protocol that can achieve the performance of explicit channel feedback for mobile users, which is critical for massive MIMO settings.} Thus, further research is necessary to understand the relationship between uplink and downlink scheduling and their associated resource allocation for TDD-based massive MIMO. \n      \n\n}}\n\n\n\n\n\n\n\n\\section{MU-MIMO Scheduling with partial CSIT}\n\\label{section:algorithms:selection-partial-csit}\n\nIn MU-MIMO systems with partial CSIT, resource allocation algorithms can multiplex up to $M$ users accounting for the feedback of one scalar (CQI) and one index (CDI) \\cite{Sharif2005}, [cf. Section~\\ref{section:introduction:needof-csit}]. The feedback information load is proportional to the number of deployed users and antennas, but scheduling usually requires rough quantization resolution in the CQI to differentiate between high and low rate users \\cite{Gesbert2004}. The CDI plays a more relevant role to achieve spatial multiplexing, thus, it requires a higher quantization granularity.\nUser scheduling requires two main steps: \\textit{i}) the transmitter sends pilots for CSI acquisition, the users quantize their channels and feed back the CQI and CDI; \\textit{ii}) subsets of users and beams are selected for data transmission based on a particular performance metric \\cite{Wagner2008}.\nAs discussed in Section~\\ref{section:precoding:partial-csit}, there are two approaches to acquire channel information using codebooks, i.e., quantizing either the channel, or the precoder that better fits the channel. In the following, we classified different scheduling approaches, based on the type of quantization.\n\n\n\\subsection{Scheduling using quantized channels}\n\\label{section:algorithms:selection-partial-csit:qunatized-csi}\n\nFor the sake of exposition, assume a MISO scenario ($N=1$), linear precoding, and equal power allocation ($\\rho=P_k = P\/M$,  $\\forall k$). The $\\text{CQI}_{k}$ of the $k$-th user can be given by its SINR defined as \\cite{Jindal2006}:\n\n\\begin{equation}\\label{eq:pcsit_sinr_user_k}\n\\text{SINR}_{k} = \\frac{  \\rho \\| \\mathbf{H}_{k} \\|^{2} | \\hat{\\mathbf{c}}_{k} \\mathbf{W}_{k} |^{2} }\n{ 1 + \\rho \\| \\mathbf{H}_{k} \\|^{2} \\sum_{j\\neq k} |  \\hat{\\mathbf{c}}_{k}\\mathbf{W}_{j} |^{2}   }, \n\\end{equation}\nwhere $\\{\\mathbf{W}_{i}\\}_{i=1}^{M}$, are the precoding vectors extracted from the CDIs (e.g. using the ZFBF).  $\\hat{\\mathbf{c}}_{k}$ is the actual quantized unit-norm channel or $\\text{CDI}_{k}$ given by\n\\begin{equation}\\label{eq:pcsit_channel_quantization_best_beam_user_k}\n\\hat{\\mathbf{c}}_{k} = \\underset{\\mathbf{c}_{b} \\in \\mathcal{C}}{ \\arg \\max} \\ \\  \\cos (\\measuredangle(\\mathbf{H}_{k},\\mathbf{c}_{b})),\n\\end{equation}\nwhere $\\mathcal{C}=\\{ \\mathbf{c}_{1}, \\ldots, \\mathbf{c}_{b}, \\ldots, \\mathbf{c}_{2^B} \\}$ is a predefined codebook, [cf. Section~\\ref{section:precoding:partial-csit}]. The $k$-th user determines its CDI using (\\ref{eq:pcsit_channel_quantization_best_beam_user_k}), according to a minimum distance criterion, see \\cite{Mukkavilli2003} for MISO or \\cite{Love2003,Chae2008} for MIMO setting. The solution of (\\ref{eq:pcsit_channel_quantization_best_beam_user_k}) has the following geometrical interpretation: the user $k$ selects the most co-linear or correlated codeword to its channel, which is equivalent to find the cone containing $\\mathbf{H}_{k}$, as illustrated in Fig.~\\ref{fig:hyperslab_cones}. \nNotice that the $\\text{SINR}_{k}$ takes into account channel magnitudes, quantization errors of the CDI, and the spatial compatibility of channel $\\mathbf{H}_{k}$ regarding $\\mathcal{C}$. However, to compute the precoders $\\{\\mathbf{W}_{i}\\}_{i=1}^{M}$, it is necessary to know a priory $\\mathcal{K}$, and the associated CDIs $\\{\\hat{\\mathbf{c}}_{i}\\}_{i=1}^{|\\mathcal{K}|}$, which results in a chicken-and-egg problem. Therefore, the exact value of (\\ref{eq:pcsit_sinr_user_k}) is unknown at the transmitter or receiver sides. Several works have proposed different ways to approximate (\\ref{eq:pcsit_sinr_user_k}), mainly by its upper\/lower bounds or expected value, cf. \\cite{Kountouris2007,Yoo2007,Trivellato2008,Conte2010,Xu2010,Ravindran2012,Sohn2012,Wang2013a,Huang2009a}. If the transmitter has statistical CSI knowledge, the formulation of (\\ref{eq:pcsit_sinr_user_k}) must take into account the covariance matrices, $\\mathbf{\\Sigma}_{k}$, $\\forall k$, to have a more accurate estimate of the interference powers \\cite{Hammarwall2008a}.\n\n\nThe CQI can also be given by the channel magnitude, i.e., $\\|\\mathbf{H}_{k}\\|^2$, but this measure ignores the valuable information contained in the codebook (spatial compatibility), and neglects quantization errors. The number of competing users, the feedback load, and scheduling complexity can be reduced by setting thresholds to the CQIs. This means that only users with strong channels will be considered for scheduling \\cite{Conte2010,Min2013}. Multiplexing and MUDiv gains are realized by carefully tuning the threshold according to statistics (e.g. \\cite{Xu2010}) or numerical characterization (e.g., \\cite{Yoo2007,Sohn2010,Min2013}) of the CQI, for a set of fixed parameters $B$, $M$, $K$, and $\\epsilon$, [cf. Section~\\ref{section:metrics-spatial-compatibility:spatial-clustering}]. \nIn some scenarios where the maximum number of spatial streams per user is at most one, each receiver uses either a weight vector \\cite{Trivellato2008} or a combining technique \\cite{Jindal2008,Bjornson2013a,Schellmann2010}  to transform its MIMO channel matrix into an \\textit{effective} MISO channel vector.\n\n\n\nDifferent approaches to reduce feedback load and  compute  the CQI and CDI can be implemented (e.g. \\cite{Wang2013a}). \nIn general, the CQI requires less bits than the CDI, but an optimized bit allocation must take into account the SNR regime, the number of competing users $K$ \\cite{Xu2010,Yoo2007}, and practical quantization levels (e.g. MCSs) \\cite{Vicario2008}, [cf. Section~\\ref{section:algorithms:selection-partial-csit:two-stage-feedback}].\nAssume that the CQI and CDI are fed back by all competing users, so that the quantized channels can be reconstructed at the transmitter. Resource allocation can be performed using the methods described in  Section~\\ref{section:algorithms} or by low-complexity approaches based on antenna selection, e.g. \\cite{Khoshnevis2013,Castaneda2015}. \nZF precoding is the most common technique used in scenarios with quantized channels, hence, scheduling based on metrics of spatial compatibility [cf. Section~\\ref{section:metrics-spatial-compatibility}], can be directly applied, e.g., \\cite{Yoo2007, Kountouris2007,Wang2007a,Chae2008}. The user selection for the general MIMO scenario can be performed by computing effective MISO channels or by treating each receive antenna as a virtual user \\cite{Xu2010}. \nIt is worth mentioning that ZF becomes interference-limited in the high SNR regime with fixed $B$. However, if $K$ or $B$ scale with the SNR, the achievable rates can be improved and the quantization errors can be mitigated \\cite{Jindal2006,Yoo2007}. \n\n\n\nA common approach to combat IUI (generated by quantization errors), is by reducing the multiplexing gain. Finding the optimal number of data streams is an optimization problem that depends on the system parameters, e.g. SNR regime, mobility, $K$, $B$, and $M$, see \\cite{Wang2007a,Dai2008,Kountouris2012,Zhang2011}. If $K \\approx M$,  it is likely to have a codebook with limited granularity (fixed $B$) yielding irreducible IUI. In such a case, the subset of selected users $\\mathcal{K}$, must have cardinality strictly less than $M$, and their CDIs must have high resolution. In that way, the precoding performance can be improved and additional spatial DoF can be used to cancel interference out \\cite{Ravindran2012}. The results in \\cite{Wang2007a,Dai2008,Zhang2011} show that, the number of selected users highly depends on $B$, whose optimal value is a function of the SNR regime. For  extreme values of the SNR (very low or high), the optimal transmission schemes are TDMA or SU-MIMO, which completely avoid IUI caused by inaccurate CSIT. \nExperimental results in \\cite{Jones2015} show that selecting about $\\frac{3}{4}$ of the total available beams maximizes performance in different MU-MIMO WLAN configurations.\n\n\n\nIt is worth noting  that the SNR regime plays an important role in the scheduling rule design. Authors in \\cite{Huang2009a} suggested the following guidelines to improve performance: \n\\begin{enumerate\n\t\\item In the high SNR regime with fixed codebook size $2^B$, the system becomes interference limited [cf. Definition~\\ref{defn:interference-limited-system}]. The scheduling rules should prioritize users whose fed back channels reduce the quantization error, i.e., the CDI and the available spatial DoF defined the attainable performance. Results in \\cite{Yoo2007} show that the error quantization can be mitigated either in the large user regime ($K \\rightarrow \\infty$), or the high resolution regime ($B \\rightarrow \\infty$).  \n\t\\item If the variance of the noise and the IUI are comparable, both the CQI and CDI should be taken into account for user selection.\n\t\\item In the low SNR regime,  the system is noise limited and the scheduling rules should prioritize the CQI, since the CDI or error quantization play a negligible role.\n\\end{enumerate}\n\nThese  guidelines for scheduling design can be applied to MU-MIMO scenarios with full CSIT, and they have been extended to multi-cellular cooperative systems in \\cite{Castaneda2015}.\n\n\n\n\\subsection{Scheduling using RBF}\n\\label{section:algorithms:selection-partial-csit:rbf}\n\nChannel information acquisition based on RBF uses a codebook $\\mathcal{F} = \\{ \\mathbf{f}_{1}, \\ldots, \\mathbf{f}_{b}, \\ldots, \\mathbf{f}_{2^B} \\}$, to define the precoders and provide high flexibility for  scheduling, [cf. Section~\\ref{section:precoding:partial-csit}]. In contrast to the methods described in Section~\\ref{section:algorithms}, where the precoders are unknown before user selection, the approach using RBF can efficiently compute a performance metric that includes the effective channel gain and interference due to quantization errors. However, the accuracy of the quantized channel information is limited by the basis (RBF) or bases (PU2RC) that comprise the codebook \\cite{Sharif2005}.\nIn general, user scheduling is performed by joint user selection and precoding allocation, and the overall complexity depends on the parameters $M$, $B$, and $K$.  \nThe set $\\mathcal{K}$ that solves problem (\\ref{eq:inner-outer-problem}), can be found as illustrated in Phase 1 of Fig.~\\ref{fig:beam_selection}, which is based on the algorithm proposed in \\cite{Sharif2005} assuming that $K>M$ and $M\\geq N$.\n\n\n$\\bullet$  \\textit{Training Phase}: The precoding vectors in $\\mathcal{F}$ can be generated according to a known distribution and chosen randomly. The transmitter sends pilots on all the spatial beams $\\mathbf{f}_{m}, \\forall m \\in \\{1, \\ldots, 2^B\\}$, so that all users can estimate their channels. The codebook design can be simplified by defining codewords as the orthonormal basis of $\\mathbb{R}^{M \\times 1}$, as is \\cite{Zhang2007a}. For this codebook design, antenna selection is implicitly performed at the receiver side. This means that each receive antenna is treated as an individual user, which can reduce scheduling complexity and feedback load.\nAnother approach to construct $\\mathcal{F}$, is by eigen-codebook design \\cite{Kountouris2006,Jindal2008}. Each user computes eigen-decomposition of its covariance matrix $\\mathbf{\\Sigma}_{k}$, $\\forall k$, and feeds back the eigenvector associated to its maximum eigenvalue. This approach requires extra feedback load, but can provide flexibility to the scheduler.\nThe quantization granularity of $\\mathcal{F}$ can be enhanced if the codebook comprises multiple bases, as in PU2RC \\cite{Ravindran2012}.\n\n\n$\\bullet$ \\textit{CSI Feedback}: The users compute a CQI metric per beam, and report their largest CQI to the transmitter. The \\textit{effective SINR} of the $k$-th user in the $m$-th beam can be defined as \\cite{Sharif2005}:\n\n\\begin{equation}\\label{eq:pcsit_rbf_sinr_user_k}\n\\text{SINR}_{k,m} =  \\frac{  \\rho  | \\mathbf{H}_{k} \\mathbf{f}_{m} |^{2} }\n{ 1 + \\rho \\sum_{n\\neq m} |  \\mathbf{H}_{k}\\mathbf{f}_{n} |^{2}   },\n\\end{equation}\nand the index of the best precoder is given by\n\\begin{equation}\\label{eq:pcsit_rbf_best_beam_user_k}\ni_{k} =  \\underset{m \\in \\mathcal{B}}{\\arg \\max} \\ \\  \\text{SINR}_{k,m}\n\\end{equation}\nwhere $\\mathcal{B} \\subseteq \\{1,\\ldots,2^B\\}$, is an index subset of active codewords defined by the transmitter so that $|\\mathcal{B}| \\leq M$. The user designates its CDI$_{k}=i_{k}$ and CQI$_{k}=Q(\\text{SINR}_{k,i_{k}})$ with $i_k \\in \\mathcal{B}$, and $Q(\\cdot)$ is a quantization function, see \\cite{Zhang2007a,Vicario2008} and discussion in Section~\\ref{section:algorithms:selection-partial-csit:two-stage-feedback}. \nFor $N>1$, evaluating (\\ref{eq:pcsit_rbf_sinr_user_k}) and (\\ref{eq:pcsit_rbf_best_beam_user_k}) can be performed in different ways \\cite{Schellmann2010}: using receiver combining techniques, e.g. MRC, if one beam is assigned per user; or equalization techniques, e.g. MMSE \\cite{Tse2005}, when multiple beams are assigned per user.\nThe CQI computation is not limited to the achievable SINR or peak rate, e.g., it can be computed from sufficient statistics of the channel conditions and past CQIs \\cite{Wagner2008}. For instance, authors in \\cite{Simon2011} defined the CQI as an estimation of the minimum power required to achieve a target SINR. In \\cite{Kountouris2008a}, the CQI is defined as a function of the current and previous channel conditions, which is an approach similar to the CDF-based scheduling \\cite{Nguyen2015}. This probabilistic method uses the channel or rate distributions\\footnote{Observe that the nature of the objective function modifies the statistics of the CQI. For instance, if we want to maximize the sum rate, then users that experience better channel conditions are more likely to be selected \\cite{Bjornson2009a}.} to schedule the users that are more likely to achieve high performance \\cite{Kountouris2008a}.\n\n\n\nOne can preselect the competing users based on a performance threshold, which reduces feedback and scheduling complexity. Accounting for i.i.d. channels allows to simplify the threshold design, since the channel directions follow a uniform distributions \\cite{Huang2007}.\nThe $k$-th user feeds back information only if its associated CQI is above a predefined threshold, $\\gamma_{th}$, a scheme called \\textit{selective multiuser diversity}, see \\cite{Gesbert2004,Sharif2005,Huang2007,Zhang2007a,Kountouris2008a,Xu2010}. \nAn extension of such a method is the multi-threshold selection, where the scheduling rule also takes into account the MCSs supported over each link, e.g. \\cite{Vicario2008,Tang2010}. If the system optimization involves queue stability constraints, other thresholds based on QSI can be imposed on top of $\\gamma_{th}$, provided that the users have knowledge of their respective queue lengths \\cite{Chen2013,Destounis2015}.\n\n\nPreselection can be performed in the spatial domain [cf. Section~\\ref{section:metrics-spatial-compatibility:spatial-clustering}], where the users meet a constrained in the quantization error. The set of competing users can be defined as $\\{k \\in \\bar{\\mathcal{K}}: \\cos (\\measuredangle(\\mathbf{H}_{k},\\mathbf{f}_{i_{k}})) < \\epsilon\\}$, for a predefined threshold $\\epsilon$, see \\cite{Huang2007,Kountouris2008a}.\nIn the large user regime ($K \\rightarrow \\infty$) with fixed $M$ and $N$, the MUDiv\\footnote{Numerical analysis of capacity versus the number of users $K$, for both full and partial CSIT, e.g. \\cite{Yoo2006,Shen2006,Yi2011,Nam2014a,Ko2012,Lee2014,Tran2010, Choi2007c,Kobayashi2007,Park2010,Huang2009a}, show diminishing returns of MUDiv over i.i.d. Rayleigh fading channels, i.e., the capacity gain  flattens as $K\\rightarrow \\infty$, see  \\cite{Viswanath2002,Hassibi2007}.} scales as $\\log(\\log(KN))$ \\cite{Sharif2007}, which suggests that preselection based on channel statistics, long-term throughput, or even randomly (e.g., \\cite{Huh2012}), can reduce the feedback load and achieve MUDiv gains. \n\n\n$\\bullet$ \\textit{User Selection}: The transmitter must find the set of users that maximizes the performance metric, e.g. WSR. There are several scheduling approaches, e.g., treating each receive antenna as a single user,  assigning at most one beam per user, or allocating multiple beams per user. \nThe simplest selection is assigning the $m$-th beam to user $k_m$, which is defined as\n\\begin{equation}\\label{eq:pcsit_rbf_best_user_per_beam}\nk_{m} =  \\underset{k \\in \\bar{\\mathcal{K}} \\ : \\ \\text{CDI}_{k} = m }{\\arg \\max} \\ \\  \\text{CQI}_{k} \n\\end{equation}\nIf the quantization granularity of the function $Q(\\cdot)$ is low, it may happen that (\\ref{eq:pcsit_rbf_best_user_per_beam}) accepts more than one solution, in which case $k_{m}$ can be chosen randomly among the best user for beam $m$ \\cite{Zhang2007a}. \nIf the elements of $\\mathbf{H}_{k}$ are i.i.d., each selected user cannot achieve maximum SINR for more than two beams on one antenna provided that $K > M$. This is a valuable design guideline that allows to reduce scheduling complexity \\cite{Sharif2005,Chen2013}. Analytical results in \\cite{Hassibi2007} show that $M$ must be scaled proportional to $\\log(K)$, so that user starvation is avoided.\nOnce that the set $\\mathcal{K} = \\{k_{m}\\}_{m=1}^{M}$ has been found, power allocation and link adaptation can be performed. \n\t\n\n\n\nOwing to the fact that the quantization granularity is fixed and bounded ($B < \\infty$), the precoders $\\mathbf{f}_{m} \\in \\mathcal{F}$ cannot fully separate users in the spatial domain. Hence, IUI is unavoidable and particularly harmful in high SNR. Moreover, in the sparse user regime, $K \\approx M$, RBF cannot benefit from MUDiv and performs poorly. Several user scheduling algorithms have been proposed in \\cite{Lee2016} to handle sparsity and limited MUDiv over mmWave channels.\nThe optimum number of beams that maximizes the WSR is a function of the number of competing users and the SNR regime \\cite{Wagner2008,Kim2005a,KountourisThesis}. Analytical results establish that for RBF with fixed $K$, and extreme values of the SNR regime, the optimal number of active beams is one\\footnote{These results are equivalent for the case where channel are quantized in Section~\\ref{section:algorithms:selection-partial-csit:qunatized-csi}, see e.g., \\cite{Wang2007a,Dai2008,Zhang2011}.} \\cite{Wagner2008,KountourisThesis}.\nDepending on the channel conditions and the system constraints, user scheduling might yield a dynamic switching between TDMA (time-sharing) and MU-MIMO transmission. This can take place if IUI is very high, or if the individual rate or power constraints are violated \\cite{Kountouris2005a,Kountouris2008}. \nTherefore, efficient operation in an arbitrary SNR requires a reduced number of scheduled users and active beams, i.e., $|\\mathcal{B}| < M$. Analytical results in \\cite{Wagner2008} show that $|\\mathcal{K}| = M$ can be attained if $K \\rightarrow \\infty$ for a fixed $B$. \nIn general, the optimal set of selected users is such that $|\\mathcal{K}|<M$,  \\cite{Vicario2008}. \nFinding the best subset of users and beams is a combinatorial problem whose optimal solution is found by ExS \\cite{Wagner2008}. One has to enumerate all user combinations, compute their associated performance metrics $U(\\cdot)$, and select the subset with maximum $U(\\cdot)$, as illustrated in Phase 2 of Fig.~\\ref{fig:beam_selection}. One alternative is to formulate an integer programming optimization, as in \\cite{Kim2005a}, or to perform greedy selection, see \\cite{Vicario2008} and  Section~\\ref{section:algorithms:aggregated-utility}.\n\n\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=0.98\\linewidth]{figure11.pdf}\\\\\n\t\\caption{Given the user ordering $\\pi(\\cdot)$, the user selection is done by finding the strongest user per beam\/antenna\/spatial dimension associated with $\\mathcal{F}$. Phase 1 matches a user with a beam based on the CQI.  Phase 2 maximizes the utility function by selecting a subset of users and their associated beams according to some enumeration or search technique.}\n\t\\label{fig:beam_selection}\n\\end{figure}\n\n\n\nSeveral extension to the approach presented in Fig.~\\ref{fig:beam_selection} have been developed: improving codebook resolution, e.g. \\cite{Huang2007,Choi2007c}; dynamically updating multi-bases codebooks \\cite{Kountouris2005}; computing the CQI by different functions, e.g. \\cite{Kountouris2008a}; combining RBF with deterministic precoding schemes, e.g. \\cite{Zorba2008}; dynamically adjusting the number of active beams, e.g. \\cite{Wagner2008,Kountouris2008}, etc. \nTo enhanced the quantization granularity, authors in \\cite{Choi2007c} generate several sets of $2^B$ beams, and the users report the best CQI and CDI per set.  \nThe user selection in \\cite{Zorba2008} combines RBF and SZF for the MISO scenario: given an initial random precoding $\\mathbf{f}_{1}$, its associated user $k_1$ solves $\\max_{k \\in \\bar{\\mathcal{K}}} \\cos (\\measuredangle(\\mathbf{H}_{k},\\mathbf{f}_{1}))$, and feeds back a finer CDI that is used to generate the precoder $\\mathbf{f}_{2}$ for the second user, $k_2$, and so on. In this way, the $k$-th selected user will only receive interference from the previous $k-1$ selected users. This algorithm fits particularly well in heterogeneous systems, where users can be clustered according to their locations or channel statistics, e.g., \\cite{Ho2009,Nam2014,Moretti2013}. \n\n\n\n\nIn systems with QoS constraints [cf. Section~\\ref{section:performance-uf:wsr-with-qos}], the opportunistic nature of the algorithm in Fig.~\\ref{fig:beam_selection} changes and plays a secondary role, which has a direct impact in the MUDiv gains. This is because the competing users are preselected based on QSI, since scheduling exclusively based on CQIs incurs in delay penalties \\cite{Hassibi2007}. In \\cite{Chen2013}, a subset of user is preselected using QSI, and only those users feed back their CSI. The final selection is done as shown in Fig.~\\ref{fig:beam_selection}, where the CQIs are computed as the effective SINRs weighted by their respective QSIs. Notice that the parameters of the deployment may affect the latency experienced by the users. For instance,  large values of $B$ might result in more scheduled users since IUI can be suppressed more efficiently. \n\n\n\n\\subsection{Two-stage Feedback Scheduling}\n\\label{section:algorithms:selection-partial-csit:two-stage-feedback}\n\nSeveral works in the literature (cf.  \\cite{Kountouris2006a,Gesbert2007a,Zakhour2007,Wagner2008,Kountouris2008,Kountouris2008a,Xu2010,Sohn2012,Khoshnevis2013}), have studied the user selection and precoding design problems in terms of the fed back information required to implement them. The total amount of feedback bits per user, $B_{t}$, is partitioned such that $B_{t}=B_1 + B_2$, where each part is used to solve a problem, as illustrated in Fig.~\\ref{fig:two-phase-feedback}. \nNotice that the beam allocation problem in Fig.~\\ref{fig:beam_selection} and the bit allocation problem Fig.~\\ref{fig:two-phase-feedback}, bear some resemblance to the methodology described in Section~\\ref{section:algorithms:aggregated-utility:csi-mapping}. The user selection problem (\\ref{eq:csi-mapping-problem-1}), is solved using some properties of the CSIT of all competing users, and the precoding design and power allocation problem (\\ref{eq:csi-mapping-problem-2}) is solved only for the set of users previously selected or grouped. \n\n\nThe communication phases between the transmitter and the receivers in Fig.~\\ref{fig:two-phase-feedback}, are part of a training or pilot mode \\cite{Wagner2008,Kobayashi2011}, which is applied to MU-MIMO system with partial CSIT. \nTaking into account the type of implemented quantization, some remarks are in order \\cite{Sohn2012,Khoshnevis2013}:\n\\begin{enumerate\n\t\\item for high-rate feedback systems, a large value of $B_{t}$ for channel quantization might be more efficient. The accurate CDI can be used for precoding design and to enhance of the multiplexing gain at high SNR \\cite{Hassibi2007}.\n\t\\item for low-rate feedback systems with small values of $B_{t}$, the RBF or PU2RC schemes, [cf. Section~\\ref{section:algorithms:selection-partial-csit:rbf}], are more suitable since they can meet rate constraints and simplify the scheduling.\n\\end{enumerate}\n\n\n\n\nIn scenarios with limited feedback rates, $B_t < \\infty$, it is desirable to: \\textit{i}) achieve significant MUDiv using $B_1$ bits for the CQI; \\textit{ii}) design codebooks using $B_2$ bits of resolution for the CDI, to efficiently mitigate quantization errors, IUI, and achieve multiplexing gains; and \\textit{iii}) reduce the scheduling complexity and limit the number of users that feedback $B_2$ bits.  \nThe optimal partition of $B_{t}$ depends on system parameters ($N$, $M$, and $K$), SNR regime, and type of precoding scheme \\cite{Kountouris2006a,Zakhour2007}. Moreover, practical systems not only limit the amount of bits per user,\\footnote{In LTE-Advanced, one physical uplink control channel (PUCCH) is assigned per user \\cite{Dahlman2013}. The control information such as CQI, CDI, channel rank, scheduling request, and hybrid-ARQ ACK\/NAK are conveyed though the PUCCH, which has few bits dedicated to CSI reporting (format 2).} $B_{t}$, but also the total feedback rate, $KB_{t}$, e.g., if the users share the feedback link, see \\cite{Khoshnevis2013,Huang2007,Xu2010,Sohn2012}. \nObserve that the multiuser scheduling in Fig.~\\ref{fig:two-phase-feedback} requires CSI feedback and  signaling of the scheduling decisions to the selected users. These feedback and control loop introduce a non-negligible overhead, $KB_{1}$, and latency in the system that must be carefully considered in the scheduling rule design \\cite{Gesbert2007a}.\n\n\n\n\nIn the first feedback phase in Fig.~\\ref{fig:two-phase-feedback}, the competing users send a coarse quantized version of their achievable performance metric. The $\\text{CQI}_{k} \\in \\{q_{1},\\ldots,q_{2^{B_1}}\\}$ is quantized using $B_1$ bits, where $q_{i}$ is the $i$-th quantization level. The set of users in $\\mathcal{K}$, can be chosen in a ranking-based per-beam selection, as in Phase 1 of Fig.~\\ref{fig:beam_selection}, or per-antenna selection as in \\cite{Zhang2007a,Khoshnevis2013}. \nA number of  works  have shown that depending on the system parameters, choosing $B_1 \\in \\{1,2,3,4\\}$ can be enough to attain the MUDiv gain of the unquantized CQI, see \\cite{Xu2010,Zhang2007a,Ravindran2012}. Results in \\cite{Khoshnevis2013} and references therein, show that efficient precoding design is attained by assigning roughly $1\/M$ out of $B_{t}$ bits to the CQI, and allocating the remaining bits to the CDI. \nIf the system operates in the large user regime ($K \\rightarrow \\infty$), small values of $B_1$ are enough to achieve asymptotically optimal system performance, see \\cite{Xu2010} and references therein. \nMUDiv is defined by users with SINR above a threshold, and constructing $\\mathcal{K}$ depends on CDIs, since spatially compatible users will be grouped with high probability \\cite{Chen2013,Sharif2005}. In the high resolution regime ($B_t \\rightarrow \\infty$), an efficient partition meets $B_1 > B_2$, and the fraction of bits dedicated for CDI scales proportionally to $\\log(B_t)\/B_t$, see  \\cite{Khoshnevis2013}.\n\n\n\nIf $K \\approx M$, larger values of $B_1$ are required to circumvent the lack of MUDiv and properly identify strong users. However, in the high SNR regime is more beneficial to have $B_1 < B_2$, since quantization errors in the CDI are the main performance-limiting factor \\cite{Zhang2011,Ravindran2012}.\nMUDiv clearly affects the optimal partition of $B$. Numerical results in \\cite{Kobayashi2011} show the effects of $K$ over the feedback rates, and the interdependence between MUDiv and $B_2$.\nAccounting for link adaptation, $B_1$ must be chosen so that the CQI fully exploits the granularity of the available MCSs, either for channel quantization \\cite{Trivellato2008} or RBF \\cite{Vicario2008}. \nThe value of $B_1$ depends on the type of metric that defines the CQI. For example, the SINR and the achievable rate are real numbers, whereas other metrics might already be defined as integer numbers, see \\cite{Kountouris2008a}. \n\n\n\n\nIn the second feedback phase in Fig.~\\ref{fig:two-phase-feedback}, each user in $\\mathcal{K}$ uses $B_2$ bits to report its CDI. This mean knowledge of refined channel information at the transmitter, e.g., more accurate spatial directions or effective channel gains.\nIf the system is constrained in the total feedback bandwidth, $|\\mathcal{K}|B_2$, heterogeneous and dynamically allocated bits can be used to quantize the CDI. By assigning different number of bits per user, i.e., $B_{2(k)}$, $\\forall k \\in \\mathcal{K}$, efficiency at the CDI feedback phase can be achieved \\cite{Sohn2012}. The dynamic assignment of $B_{2(k)}$ is more effective in scenarios where the users have heterogeneous average long-term channel gains. The bit allocation rules might consider the following guidelines \\cite{Xu2009,Sohn2012,Khoshnevis2013}:\n\n\\begin{enumerate\n\t\\item for high SNR users, large values of $B_{2(k)}$ reduce quantization errors and mitigate IUI. This is particularly important in scenarios where the CDI is used to compute the precoder weights (e.g., MMSE \\cite{Kountouris2005a} or ZFBF \\cite{Xu2010,Khoshnevis2013}), [cf. Section~\\ref{section:algorithms:selection-partial-csit:qunatized-csi}].  \n\t\\item for low SNR users, the performance is noise limited, i.e., the noise variance is larger than the interference, and the value of $B_{2(k)}$ is relatively less important to optimize performance.  \n\t\\item dynamic assignment of $B_{2(k)}$ can be extended to sequential transmissions with error correction mechanisms (e.g. ARQ \\cite{Xu2009}). Such an assignment can be used to further refine $\\mathcal{K}$, by temporarily dropping users retransmitting the same packet multiple times.\n\t\\item in multiple-transmitter scenarios, the level of coordination and transmission scheme [cf. Section~\\ref{section:algorithms:multi-cell-selection-classification}], define the methodology to assign the value of $B_{2(k)}$ at each transmitter \\cite{Khoshnevis2013}. \n\\end{enumerate}\n\n\n\nResults in \\cite{Sohn2012} suggest that grouping users based on their locations or long-term channel gains (e.g.,  \\cite{Ho2009,Nam2014,Moretti2013}), yields a more efficient assignment of the feedback bandwidth and simplify the user grouping. \nAnother parameter to be considered when defining $B_1$ and $B_2$, is the time used for hand-shaking between feedback phases. The total time used for training and data transmission must be kept below the coherence time of the channel, so that the impact of delayed\/outdated CSI is minimized \\cite{Kountouris2008,Sohn2012}. Otherwise, increasing $B_2$ would not be enough to achieve MIMO gains \\cite{Zhang2011}. \nAnalytical results in \\cite{Ravindran2012} show that for multi-basis codebooks  (e.g., PU2RC), large values of $B_2$ usually do not provide considerable capacity gains, and the overall performance is highly sensitive to MUDiv. A design rule to bear in mind is that the accuracy of the CSIT is more valuable than MUDiv in practical MU-MIMO scenarios \\cite{Ravindran2012,Xu2010}. \n\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=0.8\\linewidth]{figure12.pdf}\\\\\n\t\\caption{Two-stage feedback MU-MIMO scheduling}\n\t\\label{fig:two-phase-feedback}\n\\end{figure}\n\n\n\n\n\\subsection{Implementation in Cellular Scenarios}\n\\label{section:algorithms:multi-cell-selection-classification:discussion}\n\n\nThe scheduling algorithms for MU-MIMO in LTE-Advanced rely on quantized CSI: \\textit{i}) the RI\/PMI metrics contain information regarding the SU-MIMO channel (due to backward compatibility\\footnote{The limitations due to PMI feedback in SU-MIMO mode can be overcome for certain scenarios by means of efficient estimations of the CDI\/CQI. One can take into account spatial compatibility between the MIMO channels and the codebooks, i.e., estimating the effective channel gains \\cite{Wang2013a}.}) projected onto the subspace defined by the codebook; \\textit{ii}) the CQI metric  indicates the energy of the projection, and it might include ICI and noise \\cite{Liu2012}. Since the aggregated multiuser channel can be constructed by concatenating the users PMI metrics, and the CQIs are known at the transmitter, the scheduling algorithms described in the subsections above can be directly used in cellular scenarios. \n\n\nIn general, the scheduling algorithms in LTE-Advanced are proprietary and implementation-specific and there is no standardized procedure to define them \\cite{Liu2012}. It is worth noting that the amount of available CSIT defines the transmission mode (SU- or MU-MIMO), and the scheduling decisions. Practical scheduling rules with switching mode can be defined by evaluating the achievable performance of SU-MIMO and MU-MIMO (with multi-rank transmission) modes, and simply choosing the one with better performance \\cite{Tang2010,Wang2012d,Fan2014}. However, robust switching between these modes still requires research to guarantee adaptability to changes in channel and traffic conditions, as well as viable computational complexity and scalability \\cite{Li2014}.\n\nAuthors in \\cite{Wang2012d}, highlight the fact that for known precoding matrices at the transmitter, it is possible to generate look-up-tables (LUTs) that contain information regarding spatial compatibility and potential interference. Resource allocation based on LUTs could be used to significantly reduce the complexity of the scheduling algorithms. There are several factors that must be considered for CS\/CBF design in cellular systems \\cite{Zhang2009,Huh2012,Marzetta2010,Marsch2011,Gesbert2010,Bjornson2013}: dynamically determining whether or not coordination is required, switching between JT and CS\/CBF, optimizing cluster formation and sizing, and scalability issues. The resource allocation strategies must also include constraints on backhaul bandwidth, CSI acquisition, latency, QoS, mobility, and synchronization. \n\n\n\n\\subsection{Implementation in WLAN Scenarios}\n\\label{section:algorithms:wlan-selection}\n\nIn the IEEE 802.11ac standard, MU-MIMO transmission consists of two tasks \\cite{Aboul-Magd2013}: \\textit{i}) to identify the users that belong to the transmission set, which is implementation-specific and might depend on priority weights or traffic related parameters [cf. Section~\\ref{section:performance-uf}]; \\textit{ii}) the assignment of a Group ID for each set of co-scheduled users, and the downlink transmission signaling. For a given time instance, the AP constructs a Group ID assignment table. The number of rows is defined by the available number of ID groups, and the columns are given by the number of associated users. The condition to have a proper Group ID, is to have different users assigned to each one of the available stream positions. \nSelecting the proper Group ID and its associated scheduled users is equivalent to a coloring problem over a two dimensional array, whose optimal solution can be found by ExS. A heuristic algorithm was proposed in \\cite{Aboul-Magd2013}, where the positions of the users in a particular Group ID are determined according to the probability of occurrence. \nThe Group ID selection and success allocation probability can be improved by modeling the discrete searching process as a linear sum assignment problem, for which there exist reliable polynomial-time algorithms \\cite{Burkard2009}.\n\n \n\nFor the general MU-MIMO scenario, a selected user can receive more than one data stream, and all of them must use the same MCS \\cite{Arubanetworks2014}. The authors in \\cite{Cheng2014} proposed a user selection algorithm inspired by \\cite{Fuchs2007}, originally designed for cellular scenarios. The scheme applied BD with geometric-mean decomposition to guarantee equal MCS allocation over all spatial streams assigned to a single user. That work highlights the need of careful stream allocation per selected user, where the best number of streams is not necessarily equal to rank of the channel.\nAnother approach to guarantee maximum sum rate with joint link adaptation (MCS selection), BD, and user\/stream selection was proposed in \\cite{Rico-Alvarino2014}. The authors extended the scheduling algorithm in \\cite{Shen2006}, originally designed for cellular scenarios, to WLANs with partial CSIT. The link adaptation is performed by a machine learning classifier that provides robustness to CSI inaccuracy. Numerical results suggest that estimation of the IUI is required to allocate MCS more efficiently. \n{\\color{black}{The next generation of 802.11 standard will define the conditions and methods for user grouping\/association  \\cite{Liao2016}. A critical open problem is to build scheduling algorithms to balance overhead, fairness and individual priorities.}}\n\n\n\\subsection{Scheduling Complexity}\n\\label{section:algorithms:complexity}\n\nThe scheduling complexity can be classified in two types \\cite{Lee2014b}: implementation complexity and computational complexity. The former, refers to the amount of signaling overhead and information exchanged among different network entities. The latter refers to the processing time required to execute a certain algorithm at the transmitter or CU. \nThe implementation complexity is assumed to be reduced whenever the channel information is quantized [cf. Section~\\ref{section:precoding:partial-csit}]. This type of complexity can be estimated as a function of the system parameters $K$, $B$, $N$, $M$, and the number of clustered transmitters. Nevertheless, to the best of our knowledge, there is no published work or reference framework that analyze the complexity in a fundamental and general way, at least for the MU-MIMO systems considered here. Most research in the field of limited CSI feedback has focused on the following issues related to complexity \\cite{Gesbert2007a,Marsch2011}: \\textit{i}) reducing the number of parameters to be quantized, e.g., bits to quantize the CQI and CDI; \\textit{ii}) reducing the required quantized feedback resolution; and \\textit{iii}) constraining the signaling and message exchange within a cluster, and minimizing the overall coordination.\n\n\n\nIn the MU-MIMO literature, most works focus on the computational complexity required to select users, to extract the precoders, and to perform power allocation. There are several metrics to estimate the computational complexity: \\textit{i}) number of real or complex operations. \\textit{ii}) number of flops, where a flop is defined as a floating point operation \\cite{Shen2006}, e.g., a real addition, multiplication, or division. \\textit{iii}) using the big-Oh notation, ${O}(\\cdot)$, which is proportional to the term that dominates the number of elementary operations needed to execute an algorithm \\cite{Moretti2013}. \\textit{iv}) other metrics, such as the number of vector\/matrix operations, the number of iterations needed for convergence, or results based on simulations and execution time.\nTherefore, there is no unified approach to characterize the computational complexity for scheduling algorithms, but the aforementioned metrics, summarized in Table~\\ref{tab:complexity}, provide a coarse assessment of the computational load order. \n\nWe hasten to say that another sort of complexity analysis can be performed for MU-MIMO systems, in which precoders and powers are jointly optimized for a fixed set of users $\\mathcal{K}$, [cf. Section~\\ref{section:performance-uf:wsr-with-qos}]. The resource allocation problem only involves $\\mathbf{W}$, $\\mathbf{P}$, and is formulated as the optimization of a certain system-level utility function, $U(\\cdot)$, [cf. Section~\\ref{section:performance-uf:weighted-sum-rate}], subject to resource budget constraints, see \\cite{Hong2012} and references therein. In these cases, the set of scheduled users is not optimized, and the complexity is normally defined as a function of $M$, $N$, and the characteristics of $U(\\cdot)$, e.g., convexity or non-convexity.  \n\n\n\\begin{table}[t!]\n\t\\renewcommand{\\arraystretch}{1.3}\t\n\t\\caption{Summary of Computational Complexity Metrics for MU-MIMO Scheduling Algorithms}\n\t\\label{tab:complexity\n\t\\centering\n\t\\begin{tabularx}{0.48\\textwidth}{l X}\n\t\t\\toprule\n\t\t\\bfseries\tMetric & \\bfseries\tReferences \\\\\n\t\t\\midrule\t\n\t\tFlop-based & \\cite{Shen2006, Chen2008, Tran2010,   Dai2009, Mao2012, Ko2012, Tran2012, Nam2014a,  Elliott2012, Hei2009} \\\\\n\t\t\\midrule\n\t\tReal\/complex Operations & \\cite{Huang2013, Dimic2005, Huang2012b, Chen2007a, Wang2010, Conte2010, Wang2008a, Maciel2010, Mao2012, Shi2012}  \\\\\n\t\t\\midrule\n\t\t${O}(\\cdot)$ & \\cite{Shen2006, Moon2013, Lim2009, Dartmann2013,  Xia2009, Sohn2012, Aniba2007,   Chan2007, Moretti2013,Wagner2008,Liu2014a,YiXu2014,Liu2015a}  \\\\\n\t\t\\midrule\n\t\tOther Metrics & \\cite{Cheng2014, Vicario2008, Tsai2008, Park2010,  Yoo2006, Razi2010, Wang2010, Fuchs2007, Bayesteh2008, Chung2010, Driouch2012, Yoo2005a, Sigdel2009a, Yi2011, Xia2009, Castaneda2015,  Stridh2006a, Matskani2008, Moretti2013, Elliott2009, Lau2005a,Liu2014a}  \\\\\n\t\t\\bottomrule\t\n\t\\end{tabularx}%\n\\end{table}%\n\n\n\n\n\n{\\color{black}{\n\t\t\n\\subsection{Discussion}\n\\label{section:algorithms:selection-partial-csit:discussion}\t\n\n\nThe two approaches in Section~\\ref{section:algorithms:selection-partial-csit:qunatized-csi} and \\ref{section:algorithms:selection-partial-csit:rbf} are used in scenarios where the transmitters have different capabilities and constraints. The former approach is applied when the transmitter can compute linear precoding using the quantized channels [cf. Section~\\ref{section:precoding:linear}]. The latter approach is used when the precoders are predefined and cannot be modified based on the instantaneous CSI. The bit allocation for the CQI and CDI, described in Section~\\ref{section:algorithms:selection-partial-csit:two-stage-feedback}, can be used to analyze both approaches, channel quantization and RBF.  \nPractical resource allocation algorithms work with partial CSI, whose computational complexity depends on the codebook resolution, the number of deployed antennas and active users. On the one hand, high codebook resolution improves peak rates and reduces IUI, but it is a bottleneck for the uplink in FDD mode. On the other hand, dynamic channel-based codebook designs speed up the resource allocation, reduce user-pairing complexity, and mitigate interference \\cite{Li2014}. \nThe CQI and CDI reports can be optimized to enhance centralize or distribute resource allocation, and they are also functions of the operative wide band and feedback periodicity \\cite{Ku2014}. The main issue related to CSI feedback is to find a good trade-off between signaling overhead and accurate channel quality estimation. Enhanced scheduling algorithms for multiuser systems will depend on the type of CSI available at the BSs (e.g. channel quantization or RBF), the deployment configuration, and more complex decision-making rules that can maximize the overall throughput \\cite{Capozzi2013}.\n\n\nIn practical massive MIMO scenarios, hybrid precoding will be used for MU-MIMO communication, whose performance will depend on the number of RF chains at the transmitter and the accuracy of the fed back CSI. Moreover, the type of transceiver architecture and precoding scheme must be defined according to the objective function (e.g. sum-rate or EE) and the user sparsity \\cite{Gao2016}.  Further research is needed to understand the joint optimization of bit and stream allocation using hybrid precoding, taking into account hardware resolution and channel estimation accuracy \\cite{Kutty2016}. \n\n\n\t\n}}\n\n\n\n\n\n\n\\section{Power Allocation}\n\\label{section:power-allocation}\n\n\nThe optimal power allocation $\\mathbf{P}$ in (\\ref{eq:inner-outer-problem}) depends on the CSIT availability, the type of precoding scheme implemented at the transmitter, the individual user priorities, and the objective function. Accounting for full CSIT in single-transmitter scenarios, the optimal power allocation is usually known in analytical closed-form for several performance metrics, e.g., BER, SINR, WSR, or fairness \\cite{Bartolome2006,Vu2007}. \nAssuming that $\\mathcal{K}$ is fixed, and linear precoding is implemented at the transmitter, a network-centric power control algorithm finds the optimal $\\mathbf{P}$ for WSR maximization through convex optimization, i.e., using the water-filling principle, see \\cite{Palomar2005,Cover2006,Boyd2004}. The linear precoding decouples the signals into orthogonal spatial directions mitigating the IUI, and water-filling allocates powers according to the effective channel gains [cf. Definition~\\ref{defn:effective_channel_gains}]. \n\nIn scenarios with a fixed feasible set of users $\\mathcal{K}$, [cf. Definition~\\ref{defn:feasible_set_of_users}], and assuming that $\\mathbf{W}$ is coupled with $\\mathbf{P}$, the power allocation may involve, for example, a WSR maximization with rate control, sum power minimization with individual SINR constraints, or a max-min SINR problem, see \\cite{Schubert2004,Tan2012,Koutsopoulos2008,Bjornson2013}. For this kind of problems, power control algorithms are designed based on optimization theory (see \\cite{Hong2012,Bjornson2013}), and the Perron-Frobenius theory\\footnote{The Perron-Frobenius theory is a fundamental tool to solve congestion control problems and optimize interference limited systems [cf. Definition~\\ref{defn:interference-limited-system}], such as wireless sensor or multi-hop networks \\cite{Stanczak2009}.} for non-negative matrices (see \\cite{Stanczak2009,Chiang2008}). \nIn contrast to WSR maximization, where water-filling assigns more power to stronger channels, other objective functions require to allocate powers in a different way. For instance, balancing the power over all users so that weak users are assigned more power to improve their error rates  \\cite{Cheng2014}, assigning powers according to target SINRs \\cite{Chung2010,Tsai2008,Simon2011,Stridh2006a,Matskani2008,Ku2015}, or considering queue stability constraints \\cite{Huang2012a}. Some works model $\\mathbf{W}$ so that the power is absorbed into it, and optimizing the directions and magnitudes of the precoder weights implicitly performs power control, e.g., \\cite{Dartmann2013,Zhang2007b}.   \nThe interested reader is referred to \\cite{Chiang2008} for a comprehensive survey on theory and algorithms for joint optimization of $\\mathbf{W}$ and $\\mathbf{P}$.  \n\nEqual power allocation (EPA) is a suboptimal strategy used to simplify the evaluation of the utility function $U(\\cdot)$, especially when the transmitter does not have full CSI knowledge \\cite{Jindal2006}. In certain scenarios, EPA allows a more tractable system performance analysis or derivation of statistics based on $K$, $M$, $N$, or $B$, which generally yields closed-form expressions. Assuming full CSIT, if the WSR maximization problem [cf. Section~\\ref{section:performance-uf:weighted-sum-rate}] is optimized using ZF-based or BD-based precoding schemes, the EPA (directly proportional to the individual weights $\\omega_{k}$, $\\forall k \\in \\mathcal{K}$), asymptotically achieves the performance of the optimal water-filling power allocation at high SNR \\cite{Lee2007}. \t\nThe authors in \\cite{Song2008} discussed the secondary role of power allocation when optimizing problem (\\ref{eq:inner-outer-problem}). The paper suggested that more efforts should to be taken to find $\\mathcal{K}$ and the corresponding $\\mathbf{W}$, which may justify the adoption of EPA under specific SNR conditions.\n\n\n\nIn systems with partial CSIT, power allocation requires numerical methods that depend on the optimized performance metric \\cite{Vu2007}. Consider a practical scenario where $B$ is finite and fixed, and $\\mathcal{K}$ has been found by one of the methods described in Section~\\ref{section:algorithms:selection-partial-csit}. In such a scenario, the system is limited by interference [cf. Definition~\\ref{defn:interference-limited-system}], at the moderate and high SNR regimes, since the interference components in the denominator of (\\ref{eq:pcsit_sinr_user_k}) and (\\ref{eq:pcsit_rbf_sinr_user_k}) are non-zero. Therefore, one of the main objectives of power control is to perform interference management. However, to compute the optimal $\\mathbf{P}$ for WSR maximization, the transmitter must have knowledge of all interference components for all users. This results in non-convex optimization problems, for which efficient algorithms (depending on the operating SNR regime) can provide suboptimal solutions \\cite{Kountouris2008}. Moreover, global knowledge of the interference components at the transmitter may not be attainable in limited feedback systems (e.g. only CQIs are known), and adopting EPA is a common practice in the reviewed MU-MIMO literature. \n\n\n\n\\begin{table*}[t] \\scriptsize\n\t\\renewcommand{\\arraystretch}{1.3}\t\n\t\\caption{Scheduling Approaches and their associated Power Allocation Methods}\n\t\\label{tab:power-allocation\n\t\\centering\t\t\n\t\\begin{tabularx}{\\textwidth}{l *{3}{>{\\centering\\arraybackslash}X} }  \n\t\t\\toprule\n\t\t\\textbf{Method} & \\textbf{EPA} & \\textbf{Water-filling} & \\textbf{Adaptive} \\\\\n\t\t\\midrule\n\t\t\\textbf{Utility-based} & \\cite{Moon2013, Chen2007a, Chen2008, Xia2009, Rico-Alvarino2014, Fan2014, Wang2015, Tang2010, Vicario2008, Chae2008, Kountouris2007, Kountouris2006, Shirani-Mehr2010, Zhang2011, Ravindran2012, Jindal2008, Torabzadeh2010, Cui2011a, Seifi2011, Park2010, Wang2012d, Lossow2013,Hammarwall2008,YiXu2014,Bogale2015} & \\cite{Dimic2005, Huang2013, Wang2008a, Shen2006, Huang2012b, Chen2008, Ko2012, Tran2012, Chan2007, Lim2009, Tran2010, Boccardi2006, Huh2012,Bogale2015} & \\cite{Zhang2005a, Cheng2014, Koutsopoulos2008, Kobayashi2007, Tsai2008, Hammarwall2007, Dartmann2013, Zhang2009, Yu2013,Liu2014a} \\\\\n\t\t\\midrule\n\t\t\\textbf{CSI-Mapping} & \\cite{Kountouris2006b,Nam2014, Jiang2006, Bayesteh2008, Adhikary2013, Adhikary2013a, Chen2007a, Wang2010, Fuchs2007, Bayesteh2008, Wang2006, Xia2009, Lim2009, Tran2010, Zorba2008, Cheng2014, Yang2011, Zakhour2007, Huang2009a, Chae2008, Yoo2007, Kountouris2007, Kountouris2008, Chen2013, Zhang2007a, Kountouris2006a, Wang2007a, Trivellato2008, Choi2007c, Min2013, Sohn2010, Nam2014a, Kountouris2008a, Kountouris2005, Huang2007, Xu2010, Sohn2012, Khoshnevis2013, VanRensburg2009, Hosein2009, Castaneda2015, Jang2011, Schellmann2010, Aniba2007, Christopoulos2013,Wagner2008,Xu2009a,Liu2015a} & \\cite{Tu2003, Yoo2006, Shen2006, Dai2009, Lee2014, Wang2010, Mao2012, Shi2012, Fuchs2007, Tejera2006, Maciel2010, Driouch2012, Yoo2005a, Wang2006, Sigdel2009a, Yi2011, Sun2010, Lau2009, Kountouris2008, Wang2010a, Souihli2010, Swannack2004, Shirani-Mehr2011, Kountouris2005a,Lee2014c} & \\cite{Jagannathan2007, Razi2010, Zhang2005a, Chung2010, Christopoulos2013, Huang2012a, Kountouris2008, Zhang2007b, Simon2011} \\\\\n\t\t\\midrule\n\t\t\\textbf{Metaheuristic (stochastic)} & \\cite{Wang2015, Zhang2011, Park2014} & \\cite{Elliott2012, Elliott2009, Lau2005a, Hei2009} & \\cite{Bjornson2014, Cottatellucci2006} \\\\\n\t\t\\midrule\n\t\t\\textbf{Classic Optimization} & \\cite{Maciel2010} & \\cite{Chan2007, Moretti2013} & \\cite{Stridh2006a, Matskani2008, Ku2015, Song2008} \\\\\n\t\t\\midrule\n\t\t\\textbf{Exhaustive Search} & \\cite{Zakhour2007, Destounis2015} & \\cite{Chan2007,Swannack2004} & - \\\\\n\t\t\\bottomrule\n\t\\end{tabularx}%\n\\end{table*}%\n\n\n \nIn multi-transmitter scenarios, the level of coordination, the availability of user data and CSIT, and the precoding scheme determine the best power allocation policy. In full coordinated scenarios, all transmitters belong to the same infrastructure, and a CU determines the power allocation across the cluster. Assuming global CSIT, a fixed set $\\mathcal{K}$, and linear precoding schemes, the power allocation that optimizes a global performance metric subject to per-transmitter power constraints can be realized numerically or through water-filling, see \\cite{Zhang2009,Huh2012}. \nFor scenarios with partial coordination between transmitters, assuming that $\\mathcal{K}$ is globally known, the power allocation depends on the precoding schemes computed from local CSI and individual priorities $\\omega_{k}$, $\\forall k \\in \\mathcal{K}$, known by each transmitter, see \\cite{Jorswieck2008,Park2012a,Bjornson2010}.\n\n\nTable~\\ref{tab:power-allocation} summarizes the reviewed scheduling methods and their associated power allocation strategies: EPA, water-filling, and adaptive methods. The references listed under the \\textit{adaptive} category depend on the particular objective function and system constraints, see Tables \\ref{tab:precoding} and \\ref{tab:optimization-criteria}. Those references jointly optimize $\\mathbf{P}$ and $\\mathbf{W}$ via conventional optimization methods \\cite{Boyd2004} and iterative algorithms.  \n\n\n\n\n\n\n\n\n\n{\\color{black}{   \n\\subsection{Summary and Future Directions}\n\\label{section:power-allocation:ee}\n\n\nPower allocation is a dynamic process that compensates instantaneous channel variations to maintain or adjust the peak rates \\cite{Dahlman2013}. In MU-MIMO scenarios, power control is required to optimize an objective function, mitigate interference, and satisfy system constraints. Several problem formulations deal with convex objectives and linear constraints, for which water-filling can compute the optimal power allocation. Alternatively, EPA simplifies the optimization by transforming $\\mathbf{P}$ into a constant vector, which becomes close-to-optimal for ZF at  high SNR. In multi-cell scenarios, power allocation requires exchange of control messages to mitigate ICI. The amount of signaling overhead depends on the type cooperation and coordination between transmitters, and the coordinated signaling (centralized or distributed). Results in \\cite{Hamed2016} show that hardware calibration and distributed power allocation are critical to achieve joint transmission in MU-MIMO settings. \n\n\nRecently, energy consumption has become a central concern in academia and industry \\cite{Tombaz2011,Feng2013}. One of the objectives of 5G is to enhance EE by orders of magnitude \\cite{Andrews2014}, tanking into account power consumption at the access (transmit and circuitry power) and core (backhaul) networks. EE depends on the particular network planning and objective function [cf. Section~\\ref{section:performance-uf}]. For instance, in dense heterogeneous networks the distance between transmitters and receivers determine EE \\cite{Feng2013}. Dense deployments can provide high EE, but the number of transmitter per serving area should be carefully planned to avoid ICI and waste of resources due to handovers \\cite{Shokri-Ghadikolaei2015}. \nThe EE is expressed as a benefit-cost ratio and measured in bits per Joule \\cite{Feng2013,Bjornson2016a}. Complementary metrics such as Gflops per Watt (to calculate the typical computational efficiency \\cite{Bjornson2014a}) can be used to assess the power consumption per processing block at the transceivers, or to evaluate the EE of a particular algorithm.  \n}}\n\n\n\n\n\n\n\n\n\n\n\\section{Final Remarks}\n\\label{section:remarks}\n\n\nThere are many challenges to be solved for resource management in MU-MIMO systems. Identify and standardize the best practices to group and schedule users is fundamental to profit from MUDiv. This is tightly related to affordable-complexity algorithm design to efficiently manage the allocation of multiple users, carriers, time slots, antennas, and transmitters. One of the main requirements to obtain MUDiv and multiplexing gains is knowledge of the CSI. In practical systems, this is challenging because the feedback load rapidly increases with $K$ and $M$. Multiple transmit antennas require additional pilot overhead proportional to $M$, if the users need to learn the complete MIMO channel \\cite{Vu2007}. Further research is required to completely understand  performance bounds advance transmission schemes, e.g., dynamic SU-\/MU-MIMO switching and multi-\/single-stream allocation.\nThe viability of user scheduling and simultaneous transmission in multi-cell deployments must be further investigated. Several topics must be included in future studies: synchronization issues in MU-MIMO, metrics for user grouping in HetNets, scheduling signaling load, coordinated power control, latency and delay due to scheduling, mobility, the impact of realistic traffic models, backhaul constraints, and semi-distributed control algorithms. The scheduling policies for the next generation of wireless technologies must balance between the high cost of CSI acquisition and the benefits of cooperative transmission.\n\n\nThe current performance of MU-MIMO processing is still limited in 4G cellular communication systems. This is because user terminals do not perform interference estimation, and in all cases only CSI feedback for the SU-MIMO mode (SU-RI, PMI, and CQI) is employed \\cite{Li2014}. Most of the reviewed works assume continuous power control, but LTE only supports discrete power control in the downlink, through a user-specific data-to-pilot-power offset parameter \\cite{Dahlman2013}. These facts must be considered to properly assess current and emerging NOMA schemes in multiuser scenarios, as well as their limitations, see \\cite{Li2014a,Liu2015,Dai2015}. Resource management rule have a fundamental role to play in future generations of wireless networks, where new and evolving technologies such as mmWave and massive MIMO have already been considered \\cite{Kim2014,Nam2015,Zheng2015,Kutty2016}. Such rule will require novel precoding designs and an assertive usage of the multiuser dimension to provide substantial spectral efficiency, energy efficiency and user satisfaction.\n\n\n\n{\\color{black}{\n\n\n\\subsection{Heterogeneous Networks}\nCurrent cellular networks are facing immense challenges to cope with the ever increasing demand for throughput and coverage owing to the growing amount of mobile traffic. Network densification through\ndeployment of heterogeneous infrastructure, e.g., pico BSs and small cells, is envisioned as a\npromising solution to improve area spectral efficiency and network coverage. Nevertheless, heterogeneous networks and small cell deployments will bring significant challenges and new problems in resource allocation and scheduling for MU-MIMO system. First, network densification reduces MUDiv, since few users are associated to the closest BSs \\cite{Lopez-Perez2015}. This can reduce the net throughput and may challenge precoding techniques with imperfect CSI that rely on MUDiv to compensate for the imperfections. Furthermore, the selection metrics may need to change in HetNets as scheduling may be used not for boosting the received signal exploiting MUDiv but for serving users with low ICI. Moreover, putting a large number of antennas in small cells is not envisioned mainly due to excess cost and processing capabilities, thus the promising gains of massive MIMO may not be realized in HetNets. The same applies for cooperative and network MU-MIMO schemes (e.g. CoMP) among small cells, which may require additional signaling and communication among small cells. Access network architectures enhanced with distributed antennas \\cite{Tolstrup2015} can complement small cells and increase system throughput, but only if large backhaul capacity is available \\cite{Ngo2016}. The existence of signaling interfaces (e.g. X2 in LTE) seem to be sufficient for current deployments, but changes may be required for more advanced coordinated MIMO techniques and joint resource allocation. \nNetwork densification will result in major challenges in indoor coverage and services, for which the channel models used in existing MU-MIMO literature may not be relevant. \n\n \n\n\n\\subsection{Massive MIMO and Millimeter Waves}\n\nMassive MIMO and mmWave  technologies have a fundamental role to play in 5G \\cite{Boccardi2014,Andrews2014}, but there are many challenges and open problems. More research is needed to enable FDD mode with acceptable overhead and mitigation of pilot contamination, which will speed up the standardization and commercial adoption of massive MIMO technology. \nOperating in wideband channels at mmWaves will require efficient hybrid precoding and fast beam adaptation. Low-resolution and cost effective transceiver architectures must be designed to cope with these goals, which will bring new techniques to calibrate the antenna array, define the most efficient array geometry, estimate CSI, and mitigate hardware impairments \\cite{Puglielli2016,Hamed2016}. \nAlthough massive MIMO and mmWave will be implemented in cellular networks \\cite{Shokri-Ghadikolaei2015}, currently, they are jointly applied for indoor short-range \\cite{Kutty2016} and outdoor point-to-point applications \\cite{Taori2015,Bogale2016}.  Signal processing and medium access techniques for massive MIMO may provide a cost-effective alternative to dense HetNets \\cite{Lopez-Perez2015}, and full understanding of how these technologies complement each other is matter of future research \\cite{Bjornson2016a}.  \n\n\n\n\nInitially, synchronous massive MIMO systems have been implemented with capabilities for joint signal processing \\cite{Vieira2014}. However, coordinated per-user allocation and time-synchronous transmission may be hard to achieve in cellular systems \\cite{Bjornson2015a}. Therefore, further research is needed to design, prototype, and assess distributed and asynchronous massive MIMO systems, specially in challenging high mobility scenarios. \nAdditionally, the data buses and interfaces at the transmitters must be scaled orders of magnitude to support the traffic generated by many concurrent users. Research on intermittent user activity (bursty traffic) has been presented in \\cite{Bjornson2015a}, but more work is needed to optimize resource allocation for users with heterogeneous data rate requirements and services. These challenges will demand enhanced physical and media access control layer interactions, in order to support fast user\/channel tracking and dynamic resource allocation.\n\n\n\n}}\n\n\n\n\n \n\n\n\n\n\\subsection{Summary of Asymptotic Analysis and Scaling Laws}\n\\label{section:remarks:analytical-results}\n\n\n\n\n\n\n\n\\begin{table*}[t] \\scriptsize\n\t\\renewcommand{\\arraystretch}{1.3}\t\n\t\\caption{Asymptotic Regimes in Multiuser Systems for MISO and MIMO configurations, with full ($B=\\infty$) and partial ($B<\\infty$) CSIT}\n\t\\label{tab:analytical-results\n\t\\centering\t\t\n\t\\begin{tabularx}{\\textwidth}{l *{4}{>{\\centering\\arraybackslash}X} } \n\t\t\\toprule\n\t\t\\textbf{Parameter} & \\textbf{MISO}, $B=\\infty$ & \\textbf{MISO}, $B<\\infty$ & \\textbf{MIMO}, $B=\\infty$ & \\textbf{MIMO}, $B<\\infty$ \\\\\n\t\t\\midrule\n\t\t\\textbf{Capacity} & \\cite{Wang2015, Yang2011, Sohn2010,Lee2007} & \\cite{Jindal2006, Yang2011, Huang2009a, Vicario2008, Al-naffouri2009, Zhang2011, Min2013, Sohn2010, Au-Yeung2007, Kountouris2008a, Huang2007} &  \\cite{Sharif2007, Hassibi2007, Lee2007} & \\cite{Sharif2007, Al-naffouri2009} \\\\\n\t\t\\midrule\n\t\t\\textbf{Queues} & \\cite{Chung2010, Lau2006, Destounis2015} & \\cite{Huang2012a, Shirani-Mehr2010} & \\cite{Hassibi2007} & \\cite{Chen2013} \\\\\n\t\t\\midrule\n\t\t\\textbf{High SNR} & \\cite{Caire2003,Lee2007} & \\cite{Jindal2006, Xia2009, Wagner2008, Conte2010, Tang2010, Huang2009a, Vicario2008, Zhang2011, Yoo2007, Kountouris2008a}  & \\cite{Wang2010, Lim2009, Tran2010, Hassibi2007,Lee2007}, \\cite{Chen2008, Sun2010, Tran2012} & \\cite{Vu2007} \\\\\n\t\t\\midrule\n\t\t\\textbf{Low SNR} & \\cite{Caire2003} & \\cite{Wagner2008, Huang2009a, Vicario2008, Zhang2011, Kountouris2008a} & \\cite{Wang2010, Tran2010} \\cite{Sun2010} & \\cite{Vu2007} \\\\\n\t\t\\midrule\n\t\t\\textbf{Large} $K$ & \\cite{Viswanath2002,Wang2008a, Yoo2006, Jiang2006, Lee2014, Bjornson2014, Yoo2005a, Sohn2010, Huh2012, Jang2011,Lee2014c}, \\cite{Jagannathan2007} & \\cite{Wagner2008, Nam2014, Xia2009, Tang2010, Huang2009a, Lau2009, Yoo2007, Kountouris2007,  Dai2008, Huang2007, Xu2010} & \\cite{Sharif2007, Bayesteh2008, Tran2010, Hassibi2007}, \\cite{Sun2010, Tran2012} & \\cite{Al-naffouri2009, Trivellato2008}, \\cite{Sharif2005}  \\\\\n\t\t\\midrule\n\t\t\\textbf{Large} $M$ & \\cite{Huang2012b, Bjornson2014, Huh2012,Ngo2016}  & \\cite{Nam2014, Adhikary2013, Adhikary2013a, Lau2009, Dai2008} & \\cite{Tran2010}, \\cite{ Chen2007a} & \\cite{Sharif2005} \\\\\n\t\t\\midrule\n\t\t\\textbf{Bits} $B_1$, $B_2$ & \\cite{Zakhour2007} & \\cite{Jindal2006, Xia2009, Wagner2008, Huang2012a, Vicario2008, Yoo2007, Dai2008, Zhang2011, Ravindran2012, Kountouris2006a, Min2013, Sohn2010, Au-Yeung2007, Kountouris2008a,  Khoshnevis2013} & -  & \\cite{Trivellato2008, Jindal2008}, \\cite{Zhang2007a} \\\\ \n\t\t\\bottomrule\t\n\t\\end{tabularx}%\n\\end{table*}%\n\n\n\n\n\n\\begin{table*}[t!]\\scriptsize\n\t\\centering\n\t\\caption{Abbreviations}\n\t\\label{table:abbreviations}\n\t\\begin{tabular}{@{}llllllll@{}}\n\t\t\\toprule\n\t\t\\textbf{3GPP}     & \\begin{tabular}[c]{@{}l@{}}3rd Generation Partnership\\\\   Project\\end{tabular} &  & \\textbf{FDD}     & Frequency-division-duplex                      &  & \\textbf{PMI}     & Precoding matrix index             \\\\\n\t\t\\textbf{ADC}      & Analog-to-digital converter                                                    &  & \\textbf{GA}      & Genetic algorithms                             &  & \\textbf{QoS}     & Quality of service                 \\\\\n\t\t\\textbf{AP}       & Access point                                                                   &  & \\textbf{GUS}     & Greedy user selection                          &  & \\textbf{QCA}     & Quantization cell approximation    \\\\\n\t\t\\textbf{ACK\/NACK} & Acknowledgement handshake                                                      &  & \\textbf{HetNet}  & Heterogeneous network                          &  & \\textbf{QSI}     & Queue state information            \\\\\n\t\t\\textbf{AWGN}     & Additive white Gaussian noise                                                  &  & \\textbf{ICI}     & Inter-cell interference                        &  & \\textbf{RF}      & Radio frequency                    \\\\\n\t\t\\textbf{ARQ}      & Automatic repeat request                                                       &  & \\textbf{IFC}     & Interference channel                           &  & \\textbf{RBF}     & Random beamforming                 \\\\\n\t\t\\textbf{BS}       & Base station                                                                   &  & \\textbf{IUI}     & Inter-user interference                        &  & \\textbf{RI}      & Rank indicator                     \\\\\n\t\t\\textbf{BER}      & Bit error rate                                                                 &  & \\textbf{JT}      & Joint signal transmission\/processing           &  & \\textbf{RSS}     & Received signal strength           \\\\\n\t\t\\textbf{BD}       & Block diagonalization                                                          &  & \\textbf{LoS}     & Line-of-sight                                  &  & \\textbf{SUS}     & Semi-orthogonal user selection     \\\\\n\t\t\\textbf{BB}       & Branch and bound                                                               &  & \\textbf{LTE}     & Long term evolution                            &  & \\textbf{SLNR}    & Signal to leakage plus noise ratio \\\\\n\t\t\\textbf{BC}       & Broadcast channel                                                              &  & \\textbf{MRT}     & Maximum ratio transmission                     &  & \\textbf{SNR}     & Signal-to-noise ratio              \\\\\n\t\t\\textbf{CU}       & Central processing unit                                                        &  & \\textbf{MSE}     & Mean-square-error                              &  & \\textbf{ST}      & Single transmission                \\\\\n\t\t\\textbf{CDI}      & Channel direction information                                                  &  & \\textbf{MMSE}    & Minimum MSE filter                             &  & \\textbf{SU-MIMO} & Single user MIMO                   \\\\\n\t\t\\textbf{CQI}      & Channel quality information                                                    &  & \\textbf{MCS}     & Modulation and coding scheme                   &  & \\textbf{SISO}    & Single-input single-output         \\\\\n\t\t\\textbf{CSI}      & Channel state information                                                      &  & \\textbf{MAC}     & Multiple access channel                        &  & \\textbf{SDV}     & Singular value decomposition       \\\\\n\t\t\\textbf{CDMA}     & Code division multiple access                                                  &  & \\textbf{MIMO}    & Multiple-input multiple-output                 &  & \\textbf{SDMA}    & Space-division multiple access     \\\\\n\t\t\\textbf{CIZF}     & Constructive interference ZF                                                   &  & \\textbf{MISO}    & Multiple-input single-output                   &  & \\textbf{SZF}     & Successive ZF                      \\\\\n\t\t\\textbf{CBF}      & Coordinated beamforming                                                        &  & \\textbf{MUDiv}   & Multiuser diversity                            &  & \\textbf{TDMA}    & Time division multiple access      \\\\\n\t\t\\textbf{CoMP}     & Coordinated multi-point                                                        &  & \\textbf{MU-MIMO} & Multiuser MIMO                                 &  & \\textbf{TDD}     & Time-division-duplex               \\\\\n\t\t\\textbf{CS}       & Coordinated scheduling                                                         &  & \\textbf{NOMA}    & Non-orthogonal multiple access                 &  & \\textbf{THP}     & Tomlinson-Harashima precoding      \\\\\n\t\t\\textbf{CSIR}     & CSI at the receiver                                                            &  & \\textbf{NP}      & Non-deterministic polynomial time problem      &  & \\textbf{Wi-Fi}   & Trademark of IEEE 802.11           \\\\\n\t\t\\textbf{CSIT}     & CSI at the transmitter                                                         &  & \\textbf{NP-C}    & NP complete                                    &  & \\textbf{VP}      & Vector perturbation                \\\\\n\t\t\\textbf{CDF}      & Cumulative distribution function                                               &  & \\textbf{NSP}     & Null space projection                          &  & \\textbf{VQ}      & Vector quantization                \\\\\n\t\t\\textbf{DoF}      & Degrees-of-freedom                                                             &  & \\textbf{OFDMA}   & Orthogonal frequency-division multiple access  &  & \\textbf{WSR}     & Weighted sum rate                  \\\\\n\t\t\\textbf{DPC}      & Dirty paper coding                                                             &  & \\textbf{OFDM}    & Orthogonal frequency-division multiplexing     &  & \\textbf{WLAN}    & Wireless local area network        \\\\\n\t\t\\textbf{EE}       & Energy efficiency                                                              &  & \\textbf{PSO}     & Particle swarm optimization                    &  & \\textbf{ZF}      & Zero forcing                       \\\\\n\t\t\\textbf{EPA}      & Equal power allocation                                                         &  & \\textbf{PU2RC}   & Per unitary basis stream user and rate control &  & \\textbf{ZFBF}    & ZF beamforming                     \\\\\n\t\t\\textbf{ExS}      & Exhaustive search                                                              &  & \\textbf{PHY}     & Physical layer                                 &  & \\textbf{ZFDP}    & ZF dirty paper                    \\\\\n\t\t\\bottomrule                     \n\t\\end{tabular}\n\\end{table*}\n\n\n\nSeveral authors have focused their work on asymptotic analysis and scaling laws of performance metrics or utility functions $U(\\cdot)$. The analytical results depend on the \\textit{system parameters}, $K$, $N$, $M$, $B$, $P$, and $q_k$ $\\forall k \\in \\mathcal{K}$. The information-theoretic results derived in several works provide fundamental limits of achievable values of $U(\\cdot)$ from the user and system perspectives. They also shed light on the relevance of each parameter, the conditions where the parameters are interchangeable, the potential and limitations of MU-MIMO systems, and judicious guidelines for the overall system design. For each utility function $U(\\cdot)$, and user priorities $\\omega_{k}$ $\\forall k$, there exist optimal and suboptimal operation points that can be characterized according to the system parameters and their respective regimes. \nThe capacity of MU-MIMO systems has been assessed in various asymptotic regimes: high SNR ($P\\rightarrow \\infty$), low SNR ($P\\rightarrow 0$), large number of users ($K\\rightarrow \\infty$), large number of transmit antennas ($M\\rightarrow \\infty$), and large codebook resolution ($B \\rightarrow \\infty$). \n\n\nTable~\\ref{tab:analytical-results} summarizes the system parameters and the antenna configurations, i.e., MISO ($N=1$) or MIMO ($N>1$), and $M \\geq N$. Every single reference in the table has its own system model, assumptions, and constraints, studying one parameter and the corresponding effects on the performance $U(\\cdot)$, and other fixed parameters. The table is by no means exhaustive; a comprehensive taxonomy of the asymptotic analytical results is out of the scope of this paper. Our aim is to provide a list of organized results from the reviewed paper, so that the interested reader may use each reference for further studies. \nNotice that the first row in Table~\\ref{tab:analytical-results} points to references that analyze the capacity as a function of different parameters, while the second row refers to works that provide analytical results of MU-MIMO systems with queue constraints.\n\n\n\n\n\\bibliographystyle{IEEEtran}\n\\scriptsize\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{s:intro}\n\n\\subsection{Interval and circular-arc hypergraphs}\n\nAn \\emph{interval ordering} of a hypergraph $\\ensuremath{\\mathcal{H}}$ with a finite vertex set $V=V(\\ensuremath{\\mathcal{H}})$ is\na linear ordering $v_1,\\ldots,v_n$ of $V$ such that every hyperedge of $\\ensuremath{\\mathcal{H}}$ is an interval of\nconsecutive vertices. This notion admits generalization to an \\emph{arc ordering} where\n$v_1,\\dots,v_n$ is \\emph{circularly ordered} (i.e., $v_1$ succeeds $v_n$) \nso that every hyperedge is an \\emph{arc} of consecutive vertices.\n\nAn \\emph{interval hypergraph} is a hypergraph\nadmitting an interval ordering.\nSimilarly, if a hypergraph admits an arc ordering, we call it \\emph{circular-arc} \n(using also the shorthand \\emph{CA}).\nIn the terminology stemming from computational genomics,\ninterval hypergraphs are exactly those hypergraphs\nwhose incidence matrix has the \\emph{consecutive ones property}; e.g.,~\\cite{Dom09}.\nSimilarly, a hypergraph is CA exactly when its\nincidence matrix has the \\emph{circular ones property}; e.g.,~\\cite{HM03,GPZ08,OBS11}.\n\n\nOur goal is to study the conditions under which interval and circular-arc hypergraphs\nare \\emph{rigid} in the sense that they have a unique interval or, respectively, arc\nordering. Since any interval (or arc) ordering can be changed to another interval (or arc) \nordering by reversing, we always mean uniqueness \\emph{up to reversal}.\nAn obvious necessary condition of the uniqueness\nis that a hypergraph has no \\emph{twins}, that is, no two vertices such that\nevery hyperedge contains either both or none of them. \n\nWe say that two sets~$A$ and~$B$ \\emph{overlap} and write $A\\between B$ if\n$A$ and $B$ have nonempty intersection and neither of the two sets includes the other.\nTo facilitate notation, we use the same character $\\ensuremath{\\mathcal{H}}$ to denote a hypergraph\nand the set of its hyperedges.\nWe call $\\ensuremath{\\mathcal{H}}$ \\emph{overlap-connected} if it has  no isolated vertex\n(i.e., every vertex is contained in a hyperedge) and the graph\n$(\\ensuremath{\\mathcal{H}},\\between)$ is connected.\nAs a starting point, we refer to the following rigidity result.\n\n\\begin{theorem}[Chen and Yesha~\\cite{ChenY91}]\\label{thm:unique-overlap-1}\nA twin-free, overlap-connected interval hypergraph has, \nup to reversal, a unique interval ordering.\n\\end{theorem}\n\nIf we want to extend this result to CA hypergraphs,\nthe overlap-connectedness obviously does not suffice.\nFor example, the twin-free overlap-connected hypergraph $\\ensuremath{\\mathcal{H}}=\\big\\{\\{a,b\\},\\{a,b,c\\},\\{b,c,d\\}\\big\\}$\nhas essentially different arc orderings. \nWe, therefore, need a stronger\nnotion of connectedness. When $A$ and $B$ are overlapping subsets of $V$\n(i.e.,~$A\\between B$) that additionally satisfy $A\\cup B\\ne V$, we say that \n$A$ and~$B$ \\emph{strictly overlap} and write $A\\between^* B$. \n\nQuilliot~\\cite{Quilliot84} proves that\na CA hypergraph~$\\ensuremath{\\mathcal{H}}$ on~$n$ vertices has a unique\narc ordering if and only if for every set $X\\subset V(\\ensuremath{\\mathcal{H}})$ with\n$1<|X|< n-1$ there exists a hyperedge $H\\in\\ensuremath{\\mathcal{H}}$ such that $H\\between^* X$.\nNote that this criterion does not admit efficient verification\nas it involves quantification over all subsets~$X$.\n\nWe call a hypergraph~$\\ensuremath{\\mathcal{H}}$ \n\\emph{strictly overlap-connected} if it has  no isolated vertex and the graph\n$(\\ensuremath{\\mathcal{H}},\\between^*)$ is connected.\nWe prove the following analog of Theorem~\\ref{thm:unique-overlap-1}\nfor CA hypergraphs.\n\n\\begin{theorem}\\label{thm:unique-overlap-2}\nA twin-free, strictly overlap-connected CA hypergraph has, \nup to reversal, a unique arc representation.\n\\end{theorem}\n\n\n\n\\subsection{Tight orderings}\n\nLet us use notation $A\\bowtie B$ to say that sets $A$ and $B$\nhave a non-empty intersection. By the standard terminology,\na hypergraph $\\ensuremath{\\mathcal{H}}$ is \\emph{connected} if it has no isolated vertex\nand the graph $(\\ensuremath{\\mathcal{H}},\\bowtie)$ is connected.\nNote that the assumption made in Theorem~\\ref{thm:unique-overlap-1}\ncannot be weakened just to connectedness;\nconsider $\\ensuremath{\\mathcal{H}}=\\big\\{\\{a\\},\\{a,b,c\\}\\big\\}$ as the simplest example.\nThus, if we want to weaken the assumption,\nwe have also to weaken the conclusion.\n\nCall an arc ordering of a hypergraph $\\ensuremath{\\mathcal{H}}$ \\emph{tight} if,\nfor any two hyperedges $A$ and $B$ such that\n$\\emptyset\\ne A\\subseteq B\\ne V$,\nthe corresponding arcs share an endpoint.\nThe definition of a \\emph{tight interval ordering}\nis similar: We require that the arcs corresponding\nto hyperedges $A$ and $B$ share an endpoint whenever $\\emptyset\\ne A\\subseteq B$\n(the condition $B\\ne V$ is now dropped as the complete interval $V$ has two endpoints, while the\ncomplete arc $V$ has none).\\footnote{%\nThe class of hypergraphs admitting a tight interval ordering\nis characterized in terms of forbidden subhypergraphs in~\\cite{Moore77};\nsuch a characterization of interval hypergraphs is given in~\\cite{TrotterM76}.}\n\n\nFor nonempty $A$ and~$B$, note that\n$A\\bowtie B$ iff $A\\between B$ or $A\\subseteq B$ or $A\\supseteq B$.\nBy similarity, we define\n\\[\nA\\bowtie^* B\\text{ iff }A\\between^* B\\text{ or }A\\subseteq B\\text{ or }A\\supseteq B\n\\]\nand say that such two nonempty sets \\emph{strictly intersect}.\nWe call a hypergraph~$\\ensuremath{\\mathcal{H}}$ \n\\emph{strictly connected} if it has  no isolated vertex and the graph\n$(\\ensuremath{\\mathcal{H}},\\bowtie^*)$ is connected.\nIn Section~\\ref{s:hgs} we establish the following result.\n\n\n\\begin{theorem}\\label{thm:unique}\n\\begin{bfenumerate}\n\\item \nA twin-free, connected hypergraph has, up to reversal,\nat most one tight interval ordering.\n\\item \nA twin-free, strictly connected hypergraph has, up to reversal, \nat most one tight arc ordering.\n\\end{bfenumerate}\n\\end{theorem}\n\n\n\\subsection{The neighborhood hypergraphs of proper interval and proper circular-arc graphs}\n\nFor a vertex $v$ of a graph $G$, the set of vertices adjacent to $v$\nis denoted by $N(v)$. Furthermore, $N[v]=N(v)\\cup\\{v\\}$.\nWe define the \\emph{closed neighborhood hypergraph} of~$G$ \nby $\\ensuremath{\\mathcal{N}}[G]=\\{N[v]\\}_{v\\in V(G)}$. \n\nRoberts~\\cite{Roberts71} discovered that $G$ is a proper interval graph \nif and only if $\\ensuremath{\\mathcal{N}}[G]$ is an interval hypergraph.\nThe case of proper circular-arc (PCA) graphs is more complex.\\footnote{For a\n  definition of proper interval and PCA graphs, see the beginning of\n  Section~\\ref{s:Nhgs}.}\nIf $G$ is a PCA graph, then $\\ensuremath{\\mathcal{N}}[G]$ is a CA hypergraph.\nThe converse is not always true.\nThe class of graphs with circular-arc closed neighborhood hypergraphs,\nknown as \\emph{concave-round graphs}~\\cite{Bang-JHY00},\ncontains PCA graphs as a proper subclass.\nTaking a closer look at the relationship between PCA graphs\nand CA hypergraphs, Tucker~\\cite{Tucker71}\\footnote\nTucker~\\cite{Tucker71} uses an equivalent language of matrices.}\ndistinguishes the case when\nthe complement graph $\\overline{G}$ is non-bipartite and shows that then\n$G$ is PCA exactly when $\\ensuremath{\\mathcal{N}}[G]$ is~CA.\n\n \nOur interest in tight orderings has the following motivation.\nIn fact, $G$ is a proper interval graph \nif and only if the hypergraph $\\ensuremath{\\mathcal{N}}[G]$ has a tight interval ordering\n(this follows from the Roberts theorem and Lemma~\\ref{lem:geomistight} in\nSection~\\ref{s:Nhgs}). Moreover, $G$ is a PCA graph \nif and only if $\\ensuremath{\\mathcal{N}}[G]$ has a tight arc ordering\n(we observed this in~\\cite{fsttcs} based on Lemma~\\ref{lem:geomistight}\nand Tucker's analysis in~\\cite{Tucker71}).\n\nNow, it is natural to consider the connectedness properties of $\\ensuremath{\\mathcal{N}}[G]$\nfor proper interval and PCA graphs and derive from here\nrigidity results. For proper interval graphs this issue has been\nstudied in the literature earlier, but we discuss also this class of graphs for\nexpository purposes.\n\nWe call two vertices~$u$ and~$v$ of a graph $G$ \\emph{twins} if $N[u]=N[v]$. \nNote that $u$ and~$v$ are twins in the graph $G$ if\nand only if they are twins in the hypergraph $\\ensuremath{\\mathcal{N}}[G]$.\nThus, the absence of twins in $G$ is a necessary condition for rigidity of $\\ensuremath{\\mathcal{N}}[G]$.\nAnother obvious necessary condition is the connectedness of $G$ (and, hence, of\n$\\ensuremath{\\mathcal{N}}[G]$).\\footnote{Small graphs are an exception, as all interval orderings\n  of at most two vertices are the same up to reversal, and all arc orderings of\n  up to three vertices are the same up to reversal and rotation.}\nBy Theorem~\\ref{thm:unique}.1, if a proper interval graph $G$ is\ntwin-free and connected, then $\\ensuremath{\\mathcal{N}}[G]$ has a unique tight interval ordering. \nMaking the same assumptions, Roberts~\\cite{Roberts71} proves that\neven an interval ordering of $\\ensuremath{\\mathcal{N}}[G]$ is unique.\n\nSuppose now that $G$ is a PCA graph. Consider first the case \nwhen $\\overline{G}$ is non-bipartite. In Section~\\ref{s:Nhgs} we prove\nthat then $\\ensuremath{\\mathcal{N}}[G]$ is strictly connected. Theorem~\\ref{thm:unique}.2\napplies and shows that, if $G$ is also twin-free and connected, then\n$\\ensuremath{\\mathcal{N}}[G]$ has a unique tight arc ordering.\nMoreover, we prove that any arc ordering of $\\ensuremath{\\mathcal{N}}[G]$\nis tight and, hence, unique as well.\n\nIf $\\overline{G}$ is bipartite, it is convenient to switch to\nthe complement hypergraph $\\overline{\\ensuremath{\\mathcal{N}}[G]}=\\{V(G)\\setminus N[v]\\}_{v\\in V(G)}$.\nThis hypergraph is interval. Applying Theorem~\\ref{thm:unique}.1\nto the connected components of $\\overline{\\ensuremath{\\mathcal{N}}[G]}$,\nwe conclude that $\\ensuremath{\\mathcal{N}}[G]$ has, up to reversing, exactly two tight arc orderings\nprovided $\\overline{G}$ is connected.\n\nIn~\\cite{KoeblerKLV11} we noticed that,\nif a proper interval graph $G$ is connected, then the hypergraph\n$\\ensuremath{\\mathcal{N}}[G]\\setminus\\{V(G)\\}$ is overlap-connected.\nThis allows to derive Roberts' aforementioned rigidity result\nfrom Theorem~\\ref{thm:unique-overlap-1}.\nIn Section~\\ref{s:ov-conn}, we use Theorem~\\ref{thm:unique-overlap-2}\nto obtain a similar result for PCA graphs:\nIf $G$ is an $n$-vertex connected PCA graph with non-bipartite complement,\nthen removal of all $(n-1)$-vertex hyperedges from $\\ensuremath{\\mathcal{N}}[G]$\ngives a strictly overlap-connected hypergraph.\n\n\n\n\\subsection{Intersection representations of graphs}\n\nA proper interval representation $\\alpha$ of a graph $G$\ndetermines a linear ordering of $V(G)$\naccordingly to the appearance of the left (or, equivalently, right)\nendpoints of the intervals $\\alpha(v)$, $v\\in V(G)$, in the intersection model.\nWe call it the \\emph{geometric order} associated with $\\alpha$.\nSimilarly, a PCA representation of $G$ determines the \\emph{geometric}\ncircular order on the vertex set. Any geometric order is a tight\ninterval or, respectively, arc ordering of $\\ensuremath{\\mathcal{N}}[G]$\n(see Lemma~\\ref{lem:geomistight}). The rigidity results\noverviewed above imply that the geometric order is unique\nfor twin-free, connected proper interval graphs and\ntwin-free, connected PCA graphs with non-bipartite complement.\nIn Section~\\ref{s:repr} we show that this holds true also\nin the case of PCA graphs with bipartite connected complement.\n\nLet us impose reasonable restrictions on proper interval and PCA\nmodels of graphs. Specifically, we always suppose that a model of an $n$-vertex\ngraph has $2n$ points and consists of intervals\/arcs that never\nshare an endpoint. It turns out that such intersection representations\nare determined by the associated geometric order uniquely up to\nreflection (and rotation in the case of arcs representations).\nThis implies that any twin-free, connected proper interval\nor PCA graph has a unique intersection representation.\n\nThe last result is implicitly contained in the work by Deng, Hell, and Huang~\\cite{DengHH96}, that relies on\na theory of local tournaments~\\cite{Huang95}; see the discussion in the end of Section~\\ref{s:repr}.\n\n\n\n\n\n\n\\section{Interval and circular-arc hypergraphs}\\label{s:hgs}\n\nLet $V=\\{v_1,\\dots,v_n\\}$.\nSaying that the sequence $v_1,\\dots,v_n$\nis \\emph{circularly ordered}, \nwe mean that $V$ is endowed with\nthe circular successor relation~$\\prec$ under which\n$v_i\\prec v_{i+1}$ for $i<n$ and $v_n\\prec v_1$.\nAn ordered pair of elements $a^-,a^+\\in V$ determines an \\emph{arc}~$A=[a^-,a^+]$\nthat consists of the vertices\nappearing in the directed path from~$a^-$ to~$a^+$. \nThis notation will be used under the assumption that $A\\ne V$, \nthough we also allow the \n\\emph{complete arc} $A=V$ and the \\emph{empty arc} $A=\\emptyset$.\nBy $\\mathbb{C}_n$ we denote the set $\\{1,\\ldots,n\\}$\nendowed with the circular order $1\\prec 2\\prec \\ldots\\prec n\\prec 1$.\n\nWe here prove Theorems~\\ref{thm:unique-overlap-2} and~\\ref{thm:unique}.\nSince the proofs are very similar, we treat in detail the most\ncomplicated of these statements, namely part~2 of Theorem~\\ref{thm:unique},\nand argue that the argument also covers the other statements (including also\nTheorem~\\ref{thm:unique-overlap-1}).\n\nWe will use an inductive argument. To establish the rigidity of\na connected hypergraph $\\ensuremath{\\mathcal{H}}$, we prove this property for every\nconnected subhypergraph $\\ensuremath{\\mathcal{K}}\\subseteq\\ensuremath{\\mathcal{H}}$ by induction\non the number of hyperedges in $\\ensuremath{\\mathcal{K}}$.\nSince subhypergraphs of a twin-free hypergraph $\\ensuremath{\\mathcal{H}}$\ncan contain twins, we need an appropriate generalization\nof our rigidity concept. To this end, we switch to an equivalent language.\n\nAn \\emph{arc representation of a hypergraph~$\\ensuremath{\\mathcal{H}}$} is a hypergraph isomorphism~$\\rho$\nfrom~$\\ensuremath{\\mathcal{H}}$ to an arc system~$\\ensuremath{\\mathcal{A}}$ on the circle~$\\mathbb{C}_n$.\nNote that $\\ensuremath{\\mathcal{H}}$ has an arc representation exactly when it admits an arc ordering~$\\prec$.\nIndeed, if $\\rho\\function{V(\\ensuremath{\\mathcal{H}})}{\\{1,\\ldots,n\\}}$ is an arc representation of~$\\ensuremath{\\mathcal{H}}$,\nwe can define $\\prec_\\rho$ by $\\rho^{-1}(1)\\prec_\\rho\\rho^{-1}(2)\\prec_\\rho\\ldots\\prec_\\rho\\rho^{-1}(n)\\prec_\\rho\\rho^{-1}(1)$.\nConversely, if $v_1\\prec v_2\\prec\\ldots\\prec v_n\\prec v_1$ is an arc ordering of~$\\ensuremath{\\mathcal{H}}$,\nthen $\\rho_\\prec(v_i)=i$ is an arc representation of~$\\ensuremath{\\mathcal{H}}$. \nFurthermore, we call $\\rho$ \\emph{tight} if the circular order of $\\mathbb{C}_n$\nis tight for $\\ensuremath{\\mathcal{A}}$. Obviously, tight arc representations correspond to\ntight arc orderings and vice versa.\nThe notions of an \\emph{interval representation},\ncorresponding to an interval ordering, is introduced similarly.\n\nThe rotations $x\\mapsto(x+s)\\bmod n+1$ and the reflection $x\\mapsto n+1-x$\nwill be considered \\emph{symmetries} of the circle~$\\mathbb{C}_n$.\nThe linearly ordered segment $\\{1,\\ldots,n\\}$ has a unique symmetry, namely the reflection.\nNote that, if $\\rho$~is an interval or arc representation of~$\\ensuremath{\\mathcal{H}}$\nand $\\sigma$~is a symmetry of the circle or the interval, respectively,\nthen the composition $\\sigma\\circ\\rho$ is an \ninterval or arc representation of~$\\ensuremath{\\mathcal{H}}$ as well.\nTurning back to the equivalence between arc representations and orderings,\nnote that while $\\rho$ determines $\\prec_\\rho$ uniquely, $\\prec$ determines $\\rho_\\prec$\nup to a rotation.\n\nNow, notice that $\\ensuremath{\\mathcal{H}}$ admits a unique, up to reversing, (tight) interval ordering\nif and only if $\\ensuremath{\\mathcal{H}}$ has a unique, up to reflection, (tight) interval representation.\nFurthermore, $\\ensuremath{\\mathcal{H}}$ admits a unique, up to reversing, (tight) arc ordering\nif and only if $\\ensuremath{\\mathcal{H}}$ has a unique, up to reflection and rotation, (tight) arc representation.\n\nA \\emph{transposition of twins} in a hypergraph $\\ensuremath{\\mathcal{H}}$\nis a one-to-one map of~$V(\\ensuremath{\\mathcal{H}})$ onto itself\nthat fixes each vertex except, possibly, a pair of twins of~$\\ensuremath{\\mathcal{H}}$.\nAny composition~$\\pi$ of transpositions of twins is called a \\emph{permutation of twins}.\nNote that, if $\\rho$~is an interval or arc representation of~$\\ensuremath{\\mathcal{H}}$,\nthen the composition $\\rho\\circ\\pi$ is, respectively, an \ninterval or arc representation of~$\\ensuremath{\\mathcal{H}}$ as well.\nWe call two representations~$\\rho$ and~$\\rho'$ equivalent \\emph{up to permutation of twins}\nif $\\rho'=\\rho\\circ\\pi$ for some permutation of twins $\\pi$.\nFor twin-free hypergraphs this relation just coincides with equality of representations.\n\n\\begin{lemma}\\label{lem:arcreprequi}\nArc representations~$\\rho$ and~$\\rho'$ are equivalent up to permutation of twins\niff $\\rho(H)=\\rho'(H)$ for every hyperedge~$H\\in\\ensuremath{\\mathcal{H}}$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $\\rho'=\\rho\\circ\\pi$ for a permutation of twins $\\pi$.\nLet $H$ be an arbitrary hyperedge of~$\\ensuremath{\\mathcal{H}}$. Since $\\tau(H)=H$ for any\ntransposition of twins $\\tau$, we have $\\pi(H)=H$ and, hence, $\\rho(H)=\\rho'(H)$.\n\nTo prove the claim in the backward direction, define\na \\emph{twin-class} of a hypergraph~$\\ensuremath{\\mathcal{H}}$ to be an inclusion-maximal subset~$S$ of~$V(\\ensuremath{\\mathcal{H}})$ such that\neach hyperedge~$H\\in\\ensuremath{\\mathcal{H}}$ contains either all of~$S$ or none of it.\nThus, two vertices of~$\\ensuremath{\\mathcal{H}}$ are twins iff they are in the same twin-class.\nIt follows that $\\pi$~is a permutation of twins exactly when \n$\\pi(S)=S$ for every twin-class $S$ of~$\\ensuremath{\\mathcal{H}}$.\n\nSuppose that $\\rho(H)=\\rho'(H)$ for every hyperedge~$H\\in\\ensuremath{\\mathcal{H}}$.\nSince a twin-class is a Boolean combination of hyperedges,\nwe have $\\rho(S)=\\rho'(S)$ for every twin-class $S$ of~$\\ensuremath{\\mathcal{H}}$ and,\ntherefore, $\\rho'\\circ\\rho^{-1}(S)=S$ for all twin-classes.\nIt follows that $\\rho'\\circ\\rho^{-1}$ is a permutations of twins,\nhence $\\rho$ and~$\\rho'$ are equivalent up to permutation of twins.\n\\end{proof}\n \n\nLet $\\rho\\function{V(\\ensuremath{\\mathcal{H}})}{\\{1,\\ldots,n\\}}$ and $\\rho'\\function{V(\\ensuremath{\\mathcal{H}})}{\\{1,\\ldots,n\\}}$\nbe arc or interval representations of~$\\ensuremath{\\mathcal{H}}$. We call $\\rho$ and~$\\rho'$\nequivalent \\emph{up to symmetry and permutation of twins} if \n$\\rho'=\\sigma\\circ\\rho\\circ\\pi$ for some symmetry $\\sigma$ and\npermutation of twins $\\pi$. This is an equivalence relation between\nrepresentations because both symmetries and permutations of twins form groups.\n\nThe following lemma translates Theorems~\\ref{thm:unique-overlap-1},~\\ref{thm:unique-overlap-2}, and~\\ref{thm:unique}\ninto the language of interval\/arc representations and generalizes these results\nto hypergraphs with twins. Theorems~\\ref{thm:unique-overlap-1},~\\ref{thm:unique-overlap-2}, and~\\ref{thm:unique}\nfollow from here immediately by the equivalence between representations and orderings.\n\n\n\\begin{lemma}\\label{lem:unique}\n\\begin{bfenumerate}\n\\item\nAn overlap-connected interval hypergraph~$\\ensuremath{\\mathcal{H}}$~has, \nup to symmetry (i.e.\\ reflection) and permutation of twins, \na unique interval representation.\n\\item\nA strictly overlap-connected CA hypergraph~$\\ensuremath{\\mathcal{H}}$~has, \nup to symmetry (i.e., reflection and rotation) and permutation of twins, \na unique arc representation.\n\\item \nA connected hypergraph has, up to symmetry and permutation of twins,\nat most one tight interval representation.\n\\item \nA strictly connected hypergraph has, up to symmetry\nand permutation of twins, at most one tight arc representation.\n\\end{bfenumerate}\n\\end{lemma}\n\n\n\\begin{proof}\nWe will prove part~4. The proof of the interval variant (part~3) is virtually the same\nand even somewhat simpler as not all arc configurations can occur on the line (like one of the\ndashed configurations in Fig.~\\ref{fig:proof:unique2} that closes the circle).\nParts 1 and 2 are actually covered by the argument as well;\nthey correspond exactly to the case shown in Fig.~\\ref{fig:proof:unique2}\nas all the other cases involve inclusions.\n\n\n\\begin{figure}\n  \\centering\n  \\begin{tikzpicture}\n    \\begin{camodel}[every carc\/.style={}]\n     \n      \\carc[labelpos=.33]{180}{60}{3}{$\\rho(A)$}\n      \\carc[labelpos=.66]{115}{-20}{2}{$\\rho(B)$}\n      \\carc{150}{85}{1}{}\n      \\carc[dashed]{85}{20}{1}{}\n      \\carc[dashed,labelangle=-150]{-40}{-230}{1}{$\\rho(H)$}\n    \\end{camodel}\n    \\begin{camodel}[xshift=7cm,every carc\/.style={}]\n     \n      \\carc[labelpos=.33]{180}{60}{3}{$\\rho(A)$}\n      \\carc[labelpos=.66]{115}{-20}{2}{$\\rho(B)$}\n      \\carc[dashed]{85}{20}{1}{}\n      \\carc{20}{-30}{1}{}\n      \\carc[dashed,labelangle=-150]{-30}{-220}{1}{$\\rho(H)$}\n    \\end{camodel}\n  \\end{tikzpicture}\n  \\caption{Proof of Lemma~\\protect\\ref{lem:unique}.4, case 1: $A\\between^*B$ and\n$\\rho(B)$~intersects~$\\rho(A)$ clockwise; $B\\between^*H$.\nOn the left side:\n $\\rho(H)$~intersects~$\\rho(B)$ counter-clockwise.\nOn the right side:\n $\\rho(H)$~intersects~$\\rho(B)$ clockwise.}\\label{fig:proof:unique2}\n\\end{figure}\n\nLet us first explain the strategy of the proof.\nA set of hyperedges $\\ensuremath{\\mathcal{K}}\\subseteq\\ensuremath{\\mathcal{H}}$ will be regarded as\na \\emph{spanning subhypergraph} of $\\ensuremath{\\mathcal{H}}$, that is, $V(\\ensuremath{\\mathcal{K}})=V(\\ensuremath{\\mathcal{H}})$\n(note that $\\ensuremath{\\mathcal{K}}$ can have isolated vertices).\nGiven a hypergraph~$\\ensuremath{\\mathcal{H}}$ with $n$ vertices,\nwe will prove that any strictly connected subhypergraph $\\ensuremath{\\mathcal{K}}\\subseteq\\ensuremath{\\mathcal{H}}$ with $k$ hyperedges\nhas, up to symmetry and permutation of twins, a unique representation on the circle of size~$n$. \nThis will be done by induction on~$k$. The base cases are $k=1,2$.\nIn order to make the inductive step, it suffices to show that, whenever $k\\ge2$,\nany representation~$\\rho$ of~$\\ensuremath{\\mathcal{K}}$ has, up to permutation of twins, a unique\nextension to a representation of $\\ensuremath{\\mathcal{K}}\\cup\\{H\\}$,\nfor any $H\\in\\ensuremath{\\mathcal{H}}\\setminus\\ensuremath{\\mathcal{K}}$ such that $\\ensuremath{\\mathcal{K}}\\cup\\{H\\}$ is strictly connected.\nBy Lemma~\\ref{lem:arcreprequi} this actually means to show that the whole arc~$\\rho(H)$\n(though not necessarily each point~$\\rho(v)$ for $v\\in H$) is uniquely determined.\nMoreover, it suffices to do this job in the case of $k=2$.\nThe reason is that $\\ensuremath{\\mathcal{K}}$ always contains two hyperedges~$A$ and~$B$ such that\nthe sequence $A,B,H$ forms a strictly connected path.\n\nBefore going into detail, we introduce some terminology.\nConsider two arcs $[a^-,a^+]$ and $[b^-,b^+]$ and suppose that\n$[b^-,b^+]\\between^*[a^-,a^+]$ or $[b^-,b^+]\\subset[a^-,a^+]$.\nWe will say that $[b^-,b^+]$ \\emph{intersects} $[a^-,a^+]$ \\emph{clockwise}\nif $a^+\\in[b^-,b^+]$ and \\emph{counter-clockwise} if $a^-\\in[b^-,b^+]$.\n\nAll possible positions of a single hyperedge~$A$ on the circle are congruent by rotation.\nAll possible positions of two strictly overlapping hyperedges~$A$ and~$B$\nare congruent by rotation and reflection because the intersection of the corresponding\narcs~$\\rho(A)$ and~$\\rho(B)$ is always an arc of length $|A\\cap B|$. \nIf $A$ and~$B$ are comparable under\ninclusion, recall that we only consider tight representations.\n\nFor the inductive step, consider three hyperedges~$A$, $B$, and~$H$\nsuch that $A\\bowtie^*B$ and $B\\bowtie^*H$.\nWe have to show that the arc~$\\rho(H)$ is completely determined by\nthe arcs~$\\rho(A)$ and~$\\rho(B)$.\nThe relation $B\\bowtie^*H$ fixes the length of the intersection $\\rho(H)\\cap\\rho(B)$ \nand, hence, leaves for~$\\rho(H)$ exactly two possibilities depending on whether\nthis intersection is clockwise or counter-clockwise.\n\nWe split our analysis into three cases.\n\n\n\\Case1{$A\\between^*B$.}\nWithout loss of generality, we suppose that $\\rho(B)$ intersects~$\\rho(A)$\nclockwise; the case of counter-clockwise intersection is symmetric. \nConsider first the subcase in which $B\\between^*H$.\nLooking at the possible configurations\nfor the arc system $\\{\\rho(A),\\rho(B),\\rho(H)\\}$, \nall shown in Fig.~\\ref{fig:proof:unique2},\nwe see that $\\rho(H)$ intersects~$\\rho(B)$ counter-clockwise\nexactly if the sets $A\\setminus B$ and $H\\setminus B$ are comparable under inclusion, i.e.,\n\\begin{equation}\n  \\label{eq:ccw}\nA\\setminus B\\subseteq H\\setminus B\\text{\\ \\ or\\ \\ }H\\setminus B\\subseteq A\\setminus B.\n\\end{equation}\n\n\n\\begin{figure}\n  \\centering\n  \\begin{tikzpicture}[baseline=0cm]\n    \\begin{camodel}[every carc\/.style={},cabase=.75cm]\n      \\node[anchor=west] at (-2.5cm,1.75cm) {$B\\subset H$:};\n      \\carc[nodepos=.33]{180}{60}{3}{$\\rho(A)$}\n      \\carc[nodepos=.66]{115}{-20}{2}{$\\rho(B)$}\n      \\carc{115}{-40}{1}{}\n      \\carc[dashed,nodepos=.66]{-40}{-210}{1}{$\\rho(H)$}\n    \\end{camodel}\n    \\begin{camodel}[yshift=-2.75cm,every carc\/.style={},cabase=.75cm]\n      \\carc[nodepos=.33]{180}{60}{3}{$\\rho(A)$}\n      \\carc[nodepos=.66]{115}{-20}{2}{$\\rho(B)$}\n      \\carc{150}{-20}{1}{}\n      \\carc[dashed]{-80}{-230}{1}{$\\rho(H)$}\n    \\end{camodel}\n  \\end{tikzpicture}\\hfil\n  \\begin{tikzpicture}[baseline=0cm]\n    \\begin{camodel}[every carc\/.style={},cabase=.75cm]\n      \\node[anchor=west] at (-2.5cm,1.75cm) {$H\\subset B$:};\n      \\carc[nodepos=.33]{180}{60}{3}{$\\rho(A)$}\n      \\carc[nodepos=.66]{115}{-20}{2}{$\\rho(B)$}\n      \\carc{115}{60}{1}{}\n      \\carc[dashed,swap,nodepos=.25]{60}{15}{1}{$\\rho(H)$}\n    \\end{camodel}\n    \\begin{camodel}[yshift=-2.75cm,every carc\/.style={},cabase=.75cm]\n      \\carc[nodepos=.33]{180}{60}{3}{$\\rho(A)$}\n      \\carc[nodepos=.66]{115}{-20}{2}{$\\rho(B)$}\n      \\carc[swap]{25}{-20}{1}{$\\rho(H)$}\n      \\carc[dashed]{90}{20}{1}{}\n    \\end{camodel}\n  \\end{tikzpicture}\n  \\caption{Proof of Lemma~\\protect\\ref{lem:unique}.4, case 1: $A\\between^*B$ and\n    $\\rho(B)$ intersects~$\\rho(A)$ clockwise;\n$B$ and~$H$ are comparable under inclusion.}\\label{fig:proof:unique4a}\n\\end{figure}\n\nFor the remaining subcases, when $B$ and~$H$ are comparable under inclusion,\nall possible configurations for the arc system $\\{\\rho(A),\\rho(B),\\rho(H)\\}$\nare shown in Fig.~\\ref{fig:proof:unique4a}. If $B\\subset H$, \nwe see that $\\rho(B)$ intersects~$\\rho(H)$ clockwise\nexactly under the condition~\\refeq{ccw}.\nIf $H\\subset B$, then $\\rho(H)$ intersects~$\\rho(B)$ counter-clockwise iff\n$A\\cap B\\subseteq H$.\n\n\\begin{figure}\n  \\centering\n  \\begin{tikzpicture}[baseline=0cm]\n    \\begin{camodel}[every carc\/.style={},cabase=.75cm]\n      \\node[anchor=west] at (-2.25cm,1.75cm) {$B\\between^*H$:};\n      \\carc[nodepos=.25]{180}{0}{2}{$\\rho(A)$}\n      \\carc{90}{0}{3}{$\\rho(B)$}\n      \\carc{135}{45}{1}{}\n      \\carc[dashed]{280}{135}{1}{$\\rho(H)$}\n    \\end{camodel}\n    \\begin{camodel}[yshift=-2.75cm,every carc\/.style={},cabase=.75cm]\n      \\carc[nodepos=.25]{180}{0}{2}{$\\rho(A)$}\n      \\carc{90}{0}{3}{$\\rho(B)$}\n      \\carc[labelangle=-30]{45}{-45}{1}{$\\rho(H)$}\n      \\carc[dashed]{-45}{-215}{1}{}\n    \\end{camodel}\n  \\end{tikzpicture}\\hfill\n  \\begin{tikzpicture}[baseline=0cm]\n    \\begin{camodel}[xshift=5cm,every carc\/.style={},cabase=.75cm]\n      \\node[anchor=west] at (-2.25cm,1.75cm) {$B\\subset H$:};\n      \\carc[nodepos=.25]{180}{0}{2}{$\\rho(A)$}\n      \\carc{90}{0}{3}{$\\rho(B)$}\n      \\carc{135}{0}{1}{}\n      \\carc[dashed]{280}{135}{1}{$\\rho(H)$}\n    \\end{camodel}\n    \\begin{camodel}[xshift=5cm,yshift=-2.75cm,every carc\/.style={},cabase=.75cm]\n      \\carc[nodepos=.25]{180}{0}{2}{$\\rho(A)$}\n      \\carc{90}{0}{3}{$\\rho(B)$}\n      \\carc[labelangle=-30]{90}{-45}{1}{$\\rho(H)$}\n      \\carc[dashed]{-45}{-215}{1}{}\n    \\end{camodel}\n  \\end{tikzpicture}\\hfill\n  \\begin{tikzpicture}[baseline=0cm]\n    \\begin{camodel}[xshift=10cm,every carc\/.style={},cabase=.75cm]\n      \\node[anchor=west] at (-2.25cm,1.75cm) {$H\\subset B$:};\n      \\carc[nodepos=.25]{180}{0}{2}{$\\rho(A)$}\n      \\carc{90}{0}{3}{$\\rho(B)$}\n      \\carc[swap]{55}{0}{1}{$\\rho(H)$}\n    \\end{camodel}\n  \\end{tikzpicture}\n  \\caption{Proof of Lemma~\\protect\\ref{lem:unique}.4, case 2: $B\\subset A$ and\n    $\\rho(B)$ intersects~$\\rho(A)$ clockwise.}\\label{fig:proof:unique4b}\n\\end{figure}\n\n\n\\Case2{$A\\supset B$.}\nWithout loss of generality, we suppose that $\\rho(B)$ intersects~$\\rho(A)$\nclockwise; see Fig.~\\ref{fig:proof:unique4b}.\nIf $B\\between^*H$, then $\\rho(H)$ intersects~$\\rho(B)$ counter-clockwise\nexactly when the familiar condition~\\refeq{ccw} holds true.\nIf $B\\subset H$, then $\\rho(B)$ intersects~$\\rho(H)$ clockwise\nexactly under the same condition. Thus, these two subcases do not differ much\nfrom the corresponding subcases of Case 1.\nIf $H\\subset B$, then $\\rho(H)$~is forced to intersect~$\\rho(B)$ clockwise\nby the condition that $\\{\\rho(A),\\rho(B),\\rho(H)\\}$ is a tight arc system.\n\n\\begin{figure}\n  \\centering\n  \\begin{tikzpicture}[baseline=0pt]\n    \\begin{camodel}[every carc\/.style={},cabase=.75cm]\n      \\node[anchor=west] at (-2.25cm,1.75cm) {$B\\between^*H$:};\n      \\carc[nodepos=.25]{180}{0}{2}{$\\rho(B)$}\n      \\carc{90}{0}{3}{$\\rho(A)$}\n      \\carc[labelpos=.4]{280}{135}{1}{$\\rho(H)$}\n      \\carc[dashed]{135}{45}{1}{}\n    \\end{camodel}\n    \\begin{camodel}[yshift=-2.75cm,every carc\/.style={},cabase=.75cm]\n      \\carc[nodepos=.25]{180}{0}{2}{$\\rho(B)$}\n      \\carc{90}{0}{3}{$\\rho(A)$}\n      \\carc{50}{-100}{1}{$\\rho(H)$}\n      \\carc[dashed]{90}{50}{1}{}\n    \\end{camodel}\n  \\end{tikzpicture}\\hfill\n  \\begin{tikzpicture}[baseline=0pt]\n    \\begin{camodel}[every carc\/.style={},cabase=.75cm]\n      \\node[anchor=west] at (-2.25cm,1.75cm) {$H\\subset B$:};\n      \\carc[nodepos=.25]{180}{0}{2}{$\\rho(B)$}\n      \\carc{90}{0}{3}{$\\rho(A)$}\n      \\carc[swap]{180}{130}{1}{$\\rho(H)$}\n      \\carc[dashed]{130}{45}{1}{}\n    \\end{camodel}\n    \\begin{camodel}[yshift=-2.75cm,every carc\/.style={},cabase=.75cm]\n      \\carc[nodepos=.25]{180}{0}{2}{$\\rho(B)$}\n      \\carc{90}{0}{3}{$\\rho(A)$}\n      \\carc[swap]{50}{0}{1}{$\\rho(H)$}\n      \\carc[dashed]{135}{50}{1}{}\n    \\end{camodel}\n  \\end{tikzpicture}\\hfill\n  \\begin{tikzpicture}[baseline=0pt]\n    \\begin{camodel}[every carc\/.style={},cabase=.75cm]\n      \\node[anchor=west] at (-2.25cm,1.75cm) {$B\\subset H$:};\n      \\carc[nodepos=.25]{180}{0}{2}{$\\rho(B)$}\n      \\carc{90}{0}{3}{$\\rho(A)$}\n      \\carc[nodepos=.25]{270}{0}{1}{$\\rho(H)$}\n    \\end{camodel}\n  \\end{tikzpicture}\n  \\caption{Proof of Lemma~\\protect\\ref{lem:unique}.4, case 3: $A\\subset B$ and\n    $\\rho(A)$ intersects~$\\rho(B)$ clockwise.}\\label{fig:proof:unique4c}\n\\end{figure}\n\n\n\\Case3{$A\\subset B$.}\nWithout loss of generality, we suppose that $\\rho(A)$ intersects~$\\rho(B)$\nclockwise; see Fig.~\\ref{fig:proof:unique4c}.\nIf $B\\between^*H$, then $\\rho(H)$ intersects~$\\rho(B)$ clockwise\niff $H\\cap B\\subseteq A$.\nIf $H\\subset B$, then $\\rho(H)$ intersects~$\\rho(B)$ clockwise\niff the sets~$H$ and~$A$ are comparable under inclusion.\nFinally, if $B\\subset H$, then $\\rho(B)$~is forced to intersect~$\\rho(H)$ clockwise\nby the tightness condition.\n\\end{proof}\n\n\n\n\n\n\n\\section{The neighborhood hypergraphs}\n\\label{s:Nhgs}\n\n\nLet $\\ensuremath{\\mathcal{A}}$ be a family of arcs on the circle $\\mathbb{C}_m$.\nA bijection $\\alpha\\function{V(G)}{\\ensuremath{\\mathcal{A}}}$ is an \\emph{arc representation\nof a graph $G$} if two vertices $u$ and $v$ are adjacent in $G$ exactly when\nthe arcs $\\alpha(u)$ and $\\alpha(v)$ intersect.\nA representation $\\alpha$ is \\emph{proper} if $\\alpha(u)\\subseteq\\alpha(v)$\nfor no two vertices $u$ and $v$. Restriction to \\emph{intervals}\nin the linearly ordered set $\\{1,\\ldots,m\\}$ gives the notion of\na \\emph{proper interval representation} of $G$.\nGraphs having such intersection representations\nare known as \\emph{proper interval} and \\emph{proper circular-arc (PCA) graphs}.\n\nThe aim of this section is to prove the rigidity properties for the closed\nneighborhood hypergraphs of PCA graphs that will be given in\nTheorem~\\ref{thm:Nrigid}.\n\nRecall that a proper arc (resp.\\ interval)\nrepresentation $\\alpha\\function{V(G)}{\\ensuremath{\\mathcal{A}}}$ of a graph~$G$\ndetermines the circular (resp.\\ linear) geometric order~$\\prec_\\alpha$ on the vertex set $V(G)$\naccordingly to the appearance of the left (or, equivalently, right)\nendpoints of the arcs $\\alpha(v)$, $v\\in V(G)$, in the circle $\\mathbb{C}_m$. \nThe following lemma implies that the closed neighborhood hypergraph of any PCA\ngraph admits a tight arc ordering.\n\n\\begin{lemma}[see~\\cite{fsttcs}]\\label{lem:geomistight}\n  The geometric order~$\\prec_\\alpha$ on~$V(G)$ associated with a proper\n  arc (resp.\\ interval) representation~$\\alpha$ of a graph~$G$ is a tight arc\n(resp.\\ interval) ordering of the hypergraph~$\\ensuremath{\\mathcal{N}}[G]$.\n\\end{lemma}\n\n\n\n\nFor the remainder of this section, we will consider two subclasses of PCA\ngraphs, namely those with bipartite and those with non-bipartite complement.\n The \\emph{complement of a graph~$G$} is the graph~$\\overline{G}$ with\n $V(\\overline{G})=V(G)$\n such that two vertices are adjacent in~$\\overline{G}$\n if and only if they are not adjacent in~$G$. \n\nIn what follows, we will repeatedly need the following property of\nnon-co-bipartite PCA graphs.\nA vertex~$u$ is \\emph{universal} if $N[u]=V(G)$.\n\n\\begin{lemma}\\label{lem:nouniv}\n  A PCA graph $G$ with non-bipartite complement\ncontains no universal vertex.\n\\end{lemma}\n\n\\begin{proof}\nLet~$\\alpha\\colon V(G)\\to\\ensuremath{\\mathcal{A}}$ be a\n    proper arc representation of~$G$ and \ndenote~$\\alpha(u)=[a^-,a^+]$. \nNotice now that $N[u]$ is\n    covered by two cliques $\\setdef{x\\in\n      N[u]}{a^-\\in\\alpha(x)}$ and $\\setdef{x\\in\n      N(u)}{a^+\\in\\alpha(x)}$,\nwhich excludes $N[u]=V(G)$ as $\\overline{G}$ is not bipartite.  \n\\end{proof}\n\nIf $\\prec$ is an arc ordering of the hypergraph $\\ensuremath{\\mathcal{N}}[G]$ for\na non-co-bipartite PCA graph $G$, the closed neighborhood $N[u]$\nof a vertex $u$ is an arc on the directed cycle $(V(G),\\prec)$.\nLemma~\\ref{lem:nouniv} allows us to use notation $N[u]=[u^-,u^+]$,\nsince the left endpoint $u^-$ and the right endpoint $u^+$\nare uniquely determined.\n\nIn the following three lemmas, we establish several facts about arc orderings of\nnon-co-bipartite PCA graphs.\n\n\\begin{lemma}\\label{lem:twocliques}\nLet $G$ be a PCA graph with non-bipartite complement.\nFor any arc ordering $\\prec$ of~$\\ensuremath{\\mathcal{N}}[G]$,\nevery vertex $u\\in V(G)$ has the following property:\n$u$ divides $N[u]=[u^-,u^+]$ into two parts $[u^-,u]$~and~$[u,u^+]$\nthat both are cliques in~$G$.\n\\end{lemma}\n\n\\begin{proof}\n    \\begin{figure} \\centering\n      \\begin{tikzpicture}[baseline=0cm]\n  \\node at (-1.5cm,1.5cm) {(a)};\n    \\begin{camodel}[cabase=.75cm,castep=.15cm,capoints=true]\n          \\carc[swap,nodeangle=20]{150}{410}{2}{$y$}\n          \\carc[swap,nodeangle=180]{-20}{200}{4}{$u$}\n          \\carc[nodeangle=0]{120}{390}{5}{$x$}\n        \\end{camodel}\n      \\end{tikzpicture}\n\\qquad\\qquad\n  \\begin{tikzpicture}[baseline=0cm]\n  \\node at (-1.5cm,1.5cm) {(b)};\n    \\begin{camodel}[cabase=.75cm,castep=.15cm]\n      \\carc[startlabel=$u^-$,endlabel=$u^+$,startlabelpos=inside,endlabelpos=inside]{180}{0}{1}{}\n      \\carc[swap,nodeangle=90,every label\/.style={inner sep=5pt}]{90}{90}{1}{$u$}\n       \\carc[swap,every label\/.style={inner sep=5pt,anchor=80}]{70}{70}{1}{$v$}\n      \\carc{180}{-80}{2}{}\n      \\carc{120}{0}{3}{}\n      \\carc{120}{-80}{4}{}\n    \\end{camodel}\n  \\end{tikzpicture}\n  \\caption{(a) Proof of Lemma~\\protect\\ref{lem:twocliques}: Vertices\n        $u\\in V(G)$, $x\\in D^+[u]$, and $y\\in D^+[u]\\cap[u,x]$ along\n        with their neighborhoods in $(V(G),\\prec)$.\n\\quad\n(b) Proof of Lemma~\\protect\\ref{lem:CA-order}.3:\nPossible mutual positions of~$N[u]$ and~$N[v]$. The most\n    inward arc $[u^-,u^+]$ represents~$N[u]$; the other four arcs show\n    possible positions of $[v^-,v^+]=N[v]$.}\n\\label{fig:geomistight+proof}\n    \\end{figure}\n Call a vertex $x\\in N[u]$ a\n    \\emph{close neighbor} of~$u$ if\n    \\[\n    x\\in [u^-,u]\\text{ and }u\\in [x,x^+]\\text{ or if }x\\in [u,u^+]\\text{ and\n    }u\\in [x^-,x]\n    \\]\n    and a \\emph{distant neighbor} otherwise. \n    Denote the sets of close neighbors of~$u$ in~$[u^-,u]$\n    and~$[u,u^+]$ by~$C^-[u]$ and~$C^+[u]$,\n    respectively. \nSimilarly,  the sets of distant neighbors of~$u$ in~$[u^-,u]$\n    and~$[u,u^+]$ will be denoted by~$D^-[u]$ and~$D^+[u]$. \nEach of the four sets~$C^-[u]$, $C^+[u]$, $D^-[u]$,\n    and~$D^+[u]$ is a clique.  Indeed, if for example $x$ and~$y$ are\n    two vertices in~$D^+(u)$, then $u$ belongs to both $[x,x^+]$\n    and~$[y,y^+]$. Since $x$ and $y$ are both in~$[u,u^+]$, this\n    implies that either $x$ belongs to~$[y,y^+]$ or $y$ belongs\n    to~$[x,x^+]$ (depending on whether $y\\in[u,x]$ or\n    $x\\in[u,y]$; see Fig.~\\ref{fig:geomistight+proof}). \nHence, $x$ and $y$ are adjacent and~$D^+(u)$ is a\n    clique. The other three cases are similar.  \n\n    To complete the proof we show that,\nif $\\overline{G}$ is non-bipartite, then\n    $D^-[u]=D^+[u]=\\emptyset$.  Assume to the contrary that $D^+[u]$\n    contains a vertex~$x$; see Fig.~\\ref{fig:geomistight+proof} (the case $x\\in\n    D^-[u]$ is similar).  Since $u\\in [x,x^+]$, the sets\n    $[u,u^+]\\cap[u,x]=[u,x]$ and $[x,x^+]\\cap[x,u]=[x,u]$ cover $V(G)$. Splitting\n    the former set into $C^+[u]\\cap[u,x]$ and $D^+[u]\\cap[u,x]$ and\n    the latter into $C^+[x]\\cap[x,u]$ and $D^+[x]\\cap[x,u]$, consider\n    the cover of~$V(G)$ by two sets\n    $(C^+[u]\\cap[u,x])\\cup(D^+[x]\\cap[x,u])$ and\n    $(D^+[u]\\cap[u,x])\\cup(C^+[x]\\cap[x,u])$ and show that they are\n    cliques.  This will give us a contradiction since $\\overline{G}$~is not\n    bipartite.  By symmetry, it suffices to prove that\n    $(D^+[u]\\cap[u,x])\\cup(C^+[x]\\cap[x,u])$ is a clique. Since both\n    $D^+[u]$ and~$C^+[x]$ are cliques, we have to show that any\n    vertex~$y$ in $D^+[u]\\cap[u,x]$ is adjacent to all vertices in\n    $C^+[x]\\cap[x,u]$.  This is true because we have $\n    N[y]\\supseteq[y,u]\\supseteq[x,u] $ by the definition of~$D^+[u]$.\n\\end{proof}\n\nLemma~\\ref{lem:twocliques} allows us to derive the following lemma, which will\nbe needed for the main rigidity result of this section and also is of\nindependent interest.\n\n\\begin{lemma}\\label{lem:anyistight}\nIf $G$ is a PCA graph with non-bipartite complement, \nthen any arc ordering $\\prec$ of~$\\ensuremath{\\mathcal{N}}[G]$ is tight.\n\\end{lemma}\n\nThis complements Lemma~\\ref{lem:geomistight}, which implies that there are tight\narc orderings of $\\ensuremath{\\mathcal{N}}[G]$.\n\n\\begin{proof}\nLet $N[u]\\subseteq N[v]$.\nConsider the arcs $N[u]=[u^-,u^+]$ and $N[v]=[v^-,v^+]$ w.r.t.~$\\prec$.\nSuppose that $u\\in [v,v^+]$ (the case that $u\\in [v^-,v]$ is symmetric).\nBy Lemma~\\ref{lem:twocliques}, $[v,v^+]$ is a clique in $G$.\nTherefore, this arc is contained in~$N[u]$, which implies\nthat $u^+=v^+$. \n\\end{proof}\n\nWith the next lemma we show that all arc orderings for closed neighborhood\nhypergraph of non-co-bipartite PCA graphs follow several simple patterns.\n\n\\begin{lemma}\\label{lem:CA-order}\nLet $G$ be a PCA graph with non-bipartite complement.\nFor any arc ordering $\\prec$ of~$\\ensuremath{\\mathcal{N}}[G]$ and any vertices $u,v\\in V(G)$,\nthe following conditions are met.\n  \\begin{bfenumerate}\n  \\item $v\\in [u,u^+]$ if and only if $u\\in [v^-,v]$.\n  \\item If $v\\in [u,u^+]$, then $v^-\\in [u^-,u]$ and $u^+\\in [v,v^+]$.\n  \\item If $u\\prec v$ and these vertices are adjacent, then $u^-$, $v^-$,\n    $u$, $v$, $u^+$, and $v^+$ occur under the\n    order~$\\prec$ exactly in this circular sequence, where some\n    of the neighboring vertices except $u^-$~and~$v^+$ may coincide.\n  Moreover, it is impossible that $v^+\\prec u^-$.\n  \\end{bfenumerate}\n\\end{lemma}\n\n\\begin{proof}\n {\\bf 1.} Let $u\\ne v$ and assume that $v\\in [u,u^+]$. If $u\\in\n    [v,v^+]$, then Lemma~\\ref{lem:twocliques} shows that \n$V(G)$ is covered by two cliques\n    $[u,u^+]$~and~$[v,v^+]$, contradicting the assumption\n    that~$\\overline{G}$~is not bipartite. Therefore, $u\\in [v^-,v]$. The\n    other implication follows by symmetry.\n  \n{\\bf 2.} If the two conditions in~part~1 are true, then $v^-$,\n    $u$, and $v$ occur in this circular order. Since $[v^-,v]$ is a\n    clique, all vertices in $[v^-,u)$ are adjacent to~$u$ and hence,\n    $v^-\\in [u^-,u]$. The second containment follows\n    by symmetry.\n  \n{\\bf 3.} Parts~1~and~2 imply that $u^-$, $v^-$, $u$, $v$,\n    $u^+$, and $v^+$ occur in this circular order; see Fig.~\\ref{fig:geomistight+proof}.\nHowever, it is still not excluded $v^+$~and~$u^-$  can coincide or be swapped.\nWe have to show that the condition $u\\prec v$ rules out the last two possibilities\nas well as the possibility of $v^+\\prec u^-$. Indeed, any of these configurations\nwould give covering of $V(G)$ by two cliques $[u^-,u]$ and~$[v,v^+]$. \n\\end{proof}\n\nThe following theorem allows us to invoke Theorem~\\ref{thm:unique}.2 for the\nclosed neighborhood hypergraphs of connected non-co-bipartite PCA graphs,\nproving that these hypergraphs have a unique tight arc ordering.\n\n\\begin{theorem}\\label{thm:strict-conn}\n  If $G$~is a connected PCA graph with non-bipartite complement,\nthen $\\ensuremath{\\mathcal{N}}[G]$~is strictly connected.\n\\end{theorem}\n\n\\begin{proof}\n  Let $\\alpha$ be a proper arc representation of~$G$.\nBy Lemma~\\ref{lem:geomistight}, $\\ensuremath{\\mathcal{N}}[G]$~has an arc \n(geometric) ordering $\\prec_\\alpha$.\nSince the complement of $G$ is not bipartite,\n$G$ has at least two vertices and, by Lemma~\\ref{lem:nouniv}, no universal vertex.\nSince $G$~is connected, there is at most one pair of non-adjacent vertices~$x$ and~$y$\nsatisfying the relation $x\\prec_\\alpha y$. Therefore, all vertices of~$G$\ncan be arranged into a path $v_1,\\ldots,v_n$ such that $v_i$ and~$v_{i+1}$\nare adjacent and $v_i\\prec_\\alpha v_{i+1}$ for every $1\\le i<n$.\nBy Lemma~\\ref{lem:CA-order}.3, we have $N[v_i]\\bowtie^* N[v_{i+1}]$,\nwhich gives us a strictly connected path passing through all hyperedges of~$\\ensuremath{\\mathcal{N}}[G]$.\n\\end{proof}\n\nNow we are ready to prove our rigidity result for the closed neighborhood\nhypergraphs of PCA graphs.\n\n\\begin{theorem}\\label{thm:Nrigid}\nLet $G$ be a twin-free, connected PCA graph.\n\\begin{bfenumerate}\n\\item \nIf $\\overline{G}$ is non-bipartite, then $\\ensuremath{\\mathcal{N}}[G]$ has, up to reversal,\na unique arc ordering.\n\\item \nIf $\\overline{G}$ is bipartite and connected, then $\\ensuremath{\\mathcal{N}}[G]$ has, up to reversal,\nexactly two tight arc orderings.\n\\end{bfenumerate}\n\\end{theorem}\n\n\\begin{proof}\n  {\\bf 1.}\n$\\ensuremath{\\mathcal{N}}[G]$ has an arc ordering by Lemma~\\ref{lem:geomistight}.\nBy Lemma~\\ref{lem:anyistight}, any arc ordering of $\\ensuremath{\\mathcal{N}}[G]$\nis tight. The uniqueness follows from Theorem~\\ref{thm:strict-conn}\nby Theorem~\\ref{thm:unique}.2.\n\n{\\bf 2.} \nThe \\emph{open neighborhood hypergraph} of a graph $G$ is defined by\n$\\ensuremath{\\mathcal{N}}(G)=\\{N(v)\\}_{v\\in V(G)}$.\nIn place of $\\ensuremath{\\mathcal{N}}[G]$, it is now practical to consider\nthe complement hypergraph $\\overline{\\ensuremath{\\mathcal{N}}[G]}=\\Set{V(G)\\setminus N[v]}_{v\\in V(G)}$.\nNote that $\\overline{\\ensuremath{\\mathcal{N}}[G]}=\\ensuremath{\\mathcal{N}}(\\overline{G})$.\nThis hypergraph is disconnected, and any (tight) arc ordering\nof $\\ensuremath{\\mathcal{N}}[G]$ induces a (tight) interval ordering of\neach connected component of $\\ensuremath{\\mathcal{N}}(\\overline{G})$.\nConversely, arbitrary (tight) interval orderings\nof the components of $\\ensuremath{\\mathcal{N}}(\\overline{G})$ can be merged\ninto a (tight) arc ordering of $\\ensuremath{\\mathcal{N}}[G]$.\nSince $\\overline{G}$ is connected, $\\ensuremath{\\mathcal{N}}(\\overline{G})$ has exactly two components.\nApplying Theorem~\\ref{thm:unique}.1 to each of them, we \nconclude that each of the two components has, up to reversing,\na single interval ordering. Since there are exactly two essentially\ndifferent ways to merge them, we see \nthat $\\ensuremath{\\mathcal{N}}[G]$ has, up to reversing, exactly two tight arc orderings.\n\\end{proof}\n\nLet us stress that part 2 of Theorem~\\ref{thm:Nrigid} concerns \nonly tight orderings. To show that it\ncannot be strengthened to the class of all arc orderings,\nconsider the \\emph{half-graph}~$H_m$ that is the bipartite graph\nwith vertex classes $\\{u_1,\\ldots,u_m\\}$ and $\\{v_1,\\ldots,v_m\\}$\nwhere $u_i$ is adjacent to $v_j$ if $i\\le j$.\nThe complement $G_m=\\overline{H_m}$ is a twin-free, connected PCA graph.\nNote that, besides two pairs of mutually reversed tight interval orderings,\nthe hypergraph $\\ensuremath{\\mathcal{N}}(H_3)$ has another interval ordering\nand, hence, $\\ensuremath{\\mathcal{N}}[G_3]$ has a non-tight arc ordering.\nIf we increase the parameter $m$, the number\nof non-tight arc orderings of $\\ensuremath{\\mathcal{N}}[G_m]$ grows exponentially.\n\n\n\n\\section{Overlap-connectedness}\\label{s:ov-conn}\n\nLet $G$ be a twin-free, connected PCA graph with non-bipartite complement.\nIn Theorem~\\ref{thm:strict-conn} we established that $\\ensuremath{\\mathcal{N}}[G]$\nis strictly connected. This implies the uniqueness of a tight arc\nordering of this hypergraph by Theorem~\\ref{thm:unique}.2.\nSince by Lemma~\\ref{lem:anyistight} any arc ordering is actually tight,\nwe obtain the uniqueness for the class of all arc orderings.\n\n\\begin{figure}\n  \\centering\n  \\begin{tikzpicture}[baseline=0cm]\n    \\node at (-1.75cm,1.75cm) {(a)};\n    \\begin{camodel}[cabase=.75cm,castep=.175cm]\n      \\carc{140}{40}{1}{$A$}\n      \\carc{-40}{-140}{2}{\\strut$X$}\n      \\carc[labelpos=.67]{50}{-50}{3}{$\\!C$}\n      \\carc[labelpos=.33]{230}{130}{3}{$Y$}\n      \\carc[swap,labelpos=.67]{60}{-5}{2}{$B$}\n      \\carc[swap,labelpos=.33]{185}{120}{2}{$Z$}\n    \\end{camodel}\n  \\end{tikzpicture}\\hfil\n  \\begin{tikzpicture}[baseline=0cm]\n    \\node at (-1.5cm,1.75cm) {(b)};\n    \\path[every node\/.style={circle,fill,inner sep=1pt},every label\/.append style=rectangle]\n        (90:1cm) node[label=above:$a$] (a)   {}\n        (30:1cm) node[label=right:\\strut$b$] (b)   {} edge (a)\n        (-30:1cm) node[label=right:$c$] (c) {} edge (a) edge (b)\n        (-90:1cm) node[label=below:$x$] (x) {} edge (c)\n        (-150:1cm) node[label=left:\\strut$y$] (y) {} edge (a) edge (x)\n        (150:1cm) node[label=left:$z$] (z) {} edge (a) edge (y);\n  \\end{tikzpicture}\\hfil\n  \\begin{tikzpicture}[baseline=0cm]\n    \\node at (-2cm,1.75cm) {(c)};\n    \\begin{camodel}[capoints=true,cabase=.75cm,castep=.175cm,\n      spaced\/.style={every label\/.append style={inner sep=4pt}},\n      close\/.style={every label\/.append style={inner sep=1pt}}]\n      \\carc[nodeangle=90,thick,swap,spaced]{210}{-30}{1}{$a$}\n      \\carc[nodeangle=-90,swap,spaced]{-30}{-150}{2}{$x$}\n      \\carc[nodeangle=150,swap,close]{210}{90}{3}{$z$}\n      \\carc[nodeangle=30,swap,close]{90}{-30}{4}{$b$}\n      \\carc[nodeangle=-150]{-90}{-270}{5}{$y$}\n      \\carc[nodeangle=-30]{90}{-90}{6}{$c$}\n    \\end{camodel}\n  \\end{tikzpicture}%\n  \\caption{An example: (a) A proper arc system; (b) The corresponding intersection graph~$G$.\nIts complement $\\overline{G}$~is non-bipartite; (c) The closed neighborhood hypergraph~$\\ensuremath{\\mathcal{N}}[G]$ is not strictly\noverlap-connected: the hyperedge~$N[a]$ forms a single component.\nNevertheless, since after removal of~$N[a]$ the hypergraph~$\\ensuremath{\\mathcal{N}}[G]$ stays twin-free\nand becomes strictly overlap-connected, it has a unique, up to reversal, arc ordering.}\\label{fig:example}\n\\end{figure}\n\nThe same conclusion could be derived more directly\nby Theorem~\\ref{thm:unique-overlap-2} when $\\ensuremath{\\mathcal{N}}[G]$\nwould be strictly overlap-connected.\nHowever, the last condition is not always true\nbecause $\\ensuremath{\\mathcal{N}}[G]$ can have hyperedges of size $n-1$ and\neach such hyperedge forms a separate\nstrictly overlap-connected component; see an example in Fig.~\\ref{fig:example}.\nNevertheless, if we remove the $(n-1)$-element hyperedge\nfrom $\\ensuremath{\\mathcal{N}}[G]$ in this example, the remaining hypergraph becomes\nstrictly overlap-connected and stays twin-free.\nIt turns out that this is a general phenomenon.\nIn fact, we derive this result from the uniqueness of an arc ordering, \nthat we already established for $\\ensuremath{\\mathcal{N}}[G]$ in the preceding section,\nand a criterion of uniqueness given below.\n\n\n\\begin{lemma}\\label{lem:ov-conn-gen}\nGiven a hypergraph~$\\ensuremath{\\mathcal{H}}$ on $n\\ge4$ vertices, let $\\ensuremath{\\mathcal{H}}'$ \nbe the hypergraph on the same vertex set \nobtained from~$\\ensuremath{\\mathcal{H}}$ by removing all hyperedges of size $0$, $1$, $n-1$, and~$n$.\nThen $\\ensuremath{\\mathcal{H}}$~has a unique, up to reversing, arc ordering if and only if \n$\\ensuremath{\\mathcal{H}}'$~is twin-free and either is strictly overlap-connected\nor has a single isolated vertex and becomes strictly overlap-connected\nafter its removal.\n\\end{lemma}\n\n\n\n\\begin{proof}\nAssume that $\\ensuremath{\\mathcal{H}}'$~is twin-free and either is strictly overlap-connected\nor becomes so after removing a single isolated vertex.\nBy Theorem~\\ref{thm:unique-overlap-2}, $\\ensuremath{\\mathcal{H}}'$~has a unique arc ordering. \nRecall that uniqueness is always meant up to reversal.\nThis holds also for~$\\ensuremath{\\mathcal{H}}$ because the two hypergraphs have\nthe same arc orderings.\n\nLet us prove the lemma in the other direction.\nAssume that $\\ensuremath{\\mathcal{H}}$~has a unique arc ordering.\nSince $n\\ge4$, $\\ensuremath{\\mathcal{H}}$~has no twins (for else transposition of two twins would give\nus another arc ordering).\n\nFix an arbitrary vertex~$x$ of~$\\ensuremath{\\mathcal{H}}$. For a hyperedge $H\\in\\ensuremath{\\mathcal{H}}$,\nset $H_x=H$ if $x\\notin H$ and $H_x=V(\\ensuremath{\\mathcal{H}})\\setminus H$ otherwise.\nDefine the interval hypergraph $\\ensuremath{\\mathcal{H}}_x=\\Set{H_x}_{H\\in\\ensuremath{\\mathcal{H}}}$ \nwhere any empty hyperedge $H_x=\\emptyset$ is removed.\nIf vertices~$u$ and~$v$ are distinguished by the incidence relation to a hyperedge~$H$, \nthey are distinguished as well by the complement of $H$. This shows that $\\ensuremath{\\mathcal{H}}_x$~is, like $\\ensuremath{\\mathcal{H}}$, twin-free.\nIn particular, $x$~is the only isolated vertex of~$\\ensuremath{\\mathcal{H}}_x$.\nLet $\\ensuremath{\\mathcal{H}}_x^\\circ$ denote the hypergraph obtained from~$\\ensuremath{\\mathcal{H}}_x$ by removing the vertex~$x$.\n\nAny arc ordering for $\\ensuremath{\\mathcal{H}}_x$~is also an arc ordering for~$\\ensuremath{\\mathcal{H}}$.\nTherefore $\\ensuremath{\\mathcal{H}}_x$~has a unique arc ordering.\nIt readily follows that $\\ensuremath{\\mathcal{H}}_x^\\circ$~has a unique interval ordering.\nThe set of all interval orderings of a given hypergraph is described in~\\cite{KoeblerKLV11}\nin terms of the tree of its overlap-connected components, which is an analog of\nthe classical structure known as a \\emph{PQ-tree}~\\cite{BoothL76}. \nThis description is based on an observation that any two overlap-connected components\nof a hypergraph either are vertex-disjoint or one is contained in a twin-class of the other.\nWith respect to this containment relation, the overlap-connected components\nform a directed forest. By Theorem~\\ref{thm:unique-overlap-1}, every overlap-connected component\nadmits a unique interval ordering, and we have a freedom to reverse it within each of the\ncomponents. It follows that $\\ensuremath{\\mathcal{H}}_x^\\circ$~is connected \nand that the root overlap-connected component $\\ensuremath{\\mathcal{R}}$ of~$\\ensuremath{\\mathcal{H}}_x^\\circ$ is twin-free.\nTherefore, any other overlap-connected component of~$\\ensuremath{\\mathcal{H}}_x^\\circ$\nmust consist of a single one-vertex hyperedge (possibly obtained by complementing\na $(n-1)$-vertex hyperedge of~$\\ensuremath{\\mathcal{H}}$).\nIt follows that $\\ensuremath{\\mathcal{R}}$\nis equal to the hypergraph~$(\\ensuremath{\\mathcal{H}}')_x^\\circ$ obtained from~$\\ensuremath{\\mathcal{H}}'$ by complementing\nall hyperedges that contain~$x$ and removing~$x$.\nThus, $(\\ensuremath{\\mathcal{H}}')_x^\\circ$ is twin-free and overlap-connected.\nSince $(\\ensuremath{\\mathcal{H}}')_x$~is obtainable from the connected hypergraph~$(\\ensuremath{\\mathcal{H}}')_x^\\circ$ by adding\nan isolated vertex, $(\\ensuremath{\\mathcal{H}}')_x$~is twin-free as well.\nBy the same argument as used above for~$\\ensuremath{\\mathcal{H}}$ and~$\\ensuremath{\\mathcal{H}}_x$, the hypergraph~$\\ensuremath{\\mathcal{H}}'$ must be twin-free too.\nIn particular, $\\ensuremath{\\mathcal{H}}'$~has at most one isolated vertex.\n\nNote that if hyperedges~$H_x$ and~$K_x$ of the hypergraph~$(\\ensuremath{\\mathcal{H}}')_x$\noverlap, then they strictly overlap in the full circle $V(\\ensuremath{\\mathcal{H}})$ and, therefore,\nthe corresponding hyperedges~$H$ and~$K$ of the hypergraph~$\\ensuremath{\\mathcal{H}}'$\nstrictly overlap. It now follows from the overlap-connectedness of~$(\\ensuremath{\\mathcal{H}}')_x^\\circ$ that\nany two hyperedges in~$\\ensuremath{\\mathcal{H}}'$ are connected by a $\\between^*$-path.\nThis readily implies that $\\ensuremath{\\mathcal{H}}'$~is strictly overlap-connected if it has no isolated\nvertex or becomes such after removal of the (single) isolated vertex.\n\\end{proof}\n\nRecall that $\\ensuremath{\\mathcal{K}}$ is a \\emph{spanning subhypergraph} of\na hypergraph~$\\ensuremath{\\mathcal{H}}$ if\n$V(\\ensuremath{\\mathcal{K}})=V(\\ensuremath{\\mathcal{H}})$ and any hyperedge of~$\\ensuremath{\\mathcal{K}}$ is a hyperedge of~$\\ensuremath{\\mathcal{H}}$.\n\n\\begin{theorem}\\label{thm:ov-conn-str}\nIf $G$~is a twin-free, non-co-bipartite, connected PCA graph on~$n$ vertices, then\n the spanning subhypergraph of~$\\ensuremath{\\mathcal{N}}[G]$ obtained by removing\nall hyperedges of size $n-1$ is twin-free and strictly overlap-connected.\n\\end{theorem}\n\n\\begin{proof}\nAssume that a graph~$G$ satisfies all the assumptions.\nNote that then $G$~has $n\\ge4$ vertices.\nDenote $\\ensuremath{\\mathcal{H}}=\\ensuremath{\\mathcal{N}}[G]$. \nSince $G$~is connected, $\\ensuremath{\\mathcal{H}}$~has no hyperedge of size~$1$. \nIt also has no hyperedge of size $n$\nbecause $G$~has no universal vertex by Lemma~\\ref{lem:nouniv}.\nRemove all hyperedges of size $n-1$ and denote the result by~$\\ensuremath{\\mathcal{H}}'$.\nWe have to show that $\\ensuremath{\\mathcal{H}}'$~is twin-free and strictly overlap-connected.\n\nBy Theorem~\\ref{thm:Nrigid}.1,\n$\\ensuremath{\\mathcal{H}}$~has a unique, up to reversing, arc ordering.\nBy Lemma~\\ref{lem:ov-conn-gen},\n$\\ensuremath{\\mathcal{H}}'$~is twin-free and strictly overlap-connected unless it has an isolated vertex.\nIt remains to exclude the last possibility.\n\nSuppose that all hyperedges of~$\\ensuremath{\\mathcal{H}}$ containing a vertex~$v$ are of size $n-1$.\nIn particular, $N[v]=V(G)\\setminus\\{v'\\}$, where $v'$~is a unique vertex non-adjacent\nto~$v$. Note that $v$~is contained in the hyperedge~$N[u]$ for any $u\\ne v'$.\nFor each of these hyperedges we, therefore, have $|N[u]|=n-1$.\nMoreover, $v'\\in N[u]$ for else $u$ and~$v$ would be twins.\nIt follows that $|N[v']|=n-1$ as well. We conclude that $\\overline{G}$~is a perfect matching,\ncontradicting the assumptions that $\\overline{G}$~is not bipartite.  \n\\end{proof}\n\nNote that Theorem~\\ref{thm:ov-conn-str} gives us a complete description of the decomposition of~$\\ensuremath{\\mathcal{N}}[G]$\ninto strictly overlap-connected components because each hyperedge  of size $n-1$ forms such a \ncomponent alone. Figure~\\ref{fig:manylargehyperedges} shows that the number of\nsuch single-hyperedge components can be linear.\n\n\\begin{figure}\n  \\centering\n  \\numdef\\k{3}\n  \\numdef\\n{3*\\k-1}\n  \\numdef\\p{2*\\k}\n  \\numdef\\km{\\k-1}\n  \\numdef\\rot{-180\/\\p}\n  \\dimdef\\base{1.2cm}\n  \\begin{tikzpicture}\n    \\begin{camodel}[cabase=\\base,caanglestep=360\/\\p,castep=.1cm]\n      \\node[anchor=east] at (-2,2) {$\\ensuremath{\\mathcal{A}}_3$};\n      \\begin{scope}[rotate=\\rot]\n        \\begin{pgfonlayer}{background}\n          \\draw[black!25,line width=3mm,line cap=round]\n          (360\/\\p:1cm+\\base) -- (360\/\\p:\\base)\n          node[black,anchor=\\rot+360\/\\p,inner sep=1pt] {$u_1$};\n        \\end{pgfonlayer}\n        \\CArc{start=1,end=(\\k-1),startlevel=2,endlevel=5.5}\n        \\foreach \\i in {2,...,\\k} {%\n          \\begin{pgfonlayer}{background}\n            \\draw[black!25,line width=3mm,line cap=round]\n            (\\i*360\/\\p:1cm+\\base) -- (\\i*360\/\\p:\\base)\n            node[black,anchor=\\rot+\\i*360\/\\p,inner sep=1pt] {$u_\\i$};\n          \\end{pgfonlayer}\n          \\CArc{start=\\i,end=(\\k+\\i-1),startlevel=2,endlevel=(3*\\k)}\n        }\n        \\foreach \\i in {1,...,\\k} {%\n          \\begin{pgfonlayer}{background}\n            \\draw[black!25,line width=3mm,line cap=round]\n            ({(\\k+\\i)*360\/\\p}:1cm+\\base) -- ({(\\k+\\i)*360\/\\p}:\\base)\n            node[black,anchor=\\rot+(\\k+\\i)*360\/\\p,inner sep=1pt] {$v_\\i$};\n          \\end{pgfonlayer}\n          \\CArc{start=(\\k+\\i),end=(\\p+\\i-1),startlevel=2,endlevel=(3*\\k)}\n        }\n        \\foreach \\i in {1,...,\\km} {%\n          \\begin{pgfonlayer}{background}\n            \\draw[black!25,line width=3mm,line cap=round]\n            ({(\\k+\\i+1\/2)*360\/\\p}:1cm+\\base) -- ({(\\k+\\i+1\/2)*360\/\\p}:\\base)\n            node[black,anchor=\\rot+(\\k+\\i+1\/2)*360\/\\p,inner sep=1pt] {$w_\\i$};\n          \\end{pgfonlayer}\n          \\CArc{start=(\\k+\\i+1\/2),end=(\\p+\\i-1),startlevel=2,endlevel=(3*\\k-1.5)}\n        }\n      \\end{scope}\n    \\end{camodel}\n    \\begin{camodel}[cabase=\\base,xshift=7cm,caanglestep=360\/\\p,castep=.1cm,capoints=true]\n      \\node[anchor=east] at (-2,2) {$\\ensuremath{\\mathcal{N}}[G_3]$};\n      \\begin{scope}[rotate=\\rot]\n        \\begin{pgfonlayer}{background}\n          \\draw[black!25,line width=3mm,line cap=round]\n          (360\/\\p:1.25cm+\\base) -- (360\/\\p:\\base)\n          node[black,anchor=\\rot+360\/\\p,inner sep=1pt] {$u_1$};\n        \\end{pgfonlayer}\n        \\CArc{start=(1+1-\\k),end=(\\k-1),startlevel=2,endlevel=(4*\\k-2.5),labelangle=1}\n        \\foreach \\i in {2,...,\\k} {%\n          \\begin{pgfonlayer}{background}\n            \\draw[black!25,line width=3mm,line cap=round]\n            (\\i*360\/\\p:1.25cm+\\base) -- (\\i*360\/\\p:\\base)\n            node[black,anchor=\\rot+\\i*360\/\\p,inner sep=1pt] {$u_\\i$};\n          \\end{pgfonlayer}\n          \\CArc{start=(\\i+1-\\k),end=(\\k+\\i-1),startlevel=2,endlevel=(4*\\k)} }\n        \\foreach \\i in {1,...,\\k} {%\n          \\begin{pgfonlayer}{background}\n            \\draw[black!25,line width=3mm,line cap=round]\n            ({(\\k+\\i)*360\/\\p}:1.25cm+\\base) -- ({(\\k+\\i)*360\/\\p}:\\base)\n            node[black,anchor=\\rot+(\\k+\\i)*360\/\\p,inner sep=1pt] {$v_\\i$};\n          \\end{pgfonlayer}\n          \\CArc{gray,start=(\\i+1),end=(\\p+\\i-1),startlevel=2,endlevel=(4*\\k)} }\n        \\foreach \\i in {1,...,\\km} {%\n          \\begin{pgfonlayer}{background}\n            \\draw[black!25,line width=3mm,line cap=round]\n            ({(\\k+\\i+1\/2)*360\/\\p}:1.25cm+\\base) -- ({(\\k+\\i+1\/2)*360\/\\p}:\\base)\n            node[black,anchor=\\rot+(\\k+\\i+1\/2)*360\/\\p,inner sep=1pt] {$w_\\i$};\n          \\end{pgfonlayer}\n          \\CArc{start=(\\i+2),end=(\\p+\\i-1),startlevel=3.25,endlevel=(4*\\k-1.25)}\n        }\n      \\end{scope}\n    \\end{camodel}\n  \\end{tikzpicture}\n  \\caption{A non-co-bipartite, twin-free, connected PCA graph on~$n$ vertices\n    can have more than $n\/3$ vertices of degree $n-2$ (a vertex~$v$ has degree $n-2$ iff\n    $\\left|N[v]\\right|=n-1$). For each $k\\ge2$, define a graph~$G_k$ on $3k-1$\n    vertices by its arc model~$\\ensuremath{\\mathcal{A}}_k$ such that $k$ of the vertices\nwill have degree $n-2$. On the circle\n    $\\mathbb{C}=\\{u_1,u_2,\\dotsc,u_k,v_1,w_1,v_2,w_2,\\dotsc,v_{k-1},w_{k-1},v_k\\}$,\n    whose points go in this circular order, consider arcs\n    $U_1=[u_1,u_{k-1}]$, $U_i=[u_i,v_{i-1}]$ for $2\\le i\\le k$, \n    $V_1=[v_1,v_k]$, $V_i=[v_i,u_{i-1}]=\\mathbb{C}\\setminus U_i$ for $2\\le i\\le k$, and\n    $W_i=V_i\\setminus\\{v_i\\}$ for $i\\le k-1$. As $\\ensuremath{\\mathcal{A}}_k$~is tight, it can easily\nbe made proper and, hence, $G_k$~is PCA. Since the arcs have pairwise\ndifferent left endpoints, we can identify $V(G_k)=\\mathbb{C}$. The graph can be described\nby listing the pairs of non-adjacent vertices, namely $v_iu_i$, $w_iu_i$, $w_iu_{i+1}$,\nand~$u_1u_k$. There are no twins, and $G_k$~is not co-bipartite\nbecause its complement contains an odd cycle, namely $u_1w_1u_2w_2\\ldots u_k$.\nIn accordance with Theorem~\\protect\\ref{thm:ov-conn-str}, \nwhen we remove from~$\\ensuremath{\\mathcal{N}}[G_k]$ the hyperedges $N[v_i]=\\mathbb{C}\\setminus\\{u_i\\}$,\nthe hypergraph stays twin-free and becomes strictly overlap-connected.\nThis can be seen by looking at the following path in the complementing hypergraph:\n$\\overline{N[u_1]}=\\{u_k,v_1,w_1\\}$, $\\overline{N[u_2]}=\\{w_1,v_2,w_2\\}$, \\ldots,\n$\\overline{N[u_{k-1}]}=\\{w_{k-2},v_{k-1},w_{k-1}\\}$, $\\overline{N[u_k]}=\\{w_{k-1},v_k,u_1\\}$,\n$\\overline{N[w_1]}=\\{u_1,u_2\\}$, \\ldots, $\\overline{N[w_{k-1}]}=\\{u_{k-1},u_k\\}$.\nThe figure shows~$\\ensuremath{\\mathcal{A}}_3$\n    and an arc model for~$\\ensuremath{\\mathcal{N}}[G_3]$ which has the arcs of size $n-1$ grayed\n    out.}\\label{fig:manylargehyperedges}\n\\end{figure}\n\n\n\n\n\\section{Intersection representations of graphs}\\label{s:repr}\n\nPart 1 of Theorem~\\ref{thm:Nrigid} implies that a twin-free, connected PCA graph $G$\nwith non-bipartite complement has a unique geometric order.\nIf $\\overline{G}$ is bipartite and connected, part~2 (combined with Lemma~\\ref{lem:geomistight})\nleaves two different possibilities. It turns out that, nevertheless, \na geometric order is unique also in this case. We state this fact\nin Theorem~\\ref{thm:Aunique2} below, proving an auxiliary lemma\nbeforehand.\n\n\\begin{lemma}\\label{lem:one-endpoint}\nIf a graph $G$ contains at most one universal vertex, then\nany proper arc representation $\\alpha$ of $G$ has the following property:\nIf $v$ and $v'$ are adjacent vertices of $G$,\nthen the arcs~$\\alpha(v)$ and~$\\alpha(v')$\nstrictly overlap, that is, contain exactly one endpoint of each other.  \n\\end{lemma}\n\n\\begin{proof}\nOtherwise we would have either $\\alpha(v)\\subset\\alpha(v')$,\nor $\\alpha(v)\\supset\\alpha(v')$, or the union $\\alpha(v)\\cup\\alpha(v')$\nwould cover the whole circle.\nThe first two conditions would contradict the assumption that $\\alpha$~is proper,\nwhile the last condition would imply that every $\\alpha(u)$ intersects both $\\alpha(v)$\nand~$\\alpha(v')$. Therefore, both $v$ and $v'$ were universal,\na contradiction.\n\\end{proof}\n\n\\begin{theorem}\\label{thm:Aunique2}\nLet $G$ be a twin-free, connected PCA graph.\nIf its complement $\\overline{G}$ is non-bipartite or connected,\nthen all geometric orders associated with proper arc representations of~$G$\nare equal up to reversing.\n\\end{theorem}\n\n\\begin{proof}\nAs we just discussed, it is enough to consider the case that $\\overline{G}$\nis bipartite and connected.\n  Let $V(G)=U\\cup W$ be the bipartition of~$\\overline{G}$ into two independent sets.\nNote that $U$ and $W$ span two connected components of the hypergraph $\\ensuremath{\\mathcal{N}}(\\overline{G})$.\nConsider an arbitrary proper arc representation~$\\alpha$ of~$G$ and the corresponding\ngeometric order $\\prec_\\alpha$ on~$V(G)$. \nLike in the proof of Theorem~\\ref{thm:Nrigid}.2,\nnotice that $\\prec_\\alpha$~is a tight\ninterval order for each connected component of~$\\ensuremath{\\mathcal{N}}(\\overline{G})$ and, therefore,\nthe restrictions of~$\\prec_\\alpha$ to~$U$ and~$W$,\neach considered up to reversing, do not depend on~$\\alpha$ by Theorem~\\ref{thm:unique}.1.\nThis still leaves two distinct possibilities of merging them into~$\\prec_\\alpha$.\nWe now show that one of them is actually ruled out, and this will\nprove that $\\prec_\\alpha$ does not depend on~$\\alpha$, \nif this order is considered up to reversing.\n\n\\begin{figure}\n  \\centering\n\\begin{tikzpicture}\n \\begin{camodel}\n  \\carc{140}{45}{1}{$\\alpha(w')$}\n  \\carc[endlabel=$\\alpha(u)$,endlabelpos=inside]{0}{-90}{1}{}\n  \\carc{70}{-30}{2}{$\\alpha(w)$}\n  \\carc{-45}{-140}{2}{$\\alpha(u')$}\n \\end{camodel}\n\\end{tikzpicture}\n  \\caption{Proof of Theorem~\\protect\\ref{thm:Aunique2}.}\\label{fig:wwuu}\n\\end{figure}\n\nNote that, since $\\overline{G}$ is connected, $G$ cannot have universal vertices\nand we can use Lemma~\\ref{lem:one-endpoint}.\n\nThe assumptions of the theorem imply that both $U$ and~$W$ contain at least two\nvertices. Let $w$ and~$w'$ be two vertices in~$W$. Since $w$ and~$w'$ are not twins,\nthey are distinguished by adjacency to some vertex $u\\in U$. W.l.o.g.\\\nsuppose that $w$ and~$u$ are adjacent, while $w'$ and~$u$ are not.\nSince $w$~is not universal in~$G$, there is a vertex $u'$\nin~$U$ non-adjacent to~$w$. Note that $\\alpha(w')$ and~$\\alpha(u)$\ncontain different endpoints of~$\\alpha(w)$, and $\\alpha(u')$ and~$\\alpha(w)$\ncontain different endpoints of~$\\alpha(u)$; see Fig.~\\ref{fig:wwuu}. \nIt follows that the arcs~$\\alpha(w')$, $\\alpha(w)$, $\\alpha(u)$, and~$\\alpha(u')$ occur\nin the arc model exactly in this circular order, irrespective of whether $\\alpha(w')$\nand~$\\alpha(u')$ intersect or not. \nSince the quadruple $w',w,u,u'$ was chosen in terms of the graph~$G$ alone\nand $\\alpha$ was supposed to be arbitrary, this conclusion holds true for\nany proper arc representation of~$G$. Therefore, there is a unique way\nof merging the restrictions of~$\\prec_\\alpha$ to~$U$ and~$W$ so as to obtain\nan ordering of~$V(G)$ consistent with some geometric order.\nThis proves the desired uniqueness result.\n\\end{proof}\n\n\nIf a graph has one interval or arc representation, \nwe can obtain many other representations, for example, by cloning some points\nof the circle. To disallow such trivial modifications, let us\nimpose some nonrestrictive conditions on interval and arc models.\nLet $x$ be a point in an arc system~$\\ensuremath{\\mathcal{A}}$. We call $x$ \\emph{inner}\nif no arc of~$\\ensuremath{\\mathcal{A}}$ begins or ends at~$x$. \nSuppose now that $\\ensuremath{\\mathcal{A}}$~is a proper intersection model of a graph~$G$.\nWe can modify $\\ensuremath{\\mathcal{A}}$ so that it remains a proper model of~$G$\nwhile the following two conditions are true:\n\\begin{itemize}\n\\item \nno two arcs share an endpoint point; this can always be achieved by cloning\na shared endpoint point (if $x$ the right endpoint of~$A_1$ and the left endpoint of~$A_2$,\nreplace $x$ with the pair $x_1,x_2$ so that $A_1$ ends at $x_1$, $A_2$ starts at $x_2$,\nand $x_1$~is the right neighbor of~$x_2$);\n\\item \nthere is no inner point (just remove all of them).\n\\end{itemize}\nWe call such arc models and representations \\emph{sharp}. \nSharp interval representations are defined similarly.\nNote that, if $G$~has $n$ vertices, any sharp\nmodel of~$G$~has $m=2n$ points. \n\n\nWe now show that a sharp proper representation is\nreconstructible from the associated geometric order.\nGiven two sharp arc representations $\\alpha$ and $\\alpha'$\nof a graph $G$, we say that they are equal \\emph{up to rotation}\nof the circle $\\mathbb{C}_{m}$ if there is a rotation $\\sigma$\nof $\\mathbb{C}_{m}$ such that $\\alpha'=\\sigma\\circ\\alpha$.\nIf $\\sigma$ is allowed to be also the reflection of $\\mathbb{C}_{m}$,\nthen we say that $\\alpha$ and $\\alpha'$ are equal \\emph{up to rotation and reflection}.\n\n\n\\begin{lemma}\\label{lem:ReprOrder}\n\\begin{bfenumerate}\n\\item\nIf sharp proper interval representations of a graph~$G$ determine the same\ngeometric order, then they are equal; that is, if $\\prec_\\alpha=\\prec_{\\alpha'}$,\nthen $\\alpha=\\alpha'$.\n\\item\nSuppose that $G$~has at most one universal vertex.\nIf sharp proper arc representations of~$G$ determine the same geometric order, \nthen they are equal up to rotation.\n\\end{bfenumerate}\n\\end{lemma}\n\n\\begin{proof}\n{\\bf 1.}\nLet $\\alpha$ be a sharp proper interval representation of a graph~$G$ and\n$v_1,\\ldots,v_n$ be the ordering of~$V(G)$ according to the geometric order $\\prec_\\alpha$\nassociated with $\\alpha$. It suffices to notice that\n$\\alpha$~is completely determined by this order. Indeed, let $\\alpha(v_i)=[a_i,b_i]$.\nThen we must have\n\\begin{eqnarray}\nb_i&=&a_i+1+|N(v_i)|\\text{\\ \\ and}\\label{eq:bi}\\\\      \na_i&=&i+\\left|\\setdef{j<i}{v_j\\notin N[v_i]}\\right|.\\label{eq:ai}\n\\end{eqnarray}\n\n\n{\\bf 2.}\nGiven the geometric order of the vertex set \nassociated with a sharp proper arc representation~$\\alpha$ of~$G$,\nwe have to show that it determines $\\alpha$ up to rotation.\nFix a sequence $v_1,\\ldots,v_n$ according to this order.\nFrom this sequence we can determine the clockwise neighborhood~$N^+[v_i]$\nand the counter-clockwise neighborhood~$N^-[v_i]$ of any vertex~$v_i$.\nSuppose that $\\alpha(v_i)=[a_i,b_i]$ and $a_1=1$. The latter condition\ncan be ensured by shifting (renaming) the points in the circle~$\\mathbb{C}_{2n}$,\nwhich results just in a rotation of~$\\alpha$. \n\nBy Lemma~\\ref{lem:one-endpoint}, if\n$v_i$~is adjacent to~$v_j$ then $\\alpha(v_i)$ contains exactly\none of the endpoints~$a_j$ and~$b_j$. \nWe can now see that the start points~$a_i$ \nand end points~$b_i$ are uniquely determined. Equality~\\refeq{bi} holds true\nexactly as in part~1; whenever the right hand side exceeds $2n$, it has\nto be decreased by this number. Equality~\\refeq{ai} has to be adjusted. \nIf $v_i\\notin N^+[v_1]$, then\n\\[\n[1,a_i]=[1,b_1]\\cup\\setdef{a_j}{1<j\\le i,a_j\\notin[1,b_1]}\\cup\n\\setdef{b_j}{1<j<i,b_j\\notin[a_i,b_i]}.\n\\]\nIt follows that\n\\[\na_i=|[1,a_i]|=2+|N(v_1)|+\\left|\\setdef{j}{1<j\\le i,v_j\\notin N^+[v_1]}\\right|+\n\\left|\\setdef{j}{1<j<i,v_j\\notin N^-[v_i]}\\right|.\n\\]\nIf $v_i\\in N^+[v_1]$, then\n\\[\n[1,a_i]=\\setdef{a_j}{1\\le j\\le i}\\cup\\setdef{b_j}{v_j\\in N^-[v_1]\\setminus N^-[v_i]},\n\\]\nhence\n\\[\na_i=|[1,a_i]|=i+|N^-[v_1]\\setminus N^-[v_i]|,\n\\]\ncompleting the proof.\n\\end{proof}\n\n\n\n\nCombining Lemma~\\ref{lem:ReprOrder} with Theorem~\\ref{thm:Aunique2}\n(or with Roberts' uniqueness theorem for proper interval graphs),\nwe immediately obtain the following result.\n\n\n\\begin{theorem}\\label{thm:prop-repr-unique}\\mbox{}\n  \\begin{bfenumerate}\n  \\item \nA twin-free, connected proper interval graph has, up to reflection,\na unique sharp proper interval representation.\n  \\item \nA twin-free, connected PCA graph with non-bipartite or connected complement has, \nup to rotation and reflection, a unique sharp proper arc representation.\n  \\end{bfenumerate}\n\\end{theorem}\n\n\n\n\\paragraph{Straight and round orientations.}\nWe conclude with discussing how Theorem~\\ref{thm:prop-repr-unique} is related\nto the results of Deng, Hell, and Huang~\\cite{DengHH96}.\nWe begin with an overview of some concepts introduced in~\\cite{DengHH96}.\nRecall that an \\emph{orientation} of a graph $G$ is a directed graph\nobtained from $G$ by directing each edge. A directed graph obtained\nin such a way is called \\emph{oriented}.\nLet $D$ be an orientation of a graph $G$.\nA \\emph{straight enumeration} of $D$ is an interval ordering of $\\ensuremath{\\mathcal{N}}[G]$\nsuch that every vertex $v$ splits the interval $N[v]=[v^-,v^+]$\ninto the set $[v^-,v)$ of in-neighbors of $v$ and into the set\n$(v,v^+]$ of out-neighbors of $v$. Similarly, \na \\emph{round enumeration} of $D$ is an arc ordering of $\\ensuremath{\\mathcal{N}}[G]$\nsuch that $[v^-,v)$ and $(v,v^+]$ consist of, respectively, the in- and the out-neighbors\nof $v$. If the orientation $D$ admits a straight (resp.\\ round) enumeration,\nit is called \\emph{straight (resp.\\ round)}.\nDeng, Hell, and Huang~\\cite{DengHH96} notice a close connection between\nproper interval\/arc representations of a graph $G$ and its straight\/round orientations:\n$G$ is proper interval (resp.\\ PCA) if and only if its has a straight (resp.\\ round)\norientation. This connection enables obtaining Theorem~\\ref{thm:prop-repr-unique}\nfrom the following result in~\\cite{DengHH96}.\n\n\n\\begin{theorem}[cf.\\ Deng, Hell, and Huang~\\cite{DengHH96}]\\label{thm:orient-unique}\\mbox{}\n  \\begin{bfenumerate}\n  \\item \nA twin-free, connected proper interval graph has, up to reversal,\na unique straight orientation.\n  \\item \nA twin-free, connected PCA graph with non-bipartite or connected complement has, \nup to reversal, a unique round orientation.\n  \\end{bfenumerate}\n\\end{theorem}\n\nThough obtained in our paper and in~\\cite{DengHH96} by completely different methods, \nTheorems~\\ref{thm:prop-repr-unique} and~\\ref{thm:orient-unique}\nare equivalent statements of the same nature. For completeness we now\nderive the latter result from the former.\n\n\n\\begin{proofof}{Theorem~\\ref{thm:orient-unique}}\nWe show part 2; part 1 is similar.\nFollowing~\\cite{DengHH96}, we first notice that a proper arc representation of\na twin-free graph $G$ determines a round orientation of $G$ and vice versa.\n\nGiven a proper arc representation $\\alpha$ of $G$,\ndefine an orientation $D_\\alpha$ of $G$ by directing each edge $\\{u,v\\}$\nas $uv$ if $\\alpha(u)$ contains the left endpoint of $\\alpha(v)$.\nThe definition is unambiguous by Lemma~\\ref{lem:one-endpoint}.\nNote that the geometric order $\\prec_\\alpha$ is a round enumeration of~$D_\\alpha$.\n\nNote also that, if $\\alpha'$ is a rotation of $\\alpha$, then\n$D_{\\alpha'}=D_\\alpha$. If $\\alpha'$ is the reflection of $\\alpha$, then\n$D_{\\alpha'}$ is the reversed version of $D_\\alpha$.\n\nAs observed in~\\cite[Proposition 2.6]{DengHH96} (see also~\\cite[Theorem 3.1]{HellH95}),\nevery round orientation $D$ of $G$ is obtainable in this way,\nthat is, $D=D_\\alpha$ for some proper arc representation $\\alpha$ of $G$.\nUnder the assumptions made for $G$, Theorem~\\ref{thm:prop-repr-unique}\nstates that $\\alpha$ is unique up to rotation and reflection.\nIt follows that $D=D_\\alpha$ is unique up to reversal.\n\\end{proofof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nStarting from the seminal works by \\cite{Sny1} and \\cite{Sny2},\nquantum systems with deformed Poison brackets have attracted much\nattention in various fields of theoretical physics. Of a special\ninterest is the so called space with the ``minimal length,'' which\nmeans that the deformation of the Poison bracket is quadratic in\nmomentum for both coordinates and momenta. This has first been\nintroduces in Refs.~\\cite{kempf1,kempf2}. Such a deformation leads\nto a non-zero minimal root-mean-square value of the coordinate.\nThere is an increasing number of papers on the subject, and the\nliterature review would require a separate article.\n\nThe deformation with the minimal length can in principle be\nextended to an arbitrary space of generalized coordinates and\nmomenta which are natural for the description of various models.\nIn particular, one can study the electromagnetic field represented\nas a set of oscillators with the deformed commutator relations for\ncoordinates and momenta, which generates the minimal length.\nObviously, in this case we have the deformation of the field\nitself rather then the deformation of the real space. This model\nof the electromagnetic field has been considered in\nRefs.~\\cite{camacho1,camacho2}. The Casimir effect in such a\ndeformed field has been studied in Ref.~\\cite{vakarchuk}, where\nthe Casimir energy was calculated for the one-, two, and\nthree-dimensional cases. It has been shown that the deformation\nsuppresses the interaction between the confining boundaries.\n\nIn the present work, we study the interaction of the deformed\nelectromagnetic field with atomic systems in the coordinate\nundeformed space. We shall not discuss the properties of the field\nequations (which are non-linear) or problems related to their\nLorentz and gauge invariance which will be considered in a\nseparate paper, we shall rather propose a model of interaction of\nsuch fields with atomic systems. We shall also calculate the\nintensity of spontaneous radiation as a function of the\ndeformation parameter.\n\n\\section{Basic equations}\n\nWe study the interaction of an atomic system with the\nelectromagnetic field, which Hamiltonian after the decomposition\ninto a set of oscillators is given by:\n\\begin{eqnarray}\n\\hat H=\\sum_{\\bf k}\\sum_\\alpha\\left(\n    \\frac{\\hat P_{{\\bf k},\\alpha}^2}{2} +\n    \\frac{\\omega_k^2 \\hat Q_{{\\bf k},\\alpha}^2}{2}\\right),\n\\label{v2.1} \\end{eqnarray} where ${\\bf k}$ is the wave-vector, $\\alpha$\ndenotes the polarization, $\\omega_k=ck$ is the frequency (where\n$c$ is the speed of light in the vacuum). $\\hat Q_{{\\bf\nk},\\alpha}$, $\\hat P_{{\\bf k},\\alpha}$ are the generalized\ncoordinate and momenta operators with the deformed Poison\nbrackets:\n\\begin{eqnarray}\n\\hat Q_{{\\bf k},\\alpha} \\hat P_{{\\bf k},\\alpha}- \\hat P_{{\\bf\nk},\\alpha}\\hat Q_{{\\bf k},\\alpha}= i\\hbar \\Big(1+\\beta \\hat\nP_{{\\bf k},\\alpha}^2\\Big), \\label{1.2} \\end{eqnarray} where $\\beta \\geq 0$ is\nthe deformation parameter, and all remaining commutators are equal\nzero. As it is well-known, such commutation relations lead to the\n``minimal length'' in the coordinate space, $\\sqrt{\\langle\\hat Q_{{\\bf\nk},\\alpha}^2\\rangle}=\\hbar\\sqrt \\beta$ \\cite{kempf1,kempf2}. Because\nin this case we do not deform the ordinary ``physical'' space (in\nwhich our field lives) but rather the commutation relations of the\ndynamical variables (fields), we have the deformation of the field\nitself rather then the space-deformation.\n\nLet us introduce new operators $\\hat q_{{\\bf k},\\alpha}$ and $\\hat\np_{{\\bf k},\\alpha}$ as follows:\n\\begin{eqnarray}\n\\label{eq:new_operators} \\hat q_{{\\bf k},\\alpha} = \\hat Q_{{\\bf\nk},\\alpha},\\ \\ \\ \\ \\ \\hat P_{{\\bf k},\\alpha} = \\frac{1}{\\sqrt\n\\beta}\\tan (\\hat p_{{\\bf k},\\alpha}\\sqrt \\beta). \\end{eqnarray} It is easy to\nshow that they are canonically conjugate operators:\n\\begin{eqnarray}\n\\hat q_{{\\bf k},\\alpha}\\hat p_{{\\bf k},\\alpha}-\\hat p_{{\\bf\nk},\\alpha}\\hat q_{{\\bf k},\\alpha}=i\\hbar. \\end{eqnarray} The Hamiltonian\n(\\ref{v2.1}) becomes\n\\begin{eqnarray}\n\\hat H =\\sum\\limits_{\\bf k}\\sum\\limits_\\alpha \\left(\n\\frac{\\omega_k^2 \\hat q_{{\\bf k},\\alpha}^2}{2}\n    + \\frac{\\tan^2(\\hat p_{{\\bf k},\\alpha}\\sqrt \\beta)}{2\\beta}\\right).\n\\label{1.5} \\end{eqnarray}\n\nThe equation of motion of such a field have been studied in\nRef.~\\cite{camacho1,camacho2}, where the Hamiltonian was presented\nas an expansion in ordinary annihilation and creation operators.\nThe field equations are non-linear but can be treated\nperturbatively.\n\nAs it is usual for oscillatory systems, we present the magnetic\nvector potential ${\\bf A}$ as follows\n\\begin{eqnarray}\n{\\bf A} = \\sqrt{\\frac{4\\pi c^2}{V}}\\sum_{\\bf k}\n    \\sum_\\alpha {\\bf e}_{{\\bf k},\\alpha}\n      \\left[\\frac{1}{2} \\left(\\hat Q_{{\\bf k},\\alpha} - \\frac{\\hat P_{{\\bf k},\\alpha}}{i\\omega_k}\\right) e^{i{\\bf kr}}\n      + \\frac{1}{2} \\left(\\hat Q_{{\\bf k},\\alpha} + \\frac{\\hat P_{{\\bf k},\\alpha}}{i\\omega_k}\\right)\n                    e^{-i{\\bf kr}}\\right].\n\\end{eqnarray} In terms of the new, canonically conjugate operators\n(\\ref{eq:new_operators}) we have:\n\\begin{eqnarray}\n{\\bf A} = \\sqrt{\\frac{4\\pi c^2}{V}}\\sum_{\\bf k}\n    \\sum_\\alpha {\\bf e}_{{\\bf k},\\alpha}\n    \\left[\\hat q_{{\\bf k},\\alpha}\\cos({\\bf kr})\n    - \\frac{\\tan (\\hat p_{{\\bf k},\\alpha}\\sqrt\\beta)}{\\omega_k\\sqrt\\beta}\n    \\sin({\\bf kr})\\right].\n\\end{eqnarray}\n\nAs usual, the operator of the interaction of the field with an\natom is:\n\\begin{eqnarray}\n\\hat V = -\\frac{e}{mc}({\\bf A}\\hat{\\bf p}) + \\frac{e^2}{2mc^2}{\\bf\nA}^2, \\label{v2.8} \\end{eqnarray} where $\\hat {\\bf p}$ is the momentum\noperator (of the electron in an atom), $e=-|e|$ is the electron\ncharge and $m$ the mass.\n\nTherefore, we propose the following model of the interaction of\nthe deformed field with an atomic system: we assume that all the\nexpressions of the ordinary theory of the electromagnetic field\nare valid, except that there are deformed commutation relations\nfor the operators $\\hat Q_{{\\bf k},\\alpha}$ and $\\hat P_{{\\bf\nk},\\alpha}$. Such a non-linear field is a simple model which\nallows us to study the effects of the deformation of the Poison\nbrackets on the properties of physical systems.\n\n\\section{Wave function and energy levels of the deformed field}\n\nThe eigenvalue problem for the Hamiltonian (\\ref{1.5}) has an\nanalytic solution. In momentum representation the wave function\nis:\n\\begin{eqnarray}\n\\psi_{\\ldots,N_{{\\bf k},\\alpha},\\ldots}(\\ldots,p_{{\\bf\nk},\\alpha},\\ldots)=\\prod_{\\bf k}\\prod_\\alpha \\psi_{N_{{\\bf\nk},\\alpha}}(p_{{\\bf k},\\alpha}), \\end{eqnarray} where $\\psi_{N_{{\\bf\nk},\\alpha}}(p_{{\\bf k},\\alpha})$ is the eigenfunction for the\n$({\\bf k},\\alpha)$ mode of the Hamiltonian\n\\begin{eqnarray}\n\\hat H = \\frac{\\omega^2\\hat q^2}{2} + \\frac{\\tan^2(\\hat\np\\sqrt\\beta)}{2\\beta}, \\end{eqnarray} which can be calculated using the\nstandard methods, for instance, the factorization method\n\\cite{cooper}. For brevity, we drop indexes ${\\bf k},\\alpha$ from\noperators $\\hat q_{{\\bf k},\\alpha}$, $\\hat p_{{\\bf k},\\alpha}$,\nfrequency $\\omega_{k}$, quantum number $N_{{\\bf\nk},\\alpha}=0,1,2,\\ldots$ and other quantities:\n\\begin{eqnarray}\np_{{\\bf k},\\alpha}\\to p,\\quad \\hat q_{{\\bf k},\\alpha} =\ni\\hbar{\\partial\\over \\partial p_{{\\bf k},\\alpha}}\\to i\\hbar\n{\\partial\\over \\partial p},\\quad \\omega_k \\to \\omega, \\quad\nN_{{\\bf k},\\alpha}\\to n=0,1,2,\\cdots . \\end{eqnarray}\n\nWe have therefore for the wave function in the momentum\nrepresentation:\n\\begin{eqnarray}\n&&\\psi_n(p)=\\beta^{1\/4}\\sqrt{\\Gamma(\\alpha+n+1)\\Gamma(n+2\\alpha)\\over\nn!\\Gamma(1\/2)\\Gamma\\left(\\alpha+n+{1\/2}\\right)\\Gamma(2n+2\\alpha)}\\nonumber\\\\\n&&\\times \\left(-{d\\over d\\bar p}+\\alpha\\tan{\\bar\np}\\right)\\ldots\\left(-{d\\over d\\bar p}+(\\alpha+n-1)\\tan{\\bar\np}\\right)\\cos^{\\alpha+n}\\bar p, \\label{v2.12} \\end{eqnarray} where the\ndimensionless momentum $\\bar p=p\\sqrt \\beta$, with the domain of\ndefinition of the momentum $p$ being\n\\begin{eqnarray}\n-\\pi\/2\\sqrt\\beta\\le p\\le \\pi\/2\\sqrt\\beta. \\end{eqnarray} The quantum number\nis $n=0,1,2,\\ldots$, and\n\\begin{eqnarray}\n\\alpha={1\\over 2}\\left[1+{2\\over \\beta\\hbar\n\\omega}\\sqrt{1+\\left({\\beta\\hbar \\omega\\over 2}\\right)^2}\\right].\n\\label{v14}\n\\end{eqnarray}\n\nThe wave-function (\\ref{v2.12}) is normalized as usually,\n\\begin{eqnarray}\n\\int\\limits_{-\\pi\/2\\sqrt\\beta}^{\\pi\/2\\sqrt\\beta}\n\\psi_{n'}(p)\\psi_n(p)\\,dp=\\delta_{n'n}. \\end{eqnarray}\n\nA few first wave-functions read:\n\\begin{eqnarray}\n&&\\psi_0(p)=\\beta^{1\/4}\\sqrt{\\Gamma(\\alpha+1)\\over\n\\Gamma(1\/2)\\Gamma(\\alpha+1\/2)}\\cos^\\alpha\\bar p,\\nonumber\\\\\n&&\\psi_1(p)=\\beta^{1\/4}\\sqrt{2\\Gamma(\\alpha+2)\\over\n\\Gamma(1\/2)\\Gamma(\\alpha+1\/2)}\\cos^\\alpha \\bar p\\sin\\bar\np,\\label{v3.13}\\\\\n&&\\psi_2(p)=\\beta^{1\/4}\\sqrt{2\\Gamma(\\alpha+3)\\over\n2(\\alpha+1)\\Gamma(1\/2)\\Gamma(\\alpha+3\/2)}\\left[(2\\alpha+1)-2(\\alpha+1)\\cos^2\\bar\np\\right] \\cos^\\alpha \\bar p.\\nonumber \\end{eqnarray}\n\n\nLet us calculate the wave-functions for $\\beta=0$. In this case\nthere is no deformation and the Hamiltonian (\\ref{1.5}) reduces to\nthe Hamiltonian of a harmonic oscillator with the mass equal to\nunity. Because $\\alpha\\to 1\/\\beta\\hbar\\omega\\to \\infty$ for\n$\\beta\\to 0$, we obtain\n\\begin{eqnarray}\n\\cos^\\alpha(p\\sqrt\\beta)\\mathop{=}\\limits_{\\beta\\to 0}\n\\left(1-{p^2\\beta\\over\n2}+\\ldots\\right)^{1\/\\beta\\hbar\\omega}\\mathop{=}\\limits_{\\beta\\to\n0}\n e^{-p^2\/2\\hbar\\omega}.\n\\end{eqnarray} Noticing that $\\Gamma(z +a)\\sim\\sqrt{2\\pi}e^{-z}z^{z+a-1\/2}$\nfor $z\\to \\infty$, we obtain from Eq.~(\\ref{v2.12}), as expected,\nthe wave-fucntion of the harmonic oscillator in the momentum\nrepresentation:\n\\begin{eqnarray}\n\\psi_n(p)={1\\over (\\pi\\hbar\\omega)^{1\/4}} {1\\over\n\\sqrt{n!2^n}}\\left(-{d\\over d\\eta}+\\eta\\right)^ne^{-\\eta^2\/2}, \\end{eqnarray}\nwhere the dimensionless variable  $\\eta=p\/\\sqrt{\\hbar\\omega}$,\nwith $-\\infty<p<\\infty$.\n\nIt is convenient to rewrite Eq.~(\\ref{v2.12}) using the Gegenbauer\npolynomials $C_n^\\alpha(x)$ \\cite{janke,gradshteyn}. To do this,\nwe change the variable $x=\\sin\\bar p$ to obtain, after simple\ntransformations,\n\\begin{eqnarray}\n\\bar\\psi_n(x)=\\sqrt{n!(\\alpha+n)\\over 2\\pi\n\\Gamma(2\\alpha+n)}2^\\alpha\n\\Gamma(\\alpha)(1-x^2)^{\\alpha\/2-1\/4}C_n^\\alpha(x), \\end{eqnarray}\n\n\\begin{eqnarray}\nC_n^\\alpha(x)&=&(-)^n{\\sqrt\\pi \\Gamma(2\\alpha+n)\\over\nn!2^{2\\alpha+n-1}\\Gamma(\\alpha)\\Gamma(\\alpha+n+1\/2)}(1-x^2)^{1\/2-\\alpha}\\left({d\\over\ndx}\\right)^n\\nonumber\\\\\n&\\times& \\left[(1-x^2)^{n+\\alpha-1\/2}\\right]. \\end{eqnarray}\nThe wave-functions have the following normalization\n\\begin{eqnarray}\n\\int\\limits_{-1}^1 \\bar\\psi_{n'}(x)\\bar\\psi_n(x)\\,\ndx=\\delta_{n'n}. \\end{eqnarray} Our change of the variable is not a unitary\ntransformation; the Jacobian of the transformation is $1\/\\cos \\bar\np=(1-x^2)^{-1\/2}$. Hence, for instance, the coordinate operator in\nthe coordinate representation is\n\\begin{eqnarray}\n\\hat q={i\\hbar \\sqrt \\beta} (1-x^2)^{1\/4}{d\\over dx}(1-x^2)^{1\/4}.\n\\end{eqnarray}\n\nThe energy levels of the harmonic oscillator with the commutation\nrelations (\\ref{1.2}) are eigenvalues of the Hamiltonian\n(\\ref{1.5}). They are well-known\\cite{kempf2,vak}. Then the energy\nlevels of the deformed field read:\n\\begin{eqnarray}\n\\label{eq:energy_deformed} E_{\\ldots,N_{{\\bf\nk},\\alpha},\\ldots}&=&\\sum_{\\bf k}\\sum_\\alpha \\hbar\n\\omega_k\\Bigg[\\left(N_{{\\bf k},\\alpha}+{1\\over 2}\\right)\n\\sqrt{1+\\left({\\beta \\hbar \\omega_k\\over 2}\\right)^2}\\nonumber\n\\\\\n&+&{\\beta \\hbar \\omega_k\\over 2} \\left(N_{{\\bf\nk},\\alpha}^2+N_{{\\bf k},\\alpha}+{1\\over 2}\\right)\\Bigg], \\end{eqnarray} where\nthe quantum numbers $N_{{\\bf k},\\alpha}=0,1,2,\\ldots\\;$. We note\nthat the deformation parameter may depend on ${\\bf k}$ and\n$\\alpha$. In our work, however, we restrict our considerations to\nthe case $\\beta=\\mathrm{const}$.\n\n\\section {Spontaneous radiation}\n\nLet us assume that an atom is in the excited state $|2\\rangle$ (with\nthe energy $E_2$) while the field is in the ground state\n$|\\ldots,0,\\ldots\\rangle$. As a result of the interaction of the atom\nwith the field, the atom jumps to a level with the lower energy\n$E_1$ and radiates a light quantum with the energy $\\hbar\n\\omega=E_2-E_1$. The field, therefore, jumps to the state with one\nphonon, i.e., $|\\ldots,0,N_{{\\bf k},\\alpha}=1,0\\ldots\\rangle$. The\ntransition probability rate of a system ``field-plus-atom'' to\njump from the initial state $|i\\rangle=|2\\rangle|\\ldots,0,\\ldots\\rangle$ to\nthe final state $|f\\rangle=|1\\rangle|\\ldots,0,N_{{\\bf\nk},\\alpha},0,\\ldots\\rangle$  is given by: \\begin{eqnarray} w_{k\\to f}={2\\pi\\over\n\\hbar}\\left({e\\over mc}\\right)^2\\left|\\langle f|{\\bf A}\\hat {\\bf\np}|i\\rangle\\right|^2\\delta (E_2-E_1-\\hbar \\Omega_k), \\label{v4.20} \\end{eqnarray}\nwhere the energy of the field quantum, according to\nEq.~(\\ref{eq:energy_deformed}), is\n\\begin{eqnarray}\n\\hbar \\Omega_k&=&E_{\\ldots,0,N_{{\\bf\nk},\\alpha}=1,0,\\ldots}-E_{\\ldots,0,\\ldots},\\nonumber\\\\ \\nonumber\n\\\\ \\Omega_k&=&\\omega_k\\left[\\sqrt{1+\\left(\\beta{\\hbar\n\\omega_k\\over 2}\\right)^2}-\\beta\\hbar \\omega_k\\right].\n\\label{v4.21} \\end{eqnarray} We take into account the first term of the\ninteraction operator (\\ref{v2.8}), which is linear in the vector\npotential.\n\nThe matrix element $\\langle f|{\\bf A\\hat p}|i\\rangle$ in Eq.~(\\ref{v4.20})\nis \\begin{eqnarray*} \\langle f|{\\bf A\\hat p}|i\\rangle=\\sqrt{4\\pi c^2\\over V}\\left(\\langle\nN_{{\\bf k},\\alpha}=1|\\hat q_{{\\bf k},\\alpha}|0\\rangle p_{12}^c-\n{1\\over \\omega_k\\sqrt\\beta}\\langle N_{{\\bf k},\\alpha}=1\\left|\\tan\n(p_{{\\bf k},\\alpha}\\sqrt\\beta)\\right|0\\rangle p_{12}^s\\right), \\end{eqnarray*}\n\n\\begin{eqnarray} p_{12}^c=\\langle 1|\\cos({\\bf kr}){\\bf e}_{{\\bf k},\\alpha}\\hat {\\bf\np}|2\\rangle,\\quad p_{12}^s=\\langle 1|\\sin({\\bf kr}){\\bf e}_{{\\bf\nk},\\alpha}\\hat {\\bf p}|2\\rangle. \\label{v4.22} \\end{eqnarray}\n\nUsing the wave functions (\\ref{v3.13}), we get for the matrix\nelements of the field operators:\n\\begin{eqnarray}\n\\langle N_{{\\bf k},\\alpha}=1|\\hat q_{{\\bf k},\\alpha}|0\\rangle=-i\\hbar\n{\\sqrt{2\\beta(\\alpha_k+1)}\\over\n2\\alpha_k+1}\\left[{\\Gamma(\\alpha_k+1)\\over \\Gamma(\\alpha_k+1\/2)}\n\\right]^2, \\nonumber\\\\ \\label{v4.23}\n\\\\ \\nonumber\n\\langle N_{{\\bf k},\\alpha}=1\\left|\\tan (p_{{\\bf\nk},\\alpha}\\sqrt\\beta)\\right|0\\rangle ={\\sqrt{2(\\alpha_k+1)}\\over\n\\alpha_k(2\\alpha_k+1)}\\left[{\\Gamma(\\alpha_k+1)\\over\n\\Gamma(\\alpha_k+1\/2)} \\right]^2, \\end{eqnarray} here $\\alpha_k=\\alpha$,\nwhich is defined by (\\ref{v14}) at $\\omega\\to \\omega_k$.\n\nThe intensity of spontaneous radiation $I_{{\\bf k},\\alpha}$ is\ndefined in a usual way as the amount of energy with the given\npolarization $\\alpha$ radiated by an atom in unit time on the\nresonance frequency $\\omega=(E_2-E_1)\/\\hbar$ per space angle,\ni.e.,\n\\begin{align}\n\\label{eq:intensity:def} I_{{\\bf k},\\alpha} =\n    {V\\over (2\\pi)^3}\\int_0^\\infty k^2\\hbar \\Omega_k w_{i\\to f}\\, dk.\n\\end{align}\n\nBy inserting Eqs.~(\\ref{v4.20})--(\\ref{v4.23}) into\nEq.~(\\ref{eq:intensity:def}) we obtain, after integration: \\begin{eqnarray*}\nI_{{\\bf k},\\alpha}={e^2\\hbar\\over m^2c^3\\pi}\\left(\n{2\\beta(\\alpha_k+1)\\over\n(2\\alpha_k+1)^2}\\left[{\\Gamma(\\alpha_k+1)\\over\n\\Gamma(\\alpha_k+1\/2)} \\right]^4\\omega_k^2\\Omega_k{d\\omega_k\\over\nd\\Omega_k}\\left|p_{12}^c-{i\\over \\beta\\alpha_k\\hbar \\omega_k}\np_{12}^s\\right|^2\\right)_{\\Omega_k=\\omega}. \\end{eqnarray*}\n\nTaking into account Eq.~(\\ref{v4.21}), we finally get, after\nsimple transformations,\n\\begin{eqnarray}\nI_{{\\bf k},\\alpha}={e^2\\omega^2\\over 2\\pi m^2c^3} g(\\bar\\omega)\n\\left|p_{12}^c-ip_{12}^s{1\\over \\bar\\alpha}\\sqrt{2\\bar\n\\alpha+1\\over 4\\bar\\omega}\\right|^2, \\label{v4.24} \\end{eqnarray} where\n\\begin{eqnarray}\ng(\\bar\\omega)={8(\\bar\\alpha+1)(2\\bar\\alpha-1)\\bar\\omega^{1\/2}\\over\n[2\\bar\\alpha+1+2\\bar\\omega(4\\bar\\alpha-1)](2\\bar\\alpha+1)^{5\/2}}\n\\left[{\\Gamma(\\bar\\alpha+1)\\over \\Gamma(\\bar\\alpha+1\/2)}\n\\right]^4, \\end{eqnarray} \\begin{eqnarray*} \\bar\\alpha={3\\bar \\omega\\over\n1+4\\bar\\omega-\\sqrt{1+8\\bar\\omega+4\\bar\\omega^2}}-{1\\over 2}, \\end{eqnarray*}\nand the dimensionless deformation parameter is\n\\begin{eqnarray}\n\\bar \\omega={\\beta \\hbar \\omega\\over 2}. \\label{v4.26} \\end{eqnarray}\n\nIn the limiting case of no deformation ($\\beta=0$) we have\n$g(0)=1$ (see Eq.~(\\ref{v4.24})), and therefore one gets for the\nintensity\n\\begin{eqnarray}\nI_{{\\bf k},\\alpha}={e^2\\omega^{2}\\over 2\\pi\nm^2c^3}|p_{12}|^2,\\quad p_{12}=\\langle 1|e^{i{\\bf kr}}({\\bf e}_{{\\bf\nk},\\alpha} \\hat {\\bf p})|2\\rangle, \\label{v4.27} \\end{eqnarray} which is the\nwell-known expression for the intensity of radiation for the\nnon-deformed field.\n\nIn the opposite limit of large deformation parameter ($\\bar\n\\omega\\gg 1$), the term $\\sim \\sin({\\bf kr})$ vanishes (see\nEq.~(\\ref{v4.24})), and hence the quadrupole radiation vanishes\ntoo. The reason is that the $\\sin({\\bf kr})$ term is the main\ncontribution to the quadrupole radiation in the long wavelength\nlimit. Finally we note that the function $g(\\bar \\omega)$ in the\ncase of considerable deformations ($\\bar\\alpha=1$) has the\nfollowing asymptotic form:\n\\begin{eqnarray}\n\\label{eq:g(w)} g(\\bar\\omega)\\mathop{=}\\limits_{\\bar\\omega\\to\n\\infty}{128\\over 27\\pi^2}{1\\over \\sqrt {3\\bar\\omega}}. \\end{eqnarray}\n\n\\section{Intensity of the dipole radiation}\n\nWithin the dipole-transition approximation, the wave-vector ${\\bf\nk}\\to0$ ($\\cos({\\bf k r})\\to 1$ and $\\sin({\\bf kr})\\to 0$), hence\n$p_{12}^c=({\\bf e}_{{\\bf k},\\alpha} {\\bf p}_{12})$ and\n$p_{12}^s=0$ (see Eq.~(\\ref{v4.22})). We therefore get:\n\\begin{eqnarray}\nI_{{\\bf k},\\alpha}={e^2\\omega^2\\over 2\\pi m^2c^3}| {\\bf e}_{{\\bf\nk},\\alpha} {\\bf p}_{12}|^2g(\\bar\\omega), \\label{v5.29} \\end{eqnarray}\nEquation (\\ref{v5.29}) leads to Eq.~(\\ref{v4.27}) for\n$\\bar\\omega=0$ with ${\\bf k}=0$, while for large values of the\ndeformation parameter the intensity of the dipole radiation is\n(after making use of Eq.~(\\ref{eq:g(w)}))\n\\begin{eqnarray}\nI_{{\\bf k},\\alpha}={e^2\\omega^{3\/2}\\over 2\\pi m^2c^3}|{\\bf\ne}_{{\\bf k},\\alpha} {\\bf p}_{12}|^2 \\left({2\\over\n3}\\right)^{7\/2}{1\\over \\sqrt{\\beta\\hbar}}\\left({4\\over\n\\pi}\\right)^2. \\end{eqnarray} We therefor conclude that with increasing of\nthe deormation parameter the intensity decreases as\n$\\beta^{-1\/2}$.\n\n\\begin{figure}[htb]\n\\centerline{\\includegraphics[width=0.7\\textwidth]{Radiation1.eps}}\n\\caption{The intensity of the spontaneous dipole radiation of the\ndeformed field (see Eq.~(\\ref{v5.29})), divided by the intensity\nfor the undeformed field, as a function of the dimensionless\ndeformation parameter~(\\ref{v4.26}).}  \\label{fig:intensity}\n\\end{figure}\n\n\nThe intensity of the dipole interaction as a function of the\ndimensionless deformation parameter $\\bar\\omega$ is shown in\nFig.~\\ref{fig:intensity}\n\n\n\\section{Conclusion}\n\nWe have considered a theory of spontaneous radiation taking into\naccount the creation of one photon. Because the field is\nnon-linear, the transitions between any two states are allowed,\nnot only between the two neighboring, as in the case of the\nordinary oscillator. In order to study such processes in the case\nof more then one photon, but in the linear approximation (in the\nvector potential ${\\bf A}$ in the interaction operator\n(\\ref{v2.8})) and neglecting all the higher order terms in an\nexpression for the quantum transition rates, we take the following\nmatrix elements instead of (\\ref{v4.23}):\n\\begin{align}\n\\label{eq:matrix_elements2} q_{n,n'} = {i\\hbar\\sqrt\\beta}\n\\int_{-1}^1 \\bar \\psi_{n}(x)(1-x^2)^{1\/4}\\,{d\\over dx}\n    \\left[\\bar\\psi_{n'}(x)(1-x^2)^{1\/4}\\right]\\,dx,\\\\\n\\left(\\tan\\,\\bar\np\\right)_{n,n'}=\\int_{-1}^1\\bar\\psi_{n}(x)(1-x^2)^{1\/4}{x\\over\n\\sqrt{1-x^2}}\\bar\\psi_{n'}(x)(1-x^2)^{1\/4}\\,dx.\n\\end{align}\nIt is obviously that these integrals are not equal zero only at\neven $n+n'$.\n\nIn general case the integrals in Eq.~(\\ref{eq:matrix_elements2})\ncannot be expressed in terms of elementary functions. For some\nparticular values of $n$ and $n'$, however, the integrals can be\neasily evaluated, as for instance for $n=1$ and $n'=0$ (see\nEq.~(\\ref{v4.23})). Evidently, many-photon processes are absent\nfor $\\beta = 0$ while their role increases with increasing the\ndeformation parameter $\\beta$.\n\n\n\n\\section*{Acknowledgements}\nI wish to thank Prof. I.~Stasyuk and Dr. V.~Tkachuk for\nstimulating discussions, especially during the 12th {\\it Christmas\nDiscussions} (2008) held at the Department of Theoretical Physics\nof the Lviv National University.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Abstract}\n{\\bf The last few years have seen community-wide excitement in the study of jet substructure derived from the inner workings of clustering algorithms. Such efforts have resulted in the design of new observables which are related to partonic processes from final state hadrons. Since jets are multi-scale objects, they necessarily encode information about both the perturbative (pQCD) parton shower and non-perturbative (npQCD) physics including hadronization. Recent high precision measurements of jet substructure in proton-proton (pp) collisions have pushed the theoretical community into extending their predictions to higher orders resulting in the observation of large theoretical uncertainties from the non-perturbative regime of the calculations. We emphasize the importance of understanding a jet shower from a multi-dimensional point of view and highlight a recent measurement focused on distinguishing the pQCD vs. npQCD regimes within a jet shower. We introduce and discuss the utility of the formation time evaluated at varying stages of the jet shower in pp collisions. Finally, we present a monte-carlo study of the formation time of charged particles within the jet to gain a handle on hadronization mechanisms including string-breaking, and outline a path forward for such observables in heavy ion collisions.  \n}\n\n\n\\section{Introduction}\nJets are an algorithmic representation of groups of hadron originating from the fragmentation and hadronization of a hard scattered quark\/gluon (parton) in high energy collisions. The latest implementation of jet reconstruction algorithms iteratively clusters objects~\\cite{FastJet, Marzani:2019hun}; charged tracks or calorimeter towers in experiment and particles in event generators, within an user-defined angular size commonly referred to as a resolution parameter or radius. These clustering algorithms have proven themselves immensely useful in studying quantum chromodynamics (QCD) at both the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC). Measured jet cross-sections as a function of the transverse momentum have shown to be adequately reproduced by perturbative calculations (pQCD) across several orders of magnitude~\\cite{Britzger:2012bs}. The accuracy of the pQCD predictions at the parton level can be improved with non-perturbative corrections (due to underlying event, multi-parton interactions and hadronization) predominantly estimated from monte-carlo (MC) event generators. Therefore, reconstructed jets are designated as proxies for the out-going hard-scattered partons, and now we study the evolution of the parton via the hadrons associated with the jet. There are two fundamental scales to consider in a parton shower, or simply a splitting process (e.g $q \\rightarrow q+g)$, which are the momentum fraction carried by the emitted particle, a gluon in this case, and the emission angle. The splitting probabilities, which are inversely related to the opening angle and momentum fractions, can be converted into the well known DGLAP evolution equations for high energy partons. These evolution equations are utilized in MC to evolve the parton, and once the shower is fully completed and all the remaining partons end up on the mass shell, hadronization mechanisms take over and convert these color charged partons into color neutral hadrons (and decay them as applicable) which can be detected in experiments. The current era of jet measurements are focused on exploiting the information contained within jet clusterings to extract the structure and evolution of the jets.  \n\nThere are two main classes of observables related to jet structure measured thus far in high energy collisions; momentum fraction and an opening angle similar to the scales in the parton shower. Canonical jet structure measurements include the jet fragmentation function, i.e. the momentum carried by a hadron (charged particles, for ease of measurement) w.r.t a jet, and the jet shape which measures the (charged) particle density at a certain distance from the jet axis. Both these observables describe the structure of the jet and its fragmentation into hadrons, but do not provide any information related to its evolution. See ~\\cite{Cunqueiro:2021wls} for experimental and ~\\cite{Blaizot:2015lma} for theoretical review articles. Recently, experimentalists and theorists have come together to expand on the idea of jet clustering and started studying the possibility of translating an ordered collection of particles into a probe of the multi-scale dynamics of the parton shower evolution. One such example of an analysis technique is SoftDrop (SD) grooming~\\cite{Larkoski:2014wba, Larkoski:2015lea}, which re-clusters all constituents of a jet into an angular ordered tree via the Cambridge\/Aachen (C\/A) algorithm. In vacuum, angular ordering follows the emission spectrum wherein subsequent emissions occur at angles smaller than the previous emission. As a consequence, early time emissions can be associated with wide angles and later emission with narrow angles. It has been shown that jet substructure observables at the first hard splitting, such as the momentum fraction ($z_{g}$) and the groomed jet radius ($R_{g}$), can be described sufficiently by pQCD calculations, without the need for large hadronization corrections at sufficiently large jet radii and momenta~\\cite{STAR:2020ejj, STAR:2021lvw}. This provides evidence that the early time dynamics of a jet shower can be described in a perturbative fashion while one expects the later stages to be dominated by non-perturbative effects since the end result of a jet shower is hadronized particles measured in the detector. \n\n\\begin{equation}\nz_{g} = z = \\frac{p_{T, 2}}{p_{T, 1} - p_{T, 2}}; R_{g} = \\theta = \\Delta R(1, 2)\n\\label{eq1}\n\\end{equation}\n\nThe definition of the momentum fraction and groomed jet radius as shown in Eq~\\ref{eq1} calculated using the transverse momenta ($p_{T}$) and the $y$ (rapidity), $\\phi$ (azimuthal angle) of the two objects $(1, 2)$ which are numbered such that $p_{T, 2} < p_{T, 1}$. The opening angle is commonly referred to using the $\\theta$ variable, and in this case, is calculated as the distance between the two prongs in the $y-\\phi$ space ($\\delta R$). For SD, these are the prongs in the C\/A clustering that satisfy the grooming criteria, but in principle this definition can be used for any two quantities such as charged particles or other splittings within the clustering tree as we will explore in this study.  \n\n\\section{Jet substructure and formation times}\n\nA vacuum parton shower is inherently a quantum-mechanical process and as such the Heisenberg uncertainty principle can be applied at a particular splitting to define a formation time given the energy and angle of the emission. Since the energy scales in discussion are on the order of a few GeVs, we should also include relativistic effects of time dilation and as a consequence, one can define as formation time~\\cite{Apolinario:2020uvt} as follows - \n\n\\begin{equation}\n\\tau_f = \\frac{1}{z \\cdot (1-z) \\cdot \\theta^{2} E} [\\rm{fm}\/c],\n\\label{eq2}\n\\end{equation}\n\nwhere $z$ is the momentum fraction, $\\theta$ is the opening angle and $E$ is the energy of the radiator. The impact of such a formation time observable is to associate a time scale to different parts of the jet evolution and to quantify the varying physics regimes in the multi-scale jet shower. We first begin with the perturbative part of the jet shower, i.e the SD first split.  \n\n\\begin{figure}\n \\includegraphics[width=0.9\\linewidth]{Fig1.png}\n \\caption{Correlation between the formation time at the first SD emission (y-axis) and the invariant jet mass (x-axis) for jets from PYTHIA 6. The z-axis color scale shows the density of the correlation normalized per total number of jets.}\n \\label{fig1}\n\\end{figure}\n\nAt the SD first split, the necessary ingredients to calculate the formation time, such as the momentum fraction ($z = z_{g}$), the opening angle ($\\theta = R_{g}$), the mother prong energy ($E$, proxy for jet energy) have been shown in pervious publications~\\cite{STAR:2020ejj, STAR:2021lvw}. In order to gain a conceptual understanding of a jet's formation time at the first SD split, we show the correlation between $ln(\\tau^{SD}_{f})$ and the invariant jet mass $M = \\sqrt{E^{2} - p^{2}}$ in Fig~\\ref{fig1}. This correlation is estimated from the PYTHIA 6 event generator~\\cite{Sjostrand:2006za} with the STAR tune~\\cite{STAR:2018yxi} for anti-$k_{T}~ R=0.4$ jets with $20 < p_{T} < 30$ GeV\/$c$. We observe a strong anti-correlation whereby large\/late formation times correspond to small jet masses and conversely, those with small\/early SD formation times, end up as jets with very large masses. We understand this behavior primarily based on the dependence of both the jet mass and formation time to the opening angle. Early time emissions predominantly occur via a large opening angle and correspond to jets of large virtuality and mass. Jets with a late $\\tau^{SD}_{f}$ have a shower topology characterized by very narrow opening angles and subsequently the overall virtuality and mass are smaller on average. \n\n\\begin{figure}\n \\includegraphics[width=0.9\\linewidth]{Fig2.png}\n \\caption{Cartoon representation of the three different stages of calculating the formation time for a particular jet. The left drawing highlights the SD first split in the thick black lines. The right panel shows the leading and subleading charged particles within the jet in the blue solid lines and the middle panel emphasizes the resolved splitting in red, where the charged particles are first resolved by the C\/A de-clustering tree into two separate prongs.}\n \\label{fig2}\n\\end{figure}\n\nNow we define, for the first time, three different formation times associated with the jet. The first one is the SD formation time as we discussed earlier, which we expect to follow a pQCD prescription. In order to signify a later segment of the jet shower, we shall consider the leading and sub-leading (in terms of their $p_{T}$) charged particles within the jet as the two objects $1, 2$ and use Eq~\\ref{eq2} to calculate a second formation time $\\tau^{ch}_f$. This is not a realistic splitting in a theoretical sense, since we know that the charged particles are produced post hadronization. Thus, one can expect that this $\\tau^{ch}_f$ will have predominantly later or larger mean values of the formation times as compared to SD first split, and as a consequence, we expect this particular quantity to have large npQCD corrections from hadronization effects~\\cite{Chien:2021yol}. Lastly, we look at a third formation time from an intermediate stage within the jet shower which we call the resolved splitting $\\tau^{res}_{f}$. The resolved splitting is the first instance along the C\/A de-clustering tree wherein the leading and subleading charged particles are separated into their own prongs. This signifies a specific point in time along the jet shower where the high energy hadronized remnants of the jet are resolved in the jet clustering step. All the three types of formation times are shown in Fig~\\ref{fig2} where the black markers represent the SD, red for the resolved splittings and blue for the charged particles. \n\n\\begin{figure}\n \\includegraphics[width=0.9\\linewidth]{Fig3.pdf}\n \\caption{Fully corrected preliminary measurement from the STAR collaboration of the formation time for the SD first splitting (black), resolved splitting (red) and the leading and subleading charged particle (blue). The shaded regions behind the data markers represent the total systematic uncertainties. The formation times are calculated for anti$-k_{T}~ R=0.4$ jets with $20 < p_{T} < 30$ GeV\/$c$ and the bottom panel shows the ratio of SD and resolved splittings to the charged-particle split.\n }\n \\label{fig3}\n\\end{figure}\n\nThe first measurement of these formation time observables within a jet shower was done by the STAR collaboration at RHIC and presented recently for anti$-k_{T}~ R=0.4$ jets with $20 < p_{T} < 30$ GeV\/$c$ in proton-proton collisions at $\\sqrt{s} = 200$ GeV. The top panel shows the fully corrected distributions of the $\\tau^{SD}_{f}$ in black markers, $\\tau^{res}_{f}$ in red markers and $\\tau^{ch}_{f}$ in blue markers, where each distribution is normalized to its integral. As expected, we observe a trend in the three different formation times with $\\langle \\tau^{SD}_{f} \\rangle < \\langle \\tau^{res}_{f} \\rangle < \\langle \\tau^{ch}_{f} \\rangle$. In order to quantify the shapes of these formation times, the bottom panel in Fig~\\ref{fig3} presents the ratio of both the $\\tau^{SD}_{f}$ and $\\tau^{res}_{f}$ with $\\tau^{ch}_{f}$. At early formation times, we see a large negative slope for the $\\tau^{SD}_{f}$ splittings indicating the early time (perturbative) peak of the SD selected splits. At later formation times ($\\tau > 3 - 4 $ fm\/$c$), the shape of the resolved splittings are nearly consistent with the charged hadron splittings. With such specially selected resolved splitting, we can now identify and specify a time within the jet clustering shower where all the splittings that come afterwards, can be described as the splitting one calculated using hadrons as opposed to a partonic splitting. This measurement now provides a unique method whereby one can use the resolved splittings within a jet shower, to select a specific jet topology entirely based on the time a jet spends in the perturbative sector ($\\tau^{res}_{f} < 2$ fm\/$c$), or in the non-perturbative regime ($\\tau^{res}_{f} > 5$ fm\/$c$). These results will serve as a baseline in proton-proton collisions to first identify the region of perturbative calculability and then compare different hadronization mechanisms to explore the non-perturbative sector in QCD. \n\nSpace-time tomography of the quark-gluon plasma (QGP) via jets produced in heavy ion collisions is one of the standard methods of studying the transport properties of deconfined quarks and gluons. Since the jets evolve concurrently with the QGP, selecting jet topologies with a particular formation time enables a time-dependent study of the jet-QGP interactions. Since there is a large heavy ion underlying event there is a requirement that the selection of the formation time observable in a jet be robust to the background and still be sensitive to the kinematics of the jet shower. The first SD split for jets in heavy ion collisions, with the nominal value of the grooming parameters, have been shown to be extremely sensitive to the heavy ion background~\\cite{STAR:2021kjt}. This can be overcome by utilizing a stricter grooming criterion~\\cite{Mulligan:2020tim} which affects the selection bias of the surviving jet population. Since the leading and subleading charged particles in a jet are predominantly higher momentum, their reconstruction is unaffected by the presence of the underlying event.  Thus, the resolved splitting is expected to be a robust guide to selecting specific jet topologies and a formation time which can be translated to their path-length traversed in the QGP medium. \n\n\\section{Hadronization and jet substructure}\n\nIn order to study the impact of selecting on the charged hadron formation time, we look at the charge of the particles which make up the first, second and third (ordered in $p_{T}$) charged hadrons in the jet constituents. The charged particle formation time is then plotted for same charged leading and subleading hadrons with the third particle being either positively or negatively charged. The ratios of these formation times are shown in Fig~\\ref{fig4} for negatively (top) and positively (bottom) charged leading and subleading charged particles. The red circle (black box) markers in each of the panels correspond to the third charged particle being positively (negatively) charged. In each of the cases, we observe a clear separation of the formation times based on the charge of the third particle predominantly expected to oppositely charged to the leading and subleading particles. There is no dependence to the formation time on this ratio estimated from PYTHIA 8 simulations~\\cite{Bierlich:2022pfr} at jet kinematics accessible at RHIC energies. This particular separation shown in Fig~\\ref{fig4} is a feature of the Lund string breaking hadronization mechanism implemented in PYTHIA where one expects an overall balancing of electric charge in the hadrons produced due to the jet fragmentation. Ongoing experimental measurements at both RHIC and LHC are expected to highlight this feature and aim to understand hadronization mechanism via data-driven approaches. \n\n\\begin{figure}\n \\includegraphics[width=0.49\\linewidth]{Fig4a.png}\n \\includegraphics[width=0.49\\linewidth]{Fig4b.png}\n \\caption{Top: Normalized ratio of ch-particle formation time when the leading and subleading charged particles are negatively charged and the third particle, ordered in $p_{T}$ is either positive (red circles) or negative (black boxes) as a function of formation time. Bottom: Similar ratio of ch-particle formation time in the scenario that the leading and subleading charged particles are positively charged and the third highest $p_{T}$ particle is either positive (red) or negative (black).}\n \\label{fig4}\n\\end{figure}\n\n\\section{Conclusions}\n\nWe defined the concept of jet substructure and introduced many of the recently studied analysis techniques such as SoftDrop and iterative clustering. We also introduced the idea of formation times at varying stages of the jet shower and presented the selection of the resolved splittings. Recent STAR data of the formation times corresponding to the SD first splits, leading and subleading charged hadron and the resolved splits was discussed. The resolved splitting is shown to identify a particular time within the jet clustering tree wherein the splittings are consistent with the formation time estimated via hadronized charged particles. For jets at RHIC energies, the transition region from pQCD to npQCD is expected to occur $\\tau \\approx 3-5$ fm\/$c$. We finally highlighted the ability of such formation time observables to potentially study hadronization mechanisms in data by looking at the dependence of the electric charge of the particles considered in the calculation of the charged particle formation time. This work was supported by the Office of Nuclear Physics of the U.S. Department of Energy under award number DE-SC004168. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n{\nMultivariate time-series (temporal signals) have been studied in the statistics and signal processing societies for many years (e.g., see \\cite{tsay2013multivariate,gomez2016multivariate} for a non-exhaustive literature survey), and traditional analysis methods usually highly depend on some predefined models and do not consider the unknown nonlinear structure of the variables that often exists underlying the high-dimensional time samples.\nIn order to accommodate contemporary data acquisitions and collections, large research activity has been devoted to developing spatiotemporal analysis that is specifically-designed to infer this underlying structure and\/or take it explicitly into account.\nIn the last decade, perhaps the most notable attempts to handle such signals with a geometry defined by graphs are graph signal processing \\cite{shuman2013emerging,sandryhaila2013discrete,sandryhaila2014discrete,ortega2018graph}, graph neural networks \\cite{kipf2016semi,scarselli2008graph}, and geometric deep learning \\cite{bronstein2017geometric}.\nAnother prominent line of work is based on an operator-theoretic approach for dynamical systems analysis \\cite{schmid2010dynamic,budivsic2012applied,tu2014dynamic,kutz2016dynamic}, where the time samples have a manifold structure.\nStill, despite these recent efforts, to the best of our knowledge, when there is a time-varying manifold structure underlying the time samples, only a few works are available, e.g. \\cite{froyland2015dynamic,banisch2017understanding,froyland2020dynamic}.\n\n \nIn this work, we propose a new multi-resolution spatiotemporal analysis of multivariate time-series. In contrast to standard multi-resolution analysis using wavelets defined on Euclidean space \\cite{daubechies1992ten,mallat1999wavelet}, we present an operator-based analysis approach combining manifold learning and Riemannian geometry, which we term {\\em Riemannian multi-resolution analysis} (RMRA).\nConcretely, consider a multivariate time-series $\\{\\mathbf{x}_t\\}$. Suppose the temporal propagation of the time-series at time step $t$ can be modelled by two diffeomorphic manifolds $f_t:\\mathcal{M}_t \\rightarrow \\mathcal{M}_{t+1}$, and suppose the corresponding pairs of time samples $(\\mathbf{x}_t, \\mathbf{x}_{t+1})$ are given by $\\mathbf{x}_t[i] \\in \\mathcal{M}_t$ and $\\mathbf{x}_{t+1}[i] = f_t(\\mathbf{x}_t[i]) \\in \\mathcal{M}_{t+1}$, where $\\mathbf{x}_t[i]$ is the $i$th entry of the sample $\\mathbf{x}_t$ for $i=1,\\ldots,N$. \nNote that the entries of the samples $\\mathbf{x}_t$ lie on a manifold, and therefore, each entry is typically high-dimensional. In other words, at each time $t$, we have $N$ high-dimensional points that are distributed on the manifold $\\mathcal{M}_t$.\nOur RMRA consists of the following steps. First, we construct a diffusion operator for each time sample $\\mathbf{x}_t$, characterizing its underlying manifold $\\mathcal{M}_t$. This step is performed using a manifold learning technique, diffusion maps \\cite{Coifman2006}, that facilitates a finite-dimensional matrix approximation of the Laplacian operator of the manifold based on the time sample. This approximation is informative because the Laplacian operator is known to bear the geometric information of the manifold \\cite{berard1994embedding,jones2008manifold}. Then, for each pair of temporally consecutive time frames $(\\mathbf{x}_t,\\mathbf{x}_{t+1})$, we present two composite operators based on ``Riemannian combinations'' of the two respective diffusion operators. \nTypically, diffusion operators are not symmetric, but they are similar to symmetric positive-definite (SPD) matrices. We could thus define diffusion operators as SPD matrices, whose space is endowed with a Riemannian structure.\nTherefore, taking into account this Riemannian manifold structure for the composition of the operators is natural. \nIndeed, we show, both theoretically and in practice, that one operator enhances common components that are expressed similarly in $\\mathcal{M}_t$ and $\\mathcal{M}_{t+1}$, while the other enhances common components that are expressed differently. These properties could be viewed as analogous to low-pass and high-pass filters in this setting, leading to a spatiotemporal decomposition of the multivariate time series into ``low frequency'' and ``high frequency'' components, by considering the common components expressed similarly (resp. differently) as the slowly (resp. rapidly) varying components.\n\nTo facilitate the multi-resolution analysis of the entire temporal sequence, the construction of the composite operators is recursively repeated at different time scales. Since the composite operators are viewed as low-pass and high-pass filters, the proposed framework can be viewed as analogous to the wavelet decomposition for time-varying manifolds in the following sense. At each iteration, the two consecutive time samples are ``fused'' using the composite operators, ``decomposing'' the multivariate time-series into two components: one that varies slowly and one that varies rapidly. The fast varying component is viewed as the ``spectral feature'' of the ``first layer'', and the slowly varying component is ``downsampled'', decomposed again in the next iteration using the composite operators into a slow component and a fast component. Again, the fast component leads to the ``spectral feature'' of the ``second layer''. By iterating this procedure, the multivariate time series is decomposed in multiple resolutions.\n\nBroadly, the basic building block of our analysis, focusing on one time step, consists of two construction steps. First, given two consecutive time samples $(\\mathbf{x}_t, \\mathbf{x}_{t+1})$, we learn the underlying manifolds $\\mathcal{M}_t$ and $\\mathcal{M}_{t+1}$, and then, we study the (unknown) diffeomorphism $f_t$.\nWe posit that this building block can serve as an independent analysis module by itself.\nIndeed, the setting of one time step we consider can be recast as a related multiview data analysis problem (see Section \\ref{sec:related_work}). Consider two diffeomorphic manifolds $\\mathcal{M}_1$ and $\\mathcal{M}_2$ and the diffeomorphism $f:\\mathcal{M}_1 \\rightarrow \\mathcal{M}_2$. Let $x \\in \\mathcal{M}_1$ and $y=f(x)\\in \\mathcal{M}_2$. The pair $(x,y)$ could be considered as two views of some object of interest, providing distinct and complementary information. Applying the proposed two-step procedure to this case first learns the manifold of each view, and then, studies the diffeomorphism representing the relationship between the two views.\nIn \\cite{shnitzer2019recovering}, the diffeomorphism was analyzed in terms of common and unique components, which were represented by the eigenvectors of the diffusion operators.\nHere, we further characterize these spectral common components. Roughly, the common components are classified into two kinds: components that are expressed similarly in the two manifolds in the sense that they have similar eigenvalues in both manifolds, and components that are expressed differently in the sense that they have different eigenvalues. Furthermore, we refine the analysis and in addition to considering\n{\\em strictly common components}, i.e., the same eigenvectors in both manifolds, we also consider {\\em weakly common components}, i.e., similar but not the same eigenvectors. \nIn contrast to the local analysis presented in \\cite{shnitzer2019recovering}, we provide global and spectral analyses, showing that our method indeed extracts and identifies these different components.\n\nWe demonstrate the proposed RMRA on a dynamical system with a transitory double gyre configuration. We show that this framework is sensitive to the change rate of the dynamical system at different time-scales. Such a framework may be especially suitable for studying non-stationary multivariate time-series, particularly when there is a nontrivial geometric relationship among the multivariate coefficients.\nIn addition, for the purpose of multimodal data analysis, we demonstrate that the proposed Riemannian composite operators enhance common structures in remote sensing data captured using hyperspectral and LiDAR sensors.\n\nThe remainder of this paper is organized as follows. In Section \\ref{sec:related_work}, we review related work. In Section \\ref{sec:prelim}, we present preliminaries. In Section \\ref{sec:rmra}, we present the proposed approach for multi-resolution spatiotemporal analysis using Riemannian composition of diffusion operators. Section \\ref{sec:results} shows experimental results. In Section \\ref{sec:analysis}, we present spectral analysis of the proposed composite operators. Finally, in Section \\ref{sec:conc}, we conclude the paper.\n}\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\\section{Related work}\n\\label{sec:related_work}\n\n\\subsection{Manifold learning, diffusion maps, and diffusion wavelets}\n\n{\nManifold learning is a family of methods that consider data lying on some inaccessible manifold and provide a new low-dimensional representation of the data based on intrinsic patterns and similarities in the data \\cite{tenenbaum2000global,Roweis2000,Belkin2003,Coifman2006}.\nFrom an algorithmic viewpoint, manifold learning techniques are broadly based on two stages. The first stage is the computation of a typically positive kernel that provides a notion of similarity between the data points. The second stage is the spectral analysis of the kernel, giving rise to an embedding of the data points into a low-dimensional space.\nSuch a two-stage procedure results in aggregation of multiple pairwise similarities of data points, facilitating the extraction of the underlying manifold structure. This procedure was shown to be especially useful when there are limited high-dimensional data, plausibly circumventing the curse of dimensionality.\n\nWhile the spectral analysis of the kernels has been the dominant approach and well investigated, recent work explores different directions as well.\nOne prominent direction employs an operator-based analysis, which has led to the development of several key methods.\nArguably the first and most influential is diffusion maps \\cite{Coifman2006}\\footnote{Laplacian eigenmaps could also be considered if the diffusion time is not taken into account \\cite{Belkin2003}.}, where a transition matrix is constructed based on the kernel, forming a random walk on the dataset; such transition matrix is viewed as a diffusion operator on the data.\nThere has been abundant theoretical support for diffusion maps. For example, it was shown in \\cite{Belkin2003,hein2006uniform,singer2006graph} that the operator associated with diffusion maps converges point-wisely to the Laplace-Beltrami operator of the underlying manifold, which embodies the geometric properties of the manifold, and its eigenfunctions form a natural basis for square integrable functions on the manifold. The spectral convergence of the eigenvalues and eigenvectors of the operator associated with diffusion maps to the eigenvalues and eigenfunctions of the Laplace-Beltrami operator was first explored in \\cite{belkin2007convergence}, and recently, the $L^\\infty$ spectral convergence with convergence rate was reported in \\cite{dunson2019spectral}. See \\cite{dunson2019spectral} and references therein for additional related work in this direction. The robustness of the diffusion maps operator was studied in \\cite{el2010information,el2016graph}, and recently, its behavior under different noise levels and kernel bandwidths was explored using random matrix theory \\cite{ding2021impact}.\n\nThe propagation rules associated with this diffusion operator are in turn used for defining a new distance, the so-called diffusion distance, which was shown to be useful and informative in many domains and applications \\cite{Lafon2006,li2017efficient,TalmonMagazine,wu2014assess}. \nThis notion of diffusion promoted the development of well-designed and controlled anisotropic diffusions for various purposes, e.g., nonlinear independent component analysis \\cite{singer2008non}, intrinsic representations \\cite{talmon2013empirical}, reduction of stochastic dynamical systems \\cite{singer2009detecting,dsilva2016data}, and time-series forecasting \\cite{zhao2016analog} and filtering \\cite{shnitzer2016manifold}, to name but a few.\nIn another line of work, the combination and composition of diffusion operators led to the development of new manifold learning techniques for learning multiple manifolds  \\cite{lederman2015alternating,shnitzer2019recovering,lindenbaum2020multi} as well as for time-series analysis \\cite{froyland2015dynamic}.\n\nA related line of work that considers multivariate time-series (high-dimensional temporal signals) introduces ways to define wavelets on graphs and manifolds, e.g., \\cite{coifman2006wavelets,hammond2011wavelets,ram2011generalized}. These techniques extend the classical wavelet analysis \\cite{mallat1999wavelet} from one or two dimensional Euclidean space to high-dimensional non-Euclidean spaces represented by graphs and manifolds. Specifically, diffusion wavelets \\cite{coifman2006wavelets} makes use of a hierarchy of diffusion operators with multiple well-designed diffusion scales organized in a dyadic tree.\nImportantly, none of these methods addresses an underlying manifold with a time-varying metric, but rather a fixed metric that exhibits different characteristics in different scales.\n}\n\n\\subsection{Manifold learning for sensor fusion}\n\n{\nThe basic building block of our RMRA is based on two diffeomorphic manifolds $\\mathcal{M}_t$ and $\\mathcal{M}_{t+1}$, which represent the temporal evolution at time $t$.\nA similar setting consisting of two diffeomorphic manifolds, say $\\mathcal{M}_1$ and $\\mathcal{M}_2$, has recently been investigate in the context of multimodal data analysis and sensor fusion.\n\nThe sensor fusion problem typically refers to the problem of harvesting useful information from data collected from multiple, often heterogeneous, sensors. \nSensor fusion is a gigantic field. One line of work focuses on the extraction, analysis and comparison of the components expressed by the different sensors for the purpose of gaining understanding of the underlying scene \\cite{murphy2018diffusion,swatantran2011mapping,czaja2016fusion}. However, due to the complex nature of such data, finding informative representations and metrics of these components by combining the information from the different sensors is challenging.\nRecently, several papers propose data fusion methods relying on manifold learning techniques and operator-based data analysis \\cite{de2005spectral,eynard2015multimodal,lederman2015alternating,shnitzer2019recovering,talmon2019latent,katz2019alternating,lindenbaum2020multi}.\nThe basic idea is that data from different modalities or views are fused by constructing kernels that represent the data from each view and operators that combine those kernels.\nDifferent approaches are considered for the combination of kernels. \nPerhaps the most relevant to the present work is the alternating diffusion operator, which was introduced in \\cite{lederman2015alternating,talmon2019latent} and shown to recover the common latent variables from multiple modalities.\nThis operator is defined based on a product of two diffusion operators and then used for extracting a low dimensional representation of the common components shared by the different sensors.\nOther related approaches include different combinations of graph Laplacians  \\cite{de2005spectral,eynard2015multimodal}, product of kernel density estimators and their transpose for nonparametric extension of canonical correlation analysis \\cite{michaeli2016nonparametric}, and various other combinations of diffusion operators \\cite{katz2019alternating,shnitzer2019recovering,lindenbaum2020multi}. For a more comprehensive review of the different approaches, see \\cite{shnitzer2019recovering} and references therein.\n\nLargely, most existing sensor fusion algorithms, and particularly those based on kernel and manifold learning approaches, focus on the extraction and representation of the common components, in a broad sense. The sensor fusion framework proposed in \\cite{shnitzer2019recovering} extends this scope and considers both the common components and the unique components of each sensor. \nTherefore, in the context of the present work, it could be used for the analysis of the basic building block consisting of two diffeomorphic manifolds. \nHowever, similarly to the other methods described above, the kernel combination in \\cite{shnitzer2019recovering} is achieved through linear operations, thereby ignoring the prototypical geometry of the kernels.\nConversely, in this work, by taking the Riemannian structure of SPD matrices into account, we propose a new geometry-driven combination of kernels and a systematic analysis.\nThis aspect of our work could be viewed as an extension of \\cite{shnitzer2019recovering} for the purpose of sensor fusion and multimodal data analysis, in addition to the new utility for multivariate time-series analysis.\n}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Preliminaries}\\label{sec:prelim}\nIn this section we briefly present the required background for our method. \nFor further details on the theory and derivations we refer the readers to \\cite{bhatia2009positive} and \\cite{Coifman2006}.\n\n\\subsection{Riemannian {Structure} of SPD Matrices\\label{sub:bg_spd}}\n\n{\nIn many recent studies, representing the raw data using SPD matrices and taking into account their specific Riemannian geometry have shown promising results, e.g. in computer vision \\cite{bergmann2018priors,tuzel2008pedestrian}, for domain adaptation \\cite{yair2019parallel}, on medical data \\cite{barachant2013classification} and in recognition tasks \\cite{harandi2014manifold}.\nFor example, Barachant et al. \\cite{barachant2013classification} proposed a support-vector-machine (SVM) classifier that takes into account the Riemannian geometry of the features, which are SPD covariance matrices, representing Electroencephalogram (EEG) recordings. They showed that their ``geometry-aware'' classifier obtains significantly better results compared with a classifier that simply vectorizes the covariance matrices.\n\nHere, we consider the space of SPD matrices endowed with the so-called affine-invariant metric \\cite{pennec2006riemannian}. Using this particular Riemannian geometry results in closed-form expressions for useful properties and operations, such as the geodesic path connecting two points on the manifold \\cite{bhatia2009positive} and the logarithmic map and the exponential map \\cite{pennec2006riemannian}, which locally project SPD matrices onto the tangent space and back. \nWhile the focus is on the affine-invariant metric, which is arguably the most widely used, we remark that other geometries of SPD matrices exist, e.g., the log-Euclidean \\cite{arsigny2007geometric,quang2014log}, the log-det \\cite{sra2012new,chebbi2012means}, the log-Cholesky \\cite{lin2019riemannian}, and the Bures-Wasserstein \\cite{malago2018wasserstein,bhatia2019bures}, which could be considered as well.\nIn the context of this work, since diffusion operators are strictly positive in principal but in practice often have negligible eigenvalues, one particular advantage of the affine-invariant geometry is its existing extensions to symmetric positive semi-definite (SPSD) matrices (see Section \\ref{subsec:SPSD}).\n}\n\nConsider the set of symmetric matrices in $\\mathbb{R}^{N\\times N}$, denoted by $\\mathcal{S}_N$.\nA symmetric matrix $\\mathbf{W}\\in\\mathcal{S}_N$ is an SPD matrix if it has strictly positive eigenvalues.\nLet $\\mathcal{P}_N$ denote the set of all $N\\times N$ SPD matrices.\nThe tangent space at any point in this set is the space of symmetric matrices $\\mathcal{S}_N$. \nWe denote the tangent space at $\\mathbf{W}\\in\\mathcal{P}_N$ by $\\mathcal{T}_\\mathbf{W}\\mathcal{P}_N$.\nIn this work we consider the following affine-invariant metric in the tangent space at each matrix $\\mathbf{W}\\in\\mathcal{P}_N$, which forms a differentiable Riemannian manifold \\cite{moakher2005differential}:\n\\begin{equation}\n    \\left\\langle \\mathbf{D}_1,\\mathbf{D}_2\\right\\rangle_\\mathbf{W} = \\left\\langle \\mathbf{W}^{-1\/2}\\mathbf{D}_1\\mathbf{W}^{-1\/2}, \\mathbf{W}^{-1\/2}\\mathbf{D}_2\\mathbf{W}^{-1\/2} \\right\\rangle\\label{eq:riemann_spd_metric}\n\\end{equation}\nwhere $\\mathbf{D}_1,\\mathbf{D}_2\\in\\mathcal{T}_\\mathbf{W}\\mathcal{P}_N$ denote matrices in the tangent space at $\\mathbf{W}\\in\\mathcal{P}_N$ and $\\left\\langle\\cdot,\\cdot\\right\\rangle$ is given by the standard Frobenius inner product $\\left\\langle \\mathbf{A},\\mathbf{B}\\right\\rangle=\\mathrm{Tr}\\left(\\mathbf{A}^T\\mathbf{B}\\right)$.\nUsing this metric, there is a unique geodesic path connecting any two matrices $\\mathbf{W}_1,\\mathbf{W}_2\\in\\mathcal{P}_N$ \\cite{bhatia2009positive}, which is explicitly given by:\n\\begin{equation}\n\\gamma_{\\mathbf{W}_1\\rightarrow\\mathbf{W}_2}(p)=\\mathbf{W}_1^{1\/2}\\left(\\mathbf{W}_1^{-1\/2}\\mathbf{W}_2\\mathbf{W}_1^{-1\/2}\\right)^p\\mathbf{W}_1^{1\/2}, \\ \\ p\\in[0,1],\\label{eq:riemann_spd_geodesic}\n\\end{equation}\n{The arc-length of this geodesic path defines the Riemannian distance on the manifold, \n\\begin{equation}\nd^2_R\\left(\\mathbf{W}_1,\\mathbf{W}_2\\right)=\\left\\Vert \\log\\left(\\mathbf{W}_1^{-1\/2}\\mathbf{W}_2\\mathbf{W}_1^{-1\/2}\\right)\\right\\Vert^2_F.  \n\\end{equation}\nUsing the Fr\\'echet mean, we define the Riemannian mean of a set of matrices, {$\\mathbf{W}_1,\\ldots,\\mathbf{W}_n$,} by $\\arg\\min_{\\mathbf{W}\\in\\mathcal{P}_N}{\\sum_{i=1}^n} d^2_R\\left(\\mathbf{W},\\mathbf{W}_i\\right)$. The Riemannian mean of two matrices is a special case, which coincides with the mid-point of the geodesic path connecting them and has the following closed form: }\n\\begin{equation}\n    \\gamma_{\\mathbf{W}_1\\rightarrow\\mathbf{W}_2}(1\/2)=\\mathbf{W}_1^{1\/2}\\left(\\mathbf{W}_1^{-1\/2}\\mathbf{W}_2\\mathbf{W}_1^{-1\/2}\\right)^{1\/2}\\mathbf{W}_1^{1\/2}\n\\end{equation}\n\nThe mapping between the Riemannian manifold of SPD matrices and its tangent space is given by the exponential map and the logarithmic map.\nEach matrix $\\mathbf{D}$ in the tangent space at $\\mathbf{W}\\in\\mathcal{P}_N$ \ncan be seen as the derivative of the geodesic connecting $\\mathbf{W}$ and $\\tilde{\\mathbf{W}}=\\mathrm{Exp}_\\mathbf{W}(\\mathbf{D})$, i.e., $\\gamma_{\\mathbf{W}\\rightarrow\\tilde{\\mathbf{W}}}(p)$, at $p=0$.\nThe exponential map in this setting has a known closed-form given by \\cite{moakher2005differential}:\n\\begin{equation}\n    \\mathrm{Exp}_{\\mathbf{W}}\\left(\\mathbf{D}\\right)=\\mathbf{W}^{1\/2}\\exp\\left(\\mathbf{W}^{-1\/2}\\mathbf{D}\\mathbf{W}^{-1\/2}\\right)\\mathbf{W}^{1\/2}\\label{eq:spd_expmap},\n\\end{equation}\nwhere $\\mathrm{Exp}_{\\mathbf{W}}(\\mathbf{D})\\in\\mathcal{P}_N$ and $\\exp(\\cdot)$ is applied to the eigenvalues.\nThe inverse of the exponential map is the logarithmic map, which is explicitly given by:\n\\begin{equation}\n    \\mathrm{Log}_{\\mathbf{W}}(\\tilde{\\mathbf{W}})=\\mathbf{W}^{1\/2}\\log\\left(\\mathbf{W}^{-1\/2}\\tilde{\\mathbf{W}}\\mathbf{W}^{-1\/2}\\right)\\mathbf{W}^{1\/2},\\label{eq:spd_logmap}\n\\end{equation}\nwhere $\\log(\\cdot)$ is applied to the eigenvalues, $\\tilde{\\mathbf{W}}\\in\\mathcal{P}_N$, and $\\mathrm{Log}_{\\mathbf{W}}(\\tilde{\\mathbf{W}})\\in\\mathcal{T}_\\mathbf{W}\\mathcal{P}_N$.\n\n\n\\subsection{An extension to SPSD Matrices\\label{subsec:SPSD}}\n\nIn practice, the matrices of interest are often not strictly positive, but rather symmetric positive \\emph{semi}-definite (SPSD) matrices, that is symmetric matrices with non-negative eigenvalues.\nBelow is a summary of a Riemannian geometry introduced in \\cite{bonnabel2010riemannian} that extends the affine-invariant metric. We remark that the existence of such an extension serves as an additional motivation to particularly consider the affine-invariant metric over the alternatives.\n\nLet $\\mathcal{S}^+(r,N)$ denote the set of $N\\times N$ SPSD matrices of rank $r<N$, and let $\\mathcal{V}_{N,r}$ denote the set of $N\\times r$ matrices with orthonormal columns, i.e. $\\mathbf{V}^T\\mathbf{V}=\\mathrm{I}_{r\\times r}, \\ \\forall\\mathbf{V}\\in\\mathcal{V}_{N,r}$.\nGiven an SPSD matrix $\\mathbf{W}\\in\\mathcal{S}^+(r,N)$, we consider the following decomposition:\n\\begin{equation}\n    \\mathbf{W} = \\mathbf{V}\\mathbf{\\Lambda}\\mathbf{V}^T\\,,\n\\end{equation}\nwhere $\\mathbf{V}\\in\\mathcal{V}_{N,r}$ and $\\mathbf{\\Lambda}\\in\\mathcal{P}_r$.\nConsequently, the pair $\\left(\\mathbf{V},\\mathbf{\\Lambda}\\right)\\in\\mathcal{V}_{N,r}\\times\\mathcal{P}_r$ can be considered as a representation of the matrix $\\mathbf{W}$.\nBased on this representation, the authors of \\cite{bonnabel2010riemannian} proposed to present vectors in the tangent space of $\\mathcal{S}^+\\left(r,N\\right)$ at $\\mathbf{W}$, denoted by $\\mathcal{T}_\\mathbf{W}\\mathcal{S}^+\\left(r,N\\right)$, by the infinitesimal variation $\\left(\\mathbf{\\Delta},\\mathbf{D}\\right)$, where $\\mathbf{\\Delta}=\\mathbf{V}_\\perp\\mathbf{B}$, $\\mathbf{V}_\\perp\\in\\mathcal{V}_{N,N-r}$, $\\mathbf{V}_\\perp\\mathbf{V}=0$, $\\mathbf{B}\\in\\mathbb{R}^{(N-r)\\times r}$ and $\\mathbf{D}\\in\\mathcal{T}_{\\mathbf{\\Lambda}}\\mathcal{P}_r=\\mathcal{S}_r$.\nUsing these tangent vectors, the metric on the tangent space of the SPSD manifold can be defined as a generalization of the metric on the manifold of SPD matrices:\n\\begin{equation}\n    \\left\\langle\\left(\\mathbf{\\Delta}_1,\\mathbf{D}_1\\right),\\left(\\mathbf{\\Delta}_2,\\mathbf{D}_2\\right)\\right\\rangle_{\\left(\\mathbf{V},\\mathbf{\\Lambda}\\right)} = \\mathrm{Tr}\\left(\\mathbf{\\Delta}^T_1\\mathbf{\\Delta}_2\\right) + k\\left\\langle\\mathbf{D}_1,\\mathbf{D}_2\\right\\rangle_\\mathbf{\\Lambda}\\label{eq:SPSDmetric}\n\\end{equation}\nfor some $k>0$, where $\\left(\\mathbf{\\Delta}_1,\\mathbf{D}_1\\right),\\left(\\mathbf{\\Delta}_2,\\mathbf{D}_2\\right)\\in\\mathcal{T}_\\mathbf{W}\\mathcal{S}^+\\left(r,N\\right)$ and $\\left\\langle\\cdot,\\cdot\\right\\rangle_\\mathbf{\\Lambda}$ denotes the metric defined in \\eqref{eq:riemann_spd_metric} for SPD matrices.\n\nLet $\\mathbf{W}_1=\\mathbf{V}_1\\mathbf{\\Lambda}_1\\mathbf{V}_1^T$ and $\\mathbf{W}_2=\\mathbf{V}_2\\mathbf{\\Lambda}_2\\mathbf{V}_2^T$ denote the decompositions of two SPSD matrices $\\mathbf{W}_1,\\mathbf{W}_2\\in\\mathcal{S}^+\\left(r,N\\right)$, where $\\mathbf{V}_1,\\mathbf{V}_2\\in\\mathcal{V}_{N,r}$ and $\\mathbf{\\Lambda}_1,\\mathbf{\\Lambda}_2\\in\\mathcal{P}_r$.\nThe closed-form expression of the geodesic path connecting any two such matrices in $\\mathcal{S}^+\\left(r,N\\right)$ using the metric in \\eqref{eq:SPSDmetric} is unknown. \nHowever, the following approximation of it was proposed in \\cite{bonnabel2010riemannian}.\nDenote the singular value decomposition (SVD) of $\\mathbf{V}_2^T\\mathbf{V}_1$ by $\\mathbf{O}_2\\mathbf{\\Sigma}\\mathbf{O}_1^T$, where $\\mathbf{O}_1,\\mathbf{O}_2\\in\\mathbb{R}^{r\\times r}$ and $\\mathrm{diag}(\\mathbf{\\Sigma})$ are the cosines of the principal angles between $\\mathrm{range}(\\mathbf{W}_1)$ and $\\mathrm{range}(\\mathbf{W}_2)$, where $\\mathrm{range}(\\mathbf{W})$ denotes the column space of $\\mathbf{W}$.\nDefine $\\mathbf{\\Theta}=\\arccos\\left(\\mathbf{\\Sigma}\\right)$, which is a diagonal matrix of size $r\\times r$ with the principal angles between the two subspaces on its diagonal.\nThe approximation of the geodesic path connecting two points in $\\mathcal{S}^+\\left(r,N\\right)$ is then given by:\n\\begin{equation}\n    \\tilde{\\gamma}_{\\mathbf{W}_1\\rightarrow\\mathbf{W}_2}(p)=\\mathbf{U}_{\\mathbf{W}_1\\rightarrow\\mathbf{W}_2}(p)\\mathbf{R}_{\\mathbf{W}_1\\rightarrow\\mathbf{W}_2}(p)\\mathbf{U}^T_{\\mathbf{W}_1\\rightarrow\\mathbf{W}_2}(p), \\ \\ p\\in[0,1]\\label{eq:riemann_spsd_geodesic}\n\\end{equation}\nwhere $\\mathbf{R}_{\\mathbf{W}_1\\rightarrow\\mathbf{W}_2}(p)$ is the geodesic path connecting SPD matrices as defined in \\eqref{eq:riemann_spd_geodesic} calculated between the matrices $\\mathbf{R}_1=\\mathbf{O}_1^T\\mathbf{\\Lambda}_1\\mathbf{O}_1$ and $\\mathbf{R}_2=\\mathbf{O}_2^T\\mathbf{\\Lambda}_2\\mathbf{O}_2$, i.e. $\\mathbf{R}_{\\mathbf{W}_1\\rightarrow\\mathbf{W}_2}(p)=\\gamma_{\\mathbf{R}_1\\rightarrow\\mathbf{R}_2}(p)$, and $\\mathbf{U}_{\\mathbf{W}_1\\rightarrow\\mathbf{W}_2}(p)$ is the geodesic connecting $\\mathrm{range}(\\mathbf{W}_1)$ and $\\mathrm{range}(\\mathbf{W}_2)$ on the Grassman manifold (the set of $r$ dimensional subspaces of $\\mathbb{R}^N$), defined by:\n\\begin{equation}\n    \\mathbf{U}_{\\mathbf{W}_1\\rightarrow\\mathbf{W}_2}(p) = \\mathbf{U}_1\\cos\\left(\\mathbf{\\Theta}p\\right)+\\mathbf{X}\\sin\\left(\\mathbf{\\Theta}p\\right)\\label{eq:grassman_geodesic}\n\\end{equation}\nwhere $\\mathbf{U}_1=\\mathbf{V}_1\\mathbf{O}_1$, $\\mathbf{U}_2=\\mathbf{V}_2\\mathbf{O}_2$ and $\\mathbf{X}=\\left(\\mathrm{I}-\\mathbf{U}_1\\mathbf{U}_1^T\\right)\\mathbf{U}_2\\left(\\sin\\left(\\mathbf{\\Theta}\\right)\\right)^{\\dagger}$, with $(\\cdot)^\\dagger$ denoting the Moore-Penrose pseudo-inverse.\n\n\n\n\n\\subsection{Diffusion Operator\\label{sub:meas_rep}}\n{As described in Section \\ref{sec:related_work}, most manifold learning methods, and particularly diffusion maps \\cite{Coifman2006}, are based on positive kernel matrices. Here, we briefly present the construction of such a kernel, which we term the diffusion operator, as proposed in \\cite{Coifman2006}. In the sequel, we employ this diffusion operator in our framework {to recover the geometry} underlying each modality.}\n\nGiven a set of $N$ points, $\\{\\mathbf{x}[i]\\}_{i=1}^N$, which are sampled from some hidden manifold $\\mathcal{M}$ embedded in $\\mathbb{R}^n$, consider the following affinity kernel matrix $\\mathbf{K}\\in\\mathbb{R}^{N\\times N}$, whose $(i,j)$th entry is given by:\n\\begin{equation}\n\\mathrm{K}[i,j] = \\exp\\left(-\\frac{\\left\\Vert\\mathbf{x}[i]-\\mathbf{x}[j]\\right\\Vert_2^2}{\\sigma^2}\\right),\\label{eq:SPDkernK}\n\\end{equation}\nwhere $\\left\\Vert\\cdot\\right\\Vert_2$ denotes the $\\ell_2$ norm and $\\sigma$ denotes the kernel scale, typically set to the median of the Euclidean distances between the sample points multiplied by some scalar. {By Bochner's theorem, $\\mathbf{K}$ is an SPD matrix.}\nThe kernel is normalized twice according to:\n\\begin{eqnarray}\n\\widehat{\\mathbf{W}} & = & \\widehat{\\mathbf{D}}^{-1}\\ \\mathbf{K}\\ \\widehat{\\mathbf{D}}^{-1}\\nonumber\\\\\n\\mathbf{W} & = & \\mathbf{D}^{-1\/2}\\ \\widehat{\\mathbf{W}}\\ \\mathbf{D}^{-1\/2},\\label{eq:SPDKern}\n\\end{eqnarray}\nwhere $\\widehat{\\mathbf{D}}$ and $\\mathbf{D}$ are diagonal matrices with $\\widehat{\\mathrm{D}}[i,i]=\\sum_{j=1}^N \\mathrm{K}[i,j]$ and $\\mathrm{D}[i,i]=\\sum_{j=1}^N \\widehat{\\mathrm{W}}[i,j]$, respectively.\n\nThe matrix $\\mathbf{W}$ defined in \\eqref{eq:SPDKern} is similar to the diffusion {operator considered in} \\cite{Coifman2006}{, which we call the diffusion maps operator for simplicity,} with a normalization that removes the point density influence, given by $\\mathbf{W}_{\\texttt{DM}}=\\mathbf{D}^{-1}\\widehat{\\mathbf{W}}$.\nDue to this similarity, the matrix $\\mathbf{W}$ and the diffusion maps operator $\\mathbf{W}_{\\texttt{DM}}$ share the same eigenvalues and their eigenvectors are related by $\\psi_{\\texttt{DM}}=\\mathbf{D}^{-1\/2}\\psi$ and $\\tilde{\\psi}_{\\texttt{DM}}=\\mathbf{D}^{1\/2}\\psi$, where $\\psi$ denotes an eigenvector of $\\mathbf{W}$ and $\\psi_{DM}$ and $\\tilde{\\psi}_{\\texttt{DM}}$ denote the right and left eigenvectors of $\\mathbf{W}_{\\texttt{DM}}$, respectively.\n\n\n\n\n\n\n\n\n\\section{Riemannian Multi-resolution Analysis}\\label{sec:rmra}\n\nWe are ready to present our multi-resolution framework for multivariate time-series analysis from a manifold learning perspective. In this section, we focus on the algorithmic aspect, and in Section \\ref{sec:analysis} we present the theoretical justification. We start by introducing the Riemannian composition of two operators that capture the relationship between two datasets sampled from two underlying diffeomorphic manifolds. Then, we generalize the setting to a sequence of datasets in time by presenting a wavelet-like analysis using the composite operators. Finally, we conclude this section with important implementation remarks.\n\n\n\\subsection{Riemannian composition of two operators\\label{sub:op_def}}\n\n{Consider two datasets of $N$ points denoted by $\\{\\mathbf{x}_1[i]\\}_{i=1}^N,\\{\\mathbf{x}_2[i]\\}_{i=1}^N$. Suppose there is some correspondence between the datasets and that they are ordered according to this correspondence, i.e., the two points $\\mathbf{x}_1[i]$ and $\\mathbf{x}_2[i]$ correspond. Such a correspondence could be the result of simultaneous recording from two, possibly different, sensors.\nWe aim to recover the common structures in these datasets and characterize their expression in each dataset. Specifically, we consider two types of common components: common components that are expressed similarly in the two datasets and common components that are expressed differently.}\n\nTo this end, we propose a two-step method.\nFirst, we assume that each dataset lies on some manifold and we characterize its underlying geometry using a diffusion operator constructed according to \\eqref{eq:SPDKern}. \nThis results in two SPD matrices denoted by $\\mathbf{W}_1$ and $\\mathbf{W}_2$ (see Section \\ref{sub:bg_spd}).\nThen, we propose to ``fuse'' the two datasets by considering compositions of $\\mathbf{W}_1$ and $\\mathbf{W}_2$ based on Riemannian geometry.\nIn contrast to previous studies that consider linear combinations involving addition, subtraction, and multiplication, e.g., \\cite{lederman2015alternating,shnitzer2019recovering,lindenbaum2020multi}, which often results in non-symmetric or non-positive matrices violating the fundamental geometric structure of diffusion operators, our Riemannian compositions yield symmetric and SPD matrices. \n{{Specifically, we define} two new operators by:\n\\begin{eqnarray}\n\\mathbf{S}_{p} & = & \\mathbf{W}_1\\#_{p}\\mathbf{W}_2=\\mathbf{W}_1^{1\/2}\\left(\\mathbf{W}_1^{-1\/2}\\mathbf{W}_2\\mathbf{W}_1^{-1\/2}\\right)^{p}\\mathbf{W}_1^{1\/2},\\label{eq:Sr}\\\\\n\\mathbf{F}_{p} & = & \\mathrm{Log}_{\\mathbf{S}_{p}}\\left(\\mathbf{W}_1\\right)=\\mathbf{S}_{p}^{1\/2}\\log\\left(\\mathbf{S}_{p}^{-1\/2}\\mathbf{W}_1\\mathbf{S}_{p}^{-1\/2}\\right)\\mathbf{S}_{p}^{1\/2},\\label{eq:Ar}\n\\end{eqnarray}\nwhere $0 \\le p \\le 1$ denotes the position along the geodesic path connecting $\\mathbf{W}_1$ and $\\mathbf{W}_2$ on the SPD manifold, $\\mathbf{W}_1\\#_{p}\\mathbf{W}_2$ with $p=1\/2$ denotes the midpoint on this geodesic, and $\\mathrm{Log}_{\\mathbf{S}_{p}}\\left(\\mathbf{W}_1\\right)$ denotes the logarithmic map, projecting the matrix $\\mathbf{W}_1$ onto the tangent space of the SPD manifold at point $\\mathbf{S}_{p}=\\mathbf{W}_1\\#_{p}\\mathbf{W}_2$. Figure \\ref{fig:SA_vis} presents an illustration of the definitions of the operators on the Riemannian manifold of SPD matrices and its tangent space.\n\n\nIntuitively, {$\\mathbf{S}$} describes the mean of the two matrices, so it enhances the components that are expressed similarly; that is, common eigenvectors with similar eigenvalues. Conversely, $\\mathbf{F}_{p}$ can be seen as the difference of $\\mathbf{S}_{p}$ and $\\mathbf{W}_1$ along the geodesic connecting {them}, and therefore, it is related to the components expressed differently; that is, the common eigenvectors with different eigenvalues. In Section \\ref{sec:analysis} we {provide a theoretical justification for the above statements}.\n\nWe remark that $\\mathbf{F}_{p}$ is symmetric but not positive-definite, since it is defined as a projection of an SPD matrix onto the tangent space, and that using $\\mathbf{W}_2$ instead of $\\mathbf{W}_1$ in the projection leads only to a change of sign.\nIn addition, given $\\mathbf{S}_p$ and $\\mathbf{F}_p$, both SPD matrices $\\mathbf{W}_1$ and $\\mathbf{W}_2$ can be reconstructed using the exponential map $\\mathrm{Exp}_{\\mathbf{S}_p}\\left(\\pm\\mathbf{F}_p\\right)$ (as defined in \\eqref{eq:spd_expmap}).}\nFor simplicity of notations, we focus in the following on $\\mathbf{F}_p$ and $\\mathbf{S}_p$ with $p=0.5$ and omit the notation of $p$. The extension to other values of $p$ is straightforward.\n\n\n\\begin{figure}[t]\n\\centering\n\\subfloat[]{\\includegraphics[trim=30 10 30 20,clip,width=0.5\\textwidth]{figures\/SA_vis\/S_operator_vis-eps-converted-to.pdf}}\n\\subfloat[]{\\includegraphics[trim=30 60 30 70,clip,width=0.5\\textwidth]{figures\/SA_vis\/F_operator_vis-eps-converted-to.pdf}}\n\\caption{Illustration of the definitions of the operators $\\mathbf{S}$ and $\\mathbf{F}$. (a) Illustration of the operator $\\mathbf{S}$ (blue point) on the geodesic path (solid line in magenta) connecting $\\mathbf{W}_1$ and $\\mathbf{W}_2$ (magenta points) on the manifold of SPD matrices (gray surface represents one level-set of the Riemannian manifold of SPD matrices). The dashed gray line denotes the shortest Euclidean path connecting the two matrices. (b) Illustration of the operator $\\mathbf{F}$ (cyan point) on the tangent space at $\\mathbf{S}$ (colored plane). Both plots present the same region in different orientations. \n\\label{fig:SA_vis}}\n\\end{figure}\n\n\n{In the second step, we} propose new embeddings of the data points, representing the common components between the two datasets based on the operators $\\mathbf{S}$ and $\\mathbf{F}$.\nSince both operators are symmetric, their eigenvalues and eigenvectors are real, and the eigenvectors are orthogonal. \nDenote the eigenvalues and eigenvectors of the operator $\\mathbf{S}$ by $\\lambda_n^{(\\mathbf{S})}$ and $\\psi_n^{(\\mathbf{S})}$, respectively, and the eigenvalues and eigenvectors of the operator $\\mathbf{F}$ by $\\lambda_n^{(\\mathbf{F})}$ and $\\psi_n^{(\\mathbf{F})}$, respectively, where $n=1,\\dots,N$.\nThe new embeddings are constructed based on the eigenvectors of $\\mathbf{S}$ and $\\mathbf{F}$ by taking the $M\\leq N$ leading eigenvectors, i.e. eigenvectors that correspond to the $M$ largest eigenvalues (in absolute value for $\\mathbf{F}$), which are organized in decreasing order $\\lambda^{(\\mathbf{S})}_1\\geq\\lambda^{(\\mathbf{S})}_2\\geq\\dots\\geq\\lambda^{(\\mathbf{S})}_M$ and $\\lambda^{(\\mathbf{F})}_1\\geq\\lambda^{(\\mathbf{F})}_2\\geq\\dots\\geq\\lambda^{(\\mathbf{F})}_M$.\nThe new embeddings are defined by:\n\\begin{eqnarray}\n(\\mathbf{x}_1[i],\\mathbf{x}_2[i]) & \\mapsto & \\Psi^{(\\mathbf{S})}[i]=\\left\\lbrace\\psi^{(\\mathbf{S})}_1[i],\\dots,\\psi^{(\\mathbf{S})}_M[i]\\right\\rbrace\\\\\n(\\mathbf{x}_1[i],\\mathbf{x}_2[i]) & \\mapsto & \\Psi^{(\\mathbf{F})}[i]=\\left\\lbrace\\psi^{(\\mathbf{F})}_1[i],\\dots,\\psi^{(\\mathbf{F})}_M[i]\\right\\rbrace.\n\\end{eqnarray}\n\n{Algorithm \\ref{alg:twomod} summarizes the above two-step operator and embedding construction.}\nIn Section \\ref{subsub:ills_exmp}, we demonstrate the properties of the operators $\\mathbf{S}$ and $\\mathbf{F}$ and the proposed embeddings on an illustrative toy example. In Section \\ref{sec:analysis}, we present some analysis.\n\n{As a final remark, we note that} other SPD kernels and matrices {may be considered} instead of the proposed diffusion operators, e.g. covariance or correlation matrices, which can simply substitute $\\mathbf{W}_1$ and $\\mathbf{W}_2$ in the above definitions.\n\n\\begin{algorithm}[hbt!]\n\\caption{{Operator composition and spectral embedding based on Riemannian geometry}}\n\\label{alg:twomod}\n\\textbf{Input:} Two datasets $\\{\\mathbf{x}_1[i]\\}_{i=1}^N,\\{\\mathbf{x}_2[i]\\}_{i=1}^N$; {the embedding dimension $M$}\\\\\n\\textbf{Output:} Operators $\\mathbf{S}$ and $\\mathbf{F}$ and new embeddings\n$\\mathbf{\\Psi}^{(\\mathbf{S})}$ and $\\mathbf{\\Psi}^{(\\mathbf{F})}$\n\\begin{algorithmic}[1]\n\\Statex\n\\State{Construct a diffusion operator for each dataset, $\\mathbf{W}_\\ell\\in\\mathbb{R}^{N\\times N}$, $\\ell=1,2$, according to \\eqref{eq:SPDkernK} and \\eqref{eq:SPDKern}}\n\\Statex\n\\State{Build operators $\\mathbf{S}$ and $\\mathbf{F}$:}\n\\State{$\\ \\ \\ \\ \\ \\mathbf{S}=\\mathbf{W}_1^{1\/2}\\left(\\mathbf{W}_1^{-1\/2}\\mathbf{W}_2\\mathbf{W}_1^{-1\/2}\\right)^{1\/2}\\mathbf{W}_1^{1\/2}$}\n\\State{$\\ \\ \\ \\ \\ \\mathbf{F}=\\mathbf{S}^{1\/2}\\log\\left(\\mathbf{S}^{-1\/2}\\mathbf{W}_1\\mathbf{S}^{-1\/2}\\right)\\mathbf{S}^{1\/2}$}\n\\Statex\n\\State{Compute the eigenvalue decomposition of the operators $\\mathbf{S}$ and $\\mathbf{F}$}\n\\Statex\n\\State{Take the $M$ largest eigenvalues (in absolute value) and order them such that $\\lambda^{(\\mathbf{S})}_1\\geq\\lambda^{(\\mathbf{S})}_2\\geq\\dots\\geq\\lambda^{(\\mathbf{S})}_M$ and $\\lambda^{(\\mathbf{F})}_1\\geq\\lambda^{(\\mathbf{F})}_2\\geq\\dots\\geq\\lambda^{(\\mathbf{F})}_M$}\n\\Statex\n\\State{Take the corresponding $M$ eigenvectors of $\\mathbf{S}$ and define:\\\\\n$\\mathbf{\\Psi}^{(\\mathbf{S})}=\\left\\lbrace\\psi^{(\\mathbf{S})}_1,\\dots,\\psi^{(\\mathbf{S})}_M\\right\\rbrace\\in\\mathbb{R}^{N\\times M}$}\n\\Comment{{Capture} similarly expressed common components}\n\\Statex\n\\State{Take the corresponding $M$ eigenvectors of $\\mathbf{F}$ and define:\\\\\n$\\mathbf{\\Psi}^{(\\mathbf{F})}=\\left\\lbrace\\psi^{(\\mathbf{F})}_1,\\dots,\\psi^{(\\mathbf{F})}_M\\right\\rbrace\\in\\mathbb{R}^{N\\times M}$}\n\\Comment{{Capture} differently expressed common components}\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\n\\subsection{Operator-based analysis of a sequence of datasets\\label{sub:rmra}}\n\n\n{Let $\\{\\mathrm{x}_t[i]\\}_{i=1}^{N}$ denote a temporal sequence of datasets, where $t=1,\\dots,T=2^m$, $m\\in\\mathbb{N}$, denotes time, and $\\mathrm{x}_t[i]\\in\\mathbb{R}^d$ is the $i$-th point sampled at time $t$.\nConsidering first just two consecutive datasets $\\{\\mathrm{x}_t[i]\\}_{i=1}^{N}$ and $\\{\\mathrm{x}_{t+1}[i]\\}_{i=1}^{N}$ is analogous to the setting presented in Section \\ref{sub:op_def}. Applying the same analysis gives rise to the operators $\\mathbf{S}$ and $\\mathbf{F}$ corresponding to \n$\\{\\mathrm{x}_t[i]\\}_{i=1}^{N}$ and $\\{\\mathrm{x}_{t+1}[i]\\}_{i=1}^{N}$, which facilitate the extraction of the two types of underlying common components. Unlike the general setting in Section \\ref{sub:op_def}, the temporal order of the two datasets considered here allows us to view the common components that are expressed similarly and extracted by $\\mathbf{S}$ as the slowly changing components. Similarly, the common components that are expressed differently and extracted by $\\mathbf{F}$ are considered as rapidly changing components.}\n\n\nThe above description constitutes the basic building block of our analysis. \nWith that in mind, we proceed to the construction of the proposed multi-resolution analysis of the entire sequence.\nAt the first step, we build a diffusion operator according to \\eqref{eq:SPDKern} for the dataset $\\{\\mathrm{x}_t[i]\\}_{i=1}^{N}$ at each time point $t$, resulting in $T$ kernels $\\mathbf{W}_t\\in\\mathbb{R}^{N\\times N}$, $t=1,\\dots,T$.\nThen, for every pair of consecutive time-points, $(2t-1,2t)$, $t=1,\\dots,T\/2$, we construct the two operators\\footnote{Note that the operator underscore notation now denotes the time index rather than the geodesic curve parameter $p$ as in Section \\ref{sub:op_def}.} \n$\\mathbf{S}_t^{(1)}$ and $\\mathbf{F}_t^{(1)}$ according to \\eqref{eq:Sr} and \\eqref{eq:Ar} with $p=0.5$. \nThese $2\\times T\/2$ operators represent the fine level, denoted $\\ell=1$, of the multi-resolution framework and recover components which are common to consecutive time-frames.\nAt coarser (higher) levels, i.e. for $\\ell>1$, the operators are constructed according to \\eqref{eq:Sr} and \\eqref{eq:Ar} (with $p=0.5$) using the operators from the previous level as input. Specifically, at level $\\ell>1$, the operators are given by\n\\begin{eqnarray}\n\\mathbf{S}^{(\\ell)}_t&=&\\mathbf{S}_{2t-1}^{(\\ell-1)}\\#\\mathbf{S}_{2t}^{(\\ell-1)}\\label{eq:Swav}\\\\\n\\mathbf{F}_t^{(\\ell)}&=&\\mathrm{Log}_{\\mathbf{S}_t^{(\\ell)}}\\left(\\mathbf{S}_{2t-1}^{(\\ell-1)}\\right),\\label{eq:Awav}\n\\end{eqnarray}\nwhere $t=1,\\dots,T\/2^\\ell$ and $\\ell=2,\\dots,\\log_2T$.\nAt each level, only the operator $\\mathbf{S}$ is used to construct the operators of the next level. The reason for this construction choice is that $\\mathbf{S}$ enhances similarly expressed common components. \nIn the present setting, the similarly expressed common components of consecutive time frames are in fact components that change slowly in time.\nTherefore, using the operators $\\mathbf{S}^{(\\ell-1)}_t$, $t=1,\\dots,T\/2^{\\ell-1}$, to construct the operators of the coarser level, $\\ell$, has a smoothing effect.\nThis is analogous to the construction of the ordinary wavelet decomposition, where the outputs of the low-pass filters at each level are used as inputs to the coarser level.\nIn this analogy, the operator $\\mathbf{F}^{(\\ell)}_t$ can be viewed as a high-pass filter, since it enhances components that are expressed significantly different in consecutive time frames, i.e. rapidly changing components.\n\nSimilarly to the embedding defined based on $\\mathbf{S}$ and $\\mathbf{F}$ in Subsection \\ref{sub:op_def}, we define an embedding based on the eigenvectors of the operators $\\mathbf{S}^{(\\ell)}_t$ and $\\mathbf{F}^{(\\ell)}_t$ at different levels and time-frames.\nDenote the eigenvalues and eigenvectors of the operator $\\mathbf{S}^{(\\ell)}_t$ by $\\lambda_n^{(\\mathbf{S}^{(\\ell)}_t)}$ and $\\psi_n^{(\\mathbf{S}^{(\\ell)}_t)}$, respectively, and the eigenvalues and eigenvectors of the operator $\\mathbf{F}^{(\\ell)}_t$ by $\\lambda_n^{(\\mathbf{F}^{(\\ell)}_t)}$ and $\\psi_n^{(\\mathbf{F}^{(\\ell)}_t)}$, respectively, where $n=1,\\dots,N$ and the eigenvalues are ordered in decreasing magnitude.\nThe embedding at each level and each time frame is then defined by taking the $M\\leq N$ leading eigenvectors of the operators as follows:\n\\begin{eqnarray}\n\\mathbf{x}_t[i] & \\rightarrow & \\Psi^{(\\mathbf{S}^{(\\ell)}_t)}=\\left\\lbrace\\psi^{(\\mathbf{S}^{(\\ell)}_t)}_1[i],\\dots,\\psi^{(\\mathbf{S}^{(\\ell)}_t)}_M[i]\\right\\rbrace\\\\\n\\mathbf{x}_t[i] & \\rightarrow & \\Psi^{(\\mathbf{F}^{(\\ell)}_t)}=\\left\\lbrace\\psi^{(\\mathbf{F}^{(\\ell)}_t)}_1[i],\\dots,\\psi^{(\\mathbf{F}^{(\\ell)}_t)}_M[i]\\right\\rbrace.\n\\end{eqnarray}\nwhere $t=1,\\dots,T\/2^\\ell$ and $\\ell=2,\\dots,\\log_2T$.\nThese embedding coordinates capture the slowly varying components ($\\Psi^{(\\mathbf{S}^{(\\ell)}_t)}$) and fast varying components ($\\Psi^{(\\mathbf{F}^{(\\ell)}_t)}$) at each time frame $t$ and each level $\\ell$.\n\nThe proposed algorithm is summarized in Algorithm \\ref{alg:wavelet}.\n\n\\begin{algorithm}[hbt!]\n\\caption{Riemannian multi-resolution analysis algorithm}\n\\label{alg:wavelet}\n\\textbf{Input:} A time-varying dataset $\\{\\mathrm{x}_t[i]\\}_{i=1}^N$, $\\mathrm{x}_t[i]\\in\\mathbb{R}^d$, $t=1,\\dots,T$.\\\\\n\\textbf{Output:} Operators $\\{\\mathbf{S}_t^{(\\ell)},\\mathbf{F}_t^{(\\ell)}\\}_{t=1}^{T\/2^\\ell}$ and new representations for each level\\\\\n$\\{\\mathbf{\\Psi}^{(\\mathbf{S}_t^{(\\ell)})},\\mathbf{\\Psi}^{(\\mathbf{F}_t^{(\\ell)})}\\}_{t=1}^{T\/2^\\ell}$, where $\\ell=1,\\dots,\\log_2T$\n\\begin{algorithmic}[1]\n\\Statex\n\\State{Construct an SPD kernel representing each time point $t$, denoted by $\\{\\mathbf{W}_t\\}_{t=1}^T$, according to \\eqref{eq:SPDKern}.}\n\\Statex\n\\For{$t=1:T\/2$} \\Comment{Construct the operators for level $1$}\n\t\\State{$\\mathbf{S}_t^{(1)}=\\mathbf{W}_{2t-1}\\#_{0.5}\\mathbf{W}_{2t}$}\n\t\\State{$\\mathbf{F}_t^{(1)}=\\mathrm{Log}_{\\mathbf{S}_t^{(1)}}\\left(\\mathbf{W}_{2t-1}\\right)$}\n\\EndFor\n\\For{$\\ell=2:\\log_2T$} \\Comment{Construct the operators for level $\\ell$}\n\\For{$t=1:T\/2^\\ell$}\n\t\\State{$\\mathbf{S}_t^{(\\ell)}=\\mathbf{S}_{2t-1}^{(\\ell-1)}\\#_{0.5}\\mathbf{S}_{2t}^{(\\ell-1)}$}\n\t\\State{$\\mathbf{F}_t^{(\\ell)}=\\mathrm{Log}_{\\mathbf{S}_t^{(\\ell)}}\\left(\\mathbf{S}_{2t-1}^{(\\ell-1)}\\right)$}\n\\EndFor\n\\EndFor\n\\Statex\n\\For{$\\ell=1:\\log_2T$} \\Comment{Construct the new representations}\n\\For{$t=1:T\/2^\\ell$}\n\t\\State{$\\Psi^{(\\mathbf{S}_t^{(\\ell)})}=\\left[\\psi^{(\\mathbf{S}_t^{(\\ell)})}_1,\\dots,\\psi^{(\\mathbf{S}_t^{(\\ell)})}_M\\right]$}\n\t\\State{$\\Psi^{(\\mathbf{F}_t^{(\\ell)})}=\\left[\\psi^{(\\mathbf{F}_t^{(\\ell)})}_1,\\dots,\\psi^{(\\mathbf{F}_t^{(\\ell)})}_M\\right]$}\n\t\\State{where $\\psi^{(\\mathbf{S}_t^{(\\ell)})}$ and $\\psi^{(\\mathbf{F}_t^{(\\ell)})}$ are the eigenvectors of $\\mathbf{S}_t^{(\\ell)}$ and $\\mathbf{F}_t^{(\\ell)}$.}\n\\EndFor\n\\EndFor\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\\subsection{Implementation Remarks\\label{sec:implement}}\n\n{The numerical implementation of the proposed algorithm, particularly the Riemannian composition of operators, needs some elaboration. While the diffusion operators we consider in \\eqref{eq:SPDKern} are  SPD matrices by definition,} in practice, some of their eigenvalues could be {close to zero numerically,} forming in effect SPSD matrices instead.\nIn order to address this issue, we propose an equivalent definition of the operators $\\mathbf{S}$ and $\\mathbf{F}$ for SPSD matrices of fixed rank, based on the Riemannian metric and mean that were introduced in \\cite{bonnabel2010riemannian}.\n\n\nBased on the approximated geodesic path in \\eqref{eq:riemann_spsd_geodesic}, we define the operators $\\mathbf{S}$ and $\\mathbf{F}$ for SPSD matrices as follows. First, define\n\\begin{eqnarray}\n    \\mathbf{S} & = & \\tilde{\\gamma}_{\\mathbf{W}_1\\rightarrow\\mathbf{W}_2}(0.5)=\\mathbf{U}_{\\mathbf{W}_1\\rightarrow\\mathbf{W}_2}(0.5)\\left(\\mathbf{R}_1\\#_{0.5}\\mathbf{R}_2\\right)\\mathbf{U}^T_{\\mathbf{W}_1\\rightarrow\\mathbf{W}_2}(0.5)\\,.\n\\end{eqnarray}\n{Next, evaluate $\\mathbf{V}_1^T\\mathbf{V}_\\mathbf{S}=\\mathbf{O}_\\mathbf{S}\\tilde{\\mathbf{\\Sigma}}\\tilde{\\mathbf{O}}^T_1$ by SVD, where $\\mathbf{\\Lambda}_\\mathbf{S}$ and $\\mathbf{V}_\\mathbf{S}$ denote the eigenvalues and eigenvectors of $\\mathbf{S}$, respectively. Also, define $\\mathbf{R}_\\mathbf{S}:=\\mathbf{O}_\\mathbf{S}^T\\mathbf{\\Lambda}_\\mathbf{S}\\mathbf{O}_\\mathbf{S}$.\nThen, define}\n\\begin{eqnarray}\n    \\mathbf{F} & = & \\mathbf{U}_{\\mathbf{S}\\rightarrow\\mathbf{W}_1}(1)\\mathrm{Log}_{\\mathbf{R}_\\mathbf{S}}\\left(\\tilde{\\mathbf{O}}_1^T\\mathbf{\\Lambda}_1\\tilde{\\mathbf{O}}_1\\right)\\mathbf{U}^T_{\\mathbf{S}\\rightarrow\\mathbf{W}_1}(1)\n\\end{eqnarray}\nwhere $\\mathbf{U}_{\\mathbf{S}\\rightarrow\\mathbf{W}_1}(p)$ is defined for matrices $\\mathbf{S}$ and $\\mathbf{W}_1$ correspondingly to \\eqref{eq:grassman_geodesic} and the derivations leading to it.\nIntuitively, this can be viewed as applying the operators $\\mathbf{S}$ and $\\mathbf{F}$ to matrices expressed in the bases associated with the non-trivial SPD matrices of rank $r$, and then projecting the resulting operators back to the original space of SPSD matrices by applying $\\mathbf{U}_{\\mathbf{W}_1\\rightarrow\\mathbf{W}_2}(p)$.\n\n\n\nA summary of this derivation is presented in Algorithm \\ref{alg:SA_implementation}.\nA demonstration of the properties of these new operators for SPSD matrices using a simple simulation of $4\\times 4$ matrices, similar to the one presented in Subsection \\ref{subsub:ills_exmp} but with matrices of rank $3$, could be found in Section \\ref{subsub:mat4x4_exmp_fxd_rnk}.\n\n\\begin{algorithm}\n\\caption{Implementation of the operators for SPSD matrices}\n\\textbf{Input:} Two datasets with point correspondence $\\{\\mathbf{x}_1[i],\\mathbf{x}_2[i]\\}_{i=1}^N$\\\\\n\\textbf{Output:} Operators $\\mathbf{S}$ and $\\mathbf{F}$ and their eigenvectors:  $\\mathbf{\\Psi}^{(\\mathbf{S})},\\mathbf{\\Psi}^{(\\mathbf{F})}$\\\\\n\\begin{algorithmic}[1]\n\\Statex\n\\Function{SPSD-Geodesics}{$\\mathbf{G}_1,\\mathbf{G}_2,p$}\n\\Comment{As defined in \\cite{bonnabel2010riemannian}}\n\\State{Set $r=\\min\\left\\lbrace\\textrm{rank}\\left(\\mathbf{G}_1\\right),\\textrm{rank}\\left(\\mathbf{G}_2\\right)\\right\\rbrace$.}\n\\For{$i\\in[1,2]$}\n\\State{Set $\\mathbf{G}_i=\\mathbf{V}_i\\mathbf{\\Lambda}_i\\mathbf{V}_i^T$} \\Comment{Eigenvalue decomposition}\n\\State{Set $\\tilde{\\mathbf{V}}_i=\\mathbf{V}_i(:,1:r)$}\n\\State{Set $\\tilde{\\mathbf{\\Lambda}}_i=\\mathbf{\\Lambda}_i(1:r,1:r)$}\n\\EndFor\n\\State{Set $[\\mathbf{O}_1,\\mathbf{\\Sigma},\\mathbf{O}_2]=\\mathrm{SVD}\\left(\\tilde{\\mathbf{V}}_2^T\\tilde{\\mathbf{V}}_1\\right)$}\n\\State{Set $\\mathbf{\\Theta}=\\arccos(\\mathbf{\\Sigma})$}\n\\For{$i\\in[1,2]$}\n\\State{Set $\\mathbf{U}_i=\\tilde{\\mathbf{V}}_i\\mathbf{O}_i$}\n\\State{Set $\\mathbf{R}_{\\mathbf{G}_i}=\\mathbf{O}_i^T\\tilde{\\mathbf{\\Lambda}}_i\\mathbf{O}_i$}\n\\EndFor\n\\State{Compute $\\mathbf{U}_{\\mathbf{G}_1\\rightarrow\\mathbf{G}_2}(p)$} \\Comment{According to \\eqref{eq:grassman_geodesic}}\n\\State{Compute $\\mathbf{R}_{\\mathbf{G}_1\\rightarrow\\mathbf{G}_2}(p)=\\mathbf{R}_1^{1\/2}\\left(\\mathbf{R}_1^{-1\/2}\\mathbf{R}_2\\mathbf{R}_1^{-1\/2}\\right)^{p}\\mathbf{R}_1^{1\/2}$}\n\\State{\\textbf{return} $\\mathbf{U}_{\\mathbf{G}_1\\rightarrow\\mathbf{G}_2}(p)$, $\\mathbf{R}_{\\mathbf{G}_1\\rightarrow\\mathbf{G}_2}(p)$, $\\mathbf{R}_{\\mathbf{G}_1}$, $\\mathbf{R}_{\\mathbf{G}_2}$}\n\\EndFunction\n\\Statex\n\\Function{Main}{}\n\\State{Construct SPSD matrices for the two datasets $\\mathbf{W}_1$ and $\\mathbf{W}_2$} \\Comment{According to \\eqref{eq:SPDKern}}\n\\Statex\n\\State{\\Call{SPSD-Geodesics}{$\\mathbf{W}_1,\\mathbf{W}_2,0.5$}}\n\\State{$\\mathbf{S}=\\mathbf{U}_{\\mathbf{W}_1\\rightarrow\\mathbf{W}_2}(0.5)\\mathbf{R}_{\\mathbf{W}_1\\rightarrow\\mathbf{W}_2}(0.5)\\mathbf{U}^T_{\\mathbf{W}_1\\rightarrow\\mathbf{W}_2}(0.5)$}\n\\Statex\n\\State{\\Call{SPSD-Geodesics}{$\\mathbf{S},\\mathbf{W}_1,1$}}\n\\State{$\\mathbf{F}=\\mathbf{U}_{\\mathbf{S}\\rightarrow\\mathbf{W}_1}(1)\\mathrm{Log}_{\\mathbf{R}_{\\mathbf{S}}}\\left(\\mathbf{R}_{\\mathbf{W}_1}\\right)\\mathbf{U}^T_{\\mathbf{S}\\rightarrow\\mathbf{W}_1}(1)$} \\Comment{$\\mathrm{Log}_\\cdot(\\cdot)$ is defined as in \\eqref{eq:spd_logmap}}\n\\EndFunction\n\\end{algorithmic}\n\\label{alg:SA_implementation}\n\\end{algorithm}\n\n\n\n\n\n\\section{Experimental Results\\label{sec:results}}\n\n\n\n\\subsection{Illustrative Toy Example: SPD Case\\label{subsub:ills_exmp}}\nWe demonstrate the properties of the composite operators $\\mathbf{S}$ and $\\mathbf{F}$, constructed in Algorithm \\ref{alg:twomod}, using a simple simulation of $4\\times4$ matrices.\nDefine two matrices, $\\mathbf{M}_1=\\mathbf{\\Psi}\\mathbf{\\Lambda}^{(1)}\\mathbf{\\Psi}^T$ and $\\mathbf{M}_2=\\mathbf{\\Psi}\\mathbf{\\Lambda}^{(2)}\\mathbf{\\Psi}^T$, with the following common eigenvectors:\n\\begin{eqnarray}\\label{toy example eigenvector matrix}\n\\mathbf{\\Psi} = \\left[\n\\begin{matrix}\n\\psi_1, \\psi_2, \\psi_3, \\psi_4\n\\end{matrix}\n\\right] = \\frac{1}{2}\\left[\n\\begin{matrix}\n1,\\ \\ 1,\\ \\ 1,\\ \\ 1\\\\\n1,\\ \\ 1, -1, -1\\\\\n1, -1, -1,\\ \\ 1\\\\\n1, -1,\\ \\ 1, -1\n\\end{matrix}\n\\right]\n\\end{eqnarray}\nand the following eigenvalues:\n\\begin{eqnarray}\n\\mathbf{\\Lambda}^{(1)} = \\mathrm{diag}\\left(\\left[\\lambda^{(1)}_1,\\lambda^{(1)}_2,\\lambda^{(1)}_3,\\lambda^{(1)}_4\\right]\\right) = \\mathrm{diag}(\\left[ 0.5,\\ \\ 1,\\ 0.01,\\ 0.2 \\right])\\\\\n\\mathbf{\\Lambda}^{(2)} = \\mathrm{diag}\\left(\\left[\\lambda^{(2)}_1,\\lambda^{(2)}_2,\\lambda^{(2)}_3,\\lambda^{(2)}_4\\right]\\right) = \\mathrm{diag}(\\left[ 0.01, \\ 1,\\ 0.5,\\ \\ 0.2\\right])\n\\end{eqnarray}\nIn this example, $\\psi_1$ is a common eigenvector that is dominant in $\\mathbf{M}_1$ and weak in $\\mathbf{M}_2$, $\\psi_3$ is a common eigenvector that is dominant in $\\mathbf{M}_2$ and weak in $\\mathbf{M}_1$ and $\\psi_2$ and $\\psi_4$ are common eigenvectors that are similarly expressed in both $\\mathbf{M}_1$ and $\\mathbf{M}_2$.\n\nWe construct the operators $\\mathbf{S}=\\mathbf{M}_1\\#\\mathbf{M}_2$ and $\\mathbf{F}=\\mathrm{Log}_{\\mathbf{S}}\\left(\\mathbf{M}_1\\right)$ and compute their eigenvalues and eigenvectors. \nFigure \\ref{fig:4x4mat_full} presents the $4$ eigenvalues of $\\mathbf{M}_1$, $\\mathbf{M}_2$, $\\mathbf{S}$ and $\\mathbf{F}$, denoted by $\\{\\lambda^{(1)}_n\\}_{n=1}^4$, $\\{\\lambda^{(2)}_n\\}_{n=1}^4$, $\\{\\lambda^{(\\mathbf{S})}_n\\}_{n=1}^4$ and $\\{\\lambda^{(\\mathbf{F})}_n\\}_{n=1}^4$, respectively, in the left plots, and the corresponding eigenvectors in the right plots. \nThis figure depicts that the two matrices $\\mathbf{M}_1$ and $\\mathbf{M}_2$ share the same $4$ eigenvectors (as defined) and that the resulting eigenvectors of $\\mathbf{S}$ and $\\mathbf{F}$ are similar to these $4$ eigenvectors.\nNote that eigenvectors $2$ and $4$ of operator $\\mathbf{F}$ are not identical to the eigenvectors of $\\mathbf{M}_1$ and $\\mathbf{M}_2$ due to numerical issues, which arise since these eigenvectors in $\\mathbf{F}$ correspond to negligible eigenvalues.\nThe left plots show that the eigenvalues of $\\mathbf{S}$ and $\\mathbf{F}$ capture the similarities and differences in the expression of the spectral components of $\\mathbf{M}_1$ and $\\mathbf{M}_2$.\nSpecifically, since $\\lambda^{(1)}_2=\\lambda^{(2)}_2$ and $\\lambda^{(1)}_4=\\lambda^{(2)}_4$, the corresponding eigenvalues of $\\mathbf{S}$ assume the same magnitude.\nIn contrast, due to this equality, these eigenvalues correspond to negligible eigenvalues of $\\mathbf{F}$.\nThe two other eigenvectors, $\\psi_1$ and $\\psi_3$, correspond to eigenvalues that differ by an order of magnitude in the two matrices and are therefore the most dominant components in $\\mathbf{F}$.\nIn addition, note the opposite sign of eigenvalues $\\lambda^{(\\mathbf{F})}_1$ and $\\lambda^{(\\mathbf{F})}_3$, which indicates the source of the more dominant component, i.e. whether $\\lambda^{(1)}_n>\\lambda^{(2)}_n$ or $\\lambda^{(1)}_n{<}\\lambda^{(2)}_n$.\nThese properties are proved and explained in more detail in Section \\ref{sec:analysis}.\n\n\\begin{figure}[bhtp!]\n\\centering\n\\includegraphics[width=1\\textwidth]{figures\/matrices4x4\/full_rank_F-eps-converted-to.pdf}\n\\caption{Application of $\\mathbf{S}$ and $\\mathbf{F}$ to two $4\\times 4$ matrices with identical eigenvectors.\\label{fig:4x4mat_full}}\n\\end{figure}\n\n\\subsection{Illustrative Toy Example: SPSD case\\label{subsub:mat4x4_exmp_fxd_rnk}}\n\nConsider two matrices, $\\mathbf{M}_1=\\mathbf{\\Psi}\\mathbf{\\Lambda}^{(1)}\\mathbf{\\Psi}^T$ and $\\mathbf{M}_2=\\mathbf{\\Psi}\\mathbf{\\Lambda}^{(2)}\\mathbf{\\Psi}^T$, {with $\\mathbf{\\Psi}$ defined in \\eqref{toy example eigenvector matrix}}\nand the following eigenvalues:\n\\begin{eqnarray}\n\\mathbf{\\Lambda}^{(1)} = \\mathrm{diag}\\left(\\left[\\lambda^{(1)}_1,\\lambda^{(1)}_2,\\lambda^{(1)}_3,\\lambda^{(1)}_4\\right]\\right) = \\mathrm{diag}(\\left[ 0.5,\\ \\ 1,\\ 0.01,\\ 0 \\right])\\\\\n\\mathbf{\\Lambda}^{(2)} = \\mathrm{diag}\\left(\\left[\\lambda^{(2)}_1,\\lambda^{(2)}_2,\\lambda^{(2)}_3,\\lambda^{(2)}_4\\right]\\right) = \\mathrm{diag}(\\left[ 0.01, \\ 1,\\ 0.5,\\ \\ 0\\right])\n\\end{eqnarray}\nNote that the $4$th eigenvalue is zero in both matrices resulting in SPSD matrices of rank $3$.\n\nWe construct the operators $\\mathbf{S}$ and $\\mathbf{F}$ according to Algorithm \\ref{alg:SA_implementation} and compute their eigenvalues and eigenvectors. The results are presented in Figure \\ref{fig:4x4mat_rank3}. Same as Figure \\ref{fig:4x4mat_full},\nFigure \\ref{fig:4x4mat_rank3} presents in the left plots the $4$ eigenvalues of $\\mathbf{M}_1$, $\\mathbf{M}_2$, $\\mathbf{S}$ and $\\mathbf{F}$, denoted by $\\{\\lambda^{(1)}_n\\}_{n=1}^4$, $\\{\\lambda^{(2)}_n\\}_{n=1}^4$, $\\{\\lambda^{(\\mathbf{S})}_n\\}_{n=1}^4$ and $\\{\\lambda^{(\\mathbf{F})}_n\\}_{n=1}^4$, respectively, and the corresponding eigenvectors in the right plots.\nBoth matrices, $\\mathbf{M}_1$ and $\\mathbf{M}_2$, share the same $4$ eigenvectors as depicted in the right plots, and the resulting eigenvectors of $\\mathbf{S}$ and $\\mathbf{F}$ are similar to these $4$ eigenvectors.\nIn this example, $\\psi_2$ is a dominant component in both $\\mathbf{M}_1$ and $\\mathbf{M}_2$ with the same large eigenvalue. \nTherefore, similarly to the SPD case, the eigenvalue of $\\mathbf{S}$ associated with this eigenvector remains large, whereas the eigenvalue of $\\mathbf{F}$ associated with this eigenvector is negligible, as expected.\nIn contrast, $\\psi_1$ and $\\psi_3$ are eigenvectors that are differently expressed in the two matrices (corresponding to eigenvalues $0.5$ and $0.01$), and therefore, they correspond to dominant eigenvalues in $\\mathbf{F}$.\nThe left plot demonstrates that this behavior is indeed captured by the operators $\\mathbf{S}$ and $\\mathbf{F}$ for SPSD matrices.\nMoreover, the eigenvalues of the operators $\\mathbf{S}$ and $\\mathbf{F}$ for SPSD matrices are equal to the eigenvalues that were obtained by the operators for SPD matrices in a corresponding toy example, presented in Figure \\ref{fig:4x4mat_full}.\nNote that all eigenvalues that correspond to $\\psi_4$ are very close to zero, as expected due to the definition of the matrices $\\mathbf{M}_1$ and $\\mathbf{M}_2$.\n\n\\begin{figure}[bhtp!]\n\\centering\n\\includegraphics[width=1\\textwidth]{figures\/matrices4x4\/rank_3_F-eps-converted-to.pdf}\n\\caption{Application of the operators $\\mathbf{S}$ and $\\mathbf{F}$ for SPSD matrices to two $4\\times 4$ matrices of rank $3$ with identical eigenvectors.\\label{fig:4x4mat_rank3}}\n\\end{figure}\n\n\n\n\\subsection{Transitory Double Gyre Flow\\label{sub:2gyre}}\n\nTo demonstrate the proposed Riemannian multi-resolution analysis described in Section \\ref{sec:rmra}, we consider a variation of the transitory double gyre flow presented in \\cite{mosovsky2011transport,froyland2014almost}.\n\nWe simulate a 2D dynamical system with coordinates $(x_t,y_t)$ using the following equations:\n\\begin{eqnarray}\n\\dot{x}_t & = & -\\frac{\\partial}{\\partial y_t}H\\left(x_t,y_t,t\\right)\\\\\n\\dot{y}_t & = & \\frac{\\partial}{\\partial x_t}H\\left(x_t,y_t,t\\right)\n\\end{eqnarray}\nwith the function:\n\\begin{eqnarray}\nH\\left(x_t,y_t,t\\right) & = & (1-g(t))H_1\\left(x_t,y_t\\right) + g(t)H_2\\left(x_t,y_t\\right)\\\\\nH_1\\left(x_t,y_t\\right) & = & c_1\\sin(2\\pi x_t)\\sin(\\pi y_t)\\\\\nH_2\\left(x_t,y_t\\right) & = & c_2\\sin(\\pi x_t)\\sin(2\\pi y_t)\\\\\ng(t) & = & t^2(3-2t),\n\\end{eqnarray}\nwhere $c_1=2$, $c_2=10$, $i=1,...,N$ and $t\\in[0,1]$.\n\nThese equations describe a double gyre pattern, which is horizontal at $t=0$ and transitions into a vertical double gyre pattern at $t=1$.\nNote that in our simulations we add the parameters $c_1$ and $c_2$ to the dynamics, which lead to a change in rate between the (slower) horizontal and (faster) vertical double gyre patterns. \nThese parameters are added in order to demonstrate the time-varying multi-resolution properties of our analysis.\n\nWe generate $N=2500$ trajectories with initial values uniformly distributed in $\\left(x_0,y_0\\right)\\in [0,1]\\times[0,1]$, where each trajectory has $T=256$ time points on a discrete uniform time-grid with a step size of $\\Delta t =1\/256$.\nWe denote each of these trajectories with an index $i=1,\\ldots, N$ by a matrix $\\mathbf{x}[i] \\in \\mathbb{R}^{2 \\times T}$, whose columns are the pair of time samples $(x_t[i],y_t[i])^T$.\nA short GIF file demonstrating the resulting trajectories is available on \\href{https:\/\/github.com\/shnitzer\/Manifold-based-temporal-analysis\/blob\/main\/Transitory_Double_Gyre_Flow_Data.gif}{GitHub}, where each point is colored according to its initial location along the x-axis to illustrate the point movement in time.\nThe point movement demonstrated in this GIF exhibits two main structures: (i) points that rotate in two circular structures (transitioning from a horizontal setting into a vertical setting), which can be described as almost-invariant (coherent) sets as defined and captured by \\cite{froyland2014almost}, and (ii) points that are located on the boundary of these almost-invariant sets and their movement changes significantly over time.\nOur goal in this example is to analyze these two movement types and to recover their different trajectories over time.\n\nFor this purpose, we construct an SPD kernel for each time frame $t$, denoted by $\\mathbf{W}_t\\in\\mathbb{R}^{N\\times N}$, according to \\eqref{eq:SPDKern} based on the distances between the points in that time frame, i.e., $\\left\\Vert(x_t[i]-x_t[j],y_t[i]-y_t[j])\\right\\Vert_2^2$, $i,j=1,\\dots,N$, with $\\sigma$ set to $0.5$ times the median of these distances.\nWe then apply Algorithm \\ref{alg:wavelet} and obtain the multi-resolution representation and $\\ell=\\log_2(T)=8$ levels of operators.\nWe denote the operators of different levels and different time frames by $\\mathbf{S}_r^{(\\ell)}\\in\\mathbb{R}^{N\\times N}$ and $\\mathbf{F}_r^{(\\ell)}\\in\\mathbb{R}^{N\\times N}$, where $\\ell=1,\\dots,8$ and $r=1,\\dots,T\/2^\\ell$. Note that $r$ is associated with the time-frame indices, e.g., at level $\\ell=6$, $r=\\lceil t\/2^\\ell\\rceil=3$ corresponds to time points $t=129,\\dots,192$.\n\nIn the following, we focus on the second eigenvector of $\\mathbf{S}_r^{(\\ell)}$ and show that it indeed captures the common components and common trajectory behavior at the different operator levels.\nFigure \\ref{fig:DG_S} presents the data-points colored according to the second eigenvector of $\\mathbf{S}_r^{(\\ell)}$, denoted by $\\psi_2^{(\\mathbf{S}_r^{(\\ell)})}$ at levels $\\ell=8$, $\\ell=4$ and $\\ell=3$ and with $r$ values corresponding to different time frames.\nMore specifically, the locations of all $N=2500$ points are presented at $8$ different time-instances along the trajectory. \nEach point is colored according to its value in: (a) $\\psi^{(\\mathbf{S}_1^{(8)})}_2$, (b) $\\psi^{(\\mathbf{S}_4^{(4)})}_2$, (c) $\\psi^{(\\mathbf{S}_{10}^{(4)})}_2$, (d) $\\psi_2^{(\\mathbf{S}_7^{(3)})}$ and (e) $\\psi_2^{(\\mathbf{S}_8^{(3)})}$.\nNote that Figure \\ref{fig:DG_S} (d) and Figure \\ref{fig:DG_S} (e) present the preceding time frames of Figure \\ref{fig:DG_S} (b).\nWe maintained a consistent color coding in all time-instances, i.e., each point has the same color throughout its trajectory in time, and the most significant values (largest in absolute value) are colored in either yellow or blue.\n\n\n\n\\begin{landscape}\n\\begin{figure}[bhtp!]\n\\centering\n\\includegraphics[height=0.8\\textheight]{figures\/DoubleGyre\/S_tree.png}\n\\caption{Data points colored according to the second eigenvector of the operator $\\mathbf{S}_r^{(\\ell)}$ at different levels and time frames: (a) $\\mathbf{S}_1^{(8)}$, (b) $\\mathbf{S}_4^{(4)}$, (c) $\\mathbf{S}_{10}^{(4)}$, (d) $\\mathbf{S}_7^{(3)}$ and (e) $\\mathbf{S}_8^{(3)}$. Plots (d) and (e) present the same $8$ time points as in plot (b), where the points are colored according to a different eigenvector in each plot.\n    \\label{fig:DG_S}}\n\\end{figure}\n\\end{landscape}\n\n\n\nIn the figure we see the multi-resolution properties of the proposed framework.\nAt the highest level, $\\ell=8$ in plot (a), the eigenvector of $\\mathbf{S}_1^{(8)}$ captures the coherent circular structures, i.e. the almost-invariant sets, which change from a horizontal orientation at the beginning of the trajectory to a vertical orientation at the end. These structures are consistent with the ones described by \\cite{froyland2014almost}.\nIn contrast, in plots (b)-(c) (level $\\ell=4$), the effect of the velocity change over time is apparent, demonstrating that our framework is capable of detecting such properties. \nThese plots present two equal-length sub-segments of the trajectory: $t\\in\\{49,\\ldots,64\\}$ in plot (b) and $t\\in\\{145,\\ldots,160\\}$ in plot (c).\nPlot (c), which corresponds to the faster regime closer to the end of the trajectory, depicts that the circular structures are captured by the eigenvector, whereas in plot (b), which corresponds to the slower regime, these structures are not visible. \nDue to the increase in point movement velocity over time, the components that are similarly expressed over time in the sub-segment that is closer to the end of the trajectory (plot (c)) are mainly the almost-invariant sets, as captured by the operator from the highest level in plot (a). \nConversely, in the slower regime, there are other components that are similarly expressed over short sub-segments in time, as captured by the eigenvector presented in plot (b).\nPlots (d) and (e) correspond to the two sub-segments $t\\in\\{49,\\ldots,56\\}$ and $t\\in\\{57,\\ldots,64\\}$, respectively, whose union is the sub-segment presented in plot (b). \nNote the similarity between the captured point dynamics in the two sub-segments of plots (d) and (e). This similarity explains the component emphasized by the eigenvector in plot (b), which is constructed based on these two sub-segments.\n\nNote that the leading eigenvector of the operator $\\mathbf{S}_r^{(\\ell)}$ was omitted throughout this example since it mostly captures the common point distribution at the different time-frames.  \nThe point distribution is of less interest in this example since it provides a general geometric description of the problem setting rather than the common trajectory properties.\n\nIn the following we present the eigenvectors of the operators $\\mathbf{F}$ and show that they indeed capture the time-varying trajectory behavior in consecutive time-frames.\n\nFigure \\ref{fig:DG_A} presents the data-points colored according to the leading eigenvectors of the respective operators $\\mathbf{F}$ corresponding to the largest positive and negative (in absolute value) eigenvalues. \nIn this setting, the eigenvectors corresponding to negative eigenvalues describe the components that are significantly more dominant in the first half of the time segment and the eigenvectors corresponding to positive eigenvalues describe components that are significantly more dominant in the second half of the segment, as we will show in Section \\ref{sec:analysis} (Theorem \\ref{theo:A_eigs}).\n\nFigure \\ref{fig:DG_A} (a) and Figure \\ref{fig:DG_A} (b) present the eigenvectors of the operator $\\mathbf{F}_1^{(8)}$ corresponding to the smallest negative eigenvalue in plot (a) and to the largest positive eigenvalue in plot (b).\nThese plots depict that $\\mathbf{F}_1^{(8)}$ captures the differences between the slower point movement in $t\\in\\{1,\\ldots,128\\}$ and the faster point movement in $t\\in\\{129,\\ldots,256\\}$. \nDue to the change in point movement velocity over time, the component describing the circular structures (the almost-invariant sets) is significantly more dominant in the sub-segment from the faster regime ($t\\in\\{129,\\ldots,256\\}$) than the slower regime ($t\\in\\{1,\\ldots,128\\}$), as captured by the eigenvector in plot (b). In contrast, in the slower regime other components are dominant (as demonstrated also by Figure \\ref{fig:DG_S} (b)), leading to different structures being emphasized in Figure \\ref{fig:DG_A} (a), which mostly captures the boundary points.\nFigure \\ref{fig:DG_A} (c) and Figure \\ref{fig:DG_A} (d) correspond to the time segment $t\\in\\{129,\\ldots,256\\}$.\nPlot (c) presents the leading eigenvector of the operator $\\mathbf{F}_2^{(7)}$ with a negative eigenvalue, describing the components that are more dominant in the slower regime ($t\\in\\{129,\\ldots,192\\}$) and plot (d) presents the leading eigenvector with a positive eigenvalue, describing the components that are more dominant in the faster regime ($t\\in\\{193,\\ldots,256\\}$).\nNote that both plots (c) and (d) emphasize circular structures, however, the structures in plot (d) are smaller than the ones in plot (b) and are approximately complemented by the structures in plot (c).\nThis behavior implies that our framework decomposes the almost-invariant sets into smaller components in short sub-segments (at lower operator-tree levels), and therefore, indicates that the proposed method indeed captures meaningful dynamical information in different time-scales.\nFigure \\ref{fig:DG_A} (e) presents the eigenvector of $\\mathbf{F}_1^{(6)}$ (describing $t\\in[\\{1,\\ldots,64\\}$) with the largest negative eigenvalue.\nThis plot depicts that in the slower regime (at the beginning of the trajectory) the operator $\\mathbf{F}$ highlights high-resolution fine components of the point movement dynamics.\n\n\\begin{landscape}\n\\begin{figure}[bhtp!]\n\\centering\n\\includegraphics[height=0.7\\textheight]{figures\/DoubleGyre\/F_tree.png}\n\\caption{Data points colored according to the eigenvector of (a-b) $\\mathbf{F}_1^{(8)}$ with the smallest negative and largest positive eigenvalues, (c-d) $\\mathbf{F}_2^{(7)}$ with the smallest negative and largest positive eigenvalues, and (e) $\\mathbf{F}_1^{(6)}$ with the smallest negative eigenvalue.\n\\label{fig:DG_A}}\n\\end{figure}\n\n\\end{landscape}\n\n\nWe remark that different choices of the kernel scale in \\eqref{eq:SPDKern} lead to different resolutions. \nFor example, taking a smaller kernel scale leads to a slower ``convergence'' to the almost-invariant sets of the representations obtained at the different levels of the operator $\\mathbf{S}_r^{(\\ell)}$, as well as an enhancement of finer structures captured by the operator $\\mathbf{F}_r^{(\\ell)}$.\n\n\n\n\n\n    \n    \n\n\n\nIn order to evaluate our framework with respect to previous work, we compare the operators $\\mathbf{S}$ and $\\mathbf{F}$, which serve as the building blocks of our algorithm, with related operators: (i) the dynamic Laplacian \\cite{froyland2015dynamic}, which was shown to recover coherent sets from multiple time-frames of dynamical systems, and (ii) symmetric and anti-symmetric diffusion operators that were shown to recover similar and different components in multimodal data \\cite{shnitzer2019recovering}. \nIn \\cite{froyland2015dynamic}, with a slight abuse of notation and in analogy to \\eqref{eq:Sr}, the dynamic Laplacian is defined by $\\mathbf{L}^\\top\\mathbf{L}$, where $\\mathbf{L}=\\mathbf{W}_{1}\\mathbf{W}_{2}$. The common and difference operators in \\cite{shnitzer2019recovering} are defined by $\\mathbf{\\hat{S}}=\\mathbf{W}_{1}\\mathbf{W}_{2}^\\top+\\mathbf{W}_{2}\\mathbf{W}_{1}^\\top$ and $\\mathbf{\\hat{A}}=\\mathbf{W}_{1}\\mathbf{W}_{2}^\\top-\\mathbf{W}_{2}\\mathbf{W}_{1}^\\top$, respectively, which are analogous to the operators $\\mathbf{S}$ and $\\mathbf{F}$ in \\eqref{eq:Sr} and \\eqref{eq:Ar}.\n\nFigure \\ref{fig:op_compare_S} presents point clustering using k-means applied to the second eigenvector of the $3$ operators used for recovering similarities: the proposed operator $\\mathbf{S}$ in plot (a), the operator $\\mathbf{\\hat{S}}$ from \\cite{shnitzer2019recovering} in plot (b) and the operator $\\mathbf{L}^\\top\\mathbf{L}$ from \\cite{froyland2015dynamic} in plot (c).\nAll $3$ operators were constructed from time frames $t=250$ and $t=256$.\nWe see in this figure that the proposed formulation of the operator is significantly better at capturing the almost-invariant sets (the circular structures). \nNote that the results presented here for $\\mathbf{L}^\\top\\mathbf{L}$ are different than those in \\cite{froyland2015dynamic}, since we take into account only two close time frames, whereas in \\cite{froyland2015dynamic} the operator is constructed using all the points along the trajectory.\n\n\\begin{figure}[t]\n    \\centering\n    \\subfloat[$\\psi^{(\\mathbf{S})}_2$]{\\includegraphics[width=0.33\\textwidth]{figures\/DoubleGyre\/operator_comparison\/S2_rie_250_256_kmeans-eps-converted-to.pdf}}\n    \\subfloat[$\\psi^{(\\mathbf{\\hat{S}})}_2$ \\cite{shnitzer2019recovering}]{\\includegraphics[width=0.33\\textwidth]{figures\/DoubleGyre\/operator_comparison\/S2_old_250_256_kmeans-eps-converted-to.pdf}}\n    \\subfloat[$\\psi^{(\\mathbf{L}^T\\mathbf{L})}_2$ \\cite{froyland2015dynamic}]{\\includegraphics[width=0.33\\textwidth]{figures\/DoubleGyre\/operator_comparison\/S2_dlp_250_256_kmeans-eps-converted-to.pdf}}\n    \\caption{Clustering of trajectory points based on eigenvectors of the proposed operator $\\mathbf{S}$ in plot (a), and the operators from \\cite{shnitzer2019recovering} in plot (b) and from \\cite{froyland2015dynamic} in plot (c). All operators are constructed by combining two time frames at $t=250$ and $t=256$.}\n    \\label{fig:op_compare_S}\n\\end{figure}\n\nIn Figure \\ref{fig:op_compare_A}, we present the point clustering obtained by applying k-means to the second eigenvector of the $2$ operators used for recovering differences: the proposed operator $\\mathbf{F}$ in plot (a) and the operator $\\mathbf{\\hat{A}}$ from \\cite{shnitzer2019recovering} in plot (b). In contrast to plot (b), in plot (a) we clearly see the swirl of the flow from and to the invariant sets (outward and inward).\n\n\\begin{figure}[bhtp!]\n    \\centering\n    \\subfloat[$\\psi^{(\\mathbf{F})}_2$]{\\includegraphics[width=0.33\\textwidth]{figures\/DoubleGyre\/operator_comparison\/A2_rie_250_256_kmeans-eps-converted-to.pdf}}\n    \\subfloat[$\\psi^{(\\mathbf{\\hat{A}})}_2$ \\cite{shnitzer2019recovering}]{\\includegraphics[width=0.33\\textwidth]{figures\/DoubleGyre\/operator_comparison\/A2_euc_250_256_kmeans-eps-converted-to.pdf}}\n    \n    \\caption{Clustering of trajectory points based on eigenvectors of the proposed operator $\\mathbf{F}$ in plot (a) and operator $\\mathbf{\\hat{A}}$ from \\cite{shnitzer2019recovering} in plot (b).}\n    \\label{fig:op_compare_A}\n\\end{figure}\n\nIn addition to the differences demonstrated in Figure \\ref{fig:op_compare_S} and Figure \\ref{fig:op_compare_A}, another crucial advantage of our formulation relates to the construction of the multi-resolution framework and its theoretical justification presented in Section \\ref{sec:analysis}.\nThe operators in \\cite{shnitzer2019recovering} and \\cite{froyland2015dynamic} do not have any theoretical guarantees in such an operator-tree construction and may not be suitable to such a setting. \nIndeed, we report that a similar operator-tree constructed using the operators from \\cite{shnitzer2019recovering} did not exhibit the expected behavior and no meaningful representations were obtained.\n\nWe conclude by noting that such a multi-resolution analysis of the dynamics may be especially useful in applications where the parameters of interest are inaccessible, e.g., for oceanic current analysis based on ocean drifters data \\cite{froyland2015rough,banisch2017understanding}, since the data is represented using non-linear kernels.\n\n\n\\subsection{Hyperspectral and LiDAR Imagery\\label{sec:HSI_LIDAR}}\n\nIn Section \\ref{sub:op_def}, we consider two datasets and present the two Riemannian composite operators $\\mathbf{S}$ and $\\mathbf{F}$.\nLater, in Section \\ref{sub:rmra}, these two datasets are considered as two consecutive sets in a temporal sequence of datasets, and the two Riemannian composite operators are used as a basic building block for our Riemannian multi-resolution analysis. \nAlternatively, similarly to the setting in \\cite{lederman2015alternating,shnitzer2019recovering}, the two datasets could arise from simultaneous observations from two views or modalities. \nHere, we demonstrate the properties of the Riemannian composite operators $\\mathbf{S}$ and $\\mathbf{F}$ on real remote sensing data obtained by two different modalities. \n\nWe consider data from the 2013 IEEE GRSS data fusion contest\\footnote{\\url{http:\/\/www.grss-ieee.org\/community\/technical-committees\/data-fusion\/2013-ieee-grss-data-fusion-contest\/}}, which includes a hyperspectral image (HSI) with 144 spectral bands ($380-1050$nm range) and a LiDAR Digital Surface Model of the University of Houston campus and its neighboring urban area in Houston, Texas. \nThe data from both modalities have the same spatial resolution of $2.5$m.\nThis data was previously considered in the context of manifold learning in \\cite{murphy2018diffusion}.\n\nWe focus on two $60\\times 90$ image patches from the full dataset, in order to reduce computation time of the operators and their eigenvalue decomposition.\nWe first preprocess the $60\\times 90$ LiDAR image and each image in the $60\\times 90\\times 144$ HSI data.\nThe preprocessing stage includes dividing each $60\\times 90$ image by its standard deviation and removing outliers: for the LiDAR image, pixel values larger than the $99$th percentile were removed, and for the HSI data, in each image, pixel values larger than the $95$th percentile or smaller than the $5$th percentile were removed. In both modalities the outliers were replaced by their nearest non-outlier values.\nFigure \\ref{fig:HSI_LDR_1_all} (a) and Figure \\ref{fig:HSI_LDR_2_allEV} (a) present the two LiDAR image patches after preprocessing, and Figure \\ref{fig:HSI_LDR_1_all} (b) and Figure \\ref{fig:HSI_LDR_2_allEV} (b) present the two average HSI image patches after preprocessing.\n\n\nWe apply the operators $\\mathbf{S}$ and $\\mathbf{F}$ to this data in order to analyze the scene properties captured by both the LiDAR and the HSI sensors and extract the similarities and differences between them.\nThis can be viewed as a manifold-driven component analysis.\n\nWe construct the operators according to Algorithm \\ref{alg:SA_implementation}, where the LiDAR image and the HSI images are defined as two datasets with point correspondence between them given by the pixel location. \nWe reshape both datasets such that $\\mathbf{x}_1\\in\\mathbb{R}^{N\\times 1}$ is the reshaped LiDAR image and $\\mathbf{x}_2\\in\\mathbb{R}^{N\\times 144}$ is the reshaped HSI images, where $N=5400$. \nThe resulting kernels, $\\mathbf{W}_1$ and $\\mathbf{W}_2$, and operators, $\\mathbf{S}$ and $\\mathbf{F}$, are matrices of size $N\\times N$.\n\nWe begin with an analysis of the first chosen image patch, shown in Figure \\ref{fig:HSI_LDR_1_all} (a) and (b).\nTo depict the advantages of applying the proposed operators, we visually compare the eigenvectors of the kernels, $\\mathbf{W}_1$ and $\\mathbf{W}_2$, with the eigenvectors of the operators $\\mathbf{S}$ and $\\mathbf{F}$.\n\nFigure \\ref{fig:HSI_LDR_1_all} (c-k) presents the absolute values of the leading eigenvectors of $\\mathbf{S}$ in (c), $\\mathbf{W}_1$ in (d-e), $\\mathbf{W}_2$ in (h-i), and of $\\mathbf{F}$ that correspond to the largest positive eigenvalues in (f-g) and largest negative (in absolute value) eigenvalues in (j-k).\nAll eigenvectors are reshaped into images of size $60\\times 90$.\nThe absolute value of the eigenvectors is presented in order to emphasize the dominant structures in the images and the differences between the leading eigenvectors of the two kernels and the leading eigenvectors of the operators $\\mathbf{S}$ and $\\mathbf{F}$.\n\nFigure \\ref{fig:HSI_LDR_1_all} (c) presents the absolute values of the leading eigenvector of $\\mathbf{S}$ and depicts that the operator $\\mathbf{S}$ indeed recovers common structures strongly expressed in both images.\nSpecifically, this figure mostly highlights an `L'-shaped building at the top of the image, which is the most dominant structure (represented by the high pixel values) in both modalities.\nFigure \\ref{fig:HSI_LDR_1_all} (d-k) depicts that the eigenvectors of the operator $\\mathbf{F}$ capture and enhance differently expressed common structures.\nConsider for example the most dominant structures (with highest absolute values) in the LiDAR image presented in Figure \\ref{fig:HSI_LDR_1_all} (a).\nThese structures include the `L'-shaped building at the top of the image and trees at the bottom.\nBoth structures are represented by high values in the eigenvectors of $\\mathbf{W}_1$ in Figure \\ref{fig:HSI_LDR_1_all} (d-e).\nHowever, in Figure \\ref{fig:HSI_LDR_1_all} (f-g), which presents the leading eigenvectors of $\\mathbf{F}$ with positive eigenvalues, only the trees are significantly highlighted, whereas the `L'-shaped building is significantly attenuated.\nThis is due to the differences between the two modalities, since the HSI image highlights this `L' shaped building but not the trees.\nOther structures exhibiting such properties are marked by black arrows in Figure \\ref{fig:HSI_LDR_1_all} (f-g).\nIn addition, Figure \\ref{fig:HSI_LDR_1_all} (h-k) depicts that the structures which are dominant only in the HSI images are emphasized by the eigenvectors of $\\mathbf{F}$ corresponding to negative eigenvalues, whereas structures that are dominant in both modalities are significantly attenuated.\nExamples for such structures are marked by black arrows in Figure \\ref{fig:HSI_LDR_1_all} (j-k).\n\n\\begin{figure}[bhtp!]\n\\centering\n\\subfloat[]{\\includegraphics[width=0.29\\textwidth]{figures\/HSI_LDR\/example1\/data1_ldr-eps-converted-to.pdf}}\n\\subfloat[]{\\includegraphics[width=0.29\\textwidth]{figures\/HSI_LDR\/example1\/data1_hsi-eps-converted-to.pdf}}\n\\subfloat[]{\\includegraphics[width=0.25\\textwidth]{figures\/HSI_LDR\/example1\/S_psi_2_arrow-eps-converted-to.pdf}}\n\n\\subfloat[]{\\includegraphics[width=0.25\\textwidth]{figures\/HSI_LDR\/example1\/W2_psi_1-eps-converted-to.pdf}}\n\\subfloat[]{\\includegraphics[width=0.25\\textwidth]{figures\/HSI_LDR\/example1\/W2_psi_2-eps-converted-to.pdf}}\n\\subfloat[]{\\includegraphics[width=0.25\\textwidth]{figures\/HSI_LDR\/example1\/Fpos_psi_1_arrow-eps-converted-to.pdf}}\n\\subfloat[]{\\includegraphics[width=0.25\\textwidth]{figures\/HSI_LDR\/example1\/Fpos_psi_2_arrow-eps-converted-to.pdf}}\n\n\\subfloat[]{\\includegraphics[width=0.25\\textwidth]{figures\/HSI_LDR\/example1\/W1_psi_1-eps-converted-to.pdf}}\n\\subfloat[]{\\includegraphics[width=0.25\\textwidth]{figures\/HSI_LDR\/example1\/W1_psi_2-eps-converted-to.pdf}}\n\\subfloat[]{\\includegraphics[width=0.25\\textwidth]{figures\/HSI_LDR\/example1\/Fneg_psi_1_arrow-eps-converted-to.pdf}}\n\\subfloat[]{\\includegraphics[width=0.25\\textwidth]{figures\/HSI_LDR\/example1\/Fneg_psi_2_arrow-eps-converted-to.pdf}}\n\\caption{The two chosen image patches of (a) the LiDAR image and (b) the HSI image after preprocessing, along with the leading eigenvectors of (c) $\\mathbf{S}$, (d-e) $\\mathbf{W}_1$ (LiDAR), (f-g) $\\mathbf{F}$ corresponding to its $2$ largest positive eigenvalues,\n(h-i) $\\mathbf{W}_2$ (HSI), (j-k) $\\mathbf{F}$ corresponding to its two smallest negative eigenvalues.\n\\label{fig:HSI_LDR_1_all}}\n\\end{figure}\n\nWe repeat the presentation for the second image patch, shown in Figure \\ref{fig:HSI_LDR_2_allEV} (a) and (b).\nFigure \\ref{fig:HSI_LDR_2_allEV} (c-g) presents the absolute value of the leading eigenvector of $\\mathbf{W}_1$ in plot (c), of $\\mathbf{W}_2$ in plot (d), of $\\mathbf{F}$ with a positive eigenvalue in plot (e), of $\\mathbf{F}$ with a negative eigenvalue in plot (f) and of $\\mathbf{S}$ in plot (g).\n\\begin{figure}[bhtp!]\n\\centering\n\\subfloat[]{\\includegraphics[width=0.33\\textwidth]{figures\/HSI_LDR\/example2\/data2_ldr-eps-converted-to.pdf}}\n\\subfloat[]{\\includegraphics[width=0.33\\textwidth]{figures\/HSI_LDR\/example2\/data2_hsi-eps-converted-to.pdf}}\n\n\\subfloat[]{\\includegraphics[width=0.33\\textwidth]{figures\/HSI_LDR\/example2\/W1_psi1-eps-converted-to.pdf}}\n\\subfloat[]{\\includegraphics[width=0.33\\textwidth]{figures\/HSI_LDR\/example2\/W2_psi1-eps-converted-to.pdf}}\n\n\\subfloat[]{\\includegraphics[width=0.33\\textwidth]{figures\/HSI_LDR\/example2\/Fpos_psi1-eps-converted-to.pdf}}\n\\subfloat[]{\\includegraphics[width=0.33\\textwidth]{figures\/HSI_LDR\/example2\/Fneg_psi1_arrow-eps-converted-to.pdf}}\n\\subfloat[]{\\includegraphics[width=0.33\\textwidth]{figures\/HSI_LDR\/example2\/S_psi1-eps-converted-to.pdf}}\n\\caption{The two chosen image patches of (a) the LiDAR image and (b) the HSI image after preprocessing along with the leading eigenvectors of (c) $\\mathbf{W}_1$ (LiDAR data), (d)  $\\mathbf{W}_2$ (HSI data), (e) $\\mathbf{F}$ corresponding to the largest positive eigenvalue, (f) $\\mathbf{F}$ corresponding to the smallest negative eigenvalue and (g) $\\mathbf{S}$.\n\\label{fig:HSI_LDR_2_allEV}}\n\\end{figure}\n\nNote the dominant structures (with high absolute values) in the leading eigenvectors of the two modalities in Figure \\ref{fig:HSI_LDR_2_allEV} (c) and (d). \nThe dominant structures of the eigenvector representing the LiDAR image in plot (c) include buildings and trees, whereas in plot (d), which relates to the HSI image, only some of the building appear with high intensities and the trees are not clearly visible. \nThis corresponds to the data presented in Figure \\ref{fig:HSI_LDR_2_allEV} (a) and (b).\nThe leading eigenvector of $\\mathbf{F}$ with a positive eigenvalue, presented in Figure \\ref{fig:HSI_LDR_2_allEV} (e), captures buildings and trees that are expressed more dominantly in the LiDAR image compared with the HSI image. \nIn addition, the structures that are dominant in both modalities appear to be less dominant in this plot. \nFor example, the ``attenuated'' structures in plot (e) include the small rectangular roof part around pixels $(x,y)=(10,25)$ (where $x$ denotes the horizontal axis and $y$ denotes the vertical axis) and the building around pixels $(x,y)=(80,10)$.\nConversely, the leading eigenvector of $\\mathbf{F}$ with a negative eigenvalue, presented in Figure \\ref{fig:HSI_LDR_2_allEV} (f), significantly enhances a specific location, marked by a black arrow in this plot, that is clearly visible in the HSI image presented in Figure \\ref{fig:HSI_LDR_2_allEV} (b) but barely visible in Figure \\ref{fig:HSI_LDR_2_allEV} plot (a).\nNote that this building is not represented by high pixel values in the raw HSI average image and therefore a simple subtraction between the two images will not lead to a similar emphasis of the building.\n\nThe leading eigenvector of $\\mathbf{S}$, presented in Figure \\ref{fig:HSI_LDR_2_allEV} (g), captures some combination of the structures that are dominant in both modalities.\n\nTo summarize this example, we showed that the operator $\\mathbf{F}$ captures common components that are expressed strongly only by one of the modalities and that the sign of the eigenvalues of $\\mathbf{F}$ indicates in which modality the component is stronger. \nIn addition, we showed that the operator $\\mathbf{S}$ captures some combination of the dominant components in both modalities.\n\n\n\n\n\n\n\n\n\n\\section{Spectral Analysis\\label{sec:analysis}}\n\nTo provide theoretical justification to the proposed RMRA framework for spatiotemporal analysis presented in Section \\ref{sec:rmra}, we analyze the operators $\\mathbf{S}$ and $\\mathbf{F}$ defined in \\eqref{eq:Sr} and \\eqref{eq:Ar} and show that they admit the desired properties. \nSpecifically, we show that the operator $\\mathbf{S}$ enhances common eigenvectors that are expressed similarly in two consecutive time frames in the sense that they have similar eigenvalues. In addition, we show that the operator $\\mathbf{F}$ enhances common eigenvectors that are expressed differently in two consecutive time frames in the sense that they have different eigenvalues.\n\n\n\n\n\n\n\n\n\n\nIn the following theoretical analysis we focus on two cases: (i) $\\mathbf{W}_1$ and $\\mathbf{W}_2$ have \\emph{strictly} common components, i.e., some of the eigenvectors of the two matrices are identical, and (ii) $\\mathbf{W}_1$ and $\\mathbf{W}_2$ have \\emph{weakly} common components, i.e. some of the eigenvectors of the two matrices differ by a small perturbation.\n\n\n\\subsection{Strictly Common Components\\label{sub:ident_eig}}\n\nGiven that some of the eigenvectors of matrices $\\mathbf{W}_1$ and $\\mathbf{W}_2$ are identical, we show in the following that for these identical eigenvectors, the operator $\\mathbf{S}$ enhances the eigenvectors that have similar dominant eigenvalues and the operator $\\mathbf{F}$ enhances the eigenvectors that have significantly different eigenvalues.\n\nWe begin by reiterating a theorem from \\cite{ok19} along with its proof, which shows that the eigenvectors that are similarly expressed in both matrices, $\\mathbf{W}_1$ and $\\mathbf{W}_2$, correspond to the largest eigenvalues of $\\mathbf{S}$.\n\\begin{theorem}\\label{theo:S_eigs}\nConsider a vector $\\psi$, which is an eigenvector of both $\\mathbf{W}_1$ and $\\mathbf{W}_2$ with possibly different eigenvalues: $\\mathbf{W}_1\\psi=\\lambda^{(1)}\\psi$ and $\\mathbf{W}_2\\psi=\\lambda^{(2)}\\psi$.\nThen, $\\psi$ is also an eigenvector of $\\mathbf{S}$ with the corresponding eigenvalue:\n\\begin{equation}\n\\lambda^{(\\mathbf{S})}=\\sqrt{\\lambda^{(1)}\\lambda^{(2)}}    \n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFrom \\eqref{eq:Sr} we have:\n\\begin{align*}\n\\mathbf{S}\\psi=&\\,\\mathbf{W}_1^{1\/2}\\left(\\mathbf{W}_1^{-1\/2}\\mathbf{W}_2\\mathbf{W}_1^{-1\/2}\\right)^{1\/2}\\mathbf{W}_1^{1\/2}\\psi`\\\\\n=&\\,\\mathbf{W}_1^{1\/2}\\left(\\mathbf{W}_1^{-1\/2}\\mathbf{W}_2\\mathbf{W}_1^{-1\/2}\\right)^{1\/2}\\sqrt{\\lambda^{(1)}}\\psi\\\\\n=&\\,\\mathbf{W}_1^{1\/2}\\sqrt{\\lambda^{(2)}\/\\lambda^{(1)}}\\sqrt{\\lambda^{(1)}}\\psi\\\\\n=&\\,\\sqrt{\\lambda^{(1)}\\lambda^{(2)}}\\psi\n\\end{align*}\nwhere the transition before last is due to $\\mathbf{W}_1^{-1\/2}\\mathbf{W}^{(2)}\\mathbf{W}_1^{-1\/2}\\psi=(\\lambda^{(2)}\/\\lambda^{(1)})\\psi$.\n\\end{proof}\nThis result implies that strictly common components that are dominant and similarly expressed in both datasets (with similar large eigenvalues) are dominant in $\\mathbf{S}$ (have a large eigenvalue $\\lambda^{(\\mathbf{S})}$), i.e. if $\\lambda^{(1)}\\approx\\lambda^{(2)}$ then $\\lambda^{(\\mathbf{S})}\\approx\\lambda^{(1)},\\lambda^{(2)}$.\n\nWe derive a similar theoretical analysis for the operator $\\mathbf{F}$.\n\\begin{theorem}\\label{theo:A_eigs}\nConsider a vector $\\psi$, which is an eigenvector of both $\\mathbf{W}_1$ and $\\mathbf{W}_2$ with possibly different eigenvalues: $\\mathbf{W}_1\\psi=\\lambda^{(1)}\\psi$ and $\\mathbf{W}_2\\psi=\\lambda^{(2)}\\psi$.\nThen $\\psi$ is also an eigenvector of $\\mathbf{F}$ with the corresponding eigenvalue:\n\\begin{equation}\n    \\lambda^{(\\mathbf{F})}=\\frac{1}{2}\\sqrt{\\lambda^{(1)}\\lambda^{(2)}}(\\log(\\lambda^{(1)})-\\log(\\lambda^{(2)}))\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFrom \\eqref{eq:Ar} we have:\n\\begin{align*}\n\\mathbf{F}\\psi=&\\,\\mathbf{S}^{1\/2}\\log\\left(\\mathbf{S}^{-1\/2}\\mathbf{W}_1\\mathbf{S}^{-1\/2}\\right)\\mathbf{S}^{1\/2}\\psi\\\\\n=&\\,\\mathbf{S}^{1\/2}\\log\\left(\\mathbf{S}^{-1\/2}\\mathbf{W}_1\\mathbf{S}^{-1\/2}\\right)(\\lambda^{(1)}\\lambda^{(2)})^{0.25}\\psi\\\\\n=&\\,\\mathbf{S}^{1\/2}(0.5\\log(\\lambda^{(1)})-0.5\\log(\\lambda^{(2)}))(\\lambda^{(1)}\\lambda^{(2)})^{0.25}\\psi\\\\\n=&\\,\\frac{1}{2}\\sqrt{\\lambda^{(1)}\\lambda^{(2)}}(\\log(\\lambda^{(1)})-\\log(\\lambda^{(2)}))\\psi\n\\end{align*}\nwhere the transition before last is due to $\\mathbf{S}^{-1\/2}\\mathbf{W}_1\\mathbf{S}^{-1\/2}\\psi=\\sqrt{\\lambda^{(1)}\/\\lambda^{(2)}}\\psi$ and the application of $\\log$ to the matrix multiplication, which is equivalent to applying $\\log$ to its eigenvalues, leading to $\\log(\\sqrt{\\lambda^{(1)}\/\\lambda^{(2)}})=0.5\\log(\\lambda^{(1)})-0.5\\log(\\lambda^{(2)})$. \n\\end{proof}\nThis result indicates that the strictly common components of the two datasets are also expressed by $\\mathbf{F}$ and that their order is determined by their relative expression in each dataset.\nFor example, if $\\psi$ is an eigenvector of both $\\mathbf{W}_1$ and $\\mathbf{W}_2$ and corresponds to equal eigenvalues, $\\lambda^{(1)}=\\lambda^{(2)}$, then this component is part of the null space of $\\mathbf{F}$;\nif $\\lambda^{(1)}\\neq\\lambda^{(2)}$, then $\\psi$ corresponds to a nonzero eigenvalue in $\\mathbf{F}$. \nNote that $\\lambda^{(\\mathbf{F})}$ depends also on the multiplication by $\\sqrt{\\lambda^{(1)}\\lambda^{(2)}}$. \n\n\nAnother notable result of Theorem \\ref{theo:A_eigs} is that the sign of the eigenvalues of $\\mathbf{F}$ indicates in which dataset their corresponding eigenvector is more dominant. \nFor example, if $\\psi$ is an eigenvector of both $\\mathbf{W}_1$ and $\\mathbf{W}_2$ that has a large corresponding eigenvalue in $\\mathbf{W}_1$ but a small eigenvalue in $\\mathbf{W}_2$ ($\\lambda^{(1)}\\gg\\lambda^{(2)}$), then the corresponding eigenvalue in $\\mathbf{F}$ is large and positive.\nConversely, if $\\psi$ is more dominant in $\\mathbf{W}_2$, then its corresponding eigenvalue in $\\mathbf{F}$ is large (in absolute value) and negative.\n\nAn example depicting these properties is presented in Subsection \\ref{subsub:ills_exmp}, which demonstrates that the eigenvalues of the operators $\\mathbf{S}$ and $\\mathbf{F}$ are indeed equal to the expected values based on Theorem \\ref{theo:S_eigs} and Theorem \\ref{theo:A_eigs}.\n\n\n\\subsection{Weakly Common Components}\n\n{\nTo further demonstrate the power of the operators $\\mathbf{S}$ and $\\mathbf{F}$, we provide stability analysis by investigating how a small variation of the common eigenvector affects the results. For this purpose, we make use of the concept of a pseudo-spectrum \\cite{trefethen2005spectra}. While  pseudo-spectra is typically used to provide an analytic framework for investigating non-normal matrices and operators, here we apply it to symmetric matrices for the purpose of analysis of nearly but non-common eigenvectors.\nWe begin by recalling three equivalent definitions of the $\\epsilon$-pseudo-spectrum as presented in \\cite{trefethen2005spectra}. \n\n\n\\begin{definition}[Pseudo-spectrum]\\label{def:pseudo}\nGiven a matrix $\\mathbf{M}\\in\\mathbb{R}^{N\\times N}$, the following definitions of the $\\epsilon$-pseudo-spectrum are equivalent \nfor a small $\\epsilon>0$:\n\\begin{enumerate}\n    \\item $\\sigma_\\epsilon(\\mathbf{M}) = \\left\\lbrace\\lambda\\in\\mathbb{R}:\\left\\Vert(\\lambda\\mathrm{I}-\\mathbf{M})^{-1}\\right\\Vert\\geq\\epsilon^{-1}\\right\\rbrace$\n    \\item $\\sigma_\\epsilon(\\mathbf{M}) = \\left\\lbrace\\lambda\\in\\mathbb{R}:\\lambda\\in\\sigma(\\mathbf{M}+\\mathbf{E})\\text{ for a }\\mathbf{E}\\text{ with }\\left\\Vert\\mathbf{E}\\right\\Vert\\leq\\epsilon\\right\\rbrace$\n    \\item $\\sigma_\\epsilon(\\mathbf{M}) = \\left\\lbrace\\lambda\\in\\mathbb{R}:\\exists v\\in\\mathbb{R}^N\\text{ with }\\left\\Vert v\\right\\Vert_2=1\\text{ s.t. }\\left\\Vert(\\mathbf{M}-\\lambda\\mathrm{I})v\\right\\Vert_2\\leq\\epsilon\\right\\rbrace$\n\\end{enumerate}\nwhere $\\sigma(\\mathbf{M})$ denotes the set of eigenvalues of $\\mathbf{M}$, $\\mathrm{I}$ denotes the identity matrix, $\\left\\Vert\\cdot\\right\\Vert_2$ denotes the $\\ell_2$ norm, and $\\left\\Vert\\cdot\\right\\Vert$ denotes the induced operator norm. Moreover, we term a vector $v$ that adheres to definition 3 an $\\epsilon$-pseudo-eigenvector.\n\\end{definition}\n\nThe following theorem is the counterpart of Theorem \\ref{theo:S_eigs} for the case where the eigenvector is slightly perturbed.\n\n\\begin{theorem}\\label{prop:pseudo_sa}\n\nSuppose there exists an eigenpair $\\lambda^{(k)}$ and $\\psi^{(k)}$ of $\\mathbf{W}_k$ for $k=1,2$ so that $\\psi^{(1)}=\\psi^{(2)}+\\psi_{\\epsilon_{\\mathbf{S}}}$, where $\\left\\Vert \\psi_{\\epsilon_{\\mathbf{S}}}\\right\\Vert_2\\leq \\frac{\\sqrt{\\lambda^{(2)}}}{\\tilde{\\lambda}^{(2)}_{max}\\sqrt{\\lambda^{(1)}}}\\epsilon_{\\mathbf{S}}$ for a small $\\epsilon_{\\mathbf{S}}>0$, where $\\tilde{\\lambda}^{(2)}_{max} = \\Vert\\mathbf{W}_2-\\lambda^{(2)}\\mathrm{I}\\Vert$. Then, we have\n\\begin{equation}\n\\sqrt{\\lambda^{(1)}\\lambda^{(2)}}\\in \\sigma_{\\epsilon_{\\mathbf{S}}}(\\mathbf{S})\\,.\\label{pseudospectrum of S}\n\\end{equation}\nSpecifically, we have \n\\[\n\\left\\Vert(\\mathbf{S}-\\sqrt{\\lambda^{(1)}\\lambda^{(2)}}\\mathrm{I})\\psi^{(1)}\\right\\Vert_2\\leq\\epsilon_{\\mathbf{S}}\\,.\n\\]\nand $\\psi^{(1)}$ is a corresponding $\\epsilon_{\\mathbf{S}}$-pseudo-eigenvector of $\\mathbf{S}$.\n\\end{theorem}\n\nInformally, this theorem implies that when $\\psi^{(1)}$ is a slight perturbation of $\\psi^{(2)}$, then $\\psi^{(1)}$ is ``almost'' an eigenvector of $\\mathbf{S}$ with a corresponding eigenvalue $\\sqrt{\\lambda^{(1)}\\lambda^{(2)}}$.\nEquivalently, $\\psi^{(2)}$ can also be shown to be an $\\epsilon_{\\mathbf{S}}$-pseudo-eigenvector of $\\mathbf{S}$ with the same corresponding eigenvalue, which suggests that the operator $\\mathbf{S}$ is ``stable'' to finite perturbations.\nWe remark that since $\\|\\mathbf{W}_2\\|=1$, we have that $\\tilde{\\lambda}^{(2)}_{max}=\\max(1-\\lambda^{(2)}, \\lambda^{(2)})\\in[0.5,1)$, guaranteeing that the perturbation of the eigenvector is small.\nWe also remark that our numerical study shows that the bound for the $\\epsilon$-pseudo-eigenvalues and $\\epsilon$-pseudo-eigenvectors of $\\mathbf{S}$ is tight.\n\n\n\n\\begin{proof}\nBy Proposition \\ref{prop_app:sa_equiv_forms} (see Appendix \\ref{app:add_state}), we have\n\\begin{equation}\n\\mathbf{S}=\\mathbf{W}_1^{1\/2}\\left(\\mathbf{W}_1^{-1\/2}\\mathbf{W}_2\\mathbf{W}_1^{-1\/2}\\right)^{1\/2}\\mathbf{W}_1^{1\/2}=(\\mathbf{W}_2\\mathbf{W}_1^{-1})^{1\/2}\\mathbf{W}_1.\\label{eq:equivexp}\n\\end{equation}\nSince $\\psi^{(1)}$ is an eigenvector of $\\mathbf{W}_1$ with an eigenvalue $\\lambda^{(1)}$, we have \n\\begin{align}\n    \\mathbf{S}\\psi^{(1)} =  \\left(\\mathbf{W}_2\\mathbf{W}_1^{-1}\\right)^{1\/2}\\mathbf{W}_1\\psi^{(1)}\n    = \\lambda^{(1)}\\left(\\mathbf{W}_2\\mathbf{W}_1^{-1}\\right)^{1\/2}\\psi^{(1)}\\label{eq:pseudo_s_psi}.\n\\end{align}\nTherefore, it is sufficient to show that $\\psi^{(1)}$ is an $\\epsilon$-pseudo-eigenvector of $\\left(\\mathbf{W}_2\\mathbf{W}_1^{-1}\\right)^{1\/2}$. \nBy a direct expansion, we have\n\\begin{align}\n\\mathbf{W}_2\\mathbf{W}_1^{-1}\\psi^{(1)}  = & \\, \\frac{1}{\\lambda^{(1)}}\\mathbf{W}_2 \\psi^{(1)}\n = \\frac{1}{\\lambda^{(1)}}\\mathbf{W}_2 (\\psi^{(2)}+\\psi_{\\epsilon_{\\mathbf{S}}})\\nonumber\\\\\n = & \\, \\frac{\\lambda^{(2)}}{\\lambda^{(1)}}\\psi^{(2)}+\\frac{1}{\\lambda^{(1)}}\\mathbf{W}_2 \\psi_{\\epsilon_{\\mathbf{S}}} =  \\frac{\\lambda^{(2)}}{\\lambda^{(1)}}\\psi^{(1)}+\\frac{1}{\\lambda^{(1)}}(\\mathbf{W}_2-\\lambda^{(2)}\\mathrm{I}) \\psi_{\\epsilon_{\\mathbf{S}}}\\,,\\label{eq:pseudo_s_1}\n\\end{align}\nwhere the last transition is obtained by replacing $\\psi^{(2)}$ with $\\psi^{(1)}-\\psi_{\\epsilon_{\\mathbf{S}}}$.\nBy reorganizing the elements in \\eqref{eq:pseudo_s_1}, we have\n\\begin{align}\n&\\left(\\mathbf{W}_2\\mathbf{W}_1^{-1}-\\frac{\\lambda^{(2)}}{\\lambda^{(1)}}\\mathrm{I}\\right)\\psi^{(1)} = \\frac{1}{\\lambda^{(1)}}(\\mathbf{W}_2-\\lambda^{(2)}\\mathrm{I}) \\psi_{\\epsilon_{\\mathbf{S}}}\\,,\n\\end{align}\nand applying the $\\ell_2$ norm leads to:\n\\begin{align}\n&\\left\\Vert\\left(\\mathbf{W}_2\\mathbf{W}_1^{-1}-\\frac{\\lambda^{(2)}}{\\lambda^{(1)}}\\mathrm{I}\\right)\\psi^{(1)}\\right\\Vert_2  =  \\left\\Vert\\frac{1}{\\lambda^{(1)}}(\\mathbf{W}_2-\\lambda^{(2)}\\mathrm{I}) \\psi_{\\epsilon_{\\mathbf{S}}}\\right\\Vert_2\n \\leq  \\frac{1}{\\lambda^{(1)}}\\left\\Vert \\mathbf{W}_2-\\lambda^{(2)}\\mathrm{I}\\right\\Vert\\left\\Vert \\psi_{\\epsilon_{\\mathbf{S}}}\\right\\Vert_2\\label{eq:pseudo_s_2}\\\\\n&\\qquad\\qquad \\leq  \\,\\frac{1}{\\lambda^{(1)}}\\left\\Vert\\mathbf{W}_2-\\lambda^{(2)}\\mathrm{I}\\right\\Vert\\frac{\\sqrt{\\lambda^{(2)}}}{\\tilde{\\lambda}^{(2)}_{max}\\sqrt{\\lambda^{(1)}}}\\epsilon_{\\mathbf{S}} =  \\frac{\\tilde{\\lambda}^{(2)}_{max}}{\\lambda^{(1)}}\\frac{\\sqrt{\\lambda^{(2)}}}{\\tilde{\\lambda}^{(2)}_{max}\\sqrt{\\lambda^{(1)}}}\\epsilon_{\\mathbf{S}}=\\frac{\\sqrt{\\lambda^{(2)}}}{\\lambda^{(1)}\\sqrt{\\lambda^{(1)}}}\\epsilon_{\\mathbf{S}}\\,.\\nonumber\n\\end{align}\nThis derivation shows that $\\psi^{(1)}$ is a pseudo-eigenvector of $\\mathbf{W}_2\\mathbf{W}_1^{-1}$ with a pseudo-eigenvalue $\\frac{\\lambda^{(2)}}{\\lambda^{(1)}}$, i.e. $\\left\\Vert\\left(\\mathbf{W}_2\\mathbf{W}_1^{-1}-\\frac{\\lambda^{(2)}}{\\lambda^{(1)}}\\mathrm{I}\\right)\\psi^{(1)}\\right\\Vert_2\\leq\\frac{\\sqrt{\\lambda^{(2)}}}{\\lambda^{(1)}\\sqrt{\\lambda^{(1)}}}\\epsilon_{\\mathbf{S}}$.\n{Thus, by definition (see Proposition \\ref{prop_app:pseudo_explicit} in Appendix \\ref{app:add_state})}, there exists a matrix $\\mathbf{E}$ such that $(\\mathbf{W}_2\\mathbf{W}_1^{-1}+\\mathbf{E})\\psi^{(1)}=\\frac{\\lambda^{(2)}}{\\lambda^{(1)}}\\psi^{(1)}$ and $\\left\\Vert\\mathbf{E}\\right\\Vert\\leq\\frac{\\sqrt{\\lambda^{(2)}}}{\\lambda^{(1)}\\sqrt{\\lambda^{(1)}}}\\epsilon_{\\mathbf{S}}$. \nTherefore, we can write the following:\n\\begin{align}\n\\mathbf{E}\\psi^{(1)}  = &\\, -\\left(\\mathbf{W}_2\\mathbf{W}_1^{-1}-\\frac{\\lambda^{(2)}}{\\lambda^{(1)}}\\mathrm{I}\\right)\\psi^{(1)}\\label{eq:pseudo_s_3}\\\\\n = &\\, -\\left((\\mathbf{W}_2\\mathbf{W}_1^{-1})^{1\/2}+\\sqrt{\\frac{\\lambda^{(2)}}{\\lambda^{(1)}}}\\mathrm{I}\\right)\\left((\\mathbf{W}_2\\mathbf{W}_1^{-1})^{1\/2}-\\sqrt{\\frac{\\lambda^{(2)}}{\\lambda^{(1)}}}\\mathrm{I}\\right)\\psi^{(1)}.\\nonumber\n\\end{align}\nSince $(\\mathbf{W}_2\\mathbf{W}_1^{-1})^{1\/2}+\\sqrt{\\lambda^{(2)}\/\\lambda^{(1)}}\\mathrm{I}$ is positive definite, \\eqref{eq:pseudo_s_3} leads to\n\\begin{equation}\n\\left((\\mathbf{W}_2\\mathbf{W}_1^{-1})^{1\/2}-\\sqrt{\\frac{\\lambda^{(2)}}{\\lambda^{(1)}}}\\mathrm{I}\\right)\\psi^{(1)}= -    \\left((\\mathbf{W}_2\\mathbf{W}_1^{-1})^{1\/2}+\\sqrt{\\frac{\\lambda^{(2)}}{\\lambda^{(1)}}}\\mathrm{I}\\right)^{-1}\\mathbf{E}\\psi^{(1)}. \n\\label{eq:pseudo_s_sqrt_eq}\n\\end{equation}\nWith the above preparation, we have \n\\begin{align}\n\\left(\\mathbf{S}-\\sqrt{\\lambda^{(1)}\\lambda^{(2)}}\\mathrm{I}\\right)\\psi^{(1)} \\underset{\\eqref{eq:pseudo_s_psi}}{=} & \\lambda^{(1)}(\\mathbf{W}_2\\mathbf{W}_1^{-1})^{1\/2}\\psi^{(1)}-\\sqrt{\\lambda^{(1)}\\lambda^{(2)}}\\psi^{(1)} \\nonumber \\\\ \n= \\,& \\lambda^{(1)}\\left((\\mathbf{W}_2\\mathbf{W}_1^{-1})^{1\/2}-\\sqrt{\\frac{\\lambda^{(2)}}{\\lambda^{(1)}}}\\mathrm{I}\\right)\\psi^{(1)} \\nonumber \\\\\n\\underset{\\eqref{eq:pseudo_s_sqrt_eq}}{=} & -\\lambda^{(1)}\\left((\\mathbf{W}_2\\mathbf{W}_1^{-1})^{1\/2}+\\sqrt{\\frac{\\lambda^{(2)}}{\\lambda^{(1)}}}\\mathrm{I}\\right)^{-1}\\mathbf{E}\\psi^{(1)}.\\label{eq:pseudo_s_eq}\n\\end{align}\nTaking the norm gives\n\\begin{align}\n\\left\\Vert \\left(\\mathbf{S}-\\sqrt{\\lambda^{(1)}\\lambda^{(2)}}\\mathrm{I}\\right)\\psi^{(1)} \\right\\Vert_2 \\le \\, &\\lambda^{(1)}\\left\\Vert\\left((\\mathbf{W}_2\\mathbf{W}_1^{-1})^{1\/2}+\\sqrt{\\frac{\\lambda^{(2)}}{\\lambda^{(1)}}}\\mathrm{I}\\right)^{-1}\\right\\Vert\\left\\Vert\\mathbf{E}\\right\\Vert\\\\\n\\leq \\,&  \\frac{\\sqrt{\\lambda^{(2)}\/\\lambda^{(1)}}}{\\sigma_{\\min}\\left((\\mathbf{W}_2\\mathbf{W}_1^{-1})^{1\/2}\\right)+\\sqrt{\\lambda^{(2)}\/\\lambda^{(1)}}}\\epsilon_{\\mathbf{S}} \\nonumber \\\\\n\\leq \\,& \\frac{\\sqrt{\\lambda^{(2)}\/\\lambda^{(1)}}}{\\sqrt{\\lambda^{(2)}\/\\lambda^{(1)}}}\\epsilon_{\\mathbf{S}}=\\epsilon_{\\mathbf{S}} \\nonumber, \\label{eq:pseudo_s_sqrt_norm}\n\\end{align}\nwhere $\\sigma_{\\min}$ denotes the minimum eigenvalue. \nThus, $\\sqrt{\\lambda^{(1)}\\lambda^{(2)}}$ is an $\\epsilon_{\\mathbf{S}}$-pseudo-eigenvalue of $\\mathbf{S}$, where $\\psi^{(1)}$ is a corresponding $\\epsilon_{\\mathbf{S}}$-pseudo-eigenvector.\n\\end{proof}\n\nNote that the above proof shows that $\\psi^{(1)}$ is a pseudo-eigenvector of $\\mathbf{S}$, when $\\mathbf{S}$ is defined as the midpoint of the geodesic curve connecting $\\mathbf{W}_1$ and $\\mathbf{W}_2$ (by setting $p=0.5$).\nHowever, due to the decomposition in \\eqref{eq:pseudo_s_3}, the proof is not compatible with definitions of $\\mathbf{S}$ at other points $p \\in (0,1)$ along the geodesic path.\nFor such cases, a different proof is required, specifically, without using the algebraic relationship in \\eqref{eq:pseudo_s_3} that leads to \\eqref{eq:pseudo_s_sqrt_eq}. In the following statement, which is the counterpart of Theorem \\ref{theo:A_eigs} for the case where the eigenvector is not strictly common, we control the ``pseudo'' part by a straightforward perturbation argument.\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{theorem}\\label{prop:pseudo_apart}\nFor $k=1,2$, consider the eigendecomposition $\\mathbf{W}_k=\\mathbf{U}_k \\mathbf{L}_k \\mathbf{U}_k^\\top\\in \\mathbb{R}^{N\\times N}$, where $\\mathbf{L}_k:=\\text{diag}(\\lambda_1^{(k)},\\ldots,\\lambda_N^{(k)})$ so that $\\lambda_1^{(1)}\\geq\\ldots\\geq\\lambda_N^{(1)}$ and $\\mathbf{U}_k:=\\begin{bmatrix}\\psi_1^{(k)}&\\ldots&\\psi_N^{(k)}\\end{bmatrix}\\in \\mathcal{V}_{N,N}$. \nAssume the above eigendecomposition satisfies $\\mathbf{U}_2=\\mathbf{U}_1+\\epsilon \\mathbf{A}$, where $\\|\\mathbf{A}\\|=1$, $\\epsilon>0$ is a small constant, and $c^{-1}\\leq \\ell_i:= \\lambda_i^{(2)}\/\\lambda_i^{(1)}\\leq c$ for some constant $c\\geq1$ for all $i=1,\\ldots,N$. For any $i=1,\\ldots,N$, denote the spectral gap $\\gamma_i:=\\min_{k,\\,\\ell_k\\neq \\ell_i}|\\ell_i-\\ell_k|$.\n\nFix $j$. Then, for the $j$-th eigenpair of $\\mathbf{W}_1$, when $\\epsilon$ is sufficiently small, we have\n\\[\n\\left\\Vert\\left(\\mathbf{F}-0.5\\sqrt{\\lambda_j^{(1)}\\lambda_j^{(2)}}\\log\\left(\\frac{\\lambda_j^{(1)}}{\\lambda_j^{(2)}}\\right)\\mathrm{I}\\right)\\psi_j^{(1)}\\right\\Vert_2=O(\\epsilon)\\,,\n\\]\nwhere the implied constant depends on \n$\\frac{\\sqrt{c}\\ln c}{\\min_i\\left(\\gamma_i\\sqrt{\\lambda_i^{(1)}}\\right)}$.\n\n\\end{theorem}\n\n\n\n\\begin{proof}  \n\nBy Proposition \\ref{prop_app:sa_equiv_forms} (see Appendix \\ref{app:add_state}), we rewrite the operator $\\mathbf{F}=\\mathbf{S}^{1\/2}\\log\\left(\\mathbf{S}^{-1\/2}\\mathbf{W}_1\\mathbf{S}^{-1\/2}\\right)\\mathbf{S}^{1\/2}$ as $\\mathbf{F} = \\log\\left(\\mathbf{W}_1\\mathbf{S}^{-1}\\right)\\mathbf{S}$. \nDenote \n\\[\n\\hat{\\lambda}_{\\mathbf{F}}=0.5\\sqrt{\\lambda_j^{(1)}\\lambda_j^{(2)}}\\log\\left(\\frac{\\lambda_j^{(1)}}{\\lambda_j^{(2)}}\\right)\\,.\n\\]\n{By Theorem \\ref{prop:pseudo_sa}, there exists $\\mathbf{E}_{\\mathbf{S}}$ with a sufficiently small norm, such that\n\\[\n    \\mathbf{S}\\psi_j^{(1)} = \\left(\\sqrt{\\lambda_j^{(1)}\\lambda_j^{(2)}}\\mathrm{I}-\\mathbf{E}_{\\mathbf{S}}\\right)\\psi_j^{(1)}, \\\n\\]\nand therefore, we have}\n\\begin{align}\n\\left(\\mathbf{F}-\\hat{\\lambda}_{\\mathbf{F}}\\mathrm{I}\\right)\\psi_j^{(1)} = &\\, \\left(\\log\\left(\\mathbf{W}_1\\mathbf{S}^{-1}\\right)\\mathbf{S}-\\hat{\\lambda}_{\\mathbf{F}}\\mathrm{I}\\right)\\psi_j^{(1)}\n\\nonumber\\\\\n=&\\,\\sqrt{\\lambda_j^{(1)}\\lambda_j^{(2)}}\\left(\\log\\left(\\mathbf{W}_1\\mathbf{S}^{-1}\\right)-\\frac{\\hat{\\lambda}_{\\mathbf{F}}}{\\sqrt{\\lambda_j^{(1)}\\lambda_j^{(2)}}}\\mathrm{I}\\right)\\psi_j^{(1)}-\\log\\left(\\mathbf{W}_1\\mathbf{S}^{-1}\\right)\\mathbf{E}_{\\mathbf{S}}\\psi_j^{(1)}.\n\\label{eq:pseudo_a_1}\n\\end{align}\n\nSince both $\\mathbf{W}_1$ and $\\mathbf{W}_2$ are positive definite, they are invertible and also $\\left(\\mathbf{W}_2\\mathbf{W}_1^{-1}\\right)^{1\/2}$ is invertible. \nTherefore, by \\eqref{eq:equivexp}, $\\mathbf{W}_1\\mathbf{S}^{-1}=\\mathbf{W}_1\\mathbf{W}_1^{-1}\\left(\\mathbf{W}_2\\mathbf{W}_1^{-1}\\right)^{-1\/2}=\\left(\\mathbf{W}_2\\mathbf{W}_1^{-1}\\right)^{-1\/2}$. \nSubstituting this relationship into \\eqref{eq:pseudo_a_1} yields\n\\begin{align}\n\\left(\\mathbf{F}-\\hat{\\lambda}_{\\mathbf{F}}\\mathrm{I}\\right)\\psi^{(1)}_j\n= - \\frac{\\sqrt{\\lambda_j^{(1)}\\lambda_j^{(2)}}}{2}\n\\left(\\log\\left(\\mathbf{W}_2\\mathbf{W}_1^{-1}\\right)-\\log\\left(\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}\\right)\\mathrm{I}\\right)\\psi_j^{(1)}-\\frac{1}{2}\\log\\left(\\mathbf{W}_2\\mathbf{W}_1^{-1}\\right)\\mathbf{E}_{\\mathbf{S}}\\psi_j^{(1)}\\,. \\label{eq:pseudo_a_2}\n\\end{align}\nNow we control the right hand side term by term. \nRecall that for any analytic function $f$ over an open set in $\\mathbb{R}$ that contains the spectrum of $\\mathbf{W}_2\\mathbf{W}_1^{-1}$, we can define $f(\\mathbf{W}_2\\mathbf{W}_1^{-1})$.\nSince $\\mathbf{W}_2\\mathbf{W}_1^{-1}$ and $\\mathbf{W}_1^{-1\/2}\\mathbf{W}_2 \\mathbf{W}_1^{-1\/2}$ are similar, we have\n\\begin{equation}\nf(\\mathbf{W}_2\\mathbf{W}_1^{-1})=\\mathbf{W}_1^{1\/2}f\\left(\\mathbf{W}_1^{-1\/2}\\mathbf{W}_2 \\mathbf{W}_1^{-1\/2}\\right)\\mathbf{W}_1^{-1\/2}\\,,\\label{eq:pseudo_a_6}\n\\end{equation}\nand hence\n\\begin{align}\nf\\left(\\mathbf{W}_1^{-1\/2}\\mathbf{W}_2\\mathbf{W}_1^{-1\/2}\\right)\\psi_j^{(1)}= \\mathbf{W}_1^{-1\/2}f(\\mathbf{W}_2\\mathbf{W}_1^{-1})\\mathbf{W}_1^{1\/2}\\psi_j^{(1)}=\\sqrt{\\lambda_j^{(1)}}\\mathbf{W}_1^{-1\/2}f(\\mathbf{W}_2\\mathbf{W}_1^{-1})\\psi_j^{(1)}.\\label{eq:pseudo_a_6q}\n\\end{align}\n\nLet $\\mu_i$ and $v_i$ denote the eigenvalues and eigenvectors of the matrix $\\mathbf{W}_1^{-1\/2}\\mathbf{W}_2\\mathbf{W}_1^{-1\/2}$, respectively, for $i=1,\\dots,N$.  %\nSince $\\{\\psi_j^{(1)}\\}_{j=1}^N$ and $\\{v_j\\}_{j=1}^N$ are both orthonormal bases of $\\mathbb{R}^N$, we have $\\psi_j^{(1)}=\\sum_i \\alpha_{ji} v_i$, where $\\alpha_{ji}\\in\\mathbb{R}$ and $\\sum_i \\alpha_{ji}^2=1$ for all $j$.\nBy \\eqref{eq:pseudo_a_6q}, we have \n\n\\begin{align}\n&\\frac{\\sqrt{\\lambda_j^{(1)}\\lambda_j^{(2)}}}{2}\\left(\\log(\\mathbf{W}_2\\mathbf{W}_1^{-1})-\\log\\left(\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}\\right)\\mathrm{I}\\right)\\psi_j^{(1)}\\nonumber\\\\\n=&\\,\\frac{\\sqrt{\\lambda_j^{(2)}}}{2}\\mathbf{W}_1^{1\/2}\\left(\\log(\\mathbf{W}_1^{-1\/2}\\mathbf{W}_2\\mathbf{W}_1^{-1\/2})-\\log\\left(\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}\\right)\\mathrm{I}\\right)\\psi_j^{(1)}\\nonumber\\\\\n=&\\,\\frac{\\sqrt{\\lambda_j^{(2)}}}{2}\\mathbf{W}_1^{1\/2}\\sum_{i=1}^N\\left(\\log(\\mathbf{W}_1^{-1\/2}\\mathbf{W}_2\\mathbf{W}_1^{-1\/2})-\\log\\left(\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}\\right)\\mathrm{I}\\right)\\alpha_{ji}v_i.\n\\end{align}\nUsing the fact that \n\\begin{eqnarray}\n \\left(\\log\\left(\\mathbf{W}_1^{-1\/2}\\mathbf{W}_2\\mathbf{W}_1^{-1\/2}\\right)-\\log\\left(\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}\\right)\\mathrm{I}\\right)v_i = \\left(\\log\\mu_i-\\log\\left(\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}\\right)\\right)v_i\\,,\\label{eq:pseudo_a_5}\n\\end{eqnarray}\nyields\n\\begin{align}\n&\\frac{\\sqrt{\\lambda_j^{(2)}}}{2}\\mathbf{W}_1^{1\/2}\\sum_{i=1}^N\\left(\\log(\\mathbf{W}_1^{-1\/2}\\mathbf{W}_2\\mathbf{W}_1^{-1\/2})-\\log\\left(\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}\\right)\\mathrm{I}\\right)\\alpha_{ji}v_i\\nonumber\\\\\n=&\\,\\frac{\\sqrt{\\lambda_j^{(2)}}}{2}\\mathbf{W}_1^{1\/2}\\sum_{i=1}^N\\left(\\log\\mu_i-\\log\\left(\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}\\right)\\right)\\alpha_{ji}v_i\\label{eq to control 00}\n\\end{align}\nTherefore, the squared $L^2$ norm of the first term in the right hand side of \\eqref{eq:pseudo_a_2} becomes\n\\begin{align}\n&\\left\\| \\frac{\\sqrt{\\lambda_j^{(1)}\\lambda_j^{(2)}}}{2}\\left(\\log(\\mathbf{W}_2\\mathbf{W}_1^{-1})-\\log\\left(\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}\\right)\\mathrm{I}\\right)\\psi_j^{(1)}\\right\\|^2_2\\nonumber\\\\\n=&\\left\\|\\frac{\\sqrt{\\lambda_j^{(2)}}}{2}\\mathbf{W}_1^{1\/2}\\sum_{i=1}^N\\left(\\log\\mu_i-\\log\\left(\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}\\right)\\right)\\alpha_{ji}v_i\\right\\|^2_2\n\\leq \\frac{\\lambda_j^{(2)}}{4}\\left\\|\\mathbf{W}_1^{1\/2}\\right\\|^2\\left\\|\\sum_{i=1}^N\\left(\\log\\mu_i-\\log\\left(\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}\\right)\\right)\\alpha_{ji}v_i\\right\\|^2_2\\nonumber\\\\\n=&\\frac{\\lambda_j^{(2)}}{4}  \\sum_{i=1}^N\\alpha_{ji}^2\n\\left(\\log\\mu_i-\\log\\left(\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}\\right)\\right)^2\\, ,\\label{eq to control 1}\n\\end{align}\nwhere we use the fact that $\\{v_j\\}$ form an orthonormal basis and that the operator norm $\\left\\|\\mathbf{W}_1^{1\/2}\\right\\|^2=1$, since $\\mathbf{W}_1$ is normalized.\n\n\nNext, as in \\eqref{eq:pseudo_s_eq}, we set\n\\begin{align}\n\\mathbf{E}_{\\mathbf{S}}\\psi_j^{(1)}\n\\lambda_j^{(1)}\\left((\\mathbf{W}_2\\mathbf{W}_1^{-1})^{1\/2}+\\sqrt{\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}}\\mathrm{I}\\right)^{-1}\\mathbf{E}\\psi_j^{(1)} \n\\underset{\\eqref{eq:pseudo_s_sqrt_eq}}{=}-\\lambda_j^{(1)}\\left((\\mathbf{W}_2\\mathbf{W}_1^{-1})^{1\/2}-\\sqrt{\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}}\\mathrm{I}\\right)\\psi_j^{(1)}\n\\end{align}\nfor some $\\mathbf{E}$ with a sufficiently small norm.\nSince $f(x)=\\log(x)\\sqrt{x}$ is analytic over an open set that contains the spectrum of $\\mathbf{W}_1\\mathbf{W}_2^{-1}$, by the same argument as that for \\eqref{eq to control 00}, we have \n\\begin{align}\n\\log\\left(\\mathbf{W}_2\\mathbf{W}_1^{-1}\\right)\\mathbf{E}_{\\mathbf{S}}\\psi_j^{(1)}&\\,=-\\lambda_j^{(1)}\\log\\left(\\mathbf{W}_2\\mathbf{W}_1^{-1}\\right)\\left((\\mathbf{W}_2\\mathbf{W}_1^{-1})^{1\/2}-\\sqrt{\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}}\\mathrm{I}\\right)\\psi_j^{(1)}\\nonumber\\\\\n&\\,=-\\sqrt{\\lambda_j^{(1)}}  \\mathbf{W}_1^{1\/2}\\sum_{i=1}^N\\alpha_{ji}\\log(\\mu_i)\\left(\\sqrt{\\mu_i}-\\sqrt{\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}}\\right)v_i\\,,\\nonumbe\n\\end{align}\nand hence the squared $L^2$ norm of the second term in the right hand side of \\eqref{eq:pseudo_a_2} becomes\n\\begin{align}\n\\left\\|\\log\\left(\\mathbf{W}_2\\mathbf{W}_1^{-1}\\right)\\mathbf{E}_{\\mathbf{S}}\\psi_j^{(1)}\\right\\|_2^2 \\leq \\lambda_j^{(1)} \\sum_{i=1}^N \n\\alpha^2_{ji}(\\log \\mu_i)^2\\left(\\sqrt{\\mu_i}-\\sqrt{\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}}\\right)^2\\,,\\label{eq to control 2}\n\\end{align}\nwhere we again use the fact that $\\{v_j\\}$ form an orthonormal basis and that the operator norm $\\left\\|\\mathbf{W}_1^{1\/2}\\right\\|^2=1$.\nTo finish the proof, we control $\\alpha_{ji}$ and the relationship between $\\mu_i$ and $\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}$ in \\eqref{eq to control 1} and \\eqref{eq to control 2} using matrix perturbation theory.\nBy a direct expansion, we have\n\\begin{align}\n\\mathbf{W}_1^{-1\/2}\\mathbf{W}_2\\mathbf{W}_1^{-1\/2}=\\mathbf{U}_1 \\mathbf{L}_2 \\mathbf{L}_1^{-1} \\mathbf{U}_1^\\top+\\epsilon \\mathbf{B}\\,,\n\\end{align}\nwhere $\\mathbf{B}=\\mathbf{U}_1 \\mathbf{L}_1^{-1\/2} \\mathbf{U}_1^\\top(\\mathbf{A}\\mathbf{L}_2\\mathbf{U}_1^\\top+\\mathbf{U}_1\\mathbf{L}_2\\mathbf{A}^\\top)\\mathbf{U}_1\\mathbf{L}_1^{-1\/2}\\mathbf{U}_1^\\top+\\epsilon \\mathbf{U}_1\\mathbf{L}_1^{-1\/2}\\mathbf{U}_1^\\top \\mathbf{A} \\mathbf{L}_2 \\mathbf{A}^\\top \\mathbf{U}_1 \\mathbf{L}_1^{-1\/2}\\mathbf{U}_1^\\top$. \nTo simplify the notation, we assume that the diagonal entries of $\\mathbf{L}_2\\mathbf{L}_1^{-1}=\\text{diag}\\left(\\frac{\\lambda_1^{(2)}}{\\lambda_1^{(1)}},\\ldots,\\frac{\\lambda_N^{(2)}}{\\lambda_N^{(1)}}\\right)=\\text{diag}\\left(\\ell_1,\\ldots,\\ell_N\\right)$ are all distinct, or the following argument could be carried out with eigenprojections. Thus, by a standard perturbation argument, when $\\epsilon$ is sufficiently small, we have \n\\begin{equation}\nv_i=\\psi_i^{(1)}+\\epsilon\\sum_{k\\neq i}\\frac{\\langle \\psi_k^{(1)}, \\mathbf{B}\\psi_i^{(1)}\\rangle}{\\lambda_i^{(2)}\/\\lambda_i^{(1)}-\\lambda_k^{(2)}\/\\lambda_k^{(1)}}\\psi_k^{(1)}+O(\\epsilon^2),\\ \\ \n\\mu_i=\\frac{\\lambda_i^{(2)}}{\\lambda_i^{(1)}}+\\epsilon \\langle \\psi_i^{(1)}, \\mathbf{B}\\psi_i^{(1)}\\rangle +O(\\epsilon^2)\\label{relationship vi and psii}\n\\end{equation}\nfor each $i=1,\\ldots,N$.\nNote that\n\\[\nI_N=\\mathbf{U}_2^\\top\\mathbf{U}_2=(\\mathbf{U}_1+\\epsilon\\mathbf{A})^\\top(\\mathbf{U}_1+\\epsilon\\mathbf{A})=I_N+\\epsilon(\\mathbf{A}^\\top\\mathbf{U}_1+\\mathbf{U}_1^\\top\\mathbf{A}) +\\epsilon^2 \\mathbf{A}^\\top \\mathbf{A}\\,,\n\\]\nso we have\n\\begin{equation}\n\\mathbf{A}^\\top\\mathbf{U}_1=-\\mathbf{U}_1^\\top\\mathbf{A} -\\epsilon \\mathbf{A}^\\top \\mathbf{A}\\,.\\label{AU=-UA-eps AA}\n\\end{equation}\nThus, by a direct expansion with the definition of $\\mathbf{B}$, we have \n\\begin{align}\n\\langle \\psi_k^{(1)}, \\mathbf{B}\\psi_i^{(1)}\\rangle\n=&\\,(\\lambda_i^{(1)}\\lambda_k^{(1)})^{-1\/2}e_k^\\top \\mathbf{U}_1^\\top(\\mathbf{A}\\mathbf{L}_2\\mathbf{U}_1^\\top+\\mathbf{U}_1\\mathbf{L}_2\\mathbf{A}^\\top)\\mathbf{U}_1e_i +O(\\epsilon)\\nonumber\\\\\n=&\\, (\\lambda_i^{(1)}\\lambda_k^{(1)})^{-1\/2} (\\lambda_i^{(2)}e_k^\\top \\mathbf{U}_1^\\top\\mathbf{A} e_i + \\lambda_k^{(2)}e_k^\\top \\mathbf{A}^\\top\\mathbf{U}_1 e_i)\n+O(\\epsilon)\\nonumber\\\\\n=&\\,(\\lambda_i^{(1)}\\lambda_k^{(1)})^{-1\/2}\\left((\\lambda_i^{(2)}-\\lambda_k^{(2)})e_k^\\top\\mathbf{U}_1^\\top\\mathbf{A}e_i-\\epsilon \\lambda_k^{(2)} e_k\\mathbf{A}^\\top\\mathbf{A}e_i\\right)+(\\lambda_i^{(1)}\\lambda_k^{(1)})^{-1\/2}O(\\epsilon)\\nonumber\\\\\n=&\\,(\\lambda_i^{(1)}\\lambda_k^{(1)})^{-1\/2}(\\lambda_i^{(2)}-\\lambda_k^{(2)})e_k^\\top\\mathbf{U}_1^\\top\\mathbf{A}e_i+O(\\epsilon)\\,,\\nonumber\n\\end{align}\nwhere the equality before last\ncomes from \\eqref{AU=-UA-eps AA}, the last equality is due to $|e_k^\\top \\mathbf{A}^\\top \\mathbf{A}e_i|\\leq 1$, since $\\|A\\|=1$, and the constant in this derivation depends on $(\\lambda_i^{(1)}\\lambda_k^{(1)})^{-1\/2}$.\nThus, for the $j$-th eigenpair we are concerned with, we have $\\mu_j=\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}+O(\\epsilon^2)$, since $\\langle \\psi_j^{(1)}, \\mathbf{B}\\psi_j^{(1)}\\rangle=O(\\epsilon)$.\nFor the eigenvector, when $k\\neq i$,\nwe have\n\\begin{align}\n\\left|\\frac{\\langle \\psi_k^{(1)}, \\mathbf{B}\\psi_i^{(1)}\\rangle}{\\lambda_i^{(2)}\/\\lambda_i^{(1)}-\\lambda_k^{(2)}\/\\lambda_k^{(1)}}\\right|\n\\leq &\\,\\frac{(\\lambda_i^{(1)}\\lambda_k^{(1)})^{-1\/2}\\left|\\lambda_i^{(2)}-\\lambda_k^{(2)}\\right|}{\\left|\\lambda_i^{(2)}\/\\lambda_i^{(1)}-\\lambda_k^{(2)}\/\\lambda_k^{(1)}\\right|}\\left|e_k^\\top \\mathbf{U}_1^\\top\\mathbf{A}e_i\\right|+O(\\epsilon)\\nonumber\\\\\n\\leq &\\,\\frac{1}{\\gamma_i} \\frac{\\left|(\\lambda_i^{(2)}-\\lambda_k^{(2)})\\right|}{\\sqrt{\\lambda_i^{(1)}\\lambda_k^{(1)}}} \\left|e_k^\\top \\mathbf{U}_1^\\top\\mathbf{A}e_i\\right|+O(\\epsilon\n\\nonumber\\,,\n\\end{align}\nwhere the constant depends on $\\frac{1}{\\sqrt{\\lambda_i^{(1)}\\lambda_k^{(1)}} \\gamma_i}$.\n\nBy \\eqref{relationship vi and psii} we have for $j\\neq i$:\n\\begin{align}\n\t\\alpha_{ji} &= \\langle \\psi_j^{(1)}, v_i \\rangle = \\langle \\psi_j^{(1)}, \\psi_i^{(1)}+\\epsilon\\sum_{k\\neq i}\\frac{\\langle \\psi_k^{(1)}, \\mathbf{B}\\psi_i^{(1)}\\rangle}{\\lambda_i^{(2)}\/\\lambda_i^{(1)}-\\lambda_k^{(2)}\/\\lambda_k^{(1)}}\\psi_k^{(1)}+O(\\epsilon^2) \\rangle\\nonumber \\\\\n\t&= \\epsilon\\sum_{k\\neq i}\\frac{\\langle \\psi_k^{(1)}, \\mathbf{B}\\psi_i^{(1)}\\rangle}{\\lambda_i^{(2)}\/\\lambda_i^{(1)}-\\lambda_k^{(2)}\/\\lambda_k^{(1)}}\\langle \\psi_j^{(1)}, \\psi_k^{(1)}\\rangle +O(\\epsilon^2)\\nonumber \\\\\n\t&= \\epsilon \\frac{\\langle \\psi_j^{(1)}, \\mathbf{B}\\psi_i^{(1)}\\rangle}{\\lambda_i^{(2)}\/\\lambda_i^{(1)}-\\lambda_j^{(2)}\/\\lambda_j^{(1)}} +O(\\epsilon^2)\\, .\\nonumber\n\\end{align}\nCombining this with the inequality above, we have for $j\\neq i$:\n\\begin{align}\n\t\\left| \\alpha_{ji} \\right| \\leq  \\epsilon\n\t\\frac{1}{\\gamma_i} \\frac{\\left|(\\lambda_i^{(2)}-\\lambda_j^{(2)})\\right|}{\\sqrt{\\lambda_i^{(1)}\\lambda_j^{(1)}}}\n\n\t\\left|e_j^\\top\\mathbf{U}_1^\\top\\mathbf{A}e_i \\right| + O(\\epsilon^2)\\, ,\n\\end{align}\nwhere the constant depends on $\\frac{1}{\\sqrt{\\lambda_i^{(1)}\\lambda_j^{(1)}} \\gamma_i}$.\n\nWe thus have for \\eqref{eq to control 1}:\n\\begin{align}\n &\\frac{\\lambda_j^{(2)}}{4}\\sum_{i=1}^N\\alpha_{ji}^2\n \\left(\\log\\mu_i-\\log\\left(\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}\\right)\\right)^2\\nonumber\\\\\n=&\\,\\frac{\\lambda_j^{(2)}}{4}\\alpha_{jj}^2\n\\left(\\log\\mu_j-\\log\\left(\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}\\right)\\right)^2+ \\frac{\\lambda_j^{(2)}}{4}\\sum_{i\\neq j}\\alpha_{ji}^2\n\\left(\\log\\mu_i-\\log\\left(\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}\\right)\\right)^\n=O(\\epsilon^2)\\,,\\label{final bound part 1}\n\\end{align}\nwhere the constant depends on $\\frac{c(\\ln c)^2}{\\min_i\\{\\gamma_i^2\\lambda_i^{(1)}\\}}$.\n\nThis is because the first term is $O(\\epsilon^2)$ by\n\\begin{align}\n\t\\alpha_{jj}^2 \\left(\\log\\mu_j-\\log\\left(\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}\\right)\\right)^2 = &\\, \\alpha_{jj}^2 \\left(\\epsilon \\langle \\psi_j^{(1)}, \\mathbf{B}\\psi_j^{(1)}\\rangle \\frac{\\lambda_j^{(1)}}{\\lambda_j^{(2)}}\\right)^2 + O(\\epsilon^2) \\\\\n\t= &\\, \\epsilon^2 \\alpha_{jj}^2 \\left( \\frac{\\lambda_j^{(1)}}{\\lambda_j^{(2)}} \\right)^2 (\\lambda_j^{(1)}\\lambda_j^{(1)})^{-1} \\left((\\lambda_j^{(2)}-\\lambda_j^{(2)})e_j^\\top\\mathbf{U}_1^\\top\\mathbf{A}e_j+O(\\epsilon)\\right) ^2 + O(\\epsilon^2)\n\\end{align}\nusing Taylor expansion $\\log (x+h)=\\log(x)+h\/x+O(h^2)$ for $x>0$ and sufficiently small $h$.\nIn addition, noting that $\\left(\\log\\mu_i-\\log\\left(\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}\\right)\\right)^2\\leq 5(\\ln c)^2$ we have that the second term is bounded by:\n\\begin{align}\n\t\\frac{\\lambda_j^{(2)}}{4}\\sum_{i\\neq j}\\alpha_{ji}^2 \\left(\\log\\mu_i-\\log\\left(\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}\\right)\\right)^2 & \\leq \\frac{\\lambda_j^{(2)}}{4}5(\\ln c)^2 \\sum_{i\\neq j}\\alpha_{ji}^2 \\nonumber \\\\\n\t& \\leq \\frac{5\\lambda_j^{(2)}}{4}(\\ln c)^2 \\sum_{i\\neq j} \\epsilon^2 \n\t\\frac{1}{\\gamma_i^2} \\frac{(\\lambda_i^{(2)}-\\lambda_j^{(2)})^2}{\\lambda_i^{(1)}\\lambda_j^{(1)}}\n\n\t\\left|e_j^\\top\\mathbf{U}_1^\\top\\mathbf{A}e_i \\right|^2 \\nonumber\\\\\n\n\t& \\leq \\frac{5}{4}\\epsilon^2(\\ln c)^2\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}} \\sum_{i\\neq j} \n\t\\frac{1}{\\gamma_i^2} \\frac{(\\lambda_i^{(2)}-\\lambda_j^{(2)})^2}{\\lambda_i^{(1)}}\\left|e_j^\\top\\mathbf{U}_1^\\top\\mathbf{A}e_i \\right|^2 \\, .\n\\end{align}\nContinuing with a few coarse steps:\n\\begin{align}\n    \\frac{\\lambda_j^{(2)}}{4}\\sum_{i\\neq j}\\alpha_{ji}^2 \\left(\\log\\mu_i-\\log\\left(\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}\\right)\\right)^2 & \\leq \\frac{5}{4}\\epsilon^2\\frac{c(\\ln c)^2}{\\min_i\\{\\gamma_i^2\\lambda_i^{(1)}\\}} \\sum_{i\\neq j} \n\t\\left|e_j^\\top\\mathbf{U}_1^\\top\\mathbf{A}e_i \\right|^2\\nonumber\\\\\n\t& \\leq \\frac{5}{2}\\epsilon^2\\frac{c(\\ln c)^2}{\\min_i\\{\\gamma_i^2\\lambda_i^{(1)}\\}}\n\\end{align}\nwhere $\\left|\\lambda_i^{(2)}-\\lambda_j^{(2)}\\right|^2\\leq 1$ due to the normalization of $\\mathbf{W}_1$ and $\\mathbf{W}_2$ and $\\sum_{i\\neq j} \\left|e_j^\\top\\mathbf{U}_1^\\top\\mathbf{A}e_i \\right|^2 \\leq 2 \\| \\mathbf{A} e_j \\|^2 \\leq 2$ due to $\\| \\mathbf{A}\\|=1$.\n\n\nSimilarly, it can be shown for \\eqref{eq to control 2} that\n\\[\n\\sum_{i=1}^N  \\alpha^2_{ji}(\\log \\mu_i)^2\\left(\\sqrt{\\mu_i}-\\sqrt{\\frac{\\lambda_j^{(2)}}{\\lambda_j^{(1)}}}\\right)^2=O(\\epsilon^2)\\,,\n\\]\nand the proof is concluded.\n\\end{proof}\n\n\\begin{remark}\nNote that the implied constant $\\frac{\\sqrt{c}\\ln c}{\\min_i\\left(\\gamma_i\\sqrt{\\lambda_i^{(1)}}\\right)}$ might be large, narrowing the scope of Theorem \\ref{prop:pseudo_apart}. Particularly in our context, the matrix $\\mathbf{W}_1$ (and $\\mathbf{W}_2$) tends to be close to low rank, for which $\\min_i\\left(\\gamma_i\\sqrt{\\lambda_i^{(1)}}\\right)$ is small.\n\\end{remark}\n\\begin{remark}\nEmpirically, we observe that $\\psi_j^{(1)}=\\sum_i \\alpha_{ji} v_i \\simeq \\sum_{i \\sim j} \\alpha_{ji} v_i$, i.e., only a small number of expansion coefficients $\\alpha_{ji}$ are non-negligible, for which $\\lambda_i^{(1)}$ is close to $\\lambda_j^{(1)}$. \nTherefore, in practice, the implied constant depends on $1\/\\min_{i \\sim j}\\left(\\gamma_i\\sqrt{\\lambda_i^{(1)}}\\right)$.\nSince we are usually interested in principal components $\\psi_j^{(1)}$ (i.e., with large $\\lambda_j^{(1)}$), the implied constant is typically sufficiently large. \n\\end{remark}\n}\n\n\n\n\n\n\\section{Conclusions\\label{sec:conc}}\n\n\nIn this work, we introduce a new multi-resolution analysis of temporal high-dimensional data with an underlying time-varying manifold structure. Our analysis is based on the definition of two new composite operators that represent the relation of two aligned datasets jointly sampled from two diffeomorphic manifolds in terms of their spectral components. Specifically, we showed that these operators not only recover but also distinguish different types of common spectral components of the underlying manifolds and that each operator emphasizes different properties. \nOne operator was shown to emphasize common components that are similarly expressed in the two manifolds, and the other operator was shown to emphasize the common components that are expressed with significantly different eigenvalues.\nIn the context of spatiotemporal data analysis, the application of the new operators is analogous to low-pass and high-pass filters. Therefore, by applying them in multiple resolutions, we devise a wavelet-like analysis framework.\nWe demonstrated this framework on a dynamical system describing a transitory double-gyre flow, showing that such a framework can be used for the analysis of non-stationary multivariate time-series.\n\nIn addition to spatiotemporal analysis, we showed that the new composite operators may be useful for multimodal data analysis as well. Specifically, we showed application to remote sensing, demonstrating the recovery of meaningful properties expressed by  different sensing modalities.\n\n\nIn the future, we plan to extend the definition of the operators $\\mathbf{S}$ and $\\mathbf{F}$ from two to more time frames (datasets). \nIn addition, since our analysis results in a large number of vectors representing the common components at different scales and time-points, we plan to develop compact representations of these components, which may lead to improved, more conclusive results for highly non-stationary time-series.\n\nFinally, we remark that in our model, we represent each sample by an undirected weighted graph, and then, analyze the temporal sequence of graphs. Another interesting future work would be to investigate our Riemannian composite operators in the context of graph neural networks (GNNs) and graph convolutional networks (GCNs) \\cite{scarselli2008graph,kipf2016semi,bronstein2017geometric}.\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nNeural machine translation (NMT) systems \nare sensitive to the data they are trained on. \nThe available parallel corpora come from various genres and have different stylistic variations and \nsemantic ambiguities. \nWhile\nsuch data is often beneficial for a general purpose machine translation system, a problem arises when building systems for specific domains such as lectures \\cite{guzman-sajjad-etal:iwslt13,cettolo2014report}, patents \\cite{Fujii10overviewof} or medical text \\cite{bojar-EtAl:2014:W14-33}, where either the in-domain bilingual text does not exist or is available in small \nquantities.\n\nDomain adaptation aims to preserve the identity of the in-domain data while exploiting the out-of-domain data in favor of the in-domain data and avoiding possible drift towards out-of-domain jargon and style. \nThe most commonly used approach to train a domain-specific neural MT system is to fine-tune an existing model (trained on generic data) with the new domain\n\\cite{luong-manning:iwslt15,FreitagA16,ServanCS16,ChuDK17} or \nto add domain-aware tags in building a concatenated system \\cite{KobusCS16}. \n\\cite{wees:da:emnlp17} proposed a gradual fine-tuning method that starts training with complete in- and out-of-domain data and gradually reduces the out-of-domain data for next epochs. Other approaches that have been recently proposed for domain adaptation of neural machine translation are instance weighting \\cite{wang:da:emnlp17,chen:da:wnmt17} and data selection \\cite{wang:da:acl17}. \n\n\n\nIn this paper we explore NMT in a multi-domain scenario. \nConsidering a small in-domain corpus and a number of out-of-domain corpora, we target questions like: \n\n\\begin{itemize}\n\n\\item What are the different ways to combine multiple domains during a training process?\n\\item What is the best strategy to build an optimal in-domain system?\n\\item Which training strategy results in a robust system?\n\\item Which strategy should be used to build a decent in-domain system given limited time?\n\\end{itemize}\nTo answer these, we try the following approaches: \\textbf{i) data concatenation:} train a system by concatenating all the available in-domain and out-of-domain data; \\textbf{ii) model stacking:} build NMT in an online fashion starting from the most distant domain, fine-tune on the closer domain and finish by fine-tuning the model on the in-domain data; \\textbf{iii) data selection:} select a certain percentage of the available out-of-domain corpora that is closest to the in-domain data and use it for training the system; \\textbf{iv) multi-model ensemble:} separately train models for each available domain and combine them during decoding using balanced or weighted averaging.\nWe experiment with Arabic-English and German-English language pairs. Our results demonstrate the following findings: \n\\begin{itemize}\n\\item A concatenated system fine-tuned on the in-domain data achieves the most optimal in-domain system.\n\\item Model stacking works best when\nstarting from the furthest domain, fine-tuning on closer domains and then\nfinally fine-tuning on the in-domain data.\n\\item A concatenated system on all available data results in the most robust system.\n\\item Data selection\ngives a decent trade-off between translation quality and training time.\n\\item Weighted ensemble is helpful when several individual models have been already\ntrained\nand there is no time for retraining\/fine-tuning.\n\\end{itemize}\n\n\n\n\n\nThe paper is organized as follows:\nSection \\ref{sec:approaches} describes the adaptation approaches explored in this work. We present experimental design in Section \\ref{sec:experiments}. Section \\ref{sec:results} summarizes the results and Section \\ref{sec:conclusion} concludes.\n\n\n\n\n\n\n\\section{Approaches}\n\\label{sec:approaches}\n\\begin{figure} \n\t\\centering\n\t\\includegraphics[width=\\linewidth]{data-approaches.png}\n\t\\caption{Multi-domain training approaches}\n\t\\label{fig:data-approaches}\n\\end{figure}\n\nConsider an in-domain data D$_i$ and a set of out-of-domain data D$_o$ = {D$_{o_1}$, D$_{o_2}$, ..D$_{o_n}$}. We explore several methods to benefit from the available data with an aim to optimize translation quality on the in-domain data. Specifically, we try data concatenation, model stacking, data selection and ensemble. Figure \\ref{fig:data-approaches} presents them graphically. In the following, we describe each approach briefly.\n\n\\subsection{Concatenation}\n\nA na\\\"ive yet commonly used method when training both statistical \\cite{williams-EtAl:2016:WMT}\\footnote{State-of-the-art baselines are trained on plain concatenation of the data with MT feature functions (such as Language Model) skewed towards in-domain data, through interpolation.} and neural machine translation systems \\cite{sennrich-haddow-birch:2016:WMT} is to simply concatenate all the bilingual parallel data before training the system. During training an in-domain validation set is used to guide the training loss.\nThe resulting system has an advantage of seeing a mix of all available data at every time interval, and is thus robust to handle heterogeneous test data.\n\n\n\\subsection{Fine Tuning and Model Stacking}\n\nNeural machine translation follows an online training strategy. It sees only a small portion of the data in every training step and estimates the value of network parameters based on that portion. Previous work has exploited this strategy in the context of domain adaptation. \\cite{luong-manning:iwslt15} trained an initial model on an out-of-domain data and later extended the training on in-domain data. In this way the final model parameters are tuned towards the in-domain data. The approach is referred as \\emph{fine-tuning} later on.\n\nSince in this work we deal with several domains, we propose a stacking method that uses \nmulti-level\nfine-tuning to train a system.\nFigure \\ref{fig:data-approaches} \n(second row) shows the complete procedure: first, \nthe\nmodel is trained on the out-of-domain data D$_{o_1}$ for $N$ epochs; training is resumed from $N+1$-th epoch \nto the $M$-th epoch\nbut using the \nnext \navailable out-of-domain data D$_{o_2}$; repeat the process till all of the available out-of-domain corpora have been used; in the last step, resume training on the in-domain data D$_i$ for a few \nepochs.\nThe resulting model has seen all of the available data as in the case of \nthe \ndata concatenation approach. However, here the system learns from the data domain by domain. We call this technique \\emph{model stacking}. \n\nThe model stacking and fine-tuning approaches \nhave the \nadvantage of seeing the in-domain data in the end \nof training, \nthus making the system parameters more optimized for the in-domain data. They also provide flexibility in extending an existing model to any new domain without having to retrain the complete system again on the available corpora.\n\n\n\\subsection{Data Selection}\n\n\nBuilding a model, whether concatenated or stacked, on all the available data is computationally expensive. \nAn alternative approach is \\emph{data selection}, where we select a part of the out-of-domain data which is close to the in-domain data for training. The intuition here is two \nfold:\ni) \nthe \nout-of-domain data is huge and takes a lot of time to train on, and ii) not all parts of the out-of-domain data are beneficial for the in-domain data. \nTraining only on a selected part of the out-of-domain data reduces the training time significantly \nwhile at the same time creating\na model closer to the in-domain.  \n\nIn this work, we use \nthe\nmodified Moore-Lewis \\cite{Axelrod_2011_emnlp} for data selection.\nIt trains in- and out-of-domain n-gram models and then ranks sequences in the out-of-domain data based on cross-entropy difference. The out-of-domain sentences below a certain threshold \nare selected for training. Since we are dealing with several out-of-domain corpora, we apply data selection separately on each of them and build a concatenated system using \nin-domain\nplus selected out-of-domain data as shown in Figure \\ref{fig:data-approaches}.\nData selection significantly reduces data size thus \nimproving training time for NMT. However, finding the optimal threshold to filter data is a cumbersome process. Data selection using joint neural networks has been explored in \\cite{durraniEtAl:MT-Summit2015}. We explore data selection as an alternative to the above mentioned techniques.\n\n\\subsection{Multi-domain Ensemble}\n\nOut-of-domain data is generally available in larger quantity. Training a concatenated system whenever a new in-domain becomes available \nis expensive in terms of both time and computation. An alternative to fine-tuning the system with new in-domain is to do ensemble of the new model with the existing model.\nThe ensemble approach brings the flexibility to use them\nduring decoding without a need of retraining and fine-tuning.\n\nConsider $N$ models that we would like to use to generate translations. For each decoding step, we use the scores over the vocabulary from each of these $N$ models and combine them by averaging. We then use these averaged scores to choose the output word(s) for each hypothesis in our beam. The intuition is to combine the knowledge of the $N$ models to generate a translation. We refer to this approach as \\emph{balanced ensemble} later on. Since \nhere\nwe deal with several different domains, averaging scores of all the models equally may not result in optimum performance. We explore a variation of balanced ensemble called \\emph{weighted ensemble} that performs a weighted average of these scores, where the weights can be pre-defined or learned on a development set.\n \nBalanced ensemble using several models of a single training run saved at different iterations has shown to improve performance by 1-2 BLEU points \\cite{sennrich-haddow-birch:2016:WMT}. Here our goal is not to improve the best system but to benefit from individual models built using several domains during a single decoding process. We experiment with both balanced and weighted ensemble under \nthe\nmulti-domain condition only.\\footnote{Weighted fusion of Neural Networks trained on different domains has been explored in \\cite{durrani-EtAl:2016:COLING} for phrase-based SMT. Weighted training for Neural Network Models has been proposed in \\cite{joty-etAL:2015:EMNLP}.} \n\n\n\n\\section{Experimental Design}\n\\label{sec:experiments}\n\n\\subsection{Data}\nWe experiment with Arabic-English and German-English language pairs \nusing \nthe \nWIT$^3$ TED corpus\n\\cite{cettolol:SeMaT:2016} made available for IWSLT 2016 as our in-domain data. For Arabic-English, we take the UN corpus \\cite{ZiemskiJP16}  and the OPUS corpus \\cite{LISON16.947} as out-of-domain corpora.\nFor German-English, we use the Europarl (EP), and the Common Crawl (CC) corpora made available for the {\\it $1^{st}$} Conference on Statistical Machine Translation\\footnote{http:\/\/www.statmt.org\/wmt16\/translation-task.html} as out-of-domain corpus. We tokenize Arabic, German and English using the default \\emph{Moses} tokenizer. We did not do morphological segmentation of Arabic. Instead we apply sub-word based segmentation \\cite{sennrich-haddow-birch:2016:P16-12} that implicitly segment as part of the compression process.\\footnote{\\cite{sajjad-etal:2017:ACLShort} showed that using BPE performs comparable to\nmorphological tokenization \\cite{abdelali-EtAl:2016:N16-3} in Arabic-English machine translation.} \nTable \\ref{tab:corpusstats} shows the data statistics after running the Moses tokenizer.\n\nWe use a concatenation of dev2010 and tst2010 sets for validation during\ntraining. Test sets tst2011 and tst2012 served as development sets\nto find the best model for fine-tuning and tst2013 and tst2014 are used for evaluation. We use BLEU \\cite{Papineni:Roukos:Ward:Zhu:2002} to measure performance. \n\n\\begin{table}\n\\centering\n\\begin{tabular}{lrrr}\n\\toprule\n\n\\multicolumn{4}{c}{\\bf Arabic-English} \\\\\nCorpus & Sentences & Tok${_{ar}}$ & Tok${_{en}}$ \\\\\n\\midrule\nTED & 229k & 3.7M & 4.7M  \\\\\nUN & 18.3M & 433M & 494M   \\\\\nOPUS & 22.4M & 139M & 195M \\\\\n\\midrule\n\\multicolumn{4}{c}{\\bf German-English} \\\\\nCorpus & Sentences & Tok${_{de}}$ & Tok${_{en}}$ \\\\\n\\midrule\nTED & 209K & 4M & 4.2M \\\\\nEP  & 1.9M & 51M & 53M \\\\\nCC  & 2.3M & 55M & 59M \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{\\label{tab:corpusstats} Statistics of the Arabic-English and German-English training corpora in terms of Sentences and Tokens. EP = Europarl, CC = Common Crawl, UN = United Nations.}\n\\end{table}\n\n\\subsection{System Settings}\n\nWe use the Nematus tool \\cite{sennrich-EtAl:2017:EACLDemo} to train a 2-layered LSTM encoder-decoder with attention \\cite{bahdanau:ICLR:2015}. We use the default settings: embedding layer size: 512, hidden layer size: 1000. We limit the vocabulary to 50k words\nusing BPE \\cite{sennrich-haddow-birch:2016:P16-12} with 50,000 operations. \n\n\\section{Results}\n\\label{sec:results}\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{ar-curves-shorthand.png}\n\t\\caption{Arabic-English system development life line evaluated on development set tst-11 and tst-12. Here, \\texttt{ALL} refers to \\texttt{UN+OPUS+TED}, and \\texttt{OD} refers to \\texttt{UN+OPUS}}\n\t\\label{fig:ar-curves}\n\\end{figure*}\n\nIn this section, we empirically compare several approaches to combine in- and out-of-domain data to train an NMT system. Figure \\ref{fig:ar-curves} and Figure \\ref{fig:de-curves} show the learning curve on development sets using various approaches mentioned in this work. We will go through them individually later in this section.\n\n\n\n\\subsection{Individual Systems}\n\nWe trained systems on each domain individually (for 10 epochs)\\footnote{For German-English, we ran the models until they converged because the training data is much smaller compared to Arabic-English direction} and chose the best model using the development set. We tested every model on the in-domain testsets. Table \\ref{tab:baseline} shows the results. On Arabic-English, the system trained on the out-of-domain data OPUS performed the best. This is due to the large size of the corpus and its spoken nature which makes it close to TED in style and genre.\nHowever, despite the large size of UN, the system trained using UN performed poorly. The reason is the difference in genre of UN from the TED corpus where \nthe former consists of United Nations proceedings and \nthe latter is based on talks. \n\nFor German-English, the systems built using out-of-domain corpora performed better than the in-domain corpus. \nThe CC corpus appeared to be very close to the TED domain.\nThe system trained on it performed even better than the in-domain system by an average of 2 BLEU points.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lrrrr}\n\\toprule\n\\multicolumn{4}{c}{\\bf Arabic-English} \\\\\n& TED & UN & OPUS & \\\\\n\\midrule\ntst13 & 23.6 & 22.4 & {\\bf 32.2}   \\\\\ntst14 & 20.5 & 17.8 & {\\bf 27.3}   \\\\\navg. & 22.1 & 20.1 & {\\bf 29.7}   \\\\\n\\midrule\n\\multicolumn{4}{c}{\\bf German-English} \\\\\n& TED  & CC & EP \\\\\n\\midrule\ntst13 & 29.5 & {\\bf 29.8} & 29.1 \\\\\ntst14 & 23.3  & {\\bf 25.7} & 25.1 \\\\\navg. & 26.4 & {\\bf 27.7} & 27.1 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{\\label{tab:baseline} Individual domain models evaluated on TED testsets}\n\\end{table}\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{de-curves-shorthand.png}\n\t\\caption{German-English system development life line evaluated on development set tst-11 and tst-12. Here, \\texttt{ALL} refers to \\texttt{EP+CC+TED}, and \\texttt{OD} refers to \\texttt{EP+CC}}\n\t\\label{fig:de-curves}\n\\end{figure*}\n\n\\subsection{Concatenation and Fine-tuning}\nNext we evaluated how the models performed when trained on concatenated data. We mainly tried two variations:\ni) concatenating all the available data (\\emph{ALL}) ii) combine only the available out-of-domain data (\\emph{OD}) and later fine-tune the model on the in-domain data. Table \\ref{tab:concatenate} shows the results. The fine-tuned system outperformed a full concatenated system by 1.8 and 2.1 average BLEU points in Arabic-English and German-English systems respectively.\n\nLooking at the development life line of these systems (Figures \\ref{fig:ar-curves}, \\ref{fig:de-curves}), since \\emph{ALL} has seen all of the data, it is better than \\emph{OD } till the point \\emph{OD} is fine-tuned on the in-domain corpus. Interestingly, at that point \\emph{ALL} and \\emph{OD}$\\rightarrow$TED have seen the same amount of data but the parameters of the latter model are fine-tuned towards the in-domain data. This gives it \naverage improvements of up to 2 BLEU points over \\emph{ALL}.   \n\nThe \\emph{ALL} system does not give any explicit weight to any domain \\footnote{other than the data size itself} \nduring training. In order to revive the in-domain data, we fine-tuned it on the in-domain data. We achieved comparable results to that of the OD$\\rightarrow$TED model which means that one can adapt an already trained model on all the available data to a specific domain by fine tuning it on the domain of interest. This can be helpful in cases where in-domain data is not known beforehand. \n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lccc|c}\n\\toprule\n\\multicolumn{5}{c}{\\bf Arabic-English} \\\\\n& TED & ALL & OD$\\rightarrow$TED & ALL$\\rightarrow$TED \\\\\n\\midrule\ntst13 & 23.6 & 36.1 & 37.9 & 38.0 \\\\\ntst14 & 20.5 & 30.2 & 32.1 & 32.2 \\\\\navg. & 22.1 & 33.2 & 35.0 & 35.1 \\\\\n\\midrule\n\\multicolumn{5}{c}{\\bf German-English} \\\\\n& TED & ALL & OD$\\rightarrow$TED & ALL$\\rightarrow$TED\\\\\n\\midrule\ntst13 & 29.5 & 35.7 &  38.1 & 38.1 \\\\\ntst14 & 23.3 & 30.8 &  32.8 & 32.9 \\\\\navg. & 28.0 & 33.3 &  35.4 & 35.5 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{\\label{tab:concatenate} Comparing results of systems built on a concatenation of the data. OD represents a concatenation of the out-of-domain corpora and ALL represents a concatenation of OD and the in-domain data. $\\rightarrow$ sign means fine-tuning}\n\\end{table}\n\n\\subsection{Model Stacking}\n\nPreviously we concatenated all out-of-domain data and fine-tuned it with the in-domain TED corpus. \nIn this approach,\nwe picked one out-of-domain corpus at a time, trained a model and fine-tuned it with the other available domain. We repeated this process till all out-of-domain data had been used. In the last step, we fine-tuned the model on the in-domain data. \nSince we have a number of out-of-domain corpora available, we experimented with using them in different permutations for training and analyzed their effect on the development sets. Figure \\ref{fig:ar-curves} and Figure \\ref{fig:de-curves} show the results. It is interesting to see that the order of stacking has a significant effect \non achieving a high quality system. The best combination for \nthe Arabic-English language pair started with the UN data, fine-tuned on OPUS and then fine-tuned on TED. \nWhen we started with OPUS and fine-tuned the model on UN, the results dropped drastically as shown in Figure \\ref{fig:ar-curves} (see OPUS$\\rightarrow$UN). The model started forgetting the previously used data and focused on the newly provided data which is very distant from the in-domain data. We saw similar trends in the case of German-English language pair where CC$\\rightarrow$EP dropped the performance drastically. We did not fine-tune CC$\\rightarrow$EP and OPUS$\\rightarrow$UN on TED since there was no better model to fine-tune than to completely ignore the second corpus i.e. UN and EP for Arabic and German respectively and fine-tune OPUS and CC on TED. The results of OPUS$\\rightarrow$TED and CC$\\rightarrow$TED are shown in Figures.\n\nComparing the OPUS$\\rightarrow$TED system with the UN$\\rightarrow$OPUS$\\rightarrow$TED system, \nthe result of OPUS$\\rightarrow$TED are lowered by 0.62 BLEU points from the UN$\\rightarrow$OPUS$\\rightarrow$TED system. \nSimilarly, we saw a drop of 0.4 BLEU points for German-English language pair when we did not use EP and directly fine-tuned CC on TED. \nThere are two ways to look at these results,\nconsidering\nquality vs. time: i) by using UN and EP in model stacking, \nthe\nmodel learned to remember only those parts of the data that are beneficial \nfor achieving\nbetter translation quality on the in-domain development sets. Thus using them as part of the training pipeline is helpful \nfor building a\nbetter system. ii) training on UN and EP is expensive. Dropping them from the pipeline significantly reduced the training time and resulted in a loss of \n0.62 and 0.4 BLEU points only.\n\nTo summarize, model stacking performs best when it starts from the \ndomain furthest \nfrom the in-domain data. In the following, we compare it with the data concatenation approach. \n\n\n\\subsection{Stacking versus Concatenation}\n\nWe compared model stacking with different forms of concatenation. In terms of data usage, all models are exposed to identical data. Table \\ref{tab:stackvscat} shows the results. The best systems are achieved using a concatenation of all of the out-of-domain data for initial model training and then fine-tuning the trained model on the in-domain data. The concatenated system \\emph{ALL} performed the lowest among all.  \n\n\\emph{ALL} learned a generic model from all the available data without giving explicit weight to any particular domain whereas model stacking resulted in a specialized system for the in-domain data. In order to confirm the generalization ability of \\emph{ALL} vs. model stacking, we tested them on a new domain, News. \\emph{ALL} performed 4 BLEU points better than model stacking in translating the news NIST MT04 testset. \nThis concludes that a concatenation system is not an optimum solution for one particular domain but is robust enough to perform well in new testing conditions.\n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lccc}\n\\toprule\n\\multicolumn{4}{c}{\\bf Arabic-English} \\\\\n&  ALL & OD$\\rightarrow$TED & UN$\\rightarrow$OPUS$\\rightarrow$TED \\\\\n\\midrule\ntst13 & 36.1 & 37.9 & 36.8\\\\\ntst14 & 30.2 & 32.1  & 31.2\\\\\navg. & 33.2 & 35.0  & 34.0 \\\\\n\\midrule\n\\multicolumn{4}{c}{\\bf German-English} \\\\\n& ALL & OD$\\rightarrow$TED & EP$\\rightarrow$CC$\\rightarrow$TED \\\\\n\\midrule\ntst13 & 35.7 & 38.1 &  36.8 \\\\\ntst14 & 30.8 & 32.8 &  31.7 \\\\\navg. & 33.3 & 35.4 &  34.3 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{\\label{tab:stackvscat} Stacking versus concatenation}\n\\end{table}\n\n\\subsection{Data Selection}\nSince training on large out-of-domain data is time inefficient, we selected a small portion of \nout-of-domain data that is closer to the in-domain data. For Arabic-English, we selected 3\\% and 5\\% from the UN and OPUS data respectively which constitutes roughly ~2M sentences. For German-English, we selected 20\\% from a concatenation of EP and CC, which roughly constitutes 1M training sentences.\\footnote{These data-selection percentages have been previously found to be optimal when training phrase-based systems using the same data. For example see \\cite{sajjad-etal:iwslt13}.}\n \nWe concatenated the selected data and the in-domain data to train an NMT system.\nTable \\ref{tab:selection} presents the results. \nThe selected system is worse than the \\emph{ALL} system. This is \nin\ncontrary to the results mentioned in the literature \non\nphrase-based machine translation where data selection on UN improves translation quality \\cite{sajjad-etal:iwslt13}. This shows that NMT is not as sensitive as phrase-based to the presence of the out-of-domain data. \n\nData selection comes with a cost of reduced translation quality. \nHowever, the selected system is better than all individual systems shown in Table \\ref{tab:baseline}. Each of these out-of-domain systems take more time to train than a selected system. For example, compared to individual UN system, the selected system took approximately 1\/10th of the time to train.\nOne can look at data selected system as a decent trade-off between training time and translation quality.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lcc|cc}\n\\toprule\n&\\multicolumn{2}{c}{\\bf Arabic-English}&\\multicolumn{2}{c}{\\bf German-English} \\\\\n&  ALL & Selected & ALL & Selected \\\\\n\\midrule\ntst13 & 36.1 & 32.7   & 35.7 & 34.1 \\\\\ntst14 & 30.2 & 27.8   & 30.8 & 29.9 \\\\\navg. & 33.2 & 30.3    & 33.3 & 32.0 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{\\label{tab:selection} Results of systems trained on a concatenation of selected data and on a concatenation of all available data}\n\\end{table}\n\n\n\\subsection{Multi-domain Ensemble}\n\nWe took \nthe \nbest model \nfor every domain according to the average BLEU on the development sets and ensembled them during decoding. \nFor weighted ensemble, we did a grid search and selected the weights using the development set.\nTable \\ref{tab:ensemble} presents the results of\nan ensemble on the\nArabic-English language pair and compares them with the individual best model, OPUS, \nand a model built on \\emph{ALL}. As expected, balanced ensemble (\\emph{ENS$_b$}) dropped results \ncompared to \nthe best individual model. Since \nthe\ndomains are very distant, giving equal weights to them hurts the overall performance. The weighted ensemble (\\emph{ENS$_w$}) improved from the best individual model by 1.8 BLEU points but is still lower than the concatenated system by 1.7 BLEU points. The weighted ensemble approach is beneficial when individual domain specific models are already available for testing. Decoding with multiple models is more efficient compared to training a system from scratch on a concatenation of the entire data.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lcc|cc}\n\\toprule\n\\multicolumn{5}{c}{\\bf Arabic-English} \\\\\n& OPUS & ALL & ENS$_b$ & ENS$_w$ \\\\\n\\midrule\ntst13 & 32.2 & 36.1 & 31.9 & 34.3\\\\\ntst14 & 27.3 & 30.2 & 25.8  & 28.6\\\\\navg. & 29.7 & 33.2 & 28.9 & 31.5 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{\\label{tab:ensemble} Comparing results of balanced ensemble (ENS$_b$) and weighted ensemble (ENS$_w$) with the best individual model and the concatenated model}\n\\end{table}\n\n\\subsection{Discussion}\n\nThe concatenation system showed robust behavior in translating new domains. \\cite{KobusCS16} proposed a domain aware concatenated system by introducing domain tags for every domain. We trained a system using their approach and compared the results with simple concatenated system. The domain aware system performed slightly better than the concatenated system (up to 0.3 BLEU points) when tested on the in-domain TED development sets. However, domain tags bring a limitation to the model since it can only be tested on the domains it is trained on. Testing on an unknown domain would first require to find its closest domain from the set of domains the model is trained on. The system can then use that tag to translate unknown domain sentences.\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nWe explored several approaches to train a neural machine translation system under multi-domain conditions and evaluated them based on three metrics: translation quality, training time and robustness. Our results showed that an optimum in-domain system can be built using a concatenation of the out-of-domain data and then fine-tuning it on the in-domain data. A system built on the concatenated data resulted in a generic system that is robust to new domains. Model stacking is sensitive to the order of domains it is trained on. Data selection and weighted ensemble resulted in a less \noptimal\nsolution. \nThe\nformer is efficient to train in a short time and \nthe\nlatter is useful when different individual models are available for testing. It provides a mix of all\ndomains\nwithout retraining or fine-tuning the system. \n\n\\section{Acknowledgments}\nThe research presented in this paper is partially conducted as part of the European Union's Horizon 2020\nresearch and innovation programme under grant\nagreement 644333 (SUMMA).\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\\begin{table*}\n\\begin{center}\n\\caption{Details of the observations made of each target. Standard stars\n(HD49798, EG 274 and EG 21) for photometric calibration were observed with the\nsame instrumental configuration.}\\label{tab:obs}\n\\begin{tabular}{cccccccl}\n\\hline\nTarget & Redshift & \\multicolumn{2}{c}{Position (J2000)} & Date & Exposure & Axial            & Comment \\\\\n       &          & RA & Dec                             &      & Time (s) & Wavelength (\\AA) &\\\\\n\\hline\nMRC B1256-243 & 2.263 & \\ra{12}{59}{12.6} & \\dec{-24}{36}{05} & 2003 July 27 & 15 $\\times$ 60  & 3957.2 & Repeated \\\\\n& & & & & & 3967.1 & twice. \\\\\n& & & & & & 3977.1 & \\\\\n& & & & & & 3987.1 & \\\\\n\\\\\nMRC B2158-206 & 2.249 & \\ra{22}{01}{27.0} & \\dec{-20}{25}{36} & 2003 July 27 & 15 $\\times$ 60  & 3959.1 & Repeated \\\\\n& & & & & & 3969.1 & four \\\\\n& & & & & & 3979.1 & times. \\\\\n& & & & & & 3989.0 & \\\\\n& & & & & & 3999.0 & \\\\\n\\\\\nBR B0019-1522 & 4.528 & \\ra{00}{22}{08.0} & \\dec{-15}{05}{39} & 1997 Nov. 6 & 600 & 6709.5 & Repeated \\\\\n& & & & & & 6725.9 & eight \\\\\n& & & & & & 6742.3 & times. \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\nThe evolution of clustering with cosmic time is widely recognised as one of\nthe most rigid tests of the cold dark matter paradigm \\citep{Kaiser91,\nSpringel05}. However, locating high redshift clusters is challenging. The\ntraditional methods of X-ray and blind optical searches are limited: X-ray\nsurveys can detect only the most luminous sources at high-$z$, while optical\nsearches are highly vulnerable to projection effects. In order to overcome\nthese limitations, a way of targeting the search is needed.\n\nSince the earliest studies, it has been established that quasars are\nassociated with groups and clusters of galaxies \\citep{Bahcall69, Oemler72}.\nMore recently, \\citet{McLure01} argued that a close match between the space\ndensity of clusters and that of quasars indicates that practically all\nclusters contained an AGN at high redshift. Further, \\citet{Rawlings04}\npropose that radio jets from AGN are a major influence on cluster evolution.\nThey suggest that a galaxy merger within the cluster triggers a radio-jet\nepisode; the jets then delivery energy to the intracluster medium, heating it\nand preventing it from falling into the other developing cluster galaxies.\nThese galaxies are thus starved of fuel, and star formation within the cluster\nwill effectively shut down. \\citeauthor{Rawlings04} speculate that every\nprotocluster undergoes such an episode, strengthening the link postulated by\n\\citeauthor{McLure01}.\n\nThis relationship between galaxy overdensities and AGN suggests a method for\nlocating high-$z$ clusters: we can use quasars as convenient `anchors' for our\nsearch.  This technique has already been exploited by others with notable\nsuccess: for example, \\citet{Stiavelli05} tentatively report the detection of\nclustering around a radio-quiet quasar at $z = 6.28$.\n\nTo date most galaxy clusters detected around AGN have been identified based on\nstatistical overdensities of objects observed in their vicinity. A better\nstrategy for overcoming foreground contamination is to identify individual\nstar forming galaxies in the AGN field by their characteristic redshift\ndependent features. In particular, Lyman $\\alpha$ emission has been used to\nidentify high redshift galaxies for some time. Among the first high redshift\nobjects identified by emission lines were the $z = 4.55$ Ly $\\alpha$ emitters\nobserved in the field of the quasar BR B2237-0607 by \\citet{Hu96}. Since then,\na series of highly profitable observations of Ly $\\alpha$ emitters in AGN\nfields have been carried out. \\citet{Kurk00} and \\citet{Pentericci00} used a\ncombination of narrow- and broad-band imaging with follow-up spectroscopy to\nidentify a galaxy overdensity within 1.5 Mpc of the $z = 2.156$ radio galaxy\nPKS B1138-262. Similar results have been achieved for the radio galaxies TN\nJ1338-1942 \\citep[$z=4.1$;][]{Venemans02}, TN J0924-2201\n\\citep[$z=5.2$;][]{Venemans04, Overzier06} and MRC B0316-257\n\\citep[$z=3.13$;][]{Venemans05} and 6C0140+326 \\citep[$z=4.413$;][]{Kuiper11}.\n\nWhile this combination of broad and narrowband imaging has produced\ndemonstrably successful results, the more direct antecedents of this work have\nadopted an alternative approach. The \\textit{Taurus Tunable Filter} (TTF)\ninstrument, installed on the Anglo-Australian Telescope, provided a powerful\nmethod of narrow-band (of order 10 \\AA) imaging over a large range of\nwavelengths \\citep{BH982}. \\citet{Bremer99} introduced the strategy used to\nsearch for line emitters at a given redshift with TTF: broadly, the tunable\nfilter is stepped across a range of wavelengths around the expected redshifted\nposition of the emission. Emission line galaxies then appear brighter in those\nframes centred on the spectral line.\n\nConsiderable success has been achieved at lower redshifts with this technique.\n\\citet{Baker01} located a cluster around the $z = 0.9$ radio-loud quasar MRC\nB0450-221 using TTF to search for $[$O\\,{\\sc ii}$]$ 3727 \\AA{} emission. The\nsame technique was used by \\citet{Barr04}, who examined six radio-loud quasars\nat redshifts $0.8 < z < 1.3$, identifying a total of 47 candidate emission\nline galaxies (ELGs), at an average space density around 100 times higher than\nthat found locally.\n\nFurther work with TTF was performed by \\citet{Francis04}, who targeted Ly\n$\\alpha$ emitters within 1 Mpc of the $z=2.159$ radio loud quasar PKS\nB0424-131 without making {\\it any} detections. These authors selected this\nextremely luminous UV source with the expectation of finding Ly $\\alpha$\nfluorescent clouds in the vicinity of the quasar but these were not detected.\nWith specific application to PKS B0424-131, \\citet{Bruns11} demonstrated that\nthe most intrinsically UV-luminous quasars observed beyond $z=1$ suppress star\nformation in low-mass haloes ($M_{\\rm vir} \\lesssim 10^{12}$ M$_\\odot$) within\na megaparsec of the quasar. The intense UV radiation field is expected to\nphoto-evaporate HI clouds which presumably accounts for the lack of\ndetections.  We return to this point in our conclusion\n(\\S~\\ref{sec:conclusion}).\n\nThe present work continues to push TTF to higher redshifts, searching three\nquasar fields at redshifts up to $z \\sim 4.5$. The objects selected include\nexamples of both radio-loud and radio-quiet quasars, and their environments\nare compared. Section \\ref{sec:obs} of this paper describes the observations,\nincluding target selection, instrumental characteristics and a note on data\nreduction. Section \\ref{sec:sim} describes simulations performed to examine\nstatistical properties and completeness of our sample. Section \\ref{sec:id}\ndescribes how candidate ELGs were identified and presents details on the\ndetections, as well as considering the possible sources of mis-identified\n`interloper' objects. Section \\ref{sec:properties} analyses the distribution\nand properties of the sample. Our conclusions are summarised in Section\n\\ref{sec:conclusion}.  Throughout, we assume an $H_0 = 70$ km s$^{-1}$\nMpc$^{-3}$, $\\Omega_{\\Lambda} = 0.7$, $\\Omega_{\\mathrm{M}} = 0.3$ cosmology.\n\n\\section{Observations}\n\\label{sec:obs}\n\n\n\\subsection{Target selection}\n\nTwo data sources were used for this analysis. The authors used TTF to observe\nobjects drawn from the Molonglo Quasar Sample \\citep[MQS;][]{Kapahi98} of\nlow-frequency-selected radio-loud quasars in July 2003. Six targets had been\nselected from the MQS on the basis of observability, suitable redshifts being\nlimited by the necessity to place Lyman $\\alpha$ within the wavelength ranges\naccessible to TTF's order-blocking filters. Due to weather constraints, only\ntwo quasars were observed: MRC B1256-243 ($z = 2.263$) and MRC B2158-206 ($z =\n2.249$). Immediately following each quasar observation, a standard star was\nobserved with the same instrumental settings for flux calibration.  In\naddition, observations of BR B0019-1522, a $z = 4.528$ radio-quiet quasar,\nwere drawn from the Anglo-Australian Observatory archive. These data were\ntaken on 1997 November 6 by Bland-Hawthorn, Boyle and Glazebrook, and were\naccompanied by companion observations of a standard star.  Details of each\ntarget are given in Table \\ref{tab:obs}.\n\n\\subsection{Instrumental setup and characteristics}\n\nThroughout this work, a distinction is drawn between a \\textit{frame}\n(corresponding to one set of data read from the CCD), an \\textit{image} (a\nnumber of frames at the same etalon settings which have been combined for\nanalysis) and a \\textit{field}, or stack of images of the same area of sky at\ndifferent etalon settings.\n\n\\subsubsection{Wavelength variation and the optical axis}\n\\label{sec:wlvariation}\n\nFabry-P\\'erot images have a quadratic radial wavelength dependence of the form\n$\\lambda_\\theta = \\lambda_{centre}(1 - \\theta^2\/2)$ \\citep{Bland89}, where\n$\\theta$ is the off-axis angle at the etalon. In a typical observation, the\nwavelength varies across the field by around 1\\% of $\\lambda_{centre}$.\nWavelength calibration is performed with respect to the axial wavelength; for\nany given pixel position on the image, it is then possible to calculate the\nwavelength observed at that point.\n\n\\subsubsection{Objects at $z \\sim 2.2$}\n\nThe TTF was used at $f\/8$ on the AAT in combination with the EEV2 CCD. This\nresulted in a scale of 0.33'' per pixel. After processing, the total useful\nrectangular field of view in the observations was around 7' by 5'. The radial\nwavelength variation described in Section \\ref{sec:wlvariation} resulted in a\nshift of 1.4~\\AA{} at 2' from the optical axis and 6.7~\\AA{} at 4' from the axis.\nConditions were photometric, and seeing was on the order of 1.5''. The full\nwidth at half maximum of the etalon transmission band was 7.5~\\AA.\n\nThe targets were scanned at etalon plate spacings corresponding to a series of\nwavelength steps of approximately 10~\\AA, the aim being to straddle the\nredshifted Ly $\\alpha$. However, an intermediate-band order-blocking filter is\nnecessary to eliminate unwanted wavelengths and other orders of interference.\nIn this case, the AAT's B1 filter was the best available.  Unfortunately, the\nobserved wavelengths were at the very edge of the filter transmission, as\nshown in Fig. \\ref{fig:trans}: the signal to noise ratio therefore decreases\nsignificantly with wavelength. Table \\ref{tab:obs} and Fig. \\ref{fig:trans}\nrecord observations of MRC B1256-243 at 3987.1 \\AA.  When these data were\nanalysed, it was clear that the reduced filter transmission had resulted in no\nuseful results at this wavelength. These data are not considered further in\nthis work. The MRC B2158-206 observations at 3989.0 \\AA{} and 3999.0 \\AA{} are\nincluded hereafter, but did not include any useful detections.\n\nEach CCD frame contained a total of 30 minutes of observations, taken at two\nseparate axial wavelengths. Each wavelength was exposed for 60 seconds a total\nof 15 times. This procedure was repeated twice in the case of MRC B1256-243\nand four times for MRC B2158-206; the total exposure times at each wavelength\nare thus 30 minutes and 1 hour, respectively. Between each image, the\ntelescope pointing was shifted slightly: this enabled the easy identification\nand subsequent elimination of diametric ghosts in the data.\n\n\\subsubsection{Objects at $z \\sim 4.5$}\n\nThe TTF was used at $f\/8$ on the AAT in combination with the MITLL2 CCD. This\nresulted in a scale of 0.37'' per pixel. After processing, the total useful\nrectangular field of view in the observations was 9'17'' by 4'10''.  The\nradial wavelength variation described in Section \\ref{sec:wlvariation}\nresulted in a shift of 5.1~\\AA{} at 2' from the optical axis and 20.3~\\AA{} at\n4' from the axis.  Conditions were photometric, and the seeing was on the\norder of 1.5\". The full width at half maximum of the etalon transmission band\nwas 9.5~\\AA. The AAT's R0 intermediate-band order-blocking filter was used:\nthis provided effectively constant transmission across the wavelength range\nunder consideration.\n\nEach CCD frame contained a total of 30 minutes of observations: ten at each of\nthree axial wavelengths. Eight CCD frames were recorded, resulting in a total\nof 80 minutes exposure for each axial wavelength. As before, the telescope\nposition was shifted slightly between images.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig1}\n\\caption{On-axis etalon transmission bands for each of the three fields\nobserved shown relative to the relevant order-blocking filter used on the\ntelescope. Away from the optical axis the etalon transmission shifts to\nshorter wavelengths (\\S\\ref{sec:wlvariation}).}\\label{fig:trans}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Data reduction and catalogue construction}\n\nData reduction proceeds broadly as for standard broadband imaging. A full\nconsideration of the issues surrounding tunable filter data is given by\n\\citet{Jones012} and \\citet{Jones02}. The various different images of each\nfield at the same axial wavelengths were aligned by a marginal centroid fit on\nbright stars and then combined. Wavelength calibration was performed through\nan emission line, as described by \\citeauthor{Jones02}; xenon and\ncopper-helium arc lamps were used for the $z \\sim 2.2$ fields, and a neon arc\nlamp for BR B0019-1522.\n\nAfter the data had been reduced, object detection and fixed aperture\nphotometry were performed on each image using {\\sc SExtractor}\n\\citep{Bertin96}. The object detection parameters were defined as described in\nthe next section.\n\n\\subsection{Photometry}\n\\label{sec:photo}\n\nThe observations of the standard stars were reduced in the same way. For each\nstar, {\\sc SExtractor} was used to perform aperture photometry yielding a\ncount $C_\\mathrm{s}$. This corresponds to a known magnitude $m_\\mathrm{s}$,\nbased on \\citet{Hamuy92} for the lower redshift fields or from the ESO\nStandard Star Catalogue for that of BR B0019-1522. If the exposure time on the\nstandard is $t_\\mathrm{s}$ and that on an object in the field is\n$t_\\mathrm{Obj}$, the AB magnitude of the object is\n\n\\begin{equation}\nm_\\mathrm{AB} = m_\\mathrm{s} - 2.5 \\log_{10} (C_\\mathrm{Obj}t_\\mathrm{s})\/(C_\\mathrm{s}t_\\mathrm{Obj}).\n\\end{equation}\n\nThe AB magnitude system \\citep{Oke74} is defined by $m_\\mathrm{AB} = -2.5\n\\log_{10} f_\\nu - 48.60$ where $f_\\nu$ is the flux in units of \\mbox{ergs\ncm$^{-2}$ s$^{-1}$ Hz$^{-1}$}.  The monochromatic flux $f_\\lambda$, in units of\n\\mbox{ergs cm$^{-2}$ s$^{-1}$ \\AA$^{-1}$}, is then\n\n\\begin{equation}\n\\label{eq:abtoflux}\nf_\\lambda = (c \\times 10^{-\\left(m_{\\mathrm{AB}} + 48.60\\right)\/2.5})\/\\lambda^2.\n\\end{equation}\n\nConversion from $f_\\lambda$ to the total flux in the band, $f_\\mathrm{total}$\nis performed by multiplying by the effective width of the etalon transmission.\nThe etalon transmission band may be taken as Lorentzian, normalised to 1 at\nthe wavelength of peak transmission, thus:\n\n\\begin{equation}\n\\label{eq:ttfpass}\nT(\\lambda) = (\\lambda_{\\nicefrac{1}{2}}^2 \/ 4)\/((\\lambda - \\lambda_\\mathrm{c})^2 + \\lambda_{\\nicefrac{1}{2}}^2 \/ 4)\n\\end{equation}\n\nwhere $\\lambda$ is the wavelength, $\\lambda_c$ the central wavelength of the\nband and $\\lambda_{\\nicefrac{1}{2}}$ its full width at half maximum. Assuming\nthat $\\lambda_\\mathrm{c} \\gg \\lambda_{\\nicefrac{1}{2}}$, Equation\n\\ref{eq:ttfpass} may be integrated over $0 \\le \\lambda \\le \\infty$ to yield a\nwidth of $\\pi \\lambda_{\\nicefrac{1}{2}}\/2$. Combining this with Equation\n\\ref{eq:abtoflux} yields a total flux in the band of\n\n\\begin{equation}\n\\label{eq:fluxinband}\nf_{\\mathrm{total}} = (\\pi c \\lambda_{\\nicefrac{1}{2}} \\times 10^{-\\left(m_\\mathrm{AB} + 48.60\\right)\/2.5})\/2 \\lambda_\\mathrm{c}^2\n\\end{equation}\n\nwith units \\mbox{ergs cm$^{-2}$ s$^{-1}$}.\n\nIt is worth noting that this measures the flux received in the etalon\npassband, and is thus a lower limit of the line flux of the ELG: variations of\nline shapes and widths, and their positions relative to the etalon passband,\nwill cause the fluxes measured to be systematically underestimated. They\nshould therefore be regarded as lower limits.\n\n\\section{Simulations}\n\\label{sec:sim}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics{fig2}\n\\caption{Depths of each of the three fields as determined by the simulations\ndescribed in Section \\ref{sec:dof}. On the left, the data is plotted in terms\nof simulation inputs; on the right, in terms of the measurements made from\nthe simulated images. Note that the effects of the blocking filter are clearly\nseen in the two upper (lower redshift) fields, as the completeness tails off\nat higher wavelength. The higher redshift BR B0019-1522 field falls well\nwithin the blocking filter, so the depth is relatively constant with\nwavelength across the observed range.}\n\\label{fig:simresults}\n\\end{center}\n\\end{figure*}\n\nWe constructed a series of simulated images: data with properties similar to\nour observations, but containing a known population of objects. The analysis\nof these enables us to address the following questions:\n\n\\begin{itemize}\n\\item What are the most appropriate {\\sc SExtractor} parameters for\nextracting useful data from the images?\n\\item To what depth is each field complete--and how does that vary over the\nfield?\n\\item To what extent is our analysis prone to mis-identifying spurious `noisy'\nfeatures in an image as candidate emission line galaxies?\n\\end{itemize}\n\n\\subsection{Construction of simulated images}\n\nImages were simulated in two stages: first, a background was generated, then\nobjects were superimposed on top of it.\n\nDue to the properties of the blocking filter and the variation of wavelength\nacross the image, the background signal is not constant across the image. Each\ndata image was therefore divided into 100 by 100 pixel blocks, and the mean\nbackground signal and associated noise was measured in each block. Simulated\nblocks were then generated matching each of these, and then recombined to form\nan overall simulated background of the same shape as the data.\n\nA Ruby\\footnote{\\url{http:\/\/www.ruby-lang.org\/}} code was written to simulate\nthe expected properties of objects we might observe. Objects were simulated at\nrandom redshifts (over the range the observations might be expected to cover)\nand pixel positions within the images. Based on the work of\n\\citet{LeDelliou06}, our observations were not expected to be sensitive to\ncontinuum emission from ELGs, so this was not considered. Further, the ELGs\nare spatially unresolved, so were simulated with a Gaussian point spread\nfunction equal to the measured seeing. An emission line model was developed\nbased on the widths and profiles of high-$z$ Lyman $\\alpha$ emitters based\nchiefly on the $z \\sim 4.5$ objects observed by \\citet{Dawson04}.\nExperimentation suggested that the results obtained were not sensitive to line\nprofile; velocity widths in the range 100--1000 km\\,s$^{-1}$ were chosen\nbased on both \\citet{Dawson04} and the more extreme example documented by\n\\citet{Tapken04}.\n\nThe effects of the instrument on the objects' detectabilty were then\nconsidered before they were added to the background images. First a correction\nfor the order-blocking filter transmission was applied, using the position of\nthe object within the field to determine the observed wavelength and hence\nfilter transmission.  The line profile was then multiplied by the transmission\nprofile of the etalon for the image under construction.\n\n\\subsection{Results of simulations}\n\nFollowing the procedure above, simulations were run of all three fields. For\neach data image, a total of 500 simulated images were constructed, each\ncontaining 500 simulated sources.\n\n\\subsubsection{Detection parameters}\n\\label{sec:detpar}\n\nSource extraction was run multiple times on each image with different\n{\\sc SExtractor} configuration parameters. In each case, the results were\ncompared with the catalogue of simulated objects in the image. The combination\nof parameters that produced the greatest number of detections of known objects\ncombined with the smallest number of spurious detections of noise were then\nused for the analysis of both the simulations and the observed data. These\nparameters are listed in Table \\ref{tab:sextractor}.\n\n\\begin{table}\n\\begin{center}\n\\caption{Optimal {\\sc SExtractor} parameters determined by simulations and\nused throughout this work.}\\label{tab:sextractor}\n\\begin{tabular}{ccp{4.1cm}}\n\\hline\nParameter & Value & Description \\\\\n\\hline\n{\\sc detect\\_minarea} & \\phantom{0}6\\phantom{.0} & Minimum number of pixels per detection. \\\\\n{\\sc detect\\_thresh} & \\phantom{0}1.3 & Detection threshold in $\\sigma$ above local background. \\\\\n{\\sc back\\_size} & 64\\phantom{.0} & Size in pixels of mesh used for background estimation. \\\\\n{\\sc phot\\_apertures} & \\phantom{0}6\\phantom{.0} & Aperture diameter (pixels). \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsubsection{Depths of fields}\n\\label{sec:dof}\n\nAs in the previous section, a source detection procedure was run on each\nimage and the results compared with the known simulation inputs. This time,\nthe fraction of the objects at each wavelength and magnitude which were\ndetected was recorded. The results are shown Fig. \\ref{fig:simresults}.\n\nNote that this data can be recorded both in terms of the \\textit{simulated}\nwavelength and magnitude and and their \\textit{detected} equivalents. For any\ngiven pixel position in a field, an object can only be detected as peaking at\none of a limited range of wavelengths, since its peak will be seen to appear\nat the wavelength of the image in which it occurs (of which there are at most\n5). Hence, an object which is simulated with a very bright magnitude, but at a\nwavelength far from the peak transmission of any of the filters, will be\ndetected with a somewhat dimmer magnitude at a wavelength corresponding to the\nimage in which it is brightest. Fig. \\ref{fig:simresults} shows both the\nsimulated (on the left) and detected (on the right) quantities for each of\nthe three fields.\n\n\\section{Identification of candidate ELGs}\n\\label{sec:id}\n\n\\begin{table*}\n\\begin{center}\n\\caption{ELG candidates in the field of BR B0019-1522. The AB magnitude given\nis that measured in the peak from with no correction for galactic extinction\nor etalon transmission; the flux is calculated from that magnitude via Equation\n\\ref{eq:fluxinband}.}\\label{tab:elgresults}\n\\begin{tabular}{lccccccc}\n\\hline\nField & ELG & \\multicolumn{2}{c}{Position (J2000)} & Projected distance & Lyman $\\alpha$ Peak & AB   & Flux in band \\\\\n      & Id. &  R.A. & Decl.                        & from Quasar (Mpc)  & Wavelength (\\AA)    & mag. & (ergs cm$^{-2}$ s$^{-1} \\times 10^{18}$)\\\\\n\\hline\nMRC B1256 & A & \\ra{12}{59}{23.2} & \\dec{-24}{37}{32.9} & 1.428 & 3966 & 20.9 & 371 \\\\\n          & B & \\ra{12}{59}{15.7} & \\dec{-24}{37}{40.7} & 0.871 & 3966 & 21.1 & 293 \\\\\n          & C & \\ra{12}{59}{02.7} & \\dec{-24}{37}{15.1} & 1.257 & 3957 & 20.9 & 363 \\\\\n          & D & \\ra{12}{59}{05.3} & \\dec{-24}{37}{31.3} & 1.085 & 3960 & 20.7 & 424 \\\\\n\\\\\nMRC B2158 & A & \\ra{22}{01}{26.0} & \\dec{-20}{25}{08.0} & 0.263 & 3956 & 21.8 & 161 \\\\\n          & B & \\ra{22}{01}{41.7} & \\dec{-20}{24}{03.5} & 1.986 & 3971 & 21.7 & 192 \\\\\n\\\\\nBR B0019 & A & \\ra{0}{21}{56.9} & \\dec{-15}{04}{04.3} & 1.229 & 6673 & 22.5 & \\phantom{0}37 \\\\\n         & B & \\ra{0}{22}{03.8} & \\dec{-15}{07}{41.2} & 0.898 & 6706 & 22.5 & \\phantom{0}37 \\\\\n         & C & \\ra{0}{22}{08.8} & \\dec{-15}{06}{58.8} & 0.531 & 6705 & 22.0 & \\phantom{0}57 \\\\\n         & D & \\ra{0}{22}{08.8} & \\dec{-15}{06}{56.3} & 0.515 & 6704 & 21.7 & \\phantom{0}71 \\\\\n         & E & \\ra{0}{21}{57.8} & \\dec{-15}{06}{58.7} & 1.105 & 6697 & 22.7 & \\phantom{0}31 \\\\\n         & F & \\ra{0}{22}{14.5} & \\dec{-15}{06}{42.6} & 0.748 & 6717 & 22.1 & \\phantom{0}52 \\\\\n         & G & \\ra{0}{22}{12.4} & \\dec{-15}{06}{17.8} & 0.491 & 6716 & 22.1 & \\phantom{0}51 \\\\\n         & H & \\ra{0}{22}{12.7} & \\dec{-15}{06}{01.4} & 0.471 & 6697 & 22.5 & \\phantom{0}37 \\\\\n         & I & \\ra{0}{22}{07.6} & \\dec{-15}{05}{27.1} & 0.087 & 6694 & 22.4 & \\phantom{0}39 \\\\\n         & J & \\ra{0}{21}{58.6} & \\dec{-15}{04}{56.2} & 0.940 & 6701 & 22.3 & \\phantom{0}43 \\\\\n         & K & \\ra{0}{22}{14.2} & \\dec{-15}{04}{20.6} & 0.785 & 6680 & 22.6 & \\phantom{0}32 \\\\\n         & L & \\ra{0}{22}{14.8} & \\dec{-15}{07}{22.1} & 0.939 & 6719 & 22.5 & \\phantom{0}37 \\\\\n         & M & \\ra{0}{22}{15.3} & \\dec{-15}{06}{52.7} & 0.849 & 6716 & 22.2 & \\phantom{0}48 \\\\\n         & N & \\ra{0}{22}{11.5} & \\dec{-15}{05}{04.1} & 0.405 & 6706 & 22.3 & \\phantom{0}43 \\\\\n         & O & \\ra{0}{22}{18.0} & \\dec{-15}{04}{36.8} & 1.038 & 6694 & 22.4 & \\phantom{0}39 \\\\\n         & P & \\ra{0}{21}{53.9} & \\dec{-15}{05}{58.2} & 1.351 & 6685 & 22.4 & \\phantom{0}40 \\\\\n         & Q & \\ra{0}{22}{13.9} & \\dec{-15}{05}{08.8} & 0.597 & 6689 & 22.5 & \\phantom{0}35 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{fig3}\n\\caption{Relative positions of the ELG candidates detected in each of the\nthree fields. The dimensions of the plots indicate the size of the observed\nfields. The quasars are located at the origin. The letters refer to the ELG\ndesignations used throughout the text.}\\label{fig:elgcandidates}\n\\end{center}\n\\end{figure}\n\n{\\sc SExtractor} was used with the parameters determined in Section\n\\ref{sec:detpar} and a detection threshold of 5$\\sigma$ to build a catalogue\nof sources for each image. Within each field, the catalogues from each image\nwere cross-matched: objects were associated by position, with a three pixel\nthreshold.\n\nThese observations are not deep enough to observe continuum flux from a\ntypical Lyman $\\alpha$ emitting galaxy \\citep{LeDelliou06}. Given the likely\nrange of line widths \\citep{Dawson04, Tapken04}, we do not expect to observe\nLyman $\\alpha$ emitters in more than two adjacent passbands. Objects which\nwere identified in either one or two bands were therefore flagged for further\ninvestigation.\n\nIn order to minimise the risk of contamination by noisy artefacts, all\nflagged objects were examined eye, and those which appeared unphysical or\ncorresponded to sites of corruption by (for example) heavy cosmic ray\nor charge trapping activity in the original images were rejected.\n\n\\subsection{MRC B1256-243}\n\nFour candidate emission line galaxies were identified in the field of MRC\nB1256-243. Details are given in Table \\ref{tab:elgresults}, and their\nlocations are shown in Fig. 3(a). Thumbnail images of the\ncandidate galaxies from each field, together with the measured fluxes, are\nshown in Fig. \\ref{fig:1256objects}.\n\n\\subsection{MRC B2158-206}\n\nTwo candidate emission line galaxies were identified in the field of MRC\nB2158-206. Details are given in Table \\ref{tab:elgresults}, and their\nlocations are shown in Fig. 3(b). Thumbnail images of the\ncandidate galaxies from each field, together with the measured fluxes, are\nshown in Fig. \\ref{fig:2158objects}.\n\n\\subsection{BR B0019-1522}\n\nSeventeen candidate emission line galaxies were identified in the field of BR\nB0019-1522. Details are given in Table \\ref{tab:elgresults}, and their\nlocations are shown in Fig. 3(c). Thumbnail images of the\ncandidate galaxies from each field, together with the measured fluxes, are\nshown in Fig. \\ref{fig:0019objects}.\n\n\\subsection{Contaminants}\n\nThis section briefly addresses the likelihood that our method might\nincorrectly identify another sort of object as an ELG.\n\n\\subsubsection{Continuum objects}\n\nAs per Figs. \\ref{fig:trans} and \\ref{fig:simresults}, the sensitivity of\nour instrument varies from image to image. Therefore, it is possible that a\nflat-spectrum continuum object may be detected in some images but not others,\nthereby appearing to be a potential ELG.\n\nWe use the results of Section \\ref{sec:sim} to estimate the probability of\nthis occurring. Each of the 250,000 simulated objects was sorted into one of\n3,600 bins by wavelength and magnitude (each bin covering 1 \\AA{} and 0.1\nmagnitudes). It is then possible to calculate the completeness of the bin\n(i.e. the fraction of simulated objects which were recovered). Each candidate\nELG is assigned to a bin, and we then check the corresponding bins in adjacent\nimages for completeness. A low completeness value in these bins indicates that\na flat-spectrum object may have been `lost'.\n\nThis procedure calls into question four objects: A in the field of MRC\nB2158-206, B in the field of MRC B2156-243 and E and K in the field of BR\nB0019-1522. These sources were examined by eye, but there is no indication of\na faint detection in the crucial frame. They have not, therefore, been\nexcluded from this analysis.\n\n\\subsubsection{Lower redshift interlopers}\n\nAnother possibility is other emission lines at lower redshift may appear in\nour observations. The lines which might be observed are listed in Table\n\\ref{tab:interlopers}.\n\n\\begin{table*}\n\\begin{center}\n\\caption{Potential low-redshift `interloper' emission lines, together with the\nredshifts at which they appear and the estimated number observed in each of\nthe fields. The flux of each line relative to \\mbox{H\\,$\\alpha$}{} in\na ``typical'' galaxy is given, based on \\citet{Kennicutt92}.}\\label{tab:interlopers}\n\\begin{tabular}{ccccccccccc}\n\\hline\nLine & \\AA           & Flux  & \\multicolumn{2}{c}{MRC B2158-206} & \\multicolumn{2}{c}{MRC B1256-243} & \\multicolumn{2}{c}{BR B0019-1522} \\\\\n     & (rest)        & ratio & $z$ & Number                      & $z$ & Number                      & $z$ & Number \\\\\n\\hline\n\\fline{O}{ii} & 3727 & $0.41\\pm0.21$  & 0.065 & \\phantom{$^*$}0.05\\phantom{$^*$} & 0.060 & 0.02 & 0.803 & 1.93 \\\\\n\\mbox{H\\,$\\beta$} & 4860 & $0.14\\pm0.06$            & -     & -        & -     & -    & 0.383 & 1.68 \\\\\n\\fline{O}{iii} & 5007 & $0.20\\pm0.15$ & -     & -        & -     & -    & 0.342 & 1.41 \\\\\n\\mbox{H\\,$\\alpha$} & 6548 & $1.00\\pm0.00$            & -     & -        & -     & -    & 0.027 & \\phantom{$^*$}0.01\\phantom{$^*$} \\\\\n\\fline{N}{ii} & 6583 & $0.43\\pm0.16$  & -     & -        & -     & -    & 0.021 & \\phantom{$^*$}0.01\\phantom{$^*$} \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\citet{Cowie97} and \\citet{Gallego95} provide number density counts for star\nforming galaxies at a range of redshifts. Both adopt a \\mbox{$H_0 =\n50$ km\\,s$^{-1}$\\,Mpc$^{-3}$}, $\\Omega_{\\Lambda} = 0$, $\\Omega_{\\mathrm{M}} =\n1$ cosmology, which we converted to match that used in this work (Section\n\\ref{sec:intro}). In addition, \\citeauthor{Gallego95} assume a \\citet{Scalo86}\nIMF; \\citeauthor{Cowie97} provide a conversion to a \\citet{Salpeter55} IMF,\nand it is these results we adopt in this work. Based on these, we can estimate\nthe number density of star forming galaxies along our line of sight: see\nFig. \\ref{fig:sfgs}.\n\n\\begin{figure}\n\\begin{center}\n\\rotatebox{270}{\\resizebox{!}{\\columnwidth}{\\includegraphics{fig4}}}\n\\caption{Variation of galaxy number density with star formation rate for a\nrange of redshifts. Based on data from \\citet{Cowie97} and \\citet{Gallego95}.}\n\\label{fig:sfgs}\n\\end{center}\n\\end{figure}\n\n\\citet{Kennicutt98} provides a conversion between star formation rate in a\ngalaxy and \\mbox{H\\,$\\alpha$}{} luminosity; the ratios given in Table \\ref{tab:interlopers}\nmake it possible to convert that into expected luminiosities for the other\nlines. After applying a correction for instrumental effects and galactic\nextinction \\citep{Schlegel98}, a locus of points in the magnitude-wavelength\ncompleteness diagrams (Fig. \\ref{fig:simresults}) on which each line at a\ngiven redshift might be detected is determined. This locus is then integrated\nto estimate the total volume over which the line might be observed at this\nredshift. This procedure is then repeated along the full length of the curves\nshown in Fig. \\ref{fig:sfgs}. In this way, the total number of interlopers\nwhich might be observed is estimated. The results are shown in Table\n\\ref{tab:interlopers}.\n\nIt is clear that the estaimted number of interlopers is negligible in the case\nof the two lower-redshift fields. However, it is possible that as many as five\nof the candidate ELGs in the BR B0019-1522 field are, in fact, low redshift\ninterlopers. This could only be confirmed by further observations.\n\n\\section{Properties of candidate ELGs}\n\\label{sec:properties}\n\nIn this section, we consider the distribution of candidate ELGs around the\nquasars to determine whether the quasar lies in an identifiable overdensity\nrelative to the field.\n\nThe small number of candidates around the lower-$z$ quasars renders a\nmeaningful statistical analysis of the individual fields unreliable. In an\nattempt to mitigate this, and given the apparent similarity of the fields,\nthey are both considered as one unit in this section.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics{fig5}\n\\caption{Distribution of ELG candidates around the quasars. On the left, the\nprojected distance seen on the sky for both the ELG candidates (boxes) and all\nthe objects observed (crosses); at right, the relative velocities.}\n\\label{fig:distribution}\n\\end{center}\n\\end{figure*}\n\nThe distribution of ELG candidates around the quasar is shown in both\nprojection on the sky (left) and velocity distribution (right) in Fig.\n\\ref{fig:distribution}.  When calculating the projection on the sky, we have\nnormalised the total visible area on the sky in each distance bin. We also\nplot the distribution of all objects detected by {\\sc SExtractor} in the field\nfor comparison.\n\nBased on these figures, there is little evidence of projected clustering in\nthe low-$z$ fields. However, there is a notably higher density of objects\nwithin 1 Mpc (projected) of BR B0019-1522. This is consistent with what one\nmight expect from an examination of Fig. \\ref{fig:elgcandidates}: note the\nlarge number of objects to the east of the quasar in Fig. 3(c).  It is also\nin-line with the scale lengths observed in clusters around other AGN\n\\citep{Venemans02, Bremer02, Barr04}.\n\nThere is no suggestion of clustering in velocity space in Fig.\n\\ref{fig:distribution}. In part, this may be due to the low number of\ndetections in the low-$z$ fields. In the field of BR B0019-1522, we note that\nall candidates were observed as bluer than the quasar itself; this is\nnoteworthy, but not implausible given the wavelength range probed (6650--6740\n\\AA, with the quasar at 6722 \\AA).  Although the bluest velocity bins show a\nlower number of total counts, this can be attributed to the reduced\ninstrumental sensitivity at the relevant wavelengths (see Fig. 3(c)).\n\nThe space density of galaxies in the three fields may also be estimated.  As\nalluded to in the previous section, the comoving volume being probed by our\nmeasurements varies with wavelength and magnitude. Consider for example Fig.\n2(a): a bright object--magnitude 19, say--may be detected at a range of\nwavelengths, from around 3920 \\AA{} to 4010 \\AA. A fainter object at, for\ninstance, magnitude 22 is only detected if it lies within a much smaller\nwavelength range: around 3940 \\AA{} to 3960 \\AA. Therefore, we define an\n`accessible volume', $\\mathcal{V}_n$, for each detected object $n$ within the\nfield. $\\mathcal{V}_n$ is calculated by taking the locus of points in Fig.\n\\ref{fig:simresults} occupied by a source with the observed properties and\nintegrating over all wavelengths. The density is taken as $\\rho =\n1\/\\mathcal{V}_1 + 1\/\\mathcal{V}_2 + ... + 1\/\\mathcal{V}_n$. The results for\nour fields are given in Table \\ref{tab:density}.\n\n\\begin{table}\n\\begin{center}\n\\caption{Estimated space and star formation rate densities, together with the\ntotal number of ELG candidates (\\#), for each of the fields\nobserved. Note that our observations are valid only to an approximately\ndefined lower limit of star formation.}\\label{tab:density}\n\\begin{tabular}{cccc}\n\\hline\nField & \\# & Number density             & SFR density \\\\\n      &   & (Mpc$^{-3}\\,\\times\\,10^4$) & (M$_\\odot\\;$\\,yr$^{-1}$\\,Mpc$^{-3}$) \\\\\n\\hline\nMRC B1256 & \\phantom{0}4 & $22.48 \\pm 11.64$ & $0.0346 \\pm 0.0174$ \\\\\nMRC B2158 & \\phantom{0}2 & $\\phantom{0}9.09 \\pm \\phantom{0}6.52$ & $0.0070 \\pm 0.0049$ \\\\\nBR B0019  &           17 & $49.09 \\pm 12.21$ & $0.0484 \\pm 0.0117$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nIt is also instructive to estimate the star formation rates found in these\nfields. Based on \\citet{Kennicutt94} combined with \\citet{Brocklehurst71} and\n\\citet{Hu96}, we arrive at the relationship:\n\n\\begin{equation}\n\\mathrm{SFR}(\\mathrm{M}_\\odot\\,\\mathrm{yr^{-1}}) = 0.91 \\times 10^{-42} L(\\mathrm{Ly} \\alpha) (\\mathrm{erg\\,s^{-1}})\n\\label{eq:sfr}\n\\end{equation}\n\nIt should be noted that \\mbox{Ly $\\alpha$}{} is a very poor indicator of star formation\nrate. It is resonantly scattered by neutral hydrogen, and hence has a high\nchance of absorption either before leaving the galaxy or by clouds in the\nintergalactic medium \\citep{Haiman99}. Further, \\citet{VG93} argues that \\mbox{Ly $\\alpha$}{}\nemission in starbursts is strongly dependent on the age of the burst,\nrendering the calibration of Equation \\ref{eq:sfr} unreliable from around\n$10^7$ years after the burst start. Nevertheless, \\mbox{Ly $\\alpha$}{} is the only\ndiagnostic available to us, so we persist in these estimates with caution.\n\nWe take the star formation rate density as $\\rho_{SFR} = SFR_1\/\\mathcal{V}_1 +\nSFR_2\/\\mathcal{V}_2 + ... + SFR_n\/\\mathcal{V}_n$, where $SFR_n$ is the star\nformation rate associated with ELG candidate $n$ as calculated by Equation\n\\ref{eq:sfr}. Recall from Section \\ref{sec:photo} that the line fluxes are\nsystematically underestimated since objects will fall outside the peaks of the\netalon passpands. Making the approximation that objects are evenly spread in\nwavelength around the etalon peaks, we apply a correction to the observed\nmagnitudes of 0.23 (in the low-$z$ field) or 0.27 (BR B0019-1522 field) to\naccount for this.  We correct the results for completeness based on Fig.\n\\ref{fig:simresults}: a single detection in an area with a low detection rate\nis taken as representative of a larger population.\n\nThe results are shown in Table \\ref{tab:density}. Note that our observations\nare sensitive to galaxies only down to some minimum level of star formation\n(\\sfr{9} in the case of MRC B2158-206 and BR B0019-1522; \\sfr{25} in the case\nof MRC B1256-243): there may be a fainter population which we do not probe.\n\nIt is noteworthy that the star formation rate in the field of MRC B1256-243 is\nanomalously high, but the large uncertainties in the field and the higher\nminimum detectable rate render this result questionable. The most well\nconstrained result is that for BR B0019-1522; our results there are broadly\nsimilar to those reported by \\citet{Venemans02} around the $z = 4.1$ radio\ngalaxy TN J1338-1942. In all three fields, the number of objects detected is\nhigher than that which might be expected in the absence of any clustering.\nBased on \\citet{Cowie97}, we might expect on average 0.86 galaxies in the\nfield of MRC B2158-206, 0.25 in that of MRC B1256-243, and 1.3 in that of BR\nB0019-1522, while an extrapolation from the results of the LALA \\citep[`Large\nArea Lyman $\\alpha$';][]{Rhoads00} survey suggests we should observe 1.1\nobjects in the field of MRC B2158-206, 0.8 in that of MRC B1256-243 and 2.1 in\nthat of BR B0019-1522 (assuming that the density of \\mbox{Ly $\\alpha$}{} emitters is similar\nat $z \\sim 2.2$ to that observed at $z \\sim 4.5$).\n\n\\section{Conclusions}\n\\label{sec:conclusion}\n\nUntil recently, it has proved difficult to find high-redshift clusters and,\nindeed, there are very few known beyond $z \\sim 1$.  The detection of hot\nX-ray emission from intracluster gas followed by optical imaging and\/or\nspectroscopic confirmation becomes inefficient for detecting more distant\nclusters; a manifestly higher success rate is achieved by targeting the\nvicinity of high redshift radio galaxies and quasars.\n\nWe have used tunable filter observations to identify a galaxy overdensity in\nthe field of BR B0019-1522, with a local number density an order of magnitude\nhigher than that which might be expected in the field. This is among the\nhighest-redshift clusters detected around a radio quiet quasar. We have also\nidentified potential overdensities in the fields of and MRC B1256-243 and MRC\nB2158-208, although deeper observations are required to confirm these\ndetections.\n\nThe current observations were made with the Taurus Tunable Filter, an\ninstrument which has now been decommissioned, on the 4 metre class\nAnglo-Australian Telescope. These observations have clearly demonstrated the\nsuccess of the tunable imaging technique. The prospects for further progress\nin this area are strong, as the next generation of tunable filter instruments\nare now available or becoming available on telescopes such as the GTC 10-m\n\\citep[OSIRIS;][]{Cepa00}, SOAR 4-m \\citep[BTFI;][]{Taylor10}, SALT 11-m\n\\citep[PFIS;][]{Smith06}, NTT 3.5-m \\citep[3D-NTT;][]{Marcelin08} and the\nMagellan 6.5-m \\citep[MMTF;][]{Veilleux10}.\n\nWith existing telescopes, it is very difficult to extract more information\nthan a few emission lines and broadband photometry for the host galaxies in\nthese high-redshift environments. More detailed spectral information will not\nbe possible until the next generation of extremely large telescopes or the\nJames Webb Space Telescope come on line.  But there are other uses for these\nobservations: in particular, \\citet{Bruns11} have shown that quasar\nenvironments may act as a surrogate for studying the radiative suppression of\ngalaxy formation during the epoch of reionization. Interestingly, the UV\nsuppression reduces the star-forming galaxy counts by a factor of 2--3 but\ndoes not suppress them altogether.  The time is therefore ripe to further\ndevelop this promising method of investigation in order to learn about the\noccurrence of high-redshift, star forming groups and the impact on these\ngroups by quasar activity.\n\n\\bibliographystyle{mn2e}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nScientists and engineers often study a physical system with the goal of making spatio-temporal predictions (e.g., temperature or glacier thickness) and inferring unknown quantities governing the system (e.g., atmospheric density or ice viscosity). This system's dynamics can often be phrased in terms of spatio-temporal partial differential equations (PDEs) that are based on approximations. The scientist or engineer may also be able to simulate the physical system with a computer simulator, such as a numerical PDE solver, which is subject to imperfections (e.g., numerical error). Moreover, the scientific constants entering into the system's dynamical equations such as density, friction, or viscosity may not be known precisely, but their range can be constrained to some set of plausible values. Additionally field data, though potentially scarce and noisy, can be incorporated into the analysis.\n\nSuch scenarios can be modeled with a variant of a Bayesian hierarchical spatio-temporal model that was introduced in \\citet{gopalan2018bayesian} for glacial dynamics, if considered more generally. We delineate three methods to make posterior inference efficient: the first is to utilize bandwidth limited linear-algebraic routines for likelihood evaluation \\citep{rue2001fast}, the second is to utilize an embarrassingly parallel approximation to the likelihood, and the third is to use first-order emulators \\citep{Hooten2011} for speeding up computer simulators. Though our modeling and numerical results are still within a glaciology context, we conclude with a discussion of how the model can be applied to other physical scenarios. Before introducing the Bayesian hierarchical model and associated methodology for computationally efficient posterior inference, it is appropriate to summarize relevant statistical literature developed over the last two decades.\n\nBayesian hierarchical modeling for geophysical problems was introduced in \\citet{10.1007\/978-94-011-5430-7_3} and \\citet{Wikle1998}, and summarized in \\citet{Berliner}, \\citet{cressie2011statistics}, and \\citet{Wikle2016}. In this modeling approach, prior distributions are specified for physical parameters of interest, a physical process is modeled at the intermediary, latent level (conditional on the physical parameters), and the data collection process is modeled conditional on the latent physical process values. Both numerical error and model uncertainty can be incorporated at the process level, while measurement errors can be modeled at the data level. This approach has been applied in a variety of scientific contexts, including the study of ozone concentrations \\citep{berrocal2014assessing}, sediment loads at the Great Barrier Reef \\citep{pagendam2014assimilating}, precipitation in Iceland \\citep{RePEc:wly:envmet:v:27:y:2016:i:1:p:27-41}, Antarctic contributions to sea level rise \\citep{zammit2014resolving}, and tropical ocean surface winds \\citep{wikle2001spatiotemporal} (among many others). In \\citet{gopalan2018bayesian}, the motivating example for the work in this paper, a Bayesian hierarchical model for shallow glaciers based on the shallow ice approximation (SIA) PDE was developed and evaluated. \n\n\\citet{kennedy2001bayesian} suggest constructing Bayesian statistical models that incorporate the output of a computer simulator of a physical process, such as a numerical solver for the underlying system of PDEs. Fundamental to their approach is the inclusion of a specific term that represents the deviation between the output of a computer simulator and the actual process values, known as \\textit{model discrepancy} or \\textit{model inadequacy}. This framework is developed in \\citet{higdon2004combining}, \\citet{higdon2008computer}, and \\citet{discrepancy}. In particular, \\citet{higdon2008computer} use a Bayesian model along with a principal components based approach for reducing the computational overhead of running a computer simulation with high dimensional output multiple times (an approach termed as \\textit{emulation}). \\citet{discrepancy} note that the prior for model discrepancy must be chosen carefully to mitigate bias of physical parameters and predictions. In particular, as more prior information is incorporated into a model discrepancy term through a constrained Gaussian process (GP) prior over a space of functions, the less biased inferences and predictions tend to become. The notions of an emulator, a computer simulator, and model discrepancy enter naturally into the aforementioned Bayesian hierarchical framework. Conditional on physical parameters coupled with initial and\/or boundary conditions, the physical process values at the latent level can be written as the sum of a computer simulator or emulator term and a model discrepancy term. \n\nTo be precise, let us assume that the physical process $\\bm{S}$ can be indexed through time, i.e., as $\\bm{S}_j$, and $\\bm{S}_j$ is a vector where each element corresponds to a distinct spatial location. One can specify the process level conditional on physical parameter $\\bm{\\theta}$ as \n\\begin{eqnarray}\n\\bm{S}_j &=& \\bm{f}(\\bm{\\theta},j)+\\bm{\\delta}(j)\n\\end{eqnarray}\nwhere $\\bm{\\delta}(.)$ is a vector valued model discrepancy function that is independent of $\\bm{\\theta}$, and $\\bm{f}(\\bm{\\theta},j)$ is the output of a computer simulation or emulator for physical parameter $\\bm{\\theta}$ at time index $j$. If, for instance, at each time point $j$ an observation $\\bm{Y}_j$ of $\\bm{S}_j$ is made with associated measurement error $\\bm{\\eta}_j$, then observations can be written as\n\\begin{eqnarray}\n\\bm{Y}_j &=& \\bm{f}(\\bm{\\theta},j)+\\bm{\\delta}(j)+\\bm{\\eta}_j,\n\\end{eqnarray}\nwhich is analogous to Eq. 5 of \\citet{kennedy2001bayesian}.\n\nIn \\citet{kennedy2001bayesian}, $\\bm{\\delta}(.)$ is a fixed but unknown function independent of $\\bm{\\theta}$ that is learned with a GP prior distribution. Similarly, $\\bm{\\delta}(.)$ has a constrained GP prior in \\citet{discrepancy}. The approach in this paper instead assumes a temporally indexed stochastic process (with spatial correlation) that follows a multivariate random walk, rather than a deterministic function. Additionally, in \\citet{liu2009}, the authors frame a computer emulator of time series run under multiple inputs as a dynamic linear model (DLM). As part of their approach, they allow for time varying auto-regressive coefficients that follow a random walk process, to embed non-stationarity into the model.\n\nWhile the approach taken in this paper most closely follows the above literature (i.e., Bayesian hierarchical modeling, model discrepancy, and emulation), we briefly review literature in probabilistic numerics and Bayesian numerical analysis; the emphasis in Bayesian numerical analysis is to use probabilistic methods to solve numerical problems, whereas, in the Bayesian hierarchical setup, one is also interested in inference of scientifically relevant parameters and predictions of the physical process. In \\citet{conrad2017statistical}, a probabilistic ordinary differential equation (ODE) solver is developed that adds stochasticity at each iteration; conditions for the convergence of this method to the ODE solution are given. \\citet{chkrebtii2016bayesian} utilize GPs for solving ODEs; moreover, \\citet{Calderhead:2008:ABI:2981780.2981808} use a GP regression based method to avoid explicitly solving nonlinear ODEs when performing inference for parameters that provides computational speed ups; additionally, \\citet{Owhadi} present a gamblet based solver that comes with provably computationally efficient solutions to PDEs. The approach is derived from a game theoretic and stochastic PDE framework.\n\nIn the spatio-temporal model described in this paper, stochasticity is induced with an error-correcting process that is separated from the numerical solution. In general, another way to achieve this is to define a stochastic process by equating a PDE to a white noise term -- that is, the solution $\\bm{X}$ to a stochastic partial differential equation (SPDE) $L[\\bm{X}] = \\bm{W}$, where $\\bm{L}$ is a differential operator and $\\bm{W}$ is a white noise process (indexed by spatio-temporal coordinates). For instance, a fractional Laplacian operator yields the Mat\\'{e}rn covariance function \\citep{whittle1954stationary,Whittle63,lindgren2011explicit}. We employ the former approach mainly because it is difficult to derive exact covariance functions for arbitrary differential equations (e.g., in the presence of nonlinearities), though we highlight the utility of the latter approach in situations where an analytical covariance function can be derived exactly.\n\n A major feature of this work is to represent the discrepancy between real physical process values and the output of a computer simulator for these physical process values as a multivariate random walk; typically, model discrepancy is endowed with a GP prior or a constrained GP prior over a space of functions as in \\citet{kennedy2001bayesian} and \\citet{discrepancy}. Along with this model is the development of two ways for making computations faster: the first is harnessing first-order emulator inference \\citep{Hooten2011} for speeding up the computation of a numerical solver, and the second is the use of bandwidth limited numerical linear algebra \\citep{rue2001fast} for computing the likelihood efficiently. Moreover, in the regime of a high signal-to-noise ratio, an embarrassingly parallel approximation to the likelihood can be employed. Finally, methodology to fit a spatial Gaussian field for the log of the scale of numerical errors is discussed.\n \nWe must also be clear about what distinguishes this work from its predecessor, \\citet{gopalan2018bayesian}. This includes the use of emulators, probing higher order random walks besides order 1, derivation of sparsity and computational complexity of log-likelihood evaluation, empirical run time results, and methodology to fit an error-correcting process when little prior information is available. The structure of this paper is as follows: First a test system from glaciology is described. Then the statistical model that is the focus of this work is presented in detail (in the context of the glaciology test case), followed by the exact and approximate likelihood. Then this model is analyzed in terms of computational run time and accuracy of inference, based on the test system from glaciology; moreover, the random walk error-correcting process is assessed with residual analysis. Afterward, we discuss how the model and associated methodology can be applied to other physical scenarios, and conclude by summarizing the model, method, and limitations of the approach.\n\n \\section{Description of a test system from glaciology}\nBefore delving into the specifics of the Bayesian hierarchical model and computational subtleties, we begin with a brief discussion of glaciology. Glaciology is the study of physical systems consisting mostly of ice and snow. This broad definition includes the study of the crystalline nature of ice, the transformation and compaction of snow into ice, the dynamics of the flow of ice and water in a glacier, the relationships between fundamental quantities like viscosity, temperature, and pressure, the relationships between precipitation and meteorology with said ice systems, the interaction of ice systems with other geological systems such as volcanoes and bedrock, and so on. As such, glaciology synthesizes elements from a multitude of scientific disciplines including continuum mechanics, fluid mechanics, hydraulics, chemistry, and meteorology.\n\n\\cite{Bueler} introduce analytical solutions for the SIA PDE, a commonly used model for the dynamics of glaciers \\citep{10.2307\/79748, doi:10.1080\/03091928208209013, flowers2005sensitivity,Paterson,vanderveen, Brinkerhoff, 2016arXiv161201454G, gopalan2018bayesian}. Based on the principle of conservation of mass, the SIA dictates that glacier flow is in the direction of the (negative) gradient of the glacier surface and is due to gravity and basal sliding (also referred to as friction or drag if in the direction of the positive gradient). While an explanation of the SIA PDE is given in \\cite{gopalan2018bayesian}, our focus is on ice viscosity, $B$. Intuitively, this parameter controls the softness of the ice. The other main physical parameter, which is not the subject of this paper, is $C_0\\gamma$. This controls basal sliding or friction. \n\nFor the analysis that follows, we focus on a periodic solution to the SIA in which the thickness of the glacier oscillates through time; $H(r,t)$, the thickness of the glacier as a function of two dimensional space (in polar coordinates) and time, is \n\\begin{eqnarray}\nH(r,t) &=& H_s(r)+P(r,t), \\\\\nP(r,t)  &=& C_p\\sin(2\\pi t\/T_p)\\cos^2\\left[\\frac{\\pi(r-0.6L)}{.6L}\\right]; \\textrm{if } 0.3L < r < .9L, \\\\\nP(r,t) &=& 0; \\textrm{if } 0 \\leq r \\leq 0.3L \\textrm{ or if } r \\geq 0.9L. \n\\end{eqnarray}\nIn Eq. 3, $H_s$ is a static initial profile of the glacier (i.e., a dome as in Eq. 21 of \\citet{Bueler}), $P$ is a perturbation (e.g., precipitation) function, $L$ is the margin length, $C_p$ is the magnitude of the periodic perturbation, and $T_p$ is the period of the perturbation. \\citet{Bueler} derive a mass balance function that achieves this periodic solution for the SIA PDE. Qualitatively, this test case appears like a dome with a periodic oscillation in thickness around an annulus defined by $0.3L < r < .9L$. In Figure 1, an illustration of the oscillations of glacier thickness through time is displayed.\n\nThe value of each surface elevation measurement is the value of the exact analytical solution above added to a zero-mean Gaussian random variable with standard deviation of 1 meter, larger than errors of the digital-GPS instruments employed by the UI-IES. We use the same values of parameters as in \\citet{Bueler} to make for easier comparison to that work and the EISMINT experiment. In particular, $H_0 = 3600$ m, $L = 750$ km, $C_p = 200$ m, and $T_p = 5000$ years.\n\nEmploying the same set up as \\citet{gopalan2018bayesian}, glacial surface elevation measurements are assumed to be collected for 20 years, twice a year, and at 25 fixed spatial locations across the glacier, to emulate how the glaciology team at the University of Iceland Institute of Earth Sciences (UI-IES) collects data at Icelandic glaciers (e.g., see Figure 2 illustrating Langj\\\"{o}kull and the mass balance measurement sites). \n\n\\begin{figure*}[h!]\n  \\centering\n    \\includegraphics[width=0.5\\textwidth]{glacier_plot.jpeg}\n\\caption{An illustration of the periodic oscillatory exact solution to the SIA PDE that is used for the analysis. Since the solution is radially symmetric, only a radial cross section is illustrated. This solution is stationary except for an annulus defined by $0.3L < r < .9L$, where $L$ is 750 $km$; in the annulus, the glacier thickness vibrates back and forth periodically, as illustrated.}\n\\end{figure*}\n\n\\begin{figure*}[h!]\n  \\centering\n    \\includegraphics[width=0.5\\textwidth]{lang.pdf}\n\\caption{A digital elevation map of Langj\\\"{o}kull along with measurement sites demarcated on the right, provided by the University of Iceland Institute of Earth Sciences (UI-IES). Langj\\\"{o}kull is Iceland's second largest glacier by area, at $900$ sq. km, and its mean thickness is 210 meters above sea level \\citep{bjornsson2008icelandic}, so Langj\\\"{o}kull is shallow.}\n\\end{figure*}\n\n\\section{The hierarchical spatio-temporal model and its properties}\nNow that we have acquainted the reader with some facts about glaciology and the particular test case used for the analysis in this paper, we next delineate the hierarchical spatio-temporal model that is the focus of this work, by specifying its variables, parameters, and properties, including efficient computation of the likelihood and connections to other modeling frameworks. For the sake of specificity of the presentation, the glaciology example is referenced, similarly to the set up in \\citet{gopalan2018bayesian}. We assume that $n$ spatial locations are modeled at the latent level, and $m$ of those locations are observed, where $m$ is typically much smaller than $n$. We use the index $j$ to refer to time indices and $i$ to refer to spatial indices; while space and time are discretized, the differences between successive time and spatial points can be made as small as desired depending on the context of the application and computational resources available. Throughout, we use bolded notation for vectors and uppercase, unbolded, and non-italic notation for matrices. All other mathematical symbols are scalars.\n\nWe introduce the Bayesian hierarchical model in the parameter, process, data level framework of \\citet{10.1007\/978-94-011-5430-7_3}. We denote the physical parameters as $\\bm{\\theta}$ and initial and\/or boundary conditions for the physical process as $\\bm{\\phi}$. At the parameter level, one possibility is to use a truncated normal distribution for $\\bm{\\theta}$ if the support of the parameter value can be constrained, as was done in \\citet{gopalan2018bayesian}, where $\\bm{\\theta}$ represented ice viscosity. However, more generally, the distribution can be specified based on domain knowledge or expertise. We denote the output of a computer simulator, which could be either a numerical solver or an emulator, at time $j$ with the notation $\\bm{f}(\\bm{\\theta},\\bm{\\phi}, j)$, which, in full generality, is an element of $\\mathbb{R}^n$. While some values could be negative (e.g., temperature), in many cases the computer simulator output can be restricted to the nonnegative real numbers. For a specific example, in Appendix A of \\citet{gopalan2018bayesian}, $\\bm{f}(\\bm{\\theta},\\bm{\\phi}, j)$ is a second-order finite difference solver for glacier thickness, which is constrained to be nonnegative based on a boundary condition. Evidence for a nonnegative support for the physical process, in glaciology, can be found in \\cite{gopalan2018bayesian}. Particularly, this is evident in Figure 6 of that paper, which shows the process (i.e., glacier thickness) predictions across the glacier, and the distributions are all greater than zero. Specifically, the minimum of the smallest box-plot is more than 750 m. Nonetheless, the reader is suggested to think carefully about whether a negligible amount of probability mass is below zero in different applications (e.g., temperature models).\n\nThe process level of the model, conditional on $\\bm{\\theta}$ and $\\bm{\\phi}$, can be written as:\n\n\\begin{eqnarray}\n\\bm{X}_{j} &=& \\bm{X}_{j-1} +\\bm{\\epsilon}_{j}, \\\\\n\\bm{S}_{j} &=& \\bm{f}(\\bm{\\theta},\\bm{\\phi},j) + \\bm{X}_j,\n\\end{eqnarray}\nwhere $\\bm{X}_{0}$ is a vector of zeros.\n\nIn the above expressions, $\\bm{\\epsilon}_{j}$ is $\\textrm{MVN}(0,\\Sigma)$ and independent of $\\bm{\\epsilon}_{l}$ for $j \\neq l$. Furthermore, $\\bm{X}_{j}$, $\\bm{\\epsilon}_{j}$,  $\\bm{f}(\\bm{\\theta},\\bm{\\phi},j)$, and (consequently) $\\bm{S}_{j}$ are members of $\\mathbb{R}^n$. In \\citet{gopalan2018bayesian},  $\\{\\bm{X}_1, \\bm{X}_2, ..., \\}$ was referred to as an error-correcting process because it was meant to represent the difference between the numerical solver and  the exact solution to the SIA PDE. Note that in \\citet{gopalan2018bayesian}, $\\bm{S}_j$ referred to glacial thickness at a particular time point, where each component referred to the glacial thickness at a particular grid point. In more generality, the error-correcting statistical process can be a random walk of higher order; a multivariate RW process of order $q$ ($RW(q)$) is given by:\n\\begin{eqnarray}\n\\bm{X}_{j}+ \\sum_{p=1}^q(-1)^{p}{q \\choose p}\\bm{X}_{j-p} &=& \\bm{\\epsilon}_j\n\\end{eqnarray}\nwhere $\\bm{\\epsilon}_1$,..., $\\bm{\\epsilon}_q$ are independent and marginally $\\textrm{MVN}(0,\\Sigma)$. This form of a higher order random walk is a multivariate extension of the integrated auto-regressive process given in Chapter 5.6 of \\citet{madsen2007time}. For $q=2$, this corresponds to RW(2) of \\citet{rue2005gaussian}.\n\nAt the data level, it is assumed that data are regularly sampled at every $k$-th time point, so that one observes $\\bm{Y}_k, \\bm{Y}_{2k},..., \\bm{Y}_{Nk} \\in \\mathbb{R}^m$; in the glaciology test case, the variables $\\bm{Y}$ referred to glacial surface elevation measurements, and $k$ was set to 5, to represent the fact that the glaciologists take a set of measurements in the summer and winter, or twice a year. The corresponding observation errors  $\\bm{\\eta}_k, \\bm{\\eta}_{2k},..., \\bm{\\eta}_{Nk}$ are IID $\\textrm{MVN}(0,\\sigma^2\\textrm{I})$, and represent digitial-GPS measurement errors in the glaciology example. We define the matrix  A $\\in \\mathbb{R}^{m \\times n}$ to be such that its rows are unit basis vectors (i.e., an incidence matrix as in \\citet{cressie2011statistics}). That is, $\\textrm{A}_{ab} = 1$ if and only if the $b$th index of the process level vector $\\bm{S}$ has been observed, and $\\textrm{A}_{ab} = 0$ for all other entries. Then the data level model, conditional on the process $\\bm{S}$, is\n\n\\begin{eqnarray}\n\\bm{Y}_{ck} &=& \\textrm{A}\\bm{S}_{ck}+\\bm{\\eta}_{ck}, \n\\end{eqnarray}\nwhere we assume that  $j \\in \\{1,2,..., T\\}$ and $c \\in \\{1,2,..., N\\}$, so there are $N$ total observed spatial vectors, observed with a period of length $k$. \n\nConditional on $\\bm{\\theta}$, $\\bm{\\phi}$, and a computer simulator, the model can be thought of as a hidden Markov model (HMM) \\citep{baum1966}; the latent physical process evolves according to a RW(1) process added to a numerical solution, and it is observed indirectly with Gaussian noise. It can also be thought of as a conditional general state space model. This is because, conditioning on  $\\bm{\\theta}$, $\\bm{\\phi}$, and a computer simulator, one can write:\n\n\\begin{eqnarray}\n\\bm{S}_{j} &=& \\bm{S}_{j-1}+[-\\bm{f}(\\bm{\\theta},\\bm{\\phi},j-1)+\\bm{f}(\\bm{\\theta},\\bm{\\phi},j)] + \\bm{\\epsilon}_j,\\\\\n\\bm{Y}_{ck} &=& \\textrm{A}\\bm{S}_{ck}+\\bm{\\eta}_{ck}.\n\\end{eqnarray}\nHere, the state evolves linearly with a time dependent offset term: $[-\\bm{f}(\\bm{\\theta},\\bm{\\phi},j-1)+\\bm{f}(\\bm{\\theta},\\bm{\\phi},j)]$. The notation $ck$ is used in Eq. 11 to indicate that observations of the process are only observed every $k$th time point, whereas the latent process evolves at every time step $j$. The reader who is interested in further understanding the connection between Gaussian processes and state space models may consult \\cite{pmlr-v33-solin14}. \n\n\\subsection{Exact likelihood}\nAn advantage of using this model is that the likelihood, $p(\\bm{Y}_k, \\bm{Y}_{2k},..., \\bm{Y}_{Nk}|\\bm{\\theta},\\bm{\\phi})$, can be computed exactly in an efficient manner. It can also be approximated in a way that leads to embarrassingly parallel computation when the signal-to-noise ratio is high. The next several sections provide more details for these considerations. The likelihood of the model, $L(\\bm{\\theta},\\bm{\\phi}) = p(\\bm{Y}_k, \\bm{Y}_{2k},..., \\bm{Y}_{Nk}|\\bm{\\theta},\\bm{\\phi})$, has a multivariate normal PDF form:\n\\begin{eqnarray}\nL(\\bm{\\bm{\\theta}},\\bm{\\phi}) &=& \\frac{1}{(2\\pi)^{(mN)\/2}|\\Sigma_l|^{1\/2}}\\exp^{-(\\bm{Y-\\mu_l})^T\\Sigma_l^{-1}(\\bm{Y-\\mu_l})\/2},\n\\end{eqnarray}\nwhere the mean is:\n\\begin{eqnarray}\n\\bm{\\mu}_l &=& (\\textrm{A}\\bm{f}(\\bm{\\theta},\\bm{\\phi},k),...,\\textrm{A}\\bm{f}(\\bm{\\theta},\\bm{\\phi},Nk)),\n\\end{eqnarray}\nand the covariance matrix is:\n\\begin{eqnarray}\n\\Sigma_l &=& \\textrm{U} \\otimes \\textrm{V} + \\sigma^2\\textrm{I},\n\\end{eqnarray}\nwhere $\\textrm{U}_{ab} = k \\min(a,b)$ with $\\textrm{U} \\in \\mathbb{R}^{N \\times N}$, and $\\textrm{V} = \\textrm{A}\\Sigma \\textrm{A}^{\\intercal}$. Also, the symbol $\\otimes$ stands for the Kronecker product. $\\bm{Y}_{ck}$ is multivariate normal (conditioning on $\\bm{\\theta}$ and $\\bm{\\phi}$) as a direct consequence of equations 7 and 9, noting that $\\bm{X}_{ck}$ and $\\bm{\\eta}_{ck}$ are independent conditional on $\\bm{\\theta}$ and $\\bm{\\phi}$. Moreover, the linearity property of expectations can be used to show that the mean of $\\bm{Y}_{ck}$ is $E[\\textrm{A}\\bm{S}_{ck}+\\bm{\\eta}_{ck}] = E[\\textrm{A}\\bm{S}_{ck}]+E[\\bm{\\eta}_{ck}] = E[\\textrm{A}\\bm{f}(\\bm{\\theta},\\bm{\\phi},ck)+\\textrm{A}\\bm{X}_{ck}]+E[\\bm{\\eta}_{ck}] = E[\\textrm{A}\\bm{f}(\\bm{\\theta},\\bm{\\phi},ck)]+E[\\textrm{A}\\bm{X}_{ck}]+E[\\bm{\\eta}_{ck}] =  \\textrm{A}\\bm{f}(\\bm{\\theta},\\bm{\\phi},ck) + \\bm{0} + \\bm{0}$ (again, conditional on fixed $\\bm{\\theta}$ and $\\bm{\\phi}$ fixed). Appendix A contains more details of the covariance matrix.\n\nSince evaluating Eq. 12 requires the calculation of the inverse of the matrix $\\Sigma_l$ and its determinant, these must be calculated efficiently (generally this takes $O(N^3m^3)$ operations, which can grow very quickly with more space and time observations). Since $\\textrm{U}^{-1}$ is tridiagonal, the bandwidth of $\\textrm{U}^{-1}$ is 1, and the band-limited nature of $\\textrm{U}^{-1}$ allows us to compute $\\Sigma_l^{-1}$ and $|\\Sigma_l|$ in $O(Nm^3)$ time \\citep{rue2001fast, golub2012matrix}. More details for this derivation are given in Appendix A. While using band-limited linear algebra routines can improve computation, in the next subsection we derive an approximation to the likelihood that is embarrassingly parallel and can therefore accelerate computation even more. \n\n\\subsection{An approximate likelihood}\nHere we show how to approximate the likelihood in a way that leads to embarrassingly parallel computation. The likelihood $p(\\bm{Y_k,...,Y_{Nk}}|\\bm{\\bm{\\theta}},\\bm{\\phi})$ can be equivalently written as $p(\\bm{Y_k}|\\bm{\\bm{\\theta}},\\bm{\\phi})p(\\bm{Y_{2k}|Y_k},\\bm{\\bm{\\theta}},\\bm{\\phi})... \\\\p(\\bm{Y_{Nk}|Y_k,..,Y_{(N-1)k}},\\bm{\\bm{\\theta}},\\bm{\\phi})$.\nFirst note that:\n\\begin{eqnarray}\n\\bm{Y}_{k} &=& \\textrm{A}\\bm{f}(\\bm{\\theta},\\bm{\\phi},k)+\\bm{\\eta}_{k}+\\sum_{j=1}^{k} \\textrm{A}\\bm{\\epsilon}_{j}.\n\\end{eqnarray}\nHence, $p(\\bm{Y}_{k}|\\bm{\\theta},\\bm{\\phi})$ is multivariate normal with mean $\\textrm{A}\\bm{f}(\\bm{\\theta},\\bm{\\phi},k)$ and covariance matrix $\\textrm{A}(k\\Sigma)\\textrm{A}^{\\intercal}+\\sigma^2\\textrm{I}$.\nMore generally, we have the relationship:\n\\begin{eqnarray}\n\\label{eq:rec}\n\\bm{Y}_{ck} &=& \\bm{Y}_{(c-1)k} + \\textrm{A}[\\bm{f}(\\bm{\\theta},\\bm{\\phi},ck)-\\bm{f}(\\bm{\\theta},\\bm{\\phi},(c-1)k)] + \\bm{\\eta}_{ck} - \\bm{\\eta}_{(c-1)k} + \\sum_{j=(c-1)k+1}^{ck} \\textrm{A}\\bm{\\epsilon}_{j}.\n\\end{eqnarray}\nThus we can approximate  $p(\\bm{Y}_{ck}|\\bm{Y}_k,..,\\bm{Y}_{(c-1)k},\\bm{\\theta},\\bm{\\phi})$ as a MVN distribution with mean $\\bm{Y}_{(c-1)k} + \\textrm{A}[\\bm{f}(\\bm{\\theta},\\bm{\\phi},ck)-\\bm{f}(\\bm{\\theta},\\bm{\\phi},(c-1)k)]$ and covariance matrix $\\textrm{A}(k\\Sigma)\\textrm{A}^{\\intercal} +2\\sigma^2\\textrm{I}$. Nonetheless, to clarify,  $p(\\bm{Y}_{ck}|\\bm{Y}_k,..,\\bm{Y}_{(c-1)k},\\bm{\\theta},\\bm{\\phi})$ is not exactly a MVN with mean $\\bm{Y}_{(c-1)k} + \\textrm{A}[\\bm{f}(\\bm{\\theta},\\bm{\\phi},ck)-\\bm{f}(\\bm{\\theta},\\bm{\\phi},(c-1)k)]$ and covariance matrix $\\textrm{A}(k\\Sigma)\\textrm{A}^{\\intercal} +2\\sigma^2\\textrm{I}$ because $\\bm{Y}_{(c-1)k}$ and $\\bm{\\eta}_{(c-1)k}$ are dependent. However, when the magnitude of the observation error $\\bm{\\eta}_{(c-1)k}$ is much smaller in comparison to the magnitude of the observation $\\bm{Y}_{(c-1)k}$, and for $\\bm{Z} \\sim \\textrm{MVN}(0,\\sigma^2\\textrm{I})$ with $\\bm{Z}$ independent of $\\bm{Y}_{(c-1)k}$, $\\bm{Y}_{(c-1)k} - \\bm{\\eta}_{(c-1)k}  \\approx \\bm{Y}_{(c-1)k} - \\bm{Z}$. \n \nThis approximation is embarrassingly parallel because each of the $N$ terms in the product form of the likelihood $p(\\bm{Y}_k,...,\\bm{Y}_{T}|\\bm{\\theta},\\bm{\\phi}) = p(\\bm{Y}_k|\\bm{\\theta},\\bm{\\phi})p(\\bm{Y}_{2k}|\\bm{Y}_k,\\bm{\\theta},\\bm{\\phi})...p(\\bm{Y}_{Nk}|\\bm{Y}_k,..,\\bm{Y}_{(N-1)k},\\bm{\\theta},\\bm{\\phi})$ (or sum, if computing the log-likelihood) can be evaluated independently of each other. Therefore, in parallel, the computation comes down to evaluating a multivariate normal PDF of dimension $m$ -- this can be done in $O(m^3)$.  \n\n\\subsection{Computational complexity summary}\nIf no attention is paid to the structure of $\\Sigma_l$, the cost of evaluating $L(\\bm{\\theta},\\bm{\\phi})$ is limited by the evaluation of $\\Sigma_l^{-1}$ and $|\\Sigma_l|$, which generally takes $O(N^3m^3)$ operations. However, the exact likelihood evaluation can be reduced to $O(Nm^3)$ using band-limited numerical linear algebra. The computational complexity of the approximation is also $O(Nm^3)$ (if no parallelism is used). While an exact likelihood is preferred to an approximation, a benefit of the approximation is that it is embarrassingly parallel -- if parallelized, the time complexity is that of evaluating a multivariate normal PDF of dimension $m$, which is $O(m^3)$. Nonetheless, there also exist parallel versions of sparse Cholesky decomposition, for instance in \\citet{Gupta:1994:SPA:602770.602898}. Empirical comparisons of the exact and approximate likelihood computations are presented in Section 4. \n\n\n\\section{Analysis of the model and associated methodology}\nThe purpose of this section is to motivate the various modeling choices introduced in this paper using the previously described test system from glaciology, both in terms of computational run time and quality of inferences. In particular, we compare a posterior based on an emulator to a posterior based on a numerical PDE solver, motivate the use of the random walk error-correcting process with residual analysis, examine the impact of prior information encoded into the error-correcting process on the bias of posterior distributions for physical parameters, and compare the run-time and accuracy of the likelihood approximation versus the exact likelihood. The physical parameter of interest in these examples is ice viscosity, $B$, whose actual value is the same as \\citet{Bueler}, \\citet{EISMINT}, and \\citet{gopalan2018bayesian}:  $31.7 \\times 10^{-25}$ in units of $s^{-1}Pa^{-3}$.\n\nConsistent with \\citet{gopalan2018bayesian} is the choice of settings for the numerical PDE solver: a 21 by 21 grid (so $n = 441$) is used with $\\Delta_x = \\Delta_y = 10^5$ m and $\\Delta_t = .1$ years. Note that, consequently, the number of simulator runs (25) is much smaller than the dimensionality of the output of the solver (441). \n\n\\subsection{Posterior inference of the ice viscosity parameter with an emulator compared to a numerical PDE solver}\nIn this section, we conduct an empirical study to examine how a first-order spatio-temporal emulator (i.e., an emulator based on the method in Appendix B) compares to a numerical solver of the PDE, both in terms of run-time of computations and posterior inference of ice viscosity. While the precise technical details for constructing a first-order spatio-temporal emulator are given in Appendix B, the idea is to approximate the numerical solver output for each time point that there is collected data. To do this, we train an emulator using the following values for ice viscosity: $\\{10, 12.5, 15.0,...,70.0\\}$ in units of $10^{-25} s^{-1}Pa^{-3}$, a grid of values that is intentionally coarser than the values used for posterior computation, since in this case the emulator must be used for parameter values not in the training set. We used the \\texttt{rbenchmark} \\citep{rbenchmark} package to benchmark the run-time of the log-likelihood of the model evaluated at the actual parameter value computed with a numerical solver versus a first-order spatio-temporal emulator, using a MacBook Pro early 2015 model with a 2.7 GHz Intel Core i5 processor and 8 GB 1867 MHz DDR3 memory. The emulator version performs 14.5 times faster (.354 seconds for the emulator based log-likelihood versus 5.148 seconds for the numerical solver based log-likelihood). We also generated samples from the posterior distribution of ice viscosity with grid sampling (grid [10,70] inclusive with grid width .50 in units of  $10^{-25} s^{-1}Pa^{-3}$), using both the numerical PDE version and the emulated version. The summary statistics of $10^6$ posterior samples for ice viscosity using both the emulator and numerical solver are given in Table 1. Qualitatively, the summary statistics are similar. \n\nThe principle behind choosing the ice viscosity parameter values in the training set is to fill the space of the support for ice viscosity, but not to choose a grid as fine as the one used for posterior sampling. (Such an approach would be circular, in that the emulator would just be generating predictions inside of the training set.) However, such a heuristic will not be feasible as the number of parameters grows beyond one parameter (the number of design points would need to grow exponentially in the number of dimensions). In such cases, we suggest using other space-filling designs: notably, a latin hypercube design has been used extensively in the computer experiments literature, for instance in \\cite{higdon2008computer}. \n\n\n\\begin {table}[H]\n\\begin{center}\n\\begin{tabular}{ |c|c|c|c|c|c|c| }\n\\hline\nTest Case & Min & 1st Quartile & Median & Mean & 3rd Quartile & Max \\\\ \\hline\nEmulator SIA solver & $15.0$ & $26.5$  & $27.0$ & $27.4$ & $29.0$ & $38.5$\\\\ \\hline\nNumerical SIA solver & $15.0$ & $24.5$  & $26.5$ & $26.3$ & $28.0$ & $37.5$\\\\ \\hline\n \\hline\n  \\end{tabular}\n\\caption{Summary statistics of $10^6$ posterior samples of the ice viscosity parameter using an emulator for the SIA and a numerical solver for the SIA; qualitatively, these posterior samples are similar. Units are in $10^{-25} s^{-1}Pa^{-3}$.}\n\\end{center}\n\\end {table}\n\n\n\\subsection{Assessing a random walk for representing model discrepancy}\nThe choice of using a random walk to correct for deviations between the output of a computer simulator and the actual physical process values has a few important motivations:\n\n\\begin{enumerate}\n    \\item The inaccuracy of a spatio-temporal computer simulation is most likely going to increase as it is run further into the future. Conveniently, a random walk's variance increases with time -- for example a RW(1) has marginal variance $j\\Sigma$ at time $j$.\n    \\item As shown in Appendix A and Section 3.1, the likelihood involves band-limited matrices, for which there exist specialized numerical linear algebra routines. However, there is a trade-off in bandwidth and the order of the random walk utilized.\n    \\item Spatial correlations in the inaccuracies of a computer simulation can be captured with the covariance matrix $\\Sigma$.\n\\end{enumerate}\n\nIn addition to these motivations, the purpose of this section is to empirically assess how a random walk model performs for correcting the output of a numerical SIA PDE solver. To do this, we use the analytical SIA solution as a gold standard. This is a simplification in the sense that the real glacial dynamics will not follow the SIA PDE and therefore the analytical SIA solution exactly, but nonetheless this is a way to check the veracity of the random walk error model in some capacity -- at the very least, as a model for numerical error but not model uncertainty.\n\nFigure 3 displays the differences between the analytical SIA PDE solution for glacial thickness and the numerical SIA PDE solution for glacial thickness at all of the glacier grid points, run forward for 5000 time steps (i.e., 500 years). More precisely, the points in blue are at the margin of the glacier, the points in red are at the interior, and the points filled in black are close to the top (also referred to as the dome) of the glacier. Recall from Figure 1 that the glacier looks like a shallow ellipsoid sliced in half (in the x-y plane), and so the panel on the right of this figure is a top view of the glacier grid points, which looks like a circle of radius 750 km projected onto the x-y plane. In comparison, the height is 3600 m.\n\n\\begin{figure*}[h!]\n  \\centering\n    \\includegraphics[width=0.5\\textwidth]{discrepancy.jpeg}\n\\caption{An illustration of the difference between the exact analytical solution and the numerical solution for the SIA PDE. On the \\textit{right} panel is a top view of the glacier, whose shape looks like a dome, and therefore the projection on to the x-y plane is a circle. The {\\color{blue}blue points} signify the margin of the glacier (where it drops down to zero thickness), the {\\color{red}red points} are at the interior of the glacier, and the {\\color{black}black points} are towards the top of the glacier. The points that are not filled in signify the border of the glacier, where there is no ice thickness. On the \\textit{left} panel the discrepancies between the analytical SIA PDE solution and the numerical SIA PDE solution for all grid points are shown. Specifically, the color of each path corresponds to the grid points on the right panel. Additionally, the paths are shown for 500 years, or 5000 time steps.\n}\n\\end{figure*}\n\n\nThe differences are all very smooth (i.e., continuous) functions of time, implying that the numerical SIA PDE solver is producing continuous output as well -- we know that the analytical solution is continuous based on the functional form in Eqs. 3-5. Thus, it appears that a random walk of at least a few orders is necessary to represent these differences. Moreover, as expected from \\citet{Bueler}, the largest errors occur at the margin, whereas the interior and dome differences are less extreme.\n\nTo assess if a random walk model is appropriate, for each time point $j$ and for orders 1-7, we computed residuals, in other words, the left hand side of Eq. 8, which should theoretically be distributed like $\\bm{\\epsilon_j}$ (i.e., independent $\\textrm{MVN}(0, \\Sigma)$ random variables). To compute $\\bm{X}_j$, we take the difference $\\bm{S}_j-\\bm{f}(\\bm{\\theta},j)$, where $\\bm{S}_j$ is the analytical glacial thickness solution to the SIA PDE at time $j$ (i.e., the real physical process for the purpose of this analysis), and $\\bm{f}(\\bm{\\theta},j)$ is the numerical glacial thickness solution to the SIA PDE at time $j$. We examine the residuals for two randomly selected grid points of the glacier (one at the interior and one at the margin) in Figures 4 and 5.\n\n\\begin{figure*}[h!]\n  \\centering\n    \\includegraphics[width=.6\\textwidth]{interior_RW_new.jpeg}\n\\caption{This figure displays residuals in units of meters (i.e., the term $\\bm{\\epsilon}_j$ in Eq. 8) for RW(q) of orders 1-7 for a randomly selected interior grid point. The first four panels display values on different scaled y-axes to better show the shapes, whereas the bottom four panels have the same scaling for the y-axis to be able to compare across the figures. RW(5) and above look like white noise processes, though RW(5) has the smallest variance. \n}\n\\end{figure*}\n\n\\begin{figure*}[h!]\n  \\centering\n    \\includegraphics[width=.6\\textwidth]{margin_RW_new.jpeg}\n\\caption{This figure also displays residuals in units of meters (i.e., the term $\\bm{\\epsilon}_j$ in Eq. 8) for RW(q) of orders 1-7 for a randomly selected margin grid point. Just like the previous figure, the first four panels display values on different scaled y-axes to better show the shapes, whereas the bottom four panels have the same scaling for the y-axis to be able to compare across the figures. Just as in the previous figure, RW(5) and above look like white noise processes, though RW(5) has the smallest variance.\n}\n\\end{figure*}\n\n\nA few important observations should be emphasized based on the empirical analysis displayed in these figures. The first is that a single order random walk substantially filters the discrepancy; for the interior grid point, it is reduced from the order of 10 m to the order of .01 m (1000 times reduction in magnitude), and for the margin grid point from the order of 100 m to .05 m (more than 1000 times reduction). Additionally, for both the interior and margin grid points, it appears that RW(5) is optimal in the sense that the residuals  closely resemble a white noise process and have the smallest variance. While the residuals from higher order RW processes also resemble white noise, the magnitude of the noise is larger. Nonetheless, we believe that real physical processes will not always be as smooth as the analytical SIA PDE solutions, and hence it is likely that a lower order RW process will be preferred for real scenarios. \n\n\\subsection{Reducing bias for the posterior distribution of $\\bm{\\theta}$}\nIn \\citet{discrepancy}, when prior information about the model discrepancy term is introduced in a simple physical system (i.e., a constrained GP over a space of functions), the bias of a posterior distribution of a physically relevant parameter reduces. We have found that a very similar phenomenon occurs in the glaciology test case, a result that was pointed out in \\citet{gopalan2018bayesian}. Specifically, in \\citet{Bueler}, it is shown that there is large spatial variation in the scale of deviations between the exact solution to the SIA and a numerical finite difference solver of the SIA. Specifically, there is spatial variation between the dome, interior, and margin of a glacier, with deviations at the margin being markedly larger than at the interior and dome. To investigate the effect of such prior information, we choose the matrix $\\Sigma$ to be such that it is block diagonal with 3 blocks, $\\Sigma_{int}$, $\\Sigma_{dome}$, and $\\Sigma_{margin} $. Each of these blocks is derived from a square exponential covariance kernel with the same length-scale parameter $\\bm{\\phi}$ = 70 km, but differing variance parameters $\\sigma^2_{int}$, $\\sigma^2_{dome}$, and  $\\sigma^2_{margin}$. If we ignore prior information from \\citet{Bueler}, we assume that there is an equal prior probability that each of $\\sigma^2_{int}$, $\\sigma^2_{dome}$, and $\\sigma^2_{margin}$ is in the set $\\{.1, 1, 10, 100\\}$ in units of $m^2$. If we use prior information from \\citet{Bueler}, we instead assume equal prior probability on $\\{.1,1\\}$ for $\\sigma^2_{int}$, $\\{1,10\\}$ for $\\sigma^2_{dome}$, and $\\{10,100\\}$ for $\\sigma^2_{margin}$ (again all units are $m^2)$. As shown in \\citet{gopalan2018bayesian}, the posterior for ice viscosity is less biased in the case that incorporates prior information for the scale of errors; this phenomenon is explored again in the next section.\n\nWhile in the above discussion we have not been precise about the term bias, the following ought to make this notion more rigorous. Let $\\bm{\\theta}_0$ be the true parameter, and $\\hat{\\bm{\\theta}}$ be an estimator of $\\bm{\\theta}_0$. The frequentist definition of bias is usually $E[\\hat{\\bm{\\theta}} - \\bm{\\theta}_0]$, where the expectation (i.e., average) is taken over the sampling distribution, $p(\\bm{Y}|\\bm{\\bm{\\theta}}_0)$. The Bayesian notion of bias used informally in the preceding paragraph (and essentially the same notion as in \\citet{discrepancy}) is $b(\\bm{Y},\\bm{\\bm{\\theta}}_0) = E[\\bm{\\bm{\\theta}} - \\bm{\\bm{\\theta}}_0]$, where the expectation (i.e., average) is taken with respect to the posterior distribution of $\\bm{\\bm{\\theta}}$, $p(\\bm{\\theta}|\\bm{Y})$. Consider $E[b(\\bm{Y},\\bm{\\bm{\\theta}}_0)]$, where the (outer) expectation is taken with respect to the sampling distribution. Then $E[b(\\bm{Y},\\bm{\\bm{\\theta}}_0)] = E[E[\\bm{\\bm{\\theta}}-\\bm{\\bm{\\theta}}_0]] = E[E[\\bm{\\bm{\\theta}}]-\\bm{\\bm{\\theta}}_0] = E[\\hat{\\bm{\\bm{\\theta}}}-\\bm{\\bm{\\theta}}_0]$, which is the frequentist bias. In other words, the frequentist bias is equivalent to the average of $b(\\bm{Y},\\bm{\\bm{\\theta}}_0)$ over the sampling distribution, if the posterior mean is chosen as an estimator. In the glaciology test case, we have (informally) not noticed much variability in the posterior for ice viscosity over repeated sampling of the data, and hence the distinction between Bayesian bias and frequentist bias is not significant.\n\nThe reader may wonder why a fixed $\\bm{\\theta}_0$ was assumed in the preceding paragraph, despite that a Bayesian model has been presented in this paper. In fact, it is typical to assume that the actual value of a parameter is fixed, despite ascribing a probability distribution to it in the form of a prior or posterior. Conceptually, such a probability distribution is a representation of a modeler's uncertainty regarding the fixed, unknown value of the parameter. For more on this interpretation of Bayesian statistics, the reader can consult results of statistical decision theory (e.g., on admissibility) in \\cite{lehmann2003theory} and \\cite{robert2007bayesian}. This viewpoint is also taken in Bayesian asymptotic analysis, such as the Bernstein-von Mises theorem \\citep{van2000asymptotic, shen2001}.\n\n\\subsection{Inferring $\\Sigma$}\n\nThe covariance matrix $\\Sigma$, first introduced after Equation 7 in Section 3, determines the spatial correlation inherent in the error-correcting process, $\\bm{X}$. Since spatial correlation in the error-correcting process is important to model (which is particularly evident in the glaciology example of \\citet{Bueler}), we need to discuss how $\\Sigma$ ought to be specified. Choosing $\\Sigma$ can be difficult if no or little prior information is available, and in such a case, we suggest:\n\n\\begin{eqnarray*}\n\\Sigma &=& \\textrm{diag}(\\bm{v})\\textrm{R} \\ \\textrm{diag}(\\bm{v}), \n\\end{eqnarray*}\nwhere $\\textrm{log}(\\bm{v}) \\sim \\textrm{MVN}(\\bm{\\mu}_v,\\Sigma_v)$,  $\\Sigma_v$ is derived from a GP kernel such as squared-exponential or Mat\\'{e}rn kernel, and $\\textrm{R}$ is a correlation matrix also derived from a GP kernel. To avoid non-identifiability and complexity of inference, it is suggested to pre-specify the parameters of these GP kernels.  This approach is similar to the modeling strategy employed in \\citet{doi:10.1002\/env.2343}. The intuition behind this approach is that the term $\\bm{v}$ encodes spatial variability in the scale of deviations between the output of a computer simulator and the true physical process, and spatial correlation in these deviations is strongly enforced with non-diagonal terms in both $\\Sigma_v$ and $\\textrm{R}$.\n\nFigure 6 illustrates a map of the mean posterior field for the variances of the error-correcting process, where the area of each circle is proportional to the inferred posterior mean of variance; due to a multivariate normal prior on $\\textrm{log}(\\bm{v})$, elliptical slice sampling is used as the method for posterior sampling \\citep{murray2010elliptical}. Consistent with \\citet{Bueler}, the variances tend to increase at the margins and are smaller at the interior. Additionally, the scaled differences between the analytical solution and numerical solver at the final time point the simulator is run (where scaling is inverse of the posterior mean of standard deviation) should theoretically approach a mean zero normal distribution according to the model. The p-value for an Anderson-Darling test is .436, suggesting that the scaled differences between the analytical solution and numerical solver are consistent with a normal distribution. Moreover, the sample mean for these scaled differences is .079 and the sample standard deviation is .409.\n\n\\begin{figure*}[h!]\n  \\centering\n    \\includegraphics[width=0.5\\textwidth]{post_var.jpeg}\n\\caption{Inferred posterior variance field of the error-correcting process, where the area of each circle is proportional to the variance at the grid point centered at the circle. Qualitatively, this field behaves as one would expect from the the work of \\citet{Bueler}, where the authors demonstrate that numerical inaccuracies for the SIA PDE are greatest toward the margin, but much smaller at the interior of the glacier.}\n\\end{figure*}\n\nAs is discussed in the previous subsection, prior information for $\\Sigma$ has an effect on the inference of physical parameters (i.e., ice viscosity), and in particular, a lack of prior information can lead to a very biased posterior distribution for physical parameters.  To compare the fitted $\\Sigma$ using a GP field against the $\\Sigma$ matrices discussed in the previous section, we show in Table 2 a comparison of posterior inference for the ice viscosity parameter for three choices of $\\Sigma$. The first choice of $\\Sigma$ is the posterior mean of samples assuming the structure $\\Sigma = \\textrm{diag}(\\bm{v})\\textrm{R} \\textrm{diag}(\\bm{v})$, with $\\textrm{log}(\\bm{v}) \\sim \\textrm{MVN}(\\bm{\\mu_v},\\Sigma_v)$. In the second and third scenarios, $\\Sigma$ is block diagonal with three variance parameters for each of the three blocks. A weakly informative case assumes that $\\sigma^2_{int} = \\sigma^2_{dome} = \\sigma^2_{margin} = .1$, whereas a more informative case (using prior information from \\citet{Bueler}) has $\\sigma^2_{int} = \\sigma^2_{dome} = .1$ and $\\sigma^2_{margin} =  10$ (all units are $m^2$). The scenario for weak prior information for $\\Sigma$ results in a very biased posterior distribution whose support does not cover the actual parameter value ($31.7 \\times 10^{-25}$ in units of $s^{-1}Pa^{-3}$) -- the maximum in this case is $26.5 \\times 10^{-25}$ in units of $s^{-1}Pa^{-3}$.  While the (absolute) biases of the posterior for ice viscosity for the GP field version compared to the prior information from \\citet{Bueler} are comparable (5.09 versus 4.01 in units of  $10^{-25} s^{-1}Pa^{-3}$), the posterior variance is markedly larger in the former case. This result suggests that prior knowledge from a domain expert is likely to be useful in determining $\\Sigma$, though in a case when that does not exist, the methodology described in this section is an adequate alternative.\n\n\\begin {table}[H]\n\\begin{center}\n\\begin{tabular}{ |c|c|c|c|c|c|c| }\n\\hline\nTest Case & Min & 1st Quartile & Median & Mean & 3rd Quartile & Max \\\\ \\hline\n$\\Sigma$ with GP field& $10.0$ & $21.0$  & $36.0$ & $35.7$ & $50.5$ & $70.0$\\\\ \\hline\n$\\Sigma$ with strong prior information & $18.0$ & $25.0$  & $26.5$ & $26.6$ & $28.0$ & $35.5$\\\\ \\hline\n$\\Sigma$ with weak prior information & $12.5$ & $18.5$  & $19.5$ & $19.5$ & $20.5$ & $26.5$\\\\ \\hline\n \\hline\n  \\end{tabular}\n\\caption{Summary statistics of $10^6$ posterior samples of the ice viscosity parameter under three versions of $\\Sigma$. While the weakly-informative case leads to a very biased posterior, the biases for the ice viscosity posterior in the first two $\\Sigma$ matrices are comparable. Nonetheless, the posterior variance is much less in the case with prior information from \\citet{Bueler}.}\n\\end{center}\n\\end {table}\n\n\n\\subsection{Exact versus approximate likelihood}\nIn Section 3.1, we showed an exact way to calculate the model likelihood as well as an approximation in Section 3.2. In this subsection, our purpose is to compare these two methods of likelihood computation in terms of run-time and posterior inference. Using a MacBook Pro early 2015 model with a 2.7 GHz Intel Core i5 processor and 8 GB 1867 MHz DDR3 memory (as before), one component of the log-likelihood approximation (which can be computed in an embarrassingly parallel fashion with the other components of the sum) takes .0179 s, whereas the full log-likelihood calculation, as in Section 3.1, is .354 seconds (in both cases, using a first-order emulator). The results of comparing posterior samples for the ice viscosity parameter are given in Table 3 -- thus, while the mean, median, first, and third quartiles are comparable, the approximate version has larger posterior uncertainty than the exact version as is evidenced by the wider tails. These results suggest that, while there is likely a computational speed-up afforded by using the approximation (i.e., at least an order of magnitude), the price to pay is increased posterior uncertainty.\n\\begin {table}[H]\n\\begin{center}\n\\begin{tabular}{ |c|c|c|c|c|c|c| }\n\\hline\nTest Case & Min & 1st Quartile & Median & Mean & 3rd Quartile & Max \\\\ \\hline\nExact likelihood & $15.0$ & $26.5$  & $27.0$ & $27.4$ & $29.0$ & $39.5$\\\\ \\hline\nLikelihood approximation & $10.0$ & $26.0$  & $28.0$ & $27.7$ & $31.0$ & $52.5$\\\\ \\hline\n \\hline\n  \\end{tabular}\n\\caption{Summary statistics of $10^6$ posterior samples of the ice viscosity parameter using an exact likelihood and a likelihood approximation (units are in $10^{-25} s^{-1}Pa^{-3}$). While the 1st quartile, median, mean, and 3rd quartile are similar, the tails in the approximation are much wider.}\n\\end{center}\n\\end {table}\n\n\\section{Generality of the model and methodology}\nThough we have tested the model and methodology in the previous sections in the context of a glaciology example, it should be noted that they can be used in other physical systems with similar components. In essence, this modeling and methodology can be applied in scenarios where:\n\\begin{enumerate}\n\\item A computer program (i.e., \\textit{computer simulator}) is available to simulate a continuous physical process through space and time, but there is a deviation between the output of the computer simulator and the actual physical process.\n\\item The deviations between the computer simulator output and the actual physical process values tend to grow with time and exhibit spatial correlation structure.\n\\item Measurements of the physical process are available, but they are potentially scarce both in space and time.\n\\item Physical parameters governing the physical process are uncertain but can be constrained with domain knowledge for the random walk error covariance (i.e., $\\Sigma$).\n\\end{enumerate}\nRecall that at the process level, the model stipulates that:\n\\begin{eqnarray}\n\\bm{S}_{j} &=& \\bm{f}(\\bm{\\theta},\\bm{\\phi},j) + \\bm{X}_j.\n\\end{eqnarray}\nTo apply the same setup to another physical scenario, a different version of $\\bm{f}(.,.,.)$, such as a numerical PDE solver for another system of spatio-temporal PDEs besides the SIA, can be used. However, while $\\bm{f}(.,.,.)$ will need to be tailored to another physical scenario based on a different numerical scheme or physical model, the $\\bm{X}_j$ term would be modeled in the same way (i.e., with a random walk).\n\n\\section{Conclusion}\nThe objective of this work has been to set forth a versatile physical-statistical model in the Bayesian hierarchical framework that incorporates a computer simulator for a physical process, such as a numerical solver for a system of PDEs. Posterior inference for physical parameters (and, consequently, posterior predictions of the physical process) can be computationally demanding within this model, since each evaluation of the likelihood requires a full PDE solve and computing the inverse and determinant of a large covariance matrix. Therefore, we have set forth two main ways to speed up computation: first is the use of bandwidth limited linear algebra in a manner similar to \\citet{rue2001fast} for quickly handling the covariance matrix in the likelihood, and the second is the use of spatio-temporal emulation in a manner similar to \\citet{Hooten2011} to emulate a PDE solver that is expensive to evaluate. An additional method for speeding up computation is to approximate the likelihood in a way that leads to embarrassingly parallel computation. The utility of this model and corresponding inference methodology is demonstrated with a test example from glaciology.\n\nA unique feature of this work is how we represent the discrepancy between a computer simulator for a physical process and the real physical process values. One approach, as in  \\citet{kennedy2001bayesian} and \\citet{discrepancy}, is to assume that this is a fixed yet unknown function that can be learned with a GP (or constrained GP) prior distribution over a space of functions. Instead, we assume that this discrepancy is a spatio-temporal stochastic process (i.e., a random walk), which is motivated by the fact that a computer simulation is likely to become less accurate as it is run forward in time, as well as exhibit some degree of spatial correlation in inaccuracies. An interesting consequence of this modeling decision is that linear algebraic routines for band-limited matrices can be utilized for evaluating the likelihood of the model in an efficient manner. Another interesting artifact of this approach is that when prior information is used for the random walk's error term (i.e., in $\\Sigma$), the bias for the posterior distribution of $\\bm{\\theta}$ is reduced. The same phenomenon is exhibited in the work of \\citet{discrepancy}, where a constrained GP prior over a space of functions ends up reducing the bias of the physical parameter posterior distribution. \n\nDespite that the model and methodology appear to perform well in the analysis of this paper, it is important to comment on some potential drawbacks of the approach, particularly when applied to other physical contexts. In this paper, emulation works adequately with a single parameter, though emulators do not always work well in other applications or higher dimensional parameter spaces. For example, \\cite{doi:10.1080\/01621459.2018.1514306} document some shortcomings of a principal components based emulator in climate modeling. The second main computational advantages stem from log-likelihood evaluation speed-ups. The use of bandwidth limited matrix algebra for the exact log-likelihood can be used so long as the model holds, which may not always be the case (e.g., with a non-Gaussian data distribution). Additionally, the log-likelihood approximation holds when the measurement errors are small relative to the signal modeled, which depends on the measurement instruments used to collect the data. For instance, on common geophysical scales of thousands of meters, light detection and ranging (LIDAR) or digital-GPS data have maximum errors on the order of a meter.\n\nAdditionally, if it is not possible to program the computer simulator to produce output at the data measurement locations, there are essentially two main ways to handle such a scenario. The first is to use spatial kriging to predict the value of the computer simulator at the spatial locations where data are collected, given the output of the computer simulator at the grid points. A simpler approach is to use inverse-distance weighting of the simulator output at the nearest neighbors; that is, take a weighted average of the four nearest grid points of the simulator, where the weights are proportional to the inverse of distance. Such an approach, for example, has been used in \\citet{doi:10.1002\/env.2343}.\n\nFuture research will include predicting Langj\\\"{o}kull glacier surface elevation using the modeling and methodology within this paper, based on actual data collected by the UI-IES. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nExpander graphs are now ubiquitous in both mathematics and computer science. The problem of explicitly constructing these highly connected sparse graphs has drawn the attention of researchers from across both disciplines, who have uncovered deep and surprising connections to topics as diverse as Kazhdan's property (T) and the Ramanujan conjecture. Their usefulness has been known to computer scientists for some time, who have applied them to complexity theory, derandomisation, coding theory, cryptography and more, but they are now seeing increasing use in disparate areas of mathematics. We refer the interested reader to the excellent surveys~\\cite{HLW06} and~\\cite{L12} for further information. \n\nGiven these successes, there has been a strong push in recent years towards defining and constructing high-dimensional, or hypergraph, expanders. There has already been a great deal of interesting work in this area (see, for example, the survey~\\cite{L14}), but much more remains to be done. In particular, there are only a small number of examples known which satisfy the strongest notions of expansion. In the bounded-degree case, there is essentially only one such construction~\\cite{EK17, KKL16}, arising from the so-called Ramanujan complexes~\\cite{LSV05}, defined in analogy to Ramanujan graphs~\\cite{LPS88, M88} as finite quotients of certain affine buildings. The main result of this paper is a comparatively simple mechanism for constructing $3$-uniform expanders of low degree which satisfy most, and perhaps even all, of the expansion properties discussed in the literature. To say more, we first describe the mechanism.\n\nLet $S$ be a subset of the finite abelian group $\\mathbb{Z}_2^t$. We then let $H \\coloneqq H(\\mathbb{Z}_2^t, S)$ be the $3$-uniform hypergraph with vertex set $\\mathbb{Z}_2^t$ and edge set consisting of all triples of the form $(x + s_1, x+ s_2, x+ s_3)$, where $x \\in \\mathbb{Z}_2^t$ and $s_1, s_2, s_3$ are distinct elements of $S$.  A useful alternative perspective on $H$ is to consider the Cayley graph Cay$(\\mathbb{Z}_2^t, S')$, where $S'$ is the set $\\{s_1 + s_2 : s_1, s_2 \\in S, s_1 \\neq s_2\\}$. This is the graph with vertex set $\\mathbb{Z}_2^t$ where $x, y \\in \\mathbb{Z}_2^t$ are joined if and only if $x + y \\in S'$. Then $H$ is the $3$-uniform hypergraph on the same vertex set whose triples are the triangles of Cay$(\\mathbb{Z}_2^t, S')$. Note that if $S$ contains no non-trivial solutions to the equation $s_1 + s_2 = s'_1 + s'_2$, then every vertex in $H$ is contained in exactly $3\\binom{|S|}{3}$ edges and every pair of vertices is contained in either $0$ or $2|S| - 4$ edges.\n\nWe will see that this hypergraph inherits its expansion properties from the Cayley graph Cay$(\\mathbb{Z}_2^t, S)$. However, when $|S| < t$, the Cayley graph Cay$(\\mathbb{Z}_2^t, S)$ is not even connected, showing that $|S|$ will have to be at least logarithmic in the number of vertices. On the other hand, a celebrated result of Alon and Roichman~\\cite{AR94} shows that logarithmic size will also suffice, in that if $S$ is a randomly chosen subset of $\\mathbb{Z}_2^t$ of size $C t$, for $C$ sufficiently large, then Cay$(\\mathbb{Z}_2^t, S)$ will, with high probability as $t \\rightarrow \\infty$, be an expander. The set $S$ may also be chosen explicitly, but the fact that a random choice works partially addresses a question raised repeatedly in the literature~\\cite{EK17, P14, P17} as to whether there are random models for sparse high-dimensional expanders. \n\nSince random Cayley graphs over many other groups are known to have much better expansion properties (see~\\cite{BGGT15} and its references), one might ask why we use $\\mathbb{Z}_2^t$. To see why, we define, for each $x \\in \\mathbb{Z}_2^t$, the set $C_x := \\{x + s : s \\in S\\}$. The hypergraph $H$ is then a union of $3$-uniform cliques, one on each set $C_x$. The key observation about $H$, and the reason why it serves as a hypergraph expander, is that each edge in Cay$(\\mathbb{Z}_2^t, S')$ appears in at least two different sets of the form $C_x$. More specifically, if $(x, x + s_1 + s_2)$ is an edge of Cay$(\\mathbb{Z}_2^t, S')$, then this edge is contained in both $C_{x+s_1}$ and $C_{x+s_2}$. This property, for which the fact that $\\mathbb{Z}_2^t$ is abelian is crucial, means, for instance, that a random walk on the edges of $H$ never becomes trapped inside a particular set $C_x$.\n\nTo say more about the notions of expansion satisfied by our construction, we require some notation. Given a $3$-uniform hypergraph $H$, we let $E$ be the collection of pairs of distinct vertices $(u, v)$ for which there exists an edge $(u, v, w)$ of $H$ containing $u$ and $v$. The corresponding graph will be referred to as the {\\it skeleton} of $H$. To distinguish the edges of $H$ from the set $E$, we will write $T$ for these edges and refer to them as the triples of $H$, reserving the term edges for the elements of $E$.\n\nFor any subset $F$ of the skeleton $E$ of a hypergraph $H$, we may define two notions of neighbourhood, the \\emph{edge neighbourhood $N_E(F)$} and the \\emph{triple neighbourhood $N_T(F)$}, by\n\\[N_E(F) = \\{e \\in E \\setminus F: e \\cup f \\in T \\textrm{ for some } f \\in F\\}\\]\nand, writing $\\Delta(F)$ for the set of triangles in $F$,\n\\[N_T(F) = \\{t \\in T: t \\supset f \\textrm{ for some } f \\in F \\textrm{ and } t \\notin \\Delta(F)\\}.\\]\nWe then let\n\\[h_E(H) = \\min_{\\{F : |F| \\leq |E|\/2\\}} \\frac{|N_E(F)|}{|F|} \\textrm{ and } h_T(H) = \\min_{\\{F : |F| \\leq |E|\/2\\}} \\frac{|N_T(F)|}{|F|}\\]\nand say that $H$ is an {\\it $\\epsilon$-edge-expander} if $h_E(H) \\geq \\epsilon$ and an {\\it $\\epsilon$-triple-expander} if $h_T(H) \\geq \\epsilon$. Note that these notions are the direct analogues of the standard notions of vertex and edge expansion in graphs. Our main result is that under suitable conditions on Cay$(\\mathbb{Z}_2^t, S)$, the hypergraph $H(\\mathbb{Z}_2^t, S)$ is an $\\epsilon$-edge-expander and an $\\epsilon |S|$-triple-expander for some $\\epsilon > 0$. \n\nTo state the result formally, recall that the eigenvalues of an $n$-vertex graph $G$ are the eigenvalues $\\lambda_1 \\geq \\dots \\geq \\lambda_n$ of its adjacency matrix $A$. When $G$ is $d$-regular, $\\lambda_1 = d$. One of the key facts about $d$-regular expander graphs is that if $\\lambda(G) = \\max_{i \\neq 1} |\\lambda_i|$ is bounded away from $d$, then the graph $G$ exhibits strong expansion properties. We show that this behaviour carries over to the derived hypergraph $H$.\n\n\\begin{theorem} \\label{thm:main}\nSuppose that $S \\subseteq \\mathbb{Z}_2^t$ contains no non-trivial solutions to the equation $s_1 + s_2 = s'_1 + s'_2$ and $\\lambda(Cay(\\mathbb{Z}_2^t, S)) \\leq (1 - \\epsilon)|S|$. Then the hypergraph $H(\\mathbb{Z}_2^t, S)$ is an $\\frac{\\epsilon^2}{128}$-edge-expander and an $\\frac{\\epsilon^2}{64} |S|$-triple-expander.\n\\end{theorem}\n\nBy the Alon--Roichman theorem~\\cite{AR94}, a random set $S \\subseteq \\mathbb{Z}_2^t$ of order $Ct$, for $C$ a sufficiently large constant, will a.a.s.~have the required properties with $\\epsilon = 1\/2$. Therefore, writing $n = |\\mathbb{Z}_2^t|$, the theorem implies there exists a hypergraph with $n$ vertices and $O(n \\log^3 n)$ triples which is a $2^{-9}$-edge-expander and a $(2^{-8} \\log n)$-triple-expander. \n\nGetting back to random walks, we follow Kaufman and Mass~\\cite{KM16, KM162} in defining the {\\it random walk} on a $3$-uniform hypergraph $H$ to be a sequence of edges $e_0, e_1, \\dots \\in E$ such that\n\n\\begin{enumerate}\n\n\\item\n$e_0$ is chosen from some initial probability distribution $\\mathbf{p}_0$ on $E$,\n\n\\item\nfor every $i \\geq 1$, $e_i$ is chosen uniformly at random from the neighbours of $e_{i-1}$, that is, the set of $f \\in E$ such that $e_{i-1} \\cup f$ is an edge of $H$.\n\n\\end{enumerate}\n\nWe say that the random walk is {\\it $\\alpha$-rapidly mixing} if, for any initial probability distribution $\\mathbf{p}_0$ and any $i \\in \\mathbb{N}$,\n\\[\\|\\mathbf{p}_i - \\mathbf{u}\\|_2 \\leq \\alpha^i,\\]\nwhere $\\mathbf{p}_i$ is the probability distribution on $E$ after $i$ steps of the walk, $\\mathbf{u}$ is the uniform distribution on $E$ and $\\|\\mathbf{x}\\|_2 = (\\sum_i x_i^2)^{1\/2}$ for $\\mathbf{x} = (x_1, x_2, \\dots, x_n) \\in \\mathbb{R}^n$. As a corollary of Theorem~\\ref{thm:main}, we can show that under appropriate conditions on Cay$(\\mathbb{Z}_2^t, S)$ the hypergraph $H(\\mathbb{Z}_2^t, S)$ is $\\alpha$-rapidly mixing for some $\\alpha < 1$. \n \n\\begin{corollary} \\label{cor:mixing}\nSuppose that $S \\subseteq \\mathbb{Z}_2^t$ contains no non-trivial solutions to the equation $s_1 + s_2 = s'_1 + s'_2$ and $\\lambda(Cay(\\mathbb{Z}_2^t, S)) \\leq (1 - \\epsilon)|S|$.\nThen $H(\\mathbb{Z}_2^t, S)$ is $\\alpha$-rapidly mixing with $\\alpha = 1 - \\Omega(\\epsilon^4)$.\n\\end{corollary}\n\nThe hypergraph $H(\\mathbb{Z}_2^t, S)$ also satisfies several other properties, one notable example being the following pseudorandomness condition.\n\n\\begin{theorem} \\label{thm:pseudo}\nSuppose that $S_t \\subseteq \\mathbb{Z}_2^t$ is a sequence of sets with $t \\rightarrow \\infty$, where $S_t$ contains no non-trivial solutions to the equation $s_1 + s_2 = s'_1 + s'_2$ and $\\lambda(Cay(\\mathbb{Z}_2^{t}, S_t)) = o(|S_t|)$. Then, for any sets $A, B, C \\subseteq \\mathbb{Z}_2^{t}$ with $|A| = \\alpha |\\mathbb{Z}_2^{t}|$, $|B| = \\beta |\\mathbb{Z}_2^{t}|$ and $|C| = \\gamma |\\mathbb{Z}_2^{t}|$, where $\\alpha, \\beta$ and $\\gamma$ are fixed positive constants, the number of triples $T(A, B, C)$ in $H(\\mathbb{Z}_2^{t}, S_t)$ with one vertex in each of $A, B$ and $C$ is $(1 + o(1)) \\alpha \\beta \\gamma |S_t|^3 |\\mathbb{Z}_2^{t}|$.\n\\end{theorem}\n\nThat is, provided $\\lambda(Cay(\\mathbb{Z}_2^{t}, S))$ is small, the density of edges between any three large vertex subsets $A$, $B$ and $C$ in $H(\\mathbb{Z}_2^t, S)$ is asymptotic to the expected value, as it would be in a random hypergraph of the same density. It is worth noting that this pseudorandomness property is not known to hold in Ramanujan triangle complexes. Though a result of Parzanchevski~\\cite{P17} (see also~\\cite{PRT16}) says that a sequence of hypergraphs with good enough spectral properties will satisfy this condition, the spectrum of Ramanujan complexes~\\cite{GP14} is not sufficiently well-behaved for this result to apply.\n\nAs noted in~\\cite{P17, PRT16}, any sequence of hypergraphs satisfying the conclusion of Theorem~\\ref{thm:pseudo} will also satisfy Gromov's geometric overlap property~\\cite{G10}. We say that a $3$-uniform hypergraph $H$ has the {\\it $c$-geometric overlap property} if, for every embedding $\\varphi: V(H) \\rightarrow \\mathbb{R}^2$ of the vertices of $H$ in the plane, there is a point $x \\in \\mathbb{R}^2$ which is contained in the convex hull of at least a $c$-proportion of the triples of $H$. A result of Boros and F\\\"uredi~\\cite{BF84} (which was generalised to higher dimensions by B\\'ar\\'any~\\cite{B82}) says that the family of complete $3$-uniform hypergraphs has the geometric overlap property with $c = \\frac{2}{9} - o(1)$ and this constant is known to be sharp~\\cite{BMN10}. Much more recently, answering a question of Gromov~\\cite{G10}, Fox, Gromov, Lafforgue, Naor and Pach~\\cite{FGLNP12} found a number of families of bounded-degree hypergraphs with the geometric overlap property. Indeed, one of their constructions~\\cite[Section 4.1]{FGLNP12} is a close relative of ours and satisfies a version of Theorem~\\ref{thm:pseudo}, but lacks the intersection property between the sets $C_x$ which is necessary to guarantee the properties encapsulated in Theorem~\\ref{thm:main}. \n\nTo see how the geometric overlap property follows from the conclusion of Theorem~\\ref{thm:pseudo}, we recall Pach's selection theorem~\\cite{P98}, which gives a stronger guarantee than the Boros--F\\\"uredi result, saying that for any set of $n$ points in the plane there exist three sets $A$, $B$ and $C$, each of order at least $cn$, and a point $x \\in \\mathbb{R}^2$ such that every triangle $abc$ with $a \\in A$, $b \\in B$ and $c \\in C$ has $x$ in its convex hull. Suppose now that $\\varphi : V(H) \\rightarrow \\mathbb{R}^2$ is an embedding of the vertices of $H(\\mathbb{Z}_2^t, S_t)$ in the plane and let $A$, $B$ and $C$ be the sets and $x$ the point guaranteed by applying Pach's theorem to this point set. By Theorem~\\ref{thm:pseudo}, the number of triples of $H$ with one vertex in each of $A$, $B$ and $C$ is itself a positive proportion of the total number of triples in $H$. Since each one of these triples has $x$ in its convex hull, we have deduced the following corollary.\n\n\\begin{corollary} \\label{cor:geom}\nSuppose that $S_t \\subseteq \\mathbb{Z}_2^t$ is a sequence of sets with $t \\rightarrow \\infty$, where $S_t$ contains no non-trivial solutions to the equation $s_1 + s_2 = s'_1 + s'_2$ and $\\lambda(Cay(\\mathbb{Z}_2^{t}, S_t)) = o(|S_t|)$. Then the family of $3$-uniform hypergraphs $H(\\mathbb{Z}_2^t, S_t)$ has the $(c - o(1))$-geometric overlap property for some $c > 0$.\n\\end{corollary}\n\nAs in~\\cite{FGLNP12}, we can also recover the sharp constant $c = \\frac{2}{9}$ by following Bukh's proof~\\cite{B06} of the Boros--F\\\"uredi result. However, we omit the proof of this result, referring the reader instead to~\\cite{FGLNP12}. \n\nIt remains an open problem to determine whether an analogue of Corollary~\\ref{cor:geom} holds for the stronger topological overlap property. We say that a $3$-uniform hypergraph $H$ has the {\\it $c$-topological overlap property} if, for every continuous map $\\varphi: X \\rightarrow \\mathbb{R}^2$ from the simplicial complex $X = (V, E, T)$ of the hypergraph $H$ to the plane, there is a point $x \\in \\mathbb{R}^2$ which is contained in the image of at least a $c$-proportion of the triples of $H$. Gromov~\\cite{G10} generalised the Boros--F\\\"uredi result (and B\\'ar\\'any's result), showing that the family of complete $3$-uniform hypergraphs has the topological overlap property with $c = \\frac{2}{9} - o(1)$. The difficult problem of constructing $3$-uniform hypergraphs of bounded degree with the topological overlap property was only solved recently by Kaufman, Kazhdan and Lubotzky~\\cite{KKL16} and then extended to higher uniformities by Evra and Kaufman~\\cite{EK17}. Their work relies on the properties of Ramanujan complexes, but we conjecture that our construction gives another simpler example, albeit one with polylogarithmic rather than constant degree.\n\n\\section{Proofs}\n\nWe begin with a simple lemma relating the eigenvalues of Cay$(\\mathbb{Z}_2^t, S')$ to those of Cay$(\\mathbb{Z}_2^t, S)$. In particular, this means that the skeleton Cay$(\\mathbb{Z}_2^t, S')$ of our hypergraph $H(\\mathbb{Z}_2^t, S)$ is an expander whenever Cay$(\\mathbb{Z}_2^t, S)$ is. Here and throughout this section, we will use the shorthands $n = |\\mathbb{Z}_2^t|$ and $d = |S|$.\n\n\\begin{lemma} \\label{lem:eig}\nSuppose that $S \\subseteq \\mathbb{Z}_2^t$ contains no non-trivial solutions to the equation $s_1 + s_2 = s'_1 + s'_2$ and that the eigenvalues of the Cayley graph Cay$(\\mathbb{Z}_2^t, S)$ are $\\lambda_i$ for $i = 1, \\dots, n$. Then the eigenvalues of the Cayley graph Cay$(\\mathbb{Z}_2^t, S')$, where $S' = \\{s_1 + s_2 : s_1, s_2 \\in S, s_1 \\neq s_2\\}$, are $\\frac{1}{2}(\\lambda_i^2 - d)$ for $i = 1, \\dots, n$.\n\\end{lemma}\n\n\\begin{proof}\nLet $A$ be the adjacency matrix of Cay$(\\mathbb{Z}_2^t, S)$. It will suffice to show that the adjacency matrix of Cay$(\\mathbb{Z}_2^t, S')$ is $\\frac{1}{2}(A^2 - d I)$. To see this, note that $A^2_{xy}$ is the number of solutions to $x + y = s_1 + s_2$ with $s_1, s_2 \\in S$. When $x \\neq y$, the assumption that there are no non-trivial solutions to $s_1 + s_2 = s'_1 + s'_2$ tells us that $A^2_{xy} = 2$ or $0$ depending on whether or not $x + y$ are joined in Cay$(\\mathbb{Z}_2^t, S')$ or not. When $x = y$, $A_{xx}^2 = d$, corresponding to the $d$ solutions $x + x = s + s = 0$ for all $s \\in S$. The result follows. \n\\end{proof}\n\nGiven two multisets $V$ and $W$ taken from the vertex set of a graph with edge set $E$, we write $e(V,W)$ to denote $\\sum_{v \\in V, w \\in W} 1_E(vw)$. We will also use $v_x$ and $w_y$ to denote the multiplicity of $x$ in $V$ and $y$ in $W$, respectively. In what follows, we will need a slight variant of the expander mixing lemma which applies to multisets. Since the proof is identical to the usual expander mixing lemma, we omit it.\n\n\\begin{lemma}[Expander mixing lemma] \\label{lem:eml}\nSuppose that $G$ is an $(n, d, \\lambda)$-graph, that is, $G$ has $n$ vertices of degree $d$ and all eigenvalues, save the largest, have absolute value at most $\\lambda$. Then, for any two multisets $V, W \\subseteq V(G)$,\n\\[\\left|e(V, W) - \\frac{d}{n} |V||W|\\right| \\leq \\lambda \\sqrt{\\left(\\sum_{x \\in V} v_x^2 - \\frac{|V|^2}{n}\\right) \\left(\\sum_{y \\in W} w_y^2 - \\frac{|W|^2}{n}\\right)}.\\]\n\\end{lemma}\n\nWe are already in a position to show that $H(\\mathbb{Z}_2^t, S)$ satisfies the pseudorandomness property encapsulated in Theorem~\\ref{thm:pseudo}. This is a simple corollary of the following more precise result.\n\n\\begin{theorem} \\label{lem:comb}\nSuppose that $S \\subseteq \\mathbb{Z}_2^t$ contains no non-trivial solutions to the equation $s_1 + s_2 = s'_1 + s'_2$ and the eigenvalues of the Cayley graph Cay$(\\mathbb{Z}_2^t, S)$ satisfy $|\\lambda_i| \\leq \\lambda$ for all $i = 2, \\dots, n$. Then, for any sets $A, B, C \\subseteq \\mathbb{Z}_2^t$ with $|A| = \\alpha n$, $|B| = \\beta n$ and $|C| = \\gamma n$, the number of triples $e(A, B, C)$ in $H(\\mathbb{Z}_2^t, S)$ with one vertex in each of $A, B$ and $C$ is\n\\[(d^3 - d^2) \\alpha \\beta \\gamma n \\pm 2 \\mu d \\sqrt{\\alpha \\beta} \\gamma n \\pm \\lambda d^2 \\sqrt{\\alpha \\beta \\gamma} n \\pm \\lambda \\sqrt{\\mu} d (\\alpha \\beta)^{1\/4} \\sqrt{\\gamma} n,\\]\nwhere $\\mu = \\frac{1}{2} (\\lambda^2 + d)$.\n\\end{theorem}\n\n\\begin{proof}\nSince Cay$(G, S')$ is $\\binom{d}{2}$-regular and, by Lemma~\\ref{lem:eig}, has all eigenvalues, except the largest, at most $\\mu = \\frac{1}{2}(\\lambda^2 + d)$ in absolute value, the number of pairs in the skeleton of $H(\\mathbb{Z}_2^t, S)$ that have one vertex in $A$ and one vertex in $B$ is, by the expander mixing lemma,\n\\begin{equation} \\label{eqn:comexp1} \n\\binom{d}{2} \\alpha \\beta n \\pm \\mu \\sqrt{\\alpha \\beta} n.\n\\end{equation}\nGiven this set of edges $E(A, B)$, let $W(A, B)$ be the multiset of corresponding centres, that is, $w$ appears in $W(A, B)$ once for each edge $e \\in E(A, B)$ in the induced subgraph of Cay$(G, S')$ on $C_w$. We claim that $|W(A, B)| = 2 |E(A, B)|$. To see this, write $e = (u, v)$ and note that if $e \\subset C_{w_1}, C_{w_2}$, then $u = w_1 + s_1$, $v = w_1 + s_2$, $u = w_2 + s'_1$ and $v = w_2 + s'_2$ for $s_1, s_2, s'_1, s'_2 \\in S$. This implies that $s_1 + s_2 = s'_1 + s'_2$, but since there are no non-trivial solutions to this equation, we must have $s'_1 = s_2$ and $s'_2 = s_1$. Therefore, $e$ is contained in at most two sets of the form $C_w$. To see that it is exactly two, note that $e \\subset C_{u + s_1}, C_{u + s_2}$. \n\nThe number of triples containing a pair from $E(A, B)$ and a vertex from $C$ is now the number of edges between $W := W(A, B)$ and $C$ in the graph $S$. Therefore, by Lemma~\\ref{lem:eml},\n\\begin{equation} \\label{eqn:comexp2}\ne(A, B, C) = e(W, C) = \\frac{d}{n} |W||C| \\pm \\lambda \\sqrt{\\sum_{x \\in W} w_x^2 |C|} .\n\\end{equation}\nSince $w_x \\leq \\binom{d}{2}$ for all $x$, we have \n$$\\sum_{x \\in W} w_x^2 \\leq \\binom{d}{2} \\sum_x w_x \\leq \\binom{d}{2} |W| \\leq d^2 |E(A,B)|.$$ \nSubstituting \\eqref{eqn:comexp1} into \\eqref{eqn:comexp2} using this fact yields the required result.\n\\end{proof}\n\nWe now prove our main theorem, Theorem~\\ref{thm:main}, beginning with the triple expansion property. We will make use of the following result of de Caen~\\cite{dC98} putting an upper bound on the sum of the squares of the degrees of a graph.\n\n\\begin{lemma} \\label{lem:deC}\nFor any graph with $n$ vertices, $m$ edges and vertex degrees $d_1, d_2, \\dots, d_n$, \n\\[\\sum_{i=1}^n d_i^2 \\leq m \\left(\\frac{2m}{n-1} + (n-2)\\right).\\]\n\\end{lemma}\n\n\\begin{theorem} \\label{thm:tripexp}\nSuppose that $S \\subseteq \\mathbb{Z}_2^t$ contains no non-trivial solutions to the equation $s_1 + s_2 = s'_1 + s'_2$ and $\\lambda(\\textrm{Cay}(\\mathbb{Z}_2^t, S)) \\leq (1 - \\epsilon)d$. Then, for all subsets $F$ of the skeleton $E$ of $H(\\mathbb{Z}_2^t, S)$ with $|F| \\leq |E|\/2$, $|N_T(F)| \\geq \\frac{\\epsilon^2}{64} d|F|$.\n\\end{theorem}\n\n\\begin{proof}\nFor each $x$, let $F_x = \\{e \\in F : e \\subset C_x\\}$. As in the proof of Lemma~\\ref{lem:comb}, the assumption that $S$ contains no non-trivial solutions to the equation $s_1 + s_2 = s'_1 + s'_2$ implies that each edge in $F$ is contained in precisely two sets $C_x$. Therefore, $\\sum_{x \\in \\mathbb{Z}_2^t} |F_x| = 2 |F|$. Suppose now that $X = \\{x \\in \\mathbb{Z}_2^t : |F_x| \\geq (1 - \\delta) \\binom{d}{2}\\}$, where $\\delta = \\frac{\\epsilon}{8}$. We claim that $\\sum_{x \\in X^c} |F_x| \\geq \\frac{\\epsilon}{4}|F|$.\n\nTo prove the claim, we may clearly assume that $\\sum_{x \\in X} |F_x| \\geq |F|$, for otherwise, \n$$\\sum_{x \\in X^c} |F_x| \\geq 2|F| - \\sum_{x \\in X} |F_x| \\geq |F|.$$  \nConsider now the number of edges $N(X, X^c)$ in Cay$(G, S')$ between $X$ and its complement $X^c$. By Lemma~\\ref{lem:eig} and the expander mixing lemma, this number is at least\n\\[\\binom{d}{2}\\frac{|X||X^c|}{n} - \\frac{1}{2} (\\lambda^2 - d) \\frac{|X||X^c|}{n} = \\frac{1}{2} (d^2 - \\lambda^2) \\frac{|X||X^c|}{n}\\]\nfor some $\\lambda \\leq (1 - \\epsilon)d$. Suppose now that $x \\in X$ and $x + s_1 + s_2 \\in X^c$. If $e = (x + s_1, x + s_2)$ is in $F$, we see that $e \\in F_{x + s_1 + s_2}$, contributing one to $\\sum_{x \\in X^c} |F_x|$. If it is not in $F$, it contributes one to $\\sum_{x \\in X} |F_x^c|$. Therefore,\n\\[\\sum_{x \\in X^c} |F_x| + \\sum_{x \\in X} |F_x^c| \\geq N(X, X^c)\\]\nand, since $|F_x^c| \\leq \\delta \\binom{d}{2}$ for each $x \\in X$,\n\\[\\sum_{x \\in X^c} |F_x| \\geq N(X, X^c) - \\delta \\binom{d}{2}|X| \\geq \\frac{1}{2} (d^2 - \\lambda^2) \\frac{|X||X^c|}{n} - \\delta \\binom{d}{2}|X|.\\]\nNote now, since $|F| \\leq \\frac{1}{2} |E| \\leq \\frac{1}{4} \\binom{d}{2} n$, that \n\\[|X| \\leq \\frac{2|F|}{(1 - \\delta) \\binom{d}{2}} \\leq  \\frac{n}{2(1 - \\delta)} \\leq (1 + 2\\delta)\\frac{n}{2},\\]\nso $|X^c| \\geq (1 - 2\\delta)\\frac{n}{2}$. Therefore, since $\\lambda^2 \\leq (1 - \\epsilon) d^2$ and $\\delta = \\frac{\\epsilon}{8}$,\n\\begin{align*}\n\\sum_{x \\in X^c} |F_x| & \\geq \\frac{1}{2} (d^2 - \\lambda^2) \\frac{|X||X^c|}{n} - \\delta \\binom{d}{2}|X| \\\\\n& \\geq \\frac{1}{4} \\left((d^2 - \\lambda^2) (1 - 2\\delta) - 2 \\delta d^2\\right) |X| \\\\\n& \\geq \\frac{1}{4} \\left(\\epsilon (1 - 2 \\delta) - 2 \\delta\\right) d^2 |X| \\geq \\frac{\\epsilon}{8} d^2 |X|.\n\\end{align*}\nSince $\\binom{d}{2}|X| \\geq \\sum_{x \\in X}|F_x| \\geq |F|$, the required claim, that $\\sum_{x \\in X^c} |F_x| \\geq \\frac{\\epsilon}{4}|F|$, now follows.\n\nGiven $x \\in X^c$, let $N_T(F_x)$ denote the set of triples $t \\subset C_x$ such that $f \\subset t$ for some $f \\in F_x$ and $t \\notin \\Delta(F_x)$. Then\n\\[|N_T(F_x)| = \\frac{1}{2} \\sum_{y \\in C_x} d(y) (d - 1 - d(y)),\\]\nwhere $d(y)$ is the degree of $y$ in the graph whose edges are $F_x$. This is because $N_T(F_x)$ includes every triple which contains an edge of both $F_x$ and $E_x \\setminus F_x$ and the factor of $1\/2$ accounts for the fact that we will include any admissible triple twice in this count. Now, by Lemma~\\ref{lem:deC},\n\\[\\sum_{y \\in C_x} d^2(y) \\leq |F_x| \\left( \\frac{2 |F_x|}{d-1} + d - 2\\right),\\]\nso that, since $\\sum_{y \\in C_x} d(y) = 2|F_x|$,\n\\[|N_T(F_x)| \\geq (d-1) |F_x| - \\frac{|F_x|^2}{d-1} - \\frac{1}{2} (d-2) |F_x| = \\frac{d}{2} |F_x| - \\frac{|F_x|^2}{d-1}.\\]\nTherefore, since $|F_x| \\leq (1 - \\delta) \\binom{d}{2}$ for $x \\in X^c$, $|N_T(F_x)| \\geq \\frac{\\delta}{2} d |F_x|$. Since no triple appears in more than one $C_x$, it follows that\n\\[|N_T(F)| \\geq \\sum_{x \\in X^c} |N_T(F_x)| \\geq \\frac{\\delta}{2} d \\sum_{x \\in X^c} |F_x| \\geq \\frac{\\epsilon^2}{64} d |F|,\\]\nas required.\n\\end{proof}\n\nThe edge expansion property from Theorem~\\ref{thm:main} now follows as a simple corollary.\n\n\\begin{corollary} \\label{cor:edgeexp}\nSuppose that $S \\subseteq \\mathbb{Z}_2^t$ contains no non-trivial solutions to the equation $s_1 + s_2 = s'_1 + s'_2$ and $\\lambda(\\textrm{Cay}(\\mathbb{Z}_2^t, S)) \\leq (1 - \\epsilon)d$. Then, for all subsets $F$ of the skeleton $E$ of $H(\\mathbb{Z}_2^t, S)$ with $|F| \\leq |E|\/2$, $|N_E(F)| \\geq \\frac{\\epsilon^2}{128} |F|$.\n\\end{corollary}\n\n\\begin{proof}\nBy Theorem~\\ref{thm:tripexp}, there are at least $\\frac{\\epsilon^2}{64} d|F|$ triples which contain an edge from both $F$ and $E \\setminus F$. Since each edge in $E \\setminus F$ is contained in at most $2d$ of these triples, the result follows by division.\n\\end{proof}\n\nWe will prove Corollary~\\ref{cor:mixing} on the rapid mixing of the random walk on the edges of $H(\\mathbb{Z}_2^t, S)$ by constructing an auxiliary graph $G$ and then appealing to the following result~\\cite[Theorem 3.3]{HLW06}.\n\n\\begin{lemma} \\label{lem:mix}\nLet $G$ be an $N$-vertex $D$-regular graph with $\\lambda = \\max_{i \\neq 1} |\\lambda_i(G)|$ and $\\mathbf{p}_0$ a probability distribution on $V(G)$. Then the random walk on $G$ starting from the initial distribution $\\mathbf{p}_0$ satisfies\n\\[\\|\\mathbf{p}_i - \\mathbf{u}\\|_2 \\leq \\left(\\frac{\\lambda}{D}\\right)^i,\\]\nwhere $\\mathbf{p}_i$ is the probability distribution on $V(G)$ after $i$ steps of the walk, $\\mathbf{u}$ is the uniform distribution on $V(G)$ and $\\|\\mathbf{x}\\|_2 = (\\sum_i x_i^2)^{1\/2}$ for $\\mathbf{x} = (x_1, x_2, \\dots, x_N) \\in \\mathbb{R}^N$.\n\\end{lemma}\n\nTo apply this lemma, we need to estimate $\\lambda$. Recall that the \\emph{edge expansion ratio} of a graph $G$ is \n\\[h(G) = \\min_{\\{U : |U| \\leq |V|\/2\\}} \\frac{e(U, U^c)}{|U|}.\\]\nThe following discrete analogue of the Cheeger inequality, due to Dodziuk~\\cite{D84} and Alon and Milman~\\cite{AM85}, places an upper bound on the second eigenvalue $\\lambda_2$ of a graph $G$ in terms of its edge expansion ratio.\n\n\\begin{lemma} \\label{lem:Cheeger}\nIf $G$ is an $N$-vertex $D$-regular graph with eigenvalues $\\lambda_1 \\geq \\dots \\geq \\lambda_N$, then\n\\[\\lambda_2 \\leq D - \\frac{h(G)^2}{2D}.\\]\n\\end{lemma}\n\nTo estimate $\\lambda_N$, we use a result of Desai and Roy~\\cite{DR94}. To state their result, for any subset $U$ of the vertex set of a graph $G$, we define $b(U)$ to be the minimum number of edges that need to be removed from the induced subgraph $G[U]$ to make the graph bipartite.\n\n\\begin{lemma} \\label{lem:DR}\nIf $G$ is an $N$-vertex $D$-regular graph with eigenvalues $\\lambda_1 \\geq \\dots \\geq \\lambda_N$, then\n\\[\\lambda_N \\geq -D + \\frac{\\Psi^2}{4D},\\]\nwhere\n\\[\\Psi = \\min_{U \\neq \\emptyset} \\frac{b(U) + e(U, U^c)}{|U|}.\\]\n\\end{lemma}\n\n{\\bf Proof of Corollary~\\ref{cor:mixing}:}\nConsider the auxiliary graph $G$ whose vertices $V$ are the edges of the skeleton $E$ and where two vertices are joined if the union of the corresponding edges $e_1, e_2 \\in E$ is in $T$. Note that $G$ is an $N$-vertex $D$-regular with $N = \\frac{1}{2} \\binom{d}{2} n$ and $D = 2d-4$. The random walk on $G$ is in one-to-one correspondence with the random walk on the original hypergraph $H(\\mathbb{Z}_2^t, S)$ and the fact that $H$ is an $\\frac{\\epsilon^2}{64}d$-triple-expander easily implies that for any $U \\subseteq V$ with $|U| \\leq |V|\/2$ the number of edges between $U$ and $U^c$ is at least $\\frac{\\epsilon^2}{64}d |U|$. Therefore, $h(G) \\geq \\frac{\\epsilon^2}{64} d \\geq \\frac{\\epsilon^2}{128}D$. By Lemma~\\ref{lem:Cheeger}, this implies that \n\\[\\lambda_2(G) \\leq D - \\frac{h(G)^2}{2D} \\leq \\left(1 - \\frac{\\epsilon^4}{2^{15}}\\right)D.\\]\nTo estimate $\\Psi$, we split into two cases. If $|U| < \\frac{15}{16}n$, we use the fact that \n\\[e(U, U^c) \\geq \\frac{\\epsilon^2}{64}d \\min(|U|, |U^c|) \\geq \\frac{\\epsilon^2}{2^{10}} d |U|\\] \nto conclude that $e(U, U^c)\/|U| \\geq \\frac{\\epsilon^2}{2^{10}} d$. On the other hand, if $|U| \\geq \\frac{15}{16} N$, the corresponding edge set $F$ in the skeleton $E$ of $H$ has at least $\\frac{3}{4} \\binom{d}{2}$ edges in at least $\\frac{3}{4} n$ of the sets $C_x$. By supersaturation, there exists a constant $c > 0$ such that each $C_x$ with at least $\\frac{3}{4} \\binom{d}{2}$ edges has at least $cd^3$ triangles with all edges in $F$. As there are at least $\\frac{3}{4} n$ sets $C_x$ with this property, $F$ contains at least $\\frac{3}{4} c d^3 n$ triangles, which in turn implies that $G[U]$ contains at least $\\frac{3}{4} c d^3 n$ triangles. Since $G$ is an edge-disjoint union of triangles, the number of edges which must be removed to make $G[U]$ bipartite is at least the number of triangles in $G[U]$. That is, $b(U)$ is at least $\\frac{3}{4} c d^3 n$, so $b(U)\/|U| \\geq c d$. In either case, we see that $\\Psi = \\Omega(\\epsilon^2 D)$ and, hence, by Lemma~\\ref{lem:DR}, there is a positive constant $c$, which we may assume is at most $2^{-15}$, such that \n\\[\\lambda_N(G) \\geq -D + \\frac{\\Psi^2}{4D} \\geq (-1 + c \\epsilon^4) D.\\]\nPutting everything together, we see that $\\lambda = \\max(|\\lambda_2(G)|, |\\lambda_N(G)|) \\leq (1 - c \\epsilon^4) D$. Therefore, applying Lemma~\\ref{lem:mix},\n\\[\\|\\mathbf{p}_i - \\mathbf{u}\\|_2 \\leq \\left(\\frac{\\lambda}{D}\\right)^i \\leq (1 - c \\epsilon^4)^i,\\]\nas required.\n\\ifvmode\\mbox{ }\\else\\unskip\\fi\\hskip 1em plus 10fill$\\Box$\n\n\\section{Further remarks}\n\n{\\bf Generalised constructions.}\n\nFor simplicity, we have worked throughout with the group $\\mathbb{Z}_2^t$. However, a similar construction works over any abelian group $G$. Indeed, given a subset $S$ of $G$, we can let $H(G, S)$ be the $3$-uniform hypergraph with vertex set $G$ and edge set consisting of all triples of the form $(x + s_1, x+ s_2, x+ s_3)$, where $x \\in G$ and $s_1, s_2, s_3 \\in S \\cup (-S)$ with $s_i \\neq \\pm s_j$ for $i \\neq j$.  Alternatively, $H$ is the $3$-uniform hypergraph on the same vertex set whose triples are the triangles of Cay$(G, S')$, where\n\\[S' = \\{s_1 + s_2 : s_1, s_2 \\in S \\cup (-S), s_1 \\neq \\pm s_2\\}.\\]\nIt is worth noting that we omit edges of the form $(x, x + s + s)$, where they exist, since they will typically only be contained in one set of the form $C_x = \\{x + s: s \\in S \\cup (-S)\\}$. Over $\\mathbb{Z}_2^t$, such edges do not exist, so this issue does not arise.\n\nIt is also possible to define a variant of our construction by using longer sums. For instance, given a subset $S$ of $\\mathbb{Z}_2^t$, we may let\n\\[S' = \\{s_1 + \\dots + s_{2\\ell} : s_i \\in S, s_i \\neq s_j\\}\\]\nand take $H$ to again be the $3$-uniform hypergraph whose triples are the triangles of Cay$(\\mathbb{Z}_2^t, S')$. However, this generalised construction seems to have few tangible benefits over the $\\ell = 1$ case, so we have not pursued it further.\n\n\\vspace{3mm}\n\n{\\bf Vertex expansion.}\n\nGiven a subset $F$ of the skeleton $E$ of a hypergraph $H$, one may also define its \\emph{vertex neighbourhood}\n\\[N_V(F) = \\{v \\in V(H) : v \\cup f \\in T \\textrm{ for some } f \\in F \\textrm{ and } v \\notin V(F)\\},\\]\nwhere $V(F)$ is the set of vertices of $H$ which are contained in some edge of $F$. Assuming $H$ has $n$ vertices, we then let\n\\[h_V(H) = \\min_{\\{F: |V(F)| \\leq n\/2\\}} \\frac{|N_V(F)|}{|V(F)|}\\]\nand say that $H$ is an \\emph{$\\epsilon$-vertex-expander} if $h_V(H) \\geq \\epsilon$. One might now ask if our construction $H(\\mathbb{Z}_2^t, S)$ has this vertex expansion property for a random choice of $S$. This problem seems surprisingly delicate and we were unable to decide in which direction the truth lies. A positive solution would be of substantial interest and is likely to facilitate applications to extremal combinatorics, such as to the determination of the size Ramsey number of tight paths (see~\\cite{DLMR17} for the current status of this problem). It would also be of great interest to find alternative constructions, preferably of bounded degree, with this vertex expansion property. \n\n\\vspace{3mm}\n\n{\\bf Coboundary and cosystolic expansion.}\n\nThe progress by Evra, Kaufman, Kazhdan and Lubotzky~\\cite{EK17, KKL16} on constructing bounded-degree hypergraphs with the topological overlap property stems from a connection to another, more combinatorial, expansion property known as coboundary expansion~\\cite{DK12, LM06}. We will not attempt to describe this property here, but suffice to say that coboundary expansion and a slightly weaker notion known as cosystolic expansion are both known to imply the topological overlap property~\\cite{DKW16, G10}. The papers~\\cite{EK17, KKL16} (and the related work in~\\cite{LLR15, LM15}) then proceed by showing that the constructions under consideration are cosystolic expanders, from which the desired topological overlap property follows.\n\nIn assessing whether our construction gives cosystolic expanders, it is tempting to appeal to a criterion established by Evra and Kaufman~\\cite{EK17}. In the $3$-uniform case, this roughly says that if a hypergraph $H$ has the property that the skeleton graph and the link of each vertex are good expander graphs, then $H$ is a cosystolic expander. Unfortunately, this result does not apply in our situation, since the links in our construction are not good expanders. Nevertheless, we are still willing to conjecture that our construction yields cosystolic, and perhaps even coboundary, expanders.\n\n\\vspace{3mm}\n\n{\\bf Higher uniformities.}\n\nThe construction given in this paper does not seem to generalise to higher uniformities. To see why, recall that, given a set $S$ and $x \\in \\mathbb{Z}_2^t$, we define $C_x = \\{x + s : s \\in S\\}$. The principal reason our $3$-uniform construction goes through is that any edge $(x+s_1, x+s_2)$ in $C_x$ also appears in $C_{x+s_1+s_2}$ as $((x+s_1+s_2) + s_2, (x+s_1+s_2) + s_1)$. The natural analogue of this observation in the $4$-uniform case is to consider a triple $(x+s_1, x+s_2, x+s_3)$ in $C_x$ and to note that this triple can be rewritten as $((x+s_1+s_2 + s_3) + s_2 + s_3, (x+s_1+s_2 + s_3) + s_3 + s_1, (x+s_1+s_2 + s_3) + s_1 + s_2)$. Therefore, if $s_i + s_j$ is in $S$ for all $i \\neq j$, we see that the triple $(x+s_1, x+s_2, x+s_3)$ is also in $C_{x + s_1 + s_2 + s_3}$. However, the requirement that $s + s'$ is in $S$ for any distinct $s, s' \\in S$ is a very strong one, implying that $S$ contains every non-zero element in its span. Since $S$ needs to span all of $\\mathbb{Z}_2^t$ for Cay$(\\mathbb{Z}_2^t, S)$ to even be connected, it would need to contain all non-zero elements of $\\mathbb{Z}_2^t$. The construction would then reduce to taking the complete $4$-uniform hypergraph on $\\mathbb{Z}_2^t$. However, despite the failure of this particular mechanism, it remains a very interesting problem to find simple constructions of sparse expanders in higher uniformities.\n\n\\vspace{5mm}\n\\noindent\n{\\bf Acknowledgements.} The author gratefully acknowledges the support of the Simons Institute for the Theory of Computing during part of the period when this paper was written. The author is also indebted to Noga Alon, who brought the problem of constructing high-dimensional expanders to his attention, and to Rajko Nenadov for several valuable discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nGeometric Langlands correspondence was proposed by V.~G.~Drinfeld [1] as a geometric analog\nof the Langlands conjecture relating Galois representations of a global field with\nrepresentations of adelic algebraic group. In the geometric Langlands correspondence\nthe global field is replaced by the field of functions on a complete nonsingular\nalgebraic curve $C$ defined over the field of complex numbers $\\mathbb C$; Galois representations\nare replaced by local systems on the curve $C$;\nfinally, representations of adelic algebraic group\nare replaced by $\\mathcal D$-modules (or constructible sheaves) on the moduli space of bundles on $C$.\nThus, the conjectural geometric Langlands correspondence is a correspondence between\n(certain) $G^\\vee$-local systems on $C$ (here $G^\\vee$ is a semisimple algebraic group over\n$\\mathbb C$) and between (certain) $\\mathcal D$-modules on the moduli space $\\mathcal Bun_G$ of principal $G$-bundles\non $C$ (here $G$ is the group Langlands dual to $G^\\vee$).\n\nThe goal of this paper is to state a conjecture on deformation of the geometric Langlands\ncorrespondence, and to relate it with constructions of algebraic conformal field theory [2].\nThis conjecture, originally proposed also by V.~G.~Drinfeld, looks more simple and\nsymmetric than the conjecture on the geometric Langlands correspondence itself. Its\nrelation with conformal field theory found by the author is a refinement and an argument\nin favour of these conjectures, because it unifies many notions into a self-consistent picture.\n\nFor a more detailed exposition of the material of this paper, see [3].\n\nThe author is deeply grateful to B.~L.~Feigin and E.~V.~Frenkel; the main constructions of the\npaper arose from discussions with them. The author is also grateful to \nA.~A.~Beilinson and V.~G.~Drinfeld for permission to cite their \nunpublished conjectures.\n\n\n\n\\section{The main conjectures}\n\n\\subsection{} Let $G$ be a simple algebraic group over $\\mathbb C$, let $C$ be a complete smooth\nalgebraic curve of genus $g$, and let $\\mathcal Bun_G$ be the moduli space of principal $G$-bundles\non the curve $C$. It is known [3] that for any affine algebraic group $A$ the moduli space\n$\\mathcal Bun_A$ of principal $A$-bundles on $C$ is a smooth algebraic stack [4]. Moreover, this\nstack can be covered by open substacks of the form $X_n\/G_n$, where $X_n$ is a smooth\nalgebraic variety, and $G_n$ is an affine algebraic group acting on $X_n$. Hence various\nobjects (functions, $\\mathcal D$-modules, sheaves, etc.) on the stack $\\mathcal Bun_A$ can be\nobtained by glueing corresponding $G_n$-equivariant objects on the varieties $X_n$.\n\nDenote by $G^\\vee$ the group Langlands dual to $G$. For a principal $G^\\vee$-bundle\n$P^\\vee$ on the curve $C$, the space of (algebraic) connections on $P^\\vee$ is an affine space,\nwhose associated vector space is the cotangent space\n$$\nT_{P^\\vee}^*\\mathcal Bun_{G^\\vee}\\simeq\\Gamma(C,\\Omega^1_C\\otimes\\mathop{\\rm ad}\\nolimits P^\\vee)\n$$\nto the stack $\\mathcal Bun_{G^\\vee}$ at the point $P^\\vee$.\nHence we call the moduli space of $G^\\vee$-local systems on $C$,\ni.~e. principal $G^\\vee$-bundles with a connection, by the {\\it twisted cotangent bundle}\nto the stack $\\mathcal Bun_{G^\\vee}$, and denote it by $\\widetilde T^*\\mathcal Bun_{G^\\vee}$. It is known [3]\nthat the cocycle of the affine bundle $\\widetilde T^*\\mathcal Bun_{G^\\vee}$ is obtained from the\ncocycle of the canonical line bundle $\\omega_{\\mathcal Bun_{G^\\vee}}$ on the stack $\\mathcal Bun_{G^\\vee}$\nvia the homomorphism\n$$\nd\\log:H^1(\\mathcal Bun_{G^\\vee}, \\mathcal O^*_{\\mathcal Bun_{G^\\vee}})\\to H^1(\\mathcal Bun_{G^\\vee}, \n\\Omega^1_{\\mathcal Bun_{G^\\vee}}).\n$$\n\n\n\\subsection{} {\\bf Conjecture 1.} [6] The derived category of $\\mathcal D_{\\mathcal Bun_G}$-modules\nis equivalent to the derived category of quasicoherent\n$\\mathcal O_{\\widetilde T^*\\mathcal Bun_{G^\\vee}}$-modules.\n\nThis conjecture (as well as definition of the derived categories [5]) is due to Beilinson\nand Drinfeld. They refine it in the following way. The required equivalence of derived\ncategories should be given by the ``kernel'' $\\mathcal L_0$, an object of derived \ncategory of\n$\\mathcal D_{\\mathcal Bun_G}\\boxtimes\\mathcal O_{\\widetilde T^*\\mathcal Bun_{G^\\vee}}$-modules. The \nrestriction of this object\nto the open substack $\\widetilde T^*\\mathcal Bun_{G^\\vee}^\\circ$ of irreducible $G^\\vee$-local\nsystems should be a $\\mathcal D_{\\mathcal Bun_G}\\boxtimes\\mathcal O_{\\widetilde \nT^*\\mathcal Bun_{G^\\vee}^\\circ}$-module,\nflat as an $\\mathcal O_{\\widetilde T^*\\mathcal Bun_{G^\\vee}^\\circ}$-module, whose fiber \nover the local\nsystem\n$$\n\\P^\\vee=(P^\\vee,\\nabla)\\in\\widetilde T^*\\mathcal Bun_{G^\\vee}^\\circ\n$$\nshould be the unique, up to isomorphism, holonomic $\\mathcal D_{\\mathcal Bun_G}$-module $\\mathcal F_{\\P^\\vee}$,\nwhich is a Hecke eigen-$\\mathcal D_{\\mathcal Bun_G}$-module with eigenvalue $\\P^\\vee$. (For a definition of\nthis notion, see [5].) The correspondence $\\P^\\vee\\to\\mathcal F_{\\P^\\vee}$ is called the geometric\nLanglands correspondence.\n\n\\subsection{} Let us now proceed to the conjecture on deformation of the geometric\nLanglands correspondence. Denote\n$$\n\\xi=\\omega_{\\mathcal Bun_G}^{\\otimes\\left(-\\frac1{2h^\\vee}\\right)}\\in\\mathop{\\rm Pic}\\nolimits\\mathcal Bun_G\\otimes_\\mathbb Z\\mathbb Q\n$$\n($\\mathop{\\rm Pic}\\nolimits$ is the Picard group), where $h^\\vee$ is the dual Coxeter number of the group $G$.\nIn the case when $G$ is simply connected, it is known [5] that $\\xi$ is the positive generator\nof the group $\\mathop{\\rm Pic}\\nolimits\\mathcal Bun_G\\simeq\\mathbb Z$. Similarly, introduce the element\n$$\n\\xi^\\vee=\\omega_{\\mathcal Bun_{G^\\vee}}^{\\otimes\\left(-\\frac1{2h}\\right)}\\in\\mathop{\\rm Pic}\\nolimits\\mathcal Bun_{G^\\vee}\n\\otimes_\\mathbb Z\\mathbb Q.\n$$\n\n{\\bf Conjecture 2.} [6] The derived category of twisted\n$\\mathcal D_{\\mathcal Bun_G}(\\xi^{\\otimes\\kappa})$-modules is equivalent to the derived category of\ntwisted $\\mathcal D_{\\mathcal Bun_{G^\\vee}}(\\xi^{\\vee\\otimes\\kappa^\\vee})$-modules\nfor any $\\kappa\\in\\mathbb C$, $\\kappa\\ne0$, where $\\kappa^\\vee=1\/(r\\kappa)$, and $r$ is the maximal\nmultiplicity of an edge in the Dynkin diagram of the group $G$ (or $G^\\vee$);\n$r=1$, $2$, or $3$.\n\nThe idea of this conjecture is due to V.~G.~Drinfeld. This conjecture also has several\nrefinements. We are going to state them in the rest of this paper.\n\nThe required equivalence of derived categories should be given by a kernel\n$\\mathcal L_\\kappa$ which is an object of derived category of\n$\\mathcal D_{\\mathcal Bun_G}(\\xi^{\\otimes\\kappa})\n\\boxtimes\\mathcal D_{\\mathcal Bun_{G^\\vee}}(\\xi^{\\vee\\otimes\\kappa^\\vee})$-modules. All refinements of\nConjecture~2 stated below deal with the properties of this kernel.\n\n\\subsection{} {\\bf Property 1. Dependence on the parameter $\\kappa$; classical limits.}\n\nLet us first define the notion of asymptotic twisted $\\mathcal D$-module, or\n$\\mathcal D_X^{\\mathop{\\rm asym}\\nolimits}(\\xi^{\\otimes t})$-module, on a smooth variety $X$ with a line bundle $\\xi$.\nThis notion is introduced in [3]; let us briefly discuss it.\n\nThere exists a natural sheaf $\\mathcal D_X^{\\mathop{\\rm asym}\\nolimits}(\\xi^{\\otimes t})$ of quasicoherent\n$\\mathcal O_{\\mathbb P^1}$-algebras on the product $\\mathbb P^1\\times X$, flat as an \n$\\mathcal O_{\\mathbb P^1}$-module,\nwhose fiber at the point $\\kappa\\in\\mathbb P^1$, $\\kappa\\ne\\infty$, is isomorphic to the sheaf\n$\\mathcal D_X(\\xi^{\\otimes\\kappa})$ of twisted differential operators, and the fiber over the point\n$\\infty\\in\\mathbb P^1$ is isomorphic to the sheaf $\\pi_*\\mathcal O_{\\widetilde T^*X}$, \nwhere\n$\\pi:\\widetilde T^*X\\to X$ is the twisted cotangent affine bundle over $X$, whose\ncocycle coresponds to the cocycle of the bundle $\\xi$ under the homomorphism\n$$\nd\\log:H^1(X,\\mathcal O_X^*)\\to H^1(X,\\Omega^1_X).\n$$\nSections of the sheaf $\\mathcal D_X^{\\mathop{\\rm asym}\\nolimits}(\\xi^{\\otimes t})$ are called twisted asymptotic\ndifferential operators, cf. [7]. On a sufficiently small open subset $U\\subset X$,\non which the bundle $\\xi$ is trivial, we have\n$$\n\\Gamma((\\mathbb P^1\\setminus\\{0\\})\\times U, \\mathcal D_X^{\\mathop{\\rm asym}\\nolimits}(\\xi^{\\otimes t}))\\simeq\n\\mathcal O_U[t^{-1},t^{-1}\\partial_1,\\ldots,t^{-1}\\partial_n],\n$$\nwhere $\\partial_1$, $\\ldots$, $\\partial_n$ is a basis of vector fields on $U$, $t$ is the\nparameter on the line $\\mathbb P^1$.\n\nBy definition, a $\\mathcal D_X^{\\mathop{\\rm asym}\\nolimits}(\\xi^{\\otimes t})$-module is a sheaf of\n$\\mathcal D_X^{\\mathop{\\rm asym}\\nolimits}(\\xi^{\\otimes t})$-modules on the product $\\mathbb P^1\\times X$ quasicoherent as an\n$\\mathcal O_{\\mathbb P^1\\times X}$-module.\n\nLet us return to the kernel $\\mathcal L_\\kappa$. The property of dependence of \n$\\mathcal L_\\kappa$ on the\nparameter $\\kappa$ states that the objects $\\mathcal L_0$, $\\mathcal L_\\kappa$ are the \nfibers of an object\n$\\mathcal L_t$ of the derived category of\n$\\mathcal D_{\\mathcal Bun_G}^{\\mathop{\\rm asym}\\nolimits}(\\xi^{\\otimes t})\n\\boxtimes_{\\mathcal O_{\\mathbb P^1}}\\mathcal D_{\\mathcal Bun_{G^\\vee}}^{\\mathop{\\rm asym}\\nolimits}(\\xi^{\\vee\\otimes \nt^\\vee})$-modules\non the product $\\mathbb P^1\\times\\mathcal Bun_G\\times\\mathcal Bun_{G^\\vee}$, flat as an \n$\\mathcal O_{\\mathbb P^1}$-module.\nHere $t^\\vee=1\/(rt)$. The fiber of this object at $t=\\kappa$ is the object \n$\\mathcal L_\\kappa$,\nthe fiber at $t=0$ is the object $\\mathcal L_0$ which is the kernel of the \ngeometric Langlands\ncorrespondence defined in 1.2 above, and the fiber at $t=\\infty$ is the \nobject $\\mathcal L_\\infty$\nwhich is the kernel of the geometric Langlands correspondence for the group $G^\\vee$, i.~e.,\nthe object $\\mathcal L_\\infty$ is obtained from $\\mathcal L_0$ by exchanging the roles \nof the groups $G$ and\n$G^\\vee$.\n\n\\subsection{} {\\bf Property 2: singular support of the kernel \n$\\mathcal L_\\kappa$}.\nOne has the Hitchin map [5]\n$$\n\\chi_G: T^*\\mathcal Bun_G\\to\\oplus_{i=1}^{\\mathop{\\rm rk}\\nolimits G}\\Gamma(C, \\omega_C^{\\otimes d_i}),\n$$\nwhere $d_i$ are the exponents of the group $G$, $\\mathop{\\rm rk}\\nolimits G$ is the rank of $G$. After restriction\nto certain open dense subset $U\\subset\\mathcal Bun_G\\times\\mathcal Bun_{G^\\vee}$, the \nkernel $\\mathcal L_\\kappa$\nshould be a coherent $\\mathcal D_{\\mathcal Bun_G}(\\xi^{\\otimes\\kappa})\n\\boxtimes\\mathcal D_{\\mathcal Bun_{G^\\vee}}(\\xi^{\\vee\\otimes\\kappa^\\vee})$-module whose singular support\ncoincides with the preimage of the diagonal\n$$\n\\Delta_\\kappa=\\{v_i,v_i^\\vee\\in\\Gamma(C,\\omega_C^{\\otimes d_i}):\nv_i^\\vee=\\kappa^{d_i}v_i\\}_{i=1}^{\\mathop{\\rm rk}\\nolimits G}\n$$\nunder the product of Hitchin maps\n$$\n\\chi_G\\times\\chi_{G^\\vee}: T^*\\mathcal Bun_G\\times T^*\\mathcal Bun_{G^\\vee}\n\\to\\left(\\oplus_{i=1}^{\\mathop{\\rm rk}\\nolimits G}\\Gamma(C, \\omega_C^{\\otimes d_i})\\right)^{\\oplus2}.\n$$\nThe classical limit of this property as $\\kappa\\to0$ yields the conjecture that\nthe singular support of $\\mathcal D_{\\mathcal Bun_G}$-modules $\\mathcal F_{\\P^\\vee}$ from the geometric Langlands\ncorrespondence is contained in the global nilpotent cone, which is the preimage of zero\nunder the Hitchin map.\n\n\n\\section{Relation with conformal field theory}\n\n\\subsection{} For simplicity assume in this Section that the group $G$ is adjoint, and the\ngroup $G^\\vee$ is simply connected. For the general case, see [3].\n\nLet us fix a Borel subgroup $B\\subset G$ with the unipotent radical $N$; let $H=B\/N$ be the\nCartan group. Consider the diagram\n$$\n\\leqno{(*)}\n\\xymatrix{\n       & \\mathcal Bun_B^{>0}\\ar[dl]_{\\sigma}\\ar[dr]^{\\rho} &   &   &   \\\\\n\\mathcal Bun_G & & \\mathcal Bun^{>0}_{B\/[N,N]}\\ar[dr]^{\\beta} & & \n\\mathcal Bun_{\\omega,H}\\stackrel{\\iota}{\\hookleftarrow}\\mathcal Conf_{G^\\vee}\\ar[dl]_{\\alpha} \n\\\\\n       &   &   & \\mathcal Bun_H^{>0} &   \\\\\n}\n$$\nThe only thing to be explained in this diagram is what are the spaces $\\mathcal Bun_{\\omega,H}$,\n$\\mathcal Conf_{G^\\vee}$, and what does the sign $>0$ mean. To explain this, note that\n$$\nB\/[N,N]\\simeq\\prod_{i=1}^{\\mathop{\\rm rk}\\nolimits G}B_i,\n$$\nwhere $B_i$ is a copy of the upper triangular Borel subgroup in $PGL(2)$. Hence\n$\\mathcal Bun_{B\/[N,N]}$ is identified with the moduli space of sets of exact triples\n$$\n\\leqno{(**)}\\qquad\\qquad\\qquad\\qquad 0\\to\\mathcal O_C\\to E_i\\to L_i\\to0,\n$$\nwhere $L_i$ is a line bundle on the curve $C$, $1\\le i\\le\\mathop{\\rm rk}\\nolimits G$. The projection\n$\\mathcal Bun_{B\/[N,N]}\\to\\mathcal Bun_H$ takes a set of triples $(**)$ to the set of line bundles $(L_i)\\in\\mathcal Bun_H$.\nBy definition,\n$$\n\\mathcal Bun_H^{>0}=\\{(L_i), \\deg L_i>0, 1\\le i\\le\\mathop{\\rm rk}\\nolimits G\\};\n$$\n$\\mathcal Bun_{B\/[N,N]}^{>0}$ and $\\mathcal Bun_B^{>0}$ are the preimages of $\\mathcal Bun_H^{>0}$ under the natural\nprojections. The projection\n$$\n\\beta:\\mathcal Bun_{B\/[N,N]}^{>0}\\to\\mathcal Bun_H^{>0}\n$$\nis a vector bundle whose fiber over a point $(L_i)\\in\\mathcal Bun_H^{>0}$ is the vector space\n$$\n\\oplus\\mathop{\\rm Ext}\\nolimits_C^1(L_i,\\mathcal O)\\simeq\\oplus H^1(C,L_i^{-1}).\n$$\nBy definition, $Bun_{\\omega,H}$ is the vector bundle dual to the vector bundle $\\beta$, and\n$\\mathcal Conf_{G^\\vee}$ is an open substack of this vector bundle defined as follows:\n$$\n\\begin{aligned}{}\n\\mathcal Bun_{\\omega,H}&=\\{(L_i,s_i),\\deg L_i>0,s_i\\in\\Gamma(C,\\omega_C\\otimes L_i)\\},\\\\\n\\mathcal Conf_{G^\\vee}&=\\{(L_i,s_i),s_i\\ne0\\}\\simeq\\{D_i: \\deg D_i>2g-2\\},\n\\end{aligned}\n$$\ni.~e., $\\mathcal Conf_{G^\\vee}$ is isomorphic to the space of sets of effective divisors $(D_i)$\non the curve $C$, i.~e. to the space of divisors with values in the semigroup $\\Gamma_+$\nof dominant weights of the group $G^\\vee$. Let us call these sets of divisors by\n{\\it coloured divisors}. As a variety $\\mathcal Conf_{G^\\vee}$ is the disjoint union of products of\nsymmetric powers of the curve $C$. The fact that an open dense substack of a\nvector bundle is a disjoint union of projective varieties, is due to the fact that each\npoint of the stack $\\mathcal Bun_H$ has the group of automorphisms $(\\mathbb C^*)^{\\mathop{\\rm rk}\\nolimits G}$.\n\nThe space $\\mathcal Conf_{G^\\vee}$ is naturally stratified:\n$$\n\\mathcal Conf_{G^\\vee}=\\sqcup_{\\lambda^\\vee}\\mathcal Conf_{G^\\vee}^{\\lambda^\\vee},\n$$\nwhere $\\lambda^\\vee=(\\lambda^\\vee_1,\\ldots,\\lambda^\\vee_N)$, $\\lambda^\\vee_i\\in\\Gamma_+$, and\n$$\n\\mathcal Conf_{G^\\vee}^{\\lambda^\\vee}=\\{\\lambda_1^\\vee x_1+\\ldots+\\lambda_N^\\vee x_N, x_i\\in C,\nx_i\\ne x_j\\text{ for }i\\ne j\\}\/\\mathfrak G_{\\lambda^\\vee},\n$$\nwhere $\\mathfrak G_{\\lambda^\\vee}$ is the group of permutations of indices $i$ preserving the weights\n$\\lambda_i^\\vee$. Denote the inclusion\n$\\mathcal Conf_{G^\\vee}^{\\lambda^\\vee}\\hookrightarrow\\mathcal Conf_{G^\\vee}$ by $j_{\\lambda^\\vee}$.\n\n\\subsection{} {\\bf Property 3 of the kernel $\\mathcal L_\\kappa$.} This property \ndescribes the object of\nderived category of twisted $\\mathcal D_{\\mathcal Conf_{G^\\vee}^{\\lambda^\\vee}\\times\\mathcal Bun_{G^\\vee}}$-modules\n$$\n\\leqno(***)\\qquad\\qquad\\qquad\\qquad \nj_{\\lambda^\\vee}^!\\iota^!F\\rho_*\\sigma^!\\mathcal L_\\kappa,\n$$\nwhere all the spaces in the diagram $(*)$ are multiplied by $\\mathcal Bun_{G^\\vee}$; $F$ denotes the\nFourier--Laplace transform of a $\\mathcal D$-module on the vector bundle $\\mathcal Bun_{B\/[N,N]}^{>0}$.\nThe object $(***)$ should be isomorphic to the twisted\n$\\mathcal D_{\\mathcal Conf_{G^\\vee}^{\\lambda^\\vee}\\times\\mathcal Bun_{G^\\vee}}$-module\n$KZ_{G^\\vee,\\kappa^\\vee}^{\\lambda^\\vee}$ constructed in conformal field theory.\nAs an\n$\\mathcal O_{\\mathcal Conf_{G^\\vee}^{\\lambda^\\vee}}\n\\boxtimes\\mathcal D_{\\mathcal Bun_{G^\\vee}}(\\xi^{\\vee\\otimes\\kappa^\\vee})$-module,\n$KZ_{G^\\vee,\\kappa^\\vee}^{\\lambda^\\vee}$ coincides with the induced module from the\n$\\mathcal O_{\\mathcal Conf_{G^\\vee}^{\\lambda^\\vee}\\times\\mathcal Bun_{G^\\vee}}$-module whose \nfiber at the point\n$$\n(\\lambda^\\vee_1x_1+\\ldots+\\lambda_N^\\vee x_N, P^\\vee)\n\\in\\mathcal Conf_{G^\\vee}^{\\lambda^\\vee}\\times\\mathcal Bun_{G^\\vee}\n$$\nis the tensor product\n$(V_{P^\\vee}^{\\lambda^\\vee_1})_{x_1}\\otimes\\ldots\\otimes (V_{P^\\vee}^{\\lambda^\\vee_N})_{x_N}$,\nwhere $V_{P^\\vee}^{\\lambda^\\vee_i}$ is the vector bundle on $C$ associated with the\nprincipal $G^\\vee$-bundle $P^\\vee$ and with the $G^\\vee$-module $V^{\\lambda^\\vee_i}$\nwith highest weight $\\lambda_i^\\vee$. Further, this\n$\\mathcal O_{\\mathcal Conf_{G^\\vee}^{\\lambda^\\vee}}\n\\boxtimes\\mathcal D_{\\mathcal Bun_{G^\\vee}}(\\xi^{\\vee\\otimes\\kappa^\\vee})$-module has a natural structure\nof a twisted $\\mathcal D_{\\mathcal Conf_{G^\\vee}^{\\lambda^\\vee}}$-module [2], which is a direct\ngeneralization of the Knizh\\-nik--Za\\-mo\\-lod\\-chi\\-kov connection to curves of genus $g$.\nA direct check with the use of Riemann-Roch theorem shows that the object $(***)$ has the same\ntwist. The property~3 of the kernel $\\mathcal L_\\kappa$ states that these two \nobjects are isomorphic.\n\n\\subsection{} {\\bf Classical limits of the property 3.}\n\na) {\\it The limit $\\kappa\\to0$} amounts to the geometric analog of the\nCasselman--Shalika--Shintani formula for the Whittaker function of an automorphic form. This\nstatement is that the $\\mathcal D_{\\mathcal Conf_{G^\\vee}^{\\lambda^\\vee}}$-module\n$$\nj_{\\lambda^\\vee}^!\\iota^!F\\rho_*\\sigma^!\\mathcal F_{\\P^\\vee}\n$$\nis isomorphic to the local system whose fiber over the point\n$\\lambda^\\vee_1x_1+\\ldots+\\lambda^\\vee_Nx_N\\in\\mathcal Conf_{G^\\vee}^{\\lambda^\\vee}$ equals\n$(V_{\\P^\\vee}^{\\lambda^\\vee_1})_{x_1}\\otimes\\ldots\\otimes (V_{\\P^\\vee}^{\\lambda^\\vee_N})_{x_N}$.\n\nb) {\\it The limit $\\kappa\\to\\infty$} amounts to the construction of the\n$\\mathcal D_{\\mathcal Bun_{G^\\vee}}$-module $\\mathcal F_\\P$, for $\\P\\in\\widetilde T^*\\mathcal Bun_G^\\circ$,\nby means of a $G$-oper with regular singularities with trivial monodromy [1].\n\nFor details, see [3].\n\n\\subsection{} The following natural question arises: what is the result of applying the same\noperation $j_{\\lambda^\\vee}^!\\iota^!F\\rho_*\\sigma^!$ to the twisted $\\mathcal D$-module\n$KZ_{G,\\kappa}^\\lambda$ on the product $\\mathcal Conf_G^\\lambda\\times\\mathcal Bun_G$? The arising twisted\n$\\mathcal D$-module on the product $\\mathcal Conf_G^\\lambda\\times\\mathcal Conf_{G^\\vee}^{\\lambda^\\vee}$ should\nbe related with the $W$-algebra $W_G^\\kappa\\simeq W_{G^\\vee}^{\\kappa^\\vee}$ [8]. In the case\n$G=SL(2)$ it is the Virasoro algebra.\n\nFor a statement of this kind for a curve $C$ of genus $g=0$ with marked points and for\n$G=SL(2)$, see [9].\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nRecognising people by the way they walk (also known as gait recognition or gait-based person identification) is a relatively new field of research.\nMost of previously studied methods work in the visual domain, where this topic is an active field of research since the last decade~\\cite{lee2002gait}.\nHowever, acoustic information can also be used for gait recognition.\nEven though the focus on this modality has so far been significantly less, results are promising.\nWhile in the visual domain, identification systems can rely on analysing the silhouette~\\cite{wang2003silhouette},\nthe task is much more difficult for systems working only with audio information.\nThe relevant information which can be exploited by such systems consists not only of the sounds of the steps,\nbut also adjacent sounds produced by the clothes of moving arms and legs.\nThese sounds are influenced by the gait pattern of the walking person,\nmaking them suitable to be used for person identification.\nFurthermore, the sounds produced during walking are highly dependent on factors such as the floor type, type of shoes and clothes.\n\nIn a user study~\\cite{makela2003use}, the potential of humans to recognise others by their walking sounds was evaluated.\nAfter a training phase, twelve subjects were able to identify their co-workers by their walking sounds with an accuracy of 66\\,\\%.\nThis result shows that sounds produced by walking persons convey characteristic information about the subject and can thus be used for person identification.\n\nPotential applications of gait-based person identification using audio information are smart homes for ambient assisted living, indoor surveillance scenarios, or access control systems.\nSuch an audio-based system can be used to enhance visual surveillance and facilitate multimodal approaches.\nAs compared to video-based person identification, acoustic systems will also work in the darkness, require less expensive hardware and often lower sensor density and are less obtrusive.\nAcoustic gait-based person identification is also known as \\textit{acoustic gait recognition}.\n\n\n\n\n\\vspace{-0.1cm}\n\\subsection{Contribution}\n\nThe contribution of this paper is a system for acoustic gait-based person identification that is based on hidden Markov models (HMMs).\nTo our knowledge, this is the first time that HMMs are applied for this task.\nWe use Mel-frequency cepstral coefficients (MFCCs) as audio features and HMMs with a cyclic topology for dynamic classification, in order to model the dynamics of gait patterns.\nWith the cyclic topology, one pass through the model corresponds to a half gait cycle containing one step.\nThus, the system is capable of detecting the individual steps in a recording and using them for person identification. \nExperiments are conducted using the TUM GAID corpus, which contains 3\\,050 recordings of 305 subjects in three walking variations in a realistic setup.\nThe recognition system is trained with normal walking style recordings and evaluated on other recordings of normal walking style as well as variations including a backpack and shoe covers.\nOur experimental results show that the developed system is capable of achieving excellent recognition rates compared to previous work.\n\n\n\n\n\\vspace{-0.1cm}\n\\subsection{Related Work}\n\nThe most-widespread approach for video-based gait recognition is the Gait Energy Image (GEI)~\\cite{han2006}, which is a simple silhouette-based approach.\nIt can be combined with face recognition~\\cite{hofmann2012combined} or with depth information~\\cite{hofmann2012gait}.\nFurthermore, model-based approaches have been proposed for visual gait recognition~\\cite{yam2004}.\nBesides using video or audio information, other methods to identify walking persons include using acoustic Doppler sonar~\\cite{kalgaonkar2007acoustic} or pressure sensors in the floor~\\cite{yun2003user}.\n\n\nUsing audio information for the task of gait-based person identification is a relatively new research field.\nIn~\\cite{she2004framework}, footstep sounds were detected in a corpus of various environmental sounds.\nA system for person identification using footstep detection was introduced in~\\cite{shoji2004personal}.\nThe system was tested with a database of five persons.\nThis work was extended in~\\cite{itai2006footstep} by adding psychoacoustic features such as loudness, sharpness, fluctuation strength and roughness.\nFinally, in~\\cite{itai2008footstep}, dynamic time warping was used for classification and the database was extended to contain ten persons. \nThe system achieves almost 100\\,\\% perfect classification rates (using ten persons).\nHowever, the task is simplified by reducing it to classification of pre-segmented footsteps.\nA similar task is addressed in the recently published study by Altaf et al.~\\cite{Altaf13-PIU}.\nThere, a database of segmented footstep sounds from ten persons is used.\nInstead of extracting spectral features,  \nthe shape and properties of a footstep sound are examined in a temporal energy domain.\nAs a result, an identification accuracy above 90\\,\\% is achieved by using a large number of footsteps during testing.\nWhen using only three consecutive footsteps, which is more comparable to our work, an accuracy of 45\\,\\% is obtained.\nOther studies on acoustic gait-based person identification were presented in ~\\cite{deCarvalho2010identification,alpert2010acoustic}.\nThe weakness of all previous studies about acoustic gait-based person identification that are mentioned here is the fact that only small databases (mostly no more than ten subjects) that are overly prototypical have been employed.\nIn addition, very often, classification is performed using pre-segmented footsteps.\nIn our previous work~\\cite{13hof1}, we investigated the potential of spectral, cepstral and energy-related audio features in combination with support vector machines (SVM) for acoustic gait-based person identification.\nThis work was continued in~\\cite{Geiger13GBP}, where a feature analysis method was used to select relevant audio features.\nIn~\\cite{11wen2}, we had also employed cyclic HMMs, for animal sound classification.\nThe cyclic model topology proved to be efficient to model the repetitive structure of these sounds.\n\n\n\nThe remainder of this paper is structured as follows:\nIn Section~\\ref{sec:database}, we introduce the TUM GAID database which is used in the experiments.\nThe employed system is described in Section~\\ref{sec:system}, followed by the experimental setup and results in Section~\\ref{sec:experiments}.\nSome concluding remarks are given in Section~\\ref{sec:conclusions}.\n\n\\section{The TUM GAID Database}\n\\label{sec:database}\n\nFor our experiments, we use our freely available\\footnote{\\url{www.mmk.ei.tum.de\/tumgaid}} TUM Gait from Audio, Image and Depth (GAID) database~\\cite{13hof1}.\nThe motivation behind the TUM GAID database is to foster research in multimodal gait recognition.\nTherefore, data was recorded with an RGB-D sensor, as well as with a four-channel microphone array.\nThus, a typical colour video stream, a depth stream and an audio stream are simultaneously available.\nThe database contains recordings of 305 subjects walking perpendicular to the recording device in a 3.5\\,m wide hallway corridor with a solid floor.\nIn each recorded sequence, the subject walks for roughly 4\\,m, typically performing between 1.5 and 2.5 gait cycles (each of them consisting of two steps).\nMost of the sequences have a length of approximately 2 -- 3\\,s.\nThree variations are recorded for each subject: Normal walking ($\\mathcal{N}$), walking with a backpack ($\\mathcal{B}$), and walking with shoe covers ($\\mathcal{S}$).\nFor each subject, all recordings of the $\\mathcal{N}$ condition were recorded directly after each other.\nThis means that the same shoes and clothes are used, which corresponds more to a re-identification scenario.\nThe backpack constitutes a significant variation in gait pattern and sound, and the shoe covers pose a considerable change in acoustic condition.\nFigure~\\ref{fig:screenshots} shows screenshots of the three different walking conditions for one subject.\n\\begin{figure}[htb]\n\\vspace{-0.3cm}\n\\center{\n  \\subfloat[Normal recording]{\\includegraphics[width=0.17\\linewidth]{Bild1.png}}\\hspace{0.1cm}\n  \\subfloat[Backpack recording]{\\includegraphics[width=0.17\\linewidth]{Bild2.png}}\\hspace{0.1cm}\n  \\subfloat[Shoe cover recording]{\\includegraphics[width=0.17\\linewidth]{Bild3.png}\n\n \n \n\n  \\caption{Screenshots of three recordings in the TUM GAID database}\n\t\\label{fig:screenshots}\n\\vspace{-0.2cm}\n\\end{figure}\nFor each subject, there are six recordings of the $\\mathcal{N}$ setup, and two each of the $\\mathcal{B}$ and $\\mathcal{S}$ setups.\nThis sums to a total number of 3\\,050 recordings.\nThe metadata distribution of the database is well-balanced with a female proportion of 39\\,\\% and ages ranging from 18 to 55 years (average 24.8 years and standard deviation 6.3 years).\nMore than half of the subjects are wearing sneakers while other commonly-used types of shoes are boots and loafers.\n\nTo allow for a proper scientific evaluation and to prevent overfitting on the test data, the database is divided into a \\textit{development set} and a \\textit{test set}.\nThe two sets are person-disjunct and contain 150 and 155 subjects, respectively.\nBoth for the development and for the test set, the first four $\\mathcal{N}$ recordings of each subject are used for the enrollment process.\nThe other two $\\mathcal{N}$ recordings as well as the $\\mathcal{B}$ and $\\mathcal{S}$ recordings are used to perform the identification experiments.\nThis means that models are learnt only using the $\\mathcal{N}$ recordings, while the $\\mathcal{B}$ and $\\mathcal{S}$ conditions constitute previously unseen variations during the identification experiments and will therefore deteriorate the identification performance.\nThe partition of the database is shown in Table~\\ref{Tab:databaseid}.\n\\begin{table}[t]\n\\centering\n\\caption{Partition of the TUM GAID database}\n\\vspace{-0.2cm}\n\\begin{tabular}{lcc}\n& \\textbf{Development} & \\textbf{Test} \\\\\n& (150 subj.) &(155 subj.)\\\\\n\\midrule\n$\\mathcal{N}$1 -- $\\mathcal{N}$4 & Enrollment &  Enrollment\\\\\n$\\mathcal{N}$5 -- $\\mathcal{N}$6 & Identification  &  Identification\\\\\n$\\mathcal{B}$1 -- $\\mathcal{B}$2 & Identification &  Identification\\\\\n$\\mathcal{S}$1 -- $\\mathcal{S}$2 & Identification &  Identification\\\\\n\\end{tabular}\n\\label{Tab:databaseid}\n\\end{table}\n\n\n\n\n\n\\section{System Description}\n\\label{sec:system}\n\n\n\n\nWe use an HMM system for classification.\nEach individual subject is modelled by one HMM.\nWhile we started with using system settings from a simple word-based speech recognition system, we modified and improved the system properties to fit to the problem of acoustic gait recognition.\n\n\n\n\\subsection{Audio Features}\n\n\nIn our previous work we focussed on exploring the suitability of different audio features for the problem of acoustic gait-based person identification~\\cite{Geiger13GBP}.\nUsing SVMs for classification, we evaluated different feature sets containing MFCCs and other spectral or energy-related features.\nSince SVMs are relatively robust (in contrast to HMMs) with regard to the number of employed features, we were able to improve the average identification accuracy (on the test set of the TUM GAID database) from 23.9\\,\\% (only MFCCs) to 28.2\\,\\% by adding and selecting relevant features.\nIn the present work, the focus is not on the front-end processing but rather on the back-end recognition system.\nTherefore we keep the front-end fixed to using only MFCCs.\nWe use MFCC features in the standard configuration: MFCCs 0--12 including their delta and acceleration coefficients, computed every 10 $ms$ from a 25 $ms$ Hamming window, resulting in 39 features in total.\nWhile the database provides four-channel audio recordings, we extract features from monaural recordings, which are obtained by averaging over the four channels.\nIn addition, we obtained slight improvements by processing the audio features with principal component analysis (PCA), without reducing the number of components.\nHere, the transformations are computed only on the enrollment data, and applied on both the enrollment and identification data.\n\nFigure~\\ref{fig:feat_spec} shows the spectrograms and corresponding first MFCC coefficients for two exemplary recordings ($\\mathcal{N}$ setup) of two different subjects.\n\\begin{figure}\n\t\\centering\n\n\t\\subfloat{\n\t\t\\includegraphics[trim = 11mm 80mm 30mm 80mm, clip, scale=0.35]{feature_spec_n01_p010}\n\t\t\\label{fig:feat_spec10}\n\t}\\\\\n\t\\vspace{-0.8cm}\n\n\t\\subfloat{\n\t\t\\includegraphics[trim = 11mm 80mm 30mm 80mm, clip, scale=0.35]{feature_spec_n01_p020}\n\t\t\\label{fig:feat_spec20}\n\t}\n\t\\caption{Spectrograms (top) and corresponding first MFCC coefficients (bottom), each, for a normal-type recording of two different subjects. Temporal position of footsteps is marked with a vertical line.}\n\t\n\t\\label{fig:feat_spec}\n\t\\vspace{-0.2cm}\n\\end{figure}\nThe spectrograms reveal a considerable static background noise, which is due to the recording environment.\nSeveral spectral peaks can be identified which correspond to the footsteps and the sounds between the steps, which are mostly made by the legs of the trousers or skirts rubbing against each other.\nIn the plot of the MFCCs, the temporal position of the steps are marked.\nThe behaviour of the MFCC features indicates that they are useful to detect the position of the steps and to distinguish between different persons.\n\n\\subsection{HMM System}\n\\label{ssec:basichmm}\n\nOur starting point is a simple HMM system that can be compared to a whole-word recognition system (each person representing one \\textit{word}) in speech recognition.\nEach subject in the dataset is represented by an HMM.\nThe models are equipped with a linear left-right topology.\nWith such a model topology, the HMM has to pass through all of its states sequentially without skipping a state.\nBefore introducing an appropriate step modelling method, which will be described in the next subsection,\nwe apply an approach where each recording containing several steps is modelled by one pass through an HMM.\nAs a result, rather large numbers of states (generally more than ten) are required to be able to model the dynamic sequence of sounds during walking.\n\nIn a standard HMM system, the observations are modelled with a mixture of Gaussians.\nHowever, our first experiments showed that the best results are obtained by using HMMs with a single Gaussian state model,\nas the amount of training data is very small and hence probably not sufficient to train a more fine-grained distribution of the features.\nAnother reason could be that a higher number of components leads to overfitting, modelling also the noise in the recordings.\n\nDuring decoding, a grammar controls the possible recognition output.\nOur most simple employed grammar follows the basic HMM system setup where exactly one pass through a model is allowed for each recording.\nA multi-step grammar is then introduced to let the system automatically segment the recording:\nAny number of repetitions of the same model (subject) is allowed.\nIn order to train the HMMs to model the separate steps, an approach using a cyclic HMM topology is employed as described in the following.\n\n\\subsection{Step Modelling}\n\\label{sec:stepmodeling}\n\nTo be able to model the individual steps in each recording, we use cyclic HMMs.\nIn our basic HMM system, each recording (containing several gait cycles) is modelled by one pass through the HMM.\nThe strategy of representing each gait cycle separately by one pass through all states of the HMM is better suited to model the observations.\nWe consider the two halves of each gait cycle to be equivalent (although in fact, there is a person-dependent assymetry~\\cite{nixon2006human}),\nand therefore the system is designed to model half a gait cycle (containing one step) by each HMM.\nIn this way, one pass through the HMM models the sounds of one step and adjacent sounds (produced by the moving arms and legs). \nThis method of step modelling is implemented in the system configuraton and training in the following way:\nThe state transition matrix of each HMM has a left-right topology, and jumps from the last state to the first state are allowed.\nModels are trained with embedded re-estimation, where the number of steps is known (as determined by simple video processing methods).\nAs a result, the position of the steps in the training data is automatically estimated during model training.\nTogether with the introduced multi-step decoding grammar, the developed system is then capable of detecting, segmenting and recognising the steps occurring in the recordings.\n\n\n\\section{Experiments}\n\\label{sec:experiments}\n\nExperiments are performed with the TUM GAID database that was described in Section~\\ref{sec:database},\nusing the development set for system design and tuning.\nFinally, we use the test set to evaluate our best system configuration.\nFor all systems evaluated using the development set, 15 HMM states appeared to be the optimal configuration. \nIn addition, the best results were obtained with six training iterations.\nFor each system setup, we report experimental results (identification accuracy) separately for the three different recording conditions (normal, backpack, shoe covers).\nIn addition, the average accuracy over these three conditions is included.\n\n\n\n\n\n\n\\vspace{-0.2cm}\n\\subsection{Development set}\n\nTable~\\ref{tab:resultsdev} shows the results on the development set for different system configurations.\n\\begin{table}[t!]\n\\begin{center}\n\\caption{\\em Development set (150 subjects) evaluation of different audio features, for the normal ($\\mathcal{N}$), backpack ($\\mathcal{B}$) and shoe cover ($\\mathcal{S}$) recording conditions.}\n\\begin{tabular}{c|ccc|c}\n  & \\multicolumn{3}{|c|}{\\textbf{Condition}} & \\\\\n{\\textbf{Accuracy [\\%]}} & $\\mathcal{N}$ & $\\mathcal{B}$ & $\\mathcal{S}$ & \\bf average \\\\\n\\toprule\nbasic HMM & 53.3 & 30.7 & 7.0 & 28.2 \\\\\n+ multi-step decoding & 56.3 & 31.3 & 7.3 & 31.6 \\\\\n+ PCA & 57.7 & 34.3 & 9.7 & 33.9 \\\\\n+ step modelling & 69.7 & 44.7 & 9.3 & 41.2 \\\\\n\\end{tabular}\n\\label{tab:resultsdev}\n\\end{center}\n\\vspace{-0.2cm}\n\\end{table}\nThe basic HMM system without explicit modelling of separate steps (cf. Section~\\ref{ssec:basichmm}) is the first evaluated system.\nIn the normal recording condition, slightly more than half of the testing samples are classified correctly.\nAveraging over the three different conditions, an accuraccy of 28.2\\,\\% is obtained, which serves as a baseline for further experiments.\nThe first step towards the improved recognition system is the introduction of a decoding grammar which allows to recognise multiple sequential instances of the same subject in the recordings.\nThis modification improves the average accuracy to 31.6\\,\\% (mostly due to improvements in the $\\mathcal{N}$ setup).\nApplying PCA to the features improves the accuracy for all three recording conditions.\nTraining the system to model each step by one pass through an HMM (cf. Section~\\ref{sec:stepmodeling}) leads to the largest improvement in accuracy.\nIn the normal walking condition, more than two thirds of the samples are now identified correctly.\nThe accuracy in the backpack walking condition is also greatly improved, whereas the performance in the shoe cover condition remains largely unaffected.\nWhile the improvements obtained with the multi-step grammar and PCA are not significant, improved step modelling leads to a significant improvement in the $\\mathcal{N}$ and $\\mathcal{B}$ conditions and for the average accuracy (evaluated with a one-tailed t-test with a significance level of $\\alpha=0.05$).\n\n\n\nWith a simple analysis we examined the system's ability to correctly detect the individual steps.\nTo this end, we use the best-performing developed system (row four in Table~\\ref{tab:resultsdev}).\nFor the test samples of the normal walking conditions, we observe the number of steps detected by the system.\nThe average number of steps in these test recordings is 5.3, while the system predicts 4.3 steps, on average.\nFor correctly identified \\textit{subjects}, the average number of predicted \\textit{steps} is 5.0, while for incorrectly identified subjects it is 3.5.\nThis shows that when the subjects are identified correctly, the step segmentation works very well.\n\n\\vspace{-0.1cm}\n\\subsection{Test set}\n\nIn Table~\\ref{tab:resultstest}, we show the results on the test set, for our baseline system and the best system configuration.\nFor comparison, we include our previously published results on the same dataset.\n\\begin{table}[t!]\n\\begin{center}\n\\caption{\\em Test set (155 subjects) evaluation of our system compared to our previously published results, for the normal ($\\mathcal{N}$), backpack ($\\mathcal{B}$) and shoe cover ($\\mathcal{S}$) recording conditions.}\n\\begin{tabular}{c|ccc|c}\n  & \\multicolumn{3}{|c|}{\\textbf{Condition}} & \\\\\n{\\textbf{Accuracy [\\%]}} & $\\mathcal{N}$ & $\\mathcal{B}$ & $\\mathcal{S}$ & \\bf average \\\\\n\\toprule\nvideo (GEI) \\cite{13hof1} & 99.4 & 27.1 & 52.6 & 59.7 \\\\\nbaseline SVM \\cite{13hof1} & 44.5 & 27.4 & 4.8 & 25.6 \\\\\nSVM + feat. sel. \\cite{Geiger13GBP} & 51.9 & 28.4 & 4.2 & 28.2 \\\\\n\\midrule\nbasic HMM & 41.0 & 24.2 & 7.1 & 24.1 \\\\\nimproved HMM & 65.5 & 36.5 & 9.0 & 37.0 \\\\\n\\end{tabular}\n\\label{tab:resultstest}\n\\end{center}\n\\vspace{-0.2cm}\n\\end{table}\nThe first row shows results of a state-of-the-art gait recognition method working with video data, namely the GEI~\\cite{13hof1}.\nThis method achieves almost perfect results in the normal walking condition, while especially the backpack and also the shoe variation constitute a real difficulty for the system (59.7\\,\\% on average).\nHowever, these results have to be interpreted carefully, since the GEI utilises mainly the appearance (the silhouette of a person) and not the behaviour (the gait pattern).\nUsing a large set of different audio features (1\\,625 static features per recording) and SVMs for classification (second row) was our first audio-domain baseline system~\\cite{13hof1}.\nNaturally, the addressed task is much more difficult when dealing only with audio data (average accuracy 25.6\\,\\%).\nHowever, this system can compete with the GEI in the backpack recording variation.\nIn~\\cite{Geiger13GBP}, we improved the SVM system by employing a feature-selection technique to chose relevant features for the task,\nobtaining an average identification accuracy of 28.2\\,\\%.\nNow, with our basic HMM setup, the resulting accuracy of 24.1\\,\\% is comparable to the baseline SVM system.\nThe methods introduced in this work (primarily modelling each step separately during model training and decoding)\nare able to bring a large improvement, reaching 37.0\\,\\%.\nIn the $\\mathcal{N}$ and $\\mathcal{B}$ recording conditions, the accuracy is improved significantly, by more than one third.\nThe accuracy of the video-processing method (GEI) in the backpack recording condition is surpassed by 26\\,\\% relatively.\nCompared to the previous best-performing audio system (the SVM system including feature selection) the average accuracy is improved by 24\\,\\%, relatively (significant in all recording conditions).\n\n\n\n\n\\vspace{-0.2cm}\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\n\n\nWe developed a model-based system for recognising people from walking sounds.\nThe system uses HMMs in a cyclic topology to automatically segment the recordings according to separate steps.\nExperiments were conducted using the TUM GAID database containing recordings of 305 subjects (150 in the development set and 155 in the test set) in three different recording conditions:\nnormal walking, walking with a backpack, and walking with shoe covers.\nThe results show that a basic HMM system (without explicit modelling of separate steps) achieves a similar performance in comparison to the SVM system presented in our previous work.\nImproving the system with the methods introduced in this work results in large performance gains in identification accuracy.\nWith this system, each half gait cycle is modelled by one pass through a cyclic HMM.\nThis covers the sound of one step and adjacent sounds, which are mainly produced by moving arms and legs.\nThus, it is clear that the backpack or shoe cover variation influence the identification performance in a negative way.\nHowever, when identification experiments are carried out with the same walking style and shoe type as the model was trained with (normal walking condition), almost two thirds of the subjects are identified\ncorrectly from the test set containing 155 individuals.\n\n\n\nGiven the challenging but application-friendly enrollment of only four examples per walking subject\nand in order to improve the robustness of the system, adopting approaches \nfrom speaker recognition like creation of models through adaption from a background model~\\cite{reynolds2000speaker} could be a promising strategy in the future.\nFurthermore, we will work on improving the system's robustness to variations.\nThis includes better coping with the backpack and shoe cover recording conditions.\nIn addition, the TUM GAID database contains a set of subjects with recordings made on two different dates in time (with three months in between).\nTherewith, the influence of changing types of shoes and clothes as well as possibly higher variation of the walking style on the system performance can be evaluated.\nIn order to improve the system in this direction, we want to test approaches to address session variability known from speaker recognition (such as joint factor analysis~\\cite{kenny2007joint}) as well as methods for model adaptation or feature transformation adopted from speech recognition systems.\n\n\n\\vfill\\pagebreak\n\n\n\\balance\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Diffusion and opacities in cool B stars}\n\nSlowly rotating B stars on the main sequence are thought to be extremely stable as\nthere is only limited convection in the outer envelope and none at the surface, mass\nloss is expected to be small and rotational mixing is also negligible.\nIn such a stable environment, diffusion should proceed with few impediments.\nConsequently, the large abundance anomalies observed in HgMn stars are understood to be \nthe result of diffusion in the atmosphere of such stars.\n\n\nThat diffusion occurs in the atmosphere suggests that diffusion also occurs in the interior\nin the absence of mixing processes. \nRicher, Michaud \\&~Turcotte~(2000) and Richard, Michaud \\&~Richer~(2002) (see also Richard \nin these proceedings) have shown that\nradiative levitation pushes iron-peak elements up in the envelope of hot A and \ncool B stars. While these elements are radiatively supported throughout\nthe outer envelope of these stars, they tend to accumulate at a temperature of \nroughly 200\\,000~K because of a local reduction of the outward flux there.\nAt such a temperature, iron-peak elements are the dominant contributors to the opacity \nand, naturally, as they accumulate, the opacity also increases locally. \nFigure~1 shows the abundance profiles of two models, one with significant diffusion and\nanother where the effect of diffusion is only marginal. An overabundance of the order \nof a factor of ten is achieved in the former.\n\nWith the notable exception of Hydrogen, lighter elements that play an important \nrole in the opacity at lower temperature \nare generally not supported by radiative pressure and therefore sink out of the superficial\nregions toward the core. Consequently, their contribution to \nthe opacity diminishes.\n\nThe combined effect of the evolution of the chemical composition as a result of \ndiffusion yields a marked increase\nin the opacity at around 200\\,000~K and, perhaps somewhat surprisingly, an increase\nof the opacity at lower temperatures, as shown in Figure~1, due to the combined increase\nin the opacity due to hydrogen and iron. \n\nNotice that the abundances are homogeneous from the surface to a point deeper than\n200\\,000~K ($\\log T =5.3$). These low temperature regions are artificially homogenized\nwith an ad hoc turbulent mixing coefficient. In one respect this allows us to tweak the\nlevel of chemical anomaly in order to investigate the effect of diffusion in the star's\ninterior. Unfortunately, it also means that the structure of the cool regions of\nthe envelope may be significantly inaccurate.\n\n\\section{Pulsations in cool B stars}\n\nThe kappa-mechanism due to the opacity of iron-peak elements\nis responsible for variability on the lower main sequence (Pamyatnykh 1999). \nThe SPB stars (Slowly Pulsating B stars; see Pamyatnykh) are long-period pulsators\nfound in chemically normal young main-sequence stars earlier than B8. \nThe distribution of these stars overlap in the H.-R. diagram those of\nthe chemically peculiar but seemingly stable HgMn stars.\n\nApart from variability and surface chemical composition these two classes of\nstars are very similar. Interestingly, SPB stars are found to be mostly slowly rotating\nstars,  as are the HgMn stars, but the lack of rapidly rotating SPB stars may well be\nonly a selection effect.\nThis suggests that there might well be a correlation between the chemical composition \nand the excitation of the pulsations, as in Am stars where diffusion leads to\nstability, or that the conditions that allow diffusion are\nnot conducive to pulsations occurring.\n\nThe best models currently available (Turcotte \\& Richard, submitted) do not however \nsupport the hypothesis that diffusion can undermine the excitation of pulsations.  \nAs the opacity bump due to iron-peak is enhanced as a result of diffusion\nin those models, they suggest that the excitation of pulsations in HgMn stars should\nbe at least as high than in chemically normal SPB stars. Again in Figure~1, the differential work \nfor a given mode of pulsation is shown in a model with nearly normal composition \nand one with strongly enhanced\niron and opacity in the driving region. The net normalized growth rate, which must be positive\nfor a mode to be unstable and for the star to become variable, for this\nmode is 0.08 in the ``normal'' model and 0.22 in the ``peculiar'' model. The \npeak in the driving region is higher, but there is also more damping on the \nhot side of the peak.  In this mode the net driving is in fact considerably enhanced,\nbut in many modes, especially in more evolved models, the net excitation (the value \nof the normalized growth rate) is surprisingly insensitive to the \nmagnitude of the abundance anomalies.\nNevertheless, one must conclude that the models are lacking the necessary ingredient \nto explain the lack of observed pulsations in HgMn stars.\n\\begin{figure}\n  \\includegraphics[width=13cm]{turcotte_fig.eps}\n\\caption{The figure illustrates the effect of diffusion on the excitation of pulsations\nin the model of a 10~Myr old 4~M$_\\odot$ star. Two models of the same age and mass are compared,\none with only marginal change in abundances (dashed line) and one with more efficient\ndiffusion (solid line). The top panel shows the iron abundance profiles;\nthe middle panel the mean Rosseland opacity; and the bottom panel shows the \ndifferential work for a $\\ell=1, n=16$ mode with a period of 1.2 days. A positive work indicates\nmode excitation while a negative value indicates damping. By integrating the differential work\nover the whole star we obtain the total work integral which is used to calculate the \nnormalized growth rate (see text).}\n\\end{figure}\n\n\n\\section{What does this tell us about cool B stars?}\n\nWe can speculate as to what is the missing ingredient in the models.\n \nWe can first argue that adding mixing in the interior of HgMn stars \nwould not resolve the discrepancy as the result of mixing would be to homogenize\nthe composition to its initial, here solar, value. This would still leave too much iron-peak\nelements in the pulsations' driving region, leading to the expectation\nof pulsations in HgMn stars as in SPB stars.\n\nA possible solution may be the selective mass loss of certain elements from radiation\npressure in the atmosphere but not others (Babel 1995).  It is possible that this\nmay lead to the depletion of some elements in the driving region. \nThe detailed process by which this depletion would occur, if indeed it can,\nhas not been worked out yet. \n\nAnother possibility is that the issues of mode selection, interference or\nvisibility that often befall the asteroseismology of pulsating stars obscures\nany direct conclusions we can hope to make on models and stellar physics \nfrom the observations. \n\nFinally, our models are lacking in one crucial aspect.\nOur current models cannot model the region cooler than 200\\,000~K consistently \nbecause of numerical problems. Therefore the structure of the models there may not\nbe appropriate. Though the work integrals seem rather insensitive to those regions,\na substantial change in structure there may lead to smaller predicted excitations.\n\nThe major stumbling block to improved models is the lack of\nopacity spectra appropriate to model diffusion consistently at low temperatures \n(Leblanc, Michaud \\&~Richer~2000).  Only when this will be possible will the \nfull picture of mode driving in HgMn stars be achieved. Before then, the models \nremain informative of the processes that occur in the interior, but speculative\nas to the net effect of diffusion of mode damping.\n\nObservationally, the advent of space-based experiments dedicated to asteroseismology\nwill eventually resolve the question of whether HgMn stars are really stable or if \nthey undergo undetected low-amplitude variations. Observations are underway to \nidentify faint HgMn stars at the VLT so they can thereafter be observed in the\nplanetary field of CoRot. Whether very-low amplitude modes are detected or not, these \nobservations will pose important new constraints on the models.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:introduction}\n\nCoreference annotations are costly and difficult to obtain, since trained annotators with sufficient world knowledge are necessary for reliable annotations. This paper presents a way to {\\em simulate} annotators using reinforcement learning. To motivate our approach, we rely on the following example from \\newcite[colors added to mark entity mentions]{strube-etal}:\n\\begin{itemize}\n    \\item[(1)] [\\clusterAbf{Lynyrd Skynyrd}]$_1$ was formed in \\clusterBbf{Florida$_2$}. Other bands from [\\clusterBbf{the Sunshine State}]$_2$ include \\clusterCbf{Fireflight} and \\clusterDbf{Marilyn Manson}.\n\\end{itemize}\n\n\\newcite{strube-etal} cite the association between \\clusterB{Florida} and \\clusterB{the Sunshine State} as an example of a common source of name-name recall error for state-of-the-art coreference resolution systems. The challenge is that the two names co-occur relatively infrequently and are unlikely to do so in a moderate-sized, manually annotated training corpus. A state-of-the-art system may be able to infer the relation using distributional information about the phrase \\clusterB{the Sunshine State}, but is likely to have limited evidence for the decision that it is coreferential with \\clusterB{Florida} rather than \\clusterA{Lynyrd Skynyrd}.\n\nWhile coreference-annotated data is scarce, knowledge bases including factual information (such as that \\clusterC{Fireflight} is from \\clusterB{Florida}) are increasingly available. For a human annotator unaware that \\clusterB{Florida} is sometimes referred to as \\clusterB{the Sunshine State}, the information that \\clusterC{Fireflight} is from \\clusterB{Florida} is sufficient to establish that \\clusterB{Florida} and \\clusterB{the Sunshine State} are (with high probability) coreferential. This paper explores a novel architecture for making use of such information from knowledge bases by tying a coreference resolution system to a relation extraction system, enabling us to reward the coreference system for making predictions that lead us to infer facts that are consistent with such knowledge bases. This potentially provides us with more evidence for resolving coreference such as (1). \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{images\/arch.pdf}\n    \\caption{\\label{fig:sys_arch}Our strategy for training a coreference resolver using reward from relation extraction.}\n\\end{figure}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{images\/overall-newest.pdf}\n    \\caption{\\label{fig:overall}The columns show the different pipelines used to obtain data for training the reward models. The pipeline for: (i) RE-KG directly extracts triples from Wikidata, (ii) RE-Text runs Wikipedia summaries through OpenRE to generate triples, and (iii) RE-Joint adds an additional verification step by checking if the generated triples exist in Wikidata.}\n\\end{figure*}\n\n\nWe propose a training strategy (Figure~\\ref{fig:sys_arch}) in which we pass on the predictions of a neural coreference resolver to an open relation extraction (OpenRE) system, matching relations extracted from resolved sentences with a knowledge base. We show how checking the produced relationships for consistency against the knowledge base produces a reward that is, indirectly, a signal about the quality of the coreference resolution. In order to generalize this signal beyond the coverage of the knowledge base, we train a Universal Schema model \\cite{riedel-etal-2013-relation} and use its confidence as our reward function.  With this reward function, we do policy-gradient fine-tuning of our coreference resolver, effectively optimizing its predictions' consistency with world knowledge. \n\n\\paragraph{Contributions}\nWe demonstrate that training a coreference resolver by reinforcement learning with rewards from a relation extraction system, results in improvements for coreference resolution.\nOur code is made publicly available at \\url{https:\/\/github.com\/rahular\/coref-rl}\n\n\\section{Consistency Reward for Coreference Resolution}\\label{sec:strategy}\n\nIn order to reward a coreference resolver for being consistent with world knowledge, we propose a simple training strategy based on relation extraction: (i) Sample a Wikipedia\\footnote{\\url{https:\/\/www.wikipedia.org}} document at random, (ii)  Replace mentions with their antecedents using a coreference resolver, (iii)  Apply an off-the-shelf openRE system to each rewritten document, (iv) Score relationships that include coreferent mentions using Universal Schema, and (v)  Use the score as a reward for training the coreference resolvers. \n\n\\paragraph{Reward functions} To model consistency with world knowledge, we train different Universal Schema models \\cite{riedel-etal-2013-relation,Verga:McCallum:16}, resulting in three reward functions (Figure~\\ref{fig:overall}): \\textbf{RE-KG} (Knowledge Graph Universal Schema) is trained to predict whether two entities are linked in Wikidata\\footnote{\\url{https:\/\/www.wikidata.org}}; \\textbf{RE-Text} (Text-based Universal Schema) is trained to predict whether two entities co-occur in Wikipedia; and \\textbf{RE-Joint} (Joint Universal Schema) is trained to predict whether two entities are linked {\\it and} co-occur. The three rewards focus on different aspects of relationships between entities, giving complimentary views of what entities are related.\n\nSimilar to \\newcite{verga-etal-2016-multilingual}, we parameterize candidate relation phrases with a BiLSTM \\cite{graves2005framewise}, and use pre-trained Wikidata BigGraph embeddings \\cite{pbg} as the entity representations. We apply a one-layer MLP on the concatenated representations to get the reward value.\n\n\\paragraph{Updating the coreference resolver} Each resolved document is converted into $n$ subject-relation-object (SRO) triples by an open information retrieval system \\cite{openIE}. Each triple $t_i$ is then scored using a reward function to obtain a reward $r_i$ for $i \\in \\{1, \\ldots, n\\}$. The final document-level reward is the normalized sum of the individual rewards as shown in Equation~\\ref{eqn:reward}, where $R_h$ is a moving window containing the previous $h=100$ normalized reward values.\n\n\\begin{equation}\n    R = \\frac{\\sum_{i} r_i - mean(R_h)}{stddev(R_h)}\n    \\label{eqn:reward}\n\\end{equation}\n\nSince $R$ is not differentiable with respect to the coreference resolver's parameters, we use policy gradient training to update the coreference resolver. We select the best action according to the current policy, using random exploration of the alternative solutions with $p=\\frac{1}{10}$.\n\n\\paragraph{Multi-task reinforcement learning} Our overall  training procedure is presented in Algorithm~\\ref{alg:distral}. After training the three aforementioned reward models, we create \\textbf{RE-Distill} by interpolating their trained weights. Next, we pre-train a coreference resolver using supervised learning, and fine-tune it using each of the three reward functions to get three different coreference policies: \\textbf{Coref-KG}, \\textbf{Coref-Text} and \\textbf{Coref-Joint}, respectively. We then use multi-task reinforcement learning to combine these three policies to get \\textbf{Coref-Distill}. Our approach is a particular instance of DisTraL \\cite{distral}, using policy gradient and model interpolation.  Finally, \\textbf{Coref-Distill} is fine-tuned with rewards from {\\bf RE-Distill}. \n\n\\begin{algorithm}\n\\begin{small}\n\\caption{\\label{alg:distral} Multi-task Reinforcement Learning}\n\\begin{algorithmic}\n\\REQUIRE  Baseline initialized policies $\\theta_n$ for $n\\in \\{1,2,3\\}$ \\label{alg:pretrained-policies}\\\\\n\\REQUIRE Reward functions \\texttt{reward$_n$} for $n\\in \\{1,2,3\\}$\n\\REQUIRE Distilled reward function \\texttt{reward$_*$}\n\\WHILE{stopping criterion not met}\n\\STATE Sample $k$ documents $D^k$\n\\FOR{$d\\in D^k$}\n\\FOR{$n\\in\\{1,2,3\\}$}\n\\STATE $\\mathcal{C}_d$ = entity clusters with $\\theta_n$\n\\STATE $d'$ = resolve $d$ with $\\mathcal{C}_d$\n\\STATE $\\mathcal{T}$ = obtain OpenIE triples for $d'$ \n\\STATE $r$ = reward$_n$($d'$)\n\\STATE $\\hat{g}_k$ = policy gradient for $\\theta_n$ with reward $r$\n\\STATE\n$\\theta_n^{k+1}=\\theta_n^k+\\alpha_k\\hat{g}_k$\n\\ENDFOR\n\\ENDFOR\n\\ENDWHILE \n\\STATE Distilled policy $\\theta_*=\\frac{\\theta_1+\\theta_2+\\theta_3}{3}$\n\\STATE Sample $k$ documents $D^k$\n\\FOR{$d\\in D^k$}\n\\STATE $d'$ = resolve $d$ with $\\mathcal{C}_d$\n\\STATE $\\mathcal{T}$ = obtain OpenIE triples for $d'$ \n\\STATE $r$ = reward$_*$($d'$)\n\\STATE $\\hat{g}_k$ = policy gradient for $\\theta_*$ with reward $r$\n\\STATE\n$\\theta_*^{k+1}=\\theta_*^k+\\alpha_k\\hat{g}_k$\n\\ENDFOR\n\\RETURN Distilled policy $\\theta_*$\n\\end{algorithmic}\n\\end{small}\n\\end{algorithm}\n\n\\section{Experiments}\\label{sec:experiments}\n\nWe use a state-of-the-art neural coreference resolution model \\cite{lee2018higher} as our baseline coreference resolver.\\footnote{\\url{https:\/\/github.com\/kentonl\/e2e-coref}} This model extends \\citet{lee2017end} with coarse-to-fine inference and ELMo pretrained embeddings \\cite{peters2018deep}.\n\n\n\\paragraph{Data} We use the standard training, validation, and test splits from the English OntoNotes.\\footnote{\\url{https:\/\/catalog.ldc.upenn.edu\/LDC2013T19}}\nWe also evaluate on the English WikiCoref \\cite{wikicoref}, with a validation and test split of 10 and 20 documents respectively.\n\n\\paragraph{Reward model training} We use data from English Wikipedia and Wikidata to train our three reward models.\nFor training \\textbf{RE-KG}, we sample 1 million Wikidata triples, and expand them to 12 million triples by replacing relation phrases with their aliases.\nFor \\textbf{RE-Text}, we pass the summary paragraphs from 50,000 random Wikipedia pages to Stanford's OpenIE extractor \\cite{corenlp}, creating 2 million triples.\nFor \\textbf{RE-Joint}, we only use Wikipedia triples that are grounded in Wikidata, resulting in 60,000 triples.\\footnote{That is, we retain only those triples whose subject and object can be linked to an entity in Wikidata.}\nWe further sample 200,000 triples from Wikidata and Wikipedia for validation, and train the reward models with early stopping based on the F$_1$ score of their predictions.\n\n\\paragraph{Evaluation}\nAll models are evaluated using the standard CoNLL metric, which is the average F$_1$ score over MUC, CEAFe, and $B^3$ \\cite{denis2009global}.\n\n\\section{Results}\\label{sec:results}\n\nSince the quality of our reward models is essential to the performance of the coreference resolver adaptations, we first report the validation accuracy and F$_1$ scores of the four reward models used, in Table~\\ref{tab:reward_results}. We clearly see the advantage of distillation, with a 5\\% absolute difference between the best single model ({\\bf RE-Text}) and {\\bf RE-Distill}.\n\n\\begin{table}[t]\n    \\centering\n    \\begin{tabular}{l|ccc}\n        \\toprule\n        {\\bf System} & {\\bf Data} & {\\bf Accuracy} & {\\bf F$_1$ score} \\\\\n        \\midrule\n        RE-KG & 12M & 0.64 & 0.78 \\\\\n        RE-Text & 2M & 0.71 & 0.83 \\\\\n        RE-Joint & 60K & 0.58 & 0.73 \\\\\n        \\midrule\n        RE-Distill & --- & \\textbf{0.78} & \\textbf{0.88} \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{Training data size, accuracy and F$_1$ scores of the reward models on the 200,000 validation triples.\\label{tab:reward_results}}\n\\end{table}\n\n\\begin{table}[t]\n    \\centering\n    \\begin{tabular}{l|cc}\n        \\toprule\n        {\\bf System} & {\\bf OntoNotes}&{\\bf WikiCoref} \\\\\n        \\midrule\n        \\newcite{lee2018higher} & 72.60 & 57.49 \\\\\n        \\midrule\n        Coref-KG & 72.96 & 57.84 \\\\\n        Coref-Text & 72.99 & 57.54 \\\\\n        Coref-Joint & 72.77 & 57.51 \\\\\n        \\midrule\n        Coref-Distill & \\textbf{73.10} & \\textbf{58.14} \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{Coreference results: average F$_1$ scores on the OntoNotes and WikiCoref test sets. Differences are significant w.r.t. $B^3$ (bootstrap test, $p<0.05$).\\label{tab:coref_results}} \n\\end{table}\n\nTable~\\ref{tab:coref_results} presents the downstream effects of applying these reward functions to our baseline coreference policy.\\footnote{The models were re-trained from scratch, and the scores are slightly different from those reported in \\newcite{lee2018higher}.}\n\nThe coreference resolution results are similar to the relation extraction results: using a distilled policy, learned through multi-task reinforcement learning, leads to better results on both datasets.\\footnote{We repeat this experiment three times with different random seeds and observed the same pattern and very robust performance across the board.}\n\nWhile improvements over the current state of the art are relatively small, they reflect significant progress, as they demonstrate the ability to successfully augment coreference resolvers with ``free\" data from large-scale KB like Wikidata. For relation extraction, this could have positive downstream effects, and also ensure that relations are consistent with real world knowledge. Moreover, this approach has the potential to also be beneficial for coreference resolution in low resource languages, where less annotated data is available, as Wikidata triples are abundant for many languages.\n\n\\section{Analysis}\\label{sec:analysis}\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{images\/analysis.pdf}\n    \\caption{\\label{fig:analysis}Mention detection and linking examples by the baseline system from \\newcite{lee2018higher}, and the best performing fine-tuned system (Coref-Distill). Mentions of the same color are linked to form a coreference cluster.}\n\\end{figure*}\n\nEmpirically, we find that fine-tuning the coreference resolver on Wikidata results in two kinds of improvements: \n\n\\paragraph{Better mention detection} Since the model is rewarded if the SRO triples produced from the resolved document are present in Wikidata, the model can do well only if it correctly resolves the subject and object, which are usually named entities (more generally, noun phrases). Indeed, we see an improvement in mention detection as exemplified in the first example of Figure~\\ref{fig:analysis}. Compared to the baseline, the fine-tuned model identifies a larger number of entities, including ``southern hemisphere'', ``Cambridge'' and ``Oxford'', which are missed by the baseline model.\n\n\\paragraph{Better linking}  As a direct consequence of the above, the model is inclined to also link noun phrases that are not entities. In the second example of Figure~\\ref{fig:analysis}, we see that ``This attempt'' is linked to ``releasing'' by the fine-tuned model. Interestingly, we do not see this type of \\textit{eventive} noun phrase linking either in OntoNotes or in the predictions of the baseline model. \n\nThis phenomenon, however, also has a side-effect of producing singleton clusters and spurious linking, which adversely affect the recall. On the OntoNotes test data, while the average precision of the best performing fine-tuned model is higher than the baseline (75.62 vs. 73.80), a drop in recall (70.75 vs. 71.34) causes the final F$_1$ score to only marginally improve.\n\n\\section{Related Work}\\label{sec:related}\n\\paragraph{Coreference resolution} Among neural coreference resolvers \\cite{Wu2017ADL, Meng2018TriadbasedNN}, \\citet{lee2017end} were the first to propose an end-to-end resolver which did not rely on hand-crafted rules or a syntactic parser. Extending this work, \\citet{lee2018higher} introduced a novel attention mechanism for iteratively ranking spans of candidate coreferent mentions, thereby improving the identification of long distance coreference chains. \\newcite{zhang-etal} improve pronoun coreference resolution by 2.2 F1 using linguistic features (gender, animacy and plurality) and a frequency based predicate-argument selection preference as external knowledge. \\newcite{emami-etal} incorporate knowledge into coreference resolution by means of information retrieval, finding sentences that are syntactically similar to a given instance, and improving F1 by 0.16.\n\n\\paragraph{Reinforcement learning} RL has been used for many NLP tasks, including coreference resolution \\cite{clark2016deep} and relation extraction \\cite{Zeng2018LargeSR}. \\citet{clark2016deep} use RL to improve coreference resolution by optimizing their mention ranking model and directly use the standard evaluation metrics as the rewards. We, on the other hand, perform end-to-end optimization by rewarding the model's consistency with real world knowledge using relation extraction. To our knowledge, we are the first to use consistency with world knowledge as a reward for tasks other than knowledge base construction.\\footnote{\\newcite{Mao:ea:18}, for example, use reinforcement learning with consistency-like reward to induce lexical taxonomies.}  \n\n\\paragraph{Knowledge bases} Knowledge bases have been leveraged across multiple tasks across NLP \\cite{Bordes2011LearningSE,Chang2014TypedTD, Lin2015ModelingRP, Toutanova2015RepresentingTF, Yang2017LeveragingKB}. Specifically for coreference resolution, \\citet{Prokofyev2015SANAP} implement a resolver that ensures semantic relatedness of resulting coreference clusters by leveraging Semantic Web annotations. Their work incorporates knowledge graph information only in the final stage of the resolver's pipeline, and not during training. In contrast, our work augments information from the knowledge base directly into the training pipeline. Also, they use DBpedia \\cite{dbpedia07} as the ontology. Although both Wikidata and DBpedia are designed to support working with Wikipedia articles, DBpedia can be considered as a subset of Wikidata as Wikipedia infoboxes are its main data source. The advantage of Wikidata over DBpedia is its size, and the fact that it is multilingual, which will allow applying our method to other languages in the future. \n\n\\section{Conclusion}\\label{sec:conclusion}\n\nWe presented an architecture for adapting coreference resolvers by rewarding them for being consistent with world knowledge. Using simple multi-task reinforcement learning and a knowledge extraction pipeline, we achieved  improvements over the state of the art across two datasets. We believe this is an important first step in exploring the usefulness of knowledge bases in the context of coreference resolution and other discourse-level phenomena. In this area, manually annotated data is particularly expensive, and we believe leveraging knowledge bases will eventually reduce the need for manual annotation. \n\n\\section*{Acknowlegments}\n\nWe thank the reviewers for their valuable comments.\nRahul Aralikatte, Daniel Hershcovich, Heather Lent, and Anders S{\\o}gaard are funded by a Google Focused Research Award. Heather Lent is also funded by the European Union's Horizon 2020 research and innovation programme under the Marie Sk{\\l}odowska-Curie grant agreement No. 801199. Chen Qiu is funded in part by the National Natural Science Foundation of China under grant No. 61773355 and the China Scholarship Council.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n The field of communication complexity studies the amount of communication required to solve the problem of computing discrete functions when the input is split between two or more parties. In the most commonly studied framework, there are two parties, often called Alice and Bob, and a communication problem is defined by a Boolean matrix $M=[m_{ij}]_{n \\times n}$, where   \\emph{Boolean} means that the entries are either $0$ or $1$.   Alice receives a row number, and Bob receives a column number $j$. Together, they should both compute the entry $m_{ij}$ by exchanging bits of information in turn, according to a previously agreed-on protocol. There is no restriction on their computational power; the only measure we care to minimize is the number of exchanged bits.\n\nA deterministic protocol $\\pi$ specifies how the communication proceeds. It specifies what bit a player sends at each step. This bit depends on the input of the player and the history of the communication so far. It is often assumed that the last communicated bit must be the output of the protocol.   A protocol naturally corresponds to a binary tree as follows. Every internal node of the tree is associated with either Alice or Bob.  If an  internal node $v$ is associated with  Alice, then it is labeled with a Boolean function $a_v$, which prescribes the bit sent by Alice at this node as a function of her input $i$. Similarly, the nodes associated with Bob are labeled with Boolean functions of $j$.  Each leaf is labeled by $0$ or $1$ which corresponds to the output of the protocol at that leaf.\n\nWe denote the number of bits exchanged on the input $(i,j)$ by $\\cost_\\pi(i,j)$, which is precisely the length of the path from the root to the corresponding leaf. The \\emph{communication cost} of the protocol is simply the depth of the protocol tree, which is the maximum of $\\cost_\\pi(i,j)$  over all inputs $(i,j)$. \n$$\\cc(\\pi) \\coloneqq \\max_{i,j}  \\cost_\\pi(i,j).$$\n\n\n We say that $\\pi$ computes $M$  if $\\pi(i,j)=m_{ij}$ for all $(i,j)$, where $\\pi(i,j)$ denotes the protocol's output on the input $(i,j)$.   The \\emph{deterministic communication complexity} of $M$, denoted by $\\DD(M)$, is the smallest communication cost of a protocol that computes $M$.  It is easy to see that $\\DD(M)\\le \\ceil{\\log(n)}+1$ as Alice can use $\\ceil{\\log(n)}$ bits to send her entire input to Bob, and Bob knowing the values of both $i$ and $j$, can send back $m_{ij}$. \n\nIt is well-known that if the deterministic communication complexity of a  matrix is bounded by a constant $c$, then the matrix is highly structured -- its rank is at most $2^c$, and it can be partitioned into at most $2^c$  all-zero and all-one submatrices~\\cite{MR1426129}. These facts characterize the family of matrices that satisfy $\\DD(M)=O(1)$.  A fundamental problem in communication complexity, with connections to harmonic analysis and operator theory~\\cite{Hat21}, is to obtain a  characterization of families of matrices that have $O(1)$ \\emph{randomized} communication complexity. \n\nA (public-coin) \\emph{randomized protocol} $\\pi_R$ of cost $c$ is simply a probability distribution  over the deterministic protocols of cost $c$. Given an input $(i,j)$, to compute $m_{ij}$, Alice and Bob use their shared randomness to sample a deterministic protocol from this distribution, and execute it. \n\nWe say that the error probability of $\\pi_R$ is at most $\\epsilon$ if $\\Pr[\\pi_R(i,j) \\neq m_{ij}] \\le \\epsilon$ for every input $(i,j)$.  For $\\epsilon \\in (0,1\/2)$, let $\\RR_\\epsilon(M)$ denote the smallest cost of a randomized protocol that computes $M$ with error probability at most $\\epsilon$. Note that $\\epsilon=1\/2$ can be easily achieved by  outputting a random bit; hence it is crucial that $\\epsilon$ is defined to be strictly less than $1\/2$. It is common to take $\\epsilon=\\frac{1}{3}$. Indeed, the choice of $\\epsilon$ is not important as long as $\\epsilon \\in (0,1\/2)$, since the probability of   error can be reduced to any  constant $\\epsilon'>0$ by repeating the same protocol  independently for some $O(1)$ times, and outputting the most frequent output.   We denote $\\RR(M) \\coloneqq \\RR_{1\/3}(M).$\n\nIt is well-known that the $n \\times n$ identity matrix $\\mathtt{I}_n$ satisfies $\\RR(\\mathtt{I}_n)\\le 3$ and $\\DD(\\mathtt{I}_n) =\\ceil{\\log(n)}+1$.  Hence, in contrast to the deterministic case, there are matrices with $\\RR(M)=O(1)$ that have arbitrarily large rank. \n\nThere are very few known examples of matrix classes that have  randomized communication complexity $O(1)$~\\cite{Hat21,harms2021randomized}. Let $\\mathcal M=(M_n)_{n \\in \\mathbb{N}}$ be a sequence of $n \\times n$ Boolean matrices $M_n$, and define $\\RR(\\mathcal M): n \\mapsto \\RR(M_n)$.   Let us look at some necessary conditions for $\\mathcal M$ to satisfy $\\RR(\\mathcal M)=O(1)$. \n\nLet $\\cl(\\mathcal M)$ denote  the \\emph{closure} of $\\mathcal M$, defined as the set of all square matrices that are a submatrix of some $M_n$. Note that $\\cl(\\mathcal M)$ is the smallest hereditary property that contains all the matrices in $\\mathcal M$, where a set of square matrices is called \\emph{hereditary} if it is closed under taking square submatrices. \n\n\nLet $\\GT_k$ denote the $k \\times k$ \\emph{Greater-Than}   matrix defined as $\\GT_k(i,j)=1$ if and only if $i \\le j$. The sequence $\\mathcal M$ is called \\emph{stable}, if there exists $k\\in \\mathbb{N}$ such that   $\\GT_k \\not\\in \\cl(\\mathcal M)$. It is well-known~\\cite{MR3439794,RamamoorthyS15} that $\\RR(\\GT_k)= \\Omega(\\log \\log k)$ which tends to infinity as $k$ grows. Hence, if $\\RR(\\mathcal M)=O(1)$, then $\\mathcal M$ must be stable. The term stability is coined due to Shelah's unstable formula theorem in model theory which characterizes unstable theories by the nonexistence of countably infinite half-graphs~\\cite{MR1083551}, where half-graphs are the graphs with biadjacency matrix $\\GT_k$ for some $k$. Stable graph families are known to have useful properties such as strong regularity lemmas~\\cite{malliaris2014regularity} and the Erd\\H{o}s-Hajnal property~\\cite{chernikov2018note}. \n\nThe second necessary condition for $\\RR(\\mathcal M)=O(1)$ follows from a bound on the number of matrices with $O(1)$ randomized communication complexity.  A standard derandomization argument, \\cref{prop:CountR}, shows that the number of such $n\\times n$ matrices  is bounded by $2^{O(n \\log n)}$. Consequently, if $\\RR(\\mathcal M)=O(1)$, then  $|\\cl(\\mathcal M)_n|\\leq 2^{O(n \\log n)}$, where  $\\cl(\\mathcal M)_n$ denotes the set of $n \\times n$ matrices in $\\cl(\\mathcal M)$. Thus, in the terminology of graph theory~\\cite{MR1490438,MR1769217}, the speed of growth of $|\\cl(\\mathcal M)_n|$ is at most \\emph{factorial}. \n\nInspired by the \\emph{Implicit Graph Conjecture}~\\cite{MR1186827} and its connection to the growth rate of hereditary graph properties,  Harms, Wild and Zamaraev~\\cite{harms2021randomized} formulated a probabilistic version of the  Implicit Graph Conjecture, which translates to the following statement in communication complexity (See \\cite[Conjecture 1.2 and Proposition 1.6]{harms2021randomized}). \n\n\\begin{conjecture}[Probabilistic Universal Graph Conjecture~\\cite{harms2021randomized}]\n\\label{conj:PUG}\nLet $\\mathcal M$  be a sequence of $n \\times n$ Boolean matrices. Then $\\RR(\\mathcal M)=O(1)$ if and only if $\\mathcal M$ is stable and  $|\\cl(\\mathcal M)_n|\\leq 2^{O(n \\log n)}$. \n\\end{conjecture}\n\n\\cref{conj:PUG}  speculates that the  two necessary conditions for $\\RR(\\mathcal M)=O(1)$ that we discussed above are also sufficient. In other words, they  characterize Boolean matrices  that have randomized communication complexity $O(1)$. It is shown in \\cite{harms2021randomized} that \\cref{conj:PUG} is true for matrix sequences corresponding to restricted classes of hereditary graph families such as monogenic bipartite families, interval graphs, and families of bounded twin-width.\n\n\n\nIn this article, we prove the following theorem which refutes~\\cref{conj:PUG}. \n\n\\begin{theorem}[Main Theorem]\n\\label{thm:main}\nThere exists a stable sequence $\\mathcal M$ of  Boolean matrices $(M_n)_{n \\in \\mathbb{N}}$ such that $\\RR(M_n)=\\Theta(\\log(n))$ and $|\\cl(\\mathcal M)_n|\\leq 2^{O(n \\log n)}$. \n\\end{theorem}\nNote that every $n\\times n$ matrix $M$ satisfies $\\RR(M)= O(\\log n)$. In particular, the above construction shows that this maximum is achievable even for stable hereditary matrix families of speed at most factorial. \n\n\n\nFurthermore, as a consequence of \\cref{conj:PUG},   \\cite{harms2021randomized} speculates that the randomized communication complexity of every hereditary property of Boolean matrices $\\mathcal M$ with at most factorial speed has a gap behavior, either $\\RR(\\mathcal M)=O(1)$ or $\\RR(\\mathcal M)=\\Omega(\\log \\log n)$. We refute this weaker conjecture as well. In particular, \\cref{main:thm2}, proved in \\cref{sec:mainthm2}, shows that for every growing function $w(n)<10^{-3}\\log n$, there exists a matrix sequence $\\mathcal M=(M_n)_{n\\in \\mathbb{N}}$ such that $\\RR(M_n)=w(n)$, and every $n^{1\/4}\\times n^{1\/4}$ submatrix $F$ of $M_n$ satisfies $\\RR(F)=O(1)$. As the proof of \\cref{thm:main} demonstrates, if we take $w(n)$ to be any function that is $\\omega(1)$ and $o(\\log \\log (n))$, then  $\\cl(\\mathcal M)$ is a hereditary matrix property with factorial speed and $\\RR(\\mathcal M)=\\Theta(w(n))$. \n\nWe present the proof of \\cref{thm:main}, which builds on \\cref{main:thm2}, in \\cref{sec:thmmain}. \n\n\\section{Preliminaries}\n\nAll logarithms in this article are in base $2$. For a positive integer $n$, we denote $[n]=\\{1,\\ldots,n\\}$.  We use the standard Bachmann-Landau asymptotic notations: $O(\\cdot)$, $\\Omega(\\cdot)$, $\\Theta(\\cdot)$, $o(\\cdot)$, and $\\omega(\\cdot)$. \n\nWe will need the following concentration inequality. \n\n\\begin{theorem}[Bernstein's inequality (See~\\cite{MR3185193})]\\label{thm:bernstein}\nLet $X$ be the sum of $n$ independent random variables $X_1,\\ldots,X_n$ each taking values in the interval $[0,c]$. Then for any $\\delta \\ge 0$, we have\n$$\\Pr[|X-\\Ex[X]| \\ge \\delta] \\le 2 e^{-\\frac{\\delta^2\/2}{\\Var(X)+(c \\delta\/3)}}.  $$\n\\end{theorem}\n\nThe Cartesian product $A \\times B$ of two sets $A,B \\subseteq [n]$ is called a \\emph{combinatorial rectangle}. We will need the following lower bound on randomized communication complexity. \n\n\\begin{definition}\nLet $M$ be an $n \\times n$ Boolean matrix, and let $\\mu$ be a probability distribution on $[n] \\times [n]$.  The discrepancy of a combinatorial rectangle $R  \\subseteq [n] \\times [n]$  under  $\\mu$ is defined as\n$$ \\Disc_{\\mu}(M,R) = \\left|\\Pr_{\\mu}[m_{ij}=1 \\text{ and } (i,j) \\in R] - \\Pr_{\\mu}[m_{ij}=0 \\text{ and } (i,j) \\in R]\\right|.$$\nThe discrepancy of $M$ under $\\mu$ is defined as \n$\\Disc_{\\mu}(f) = \\max_{R}\\{\\Disc_{\\mu}(M,R)\\}$,\nwhere the maximum is over all combinatorial rectangles  $R$.\n\\end{definition}\n\n\n\\begin{theorem}\\cite[Proposition 3.28]{MR1426129}\n\\label{thm:DiscLower}\nLet $M$ be an $n \\times n$ Boolean matrix, and let $\\mu$ be a probability distribution on $[n] \\times [n]$. Then for every $\\epsilon>0$, \n$$\\RR_{\\frac{1}{2}-\\epsilon}(M) \\ge  \\log \\frac{2\\epsilon}{ \\Disc_\\mu(M)}. $$\nIn particular, \n\\begin{equation}\n\\label{eq:discLow}\n\\RR(M) \\ge  \\log \\frac{1}{3 \\Disc_\\mu(M)}. \n\\end{equation}\n\\end{theorem}\n\\iffalse \nWe start by giving examples of matrix families that satisfy $\\RR(\\mathcal M)=O(1)$. The following lemma generalizes the well-known example of $\\mathtt{I}_n$  to any matrix that has $O(1)$ 1-entries on each row. \n\\begin{lemma}\n\\label{lem:BoundedDegree}\nLet $M$ be a finite Boolean matrix where the number of $1$-entries in each row is at most $r$. Then \n$$\\RR(M) \\le \\lceil \\log(r) \\rceil +3.$$\n\\end{lemma}\n\\begin{proof}\nThe proof is a standard argument based on hashing. Let $m_{ij}$ denote the $ij$-th entry of $M$. Suppose that Alice receives the row number $i$, and Bob receives the column number $j$.   Alice knows the set $O_i=\\{j : m_{ij}=1\\}$, and we have $|O_i| \\le r$. Let $k=\\lceil \\log(r) \\rceil +2$. Using the public randomness, together Alice and Bob sample $S_1,\\ldots,S_{k} \\subseteq [n]$ uniformly at random  and independently. Then Bob sends a string of $k$ bits indicating which of these sets include $j$. Alice then checks to see if any element in $O_i$ matches this profile. If there is such an element, then Alice declares ``$m_{ij}=1$'', and otherwise she declares ``$m_{ij}=0$.''  \n\nThe number of communicated bits is $k+1 = \\lceil \\log(r)\\rceil + 3$. \nNote that an error can occur only when $m_{ij}=0$ and Alice mistakes $j$ for an element in $O_i$. By applying the union bound over all elements in $O_i$,  the probability of error is at most  $r 2^{-k} \\le \\frac{1}{4}$.\n\\end{proof}\n\\fi \n\n\nAs discussed in the introduction, stability is a necessary condition for a matrix sequence to satisfy $\\RR(\\mathcal M)=O(1)$. The next proposition proves a second necessary condition: an upper bound on $|\\cl(\\mathcal M)_n|$. \n\n\\begin{proposition}\n\\label{prop:CountR}\nThe number of $n\\times n$ matrices $M$ with $\\RR(M) \\le c$ is  $2^{O(2^cn \\log n)}$. \n\\end{proposition}\n\\begin{proof}\nLet $M$ be an $n\\times n$ Boolean matrix with $\\RR(M) \\leq c$. For every such $M$, there is a distribution $\\mu_M$ over deterministic protocols $\\pi$ of cost $c$ such that \n$$\n\\Pr_{\\pi\\sim \\mu_M}[M(i,j)=\\pi(i,j)]\\geq \\frac{2}{3} \\qquad \\text{for all $i,j$.}\n$$\nBy the Chernoff bound, the error probability of the protocol can be reduced to strictly less than $\\frac{1}{n^2}$ by sampling $O(\\log n)$ independent samples from $\\mu_M$ and outputting the majority outcome. Thus by union bound, there exists $t=O(\\log n)$ deterministic protocols $\\pi_1,\\ldots, \\pi_t$, each of cost $c$, such that for every $i$ and $j$, \n\\begin{equation}\\label{eq:derandcounting}\n    M(i,j)= \\mathsf{majority}\\{\\pi_1(i,j), \\ldots, \\pi_t(i,j)\\}. \n\\end{equation}\n\nNext, we show that the number of deterministic protocols of cost $c$ is at most $2^{O(2^c n)}$. Every such protocol corresponds to a binary tree of depth at most $c$, which has $O(2^c)$ nodes. Every node is associated with one of the two players, and the communicated bit is determined by the input of the corresponding player according to a function $[n] \\to \\{0,1\\}$. Thus there are $2^{n+1}$ possible choices for each node of the tree. Overall, this bounds the number of such protocols by $2^{O(2^cn)}$. \n\nFinally, since every matrix $M$ can be described in the form of \\cref{eq:derandcounting}, and there are $2^{O(2^cn)}$ choices for each $\\pi_i$, the number of such matrices is at most $2^{O(2^c n \\log n)}$. \n\\end{proof}\n\n\n\n\n\\section{Proof of \\cref{thm:main}}\\label{sec:thmmain}\nThe proof will rely on the following theorem, which involves a probabilistic argument presented in \\cref{sec:mainthm2}.\n\\begin{theorem}\n\\label{main:thm2}\nLet $w:\\mathbb{N}\\to \\mathbb{N}$ be a non-decreasing function satisfying $ w(n) \\to \\infty$ and $w(n)\\le 10^{-3} \\log(n)$. For every sufficiently large $n$, there exists an $n \\times n$ Boolean matrix $M$ with the following properties.  \n\\begin{enumerate}[label=(\\roman*)]\n    \\item $ \\RR(M)  = w(n).$\n    \\item  Every $n^{1\/4} \\times n^{1\/4}$ submatrix $F$ of $M$  satisfies    $\\RR(F) =O(1).$ \n\\end{enumerate}\n\\end{theorem}\n\n\nLet $w(n)= \\floor{10^{-3}\\log(n)}$, and for every sufficiently large $n$, let $M_n$ be the matrix that is  guaranteed to exist by \\cref{main:thm2}. For smaller values of $n$, let $M_n$ be an arbitrary $n \\times n$ Boolean matrix and let $\\mathcal M$ denote the corresponding sequence. By \\cref{main:thm2}~(i), we have $\\RR(\\mathcal M)=\\Theta(\\log(n))$, and by \\cref{main:thm2}~(ii), $\\mathcal M$ is stable. \n\nIt remains to bound $|\\cl(\\mathcal M)|_n$. Let $F$ be an $n\\times n$ matrix in $\\cl(\\mathcal M)$. There are two cases to consider:\n\n\\begin{enumerate}\n    \\item $F$ is a submatrix of an $M_k$ for $k > n^4$. In this case, by \\cref{main:thm2}~(ii), $\\RR(F) =O(1)$. So by~\\cref{prop:CountR}, the number of such matrices is bounded by $2^{O(n \\log n)}$.\n    \\item $F$ is a submatrix of an $M_k$ with $n \\le k \\le n^4$. The number of such matrices is at most \n    $$n^4 {n^4 \\choose n}^2 = 2^{O(n \\log n)}.$$\n\\end{enumerate}\n\nWe conclude that the total number of $n\\times n$ matrices in $\\cl(\\mathcal M)$ is $ 2^{O(n \\log n)}$ as desired. \n\n\\section{Proof of \\cref{main:thm2}}\\label{sec:mainthm2}\nWe will use a probabilistic argument to show the existence of an $n \\times n$ matrix $M$ that satisfies $\\RR(M) \\ge w(n)$, and the property (ii).  Note that modifying a row of a matrix can change its randomized communication complexity by at most $1$. Hence, to guarantee $\\RR(M) = w(n)$, we can replace the rows of $M$ to all-zero rows, one by one, until we achieve $\\RR(M) = w(n)$. We will also show that for our construction, (ii) will remain valid under such modifications. \n\nLet $M=[m_{ij}]_{n \\times n}$ be selected uniformly at random from the set of all Boolean $n \\times n$ matrices that have  exactly $r=2^{3 w(n)}<n^{0.01}$ number of 1's in each row. Denote $p=\\frac{r}{n}$.\n\n\n\\paragraph{Lower-bounding $\\RR(M)$:} To prove a lower bound on $\\RR(M)$, we show that, with high probability, the discrepancy of $M$ is small. Let $M_0=\\{(i,j) \\ | \\ m_{ij}=0\\}$ and  $M_1=\\{(i,j) \\ | \\ m_{ij}=1\\}$. Consider a fixed $a \\times b$ combinatorial rectangle $R \\subseteq [n] \\times [n]$. We first obtain an upper bound on $\\Pr \\left[\\left| |R \\cap M_1|-abp \\right| > t \\right]$ for $0 \\leq t \\leq nr$.\n\n\nNote that\n$$\\Ex[m_{ij}]=  \\frac{r}{n}=p,$$\nand for $(i,j) \\neq (i',j')$,  \n$$\\Ex[m_{ij} m_{i'j'}] \\le \\frac{{n-2 \\choose r-2}}{{n \\choose r}} \\le p^2=\\Ex[m_{ij}]\\Ex[m_{i'j'}].$$ \n\n\nIt follows that \n$$\\Ex [ |R \\cap M_1| ]= pab,$$\nand\n$$\\Var(|R \\cap M_1|) \\le abp(1-p) \\le abp \\le nr.$$ \n\nApplying Bernstein's inequality (\\cref{thm:bernstein}), for every $0 \\le t \\le nr$, we have \n\\begin{equation}\n\\label{eq:Bernstein}\n\\Pr \\left[\\left| |R \\cap M_1|-abp \\right| > t \\right] \\le 2 e^{-\\frac{t^2\/2}{nr+ t}} \\le 2 e^{-\\frac{t^2}{4nr}}.  \n\\end{equation}\n\nDefine the probability distribution $\\mu$ on $[n] \\times [n]$   as \n\t$$\\mu:(i,j) \\mapsto \\left\\{ \n\t\\begin{array}{lcr}\n\t\\frac{1}{2rn} & \\qquad & m_{ij}=1 \\\\\n\t\\frac{1}{2(n-r)n}& &  m_{ij}=0 \\\\\n\t\\end{array}\n\t\\right. .$$\n\nNote that $\\mu$ is defined so that it assigns the total measure of $\\frac{1}{2}$ uniformly to each of $M_0$ and $M_1$. Then, \n\\begin{eqnarray*}\n\\Disc_\\mu(R)&=& \\left|\\frac{|R \\cap M_1|}{2rn}- \\frac{|R \\cap M_0|}{2(n-r)n}\\right|= \\left|\\frac{|R \\cap M_1|}{2rn}- \\frac{|R|-|R \\cap M_1|}{2(n-r)n}\\right|  \\\\\n&=&\\left|\\frac{n|R \\cap M_1|- r|R|}{2nr(n-r)}\\right|= \\left|\\frac{|R \\cap M_1|- p|R|}{2r(n-r)}\\right|\\le \\left|\\frac{|R \\cap M_1|- abp}{rn}\\right|.\n\\end{eqnarray*}\nBy substituting $t=\\epsilon rn$ in \\cref{eq:Bernstein}, we obtain \n$$\\Pr\\left[\\Disc_\\mu(R) \\ge \\epsilon \\right] \\le 2 e^{\\frac{-r^2n^2 \\epsilon^2}{4nr}}\\le  2 e^{\\frac{-rn \\epsilon^2}{4}}.$$\nBy applying the union bound over all the $2^{2n}$ possible rectangles, and taking $\\epsilon=\\frac{3}{\\sqrt{r}}$, we obtain that for sufficiently large $n$, \n$$\\Pr[\\Disc_\\mu(M) \\ge \\epsilon] \\le 2^{2n}\\times  2 e^{-\\frac{nr\\epsilon^2}{4}} \\le 2^{\\frac{-n}{4}+1} \\le \\frac{1}{10}. $$\nSubstituting $r=2^{3w(n)}$, and applying the discrepancy lower bound of \\cref{eq:discLow}, we obtain that for sufficiently large $n$, \n$$\\Pr\\left[\\RR(M) \\le w(n)\\right] \\le  \\Pr\\left[\\RR(M) \\le \n\\log\\frac{\\sqrt{r}}{9}\\right]=\\Pr\\left[\\RR(M) \\le \n\\log\\frac{1}{3\\epsilon}\\right] \\le \\Pr[\\Disc_\\mu(M) \\ge \\epsilon]  \\le  \\frac{1}{10}. $$ \n \n\n\n\\paragraph{Verifying (ii):}   Let $k=n^{1\/4}$.  We first prove that with probability $1-o(1)$, for every $a,b\\le k$, every $a \\times b$ submatrix of $M$ contains a row or a column with at most two $1$'s. Note that the statement is trivial when $\\min(a,b)\\le 2$, and hence, we fix  $a,b> 2$. \n\nIf $a \\le b$, then the probability that there is an $a \\times b$ submatrix such that each of its $b$ columns contains at least three $1$'s is bounded by\n$$\n\\binom{n}{a}\\binom{n}{b} \\left(\\binom{a}{3} p^3 \\right)^b  \\le n^a n^b (a^3p^3)^{b} \\le   (n^2 p^3 b^3)^b\\le  \\left(\\frac{r^3}{n^{1\/4}} \\right)^b\\le \\left(\\frac{n^{0.03}}{n^{1\/4}} \\right)^b \\leq o(n^{-1\/2}),\n$$ \nwhere we used $r=2^{3 w(n)}<n^{0.01}$. Similarly if $a > b$, then the probability that there is an $a \\times b$ submatrix such that each of its $a$ rows contains at least three $1$'s is bounded by \n$$\n\\binom{n}{a}\\binom{n}{b} \\left(\\frac{\\binom{b}{3}  {n-3 \\choose r-3}}{{n \\choose r}} \\right)^a  \\le n^a n^b (b^3p^3)^{a} \\le   (n^2 p^3 b^3)^a\\le  \\left(\\frac{r^3}{n^{1\/4}} \\right)^a\\leq o(n^{-1\/2}). \n$$ \nThus by a union bound over all choices of $a,b\\leq k$, the probability that there is  $a,b \\in [k]$ and an $a \\times b$  submatrix where every column or row contains at least three $1$'s  is bounded by $o(k^2n^{-1\/2})$ which is $o(1)$ as desired. \n\n\nNow suppose that every $a \\times b$ submatrix $F$ of $M$ contains a row or a column with at most two $1$'s. We will show that in this case, every such $F$ corresponds to the  biadjacency matrix of a disjoint union of two bipartite graphs that are both forests.  Consider a row (or a column) with at most two $1$'s, and let $e_1$ and $e_2$ be the edges corresponding to these (at most) two entries.   Removing this row from $F$ will result in a smaller submatrix, which by induction hypothesis, can be written as the union of two forests $\\mathcal F_1$ and $\\mathcal F_2$. Now $F$ can be decomposed into the union of  two forests $\\mathcal F_1 \\cup \\{e_1\\}$ and $\\mathcal F_2 \\cup \\{e_2\\}$. \n\nThe bound $\\RR(F)=O(1)$ follows by first observing that each forest is an edge-disjoint union of two graphs, each   a vertex-disjoint union of stars. Hence, it suffices to show that the biadjacency matrix of any vertex-disjoint union of stars has $O(1)$  randomized communication complexity. Suppose that $G$ is a union of vertex-disjoint stars $S_1,\\ldots, S_k$. Alice receives $u\\in V(G)$ and Bob receives $v\\in V(G)$, and they want to decide whether $(u,v)\\in E(G)$, which is equivalent to whether $u$ and $v$ belong to the same star. To solve this problem, Alice maps her input $u$ to the index $i$ such that $u \\in S_i$. Similarly, Bob maps $v$ to $j$ such that $v \\in S_j$. Now they can use the randomized communication protocol for $\\mathtt{I}_k$ to check whether $i=j$. This verifies (ii).\n\nFinally, note that if $F$ is a union of two forests, then replacing a row of $F$ with an all-zero row   will not violate this property.  \n\\bibliographystyle{alpha}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nInteresting phenomena emerge in the population dynamics\n in heterogeneous environments. For example, experimental and theoretical\n studies have shown that spatial heterogeneity accelerates\n the emergence of drug resistance \n\\cite{PhysRevLett.105.248104,zhang2011acceleration} and solid tumor evolution in heterogeneous  microenvironments\n \\cite{zhang2012physics}. On a larger scale, heterogeneity\n plays a central role in population biology of infectious \ndiseases \\cite{PhysRevLett.99.148701} \nand emerges in the development\n of large physics projects, such as ATLAS, CERN \\cite{Turtscher}. Finally, in heterogeneous environments,\nevolved networks are modular when there are local extinctions\n \\cite{kashtan2009extinctions}.\n\nPopulations experience heterogeneous environments during\n migration.  Migration can occur in different dimensions:\n for example, cells undergo one-dimensional, \ntwo-dimensional, or three-dimensional migration \\cite{doyle2009one}.\n In two-dimensional or three-dimensional migration, \nthe environmental gradient can additionally\nbe distinct in the different directions.\n For example, in the case of human migration, \nthe north-south direction has a greater environmental \ngradient than does the east-west direction \\cite{diamond1997guns}.\nThe heterogeneity is important in simulating human dispersal in \nthe Americas \\cite{steele2009human}.\n In the east-west direction, food production spread from southwest Asia \nto Egypt and Europe at about $0.7$ miles per year around 5000 BC, while in \nthe north-south direction, it spread \nnorthward in the American continent at about $0.2$ to $0.5$\n miles per year around 2000 BC \\cite{diamond1997guns}. This spread is on the same\n order as the velocity of human migration, so we estimate \nthat the human migration velocity in the east-west direction is\n about $2$ to $3$ times faster than in the north-south direction.\nPrevious work has generated detailed migration paths using geographical \ndata \\cite{anderson2000paleoindian} as well as results that match existing archaeological evidences \nwell after considering spatial and temporal variations \\cite{steele2009human}. \nWe do not try to generate a detailed map of human migration in this paper. \nInstead, we use a general model to generate east-west north-south asymmetry \nand study the role of a modular knowledge system.\n\nKnowledge of local environments, such as \neffective agricultural or animal husbandry techniques,\nwas vital to the survival of these early migrants \\cite{diamond1997guns}.\nEvolutionary epistemology views the \ngaining of knowledge as an adaptive process with blind \nvariation and selective retention \\cite{campbell1960blind}. \nCommunication of\n knowledge between individuals is also \nan efficient means to spread this discovered, locally\nadapted knowledge \\cite{mithen1996preshistory}. Similarly, models of social \nlearning theory stress the importance of social learning in the spread of \ninnovations \\cite{kandler2010social}. Here we\nmodel the adaptation of a population to the local environment\n using an evolutionary model with natural selection, mutation and communication. \nThe knowledge of an individual\n determines his or her fitness. Evolutionary psychology \nand archeology posit that the human mind is modular \\cite{steele1996weak},\n and that this modularity is shaped by evolution \n\\cite{tooby1995psychological} and facilitates  understanding of local environments\n\\cite{mithen1996preshistory}. Conjugate to this modularity\n must be dynamical exchange of corpora of knowledge between individuals \\cite{Goldenfeld2011,AR}. \n\n\\section{Methods}\n\n\\begin{table}[h]\n\\caption{Symbols used in this paper}\\label{Table1}\n\\begin{tabular}{|c|c|}\n\\hline\nSymbol & Meaning \\\\\\hline\n$\\chi$ & Similarity between adjacent environments \\\\\\hline\n $v$ &Emigration velocity \\\\\\hline\n $t$ & Emigration time\\\\ \\hline\n$N$ &Number of individuals in one environment\\\\\\hline\n$N^*$ &Carrying capacity of one environment\\\\\\hline\n$N_0$ &Initial population size of one environment\\\\\\hline\n$f$ & Fitness\\\\\\hline\n$f^*$ &Fitness threshold\\\\\\hline\n$J$ & Interaction matrix\\\\\\hline\n$\\Delta$ & Connection matrix\\\\\\hline\n $K$ & Number of modules in a sequence\\\\\\hline\n$l$ & Module size \\\\\\hline\n$\\mu$ & Mutational rate\\\\\\hline\n$\\nu$ & Knowledge transfer rate\\\\\\hline\n$d$ &Genetic distance\\\\\\hline\n $S$ & A whole sequence\\\\\\hline\n$s$ & One locus in a sequence \\\\\\hline\n$L$ & Length of one sequence\\\\\\hline\n$M$ & Modularity\\\\\\hline\n\\end{tabular}\n\\end{table}\n\nTable \\ref{Table1} shows the symbols in this paper. \nThe observed emigration time and asymmetry of emigration \ntime are critical in the determination of the values of \nthese parameters.\nWe consider migration in random, asymmetric, \nmodularly correlated environments.  \nWe use $9 \\times 25$ correlated, random environments, \nwhere $25$ is the number of environments\n in the north-south direction at the same longitude \\cite{kottek2006world}, \nand $9$ is chosen so that \n$9\/25$ is approximately the ratio of the east-west to \nnorth-south dimension of the\n Americas. See Fig.\\ \\ref{Fig1} for an illustration,\n where each square block corresponds to an environment.\n\nEach individual \n$\\textbf{a}$ has a fitness $f_\\textbf{a}$, as well as a sequence\n $S^\\textbf{a}$ that is composed of $L$\nloci, $s_i^\\textbf{a}$, representing the knowledge of the \nindividual. Fitness describes reproductive success and is proportional to\nthe reproduction rate. For simplicity, we take $s_i^\\textbf{a} = \\pm 1$.\nWe first consider a linear fitness landscape, later\ngeneralizing to an interacting landscape:\n\\begin{eqnarray}\nf[S] &=& 2 L + H[S]\n\\nonumber \\\\\nH[S] &=& \\sum_{i} s_i J_{i} \n\\label{Eq2}\n\\end{eqnarray}\nwhere $J_{i}$ is a quenched, Gaussian random interaction parameter, \nwith variance $\\sqrt 2$, and the offset $2L$ is chosen\nso that fitness is non-negative, since $H_{\\min}$ is $-2 L \/\\sqrt\\pi$. \nFor a given instance of the model, the interaction parameters $J_{i}$ \nare randomly chosen and then fixed for that instance of the model.\nWhen for each $i$ from $1$ to $L$, $s_i J_i>0$, the fitness reaches \nits highest value, and natural selection selects the sequence with \nthe best configuration.\n\nThe fitness of the population is influenced by the environment, \nquantified by interaction parameters $J$, describing the interaction between \n knowledge element $i$ of the individuals and the environment\n(see also Eq.\\ \\ref{Eq2} above). The \ninteraction parameters $J$\nin two adjacent environments, $J$ and $J'$, are correlated,\n\\begin{equation}\\label{Eq1}\n\\langle J_{i} J'_{i}\\rangle\/\\langle J_i^2\\rangle = \\chi\n\\end{equation}\nwhere $\\chi = \\chi_{\\rm{EW}}$ if the two have the same latitude,\n and $\\chi = \\chi_{\\rm{NS}}$ if they have the same longitude. The\n smaller the $\\chi$, the bigger the environmental gradient is. Here $0<\\chi<1$,\nand $\\chi_{\\rm{NS}} < \\chi_{\\rm{EW}}$, since the gradient of environment in the\nnorth-south direction is more dramatic \\cite{diamond1997guns}. \n\nIn each environment, we use a Markov process to describe the\n evolutionary dynamics, including replication with rate $f$, mutation\nwith rate $\\mu$ coming from discovering new knowledge\n through trial and error, and transfer of a corpus of knowledge\nof length $L\/K$ with rate $\\nu$. When individuals reproduce, \nthey inherent the knowledge and genes from their parent without error.\nBoth mutations and knowledge transfers are random, and they do not\ndepend on the fitness of individuals.\n The relative rates of replication, mutation, and transfer\n are $f$, $\\mu L$, and $\\nu K$, respectively, so on average each individual\n makes $\\mu L\/f \\approx \\mu\/2$ mutations, \nas $f \\approx 2L$ at short times for which these populations evolve,\nand $\\nu K\/f \\approx \\nu K\/(2L)$ \nknowledge transfers per lifetime of an individual.\nWe set the information sequence length $L=100$.\nDiscovery of new facts, represented by mutation, changes\none site, or 1\\% of the knowledge of an individual,\nwhereas knowledge transfer changes $1\/K$ of the knowledge.\nDiscovery of new facts should be rare,\nand in our simulation \nwe set $\\mu=0.5$, so that approximately one-quarter of the \nindividuals attempt to make a \ndiscovery through trial and error during his or her \nlifetime.\nWe consider $K=5$ corpora \nof knowledge. Transfer of one corpus, for example, could be one farmer\nattempting to communicate to another farmer how to grow a new crop in a new environment. \nKnowledge transfer must be\nrare, so we set $\\nu=6$, so that roughly $\\nu K \/ (2 L) \\approx 1\/7$ \nof the individuals attempt a\nknowledge transfer process during his or her lifetime. \nWe additionally consider various values of $\\nu$ in \nthis work to investigate the coupling of $\\nu$ to modularity.\nSelection is based on the fitness of the knowledge and\nit determines the the utility of theses mutation and knowledge transfer\nevents.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.85]{fig1.eps}\n\\end{center}\n\\caption{\\label{Fig1}\nPopulation distribution half way through \nthe migration. Color\n indicates density of the population in each environment. The maximum \ncapacity of each environments is $N^*=10000$. Initially there are \n$1000$ individuals in the top center environment  $(1,5)$, and no individuals in\n other environments. Here $\\chi_{\\text{EW}}=0.8$, \n$\\chi_{\\text{NS}}=0.4$, $f^*-2L=0.3 L$, $L=100$, $\\mu=0.5$, \n$\\nu=6$, and $K=5$. Density was averaged over $24$ runs.\n}\n\\end{figure}\n\nThis dynamics of migration is described by a\nMarkov process, whose master equation\nis detailed in the Appendix.\nInitially, one of the \nenvironments with the highest latitude is occupied by $1000$ \nindividuals with random sequences, as Native Americans are believed\n to have entered the Americas through Alaska in the north. Since\n the population migrates from north down to south, we only allow \nmigration to the east, west, and south. In each environment, the \npopulation evolves according to the Markov dynamics.\n\nThe qualitative behavior of the migration depends on the\ncarrying capacity, $N^*$, and the fitness threshold, $f^*$. The carrying capacity is defined \nas the maximum population load of an environment \\cite{hui2006carrying}. \nAfter the population size reaches $N^*$, we randomly kill an individual every time another \nindividual reproduces, as described in detail in Eq.\\ \\ref{Eq8}. As a result, the total \nnumber of individuals does not exceed $N^*$.\nThe initial colonization of the Americas \noccurred before the Common Era, for which there are no reliable population data. \n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.7]{fig2.eps}\n\\end{center}\n\\caption{\\label{Fig2}\nAsymmetry in emigration times $t_{\\text{NS}}\/t_{\\text{EW}}$ for different $N^*$\n and $f^*$.\nUpper left, phase diagram of the linear model, Eq.\\ \\ref{Eq2}.\nUpper right, phase diagram of the quadratic model, Eq.\\ \\ref{Eq5}, with $M=1$.\nLower left, phase diagram of the quadratic model, with $M=1\/2$.\nLower right, phase diagram of the quadratic model, with $M=0$.\nOther parameters are as in Fig.\\ \\ref{Fig1}.\nThe color indicates the asymmetry in emigration times:\n$t_{\\rm NS} \/ t_{\\rm EW}$. \nThere are three phases, with the boundaries denoted by the two curves.\nThe steady-state fitness dynamics, $f(t)$ vs $t$, of the right \nphase and the left phase are shown in inset. The fitness dynamics of the middle\nphase in the north-south direction follows that of the upper inset, \nand in the east-west direction\n follows that of the lower inset.  The phase boundaries are given approximately by\nequating the times in Eq.\\ \\ref{Eq3} for the north-south (left) or \neast-west (right) migration directions for linear model and $M=1$ quadratic model.\nThe model for human migration has $N^* = 10000$ and $f^*-2L = 30$.\n}\n\\end{figure}\n\nIt is estimated that there were seven million people in the Americas at the start \nof the Common Era \\cite{maddison2007world}, corresponding to $7000000\/(25\\times 9)=31111$ \nindividuals in each environment. We choose the carrying capacity to be $N^*=10000$, less\nthan $31111$, reflecting that the population size was smaller the earlier time of initial \npopulation expansion. \nWe show the results for various $N^*$ in Fig.\\ \\ref{Fig2}. \nWe introduce the fitness threshold, $f^*$, \nbecause individuals need to be well prepared before emigrating to the next environment. \nFor example, young male ground squirrels appear to disperse after attaining a threshold \nbody mass \\cite{nunes1996mass}, and dispersing males tend to have greater fat percentage for \ntheir bodies \\cite{nunes1996mass}. The increased body mass and fat \npercentage are thresholds required for migration. Similarly, naked mole-rats migrate more \nfrequently after body mass reaches a certain value \\cite{o1996dispersive}. \nIt is possible that some individuals try to emigrate without reaching the fitness \nthreshold when the local population size reach environmental capacity. \nHowever, they are not fit enough to colonize the new environment. \nThus, we employ a fitness threshold in our approach, and allow no emigration \nbefore the average fitness value reaches $f^*$. \nWhen the population size reaches $N^*$ and \nthe average fitness reaches $f^*$ in an environment, we \nmove $N_0=1000$ randomly chosen individuals to one of the unoccupied adjacent\n environments. Fitter individuals\nmay be more likely to migrate since they are physically better prepared to\nmigrate, while on the other hand less fit individuals may have more desire\nto migrate since they do not live well in the current environment. We randomly \nchoose individuals to migrate because of this ambiguous relationship between \nfitness and migration.\nIf we move fitter individuals instead of \nrandomly chosen individuals, the\ninitial fitness of the individuals in the new environment will be higher.\nThus, effectively the $\\chi$ would be higher. \nThe time required for a population to emigrate from\n an environment is denoted by the emigration time, $t$, \n and the emigration velocity $v$ is defined as $v=1\/t$. The emigration time \nof an environment is the time from the arrival of the first individuals to \nthe departure of the first individuals.\n\nTo compare our results with current human genetic data, we assign to each \nindividual another sequence $S'$, also composed of $L$ loci, and each locus\ncan take values $\\pm 1$. These sites correspond to automosal microsatellite \nmarker genotype data \\cite{wang2007genetic}, which we will compare with later in this paper. \nThe traits of the genetic data are neutral in \nour model. That is, the values of the loci in the sequence $S'$ have no \neffect on the fitness. The genetic sequence mutates at a rate $\\mu'$. \nWhen an individual reproduces, both the knowledge sequence and the \ngenetic sequence reproduce.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{fig3.eps}\n\\end{center}\n\\caption{\\label{Fig3} Emigration time versus $1-\\chi$ \n for the linear model and the quadratic model with various modularities.\n Here $\\chi_{\\rm EW} = \\chi_{\\rm NS} = \\chi$, and other\n parameters are as in Fig.\\ \\ref{Fig1}. }\n\\end{figure}\n\nWe set the time scale in our simulation by the observation that\n Native Americans spent about $10000$ years to migrate from the north\n tip of the American continent to the south tip \\cite{goebel2008late}, experiencing \nabout $25$ climate zones \\cite{kottek2006world}, so migration\n to a new environment occurred roughly every $400$ years, i.e. roughly every $20$ \ngenerations. In our simulation, we define a generation as the time\n period during which on average, each individual is replaced by \nanother individual. We find that the population migrates approximately\n once per $20$ generations when $\\chi_{\\text{EW}}=0.8$, \n$\\chi_{\\text{NS}}=0.4$, and $f^*-2L=0.3L$. One can estimate \nhow many generations it takes to migrate to the next environment. The rate of \nchange of fitness at short time roughly follows \\cite{park2015modularity},\n\\begin{equation}\ndf\/dt = 2L\n\\end{equation}\nSince $\\Delta f=0.6\\times(f^*-2L)=0.18\\times 2L$ for migration from the north \nor \n$\\Delta f=0.2\\times(f^*-2L)=0.06\\times 2L$ for migrating \nfrom the east or west, the emigration time is $0.18$ or $0.06$ \ndepending on the origin of migration, and this is consistent with Fig.\\ \\ref{Fig3}.\nWe use $\\Delta t=0.1$ as a rough estimate for emigration time.\nTo convert this time in our simulation to number of human replications, we consider \nthat one replication takes around $dt=1\/f=1\/2L$ time, so one emigration takes \n$\\Delta t\/dt=20$ generations.\nTo compare the genetic data with \ncurrent human data, we allow all \nenvironments to evolve for another $10000$ \nyears after all environments are occupied, without migration between environments. \nWe assume no gene flows between these environments, as previous work \\cite{ramachandran2011test}\nassumes that the asymmetry\nin the genetic distance originates from the asymmetry of gene flows in\ndifferent directions. Here we investigate another possible origin of the\nasymmetry of genetic distance, that is, the asymmetry already exists\nwhen the population colonized the Americas. It is quite possible\nthat both mechanisms help to create this asymmetry, but in order to show that the\ninitial colonizing process itself could generate this asymmetry, we \nsuppress the possibly asymmetric genetic flows.\n\n\\section{Results}\\label{sec:results}\n\nIn Fig.\\ \\ref{Fig1} we show a snapshot of population \ndistribution, approximately half way through the migration. \nMigration sweeps south\n and spreads both to the east and west. Migration forms a tilted \nfront, with slope magnitude equal to $v_{\\text{NS}}\/v_{\\text{EW}} = 0.35$,\nindicating the velocities of migration in \ndifferent directions are different. \n\nIn Fig.\\ \\ref{Fig2}\nwe show the three possible phases for different carrying capacity, $N^*$, and \nfitness threshold, $f^*$. Different phases correspond to whether \nthe migration is limited by the fitness\nthreshold or the population size threshold. In the left phase, the\npopulation is limited by the population size threshold, and there is\nno east-west north-south asymmetry. In addition, as the population\nmigrates, the maximum fitness value increases since the population is\nallowed to evolve further after reaching the fitness threshold, as shown\nin the left inset of the upper right figure. In the middle phase, the\nmigration in the east-west direction is limited by population size\nthreshold while the migration in the north-south direction is limited\nby the fitness threshold. The maximum fitness value increase as the\npopulation migrates in the east-west direction, but in the north-south\ndirection, the maximum fitness value is $f^*$. The degree of the\neast-west north-south asymmetry increases in this phase from the boundary\nwith the left phase to the boundary with the right phase. In the right\nphase, migrations in both directions are limited by the fitness\nthreshold, and the maximum fitness value remains the same as the\npopulation migrates. The east-west north-south asymmetry is approximately\nunchanged in this phase.\nThe boundaries of these phases are determined by\nnoting the times to reach the carrying capacity\nand the fitness threshold:\n\\begin{eqnarray}\nt_{N^*} &=& \\frac{\\ln (N^* \/ N_0) }{2 L}\n\\nonumber \\\\\nt_f &=& \\frac{(f^*-2L) (1-\\chi) }{2 L}\n\\label{Eq3}\n\\end{eqnarray}\nwhere $N_0$ is the initial population of one environment. \nHere we have used that the evolution  of the\nfitness in one generation is small compared to the\noffset $2 L$, and that the evolution within one\nenvironment at steady state is from $\\chi (f^*-2L)+2L$\nto $f^*$ in the rightmost phase.\nThe left phase boundary in Fig.\\ \\ref{Fig2} is\ngiven by the condition $t_{N^*}  = t_f$ in the north-south direction, and\nthe right phase boundary is given by $t_{N^*}  = t_f$ in the east-west direction. \nWe note that our current choice of parameters is deep in the right phase, \nindicating that the east-west north-south asymmetry is robust to the change of \n$f^*$ or the ratio $N^*\/N_0$.\n\n\nWe determine quantitatively how the environmental gradient \ninfluences the velocity of migration. In Fig.\\ \\ref{Fig3} we show the\n emigration time versus $1-\\chi$, the change between adjacent \nenvironments.\n It is interesting that the emigration time is approximately \nproportional to $1-\\chi$. This occurs because in our simulation \nfor these parameters,\n the population reaches $N^*$ earlier than $f^*$, so the emigration\n time is the time required to reach $f^*$. \nFor our model,\n $f^*-2L = 0.3L$, while $\\max(f-2L)\\approx 2L$, \nso $f^*$ is still far from optimal, and the\n fitness increases linearly with time in the regime we are discussing.\n So $t=\\Delta f \/ v_f = (1-\\chi) f^* \/v_f$, where $v_f$ is a \nconstant for a fixed modularity. So $t\\propto 1-\\chi$, and we \nquantify the ratio of velocity in the two different directions as\n\\begin{equation}\\label{Eq4}\n\\frac{v_{\\text{EW}}(M)}{ v_{\\text{NS}}(M)}=\\frac{1\/t_{\\text{EW}}(M)}\n{1\/t_{\\text{NS}}(M)}=\\frac{1-\\chi_{\\text{NS}}}{1-\\chi_{\\text{EW}}}\n\\end{equation}\n\n\nIn the linear model, it is quite easy to evolve the optimal\npieces of knowledge, while in reality, finding the best knowledge\nis difficult at the individual level. \nWe now show that these results are robust to considering an\ninteracting model, while also demonstrating the significance\nof the modularity order parameter in the interacting model.\nAs finding optimal knowledge\n for a local environment is difficult, the fitness \nlandscape is rugged \\cite{PhysRevLett.99.228107}, and we use\na spin glass to represent the fitness:\n\\begin{eqnarray}\\label{Eq5}\nf[S] &=& 2 L + H[S]\n\\nonumber \\\\\nH[S] &=& \\sum_{ij} s_i s_j J_{ij} \\Delta_{ij}\n\\end{eqnarray}\nwhere $J_{ij}$ is a Gaussian random matrix, \nwith variance $1\/C$.\nThe offset value $2 L$ is chosen by Wigner's semicircle law\n \\cite{wigner1958distribution} so that the\nminimum eigenvalue of $f$ is non-negative. \nThe entries in the matrix $\\Delta$ are zero or one, with probability\n$C\/L$ per entry, so that the average number of connections\n per row is $C$. The optimization of this fitness model is hard\nwhen $L$ is large, and here we give a simple example to show why. \nConsider a case when $J_{ij}>0$, $J_{ik} >0$ and $J_{jk}<0$ \ngiven $i$, $j$ and $k$. To make $J_{ij}s_is_j$ positive and fitness value \nlarger, $s_i$ and $s_j$ must have the same sign. Similarly, to make $J_{ik}s_is_k$ \nand $J_{jk}s_js_k$ positive, we need $s_k$ to have the same sign with $s_i$, \nand $s_k$ to have different sign with $s_j$. This indicates that $s_i$ and $s_j$ \nhave different signs, contradicting that $s_i$ and $s_j$ having the same sign. This \nphenomena is called frustration in physics \\cite{sadoc2006geometrical}, making the  \nfitness hard to optimize. Let us exemplify this using an example from the human knowledge \nsystem. Humans developed three pieces of knowledge: the knowledge of toxicity of \nmushrooms, the knowledge of red food, and the knowledge of apples. The interaction \nbetween the first two pieces of knowledge implies that red food is bad and \nundesirable, while the later two pieces of knowledge implies something on the \ncontrary. As a result the human knowledge system can be difficult to optimize.  \nWe will discuss how modularity helps to reduce this frustration and thus \nmakes it easier to optimize the fitness in section \\ref{sec:discussion}.\n\nWe introduce modularity by an excess of interactions in\n $\\Delta$ along the\n$l \\times l$ block diagonals of the $L \\times L$\nconnection matrix. There are $K$ of these\nblock diagonals, and $K=L\/l$.  Thus,\nthe probability of a connection is\n$C_0\/L$ when\n$ \\lfloor i\/l \\rfloor \\ne \\lfloor j\/l \\rfloor$\nand $C_1\/L$ when\n$ \\lfloor i\/l \\rfloor = \\lfloor  j\/l \\rfloor$.  The number of\nconnections is $C = C_0   + (C_1 - C_0) \/K$, and modularity \nis defined by $M = (C_1 - C_0)  \/ (K C)$. In Fig.\\ \\ref{Fig6} we illustrate \nthree $20\\times 20$ matrices with modularities $1$, $0.5$ and $0$ and $C=9$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.7]{fig6.eps}\n\\end{center}\n\\caption{\\label{Fig6} Illustration of $L=20$ connection matrices with \ndifferent modularities. Left, a completely modular connection matrix, $M=1$. Middle, \na moderately modular connection matrix, $M=0.5$. Right, a non-modular \nconnection matrix, $M=0$.}\n\\end{figure}\n\n\nModularity, coupled with knowledge transfer, accelerates the \nevolution of a population in a new environment \\cite{park2015modularity}.\n We now check how modularity and knowledge transfer influence the\n velocity of migration. For different $M$ and\n $\\nu$, the results are shown in Fig.\\ \\ref{Fig4}. For small $M$, a larger\n $\\nu$ implies a smaller migrating velocity, indicating that the transfer of\n(non-useful) knowledge slows down evolution. As modularity increases,\n the migration velocity at larger $\\nu$ catches up with that of smaller\n $\\nu$. At $M=1$, in the range of $\\nu$ shown, \nthe faster the population migrates faster for larger $\\nu$. \n\nWe fit the curve of $v_{\\rm NS}$-$M$ for $\\nu\\leqslant 4$ in Fig.\\ \\ref{Fig4} with \nlinear regression, observing $R^2\\geqslant 0.95$, except for\n $\\nu =0$, which has a zero slope and larger noise. We also fit the data\n for $\\nu=1$ and $\\nu=3$, not shown in Fig.\\ \\ref{Fig4}. We show \n$dv_{\\text{NS}}\/dM$ versus modularity for different $\\nu$\n in the inset to\nFig.\\ \\ref{Fig4}. For $\\nu\\leqslant 4$, the slope is\n proportional to $\\nu$. So, $d v_{\\text{NS}}\/dM=\\alpha_{\\text{NS}} \\nu$, \nand after integration we have,\n\\begin{equation}\\label{Eq6}\nv_{\\text{NS}}=\\alpha_{\\text{NS}} \\nu M + v_{\\text{NS}}^0(\\nu)\n\\end{equation}\nwhere $v_{\\text{NS}}^0(\\nu)$ is determined by the evolutionary load of knowledge transfer. \nLinearity originates from perturbation of knowledge transfer when $\\nu$ is small. \nNote that for $\\nu=6$, the value used in most part of this paper, the linear relationship \nno longer holds, indicating that $\\nu=6$ is large enough to break the linearity.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.62]{fig4.eps}\n\\end{center}\n\\caption{\\label{Fig4} North-south Emigration velocity versus modularity for \ndifferent $\\nu$. The lines are linear fit of the data of the corresponding \nhorizontal gene transfer rate.\nThe inset shows $dv_{\\rm NS}\/dM$ versus $\\nu$.\n The dots are data points and the line is a linear fit to the data. \nOther parameters are as in Fig.\\ \\ref{Fig1}. }\n\\end{figure}\n\n\nFrom our model, we make a prediction by calculating \ngenetic distances between populations in different \nenvironments, using the genetic sequence $S'$. For each pair of environments, we calculate the \nfixation index $F_{\\text{ST}}$ between them using Eq.\\ 5.12 from \\cite{weir1996genetic}:\n\n\\begin{equation}\\label{Eq7}\nF_{\\text{ST}}=\\frac{\\sum_{i=1}^{L}\\left[\\frac{1}{2}\\sum_{j=1}^{2}(p_{ij}-p_{ij}')^2-\\frac{1}{2(2n-1)}\\left(2-\\sum_{j=1}^{2}(p_{ij}^2+p_{ij}'^2)\\right)\\right]}{\\sum_{i=1}^{L}(1-\\sum_{j=1}^{2}p_{ij}p_{ij}')}\n\\end{equation}\nwhere $p_{i1}$ is the probability of the value of locus $i$ being $+1$, and $p_{i2}$ \nis the probability of the value of locus $i$ being $-1$ in the first environment. \n$p_{ij}'$ is the corresponding \nprobability in the other environment. Here $n$ is the sample size drawn from \nthe population to estimate $F_{\\text{ST}}$, and in our case $n=18$, in accordance with the \naverage sample size used in \\cite{ramachandran2011test}.\n\n The east-west distance between \nenvironments $(x_1,y_1)$ and $(x_2,y_2)$ is $d_{\\text{EW}}=|x_1-x_2|$, and the north-south distance is \n$d_{\\text{NS}}=|y_1-y_2|$. We also calculated heterozygosities of the population\n of environment a, defined as\n\n\\begin{equation}\n\\text{het}_{a}=1-\\frac{1}{L}\\sum_{i=1}^L\\sum_{j=1}^2 p_{ij}^2\n\\end{equation}\nwhere $p_{i1}$ and $p_{i2}$ have the same meanings as those in Eq.\\ \\ref{Eq7}. \nEach fixation index $F_{\\text{ST}}$ \nwas regressed onto the sum of mean heterozygosity and geographic distance, which \ncan be either east-west distance or north-south distance. The $R^2$ of the regression \nis around $0.9$. For each pair of environments \n$a$ and $b$, we express the $F_{\\text{ST}}$ as,\n\n\\begin{eqnarray}\nF_{\\text{ST}}&=&c_{\\text{EW}} d_{\\text{EW}}+c_1\\frac{\\text{het}_a+\\text{het}_b}{2}+c_0\\\\\nF_{\\text{ST}}&=&c_{\\text{NS}} d_{\\text{NS}}+c_1'\\frac{\\text{het}_a+\\text{het}_b}{2}+c_0'\n\\end{eqnarray}\n\n The coefficient of \ngeographic distance term using east-west distance is $c_{\\text{EW}}$, and $c_{\\text{NS}}$ \nusing north-south distance. The ratio of them, $r=c_{\\text{NS}}\/c_{\\text{EW}}$, \nindicates the asymmetry of rate of change of genetic distance.\nFor humans in the Americas, the \nratio is approximately $1.26$ \\cite{ramachandran2011test}. The mutational rates of \ngenetic sequences at which the $F_{\\text{ST}}$ ratio is $1.26$ depend on modularity, \nas shown in Fig.\\ \\ref{Fig5}. The estimated mutation rate of human automosal microsatellites \nrange from $10^{-4}$ to $10^{-2}$ \\cite{kayser2000characteristics}. In our model, we can \ncalculate the mutational rate per generation $\\mu_g = \\mu'\\times L \/2L$. So for $M=1$ the \nmutational rate is $0.005$ per\nlocus per generation, and for $M=0$ the mutational rate is $0.025$\nper locus per generation. Thus the mutational rate for the $M=1$ case falls within the range \nof experimental results, indicating that human knowledge system is probably modular.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.62]{fig5.eps}\n\\end{center}\n\\caption{\\label{Fig5} Mutation rate of genetic sequence at which the $F_{\\text{ST}}$ \nratio is $1.26$ versus modularity . Other parameters are as in Fig.\\ \\ref{Fig1}. }\n\\end{figure}\n\n\\section{Discussion}\\label{sec:discussion}\n\nSo why is having a modular knowledge system so helpful in the human migration\nprocess? A migrating human population must adapt knowledge quickly. \nNew knowledge is generated through trial\n and error (mutation, $\\mu$). Communication (knowledge corpus transfer, $\\nu$)\n propagates useful new knowledge in the population. \nIf the knowledge system is non-modular, however, communication \ncauses confusion. This is because transfer of simply a $L\/K$ segment \n does not transfer useful information in a non-modular knowledge system. \nFor example, a hunter can teach\n a wood gatherer how to hunt, including how to make stone arrowheads. If the knowledge \nsystem of the wood gatherer is non-modular, the hunting module can interact with \nthe wood gathering module, and the wood gatherer may wrongly believe that arrow-shaped \ntools could also work for cutting trees, and replace his or her ax with arrows. For a \nmodular knowledge system, this frustrating confusion will not happen, and modularity \nreduces frustration.\nSo if the knowledge system is modular, the population can take advantage\n of faster knowledge communication, while if the system is non-modular,\n knowledge communication can cause confusion and is deleterious\nbetween individuals with different specializations.\n\nFor a population with a modular knowledge \nsystem, a smaller mutational rate of genes creates the same $F_{\\text{ST}}$ ratio, \nso the evolutionary rate is higher than the non-modular counterpart when the mutational \nrates are the same. The population \nwith a modular knowledge system evolves faster, and from Fisher's fundamental theorem of \nnatural selection \\cite{fisher1930genetical} we expect that the genetic diversity is\n higher in the more rapidly evolving population.\n\nWhy does environmental heterogeneity create an asymmetry of genetic distance in \ndifferent directions, even if environmental change does not directly \ninfluence genes in our model? For a\npopulation migrating in the north-south direction, the new environment poses severe challenges \nto the immigrants, and fewer founders may survive compared to a east-west migration. \nThis founder effect \nincreases the genetic distance between the immigrant population and the population \nthey originate from \\cite{hedrick2011genetics}. \nFor the population migrating in east-west direction, much milder \nenvironmental changes largely reduce the founder effect, thus reducing the genetic \ndistance from the original population. \n\nIn addition to spatial heterogeneity, our stochastic model \nnaturally creates temporal inhomogeneity. Even though the average fitness of a \npopulation changes smoothly, fitness spikes appear occasionally, corresponding \nto knowledgeable people or \"heroes\" in human history. Immediately after the initial \ncolonization of one environment, the highest individual fitness value is more than five \ntimes the average fitness value of the population in our model. \nAfter evolution of the population for approximately $400$ generations, \nthe fitness is \"saturated\", and the highest fitness is only 50\\% better than \nthe average fitness. This is consistent with our impression that more heroes emerge \nin a fast-changing society than a stagnant one.\n\n\n\\section{Conclusion}\n\nIn conclusion, we built a model of population migration in an asymmetric,\n two dimensional system. We have shown the vital role that modularity\n plays in the migration rates and gene flows. We have shown that a modular knowledge \nsystem coupled with knowledge transfer accelerates human migration. \nOur results demonstrate an east-west and\n north-south migration rate difference, and we have related environmental\n variation with longitude and latitude \nto migration rate. We have shown that the asymmetry of migration velocity \noriginates from asymmetric environmental gradients. \nThe asymmetry of migration velocity exists only if migration is limited by fitness. \nPredictions for asymmetry of genetic variation are in agreement \nwith patterns of human gene flow in the Americas. \nOur model may be applied to other systems such as the spread of invasive species, \ncancer cells migration, and bacterial migration.\n\n\\section*{Authors' contribution}\nD.W. wrote the codes and collected and analyzed the data. Both D.W. and M.W.D. \ndeveloped and analyzed the model and drafted the manuscript. All authors gave final approval for publication.\n\n\\section*{Competing interest}\nWe declare we have no competing interests.\n\n\\section*{Funding}\nWe received no funding for this study.\n\n\\section*{Appendix}\n\nThe dynamics of evolution in one environment is described by a master equation:\n\n\\begin{eqnarray}\\label{Eq8}\n\\frac{d P ( \\{ n_{\\bf a} \\}; t)}{d t} &=&\n\\sum_{ \\{ {\\bf a} \\} }\n\\bigg[\nf(S_{\\bf a}) (n_{\\bf a}-1) \\sum_{ \\{ {\\bf b} \\ne {\\bf a} \\} }\n \\frac{n_{\\bf b}+1}{N} P(n_{\\bf a}-1, n_{\\bf b}+1; t)\\nonumber\n\\\\&&-f(S_{\\bf a}) n_{\\bf a} \\sum_{ \\{ {\\bf b} \\ne {\\bf a} \\} } \n\\frac{n_{\\bf b}}{N} P(n_{\\bf a}, n_{\\bf b}; t)\n\\bigg]\\delta_{N,N^*}\n\\nonumber \n\\\\&& \\nonumber +\\sum_{ \\{ {\\bf a} \\} }\n\\bigg[\nf(S_{\\bf a}) (n_{\\bf a}-1)\n  P(n_{\\bf a}-1 ; t) -f(S_{\\bf a}) n_{\\bf a} \n P(n_{\\bf a}; t)\n\\bigg](1-\\delta_{N,N^*})\n\\\\ &&\n+ \\mu\n\\sum_{ \\{ {\\bf a} \\} }\n\\sum_{ \\{ {\\bf b}=\\partial {\\bf a} \\} }\n\\bigg[\n(n_{\\bf b}+1)  P(n_{\\bf a}-1, n_{\\bf b}+1; t) -\nn_{\\bf b}  P(n_{\\bf a}, n_{\\bf b}; t)\n\\bigg]\n\\nonumber \\\\ &&\n+ \\nu\n\\sum_{ \\{ {\\bf a} \\} }\n\\sum_{k=1}^K\n\\sum_{ \\{ {\\bf b}, {\\bf b}_k \\ne {\\bf a}_k  \\} }\n\\bigg[\n(n_{ {\\bf a} \/ {\\bf b}_k } +1)\n \\frac{ n_ { {\\bf b} \/ {\\bf a}_k }  }{N} P(n_{\\bf a}-1, \nn_{ {\\bf a} \/ {\\bf b}_k } +1; t)\\nonumber\n\\\\&&-n_{ {\\bf a} \/ {\\bf b}_k }\n \\frac{ n_ { {\\bf b} \/ {\\bf a}_k }  }{N} P(n_{\\bf a}, \nn_{ {\\bf a} \/ {\\bf b}_k } ; t)\n\\bigg]\n\\end{eqnarray}\n\nHere $n_{\\bf a}$ is the number of individuals with sequence $S_{\\bf a}$, with\nthe vector index ${\\bf a}$ used to label the $2^L$ sequences.\nThe notation $\\partial {\\bf a}$ means the $L$ sequences created by\na single mutation from sequence $S_{\\bf a}$.\nThe notation ${\\bf a} \/ {\\bf b}_k$ means the sequence created by\ntransferring module $k$ from sequence \n$S_{\\bf b}$ into sequence $S_{\\bf a}$. \nHere $N^*$ is the environmental capacity of the environment.\n\n\\bibliographystyle{vancouver}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAs introduced in \\cite{coffee-time} and \\cite{geodetic}, the $P_3$-hull number of a simple connected graph is the minimum cardinality of a set $U$ of initially infected vertices that will eventually infect the entire graph where an uninfected node becomes infected if two or more of its neighbors are infected. There has been much work on formulas for the $P_3$-hull numbers of various types of graphs, \\cite{p_3-Hamming, MR3040145, p_3-product, p3Kneser}, as well as with the closely related notion of the $2$-neighbor bootstrap percolation problem,\n\\cite{2-neighbour_bootstrap_percolation, Marcilon_Thiago, Przykucki_Michal}.\n\nImportant to this paper is the decycling number. Given a graph $G$, its decycling number, $\\nabla(G)$, is the minimum cardinality of a set $U$ of vertices such that $G-U$ is acyclic. In general, it is very hard to compute a graph's decycling number. In fact, it has been shown to be NP-complete \\cite{DecyclingNumberIsNPComplete}. However, results in special cases have been obtained, \\cite{decycleOfGraphs, decylceRandom, DecyclingNumberOfGeneralizedPetersenGraphs, decycleCubic, decycleBoxProduct}.\n\nIn this paper, after initial definitions, we show that for a cubic graph, the $P_3$-hull number and the decycling number coincide, Theorem \\ref{Main Theorem}. By \\cite{DecyclingNumberOfGeneralizedPetersenGraphs}, it follows that the $P_3$-hull number of the generalized Petersen graph, $G(n,k)$, is $\\left\\lceil\\frac{n+1}{2}\\right\\rceil$, Corollary \\ref{p3_hull_GP}. Furthermore, the complement of any initial infecting set is a forest. In Theorem \\ref{components thm}, it is show that for any infecting set of minimum cardinality this forest always has exactly one or two components.\n\nIn addition, we introduce the notion of the infecting time of an infecting set and study it for the Petersen graph. Explicit times are computed for special infecting sets, Theorem \\ref{dan_time}. Giving explicit formulas for the minimal and maximal infecting times is a very difficult problem. However, we give complete answers for the special case of $G(n,1)$, Theorem \\ref{full_time}.\n\nFinally, we introduce a number of graphs related to the generalized Petersen graph. For a type of surgery, $G(n,k)\\# G(n,k)$, the $P_3$-hull number is computed in Theorem \\ref{thm_surgery}. Associated to a permutation $\\sigma$ of $S_n$, a generalization of $G(n,k)$, called $GG(n,\\sigma)$, is introduced. General bounds for its $P_3$-hull number are given in Theorem \\ref{odd cycles upper bound}. An exact answer is computed under an additional hypothesis in Theorem \\ref{thm:minGGP}.\n\n\n\n\\section{Initial Definitions}\n\n    Throughout the paper, let $G = (V,E)$ be a finite, simple, connected graph and let $S\\subseteq V$. We write $G[S]$ for the corresponding \\textit{induced subgraph} of $G$ on $S$. We say $G$ is \\textit{cubic} if each vertex of $G$ has degree $3$.\n\n    Following \\cite{p3Kneser}, the \\textit{$P_3$-interval}, $I[S]$, is the set $S$ together with all vertices in $G$ that have two or more neighbors in $S$.\n     If $I[S] = S$, then the set $S$ is called \\textit{$P_3$-convex}.\n     The $P_3$\\textit{-convex hull}, $H_\\mathcal C(S)$ of $S$, is the smallest $P_3$-convex set containing $S$.\n     Iteratively, define $I^0[S] = S$ and $I^p[S] = I[I^{p-1}[S]]$ for any positive integer $p$. Then $H_\\mathcal C(S)$ is the union of all $I^p[S]$.\n\n     If $H_\\mathcal C(S) = V$, we say that $S$ is a \\textit{$P_3$-hull set} of $G$. We also refer to a $P_3$-hull set as an \\textit{infecting set}. The minimum cardinality, $h_{P_3}(G)$, of a $P_3$-hull set in G is called the \\textit{$P_3$-hull number} of G. This will be the main object of our study, and we will refer to $P_3$-hull sets of cardinality $h_{P_3}(G)$ as \\textit{minimum size infecting sets}. For a $P_3$-hull set $S$, we say that the \\textit{infecting time} of $S$, denoted $T_I(S)$, is the smallest integer $p$ such that $I^p[S] = V$.\n\n    Relevant for this paper, we say that $S$ is a \\textit{decycling set} of $G$ if the induced subgraph $G[V-S]$ is acyclic, \\cite{decycleOfGraphs, decylceRandom}. The minimum cardinality of a decycling set of $G$ is called the \\textit{decycling number} of $G$ and denoted by $\\nabla(G)$.\n\nRecall that a \\textit{generalized Petersen graph}, $G(n,k)$ with $1\\le k< \\frac n2$, has vertex set\n\\[V=\\{{u}_0,{u}_1,\\ldots,{u}_{n-1},\\,v_0,v_1,\\dots,v_{n-1}\\}\\]\nand edge set (interpreting each index modulo $n$)\n\\[E=\\{{u}_i{u}_{i+1},\\,{u}_iv_i, \\,v_iv_{i+k}:0\\leq i\\leq n-1\\}.\\]\n\n\n\n\\section{$P_3$-Hull Numbers and Minimum Infecting Sets for $G(n,k)$}\n\n\n\\begin{theorem}\\label{Main Theorem}\n    Let $G = (V,E)$ be a cubic graph and $S\\subseteq V$. Then $S$ is an infecting set of $G$ if and only if it is a decycling set of $G$. In particular, $h_{P_3}(G) = \\nabla(G)$.\n\\end{theorem}\n\n\\begin{proof}\n\tWe show first that an infecting set is a decycling set via the contrapositive. Assume that $S$ is not a decycling set. Then there exists some nonempty $W \\subseteq V- S$ so that $G[W]$ is a cycle. Hence each $w\\in W$ has exactly two neighbors in $W$ and one neighbor in $V- W$.\n\t\n\tSuppose now that $S$ is an infecting set. Since $S\\cap W=\\emptyset$, there is a minimal integer $n\\ge 0$ so that $I^{n}[S] \\cap W = \\emptyset$ and $I^{n+1}[S] \\cap W \\not= \\emptyset$. But then for $w\\in I^{n+1}[S] \\cap W$, there must have been two neighbors of $w$ in $I^{n}[S]$. As $I^{n}[S] \\cap W = \\emptyset$, these two neighbors must lie in $V-W$ which is a contradiction.\n\t\n    Next we show that a decycling set is an infecting set.\n    Suppose that $S$ is a decycling set of $G$. If $S$ is not infecting, then $H_\\mathcal C(S)$ is a proper subset of $V$ that is still decycling. As the subgraph induced by $V-H_\\mathcal C(S)$ is a forest, there exists some $v\\in V-H_\\mathcal C(S)$ with degree $1$. This implies two neighbors of $v$ lie in $H_\\mathcal C(S)$ which is a contradiction.\n\\end{proof}\n\n\\begin{corollary} \\label{p3_hull_GP}\n$h_{P_3}(G(n,k)) = \\lceil\\frac{n+1}{2}\\rceil$.\n\\end{corollary}\n\n\\begin{proof}\n    This follows from Theorem \\ref{Main Theorem} and  \\cite[Theorem 3.1]{DecyclingNumberOfGeneralizedPetersenGraphs}, where it is shown that  $\\nabla(G(n,k)) = \\left\\lceil\\frac{n+1}{2}\\right\\rceil$.\n\\end{proof}\n\nAs a prelude to infecting time calculations, we present an explicit minimum size infecting set for $G(n,k)$ (examples can be seen in Figure \\ref{ExamplesFigure}). An alternate example may be found in \\cite[Lemma 3.3]{DecyclingNumberOfGeneralizedPetersenGraphs}.\n\n\\begin{corollary}\\label{infecting_set}\n\tLet $c = \\gcd(n,k)$,  $l=\\frac{n}{c}$, and\n\t\\[ \\mathcal S_v = \\{v_{j+ik}:0\\leq i\\leq l-1,\\, 0\\leq j\\leq c-1,\\, i\\text{ odd }\\}. \\]\n\tFor $l$ even, let\n\t\\[ \\mathcal S_u = \\{u_0\\}\\]\n\tand for $l$ odd, let\n\t\\[ \\mathcal S_u = \\{u_{c-1}\\} \\cup \\{u_j:0\\leq j\\leq c-1,\n\t\\, j\\text{ even }\\}. \\]\n\tThen \\[ \\mathcal S = \\mathcal S_v \\cup \\mathcal S_u \\] is a minimum size infecting set for $G(n,k)$.\n\\end{corollary}\n\n\\begin{figure}[H]\n\t\\centering\n\t\\begin{tikzpicture}\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (a) at (4*1,4*0) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (b) at (4*.866,.5*4) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (c) at (4*.5,.866*4) {};\n\t\t\\node[fill,circle,inner sep=3pt,minimum size=1pt, label = \\hspace{0em}$u_0$] (d) at (4*0,1*4) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (g) at (4*-1,0*4) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (f) at (4*-.866,.5*4) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (e) at (4*-.5,.866*4) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (j) at (4*0,-1*4) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (h) at (4*-.866,-.5*4) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (i) at (4*-.5,-.866*4) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (l) at (4*.866,-.5*4) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (k) at (4*.5,-.866*4) {};\n\t\t\n\t\t\\draw[black] (a) -- (b);\n\t\t\\draw[black] (b) -- (c);\n\t\t\\draw[black] (c) -- (d);\n\t\t\\draw[black] (d) -- (e);\n\t\t\\draw[black] (e) -- (f);\n\t\t\\draw[black] (f) -- (g);\n\t\t\\draw[black] (g) -- (h);\n\t\t\\draw[black] (h) -- (i);\n\t\t\\draw[black] (i) -- (j);\n\t\t\\draw[black] (j) -- (k);\n\t\t\\draw[black] (k) -- (l);\n\t\t\\draw[black] (l) -- (a);\n\t\t\n\t\t\\node[fill,circle,inner sep=3pt,minimum size=1pt, label = \\hspace{1.5em}$v_3$] (m) at (2.5*1,2.5*0) {};\n\t\t\\node[fill,circle,inner sep=3pt,minimum size=1pt, label = \\hspace{1.5em}$v_2$] (n) at (2.5*.866,.5*2.5) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (o) at (2.5*.5,.866*2.5) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (p) at (2.5*0,1*2.5) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (s) at (2.5*-1,0*2.5) {};\n\t\t\\node[fill,circle,inner sep=3pt,minimum size=1pt, label = \\hspace{1.5em}$v_{10}$] (r) at (2.5*-.866,.5*2.5) {};\n\t\t\\node[fill,circle,inner sep=3pt,minimum size=1pt, label = \\hspace{1.5em}$v_{11}$] (q) at (2.5*-.5,.866*2.5) {};\n\t\t\\node[fill,circle,inner sep=3pt,minimum size=1pt, label = \\hspace{1.5em}$v_6$] (v) at (2.5*0,-1*2.5) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (t) at (2.5*-.866,-.5*2.5) {};\n\t\t\\node[fill,circle,inner sep=3pt,minimum size=1pt, label = \\hspace{1.5em}$v_7$] (u) at (2.5*-.5,-.866*2.5) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (x) at (2.5*.866,-.5*2.5) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (w) at (2.5*.5,-.866*2.5) {};\n\t\t\n\t\t\\draw[black] (n) -- (p);\n\t\t\\draw[black] (p) -- (r);\n\t\t\\draw[black] (r) -- (t);\n\t\t\\draw[black] (t) -- (v);\n\t\t\\draw[black] (v) -- (x);\n\t\t\\draw[black] (x) -- (n);\n\t\t\n\t\t\\draw[black, dashed] (m) -- (o);\n\t\t\\draw[black, dashed] (o) -- (q);\n\t\t\\draw[black, dashed] (q) -- (s);\n\t\t\\draw[black, dashed] (s) -- (u);\n\t\t\\draw[black, dashed] (u) -- (w);\n\t\t\\draw[black, dashed] (w) -- (m);\n\t\t\n\t\t\\draw[black] (m) -- (a);\n\t\t\\draw[black] (n) -- (b);\n\t\t\\draw[black] (o) -- (c);\n\t\t\\draw[black] (p) -- (d);\n\t\t\\draw[black] (q) -- (e);\n\t\t\\draw[black] (r) -- (f);\n\t\t\\draw[black] (s) -- (g);\n\t\t\\draw[black] (t) -- (h);\n\t\t\\draw[black] (u) -- (i);\n\t\t\\draw[black] (v) -- (j);\n\t\t\\draw[black] (w) -- (k);\n\t\t\\draw[black] (x) -- (l);\n\t\t\n\t\\end{tikzpicture}\n\t\\qquad\n\t\\begin{tikzpicture}\n\t\t\\node[fill,circle,inner sep=3pt,minimum size=1pt, label = \\hspace{1.5em}$u_3$] (a) at (4*1,4*0) {};\n\t\t\\node[fill,circle,inner sep=3pt,minimum size=1pt, label = \\hspace{1.5em}$u_2$] (b) at (4*.866,.5*4) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (c) at (4*.5,.866*4) {};\n\t\t\\node[fill,circle,inner sep=3pt,minimum size=1pt, label = \\hspace{0em}$u_0$] (d) at (4*0,1*4) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (g) at (4*-1,0*4) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (f) at (4*-.866,.5*4) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (e) at (4*-.5,.866*4) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (j) at (4*0,-1*4) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (h) at (4*-.866,-.5*4) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (i) at (4*-.5,-.866*4) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (l) at (4*.866,-.5*4) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (k) at (4*.5,-.866*4) {};\n\t\t\n\t\t\\draw[black] (a) -- (b);\n\t\t\\draw[black] (b) -- (c);\n\t\t\\draw[black] (c) -- (d);\n\t\t\\draw[black] (d) -- (e);\n\t\t\\draw[black] (e) -- (f);\n\t\t\\draw[black] (f) -- (g);\n\t\t\\draw[black] (g) -- (h);\n\t\t\\draw[black] (h) -- (i);\n\t\t\\draw[black] (i) -- (j);\n\t\t\\draw[black] (j) -- (k);\n\t\t\\draw[black] (k) -- (l);\n\t\t\\draw[black] (l) -- (a);\n\t\t\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (m) at (2.5*1,2.5*0) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (n) at (2.5*.866,.5*2.5) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (o) at (2.5*.5,.866*2.5) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (p) at (2.5*0,1*2.5) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (s) at (2.5*-1,0*2.5) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (r) at (2.5*-.866,.5*2.5) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (q) at (2.5*-.5,.866*2.5) {};\n\t\t\\node[fill,circle,inner sep=3pt,minimum size=1pt, label = \\hspace{1.5em}$v_6$] (v) at (2.5*0,-1*2.5) {};\n\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (t) at (2.5*-.866,-.5*2.5) {};\n\t\t\\node[fill,circle,inner sep=3pt,minimum size=1pt, label = \\hspace{1.5em}$v_7$] (u) at (2.5*-.5,-.866*2.5) {};\n\t\t\\node[fill,circle,inner sep=3pt,minimum size=1pt, label = \\hspace{1.5em}$v_4$] (x) at (2.5*.866,-.5*2.5) {};\n\t\t\\node[fill,circle,inner sep=3pt,minimum size=1pt, label = \\hspace{1.5em}$v_5$] (w) at (2.5*.5,-.866*2.5) {};\n\t\t\n\t\t\n\t\t\\draw[black] (p) -- (t);\n\t\t\\draw[black] (t) -- (x);\n\t\t\\draw[black] (x) -- (p);\n\t\t\n\t\t\\draw[black, dashed] (q) -- (u);\n\t\t\\draw[black, dashed] (u) -- (m);\n\t\t\\draw[black, dashed] (m) -- (q);\n\t\t\n\t\t\\draw[black, densely dotted] (r) -- (v);\n\t\t\\draw[black, densely dotted] (v) -- (n);\n\t\t\\draw[black, densely dotted] (n) -- (r);\n\t\t\n\t\t\\draw[black, dashdotted] (s) -- (w);\n\t\t\\draw[black, dashdotted] (w) -- (o);\n\t\t\\draw[black, dashdotted] (o) -- (s);\n\t\t\n\t\t\\draw[black] (m) -- (a);\n\t\t\\draw[black] (n) -- (b);\n\t\t\\draw[black] (o) -- (c);\n\t\t\\draw[black] (p) -- (d);\n\t\t\\draw[black] (q) -- (e);\n\t\t\\draw[black] (r) -- (f);\n\t\t\\draw[black] (s) -- (g);\n\t\t\\draw[black] (t) -- (h);\n\t\t\\draw[black] (u) -- (i);\n\t\t\\draw[black] (v) -- (j);\n\t\t\\draw[black] (w) -- (k);\n\t\t\\draw[black] (x) -- (l);\n\t\t\n\t\\end{tikzpicture}\n\t\\centering\n\t\\caption{Minimum size infecting sets for $G(12,2)$ and $G(12, 4)$, respectively.}\n\t\\label{ExamplesFigure}\n\\end{figure}\n\n\\begin{proof}\n\tNote that the subgraph of $G(n,k)$ which is induced on the vertex set $\\{v_0,v_1,\\ldots,v_{n-1}\\}$ is the disjoint union of $c$ cycles of length $l$. The corresponding vertex set of each cycle is given by \t\\[ V_j = \\{v_{j+ik}:0\\leq i\\leq l-1\\}\\]\n\twith $0\\leq j \\leq c-1$.\n\t\n\tWe will distinguish two cases. The first is when $l$ is even.\n\tGiven $\\mathcal S$ as an initial set of infected points, the infection will spread to infect every vertex set $V_j$, $0\\leq j\\leq c-1$. From there with ${u}_0$ the infection spreads to all of $\\{{u}_0, {u}_1, \\ldots, {u}_{n-1}\\}.$ Note that $n=cl$ is even and\n\t\\[|\\mathcal S|=1+c\\frac{l}{2}=\\frac{n}{2}+1=\\bigg\\lceil\\frac{n+1}{2}\\bigg\\rceil\\]\n\tso that $\\mathcal S$ is a minimum size infecting set by Corollary \\ref{p3_hull_GP}.\n\t\n\tTurn now to the case of $l$ odd.\n\tGiven $\\mathcal S$ as an initial set of infected points, the infection will spread to $\\{{u}_j:0\\leq j\\leq c-1\\}.$ With ${u}_j$ and $v_{j+k}$ infected, also $v_j$ will be infected, and the infection will spread to infect every vertex set $V_j,\\hspace{1mm} 0\\leq j\\leq c-1$. From there with $\\{{u}_j:0\\leq j\\leq c-1\\}$ the infection spreads to all of $\\{{u}_0,{u}_1,\\ldots,{u}_{n-1}\\}.$ As\n\t\\[|\\mathcal S|=\\bigg\\lceil\\frac{c+1}{2}\\bigg\\rceil +c\\frac{l-1}{2}=\\bigg\\lceil c\\frac{l-1}{2}+\\frac{c+1}{2}\\bigg\\rceil=\\bigg\\lceil\\frac{n+1}{2}\\bigg\\rceil,\\]\n\tit follows that $\\mathcal S$ is a minimum size infecting set.\n\\end{proof}\n\nBy Theorem \\ref{Main Theorem}, the complement of a minimum size infecting set of a cubic graph is a forest. The next theorem constrains the number of connected components of this forest for $G(n,k)$.\n\n\\begin{theorem}\\label{components thm}\n\tLet $S$ be a minimum size infecting set of the generalized Petersen graph $G=G(n,k)=(V,E)$.\n\n\\begin{itemize}\t\n\t\\item If $n$ is odd, then $G[V-S]$ is a tree.\n\t\n\t\\item If $n$ is even, then the forest $G[V-S]$ may have two or one connected components. It will have two connected components if and only if $S$ has no neighboring points and one connected component if and only if $S$ has exactly one pair of points that are neighbors.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proof}\n\tRecall for $G$ that $|V|=2n$ and $|E|=3n$. Write $\\nu$ and $\\epsilon$ for the number of vertices and edges, respectively, of $G[V-S]$. As $G[V-S]$ is a forest, $\\nu-\\epsilon$ is the number of trees in the forest.\n\t\n\tWhen $n$ is odd, write $n=2k+1$ so that $|S|=\\lceil \\frac{n+1}{2} \\rceil= k+1$ and $\\nu=2(2k+1) - (k+1) = 3 k +1$. At most, if $S$ has no neighboring points, passing from $G$ to $G[V-S]$ would remove $3|S|$ edges. Therefore, $\\epsilon\n\t\\geq 3(2k+1) - 3(k+1) = 3k$. It follows that $\\nu- \\epsilon \\le 1$, and $G[V-S]$ must be a single tree.\n\t\n\tWhen $n$ is even, write $n=2k$ so $|S|=\\lceil \\frac{n+1}{2} \\rceil= k+1$ and $\\nu = 2(2k) - (k+1) = 3 k -1$. As in the previous paragraph, we get $\\epsilon\n\t\\geq 3(2k) - 3(k+1) = 3k-3$. It follows that $\\nu- \\epsilon \\le 2$, and $G[V-S]$ has either one or two connected components. In addition, $G[V-S]$ is a tree exactly when $S$ has no neighboring points, and $G[V-S]$ has two trees exactly when $S$ has exactly one pair of neighboring points.\n\\end{proof}\n\n\n\n\\section{Infecting Times}\n\n\\begin{theorem}\\label{dan_time}\n\tLet $c = \\gcd(n,k)$,  $l=\\frac{n}{c}$, and $\\mathcal S$ be the infecting set for $G(n,k)$ from Corollary \\ref{infecting_set}.\n\\begin{itemize}\t\n\t\\item When $l$ is even, the infecting time for $\\mathcal S$ is $\\frac{n}{2}$.\n    \\item When $l$ is odd, the infecting time for $\\mathcal S$ is $\\frac{n-c}{2}+1$.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proof}\nNote that $n = cl$ with $1\\le c \\le k < \\frac n2$. Thus $l \\ge 3$.\n\nBegin with the case of $l\\ge 4$ even. Obviously, $\\{u_0\\} \\cup \\{v_k : 0\\leq k \\leq n-1\\}\\subseteq I^1[\\mathcal S]$ by definition. However, also $u_{n-1} \\in I^1[\\mathcal S]$. To see this, note that $v_{n-1} \\in V_{c-1}$, and $(c-1)+ik \\equiv n-1 \\mbox{ mod } n$ holds iff $i\\frac kc \\equiv -1 \\mbox{ mod } l$. As $l$ is even, the last congruency can only hold for $i$ odd. Thus, $v_{n-1} \\in \\mathcal S_v \\subseteq \\mathcal S$ and $u_{n-1} \\in I^1[\\mathcal S]$.\n\nIn addition to $u_{n-1} \\in I^1[\\mathcal S]$, $u_{1} \\in I^1[\\mathcal S]$ happens if and only if $c=1$. In any case, another $\\frac n2 -1$ steps are necessary for the infection to spread from $I^1[\\mathcal S]$ to the rest of the graph.\n\n\n\n\n\nTurn now to the case of $l \\ge 3$ odd. Obviously, \\[\\{u_j: 0\\le j\\le c-1\\} \\cup \\{v_{j+ik} : 0< i < l-1, 0\\le j\\le c-1\\}\\subseteq I^1[\\mathcal S]\\] by definition, where some of the $v_{j+ik}$ for $i\\in \\{0,l-1\\}$ will still be missing. In addition, exactly one of $u_c$ and $u_{n-1}$ will be in $I^1[\\mathcal S]$. To see this, note that $v_{c} \\in V_{0}$, and $i_1k \\equiv c \\mbox{ mod } n$ holds iff $i_1\\frac kc \\equiv 1 \\mbox{ mod } l$. Similarly,\n$v_{n-1} \\in V_{c-1}$, and $(c-1)+i_2k \\equiv n-1 \\mbox{ mod } n$ holds iff $i_2\\frac kc \\equiv -1 \\mbox{ mod } l$. Thus, $i_1\\frac kc + i_2\\frac kc\\equiv 0 \\mbox{ mod } l$, and $i_1+i_2$ is a multiple of $l$. With $0<i_1,i_2< l$, we must have $i_1+i_2 = l$ odd. Thus, exactly one of $i_1,i_2$ is odd, exactly one of $v_c, v_{n-1}$ is in $\\mathcal S_v \\subseteq \\mathcal S$, and exactly one of $u_c,u_{n-1}$ is in $I^1[\\mathcal S]$. In particular, \\[\\mbox{$I^1[\\mathcal S]$ contains exactly $c+1$ of the points $u_j$. These $c+1$ points are the vertices of a path of length $c$ on the exterior cycle.}\\]\n\nContinuing with $v_c$ and $v_{n-1}$, we have $v_c \\in I^1[\\mathcal S]$ and $u_c \\in I^2[\\mathcal S]$. To see this, note that $v_c \\in I^1[\\mathcal S]$ fails iff $i_1 = l-1$. However, $i_1 = l-1$ implies $\\frac kc \\equiv -1 \\mbox{ mod } l$ and $k+c \\equiv 0 \\mbox{ mod } n$ which is impossible for $1\\le c \\le k < \\frac n2$. A similar argument shows that $u_{n-1} \\in I^2[\\mathcal S]$ unless $k=c =\\gcd(n,k)$, i.e., unless $n$ is a multiple of $k$. Thus, $u_c, u_{n-1} \\in I^2[\\mathcal S]$ for $k \\nmid n$ while a direct inspection shows $u_c, u_{c+1} \\in I^2[\\mathcal S]$ for $k \\mid n$. In any case,\n\\[\\mbox{$I^2[\\mathcal S]$ will add at least one new point $u_j$ to $I^1[\\mathcal S]$ but not more than two.}\\]\n\nFor $c\\in \\{1,2\\}$, note $\\{v_k : 0\\leq k \\leq n-1\\}\\subseteq I^2[\\mathcal S]$. If $c=1$, then $n=l$ is odd and the infection will need another $\\frac {n+1}2 -2$ steps to spread from $I^2[\\mathcal S]$ to the rest of the graph.\nIf $c=2$, then $n=2l$ is even and the infection will need another $\\frac {n}2 -2$ steps to spread from $I^2[\\mathcal S]$ to the rest of the graph.\n\nFor $c \\ge 3$, we have $\\{v_k : 0\\leq k \\leq n-1\\}\\subseteq I^3[\\mathcal S]$. If $k\\nmid n$ and $u_c \\in I^1[\\mathcal S]$, then $u_{c+1}, u_{n-1} \\in I^2[\\mathcal S]$, $u_{c+2}, u_{n-2} \\in I^3[\\mathcal S]$, and the infection will need a total of $\\frac {n-c}2 +1$ steps to spread from $\\mathcal S$ to the rest of the graph. A similar argument is true for $k\\nmid n$ and $u_{n-1} \\in I^1[\\mathcal S]$.\nIf $k\\mid n$, then $u_{c+1} \\in I^2[\\mathcal S]$, $u_{c+2}, u_{n-1} \\in I^3[\\mathcal S]$, and the infection will need again a total of $\\frac {n-c}2 +1$ steps to spread from $\\mathcal S$ to the rest of the graph.\n\\end{proof}\n\n\n\\begin{lemma}\\label{diam_lemma}\n\tFor a cubic graph, $G=(V,E)$, and infecting set $S$, the infecting time is $\\lceil \\frac{d+1}{2}\\rceil$ where $d$ is the largest diameter of a tree of the forest $G[V-S]$.\n\\end{lemma}\n\n\\begin{proof}\n\tFrom Theorem \\ref{Main Theorem}, $G[V-S]$ is a forest. The result follows by observing that since $G$ is cubic, the infection will first infect the leaves of $G[V-S]$ and progress in a similar fashion iteratively.\n\\end{proof}\n\nIn general, determining the minimal or maximal infecting time of a minimum size infecting set for $G(n,k)$ is very difficult. We give next full results for the special case of $G(n,1)$, including an exhaustive list of all possible infecting times. By Lemma \\ref{diam_lemma}, it suffices to calculate the maximal diameter for the connected components of the complement of a minimum size infecting set. We give these maximal diameters below as it is finer information.\n\n\n\\begin{theorem}\\label{full_time}\n\tFor $G(n,1)$, the set of possible maximal diameters of connected components for the complement of a minimum size infecting set $S$ is\n\t\\[ \\bigg\\{n, n+2, n+4, \\ldots, n +2 \\bigg\\lfloor\\frac{n-3}{4}\\bigg\\rfloor \\bigg\\} \\]\n\twhen $n\\ge 3$ is odd and\n\t\\[ \\bigg\\{d : \\bigg\\lfloor \\frac{n}{2} \\bigg\\rfloor \\leq d \\leq \\frac 32 n -2 \\text{ with }d \\not= n-1 \\bigg\\} \\]\n\twhen $n\\ge 4$ is even.\n\\end{theorem}\n\n\\begin{proof}\n\t\n\tBegin by viewing $G=G(n,1)$ as a discrete annulus as in Figure \\ref{fig:ladder}.\n\t\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\t\t\\tikzstyle{hollow node}=[draw,circle, fill=white, outer sep=-4pt,inner sep=0pt,\n\t\t\tminimum width=8pt, above]\n\t\t\t\\foreach \\i in {0,...,5,7,8}{\n\t\t\t\t\\foreach \\j in {0,1}{\n\t\t\t\t\t\\draw[black,thick] (\\i,\\j) -- (\\i+1,\\j);\n\t\t\t\t}\n\t\t\t}\n\t\t\t\\foreach \\i in {0,...,9}{\n\t\t\t\t\\draw[black,thick] (\\i,0) -- (\\i,1);\n\t\t\t}\n\t\t\t\\draw[black,thick] (6,0) -- (6.25,0);\n\t\t\t\\draw[black,thick] (6,1) -- (6.25,1);\n\t\t\t\\draw[black,thick] (6.75,0) -- (7,0);\n\t\t\t\\draw[black,thick] (6.75,1) -- (7,1);\n\t\t\t\\draw[black,thick] (-.4,0) -- (0,0);\n\t\t\t\\draw[black,thick] (-.4,1) -- (0,1);\n\t\t\t\\draw[black,thick] (9,1) -- (9.4,1);\n\t\t\t\\draw[black,thick] (9,0) -- (9.4,0);\n\t\t\t\n\t\t\t\\foreach \\i in {0,1,2}{\n\t\t\t\t\\foreach \\j in {0,1}{\n\t\t\t\t\t\\draw  node[fill,circle,inner sep=0pt,minimum size=2pt] at (6.36 + .14*\\i,\\j){};\n\t\t\t\t\t\\draw  node[fill,circle,inner sep=0pt,minimum size=2pt] at (9.6 + .14*\\i,\\j){};\n\t\t\t\t\t\\draw  node[fill,circle,inner sep=0pt,minimum size=2pt] at (-.6 - .14*\\i,\\j){};\n\t\t\t\t}\n\t\t\t}\n\t\t\t\\foreach \\i in {0,...,9}{\n\t\t\t\t\\node[hollow node] (\\i0) at (\\i,0){};\n\t\t\t\t\\node[hollow node] (\\i1) at (\\i,1){};\n\t\t\t}\n\t\t\t\\foreach \\i in {0,...,6}{\n\t\t\t\t\\node at (\\i,-.4){$v_{\\i}$};\n\t\t\t\t\\node at (\\i,1.4){$u_{\\i}$};\n\t\t\t}\n\t\t\t\\foreach \\i in {1,2,3}{\n\t\t\t\t\\node at (10-\\i,-.4){$v_{n-\\i}$};\n\t\t\t\t\\node at (10-\\i,1.4){$u_{n-\\i}$};\n\t\t\t}\n\t\t\\end{tikzpicture}\n\t\t\\caption{}\n\t\t\\label{fig:ladder}\n\t\\end{figure}\n\t\n\tStart with the case where $n$ is odd ($n\\geq 3$) and write $n=2k+1$.\n\tBy Corollary \\ref{p3_hull_GP} and Theorem \\ref{components thm}, every minimum size infecting set, $S$, contains $k+1$ vertices, the remaining vertices constitute a tree, $T$, and no two vertices of $S$ are neighbors. To prevent a cycle from appearing in $T$, there must exist a (square) face of the annulus whose top and bottom edges both are not in $T$. From this it follows, possibly after turning the annulus over, that there is a face with exactly two vertices of $S$ configured as in Figure~\\ref{fig:1}, where marked points belong to $S$.\n\t\n\t\t\n\t\t\n\\begin{figure}[H]\n\t\\centering\n\t\\begin{tikzpicture}\n\t\t\\tikzstyle{hollow node}=[draw,circle, fill=white, outer sep=-4pt,inner sep=0pt,\n\t\tminimum width=8pt, above]\n\t\t\\foreach \\i in {0,...,0}{\n\t\t\t\\foreach \\j in {0,1}{\n\t\t\t\t\\draw[black,thick] (\\i,\\j) -- (\\i+1,\\j);\n\t\t\t}\n\t\t}\n\t\t\\foreach \\i in {0,...,1}{\n\t\t\t\\draw[black,thick] (\\i,0) -- (\\i,1);\n\t\t}\n\t\t\\draw[black,thick] (-.4,0) -- (0,0);\n\t\t\\draw[black,thick] (-.4,1) -- (0,1);\n\t\n\t\\draw[black,thick] (1.4,0) -- (1,0);\n\t\\draw[black,thick] (1.4,1) -- (1,1);\n\t\t\n\t\t\\foreach \\i in {0,1,2}{\n\t\t\t\\foreach \\j in {0,1}{\n\t\t\t\t\\draw  node[fill,circle,inner sep=0pt,minimum size=2pt] at (1.56 + .14*\\i,\\j){};\n\t\t\n\t\t\t\t\\draw  node[fill,circle,inner sep=0pt,minimum size=2pt] at (-.6 - .14*\\i,\\j){};\n\t\t\t}\n\t\t}\n\t\t\\foreach \\i in {0,1}{\n\t\t\t\\node[hollow node] (\\i0) at (\\i,0){};\n\t\t\t\\node[hollow node] (\\i1) at (\\i,1){};\n\t\t}\n\n\n\t\\tikzstyle{red node}=[draw,circle, fill=red, outer sep=-4pt,inner sep=0pt, minimum width=8pt, above]\n\t\n\t\\node[red node] at (0,0){};\n\t\\node[red node] at (1,1){};\n\n\t\\end{tikzpicture}\n\t\\caption{}\n\t\\label{fig:1}\n\\end{figure}\n\n\n\tAfter relabeling, the top left vertex can be labeled as $u_{n-1}$ and the bottom right as $v_0$. Cut the annulus now through the middle of this square and unfold it. The tree $T$ must connect $v_0$ to $u_{2k}$ and is forced to contain $u_1,v_1,u_{2k-1},$ and $v_{2k-1}$, see Figure \\ref{fig:ladder}. Recalling that $S$ has no neighboring vertices, to prevent $T$ from having a cycle around the face with indices $1$ and $2$, $S$ must contain exactly one element of $\\{u_2,v_2\\}$. Arguing iteratively, it follows that each set $\\{u_{2i}, v_{2i}\\}$, $0\\leq i \\leq k$ contains exactly one element of $S$.\n\t\n\tThe shortest diameter for such a tree $T$ is pictured in Figure \\ref{fig:narrowest odd tree}. It has diameter $n$ and, by Lemma \\ref{diam_lemma}, a corresponding infecting time of $\\lceil \\frac{n+1}{2} \\rceil$.\n\t\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\t\t\\tikzstyle{hollow node}=[draw,circle, fill=white, outer sep=-4pt,inner sep=0pt,\n\t\t\tminimum width=8pt, above]\n\t\t\t\\tikzstyle{red node}=[draw,circle, fill=red, outer sep=-4pt,inner sep=0pt,\n\t\t\tminimum width=8pt, above]\n\t\t\t\\foreach \\i in {0,...,5,7,8,9}{\n\t\t\t\t\\foreach \\j in {0,1}{\n\t\t\t\t\t\\draw[black,thick] (\\i,\\j) -- (\\i+1,\\j);\n\t\t\t\t}\n\t\t\t}\n\t\t\t\\foreach \\i in {0,...,10}{\n\t\t\t\t\\draw[black,thick] (\\i,0) -- (\\i,1);\n\t\t\t}\n\t\t\t\\draw[black,thick] (6,0) -- (6.25,0);\n\t\t\t\\draw[dark green, line width = 4pt] (6,1) -- (6.25,1);\n\t\t\t\\draw[black,thick] (6.75,0) -- (7,0);\n\t\t\t\\draw[dark green, line width = 4pt] (6.75,1) -- (7,1);\n\t\t\t\\draw[black,thick] (-.4,0) -- (0,0);\n\t\t\t\\draw[black,thick] (-.4,1) -- (0,1);\n\t\t\t\\draw[black,thick] (10,1) -- (10.4,1);\n\t\t\t\\draw[black,thick] (10,0) -- (10.4,0);\n\t\t\t\n\t\t\t\\foreach \\i in {1,...,5,7,8,9}{\n\t\t\t\t\\draw[dark green, line width = 4pt](\\i,1) -- (\\i+1,1);\n\t\t\t}\n\t\t\t\\draw[dark green, line width = 4pt](0,0) -- (1,0);\n\t\t\t\\draw[dark green, line width = 4pt](1,0) -- (1,1);\n\t\t\t\\foreach \\i in {0,1,2}{\n\t\t\t\t\\foreach \\j in {0,1}{\n\t\t\t\t\t\\draw  node[fill,circle,inner sep=0pt,minimum size=2pt] at (6.36 + .14*\\i,\\j){};\n\t\t\t\t\t\\draw  node[fill,circle,inner sep=0pt,minimum size=2pt] at (10.6 + .14*\\i,\\j){};\n\t\t\t\t\t\\draw  node[fill,circle,inner sep=0pt,minimum size=2pt] at (-.6 - .14*\\i,\\j){};\n\t\t\t\t}\n\t\t\t}\n\t\t\t\\foreach \\i in {3,5,7,9}{\n\t\t\t\t\\draw[dark green, line width = 4pt](\\i,0) -- (\\i,1);}\n\t\t\t\n\t\t\t\\foreach \\i in {0,...,10}{\n\t\t\t\t\\node[hollow node] (\\i0) at (\\i,0){};\n\t\t\t\t\\node[hollow node] (\\i1) at (\\i,1){};\n\t\t\t}\n\t\t\t\\foreach \\i in {0,...,6}{\n\t\t\t\t\\node at (\\i,-.4){$v_{\\i}$};\n\t\t\t\t\\node at (\\i,1.4){$u_{\\i}$};\n\t\t\t}\n\t\t\t\\foreach \\i in {1,2,3}{\n\t\t\t\t\\node at (10-\\i,-.4){$v_{2k-\\i}$};\n\t\t\t\t\\node at (10-\\i,1.4){$u_{2k-\\i}$};\n\t\t\t}\n\t\t\t\\node at (10,-.4){$v_{2k}$};\n\t\t\t\\node at (10,1.4){$u_{2k}$};\n\t\t\t\\node[red node] at (01){};\n\t\t\t\\foreach \\i in {2,4,6,8,10}{\n\t\t\t\t\\node[red node] at (\\i0){};\n\t\t\t}\n\t\t\t\n\t\t\\end{tikzpicture}\n\t\t\\caption{Minimal Diameter Infecting Set, I}\n\t\t\\label{fig:narrowest odd tree}\n\t\t\n\t\\end{figure}\n\t\n\tLarger diameters may be obtained by ``weaving in and out.'' For example, exchanging $v_4$ for $u_4$ in Figure \\ref{fig:narrowest odd tree} to ``weave around'' $u_4$ gives a diameter of $n+2$ (see Figure \\ref{fig:middle odd tree}). Additional bumps may be added and each increases the diameter of $T$ by $2$, to a maximum diameter of $n+2\\lfloor\\frac{n-3}{4}\\rfloor$ (see Figure \\ref{fig:widest odd tree} for $k$ odd).\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\t\t\\tikzstyle{hollow node}=[draw,circle, fill=white, outer sep=-4pt,inner sep=0pt,\n\t\t\tminimum width=8pt, above]\n\t\t\t\\tikzstyle{red node}=[draw,circle, fill=red, outer sep=-4pt,inner sep=0pt,\n\t\t\tminimum width=8pt, above]\n\t\t\t\\foreach \\i in {0,...,5,7,8,9}{\n\t\t\t\t\\foreach \\j in {0,1}{\n\t\t\t\t\t\\draw[black,thick] (\\i,\\j) -- (\\i+1,\\j);\n\t\t\t\t}\n\t\t\t}\n\t\t\t\\foreach \\i in {0,...,10}{\n\t\t\t\t\\draw[black,thick] (\\i,0) -- (\\i,1);\n\t\t\t}\n\t\t\t\\draw[black,thick] (6,0) -- (6.25,0);\n\t\t\t\\draw[dark green, line width = 4pt] (6,1) -- (6.25,1);\n\t\t\t\\draw[black,thick] (6.75,0) -- (7,0);\n\t\t\t\\draw[dark green, line width = 4pt] (6.75,1) -- (7,1);\n\t\t\t\\draw[black,thick] (-.4,0) -- (0,0);\n\t\t\t\\draw[black,thick] (-.4,1) -- (0,1);\n\t\t\t\\draw[black,thick] (10,1) -- (10.4,1);\n\t\t\t\\draw[black,thick] (10,0) -- (10.4,0);\n\t\t\t\n\t\t\t\\foreach \\i in {1,2,5,7,8,9}{\n\t\t\t\t\\draw[dark green, line width = 4pt](\\i,1) -- (\\i+1,1);\n\t\t\t}\n\t\t\t\\foreach \\i in {0,3,4}{\n\t\t\t\t\\draw[dark green, line width = 4pt](\\i,0) -- (\\i+1,0);}\n\t\t\t\n\t\t\t\\foreach \\i in {0,1,2}{\n\t\t\t\t\\foreach \\j in {0,1}{\n\t\t\t\t\t\\draw  node[fill,circle,inner sep=0pt,minimum size=2pt] at (6.36 + .14*\\i,\\j){};\n\t\t\t\t\t\\draw  node[fill,circle,inner sep=0pt,minimum size=2pt] at (10.6 + .14*\\i,\\j){};\n\t\t\t\t\t\\draw  node[fill,circle,inner sep=0pt,minimum size=2pt] at (-.6 - .14*\\i,\\j){};\n\t\t\t\t}\n\t\t\t}\n\t\t\t\\foreach \\i in {1,3,5,7,9}{\n\t\t\t\t\\draw[dark green, line width = 4pt](\\i,0) -- (\\i,1);}\n\t\t\t\n\t\t\t\\foreach \\i in {0,...,10}{\n\t\t\t\t\\node[hollow node] (\\i0) at (\\i,0){};\n\t\t\t\t\\node[hollow node] (\\i1) at (\\i,1){};\n\t\t\t}\n\t\t\t\\foreach \\i in {0,...,6}{\n\t\t\t\t\\node at (\\i,-.4){$v_{\\i}$};\n\t\t\t\t\\node at (\\i,1.4){$u_{\\i}$};\n\t\t\t}\n\t\t\t\\foreach \\i in {1,2,3}{\n\t\t\t\t\\node at (10-\\i,-.4){$v_{2k-\\i}$};\n\t\t\t\t\\node at (10-\\i,1.4){$u_{2k-\\i}$};\n\t\t\t}\n\t\t\t\\node at (10,-.4){$v_{2k}$};\n\t\t\t\\node at (10,1.4){$u_{2k}$};\n\t\t\t\\node[red node] at (01){};\n\t\t\t\\foreach \\i in {2,6,8,10}{\n\t\t\t\t\\node[red node] at (\\i0){};\n\t\t\t}\n\t\t\t\\node[red node] at (41){};\n\t\t\t\n\t\t\\end{tikzpicture}\n\t\t\\caption{Increasing the Diameter by $2$}\n\t\t\\label{fig:middle odd tree}\n\t\t\n\t\\end{figure}\n\t\n\t\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\t\t\\tikzstyle{hollow node}=[draw,circle, fill=white, outer sep=-4pt,inner sep=0pt,\n\t\t\tminimum width=8pt, above]\n\t\t\t\\tikzstyle{red node}=[draw,circle, fill=red, outer sep=-4pt,inner sep=0pt,\n\t\t\tminimum width=8pt, above]\n\t\t\t\\foreach \\i in {0,...,5,7,8,9}{\n\t\t\t\t\\foreach \\j in {0,1}{\n\t\t\t\t\t\\draw[black,thick] (\\i,\\j) -- (\\i+1,\\j);\n\t\t\t\t}\n\t\t\t}\n\t\t\t\\foreach \\i in {0,...,10}{\n\t\t\t\t\\draw[black,thick] (\\i,0) -- (\\i,1);\n\t\t\t}\n\t\t\t\\draw[black,thick] (6,0) -- (6.25,0);\n\t\t\t\\draw[dark green, line width = 4pt] (6,1) -- (6.25,1);\n\t\t\t\\draw[black,thick] (6.75,0) -- (7,0);\n\t\t\t\\draw[dark green, line width = 4pt] (6.75,1) -- (7,1);\n\t\t\t\\draw[black,thick] (-.4,0) -- (0,0);\n\t\t\t\\draw[black,thick] (-.4,1) -- (0,1);\n\t\t\t\\draw[black,thick] (10,1) -- (10.4,1);\n\t\t\t\\draw[black,thick] (10,0) -- (10.4,0);\n\t\t\t\n\t\t\t\\foreach \\i in {1,2,5,9}{\n\t\t\t\t\\draw[dark green, line width = 4pt](\\i,1) -- (\\i+1,1);\n\t\t\t}\n\t\t\t\\foreach \\i in {0,3,4,7,8}{\n\t\t\t\t\\draw[dark green, line width = 4pt](\\i,0) -- (\\i+1,0);}\n\t\t\t\n\t\t\t\\foreach \\i in {0,1,2}{\n\t\t\t\t\\foreach \\j in {0,1}{\n\t\t\t\t\t\\draw  node[fill,circle,inner sep=0pt,minimum size=2pt] at (6.36 + .14*\\i,\\j){};\n\t\t\t\t\t\\draw  node[fill,circle,inner sep=0pt,minimum size=2pt] at (10.6 + .14*\\i,\\j){};\n\t\t\t\t\t\\draw  node[fill,circle,inner sep=0pt,minimum size=2pt] at (-.6 - .14*\\i,\\j){};\n\t\t\t\t}\n\t\t\t}\n\t\t\t\\foreach \\i in {1,3,5,7,9}{\n\t\t\t\t\\draw[dark green, line width = 4pt](\\i,0) -- (\\i,1);}\n\t\t\t\n\t\t\t\\foreach \\i in {0,...,10}{\n\t\t\t\t\\node[hollow node] (\\i0) at (\\i,0){};\n\t\t\t\t\\node[hollow node] (\\i1) at (\\i,1){};\n\t\t\t}\n\t\t\t\\foreach \\i in {0,...,6}{\n\t\t\t\t\\node at (\\i,-.4){$v_{\\i}$};\n\t\t\t\t\\node at (\\i,1.4){$u_{\\i}$};\n\t\t\t}\n\t\t\t\\foreach \\i in {1,2,3}{\n\t\t\t\t\\node at (10-\\i,-.4){$v_{2k-\\i}$};\n\t\t\t\t\\node at (10-\\i,1.4){$u_{2k-\\i}$};\n\t\t\t}\n\t\t\t\\node at (10,-.4){$v_{2k}$};\n\t\t\t\\node at (10,1.4){$u_{2k}$};\n\t\t\t\\node[red node] at (01){};\n\t\t\t\\foreach \\i in {2,6,10}{\n\t\t\t\t\\node[red node] at (\\i0){};\n\t\t\t}\n\t\t\t\\node[red node] at (41){};\n\t\t\t\\node[red node] at (81){};\n\t\t\t\n\t\t\t\n\t\t\\end{tikzpicture}\n\t\t\\caption{Maximal Diameter Infecting Set, $k$ Odd}\n\t\t\\label{fig:widest odd tree}\n\t\t\n\t\\end{figure}\n\t\n\t\n\tTurn now to the case of $n$ even and write $n = 2k$ $(n\\geq4)$. Again a minimum size infecting set $S$ has $k+1$ vertices. This case has several subcases, but as each is handled similarly to the previous case, we only sketch the major steps.\n\t\n\tBy Theorem \\ref{components thm}, $G[V-S]$ may be a forest with one or two connected components. Consider first the case where it is a single tree. In this case there now exists precisely one pair of neighboring vertices in $S$. We can still find a square face $F$ of the annulus with both horizontal edges not in $T$. As before, split the annulus at this face. There will be two subcases to examine.\t\n\t\n\tThe first subcase is when $F$ has the same form as Figure \\ref{fig:1}.  The minimal diameter tree will look similar to Figure \\ref{fig:narrowest odd tree}, except that $T$ will connect $v_1$ to $v_2$ before moving up to $u_2$ and continuing horizontally, and the indices will have a maximal label of $2k-1$. Weaving will increase the diameter by $2$ and the possible diameters of $T$ are $n, n+2, \\ldots, n+ 2\\lfloor\\frac{n-4}{4}\\rfloor$.\n\t\n\t\n\t\n\n\tThe second subcase is when the neighboring vertices of $F$ are, after relabeling, $u_0$ and $v_0$. In this case, the tree $T$ with minimal diameter is pictured in Figure\n\t\\ref{fig:narrowest even tree 2}. Weaving allows incrementing the diameter by $1$ and the possible diameters of $T$ are $n, n+1, \\ldots, \\frac 32 n -2$.\n\t\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\t\t\\tikzstyle{hollow node}=[draw,circle, fill=white, outer sep=-4pt,inner sep=0pt,\n\t\t\tminimum width=8pt, above]\n\t\t\t\\tikzstyle{red node}=[draw,circle, fill=red, outer sep=-4pt,inner sep=0pt,\n\t\t\tminimum width=8pt, above]\n\t\t\t\\foreach \\i in {0,...,5,7,8,9}{\n\t\t\t\t\\foreach \\j in {0,1}{\n\t\t\t\t\t\\draw[black,thick] (\\i,\\j) -- (\\i+1,\\j);\n\t\t\t\t}\n\t\t\t}\n\t\t\t\\foreach \\i in {0,...,10}{\n\t\t\t\t\\draw[black,thick] (\\i,0) -- (\\i,1);\n\t\t\t}\n\t\t\t\\draw[black,thick] (6,0) -- (6.25,0);\n\t\t\t\\draw[dark green, line width = 4pt] (6,1) -- (6.25,1);\n\t\t\t\\draw[black,thick] (6.75,0) -- (7,0);\n\t\t\t\\draw[dark green, line width = 4pt] (6.75,1) -- (7,1);\n\t\t\t\\draw[black,thick] (-.4,0) -- (0,0);\n\t\t\t\\draw[black,thick] (-.4,1) -- (0,1);\n\t\t\t\\draw[black,thick] (10,1) -- (10.4,1);\n\t\t\t\\draw[black,thick] (10,0) -- (10.4,0);\n\t\t\t\n\t\t\t\\foreach \\i in {0,1,2}{\n\t\t\t\t\\foreach \\j in {0,1}{\n\t\t\t\t\t\\draw  node[fill,circle,inner sep=0pt,minimum size=2pt] at (6.36 + .14*\\i,\\j){};\n\t\t\t\t\t\\draw  node[fill,circle,inner sep=0pt,minimum size=2pt] at (10.6 + .14*\\i,\\j){};\n\t\t\t\t\t\\draw  node[fill,circle,inner sep=0pt,minimum size=2pt] at (-.6 - .14*\\i,\\j){};\n\t\t\t\t}\n\t\t\t}\n\t\t\t\n\t\t\t\\foreach \\i in {1,...,5,7,8}{\n\t\t\t\t\\draw[dark green, line width = 4pt](\\i,1) -- (\\i+1,1);\n\t\t\t}\n\t\t\t\n\t\t\t\\foreach \\i in {1,3,5,7,9}{\n\t\t\t\t\\draw[dark green, line width = 4pt](\\i,0) -- (\\i,1);\n\t\t\t}\n\t\t\t\n\t\t\t\n\t\t\t\\foreach \\i in {0,...,10}{\n\t\t\t\t\\node[hollow node] (\\i0) at (\\i,0){};\n\t\t\t\t\\node[hollow node] (\\i1) at (\\i,1){};\n\t\t\t}\n\t\t\t\\foreach \\i in {0,...,6}{\n\t\t\t\t\\node at (\\i,-.4){$v_{\\i}$};\n\t\t\t\t\\node at (\\i,1.4){$u_{\\i}$};\n\t\t\t}\n\t\t\t\\foreach \\i in {1,2,3}{\n\t\t\t\t\\node at (10-\\i,-.4){$v_{n-\\i}$};\n\t\t\t\t\\node at (10-\\i,1.4){$u_{n-\\i}$};\n\t\t\t}\n\t\t\t\t\\node at (10,-.4){$v_{0}$};\n\t\t\t\t\\node at (10,1.4){$u_{0}$};\n\t\t\t\n\t\t\t\\foreach \\i in {2,4,6,8}{\n\t\t\t\t\\node[red node] at (\\i,0){};\n\t\t\t}\n\t\t\t\\foreach \\i in {0,1}{\n\t\t\t\t\\node[red node] at (0,\\i){};\n\t\t\t\t\\node[red node] at (10,\\i){};\n\t\t\t}\n\t\t\t\n\t\t\\end{tikzpicture}\n\t\t\\caption{Minimal Diameter Infecting Set, II}\n\t\t\\label{fig:narrowest even tree 2}\n\t\\end{figure}\n\t\n\tThe final case is when $G[V-S]$ constitutes $2$ connected components and $S$ has no neighboring vertices. The infecting time is controlled by the diameter of the larger tree. The analysis is similar, but now the annulus may be cut in two places. After relabeling, there are two possibilities. The first is that $S$ contains $\\{u_0,u_m, v_{m+1}, v_{2k-1}\\}$ and the first tree connects $v_0$ to $v_m$ and the second tree connects $u_{m+1}$ to $u_{2k-1}$. The other is that $S$ contains $\\{u_0,v_m, u_{m+1}, v_{2k-1}\\}$ and the first tree connects $v_0$ to $u_m$ and the second tree connects $v_{m+1}$ to $u_{2k-1}$. A direct inspection shows that the smallest possible diameter for the larger tree is $\\lfloor \\frac{n}{2} \\rfloor$. Weaving allows incrementing the diameter by $1$ and the possible diameters for the larger tree are $\\lfloor \\frac{n}{2} \\rfloor, \\lfloor \\frac{n}{2} \\rfloor +1, \\ldots, n-2$.\n\t\n\tWe now summarize the possible maximal diameters of connected components for the complement of a minimum size infecting set $S$ for the case of $n$ even. The diameter $d$ may be achieved for each $d$ satisfying\n\t$\\lfloor \\frac{n}{2} \\rfloor \\leq d \\leq \\frac 32 n -2$ with $d \\not= n-1$.\t\n\\end{proof}\n\n\n\n\\section{Further Results}\n\nIn this section, we examine some closely related variants of $G(n,k)$.\n\nWhen working with two copies of $G(n,k)$, we will write $\\{u_k',v_k':0\\leq k \\leq n-1\\}$ for the second set of vertices. We then write $G(n,k)\\#G(n,k)$ for the graph obtained by surgery on the two copies of $G(n,k)$ along the edges $u_0 u_1$ and $u_0' u_1'$. By this we mean that $G(n,k)\\#G(n,k)$ is the union of the two graphs with the edges $u_0 u_1$ and $u_0' u_1'$ replaced by $u_0 u_0'$ and $u_1 u_1'$. The following result calculates the $P_3$-hull number of $G(n,k)\\#G(n,k)$. In fact, the result generalizes to most types of surgery along other edges.\n\n\\begin{theorem}\\label{thm_surgery}\n $h_{P_3}(G(n,k)\\#G(n,k))=n+1$.\n\\end{theorem}\n\n\\begin{proof}\n\tWrite $G=G(n,k)\\# G(n,k)$. Then $G$ is cubic with $4n$ vertices and $6n$ edges, so that, by Theorem \\ref{Main Theorem} and\n\t\\cite[Corollary~2]{Betti-Deficiancy-Thing}, its $P_3$-hull number is at least $n+1$. We distinguish two cases. For each recall that $G(n,k)$ has a minimum size infecting set of size $\\lceil\\frac{n+1}{2}\\rceil$ with an explicit realization in Corollary \\ref{infecting_set}. We will use notation from that Corollary.\n\t\nConsider first the case where $n$ is odd so that $l$ (and $c$) are also odd. Apply an index shift to $G(n,k)\\# G(n,k)$, so that the surgery happens along the edges $u_{c-1} u_c$ and $u_{c-1}' u_c'$. By inspection, the union of our infecting sets for each $G(n,k)$ infects $G$. As the order of this union is $n+1$, we are done.\n\nConsider next the case where $n$ is even. When $l$ is also even, a slight modification of $\\mathcal S$ provides the infecting set\n\\[ \\{u_{0}\\} \\cup \\{v_{j+ik}, v_{j+ik}':0\\leq i\\leq l-1,\\, 0\\leq j\\leq c-1,\\, i\\text{ even }\\} \\]\nof size $n+1$.\nWhen $l$ is odd, then $c$ is even. Apply an index shift to $G(n,k)\\# G(n,k)$, so that the surgery happens along the edges $u_{c-1} u_c$ and $u_{c-1}' u_c'$.\nThis time, an infecting set is given by the union of the two infecting sets for $G(n,k)$ minus $\\{u'_{c-1}\\}$. The size of this set is $(\\frac{n}{2}+1) +\\frac{n}{2} = n+1$, and we are done.\n\\end{proof}\n\nThe following is a generalization of $G(n,k)$.\n\n\\begin{definition}\n\tLet $\\sigma$ be a nontrivial permutation in $S_n$ and write it as a product of disjoint, nontrivial cycles, $\\sigma = \\prod_{i=1}^m \\sigma_i$. Assume $|\\sigma_i| \\geq 3$ for $1 \\leq i \\leq m$.\n\t\n\tThe \\textit{GGP graph} for $\\sigma$, $GGP(n,\\sigma)$ is a graph on $2n$ vertices with vertex set\n\t$$V=\\{u_0, u_1,\\dots, u_{n-1}, v_0, v_1, \\dots, v_{n-1}\\}$$\n\tand edge set\n\t\\[ E= \\{u_i u_{i+1}, \\, u_i v_i, \\, v_i v_{\\sigma(i)}: 0\\leq i \\leq n-1\\} \\]\n\twith indices interpreted modulo $n$.\n\\end{definition}\n\nThe assumption above that each $|\\sigma_i| \\geq 3$ guarantees that $GGP(n,\\sigma)$ is a simple cubic graph. Note that if $\\sigma(i)=i+k$, then $GGP(n,\\sigma) = G(n,k)$.\n\n\n\n\\newcount\\CircNum\n\\newcommand\\Clr{black}\n\\begin{example} \\label{3 inner triangles}\n\tThe following is an example of a GGP graph.\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\t\n\t\t\t\\node [circle, draw, minimum size=4cm] (c) {};\n\t\t\t\n\t\t\t\\foreach \\a in {1,2,...,18}{\n\t\t\t\t\\ifnum\\a=\\the\\CircNum\n\t\t\t\t\\renewcommand\\Clr{blue}\n\t\t\t\t\\else\n\t\t\t\t\\renewcommand\\Clr{black}\n\t\t\t\t\\fi\n\t\t\t\t\\node[circle,inner sep=0pt,minimum size=1pt] at (c.\\a*360\/18) {}\n\t\t\t\t{};\n\t\t\t};\n\t\t\t\n\t\t\t\\newlength\\Radius\n\t\t\t\\setlength\\Radius{2cm}\n\t\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt](A1) at (-1,0) {};\n\t\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt](A2) at (-0.5,-0.5) {};\n\t\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt](A3) at (-0.5,0.5) {};\n\t\t\t\n\t\t\t\\draw (A1) -- (A2) -- (A3) --(A1);\n\t\t\t\n\t\t\t\\draw (c.9*360\/18) -- (A1);\n\t\t\t\\draw (c.10*360\/18) -- (A2);\n\t\t\t\\draw (c.8*360\/18) -- (A3);\n\t\t\t\n\t\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt](A1) at (0.8,0.8) {};\n\t\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt](A2) at (0.7,0.1) {};\n\t\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt](A3) at (0.1,0.9) {};\n\t\t\t\n\t\t\t\\draw (A1) -- (A2) -- (A3) --(A1);\n\t\t\t\n\t\t\t\\draw (c.2*360\/18) -- (A1);\n\t\t\t\\draw (c.1*360\/18) -- (A2);\n\t\t\t\\draw (c.3*360\/18) -- (A3);\n\t\t\t\n\t\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt](A1) at (0.8,-0.8) {};\n\t\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt](A2) at (0.7,-0.1) {};\n\t\t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt](A3) at (0.1,-0.9) {};\n\t\t\t\n\t\t\t\\draw (A1) -- (A2) -- (A3) --(A1);\n\t\t\t\n\t\t\t\\draw (c.16*360\/18) -- (A1);\n\t\t\t\\draw (c.17*360\/18) -- (A2);\n\t\t\t\\draw (c.15*360\/18) -- (A3);\n\t\t\\end{tikzpicture}\n\t\t\\caption{Example of $GGP(9,\\sigma)$}\n\t\t\\label{fig:GGP1}\n\t\\end{figure}\n\\end{example}\n\n\n\n\tNote that the example of $GGP(9,\\sigma)$ in Figure \\ref{fig:GGP1} has a decycling number of 6, but that $\\lceil\\frac{9+1}{2}\\rceil = 5$. It follows that Corollary \\ref{p3_hull_GP} does not hold for all $GGP$ graphs. It is expected that a general formula for the $P_3$-hull number for all $GGP$ graphs is difficult. We can, however, provide some bounds.\n\n\n\\begin{theorem}\\label{odd cycles upper bound}\n\tLet $\\sigma$ be a nontrivial permutation in $S_n$ written as a product of disjoint, nontrivial cycles, $\\sigma = \\prod_{i=1}^m \\sigma_i$, with $n_i = |\\sigma_i| \\geq 3$ for $1 \\leq i \\leq m$. Write $k$ for the number of $n_i$ that are odd and let $G=GGP(n,\\sigma)$.\n\t\n\tThen \\[h_{P_3}(G) \\geq \\bigg\\lceil \\frac{n+1}{2} \\bigg\\rceil.\\] If each $n_i$ is even, then  \\[h_{P_3}(G)  = \\frac{n+2}{2}.\\] If there are odd $n_i$, then \\[h_{P_3}(G)  \\leq \\frac{n+k}{2}.\\]\n\nNote, in particular, $h_{P_3}(G)  = \\frac{n+2}{2}$ for $k=0$, $h_{P_3}(G)  = \\frac{n+1}{2}$ for $k=1$, and $h_{P_3}(G)  = \\frac{n+2}{2}$ for $k=2$.\n\\end{theorem}\n\n\\begin{proof}\n    Since $G=(V,E)$ is cubic, Theorem \\ref{Main Theorem} and\n    \\cite[Corollary 2]{Betti-Deficiancy-Thing} provide the lower bound.\n    In addition, \\cite[Theorem 3]{Betti-Deficiancy-Thing} shows that\n    $h_{P_3}(G) = \\gamma_M(G) + \\xi(G)$, where $\\gamma_M(G)$ is the maximum genus of $G$ and $\\xi(G)$ is its Betti deficiency. As $G$ is connected, $\\gamma_M(G) = \\frac{\\beta(G) - \\xi(G)}{2}$, where $\\beta(G)$ is the cycle rank of $G$, which is $|E|-|V|+1 = n + 1$ for a cubic connected graph on $2n$ vertices. Thus $h_{P_3}(G) = \\frac{\\beta(G) + \\xi(G)}{2}=\\frac{n+1+\\xi(G)}{2}$.\n\n    Consider first the case of $k = 0$. Recall that the Betti deficiency, $\\xi(G)$, is the minimum over all co-trees of $G$ of the number of components with an odd number of edges. So, we only need to construct a co-tree of $G$ with at most $1$ odd component. We can construct such a tree $T$ by taking all exterior edges $u_iu_{i+1}$ except $u_0u_1$, plus all in-between edges, $u_i v_i$. Thus the components of the co-tree corresponding to $T$ are the interior cycles and $u_0u_1$. There are no odd interior cycles so $u_0u_1$ is the only odd component. Thus $\\xi(G)\\leq 1$ and $h_{P_3}(G)  \\le \\frac{n+2}{2}$. However, note that $n =\\sum_{i=1}^m n_i\\equiv k \\mbox{ mod } 2$ is even and $h_{P_3}(G) \\geq \\lceil \\frac{n+1}{2} \\rceil = \\frac{n+2}{2}$ by our lower bound.\n\n    Turn now to the case of $k \\geq 1$. To show that $\\xi(G)\\leq k-1$, it suffices to construct a co-tree, $T$, of $G$ with $k-1$ odd components. As a visual guide for the construction, see Figure \\ref{fig:my_label}.\n\n    \\begin{figure}[H]\\label{co-tree picture}\n    \t\\centering\n    \t\\begin{tikzpicture}\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (a) at (4*1,4*0) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (b) at (4*.866,.5*4) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (c) at (4*.5,.866*4) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt, label = \\hspace{0em}$u^*$] (d) at (4*0,1*4) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (g) at (4*-1,0*4) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (f) at (4*-.866,.5*4) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt, label = $x^*$] (e) at (4*-.5,.866*4) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (j) at (4*0,-1*4) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (h) at (4*-.866,-.5*4) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (i) at (4*-.5,-.866*4) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (l) at (4*.866,-.5*4) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (k) at (4*.5,-.866*4) {};\n    \t\t\n    \t\t\\draw[black, ultra thick] (a) -- (b);\n    \t\t\\draw[black, ultra thick] (b) -- (c);\n    \t\t\\draw[black, ultra thick] (c) -- (d);\n    \t\t\\draw[black, dashed] (d) -- (e);\n    \t\t\\draw[black, ultra thick] (e) -- (f);\n    \t\t\\draw[black, ultra thick] (f) -- (g);\n    \t\t\\draw[black, ultra thick] (g) -- (h);\n    \t\t\\draw[black, ultra thick] (h) -- (i);\n    \t\t\\draw[black, ultra thick] (i) -- (j);\n    \t\t\\draw[black, ultra thick] (j) -- (k);\n    \t\t\\draw[black, ultra thick] (k) -- (l);\n    \t\t\\draw[black, ultra thick] (l) -- (a);\n    \t\t\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (m) at (2.5*1,2.5*0) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (n) at (2.5*.866,.5*2.5) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (o) at (2.5*.5,.866*2.5) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt, label = \\hspace{1.5em}$v^*$] (p) at (2.5*0,1*2.5) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (s) at (2.5*-1,0*2.5) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (r) at (2.5*-.866,.5*2.5) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (q) at (2.5*-.5,.866*2.5) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (v) at (2.5*0,-1*2.5) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (t) at (2.5*-.866,-.5*2.5) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (u) at (2.5*-.5,-.866*2.5) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (x) at (2.5*.866,-.5*2.5) {};\n    \t\t\\node[fill,circle,inner sep=0pt,minimum size=1pt] (w) at (2.5*.5,-.866*2.5) {};\n    \t\t\n    \t\t\\draw[black, ultra thick] (p) -- (q);\n    \t\t\\draw[black, dashed] (q) -- (r);\n    \t\t\\draw[black, dashed] (r) -- (n);\n    \t\t\\draw[black, dashed] (n) -- (o);\n    \t\t\\draw[black, dashed] (o) -- (p);\n    \t\t\n    \t\t\\draw[black, dashed] (t) -- (u);\n    \t\t\\draw[black, dashed] (u) -- (x);\n    \t\t\\draw[black, dashed] (x) -- (w);\n    \t\t\\draw[black, dashed] (w) -- (t);\n    \t\t\\draw[black, dashed] (s) -- (v);\n    \t\t\\draw[black, dashed] (v) -- (m);\n    \t\t\\draw[black, dashed] (m) -- (s);\n    \t\t\n    \t\t\\draw[black, ultra thick] (m) -- (a);\n    \t\t\\draw[black, ultra thick] (n) -- (b);\n    \t\t\\draw[black, ultra thick] (o) -- (c);\n    \t\t\\draw[black, dashed] (p) -- (d);\n    \t\t\\draw[black, ultra thick] (q) -- (e);\n    \t\t\\draw[black, ultra thick] (r) -- (f);\n    \t\t\\draw[black, ultra thick] (s) -- (g);\n    \t\t\\draw[black, ultra thick] (t) -- (h);\n    \t\t\\draw[black, ultra thick] (u) -- (i);\n    \t\t\\draw[black, ultra thick] (v) -- (j);\n    \t\t\\draw[black, ultra thick] (w) -- (k);\n    \t\t\\draw[black, ultra thick] (x) -- (l);\n    \t\t\n    \t\\end{tikzpicture}\n    \t\\centering\n    \t\\caption{A $GGP$ on 24 Vertices With 2 Odd Inner Cycles}\n    \t\\label{fig:my_label}\n    \\end{figure}\n\n    Begin with any interior cycle, $C$, of odd length in $G$. Pick an edge $e^* = u^*v^*$ between $v^*\\in C$ and $u^*\\notin C$. Include all edges $u_i v_i$, except $e^*$, in $T$. Label the neighbors of $u^*$ on the exterior cycle as $x^*$ and $y^*$. Also include every exterior edge, $u_iu_{i+1}$ except $u^*x^*$, in $T$ (this includes $u^*y^*$). Finally, add to $T$ an edge of $C$ that is incident to $v^*$. This results in a spanning tree of~$G$.\n\n    Write $H$ for the co-tree of $G$ corresponding to $T$. Observe that the remaining edges of $C$, together with $e^*$ and $u^*x^*$, form an even component of $H$. The only other components of $H$ are the rest of the interior cycles. Thus the number of odd components of $H$ is one less than the number of odd interior cycles of $G$, so $\\xi(G)\\leq k-1$.\n\\end{proof}\n\n\nThe upper bound in Theorem \\ref{odd cycles upper bound} is achieved for certain combinations of $n$ and $k$. For instance, Figure \\ref{fig:GGP1} has $n = 9, k = 3$ and $h_{P_3}(G) = 6$.\nMore generally, $n=3k$ and $h_{P_3}(G) = 2k$ holds for the corresponding graph with $k$ triangles in the interior achieving the upper bound.\nHowever, there are quite general cases where the lower bound is achieved. The next theorem gives a sufficient condition based on an interleaving principle.\n\n\\begin{theorem}\\label{thm:minGGP}\n\tLet $\\sigma$ be a nontrivial permutation in $S_n$ written as a product of disjoint, nontrivial cycles, $\\sigma = \\prod_{i=1}^m \\sigma_i$, with $n_i = |\\sigma_i| \\geq 3$ for $1 \\leq i \\leq m$.\n\tLet $G=GGP(n,\\sigma)$. If there are $k$ odd $n_i$, assume there is a path of $k$ exterior vertices $u_j$, each of which connects to a different odd internal cycle.\nThen $h_{P_3} (G) = \\lceil\\frac{n+1}{2}\\rceil$.\n\\end{theorem}\n\n\\begin{proof}\n\tDefine an infecting set by following a similar strategy to Corollary \\ref{infecting_set}. Relabel so that the vertices of our path of $k$ exterior vertices are $u_0, \\ldots, u_{k-1}$. Each odd internal cycle connects uniquely to this path by some edge $u_i v_i$, $0\\le i\\le k-1$.\n\t\n\tTo define the initial infecting set, $\\mathcal S$, for each internal cycle, $C_i$, add to $\\mathcal S$ every other vertex of the cycle as was done with $\\mathcal S_v$, avoiding the points $v_0, \\ldots, v_{k-1}$ on the odd internal cycles. If there are only even cycles, add one external vertex for a total of $\\frac n2+1 =\\lceil\\frac{n+1}{2}\\rceil$ points. If there are odd cycles, add vertices from our path of $k$ exterior vertices $u_0, \\ldots, u_{k-1}$ as was done with $\\mathcal S_u$ in Corollary \\ref{infecting_set} in the case of $l$ odd. It is straightforward to verify this is an infecting set of size $\\lceil\\frac{n+1}{2}\\rceil$.\n\\end{proof}\n\nThere are numerous generalizations of Theorem \\ref{thm:minGGP} that can be made and it would be very interesting to determine exactly when the possible minimal and maximal $P_3$-hull numbers are obtained for $GGP$ graphs.\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:1}\n\nSpecial functions of mathematical physics are usually defined either in form of power series, or as solutions to some differential equations, or via integral representations. Of course, for a given function, these three (and possibly other) forms coincide for all arguments and parameters values for that they do exist. However, the validity domains of different representations can be unequal. Very often the series representations of the special functions hold valid only on some restricted domains. In order to define the corresponding functions for other values of their arguments and parameters, analytical continuation of the series in form of integral representations is usually employed.  \n\nFor the special functions of Fractional Calculus (FC), the situation is very similar to the one described above. For instance, one of the most important FC special function - the two-parameters Mittag-Leffler function - is usually defined in form of a power series:\n\\begin{equation}\n\\label{ML}\nE_{\\alpha,\\beta}(z) = \\sum_{k=0}^{+\\infty} \\frac{z^k}{\\Gamma(\\alpha\\, k + \\beta)},\\ \\alpha >0,\\ \\beta,z\\in \\Com.\n\\end{equation}\nBecause the series is convergent for all $z\\in \\Com$, this definition can be used for all $z\\in \\Com$ without any analytical continuation. Still, the integral representations of the  Mittag-Leffler function are very important, say, for derivation of its asymptotic behavior (\\cite{Dzh}) and for its numerical calculation (\\cite{GLL}). For \n$0<\\alpha <2$ and $\\Re(\\beta)>0$,  the following integral representations of the Mittag-Leffler function in terms of the integrals over the Hankel-type contours were presented in \\cite{Dzh}: \n$$\nE_{\\alpha,\\beta}(z) ={1 \\over 2\\pi\\alpha i}\\int_{\\gamma(\\epsilon;\\delta)}\n{e^{\\zeta^{1\/\\alpha}}\\zeta^{(1-\\beta)\/\\alpha} \\over \\zeta -z}\\, d\\zeta, \\ z\\in G^{(-)}(\\epsilon;\\delta),\n$$\n$$\nE_{\\alpha,\\beta}(z) = \\frac{1}{\\alpha}z^{(1-\\beta)\/\\alpha}e^{z^{1\/\\alpha}} + {1\\over 2\\pi\\alpha i}\\int_{\\gamma(\\epsilon;\\delta)}\n{e^{\\zeta^{1\/\\alpha}}\\zeta^{(1-\\beta)\/\\alpha} \\over \\zeta -z} d\\zeta, \\ z\\in G^{(+)}(\\epsilon;\\delta),\n$$\nwhere   the integration contour $\\gamma(\\epsilon;\\delta) \\ (\\epsilon >0, \\ 0<\\delta\\le \\pi)$  with\nnon-decreasing $\\arg \\zeta$ consists of the following parts:\n\\par \\noindent\n1) the ray $\\arg \\zeta =-\\delta, \\ |\\zeta|\\ge \\epsilon$;\n\\par \\noindent\n2) the arc $-\\delta \\le \\arg \\zeta \\le \\delta$ of the circumference $|\\zeta|=\\epsilon$;\n\\par \\noindent\n3) the ray  $\\arg \\zeta =\\delta, \\ |\\zeta|\\ge \\epsilon$.\n\\par \\noindent\nFor $0<\\delta< \\pi$, the domain $G^{(-)}(\\epsilon;\\delta)$ is to the left\n of the contour $\\gamma(\\epsilon;\\delta)$ and the domain  $G^{(+)}(\\epsilon;\\delta)$ is to the right of this contour. If $\\delta\n=\\pi $, the contour $\\gamma(\\epsilon;\\delta)$  consists of the circumference $|\\zeta|=\\epsilon $ and of the cut\n$-\\infty <\\zeta \\le -\\epsilon$. In this case, the domain  $G^{(-)}(\\epsilon;\\delta)$ is the circle $|\\zeta|<\n\\epsilon$ and  $G^{(+)}(\\epsilon;\\alpha)=\\{\\zeta: |\\arg\\zeta| <\\pi, \\ |\\zeta|>\\epsilon \\}$.\n\nFor some parameters values, the Mittag-Leffler function can be also introduced in terms of solutions to the fractional differential equations with the Riemann-Liouville or Caputo fractional derivatives. For instance, for $0<\\alpha \\le 1$, the equation \n\\begin{equation}\n\\label{eqRL}\n(D^\\alpha_{0+} y)(t)  = \\lambda\\, y(t)\n\\end{equation}\nhas the general solution (\\cite{LucSri})\n\\begin{equation}\n\\label{eqRL_sol}\ny(t) = C\\, t^{\\alpha -1} E_{\\alpha, \\alpha}(\\lambda\\, t^\\alpha),\\ C\\in \\R.\n\\end{equation}\nIn the equation \\eqref{eqRL}, the Riemann-Liouville fractional derivative $D^\\alpha_{0+}$ is defined by \n\\begin{equation}\n\\label{RLD}\n(D^\\alpha_{0+} \\, f)(t) = \\frac{d}{dt}(I^{1-\\alpha}_{0+} f)(t),\\ t>0,\n\\end{equation}\nwhere $I^\\alpha_{0+}$ is the Riemann-Liouville fractional integral of order $\\alpha\\ (\\alpha >0)$:\n\\begin{equation}\n\\label{RLI}\n\\left(I^\\alpha_{0+} f \\right) (t)=\\frac{1}{\\Gamma (\\alpha )}\\int\\limits_0^t(t -\\tau)^{\\alpha -1}f(\\tau)\\,d\\tau,\\ t>0.\n\\end{equation}\nThe general solution to the equation \n\\begin{equation}\n\\label{eqC}\n(\\,_*D^\\alpha_{0+} y)(t)  = \\lambda\\, y(t)\n\\end{equation}\nwith the Caputo fractional derivative\n\\begin{equation}\n\\label{CD}\n(\\,_*D^\\alpha_{0+}\\, f)(t)  = (D^\\alpha_{0+} \\, f)(t) - f(0)\\frac{t^{-\\alpha}}{\\Gamma(1-\\alpha)},\\ t>0 \n\\end{equation}\nhas the form  (\\cite{LucGor})\n\\begin{equation}\n\\label{eqC_sol}\ny(t) = C\\,  E_{\\alpha, 1}(\\lambda\\, t^\\alpha),\\ C\\in \\R.\n\\end{equation}\nAs we see, the solutions to the fractional differential equations \\eqref{eqRL} and \\eqref{eqC} are expressed in terms of the Mittag-Leffler functions. However, the arguments of these functions are  $\\lambda\\, t^\\alpha$ and not just $\\lambda\\, t$. Thus, these solutions are represented in form of power series with the fractional and not integer exponents. For more advanced properties and applications of the Mittag-Leffler type functions see \\cite{Dzh} and the recent book \\cite{GKRM}.  \n\nIn \\cite{Luc21b}, the single- and multi-terms fractional differential equations with the general fractional derivatives of Caputo type have been studied. By definition, their solutions belong to the class of the FC special functions (as the ones represented in form of solutions to the fractional differential equations). Moreover, in \\cite{Luc21b}, another representation of these new FC special functions was derived, namely, in form of the convolution series generated by the Sonine kernels.  \n\nThe convolution series are a far reaching generalization of the conventional power series and the power series with the fractional exponents including the Mittag-Leffler functions \\eqref{eqRL_sol} and \\eqref{eqC_sol}. They represent a new type of the FC special functions worth for investigation. In \\cite{Luc21d}, the convolution series were employed for derivation of two different forms of a generalized convolution Taylor formula for representation of a function as a convolution polynomial with a remainder in form of a composition of the $n$-fold general fractional integral and the $n$-fold general sequential fractional derivative of the Riemann-Liouville and the Caputo types, respectively.  In \\cite{Luc21d}, the generalized Taylor series in form of convolution series were also discussed. In this paper, we employ the convolution series for derivation of analytical solutions to the single- and multi-terms fractional differential equations with the general fractional derivatives in the Riemann-Liouville  sense.  This type of the fractional differential equations was not yet considered in the FC literature.\n\nThe rest of the paper is organized as follows. In the 2nd section, we introduce the general fractional derivatives of the Riemann-Liouville and Caputo types with the Sonine kernels from a special class of kernels and discuss some of their properties needed for the further discussions. In the 3rd section, we first provide some results  regarding the convolution series generated by the Sonine kernels. Then the convolution series are applied for derivation of analytical solutions to the single- and multi-terms fractional differential equations with the general fractional derivatives in the Riemann-Liouville  sense. For a treatment of the single- and multi-terms fractional differential equations with the general fractional derivatives in the Caputo  sense, we refer the interested readers to \\cite{Luc21b}.  \n\n\n\\section{General Fractional Integrals and Derivatives}\n\nThe general fractional derivatives (GFDs) with the kernel $k$ in the Riemann-Liouville and in the Caputo sense, respectively, are defined as follows (\\cite{Koch11,LucYam16,Koch19_1,LucYam20,Luc21a,Luc21c}): \n\\begin{equation}\n\\label{FDR-L}\n(\\D_{(k)}\\, f) (t) = \\frac{d}{dt}(k\\, *\\, f)(t)= \\frac{d}{dt}\\int_0^t k(t-\\tau)f(\\tau)\\, d\\tau,\n\\end{equation}\n\\begin{equation}\n\\label{FDC}\n( _*\\D_{(k)}\\, f) (t) =  (\\D_{(k)}\\, f) (t) - f(0)k(t),\n\\end{equation}\nwhere by $*$ the Laplace convolution is denoted:\n\\begin{equation}\n\\label{2-2}\n(f\\, *\\, g)(t) = \\int_0^{t}\\, f(t-\\tau)g(\\tau)\\, d\\tau.\n\\end{equation}\n\nThe Riemann-Liouville and the Caputo  fractional derivatives of order $\\alpha,\\ 0< \\alpha < 1$, defined by \\eqref{RLD} and \\eqref{CD}, respectively, are particular cases of the GFDs \\eqref{FDR-L} and \\eqref{FDC} with the kernel\n\\begin{equation}\n\\label{single}\nk(t) = h_{1-\\alpha}(t),\\ 0 <\\alpha <1,\\ h_{\\beta}(t) := \\frac{t^{\\beta -1}}{\\Gamma(\\beta)},\\ \\beta >0.\n\\end{equation}\n\nThe multi-term fractional derivatives and the fractional derivatives of the\ndistributed order are also particular cases of the GFDs \\eqref{FDR-L} and \\eqref{FDC} with the kernels \n\\begin{equation}\n\\label{multi}\nk(t) = \\sum_{k=1}^n a_k\\, h_{1-\\alpha_k}(t),\n\\ \\  0 < \\alpha_1 <\\dots < \\alpha_n < 1,\\ a_k \\in \\R,\\ k=1,\\dots,n, \n\\end{equation}\n\\begin{equation}\n\\label{distr}\nk(t) = \\int_0^1 h_{1-\\alpha}(t)\\, d\\rho(\\alpha),\n\\end{equation}\nrespectively, where $\\rho$ is a Borel measure defined on the interval $[0,\\, 1]$.\n\nSeveral useful properties of the Riemann-Liouville fractional integral and the Riemann-Liouville and Caputo fractional derivatives are based on the formula\n\\begin{equation}\n\\label{2-9}\n(h_{\\alpha} \\, * \\, h_\\beta)(t) \\, = \\, h_{\\alpha+\\beta}(t),\\ \\alpha,\\beta >0,\\ t>0\n\\end{equation}\nthat immediately follows from the well-known representation of the Euler Beta-function in terms of the Gamma-function:\n$$\nB(\\alpha,\\beta) := \\int_0^1 (1-\\tau)^{\\alpha -1}\\, \\tau^{\\beta -1}\\, d\\tau \\, = \\, \\frac{\\Gamma(\\alpha)\\Gamma(\\beta)}{\\Gamma(\\alpha+\\beta)},\\ \\alpha,\\beta>0.\n$$ \nIn the formula \\eqref{2-9} and in what follows,  the power function $h_{\\alpha}$ is defined as in \\eqref{single}. \n\nIn our discussions, we employ the integer order convolution powers that for a function $f=f(t),\\ t>0$ are defined by the expression\n\\begin{equation}\n\\label{I-6}\nf^{<n>}(t):= \\begin{cases}\n1,& n=0,\\\\\nf(t), & n=1,\\\\\n(\\underbrace{f* \\ldots * f}_{n\\ \\mbox{times}})(t),& n=2,3,\\dots .\n\\end{cases}\n\\end{equation}\nFor the kernel $\\kappa(t) = h_\\alpha(t)$ of the Riemann-Liouville fractional integral, we apply the formula \\eqref{2-9} and arrive at the important representation\n\\begin{equation}\n\\label{2-13}\nh_{\\alpha}^{<n>}(t) = h_{n\\alpha}(t),\\ n\\in \\N.\n\\end{equation}\nA well-known particular case of \\eqref{2-13} is the formula\n\\begin{equation}\n\\label{2-13-1}\n\\{1\\}^n(t) = h_{1}^n(t)= h_{n}(t)=\\frac{t^{n-1}}{\\Gamma(n)} = \\frac{t^{n-1}}{(n-1)!},\\ n\\in \\N,\n\\end{equation}\nwhere by $\\{1\\}$ we denoted a function that is identically equal to 1 for $t\\ge 1$. \n\nNow let us write down the formula \\eqref{2-9} for $\\beta = 1-\\alpha,\\ 0<\\alpha <1$: \n\\begin{equation}\n\\label{3-1}\n(h_{\\alpha}\\, * \\, h_{1-\\alpha})(t) = h_1(t) = \\{1 \\},\\ 0<\\alpha<1,\\  t>0.\n\\end{equation}\n\nIn \\cite{Abel1,Abel2}, Abel employed the relation \\eqref{3-1} to derive an inversion formula for the operator that is nowadays referred to as the Caputo fractional derivative and obtained it in form of the Riemann-Liouville fractional integral (solution to the Abel model for the tautochrone problem). \n\nBy an attempt to extend the Abel solution  method to more general integral equations of convolution type, \nSonine introduced in \\cite{Son} the relation\n\\begin{equation}\n\\label{3-2}\n(\\kappa \\, *\\, k )(t) = \\{1 \\},\\  t>0\n\\end{equation}\nthat  is nowadays referred to as the Sonine condition. The functions that satisfy the Sonine condition are called the Sonine kernels. For a Sonine kernel $\\kappa$, the kernel $k$ that satisfies the Sonine condition \\eqref{3-2} is called an associated kernel to $\\kappa$. Of course, $\\kappa$ is then an associated kernel to $k$. In what follows, we denote the set of the Sonine kernels by $\\mathcal{S}$.   \n\nIn \\cite{Son}, Sonine introduced a class of the Sonine kernels in the form\n\\begin{equation}\n\\label{3-3}\n\\kappa(t) = h_{\\alpha}(t) \\cdot \\, \\kappa_1(t),\\ \\kappa_1(t)=\\sum_{k=0}^{+\\infty}\\, a_k t^k, \\ a_0 \\not = 0,\\ 0<\\alpha <1,\n\\end{equation}\n\\begin{equation}\n\\label{3-4}\nk(t) = h_{1-\\alpha}(t) \\cdot k_1(t),\\ k_1(t)=\\sum_{k=0}^{+\\infty}\\, b_k t^k, \n\\end{equation}\nwhere $\\kappa_1=\\kappa_1(t)$ and $k_1=k_1(t)$ are analytical functions and the coefficients $a_k,\\ b_k,\\ k\\in \\N_0$  satisfy the following triangular system of linear equations:\n\\begin{equation}\n\\label{3-5}\na_0b_0 = 1,\\ \\sum_{k=0}^n\\Gamma(k+1-\\alpha)\\Gamma(\\alpha+n-k)a_{n-k}b_k = 0,\\ n\\ge 1.\n\\end{equation}\nAn important example of the kernels from $\\mathcal{S}$ in form \\eqref{3-3}, \\eqref{3-4} was derived in \\cite{Son}  in terms of the Bessel function $J_{\\nu}$ and the modified Bessel function $I_{\\nu}$: \n\\begin{equation}\n\\label{Bess}\n\\kappa(t) = (\\sqrt{t})^{\\alpha-1}J_{\\alpha-1}(2\\sqrt{t}),\\ \nk(t) = (\\sqrt{t})^{-\\alpha}I_{-\\alpha}(2\\sqrt{t}),\\ 0<\\alpha <1,\n\\end{equation}\nwhere \n$$\nJ_\\nu (t) = \\sum_{k=0}^{+\\infty} \\frac{(-1)^k(t\/2)^{2k+\\nu}}{k!\\Gamma(k+\\nu+1)},\\ \nI_\\nu (t) = \\sum_{k=0}^{+\\infty} \\frac{(t\/2)^{2k+\\nu}}{k!\\Gamma(k+\\nu+1)}.\n$$\n\nFor other examples of the Sonine kernels we refer the readers to \\cite{Luc21c,Koch11,Luc21a,Sam}.\n\nIn this paper, we deal with the general fractional integrals (GFIs) with the kernels $\\kappa \\in \\mathcal{S}$ \ndefined by the formula\n\\begin{equation}\n\\label{GFI}\n(\\I_{(\\kappa)}\\, f)(t) := (\\kappa\\, *\\, f)(t) = \\int_0^t \\kappa(t-\\tau)f(\\tau)\\, d\\tau,\\ t>0\n\\end{equation}\nand with the GFDs with the associated Sonine kernels $k$ in the Riemann-Liouville and Caputo senses defined by \\eqref{FDR-L} and \\eqref{FDC}, respectively.  \n\n\nIn our discussions, we restrict ourselves to a class of the Sonine kernels from the space $C_{-1,0}(0,+\\infty)$ that is an important particular case of the following two-parameters family of spaces (\\cite{Luc21b,Luc21a,Luc21c}):\n\\begin{equation}\n\\label{subspace}\n C_{\\alpha,\\beta}(0,+\\infty) \\, = \\, \\{f:\\ f(t) = t^{p}f_1(t),\\ t>0,\\ \\alpha < p < \\beta,\\ f_1\\in C[0,+\\infty)\\}.\n\\end{equation}\nBy $C_{-1}(0,+\\infty)$ we mean the space $C_{-1,+\\infty}(0,+\\infty)$.\n\nThe set of such Sonine kernels will be denoted by $\\mathcal{L}_{1}$ (\\cite{Luc21c}):\n\\begin{equation}\n\\label{Son}\n(\\kappa,\\, k \\in \\mathcal{L}_{1} ) \\ \\Leftrightarrow \\ (\\kappa,\\, k \\in C_{-1,0}(0,+\\infty))\\wedge ((\\kappa\\, *\\, k)(t) \\, = \\, \\{1\\}).\n\\end{equation}\n\nIn the rest of this section, we present some important results for the GFIs and the GFDs  with the Sonine kernels from $\\mathcal{L}_{1}$ on the space $C_{-1}(0,+\\infty)$ and its sub-spaces. \n\nThe basic properties of the GFI \\eqref{GFI} on the space $C_{-1}(0,+\\infty)$ easily follow from the known properties of the Laplace convolution:\n\\begin{equation}\n\\label{GFI-map}\n\\I_{(\\kappa)}:\\, C_{-1}(0,+\\infty)\\, \\rightarrow C_{-1}(0,+\\infty),\n\\end{equation}\n\\begin{equation}\n\\label{GFI-com}\n\\I_{(\\kappa_1)}\\, \\I_{(\\kappa_2)} = \\I_{(\\kappa_2)}\\, \\I_{(\\kappa_1)},\\ \\kappa_1,\\, \\kappa_2 \\in \\mathcal{L}_{1},\n\\end{equation}\n\\begin{equation}\n\\label{GFI-index}\n\\I_{(\\kappa_1)}\\, \\I_{(\\kappa_2)} = \\I_{(\\kappa_1*\\kappa_2)},\\  \\kappa_1,\\, \\kappa_2 \\in \\mathcal{L}_{1}.\n\\end{equation}\n\nFor the functions $f\\in C_{-1}^1(0,+\\infty):=\\{f:\\ f^\\prime\\in C_{-1}(0,+\\infty)\\}$, the GFDs of the Riemann-Liouville type can be represented as follows (\\cite{Luc21a}):\n\\begin{equation}\n\\label{GFDL-1}\n(\\D_{(k)}\\, f) (t) = (k\\, * \\, f^\\prime)(t) + f(0)k(t),\\ t>0.\n\\end{equation}\nThus, for $f\\in C_{-1}^1(0,+\\infty)$, the GFD \\eqref{FDC} of the Caputo type takes the form\n\\begin{equation}\n\\label{GFDC_1}\n( _*\\D_{(k)}\\, f) (t) =  (k\\, * \\, f^\\prime)(t),\\ t>0. \n\\end{equation}\n\nIt is worth mentioning that in the FC publications, the Caputo fractional derivative \\eqref{CD} is often defined as in the formula \\eqref{GFDC_1}:\n\\begin{equation}\n\\label{CD-1}\n(\\,_*D^\\alpha_{0+}\\, f)(t)  = (h_{1-\\alpha}\\, * \\, f^\\prime)(t) = (I^{1-\\alpha}_{0+} f^\\prime)(t),\\ t>0. \n\\end{equation}\n\nNow, following \\cite{Luc21a,Luc21d}, we define the $n$-fold GFI and the $n$-fold sequential GFDs in the Riemann-Liouville and Caputo sense. \n\n\\begin{definition}[\\cite{Luc21a}]\n\\label{d1}\nLet $\\kappa \\in \\mathcal{L}_{1}$. The $n$-fold GFI ($n \\in \\N$) is  a composition of $n$ GFIs with the kernel $\\kappa$:\n\\begin{equation}\n\\label{GFIn}\n(\\I_{(\\kappa)}^{<n>}\\, f)(t) := (\\underbrace{\\I_{(\\kappa)} \\ldots \\I_{(\\kappa)}}_{n\\ \\mbox{times}}\\, f)(t),\\  t>0.\n\\end{equation}\n\\end{definition}\n\nIt is worth mentioning that  the index law \\eqref{GFI-index} leads to a representation of the $n$-fold GFI \\eqref{GFIn} in form of the GFI with the kernel $\\kappa^{<n>}$:\n\\begin{equation}\n\\label{GFIn-1}\n(\\I_{(\\kappa)}^{<n>}\\, f)(t) = (\\kappa^{<n>}\\, *\\, f)(t) = (\\I_{(\\kappa)^{<n>}}\\, f)(t),\\  t>0.\n\\end{equation}\n\nThe kernel $\\kappa^{<n>},\\ n\\in \\N$ belongs to the space $C_{-1}(0,+\\infty)$, but it is not always a Sonine kernel. \n\n\\begin{definition}[\\cite{Luc21d}]\n\\label{d2}\nLet $\\kappa \\in \\mathcal{L}_{1}$ and $k$ be its associated Sonine kernel. The $n$-fold sequential GFDs in the Riemann-Liouville and in the Caputo sense, respectively, are defined as follows:\n\\begin{equation}\n\\label{GFDLn}\n(\\D_{(k)}^{<n>}\\, f)(t) := (\\underbrace{\\D_{(k)} \\ldots \\D_{(k)}}_{n\\ \\mbox{times}}\\, f)(t),\\  t>0,\n\\end{equation}\n\\begin{equation}\n\\label{GFDLn-C}\n(\\,_*\\D_{(k)}^{<n>}\\, f)(t) := (\\underbrace{\\,_*\\D_{(k)} \\ldots \\,_*\\D_{(k)}}_{n\\ \\mbox{times}}\\, f)(t),\\  t>0.\n\\end{equation}\n\\end{definition}\n\nIt is worth mentioning that in \\cite{Luc21b,Luc21a}, the $n$-fold  GFDs ($n \\in \\N$) were defined in a different form:\n\\begin{equation}\n\\label{GFDLn-1}\n(\\D_{(k)}^n\\, f)(t) := \\frac{d^n}{dt^n} ( k^{<n>} * f)(t),\\ t>0,\n\\end{equation}\n\\begin{equation}\n\\label{GFDLn-1_C}\n(\\,_*\\D_{(k)}^n\\, f)(t) := ( k^{<n>} * f^{(n)})(t),\\ t>0.\n\\end{equation}\n\nThe $n$-fold sequential GFDs \\eqref{GFDLn} and \\eqref{GFDLn-C} are a far reaching generalization of the Riemann-Liouville and the Caputo sequential fractional derivatives to the case of the Sonine kernels from $\\mathcal{L}_{1}$.\n\nSome important connections between the $n$-fold GFI \\eqref{GFIn} and the $n$-fold sequential GFDs \\eqref{GFDLn} and \\eqref{GFDLn-C} in the Riemann-Liouville and Caputo senses are provided in the so-called first and second fundamental theorems of FC (\\cite{Luc20}) formulated below.  \n\n\\begin{theorem}[\\cite{Luc21d}]\n\\label{t3-n}\nLet $\\kappa \\in \\mathcal{L}_{1}$ and $k$ be its associated Sonine kernel. \n\nThen,  the $n$-fold sequential GFD \\eqref{GFDLn} in the Riemann-Liouville sense  is a left inverse operator to the $n$-fold  GFI \\eqref{GFIn} on the space $C_{-1}(0,+\\infty)$: \n\\begin{equation}\n\\label{FTLn}\n(\\D_{(k)}^{<n>}\\, \\I_{(\\kappa)}^{<n>}\\, f) (t) = f(t),\\ f\\in C_{-1}(0,+\\infty),\\ t>0,\n\\end{equation}\nand the $n$-fold sequential GFD \\eqref{GFDLn-C} in the Caputo sense  is a left inverse operator to the $n$-fold  GFI \\eqref{GFIn} on the space $C_{-1,(k)}^n(0,+\\infty)$: \n\\begin{equation}\n\\label{FTLn-C}\n(\\,_*\\D_{(k)}^{<n>}\\, \\I_{(\\kappa)}^{<n>}\\, f) (t) = f(t),\\ f\\in C_{-1,(k)}^n(0,+\\infty),\\ t>0,\n\\end{equation}\nwhere $C_{-1,(k)}^n(0,+\\infty) := \\{f:\\ f(t)=(\\I_{(k)}^{<n>}\\, \\phi)(t),\\ \\phi\\in C_{-1}(0,+\\infty)\\}$.\n\\end{theorem}\n\n\\begin{theorem}[\\cite{Luc21d}]\n\\label{tgcTfn}\nLet $\\kappa \\in \\mathcal{L}_{1}$ and $k$ be its associated Sonine kernel. \n\nFor a function $f\\in C_{-1,(k)}^{(n)}(0,+\\infty) = \\{ f:\\ (\\D_{(k)}^{<j>}\\, f)\\in C_{-1}(0,+\\infty),\\ j=0,1,\\dots,n\\}$, the formula\n\\begin{equation}\n\\label{sFTLn}\n(\\I_{(\\kappa)}^{<n>}\\, \\D_{(k)}^{<n>}\\, f) (t) = f(t) - \\sum_{j=0}^{n-1}\\left( k\\, * \\, \\D_{(k)}^{<j>}\\, f\\right)(0)\\kappa^{<j+1>}(t) = \n\\end{equation}\n$$\nf(t) - \\sum_{j=0}^{n-1}\\left( \\I_{(k)}\\, \\D_{(k)}^{<j>}\\, f\\right)(0)\\kappa^{<j+1>}(t),\\ t>0\n$$\nholds valid, where $\\I_{(\\kappa)}^{<n>}$ is the $n$-fold GFI \\eqref{GFIn} and $\\D_{(k)}^{<n>}$ is the $n$-fold sequential GFD \\eqref{GFDLn} in the Riemann-Liouville sense. \n\nFor a function $f\\in C_{-1}^{n}(0,+\\infty):=\\{f:\\ f^{(n)}\\in C_{-1}(0,+\\infty)\\}$, the formula\n\\begin{equation}\n\\label{sFTLn-C}\n(\\I_{(\\kappa)}^{<n>}\\,_*\\D_{(k)}^{<n>}\\, f) (t) = f(t) - f(0) - \\sum_{j=1}^{n-1}\\left(\\,_*\\D_{(k)}^{<j>}\\, f\\right)(0)\\left( \\{1\\} \\, *\\, \\kappa^{<j>}\\right)(t)\n\\end{equation}\nholds valid, where $\\I_{(\\kappa)}^{<n>}$ is the $n$-fold GFI \\eqref{GFIn} and $\\,_*\\D_{(k)}^{<n>}$ is the $n$-fold sequential GFD \\eqref{GFDLn-C}.\n\\end{theorem}\n\nFor the proofs of Theorems \\ref{t3-n} and \\ref{tgcTfn} and their particular cases we refer the interested readers to \\cite{Luc21d}. \n\n\\section{Solutions to the Fractional Differential Equations with the GFDs in the Riemann-Liouville Sense in Terms of the Convolution Series}\n\nFirst, we introduce the convolution series and treat some of their properties needed for the further discussions. \n\nFor a Sonine kernel $\\kappa \\in \\mathcal{L}_{1}$,  a convolution series in form \n\\begin{equation}\n\\label{conser}\n\\Sigma_\\kappa(t) = \\sum_{j=0}^{+\\infty} a_j\\, \\kappa^{<j+1>}(t),\\ a_j\\in \\R\n\\end{equation}\nwas introduced in \\cite{Luc21c} for analytical treatment of the fractional differential equations with the $n$-fold GFDs of the Caputo type by means of an operational calculus developed for these GFDs. In \\cite{Luc21d}, some of the results presented in \\cite{Luc21c} were extended to convolution series in form \\eqref{conser} generated by any function $\\kappa \\in C_{-1}(0,+\\infty)$ (that is not necessarily a Sonine kernel). \n\nA very important question regarding convergence of the convolution series \\eqref{conser} was answered in \\cite{Luc21b,Luc21d}. \n\n\\begin{theorem}[\\cite{Luc21d}]\n\\label{t11}\nLet a function $\\kappa \\in C_{-1}(0,+\\infty)$  be represented in the form\n\\begin{equation}\n\\label{rep}\n\\kappa(t) = h_{p}(t)\\kappa_1(t),\\ t>0,\\ p>0,\\ \\kappa_1\\in C[0,+\\infty)\n\\end{equation} \nand the convergence radius of the power  series\n\\begin{equation}\n\\label{ser}\n\\Sigma(z) = \\sum^{+\\infty }_{j=0}a_{j}\\, z^j,\\ a_{j}\\in \\Com,\\ z\\in \\Com\n\\end{equation}\nbe non-zero. Then the convolution series \n\\eqref{conser}\nis convergent for all $t>0$ and defines a function  from the space $C_{-1}(0,+\\infty)$. \nMoreover, the series \n\\begin{equation}\n\\label{conser_p}\nt^{1-\\alpha}\\, \\Sigma_\\kappa(t) = \\sum^{+\\infty }_{j=0}a_{j}\\, t^{1-\\alpha}\\, \\kappa^{<j+1>}(t),\\ \\ \\alpha = \\min\\{p,\\, 1\\}\n\\end{equation}\nis uniformly convergent for $t\\in [0,\\, T]$ for any $T>0$.\n\\end{theorem}\n\nIn what follows, we always assume that the coefficients of the convolution series satisfy the condition that the convergence radius of the corresponding power series is non-zero and thus Theorem \\ref{t11} is applicable for these convolution series. \n\nAs an example, let us consider  the geometric series\n\\begin{equation}\n\\label{geom}\n\\Sigma(z) = \\sum_{j=0}^{+\\infty} \\lambda^{j}z^{j},\\ \\lambda \\in \\Com,\\ z\\in \\Com.\n\\end{equation}\nFor $\\lambda \\not =0$,  the convergence radius $r$ of this series is equal to $1\/|\\lambda|$ and thus we can apply  Theorem \\ref{t11} that says  that the convolution series generated by a function $\\kappa \\in C_{-1}(0,+\\infty)$ in form\n\\begin{equation}\n\\label{l}\nl_{\\kappa,\\lambda}(t) = \\sum_{j=0}^{+\\infty} \\lambda^{j}\\kappa^{<j+1>}(t),\\ \\lambda \\in \\Com\n\\end{equation}\nis convergent for all $t>0$ and defines a function from the space $C_{-1}(0,+\\infty)$.\n\nThe convolution series $l_{\\kappa,\\lambda}$ defined by \\eqref{l} plays a very important role in the operational calculus for the GFD of Caputo type developed in \\cite{Luc21b}. It provides a far reaching generalization of both the exponential function and the two-parameters Mittag-Leffler function in form \\eqref{eqRL_sol}. \n\nIndeed, let us consider the convolution series \\eqref{l} in the case of the kernel function $\\kappa=\\{ 1\\}$. Due to  the formula $\\kappa^{<j+1>}(t) = \\{ 1\\}^{<j+1>}(t) = h_{j+1}(t)$ (see \\eqref{2-13}), the convolution series \\eqref{l} is reduced to the power series for the exponential function:\n\\begin{equation}\n\\label{l-Mic}\nl_{\\kappa,\\lambda}(t) = \\sum_{j=0}^{+\\infty} \\lambda^{j}h_{j+1}(t) =\n\\sum_{j=0}^{+\\infty} \\frac{(\\lambda\\, t)^j}{j!} = e^{\\lambda\\, t}.\n\\end{equation}\n\nFor the kernel $\\kappa(t) = h_{\\alpha}(t)$ of the Riemann-Liouville fractional integral,  the formula  $\\kappa^{<j+1>}(t) = h_{\\alpha}^{<j+1>}(t) = h_{(j+1)\\alpha}(t)$ (see \\eqref{2-13}) holds valid. Thus, the convolution series \\eqref{l} takes the form\n\\begin{equation}\n\\label{l-Cap}\nl_{\\kappa,\\lambda}(t) = \\sum_{j=0}^{+\\infty} \\lambda^{j}h_{(j+1)\\alpha}(t) =\nt^{\\alpha-1}\\sum_{j=0}^{+\\infty} \\frac{\\lambda^j\\, t^{j\\alpha}}{\\Gamma(j\\alpha+\\alpha)} = t^{\\alpha -1}E_{\\alpha,\\alpha}(\\lambda\\, t^{\\alpha})\n\\end{equation}\nthat is the same as the two-parameters Mittag-Leffler function \\eqref{eqRL_sol}. \n\nFor $\\kappa \\in \\mathcal{L}_{1}$, another important convolution series was introduced in  \\cite{Luc21b}  as follows:\n\\begin{equation}\n\\label{L}\nL_{\\kappa,\\lambda}(t) = (k \\, *\\ l_{\\kappa,\\lambda})(t) = 1 + \\left(\\{ 1 \\} * \\sum_{j=1}^{+\\infty} \\lambda^{j}\\kappa^{<j>}(\\cdot)\\right)(t),\\ \\lambda \\in \\Com,\n\\end{equation}\nwhere $k$ is the Sonine kernel associated to the kernel $\\kappa$. It is easy to see that in the case $\\kappa=\\{ 1\\}$, the convolution series \\eqref{L} coincides with the exponential function: \n\\begin{equation}\n\\label{L-Mic}\nL_{\\kappa,\\lambda}(t) = 1 + \\left(\\{ 1 \\} * \\sum_{j=1}^{+\\infty} \\lambda^{j}h_j(\\cdot) \\right)(t)=\n1+ \\sum_{j=1}^{+\\infty} \\lambda^{j}h_{j+1}(t) = e^{\\lambda\\, t}.\n\\end{equation}\n\nIn the case of the kernel $\\kappa(t) = h_{\\alpha}(t),\\ t>0,\\ 0<\\alpha <1$,  the convolution series $L_{\\kappa,\\lambda}$ is reduced to the two-parameters Mittag-Leffler function \\eqref{eqC_sol}:\n\\begin{equation}\n\\label{L-Cap-op}\nL_{\\kappa,\\lambda}(t)\\, = \\, 1 + \\left(\\{ 1 \\} * \\sum_{j=1}^{+\\infty} \\lambda^{j}h_{j\\alpha}(\\cdot)\\right)(t) =\n1 + \\sum_{j=1}^{+\\infty} \\lambda^{j}h_{j\\alpha+1}(t) = E_{\\alpha,1}(\\lambda\\, t^{\\alpha}).\n\\end{equation}\n\nAnalytical solutions to the single- and multi-terms  fractional differential equations with the $n$-fold GFDs of the Caputo type  were presented in \\cite{Luc21b} in terms of the convolution series $l_{\\kappa,\\lambda}$ and $L_{\\kappa,\\lambda}$. In the rest of this section, we treat the linear single- and multi-terms  fractional differential equations with the $n$-fold GFDs in the Riemann-Liouville sense.\n\nWe start with the following auxiliary result:\n\n\\begin{theorem}\n\\label{eqconv}\nTwo convolution series generated by the same Sonine kernel $\\kappa  \\in \\mathcal{L}_{1}$ coincide for all $t>0$, i.e., \n\\begin{equation}\n\\label{ser12}\n\\sum_{j=0}^{+\\infty} b_{j}\\, \\kappa^{<j+1>}(t) \\equiv  \\sum_{j=0}^{+\\infty} c_{j}\\, \\kappa^{<j+1>}(t),\\ t>0\n\\end{equation}\nif and only if the corresponding coefficients of these series are equal: \n\\begin{equation}\n\\label{ser13}\na_j = b_j,\\ j=0,1,2,\\dots .\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof} \nIf the corresponding coefficients of two convolution series generated by the same Sonine kernel $\\kappa  \\in \\mathcal{L}_{1}$ are equal, then we have just one series and evidently the identity \\eqref{ser12} holds valid. \n\nThe idea of the proof of the second part of this theorem is the same as the one for the proof of the analogous calculus result for the power series, i.e., under the condition that the identity \\eqref{ser12} holds valid we first show that $b_0=c_0$ and then apply the same arguments to prove that $b_1=c_1$, $b_2=c_2$, etc. \n\nAccording to Theorem \\ref{t11}, the convolution series in form \\eqref{conser} is uniformly convergent on any interval $[\\epsilon,\\ T]$, and thus we can apply the GFI $\\I_{(k)}$ to this series term by term:\n$$\n\\left( \\I_{(k)}\\, \\sum_{j=0}^{+\\infty} a_{j}\\, \\kappa^{<j+1>}(\\cdot )\\right)(t)  = \n \\sum_{j=0}^{+\\infty} \\left( \\I_{(k)}\\, a_j\\, \\kappa^{<j+1>}(\\cdot)\\right)(t) = \n $$\n $$\n\\sum_{j=0}^{+\\infty} \\left(  a_j\\, (k(\\cdot)\\, *\\, \\kappa^{<j+1>}(\\cdot )\\right)(t) = \na_0 + \\sum_{j=1}^{+\\infty} a_{j}\\, \\left(\\{ 1\\} \\, *\\, \\kappa^{<j>}(\\cdot)\\right)(t) =  \n$$\n$$\na_0 + \\left( \\{1\\}\\, *\\, \\sum_{j=1}^{+\\infty} a_j\\, \\kappa^{<j>}(\\cdot)\\right)(t) = a_0 + (\\{1\\}\\, *\\, f_1)(t),\n$$\nwhere $f_1$ is the following convolution series:\n\\begin{equation}\n\\label{f1}\nf_1(t) = \\sum_{j=1}^{+\\infty} a_j\\, \\kappa^{<j>}(t)=\\sum_{j=0}^{+\\infty} a_{j+1}\\, \\kappa^{<j+1>}(t).\n\\end{equation}\nSummarizing the calculations from above, for the convolution series in form \\eqref{conser}, the formula\n\\begin{equation}\n\\label{Intcon} \n\\left( \\I_{(k)}\\, \\sum_{j=0}^{+\\infty} a_{j}\\, \\kappa^{<j+1>}(\\cdot )\\right)(t) =  a_0 + \\left( \\{1\\}\\, *\\, \\sum_{j=0}^{+\\infty} a_{j+1}\\, \\kappa^{<j+1>}(\\cdot)\\right)(t)\n\\end{equation}\nholds valid.\n\nBecause the convergence radius of the power series $\\Sigma_1(t) = \\sum_{j=0}^{+\\infty} a_{j+1}\\, z^j$ is the same as the convergence radius of the power series $\\Sigma(t) = \\sum_{j=0}^{+\\infty} a_{j}\\, z^j$, Theorem  \\ref{t11} ensures the inclusion $f_1 \\in C_{-1}(0,+\\infty)$, where $f_1$ is defined by the formula \\eqref{f1}.  As have been shown in \\cite{LucGor}, the definite integral of a function from $C_{-1}(0,+\\infty)$ is a continuous function on the whole interval $[0,\\, +\\infty)$ that takes the value zero at the point zero:\n\\begin{equation}\n\\label{a0_2} \n\\left( \\{1\\}\\, * \\, f_1 \\right) (x) = (I_{0+}^1\\, f_1)(x) \\in C[0,\\ +\\infty),\\ \\ (I_{0+}^1\\, f_1)(0) = 0.\n\\end{equation}\n\nNow we act with the GFI $\\I_{(k)}$ on the equality \\eqref{ser12} and apply the formula \\eqref{Intcon} to get the relation\n\\begin{equation}\n\\label{ser12_1}\nb_0 + \\left( \\{1\\}\\, *\\, \\sum_{j=0}^{+\\infty} b_{j+1}\\, \\kappa^{<j+1>}(\\cdot)\\right)(t) \\equiv  c_0 + \\left( \\{1\\}\\, *\\, \\sum_{j=0}^{+\\infty} c_{j+1}\\, \\kappa^{<j+1>}(\\cdot)\\right)(t),\\ t>0.\n\\end{equation}\nSubstituting the point $t=0$ into the equality \\eqref{ser12_1} and using the formula \\eqref{a0_2}, we deduce that $b_0 = c_0$. Now we differentiate the equality \\eqref{ser12_1} and get the following identity:\n\\begin{equation}\n\\label{ser12_2}\n\\sum_{j=0}^{+\\infty} b_{j+1}\\, \\kappa^{<j+1>}(t) \\equiv  \\sum_{j=0}^{+\\infty} c_{j+1}\\, \\kappa^{<j+1>}(t),\\ t>0.\n\\end{equation}\nThis identity has exactly same structure as the identity \\eqref{ser12} from Theorem \\ref{eqconv}. Thus we can apply the same arguments as above and derive the relation $b_1 = c_1$. By repeating the same reasoning again and again we arrive at the formula \\eqref{ser13} that we wanted to prove.\n\\end{proof}\n\nNow we are ready to apply the method of convolution series for derivation of solutions to the fractional differential equations with the GFDs and start with the fractional relaxation equation with the GFD of the Riemann-Liouville type:\n\\begin{equation}\n\\label{eq-1-1}\n( \\D_{(k)}\\, y)(t)  = \\lambda y(t), \\ \\lambda \\in \\R,\\ t>0. \n\\end{equation}\n\nAs in the case of the power series, we look for a general solution to this equation in form of a convolution series generated by the Sonine kernel $\\kappa$ that is an associated kernel to the kernel $k$ of the GFD from the equation \\eqref{eq-1-1}:\n\\begin{equation}\n\\label{sol-1-1}\ny(t) = \\sum_{j=0}^{+\\infty} b_j\\, \\kappa^{<j+1>}(t),\\ b_j\\in \\R.\n\\end{equation}\n\nTo proceed, let as first calculate the image of the convolution series \\eqref{sol-1-1} by action of the GFD $\\D_{(k)}$: \n$$\n( \\D_{(k)}\\, y)(t)  = \\left( \\D_{(k)}\\, \\sum_{j=0}^{+\\infty} b_j\\, \\kappa^{<j+1>}(\\cdot)\\right)(t) = \n\\frac{d}{dt}\\left( \\I_{(k)}\\, \\sum_{j=0}^{+\\infty} b_j\\, \\kappa^{<j+1>}(\\cdot)\\right)(t).\n$$\n\nIn the proof of Theorem \\ref{eqconv} we already calculated the image of the convolution series \\eqref{sol-1-1} by action of the GFI $\\I_{(k)}$ (formula \\eqref{Intcon}). Applying this formula, we arrive at the representation \n\\begin{equation}\n\\label{term1}\n( \\D_{(k)}\\, y)(t)  = \\frac{d}{dt}\\left( b_0 + \\left( \\{1\\}\\, *\\, \\sum_{j=0}^{+\\infty} b_{j+1}\\, \\kappa^{<j+1>}\n(\\cdot ) \\right)(t)\\right) = \\sum_{j=0}^{+\\infty} b_{j+1}\\, \\kappa^{<j+1>}(t).\n\\end{equation}\n\nIn the next step, we substitute the right-hand side of \\eqref{term1} into the equation \\eqref{eq-1-1} and get an equality of two convolution series generated by the same kernel $\\kappa$:\n$$\n\\sum_{j=0}^{+\\infty} b_{j+1}\\, \\kappa^{<j+1>}(t) = \\sum_{j=0}^{+\\infty} \\lambda\\ b_{j}\\, \\kappa^{<j+1>}(t),\\ t>0.\n$$\nApplication of Theorem \\ref{eqconv} to the above identity leads to the following relations for the coefficients of the convolution series \\eqref{sol-1-1}:\n\\begin{equation}\n\\label{coef1}\nb_{j+1} = \\lambda\\, b_j,\\ j=0,1,2,\\dots .\n\\end{equation}\nThe infinite system \\eqref{coef1} of linear equations can be easily solved step by step and we arrive at the explicit solution in form\n\\begin{equation}\n\\label{coef2}\nb_{j} = b_0\\, \\lambda^j,\\ j=1,2,\\dots ,\n\\end{equation}\nwhere $b_0\\in \\R$ is an arbitrary constant. Summarizing the arguments presented above, we proved the following theorem:\n\n\\begin{theorem}\n\\label{t-eq1}\nThe general solution to the fractional relaxation equation  \\eqref{eq-1-1} with  the GFD \\eqref{FDR-L} in the Riemann-Liouville sense \ncan be represented  as follows:\n\\begin{equation}\n\\label{eig}\ny(t) = \\sum_{j=0}^{+\\infty} b_0 \\, \\lambda^j\\,  \\kappa^{<j+1>}(t) = b_0\\, l_{\\kappa,\\lambda}(t),\\ b_0\\in \\R,\n\\end{equation}\nwhere $l_{\\kappa,\\lambda}$ is the convolution series \\eqref{l}.\n\\end{theorem}\n\n\n\\begin{remark}\n\\label{r1}\nThe constant $b_0$ in the general solution \\eqref{eig} to the equation \\eqref{eq-1-1} can be determined from a suitably posed initial condition. The form of this initial condition is prescribed  by Theorem \\ref{tgcTfn} (see also the formula \\eqref{Intcon}). Indeed, setting $n=1$ in the relation \\eqref{sFTLn}, we get the following representation of the projector operator of the GFD \\eqref{FDR-L} in the Riemann-Liouville sense:\n\\begin{equation}\n\\label{proj1}\n(P\\, f)(t) = f(t) - (\\I_{(\\kappa)}\\, \\D_{(k)}\\, f) (t) = \\left(\\I_{(k)}\\, f\\right) (0)\\kappa(t),\\  f\\in C_{-1,(k)}^{(1)}(0,+\\infty).\n\\end{equation}\nThus, the initial-value problem \n\\begin{equation}\n\\label{eq-1-1-iv}\n\\begin{cases}\n( \\D_{(k)}\\, y)(t)  = \\lambda y(t), \\ \\lambda \\in \\R,\\ t>0,\\\\\n\\left(\\I_{(k)}\\, y\\right) (0) = b_0\n\\end{cases}\n\\end{equation}\nhas a unique solution given by the formula \\eqref{eig}.\n\\end{remark}\n\n\nIn the case of the Sonine kernel $k(t) = h_{1-\\alpha}(t),\\ 0<\\alpha<1$, the  equation \\eqref{eq-1-1} is reduced to the equation \\eqref{eqRL} with the Riemann-Liouville fractional derivative and its solution \\eqref{eig} is exactly the solution \\eqref{eqRL_sol} of the equation \\eqref{eqRL} in terms of the two-parameters Mittag-Leffler function (see the formula \\eqref{l-Cap}). The initial-value problem \\eqref{eq-1-1-iv} takes the well-known form\n\\begin{equation}\n\\label{eq-1-1-iv-RL}\n\\begin{cases}\n( D_{0+}^\\alpha\\, y)(t)  = \\lambda y(t), \\ \\lambda \\in \\R,\\ t>0,\\\\\n\\left(I_{0+}^{1-\\alpha}\\, y\\right) (0) = b_0.\n\\end{cases}\n\\end{equation}\nIts unique solution is given by the formula $y(t) = b_0\\, t^{\\alpha -1} E_{\\alpha, \\alpha}(\\lambda\\, t^\\alpha)$. \n\nNow we proceed with the inhomogeneous equation of type \\eqref{eq-1-1}\n\\begin{equation}\n\\label{eq-1-2}\n( \\D_{(k)}\\, y)(t)  = \\lambda y(t) + f(t), \\ \\lambda \\in \\R,\\ t>0, \n\\end{equation} \nwhere the source function $f$ is represented in form of a convolution series\n\\begin{equation}\n\\label{f}\nf(t) = \\sum_{j=0}^{+\\infty} a_j\\, \\kappa^{<j+1>}(t),\\ a_j\\in \\R.\n\\end{equation} \nAgain, we look for solutions to the equation \\eqref{eq-1-2} in form of the convolution series \\eqref{sol-1-1}. Applying exactly the same reasoning as above, we arrive at the following infinite system of linear equations for the coefficients of the convolution series \\eqref{sol-1-1}:\n\\begin{equation}\n\\label{coef1-1}\nb_{j+1} = \\lambda\\, b_j + a_j,\\ j=0,1,2,\\dots .\n\\end{equation}\nThe explicit form of solutions to this system of equations is as follows:\n\\begin{equation}\n\\label{coef2-1}\nb_{j} = b_0\\, \\lambda^j\\,  +\\,  \\sum_{i=0}^{j-1}a_i\\, \\lambda^{j-i-1}, \\ j=1,2,\\dots,\n\\end{equation}\nwhere $b_0\\in \\R$ is an arbitrary constant. Then the general solution to the equation \\eqref{eq-1-2} takes the form\n$$\ny(t) = b_0\\, \\kappa(t)+ \\sum_{j=1}^{+\\infty} \\left(b_0\\, \\lambda^j\\,  +\\,  \\sum_{i=0}^{j-1}a_i\\, \\lambda^{j-i-1}\\right) \\, \\kappa^{<j+1>}(t) = \n$$\n$$\nb_0\\, \\sum_{j=0}^{+\\infty} \\lambda^j\\,\\kappa^{<j+1>}(t)  + \\sum_{j=1}^{+\\infty} \\sum_{i=0}^{j-1} a_i\\, \\lambda^{j-i-1}\\, \\kappa^{<j+1>}(t).\n$$\nBy direct calculations, we  verify that the second sum in the last formula can be written in a more compact form:\n$$\n\\sum_{j=1}^{+\\infty} \\sum_{i=0}^{j-1} a_i\\,  \\lambda^{j-i-1}\\, \\kappa^{<j+1>}(t) = \\sum_{i=0}^{+\\infty}  a_i \\sum_{j=1}^{+\\infty}  \\lambda^{j-1}\\, \\kappa^{<j+i+1>}(t) = (f\\, *\\, l_{\\kappa,\\lambda})(t),\n$$\nwhere the convolution series $l_{\\kappa,\\lambda}$ is defined by \\eqref{l}. We thus proved the following result:\n\n\\begin{theorem}\n\\label{t-eq2}\nThe general solution to the inhomogeneous equation \\eqref{eq-1-2} has the form\n\\begin{equation}\n\\label{eig1}\ny(t) = b_0\\, l_{\\kappa,\\lambda}(t) +(f\\, *\\, l_{\\kappa,\\lambda})(t),\\ b_0\\in \\R,\n\\end{equation}\nwhere the convolution series $l_{\\kappa,\\lambda}$ is defined by \\eqref{l}. \n\nThe constant $b_0$ is uniquely determined by the initial condition \n\\begin{equation}\n\\label{ic1}\n\\left(\\I_{(k)}\\, y\\right) (0) = b_0.\n\\end{equation}\n\\end{theorem}\n\nApplying Theorem \\ref{t-eq2} to the case of the Riemann-Liouville fractional derivative (kernel $k(t) = h_{1-\\alpha}(t),\\ 0<\\alpha<1$), we obtain the well-known result (\\cite{LucSri}):\n\nThe unique solution to the initial-value problem \n$$\n\\begin{cases}\n( D_{0+}^\\alpha\\, y)(t)  = \\lambda y(t)+f(t), \\ \\lambda \\in \\R,\\ t>0,\\\\\n\\left(I_{0+}^{1-\\alpha}\\, y\\right) (0) = b_0\n\\end{cases}\n$$\nis given by the formula\n$$\ny(t) = b_0\\, t^{\\alpha -1} E_{\\alpha, \\alpha}(\\lambda\\, t^\\alpha) + \\int_{0}^t \\tau^{\\alpha -1} E_{\\alpha, \\alpha}(\\lambda\\, \\tau^\\alpha)\\, f(t-\\tau)\\, d\\tau.\n$$\n\nLet us now consider a linear inhomogeneous multi-term fractional differential equation with the sequential GFDs \\eqref{GFDLn} of the Riemann-Liouville type and with the constant coefficients:\n\\begin{equation}\n\\label{eq-1-3}\n\\sum_{i=0}^n\\lambda_i(\\D_{(k)}^{<i>}\\, y)(t)  =  f(t), \\ \\lambda_i \\in \\R,\\ i=0,1,\\dots,n,\\ \\lambda_n \\not = 0,\\  t>0, \n\\end{equation} \nwhere the source function $f$ is represented in form of the  convolution series \\eqref{f}. \n\nAs  in the case of the single-term equation \\eqref{eq-1-2},  we look for solutions to the multi-term equation \\eqref{eq-1-3} in form of the convolution series \\eqref{sol-1-1}. First we determine the images of the convolution series \\eqref{sol-1-1} by action of the sequential GFDs $\\D_{(k)}^{<i>}$, $i=1,2,\\dots,n$. For $i=1$, the image is provided by the formula \\eqref{term1}. For $i=2,\\dots,n$, the formula \\eqref{term1} is applied iterative and we arrive at the following result:\n\\begin{equation}\n\\label{termi}\n( \\D_{(k)}^{<i>}\\, y)(t)   = \\sum_{j=0}^{+\\infty} b_{j+i}\\, \\kappa^{<j+1>}(t),\\ i=1,2,\\dots,n.\n\\end{equation}\nNow we substitute the convolution series \\eqref{sol-1-1}, its images by action of the sequential GFDs $\\D_{(k)}^{<i>}$, $i=1,2,\\dots,n$ provided by the formula \\eqref{termi}, and the convolution series \\eqref{f} for the source function into the equation \\eqref{eq-1-3} and arrive at the following identity:\n$$\n \\sum_{i=0}^n\\lambda_i\\, \\left(\\sum_{j=0}^{+\\infty} b_{j+i}\\, \\kappa^{<j+1>}(t)\\right)  =  \\sum_{j=0}^{+\\infty} a_{j}\\, \\kappa^{<j+1>}(t),\\ t>0.\n$$\nApplication of Theorem \\ref{eqconv} to the above identity leads to the following infinite triangular system of linear equations for the coefficients of the convolution series \\eqref{sol-1-1}:\n\\begin{equation}\n\\label{coef1-3}\n\\begin{cases}\n\\lambda_0\\, b_0 +\\lambda_1\\, b_1 +\\dots + \\ \\lambda_n\\, b_n = a_0,\\\\\n\\lambda_0\\, b_1 +\\lambda_1\\, b_2 +\\dots + \\ \\lambda_n\\, b_{n+1} = a_1,\\\\\n\\dots \\\\\n\\lambda_0\\, b_n +\\lambda_1\\, b_{n+1} +\\dots + \\ \\lambda_n\\, b_{2n} = a_n,\\\\\n\\lambda_0\\, b_{n+1} +\\lambda_1\\, b_{n+2} +\\dots + \\ \\lambda_n\\, b_{2n+1} = a_{n+1} \\\\\n\\dots\n\\end{cases}\n\\end{equation}\nIn this system, the first $n$ coefficients ($b_0,\\ b_1,\\dots,b_{n-1}$) can be chosen arbitrary and all other coefficients are determined step by step as solutions to the infinite triangular system \\eqref{coef1-3} of linear equations:\n\\begin{equation}\n\\label{coef1-4}\nb_{n+l}=(a_l - \\lambda_0\\, b_l- \\dots - \\lambda_{n-1} b_{n+l-1})\/\\lambda_n,\\ l=0,1,2,\\dots\n\\end{equation}\n\nWe thus proved the following result:\n\n\\begin{theorem}\n\\label{t-eq3}\nThe general solution to the inhomogeneous multi-term fractional differential equation \\eqref{eq-1-3} can be represented as the convolution series \\eqref{sol-1-1}, where the first $n$ coefficients ($b_0,\\ b_1,\\dots,b_{n-1}$) are arbitrary real constants and other coefficients are calculated according to the formula \\eqref{coef1-4}.\n\\end{theorem}\n\nThe constants $b_0,\\ b_1,\\dots,b_{n-1}$ in the general solution to the equation \\eqref{eq-1-3} presented in Theorem \\eqref{t-eq3}  can be determined based on the suitably posed initial conditions.  The form of these initial conditions is prescribed by Theorem \\ref{tgcTfn}. Indeed,  for a function $f\\in C_{-1,(k)}^{(n)}(0,+\\infty)$, the formula \\eqref{sFTLn} can be rewritten as follows: \n\\begin{equation}\n\\label{projn}\n(P\\, f)(t) = f(t) -  (\\I_{(\\kappa)}^{<n>}\\, \\D_{(k)}^{<n>}\\, f) (t) = \\sum_{j=0}^{n-1}\\left( \\I_{(k)}\\, \\D_{(k)}^{<j>}\\, f\\right)(0)\\kappa^{<j+1>}(t),\\  t>0,\n\\end{equation}\nwhere $P$ is the projector operator of the $n$-fold sequential GFD of the Riemann-Liouville type. Thus, to uniquely determine the constants $b_0,\\ b_1,\\dots,b_{n-1}$ in the general solution, the equation \\eqref{eq-1-3} has to be equipped with the initial conditions in the form\n\\begin{equation}\n\\label{icm}\n\\left( \\I_{(k)}\\, \\D_{(k)}^{<j>}\\, y\\right)(0) = b_j,\\ j=0,1,\\dots,n-1.\n\\end{equation}\n\nFinally, we mention that the inhomogeneous multi-term fractional differential equation of type \\eqref{eq-1-3} with the sequential Riemann-Liouville fractional derivatives (the case of the kernel $k(t)=h_{1-\\alpha}(t)$ in the equation \\eqref{eq-1-3}) was treated in \\cite{LucSri,Luc99} my using operational calculus of the Mikusi\\'nski type for the Riemann-Liouville fractional derivative. \n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn comparison to the metallic and semiconducting material, the thermoelectric effects are strongly suppressed in \nsuperconductors~\\cite{chandrasekhar2009,machon}. One of the reasons behind this is the interference of temperature \ndependent super current with the thermal current. The electron-hole symmetry present in the superconducting density \nof states (DOS) makes the opposite directional electron and hole thermo-currents (generated due to the thermal \ngradient) nullify each other~\\cite{ozaeta2014}. \n\nRecently, superconducting hybrid structures, especially ferromagnet-superconductor (FS) junctions, have attracted a \nlot of research interests due to the dramatic boosts of thermoelectric effects in them~\\cite{chandrasekhar2009,\nkalenkov2012theory,ozaeta2014,machon,machon2,kolenda2,kolenda2016}. \nInducing spin-triplet correlation within the superconductor and the asymmetric DOS profiles corresponding to the two spin \nsub-bands of the ferromagnet are the key features to be utilized in order to make such FS junction suitable in the context \nof thermal transport. The asymmetry in the two spin sub-bands according to the polarization of the ferromagnet can manipulate \nthe Andreev reflection (AR) which occurs when an incoming electron reflects back as a hole from the FS interface resulting \nin a cooper pair transmission into the superconductor within the sub-gapped regime~\\cite{andreev1}. Mixing of electron and \nhole-like excitations due to Andreev reflection may yield large electron-hole asymmetry. This asymmetry makes the expression \nof the thermoelectric coefficient to get rid of $E\/T_F$ factor which is responsible for the low value of the thermoelectric \ncoefficient in the normal state of the material~\\cite{abrikosov1988fundamentals}.\n\nIn order to investigate the thermoelectric properties of a material or hybrid junction it is customary to derive the thermal \nconductance (TC) or thermal current generated by the temperature gradient~\\cite{yokoyama2008heat,beiranvand2016spin,\nalomar2014thermoelectric}. Particularly in case of superconducting hybrid junction, the information of the superconducting gap \nparameter like its magnitude, pairing symmetry etc. can be extracted from the behavior of the thermal conductance~\\cite{yokoyama2008heat}. \nFrom the application perspective, it is more favorable to compute the Seebeck coefficient (SkC), known as \nthermopower, which is the open circuit voltage developed across the junction due to the electron flow caused by the thermal \ngradient~\\cite{blundell2009concepts}. Enhancement of \\sbk can pave the way of promising application to make an efficient \nheat-to-energy converter which may be a step forward to the fulfilment of the global demand of \nenergy~\\cite{alomar2014thermoelectric}. Since last few decades intense research is being carried out in search of newer and \nefficient energy harvesting devices that convert waste heat into electricity~\\cite{hwang2015large,snyder2008complex}. Usage \nof good thermoelectric material is one of the ways of making those devices more efficient. Now in order to determine how \ngood thermoelectric a system is, one can calculate \\sbk as well as the dimensionless parameter called figure of merit (FOM) \nwhich is naively the ratio of the power extracted from the device to the power we have to continually provide in order to \nmaintain the temperature difference~\\cite{sevinccli2010enhanced,zebarjadi2012perspectives}. It provides us an estimate of \nthe efficiency of a mesoscopic thermoelectric device like refrigerator, generator etc. based on thermoelectric \neffects~\\cite{giazotto2006opportunities}. Improving this thermoelectric \\fom with enhanced \\sbk so that the heat-electricity\nconversion is more efficient~\\cite{goldsmid3electronic,xu2014enhanced,liu2010enhancement,dragoman2007giant,ohta2007giant}\nis one of the greatest challenges in material science. Particularly, \nenhancement of the performance of any superconducting hybrid junction is much more challenging due to the above-mentioned reasons. \n\nThe prospects of FS junction, as far as thermoelectric property is concerned, depend on the new ingredients to manipulate \nthe spin dependent particle-hole symmetry. The latter has been implemented using external magnetic \nfield~\\cite{linder2016,ozaeta2014,bathen2016,kolenda2016,linder2007spin}, quantum dot at the junction~\\cite{hwang}, non-uniform \nexchange field~\\cite{alidoust2010phase}, phase modulation~\\cite{zhao2003phase}, magnetic impurities~\\cite{kalenkov2012theory} \nor internal properties like inverse proximity effect~\\cite{peltonen2010} etc. Recently, Machon {\\it et al. }~have considered simultaneous \neffects of spin splitting and spin polarized transport~\\cite{machon} in order to obtain enhanced thermoelectric effects in FS \nhybrid structure. In addition to these effects, presence of spin-orbit field~\\cite{linder2016,alomar2014thermoelectric} can \nplay a vital role in this context. \n\nStudy of interfacial spin-orbit coupling effect on transport phenomena has become a topic of intense research interest during \npast few decades due to the spin manipulation~\\cite{datta1990electronic}. Interplay of the polarization and the interfacial \nfield may lead to marked anisotropy in the junction electrical conductance~\\cite{hogl} and Josephson current~\\cite{costa16}. \nInterfacial spin-orbit field, especially Rashba spin-orbit field~\\cite{rashba,rashba2} arising due to the confinement potential \nat the semiconductor or superconductor hybrid structure, can also be the key ingredient behind such spin \nmanipulation~\\cite{sun2015general}. \n\nThe aspect of thermal transport in FS hybrid junction incorporating the role of interfacial spin-orbit interaction has not \nbeen studied in detail so far in case of ordinary ferromagnet. A few groups have performed their research in this direction \nin graphene~\\cite{alomar2014thermoelectric,beiranvand2016spin}. Motivated by these facts, in this article we study \nthermoelectric properties of a FS structure with Rashba spin-orbit interaction (RSOI)~\\cite{rashba,review} at the interfacial \nlayer. We employ Blonder-Tinkham-Klapwijk~\\cite{blonder} (BTK) formalism to compute the \\tc, \\sbk and \\fom therein. We \ninvestigate the role of RSOI on the thermoelectric properties. The interfacial scalar barrier at the FS interface reduces \n\\tc. On the other hand, the presence of RSOI at the FS interface can stimulate enhancement of \\tc driven by the thermal \ngradient across the junction. In order to reveal the local thermoelectric response we investigate the behavior of the \nthermopower with the polarization, temperature as well as the barrier strength. \\sbk is enhanced when the polarization of \nthe ferromagnet increases towards the half-metallic limit. In presence of finite barrier at the junction, it could be higher \neven for low polarization. Presence of RSOI at the interface may reduce or enhance it depending on the barrier strength, \ntemperature and the polarization. For higher barrier strength it always shows non-monotonic behavior with the temperature \nboth in presence and absence of RSOI. Similar non-monotonic behavior is obtained for \\fom with the rise of temperature and \nRashba strength. We predict that FOM can exceed the value $1$ with higher polarization of the ferromagnet. The magnitude \ncan even be more than $5$ for higher strength of barrier potential at the junction. It is also true in presence of weak \nRSOI. On the contrary, strong Rashba interaction can reduce it irrespective of the polarization and temperature. \n\nThe remainder of the paper is organized as follows. In Sec.~\\ref{modeltheory} we describe our model and the theoretical background. \nWe discuss our results for thermal conductance, thermopower and Figure of merit in Sec.~\\ref{result}. \nFinally, we summarise and conclude in Sec.~\\ref{conclu}.\n\n\\section{Model and theoretical background}\\label{modeltheory}\nWe consider a model comprising of a ferromagnet F ($z>0$) and a $s$-wave superconductor S ($z<0$) hybrid structure as shown in \nFig.~\\ref{geometry}. The flat interface of semi-infinite ferromagnet-superconductor (FS) junction located at $z=0$ is modelled \nby a $\\delta$-function potential with dimensionless barrier strength $Z$~\\cite{vzutic2000tunneling,blonder} and Rashba spin-orbit \ninteraction (RSOI) with strength $\\lambda_{rso}$. The FS junction can be described by the Bogoliubov-deGennes (BdG) \nequation~\\cite{de1999superconductivity} as,\n\\begin{eqnarray}\n\\begin{bmatrix}\n[\\hat{H}_e-\\mu]\\hat{\\sigma}_0 & \\hat{\\Delta}\\\\\n\\hat{\\Delta}^\\dagger & [\\mu-\\hat{H}_h]\\hat{\\sigma}_0 \n\\end{bmatrix} \\Psi(\\mathbf{r})=E \\Psi(\\mathbf{r})\n\\end{eqnarray}\nwhere the single-particle Hamiltonian for the electron is given by,\n\\beq\n\\hat{H_e}=-(\\hbar^2\/2m)\\nabla^2-(\\Delta_{xc}\/2) \\Theta(z) \\mathbf{m}.\\hat{\\mathbf{\\sigma}}+\\hat{H}_{int}.\n\\end{equation}\nSimilarly, for hole the Hamiltonian reads $\\hat{H}_h=\\hat{\\sigma}_2 \\hat{H}_e^*\\hat{\\sigma}_2$. The excitations of the electrons \nwith effective mass $m$ are measured with respect to the chemical potential $\\mu$. We set $m=1$ and $\\mu=0$ throughout our \ncalculation. The interfacial barrier is described by the Hamiltonian \n$\\hat{H}_{int}=(V d\\hat{\\sigma_0}+ \\mathbf{\\omega}\\cdot\\hat{\\sigma})\\delta(z)$~\\cite{hogl} with the \n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=6.5cm,height=4.6cm]{Fig1.pdf}\n\\caption{(Color online) Cartoon of the FS junction with the magnetization vector $\\mathbf{m}$. The dark red (dark grey) color is \nused to highlight the interfacial region of the FS hybrid structure. The F-region is kept at higher temperature ($T_F=T+\\delta T\/2$) \ncompared to the S-region ($T_S=T-\\delta T\/2$) in order to maintain a temperature gradient ($\\delta T=T_F-T_S$) across the junction.}\n\\label{geometry}\n\\end{center}\n\\end{figure}\nheight $V$, width $d$ and Rashba field $\\omega$ $=\\lambda[k_y,-k_x,0]$, $\\lambda$ being the effective strength of the RSOI. \nThe Stoner band model~\\cite{stoner1939collective}, characterized by exchange spin splitting $\\Delta_{xc}$, is employed to \ndescribe the F-region with the magnetization vector $\\mathbf{m}=[\\sin{\\theta}\\cos{\\phi},\\sin{\\theta}\\sin{\\phi},\\cos{\\theta}]$. \nHere $\\hat{\\sigma}$ is the Pauli spin matrix. Note that, the growth direction ($z$-axis) of the heterostructure is chosen \nalong [001] crystallographic axis~\\cite{matos2009angular}. The superconducting pairing potential is expressed as \n$\\hat{\\Delta}=\\Delta_s \\Theta(z)\\hat{\\sigma_0}$. We assume it to be a spatially independent positive constant following \nRef.~\\onlinecite{hogl}.\n\nDepending on the incoming electron energy there are four scattering processes possible at the FS interface. For electron \nwith a particular spin, say $\\sigma$, there can be normal reflection (NR), Andreev reflection (AR), tunneling as electron \nlike (TE) or hole like (TH) quasi-particles. In addition to these phenomena there may be spin-flip scattering processes \ndue to the interfacial spin-orbit field. Accordingly, we can have spin-flip counter parts of the above-mentioned four \nscattering processes namely, spin-flip NR (SNR), spin-flip AR (SAR), spin-flip TE (STE) and spin-flip TH \n(STH)~\\cite{de1995andreev,cao2004spin}. The above mentioned scattering processes are schematically displayed in \nFig.~\\ref{scattering} \n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=7.7cm,height=4.5cm]{Fig2.pdf}\n\\caption{(Color online) Schematic diagram for the quantum mechanical scattering processes taking place at FS interface. The solid and \nhollow spheres are used to denote electron (e) and hole (h), respectively. The letters `R (L)' indicates the right (left)-moving \nparticles. Corresponding spin states are denoted by `up' ($\\up$) and `down' ($\\dn$), respectively.}\n\\label{scattering}\n\\end{center}\n\\end{figure}\nfor a right-moving electron with spin $\\up$ (eRup). Note that, due to the possibility of spin-flip scattering processes in presence of\nRSOI at the FS interface, spin-triplet~\\cite{eschrig2011spin} superconducting correlation ($\\up\\up$ or $\\dn\\dn$) can be induced \nin addition to the conventional singlet pairing ($\\up\\dn$ or $\\dn\\up$)~\\cite{hogl}.\n\nThe solution of the \\bdg equations for the F-region, describing electrons and holes with spin $\\sigma$, can be written as~\\cite{hogl},\n\\begin{eqnarray}\n\\Psi_{\\sigma}^F(z)&=&\\frac{1}{\\sqrt{k_{\\sigma}^e}} e^{i k_{\\sigma}^e z}\\psi_{\\sigma}^e + r_{\\sigma,\\sigma}^e\ne^{-ik^e_{\\sigma}z}\\psi_{\\sigma}^e+r_{\\sigma,\\sigma}^h e^{ik^e_{\\sigma}z}\\psi_{\\sigma}^h \\non \\\\\n&&+r_{\\sigma,-\\sigma}^e e^{-ik^e_{-\\sigma}z}\\psi_{-\\sigma}^e \n+r_{\\sigma,-\\sigma}^e e^{ik^e_{-\\sigma}z}\\psi_{-\\sigma}^e\n\\label{enormal}\n\\end{eqnarray}\nwhere {\\small $k^{e(h)}_{\\sigma}=\\sqrt{k_F^2-k_{||}^2+2m(\\sigma\\Delta_{xc}\/2+(-) E)\/\\hbar^2}$} is the electron (hole)-like wave vector. \n$\\sigma$ may be $\\pm 1$ depending on whether the spin is parallel or anti-parallel to the vector $\\mathbf{m}$. $k_F$ and $k_{||}$ are \nthe Fermi and in-plane wave vector, respectively. The spinors for the electron-like and hole-like quasi-particles are respectively \n$\\psi_{\\sigma}^e=[\\psi_{\\sigma},0]^T$ and $\\psi_{\\sigma}^h=[0,\\psi_{\\sigma}]^T$ with \n{\\small $\\psi_{\\sigma}^T=[\\sigma\\sqrt{1+\\sigma\\cos{\\theta}}e^{-i \\phi},\\sqrt{1-\\sigma\\cos{\\theta}}]\/\\sqrt{2}$}.\nHere, $r^{e(h)}_{\\sigma,\\sigma^{\\prime}}$ corresponds to the amplitude of normal (Andreev) reflection from the FS interface. $\\sigma$ and \n$\\sigma^{\\prime}$ are the spin states for the incident and reflected electron or hole depending on the spin-conserving or spin-flipping \nprocess. Similarly, inside the superconducting region the solutions for the electron-like and hole like quasiparticles read~\\cite{hogl} \n\\begin{eqnarray}\n\\Psi_{\\sigma}^{S}=t^e_{\\sigma,\\sigma}\\left[\\begin{array}{c} u\n\\\\ 0 \\\\v\\\\0\\end{array} \\right]{e}^{iq_{e}z}+ t^e_{\\sigma,-\\sigma}\\left[\\begin{array}{c} 0\n\\\\ u \\\\0\\\\v\\end{array} \\right]{e}^{iq_{e}z}\\non \\\\\n+t^h_{\\sigma,\\sigma}\\left[\\begin{array}{c} u\n\\\\ 0 \\\\v\\\\0\\end{array} \\right]{e}^{-iq_{h}z}+ t^h_{\\sigma,-\\sigma}\\left[\\begin{array}{c} 0\n\\\\ u \\\\0\\\\v\\end{array} \\right]{e}^{-iq_{h}z},\n\\label{esupcon}\n\\end{eqnarray}\nwhere the $z$-components of the quasi-particle wave vectors can be expressed as, \n{\\small $q_{e(h)}=\\sqrt{q_F^2-k_{||}^2+(-)2 m \\sqrt{E^2-\\Delta^2}\/\\hbar^2}$} and the superconducting coherence factors are \n{\\small $u(v)=\\sqrt{[1\\pm\\sqrt{1-\\Delta^2\/E^{2}}]\/2}$}. We set the Fermi wave vector in both the F and S-regions to be the same \n\\ie $q_F=k_F$~\\cite{hogl}. Note that, we have written only the $z$ component of the wave functions. In the $x-y$ plane the wave \nvector is conserved giving rise to the planar wave function as,  $\\Psi_{\\sigma}(\\mathbf{r})=\\Psi_{\\sigma}(z) e^{i (k_{x} x+k_{y} y)}$ \nwhere $k_x$ and $k_y$ are the components of $k_{||}$. Here, $t_{\\sigma,\\sigma^{\\prime}}^{e(h)}$ denotes the amplitude of spin-conserving \nor spin-flipping transmitted electron (hole) like quasi-particles in the S region. We obtain the reflection and transmission amplitudes \nusing the boundary conditions as,\n\\begin{eqnarray}\n\\Psi^F_{\\sigma}|_{z=0^+}&=&\\Psi_{\\sigma}^S|_{z=0^-}~,\\non \\\\\n\\frac{\\hbar^2}{2m}\\left(\\frac{d}{dz}\\Psi_{\\sigma}^S|_{z=0^-}-\\frac{d}{dz}\\zeta\\Psi_{\\sigma}^F|_{z=0^+}\\right)\n&=&V d~\\zeta\\Psi_{\\sigma}^F|_{z=0^+} \\non \\\\\n+\\begin{bmatrix} \\mathbf{\\omega}.\\hat{\\sigma} & 0 \\\\\n                  0 & -\\mathbf{\\omega}.\\hat{\\sigma}\n\\end{bmatrix}\n\\Psi_{\\sigma}^F|_{z=0^+}\n\\end{eqnarray}\nwhere $\\zeta=diag(1,1,-1,-1)$. We describe our results in terms of the dimensionless barrier strength $Z=\\frac{V \\ d \\ m}{\\hbar^2 k_F}$, \nRSOI strength $\\lambda_{rso}=\\frac{2m\\lambda}{\\hbar^2}$ and spin polarization $P=\\frac{\\Delta_{xc}}{2 E_F}$.\\\\\n\nIn presence of thermal gradient across the junction with no applied bias voltage, the electronic contribution to the thermal conductance,\n(see appendix~\\ref{appndx1} for details), in terms of the scattering processes is given by~\\cite{blonder,yokoyama2008heat},\n\\begin{eqnarray}\n\\kappa&=&\\sum\\limits_{\\sigma}\\int\\limits_0^{\\infty}\\int\\limits_{s}\\frac{d^2k_{||}}{2 \\pi k_F^2}\\left[1-R^h_{\\sigma}-R^e_{\\sigma} \\right] \\non \\\\\n&&~~~~~~~~~~~~~\\left[\\frac{(E-E_F)^2}{T^2\\cosh^2{(\\frac{E-E_F}{2k_B T})}}\\right]dE \n\\label{kappa_form}\n\\end{eqnarray}\nwhere the NR and AR probability can be defined as \n$ R_{\\sigma}^{e(h)}(E,k_{||})=Re[k_{\\sigma}^{e(h)}|r_{\\sigma}^{e(h)}|^2+k_{-\\sigma}^{e(h)}|r_{-\\sigma}^{e(h)}|^2]$ satisfying the current conservation. \nHere, the integration with respect to $k_{||}$ is performed over the entire plane $x-y$ of the interface. It is convenient to define a dimensionless \nwave vector $k=k_{||}\/k_F$ and compute the integration in terms of it while calculating the TC. The Boltzmann constant is\ndenoted by $k_B$. $T$ is scaled by $T_c$, \nwhich is the critical temperature of the conventional singlet superconductor. \n\nWithin the linear response regime, we obtain the expression for the thermopower or \\sbk in unit of $k_B\/e$ as follows~\\cite{wysokinski2012thermoelectric},\n\\beq\nS =-\\left(\\frac{V}{\\delta T}\\right)_{I=0}=-\\frac{1}{T} \\frac{\\alpha}{G}\n\\label{sbk_exp}\n\\end{equation}\nwhere the thermoelectric coefficient $\\alpha$ and the electrical conductance $G$, in unit of $G_0$ ($e^2\/h$), are represented as,\n\\begin{eqnarray}\n\\alpha&=&\\sum\\limits_{\\sigma}\\int\\limits_0^{\\infty}\\int\\limits_{s}\\frac{d^2k_{||}}{2 \\pi k_F^2}\\left[1-R^h_{\\sigma}-R^e_{\\sigma} \\right] \\non \\\\\n&&~~~~~~~~~~~~~~~~~~\\left[\\frac{(E-E_F)}{T \\cosh^2(\\frac{E-E_F}{2 T})}\\right]dE  \n\\label{alphaform}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\nG&=&\\sum\\limits_{\\sigma}\\int\\limits_0^{\\infty}\\int\\limits_{s}\\frac{d^2k_{||}}{2 \\pi k_F^2}\\frac{\\left[1+R^h_{\\sigma}-R^e_{\\sigma}\\right]}\n{\\left[T\\cosh^2{(\\frac{E-E_F}{2 T})}\\right]}dE .\n\\label{intG}\n\\end{eqnarray}\nHere $\\alpha$ is expressed in unit of $G_0 k_B T\/e$ ($\\equiv k_B e T \/h$). In terms of SkC, electrical conductance and thermal \nconductance, the \\fom $zT$ is given by,\n\\begin{eqnarray}\nzT=\\frac{S^2 G T}{K}\n\\label{merit}\n\\end{eqnarray}\nwhere $K=\\kappa-\\frac{\\alpha^2}{TG}$ is expressed in unit of $\\kappa_0$ ($\\equiv k_B^2T\/h$). After applying the temperature difference \nbetween the two sides of the junction we obtain thermal current which essentially develops a voltage difference between them following \nthe Peltier effect. This causes a correction to the thermal conductance as well. We consider such correction while defining the \\fom of \nthe system as every material manifesting Seebeck effect must exhibit the Peltier effect~\\cite{bardas1995peltier}.\n\n\\section{Results and Discussion}\\label{result}\nIn this section we present our numerical results for \\tc, \\sbk and \\fom of the ferromagnet-superconductor junction, both in absence \nand presence of interfacial RSOI, in three different sub-sections. We discuss our results in terms of the scattering processes \nthat occur at the interface of the FS hybrid structure and various parameters of the system. \n\\subsection{Thermal conductance}\nIn this subsection we discuss the effect of polarization and RSOI, both in absence and presence of finite scalar barrier, on the behavior of \nthe \\tc throughout the temperature regime from low to high.\n\\subsubsection{Effect of polarization and barrier in absence of RSOI}\nIn Fig.~\\ref{cond_T} we show the variation of \\tc $\\kappa$ as a function of temperature $T\/T_c$ in absence of RSOI \nfor various polarization strength $P$ of the ferromagnet, starting from the unpolarized ($P=0$) towards the half-metallic ($P=0.9$) \nlimit. Fig.~\\ref{cond_T}[(a), (b), (c) and (d)] correspond to the interfacial scalar barrier strength $Z=0$, $1$, $2$ and $4$, respectively. \nFrom all the four figures it is apparent that \\tc increases exponentially with temperature. This behavior, being independent of the barrier \nstrength, is true for conventional normal metal-superconductor junction ($P=0$) as well as for any finite value of polarization ($P\\ne 0$) \nof the ferromagnet. The fully developed gap of the superconductor is responsible for the exponential increase of the thermal \nconductance~\\cite{andreev,andreev1} \nWith the increase of temperature, the superconducting gap decreases resulting in reduction of AR amplitude and simultaneous increase of \ntunneling as electron-like quasi-particles. Thermal resistance of the superconductor falls off exponentially as the temperature is \nincreased~\\cite{andreev}. As a consequence, $\\kappa$ rises with the temperature following an exponential nature. \n\nHowever, the rate of increase of $\\kappa$ completely depends on the polarization $P$ of the ferromagnet. Gradual tunability of the \npolarization $P$ does not ensure any monotonic \n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=8.7cm,height=7.5cm]{Fig3.pdf}\n\\caption{(Color online) The behavior of thermal conductance ($\\kappa$), in unit of $k_B^2\/h$, is shown as a function of temperature \n($T\/T_c$) in absence of RSOI ($\\lambda_{rso}=0$) for different values of barrier strength ($Z$) and polarization ($P$) of the ferromagnet.}\n\\label{cond_T}\n\\end{center}\n\\end{figure}\nbehavior of the TC. It depends on both the temperature and barrier strength. To illustrate this, we discuss the scenarios for \ndifferent values of $Z$ one by one. When $Z=0$, the rate of increase of $\\kappa$ is very slow with the increase of polarization for \na particular value of $T\/T_c$ (see Fig.~\\ref{cond_T}(a)). This is true as long as $T\/T_c<0.3$. On the other hand, for $T\/T_c>0.3$ the scenario \nbecomes opposite \\ie~$\\kappa$ starts decreasing with the change of polarization for a fixed $T\/T_c$. There is a cross-over temperature \n$T_x$ ($\\sim 0.3$ in this case) separating the two different behaviors of the \\tc with polarization. We explain this phenomenon as \nfollows. For very low $T\/T_c$ \\ie~$T<T_x$, superconductor gap parameter does not change by appreciable amount. In this situation, \nincrease of polarization causes reduction of AR due to minority spin sub-band. This results in enhancement of \\tc. Such enhancement is \nmaximum towards the half-metallic limit ($P=0.9$) when AR vanishes due to the absence of minority spin-band. After a certain cross-over \ntemperature, the gap decreases significantly with $T\/T_c$ and the tunneling increases accordingly as long as $T \\simeq T_c$. On top \nof that, if we increase the polarization, tunneling due to the minor spin band decreases leaving the major spin band contribution \nunchanged. As a whole, the change of behaviors of all the scattering processes results in reduction of \\tc with polarization in the \nhigh temperature regime ($T>T_x$). \n\nNow let us consider finite $Z$ at the interface. In presence of the barrier, incident electrons encounter NR along with AR from the \ninterface. NR reduces $\\kappa$. Hence, the higher is the barrier strength $Z$, the lower is $\\kappa$ for a particular temperature and \npolarization. This is apparent by comparing all the four figures of Fig.~\\ref{cond_T}. The cross-over temperature $T_x$, separating the \nbehaviors of the \\tc with $P$, decreases as soon as we consider finite $Z$ as depicted in Fig.~\\ref{cond_T}(b). It becomes $\\sim$ 0.2 for \n$Z=1$. However, $T_x$ translates towards the high temperature limit with the increase of barrier strength (see Fig.~\\ref{cond_T}(c) and (d)). \nFor low $Z$, \\tc does not change by appreciable amount with the increase of $P$ because of NR. In the low temperature regime, enhancement \nof $P$ causes reduction of AR. This does not ensure the increase of \\tc as tunneling decreases due to the reflection from the interface. \nAs $Z$ is enhanced, NR starts dominating over the other processes. This not only causes reduction of \\tc but also translates $T_x$ towards \nthe high temperature regime. For example, $T_x \\sim 0.5$ (see Fig.~\\ref{cond_T}(c)) and $0.8$ (see Fig.~\\ref{cond_T}(d)) for $Z=2$ and \n$Z=4$, respectively. More over, there is a tendency of saturation of $\\kappa$ when $T\\rightarrow T_c$ irrespective of $P$ for higher barrier \nstrength associated with very small change of $\\kappa$ with $P$. For higher $Z$, AR, TE and TH are dominated by NR. Therefore, tuning \npolarization does not cause appreciable variation in the tunneling as well as AR resulting in very small change of \\tc leading towards \nits saturation.\n\nTherefore, the effect of polarization of the ferromagnet cannot be uniquely determined. The behavior of \\tc with \npolarization changes depending on the temperature and the barrier strength as well.\n\nSo far, we have not discussed about the orientation of the magnetization. We present all of our results for $\\theta=\\pi\/2$ and $\\phi=0$. \nVery recently, H\\\"{o}gl {\\it et al. }~have revealed the fact that electronic conductance shows an anisotropy with the rotation of the magnetization \n$\\mathbf{m}$~\\cite{hogl}. However, in case of thermal transport, contributions from all the energy values are taken into consideration. \nTherefore, with the change of $\\mathbf{m}$ there is no appreciable change in \\tc as all contributions due to different \norientations of $\\mathbf{m}$ are averaged out during integration over the energy (see Eq.(\\ref{kappa_form})). This fact remains unchanged \nfor any temperature ($T<T_c$) and polarization $P$.\n\\subsubsection{Interplay of polarization, barrier and interfacial RSOI}\nHere we incorporate RSOI at the interface of the FS junction. The behavior of the \\tc with the polarization and temperature in presence \nof RSOI remain qualitatively similar to that in absence of RSOI.  We refer to Appendix~\\ref{appndx2} for more details manifesting this qualitative \nsimilarity. On the other hand, the magnitude of $\\kappa$ may increase or decrease in presence of RSOI. It completely depends on the \nstrength of RSOI as well as the strength of the barrier at the interface. \n\nFor illustration, we present in Fig.~\\ref{cond_R} the behavior of \\tc as a function of the RSOI strength for a particular temperature \n$T\/T_c=0.7$. Here, (a), (b), (c) and (d) represent different barrier strength, $Z=0$, $1$, $2$ and $3$, respectively. In absence of scalar \nbarrier ($Z=0$), when we increase the RSOI strength \\tc monotonically decreases irrespective of the polarization (see Fig.~\\ref{cond_R}(a)). \nFerromagnet-superconductor structure with $P=0$ is exactly equivalent to a conventional normal metal-superconductor \njunction~\\cite{chandrasekhar2009} for which the sub-gapped contribution to $\\kappa$ is zero. The reason is that within the sub-gapped regime \ntotal AR probability is exactly equal to $1$ \\ie~$R_{\\sigma}^h=1$ with the reflection probability $R_{\\sigma}^e=0$ as $Z=0$~\\cite{blonder}. \nTherefore, the sub-gap contribution to the \\tc is zero for this particular case. This is also evident from Eq.(\\ref{kappa_form}). Only \nquasi-particle tunneling contributes to the \\tc above the gap. Nevertheless, in presence of finite polarization of the ferromagnet the \nsub-gap contribution is finite. \nWith the inclusion of interfacial RSOI, the spin-flip counter-parts of both types of scattering processes, AR and tunneling as electron or \nhole like quasi-particles, occur at the interface in addition to the spin-conserving processes. The relative amplitudes corresponding to the spin-conserving and \nspin-flipping processes will be completely determined by the strength of RSOI. As we increase $\\lambda_{rso}$ spin-flip counter part starts \ndominating. Hence, SAR becomes higher in magnitude with the enhancement of $\\lambda_{rso}$. With the increase of SAR, $\\kappa$ decreases. \nSpin-flip counter parts of NR will increase in this case but its effect on \\tc is less dominating compared to AR. As a consequence we \nget monotonically decreasing behavior of $\\kappa$. Now with a particular RSOI strength, increasing polarization means reduction of AR due \nto the minority spin sub-band. SAR probability increases keeping the total AR probability $1$ and maintaining the usual result for the \nsub-gapped energy regime whereas tunneling corresponding to the minor spin sub-band reduces. This effectively reduces $\\kappa$ by considerable \namount as during the evaluation of $\\kappa$,  all the electrons corresponding to all the energy values are taken into account. \n\nComparing the magnitudes of $\\kappa$ as shown in four figures of Fig.~\\ref{cond_R}, it is evident that introduction of finite barrier $Z$ \neffectively reduces TC, being independent of the value of RSOI strength, due to the finite NR from the barrier.\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=8.7cm,height=7.3cm]{Fig5.pdf}\n\\caption{The behavior of thermal conductance ($\\kappa$), in unit of $k_B^2\/h$, is displayed as a function of RSOI strength $\\lambda_{rso}$ \nfor a particular temperature ($T\/T_c=0.7$) and different values of the barrier strength ($Z$) and polarization ($P$) of the ferromagnet.}\n\\label{cond_R}\n\\end{center}\n\\end{figure}\nIn presence of barrier when we increase RSOI, initially $\\kappa$ does not show any appreciable change as long as $\\lambda_{rso}$ is very \nsmall compared to $Z$. If we gradually increase RSOI strength, \\tc exhibits non-monotonic behavior. $\\kappa$ increases gradually, attains \na maxima and then again decreases with the increase of $\\lambda_{rso}$ as depicted in Fig.~\\ref{cond_R}(b). This is one of the main results\nof the present article. The reason behind this non-monotonic behavior can be attributed to the interplay of all the six scattering processes \noccurring at the interface of FS structure. With the increase of $\\lambda_{rso}$, probabilities of all the spin-flip scattering processes \nincrease. However, for a low barrier strength ($Z=1$) SNR cannot dominate $\\kappa$ significantly compared to the spin-flip tunneling. \nInterplay of these processes results in enhancement of $\\kappa$ with the rise of $\\lambda_{rso}$ accordingly. For sufficiently higher \n$\\lambda_{rso}$ all the spin-flip processes start dominating over the spin-conserving processes. After a certain enhancement of RSOI, the \nprobability of spin-flip scattering processes do not dominate further. Instead the spin-conserving scattering probabilities decrease with \nthe increase of $\\lambda_{rso}$. As a consequence, TC decreases. Behavior of $\\kappa$ with the change of polarization remains monotonically \ndecreasing similar to the case before.\n\nIf we further increase $Z$, $\\kappa$ maintains its non-monotonic behavior with the rise of RSOI strength. However, the maxima moves towards \nthe higher value of $\\lambda_{rso}$. To illustrate this, we refer to Fig.~\\ref{cond_R}(c) and (d). For higher values of $Z$, magnitudes of \n$\\kappa$ decreases as SNR and NR starts dominating. In this situation, to obtain the maxima of $\\kappa$ we have to tune RSOI strength\naccordingly. This results in shifting of the peaks of $\\kappa$ towards higher $\\lambda_{rso}$. \n\nNote that, we present our results of $\\kappa$ as a function of $\\lambda_{rso}$ for $T\/T_c=0.7$. If we consider the low temperature regime, \nparticularly lower than the cross-over value $T_{x}$, we expect similar non-monotonic behavior of the \\tc with RSOI strength. There will\nbe only one change in the nature of $\\kappa$. Following the discussion in the previous subsection, at a particular RSOI strength $\\kappa$ \nwill rise with the increase of $P$ when temperature is lower than the cross-over value. However, for very low temperature such as \n$T\/T_c \\sim 0.1$, the amount of change in magnitude of the \\tc is vanishingly small.\n\\subsection{Seebeck coefficient}\nIn this sub-section we study the phenomenon of Seebeck effect which is a direct measure of the local thermoelectric response. The behavior \nof \\sbk as a function of temperature and RSOI for different values of $P$ and $Z$ are presented. All the results, we have presented here, \n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=8.7cm,height=7.3cm]{Fig6.pdf}\n\\caption{(Color online) The behavior of Seebeck coefficient ($S$), in unit of $k_B\/e$, is shown as a function of temperature ($T\/T_c$) in \nabsence of RSOI ($\\lambda_{rso}=0$) for different values of the barrier strength ($Z$) and polarization ($P$) of the ferromagnet.}\n\\label{sbk_T}\n\\end{center}\n\\end{figure}\ncorrespond to the particular orientation of the magnetization vector as mentioned in the previous sub-section. We divide our discussion \nin two different parts in order to highlight the effect of polarization of the ferromagnet and the effect of the interfacial RSOI on \nthermopower as follows.\n\\subsubsection{Effect of polarization and barrier in absence of RSOI}\nIn Fig.~\\ref{sbk_T}[(a), (b), (c) and (d)], we show the behavior of $S$ as a function of $T\/T_c$ corresponding to the scalar barrier potential \n$Z=0$, $1$, $2$ and $4$, respectively. In all of the four figures we observe that \\sbk is negative throughout the window for all values of the \npolarization of the ferromagnet irrespective of the value of $Z$. It is solely due to the contributions arising from the electrons. So, we \ndiscuss only the magnitude of the \\sbk throughout the rest of the manuscript. \\sbk increases with $T\/T_c$ in absence of the barrier \\ie $Z=0$. \nThis phenomenon is true for all values of the polarization $P$. On the other hand, for a particular value of temperature, it increases with the enhancement\nof $P$. We can explain this phenomenon as follows. For $Z=0$, the probability of NR is zero. With the increase of temperature, superconducting \ngap decreases minimizing the phenomenon of AR to occur. This results in enhancement of transmission of energy with temperature. It is evident \nfrom the expression of \\sbk (see Eq.(\\ref{sbk_exp})). Hence, we can say that reduction of $R^h_{\\sigma}$ causes enhancement of the magnitude of \nthe numerator in Eq.(\\ref{sbk_exp}). The latter essentially describes the thermal coefficient. At the same time denominator or the electronic \nconductance decreases. This effectively enhances \\sbk magnitude. In fact, for the sub-gapped energy regime we always have zero thermal \ncoefficient resulting in vanishing contribution to the \\sbk. There are finite contributions arising only from the energy regime above the gap.\n\nNow we consider barrier at the junction ($Z \\ne 0$) and investigate how the thermopower behaves with the change of polarization in presence \nof finite barrier. In presence of finite $Z$, thermopower increases monotonically for low values of $P$. The amount of enhancement is higher \nfor the higher values of $P$. However there is a smooth transition from monotonic to non-monotonic behavior of \\sbk with temperature as the \npolarization tends to the half-metallic limit \\ie $P=0.9$. It initially rises with $T\/T_c$ and then decreases in high temperature regime with \nan extremum in the Seebeck profile. According to the definition we can describe this feature of \\sbk in terms of $\\alpha$ and $G$. In presence \nof finite $Z$, NR reduces both $\\alpha$ and $G$. Now change in \\sbk occurs depending on their relative magnitudes. As a consequence, $S$ \nincreases in presence of finite $Z$ in comparison to the case for $Z=0$ (comparing Figs.~\\ref{sbk_T}(a) and (b)). Moreover, there is a tendency \nof saturation of the thermopower magnitude in the higher temperature regime irrespective of the values of $P$. For $Z=1$, it is more clear for \nhigher values of $P$. For sufficiently higher temperature the gap reduces significantly resulting in enhancement of conductance. However, for \nhigher polarization, $G$ does not increase with $T\/T_c$ by appreciable amount after a certain temperature due to the absence of minority spin \nband. As a consequence, \\sbk decreases resulting in a non-monotonic behavior especially for higher polarization. Such non-monotonic nature is \nmuch more pronounced for higher values of $Z$ (see Fig.~\\ref{sbk_T}(c) and (d)). Additionally, the saturation regime is achieved faster for higher $Z$ \nwith the enhanced NR. For higher barrier strength, \\sbk changes by very small amount with $Z$ for a particular $P$ and $T\/T_c$ due to the \nsuppression of all other scattering processes by NR.\n\nNote that, our results does not imply that the \\sbk approaches unity in the limit of zero RSOI and zero polarization. All these values are as small as in agreement with those found in the existing literature. We present a quantitative comparison in Appendix~\\ref{appndx3}.\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=8.7cm,height=7.2cm]{Fig8.pdf}\n\\caption{(Color online) The behavior of Seebeck coefficient ($S$), in unit of $k_B\/e$, at temperature $T\/T_c=0.5$, is shown as a function of RSOI \nstrength ($\\lambda_{rso}$) for different values of barrier strength ($Z$) and polarization ($P$) of the ferromagnet.}\n\\label{sbk_R}\n\\end{center}\n\\end{figure}\n\n\\subsubsection{Interplay of polarization, barrier and interfacial RSOI}\nIn this sub-subsection, we investigate what happens to \\sbk when RSOI is taken into account at the interfacial layer, both in absence and presence \nof finite scalar barrier at the junction. To understand the role of RSOI, we illustrate the behavior of \\sbk ($S$) with respect to RSOI ($\\lambda_{rso}$) \nfor a fixed value of temperature $T\/T_c=0.5$ in Fig.~\\ref{sbk_R}[(a), (b), (c) and (d)] for four different values of the barrier strength \nas before. For $Z=0$, behavior of \\sbk with $\\lambda_{rso}$ are different for different $P$. For zero or lower values of polarization, \nit rises with the increase of RSOI strength. The rate of increase, however, changes when $P$ becomes high. There is a transition from \nincreasing to decreasing nature of $S$ vs. $\\lambda_{rso}$ curves with the increase of polarization. As soon as we incorporate $Z$, \nthe situation changes dramatically. \\sbk sharply falls with the temperature in presence of barrier strength due to the large boosts of NR \nand SNR scattering phenomenon. The rate of decrease of \\sbk becomes lower with the increase of barrier strength. With very high RSOI it becomes\nsaturated with the saturation region shifting towards the higher RSOI strength as one increases the barrier strength $Z$.  This is clear by \ncomparing Figs.~\\ref{sbk_R}[(b)-(d)]. Most interestingly, \n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=8.7cm,height=7.5cm]{Fig9.pdf}\n\\caption{(Color online) The feature of Figure of merit ($zT$) is depicted as a function of temperature ($T\/T_c$) in absence of RSOI \n($\\lambda_{rso}=0$) for different values of barrier strength ($Z$) and polarization ($P$) of the ferromagnet.}\n\\label{fom_T}\n\\end{center}\n\\end{figure}\nwe observe that for finite and low value of $Z$, \\sbk always increases with polarization for all values of $\\lambda_{rso}$. On the contrary, \nfor higher $Z$ ({\\it e.g., } $Z=4$) we obtain exactly opposite scenario where $S$ decreases with the increase of $P$. There is a transition of the \nbehavior of the \\sbk with the polarization at some certain strength of $Z$. However, it is always possible to obtain large thermopower in \npresence of low but finite RSOI strength for all values of polarization. Note that, the behavior of $S$ with temperature remains quite\nsimilar even in presence of fixed RSOI. For detailed discussion with more plots see Appendix~\\ref{appndx2}.\n\n\nThroughout our manuscript, we have focused on the low temperature regime for which our results are well justified. For higher temperature, close to or above $T_c$, \none has to take the microscopic details of the self-consistency of the superconducting gap parameter into account which we have neglected in our toy model. \n\\subsection{Figure of merit ($zT$)}\nIn order to understand the efficiency of our FS junction as a thermoelectric material, we explore the behavior of \\fom \nboth in presence and absence of RSOI at the interface. \n\n\\subsubsection{Effect of polarization and barrier in absence of RSOI}\nWe discuss the role of polarization when we vary the FOM with temperature in absence of RSOI at the junction \\ie $\\lambda_{rso}=0$ as follows. \nIn Fig.~\\ref{fom_T} we display the variation of $zT$ with respect to $T\/T_c$ for various polarizations and barrier strengths. Here, (a), (b), \n(c) and (d) represent the same parameter values as mentioned in the previous cases. The value of $zT$ is very small for $P=0$ and in the \nabsence of barrier (see Fig.~\\ref{fom_T}(a)). $zT$ is exactly zero if we consider only energy values within the subgapped regime. \nHowever, it increases with the increase of polarization as well as temperature. We can obtain FOM $\\sim1.3$ when the polarization of the ferromagnet \ntends to half-metallic limit ($P=0.9$). The rate of increase of $zT$ with $P$, for a particular temperature, rises with the increase of polarization of the ferromagnet. \nIn general, FOM can be defined in terms of the three parameters namely thermal coefficient, \\sbk and electronic conductance (see Eq.(\\ref{merit})). \nThe nature of \\fomd, whether it is increasing or decreasing, completely depends on their relative magnitudes. Similar enhancement of $zT$ has also been \nproposed in literature by tuning exchange field~\\cite{machon}. Search for FOM more than $1$, in order to have an efficient thermoelectric material, has been always \nremained in the focus of material science. In our case FOM can be enhanced not only by tuning the polarization $P$ but also setting \n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=8.7cm,height=7.5cm]{Fig11.pdf}\n\\caption{(Color online) The behavior of Figure of merit ($zT$) is illustrated as a function of RSOI strength ($\\lambda_{rso}$) at $T\/T_c=0.5$ \nfor different values of the barrier strength ($Z$) and polarization ($P$) of the ferromagnet.}\n\\label{fom_R}\n\\end{center}\n\\end{figure}\nfinite barrier potential at the junction as shown in Figs.~\\ref{fom_T}[(b)-(d)]. For higher values of $Z$, the behavior of $zT$ becomes \nnon-monotonic. It initially rises with the temperature and after a certain temperature it decreases. Remarkably, unlike \\sbk, the point of \nmaxima does not shift by appreciable amount with the change of barrier strength. More over, the sensitivity to the polarization reduces \nfor higher strength of the barrier potential. \n\nNote that, in the limit of $\\lambda_{rso}=0$, $Z=0$ and $P=0$, a comparison of our calculated values of \\fom with the existing literature can be found \nin Appendix~\\ref{appndx3}.\n\n\\subsubsection{Interplay of polarization, barrier and interfacial RSOI}\nIn presence of finite interfacial RSOI, the behavior of $zT$ with temperature is quite similar to that in absence of Rashba. A discussion on \nthis issue can be found in Appendix~\\ref{appndx2}. However, the magnitude of $zT$ depends on the strength of RSOI both in absence and presence \nof $Z$. We demonstrate the behavior of $zT$ as a function of $\\lambda_{rso}$ in Fig.~\\ref{fom_R} where (a)-(d) correspond to the same values of \n$Z$ as in the previous figures. \n\nIn absence of any barrier ($Z=0$) and for low polarization of the ferromagnet, $zT$ rises with the enhancement of $\\lambda_{rso}$ almost linearly\nas shown in Fig.~\\ref{fom_R}(a). Such linear behavior changes when the polarization of the ferromagnet is considered in the limit of half-metal, \n\\ie $P=0.9$. For a particular choice of $\\lambda_{rso}$, \\fom increases with the increase of polarization too as in absence of RSOI (see \nAppendix~\\ref{appndx2} for more details). This scenario reverses its character in presence of low $Z$. It is clear from Fig.~\\ref{fom_R}(b). Under such \ncircumstances, $zT$ falls off rapidly with the increase of $\\lambda_{rso}$. As mentioned earlier, presence of RSOI induces \nspin-flip scattering process which reduces the \\sbk and enhances \\tc at the same time. The electronic conductance increases by \nsmall amount as shown in Ref.~\\onlinecite{hogl}. When we further increase $Z$, the behavior of $zT$ changes slowly with the RSOI strength. Depending \non the value of $Z$, (see Figs.~\\ref{fom_R}(c) and (d)) \\fom can even reach the value $\\sim 5$ by using half-metal for weak or moderate RSOI. After a \ncritical Rashba strength it falls off rapidly and becomes vanishingly small. However, the critical value of Rashba strength translates towards the \nhigher regime with the increase of $Z$.\n\nThe main result of our article is that we have obtained large values of the thermoelectric coefficient in presence of a scalar barrier and\/or \nRashba spin-orbit interaction (RSOI) at the interface when polarization of the ferromagnet tends towards the half-metallic limit. Note that, \nthe maximum value of $zT$ appears at a temperature scale which is well below the critical temperature of the superconductor. The physical reason \nbehind such large values of the thermoelectric coefficients can be attributed to the interplay between scalar barrier strength, RSOI strength at \nthe interface, polarization of the ferromagnet and quasiparticle tunneling, either spin-conserving or spin-flipping, through the interfacial \nbarrier. Enhancement of thermoelectric coefficient due to the quasiparticle tunneling has been already established in literature but they \ninvolve either spin-split superconductor or spin-active interface~\\cite{machon} corresponding to spin-dependent phase-shift~\\cite{seviour2000giant}. \nIn our case, the significant enhancement of the thermoelectric effect lies in the quasi-particle tunneling through the barrier breaking the \nspin-symmetry by Rashba spin-orbit interaction.\n\nTherefore, we can say that \\fom can be enhanced by increasing any one of the parameters, \\ie polarization, barrier strength and temperature keeping \nthe other two fixed both in absence and presence of RSOI at the interfacial layer of our FS junction. Note that, all these phenomena are true for weak or moderate Rashba interaction.\n\n\\section{Summary and Conclusions} \\label{conclu}\nTo summarise, in this article, we have explored thermoelectric properties of a ferromagnet-superconductor junction driven by a thermal gradient. \nAt the interfacial layer, we have incorporated RSOI along with scalar potential barrier. We have analyzed our results for the thermal conductance,\nSeebeck coefficient and figure of merit as a function of temperature, polarization of the ferromagnet and RSOI strength. Thermal conductance \n$\\kappa$ exhibits exponential rise with temperature following the fully developed gap feature. We have observed a cross-over temperature $T_{x}$ \nseparating the two different regimes corresponding to opposite behaviors of $\\kappa$ with the polarization of the ferromagnet. Below the \ncross-over temperature, $\\kappa$ increases with the increase of polarization, whereas for all the temperatures above the cross-over temperature \nit decreases with polarization. This phenomenon is true both in absence and presence of barrier strength. With the increase of barrier strength \nthe cross-over temperature moves towards the higher temperature value. Inclusion of interfacial Rashba spin-orbit field causes reduction of \nthermal conductance due to the appearance of spin-flipped SAR process, in absence of any barrier. On the other hand, interfacial RSOI can enhance \nthe \\tc in presence of finite barrier due to the interplay of both of them. We have obtained a non-monotonic behavior of the thermal conductance \n$\\kappa$ as we vary the interfacial RSOI strength. The maxima of $\\kappa$ moves towards the critical temperature $T_c$ with the enhancement of \nbarrier strength $Z$. Throughout the manuscript we have considered only the electronic contribution to the \\tc neglecting the contribution due \nto phonons which is a valid approximation for very small temperature gradient.\n\nWe have investigated the Seebeck coefficient in absence and presence of interfacial RSOI and scalar barrier. It can be enhanced by using \nferromagnet with higher polarization and by increasing the temperature (below the critical temperature $T_{c}$) as well. A non-linear behavior \nof the \\sbk with the temperature can be obtained in presence of barrier potential at the junction. Whereas presence of RSOI can reduce or enhance \nit depending on the barrier strength. For low barrier strength, \\sbk increases with the increase of polarization. However, an exactly opposite \nbehavior is observed when barrier strength becomes high. \n\nTo quantify the thermoelectric power efficiency of the FS structure, we compute the figure of merit $zT$ both in presence and absence \nof RSOI and scalar potential barrier. We show that the \\fom can be enhanced to more than $1$ by setting the polarization of the ferromagnet very \nhigh (towards the half-metallic limit). It can further be enhanced by considering finite potential barrier at the junction. In particular, the \nhigher the barrier strength the higher the \\fom turns out to be while inclusion of RSOI can reduce it for low barrier strength. For\nhigher barrier strength, value of $zT$ can again be more than $1$ ($zT\\sim 4-5$) depending on the polarization of the ferromagnet. These phenomena are \nvalid for weak Rashba spin-orbit interaction. Presence of strong RSOI can highly reduce it irrespective of the strength of the barrier potential.\n\nAs far as the practical realization of our model is concerned, it may be possible to fabricate such a FS hybrid structure by growing thin layers \nof a spin singlet superconductor (e.g. $\\rm Nb$ material) and a ferromagnetic insulator (EuO)~\\cite{matthias1961ferromagnetic,tokuyasu1988proximity} \non top of each other. To design the interfacial layer, responsible for the spin-orbit field, one can use a thin layer of \nzinc-blende semiconductor~\\cite{deng2012anomalous}. An additional gate \nvoltage can be implemented to create the scalar potential barrier at the interface. The magnetization vector of the ferromagnet can be rotated \nby some external magnetic field. Additional effects of such external field can be avoided by using dysprosium magnets~\\cite{hogl}. \nFrom an experimental point of view, the polarization of the ferromagnet can be a more controllable parameter than the interfacial \nRSOI strength. For a typical $s$-wave superconductor, for {\\it e.g., } $\\rm Nb$ with $T_{c}=9.3~\\rm{K}~(\\Delta_{s}\\sim 3 \\rm~{meV})$, to attain $zT\\sim 4.5$ \none needs a ferromagnetic exchange coupling $\\Delta_{xc}\\sim 0.6 \\rm~{eV}$~\\cite{Steeneken2002,Tiffany2004}, temperature $T\\sim 4 \\rm~{K}$, \ninterfacial Rashba parameter $\\sim 60-80 \\rm~{meV \\cdot \\AA{}}$~\\cite{manchon3}. A scalar potential barrier with height $V\\sim 0.5 \\rm~{eV}$\nand width $d \\sim 2-4 \\rm~{nm}$~\\cite{Huang2011} is also required. \n\nIn conclusion, we have shown the possibility of obtaining enhanced thermoelectric properties of a FS hybrid structure in presence of finite barrier \nand interfacial RSOI. In the context of thermal transport, the idea of exploring interfacial RSOI in FS junction is unique. The enhancement of the \nthermoelectric properties caused by the interplay of RSOI along with potential barrier at the interface of the heterostructure and the polarization \nof the ferromagnet elevate the potential of the FS structure as a thermoelectric. The proposed set-up, within the experimentally achievable parameter \nregime, may be utilized in thermoelectric devices. \n\n\n\\vspace{.2cm}\n\\acknowledgments{PD thanks Department of Science and Technology (DST), India for the financial support through SERB NPDF \n(File no. PDF\/2016\/001178). We acknowledge S. D. Mahanti for helpful discussions and encouragement. \nAMJ also thanks DST, India for financial support.}\n\n\n\\begin{appendix}\n\\section{Outline of derivation of the thermoelectric coefficients}\n\\label{appndx1}\n\nIn presence of either a small bias or small temperature gradient across the junction, the linear response for the charge \nand heat currents can be expressed following the Onsager matrix equation~\\cite{onsager1931reciprocal1,onsager1931reciprocal2} given as,\n\\begin{eqnarray}\n\\left( \\begin{array}{c}\nI \\\\\nI_Q \n\\end{array}\\right)\n&=&\n\\left( \\begin{array}{c c }\nL_{11} & L_{12} \\\\\nL_{12} & L_{22} T\n\\end{array}\\right)\n\\left( \\begin{array}{c}\nV \\\\\n\\Delta T\/T \n\\end{array}\\right)\\ \n\\end{eqnarray}\nwhere $L_{11}$ is the electrical conductance describing the charge current flowing due to the applied bias, $L_{22}$ is the thermal conductance\ndescribing the heat flow driven by the temperature gradient and $L_{12}$ is the thermoelectric coefficient corresponding to heat flow due to the \nbias or charge current due to temperature gradient. \n\nFollowing BTK approach~\\cite{blonder,wysokinski2012thermoelectric}, in presence of bias $V$ and temperature gradient $\\delta T$, \nwe can express the net current flowing through the FS junction as,\n\\begin{eqnarray}\nI_{\\rm FS}&=&2 N(0) e v_F \\mathcal{A} \\int\\limits_{-\\infty}^{\\infty} \\Big[ \\{1-R^e(E)\\} f_0(E-eV,T+\\delta T\/2)\\Big.\\non\\\\\n&&~~~~~~~~~~~~~~~~~~~~~~~~ -R^h(E) f_0 (E+eV,T+\\delta T\/2) \\Big. \\non \\\\\n&& -\\{1-R^e(E)-R^h(E)\\}f_0(E,T-\\delta T\/2) \\Big]dE \\ \n\\label{btk}\n\\end{eqnarray} \nwhere $N(0)$ is the density of states at the Fermi level, $e$ is the electronic charge, $v_F$ is the Fermi velocity and $\\mathcal{A}$ \nis the area of contact. For very small applied bias voltage and temperature gradient, we can expand the Fermi distribution functions \nin Taylor series as follows,\n\\begin{eqnarray}\nf_0(E-eV,T+\\delta T\/2)&=&f_0 (E,T)-eV \\frac{\\partial f_0}{\\partial E} +\\frac{\\delta T}{2} \\frac{\\partial f_0}{\\partial T} \\non \\\\\nf_0(E+eV,T+\\delta T\/2)&=&f_0 (E,T)+eV \\frac{\\partial f_0}{\\partial E} +\\frac{\\delta T}{2} \\frac{\\partial f_0}{\\partial T} \\non \\\\\nf_0(E,T-\\delta T\/2)&=&f_0 (E,T)-\\frac{\\delta T}{2} \\frac{\\partial f_0}{\\partial T}.\n\\end{eqnarray}\nSubstituting the expressions of the distribution functions, we simplify Eq.(\\ref{btk}) for finite but small (linear response regime) temperature gradient \nin absence of bias as,\n\\begin{eqnarray}\nI_{\\rm FS}^{\\prime} &=& 2 N(0) e v_F \\mathcal{A} \\int \\limits_{-\\infty}^{\\infty} \\Big[1-R^e(E)-R^h(E)\\Big] \\delta T \\frac{\\partial f_0}{\\partial T} dE \\ .\\non \\\\\n\\end{eqnarray}\nTherefore the current flow through the junction per unit temperature difference is obtained as\n\\begin{eqnarray}\n\\frac{I_{\\rm FS}^{\\prime}}{I_0}=\\int \\limits_{0}^{\\infty}\\frac{\\Big[ 1-R^e(E)-R^h(E) \\Big]}{k_B T^2 \\cosh^2(\\frac{E-E_F}{2 k_B T})} \\delta T (E-E_F)~dE \\  \\non\\\\\n\\label{IFS2}\n\\end{eqnarray}\nwhere the normalization constant is given by $I_0=N(0)e v_F\\mathcal{A}$. Similarly, if we consider only the bias instead of the temperature gradient\nwe obtain the expression for current as,\n\\begin{eqnarray}\n\\frac{I_{\\rm FS}^{\\prime\\prime}}{I_0}=\\int \\limits_{0}^{\\infty}\\frac{\\Big[ 1-R^e(E)+R^h(E) \\Big]}{k_B T^2 \\cosh^2(\\frac{E-E_F}{2 k_B T})} eV ~dE \\ . \\non\\\\\n\\label{IFS3}\n\\end{eqnarray}\nFrom Eq.(\\ref{IFS3}) we obtain the mathematical expression of electrical conductance or the well-known BTK formula for the electrical conductance~\\cite{blonder} \nin the linear response regime as,\n\\begin{eqnarray}\nL_{11}= \\int \\limits_{0}^{\\infty}\\frac{\\Big[1-R^e(E)+R^h(E) \\Big]}{k_B T \\cosh^2(\\frac{E-E_F}{2 k_B T})} dE.\n\\end{eqnarray}\nAfter performing the wave vector integration we can get back Eq.(\\ref{intG}) for the electrical conductance $G$ normalized by the Sharvin conductance $G_0$ \n(in unit of $e^2\/h$) for a perfect contact as mentioned in Ref.~\\onlinecite{hogl}.\n\nOn the other hand, thermal conductance can be defined as the heat current flowing across the junction per unit temperature difference and given \nby~\\cite{xu2014enhanced,linder2016},\n\\begin{eqnarray}\nL_{22}=\\int \\limits_{0}^{\\infty}\\frac{\\Big[1-R^e(E)-R^h(E) \\Big]}{k_B T^2 \\cosh^2(\\frac{E-E_F}{2 k_B T})} \\frac{(E-E_F)^2}{k_B e^2} dE.\n\\end{eqnarray}\nIn our case additional summation for the electron spin as well as an integration over the wave vector parallel to the interface have to \nbe taken into account following Ref.~\\onlinecite{hogl} as written in the expression of normalized thermal conductance $\\kappa$ in \nEq.(\\ref{kappa_form}). The normalization constant $k_0$ is $G_0 k_B^2 T\/e^2$ ($\\equiv k_B^2 T\/h$).\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=8.7cm,height=7.cm]{Fig4.pdf}\n\\caption{(Color online) The variation of thermal conductance ($\\kappa$), in unit of $k_B^2\/h$, with respect to temperature ($T\/T_c$) \nis depicted in presence of RSOI ($\\lambda_{rso}=2$) for different values of the barrier strength ($Z$) and polarization ($P$) of the \nferromagnet.}\n\\label{rashbacond_T}\n\\end{center}\n\\end{figure}\nUsing Eq.(\\ref{IFS2}) we can also write the expression for $L_{12}$ as,\n\\begin{eqnarray}\nL_{12}=\\int \\limits_{0}^{\\infty}\\frac{\\Big[1-R^e(E)-R^h(E) \\Big]}{k_B T \\cosh^2(\\frac{E-E_F}{2 k_B T})} \\frac{(E-E_F)}{k_B e} dE \\ .\n\\end{eqnarray} \nFollowing the route of electrical and thermal conductance, we perform the summation over spin and integration over wave vector to obtain \n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=8.7cm,height=7.2cm]{Fig7.pdf}\n\\caption{(Color online) The variation of Seebeck coefficient ($S$), in unit of $k_B\/e$, with respect to temperature ($T\/T_c$) is displayed \nin presence of RSOI ($\\lambda_{rso}=2$) for different values of barrier strength ($Z$) and polarization ($P$) of the ferromagnet.}\n\\label{rashbasbk_T}\n\\end{center}\n\\end{figure}\nthe exact form of the thermoelectric coefficient as provided in Eq.(\\ref{alphaform}).\n\nThus, from Onsager matrix equation we write\n\\beq\nI=L_{11} V + L_{12}\\Delta T\/T.\n\\end{equation}\nIn open circuit condition ($I=0$), the induced voltage per unit temperature gradient or the so-called Seebeck coefficient can be found \nin terms of $L_{12}$ and $L_{22}$ \n\\begin{eqnarray}\nS=-\\frac{1}{T} \\frac{L_{12}}{L_{11}}.\n\\label{seebeck_L12}\n\\end{eqnarray}\nIt is measured in the unit of $k_B\/e$ (V\/K). Note that, we can obtain Eq.~(\\ref{sbk_exp}) from Eq.~(\\ref{seebeck_L12}) by replacing \n$L_{12}$ by $\\alpha$ and $L_{11}$ by $G$ as given in Eq.(\\ref{sbk_exp}) in unit of $k_B\/e$.\n\nHence, incorporation of the units of $G$ ($e^2\/h$), $S$ ($k_B\/e$), and $K$ ($k_B^2 T \/h$) will make the figure of \\\nmerit $zT$ dimensionless.\n\n\\section{Behavior of the thermal conductance, Seebeck coefficient and figure of merit with temperature at fixed RSOI}\n\\label{appndx2}\n\nIn Fig.~\\ref{rashbacond_T}, we show the variation of the \\tc as a function of temperature ($T\/T_c$) for finite RSOI strength. Here, \nFig.~\\ref{rashbacond_T}[(a), (b), (c) and (d)] represent the scalar barrier strength $Z=0$, $1$, $2$ and $4$, respectively. Comparing \nFig.~\\ref{cond_T}(a) and Fig.~\\ref{rashbacond_T}(a) we observe that in presence of finite RSOI, \\tc decreases irrespective of the degree \nof polarization of the ferromagnet for $Z=0$. Now incorporation of finite barrier strength $Z$ makes $\\kappa$ behave differently from the \n$Z=0$ \n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=8.7cm,height=7.2cm]{Fig10.pdf}\n\\caption{(Color online) The variation of Figure of merit ($zT$) is shown as a function of temperature ($T\/T_c$) in presence of finite RSOI \n($\\lambda_{rso}=2$) for different values of barrier strength ($Z$) and polarization ($P$) of the ferromagnet.}\n\\label{rashbafom_T}\n\\end{center}\n\\end{figure}\ncase [see Fig.~\\ref{rashbacond_T}(b)-(d)]. The scenario becomes much more interesting when we compare Fig.~\\ref{cond_T}(b) and \nFig.~\\ref{rashbacond_T}(b). We notice that in presence of RSOI, behavior of $\\kappa$ with RSOI is non-monotonic. It is clear from the fact \nthat for $Z=1$, \\tc increases while we introduce RSOI at the interface. On the other hand for higher values of $Z$ ($Z=2$ and $Z=4$), \n$\\kappa$ decreases with the incorporation of RSOI. It is evident from the comparison of Fig.~\\ref{cond_T} with Fig.~\\ref{rashbacond_T} for \nall values of $Z$. Note that, the cross-over temperature separating the two opposite behaviors of the \\tc with polarization also exists in \npresence of RSOI. Also the magnitude of $T_x$ shifts towards the high temperature regime with the enhancement of $Z$ similar to the case of \nin absence of barrier at the junction (see Fig.~\\ref{rashbacond_T}(c) and (d)).\n\nIn Fig.~\\ref{rashbasbk_T} we display the behavior of \\sbk with $T\/T_c$ in presence of RSOI for different values of $Z$ and $P$. Here, (a), \n(b), (c) and (d) represent $Z=0$, $1$, $2$ and $4$, respectively similar to the previous figures. Comparing Fig.~\\ref{sbk_T}(a) and \nFig.~\\ref{rashbasbk_T}(a), it is clear that for $Z=0$ the nature of the curves for $S$ as a function of $T\/T_c$, in absence and presence of RSOI, \nare quite similar to each other except the slopes. Note that, \\sbk changes much faster with $T\/T_c$ in presence of RSOI whereas for a particular\ntemperature, \\sbk increases with the increase of polarization $P$. However, the behavior of \\sbk changes significantly when we incorporate \nbarrier strength at the FS interface. For low $Z$ (for {\\it e.g., } $Z=1$), \\sbk gets reduced as soon as we incorporate RSOI at the interface (compare \nFigs.~\\ref{sbk_T}(b) and \\ref{rashbasbk_T}(b)). However, we recover the non-monotonic behavior of \\sbk when $Z$ becomes high ($Z=2$ and $4$) \nas demonstrated in Figs.~\\ref{rashbasbk_T}(c) and (d). \n\n\nThe behavior of \\fom with temperature in presence of RSOI ($\\lambda_{rso}=2$) is shown in Fig.~\\ref{rashbafom_T}. In absence of any barrier \nat the interface, FOM increases with the rise of temperature, for a particular value of polarization. The behavior of $zT$ with temperature \nand also with polarization in presence of $\\lambda_{rso}$ is very much similar to that in absence of RSOI which is clear by comparing \nFig.~\\ref{fom_T}(a) and Fig.~\\ref{rashbafom_T}(a). However, the scenario changes drastically when we consider finite barrier at the junction. \nIn presence of low $Z$, $zT$ decreases for all values of $P$ as shown in Fig.~\\ref{rashbafom_T}(b). Moreover, the sensitivity of $zT$ to $P$ \nis very small for low $Z$. For higher barrier strength, exactly opposite situation occurs. The interplay of high potential barrier, polarization \nand RSOI (weak or moderate) causes enhancement of $zT$ with the increase of $Z$ (see Figs.~\\ref{rashbafom_T}(c)-(d)) for the weak strength of \nRSOI ($\\lambda_{rso}$). \n\n\n\\section{Comparison of our results with the existing literature for $\\lambda_{rso}=0$, $P=0$ and $Z=0$}\n\\label{appndx3}\n\\end{appendix}\nWe compare the magnitudes of the \\sbk and \\fom with the results existing in the literature for the limit of zero interfacial RSOI ($\\lambda_{rso}=0$) and zero exchange field ($P=0$).  Within this limit, we have found \\sbk $\\sim 0.3 \\times 10^{-4}$ V\/K at $T\/T_c=0.5$ in absence of barrier ($Z=0$) (see orange curve of Fig.~\\ref{sbk_T}(a)). In Ref.~\\onlinecite{linder2016}, we observe that \\sbk is zero for the same parameter regime. However, for other values of polarization and temperature, $S$ $\\sim 10^{-4}$  V\/K or $10^{-3}$ V\/K depending on the values of $T$ and $P$ (see Ref.~\\onlinecite{linder2016} for details). \nFor $\\lambda_{rso}=0$, $P=0$ and $Z=0$, our system basically converts to a normal metal-superconductor junction and we can compute our results directly from the BTK formalism using Eq.~(\\ref{sbk_exp}). \\sbk should be zero in this parameter regime only if we consider the energy values within the superconducting sub-gapped regime. Within this energy window, integration over energy gives rise to exactly zero thermopower. However, we have considered all the energy levels in our analysis and there are some finite contributions arising from the energy levels above the superconducting gap. As a consequence, we obtain low but finite $S$. Now, zero \\sbk results in zero \\fom (see Eq.~(\\ref{merit})) as shown in Ref.~\\onlinecite{linder2016}. However, due to the finite contributions arising from the energy levels above the \nsuperconducting gap, we have obtained very low but finite value of figure of merit. Note that, our system is not exactly equivalent to the system considered in Ref.~\\onlinecite{linder2016}. Instead of two spin-split superconductors separated by a magnetic barrier, we have only one superconductor to form the junction with a ferromagnet. However, we have compared the values of $S$ and $zT$ with this reference only in absence of exchange interaction and found that the order of magnitude are well in agreement.\n\n\\end{document}\n\\acknowledgments{PD thanks Department of Science and Technology (DST), India for the financial support through SERB NPDF (File no. PDF\/2016\/001178). \nWe acknowledge S. D. Mahanti for helpful discussions and encouragement. AMJ also thanks DST, India for financial support. We thank the anonymous \nreferee for his\/her constructive comments and suggestions for the improvement of our manuscript. Finally, we acknowledge Indian citizens for supporting \nresearch in basic science.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec1}\n\nGEANT4 \\citep{Agostinelli,G4App,G4Toolkit} is an object-oriented Monte Carlo simulation toolkit developed for simulating the passage of particles through matter. Although originally developed for nuclear and particle physics experiments, its application domain has spread to fields such as space science, accelerator, and medical physics. The toolkit is regularly updated, developed, and maintained by a worldwide collaboration of scientists. In recent years, it has become the most preferred detector simulation software in the high energy and nuclear physics communities. \n\nThe classes comprising the Geant4 software are categorized based on their logical content. Each category is designed as independently as possible. This allows each to be developed in parallel without serious interference. This work falls under the geometry and material category of Geant4 and proposes a new detector construction scheme for achieving more flexible detector setups.\n\nGeant4 provides \\texttt{G4VUserDetectorConstruction} class that is responsible for the entire detector construction. Users inherit from this class and implement its \\texttt{Construct()} and \\texttt{ConstructSDAndField()} methods. All operations related to the geometry definition are performed within these two methods. The method \\texttt{Construct()} returns a pointer of type \\texttt{G4VPhysicalVolume}, also known as world volume, representing the entire detector geometry.\n\nWhile this approach is effective for simple detector constructions, using a single class for building a complex detector causes complexity in the code. Therefore, splitting the implementation into more methods or classes is advised in many of Geant4's official training courses \\citep{Makoto,Giada,Sebastian,Anton}; however, how this is to be applied is not demonstrated. \n\nThe aforementioned issue becomes more noticeable in the construction of scintillation detectors. With the inclusion of the optical properties of the materials in the implementation process, the responsibility of the detector construction class increases further, and refactoring becomes inevitable. \n\nFurthermore, detector designs may undergo numerous changes during the detector development process. Some components, for instance, can be upgraded, replaced by other components, or totally removed from the system. More importantly, such change requests should be made by extending the code rather than breaking it, so that different versions of detectors become comparable. This necessitates a highly flexible detector production code.\n\nIn order to circumvent all these complications and have a clear implementation structure, this work proposes a component-based detector construction strategy for flexible detector setups. The strategy involves applying the \"Builder design pattern\", which is commonly used in object-oriented programming for creating complex objects step-by-step, to the detector construction phase. However, in the context of Geant4 geometry construction, I call it \"Geant4 Detector Construction Pattern\"\\footnote{This is also where the package gets its name.}. The following section publicizes participants in this pattern and describes its implementation in detail. \n\n\n\n\\section{Implementation}\n\\label{sec2}\n\n\\subsection{Utility classes for geometry construction}\n\\label{sec2.1}\n\nThe Builder design pattern is applied to the detector construction process in two stages. The first stage involves creating detector components as single or multiple parts, and the second involves assembling those components in a volume that represents the entire detector geometry. The following introduces elements of this pattern: \n\n\n\n\n\n\n\n\n\n\n\n\\begin{itemize}\n\\setlength\\itemsep{0.em}\n\n\\item Abstract builder classes:~\\texttt{VComponentBuilder} and \\texttt{VMultiComponentBuilder}. The first is used to build a single detector component, and the second is used to build a multiple detector component. These two classes declare product construction steps.  \n\t\n\\item User builder classes: These are the concrete classes to be derived by users. Two kinds of builder classes are derived, one for detector components and one for the detector itself. I further provide ready-to-use concrete component builder classes for the most commonly used shapes in Geant4: \\texttt{BoxComponentBuilder}, \\texttt{TubsComponentBuilder} and \\texttt{TrdComponentBuilder}. When an object of one of these classes is created, a few user interface commands are defined automatically for the modification of the material and size of that component. \n\n\\item Director class:  \\texttt{DetectorDirector}. This class defines the order in which to execute the building steps, while the detector builder class provides the implementation for those steps. The usage of this class is optional.\n\t\n\\item Product class:  \\texttt{SmartComponent}. All builder classes produce the same type of product. A product is an object of type \\texttt{SmartComponent} encapsulating \\texttt{G4LogicalVolume} and \\texttt{G4VPhysicalVolume} objects. So, depending on its content, it may represent a single detector component, a multi-component, or the detector itself. It also keeps a list of locations where it is positioned in the detector geometry.\n \n\\end{itemize} \n\nFig.~\\ref{fig:fig1} summarizes in a UML \\cite{Fowler:2015} diagram the structure of the classes provided for applying the Builder design pattern to the detector construction process of a Geant4 application.\n\n\\begin{figure*}[!htp]\n \\centering\n    \\includegraphics[width=0.9\\textwidth]{Figures\/Fig1}\n     \\caption{ Unified Modeling Language diagram of the classes provided for applying the Builder design pattern to the detector build phase of a Geant4 application.}\n    \\label{fig:fig1}\n\\end{figure*}\n\nThe following steps explain how to set up a complex detector in a highly flexible manner using the supplied classes:\n\n\\begin{itemize}\n\\setlength\\itemsep{0.em}\n\t\n\\item Make a list of all the components to be employed in the detector construction.  \n\t\n\\item Create a builder class for each detector component and implement their construction steps. A component can be composed of only one detector element or multiple detector elements. In the first case, the builder class is derived from \\texttt{VComponetBuilder}, and in the second case, it is derived from \\texttt{VMultiComponentBuilder}. Moreover, if a component's physical shape is a box, cylinder, or trapezoid, and if it is not going to be made sensitive, there is no need to create a new class for that component. \\texttt{BoxComponentBuilder}, \\texttt{TubsComponentBuilder}, and \\texttt{TrdComponentBuilder} are made available for this purpose.\n\n\\item Create a concrete material factory class as described in Sec. \\ref{sec2.3}. If a material is employed just for one component, it can be defined simply within the \\texttt{CreateMaterial()} method of the relevant builder class. If employed for more than one component, it can be defined either in a member method within the concrete material factory class or in a new material builder class created for that material. In the second case, the object of the material is called from the \\texttt{CreateMaterial()} method.\n\n\\item After the above three steps are completed, proceed to the assembly phase of the components. For this, derive the detector builder class from \\texttt{VMultiComponentBuilder}\\footnote{Since the detector is multi-component, its builder class derives from \\texttt{VMultiComponentBuilder} class.} and implement its construction steps. Listing \\ref{lst1} shows an example derivation of the detector builder class. \t \n\t  \n\\item The last step is the implementation of the Geant4 detector construction class. This class makes use of \\texttt{UserDetectorBuilder} and optionally \\texttt{DetectorDirector} class for the construction of the detector. Listing \\ref{lst2} shows an example implementation of the detector construction class. \n\n\\end{itemize}\t\n\t\t\n\\lstinputlisting[float=*, label=lst1, language={C++}, caption={An example of a concrete implementation of VMultiComponentBuilder class.}]{Listings\/Listing1.cc}\t\n\n\\lstinputlisting[float=*, label=lst2, language={C++}, caption={An example implementation of the Geant4 detector construction class.}]{Listings\/Listing2.cc}\n\nThe advantages of using this pattern are summarized below:\n\n\\begin{itemize}\n\\setlength\\itemsep{0.em}\n\t\n\\item It provides flexibility for change requests that may arise during the detector development process. \n\t \n\\item It provides reusability of detector components. As detector components are treated as self-contained pieces, a component developed for a particular detector could be used in various detector constructions.\t\n \n\\item It allows developers to create a messenger class per component. \n\t\n\\end{itemize}\n\n\\subsection{Utility classes for scintillation detectors}\n\\label{sec2.2}  \n\nThis section concerns the construction of scintillation detectors. One of the crucial steps in the implementation process of scintillation detectors is to specify the optical properties of the materials employed and the optical surfaces defined. Geant4 provides \\texttt{G4MaterialPropertiesTable} and \\texttt{G4OpticalSurface} classes for this purpose. The former is used to assign new properties to pre-defined materials or surfaces, while the latter is used to describe optical properties of medium boundaries.\n\nOptical properties are stored as entries in a \\texttt{G4MaterialPropertiesTable}, and each entry consists of a key and a value pair. These properties may be energy-independent or energy-dependent. In the first case, the value is a constant, and in the second case, it is a list of energy-value pairs.\n\nThe usual approach is to put these energy-value pairs inside the detector construction file as elements of \\texttt{std::vector} or C-style arrays. However, the presence of various datasets in a typical application and their direct embedding into the source file results in long lines of code and reduces code readability.\n\nMoreover, the parameters defining the properties of materials and surfaces are frequently changed in studies where the effect of these parameters is investigated. This requires changing the source code repeatedly.\n\nTo overcome these challenges and perform the entire implementation process through user interface commands, I propose using two new classes. The first one is \\texttt{OpticalSurface} class\\footnote{I did not put a prefix in front of the class name. The user can add a prefix according to his\/her own content.}, which is derived from \\texttt{G4OpticalSurface}. The only functionality added to this subclass is that its implementation can be conducted by user interface commands. The second one is \\texttt{MaterialPropertiesHelper} class, which is a subclass of \\texttt{G4MaterialPropertiesTable}. This class provides additional methods to receive data from users. Fig.~\\ref{fig:fig2} summarizes, in a UML class diagram, the provided classes and their relationships with the Geant4 classes.\n\n\\begin{figure*}\n \\centering\n    \\includegraphics[width=0.9\\textwidth]{Figures\/Fig2}\n     \\caption{ Unified Modelling Language diagram of the classes provided for the implementation of optical components in scintillation detectors.}\n    \\label{fig:fig2}\n\\end{figure*}\n\nTo illustrate, I take the code snippet shown in Listing \\ref{lst3} from the examples of Geant4\\footnote{It is taken from the examples\/extented\/optical\/opNovice\/src\/OpNoviceDetectorConstruction.cc} and compare it to the implementations of \\texttt{MaterialPropertiesHelper} and \\texttt{OpticalSurface} classes ( Listings \\ref{lst4} and \\ref{lst5}), which fulfill the same task.\n\n\\begin{figure*}\n\\lstinputlisting[label=lst3, language=C++, caption={An example implementation of G4MaterialPropertiesTable and G4OpticalSurface classes.} ]{Listings\/Listing3.cc}\n\n\\lstinputlisting[label=lst4, language=C++ ,caption={An example implementation of MaterialPropertiesHelper and OpticalSurface classes.} ]{Listings\/Listing4.cc}\n\n\\end{figure*}\n\n\\begin{figure*}\n\\lstinputlisting[label=lst5, language={bash}, caption={Implementation of optical components through user interface commands. }]{Listings\/Listing5.mac}\n\\end{figure*}\n\n\\subsection{Utility classes for material construction}\n\\label{sec2.3}\n\nThis section introduces two new abstract classes, \\texttt{VMaterialBuilder} and \\texttt{VMaterialFactory}, that developers can utilize in the detector construction phase of a Geant4 application. The first offers an interface for developers to create custom material classes, while the second serves as interface for creating material objects from a single place in an application. Fig. \\ref{fig:fig3} shows the UML class diagram of these two abstract classes. \n\n\\begin{figure*}\n \\centering\n    \\includegraphics[width=0.9\\textwidth]{Figures\/Fig3}\n     \\caption{ Diagram of \\texttt{VMaterialBuilder} and \\texttt{VMaterialFactory} classes in Unified Modelling Language.}\n    \\label{fig:fig3}\n\\end{figure*}\n\nTo create a concrete material builder class, users inherit from \\texttt{VMaterialBuilder} and implement its \\texttt{CreateMaterial()} method. This class utilizes \\texttt{MaterialPropertiesHelper} to assign additional properties to the materials to be defined. To get the resulting material object from the builder, the method \\texttt{GetProduct()} is invoked. Listing \\ref{lst6} shows an example concrete implementation of \\texttt{VMaterialBuilder} class. \n\n\n\n\n\\lstinputlisting[float=*, label=lst6, language={C++}, caption={An example of a concrete implementation of \\texttt{VMaterialBuilder} class. EJ-200 \\cite{Eljen} scintillator is chosen as an example. }]{Listings\/Listing6.cc}\n\nTo derive a concrete material factory class, users inherit from \\texttt{VMaterialFactory} and implement its \\texttt{GetMaterial()} method. This method is used to create objects of user-defined materials. However, it is not invoked directly by users to obtain a material object; instead, the \\texttt{FindOrCreateMaterial()} method is invoked. This method first searches the desired material in GEANT4's material database. If the material is not found, it invokes the \\texttt{GetMaterial()} to search among the user-defined materials. If it is not found there as well, it throws an exception. Listing \\ref{lst7} depicts an example usage of \\texttt{VMaterialFactory} class.\n\n\\lstinputlisting[float=*, label=lst7, language={C++}, caption={An example of a concrete implementation of VMaterialFactory class.}]{Listings\/Listing7.cc} \n\nThe benefits of these two classes can be summarized as:\n\n\\begin{itemize}\n\\setlength\\itemsep{0.em}\n\n\\item Providing reusability of materials. Encapsulating a material construction code into a class makes it available to other developers. This is particularly significant for materials that are troublesome to implement. For example, to implement a scintillator, its atomic composition, scintillation character, and all-optical properties should be defined. This requires an extensive literature review followed by conversion of the collected data into the desired format. Once a scintillator is implemented by a developer, it can be used by others as well. \\texttt{VMaterialBuilder} class acts as an interface for producing ready-to-use material classes.\n\n\\item Improving code readability by encapsulating material construction code.\n\n\\item Preventing code duplication by allowing users to access material objects from different sections of the code.\n    \n\\end{itemize}\n\n\\section{Structure of GDCP}\n\\label{sec3}\n\nThe GDCP package is a collection of C++ classes developed for designing flexible detector setups in Geant4 applications. The provided classes are collected and organized in directories according to their dependencies and functionalities. The package consists of the following three directories:\n\n\\begin{itemize}\n\\setlength\\itemsep{0.em}\n\n\\item \\texttt{gdcp\/}: This directory includes header and implementation files of GDCP. \n\\item \\texttt{scntDetector\/}: This directory includes various utility classes for the implementation of scintillation detectors.\n\\item \\texttt{g4Example\/}: This directory includes an advanced geant4 application that demonstrates the usage of the provided classes. The application also uses the NuSD framework\\cite{NuSD} that we previously developed for modeling segmented scintillation detectors. The detector construction interface in this framework is updated using the GDCP in order to enhance the framework's flexibility and maintainability. The application includes the installation of PANDA \\citep{Panda:2012,Panda:2014} and SOLID \\citep{Solid:2015,Solid:2017,Solid2:2017} experiments as examples.\n\n\\end{itemize}\n\n\\section{Conclusions}\n\nIn this study, a new custom package GDCP developed to create flexible detector installations in Geant4 applications is introduced. The classes in the package are grouped into three categories based on their roles and responsibilities. The first set of classes offers a detector construction interface based on independent detector components for flexible detector setups. This interface forces to apply the Builder design pattern to the detector construction phase in two steps: 1) for the construction of the detector components and 2) for the assembly of those components. For this, the elements of the Builder design pattern are introduced and made available for users. \n\nThe second set of classes provides an interface to obtain custom material classes. This is particularly significant for materials that implementation necessitates extensive data collection from the literature. Including a material construction code in a class allows other developers to easily use that material.  \n\nThe third group of classes is related to scintillation detectors and makes a dual contribution to the implementation process of optical components. This both provides an easier interface for defining properties of optical materials and allows that interface to be implemented through the user interface commands.\n\n\n\n\n\n\\clearpage\n\\clearpage\n\n\n\n\\bibliographystyle{elsarticle-num}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nThe cosmological relevance of local starbursts is due to their\nsimilarities with their analogs at high redshift. They may therefore\nprovide extremely valuable laboratories to study with high resolution\nand sensitivity relevant physical processes associated with the\nstarburst activity in much better detail.  One of the most suitable\nenvironments for these purposes can be the intriguing class of very\ncompact, extremely star-forming galaxy (SFG) at low redshift ($0.11 \\la z\n\\la 0.35$) recently discovered by volunteers in the ``Galaxy Zoo''\nproject \\citep{Lintott}.  These galaxies, popularly referred to as\n``green peas'' (hereafter GPs), were first reported and studied at\nsome length by \\citet[][hereafter C09]{C09}, who classified more than\none hundred of them.  The GPs are rare objects, located in\nlower-density environments and spectroscopically characterized by very\nfaint continuum emission and strong optical emission lines, namely\n[O{\\sc iii}]$\\lambda$5007, with an unusually large equivalent width\nof up to $\\sim$1000\\AA.  These properties along with their extreme\ncompactness (unresolved in Sloan Digital Sky Survey (SDSS) imaging) \nare the reason for their green colors and point-like appearance in \nthe SDSS plates. According to C09, GPs are low-mass galaxies (stellar masses\n$M_{\\star}$$\\sim$10$^{8.5}$ to 10$^{10}$$M_{\\odot}$) with extremely high\nstar formation rates (SFRs up to 30 $M_{\\odot}$ yr$^{-1}$), as derived\nfrom their emission lines and UV luminosity.  Therefore, their\nspecific SFRs (SSFRs, i.e., SFR per unit of mass) are among the highest\ninferred in the nearby universe.  The GP properties are then\nconsistent with the picture of galaxy downsizing \\citep{Cowie96},\nwhere the sites of active star formation shift from high-mass galaxies\nat early times to lower mass systems at later epoch \\citep[see\n  also][]{Bundy06,Cowie08,Noeske07}.\n\nThe motivation for this Letter was one of the most striking results\nobtained by C09: the nearly constant solar metallicity of the GPs\n(log[O\/H]+12$\\sim$8.7 in median) over more than two orders of\nmagnitudes in stellar mass.  These results would imply strong\nconstrains to the nature of GPs and their evolutionary status.  In\nthis Letter we examine in more detail the GP chemical abundances,\nnamely oxygen and nitrogen, and their relation with the stellar mass,\nusing the same SDSS spectra but a different methodology than in\nC09. This analysis indicates instead, that GPs can be considered as a\ngenuine population of metal-poor galaxies.  The results are compared\nwith a larger sample of local SFGs from the SDSS and with a smaller \nsample of super compact UV-luminous starburst\ngalaxies \\citep{Heckman05} placed at the same redshift range of the\nGPs.  In particular, the position of GPs in the fundamental relation\nbetween mass and metallicity is discussed in the context of their\nstarburst activity and feedback processes.\n\n\\section{THE DATA}\n\nThe sample of galaxies was taken from the list of C09.  They compiled\nand analyzed by means of SDSS Data Release 7\n\\citep[SDSS DR7;][]{York} imaging and spectroscopy, {\\em Hubble Space \nTelescope (HST)} optical and {\\em GALEX} UV imaging, a well-defined \nsample of GPs in the redshift range $0.11 \\leq z \\leq 0.36$ based on \ncolor selection criteria.  From\ntheir list we selected a sub-sample of 79 GPs spectroscopically\nclassified as star-forming systems (their Table~4), ruling out active\ngalactic nuclei (AGNs) according to the diagnostic diagram\n[\\relax \\ifmmode {\\mbox N\\,{\\scshape ii}}\\else N\\,{\\scshape ii}\\fi]\/\\relax \\ifmmode {\\mbox H}\\alpha\\else H$\\alpha$\\fi\\ versus [\\oiii]\/\\relax \\ifmmode {\\mbox H}\\beta\\else H$\\beta$\\fi\\ \\citep[][]{BPT}.  We refer the reader to\nC09 for a complete description of the sample selection.\n\nWe obtained the full set of calibrated one-dimensional spectra, \ni.e., {\\sl spSpec fits} files, from the SDSS archive using the SDSS DR7 query\ntools\\footnote{\\url{http:\/\/cas.sdss.org\/dr7\/en\/tools\/search\/sql.asp}\n  and \\url{http:\/\/das.sdss.org}}. The SDSS spectra cover\n3800\\AA$<\\lambda<$9200\\AA\\ and the fiber has a 3$''$ diameter which, due\nto the compactness of the galaxies, include $\\ga$70\\% of the GP\n$r-$band emission\\footnote{We used $f_{fib}=10^{-0.4(m_{r,\n     \\rm{fiber}}-m_{r, \\rm{Petro}})}$, which is the fraction of the fiber total\n  light (line+continuum) within the fiber aperture}.  Emission-line\nintegrated fluxes were consistently measured for each galaxy spectrum\nusing the {\\sc IRAF} task {\\sl splot}, by summing all the pixels of\nthe emission line after a linear subtraction of the continuum. The\nflux uncertainties are then calculated as\n\n\\begin{equation}\n\\sigma_{l} = \\sigma_{c} \\sqrt{N + \\frac{EW}{\\Delta}}\n\\end{equation}\n\\citep{PMD03}, where $\\sigma_{l}$ is the line flux uncertainty,\n$\\sigma_{c}$ is the uncertainty in the position of the continuum, $N$\nis the number of pixels in the flux measurement, $EW$ is the\nequivalent width of the line and $\\Delta$ is the wavelength\ndispersion.  In most cases the measurements are in agreement with\nthose based on Gaussian fitting.\n\n\\section{RESULTS}\n\n\\subsection{Green Peas: A Genuine Population of Metal-poor Galaxies}\n\nPhysical properties and oxygen and nitrogen ionic abundances were\ncalculated from the measured emission lines using the direct method,\nwhich is expected to provide more accurate abundances compared to\nstrong-line methods \\citep[see][hereafter PMC09]{Enrique09}.  First,\nwe have estimated the reddening constant from the \\relax \\ifmmode {\\mbox H}\\alpha\\else H$\\alpha$\\fi\/\\relax \\ifmmode {\\mbox H}\\beta\\else H$\\beta$\\fi\\ ratio using\nthe extinction curve of \\citet{Cardelli89}.  Then, for those objects\nwith a reliable measurement of the [\\oiii] $\\lambda$ 4363 emission\nline ($\\sim$ 70\\% of the sample), we have derived the [\\oiii] electron\ntemperature.  The [\\relax \\ifmmode {\\mbox O\\,{\\scshape ii}}\\else O\\,{\\scshape ii}\\fi] electron temperature has been derived from it\nusing the relations proposed by PMC09. O$^+$, O$^{2+}$, and N$^+$ ionic\nabundances have been calculated using these temperatures and the\nrelative intensities of the corresponding brightest emission lines.\nFinally, total O\/H and N\/O abundance ratios have been obtained with\nthe usual expressions (see PMC09).\n\nFor the full sample, including those galaxies with no direct\ndetermination of the temperature, oxygen total abundances have \nalso been estimated from the strong-line calibrator N2 $\\equiv$\n\\mbox{log([\\relax \\ifmmode {\\mbox N\\,{\\scshape ii}}\\else N\\,{\\scshape ii}\\fi] $\\lambda$6584\/\\relax \\ifmmode {\\mbox H}\\alpha\\else H$\\alpha$\\fi)}.  We have used the empirical\nrelation derived by PMC09, which gives oxygen abundances fully\nconsistent with those derived from $T_{\\rm e}$-sensitive methods over\nthe whole observed range of metallicity.  According to these authors,\nN2 has a non-negligible dependence on the N\/O ratio. Nevertheless, our\ncalculations indicate that correction for this dependence in this\nsample does not reduce the dispersion, so we have preferred to keep\nthe calculated abundances in the non-corrected form. Regarding N\/O,\nthis can be derived using the N2S2 parameter (i.e., [\\relax \\ifmmode {\\mbox N\\,{\\scshape ii}}\\else N\\,{\\scshape ii}\\fi]\/[\\relax \\ifmmode {\\mbox S\\,{\\scshape ii}}\\else S\\,{\\scshape ii}\\fi]),\nminimizing its dependence on reddening or flux calibration.\n\n\\begin{figure}[t!]\n\\begin{tabular}{l l}\n\\includegraphics[scale=.35]{figura_1a.eps}&\n\\includegraphics[scale=.35]{figura_1b.eps}\n\\end{tabular}\n\\caption{Oxygen and nitrogen abundances.  On the left, the lined \nand the non-lined histograms indicate the number of GPs with metallicities \ncalculated from the direct method and using the N2 index, respectively. \nFor comparison, metallicities derived by C09 are shown in the filled \nhistogram. On the right, the lined and the non-lined histograms show the\ncorresponding N\/O distributions. (A color version of this figure is available \nin the online journal.)\\label{fig1}}\n\\end{figure}\n\nWe have compared the derived properties of the GPs with two different\nsamples of SFGs.  First, we have taken the large\nsample of emission-line galaxies listed in the Max Planck\nInstitute for Astrophysics\/Johns Hopkings University (MPA\/JHU) Data \ncatalog of the SDSS DR 7\\footnote {Available at\n  http:\/\/www.mpa-garching.mpg.de\/SDSS\/ }, which covers a redshift\nrange 0.03 $\\leq z \\leq$ 0.37.  We have kept the non-duplicated galaxies\nwith a signal-to-noise ratio (S\/N) of at least 5 in all the involved\nlines.  Then, we have discarded those objects classified as AGNs,\nusing the diagnostic diagram [\\relax \\ifmmode {\\mbox N\\,{\\scshape ii}}\\else N\\,{\\scshape ii}\\fi]\/\\relax \\ifmmode {\\mbox H}\\alpha\\else H$\\alpha$\\fi\\ versus [\\oiii]\/\\relax \\ifmmode {\\mbox H}\\beta\\else H$\\beta$\\fi.  Finally, we\nhave calculated oxygen and nitrogen abundances in the same way as\ndescribed above for the sample of GPs.  Moreover, using the SDSS SFGs\nwith direct estimation of the ionic abundances, we have obtained the\nempirical relation between N\/O and the N2S2 parameter (PMC09) that\nwas consistently applied to estimate N\/O for all the galaxies used in\nour analysis. Our polynomial fit:\n\\begin{equation}\n\\log ({\\mathrm N}\/{\\mathrm O}) = -0.76 + 1.94\\, {\\mathrm N}2{\\mathrm\n  S}2 + 0.55\\, ({\\mathrm N}2{\\mathrm S}2)^2\n\\end{equation}\nwhere N2S2 $=$ log([\\relax \\ifmmode {\\mbox N\\,{\\scshape ii}}\\else N\\,{\\scshape ii}\\fi]\/[\\relax \\ifmmode {\\mbox S\\,{\\scshape ii}}\\else S\\,{\\scshape ii}\\fi]), gives a slightly higher slope and a\nmuch lower dispersion ($\\sim$0.1 dex) than the calibration of PMC09.\n\nIn addition, we have taken for comparison a sample of 31 nearby ($z <\n0.3$) UV-luminous starburst galaxies studied by \\citep{Overzier08,\n  Overzier09a, Overzier09b}.  They showed that these galaxies exhibit\nstrong similarities in their properties with the Lyman-break galaxies\nat high--$z$, and therefore were proposed as good local Lyman-break\nanalogs (LBAs).  Most massive LBAs display a resolved morphology,\ncharacterized by the presence of a dominant central object\n\\citep[DCO,][]{Overzier09a}.  In contrast, low-mass LBAs are\nmetal-poor galaxies showing close similarities (e.g., morphology,\nredshift, UV luminosities, size, SSFRs, and environment) to GPs (C09).\nIn fact, three LBAs are also included in the GP sample.  These three\ngalaxies are those for which {\\em HST} optical imaging was used to analyze\nthe morphology of the GPs (see Figure~7 in C09).  To be consistent, the\noxygen and nitrogen abundances for the sample of LBAs were\nre-calculated using the line ratios measured from SDSS spectra\n\\citep{Overzier09a}, and the same empirical calibrations described\nabove.\n\nIn Figure~\\ref{fig1}, we show histograms with our metallicity estimates\nfor the GPs and those derived by C09 for comparison.  The GPs show\ndirect metallicities $7.7 \\la$ 12$+$log(O\/H)$\\la 8.4$, with a mean\nvalue of 8.05 $\\pm$ 0.14, i.e., $\\sim$20\\% the solar value\\footnote{We\n  adopted $12+$log(O\/H)$_{\\odot}=8.69$ \\citep{Allende01}}, whereas\ntheir measured C$($\\relax \\ifmmode {\\mbox H}\\beta\\else H$\\beta$\\fi$)$ and \\relax \\ifmmode {\\mbox H}\\alpha\\else H$\\alpha$\\fi\/\\relax \\ifmmode {\\mbox H}\\beta\\else H$\\beta$\\fi\\ mean values are 0.23 and 3.3,\nrespectively.  Our direct estimates and those derived using the N2\nparameter agree well in $\\sim$0.05 dex. While\nextinction values are in agreement with those obtained by C09,\nFigure~\\ref{fig1} shows a clear offset of $\\sim$0.65 dex between their\nmetallicity estimates (mean log(O\/H) $+ 12 =$ 8.7) and ours.  The C09\nvalues were obtained using the \\mbox{[\\ion{N}{2}]$\\lambda6584$}\\, \\AA\/\\mbox{[\\ion{O}{2}]$\\lambda\\lambda 3726, 3729$\\,\\AA}\\ ratio to estimate log(O\/H)\n$+$ 12 via the theoretical calibration based on photoionization\nmodels by \\citet{KD02}.  Systematical discrepancies are usually found\nin the literature between metallicities derived from theoretical and\nempirical methods \\citep[][and references therein]{KE08}. We have used\nthe conversion constants provided by \\citet{KE08} to scale our\nN2-based oxygen abundance estimates to those expected from the\ntheoretical calibration (their Table 3).  In doing so, we obtained a\nmean difference of 0.23 dex, which is not enough to explain the\nlargest offset in O\/H shown in Figure~\\ref{fig1}. We conclude that GPs\nare genuine metal-poor galaxies with low extinction values.\n\n\\begin{figure}[t!]\n\\includegraphics[scale=.46]{figura_2_rev.eps}\n\\caption{N\/O vs. O\/H. The two-dimensional histogram shows the number \ndistribution of SDSS SFGs in the logarithmic gray scale. \nGPs are indicated by green circles, while LBAs are indicated with open \nred squares. LBAs with DCOs are marked with open red squares with plus \nsign, whereas those LBAs also belonging to the GP sample are indicated \nwith filled red squares. (A color version of this figure is available \nin the online journal.)\\label{fig2}}\n\\end{figure}\n\n\\subsection{N\/O Ratios and Metallicity}\n\nThe nitrogen-to-oxygen ratio is a powerful evolutionary indicator in\ngalaxies \\citep{Pilyugin04,Molla06}.  We have investigated the\nrelation between the N\/O ratio and the oxygen abundance for the sample\nof GPs, LBAs, and the SFGs taken from the SDSS in Figure~\\ref{fig2}.\nFor the SFGs, we found a positive trend with an increasing scatter\ntoward the metal-rich regime reflecting both primary and secondary\nproduction of nitrogen in the same metallicity range.  At low\nmetallicities the trend flattens out, possibly a consequence of\nnitrogen being of primary origin, coming mainly from massive stars.\nIn the low-to-intermediate metallicity range, most GPs and some LBAs\ndisplay systematically larger N\/O ratios for a given metallicity\ncompared to the SDSS SFGs.  Possible reasons for this include extra\nproduction of primary nitrogen, coming from low-metallicity\nintermediate-mass stars \\citep{Renzini81,Gavilan06,Molla06}, pollution\nby Wolf$-$Rayet stars \\citep[e.g.,][]{Brinchmann08,Angel10,AnaMon10}, or\nhydrodynamical effects involving outflow and inflow of gas\n\\citep{Vanzee98,Koppen05}.  This systematically larger N\/O ratios\ncould explain the difference between our metallicity and the higher\nestimates obtained by C09 using the calibration by \\cite{KD02}.  The\nlatter assumes that N\/O is roughly constant below\n12$+$log(O\/H)$\\sim$8.4, and proportional to O\/H above this value.\nThis implies that larger values of [\\relax \\ifmmode {\\mbox N\\,{\\scshape ii}}\\else N\\,{\\scshape ii}\\fi]\/[\\relax \\ifmmode {\\mbox O\\,{\\scshape ii}}\\else O\\,{\\scshape ii}\\fi] lead to higher\nmetallicities, but this calibration does not take into account the\ndependence of [\\relax \\ifmmode {\\mbox N\\,{\\scshape ii}}\\else N\\,{\\scshape ii}\\fi]\/[\\relax \\ifmmode {\\mbox O\\,{\\scshape ii}}\\else O\\,{\\scshape ii}\\fi] on the variation of the N\/O ratio at a\ngiven O\/H. According to PMC09, large values of the N\/O ratio can\nenhance the [\\relax \\ifmmode {\\mbox N\\,{\\scshape ii}}\\else N\\,{\\scshape ii}\\fi]\/[\\relax \\ifmmode {\\mbox O\\,{\\scshape ii}}\\else O\\,{\\scshape ii}\\fi] ratio even in the low metallicity regime.\n\n\\subsection{Relation Between Stellar Mass, Metallicity and the N\/O Ratio}\n\n\\begin{figure*}[t!]\n\\begin{tabular}{c c}\n\\includegraphics[scale=.5]{figura_3a_rev.eps}&\n\\includegraphics[scale=.5]{figura_3b.eps}\n\\end{tabular}\n\\caption{$M_{\\star}$ vs. O\/H (left) and $M_{\\star}$ vs. N\/O \n(rigth). \nSymbols stand for the same type of objects as described in Figure~\\ref{fig2}.\nFor better comparison the trends observed in the $M_{\\star}-Z$ plane, green \nand black lines indicating second-order polynomial fits to the \nGPs and SFGs, respectively, have been overplotted.\n (A color version of this figure is available in the online journal.)\n\\label{fig3}}\n\\end{figure*}\n\nIn Figure~\\ref{fig3}, we studied the stellar mass$-$metallicity relation\n\\citep[$M_{\\star}-Z$, MZR;][]{Lequeux79} and the relation between N\/O\nand the stellar mass ($M_{\\star}-$N\/O; PMC09) for the GPs, LBAs, and\nSDSS SFGs.  Estimates of $M_{\\star}$ for the three galaxy samples have\nbeen taken from C09 (GPs), \\citet[][LBAs]{Overzier09a}, and the MPA$-$JHU\ncatalog (SFGs). They were derived using spectral energy distribution \nfitting and SDSS photometry. \nThus, stellar masses are expected to to be consistent within a typical \ndispersion of $\\sim$ 0.20 dex \\citep[e.g.,][]{Drory,vanderwel06}. \nThis value, is lower than the expected uncertainties of 0.3 dex quoted \nfor GPs and LBAs as due to uncertainties in their star formation history\n\\citep[C09;][]{Overzier09a}.\n\nIn the $M_{\\star}-$Z plane, SDSS SFGs show a clear positive trend which\nflattens toward higher masses and show a relatively large scatter at\nlower masses, in agreement with previous findings for local galaxies\n\\citep[e.g.,][]{Tremonti}.  Similarly, GPs and LBAs increase their\nmetallicity with increasing masses.  Nevertheless, GPs lie more than a\nfactor of 2 ($\\ga$ 0.3 dex) below the MZR of SDSS SFGs, i.e. {\\it at\n  a fixed stellar mass GPs are systematically more metal-poor than\n  normal SFGs}. A similar shift in the MZR has been observed for the\nless massive (log(M$_{\\star}) \\la 10$) LBAs \\citep[][this\n  work]{Hoopes07,Overzier09b}.  In terms of stellar mass, the observed\noffset (as large as 1 dex) largely exceeds the typical uncertainty\nquoted for the GP masses.\n\nOn the other hand, the SDSS SFGs display a correlation between N\/O and\n$M_{\\star}$, with higher N\/O ratios at higher stellar masses, in\nagreement with PMC09 for a different sample of local SFGs.  The\nexistence of the $M_{\\star}-$N\/O relation reflects the fact that the most\nmassive galaxies evolve more quickly and, hence, they should have on\naverage higher metallicities and N\/O ratios. Though the scatter in the\n$M_{\\star}-$N\/O relation is large, most GPs and low-mass LBAs are\nroughly consistent with the trend of SDSS SFGs, and no systematic\noffset is observed.\n\n\\section{DISCUSSION}\n\nThe shape of the MZR in galaxies has been found to depend on several\nkey processes in galaxy evolution, including the efficiency of star\nformation \\citep[e.g.,][]{Lequeux79,MollaDiaz05,Brooks07,Calura09},\nthe action of metal-rich selective outflows and metal-poor gas inflows\n\\citep[e.g.,][]{Larson74,Garnett02,Tremonti,Finlator08}, and the\npossible variations in the initial mass function \\citep{Koppen07}.\nMoreover, different amounts of dark matter can also assist in some of\nthese mechanisms \\citep{Dekel86}.\n\nThe GPs follow a relation between mass and metallicity that parallels\nthe MZR defined by the SDSS SFGs, but is offset $\\ga$0.3~dex to lower\nmetallicities.  Interestingly, we find some remarkable similarities\nbetween the GPs and the population of [\\oiii]-selected galaxies of\nsimilar luminosity in the $z$ range 0.29--0.42 recently found by\n\\citet{Salzer09}.  These galaxies follow a luminosity$-$metallicity\nrelation that parallels the one defined by SFGs, but is offset by a\nfactor of more than 10 to lower abundances.  On the other hand, since\nGPs and low-mass LBAs lie in a similar offset position in the MZR\nrelative to normal SFGs, their mentioned similarities appear even\ngreater. One possibility to explain their offset position in the MZR\nis that these galaxies could be still converting a large amount of\ntheir cold gas reservoirs into stars.  In that case, their low\nabundances could be due to their relatively young ages compared to\nnormal SFGs. In the same range of masses, both GPs and LBAs have much\nhigher SSFR (typically $>$10$^{-9}$yr$^{-1}$) compared to other SFGs\nof similar mass (C09).  Recent studies show that galaxies with higher\nSSFRs or larger half-light radii for their stellar mass have\nsystematically lower metallicities \\citep[e.g.,][]{Tremonti,\n  Ellison08}.  However, we have found even greater under-abundances in\nthe GPs, which have high SSFRs but are extremely compact.\n\nSome models show that in highly concentrated (typical sizes $<$3~kpc)\nlow-mass galaxies such as the GPs, galactic winds induced by their\nlarge SSFR are strong enough to escape from their weak potential\nwells, diminishing the observed global abundances\n\\citep[e.g.,][]{Finlator08}. In contrast, analytical models by\n\\citet{Dalcanton07} show that any subsequent star formation to the\noutflow will remove their effects on metallicity, unless galaxies have\nan inefficient star formation. Smoothed particle hydrodynamics (SPH) \nplus $N$-body simulations have\nshown that low star formation efficiencies, regulated by supernova\n(SN) feedback, could be primarily responsible for the lower\nmetallicities of low-mass galaxies and their overall trend in the MZR\n\\citep{Brooks07}.  As shown by \\citet{Erb06} for SFGs at $z \\sim 2$,\nthe constancy in the offset of the MZR suggests the presence of\nselective metal-rich gas loss driven by SN winds.\n\nInflow of metal-poor gas, either from the outskirts of the galaxy or\nbeyond, can dilute metals in the galaxy centers, explaining an offset\nto lower abundances in both the MZR\n\\citep{Mihos96,Barnes96,Finlator08} and the N\/O -- O\/H diagram. In\nstarburst galaxies, a recent cold gas accretion can be due to\ninteractions, which eventually increases the gas surface density and\nconsequently the star formation.  As explained by \\citet{Ellison08},\nthe dilution of metals due to an inflow can be restored by the effects\nof star formation depending on the dilution-to-dynamical timescale\nratio.  Since this ratio depends inversely with galaxy radius,\ngalaxies with smaller radius, such as the GPs, may be expected to take\nlonger time to enhance their oxygen abundances to the values expected\nfrom the MZR.  In this line, the position of GPs in the\n$M_{\\star}-$N\/O relation and the offset observed in the N\/O -- O\/H\nplane may favor this scenario.  Models by \\citet{Koppen05} have shown\nthat the rapid decrease of the oxygen abundance during an episode of\nmassive and rapid accretion of metal-poor gas is followed by a slower\nevolution which leads to the closed-box relation, thus forming a loop\nin the N\/O -- O\/H diagram.\n\nThe inflow hypothesis is also strongly suggested by the disturbed\nmorphologies and close companions observed in spatially resolved {\\em HST}\nimages for three GPs and most LBAs \\citep[C09;][]{Overzier08}.  Recent\nresults revealed that galaxies involved in galaxy interactions fall\n$\\ga$0.2 dex below the MZR of normal galaxies due to tidally induced\nlarge-scale gas inflow to the galaxies' central regions\n\\citep[e.g.,][]{Kewley06,Michel-Dansac08,Peeples09}.  Several\n$N$-body\/SPH simulations have shown that major interactions drive\nstarbursts and gas inflow from the outskirts of the H {\\sc i}\nprogenitor disks \\citep[e.g.,][]{Mihos96,Rupke10}, also supporting\nthis scenario.\n\nWe conclude arguing that recent interaction-induced inflow of gas,\npossibly coupled with a selective metal-rich gas loss driven by SN\nwinds, may explain our findings and the known galaxy properties.\nNevertheless, further work is needed to constrain this possible\nscenario.  In particular, future assessment of the H {\\sc i} gas\nproperties and the star formation efficiency of GPs, as well as the\nbehavior of effective yields with mass compared with models of\nchemical evolution, will shed new light on the relative importance of\nthe above processes.\n\nOur results allow us to further constrain the evolutionary status of the\nGPs. These galaxies, as well as the low-mass LBAs and the\n[\\oiii]-selected galaxies by \\citet{Salzer09} should be analyzed and\ncompared in more detail to elucidate whether these rare objects are\nsharing similar evolutionary pathways.  Even if this is not the case,\ntheir properties suggest that these galaxies are snapshots of an\nextreme and short phase in their evolution.  They therefore offer the\nopportunity of studying in great detail the physical processes\ninvolved in the triggering and evolution of massive star formation and\nthe chemical enrichment processes, under physical conditions\napproaching those in galaxies at higher redshifts.\n\nForthcoming analysis, based on high S\/N intermediate\/high-resolution\nspectroscopy and deep NIR imaging with the Gran Telescopio Canarias (GTC), \nwill be used to better constrain the evolutionary status of the GPs.\n\n\\acknowledgments\n\nWe are very grateful to P. Papaderos, M. Moll\\'a and Y. Tsamis for\nvery stimulating discussions and useful suggestions to improve this\nmanuscript. We also thank the anonymous referee for a useful and prompt \nreport. This work has been funded by grants AYA2007-67965-C03-02,\nand CSD2006-00070: First Science with the GTC ({\\url\n  http:\/\/www.iac.es\/consolider-ingenio-gtc\/}) of the\nConsolider-Ingenio 2010 Program, by the Spanish MICINN.\n\n{\\it Facility:} \\facility{Sloan}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Analysis}\n\\label{sec:analysis}\n\n\\emph{How ``easier'' for CG is $\\mathcal{W} - U_1 \\Lambda_1 U_1^{\\!\\top}$ in~(\\ref{eq:h-temporal}) compared to $\\mathcal{W}$ in~(\\ref{eq:temporal})?}\n\nIn solving a linear system $A \\mathbf{x} = \\mathbf{b}$ where $A$ is an $n \\times n$ symmetric positive-definite matrix, CG generates a unique sequence of iterates $\\mathbf{x}_i$, $i=0,1,\\dots$, such that the $A$-norm $\\norm{\\mathbf{e}_i}_A$ of the error $\\mathbf{e}_i \\mathrel{\\operatorname{:=}} \\mathbf{x}^* - \\mathbf{x}_i$ is minimized over the Krylov subspace $\\mathcal{K}_i \\mathrel{\\operatorname{:=}} \\gen{\\mathbf{b}, A \\mathbf{b}, \\dots, A^i \\mathbf{b}}$ at each iteration $i$, where $\\mathbf{x}^* \\mathrel{\\operatorname{:=}} A^{-1} \\mathbf{b}$ is the exact solution and the $A$-norm is defined by $\\norm{\\mathbf{x}}_A \\mathrel{\\operatorname{:=}} \\sqrt{\\mathbf{x}^{\\!\\top} A \\mathbf{x}}$ for $\\mathbf{x} \\in \\mathbb{R}^n$.\n\nA well-known result on the rate of convergence of CG that assumes minimal knowledge of the eigenvalues of $A$ states that the $A$-norm of the error at iteration $i$, relative to the $A$-norm of the initial error $\\mathbf{e}_0 \\mathrel{\\operatorname{:=}} \\mathbf{x}^*$, is upper-bounded by~\\cite{Tref97}\n\\begin{equation}\n\t\\frac{\\norm{\\mathbf{e}_i}_A}{\\norm{\\mathbf{e}_0}_A} \\le \\phi_i(A) \\mathrel{\\operatorname{:=}} 2 \\left(\n\t\t\t\\frac{\\sqrt{\\kappa(A)} - 1}{\\sqrt{\\kappa(A)} + 1}\n\t\t\\right)^i,\n\\label{eq:bound}\n\\end{equation}\nwhere $\\kappa(A) \\mathrel{\\operatorname{:=}} \\norm{A} \\norm{A^{-1}} = \\lambda_1(A) \/ \\lambda_n(A)$ is the $2$-norm \\emph{condition number} of $A$, and $\\lambda_j(A)$ for $j = 1,\\dots,n$ are the eigenvalues of $A$ in descending order.\n\nIn our case, matrix $\\mathcal{L}_\\alpha(\\mathcal{W})$ of linear system~(\\ref{eq:temporal}) has condition number\n\\begin{equation}\n\t\\kappa(\\mathcal{L}_\\alpha(\\mathcal{W}))\n\t\t= \\frac{1 - \\alpha \\lambda_n(\\mathcal{W})}{1 - \\alpha \\lambda_1(\\mathcal{W})}\n\t\t= \\frac{1 - \\alpha \\lambda_n(\\mathcal{W})}{1 - \\alpha}.\n\\label{eq:cond}\n\\end{equation}\nThe first equality holds because for each eigenvalue $\\lambda$ of $\\mathcal{W}$ there is a corresponding eigenvalue $(1 - \\alpha \\lambda) \/ (1 - \\alpha)$ of $\\mathcal{L}_\\alpha(\\mathcal{W})$, which is a decreasing function. The second holds because $\\lambda_1(\\mathcal{W}) = 1$~\\cite{Chun97}.\n\nNow, let $\\mathcal{W}_r \\mathrel{\\operatorname{:=}} \\mathcal{W} - U_1 \\Lambda_1 U_1^{\\!\\top}$ for $r = 0,1,\\dots,n-1$, where $\\Lambda_1$, $U_1$ represent the largest $r$ eigenvalues and the corresponding eigenvectors of $\\mathcal{W}$ respectively. Clearly, $\\mathcal{W}_r$ has the same eigenvalues as $\\mathcal{W}$ except for the largest $r$, which are replaced by zero. That is, $\\lambda_1(\\mathcal{W}_r) = \\lambda_{r+1}(\\mathcal{W})$ and $\\lambda_n(\\mathcal{W}_r) = \\lambda_n(\\mathcal{W})$. The latter is due to the fact that $\\lambda_n(\\mathcal{W}) \\le -1 \/ (n-1) \\le 0$~\\cite{Chun97}, so the new zero eigenvalues do not affect the smallest one. Then,\n\\begin{equation}\n\t\\kappa(\\mathcal{L}_\\alpha(\\mathcal{W}_r))\n\t\t= \\frac{1 - \\alpha \\lambda_n(\\mathcal{W})}{1 - \\alpha \\lambda_{r+1}(\\mathcal{W})}\n\t\t\\le \\kappa(\\mathcal{L}_\\alpha(\\mathcal{W})).\n\\label{eq:cond-r}\n\\end{equation}\nThis last expression generalizes~(\\ref{eq:cond}). Indeed, $\\mathcal{W} = \\mathcal{W}_0$. Then, our hybrid spectral-temporal filtering involves CG on $\\mathcal{L}_\\alpha(\\mathcal{W}_r)$ for $r \\ge 0$, compared to the baseline temporal filtering for $r = 0$. The inequality in~(\\ref{eq:cond-r}) is due to the fact that $|\\lambda_j(\\mathcal{W})| \\le 1$ for $j = 1,\\dots,n$~\\cite{Chun97}. Removing the largest $r$ eigenvalues of $\\mathcal{W}$ clearly improves (decreases) the condition number of $\\mathcal{L}_\\alpha(\\mathcal{W}_r)$ relative to $\\mathcal{L}_\\alpha(\\mathcal{W})$. The improvement is dramatic given that $\\alpha$ is close to $1$ in practice. For $\\alpha = 0.99$ and $\\lambda_{r+1}(\\mathcal{W}) = 0.7$ for instance, $\\kappa(\\mathcal{L}_\\alpha(\\mathcal{W}_r)) \/ \\kappa(\\mathcal{L}_\\alpha(\\mathcal{W})) = 0.0326$.\n\n\\begin{figure}[b!]\n\\vspace{-6pt}\n\\begin{tabular}{cc}\n\\begin{tikzpicture}\n\\begin{axis}[\n\tgrid=both,\n\twidth=.45\\textwidth,\n\theight=.4\\textwidth,\n\tenlargelimits=false,\n\txlabel={order $j$},\n\tylabel={eigenvalue $\\lambda_j(\\mathcal{W})$},\n]\n\\addplot[blue] table{figs\/rate\/eig.txt};\n\\addplot[red] coordinates {(300,-1) (300,1)};\n\\end{axis}\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}\n\\begin{axis}[\n\twidth=.55\\textwidth,\n\theight=.4\\textwidth,\n\tenlargelimits=false,\n\txlabel={rank $r$ (space)},\n\tylabel={iteration $i$ (time)},\n]\n\\addplot[\n\tcontour prepared={\n\t\tlabels over line,\n\t\tlabel distance=70pt,\n\t\tcontour label style={font=\\tiny},\n\t},\n\tcontour prepared format=matlab,\n] table{figs\/rate\/contour.txt};\n\\end{axis}\n\\end{tikzpicture}\n\\\\\n(a) & (b)\n\\end{tabular}\n\\vspace{-6pt}\n\\caption{(a) In descending order, eigenvalues of adjacency matrix $\\mathcal{W}$ of Oxford5k dataset of $n = 5,063$ images with global GeM features by ResNet101\nand $k = 50$ neighbors per point (see section~\\ref{sec:exp}). Eigenvalues on the left of vertical red line at $j=300$ are the largest $300$ ones, candidate for removal. (b) Contour plot of upper bound $\\phi_i(\\mathcal{L}_\\alpha(\\mathcal{W}_r))$ of CG's relative error as a function of rank $r$ and iteration $i$ for $\\alpha = 0.99$, illustrating the space($r$)-time($i$) trade-off for constant relative error.}\n\\label{fig:analysis}\n\\end{figure}\n\nMore generally, given the eigenvalues $\\lambda_{r+1}(\\mathcal{W})$ and $\\lambda_n(\\mathcal{W})$, the improvement can be estimated by measuring the upper bound $\\phi_i(\\mathcal{L}_\\alpha(\\mathcal{W}_r))$ for different $i$ and $r$. A concrete example is shown in Figure~\\ref{fig:analysis}, where we measure the eigenvalues of the adjacency matrix $\\mathcal{W}$ of a real dataset, remove the largest $r$ for $0 \\le r \\le 300$ and plot the upper bound $\\phi_i(\\mathcal{L}_\\alpha(\\mathcal{W}_r))$ of the relative error as a function of rank $r$ and iteration $i$ given by~(\\ref{eq:bound}) and~(\\ref{eq:cond-r}). Clearly, as more eigenvalues are removed, less CG iterations are needed to achieve the same relative error; the approximation error represented by the temporal term decreases and at the same time the linear system becomes easier to solve. Of course, iterations become more expensive as $r$ increases; precise timings are given in section~\\ref{sec:exp}.\n\n\n\\section{Conclusions} \\label{sec:conclusions}\n\nIn this work we have tested the two most successful manifold ranking methods of temporal filtering~\\cite{ITA+17} and spectral filtering~\\cite{IAT+18} on the very challenging new benchmark of Oxford and Paris datasets~\\cite{RIT+18}. It is the first time that such methods are evaluated at the scale of one million images. \\emph{Spectral filtering}, with both its FSR and FSRw variants, fails at this scale, despite the significant space required for additional vector embeddings. It is possible that a higher rank would work, but it wouldn't be practical in terms of space. In terms of query time, \\emph{temporal filtering} is only practical with its truncated variant at this scale. It works pretty well in terms of performance, but the query time is still high.\n\nOur new \\emph{hybrid filtering} method allows for the first time to strike a reasonable balance between the two extremes. Without truncation, it outperforms temporal filtering while being significantly faster, and its memory overhead is one order of magnitude less than that of spectral filtering. Unlike spectral filtering, it is possible to extremely sparsify the dataset embeddings with only negligible drop in performance. This, together with its very low rank, makes our hybrid method even faster than spectral, despite being iterative. More importantly, while previous methods were long known in other fields before being applied to image retrieval, to our knowledge our hybrid method is novel and can apply \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot to any field where graph signal processing applies and beyond. Our theoretical analysis shows exactly why our method works and quantifies its space-time-accuracy trade-off using simple ideas from numerical linear algebra.\n\\subsection{Derivation \\alert{(version 1: inverse only)}}\n\nWe begin with the eigenvalue decomposition $\\mathcal{W} = U \\Lambda U^{\\!\\top}$, which we partition as $\\Lambda = \\operatorname{diag}(\\Lambda_1, \\Lambda_2)$ and $U = (U_1 \\ U_2)$. Matrices $\\Lambda_1$ and $\\Lambda_2$ are diagonal $r \\times r$ and $(n-r) \\times (n-r)$, respectively. Matrices $U_1$ and $U_2$ are $n \\times r$ and $n \\times (n-r)$, respectively, and have the following orthogonality properties, all due to the orthogonality of $U$ itself:\n\\begin{equation}\n\tU_1^{\\!\\top} U_1 = I_r, \\quad U_2^{\\!\\top} U_2 = I_{n-r}, \\quad\n\tU_1^{\\!\\top} U_2 = \\mathbf{O}, \\quad U_1 U_1^{\\!\\top} + U_2 U_2^{\\!\\top} = I_n.\n\\label{eq:ortho}\n\\end{equation}\nThen, $\\mathcal{W}$ is decomposed as\n\\begin{equation}\n\t\\mathcal{W} = U_1 \\Lambda_1 U_1^{\\!\\top} + U_2 \\Lambda_2 U_2^{\\!\\top}.\n\\label{eq:w-decomp}\n\\end{equation}\nSimilarly, $h_\\alpha(\\mathcal{W})$ is decomposed as\n\\begin{align}\n\th_\\alpha(\\mathcal{W})\n\t\t& = U h_\\alpha(\\Lambda) U^{\\!\\top}\n\t\t\t\\label{eq:h-decomp-1} \\\\\n\t\t& = U_1 h_\\alpha(\\Lambda_1) U_1^{\\!\\top} + U_2 h_\\alpha(\\Lambda_2) U_2^{\\!\\top},\n\t\t\t\\label{eq:h-decomp-2}\n\\end{align}\nwhich is due to the fact that diagonal matrix $h_\\alpha(\\Lambda)$ is obtained by element-wise application, hence decomposed as $h_\\alpha(\\Lambda) = \\operatorname{diag}(h_\\alpha(\\Lambda_1), h_\\alpha(\\Lambda_2))$. Here the first term is exactly the low-rank approximation that is used by spectral filtering, and the second is the approximation error\n\\begin{align}\n\te_\\alpha(\\mathcal{W})\n\t\t& \\mathrel{\\operatorname{:=}} U_2 h_\\alpha(\\Lambda_2) U_2^{\\!\\top}\n\t\t\t\\label{eq:error-1} \\\\\n\t\t& = (1 - \\alpha) U_2 (I_{n-r} - \\alpha \\Lambda_2)^{-1} U_2^{\\!\\top}\n\t\t\t\\label{eq:error-2} \\\\\n\t\t& = (1 - \\alpha) \\left(\n\t\t\t\t\\left( I_n - \\alpha U_2 \\Lambda_2 U_2^{\\!\\top} \\right)^{-1} - U_1 U_1^{\\!\\top}\n\t\t\t\\right)\n\t\t\t\\label{eq:error-3} \\\\\n\t\t& = h_\\alpha(U_2 \\Lambda_2 U_2^{\\!\\top}) - (1 - \\alpha) U_1 U_1^{\\!\\top}.\n\t\t\t\\label{eq:error-4}\n\\end{align}\nWe have used the definition~(\\ref{eq:transfer}) of $h_\\alpha$ in~(\\ref{eq:error-2}) and~(\\ref{eq:error-4}). Equation~(\\ref{eq:error-3}) can be verified by direct multiplication\n\\begin{align}\n\t& \\left( U_1 U_1^{\\!\\top} + U_2 (I_{n-r} - \\alpha \\Lambda_2)^{-1} U_2^{\\!\\top} \\right)\n\t\t(I_n - \\alpha U_2 \\Lambda_2 U_2^{\\!\\top})\n\t\t\t\\label{eq:verify-1} \\\\\n\t& = U_1 U_1^{\\!\\top} +\n\t    U_2 (I_{n-r} - \\alpha\\Lambda_2)^{-1} U_2^{\\!\\top} -\n\t    U_2 (I_{n-r} - \\alpha\\Lambda_2)^{-1} \\alpha\\Lambda_2 U_2^{\\!\\top}\n\t\t\t\\label{eq:verify-2} \\\\\n\t& = U_1 U_1^{\\!\\top} +\n\t    U_2 (I_{n-r} - \\alpha\\Lambda_2)^{-1} (I_{n-r} - \\alpha\\Lambda_2) U_2^{\\!\\top}\n\t\t\t\\label{eq:verify-3} \\\\\n\t& = U_1 U_1^{\\!\\top} + U_2 U_2^{\\!\\top}\n\t\t\t\\label{eq:verify-4} \\\\\n\t& = I_n,\n\t\t\\label{eq:verify-6}\n\\end{align}\nwhere we have used the orthogonality properties~(\\ref{eq:ortho}) in the expansion~(\\ref{eq:verify-2}) and in~(\\ref{eq:verify-6}). A similar verification holds for the opposite order of multiplication.\n\nNow, combining~(\\ref{eq:h-decomp-2}),~(\\ref{eq:error-4}) and~(\\ref{eq:w-decomp}), we have proved the following.\n\n\\begin{theorem}\n\tAssuming the definition~(\\ref{eq:transfer}) of transfer function $h_\\alpha$ and the eigenvalue decomposition~(\\ref{eq:w-decomp}) of the symmetrically normalized adjacency matrix $\\mathcal{W}$, $h_\\alpha(\\mathcal{W})$ is decomposed as\n\t%\n\t\\begin{equation}\n\t\th_\\alpha(\\mathcal{W}) = U_1 g_\\alpha(\\Lambda_1) U_1^{\\!\\top} + h_\\alpha(\\mathcal{W} - U_1 \\Lambda_1 U_1^{\\!\\top}),\n\t\\label{eq:main}\n\t\\end{equation}\n\t%\n\twhere\n\t%\n\t\\begin{equation}\n\t\tg_\\alpha(A) \\mathrel{\\operatorname{:=}} (1 - \\alpha) \\left( (I_n - \\alpha A)^{-1} - I_n \\right)\n\t\\label{eq:aux}\n\t\\end{equation}\n\t%\n\tfor $n \\times n$ real symmetric matrix $A$. For $x \\in \\mathbb{R}$ in particular, $g_\\alpha(x) \\mathrel{\\operatorname{:=}}  (1 - \\alpha) \\alpha x \/ (1 - \\alpha x)$.\n\\end{theorem}\n\n\n\\subsection{Derivation}\n\\label{sec:deriv}\n\nWe begin with the eigenvalue decomposition $\\mathcal{W} = U \\Lambda U^{\\!\\top}$, which we partition as $\\Lambda = \\operatorname{diag}(\\Lambda_1, \\Lambda_2)$ and $U = (U_1 \\ U_2)$. Matrices $\\Lambda_1$ and $\\Lambda_2$ are diagonal $r \\times r$ and $(n-r) \\times (n-r)$, respectively. Matrices $U_1$ and $U_2$ are $n \\times r$ and $n \\times (n-r)$, respectively, and have the following orthogonality properties, all due to the orthogonality of $U$ itself:\n\\begin{equation}\n\tU_1^{\\!\\top} U_1 = I_r, \\quad U_2^{\\!\\top} U_2 = I_{n-r}, \\quad\n\tU_1^{\\!\\top} U_2 = \\mathbf{O}, \\quad U_1 U_1^{\\!\\top} + U_2 U_2^{\\!\\top} = I_n.\n\\label{eq:ortho}\n\\end{equation}\nThen, $\\mathcal{W}$ is decomposed as\n\\begin{equation}\n\t\\mathcal{W} = U_1 \\Lambda_1 U_1^{\\!\\top} + U_2 \\Lambda_2 U_2^{\\!\\top}.\n\\label{eq:w-decomp}\n\\end{equation}\nSimilarly, $h_\\alpha(\\mathcal{W})$ is decomposed as\n\\begin{align}\n\th_\\alpha(\\mathcal{W})\n\t\t& = U h_\\alpha(\\Lambda) U^{\\!\\top}\n\t\t\t\\label{eq:h-decomp-1} \\\\\n\t\t& = U_1 h_\\alpha(\\Lambda_1) U_1^{\\!\\top} + U_2 h_\\alpha(\\Lambda_2) U_2^{\\!\\top},\n\t\t\t\\label{eq:h-decomp-2}\n\\end{align}\nwhich is due to the fact that diagonal matrix $h_\\alpha(\\Lambda)$ is obtained element-wise, hence decomposed as $h_\\alpha(\\Lambda) = \\operatorname{diag}(h_\\alpha(\\Lambda_1), h_\\alpha(\\Lambda_2))$. Here the first term is exactly the low-rank approximation that is used by spectral filtering, and the second is the approximation error\n\\begin{align}\n\te_\\alpha(\\mathcal{W})\n\t\t& \\mathrel{\\operatorname{:=}} U_2 h_\\alpha(\\Lambda_2) U_2^{\\!\\top}\n\t\t\t\\label{eq:error-1} \\\\\n\t\t& = (1 - \\alpha) \\left(\n\t\t\t\tU_2 (I_{n-r} - \\alpha \\Lambda_2)^{-1} U_2^{\\!\\top} + U_1 U_1^{\\!\\top} - U_1 U_1^{\\!\\top}\n\t\t\t\\right)\n\t\t\t\\label{eq:error-2} \\\\\n\t\t& = (1 - \\alpha) \\left(\n\t\t\t\t\\left(U_2 (I_{n-r} - \\alpha \\Lambda_2) U_2^{\\!\\top} + U_1 U_1^{\\!\\top} \\right)^{-1}\n\t\t\t\t- U_1 U_1^{\\!\\top}\n\t\t\t\\right)\n\t\t\t\\label{eq:error-3} \\\\\n\t\t& = (1 - \\alpha) \\left(\n\t\t\t\t\\left( I_n - \\alpha U_2 \\Lambda_2 U_2^{\\!\\top} \\right)^{-1} - U_1 U_1^{\\!\\top}\n\t\t\t\\right)\n\t\t\t\\label{eq:error-4} \\\\\n\t\t& = h_\\alpha(U_2 \\Lambda_2 U_2^{\\!\\top}) - (1 - \\alpha) U_1 U_1^{\\!\\top}.\n\t\t\t\\label{eq:error-5}\n\\end{align}\nWe have used the definition~(\\ref{eq:transfer}) of $h_\\alpha$ in~(\\ref{eq:error-2}) and~(\\ref{eq:error-5}). Equation~(\\ref{eq:error-4}) is due to the orthogonality properties~(\\ref{eq:ortho}). Equation~(\\ref{eq:error-3}) follows from the fact that for any invertible matrices $A$, $B$ of conformable sizes,\n\\begin{equation}\n\t\\left( U_1 A U_1 + U_2 B U_2 \\right)^{-1} = U_1 A^{-1} U_1 + U_2 B^{-1} U_2,\n\\label{eq:inversion}\n\\end{equation}\nwhich can be verified by direct multiplication, and is also due to orthogonality.\n\nNow, combining~(\\ref{eq:h-decomp-2}),~(\\ref{eq:error-5}) and~(\\ref{eq:w-decomp}), we have proved the following.\n\n\\begin{theorem}\n\tAssuming the definition~(\\ref{eq:transfer}) of transfer function $h_\\alpha$ and the eigenvalue decomposition~(\\ref{eq:w-decomp}) of the symmetrically normalized adjacency matrix $\\mathcal{W}$, $h_\\alpha(\\mathcal{W})$ is decomposed as\n\t%\n\t\\begin{equation}\n\t\th_\\alpha(\\mathcal{W}) = U_1 g_\\alpha(\\Lambda_1) U_1^{\\!\\top} + h_\\alpha(\\mathcal{W} - U_1 \\Lambda_1 U_1^{\\!\\top}),\n\t\\label{eq:main}\n\t\\end{equation}\n\t%\n\twhere\n\t%\n\t\\begin{equation}\n\t\tg_\\alpha(A)\n\t\t\t\\mathrel{\\operatorname{:=}} h_\\alpha(A) - h_\\alpha(\\mathbf{O})\n\t\t\t= (1 - \\alpha) \\left( (I_n - \\alpha A)^{-1} - I_n \\right)\n\t\\label{eq:aux}\n\t\\end{equation}\n\t%\n\tfor $n \\times n$ real symmetric matrix $A$. For $x \\in [-1,1]$ in particular, $g_\\alpha(x) \\mathrel{\\operatorname{:=}} h_\\alpha(x) - h_\\alpha(0) = (1 - \\alpha) \\alpha x \/ (1 - \\alpha x)$.\n\\end{theorem}\n\nObserve that $\\Lambda_2, U_2$ do not appear in~(\\ref{eq:main}) and indeed it is only the largest $r$ eigenvalues $\\Lambda_1$ and corresponding eigenvectors $U_1$ of $\\mathcal{W}$ that we need to compute. The above derivation is generalized from $h_\\alpha$ to a much larger class of functions in appendix~\\ref{sec:deriv2}.\n\n\\subsection{Derivation \\alert{(version 2: series)}}\n\nWe begin with the eigenvalue decomposition $\\mathcal{W} = U \\Lambda U^{\\!\\top}$, which we partition as $\\Lambda = \\operatorname{diag}(\\Lambda_1, \\Lambda_2)$ and $U = (U_1 \\ U_2)$. Matrices $\\Lambda_1$ and $\\Lambda_2$ are diagonal $r \\times r$ and $(n-r) \\times (n-r)$, respectively. Matrices $U_1$ and $U_2$ are $n \\times r$ and $n \\times (n-r)$, respectively, and have the following orthogonality properties, all due to the orthogonality of $U$ itself:\n\\begin{equation}\n\tU_1^{\\!\\top} U_1 = I_r, \\quad U_2^{\\!\\top} U_2 = I_{n-r}, \\quad\n\tU_1^{\\!\\top} U_2 = \\mathbf{O}, \\quad U_1 U_1^{\\!\\top} + U_2 U_2^{\\!\\top} = I_n.\n\\label{eq:2-ortho}\n\\end{equation}\nThen, $\\mathcal{W}$ is decomposed as\n\\begin{equation}\n\t\\mathcal{W} = U_1 \\Lambda_1 U_1^{\\!\\top} + U_2 \\Lambda_2 U_2^{\\!\\top}.\n\\label{eq:2-w-decomp}\n\\end{equation}\nNow, as in~\\cite{IAT+18}, we generalize $h_\\alpha$ to any function $h$ that has a series expansion\n\\begin{equation}\n\th(A) = \\sum_{i=0}^\\infty c_i A^i.\n\\label{eq:series}\n\\end{equation}\nThen, assuming that $h(\\mathcal{W})$ converges absolutely, it is similarly decomposed as\n\\begin{align}\n\th(\\mathcal{W})\n\t\t& = U h(\\Lambda) U^{\\!\\top}\n\t\t\t\\label{eq:2-h-decomp-1} \\\\\n\t\t& = U_1 h(\\Lambda_1) U_1^{\\!\\top} + U_2 h(\\Lambda_2) U_2^{\\!\\top},\n\t\t\t\\label{eq:2-h-decomp-2}\n\\end{align}\nwhich is due to the fact that diagonal matrix $h_\\alpha(\\Lambda)$ is obtained by element-wise application, hence decomposed as $h_\\alpha(\\Lambda) = \\operatorname{diag}(h_\\alpha(\\Lambda_1), h_\\alpha(\\Lambda_2))$. Here the first term is exactly the low-rank approximation that is used by spectral filtering, and the second is the approximation error\n\\begin{align}\n\te_\\alpha(\\mathcal{W})\n\t\t& \\mathrel{\\operatorname{:=}} U_2 h(\\Lambda_2) U_2^{\\!\\top}\n\t\t\t\\label{eq:2-error-1} \\\\\n\t\t& = \\sum_{i=0}^\\infty c_i U_2 \\Lambda_2^i U_2^{\\!\\top}\n\t\t\t\\label{eq:2-error-2} \\\\\n\t\t& = \\sum_{i=0}^\\infty c_i \\left( U_2 \\Lambda_2 U_2^{\\!\\top} \\right)^i - c_0 U_1 U_1^{\\!\\top}\n\t\t\t\\label{eq:2-error-3} \\\\\n\t\t& = h(U_2 \\Lambda_2 U_2^{\\!\\top}) - h(0) U_1 U_1^{\\!\\top}.\n\t\t\t\\label{eq:2-error-4}\n\\end{align}\nWe have used the series expansion~(\\ref{eq:series}) of $h$ in~(\\ref{eq:2-error-2}) and~(\\ref{eq:2-error-4}). Equation~(\\ref{eq:2-error-3}) is due to the fact that\n\\begin{equation}\n\t(U_2 \\Lambda_2 U_2^{\\!\\top})^i = U_2 \\Lambda_2^i U_2^{\\!\\top}\n\\label{eq:term-pos}\n\\end{equation}\nfor $i \\ge 1$, as can be verified by induction, while for $i = 0$,\n\\begin{equation}\n\tU_2 \\Lambda_2^0 U_2^{\\!\\top} = U_2 U_2^{\\!\\top} = I_n - U_1 U_1^{\\!\\top} = (U_2 \\Lambda_2 U_2^{\\!\\top})^0 - U_1 U_1^{\\!\\top}.\n\\label{eq:term-zero}\n\\end{equation}\nIn both~(\\ref{eq:term-pos}) and~(\\ref{eq:term-zero}) we have used the orthogonality properties~(\\ref{eq:2-ortho}).\n\nNow, combining~(\\ref{eq:2-h-decomp-2}),~(\\ref{eq:2-error-4}) and~(\\ref{eq:2-w-decomp}), we have proved the following.\n\n\\begin{theorem}\n\tAssuming the series expansion~(\\ref{eq:series}) of transfer function $h$ and the eigenvalue decomposition~(\\ref{eq:2-w-decomp}) of the symmetrically normalized adjacency matrix $\\mathcal{W}$, and given that $h(\\mathcal{W})$ converges absolutely, it is decomposed as\n\t%\n\t\\begin{equation}\n\t\th(\\mathcal{W}) = U_1 g(\\Lambda_1) U_1^{\\!\\top} + h(\\mathcal{W} - U_1 \\Lambda_1 U_1^{\\!\\top}),\n\t\\label{eq:2-main}\n\t\\end{equation}\n\t%\n\twhere\n\t%\n\t\\begin{equation}\n\t\tg(A) \\mathrel{\\operatorname{:=}} h(A) - h(\\mathbf{O})\n\t\\label{eq:2-aux}\n\t\\end{equation}\n\t%\n\tfor $n \\times n$ real symmetric matrix $A$. For $h = h_\\alpha$ and for $x \\in [-1,1]$ in particular, $g_\\alpha(x) \\mathrel{\\operatorname{:=}} h_\\alpha(x) - h_\\alpha(0) = (1 - \\alpha) \\alpha x \/ (1 - \\alpha x)$.\n\\end{theorem}\n\nObserve that $\\Lambda_2, U_2$ do not appear in~(\\ref{eq:main}) and indeed it is only the largest $r$ eigenvalues $\\Lambda_1$ and corresponding eigenvectors $U_1$ of $\\mathcal{W}$ that we need to compute.\n\n\\section{General derivation}\n\\label{sec:deriv2}\n\nThe derivation of our algorithm in section~\\ref{sec:deriv} applies only to the particular function (filter) $h_\\alpha$~(\\ref{eq:transfer}). Here, as in~\\cite{IAT+18}, we generalize to a much larger class of functions, that is, any function $h$ that has a series expansion\n\\begin{equation}\n\th(A) = \\sum_{i=0}^\\infty c_i A^i.\n\\label{eq:series}\n\\end{equation}\nWe begin with the same eigenvalue decomposition~(\\ref{eq:w-decomp}) of $\\mathcal{W}$ and, assuming that $h(\\mathcal{W})$ converges absolutely, its corresponding decomposition\n\\begin{equation}\n\th(\\mathcal{W}) = U_1 h(\\Lambda_1) U_1^{\\!\\top} + U_2 h(\\Lambda_2) U_2^{\\!\\top},\n\\label{eq:2-h-decomp}\n\\end{equation}\nsimilar to~(\\ref{eq:h-decomp-2}), where $U_1$, $U_2$ have the same orthogonality properties~(\\ref{eq:ortho}).\n\nAgain, the first term is exactly the low-rank approximation that is used by spectral filtering, and the second is the approximation error\n\\begin{align}\n\te_\\alpha(\\mathcal{W})\n\t\t& \\mathrel{\\operatorname{:=}} U_2 h(\\Lambda_2) U_2^{\\!\\top}\n\t\t\t\\label{eq:2-error-1} \\\\\n\t\t& = \\sum_{i=0}^\\infty c_i U_2 \\Lambda_2^i U_2^{\\!\\top}\n\t\t\t\\label{eq:2-error-2} \\\\\n\t\t& = \\sum_{i=0}^\\infty c_i \\left( U_2 \\Lambda_2 U_2^{\\!\\top} \\right)^i - c_0 U_1 U_1^{\\!\\top}\n\t\t\t\\label{eq:2-error-3} \\\\\n\t\t& = h(U_2 \\Lambda_2 U_2^{\\!\\top}) - h(0) U_1 U_1^{\\!\\top}.\n\t\t\t\\label{eq:2-error-4}\n\\end{align}\nAgain, we have used the series expansion~(\\ref{eq:series}) of $h$ in~(\\ref{eq:2-error-2}) and~(\\ref{eq:2-error-4}). Now, equation~(\\ref{eq:2-error-3}) is due to the fact that\n\\begin{equation}\n\t(U_2 \\Lambda_2 U_2^{\\!\\top})^i = U_2 \\Lambda_2^i U_2^{\\!\\top}\n\\label{eq:term-pos}\n\\end{equation}\nfor $i \\ge 1$, as can be verified by induction, while for $i = 0$,\n\\begin{equation}\n\tU_2 \\Lambda_2^0 U_2^{\\!\\top} = U_2 U_2^{\\!\\top} = I_n - U_1 U_1^{\\!\\top} = (U_2 \\Lambda_2 U_2^{\\!\\top})^0 - U_1 U_1^{\\!\\top}.\n\\label{eq:term-zero}\n\\end{equation}\nIn both~(\\ref{eq:term-pos}) and~(\\ref{eq:term-zero}) we have used the orthogonality properties~(\\ref{eq:ortho}).\n\nFinally, combining~(\\ref{eq:2-h-decomp}),~(\\ref{eq:2-error-4}) and~(\\ref{eq:w-decomp}), we have proved the following.\n\n\\begin{theorem}\n\tAssuming the series expansion~(\\ref{eq:series}) of transfer function $h$ and the eigenvalue decomposition~(\\ref{eq:w-decomp}) of the symmetrically normalized adjacency matrix $\\mathcal{W}$, and given that $h(\\mathcal{W})$ converges absolutely, it is decomposed as\n\t%\n\t\\begin{equation}\n\t\th(\\mathcal{W}) = U_1 g(\\Lambda_1) U_1^{\\!\\top} + h(\\mathcal{W} - U_1 \\Lambda_1 U_1^{\\!\\top}),\n\t\\label{eq:2-main}\n\t\\end{equation}\n\t%\n\twhere\n\t%\n\t\\begin{equation}\n\t\tg(A) \\mathrel{\\operatorname{:=}} h(A) - h(\\mathbf{O})\n\t\\label{eq:2-aux}\n\t\\end{equation}\n\t%\n\tfor $n \\times n$ real symmetric matrix $A$. For $h = h_\\alpha$ and for $x \\in [-1,1]$ in particular, $g_\\alpha(x) \\mathrel{\\operatorname{:=}} h_\\alpha(x) - h_\\alpha(0) = (1 - \\alpha) \\alpha x \/ (1 - \\alpha x)$.\n\\end{theorem}\n\nThis general derivation explains where the general definition of function $g$~(\\ref{eq:2-aux}) is coming from in~(\\ref{eq:aux}) corresponding to our treatment of $h_\\alpha$ in section~\\ref{sec:deriv}.\n\\section{Experiments}\n\\label{sec:exp}\n\nIn this section we evaluate our hybrid method on popular image retrieval benchmarks.\nWe provide comparisons to baseline methods, analyze the trade-off between runtime complexity, memory footprint and search accuracy, and compare with the state of the art.\n\n\\subsection{Experimental setup}\n\\label{sec:expSetup}\n\n\\head{Datasets.}\nWe use the revisited retrieval benchmark~\\cite{RIT+18} of the popular Oxford buildings~\\cite{PCISZ07} and Paris~\\cite{PCISZ08} datasets, referred to as $\\mathcal{R}$Oxford\\xspace and $\\mathcal{R}$Paris\\xspace, respectively.\nUnless otherwise specified, we evaluate using the \\emph{Medium} setup and always report mean Average Precision (mAP).\nLarge-scale experiments are conducted on $\\mathcal{R}$Oxford\\xspace+$\\mathcal{R}$1M\\xspace and $\\mathcal{R}$Paris\\xspace+$\\mathcal{R}$1M\\xspace by adding the new 1M challenging distractor set~\\cite{RIT+18}.\n\n\\head{Image representation.}\nWe use GeM descriptors~\\cite{RTC18} to represent images.\nWe extract GeM at 3 different image scales, aggregate the 3 descriptors, and perform whitening, exactly as in~\\cite{RTC18}.\nFinally, each image is represented by a single vector with $d = 2048$ dimensions, since ResNet-101 architecture is used.\n\n\\head{Baseline methods.} We consider the two baseline methods described in Section~\\ref{sec:background}, namely temporal and spectral filtering.\n\\emph{Temporal filtering} corresponds to solving a linear system with CG~\\cite{ITA+17} and is evaluated for different numbers of CG iterations. It is used with truncation at large scale to speed up the search~\\cite{ITA+17} and is denoted by \\emph{Temporal}$\\dagger$.\n\\emph{Spectral filtering} corresponds to FSR and its FSRw variant~\\cite{IAT+18}.\nBoth FSR variants are parametrized by the rank $r$ of the approximation, which is equal to the dimensionality of the spectral embedding.\n\n\\head{Implementation details.}\nTemporal ranking is performed with the implementation\\footnote{\\url{https:\/\/github.com\/ahmetius\/diffusion-retrieval\/}} provided by Iscen~\\emph{et al}\\onedot~\\cite{ITA+17}.\nThe adjacency matrix is constructed by using top $k=50$ reciprocal neighbors.\nPairwise similarity between descriptors $\\mathbf{v}$ and $\\mathbf{z}$ is estimated by $( \\mathbf{v}^{\\top}\\mathbf{z} )_+^3$.\nParameter $\\alpha$ is set to $0.99$, while the observation vector $\\mathbf{y}$ includes the top $5$  neighbors.\nThe eigendecomposition is performed on the largest connected component, as in~\\cite{IAT+18}.\nIts size is 933,412 and 934,809 for $\\mathcal{R}$Oxford\\xspace+$\\mathcal{R}$1M\\xspace and $\\mathcal{R}$Paris\\xspace+$\\mathcal{R}$1M\\xspace, respectively.\nTimings are measured with Matlab implementation on a 4-core Intel Xeon 2.60GHz CPU with 200 GB of RAM.\nWe only report timings for the diffusion part of the ranking and exclude the initial nearest neighbor search used to construct the observation vector.\n\n\\begin{figure}[t]\n\\vspace{-5pt}\n\\input{fig_main}\n\\vspace{-10pt}\n\\caption{mAP \\vs CG iteration $i$ and mAP \\vs time for temporal, spectral, and hybrid filtering.\nSparsified hybrid is used with sparsity $99$\\%.\n\\label{fig:mainexp}}\n\\end{figure}\n\n\\begin{figure}[t]\n\\input{fig_tradeoff2}\n\\vspace{-10pt}\n \\caption{mAP \\vs CG iteration $i$ for different rank $r$ for our hybrid method, where $r=0$ means temporal only.\n \\label{fig:tradeoff} }\n\\vspace{-10pt}\n\\end{figure}\n\n\\subsection{Comparisons}\n\\head{Comparison with baseline methods.} We first compare performance, query time and required memory for temporal, spectral, and hybrid ranking.\nWith respect to the memory, all methods store the initial descriptors, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot one $2048$-dimensional vector per image.\nTemporal ranking additionally stores the sparse regularized Laplacian. \nSpectral ranking stores for each vector an additional embedding of dimensionality equal to rank $r$, which is a parameter of the method.\nOur hybrid method stores both the Laplacian and the embedding, but with significantly lower rank $r$.\n\nWe evaluate on $\\mathcal{R}$Oxford\\xspace+$\\mathcal{R}$1M\\xspace and $\\mathcal{R}$Paris\\xspace+$\\mathcal{R}$1M\\xspace with global image descriptors, which  is a  challenging large scale problem, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot large adjacency matrix, where prior methods fail or have serious drawbacks.\nResults are shown in Figure~\\ref{fig:mainexp}.\nTemporal ranking with CG is\nreaching saturation near $20$ iterations as in~\\cite{ITA+17}.\nSpectral ranking is evaluated for a decomposition of rank $r$ whose computation and required memory are reasonable and feasible on a single machine.\nFinally, the proposed hybrid method is evaluated for\nrank $r=400$, which is a good compromise of speed and memory, as shown below.\n\nSpectral ranking variants (FSR, FSRw) are not performing well despite requiring about $250\\%$ additional memory compared to nearest neighbor search in the original space. Compared to hybrid, more than one order of magnitude higher rank $r$ is required for problems of this scale.\nTemporal ranking achieves good performance but at much more iterations and higher query times.\nOur hybrid solution provides a very reasonable space-time trade-off.\n\n\\head{Runtime and memory trade-off.}\nWe report the trade-off between number of iterations and rank $r$, representing additional memory, in more detail in Figure~\\ref{fig:tradeoff}.\nIt is shown that the number of iterations to achieve the top mAP decreases as the rank increases.\nWe achieve the optimal trade-off at $r=400$ where we only need 5 or less iterations.\nNote that, the rank not only affects the memory and the query time of the spectral part in a linear manner, but the query time of the temporal part too~(\\ref{eq:box}).\n\n\\head{Sparsification} of spectral embeddings is exploited in prior work~\\cite{IAT+18}.\nWe sparsify the embeddings of our hybrid method by setting the smallest values of $U_1$ to zero until a desired level of sparsity is reached.\nWe denote this method by \\emph{SHybrid}.\nThis sparse variant provides memory savings and an additional speedup due to the computations with sparse matrices.\nFigure~\\ref{fig:sparse} shows that performance loss remains small even for extreme sparsity \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot $99$\\%, while the results in Figure~\\ref{fig:mainexp} show that it offers a significant speedup in the global descriptor setup.\n\n\\begin{figure}\n\\input{fig_sparse}\n\\vspace{-10pt}\n \\caption{mAP \\vs level of sparsification on our hybrid method for $r = 400$. Dashed horizontal line indicates no sparsification. \\label{fig:sparse} }\n\\end{figure}\n\n\\begin{table}[t]\n\\begin{center}\n\\input{tab_res}\n\\vspace{5pt}\n\\caption{Performance, memory and query time comparison on $\\mathcal{R}$Oxford\\xspace +$\\mathcal{R}$1M\\xspace with GeM descriptors for temporal (20 iterations), truncated temporal (20 iterations, 75k images in shortlist), spectral ($r=5k$), and hybrid ranking ($r=400$, 5 iterations). Hybrid ranking is sparsified by setting the 99\\% smallest values to 0. Reported memory excludes the initial descriptors requiring 8.2 GB. $U_1$ is stored with double precision.\n\t\\label{tab:ptm}}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[t]\n\\input{fig_large_trunc}\n\\vspace{-5pt}\n\\caption{Time (s) - memory (MB) - performance\n(mAP) for different methods.\nWe show mAP \\vs time, memory \\vs time, and memory \\vs mAP on $\\mathcal{R}$Oxford\\xspace+$\\mathcal{R}$1M\\xspace.\nMethods in the comparison: temporal for 20 iterations, truncated temporal for 20 iterations and shortlist of size 50k, 75k, 100k, 200k and 300k, spectral (FSRw) with $r=5k$, hybrid with $r \\in \\{100, 200, 300, 400\\}$ and 5 iterations, sparse hybrid with 99\\% sparsity, $r \\in \\{100, 200, 300, 400\\}$ and 5 iterations.\nText labels indicate the shortlist size (in thousands) for truncated temporal and rank for hybrid. Observe that the two plots on the left are aligned horizontally with respect to time, while the two at the bottom vertically with respect to memory.\n \\label{fig:large_trunc} }\n\\end{figure}\n\n\n\\head{Performance-memory-speed comparison}\nwith the baselines is shown in Table~\\ref{tab:ptm}.\nOur hybrid approach enjoys query times lower than those of temporal with truncation or spectral with FSRw, while at the same time achieves higher performance and requires less memory than the spectral-only approach.\n\nWe summarize our achievement in terms of mAP, required memory, and query time in Figure~\\ref{fig:large_trunc}.\nTemporal ranking achieves high performance at the cost of high query time and its truncated counterpart saves query time but sacrifices performance.\nSpectral ranking is not effective at this scale, while our hybrid solution achieves high performance at low query times.\n\n\\head{Comparison with the state of the art.}\nWe present an extensive comparison with existing methods in the literature for global descriptors at small and large scale  (1M distractors).\nWe choose $r=400$ and $5$ iterations for our hybrid method, $20$ iterations for temporal ranking, $r=2k$ and $r=5k$ for spectral ranking at small and large scale, respectively.\nTemporal ranking is also performed with truncation on a shortlist size of $75k$ images  at large scale.\nThe comparison is presented in Table~\\ref{tab:soa}.\nOur hybrid approach performs the best or second best right after the temporal one, while providing much smaller query times at a small amount of additional required memory.\n\n\\begin{table}[t]\n\\vspace{10pt}\n\\begin{center}\n\\input{tab_soa}\n\\caption{mAP comparison with existing methods in the literature on small and large scale datasets, using Medium and Hard setup of the revisited benchmark.\n\t\\label{tab:soa}\n}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\nMost image retrieval methods obtain their initial ranking of the database images by computing similarity between the query descriptor and descriptors of the database images. Descriptors based on local features~\\cite{SZ03,PCISZ07} have been largely replaced by more efficient CNN-based image descriptors~\\cite{GARL17,RTC18}.\nRegardless of the initial ranking, the retrieval performance is commonly boosted by considering the manifold structure of the database descriptors, rather than just independent distances of query to database images.\nExamples are query expansion~\\cite{CPSIZ07,AZ12} and diffusion~\\cite{DB13,ITA+17,IAT+18}. \\emph{Query expansion} uses the results of initial ranking to issue a novel, enriched, query~\\cite{CPSIZ07} on-line only. \\emph{Diffusion} on the other hand, is based on the $k$-NN graph of the dataset that is constructed off-line, so that, assuming novel queries are part of the dataset, their results are essentially pre-computed. Diffusion can then be seen as infinite-order query expansion~\\cite{ITA+17}.\n\nThe significance of the performance boost achieved by diffusion has been recently demonstrated at the ``Large-Scale Landmark Recognition''\\footnote{\\url{https:\/\/landmarkscvprw18.github.io\/}} challenge in conjunction with CVPR 2018. The vast majority of top-ranked teams have used query expansion or diffusion\nas the last step of their method.\n\nRecently, efficient diffusion methods have been introduced to the image retrieval community. Iscen \\emph{et al}\\onedot~\\cite{ITA+17} apply diffusion to obtain the final ranking, in particular by solving a large and sparse system of linear equations. Even though an efficient \\emph{conjugate gradient} (CG)~\\cite{Hack94} solver is used, query times on large-scale datasets are in a range of several seconds. A significant speed-up is achieved by \\emph{truncating} the system of linear equations. Such an approximation, however, brings a slight degradation in the retrieval performance. Their method can be interpreted as graph\nfiltering in the {\\em temporal} domain.\n\nIn the recent work of Iscen \\emph{et al}\\onedot~\\cite{IAT+18}, more computation is shifted to the off-line phase to accelerate the query. The solution of the linear system is estimated by low-rank approximation of the $k$-NN graph Laplacian. Since the eigenvectors of the Laplacian represent a Fourier basis of the graph, this is interpreted as graph filtering in the {\\em spectral} domain.\nThe price to pay\nis increased space complexity to store the embeddings of the database descriptors. For comparable performance, a 5k-10k dimensional vector is needed per image.\n\n\nIn this paper, we introduce a \\emph{hybrid} method that combines spectral filtering~\\cite{IAT+18} and temporal filtering%\n~\\cite{ITA+17}. This hybrid method offers a trade-off between speed (\\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, the number of iterations of CG) and the additional memory required\n(\\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, the dimensionality of the embedding). The two approaches~\\cite{ITA+17,IAT+18} are extreme cases of our hybrid method. We show that the proposed method pairs or outperforms the previous methods while either requiring less memory or being significantly faster -- only three to five iterations of CG are necessary for embeddings of 100 to 500 dimensions.\n\nWhile both temporal and spectral filtering approaches were known in other scientific fields before being successfully applied to image retrieval, to our knowledge the proposed method is novel and can be applied to other domains.\n\nThe rest of the paper is organized as follows. Related work is reviewed in Section~\\ref{sec:related}. Previous work on temporal and spectral filtering is detailed in Sections~\\ref{sec:temporal} and~\\ref{sec:spectral} respectively, since the paper builds on this work. The proposed method is described in Section~\\ref{sec:hybrid} and its behavior is analyzed in Section~\\ref{sec:analysis}. Experimental results are provided in Section~\\ref{sec:exp}. Conclusions are drawn in Section~\\ref{sec:conclusions}.\n\\section{Hybrid spectral-temporal filtering}\n\\label{sec:hybrid}\n\nTemporal filtering~(\\ref{eq:temporal}) is performed once for every new query represented by $\\mathbf{y}$, but $\\mathcal{W}$ represents the dataset and is fixed. Could CG be accelerated if we had some very limited additional information on $\\mathcal{W}$?\n\nOn the other extreme, spectral filtering~(\\ref{eq:spectral}) needs a large number of eigenvectors and eigenvalues of $\\mathcal{W}$ to provide a high quality approximation, but always leaves some error. Could we reduce this space requirement by allocating some additional query time to recover the approximation error?\n\nThe answer is positive to both questions and in fact these are the two extreme cases of \\emph{hybrid spectral-temporal filtering}, which we formulate next.\n\n\\input{deriv-inv-v2}\n\n\\subsection{Algorithm}\n\n\\emph{Why is decomposition~(\\ref{eq:main}) of $h_\\alpha(\\mathcal{W})$ important?} Because given an observation vector $\\mathbf{y}$ at query time, we can express the ranking vector $\\mathbf{x}$ as\n\\begin{equation}\n\t\\mathbf{x} = \\mathbf{x}^s + \\mathbf{x}^t,\n\\label{eq:hybrid}\n\\end{equation}\nwhere the first, \\emph{spectral}, term $\\mathbf{x}^s$ is obtained by spectral filtering\n\\begin{equation}\n\t\\mathbf{x}^s = U_1 g_\\alpha(\\Lambda_1) U_1^{\\!\\top} \\mathbf{y},\n\\label{eq:h-spectral}\n\\end{equation}\nas in~\\cite{IAT+18}, where $g_\\alpha$ applies element-wise, while the second, \\emph{temporal}, term $\\mathbf{x}^t$ is obtained by temporal filtering, that is, solving the linear system\n\\begin{equation}\n\t\\mathcal{L}_\\alpha(\\mathcal{W} - U_1 \\Lambda_1 U_1^{\\!\\top}) \\mathbf{x}^t = \\mathbf{y},\n\\label{eq:h-temporal}\n\\end{equation}\nwhich we do by a few iterations of CG as in~\\cite{ITA+17}. The latter is possible because $\\mathcal{L}_\\alpha(\\mathcal{W} - U_1 \\Lambda_1 U_1^{\\!\\top})$ is still positive-definite, like $\\mathcal{L}_\\alpha(\\mathcal{W})$.\nIt's also possible without an explicit dense representation of $U_1 \\Lambda_1 U_1^{\\!\\top}$ because CG, like all Krylov subspace methods, only needs \\emph{black-box} access to the matrix $A$ of the linear system, that is, a mapping $\\mathbf{z} \\mapsto A \\mathbf{z}$ for $\\mathbf{z} \\in \\mathbb{R}^n$. For system~(\\ref{eq:h-temporal}) in particular, according to the definition~(\\ref{eq:laplacian}) of $\\mathcal{L}_\\alpha$, we use the mapping\n\\begin{equation}\n\t\\mathbf{z} \\mapsto \\left(\n\t\t\t\\mathbf{z} - \\alpha \\left( \\mathcal{W} \\mathbf{z} - U_1 \\Lambda_1 U_1^{\\!\\top} \\mathbf{z} \\right)\n\t\t\\right) \/ (1 - \\alpha),\n\\label{eq:box}\n\\end{equation}\nwhere product $\\mathcal{W} \\mathbf{z}$ is efficient because $\\mathcal{W}$ is sparse as in~\\cite{ITA+17}, while $U_1 \\Lambda_1 U_1^{\\!\\top} \\mathbf{z}$ is efficient if computed right-to-left because $U_1$ is an $n \\times r$ matrix with $r \\ll n$ and $\\Lambda_1$ is diagonal as in~\\cite{IAT+18}.\n\n\\subsection{Discussion}\n\n\\emph{What is there to gain from spectral-temporal decomposition~(\\ref{eq:hybrid}) of $\\mathbf{x}$?}\n\nFirst, since the temporal term~(\\ref{eq:h-temporal}) can recover the spectral approximation error, the rank $r$ of $U_1$, $\\Lambda_1$ in the spectral term~(\\ref{eq:h-spectral}) can be chosen as small as we like. In the extreme case $r = 0$, the spectral term vanishes and we recover temporal filtering~\\cite{ITA+17}. This allows efficient computation of only a few eigenvectors\/values, even with the Lanczos algorithm rather than the approximate method of~\\cite{IAT+18}. Most importantly, it reduces significantly the space complexity, at the expense of query time.\nLike spectral filtering, it is possible to \\emph{sparsify} $U_1$ to compress the dataset embeddings and accelerate the queries online. In fact, we show in section~\\ref{sec:exp} that sparsification is much more efficient in our case.\n\nSecond, the matrix $\\mathcal{W} - U_1 \\Lambda_1 U_1^{\\!\\top}$ is effectively like $\\mathcal{W}$ with the $r$ largest eigenvalues removed. This improves significantly the condition number of matrix $\\mathcal{L}_\\alpha(\\mathcal{W} - U_1 \\Lambda_1 U_1^{\\!\\top})$ in the temporal term~(\\ref{eq:h-temporal}) compared to $\\mathcal{L}_\\alpha(\\mathcal{W})$ in the linear system~(\\ref{eq:temporal}) of temporal filtering~\\cite{ITA+17}, on which the convergence rate depends. In the extreme case $r = n$, the temporal term vanishes and we recover spectral filtering~\\cite{IAT+18}. In turn, even with small $r$, this reduces significantly the number of iterations needed for a given accuracy, at the expense of computing and storing $U_1$, $\\Lambda_1$ off-line as in~\\cite{IAT+18}. The improvement is a function of $\\alpha$ and the spectrum of $\\mathcal{W}$, and is quantified in section~\\ref{sec:analysis}.\n\nIn summary, for a given desired accuracy, we can choose the rank $r$ of the spectral term and a corresponding number of iterations of the temporal term, determining a trade-off between the space needed for the eigenvectors (and the off-line cost to obtain them) and the (online) query time. Such choice is not possible with either spectral or temporal filtering alone: at large scale, the former may need too much space and the latter may be too slow.\n\n\n\n\n\\section{Related work}\n\\label{sec:related}\nQuery expansion (QE) has been a standard way to improve recall of image retrieval since the work of Chum \\emph{et al}\\onedot~\\cite{CPSIZ07}.\nA variety of approaches exploit local feature matching and perform various types of verification.\nSuch matching ranges from selective kernel matching~\\cite{TJ14} to geometric consensus~\\cite{CPSIZ07,CMPM11,JB09} with RANSAC-like techniques.\nThe verified images are then used to refine the global or local image representation of a novel query.\n\nAnother family of QE methods are more generic and simply assume a global image descriptor~\\cite{SLBW14,JHS07,DJAH14,DGBQG11,ZYCYM12,DB13,AZ12}.\nA simple and popular one is average-QE~\\cite{CPSIZ07}, recently extended to $\\alpha$-QE~\\cite{RTC18}.\nAt some small extra cost, recall is significantly boosted.\nThis additional cost is restricted to the on-line query phase. This is in contrast to another family of approaches that considers an off-line pre-processing of the database.\nGiven the nearest neighbors list for database images, QE is performed by adapting the local similarity measure~\\cite{JHS07}, using reciprocity constraints~\\cite{DJAH14,DGBQG11} or graph-based similarity propagation~\\cite{ZYCYM12,DB13,ITA+17}. The graph-based approaches, also known as diffusion, are shown to achieve great performance~\\cite{ITA+17} and to be a good way for feature fusion~\\cite{ZYCYM12}.\nSuch on-line re-ranking is typically orders of magnitude more costly than simple average-QE.\n\nThe advent of CNN-based features, especially global image descriptors, made QE even more attractive.\nAverage-QE or $\\alpha$-QE are easily applicable and very effective with a variety of CNN-based descriptors~\\cite{RTC18,GARL17,TSJ16,KMO15}.\nState-of-the-art performance is achieved with diffusion on global or regional  descriptors~\\cite{ITA+17}.\nThe latter is possible due to the small number of regions that are adequate to represent small objects, in contrast to thousands in the case of local features.\n\nDiffusion based on tensor products can be attractive in terms of performance~\\cite{BBT+18,BZW+17}. However, in this work, we focus on the page-rank like diffusion~\\cite{ITA+17,ZWG+03} due to its reasonable query times. An iterative on-line solution was commonly preferred~\\cite{DB13} until the work of Iscen \\emph{et al}\\onedot~\\cite{ITA+17}, who solve a linear system to speed up the process.\nAdditional off-line pre-processing and the construction and storage of additional embeddings reduce diffusion to inner product search in the spectral ranking of Iscen \\emph{et al}\\onedot~\\cite{IAT+18}. This work lies exactly in between these two worlds and offers a trade-off exchanging memory for speed and vice versa.\n\nFast random walk with restart~\\cite{ToFP06} is very relevant in the sense that it follows the same diffusion model as~\\cite{ITA+17,ZWG+03} and is a hybrid method like ours. It first disconnects the graph into distinct components through clustering and then obtains a low-rank spectral approximation of the residual error. Apart from the additional complexity, parameters \\emph{etc}\\onedot} \\def\\vs{\\emph{vs}\\onedot of the off-line clustering process and the storage of both eigenvectors and a large inverse matrix, its online phase is also complex, involving the Woodbury matrix identity and several dense matrix-vector multiplications. Compared to that, we first obtain a \\emph{very} low-rank spectral approximation of the original graph, and then solve a sparse linear system of the residual error. Thanks to orthogonality properties, the online phase is nearly as simple as the original one and  significantly faster.\n\\section{Problem formulation and background}\n\\label{sec:background}\n\n\nThe methods we consider are based on a nearest neighbor graph of a dataset of $n$ items, represented by $n \\times n$ \\emph{adjacency matrix} $W$. The graph is undirected and weighted according to similarity: $W$ is sparse, symmetric, nonnegative and zero-diagonal. We symmetrically normalize $W$ as $\\mathcal{W} \\mathrel{\\operatorname{:=}} D^{-1\/2} W D^{-1\/2}$, where $D \\mathrel{\\operatorname{:=}} \\operatorname{diag}(W \\mathbf{1})$ is the diagonal \\emph{degree matrix}, containing the row-wise sum of $W$ on its diagonal.\nThe eigenvalues of $\\mathcal{W}$ lie in $[-1,1]$~\\cite{Chun97}.\n\nAt query time, we are given a sparse $n \\times 1$ \\emph{observation vector} $\\mathbf{y}$, which is constructed by searching for the nearest neighbors of a query item in the dataset and setting its nonzero entries to the corresponding similarities. The problem is to obtain an $n \\times 1$ \\emph{ranking vector} $\\mathbf{x}$ such that retrieved items of the dataset are ranked by decreasing order of the elements of $\\mathbf{x}$. Vector $\\mathbf{x}$ should be close to $\\mathbf{y}$ but at the same time similar items are encouraged to have similar ranks in $\\mathbf{x}$, essentially by exploring the graph to retrieve more items.\n\n\\subsection{Temporal filtering} \\label{sec:temporal}\n\nGiven a parameter $\\alpha \\in [0,1)$, define the $n \\times n$ \\emph{regularized Laplacian} function by\n\\begin{equation}\n\t\\mathcal{L}_\\alpha(A) \\mathrel{\\operatorname{:=}} (I_n - \\alpha A) \/ (1 - \\alpha)\n\\label{eq:laplacian}\n\\end{equation}\nfor $n \\times n$ real symmetric matrix $A$, where $I_n$ is the $n \\times n$ identity matrix. Iscen \\emph{et al}\\onedot~\\cite{ITA+17} then define $\\mathbf{x}$ as the unique solution of the linear system\n\\begin{equation}\n\t\\mathcal{L}_\\alpha(\\mathcal{W}) \\mathbf{x} = \\mathbf{y},\n\\label{eq:temporal}\n\\end{equation}\nwhich they obtain approximately in practice by a few iterations of the \\emph{conjugate gradient} (CG) method, since $\\mathcal{L}_\\alpha(\\mathcal{W})$ is positive-definite. At large scale, they truncate $\\mathcal{W}$ by keeping only the rows and columns corresponding to a fixed number of neighbors of the query, and re-normalize. Then,~(\\ref{eq:temporal}) only performs re-ranking within this neighbor set.\n\nWe call this method \\emph{temporal\\footnote{``Temporal'' stems from conventional signal processing where signals are functions of ``time''; while ``spectral'' is standardized also in graph signal processing.} filtering} because if $\\mathbf{x}$, $\\mathbf{y}$ are seen as signals, then $\\mathbf{x}$ is the result of applying a linear graph filter on $\\mathbf{y}$, and CG iteratively applies a set of recurrence relations that determine the filter. While $\\mathcal{W}$ is computed and stored off-line,~(\\ref{eq:temporal}) is solved online (at query time), and this is expensive.\n\n\\subsection{Spectral filtering} \\label{sec:spectral}\n\nLinear system~(\\ref{eq:temporal}) can be written as $\\mathbf{x} = h_\\alpha(\\mathcal{W}) \\mathbf{y}$, where the \\emph{transfer function} $h_\\alpha$ is defined by\n\\begin{equation}\n\th_\\alpha(A) \\mathrel{\\operatorname{:=}} \\mathcal{L}_\\alpha(A)^{-1} = (1 - \\alpha) (I_n - \\alpha A)^{-1}\n\\label{eq:transfer}\n\\end{equation}\nfor $n \\times n$ real symmetric matrix $A$. Given the eigenvalue decomposition $\\mathcal{W} = U \\Lambda U^{\\!\\top}$ of the symmetric matrix $\\mathcal{W}$, Iscen \\emph{et al}\\onedot~\\cite{IAT+18} observe that $h_\\alpha(\\mathcal{W}) = U h_\\alpha(\\Lambda) U^{\\!\\top}$, so that~(\\ref{eq:temporal}) can be written as\n\\begin{equation}\n\t\\mathbf{x} = U h_\\alpha(\\Lambda) U^{\\!\\top} \\mathbf{y},\n\\label{eq:spectral}\n\\end{equation}\nwhich they approximate by keeping only the largest $r$ eigenvalues and the corresponding eigenvectors of $\\mathcal{W}$. This defines a low-rank approximation $h_\\alpha(\\mathcal{W}) \\approx U_1 h_\\alpha(\\Lambda_1) U_1^{\\!\\top}$ instead. This method is referred to as \\emph{fast spectral ranking} (FSR) in~\\cite{IAT+18}. Crucially, $\\Lambda$ is a diagonal matrix, hence $h_\\alpha$ applies element-wise, as a scalar function $h_\\alpha(x) \\mathrel{\\operatorname{:=}} (1 - \\alpha) \/ (1 - \\alpha x)$ for $x \\in [-1,1]$.\n\nAt the expense of off-line computing and storing the $n \\times r$ matrix $U_1$ and the $r$ eigenvalues in $\\Lambda_1$, filtering is now computed as a sequence of low-rank matrix-vector multiplications, and the query is accelerated by orders of magnitude compared to~\\cite{ITA+17}. However, the space overhead is significant. To deal with approximation errors when $r$ is small, a heuristic is introduced that gradually falls back to the similarity in the original space for items that are poorly represented, referred to as FSRw~\\cite{IAT+18}.\n\nWe call this method \\emph{spectral filtering} because $U$ represents the Fourier basis of the graph and the sequence of matrix-vector multiplications from right to left in the right-hand side of~(\\ref{eq:spectral}) represents the Fourier transform of $\\mathbf{y}$ by $U^{\\!\\top}$, filtering in the frequency domain by $h_\\alpha(\\Lambda)$, and finally the inverse Fourier transform to obtain $\\mathbf{x}$ in the time domain by $U$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Introduction}\n\n\nOne of the foundations of Einstein's theory of General Relativity is that matter\n  curves the surrounding space-time. For the rare cases of nearly perfect alignment between\n  an astronomical source, an intervening massive object and the observer, \n  multiple images of a single source can be detected, \n a phenomenon known as strong gravitational lensing.\n  \n Although many strongly lensed galaxies and quasars have been detected\n to date, it has proved extremely difficult  to find multiply-imaged lensed  supernova (SN\\xspace) explosions. \n  Type Ia supernovae (SNe~Ia\\xspace) \n  are particularly interesting sources due to their\n   ``standard candle'' nature.  These explosions have nearly identical peak luminosity\n   which makes them excellent distance indicators in\n cosmology \\cite{2011ARNPS..61..251G}. \n For lensed SNe~Ia\\xspace, the standard candle property allows the flux magnification to be estimated directly, independent\n of any model related to the lensing galaxy \\cite{Kolatt:1997zh,Oguri:2002ku}. This removes important \n degeneracies in gravitational lensing measurements, the  mass-sheet degeneracy \\cite{1985ApJ...289L...1F} \n and the source-plane degeneracy \\cite{2013A&A...559A..37S}.\n  \n  \n   A lensed SN~Ia\\xspace  at redshift $z=1.388$ with a large amplification ($\\mu\\sim 30$) , PS1-10afx,  where multiple images could have been expected, has been reported earlier \\cite{2013ApJ...768L..20Q}. A foreground lens was later identified at $z=1.117$ \\cite{2014Sci...344..396Q}. However, at the time of the discovery several interpretations were discussed,\n  including a super-luminous supernova \\cite{2013ApJ...767..162C}. \nSince the lensed SN~Ia\\xspace hypothesis was only\n  accepted long after the explosion had faded, no high spatial\n  resolution imaging could be carried out in that case to verify the strong lensing nature of the system.\n  Multiple-images  of  another supernova, SN\\xspace Refsdal \\cite{2015Sci...347.1123K}, were discovered in \n   a  Hubble Space Telescope (HST) survey of the massive galaxy cluster MACS J1149.6+2223. As the source was identified as a core-collapse supernova it could not be used to measure the\n  lensing magnification directly.\n\n\n\nThanks to the well-known characteristics of their time-dependent brightness in optical and near-infrared filters  (the SN\\xspace  lightcurves), multiply-imaged SNe~Ia\\xspace\n  are also ideally suited to measure time-delays in the arrival of the\n  images. This  provides a direct probe of the Hubble constant, the cosmological parameter measuring  the expansion rate of the universe\\cite{1964MNRAS.128..307R}, as well as\n  leverage for studies of dark energy \n  \\cite{2002A&A...393...25G,2013ApJ...766...70S}, the cosmic constituent responsible for the accelerated expansion of the universe. \n\nThe intermediate Palomar Transient Factory (iPTF) searches the sky for new transient phenomena at optical\nwavelengths. It uses image differencing between repeated observations\n\\cite{cnk16} with a large field-of-view camera\n(7.3 sq.deg) at the 48-inch telescope (P48) at the Palomar\nObservatory \\cite{2009PASP..121.1395L}.  \nThe first detection of iPTF16geu, with a statistical significance of five standard deviations (5$\\sigma$), is from 2016 September 5. The new\nsource was first recognized by a human scanner on September 11  \\cite {ATEL9603}.  iPTF16geu (also known as SN 2016geu) was found near the center of the galaxy\nSDSS\\,J$210415.89$-$062024.7$, at\nright ascension $21^h$$4^m$$15.86^s$  and declination \\ang{-06;20;24.5} (J2000)\n\nSpectroscopic identification was carried out with the Spectral Energy Distribution (SED) Machine \n\\cite{2014CoSka..43..209R} at the Palomar 60-inch telescope (P60) on 2016 October 2 and iPTF16geu was found to be spectroscopically consistent with a normal\nSN~Ia\\xspace at $z\\approx0.4$ (see Fig.~\\ref{fig:spec}).  Further spectroscopic observations from the\nPalomar~200-inch telescope (P200) and the 2.5-meter Nordic\nOptical Telescope (NOT) were used to confirm the  SN~Ia\\xspace\nidentification and to establish the redshift of the host galaxy from\nnarrow sodium (Na~I~D) absorption lines, as $z=0.409$. The P200 and NOT spectra also\nshow absorption features from the foreground lensing galaxy at\n$z=0.216$.  To estimate the velocity dispersion of the lensing galaxy, we fit two Gaussian functions with a common width to the H${\\alpha}$ and [N~{\\sc ii}] emission lines in the P200 spectrum in Fig~\\ref{fig:spec}D. After taking the instrumental resolution into account, we measure $\\sigma = 3.6^{+0.9}_{-0.6}$ \\AA, corresponding to a velocity dispersion of $\\sigma_{v} = 163^{+41}_{-27}$ km s$^{-1}$.\n\nPhotometric observations  of iPTF16geu collected at P48 and \nwith the SED Machine Rainbow Camera (RC) at P60, between 2016 September 5 and October 13  (see Fig.~\\ref{fig:lc}), were \nused to estimate the peak flux and lightcurve properties of the SN\\xspace with the SALT2 \nlightcurve fitting tool \\cite{Guy:2007js}.\nThe best fit lightcurve template, also shown in Fig.~\\ref{fig:lc}, confirms that the observed \nlightcurve shapes are consistent with a SN~Ia\\xspace at $z=0.409$.  These fits also indicate \nsome reddening of the supernova, suggesting that iPTF16geu suffers from moderate extinction \nby dust. This produces dimming at optical wavelengths of 20-40\\%,  whith the largest losses \nin the $g$-band observations.  Thanks to the standard candle nature of SNe~Ia\\xspace, after correcting the peak magnitude for lightcurve properties \\cite{1993ApJ...413L.105P,1998A&A...331..815T},\nthe flux of the SN\\xspace was found to be $\\sim$30 standard deviations brighter than expected for the measured\nredshift.\nThis suggested that iPTF16geu was gravitationally lensed and we estimated the lensing amplification to be $\\mu \\sim 52$. Expressed in astronomical magnitudes, \n$\\Delta  m = -4.3\\pm {0.2}$~mag, where the uncertainty is dominated by the brightness dispersion of normal SNe~Ia\\xspace.\nSince the magnification is derived from comparing the observed brightness of iPTF16geu to other \nSNe~Ia\\xspace \\cite{Betoule:2014iz} within a narrow redshift range around $z=0.409$, the measurement of the lensing \nmagnification is independent of any assumptions on cosmology, e.g., the value of the Hubble constant or \nother cosmological parameters. The lensing magnification is also independent of a lens model, which is the only \nway to determine the magnification for almost all other strong lensing systems.\n\nThe optical observations from Palomar, with a typical angular resolution (atmospheric seeing) of \\ang{;;2},  were \ninsufficient to spatially resolve any multiple images that could result from the strong lensing nature of the system \n(Fig.~\\ref{fig:zoom}A). We therefore obtained $K_{\\mathrm{s}}$-band (2.2\\,$\\mu$m) observations from the European Southern Observatory (ESO) with the Nasmyth \nAdaptive Optics System  Near-Infrared Imager and Spectrograph (NACO) at the Very Large Telescope (VLT).\nAn angular resolution of $\\sim$\\ang{;;0.3} (full-width half-max, FWHM) was obtained at the location of the target. Adaptive optics (AO) \ncorrections of the seeing were performed using a natural bright star,  $\\sim$\\ang{;;30} south-east of the SN\\xspace location,  \nindicated in  Fig.~\\ref{fig:zoom} along with the SDSS pre-explosion image of the field \\cite{2015ApJS..219...12A}. \n\n\nThe near-IR  image from VLT  indicated the structure expected in a  strongly lensed system, with higher flux in the  northeastern and southwestern regions of the system, compared to the center (Fig.~\\ref{fig:zoom}B).\nMultiple images of the system were first resolved with observations from the Keck observatory at near-infrared wavelengths, using the Laser Guide Star aided Adaptive Optics (LGSAO) with the\nOH-Suppressing Infra-Red Imaging Spectrograph (OSIRIS) instrument, yielding an image quality of \\ang{;;0.07}\nFWHM in  the $H$-band  centered at 1.6 $\\mu$m (Fig.~\\ref{fig:zoom}C). \n\nLGSAO observations of iPTF16geu using the Near-InfraRed Camera 2 (NIRC2) at the Keck telescope on 2016 October 22 and November 5,  in  $K_{\\mathrm{s}}$-band and \n$J$-band (1.1$\\mu$m) respectively, and optical images  obtained with the Hubble Space Telescope (HST) on 2016 October 25, are shown in Fig.~\\ref{fig:combo}. The HST observations were carried out through the $F475W$, $F625W$ and\n$F814W$ filters, where the names correspond to the approximate location of the central wavelength in nanometers.\n\nThe observations exhibit four images of  iPTF16geu, \\ang{;;0.26}--\\ang{;;0.31} from the\ncenter of the lensing galaxy,  with nearly 90$^\\circ$ azimuthal\nseparations. The extended host galaxy, warped by the lens to form a partial\nEinstein ring, is brighter in the near-IR compared to the observations through optical filters. Thus,  the fainter individual SN\\xspace images are poorly resolved\nfor the observations with the longest wavelengths in Fig.~\\ref{fig:combo}.  Furthermore, the SN~Ia\\xspace \nspectral energy distribution (redshifted to $z=0.4$) peaks within the $F625W$ and $F814W$ filters, see e.g. \\cite{2014MNRAS.439.1959M}. \nDimming by interstellar dust in the line of sight is roughly inversely proportional to wavelength in the optical and near-IR \\cite{1989ApJ...345..245C}.\nThe biggest impact from extinction by dust is therefore expected for the shortest wavelength, in $F475W$ filter observations, where the two faintest SN\\xspace images cannot be detected above the\nbackground light.\n\\newcommand{\\mathcal{C}}{\\mathcal{C}}\nThe low  spatial resolution lightcurves in Fig.~\\ref{fig:lc} are  dominated by the two brightest SN\\xspace images, \nlabelled 1 and 2 in Fig.~\\ref{fig:combo}D. The $F625W$-$F814W$ magnitude difference (color) of the resolved images \nmeasured with HST indicate small differences in relative extinction between the SN\\xspace images, except for image 4, \nwhich appears to have about two magnitudes of additional dimming in $F814W$. \n\nUnaccounted dimming of light by scattering on dust grains in the line of sight would lead to an underestimation \nof the lensing amplification. Including corrections for differential extinction in the intervening lensing galaxy \nbetween the SN\\xspace images suggest a wider range for the lensing magnification of iPTF16geu, between $-4.1$ \nand $-4.8$ mag \\cite{sup}.\n\n\nThe SN\\xspace multiple-image positions in Fig~\\ref{fig:combo} were used to construct a lensing model,  with an isothermal ellipsoid galaxy\nlens \\cite{1993LIACo..31..571K,1994A&A...284..285K} with ellipticity $\\epsilon_e=0.15\\pm 0.07$ and mass\n$M=(1.70 \\pm 0.06)\\cdot 10^{10}\\,M_\\odot$ inside an ellipse with major axis $1.13$ kpc and minor axis $0.97$ kpc.  Details of the lensing model are presented in the Supplementary Material \\cite{sup}. The lens model can be independently verified through comparisons between the model-predicted and observed velocity dispersion of the lensing galaxy. From the model \nwe derive an estimate,  $\\sigma^{\\rm mod}_v=156\\pm 4$ km s$^{-1}$,  in good agreement with the measured value of the velocity dispersion (Fig.~\\ref{fig:spec}D).\n\n\nHowever, the adopted smooth isothermal ellipsoid lens model predicts brightness differences\nbetween the multiple SN\\xspace images that are in disagreement with the observations. Including corrections for \nextinction in the resolved SN\\xspace images in the $F814W$ filter, we find large discrepancies between the model and \nmeasured magnitude differences for the multiple images of iPTF16geu:\n   $\\Delta m^{obs}_{1j}-\\Delta m^{mod}_{1j}$  = ($-0.3,  -1.6, -1.5$) mag for $j=2,3$ and $4$, where the indices follow the numbering scheme adopted in Fig.~\\ref{fig:combo}.\nThe observed discrepancy between the smooth model predictions for the  SN\\xspace images 1 and 2  compared to 3 and 4 (brighter by a factor 4 and 3, respectively),  cannot be accounted for by time-delays between the images, as they are \npredicted to be $<35$ hours \\cite{sup}.  Graininess of the stellar distribution and dark matter sub-halos in the lens\ngalaxy, in addition to the smooth mass profile, can cause variations to magnification without altering image locations. \nThese milli- and micro-lensing effects \\cite{1989Natur.338..745K,1994ApJ...429...66W}, small enough not to cause additional resolved image separations, offer a plausible explanation for the deviation from the smooth lens model.\n  \n  \n    \n\n  \n \n\nAvailable forecasts for wide-field surveys \\cite{2010MNRAS.405.2579O} suggest that  about one strongly lensed SN~Ia\\xspace could be expected in our survey, \nirrespectively of redshift and magnification, with approximately a 30\\% chance to be in a quad configuration.  For an average ellipticity of the lenses $e = 0.3$ \\cite{2010MNRAS.405.2579O}, only about  1\\% of the lensed SNe\\xspace are expected to \nhave $\\mu \\raise0.3ex\\hbox{$>$}\\kern-0.75em{\\lower0.65ex\\hbox{$\\sim$}} 50$ \\cite{Chae:2002uf}.\n \n \n \nWe have performed an independent rate estimate, with a somewhat simplified lensing simulation but including survey specific parameters, and confirm that the probability to detect and classify a\nhighly magnified SN~Ia\\xspace like iPTF16geu does not exceed the few percent level \\cite{sup}.\n\n  iPTF16geu appears to be a rather unlikely event, unless the actual rate of very magnified SNe\\xspace is higher\n  than anticipated, e.g., if the contribution from lensing by any kind of sub-structures in galaxies is underestimated, or if\n  we are otherwise lacking an adequate description of gravitational lensing at the $\\sim$ 1 kpc scale. The  physical scale probed by the resolved images of iPTF16geu is comparable to the smallest of the 299 multiply-imaged lensed systems in the Master \nLens Database \\cite{master}. Using the  standard candle nature of SNe~Ia\\xspace we can more easily detect strongly lensed systems with sub-arcsecond angular separations,\nallowing exploration of the bending of light at  scales $\\raise0.3ex\\hbox{$<$}\\kern-0.75em{\\lower0.65ex\\hbox{$\\sim$}}$ 1 kpc, an otherwise challengingly small distance in studies of gravitational lensing \\cite{2017ApJ...834L...5G}.  \nAs demonstrated with iPTF16geu, discovered while still brightening with a modest size telescope and sub-optimal atmospheric conditions, the locations of these rare systems can be identified\nin advance of  extensive follow-up imaging at high spatial resolution. \n\\bibliographystyle{Science}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe relativistic transformation of temperature is a problem which has\nbeen controvertial for almost a century. There has been many proposals,\nstarting by pure classical\nthermodynamics~\\cite{Einstein,planck,tolman,Ott,landsvarianza,newburgh} until\nclassical and quantum statistical mechanics~\\cite{bors,impos,kania1,kania2,cubero}. Starting from different\npostulates, each one of this works had tried to establish how the \ndifferent thermodynamics quantities change under Lorentz\ntransformations,\nbut they have obtained incompatible results. For example, in\nRefs.~\\cite{kampen,yuen} a review of different formalisms is done. In\nparticular, in Ref.~\\cite{kampen} it is established that the different\nformalisms are mathematically equivalent to each other, because there\nis a one to one correspondence \nbetween the quantities defined in every formalism.\n\n\n\n\nThe idea to generalize the statistical mechanics to relativistic\nsystems dates from J\\\"uttner~\\cite{juttner,juttner2}, who proposed a relativistic form of\nMaxwell-Boltzmann velocities distribution. Other attempts to get the\ncorrect relativistic distribution function that fits correctly\nexperimental data have been recently done. For example, in\nRefs.~\\cite{kania1,kania2} a new mathematical formalism was created in\norder to develope a non-extensive relativistic statistical mechanics under a\ncanonical ensemble, which works to fit data of cosmic rays. Recently, it has been shown through\nnumerical simulation that J\\\"uttner's distribution function is the\ndistribution in special relativity that produces the best fit for a dilute gas of two components mixture with collisions\nin one dimension~\\cite{cubero}. \n\nIn other hand, other works conduct to others distribution functions\nthan J\\\"uttner one. For instance, in Ref.~\\cite{lapenta}, authors perfom\nnumerical simulations of electrons accelerated to relativistic\nenergies due to its interaction with waves generated by longitudinal\nstreaming plasma instabilities. They found an equilibrium\ndistribution which present power-law tails at high energies. Although\nRefs.~\\cite{cubero,lapenta} consider different systems, both show that\nthe old problems of transformation of temperature and pressure, and\nthe form of distribution function in theoretical relativistic\nstatistical mechanics appears in numerical simulation.  \n\nHowever, in many of these works the temperature transformation are\nassumed and not derived from the theory itself. This happen because in\ncanonical ensembles the temperature $\\beta=T^{-1}$ is a system\nvariable (we set Boltzmann constant $k_B=1$). Therefore, it is not easy\nto find a temperature transformation between two inertial frames\nmoving with relative velocities by direct calculation. \n\nTo overcome this problem, in this article we calculate the temperature\nin the microcanonical ensemble of a relativistic ideal gas of\nbradyons, luxons or tachyons. In this ensemble the intensive quantities are not\nvariables and it is possible to find the temperature only by taking\nderivatives. Thus, the calculations are\nsimpler than in a canonical ensemble because we only need to fix the energy of these\nparticles. In addition, according to Gibbs' postulate, the results\nshould be independent from the ensemble used to calculate it. This\npostulate allow us to obtain a result that is equivalent to the one\nobtained in any other ensemble~\\cite{gibbs}. \n\nWe are extending the old problem of how the temperature of\nbradyons transform in different frames to luxons and tachyons. The reason to\ninclude tachyons under this study is the wide range of relativistic\nsystems in which they can be included. They play an important role in\nrecent developments in inflationary cosmological\nmodels~\\cite{balart,frey,xiong}, string theory black holes\nmodels~\\cite{atish,rama} and there are, even, proposed procedures to\nmeasure tachyonic states~\\cite{chiao}.\n\nTo find the temperature transformation we first derive the\nmicrocanonical entropy of the systems. Then we calculate the\ntemperature in a thermodynamic way showing how it\ntransforms. Futhermore, we show that the entropy thermodinamic $dS$\nelement is Lorentz invariant for each particle specie.\n\n\n\\section{Entropy calculation}\n\nConsider an ideal gas (of bradyons, luxons or\ntachyons) which is at rest in a inertial\nframe $I$. Let us suppose other inertial frame $I'$ moving with constant velocity\n$\\mathbf{w}=w\\hat x$ respective to $I$. \nSetting $c=1$, we choose the magnitude $w\\leq1$ if the particles of the\nsystems are bradyons or luxons, and $w>1$ if the particle system are\ntachyons. \n\nA bradyon is a particle with rest mass $m$\nwhich moves slower than speed of light. Its dispersion relation is\ngiven by  \n\\begin{equation}\n  \\label{erel}\n  p_\\mu p^\\mu= m^2\\, ,\n\\end{equation}\nwhere $p_\\mu=(\\epsilon,\\mathbf{p})$ is the 4-momentum of the particle with energy $\\epsilon$\nand momentum $\\mathbf{p}$. We use the signature $(+,-,-,-)$ for our\ncalculations.\n\nA luxon is a particle with null mass which moves\nat the speed of light. Its dispersion relation has the form\n \\begin{equation}\n   \\label{erell}\n   p_\\mu p^\\mu=0\\, . \\end{equation}\n\nFinally, a tachyon is a particle with imaginary mass $M=im$ (with $m$\na real quantity) which moves faster than speed of\nlight~\\cite{feinberg,eve,mariwalla,maccarrone,feinberg2,kowa,antippa}.\nIts dispersion relation is\n\\begin{equation}\n  \\label{erel2}\n  p_\\mu p^\\mu=-m^2\\, .\n\\end{equation} \n\n\nWe calculate the number of states $\\Omega$ using the microcanonical\nensemble. The three-vector phase-space $d^3\\mathbf{q}d^3\\mathbf{p}$\nis Lorentz invariant for bradyons, luxons and tachyons ~\\cite{kowa}.\n\nWe consider an ideal gas of bradyons, luxons or tachyons,\nconsisting of $N$ particles ($N\\gg1$) contained in a volume $V$. The Hamiltonian of $N$ bradyons is\n\\begin{equation}\nH(p_i)=\\sum_{i=1}^N\\sqrt{|\\mathbf{p}_i|^2+m^2}\\, ,\n\\end{equation}\nwhere $|\\mathbf{p}_i|=(p_{x,i}^2+p_{y,i}^2+p_{z_i}^2)^{1\/2}$. The Hamiltonian for $N$ luxons is\n\\begin{equation}\nH(p_i)=\\sum_{i=1}^N|\\mathbf{p}_i|\\, ,\n\\end{equation}\nand the Hamiltonian for $N$ tachyons is\n\\begin{equation}\nH(p_i)=\\sum_{i=1}^N\\sqrt{|\\mathbf{p}_i|^2-m^2}\\, .\n\\end{equation}\n\nSetting $h=1$, the microcanonical number of states for each specie is given by \n\\begin{eqnarray}\n  \\label{numestados}\n  \\Omega&=&\\frac{1}{N!}\\int_{E\\leq H(p_i)\\leq E+\\Delta E} d^3\\mathbf{q}_1\\ldots d^3\\mathbf{q}_Nd^3\\mathbf{p}_1\\ldots d^3\\mathbf{p}_{N}\\nonumber\\\\\n&=&\\frac{V^N}{N!}\\int_{E\\leq H(p_i)\\leq E+\\Delta E} d^3\\mathbf{p}_1\\ldots d^3\\mathbf{p}_{N}\\, .\n\\end{eqnarray}\n\nFor simplicity, we first calculate $\\Sigma$ instead $\\Omega$, where\n\\begin{equation}\n\\label{sigma}\n\\Sigma=\\frac{V^N}{N!}\\int_{H(p_i)\\leq E} d^3\\mathbf{p}_1\\ldots d^3\\mathbf{p}_{N}\\, .\n\\end{equation}\n\nThe number of states in a energy interval can be calculated from $\\Omega=(\\partial\n\\Sigma\/\\partial E)\\Delta E$. Thus, we must write the condition $H(p_i)\\leq E$ in a\n$3N$-dimensional momentum space. For photons $m=0$, and then \n\\begin{equation}\nH=\\sum_i |{\\bf p}_i|\\leq E\n\\label{condmom0}\n\\end{equation}\n\nNow, we seek for the condition for bradyons. Since no direction in space is preferred,\nlet us start supposing $n$ particles with the same momentum ${\\bf p_0},$\nand $N-n$ particles without momentum, with $n\\leq N$. In this way, the\ncondition for the Hamiltonian is $H=n\\left(|{\\bf p_0}|^2+m^2\\right)^{1\/2}+(N-n)m\\leq E$. Using this, we can obtain\n$$\\sum_i|{\\bf p}_i|=n|{\\bf p_0}|\\leq\\left((E-(N-n)m)^2-n^2m^2\\right)^{1\/2}$$\nHowever, the factor $(E-(N-n)m)^2-n^2m^2=(E-Nm)(E-Nm+2nm)\\leq E^2-N^2 m^2$, and then, it is fulfilled \n\\begin{equation}\n\\sum_i|{\\bf p}_i|\\leq \\left(E^2-N^2m^2\\right)^{1\/2}\n\\label{condmom}\n\\end{equation}\neven when $n=N$.\n\nNow, we should study what happen when we have different momenta for\neach particle. An illustrative example is the next case. If we have\n$N-1$ particles with the same momentum and one particle with a\ndifferent momentum, its sum of the norm of all momenta will be always\nless than the sum of momenta in Eq.~(\\ref{condmom}) owing to the\nparticles obey the condition $H\\leq E$. Following this example, the\ndifferent cases of momentum of each particle will produced a sum which\nwill be less than Eq.~(\\ref{condmom}). So, the condition Eq.~(\\ref{condmom})\nis always valid for bradyons.\n\nUsing an analogue argument we can obtain the condition for momentum space for tachyons\n\\begin{equation}\n\\sum_i|{\\bf p}_i|\\leq \\left(E^2+N^2m^2\\right)^{1\/2}\n\\label{condmom2}\n\\end{equation}\nwhich is fulfilled always.\n\nAll these conditions can be easily written for the momentum\ncomponents. Thus, the sum will go from 1 to $3N$. Written in that\nform, they will represent a regular geometric body in $3N$ dimensions,\nwhich would be a sphere in the case of classical ideal gas. Then, the\nproblem of calculate the integral Eq.~(\\ref{sigma}) is reduced to find the\nvolume of this regular geometric body. Following the procedure\ndescribed in Ref.~\\cite{greiner}, we obtain the number of states for\nbradyons as  \n\\begin{equation}\n  \\label{numestadosb}\n  \\Omega=\\frac{V^N}{N!}\\left(2\\sqrt 3\\right)^{3N}\\frac{\\left(E^2- N^2m^2\\right)^{3N\/2}}{(3N)!}\\, ,\n\\end{equation}\nthe number of states for luxons as\n\\begin{equation}\n  \\label{numestadosl}\n  \\Omega=\\frac{V^N}{N!}\\left(2\\sqrt 3\\right)^{3N}\\frac{E^{3N}}{(3N)!}\\, ,\n\\end{equation}\nand the number of states of tachyons as\n\\begin{equation}\n  \\label{numestadost}\n  \\Omega=\\frac{V^N}{N!}\\left(2\\sqrt 3\\right)^{3N}\\frac{\\left(E^2+ N^2m^2\\right)^{3N\/2}}{(3N)!}\\, .\n\\end{equation}\n\nIt is straigthforward to obtain the entropy as $S=\\ln \\Omega$ in a\nmicrocanonical ensemble. For bradyons the entropy is\n\\begin{equation}\n  \\label{entropiab}\n  S=N\\ln\\left[\\frac{V(E^2- N^2m^2)^{3\/2}}{27N^4}\\right]+3N\\ln \\left[2\\sqrt 3 e^{4\/3}\\right]\\, .\n\\end{equation}\n\nIn the same way, the entropy for luxons is\n\\begin{equation}\n  \\label{entropial}\n  S=N\\ln\\left[\\frac{VE^3}{27N^4}\\right]+3N\\ln \\left[2\\sqrt 3 e^{4\/3}\\right]\\, ,\n\\end{equation}\nand the entropy for tachyons\n\\begin{equation}\n  \\label{entropiat}\n  S=N\\ln\\left[\\frac{V(E^2+ N^2m^2)^{3\/2}}{27N^4}\\right]+3N\\ln \\left[2\\sqrt 3 e^{4\/3}\\right]\\, .\n\\end{equation}\n\n\\section{Temperature transformation}\n\nTo find the relation between the\ntemperature of the system in the $I$ frame and the temperature in the\n$I'$ frame we need to find how to calculate the number of\nstates in $I'$. According to Liouville theorem~\\cite{misner} the \ndimensional phase space $d^3\\mathbf{p}'d^3\\mathbf{q}' = d^3\\mathbf{p}d^3\\mathbf{q}$ is Lorentz\ninvariant. Using this, the number of\nstates $\\Omega'$ in the $I'$ frame can be written using the phase space of $I$ frame\n\\begin{multline}\n\\label{volps}\nN!\\, \\Omega= \\int_Id^3\\mathbf{p}d^3\\mathbf{q} \\\\\\to N!\\, \\Omega'= \\int_{I'}d^3\\mathbf{p}'d^3\\mathbf{q}' = \\int_{I'}d^3\\mathbf{p}d^3\\mathbf{q}\\, ,\n\\end{multline}\nwhere the $I'$ subindex means that now the integration is for all\n$\\mathbf{p'}_j$ that satisfy $\\sum_{i=1}^N |\\mathbf{p'}_i|\\leq ({E'^2-N^2\n  m'^2})^{1\/2}$ for bradyons, $\\sum_{i=1}^N|\\mathbf{p'}_i|\\leq E'$ for\nluxons and $\\sum_{i=1}^N |\\mathbf{p'}_i|\\leq ({E'^2+N^2 m'^2})^{1\/2}$ for\ntachyons in the $I'$ frame.  \n\n\n\n\nDue to Eq.~(\\ref{volps}), the entropy $S'$ calculated in the $I'$\nframe has the same form of the entropy $S$ of Eq.~(\\ref{entropiab}), Eq.~(\\ref{entropial}), and\nEq.~(\\ref{entropiat}), but\nchanging the energy $E$ by the energy $E'$, and the volume $V$ by the volume $V'$.\n\nFor bradyons and luxons the energy transforms as $E'=\\gamma E$, the momentum\ntransforms as $p'=\\gamma p$ and the\nvolume transform as $V=\\gamma V'$ since the relative\nmovement is in one dimension. The relativistic factor is $\\gamma=(1-w^2)^{-1\/2}$ with $w\\leq\n1$. For this particles we are considering positive energies. \n\nIn the case of tachyons, the energy and momentum transformations are $E'=\\zeta E$ and\n$p'=\\zeta p$ respectively, where $\\zeta=(w^2-1)^{-1\/2}$ with\n$w>1$~\\cite{feinberg,maccarrone}. For simplicity, we consider the\npositive momentum tachyons. Similarly, the volume transformation for tachyons is $V=\\zeta V'$~\\cite{eve}. Note\nthat if in the $I$ frame the energy, the momentum and the volume of tachyons are real\nquantities, then in $I'$ frame these quantities are still real. \n\nThe above energy, momentum and volume transformations are one of\nthe multiple transformations that can be constructed for a Lorentz\ninvariant tachyon-theory~\\cite{eve,maccarrone,mariwalla,feinberg2}. Although the present\nanalysis can be done with other transformations, the election of the above one gives back\nthe usual and simpler energy and momentum relations for\ntachyons~\\cite{eve}. They ensure that the tachyon three-vector\nphase-space $d^3{\\bf q}d^3{\\bf p}$ is invariant.\n\nIn order to obtain the temperature, we calculate the thermodynamical variation of\nentropy. The variation is $dS=dE\/T+(P\/T)dV$, where the temperature $T$\nand the pressure $P$ are defined by~\\cite{greiner}\n\\begin{equation}\n  \\label{entrp11}\n  \\frac{1}{T}=\\left(\\frac{\\partial S}{\\partial E}\\right)_V\\, \\quad,\\quad \\frac{P}{T}=\\left(\\frac{\\partial S}{\\partial V}\\right)_E .\n\\end{equation}\n\nIn this way, the calculation of the temperature for bradyons, from\nEq.~(\\ref{entrp11}), is\n\\begin{equation}\n  \\label{tempb}\n  \\frac 1 T =\\frac{3NE}{E^2- N^2m^2}\\, .\n\\end{equation}\n\nThe temperature for luxons is\n\\begin{equation}\n  \\label{templ}\n  \\frac 1 T =\\frac{3N}{E}\\, ,\n\\end{equation}\nand the temperature for tachyons is\n\\begin{equation}\n  \\label{temt}\n  \\frac 1 T =\\frac{3NE}{E^2+ N^2m^2}\\, .\n\\end{equation}\n\nLikewise, we can calculate the pressure for bradyons, luxons and tachyons from\nEq.~(\\ref{entrp11}). This is\n\\begin{equation}\n  \\label{pressure}\n  \\frac P T =\\frac{N}{V}\\, ,\n\\end{equation}\nfor the three species. It corresponds to the state equation for an ideal gas.\n\nCalculation of the temperature $T'$ for bradyons, luxons or tachyons in the $I'\n$ frame\n can be done using Eq.~(\\ref{volps}) to\n evaluate the entropy $S'$. We can write Eq.~(\\ref{entrp11}) for intensive quantities in the\n $I'$ frame. This allow us to express Eq.~(\\ref{tempb}) for bradyons,\n Eq.~(\\ref{templ}) for luxons, and Eq.~(\\ref{temt}) for tachyons in\n $I'$. Thus, we obtain how the temperature $T'$ from $I'$ frame\n transforms to temperature $T$ in the $I$ frame for bradyon and luxon ideal gas under the transformations\n for energy and momentum previously established\n\\begin{equation}\n  \\label{transT}\n  {T'}={\\gamma T}\\, ,\n\\end{equation}\nand for tachyon gas\n\\begin{equation}\n  \\label{transTt}\n  {T'}={\\zeta T}\\, .\n\\end{equation}\n\nThe transformations in Eq.~(\\ref{transT}) and in Eq.~(\\ref{transTt}) implies that the\ntemperature is not a Lorentz invariant. The temperature transformation\nfor bradyons and luxons (\\ref{transT}) is in coincidence with Ott's\ntemperature transformation~\\cite{Ott} and other previous\nworks~\\cite{newburgh,bors,sutcliffe,moya}, and it is in disagreement\nwith Planck's formalism~\\cite{Einstein,planck,tolman,impos,kania2,kowa}. This means\nthat a moving gas of bradyons or luxons appears hotter. The temperature transformation for\ntachyons~(\\ref{transTt}) is derived for the first time in the\nknowledge of the authors. \n\nThe difference between our approach and other approaches is the\ndefinition of temperature. We emphasize that temperature is defined in\na thermodynamic and statistical form by Eq.~(\\ref{entrp11}). Thus, the\ncorrect definition of Eq.~(\\ref{entrp11}) leads naturally to the above\ncorrect temperature transformations. \n\nWe can do the same analysis for the pressure $P'$ in the $I'$ frame\nusing the transformation for the energy and the momentum. In the same\nway, according to Liouville theorem and using Eq.~(\\ref{entrp11}) and\nEq.~(\\ref{pressure}) for $I'$, we can get the transformation of\npressure $P'$ from $I'$ frame to pressure $P$ in the $I$ frame for\nbradyons and luxons as\n\\begin{equation}\n  \\label{transp}\n  {P'}={\\gamma^2 P}\\, .\n\\end{equation}\n\nSimilarly, for tachyons, the pressure transformation is\n\\begin{equation}\n  \\label{transpt}\n  {P'}={\\zeta^2 P}\\, .\n\\end{equation}\n\nWe also see that pressure is not Lorentz invariant. The result for bradyons and luxons coincide with the\nsame one found previously in Refs.~\\cite{moya,sutcliffe}. This\ntransformation for pressure goes in contradiction with some previous\nworks~\\cite{Einstein,planck,tolman,impos,kowa}. The tachyon\npressure transformation is derived for first time. Our thermodynamic and\nstatistical definition for pressure transformations~(\\ref{transp})\nand~(\\ref{transpt}) preserves the properties of ideal gases. Thus, the\ntemperature and pressure transformation are necessary to get \nan ideal gas of any of these particles in both frames. After taking\ninto account Eq.~(\\ref{transT}) and Eq.~(\\ref{transp}), we can write\nEq.~(\\ref{pressure}) in the $I'$ frame as\n\\begin{equation}\n\\label{gasI'}\nP'\\, V' = N\\, T'\\,.\n\\end{equation}\n\nFrom this we can conclude that in every inertial frame an ideal\ngas behaves like an ideal gas under Lorentz transformations as one would\nexpect due to special relativity principle. \n\nIn the same token, the transformations of Eq.~(\\ref{transT}),\nEq.~(\\ref{transTt}), Eq.~(\\ref{transp}) and Eq.~(\\ref{transpt})\nfor intensive quantities $T$ and $p$, for bradyons, luxons\nor tachyons satisfy\n\\begin{equation}\n  \\label{dS\/dE}\n   \\left(\\frac{\\partial S'}{\\partial E'}\\right)_{V'} dE' = \\left(\\frac{\\partial S}{\\partial E}\\right)_{V} dE\\,,\n\\end{equation}\nand\n\\begin{equation}\n  \\label{dS\/dV}\n   \\left(\\frac{\\partial S'}{\\partial V'}\\right)_{E'} dV' = \\left(\\frac{\\partial S}{\\partial V}\\right)_{E} dV\\,.\n\\end{equation}\n\nTherefore, the variation of the entropy is the same in both frames, which means\n\\begin{equation}\n  \\label{dS}\n  dS = dS'\\,,\n\\end{equation}\nfor any of the three species, wich is according to all previous works. \n\n\nFinally, it is easy to get from Eq.~(\\ref{templ}) the correct energy for\nan ideal gas of luxons as $E=3NT$. From Eq.~(\\ref{tempb}) we can\nobtain the correct non-relativistic energy for an ideal gas of\nbradyons when $m\\gg T$ \n\\begin{equation}\n  \\label{Ebnorel}\n E\\simeq\\frac{3}{2}NT+Nm\\, .\n\\end{equation}\n\nFor an ideal gas of tachyons, it is possible to obtain the energy in\nthe very-high temperature limit when $m\\ll T$. From Eq.~(\\ref{temt}) we obtain\n\\begin{equation}\n  \\label{Etnorel}\n E\\simeq\\frac{Nm^2}{3T}\\, .\n\\end{equation}\n\nFrom~(\\ref{Etnorel}) we can see that the energy become null when the tachyon velocity and temperature goes to\ninfinite as expected~\\cite{feinberg}.\n\n\n\\section{Conclusions}\n\nWe have shown a new path to get some known results of temperature\ntransformation for a gas of non-interacting particles. Our treatment is from statistical first principles, only\nassuming the known space-time and\nenergy-momentum Lorentz transformation along with Liouville theorem and Gibss' postulate.\n\nThe temperature transformation for a classical ideal gas composed of\nparticles which moves slower, equal or faster than light was shown\nexplicitely. These transformations are the correct ones, at least for\nmicrocanonical ensemble, because they were derived using \nonly the known statistical properties for each particle. In addition,\nLiouville theorem allow us to work in the microcanonical ensemble for\nany inertial frame, so the transformations obtained preserve the form\nof the first and second laws of thermodynamics in all inertial\nframes. This is a difference with the usual relativistic\nthermodynamics treatment, where the forms of the first and second laws\nare choosen in a more arbitrary way. \n\n\nAn interesting consecuence of the transformations that we\nfound for temperature and pressure is that the equation of\nstate of an ideal gas is a Lorentz invariant. This is in agreement\nwith the first postulate of special relativity as one would expect.\n\nFor tachyons, Eq.~(\\ref{Etnorel}) is correct in the high temperature\nlimit, when their velocity goes to infinite and their\nenergy $E\\to 0$. However, when the relative speed between frames\n$w$ goes to infinite, the temperature $T'$ goes to zero from\nEq.~(\\ref{transTt}). This is due to a tachyon in $I$ frame is a bradyon\nin a $I'$ frame which moves with speed greater than $1$ relative to the $I$\nframe~\\cite{antippa}. The behavior of the tachyon temperature transformation~(\\ref{transTt}) is\nequivalent  to temperature transformation of bradyons when $w\\to\n1$. This shows the duality bradyon-tachyon between frames moving at\nrelative speeds greater than light.\n\n\n\\acknowledgments\n\nWe thank to Dr. Gonzalo Guti\\'errez, Dr. J. Alejandro Valdivia and\nMSc. Andr\\'es Gom\\'ez for useful discussions and their enlightening\ncomments. \n\nF. A. is grateful to Programa MECE Educaci\\'on Superior for a Doctoral\nFellowship, C. A. F. is grateful to CONICyT Master Fellowship and\nP. S. M. is grateful to CONICyT Doctoral Fellowship.  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\setcounter{equation}{0}A major direction in differential geometry is the\nstudy of Riemannian manifolds with exceptional holonomy, i.e. $7\n-dimensional $G_{2}$-manifolds and $8$-dimensional $\\func{Spin}\\left(\n7\\right) $-manifolds, as well as more generally, $G_{2}$-structures and \n\\func{Spin}\\left( 7\\right) $-structures. As it turns out, both of these\ngroups are closely related to the octonions \\cite{Harvey}, which is the \n8$-dimensional nonassociative normed division algebra $\\mathbb{O}$ over \n\\mathbb{R}.$ A number of properties of $G_{2}$-structures and $\\func{Spin\n\\left( 7\\right) $-structure are hence artifacts of the octonionic origin of\nthese groups. In particular, in \\cite{GrigorianOctobundle}, the author has\nexplicitly used an octonion formalism to investigate properties of isometric \n$G_{2}$-structures. In that setting, it emerged that objects such as the\ntorsion of a $G_{2}$-structure are naturally expressed in terms of sections\nof a unit octonion bundle. The set of unit octonions $U\\mathbb{O}\\cong\nS^{7}, $ has the algebraic structure of a \\emph{Moufang loop}. Indeed, a\ncloser look shows that in the context of $G_{2}$-structure, the algebra\nstructure of $\\mathbb{O}$ played a secondary role to the loop structure on $\n\\mathbb{O} $ and the corresponding cross-product structure on the tangent\nspace at the identity $T_{1}U\\mathbb{O\\cong }\\func{Im}\\mathbb{O}$, the pure\nimaginary octonions. This suggests that there is room for generalization by\nconsidering bundles of other smooth loops. As far as possible, we will\nminimize assumptions made on the loops. Generally, there is a large supply\nof smooth loops, because given a Lie group $G,$ a Lie subgroup $H$, and a\nsmooth section $\\sigma :G\/H\\longrightarrow G$ (i.e. a smooth collection of\ncoset representatives), we may define a loop structure on $G\/H$ if $\\sigma $\nsatisfies certain conditions, such as $\\sigma \\left( H\\right) =1$, and for\nany cosets $xH$ and $yH,$ there exists a unique element $z\\in \\sigma \\left(\nG\/H\\right) $ such that $zxH=yH$ \\cite{NagyStrambachBook}. Conversely, any\nsmooth loop can also be described in terms of a section of a quotient of Lie\ngroups. Special kinds of smooth loops, such as Moufang loops have been\nclassified \\cite{NagyStrambachBook}$,$ however for a broader classes, such\nas Bol loops, there exists only a partial classification \\cite{FigulaBol}.\n\nIn \\cite{GrigorianOctobundle}, the octonion bundle is constructed out of the\ntangent bundle, and is hence very specific, one could say canonical. However\nto understand properties of the bundle, it is helpful to decouple the bundle\nstructure and the properties of the base manifold. Hence, another direction\nfor generalization is to consider loop bundles over arbitrary manifolds. In\nparticular, such an approach will also make it more clear which properties\nof the octonion bundle in the $G_{2}$ setting are generic and which are\nintrinsic to the $G_{2}$-structure.\n\nThe purpose of this paper is two-fold. One is to carefully build up the\ntheory of loop bundles starting with all the necessary algebraic\npreliminaries and properties of smooth loops. The second is to define a\nunified framework through which geometric structures based on certain\nalgebraic structures may be studied. In this sense, this can be considered\nas an extension of the normed division algebra approach to various\nstructures in Riemannian geometry as developed by Leung \\cite{LeungDivision\n. The long-term goal in $G_{2}$-geometry is to obtain some kind of analog of\nYau's celebrated theorem on existence of Calabi-Yau metrics \\cite{CalabiYau\n, and thus a key theme in the study of $G_{2}$-manifolds is to try to\ncompare and contrast the corresponding theory of K\\\"{a}hler and Calabi-Yau\nmanifolds. This requires putting the complex and octonionic geometries into\nthe same framework. However, a certain amount of generalization allows to\nsee clearer some aspects of the theory.\n\nIn Section \\ref{sectLoop} we give an overview of the key algebraic\nproperties of loops. While many basic properties of loops may be known to\nalgebraists, they may be new to geometers. Moreover, we adopt a point of\nview where we emphasize the pseudoautomorphism group of a loop, which is a\ngeneralization of the automorphism group, and properties of modified\nproducts defined on loops. These are the key objects that are required to\ndefine loop bundles, however in algebraic literature they typically take the\nbackstage. In particular, we show how the pseudoautomorphism group, the\nautomorphism group, the nucleus of a loop are related and how these\nrelationships manifest themselves in the octonion case as well-known\nrelationships between the groups $\\func{Spin}\\left( 7\\right) ,$ $SO\\left(\n7\\right) $, and $G_{2}$.\n\nIn Section \\ref{sectSmooth}, we then restrict attention to smooth loops,\nwhich are the not necessarily associative analogs of Lie groups. We also\nmake the assumption that the pseudoautomorphism group acts on the smooth\nloop via diffeomorphisms and is hence itself a Lie group. This is an\nimportant assumption and it is not known whether this is always true. The\nkey example of a non-associative smooth loop is precisely the loop of unit\noctonions. We first define the concept of an exponential function, which is\nsimilar to that on Lie groups. This is certainly not a new concept - it\nfirst defined by Malcev in 1955 \\cite{Malcev1955}, but here we show that in\nfact, generally, there may be different exponential maps, based on the\ninitial conditions of the flow equation. This then relates to the concept of\nthe modified product as defined in Section \\ref{sectLoop}. Then, in Section\n\\ref{secTangent}, we define an algebra structure on tangent spaces of the\nloop. The key difference with Lie algebras is that in the non-associative\ncase, there is a bracket defined at each point of the loop. Indeed, as shown\nin Section \\ref{sectStruct}, the differential of the bracket depends on the\nassociator, which of course vanishes on Lie algebras, but is non-trivial on\ntangent algebras of non-associative loops. Moreover, in Section \\re\n{sectStruct}, we prove a loop version of the Maurer-Cartan structural\nequation. Namely, for any point $p$ in the loop, the right Maurer-Cartan\nform satisfies the following equation\n\\begin{equation}\n\\left( d\\theta \\right) _{p}-\\frac{1}{2}\\left[ \\theta ,\\theta \\right]\n^{\\left( p\\right) }=0,\n\\end{equation\nwhere $\\left[ \\cdot ,\\cdot \\right] ^{\\left( p\\right) }$ is the bracket at\npoint $p$. In Lie theory, the Jacobi identity is the integrability condition\nfor the Maurer-Cartan equation, however in the non-associative case, the\ncorresponding equation is known as the Akivis identity \\cit\n{HofmannStrambach}, and involves the associator.\n\nIn Section \\ref{sectStruct} we define another key component in the theory of\nsmooth loops. As discussed above, each element $s$ of the loop $\\mathbb{L}$\ndefines a bracket $b_{s}$ on the tangent algebra $\\mathfrak{l}$. Moreover,\nwe also define a map $\\varphi _{s}$ that maps the Lie algebra $\\mathfrak{p}$\nof the pseudoautomorphism group to the loop tangent algebra. The kernel of\nthis map is precisely the Lie algebra $\\mathfrak{h}_{s}$ of the stabilizer\nof $s$ in the pseudoautomorphism group. In the case of unit octonions, we\nknow $\\mathfrak{p\\cong so}\\left( 7\\right) \\cong \\Lambda ^{2}\\left( \\mathbb{R\n^{7}\\right) ^{\\ast }$ and $\\mathfrak{l=}\\func{Im}\\mathbb{O}\\cong \\mathbb{R\n^{7}$ , so $\\varphi _{s}$ can be regarded as an element of $\\mathbb{R\n^{7}\\otimes $ $\\Lambda ^{2}\\mathbb{R}^{7},$ and this is (up to a constant\nfactor) a dualized version of the $G_{2}$-invariant $3$-form $\\varphi $, as\nused to project from $\\Lambda ^{2}\\left( \\mathbb{R}^{7}\\right) ^{\\ast }$ to \n\\mathbb{R}^{7}.$ The kernel of this map is then the Lie algebra $\\mathfrak{g\n_{2}.$ The $3$-form $\\varphi $ also defines the bracket on $\\func{Im}\\mathbb\nO}$, so in this case, both $b_{s}$ and $\\varphi _{s}$ are determined by the\nsame object, but in general they have different roles. By considering the\naction of $U\\left( n\\right) $ on $U\\left( 1\\right) $ (i.e. the unit complex\nnumbers) and $Sp\\left( n\\right) Sp\\left( 1\\right) $ on $Sp\\left( 1\\right) $\n(i.e. the unit quaternions), we find that Hermitian and hyperHermitian\nstructures fit into the same framework. Namely, a complex Hermitian form, a\nquaternionic triple of Hermitian forms, and the $G_{2}$-invariant $3$-form\nhave the same origin as $2$-forms with values in imaginary complex numbers,\nquaternions, and octonions, respectively.\n\nIn Section \\ref{sectKilling} we define an analog of the Killing form on \n\\mathfrak{l}$ and give conditions for it to be invariant under both the\naction of $\\mathfrak{p}$ and the bracket on $\\mathfrak{l}.$ In particular,\nusing the Killing form, we define the adjoint $\\varphi _{s}^{t}$ of $\\varphi\n_{s}$. This allows to use the Lie bracket on $\\mathfrak{p}$ to define\nanother bracket on $\\mathfrak{l}.$ In the case of octonions, it's\nproportional to the standard bracket on $\\mathfrak{l},$ but in general could\nbe a distinct object.\n\nIn Section \\ref{sectDarboux}, we consider maps from some smooth manifold $M$\nto a smooth loop. Given a fixed map $s$, we can then define the\ncorresponding products of loop-valued maps and correspondingly a bracket of \n\\mathfrak{l}$-valued maps. Similarly as for maps to Lie groups, we define\nthe Darboux derivative \\cite{SharpeBook} of $s$ - this is just $s^{\\ast\n}\\theta $ - the pullback of the Maurer-Cartan form on $\\mathbb{L}.$ This now\nsatisfies a structural equation, which is just the pullback of the loop\nMaurer-Cartan equation, as derived in Section \\ref{sectStruct}, with respect\nto the bracket defined by $s$. For maps to Lie groups, there holds a\nnon-abelian \\textquotedblleft Fundamental Theorem of\nCalculus\\textquotedblright\\ \\cite[Theorem 7.14]{SharpeBook}, namely that if\na Lie algebra-valued $1$-form on $M$ satisfies the structural equation, then\nit is the Darboux derivative of some Lie group-valued function. Here, we\nprove an analog for $\\mathfrak{l}$-valued $1$-forms (Theorem \\re\n{thmLoopCartan}). However, since in the non-associative case, the bracket in\nthe structural equation depends on $s,$ Theorem \\ref{thmLoopCartan} requires\nthat such a map already exists and some additional conditions are also\nneeded, so as expected, it's not as powerful as for Lie groups. However, in\nthe case the loop is associative, it does reduce to the theorem for Lie\ngroups.\n\nFinally, in Section \\ref{sectBundle}, we turn our attention to loop bundles\nover a smooth manifold $M$. In fact, since it's not a single bundle, it's\nbest to refer to a \\emph{loop structure} over a manifold. The key component\nis $\\Psi $-principal bundle $\\mathcal{P}$ where $\\Psi $ is a group that acts\nvia pseudoautomorphisms on the loop $\\mathbb{L}.$ Then, several bundles\nassociated to $\\mathcal{P}$ are defined: two bundles $\\mathcal{Q}$ and \n\\mathcal{\\mathring{Q}}$ with fibers diffeomorphic to $\\mathbb{L}$, but with\nthe bundle structure with respect to different actions of $\\Psi $; the\nvector bundle $\\mathcal{A}$ with fibers isomorphic to $\\mathfrak{l},$ as\nwell as some others. Crucially, a section $s$ of the bundle $\\mathcal\n\\mathring{Q}}$ then defines a fiberwise product structure on sections of \n\\mathcal{Q}$, a fiberwise bracket structure, and a map $\\varphi _{s}$ from\nsections of the adjoint bundle $\\mathfrak{p}_{\\mathcal{P}}$ to sections of \n\\mathcal{A}.$ In the key example of a $G_{2}$-structure on a $7$-manifold $M\n, the bundle $\\mathcal{P}$ is then the $Spin\\left( 7\\right) $-bundle that is\nthe lifting of the orthonormal frame bundle. The bundles $\\mathcal{Q}$ and \n\\mathcal{\\mathring{Q}}$ are unit octonion bundles, similarly as defined in \n\\cite{GrigorianOctobundle}, but $\\mathcal{Q}$ transforms under $SO\\left(\n7\\right) ,$ and hence corresponds the the unit subbundle of $\\mathbb{R\n\\oplus TM,$ while $\\mathcal{\\mathring{Q}}$ transforms under $Spin\\left(\n7\\right) $, and hence corresponds to the unit subbundle of the spinor\nbundle. The section $s$ then defines a global unit spinor, and hence defines\na reduction of the $Spin\\left( 7\\right) $ structure group to $G_{2}$, and\nthus defines a $G_{2}$-structure. In the complex and quaternionic examples,\nthe corresponding bundle $\\mathcal{P}$ then has $U\\left( n\\right) $ and \nSp\\left( n\\right) Sp\\left( 1\\right) $ structure group, respectively, and the\nsection $s$ defines a reduction to $SU\\left( n\\right) $ and $Sp\\left(\nn\\right) ,$ respectively. Thus, as noted in \\cite{LeungDivision}, indeed the\noctonionic analog of a reduction from K\\\"{a}hler structure to Calabi-Yau\nstructure and from quaternionic K\\\"{a}hler to HyperK\\\"{a}hler, is the\nreduction from $Spin\\left( 7\\right) $ to $G_{2}.$\n\nUsing the equivalence between sections of bundles associated to $\\mathcal{P}$\nand corresponding equivariant maps, we generally work with equivariant maps.\nIndeed, in that case, $s:\\mathcal{P}\\longrightarrow \\mathbb{L}$ is an\nequivariant map, and given a connection $\\omega $ on $\\mathcal{P}$, we find\nthat the Darboux derivative of $s$ decomposes as \n\\begin{equation}\ns^{\\ast }\\theta =T^{\\left( s,\\omega \\right) }-\\hat{\\omega}^{\\left( s\\right) \n\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{,}  \\label{sTom}\n\\end{equation\nwhere $\\hat{\\omega}^{\\left( s\\right) }=\\varphi _{s}\\left( \\omega \\right) $\nand $T^{\\left( s,\\omega \\right) }$ is the \\emph{torsion of }$s$ \\emph{with\nrespect to the connection }$\\omega $, which is defined as the horizontal\npart of $s^{\\ast }\\theta .$ The quantity $T^{\\left( s,\\omega \\right) }$ is\ncalled the torsion because in the case of $G_{2}$-structures on a $7\n-manifold, if we take $\\mathcal{P}$ to be the spin bundle and $\\omega $ the\nLevi-Civita connection for a fixed metric, then $T^{\\left( s,\\omega \\right)\n} $ is precisely (up to the chosen sign convention) the torsion of the \nG_{2} $-structure defined by the section $s$. Moreover, vanishing of \nT^{\\left( s,\\omega \\right) }\\ $implies a reduction of the holonomy group of \n\\omega $. As shown in \\cite{GrigorianOctobundle}, the torsion of a $G_{2}\n-structure may be considered as a $1$-form with values in the bundle of\nimaginary octonions. Indeed, in general, $T^{\\left( s,\\omega \\right) }$ is a\nbasic (i.e. horizontal and equivariant) $\\mathfrak{l}$-valued $1$-form on \n\\mathcal{P}$, so it corresponds to an $\\mathcal{A}$-valued $1$-form on $M$.\nIt also enters expressions for covariant derivatives of products of sections\nof $\\mathcal{Q}$ and the bracket on $\\mathcal{A}.$\n\nThe relation (\\ref{sTom}) is significant because it shows that the torsion\nvanishes if and only if $-\\hat{\\omega}^{\\left( s\\right) }$ is equal to the \n\\mathfrak{l}$-valued Darboux derivative $s^{\\ast }\\theta $. In particular, a\nnecessary condition is then that $-\\hat{\\omega}^{\\left( s\\right) }$\nsatisfies the loop structural equation. In Theorem \\ref{thmTNucl}, we give a\npartial converse under certain assumptions on $\\mathbb{L}.$\n\nIn Section \\ref{sectCurv}, we then also consider the projection of the\ncurvature $F$ of $\\omega $ to $\\mathfrak{l}.$ We define $\\hat{F}=\\varphi\n_{s}\\left( F\\right) $, which is then equal to the horizontal part of $d\\hat\n\\omega},$ and show in Theorem \\ref{thmFTstruct} that $\\hat{F}$ and $T$ are\nrelated via a structural equation\n\\begin{equation}\n\\hat{F}=d^{\\mathcal{H}}T-\\frac{1}{2}\\left[ T,T\\right] ^{\\left( s\\right) },\n\\end{equation\nwhere $\\left[ \\cdot ,\\cdot \\right] ^{\\left( s\\right) }$ is the bracket\ndefined by $s$. Again, such a relationship is recognizable from $G_{2}\n-geometry, where the projection $\\pi _{7}\\func{Riem}$ of the Riemann\ncurvature to the $7$-dimensional representation of $G_{2}$ satisfy the\n\\textquotedblleft $G_{2}$ Bianchi identity\\textquotedblright\\ \\cit\n{GrigorianOctobundle,karigiannis-2007}. We also consider gauge\ntransformations. In this setting, we have two quantities - the connection\nand the section $s.$ We show that under a simultaneous gauge transformation\nof the pair $\\left( s,\\omega \\right) ,$ $\\hat{F}$ and $T$ transform\nequivariantly.\n\nFinally, in Section \\ref{sectVar}, we consider several functionals and the\ncorresponding critical points, at least under some assumptions on the loop \n\\mathbb{L}.$ Indeed, if we consider the loop bundle structure over a $3\n-dimensional manifold, then we can write down an analog of the Chern-Simons\nfunctional. The critical points over the space of connections, but with a\nfixed section $s$, are connections for which $\\hat{F}=0$, i.e. the curvature\nlies in $\\mathfrak{h}_{s}$ everywhere. If we moreover consider the critical\npoints over pairs $\\left( s,\\omega \\right) $, then we get an additional\ncondition on the torsion, namely that $\\left[ T,T,T\\right] ^{\\left( s\\right)\n}=0$, where $\\left[ \\cdot ,\\cdot ,\\cdot \\right] ^{\\left( s\\right) }$ is the\nassociator defined by $s$ and wedge products of $1$-forms are implied.\n\nAnother functional that we consider is the $L^{2}$-norm squared of the\ntorsion $\\int_{M}\\left\\vert T\\right\\vert ^{2}$. In this case, we fix the\nconnection, and consider critical points over the space of sections $s$, or\nequivalently, equivariant loop-valued maps from $\\mathcal{P}.$ In the $G_{2}$\nsetting, similar functionals have been considered in \\cit\n{Bagaglini2,DGKisoflow,GrigorianOctobundle, GrigorianIsoflow,GrigorianIsoFlowSurvey,SaEarpLoubeau}.\nThis is then closely related to the Dirichlet energy functional, but\nrestricted to equivariant maps. The critical points then are maps $s$, for\nwhich the torsion is divergence-free.\n\n\\subsection*{Acknowledgements}\n\nThis work was supported by the National Science Foundation [DMS-1811754].\nThe author also thanks Henrique S\\'{a} Earp and Jonathan D.H. Smith for some\nhelpful suggestions.\n\n\\section{Loops}\n\n\\setcounter{equation}{0}\\label{sectLoop}\n\n\\subsection{Definitions}\n\nThe main object of study in this paper is a \\emph{loop}. Roughly, this can\nbe thought of as a non-associative analog of a group, but with a few\ncaveats. According to \\cite{PflugHistorical}, this term was coined by the\ngroup of Abraham Albert in Chicago in 1940's, as rhyming with \\emph{group \nand also referring to the Chicago Loop. Unfortunately however, for\nnon-algebraists, and especially in geometry and topology, this term may\ncause confusion. A less ambiguous term would be something like a \\emph\nunital quasigroup }or \\emph{quasigroup with identity}, however this would be\nnonstandard terminology and also much longer than a loop. In general,\nnon-associative algebra requires a large number of definitions and concepts\nthat become unnecessary in the more standard associative setting. In this\nsection we go over some of the terminology and notation that we will be\nusing. The reader can also refer to \\cit\n{HofmannStrambach,KiechleKloops,NagyStrambachBook,SabininBook,SmithJDHQuasiReps}\nfor the various concepts, although, as far as the author knows, much of the\nnotation in this setting is not standardized.\n\n\\begin{definition}\nA \\emph{quasigroup }$\\mathbb{L}$ is a set together with the following\noperations $\\mathbb{L}\\times \\mathbb{L}\\longrightarrow \\mathbb{L}$\n\n\\begin{enumerate}\n\\item Product $\\left( p,q\\right) \\mapsto pq$\n\n\\item Right quotient $\\left( p,q\\right) \\mapsto p\\backslash q$\n\n\\item Left quotient $\\left( p,q\\right) \\mapsto q\\backslash p$,\n\\end{enumerate}\n\nthat satisfy the following properties\n\n\\begin{enumerate}\n\\item $\\left( p\\backslash q\\right) q=p$\n\n\\item $q\\left( q\\backslash p\\right) =p$\n\n\\item $\\faktor{pq}{q}=p$\n\n\\item $\\scalebox{-1}[1]{\\nicefrac{\\scalebox{-1}[1]{$pq$}}\n\\scalebox{-1}[1]{$p$}}} =q.$\n\\end{enumerate}\n\\end{definition}\n\nWe will interchangeably denote the product operation by $p\\cdot q.$ To avoid\nmultiple parentheses, at times we will use the convention $a\\cdot bc=a\\left(\nbc\\right) $ and $ab\/c=\\left( ab\\right) \/c$. If the same underlying set \n\\mathbb{L}$ is equipped with a different product operation $\\circ _{r}$(to\nbe defined later), then the corresponding quasigroup will be denoted by \n\\left( \\mathbb{L},\\circ _{r}\\right) $ and the corresponding quotient\noperation by $\\backslash _{r}$.\n\n\\begin{definition}\nLet $\\mathbb{L}$ be a quasigroup. The \\emph{right nucleus }$\\mathcal{N\n^{R}\\left( \\mathbb{L}\\right) $ \\emph{of }$\\mathbb{L}$ is the set of all \nr\\in \\mathbb{L},$ such that for any $p,q\\in \\mathbb{L}$, \n\\begin{equation}\npq\\cdot r=p\\cdot qr.  \\label{assoc}\n\\end{equation\nSimilarly, define the left nucleus $\\mathcal{N}^{L}\\left( \\mathbb{L}\\right) $\nand the middle nucleus $\\mathcal{N}^{M}\\left( \\mathbb{L}\\right) $.\n\\end{definition}\n\nElements of $\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) $ satisfy several other\nuseful properties.\n\n\\begin{lemma}\n\\label{LemAssoc} If $r\\in \\mathcal{N}^{R}\\left( \\mathbb{L}\\right) $, then\nfor any $p,q\\in \\mathbb{L}$,\n\n\\begin{enumerate}\n\\item $\\faktor{pr}{qr}=p\/q$\n\n\\item $p\\cdot q\\slash r=\\faktor{pq}{r}$\n\n\\item $\\scalebox{-1}[1]{\\nicefrac{\\scalebox{-1}[1]{$qr$}}\n\\scalebox{-1}[1]{$p$}}}=p\\backslash q\\cdot r.$\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{lemma}\nThe first property follows from (\\ref{assoc}) using \n\\begin{equation*}\np\/q\\cdot qr=\\left( p\/q\\cdot q\\right) r.\n\\end{equation*\nThe second property follows similarly using \n\\begin{equation*}\np\\left( q\/r\\cdot r\\right) =\\left( p\\cdot q\/r\\right) r.\n\\end{equation*\nThe third property follows using \n\\begin{equation*}\n\\left( p\\cdot p\\backslash q\\right) r=p\\left( p\\backslash q\\cdot r\\right) .\n\\end{equation*}\n\\end{lemma}\n\nIn group theory the only reasonable morphism between groups is a group\nhomomorphism, however for quasigroups there is significantly more\nflexibility.\n\n\\begin{definition}\nSuppose $\\mathbb{L}_{1},\\mathbb{L}_{2}$ are quasigroups. Then a triple \n\\left( \\alpha ,\\beta ,\\gamma \\right) $ of maps from $\\mathbb{L}_{1}$ to \n\\mathbb{L}_{2}$ is a \\emph{homotopy} from $\\mathbb{L}_{1}$ to $\\mathbb{L\n_{2} $ if for any $p,q\\in \\mathbb{L}_{1}$, \n\\begin{equation}\n\\alpha \\left( p\\right) \\beta \\left( q\\right) =\\gamma \\left( pq\\right) .\n\\label{Qhom}\n\\end{equation\nIf $\\left( \\alpha ,\\alpha ,\\alpha \\right) $ is a homotopy, then $\\alpha $ is\na \\emph{quasigroup homomorphism}. If each of the maps $\\alpha ,\\beta ,\\gamma \n$ is a bijection, then $\\left( \\alpha ,\\beta ,\\gamma \\right) $ is an \\emph\nisotopy}. An isotopy from a quasigroup to itself is an \\emph{autotopy}. The\nset of all autotopies of a quasigroup $\\mathbb{L}$ is clearly a group under\ncomposition. If $\\left( \\alpha ,\\alpha ,\\alpha \\right) \\ $is an autotopy,\nthen $\\alpha $ is an automorphism of $\\mathbb{L}$, and the group of\nautomorphisms is denoted by $\\func{Aut}\\left( \\mathbb{L}\\right) $.\n\\end{definition}\n\nWe will only be concerned with quasigroups that have an identity element,\ni.e. loops.\n\n\\begin{definition}\nA \\emph{loop} $\\mathbb{L}$ is a quasigroup that has a unique identity\nelement $1\\in \\mathbb{L}$ such that for any $q\\in \\mathbb{L}$, \n\\begin{equation}\n1\\cdot q=q\\cdot 1=q.  \\label{idelem}\n\\end{equation}\n\\end{definition}\n\n\\begin{definition}\nLet $\\mathbb{L}$ be a loop. Then, for any $q\\in \\mathbb{L}$ define\n\n\\begin{enumerate}\n\\item The \\emph{right inverse }$q^{\\rho }=q\\backslash 1.$\n\n\\item The \\emph{left inverse }$q^{\\lambda }=1\/q.$\n\nIn particular, they satisfy \n\\begin{equation}\nqq^{\\rho }=q^{\\lambda }q=1.\n\\end{equation}\n\\end{enumerate}\n\\end{definition}\n\nFor a general quasigroup, the nuclei may be empty, however if $\\mathbb{L}$\nis a loop, the identity element $1$ associates with any other element, so\nthe nuclei are non-empty. Moreover, it is easy to show that $\\mathcal{N\n^{R}\\left( \\mathbb{L}\\right) $ (and similarly, $\\mathcal{N}^{L}\\left( \n\\mathbb{L}\\right) $ and $\\mathcal{N}^{M}\\left( \\mathbb{L}\\right) $) is a\ngroup.\n\n\\begin{theorem}\nLet $\\mathbb{L}$ be a loop, then the right nucleus $\\mathcal{N}^{R}\\left( \n\\mathbb{L}\\right) $ is a group.\n\\end{theorem}\n\n\\begin{proof}\nClearly, $1\\in \\mathcal{N}^{R}\\left( \\mathbb{L}\\right) $. Also, suppose \na,b\\in \\mathcal{N}^{R}\\left( \\mathbb{L}\\right) .$ Then, for any $p,q\\in \n\\mathbb{L}$, \n\\begin{eqnarray*}\npq\\cdot ab &=&\\left( pq\\cdot a\\right) b=\\left( p\\cdot qa\\right) b \\\\\n&=&p\\left( qa\\cdot b\\right) =p\\left( q\\cdot ab\\right)\n\\end{eqnarray*\nand hence, $ab\\in \\mathcal{N}^{R}\\left( \\mathbb{L}\\right) $. Moreover, it is\nclear that the product on $\\mathbb{L}$ restricted to $\\mathcal{N}^{R}\\left( \n\\mathbb{L}\\right) $ is associative.\n\nIf $a\\in \\mathcal{N}^{R}\\left( \\mathbb{L}\\right) $, then \n\\begin{equation*}\na=a\\cdot a^{\\lambda }a=aa^{\\lambda }\\cdot a\n\\end{equation*\nand thus, $\\alpha ^{\\lambda }=a^{\\rho }$, so $a$ has a well-defined inverse \na^{-1}=a^{\\lambda }=a^{\\rho }.$ Moreover, since for any $p\\in \\mathbb{L}$, \n\\left( pa^{-1}\\right) a=p$, we see that $pa^{-1}=p\/a$. Now, for $p,q\\in \n\\mathbb{L}$ we have \n\\begin{equation*}\n\\left( p\\cdot qa^{-1}\\right) a=p\\left( qa^{-1}\\cdot a\\right) =pq\n\\end{equation*\nand hence \n\\begin{equation*}\np\\cdot qa^{-1}=\\left( pq\\right) \/a=pq\\cdot a^{-1}.\n\\end{equation*\nThus, $a^{-1}\\in \\mathcal{N}^{R}\\left( \\mathbb{L}\\right) .$\n\\end{proof}\n\nLoops may be endowed with additional properties that bestow various weaker\nforms of associativity and inverse properties.\n\n\\begin{enumerate}\n\\item \\emph{Two-sided inverse}: for any $p\\in \\mathbb{L}$, $p^{\\rho\n}=p^{\\lambda }.$ Then we can define a unique two-sided inverse $p^{-1}.$\n\n\\item \\emph{Right inverse property}: for any $p,q\\in \\mathbb{L}$, $pq\\cdot\nq^{\\rho }=p.$ In particular, this implies that the inverses are two-sided,\nso we can set $p^{-1}=p^{\\rho }=p^{\\lambda }$, and moreover $p\/q=pq^{-1}$.\nThe \\emph{left} inverse property is defined similarly. A loop with both the\nleft and right inverse properties is said to be an \\emph{inverse loop}.\n\n\\item \\emph{Power-associativity }(or\\emph{\\ monoassociativity}): any element \n$p\\in \\mathbb{L}$ generates a subgroup of $\\mathbb{L}.$ In particular, this\nimplies that $\\mathbb{L}$ has two-sided inverses. Power-associativity allows\nto unambiguously define integer powers $p^{n}$ of elements. Note that some\nauthors use monoassociativity as a more restrictive property, namely only\nthat $pp\\cdot p=p\\cdot pp$.\n\n\\item \\emph{(Left)-alternative}: for any $p,q\\in \\mathbb{L}$, $p\\cdot\npq=pp\\cdot q.$ Similarly we can define the right-alternative property (i.e. \nq\\cdot pp=qp\\cdot p$). In each of these cases, $\\mathbb{L}$ has two-sided\ninverses. If $\\mathbb{L}$ is both left-alternative and right-alternative,\nthen it is said to be \\emph{alternative. }A loop with a similar property\nthat $p\\cdot qp=pq\\cdot p$ is known as a \\emph{flexible loop}.\n\n\\item \\emph{Diassociative: }any two elements $p,q\\in \\mathbb{L}$ generate a\nsubgroup of $\\mathbb{L}$. Clearly, a diassociative loop has the inverse\nproperty, is power-associative, alternative, and flexible.\n\n\\item \\emph{(Left) Bol loop}: for any $p,q,r\\in \\mathbb{L}$, \n\\begin{equation}\np\\left( q\\cdot pr\\right) =\\left( p\\cdot qp\\right) r.  \\label{leftBol}\n\\end{equation\nIt is easy to see that a left Bol loop has the left inverse property and is\nleft-alternative and flexible \\cite{RobinsonBol}. It is also\npower-associative. Similarly, define a right Bol loop: for any $p,q,r\\in \n\\mathbb{L}\n\\begin{equation}\n\\left( pq\\cdot r\\right) q=p\\left( qr\\cdot q\\right) .  \\label{rightBol}\n\\end{equation}\n\n\\item \\emph{Moufang loop: }a loop is a Moufang loop if it satisfies both the\nleft and right Bol identities. In particular, Moufang loops are\ndiassociative.\n\n\\item \\emph{Group}: clearly any associative loop is a group.\n\\end{enumerate}\n\n\\begin{example}\nThe best-known example of a non-associative loop is the Moufang loop of unit\noctonions.\n\\end{example}\n\n\\subsection{Pseudoautomorphisms}\n\nSuppose now $\\mathbb{L}$ is a loop and $\\left( \\alpha ,\\beta ,\\gamma \\right)\n\\ $is an autotopy of $\\mathbb{L}.$ Let $B=\\alpha \\left( 1\\right) ,$ $A=\\beta\n\\left( 1\\right) $, $C=\\gamma \\left( 1\\right) $. It is clear that $BA=C$.\nMoreover, from (\\ref{Qhom}) we see that \n\\begin{eqnarray*}\n\\alpha \\left( p\\right) &=&\\gamma \\left( p\\right) \/A \\\\\n\\beta \\left( p\\right) &=&B\\backslash \\gamma \\left( p\\right) .\n\\end{eqnarray*\nWe can rewrite (\\ref{Qhom}) as \n\\begin{equation*}\n\\alpha \\left( p\\right) \\cdot \\scalebox{-1}[1]{\\nicefrac{\\scalebox{-1}[1]{$\n\\left( q\\right) A$}}{\\scalebox{-1}[1]{$B$}}} =\\alpha \\left( pq\\right) A\n\\end{equation*\nIf $B=1,$ then, we obtain a \\emph{right pseudoautomorphism }$\\alpha $\\emph{\\\nof }$\\mathbb{L}$\\emph{\\ with companion }$A$, which we'll denote by the pair \n\\left( \\alpha ,A\\right) ,$ and which satisfies \n\\begin{equation}\n\\alpha \\left( p\\right) \\cdot \\alpha \\left( q\\right) A=\\alpha \\left(\npq\\right) A.  \\label{PsAutoPair}\n\\end{equation\nWe have the following useful relations for quotients: \n\\begin{subequations\n\\label{PsAutquot} \n\\begin{eqnarray}\n\\alpha \\left( q\\backslash p\\right) A &=& \\scalebox{-1}[1]{\\nicefrac\n\\scalebox{-1}[1]{$\\alpha \\left( p\\right) A$}}{\\scalebox{-1}[1]{$\\alpha\n\\left( q\\right)$}}} \\\\\n\\alpha \\left( p\/q\\right) \\cdot \\alpha \\left( q\\right) A &=&\\alpha \\left(\np\\right) A\n\\end{eqnarray\n\\end{subequations\nThere are several equivalent ways of characterizing \\emph{right\npseudoautomorphisms}$.$\n\n\\begin{theorem}\nLet $\\mathbb{L}$ be a loop and suppose $\\alpha :\\mathbb{L}\\longrightarrow \n\\mathbb{L}$. Also, let $A\\in \\mathbb{L}$ and $\\gamma =R_{A}\\circ \\alpha $.\nThen the following are equivalent:\n\n\\begin{enumerate}\n\\item $\\left( \\alpha ,A\\right) $ is a \\emph{right pseudoautomorphism of }\n\\mathbb{L}$\\emph{\\ with companion }$A$.\n\n\\item $\\left( \\alpha ,\\beta ,\\gamma \\right) $ is an autotopy of $\\mathbb{L}$\nwith $\\alpha \\left( 1\\right) =1$ and $\\beta \\left( 1\\right) =\\gamma \\left(\n1\\right) =A$.\n\n\\item $\\gamma \\left( 1\\right) =A$ and $\\gamma $ satisfies \n\\begin{equation}\n\\gamma \\left( p\\right) \\gamma \\left( q\\gamma ^{-1}\\left( 1\\right) \\right)\n=\\gamma \\left( pq\\right) .  \\label{PsAutosingle}\n\\end{equation}\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark}\nSimilarly, if $A=1,$ then we can rewrite (\\ref{Qhom}) a\n\\begin{equation*}\nB\\beta \\left( p\\right) \\cdot \\beta \\left( q\\right) =B\\beta \\left( pq\\right)\n\\end{equation*\nand in this case, $\\beta $ is a \\emph{left pseudoautomorphism} with\ncompanion $B$. Finally, suppose $C=1,$ so that then $A=B^{\\rho },$ and we\ncan rewrite (\\ref{Qhom}\n\\begin{equation*}\n\\gamma \\left( p\\right) \/B^{\\rho }\\cdot B\\backslash \\gamma \\left( q\\right)\n=\\gamma \\left( pq\\right)\n\\end{equation*\nso that in this case, $\\gamma $ is a \\emph{middle pseudoautomorphism} with\ncompanion $B$.\n\\end{remark}\n\n\\begin{example}\nIn a Moufang loop, consider the map $\\func{Ad}_{q},$ given by $p\\longmapsto\nqpq^{-1}.$ Note that this can be written unambiguously due to\ndiassociativity. Then, this is a right pseudoautomorphism with companion \nq^{3}$ \\cite[Lemma 1.2]{NagyStrambachBook}. Indeed, using diassociativity\nfor $\\left\\{ q,xy\\right\\} $, we have \n\\begin{equation*}\nq\\left( xy\\right) q^{-1}\\cdot q^{3}=q\\left( xy\\right) q^{2}.\n\\end{equation*\nOn the other hand, \n\\begin{eqnarray*}\nqxq^{-1}\\cdot qyq^{2} &=&q\\left( xq^{-1}\\right) \\cdot \\left( qyq\\right) q \\\\\n&=&\\left( q\\left( xq^{-1}\\cdot qyq\\right) \\right) q \\\\\n&=&\\left( q\\left( xy\\cdot q\\right) \\right) q \\\\\n&=&q\\left( xy\\right) q^{2},\n\\end{eqnarray*\nwhere we have use appropriate Moufang identities. Hence, indeed, \n\\begin{equation*}\nq\\left( xy\\right) q^{-1}\\cdot q^{3}=\\left( qxq^{-1}\\right) \\left(\nqyq^{-1}\\cdot q^{3}\\right) .\n\\end{equation*\nIn general, the adjoint map on a loop is \\emph{not} a pseudoautomorphism or\na loop homomorphism. For each $q\\in \\mathbb{L}$, $\\func{Ad}_{q}$ is just a\nbijection that preserves $1\\in \\mathbb{L}$. However, as we see above, it is\na pseudoautomorphism if the loop is Moufang. Keeping the same terminology as\nfor groups, we'll say that $\\func{Ad}$ defines an adjoint action of $\\mathbb\nL}$ on itself, although for a non-associative loop, this is not an action in\nthe usual sense of a group action.\n\\end{example}\n\nWe can easily see that the right pseudoautomorphisms of $\\mathbb{L}$ form a\ngroup under composition. Denote this group by $\\func{PsAut}^{R}\\left( \n\\mathbb{L}\\right) $. Clearly, $\\func{Aut}\\left( \\mathbb{L}\\right) \\subset \n\\func{PsAut}^{R}\\left( \\mathbb{L}\\right) $. Similarly for left and middle\npseudoautomorphisms. More precisely, $\\alpha \\in $ $\\func{PsAut}^{R}\\left( \n\\mathbb{L}\\right) $ if there exists $A\\in \\mathbb{L}$ such that (\\re\n{PsAutoPair}) holds. Here we are not fixing the companion. On the other\nhand, consider the set $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ of all pairs \n\\left( \\alpha ,A\\right) $ of\\emph{\\ right pseudoautomorphisms with fixed\ncompanions}. This then also forms a group.\n\n\\begin{lemma}\nThe set $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ of all pairs $\\left( \\alpha\n,A\\right) $, where $\\alpha \\in \\func{PsAut}^{R}\\left( \\mathbb{L}\\right) $\nand $A\\in \\mathbb{L}$ is its companion, is a group with identity element \n\\left( \\func{id},1\\right) $ and the following group operations\n\\begin{subequations\n\\begin{eqnarray}\n\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{product}\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{:\\ } &&\\left( \\alpha _{1},A_{1}\\right) \\left( \\alpha\n_{2},A_{2}\\right) =\\left( \\alpha _{1}\\circ \\alpha _{2},\\alpha _{1}\\left(\nA_{2}\\right) A_{1}\\right)  \\label{PsAutprod} \\\\\n\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{inverse}\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{: } &&\\left( \\alpha ,A\\right) ^{-1}=\\left( \\alpha\n^{-1},\\alpha ^{-1}\\left( A^{\\lambda }\\right) \\right) =\\left( \\alpha\n^{-1},\\left( \\alpha ^{-1}\\left( A\\right) \\right) ^{\\rho }\\right) .\n\\label{PsAutInv}\n\\end{eqnarray\n\\end{subequations\n\\end{lemma}\n\n\\begin{proof}\nIndeed, it is easy to see that $\\alpha _{1}\\left( A_{2}\\right) A_{1}$ is a\ncompanion of $\\alpha _{1}\\circ \\alpha _{2}$, that (\\ref{PsAutprod}) is\nassociative, and that $\\left( \\func{id},1\\right) $ is the identity element\nwith respect to it. Also, it is easy to see that \n\\begin{equation*}\n\\left( \\alpha ,A\\right) \\left( \\alpha ^{-1},\\alpha ^{-1}\\left( A^{\\lambda\n}\\right) \\right) =\\left( \\func{id},1\\right) .\n\\end{equation*\nOn the other hand, setting $B=\\alpha ^{-1}\\left( A^{\\lambda }\\right) $, we\nhave \n\\begin{eqnarray*}\nB &=&\\alpha ^{-1}\\left( 1\\right) B=\\alpha ^{-1}\\left( A^{\\lambda }A\\right) B\n\\\\\n&=&\\alpha ^{-1}\\left( A^{\\lambda }\\right) \\cdot \\alpha ^{-1}\\left( A\\right) B\n\\\\\n&=&B\\cdot \\alpha ^{-1}\\left( A\\right) B.\n\\end{eqnarray*\nCancelling $A$ on both sides on the left, we see that $B=\\left( \\alpha\n^{-1}\\left( A\\right) \\right) ^{\\rho }.$\n\\end{proof}\n\nLet $\\mathcal{C}^{R}\\left( \\mathbb{L}\\right) $ be the set of elements of \n\\mathbb{L}$ that are a companion for a right pseudoautomorphism. Then, (\\re\n{PsAutprod}) shows that there is a left action of $\\Psi ^{R}\\left( \\mathbb{L\n\\right) $ on $\\mathcal{C}^{R}\\left( \\mathbb{L}\\right) $ given by\n\\begin{subequations\n\\label{PsiLeftaction\n\\begin{eqnarray}\n\\Psi ^{R}\\left( \\mathbb{L}\\right) \\times \\mathcal{C}^{R}\\left( \\mathbb{L\n\\right) &\\longrightarrow &\\mathcal{C}^{R}\\left( \\mathbb{L}\\right) \\\\\n\\left( \\left( \\alpha ,A\\right) ,B\\right) &\\mapsto &\\left( \\alpha ,A\\right)\nB=\\alpha \\left( B\\right) A.\n\\end{eqnarray\n\\end{subequations\nThis action is transitive, because if $A,B\\in \\mathcal{C}^{R}\\left( \\mathbb{\n}\\right) $, then exist $\\alpha ,\\beta \\in \\func{PsAut}^{R}\\left( \\mathbb{L\n\\right) $, such that $\\left( \\alpha ,A\\right) ,\\left( \\beta ,B\\right) \\in\n\\Psi ^{R}\\left( \\mathbb{L}\\right) $, and hence $\\left( \\left( \\beta\n,B\\right) \\left( \\alpha ,A\\right) ^{-1}\\right) A=B.$ Similarly, $\\Psi\n^{R}\\left( \\mathbb{L}\\right) $ also acts on all of $\\mathbb{L}$. Let \nh=\\left( \\alpha ,A\\right) \\in \\Psi ^{R}\\left( \\mathbb{L}\\right) $, then for\nany $p\\in \\mathbb{L}$, $h\\left( p\\right) =\\alpha \\left( p\\right) A.$ This is\nin general non-transitive, but a faithful action (assuming $\\mathbb{L}$ is\nnon-trivial). Using this, the definition of (\\ref{PsAutoPair}) can be\nrewritten as \n\\begin{equation}\nh\\left( pq\\right) =\\alpha \\left( p\\right) h\\left( q\\right)\n\\label{PsAutProd2}\n\\end{equation\nand hence the quotient relations (\\ref{PsAutquot}) may be rewritten as \n\\begin{subequations\n\\label{PsAutquot2} \n\\begin{eqnarray}\nh\\left( q\\backslash p\\right) &=&\\alpha \\left( q\\right) \\backslash h\\left(\np\\right)  \\label{PsAutquot2b} \\\\\n\\alpha \\left( p\/q\\right) &=&h\\left( p\\right) \/h\\left( q\\right) .\n\\label{PsAutquot2a}\n\\end{eqnarray\n\\end{subequations}\nIf $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ acts transitively on $\\mathbb{L},$\nthen $\\mathcal{C}^{R}\\left( \\mathbb{L}\\right) \\cong \\mathbb{L}$, since every\nelement of $\\mathbb{L}$ will be a companion for some right\npseudoautomorphism. In that case, $\\mathbb{L}$ is known as a (\\emph{right)} \n\\emph{G}$\\emph{-loop}. Note that usually a loop is known as a $G$-loop is\nevery element of $\\mathbb{L}$ is a companion for a right pseudoautomorphism\nand for a left pseudoautomorphism \\cite{KunenGloops}. However, in this paper\nwe will only be concerned with right pseudoautomorphisms, so for brevity we\nwill say $\\mathbb{L}$ is a $G$-loop if $\\Psi ^{R}\\left( \\mathbb{L}\\right) $\nacts transitively on it.\n\nThere is another action of $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ on $\\mathbb{\n}$ - which is the action by the pseudoautomorphism. This is a non-faithful\naction of $\\Psi ^{R}\\left( \\mathbb{L}\\right) $, but corresponds to a\nfaithful action of $\\func{PsAut}^{R}\\left( \\mathbb{L}\\right) $. Namely, let \nh=\\left( \\alpha ,A\\right) \\in \\Psi ^{R}\\left( \\mathbb{L}\\right) $, then $h\\ \nacts on $p\\in \\mathbb{L}$ by $p\\mapsto \\alpha \\left( p\\right) $. To\ndistinguish these two actions, we make the following definitions.\n\n\\begin{definition}\nLet $\\mathbb{L}$ be a loop and let $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ the\ngroup of right pseudoautomorphism pairs. $\\mathbb{L}$ admits two left\nactions of $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ on itself. Let $h=\\left(\n\\alpha ,A\\right) \\in \\Psi ^{R}\\left( \\mathbb{L}\\right) $ and $p\\in \\mathbb{L\n.$\n\n\\begin{enumerate}\n\\item The \\emph{full} action is given by $\\left( h,p\\right) \\mapsto h\\left(\np\\right) =\\alpha \\left( p\\right) A.$ The set $\\mathbb{L}$ together with this\naction of $\\mathbb{\\Psi }^{R}\\left( \\mathbb{L}\\right) $ will be denoted by \n\\mathbb{\\mathring{L}}.$\n\n\\item The \\emph{partial} action, given by $\\left( h,p\\right) \\mapsto\nh^{\\prime }\\left( p\\right) =\\alpha \\left( p\\right) .$ The set $\\mathbb{L}$\ntogether with this action of $\\mathbb{\\Psi }^{R}\\left( \\mathbb{L}\\right) $\nwill be denoted by $\\mathbb{L}$ again.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark}\nFrom (\\ref{PsAutProd2}), these definitions suggest that the loop product on \n\\mathbb{L}$ can be regarded as a map $\\cdot :\\mathbb{L\\times \\mathring{L}\n\\longrightarrow \\mathbb{\\mathring{L}}$. This bears some similarity to\nClifford product structure on spinors, however without the linear structure,\nbut instead with the constraint that $\\mathbb{L}$ and $\\mathbb{\\mathring{L}}$\nare identical as sets. This however allows to define left and right\ndivision. '\n\\end{remark}\n\nNow let us consider several relationships between the different groups\nassociated to $\\mathbb{L}.$ First of all define the following maps\n\\begin{eqnarray}\n\\iota _{1} &:&\\func{Aut}\\left( \\mathbb{L}\\right) \\hookrightarrow \\Psi\n^{R}\\left( \\mathbb{L}\\right)  \\label{i1map} \\\\\n\\gamma &\\mapsto &\\left( \\gamma ,1\\right)  \\notag\n\\end{eqnarray\nand \n\\begin{eqnarray}\n\\iota _{2} &:&\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) \\hookrightarrow \\Psi\n^{R}\\left( \\mathbb{L}\\right)  \\notag \\\\\nC &\\mapsto &\\left( \\func{id},C\\right) ,  \\label{i2map}\n\\end{eqnarray\nThe map $\\iota _{1}$ is clearly injective and is a group homomorphism, so \n\\iota _{1}\\left( \\func{Aut}\\left( \\mathbb{L}\\right) \\right) $ is a subgroup\nof $\\Psi ^{R}\\left( \\mathbb{L}\\right) .$ On the other hand, if $A,B\\in \n\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) $, then in $\\Psi ^{R}\\left( \\mathbb{\n}\\right) $, $\\left( \\func{id},A\\right) \\left( \\func{id},B\\right) =\\left( \n\\func{id},BA\\right) ,$ so $\\iota _{2}$ is an antihomomorphism from $\\mathcal\nN}^{R}\\left( \\mathbb{L}\\right) $ to $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ and\nthus a homomorphism from the opposite group $\\mathcal{N}^{R}\\left( \\mathbb{L\n\\right) ^{\\func{op}}.$ So, $\\iota _{2}\\left( \\mathcal{N}^{R}\\left( \\mathbb{L\n\\right) \\right) $ is a subgroup of $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ that\nis isomorphic to $\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) ^{\\func{op}}.$\n\nUsing (\\ref{i1map}) let us define a right action of $\\func{Aut}\\left( \n\\mathbb{L}\\right) $ on $\\Psi ^{R}\\left( \\mathbb{L}\\right) .$ Given $\\gamma\n\\in \\func{Aut}\\left( \\mathbb{L}\\right) $ and $\\left( \\alpha ,A\\right) \\in\n\\Psi ^{R}\\left( \\mathbb{L}\\right) $, we define \n\\begin{equation}\n\\left( \\alpha ,A\\right) \\cdot \\gamma =\\left( \\alpha ,A\\right) \\iota\n_{1}\\left( \\gamma \\right) =\\left( \\alpha \\circ \\gamma ,A\\right) .\n\\label{AutRAct}\n\\end{equation\nSimilarly, (\\ref{i2map}) allows to define a left action of $\\mathcal{N\n^{R}\\left( \\mathbb{L}\\right) ^{\\func{op}}$,and hence a right action of \n\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) $, on $\\Psi ^{R}\\left( \\mathbb{L\n\\right) $\n\\begin{equation}\nC\\cdot \\left( \\alpha ,A\\right) =\\iota _{2}\\left( C\\right) \\left( \\alpha\n,A\\right) =\\left( \\alpha ,AC\\right) .  \\label{NLAct}\n\\end{equation\nThe actions (\\ref{AutRAct}) and (\\ref{NLAct}) commute, so we can combine\nthem to define a left action of $\\func{Aut}\\left( \\mathbb{L}\\right) \\times \n\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) ^{\\func{op}}.$ Indeed, given $\\gamma\n\\in \\func{Aut}\\left( \\mathbb{L}\\right) $ and $C\\in \\mathcal{N}^{R}\\left( \n\\mathbb{L}\\right) $, \n\\begin{equation}\n\\left( \\alpha ,A\\right) \\cdot \\left( \\gamma ,C\\right) =\\iota _{2}\\left(\nC\\right) \\left( \\alpha ,A\\right) \\iota _{1}\\left( \\gamma \\right) =\\left(\n\\alpha \\circ \\gamma ,AC\\right) .  \\label{AutNAct}\n\\end{equation}\n\n\\begin{remark}\nSince any element of $\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) $ is a right\ncompanion for any automorphism, we can also define the semi-direct product\nsubgroup $\\iota _{1}\\left( \\func{Aut}\\left( \\mathbb{L}\\right) \\right)\n\\ltimes \\iota _{2}\\left( \\mathcal{N}^{R}\\left( \\mathbb{L}\\right) \\right)\n\\subset \\Psi ^{R}\\left( \\mathbb{L}\\right) $. Suppose $\\beta ,\\gamma \\in \n\\func{Aut}\\left( \\mathbb{L}\\right) $ and $B,C\\in \\mathcal{N}^{R}\\left( \n\\mathbb{L}\\right) ,$ then in this semi-direct product, \n\\begin{equation*}\n\\left( \\beta ,B\\right) \\left( \\gamma ,C\\right) =\\left( \\beta \\circ \\gamma\n,\\beta \\left( C\\right) B\\right) .\n\\end{equation*}\n\\end{remark}\n\n\\begin{lemma}\nGiven the actions of $\\func{Aut}\\left( \\mathbb{L}\\right) $ and $\\mathcal{N\n^{R}\\left( \\mathbb{L}\\right) $ on $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ as in\n(\\ref{AutRAct}) and (\\ref{NLAct}), respectively, we have the following\nproperties.\n\n\\begin{enumerate}\n\\item \n\\faktor{\\Psi ^{R}\\left( \\mathbb{L}\\right)}{\\func{Aut}\\left(\n\\mathbb{L}\\right)} \\cong \\mathcal{C}^{R}\\left( \\mathbb{L}\\right) $ as $\\Psi\n^{R}\\left( \\mathbb{L}\\right) $-sets.\n\n\\item The image $\\iota _{2}\\left( \\mathcal{N}^{R}\\left( \\mathbb{L}\\right)\n\\right) $ is a normal subgroup of $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ and\nhence \n\\begin{equation*}\n\\faktor{\\Psi ^{R}\\left( \\mathbb{L}\\right)}{\\mathcal{N}^{R}\\left(\n\\mathbb{L}\\right)} \\cong \\func{PsAut}^{R}\\left( \\mathbb{L}\\right) .\n\\end{equation*}\n\n\\item Moreover, \n\\begin{equation*}\n\\faktor{\\Psi ^{R}\\left( \\mathbb{L}\\right)}{\\func{Aut}\\left(\n\\mathbb{L}\\right) \\times \\mathcal{N}^{R}\\left( \\mathbb{L}\\right)} \\cong\n\\faktor{\\func{PsAut}^{R}\\left( \\mathbb{L}\\right)}{\\func{Aut}\\left(\n\\mathbb{L}\\right)} \\cong \\faktor{\\mathcal{C}^{R}\\left(\n\\mathbb{L}\\right)}{\\mathcal{N}^{R}\\left( \\mathbb{L}\\right)}\n\\end{equation*\nwhere equivalence is as $\\func{Aut}\\left( \\mathbb{L}\\right) \\times \\mathcal{\n}^{R}\\left( \\mathbb{L}\\right) $-sets.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nSuppose $\\mathbb{L}$ is a loop.\n\n\\begin{enumerate}\n\\item Consider the projection on the second component $\\func{prj}_{2}:\\Psi\n^{R}\\left( \\mathbb{L}\\right) \\longrightarrow \\mathcal{C}^{R}\\left( \\mathbb{L\n\\right) $ under which $\\left( \\alpha ,A\\right) \\mapsto A.$ Both $\\Psi\n^{R}\\left( \\mathbb{L}\\right) \\ $and $\\mathcal{C}^{R}\\left( \\mathbb{L}\\right) \n$ are left $\\Psi ^{R}\\left( \\mathbb{L}\\right) $-sets, since both admit a\nleft $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ action - $\\Psi ^{R}\\left( \\mathbb{\n}\\right) $ acts on itself by left multiplication and acts on $\\mathcal{C\n^{R}\\left( \\mathbb{L}\\right) $ via the action (\\ref{PsiLeftaction}). Hence, \n\\func{prj}_{2}$ is a $\\Psi ^{R}\\left( \\mathbb{L}\\right) $-equivariant map\n(i.e. a $G$-set homomorphism). On the other hand, given the action (\\re\n{AutRAct}) of $\\func{Aut}\\left( \\mathbb{L}\\right) $ on $\\Psi ^{R}\\left( \n\\mathbb{L}\\right) ,$ we easily see that two pseudoautomorphisms have the\nsame companion if and only if they lie in the same orbit of $\\func{Aut\n\\left( \\mathbb{L}\\right) $. Thus, $\\func{prj}_{2}$ descends to a $\\Psi\n^{R}\\left( \\mathbb{L}\\right) $-equivariant bijection $\\Psi ^{R}\\left( \n\\mathbb{L}\\right) \/\\func{Aut}\\left( \\mathbb{L}\\right) \\longrightarrow \n\\mathcal{C}^{R}\\left( \\mathbb{L}\\right) $, so that $\\Psi ^{R}\\left( \\mathbb{\n}\\right) \/\\func{Aut}\\left( \\mathbb{L}\\right) \\cong \\mathcal{C}^{R}\\left( \n\\mathbb{L}\\right) $ as $\\Psi ^{R}\\left( \\mathbb{L}\\right) $-sets.\n\n\\item It is clear that $C\\in $ $\\mathcal{C}^{R}\\left( \\mathbb{L}\\right) $ is\na right companion of the identity map $\\func{id}$ if and only if $C\\in $ \n\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) $. Now, let $\\nu =\\left( \\func{id\n,C\\right) \\in $ $\\iota _{2}\\left( \\mathcal{N}^{R}\\left( \\mathbb{L}\\right)\n\\right) $ and $g=\\left( \\alpha ,A\\right) \\in $ $\\Psi ^{R}\\left( \\mathbb{L\n\\right) .$ Then, \n\\begin{equation}\ng\\nu g^{-1}=\\left( \\alpha ,A\\right) \\left( \\func{id},C\\right) \\left( \\alpha\n^{-1},\\alpha ^{-1}\\left( A^{\\lambda }\\right) \\right) =\\left( \\func{id\n,A^{\\lambda }\\cdot \\alpha \\left( C\\right) A\\right) .  \\label{PsiAdjN}\n\\end{equation\nIn particular, this shows that $g\\nu g^{-1}\\in \\iota _{2}\\left( \\mathcal{N\n^{R}\\left( \\mathbb{L}\\right) \\right) $ since $A^{\\lambda }\\cdot \\alpha\n\\left( C\\right) A$ is the right companion of $\\func{id}$. Thus indeed, \n\\iota _{2}\\left( \\mathcal{N}^{R}\\left( \\mathbb{L}\\right) \\right) $ is a\nnormal subgroup of $\\Psi ^{R}\\left( \\mathbb{L}\\right) .$ Now consider the\nprojection on the first component $\\func{prj}_{1}:\\Psi ^{R}\\left( \\mathbb{L\n\\right) \\longrightarrow \\func{PsAut}^{R}\\left( \\mathbb{L}\\right) $ under\nwhich $\\left( \\alpha ,A\\right) \\mapsto \\alpha .$ This is clearly a group\nhomomorphism with kernel $\\iota _{2}\\left( \\mathcal{N}^{R}\\left( \\mathbb{L\n\\right) \\right) .$ Thus, $^{R}\\left( \\mathbb{L}\\right) ^{\\func{op\n}\\backslash \\Psi ^{R}\\left( \\mathbb{L}\\right) \\cong \\Psi ^{R}\\left( \\mathbb{\n}\\right) \/\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) \\cong \\func{PsAut\n^{R}\\left( \\mathbb{L}\\right) $.\n\n\\item Since the actions of $\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) $ and \n\\func{Aut}\\left( \\mathbb{L}\\right) $ on $\\Psi ^{R}\\left( \\mathbb{L}\\right) $\ncommute, the action of $\\func{Aut}\\left( \\mathbb{L}\\right) $ descends to \n\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) ^{\\func{op}}\\backslash \\Psi\n^{R}\\left( \\mathbb{L}\\right) \\cong \\func{PsAut}^{R}\\left( \\mathbb{L}\\right) $\nand the action of $\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) ^{\\func{op}}$\ndescends to $\\Psi ^{R}\\left( \\mathbb{L}\\right) \/\\func{Aut}\\left( \\mathbb{L\n\\right) \\cong \\mathcal{C}^{R}\\left( \\mathbb{L}\\right) .$ Since the left\naction of $\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) ^{\\func{op}}$ on $\\Psi\n^{R}\\left( \\mathbb{L}\\right) $ corresponds to an action by right\nmultiplication on $\\mathcal{C}^{R}\\left( \\mathbb{L}\\right) $, we find that\nthere is a bijection $\\func{PsAut}^{R}\\left( \\mathbb{L}\\right) \/\\func{Aut\n\\left( \\mathbb{L}\\right) \\longrightarrow \\mathcal{C}^{R}\\left( \\mathbb{L\n\\right) \/\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) .$\n\nSuppose $\\left( \\alpha ,A\\right) \\in \\Psi ^{R}\\left( \\mathbb{L}\\right) $ and\nlet $\\left[ \\alpha \\right] _{\\func{Aut}\\left( \\mathbb{L}\\right) }\\in $ \n\\func{PsAut}^{R}\\left( \\mathbb{L}\\right) \/\\func{Aut}\\left( \\mathbb{L}\\right) \n$ be the orbit of $\\alpha $ under the action of $\\func{Aut}\\left( \\mathbb{L\n\\right) $ and let $\\left[ A\\right] _{\\mathcal{N}^{R}\\left( \\mathbb{L}\\right)\n}\\in \\mathcal{C}^{R}\\left( \\mathbb{L}\\right) \/\\mathcal{N}^{R}\\left( \\mathbb{\n}\\right) $ be the orbit of $A$ under the action of $\\mathcal{N}^{R}\\left( \n\\mathbb{L}\\right) $. Then the bijection is given by $\\left[ \\alpha \\right] _\n\\func{Aut}\\left( \\mathbb{L}\\right) }\\mapsto \\left[ A\\right] _{\\mathcal{N\n^{R}\\left( \\mathbb{L}\\right) }$. Moreover, each of these orbits also\ncorresponds to the orbit of $\\left( \\alpha ,A\\right) $ under the right\naction of $\\func{Aut}\\left( \\mathbb{L}\\right) \\times \\mathcal{N}^{R}\\left( \n\\mathbb{L}\\right) $ on $\\Psi ^{R}\\left( \\mathbb{L}\\right) .$ These quotients\npreserve actions of $\\func{Aut}\\left( \\mathbb{L}\\right) \\times \\mathcal{N\n^{R}\\left( \\mathbb{L}\\right) $ on corresponding sets and thus these coset\nspaces are equivalent as $\\func{Aut}\\left( \\mathbb{L}\\right) \\times \\mathcal\nN}^{R}\\left( \\mathbb{L}\\right) $-sets.\n\\end{enumerate}\n\\end{proof}\n\nThe above relationships between the different groups are summarized in\nFigure \\ref{tikGroup}.\n\n\\begin{figure}[tbp]\n\\begin{tikzcd} \\func{Aut}(\\mathbb{L}) \\arrow[r,hookrightarrow,bend\nleft=20,\"{(\\cdot,1)}\"] \\arrow[d,hook] & \\Psi ^ {R} (\\mathbb{L})\n\\arrow[dl,swap,\"\\func{prj}_1\"] \\arrow[dr,\"\\func{prj}_2\"] &\n\\mathcal{N}^{R}(\\mathbb{L})^{\\func{op}}\\arrow[d,hook'] \\arrow[l,hook', bend\nright,swap,\"{(\\func{id},\\cdot)}\"]\\\\ \\func{PsAut} ^ {R} (\\mathbb{L}) \\cong\n\\scalebox{-1}[1]{\\nicefrac{\\scalebox{-1}[1]{$\\Psi ^\n{R}(\\mathbb{L})$}}{\\scalebox{-1}[1]{$\\mathcal{N}^{R}(\\mathbb{L})^\n\\func{op}}$}}} \\arrow[d]& &\\faktor{ \\Psi ^\n{R}(\\mathbb{L})}{\\func{Aut}(\\mathbb{L})} \\cong \\mathcal{C}^{R}(\\mathbb{L})\n\\arrow[d] \\\\ \\faktor{\\func{PsAut} ^ {R}\n(\\mathbb{L})}{\\func{Aut}(\\mathbb{L})} \\arrow[rr,bend right=15,\"\\cong\"] &\n&\\faktor{\\mathcal{C}^{R}(\\mathbb{L})}{\\mathcal{N}^{R}(\\mathbb{L})}\n\\end{tikzcd}\n\\caption{Groups related to the loop $\\mathbb{L}$}\n\\label{tikGroup}\n\\end{figure}\n\n\\begin{example}\n\\label{ExPsQuat}Suppose $\\mathbb{L=}U\\mathbb{H\\cong }S^{3}$ - the group of\nunit quaternions. Then, since this is associative, $\\mathcal{N}^{R}\\left( \n\\mathbb{H}\\right) =U\\mathbb{H\\cong }Sp\\left( 1\\right) .$ We also know that \n\\func{Aut}\\left( U\\mathbb{H}\\right) \\cong SO\\left( 3\\right) .$ Now however, \n\\Psi ^{R}\\left( U\\mathbb{H}\\right) $ consists of all pairs $\\left( \\alpha\n,A\\right) \\in SO\\left( 3\\right) \\times U\\mathbb{H}$ with the group structure\ndefined by (\\ref{PsAutprod}),which is the semi-direct product \n\\begin{equation}\n\\Psi ^{R}\\left( U\\mathbb{H}\\right) \\cong SO\\left( 3\\right) \\ltimes Sp\\left(\n1\\right) \\cong Sp\\left( 1\\right) Sp\\left( 1\\right) \\cong SO\\left( 4\\right) .\n\\end{equation\nIn this case, $\\func{PsAut}^{R}\\left( U\\mathbb{H}\\right) \\cong \\func{Aut\n\\left( U\\mathbb{H}\\right) \\cong SO\\left( 3\\right) .$ Here $\\left( p,q\\right)\n\\sim \\left( -p,-q\\right) $ acts on $U\\mathbb{H}$ via $r\\mapsto prq^{-1}$.\n\\end{example}\n\n\\begin{example}\n\\label{exGroup}More generally, suppose $\\mathbb{L=}G$ is a group. Then, \n\\func{PsAut}^{R}\\left( G\\right) \\cong \\func{Aut}\\left( G\\right) $ and $\\Psi\n^{R}\\left( G\\right) \\cong \\func{Aut}\\left( G\\right) \\ltimes G^{\\func{op}}$,\nwith $h=\\left( \\alpha ,A\\right) \\in \\Psi ^{R}\\left( G\\right) $ acting on $G$\nby \n\\begin{equation}\nh\\left( g\\right) =\\alpha \\left( g\\right) A  \\label{hG}\n\\end{equation\nNote that the group $\\func{Aut}\\left( G\\right) \\ltimes G$ is known as the \n\\emph{holomorph} of $G.$\n\\end{example}\n\n\\begin{example}\n\\label{ExPsOcto}Suppose $\\mathbb{L=}U\\mathbb{O}$ - the Moufang loop of unit\noctonions, which is homeomorphic to the $7$-sphere $S^{7}.$ From \\cite[Lemma\n14.61]{Harvey} we know that $g\\in O\\left( \\mathbb{O}\\right) $ belongs to \nSpin\\left( 7\\right) $ if and only i\n\\begin{equation}\ng\\left( uv\\right) =\\chi _{g}\\left( u\\right) g\\left( v\\right)  \\label{spin7}\n\\end{equation\nfor all $u,v\\in \\mathbb{O}$ where $\\chi _{g}\\left( u\\right) =g\\left(\nug^{-1}\\left( 1\\right) \\right) $ gives the vector representation of \nSpin\\left( 7\\right) $ on $\\func{Im}\\mathbb{O}$. We may as well restrict\neverything to the non-zero octonions $\\mathbb{O}^{\\ast }$ or the unit\noctonions $U\\mathbb{O}$, so that we have a loop. Now, \n\\begin{eqnarray*}\ng\\left( u\\right) &=&g\\left( u\\cdot 1\\right) =\\chi _{g}\\left( u\\right)\ng\\left( 1\\right) \\\\\ng\\left( uv\\right) &=&g\\left( uv\\cdot 1\\right) =\\chi _{g}\\left( uv\\right)\ng\\left( 1\\right)\n\\end{eqnarray*\nHence, we find that (\\ref{spin7}) implies \n\\begin{equation*}\n\\chi _{g}\\left( uv\\right) g\\left( 1\\right) =\\chi _{g}\\left( u\\right) \\cdot\n\\chi _{g}\\left( v\\right) g\\left( 1\\right) .\n\\end{equation*\nThus, $\\left( \\chi _{g},g\\left( 1\\right) \\right) $ is a right\npseudoautomorphism of $U\\mathbb{O}$ with companion $g\\left( 1\\right) $.\nThus, in this case we find that $\\Psi ^{R}\\left( U\\mathbb{O}\\right) \\cong\nSpin\\left( 7\\right) $. We also know that $\\mathcal{N}^{R}\\left( U\\mathbb{O\n\\right) =\\left\\{ \\pm 1\\right\\} \\cong \\mathbb{Z}_{2}$ and thus the projection \n$\\left( \\chi ,A\\right) \\mapsto \\chi $ corresponds to the double cover \nSpin\\left( 7\\right) \\longrightarrow SO\\left( 7\\right) $. Hence, $\\func{PsAut\n^{R}\\left( U\\mathbb{O}\\right) \\cong SO\\left( 7\\right) $ and as we know, \n\\func{Aut}\\left( U\\mathbb{O}\\right) \\cong G_{2}$. Since $U\\mathbb{O}$ is a\nMoufang loop, and we know that for any $q$, the map $\\func{Ad}_{q}$ is a\nright pseudoautomorphism with companion $q$, we see that $\\mathcal{C\n^{R}\\left( U\\mathbb{O}\\right) =U\\mathbb{O},$ and indeed as we know, \nSpin\\left( 7\\right) \/G_{2}\\cong S^{7}.$\n\\end{example}\n\n\\begin{remark}\nWe have defined the group $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ as the set of \n\\emph{all} right pseudoautomorphism pairs $\\left( \\alpha ,A\\right) ,$\nhowever we could consistently truncate $\\Psi ^{R}\\left( \\mathbb{L}\\right) $\nto a subgroup, or more generally, if $G$ is some group with a homomorphism \n\\rho :G\\longrightarrow \\Psi ^{R}\\left( \\mathbb{L}\\right) $, we can use this\nhomomorphism to define a \\emph{pseudoautomorphism action} of $G$ on $\\mathbb\nL}.$ For example, if $G=\\func{Aut}\\left( \\mathbb{L}\\right) \\ltimes \\mathcal{\n}^{R}\\left( \\mathbb{L}\\right) ^{\\func{op}},$ then we know that $\\iota\n_{1}\\times \\iota _{2}:G\\longrightarrow \\Psi ^{R}\\left( \\mathbb{L}\\right) $\nis a homomorphism. With respect to the action of $G,$ the companions would\nbe just the elements of $\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) .$\n\\end{remark}\n\n\\begin{example}\n\\label{ExNormedDiv}In \\cite{LeungDivision}, Leung developed a general\nframework for structures in Riemannian geometry based on division algebras - \n$\\mathbb{R},\\mathbb{C},\\mathbb{H},\\mathbb{O}$. As a first step, this\ninvolved representations of unitary groups with values in each of these\nalgebras on the algebras themselves. The unitary groups, $O\\left( n\\right) \n, $U\\left( n\\right) $, $Sp\\left( n\\right) Sp\\left( 1\\right) $, and \nSpin\\left( 7\\right) ,$ as well as the corresponding special unitary groups \nSO\\left( n\\right) ,\\ SU\\left( n\\right) ,$ $Sp\\left( n\\right) $, and $G_{2},$\nare precisely the possible Riemannian holonomy groups for irreducible, not\nlocally symmetric smooth manifolds \\cite{Berger1955}. By considering the\ncorresponding loops (groups for the associative cases) we can look at the\npseudoautomorphism actions. The octonionic case is already covered in\nExample \\ref{ExPsOcto}.\n\n\\begin{enumerate}\n\\item In the case of $\\mathbb{R},$ consider instead the group of\n\\textquotedblleft unit reals\\textquotedblright\\ $U\\mathbb{R}=\\left\\{ \\pm\n1\\right\\} \\cong \\mathbb{Z}_{2}.$ Then, $\\Psi ^{R}\\left( U\\mathbb{R}\\right)\n=\\left\\{ 1\\right\\} \\ltimes \\left\\{ \\pm 1\\right\\} \\mathbb{\\cong }\\mathbb{Z\n_{2},$ however consider now for some positive integer $n,$ the homomorphism \n\\det :O\\left( n\\right) \\longrightarrow \\mathbb{Z}_{2}.$ Thus, $O\\left(\nn\\right) $ acts on $\\mathbb{Z}_{2}$ via this homomorphism: $\\left(\ng,x\\right) \\mapsto x\\det g$, where $x\\in \\mathbb{Z}_{2}$ and $g\\in O\\left(\nn\\right) .$ The preimage $\\func{Aut}\\left( \\mathbb{Z}_{2}\\right) =\\left\\{\n1\\right\\} $ is then just $\\ker \\det =SO\\left( n\\right) .$ Thus, we can now\ndefine the group $\\Psi _{n}^{R}\\left( U\\mathbb{R}\\right) =O\\left( n\\right) .$\nThe full action of $\\Psi _{n}^{R}\\left( U\\mathbb{R}\\right) $ on $U\\mathbb{R}$\nis transitive, while the partial action is trivial. Similarly, we can also\ndefine $\\func{Aut}_{n}\\left( U\\mathbb{R}\\right) =SO\\left( n\\right) .$\n\n\\item In the complex case, the group of unit complex numbers $U\\mathbb{C\n=U\\left( 1\\right) \\cong S^{1}.$ Similarly, as above, $\\Psi ^{R}\\left( \n\\mathbb{C}\\right) =\\left\\{ 1\\right\\} \\ltimes U\\left( 1\\right) \\mathbb{\\cong \nU\\left( 1\\right) .$ Now however, we also have the homomorphism $\\det_\n\\mathbb{C}}:U\\left( n\\right) \\longrightarrow U\\left( 1\\right) .$ Then, \nU\\left( n\\right) $ acts on $U\\left( 1\\right) $ via $\\left( g,z\\right)\n\\mapsto z\\det g$, where $z\\in U\\left( 1\\right) $ and $g\\in U\\left( n\\right)\n. $ The preimage of $\\func{Aut}\\left( U\\left( 1\\right) \\right) =\\left\\{\n1\\right\\} $ is then just $\\ker \\det_{\\mathbb{C}}=SU\\left( n\\right) .$ Thus,\nsimilarly as above, we can now define the group $\\Psi _{n}^{R}\\left( \n\\mathbb{C}\\right) =U\\left( n\\right) .$ The full action of $\\Psi\n_{n}^{R}\\left( U\\mathbb{R}\\right) $ on $U\\mathbb{C}$ is transitive, while\nthe partial action is trivial. Similarly, we can also define $\\func{Aut\n_{n}\\left( U\\mathbb{C}\\right) =SU\\left( n\\right) .$\n\n\\item In the quaternionic case, we have already seen the case $n=1$ in\nExample \\ref{ExPsQuat}. The $n$-dimensional quaternionic unitary group is in\ngeneral $Sp\\left( n\\right) Sp\\left( 1\\right) $, where $Sp\\left( n\\right) $\nis the compact symplectic group or equivalently, the quaternion special\nunitary group. The group $Sp\\left( n\\right) Sp\\left( 1\\right) $ acts on \n\\mathbb{H}^{n}$ by $Sp\\left( n\\right) $ on the left, and multiplication by a\nunit quaternion on the right, and hence can be represented by pairs \nh=\\left( \\alpha ,q\\right) \\in Sp\\left( n\\right) \\times Sp\\left( 1\\right) ,$\nwith the identification $\\left( -\\alpha ,-q\\right) \\sim \\left( \\alpha\n,q\\right) $. For $n\\geq 2$, define the homomorphism $\\rho _{\\mathbb{H\n}:Sp\\left( n\\right) Sp\\left( 1\\right) \\longrightarrow Sp\\left( 1\\right)\nSp\\left( 1\\right) $ given by $\\left[ \\alpha ,q\\right] \\mapsto \\left[ 1,\n\\right] .$ The image of this homomorphism simply corresponds to elements of \n\\Psi ^{R}\\left( U\\mathbb{H}\\right) $ that are of the form $\\left( \\func{id\n,q\\right) ,$ i.e. act by right multiplication of $U\\mathbb{H}$ on itself.\nThe preimage of $\\func{Aut}\\left( U\\mathbb{H}\\right) \\cong SO\\left( 3\\right) \n$ is then $\\ker \\rho _{\\mathbb{H}}\\cong Sp\\left( n\\right) .$ Overall, we may\ndefine the group $\\Psi _{n}^{R}\\left( U\\mathbb{H}\\right) =Sp\\left( n\\right)\nSp\\left( 1\\right) $ and $\\func{Aut}_{n}\\left( U\\mathbb{H}\\right) =Sp\\left(\nn\\right) .$ As in the previous examples, the full action of $\\Psi\n_{n}^{R}\\left( U\\mathbb{H}\\right) $ on $U\\mathbb{H}$ is transitive, whereas\nthe partial action is again trivial. We will refer to this example later on,\nwith the assumption that $n\\geq 2$.\n\\end{enumerate}\n\nThus, in each of the above cases, we may regard $\\Psi _{n}^{R}$ ($O\\left(\nn\\right) ,U\\left( n\\right) ,$ or $Sp\\left( n\\right) Sp\\left( 1\\right) $) as \n\\emph{a }group of\\emph{\\ }pseudoautomorphism pairs acting on the unit real\nnumbers, unit complex numbers, and unit quaternions with a trivial partial\naction and will the full action just given by right multiplication. The\ncorresponding automorphism subgroups are then the \\textquotedblleft\nspecial\\textquotedblright\\ unitary subgroups $SO\\left( n\\right) ,$ $SU\\left(\nn\\right) ,$ $Sp\\left( n\\right) .$\n\\end{example}\n\n\\subsection{Modified product}\n\nLet $r\\in \\mathbb{L}$, and define the modified product $\\circ _{r}$ on \n\\mathbb{L}$ vi\n\\begin{equation}\np\\circ _{r}q=\\faktor{\\left( p\\cdot qr\\right)}{r}.  \\label{rprod}\n\\end{equation\nThen, $p\\circ _{r}q=p\\cdot q$ if and only if $p\\cdot qr=pq\\cdot r.$ This is\ntrue for all $p,q$ if and only if $r\\in \\mathcal{N}^{R}\\left( \\mathbb{L\n\\right) $. However, this will not hold for all $r$ unless $\\mathbb{L}$ is\nassociative (and is thus a group). If $\\mathbb{L}$ is a right Bol loop, and \na\\in \\mathbb{L}$, setting $r=q\\backslash a$ in the right Bol identity (\\re\n{rightBol}), gives us \n\\begin{equation}\npq\\cdot q\\backslash a=\\faktor{\\left( p\\cdot aq\\right)}{q}=p\\circ _{q}a.\n\\label{midprod}\n\\end{equation\nOn octonions, the left-hand side of (\\ref{midprod}) is precisely the\n\\textquotedblleft modified octonion product\\textquotedblright\\ defined in \n\\cite{GrigorianOctobundle} and also used in \\cite{GrigorianOctoSUSY}. Since\nunit octonions are in particular a right Bol loop, the two products are\nequal on octonions.\n\nThe product (\\ref{rprod}) gives us a convenient definition of the \\emph{loop\nassociator}.\n\n\\begin{definition}\nGiven $p,q,r\\in \\mathbb{L}$, the \\emph{loop associator }of $p,q,r$ is given\nby \n\\begin{equation}\n\\left[ p,q,r\\right] =\\faktor{\\left( p\\circ _{r}q\\right)}{pq}.\n\\label{loopassoc}\n\\end{equation\nThe \\emph{loop commutator }of $p$ and $q$ is given by \n\\begin{equation}\n\\left[ p,q\\right] =\\faktor{\\left( pq\/p\\right)}{q}.  \\label{loopcomm}\n\\end{equation}\n\\end{definition}\n\nFrom the definition (\\ref{loopassoc}), we see that $\\left[ p,q,r\\right] =1$\nif and only if $p\\left( qr\\right) =\\left( pq\\right) r.$ There are several\npossible equivalent definitions of the associator, but from our point of\nview, (\\ref{loopassoc}) will be the most convenient. Similarly, the loop\ncommutator can be defined in different ways, however (\\ref{loopcomm}) has an\nadvantage, because if we define $\\func{Ad}_{p}\\left( q\\right) =pq\/p$, then \n\\left[ p,q\\right] =\\left( \\func{Ad}_{p}\\left( q\\right) \\right) \/q,$ which is\na similar relation as for the group commutator.\n\nWe can easily see that $\\left( \\mathbb{L},\\circ _{r}\\right) $ is a loop.\n\n\\begin{lemma}\nConsider the pair $\\left( \\mathbb{L},\\circ _{r}\\right) $ of the set $\\mathbb\nL}$ equipped with the binary operation $\\circ _{r}$.\n\n\\begin{enumerate}\n\\item The right quotient $\/_{r}$ and the left quotient $\\backslash _{r}$ on \n\\left( \\mathbb{L},\\circ _{r}\\right) $ are given by \n\\begin{subequations\n\\label{rprodq} \n\\begin{eqnarray}\np\/_{r}q &=&\\faktor{pr}{qr}  \\label{rprodqright} \\\\\np\\backslash _{r}q &=&\\faktor{\\left( p\\backslash qr\\right)}{r},\n\\label{rprodqleft}\n\\end{eqnarray\n\\end{subequations\nand hence, $\\left( \\mathbb{L},\\circ _{r}\\right) $ is a quasigroup.\n\n\\item $1\\in \\mathbb{L}$ is the identity element for $\\left( \\mathbb{L},\\circ\n_{r}\\right) ,$ and hence $\\left( \\mathbb{L},\\circ _{r}\\right) $ is a loop.\n\n\\item Let $q\\in \\mathbb{L}$, the left and right inverses with respect to \n\\circ _{r}$ are given by \n\\begin{subequations}\n\\begin{eqnarray}\nq^{\\lambda _{\\left( r\\right) }} &=&\\faktor{r}{qr}  \\label{linvr} \\\\\nq^{\\rho _{\\left( r\\right) }} &=&\\faktor{\\left( q\\backslash r\\right)}{r}.\n\\label{rinvr}\n\\end{eqnarray\n\\end{subequations\n\n\\item $\\left( \\mathbb{L},\\circ _{r}\\right) $ is isomorphic to $\\left( \n\\mathbb{L},\\cdot \\right) $ if and only if $r\\in \\mathcal{C}^{R}\\left( \n\\mathbb{L}\\right) $. In particular, $\\alpha :\\left( \\mathbb{L},\\cdot \\right)\n\\longrightarrow \\left( \\mathbb{L},\\circ _{r}\\right) $ is an isomorphism,\ni.e. for any $p,q\\in \\mathbb{L},\n\\begin{equation}\n\\alpha \\left( pq\\right) =\\alpha \\left( p\\right) \\circ _{r}\\alpha \\left(\nq\\right) ,  \\label{alpharcirc}\n\\end{equation\nif and only if $\\alpha $ is a right pseudoautomorphism on $\\left( \\mathbb{L\n,\\cdot \\right) $ with companion $r$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nLet $x,p,q,r\\in \\mathbb{L}.$\n\n\\begin{enumerate}\n\\item Suppose \n\\begin{equation*}\nx\\circ _{r}q=p.\n\\end{equation*\nUsing (\\ref{rprod}), \n\\begin{equation*}\nx\\cdot qr=pr,\n\\end{equation*\nand thus \n\\begin{equation*}\nx=pr\/qr:=p\/_{r}q.\n\\end{equation*\nSimilarly, suppos\n\\begin{equation*}\np\\circ _{r}x=q,\n\\end{equation*\nso that \n\\begin{equation*}\np\\cdot xr=qr,\n\\end{equation*\nand thus \n\\begin{equation*}\nx=\\left( p\\backslash \\left( qr\\right) \\right) \/r:=p\\backslash _{r}q.\n\\end{equation*\nSince the left and right quotients are both defined, $\\left( \\mathbb{L\n,\\circ _{r}\\right) $ is a quasigroup.\n\n\\item We have \n\\begin{eqnarray*}\np\\circ _{r}1 &=&\\left( p\\cdot r\\right) \/r=p \\\\\n1\\circ _{r}p &=&\\left( 1\\cdot pr\\right) \/r=p.\n\\end{eqnarray*\nHence, $1$ is indeed the identity element for $\\left( \\mathbb{L},\\circ\n_{r}\\right) ,$ and thus $\\left( \\mathbb{L},\\circ _{r}\\right) $ is a loop.\n\n\\item Setting $p=1$ in (\\ref{rprodq}) we get the desired expressions.\n\n\\item Suppose $\\left( \\alpha ,r\\right) \\in \\Psi ^{R}\\left( \\mathbb{L}\\right) \n$. Then, by definition, for any $p,q\\in \\mathbb{L}$, \n\\begin{equation*}\n\\alpha \\left( pq\\right) =\\faktor{\\left( \\alpha \\left( p\\right) \\cdot \\alpha\n\\left( q\\right) r\\right)}{r}\n\\end{equation*\nHence, from (\\ref{rprod}), \n\\begin{equation}\n\\alpha \\left( pq\\right) =\\alpha \\left( p\\right) \\circ _{r}\\alpha \\left(\nq\\right) ,\n\\end{equation\nThus, $\\alpha $ is an isomorphism\\emph{\\ }from $\\left( \\mathbb{L},\\cdot\n\\right) $ to $\\left( \\mathbb{L},\\circ _{r}\\right) $. Clearly the converse is\nalso true: if $\\alpha $ is an isomorphism from $\\left( \\mathbb{L},\\cdot\n\\right) $ to $\\left( \\mathbb{L},\\circ _{r}\\right) $, then $r$ is companion\nfor $\\alpha $. Hence, $\\left( \\mathbb{L},\\cdot \\right) $ and $\\left( \\mathbb\nL},\\circ _{r}\\right) $ are isomorphic if and only if $r$ is a companion for\nsome right pseudoautomorphism.\n\\end{enumerate}\n\\end{proof}\n\nSuppose $r,x\\in \\mathbb{L}$, then the next lemma shows the relationship\nbetween products $\\circ _{x}$ and $\\circ _{rx}$.\n\n\\begin{lemma}\n\\label{lemxrprod}Let $r,x\\in \\mathbb{L}$, then \n\\begin{equation}\np\\circ _{rx}q=\\left( p\\circ _{x}\\left( q\\circ _{x}r\\right) \\right) \/_{x}r.\n\\label{xrprod}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nLet $r,x\\in \\mathbb{L}$, and suppose $y=rx.$ Then, by (\\ref{rprod}), \n\\begin{eqnarray*}\np\\cdot qy &=&\\left( p\\circ _{y}q\\right) \\cdot y \\\\\n&=&\\left( p\\circ _{y}q\\right) \\cdot rx \\\\\n&=&\\left( \\left( p\\circ _{y}q\\right) \\circ _{x}r\\right) \\cdot x.\n\\end{eqnarray*\nOn the other hand, using (\\ref{rprod}) in a different way, we get \n\\begin{eqnarray*}\np\\cdot qy &=&p\\cdot q\\left( rx\\right) \\\\\n&=&p\\cdot \\left( \\left( q\\circ _{x}r\\right) x\\right) \\\\\n&=&\\left( p\\circ _{x}\\left( q\\circ _{x}r\\right) \\right) \\cdot x\n\\end{eqnarray*\nHence, \n\\begin{equation*}\n\\left( p\\circ _{y}q\\right) \\circ _{x}r=p\\circ _{x}\\left( q\\circ _{x}r\\right)\n.\n\\end{equation*\nDividing by $r$ on the right using $\/_{x}$ gives (\\ref{xrprod}).\n\\end{proof}\n\n\\begin{remark}\nLemma \\ref{lemxrprod} shows that the $rx$-product is equivalent to the $r\n-product, \\emph{but defined on }$\\left( \\mathbb{L},\\circ _{x}\\right) .$ That\nis, if we start with $\\circ _{x}$ define the $r$-product using $\\circ _{x}$,\nthen we obtain the $rx$-product \\emph{on }$\\left( \\mathbb{L},\\cdot \\right) \n. If $x\\in \\mathcal{C}^{R}\\left( \\mathbb{L},\\cdot \\right) $, then $\\left( \n\\mathbb{L},\\circ _{x}\\right) $ is isomorphic to $\\left( \\mathbb{L},\\cdot\n\\right) $. Similarly, if $r\\in \\mathcal{C}^{R}\\left( \\mathbb{L},\\circ\n_{x}\\right) $, then $\\left( \\mathbb{L},\\circ _{rx}\\right) $ is isomorphic to \n$\\left( \\mathbb{L},\\circ _{x}\\right) .$\n\\end{remark}\n\nOn $\\left( \\mathbb{L},\\circ _{x}\\right) $ we can define the associator and\ncommutator. Given $p,q,r\\in \\mathbb{L}$, the \\emph{loop associator }on\\emph\n\\ } $\\left( \\mathbb{L},\\circ _{x}\\right) $ is given by \n\\begin{equation}\n\\left[ p,q,r\\right] ^{\\left( x\\right) }=\\left( p\\circ _{rx}q\\right)\n\/_{x}\\left( p\\circ _{x}q\\right) .  \\label{loopassoc2}\n\\end{equation\nThe \\emph{loop commutator }on $\\left( \\mathbb{L},\\circ _{x}\\right) $ is\ngiven by \n\\begin{equation}\n\\left[ p,q\\right] ^{\\left( x\\right) }=\\left( \\left( p\\circ _{x}q\\right)\n\/_{x}p\\right) \/_{x}q.  \\label{loopcomm2}\n\\end{equation\nFor any $x\\in \\mathbb{L}$, the adjoint map $\\func{Ad}^{\\left( x\\right) }:$ \n\\mathbb{L\\times L}\\longrightarrow \\mathbb{L}$ with respect to $\\circ _{x}$\nis given by \n\\begin{equation}\n\\func{Ad}_{p}^{\\left( x\\right) }\\left( q\\right) =\\left( \\left( R_{p}^{\\left(\nx\\right) }\\right) ^{-1}\\circ L_{p}^{\\left( x\\right) }\\right) q=\\left( p\\circ\n_{x}q\\right) \/_{x}p  \\label{Adpx}\n\\end{equation\nfor any $p,q\\in \\mathbb{L},$ and its inverse for a fixed $p$ i\n\\begin{equation}\n\\left( \\func{Ad}_{p}^{\\left( x\\right) }\\right) ^{-1}\\left( q\\right) =\\left(\n\\left( L_{p}^{\\left( x\\right) }\\right) ^{-1}\\circ R_{p}^{\\left( x\\right)\n}\\right) q=p\\backslash _{x}\\left( q\\circ _{x}p\\right) .\n\\end{equation}\n\nLet us now consider how pseudoautomorphisms of $\\left( \\mathbb{L},\\cdot\n\\right) $ act on $\\left( \\mathbb{L},\\circ _{r}\\right) $.\n\n\\begin{lemma}\n\\label{lemPseudoHom}Let $h=\\left( \\beta ,B\\right) \\in \\Psi ^{R}\\left( \n\\mathbb{L},\\cdot \\right) $. Then, for any $p,q,r\\in \\mathbb{L},$ \n\\begin{equation}\n\\beta \\left( p\\circ _{r}q\\right) =\\beta \\left( p\\right) \\circ _{h\\left(\nr\\right) }\\beta \\left( q\\right)  \\label{PsiActcircr}\n\\end{equation\nand $\\beta $ is a right pseudoautomorphism of $\\left( \\mathbb{L},\\circ\n_{r}\\right) $ with companion $h\\left( r\\right) \/r$. It also follows that \n\\begin{equation}\n\\beta \\left( p\/_{r}q\\right) =\\beta \\left( p\\right) \/_{h\\left( r\\right)\n}\\beta \\left( q\\right) .  \\label{PsiActQuot}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nConsider $\\beta \\left( p\\circ _{r}q\\right) $. Then, using (\\ref{PsAutprod})\nand (\\ref{PsAutquot2}), \n\\begin{eqnarray*}\n\\beta \\left( p\\circ _{r}q\\right) &=&\\beta \\left( \\left( p\\cdot qr\\right)\n\/r\\right) \\\\\n&=&h\\left( p\\cdot qr\\right) \/h\\left( r\\right) \\\\\n&=&\\left( \\beta \\left( p\\right) \\cdot h\\left( qr\\right) \\right) \/h\\left(\nr\\right) \\\\\n&=&\\left( \\beta \\left( p\\right) \\cdot \\beta \\left( q\\right) h\\left( r\\right)\n\\right) \/h\\left( r\\right) \\\\\n&=&\\beta \\left( p\\right) \\circ _{h\\left( r\\right) }\\beta \\left( q\\right) ,\n\\end{eqnarray*\nand hence we get (\\ref{PsiActcircr}). Alternatively, using (\\ref{rprodqright\n), \n\\begin{eqnarray*}\n\\beta \\left( p\\circ _{r}q\\right) &=&\\faktor{\\left( \\beta \\left( p\\right)\n\\cdot \\beta \\left( q\\right) h\\left( r\\right) \\right)}{h\\left( r\\right)} \\\\\n&=&\\left(\\faktor{\\left( \\beta \\left( p\\right) \\cdot \\beta \\left( q\\right)\nh\\left( r\\right) \\right)}{r}\\right) \/_{r}\\left(\\faktor{ h\\left( r\\right)}{r\n\\right) .\n\\end{eqnarray*\nNow, let $C=h\\left( r\\right) \/r$. Thus, \n\\begin{eqnarray*}\n\\beta \\left( p\\circ _{r}q\\right) &=&\\left( \\faktor{\\left( \\beta \\left(\np\\right) \\left( \\beta \\left( q\\right) \\cdot Cr\\right) \\right)}{r}\\right)\n\/_{r}C \\\\\n&=&\\left( \\beta \\left( p\\right) \\circ _{r}\\left( \\beta \\left( q\\right) \\circ\n_{r}C\\right) \\right) \/_{r}C\n\\end{eqnarray*\nThus, indeed, $\\beta $ is a right pseudoautomorphism of $\\left( \\mathbb{L\n,\\circ _{r}\\right) $ with companion $C=h\\left( r\\right) \/r$.\n\nNow using (\\ref{PsiActcircr}) with $p\/_{r}q$ and $q$, we find \n\\begin{equation*}\n\\beta \\left( p\\right) =\\beta \\left( p\/_{r}q\\circ _{r}q\\right) =\\beta \\left(\np\/_{r}q\\right) \\circ _{h\\left( r\\right) }\\beta \\left( q\\right)\n\\end{equation*\nand hence we get (\\ref{PsiActQuot}).\n\\end{proof}\n\n\\begin{remark}\nWe will use the notation $\\left( \\beta ,C\\right) _{r}$ to denote that \n\\left( \\beta ,C\\right) _{r}$ is considered as a pseudoautomorphism pair on \n\\left( \\mathbb{L},\\circ _{r}\\right) $, i.e. $\\left( \\beta ,C\\right) _{r}\\in\n\\Psi ^{R}\\left( \\mathbb{L},\\circ _{r}\\right) $. Of course, the product of $C$\nwith any element in $\\mathcal{N}^{R}\\left( \\mathbb{L},\\circ _{r}\\right) $ on\nthe right will also give a companion of $\\beta $ on $\\left( \\mathbb{L},\\circ\n_{r}\\right) $. Any right pseudoautomorphism of $\\left( \\mathbb{L},\\cdot\n\\right) $ is also a right pseudoautomorphism of $\\left( \\mathbb{L},\\circ\n_{r}\\right) $, however their companions may be different. In particular, \n\\func{PsAut}^{R}\\left( \\mathbb{L},\\cdot \\right) =\\func{PsAut}^{R}\\left( \n\\mathbb{L},\\circ _{r}\\right) $. For $\\Psi ^{R}\\left( \\mathbb{L},\\cdot\n\\right) $ and $\\Psi ^{R}\\left( \\mathbb{L},\\circ _{r}\\right) $ we have a\ngroup isomorphism \n\\begin{eqnarray}\n\\Psi ^{R}\\left( \\mathbb{L},\\cdot \\right) &\\longrightarrow &\\Psi ^{R}\\left( \n\\mathbb{L},\\circ _{r}\\right)  \\notag \\\\\nh &=&\\left( \\beta ,B\\right) \\mapsto h_{r}=\\left( \\beta ,\\faktor{h\\left(\nr\\right)} {r}\\right) _{r}.  \\label{PsAutoriso}\n\\end{eqnarray}\nConversely, if we have $h_{r}=\\left( \\beta ,C\\right) _{r}\\in \\Psi ^{R}\\left( \n\\mathbb{L},\\circ _{r}\\right) $, then this corresponds to $h=\\left( \\beta\n,B\\right) \\in \\Psi ^{R}\\left( \\mathbb{L},\\cdot \\right) $ where \n\\begin{equation}\nB=\\beta \\left( r\\right) \\backslash \\left( Cr\\right) .  \\label{PsAutorisorev}\n\\end{equation}\n\\end{remark}\n\nThe group isomorphism (\\ref{PsAutoriso}) together with $R_{r}^{-1}$ (right\ndivision by $r$) induces a $G$-set isomorphism between $\\left( \\mathbb\n\\mathring{L}},\\cdot \\right) $with the action of $\\Psi ^{R}\\left( \\mathbb{L\n,\\cdot \\right) $ and $\\left( \\mathbb{\\mathring{L}},\\circ _{r}\\right) $ with\nthe action of $\\Psi ^{R}\\left( \\mathbb{L},\\circ _{r}\\right) $.\n\n\\begin{lemma}\nLet $r\\in \\mathbb{L}$, then the mapping (\\ref{PsAutoriso}) $h\\mapsto h_{r}$\nfrom $\\Psi ^{R}\\left( \\mathbb{L},\\cdot \\right) \\ $to $\\Psi ^{R}\\left( \n\\mathbb{L},\\circ _{r}\\right) $ together with the map $R_{r}^{-1}:\\left( \n\\mathbb{\\mathring{L}},\\cdot \\right) \\longrightarrow \\left( \\mathbb{\\mathring\nL}},\\circ _{r}\\right) $ gives a $G$-set isomorphism. In particular, for any \nA\\in \\mathbb{\\mathring{L}}$ and $h\\in \\Psi ^{R}\\left( \\mathbb{L},\\cdot\n\\right) ,$ \n\\begin{equation}\nh\\left( A\\right) \/r=h_{r}\\left( A\/r\\right) .  \\label{Gsetiso}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nSuppose $h=\\left( \\beta ,B\\right) $ and correspondingly, from (\\re\n{PsAutoriso}), $h_{r}=\\left( \\beta ,\\faktor{h\\left( r\\right)} {r}\\right) $.\nThen, we have, \n\\begin{eqnarray*}\nh_{r}\\left( A\/r\\right) &=&\\beta \\left( A\/r\\right) \\circ _{r}\\faktor{h\\left(\nr\\right)} {r} \\\\\n&=&\\faktor{\\left( h\\left( A\\right) \/h\\left( r\\right) \\cdot h\\left( r\\right)\n\\right)}{ r } \\\\\n&=&h\\left( A\\right) \/r,\n\\end{eqnarray*\nwhere we have also used (\\ref{PsAutquot2a}).\n\\end{proof}\n\nUsing (\\ref{PsAutoriso}), we now have the following characterizations of \n\\mathcal{C}^{R}\\left( \\mathbb{L},\\circ _{r}\\right) ,$ $\\mathcal{N}^{R}\\left( \n\\mathbb{L},\\circ _{r}\\right) $, and $\\func{Aut}\\left( \\mathbb{L},\\circ\n_{r}\\right) $.\n\n\\begin{lemma}\nLet $r,C\\in \\mathbb{L}$, then \n\\begin{subequations}\n\\begin{eqnarray}\nC &\\in &\\mathcal{C}^{R}\\left( \\mathbb{L},\\circ _{r}\\right) \\iff C=A\/r\\ \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi\nfor some }A\\in \\func{Orb}_{\\Psi ^{R}\\left( \\mathbb{L},\\cdot \\right) }\\left(\nr\\right)  \\label{CRrdef} \\\\\nC &\\in &\\mathcal{N}^{R}\\left( \\mathbb{L},\\circ _{r}\\right) \\iff C=\\func{Ad\n_{r}\\left( A\\right) \\ \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{for some }A\\in \\mathcal{N}^{R}\\left( \\mathbb{L\n,\\cdot \\right)  \\label{CRNucl}\n\\end{eqnarray\n\\end{subequations\nand \n\\begin{equation}\n\\func{Aut}\\left( \\mathbb{L},\\circ _{r}\\right) \\cong \\func{Stab}_{\\Psi\n^{R}\\left( \\mathbb{L},\\cdot \\right) }\\left( r\\right) .  \\label{AutLr}\n\\end{equation\nIf $r\\in \\mathcal{C}^{R}\\left( \\mathbb{L},\\cdot \\right) $, so that there\nexists a right pseudoautomorphism pair $h=\\left( \\alpha ,r\\right) \\in \\Psi\n^{R}\\left( \\mathbb{L},\\cdot \\right) $, then $\\func{Aut}\\left( \\mathbb{L\n,\\circ _{r}\\right) \\cong h\\func{Aut}\\left( \\mathbb{L},\\cdot \\right) h^{-1}.$\n\\end{lemma}\n\n\\begin{proof}\nFrom (\\ref{PsAutoriso}) we see that any companion in $\\left( \\mathbb{L\n,\\circ _{r}\\right) $ is of the form $h\\left( r\\right) \/r$ for some $h\\in\n\\Psi ^{R}\\left( \\mathbb{L},\\cdot \\right) $. Therefore, $C\\in \\mathbb{L}$ is\na companion in $\\left( \\mathbb{L},\\circ _{r}\\right) $ if and only if it is\nof the form $C=A\/r\\ $for some $A\\in \\func{Orb}_{\\Psi ^{R}\\left( \\mathbb{L\n,\\cdot \\right) }\\left( r\\right) .$\n\nThe right nucleus $\\mathcal{N}^{R}\\left( \\mathbb{L},\\circ _{r}\\right) $\ncorresponds to the companions of the identity map $\\func{id}$ on $\\mathbb{L}\n, hence taking $\\beta =\\func{id}$ in (\\ref{PsAutoriso}), we find that\ncompanions of $\\func{id}$ in $\\left( \\mathbb{L},\\circ _{r}\\right) $ must be\nof the form $C=\\left( rA\\right) \/r=\\func{Ad}_{r}\\left( A\\right) \\ $for some \nA\\in \\mathcal{N}^{R}\\left( \\mathbb{L},\\cdot \\right) $. Conversely, suppose \nC=\\left( rA\\right) \/r\\ $for some $A\\in \\mathcal{N}^{R}\\left( \\mathbb{L\n,\\cdot \\right) $, then we can explicitly check that for any $p,q\\in \\mathbb{\n}$, we have \n\\begin{eqnarray*}\n\\left( p\\circ _{r}q\\right) \\circ _{r}C &=&\\left( \\left( p\\cdot qr\\right)\n\/r\\cdot rA\\right) \/r \\\\\n&=&\\left( \\left( p\\cdot qr\\right) \\cdot A\\right) \/r \\\\\n&=&\\left( p\\cdot \\left( qr\\cdot A\\right) \\right) \/r=\\left( p\\cdot \\left(\nq\\cdot rA\\right) \\right) \/r \\\\\n&=&\\left( p\\cdot \\left( q\\cdot Cr\\right) \\right) \/r=\\left( p\\cdot \\left(\nq\\circ _{r}C\\right) r\\right) \/r \\\\\n&=&p\\circ _{r}\\left( q\\circ _{r}C\\right)\n\\end{eqnarray*\nand hence, $C\\in \\mathcal{N}^{R}\\left( \\mathbb{L},\\circ _{r}\\right) $.\n\nThe group $\\func{Aut}\\left( \\mathbb{L},\\circ _{r}\\right) $ is isomorphic to\nthe preimage $\\func{prj}_{2}^{-1}\\left( 1\\right) $ with respect to the\nprojection map $\\func{prj}_{2}:$ $\\Psi ^{R}\\left( \\mathbb{L},\\circ\n_{r}\\right) \\longrightarrow \\mathcal{C}^{R}\\left( \\mathbb{L},\\circ\n_{r}\\right) $. From (\\ref{PsAutoriso}), this corresponds precisely to the\nmaps $h\\in \\Psi ^{R}\\left( \\mathbb{L},\\cdot \\right) $ for which $h\\left(\nr\\right) =r$. If $r$ is in the $\\Psi ^{R}\\left( \\mathbb{L},\\cdot \\right) \n-orbit of $1$, then clearly $\\func{Aut}\\left( \\mathbb{L},\\circ _{r}\\right) $\nis conjugate to $\\func{Aut}\\left( \\mathbb{L},\\cdot \\right) .$\n\\end{proof}\n\n\\begin{remark}\nSuppose $r\\in \\mathcal{C}^{R}\\left( \\mathbb{L}\\right) $, then from (\\re\n{CRrdef}), we see that if $A\\in \\mathcal{C}^{R}\\left( \\mathbb{L},\\circ\n_{r}\\right) $, then $Ar\\in \\mathcal{C}^{R}\\left( \\mathbb{L}\\right) .$ Also,\nusing the isomorphism (\\ref{PsAutoriso}), we can define the left action of \n\\Psi ^{R}\\left( \\mathbb{L},\\circ _{r}\\right) $ on $\\Psi ^{R}\\left( \\mathbb{L\n,\\cdot \\right) $ just by composition on the left by the corresponding\nelement in $\\Psi ^{R}\\left( \\mathbb{L},\\cdot \\right) $. Now recall that \n\\begin{equation*}\n\\mathcal{C}^{R}\\left( \\mathbb{L},\\circ _{r}\\right) \\cong \\faktor{\\Psi\n^{R}\\left( \\mathbb{L},\\circ _{r}\\right)}{\\func{Aut}\\left( \\mathbb{L},\\circ\n_{r}\\right)}\\ \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{and\\ }\\mathcal{C}^{R}\\left( \\mathbb{L}\\right) \\cong\n\\faktor{\\Psi ^{R}\\left( \\mathbb{L},\\cdot \\right)}{\\func{Aut}\\left(\n\\mathbb{L},\\cdot \\right)}.\n\\end{equation*\nThen, for any equivalence classes $\\left\\lfloor \\alpha ,A\\right\\rfloor\n_{r}\\in \n\\faktor{\\Psi ^{R}\\left( \\mathbb{L},\\circ _{r}\\right)}{\\func{Aut}\\left(\n\\mathbb{L},\\circ _{r}\\right)}$ and $\\left\\lfloor \\beta ,r\\right\\rfloor \\in \n\\faktor{\\Psi\n^{R}\\left( \\mathbb{L},\\cdot \\right)}{\\func{Aut}\\left( \\mathbb{L},\\cdot\n\\right)}$, we find that \n\\begin{equation}\n\\left\\lfloor \\alpha ,A\\right\\rfloor _{r}\\cdot \\left\\lfloor \\beta\n,r\\right\\rfloor =\\left\\lfloor \\alpha \\circ \\beta ,Ar\\right\\rfloor .\n\\label{CCaction}\n\\end{equation\nAnother way to see this is the following. From (\\ref{PsAutorisorev}), the\nelement in $\\Psi ^{R}\\left( \\mathbb{L},\\cdot \\right) $ that corresponds to \n\\left( \\alpha ,A\\right) _{r}\\in \\Psi ^{R}\\left( \\mathbb{L},\\circ _{r}\\right) \n$ is $\\left( \\alpha \n\\scalebox{-1}[1]{\\nicefrac{\\scalebox{-1}[1]{$\nAr$}}{\\scalebox{-1}[1]{$\\alpha \\left( r\\right)$}}}\\right) .$ The composition\nof this with $\\left( \\beta ,r\\right) $ is then $\\left( \\alpha \\circ \\beta\n,Ar\\right) .$ Then, it is easy to see that this reduces to cosets.\n\\end{remark}\n\n\\begin{example}\n\\label{exMouf}Recall that in a Moufang loop $\\mathbb{L}$, the map $\\func{Ad\n_{q}$ is a right pseudoautomorphism with companion $q^{3}$. The relation \n\\ref{CCaction}) then shows that for any $r\\in \\mathbb{L},$ \n\\begin{equation}\n\\func{Ad}_{q}^{\\left( r^{3}\\right) }\\circ \\func{Ad}_{r}=\\func{Ad}_{\\left(\nq^{3}r^{3}\\right) ^{\\frac{1}{3}}}\\circ h\n\\end{equation\nwhere $h\\in \\func{Aut}\\left( \\mathbb{L}\\right) $. This follows because \n\\func{Ad}_{q}^{\\left( r^{3}\\right) }$ has companion $q^{3}$ in $\\Psi\n^{R}\\left( \\mathbb{L},\\circ _{r^{3}}\\right) $ and $\\func{Ad}_{r}$ has\ncompanion $r^{3}$ in $\\Psi ^{R}\\left( \\mathbb{L}\\right) $, thus the\ncomposition has companion $q^{3}r^{3}$, so up to composition with $\\func{Aut\n\\left( \\mathbb{L}\\right) ,$ it is given by $\\func{Ad}_{\\left(\nq^{3}r^{3}\\right) ^{\\frac{1}{3}}}.$ A similar expression for octonions has\nbeen derived in \\cite{GrigorianOctobundle}.\n\\end{example}\n\nAs we have seen, $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ acts transitively on \n\\mathcal{C}^{R}\\left( \\mathbb{L}\\right) $ and moreover, for each $r\\in \n\\mathcal{C}^{R}\\left( \\mathbb{L}\\right) $, the loops $\\left( \\mathbb{L\n,\\circ _{r}\\right) $ are all isomorphic to one another, and related via\nelements of $\\Psi ^{R}\\left( \\mathbb{L}\\right) $. Concretely, consider \n\\left( \\mathbb{L},\\circ _{r}\\right) $ and suppose $h=\\left( \\alpha ,A\\right)\n\\in \\Psi ^{R}\\left( \\mathbb{L}\\right) $. Then, define the map \n\\begin{equation*}\nh:\\left( \\mathbb{L},\\circ _{r}\\right) \\longrightarrow \\left( \\mathbb{L\n,\\circ _{h\\left( r\\right) }\\right) ,\n\\end{equation*\nwhere $h$ acts on $\\mathbb{L}$ via the partial action (i.e. via $\\alpha $).\nIndeed, from (\\ref{alpharcirc}), we have for $p,q\\in h\\left( \\mathbb{L\n\\right) $ \n\\begin{equation}\n\\alpha \\left( \\alpha ^{-1}\\left( p\\right) \\circ _{r}\\alpha ^{-1}\\left(\nq\\right) \\right) =p\\circ _{h\\left( r\\right) }q.  \\label{alphaprod}\n\\end{equation\nMoreover, if we instead consider the action of $\\Psi ^{R}\\left( \\mathbb{L\n,\\circ _{r}\\right) ,$ then given $h_{r}=\\left( \\alpha ,x\\right) _{r}\\in \\Psi\n^{R}\\left( \\mathbb{L},\\circ _{r}\\right) $, $h_{r}\\left( \\mathbb{L}\\right)\n\\cong \\left( \\mathbb{L},\\circ _{xr}\\right) .$ This is summarized in the\ntheorem below.\n\n\\begin{theorem}\n\\label{thmLeftProd}Let $\\mathbb{L}$ be a loop with the set of right\ncompanions $\\mathcal{C}^{R}\\left( \\mathbb{L}\\right) .$ For every $r\\in \n\\mathcal{C}^{R}\\left( \\mathbb{L}\\right) $ and every $h\\in \\Psi ^{R}\\left( \n\\mathbb{L}\\right) $, the loop $\\left( \\mathbb{L},\\circ _{r}\\right) $ is\nisomorphic to $\\left( \\mathbb{L},\\circ _{h\\left( r\\right) }\\right) .$\nMoreover, if instead, the action of $\\Psi ^{R}\\left( \\mathbb{L},\\circ\n_{r}\\right) $ is considered, then an element of $\\Psi ^{R}\\left( \\mathbb{L\n,\\circ _{r}\\right) $ with companion $x$ induces a loop isomorphism from \n\\left( \\mathbb{L},\\circ _{r}\\right) $ to $\\left( \\mathbb{L},\\circ\n_{xr}\\right) .$\n\\end{theorem}\n\nNow again, let $h=\\left( \\alpha ,A\\right) \\in \\Psi ^{R}\\left( \\mathbb{L\n\\right) $, and we will consider the action of $h$ on the nucleus. It is easy\nto see how the loop associator transforms under this map. Using (\\re\n{loopassoc2}) and (\\ref{PsiActQuot}), we hav\n\\begin{eqnarray}\n\\alpha \\left( \\left[ p,q,r\\right] ^{\\left( x\\right) }\\right) &=&\\alpha\n\\left( \\left( p\\circ _{rx}q\\right) \/_{x}\\left( p\\circ _{x}q\\right) \\right) \n\\notag \\\\\n&=&\\left( \\alpha \\left( p\\right) \\circ _{\\alpha \\left( r\\right) h\\left(\nx\\right) }\\alpha \\left( q\\right) \\right) \/_{h\\left( x\\right) }\\left( \\alpha\n\\left( p\\right) \\circ _{h\\left( x\\right) }\\alpha \\left( q\\right) \\right) \n\\notag \\\\\n&=&\\left[ \\alpha \\left( p\\right) ,\\alpha \\left( q\\right) ,\\alpha \\left(\nr\\right) \\right] ^{\\left( h\\left( x\\right) \\right) }.  \\label{alphaassoc}\n\\end{eqnarray\nSo in particular, taking $x=1$, $C\\in \\mathcal{N}^{R}\\left( \\mathbb{L\n\\right) $ if and only if $\\alpha \\left( C\\right) \\in \\mathcal{N}^{R}\\left( \n\\mathbb{L},\\circ _{A}\\right) .$ However from (\\ref{CRNucl}), we know that \nC\\in \\mathcal{N}^{R}\\left( \\mathbb{L}\\right) $ if and only if $\\left( \\func\nAd}_{A}\\right) C\\in \\mathcal{N}^{R}\\left( \\mathbb{L},\\circ _{A}\\right) .$ In\nparticular, this means that $C\\in \\mathcal{N}^{R}\\left( \\mathbb{L}\\right) $\nif and only if $\\alpha ^{-1}\\left( \\func{Ad}_{A}C\\right) \\in \\mathcal{N\n^{R}\\left( \\mathbb{L}\\right) .$ This defines a left action of $\\Psi\n^{R}\\left( \\mathbb{L}\\right) $ on $\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) \n: \n\\begin{equation}\nh^{\\prime \\prime }\\left( C\\right) =\\func{Ad}_{A}^{-1}\\left( \\alpha \\left(\nC\\right) \\right) =\\scalebox{-1}[1]{\\nicefrac{\\scalebox{-1}[1]{$ h\\left(\nC\\right)$}}{\\scalebox{-1}[1]{$A$}}}  \\label{nuclearaction}\n\\end{equation\nfor $h=\\left( \\alpha ,A\\right) \\in \\Psi ^{R}\\left( \\mathbb{L}\\right) $ and \nC\\in \\mathcal{N}^{R}\\left( \\mathbb{L}\\right) .$ The action (\\re\n{nuclearaction}) can be seen from the following considerations. Recall \n\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) ^{\\func{op}}$ embeds in $\\Psi\n^{R}\\left( \\mathbb{L}\\right) $ via the map $C\\mapsto \\iota _{2}\\left(\nC\\right) =\\left( \\func{id},C\\right) .$ The group $\\Psi ^{R}\\left( \\mathbb{L\n\\right) $ acts on itself via the adjoint action, so let $h=\\left( \\alpha\n,A\\right) \\in \\Psi ^{R}\\left( \\mathbb{L}\\right) $, then from (\\ref{PsiAdjN})\nrecall, \n\\begin{equation}\nh\\left( \\iota _{2}\\left( C\\right) \\right) h^{-1}=\\left( \\alpha ,h\\left(\nC\\right) \\right) h^{-1}=\\left( \\func{id},A^{\\lambda }\\cdot h\\left( C\\right)\n\\right) .\n\\end{equation\nOn the other hand, suppose \n\\begin{equation*}\n\\left( \\alpha ,h\\left( C\\right) \\right) h^{-1}=\\left( \\func{id},x\\right) ,\n\\end{equation*\nso that \n\\begin{equation*}\n\\left( \\alpha ,h\\left( C\\right) \\right) =\\left( \\func{id},x\\right) \\left(\n\\alpha ,A\\right) =\\left( \\alpha ,Ax\\right)\n\\end{equation*\nTherefore, $x=A\\backslash h\\left( C\\right) .$ In particular, $A\\backslash\nh\\left( C\\right) \\in \\mathcal{N}^{R}\\left( \\mathbb{L}\\right) .$ Thus the\ninduced action on $\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) $ is precisely \nC\\mapsto A\\backslash h\\left( C\\right) =\\func{Ad}_{A}^{-1}\\left( \\alpha\n\\left( C\\right) \\right) $. Moreover, right multiplication of elements in \n\\mathbb{\\mathring{L}}$ by elements of $\\mathcal{N}^{R}\\left( \\mathbb{L\n\\right) $ is compatible with the corresponding actions of $\\Psi ^{R}\\left( \n\\mathbb{L}\\right) $.\n\n\\begin{lemma}\nFor any $s\\in \\mathbb{\\mathring{L}},C\\in \\mathcal{N}^{R}\\left( \\mathbb{L\n\\right) $, and $h\\in \\Psi ^{R}\\left( \\mathbb{L}\\right) $, we have \n\\begin{equation}\nh\\left( sC\\right) =h\\left( s\\right) h^{\\prime \\prime }\\left( C\\right) ,\n\\label{nuclearaction1}\n\\end{equation\nwhere $h^{\\prime \\prime }$ is the action (\\ref{nuclearaction}).\n\\end{lemma}\n\n\\begin{proof}\nIndeed, to show (\\ref{nuclearaction1}), we have \n\\begin{eqnarray*}\nh\\left( sC\\right) &=&\\alpha \\left( s\\right) h\\left( C\\right) \\\\\n&=&h\\left( s\\right) \/A\\cdot Ah^{\\prime \\prime }\\left( C\\right) \\\\\n&=&\\left( h\\left( s\\right) \/A\\cdot A\\right) h^{\\prime \\prime }\\left( C\\right)\n\\\\\n&=&h\\left( s\\right) \\cdot h^{\\prime \\prime }\\left( C\\right) ,\n\\end{eqnarray*\nsince $h^{\\prime \\prime }\\left( C\\right) \\in \\mathcal{N}^{R}\\left( \\mathbb{L\n\\right) .$\n\\end{proof}\n\n\\section{Smooth loops}\n\n\\setcounter{equation}{0}\\label{sectSmooth}Suppose the loop $\\mathbb{L}$ is a\nsmooth finite-dimensional manifold such that the loop multiplication and\ndivision are smooth functions. Define maps\n\n\\begin{equation}\n\\begin{array}{c}\nL_{r}:\\mathbb{L}\\longrightarrow \\mathbb{L} \\\\ \nq\\longmapsto r\n\\end{array}\n\\label{lprod}\n\\end{equation\nand \n\\begin{equation}\n\\begin{array}{c}\nR_{r}:\\mathbb{L}\\longrightarrow \\mathbb{L} \\\\ \nq\\longmapsto qr\n\\end{array}\n\\label{rprod0}\n\\end{equation\nThese are diffeomorphisms of $\\mathbb{L}$ with smooth inverses $L_{r}^{-1}$\nand $R_{r}^{-1}$ that correspond to left division and right division by $r$,\nrespectively. Also, assume that $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ acts\nsmoothly on $\\mathbb{L}$ (as before, $\\mathbb{L}$ together with the full\naction of $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ will be denoted by $\\mathbb\n\\mathring{L}}$). Thus, the action of $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ is\na group homomorphism from $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ to $\\func{Dif\n}\\left( \\mathbb{L}\\right) .$ In particular, this allows to induce a Lie\ngroup structure on $\\Psi ^{R}\\left( \\mathbb{L}\\right) .$ Similarly, $\\func\nPsAut}^{R}\\left( \\mathbb{L}\\right) $ is then also a Lie group, and for any \ns\\in \\mathbb{\\mathring{L}}$, $\\func{Aut}\\left( \\mathbb{L},\\circ _{s}\\right)\n\\cong \\func{Stab}_{\\Psi ^{R}\\left( \\mathbb{L}\\right) }\\left( s\\right) $ is\nthen a Lie subgroup of $\\Psi ^{R}\\left( \\mathbb{L}\\right) $, and indeed of \n\\func{PsAut}^{R}\\left( \\mathbb{L}\\right) $ as well. The assumption that\npseudoautomorphisms acts smoothly on $\\mathbb{L}$ may be nontrivial. To the\nbest of the author's knowledge, it is an open question whether this is\nalways true. However, for the loop $U\\mathbb{O}$ of unit octonions, this is\nindeed true, as can be seen from Example \\ref{ExPsOcto}.\n\nDefine $X$ to be a \\emph{right fundamental vector field}\\textbf{\\ }if for\nany $q\\in \\mathbb{L},$ it is determined by a tangent vector at $1$ via right\ntranslations. That is, given a tangent vector $\\xi \\in T_{1}\\mathbb{L}$, we\ndefine a corresponding right fundamental vector field $\\rho \\left( \\xi\n\\right) $ given by \n\\begin{equation}\n\\rho \\left( \\xi \\right) _{q}=\\left( R_{q}\\right) _{\\ast }\\xi\n\\end{equation\nat any $p\\in \\mathbb{L}$. If $\\mathbb{L}$ is a Lie group, then this\ndefinition is equivalent to the standard definition of a right-invariant\nvector field $X$ such that $\\left( R_{q}\\right) _{\\ast }X_{p}=X_{pq}$,\nhowever in the non-associative case, $R_{q}\\circ R_{p}\\neq R_{pq},$ so the\nstandard definition wouldn't work, so a right fundamental vector field is\nnot actually right-invariant in the usual sense. We can still say that the\nvector space of right fundamental vector fields has dimension $\\dim \\mathbb{\n}$, and at any point, they still form a basis for the tangent space. In\nparticular, any smooth loop is parallelizable. However this vector space is\nnow in general not a Lie algebra under the Lie bracket of vector fields,\nwhich is to be expected, since $T_{1}\\mathbb{L}$ doesn't necessarily have\nthe Lie algebra structure either.\n\nInstead of right invariance, we see that given a right fundamental vector\nfield $X=\\rho \\left( \\xi \\right) $, \n\\begin{eqnarray}\n\\left( R_{p}^{-1}\\right) _{\\ast }X_{q} &=&\\left( R_{p}^{-1}\\circ\nR_{q}\\right) _{\\ast }\\xi  \\notag \\\\\n&=&\\left( R_{q\/p}^{\\left( p\\right) }\\right) _{\\ast }\\xi  \\label{rightvect}\n\\end{eqnarray\nwhere $R^{\\left( p\\right) }$ is the right product with respect to the\noperation $\\circ _{p}.$ This is because \n\\begin{eqnarray}\n\\left( R_{p}^{-1}\\circ R_{q}\\right) r &=&\\left( rq\\right) \/p  \\notag \\\\\n&=&\\left( r\\cdot \\left( q\/p\\cdot p\\right) \\right) \/p  \\notag \\\\\n&=&r\\circ _{p}\\left( q\/p\\right) =R_{q\/p}^{\\left( p\\right) }r,  \\label{RinvR}\n\\end{eqnarray\nwhere we have used (\\ref{rprod}).\n\n\\subsection{Exponential map}\n\n\\label{secExpMap}For some $\\xi \\in T_{1}\\mathbb{L},$ define a flow $p_{\\xi }$\non $\\mathbb{L}$ given by \n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n\\frac{dp_{\\xi }\\left( t\\right) }{dt}=\\left( R_{p_{\\xi }\\left( t\\right)\n}\\right) _{\\ast }\\xi \\\\ \np_{\\xi }\\left( 0\\right) =\n\\end{array\n\\right.  \\label{floweq}\n\\end{equation\nThis generally has a solution for some sufficiently small time interval \n\\left( -\\varepsilon ,\\varepsilon \\right) $, and is only a local 1-parameter\nsubgroup. However it is shown in \\cite{Kuzmin1971,Malcev1955} that if \n\\mathbb{L}$ is at least power-associative, then $p_{\\xi }\\left( t+s\\right)\n=p_{\\xi }\\left( t\\right) p_{\\xi }\\left( s\\right) $ for all $t,s$, and hence\nthe solution can extended for all $t$. The weakest power-associativity\nassumption is required in order to be able to define $p_{\\xi }\\left(\nnh\\right) =p_{\\xi }\\left( h\\right) ^{n}$ unambiguously.\n\nThe solutions of (\\ref{floweq}) define the (local) exponential map: $\\exp\n\\left( t\\xi \\right) :=p_{\\xi }\\left( t\\right) $. The corresponding\ndiffeomorphisms are then the right translations $R_{\\exp \\left( t\\xi \\right)\n}$. We will generally only need this locally, so the power-associativity\nassumption will not be necessary. Now consider a similar flow but with a\ndifferent initial condition: \n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n\\frac{dp_{\\xi ,q}\\left( t\\right) }{dt}=\\left( R_{p_{\\xi ,q}\\left( t\\right)\n}\\right) _{\\ast }\\xi \\\\ \np_{\\xi ,q}\\left( 0\\right) =\n\\end{array\n\\right.  \\label{floweq2}\n\\end{equation\nwhere $q\\in \\mathbb{L}$. Applying $R_{q}^{-1}$, and setting $\\tilde{p}\\left(\nt\\right) =\\faktor{p_{\\xi ,q}\\left( t\\right)}{q}$, we obtain \n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n\\frac{d\\tilde{p}\\left( t\\right) }{dt}=\\left( R_{q}^{-1}\\circ R_{p_{\\xi\n,q}\\left( t\\right) }\\right) _{\\ast }\\xi \\\\ \n\\tilde{p}\\left( 0\\right) =\n\\end{array\n\\right. .  \\label{floweq2a}\n\\end{equation\nIf $\\mathbb{L}$ is associative, then $R_{q}^{-1}\\circ R_{p_{\\xi ,q}\\left(\nt\\right) }=R_{\\left( p_{\\xi ,q}\\left( t\\right) \\right) \/q},$ and thus \n\\tilde{p}\\left( t\\right) $ would satisfy (\\ref{floweq}), and we could\nconclude that $p_{\\xi ,q}\\left( t\\right) =\\exp \\left( t\\xi \\right) q.$\nHowever, in the general case, we have (\\ref{RinvR}) and hence, $\\tilde{p\n\\left( t\\right) $ satisfies the following equatio\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n\\frac{d\\tilde{p}\\left( t\\right) }{dt}=\\left( R_{\\tilde{p}\\left( t\\right)\n}^{\\left( q\\right) }\\right) _{\\ast }\\xi \\\\ \n\\tilde{p}\\left( 0\\right) =\n\\end{array\n\\right. .  \\label{floweq3}\n\\end{equation\nThis is now an integral curve equation for $\\xi $ on $\\left( \\mathbb{L\n,\\circ _{q}\\right) $, and hence for sufficiently small $t$ we can define a\nlocal exponential map $\\exp _{q}$ for $\\left( \\mathbb{L},\\circ _{q}\\right) $\n\\begin{equation}\n\\tilde{p}\\left( t\\right) =\\exp _{q}\\left( t\\xi \\right) ,  \\label{ptildesol}\n\\end{equation\nso, that \n\\begin{equation}\np_{\\xi ,q}\\left( t\\right) =\\exp _{q}\\left( t\\xi \\right) q.  \\label{pxiqsol}\n\\end{equation\nIf $q\\in \\mathcal{C}^{R}\\left( \\mathbb{L}\\right) $, then $\\left( \\mathbb{L\n,\\circ _{q}\\right) $ is isomorphic to $\\mathbb{L}$, so if $\\mathbb{L}$ is\npower-associative, then so is $\\left( \\mathbb{L},\\circ _{q}\\right) $, and\nhence, the solutions (\\ref{ptildesol}) are defined for all $t.$\n\nSuppose $h=\\left( \\alpha ,q\\right) \\in \\Psi ^{R}\\left( \\mathbb{L}\\right) ,$\nthen let $\\hat{p}\\left( t\\right) =\\alpha ^{-1}\\left( \\tilde{p}\\left(\nt\\right) \\right) .$ This then satisfies $\\hat{p}\\left( 0\\right) =1$ and \n\\begin{equation}\n\\frac{d\\hat{p}\\left( t\\right) }{dt}=\\left( \\alpha ^{-1}\\right) _{\\ast\n}\\left( R_{\\tilde{p}\\left( t\\right) }^{\\left( q\\right) }\\right) _{\\ast }\\xi .\n\\label{dphat1}\n\\end{equation\nHowever, let $r\\in \\mathbb{L}$ and consider $R_{p}^{\\left( q\\right) }$\n\\begin{eqnarray*}\nR_{p}^{\\left( q\\right) }r &=&r\\circ _{q}p=\\alpha \\left( \\alpha ^{-1}\\left(\nr\\right) \\cdot \\alpha ^{-1}\\left( p\\right) \\right) \\\\\n&=&\\left( \\alpha \\circ R_{\\alpha ^{-1}\\left( p\\right) }\\circ \\alpha\n^{-1}\\right) \\left( r\\right) .\n\\end{eqnarray*\nThus, \n\\begin{equation}\nR_{p}^{\\left( q\\right) }=\\alpha \\circ R_{\\alpha ^{-1}\\left( p\\right) }\\circ\n\\alpha ^{-1},  \\label{Rpqalpha}\n\\end{equation\nand hence, (\\ref{dphat1}) becomes \n\\begin{equation}\n\\frac{d\\hat{p}\\left( t\\right) }{dt}=\\left( R_{\\hat{p}\\left( t\\right)\n}\\right) _{\\ast }\\left( \\left( \\alpha ^{-1}\\right) _{\\ast }\\xi \\right) .\n\\end{equation\nThis shows that $\\hat{p}$ is a solution of (\\ref{floweq}) with initial\nvelocity vector $\\left( \\alpha ^{-1}\\right) _{\\ast }\\xi \\in T_{1}\\mathbb{L}\n, and is hence given by $\\hat{p}=\\exp \\left( t\\left( \\alpha ^{-1}\\right)\n_{\\ast }\\xi \\right) .$ Comparing with (\\ref{ptildesol}) we see that in this\ncase, \n\\begin{equation}\n\\exp _{q}\\left( t\\xi \\right) =\\alpha \\left( \\exp \\left( t\\left( \\alpha\n^{-1}\\right) _{\\ast }\\xi \\right) \\right) ,  \\label{expqtalpha}\n\\end{equation\nand hence the solution $p_{\\xi ,q}\\left( t\\right) $ of (\\ref{floweq2}) can\nbe written as \n\\begin{equation}\np_{\\xi ,q}\\left( t\\right) =h\\left( \\exp \\left( t\\left( \\alpha ^{-1}\\right)\n_{\\ast }\\xi \\right) \\right) .  \\label{expqtalpha2}\n\\end{equation\nWe can summarize these findings in the theorem below.\n\n\\begin{theorem}\n\\label{thmLoopflow}Suppose $\\mathbb{L}$ is a smooth loop and suppose $q\\in \n\\mathcal{C}^{R}\\left( \\mathbb{L}\\right) .$ Then, given $\\xi \\in T_{1}\\mathbb\nL},$ the equation \n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n\\frac{dp\\left( t\\right) }{dt}=\\left( R_{p\\left( t\\right) }\\right) _{\\ast }\\xi\n\\\\ \np\\left( 0\\right) =\n\\end{array\n\\right.  \\label{floweq4}\n\\end{equation\nhas the solution \n\\begin{equation}\np\\left( t\\right) =\\exp _{q}\\left( t\\xi \\right) q\n\\end{equation\nfor sufficiently small $t$, where \n\\begin{equation*}\n\\exp _{q}\\left( t\\xi \\right) =\\alpha \\left( \\exp \\left( t\\left( \\alpha\n^{-1}\\right) _{\\ast }\\xi \\right) \\right)\n\\end{equation*\nwhere $\\alpha $ is a right pseudoautomorphism of $\\mathbb{L}$ that has\ncompanion $q$ and $\\exp \\left( t\\xi \\right) $ is defined as the solution of \n\\ref{floweq4}) with initial condition $p\\left( t\\right) =1$. In particular, \n\\xi $ defines a flow $\\Phi _{\\xi ,t}$, given by \n\\begin{equation}\n\\Phi _{\\xi ,t}\\left( q\\right) =\\exp _{q}\\left( t\\xi \\right) q.\n\\label{flowPhi}\n\\end{equation}\n\\end{theorem}\n\n\\begin{remark}\nThe expression (\\ref{expqtalpha}) can be made a bit more general. Suppose \n\\mathbb{L}_{1}$ and $\\mathbb{L}_{2}$ are two loops and $\\alpha :\\mathbb{L\n_{1}\\longrightarrow \\mathbb{L}_{2}$ is a loop homomorphism. If we suppose \n\\exp _{\\left( 1\\right) }$ and $\\exp _{\\left( 2\\right) }$ are the exponential\nmaps on $\\mathbb{L}_{1}$ and $\\mathbb{L}_{2}$, respectively, then the\nfollowing diagram in Figure \\ref{loopexp}.\n\\end{remark}\n\n\\begin{center}\n\\begin{tikzcd}[sep=large] & T_{1}\\mathbb{L}_{1} \\arrow[r,\"\\alpha_{*}\"]\n\\arrow[d,\"\\func{exp}_{(1)}\"] & T_{1}\\mathbb{L}_{2}\n\\arrow[d,\"\\func{exp}_{(2)}\"] & \\\\ & \\mathbb{L}_{1} \\arrow[r,\"\\alpha\"] &\n\\mathbb{L}_{2} & \n\\end{tikzcd} \n\\captionof{figure}{Loop exponential maps.} \\label{loopexp}\n\\end{center}\n\n\\begin{remark}\nThe action of $\\Phi _{\\xi ,t}$ given by (\\ref{flowPhi}) looks like it\ndepends on $q$, however we easily see that for sufficiently small $t$, $\\exp\n_{q}\\left( t\\xi \\right) =\\exp _{r}\\left( t\\xi \\right) $ whenever $q$ and $r$\nare on the same integral curve generated by $\\xi $ (equivalently in the same\norbit of $\\Phi _{\\xi }$). This is consistent with the $1$-parameter subgroup\nproperty $\\Phi _{\\xi ,t}\\left( \\Phi _{\\xi ,s}\\left( q\\right) \\right) =\\Phi\n_{\\xi ,t+s}\\left( q\\right) $.\n\nIndeed, consider $r=\\exp _{q}\\left( s\\xi \\right) q$ and $\\tilde{r}=\\exp\n_{q}\\left( \\left( t+s\\right) \\xi \\right) q.$ These are points that lie along\nthe solution curve of (\\ref{floweq4}). On the other hand, consider the\nsolution of (\\ref{floweq4}) at with $p\\left( 0\\right) =r.$ This is then\ngiven by $\\hat{r}=\\exp _{r}\\left( t\\xi \\right) r.$ However, clearly by\nuniqueness of solutions of ODEs, $\\hat{r}=\\tilde{r}.$ So now, \n\\begin{eqnarray*}\n\\hat{r} &=&\\tilde{r} \\\\\n&=&\\exp _{q}\\left( \\left( t+s\\right) \\xi \\right) q=\\left( \\exp _{q}\\left(\nt\\xi \\right) \\circ _{q}\\exp _{q}\\left( s\\xi \\right) \\right) q \\\\\n&=&\\exp _{q}\\left( t\\xi \\right) \\left( \\exp _{q}\\left( s\\xi \\right) q\\right)\n\\\\\n&=&\\exp _{q}\\left( t\\xi \\right) r\n\\end{eqnarray*\nHence, we conclude that $\\exp _{q}\\left( t\\xi \\right) =\\exp _{r}\\left( t\\xi\n\\right) .$\n\\end{remark}\n\n\\begin{remark}\nSuppose $\\left( \\mathbb{L},\\cdot \\right) $ power left-alternative, i.e. \nx^{k}\\left( x^{l}q\\right) =x^{k+l}q$ for all $x,q\\in \\mathbb{L}$ and any\nintegers $k,l$. In particular this also means that $\\left( \\mathbb{L},\\cdot\n\\right) $ is power-associative and has the left inverse property. In\nparticular, powers of $x\\in \\mathbb{L}$ with respect to $\\circ _{q}$ are\nequal to powers of $x$ with respect to $\\cdot $. For any $q\\in \\mathbb{L}$, \n\\left( \\mathbb{L},\\circ _{q}\\right) $ is then also power left-alternative.\nNow the right-hand side of (\\ref{floweq3}) can be written as \n\\begin{equation}\n\\left( R_{\\tilde{p}\\left( t\\right) }^{\\left( q\\right) }\\right) _{\\ast }\\xi\n=\\left. \\frac{d}{ds}\\left( r\\left( s\\right) \\circ _{q}\\tilde{p}\\left(\nt\\right) \\right) \\right\\vert _{s=0}  \\label{floweq3a}\n\\end{equation\nwhere $r\\left( s\\right) $ is a curve with $r\\left( 0\\right) =1$ and \nr^{\\prime }\\left( 0\\right) =\\xi $, so we may take $r\\left( s\\right) =\\tilde{\n}\\left( s\\right) .$ Suppose there exist integers $n,k$ and a real number $h\n, such that $t=nh$ and $s=kh$. Then \n\\begin{eqnarray*}\n\\tilde{p}\\left( s\\right) \\circ _{q}\\tilde{p}\\left( t\\right) &=&\\tilde{p\n\\left( kh\\right) \\circ _{q}\\tilde{p}\\left( nh\\right) \\\\\n&=&\\left( \\tilde{p}\\left( h\\right) ^{k}\\cdot \\tilde{p}\\left( h\\right)\n^{n}q\\right) \/q \\\\\n&=&\\tilde{p}\\left( h\\right) ^{k+n}=\\tilde{p}\\left( kh\\right) \\tilde{p}\\left(\nnh\\right) \\\\\n&=&\\tilde{p}\\left( s\\right) \\tilde{p}\\left( t\\right) .\n\\end{eqnarray*\nThis is independent of $n$ and $k$, and is hence true for any $s,t$. Thus we\nfind that (\\ref{floweq3a}) is equal to the right-hand side of (\\ref{floweq\n), so $\\tilde{p}$ actually satisfies the same equation as $p,$ so by\nuniqueness of solutions $\\tilde{p}=p$. Hence, in this case, $\\exp _{q}=\\exp \n. In general however, the exponential map will not be unique and will depend\non the choice of $q.$\n\\end{remark}\n\n\\subsection{Tangent algebra}\n\n\\label{secTangent}Suppose $\\xi ,\\gamma \\in T_{1}\\mathbb{L}$ and let $X=\\rho\n\\left( \\xi \\right) $ and $Y=\\rho \\left( \\gamma \\right) $ be the\ncorresponding right fundamental vector fields on $\\mathbb{L}$. Then, recall\nthat the vector field Lie bracket of $X$ and $Y$ is given by \n\\begin{equation}\n\\left[ X,Y\\right] _{p}=\\left. \\frac{d}{dt}\\left( \\left( \\Phi\n_{t}^{-1}\\right) _{\\ast }\\left( Y_{\\Phi _{t}\\left( p\\right) }\\right) \\right)\n\\right\\vert _{t=0},  \\label{vecbracket}\n\\end{equation\nwhere $\\Phi _{t}=\\Phi \\left( \\xi ,t\\right) $ is the flow generated by $X$.\nFor sufficiently small $t$, we have $\\Phi _{t}\\left( p\\right) =\\exp\n_{p}\\left( t\\xi \\right) p,$ and thus \n\\begin{equation*}\nY_{\\Phi _{t}\\left( p\\right) }=\\left( R_{\\exp _{p}\\left( t\\xi \\right)\np}\\right) _{\\ast }\\gamma .\n\\end{equation*\nHence \n\\begin{equation}\n\\left( \\Phi _{t}^{-1}\\right) _{\\ast }\\left( Y_{\\Phi _{t}\\left( p\\right)\n}\\right) =\\left( L_{\\exp _{p}\\left( t\\xi \\right) }^{-1}\\circ R_{\\exp\n_{p}\\left( t\\xi \\right) p}\\right) _{\\ast }\\gamma .  \\label{Phinegt}\n\\end{equation\nNow right translating back to $T_{1}\\mathbb{L}$, we obtain \n\\begin{equation}\n\\left( R_{p}^{-1}\\right) _{\\ast }\\left[ X,Y\\right] _{p}=\\left. \\frac{d}{dt\n\\left( \\left( R_{p}^{-1}\\circ L_{\\exp _{p}\\left( t\\xi \\right) }^{-1}\\circ\nR_{\\exp _{p}\\left( t\\xi \\right) p}\\right) _{\\ast }\\gamma \\right) \\right\\vert\n_{t=0}.  \\label{Rpbrack0}\n\\end{equation\nIn general, let $q,x,y\\in \\mathbb{L},$ then \n\\begin{eqnarray*}\n\\left( R_{p}^{-1}\\circ L_{x}^{-1}\\circ R_{yp}\\right) q &=&\\faktor{\\left(\nx\\backslash \\left( q\\cdot yp\\right) \\right)} {p} \\\\\n&=&\\faktor{\\left( x\\backslash \\left( \\left( q\\cdot yp\\right) \/p\\cdot\np\\right) \\right)}{p} \\\\\n&=&x\\backslash _{p}\\left( q\\circ _{p}y\\right) \\\\\n&=&\\left( \\left( L_{x}^{\\left( p\\right) }\\right) ^{-1}\\circ R_{y}^{\\left(\np\\right) }\\right) q,\n\\end{eqnarray*\nwhere we have used (\\ref{rprodqleft}). Hence (\\ref{Rpbrack0}) become\n\\begin{eqnarray}\n\\left( R_{p}^{-1}\\right) _{\\ast }\\left[ X,Y\\right] _{p} &=&\\left. \\frac{d}{d\n}\\left( \\left( \\left( L_{\\exp _{p}\\left( t\\xi \\right) }^{\\left( p\\right)\n}\\right) ^{-1}\\circ R_{\\exp _{p}\\left( t\\xi \\right) }^{\\left( p\\right)\n}\\right) _{\\ast }\\gamma \\right) \\right\\vert _{t=0}  \\notag \\\\\n&=&\\left. \\frac{d}{dt}\\left( \\left( \\func{Ad}_{\\exp _{p}\\left( t\\xi \\right)\n}^{\\left( p\\right) }\\right) _{\\ast }^{-1}\\gamma \\right) \\right\\vert ;_{t=0} \n\\notag \\\\\n&=&-\\left. \\frac{d}{dt}\\left( \\left( \\func{Ad}_{\\exp _{p}\\left( t\\xi \\right)\n}^{\\left( p\\right) }\\right) _{\\ast }\\gamma \\right) \\right\\vert _{t=0}  \\notag\n\\\\\n&=&-\\left. d_{\\xi }\\left( \\func{Ad}^{\\left( p\\right) }\\right) _{\\ast\n}\\right\\vert _{1}\\left( \\gamma \\right)  \\label{brackdtAd}\n\\end{eqnarray\nHere, $\\left( \\func{Ad}_{x}^{\\left( p\\right) }\\right) _{\\ast }$ denotes the\ninduced adjoint action of $\\mathbb{L}$ on $T_{1}\\mathbb{L}.$ As remarked\nearlier, this is not an action in the sense of group actions. Similarly, as\nfor Lie groups and Lie algebras, we can also think of $\\left( \\func{Ad\n^{\\left( p\\right) }\\right) _{\\ast }:\\mathbb{L}\\longrightarrow \\func{End\n\\left( T_{1}\\mathbb{L}\\right) $, and then (\\ref{brackdtAd}) is just the\ndifferential of this map at $1\\in \\mathbb{L}$ in the direction $\\xi \\in T_{1\n\\mathbb{L}$. The differential of $\\left( \\func{Ad}^{\\left( p\\right) }\\right)\n_{\\ast }$ at an arbitrary point in $\\mathbb{L}$ is given in Lemma \\re\n{lemdtAd}. This now allows us to define the tangent adjoint map $\\func{ad\n^{\\left( p\\right) }$ on $T_{1}\\mathbb{L}.$\n\n\\begin{definition}\nFor any $\\xi ,\\gamma \\in T_{1}\\mathbb{L},$ the tangent adjoint map $\\func{ad\n_{\\xi }^{\\left( p\\right) }:T_{1}\\mathbb{L}\\longrightarrow T_{1}\\mathbb{L}$\nis defined as \n\\begin{equation}\n\\func{ad}_{\\xi }^{\\left( p\\right) }\\left( \\gamma \\right) =\\left. d_{\\xi\n}\\left( \\func{Ad}^{\\left( p\\right) }\\right) _{\\ast }\\right\\vert _{1}\\left(\n\\gamma \\right) =-\\left( R_{p}^{-1}\\right) _{\\ast }\\left[ X,Y\\right] _{p}.\n\\label{ladpx}\n\\end{equation}\n\\end{definition}\n\nThe negative sign in (\\ref{ladpx}) is there to be consistent with the\ncorresponding definitions for Lie groups for right-invariant vector fields.\nWe then define the $p$-bracket $\\left[ \\cdot ,\\cdot \\right] ^{\\left(\np\\right) }$ on $T_{1}\\mathbb{L}$ as \n\\begin{equation}\n\\left[ \\xi ,\\gamma \\right] ^{\\left( p\\right) }=\\func{ad}_{\\xi }^{\\left(\np\\right) }\\left( \\gamma \\right) .  \\label{T1Lbrack}\n\\end{equation\nFrom (\\ref{ladpx}) it is clear that it's skew-symmetric in $\\xi $ and \n\\gamma $. Equivalently, we can say \n\\begin{equation}\n\\left[ \\left( R_{p}^{-1}\\right) _{\\ast }X_{p},\\left( R_{p}^{-1}\\right)\n_{\\ast }Y_{p}\\right] ^{\\left( p\\right) }=-\\left( R_{p}^{-1}\\right) _{\\ast \n\\left[ X,Y\\right] _{p}.  \\label{T1Lbrack2}\n\\end{equation}\n\n\\begin{definition}\nThe vector space $T_{1}\\mathbb{L}$ together with the bracket $\\left[ \\cdot\n,\\cdot \\right] ^{\\left( p\\right) }$ is the \\emph{tangent algebra }or \n\\mathbb{L}$\\emph{-algebra }$\\mathfrak{l}^{\\left( p\\right) }$ of $\\left( \n\\mathbb{L},\\circ _{p}\\right) $.\n\\end{definition}\n\nThis is obviously a generalization of a Lie algebra. However, since now\nthere is a bracket $\\left[ \\cdot ,\\cdot \\right] ^{\\left( p\\right) }$\ncorresponding to each point $p\\in \\mathbb{L},$ it does not make sense to try\nand express $\\left[ \\left[ \\cdot ,\\cdot \\right] ^{\\left( p\\right) },\\cdot\n\\right] ^{\\left( p\\right) }$ in terms of Lie brackets of corresponding\nvector fields. Hence, the Jacobi identity for $\\left[ \\cdot ,\\cdot \\right]\n^{\\left( p\\right) }$ cannot be inferred, as expected. From (\\ref{T1Lbrack2\n), we cannot even infer that the bracket of two right fundamental vector\nfields is again a right fundamental vector field. In fact, at each point $p$\nit will be the pushforward of the bracket on $T_{1}\\mathbb{L}$ with respect\nto $p.$ Overall, we can summarize properties of the bracket in the theorem\nbelow.\n\n\\begin{theorem}\nLet $\\xi ,\\gamma \\in T_{1}\\mathbb{L}$ and suppose $X=\\rho \\left( \\xi \\right) \n$ and $Y=\\rho \\left( \\gamma \\right) $ are the corresponding right\nfundamental vector fields on $\\mathbb{L}$. Then, for any $p\\in \\mathbb{L}$, \n\\begin{equation}\n\\left[ \\xi ,\\gamma \\right] ^{\\left( p\\right) }=\\func{ad}_{\\xi }^{\\left(\np\\right) }\\left( \\gamma \\right) =\\left. \\frac{d}{dt}\\left( \\left( \\func{Ad\n_{\\exp \\left( t\\xi \\right) }^{\\left( p\\right) }\\right) _{\\ast }\\gamma\n\\right) \\right\\vert _{t=0}=-\\left( R_{p}^{-1}\\right) _{\\ast }\\left[ X,\n\\right] _{p},  \\label{Rpbrack}\n\\end{equation\nand moreover, \n\\begin{eqnarray}\n\\left[ \\xi ,\\gamma \\right] ^{\\left( p\\right) } &=&\\left. \\frac{d^{2}}\ndtd\\tau }\\left[ \\exp \\left( t\\xi \\right) ,\\exp \\left( \\tau \\gamma \\right)\n\\right] ^{\\left( \\mathbb{L},\\circ _{p}\\right) }\\right\\vert _{t,\\tau =0} \n\\notag \\\\\n&=&\\left. \\frac{d^{2}}{dtd\\tau }\\exp \\left( t\\xi \\right) \\circ _{p}\\exp\n\\left( \\tau \\gamma \\right) \\right\\vert _{t,\\tau =0}  \\label{brack2deriv} \\\\\n&&-\\left. \\frac{d^{2}}{dtd\\tau }\\exp \\left( \\tau \\gamma \\right) \\circ\n_{p}\\exp \\left( t\\xi \\right) \\right\\vert _{t,\\tau =0}.  \\notag\n\\end{eqnarray\nHere $\\left[ \\cdot ,\\cdot \\right] ^{\\left( p\\right) }$ is the $\\mathbb{L}\n-algebra bracket on $\\mathfrak{l}^{\\left( p\\right) }$, $\\left[ \\cdot ,\\cdot\n\\right] _{p}$ refers to the value of the vector field Lie bracket at $p\\in \n\\mathbb{L}$, and $\\left[ \\cdot ,\\cdot \\right] ^{\\left( \\mathbb{L},\\circ\n_{p}\\right) }$ is the loop commutator (\\ref{loopcomm2}) on $\\left( \\mathbb{L\n,\\circ _{p}\\right) .$\n\\end{theorem}\n\n\\begin{proof}\nWe have already shown (\\ref{Rpbrack}), so let us prove (\\ref{brack2deriv}).\nRecall from (\\ref{loopcomm2}) that \n\\begin{equation}\n\\left[ \\exp \\left( t\\xi \\right) ,\\exp \\left( \\tau \\gamma \\right) \\right]\n^{\\left( \\mathbb{L},\\circ _{p}\\right) }=\\func{Ad}_{\\exp \\left( t\\xi \\right)\n}^{\\left( p\\right) }\\left( \\exp \\left( \\tau \\gamma \\right) \\right) \/_{p}\\exp\n\\left( \\tau \\gamma \\right) .  \\label{commexp}\n\\end{equation\nDifferentiating (\\ref{commexp}) with respect to $\\tau $ and evaluating at \n\\tau =0$ using Lemma \\ref{lemQuotient} gives \n\\begin{eqnarray}\n\\left. \\frac{d}{d\\tau }\\left[ \\exp \\left( t\\xi \\right) ,\\exp \\left( \\tau\n\\gamma \\right) \\right] ^{\\left( \\mathbb{L},\\circ _{p}\\right) }\\right\\vert\n_{\\tau =0} &=&\\left. \\frac{d}{d\\tau }\\func{Ad}_{\\exp \\left( t\\xi \\right)\n}^{\\left( p\\right) }\\left( \\exp \\left( \\tau \\gamma \\right) \\right)\n\\right\\vert _{\\tau =0}  \\notag \\\\\n&&-\\left. \\frac{d}{d\\tau }\\exp \\left( \\tau \\gamma \\right) \\right\\vert _{\\tau\n=0}  \\notag \\\\\n&=&\\left( \\func{Ad}_{\\exp \\left( t\\xi \\right) }^{\\left( p\\right) }\\right)\n_{\\ast }\\gamma -\\tau\n\\end{eqnarray\nwhere we have also used the definition of $\\exp _{p}$ via (\\ref{floweq3}).\nThis gives us the first part of (\\ref{brack2deriv}). Now, using Lemma \\re\n{lemQuotient} again, we can differentiate $\\left( \\func{Ad}_{\\exp \\left(\nt\\xi \\right) }^{\\left( p\\right) }\\right) _{\\ast }\\gamma $ with respect to $t$\nto get the second part\n\\begin{eqnarray*}\n\\left. \\frac{d}{dt}\\left( \\left( \\func{Ad}_{\\exp \\left( t\\xi \\right)\n}^{\\left( p\\right) }\\right) _{\\ast }\\gamma \\right) \\right\\vert _{t=0}\n&=&\\left. \\frac{d^{2}}{dtd\\tau }\\left( \\left( \\exp \\left( t\\xi \\right) \\circ\n_{p}\\exp \\left( \\tau \\gamma \\right) \\right) \/_{p}\\exp \\left( t\\xi \\right)\n\\right) \\right\\vert _{t,\\tau =0} \\\\\n&=&\\left. \\frac{d^{2}}{dtd\\tau }\\left( \\exp \\left( t\\xi \\right) \\circ\n_{p}\\exp \\left( \\tau \\gamma \\right) \\right) \\right\\vert _{t,\\tau =0} \\\\\n&&-\\left. \\frac{d^{2}}{dtd\\tau }\\exp \\left( \\tau \\gamma \\right) \\circ\n_{p}\\exp \\left( t\\xi \\right) \\right\\vert _{t,\\tau =0}.\n\\end{eqnarray*}\n\\end{proof}\n\n\\begin{remark}\nApplying (\\ref{brack2deriv}) to the Moufang loop of unit octonions and the\ncorresponding $\\mathbb{L}$-algebra of imaginary octonions shows that as\nexpected, the bracket on the $\\mathbb{L}$-algebra coincides with the\ncommutator of imaginary octonions in the algebra of octonions.\n\\end{remark}\n\nAlthough $\\mathbb{L}$ and $\\mathfrak{l}$ are not in general a Lie group and\na Lie algebra, there are analogs of actions of these spaces on one another,\nwhich we summarize below.\n\nLet $s\\in \\mathbb{\\mathring{L}},$ $A\\in \\mathbb{L}$, and $\\xi ,\\eta \\in \n\\mathfrak{l},$ then we have the following:\n\n\\begin{enumerate}\n\\item Action of $\\mathbb{L}$ on $\\mathbb{\\mathring{L}}$: $A\\cdot s=As.$\n\n\\item Adjoint action of $\\left( \\mathbb{L},\\circ _{s}\\right) $ on $\\mathbb{L}\n$: $A\\cdot B=\\func{Ad}_{A}^{\\left( s\\right) }\\left( B\\right) =\\left( A\\circ\n_{s}B\\right) \/_{s}A.$\n\n\\item Action of $\\left( \\mathbb{L},\\circ _{s}\\right) $ on $\\mathfrak{l}$: \nA\\cdot \\xi =\\left( \\func{Ad}_{A}^{\\left( s\\right) }\\right) _{\\ast }\\xi .$\n\n\\item Action of $\\mathfrak{l}^{\\left( s\\right) }$ on itself: $\\xi \\cdot\n_{s}\\eta =\\left[ \\xi ,\\eta \\right] ^{\\left( s\\right) }.$\n\n\\item Action of $\\mathfrak{l}$ on $\\mathbb{\\mathring{L}}$: $\\xi \\cdot\ns=\\left( R_{s}\\right) _{\\ast }\\xi =\\left. \\frac{d}{dt}\\exp _{s}\\left( t\\xi\n\\right) s\\right\\vert _{t=0}.$\n\\end{enumerate}\n\n\\begin{remark}\nThere may be some confusion about notation because we will sometimes\nconsider the same objects but in different categories. Generally, for the\nloop $\\mathbb{L}$, the notation \\textquotedblleft $\\mathbb{L}\n\\textquotedblright\\ will denote the underlying set, the underlying smooth\nmanifold, the loop, and the $G$-set with the partial action of $\\Psi\n^{R}\\left( \\mathbb{L}\\right) .$ Similarly, $\\mathbb{\\mathring{L}}$ will\ndenote the same underlying set, the same underlying smooth manifold, but\nwill be different as a $G$-set - it has the full action of $\\Psi ^{R}\\left( \n\\mathbb{L}\\right) .$ Since $\\mathbb{L}$ and $\\mathbb{\\mathring{L}}$ are\nidentical as smooth manifolds, they have the same tangent space at $1$.\nGenerally, we will only refer to $\\mathbb{\\mathring{L}}$ if we need to\nemphasize the group action. For the $\\mathbb{L}$-algebra, the notation\n\\textquotedblleft $\\mathfrak{l}$\\textquotedblright\\ will denote both the\nunderlying vector space, and the vector space with the algebra structure on \nT_{1}\\mathbb{L}$ induced from the loop $\\mathbb{L}.$ For different values of \n$p\\in \\mathbb{L}$, $\\mathfrak{l}^{\\left( p\\right) }$ is identical to \n\\mathfrak{l}$ as a vector space, but has a different algebra structure. We\nwill use the notation $\\mathfrak{l}^{\\left( p\\right) }$ to emphasize the\nalgebra structure.\n\\end{remark}\n\n\\subsection{Structural equation}\n\n\\label{sectStruct}Let us now define an analog of the Maurer-Cartan form on\nright fundamental vector fields. Given $p\\in \\mathbb{L}$ and and $\\xi \\in \n\\mathfrak{l},$ define $\\theta _{p}$ to be \n\\begin{equation}\n\\theta _{p}\\left( \\rho \\left( \\xi \\right) _{p}\\right) =\\left(\nR_{p}^{-1}\\right) _{\\ast }\\rho \\left( \\xi \\right) _{p}=\\xi .  \\label{MCloop}\n\\end{equation\nThus, this is an $\\mathfrak{l}$-valued $1$-form. The right fundamental\nvector fields still form a global frame for $T\\mathbb{L},$ so this is\nsufficient to define the $1$-form $\\theta .$ Just as the right fundamental\nvector field $\\rho \\left( \\xi \\right) $ is generally not right-invariant,\nneither is $\\theta .$ Indeed, let $q\\in \\mathbb{L}$ and consider $\\left(\nR_{q}^{-1}\\right) ^{\\ast }\\theta .$ Then, given $X_{p}=\\left( R_{p}\\right)\n_{\\ast }\\xi \\in T_{p}\\mathbb{L}$ \n\\begin{eqnarray}\n\\left( \\left( R_{q}^{-1}\\right) ^{\\ast }\\theta \\right) _{p}\\left(\nX_{p}\\right) &=&\\theta _{p\/q}\\left( \\left( R_{q}^{-1}\\circ R_{p}\\right)\n_{\\ast }\\xi \\right)  \\notag \\\\\n&=&\\left( R_{p\/q}^{-1}\\circ R_{q}^{-1}\\circ R_{p}\\right) _{\\ast }\\xi  \\notag\n\\\\\n&=&\\left( R_{p\/q}^{-1}\\circ R_{p\/q}^{\\left( q\\right) }\\right) _{\\ast }\\xi\n\\label{thetarighttr}\n\\end{eqnarray\nwhere same idea as in (\\ref{rightvect}) was used.\n\nNow consider $d\\theta .$ Generally, for a $1$-form$,$ we have \n\\begin{equation}\nd\\theta \\left( X,Y\\right) =X\\theta \\left( Y\\right) -Y\\theta \\left( X\\right)\n-\\theta \\left( \\left[ X,Y\\right] \\right) .\n\\end{equation\nSuppose $X,$ $Y$ are right fundamental, then from (\\ref{T1Lbrack2}), we get \n\\begin{equation}\n\\left( d\\theta \\right) _{p}\\left( X,Y\\right) -\\left[ \\theta \\left( X\\right)\n,\\theta \\left( Y\\right) \\right] ^{\\left( p\\right) }=0.  \\label{MCequation1}\n\\end{equation\nHowever, since right fundamental vector fields span the space of vector\nfields on $\\mathbb{L}$, (\\ref{MCequation1}) is true for any vector fields,\nand we obtain the following analogue of the Maurer-Cartan equation.\n\n\\begin{theorem}\n\\label{thmMC}Let $p\\in \\mathbb{L}$ and let $\\left[ \\cdot ,\\cdot \\right]\n^{\\left( p\\right) }$ be bracket on $\\mathfrak{l}^{\\left( p\\right) }$. Then, \n\\theta $ satisfies the following equation at $p$: \n\\begin{equation}\n\\left( d\\theta \\right) _{p}-\\frac{1}{2}\\left[ \\theta ,\\theta \\right]\n^{\\left( p\\right) }=0,  \\label{MCequation2}\n\\end{equation\n\\qquad where $\\left[ \\theta ,\\theta \\right] ^{\\left( p\\right) }$ is the\nbracket of $\\mathbb{L}$-algebra-valued $1$-forms such that for any $X,Y\\in\nT_{p}\\mathbb{L}$, $\\frac{1}{2}\\left[ \\theta ,\\theta \\right] ^{\\left(\np\\right) }\\left( X,Y\\right) =\\left[ \\theta \\left( X\\right) ,\\theta \\left(\nY\\right) \\right] ^{\\left( p\\right) }.$\n\nLet $q\\in \\mathbb{L}$ and $\\theta ^{\\left( q\\right) }=\\left( R_{q}\\right)\n^{\\ast }\\theta ,$ then $\\theta ^{\\left( q\\right) }$ satisfies \n\\begin{equation}\n\\left( d\\theta ^{\\left( q\\right) }\\right) _{p}-\\frac{1}{2}\\left[ \\theta\n^{\\left( q\\right) },\\theta ^{\\left( q\\right) }\\right] ^{\\left( pq\\right) }=0,\n\\label{MCequation3}\n\\end{equation\nwhere $\\left[ \\cdot ,\\cdot \\right] ^{\\left( pq\\right) }$ is the bracket on \n\\mathfrak{l}^{\\left( pq\\right) }.$\n\\end{theorem}\n\n\\begin{proof}\nThe first part already follows from (\\ref{MCequation1}). For the second\npart, by applying $\\left( R_{q}\\right) ^{\\ast }$ to (\\ref{MCequation2}) we\neasily see that $\\theta ^{\\left( q\\right) }$ satisfies (\\ref{MCequation2})\nwith the translated bracket $\\left[ \\cdot ,\\cdot \\right] ^{\\left( pq\\right)\n} $, and hence we get (\\ref{MCequation3}).\n\\end{proof}\n\n\\begin{remark}\nThe $1$-form $\\theta ^{\\left( q\\right) }$ can be seen as translating a\nvector in $T_{p}\\mathbb{L}$ by $R_{q}$ to $T_{pq}\\mathbb{L}$, and then by \nR_{pq}^{-1}$ back to $\\mathfrak{l}.$ However, given the identity \nxq\/pq=x\/_{q}p$, we see that $\\theta ^{\\left( q\\right) }$ is just the loop\nMaurer-Cartan form in $\\left( \\mathbb{L},\\circ _{q}\\right) .$\n\\end{remark}\n\nThe obvious key difference with the Lie group picture here is that the\nbracket in (\\ref{MCequation2}) non-constant on $\\mathbb{L},$ i.e. given a\nbasis, the structure \\textquotedblleft constants\\textquotedblright\\ would no\nlonger be constants. In particular, the Jacobi identity is the integrability\ncondition for the Maurer-Cartan equation on Lie groups, however here we see\nthat the right-hand side of the Jacobi identity is related to a ternary form\ngiven by the derivative of the bracket. For any $\\xi ,\\eta ,\\gamma \\in \n\\mathfrak{l}^{\\left( p\\right) }$, define \n\\begin{equation}\n\\func{Jac}^{\\left( p\\right) }\\left( \\xi ,\\eta ,\\gamma \\right) =\\left[ \\xi \n\\left[ \\eta ,\\gamma \\right] ^{\\left( p\\right) }\\right] ^{\\left( p\\right) }\n\\left[ \\eta ,\\left[ \\gamma ,\\xi \\right] ^{\\left( p\\right) }\\right] ^{\\left(\np\\right) }+\\left[ \\gamma ,\\left[ \\xi ,\\eta \\right] ^{\\left( p\\right) }\\right]\n^{\\left( p\\right) }.  \\label{Jac}\n\\end{equation\nWe also need the following definition.\n\n\\begin{definition}\nDefine the \\emph{bracket function }$b:\\mathbb{\\mathring{L}}\\longrightarrow \n\\mathfrak{l}\\otimes \\Lambda ^{2}\\mathfrak{l}^{\\ast }$ to be the map that\ntakes $p\\mapsto \\left[ \\cdot ,\\cdot \\right] ^{\\left( p\\right) }\\in \\mathfrak\nl}\\otimes \\Lambda ^{2}\\mathfrak{l}^{\\ast }$, so that $b\\left( \\theta ,\\theta\n\\right) $ is an $\\mathfrak{l}$-valued $2$-form on $\\mathbb{L}$, i.e. \nb\\left( \\theta ,\\theta \\right) \\in \\Omega ^{2}\\left( \\mathfrak{l}\\right) .$\n\\end{definition}\n\nLemma \\ref{lemAssoc} below will give the differential of $b$. The proof is\ngiven in Appendix \\ref{secAppendix}.\n\n\\begin{lemma}\n\\label{lemAssoc}For fixed $\\eta ,\\gamma \\in \\mathfrak{l},$ \n\\begin{equation}\n\\left. db\\right\\vert _{p}\\left( \\eta ,\\gamma \\right) =\\left[ \\eta ,\\gamma\n,\\theta _{p}\\right] ^{\\left( p\\right) }-\\left[ \\gamma ,\\eta ,\\theta _{p\n\\right] ^{\\left( p\\right) }\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{,}  \\label{db1}\n\\end{equation\nwhere $\\left[ \\cdot ,\\cdot ,\\cdot \\right] ^{\\left( p\\right) }$ is the \n\\mathbb{L}$\\emph{-algebra associator }on $\\mathfrak{l}^{\\left( p\\right) }$\ngiven by \n\\begin{eqnarray}\n\\left[ \\eta ,\\gamma ,\\xi \\right] ^{\\left( p\\right) } &=&\\left. \\frac{d^{3}}\ndtd\\tau d\\tau ^{\\prime }}\\exp \\left( \\tau \\eta \\right) \\circ _{p}\\left( \\exp\n\\left( \\tau ^{\\prime }\\gamma \\right) \\circ _{p}\\exp \\left( t\\xi \\right)\n\\right) \\right\\vert _{t,\\tau ,\\tau ^{\\prime }=0}  \\label{Lalgassoc} \\\\\n&&-\\left. \\frac{d^{3}}{dtd\\tau d\\tau ^{\\prime }}\\left( \\exp \\left( \\tau \\eta\n\\right) \\circ _{p}\\exp \\left( \\tau ^{\\prime }\\gamma \\right) \\right) \\circ\n_{p}\\exp \\left( t\\xi \\right) \\right\\vert _{t,\\tau ,\\tau ^{\\prime }=0}. \n\\notag\n\\end{eqnarray\nMoreover, \n\\begin{equation}\n\\left[ \\eta ,\\gamma ,\\xi \\right] ^{\\left( p\\right) }=\\left. \\frac{d^{3}}\ndtd\\tau d\\tau ^{\\prime }}\\left[ \\exp \\left( \\tau \\eta \\right) ,\\exp \\left(\n\\tau ^{\\prime }\\gamma \\right) ,\\exp \\left( t\\xi \\right) \\right] ^{\\left( \n\\mathbb{L},\\circ _{p}\\right) }\\right\\vert _{t,\\tau ,\\tau ^{\\prime }=0}\n\\label{Lalgassoc2}\n\\end{equation\nwhere $\\left[ \\cdot ,\\cdot ,\\cdot \\right] ^{\\left( \\mathbb{L},\\circ\n_{p}\\right) }$ is the loop associator on $\\left( \\mathbb{L},\\circ\n_{p}\\right) $ as defined by (\\ref{loopassoc2}).\n\\end{lemma}\n\nThe skew-symmetric combination of associators, as in (\\ref{db1}) will\nfrequently occur later on, so let us define for convenience \n\\begin{equation}\na_{p}\\left( \\eta ,\\gamma ,\\xi \\right) =\\left[ \\eta ,\\gamma ,\\xi \\right]\n^{\\left( p\\right) }-\\left[ \\gamma ,\\eta ,\\xi \\right] ^{\\left( p\\right) },\n\\label{ap}\n\\end{equation\nwhich we can can call the \\emph{left-alternating associator}, so in\nparticular, (\\ref{db1}) becomes \n\\begin{equation}\n\\left. db\\right\\vert _{p}\\left( \\eta ,\\gamma \\right) =a_{p}\\left( \\eta\n,\\gamma ,\\theta _{p}\\right) .  \\label{db2}\n\\end{equation}\n\nThe loop Maurer-Cartan equation can be rewritten as \n\\begin{equation}\nd\\theta =\\frac{1}{2}b\\left( \\theta ,\\theta \\right) ,  \\label{MC3}\n\\end{equation\nand hence we see that $b\\left( \\theta ,\\theta \\right) $ is an exact form, so\nin particular, $d\\left( b\\left( \\theta ,\\theta \\right) \\right) =0$. We will\nnow use this to derive a generalization of the Jacobi identity.\n\n\\begin{theorem}\n\\label{thmJacobi}The maps $a$ and $b$ satisfy the relation \n\\begin{equation}\nb\\left( \\theta ,b\\left( \\theta ,\\theta \\right) \\right) =a\\left( \\theta\n,\\theta ,\\theta \\right) .  \\label{Jac3}\n\\end{equation\nwhere wedge products are assumed. Equivalently, if $\\xi ,\\eta ,\\gamma \\in \n\\mathfrak{l}$ and $p\\in \\mathbb{L}$, the\n\\begin{equation}\n\\func{Jac}^{\\left( p\\right) }\\left( \\xi ,\\eta ,\\gamma \\right) =a_{p}\\left(\n\\xi ,\\eta ,\\gamma \\right) +a_{p}\\left( \\eta ,\\gamma ,\\xi \\right)\n+a_{p}\\left( \\gamma ,\\xi ,\\eta \\right) .  \\label{Jac2}\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nWe know that $d\\left( b\\left( \\theta ,\\theta \\right) \\right) =0,$ and thus,\nusing (\\ref{db1}) and (\\ref{MC3}), we have \n\\begin{eqnarray*}\n0 &=&d\\left( b\\left( \\theta ,\\theta \\right) \\right) \\\\\n&=&\\left( db\\right) \\left( \\theta ,\\theta \\right) +b\\left( d\\theta ,\\theta\n\\right) -b\\left( \\theta ,d\\theta \\right) \\\\\n&=&a\\left( \\theta ,\\theta ,\\theta \\right) -b\\left( \\theta ,b\\left( \\theta\n,\\theta \\right) \\right) .\n\\end{eqnarray*\nSo indeed, (\\ref{Jac3}) holds. Now let $X,Y,Z$ be vector fields on $\\mathbb{\n}$, such that $X=\\rho \\left( \\xi \\right) ,$ $Y=\\rho \\left( \\eta \\right) ,$ \nZ=\\rho \\left( \\gamma \\right) $. Then, $a\\left( \\theta ,\\theta ,\\theta\n\\right) _{p}\\left( X,Y,Z\\right) =2\\func{Jac}^{\\left( p\\right) }\\left( \\xi\n,\\eta ,\\gamma \\right) $ and $\\frac{1}{2}b\\left( \\theta ,b\\left( \\theta\n,\\theta \\right) \\right) _{p}\\left( X,Y,Z\\right) $ gives the right-hand side\nof (\\ref{Jac2}).\n\\end{proof}\n\n\\begin{remark}\nAn algebra $\\left( A,\\left[ \\cdot ,\\cdot \\right] ,\\left[ \\cdot ,\\cdot ,\\cdot\n\\right] \\right) $ with a skew-symmetric bracket $\\left[ \\cdot ,\\cdot \\right] \n$ and ternary multilinear bracket $\\left[ \\cdot ,\\cdot ,\\cdot \\right] $ that\nsatisfies (\\ref{Jac2}) is known as an \\emph{Akivis algebra} \\cit\n{Akivis1,ShestakovAkivis1}. If $\\left( \\mathbb{L},\\circ _{p}\\right) $ is\nleft-alternative, we find from (\\ref{Lalgassoc}) that for any $\\xi ,\\eta \\in \n\\mathfrak{l},$ $\\left[ \\xi ,\\xi ,\\eta \\right] ^{\\left( p\\right) }=0$, that\nis, the $\\mathbb{L}$-algebra associator on $\\mathfrak{l}^{\\left( p\\right) }$\nis skew-symmetric in the first two entries, and thus $a_{p}=2\\left[ \\cdot\n,\\cdot ,\\cdot \\right] ^{\\left( p\\right) }.$ If the algebra is alternative,\nthen $\\func{Jac}^{\\left( p\\right) }\\left( \\xi ,\\eta ,\\gamma \\right) =6\\left[\n\\xi ,\\eta ,\\gamma \\right] ^{\\left( p\\right) }.$ It is known \\cit\n{HofmannStrambach}, that conversely, to an alternative Akivis algebra, there\ncorresponds a unique, up to local isomorphism, local analytic alternative\nloop. If $\\left( \\mathbb{L},\\circ _{p}\\right) $ is a left Bol loop (so that\nit is left-alternative) then the corresponding algebra on $\\mathfrak{l\n^{\\left( p\\right) }$ will be a \\emph{Bol algebra}, where $\\left[ \\cdot\n,\\cdot \\right] ^{\\left( p\\right) }$ and $\\left[ \\cdot ,\\cdot ,\\cdot \\right]\n^{\\left( p\\right) }$ satisfy additional identities \\cit\n{Akivis1,OnishchikVinberg,SabininMikheev1985}. If $\\left( \\mathbb{L},\\circ\n_{p}\\right) $ is a Moufang loop (so in particular it is alternative), then\nthe associator is totally skew-symmetric and the algebra on $\\mathfrak{l\n^{\\left( p\\right) }$ is then a \\emph{Malcev\\ algebra}. It then satisfies in\naddition the following identity \\cite{Kuzmin1971,Malcev1955}\n\\begin{equation}\n\\left[ \\xi ,\\eta ,\\left[ \\xi ,\\gamma \\right] ^{\\left( p\\right) }\\right]\n^{\\left( p\\right) }=\\left[ \\left[ \\xi ,\\eta ,\\gamma \\right] ^{\\left(\np\\right) },\\xi \\right] ^{\\left( p\\right) }.  \\label{MalcevId}\n\\end{equation\nMoreover, all non-Lie simple Malcev algebras have been classified \\cit\n{Kuzmin1968b} - these are either the imaginary octonions over the real\nnumber, imaginary octonions over the complex numbers, or split octonions\nover the real numbers.\n\\end{remark}\n\nWe generally will not distinguish the notation between loop associators and \n\\mathbb{L}$-algebra associators. It should be clear from the context which\nis being used. Moreover, it will be convenient to define mixed associators\nbetween elements of $\\mathbb{L}$ and $\\mathfrak{l}$. For example, an $\\left( \n\\mathbb{L},\\mathbb{L},\\mathfrak{l}\\right) $-associator is defined for any \np,q\\in \\mathbb{L}$ and $\\xi \\in \\mathfrak{l}$ as \n\\begin{equation}\n\\left[ p,q,\\xi \\right] ^{\\left( s\\right) }=\\left( L_{p}^{\\left( s\\right)\n}\\circ L_{q}^{\\left( s\\right) }\\right) _{\\ast }\\xi -\\left( L_{p\\circ\n_{s}q}^{\\left( s\\right) }\\right) _{\\ast }\\xi \\in T_{p\\circ _{s}q}\\mathbb{L}\n\\label{pqxiassoc}\n\\end{equation\nand an $\\left( \\mathbb{L},\\mathfrak{l},\\mathfrak{l}\\right) $-associator is\ndefined for an $p\\in \\mathbb{L}$ and $\\eta ,\\xi \\in \\mathfrak{l}$ as \n\\begin{eqnarray}\n\\left[ p,\\eta ,\\xi \\right] ^{\\left( s\\right) } &=&\\left. \\frac{d}{dtd\\tau \n\\left( p\\circ _{s}\\left( \\exp \\left( t\\eta \\right) \\circ _{s}\\exp \\left(\n\\tau \\xi \\right) \\right) \\right) \\right\\vert _{t=0}  \\notag \\\\\n&&-\\left. \\frac{d}{dtd\\tau }\\left( \\left( p\\circ _{s}\\exp \\left( t\\eta\n\\right) \\right) \\circ _{s}\\exp \\left( \\tau \\xi \\right) \\right) \\right\\vert\n_{t=0},  \\label{etapxiassoc}\n\\end{eqnarray\nwhere we see that $\\left[ p,\\eta ,\\xi \\right] ^{\\left( s\\right) }\\in T_{p\n\\mathbb{L}.$ Similarly, for other combinations.\n\nLet us now consider the action of loop homomophisms on $\\mathbb{L}$-algebras.\n\n\\begin{lemma}\n\\label{lemAlgHom}Suppose $\\mathbb{L}_{1}$ and $\\mathbb{L}_{2}$ are two\nsmooth loops with tangent algebras at identity $\\mathfrak{l}_{1}\\ $and \n\\mathfrak{l}_{2}$, respectively. Let $\\alpha :\\mathbb{L}_{1}\\longrightarrow \n\\mathbb{L}_{2}$ be a smooth loop homomorphism. Then, $\\alpha _{\\ast }:$ \n\\mathfrak{l}_{1}\\longrightarrow \\mathfrak{l}_{2}$ is an $\\mathbb{L}$-algebra\nhomomorphism, i.e., for any $\\xi ,\\gamma \\in $ $\\mathfrak{l}_{1}$, \n\\begin{equation}\n\\alpha _{\\ast }\\left[ \\xi ,\\gamma \\right] ^{\\left( 1\\right) }=\\left[ \\alpha\n_{\\ast }\\xi ,\\alpha _{\\ast }\\gamma \\right] ^{\\left( 2\\right) },\n\\label{algebrahom}\n\\end{equation\nwhere $\\left[ \\cdot ,\\cdot \\right] ^{\\left( 1\\right) }$ and $\\left[ \\cdot\n,\\cdot \\right] ^{\\left( 2\\right) }$ are the corresponding brackets on \n\\mathfrak{l}_{1}\\ $and $\\mathfrak{l}_{2}$, respectively. Moreover, $\\alpha\n_{\\ast }$ is an associator homomorphism, i.e., for any $\\xi ,\\gamma ,\\eta\n\\in $ $\\mathfrak{l}_{1}$, \n\\begin{equation}\n\\alpha _{\\ast }\\left[ \\xi ,\\gamma ,\\eta \\right] ^{\\left( 1\\right) }=\\left[\n\\alpha _{\\ast }\\xi ,\\alpha _{\\ast }\\gamma ,\\alpha _{\\ast }\\eta \\right]\n^{\\left( 2\\right) }  \\label{akivishom}\n\\end{equation\nwhere $\\left[ \\cdot ,\\cdot ,\\cdot \\right] ^{\\left( 1\\right) }$ and $\\left[\n\\cdot ,\\cdot ,\\cdot \\right] ^{\\left( 2\\right) }$ are the corresponding\nternary brackets on $\\mathfrak{l}_{1}\\ $and $\\mathfrak{l}_{2}$, respectively.\n\\end{lemma}\n\n\\begin{proof}\nSuppose $\\exp _{\\left( 1\\right) }:$ $\\mathfrak{l}_{1}\\longrightarrow \\mathbb\nL}_{1}$ and $\\exp _{\\left( 2\\right) }:\\mathfrak{l}_{2}\\longrightarrow \n\\mathbb{L}_{2}$ are the corresponding exponential maps. Let $\\xi ,\\gamma \\in \n$ $\\mathfrak{l}_{1}$. We know from (\\ref{loopexp}) that \n\\begin{equation}\n\\alpha \\left( \\exp _{\\left( 1\\right) }\\xi \\right) =\\exp _{\\left( 2\\right)\n}\\left( \\alpha _{\\ast }\\xi \\right) .  \\label{homexp}\n\\end{equation\nFrom (\\ref{brack2deriv}), we have \n\\begin{equation*}\n\\left[ \\xi ,\\gamma \\right] ^{\\left( 1\\right) }=\\left. \\frac{d^{2}}{dtd\\tau \n\\exp _{\\left( 1\\right) }\\left( t\\xi \\right) \\exp _{\\left( 1\\right) }\\left(\n\\tau \\gamma \\right) \\right\\vert _{t,\\tau =0}-\\left. \\frac{d^{2}}{dtd\\tau \n\\exp _{\\left( 1\\right) }\\left( \\tau \\gamma \\right) \\exp _{\\left( 1\\right)\n}\\left( t\\xi \\right) \\right\\vert _{t,\\tau =0},\n\\end{equation*\nApplying $\\alpha _{\\ast }$ to $\\left[ \\xi ,\\gamma \\right] ^{\\left( 1\\right)\n} $, we find \n\\begin{eqnarray*}\n\\alpha _{\\ast }\\left[ \\xi ,\\gamma \\right] ^{\\left( 1\\right) } &=&\\left. \n\\frac{d^{2}}{dtd\\tau }\\alpha \\left( \\exp _{\\left( 1\\right) }\\left( t\\xi\n\\right) \\exp _{\\left( 1\\right) }\\left( \\tau \\gamma \\right) \\right)\n\\right\\vert _{t,\\tau =0} \\\\\n&&-\\left. \\frac{d^{2}}{dtd\\tau }\\alpha \\left( \\exp _{\\left( 1\\right) }\\left(\n\\tau \\gamma \\right) \\exp _{\\left( 1\\right) }\\left( t\\xi \\right) \\right)\n\\right\\vert _{t,\\tau =0}.\n\\end{eqnarray*\nHowever, since $\\alpha $ is a loop homomorphism, and using (\\ref{homexp}),\nwe have, \n\\begin{eqnarray*}\n\\alpha _{\\ast }\\left[ \\xi ,\\gamma \\right] ^{\\left( 1\\right) } &=&\\left. \n\\frac{d^{2}}{dtd\\tau }\\exp _{\\left( 2\\right) }\\left( t\\alpha _{\\ast }\\xi\n\\right) \\exp _{\\left( 1\\right) }\\left( \\tau \\alpha _{\\ast }\\gamma \\right)\n\\right\\vert _{t,\\tau =0} \\\\\n&&-\\left. \\frac{d^{2}}{dtd\\tau }\\exp _{\\left( 1\\right) }\\left( \\tau \\alpha\n_{\\ast }\\gamma \\right) \\exp _{\\left( 1\\right) }\\left( t\\alpha _{\\ast }\\xi\n\\right) \\right\\vert _{t,\\tau =0} \\\\\n&=&\\left[ \\alpha _{\\ast }\\xi ,\\alpha _{\\ast }\\gamma \\right] ^{\\left(\n2\\right) }.\n\\end{eqnarray*\nSimilarly, using the definition (\\ref{Lalgassoc}) for the $\\mathbb{L}\n-algebra associator, we obtain (\\ref{akivishom}).\n\\end{proof}\n\nIn particular, if $\\left( \\alpha ,p\\right) \\in \\Psi ^{R}\\left( \\mathbb{L\n\\right) $, then $\\alpha $ induces an $\\mathbb{L}$-algebra isomorphism \n\\alpha _{\\ast }:\\left( \\mathfrak{l,}\\left[ \\cdot ,\\cdot \\right] \\right)\n\\longrightarrow \\left( \\mathfrak{l,}\\left[ \\cdot ,\\cdot \\right] ^{\\left(\np\\right) }\\right) $. This shows that as long as $p$ is a companion of some\nsmooth right pseudoautomorphism, the corresponding algebras are isomorphic.\nMore generally, we have the following.\n\n\\begin{corollary}\n\\label{corLoppalghom}Suppose $h=\\left( \\alpha ,p\\right) \\in \\Psi ^{R}\\left( \n\\mathbb{L}\\right) $, and $q\\in \\mathbb{\\mathring{L}}$, then, for any $\\xi\n,\\eta ,\\gamma \\in \\mathfrak{l}$\n\\begin{subequations\n\\label{loopalghom} \n\\begin{eqnarray}\n\\alpha _{\\ast }\\left[ \\xi ,\\eta \\right] ^{\\left( q\\right) } &=&\\left[ \\alpha\n_{\\ast }\\xi ,\\alpha _{\\ast }\\eta \\right] ^{h\\left( q\\right) }\n\\label{loopalghom1} \\\\\n\\alpha _{\\ast }\\left[ \\xi ,\\eta ,\\gamma \\right] ^{\\left( q\\right) } &=&\\left[\n\\alpha _{\\ast }\\xi ,\\alpha _{\\ast }\\eta ,\\alpha _{\\ast }\\gamma \\right]\n^{h\\left( q\\right) }.  \\label{loopalghom2}\n\\end{eqnarray\n\\end{subequations\n\\end{corollary}\n\n\\begin{proof}\nSince $h=\\left( \\alpha ,p\\right) $ is right pseudo-automorphism of $\\mathbb{\n},$ by Lemma \\ref{lemPseudoHom}, it induces a loop homomorphism $\\alpha\n:\\left( \\mathbb{L},q\\right) \\longrightarrow \\left( \\mathbb{L},h\\left(\nq\\right) \\right) ,$ and thus by Lemma \\ref{lemAlgHom}, $\\alpha _{\\ast }\n\\mathfrak{l}^{\\left( q\\right) }\\longrightarrow \\mathfrak{l}^{\\left( h\\left(\nq\\right) \\right) }$ is a loop algebra homomorphism. Thus (\\ref{loopalghom})\nfollows.\n\\end{proof}\n\n\\begin{remark}\nIn general, Akivis algebras are not fully defined by the binary and ternary\nbrackets, as shown in \\cite{ShestakovUmirbaev}. Indeed, for a fuller\npicture, a more complicated structure of a \\emph{Sabinin algebra }is needed \n\\cite{SabininBook}.\n\\end{remark}\n\nGenerally, we see that $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ acts on \n\\mathfrak{l}$ via pushforwards of the action on $\\mathbb{L}$, i.e. for $h\\in\n\\Psi ^{R}\\left( \\mathbb{L}\\right) $ and $\\xi \\in \\mathfrak{l}$, we have \nh\\cdot \\xi =\\left( h^{\\prime }\\right) _{\\ast }\\xi $.\n\nThe expressions (\\ref{loopalghom}) show that the maps $b\\in C^{\\infty\n}\\left( \\mathbb{\\mathring{L}},\\Lambda ^{2}\\mathfrak{l}^{\\ast }\\otimes \n\\mathfrak{l}\\right) $ and $a\\in C^{\\infty }\\left( \\mathbb{\\mathring{L}\n,\\left( \\otimes ^{3}\\mathfrak{l}^{\\ast }\\right) \\otimes \\mathfrak{l}\\right) $\nthat correspond to the brackets are equivariant maps with respect to the\naction of $\\Psi ^{R}\\left( \\mathbb{L}\\right) .$ Now suppose $s\\in \\mathbb\n\\mathring{L}},$ and denote $b_{s}=b\\left( s\\right) \\in \\Lambda ^{2}\\mathfrak\nl}^{\\ast }\\otimes \\mathfrak{l}$. Then the equivariance of $b$ means that the\nstabilizer $\\func{Stab}_{\\Psi ^{R}\\left( \\mathbb{L}\\right) }\\left(\nb_{s}\\right) $ in $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ of $b_{s}$ is\nequivalent to the the set of all $h\\in \\Psi ^{R}\\left( \\mathbb{L}\\right) $\nfor which $b_{h\\left( s\\right) }=b_{s}.$ In particular, $\\func{Stab}_{\\Psi\n^{R}\\left( \\mathbb{L}\\right) }\\left( b_{s}\\right) $ is a Lie subgroup of \n\\Psi ^{R}\\left( \\mathbb{L}\\right) $, and clearly $\\func{Aut}\\left( \\mathbb{L\n,\\circ _{s}\\right) =\\func{Stab}_{\\Psi ^{R}\\left( \\mathbb{L}\\right) }\\left(\ns\\right) \\subset $ $\\func{Stab}_{\\Psi ^{R}\\left( \\mathbb{L}\\right) }\\left(\nb_{s}\\right) .$ Moreover, note that if $h=\\left( \\gamma ,C\\right) \\in \\func\nAut}\\left( \\mathbb{L},\\circ _{s}\\right) \\times \\mathcal{N}^{R}\\left( \\mathbb\nL},\\circ _{s}\\right) $, then we still have $b_{h\\left( s\\right) }=b_{s}.$\nSo, we can say that the corresponding subgroup $\\iota _{1}\\left( \\func{Aut\n\\left( \\mathbb{L},\\circ _{s}\\right) \\right) \\ltimes \\iota _{2}\\left( \n\\mathcal{N}^{R}\\left( \\mathbb{L},\\circ _{s}\\right) \\right) \\subset \\Psi\n^{R}\\left( \\mathbb{L}\\right) $ is contained in $\\func{Stab}_{\\Psi ^{R}\\left( \n\\mathbb{L}\\right) }\\left( b_{s}\\right) .$ Hence, as long as $\\mathcal{N\n^{R}\\left( \\mathbb{L},\\circ _{s}\\right) $ is non-trivial, $\\func{Stab}_{\\Psi\n^{R}\\left( \\mathbb{L}\\right) }\\left( b_{s}\\right) $ is strictly greater than \n$\\func{Aut}\\left( \\mathbb{L},\\circ _{s}\\right) .$ Similarly for $a$.\n\nLet us now also consider how the bracket $\\left[ \\cdot ,\\cdot \\right] $ is\ntransformed by $\\left( \\func{Ad}_{p}^{\\left( s\\right) }\\right) _{\\ast }.$\n\n\\begin{theorem}\nSuppose $s\\in \\mathbb{\\mathring{L}}$ $,\\ p\\in \\mathbb{L}$, and $\\xi ,\\eta\n,\\gamma \\in \\mathfrak{l}.$ Then \n\\begin{eqnarray}\n\\left( \\func{Ad}_{p}^{\\left( s\\right) }\\right) _{\\ast }\\left[ \\xi ,\\eta\n\\right] ^{\\left( s\\right) } &=&\\left[ \\left( \\func{Ad}_{p}^{\\left( s\\right)\n}\\right) _{\\ast }\\xi ,\\left( \\func{Ad}_{p}^{\\left( s\\right) }\\right) _{\\ast\n}\\eta \\right] ^{\\left( ps\\right) }  \\label{Adbrack1} \\\\\n&&-\\left( R_{p}^{\\left( s\\right) }\\right) _{\\ast }^{-1}\\left[ \\left( \\func{A\n}_{p}^{\\left( s\\right) }\\right) _{\\ast }\\xi ,p,\\eta \\right] ^{\\left(\ns\\right) }+\\left( R_{p}^{\\left( s\\right) }\\right) _{\\ast }^{-1}\\left[ \\left( \n\\func{Ad}_{p}^{\\left( s\\right) }\\right) _{\\ast }\\eta ,p,\\xi \\right] ^{\\left(\ns\\right) }  \\notag \\\\\n&&+\\left( R_{p}^{\\left( s\\right) }\\right) _{\\ast }^{-1}\\left[ p,\\xi ,\\eta\n\\right] ^{\\left( s\\right) }-\\left( R_{p}^{\\left( s\\right) }\\right) _{\\ast\n}^{-1}\\left[ p,\\eta ,\\xi \\right] ^{\\left( s\\right) }.  \\notag\n\\end{eqnarray\nThe bracket $\\left[ \\cdot ,\\cdot \\right] ^{\\left( ps\\right) }$ is related to \n$\\left[ \\cdot ,\\cdot \\right] ^{\\left( s\\right) }$ via the expression \n\\begin{equation}\n\\left[ \\xi ,\\eta \\right] ^{\\left( ps\\right) }=\\left[ \\xi ,\\eta \\right]\n^{\\left( s\\right) }+\\left( R_{p}^{\\left( s\\right) }\\right) _{\\ast\n}^{-1}a_{s}\\left( \\xi ,\\eta ,p\\right) .  \\label{Adbrack1a}\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nConside\n\\begin{eqnarray}\n\\left( \\func{Ad}_{p}^{\\left( s\\right) }\\right) _{\\ast }\\left[ \\xi ,\\eta\n\\right] ^{\\left( s\\right) } &=&\\left. \\frac{d}{dtd\\tau }\\left( p\\circ\n_{s}\\left( \\exp \\left( t\\xi \\right) \\circ _{s}\\exp \\left( \\tau \\eta \\right)\n\\right) \\right) \/_{s}p\\right\\vert _{t,\\tau =0}  \\notag \\\\\n&&-\\left. \\frac{d}{dtd\\tau }\\left( p\\circ _{s}\\left( \\exp \\left( t\\eta\n\\right) \\circ _{s}\\exp \\left( \\tau \\xi \\right) \\right) \\right)\n\/_{s}p\\right\\vert _{t,\\tau =0}.  \\label{Adbrack}\n\\end{eqnarray\nFor brevity and clarity, let us suppress the derivatives and exponentials,\nthen using mixed associators such as (\\ref{etapxiassoc}), we can write \n\\begin{eqnarray*}\n\\left( p\\circ _{s}\\left( \\xi \\circ _{s}\\eta \\right) \\right) \/_{s}p &=&\\left(\n\\left( p\\circ _{s}\\xi \\right) \\circ _{s}\\eta \\right) \/_{s}f+\\left[ p,\\xi\n,\\eta \\right] ^{\\left( s\\right) }\/_{s}p \\\\\n&=&\\left( \\left( \\left( p\\circ _{s}\\xi \\right) \/_{s}p\\circ _{s}p\\right)\n\\circ _{s}\\eta \\right) \/_{s}p+\\left[ p,\\xi ,\\eta \\right] ^{\\left( s\\right)\n}\/_{s}p \\\\\n&=&\\left( \\func{Ad}_{p}^{\\left( s\\right) }\\xi \\circ _{s}\\left( p\\circ\n_{s}\\eta \\right) \\right) \/_{s}p-\\left[ \\func{Ad}_{p}^{\\left( s\\right) }\\xi\n,p,\\eta \\right] ^{\\left( s\\right) }\/_{s}p \\\\\n&&+\\left[ p,\\xi ,\\eta \\right] ^{\\left( s\\right) }\/_{s}p.\n\\end{eqnarray*\nApplying (\\ref{xrprod}), we get \n\\begin{equation}\n\\left( p\\circ _{s}\\left( \\xi \\circ _{s}\\eta \\right) \\right) \/_{s}p=\\func{Ad\n_{p}^{\\left( s\\right) }\\xi \\circ _{ps}\\func{Ad}_{p}^{\\left( s\\right) }\\eta \n\\left[ \\func{Ad}_{p}^{\\left( s\\right) }\\xi ,p,\\eta \\right] ^{\\left( s\\right)\n}\/_{s}p+\\left[ p,\\xi ,\\eta \\right] ^{\\left( s\\right) }\/_{s}p.\n\\end{equation\nSubtracting the same expression with $\\xi $ and $\\eta $ reversed, (\\re\n{Adbrack}) becomes \n\\begin{eqnarray}\n\\left( \\func{Ad}_{p}^{\\left( s\\right) }\\right) _{\\ast }\\left[ \\xi ,\\eta\n\\right] ^{\\left( s\\right) } &=&\\left[ \\left( \\func{Ad}_{p}^{\\left( s\\right)\n}\\right) _{\\ast }\\xi ,\\left( \\func{Ad}_{p}^{\\left( s\\right) }\\right) _{\\ast\n}\\eta \\right] ^{\\left( ps\\right) } \\\\\n&&-\\left( R_{p}^{\\left( s\\right) }\\right) _{\\ast }^{-1}\\left[ \\left( \\func{A\n}_{p}^{\\left( s\\right) }\\right) _{\\ast }\\xi ,p,\\eta \\right] ^{\\left(\ns\\right) }+\\left( R_{p}^{\\left( s\\right) }\\right) _{\\ast }^{-1}\\left[ \\left( \n\\func{Ad}_{p}^{\\left( s\\right) }\\right) _{\\ast }\\eta ,p,\\xi \\right] ^{\\left(\ns\\right) }  \\notag \\\\\n&&+\\left( R_{p}^{\\left( s\\right) }\\right) _{\\ast }^{-1}\\left[ p,\\xi ,\\eta\n\\right] ^{\\left( s\\right) }-\\left( R_{p}^{\\left( s\\right) }\\right) _{\\ast\n}^{-1}\\left[ p,\\eta ,\\xi \\right] ^{\\left( s\\right) }.  \\notag\n\\end{eqnarray\nTo obtain (\\ref{Adbrack1a}), using (\\ref{brack2deriv}), we can write \n\\begin{eqnarray}\n\\left[ \\xi ,\\eta \\right] ^{\\left( ps\\right) } &=&\\left. \\frac{d^{2}}{dtd\\tau \n}\\exp \\left( t\\xi \\right) \\circ _{ps}\\exp \\left( \\tau \\eta \\right)\n\\right\\vert _{t,\\tau =0}  \\label{brackps} \\\\\n&&-\\left. \\frac{d^{2}}{dtd\\tau }\\exp \\left( \\tau \\xi \\right) \\circ _{ps}\\exp\n\\left( t\\eta \\right) \\right\\vert _{t,\\tau =0}.  \\notag\n\\end{eqnarray\nHowever, from (\\ref{xrprod})\n\\begin{equation*}\n\\exp \\left( t\\xi \\right) \\circ _{ps}\\exp \\left( \\tau \\eta \\right) =\\left(\n\\exp \\left( t\\xi \\right) \\circ _{s}\\left( \\exp \\left( \\tau \\eta \\right)\n\\circ _{s}p\\right) \\right) \/_{s}p,\n\\end{equation*\nthus \n\\begin{eqnarray*}\n\\left. \\frac{d^{2}}{dtd\\tau }\\exp \\left( t\\xi \\right) \\circ _{ps}\\exp \\left(\n\\tau \\eta \\right) \\right\\vert _{t,\\tau =0} &=&\\left( R_{p}^{\\left( s\\right)\n}\\right) _{\\ast }^{-1}\\left. \\frac{d^{2}}{dtd\\tau }\\exp \\left( t\\xi \\right)\n\\circ _{s}\\left( \\exp \\left( \\tau \\eta \\right) \\circ _{s}p\\right)\n\\right\\vert _{t,\\tau =0} \\\\\n&=&\\left( R_{p}^{\\left( s\\right) }\\right) _{\\ast }^{-1}\\left[ \\xi ,\\eta ,\n\\right] ^{\\left( s\\right) } \\\\\n&&+\\left. \\frac{d^{2}}{dtd\\tau }\\exp \\left( t\\xi \\right) \\circ _{s}\\exp\n\\left( \\tau \\eta \\right) \\right\\vert _{t,\\tau =0}\n\\end{eqnarray*\nand similarly for the second term in (\\ref{brackps}). Hence, we obtain (\\re\n{Adbrack1a}).\n\\end{proof}\n\nFrom (\\ref{Adbrack1a}) and noting that for any $h\\in \\Psi ^{R}\\left( \\mathbb\nL}\\right) $, $h\\left( s\\right) =h\\left( s\\right) \/s\\cdot s,$ we find that \n\\left[ \\cdot ,\\cdot \\right] ^{\\left( s\\right) }=\\left[ \\cdot ,\\cdot \\right]\n^{\\left( h\\left( s\\right) \\right) }$ if and only if \n\\begin{equation}\na_{s}\\left( \\xi ,\\eta ,\\faktor{h\\left( s\\right)}{s}\\right) ^{\\left( s\\right)\n}=0  \\label{stabbrackcond}\n\\end{equation\nfor any $\\xi ,\\eta \\in \\mathfrak{l}.$ From (\\ref{PsAutoriso}) recall that \nh\\left( s\\right) \/s$ is the companion that corresponds to $h$ in $\\left( \n\\mathbb{L},\\circ _{s}\\right) .$\n\nAlso, note that from (\\ref{Adbrack1a}), we have \n\\begin{equation}\n\\left[ \\theta ,\\theta \\right] ^{\\left( p\\right) }=\\left[ \\theta ,\\theta\n\\right] ^{\\left( 1\\right) }+\\left( R_{p}\\right) _{\\ast }^{-1}a_{1}\\left(\n\\theta ,\\theta ,p\\right) ,  \\label{brackthetas}\n\\end{equation\nso the left-alternating associator with $p$ is the obstruction for the\nbrackets $\\left[ \\cdot ,\\cdot \\right] ^{\\left( p\\right) }$ and $\\left[ \\cdot\n,\\cdot \\right] ^{\\left( 1\\right) }$ to be equal. Moreover, the structural\nequation (\\ref{MCequation2}) can be rewritten as \n\\begin{equation}\nd\\theta -\\frac{1}{2}\\left[ \\theta ,\\theta \\right] ^{\\left( 1\\right) }=\\frac{\n}{2}\\left( R_{p}\\right) _{\\ast }^{-1}a_{1}\\left( \\theta ,\\theta ,p\\right) .\n\\end{equation\nThis makes the dependence on the associator more explicit.\n\nUsing the associator on $\\mathfrak{l}^{\\left( p\\right) }$ we can define the\nright nucleus $\\mathcal{N}^{R}\\left( \\mathfrak{l}^{\\left( p\\right) }\\right) $\nof $\\mathfrak{l}^{\\left( p\\right) }.$\n\n\\begin{definition}\nLet $p\\in \\mathbb{\\mathring{L}}$, then, the right nucleus $\\mathcal{N\n^{R}\\left( \\mathfrak{l}^{\\left( p\\right) }\\right) $ is defined as \n\\begin{equation}\n\\mathcal{N}^{R}\\left( \\mathfrak{l}^{\\left( p\\right) }\\right) =\\left\\{ \\xi\n\\in \\mathfrak{l}:a_{p}\\left( \\eta ,\\gamma ,\\xi \\right) =0\\ \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{for all \n\\eta ,\\gamma \\in \\mathfrak{l}\\right\\} .  \\label{NRl}\n\\end{equation}\n\\end{definition}\n\nIt may seem that a more natural definition of $\\mathcal{N}^{R}\\left( \n\\mathfrak{l}^{\\left( p\\right) }\\right) $ would be to be the set of all $\\xi\n\\in \\mathfrak{l}$ such that $\\left[ \\eta ,\\gamma ,\\xi \\right] ^{\\left(\np\\right) }=0$ for any $\\eta ,\\gamma \\in \\mathfrak{l}.\\ $However, the\nadvantage of (\\ref{NRl}) is that, as we will see, it will always be a Lie\nsubalgebra of $\\mathfrak{l}^{\\left( p\\right) }.$ For a left-alternative\nalgebra, the skew-symmetrization in (\\ref{NRl}) would be unnecessary of\ncourse.\n\n\\begin{theorem}\nThe right nucleus $\\mathcal{N}^{R}\\left( \\mathfrak{l}^{\\left( p\\right)\n}\\right) $ is a Lie subalgebra of $\\mathfrak{l}^{\\left( p\\right) }.$\n\\end{theorem}\n\n\\begin{proof}\nWe first need to show that $\\mathcal{N}^{R}\\left( \\mathfrak{l}^{\\left(\np\\right) }\\right) $ is closed under $\\left[ \\cdot ,\\cdot \\right] ^{\\left(\np\\right) }.$ Indeed, taking the exterior derivative of (\\ref{db2}), for\nvector fields $X,Y$ on $\\mathbb{L}$ we have \n\\begin{eqnarray*}\n0 &=&\\left( d^{2}b\\left( \\beta ,\\gamma \\right) \\right) \\left( X,Y\\right)\n=X\\left( d_{Y}b\\left( \\beta ,\\gamma \\right) \\right) -Y\\left( d_{X}b\\left(\n\\beta ,\\gamma \\right) \\right) -d_{\\left[ X,Y\\right] }b\\left( \\beta ,\\gamma\n\\right) \\\\\n&=&X\\left( a\\left( \\beta ,\\gamma ,\\theta \\left( Y\\right) \\right) \\right)\n-Y\\left( a\\left( \\beta ,\\gamma ,\\theta \\left( X\\right) \\right) \\right)\n-a\\left( \\beta ,\\gamma ,\\theta \\left( \\left[ X,Y\\right] \\right) \\right) .\n\\end{eqnarray*\nSuppose now $\\xi ,\\eta \\in \\mathfrak{l}^{\\left( p\\right) }$ and let $X=\\rho\n\\left( \\xi \\right) ,$ $Y=\\rho \\left( \\eta \\right) $ be the corresponding\nright fundamental vector fields, then using (\\ref{T1Lbrack}), we have \n\\begin{equation}\na\\left( \\beta ,\\gamma ,b\\left( \\xi ,\\eta \\right) \\right) =-X\\left( a\\left(\n\\beta ,\\gamma ,\\eta \\right) \\right) +Y\\left( a\\left( \\beta ,\\gamma ,\\xi\n\\right) \\right)  \\label{d2b}\n\\end{equation\nSuppose now $\\xi ,\\eta \\in \\mathcal{N}^{R}\\left( \\mathfrak{l}^{\\left(\np\\right) }\\right) $. Then, the right-hand side of (\\ref{d2b}) vanishes, and\nat $p\\in \\mathbb{L}$, \n\\begin{equation}\na_{p}\\left( \\beta ,\\gamma ,\\left[ \\xi ,\\eta \\right] ^{\\left( p\\right)\n}\\right) =0,  \\label{d2b2}\n\\end{equation\nand thus $\\left[ \\xi ,\\eta \\right] ^{\\left( p\\right) }\\in \\mathcal{N\n^{R}\\left( \\mathfrak{l}^{\\left( p\\right) }\\right) .$\n\nTo conclude that $\\mathcal{N}^{R}\\left( \\mathfrak{l}^{\\left( p\\right)\n}\\right) $ is a Lie subalgebra, we also need to verify that Lie algebra\nJacobi identity holds. That is, for any $\\xi ,\\eta ,\\gamma \\in \\mathcal{N\n^{R}\\left( \\mathfrak{l}^{\\left( p\\right) }\\right) $, $\\func{Jac}^{\\left(\np\\right) }\\left( \\xi ,\\eta ,\\gamma \\right) =0$. Indeed, from the Akivis\nidentity (\\ref{Jac2}), \n\\begin{equation}\n\\func{Jac}^{\\left( p\\right) }\\left( \\xi ,\\eta ,\\gamma \\right) =a_{p}\\left(\n\\xi ,\\eta ,\\gamma \\right) +a_{p}\\left( \\eta ,\\gamma ,\\xi \\right)\n+a_{p}\\left( \\gamma ,\\xi ,\\eta \\right) =0,\n\\end{equation\nby definition of $\\mathcal{N}^{R}\\left( \\mathfrak{l}^{\\left( p\\right)\n}\\right) .$\n\\end{proof}\n\nFor any smooth loop, consider the loop right nucleus $\\mathcal{N}^{R}\\left( \n\\mathbb{L},\\circ _{p}\\right) $ as a submanifold of $\\mathbb{L}.$ Then, \n\\begin{equation}\nT_{1}\\mathcal{N}^{R}\\left( \\mathbb{L},\\circ _{p}\\right) =\\left\\{ \\xi \\in \n\\mathfrak{l}:\\left[ q,r,\\xi \\right] ^{\\left( p\\right) }=0\\ \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{for all \nq,r\\in \\mathbb{L}\\right\\} ,  \\label{T1N}\n\\end{equation\nwhere here we are using the mixed associator as defined by (\\ref{pqxiassoc\n). Then, (\\ref{Lalgassoc2}) implies that $T_{1}\\mathcal{N}^{R}\\left( \\mathbb\nL},\\circ _{p}\\right) \\subset \\mathcal{N}^{R}\\left( \\mathfrak{l}^{\\left(\np\\right) }\\right) .$ It is unclear what are the conditions for the converse,\nand hence equality, of the two spaces.\n\nRecall from (\\ref{CRNucl}) that $A\\in \\mathcal{N}^{R}\\left( \\mathbb{L\n\\right) $\\ if and only if $\\func{Ad}_{p}\\left( A\\right) \\in \\mathcal{N\n^{R}\\left( \\mathbb{L},\\circ _{p}\\right) $, so in particular, $\\eta \\in T_{1\n\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) $ if and only if $\\left( \\func{Ad\n_{p}\\right) _{\\ast }\\eta \\in T_{1}\\mathcal{N}^{R}\\left( \\mathbb{L},\\circ\n_{p}\\right) .$ In (\\ref{Adbrack1}) we then see that for $\\eta ,\\gamma \\in\nT_{1}\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) $, the associators vanish, and\nwe get \n\\begin{equation}\n\\left( \\func{Ad}_{p}\\right) _{\\ast }\\left[ \\eta ,\\gamma \\right] =\\left[\n\\left( \\func{Ad}_{p}\\right) _{\\ast }\\eta ,\\left( \\func{Ad}_{p}\\right) _{\\ast\n}\\gamma \\right] ^{\\left( p\\right) }.  \\label{AdNucl}\n\\end{equation\nHence, for each $p\\in \\mathbb{\\mathring{L}},$ $T_{1}\\mathcal{N}^{R}\\left( \n\\mathbb{L}\\right) \\cong T_{1}\\mathcal{N}^{R}\\left( \\mathbb{L},\\circ\n_{p}\\right) $ as Lie algebras.\n\n\\begin{example}\nConsider the Moufang loop of unit octonions $U\\mathbb{O}.$ Then, $T_{1}\n\\mathbb{O}\\cong \\func{Im}\\mathbb{O}$ - the space of imaginary octonions,\nwith the bracket given by the commutator on $\\func{Im}\\mathbb{O}$: for any \n\\xi ,\\eta \\in \\func{Im}\\mathbb{O}$, $\\left[ \\xi ,\\eta \\right] =\\xi \\eta\n-\\eta \\xi .$ We also know that $\\mathcal{N}\\left( U\\mathbb{O}\\right) \\cong \n\\mathbb{Z}_{2}$ and $\\mathcal{N}\\left( \\func{Im}\\mathbb{O}\\right) =\\left\\{\n0\\right\\} .$ On the other hand, taking a direct product $G\\times U\\mathbb{O}$\nwith any Lie group $G$ will give a non-trivial nucleus.\n\\end{example}\n\nLet $s\\in \\mathbb{\\mathring{L}}.$ Suppose the Lie algebras of $\\Psi\n^{R}\\left( \\mathbb{L}\\right) $ and $\\func{Aut}\\left( \\mathbb{L},\\circ\n_{s}\\right) $ are $\\mathfrak{p}$ and $\\mathfrak{h}_{s}$, respectively. In\nparticular, $\\mathfrak{h}_{s}$ is a Lie subalgebra of $\\mathfrak{p}$. Define \n$\\mathfrak{q}_{s}=T_{1}\\mathcal{C}^{R}\\left( \\mathbb{L},\\circ _{s}\\right) ,$\nthen since $\\mathcal{C}^{R}\\left( \\mathbb{L},\\circ _{s}\\right) \\subset \n\\mathbb{L}$, so $\\mathfrak{q}_{s}\\mathfrak{\\subset l}^{\\left( s\\right) \n\\mathfrak{\\cong }T_{1}\\mathbb{L}.$ On the other hand, $\\mathcal{C}^{R}\\left( \n\\mathbb{L},\\circ _{s}\\right) \\cong \n\\faktor{\\Psi ^{R}\\left( \\mathbb{L}\\right)}{\\func{Aut}\\left( \n\\mathbb{L},\\circ _{s}\\right)}$, and the tangent space at the coset \n1=\\left\\lfloor \\func{Aut}\\left( \\mathbb{L},\\circ _{s}\\right) \\right\\rfloor $\nis $\\mathfrak{p\/h}_{s}.$ Hence, we see that $\\mathfrak{q}_{s}\\mathfrak{\\cong\np\/h}_{s}$, at least as vector spaces. The groups $\\Psi ^{R}\\left( \\mathbb{L\n\\right) $ and $\\func{Aut}\\left( \\mathbb{L},\\circ _{s}\\right) $ act on \n\\mathfrak{p}$ and $\\mathfrak{h}_{s}$ via their respective adjoint actions\nand hence $\\func{Aut}\\left( \\mathbb{L},\\circ _{s}\\right) $ acts on \n\\mathfrak{q}_{s}$ via a restriction of the adjoint action of $\\Psi\n^{R}\\left( \\mathbb{L}\\right) .$ Now note that given $h=\\left( \\alpha\n,A\\right) \\in \\Psi ^{R}\\left( \\mathbb{L}\\right) $ and $\\beta \\in \\func{Aut\n\\left( \\mathbb{L},\\circ _{s}\\right) $, the conjugation of $h$ by $\\beta $ is\ngiven by \n\\begin{equation*}\n\\left( \\beta ,1\\right) \\left( \\alpha ,A\\right) \\left( \\beta ^{-1},1\\right)\n=\\left( \\beta \\circ \\alpha \\circ \\beta ^{-1},\\beta \\left( A\\right) \\right)\n\\end{equation*\nand hence the corresponding action on the companion $A$ is via standard\naction of $\\beta $ on $\\mathbb{L}.$ The differentials of these actions give\nthe corresponding actions on the tangent spaces. We thus see that the\nadjoint action of $\\func{Aut}\\left( \\mathbb{L},\\circ _{s}\\right) $ on \n\\mathfrak{p\/h}_{s}$ is equivalent to the standard tangent action of $\\func\nAut}\\left( \\mathbb{L},\\circ _{s}\\right) $ on $\\mathfrak{q}_{s}.$ Hence, \n\\mathfrak{q}_{s}$ and $\\mathfrak{p\/h}_{s}$ are isomorphic as linear\nrepresentations of $\\func{Aut}\\left( \\mathbb{L},\\circ _{s}\\right) .$ We can\nmake the isomorphism from $\\mathfrak{p\/h}_{s}$ to $\\mathfrak{q}_{s}$ more\nexplicit in the following way.\n\n\\begin{definition}\nDefine the map $\\varphi :\\mathbb{\\mathring{L}}\\longrightarrow $ $\\mathfrak{l\n\\otimes \\mathfrak{p}^{\\ast }$ such that for each $s\\in \\mathbb{\\mathring{L}}$\nand $\\gamma \\in \\mathfrak{p}$, \n\\begin{equation}\n\\varphi _{s}\\left( \\gamma \\right) =\\left. \\frac{d}{dt}\\faktor{\\left( \\exp\n\\left( t\\gamma \\right) \\left( s\\right) \\right)}{s}\\right\\vert _{t=0}\\in \n\\mathfrak{l.}  \\label{phis}\n\\end{equation}\n\\end{definition}\n\nThus, for each $s\\in \\mathbb{\\mathring{L}},$ $\\varphi _{s}$ gives a map from \n$\\mathfrak{p}$ to $\\mathfrak{l}^{\\left( s\\right) }.$\n\n\\begin{theorem}\n\\label{lemGammahatsurj}The map $\\varphi $ as in (\\ref{phis}) is equivariant\nwith respect to corresponding actions of $\\Psi ^{R}\\left( \\mathbb{L}\\right)\n, $ in particular for $h\\in \\Psi ^{R}\\left( \\mathbb{L}\\right) ,$ $s\\in \n\\mathbb{\\mathring{L}}$, $\\gamma \\in \\mathfrak{p},$ we hav\n\\begin{equation}\n\\varphi _{h\\left( s\\right) }\\left( \\left( \\func{Ad}_{h}\\right) _{\\ast\n}\\gamma \\right) =\\left( h^{\\prime }\\right) _{\\ast }\\varphi _{s}\\left( \\gamma\n\\right) .  \\label{phihs}\n\\end{equation\nMoreover, the image of $\\varphi _{s}$ is $\\mathfrak{q}_{s}$ and the kernel\nis $\\mathfrak{h}_{s}$, and hence, \n\\begin{equation}\n\\mathfrak{p\\cong h}_{s}\\oplus \\mathfrak{q}_{s}.  \\label{pdecomp}\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nConsider $h\\in \\Psi ^{R}\\left( \\mathbb{L}\\right) $. Then, using (\\re\n{PsAutquot2a}), we have \n\\begin{eqnarray*}\n\\varphi _{h\\left( s\\right) }\\left( \\gamma \\right) &=&\\left. \\frac{d}{dt\n\\faktor{\\left[ \\exp \\left( t\\gamma \\right) \\left( h\\left( s\\right) \\right)\n\\right]}{ h\\left( s\\right)} \\right\\vert _{t=0} \\\\\n&=&\\left. \\frac{d}{dt}h^{\\prime }\\left[ \\faktor{\\func{Ad}_{h^{-1}}\\left(\n\\exp \\left( t\\gamma \\right) \\right) \\left( s\\right)}{s}\\right] \\right\\vert\n_{t=0} \\\\\n&=&\\left( h^{\\prime }\\right) _{\\ast }\\left. \\frac{d}{dt}\\faktor{\\exp \\left(\nt\\left( \\func{Ad}_{h^{-1}}\\right) _{\\ast }\\gamma \\right) \\left( s\\right)} {s\n\\right\\vert _{t=0}.\n\\end{eqnarray*\nSince $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ acts on $\\mathfrak{l}$ via \n\\left( h^{\\prime }\\right) _{\\ast }$ and on $\\mathfrak{p}$ via $\\left( \\func\nAd}_{h}\\right) _{\\ast }$ we see that $\\varphi $ is equivariant.\n\nSince $\\func{Aut}\\left( \\mathbb{L},\\circ _{s}\\right) $ is a Lie subgroup of \n\\Psi ^{R}\\left( \\mathbb{L}\\right) ,$ the projection map $\\pi :\\Psi\n^{R}\\left( \\mathbb{L}\\right) \\longrightarrow \n\\faktor{\\Psi ^{R}\\left(\n\\mathbb{L}\\right)}{\\func{Aut}\\left( \\mathbb{L},\\circ _{s}\\right)}\\cong \n\\mathcal{C}^{R}\\left( \\mathbb{L},\\circ _{s}\\right) $ is a smooth submersion\ngiven by $\\pi \\left( h\\right) =h\\left( s\\right) \/s$ for each $h\\in \\Psi\n^{R}\\left( \\mathbb{L}\\right) .$ Thus, $\\left. \\pi _{\\ast }\\right\\vert _\n\\func{id}}:\\mathfrak{p}\\longrightarrow \\mathfrak{q}_{s}$ is surjective.\nHowever, since $\\exp $ is a surjective map from $\\mathfrak{p}$ to a\nneighborhood of $\\func{id}\\in \\Psi ^{R}\\left( \\mathbb{L}\\right) $, we find\nthat $\\left. \\pi _{\\ast }\\right\\vert _{\\func{id}}\\left( \\gamma \\right)\n=\\varphi _{s}\\left( \\gamma \\right) .$ So indeed, the image of the map \n\\varphi _{s}$ is $\\mathfrak{q}_{s}.$ Clearly the kernel is $\\mathfrak{h\n_{s}. $ Then, (\\ref{pdecomp}) follows immediately.\n\\end{proof}\n\nTheorem \\ref{lemGammahatsurj} implies that $\\varphi :\\mathbb{\\mathring{L}\n\\longrightarrow $ $\\mathfrak{l}\\otimes \\mathfrak{p}^{\\ast }$ is equivariant\nwith respect to the action of $\\Psi ^{R}\\left( \\mathbb{L}\\right) ,$ and\nsimilarly as for $b$, we can define $\\func{Stab}_{\\Psi ^{R}\\left( \\mathbb{L\n\\right) }\\left( \\varphi _{s}\\right) =\\left\\{ h\\in \\Psi ^{R}\\left( \\mathbb{L\n\\right) :\\varphi _{h\\left( s\\right) }=\\varphi _{s}\\right\\} .$ This is then a\nLie subgroup of $\\Psi ^{R}\\left( \\mathbb{L}\\right) ,$ and $\\func{Aut}\\left( \n\\mathbb{L},\\circ _{s}\\right) \\subset $ $\\func{Stab}_{\\Psi ^{R}\\left( \\mathbb\nL}\\right) }\\left( \\varphi ^{\\left( s\\right) }\\right) .$ Suppose $h=\\left(\n\\alpha ,A\\right) \\in \\func{Stab}_{\\Psi ^{R}\\left( \\mathbb{L}\\right) }\\left(\n\\varphi ^{\\left( s\\right) }\\right) $, then \n\\begin{equation*}\n\\varphi _{s}\\left( \\gamma \\right) =\\varphi _{h\\left( s\\right) }\\left( \\gamma\n\\right) =\\left. \\frac{d}{dt}\\left[ \\exp \\left( t\\gamma \\right) \\left( \\alpha\n\\left( s\\right) A\\right) \\right] \/\\left( \\alpha \\left( s\\right) A\\right)\n\\right\\vert _{t=0}\n\\end{equation*}\n\nWe can also see the effect on $\\varphi $ of left multiplication of $s$ by\nelements of $\\mathbb{L}$.\n\n\\begin{lemma}\nSuppose $A\\in \\mathbb{L}$ and $s\\in \\mathbb{\\mathring{L}}$, then for any \n\\gamma \\in \\mathfrak{p},\n\\begin{equation}\n\\varphi _{As}\\left( \\gamma \\right) =\\left( R_{A}^{\\left( s\\right) }\\right)\n_{\\ast }^{-1}\\left( \\gamma ^{\\prime }\\cdot A\\right) +\\left( \\func{Ad\n_{A}^{\\left( s\\right) }\\right) _{\\ast }\\varphi _{s}\\left( \\gamma \\right) ,\n\\label{phiAs}\n\\end{equation\nwhere $\\gamma ^{\\prime }\\cdot A=\\left. \\frac{d}{dt}\\left( \\exp t\\gamma\n\\right) ^{\\prime }\\left( A\\right) \\right\\vert _{t=0}$ represents the\ninfinitesimal action of $\\mathfrak{p}$ on $\\mathbb{L}.$\n\\end{lemma}\n\n\\begin{proof}\nThis follows from a direct computation\n\\begin{eqnarray*}\n\\varphi _{As}\\left( \\gamma \\right) &=&\\left. \\frac{d}{dt}\\exp \\left( t\\gamma\n\\right) \\left( As\\right) \/As\\right\\vert _{t=0} \\\\\n&=&\\left. \\frac{d}{dt}\\left[ \\exp \\left( t\\gamma \\right) ^{\\prime }\\left(\nA\\right) \\exp \\left( t\\gamma \\right) \\left( s\\right) \\right] \/As\\right\\vert\n_{t=0} \\\\\n&=&\\left. \\frac{d}{dt}\\left[ A\\exp \\left( t\\gamma \\right) \\left( s\\right)\n\\right] \/As\\right\\vert _{t=0}+\\left. \\frac{d}{dt}\\left( \\left[ \\exp \\left(\nt\\gamma \\right) ^{\\prime }\\left( A\\right) \\right] s\\right) \/As\\right\\vert\n_{t=0} \\\\\n&=&\\left( \\func{Ad}_{A}^{\\left( s\\right) }\\right) _{\\ast }\\varphi _{s}\\left(\n\\gamma \\right) +\\left( R_{A}^{\\left( s\\right) }\\right) _{\\ast }^{-1}\\left(\n\\gamma ^{\\prime }\\cdot A\\right) ,\n\\end{eqnarray*\nwhere we have used (\\ref{rprodqright}).\n\\end{proof}\n\n\\begin{example}\n\\label{exOct}If $\\mathbb{L}\\ $is the loop of unit octonions, then we know \n\\mathfrak{p\\cong so}\\left( 7\\right) \\cong \\Lambda ^{2}\\left( \\mathbb{R\n^{7}\\right) ^{\\ast }$ and $\\mathfrak{l\\cong }\\mathbb{R}^{7}$ , so $\\varphi\n_{s}$ can be regarded as an element of $\\mathbb{R}^{7}\\otimes $ $\\Lambda ^{2\n\\mathbb{R}^{7},$ and this is precisely a dualized version of the $G_{2}\n-invariant $3$-form $\\varphi .$ The kernel is isomorphic to $\\mathfrak{g\n_{2}.$\n\\end{example}\n\n\\begin{example}\n\\label{exCx2}Suppose $\\mathbb{L=}U\\mathbb{C\\cong }S^{1}$ - the unit complex\nnumbers, so that $\\mathfrak{l\\cong }\\mathbb{R}.$ From Example \\re\n{ExNormedDiv}, we may take $\\Psi _{n}^{R}\\left( U\\mathbb{C}\\right) =U\\left(\nn\\right) ,$ with a trivial partial action on $U\\mathbb{C}.$ The\ncorresponding Lie algebra is $\\mathfrak{p}_{n}\\cong \\mathfrak{u}\\left(\nn\\right) \\cong \\mathfrak{su}\\left( n\\right) \\oplus i\\mathbb{R}.$ The map \n\\varphi _{s}:\\mathfrak{p}_{n}\\longrightarrow i\\mathbb{R}\\ $is then just the\nprojection $\\mathfrak{su}\\left( n\\right) \\oplus i\\mathbb{R}\\longrightarrow \n\\mathbb{R}$ (i.e. trace). It is independent of $s$. The kernel is $\\mathfrak\nsu}\\left( n\\right) .$ Suppose $V$ is a $n$-dimensional real vector space,\nand $V\\otimes \\mathbb{C}=V^{1,0}\\oplus V^{0,1}$. Then, the group $U\\left(\nn\\right) $ acts via unitary transformations on the complex vector space \nV^{1,0},$ and correspondingly $\\mathfrak{u}\\left( n\\right) \\cong V^{1,1}$\n(i.e. the space of $\\left( 1,1\\right) $-forms). Then, we see that $\\varphi\n_{s}$ is just the dualized version of a Hermitian form on $V\\otimes \\mathbb{\n}.$\n\\end{example}\n\n\\begin{example}\n\\label{exQuat2}Suppose $\\mathbb{L=}U\\mathbb{H\\cong }S^{3}$ - the unit\nquaternions, so that $\\mathfrak{l\\cong }\\mathfrak{sp}\\left( 1\\right) .$ From\nExample \\ref{ExNormedDiv}, we may take $\\Psi _{n}^{R}\\left( U\\mathbb{H\n\\right) =Sp\\left( n\\right) Sp\\left( 1\\right) ,$ with $n\\geq 2$, with a\ntrivial partial action on $U\\mathbb{H}.$ The corresponding Lie algebra is \n\\mathfrak{p}_{n}\\cong \\mathfrak{sp}\\left( n\\right) \\oplus \\mathfrak{sp\n\\left( 1\\right) .$ The map $\\varphi _{s}:\\mathfrak{p}_{n}\\longrightarrow \n\\mathfrak{sp}\\left( 1\\right) \\ $is then given by $\\left( a,\\xi \\right)\n\\mapsto \\left( \\func{Ad}_{s}\\right) _{\\ast }\\xi .$ The kernel is then \n\\mathfrak{sp}\\left( n\\right) .$ Suppose $Sp\\left( n\\right) Sp\\left( 1\\right) \n$ acts on a $4n$-dimensional real vector space \\ $V$, $\\mathfrak{sp}\\left(\nn\\right) \\oplus \\mathfrak{sp}\\left( 1\\right) \\subset \\Lambda ^{2}V^{\\ast }$.\nGiven that $\\mathfrak{sp}\\left( 1\\right) \\cong \\func{Im}\\mathbb{H},$ we can\nthen write $\\varphi _{s}=i\\omega _{1}^{\\ast }+j\\omega _{2}^{\\ast }+k\\omega\n_{3}^{\\ast },$ where the $\\omega _{i}^{\\ast }$ are dualized versions of the\n3 linearly independent Hermitian forms that space the $\\mathfrak{sp}\\left(\n1\\right) $ subspace of $\\Lambda ^{2}V^{\\ast }$ \\cite{SalamonBook}.\n\\end{example}\n\n\\begin{remark}\nThe above examples clearly show that one interpretation of the $G_{2}$\nstructure $3$-form $\\varphi $ is as $\\func{Im}\\mathbb{O}$-valued $2$-form. A\ncomplex Hermitian form is then an $\\func{Im}\\mathbb{C}$-valued $2$-form, and\na quaternionic Hermitian form is an $\\func{Im}\\mathbb{H}$-valued $2$-form.\n\\end{remark}\n\nNow let us summarize the actions of different spaces on one another. For a\nfixed $\\gamma $, define the map $\\hat{\\gamma}:\\mathbb{\\mathring{L}\n\\longrightarrow \\mathfrak{l\\ }$given by $s\\mapsto \\hat{\\gamma}^{\\left(\ns\\right) }=\\varphi _{s}\\left( \\gamma \\right) .$\n\n\\begin{theorem}\nSuppose $\\mathbb{L}$ is a smooth loop with tangent algebra $\\mathfrak{l}$\nand suppose $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ is a Lie group with Lie\nalgebra $\\mathfrak{p}.$ Let $A\\in \\mathbb{L},$ $s\\in \\mathbb{\\mathring{L}}$, \n$\\xi \\in \\mathfrak{l}$, and $\\gamma \\in \\mathfrak{p}.$ Then, denoting by \n\\cdot $ the relevant action, we have the following:\n\n\\begin{enumerate}\n\\item Infinitesimal action of $\\mathfrak{p}$ on $\\mathbb{\\mathring{L}}$: \n\\begin{equation}\n\\gamma \\cdot s=\\left. \\frac{d}{dt}\\exp \\left( t\\gamma \\right) \\left(\ns\\right) \\right\\vert _{t=0}=\\left( R_{s}\\right) _{\\ast }\\hat{\\gamma}^{\\left(\ns\\right) }\\in T_{s}\\mathbb{L}  \\label{infplring}\n\\end{equation}\n\n\\item Infinitesimal action of $\\mathfrak{p}$ on $\\mathbb{L}$, for any $s\\in \n\\mathbb{\\mathring{L}}$: \n\\begin{equation}\n\\gamma \\cdot A=\\left. \\frac{d}{dt}\\exp \\left( t\\gamma \\right) ^{\\prime\n}\\left( A\\right) \\right\\vert _{t=0}=\\left( R_{A}^{\\left( s\\right) }\\right)\n_{\\ast }\\hat{\\gamma}^{\\left( As\\right) }-\\left( L_{A}^{\\left( s\\right)\n}\\right) _{\\ast }\\hat{\\gamma}^{\\left( s\\right) }\\in T_{A}\\mathbb{L}.\n\\label{infpl}\n\\end{equation\nIn particular, if $s=1$, \n\\begin{equation}\n\\gamma \\cdot A=\\left( R_{A}\\right) _{\\ast }\\hat{\\gamma}^{\\left( A\\right)\n}-\\left( L_{A}\\right) _{\\ast }\\hat{\\gamma}^{\\left( 1\\right) }.\n\\label{infpl2}\n\\end{equation}\n\n\\item Action of $\\mathfrak{p}$ on $\\mathfrak{l\\ }$for any $s\\in \\mathbb\n\\mathring{L}}$\n\\begin{eqnarray}\n\\gamma \\cdot \\xi &=&\\left. \\frac{d}{dt}\\left( \\exp \\left( t\\gamma \\right)\n^{\\prime }\\right) _{\\ast }\\left( \\xi \\right) \\right\\vert _{t=0}  \\notag \\\\\n&=&\\left. d\\hat{\\gamma}\\right\\vert _{s}\\left( \\rho _{s}\\left( \\xi \\right)\n\\right) +\\left[ \\hat{\\gamma}^{\\left( s\\right) },\\xi \\right] ^{\\left(\ns\\right) }.  \\label{actpl}\n\\end{eqnarray\nIn particular, for $s=1$, we have \n\\begin{equation}\n\\gamma \\cdot \\xi =\\left. d\\hat{\\gamma}\\right\\vert _{1}\\left( \\xi \\right) \n\\left[ \\hat{\\gamma}^{\\left( 1\\right) },\\xi \\right] .  \\label{actpl2}\n\\end{equation}\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nLet $A,B\\in \\mathbb{L},$ $s\\in \\mathbb{\\mathring{L}}$, $\\xi ,\\eta \\in \n\\mathfrak{l}$, $h\\in \\Psi ^{R}\\left( \\mathbb{L}\\right) $, and $\\gamma \\in \n\\mathfrak{p}.$ Then we have the following.\n\n\\begin{enumerate}\n\\item The infinitesimal action of a Lie algebra is a standard definition.\n\n\\item Consider now the action of $\\mathfrak{p}$ on $\\mathbb{L}.$ Suppose now \n$\\gamma \\in \\mathfrak{p}$ and $A\\in \\mathbb{L}$ \n\\begin{equation}\n\\gamma ^{\\prime }\\cdot A=\\left. \\frac{d}{dt}\\left( \\exp \\left( t\\gamma\n\\right) ^{\\prime }\\right) \\left( A\\right) \\right\\vert _{t=0}.\n\\label{gammaprime}\n\\end{equation\nSuppose $h\\in \\Psi ^{R}\\left( \\mathbb{L},\\circ _{s}\\right) $, then by (\\re\n{PsAutoriso}), the action of $h$ on $A\\in \\mathbb{L}$ i\n\\begin{equation*}\nh\\left( A\\right) =h^{\\prime }\\left( A\\right) \\circ _{s}\\left(\n\\faktor{h\\left( s\\right)}{s}\\right)\n\\end{equation*\nThus, the partial action $h^{\\prime }\\left( A\\right) $ is given by \n\\begin{equation}\nh^{\\prime }\\left( A\\right) =\\left( \\faktor{h\\left( As\\right)}{s}\\right)\n\/_{s}\\left(\\faktor{ h\\left( s\\right)} {s}\\right) .  \\label{hprimes}\n\\end{equation\nMoreover, \n\\begin{equation}\n\\faktor{h\\left( As\\right)}{s}=\\left(\\faktor{h\\left( As\\right)}{As}\\right)\n\\circ _{s}A.  \\label{hprimes2}\n\\end{equation\nHence, substituting into (\\ref{gammaprime}), we have \n\\begin{eqnarray}\n\\gamma ^{\\prime }\\cdot A &=&\\left. \\frac{d}{dt}\\left( \\faktor{\\exp \\left(\nt\\gamma \\left( As\\right) \\right)}{As}\\circ _{s}A\\right) \/_{s} \\left(\n\\faktor{\\exp \\left( t\\gamma \\right) \\left( s\\right)}{s}\\right) \\right\\vert\n_{t=0}  \\notag \\\\\n&=&\\left. \\frac{d}{dt}\\left( \\faktor{\\exp \\left( t\\gamma \\left( As\\right)\n\\right)}{As}\\circ _{s}A\\right) \\right\\vert _{t=0}-\\left. \\frac{d}{dt}A\\circ\n_{s}\\left( \\faktor{\\exp \\left( t\\gamma \\right) \\left( s\\right)}{s}\\right)\n\\right\\vert _{t=0}  \\notag \\\\\n&=&\\left( R_{A}^{\\left( s\\right) }\\right) _{\\ast }\\hat{\\gamma}^{\\left(\nAs\\right) }-\\left( L_{A}^{\\left( s\\right) }\\right) _{\\ast }\\hat{\\gamma\n^{\\left( s\\right) }.\n\\end{eqnarray\nSetting $s=1$ immediately gives (\\ref{infpl2}).\n\n\\item Suppose now $\\gamma \\in \\mathfrak{p}$ and $\\xi \\in \\mathfrak{l}$, then\nwe have \n\\begin{eqnarray}\n\\gamma \\cdot \\xi &=&\\left. \\frac{d}{dt}\\left( \\exp \\left( t\\gamma \\right)\n^{\\prime }\\right) _{\\ast }\\left( \\xi \\right) \\right\\vert _{t=0}  \\notag \\\\\n&=&\\left. \\frac{d^{2}}{dtd\\tau }\\exp \\left( t\\gamma \\right) ^{\\prime }\\left(\n\\exp _{s}\\tau \\xi \\right) \\right\\vert _{t,\\tau =0}.  \\label{pactl1}\n\\end{eqnarray\nLet $\\Xi =\\exp _{s}\\tau \\xi \\in \\mathbb{L}$, then using (\\ref{hprimes}) and \n\\ref{hprimes2}), we can write \n\\begin{eqnarray*}\n\\exp \\left( t\\gamma \\right) ^{\\prime }\\left( \\exp _{s}\\tau \\xi \\right)\n&=&\\exp \\left( t\\gamma \\right) ^{\\prime }\\left( \\Xi \\right) \\\\\n&&\\left( \\exp \\left( t\\gamma \\right) \\left( \\Xi s\/\\Xi s\\circ _{s}\\Xi \\right)\n\\right) \/_{s}\\left( \\faktor{\\exp \\left( t\\gamma \\right) \\left( s\\right)}{s\n\\right) .\n\\end{eqnarray*\nUsing this, (\\ref{pactl1}) becomes \n\\begin{eqnarray}\n\\gamma ^{\\prime }\\cdot \\xi &=&\\left. \\frac{d^{2}}{dtd\\tau }\\faktor{\\left(\n\\exp \\left( t\\gamma \\right) \\left( \\left( \\exp _{s}\\tau \\xi \\right) s\\right)\n\\right)} {\\left( \\left( \\exp _{s}\\tau \\xi \\right) s\\circ _{s}\\exp _{s}\\tau\n\\xi \\right)}\\right\\vert _{t,\\tau =0}  \\notag \\\\\n&&-\\left. \\frac{d^{2}}{dtd\\tau }\\exp _{s}\\tau \\xi \\circ _{s}\\left(\n\\faktor{\\exp \\left( t\\gamma \\right) \\left( s\\right)}{s}\\right) \\right\\vert\n_{t,\\tau =0}  \\notag \\\\\n&=&\\left. \\frac{d^{2}}{dtd\\tau }\\exp \\left( t\\gamma \\right) \\left( \\left(\n\\exp _{s}\\tau \\xi \\right) s\\right) \/\\left( \\exp _{s}\\tau \\xi \\right)\ns\\right\\vert _{t,\\tau =0}+  \\notag \\\\\n&&+\\left. \\frac{d^{2}}{dtd\\tau }\\left( \\faktor{\\exp \\left( t\\gamma \\right)\n\\left( s\\right)}{s}\\right) \\circ _{s}\\exp _{s}\\tau \\xi \\right\\vert _{t,\\tau\n=0}  \\notag \\\\\n&&-\\left. \\frac{d^{2}}{dtd\\tau }\\exp _{s}\\tau \\xi \\circ _{s}\\left(\n\\faktor{\\exp \\left( t\\gamma \\right) \\left( s\\right)}{s}\\right) \\right\\vert\n_{t,\\tau =0}\n\\end{eqnarray\nHowever $\\hat{\\gamma}^{\\left( s\\right) }=\\left. \\frac{d}{dt}\\exp \\left(\nt\\gamma \\right) \\left( s\\right) \/s\\right\\vert _{t=0}\\in \\mathfrak{l}$, and\nthus \n\\begin{eqnarray*}\n\\left. \\frac{d}{d\\tau }\\left( L_{\\exp _{s}\\tau \\xi }^{\\left( s\\right)\n}\\right) _{\\ast }\\hat{\\gamma}^{\\left( s\\right) }\\right\\vert _{\\tau =0}\n&=&\\left. \\frac{d^{2}}{dtd\\tau }\\left( \\exp _{s}\\tau \\xi \\right) \\circ\n_{s}\\exp _{s}\\left( t\\hat{\\gamma}^{\\left( s\\right) }\\right) \\right\\vert\n_{t,\\tau =0} \\\\\n\\left. \\frac{d}{d\\tau }\\left( R_{\\exp _{s}\\tau \\xi }^{\\left( s\\right)\n}\\right) _{\\ast }\\hat{\\gamma}^{\\left( s\\right) }\\right\\vert _{\\tau =0}\n&=&\\left. \\frac{d^{2}}{dtd\\tau }\\exp _{s}\\left( t\\hat{\\gamma}^{\\left(\ns\\right) }\\right) \\circ _{s}\\exp _{s}\\tau \\xi \\right\\vert _{t,\\tau =0}.\n\\end{eqnarray*\nHence, using the expression (\\ref{brack2deriv}) for $\\left[ \\cdot ,\\cdot\n\\right] ^{\\left( s\\right) },$ we get \n\\begin{equation}\n\\gamma ^{\\prime }\\cdot \\xi =\\left. \\frac{d}{d\\tau }\\hat{\\gamma}^{\\left( \\exp\n_{s}\\tau \\xi \\right) s}\\right\\vert _{\\tau =0}+\\left[ \\hat{\\gamma}^{\\left(\ns\\right) },\\xi \\right] ^{\\left( s\\right) }.  \\label{gampri1}\n\\end{equation\nThe first term in (\\ref{gampri1}) is then precisely the differential of \n\\hat{\\gamma}$ at $s\\in \\mathbb{L}$ in the direction $\\left( R_{s}\\right)\n_{\\ast }\\xi .$ Setting $s=1$ we get (\\ref{actpl2}).\n\\end{enumerate}\n\\end{proof}\n\n\\begin{remark}\nSince the full action of $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ does not\npreserve $1,$ the pushforward of the action of some $h\\in \\Psi ^{R}\\left( \n\\mathbb{L}\\right) $ sends $T_{1}\\mathbb{L}$ to $T_{A}\\mathbb{L}$, where \nA=h\\left( 1\\right) $ is the companion of $\\mathbb{L}.$ To actually obtain an\naction on $T_{1}\\mathbb{L},$ translation back to $1$ is needed. This can be\nachieved either by right or left division by $A$. Dividing by $A$ on the\nright reduces to the partial action of $\\Psi ^{R}\\left( \\mathbb{L}\\right) ,$\ni.e. action by $h^{\\prime }$. This is how the action of $\\mathfrak{p}$ on \n\\mathfrak{l}$ in (\\ref{actpl}) is defined. Dividing by $A$ on the left,\ngives the map $h^{\\prime \\prime }=\\func{Ad}_{A^{-1}}\\circ h^{\\prime }$, as\ndefined in (\\ref{nuclearaction}). In that setting, it was defined on the\nnucleus, and hence gave an actual group action of $\\Psi ^{R}\\left( \\mathbb{L\n\\right) $, however in a non-associative setting, in general this will not be\na group action.\n\\end{remark}\n\nCombining some of the above results, we also have the following useful\nrelationship.\n\n\\begin{lemma}\nSuppose $\\xi \\in \\mathfrak{p}$ and $\\eta ,\\gamma \\in \\mathfrak{l},$ then \n\\begin{equation}\n\\xi \\cdot \\left[ \\eta ,\\gamma \\right] ^{\\left( s\\right) }=\\left[ \\xi \\cdot\n\\eta ,\\gamma \\right] ^{\\left( s\\right) }+\\left[ \\eta ,\\xi \\cdot \\gamma\n\\right] ^{\\left( s\\right) }+a_{s}\\left( \\eta ,\\gamma ,\\varphi _{s}\\left( \\xi\n\\right) \\right) .  \\label{xilbrack}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nUsing the definition (\\ref{actpl}) of the action of $\\mathfrak{p}$ on \n\\mathfrak{l},$ we have \n\\begin{eqnarray*}\n\\xi \\cdot \\left[ \\eta ,\\gamma \\right] ^{\\left( s\\right) } &=&\\left. \\frac{d}\ndt}\\left( \\exp \\left( t\\xi \\right) ^{\\prime }\\right) _{\\ast }\\left[ \\eta\n,\\gamma \\right] ^{\\left( s\\right) }\\right\\vert _{t=0} \\\\\n&=&\\left. \\frac{d}{dt}\\left[ \\left( \\exp \\left( t\\xi \\right) ^{\\prime\n}\\right) _{\\ast }\\eta ,\\left( \\exp \\left( t\\xi \\right) ^{\\prime }\\right)\n_{\\ast }\\gamma \\right] ^{\\exp \\left( t\\xi \\right) \\left( s\\right)\n}\\right\\vert _{t=0}\n\\end{eqnarray*\nwhere we have also used (\\ref{loopalghom1}). Hence, \n\\begin{equation}\n\\xi \\cdot \\left[ \\eta ,\\gamma \\right] ^{\\left( s\\right) }=\\left[ \\xi \\cdot\n\\eta ,\\gamma \\right] ^{\\left( s\\right) }+\\left[ \\eta ,\\xi \\cdot \\gamma\n\\right] ^{\\left( s\\right) }+\\left. \\frac{d}{dt}\\left[ \\eta ,\\gamma \\right]\n^{\\exp \\left( t\\xi \\right) \\left( s\\right) }\\right\\vert _{t=0}.\n\\label{xilbrack2}\n\\end{equation\nWe can rewrite the last term in (\\ref{xilbrack2}) as \n\\begin{equation*}\n\\left. \\frac{d}{dt}\\left[ \\eta ,\\gamma \\right] ^{\\exp \\left( t\\xi \\right)\n\\left( s\\right) }\\right\\vert _{t=0}=\\left. \\frac{d}{dt}\\left[ \\eta ,\\gamma\n\\right] ^{\\exp _{s}\\left( t\\varphi _{s}\\left( \\xi \\right) \\right)\ns}\\right\\vert _{t=0}=\\left. d_{\\rho \\left( \\hat{\\xi}\\right) }b\\right\\vert\n_{s}\\left( \\eta ,\\gamma \\right)\n\\end{equation*\nwhere $\\hat{\\xi}=\\varphi _{s}\\left( \\xi \\right) $. Then, from (\\ref{db1}),\nwe see that \n\\begin{equation}\n\\left. d_{\\rho \\left( \\hat{\\xi}\\right) }b\\right\\vert _{s}\\left( \\eta ,\\gamma\n\\right) =a_{s}\\left( \\eta ,\\gamma ,\\hat{\\xi}\\right)\n\\end{equation\nand overall, we obtain (\\ref{xilbrack}).\n\\end{proof}\n\nRecall that for each $s\\in \\mathbb{\\mathring{L}}$, the bracket function \nb_{s}\\ $is in$\\ \\Lambda ^{2}\\mathfrak{l}^{\\ast }\\otimes \\mathfrak{l}$, which\nis a tensor product of $\\mathfrak{p}$-modules, so (\\ref{xilbrack}) can be\nused to define the action of $\\xi \\in \\mathfrak{p}$ on $b_{s}$. Using the\nderivation property of Lie algebra representations on tensor products, we\nfind that for $\\eta ,\\gamma \\in \\mathfrak{l},$ \n\\begin{eqnarray}\n\\left( \\xi \\cdot b_{s}\\right) \\left( \\eta ,\\gamma \\right) &=&\\xi \\cdot\n\\left( b_{s}\\left( \\eta ,\\gamma \\right) \\right) -b_{s}\\left( \\xi \\cdot \\eta\n,\\gamma \\right) -b_{s}\\left( \\eta ,\\xi \\cdot \\gamma \\right)  \\notag \\\\\n&=&a_{s}\\left( \\eta ,\\gamma ,\\varphi _{s}\\left( \\xi \\right) \\right) .\n\\label{bsact}\n\\end{eqnarray}\n\n\\begin{definition}\nSuppose $\\mathfrak{g}$ is a Lie algebra with a representation on a vector\nspace $M$, so that $\\left( M,\\mathfrak{g}\\right) $ is a $\\mathfrak{g}\n-module. Then if $x\\in M$, define the \\emph{annihilator subalgebra }$\\func\nAnn}_{\\mathfrak{g}}\\left( x\\right) $ in $\\mathfrak{g}$ of $x$ as \n\\begin{equation}\n\\func{Ann}_{\\mathfrak{g}}\\left( x\\right) =\\left\\{ \\xi \\in \\mathfrak{g}:\\xi\n\\cdot x=0\\right\\} .  \\label{anng}\n\\end{equation}\n\\end{definition}\n\nFrom (\\ref{bsact}), we see that \n\\begin{equation}\n\\func{Ann}_{\\mathfrak{p}}\\left( b_{s}\\right) =\\left\\{ \\xi \\in \\mathfrak{p\n:a_{s}\\left( \\eta ,\\gamma ,\\varphi _{s}\\left( \\xi \\right) \\right) =0\\ \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi\nfor all }\\eta ,\\gamma \\in \\mathfrak{l}\\right\\} .  \\label{annpbs}\n\\end{equation\nThe definition (\\ref{annpbs}) is simply that $\\xi \\in $ $\\func{Ann}_\n\\mathfrak{p}}\\left( b_{s}\\right) \\ $if and only if $\\varphi _{s}\\left( \\xi\n\\right) \\in \\mathcal{N}^{R}\\left( \\mathfrak{l}^{\\left( s\\right) }\\right) ,$\nso that $\\func{Ann}_{\\mathfrak{p}}\\left( b_{s}\\right) =\\varphi\n_{s}^{-1}\\left( \\mathcal{N}^{R}\\left( \\mathfrak{l}^{\\left( s\\right) }\\right)\n\\right) .$ This is the Lie algebra that corresponds to the Lie group $\\func\nStab}_{\\Psi ^{R}\\left( \\mathbb{L}\\right) }\\left( b_{s}\\right) .$ Indeed, the\ncondition (\\ref{annpbs}) is precisely the infinitesimal version of (\\re\n{stabbrackcond}). If $\\mathbb{L}$ is a $G$-loop, so that $\\varphi _{s}\\left( \n\\mathfrak{p}\\right) =\\mathfrak{l}^{\\left( s\\right) },$ then $\\varphi\n_{s}\\left( \\func{Ann}_{\\mathfrak{p}}\\left( b_{s}\\right) \\right) =\\mathcal{N\n^{R}\\left( \\mathfrak{l}^{\\left( s\\right) }\\right) .$ Hence, in this case, \n\\func{Ann}_{\\mathfrak{p}}\\left( b_{s}\\right) \\cong \\mathfrak{h}_{s}\\oplus \n\\mathcal{N}^{R}\\left( \\mathfrak{l}^{\\left( s\\right) }\\right) $.\n\nUsing the definition (\\ref{phis}) of $\\varphi _{s}$, let us consider the\naction of $\\mathfrak{p}$ on $\\varphi _{s}.$\n\n\\begin{lemma}\n\\label{lempactl}Suppose $\\xi ,\\eta \\in \\mathfrak{p}$, then for any $s\\in \n\\mathbb{L}$, we have \n\\begin{equation}\n\\xi \\cdot \\varphi _{s}\\left( \\eta \\right) -\\eta \\cdot \\varphi _{s}\\left( \\xi\n\\right) =\\varphi _{s}\\left( \\left[ \\xi ,\\eta \\right] _{\\mathfrak{p}}\\right) \n\\left[ \\varphi _{s}\\left( \\xi \\right) ,\\varphi _{s}\\left( \\eta \\right)\n\\right] ^{\\left( s\\right) },  \\label{xiphi}\n\\end{equation\nwhere $\\cdot $ means the action of $\\mathfrak{p}$ on $\\mathfrak{l}.$\n\\end{lemma}\n\n\\begin{proof}\nUsing (\\ref{actpl}) and the definition (\\ref{phis}) of $\\varphi _{s}$, we\nhave \n\\begin{eqnarray}\n\\xi \\cdot \\varphi _{s}\\left( \\eta \\right) &=&\\left. \\frac{d^{2}}{dtd\\tau \n\\exp \\left( t\\xi \\right) ^{\\prime }\\left( \\faktor{\\exp \\left( \\tau \\eta\n\\right) \\left( s\\right)}{s}\\right) \\right\\vert _{t,\\tau =0}  \\notag \\\\\n&=&\\left. \\frac{d^{2}}{dtd\\tau }\\faktor{\\exp \\left( t\\xi \\right) \\left( \\exp\n\\left( \\tau \\eta \\right) \\left( s\\right) \\right)}{\\exp \\left( t\\xi \\right)\n\\left( s\\right)} \\right\\vert _{t,\\tau =0}  \\notag \\\\\n&=&\\left. \\frac{d^{2}}{dtd\\tau }\\exp \\left( t\\xi \\right) \\left( \\exp \\left(\n\\tau \\eta \\right) \\left( s\\right) \\right) \/s\\right\\vert _{t,\\tau =0}  \\notag\n\\\\\n&&-\\left. \\frac{d^{2}}{dtd\\tau }\\left( \\faktor{\\exp \\left( \\tau \\eta \\right)\n\\left( s\\right)}{s}\\cdot \\exp \\left( t\\xi \\right) \\left( s\\right) \\right)\n\/s\\right\\vert _{t,\\tau =0}  \\notag \\\\\n&=&\\left. \\frac{d^{2}}{dtd\\tau }\\left( \\exp \\left( t\\xi \\right) \\exp \\left(\n\\tau \\eta \\right) \\right) \\left( s\\right) \/s\\right\\vert _{t,\\tau =0}\n\\label{xiphi1} \\\\\n&&-\\left. \\frac{d^{2}}{dtd\\tau }\\faktor{\\exp \\left( \\tau \\eta \\right) \\left(\ns\\right)}{s}\\circ _{s}\\faktor{\\exp \\left( t\\xi \\right) \\left( s\\right)} {s\n\\right\\vert _{t,\\tau =0},  \\notag\n\\end{eqnarray\nwhere we have used (\\ref{PsAutquot2a}) and Lemma \\ref{lemQuotient}. Now\nsubtracting the same expression but with $\\xi $ and $\\eta $ switched around,\nwe obtain (\\ref{xiphi}).\n\\end{proof}\n\n\\begin{remark}\nIn terms of the Chevalley-Eilenberg complex of $\\mathfrak{p}$ with values in \n$\\mathfrak{l},$ the relation (\\ref{xiphi}) shows that if we regard $\\varphi\n_{s}\\in C^{1}\\left( \\mathfrak{p};\\mathfrak{l}\\right) $, i.e. a $1$-form on \n\\mathfrak{p}$ with values in $\\mathfrak{l}$, then the Chevalley-Eilenberg\ndifferential $d_{CE}$ of $\\varphi _{s}$ is given by \n\\begin{equation}\n\\left( d_{CE}\\varphi _{s}\\right) \\left( \\xi ,\\eta \\right) =\\left[ \\varphi\n_{s}\\left( \\xi \\right) ,\\varphi _{s}\\left( \\eta \\right) \\right] ^{\\left(\ns\\right) }  \\label{dCEphis}\n\\end{equation\nfor any $\\xi ,\\eta \\in \\mathfrak{p}.$ It is interesting that, at least on \n\\mathfrak{q}_{s},$ the bracket $\\left[ \\cdot ,\\cdot \\right] ^{\\left(\ns\\right) }$ corresponds to an exact $2$-cochain.\n\\end{remark}\n\nSimilarly, from (\\ref{xiphi}), we then see that the action of $\\xi \\in \n\\mathfrak{p}$ on $\\varphi _{s}$ as an $\\mathfrak{p}^{\\ast }\\otimes \\mathfrak\nl}$-valued map. Indeed, given $\\xi ,\\eta \\in \\mathfrak{p},$ we have \n\\begin{eqnarray}\n\\left( \\xi \\cdot \\varphi _{s}\\right) \\left( \\eta \\right) &=&\\xi \\cdot\n\\varphi _{s}\\left( \\eta \\right) -\\varphi _{s}\\left( \\left[ \\xi ,\\eta \\right]\n_{\\mathfrak{p}}\\right)  \\notag \\\\\n&=&\\eta \\cdot \\varphi _{s}\\left( \\xi \\right) -\\left[ \\varphi _{s}\\left( \\eta\n\\right) ,\\varphi _{s}\\left( \\xi \\right) \\right] ^{\\left( s\\right) }\n\\label{xiphi3}\n\\end{eqnarray\nwhere we have first used the fact that $\\mathfrak{p}$ acts on itself via the\nadjoint representation and then (\\ref{xiphi}) in the second line.\n\nLet us now consider $\\func{Ann}_{\\mathfrak{p}}\\left( \\varphi _{s}\\right) .$\nFrom (\\ref{xiphi3}), we see that we have two equivalent characterizations of \n$\\func{Ann}_{\\mathfrak{p}}\\left( \\varphi _{s}\\right) .$ In particular, $\\xi\n\\in \\func{Ann}_{\\mathfrak{p}}\\left( \\varphi _{s}\\right) $ if and only if \n\\begin{equation}\n\\xi \\cdot \\hat{\\eta}=\\varphi _{s}\\left( \\left[ \\xi ,\\eta \\right] _{\\mathfrak\np}}\\right)  \\label{xiphieta}\n\\end{equation\nor equivalently, for $\\xi \\not\\in \\mathfrak{h}_{s},$ if and only if, \n\\begin{equation}\n\\eta \\cdot \\hat{\\xi}=\\left[ \\hat{\\eta},\\hat{\\xi}\\right] ^{\\left( s\\right) },\n\\label{etaphixi}\n\\end{equation\nfor any $\\eta \\in \\mathfrak{p.}$ Here we are again setting $\\hat{\\xi\n=\\varphi _{s}\\left( \\xi \\right) $ and $\\hat{\\eta}=\\varphi _{s}\\left( \\eta\n\\right) .$ In particular, (\\ref{xiphieta}) shows that $\\mathfrak{q}_{s}$ is\na representation of $\\func{Ann}_{\\mathfrak{p}}\\left( \\varphi _{s}\\right) .$\nSuppose now, $\\xi _{1},\\xi _{2}\\in \\func{Ann}_{\\mathfrak{p}}\\left( \\varphi\n_{s}\\right) ,$ then using (\\ref{xiphieta}) and (\\ref{etaphixi}), we find\nthat \n\\begin{equation}\n\\varphi _{s}\\left( \\left[ \\xi _{1},\\xi _{2}\\right] _{\\mathfrak{p}}\\right)\n=\\xi _{1}\\cdot \\hat{\\xi}_{2}=\\left[ \\hat{\\xi}_{1},\\hat{\\xi}_{2}\\right]\n^{\\left( s\\right) }.  \\label{annpphi}\n\\end{equation\nTherefore, $\\varphi _{s}\\left( \\func{Ann}_{\\mathfrak{p}}\\left( \\varphi\n_{s}\\right) \\right) $ is a Lie subalgebra of $\\mathfrak{l}^{\\left( s\\right)\n} $ with $\\varphi _{s}$ being a Lie algebra homomorphism. The kernel \n\\mathfrak{h}_{s}=\\ker \\varphi _{s}$ is then of course an ideal of $\\func{Ann\n_{\\mathfrak{p}}\\left( \\varphi _{s}\\right) .$ Thus, the quotient $\\func{Ann}_\n\\mathfrak{p}}\\left( \\varphi _{s}\\right) \/\\mathfrak{h}_{s}$ is again a Lie\nalgebra, and hence $\\func{Ann}_{\\mathfrak{p}}\\left( \\varphi _{s}\\right) $ is\na trivial Lie algebra extension of $\\mathfrak{h}_{s}$. Moreover, note that\nthe Lie algebra $\\func{Ann}_{\\mathfrak{p}}\\left( \\varphi _{s}\\right) $\ncorresponds to the Lie group $\\func{Stab}_{\\Psi ^{R}\\left( \\mathbb{L}\\right)\n}\\left( \\varphi _{s}\\right) $, and thus if $\\func{Aut}\\left( \\mathbb{L\n,\\circ _{s}\\right) $ and $\\func{Stab}_{\\Psi ^{R}\\left( \\mathbb{L}\\right)\n}\\left( \\varphi _{s}\\right) $ are both connected, then we see that $\\func{Au\n}\\left( \\mathbb{L},\\circ _{s}\\right) $ is a normal subgroup of $\\func{Stab\n_{\\Psi ^{R}\\left( \\mathbb{L}\\right) }\\left( \\varphi _{s}\\right) .$\n\nIn the special case when $\\mathbb{L}$ is a $G$-loop, we get a nice property\nof $\\func{Ann}_{\\mathfrak{p}}\\left( \\varphi _{s}\\right) .$\n\n\\begin{theorem}\nSuppose $\\mathbb{L}$ is a $G$-loop, then $\\func{Ann}_{\\mathfrak{p}}\\left(\n\\varphi _{s}\\right) \\subset \\func{Ann}_{\\mathfrak{p}}\\left( b_{s}\\right) .$\n\\end{theorem}\n\n\\begin{proof}\nSuppose $\\xi \\in \\func{Ann}_{\\mathfrak{p}}\\left( \\varphi _{s}\\right) $ and\nlet $\\eta ,\\gamma \\in \\mathfrak{p}$. Consider \n\\begin{eqnarray*}\n\\left[ \\gamma ,\\eta \\right] _{\\mathfrak{p}}\\cdot \\hat{\\xi} &=&\\gamma \\cdot\n\\left( \\eta \\cdot \\hat{\\xi}\\right) -\\eta \\cdot \\left( \\gamma \\cdot \\hat{\\xi\n\\right) \\\\\n&=&\\gamma \\cdot \\left[ \\hat{\\eta},\\hat{\\xi}\\right] ^{\\left( s\\right) }-\\eta\n\\cdot \\left[ \\hat{\\gamma},\\hat{\\xi}\\right] ^{\\left( s\\right) } \\\\\n&=&\\left[ \\gamma \\cdot \\hat{\\eta},\\hat{\\xi}\\right] ^{\\left( s\\right) }+\\left[\n\\hat{\\eta},\\gamma \\cdot \\hat{\\xi}\\right] ^{\\left( s\\right) }+a_{s}\\left( \n\\hat{\\eta},\\hat{\\xi},\\hat{\\gamma}\\right) \\\\\n&&-\\left[ \\eta \\cdot \\hat{\\gamma},\\hat{\\xi}\\right] ^{\\left( s\\right) }-\\left[\n\\hat{\\gamma},\\eta \\cdot \\hat{\\xi}\\right] ^{\\left( s\\right) }-a_{s}\\left( \n\\hat{\\gamma},\\hat{\\xi},\\hat{\\eta}\\right) ^{\\left( s\\right) } \\\\\n&=&\\left[ \\varphi _{s}\\left( \\left[ \\gamma ,\\eta \\right] _{\\mathfrak{p\n}\\right) ,\\hat{\\xi}\\right] ^{\\left( s\\right) }+\\left[ \\left[ \\hat{\\gamma}\n\\hat{\\eta}\\right] ^{\\left( s\\right) },\\hat{\\xi}\\right] +\\left[ \\hat{\\eta}\n\\left[ \\hat{\\gamma},\\hat{\\xi}\\right] ^{\\left( s\\right) }\\right] ^{\\left(\ns\\right) } \\\\\n&&-\\left[ \\hat{\\gamma},\\left[ \\hat{\\eta},\\hat{\\xi}\\right] ^{\\left( s\\right) \n\\right] ^{\\left( s\\right) } \\\\\n&&+a_{s}\\left( \\hat{\\eta},\\hat{\\xi},\\hat{\\gamma}\\right) -a_{s}\\left( \\hat\n\\gamma},\\hat{\\xi},\\hat{\\eta}\\right) \\\\\n&=&\\left[ \\gamma ,\\eta \\right] _{\\mathfrak{p}}\\cdot \\hat{\\xi}-a_{s}\\left( \n\\hat{\\gamma},\\hat{\\xi},\\hat{\\eta}\\right)\n\\end{eqnarray*\nwhere we have used (\\ref{etaphixi}), (\\ref{xilbrack}), (\\ref{xiphi}), and\nthe Akivis identity (\\ref{Jac2}). We hence find that \n\\begin{equation}\na_{s}\\left( \\hat{\\gamma},\\hat{\\xi},\\hat{\\eta}\\right) =0.  \\label{annphicond}\n\\end{equation\nWe know that if $\\mathbb{L}$ is a $G$-loop, then $\\mathfrak{l}^{\\left(\ns\\right) }=\\varphi _{s}\\left( \\mathfrak{p}\\right) ,$ and thus the condition \n\\ref{annphicond}) is the same as (\\ref{annpbs}), that is $\\xi \\in \\func{Ann\n_{\\mathfrak{p}}\\left( b_{s}\\right) .$\n\\end{proof}\n\n\\begin{remark}\nOverall, if $\\mathbb{L}$ is a $G$-loop, we have the following inclusions of\nLie algebras \n\\begin{equation}\n\\ker \\varphi _{s}=\\mathfrak{h}_{s}\\underset{\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{ideal}}{\\subset }\\func{Ann\n_{\\mathfrak{p}}\\left( \\varphi _{s}\\right) \\subset \\func{Ann}_{\\mathfrak{p\n}\\left( b_{s}\\right) \\cong \\mathfrak{h}_{s}\\oplus \\mathcal{N}^{R}\\left( \n\\mathfrak{l}^{\\left( s\\right) }\\right) \\subset \\mathfrak{p.}  \\label{lieseq}\n\\end{equation\nIf we look at the octonion case, with $\\mathbb{L=}U\\mathbb{O},$ then \n\\mathfrak{p=so}\\left( 7\\right) $, $\\mathfrak{h}_{s}\\cong \\mathfrak{g}_{2}.$\nMoreover, in this case, $\\mathcal{N}^{R}\\left( \\mathfrak{l}\\right) =\\left\\{\n0\\right\\} $, so we must have $\\mathfrak{h}_{s}=\\func{Ann}_{\\mathfrak{p\n}\\left( \\varphi _{s}\\right) =\\func{Ann}_{\\mathfrak{p}}\\left( b_{s}\\right) .$\nThis also makes sense because in this case, $\\varphi _{s}$ and $b_{s}$ are\nessentially the same objects, and moreover, almost uniquely determine $s$\n(up to $\\pm 1$). At the other extreme, if $\\mathbb{L}$ is associative, so\nthat $\\mathcal{N}^{R}\\left( \\mathfrak{l}\\right) =\\mathfrak{l},$ then $\\func\nAnn}_{\\mathfrak{p}}\\left( b_{s}\\right) =\\mathfrak{p,}$ but $\\func{Ann}_\n\\mathfrak{p}}\\left( \\varphi _{s}\\right) $ does not have to equal $\\func{Ann\n_{\\mathfrak{p}}\\left( b_{s}\\right) .$\n\\end{remark}\n\n\\begin{example}\n\\label{ExNormedDiv2}Using the setup from Examples \\ref{ExNormedDiv}, \\re\n{exCx2}, and \\ref{exQuat2}, if $\\mathbb{L=}U\\mathbb{C}$ with $\\Psi\n_{n}^{R}\\left( U\\mathbb{C}\\right) =U\\left( n\\right) $ or $\\mathbb{L=}\n\\mathbb{H}$ with $\\Psi _{n}^{R}\\left( U\\mathbb{H}\\right) =Sp\\left( n\\right)\nSp\\left( 1\\right) $, the since the partial action of $\\Psi _{n}^{R}$ in each\ncase here is trivial, from (\\ref{pactl1}), we see that the action of each\nLie algebra $\\mathfrak{p}_{n}$ on $\\mathfrak{l}$ is trivial. In the complex\ncase, $\\mathfrak{l\\cong }\\mathbb{R},$ and is thus abelian. Hence, from (\\re\n{xiphi3}), we see that in this case $\\xi \\cdot \\varphi _{s}=0$ for each $\\xi\n\\in \\mathfrak{p}_{n}.$ This makes because in Example \\ref{exCx2} we noted\nthat $\\varphi _{s}$ does not depend on $s$ in the complex case. In the\nquaternion case, (\\ref{xiphi3}) shows that if $\\xi ,\\eta \\in \\mathfrak{sp\n\\left( n\\right) \\oplus \\mathfrak{sp}\\left( 1\\right) =\\mathfrak{p}_{n}$, then \n\\begin{eqnarray}\n\\left( \\xi \\cdot \\varphi _{s}\\right) \\left( \\eta \\right) &=&-\\varphi\n_{s}\\left( \\left[ \\xi ,\\eta \\right] _{\\mathfrak{p}_{n}}\\right)  \\notag \\\\\n&=&-\\left[ \\xi _{1},\\eta _{1}\\right] _{\\func{Im}\\mathbb{H}}\n\\label{quatbrack}\n\\end{eqnarray\nwhere $\\xi _{1},\\eta _{1}$ are the $\\mathfrak{sp}\\left( 1\\right) $\ncomponents of $\\xi $ and $\\eta ,$ and $\\left[ \\cdot ,\\cdot \\right] _{\\func{I\n}\\mathbb{H}}$ is the bracket on $\\func{Im}\\mathbb{H}$ (and equivalently on \n\\mathfrak{sp}\\left( 1\\right) $). In particular, $\\func{Ann}_{\\mathfrak{p\n_{n}}\\left( \\varphi _{s}\\right) =\\mathfrak{sp}\\left( n\\right) .$\n\\end{example}\n\nNote that, while it is known that any simple (i.e. has no nontrivial proper\nnormal subloops) Moufang loop is a $G$-loop, it is not known whether there\nare simple Bol loops that are not $G$-loops \\cite{NagyLoop}. On the other\nhand, there is an example of a Bol loop that is a $G$-loop but is not a\nMoufang loop \\cite{Robinson68}. That particular example is constructed from\nan alternative division ring, but if that is taken to be $\\mathbb{O},$ we\nobtain a smooth loop.\n\n\\subsection{Killing form}\n\n\\label{sectKilling}Similarly as for Lie groups, we may define a Killing form \n$K^{\\left( s\\right) }$ on $\\mathfrak{l}^{\\left( s\\right) }$. For $\\xi ,\\eta\n\\in \\mathfrak{l}$, we have \n\\begin{equation}\nK^{\\left( s\\right) }\\left( \\xi ,\\eta \\right) =\\func{Tr}\\left( \\func{ad}_{\\xi\n}^{\\left( s\\right) }\\circ \\func{ad}_{\\eta }^{\\left( s\\right) }\\right) ,\n\\label{Killing}\n\\end{equation\nwhere $\\circ $ is just composition of linear maps on $\\mathfrak{l}$ and \n\\func{ad}_{\\xi }^{\\left( s\\right) }\\left( \\cdot \\right) =\\left[ \\xi ,\\cdot\n\\right] ^{\\left( s\\right) },$ as in (\\ref{ladpx}). Clearly $K^{\\left(\ns\\right) }$ is a symmetric bilinear form on $\\mathfrak{l.}$ Given the form \nK^{\\left( s\\right) }$ on $\\mathfrak{l}$, we can extend it to a\n\\textquotedblleft right-invariant\\textquotedblright\\ form $\\left\\langle\n{}\\right\\rangle ^{\\left( s\\right) }$ on $\\mathbb{L}$ via right translation,\nso that for vector fields $X,Y$ on $\\mathbb{L}$, \n\\begin{equation}\n\\left\\langle X,Y\\right\\rangle _{\\mathbb{L}}^{\\left( s\\right) }=K^{\\left(\ns\\right) }\\left( \\theta \\left( X\\right) ,\\theta \\left( Y\\right) \\right) .\n\\label{Killing2}\n\\end{equation}\n\n\\begin{theorem}\n\\label{thmKillingprop}The bilinear form $K^{\\left( s\\right) }$ (\\ref{Killing\n) on $\\mathfrak{l}$ has the following properties.\n\n\\begin{enumerate}\n\\item Let $h\\in \\Psi ^{R}\\left( \\mathbb{L}\\right) $, then for any $\\xi ,\\eta\n\\in \\mathfrak{l},$ \n\\begin{equation}\nK^{\\left( h\\left( s\\right) \\right) }\\left( h_{\\ast }^{\\prime }\\xi ,h_{\\ast\n}^{\\prime }\\eta \\right) =K^{\\left( s\\right) }\\left( \\xi ,\\eta \\right) .\n\\label{Kpsi}\n\\end{equation}\n\n\\item Suppose also $\\gamma \\in \\mathfrak{l,}$ then \n\\begin{eqnarray}\nK^{\\left( s\\right) }\\left( \\func{ad}_{\\gamma }^{\\left( s\\right) }\\eta ,\\xi\n\\right) &=&-K^{\\left( s\\right) }\\left( \\eta ,\\func{ad}_{\\gamma }^{\\left(\ns\\right) }\\xi \\right) +\\func{Tr}\\left( \\func{Jac}_{\\xi ,\\gamma }^{\\left(\ns\\right) }\\circ \\func{ad}_{\\eta }^{\\left( s\\right) }\\right)  \\notag \\\\\n&&+\\func{Tr}\\left( \\func{Jac}_{\\eta ,\\gamma }^{\\left( s\\right) }\\circ \\func\nad}_{\\xi }^{\\left( s\\right) }\\right) ,  \\label{Kad}\n\\end{eqnarray\nwhere $\\func{Jac}_{\\gamma ,\\xi }^{\\left( s\\right) }:\\mathfrak{l\n\\longrightarrow \\mathfrak{l}$ is given by $\\func{Jac}_{\\eta ,\\gamma\n}^{\\left( s\\right) }\\left( \\xi \\right) =\\func{Jac}^{\\left( s\\right) }\\left(\n\\xi ,\\eta ,\\gamma \\right) .$\n\n\\item Let $\\alpha \\in \\mathfrak{p},$ then \n\\begin{eqnarray}\nK^{\\left( s\\right) }\\left( \\alpha \\cdot \\xi ,\\eta \\right) &=&-K^{\\left(\ns\\right) }\\left( \\xi ,\\alpha \\cdot \\eta \\right) +\\func{Tr}\\left( a_{\\eta \n\\hat{\\alpha}}^{\\left( s\\right) }\\circ \\func{ad}_{\\xi }^{\\left( s\\right)\n}\\right)  \\label{Klie} \\\\\n&&+\\func{Tr}\\left( a_{\\xi ,\\hat{\\alpha}}^{\\left( s\\right) }\\circ \\func{ad\n_{\\eta }^{\\left( s\\right) }\\right) ,  \\notag\n\\end{eqnarray\nwhere $a_{\\xi ,\\eta }^{\\left( s\\right) }:\\mathfrak{l}\\longrightarrow \n\\mathfrak{l}$ is given by $a_{\\xi ,\\eta }^{\\left( s\\right) }\\left( \\gamma\n\\right) =\\left[ \\gamma ,\\xi ,\\eta \\right] ^{\\left( s\\right) }-\\left[ \\xi\n,\\gamma ,\\eta \\right] ^{\\left( s\\right) }$ and $\\hat{\\alpha}=\\varphi\n_{s}\\left( \\alpha \\right) $.\n\\end{enumerate}\n\\end{theorem}\n\nThe proof of Theorem (\\ref{thmKillingprop}) is given in Appendix \\re\n{secAppendix}.\n\n\\begin{remark}\nIf $\\left( \\mathbb{L},\\circ _{s}\\right) $ is an alternative loop, we know\nthat $\\func{Jac}_{\\eta ,\\gamma }^{\\left( s\\right) }=3a^{\\left( s\\right) },$\nso in that in case, $K^{\\left( s\\right) }$ is invariant with respect to both \n$\\func{ad}^{\\left( s\\right) }$ and the action of $\\mathfrak{p}\\ $if and only\nif \n\\begin{equation}\n\\func{Tr}\\left( a_{\\eta ,\\hat{\\alpha}}^{\\left( s\\right) }\\circ \\func{ad\n_{\\xi }^{\\left( s\\right) }\\right) +\\func{Tr}\\left( a_{\\xi ,\\hat{\\alpha\n}^{\\left( s\\right) }\\circ \\func{ad}_{\\eta }^{\\left( s\\right) }\\right) =0.\n\\end{equation\nIndeed, in \\cite{SagleMalcev}, it is shown that for a Malcev algebra, the\nKilling form is $\\func{ad}$-invariant. A Malcev algebra is alternative and\nhence the Killing form is also $\\mathfrak{p}$-invariant in that case.\nMoreover, it shown in \\cite{LoosMalcev} that for a \\emph{semisimple} Malcev\nalgebra, the Killing form is non-degenerate. Here the definition of\n\\textquotedblleft semisimple\\textquotedblright\\ is the same as for Lie\nalgebras, namely that the maximal solvable ideal is zero. Indeed, given the\nalgebra of imaginary octonions on $\\mathbb{R}^{7},$ it is known that the\ncorresponding Killing form is negative-definite \\cite{BaezOcto}. Moreover,\nsince in this case, the pseudoautomorphism group is $SO\\left( 7\\right) ,$ so\n(\\ref{Kpsi}) actually shows that $K^{h\\left( s\\right) }=K^{s}$ for every $h\n, and thus is independent of $s$. General criteria for a loop algebra to\nadmit an invariant definite (or even just non-degenerate) Killing form do\nnot seem to appear in the literature, and could be the subject of further\nstudy. At least for well-behaved loops, such as Malcev loops, it is likely\nthat there is significant similarity to Lie groups.\n\\end{remark}\n\nSuppose now $K^{\\left( s\\right) }$ is nondegenerate and both $\\func{ad\n^{\\left( s\\right) }$- and $\\mathfrak{p}$-invariant, and moreover suppose \n\\mathfrak{p}$ is semisimple itself, so that it has a nondegenerate,\ninvariant Killing form $K_{\\mathfrak{p}}.$ We will use $\\left\\langle\n{}\\right\\rangle ^{\\left( s\\right) }$ and $\\left\\langle {}\\right\\rangle _\n\\mathfrak{p}}$ to denote the inner products using $K^{\\left( s\\right) }$ and \n$K_{\\mathfrak{p},}$ respectively. Then, given the map $\\varphi _{s}\n\\mathfrak{p}\\longrightarrow \\mathfrak{l}^{\\left( s\\right) }$, we can define\nits adjoint with respect to these two bilinear maps.\n\n\\begin{definition}\nDefine the map $\\varphi _{s}^{t}:\\mathfrak{l}^{\\left( s\\right)\n}\\longrightarrow \\mathfrak{p}$ such that for any $\\xi \\in \\mathfrak{l\n^{\\left( s\\right) }$\\ and $\\eta \\in \\mathfrak{p}$, \n\\begin{equation}\n\\left\\langle \\varphi _{s}^{t}\\left( \\xi \\right) ,\\eta \\right\\rangle _\n\\mathfrak{p}}=\\left\\langle \\xi ,\\varphi _{s}\\left( \\eta \\right)\n\\right\\rangle ^{\\left( s\\right) }.  \\label{phiadj}\n\\end{equation}\n\\end{definition}\n\nSince $\\mathfrak{h}_{s}\\cong \\ker \\varphi _{s}$, we then clearly have \n\\mathfrak{p\\cong h}_{s}\\oplus \\func{Im}\\varphi _{s}^{t}$, so that $\\mathfrak\nh}_{s}^{\\perp }=\\func{Im}\\varphi _{s}^{t}.$ On the other hand, we also have \n\\mathfrak{l}^{\\left( s\\right) }\\cong \\ker \\varphi _{s}^{t}\\oplus \\mathfrak{q\n_{s}$, since $\\mathfrak{q}_{s}=\\func{Im}\\varphi _{s}.$ Define the\ncorresponding projections $\\pi _{\\mathfrak{h}_{s}},\\pi _{\\mathfrak{h\n_{s}^{\\perp }}$ and $\\pi _{\\mathfrak{q}_{s}},\\pi _{\\mathfrak{q}_{s}^{\\perp\n}.}$We then have the following properties.\n\n\\begin{lemma}\n\\label{lemphisphist}Suppose $\\mathfrak{q}_{s}$ is an irreducible\nrepresentation of $\\mathfrak{h}\\ $and suppose the base field of $\\mathfrak{p}\n$ is $\\mathbb{F=R}$ or $\\mathbb{C}.$ Then, there exists a $\\lambda _{s}\\in $ \n$\\mathbb{F}$ such that \n\\begin{equation}\n\\varphi _{s}\\varphi _{s}^{t}=\\lambda _{s}\\pi _{\\mathfrak{q}^{\\left( s\\right)\n}}\\ \\ \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{and }\\varphi _{s}^{t}\\varphi _{s}=\\lambda _{s}\\pi _{\\mathfrak{h\n_{s}^{\\perp }}.  \\label{phistphis}\n\\end{equation\nMoreover, for any $h\\in \\Psi ^{R}\\left( \\mathbb{L}\\right) $, $\\lambda\n_{s}=\\lambda _{h\\left( s\\right) }$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\gamma ,\\eta \\in \\mathfrak{p}$ and $\\xi \\in \\mathfrak{l}^{\\left(\ns\\right) }$, then using (\\ref{xiphi3}), \n\\begin{eqnarray}\n\\left\\langle \\left( \\gamma \\cdot \\varphi _{s}^{t}\\right) \\left( \\xi \\right)\n,\\eta \\right\\rangle _{\\mathfrak{p}} &=&\\left\\langle \\left[ \\gamma ,\\varphi\n_{s}^{t}\\left( \\xi \\right) \\right] _{\\mathfrak{p}},\\eta \\right\\rangle _\n\\mathfrak{p}}-\\left\\langle \\varphi _{s}^{t}\\left( \\gamma \\cdot \\xi \\right)\n,\\eta \\right\\rangle _{\\mathfrak{p}}  \\notag \\\\\n&=&-\\left\\langle \\varphi _{s}^{t}\\left( \\xi \\right) ,\\left[ \\gamma ,\\eta\n\\right] _{\\mathfrak{p}}\\right\\rangle -\\left\\langle \\gamma \\cdot \\xi ,\\varphi\n_{s}\\left( \\eta \\right) \\right\\rangle ^{\\left( s\\right) }  \\notag \\\\\n&=&\\left\\langle \\xi ,\\gamma \\cdot \\varphi _{s}\\left( \\eta \\right) -\\varphi\n_{s}\\left( \\left[ \\gamma ,\\eta \\right] _{\\mathfrak{p}}\\right) \\right\\rangle\n^{\\left( s\\right) }  \\notag \\\\\n&=&\\left\\langle \\xi ,\\left( \\gamma \\cdot \\varphi _{s}\\right) \\left( \\eta\n\\right) \\right\\rangle ^{\\left( s\\right) },  \\label{gammaphit}\n\\end{eqnarray\nso in particular, $\\func{Ann}_{\\mathfrak{p}}\\left( \\varphi _{s}\\right) \n\\func{Ann}_{\\mathfrak{p}}\\left( \\varphi _{s}^{t}\\right) .$ Thus, the map \n\\varphi _{s}\\varphi _{s}^{t}:\\mathfrak{l}^{\\left( s\\right) }\\longrightarrow \n\\mathfrak{l}^{\\left( s\\right) }$ is an equivariant map of representations of\nthe Lie subalgebra $\\func{Ann}_{\\mathfrak{p}}\\left( \\varphi _{s}\\right)\n\\subset \\mathfrak{p}\\ $and is moreover self-adjoint with respect to \n\\left\\langle {}\\right\\rangle ^{\\left( s\\right) }.$ We can also restrict this\nmap to $\\mathfrak{q}_{s},$ which is also a representation of $\\func{Ann}_\n\\mathfrak{p}}\\left( \\varphi _{s}\\right) $, and in particular of $\\mathfrak{h\n_{s}.$ Hence, if $\\mathfrak{q}_{s}$ is an irreducible representation of \n\\mathfrak{h}_{s},$ since $\\varphi _{s}\\varphi _{s}^{t}$ is diagonalizable\n(in general, if $\\mathbb{C}$ is the base field, or because it symmetric if\nthe base field is $\\mathbb{R}$), by Schur's Lemma, there exists some number \n\\lambda _{s}\\neq 0$ such that \n\\begin{equation}\n\\left. \\varphi _{s}\\varphi _{s}^{t}\\right\\vert _{\\mathfrak{q}^{\\left(\ns\\right) }}=\\lambda _{s}\\func{id}_{\\mathfrak{q}^{\\left( s\\right) }}.\n\\label{phistid}\n\\end{equation\nApplying $\\varphi _{s}^{t}$ to (\\ref{phistid}), we also obtain. \n\\begin{equation}\n\\left. \\varphi _{s}^{t}\\varphi _{s}\\right\\vert _{\\mathfrak{h}_{s}^{\\perp\n}}=\\lambda _{s}\\func{id}_{\\mathfrak{h}_{s}^{\\perp }}.\n\\end{equation\nSince $\\varphi _{s}^{t}$ and $\\varphi _{s}$ vanish on $\\mathfrak{q\n_{s}^{\\perp }$ and $\\mathfrak{h}_{s}$, respectively, we obtain (\\re\n{phistphis}).\n\nLet $h\\in \\Psi ^{R}\\left( \\mathbb{L}\\right) $, then from (\\ref{phihs}),\nrecall that \n\\begin{equation}\n\\varphi _{h\\left( s\\right) }=\\left( h^{\\prime }\\right) _{\\ast }\\circ \\varphi\n_{s}\\circ \\left( \\func{Ad}_{h}^{-1}\\right) _{\\ast }.\n\\end{equation\nIt is then easy to see using (\\ref{Kpsi}) and the invariance of the Killing\nform on $\\mathfrak{p}$ that \n\\begin{equation}\n\\varphi _{h\\left( s\\right) }^{t}=\\left( \\func{Ad}_{h}\\right) _{\\ast }\\circ\n\\varphi _{s}^{t}\\circ \\left( h^{\\prime }\\right) _{\\ast }^{-1}.\n\\label{phiths}\n\\end{equation\nIn particular, we see that \n\\begin{equation*}\n\\left( h^{\\prime }\\right) _{\\ast }\\mathfrak{q}_{s}=\\mathfrak{q}_{h\\left(\ns\\right) }\\ \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{and }\\left( \\func{Ad}_{h}\\right) _{\\ast }\\mathfrak{h\n_{s}^{\\perp }=\\mathfrak{h}_{s}.\n\\end{equation*\nHence, \n\\begin{eqnarray*}\n\\left. \\varphi _{h\\left( s\\right) }\\varphi _{h\\left( s\\right)\n}^{t}\\right\\vert _{\\mathfrak{q}_{h\\left( s\\right) }} &=&\\left. \\left(\nh^{\\prime }\\right) _{\\ast }\\circ \\varphi _{s}\\varphi _{s}^{t}\\circ \\left(\nh^{\\prime }\\right) _{\\ast }^{-1}\\right\\vert _{\\mathfrak{q}_{h\\left( s\\right)\n}} \\\\\n&=&\\lambda _{s}\\func{id}_{\\mathfrak{q}_{h\\left( s\\right) }}\n\\end{eqnarray*\nand so indeed, $\\lambda _{s}=\\lambda _{h\\left( s\\right) }.$\n\\end{proof}\n\n\\begin{example}\nIn the case of octonions, suppose we set $\\varphi _{s}\\left( \\eta \\right)\n_{a}=k\\varphi _{abc}\\eta ^{bc}$ where $\\eta \\in \\mathfrak{so}\\left( 7\\right)\n\\cong \\Lambda ^{2}\\left( \\mathbb{R}^{7}\\right) ^{\\ast }$, $\\varphi $ is the\ndefining $3$-form on $\\mathbb{R}^{7},$ and $k\\in \\mathbb{R}$ is some\nconstant. Then, $\\varphi _{s}^{t}\\left( \\gamma \\right) _{ab}=k\\varphi\n_{abc}\\gamma ^{c}$ where $\\gamma \\in \\mathbb{R}^{7}\\cong \\func{Im}\\mathbb{O\n. $ Now, $\\mathbb{R}^{7}$ is an irreducible representation of $\\mathfrak{g\n_{2} $, so the hypothesis of Lemma \\ref{lemphisphist} is satisfied. In this\ncase, $\\lambda _{s}=6k^{2}$ due to the contraction identities for $\\varphi $ \n\\cite{GrigorianG2Torsion1,karigiannis-2005-57}.\n\\end{example}\n\nConsider the action of $\\varphi _{s}^{t}\\left( \\mathfrak{l}^{\\left( s\\right)\n}\\right) \\subset \\mathfrak{p}$ on $\\mathfrak{q}_{s}.$ Let $\\xi ,\\eta \\in \n\\mathfrak{q}_{s},$ then from (\\ref{xiphi})\n\\begin{equation}\n\\varphi _{s}^{t}\\left( \\xi \\right) \\cdot \\varphi _{s}\\varphi _{s}^{t}\\left(\n\\eta \\right) -\\varphi _{s}^{t}\\left( \\eta \\right) \\cdot \\varphi _{s}\\varphi\n_{s}^{t}\\left( \\xi \\right) =\\varphi _{s}\\left( \\left[ \\varphi _{s}^{t}\\left(\n\\xi \\right) ,\\varphi _{s}^{t}\\left( \\eta \\right) \\right] _{\\mathfrak{p\n}\\right) +\\left[ \\varphi _{s}\\varphi _{s}^{t}\\left( \\xi \\right) ,\\varphi\n_{s}\\varphi _{s}^{t}\\left( \\eta \\right) \\right] ^{\\left( s\\right) },\n\\end{equation\nand thus, \n\\begin{equation}\n\\varphi _{s}^{t}\\left( \\xi \\right) \\cdot \\eta -\\varphi _{s}^{t}\\left( \\eta\n\\right) \\cdot \\xi =\\frac{1}{\\lambda _{s}}\\varphi _{s}\\left( \\left[ \\varphi\n_{s}^{t}\\left( \\xi \\right) ,\\varphi _{s}^{t}\\left( \\eta \\right) \\right] _\n\\mathfrak{p}}\\right) +\\lambda _{s}\\left[ \\xi ,\\eta \\right] ^{\\left( s\\right)\n}.  \\label{phistxi}\n\\end{equation\nWe now show that $\\varphi _{s}^{t}\\left( \\xi \\right) \\cdot \\eta $ is\nskew-symmetric when restricted to $\\mathfrak{q}_{s}$ and then projected back\nto $\\mathfrak{q}_{s}.$\n\n\\begin{lemma}\n\\label{lemPhibrack}Suppose $\\mathbb{L}$ is a loop and $s\\in \\mathbb{L}$,\nsuch that the Killing form is non-degenerate and $\\func{ad}^{\\left( s\\right)\n}$- and $\\mathfrak{p}$-invariant. Then, for any $\\xi ,\\eta \\in \\mathfrak{q\n_{s}$, \n\\begin{equation}\n\\pi _{\\mathfrak{q}_{s}}\\left( \\varphi _{s}^{t}\\left( \\xi \\right) \\cdot \\eta\n\\right) =-\\pi _{\\mathfrak{q}_{s}}\\left( \\varphi _{s}^{t}\\left( \\eta \\right)\n\\cdot \\xi \\right) .  \\label{piqsphit}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nSuppose $\\xi ,\\eta \\in \\mathfrak{q}_{s},$ then using the $\\func{ad}^{\\left(\ns\\right) }$- and $\\mathfrak{p}$-invariance of the Killing form on $\\mathfrak\nl}^{\\left( s\\right) }$ and (\\ref{phistxi}) we have \n\\begin{eqnarray*}\n\\left\\langle \\varphi _{s}^{t}\\left( \\eta \\right) \\cdot \\eta ,\\xi\n\\right\\rangle ^{\\left( s\\right) } &=&-\\left\\langle \\eta ,\\varphi\n_{s}^{t}\\left( \\eta \\right) \\cdot \\xi \\right\\rangle ^{\\left( s\\right) } \\\\\n&=&-\\left\\langle \\eta ,\\varphi _{s}^{t}\\left( \\xi \\right) \\cdot \\eta -\\frac{\n}{\\lambda _{s}}\\varphi _{s}\\left( \\left[ \\varphi _{s}^{t}\\left( \\xi \\right)\n,\\varphi _{s}^{t}\\left( \\eta \\right) \\right] _{\\mathfrak{p}}\\right) -\\lambda\n_{s}\\left[ \\xi ,\\eta \\right] ^{\\left( s\\right) }\\right\\rangle ^{\\left(\ns\\right) } \\\\\n&=&-\\left\\langle \\eta ,\\varphi _{s}^{t}\\left( \\xi \\right) \\cdot \\eta\n\\right\\rangle ^{\\left( s\\right) }+\\frac{1}{\\lambda _{s}}\\left\\langle \\varphi\n_{s}^{t}\\left( \\eta \\right) ,\\left[ \\varphi _{s}^{t}\\left( \\xi \\right)\n,\\varphi _{s}^{t}\\left( \\eta \\right) \\right] _{\\mathfrak{p}}\\right\\rangle \\\\\n&&-\\lambda _{s}\\left\\langle \\left[ \\eta ,\\eta \\right] ^{\\left( s\\right)\n},\\xi \\right\\rangle ^{\\left( s\\right) } \\\\\n&=&-\\left\\langle \\eta ,\\varphi _{s}^{t}\\left( \\xi \\right) \\cdot \\eta\n\\right\\rangle ^{\\left( s\\right) }=\\left\\langle \\varphi _{s}^{t}\\left( \\xi\n\\right) \\cdot \\eta ,\\eta \\right\\rangle ^{\\left( s\\right) } \\\\\n&=&0.\n\\end{eqnarray*\nThus, we see that $\\pi _{\\mathfrak{q}_{s}}\\left( \\varphi _{s}^{t}\\left( \\eta\n\\right) \\cdot \\eta \\right) =0,$ and hence (\\ref{piqsphit}) holds.\n\\end{proof}\n\nTaking the $\\pi _{\\mathfrak{q}_{s}}$ projection of (\\ref{phistxi}) gives \n\\begin{equation}\n\\pi _{\\mathfrak{q}_{s}}\\left( \\varphi _{s}^{t}\\left( \\xi \\right) \\cdot \\eta\n\\right) =\\frac{1}{2\\lambda _{s}}\\varphi _{s}\\left( \\left[ \\varphi\n_{s}^{t}\\left( \\xi \\right) ,\\varphi _{s}^{t}\\left( \\eta \\right) \\right] _\n\\mathfrak{p}}+\\lambda _{s}\\varphi _{s}^{t}\\left( \\left[ \\xi ,\\eta \\right]\n^{\\left( s\\right) }\\right) \\right) .  \\label{piqsact}\n\\end{equation\nThe relation (\\ref{piqsact}) suggests that we can define a new bracket \n\\left[ \\cdot ,\\cdot \\right] _{\\varphi _{s}}$ on $\\mathfrak{l}^{\\left(\ns\\right) }$ using $\\varphi _{s}$.\n\n\\begin{definition}\nSuppose $\\mathbb{L}$ satisfies the assumptions of Lemma \\ref{lemPhibrack}.\nThen, for $\\xi ,\\eta \\in \\mathfrak{l}^{\\left( s\\right) }$, define \n\\begin{equation}\n\\left[ \\xi ,\\eta \\right] _{\\varphi _{s}}=\\varphi _{s}\\left( \\left[ \\varphi\n_{s}^{t}\\left( \\xi \\right) ,\\varphi _{s}^{t}\\left( \\eta \\right) \\right] _\n\\mathfrak{p}}\\right) .  \\label{phisbrack}\n\\end{equation}\n\\end{definition}\n\nThis bracket restricts to $\\mathfrak{q}_{s}$ and vanishes on $\\mathfrak{q\n_{s}^{\\perp }$, so that $\\mathfrak{q}_{s}^{\\perp }$ is an abelian ideal with\nrespect to it. We can rewrite (\\ref{piqsact}) as \n\\begin{equation}\n\\pi _{\\mathfrak{q}_{s}}\\left( \\varphi _{s}^{t}\\left( \\xi \\right) \\cdot \\eta\n\\right) =\\frac{1}{2\\lambda _{s}}\\left[ \\xi ,\\eta \\right] _{\\varphi _{s}}\n\\frac{\\lambda _{s}}{2}\\pi _{\\mathfrak{q}_{s}}\\left( \\left[ \\xi ,\\eta \\right]\n^{\\left( s\\right) }\\right) .  \\label{piqsact1}\n\\end{equation}\n\n\\begin{example}\nIn the case of octonions, if, as before, we set $\\varphi _{s}\\left( \\eta\n\\right) _{a}=k\\varphi _{abc}\\eta ^{bc}$ and $\\left( \\left[ \\xi ,\\gamma\n\\right] ^{\\left( s\\right) }\\right) _{a}=2\\varphi _{abc}\\xi ^{b}\\gamma ^{c}$,\nwe find that $\\left[ \\cdot ,\\cdot \\right] _{\\varphi _{s}}=3k^{3}\\left[ \\cdot\n,\\cdot \\right] ^{\\left( s\\right) }.$ Then, recalling that $\\lambda\n_{s}=6k^{2}$, (\\ref{piqsact1}) shows that in this case \n\\begin{equation*}\n\\varphi _{s}^{t}\\left( \\xi \\right) \\cdot \\gamma =\\left( \\frac{k}{4\n+3k^{2}\\right) \\left[ \\xi ,\\gamma \\right] ^{\\left( s\\right) },\n\\end{equation*\nand to be consistent with the standard action of $\\mathfrak{so}\\left(\n7\\right) $ on $\\mathbb{R}^{7}$, we must have \n\\begin{equation*}\nk\\varphi _{abc}\\xi ^{c}\\gamma ^{b}=\\left( \\frac{k}{2}+6k^{2}\\right) \\varphi\n_{abc}\\xi ^{b}\\gamma ^{c},\n\\end{equation*\nwhich means that $6k^{2}+\\frac{3}{2}k=0$ and therefore, $k=-\\frac{1}{4}.$\nThis also implies that $\\lambda _{s}=\\frac{3}{8}$ in this case.\n\\end{example}\n\n\\begin{example}\nIf $\\mathbb{L}$ is a Lie group, and $\\Psi ^{R}\\left( \\mathbb{L}\\right) $ is\nthe full group of pseudoautomorphism pairs, then $\\mathfrak{p\\cong aut\n\\left( \\mathbb{L}\\right) \\oplus \\mathfrak{l}$, where $\\mathfrak{aut}\\left( \n\\mathbb{L}\\right) $ is the Lie algebra of $\\func{Aut}\\left( \\mathbb{L\n\\right) $ and $\\mathfrak{l}$ is the Lie algebra of $\\mathbb{L}.$ In this\ncase, $\\varphi _{s}^{t}\\varphi _{s}$ is just the projection to $\\mathfrak\nl\\subset p},$ and thus $\\lambda _{s}=1$ and $\\left[ \\cdot ,\\cdot \\right]\n_{\\varphi _{s}}=\\left[ \\cdot ,\\cdot \\right] ^{\\left( s\\right) }.$ Then (\\re\n{piqsact1}) just shows that $\\mathfrak{l}$ acts on itself via the adjoint\nrepresentation.\n\\end{example}\n\n\\begin{remark}\nBoth of the above examples have the two brackets $\\left[ \\cdot ,\\cdot \\right]\n_{\\varphi _{s}}$and $\\left[ \\cdot ,\\cdot \\right] ^{\\left( s\\right) }$\nproportional to one another. This is really means that $\\mathfrak{l}^{\\left(\ns\\right) }$ and $\\mathfrak{h}_{s}^{\\perp }$ have equivalent $\\mathbb{L}\n-algebra structures with $\\varphi _{s}$ and $\\varphi _{s}^{t}$ (up to a\nconstant factor) being the corresponding isomorphisms. It is not clear if\nthis is always the case.\n\\end{remark}\n\nThe bracket $\\left[ \\cdot ,\\cdot \\right] _{\\varphi _{s}}$ has some\nreasonable properties.\n\n\\begin{lemma}\n\\label{lemPhibrack2}Under the assumptions of Lemma \\ref{lemPhibrack}, the\nbracket $\\left[ \\cdot ,\\cdot \\right] _{\\varphi _{s}}$ satisfies the\nfollowing properties. Let $\\xi ,\\eta ,\\gamma \\in \\mathfrak{l}$, then\n\n\\begin{enumerate}\n\\item $\\left\\langle \\left[ \\xi ,\\eta \\right] _{\\varphi _{s}},\\gamma\n\\right\\rangle ^{\\left( s\\right) }=-\\left\\langle \\eta ,\\left[ \\xi ,\\gamma\n\\right] _{\\varphi _{s}}\\right\\rangle ^{\\left( s\\right) }.$\n\n\\item For any $h\\in \\Psi ^{R}\\left( \\mathbb{L}\\right) $, $\\left[ \\xi ,\\eta\n\\right] _{\\varphi _{h\\left( s\\right) }}=\\left( h^{\\prime }\\right) _{\\ast \n\\left[ \\left( h^{\\prime }\\right) _{\\ast }^{-1}\\xi ,\\left( h^{\\prime }\\right)\n_{\\ast }^{-1}\\eta \\right] _{\\varphi _{s}}.$\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nThe first property follows directly from the definition (\\ref{phisbrack})\nand the $\\func{ad}$-invariance of the Killing form on $\\mathfrak{p}.$\nIndeed, \n\\begin{eqnarray*}\n\\left\\langle \\left[ \\xi ,\\eta \\right] _{\\varphi _{s}},\\gamma \\right\\rangle\n^{\\left( s\\right) } &=&\\left\\langle \\varphi _{s}\\left( \\left[ \\varphi\n_{s}^{t}\\left( \\xi \\right) ,\\varphi _{s}^{t}\\left( \\eta \\right) \\right] _\n\\mathfrak{p}}\\right) ,\\gamma \\right\\rangle ^{\\left( s\\right) } \\\\\n&=&\\left\\langle \\left[ \\varphi _{s}^{t}\\left( \\xi \\right) ,\\varphi\n_{s}^{t}\\left( \\eta \\right) \\right] _{\\mathfrak{p}},\\varphi _{s}^{t}\\left(\n\\gamma \\right) \\right\\rangle ^{\\left( s\\right) } \\\\\n&=&-\\left\\langle \\varphi _{s}^{t}\\left( \\eta \\right) ,\\left[ \\varphi\n_{s}^{t}\\left( \\xi \\right) ,\\varphi _{s}^{t}\\left( \\gamma \\right) \\right] _\n\\mathfrak{p}}\\right\\rangle ^{\\left( s\\right) } \\\\\n&=&-\\left\\langle \\eta ,\\left[ \\xi ,\\gamma \\right] _{\\varphi\n_{s}}\\right\\rangle ^{\\left( s\\right) }.\n\\end{eqnarray*\nNow let $h\\in \\Psi ^{R}\\left( \\mathbb{L}\\right) $, and then since $\\left( \n\\func{Ad}_{h}\\right) _{\\ast }$ is a Lie algebra automorphism of $\\mathfrak{p}\n$, we have \n\\begin{eqnarray}\n\\left[ \\xi ,\\eta \\right] _{\\varphi _{h\\left( s\\right) }} &=&\\varphi\n_{h\\left( s\\right) }\\left( \\left[ \\varphi _{h\\left( s\\right) }^{t}\\left( \\xi\n\\right) ,\\varphi _{h\\left( s\\right) }^{t}\\left( \\eta \\right) \\right] _\n\\mathfrak{p}}\\right)  \\notag \\\\\n&=&\\left( h^{\\prime }\\right) _{\\ast }\\circ \\varphi _{s}\\circ \\left( \\func{Ad\n_{h}^{-1}\\right) _{\\ast }\\left( \\left[ \\left( \\func{Ad}_{h}\\right) _{\\ast\n}\\left( \\varphi _{s}^{t}\\left( \\left( h^{\\prime }\\right) _{\\ast }^{-1}\\left(\n\\xi \\right) \\right) \\right) ,\\left( \\func{Ad}_{h}\\right) _{\\ast }\\left(\n\\varphi _{s}^{t}\\left( \\left( h^{\\prime }\\right) _{\\ast }^{-1}\\left( \\eta\n\\right) \\right) \\right) \\right] _{\\mathfrak{p}}\\right)  \\notag \\\\\n&=&\\left( h^{\\prime }\\right) _{\\ast }\\circ \\varphi _{s}\\left( \\left[ \\varphi\n_{s}^{t}\\left( \\left( h^{\\prime }\\right) _{\\ast }^{-1}\\left( \\xi \\right)\n\\right) ,\\varphi _{s}^{t}\\left( \\left( h^{\\prime }\\right) _{\\ast\n}^{-1}\\left( \\eta \\right) \\right) \\right] _{\\mathfrak{p}}\\right)  \\notag \\\\\n&=&\\left( h^{\\prime }\\right) _{\\ast }\\left[ \\left( h^{\\prime }\\right) _{\\ast\n}^{-1}\\xi ,\\left( h^{\\prime }\\right) _{\\ast }^{-1}\\eta \\right] _{\\varphi\n_{s}}.  \\label{phibrackequi}\n\\end{eqnarray\nTherefore, $\\left[ \\cdot ,\\cdot \\right] _{\\varphi _{s}}$ is equivariant with\nrespect to transformations of $s$.\n\\end{proof}\n\n\\subsection{Darboux derivative}\n\n\\label{sectDarboux}Let $M$ be a smooth manifold and suppose \ns:M\\longrightarrow \\mathbb{L}$ is a smooth map. The map $s$ can be used to\ndefine a product on $\\mathbb{L}$-valued maps from $M$ and a corresponding\nbracket on $\\mathfrak{l}$-valued maps. Indeed, let $A,B:M\\longrightarrow \n\\mathbb{L}$ and $\\xi ,\\eta :M\\longrightarrow \\mathfrak{l}$ be smooth maps,\nthen at each $x\\in M$, define \n\\begin{subequations\n\\label{maniproducts} \n\\begin{eqnarray}\n\\left. A\\circ _{s}B\\right\\vert _{x} &=&A_{x}\\circ _{s_{x}}B_{x}\\in \\mathbb{L}\n\\\\\n\\left. A\/_{s}B\\right\\vert _{x} &=&A_{x}\/_{s_{x}}B_{x}\\in \\mathbb{L} \\\\\n\\left. A\\backslash _{s}B\\right\\vert _{x} &=&A_{x}\\backslash _{s}B_{x}\\in \n\\mathbb{L} \\\\\n\\left. \\left[ \\xi ,\\eta \\right] ^{\\left( s\\right) }\\right\\vert _{x} &=&\\left[\n\\xi _{x},\\eta _{x}\\right] ^{\\left( s_{x}\\right) }\\in \\mathfrak{l.}\n\\end{eqnarray\n\\end{subequations\nIn particular, the bracket $\\left[ \\cdot ,\\cdot \\right] ^{\\left( s\\right) }$\ndefines the map $b_{s}:M\\longrightarrow \\Lambda ^{2}\\mathfrak{l}^{\\ast\n}\\otimes \\mathfrak{l.}$ We also have the corresponding associator $\\left[\n\\cdot ,\\cdot ,\\cdot \\right] ^{\\left( s\\right) }$ and the left-alternating\nassociator map $a_{s}:M\\longrightarrow \\Lambda ^{2}\\mathfrak{l}^{\\ast\n}\\otimes \\mathfrak{l}^{\\ast }\\otimes \\mathfrak{l}.$ Similarly, define the\nmap $\\varphi _{s}:M\\longrightarrow \\mathfrak{p}^{\\ast }\\otimes \\mathfrak{l}.$\n\nThen, similarly as for maps to Lie groups, we may define the (right) \\emph\nDarboux derivative} $\\theta _{s}$ of $s,$ which is an $\\mathfrak{l}$-valued \n1$-form on $M$ given by $s^{\\ast }\\theta $ \\cite{SharpeBook}. In particular,\nat every $x\\in M$, \n\\begin{equation}\n\\left. \\left( \\theta _{s}\\right) \\right\\vert _{x}=\\left( R_{s\\left( x\\right)\n}^{-1}\\right) _{\\ast }\\left. ds\\right\\vert _{x}.  \\label{Darbouxf}\n\\end{equation\nIt is then clear that $\\theta _{s}$, being a pullback of $\\theta $,\nsatisfies the loop Maurer-Cartan structural equation (\\ref{MCequation1}). In\nparticular, for any vectors $X,Y\\in T_{x}M$, \n\\begin{equation}\nd\\theta _{s}\\left( X,Y\\right) -\\left[ \\theta _{s}\\left( X\\right) ,\\theta\n_{s}\\left( Y\\right) \\right] ^{\\left( s\\right) }=0.  \\label{DarbouxMC}\n\\end{equation}\n\nWe can then calculate the derivatives of these maps. For clarity, we will\nsomewhat abuse notation, we will suppress the pushforwards of right\nmultiplication and their inverses (i.e. quotients) on $T\\mathbb{L}$, so that\nif $X\\in T_{q}\\mathbb{L}$, then we will write $X\\circ _{s}A$ for $\\left(\nR_{A}^{\\left( s\\right) }\\right) _{\\ast }X.$\n\n\\begin{theorem}\n\\label{thmmaniDeriv}Let $M$ be a smooth manifold and let $x\\in M$. Suppose \nA,B,s\\in C^{\\infty }\\left( M,\\mathbb{L}\\right) ,$ then \n\\begin{equation}\nd\\left( A\\circ _{s}B\\right) =\\left( dA\\right) \\circ _{s}B+A\\circ _{s}\\left(\ndB\\right) +\\left[ A,B,\\theta _{s}\\right] ^{\\left( s\\right) }  \\label{dAsB1}\n\\end{equation\nand \n\\begin{subequations\n\\label{dquots} \n\\begin{eqnarray}\nd\\left( A\/_{s}B\\right)  &=&dA\/_{s}B-\\left( A\/_{s}B\\circ _{s}dB\\right) \/_{s}B-\n\\left[ A\/_{s}B,B,\\theta _{s}\\right] ^{\\left( s\\right) }\/_{s}B  \\label{drquot}\n\\\\\nd\\left( B\\backslash _{s}A\\right)  &=&B\\backslash _{s}dA-B\\backslash\n_{s}\\left( dB\\circ _{s}\\left( B\\backslash _{s}A\\right) \\right) \n\\label{dlquot} \\\\\n&&-B\\backslash _{s}\\left[ B,B\\backslash _{s}A,\\theta _{s}\\right] ^{\\left(\ns\\right) }.  \\notag\n\\end{eqnarray\n\\end{subequations\nSuppose now $\\xi ,\\eta \\in C^{\\infty }\\left( M,\\mathfrak{l}\\right) $, then \n\\begin{equation}\nd\\left[ \\xi ,\\eta \\right] ^{\\left( s\\right) }=\\left[ d\\xi ,\\eta \\right]\n^{\\left( s\\right) }+\\left[ \\xi ,d\\eta \\right] ^{\\left( s\\right)\n}+a_{s}\\left( \\xi ,\\eta ,\\theta _{s}\\right) .  \\label{dbrack}\n\\end{equation}\n\nThe $\\mathfrak{l}\\otimes \\mathfrak{p}^{\\ast }$-valued map $\\varphi\n_{s}:M\\longrightarrow $ $\\mathfrak{l}\\otimes \\mathfrak{p}^{\\ast }$ satisfies \n\\begin{equation}\nd\\varphi _{s}=\\func{id}_{\\mathfrak{p}}\\cdot \\theta _{s}-\\left[ \\varphi\n_{s},\\theta _{s}\\right] ^{\\left( s\\right) },  \\label{dphis0}\n\\end{equation\nwhere $\\func{id}_{\\mathfrak{p}}\\ $is the identity map of $\\mathfrak{p}$ and \n\\cdot $ denotes the action of the Lie algebra $\\mathfrak{p}$ on $\\mathfrak{l}\n$ given by (\\ref{pactl1})\n\\end{theorem}\n\n\\begin{proof}\nLet $V\\in T_{x}M$ and let $x\\left( t\\right) $ be a curve on $M$ with \nx\\left( 0\\right) =x$ and $\\dot{x}\\left( 0\\right) =V.$ To show (\\ref{dAsB1}),\nfirst note that \n\\begin{equation}\n\\left. d\\left( A\\circ _{s}B\\right) \\right\\vert _{x}\\left( V\\right) =\\left. \n\\frac{d}{dt}\\left( A_{x\\left( t\\right) }\\circ _{s_{x\\left( t\\right)\n}}B_{x\\left( t\\right) }\\right) \\right\\vert _{t=0}.\n\\end{equation\nHowever\n\\begin{eqnarray}\n\\left. \\frac{d}{dt}\\left( A_{x\\left( t\\right) }\\circ _{s_{x\\left( t\\right)\n}}B_{x\\left( t\\right) }\\right) \\right\\vert _{t=0} &=&\\left. \\frac{d}{dt\n\\left( A_{x\\left( t\\right) }\\circ _{s_{x}}B_{x}\\right) \\right\\vert\n_{t=0}+\\left. \\frac{d}{dt}\\left( A_{x}\\circ _{s_{x}}B_{x\\left( t\\right)\n}\\right) \\right\\vert _{t=0}  \\notag \\\\\n&&+\\left. \\frac{d}{dt}\\left( A_{x}\\circ _{s_{x\\left( t\\right) }}B_{x}\\right)\n\\right\\vert _{t=0}  \\notag \\\\\n&=&\\left( R_{B_{x}}^{\\left( s_{x}\\right) }\\right) _{\\ast }\\left.\ndA\\right\\vert _{x}\\left( V\\right) +\\left( L_{A_{x}}^{\\left( s_{x}\\right)\n}\\right) _{\\ast }\\left. dB\\right\\vert _{x}\\left( V\\right)  \\label{dABprod0}\n\\\\\n&&+\\left. \\frac{d}{dt}\\left( A_{x}\\circ _{s_{x\\left( t\\right) }}B_{x}\\right)\n\\right\\vert _{t=0}  \\notag\n\\end{eqnarray\nand then, using Lemma \\ref{lemQuotient}, \n\\begin{eqnarray}\n\\left. \\frac{d}{dt}\\left( A_{x}\\circ _{s_{x\\left( t\\right) }}B_{x}\\right)\n\\right\\vert _{t=0} &=&\\left. \\frac{d}{dt}\\left( \\left( A_{x}\\cdot\nB_{x}s_{x\\left( t\\right) }\\right) \/s_{x\\left( t\\right) }\\right) \\right\\vert\n_{t=0}  \\notag \\\\\n&=&\\left. \\frac{d}{dt}\\left( \\left( A_{x}\\cdot B_{x}s_{x\\left( t\\right)\n}\\right) \/s_{x}\\right) \\right\\vert _{t=0}  \\label{dABprod} \\\\\n&&+\\left. \\frac{d}{dt}\\left( \\left( A_{x}\\cdot B_{x}s_{x}\\right) \/s_{x}\\cdot\ns_{x\\left( t\\right) }\\right) \/s_{x}\\right\\vert _{t=0}.  \\notag\n\\end{eqnarray\nLooking at each term in (\\ref{dABprod}), we have \n\\begin{eqnarray*}\n\\left( A_{x}\\cdot B_{x}s_{x\\left( t\\right) }\\right) \/s_{x} &=&\\left(\nA_{x}\\cdot B_{x}\\left( s_{x\\left( t\\right) }\/s_{x}\\cdot s_{x}\\right) \\right)\n\/s_{x} \\\\\n&=&A_{x}\\circ _{s_{x}}\\left( B_{x}\\circ _{s_{x}}\\left( s_{x\\left( t\\right)\n}\/s_{x}\\right) \\right)\n\\end{eqnarray*\nan\n\\begin{equation*}\n\\left( \\left( A_{x}\\cdot B_{x}s_{x}\\right) \/s_{x}\\cdot s_{x\\left( t\\right)\n}\\right) \/s_{x}=\\left( A_{x}\\circ _{s_{x}}B_{x}\\right) \\circ _{s_{x}}\\left(\ns_{x\\left( t\\right) }\/s_{x}\\right) .\n\\end{equation*\nOverall (\\ref{dABprod0}) becomes, \n\\begin{equation}\n\\left. \\frac{d}{dt}\\left( A_{x}\\circ _{s_{x\\left( t\\right) }}B_{x}\\right)\n\\right\\vert _{t=0}=\\left( \\left( L_{A_{x}}^{\\left( s_{x}\\right) }\\circ\nL_{B_{x}}^{\\left( s_{x}\\right) }\\right) _{\\ast }-\\left( L_{A_{x}\\circ\n_{s_{x}}B_{x}}^{\\left( s_{x}\\right) }\\right) _{\\ast }\\right) \\left(\nR_{s_{x}}^{-1}\\right) _{\\ast }\\left. ds\\right\\vert _{x}\\left( V_{x}\\right)\n\\end{equation\nand hence we get (\\ref{dAsB1}) using the definitions of $\\theta _{s}$ and\nthe mixed associator (\\ref{pxiqsol}).\n\nLet us now show (\\ref{dquots}). From Lemma \\ref{lemQuotient}$,$ we find \n\\begin{subequations\n\\label{dquots copy(1)} \n\\begin{eqnarray}\nd\\left( A\/B\\right) &=&\\left( dA\\right) \/B-\\left( A\/B\\cdot dB\\right) \/B\n\\label{dquots1} \\\\\nd\\left( B\\backslash A\\right) &=&B\\backslash \\left( dA\\right) -B\\backslash\n\\left( dB\\cdot B\\backslash A\\right) .  \\label{dquots2}\n\\end{eqnarray\n\\end{subequations\nNow if we instead have the quotient defined by $s$, using (\\ref{rprodqright\n), we have a modification: \n\\begin{eqnarray}\nd\\left( A\/_{s}B\\right) &=&d\\left( As\/Bs\\right) =d\\left( As\\right) \/\\left(\nBs\\right) -\\left( A\/_{s}B\\cdot d\\left( Bs\\right) \\right) \/\\left( Bs\\right) \n\\notag \\\\\n&=&dA\/_{s}B+A\\left( ds\\right) \/\\left( Bs\\right) -\\left( A\/_{s}B\\cdot \\left(\ndB\\right) s\\right) \/\\left( Bs\\right)  \\notag \\\\\n&&-\\left( A\/_{s}B\\cdot B\\left( ds\\right) \\right) \/\\left( Bs\\right)  \\notag \\\\\n&=&dA\/_{s}B-\\left( A\/_{s}B\\circ _{s}dB\\right) \/_{s}B+\\left( A\\circ\n_{s}\\theta _{s}\\right) \/_{s}B  \\notag \\\\\n&&-\\left( A\/_{s}B\\circ _{s}\\left( B\\circ _{s}\\theta _{s}\\right) \\right)\n\/_{s}B  \\notag \\\\\n&=&dA\/_{s}B-\\left( A\/_{s}B\\circ _{s}dB\\right) \/_{s}B-\\left[ A\/_{s}B,B,\\theta\n_{s}\\right] ^{\\left( s\\right) }\/_{s}B.\n\\end{eqnarray\nSimilarly, for the left quotient, using (\\ref{rprodqleft}), we have \n\\begin{eqnarray}\nd\\left( B\\backslash _{s}A\\right) &=&d\\left( \\left( B\\backslash As\\right)\n\/s\\right)  \\notag \\\\\n&=&d\\left( B\\backslash As\\right) \/s-\\left( \\left( \\left( B\\backslash\nAs\\right) \/s\\right) \\cdot ds\\right) \/s  \\notag \\\\\n&=&\\left( B\\backslash d\\left( As\\right) \\right) \/s-\\left( B\\backslash \\left(\ndB\\cdot B\\backslash As\\right) \\right) \/s-\\left( B\\backslash _{s}A\\right)\n\\circ _{s}\\theta _{s}  \\notag \\\\\n&=&B\\backslash _{s}dA+\\left( B\\backslash \\left( A\\left( ds\\right) \\right)\n\\right) \/s-B\\backslash _{s}\\left( \\left( dB\\cdot B\\backslash As\\right)\n\/s\\right)  \\notag \\\\\n&&-\\left( B\\backslash _{s}A\\right) \\circ _{s}\\theta _{s}  \\notag \\\\\n&=&B\\backslash _{s}dA-B\\backslash _{s}\\left( dB\\circ _{s}\\left( B\\backslash\n_{s}A\\right) \\right) +B\\backslash _{s}\\left( A\\circ _{s}\\theta _{s}\\right) \n\\notag \\\\\n&&-\\left( B\\backslash _{s}A\\right) \\circ _{s}\\theta _{s}\n\\end{eqnarray\nHowever, using the mixed associator (\\ref{pxiqsol}), \n\\begin{eqnarray}\nA\\circ _{s}\\theta _{s} &=&\\left( B\\circ _{s}\\left( B\\backslash _{s}A\\right)\n\\right) \\circ _{s}\\theta _{s}  \\notag \\\\\n&=&B\\circ _{s}\\left( \\left( B\\backslash _{s}A\\right) \\circ _{s}\\theta\n_{s}\\right) -\\left[ B,B\\backslash _{s}A,\\theta _{s}\\right] ^{\\left( s\\right)\n},\n\\end{eqnarray\nand thus, \n\\begin{equation*}\nd\\left( B\\backslash _{s}A\\right) =B\\backslash _{s}dA-B\\backslash _{s}\\left(\ndB\\circ _{s}\\left( B\\backslash _{s}A\\right) \\right) -B\\backslash _{s}\\left[\nB,B\\backslash _{s}A,\\theta _{s}\\right] ^{\\left( s\\right) }.\n\\end{equation*}\n\nTo show (\\ref{dbrack}), note that \n\\begin{eqnarray*}\n\\left. d\\left( \\left[ \\xi ,\\eta \\right] ^{\\left( s\\right) }\\right)\n\\right\\vert _{x}\\left( V\\right) &=&\\left. \\frac{d}{dt}\\left[ \\xi _{x\\left(\nt\\right) },\\eta _{x\\left( t\\right) }\\right] ^{\\left( s_{x\\left( t\\right)\n}\\right) }\\right\\vert _{t=0} \\\\\n&=&\\left[ \\left. d\\xi \\right\\vert _{x}\\left( V\\right) ,\\eta _{x}\\right]\n^{\\left( s_{x}\\right) }+\\left[ \\xi _{x},\\left. d\\eta \\right\\vert _{x}\\right]\n^{\\left( s_{x}\\right) } \\\\\n&&+\\left. \\frac{d}{dt}\\left[ \\xi _{x},\\eta _{x}\\right] ^{\\left( s_{x\\left(\nt\\right) }\\right) }\\right\\vert _{t=0}\n\\end{eqnarray*\nHowever, using (\\ref{db1}), the last term becomes \n\\begin{equation*}\n\\left. \\frac{d}{dt}\\left[ \\xi _{x},\\eta _{x}\\right] ^{\\left( s_{x\\left(\nt\\right) }\\right) }\\right\\vert _{t=0}=a_{s_{x}}\\left( \\xi _{x},\\eta\n_{x},\\left. \\theta _{s}\\right\\vert _{x}\\right)\n\\end{equation*\nand hence we obtain (\\ref{dbrack}).\n\nLet us now show (\\ref{dphis0}). From (\\ref{actpl}), given $\\gamma \\in \n\\mathfrak{p}$, setting $\\hat{\\gamma}\\left( r\\right) =\\varphi _{r}\\left(\n\\gamma \\right) $ for each $r\\in \\mathbb{L}$, we have \n\\begin{equation}\n\\left. d\\hat{\\gamma}\\right\\vert _{r}\\left( \\rho _{r}\\left( \\xi \\right)\n\\right) =\\gamma \\cdot \\xi -\\left[ \\hat{\\gamma}\\left( r\\right) ,\\xi \\right]\n^{\\left( r\\right) }\n\\end{equation\nfor some $\\xi \\in \\mathfrak{l}.$ Now for at each $x\\in M$ we have \n\\begin{eqnarray}\n\\left. d\\left( \\varphi _{s}\\left( \\gamma \\right) \\right) \\right\\vert\n_{x}\\left( V\\right) &=&\\left. d\\hat{\\gamma}\\right\\vert _{s_{x}}\\circ \\left.\nds\\right\\vert _{x}\\left( V\\right)  \\notag \\\\\n&=&\\left. d\\hat{\\gamma}\\right\\vert _{s_{x}}\\left( \\rho _{s_{x}}\\left( \\theta\n_{s}\\left( V\\right) \\right) \\right)  \\notag \\\\\n&=&\\gamma \\cdot \\theta _{s}\\left( V\\right) -\\left[ \\varphi _{s_{x}}\\left(\n\\gamma \\right) ,\\theta _{s}\\left( V\\right) \\right] ^{\\left( s_{x}\\right) }.\n\\end{eqnarray\nTherefore, $d\\varphi _{s}$ is given by \n\\begin{equation}\nd\\varphi _{s}\\left( \\gamma \\right) =\\gamma \\cdot \\theta _{s}-\\left[ \\varphi\n_{s}\\left( \\gamma \\right) ,\\theta _{s}\\right] ^{\\left( s\\right) }.\n\\label{dphis0a}\n\\end{equation}\n\\end{proof}\n\n\\begin{remark}\nSuppose $A$ and $B$ are now smooth maps from $M$ to $\\mathbb{L}$. In the\ncase when $\\mathbb{L}$ has the right inverse property, i.e. $A\/B=AB^{-1}$\nfor any $A,B\\in \\mathbb{L}$, (\\ref{dquots1}) becomes \n\\begin{equation}\nd\\left( AB^{-1}\\right) =\\left( dA\\right) B^{-1}-\\left( AB^{-1}\\cdot\ndB\\right) B^{-1}\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{.}  \\label{drquot2}\n\\end{equation\nHowever, from $d\\left( BB^{-1}\\right) =0$, we find that $d\\left(\nB^{-1}\\right) =-B^{-1}\\left( dB\\cdot B^{-1}\\right) $, and then expanding \nd\\left( AB^{-1}\\right) $ using the product rule, and comparing with (\\re\n{drquot2}), we find \n\\begin{equation}\n\\left( AB^{-1}\\cdot dB\\right) B^{-1}=A\\left( B^{-1}\\left( dB\\cdot\nB^{-1}\\right) \\right) ,  \\label{drquot3}\n\\end{equation\nwhich is an infinitesimal version of the right Bol identity (\\ref{rightBol\n). In particular, \n\\begin{equation}\n\\left( B^{-1}\\cdot dB\\right) B^{-1}=B^{-1}\\left( dB\\cdot B^{-1}\\right) .\n\\end{equation\nSimilarly, using (\\ref{dlquot}), the left inverse property then implies an\ninfinitesimal left Bol identity.\n\\end{remark}\n\nAt each point $x\\in M$, the map $s$ defines a stabilizer subgroup $\\func{Sta\n}\\left( s_{x}\\right) =\\func{Aut}\\left( \\mathbb{L},\\circ _{s}\\right) \\subset\n\\Psi ^{R}\\left( \\mathbb{L}\\right) $ $\\ $with the corresponding Lie algebra \n\\mathfrak{h}_{s_{x}}.$ Similarly, we also have the orbit of $s_{x}$ given by \n$\\mathcal{C}^{R}\\left( \\mathbb{L},\\circ _{s_{x}}\\right) \\cong \n\\faktor{\\Psi ^{R}\\left( \\mathbb{L}\\right)} {\\func{Aut}\\left( \\mathbb{L},\\circ\n_{s_{x}}\\right)}$, and the corresponding tangent space $\\mathfrak{q\n_{s_{x}}\\cong \\mathfrak{p\/h}_{s_{x}}.$ Suppose $\\left. \\theta\n_{s}\\right\\vert _{x}\\in \\mathfrak{q}_{s_{x}}$ for each $x\\in M$. This of\ncourse always holds if $\\mathbb{L}$ is a $G$-loop, in which case $\\mathfrak{\n}_{s_{x}}=\\mathfrak{l}^{\\left( s_{x}\\right) }.$ In this case, there exists a \n$\\mathfrak{p}$-valued $1$-form $\\Theta $ on $M$ such that $\\theta\n_{s}=\\varphi _{s}\\left( \\Theta \\right) .$ We can then characterize $\\Theta $\nin the following way.\n\n\\begin{theorem}\n\\label{thmThetaPhi}Suppose there exists $\\Theta \\in \\Omega ^{1}\\left( M\n\\mathfrak{p}\\right) $ such that $\\theta _{s}=\\varphi _{s}\\left( \\Theta\n\\right) $. Then, for each $x\\in M$, $\\left. d\\Theta -\\frac{1}{2}\\left[\n\\Theta ,\\Theta \\right] _{\\mathfrak{p}}\\right\\vert _{x}\\in \\mathfrak{h\n_{s_{x}}$, where $\\left[ \\cdot ,\\cdot \\right] _{\\mathfrak{p}}$ is the Lie\nbracket on $\\mathfrak{p.}$\n\\end{theorem}\n\n\\begin{proof}\nConsider $d\\theta _{s}$ in this case. Using (\\ref{dphis0a}), we have \n\\begin{eqnarray}\nd\\theta _{s} &=&d\\left( \\varphi _{s}\\left( \\Theta \\right) \\right) =\\left(\nd\\varphi _{s}\\right) \\left( \\Theta \\right) +\\varphi _{s}\\left( d\\Theta\n\\right)  \\notag \\\\\n&=&-\\Theta \\cdot \\theta _{s}+\\left[ \\varphi _{s}\\left( \\Theta \\right)\n,\\theta _{s}\\right] ^{\\left( s\\right) }.  \\label{dthetasq}\n\\end{eqnarray\nNote that the signs are switched in (\\ref{dthetasq}) because we also have an\nimplied wedge product of $1$-forms. Overall, we have \n\\begin{equation}\nd\\left( \\varphi _{s}\\left( \\Theta \\right) \\right) =\\varphi _{s}\\left(\nd\\Theta \\right) -\\Theta \\cdot \\varphi _{s}\\left( \\Theta \\right) +\\left[\n\\varphi _{s}\\left( \\Theta \\right) ,\\varphi _{s}\\left( \\Theta \\right) \\right]\n^{\\left( s\\right) },  \\label{dphistheta}\n\\end{equation\nhowever since $\\theta _{s}=\\varphi _{s}\\left( \\Theta \\right) $, it satisfies\nthe Maurer-Cartan structural equation (\\ref{DarbouxMC}), so we also have \n\\begin{equation}\nd\\left( \\varphi _{s}\\left( \\Theta \\right) \\right) =\\frac{1}{2}\\left[ \\varphi\n_{s}\\left( \\Theta \\right) ,\\varphi _{s}\\left( \\Theta \\right) \\right] .\n\\label{dphistheta2}\n\\end{equation\nEquating (\\ref{dphistheta}) and (\\ref{dphistheta2}) , we find \n\\begin{equation}\n\\varphi _{s}\\left( d\\Theta \\right) =\\Theta \\cdot \\varphi _{s}\\left( \\Theta\n\\right) -\\frac{1}{2}\\left[ \\varphi _{s}\\left( \\Theta \\right) ,\\varphi\n_{s}\\left( \\Theta \\right) \\right] ^{\\left( s\\right) }.  \\label{dphistheta3}\n\\end{equation\nHowever, from (\\ref{xiphi}), we find that \n\\begin{equation}\n\\Theta \\cdot \\varphi _{s}\\left( \\Theta \\right) -\\frac{1}{2}\\left[ \\varphi\n_{s}\\left( \\Theta \\right) ,\\varphi _{s}\\left( \\Theta \\right) \\right] =\\frac{\n}{2}\\varphi _{s}\\left( \\left[ \\Theta ,\\Theta \\right] _{\\mathfrak{p}}\\right) .\n\\end{equation\nThus, we see that \n\\begin{equation}\n\\varphi _{s}\\left( d\\Theta -\\frac{1}{2}\\left[ \\Theta ,\\Theta \\right] _\n\\mathfrak{p}}\\right) =0.  \\label{dphistheta4}\n\\end{equation}\n\\end{proof}\n\n\\begin{remark}\nIn general, we can think of $d-\\Theta $ as a connection on the trivial Lie\nalgebra bundle $M\\times \\mathfrak{p}$ with curvature contained in $\\mathfrak\nh}_{s\\left( x\\right) }$ for each $x\\in M$. In general the spaces $\\mathfrak{\n}_{s\\left( x\\right) }$ need not be all of the same dimension, and thus may\nthis may not give a vector subbundle. On the other hand, if $\\mathbb{L}$ is\na $G$-loop$,$ then we do get a subbundle.\n\\end{remark}\n\nNow consider how $\\theta _{s}$ behaves under the action of $\\Psi ^{R}\\left( \n\\mathbb{L}\\right) .$\n\n\\begin{lemma}\nSuppose $h:M\\longrightarrow \\Psi ^{R}\\left( \\mathbb{L}\\right) $ is a smooth\nmap, then \n\\begin{equation}\n\\theta _{h\\left( s\\right) }=\\left( h^{\\prime }\\right) _{\\ast }\\left( \\varphi\n_{s}\\left( \\theta _{h}^{\\left( \\mathfrak{p}\\right) }\\right) +\\theta\n_{s}\\right) ,  \\label{thetahf}\n\\end{equation\nwhere $\\theta _{h}^{\\left( \\mathfrak{p}\\right) }=$ $h^{\\ast }\\theta ^{\\left( \n\\mathfrak{p}\\right) }$ is the pullback of the left-invariant Maurer-Cartan\nform $\\theta ^{\\left( \\mathfrak{p}\\right) }$ on $\\Psi ^{R}\\left( \\mathbb{L\n\\right) .$\n\\end{lemma}\n\n\\begin{proof}\nSuppose $h:M\\longrightarrow \\Psi ^{R}\\left( \\mathbb{L}\\right) $ is a smooth\nmap, then consider $\\theta _{h\\left( s\\right) }$. We then have \n\\begin{eqnarray*}\n\\left. \\left( \\theta _{h\\left( s\\right) }\\right) \\right\\vert _{x} &=&\\left(\nR_{h\\left( s\\left( x\\right) \\right) }^{-1}\\right) _{\\ast }\\left. d\\left(\nh\\left( s\\right) \\right) \\right\\vert _{x} \\\\\n&=&\\left( R_{h\\left( s\\left( x\\right) \\right) }^{-1}\\right) _{\\ast }\\left.\n\\left( \\left( dh\\right) \\left( s\\right) +h\\left( ds\\right) \\right)\n\\right\\vert _{x}.\n\\end{eqnarray*\nConsider each term. Using simplified notation, we have \n\\begin{eqnarray*}\n\\left( dh\\right) \\left( s\\right) \/h\\left( s\\right) &=&\\left( h^{\\prime\n}\\right) _{\\ast }\\left( \\left( h^{-1}dh\\right) \\left( s\\right) \/s\\right) \\\\\n\\left( R_{h\\left( s\\left( x\\right) \\right) }^{-1}\\right) _{\\ast }\\left.\n\\left( h\\left( ds\\right) \\right) \\right\\vert _{x} &=&\\left( h^{\\prime\n}\\right) _{\\ast }\\left( \\theta _{s}\\right) .\n\\end{eqnarray*\nThus, \n\\begin{equation*}\n\\left( R_{h\\left( s\\left( x\\right) \\right) }^{-1}\\right) _{\\ast }\\left.\n\\left( dh\\right) \\left( s\\right) \\right\\vert _{x}=\\left( h\\left( x\\right)\n^{\\prime }\\right) _{\\ast }\\varphi _{s\\left( x\\right) }\\left( \\left. \\theta\n_{h}^{\\left( \\mathfrak{p}\\right) }\\right\\vert _{x}\\right) ,\n\\end{equation*\nand hence we get (\\ref{thetahf}).\n\\end{proof}\n\nIf we have another smooth map $f:M\\longrightarrow \\mathbb{L}$, using right\nmultiplication with respect to $\\circ _{s\\left( x\\right) }$, we can define a\nmodified Darboux derivative $\\theta _{f}^{\\left( s\\right) }$ with respect to \n$s$\n\\begin{equation}\n\\left. \\left( \\theta _{f}^{\\left( s\\right) }\\right) \\right\\vert _{x}=\\left(\nR_{f\\left( x\\right) }^{\\left( s\\left( x\\right) \\right) }\\right) _{\\ast\n}^{-1}\\left. df\\right\\vert _{x}.\n\\end{equation\nNote that this is now no longer necessarily a pullback of $\\theta $ and\nhence may not satisfy the Maurer-Cartan equation. Adopting simplified\nnotation, we have the following: \n\\begin{eqnarray}\nd\\left( fs\\right) \/fs &=&\\left( df\\cdot s+f\\cdot ds\\right) \/fs  \\notag \\\\\n&=&df\/_{s}f+\\func{Ad}_{f}^{\\left( s\\right) }\\theta _{s}  \\label{thetafs}\n\\end{eqnarray\nHence, \n\\begin{equation}\n\\theta _{f}^{\\left( s\\right) }=\\theta _{fs}-\\left( \\func{Ad}_{f}^{\\left(\ns\\right) }\\right) _{\\ast }\\theta _{s}.  \\label{thetafs2}\n\\end{equation}\n\n\\begin{lemma}\nSuppose $f,s\\in C^{\\infty }\\left( M,\\mathbb{L}\\right) ,$ then \n\\begin{equation}\nd\\theta _{f}^{\\left( s\\right) }=\\frac{1}{2}\\left[ \\theta _{f}^{\\left(\ns\\right) },\\theta _{f}^{\\left( s\\right) }\\right] ^{\\left( fs\\right) }-\\left(\nR_{f}^{\\left( s\\right) }\\right) _{\\ast }^{-1}\\left[ \\theta _{f}^{\\left(\ns\\right) },f,\\theta _{s}\\right] ^{\\left( s\\right) }.  \\label{dthetafs}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nApplying the exterior derivative to (\\ref{thetafs2}) and then the structural\nequation for $\\theta _{fs}$, we hav\n\\begin{equation}\nd\\theta _{f}^{\\left( s\\right) }=\\frac{1}{2}\\left[ \\theta _{fs},\\theta _{fs\n\\right] ^{\\left( fs\\right) }-d\\left( \\left( \\func{Ad}_{f}^{\\left( s\\right)\n}\\right) _{\\ast }\\theta _{s}\\right) .  \\label{dthetafs2}\n\\end{equation\nFrom Lemma \\ref{lemdtAd}, we can see that for $\\xi \\in \\mathfrak{l}$, \n\\begin{eqnarray}\nd\\left( \\func{Ad}_{f}^{\\left( s\\right) }\\right) _{\\ast }\\xi &=&\\left[ \\theta\n_{f}^{\\left( s\\right) },\\left( \\func{Ad}_{f}^{\\left( s\\right) }\\right)\n_{\\ast }\\xi \\right] ^{\\left( fs\\right) }-\\left( R_{f}^{\\left( s\\right)\n}\\right) _{\\ast }^{-1}\\left[ \\theta _{f}^{\\left( s\\right) },f,\\xi \\right]\n^{\\left( s\\right) }  \\notag \\\\\n&&+\\left( R_{f}^{\\left( s\\right) }\\right) _{\\ast }^{-1}\\left[ f,\\xi ,\\theta\n_{s}\\right] ^{\\left( s\\right) }  \\label{dAdfs1} \\\\\n&&-\\left( R_{f}^{\\left( s\\right) }\\right) _{\\ast }^{-1}\\left[ \\left( \\func{A\n}_{f}^{\\left( s\\right) }\\right) _{\\ast }\\xi ,f,\\theta _{s}\\right] ^{\\left(\ns\\right) },  \\notag\n\\end{eqnarray\nand hence \n\\begin{eqnarray}\nd\\left( \\func{Ad}_{f}^{\\left( s\\right) }\\right) _{\\ast }\\wedge \\theta _{s}\n&=&\\left[ \\theta _{f}^{\\left( s\\right) },\\left( \\func{Ad}_{f}^{\\left(\ns\\right) }\\right) _{\\ast }\\theta _{s}\\right] ^{\\left( fs\\right) }-\\left(\nR_{f}^{\\left( s\\right) }\\right) _{\\ast }^{-1}\\left[ \\theta _{f}^{\\left(\ns\\right) },f,\\theta _{s}\\right] ^{\\left( s\\right) }  \\notag \\\\\n&&-\\left( R_{f}^{\\left( s\\right) }\\right) _{\\ast }^{-1}\\left[ f,\\theta\n_{s},\\theta _{s}\\right] ^{\\left( s\\right) }  \\label{dAdfs2} \\\\\n&&+\\left( R_{f}^{\\left( s\\right) }\\right) _{\\ast }^{-1}\\left[ \\left( \\func{A\n}_{f}^{\\left( s\\right) }\\right) _{\\ast }\\theta _{s},f,\\theta _{s}\\right]\n^{\\left( s\\right) },  \\notag\n\\end{eqnarray\nwhere wedge products are implied. Now, using the structural equation and \n\\ref{Adbrack1}), we fin\n\\begin{eqnarray}\n\\left( \\func{Ad}_{f}^{\\left( s\\right) }\\right) _{\\ast }d\\theta _{s} &=&\\frac\n1}{2}\\left( \\func{Ad}_{f}^{\\left( s\\right) }\\right) _{\\ast }\\left[ \\theta\n_{s},\\theta _{s}\\right] ^{\\left( s\\right) }  \\notag \\\\\n&=&\\frac{1}{2}\\left[ \\left( \\func{Ad}_{f}^{\\left( s\\right) }\\right) _{\\ast\n}\\theta _{s},\\left( \\func{Ad}f\\right) _{\\ast }\\theta _{s}\\right] ^{\\left(\nfs\\right) }  \\notag \\\\\n&&-\\left( R_{f}^{\\left( s\\right) }\\right) _{\\ast }^{-1}\\left[ \\left( \\func{A\n}_{f}^{\\left( s\\right) }\\right) _{\\ast }\\theta _{s},f,\\theta _{s}\\right]\n^{\\left( s\\right) }  \\notag \\\\\n&&+\\left( R_{f}^{\\left( s\\right) }\\right) _{\\ast }^{-1}\\left[ f,\\theta\n_{s},\\theta _{s}\\right] ^{\\left( s\\right) }.  \\label{Adbracktheta}\n\\end{eqnarray\nCombining (\\ref{dAdfs2}) and (\\ref{Adbracktheta}), we see that \n\\begin{eqnarray}\nd\\left( \\left( \\func{Ad}_{f}^{\\left( s\\right) }\\right) _{\\ast }\\theta\n_{s}\\right) &=&d\\left( \\func{Ad}_{f}^{\\left( s\\right) }\\right) _{\\ast\n}\\wedge \\theta _{s}+\\left( \\func{Ad}_{f}^{\\left( s\\right) }\\right) _{\\ast\n}d\\theta _{s}  \\notag \\\\\n&=&\\left[ \\theta _{f}^{\\left( s\\right) },\\left( \\func{Ad}f\\right) _{\\ast\n}\\theta _{s}\\right] ^{\\left( fs\\right) }+\\frac{1}{2}\\left[ \\left( \\func{Ad\n_{f}^{\\left( s\\right) }\\right) _{\\ast }\\theta _{s},\\left( \\func{Ad}f\\right)\n_{\\ast }\\theta _{s}\\right] ^{\\left( fs\\right) }  \\notag \\\\\n&&-\\left( R_{f}^{\\left( s\\right) }\\right) _{\\ast }^{-1}\\left[ \\theta\n_{f}^{\\left( s\\right) },f,\\theta _{s}\\right] ^{\\left( s\\right) }  \\notag \\\\\n&=&\\frac{1}{2}\\left[ \\theta _{fs},\\theta _{fs}\\right] ^{\\left( fs\\right) }\n\\frac{1}{2}\\left[ \\theta _{f}^{\\left( s\\right) },\\theta _{f}^{\\left(\ns\\right) }\\right] ^{\\left( fs\\right) }  \\label{dadtheta} \\\\\n&&-\\left( R_{f}^{\\left( s\\right) }\\right) _{\\ast }^{-1}\\left[ \\theta\n_{f}^{\\left( s\\right) },f,\\theta _{s}\\right] ^{\\left( s\\right) }.  \\notag\n\\end{eqnarray\nThus, overall, substituting (\\ref{dadtheta}) into (\\ref{dthetafs2}), we\nobtain (\\ref{dthetafs}).\n\\end{proof}\n\nFor Lie groups, $\\theta _{f}$ determines $f$ up to right translation by a\nconstant element, however in the non-associative case this is not\nnecessarily true.\n\n\\begin{lemma}\n\\label{lemThetauniq}Let $M$ be a connected manifold and suppose \nA,B:M\\longrightarrow \\mathbb{L}$ be smooth maps. Then, $A=BC$\\ for some\nconstant $C\\in \\mathbb{L}$ if and only if \n\\begin{equation}\n\\theta _{A}=\\theta _{B}^{\\left( B\\backslash A\\right) }.  \\label{thetaAB}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nFrom (\\ref{thetafs2}), \n\\begin{equation*}\n\\theta _{A}-\\theta _{B}^{\\left( B\\backslash A\\right) }=\\left( \\func{Ad\n_{B}^{\\left( B\\backslash A\\right) }\\right) _{\\ast }\\theta _{B\\backslash A},\n\\end{equation*\nand thus, $B\\backslash A$ is constant if and only if (\\ref{thetaAB}) holds.\n\\end{proof}\n\nIn particular, if $B\\backslash A\\in \\mathcal{N}^{R}\\left( \\mathbb{L}\\right) \n, then $\\theta _{B}^{\\left( B\\backslash A\\right) }=\\theta _{B}$, and hence \n\\theta _{A}=\\theta _{B}$. If $\\mathbb{L}$ is associative, then of course \n\\theta _{B}^{\\left( A\\right) }=\\theta _{B}$ for any $A,B$, and we get the\nstandard result \\cite{SharpeBook}.\n\nWe can also get a version of the structural equation integration theorem. In\nparticular, the question is whether an $\\mathfrak{l}$-valued $1$-form that\nsatisfies the structural equation is the Darboux derivative of some $\\mathbb\nL}$-valued function.\n\n\\begin{lemma}\n\\label{lemAlphastruct}Suppose $M$ is a smooth manifold and $\\mathbb{L}$ a\nsmooth loop. Let $s\\in C^{\\infty }\\left( M,\\mathbb{L}\\right) $ and $\\alpha\n\\in \\Omega ^{1}\\left( M,\\mathfrak{l}\\right) $ satisfy the structural\nequation \n\\begin{equation}\nd\\alpha -\\frac{1}{2}\\left[ \\alpha ,\\alpha \\right] ^{\\left( s\\right) }=0,\n\\label{alphastructeq}\n\\end{equation\nthen \n\\begin{equation}\n\\left[ \\alpha ,\\alpha ,\\alpha -\\theta _{s}\\right] ^{\\left( s\\right) }=0,\n\\label{alphastruct2}\n\\end{equation\nwhere wedge products are implied.\n\\end{lemma}\n\n\\begin{proof}\nApplying $d$ to (\\ref{alphastructeq}) we have \n\\begin{eqnarray*}\n0 &=&d\\left[ \\alpha ,\\alpha \\right] ^{\\left( s\\right) } \\\\\n&=&\\left[ d\\alpha ,\\alpha \\right] ^{\\left( s\\right) }-\\left[ \\alpha ,d\\alpha\n\\right] ^{\\left( s\\right) }+\\left[ \\alpha ,\\alpha ,\\theta _{s}\\right]\n^{\\left( s\\right) } \\\\\n&=&\\left[ \\left[ \\alpha ,\\alpha \\right] ,\\alpha \\right] +\\left[ \\alpha\n,\\alpha ,\\theta _{s}\\right] ^{\\left( s\\right) } \\\\\n&=&-\\left[ \\alpha ,\\alpha ,\\alpha \\right] ^{\\left( s\\right) }+\\left[ \\alpha\n,\\alpha ,\\theta _{s}\\right] ^{\\left( s\\right) },\n\\end{eqnarray*\nwhere we have used (\\ref{db1}) and in the last line an analog of (\\ref{Jac3\n).\n\\end{proof}\n\n\\begin{theorem}\n\\label{thmLoopCartan}Suppose $M$ be a connected and simply-connected smooth\nmanifold and $\\mathbb{L}$ a smooth loop. Let $s\\in C^{\\infty }\\left( M\n\\mathbb{L}\\right) $ and $\\alpha \\in \\Omega ^{1}\\left( M,\\mathfrak{l}\\right) $\nis such that \n\\begin{equation}\nd\\alpha -\\frac{1}{2}\\left[ \\alpha ,\\alpha \\right] ^{\\left( s\\right) }=0,\n\\label{alphastruct}\n\\end{equation\nand \n\\begin{equation}\n\\left( \\func{Ad}_{s}^{-1}\\right) _{\\ast }\\left( \\alpha -\\theta _{s}\\right)\n\\in \\Omega ^{1}\\left( M,T_{1}\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) \\right)\n.  \\label{cartanhyp}\n\\end{equation\nThen, there exists a function $f\\in C^{\\infty }\\left( M,\\mathcal{N\n^{R}\\left( \\mathbb{L}\\right) \\right) $ such that $\\alpha =\\theta _{sf}.$\nMoreover, $f$ is unique up to right multiplication by a constant element of \n\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) .$\n\\end{theorem}\n\n\\begin{proof}\nModifying the standard technique \\cite{SharpeBook,WarnerBook}, let \nN=M\\times \\mathcal{N}^{R}\\left( \\mathbb{L}\\right) \\subset M\\times \\mathbb{L\n. $ Define the projection map $\\pi _{M}:N\\longrightarrow M$ and the map \n\\begin{eqnarray*}\nL_{s} &:&N\\longrightarrow \\mathbb{L} \\\\\n\\left( x,p\\right) &\\mapsto &s\\left( x\\right) p\n\\end{eqnarray*\nGiven the Maurer-Cartan form $\\theta $ on $\\mathbb{L}$ and $\\alpha \\in\n\\Omega ^{1}\\left( M,\\mathfrak{l}\\right) $, define $\\beta \\in \\Omega\n^{1}\\left( N,\\mathfrak{l}\\right) $ b\n\\begin{equation}\n\\beta =\\pi _{M}^{\\ast }\\alpha -\\left( L_{s}\\right) ^{\\ast }\\theta .\n\\label{beta}\n\\end{equation\nThen, at each point $\\left( x,p\\right) \\in N$, define $\\mathcal{D}_{\\left(\nx,p\\right) }=\\left. \\ker \\beta \\right\\vert _{\\left( x,p\\right) }$. We can\nthen see that this is a distribution on $N$ of rank $\\dim M$. Let $\\left(\nv,w\\right) \\in T_{\\left( x,p\\right) }N$, where we consider $w\\in $ $T_{p\n\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) \\subset T_{p}\\mathbb{L}.$ Then, \n\\begin{equation}\n\\beta _{\\left( x,p\\right) }\\left( v,w\\right) =\\alpha _{x}\\left( v\\right)\n-\\theta _{s\\left( x\\right) p}\\left( \\left( L_{s}\\right) _{\\ast }\\left(\nv,w\\right) \\right) .  \\label{betavw}\n\\end{equation\nNow, let $x\\left( t\\right) $ be a curve on $M$ with $x\\left( 0\\right) =x$\nand $\\dot{x}\\left( 0\\right) =v,$ and $p\\left( t\\right) $ a curve in \n\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) \\subset \\mathbb{L}$ with $p\\left(\n0\\right) =p$ and $\\dot{p}\\left( 0\\right) =w$. Then, using the fact that $p$\nis in the right nucleus\n\\begin{eqnarray*}\n\\theta _{s\\left( x\\right) p}\\left( \\left( L_{s}\\right) _{\\ast }\\left(\nv,w\\right) \\right) &=&\\left. \\frac{d}{dt}\\faktor{\\left( s\\left( x\\left(\nt\\right) \\right) p\\left( t\\right) \\right)} {\\left( s\\left( x\\right) p\\right)}\n\\right\\vert _{t=0} \\\\\n&=&\\left. \\frac{d}{dt}\\faktor{s\\left( x\\left( t\\right) \\right)} {s\\left(\nx\\right)} \\right\\vert _{t=0}+\\left. \\frac{d}{dt}\\faktor{\\left( s\\left(\nx\\right) \\left( \\faktor{p\\left( t\\right)}{p}\\right) \\right)} {s\\left(\nx\\right)} \\right\\vert _{t=0} \\\\\n&=&\\left. \\theta _{s}\\left( v\\right) \\right\\vert _{x}+\\left( \\func{Ad\n_{s\\left( x\\right) }\\right) _{\\ast }w.\n\\end{eqnarray*\nSo overall, \n\\begin{equation}\n\\beta _{\\left( x,p\\right) }\\left( v,w\\right) =\\left( \\alpha -\\theta\n_{s}\\right) _{x}\\left( v\\right) -\\left( \\func{Ad}_{s\\left( x\\right) }\\right)\n_{\\ast }w.\n\\end{equation\nHence, $\\left( v,w\\right) \\in \\mathcal{D}_{\\left( x,p\\right) }$ if and only\nif $\\left( \\alpha -\\theta _{s}\\right) _{x}\\left( v\\right) =\\left( \\func{Ad\n_{s\\left( x\\right) }\\right) _{\\ast }w.$ Now, consider $\\left. \\left( \\pi\n_{M}\\right) _{\\ast }\\right\\vert _{\\left( x,p\\right) }:\\mathcal{D}_{\\left(\nx,p\\right) }\\longrightarrow T_{x}M$. Suppose $\\left. \\left( \\pi _{M}\\right)\n_{\\ast }\\right\\vert _{\\left( x,p\\right) }\\left( v,w\\right) =0.$ Then, $v=0$,\nand since $\\left( \\alpha -\\theta _{s}\\right) _{x}\\left( v\\right) =\\left( \n\\func{Ad}_{s\\left( x\\right) }\\right) _{\\ast }w,$ we have $w=0$. Thus $\\left.\n\\left( \\pi _{M}\\right) _{\\ast }\\right\\vert _{\\left( x,p\\right) }$ is\ninjective on $\\mathcal{D}_{\\left( x,p\\right) }.$ On the other hand, it is\nalso clearly surjective, since if given $v\\in T_{x}M$, then $\\left( v,\\left( \n\\func{Ad}_{s\\left( x\\right) }^{-1}\\right) _{\\ast }\\left( \\left( \\alpha\n-\\theta _{s}\\right) _{x}\\left( v\\right) \\right) \\right) \\in \\mathcal{D\n_{\\left( x,p\\right) }.$ Overall, $\\left. \\left( \\pi _{M}\\right) _{\\ast\n}\\right\\vert _{\\left( x,p\\right) }$ is a bijection from $\\mathcal{D}_{\\left(\nx,p\\right) }$ to $T_{x}M$, so in particular, $\\dim \\mathcal{D}_{\\left(\nx,p\\right) }=\\dim M$ and thus $\\mathcal{D}$ is a distribution of rank $\\dim\nM.$\n\nNow let us show that $\\mathcal{D}$ is involutive. We have \n\\begin{eqnarray}\n\\left. d\\beta \\right\\vert _{\\left( x,p\\right) } &=&\\left. \\pi _{M}^{\\ast\n}d\\alpha \\right\\vert _{\\left( x,p\\right) }-\\left. \\left( L_{s}\\right) ^{\\ast\n}d\\theta \\right\\vert _{\\left( x,p\\right) }  \\notag \\\\\n&=&\\frac{1}{2}\\left. \\pi _{M}^{\\ast }\\left[ \\alpha ,\\alpha \\right] ^{\\left(\ns\\right) }\\right\\vert _{\\left( x,p\\right) }-\\frac{1}{2}\\left. \\left(\nL_{s}\\right) ^{\\ast }\\left[ \\theta ,\\theta \\right] \\right\\vert _{\\left(\nx,p\\right) }  \\notag \\\\\n&=&\\frac{1}{2}\\left[ \\left. \\pi _{M}^{\\ast }\\alpha \\right\\vert _{\\left(\nx,p\\right) },\\left. \\pi _{M}^{\\ast }\\alpha \\right\\vert _{\\left( x,p\\right) \n\\right] ^{s\\left( x\\right) }  \\label{dbeta} \\\\\n&&-\\frac{1}{2}\\left[ \\left. \\left( L_{s}\\right) ^{\\ast }\\theta \\right\\vert\n_{\\left( x,p\\right) },\\left. \\left( L_{s}\\right) ^{\\ast }\\theta \\right\\vert\n_{\\left( x,p\\right) }\\right] ^{s\\left( x\\right) p}.  \\notag\n\\end{eqnarray\nNote however that because $p\\in \\mathcal{N}^{R}\\left( \\mathbb{L}\\right) ,$\nwe have $\\left[ \\cdot ,\\cdot \\right] ^{s\\left( x\\right) }=\\left[ \\cdot\n,\\cdot \\right] ^{s\\left( x\\right) p}.$ So overall, using (\\ref{beta}), we\nget \n\\begin{equation*}\n\\left. d\\beta \\right\\vert _{\\left( x,p\\right) }=\\frac{1}{2}\\left[ \\left.\n\\beta \\right\\vert _{\\left( x,p\\right) },\\left. \\beta \\right\\vert _{\\left(\nx,p\\right) }\\right] ^{s\\left( x\\right) }+\\left[ \\left. \\beta \\right\\vert\n_{\\left( x,p\\right) },\\left. \\left( L_{s}\\right) ^{\\ast }\\theta \\right\\vert\n_{\\left( x,p\\right) }\\right] ^{s\\left( x\\right) }.\n\\end{equation*\nThus, $d\\beta =0$ whenever $\\beta =0$, and hence $\\mathcal{D=}\\ker \\beta $\nis involutive, and by the Frobenius Theorem, $\\mathcal{D}$ is integrable.\nLet $\\mathcal{L}$ be a leaf through the point $\\left( x,p\\right) \\in N.$\nThen, $\\pi _{M}$ induced a local diffeomorphism from a neighborhood to \n\\left( x,p\\right) $ to some neighborhood of $x\\in M.$ Then, let \nF:U\\longrightarrow \\mathcal{L}$ be the inverse map, such that $F\\left(\ny\\right) =\\left( y,f\\left( y\\right) \\right) $ for some $f:U\\longrightarrow \n\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) .$ By definition, $F^{\\ast }\\beta =0\n, so \n\\begin{eqnarray*}\n0 &=&F^{\\ast }\\beta \\\\\n&=&F^{\\ast }\\left( \\pi _{M}^{\\ast }\\alpha -\\left( L_{s}\\right) ^{\\ast\n}\\theta \\right) \\\\\n&=&\\alpha -\\left( L_{s}\\circ f\\right) ^{\\ast }\\theta\n\\end{eqnarray*\nHence, on $U$, $\\alpha =\\theta _{sf}$.\n\nIt is obvious that the distribution $\\mathcal{D}$ is right-invariant with\nrespect to $\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) $, then proceeding in\nthe same way as for Lie groups, we find that in fact that when $M$ is\nconnected and simply-connected, the function $f$ extends to the whole\nmanifold.\n\nNow suppose $f,g\\in C^{\\infty }\\left( M,\\mathcal{N}^{R}\\left( \\mathbb{L\n\\right) \\right) $ such that $\\theta _{sf}=\\theta _{sg}.$ Then using (\\re\n{thetafs}), but with roles of $s$ and $f$ reversed, we find \n\\begin{equation*}\n\\theta _{sf}=\\theta _{s}+\\left( \\func{Ad}_{s}\\right) _{\\ast }\\theta _{f},\n\\end{equation*\nand similarly for $g$. Hence, we see that $\\theta _{f}=\\theta _{g}.$ Using\nLemma \\ref{lemThetauniq} for Lie groups, we find that $f=gC$ for some\nconstant $C\\in \\mathcal{N}^{R}\\left( \\mathbb{L}\\right) .$\n\\end{proof}\n\n\\begin{remark}\nIn the case when $\\mathbb{L}$ is a group, Theorem \\ref{thmLoopCartan}\nreduces to the well-known analogous result for groups since the function $s$\ncan be taken to be arbitrary. In particular, the hypothesis (\\ref{cartanhyp\n) is automatically satisfied in that case. On the other hand, for the loop\nof unit octonions, this theorem becomes trivial. In this case, $\\mathcal{N\n^{R}\\left( \\mathbb{L}\\right) \\cong \\mathbb{Z}_{2},$ so the hypothesis (\\re\n{cartanhyp}) immediately implies that $\\alpha =\\theta _{s},$ even without\nusing the equation (\\ref{alphastruct}). However, under certain additional\nassumptions about $\\alpha $ and $s$, (\\ref{alphastruct}) may actually imply \n\\ref{cartanhyp}). Generally, (\\ref{cartanhyp}) is stronger than (\\re\n{alphastruct2}), which we know holds for any $\\alpha \\in \\Omega ^{1}\\left( M\n\\mathfrak{l}\\right) $ that satisfies (\\ref{alphastruct}). To bridge the gap\nbetween (\\ref{alphastruct2}) and (\\ref{cartanhyp}), additional properties of \n$\\mathbb{L}$ and $\\alpha $ are needed.\n\\end{remark}\n\n\\begin{corollary}\n\\label{corLoopCartan}Suppose $M$ be a connected and simply-connected smooth\nmanifold and $\\mathbb{L}$ a smooth loop such that $\\dim \\left( \\mathcal{N\n^{R}\\left( \\mathbb{L}\\right) \\right) =\\dim \\left( \\mathcal{N}^{R}\\left( \n\\mathfrak{l}\\right) \\right) .$ Also suppose that $s\\in C^{\\infty }\\left( M\n\\mathbb{L}\\right) $ and $\\alpha \\in \\Omega ^{1}\\left( M,\\mathfrak{l}\\right) $\nare such that\n\n\\begin{enumerate}\n\\item $d\\alpha -\\frac{1}{2}\\left[ \\alpha ,\\alpha \\right] ^{\\left( s\\right)\n}=0,$\n\n\\item $\\left. \\alpha \\right\\vert _{x}:T_{x}M\\longrightarrow \\mathfrak{l}$ is\nsurjective for every $x\\in M,$\n\n\\item $T_{x}M\\cong \\ker \\left. \\alpha \\right\\vert _{x}+\\ker \\left( \\left.\n\\theta _{s}\\right\\vert _{x}-\\left. \\alpha \\right\\vert _{x}\\right) $ for\nevery $x\\in M$,\n\n\\item $s_{x}\\in \\mathcal{C}^{R}\\left( \\mathbb{L}\\right) $ for every $x\\in M$.\n\\end{enumerate}\n\nThen, there exists a function $f\\in C^{\\infty }\\left( M,\\mathcal{N\n^{R}\\left( \\mathbb{L}\\right) \\right) $ such that $\\alpha =\\theta _{sf}$ with \n$f$ unique up to right multiplication by a constant element of $\\mathcal{N\n^{R}\\left( \\mathbb{L}\\right) .$\n\\end{corollary}\n\n\\begin{proof}\nSince $\\alpha $ satisfies (\\ref{alphastruct}), from Lemma \\re\n{lemAlphastruct} we know that it also satisfies (\\ref{alphastruct2}).\nSuppose $X,Y,Z\\in T_{x}M$, such that $Z\\in \\ker \\left. \\alpha \\right\\vert\n_{x}$. Then, from (\\ref{alphastruct2}) we obtain \n\\begin{equation}\n\\left[ \\alpha \\left( X\\right) ,\\alpha \\left( Y\\right) ,\\left( \\alpha -\\theta\n_{s_{x}}\\right) Z\\right] ^{\\left( s_{x}\\right) }-\\left[ \\alpha \\left(\nY\\right) ,\\alpha \\left( X\\right) ,\\left( \\alpha -\\theta _{s_{x}}\\right) \n\\right] ^{\\left( s_{x}\\right) }=0.  \\label{alphastructassoc}\n\\end{equation\nHowever, since $T_{x}M\\cong \\ker \\left. \\alpha \\right\\vert _{x}+\\ker \\left(\n\\left. \\theta _{s}\\right\\vert _{x}-\\left. \\alpha \\right\\vert _{x}\\right) $,\nthis is true for any $Z\\in T_{x}M.$ Since$\\left. \\alpha \\right\\vert _{x}$ is\nsurjective, we hence find that for any $\\xi ,\\eta \\in \\mathfrak{l},$ \n\\begin{equation}\n\\left[ \\xi ,\\eta ,\\left( \\alpha -\\theta _{s_{x}}\\right) Z\\right] ^{\\left(\ns_{x}\\right) }-\\left[ \\eta ,\\xi ,\\left( \\alpha -\\theta _{s_{x}}\\right) \n\\right] ^{\\left( s_{x}\\right) }=0.  \\label{alphastructassoc2}\n\\end{equation\nNow, since $s_{x}\\in \\mathcal{C}^{R}\\left( \\mathbb{L}\\right) ,$ it is the\nright companion of some $h\\in \\Psi ^{R}\\left( \\mathbb{L}\\right) $, thus\napplying $\\left( h^{\\prime }\\right) _{\\ast }^{-1}$ to (\\re\n{alphastructassoc2}), and using (\\ref{loopalghom2}), we find that for any \n\\xi ,\\eta \\in \\mathfrak{l},$ \n\\begin{equation*}\n\\left[ \\xi ,\\eta ,\\left( h^{\\prime }\\right) _{\\ast }^{-1}\\left( \\left(\n\\alpha -\\theta _{s_{x}}\\right) Z\\right) \\right] ^{\\left( 1\\right) }-\\left[\n\\eta ,\\xi ,\\left( h^{\\prime }\\right) _{\\ast }^{-1}\\left( \\left( \\alpha\n-\\theta _{s_{x}}\\right) Z\\right) \\right] ^{\\left( 1\\right) }=0.\n\\end{equation*\nThus, we see that for any $Z\\in T_{x}M$, $\\left( h^{\\prime }\\right) _{\\ast\n}^{-1}\\left( \\left( \\alpha -\\theta _{s_{x}}\\right) Z\\right) \\in \\mathcal{N\n^{R}\\left( \\mathfrak{l}\\right) .$ We know that $\\mathcal{\\ }T_{1}\\mathcal{N\n^{R}\\left( \\mathbb{L}\\right) \\subset \\mathcal{N}^{R}\\left( \\mathfrak{l\n\\right) $, however by hypothesis, their dimensions are equal, so in fact, \nT_{1}\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) =\\mathcal{N}^{R}\\left( \n\\mathfrak{l}\\right) .$ Thus, $\\left( h^{\\prime }\\right) _{\\ast }^{-1}\\left(\n\\left( \\alpha -\\theta _{s_{x}}\\right) Z\\right) \\in T_{1}\\mathcal{N\n^{R}\\left( \\mathbb{L}\\right) $ and hence, from (\\ref{nuclearaction}), \n\\left( \\func{Ad}_{s\\left( x\\right) }^{-1}\\right) _{\\ast }\\left( \\alpha\n-\\theta _{s_{x}}\\right) \\in \\Omega ^{1}\\left( M,T_{1}\\mathcal{N}^{R}\\left( \n\\mathbb{L}\\right) \\right) .$ This fulfils the hypothesis (\\ref{cartanhyp})\nfor Theorem \\ref{thmLoopCartan}, and thus there exists a function $f\\in\nC^{\\infty }\\left( M,\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) \\right) $ such\nthat $\\alpha =\\theta _{sf}.$\n\\end{proof}\n\n\\begin{remark}\nSince $\\alpha $ is assumed to be surjective in Corollary \\ref{corLoopCartan}\nand $\\alpha =\\theta _{sf},$ we see that $sf:M\\longrightarrow \\mathbb{L}$ is\na smooth submersion.\n\\end{remark}\n\n\\section{Loop bundles}\n\n\\setcounter{equation}{0}\\label{sectBundle}Let $\\mathbb{L}$ be a smooth loop\nwith the $\\mathbb{L}$-algebra $\\mathfrak{l},$ and let us define for brevity \n\\Psi ^{R}\\left( \\mathbb{L}\\right) =\\Psi $, $\\func{Aut}\\left( \\mathbb{L\n\\right) =H$, and $\\func{PsAut}^{R}\\left( \\mathbb{L}\\right) =G\\supset H$, and \n$\\mathcal{N}^{R}\\left( \\mathbb{L}\\right) =\\mathcal{N}$. Suppose $\\Psi ,H,G\n\\mathcal{N}$ are Lie groups. Recall that we also have $\\Psi \/\\mathcal{N\n\\cong G.$\n\nLet $M$ be a smooth, finite-dimensional manifold with a $\\Psi $-principal\nbundle $\\mathcal{P}.$ Then we will define several associated bundles. In\ngeneral, recall that there is a one-to-one correspondence between\nequivariant maps from a principal bundle and sections of associated bundles.\nMore precisely, suppose we have a manifold $S$ with a left action $l:\\Psi\n\\times S\\longrightarrow S.$ Consider the associated bundle $E=\\mathcal{P\n\\times _{\\Psi }S.$ Suppose we have a section $\\tilde{f}:M\\longrightarrow E$,\nthen this defines a unique equivariant map $f$ $:\\mathcal{P}\\longrightarrow\nS,$ that is, such that for any $h\\in \\Psi $, \n\\begin{equation}\nf_{ph}=l_{h^{-1}}\\left( f_{p}\\right) .  \\label{equimap}\n\\end{equation\nConversely, any equivariant map $f:\\mathcal{P}\\longrightarrow S$ defines a\nsection $\\left( \\func{id},f\\right) :\\mathcal{P}\\longrightarrow \\mathcal{P\n\\times S$, and then via the quotient map $q:\\mathcal{P}\\times\nS\\longrightarrow \\mathcal{P}\\times _{\\Psi }S=E$, it defines a section \n\\tilde{f}:M\\longrightarrow E.$ In particular, for each $x\\in M$, $\\tilde{f\n\\left( x\\right) =$ $\\left\\lfloor p,f_{p}\\right\\rfloor _{\\Psi }$ where $p\\in\n\\pi ^{-1}\\left( x\\right) \\subset \\mathcal{P}$ and $\\left\\lfloor \\cdot ,\\cdot\n\\right\\rfloor _{\\Psi }$ is the equivalence class with respect to the action\nof $\\Psi :\n\\begin{equation}\n\\left( p,f_{p}\\right) \\sim \\left( ph,l_{h^{-1}}\\left( f_{p}\\right) \\right)\n=\\left( ph,f_{ph}\\right) \\ \\ \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{for any }h\\in \\Psi .\n\\end{equation\nFor our purposes we will have the following associated bundles. Let $h\\in\n\\Psi $ and, as before, denote by $h^{\\prime }$ the partial action of $h$.\n\n\\begin{equation}\n\\begin{tabular}{l|l|l}\n\\textbf{Bundle} & \\textbf{Equivariant map} & \\textbf{Equivariance property}\n\\\\ \\hline\n$\\mathcal{P}$ & $k:\\mathcal{P}\\longrightarrow \\Psi $ & $k_{ph}=h^{-1}k_{p}$\n\\\\ \n$\\mathcal{Q}=\\mathcal{P}\\times _{\\Psi ^{\\prime }}\\mathbb{L}$ & $q:\\mathcal{P\n\\longrightarrow \\mathbb{L}$ & $q_{ph}=\\left( h^{\\prime }\\right) ^{-1}q_{p}$\n\\\\ \n$\\mathcal{\\mathring{Q}}=\\mathcal{P\\times }_{\\Psi }\\mathbb{\\mathring{L}}$ & \nr:\\mathcal{P}\\longrightarrow \\mathbb{\\mathring{L}}$ & $r_{ph}=h^{-1}\\left(\nr_{p}\\right) $ \\\\ \n$\\mathcal{N}\\cong \\mathcal{P}\\times _{\\Psi }\\left( \\Psi \/H\\right) $ & $s\n\\mathcal{P}\\longrightarrow \\Psi \/H\\cong \\mathcal{C\\subset }\\mathbb{\\mathring\nL}}$ & $s_{ph}=h^{-1}\\left( s_{p}\\right) $ \\\\ \n$\\mathcal{A}=\\mathcal{P\\times }_{\\Psi _{\\ast }^{\\prime }}\\mathfrak{l}$ & \n\\eta :\\mathcal{P}\\longrightarrow \\mathfrak{l}$ & $\\eta _{ph}=\\left(\nh^{\\prime }\\right) _{\\ast }^{-1}\\eta _{p}$ \\\\ \n$\\mathfrak{p}_{\\mathcal{P}}=\\mathcal{P\\times }_{\\left( \\func{Ad}_{\\xi\n}\\right) _{\\ast }}\\mathfrak{p}$ & $\\xi :\\mathcal{P}\\longrightarrow \\mathfrak\np}$ & $\\xi _{ph}=\\left( \\func{Ad}_{h}^{-1}\\right) _{\\ast }\\xi _{p}$ \\\\ \n$\\mathcal{G=P}\\times _{\\Psi ^{\\prime }}G$ & $\\gamma :\\mathcal{P\n\\longrightarrow G$ & $\\gamma _{ph}=\\left( h^{\\prime }\\right) ^{-1}\\gamma\n_{p} $ \\\\ \n$\\func{Ad}\\left( \\mathcal{P}\\right) =\\mathcal{P}\\times _{\\func{Ad}_{\\Psi\n}}\\Psi $ & $u:\\mathcal{P}\\longrightarrow \\Psi $ & $u_{ph}=h^{-1}u_{p}h\n\\end{tabular}\n\\label{equimap2}\n\\end{equation}\n\nThe bundle $\\mathcal{Q}$ is the loop bundle with respect to the partial\naction $\\Psi ^{\\prime }$ and the bundle $\\mathcal{\\mathring{Q}}$ is the loop\nbundle with respect to the full action of $\\Psi $. The bundle $\\mathcal{N}$\nhas fibers isomorphic to $\\Psi \/H\\cong \\mathcal{C},$ which is the set of\nright companions $\\mathcal{C}^{R}\\left( \\mathbb{L}\\right) \\subset \\mathbb\n\\mathring{L}}$. Thus it is a subbundle of $\\mathcal{\\mathring{Q}}.$\nEquivalently, $\\mathcal{N}=\\mathcal{P}\/H$ is the orbit space of the right $H\n-action on $\\mathcal{P}$. Recall that the structure group of $\\mathcal{P}$\nreduces to $H$ if and only if the bundle $N$ has a global section. If this\nis the case, then we can reduce the bundle $\\mathcal{P}$ to a principal $H\n-bundle $\\mathcal{H}$ over $M$, and then since $H\\subset G$, lift to a\nprincipal $G$-bundle $\\mathcal{G}$. Also, let $\\mathcal{Q}=\\mathcal{P}\\times\n_{\\Psi ^{\\prime }}\\mathbb{L}$ be the bundle associated to $\\mathcal{P}$ with\nfiber $\\mathbb{L}$, where $\\Psi ^{\\prime }$ denotes that the left action on \n\\mathbb{L}$ is via the partial action of $\\Psi $.\n\nWe also have some associated vector bundles - namely the vector bundle \n\\mathcal{A}$ with fibers isomorphic to the $\\mathbb{L}$-algebra $\\mathfrak{l\n\\ $with the tangent partial action of $\\Psi $ and the vector bundle \n\\mathfrak{p}_{\\mathcal{P}}$ with fibers isomorphic to the Lie algebra \n\\mathfrak{p}$, with the adjoint action of $\\Psi $.\n\n\\begin{example}\nSuppose $\\mathbb{L}=U\\mathbb{O}$ - the Moufang loop of unit octonions. In\nthis case, $\\Psi =Spin\\left( 7\\right) $, $H=G_{2}$, $G=SO\\left( 7\\right) $, \n\\mathcal{N}=\\mathbb{Z}_{2}$, and then we have the well-known relation\n\\begin{eqnarray*}\nSO\\left( 7\\right) &\\cong &Spin\\left( 7\\right) \/\\mathbb{Z}_{2} \\\\\nSpin\\left( 7\\right) \/G_{2} &\\cong &U\\mathbb{O\\cong }S^{7} \\\\\nSO\\left( 7\\right) \/G_{2} &\\cong &S^{7}\/\\mathbb{Z}_{2}.\n\\end{eqnarray*\nThen, if an orientable $7$-manifold has spin structure, we have a principal \nSpin\\left( 7\\right) $-bundle $\\mathcal{P}$ over $M$ and the corresponding \nSpin\\left( 7\\right) \/G_{2}$-bundle always has a smooth section, hence\nallowing the $Spin\\left( 7\\right) $-bundle to reduce to a $G_{2}$-principal\nbundle, which in turn lifts to an $SO\\left( 7\\right) $-bundle. The\nassociated bundle $\\mathcal{Q}$ in this case transforms under $SO\\left(\n7\\right) $, and is precisely the unit subbundle of the octonion bundle \n\\mathbb{R}\\oplus TM$ defined in \\cite{GrigorianOctobundle}. The bundle \n\\mathcal{\\mathring{Q}}$ then transforms under $Spin\\left( 7\\right) $ and\ncorresponds to the bundle of unit spinors. The associated vector bundle \n\\mathcal{A}$ in this case has fibers isomorphic to $\\func{Im}\\mathbb{O}\\cong \n\\mathbb{R}^{7}$, and then the bundle itself is isomorphic to the tangent\nbundle $TM.$\n\\end{example}\n\nLet $s:$ $\\mathcal{P}\\longrightarrow \\mathbb{\\mathring{L}}$ be an\nequivariant map. In particular, the equivalence class $\\left\\lfloor\np,s_{p}\\right\\rfloor _{\\Psi }$ defines a section of the bundle $\\mathcal\n\\mathring{Q}}.$ We will refer to $s$ as \\emph{the defining map} (or \\emph\nsection}). It should be noted that such a map may not always exist globally.\nIf $\\mathbb{L}$ is a $G$-loop\\textbf{,} then $\\mathcal{\\mathring{Q}\\cong N}$\nand hence existence of a global section of $\\mathcal{\\mathring{Q}}$ is\nequivalent to the reduction of the structure group of $\\mathcal{P}.$ There\nmay be topological obstructions for this.\n\n\\begin{example}\n\\label{exCx3}As in Example \\ref{ExNormedDiv}, let $\\mathbb{L=}U\\mathbb\nC\\cong }U\\left( 1\\right) $ - the unit complex numbers, and $\\Psi =U\\left(\nn\\right) $, $H=G=SU\\left( n\\right) .$ Then in this setting, $\\mathcal{P}$ is\na principal $U\\left( n\\right) $-bundle over $M$ and $\\mathcal{Q}$ is a\ncircle bundle. Existence of a section of $\\mathcal{Q}$ is equivalent to the\nreduction of the structure group of $\\mathcal{P}$ to $SU\\left( n\\right) .$\nThe obstruction for this is the first Chern class of $\\mathcal{Q}$ \\cit\n{MilnorStasheff}. In the quaternionic case, the structure group reduction\nfrom $Sp\\left( n\\right) Sp\\left( 1\\right) $ to $Sp\\left( n\\right) $ is less\nwell understood \\cite{BoyerGalicki}.\n\\end{example}\n\nGiven equivariant maps $q,r:\\mathcal{P}\\longrightarrow \\mathbb{L}$, we can\ndefine an equivariant product using $s$, such that for any $p\\in \\mathcal{P}\n\n\\begin{equation}\n\\left. q\\circ _{s}r\\right\\vert _{p}=q_{p}\\circ _{s_{p}}r_{p}.\n\\label{equiprod}\n\\end{equation\nIndeed, using (\\ref{PsiActcircr}), \n\\begin{eqnarray}\n\\left. q\\circ _{s}r\\right\\vert _{ph} &=&q_{ph}\\circ _{s_{ph}}r_{ph}  \\notag\n\\\\\n&=&\\left( h^{\\prime }\\right) ^{-1}q_{p}\\circ _{h^{-1}\\left( s_{p}\\right)\n}\\left( h^{\\prime }\\right) ^{-1}r_{p}  \\notag \\\\\n&=&\\left( h^{\\prime }\\right) ^{-1}\\left( \\left. q\\circ _{s}r\\right\\vert\n_{p}\\right) .  \\label{equiprod2}\n\\end{eqnarray\nIn particular, this allows to define a fiberwise product on sections of \n\\mathcal{Q}$. Similarly, we define equivariant left and right quotients, and\nthus well-defined fiberwise quotients of sections of $\\mathcal{Q}.$\n\n\\begin{remark}\nThe map $s$ is required to define an equivariant product of two $\\mathbb{L}\n-valued maps. In the $G_{2}$-structure case, as discussed above, sections of \n$\\mathcal{\\mathring{Q}}$ correspond to unit spinors, and each unit spinor\ndefines a $G_{2}$-structure, and hence a product on the corresponding\noctonion bundle \\cite{GrigorianOctobundle}. On the other hand, a product of\nan equivariant $\\mathbb{L}$-valued map and an equivariant $\\mathbb{\\mathring\nL}}$-valued map will be always equivariant, using (\\ref{PsAutprod}). In the \nG_{2}$-structure case, this corresponds to the Clifford product of a unit\noctonion, interpreted as an element of $\\mathbb{R}\\oplus T_{x}M$ at each\npoint, and a unit spinor. The result is then again a unit spinor. This does\nnot require any additional structure beyond the spinor bundle.\n\\end{remark}\n\nGiven equivariant maps $\\xi ,\\eta :\\mathcal{P}\\longrightarrow \\mathfrak{l}$,\nwe can define an equivariant bracket using $s$. For any $p\\in \\mathcal{P}$\n\\begin{equation}\n\\left. \\left[ \\xi ,\\eta \\right] ^{\\left( s\\right) }\\right\\vert _{p}=\\left[\n\\xi _{p},\\eta _{p}\\right] ^{\\left( s_{p}\\right) }.  \\label{equibracket}\n\\end{equation\nHere the equivariance follows from (\\ref{loopalghom}). Using (\\ref{phihs})\nwe then also have an equivariant map $\\varphi _{s}$ from equivariant \n\\mathfrak{p}$-valued maps to equivariant $\\mathfrak{l}$-valued maps\n\\begin{equation}\n\\left. \\varphi _{s}\\left( \\gamma \\right) \\right\\vert _{p}=\\varphi\n_{s_{p}}\\left( \\gamma _{p}\\right) .  \\label{equiphis}\n\\end{equation\nOther related objects such as the Killing form $K^{\\left( s\\right) }$ and\nthe adjoint $\\varphi _{s}^{t}$ to $\\varphi _{s}$ are then similarly also\nequivariant.\n\nOverall, we can condense the above discussion into the following definition\nand theorem.\n\n\\begin{definition}\n\\label{defLoopStructure}A \\emph{loop bundle structure} over a smooth\nmanifold $M$ is a quadruple $\\left( \\mathbb{L},\\Psi ,\\mathcal{P},s\\right) $\nwhere\n\n\\begin{enumerate}\n\\item $\\mathbb{L}$ is a finite-dimensional smooth loop with a smoothly\nacting group of right pseudoautomorphism pairs $\\Psi $.\n\n\\item $\\mathcal{P}$ is a principal $\\Psi $-bundle over $M$.\n\n\\item $s:\\mathcal{P}\\longrightarrow \\mathbb{\\mathring{L}}$ is a smooth\nequivariant map.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{theorem}\nGiven a loop bundle structure $\\left( \\mathbb{L},\\Psi ,\\mathcal{P},s\\right) $\nover a manifold $M,$ and associated bundles $\\mathcal{Q}=\\mathcal{P}\\times\n_{\\Psi ^{\\prime }}\\mathbb{L},$ $\\mathcal{\\mathring{Q}}=\\mathcal{P\\times \n_{\\Psi }\\mathbb{\\mathring{L}},$ $\\mathcal{A}=\\mathcal{P\\times }_{\\Psi _{\\ast\n}^{\\prime }}\\mathfrak{l},$ and $\\mathfrak{p}_{\\mathcal{P}}=\\mathcal{P\\times \n_{\\left( \\func{Ad}_{\\xi }\\right) _{\\ast }}\\mathfrak{p},$ where $\\mathfrak{l}$\nis the $\\mathbb{L}$-algebra of $\\mathbb{L}$ and $\\mathfrak{p}$ the Lie\nalgebra of $\\Psi ,$\n\n\\begin{enumerate}\n\\item $s$ determines a smooth section $\\sigma \\in \\Gamma \\left( \\mathcal\n\\mathring{Q}}\\right) .$\n\n\\item For any $A,B\\in \\Gamma \\left( \\mathcal{Q}\\right) $, $\\sigma $ defines\na fiberwise product $A\\circ _{\\sigma }B,$ via (\\ref{equiprod}).\n\n\\item For any $X,Y\\in \\Gamma \\left( \\mathcal{A}\\right) ,$ $\\sigma $ defines\na fiberwise bracket $\\left[ X,Y\\right] ^{\\left( \\sigma \\right) }$, via (\\re\n{equibracket}).\n\n\\item $\\sigma $ defines a fiberwise map $\\varphi _{\\sigma }:\\Gamma \\left( \n\\mathfrak{p}_{\\mathcal{P}}\\right) \\longrightarrow \\Gamma \\left( \\mathcal{A\n\\right) $, via (\\ref{equiphis}).\n\\end{enumerate}\n\\end{theorem}\n\n\\subsection{Connections and Torsion}\n\n\\label{sectTorsion}Suppose the principal $\\Psi $-bundle $\\mathcal{P}$ has a\nprincipal Ehresmann connection given by the decomposition \n\\begin{equation}\nT\\mathcal{P}=\\mathcal{HP}\\oplus \\mathcal{VP}  \\label{HPVP}\n\\end{equation\nwith $\\mathcal{H}_{ph}\\mathcal{P}=\\left( R_{h}\\right) _{\\ast }\\mathcal{H}_{p\n\\mathcal{P}$ for any $p\\in \\mathcal{P}$ and $h\\in \\Psi $ and $\\mathcal{V\n\\mathcal{P}=\\ker d\\pi $, where $\\pi :\\mathcal{P}\\longrightarrow M$ is the\nbundle projection map. Let the projection \n\\begin{equation*}\nv:T\\mathcal{P}\\longrightarrow \\mathcal{VP}\n\\end{equation*}\nbe the Ehresmann connection $1$-form. Similarly, define the projection \n\\func{proj}_{\\mathcal{H}}:T\\mathcal{P}\\longrightarrow \\mathcal{HP}.$\n\nLet $\\mathfrak{p}$ be the Lie algebra of $\\Psi $. Then, as it is well-known,\nwe have an isomorphism \n\\begin{eqnarray}\n\\sigma &:&\\mathcal{P}\\times \\mathfrak{p}\\longrightarrow \\mathcal{VP}  \\notag\n\\\\\n\\left( p,\\xi \\right) &\\mapsto &\\left. \\frac{d}{dt}\\left( p\\exp \\left( t\\xi\n\\right) \\right) \\right\\vert _{t=0}.  \\label{mapsigma}\n\\end{eqnarray\nFor any $\\xi \\in \\mathfrak{p},$ this defines a vertical vector field $\\sigma\n\\left( \\xi \\right) $ on $\\mathcal{P}$. Given the Ehresmann connection 1-form \n$v$, define the $\\mathfrak{p}$-valued connection $1$-form $\\omega $ via \n\\begin{equation*}\n\\left( \\pi ,\\omega \\right) =\\sigma ^{-1}\\circ v:T\\mathcal{P}\\longrightarrow \n\\mathcal{P}\\times \\mathfrak{p}\n\\end{equation*\nand recall that for any $h\\in \\Psi $, \n\\begin{equation*}\n\\left( R_{h}\\right) ^{\\ast }\\omega =\\func{Ad}_{h^{-1}}\\circ \\omega .\n\\end{equation*}\n\nAs previously, suppose $S$ is a manifold with a left action $l$ of $\\Psi .$\nGiven an equivariant map $f:\\mathcal{P}\\longrightarrow S$, define \n\\begin{equation}\nd^{\\mathcal{H}}f:=f_{\\ast }\\circ \\func{proj}_{\\mathcal{H}}:T\\mathcal{P\n\\longrightarrow \\mathcal{HP}\\longrightarrow TS.  \\label{dHftilde}\n\\end{equation\nThis is then a horizontal map since it vanishes on any vertical vectors.\nEquivalently, for any $X_{p}\\in T_{p}\\mathcal{P},$ if $\\gamma \\left(\nt\\right) $ is a curve on $\\mathcal{P}$ with $\\gamma \\left( 0\\right) =0$ and \n\\dot{\\gamma}\\left( 0\\right) =\\func{proj}_{\\mathcal{H}}X_{p}\\in \\mathcal{H\n_{p}\\mathcal{P}$, then \n\\begin{equation}\n\\left. d^{\\mathcal{H}}f\\right\\vert _{p}\\left( X_{p}\\right) =\\left. \\frac{d}\ndt}f\\left( \\gamma \\left( t\\right) \\right) \\right\\vert _{t=0}.\n\\end{equation\nThe map $d^{\\mathcal{H}}f$ is moreover still equivariant. The group $\\Psi $\nacts on $T\\mathcal{P}$ via pushforwards of the right action of $\\Psi $ on \n\\mathcal{P}.$ Let $h\\in \\Psi $, so that $r_{h}:\\mathcal{P}\\longrightarrow \n\\mathcal{P}$ gives the right action of $\\Psi $ on $\\mathcal{P},$ and the\ncorresponding action of $\\Psi $ on $T\\mathcal{P}$ is $\\left( r_{h}\\right)\n_{\\ast }:T\\mathcal{P}\\longrightarrow T\\mathcal{P}.$ Note that the\ncorresponding action of $\\Psi $ on $TS$ is then $\\left( l_{h^{-1}}\\right)\n_{\\ast }:TS\\longrightarrow TS.$ Now\n\\begin{eqnarray*}\nd^{\\mathcal{H}}f\\circ \\left( r_{h}\\right) _{\\ast } &=&f_{\\ast }\\circ \\func\nproj}_{\\mathcal{H}}\\circ \\left( r_{h}\\right) _{\\ast }=f_{\\ast }\\circ \\left(\nr_{h}\\right) _{\\ast }\\circ \\func{proj}_{\\mathcal{H}} \\\\\n&=&\\left( f\\circ r_{h}\\right) _{\\ast }\\circ \\func{proj}_{\\mathcal{H}}=\\left(\nl_{h^{-1}}\\circ f\\right) _{\\ast }\\circ \\func{proj}_{\\mathcal{H}} \\\\\n&=&\\left( l_{h^{-1}}\\right) _{\\ast }\\circ d^{\\mathcal{H}}f\n\\end{eqnarray*\nwhere we have used the equivariance of both $f$ and $\\func{proj}_{\\mathcal{H\n}.$ So indeed, $d^{\\mathcal{H}}f$ is equivariant. Now consider the quotient\nmap $q^{\\prime }:\\mathcal{P}\\times TS\\longrightarrow \\mathcal{P\\times \n_{\\Psi }TS,$ where $\\Psi $ acts via $r_{h}$ on $\\mathcal{P}$ and $\\left(\nl_{h^{-1}}\\right) _{\\ast }$ on $TS.$ This is a partial differential of the\nmap $q:\\mathcal{P}\\times S\\longrightarrow E.$ Since $d^{\\mathcal{H}}f$ is\nhorizontal, it vanishes on the kernel of $\\pi _{\\ast }:T\\mathcal{P\n\\longrightarrow TM$. Given $\\tilde{f}\\ $, the section of the associated\nbundle $\\mathcal{P}\\times _{\\Psi }S$ that corresponds to $f$, we can use $d^\n\\mathcal{H}}f$ to define the unique ma\n\\begin{equation}\nd^{\\mathcal{H}}\\tilde{f}:TM\\longrightarrow \\mathcal{P\\times }_{\\Psi }TS\n\\label{dHf}\n\\end{equation\nsuch that \n\\begin{equation*}\nd^{\\mathcal{H}}\\tilde{f}\\circ \\pi _{\\ast }=\\left( \\pi _{T\\mathcal{P}},d^\n\\mathcal{H}}f\\right) \\circ q^{\\prime }\n\\end{equation*\nwhere $\\pi _{T\\mathcal{P}}:T\\mathcal{P}\\longrightarrow \\mathcal{P}$ is the\nbundle projection for $T\\mathcal{P}.$ Moreover, $d^{\\mathcal{H}}\\tilde{f}$\ncovers the identity map on $M,$ and hence is a section of the fiber product \nTM\\times _{M}\\left( \\mathcal{P\\times }_{\\Psi }TS\\right) .$ This construction\nis summarized in the commutative diagram in Figure \\ref{tikCovDer}.\n\n\\begin{center}\n\\begin{tikzcd}\n\\mathcal{P} \\times TS \\arrow[rrrrr,\"q'\",bend left=10] \\arrow[ddd] & & & & & \\mathcal{P} \\times_{\\Psi} TS \\arrow[ddd]\\\\\n &T\\mathcal{P} \\arrow[ul,\"{(\\pi_{T\\mathcal{P}},d^{\\mathcal{H}} f)}\",swap] \\arrow[ddl,\"\\pi_{T\\mathcal{P}}\",swap] \\arrow[rrr,\"\\pi_{*}\"]& & &TM \\arrow[ddr,\"\\pi_{TM}\"] \\arrow[ur,\"d^{\\mathcal{H}} \\tilde{f}\"] & \\\\\n & &\\mathcal{P} \\times S \\arrow[r,\"q\"] \\arrow[dll,\"\\func{prj}_{1}\",bend right=15]&  \\mathcal{P} \\times_{\\Psi} S \\arrow[drr,\"\\pi_{E}\",bend left=15,swap]  & & \\\\\n\\mathcal{P} \\arrow[rrrrr,\"\\pi\",bend right=10] \\arrow[urr,\"({\\func{id},f)}\",bend right=15,swap]& & & & & M \\arrow[ull,\"\\tilde{f}\",bend left=15] \\\\\n\\end{tikzcd}\n\\captionof{figure}{Covariant derivative on maps and sections.} \\labe\n{tikCovDer}\n\\end{center}\n\nOf course, if $S$ is a vector space, then this reduces to the usual\ndefinition of the exterior covariant derivative of a vector bundle-valued\nfunction and $d^{\\mathcal{H}}f$ is a vector-bundle-valued $1$-form.\n\nGiven the above correspondence between equivariant maps from $\\mathcal{P}$\nand sections of associated bundles, for convenience, we will work with\nequivariant maps rather than sections. This will allow us to use the\nproperties of $\\mathbb{L}$ from the previous section more directly.\n\nGiven a $\\mathfrak{p}$-valued connection $1$-form $\\omega $ on \\QTR{cal}{P}\n, $ we can concretely work out $d^{\\mathcal{H}}f.$ Suppose $X\\in \\Gamma\n\\left( T\\mathcal{P}\\right) $ is a vector field on $\\mathcal{P}$, then using\nthe definition (\\ref{dHftilde}), we have \n\\begin{eqnarray*}\n\\left( d^{\\mathcal{H}}f\\right) \\left( X\\right) &=&df\\left( \\func{proj}_\n\\mathcal{H}}\\left( X\\right) \\right) \\\\\n&=&df\\left( X-v\\left( X\\right) \\right) \\\\\n&=&df\\left( X\\right) -df\\left( \\sigma \\left( \\pi _{TP}\\left( X\\right)\n,\\omega \\left( X\\right) \\right) \\right)\n\\end{eqnarray*\nwhere from (\\ref{mapsigma}), for $p\\in \\mathcal{P},$ \n\\begin{equation*}\n\\sigma \\left( \\pi _{TP}\\left( X\\right) ,\\omega \\left( X\\right) \\right)\n_{p}=\\left. \\frac{d}{dt}\\left( p\\exp \\left( t\\omega \\left( X_{p}\\right)\n\\right) \\right) \\right\\vert _{t=0}.\n\\end{equation*\nNow, let $\\gamma \\left( t\\right) =\\exp \\left( t\\omega \\left( X_{p}\\right)\n\\right) \\in \\Psi $, and note that $\\gamma \\left( t\\right) ^{-1}=\\gamma\n\\left( -t\\right) $, so that \n\\begin{eqnarray}\n\\left. df\\left( \\sigma \\left( \\pi _{TP}\\left( X\\right) ,\\omega \\left(\nX\\right) \\right) \\right) \\right\\vert _{p} &=&\\left. \\frac{d}{dt}\\left(\nf\\left( p\\gamma \\left( t\\right) \\right) \\right) \\right\\vert _{t=0}  \\notag \\\\\n&=&-\\left. \\frac{d}{dt}\\left( \\exp \\left( t\\omega \\left( X_{p}\\right)\n\\right) f\\left( p\\right) \\right) \\right\\vert _{t=0}  \\notag \\\\\n&=&-\\omega \\left( X_{p}\\right) \\cdot f\\left( p\\right)  \\label{omegasharp}\n\\end{eqnarray\nwhere we have used the equivariance of $f$ and where, $\\omega \\left(\nX_{p}\\right) \\cdot f\\left( p\\right) \\in T_{f\\left( p\\right) }S$ denotes the\ninfinitesimal action of $\\omega \\left( X_{p}\\right) \\in \\mathfrak{p}$ on $S$.\n\n\\begin{lemma}\nLet $s$ be a $\\Psi $-equivariant $S$-valued function on $\\mathcal{P}\\ $and\nlet $\\omega $ be a $\\mathfrak{p}$-valued connection $1$-form on $\\mathcal{P\n, $ then the covariant differential $d^{\\mathcal{H}}s:T\\mathcal{P\n\\longrightarrow TS$ is given by \n\\begin{equation}\nd^{\\mathcal{H}}s=ds+\\omega \\cdot s  \\label{dHftilde2}\n\\end{equation\nwhere $\\omega \\cdot s:T_{p}\\mathcal{P}\\longrightarrow T_{s\\left( p\\right) }S$\nfor each $p\\in \\mathcal{P}\\ $gives the infinitesimal action of $\\omega $ on \nS$.\n\\end{lemma}\n\nNow, more concretely, given a principal connection $\\omega $ on $\\mathcal{P\n, $ consider the induced covariant derivatives on equivariant $\\mathbb{L}$-\nand $\\mathbb{\\mathring{L}}$-valued maps. To avoid confusion, denote $d^\n\\mathcal{H}}$ acting on $\\mathbb{L}$-valued maps by $D$ and by $\\mathring{D}$\nwhen it is acting on $\\mathbb{\\mathring{L}}$-valued maps. Similarly,\nconsider equivariant $\\mathfrak{l}$-valued maps from $\\mathcal{P}.$ Given \n\\xi :\\mathcal{P}\\longrightarrow \\mathfrak{l}$ such that $\\xi _{ph}=\\left(\nh^{-1}\\right) _{\\ast }^{\\prime }\\left( \\xi \\right) ,$ define the covariant\nderivative $d^{\\mathcal{H}}\\xi $ via (\\ref{dHftilde2}), so overall, given \nX\\in \\Gamma \\left( T\\mathcal{P}\\right) ,$ \n\\begin{equation}\nd_{X}^{\\mathcal{H}}\\xi =d_{X}\\xi +\\omega \\left( X\\right) \\cdot \\xi\n\\label{dHxi}\n\\end{equation\nwhere $\\omega \\left( X\\right) \\cdot \\xi $ refers to the linear\nrepresentation of the Lie algebra $\\mathfrak{p}$ on $\\mathfrak{l}$ given by \n\\ref{pactl1}).\n\nWe have the following useful relation between $D$ and $\\mathring{D}.$\n\n\\begin{lemma}\n\\label{lemProdAs}Suppose $A:\\mathcal{P}\\longrightarrow \\mathbb{L}$ and $s\n\\mathcal{P}\\longrightarrow \\mathbb{\\mathring{L}}$ are equivariant, and let \np\\in \\mathcal{P}.$ Then, \n\\begin{equation}\n\\left. \\mathring{D}\\left( As\\right) \\right\\vert _{p}=\\left( R_{s_{p}}\\right)\n_{\\ast }\\left. DA\\right\\vert _{p}+\\left( L_{A_{p}}\\right) _{\\ast }\\left. \n\\mathring{D}s\\right\\vert _{p}.  \\label{DAs}\n\\end{equation\nNote that $\\left. \\mathring{D}\\left( As\\right) \\right\\vert _{p}:T_{p\n\\mathcal{P}\\longrightarrow T_{As}\\mathbb{\\mathring{L}}.$\n\\end{lemma}\n\n\\begin{proof}\nLet $X_{p}\\in T_{p}\\mathcal{P}$ and let $p\\left( t\\right) $ be a curve on \n\\mathcal{P}$ with $p\\left( 0\\right) =p$ and $\\dot{p}\\left( 0\\right) =\\func\nproj}_{\\mathcal{H}}\\left( X_{p}\\right) \\in \\mathcal{H}_{p}\\mathcal{P}.$\nConsider \n\\begin{equation}\n\\left. \\mathring{D}\\left( As\\right) \\right\\vert _{p}\\left( X_{p}\\right)\n=\\left. \\frac{d}{dt}\\left( A_{p\\left( t\\right) }s_{p\\left( t\\right) }\\right)\n\\right\\vert _{t=0}\n\\end{equation\nHowever, \n\\begin{eqnarray}\n\\left. \\frac{d}{dt}\\left( A_{p\\left( t\\right) }s_{p\\left( t\\right) }\\right)\n\\right\\vert _{t=0} &=&\\left. \\frac{d}{dt}\\left( A_{p\\left( t\\right)\n}s_{p}\\right) \\right\\vert _{t=0}+\\left. \\frac{d}{dt}\\left( A_{p}s_{p\\left(\nt\\right) }\\right) \\right\\vert _{t=0}  \\notag \\\\\n&=&\\left( R_{s_{p}}\\right) _{\\ast }\\left( DA\\right) _{p}\\left( X_{p}\\right)\n+\\left( L_{A_{p}}\\right) _{\\ast }\\left( \\mathring{D}s\\right) _{p}\\left(\nX_{p}\\right)\n\\end{eqnarray\nand thus (\\ref{DAs}) holds.\n\\end{proof}\n\nSuppose now $\\left( \\mathbb{L},\\Psi ,\\mathcal{P},s\\right) $ is a loop bundle\nstructure, as in Definition \\ref{defLoopStructure}, so that $s\\ $is an \n\\mathbb{\\mathring{L}}$-valued equivariant map. Then we have the following\nimportant definition.\n\n\\begin{definition}\n\\label{defTors}The $\\emph{torsion}$ $T^{\\left( s,\\omega \\right) }$ of \n\\left( \\mathbb{L},\\Psi ,\\mathcal{P},s\\right) $ with respect to $\\omega $ is\na horizontal $\\mathfrak{l}$-valued $1$-form on $\\mathcal{P}$ given by \n\\begin{equation}\nT^{\\left( s,\\omega \\right) }=\\theta _{s}\\circ \\func{proj}_{\\mathcal{H}}\n\\end{equation\nwhere $\\theta _{s}$ is the Darboux derivative of $s$. Equivalently, at $p\\in \n\\mathcal{P}$, we hav\n\\begin{equation}\n\\left. T^{\\left( s,\\omega \\right) }\\right\\vert _{p}=\\left(\nR_{s_{p}}^{-1}\\right) _{\\ast }\\left. \\mathring{D}s\\right\\vert _{p}.\n\\end{equation}\n\\end{definition}\n\nThus, $T^{\\left( s,\\omega \\right) }$ is the horizontal component of $\\theta\n_{s}.$ We also easily see that it is $\\Psi $-equivariant. Using the\nequivariance of $s$ and $\\mathring{D}s$, we have for $h\\in \\Psi $, \n\\begin{equation}\nT_{ph}^{\\left( s,\\omega \\right) }=\\left( h_{\\ast }^{\\prime }\\right)\n^{-1}T_{p}^{\\left( s,\\omega \\right) }.  \\label{Tequi1}\n\\end{equation\nThus, $T^{\\left( s,\\omega \\right) }$ is a \\emph{basic} (i.e. horizontal and\nequivariant) $\\mathfrak{l}$-valued $1$-form on $\\mathcal{P}$, and thus\ndefines a $1$-form on $M$ with values in the associated vector bundle \n\\mathcal{A=P\\times }_{\\Psi _{\\ast }^{\\prime }}\\mathfrak{l}.$ We also have\nthe following key property of $T^{\\left( s,\\omega \\right) }.$\n\n\\begin{theorem}\n\\label{thmTprop}Suppose $T^{\\left( s,\\omega \\right) }$ is as given in\nDefinition \\ref{defTors} and also let $\\hat{\\omega}^{\\left( s\\right) }\\in\n\\Omega ^{1}\\left( \\mathcal{P},\\mathfrak{l}\\right) $ be given by \n\\begin{equation}\n\\hat{\\omega}^{\\left( s\\right) }=\\varphi _{s}\\left( \\omega \\right) .\n\\label{omegahat}\n\\end{equation\nThen, \n\\begin{equation}\n\\theta _{s}=T^{\\left( s,\\omega \\right) }-\\hat{\\omega}^{\\left( s\\right) }.\n\\label{stheta}\n\\end{equation\nIn particular, $T^{\\left( s,\\omega \\right) }$ and the quantity $-\\hat{\\omega\n^{\\left( s\\right) }$ are respectively the horizontal and vertical components\nof $\\theta _{s}$.\n\\end{theorem}\n\n\\begin{proof}\nLet $p\\in \\mathcal{P}.$ Then, from (\\ref{dHftilde2}) we have \n\\begin{eqnarray}\n\\left( R_{s_{p}}^{-1}\\right) _{\\ast }\\left. \\mathring{D}s\\right\\vert _{p}\n&=&\\left( R_{s_{p}}^{-1}\\right) _{\\ast }\\left. ds\\right\\vert _{p}+\\left(\nR_{s_{p}}^{-1}\\right) _{\\ast }\\left( \\omega \\cdot s_{p}\\right)  \\notag \\\\\n&=&\\left. \\theta _{s}\\right\\vert _{p}+\\left. \\frac{d}{dt}\\faktor{\\left( \\exp\n\\left( t\\omega _{p}\\right) \\left( s_{p}\\right) \\right)}{ s_{p}}\\right\\vert\n_{t=0}  \\notag \\\\\n&=&\\left. \\theta _{s}\\right\\vert _{p}+\\varphi _{s_{p}}\\left( \\omega\n_{p}\\right)  \\label{Torsion1f2}\n\\end{eqnarray\nwhere we have used the definition (\\ref{phis}) of $\\varphi _{s}$. Hence we\nget (\\ref{stheta}).\n\\end{proof}\n\nSuppose $p\\left( t\\right) $ is a curve on $\\mathcal{P}$ with $p\\left(\n0\\right) =p$ and with a horizontal initial velocity vector $\\dot{p}\\left(\n0\\right) =X_{p}^{\\mathcal{H}}.$ Then, by definition, \n\\begin{equation}\n\\left. \\frac{d}{dt}s_{p\\left( t\\right) }\\right\\vert _{t=0}=\\left. \\mathring{\n}_{X}s\\right\\vert _{p}=\\left( R_{s_{p}}\\right) _{\\ast }\\left.\nT_{X_{p}}^{\\left( s,\\omega \\right) }\\right\\vert _{p},  \\label{dts}\n\\end{equation\nwhere $T_{X}^{\\left( s,\\omega \\right) }=T^{\\left( s,\\omega \\right) }\\left(\nX\\right) \\in \\mathfrak{l}.$ This observation will come in useful later on.\n\n\\begin{remark}\nIf $s_{p}\\in \\mathcal{C\\cong \\Psi }\/H$ for all $p\\in \\mathcal{P},$ then as\nwe know, the structure group of $\\mathcal{P}$ is reduced to $H.$ Moreover,\nthe reduced holonomy group of $\\omega $ is contained in $H$ if and only if\nthere exists such a map $s$ with $d^{\\mathcal{H}}s=0.$ This is equivalent to \n$T^{\\left( s,\\omega \\right) }=0$, so this is the motivation for calling this\nquantity the torsion. If $s$ is not necessarily in $\\mathcal{C},$ then we\ncan still say something about the holonomy of $\\omega $ in the case $d^\n\\mathcal{H}}s=0$. Let $p\\in \\mathcal{P}$ and suppose $\\Gamma \\left( t\\right) \n$ is the horizontal lift with respect to the connection $\\omega $ of some\nclosed curve based at $\\pi \\left( p\\right) .$ Then, the endpoint of $\\Gamma $\nis $\\Gamma \\left( 1\\right) =ph$ for some $h\\in \\Psi .$ The set of all such \nh\\in \\Psi $ form the holonomy group $\\func{Hol}_{p}\\left( \\omega \\right) $\nof $\\omega $ at $p$ \\cite{KobayashiNomizu1}. Now if we have an equivariant\nmap $s:\\mathcal{P}\\longrightarrow \\mathbb{L}$, then $s\\circ \\Gamma $ is a\ncurve on $\\mathbb{L}$ with $s\\left( \\Gamma \\left( 1\\right) \\right)\n=s_{ph}=h^{-1}s_{p}.$ However, $\\frac{d}{dt}\\left( s\\circ \\Gamma \\left(\nt\\right) \\right) =\\left( d^{\\mathcal{H}}s\\right) _{s\\circ \\Gamma \\left(\nt\\right) }\\dot{\\Gamma}\\left( t\\right) $ since the velocity vectors of \n\\Gamma \\left( t\\right) $ are horizontal. Thus, if $d^{\\mathcal{H}}s=0$\neverywhere, then the curve $s\\circ \\Gamma \\left( t\\right) $ is constant, and\nhence $h^{-1}s_{p}=s_{p}.$ By (\\ref{AutLr}), this means that $h\\in \\func{Aut\n\\left( \\mathbb{L},\\circ _{s_{p}}\\right) .$ This is true for any horizontal\nlift $\\Gamma $, hence we see that $\\func{Hol}_{p}\\left( \\omega \\right)\n\\subset \\func{Aut}\\left( \\mathbb{L},\\circ _{s_{p}}\\right) .$\n\\end{remark}\n\nThe torsion also enters expressions for covariant derivatives of the loop\nproduct, loop algebra bracket, as well as the map $\\varphi _{s}$.\n\n\\begin{theorem}\n\\label{lemProd}Suppose $A,B:\\mathcal{P}\\longrightarrow \\mathbb{L},$ and $s\n\\mathcal{P}\\longrightarrow \\mathbb{\\mathring{L}}$ are equivariant, and let \np\\in \\mathcal{P}.$ Then, \n\\begin{eqnarray}\n\\left. D\\left( A\\circ _{s}B\\right) \\right\\vert _{p} &=&\\left(\nR_{B_{p}}^{\\left( s_{p}\\right) }\\right) _{\\ast }\\left. DA\\right\\vert\n_{p}+\\left( L_{A_{p}}^{\\left( s_{p}\\right) }\\right) _{\\ast }\\left.\nDB\\right\\vert _{p}  \\label{DAsB} \\\\\n&&+\\left[ A_{p},B_{p},\\left. T^{\\left( s,\\omega \\right) }\\right\\vert _{p\n\\right] ^{\\left( s_{p}\\right) }.  \\notag\n\\end{eqnarray\nIf $\\xi ,\\eta :\\mathcal{P}\\longrightarrow \\mathfrak{l}$ are equivariant,\nthen \n\\begin{equation}\nd^{\\mathcal{H}}\\left[ \\xi ,\\eta \\right] ^{\\left( s\\right) }=\\left[ d^\n\\mathcal{H}}\\xi ,\\eta \\right] ^{\\left( s\\right) }+\\left[ \\xi ,d^{\\mathcal{H\n}\\eta \\right] ^{\\left( s\\right) }+\\left[ \\xi ,\\eta ,T^{\\left( s,\\omega\n\\right) }\\right] ^{\\left( s\\right) }-\\left[ \\eta ,\\xi ,T^{\\left( s,\\omega\n\\right) }\\right] ^{\\left( s\\right) }.  \\label{dHbrack}\n\\end{equation}\n\nThe $\\mathfrak{l}\\otimes \\mathfrak{p}^{\\ast }$-valued map $\\varphi _{s}\n\\mathcal{P}\\longrightarrow $ $\\mathfrak{l}\\otimes \\mathfrak{p}^{\\ast }$\nsatisfies \n\\begin{equation}\nd^{\\mathcal{H}}\\varphi _{s}=\\func{id}_{\\mathfrak{p}}\\cdot T^{\\left( s,\\omega\n\\right) }-\\left[ \\varphi _{s},T^{\\left( s,\\omega \\right) }\\right] ^{\\left(\ns\\right) },  \\label{dhphis}\n\\end{equation\nwhere $\\func{id}_{\\mathfrak{p}}\\ $is the identity map of $\\mathfrak{p}$ and \n\\cdot $ denotes the action of the Lie algebra $\\mathfrak{p}$ on $\\mathfrak{l}\n$ given by (\\ref{pactl1}).\n\\end{theorem}\n\n\\begin{proof}\nLet $X_{p}\\in T_{p}\\mathcal{P}$ and let $p\\left( t\\right) $ be a curve on \n\\mathcal{P}$ with $p\\left( 0\\right) =p$ and $\\dot{p}\\left( 0\\right) =\\func\nproj}_{\\mathcal{H}}\\left( X_{p}\\right) \\in \\mathcal{H}_{p}\\mathcal{P}.$ To\nshow (\\ref{DAsB}), first note that \n\\begin{equation}\n\\left. D\\left( A\\circ _{s}B\\right) \\right\\vert _{p}\\left( X_{p}\\right)\n=\\left. \\frac{d}{dt}\\left( A_{p\\left( t\\right) }\\circ _{s_{p\\left( t\\right)\n}}B_{p\\left( t\\right) }\\right) \\right\\vert _{t=0}.  \\label{dHprod}\n\\end{equation\nHowever\n\\begin{eqnarray}\n\\left. \\frac{d}{dt}\\left( A_{p\\left( t\\right) }\\circ _{s_{p\\left( t\\right)\n}}B_{p\\left( t\\right) }\\right) \\right\\vert _{t=0} &=&\\left. \\frac{d}{dt\n\\left( A_{p\\left( t\\right) }\\circ _{s_{p}}B_{p}\\right) \\right\\vert\n_{t=0}+\\left. \\frac{d}{dt}\\left( A_{p}\\circ _{s_{p}}B_{p\\left( t\\right)\n}\\right) \\right\\vert _{t=0}  \\notag \\\\\n&&+\\left. \\frac{d}{dt}\\left( A_{p}\\circ _{s_{p\\left( t\\right) }}B_{p}\\right)\n\\right\\vert _{t=0}  \\notag \\\\\n&=&\\left( R_{B_{p}}^{\\left( s_{p}\\right) }\\right) _{\\ast }\\left.\nDA\\right\\vert _{p}\\left( X_{p}\\right) +\\left( L_{A_{p}}^{\\left( s_{p}\\right)\n}\\right) _{\\ast }\\left. DB\\right\\vert _{p}\\left( X_{p}\\right)  \\notag \\\\\n&&+\\left. \\frac{d}{dt}\\left( A_{p}\\circ _{s_{p\\left( t\\right) }}B_{p}\\right)\n\\right\\vert _{t=0}  \\label{dHprod2}\n\\end{eqnarray\nand then, using Lemma \\ref{lemQuotient}, \n\\begin{eqnarray}\n\\left. \\frac{d}{dt}\\left( A_{p}\\circ _{s_{p\\left( t\\right) }}B_{p}\\right)\n\\right\\vert _{t=0} &=&\\left. \\frac{d}{dt}\\left( \\left( A_{p}\\cdot\nB_{p}s_{p\\left( t\\right) }\\right) \/s_{p\\left( t\\right) }\\right) \\right\\vert\n_{t=0}  \\notag \\\\\n&=&\\left. \\frac{d}{dt}\\left( \\left( A_{p}\\cdot B_{p}s_{p\\left( t\\right)\n}\\right) \/s_{p}\\right) \\right\\vert _{t=0}  \\label{dHprod2a} \\\\\n&&+\\left. \\frac{d}{dt}\\left( \\left( A_{p}\\cdot B_{p}s_{p}\\right) \/s_{p}\\cdot\ns_{p\\left( t\\right) }\\right) \/s_{p}\\right\\vert _{t=0}.  \\notag\n\\end{eqnarray\nLooking at each term in (\\ref{dHprod2a}), we have \n\\begin{eqnarray*}\n\\left( A_{p}\\cdot B_{p}s_{p\\left( t\\right) }\\right) \/s_{p} &=&\\left(\nA_{p}\\cdot B_{p}\\left( \\faktor{s_{p\\left( t\\right) }}{s_{p}}\\cdot\ns_{p}\\right) \\right) \/s_{p} \\\\\n&=&A_{p}\\circ _{s_{p}}\\left( B_{p}\\circ _{s_{p}}\\left( \\faktor{s_{p\\left(\nt\\right) }}{s_{p}}\\right) \\right)\n\\end{eqnarray*\nan\n\\begin{equation*}\n\\left( \\left( A_{p}\\cdot B_{p}s_{p}\\right) \/s_{p}\\cdot s_{p\\left( t\\right)\n}\\right) \/s_{p}=\\left( A_{p}\\circ _{s_{p}}B_{p}\\right) \\circ _{s_{p}}\\left\n\\faktor{ s_{p\\left( t\\right) }}{s_{p}}\\right) .\n\\end{equation*\nOverall (\\ref{dHprod2}) becomes, \n\\begin{equation}\n\\left. \\frac{d}{dt}\\left( A_{p}\\circ _{s_{p\\left( t\\right) }}B_{p}\\right)\n\\right\\vert _{t=0}=\\left( \\left( L_{A_{p}}^{\\left( s_{p}\\right) }\\circ\nL_{B_{p}}^{\\left( s_{p}\\right) }\\right) _{\\ast }-\\left( L_{A_{p}\\circ\n_{s_{p}}B_{p}}^{\\left( s_{p}\\right) }\\right) _{\\ast }\\right) \\left(\nR_{s_{p}}^{-1}\\right) _{\\ast }\\left. \\mathring{D}s\\right\\vert _{p}\\left(\nX_{p}\\right)  \\label{dHprod3}\n\\end{equation\nand hence we get (\\ref{DAsB}) using the definitions of $T^{\\left( s,\\omega\n\\right) }$ and the mixed associator (\\ref{pxiqsol}).\n\nTo show (\\ref{dHbrack}), note that \n\\begin{eqnarray*}\n\\left. d_{X}^{\\mathcal{H}}\\left( \\left[ \\xi ,\\eta \\right] ^{\\left( s\\right)\n}\\right) \\right\\vert _{p} &=&\\left. \\frac{d}{dt}\\left[ \\xi _{p\\left(\nt\\right) },\\eta _{p\\left( t\\right) }\\right] ^{\\left( s_{p\\left( t\\right)\n}\\right) }\\right\\vert _{t=0} \\\\\n&=&\\left[ \\left. d_{X}^{\\mathcal{H}}\\xi \\right\\vert _{p},\\eta _{p}\\right]\n^{\\left( s_{p}\\right) }+\\left[ \\xi _{p},\\left. d_{X}^{\\mathcal{H}}\\eta\n\\right\\vert _{p}\\right] ^{\\left( s_{p}\\right) } \\\\\n&&+\\left. \\frac{d}{dt}\\left[ \\xi _{p},\\eta _{p}\\right] ^{\\left( s_{p\\left(\nt\\right) }\\right) }\\right\\vert _{t=0}.\n\\end{eqnarray*\nHowever, using (\\ref{db1}) and (\\ref{dts}), the last term becomes \n\\begin{equation*}\n\\left. \\frac{d}{dt}\\left[ \\xi _{p},\\eta _{p}\\right] ^{\\left( s_{p\\left(\nt\\right) }\\right) }\\right\\vert _{t=0}=\\left[ \\xi _{p},\\eta _{p},\\left.\nT_{X}^{\\left( s,\\omega \\right) }\\right\\vert _{p}\\right] ^{\\left(\ns_{p}\\right) }-\\left[ \\eta _{p},\\xi _{p},\\left. T_{X}^{\\left( s,\\omega\n\\right) }\\right\\vert _{p}\\right] ^{\\left( s_{p}\\right) }\n\\end{equation*\nand hence we obtain (\\ref{dHbrack}).\n\nLet us now show (\\ref{dhphis}). From (\\ref{actpl}), given $\\gamma \\in \n\\mathfrak{p}$, setting $\\hat{\\gamma}\\left( r\\right) =\\varphi _{r}\\left(\n\\gamma \\right) $ for each $r\\in \\mathbb{L}$, we have \n\\begin{equation}\n\\left. d\\hat{\\gamma}\\right\\vert _{r}\\left( \\rho _{r}\\left( \\xi \\right)\n\\right) =\\gamma \\cdot \\xi -\\left[ \\hat{\\gamma}\\left( r\\right) ,\\xi \\right]\n^{\\left( r\\right) }\n\\end{equation\nfor some $\\xi \\in \\mathfrak{l}.$ Now for a map $s:\\mathcal{P}\\longrightarrow \n\\mathbb{L}\\ $and some vector field $X$ on $\\mathcal{P},$ we have at each \np\\in \\mathcal{P}$ \n\\begin{eqnarray}\n\\left. d\\left( \\varphi _{s}\\left( \\gamma \\right) \\right) \\right\\vert\n_{p}\\left( X\\right) &=&\\left. d\\hat{\\gamma}\\right\\vert _{s_{p}}\\circ \\left.\nds\\right\\vert _{p}\\left( X\\right)  \\notag \\\\\n&=&\\left. d\\hat{\\gamma}\\right\\vert _{s_{p}}\\left( \\rho _{s_{p}}\\left( \\theta\n_{s}\\left( X_{p}\\right) \\right) \\right)  \\notag \\\\\n&=&\\gamma \\cdot \\theta _{s}\\left( X_{p}\\right) -\\left[ \\varphi\n_{s_{p}}\\left( \\gamma \\right) ,\\theta _{s}\\left( X_{p}\\right) \\right]\n^{\\left( s_{p}\\right) }.\n\\end{eqnarray\nTherefore, $d\\varphi _{s}$ is given by \n\\begin{equation}\nd\\varphi _{s}\\left( \\gamma \\right) =\\gamma \\cdot \\theta _{s}-\\left[ \\varphi\n_{s}\\left( \\gamma \\right) ,\\theta _{s}\\right] ^{\\left( s\\right) }.\n\\label{dphis}\n\\end{equation\nTo obtain $d^{\\mathcal{H}}\\varphi _{s}$ we take the horizontal component,\nand hence using (\\ref{stheta}), we just replace $\\theta _{s}$ in (\\ref{dphis\n) by $T^{\\left( s,\\omega \\right) },$ which gives (\\ref{dhphis}).\n\\end{proof}\n\n\\begin{remark}\nIf $\\mathbb{L}$ is associative, i.e. is a group, then certainly $A\\circ\n_{s}B=AB$ and this is then an equivariant section, if $A$ and $B$ are such.\nIn (\\ref{DAsB}) the second term on the right vanishes, and thus $D$\nsatisfies the product rule with respect to multiplication on $\\mathbb{L}.$\n\\end{remark}\n\nWe can rewrite (\\ref{DAs}) as \n\\begin{eqnarray}\n\\mathring{D}\\left( As\\right) &=&\\left( DA\\right) s+A\\left( \\left( \\mathring{\n}s\\right) \/s\\cdot s\\right)  \\notag \\\\\n&=&\\left( DA\\right) s+\\left( A\\circ _{s}T^{\\left( s,\\omega \\right) }\\right)\ns.  \\label{DAs2}\n\\end{eqnarray\nUsing this, we can then define an adapted covariant derivative $D^{\\left(\ns\\right) }$ on equivariant $\\mathbb{L}$-valued maps, given by \n\\begin{equation}\n\\left. D^{\\left( s\\right) }A\\right\\vert _{p}=\\left( R_{s_{p}}^{-1}\\right)\n_{\\ast }\\left. \\mathring{D}\\left( As\\right) \\right\\vert _{p}=\\left.\nDA\\right\\vert _{p}+\\left( L_{A_{p}}^{\\left( s_{p}\\right) }\\right) _{\\ast\n}T_{p}^{\\left( s,\\omega \\right) }  \\label{Dsderiv}\n\\end{equation\nwith respect to which, \n\\begin{equation}\n\\left. D^{\\left( s\\right) }\\left( A\\circ _{s}B\\right) \\right\\vert\n_{p}=\\left( R_{B_{p}}^{\\left( s_{p}\\right) }\\right) _{\\ast }\\left.\nDA\\right\\vert _{p}+\\left( L_{A_{p}}^{\\left( s_{p}\\right) }\\right) _{\\ast\n}\\left. D^{\\left( s\\right) }B\\right\\vert _{p}.  \\label{Dsderivprod}\n\\end{equation\nThis is the precise analog of the octonion covariant derivative from \\cit\n{GrigorianOctobundle}. The derivative $D^{\\left( s\\right) }$ essentially\nconverts an $\\mathbb{L}$-valued map into an $\\mathbb{\\mathring{L}}$-valued\none using $s$ and then differentiates it using $\\mathring{D}$ before\nconverting back to $\\mathbb{L}.$ In particular, if we take $A=1$, \n\\begin{equation}\nD^{\\left( s\\right) }1=T^{\\left( s,\\omega \\right) }.  \\label{Ds1}\n\\end{equation}\n\n\\begin{remark}\nUp to the sign of $T$, (\\ref{DAsB}) and (\\ref{Dsderiv}) are precisely the\nexpressions obtained in \\cite{GrigorianOctobundle} for the covariant\nderivative with respect to the Levi-Civita connection of the product on the\noctonion bundle over a $7$-manifold. In that case, $T$ is precisely the\ntorsion of the $G_{2}$-structure that defines the octonion bundle. This\nprovides additional motivation for calling this quantity the torsion of $s$\nand $\\omega .$ In the case of $G_{2}$-structures, usually one takes the\ntorsion with respect to the preferred Levi-Civita connection, however in\nthis more general setting, we don't have a preferred connection, thus \nT^{\\left( s,\\omega \\right) }$ should also be taken to depend on the\nconnection.\n\\end{remark}\n\n\\begin{corollary}\nSuppose $\\mathbb{L}$ is an alternative loop, so that the associator is\nskew-symmetric. Suppose $\\xi ,\\eta \\longrightarrow \\mathfrak{l}$ and $s\n\\mathcal{P}\\longrightarrow \\mathbb{\\mathring{L}}$ are equivariant. Then,\ndefining a modified exterior derivative $d^{\\left( s\\right) }$ on\nequivariant maps from $\\mathcal{P}$ to $\\mathfrak{l}$ vi\n\\begin{equation}\nd^{\\left( s\\right) }\\xi =d^{\\mathcal{H}}\\xi +\\frac{1}{3}\\left[ \\xi\n,T^{\\left( s\\right) }\\right] ^{\\left( s\\right) },  \\label{dsbrack}\n\\end{equation\nit satisfies \n\\begin{equation}\nd^{\\left( s\\right) }\\left[ \\xi ,\\eta \\right] ^{\\left( s\\right) }=\\left[\nd^{\\left( s\\right) }\\xi ,\\eta \\right] ^{\\left( s\\right) }+\\left[ \\xi\n,d^{\\left( s\\right) }\\eta \\right] ^{\\left( s\\right) }.\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\nIf $\\mathbb{L}$ is alternative, then the loop Jacobi identity (\\ref{Jac2})\nbecomes \n\\begin{equation}\n\\left[ \\xi ,\\left[ \\eta ,\\gamma \\right] ^{\\left( s\\right) }\\right] ^{\\left(\ns\\right) }+\\left[ \\eta ,\\left[ \\gamma ,\\xi \\right] ^{\\left( s\\right) }\\right]\n^{\\left( s\\right) }+\\left[ \\gamma ,\\left[ \\xi ,\\eta \\right] ^{\\left(\ns\\right) }\\right] ^{\\left( s\\right) }=6\\left[ \\xi ,\\eta ,\\gamma \\right]\n^{\\left( s\\right) }.  \\label{Jacalt}\n\\end{equation\nOn the other hand, (\\ref{dHbrack}) becomes \n\\begin{equation}\nd^{\\mathcal{H}}\\left[ \\xi ,\\eta \\right] ^{\\left( s\\right) }=\\left[ d^\n\\mathcal{H}}\\xi ,\\eta \\right] ^{\\left( s\\right) }+\\left[ \\xi ,d^{\\mathcal{H\n}\\eta \\right] ^{\\left( s\\right) }+2\\left[ \\xi ,\\eta ,T^{\\left( s\\right) \n\\right] ^{\\left( s\\right) }.  \\label{dHbrackalt}\n\\end{equation\nThus, using both (\\ref{Jacalt}) and (\\ref{dHbrackalt}), we obtain \n\\begin{eqnarray*}\nd^{\\left( s\\right) }\\left[ \\xi ,\\eta \\right] ^{\\left( s\\right) } &=&d^\n\\mathcal{H}}\\left[ \\xi ,\\eta \\right] ^{\\left( s\\right) }+\\frac{1}{3}\\left[\n\\left[ \\xi ,\\eta \\right] ^{\\left( s\\right) },T^{\\left( s\\right) }\\right]\n^{\\left( s\\right) } \\\\\n&=&\\left[ d^{\\left( s\\right) }\\xi ,\\eta \\right] ^{\\left( s\\right) }+\\left[\n\\xi ,d^{\\left( s\\right) }\\eta \\right] ^{\\left( s\\right) } \\\\\n&&-\\frac{1}{3}\\left[ \\left[ \\xi ,T^{\\left( s\\right) }\\right] ^{\\left(\ns\\right) },\\eta \\right] ^{\\left( s\\right) }-\\frac{1}{3}\\left[ \\xi ,\\left[\n\\eta ,T^{\\left( s\\right) }\\right] ^{\\left( s\\right) }\\right] ^{\\left(\ns\\right) } \\\\\n&&+\\frac{1}{3}\\left[ \\left[ \\xi ,\\eta \\right] ^{\\left( s\\right) },T^{\\left(\ns\\right) }\\right] ^{\\left( s\\right) }+2\\left[ \\xi ,\\eta ,T^{\\left( s\\right) \n\\right] ^{\\left( s\\right) } \\\\\n&=&\\left[ d^{\\left( s\\right) }\\xi ,\\eta \\right] ^{\\left( s\\right) }+\\left[\n\\xi ,d^{\\left( s\\right) }\\eta \\right] ^{\\left( s\\right) }.\n\\end{eqnarray*}\n\\end{proof}\n\n\\begin{remark}\nIn the case of $G_{2}$-structures and octonions, the derivative (\\re\n{dsbrack}) exactly replicates the modified covariant derivative that\npreserves the $G_{2}$-structure that was introduced in \\cite{DGKisoflow}.\n\\end{remark}\n\n\\begin{example}\nThe map $\\varphi _{s}$ is equivariant on $\\mathcal{P}$ and hence defines a\nsection of the associated bundle $\\mathcal{A}\\otimes \\func{ad}\\left( \n\\mathcal{P}\\right) ^{\\ast }$ over $M.$ If $\\mathbb{L}$ is the loop of unit\noctonions and $\\mathfrak{l\\cong }\\func{Im}\\mathbb{O},$ and we have a $G_{2}\n-structure on $M,$ then $\\varphi _{s}$ corresponds to a section of \nTM\\otimes \\Lambda ^{2}TM,$ which up to a constant factor is a multiple of\nthe corresponding $G_{2}$-structure $3$-form $\\varphi $ with indices raised\nusing the associated metric. The torsion $T$ of $\\varphi $ with respect to\nthe Levi-Civita connection on $TM$ is then a section of $TM\\otimes T^{\\ast\n}M.$ Noting that $\\mathfrak{so}\\left( 7\\right) $ acts on $\\mathbb{R}^{7}$ by\nmatrix multiplication, if we set $\\left( \\varphi _{s}\\right) _{\\ }^{abc}=\n\\frac{1}{4}\\varphi ^{abc}$ in local coordinates, then (\\ref{dhphis})\nprecisely recovers the well-known formula for ${\\Greekmath 0272} \\varphi $ in terms of \nT.$ Indeed, suppose $\\xi \\in \\Gamma \\left( \\Lambda ^{2}T^{\\ast }M\\right) $,\nthen in a local basis $\\left\\{ e_{a}\\right\\} $, for some fixed vector field \nX$, we have \n\\begin{eqnarray*}\n\\left( {\\Greekmath 0272} _{X}\\varphi _{s}\\right) \\left( \\xi \\right) &=&\\xi \\cdot T_{X}- \n\\left[ \\varphi _{s}\\left( \\xi \\right) ,T_{X}\\right] ^{\\left( s\\right) } \\\\\n&=&\\left( \\xi _{\\ b}^{a}T_{X}^{b}+\\frac{1}{2}\\varphi _{\\ bc}^{a}\\varphi\n^{bde}\\xi _{de}T_{X}^{c}\\right) e_{a} \\\\\n&=&\\left( \\xi _{\\ b}^{a}T_{X}^{b}-\\frac{1}{2}\\left( \\psi _{\\ c}^{a\\\nde}+g^{ad}g_{c}^{\\ e}-g^{ae}g_{c}^{\\ d}\\right) \\xi _{de}T_{X}^{c}\\right)\ne_{a} \\\\\n&=&\\frac{1}{2}T_{X}^{c}\\psi _{c\\ }^{\\ ade}\\xi _{de}e_{a},\n\\end{eqnarray*\nwhere $\\psi =\\ast \\varphi $. Hence, indeed, \n\\begin{equation}\n{\\Greekmath 0272} _{X}\\varphi =-2T_{X}\\lrcorner \\psi ,  \\label{nablaXphi}\n\\end{equation\nwhich is exactly as in \\cite{GrigorianOctobundle}, taking into account that\nthe torsion here differs by a sign from \\cite{GrigorianOctobundle}. Here we\nalso used the convention that $\\left[ X,Y\\right] =2X\\lrcorner Y\\lrcorner\n\\varphi $ and also contraction identities for $\\varphi $ \\cit\n{GrigorianG2Torsion1,karigiannis-2005-57}. This is also consistent with the\nexpression (\\ref{dHbrack}) for the covariant derivative of the bracket.\nIndeed, in the case of an alternative loop, (\\ref{dHbrackalt}) shows that\nthe covariant derivative of the bracket function $b_{s}$ is given by \n\\begin{equation}\nd^{\\mathcal{H}}b_{s}=2\\left[ \\cdot ,\\cdot ,T^{\\left( s\\,,\\omega \\right) \n\\right] ^{\\left( s\\right) }.  \\label{dHbrack1a}\n\\end{equation\nTaking $b_{s}=2\\varphi $ and $\\left[ \\cdot ,\\cdot ,\\cdot \\right] ^{\\left(\ns\\right) }$ given by $\\left( \\left[ X,Y,Z\\right] ^{\\left( s\\right) }\\right)\n^{a}=2\\psi _{\\ bcd}^{a}X^{b}Y^{c}Z^{d},$ as in \\cite{GrigorianOctobundle},\nwe again recover (\\ref{nablaXphi}).\n\\end{example}\n\n\\begin{example}\nSuppose $\\mathcal{P}$ is a principal $U\\left( n\\right) $-bundle and $\\mathbb\nL}\\cong U\\left( 1\\right) $, the unit complex numbers, as in Example \\re\n{exCx2}. Then, (\\ref{dhphis}) shows that $d^{\\mathcal{H}}\\varphi _{s}=0$. If \n$V\\ $is an $n$-dimensional complex vector space with the standard action of \nU\\left( n\\right) $ on it and $\\mathcal{V=P\\times }_{U\\left( n\\right) }V$ is\nthe associated vector bundle to $\\mathcal{P}$ with fiber $V$, then $\\varphi\n_{s}$ defines a K\\\"{a}hler form on $\\mathcal{V}.$\n\\end{example}\n\n\\begin{example}\nSuppose $\\mathcal{P}$ is a principal $Sp\\left( n\\right) Sp\\left( 1\\right) \n-bundle and $\\mathbb{L}\\cong Sp\\left( 1\\right) ,$ the unit quaternions, as\nin Example \\ref{exQuat2}. Then, (\\ref{dhphis}) shows that $d^{\\mathcal{H\n}\\varphi _{s}=-\\left[ \\varphi _{s},T^{\\left( s\\,,\\omega \\right) }\\right] _\n\\func{Im}\\mathbb{H}}.$ If $V\\ $is an $n$-dimensional quaternionic vector\nspace with the standard action of $Sp\\left( n\\right) Sp\\left( 1\\right) $ on\nit and $\\mathcal{V=P\\times }_{Sp\\left( n\\right) Sp\\left( 1\\right) }V$ is the\nassociated vector bundle to $\\mathcal{P}$ with fiber $V$, then $\\varphi _{s}$\ndefines a 2-form on $\\mathcal{V}\\ $with values in $\\func{Im}\\mathbb{H}$\n(since the bundle $\\mathcal{A}\\ $is trivial). So this gives rise to $3$\nlinearly independent $2$-forms $\\omega _{1},\\omega _{2},\\omega _{3}.$ If \nT^{\\left( s,\\omega \\right) }=0$, then this reduces to a HyperK\\\"{a}hler\nstructure on $\\mathcal{V}.$ It is an interesting question whether the case \nT^{\\left( s,\\omega \\right) }\\neq 0$ is related to \\textquotedblleft Hyper\n\\\"{a}hler with torsion\\textquotedblright\\ geometry \\cit\n{GrantcharovPoonHKT,VerbitskyHKT}.\n\\end{example}\n\n\\subsection{Curvature}\n\n\\label{sectCurv}Recall that the curvature $F\\in \\Omega ^{2}\\left( \\mathcal{P\n,\\mathfrak{p}\\right) $ of the connection $\\omega $ on $\\mathcal{P}$ is given\nby \n\\begin{equation}\nF^{\\left( \\omega \\right) }=d^{\\mathcal{H}}\\omega =d\\omega \\circ \\func{proj}_\n\\mathcal{H}},  \\label{curvom}\n\\end{equation\nso that, for $X,Y\\in \\Gamma \\left( T\\mathcal{P}\\right) $, \n\\begin{equation}\nF^{\\left( \\omega \\right) }\\left( X,Y\\right) =d\\omega \\left( X^{\\mathcal{H\n},Y^{\\mathcal{H}}\\right) =-\\omega \\left( \\left[ X^{\\mathcal{H}},Y^{\\mathcal{\n}}\\right] \\right) ,  \\label{curvom2}\n\\end{equation\nwhere $X^{\\mathcal{H}},Y^{\\mathcal{H}}$ are the projections of $X,Y$ to \n\\mathcal{HP}.$\n\nSimilarly as $\\hat{\\omega}$, define $\\hat{F}^{\\left( s,\\omega \\right) }\\in\n\\Omega ^{2}\\left( \\mathcal{P},\\mathfrak{l}\\right) $ to be the projection of\nthe curvature $F^{\\left( \\omega \\right) }$ to $\\mathfrak{l}$ with respect to \n$s$, such that for any $X_{p},Y_{p}\\in T_{p}\\mathcal{P},$ \n\\begin{eqnarray}\n\\hat{F}^{\\left( s,\\omega \\right) }\\left( X_{p},Y_{p}\\right) &=&\\varphi\n_{s}\\left( F^{\\left( \\omega \\right) }\\right) \\left( X_{p},Y_{p}\\right) \n\\notag \\\\\n&=&\\left. \\frac{d}{dt}\\faktor{\\left( \\exp \\left( tF^{\\left( \\omega \\right)\n}\\left( X_{p},Y_{p}\\right) \\right) \\left( s_{p}\\right) \\right)}{ s_{p}\n\\right\\vert _{t=0}.  \\label{Fhat}\n\\end{eqnarray}\n\nWe easily see that \n\\begin{equation}\nd^{\\mathcal{H}}\\hat{\\omega}^{\\left( s\\right) }=\\hat{F}^{\\left( s,\\omega\n\\right) }.  \\label{dHom}\n\\end{equation\nIndeed, \n\\begin{equation*}\nd^{\\mathcal{H}}\\hat{\\omega}^{\\left( s\\right) }=d^{\\mathcal{H}}\\left( \\varphi\n_{s}\\left( \\omega \\right) \\right) =d^{\\mathcal{H}}\\varphi _{s}\\wedge \\left(\n\\omega \\circ \\func{proj}_{\\mathcal{H}}\\right) +\\varphi _{s}\\left( d^\n\\mathcal{H}}\\omega \\right) =\\hat{F}^{\\left( s,\\omega \\right) },\n\\end{equation*\nwhere we have used the fact that $\\omega $ is vertical.\n\nWe then have the following structure equations\n\n\\begin{theorem}\n\\label{thmFTstruct}$\\hat{F}^{\\left( s,\\omega \\right) }$ and $T^{\\left(\ns,\\omega \\right) }$ satisfy the following structure equation \n\\begin{equation}\n\\hat{F}^{\\left( s,\\omega \\right) }=d^{\\mathcal{H}}T^{\\left( s,\\omega \\right)\n}-\\frac{1}{2}\\left[ T^{\\left( s,\\omega \\right) },T^{\\left( s,\\omega \\right) \n\\right] ^{\\left( s\\right) },  \\label{dHT}\n\\end{equation\nwhere a wedge product between the $1$-forms $T^{\\left( s,\\omega \\right) }$\nis implied. Equivalently, (\\ref{dHT}) can be written as \n\\begin{equation}\nd\\hat{\\omega}^{\\left( s\\right) }+\\frac{1}{2}\\left[ \\hat{\\omega}^{\\left(\ns\\right) },\\hat{\\omega}^{\\left( s\\right) }\\right] ^{\\left( s\\right) }=\\hat{F\n^{\\left( s,\\omega \\right) }-d^{\\mathcal{H}}\\varphi _{s}\\wedge \\omega ,\n\\label{dwstruct}\n\\end{equation\nwhere $\\left( d^{\\mathcal{H}}\\varphi _{s}\\wedge \\omega \\right) \\left(\nX,Y\\right) =\\left( d_{X}^{\\mathcal{H}}\\varphi _{s}\\right) \\left( \\omega\n\\left( Y\\right) \\right) -\\left( d_{Y}^{\\mathcal{H}}\\varphi _{s}\\right)\n\\left( \\omega \\left( X\\right) \\right) $ for any vector fields $X$ and $Y$ on \n$\\mathcal{P}.$\n\\end{theorem}\n\n\\begin{proof}\nUsing (\\ref{stheta}), we have \n\\begin{eqnarray}\nd^{\\mathcal{H}}T^{\\left( s,\\omega \\right) } &=&dT^{\\left( s,\\omega \\right)\n}\\circ \\func{proj}_{\\mathcal{H}}  \\notag \\\\\n&=&\\left( d\\theta _{s}+d\\hat{\\omega}^{\\left( s\\right) }\\right) \\circ \\func\nproj}_{\\mathcal{H}}.  \\label{Torsion1f3}\n\\end{eqnarray\nNow consider the first term. Let $X_{p},Y_{p}\\in T_{p}\\mathcal{P}$, then \n\\begin{eqnarray}\nd\\theta _{s}\\left( X_{p}^{\\mathcal{H}},Y_{p}^{\\mathcal{H}}\\right) &=&\\left(\nd\\theta \\right) _{s_{p}}\\left( s_{\\ast }X_{p}^{\\mathcal{H}},s_{\\ast }Y_{p}^\n\\mathcal{H}}\\right)  \\notag \\\\\n&=&\\left( d\\theta \\right) _{s_{p}}\\left( \\mathring{D}_{X_{p}}s,\\mathring{D\n_{Y_{P}}s\\right) \\\\\n&=&\\left[ \\theta \\left( \\mathring{D}_{X_{p}}s\\right) ,\\theta \\left( \n\\mathring{D}_{Y_{P}}s\\right) \\right] ^{\\left( s_{p}\\right) }  \\notag \\\\\n&=&\\left[ T^{\\left( s,\\omega \\right) }\\left( X_{p}\\right) ,T^{\\left(\ns,\\omega \\right) }\\left( Y_{p}\\right) \\right] ^{\\left( s_{p}\\right) },\n\\label{Torsion1f3a}\n\\end{eqnarray\nwhere we have used the Maurer-Cartan structural equation for loops (\\re\n{MCequation1}). Using (\\ref{dHom}) for the second term, overall, we obtain \n\\ref{dHT}).\n\nFrom the Maurer-Cartan equation (\\ref{MCequation1}), \n\\begin{equation*}\nd\\theta _{s}-\\frac{1}{2}\\left[ \\theta _{s},\\theta _{s}\\right] ^{\\left(\ns\\right) }=0.\n\\end{equation*\nWe also have from (\\ref{stheta}) \n\\begin{equation*}\n\\left[ \\theta _{s},\\theta _{s}\\right] ^{\\left( s\\right) }=\\left[ T^{\\left(\ns,\\omega \\right) },T^{\\left( s,\\omega \\right) }\\right] ^{\\left( s\\right) }-\n\\left[ \\hat{\\omega}^{\\left( s\\right) },T^{\\left( s,\\omega \\right) }\\right]\n^{\\left( s\\right) }+\\left[ \\hat{\\omega}^{\\left( s\\right) },\\hat{\\omega\n^{\\left( s\\right) }\\right] ^{\\left( s\\right) }.\n\\end{equation*\nHenc\n\\begin{equation*}\nd\\theta _{s}=dT^{\\left( s,\\omega \\right) }-d\\hat{\\omega}^{\\left( s\\right) }\n\\frac{1}{2}\\left[ T^{\\left( s,\\omega \\right) },T^{\\left( s,\\omega \\right) \n\\right] ^{\\left( s\\right) }-\\left[ \\hat{\\omega}^{\\left( s\\right) },T^{\\left(\ns,\\omega \\right) }\\right] ^{\\left( s\\right) }+\\frac{1}{2}\\left[ \\hat{\\omega\n^{\\left( s\\right) },\\hat{\\omega}^{\\left( s\\right) }\\right] ^{\\left( s\\right)\n}.\n\\end{equation*\nNoting that \n\\begin{equation*}\ndT^{\\left( s,\\omega \\right) }=d^{\\mathcal{H}}T^{\\left( s,\\omega \\right)\n}-\\omega \\dot{\\wedge}T^{\\left( s,\\omega \\right) }\n\\end{equation*\nwe find \n\\begin{eqnarray*}\nd\\hat{\\omega}^{\\left( s\\right) }+\\frac{1}{2}\\left[ \\hat{\\omega}^{\\left(\ns\\right) },\\hat{\\omega}^{\\left( s\\right) }\\right] ^{\\left( s\\right) } &=&d^\n\\mathcal{H}}T^{\\left( s,\\omega \\right) }-\\omega \\dot{\\wedge}T^{\\left(\ns,\\omega \\right) } \\\\\n&&-\\frac{1}{2}\\left[ T^{\\left( s,\\omega \\right) },T^{\\left( s,\\omega \\right)\n}\\right] ^{\\left( s\\right) }+\\left[ \\hat{\\omega}^{\\left( s\\right)\n},T^{\\left( s,\\omega \\right) }\\right] ^{\\left( s\\right) }\n\\end{eqnarray*\nand then using (\\ref{dHT}) and (\\ref{dhphis}) we obtain (\\ref{dwstruct}).\n\\end{proof}\n\n\\begin{corollary}[Bianchi identity]\nThe quantity $\\hat{F}^{\\left( s,\\omega \\right) }$ satisfies the equation \n\\begin{eqnarray}\nd^{\\mathcal{H}}\\hat{F}^{\\left( s,\\omega \\right) } &=&d^{\\mathcal{H}}\\varphi\n_{s}\\wedge F  \\notag \\\\\n&=&F\\dot{\\wedge}T^{\\left( s,\\omega \\right) }-\\left[ \\hat{F}^{\\left( s,\\omega\n\\right) },T^{\\left( s,\\omega \\right) }\\right] ^{\\left( s\\right) }\n\\label{Bianchi}\n\\end{eqnarray\nwhere $\\cdot $ denotes the representation of $\\mathfrak{p}$ on $\\mathfrak{l}$\n\\end{corollary}\n\n\\begin{proof}\nUsing the definition (\\ref{Fhat}) of $\\hat{F}^{\\left( s,\\omega \\right) }$,\nwe have \n\\begin{equation*}\nd^{\\mathcal{H}}\\hat{F}^{\\left( s,\\omega \\right) }=d^{\\mathcal{H}}\\left(\n\\varphi _{s}\\left( F\\right) \\right) =d^{\\mathcal{H}}\\varphi _{s}\\wedge\nF+\\varphi _{s}\\left( d^{\\mathcal{H}}F\\right) ,\n\\end{equation*\nhowever using the standard Bianchi identity, $d^{\\mathcal{H}}F=0,$ and (\\re\n{dhphis}), we obtain (\\ref{Bianchi}).\n\\end{proof}\n\n\\begin{example}\nThe equation (\\ref{dHT}) is the precise analog of what is known as the\n\\textquotedblleft $G_{2}$-structure Bianchi identity\\textquotedblright\\ \\cit\n{GrigorianOctobundle,karigiannis-2007} (not to be confused with the Bianchi\nidentity (\\ref{Bianchi})). In that case, $\\hat{F}$ corresponds precisely to\nthe quantity $\\frac{1}{4}\\pi _{7}\\func{Riem}$, which is the projection of\nthe endomorphism part of $\\func{Riem}$ to the $7$-dimensional representation\nof $G_{2}.$ In local coordinates, it is given by $\\frac{1}{4}\\func{Riem\n_{abcd}\\varphi ^{cde}$.\n\\end{example}\n\n\\begin{example}\n\\label{exCx4}In the complex case, with $\\mathbb{L=}U\\mathbb{C}$ and \n\\mathcal{P}$ a principal $U\\left( n\\right) $-bundle, (\\ref{dHT}) shows that \n\\hat{F}^{\\left( s,\\omega \\right) }=dT^{\\left( s,\\omega \\right) }.$ Here $d^\n\\mathcal{H}}=d$ on $\\mathfrak{l}$-valued forms because the action of \n\\mathfrak{p}_{n}$ on $\\mathfrak{l}$ is trivial (as in Example \\ref{exCx2}).\nIf $s$ is a global section, then this shows that $\\hat{F}$ is an exact $2\n-form - and so the class $\\left[ \\hat{F}\\right] =0$. This is consistent with\na vanishing first Chern class which is a necessary condition for existence\nof a global $s$. On the other hand, if we suppose that $s$ is only a local\nsection, so that $T^{\\left( s,\\omega \\right) }$ is a local $1$-form$,$ then\nwe only get that $\\hat{F}^{\\left( s,\\omega \\right) }$ is closed, so in this\ncase it may define a non-trivial first Chern class. If $\\mathcal{P}$ is the\nunitary frame bundle over a complex manifold, it defines a K\\\"{a}hler\nmetric, and then $\\hat{F}$ precisely corresponds to the Ricci curvature, so\nthat the Ricci-flat condition for reduction to a Calabi-Yau manifold is \n\\hat{F}=0.$\n\\end{example}\n\nThe equation (\\ref{dwstruct}) is interesting because this is an analog of\nthe structure equation for the connection $1$-form $\\omega $ on $\\mathcal{P\n. $ However, in the case of $\\omega $, the quantity $d\\omega -\\frac{1}{2\n\\left[ \\omega ,\\omega \\right] $ is horizontal. However, for $\\hat{\\omega\n^{\\left( s\\right) }$, $\\hat{F}^{\\left( s,\\omega \\right) }$ gives the\nhorizontal component, while the remaining terms give mixed vertical and\nhorizontal components. The fully vertical components vanish. We also see\nthat $\\hat{\\omega}^{\\left( s\\right) }$ satisfies the loop Maurer-Cartan\nequation if and only if $\\hat{F}^{\\left( s,\\omega \\right) }=0$ and $d^\n\\mathcal{H}}\\varphi _{s}=0.$ In the $G_{2}$ case, ${\\Greekmath 0272} \\varphi =0$ of\ncourse is equivalent to $T=0$ and hence implies $\\frac{1}{4}\\pi _{7}\\func\nRiem}=0.$ More generally, this may not need to be the case.\n\n\\begin{lemma}\n\\label{lemTcond}Suppose $\\mathbb{L}$ is a left-alternative loop and suppose \n-\\hat{\\omega}^{\\left( s\\right) }$ satisfies the Maurer-Cartan equation \n\\begin{equation}\nd\\hat{\\omega}^{\\left( s\\right) }+\\frac{1}{2}\\left[ \\hat{\\omega}^{\\left(\ns\\right) },\\hat{\\omega}^{\\left( s\\right) }\\right] ^{\\left( s\\right) }=0,\n\\label{omegahatMC}\n\\end{equation\nthen for any $\\alpha ,\\beta \\in \\mathfrak{q}^{\\left( s_{p}\\right) }\\cong\nT_{1}\\mathcal{C}^{R}\\left( \\mathbb{L},\\circ _{s_{p}}\\right) $, \n\\begin{equation}\n\\left[ \\alpha ,\\beta ,T_{p}^{\\left( s,\\omega \\right) }\\right] ^{\\left(\ns_{p}\\right) }=0\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{.}  \\label{Trestrict}\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nTaking the exterior derivative of (\\ref{omegahatMC}) and applying (\\re\n{alphastructeq}), we find $\\hat{\\omega}^{\\left( s\\right) }$ satisfies \n\\begin{equation}\n0=\\left[ \\hat{\\omega}^{\\left( s\\right) },\\hat{\\omega}^{\\left( s\\right)\n},\\theta _{s}+\\hat{\\omega}^{\\left( s\\right) }\\right] ^{\\left( s\\right) }\n\\left[ \\hat{\\omega}^{\\left( s\\right) },\\hat{\\omega}^{\\left( s\\right)\n},T^{\\left( s,\\omega \\right) }\\right] ^{\\left( s\\right) }.\n\\label{omegahatassoc}\n\\end{equation\nSince $\\mathbb{L}$ is left-alternative, we know that the $\\mathbb{L}\n-algebra associator is skew in the first two entries, so if given vector\nfields $X,Y,Z$ on $\\mathcal{P}$, we have \n\\begin{eqnarray}\n0 &=&\\left[ \\hat{\\omega}^{\\left( s\\right) }\\left( X\\right) ,\\hat{\\omega\n^{\\left( s\\right) }\\left( Y\\right) ,T^{\\left( s,\\omega \\right) }\\left(\nZ\\right) \\right] ^{\\left( s\\right) }+\\left[ \\hat{\\omega}^{\\left( s\\right)\n}\\left( Y\\right) ,\\hat{\\omega}^{\\left( s\\right) }\\left( Z\\right) ,T^{\\left(\ns,\\omega \\right) }\\left( X\\right) \\right] ^{\\left( s\\right) }  \\notag \\\\\n&&+\\left[ \\hat{\\omega}^{\\left( s\\right) }\\left( Z\\right) ,\\hat{\\omega\n^{\\left( s\\right) }\\left( X\\right) ,T^{\\left( s,\\omega \\right) }\\left(\nY\\right) \\right] ^{\\left( s\\right) }.  \\label{omhatassoc2}\n\\end{eqnarray\nLet $\\xi \\in \\mathfrak{p\\ }$and let $X=\\sigma \\left( \\xi \\right) $ be a\nvertical vector field on $\\mathcal{P}$, then \n\\begin{equation*}\n\\hat{\\omega}^{\\left( s\\right) }\\left( X\\right) =\\varphi _{s}\\left( \\omega\n\\left( X\\right) \\right) =\\varphi _{s}\\left( \\xi \\right) .\n\\end{equation*\nIn (\\ref{omhatassoc2}), we take $X=\\sigma \\left( \\xi \\right) $ and $Y=\\sigma\n\\left( \\eta \\right) $ to be vertical vector fields and $Z=Z^{h}$ a\nhorizontal vector field. Then since $\\hat{\\omega}^{\\left( s\\right) }$ is\nvertical and $T^{\\left( s,\\omega \\right) }$ is horizontal, we find that for\nany $\\xi ,\\eta \\in \\mathfrak{p},$ \n\\begin{equation*}\n\\left[ \\varphi _{s}\\left( \\xi \\right) ,\\varphi _{s}\\left( \\eta \\right)\n,T^{\\left( s,\\omega \\right) }\\left( Z\\right) \\right] ^{\\left( s\\right) }=0.\n\\end{equation*\nWe know that for each $p\\in \\mathcal{P}$, the map $\\varphi _{s_{p}}$ is\nsurjective onto $\\mathfrak{q}^{\\left( s_{p}\\right) }\\subset \\mathfrak{l\n^{\\left( s_{p}\\right) }$ and thus (\\ref{Trestrict}) holds.\n\\end{proof}\n\n\\begin{theorem}\n\\label{thmTNucl}Suppose $\\mathcal{P}$ is connected and simply-connected and \n\\mathbb{L}$ a smooth loop such that\n\n\\begin{enumerate}\n\\item $\\mathfrak{l}$ is a left-alternative algebra (i.e. the associator on \n\\mathfrak{l}$ is skew-symmetric in the first two entries),\n\n\\item $\\dim \\left( \\mathcal{N}^{R}\\left( \\mathbb{L}\\right) \\right) =\\dim\n\\left( \\mathcal{N}^{R}\\left( \\mathfrak{l}\\right) \\right) .$\n\\end{enumerate}\n\nMoreover, suppose $s_{p}\\in \\mathcal{C}^{R}\\left( \\mathbb{L}\\right) $ for\nevery $p\\in \\mathcal{P}$, then $\\hat{\\omega}^{\\left( s\\right) }$ satisfies\nthe Maurer-Cartan equation (\\ref{omegahatMC}) if and only if there exists a\nmap $f:\\mathcal{P}\\longrightarrow \\mathcal{N}^{R}\\left( \\mathbb{L}\\right) $\nsuch that \n\\begin{equation}\nT^{\\left( s,\\omega \\right) }=-\\left( \\func{Ad}_{s}\\right) _{\\ast }\\theta\n_{f}.  \\label{Tsthetaf}\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nSince $s$ has values in $\\mathcal{C}^{R}\\left( \\mathbb{L}\\right) ,$ using\nLemma \\ref{lemTcond}, we see that the conditions of Corollary \\re\n{corLoopCartan} are satisfied, and hence there exists a map $f:\\mathcal{P\n\\longrightarrow \\mathcal{N}^{R}\\left( \\mathbb{L}\\right) $ such that \n\\begin{eqnarray*}\n-\\hat{\\omega}^{\\left( s\\right) } &=&\\theta _{sf} \\\\\n&=&\\theta _{s}+\\left( \\func{Ad}_{s}\\right) _{\\ast }\\theta _{f}.\n\\end{eqnarray*\nFrom (\\ref{stheta}), \n\\begin{equation*}\nT^{\\left( s,\\omega \\right) }=\\theta _{s}+\\hat{\\omega}^{\\left( s\\right)\n}=-\\left( \\func{Ad}_{s}\\right) _{\\ast }\\theta _{f}.\n\\end{equation*\nConversely, suppose (\\ref{Tsthetaf}) holds for some right nucleus-valued map \n$f$. Then, clearly $\\hat{\\omega}^{\\left( s\\right) }=-\\theta _{sf}$, and thus \n$-\\hat{\\omega}^{\\left( s\\right) }$ satisfies (\\ref{omegahatMC}).\n\\end{proof}\n\n\\begin{remark}\nTheorem \\ref{thmTNucl} shows that if $\\mathbb{L}$ has a sufficiently large\\\nnucleus, then $\\hat{F}^{\\left( s,\\omega \\right) }=0$ and $d^{\\mathcal{H\n}\\varphi _{s}=0$ do not necessarily imply that $T^{\\left( s,\\omega \\right)\n}=0$. In the case of unit octonions, the nucleus is just $\\left\\{ \\pm\n1\\right\\} $, so any nucleus-valued map is constant on connected components,\nhence in this case if $\\hat{\\omega}^{\\left( s\\right) }$ satisfies (\\re\n{omegahatMC}), then $T^{\\left( s,\\omega \\right) }=0.$\n\\end{remark}\n\n\\subsection{Deformations}\n\n\\label{sectDeform}The torsion of a loop structure is determined by the\nequivariant $\\mathbb{\\mathring{L}}$-valued map $s$ and the connection \n\\omega $ on $\\mathcal{P}.$ There are several possible deformations of $s$\nand $\\omega $. In particular, $s$ may be deformed by the action of $\\Psi $\nor by left multiplication action of $\\mathbb{L}.$ The connection $\\omega $\nmay be deformed by the affine action of $\\Omega _{basic}^{1}\\left( \\mathcal{\n},\\mathfrak{p}\\right) $ or by gauge transformations in $\\Psi .$ Moreover, of\ncourse, these deformations may be combined or considered infinitesimally.\nSince $T^{\\left( s,\\omega \\right) }$ is the horizontal part of $\\theta _{s}\n, when considering deformations of $s$ it is sufficient to consider what\nhappens to $\\theta _{s}$ and then taking the horizontal component.\n\nRecall that the space of connections on $\\mathcal{P}$ is an affine space\nmodelled on equivariant horizontal (i.e. basic) $\\mathfrak{p}$-valued $1\n-forms on $\\mathcal{P}.$ Thus, any connection $\\tilde{\\omega}=\\omega +A$ for\nsome basic $\\mathfrak{p}$-valued $1$-form $A$. Then, \n\\begin{equation}\nT^{\\left( s,\\tilde{\\omega}\\right) }=\\theta _{s}+\\varphi _{s}\\left( \\tilde\n\\omega}\\right) =T^{\\left( s,\\omega \\right) }+\\hat{A}  \\label{wtild}\n\\end{equation\nwhere $\\hat{A}=\\varphi _{s}\\left( A\\right) $. Thus, we can set $T^{\\left( s\n\\tilde{\\omega}\\right) }=0$ by choosing $A$ such that $\\hat{A}=-T^{\\left(\ns,\\omega \\right) }$ if and only if for each $p\\in P\\,,\\ T_{p}^{\\left(\ns,\\omega \\right) }\\in $ $\\mathfrak{q}^{\\left( s_{p}\\right) }=\\varphi\n_{s_{p}}\\left( \\mathfrak{p}\\right) $. Since $\\hat{\\omega}$ is always in the\nimage of $\\varphi _{s}$, we conclude there exists a connection $\\tilde{\\omeg\n}$ for which $T^{\\left( s,\\tilde{\\omega}\\right) }=0$ if and only if $\\left.\n\\theta _{s}\\right\\vert _{p}$ $\\in \\mathfrak{q}^{\\left( s_{p}\\right) }$ for\neach $p$. In that case, $\\theta _{s}=-\\varphi _{s}\\left( \\tilde{\\omega\n\\right) .$ From Theorem \\ref{thmThetaPhi}, we then see that $\\tilde{\\omega}$\nhas curvature with values in $\\mathfrak{h}_{s}.$\n\nRecall that if $\\phi :\\mathcal{P}\\longrightarrow \\mathcal{P}$ is a gauge\ntransformation, then there exists an $\\func{Ad}_{\\Psi }$-equivariant map $u\n\\mathcal{P}\\longrightarrow \\Psi $ such that for each $p\\in \\mathcal{P}$, \n\\phi \\left( p\\right) =pu_{p}$. Each such map then corresponds to a section\nof the associated bundle $\\func{Ad}\\left( \\mathcal{P}\\right) .$ The\ngauge-transformed connection $1$-form is then $\\omega ^{\\phi }=u^{\\ast\n}\\omega $, where \n\\begin{equation}\nu^{\\ast }\\omega =\\left( \\func{Ad}_{u^{-1}}\\right) _{\\ast }\\omega +u^{\\ast\n}\\theta _{\\Psi }  \\label{omgauge}\n\\end{equation\nwhere $\\theta _{\\Psi }$ is the \\emph{left}-invariant Maurer-Cartan form on \n\\Psi $. Then, \n\\begin{eqnarray}\nd^{u^{\\ast }\\mathcal{H}}s &=&\\left( l_{u}^{-1}\\right) _{\\ast }d^{\\mathcal{H\n}\\left( l_{u}s\\right)  \\notag \\\\\n&=&d^{\\mathcal{H}}s+\\left( u^{\\ast }\\theta _{\\Psi }\\right) ^{\\mathcal{H\n}\\cdot s_{p}  \\label{dHphi}\n\\end{eqnarray\nwhere at each $p\\in \\mathcal{P}.\n\\begin{equation*}\n\\left. \\left( u^{\\ast }\\theta _{\\Psi }\\right) ^{\\mathcal{H}}\\right\\vert\n_{p}=\\left( l_{u_{p}}\\right) _{\\ast }^{-1}\\circ \\left( d^{\\mathcal{H\n}u\\right) _{p}\\mathfrak{.}\n\\end{equation*\nHence, \n\\begin{equation}\nT^{\\left( s,u^{\\ast }\\omega \\right) }=\\left( R_{s}^{-1}\\right) _{\\ast\n}d^{u^{\\ast }\\mathcal{H}}s=T^{\\left( s,\\omega \\right) }+\\varphi _{s}\\left(\n\\left( u^{\\ast }\\theta _{\\Psi }\\right) ^{\\mathcal{H}}\\right) .\n\\label{Tsgauge}\n\\end{equation\nConsider the curvature $F^{u^{\\ast }\\omega }$ of the connection $u^{\\ast\n}\\omega $. It is well-known that it is given b\n\\begin{equation}\nF^{u^{\\ast }\\omega }=\\left( \\func{Ad}_{u^{-1}}\\right) _{\\ast }F.\n\\end{equation\nFrom Theorem \\ref{lemGammahatsurj}, we then have \n\\begin{equation}\n\\hat{F}^{\\left( s,u^{\\ast }\\omega \\right) }=\\varphi _{s}\\left( \\left( \\func\nAd}_{u^{-1}}\\right) _{\\ast }F\\right) =\\left( u^{-1}\\right) _{\\ast }^{\\prime \n\\hat{F}^{\\left( u\\left( s\\right) ,\\omega \\right) }.\n\\end{equation\nOn the other hand, using (\\ref{dHphi}) and (\\ref{Dsderiv}) we have \n\\begin{eqnarray*}\nT^{\\left( s,u^{\\ast }\\omega \\right) } &=&\\left( R_{s}^{-1}\\right) _{\\ast\n}\\left( u^{\\ast }\\mathring{D}\\right) \\left( s\\right) \\\\\n&=&\\left( R_{s}^{-1}\\right) _{\\ast }\\left( u^{-1}\\right) _{\\ast }\\mathring{D\n\\left( u\\left( s\\right) \\right) \\\\\n&=&\\left( u^{-1}\\right) _{\\ast }^{\\prime }\\left( R_{u\\left( s\\right)\n}^{-1}\\right) _{\\ast }\\mathring{D}\\left( u\\left( s\\right) \\right) \\\\\n&=&\\left( u^{-1}\\right) _{\\ast }^{\\prime }T^{\\left( u\\left( s\\right) ,\\omega\n\\right) }.\n\\end{eqnarray*\nSummarizing, we have the following.\n\n\\begin{theorem}\nSuppose $s:\\mathcal{P}\\longrightarrow \\mathbb{\\mathring{L}}$ and $u:\\mathcal\nP}\\longrightarrow \\Psi $ are equivariant smooth maps. Then, \n\\begin{subequations}\n\\begin{eqnarray}\nT^{\\left( s,u^{\\ast }\\omega \\right) } &=&T^{\\left( s,\\omega \\right)\n}+\\varphi _{s}\\left( \\left( u^{\\ast }\\theta _{\\Psi }\\right) ^{\\mathcal{H\n}\\right)  \\label{Tsustom} \\\\\n&=&\\left( u^{-1}\\right) _{\\ast }^{\\prime }T^{\\left( u\\left( s\\right) ,\\omega\n\\right) }  \\notag \\\\\n\\hat{F}^{\\left( s,u^{\\ast }\\omega \\right) } &=&\\left( u^{-1}\\right) _{\\ast\n}^{\\prime }\\hat{F}^{\\left( u\\left( s\\right) ,\\omega \\right) }.\n\\end{eqnarray\n\\end{subequations\nIn particular, \n\\begin{equation}\nT^{\\left( u^{-1}\\left( s\\right) ,u^{\\ast }\\omega \\right) }=\\left( u^{\\prime\n}\\right) _{\\ast }^{-1}T^{\\left( s,\\omega \\right) }\\ \\ \\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{and }\\hat{F\n^{\\left( u^{-1}\\left( s\\right) ,u^{\\ast }\\omega \\right) }=\\left(\nu^{-1}\\right) _{\\ast }^{\\prime }\\hat{F}^{\\left( s,\\omega \\right) }.\n\\label{Tuom}\n\\end{equation}\n\\end{theorem}\n\nThis shows that both $T$ and $\\hat{F}$ transform equivariantly with respect\nto a simultaneous transformation of $s$ and $\\omega $. In particular, if we\nhave a Riemannian metric on the base manifold $M$ and a $\\Psi $-covariant\nmetric on $\\mathfrak{l},$ then with respect to the induced metric on \nT^{\\ast }\\mathcal{P}\\otimes \\mathfrak{l}$, the quantities $\\left\\vert\nT\\right\\vert ^{2}$ and $\\left\\vert F\\right\\vert ^{2}$ are invariant with\nrespect to the transformation $\\left( s,\\omega \\right) \\mapsto \\left(\nu^{-1}\\left( s\\right) ,u^{\\ast }\\omega \\right) .$ In the case of $G_{2}\n-structure, the key question is regarding the holonomy of the Levi-Civita,\nso in this general setting, if we are interested in the holonomy of $\\omega \n, it makes sense to consider individual transformations $s\\mapsto As$ for\nsome equivariant $A\\in C^{\\infty }\\left( \\mathcal{P},\\mathbb{L}\\right) $ and \n$\\omega \\mapsto u^{\\ast }\\omega $ because each of these transformations\nleaves the holonomy group unchanged. We also see that every transformation \ns\\mapsto u\\left( s\\right) $ for some equivariant $u\\in C^{\\infty }\\left( \n\\mathcal{P},\\Psi \\right) $ corresponds to a transformation $s\\mapsto As,$\nwhere $A=h\\left( s\\right) \/s$. From (\\ref{PsAutoriso}), this is precisely\nthe companion of the corresponding map $u_{s}\\in \\Psi \\left( \\mathbb{L\n,\\circ _{s}\\right) .$ Moreover, this correspondence is one-to-one if and\nonly if $\\mathbb{L}$ is a $G$-loop. It is easy to see that $A$ is then an\nequivariant $\\mathbb{L}$-valued map. Thus, considering transformations \ns\\mapsto As$ is more general in some situations.\n\n\\begin{theorem}\nSuppose $A:\\mathcal{P}\\longrightarrow \\mathbb{L}$ and $s:\\mathcal{P\n\\longrightarrow \\mathbb{\\mathring{L}}$ . Then, \n\\begin{subequations}\n\\begin{eqnarray}\nT^{\\left( As,\\omega \\right) } &=&\\left( R_{A}^{\\left( s\\right) }\\right)\n_{\\ast }^{-1}DA+\\left( \\func{Ad}_{A}^{\\left( s\\right) }\\right) _{\\ast\n}T^{\\left( s,\\omega \\right) }=\\left( R_{A}^{\\left( s\\right) }\\right) _{\\ast\n}^{-1}D^{\\left( s\\right) }A  \\label{Trom} \\\\\n\\hat{F}^{\\left( As,\\omega \\right) } &=&\\left( R_{A}^{\\left( s\\right)\n}\\right) _{\\ast }^{-1}\\left( F^{\\prime }\\cdot A\\right) +\\left( \\func{Ad\n_{A}^{\\left( s\\right) }\\right) _{\\ast }\\hat{F}^{\\left( s,\\omega \\right) },\n\\label{From}\n\\end{eqnarray\n\\end{subequations\nwhere $F^{\\prime }\\cdot A$ denotes the infinitesimal action of $\\mathfrak{p}$\non $\\mathbb{L}.$\n\\end{theorem}\n\n\\begin{proof}\nRecall from (\\ref{thetafs2}), that \n\\begin{equation}\n\\theta _{As}=\\theta _{A}^{\\left( s\\right) }+\\left( \\func{Ad}_{A}^{\\left(\ns\\right) }\\right) _{\\ast }\\theta _{s}.  \\label{thetaAs}\n\\end{equation\nNow, $T^{\\left( s,\\omega \\right) }$ is just the horizontal part of $\\theta\n_{s}$, so taking the horizontal projection in (\\ref{thetaAs}), we\nimmediately get (\\ref{Trom}). To obtain (\\ref{From}), from (\\ref{phiAs}) we\nhav\n\\begin{equation}\n\\hat{F}^{\\left( As,\\omega \\right) }=\\varphi _{As}\\left( F\\right) =\\left(\nR_{A}^{\\left( s\\right) }\\right) _{\\ast }^{-1}\\left( F^{\\prime }\\cdot\nA\\right) +\\left( \\func{Ad}_{A}^{\\left( s\\right) }\\right) _{\\ast }\\varphi\n_{s}\\left( F\\right) ,\n\\end{equation\nand hence we obtain (\\ref{From}).\n\\end{proof}\n\n\\begin{remark}\nThe expression (\\ref{Trom}) precisely replicates the formula for the\ntransformation of torsion of a $G_{2}$-structure within a fixed metric\nclass, as derived in \\cite{GrigorianOctobundle}.\n\\end{remark}\n\nNow suppose $s_{t}$ is a $1$-parameter family of equivariant $\\mathbb\n\\mathring{L}}$-valued maps that satisfy \n\\begin{equation}\n\\frac{\\partial s_{t}}{\\partial t}=\\left( R_{s_{t}}\\right) _{\\ast }\\xi _{t}\n\\label{Aevol}\n\\end{equation\nwhere $\\xi _{t}$ is a $1$-parameter family of $\\mathfrak{l}$-valued maps. In\nparticular, if $\\xi \\left( t\\right) $ is independent of $t$, then $s\\left(\nt\\right) =\\exp _{s_{0}}\\left( t\\xi \\right) s_{0}.$ Then let us work out the\nevolution of $T^{\\left( s\\left( t\\right) ,\\omega \\right) }$ and $\\hat{F\n^{\\left( s\\left( t\\right) ,\\omega \\right) }.$ First consider the evolution\nof $\\theta _{s\\left( t\\right) }$ and $\\varphi _{s\\left( t\\right) }$.\n\n\\begin{lemma}\nSuppose $s\\left( t\\right) $ satisfies (\\ref{Aevol}), then \n\\begin{subequations}\n\\begin{eqnarray}\n\\frac{\\partial \\theta _{s\\left( t\\right) }}{\\partial t} &=&d\\xi \\left(\nt\\right) -\\left[ \\theta _{s\\left( t\\right) },\\xi \\left( t\\right) \\right]\n^{\\left( s\\left( t\\right) \\right) }  \\label{dtthetas} \\\\\n\\frac{\\partial \\varphi _{s\\left( t\\right) }}{\\partial t} &=&\\func{id}_\n\\mathfrak{p}}\\cdot \\xi \\left( t\\right) -\\left[ \\varphi _{s\\left( t\\right)\n},\\xi \\left( t\\right) \\right] ^{\\left( s\\left( t\\right) \\right) }.\n\\label{dtphis}\n\\end{eqnarray\n\\end{subequations\n\\end{lemma}\n\n\\begin{proof}\nFor $\\theta _{s\\left( t\\right) }$, suppressing pushforwards, we hav\n\\begin{eqnarray}\n\\frac{\\partial \\theta _{s\\left( t\\right) }}{\\partial t} &=&\\frac{\\partial }\n\\partial t}\\left( \\left( ds\\left( t\\right) \\right) \/s\\left( t\\right) \\right)\n\\notag \\\\\n&=&\\left( d\\dot{s}\\right) \/s-\\left( \\left( ds\\right) \/s\\cdot \\dot{s}\\right)\n\/s  \\notag \\\\\n&=&d\\left( \\xi s\\right) \/s-\\left( \\left( ds\\right) \/s\\cdot \\left( \\xi\ns\\right) \\right) \/s  \\notag \\\\\n&=&d\\xi -\\left[ \\theta _{s\\left( t\\right) },\\xi \\right] ^{\\left( s\\left(\nt\\right) \\right) }.\n\\end{eqnarray\nSimilarly, for $\\varphi _{s\\left( t\\right) },$ let $\\eta \\in \\mathfrak{p}$,\nthen \n\\begin{eqnarray}\n\\frac{\\partial \\varphi _{s\\left( t\\right) }\\left( \\eta \\right) }{\\partial t}\n&=&\\frac{\\partial }{\\partial t}\\left( \\left. \\frac{d}{d\\tau }\\exp \\left(\n\\tau \\eta \\right) \\left( s\\right) \/s\\right\\vert _{\\tau =0}\\right)  \\notag \\\\\n&=&\\left. \\frac{d}{d\\tau }\\exp \\left( \\tau \\eta \\right) \\left( \\left( \\xi\ns\\right) \/s\\right) \\right\\vert _{\\tau =0}-\\left. \\frac{d}{d\\tau }\\left( \\exp\n\\left( \\tau \\eta \\right) \\left( \\left( s\\right) \/s\\right) \\cdot \\left( \\xi\ns\\right) \\right) \/s\\right\\vert _{\\tau =0}  \\notag \\\\\n&=&\\left. \\frac{d}{d\\tau }\\exp \\left( \\tau \\eta \\right) ^{\\prime }\\left( \\xi\n\\right) \\right\\vert _{\\tau =0}+\\left. \\frac{d}{d\\tau }\\left( \\xi \\exp \\left(\n\\tau \\eta \\right) \\left( s\\right) \\right) \/s\\right\\vert _{\\tau =0}  \\notag \\\\\n&&-\\left. \\frac{d}{d\\tau }\\left( \\exp \\left( \\tau \\eta \\right) \\left( \\left(\ns\\right) \/s\\right) \\cdot \\left( \\xi s\\right) \\right) \/s\\right\\vert _{\\tau =0}\n\\notag \\\\\n&=&\\eta \\cdot \\xi \\left( t\\right) -\\left[ \\varphi _{s\\left( t\\right) }\\left(\n\\eta \\right) ,\\xi \\left( t\\right) \\right] ^{\\left( s\\left( t\\right) \\right)\n}.\n\\end{eqnarray}\n\\end{proof}\n\nTo obtain the evolution of $T^{\\left( s\\left( t\\right) ,\\omega \\right) }$\nand $\\hat{F}^{\\left( s\\left( t\\right) ,\\omega \\right) }$, we just take the\nhorizontal component of (\\ref{dtphis}) and substitute $F$ into (\\ref{dtphis\n).\n\n\\begin{corollary}\nSuppose $s\\left( t\\right) $ satisfies (\\ref{Aevol}), then \n\\begin{subequations\n\\label{dtTF} \n\\begin{eqnarray}\n\\frac{\\partial T^{\\left( s\\left( t\\right) ,\\omega \\right) }}{\\partial t}\n&=&d^{\\mathcal{H}}\\xi \\left( t\\right) -\\left[ T^{\\left( s\\left( t\\right)\n,\\omega \\right) },\\xi \\left( t\\right) \\right] ^{\\left( s\\left( t\\right)\n\\right) }  \\label{dtTF1} \\\\\n\\frac{\\partial \\hat{F}^{\\left( s\\left( t\\right) ,\\omega \\right) }}{\\partial \n} &=&F\\cdot \\xi \\left( t\\right) -\\left[ \\hat{F}^{\\left( s\\left( t\\right)\n,\\omega \\right) },\\xi \\left( t\\right) \\right] ^{\\left( s\\left( t\\right)\n\\right) }.  \\label{dtTF2}\n\\end{eqnarray\n\\end{subequations\n\\end{corollary}\n\nThe expression (\\ref{dtTF1}) is the analog of a similar expression for the\nevolution of the torsion of a $G_{2}$-structure, as given in \\cit\n{GrigorianIsoflow,karigiannis-2007}.\n\n\\begin{remark}\nSuppose $u_{t}$ is a $1$-parameter family of equivariant $\\Psi $-valued maps\nthat satisfy \n\\begin{equation}\n\\frac{\\partial u_{t}}{\\partial t}=\\left( l_{u_{t}}\\right) _{\\ast }\\gamma _{t}\n\\end{equation\nfor a $1$-parameter family $\\gamma _{t}$ of equivariant $\\mathfrak{p}\n-valued maps. Then, each $u_{t}$ defines a gauge transformation of the\nconnection $\\omega .$ Define \n\\begin{equation}\n\\omega _{t}=u_{t}^{\\ast }\\omega .\n\\end{equation\nThen, it is easy to see that \n\\begin{equation}\n\\frac{\\partial \\omega _{t}}{\\partial t}=d\\gamma _{t}+\\left[ \\omega\n_{t},\\gamma _{t}\\right] _{\\mathfrak{p}}=d^{\\mathcal{H}_{t}}\\gamma _{t},\n\\label{dtomt}\n\\end{equation\nwhere $d^{\\mathcal{H}_{t}}$ is the covariant derivative corresponding to \n\\omega _{t}.$ Similarly, the corresponding curvature $F_{t}$ evolves via the\nequatio\n\\begin{equation}\n\\frac{\\partial F_{t}}{\\partial t}=\\left[ F_{t},\\gamma _{t}\\right] _\n\\mathfrak{p}}.  \\label{dtFt}\n\\end{equation\nUsing (\\ref{dtomt}) together with (\\ref{dtTF1}) gives \n\\begin{equation}\n\\frac{\\partial T^{\\left( s_{t},\\omega _{t}\\right) }}{\\partial t}=d^{\\mathcal\nH}_{t}}\\xi _{t}-\\left[ T^{\\left( s_{t},\\omega _{t}\\right) },\\xi _{t}\\right]\n^{\\left( s_{t}\\right) }+\\varphi _{s_{t}}\\left( d^{\\mathcal{H}_{t}}\\gamma\n_{t}\\right) .  \\label{dTstomt}\n\\end{equation\nHowever, \n\\begin{eqnarray*}\n\\varphi _{s_{t}}\\left( d^{\\mathcal{H}_{t}}\\gamma _{t}\\right) &=&d^{\\mathcal{\n}_{t}}\\hat{\\gamma}_{t}^{\\left( s_{t}\\right) }-\\left( d^{\\mathcal{H\n_{t}}\\varphi _{s_{t}}\\right) \\left( \\gamma _{t}\\right) \\\\\n&=&d^{\\mathcal{H}_{t}}\\hat{\\gamma}_{t}^{\\left( s_{t}\\right) }-\\gamma\n_{t}\\cdot T^{\\left( s_{t},\\omega _{t}\\right) }-\\left[ T^{\\left( s_{t},\\omega\n_{t}\\right) },\\hat{\\gamma}_{t}^{\\left( s_{t}\\right) }\\right] ^{\\left(\ns_{t}\\right) }\n\\end{eqnarray*\nand thus (\\ref{dTstomt}) becomes \n\\begin{equation}\n\\frac{\\partial T^{\\left( s_{t},\\omega _{t}\\right) }}{\\partial t}=-\\gamma\n_{t}\\cdot T^{\\left( s_{t},\\omega _{t}\\right) }+d^{\\mathcal{H}_{t}}\\left( \\xi\n_{t}+\\hat{\\gamma}_{t}^{\\left( s_{t}\\right) }\\right) -\\left[ T^{\\left(\ns_{t},\\omega _{t}\\right) },\\xi _{t}+\\hat{\\gamma}_{t}^{\\left( s_{t}\\right) \n\\right] ^{\\left( s_{t}\\right) }.  \\label{dTstomt2}\n\\end{equation\nFor the curvature, using (\\ref{dtFt}) together with (\\ref{dtTF2}) gives \n\\begin{equation}\n\\frac{\\partial \\hat{F}^{\\left( s_{t},\\omega _{t}\\right) }}{\\partial t\n=F_{t}\\cdot \\xi _{t}-\\left[ \\hat{F}^{\\left( s_{t},\\omega _{t}\\right) },\\xi\n_{t}\\right] ^{\\left( s_{t}\\right) }+\\varphi _{s_{t}}\\left( \\left[\nF_{t},\\gamma _{t}\\right] _{\\mathfrak{p}}\\right) .  \\label{dFstomt}\n\\end{equation\nUsing (\\ref{xiphi}), we then ge\n\\begin{equation}\n\\frac{\\partial \\hat{F}^{\\left( s_{t},\\omega _{t}\\right) }}{\\partial t\n=-\\gamma _{t}\\cdot \\hat{F}_{t}+F_{t}\\cdot \\left( \\xi _{t}+\\hat{\\gamma\n_{t}^{\\left( s_{t}\\right) }\\right) -\\left[ \\hat{F}^{\\left( s_{t},\\omega\n_{t}\\right) },\\xi _{t}+\\hat{\\gamma}_{t}^{\\left( s_{t}\\right) }\\right]\n^{\\left( s_{t}\\right) }.  \\label{dFstomt2}\n\\end{equation\nTaking $\\xi _{t}=-\\hat{\\gamma}_{t}^{\\left( s_{t}\\right) }$ in (\\ref{dTstomt2\n) and (\\ref{dFstomt2}), we obtain the infinitesimal versions of (\\ref{Tuom}).\n\\end{remark}\n\n\\subsection{Variational principles}\n\n\\label{sectVar}In general we have seen that the loop bundle structure is\ngiven by $\\mathbb{\\mathring{L}}$-valued map $s$ as well as a connection \n\\omega $ on $\\mathcal{P}.$ We call the pair $\\left( s,\\omega \\right) $ the\nconfiguration of the loop bundle structure. Each point in the configuration\nspace gives rise to the corresponding torsion $T^{\\left( s,\\omega \\right) }$\nand curvature $\\hat{F}^{\\left( s,\\omega \\right) }.$Previously we considered \nT$ and $\\hat{F}$ as horizontal equivariant forms on $\\mathcal{P}$, but of\ncourse we can equivalently consider them as bundle-valued differential forms\non the base manifold $M$. To be able to define functionals on $M,$ let us\nsuppose $M$ has a Riemannian metric and moreover, $\\mathbb{L}$ has the\nfollowing properties:\n\n\\begin{enumerate}\n\\item For each $s\\in \\mathbb{\\mathring{L}}$, the Killing form $K^{\\left(\ns\\right) }$ is nondegenerate and invariant with respect to $\\func{ad\n^{\\left( s\\right) }$ and the action of $\\mathfrak{p}.$\n\n\\item $\\mathbb{L}$ is a $G$-loop, so that in particular, for each $s\\in \n\\mathbb{\\mathring{L}},$ $\\mathfrak{l}^{\\left( s\\right) }=\\mathfrak{q}_{s}.$\n\n\\item For each $s\\in \\mathbb{\\mathring{L}}$, the space $\\mathfrak{q}_{s}$ is\nan irreducible representation of the Lie algebra $\\mathfrak{h}_{s}$.\n\\end{enumerate}\n\nThese properties may not be strictly necessary, but they will simplify\narguments. Moreover, these are the properties satisfied by the loop of unit\noctonions, which is the key example. The first property means we can defined\nthe map $\\varphi _{s}^{t},$ and then the second and third properties\ntogether make sure that there exists a constant $\\lambda $ such that for any \n$s\\in \\mathbb{\\mathring{L}},$ $\\varphi _{s}\\varphi _{s}^{t}=\\lambda \\func{id\n_{\\mathfrak{l}}$ and $\\varphi _{s}^{t}\\varphi _{s}=\\lambda \\func{id}_\n\\mathfrak{h}_{s}^{\\perp }},$ as per Lemma \\ref{lemphisphist}. If $\\mathfrak{\n}_{s}$ is a reducible representation, then each irreducible component may\nhave its own constant. Moreover, the first and second properties together\nimply that $K^{\\left( s\\right) }$ is independent of the choice of $s$, and\nwhen extended as an inner product on sections, it is covariantly constant\nwith respect to a principal connection on $\\mathcal{P}.$\n\nLet $s\\in \\mathbb{\\mathring{L}}$ be fixed. Suppose we have a path of\nconnections on $\\mathcal{P}$ given by $\\tilde{\\omega}\\left( t\\right) =\\omega\n+tA$ for some basic $\\mathfrak{p}$-valued $1$-form $A$ and a fixed principal\nconnection $\\omega $. Then, define \n\\begin{subequations}\n\\begin{eqnarray}\nT\\left( t\\right)  &=&T^{\\left( s,\\tilde{\\omega}\\left( t\\right) \\right)\n}=\\theta _{s}+\\varphi _{s}\\left( \\tilde{\\omega}\\left( t\\right) \\right)\n=T^{\\left( s,\\omega \\right) }+t\\hat{A}. \\\\\n\\hat{F}\\left( t\\right)  &=&\\hat{F}^{\\left( s,\\hat{\\omega}\\left( t\\right)\n\\right) }=\\varphi _{s}\\left( F^{\\tilde{\\omega}\\left( t\\right) }\\right) =\\hat\nF}^{\\left( s,\\omega \\right) }+t\\varphi _{s}\\left( d^{\\mathcal{H}}A\\right)  \\\\\n&&+\\frac{1}{2}t^{2}\\varphi _{s}\\left( \\left[ A,A\\right] _{\\mathfrak{p\n}\\right) ,  \\notag\n\\end{eqnarray\n\\end{subequations\nwhere $\\hat{A}=\\varphi _{s}\\left( A\\right) $. Hence, using (\\ref{dhphis}), \n\\begin{subequations}\n\\begin{eqnarray}\n\\left. \\frac{d}{dt}T\\left( t\\right) \\right\\vert _{t=0} &=&\\hat{A} \\\\\n\\left. \\frac{d}{dt}\\hat{F}\\left( t\\right) \\right\\vert _{t=0} &=&\\varphi\n_{s}\\left( d^{\\mathcal{H}}A\\right) =d^{\\mathcal{H}}\\hat{A}-\\left( d^\n\\mathcal{H}}\\varphi _{s}\\right) \\wedge A  \\notag \\\\\n&=&d^{\\mathcal{H}}\\hat{A}+A\\cdot T-\\left[ \\hat{A},T\\right] ^{\\left( s\\right)\n},\n\\end{eqnarray\n\\end{subequations\nwhere for brevity, $T=T\\left( 0\\right) =T^{\\left( s,\\omega \\right) }$. Note\nthat if for each $p\\in \\mathcal{P}$, $A_{p}\\in \\mathfrak{h}_{s_{p}}$, then\nthe torsion is unaffected, so these deformations are not relevant for the\nloop bundle structure. Hence, let us assume that $A_{p}\\in \\mathfrak{h\n_{s_{p}}^{\\perp }$ for each $p\\in \\mathcal{P}.$ Equivalently, this means\nthat $A\\in \\varphi _{s}^{t}\\left( \\mathfrak{l}\\right) .$ So now suppose $\\xi\n\\in \\Omega _{basic}^{1}\\left( \\mathcal{P},\\mathfrak{l}\\right) $ is a basic \n\\mathfrak{l}$-valued $1$-form on $\\mathcal{P}\\ $such that $A=\\frac{1}\n\\lambda }\\varphi _{s}^{t}\\left( \\xi \\right) $, and thus, $\\hat{A}=\\xi .$\nMoreover, from (\\ref{piqsact}), we see that \n\\begin{equation}\nA\\cdot T=\\frac{1}{\\lambda }\\varphi _{s}^{t}\\left( \\xi \\right) \\cdot T=\\frac{\n}{2\\lambda ^{2}}\\left[ \\xi ,T\\right] _{\\varphi _{s}}+\\frac{1}{2}\\left[ \\xi ,\n\\right] ^{\\left( s\\right) },\n\\end{equation\nwhere the bracket $\\left[ \\cdot ,\\cdot \\right] _{\\varphi _{s}}$ on \n\\mathfrak{l}$ is given by \n\\begin{equation}\n\\left[ \\xi ,\\eta \\right] _{\\varphi _{s}}=\\varphi _{s}\\left( \\left[ \\varphi\n_{s}^{t}\\left( \\xi \\right) ,\\varphi _{s}^{t}\\left( \\eta \\right) \\right] _\n\\mathfrak{p}}\\right) ,\n\\end{equation\nas defined in (\\ref{phisbrack}). Overall, the deformations are now given by \n\\begin{subequations\n\\label{TFdeformxi} \n\\begin{eqnarray}\n\\left. \\frac{d}{dt}T\\left( t\\right) \\right\\vert _{t=0} &=&\\xi  \\\\\n\\left. \\frac{d}{dt}\\hat{F}\\left( t\\right) \\right\\vert _{t=0} &=&d^{\\mathcal{\n}}\\xi +\\frac{1}{2\\lambda ^{2}}\\left[ \\xi ,T\\right] _{\\varphi _{s}}-\\frac{1}{\n}\\left[ \\xi ,T\\right] ^{\\left( s\\right) }.\n\\end{eqnarray\n\\end{subequations\nSuppose now $M$ is a $3$-dimensional compact manifold. For a fixed section \ns\\in \\mathcal{\\mathring{Q}},$ consider now a functional $\\mathcal{F}^{\\left(\ns\\right) }$ on the space of connections on $\\mathcal{P}$ modulo $\\mathfrak{h\n_{s},$ given by \n\\begin{equation}\n\\mathcal{F}^{\\left( s\\right) }\\left( \\omega \\right) =\\int_{M}\\left\\langle T\n\\hat{F}\\right\\rangle ^{\\left( s\\right) }-\\frac{1}{6\\lambda ^{2}}\\left\\langle\nT,\\left[ T,T\\right] _{\\varphi _{s}}\\right\\rangle ^{\\left( s\\right) },\n\\label{Fsfunctional}\n\\end{equation\nwhere wedge products between forms are implicit. From the properties of $T\n\\hat{F},$ $\\left[ \\cdot ,\\cdot \\right] _{\\varphi _{s}}$, and $\\left\\langle\n{}\\right\\rangle ^{\\left( s\\right) }$, we see that that this is invariant\nunder simultaneous gauge transformation $\\left( s,\\omega \\right) \\mapsto\n\\left( u^{-1}\\left( s\\right) ,u^{\\ast }\\omega \\right) .$\n\nNow using (\\ref{TFdeformxi}) consider deformations of each piece of (\\re\n{Fsfunctional}). For the first term, using (\\ref{dHT}), we obtain \n\\begin{eqnarray}\n\\left. \\frac{d}{dt}\\int_{M}\\left\\langle T\\left( t\\right) ,\\hat{F}\\left(\nt\\right) \\right\\rangle ^{\\left( s\\right) }\\right\\vert _{t=0}\n&=&\\int_{M}\\left\\langle \\xi ,\\hat{F}\\right\\rangle ^{\\left( s\\right) }  \\notag\n\\\\\n&&+\\int_{M}\\left\\langle T,d^{\\mathcal{H}}\\xi +\\frac{1}{2\\lambda ^{2}}\\left[\n\\xi ,T\\right] _{\\varphi _{s}}-\\frac{1}{2}\\left[ \\xi ,T\\right] ^{\\left(\ns\\right) }\\right\\rangle ^{\\left( s\\right) }  \\notag \\\\\n&=&\\int_{M}\\left\\langle \\xi ,\\hat{F}+d^{\\mathcal{H}}T+\\frac{1}{2\\lambda ^{2}\n\\left[ T,T\\right] _{\\varphi _{s}}-\\frac{1}{2}\\left[ T,T\\right] ^{\\left(\ns\\right) }\\right\\rangle ^{\\left( s\\right) }  \\notag \\\\\n&=&\\int_{M}\\left\\langle \\xi ,2\\hat{F}+\\frac{1}{2\\lambda ^{2}}\\left[ T,\n\\right] _{\\varphi _{s}}\\right\\rangle ^{\\left( s\\right) },  \\label{dtFs1}\n\\end{eqnarray\nFor the second term in (\\ref{Fsfunctional}), using Lemma \\ref{lemPhibrack2},\nwe obtain \n\\begin{equation}\n-\\frac{1}{6\\lambda ^{2}}\\left. \\frac{d}{dt}\\int_{M}\\left\\langle T,\\left[ T,\n\\right] _{\\varphi _{s}}\\right\\rangle ^{\\left( s\\right) }\\right\\vert _{t=0}=\n\\frac{1}{2\\lambda ^{2}}\\int_{M}\\left\\langle \\xi ,\\left[ T,T\\right] _{\\varphi\n_{s}}\\right\\rangle ^{\\left( s\\right) }.  \\label{dtFs2}\n\\end{equation\nCombining (\\ref{dtFs1}) and (\\ref{dtFs2}), we obtain \n\\begin{equation}\n\\left. \\frac{d}{dt}\\mathcal{F}^{\\left( s\\right) }\\left( \\tilde{\\omega}\\left(\nt\\right) \\right) \\right\\vert _{t=0}=2\\int_{M}\\left\\langle \\xi ,\\hat{F\n\\right\\rangle ^{\\left( s\\right) }.  \\label{dtFs3}\n\\end{equation\nTherefore, we see that the critical points of $\\mathcal{F}^{\\left( s\\right)\n} $ are precisely the connections for which $\\hat{F}=0$. This gives a\ngeneralization of the standard Chern-Simons functional.\n\n\\begin{remark}\nThe condition $\\hat{F}=0$ means that each point, the curvature $F^{\\left(\n\\omega \\right) }$ lies in $\\mathfrak{h}_{s}.$ This is a restriction on the\nLie algebra part of the curvature. Usually instanton conditions on curvature\ngive conditions on the $2$-form part. So what we have here is a different\nkind of condition to an instanton, and there is term for this, coined by\nSpiro Karigiannis - an \\emph{extanton}. As we from Example \\ref{exCx4}, on a\nK\\\"{a}hler manifold, this just corresponds to the Ricci-flat condition.\n\\end{remark}\n\nThe above construction on $3$-manifolds can be extended to an $n\n-dimensional manifold $M$ if we have a closed $\\left( n-3\\right) \n-dimensional form. In that case, similarly as in \\cite{DonaldsonHigherDim},\nconsider the functional \n\\begin{equation}\n\\mathcal{F}^{\\left( s\\right) }\\left( \\omega \\right) =\\int_{M^{n}}\\left(\n\\left\\langle T,\\hat{F}\\right\\rangle ^{\\left( s\\right) }-\\frac{1}{6\\lambda\n^{2}}\\left\\langle T,\\left[ T,T\\right] _{\\varphi _{s}}\\right\\rangle ^{\\left(\ns\\right) }\\right) \\wedge \\psi .\n\\end{equation\nIn this case, the critical points then satisfy \n\\begin{equation}\n\\hat{F}\\wedge \\psi =0.  \\label{Extantonndim}\n\\end{equation\nFor example if $M$ is a $7$-dimensional manifold with a \\emph{co-closed }\nG_{2}$-structure, i.e. $\\psi =\\ast \\varphi $ is closed, then (\\re\n{Extantonndim}) shows that as a $2$-form, $\\hat{F}$ has a vanishing\ncomponent in the $7$-dimensional representation of $G_{2}.$ In contrast,\nDonaldson-Thomas connections \\cite{DonaldsonHigherDim} satisfy $F\\wedge \\psi\n=0$. If $F=\\func{Riem}$, is the Riemann curvature on the frame bundle, then\nequation (\\ref{Extantonndim}) shows that, in local coordinates, \n\\begin{equation}\n\\func{Riem}_{ijkl}\\varphi _{\\ \\alpha }^{ij}\\varphi _{\\ \\ \\beta }^{kl}=0.\n\\label{Extanton7dim}\n\\end{equation\nThe quantity on the left-hand side of (\\ref{Extanton7dim}), is sometimes\ndenoted as $\\func{Ric}^{\\ast }$ \\cit\n{CleytonIvanovClosed,CleytonIvanovCurv,GrigorianFlowSurvey}. The traceless\npart of $\\func{Ric}^{\\ast }$ corresponds to a component of the Riemann\ncurvature that lies in a $27$-dimensional representation of $G_{2}$, with\nanother $27$-dimensional component given by the traceless Ricci tensor \n\\func{Ric}$.\n\nNow consider the functional (\\ref{Fsfunctional}), however now as functional\non sections of $\\mathcal{\\mathring{Q}}$ for a fixed connection $\\omega $, so\nthat now we vary $s$\n\\begin{equation}\n\\mathcal{F}^{\\left( \\omega \\right) }\\left( s\\right) =\\int_{M}\\left\\langle T\n\\hat{F}\\right\\rangle ^{\\left( s\\right) }-\\frac{1}{6\\lambda ^{2}}\\left\\langle\nT,\\left[ T,T\\right] _{\\varphi _{s}}\\right\\rangle ^{\\left( s\\right) },\n\\end{equation\nSuppose we have \n\\begin{subequations\n\\label{sdeforms\n\\begin{eqnarray}\n\\frac{\\partial T^{\\left( s\\left( t\\right) ,\\omega \\right) }}{\\partial t}\n&=&d^{\\mathcal{H}}\\eta \\left( t\\right) -\\left[ T^{\\left( s\\left( t\\right)\n,\\omega \\right) },\\eta \\left( t\\right) \\right] ^{\\left( s\\left( t\\right)\n\\right) } \\\\\n\\frac{\\partial \\hat{F}^{\\left( s\\left( t\\right) ,\\omega \\right) }}{\\partial \n} &=&F\\cdot \\eta \\left( t\\right) -\\left[ \\hat{F}^{\\left( s\\left( t\\right)\n,\\omega \\right) },\\eta \\left( t\\right) \\right] ^{\\left( s\\left( t\\right)\n\\right) }.\n\\end{eqnarray\n\\end{subequations\nfor some $\\eta \\in \\Gamma \\left( \\mathcal{A}\\right) .$ Let us now make\nadditional assumptions:\n\n\\begin{enumerate}\n\\item $\\left[ \\cdot ,\\cdot \\right] _{\\varphi _{s}}=k\\left[ \\cdot ,\\cdot\n\\right] ^{\\left( s\\right) }$\n\n\\item $\\mathbb{L}$ is alternative\n\\end{enumerate}\n\nThe last assumption implies in particular, that the associator is\nskew-symmetric, and moreover, for any $\\alpha ,\\beta ,\\xi ,\\eta \\in \n\\mathfrak{l}^{\\left( s\\right) }$, \n\\begin{equation}\n\\left\\langle a_{s}\\left( \\alpha ,\\beta ,\\xi \\right) ,\\eta \\right\\rangle\n^{\\left( s\\right) }=\\left\\langle \\xi ,a_{s}\\left( \\alpha ,\\beta ,\\eta\n\\right) \\right\\rangle ^{\\left( s\\right) }.\n\\end{equation\nNow, \n\\begin{equation}\n\\mathcal{F}^{\\left( \\omega \\right) }\\left( s\\right) =\\int_{M}\\left\\langle T\n\\hat{F}\\right\\rangle ^{\\left( s\\right) }-\\frac{k}{6\\lambda ^{2}}\\left\\langle\nT,\\left[ T,T\\right] ^{\\left( s\\right) }\\right\\rangle ^{\\left( s\\right) },\n\\end{equation\nand in this case the derivative of $\\mathcal{F}^{\\left( \\omega \\right)\n}\\left( s\\right) $ is \n\\begin{eqnarray}\n\\left. \\frac{d}{dt}\\mathcal{F}^{\\left( \\omega \\right) }\\left( s\\left(\nt\\right) \\right) \\right\\vert _{t=0} &=&\\int_{M}\\left\\langle d^{\\mathcal{H\n}\\eta -\\left[ T,\\eta \\right] ^{\\left( s\\right) },\\hat{F}\\right\\rangle\n^{\\left( s\\right) }+\\int_{M}\\left\\langle T,F\\cdot \\eta -\\left[ \\hat{F},\\eta\n\\right] ^{\\left( s\\right) }\\right\\rangle ^{\\left( s\\right) }  \\notag \\\\\n&&-\\frac{k}{2\\lambda ^{2}}\\int_{M}\\left\\langle d^{\\mathcal{H}}\\eta -\\left[\nT,\\eta \\right] ^{\\left( s\\right) },\\left[ T,T\\right] ^{\\left( s\\right)\n}\\right\\rangle ^{\\left( s\\right) }  \\notag \\\\\n&&-\\frac{k}{6\\lambda ^{2}}\\int_{M}\\left\\langle T,a_{s}\\left( T,T,\\eta\n\\right) \\right\\rangle ^{\\left( s\\right) }.  \\label{dtFoms}\n\\end{eqnarray\nConsider the first two terms in (\\ref{dtFoms}). \n\\begin{subequations\n\\begin{eqnarray}\n\\int_{M}\\left\\langle d^{\\mathcal{H}}\\eta -\\left[ T,\\eta \\right] ^{\\left(\ns\\right) },\\hat{F}\\right\\rangle ^{\\left( s\\right) } &=&\\int_{M}\\left\\langle\n\\eta ,-d^{\\mathcal{H}}\\hat{F}-\\left[ \\hat{F},T\\right] ^{\\left( s\\right)\n}\\right\\rangle ^{\\left( s\\right) } \\\\\n\\int_{M}\\left\\langle T,F\\cdot \\eta -\\left[ \\hat{F},\\eta \\right] ^{\\left(\ns\\right) }\\right\\rangle ^{\\left( s\\right) } &=&\\int_{M}\\left\\langle \\eta \n\\left[ \\hat{F},T\\right] ^{\\left( s\\right) }-F\\cdot T\\right\\rangle ^{\\left(\ns\\right) }.\n\\end{eqnarray\n\\end{subequations\nThe third term in (\\ref{dtFoms}) becomes \n\\begin{eqnarray*}\n\\int_{M}\\left\\langle d^{\\mathcal{H}}\\eta -\\left[ T,\\eta \\right] ^{\\left(\ns\\right) },\\left[ T,T\\right] ^{\\left( s\\right) }\\right\\rangle ^{\\left(\ns\\right) } &=&\\int_{M}\\left\\langle \\eta ,-d^{\\mathcal{H}}\\left[ T,T\\right]\n^{\\left( s\\right) }+\\left[ T,\\left[ T,T\\right] ^{\\left( s\\right) }\\right]\n^{\\left( s\\right) }\\right\\rangle ^{\\left( s\\right) } \\\\\n&=&\\int_{M}\\left\\langle \\eta ,-2\\left[ \\hat{F},T\\right] ^{\\left( s\\right)\n}+a_{s}\\left( T,T,T\\right) \\right\\rangle ^{\\left( s\\right) }.\n\\end{eqnarray*\nThe last term in (\\ref{dtFoms}) is \n\\begin{equation*}\n\\int_{M}\\left\\langle T,a_{s}\\left( T,T,\\eta \\right) \\right\\rangle ^{\\left(\ns\\right) }=\\int_{M}\\left\\langle \\eta ,a_{s}\\left( T,T,T\\right) \\right\\rangle\n^{\\left( s\\right) }.\n\\end{equation*\nOverall, since $a_{s}\\left( T,T,T\\right) =\\left[ T,\\left[ T,T\\right]\n^{\\left( s\\right) }\\right] ^{\\left( s\\right) }$, \n\\begin{eqnarray}\n\\left. \\frac{d}{dt}\\mathcal{F}^{\\left( \\omega \\right) }\\left( s\\left(\nt\\right) \\right) \\right\\vert _{t=0} &=&-\\int_{M}\\left\\langle \\eta ,d^\n\\mathcal{H}}\\hat{F}+F\\cdot T-\\frac{k}{\\lambda ^{2}}\\left[ \\hat{F},T\\right]\n^{\\left( s\\right) }\\right\\rangle ^{\\left( s\\right) } \\\\\n&&-\\int_{M}\\,\\left\\langle \\eta ,\\frac{2k}{3\\lambda ^{2}}\\left[ T,\\left[ T,\n\\right] ^{\\left( s\\right) }\\right] ^{\\left( s\\right) }\\right\\rangle ^{\\left(\ns\\right) }.  \\notag\n\\end{eqnarray}\n\nFrom the Bianchi identity (\\ref{Bianchi}), \n\\begin{equation*}\nF\\cdot T=d^{\\mathcal{H}}\\hat{F}+\\left[ \\hat{F},T\\right] ^{\\left( s\\right) },\n\\end{equation*\nand thus, \n\\begin{eqnarray}\n\\left. \\frac{d}{dt}\\mathcal{F}^{\\left( \\omega \\right) }\\left( s\\left(\nt\\right) \\right) \\right\\vert _{t=0} &=&-\\int_{M}\\left\\langle \\eta ,2d^\n\\mathcal{H}}\\hat{F}+\\left( 1-\\frac{k}{\\lambda ^{2}}\\right) \\left[ \\hat{F},\n\\right] ^{\\left( s\\right) }\\right\\rangle ^{\\left( s\\right) } \\\\\n&&-\\int_{M}\\,\\left\\langle \\eta ,\\frac{2k}{3\\lambda ^{2}}\\left[ T,\\left[ T,\n\\right] ^{\\left( s\\right) }\\right] ^{\\left( s\\right) }\\right\\rangle ^{\\left(\ns\\right) }.\n\\end{eqnarray\nThus, the critical points with respect to deformations of $s$ satisf\n\\begin{equation}\nd^{\\mathcal{H}}\\hat{F}+\\left( \\frac{1}{2}-\\frac{k}{2\\lambda ^{2}}\\right)\n\\left[ \\hat{F},T\\right] ^{\\left( s\\right) }+\\frac{k}{3\\lambda ^{2}}\\left[ T\n\\left[ T,T\\right] ^{\\left( s\\right) }\\right] ^{\\left( s\\right) }=0.\n\\label{SecondCrit}\n\\end{equation}\n\n\\begin{example}\nIn the case when $\\mathbb{L}$ is a Lie group, $a_{s}=0$ and $k=\\lambda =1$,\nso we just obtain $d^{\\mathcal{H}}\\hat{F}=0$, which is of course the Bianchi\nidentity. This shows that we just have a reduction from a $\\Psi ^{R}\\left( \n\\mathbb{L}\\right) $ connection to an $\\mathbb{L}$-connection. In the case of \n$\\mathbb{L}$ being the loop of unit octonions, we know $\\lambda =\\frac{3}{8}$\nand $k=3\\lambda ^{3}=\\frac{81}{512}$ so (\\ref{SecondCrit}) become\n\\begin{equation}\nd^{\\mathcal{H}}\\hat{F}-\\frac{1}{16}\\left[ \\hat{F},T\\right] ^{\\left( s\\right)\n}+\\frac{3}{8}\\left[ T,\\left[ T,T\\right] ^{\\left( s\\right) }\\right] ^{\\left(\ns\\right) }=0.\n\\end{equation\nThe significance of this condition is not immediately clear.\n\\end{example}\n\nHowever combining the two variations, we find that critical points over \n\\left( s,\\omega \\right) $ satisfy \n\\begin{equation*}\n\\left\\{ \n\\begin{array}{c}\n\\hat{F}=0 \\\\ \n\\left[ T,T,T\\right] ^{\\left( s\\right) }=\n\\end{array\n\\right. .\n\\end{equation*}\n\n\\begin{remark}\nIt will be the subject of further work to understand the significance of\nthis Chern-Simons type functional $\\mathcal{F}$. In particular, given the\nnon-trivial $3$-form $\\left[ T,\\left[ T,T\\right] ^{\\left( s\\right) }\\right]\n^{\\left( s\\right) }$, there may be additional possibilities for similar\nhigher-dimensional functionals. The functional $\\mathcal{F}$ is invariant\nunder simultaneous gauge transformations of $\\left( s,\\omega \\right) ,$ but\nnot the individual ones. For the standard Chern-Simons functional in 3\ndimensions, the lack of gauge invariance causes it to be multi-valued, with\nonly the exponentiated action functional becomes truly gauge-invariant. It\nwill be interesting to see if there are any analogous properties in this\ncase.\n\\end{remark}\n\nIn the context of $G_{2}$-structures, another functional has been considered\nin several papers \\cite{Bagaglini2,DGKisoflow,GrigorianOctobundle,\nGrigorianIsoflow,SaEarpLoubeau}, namely the $L_{2}$-norm of the torsion,\nconsidered as functional on the space of isometric $G_{2}$-structures, i.e. \nG_{2}$-structures that correspond to the same metric. In the context of loop\nstructures we may define a similar functional. Given a compact Riemannian\nmanifold $\\left( M,g\\right) $ and a fixed connection $\\omega $ on $\\mathcal{\n}$, for any section $s\\in \\Gamma \\left( \\mathcal{\\mathring{Q}}\\right) $ let \nT^{\\left( s\\right) }$ be the torsion of $s$ with respect to $\\omega .$ Then\ndefine the energy functional on $\\Gamma \\left( \\mathcal{\\mathring{Q}}\\right) \n$ given by: \n\\begin{equation}\n\\mathcal{E}\\left( s\\right) =\\int_{M}\\left\\langle T^{\\left( s\\right) },\\ast\nT^{\\left( s\\right) }\\right\\rangle ^{\\left( s\\right) },  \\label{Efunc}\n\\end{equation\nwhere the wedge product is assumed. With respect to deformations of $s$\ngiven by (\\ref{Aevol}) and the corresponding deformation of $T$ given by \n\\ref{sdeforms}) we have \n\\begin{eqnarray}\n\\left. \\frac{d}{dt}\\mathcal{E}\\left( s_{t}\\right) \\right\\vert _{t=0}\n&=&2\\int_{M}\\left\\langle d^{\\mathcal{H}}\\eta -\\left[ T^{\\left( s\\right)\n},\\eta \\right] ^{\\left( s\\right) },\\ast T^{\\left( s\\right) }\\right\\rangle\n^{\\left( s\\right) }  \\notag \\\\\n&=&-2\\int_{M}\\left\\langle \\eta ,d^{\\mathcal{H}}\\ast T^{\\left( s\\right) }\n\\left[ T^{\\left( s\\right) },\\ast T^{\\left( s\\right) }\\right] ^{\\left(\ns\\right) }\\right\\rangle ^{\\left( s\\right) }  \\notag \\\\\n&=&-2\\int_{M}\\left\\langle \\eta ,d^{\\mathcal{H}}\\ast T^{\\left( s\\right)\n}\\right\\rangle ^{\\left( s\\right) },  \\label{Edeform}\n\\end{eqnarray\nwhere $\\left[ T^{\\left( s\\right) },\\ast T^{\\left( s\\right) }\\right] ^{\\left(\ns\\right) }=0$ due to symmetry considerations. Thus the critical points of \n\\mathcal{E}$ satisfy \n\\begin{equation}\n\\left( d^{\\mathcal{H}}\\right) ^{\\ast }T^{\\left( s\\right) }=0,  \\label{divT0}\n\\end{equation\nwhich is precisely the analog of the \\textquotedblleft divergence-free\ntorsion\\textquotedblright\\ condition in \\cit\n{Bagaglini2,DGKisoflow,GrigorianOctobundle, GrigorianIsoflow,SaEarpLoubeau}.\nAlso, similarly as in \\cite{SaEarpLoubeau}, if we assume $\\mathcal{P}$ is\ncompact, the functional $\\mathcal{E}$ may be related to the equivariant\nDirichlet energy functional for maps from $\\mathcal{P}$ to $\\mathbb\n\\mathring{L}}$. Given a metric $\\left\\langle \\cdot ,\\cdot \\right\\rangle\n^{\\left( s\\right) }$ on $\\mathfrak{l}$, we may extend it to a metric on all\nof $\\mathbb{L}$ via right translations: $\\left\\langle \\cdot ,\\cdot\n\\right\\rangle _{p}^{\\left( s\\right) }=\\left\\langle \\left( R_{p}\\right)\n_{\\ast }^{-1}\\cdot ,\\left( R_{p}\\right) _{\\ast }^{-1}\\cdot \\right\\rangle\n^{\\left( s\\right) }.$ Then, the Dirichlet energy functional on \\emph\nequivariant} maps from $\\mathcal{P}$ to $\\mathbb{\\mathring{L}}$ is given by \n\\begin{equation}\n\\mathcal{D}\\left( s\\right) =\\int_{\\mathcal{P}}\\left\\vert ds\\right\\vert\n^{2}=\\int_{\\mathcal{P}}\\left\\vert \\theta _{s}\\right\\vert ^{2},\n\\label{dirichletE}\n\\end{equation\nwhere we endow $T\\mathcal{P}$ with a metric such that the decomposition $\n\\mathcal{P=HP\\oplus VP}$ is orthogonal with respect to it, and moreover such\nthat it is compatible with the metrics on $M$ and $\\Psi $. Then, using (\\re\n{stheta}) \n\\begin{equation}\n\\mathcal{D}\\left( s\\right) =\\int_{\\mathcal{P}}\\left\\vert T^{\\left( s\\right)\n}\\right\\vert ^{2}+\\int_{\\mathcal{P}}\\left\\vert \\hat{\\omega}^{\\left( s\\right)\n}\\right\\vert ^{2}\n\\end{equation\nNote that given an orthogonal basis $\\left\\{ X_{i}\\right\\} $ on $\\mathfrak{p}\n$, $\\left\\vert \\hat{\\omega}^{\\left( s\\right) }\\right\\vert ^{2}=\\left\\vert \n\\hat{\\omega}^{\\left( s\\right) }\\left( \\sigma \\left( X_{i}\\right) \\right)\n\\right\\vert ^{2}=\\left\\vert \\hat{X}_{i}\\right\\vert ^{2}=\\lambda _{s}\\dim \n\\mathfrak{l}.$ With our previous assumptions, $\\lambda _{s}=\\lambda $ - does\nnot depend on $s$, so we have \n\\begin{equation*}\n\\mathcal{D}\\left( s\\right) =a\\mathcal{E}\\left( s\\right) +b\n\\end{equation*\nwhere $a=\\func{Vol}\\left( \\Psi \\right) $ and $b=\\lambda \\left( \\dim \\mathbb{\n}\\right) $ $\\func{Vol}\\left( \\mathcal{P}\\right) .$ Hence, the critical\npoints of $\\mathcal{E}\\left( s\\right) $ are precisely the critical points of \n$\\mathcal{D}\\left( s\\right) $ with respect to deformations through\nequivariant maps, i.e. equivariant harmonic maps. So indeed, to understand\nthe properties of these critical points, a rigorous equivariant harmonic map\ntheory is required, as initiated in \\cite{SaEarpLoubeau}.\n\n\\section{Concluding remarks}\n\n\\setcounter{equation}{0}\\label{sectConclusion}Given a smooth loop $\\mathbb{L}\n$ with tangent algebra $\\mathfrak{l}$ and a group $\\Psi $ that acts smoothly\non $\\mathbb{L}$ via pseudoautomorphism pairs, we have defined the concept of\na loop bundle structure $\\left( \\mathbb{L},\\Psi ,\\mathcal{P},s\\right) $ for\na principal $\\Psi $-bundle and a corresponding equivariant $\\mathbb\n\\mathring{L}}$-valued map $s$, that also defines a section of the\ncorresponding associated bundle. If we moreover have a connection $\\omega $\non $\\mathcal{P}$, then horizontal component of the Darboux derivative of $s$\ndefines an $\\mathfrak{l}$-valued $1$-form $T^{\\left( s,\\omega \\right) }$,\nwhich we called the torsion. This object $T^{\\left( s,\\omega \\right) }$ then\nsatisfies a structural equation based on the loop Maurer-Cartan equation and\ngives rise to an $\\mathfrak{l}$-valued component of the curvature $\\hat{F\n^{\\left( s,\\omega \\right) }.$ Overall, there are several possible directions\nto further this non-associative theory.\n\n\\begin{enumerate}\n\\item From a more algebraic perspective it would be interesting to construct\nadditional examples of smooth loops, in particular those that are not\nMoufang and possibly are not even $G$-loops in order to more concretely\nstudy the corresponding bundles in those situations. In fact, it may not\neven be necessary to have a full loop structure - it may be sufficient to\njust have a right loop structure, so that division is possible only on the\nright. Left division was used rarely, and it may be possible to build up a\nfull theory without needing it. New examples of loops may give rise to new\ngeometric structures.\n\n\\item In Lie theory, the Maurer-Cartan equation plays a central role. As\nwe've seen there is an analog in smooth loop theory as well. A better\nunderstanding of this equation is needed. The standard Maurer-Cartan\nequation is closely related to the concept of integrability, but it is not\nclear how to interpret the non-associative version.\n\n\\item In defining the loop bundle structure, we generally have assumed that\nthe map $s$ is globally defined. However, this may place strict topological\nrestrictions. It may be reasonable to allow $s$ to be defined only locally.\nThis would give more flexibility, but it would need to be checked carefully\nwhether other related quantities are well-defined.\n\n\\item We have defined a functional of Chern-Simons type in Section \\re\n{sectVar}. There are further properties that need to be investigated. For\nexample, is it possible to use the associator to define reasonable\nfunctionals on higher-dimensional manifolds? If the section $s$ is defined\nonly locally, are these functionals well-defined? Finally, do these\nfunctionals have any topological meaning?\n\n\\item In $G_{2}$-geometry, significant progress has been made in \\cit\n{Bagaglini2,DGKisoflow,GrigorianOctobundle, GrigorianIsoflow,SaEarpLoubeau}\nregarding the existence of critical points of the energy functional (\\re\n{Efunc}) via a heat flow approach. However, it is likely that a more direct\napproach, similar to Uhlenbeck's existence result for the Coulomb gauge \\cit\n{UhlenbeckConnection}, could also be used. This would give existence of a\npreferred section $s$ for a given connection or conversely, a preferred\nconnection in a gauge class for a fixed section $s$.\n\\end{enumerate}\n\nOverall, the framework presented in this paper may give an impetus to the\ndevelopment of a larger theory of \\textquotedblleft nonassociative\ngeometry\\textquotedblright .\n\n\n\\section*{Abstract (Not appropriate in this style!)}%\n   \\else \\small \n   \\begin{center}{\\bf Abstract\\vspace{-.5em}\\vspace{\\z@}}\\end{center}%\n   \\quotation \n   \\fi\n  }%\n }{%\n }%\n\\@ifundefined{endabstract}{\\def\\endabstract\n  {\\if@twocolumn\\else\\endquotation\\fi}}{}%\n\\@ifundefined{maketitle}{\\def\\maketitle#1{}}{}%\n\\@ifundefined{affiliation}{\\def\\affiliation#1{}}{}%\n\\@ifundefined{proof}{\\def\\proof{\\noindent{\\bfseries Proof. }}}{}%\n\\@ifundefined{endproof}{\\def\\endproof{\\mbox{\\ 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\\ifnum\\mathgroup=\\symbold\n           \\mathchoice{\\mbox{\\boldmath$\\displaystyle\\mathchar\"#1#2#3#4$}}%\n                      {\\mbox{\\boldmath$\\textstyle\\mathchar\"#1#2#3#4$}}%\n                      {\\mbox{\\boldmath$\\scriptstyle\\mathchar\"#1#2#3#4$}}%\n                      {\\mbox{\\boldmath$\\scriptscriptstyle\\mathchar\"#1#2#3#4$}}%\n        \\else\n           \\mathchar\"#1#2#3#\n        \\fi \n    \\else \n        \\FindBoldGroup\n        \\ifnum\\mathgroup=\\theboldgroup\n           \\mathchoice{\\mbox{\\boldmath$\\displaystyle\\mathchar\"#1#2#3#4$}}%\n                      {\\mbox{\\boldmath$\\textstyle\\mathchar\"#1#2#3#4$}}%\n                      {\\mbox{\\boldmath$\\scriptstyle\\mathchar\"#1#2#3#4$}}%\n                      {\\mbox{\\boldmath$\\scriptscriptstyle\\mathchar\"#1#2#3#4$}}%\n        \\else\n           \\mathchar\"#1#2#3#\n        \\fi     \t    \n\t  \\fi}\n\n\\newif\\ifGreekBold  \\GreekBoldfalse\n\\let\\SAVEPBF=\\pbf\n\\def\\pbf{\\GreekBoldtrue\\SAVEPBF}%\n\n\\@ifundefined{theorem}{\\newtheorem{theorem}{Theorem}}{}\n\\@ifundefined{lemma}{\\newtheorem{lemma}[theorem]{Lemma}}{}\n\\@ifundefined{corollary}{\\newtheorem{corollary}[theorem]{Corollary}}{}\n\\@ifundefined{conjecture}{\\newtheorem{conjecture}[theorem]{Conjecture}}{}\n\\@ifundefined{proposition}{\\newtheorem{proposition}[theorem]{Proposition}}{}\n\\@ifundefined{axiom}{\\newtheorem{axiom}{Axiom}}{}\n\\@ifundefined{remark}{\\newtheorem{remark}{Remark}}{}\n\\@ifundefined{example}{\\newtheorem{example}{Example}}{}\n\\@ifundefined{exercise}{\\newtheorem{exercise}{Exercise}}{}\n\\@ifundefined{definition}{\\newtheorem{definition}{Definition}}{}\n\n\n\\@ifundefined{mathletters}{%\n \n  \\newcounter{equationnumber}  \n  \\def\\mathletters{%\n     \\addtocounter{equation}{1}\n     \\edef\\@currentlabel{\\arabic{equation}}%\n     \\setcounter{equationnumber}{\\c@equation}\n     \\setcounter{equation}{0}%\n     \\edef\\arabic{equation}{\\@currentlabel\\noexpand\\alph{equation}}%\n  }\n  \\def\\endmathletters{%\n     \\setcounter{equation}{\\value{equationnumber}}%\n  }\n}{}\n\n\\@ifundefined{BibTeX}{%\n    \\def\\BibTeX{{\\rm B\\kern-.05em{\\sc i\\kern-.025em b}\\kern-.08em\n                 T\\kern-.1667em\\lower.7ex\\hbox{E}\\kern-.125emX}}}{}%\n\\@ifundefined{AmS}%\n    {\\def\\AmS{{\\protect\\usefont{OMS}{cmsy}{m}{n}%\n                A\\kern-.1667em\\lower.5ex\\hbox{M}\\kern-.125emS}}}{}%\n\\@ifundefined{AmSTeX}{\\def\\AmSTeX{\\protect\\AmS-\\protect\\TeX\\@}}{}%\n\n\\def\\@@eqncr{\\let\\@tempa\\relax\n    \\ifcase\\@eqcnt \\def\\@tempa{& & &}\\or \\def\\@tempa{& &}%\n      \\else \\def\\@tempa{&}\\fi\n     \\@tempa\n     \\if@eqnsw\n        \\iftag@\n           \\@taggnum\n        \\else\n           \\@eqnnum\\stepcounter{equation}%\n        \\fi\n     \\fi\n     \\global\\@ifnextchar*{\\@tagstar}{\\@tag}@false\n     \\global\\@eqnswtrue\n     \\global\\@eqcnt\\z@\\cr}\n\n\n\\def\\@ifnextchar*{\\@TCItagstar}{\\@TCItag}{\\@ifnextchar*{\\@TCItagstar}{\\@TCItag}}\n\\def\\@TCItag#1{%\n    \\global\\@ifnextchar*{\\@tagstar}{\\@tag}@true\n    \\global\\def\\@taggnum{(#1)}%\n    \\global\\def\\@currentlabel{#1}}\n\\def\\@TCItagstar*#1{%\n    \\global\\@ifnextchar*{\\@tagstar}{\\@tag}@true\n    \\global\\def\\@taggnum{#1}%\n    \\global\\def\\@currentlabel{#1}}\n\\def\\QATOP#1#2{{#1 \\atop #2}}%\n\\def\\QTATOP#1#2{{\\textstyle {#1 \\atop #2}}}%\n\\def\\QDATOP#1#2{{\\displaystyle {#1 \\atop #2}}}%\n\\def\\QABOVE#1#2#3{{#2 \\above#1 #3}}%\n\\def\\QTABOVE#1#2#3{{\\textstyle {#2 \\above#1 #3}}}%\n\\def\\QDABOVE#1#2#3{{\\displaystyle {#2 \\above#1 #3}}}%\n\\def\\QOVERD#1#2#3#4{{#3 \\overwithdelims#1#2 #4}}%\n\\def\\QTOVERD#1#2#3#4{{\\textstyle {#3 \\overwithdelims#1#2 #4}}}%\n\\def\\QDOVERD#1#2#3#4{{\\displaystyle {#3 \\overwithdelims#1#2 #4}}}%\n\\def\\QATOPD#1#2#3#4{{#3 \\atopwithdelims#1#2 #4}}%\n\\def\\QTATOPD#1#2#3#4{{\\textstyle {#3 \\atopwithdelims#1#2 #4}}}%\n\\def\\QDATOPD#1#2#3#4{{\\displaystyle {#3 \\atopwithdelims#1#2 #4}}}%\n\\def\\QABOVED#1#2#3#4#5{{#4 \\abovewithdelims#1#2#3 #5}}%\n\\def\\QTABOVED#1#2#3#4#5{{\\textstyle \n   {#4 \\abovewithdelims#1#2#3 #5}}}%\n\\def\\QDABOVED#1#2#3#4#5{{\\displaystyle \n   {#4 \\abovewithdelims#1#2#3 #5}}}%\n\\def\\tint{\\mathop{\\textstyle \\int}}%\n\\def\\tiint{\\mathop{\\textstyle \\iint }}%\n\\def\\tiiint{\\mathop{\\textstyle \\iiint }}%\n\\def\\tiiiint{\\mathop{\\textstyle \\iiiint }}%\n\\def\\tidotsint{\\mathop{\\textstyle \\idotsint }}%\n\\def\\toint{\\mathop{\\textstyle \\oint}}%\n\\def\\tsum{\\mathop{\\textstyle \\sum }}%\n\\def\\tprod{\\mathop{\\textstyle \\prod }}%\n\\def\\tbigcap{\\mathop{\\textstyle \\bigcap }}%\n\\def\\tbigwedge{\\mathop{\\textstyle \\bigwedge }}%\n\\def\\tbigoplus{\\mathop{\\textstyle \\bigoplus }}%\n\\def\\tbigodot{\\mathop{\\textstyle \\bigodot }}%\n\\def\\tbigsqcup{\\mathop{\\textstyle \\bigsqcup }}%\n\\def\\tcoprod{\\mathop{\\textstyle \\coprod }}%\n\\def\\tbigcup{\\mathop{\\textstyle \\bigcup }}%\n\\def\\tbigvee{\\mathop{\\textstyle \\bigvee }}%\n\\def\\tbigotimes{\\mathop{\\textstyle \\bigotimes }}%\n\\def\\tbiguplus{\\mathop{\\textstyle \\biguplus }}%\n\\def\\dint{\\mathop{\\displaystyle \\int}}%\n\\def\\diint{\\mathop{\\displaystyle \\iint}}%\n\\def\\diiint{\\mathop{\\displaystyle \\iiint}}%\n\\def\\diiiint{\\mathop{\\displaystyle \\iiiint }}%\n\\def\\didotsint{\\mathop{\\displaystyle \\idotsint }}%\n\\def\\doint{\\mathop{\\displaystyle \\oint}}%\n\\def\\dsum{\\mathop{\\displaystyle \\sum }}%\n\\def\\dprod{\\mathop{\\displaystyle \\prod }}%\n\\def\\dbigcap{\\mathop{\\displaystyle \\bigcap }}%\n\\def\\dbigwedge{\\mathop{\\displaystyle \\bigwedge }}%\n\\def\\dbigoplus{\\mathop{\\displaystyle \\bigoplus }}%\n\\def\\dbigodot{\\mathop{\\displaystyle \\bigodot }}%\n\\def\\dbigsqcup{\\mathop{\\displaystyle \\bigsqcup }}%\n\\def\\dcoprod{\\mathop{\\displaystyle \\coprod }}%\n\\def\\dbigcup{\\mathop{\\displaystyle \\bigcup }}%\n\\def\\dbigvee{\\mathop{\\displaystyle \\bigvee }}%\n\\def\\dbigotimes{\\mathop{\\displaystyle \\bigotimes }}%\n\\def\\dbiguplus{\\mathop{\\displaystyle \\biguplus }}%\n\n\\if@compatibility\\else\n \n  \\RequirePackage{amsmath}\n\\fi\n\n\\def\\makeatother\\endinput{\\makeatother\\endinput}\n\n\\bgroup\n\\ifx\\ds@amstex\\relax\n   \\message{amstex already loaded}\\aftergroup\\makeatother\\endinput\n\\else\n   \\@ifpackageloaded{amsmath}%\n      {\\if@compatibility\\message{amsmath already loaded}\\fi\\aftergroup\\makeatother\\endinput}\n      {}\n   \\@ifpackageloaded{amstex}%\n      {\\if@compatibility\\message{amstex already loaded}\\fi\\aftergroup\\makeatother\\endinput}\n      {}\n   \\@ifpackageloaded{amsgen}%\n      {\\if@compatibility\\message{amsgen already loaded}\\fi\\aftergroup\\makeatother\\endinput}\n      {}\n\\fi\n\\egroup\n\n\n\n\\typeout{TCILATEX defining AMS-like constructs in LaTeX 2.09 COMPATIBILITY MODE}\n\\let\\DOTSI\\relax\n\\def\\RIfM@{\\relax\\ifmmode}%\n\\def\\FN@{\\futurelet\\next}%\n\\newcount\\intno@\n\\def\\iint{\\DOTSI\\intno@\\tw@\\FN@\\ints@}%\n\\def\\iiint{\\DOTSI\\intno@\\thr@@\\FN@\\ints@}%\n\\def\\iiiint{\\DOTSI\\intno@4 \\FN@\\ints@}%\n\\def\\idotsint{\\DOTSI\\intno@\\z@\\FN@\\ints@}%\n\\def\\ints@{\\findlimits@\\ints@@}%\n\\newif\\iflimtoken@\n\\newif\\iflimits@\n\\def\\findlimits@{\\limtoken@true\\ifx\\next\\limits\\limits@true\n \\else\\ifx\\next\\nolimits\\limits@false\\else\n \\limtoken@false\\ifx\\ilimits@\\nolimits\\limits@false\\else\n \\ifinner\\limits@false\\else\\limits@true\\fi\\fi\\fi\\fi}%\n\\def\\multint@{\\int\\ifnum\\intno@=\\z@\\intdots@                         \n \\else\\intkern@\\fi                                                   \n \\ifnum\\intno@>\\tw@\\int\\intkern@\\fi                                  \n \\ifnum\\intno@>\\thr@@\\int\\intkern@\\fi                                \n \\int\n\\def\\multintlimits@{\\intop\\ifnum\\intno@=\\z@\\intdots@\\else\\intkern@\\fi\n \\ifnum\\intno@>\\tw@\\intop\\intkern@\\fi\n \\ifnum\\intno@>\\thr@@\\intop\\intkern@\\fi\\intop}%\n\\def\\intic@{%\n    \\mathchoice{\\hskip.5em}{\\hskip.4em}{\\hskip.4em}{\\hskip.4em}}%\n\\def\\negintic@{\\mathchoice\n {\\hskip-.5em}{\\hskip-.4em}{\\hskip-.4em}{\\hskip-.4em}}%\n\\def\\ints@@{\\iflimtoken@                                             \n \\def\\ints@@@{\\iflimits@\\negintic@\n   \\mathop{\\intic@\\multintlimits@}\\limits                            \n  \\else\\multint@\\nolimits\\fi                                         \n  \\eat@\n \\else                                                               \n \\def\\ints@@@{\\iflimits@\\negintic@\n  \\mathop{\\intic@\\multintlimits@}\\limits\\else\n  \\multint@\\nolimits\\fi}\\fi\\ints@@@}%\n\\def\\intkern@{\\mathchoice{\\!\\!\\!}{\\!\\!}{\\!\\!}{\\!\\!}}%\n\\def\\plaincdots@{\\mathinner{\\cdotp\\cdotp\\cdotp}}%\n\\def\\intdots@{\\mathchoice{\\plaincdots@}%\n {{\\cdotp}\\mkern1.5mu{\\cdotp}\\mkern1.5mu{\\cdotp}}%\n {{\\cdotp}\\mkern1mu{\\cdotp}\\mkern1mu{\\cdotp}}%\n {{\\cdotp}\\mkern1mu{\\cdotp}\\mkern1mu{\\cdotp}}}%\n\\def\\RIfM@{\\relax\\protect\\ifmmode}\n\\def\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi{\\RIfM@\\expandafter\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi@\\else\\expandafter\\mbox\\fi}\n\\let\\nfss@text\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi\n\\def\\RIfM@\\expandafter\\text@\\else\\expandafter\\mbox\\fi@#1{\\mathchoice\n   {\\textdef@\\displaystyle\\f@size{#1}}%\n   {\\textdef@\\textstyle\\tf@size{\\firstchoice@false #1}}%\n   {\\textdef@\\textstyle\\sf@size{\\firstchoice@false #1}}%\n   {\\textdef@\\textstyle \\ssf@size{\\firstchoice@false #1}}%\n   \\glb@settings}\n\n\\def\\textdef@#1#2#3{\\hbox{{%\n                    \\everymath{#1}%\n                    \\let\\f@size#2\\selectfont\n                    #3}}}\n\\newif\\iffirstchoice@\n\\firstchoice@true\n\\def\\Let@{\\relax\\iffalse{\\fi\\let\\\\=\\cr\\iffalse}\\fi}%\n\\def\\vspace@{\\def\\vspace##1{\\crcr\\noalign{\\vskip##1\\relax}}}%\n\\def\\multilimits@{\\bgroup\\vspace@\\Let@\n \\baselineskip\\fontdimen10 \\scriptfont\\tw@\n \\advance\\baselineskip\\fontdimen12 \\scriptfont\\tw@\n \\lineskip\\thr@@\\fontdimen8 \\scriptfont\\thr@@\n \\lineskiplimit\\lineskip\n \\vbox\\bgroup\\ialign\\bgroup\\hfil$\\m@th\\scriptstyle{##}$\\hfil\\crcr}%\n\\def\\Sb{_\\multilimits@}%\n\\def\\endSb{\\crcr\\egroup\\egroup\\egroup}%\n\\def\\Sp{^\\multilimits@}%\n\\let\\endSp\\endSb\n\\newdimen\\ex@\n\\ex@.2326ex\n\\def\\rightarrowfill@#1{$#1\\m@th\\mathord-\\mkern-6mu\\cleaders\n \\hbox{$#1\\mkern-2mu\\mathord-\\mkern-2mu$}\\hfill\n \\mkern-6mu\\mathord\\rightarrow$}%\n\\def\\leftarrowfill@#1{$#1\\m@th\\mathord\\leftarrow\\mkern-6mu\\cleaders\n \\hbox{$#1\\mkern-2mu\\mathord-\\mkern-2mu$}\\hfill\\mkern-6mu\\mathord-$}%\n\\def\\leftrightarrowfill@#1{$#1\\m@th\\mathord\\leftarrow\n\\mkern-6mu\\cleaders\n \\hbox{$#1\\mkern-2mu\\mathord-\\mkern-2mu$}\\hfill\n \\mkern-6mu\\mathord\\rightarrow$}%\n\\def\\overrightarrow{\\mathpalette\\overrightarrow@}%\n\\def\\overrightarrow@#1#2{\\vbox{\\ialign{##\\crcr\\rightarrowfill@#1\\crcr\n \\noalign{\\kern-\\ex@\\nointerlineskip}$\\m@th\\hfil#1#2\\hfil$\\crcr}}}%\n\\let\\overarrow\\overrightarrow\n\\def\\overleftarrow{\\mathpalette\\overleftarrow@}%\n\\def\\overleftarrow@#1#2{\\vbox{\\ialign{##\\crcr\\leftarrowfill@#1\\crcr\n \\noalign{\\kern-\\ex@\\nointerlineskip}$\\m@th\\hfil#1#2\\hfil$\\crcr}}}%\n\\def\\overleftrightarrow{\\mathpalette\\overleftrightarrow@}%\n\\def\\overleftrightarrow@#1#2{\\vbox{\\ialign{##\\crcr\n   \\leftrightarrowfill@#1\\crcr\n \\noalign{\\kern-\\ex@\\nointerlineskip}$\\m@th\\hfil#1#2\\hfil$\\crcr}}}%\n\\def\\underrightarrow{\\mathpalette\\underrightarrow@}%\n\\def\\underrightarrow@#1#2{\\vtop{\\ialign{##\\crcr$\\m@th\\hfil#1#2\\hfil\n  $\\crcr\\noalign{\\nointerlineskip}\\rightarrowfill@#1\\crcr}}}%\n\\let\\underarrow\\underrightarrow\n\\def\\underleftarrow{\\mathpalette\\underleftarrow@}%\n\\def\\underleftarrow@#1#2{\\vtop{\\ialign{##\\crcr$\\m@th\\hfil#1#2\\hfil\n  $\\crcr\\noalign{\\nointerlineskip}\\leftarrowfill@#1\\crcr}}}%\n\\def\\underleftrightarrow{\\mathpalette\\underleftrightarrow@}%\n\\def\\underleftrightarrow@#1#2{\\vtop{\\ialign{##\\crcr$\\m@th\n  \\hfil#1#2\\hfil$\\crcr\n \\noalign{\\nointerlineskip}\\leftrightarrowfill@#1\\crcr}}}%\n\n\\def\\qopnamewl@#1{\\mathop{\\operator@font#1}\\nlimits@}\n\\let\\nlimits@\\displaylimits\n\\def\\setboxz@h{\\setbox\\z@\\hbox}\n\n\n\\def\\varlim@#1#2{\\mathop{\\vtop{\\ialign{##\\crcr\n \\hfil$#1\\m@th\\operator@font lim$\\hfil\\crcr\n \\noalign{\\nointerlineskip}#2#1\\crcr\n \\noalign{\\nointerlineskip\\kern-\\ex@}\\crcr}}}}\n\n \\def\\rightarrowfill@#1{\\m@th\\setboxz@h{$#1-$}\\ht\\z@\\z@\n  $#1\\copy\\z@\\mkern-6mu\\cleaders\n  \\hbox{$#1\\mkern-2mu\\box\\z@\\mkern-2mu$}\\hfill\n  \\mkern-6mu\\mathord\\rightarrow$}\n\\def\\leftarrowfill@#1{\\m@th\\setboxz@h{$#1-$}\\ht\\z@\\z@\n  $#1\\mathord\\leftarrow\\mkern-6mu\\cleaders\n  \\hbox{$#1\\mkern-2mu\\copy\\z@\\mkern-2mu$}\\hfill\n  \\mkern-6mu\\box\\z@$}\n\n\n\\def\\qopnamewl@{proj\\,lim}{\\qopnamewl@{proj\\,lim}}\n\\def\\qopnamewl@{inj\\,lim}{\\qopnamewl@{inj\\,lim}}\n\\def\\mathpalette\\varlim@\\rightarrowfill@{\\mathpalette\\varlim@\\rightarrowfill@}\n\\def\\mathpalette\\varlim@\\leftarrowfill@{\\mathpalette\\varlim@\\leftarrowfill@}\n\\def\\mathpalette\\varliminf@{}{\\mathpalette\\mathpalette\\varliminf@{}@{}}\n\\def\\mathpalette\\varliminf@{}@#1{\\mathop{\\underline{\\vrule\\@depth.2\\ex@\\@width\\z@\n   \\hbox{$#1\\m@th\\operator@font lim$}}}}\n\\def\\mathpalette\\varlimsup@{}{\\mathpalette\\mathpalette\\varlimsup@{}@{}}\n\\def\\mathpalette\\varlimsup@{}@#1{\\mathop{\\overline\n  {\\hbox{$#1\\m@th\\operator@font lim$}}}}\n\n\\def\\stackunder#1#2{\\mathrel{\\mathop{#2}\\limits_{#1}}}%\n\\begingroup \\catcode `|=0 \\catcode `[= 1\n\\catcode`]=2 \\catcode `\\{=12 \\catcode `\\}=12\n\\catcode`\\\\=12 \n|gdef|@alignverbatim#1\\end{align}[#1|end[align]]\n|gdef|@salignverbatim#1\\end{align*}[#1|end[align*]]\n\n|gdef|@alignatverbatim#1\\end{alignat}[#1|end[alignat]]\n|gdef|@salignatverbatim#1\\end{alignat*}[#1|end[alignat*]]\n\n|gdef|@xalignatverbatim#1\\end{xalignat}[#1|end[xalignat]]\n|gdef|@sxalignatverbatim#1\\end{xalignat*}[#1|end[xalignat*]]\n\n|gdef|@gatherverbatim#1\\end{gather}[#1|end[gather]]\n|gdef|@sgatherverbatim#1\\end{gather*}[#1|end[gather*]]\n\n|gdef|@gatherverbatim#1\\end{gather}[#1|end[gather]]\n|gdef|@sgatherverbatim#1\\end{gather*}[#1|end[gather*]]\n\n\n|gdef|@multilineverbatim#1\\end{multiline}[#1|end[multiline]]\n|gdef|@smultilineverbatim#1\\end{multiline*}[#1|end[multiline*]]\n\n|gdef|@arraxverbatim#1\\end{arrax}[#1|end[arrax]]\n|gdef|@sarraxverbatim#1\\end{arrax*}[#1|end[arrax*]]\n\n|gdef|@tabulaxverbatim#1\\end{tabulax}[#1|end[tabulax]]\n|gdef|@stabulaxverbatim#1\\end{tabulax*}[#1|end[tabulax*]]\n\n\n|endgroup\n  \n\n  \n\\def\\align{\\@verbatim \\frenchspacing\\@vobeyspaces \\@alignverbatim\nYou are using the \"align\" environment in a style in which it is not defined.}\n\\let\\endalign=\\endtrivlist\n \n\\@namedef{align*}{\\@verbatim\\@salignverbatim\nYou are using the \"align*\" environment in a style in which it is not defined.}\n\\expandafter\\let\\csname endalign*\\endcsname =\\endtrivlist\n\n\n\n\n\\def\\alignat{\\@verbatim \\frenchspacing\\@vobeyspaces \\@alignatverbatim\nYou are using the \"alignat\" environment in a style in which it is not defined.}\n\\let\\endalignat=\\endtrivlist\n \n\\@namedef{alignat*}{\\@verbatim\\@salignatverbatim\nYou are using the \"alignat*\" environment in a style in which it is not defined.}\n\\expandafter\\let\\csname endalignat*\\endcsname =\\endtrivlist\n\n\n\n\n\\def\\xalignat{\\@verbatim \\frenchspacing\\@vobeyspaces \\@xalignatverbatim\nYou are using the \"xalignat\" environment in a style in which it is not defined.}\n\\let\\endxalignat=\\endtrivlist\n \n\\@namedef{xalignat*}{\\@verbatim\\@sxalignatverbatim\nYou are using the \"xalignat*\" environment in a style in which it is not defined.}\n\\expandafter\\let\\csname endxalignat*\\endcsname =\\endtrivlist\n\n\n\n\n\\def\\gather{\\@verbatim \\frenchspacing\\@vobeyspaces \\@gatherverbatim\nYou are using the \"gather\" environment in a style in which it is not defined.}\n\\let\\endgather=\\endtrivlist\n \n\\@namedef{gather*}{\\@verbatim\\@sgatherverbatim\nYou are using the \"gather*\" environment in a style in which it is not defined.}\n\\expandafter\\let\\csname endgather*\\endcsname =\\endtrivlist\n\n\n\\def\\multiline{\\@verbatim \\frenchspacing\\@vobeyspaces \\@multilineverbatim\nYou are using the \"multiline\" environment in a style in which it is not defined.}\n\\let\\endmultiline=\\endtrivlist\n \n\\@namedef{multiline*}{\\@verbatim\\@smultilineverbatim\nYou are using the \"multiline*\" environment in a style in which it is not defined.}\n\\expandafter\\let\\csname endmultiline*\\endcsname =\\endtrivlist\n\n\n\\def\\arrax{\\@verbatim \\frenchspacing\\@vobeyspaces \\@arraxverbatim\nYou are using a type of \"array\" construct that is only allowed in AmS-LaTeX.}\n\\let\\endarrax=\\endtrivlist\n\n\\def\\tabulax{\\@verbatim \\frenchspacing\\@vobeyspaces \\@tabulaxverbatim\nYou are using a type of \"tabular\" construct that is only allowed in AmS-LaTeX.}\n\\let\\endtabulax=\\endtrivlist\n\n \n\\@namedef{arrax*}{\\@verbatim\\@sarraxverbatim\nYou are using a type of \"array*\" construct that is only allowed in AmS-LaTeX.}\n\\expandafter\\let\\csname endarrax*\\endcsname =\\endtrivlist\n\n\\@namedef{tabulax*}{\\@verbatim\\@stabulaxverbatim\nYou are using a type of \"tabular*\" construct that is only allowed in AmS-LaTeX.}\n\\expandafter\\let\\csname endtabulax*\\endcsname =\\endtrivlist\n\n\n\n \\def\\endequation{%\n     \\ifmmode\\ifinner\n      \\iftag@\n        \\addtocounter{equation}{-1}\n        $\\hfil\n           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+{"text":"\\section{Introduction\n\t\t\\label{sec:Intro}}\n\nIt is somewhat misleading to use the terms 'New Physics' or 'Beyond Standard\nModel Physics' for results that explicitly search for signatures that would\nresult from extensions to the standard model.  In almost any precision\nmeasurement of the production, properties or decays of already-known property,\nthere is the possibility of an unusual result can be explained with an \nextension to the standard model.  In a certain sense then, nearly every result\nbeing shown here this week is a search for new physics.  In particular, \nLee Roberts will be giving a presentation on searches for 'New Physics' in\nlow energy experiments.\n\nHowever even restricting the discussion to analyses that search directly for\nextensions to the standard model at high energy colliders would create a\ndiscussion that goes on for far too long.  With great regret, I have had to\ntrim my selection of topics very sharply.  There is just too much good work\ndone here to give each study adequate coverage.  In particular, there has been\na great deal of excellent work preparing for the analysis of the imminent\nflood of data from the LHC.  I can only refer the reader to the parallel\nsessions of this conference.\n\nThe bulk of the recent results are from the TeVatron, which has been performing\nquite well.  Although most of the results described here are based on smaller\ndatasets, just under $7\\,\\mathrm {fb}^{-1}$ have been delivered to each experiment \nat this time.  Results from the CDF and D0 collaborations are collected\nat~\\cite{TeV_web}.\n\nCertain characteristics of hadron collisions are common to all or nearly all\nsearches for exotic phenomena in them.  The copious production of multijet\n``QCD\" events is suppressed with detectors designed to reject fake electrons\nand muons (hereafter referred to as leptons, $\\ell$), kinematic cuts and\nisolation cuts that require that the identified lepton not be surrounded by\nother activity in the detector.  Multijet background is rarely, if ever,\nmodeled effectively with Monte Carlo simulation techniques.  For the many\nsearches which select highly energetic leptons or momentum imbalance in the\nfinal state, the following known physics processes typically produce significant \nbackgrounds: the Drell-Yan process $p\\overline p$$\\rightarrow \\gamma^* \/ Z \\rightarrow$$\\ell^+ \\ell^- $,\n$\\gamma^* \/ Z \\rightarrow$$\\tau^+\\tau^-$, $W^\\pm \\rightarrow \\ell^\\pm\\nu$, and $t\\overline t$~production.  The\ndiboson production processes $p\\overline p$$\\rightarrow VV$ with $V \\in \\{\\gamma, Z,W\\}$  have \nlower production cross-sections but also create unusual signatures which are of\ninterest in many searches.\n\nsJust as limiting as backgrounds are the kinematic facts of life in hadron\ncolliders.  In $p\\overline p$~(or $pp$) collisions, the component of the initial-state\nmomentum along the collision axis is not known and kinematic calculations can\nonly be done in the plane perpendicular to the collision.  I will use \\mbox{$E_T\\hspace{-1.2em}\\slash\\hspace{1.0em}$}\nto indicate the opposite of the observed sum of particle momenta in this\ntransverse plane, and \\mbox{$p_T$}~to indicate the momentum of an object projected onto\nthis transverse plane.\n\n\n\\section{About Supersymmetry\n\t\t\\label{sec:SUSY}}\n\nAs many of the results discussed here are based on supersymmetric (SUSY)\nextensions of the standard model, a short introduction is appropriate.  In no\nway however can this replace the many excellent existing reviews and\nintroductions, some of which may be found in \nreference~\\cite{SUSY_reviews,Martins}.\n\nSUSY provides solutions to several existing dilemmas in the standard model.\nOne is the ``${\\mathrm M}_{H}$ problem\".  The propagator for a Higgs scalar with fermionic\ncouplings $\\mathcal{L} = - \\lambda_f H f {\\bar f}$ has one loop correction terms \nthat contribute to the mass in amount\n$\\Delta {\\mathrm M}_H^2 = -\\frac{|\\lambda_f|^2}{8\\pi^2} \\Lambda_{UV}^2$, where\n$\\Lambda_{UV}$ is a cutoff scale corresponding to the point where our\nexisting understanding of nature's particle content becomes inadequate.\nOur difficulty is that we have no clear value for $\\Lambda_{UV}$ short of the\nPlank scale, resulting in large negative contributions to $m_H$.  However,\nif for every fermion there is a corresponding scalar field $S$ with interaction\n$\\mathcal{L} = -\\lambda_s |h|^2 |S|^2$, then the corresponding scalar loop\ndiagram introduces canceling mass contributions\n$\\Delta _H^2 = \\frac{\\lambda_S}{16\\pi^2} [ \\Lambda_{UV}^2 + \\ldots ]$.\n\nA second outstanding problem in the standard model is the dark matter problem.\nThe lightest neutral sparticle often makes a good dark matter candidate.\n\nFinally, the coupling constants for the strong, weak and electromagnetic forces \nvary with the energy scale of the interaction according to the renormalization\ngroup.  In the minimal supersymmetric extension to the standard\nmodel (MSSM), the coupling constants evolve out to reach similar values at the\nscale of $10^{16}\\,\\mathrm {GeV}$, which does not happen in the standard model.\nQuoting~\\cite{Martins}, ``While the apparent unification of gauge couplings at\n[this scale] might just be an accident, it may also be taken as a strong hint\nin favor of a grand unified theory or superstring models, both of which can\nnaturally accommodate gauge coupling unification below ${\\mathrm M}_P$.\"\n\nBecause the SUSY mass spectrum evidently differs from that of the standard\nmodel particle content, there must be SUSY-breaking terms in the Lagrangian.\nThe primary constraint on these terms is that they not reintroduce ultraviolet\ndivergences of the sort we were glad to be rid of earlier.  This is not a\nvery tight constraint; there are at least 105 new free parameters in the most\ngeneral form of the symmetry breaking Lagrangian.  What this does is provide\na flexible framework in which different models of symmetry breaking can be\ninserted and investigated.  Of the many different physical concepts that can\nand have been inserted into the SUSY breaking Lagrangian, two of the most \nstudied ones are the mSUGRA and the GMSB models.  We have recent results in\nboth of these SUSY breaking models.\n\nThe generality of the SUSY breaking Lagrangian is perhaps why the SUSY\nhypothesis has had such a long run.  After all, SUSY was proposed in the early\n1970s, when the standard model was still a novel model, and much of its particle\ncontent was unknown.  For nearly 4 decades, theorists have been able to write\nLagrangians of all sorts into this framework and work out the their possible\nimplications.\n\n$R$-parity is a hypothesized quantum number which differentiates standard\nmodel particles from SUSY particles.  All of the searches presented here\nassume the conservation of $R$-parity, so that each SUSY particle is produced\nin conjunction with the corresponding SUSY anti-particle.\n\nIn SUSY, 2 Higgs doublets\n\\begin{equation}\nH_d = \\left( \\begin{array}{c} H_d^0 \\\\\n\t\t\t\t\t\t\t  H_d^- \\end{array} \\right)\n\\hspace{15mm}\nH_u = \\left( \\begin{array}{c} H_u^+ \\\\\n\t\t\t\t\t\t\t  H_u^0 \\end{array} \\right)\n\\label{Eqn:Higgses}\n\\end{equation}\ncoupling respectively to down- and up- type fermions are required in order to\nprevent triangle anomalies.  The ratio of the vacuum expectation values of\nthe two neutral fields,\n$\\tan \\beta = \\langle H_u^0 \\rangle \/ \\langle H_d^0 \\rangle$\nis one of the key parameters of supersymmetry, or indeed of any 2 doublet\nmodel.  Analyses that apply Bayesian methods to a random samplings of parameter\nspace~\\cite{Bayes} strongly favor larger values of $\\tan \\beta$ at least in the\ncontext of mSUGRA and similar SUSY-breaking models.\n\nThe SUSY partners to the Higgs fields are the spin 1\/2 Higgsinos:\n\\begin{equation}\n\\tilde{H}_d = \\left( \\begin{array}{c} \\tilde{H}_d^0 \\\\\n\t\t\t\t\t\t\t\t\t  \\tilde{H}_d^- \\end{array} \\right)\n\\hspace{20mm}\n\\tilde{H}_u = \\left( \\begin{array}{c} \\tilde{H}_u^+ \\\\\n\t\t\t\t\t\t\t\t\t  \\tilde{H}_u^0 \\end{array} \\right)\n\\label{Eqn:Higgsinos}\n\\end{equation}\nThe charged components of the Higgsino fields can form linear admixtures\nwith the wino to create 2 charginos, $\\tilde{\\chi}^\\pm$.  The\nconvention is that $m(\\tilde{\\chi}^\\pm_1) < m(\\tilde{\\chi}^\\pm_2)$.  \n$\\tilde{\\chi}$, without subscript, refers the lightest of the mixtures.\nThe neutral components of the Higgsino fields form linear admixtures\nwith the zino and photino to create 4 neutralinos, $\\tilde{\\chi}^0_i$.\n\nReturning to the scalars, after electroweak symmetry breaking, two doublet\nmodels yield 5 Higgs bosons: two $CP$-even neutral scalars $h$ and $H$, a\n$CP$-odd neutral $A$ and a pair of charged scalars, $H^\\pm$.\n\nNo discussion at length about SUSY is complete without mentioning that the\nMSSM at least is under some pressure from experimental results from the\nelectroweak symmetry breaking sector.  The lightest neutral MSSM Higgs boson\n$h$ must have a mass below 135 GeV~\\cite{Martins,Pesky} and experimental lower\nbounds~\\cite{LEP_Higgs} have come to approach this level.\n\n\n\\section{Searches for $\\tilde{t}$\n\t\t\\label{sec:stop}}\n\nWe have recent search results for the pair production of the SUSY partner to\nthe top quark in $p\\overline p$~ collisions.  There are 3 decay channels under study.\nIn all 3 cases, limits are placed in a plane where the horizontal axis is the\nmass of the pair-produced $\\tilde{t}$ and vertical axis is the mass of the\nfinal state SUSY particle.\n\n\nThe first channel is\n$\\tilde{t} \\rightarrow b \\tilde{\\chi}^+; \\tilde{\\chi}^+ \\rightarrow \\tilde{\\nu} \\ell^+$.\n$R$-parity conservation means that the charge conjugate process occurs on the\nother side of the event, where a $\\overline{\\tilde{t}}$ decays similarly.  The\nsignature is an $\\ell^+ \\ell^- $~pair with \\mbox{$E_T\\hspace{-1.2em}\\slash\\hspace{1.0em}$} from the escaping neutrinos.  There are 2\n$b$-jets in the final state, but both the D0 and CDF collaborations found\nkinematic selection sufficient.  The recent D0 result~\\cite{D0_stop_emu} uses\n$3.1\\,\\mathrm {fb}^{-1}$ in the $e-\\mu$ channel in conjunction with earlier\n$1.1\\,\\mathrm {fb}^{-1}$ $e-\\mu$ and $e-e$ results.  The CDF result~\\cite{CDF_stop_dilep} is\nbased on $1.0\\,\\mathrm {fb}^{-1}$ in all 3 dilepton channels.  Limits are drawn in the\n$m(\\tilde{\\nu})$ \\hbox{\\it vs.}~$m(\\tilde{t})$ plane and extend up to\n$m(\\tilde{\\nu}) \\simeq 120\\,\\mathrm {GeV}$.\n\nThe second channel is\n$\\tilde{t} \\rightarrow b \\tilde{\\chi}^+; \\tilde{\\chi}^+ \\rightarrow \\tilde{\\chi}^0 (W^+\/H^+\/G^+)$\nwhere the remaining charged gauge boson decays semileptonically.  This channel\nwas originally of interest when measured values of $m(t)$ seemed to be a little\nlower in the dilepton channel.  It is possible if $m(\\tilde{t}) < m(t)$ for the\nSUSY process to contaminate the $t\\overline t$~dilepton channel and pull down the \napparent $t$  quark mass.  With 4 undetected particles in the final state\n(the $\\tilde{\\chi}^0$ is taken to be stable), the kinematics are very\nunderconstrained, even in the transverse plane.  However, one may use a\nweighted sum of possible solutions to the kinematic problem to estimate\n$m(\\tilde{t})$.  CDF has set limits~\\cite{CDF_Robin} on $m(\\tilde{\\chi}^0)$\nas a function of $m(\\tilde{t})$ (up to $197\\,\\mathrm {GeV}$) and the assumed\n\\BR{$\\tilde{\\chi}^\\pm$}{$\\tilde{\\chi}^0\\nu\\ell^\\pm$} ~using $2.7\\,\\mathrm {fb}^{-1}$.\n\nThe third channel to be studied is $\\tilde{t} \\rightarrow c \\tilde{\\chi}^0$.  As the\nlifetime of charm hadrons typically is shorter than that of bottom hadrons,\nand as the transverse momentum of the charged products of charm decays\ntypically is less than that of bottom decays, obtaining a pure sample of charm\ndecays with impact parameter tagging is very difficult.  The CDF collaboration\nhas developed a 2 output, 22 input neural network that distinguishes (at one\noutput) between charm and bottom jets.  The other output distinguishes between\ncharm and light or $\\tau$ jets.  Cutting on the sum of the two outputs, they\nset limits~\\cite{CDF_underdocumented} in the $m(\\tilde{\\chi}^0)$\n\\hbox{\\it vs.}~$m(\\tilde{t})$ plane extending up to $m(\\tilde{t}) = 180\\,\\mathrm {GeV}$.\n\n\n\\section{Trifermion SUSY Searches\n\t\t\\label{sec:threebees}}\n\nSUSY allows a number of channels leading to 3 leptons in the final state,\nas shown in Figure~\\ref{Fig:trileptons}.  There are relatively few backgrounds,\nbut the cross-section for production times the branching ratio for decay into\nany particular combination of leptons is small.  Depending on the particular\nvalues of the SUSY-breaking parameters (here, the mSUGRA breaking is used) it\nmay happen that the mass of the charginos or neutralinos produced at the\n$q \\overline q$ vertex is only a little larger than the mass of the escaping \n$\\tilde{\\chi}_1^0$, in which case a low momentum lepton is produced.  For the\nhigh values of $\\tan \\beta$ that are of particular interest, $\\tau^\\pm$ leptons\nare often produced, and are detected by their decays to electrons or muons that\nare also of lower momentum.  To increase sensitivity then, it is common to not\nattempt to identify the lepton of third lowest \\mbox{$p_T$}, but rather to just ask for\na charged particle that is isolated from any jets that appear in the event.\nRobert Forrest and Todd Adams have presented the CDF~\\cite{CDF_trileptons} and\nD0~\\cite{D0_trileptons} results in this conference.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=80mm]{Fig1.eps}\n\\caption{Some of the ways in which SUSY creates trilepton signatures in\n$p\\overline p$~collisions.}\n\\label{Fig:trileptons}\n\\end{figure}\n\nAnother final state with three fermions produced via SUSY diagrams has been\ninvestigated by CDF~\\cite{CDF_lljj}.  Suppose that in the top diagram of\nFigure~\\ref{Fig:trileptons} the $W$ materializes as a $q\\overline q$ pair,\ncreating 2 jets.  The resulting event then appears as a $WZ$ pair with \\mbox{$E_T\\hspace{-1.2em}\\slash\\hspace{1.0em}$}.\nIn the standard model, hadronically decaying $W$s with $Z\\rightarrow$$\\ell^+ \\ell^- $ do not have\n\\mbox{$E_T\\hspace{-1.2em}\\slash\\hspace{1.0em}$} except as a result of mismeasurement, so this is a relatively clean final\nstate.  While it has to be admitted that the existing sensitivity is not really\ncomparable to what might reasonably be expected in SUSY, there are several\nreasons why large improvements can be expected in the future. To date, only\n$Z\\rightarrow$$e^+e^-$ has been investigated, and $b$-jet identification has not been \nemployed although a large $t\\overline t$~background is present.  Also, the present result\nis based on $2.7\\,\\mathrm {fb}^{-1}$ of data at one of the two TeVatron experiments; a final\nsample some 7 or 8 times larger than this could occur.\n\n\n\\section{Gauge Mediated Supersymmetry Breaking\n\t\t\\label{sec:GMSB}}\n\nIn order to give different mass spectra to SUSY \\hbox{\\it vs.}~standard model particles\nusing gauge interactions, one can postulate the existence of new fields, called\nmessengers, that couple the standard model and SUSY particles to an ultimate\nsource of symmetry breaking.  In these GMSB models, the lightest neutral SUSY\nparticle is nearly always the gravitino, which is an interesting dark matter\ncandidate for masses on the scale of a few $\\,\\mathrm {keV}$.  For the collider\nexperimentalist, the way to think of various versions of this model is to\ncategorize them in terms of their next-to-lightest SUSY particle (NLSP).\nWhatever particular SUSY particles might be created at the hard scattering\nvertex, they will cascade down to the NLSP (assuming $R$-parity conservation)\nwhich will after some lifetime go to an undetected gravitino.  The nature of\nthe NLSP will then determine what type of events to look for in the dataset. \n\nWhen the NLSP is the lightest neutralino and $m(\\tilde{\\chi}^0) <$${\\mathrm M}_Z$, its\ndecay produces a photon in conjunction with the gravitino.  If the \n$\\tilde{\\chi}_1^0$ lifetime is on the order of $10\\,\\mathrm {ns}$, the arrival of the\n$\\gamma$ will be delayed because of the flight path, as shown in\nFigure~\\ref{Fig:latelight}.  The CDF detector has $~0.5\\,\\mathrm {ns}$ time resolution in\nits EM calorimeter, which makes this type of search feasible.  In addition to\nthe delayed photon, the search requires a jet and \\mbox{$E_T\\hspace{-1.2em}\\slash\\hspace{1.0em}$}.  Limits up to\n$m(\\tilde {\\chi}^0) > 191\\,\\mathrm {GeV}$ for $\\tau(\\tilde {\\chi}^0) > 5\\,\\mathrm {ns}$ were obtained\nand a detailed description of the analysis was published in\n2008~\\cite{CDF_latelight}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=80mm]{Fig2.eps}\n\\caption{Why photons from $\\tilde{\\chi}^0 \\rightarrow \\gamma\\tilde{G}$ arrive late\nin the electromagnetic calorimeter of a large collider experiment.}\n\\label{Fig:latelight}\n\\end{figure}\n\nWhen the lifetime of the neutralino is on the order of a few $\\,\\mathrm {ns}$ or less,\nthe delayed photon technique will not work.  However, as a consequence of\n$R$-parity, there should be 2 SUSY cascades in the event leading to 2\nNLSP $\\tilde{\\chi}^0$ decays to $\\gamma\\tilde{G}$.  In this conference,\nEunsin Lee has reported on a search~\\cite{CDF_earlylight} for GMSB at CDF which\nrequires 2 photons with high \\mbox{$p_T$}~along with \\mbox{$E_T\\hspace{-1.2em}\\slash\\hspace{1.0em}$} from the gravitinos.  Limits\nup to  $m(\\tilde {\\chi}^0) > 149\\,\\mathrm {GeV}$ for $\\tau(\\tilde {\\chi}^0) < 1\\,\\mathrm {ns}$ were\nobtained.\n\n\n\\section{MSSM Higgs\n\t\t\\label{sec:MSSM_Higgs}}\n\nIn the large $\\tan \\beta$ limit, the mass and couplings of the $A$ boson\napproach the mass and couplings of one of the two $CP$-even bosons $h$ or $H$.\nIf $A \\rightarrow H$ and $m(H)$ is large, one has the ``decoupling\" limit, where $h$\nbecomes in many ways rather similar to the standard model Higgs.  If $A \\rightarrow h$,\n$m(A)$ would not be not too large and hadron colliders can search for the $A$\nin the modes $A \\rightarrow \\tau^+\\tau^-$, $bA \\rightarrow b\\tau^+\\tau^-$ and $bA \\rightarrow bbb$.\nThe $Abb$ and $A\\tau\\tau$ couplings are enhanced relative to the experimentally\ndifficult $Att$ and $A\\nu\\nu$ couplings by a factor of $\\tan^2 \\beta$ and so\nlimits on the maximum possible value of $\\tan \\beta$ can be set as a function\nof $m(A)$.  John Conway and Flera Rizatdinova in this conference have\ndiscussed the recent TeVatron results.  At this time, values of $\\tan \\beta$\nover $\\simeq 30$ are ruled out~\\cite{CDF_MSSM,D0_MSSM} at $m(A) \\simeq 130\\,\\mathrm {GeV}$;\nif these results are scaled by the expected final Run II luminosity and\n$\\tan^2 \\beta$, it is reasonable to guess that the TeVatron experiments will\nultimately be able to set limits as low as $\\tan \\beta \\simeq 20$.  More\ndetailed studies of the potential reach of the TeVatron and the LHC have been\ndone recently~\\cite{MSSM_future}.\n\n\n\\section{NMSSM Higgs\n\t\t\\label{sec:NMSSM_Higgs}}\n\nGiven the increasing restrictions on the available parameter space of the\nminimal supersymmetric extension of the standard model, it is natural to\nconsider a nearly-minimal SUSY extension.  In the NMSSM SUSY model, the\nsmallest possible combination of fields is added to the known standard model\nfields and their SUSY partners.  Neutral weak isospin singlet fermion and\ncorresponding complex scalar fields are introduced.  The resulting physical\ncontent of the theory includes a new light pseudoscalar, $a$, which (in the\nmanner characteristic of Higgs bosons) decays into the heaviest kinematically\navailable particles.  For $m(a)$ above $2M_\\mu$, $a \\rightarrow \\mu\\mu$ is possible\nand has a nearly 100\\% branching ratio.  If $m(a)$ is over $\\simeq 3$ times\nthe pion mass, hadronic decays become dominant; when $m(a) > 2 M_\\tau$, the\ndecay into $\\tau^+\\tau^-$~becomes the dominant mode.  Interest in this model was\nincreased~\\cite{HK_excitement} by the unusual dimuon mass spectrum observed in\n$\\Sigma \\rightarrow p\\mu^+\\mu^-$ by the HyperCP~\\cite{HK_thought} experiment.\n\nIn $e^+e^-$~colliders, the $\\Upsilon$ may decay to $a\\gamma$ and there should be\na narrow peak in the $\\gamma$ energy spectrum for events where a $\\tau^+\\tau^-$~or\n$\\mu^+\\mu^-$~pair has been identified.  A search using this method was performed\nearlier by the CLEO collaboration~\\cite{CLEO_no_a} which set limits on\n\\BR{$\\Upsilon(1S)$}{$a\\gamma$} ~$\\times$ \\BR{$a$}{$\\mu^+\\mu^-$} ~on the scale of\na few times $10^{-6}$ in the range of about $250\\,\\mathrm {MeV}$ to $3.5\\,\\mathrm {GeV}$, and also upon\n\\BR{$\\Upsilon(1S)$}{$a\\gamma$} ~$\\times$ \\BR{$a$}{$\\tau^+\\tau^-$} ~on the scale\nof a few times $10^{-5}$ in the range of about $5$ to $9\\,\\mathrm {GeV}$.  More\nrecently, BaBar~\\cite{BaBar_no_a} examined their data for evidence of this\nprocess, using the case where one $\\tau$ decayed to $e \\nu \\overline\\nu$ and\nthe other decayed to $\\mu \\nu \\overline\\nu$.  They set limits on\n\\BR{$\\Upsilon(3S)$}{$a\\gamma$} ~$\\times$ \\BR{$a$}{$\\tau^+\\tau^-$} ~on the scale\nof a few times $10^{-5}$ in the range of about $4\\,\\mathrm {GeV}$ to just under $10\\,\\mathrm {GeV}$.\n\nIn a hadron collider, a pair of $a$ bosons would be produced as the result of\nthe decay of an $h$.  From LEP II, we have a very general\nlimit~\\cite{OPAL_recoil} that the mass of any new scalar coupling to the $Z$,\nincluding the $h$, must have a mass over $82\\,\\mathrm {GeV}$, and so the $a$ is produced\nin a hadron collider with a high boost.  That in turn means that its decay\ninto, say, a $\\mu^+\\mu^-$~pair will produce particles with a small opening angle.  For\n$m(a) < 2{\\mathrm M}\\tau$ the two tracks can be difficult to resolve in the\n$r$-$\\phi$~plane.  D0~\\cite{D0_NMSSM} has searched for the $a$ in the case\n$2{\\mathrm M}\\tau < m(a)$ using the modes $aa \\rightarrow \\mu\\mu\\mu\\mu$ and\n$aa \\rightarrow \\mu\\mu\\tau\\tau$.  The branching ratios are substantially lower than for\n$aa \\rightarrow \\tau\\tau\\tau\\tau$ but the signature is clearer.  Andy Haas has\ndiscussed the special reconstruction criteria needed for these collinear\nleptons in this conference.  Limits on \n$\\sigma (p{\\overline p} \\rightarrow h)~\\times$ \\BR{$h$}{$aa$}\n~of a few $\\mathrm {pb}$ are obtained.\n\n\n\\section{Leptoquarks\n\t\t\\label{sec:GUTschmutz}}\n\nBecause silicon vertex detectors can identify jets produced by fragmenting $b$\nquarks, it is possibile to search for third generation leptoquarks at hadron\ncolliders.  An $LQ$-$\\overline {LQ}$ pair would produce events containing 2\n$b$ jets and a large \\mbox{$E_T\\hspace{-1.2em}\\slash\\hspace{1.0em}$} from the 2 $\\nu_{\\tau}$.  As Sergey Uzunan described\nat this conference, this is the same signature as that which one might expect \nfrom pair production of ${\\tilde b}$, with subsequent ${\\tilde b} \\rightarrow b \\chi^0$\ndecays.  Limits can then be set~\\cite{TwoForOne} upon both models as a result\nof what is basically a single search method.  As a search for $\\tilde b$,\nlimits up to $m(\\tilde b) > 250\\,\\mathrm {GeV}$ are obtained; as a search for leptoquarks,\n$m(\\mathrm {LQ}_3) > 252\\,\\mathrm {GeV}$ is obtained.\n\nThe best way to find a leptoquark, at least a first generation one, is to take\na lepton and accelerate it to high energy and then arrange for it to collide\nwith a quark, similarly accelerated.  This is exactly what HERA did, collecting just\nunder $0.8\\,\\mathrm {fb}^{-1}$ of $e^\\pm p$ data at $\\sqrt{s} = 300-319\\,\\mathrm {GeV}$, $0.3\\,\\mathrm {fb}^{-1}$ of which\nhad polarized $e^\\pm$.  The ZEUS~\\cite{ZEUS_LQ} collaboration measured the\n$Q^2$ distribution in their data and compared it to the standard model\nprediction.  The (very small) difference was then compared against deviations\nthat would be created by first generation leptoquarks, resulting in limits on\n$m(LQ) \/ \\lambda(LQ)$ of $0.5 - 1.9 \\,\\mathrm {TeV}$, where $\\lambda(LQ)$ is the coupling\nof the leptoquark to the fermions.  Using the same technique they were also\nable to set limits on large extra dimensions and contact interactions with the\nsame $Q^2$ distribution.  The H1 collaboration worked with different\nkinematic variables, specifically, $M$ and $y$; their results~\\cite{H1_LQ} are\nnot straight lines on the $\\lambda(LQ)$ \\hbox{\\it vs.}~$m(LQ)$ plane.  If the couplings\nare taken to be $\\lambda(LQ) = \\sqrt{4\\pi\\alpha_{em}}$, the H1 analysis rules\nout leptoquark masses below 275 to $325\\,\\mathrm {GeV}$, depending on the type of\nleptoquark.\n\n\n\\section{Hidden Valley Scenarios\n\t\t\\label{sec:DidYouFindItYet}}\n\nAs my long time friend and one of our kind hosts here in Detroit Dave Cinabro\nonce accurately pointed out, ``When somebody writes a paper that says he looked\nfor something and he did not find it, well then, you have to believe him.\"\nAnother, more common, reaction to a null search is to imagine that the imagined\nnew phenomena still actually does in fact exist, but at some higher energy\nscale which is at least for the moment experimentally inaccessible.  Hidden\nValley scenarios are predicated on a third possible response: the new phenomena\nstill does exist at a relatively low mass scale, but is so weakly coupled to\nthe standard model phenomenology as to render it invisible, or at least, hard\nto see.\n\nOne can postulate a wide range of fields that could exist in such a hidden\nsector; ``hidden valleys\" is really a class of models rather than than a\nspecific model.  In the simplest example of such a model~\\cite{HV_theory} \nthe valley is populated with two electrically neutral quarks which are\nconfined into so-called ``v-hadrons\".  Some of these particles may be stable,\nproviding dark matter candidates; big-bang nucleosynthesis considerations\nsuggest that at least one v-hadron has to have a lifetime much less than 1 sec.\nA $Z'$ that couples to both the hidden valley particles and the standard\nmodel ones is included in this model, with a mass in the $1 \\sim 6\\,\\mathrm {TeV}$ range.\n\nAndy Haas, in this conference, has presented D0's search~\\cite{D0_HV} for\nv-hadrons that are produced by mixing with a Higgs boson and have a long\nlifetime; their decay is mediated by the $Z'$ and produces a pair of $b$ jets\nthat emanate from a vertex that is between 1.6 and $20\\,\\mathrm {cm}$ distant from the\n$p\\overline p$~interaction point.  The large background from material interactions is\nsuppressed by comparing the locations of the jet vertices with the known\nmaterial distribution in the detector.  Limits on\n$\\sigma (p{\\overline p} \\rightarrow HX)~\\times$ \\BR{$H$}{$HV {\\overline{HV}}$}\n\t\t\t\t\t\t\t   ~$\\times~ \\mbox{Br}^2(HV \\rightarrow b{\\overline b})$\nas low as $1\\,\\mathrm {pb}$ are obtained.\n\n\n\\section{Supersymmetric Hidden Valley Dark Matter Model\n\t\t\\label{sec:TheWholeBallOfWax}}\n\nIn recent years, a number of experiments have reported results that could be\ninterpreted as dark matter annihilation to $e^+e^-$~pairs near the center of the\nMilky Way.  Additionally, the DAMA experiment reports an annual modulation in\ntheir NaI(Tl) detector which may be interpreted as a signal from a dark\nmatter galactic halo.  While there is no shortage of more mundane\nexplanations for these results, some authors~\\cite{AH_etal} have taken a more\nadventuresome approach.  They begin with the assumption that all of these\nresults are in fact due to new physics and then ask what would that new physics\nlook like.\n\nThey come to the surprising conclusion that dark matter is on the\n$0.5 - 0.8 \\,\\mathrm {TeV}$ mass scale and that it annihilates to standard model\nparticles with ``sizeable\" cross-sections.  With such a large mass, it is\nnatural to speculate that a new symmetry prevents the rapid decay of such\nstates.  However, these states might couple to light (${\\mathcal O} (1\\,\\mathrm {GeV})$)\nparticles, known as ``dark photons\" $(\\gamma_D)$.  They also have found that\nsuch a picture can be implemented in a SUSY framework with GMSB.  In that case\na clear signature for $p\\overline p$~collider searches occurs through processes such as\nthat shown in Figure~\\ref{Fig:ballOwax}; a high energy $\\gamma$ would appear in\nconjunction with \\mbox{$E_T\\hspace{-1.2em}\\slash\\hspace{1.0em}$} and a collinear $\\mu^+\\mu^-$~pair from the $\\gamma_D$ decay.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=80mm]{Fig3.eps}\n\\caption{A dark photon production diagram in $p\\overline p$~collisions.}\n\\label{Fig:ballOwax}\n\\end{figure}\n\nThe low mass, high boost and decay into $\\mu^+\\mu^-$~pairs of the dark photon means\nthat one may use the same reconstruction techniques as were applied in searching\nfor the NMSSM $a$ in hadron colliders.  The D0 collaboration has set\nlimits~\\cite{D0_ballOwax} on $m({\\tilde \\chi}^0)$ as a function of $m(\\gamma_D)$\nin the range $0.1 < m(\\gamma_D) < 2.5\\,\\mathrm {GeV}$\n\n\n\\section{Model Independent Searches\n\t\t\\label{sec:BruceGoneNow}}\n\nMuch of the motivation for searching for new physics beyond the standard model\nstems from our dissatisfaction with the many aspects of the standard model which\nwe find so surprising.  Indeed, were it not for such astonishments as parity\nviolation, the $J\/\\psi$ observation, the large value of ${\\mathrm M}_{t}$~and many others, the\nstandard model would surely have been easier to figure out!  While we do hope\nand expect that getting the correct extension to the standard model will somehow\nreduce our overall level of astonishment, history warns us that such an outcome\nis not at all certain.  With this in mind, it behooves us to try to conduct\nsearches for new physics without the guidance of models that are at least in\npart constructed so as to reduce our astonishment.\n\nThe basic scheme for the modern model-independent search begins by defining a\nlarge number of final states.  The definition is usually made in terms of the\nparticle content of the final state, where particles are defined by the\ndetection capabilities of the experiment's apparatus.  So for example, final\nstates with low \\mbox{$p_T$}~electrons would typically evade detection in hadron\ncolliders, and such final states can not be included.  Particles that require\nunusual reconstruction schemes are typically not included in the list of\npossible final states.  Particles that are found by vertexing their decay\nproducts (such as $\\mathrm K^0_{\\mathrm S}$~or $\\mathrm D^{*+}$) have by and large not been included to date,\nalthough there is no specific reason why they could not be.  One consequently\nshould not think of a model-independent search as being exactly the same as a\nsearch for ``everything\"; it is not quite that, at least to date.\n\nFor each entry on the list of possible final states, the standard model\nprocesses contributing to the final state are identified and modeled.  The data\nare then compared against this predicted background, and cases where the data\nappear at a higher rate than the known physics rate are flagged.  Cases where\nthe data appear at a lower rate are also interesting, both as a check on the\nmethod and in case there might be new physics amplitudes that interfere \ndestructively with known amplitudes.  In assessing the statistical significance\nof any departure of reality from prediction, it is important to allow for the\nfact that the more comparisons you make, the more likely it is that the most\ndiscrepant result will be at or beyond any particular level of significance.\n\nThere are different ways to compare the data to the predicted rates of\nknown physics.  There might be a different total number of events.\nDistributions of kinematic variables for the data and the expectation can be\ncompared with an overall quality of fit statistic, such as the Komolgov-Smirnov\nstatistic.  The distribution of a kinematic variable, such as a reconstructed\nmass, can be scanned for bumps.  Or one might scan the distributions of $\\,\\mathrm {GeV}$\ndimensioned kinematic variables such as \\mbox{$p_T$}~or reconstructed mass from low\nto high values, and look for discrepancies in the event counts above the scan\npoint.\n\nThis type of analysis has been completed at the CDF~\\cite{CDF_mis},\nD0~\\cite{D0_mis} and H1~\\cite{H1_mis} experiments, although not all three have\nutilized the full range of possible comparison methods.  Jim Linnemann, in this\nconference, has presented the D0 model independent search.\nTable~\\ref{Tab:MIScounts} shows the results of comparisons at the level of\nsimple event count comparisons of data with expected background levels.  The\nH1 collaboration chose to express their results in terms of number of seen\nevents \\hbox{\\it vs.}~the expected backgrounds; to facilitate comparison with the CDF\nand D0 results I have calculated a corresponding number of standard deviations.\n\n\\begin{table}[h]\n\\begin{center}\n\\caption{Significance of event count discrepancies in 3 model independent\nsearches.  See text regarding treatment of H1 results.\\\\}\n\\begin{tabular}{|c|c|c|} \\hline\n\\textbf{CDF ($2.0\\,\\mathrm {fb}^{-1}$)}                  &   \\textbf{H1 ($0.5\\,\\mathrm {fb}^{-1}$)}                & \\textbf{D0 ($1.1\\,\\mathrm {fb}^{-1}$)}                  \\\\\n\\hline\n$\\gamma \\tau$ \\hspace{5.5mm} $2.2\\sigma$ &  $\\nu 4j$ \\hspace{4.0mm} $<3.0\\sigma$   & $\\mu jj$\\mbox{$E_T\\hspace{-1.2em}\\slash\\hspace{1.0em}$} \\hspace{5.0mm} $9.3\\sigma$ \\\\\n\\hline\n$\\mu \\tau$    \\hspace{5.5mm} $1.7\\sigma$ &  $e   4j$ \\hspace{4.0mm} $<2.4\\sigma$   & $\\mu j\\gamma$\\mbox{$E_T\\hspace{-1.2em}\\slash\\hspace{1.0em}$} \\hspace{5.0mm} $6.6\\sigma$ \\\\\n\\hline\n$e \\tau$\\mbox{$E_T\\hspace{-1.2em}\\slash\\hspace{1.0em}$}  \\hspace{2.5mm} $1.7\\sigma$ &  $eee$ \\hspace{4.0mm} $\\sim2.0\\sigma$   & $\\mu^+\\mu^-$~\\mbox{$E_T\\hspace{-1.2em}\\slash\\hspace{1.0em}$}   \\hspace{1.5mm} $4.4\\sigma$ \\\\\n\\hline\n                                         & $\\mu\\nu$ \\hspace{4.0mm} $\\sim1.5\\sigma$ & $\\mu^+\\mu^-$~$\\gamma$ \\hspace{3.0mm} $4.4\\sigma$ \\\\\n\\hline\n\\end{tabular}\n\\label{Tab:MIScounts}\n\\end{center}\n\\end{table}\n\nThe statistically significant deviations in the channels flagged by the D0\nanalysis are attributed to defects in the modeling of the rate at which jets\nfake as photons, trigger simulation shortcomings, and \\mbox{$p_T$}~resolution effects\nin the D0 tracking system which effect muon measurement.\n\nSignificantly, there is no overlap in the channels found by all 3 experiments.\n\n\n\n\n\\bigskip\n\\begin{acknowledgments}\nI would like to thank a number of people who helped improve this presentation:\nTodd Adams, Arnaud Duperrin, Andy Hass, Katjia Kruger, Monica D'Onofrio,\nMonica Turcato, Stefan Schmitt and Tom Wright.  And I would also like very\nmuch to thank our hard-working conference organizers for this very productive\nmeeting and for their gracious hospitality.\n\\end{acknowledgments}\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Statistical models representations}\n\\label{app:chap2-graphs}\nGraphical representations have been developed to represent and efficiently exploit (in)dependencies between random variables encoded in joint probability distributions. They are useful tools to concisely present the model under scrutiny, as well as direct supports for some derivations of inference procedures. Let us briefly present two types of graphical representations. \n\n\\paragraph{Probabilistic graphical models}\nFormally, a probabilistic graphical model is a graph $G = (V, E)$ with nodes $V$ representing random variables and edges $E$ representing direct interactions between random variables. In many statistical models of interest, it is not necessary to keep track of all the possible combinations of realizations of the variables as the joint probability distribution can be broken up into factors involving only subsets of the variables. The structure of the connections $E$ reflects this factorization.  \n\nThere are two main types of probabilistic graphical models: \\emph{directed graphical models} (or Bayesian networks) and \\emph{undirected graphical models} (or Markov Random Fields). They allow to represent different independencies and factorizations. In the next paragraphs we provide intuitions and remind some useful properties of graphical models, a good reference to understand all the facets of this powerful tool is \\cite{Koller2009}. \n\n\\subparagraph{Undirected graphical models} \nIn undirected graphical models\nthe direct interaction between a subset of variables $C \\subset V$ is represented by undirected edges interconnecting each pair in $C$. This fully connected subgraph is called a \\emph{clique} and associated with a real positive \\emph{potential} function $\\psi_C$ over the variable $\\x_C=\\{x_i\\}_{i \\in C}$ carried by $C$. The joint distribution over all the variables carried by $V$, $\\x_V$  is the normalized product of all potentials\n    \\begin{gather}\n        p(\\x_V) = \\frac 1 \\cZ \\prod_{C \\in \\mathcal{C}} \\psi_C(\\x_C).\n    \\end{gather}\n\n\\begin{flushright}\n\\begin{minipage}{0.95\\linewidth}\n    \\hypertarget{item:chap2-rbm}{Example} \\hyperlink{item:chap2-rbm}{(i)}: the Restricted Boltzmann Machine,\n    \\begin{gather}\n        \\label{eq:chap2-rbm-def}\n        p(\\x, \\hid) = \\frac{1}{\\cZ} e^{\\x\\T\\W\\hidd}p_x(\\x)p_t(\\hidd)\n    \\end{gather}\n    with factorized $p_x$ and $p_t$ is handily represented using an undirected graphical model depicted in \\citefig~\\ref{fig:chap2-rbm-bis}. The corresponding set of cliques is the set of all the pairs with one input unit (indexed by $i = 1 \\cdots N$) and one hidden unit (indexed by $\\alpha = 1 \\cdots M$), joined with the set of all single units. The potential functions are immediately recovered from \\eqref{eq:chap2-rbm-def},\n    \\begin{gather}\n        \\mathcal{C} = \\{\\{i\\}, \\{\\alpha\\}, \\{i,\\alpha\\}\\,; \\,i=1 \\cdots N, \\,\\alpha = 1 \\cdots M \\} \\, , \\qquad\n       \n        \\psi_{i\\alpha}(x_i, t_\\alpha) = e^{x_i W_{i\\alpha}\\hiddv_\\alpha}\\, ,\\\\\n        p(\\x, \\hid) = \\frac{1}{\\cZ}\\prod_{\\{i, \\alpha\\} \\in \\mathcal{C}}\\psi_{i\\alpha}(x_i, t_\\alpha)\\prod_{i=1}^Np_x(x_i)\\prod_{\\alpha=1}^Mp_t(t_\\alpha).\n    \\end{gather}\n    It belongs to the subclass of \\emph{pairwise} undirected graphical models for which the size of the cliques is at most two. \n\\end{minipage}\n\\end{flushright}\n\n\nUndirected graphical models handily encode \\emph{conditional independencies}. Let $A, B, S \\subset V$ be three disjoint subsets of nodes of $G$. $A$ and $B$ are said to be independent given $S$ if $p(A,B |S) = p(A|S)p(B|S)$. In the graph representation it corresponds to cases where $S$ separates $A$ and $B$: there is no path between any node in $A$ and any node in $B$ that is not going through $S$. \n\n\\begin{flushright}\n    \\begin{minipage}{0.95\\linewidth}\n        Example \\hyperlink{item:chap2-rbm}{(i)}: In the RBM, hidden units are independent given the inputs, and conversely: \n    \\begin{gather}\n        p(\\hidd | \\x) = \\prod_{\\alpha =1}^M p(\\hiddv_\\alpha | \\x), \\qquad\n        p(\\x|\\hidd) = \\prod_{i=1}^M p(x_i | \\hidd).\n    \\end{gather}\n    This property is easily spotted by noticing that the graphical model (\\citefig~\\ref{fig:chap2-rbm-bis}) is \\emph{bipartite}.\n\\end{minipage}\n\\end{flushright}\n\n\\begin{figure}[t]\n    \\centering\n   \n   \n   \n   \n    \\captionsetup{width=.4\\linewidth}\n   \n    {\\includegraphics[width=0.3\\textwidth, valign=m]{chap2_glm.pdf}\n    }\n    \\captionsetup{width=.9\\linewidth}\n    \\caption{\n       \n        Left: Directed graphical model for $p(\\x,\\y,\\W)$ without assumptions of factorizations for the channel and priors. Right: Directed  graphical model reflecting factorization assumptions for $p(\\x,\\y|\\W)$.\\label{fig:chap2-glm}\n       \n        }\n\\end{figure}\n    \n\\subparagraph{Directed graphical model } A directed graphical model  uses a Directed Acyclic Graph (DAG), specifying directed edges $E$ between the random variables $V$. It induces an ordering of the random variables and in particular the notion of parent nodes  $\\pi_i \\subset V$  of any given vertex $i \\in V$: the set of vertices $j$ such that $j \\to i \\in E$. The overall joint probability distribution factorizes as\n\\begin{gather}\n    p(\\x) = \\prod_{i \\in V} p(x_i | \\x_{\\pi_i}).\n\\end{gather}\n\n\\begin{flushright}\n    \\begin{minipage}{0.95\\linewidth}\n        \\hypertarget{item:chap2-glm}{Example} \\hyperlink{item:chap2-glm}{(ii)}: The stochastic single layer feed forward network $\\y = g(\\W\\x ; \\eps)$, where $g( \\cdot; \\eps)$ is a function applied component-wise including a stochastic noise $\\eps$ that is equivalent to a conditional distribution $p_{\\rm out}(\\y | \\W\\x)$, and where inputs and weights are respectively drawn from distributions $p_x(\\x)$ and $p_W(\\W)$, has a joint probability distribution\n    \\begin{gather}\n        \\label{eq:chap2-glm-def}\n        p(\\y, \\x, \\W) = p_{\\rm out}(\\y | \\W\\x) p_x(\\x) p_W(\\W),\n    \\end{gather}  \n     precisely following such a factorization. It can be represented with a three-node DAG as in \\citefig~\\ref{fig:chap2-glm}. Here we applied the definition at the level of vector\/matrix valued random variables. By further assuming that $\\pout$, $p_W$ and $p_x$ factorize over their components, we keep a factorization compatible with a DAG representation \n    \\begin{gather}\n        p(\\y, \\x, \\W) = \\prod_{i=1}^{N} p_x(x_i) \\prod_{\\mu = 1}^M \\pout(y_\\mu | \\sum_{i=1}^N W_{\\mu i}x_i) \\prod_{\\mu, i} p_W(W_{\\mu i}).\n    \\end{gather}\n    For the purpose of reasoning it may be sometimes necessary to get to the finest level of decomposition, while sometimes the coarse grained level is sufficient. \n\\end{minipage}\n\\end{flushright}\n\nWhile a statistical physicist may have never been introduced to the formal  definitions of graphical models, she inevitably already has drawn a few - for instance when considering the Ising model. She also certainly found them useful to guide physical intuitions. The following second form of graphical representation is probably newer to her. \n\n\\paragraph{Factor graph representations}\nAlternatively, high-dimensional joint distributions can be represented with factor graphs, that are undirected bipartite graphs  $G = (V, F, E)$ with two subsets of nodes. The variable nodes $V$ representing the random variables as in the previous section (circles in the representation) and the factor nodes $F$ representing the interactions (squares in the representation) associated with potentials. The edge $(i \\mu)$ between a variable node $i$ and a factor node $\\mu$ exists if the variable $i$ participates in the interaction $\\mu$. We note $\\partial i$ the set of factor nodes in which variable $i$ is involved, they are the neighbors of $i$ in $G$. Equivalently we note $\\partial \\mu$ the neighbors of factor $\\mu$ in $G$, they carry the arguments $\\{x_i\\}_{i \\in \\partial \\mu}$, shortened as $\\x_{\\partial \\mu}$, of the potential $\\psi_\\mu$.  The overall distributions is recovered in the factorized form:\n\\begin{align}\np(\\x) = \\frac 1 \\cZ \\prod_{\\mu = 1}^M \\psi_\\mu(\\x_{\\partial \\mu}).\n\\end{align}\nCompared to an undirected graphical model, the cliques are represented by the introduction of factor nodes. \n\n\n\\begin{flushright}\n    \\begin{minipage}{0.95\\linewidth}\n    Examples: The factor-graph representation of the RBM \\hyperlink{item:chap2-rbm}{(i)} is not much more informative than the pairwise undirected graph (see \\citefig~\\ref{fig:chap2-rbm-bis}). \n    For the feed forward neural networks \\hyperlink{item:chap2-glm}{(ii)} we draw the factor graph of $p(\\y, \\x| \\W)$ (see \\citefig~\\ref{fig:chap2-glm-bis}). \n    \\end{minipage}\n\\end{flushright}\n\n\\section{Vector Approximate Message Passing for the GLM}\n\\label{sec:app-chap3-vamp}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{chap3_vamp_glm.pdf}\n    \\caption{Factor graph representation of the GLM for the derivation of VAMP (reproduction of \\citefig~\\ref{fig:chap3-vamp-glm} \\label{fig:app-chap3-vamp-glm}to help following the derivation described here in the \\citeapp).}\n\\end{figure}\n \nWe recall here a possible derivation of G-VAMP discussed in \\citechap~\\ref\n{sec:chap3} (\\citealg~\\ref{alg:chap3-vamp}). We consider a projection of the BP equations for the factor graph \\citefig~\\ref{fig:app-chap3-vamp-glm}.\n\n\\subparagraph{Gaussian assumptions} We start by parametrizing marginals as well as messages coming out of the Dirac factors. For $a = 1, 2$: \n\\begin{gather}\n    m_{x,{(a)}}(\\x^{(a)}) = \\cN(\\x^{(a)}, \\xh^{(a)}, \\Cx^{(a)}) \\, ,\\qquad\n    m_{z,{(a)}}(\\z^{(a)})  = \\cN(\\z^{(a)}, \\hat{\\z}^{(a)}, \\Cz^{(a)}) \\, ,\n\\end{gather}\nand\n\\begin{gather}\n    \\msgt{m}{\\psi_x}{\\x^{(a)}}(\\x^{(a)})  \\propto e^{-\\frac 1 2 {\\x^{(a)}}\\T\\A_x^{(a)} \\x^{(a)} + {\\B_x^{(a)}}\\T {\\x^{(a)}}  } \\, ,\\\\\n    \\msgt{m}{\\psi_z}{\\z^{(a)}}(\\z^{(a)})  \\propto e^{-\\frac 1 2 {\\z^{(a)}}\\T\\A_z^{(a)} \\z^{(a)} + {\\B_z^{(a)}}\\T {\\z^{(a)}}  } \\, .\n\\end{gather}\n\n\\subparagraph{Self consistency of the parametrizations at Dirac factor nodes} \nAround $\\psi_x$ the message passing equations are simply\n\\begin{gather}\n    \\label{eq:chap3-vamp-trick1}\n    \\msgt{m}{\\psi_x}{\\x^{(2)}}(\\x^{(2)}) = \\msg{m}{\\x^{(1)}}{\\psi_x}(\\x^{(2)}), \\qquad \\msgt{m}{\\psi_x}{\\x^{(1)}}(\\x^{(1)}) = \\msg{m}{\\x^{(2)}}{\\psi_x}(\\x^{(1)})\n\\end{gather}\nand similarly around $\\psi_z$. Moreover, considering that messages are marginals to which the contribution of the opposite message is retrieved we have\n\\begin{gather}\n    \\msg{m}{\\x^{(1)}}{\\psi_x}(\\x^{(1)}) \\propto m_{x,{(1)}}(\\x^{(1)}) \/ \\msgt{m}{\\psi_x}{\\x^{(1)}}(\\x^{(1)}), \\\\\n     \\msg{m}{\\x^{(2)}}{\\psi_x}(\\x^{(2)}) \\propto m_{x,{(2)}}(\\x^{(2)}) \/ \\msgt{m}{\\psi_x}{\\x^{(2)}}(\\x^{(2)}) \\, .\n\\end{gather}\nCombining this observation along with \\eqref{eq:chap3-vamp-trick1} leads to updates \\eqref{alg:chap3-vamp-Ax1}\nand \\eqref{alg:chap3-vamp-Ax2}.\nThe same reasoning can be followed for the messages around $\\psi_z$ leading to updates \\eqref{alg:chap3-vamp-Az1}\nand \\eqref{alg:chap3-vamp-Az2}.\n\n\\subparagraph{Input and output update functions}\nThe update functions of means and variances of the marginals are deduced from the parametrized message passing. For the variable $\\x^{(1)}$ taking into account the prior $p_x$, the updates are very similar to GAMP input functions: \n\\begin{align}\n    \\xh^{(1)} \n   \n     & \\propto \\int \\dd{\\x^{(1)}} \\x^{(1)} p_x(\\x^{(1)}) \\msgt{m}{\\psi_x}{\\x^{(1)}}(\\x^{(1)}) \\\\\n     & = \\frac{1}{\\cZ_x^{(1)}} \\int \\dd{\\x^{(1)}} \\x^{(1)} p_x(\\x^{(1)}) e^{-\\frac 1 2 {\\x^{(1)}}\\T\\A_x^{(1)} \\x^{(1)} + {\\B_x^{(1)}}\\T {\\x^{(1)}}} = f_1^x( {\\B_x^{(1)}}, \\A_x^{(1)}) \\, , \\\\\n     {\\Cx}^{(1)} &= \\frac{1}{\\cZ_x^{(1)}} \\int \\dd{\\x^{(1)}} \\x^{(1)}{\\x^{(1)}}\\T \n     p_x(\\x^{(1)}) e^{-\\frac 1 2 {\\x^{(1)}}\\T\\A_x^{(1)} \\x^{(1)} + {\\B_x^{(1)}}\\T {\\x^{(1)}}} \\\\\n     &\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad  - f_1^x( {\\B_x^{(1)}}, \\A_x^{(1)}) {f_1^x( {\\B_x^{(1)}}, \\A_x^{(1)}) }\\T \\notag \\\\\n     &= f_2^x( {\\B_x^{(1)}}, \\A_x^{(1)})  \\, ,\n\\end{align}\nwhere $\\cZ_x^{(1)}$ is as usual the partition ensuring the normalization.\n\nSimilarly for the variable $\\z^{(1)}$, the update functions are very similar to the GAMP output functions including the information coming from the observations:\n\\begin{align}\n    \\hat{\\z}^{(1)} & \\propto \\int \\dd{\\z^{(1)}} \\pout(\\y|\\z^{(1)}) \\msgt{m}{\\psi_z}{\\z^{(1)}}(\\z^{(1)}) \\\\\n    & = \\frac{1}{\\cZ_z^{(1)}} \\int \\dd{\\z^{(1)}} \\pout(\\y|\\z^{(1)}) e^{-\\frac 1 2 {\\z^{(1)}}\\T\\A_z^{(1)} \\z^{(1)} + {\\B_z^{(1)}}\\T {\\z^{(1)}}} = f_1^z( {\\B_z^{(1)}}, \\A_z^{(1)}) \\, , \\\\\n    {\\Cz}^{(1)} &= \\frac{1}{\\cZ_z^{(1)}} \\int \\dd{\\z^{(1)}} \\z^{(1)}{\\z^{(1)}}\\T \n     \\pout(\\y|\\z^{(1)}) e^{-\\frac 1 2 {\\z^{(1)}}\\T\\A_z^{(1)} \\z^{(1)} + {\\B_z^{(1)}}\\T {\\z^{(1)}}} \\\\\n     &\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad  - f_1^z( {\\B_z^{(1)}}, \\A_z^{(1)}) {f_1^z( {\\B_z^{(1)}}, \\A_z^{(1)}) }\\T \\notag \\\\\n     &= f_2^z( {\\B_z^{(1)}}, \\A_z^{(1)}) \\, .\n\\end{align}\n\n\\subparagraph{Linear transformation}\nFor the middle factor node we consider the vector variable concatenating $\\bar{\\x}=[\\x^{(2)} \\z^{(2)}] \\in \\R^{N + M}$. The computation of the corresponding marginal with the message passing then yields\n\\begin{gather}\n    m_{\\bar{\\x}}(\\bar{\\x}) \\propto \n   \n    \\lim\\limits_{\\Delta \\to 0}\\cN(\\z^{(2)}; \\W\\x^{(2)} , \\Delta \\mat{I}_M) e^{-\\frac 1 2 {\\x}\\T\\A_x^{(2)} \\x + {\\B_x^{(2)}}\\T {\\x}  } e^{-\\frac 1 2 {\\z}\\T\\A_z^{(2)} \\z + {\\B_z^{(2)}}\\T {\\z}}.\n\\end{gather}\nThe means of $\\x^{(2)}$ and $\\z^{(2)}$ are then updated through\n\\begin{gather}\n    \\xh^{(2)}, \\hat{\\z}^{(2)} = \\argminn{\\x, \\z}\\left[ \n        \\Vert\\W\\x - \\z \\Vert^2 \/\\Delta + {\\x}\\T\\A_x^{(2)} \\x - 2 {\\B_x^{(2)}}\\T {\\x} + {\\z}\\T\\A_z^{(2)} \\z - 2{\\B_z^{(2)}}\\T {\\z}\n         \\right],\n\\end{gather}\nat $\\Delta \\to 0$.\nAt this point it is advantageous in terms of speed to consider the singular value decomposition $\\W=\\mat{U}\\mat{S}\\mat{V}\\T$ and to simplify the form of the variance matrices by taking them proportional to the identify, i.e. $\\A_z^{(2)} = A_z^{(2)} \\mat{I}_{M}$ etc. Under this assumption the solution of the minimization problem is\n\\begin{gather}\n    \\xh^{(2)} = g^x_1({\\B_x^{(2)}}, A_x^{(2)}, {\\B_z^{(2)}}, A_z^{(2)}) = \\mat{V} \\, \\mat{D} \\left( {A^{(2)}_z}^{-2} \\mat{S} \\mat{U}\\T \\B_z^{(2)} + {A^{(2)}_x}^{-2} \\mat{V}\\T \\B_x^{(2)}   \\right) \\, , \\\\\n    \\hat{z}^{(2)} = g^z_1({\\B_x^{(2)}}, A_x^{(2)}, {\\B_z^{(2)}}, A_z^{(2)}) = \\W g^x_1({\\B_x^{(2)}}, A_x^{(2)}, {\\B_z^{(2)}}, A_z^{(2)}) \\, ,\n\\end{gather}\nwith $\\mat{D}$ a diagonal matrix with entries $D_{ii} = ({A_z^{(2)}}^{-1} S_{ii}^2+ {A_x^{(2)}}^{-1})^{-1} $.\nThe scalar variances are then updated using the traces of the Jacobians with respect to the $\\B^{(2)}$-s\n\\begin{align}\n    \\Cx^{(2)} &= \\frac{A_x^{(2)}}{N} \\mathrm{tr}\\left(\\partial g^x_2 \/ \\partial{\\B_x^{(2)}}\\right) \\mat{I}_N = \\frac{1}{N} \\sum_{i=1}^N ({A_z^{(2)}}^{-1} S_{ii}^2+ {A_x^{(2)}}^{-1})^{-1} \\mat{I}_N \\\\ &= g^x_2({\\B_x^{(2)}}, A_x^{(2)}, {\\B_z^{(2)}}, A_z^{(2)})\\\\\n    \\Cz^{(2)} &= \\frac{A_z^{(2)}}{M} \\mathrm{tr}\\left(\\partial g^z_2 \/ \\partial{\\B_z^{(2)}}\\right) \\mat{I}_M = \\frac{1}{M} \\sum_{i=1}^N S_{ii} ({A_z^{(2)}}^{-1} S_{ii}^2+ {A_z^{(2)}}^{-1})^{-1} \\mat{I}_M \\\\ &= g^z_2({\\B_x^{(2)}}, A_x^{(2)}, {\\B_z^{(2)}}, A_z^{(2)}).\n\\end{align}\n\n\\section{Multi-value AMP derivation for the GLM}\n\\label{app:chap6-vect-amp}\nWe here present the derivation of the multi-value AMP and its SE motivated in \\citesec~\\ref{sec:chap3-multivalue}, focusing on the multi-value GLM. These derivations also appear in \\cite{Gabrie2019}.\n\n\\subsection{Approximate Message Passing}\nThe systematic procedure to write AMP for a given joint probability distribution consists in first writing BP on the factor graph, second project the messages on a parametrized family of functions to obtain the corresponding relaxed-BP and third close the equations on a reduced set of parameters by keeping only leading terms in the thermodynamic limit.\n\n\\begin{figure}[t]\n    \\centering\n    {\\includegraphics[width=0.35\\textwidth]{chap6_glm_vec.pdf}\n    }\n   \n   \n   \n    \\caption{Factor graph of the Generalized Linear Model (GLM) on vector variables corresponding to the joint distribution \\eqref{eq:chap6-glm-vec-meas}. \\label{fig:chap6-vect-amp}}\n\\end{figure}\n\nFor the generic multi-value GLM the posterior measure we are interested in is\n\\begin{gather}\n    \\label{eq:chap6-glm-vec-meas}\n    p(\\X | \\Y, \\W) = \\frac{1}{\\cZ(\\Y, \\W)} \\prod_{i=1}^N p(\\x_i)\\prod_{\\mu=1}^M \\pout(\\y_\\mu | \\vect{w}_\\mu\\T\\X \/ \\sqrt{N}), \\quad \\x_i \\in \\R^P, \\quad \\y_\\mu \\in \\R^P. \n\\end{gather}\nwhere the known entries of matrix $\\W$ are drawn i.i.d from a standard normal distribution (the scaling in $1\/\\sqrt{N}$ is here made explicit). \nThe corresponding factor graph is given on \\citefig~\\ref{fig:chap6-vect-amp}. \nWe are considering the simultaneous reconstruction of $P$ signals $\\x_{0,(k)} \\in \\R^N$ and therefore write the message passing on the variables $\\x_i \\in \\R^P$.\nThe major difference with the scalar version (P=1) of AMP is that we will consider covariance matrices between variables coming from the $P$ observations instead of scalar variances. \n\n\\paragraph{Belief propagation (BP)} \nWe start with BP on the factor graph of \\citefig~\\ref{fig:chap6-vect-amp}. For all pairs of index $i-\\mu$, we define the update equations of messages function\n\\begin{gather}\n    \\label{eq:chap6-bp-calamp-1}\n    \\msg{\\tilde{m}^{(t)}}{\\mu}{i} (\\x_i) = \\frac{1}{\\msg{\\cZ}{\\mu}{i}}\n    \\int \\prod_{i'\\neq i} \\dd{\\x_{i'}}\\pout \\left(\\y_\\mu |  \\sum_j \\frac{W_{\\mu j}}{\\sqrt{N}}\\x_j\\right) \\prod_{i'\\neq i} \\msg{m^{(t)}}{i'}{\\mu}(\\x_{i'})\\\\\n    \\label{eq:chap6-bp-calamp-2}\n    \\msg{m^{(t+1)}}{i}{\\mu} (\\x_i) = \\frac{1}{\\msg{\\cZ}{i}{\\mu}}\n    p_x(\\x_i)\\prod_{\\mu' \\neq \\mu} \\msg{\\tilde{m}^{(t)}}{\\mu'}{i}(\\x_i),\n\\end{gather}\nwhere $\\msg{\\cZ}{\\mu}{i}$ and $\\msg{\\cZ}{i}{\\mu}$ are normalization function that allow to interpret messages as probability distributions.\nTo improve readability, we drop the time indices in the following derivation, and only specify them in the final algorithm.\n\n\\paragraph{Relaxed BP (r-BP)} The second step of the derivation is to develop messages keeping only terms up to order $O(1\/N)$ as we take the thermodynamic limit $N \\to + \\infty$ (at fixed $\\alpha = M\/N$). At this order, we will find that it is consistent to consider the messages to be approximately Gaussian, i.e. characterized by their means and co-variances. Thus we define\n\\begin{gather}\n\\label{eq:chap6-vect-amp-rbp-def-xh}\n\\msg{\\xh}{i}{\\mu} = \\int \\dd{\\x} \\x \\; \\msg{m}{i}{\\mu}(\\x) \\\\\n\\label{eq:chap6-vect-amp-rbp-def-Cx}\n\\msg{\\Cx}{i}{\\mu} = \\int \\dd{\\x}  \\x \\x^T \\; \\msg{m}{i}{\\mu}(\\x) - \\msg{\\xh}{i}{\\mu}\\msg{\\xh\\T}{i}{\\mu}\n\\end{gather}\nand\n\\begin{gather}\n\\label{eq:chap6-vect-amp-rbp-def-w}\n\\displaystyle\n\\msg{\\w}{\\mu}{i} = \\sum_{i' \\neq i} \\frac{\\Wuip}{\\sqrt{N}}\\msg{\\xh}{i'}{\\mu} \\\\\n\\label{eq:chap6-vect-amp-rbp-def-V}\n\\displaystyle\n\\msg{\\V}{\\mu}{i} = \\sum_{i' \\neq i}\\frac{\\Wuip^2}{N} \\msg{\\Cx}{i'}{\\mu}, \n\\end{gather}\nwhere $\\msg{\\w}{\\mu}{i}$ and $\\msg{\\V}{\\mu}{i}$ are related to the intermediate variable $\\z_\\mu = \\vect{w}_\\mu\\T \\X$.\n\n\\subparagraph{Expansion of $\\msgt{m}{\\mu}{i}$ -} \nWe defined the Fourier transform $\\hat{p}_{\\rm out}$ of $\\pout(\\y_\\mu|\\z_\\mu)$ with respect to its argument $\\z_\\mu = \\vect{w}_\\mu\\T\\X$,\n\\begin{gather}\n    \\hat{p}_{\\rm out}(\\y_\\mu|\\vect{\\xi}_\\mu) = \\int \\dd{\\z_\\mu} \\hat{p}_{\\rm out}(\\y_\\mu | \\z_\\mu) \\, e^{- i \\vect{\\xi}_\\mu\\T \\z_\\mu}.\n\\end{gather}\nUsing reciprocally the Fourier representation of $\\pout(\\y_\\mu|\\z_\\mu)$,\n\\begin{gather}\n    \\pout(\\y_\\mu|\\z_\\mu) = \\frac{1}{(2\\pi)^M} \\int \\dd{\\vect{\\xi}_\\mu} \\hat{p}_{\\rm out}(\\y_\\mu | \\vect{\\xi}_\\mu) \\, e^{i \\vect{\\xi}_\\mu\\T \\z_\\mu},\n\\end{gather}\nwe decouple the integrals over the different $\\x_{i'}$ in \\eqref{eq:chap6-bp-calamp-1},\n\\begin{align}\n    \\label{eq:chap6-deric-calamp-1}\n    \\msgt{m}{\\mu}{i} (\\x_i) &\n    \\propto  \\int \\dd{\\vect{\\xi}_\\mu} \\hat{p}_{\\rm out} \\left(\\y_\\mu | \\vect{\\xi}_\\mu\\right) e^{i \\frac{\\Wui}{\\sqrt{N}}\\vect{\\xi}_\\mu\\T\\x_i} \\prod_{i'\\neq i} \\int  \\dd{\\x_{i'}} \\msg{m}{i'}{\\mu}(\\x_{i'})e^{i \\frac{\\Wuip}{\\sqrt{N}}\\x_i \\vect{\\xi}_\\mu\\T\\x_{i'}} \\\\\n    \\label{eq:chap6-deric-calamp-2}\n    & \\propto \\int \\dd{\\vect{\\xi}_\\mu} \\hat{p}_{\\rm out} \\left(\\y_\\mu | \\vect{\\xi}_\\mu\\right) e^{i \\underline{\\xi}\\T \\left(\\frac{\\Wui}{\\sqrt{N}}\\x_i + \\msg{\\w}{\\mu}{i}\\right) - \\frac 1 2 \\underline{\\xi}\\T \\msg{{\\V}^{-1}}{\\mu}{i} \\underline{\\xi}}\n\\end{align}\nwhere developing the exponentials of the product in \\eqref{eq:chap6-bp-calamp-1}  allows to express the integrals over the $\\x_{i'}$ as a function of the definitions \\eqref{eq:chap6-vect-amp-rbp-def-w}-\\eqref{eq:chap6-vect-amp-rbp-def-V}, before re-exponentiating to obtain the final result \\eqref{eq:chap6-deric-calamp-2}.\nNow reversing the Fourier transform and performing the integral over $\\vect{\\xi}$ we can further rewrite \n\\begin{align}\n    \\msgt{m}{\\mu}{i} (\\x_i) &\n    \\propto  \\int \\dd{\\z_\\mu} \\pout \\left(\\y_\\mu | \\z_\\mu\\right) e^{- \\frac{1}{2} \n    \\left( \\z_\\mu - \\frac{\\Wui}{\\sqrt{N}}\\x_i - \\msg{\\w}{\\mu}{i}\\right)\\T\n    \\msg{{\\V}^{-1}}{\\mu}{i}\n    \\left( \\z_\\mu - \\frac{\\Wui}{\\sqrt{N}}\\x_i - \\msg{\\w}{\\mu}{i}\\right)\n    } \\\\\n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n    \\label{eq:chap6-vect-amp-rbp-1}\n    &\n    \\propto  \\int \\dd{\\z_\\mu} \\mathbb{P}_{\\rm out}(\\z_\\mu; \\msg{\\w}{\\mu}{i}, \\msg{\\V}{\\mu}{i} ) e^{ \\left( \\z_\\mu - \\msg{\\w}{\\mu}{i} \\right)\\T\n    \\msg{{\\V}^{-1}}{\\mu}{i}\n    \\frac{\\Wui}{\\sqrt{N}}\\x_i \n    - \\frac{\\Wui^2}{2N}\\x_i\\T \\msg{{\\V}^{-1}}{\\mu}{i} \\x_i\n    },\n\\end{align}\nwhere we are led to introduce the \\emph{output update functions},\n\\begin{gather}\n    \\label{eq:chap6-vect-amp-Pout}\n    \\mathbb{P}_{\\rm out}(\\z_\\mu; \\msg{\\w}{\\mu}{i}, \\msg{\\V}{\\mu}{i} ) =  \\pout \\left(\\y_\\mu | \\z_\\mu\\right) \\cN(\\z_\\mu; \\msg{\\w}{\\mu}{i}, \\msg{\\V}{\\mu}{i} ) \\, ,\\\\\n    \\label{eq:chap6-vect-amp-Zout}\n    \\Zout(\\y_\\mu , \\msg{\\w}{\\mu}{i}, \\msg{\\V}{\\mu}{i} ) = \\int \\dd{z_\\mu} \\pout \\left(\\y_\\mu | \\z_\\mu\\right) \\cN(\\z_\\mu; \\msg{\\w}{\\mu}{i}, \\msg{\\V}{\\mu}{i} ) \\, ,\\\\\n    \\label{eq:chap6-vect-amp-gout-dgout}\n    \\gout(\\y_\\mu , \\msg{\\w}{\\mu}{i}, \\msg{\\V}{\\mu}{i} ) = \\frac{1}{\\Zout} \\frac{\\partial \\Zout}{\\partial \\w} \\quad \\text{ and } \\quad\n    \\dgout = \\frac{\\partial \\gout}{\\partial \\w},\n\\end{gather}\nwhere $\\cN(\\z;\\w,\\V)$ is the multivariate Gaussian distribution of mean $\\w$ and covariance $\\V$.\nFurther expanding the exponential in \\eqref{eq:chap6-vect-amp-rbp-1} up to order $O(1\/N)$ leads to the Gaussian parametrization \n\\begin{align}\n    \\msgt{m}{\\mu}{i} (\\x_i) & \\propto 1 + \\frac{{\\Wui}}{\\sqrt{N}} \\gout \\x_i + \\frac{{{\\Wui}}^2}{2 N} {\\x_i}^T (\\gout\\gout^T + \\dgout^1) \\x_i \\\\\n    & \\propto e^{{\\msg{\\B}{\\mu}{i}}^T\\x_i - \\frac 1 2 {\\x_i}^T\\msg{\\A}{\\mu}{i}\\x_i}, \n\\end{align}\nwith\n\\begin{gather}\n    \\label{eq:chap6-vect-amp-rb-B}\n    \\msg{\\B}{\\mu}{i}  = \\frac{{\\Wui}}{\\sqrt{N}} \\gout (\\y_\\mu , \\msg{\\w}{\\mu}{i}, \\msg{\\V}{\\mu}{i} ) \\\\\n    \\label{eq:chap6-vect-amp-rb-A}\n    \\msg{\\A}{\\mu}{i}  = - \\frac{{{\\Wui}}^2}{ N} \\dgout(\\y_\\mu , \\msg{\\w}{\\mu}{i}, \\msg{\\V}{\\mu}{i} ) .\n\\end{gather}\n\n\\subparagraph{Consistency with $\\msg{m}{i}{\\mu}$ -}\nInserting the Gaussian approximation of $\\msgt{m}{\\mu}{i}$ in the definition of $\\msg{m}{i}{\\mu}$, we get the parametrization\n\\begin{align}\n    \\msg{m}{i}{\\mu}(\\x_i) & \\propto p_x(\\x_i) \\prod_{\\mu' \\neq \\mu} e^{{\\msg{\\B}{\\mu'}{i}}^T\\x_i - \\frac 1 2 {\\x_i}^T\\msg{\\A}{\\mu'}{i}\\x_i}  \\propto p_x(\\x_i) e^{-\\frac{1}{2}(\\x_i - \\msg{\\lbd}{i}{\\mu})^T \\msg{\\sig}{i}{\\mu}^{-1} (\\x_i - \\msg{\\lbd}{i}{\\mu})}\n\\end{align}\nwith\n\\begin{gather}\n    \\label{eq:chap6-vect-amp-rbp-lbd}\n\\msg{\\lbd}{i}{\\mu}  = \\msg{\\sig}{i}{\\mu}\\left( \\sum_{\\mu' \\neq \\mu} \\msg{\\B}{\\mu'}{i} \\right)  \\\\\n\\label{eq:chap6-vect-amp-rbp-sig}\n\\msg{\\sig}{i}{\\mu}  = \\left( \\sum_{\\mu' \\neq \\mu} \\msg{\\A}{\\mu'}{i} \\right)^{-1} .\n\\end{gather} \n\n\\subparagraph{Closing the equations -}\nEnsuring the consistency with the definitions \\eqref{eq:chap6-vect-amp-rbp-def-xh}-\\eqref{eq:chap6-vect-amp-rbp-def-Cx} of mean and covariance of $\\msg{m}{i}{\\mu}$ we finally close our set of equations by defining the \\emph{input update functions},\n\\begin{gather}\n    \\label{eq:chap6-vect-amp-Zx}\n    \\cZ^x = \\int \\dd{x} p_x(\\x)e^{-\\frac 1 2 (\\x-\\lbd)\\T\\sigma^{-1}(\\x-\\lbd)} \\\\\n    \\label{eq:chap6-vect-amp-f1x}\n    \\vect{f}^x_1(\\lbd, \\sig) = \\frac{1}{\\cZ^x}\\int \\dd{\\x} \\x \\, p_x(\\x)e^{-\\frac 1 2 (\\x-\\lbd)\\T\\sig^{-1}(\\x-\\lbd)} \\\\\n    \\label{eq:chap6-vect-amp-f2x}\n    \\mat{f}^x_2(\\lbd, \\sig) =  \\frac{1}{\\cZ^x} \\int \\dd{x} \\x\\x\\T \\, p_x(\\x)e^{-\\frac 1 2 (\\x-\\lbd)\\T\\sig^{-1}(\\x-\\lbd)} - \\vect{f}^x_1(\\lbd, \\sig)\\vect{f}^x_1(\\lbd, \\sig)\\T,\n\\end{gather}\nso that\n\\begin{gather}\n    \\label{eq:chap6-vect-amp-rb-xh}\n    \\msg{\\xh}{i}{\\mu}  =  \\vect{f}^x_1(\\msg{\\lbd}{i}{\\mu} , \\msg{\\sig}{i}{\\mu}) \\\\\n    \\label{eq:chap6-vect-amp-rb-Cx}\n    \\msg{\\Cx}{i}{\\mu}  = \\mat{f}^x_2(\\msg{\\lbd}{i}{\\mu} , \\msg{\\sig}{i}{\\mu}) .\n\\end{gather}\n\nThe closed set of equations \\eqref{eq:chap6-vect-amp-rbp-def-w}, \\eqref{eq:chap6-vect-amp-rbp-def-V},  \\eqref{eq:chap6-vect-amp-rb-B} \\eqref{eq:chap6-vect-amp-rb-A}, \\eqref{eq:chap6-vect-amp-rbp-lbd}, \\eqref{eq:chap6-vect-amp-rbp-sig}, \\eqref{eq:chap6-vect-amp-rb-xh} and \\eqref{eq:chap6-vect-amp-rb-Cx}, with restored time indices, defines the r-BP algorithm. At convergence of the iterations, we obtain the approximated marginals\n\\begin{align}\n    \\label{eq:chap6-vect-amp-marginal-def}\n    m_i(\\x_i) = \\frac 1 {\\cZ_i} p_x(\\x_i) e^{-\\frac 1 2 (\\x-\\lbd_i)\\T\\sig_i^{-1}(\\x-\\lbd_i)} \n\\end{align}\nwith \n\\begin{gather}\n   \n    \\lbd_i  = \\sig_i\\left( \\sum\\limits_{\\mu=1}^M \\msg{\\B}{\\mu}{i} \\right)  \\\\\n    \\sig_i  = \\left( \\sum\\limits_{\\mu}^M \\msg{\\A}{\\mu}{i} \\right)^{-1} .\n   \n\\end{gather}.\n\nAs usual, while BP requires to follow iterations over $M \\times N$ message distributions over vectors in $\\R^P$, r-BP only requires to track $O(M \\times N \\times P)$ variables, which is a great simplification. Nonetheless, r-BP can be further reduced to the more practical GAMP algorithm, given the scaling of the weights in $O(1\/\\sqrt{N})$.\n\n\\paragraph{Approximate message passing}\nWe define parameters $\\w_\\mu$, $\\V_\\mu$ and $\\xh_i$, $\\Cx_i$, likewise $\\lbd_i$ and $\\sig_i$ defined above and consider their relations to the original $\\msg{\\lbd}{i}{\\mu}$, $\\msg{\\sig}{i}{\\mu}$, $\\msg{\\w}{\\mu}{i}$, $\\msg{\\V}{\\mu}{i}$, $\\msg{\\xh}{i}{\\mu}$ and $\\msg{\\Cx}{i}{\\mu}$. As a result we obtain the vectorized AMP for the GLM presented in \\citealg~\\ref{alg:chap6-vect-amp}. \nNote that, similarly to GAMP, relaxing the Gaussian assumption on the weight matrix entries to any distribution with finite second moment yields the same algorithm using the Central Limit Theorem.\n\n\n\\subsection{State Evolution}\nWe consider the limit  $N \\to + \\infty$ at fixed $\\alpha = M\/N$ and a quenched average over the disorder (here the realizations of $\\X_0$, $s_0$, $\\Y$ and $\\W$), to derive a State Evolution analysis of the previously derived AMP.\nTo this end, our starting point will be the r-BP equations.\n\n\n\\subsubsection{State Evolution derivation in mismatched prior and channel setting}\n\\label{sec:chap6-se-cal-amp}\n\\paragraph{Definition of the overlaps}\nThe important quantities to follow the dynamic of iterations and fixed points of AMP are the overlaps. Here, they are the $P \\times P$ matrices\n\\begin{gather}\n    \\q = \\frac{1}{N} \\sum_{i=1}^N \\xh_i \\xh_i^T, \\quad \\mm = \\frac{1}{N} \\sum_{i=1}^N \\xh_i {\\x_{0,i}}^T, \\quad \\q_0 = \\frac{1}{N} \\sum_{i=1}^N \\x_{0,i} {\\x_{0,i}}^T.\n\\end{gather} \n\n\\paragraph{Output parameters}\nUnder independent statistics of the entries of $\\W$ and under the assumption of independent incoming messages, the variable $\\msg{\\w}{\\mu}{i}$ defined in \\eqref{eq:chap6-vect-amp-rbp-def-w} is a sum of independent variables and follows a Gaussian distribution by the Central Limit Theorem. Its first and second moments are\n\\begin{align}\n    \\EE{\\W}{\\msg{\\w}{\\mu}{i}} & = \\frac{1}{\\sqrt{N}} \\sum_{i'\\neq i} \\EE{\\W}{W_{\\mu i'}} \\msg{\\xh}{i'}{\\mu} = 0 \\, ,\n\\end{align}\n\\begin{align}\n\\EE{\\W}{\\msg{\\w}{\\mu}{i} \\msg{\\w}{\\mu}{i} ^T} \n    & =  \\frac{1}{N} \\sum_{i'\\neq i} \\sum_{i''\\neq i} \\EE{\\W}{W_{\\mu i''}W_{\\mu i'}}\\msg{\\xh}{i''}{\\mu}\\msg{\\xh}{i'}{\\mu}^T \\\\\n\t& =  \\frac{1}{N} \\sum_{i'\\neq i} \\EE{\\W}{W^2_{\\mu i'}} \\msg{\\xh}{i'}{\\mu}\\msg{\\xh}{i'}{\\mu}^T =  \\frac{1}{N} \\sum_{i'=1}^N \\msg{\\xh}{i'}{\\mu}\\msg{\\xh}{i'}{\\mu}^T + O\\left({1}\/{N}\\right)  \\notag \\\\\n\t& =  \\frac{1}{N} \\sum_{i=1}^N \\xh_{i'}\\xh_{i'}^T - \\partial_{\\lbd} \\vect{f}^x_1\\sig_i\\msg{B}{\\mu}{i} \\xh_i^T - \\left(\\partial_{\\lbd} \\vect{f}_1^x\\sig_i\\msg{B}{\\mu}{i} \\xh_i^T \\right)^T +O\\left({1}\/{N}\\right) \\notag\\\\\n\t& = \\frac{1}{N} \\sum_{i'=1}^N \\xh_{i'}\\xh_{'i}^T + O\\left({1}\/{\\sqrt{N}}\\right)\\,\n\\end{align}\nwhere we used the facts that the $W_{\\mu i}$-s are independent with zero mean, and that $\\msg{B}{\\mu}{i}$, defined in \\eqref{eq:chap6-vect-amp-rb-B}, is of order $O(1\/\\sqrt{N})$.\nSimilarly, the variable $\\msg{\\z}{\\mu}{i} = \\sum_{i'\\neq i} \\frac{W_{\\mu i'}}{\\sqrt{N}} \\x_{i'}$ is Gaussian with first and second moments\n\\begin{gather}\n    \\EE{\\W}{\\msg{\\z}{\\mu}{i}} \n   \n    = \\frac{1}{\\sqrt{N}} \\sum_{i'\\neq i} \\EE{\\W}{W_{\\mu i'}} \\x_{0,i'} = 0 \\, ,\n    \\\\\n   \n    \\EE{\\W}{\\msg{\\z}{\\mu}{i} \\msg{\\z}{\\mu}{i} ^T} \n       \n        = \\frac{1}{N} \\sum_{i'=1}^N \\x_{0,i'}{\\x_{0,i'}}^T + O\\left({1}\/{\\sqrt{N}}\\right). \n    \\end{gather}\nFurthermore, their covariance is \n\\begin{align}\n    \\EE{\\W}{\\msg{\\z}{\\mu}{i} \\msg{\\w}{\\mu}{i}^T} \n        & =  \\frac{1}{N} \\sum_{i'\\neq i} \\EE{\\W}{W^2_{\\mu i'}} \\x_{0,i'}\\msg{\\xh}{i'}{\\mu}^T =  \\frac{1}{N} \\sum_{i'=1}^N \\x_{0,i'}\\msg{\\xh}{i'}{\\mu}^T + O\\left({1}\/{N}\\right)  \\\\\n        & =  \\frac{1}{N} \\sum_{i'=1}^N \\x_{0,i'}\\xh_{i'}^T -  \\x_{0,i}\\partial_{\\lbd} \\vect{f}^x_1\\sig_i\\msg{\\B}{\\mu}{i}^T +O\\left({1}\/{N}\\right) \\\\\n        & = \\frac{1}{N} \\sum_{i'=1}^N \\x_{0,i'}\\xh_{i'}^T + O\\left({1}\/{\\sqrt{N}}\\right). \n    \\end{align} \nHence we find that for all $\\mu$-s and all $i$-s, $  \\msg{\\w}{\\mu}{i}$ and $ \\msg{\\z}{\\mu}{i}$ are approximately jointly Gaussian in the thermodynamic limit following a unique distribution $\\cN\\left( \\msg{\\z}{\\mu}{i}, \\msg{\\w}{\\mu}{i}; \\; \\vect{0}, \\, \\Q \\right) $ with the block covariance matrix\n\\begin{gather}\n        \\Q = \n        \\begin{bmatrix}\n        \\q_0 & \\mm \\\\\n        \\\\\n        {\\mm}\\T & \\q \\\\ \n        \\end{bmatrix}.\n\\end{gather}\nFor the variance message $\\msg{\\V}{\\mu}{i}$, defined in \\eqref{eq:chap6-vect-amp-rbp-def-V}, we have \n\\begin{align}\n    \\EE{\\W}{\\msg{\\V}{\\mu}{i}} &= \\sum_{i'\\neq i} \\EE{\\W}{\\frac{W_{\\mu i'}}{N}^2} \\msg{\\Cx}{i'}{\\mu} = \\sum_{i'=1}^N \\frac{1}{N} \\msg{\\Cx}{i'}{\\mu} + O\\left({1}\/{N}\\right) \\\\\n    &= \\sum_{i'=1}^N \\frac{1}{N} \\Cx_{i'} + O\\left({1}\/{\\sqrt{N}}\\right) ,\n\\end{align}\nwhere using the developments of $\\msg{\\lbd}{i}{\\mu}$ and $\\msg{\\sig}{i}{\\mu}$ \\eqref{eq:chap6-vect-amp-rbp-lbd}-\\eqref{eq:chap6-vect-amp-rbp-sig}, along with the scaling of $\\msg{\\B}{\\mu}{i} $ in $O({1}\/{\\sqrt{N}})$ we replaced \n\\begin{align}\n    \\msg{\\Cx}{i}{\\mu} = \\mat{f}_2^x(\\msg{\\lbd}{i}{\\mu}, \\msg{\\sig}{i}{\\mu}) = \\mat{f}_2^x(\\lbd_i, \\sig_i) - \\partial_{\\lbd}\\mat{f}^x_2 \\sig_i \\msg{\\B}{\\mu}{i}^T = \\mat{f}_2^x(\\lbd_i, \\sig_i) + O\\left({1}\/{\\sqrt{N}}\\right).\n\\end{align}\nFuthermore, we can check that \n\\begin{gather}\n    \\lim_{N\\to + \\infty} \\EE{\\W}{\\msg{\\V}{\\mu}{i}^2 - \\EE{\\W}{\\msg{\\V}{\\mu}{i}}^2} = 0 ,\n\\end{gather}\nmeaning that all $\\msg{\\V}{\\mu}{i}$ concentrate on their identical mean in the thermodynamic limit, which we note\n\\begin{gather}\n    \\V = \\sum_{i=1}^N \\frac{1}{N} \\Cx_i .\n\\end{gather}\n\n\\paragraph{Input parameters} Here we use the re-parametrization trick to express $\\y_\\mu$ as a function $g_0(\\cdot)$ taking\n a noise $\\eps_\\mu \\sim p_\\epsilon(\\eps_\\mu)$ as inputs:\n    $\\y_\\mu = g_0(\\vect{w}_\\mu\\T\\X_0, \\eps_\\mu)$.\nFollowing \\eqref{eq:chap6-vect-amp-rb-A}-\\eqref{eq:chap6-vect-amp-rb-B} and \\eqref{eq:chap6-vect-amp-marginal-def}, \n\\begin{align}\n    \\sig_i^{-1}\\lbd_i \n    & = \\sum_{\\mu=1}^M \\frac{W_{\\mu i}}{\\sqrt{N}} \\gout\\left(\\y_\\mu, \\msg{\\w}{\\mu}{i},  \\msg{\\V}{\\mu}{i}\\right) \\\\\n    & =  \\sum_{\\mu=1}^M \\frac{W_{\\mu i}}{\\sqrt{N}} \\gout\\left(g_0\\left( \\sum_{i'\\neq i} \\frac{W_{\\mu i'}}{\\sqrt{N}} \\x_{0,i'} + \\frac{W_{\\mu i}}{\\sqrt{N}} \\x_{0,i}, \\eps_\\mu \\right), \\msg{\\w}{\\mu}{i},  \\msg{\\V}{\\mu}{i}\\right) \\\\\n    & =  \\sum_{\\mu=1}^M \\frac{W_{\\mu i}}{\\sqrt{N}} \\gout\\left(g_0\\left( \\sum_{i'\\neq i} \\frac{W_{\\mu i'}}{\\sqrt{N}} \\x_{0,i'}, \\eps_\\mu \\right), \\msg{\\w}{\\mu}{i},  \\msg{\\V}{\\mu}{i}\\right)  \\notag\\\\\n    &\\qquad \\qquad + \\sum_{\\mu=1}^M \\frac{W^2_{\\mu i}}{N}  \\partial_{z} \\mat{\\gouts}\\left(g_0\\left( \\msg{\\z}{\\mu}{i}, \\eps_\\mu\\right), \\msg{\\w}{\\mu}{i}, \\msg{\\V}{\\mu}{i}\\right) \\x_{0,i}.\n\\end{align}\nThe first term is again a sum of independent random variables, given the $W_{\\mu i}$ are i.i.d. with zero mean, of which the messages of type $\\mu \\to i$ are assumed independent. The second term has non-zero mean and can be shown to concentrate. Finally recalling that all $\\msg{\\V}{\\mu}{i}$ also concentrate on $\\V$ we obtain the distribution\n\\begin{gather}\n    \\sig_i^{-1}\\lbd_i \\sim \\cN\\left(\\sig_i^{-1}\\lbd_i;\\; \\alpha \\mh \\x_{0,i}, \\sqrt{\\alpha \\qh} I_P\\right) \n\\end{gather}\nwith\n\\begin{gather}\n    \\label{eq:chap6-se-non-nishi-qh}\n    \\qh  = \\int \\dd{\\eps} p_{\\epsilon}(\\eps) \\dd{s_0}p_{s_0}(s_0) \\int \\dd{\\w} \\dd{\\z} \\cN(\\z, \\w ; \\underline{0}, \\Q) \n    \\gout(g_0\\left( \\z  ,  \\eps\\right) ,\\w,  \\V) \\times \\\\\n    \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\gout(g_0\\left( \\z  , \\eps\\right), \\w,  \\V)^T \\, ,\\notag \\\\\n\t\\label{eq:chap6-se-non-nishi-mh}\n    \\mh  = \\int \\dd{\\eps} p_{\\epsilon}(\\eps) \\dd{s_0}p_{s_0}(s_0) \\int \\dd{\\w} \\dd{\\z} \\cN(\\z, \\w ; \\underline{0}, \\Q)\n\t\\partial_{\\z} \\mat{\\gouts}(g_0\\left( \\z  ,  \\eps\\right), \\w , \\V) .\n\\end{gather}\nFor the inverse variance $\\sig_i^{-1}$ one can check again that it concentrates on its mean \n\\begin{gather}\n    \\sig_i^{-1}  = \\sum\\limits_{\\mu=1}^M \\frac{{\\Wui}^2}{ N} \\dgout(\\y_\\mu , \\msg{\\w}{\\mu}{i}, \\msg{\\V}{\\mu}{i} ) \\simeq \\alpha \\chih \\, , \\\\\n    \\label{eq:chap6-se-non-nishi-chih}\n    \\chih  = - \\int \\dd{\\eps} p_\\epsilon(\\eps) \\dd{s_0} p_{s_0}(s_0)\\int \\dd{\\eps} \\dd\\z \\cN(\\z, \\w ; \\underline{0}, \\Q) \n\t \\partial_{\\omega} \\mat{\\gouts}( g_0\\left( \\z , \\eps\\right), \\w , \\V) \\, .\n\\end{gather}\n\n\\paragraph{Closing the equations} These statistics of the input parameters must ensure that consistently\n\\begin{gather}\n    \\V = \\frac{1}{N}\\sum\\limits_{i=1}^N \\Cx_i = \\EE{\\lbd, \\sig}{\\mat{f}^x_2(\\lbd, \\sig)} ,\\\\\n    \\q = \\frac{1}{N} \\sum\\limits_{i=1}^N \\xh_i \\xh_i\\T = \\EE{\\lbd, \\sig}{\\mat{f}^x_1(\\lbd, \\sig)\\mat{f}^x_1(\\lbd, \\sig)\\T},\\\\\n    \\mm = \\frac{1}{N} \\sum\\limits_{i=1}^N \\xh_i {\\x_{0,i}}\\T = \\EE{\\lbd, \\sig}{ \\mat{f}^x_1(\\lbd, \\sig){\\x_{0,i}}\\T } ,\n\\end{gather}\nwhich gives upon expressing the computation of the expectations\n\\begin{gather}\n    \\label{eq:chap6-se-non-nishi-V}\n    \\V = \\int \\dd{\\x_0}p_{x_0}(\\x_0) \\int \\D{\\vect{\\xi}} \\mat{f}^x_2 \\left( (\\alpha \\chih)^{-1}\\left({\\sqrt{\\alpha \\qh} \\underline{\\xi} + \\alpha \\mh\\x_0}\\right); (\\alpha \\chih)^{-1}  \\right) \\, , \\\\\n    \\label{eq:chap6-se-non-nishi-m}\n    \\mm = \\int \\dd{\\x_0}p_{x_0}(\\x_0) \\int \\D{\\vect{\\xi}} \\vect{f}^x_1 \\left( (\\alpha \\chih)^{-1}\\left({\\sqrt{\\alpha \\qh} \\underline{\\xi} + \\alpha \\mh\\x_0}\\right); (\\alpha \\chih)^{-1}  \\right){\\x_0}\\T \\, , \\\\\n    \\q =  \\int \\dd{\\x_0}p_{x_0}(\\x_0) \\int \\D{\\vect{\\xi}} \\vect{f}^x_1 \\left( (\\alpha \\chih)^{-1}\\left({\\sqrt{\\alpha \\qh} \\underline{\\xi} + \\alpha \\mh\\x_0}\\right); (\\alpha \\chih)^{-1}  \\right) \\times \\notag\\\\\n    \\label{eq:chap6-se-non-nishi-q}\n    \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\vect{f}^x_1 \\left( (\\alpha \\chih)^{-1}\\left({\\sqrt{\\alpha \\qh} \\underline{\\xi} + \\alpha \\mh\\x_0}\\right); (\\alpha \\chih)^{-1}  \\right)\\T .\n\\end{gather}\nThe State Evolution analysis of the GLM on the vector variables finally consists in iterating alternatively the equations \\eqref{eq:chap6-se-non-nishi-qh}, \\eqref{eq:chap6-se-non-nishi-mh}, \\eqref{eq:chap6-se-non-nishi-chih}, and the equations \\eqref{eq:chap6-se-non-nishi-V}, \\eqref{eq:chap6-se-non-nishi-m} \\eqref{eq:chap6-se-non-nishi-q} until convergence.\n\n\n\\paragraph{Performance analysis}\nThe mean squared error (MSE) on the reconstruction of $\\X$ by the AMP algorithm is then predicted by \n\\begin{gather}\n\\MSE(\\X) = q - 2 m + q_0,\n\\end{gather}\nwhere the scalar values used here correspond to the (unique) value of the diagonal elements of the corresponding overlap matrices. This MSE can be computed throughout the iterations of State Evolution. \nRemarkably, the State Evolution MSEs follow precisely the MSE of the cal-AMP predictors along the iterations of the algorithm provided the procedures are initialized consistently. A random initialization of $\\xh_i$ in cal-AMP corresponds to an initialization of zero overlap $m = 0$, $\\nu = 0$, with variance of the priors $q = q_0$ in the State Evolution.\n\n\\subsubsection{Bayes optimal State Evolution}\nThe SE equations can be greatly simplified in the Bayes optimal setting where the statistical model used by the student (priors $p_x$ and $p_s$, and channel $\\pout$) is known to match the teacher. \nIn this case, the true unknown signal $\\X_0$ is in some sense statistically equivalent to the estimate $\\mat{\\hat{X}}$ coming from the posterior. More precisely one can prove the Nishimori identities \\cite{Opper1991, Iba1999, Nishimori2001} (or \\cite{Kabashima2016} for a concise demonstration and discussion) implying that\n\\begin{gather}\n    \\q =  \\mm, \\quad  \\V = \\q_0  - \\mm, \\quad \\qh  = \\mh =\\chih  \\quad  \\text{ and } \\quad r = \\nu.\n\\end{gather}\nAs a result the State Evolution reduces to a set of two equations\n\\begin{gather}\n   \n   \n   \n    \\label{eq:chap6-se-vect-glm-bayes-opt-m}\n    \\mm = \\int \\dd{\\x_0}p_{x_0}(\\x_0) \\int \\D{\\vect{\\xi}} \\vect{f}^x_1 \\left( (\\alpha \\mh)^{-1}\\left({\\sqrt{\\alpha \\mh} \\underline{\\xi} + \\alpha \\mh\\x_0}\\right); (\\alpha \\mh)^{-1}  \\right){\\x_0}\\T \\, \\\\\n    \\label{eq:chap6-se-vect-glm-bayes-opt-mh}\n    \\mh  = \\int \\dd{\\eps} p_{\\epsilon}(\\eps) \\dd{s_0}p_{s_0}(s_0) \\int \\dd{\\w} \\dd{\\z} \\cN(\\z, \\w ; \\underline{0}, \\Q)\n    \\gout\\left(g_0\\left( \\z  ,  \\eps\\right), \\w , \\q_0 - \\m)\\right) \\times \\\\\n    \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad  \\qquad \\qquad \\qquad \\gout\\left(g_0\\left( \\z  ,  \\eps\\right), \\w , \\q_0 - \\m)\\right) \\notag,\n\\end{gather}\nwith the block covariance matrix\n\\begin{gather}\n    \\label{eq:chap6-Q-bayes-opt}\n    \\Q = \n    \\begin{bmatrix}\n    \\q_0 & \\mm \\\\\n    \\\\\n    {\\mm}\\T & \\mm \\\\ \n    \\end{bmatrix}.\n\\end{gather}\n\\section{Mean-field identity}\nWe derive the exact identity \\eqref{eq:chap3-mf-identity} for fully connected Ising model with binary spins $\\x \\in \\{0,1\\}^N$,\n\\label{app:chap3-mf-identity}\n\\begin{align}\n    \\label{eq:app-chap3-1}\n    \\langle x_i \\rangle_p &= \\frac{1}{\\cZ}\n    \\sum_{\\x\\in \\{0,1\\}^N} \\, x_i \\, \\exp\\left(\\displaystyle\\beta \\sum_j b_j x_j + \\frac 1 2 \\sum_{jk}W_{jk}x_j x_k\\right) \\\\\n    &= \\frac{1}{\\cZ}\\sum_{\\x_{\\setminus i}\\in \\{0,1\\}^{N-1}} \\exp\\left(\\displaystyle\\beta \\sum_{j\\neq i} b_j x_j + \\frac 1 2 \\sum_{\\underset{j\\neq i}{k \\neq i}}W_{ij}x_i x_j\\right) \\sum_{x_i \\in \\{0,1\\}} x_i e^{\\beta b_i x_i + \\sum_j W_{ij}x_i x_j} \\notag\n   \n\\end{align}\nwhere $\\x_{\\setminus i}$ is the vector of $\\x$ without its $i$-th component.\nYet\n\\begin{gather}\n    \\sigm(\\beta b_i + \\sum_{j \\in \\partial i}\\beta W_{ij} x_j) = \\frac\n    {\\sum_{x_i \\in \\{0,1\\}} \\, x_i\\,  e^{\\beta b_i x_i + \\sum_j W_{ij}x_i x_j}}\n    {\\sum_{x_i \\in \\{0,1\\}} e^{\\beta b_i x_i + \\sum_j W_{ij}x_i x_j}},\n\\end{gather}\nso that multiplying and dividing \\eqref{eq:app-chap3-1} by the denominator above we obtain the identity \\eqref{eq:chap3-mf-identity} in \\citesec~\\ref{sec:chap3-nmf}\n\\begin{align}\n    \\langle x_i \\rangle_p = \\langle \\sigm(\\beta b_i + \\sum_{j \\in \\partial i}\\beta W_{ij} x_j) \\rangle_p\\ .\n\\end{align}\n\n\\section{Georges-Yedidia expansion for generalized Boltzmann machines}\n\\label{app:chap3-real-GY}\n\nWe here present a derivation of the Georges-Yedidia for real-valued degrees of freedom on the example of a Boltzmann machine as in \\cite{Tramel2018}.\nFormally we consider $\\x \\in \\R^N$ governed by the energy function and parametrized distribution\n\\begin{gather}\n    \\label{eq:chap4-real-meas-fully}\n    E(\\x) = - \\sum_{(ij)} W_{ij}x_i x_j - \\frac{1}{\\beta} \\sum_{i=1}^N\\log p_x(x_i; \\theta_i) \\,, \\quad\n    p(\\x) = \\frac{1}{\\cZ}  e^{\\frac{\\beta}{2} \\x\\T\\W\\x}  \\prod_{i=1}^N p_x(x_i ; \\theta_i) ,\n\\end{gather}\nwhere $p_x(x_i;\\theta_i)$ is an arbitrary prior distribution with parameter $\\theta_i$. For a Bernoulli prior with parameter $\\sigm(\\beta b_i)$ we recover the measure of binary Boltzmann machines. However we choose here a prior that does not depend on the temperature a priori. We now derive the expansion for this general case following the outline discussed in \\ref{sec:chap3-GY}, and highlighting the differences with the binary case.\n\nNote that inference in the generalized fully connected Boltzmann machine is somehow related to the symmetric rank-1 matrix factorization problem, which also features pairwise interactions. Similarly, inference for the bi-partite RBM maps to the asymmetric rank-1 matrix factorization. However, conversely to the Boltzmann inference, these factorizations are reconstruction problems. The mean-field techniques, derived in \\cite{Lesieur2016,Lesieur2017}, allow there to compute the MMSE estimator of unknown signals from approximate marginals. Here we focus on the evaluation of the free energy. \n\n\\paragraph{Minimization for fixed marginals}\nWhile fixing the value of the first moment is sufficient for binary variables, more than one constraint is now needed in order to minimize the Gibbs free energy at a given value of the marginals.\nIn the same spirit of the AMP algorithm we assume a Gaussian parametrization of the marginals. We note $\\am$ the first moment of $\\x$ and $\\cm$ its variance.\nWe wish to compute the constrained minimum over the distributions $q$ on $\\R^N$\n\\begin{gather}\n    G(\\am, \\cm) = \\min_{q} \\left[ \\langle E(\\x) \\rangle_{q} - H(q)\/\\beta \\; | \\;  \\langle \\x\\rangle_q = \\am \\,, \\langle \\x^2\\rangle_q = \\am^2 + \\cm  \\right],\n\\end{gather}\nwhere the notation of squared vectors corresponds here and below to the vectors of squared entries.\nIt is equivalent to an unconstrained problem with Lagrange multipliers $\\lbd(\\am,\\cm, \\beta)$ and $\\vect{\\xi}(\\am,\\cm, \\beta)$\n\\begin{gather}\n    \\label{eq:chap4-real-GY-G01}\n    G(\\am, \\cm) = \\min_{q} \\left[ \\langle E(\\x) \\rangle_{q} - H(q)\/\\beta - \\lbd\\T(\\langle \\x\\rangle_q - \\am) \/ \\beta - \\vect{\\xi}(\\langle \\x^2\\rangle_q - \\am^2 - \\cm)\/ \\beta \\right].\n\\end{gather}\n The terms depending on the distribution $q$ in the functional to minimize  above can be interpreted as a Gibbs free energy for the effective energy functional\n\\begin{gather}\n    \\tilde{E}(\\x) = E(\\x) -  \\lbd\\T\\x \/\\beta - \\vect{\\xi}\\T \\x^2\/\\beta .\n\\end{gather}\nThe solution of the minimization problem \\eqref{eq:chap4-real-GY-G01} is therefore the corresponding Boltzmann distribution \n\\begin{gather}\nq_{\\am, \\cm}(\\x) = \\frac{e^{-\\beta\\tilde{E}(\\x)}}{\\tilde{\\cZ}} = \\frac{1}{\\tilde{\\cZ}} e^{-\\beta E(\\x) + \\lbd(\\am,\\cm, \\beta)\\T\\x +  \\vect{\\xi}(\\am,\\cm, \\beta)\\T \\x^2 } \\, \n\\end{gather}\nand the minimum $G(\\am, \\cm)$  is \n\\begin{align}\n    \\label{eq:chap4-real-GY-G1}\n    -\\beta G(\\am, \\cm) \n    & = - \\lbd\\T \\am - \\vect{\\xi}\\T(\\am^2+\\cm) +  \\log \\int \\dd{\\x} e^{-\\beta E(\\x) + \\lbd\\T\\x + \\vect{\\xi}\\T \\x^2 } \\notag\\\\\n    & =  \\log \\int \\dd{\\x} e^{-\\beta E(\\x) + \\lbd\\T(\\x -\\am) + \\vect{\\xi}\\T (\\x^2 - \\am^2 -\\cm)},\n\\end{align}\nwhere the Lagrange multipliers $\\lbd(\\am,\\cm, \\beta)$ and $\\vect{\\xi}(\\am,\\cm, \\beta)$ enforcing the constraints are still implicit.\nDefining a functional $\\tilde{G}$ for arbitrary vectors $\\tilde{\\lbd} \\in \\R^N$ and $\\vect{\\tilde \\xi} \\in \\R^N$, \n\\begin{gather}\n    -\\beta \\tilde{G}(\\am,\\cm, \\tilde{\\lbd}, \\tilde{\\vect{\\xi}}) =  \\log \\int \\dd{\\x} e^{-\\beta E(\\x) + \\tilde{\\lbd}\\T(\\x -\\am) + \\tilde{\\vect{\\xi}}\\T (\\x^2 - \\am^2 -\\cm)}, \n\\end{gather}\nwe have\n\\begin{align}\n    \\label{eq:chap4-real-GY-stationary-lbd}\n    & a_i = \\langle x_i \\rangle_{q_{\\am,\\cm}} \\Rightarrow -\\beta \\left.\\frac{\\partial\\tilde{G}}{\\partial \\tilde \\lambda_i}\\right|_{\\lbd, \\vect{\\xi}} = 0, && -\\beta \\left.\\frac{\\partial^2\\tilde{G}}{\\partial \\tilde\\lambda_i^2}\\right|_{\\lbd, \\vect{\\xi}}= \\langle x_i^2 \\rangle_{q_{\\am,\\cm}} - a_i^2 > 0 ,\\\\\n    \\label{eq:chap4-real-GY-stationary-xi}\n    &c_i + a_i^2 = \\langle x_i^2 \\rangle_{q_{\\am,\\cm}}  \\Rightarrow -\\beta \\left.  \\frac{\\partial  \\tilde{G}}{\\partial \\tilde \\xi_i}\\right|_{\\lbd, \\vect{\\xi}} = 0,  && -\\beta \\left.\\frac{\\partial ^2\\tilde{G}}{\\partial \\tilde \\xi_i^2} \\right|_{\\lbd, \\vect{\\xi}}= \\langle (x^2_i)^2 \\rangle_{q_{\\am,\\cm}} - (c_i + a_i^2)^2 > 0 .\n\\end{align}\nHence the Lagrange multipliers are identified as minimizers of $-\\beta\\tilde{G}$ and\n\\begin{gather}\n    - \\beta G(\\am, \\cm) = - \\beta \\tilde{G}(\\am,\\cm, \\lbd(\\am,\\cm, \\beta), \\vect{\\xi}(\\am,\\cm, \\beta)) = \\min_{\\tilde{\\lbd}, \\tilde{\\vect{\\xi}}} - \\beta \\tilde{G}(\\am,\\cm, \\tilde{\\lbd}, \\tilde{\\vect{\\xi}}).\n\\end{gather}\nThe true free energy $F = - \\log \\cZ \/ \\beta$ would eventually be recovered by minimizing the constrained minimum $G(\\am, \\cm)$ with respect to its arguments.\nNevertheless, the computation of $G$ and $\\tilde{G}$ involves an integration over $\\x \\in \\R^N$ and remains intractable. The following step of the Georges-Yedidia derivation consists in approximating these functionals by a Taylor expansion at infinite temperature where interactions are neutralized.\n\n\\paragraph{Expansion around $\\beta=0$}\nTo perform the expansion we introduce the notation $A(\\beta, \\am, \\cm) = - \\beta G(\\am, \\cm) $.\nWe also define the auxiliary operator \n\\begin{gather}\n    U(\\x; \\beta) = -\\frac{1}{2} \\x\\T\\W\\x  + \\frac{1}{2}\\langle \\x\\T\\W\\x \\rangle_{q_{\\am, \\cm}} - \\sum_{i=1}^N \\frac{\\partial \\lambda_i}{\\partial \\beta} (x_i - a_i) - \\sum_{i=1}^N \\frac{\\partial \\xi_i}{\\partial \\beta} (x_i^2 - a_i^2 - c_i), \n\\end{gather}\nthat allows to write concisely for any observable $O$ the derivative of its average with respect to $\\beta$, \n\\begin{gather}\n    \\frac{\\partial \\langle O(\\x; \\beta) \\rangle_{q_{\\am,\\cm}}}{\\partial \\beta} = \\left\\langle \\frac{\\partial O(\\x; \\beta)}{\\partial \\beta} \\right\\rangle_{q_{\\am,\\cm}} - \\langle U(\\x;\\beta)O(\\x;\\beta) \\rangle_{q_{\\am,\\cm}}.\n\\end{gather}\nTo compute the derivatives of $\\lbd$ and $\\vect{\\xi}$ with respect to $\\beta$ we note that \n\\begin{gather}\n    \\label{eq:chap4-real-GY-derivative-A-a-c}\n    \\frac{\\partial A}{\\partial a_i} =  -\\beta  \\frac{\\partial \\tilde G}{\\partial a_i} =  - \\lambda_i(\\beta, \\am, \\cm)- 2 a_i \\xi_i(\\beta, \\am, \\cm) \\, ,  \\\\\n    \\frac{\\partial A}{\\partial c_i} = -\\beta  \\frac{\\partial \\tilde G}{\\partial c_i} = - \\xi_i(\\beta, \\am, \\cm),\n\\end{gather}\nwhere we used that $\\partial \\tilde G \/ \\partial \\tilde{\\lambda_i} = 0 $ and $\\partial \\tilde G \/ \\partial \\tilde{\\xi}_i = 0 $ when evaluated for $\\lbd(\\am,\\cm, \\beta)$ and $\\vect{\\xi}(\\am,\\cm, \\beta)$. Consequently,\n\\begin{gather}\n    \\label{eq:chap4-real-GY-deriv-lbd}\n    \\frac{\\partial \\xi_i}{\\partial \\beta} = - \\frac{\\partial}{\\partial c_i} \\frac{\\partial A }{\\partial \\beta} \\, , \\qquad \n    \\frac{\\partial \\lambda_i}{\\partial \\beta} = - \\frac{\\partial}{\\partial a_i} \\frac{\\partial A }{\\partial \\beta} + 2 a_i \\frac{\\partial \\xi_i}{\\partial \\beta}.\n\\end{gather}\nWe can now proceed to compute the first terms of the expansion that will be performed for the functional $A$. \n\n\\subparagraph{Zeroth order}\nSubstituting $\\beta=0$ in the definition of $A$ we have\n\\begin{gather}\n    A(0,\\am,\\cm) = - \\lbd(0,\\am,\\cm)\\T \\am - \\vect{\\xi}(0,\\am,\\cm)\\T (\\am^2 + \\cm) + \\log \\tilde{\\cZ}^0(\\lbd(0,\\am,\\cm), \\vect{\\xi}(0,\\am,\\cm)),\n\\end{gather}\nwith \n\\begin{align}\n    \\tilde{\\cZ}^0(\\lbd(0,\\am,\\cm), \\vect{\\xi}(0,\\am,\\cm)) &= \\int \\dd{\\x} e^{\\lbd(0,\\am,\\cm)\\T \\x+ \\vect{\\xi}(0,\\am,\\cm)\\T \\x^2}\\prod_{i=1}^N p_x(x_i ; \\theta_i) \\\\\n        & = \\prod_{i=1}^N  \\int \\dd{x_i} e^{\\lambda_i(0,\\am,\\cm) x_i+ \\xi_i(0,\\am,\\cm) x_i^2}p_x(x_i ; \\theta_i).\n\\end{align}\nAt infinite temperature the interaction terms of the energy do not contribute so that the integral in $\\tilde{\\cZ}^0$ factorizes and can be evaluated numerically in the event that it does not have a closed-form.\n\n\\subparagraph{First order} We compute the derivative of $A$ with respect to $\\beta$. We use again that $\\lbd(\\am,\\cm,\\beta)$ and  $\\vect{\\xi}(\\am,\\cm,\\beta)$ are stationary points of $\\tilde{G}$ to write\n\\begin{align}\n    \\frac{\\partial A}{\\partial \\beta} & = - \\beta \\frac{\\partial \\tilde{G}}{\\partial \\beta} = \\frac{\\partial }{\\partial \\beta} \\left[ \\log \\int \\dd{\\x} e^{-\\beta E(\\x) + \\lbd(\\am,\\cm, \\beta)\\T(\\x -\\am) + \\vect{\\xi}(\\am,\\cm, \\beta)\\T (\\x^2 - \\am^2 -\\cm)}  \\right]  \\\\\n   \n   \n   \n   \n     & = \\left\\langle \n        \\frac{\\partial }{\\partial \\beta} (-\\beta E(\\x))\n        + \\frac{\\partial \\lbd }{\\partial \\beta}\\T (\\x -\\am) \n        + \\frac{\\partial \\vect{\\xi}}{\\partial \\beta}\\T (\\x^2 - \\am^2 -\\cm)   \\right\\rangle_{q_{\\am, \\cm}} \\\\\n    & =  \\frac{1}{2} \\langle \\x\\T\\W\\x \\rangle_{q_{\\am, \\cm}}.\n\\end{align}\nAt infinite temperature the average over the product of variables becomes a product of averages so that we have\n\\begin{gather}\n   \\left. \\frac{\\partial A}{\\partial \\beta} \\right|_{\\beta = 0} =  \\frac{1}{2}\\am\\T\\W\\am =  \\sum_{(ij)} W_{ij} a_i a_j .\n\\end{gather}\n\n\\subparagraph{Second order} Using the first order derivative of $A$ we can compute the derivatives of the Lagrange parameters \\eqref{eq:chap4-real-GY-deriv-lbd} and the auxillary operator at infinite temperature, \n\\begin{gather}\n    \\left. \\frac{\\partial \\xi_i}{\\partial \\beta} \\right|_{\\beta = 0} = 0 \\, , \\qquad \\left. \\frac{\\partial \\lambda_i}{\\partial \\beta} \\right|_{\\beta = 0} = - \\sum_{j\\in\\partial i} W_{ij} a_j \\,, \\qquad  U(\\x;0) = - \\sum_{(ij)} W_{ij} (x_i - a_i)(x_j - a_j). \\notag\n\\end{gather}\nThe second order derivative is then easily computed at infinite temperature\n\\begin{align}\n    \\left. \\frac{\\partial^2 A}{\\partial \\beta^2} \\right|_{\\beta = 0} & = \\frac{1}{2}\\left. \\frac{\\partial}{\\partial \\beta} \\Big(\\langle \\x\\T\\W\\x \\rangle_{q_{\\am, \\cm}}\\Big) \\right|_{\\beta = 0}= - \\frac{1}{2} \\langle  U(\\x;0) (\\x\\T\\W\\x )\\rangle^{\\beta=0}_{q_{\\am, \\cm}} \\\\\n    & = \\sum_{(ij)} W_{ij}^2 \\langle (x_i -a_i)x_i (x_j - a_j)\\rangle^{\\beta=0}_{q_{\\am, \\cm}} = \\sum_{(ij)} W_{ij}^2 c_i c_j. \n\\end{align}\n\n\\paragraph{TAP free energy for the generalized Boltzmann machine}\n\\label{sec:chap4-tap-fe-grbm}\nStopping at the second order of the systematic expansion, and gathering the different terms derived above we have \n\\begin{align}\n    -\\beta G(\\am, \\cm) = - \\lbd(0,\\am,\\cm)\\T \\am & - \\vect{\\xi}(0,\\am,\\cm)\\T (\\am^2 + \\cm) + \\log \\tilde{\\cZ}^0(\\lbd(0,\\am,\\cm), \\vect{\\xi}(0,\\am,\\cm))  \\\\\n    &+ \\beta \\sum_{(ij)} W_{ij} a_i a_j + \\frac{\\beta^2}{2}\\sum_{(ij)} W_{ij}^2 c_i c_j, \\notag\n\\end{align}\nwhere the values of the parameters $\\lbd(0,\\am,\\cm)$ and $\\vect{\\xi}(0,\\am,\\cm)$ are implicitly defined through the stationary conditions \\eqref{eq:chap4-real-GY-stationary-lbd}-\\eqref{eq:chap4-real-GY-stationary-xi}.\nThe TAP approximation of the free energy also requires to consider the stationary points of the expanded expression as a function of $\\am$ and $\\cm$. \n\nThis second condition yields the relations \n\\begin{gather}\n    \\label{eq:chap4-real-GY-A}\n    -2 \\xi_i (0,\\am,\\cm) = -\\beta^2 \\sum_{j\\in \\partial i } W_{ij}^2 c_j = A_i \\\\\n    \\label{eq:chap4-real-GY-B}\n    \\lambda_i (0,\\am,\\cm) = A_i a_i + \\beta \\sum_{j\\in \\partial i } W_{ij} a_j = B_i \\,\n\\end{gather}\nwhere we define new variables $A_i$ and $B_i$. \nWhile the extremization with respect to the Lagrange multipliers gives\n\\begin{gather}\n    \\label{eq:chap4-real-GY-a}\n    a_i = \\frac{1}{\\cZ^x_i} \\int \\dd{x_i} x_i p_x(x_i; \\theta_i) e^{-\\frac{A_i}{2}x_i^2 + B_i x_i} = f_1^x(B_i,A_i;\\theta_i), \\\\\n    \\label{eq:chap4-real-GY-c}\n    c_i = \\frac{1}{\\cZ^x_i} \\int \\dd{x_i} x_i^2 p_x(x_i; \\theta_i) e^{-\\frac{A_i}{2}x_i^2 + B_i x_i} - a_i^2 = f_2^x(B_i,A_i;\\theta_i) ,\n\\end{gather}\nwhere we introduce update functions $f_1^x$ and $f_2^x$ with respect to the partition function \n\\begin{gather}\n    \\cZ^x_i(B_i,A_i;\\theta_i) = \\int \\dd{x_i} p_x(x_i; \\theta_i) e^{-\\frac{A_i}{2}x_i^2 + B_i x_i} .\n\\end{gather}\nFinally we can rewrite the TAP free energy as\n\\begin{align}\n    \\label{eq:chap4-real-GY-final-G}\n    -\\beta G(\\am, \\cm) =  - \\vect{B}\\T \\am  + \\vect{A}\\T (\\am^2 + \\cm)\/2  + \\sum_{i=1}^N &\\log \\tilde{\\cZ_i}^x(B_i, A_i; \\theta_i)   \\\\\n    &+ \\beta \\sum_{(ij)} W_{ij} a_i a_j + \\frac{\\beta^2}{2}\\sum_{(ij)} W_{ij}^2 c_i c_j, \\notag\n\\end{align}\nwith the values of the parameters set by the self-consistency conditions \\eqref{eq:chap4-real-GY-A}, \\eqref{eq:chap4-real-GY-B}, \\eqref{eq:chap4-real-GY-a} and \\eqref{eq:chap4-real-GY-c}, which are the TAP equations of the generalized Boltzmann machine at second order. Note that the naive mean-field equations are recovered by ignoring the second order terms in $\\beta^2$.\n\n\\subparagraph{Relation to message passing}\nThe TAP equations obtained above must correspond to the fixed points of the Approximate Message Passing (AMP) following the derivation from Belief Propagation (BP) that is presented in \\citesec~\\ref{sec:chap3-gamp}. In the Appendix B of \\cite{Tramel2018} the relaxed-BP equations are derived for the generalized Boltzmann machine: \n\\begin{gather}\n    \\msg{B}{i}{j}^{(t)} = \\sum_{k \\in \\partial i \\setminus j}\\beta W_{ik} \\msg{a^{(t)}}{k}{i} , \\quad \n    \\msg{A}{i}{j}^{(t)} = - \\sum_{k \\in \\partial i \\setminus j} \\beta^2 W^2_{ik} \\msg{c^{(t)}}{k}{i} ,\\\\\n    \\msg{a}{i}{j}^{(t)} = f_1^x(\\msg{B}{i}{j}^{(t-1)},\\msg{A}{i}{j}^{(t-1)};\\theta_i) , \\quad\n    \\msg{c}{i}{j}^{(t)} = f_2^x(\\msg{B}{i}{j}^{(t-1)},\\msg{A}{i}{j}^{(t-1)};\\theta_i).\n\\end{gather}\nTo recover the TAP equations for them we define\n\\begin{gather}\n    B_i^{(t)} = \\sum_{k \\in \\partial i}\\beta W_{ik} \\msg{a^{(t)}}{k}{i} , \\quad \n    A_i^{(t)} = - \\sum_{k \\in \\partial i} \\beta^2 W^2_{ik} \\msg{c^{(t)}}{k}{i} ,\\\\\n    \\label{eq:chap4-real-GY-TAP-ac}\n    a_i^{(t)} = f_1^x(B_i^{(t-1)},A_i^{(t-1)};\\theta_i) , \\quad\n    c_i^{(t)} = f_2^x(B_i^{(t-1)},A_i^{(t-1)};\\theta_i).\n\\end{gather}\nAs $B_i^{(t)} = \\msg{B}{i}{j}^{(t)} + \\beta W_{ij} \\msg{a^{(t)}}{j}{i}$ and $A_i^{(t)} = \\msg{A}{i}{j}^{(t)} - \\beta^2 W^2_{ij} \\msg{c^{(t)}}{j}{i}$ we have by developing $f_2^x$ that $c_i^{(t)} = \\msg{c}{i}{j}^{(t)} + O(\\beta)$ so that\n\\begin{gather}\n    \\label{eq:chap4-real-GY-TAP-A}\n    A_i^{(t)} = - \\beta^2 \\sum_{j \\in \\partial i}  W^2_{ij} c_j^{(t)} + o(\\beta^2).\n\\end{gather}\nBy developing $f_1^x$ we also have\n\\begin{align}\n    a_k^{(t)} &= f_1^x(\\msg{B}{k}{j}^{(t-1)} + \\beta W_{kj} \\msg{a^{(t-1)}}{j}{i}\\, , \\; \\msg{A}{k}{j}^{(t-1)}- \\beta^2 W^2_{kj} \\msg{c^{(t-1)}}{j}{k};\\theta_i) \\\\\n    & = \\msg{a^{(t)}}{k}{j} + \\frac{\\partial f_1^x}{\\partial B_k} \\beta W_{kj} \\msg{a^{(t-1)}}{j}{k} + O(\\beta^2),\n\\end{align}\nwith $\\displaystyle \\frac{\\partial f_1^x}{\\partial B_k}(B_k^{(t-1)},A_k^{(t-1)};\\theta_k) = c_k^{(t)}$. Finally, by replacing in the definition of $B_i$ the messages we obtain\n\\begin{gather}\n    B_i^{(t)}  = \\sum_{k \\in \\partial i}\\beta W_{ik} \\msg{a^{(t)}}{k}{i}  = \\sum_{k \\in \\partial i}\\beta W_{ik} a_k^{(t)} - \\beta W_{ki}c_k^{(t)} \\msg{a^{(t-1)}}{i}{k}  .\n\\end{gather}\nAs $\\msg{a^{(t-1)}}{i}{k} = {a^{(t-1)}_i + O(\\beta)}$ and using the definition of $A_i^{(t)}$, we finally recover\n\\begin{gather}\n    \\label{eq:chap4-real-GY-TAP-B}\n    B_i^{(t)}  =  \\sum_{k \\in \\partial i}\\beta W_{ik} a_k^{(t)} + A_i^{(t)}a^{(t-1)}_i.\n\\end{gather}\n\nHence we indeed recover the TAP equations as the AMP fixed points in \\eqref{eq:chap4-real-GY-TAP-ac}, \\eqref{eq:chap4-real-GY-TAP-A} and \\eqref{eq:chap4-real-GY-TAP-B}. Beyond the possibility to cross-check our results, the message passing derivation also specifies a scheme of updates to solve the self-consistency equations obtained by the Georges-Yedidia expansion. In the applications we consider below we should resort to this time indexing with good convergence properties \\cite{bolthausen2014iterative}.\n\n\\subparagraph{Solutions of the TAP equations}\nAs already discussed in \\citesec~\\ref{sec:chap3-tap}, the TAP equations do not necessarily admit a single solution. In practice, different fixed points are reached when considering different initializations of the iteration of the self-consistent equations. \n\n\\section{Some applications}\n\\label{sec:chapex-all}\n\\subsection{A brief pre-deep learning history}\n\\label{sec:chap1-nn-and-mf}\n\nThe application of mean-field methods of inference to machine learning, and in particular to neural networks, already have a long history and significant contributions to their records. Here we briefly review some historical connections anterior to the deep learning revival of neural networks in the 2010s.\n\n\\paragraph{Statistical mechanics of learning} In the 80s and 90s, a series of works pioneered the analysis of learning with neural networks through the statistical physics lense.\nBy focusing on simple models with simple data distributions, and relying on the mean-field method of replicas, these papers managed to predict quantitatively important properties such as \\emph{capacities}: the number of training data point that could be memorized by a model, or \\emph{learning curves}: the generalization error (or population risk) of a model as a function of the size of the training set. This effort was initiated by the study of the Hopfield model \\cite{Amit1985}, an undirected neural network providing associative memory \\cite{Hopfield1982}. The analysis of feed forward networks with simple architectures followed (among which \\cite{Gardner1987, Gardner1988, Opper1991,  Monasson1995, Opper1996,  Monasson2004}, see also the reviews \\cite{Seung1992,Watkin1993, Opper1995, Saad1999a, Engel2001}). The dynamics of simple learning problems was also analyzed through a mean-field framework (not covered in the previous sections) initially in the simplifying case of online learning with infinite training set \\cite{Saad1995, Saad1995a, Biehl1995, Saad1999a} but also with finite data \\cite{Sollich1997, Li1999}.\n\nPhysicists, accustomed to studying natural phenomena, fruitfully brought the tradition of modelling to their investigation of learning, which translated into assumptions of random data distributions or teacher-student scenarios. Their approach was in contrast to the focus of the machine learning theorists on worst case guarantees: bounds for an hypothesis class that hold for any data distribution (e.g. Vapnik-Chervonenkis dimension and Rademacher complexity). \nThe originality of the physicists approach, along with the heuristic character of the derivations of mean-field approximations, may nonetheless explain the minor impact of their theoretical contributions in the machine learning community at the time.\n\n\\paragraph{Mean-field algorithms for practictioners} \nAlong with these contributions to the statistical mechanics theory of learning, new practical training algorithms based on mean-field approximations were also proposed at the same period (see e.g.\\cite{Wong1995,Opper1996,Wong1997}). \nYet, before the deep learning era, mean-field methods probably had a greater influence in the practice of unsupervised learning through density estimation, where we saw that approximate inference is almost always necessary. In particular the simplest method of naive mean-field, our first example in \\citechap~\\ref{sec:chap3}, was easily adopted and even extended by statisticians (see e.g. \\cite{Wainwright2008} for a recent textbook and \\cite{Blei2017} for a recent review).  The belief propagation algorithm is another example of a well known mean-field methods by machine learners, as it was actually discovered in both communities. \nYet, for both methods, early applications rarely involved neural networks and rather relied on simple probabilistic models such as mixtures of elementary distributions.\nThey also did not take full advantage of the latest simultaneous developments in statistical physics of the mean-field theory of disordered systems. \n\n\n\\paragraph{Transferring advanced mean-field methods}\nIn this context, the inverse Ising problem has been a notable exception. The underlying question, rooted in theoretical statistical physics, is to infer the parameters of an Ising model given a set of equilibrium configurations. This is related to the unsupervised learning of the parameters of a Boltzmann machine (without hidden units) in the machine learning jargon, while it does not necessarily rely on a maximum likelihood estimation using gradients. The corresponding Boltzmann distribution, with pairwise interactions, is remarkable, not only to physicists. It is the least biased model under the assumption of fixed first and second moments in the sense that it maximizes the entropy. For this problem, physicists proposed dedicated developments of advanced mean-field methods for applications in other fields, and in particular in bio-physics (see \\cite{Nguyen2017} for a recent review). A few works even considered the case of Boltzmann machines with hidden units, more common in the machine learning community \\cite{Peterson1987,Galland1993}.\n\nBeyond the specific case of Boltzmann machines, the language barrier between communities is undoubtedly a significant hurdle delaying the global transfer of developments in one field to the other. \nIn machine learning, the potential of the most recent progress of mean-field approximations was advocated for in a pioneering workshop mixing communities in 1999 \\cite{opper2001advanced}. Yet the first widely-used application is possibly the Approximate Message Passing (AMP) algorithm for compressed sensing in 2009 \\cite{Donoho2009}.\nMeanwhile, in the different field of Constraint Satisfaction Problems (CSPs), there have been much tighter connections between developments in statistical physics and algorithmic solutions. The very first popular application of advanced mean-field methods outside of physics, beyond naive mean-field and belief propagation, is probably the survey propagation algorithm \\cite{Mezard2002} in 2002. It borrows from the 1RSB cavity method (not treated in the present paper) to solve efficiently certain types of CSPs.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section[Machine learning with neural networks\n]{Machine learning with neural networks \n}\n\\label{sec:chap1}\n\n\nMachine learning is traditionally divided into three classes of problems: supervised, unsupervised and reinforcement learning. For all of them, the advent of deep learning techniques, relying on deep neural networks, has brought great leaps forward in terms of performance and opened the way to new applications. \nNevertheless, the utterly efficient machinery of these algorithms remains full of theoretical puzzles.\nThis \\citechap~provides fundamental concepts in machine learning for the unfamiliar reader willing to approach the literature at the crossroads of statistical physics and deep learning.\nWe also take this \\citechap~as an opportunity to introduce the current challenges in building a strong theoretical understanding of deep learning.  \nA comprehensive reference is \\cite{Goodfellow2016}, while \\cite{Mehta2018} offers a broad introduction to machine learning specifically addressed to physicists. \n\n\n\\subsection{Supervised learning}\n\\paragraph{Learning under supervision}\nSupervised learning aims at discovering systematic input to output mappings from examples. Classification is a typical supervised learning problem: for instance, from a set of pictures of cats and dogs labelled accordingly, the goal is to find a function able to predict in any new picture the species of the displayed pet. \n\nIn practice, the \\emph{training set} is a collection of $P$ example pairs $\\cD = \\{\\x\\kk, \\y\\kk\\}_{k=1}^P$ from an input data space $\\mathcal{X} \\subseteq \\R^N$ and an output data space $\\mathcal{Y} \\subseteq \\R^M$. Formally, they are assumed to be i.i.d. samples from a joint distribution $p(\\x,\\y)$.  The predictor $h$ is chosen by a training algorithm from a \\emph{hypothesis class}, a set of functions from $\\mathcal{X}$ to  $\\mathcal{Y}$, so as to minimize the error on the training set. This error is formalized as the \\emph{empirical risk}\n\\begin{gather}\n    \\hat\\cR(h, \\ell, \\cD) = \\frac{1}{P}\\sum_{k=1}^P \\ell(\\y\\kk, h(\\x\\kk)) , \n\\end{gather}\nwhere the definition involves a loss function $\\ell: \\mathcal{Y} \\times \\mathcal{Y} \\rightarrow \\R$ measuring differences in the output space. \nThis learning objective nevertheless does not guarantee \\emph{generalization}, i.e. the ability of the predictor $h$ to be accurate on inputs $\\x$ that are not in the training set. It is a surrogate for the ideal, but unavailable, \\emph{population risk} \n\\begin{gather}\n    \\cR(h, \\ell) = \\E_{\\x, \\y} \\left[ \\ell(\\y, h(\\x))\\right] = \\int_{\\mathcal{X}, \\mathcal{Y}} \\dd \\x \\dd \\y p(\\x, \\y) \\ell(\\y, h(\\x)),\n\\end{gather}\nexpressed as an expectation over the joint distribution $p(\\x,\\y)$.\nThe different choices of hypothesis classes and training algorithms yield the now crowded zoo of supervised learning algorithms. \n\n\\paragraph{Representation ability of deep neural networks}\nIn the context of supervised learning, deep neural networks enter the picture in the quality of a parametrized hypothesis class. Let us first quickly recall the simplest network, the \\emph{perceptron}, including only a single neuron. It is formalized as a function from $\\R^{N}$ to $\\mathcal{Y} \\subset \\R$ applying an activation function $f$ to a weighted sum of its inputs shifted by a bias $b \\in \\R$, \n\\begin{gather}\n    \\label{eq:chap1-perceptron}\n    \\hat{y} = h_{\\vect{w}, b}(\\x) = f(\\vect{w}\\T \\x + b)\n\\end{gather}\nwhere the weights are collected in the vector $\\vect{w} \\in \\R^N$. From a practical standpoint, this very simple model can only solve the classification of linearly separable groups (see \\citefig~\\ref{fig:chap1-perceptron}). Yet from the point of view of learning theory, it has been the starting point of a rich statistical physics literature that will be discussed in \\citesec~\\ref{sec:chap1-nn-and-mf}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{perceptron.pdf}\n    \\caption{Let's assume we wish to classify data points $\\x \\in \\R^2$ with labels $y=\\pm 1$. We choose as an hypothesis class the perceptron sketched on the left with sign activation. For given weight vector $\\vect{w}$ and bias $b$ the plane is divided by a decision boundary assigning labels. If the training data are linearly separable, then it is possible to perfectly predict all the labels with the perceptron, otherwise it is impossible.  \\label{fig:chap1-perceptron}}\n\\end{figure}\n\nCombining several neurons into networks defines more complex functions. The universal approximation theorem \\cite{Cybenko1989, Hornik1991} proves that the following two-layer network architecture can approximate any well-behaved function with a finite number of neurons,\n\\begin{gather}\n    \\hat{y} = h_{\\theta}(\\x) = {\\vect{w}^{(2)}}\\T f(\\W^{(1)} \\x + \\vect{b}) = \\sum_{\\alpha = 1}^M w^{(2)}_\\alpha f({\\vect{w}_\\alpha^{(1)}}\\T\\x + b_\\alpha), \\qquad \\theta = \\{\\vect{w}^{(2)}, \\W^{(1)}, \\vect{b}\\}\n\\end{gather}\nfor $f$ a bounded, non-constant, continuous scalar function, acting component-wise. In the language of deep learning this network has one hidden layer of $M$ units. Input weight vectors $w^{(1)}_\\alpha \\in \\R^N$ are collected in a weight matrix $\\W^{(1)} \\in \\R^{M \\times N}$. Here, and in the following, the notation $\\theta$ is used as short for the collection of adjustable parameters. The universal approximation theorem is a strong result in terms of representative power of neural networks but it is not constructive. \nIt does not quantify the size of the network, i.e. the number $M$ of hidden units, to approximate a given function, nor does it prescribe how to obtain the values of the parameters $\\vect{w}^{(2)}, \\W^{(1)}$ and $\\vect{b}$ for the optimal approximation. While building an approximation theory is still ongoing (see e.g. \\cite{Grohs2019}). Practice, led by empirical considerations, has nevertheless demonstrated the efficiency of neural networks.\n\nIn applications, neural networks with multiple hidden layers, deep neural networks, are preferred. A generic neural network of \\emph{depth} $L$ is the function\n\\begin{gather}\n    \\label{eq:chap1-dnn}\n    \\hat{\\y} = h_{\\theta}(\\x) = f(\\W^{(L)}f(\\W^{(L-1)} \\cdots f(\\W^{(1)} \\x + \\vect{b}^{(1)}) \\cdots +\\vect{b}^{(L-1)})+ \\vect{b}^{(L)}), \\\\\n    \\label{eq:chap1-dnn-theta}\n    \\theta = \\{\\W^{(l)} \\in \\R^{N_{l} \\times N_{l-1}} , \\, \\vect{b}^{(l)} \\in \\R^{N_l} \\;; \\; l=1 \\cdots L\\},\n\\end{gather}\nwhere $N_0 = N$ is the dimension of the input and $N_L = M$ is the dimension of the output. The architecture is fixed by specifying the number of neurons, or \\emph{width}, of the hidden layers $\\{N_l\\}_{l=1}^{L-1}$. The latter can be denoted $\\hid^{(l)} \\in \\R^{N_l}$ and follow the recursion\n\\begin{gather}\n\t\\label{eq:chap1-dnn-rec1}\n\t\\hid^{(1)} = f(\\W^{(1)} \\x + \\vect{b}^{(1)}) \\, , \\\\\n\t\\label{eq:chap1-dnn-rec2}\n    \\hid^{(l)}  = f(\\W^{(l)} \\hid^{(l-1)} + \\vect{b}^{(l)}) \\, , \\quad l = 2 \\cdots L-1 \\, ,\\\\\n    \\label{eq:chap1-dnn-rec3}\n\t\\hat{\\y}  = f(\\W^{(L)} \\hid^{(L-1)} + \\vect{b}^{(L)}) \\, .\n\\end{gather}\nFixing the activation functions and the architecture of a neural network defines an hypothesis class. It is crucial that activations introduce non-linearities; the most common are the hyperbolic tangent tanh and the rectified linear unit defined as $\\mathrm{relu}(x)= \\max(0,x)$. Note that it is also possible to define stochastic neural networks by using noisy activation functions, uncommon in supervised learning applications except  at training time so as to encourage generalization \\cite{Poole2014, Srivastava2014}.\n\nAn originally proposed intuition for the advantage of depth is that it enables to treat the information in a hierarchical manner; either looking at different scales in different layers, or learning more and more abstract representations \\cite{Bengio2013}. Nevertheless, getting a clear theoretical understanding why in practice `the deeper the better' is still an ongoing direction of research (see e.g. \\cite{Telgarsky2016, Daniely2017, Safran2019}).\n\n\\paragraph{Neural network training}\nGiven an architecture defining $h_\\theta$, the supervised learning objective is to minimize the empirical risk $\\hat \\cR$ with respect to the parameters $\\theta$. This optimization problem lives in the dimension of the number of parameters which can range from tens to millions. The idea underlying the majority of training algorithms is to perform a gradient descent (GD) starting at parameters drawn randomly from an initialization distribution:\n\\begin{gather}\n    \\theta_0 \\sim p_{\\theta_0}(\\theta_0) \\\\\n    \\theta_{t+1} \\leftarrow \\theta_t - \\eta \\nabla_\\theta \\hat \\cR = \\theta_t - \\eta  \\frac{1}{P}\\sum_{k=1}^P \\nabla_\\theta \\ell\\left(\\y\\kk, h_{\\theta_t}\\left(\\x\\kk\\right)\\right) \\,.\n\\end{gather}\nThe parameter $\\eta$ is the learning rate, controlling the size of the step in the direction of decreasing gradient per iteration. The computation of the gradients can be performed in time scaling linearly with depth by applying the derivative chain-rule leading to the \\emph{back-propagation} algorithm \\cite{Goodfellow2016}. A popular alternative to gradient descent is stochastic gradient descent (SGD) where the sum over the gradients for the entire training set is replaced by the sum over a small number of samples, randomly selected at each step \\cite{RobbinsHMonro1951,Bottou2010}.\n\nDuring the training iterations, one typically monitors the \\emph{training error} (another name for the empirical risk given a training data set) and the \\emph{validation error}. The latter corresponds to the empirical risk computed on a set of points held-out from the training set, the validation set, to assess the generalization ability of the model either along the training or in order to select hyperparameters of training such as the value of the learning rate. A posteriori, the performance of the model is judged from the \\emph{generalization error}, which is evaluated on the never seen \\emph{test set}.\nWhile two different training algorithms (e.g. GD vs SGD) may achieve zero training error, they may differ in the level of generalization they typically reach.\n\n\n\\paragraph{Open questions and challenges}\nBuilding on the fundamental concepts presented in the previous paragraphs, practitioners managed to bring deep learning to unanticipated performances in the automatic processing of images, speech and text (see \\cite{LeCun2015a} for a few years old review). \nStill, many of the greatest successes in the field of neural network were obtained using ingenious tricks while many fundamental theoretical questions remain unresolved. \n\nRegarding the optimization first, (S)GD training generally discovers parameters close to zero risk. Yet, gradient descent is guaranteed to converge to the neighborhood of a global minimum only for a convex function and is otherwise expected to get stuck in a local minimum. Therefore, the efficiency of gradient-based optimization is a priori a paradox given the empirical risk $\\hat R$ is non-convex in the parameters $\\theta$. \nSecond, the generalization ability of deep neural networks trained by (S)GD is still poorly understood.\nThe size of training data sets is limited by the cost of labelling by humans,  experts or heavy computations.\nThus training a deep and wide network amounts in practice to fitting a model of millions of degrees of freedom against a somehow relatively small amount of data points. Nevertheless it does not systematically lead to \\emph{overfitting}. The resulting neural networks can have surprisingly good predictions both on inputs seen during training and on new inputs \\cite{Zhang2017}.   \nResults in the literature that relate the size and architecture of a network to a measure of its ability to generalize are too far from realistic settings to guide choices of practitioners. \nOn the one hand, traditional bounds in statistics, considering worst cases, appear overly pessimistic \\cite{Vapnik2000,Bartlett2002,Shalev-Shwartz2014,Abbara2019}. On the other hand, historical statistical physics analyses of learning, briefly reviewed in \\citesec~\\ref{sec:chap1-nn-and-mf}, only concern simple architectures and synthetic data. This lack of theory results in potentially important waste: in terms of time lost by engineers in trial and error to optimize their solution, and in terms of electrical resources used to train and re-train possibly oversized networks while storing potentially unnecessarily large training data sets.\n\n\nThe success of deep learning, beyond these apparent theoretical puzzles, certainly lies in the interplay of advantageous properties of training algorithms, the neural network hypothesis class and structures in typical data (e.g. real images, conversations). Disentangling the role of the different ingredients is a very active line of research (see \\cite{Giryes2016} for a review).\n\n\\subsection{Unsupervised learning}\n\\label{sec:chap1-unsupervised}\n\\paragraph{Density estimation and generative modelling}\nThe goal of unsupervised learning is to directly extract structure from data. \nCompared to the supervised learning setting, the training data set is made of a set of example inputs $\\cD = \\{\\vect{x}\\kk\\}_{k=1}^P$ without corresponding outputs. A simple example of unsupervised learning is clustering, consisting in the discovery of unlabelled subgroups in the training data. \nMost unsupervised learning algorithms either implicitly or explicitly adopt a probabilistic viewpoint and implement \\emph{density estimation}. The idea is to approximate the true density $p(\\x)$ from which the training data was sampled by the closest (in various senses) element among a family of parametrized distributions over the input space $\\{ p_\\theta(.),\\;  \\theta \\in \\R^{N_\\theta} \\}$. The selected $p_{\\theta}$ is then a model of the data.\nIf the model $p_{\\theta}$ is easy to sample, it can be used to generate new inputs comparable to the training data points - which leads to the terminology of \\emph{generative models}. In this context, \\emph{unsupervised deep learning} exploits the representational power of deep neural networks to create sophisticated candidate $p_\\theta$.\n\nA common formalization of the learning objective is to maximize the \\emph{likelihood}, defined as the probability of i.i.d. draws from the model $p_\\theta$ to have generated the training data $\\cD = \\{\\vect{x}\\kk\\}_{k=1}^P$, or equivalently its logarithm,\n\\begin{gather}\n    \\maxx{\\theta} \\prod_{k=1}^P p_\\theta(\\x\\kk) \\quad \\iff \\quad \\maxx{\\theta} \\sum_{k=1}^P \\log p_\\theta(\\x\\kk).\n\\end{gather} \nThe second logarithmic additive formulation is generally preferred. \nIt can be interpreted as the minimization of the Kullback-Leibler divergence between the empirical distribution $p_\\cD(\\x) = \\sum_{k=1}^P \\delta(\\x - \\x\\kk) \/ P$ and the model $p_\\theta$:\n\\begin{gather}\n    \\minn{\\theta} \\KL(p_\\cD || p_\\theta) = \\minn{\\theta} \\int \\dd{\\x}  p_\\cD(\\x) \\log \\frac{p_\\cD(\\x) }{p_\\theta(\\x)} \\quad \\iff  \\quad \\maxx{\\theta} \\sum_{k=1}^P \\log p_\\theta(\\x\\kk) \\,, \n\\end{gather}\nalthough considering the divergence with the discrete empirical measure is slightly abusive.\nThe detail of the optimization algorithm here depends on the specification of $p_\\theta$. As we will see, the likelihood in itself is often intractable and learning consists in a gradient ascent on at best a lower bound, otherwise an approximation, of the likelihood. \n\nA few years ago, an alternative strategy called adversarial training was introduced by \\cite{Goodfellow2014}. Here an additional trainable model called the discriminator, for instance parametrized by $\\phi$ and denoted $d_\\phi(\\cdot)$, computes the probability for points in the input space $\\mathcal{X}$ of belonging to the training set $\\cD$ rather than being generated by the model $p_\\theta(\\cdot)$. The parameters $\\theta$ and $\\phi$ are trained simultaneously such that, the generator learns to fool the discriminator and the discriminator learns not to be fooled by the generator. The optimization problem usually considered is\n\\begin{gather}\n    \\minn{\\theta}\\maxx{\\phi} \\EE{\\cD}{\\log(d_\\phi(\\x))}  + \\EE{p_\\theta}{\\log(1-d_\\phi(\\x))} \\, ,\n\\end{gather}\nwhere the sum of the expected log-probabilities according to the discriminator for examples in $\\cD$ to be drawn from $\\cD$ and examples generated by the model not to be drawn from $\\cD$ is maximized with respect to $\\phi$ and minimized with respect to $\\theta$.\n\nIn the following, we present two classes of generative models based on neural networks.\n\n\\paragraph{Deep Generative Models}\n\\label{sec:chap1-vae}\nA deep generative models defines a density $p_\\theta$  obtained by propagating a simple distribution through a deep neural network. It can be formalized by introducing a latent variable $\\z \\in \\R^N$ and a deep neural network $h_\\theta$ similar to \\eqref{eq:chap1-dnn} of input dimension $N$. The generative process is then \n\\begin{gather}\n    \\label{eq:chap1-dgm-1}\n    \\z \\sim p_z(\\z) \\\\\n    \\label{eq:chap1-dgm-2}\n    \\x \\sim p_\\theta(\\x |\\z) = p_{\\rm out}(\\x| h_\\theta(\\z)),\n\\end{gather}\nwhere $p_z$ is typically a factorized distribution on $\\R^N$ easy to sample (e.g. a standard normal distribution), and $p_{\\rm out}(.|h_\\theta(\\z))$ is for instance a multivariate Gaussian distribution with mean and covariance that are functions of $h_\\theta(\\z)$.\nThe motivation to consider this class of models for joint distributions is three-fold. First the class is highly expressive. Second, it follows from the intuition that data sets leave on low dimensional manifolds, which here can be spaned by varying the latent representation $\\z$ usually much smaller than the input space dimension (for further intuition see also the reconstruction objective of the first autoencoders, see e.g. Chapter 14 of \\cite{Goodfellow2016}). Third, yet perhaps more importantly, the class can be optimized over easily using back-propagation, unlike the Restricted Boltzmann Machines presented in the next paragraph largely replaced by deep generative models.\nThere are two main types of deep generative models. Generative Adversarial Networks (GAN) \\cite{Goodfellow2014} trained following the adversarial objective mentioned above, and Variational AutoEncoders (VAE) \\cite{Kingma2014, Rezende2014} trained to maximize a likelihood lower-bound.\n\n\\subparagraph{Variational AutoEncoders}\nThe computation of the likelihood of one training sample $\\x\\kk$ for a deep generative model \\eqref{eq:chap1-dgm-1}-\\eqref{eq:chap1-dgm-2} requires then the marginalization over the latent variable $\\z$, \n\\begin{gather}\n    p_\\theta(\\x) = \\int \\dd{\\z} p_{\\rm out}(\\x | h_\\theta(\\z)) p_z(\\z).\n\\end{gather}\nThis multidimensional integral cannot be performed analytically in the general case. It is also hard to evaluate numerically as it does not factorize over the dimensions of $\\z$ which are mixed by the neural network $h_\\theta$. \nYet a lower bound on the log-likelihood can be defined by introducing a tractable conditional distribution $q(\\z |\\x)$ that will play the role of an approximation of the intractable \\emph{posterior} distribution $p_{\\theta}(\\z | \\x)$ implicitly defined by the model:\n\\begin{align}\n    \\log p_{\\theta}(\\x)\n                       \n                       \n                       \n                        & \\geq \\int \\dd{\\z} q(\\z | \\x) \\left[ - \\log q(\\z | \\x) + \\log p_{\\theta}(\\x, \\z) \\right] = \\mathrm{LB}(q, \\theta, \\x) \\label{eq:chap1-vae-lb}.\n\\end{align}\nMaximum likelihood learning is then approached by the maximization of the lower bound $\\mathrm{LB}(q, \\theta, \\x)$, which requires in practice to parametrize the tractable posterior $q = q_{\\phi}$, typically with a neural network. \nUsing the so-called re-parametrization trick \\cite{Kingma2014,Rezende2014}, the gradients of $\\mathrm{LB}(q_\\phi, \\theta, \\x)$ with respect to $\\theta$ and $\\phi$ can be approximated by a Monte Carlo, so that the likelihood lower bound can be optimized by gradient ascent.\n\n\\subparagraph{Generative Adversarial Networks}\nThe principle of adversarial training was designed directly for a deep generative model \\cite{Goodfellow2014}. Using a deep neural network to parametrize the discriminator $d_\\phi(\\cdot)$ as well as the generator $p_\\theta(\\cdot)$, it leads to a remarkable quality of produced samples and is now one of the most studied generative model.\n\n\n\n\\paragraph{Restricted Boltzmann Machines}\nModels described in the preceding paragraphs comprised only \\emph{feed forward} neural networks. In feed forward neural networks, the state or value of successive layers is determined following the recursion \\eqref{eq:chap1-dnn-rec1}-\\eqref{eq:chap1-dnn-rec3}, in one pass from inputs to outputs.\nBoltzmann machines instead involve \\emph{undirected} neural networks which consist of stochastic neurons with symmetric interactions.\nThe probability law associated with a neuron state is a function of neighboring neurons, themselves reciprocally function of the first neuron. Sampling a configuration therefore requires an equilibration in the place of a simple forward pass. \n\nA Restricted Boltzmann Machine (RBM) \\cite{Ackley1985, Smolensky186} with M hidden neurons in practice defines a joint distribution over an input (or visible) layer $\\x \\in \\{0,1\\}^N$ and a hidden layer $\\hidd  \\in \\{0,1\\}^M$, \n\\begin{gather}\n\\label{eq:chap1-rbm-meas}\np_\\theta(\\x, \\hid) = \\frac{1}{\\cZ} e^{\\vect{a}\\T \\x + \\vect{b}\\T \\hidd + \\x\\T\\W\\hidd} \\, , \\qquad \\theta = \\{ \\W, \\vect{a}, \\vect{b} \\} \\, , \n\\end{gather}\nwhere $\\cZ$ is the normalization factor, similar to the partition function of statistical physics. The parametric density model over inputs is then the marginal $p_\\theta(\\x) = \\sum_{\\hidd \\in \\{0,1\\}^M} p_\\theta(\\x, \\hid)$. Although seemingly very similar to pairwise Ising models, the introduction of hidden units provides a greater representative power to RBMs as hidden units can mediate interactions between arbitrary groups of input units. Furthermore, they can be generalized to Deep Boltzmann Machines (DBMs) \\cite{Salakhutdinov2009}, where several hidden layers are stacked on top of each other. \n\nIdentically to VAEs, RBMs can represent sophisticated distributions at the cost of an intractable likelihood. Indeed the summation over $2^{M+N}$ terms in the partition function cannot be simplified by an analytical trick and is only realistically doable for small models. \nRBMs are commonly trained through a gradient ascent of the likelihood using approximated gradients. As exact Monte Carlo evaluation is a costly operation that would need to be repeated at each parameter update in the gradient ascent, several more or less sophisticated approximations are preferred: contrastive divergence (CD) \\cite{Hinton2002},  its persistent variant (PCD) \\cite{Tieleman2008} or even parallel tempering \\cite{Desjardins2010,Cho2010}.\n\nRBMs were the first effective generative models using neural networks. They found applications in various domains including dimensionality reduction \\cite{Hinton2006a}, classification \\cite{larochelle2008classification}, collaborative filtering \\cite{salakhutdinov2007restricted}, feature learning \\cite{coates2011analysis}, and topic modeling \\cite{hinton2009replicated}.\nUsed for an unsupervised pre-training of deep neural networks layer by layer \\cite{Hinton2006,Bengio2007}, they also played a crucial role in the take-off of supervised deep learning. \n\n\n\n\n\\paragraph{Open questions and challenges}\nGenerative models involving neural networks such as VAE, GANs and RBMs have great expressive powers at the cost of not being amenable to exact treatment. Their training, and sometimes even their sampling requires approximations. From a practical standpoint, whether these approximations can be either made more accurate or less costly is an open direction of research. \nAnother important related question is the evaluation of the performance of generative models \\cite{Sajjadi2018}. To start with the objective function of training is very often itself intractable (e.g. the likelihood of a VAE or a RBM), and beyond this objective, the unsupervised setting does not define a priori a test task for the performance of the model. \nAdditionally, unsupervised deep learning inherits some of the theoretical puzzles already discussed in the supervised learning section. In particular, assessing the difficulty to represent a distribution and select a sufficient minimal model and\/or training data set is an ongoing effort of research.\n\n\n\\section[Machine learning with neural networks\n]{Machine learning with neural networks \n}\n\\label{sec:chap1}\n\n\nThis \\citechap~provides the fundamental concepts in machine learning that will be used throughout this thesis. A comprehensive reference is \\cite{Goodfellow2016}. We also take this \\citechap~as an opportunity to introduce the current challenges in the understanding of deep learning.  \n\n\\input{chap1_ml.tex}\n\n\n\\section{Statistical inference and the statistical physics approach\n}\n\\label{sec:chap2}\n\nTo tackle the open questions and challenges surrounding neural networks mentioned in the previous \\citesec, we need to manipulate high-dimensional probability distributions. The generic concept of statistical inference refers to the extraction of useful information from these complicated objects.  Statistical physics, with its probabilistic interpretation of natural systems composed of many elementary components, is naturally interested in similar questions. \nWe provide in this section a few concrete examples of inference questions arising in neural networks and explicit how statistical physics enters the picture. In particular, the theory of disordered systems appears here especially relevant. \n    \n\n\\subsection{Statistical inference}\n\n\\label{sec:chap2-stat-inf}\n \n\\subsubsection{Some inference questions in neural networks for machine learning}\n\\label{sec:chap2-teacher-student}\n\n\\paragraph{Inference in generative models}\nGenerative models used for unsupervised learning are statistical models defining high-dimensional distributions with complex dependencies. As we have seen in \\citesec~\\ref{sec:chap1-unsupervised}, the most common training objective in unsupervised learning is the maximization of the log-likelihood, i.e. the log of the probability assigned by the generative model to the training set $\\{\\x\\kk\\}_{k=1}^{P}$. Computing the probability of observing a given sample $\\x\\kk$ is an inference question. It requires to marginalize over all the hidden representations of the problem. For instance in the RBM \\eqref{eq:chap1-rbm-meas},\n\\begin{gather}\n   \n    p_\\theta(\\x\\kk) = \\frac{1}{\\cZ} \\sum_{\\hidd \\in \\{0,1\\}^M} e^{\\vect{a}\\T \\x\\kk + \\vect{b}\\T \\hidd + \\x\\kk\\T\\W\\hidd}.\n    \\end{gather}\nWhile the numerator will be easy to evaluate, the partition function has no analytical expression and its exact evaluation requires to sum over all possible states of the network.\n\n\n\n\\paragraph{Learning as statistical inference: Bayesian inference and the teacher-student scenario}\n\n\nThe practical problem of training neural networks from data as introduced in \\citechap~\\ref{sec:chap1} is not in general interpreted as inference. To do so, one needs to treat the learnable parameters as random variables, which is the case in Bayesian learning. For instance in supervised learning, an underlying prior distribution $p_\\theta(\\theta)$ for the weights and biases of a neural network \\eqref{eq:chap1-dnn}-\\eqref{eq:chap1-dnn-theta} is assumed, so that Bayes rule defines a posterior distribution given the training data $\\mathcal{D}$,\n\\begin{align}\np(\\theta | \\mathcal{D}) & = \\frac{p(\\mathcal{D} | \\theta) p_\\theta(\\theta)}{p(\\mathcal{D})}. \n\\end{align}\nCompared to the single output of risk minimization, we obtain an entire distribution for the learned parameters $\\theta$, which takes into account not only the training data but also some knowledge on the structure of the parameters (e.g. sparsity) through the prior. In practice, Bayesian learning and traditional empirical risk minimization may not be so different. On the one hand, the Bayesian posterior distribution is often summarized by a point estimate such as its maximum. On the other hand risk minimization is often biased towards desired properties of the weights through regularization techniques (e.g. promoting small norm) recalling the role of the Bayesian prior. \n\nHowever, from a theoretical point of view, Bayesian learning is of particular interest in the \\emph{teacher-student} scenario. The idea here is to consider a toy model of the learning problem where parameters are effectively drawn from a prior distribution.\nLet us use as an illustration the case of the supervised learning of the perceptron model \\eqref{eq:chap1-perceptron}. We draw a weight vector $\\vect{w}_0$, from a prior distribution $p_w(\\cdot)$, along with a set of $P$ inputs $\\{\\x\\kk\\}_{k=1}^{P}$ i.i.d from a data distribution $p_x(\\cdot)$. Using this \\emph{teacher} perceptron model we also draw a set of possibly noisy corresponding outputs $y\\kk$ from a teacher conditional probability $p(.| \\vect{w}_0\\T\\x\\kk)$. From the training set of the $P$ pairs $\\mathcal{D} = \\{\\x\\kk, y\\kk\\}$, one can attempt to rediscover the teacher rule by training a \\emph{student} perceptron model.\nThe problem can equivalently be phrased as a reconstruction inference question: can we recover the value of $\\vect{w}_0$ from the observations in $\\mathcal{D}$? The Bayesian framework yields a posterior distribution of solutions\n\\begin{gather}\n    p(\\vect{w}| \\mathcal{D}) = \\prod_{k=1}^P p(y\\kk| \\vect{w}\\T\\x\\kk)p_w(\\vect{w}) \\, \/ \\, \\prod_{k=1}^P p(y\\kk | \\x\\kk).\n\\end{gather}\n\nNote that the terminology of teacher-student applies for a generic inference problem of reconstruction: the statistical model used to generate the data along with the realization of the unknown signal is called the \\emph{teacher}; the statistical model assumed to perform the reconstruction of the signal is called the \\emph{student}. When the two models are identical or matched, the inference is \\emph{Bayes optimal}. When the teacher model is not perfectly known, the statistical models can also be different (from slightly differing prior distributions to entirely different models), in which case they are said to be mismatched, and the reconstruction is suboptimal.\n\nOf course in practical machine learning applications of neural networks, one has only access to an empirical distribution of the data and it is unclear whether there should exist a formal rule underlying the input-output mapping.\nYet the teacher-student setting is a modelling strategy of learning which offers interesting possibilities of analysis and we shall refer to numerous works resorting to the setup in  \\citesec~\\ref{sec:chapex-all}.\n\n\n\\subsubsection{Answering inference questions}\n\\label{sec:chap2-challenges-inf}\n\nMany inference questions in the line of the ones mentioned in the previous \\citesec~have no tractable exact solution. When there exists no analytical closed-form, computations of averages and marginals require summing over configurations. Their number typically scales exponentially with the size of the system, then becoming astronomically large for high-dimensional models.\nHence it is necessary to design approximate inference strategies. They may require an algorithmic implementation but must run in finite (typically polynomial) time. \nAn important cross-fertilization between statistical physics and information sciences have taken place over the past decades around the questions of inference. Two major classes of such algorithms are Monte Carlo Markov Chains (MCMC), and mean-field methods. The former is nicely reviewed in the context of statistical physics in \\cite{Krauth2006}. The latter will be the focus of this short review, in the context of deep learning.\n\nNote that representations of joint probability distributions through probabilistic graphical models and factor graphs are crucial tools to design efficient inference strategies. In \\citeapp~\\ref{app:chap2-graphs}, we quickly introduce for the unfamiliar reader these two formalisms that enable to encode and exploit independencies between random variables. As examples, \\citefig~\\ref{fig:chap2-graphs} presents graphical representations of the RBM measure \\eqref{eq:chap1-rbm-meas} and the posterior distribution in the Bayesian learning of the perceptron as discussed in the previous \\citesec.\n\n \n\n\n\n\n\\begin{figure}[t]\n    \\centering\n    \\captionsetup{width=.3\\linewidth}\n    \\subfloat[Restricted Boltzmann Machine]{\\includegraphics[width=0.42\\textwidth, valign=m]{chap2_rbm_GM.pdf}\n    \\label{fig:chap2-rbm-bis}}\n   \n    \\captionsetup{width=.4\\linewidth}\n    \\hspace{0.1\\textwidth}\n    \\subfloat[Perceptron teacher and student. ]{\\includegraphics[width=0.4\\textwidth, valign=m]{chap2_perceptron.pdf}\n    \\label{fig:chap2-glm-bis}}\n    \\captionsetup{width=.9\\linewidth}\n    \\caption{\\textbf{(a)} Undirected probabilistic graphical model (left) and factor graph representation (right). \\textbf{(b)} Left: Directed graphical model of the generative model for the training data knowing the teacher weight vector $\\vect{w}_0$. Right: Factor graph representation of the posterior distribution for the student $p(\\vect{w} | \\X, \\y )$, where the vector $\\y \\in \\R^P$ gathers the outputs $y_{(k)}$ and the matrix $\\X\\in\\R^{N \\times P}$ gathers the inputs $\\x_{(k)}$.\\label{fig:chap2-graphs}}\n\\end{figure}\n\n\n\n\\subsection{Statistical physics of disordered systems, first appearance on stage}\nHere we re-introduce briefly fundamental concepts of statistical physics that will help to understand connections with inference and the origin of the methods presented in what follows.\n\n\\paragraph{The thermodynamic limit}\n\nThe equilibrium statistics of classical physical systems are described by the Boltzmann distribution.\nFor a system with $N$ degrees of freedom noted $\\x \\in \\cX^N$ and an energy functional $E(\\x)$, we have\n\\begin{align}\n    \\label{eq:chap2-boltzmann}\n    p(\\x) = \\frac{e^{-\\beta E(\\x)}}{\\cZ_N}, \\quad \\cZ_N = \\sum_{\\x \\in \\mathcal{X}^N} e^{-\\beta E(\\x)}, \\quad \\beta = 1\/k_B T ,\n\\end{align}\nwhere we defined the partition function $\\cZ_N$ and the inverse temperature $\\beta$.\nTo characterize the macroscopic state of the system, an important functional is the free energy \n\\begin{align}\n    F_N & = - \\log \\cZ_N \/ \\beta = - \\frac 1 \\beta \\log \\sum_{\\x \\in \\mathcal{X}^N} e^{-\\beta E(\\x)}. \n   \n   \n   \n\\end{align} \nWhile the number of available configurations $\\cX^N$ grows exponentially with $N$, considering the \\emph{thermodynamic} limit $N \\to \\infty$ typically simplifies computations due to concentrations. \nLet $e_N = E \/N$ be the energy per degree of freedom, the partition function can be re-written as a sum over the configurations of a given energy $e_N$\n\\begin{align}\n    \\cZ_N = \n   \n    \\sum_{e_N} e^{-N \\beta f_N(e_N)} ,\n\\end{align}\nwhere we define $f_N(e_N)$ the free energy density of states of energy $e_N$. This rewriting implies that \nat large $N$ the states of energy minimizing the free energy are exponentially more likely than any other states. Provided the following limits exist, the statistics of the system are dominated by the former states and we have the thermodynamic quantities\n\\begin{gather}\n    \\cZ = \\lim_{N \\to \\infty} \\cZ_{N} =  e^{-\\beta f}, \\text{ and } f = \\lim_{N \\to \\infty} F_N \/ N .\n\\end{gather}\nThe interested reader will also find a more detailed yet friendly presentation of the thermodynamic limit in \\citesec~2.4 of \\cite{Mezard2009}.\n\n\\paragraph{Disordered systems}\n\nRemarkably, the statistical physics framework can be applied to inhomogeneous systems with \\emph{quenched disorder}. In these systems, interactions are functions of the realization of some random variables. An iconic example is the Sherrington-Kirkpatrick (SK) model \\cite{Sherrington1975}, a fully connected Ising model with random Gaussian couplings $\\mat{J} = (J_{i j})$, that is where the $J_{ij}$ are drawn independently from a Gaussian distribution. As a result, the energy functional of disordered systems is itself function of the random variables. For instance here, the energy of a spin configuration $\\x$ is then $E(\\x; \\mat{J}) = - \\frac 1 2 \\x\\T \\mat{J} \\x$. \nIn principle, system properties depend on a given realization of the disorder. In our example, the correlation between two spins $\\langle x_i x_j \\rangle_J$ certainly does. Yet some aggregated properties are expected to be \\emph{self-averaging} in the thermodynamic limit, meaning that they concentrate on their mean with respect to the disorder as the fluctuations are averaged out.\nIt is the case for the free energy. As a result, here it formally verifies:\n\\begin{gather}\n    \\lim_{N\\to\\infty} F_{N; \\mat{J}} \/ N = \\lim_{N\\to\\infty} \\E_{\\mat{J}}[F_{N; \\mat{J}} \/ N] = f.\n\\end{gather}\n(see e.g. \\cite{Mezard1986, Castellani2005} for discussions of self-averaging in spin glasses).\nThus the typical behavior of complex systems is studied in the statistical physics framework by taking two important conceptual steps: averaging over the realizations of the disorder and considering the thermodynamic limit. These are starting points to design approximate inference methods. Before turning to an introduction to mean-field approximations, we stress the originality of the statistical physics approach to inference. \n\n\n\n\n\\paragraph{Statistical physics of inference problems}\nStatistical inference questions are mapped to statistical physics systems by interpreting general joint probability distributions as Boltzmann distributions \\eqref{eq:chap2-boltzmann}. \nTurning back to our simple examples of \\citesec~\\ref{sec:chap2-stat-inf}, the RBM is trivially mapped as it directly borrows its definition from statistical physics. We have\n    \\begin{gather}\n        E(\\x, \\hidd ; \\W) = - \\vect{a}\\T \\x - \\vect{b}\\T \\hidd - \\x\\T\\W\\hidd .\n    \\end{gather} \nThe inverse temperature parameter can either be considered equal to 1 or as a scaling factor of the weight matrix $\\W \\leftarrow \\beta \\W$ and bias vectors $\\vect{a} \\leftarrow \\beta \\vect{a}$ and $\\vect{b} \\leftarrow \\beta \\vect{b} $. The RBM parameters play the role of the disorder. Here the computational hardness in estimating the log-likelihood comes from the estimation of the log-partition function, which is precisely the free energy. In our second example, the estimation of the student perceptron weight vector,\nthe posterior distribution is mapped to a Boltzmann distribution by setting \n\\begin{gather}\n   \n    E(\\vect{w} ; \\y, \\X) = - \\log p(\\y| \\vect{w}\\T\\X)p_w(\\vect{w})\n   \n    . \n\\end{gather}The disorder is here materialized by the training data.\nThe difficulty is here to compute $p(\\y | \\X)$ which is again the partition function in the Boltzmann distribution mapping.\nRelying on the thermodynamic limit, mean-field methods will provide asymptotic results. Nevertheless, experience shows that the behavior of large finite-size systems are often well explained by the infinite-size limits. \n\nAlso, the application of mean-field inference requires assumptions about the distribution of the disorder which is averaged over. Practical algorithms for arbitrary cases can be derived with ad-hoc assumptions, but studying a precise toy statistical model can also bring interesting insights. The simplest model in most cases is to consider uncorrelated disorder: in the example of the perceptron this corresponds to random input data points with arbitrary random labels. Yet, the teacher-student scenario offers many advantages with little more difficulty. It allows to create data sets with structure (the underlying teacher rule).\nIt also allows to formalize an analysis of the difficulty of a learning problem and of the performance in the resolution. Intuitively, the definition of a ground-truth teacher rule with a fixed number of degrees of freedom sets the minimum information necessary to extract from the observations, or training data, in order to achieve perfect reconstruction. This is an \\emph{information-theoretic limit}. \n\nFurthermore, the assumption of an underlying statistical model enables the measurement of performance of different learning algorithms over the class of corresponding problems from an average viewpoint. This is in contrast with\nthe traditional approach of computer science in studying the difficulty of a class of problem based on the \\emph{worst} case. This conservative strategy yields strong guarantees of success, yet it may be overly pessimistic compared to the experience of practitioners. Considering a distribution over the possible problems (a.k.a different realizations of the disorder), the average performances are sometimes more informative of \\emph{typical} instances rather than worst ones.\nFor deep learning, this approach may prove particularly interesting as the traditional bounds, based on the VC-dimension \\cite{Vapnik2000} and Rademacher complexity \\cite{Bartlett2002,Shalev-Shwartz2014}, appear extremely loose when compared to practical examples.\n\nFinally, we must emphasize that derivations presented here are not mathematically rigorous. They are based on `correct' assumptions allowing to push further the understanding of the problems at hand, while a formal proof of the assumptions is possibly much harder to obtain.  \n\n\n\n\n\n\n\n\n\n\n\n\\section{Further extensions of interest for learning}\n\\label{sec:chap3further}\nIn the previous \\citesec~we presented the classical mean-field approximations focusing on the simple and original examples of the Boltzmann machine (a.k.a. SK model) and the GLM with Gaussian i.i.d weight matrices. Along the way, we tried to emphasize how the procedures of approximation rely on structural (e.g. connectivity) and statistical properties of the model under scrutiny. In the present \\citesec, we will see that extensions of the message passing and replica methods have now broadened the span of applicability of mean-field approximations. We focus on a selection of recent developments of particular interest to study learning problems.\n\n\\subsection{Streaming AMP for online learning}\n\\label{sec:chap3-streaming-amp}\nIn learning applications, it is sometimes advantageous for speed or generalization concerns to only treat a subset of examples at the time - making\nfor instance the SGD algorithm the most popular training algorithm in deep learning. Sometimes also, the size of the current data sets may exceed the available memory. Methods implementing a step-by-step learning as the data arrives are referred to as \\emph{online}, \\emph{streaming} or \\emph{mini-batch} learning, as opposed to \\emph{offline} or \\emph{batch} learning. \n\n\nIn \\cite{Manoel2018}, a mini-batch version of the AMP algorithm is proposed. It consists in a generalization of Assumed Density Filtering \\cite{Opper1999, Rossi2016} that are fully-online, meaning that only a single example is received at once, or mini-batches have size 1. The general derivation principle is the same. On the example of the GLM, one imagines receiving at each iteration a subset of the components of $\\y$ to reconstruct $\\x$. We denote by $\\y\\kk$ these successive mini-batches. Bayes formula gives the posterior distribution over $\\x$ after seeing $k$ mini-batches\n\\begin{align}\n    p(\\x|\\y\\kk, \\{\\y_{(k-1)}, \\cdots \\y_{(1)}\\}) = \\frac{p(\\y\\kk|\\x)p(\\x|\\{\\y_{(k-1)}, \\cdots \\y_{(1)}\\})}{\\int \\dd{\\x}p(\\y\\kk|\\x)p(\\x|\\{\\y_{(k-1)}, \\cdots \\y_{(1)}\\})} .\n\\end{align}\nThis formula suggests the iterative scheme of using as a prior on $\\x$ at iteration $k$ the posterior obtained at iteration $k-1$. This idea can be implemented in different approximate inference algorithms, as also noticed by \\cite{Broderick2013} using a variational method. In the regular version of AMP an effective factorized posterior is given at convergence by the input update functions \\eqref{eq:chap3-Zx}-\\eqref{eq:chap3-f2x}:\n\\begin{align}\np(\\x|\\y, \\W) \\simeq \\prod_{i=1}^N \\frac{1}{\\cZ_x(\\lambda_i, \\sigma_i)}p_x(x_i)e^{-\\frac{(\\lambda_i-x_i)^2}{2 \\sigma_i}}.     \n\\end{align}\nPlugging this posterior approximation in the iterative scheme yields the mini-AMP algorithm using the converged values of ${\\lambda_{(\\ell)}}_i$ and ${\\sigma_{(\\ell)}}_i$ at each anterior mini-batch $\\ell < k$ to compute the prior\n\\begin{align}\n    p_{x}^{(k)}(\\x) = p(\\x|\\{\\y_{(k-1)}, \\cdots \\y_{(1)}\\}, \\W) \\simeq \\prod_{i=1}^N \\frac{1}{\\cZ_{x,i}} \\; p_x(x_i)  \\; e^{-\\sum\\limits_{\\ell=1}^{k-1}\\frac{(\\lambda_{(\\ell), i}-x_i)^2}{2 \\sigma_{(\\ell), i}}},     \n\\end{align}\nwhere the $\\cZ_{x,i}$ normalize each marginal factor.\nCompared to a naive mean-field variational approximation of the posterior, AMP takes into account more correlations and is indeed found to perform better in experiments reported by \\cite{Manoel2018}. Another advantage of the AMP based online inference is that it is amenable to theoretical analysis by a corresponding State Evolution \\cite{Opper1999, Rossi2016, Manoel2018}.\n\n\n\\subsection{Algorithms and free energies beyond i.i.d. matrices} \n\\label{sec:chap3-ortho-invariant}\nThe derivations outlined in the previous \\citesecs~of the equivalent replica, TAP and AMP equations required the weight matrices to have Gaussian i.i.d. entries. In this case, rigorous proofs of asymptotic exactness of the mean-field solutions were found, for the SK model \\cite{Talagrand2006} and the GLM \\cite{Reeves2016, Barbier2017a}. Mean-field inference with different weight statistics is a priori feasible if one finds a way either to perform the corresponding disorder average in the replica computation, to evaluate the corresponding Onsager correction in the TAP equations, or to write a message passing where messages remain uncorrelated (even in the high-connectivity limit we may be interested in).  \n\nEfforts to broaden in practice the class of matrices amenable to such mean-field treatments lead to a series of works in statistical physics and signal processing with related propositions.\nParisi and Potters pioneered this direction by deriving mean-field equations for orthogonal weight matrices using a high-temperature expansion \\cite{Parisi1995}.\nThe adaptive TAP approach proposed by Opper and Winther \\cite{Opper2001, Opper2001prl} further allowed for inference in densely connected graphical models without prior knowledge on the weight statistics. The Onsager term of these TAP equations was evaluated using the cavity method for a given weight sample. The resulting equations were then understood to be a particular case of the Expectation Propagation (EP) \\cite{Minka2001} - belonging to the class of message passing algorithms for approximate inference - yet applied in densely connected models \\cite{Opper2005}. An associated approximation of the free energy called Expectation Consistency (EC) was additionally derived from the EP messages. Subsequently, Kabashima and collaborators \\cite{Shinzato2008, Shinzato2009, Kabashima2008} focused on the perceptron and the GLM to propose TAP equations and a replica derivation of the free energy for the ensemble of orthogonally invariant random weight matrices. In the singular value decomposition of such weight matrices, $\\W=\\mat{U}\\,\\mat{S}\\,\\mat{V}\\T \\in \\R^{M\\times N}$, the orthogonal basis matrices $\\mat{U}$ and $\\mat{V}$ are drawn uniformly at random from respectively $\\mathrm{O}(M)$ and $\\mathrm{O}(N)$, while the diagonal matrix of singular values $\\mat{S}$ has an arbitrary spectrum. The consistency between the EC free energy and the replica derivation for orthogonally invariant matrices was verified by \\cite{Kabashima2014} for signal recovery from linear measurements (the GLM without G). From the algorithmic perspective, Fletcher, Rangan and Schniter \\cite{Rangan2016, Schniter2016} applied the EP to the GLM to obtain the (Generalized) Vector-Approximate Message Passing (G-VAMP) algorithm. Remarkably, these authors proved that the behavior of the algorithm could be characterized in the thermodynamic limit, provided the weight matrix is drawn from the orthogonally invariant ensemble, by a set of scalar State Evolution equations similarly to the AMP algorithm. These equations are again related to the saddle point equations of the replica free energy. Concurrently, Opper, Cakmak and Winther proposed an alternative procedure for solving TAP equations with orthogonally invariant weight matrices in Ising spin systems relying on an analysis of iterative algorithms \\cite{Opper2016, Cakmak2019}. Finally, \\cite{Maillard2019} revisits the above cited contributions and provides detailed considerations on their connections. \n\nBelow we present the aforementioned free energy as proposed by \\cite{Shinzato2008, Shinzato2009, Kabashima2008}, and the G-VAMP algorithm of \\cite{Schniter2016}.\n\n\\subsubsection{Replica free energy for the GLM in the Bayes Optimal setting}\nConsider the ensemble of orthogonally invariant weight matrices $\\W$ with spectral density $\\sum_{i=1}^N\\dirac(\\lambda - \\lambda_i) \/ N$ of their `square' $\\W \\W\\T$ converging in the thermodynamic limit $ N \\to + \\infty$ to a given density $\\rho_\\lambda(\\lambda)$. The quenched free energy of the GLM in the Bayes optimal setting derived in \\cite{Kabashima2008, Shinzato2009} writes\n\\begin{gather}\n    \\label{eq:chap3-kaba-fe} \n    -f = \\mathrm{extr}_{q \\hat{q}} \\left[ - \\frac{1}{2} q \\hat{q} +  \\mathcal{I}_x(\\hat{q}) + \\mathcal{J}_z(q_0, q, \\alpha, \\rho_\\lambda) \\right] \\, ,\\\\\n    \\mathcal{J}_z(q_0, q, \\alpha, \\rho_\\lambda) =  \\mathrm{extr}_{u \\hat{u}} \\left[ F_{\\rho_\\lambda, \\alpha}(q_0-q, \\hat{u}\/\\lambda_0) +\\frac{\\hat{u}q_0}{2} - \\frac{\\alpha \\hat{u}u}{2 \\lambda_0} + \\alpha \\mathcal{I}_z(q_0\\lambda_0\/\\alpha, u) \\right],\n\\end{gather}\nwhere $\\mathcal{I}_x$ and $\\mathcal{I}_z$ were defined as \\eqref{eq:chap3-replica-fe-glm_Ix}-\\eqref{eq:chap3-replica-fe-glm_Iz} and the spectral density $\\rho_\\lambda(\\lambda)$  appears via its mean $\\lambda_0=\\E_{\\lambda}[\\lambda]$ and in the definition of \n\\begin{gather}\n    F_{\\rho_\\lambda, \\alpha}(q, u) = \\frac{1}{2} \\mathrm{extr}_{\\Lambda_q, \\Lambda_u} \\left[ -(\\alpha-1)\\log\\Lambda_u - \\E_{\\lambda}\\log(\\Lambda_u\\Lambda_q + \\lambda) + \\Lambda_q q + \\alpha\\Lambda_u u \\right] \\notag \\\\\n    \\qquad \\qquad \\qquad  - \\frac{1}2 (\\log q + 1) + \\frac{\\alpha}2 (\\log u + 1) .\n\\end{gather}\nGaussian random matrices are a particular case of the considered ensemble. Their singular values are characterized asymptotically by the Marcenko-Pastur distribution \\cite{Marcenko1967}. In this case, one can check that the above expression reduces to \\eqref{eq:chap3-replica-fe-glm}. More generally, note that $\\mathcal{J}_z$ generalizes $\\mathcal{I}_z$.\n\n\\subsubsection{Vector Approximate Message Passing for the GLM}\nThe VAMP algorithm consists in writing EP \\cite{Minka2001} with Gaussian messages on the factor graph representation of the GLM posterior distribution given in \\citefig~\\ref{fig:chap3-vamp-glm}. The estimation of the signal $\\x$ is decomposed onto four variables, two duplicates of $\\x$ itself and two duplicates of the linear transformation $\\z = \\W\\x$. The potential functions $\\psi_x$ and $\\psi_z$ of factors connecting copies of the same variable are Dirac distributions enforcing their equality. The factor node linking $\\z^{(2)}$ and $\\x^{(2)}$ is assumed Gaussian with variance going to zero.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{chap3_vamp_glm.pdf}\n    \\caption{Factor graph representation of the GLM for the derivation of VAMP \\label{fig:chap3-vamp-glm}}\n\\end{figure}\nThe procedure of derivation, equivalent to the projection of the BP algorithm on Gaussian messages, is recalled in \\citeapp~\\ref{sec:app-chap3-vamp} and leads to \\citealg~\\ref{alg:chap3-vamp}. Like for AMP, the algorithm features some auxiliary variables introduced along the derivation. At convergence the duplicated $\\xh_1$, $\\xh_2$ (and $\\hat{\\z}_1$, $\\hat{\\z}_2$) are equal and either can be returned by the algorithm as an estimator.\nFor readability, we omitted the time indices in the iterations that here simply follow the indicated update.\n\n\\input{chap3_vamp.tex}\n\nFor a given instance of the GLM inference problem, i.e. a given weight matrix $\\W$, one can always launch either the AMP algorithm or the VAMP algorithm to attempt the reconstruction. If the weight matrix has i.i.d. zero mean Gaussian entries, the two strategies are conjectured to be equivalent and GAMP can be provably convergent for certain settings \\cite{Rangan2014}. If the weight matrix is not Gaussian but orthogonally invariant, then only VAMP is expected to always converge. More generally, even in cases where none of these assumptions are verified, VAMP has been observed to have less convergence issues than AMP. \n\nLike for AMP, a State Evolution can also be derived for VAMP (which was actually directly proposed for the multi-layer GLM \\cite{Fletcher2018a}). It rigorously characterizes the behavior of the algorithm when $\\W$ is orthogonally invariant. One can also verify that the SE fixed points can be mapped to the solutions of the saddle point equations of the replica free energy \\eqref{eq:chap3-kaba-fe} (see \\citesec~1 of Supplementary Material of \\cite{Gabrie2018}); so that the robust algorithmic procedure can advantageously be used to compute the fixed points to be plugged in the replica potential to approximate the free energy.\n\n\n\n\n\n\\subsection{Multi-value AMP}\n\\label{sec:chap3-multivalue}\nA recent extension of AMP consists in treating the simultaneous reconstruction of multiple signals undergoing the same mixing step from the corresponding multiple observations. This is a situation of particular interest for learning appearing for instance in the teacher-student set-up of committee machines. The authors of \\cite{Aubin2018} showed that the different weight vectors of these neural networks can be inferred from the knowledge of training input-output pairs introducing this extended version of AMP. Here the same matrix of training input data mixes the teacher weight vectors to produce the training output data. For a matter of consistency with the examples used in the previous sections, we here formalize the algorithm for the GLM. Nevertheless this is just a matter of rewriting of the committee algorithm of \\cite{Aubin2018}.\n\nConcretely let's consider a GLM with $P$ pairs of signal and observations $\\{\\x\\kk, \\y\\kk\\}_{k=1}^P$, gathered in matrices $\\X \\in \\R^{N\\times P}$ and $\\Y \\in \\R^{M\\times P}$. We are interested in the posterior distribution \n\\begin{gather}\n    \\label{eq:chap6-glm-vec-meas}\n    p(\\X | \\Y, \\W) = \\frac{1}{\\cZ(\\Y, \\W)} \\prod_{i=1}^N p(\\x_i)\\prod_{\\mu=1}^M \\pout(\\y_\\mu | \\vect{w}_\\mu\\T\\X), \\quad \\x_i \\in \\R^P, \\quad \\y_\\mu \\in \\R^P. \n\\end{gather}\nCompared to the traditional GLM measure \\eqref{eq:chap3-glm-meas}, scalar variables are here replaced by vectors in $\\R^P$. In \\citeapp~\\ref{app:chap6-vect-amp} we present a derivation starting from BP of the corresponding AMP presented in \\citealg~\\ref{alg:chap6-vect-amp}.  \nThe major difference with the scalar GLM consists in the necessity of tracking covariance matrices between the $P$ different variables instead of simple variances.\n\n\\input{chap6_vect_amp}\n\nThis AMP algorithm can also be analyzed by a State Evolution. \nIn \\cite{Aubin2018}, the teacher-student matched setting of the committee machine is examined through the replica approach and the Bayes optimal State Evolution equations are obtained as the saddle point equations of the replica free energy.\nIn \\citeapp~\\ref{app:chap6-vect-amp} we present the alternative derivation of the State Evolution equations from the message passing and without assuming a priori matched teacher and student, as done in \\cite{Gabrie2019}.\n\n\n\n\n\\begin{figure}[t]\n    \\centering\n    {\\includegraphics[width=0.5\\textwidth]{chap6_mlglm_scalar.pdf}\n    }\n   \n    \\caption{Factor graph representation of a generic 2-layer GLM. \\label{fig:chap6-mlglm}}\n\\end{figure}\n\n\\subsection{Model composition and multi-layer inference}\nAnother recent and ongoing direction of extension of mean-field methods is the combination of solutions of elementary models to tackle more sophisticated inference questions. The graphical representations of probability distributions (reintroduced briefly in \\citeapp~\\ref{app:chap2-graphs}) are here of great help. In a complicated joint probability distribution, it is sometimes possible to identify well-known sub-models, such as the GLM or the RBM. Understanding how and when it is justified to plug-in different solutions is of course non-trivial and a very promising direction of research. \n\nA particularly relevant extension in this direction is the treatment of multi-layer GLMs, or in other words multi-layer neural networks. With depth $L$,  hidden layers noted $\\vect{u}^\\ell \\in \\R^{N_\\ell}$, and weight matrices $\\mat{\\Phi}^\\ell \\in \\R^{N_{\\ell +1} \\times N_\\ell}$, it formally corresponds to the statistical model\n\\begin{gather}\n    \\vect{u}^0 = \\x_0 \\sim p_{x_0}(\\x_0) \\, ,\\\\\n    \\vect{u}^\\ell \\sim \\pout^\\ell(u^\\ell | \\mat{\\Phi}^{\\ell-1} \\vect{u}^{\\ell -1}) \\quad \\forall \\ell = 1 \\cdots L -1 \\, , \\\\\n    \\y \\sim \\pout^{L}(\\y | \\mat{\\Phi}^{L-1} \\vect{u}^{L -1} ).\n\\end{gather}\nIn \\cite{Manoel2017b} a multi-layer version of AMP is derived, assuming Gaussian i.i.d weight matrices, along with a State Evolution and a replica free energy. Remarkably, the asymptotic replica prediction was mathematically proven correct in \\cite{Gabrie2018}.  In \\cite{Fletcher2018a}, the multi-layer version of the VAMP algorithm is derived with the corresponding State Evolution for orthogonally invariant weight matrices. The matching free energies were obtained independently in \\cite{Gabrie2018} by the generalization of a replica result and by \\cite{Reeves2016} from a different argument.  \n\nIn the next paragraph we sketch a derivation of the 2-layer AMP presented in \\citealg~\\ref{alg:chap6-amp-2layer}, it provides a good intuition of the composition ability of mean-field inference methods.\n\n\\paragraph{Heuristic derivation of 2-layer AMP}\n\n\nThe derivation of the multi-layer AMP follows identical steps to the derivation of the single layer presented in \\citesec~\\ref{sec:chap3-bp-to-amp},  yet for a more complicated factor graph and consequently a larger collection of messages. Without conducting the lengthy procedure, one can form an intuition for the resulting algorithm starting from the single-layer AMP. \nThe corresponding factor graph is given on \\citefig~\\ref{fig:chap6-mlglm}. Compared to the single-layer case (see \\citefig~\\ref{fig:chap3-glm}), an interface with a set of $M=N_1$ hidden variables $u_\\mu$ is inserted between the $N=N_0$ signals $x_i$ and the $Q=N_2$ observations $y_a$. In the neighborhood of the inputs $x_i$ the factor graph is identical to the single-layer and the input functions can be defined from a normalization partition identical to \\eqref{eq:chap3-Zx}, \n\\begin{gather}\n    \\cZ^x(\\lambda_i, \\sigma_i) = \\int \\dd{x_i} p_x(x_i)e^{-\\frac{ (x_i-\\lambda_i)^2}{2 \\sigma_i} },\n   \n   \n   \n   \n\\end{gather}\nyielding updates \\eqref{alg:chap6-vect-amp-2layer-xi}-\\eqref{alg:chap6-vect-amp-2layer-Cxi}. Similarly, the neighborhood of the observations $y_a$ is also unchanged and the updates \\eqref{alg:chap6-vect-amp-2layer-goutii} and \\eqref{alg:chap6-vect-amp-2layer-dgoutii} are following from the definition of\n\\begin{gather}\n    \\Zout^y(\\omega^2_a, V^2_a) =  \\int \\dd{z_a} \\; \\pout^2(\\y_a|z_a) \\;  e^{- \\frac{\\left(z_a - \\omega^2_a\\right)^2}{2V^2_a }} ,\n\\end{gather}\nidentical to the single layer \\eqref{eq:chap3-Zout}. At the interface however, the variables $u_\\mu$ play the role of outputs for the first GLM and of inputs for the second GLM, which translates into a normalization partition function of mixed form\n\\begin{gather}\n    \\label{eq:chap6-2layer-Zout-t}\n    \\Zout^u(\\omega^1_\\mu, V^1_\\mu, \\lambda^1_\\mu, \\sigma^1_\\mu )  = \\int \\dd{z_\\mu} \\int \\dd{u_\\mu} \\pout^1(u_\\mu | z_\\mu) \\; \\times \n     e^{- \\frac{\\left(u_\\mu - \\lambda^1_\\mu\\right)^2}{ 2 \\sigma^1_\\mu} } e^{-\\frac{\\left(z_\\mu - \\omega^1_\\mu\\right)^2}{2 V^1_\\mu}}. \\notag\n   \n   \n\\end{gather}\nUpdates \\eqref{alg:chap6-vect-amp-2layer-gouti} and \\eqref{alg:chap6-vect-amp-2layer-dgouti} are obtained by considering that the second layer acts as an effective channel for the first layer, i.e. from the normalization interpreted as\n\\begin{gather}\n    \\Zout^u(\\omega^1_\\mu, V^1_\\mu, \\lambda^1_\\mu, \\sigma^1_\\mu ) =  \\int \\dd{z_\\mu} \\; \\pout^{\\rm eff}(z_\\mu) \\;  e^{- \\frac{\\left(z_\\mu - \\omega^1_\\mu\\right)^2}{2 V^1_\\mu}} .\n   \n   \n   \n\\end{gather}\nFinally, update equations \\eqref{alg:chap6-vect-amp-2layer-t} and \\eqref{alg:chap6-vect-amp-2layer-Cti} are in turn derived considering the first layer defines an effective prior for the hidden variables and rewriting the normalization as\n\\begin{gather}\n    \\Zout^u=  \\int \\dd{u_\\mu}\\; p_u^{\\rm eff}(u_\\mu) \\; e^{- \\frac{\\left(u_\\mu - \\lambda^1_\\mu\\right)^2} {2 \\sigma^1_\\mu}}.\n\\end{gather}\nThe rest of the algorithm updates follows as usual from the self-consistency between the different variables introduced as they correspond to different parametrization of the same marginals. The schedule of updates and the time indexing reported in \\citealg~\\ref{alg:chap6-amp-2layer} results from the entire derivation starting from the BP messages. The generalization of the algorithm to an arbitrary number of layers is easily obtained repeating the heuristic arguments presented here. \n\n\\include{chap6_2layer_amp}\n\n\n\\section[\n    Selected overview of mean-field treatments: free energies and algorithms \n   \n    ]{\nSelected overview of mean-field treatments: \\\\ Free energies and algorithms \n}\n\\label{sec:chap3}\n\n\nMean-field methods are a set of techniques enabling to approximate marginalized quantities of joint probability distributions by exploiting knowledge on the dependencies between random variables. \nThey are usually said to be analytical - as opposed to numerical Monte Carlo methods. In practice they usually replace a summation exponentially large in the size of the system by an analytical formula involving a set of parameters, themselves solution of a closed set of non-linear equations. Finding the values of these parameters typically requires only a polynomial number of operations.  \n\nIn this \\citechap, we will give a selected overview of mean-field methods as they were introduced in the statistical physics and\/or signal processing literature. A key take away of what follows is that closely related results can be obtained from different heuristics of derivation. \nWe will start by deriving the simplest and historically first mean-field method. We will then introduce the important broad techniques that are high-temperature expansions, message-passing algorithms and the replica method. In the following \\citechap~\\ref{sec:chap3further} we will additionally cover the most recent extensions of mean-field methods presented in the present \\citechap~\\ref{sec:chap3} that are relevant to study learning problems.\n\n\n\n\\subsection{Naive mean-field}\n\\label{sec:chap3-nmf}\nThe naive mean-field method is the first and somehow simplest mean-field approximation. It was introduced by the physicists Curie  \\cite{Curie1895} and Weiss \\cite{Weiss1907} and then adopted by the different communities interested in inference \\cite{Wainwright2008}.\n\n\\subsubsection{Variational derivation}\nThe naive mean-field method consists in approximating the joint probability distribution of interest by a fully factorized distribution. Therefore, it ignores correlations between random variables. Among multiple methods of derivation, we present here the variational method: it is the best known method across fields and it readily shows that, for any joint probability distribution interpreted as a Boltzmann distribution, the rather crude naive mean-field approximation yields an upper bound on the free energy. \nFor the purpose of demonstration we consider a Boltzmann machine without hidden units (Ising model) with variables (spins) $\\x = (x_1, \\cdots ,x_N) \\in \\mathcal{X} = \\{0,1\\}^N $, and energy function\n\\begin{gather}\n    \\label{eq:chap3-ising-energy}\n    E(\\x) = - \\sum_{i=1}^N b_i x_i -  \\sum_{(ij)} W_{ij}x_i x_j = - \\vect{b}\\T \\x - \\frac{1}{2} \\x\\T \\W \\x \\, , \\quad \\vect{b} \\in \\R^N \\, , \\quad \\W \\in \\R^{N\\times N} \\, ,\n\\end{gather}\nwhere the notation $(ij)$ stands for pairs of connected spin-variables, and the weight matrix $\\W$ is symmetric. \nThe choices for $\\{0,1\\}$ rather than $\\{-1,+1\\}$ for the variable values, the notations $\\W$ for weights (instead of couplings), $\\vect{b}$ for biases (instead of local fields), as well as the vector notation, are leaning towards the machine learning conventions. We denote by $q_{\\m}$ a fully factorized distribution on $\\{0,1\\}^N$, which is a multivariate Bernoulli distribution parametrized by the mean values $\\m = (m_1, \\cdots, m_N) \\in [0,1]^N$ of the marginals (denoted by $q_{m_i}$):\n\\begin{gather}\n    q_{\\m} (\\x) = \\prod_{i=1}^N q_{m_i}(x_i) = \\prod_{i=1}^N m_i \\dirac(x_i - 1) + (1-m_i) \\dirac(x_i). \n\\end{gather}\nWe look for the optimal $q_{\\m}$ distribution to approximate the Boltzmann distribution $p(\\x) =  e^{-\\beta E(\\x)}\/\\cZ$ by minimizing the KL-divergence \n\\begin{align}\n    \\minn{\\m} \\KL(q_{\\m} || p) & = \\minn{\\m} \\sum_{\\x \\in \\mathcal{X}} q_{\\m}(\\x) \\log \\frac{q_{\\m}(\\x)}{p(\\x)} \\\\\n    & = \\minn{\\m}  \\sum_{\\x \\in \\mathcal{X}} q_{\\m}(\\x) \\log q_{\\m}(\\x) +  \\beta \\sum_{\\x \\in \\mathcal{X}} q_{\\m}(\\x) E(\\x)  + \\log \\cZ \\\\\n    & = \\minn{\\m} \\; \\beta G(q_{\\m}) - \\beta F \\geq 0, \\label{eq:chap3-variational-inequality}\n\\end{align}\nwhere the last inequality comes from the positivity of the KL-divergence. For a generic distribution $q$, $G(q)$ is the \\emph{Gibbs free energy} for the energy $E(\\x)$, \n\\begin{gather}\n    G(q) =  \\sum_{\\x \\in \\mathcal{X}} q(\\x) E(\\x) + \\frac{1}{\\beta}\\sum_{\\x \\in \\mathcal{X}} q(\\x) \\log q(\\x) = U(q) - H(q)\/\\beta \\geq F ,\n\\end{gather}\ninvolving the average energy $U(q)$ and the entropy $H(q)$.\nIt is greater than the true free energy $F$\nexcept when $q = p$, in which case they are equal. Note that this fact also means that the Boltzmann distribution minimizes the Gibbs free energy. Restricting to factorized $q_{\\m}$ distributions, we obtain the naive mean-field approximations for the mean value of the variables (or \\emph{magnetizations}) and the free energy:\n\\begin{gather}\n    \\m^* = \\argminn{\\m} G(q_{\\m}) = \\langle \\x \\rangle_{q_{\\m^*}} \\, , \\\\\n   \n    F_{\\rm NMF} = G(q_{\\m^*}) \\geq F.\n\\end{gather} \nThe choice of a very simple family of distributions $q_{\\m}$ limits the quality of the approximation but allows for tractable computations of observables, for instance the two-spins correlations $\\langle x_i x_j \\rangle_{q^*} = m^*_i  m^*_j$ or variance of one spin $\\langle x_i^2 \\rangle_{q^*} - \\langle x_i \\rangle_{q^*}^2 = m^*_i - {m^*_i}^2$. \n\nIn our example of the Boltzmann machine, it is easy to compute \nthe Gibbs free energy for the factorized ansatz, we define functions of the magnetization vector:\n\\begin{align}\n    U_{\\rm NMF}(\\m) & = \\langle E(\\x) \\rangle_{q_{\\m}} = - \\vect{b}\\T \\m - \\frac{1}{2} \\m\\T\\W\\m \\, ,\\\\\n    \\label{eq:chap3-hnmf}\n    H_{\\rm NMF}(\\m) & = - \\langle \\log q_{\\m}(\\x) \\rangle_{q_{\\m}} = - \\sum_{i=1}^N m_i \\log m_i + (1-m_i) \\log (1-m_i) \\, ,\\\\\n    G_{\\rm NMF}(\\m) &= G(q_{\\m})  = U_{\\rm NMF}(\\m) - H_{\\rm NMF}(\\m) \/ \\beta.\n\\end{align}\nLooking for stationary points we find a closed set of non linear equations for the $m^*_i$,\n\\begin{gather}\n    \\label{eq:chap3-nmf-eq}\n    \\left. \\frac{\\partial G_{\\rm NMF}}{\\partial m_i} \\right|_{\\m^*} = 0\n    \\quad \\Rightarrow \\quad m^*_i = \\sigm(\\beta b_i + \\sum_{j \\in \\partial i} \\beta W_{ij} m^*_j) \\quad \\forall i = 1 \\cdots N\\,\n\\end{gather}\nwhere $\\sigm(x) = (1  + e^{-x})^{-1}$.\nThe solutions can be computed by iterating these relations from a random initialization until a fixed point is reached.\n\nTo understand the implication of the restriction to factorized distributions,\nit is instructive to compare this naive mean-field equation with the exact identity \n\\begin{align}\n    \\label{eq:chap3-mf-identity}\n    \\langle x_i \\rangle_p = \\langle \\sigm(\\beta b_i + \\sum_{j \\in \\partial i}\\beta W_{ij} x_j) \\rangle_p\\,,\n\\end{align}\nderived in a few lines in \\citeapp~\\ref{app:chap3-mf-identity}.\nUnder the Boltzmann distribution $p(\\x) = e^{-\\beta E(\\x)}\/\\cZ$, these averages are difficult to compute. The naive mean-field method is neglecting the fluctuations of the effective field felt by the variable $x_i$:  $\\sum_{j \\in \\partial i} W_{ij} x_j$, keeping only its mean $\\sum_{j \\in \\partial i} W_{ij} m_j$. This incidentally justifies the name of mean-field methods.   \n\n\\subsubsection{When does naive mean-field hold true?}\nThe previous derivation shows that the naive mean-field approximation allows to bound the free energy. While this bound is expected to be rough in general, the approximation is reliable when the fluctuations of the local effective fields $\\sum_{j \\in \\partial i} W_{ij} x_j$ are small. This may happen in particular in the thermodynamic limit $N\\to \\infty$ in \\emph{infinite range} models, that is when weights or couplings are not only local but distributed in the entire system, or if each variable interacts directly with a non-vanishing fraction of the whole set of variables (e.g. \\cite{opper2001advanced} \\citechap~2). The influence on one given variable of the rest of the system can then be treated as an average background. Provided the couplings are weak enough, the naive mean-field method may even become asymptotically exact. This is the case of the \\emph{Curie-Weiss} model, which is the fully connected version of the model \\eqref{eq:chap3-ising-energy} with all $W_{ij} = 1\/N$ (see e.g. \\citesec~2.5.2 of \\cite{Mezard2009}). The sum of weakly dependent variables then concentrates on its mean by the central limit theorem. \nWe stress that it means that for finite dimensional models (more representative of a physical system, where for instance variables are assumed to be attached to the vertices of a lattice with nearest neighbors interactions), mean-field methods are expected to be quite poor. By contrast, infinite range models (interpreted as infinite-dimensional models by physicists) are thus traditionally called \\emph{mean-field models}. \n\nIn the next \\citesec~we will recover the naive mean-field equations through a different method. The following derivation will also allow to compute corrections to the rather crude approximation we just discussed by taking into account some of the correlations it neglects.\n\n\n\\subsection{Thouless Anderson and Palmer equations}\n\n\n\\label{sec:chap3-tap}\nThe TAP mean-field equations \\cite{Thouless1977, Morita1976} were originally derived as an exact mean-field theory for the Sherrington-Kirkpatrick (SK) model \\cite{Sherrington1975}. The emblematic \\emph{spin glass} SK model we already mentioned corresponds to a fully connected Ising model with energy \\eqref{eq:chap3-ising-energy} and disordered couplings $W_{ij}$ drawn independently from a Gaussian distribution with zero mean and variance $W_0 \/ N$. The derivation of \\cite{Thouless1977} followed from arguments specific to the SK model. Later, it was shown that the same approximation could be recovered from a second order Taylor expansion at high temperature by Plefka \\cite{Plefka1982} and that it could be further corrected by the systematic computation of higher orders by Georges and Yedidia \\cite{Georges1999}. We will briefly present this last derivation, having again in mind the example of the generic Boltzmann machine \\eqref{eq:chap3-ising-energy}. \n\n\\subsubsection{Outline of the derivation}\n\n\\label{sec:chap3-GY}\nGoing back to the variational formulation \\eqref{eq:chap3-variational-inequality}, we shall now perform a minimization in two steps. Consider first the family of distributions $q_{\\m}$ enforcing $\\langle \\x \\rangle_{q_{\\m}} = \\m$ for a fixed vector of magnetizations $\\m$, but without any factorization constraint. The corresponding Gibbs free energy is\n\\begin{gather}\n    G(q_{\\m})  = U(q_{\\m}) - H(q_{\\m}) \/ \\beta \n   \n   \n\\end{gather}  \nA first minimization at fixed $\\m$ over the $q_{\\m}$ defines another auxiliary free energy\n\\begin{align}\n    G_{\\rm TAP}(\\m) = \\minn{q_{\\m}} G(q_{\\m}).\n\\end{align} \nA second minimization over $\\m$ would recover the overall unconstrained minimum of the variational problem \\eqref{eq:chap3-variational-inequality} which is the exact free energy\n\\begin{align}\n    F = -\\log \\cZ \/ \\beta = \\minn{\\m} G_{\\rm TAP}(\\m).\n\\end{align}\nYet the actual value of $G_{\\rm TAP}(\\m)$ turns out as complicated to compute as $F$ itself. Fortunately, $\\beta G_{\\rm TAP}(\\m)$ can be easily approximated by a Taylor expansion around $\\beta = 0$ due to interactions vanishing at high temperature, as noticed by Plefka, Georges and Yedidia \\cite{Plefka1982, Georges1999}. \nAfter expanding, the minimization over $G_{\\rm TAP}(\\m)$ yields a set of self consistent equations on the magnetizations $\\m$, called the \\emph{TAP equations}, reminiscent of the naive mean-field equations \\eqref{eq:chap3-nmf-eq}. Here again, the consistency equations are typically solved by iterations. Plugging the solutions $\\m^*$ back into the expanded expression yields the \\emph{TAP free energy} $F_{\\rm TAP}=G_{\\rm TAP}(\\m^*)$.\nNote that ultimately the approximation lies in the truncation of the expansion. At first order the naive mean-field approximation is recovered. Historically, the expansion was first stopped at the second order. This choice was model dependent, it results from the fact that the mean-field theory is already exact at the second order for the SK model \\cite{Morita1976, Thouless1977, Plefka1982}.\n\n\n\n\n\\subsubsection{Illustration on binary Boltzmann machines and important remarks}\nFor the Boltzmann machine \\eqref{eq:chap3-ising-energy}, the TAP equations and TAP free energy (truncated at second order) are \\cite{Thouless1977},\n\\begin{gather}\n    m^*_i = \\sigm\\left(\\beta b_i + \\sum_{j \\in \\partial i} \\beta W_{ij} m^*_j - \\beta^2 W_{ij}^2(m^*_j - \\frac 1 2)(m^*_i - {m^*_i}^2 )\\right) \\; \\forall i \\label{eq:chap3-tap-eq}\\\\\n    \\beta G_{\\rm TAP}(\\m^*) = - H_{\\rm NMF}(\\m^*) - \\beta \\sum_{i=1}^N b_i m^*_i - \\beta \\sum_{(ij)} m_i^*W_{ij}m^*_j\\\\\n    \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad - \\frac{\\beta^2}{2} \\sum_{(ij)} W_{ij}^2 (m^*_i - {m^*_i}^2)(m^*_j - {m^*_j}^2)\\, , \\notag\n\\end{gather}\nwhere the naive mean-field entropy $H_{\\rm NMF}$ was defined in \\eqref{eq:chap3-hnmf}. For this model, albeit with $\\{+1, -1\\}$ variables instead of $\\{0,1\\}$, several references pedagogically present the details of the derivation sketched in the previous paragraph. The interested reader should check in particular \\cite{opper2001advanced, Zamponi2010}. We also present a more general derivation in \\citeapp~\\ref{app:chap3-real-GY}, see \\citesec~\\ref{sec:chap3-GY-generalized}.\n\n\\subparagraph{Onsager reaction term}\nCompared to the naive mean-field approximation the TAP equations include a correction to the effective field called the \\emph{Onsager reaction term}. The idea is that, in the effective field at variable $i$, we should consider corrected magnetizations of neighboring spins $j \\in \\partial i$, that would correspond to the absence of variable $i$. \nThis intuition echoes at two other derivations of the TAP approximation: the cavity method \\cite{Mezard1986} that will not be covered here and the message passing which will be discussed in the next \\citesec.\n\nAs far as the SK model is concerned,\nthis second order correction is enough in the thermodynamic limit as the statistics of the weights imply that higher orders will typically be subleading. Yet in general, the correct TAP equations for a given model will depend on the statistics of interactions and there is no guarantee that there exists a finite order of truncation leading to an exact mean-field theory. \nIn \\citesec~\\ref{sec:chap3-ortho-invariant} we will discuss models beyond SK where a conjectured exact TAP approximation can be derived. \n\n\n\\subparagraph{Single instance}\nAlthough the selection of the correct TAP approximation relies on the statistics of the weights, the derivation of the expansion outlined above does not require to average over them, i.e. it does not require an average over the disorder. Consequently, the approximation method is well defined for a single instance of the random disordered model and the TAP free energy and magnetizations can be computed for a given (realization of the) set of weights $\\{W_{ij}\\}_{(ij)}$ as explained in the following paragraph. \nIn other words, it means that the approximation can be used to design practical inference algorithms in finite-sized problems and not only for theoretical predictions on average over the disordered class of models. Crucially, these algorithms may provide approximations of disorder-dependent observables, such as correlations, and not only of self averaging quantities. \n\n\\subparagraph{Finding solutions}\nThe self-consistent equations on the magnetizations \\eqref{eq:chap3-tap-eq} are usually solved by turning them into an iteration scheme and looking for fixed points. This generic recipe leaves nonetheless room for interpretation: which exact form should be iterated? How should the updates for the different equations be scheduled? Which time indexing should be used? While the following scheme may seem natural\n\\begin{align}\n    {m_i}^{(t+1)} \\leftarrow \\sigm\\left(\\beta b_i + \\sum_{j \\in \\partial i} \\beta W_{ij} {m_j}^{(t)} - W_{ij}^2\\left({m_j}^{(t)} - \\frac 1 2\\right)\\left({m_i}^{\\mathbf{(t)}} - {{m_i}^{\\mathbf{(t)}}}^2 \\right)\\right),\n\\end{align}\nit typically has more convergence issues than the following alternative scheme including the time index $t-1$\n\\begin{align}\n    {m_i}^{(t+1)} \\leftarrow \\sigm\\left(\\beta b_i + \\sum_{j \\in \\partial i} \\beta W_{ij} {m_j}^{(t)} - W_{ij}^2\\left({m_j}^{(t)} - \\frac 1 2\\right)\\left({m_i}^{\\mathbf{(t-1)}} - {{m_i}^{\\mathbf{(t-1)}}}^2 \\right)\\right).\n\\end{align}\nThis issue was discussed in particular in \\cite{Kabashima2003,bolthausen2014iterative}.\nRemarkably, this last scheme, or algorithm, is actually the one obtained by the approximate message passing derivation that will be discussed in the upcoming \\citesec~\\ref{sec:chap3-bp-to-amp}.\n\n\\subparagraph{Solutions of the TAP equations}\nThe TAP equations can admit multiple solutions with either equal or different TAP free energy. \nWhile the true free energy $F$ corresponds to the minimum of the Gibbs free energy, reached for the Boltzmann distribution, the TAP derivation consists in performing an effectively unconstrained minimization in two steps, but with an approximation through a Taylor expansion in between. The truncation of the expansion therefore breaks the correspondence between the discovered minimizer and the unique Boltzmann distribution, hence the possible multiplicity of solutions. For the SK model for instance, the number of solutions of the TAP equations increases rapidly as $\\beta$ grows \\cite{Mezard1986}. While the different solutions can be accessed using different initializations of the iterative scheme, it is notably hard in phases where they are numerous to find exhaustively all the TAP solutions. In theory, they should be weighted according to their free energy density and averaged to recover the thermodynamics predicted by the replica computation \\cite{Dominicis1983}, another mean-field approximation discussed in \\citesec~\\ref{sec:chap3-replica}.\n\n\n\\subsubsection{Generalizing the Georges-Yedidia expansion}\n\\label{sec:chap3-GY-generalized}\nIn the derivation outlined above for binary variables, $x_i = 0$ or $1$, the mean of each variable $m_i$ was fixed. This is enough to parametrize the corresponding marginal distribution $q_{m_i}(x_i)$. Yet the expansion can actually be generalized to Potts variables (taking multiple discrete values) or even real valued variables by introducing appropriate parameters for the marginals. A general derivation fixing arbitrary real valued marginal distribution was proposed in \\citeapp~B of \\cite{Lesieur2017} for the problem of low rank matrix factorization. Alternatively, another level of approximation can be introduced for real valued variables by restricting the set of marginal distributions tested to a parametrized family of distributions. By choosing a Gaussian parametrization, one recovers TAP equations equivalent to the approximate message passing algorithm that will be discussed in the next \\citesec. In \\citeapp~\\ref{app:chap3-real-GY}, we present a derivation for real-valued Boltzmann machines with a Gaussian parametrization as proposed in \\cite{Tramel2018}.\n\n\\subsection{Belief propagation and approximate message passing}\n\\label{sec:chap3-bp-to-amp}\nAnother route to rediscover the TAP equations is through the approximation of message passing algorithms. Variations of the latter were discovered multiple times in different fields. In physics they were written in a restricted version  as soon as 1935 by Bethe \\cite{Bethe1935}. In statistics, they were developed by Pearl as methods for probabilistic inference \\cite{Pearl1988}. \nIn this section we will start by introducing a case-study of interest, the Generalized Linear Model. We will then proceed by steps to outline the derivation of the Approximate Message Passing (AMP) algorithm from the Belief Propagation (BP) equations.\n\n\n\n\\subsubsection{Generalized linear model}\n\\label{sec:chap3-glm}\n\\subparagraph{Definition} We introduce the \\emph{Generalized Linear Model} (GLM) which is a fairly simple model to illustrate message passing algorithms and which is also an elementary brick for a large range of interesting inference questions on neural networks. It falls under the teacher-student set up: a student model is used to reconstruct a signal from a teacher model producing indirect observations. \nIn the GLM, the product of an unknown signal $\\x_0 \\in \\R^N$ and a known weight matrix $\\W \\in \\R^{N \\times M}$ is observed as $\\y$ through a noisy channel $\\pouto$,\n\\begin{gather}\n    \\left\\{\n    \\begin{array}{l}\n    \\W \\sim p_W(\\W)\n    \\\\\n   \n    \\x_0 \\sim p_{x_0}(\\x_0) = \\prod\\limits_{i=1}^N p_{x_0}(x_{0,i})\n    \\end{array}\n    \\right. \n     \\quad \n   \n    \\Rightarrow\n     \\y \\sim \\pouto(\\y | \\W\\x_0) = \\prod_{\\mu=1}^M \\pouto(y_\\mu | \\vect{w}_\\mu\\T\\x_0). \n\\end{gather}\nThe probabilistic graphical model corresponding to this teacher is represented in \\citefig~\\ref{fig:chap3-glm}. The prior over the signal $p_{x_0}$ is supposed to be factorized, and the channel $\\pouto$ likewise. The inference problem is to produce an estimator $\\xh$ for the unknown signal $\\x_0$ from the observations $\\y$. Given the prior $p_x$ and the channel $\\pout$ of the student, not necessarily matching the teacher, the posterior distribution is \n\\begin{align}\n   \n    p(\\x|\\y, \\W) &= \\frac{1}{\\cZ(\\y, \\W)} \\, \\prod_{\\mu=1}^M \\pout(y_\\mu | \\sum_{i=1}^N W_{\\mu i} x_i)\\, \\prod_{i=1}^N p_x(x_i) \\, ,  \\label{eq:chap3-glm-meas}\\\\\n   \n    \\cZ(\\y, \\W) &= \\int \\dd{\\x} \\pout(\\y | \\x, \\W) p_x(\\x), \\label{eq:chap3-glm-Z}\n\\end{align}\nrepresented as a factor graph also in \\citefig~\\ref{fig:chap3-glm}. The difficulty of the reconstruction task  of $\\x_0$ from $\\y$ is controlled by the measurement ratio $\\alpha = M\/N$ and the amplitude of the noise possibly present in the channel. \n\n\n\\subparagraph{Applications}\nThe generic GLM underlies a number of applications. \nIn the context of neural networks of particular interest in this technical review, the channel $\\pout$ generating observations $\\y \\in \\R^M$ can equivalently be seen as a stochastic activation function $g(\\cdot; \\eps)$ incorporating a noise $\\eps \\in \\R^M$ component-wise to the output,\n\\begin{gather}\n    y_\\mu = g(\\vect{w}_\\mu\\T\\x \\,; \\, \\eps_\\mu).\n\\end{gather}\nThe inference of the teacher signal in a GLM has then two possible interpretations. \nOn the one hand, it can be interpreted as the reconstruction of the input $\\x$ of a stochastic single-layer neural network from its output $\\y$. For example, this inference problem can arise in the maximum likelihood training of a one-layer VAE (see corresponding paragraph in \\citesec~\\ref{sec:chap1-vae}).  On the other hand, the same question can also correspond to the Bayesian learning of a single-layer neural network with a single output - the perceptron - where this time $\\{\\W, \\y\\}$ are interpreted as the collection of training input-output pairs and $\\x_0$ plays the role of the unknown weight vector of the teacher (as cited as an example in \\citesec~\\ref{sec:chap2-teacher-student}). \nHowever, note that one of the most important applications of the GLM, Compressed Sensing (CS) \\cite{Donoho2006}, does not involve neural networks.\n\n\n\\subparagraph{Statistical physics treatment, random weights and scaling}\nFrom the statistical physics perspective, the effective energy functional is read from the posterior \\eqref{eq:chap3-glm-Z} seen as a Boltzmann distribution with energy\n\\begin{gather}\n    E(\\x) = - \\log \\pout(\\y | \\x, \\W) p_x(\\x) = - \\sum_{\\mu =1}^M\\log \\pout(y_\\mu | \\sum_{i=1}^N W_{\\mu i} x_i) - \\sum_{i=1}^\n    N \\log p_x(x_i) .\n\\end{gather} \nThe inverse temperature $\\beta$ has here no formal equivalent and can be thought as being equal to 1. The energy is a function of the random realizations of $\\W$ and $\\y$, playing the role of the disorder. Furthermore, the validity of the approximation presented below require additional assumptions.  Crucially, the weight matrix is assumed to have i.i.d. Gaussian entries with zero mean and variance $1\/N$, much like in the SK model. The prior of the signal is chosen so as to ensure that the $x_i$-s (and consequently the $y_\\mu$-s) remain of order 1.\nFinally, the thermodynamic limit $N \\to \\infty$ is taken for a fixed measurement ratio $\\alpha=M\/N$.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{chap3_glm.pdf}\n    \\caption{Graphical representations of the Generalized Linear Model. \\textbf{Left:} Probabilistic graphical model of the teacher. \\textbf{Middle left:} Factor graph representation of the posterior distribution on the signal $\\x$ under the student statistical model. \\textbf{Middle right and right:} Belief propagation updates \\eqref{eq:chap3-bp-glm-1} - \\eqref{eq:chap3-bp-glm-2} for approximate inference.   \\label{fig:chap3-glm}}\n\\end{figure}\n\n\n\n\n\\subsubsection{Belief Propagation} \n\\label{sec:chap3-bp}\nRecall that inference in high-dimensional problems consists in marginalizations over complex joint distributions, typically in the view of computing partition functions, averages or marginal probabilities for sampling. Belief Propagation (BP) is an inference algorithm, sometimes exact and sometimes approximate as we will see, leveraging the known factorization of a distribution, which encodes the precious information of (in)depencies between the random variables in the joint distribution. For a generic joint probability distribution $p$ over $\\x \\in \\R^N$ factorized as\n\\begin{gather}\n    \\label{eq:chap3-factorization}\n    p(\\x) = \\frac 1 \\cZ \\prod_{\\mu = 1}^M \\psi_\\mu(\\x_{\\partial \\mu}),\n\\end{gather}\n$\\psi_\\mu$ are called potential functions taking as arguments the variables $x_i$-s involved in the factor $\\mu$ \nshortened as $\\x_{\\partial \\mu}$.\n\n\n\\subparagraph{Definition of messages}\nLet us first write the BP equations and then explain the origin of these definitions. \nThe underlying factor graph of \\eqref{eq:chap3-factorization}\nhas $N$ nodes carrying variables $x_i$-s and $M$ factors associated with the potential functions $\\psi_\\mu$-s (see \\citeapp~\\ref{app:chap2-graphs} for a quick reminder). BP acts on \\emph{messages} variables which are tied to the edges of the factor graph.\nSpecifically, the sum-product version of the algorithm (as opposed to the max-sum, see e.g. \\cite{Mezard2009}) consists in the update equations\n\\begin{align}\n    \\label{eq:chap3-bp1}\n     \\msg{\\tilde{m}^{(t)}}{\\mu}{i}(x_i) & =  \\frac{1}{\\msg{\\cZ}{\\mu}{i}} \\int \\prod_{i'\\in \\partial \\mu \\setminus i} \\dd{x_{i'}} \\psi_\\mu(\\x_{\\partial \\mu})  \\prod_{i'\\in \\partial \\mu \\setminus i} \\msg{m^{(t)}}{i'}{\\mu}(x_{i'}), \\\\\n    \\label{eq:chap3-bp2}\n    \\msg{m^{(t+1)} }{i}{\\mu}(x_i)  & = \\frac{1}{\\msg{\\cZ}{i}{\\mu}} p_x(x_i)\\prod_{\\mu'\\in \\partial i \\setminus  \\mu} \\msg{\\tilde{m}^{(t)}}{\\mu'}{i}(x_i) \n\\end{align}\nwhere again the $i$-s index the variable nodes and the $\\mu$-s index the factor nodes. \nThe notation $\\partial \\mu \\setminus i$ designate the set of neighbor variables of the factor $\\mu$ except the variable $i$ (and reciprocally for $\\partial i \\setminus \\mu$).\nThe partition functions $\\msg{\\cZ}{i}{\\mu}$ and $\\msg{\\cZ}{\\mu}{i}$ are normalization factors ensuring that the messages can be interpreted as probabilities. \n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.75\\textwidth]{chap3_bp.pdf}\n    \\caption{Representations of the neighborhood of edge $i$-$\\mu$ in the factor graph and corresponding local BP updates. Factors are represented as squares and variable nodes as circles. \\textbf{Left:} In the factor graph where all factors around $x_i$ are removed (in gray) except for the factor $\\mu$, the marginal of $x_i$ (in red) is updated from the messages incoming at factor $\\mu$ (in blue) \\eqref{eq:chap3-bp1}. \\textbf{Right:} In the factor graph where factor $\\mu$ is deleted (in gray), the marginal of $x_i$ (in blue) is updated with the incoming messages (in red) from the rest of the factors \\eqref{eq:chap3-bp2}. \\label{fig:chap3-bp}}\n\\end{figure}\n\nFor acyclic (or tree-like) factor graphs, the BP updates are guaranteed to converge to a fixed point, that is a set of time independent messages $\\{\\msg{m}{i}{\\mu} , \\msg{\\tilde{m}}{\\mu}{i}\\}$ solution of the system of equations \\eqref{eq:chap3-bp1}-\\eqref{eq:chap3-bp2}. Starting at a leaf of the tree, these messages communicate beliefs of a given node variable taking a given value based on the nodes and factors already visited along the tree. More precisely, $\\msg{\\tilde{m}}{\\mu}{i}(x_i)$ is the marginal probability of $x_i$ in the factor graph before visiting the factors in $\\partial i$ except for $\\mu$, and $\\msg{m}{i}{\\mu}(x_i)$ is equal the marginal probability of $x_i$ in the factor graph before visiting the factor $\\mu$, see \\citefig~\\ref{fig:chap3-bp}. \n\nThus, at convergence of the iterations, the marginals can be computed as\n\\begin{gather}\n    \\label{eq:chap3-bp-marginal}\n    m_i(x_i) = \\frac{1}{\\cZ_i} p_x(x_i) \\prod_{\\mu \\in \\partial i} \\msgt{m}{\\mu}{i}(x_i),\n\\end{gather}\nwhich can be seen as the main output of the BP algorithm. \nThese marginals will only be exact on trees where incoming messages, computed from different part of the graph, are independent. Nonetheless, the algorithm \\eqref{eq:chap3-bp1}-\\eqref{eq:chap3-bp2}, occasionally then called \\emph{loopy-BP}, can sometimes be converged on graphs with cycles and in some cases will still provide high quality approximations. For instance, graphs with no short loops are locally tree like and BP is an efficient method of approximate inference, provided correlations decay with distance (i.e. incoming messages at each node are still effectively independent). \nBP will also appear principled for some infinite range mean-field models previously discussed; an example of which being our case-study the GLM discussed below. \nWhile this is the only example that will be discussed here in the interest of conciseness, getting fluent in BP generally requires more than one application. The interested reader could also consult \\cite{Yedidia2002} and \\cite{Mezard2009} \\citesec~14.1. for simple concrete examples.\n\n\\paragraph{The Bethe free energy}\nThe BP algorithm can also be recovered from a variational argument. Let's consider both the single variable marginals $m_i(x_i)$ and the marginals of the neighborhood of each factor $\\tilde{m}_\\mu(\\x_{\\partial \\mu}$). On tree graphs, the joint distribution \\eqref{eq:chap3-factorization} can be re-expressed as\n\\begin{gather}\n    p(\\x) = \\frac{\\prod_{\\mu=1}^M \\tilde{m}_\\mu(\\x_{\\partial \\mu})}{\\prod_{i=1}^Nm_i(x_i)^{n_i-1}},\n\\end{gather}\nwhere $n_i$ is the number of neighbor factors of the $i$-th variable. Abusively, we can use this form as an ansatz for loopy graph and plug it in the Gibbs free energy to derive an approximation of the free energy, similarly to the naive mean-field derivation of \\citesec~\\ref{sec:chap3-nmf}. This time the variational parameters will be the distributions $m_i$ and $\\tilde{m}_\\mu$ (see e.g. \\cite{Yedidia2002, Mezard2009} for additional details). The corresponding functional form of the Gibbs free energy is called the Bethe free energy:\n\\begin{gather}\n    F_{\\rm Bethe}(m_i, \\tilde{m}_\\mu) =  - \\int \\dd{\\x_{\\partial \\mu}} \\tilde{m}_\\mu(\\x_{\\partial \\mu}) \\ln \\psi_\\mu(\\x_{\\partial \\mu}) + (n_i -1) H(m_i) - H(\\tilde{m}_\\mu), \n\\end{gather}\nwhere $H(q)$ is the entropy of the distribution $q$. Optimization of the Bethe free energy with respect to its arguments under the constraint of consistency\n\\begin{gather}\n    \\int \\dd{\\x_{\\partial \\mu \\setminus i}} \\tilde{m}_\\mu(\\x_{\\partial \\mu }) = m_i(x_i)\n\\end{gather}\ninvolves Lagrange multipliers which can be shown to be related to the messages defined in \\eqref{eq:chap3-bp1}-\\eqref{eq:chap3-bp2}. Eventually, one can verify that marginals defined as \\eqref{eq:chap3-bp-marginal} and \n\\begin{gather}\n    \\tilde{m}_\\mu(\\x_{\\partial_\\mu}) = \\frac{1}{\\cZ_\\mu} \\psi(\\x_{\\partial \\mu}) \\prod_{i \\in \\partial \\mu}\\msg{m}{i}{\\mu}(x_i),\n\\end{gather}\nare stationary point of the Bethe free energy for messages that are BP solutions. In other words, the BP fixed points are consistent with the stationary point of the Bethe free energy. Using the normalizing constants of the messages, the Bethe free energy can also be re-written as\n\\begin{gather}\n    \\label{eq:chap3-bethe-fe}\n    F_{\\rm Bethe} = - \\sum_{i \\in V} \\log \\cZ_i  - \\sum_{\\mu \\in F} \\log \\cZ_\\mu + \\sum_{(i \\mu) \\in E} \\log \\cZ_{\\mu i} \\, , \n\\end{gather}\nwith\n\\begin{gather}\n    \\cZ_i = \\int \\dd{x_i} p_x(x_i) \\prod_{\\mu \\in \\partial i} \\msgt{m}{\\mu}{i}(x_i) \\, ,\\\\\n    \\cZ_\\mu = \\int \\prod_{i \\in \\partial \\mu} \\dd{x_i} \\psi(\\x_{\\partial \\mu}) \\prod_{i \\in \\partial \\mu}\\msg{m}{i}{\\mu}(x_i) \\, ,\\\\\n    \\cZ_{\\mu i } = \\int \\dd{x_i} \\msgt{m}{\\mu}{i}(x_i)  \\msg{m}{i}{\\mu}(x_i)\\, .\n \\end{gather}\n\n\nAs for the marginals, the Bethe free energy, will only be exact if the underlying factor graph is a tree. Otherwise it is an approximation of the free energy, that is not generally an upper bound.\n\n\n\\subparagraph{Belief propagation for the GLM} The writing of the BP-equations for our case-study is schematized on the right of \\citefig~\\ref{fig:chap3-glm}. There are $2 \\times N \\times M$ updates:\n\\begin{align}\n    \\label{eq:chap3-bp-glm-1}\n     \\msg{\\tilde{m}^{(t)}}{\\mu}{i}(x_i) \n     &  = \\frac{1}{\\msg{\\cZ}{\\mu}{i}} \\int \\prod_{i'\\neq i} \\dd{x_{i'}} \\pout(y_\\mu | {w_\\mu}\\T\\x)  \\prod_{i'\\neq i} \\msg{m^{(t)}}{i'}{\\mu}(x_{i'}),\\\\\n     \\label{eq:chap3-bp-glm-2}\n     \\msg{m^{(t+1)}}{i}{\\mu}(x_i)  \n    & = \\frac{1}{\\msg{\\cZ}{i}{\\mu}} p_x(x_i)\\prod_{\\mu'\\neq \\mu} \\msg{\\tilde{m}^{(t)}}{\\mu'}{i}(x_i),\n\\end{align}\nfor all $i-\\mu$ pairs. \nDespite a relatively concise formulation, running BP in practice turns out intractable since for a signal $\\x$ taking continuous values it would entail keeping track of distributions on continuous variables. In this case, BP is approximated by the (G)AMP algorithm presented in the next section.\n\n\n\n\n\\subsubsection{(Generalized) approximate message passing}\n\\label{sec:chap3-gamp}\nThe name of approximate message passing (AMP) was fixed by Donoho, Maleki and Montanari \\cite{Donoho2009} who derived the algorithm in the context of Compressed Sensing. Several works from statistical physics had nevertheless already proposed related algorithmic procedures and made connections with the TAP equations for different systems \\cite{Kabashima1998, opper2001advanced, Kabashima2003}. The algorithm was derived systematically for any channel of the GLM by Rangan \\cite{Rangan2011} and became Generalized-AMP (GAMP), yet again it seems that \\cite{Kabashima2004} proposed the first generalized derivation.\n\nThe systematic procedure to write AMP for a given joint probability distribution consists in first writing BP on the factor graph, second project the messages on a parametrized family of functions to obtain the corresponding \\emph{relaxed-BP} and third close the equations on a reduced set of parameters by keeping only leading terms in the thermodynamic limit. We will quickly review and justify these steps for the GLM. \nNote that here a relevant algorithm for approximate inference will be derived from message passing on a fully connected graph of interactions. As it tuns out, the high connectivity limit and the introduction of short loops does not break the assumption of independence of incoming messages in this specific case thanks to the small scale $O(1\/\\sqrt{N})$ and the independence of the weight entries. The statistics of the weights are here crucial.\n\n\\paragraph{Relaxed Belief Propagation}\nIn the thermodynamic limit $M, N \\to + \\infty$, one can show that the scaling $1\/\\sqrt{N}$ of the $W_{ij}$ and the extensive connectivity of the underlying factor graph imply that messages are approximately Gaussian.\nWithout giving all the details of the computation which can be cumbersome, let us try to provide some intuitions. We drop the time indices for simplicity and start with \\eqref{eq:chap3-bp-glm-1}. Consider the intermediate reconstruction variable $z_\\mu = \\vect{w}_\\mu\\T\\x = \\sum_{i'\\neq i}W_{\\mu i'}x_{i'} + W_{\\mu i}x_i$. Under the statistics of the messages $\\msg{m}{i'}{\\mu}(x_{i'})$, the $x_{i'}$ are independent such that by the central limit theorem  $z_\\mu - W_{\\mu i}x_i$ is a Gaussian random variable with respectively mean and variance\n\\begin{gather}\n    \\label{eq:chap3-rbp-om}\n    \\msg{\\omega}{\\mu}{i} = \\sum_{i'\\neq i}W_{\\mu i'}\\msg{\\hat{x}}{i'}{\\mu},\\\\\n    \\label{eq:chap3-rbp-V}\n    \\msg{V}{\\mu}{i} = \\sum_{i'\\neq i}W^2_{\\mu i'}\\msg{C^x}{i'}{\\mu},\n\\end{gather}\nwhere we defined the mean and the variance of the messages $\\msg{m}{i'}{\\mu}(x_{i'})$,\n\\begin{gather}\n    \\label{eq:chap3-rbp-xhat}\n    \\msg{\\hat{x}}{i'}{\\mu} = \\int \\dd{x_{i'}} x_{i'} \\, \\msg{m}{i'}{\\mu}(x_{i'}), \\\\\n    \\label{eq:chap3-rbp-cx} \n    \\msg{C^x}{i'}{\\mu} = \\int \\dd{x_{i'}} x^2_{i'} \\, \\msg{m}{i'}{\\mu}(x_{i'}) - \\msg{\\hat{x}}{i'}{\\mu}^2.\n\\end{gather}\nUsing these new definitions, \\eqref{eq:chap3-bp-glm-1} can be rewritten as \n\\begin{gather}\n    \\label{eq:chap3-rbp-01}\n    \\msg{\\tilde{m}}{\\mu}{i}(x_i) \\propto \\int \\dd{z_\\mu} \\pout(y_\\mu | z_\\mu)e^{-\\frac{(z_\\mu - W_{\\mu i} x_i - \\msg{\\omega}{\\mu}{i})^2}{2 \\msg{V}{\\mu}{i}}} ,\n\\end{gather}\nwhere the notation $\\propto$ omits the normalization factor for distributions. Considering that $W_{\\mu i}$ is of order $1\/\\sqrt{N}$, the development of \\eqref{eq:chap3-rbp-01} shows that at leading order $\\msg{\\tilde{m}}{\\mu}{i}(x_i)$ is Gaussian:\n\\begin{gather}\n    \\label{eq:chap3-rbp-02}\n    \\msg{\\tilde{m}}{\\mu}{i}(x_i) \\propto e^{\\msg{B}{\\mu}{i}x_i + \\frac 1 2 \\msg{A}{\\mu}{i}x_i^2 }\n\\end{gather}\nwhere the details of the computations yield\n\\begin{gather}\n    \\label{eq:chap3-rbp-B}\n    \\msg{B}{\\mu}{i} = W_{\\mu i} \\, \\gouts(y_\\mu, \\msg{\\omega}{\\mu}{i}, \\msg{V}{\\mu}{i}) \\\\\n    \\label{eq:chap3-rbp-A}\n    \\msg{A}{\\mu}{i} = - W_{\\mu i}^2 \\, \\dgouts(y_\\mu, \\msg{\\omega}{\\mu}{i}, \\msg{V}{\\mu}{i})\n\\end{gather}\nusing the \\emph{output update functions}\n\\begin{gather}\n\\label{eq:chap3-gout}\n\\gouts(y, \\omega, V) = \\frac{1}{\\Zout}\\int \\dd{z} \\frac{(z - \\omega)}{V} \\pout(y|z)  \\cN(z; \\omega, V), \\\\\n\\label{eq:chap3-dgout}\n\\dgouts(y, \\omega, V) = \\frac{1}{\\Zout}\\int \\dd{z} \\frac{(z - \\omega)^2}{V^2} \\pout(y|z)  \\cN(z; \\omega, V) - \\frac 1 V - \\gouts(y, \\omega, V)^2,\\\\ \n\\label{eq:chap3-Zout}\n\\Zout( y, \\omega, V) = \\int \\dd{z}  \\pout(y|z)  \\cN(z; \\omega, V).\n\\end{gather}\nThese arguably complicated functions, again coming out of the development of \\eqref{eq:chap3-rbp-01}, can be interpreted as the estimation of the mean and the variance of the gap between two different estimate of $z_\\mu$ considered by the algorithm: the mean estimate $\\msg{\\omega}{\\mu}{i}$ given incoming messages $\\msg{m}{i'}{\\mu}(x_{i'})$ and the same mean estimate updated to incorporate the information coming from the channel $\\pout$ and observation $y_\\mu$. \nFinally, the Gaussian parametrization \\eqref{eq:chap3-rbp-02} of $\\msg{\\tilde{m}}{\\mu}{i}(x_i)$ serves to rewrite the other type of messages $\\msg{m}{i}{\\mu}(x_i)$ \\eqref{eq:chap3-bp-glm-2},\n\\begin{gather}\n    \\label{eq:chap3-rbp-2}\n    \\msg{m}{i}{\\mu}(x_i) \\propto p_x(x_i)e^{-\\frac{(\\msg{\\lambda}{i}{\\mu}- x_i)^2}{2 \\msg{\\sigma}{i}{\\mu}}}, \n\\end{gather}\nwith\n\\begin{gather}\n    \\label{eq:chap3-rbp-sig}\n    \\msg{\\sigma}{i}{\\mu} = \\left(\\sum_{\\mu' \\neq \\mu}\\msg{A}{\\mu'}{i} \\right)^{-1} \\\\ \n    \\label{eq:chap3-rbp-lbd}\n    \\msg{\\lambda}{i}{\\mu} = \\msg{\\sigma}{i}{\\mu} \\left(\\sum_{\\mu' \\neq \\mu} \\msg{B}{\\mu'}{i}\\right).\n\\end{gather}\nThe set of equations can finally be closed by recalling the definitions \\eqref{eq:chap3-rbp-xhat}-\\eqref{eq:chap3-rbp-cx}:\n\\begin{gather}\n    \\label{eq:chap3-rbp-xhat-f1}\n    \\msg{\\hat{x}}{i}{\\mu} = f^x_1(\\msg{\\lambda}{i}{\\mu}, \\msg{\\sigma}{i}{\\mu})\\\\\n    \\label{eq:chap3-rbp-xhat-f2}\n    \\msg{C^x}{i}{\\mu} = f^x_2(\\msg{\\lambda}{i}{\\mu}, \\msg{\\sigma}{i}{\\mu})\n\\end{gather}\nwith now the \\emph{input update functions}\n\\begin{gather}\n    \\label{eq:chap3-Zx}\n    \\cZ^x = \\int \\dd{x} p_x(x)e^{-\\frac{(x-\\lambda)^2}{2\\sigma}}, \\\\\n    \\label{eq:chap3-f1x}\n    f^x_1(\\lambda, \\sigma) = \\frac{1}{\\cZ^x}\\int \\dd{x} x \\, p_x(x)e^{-\\frac{(x-\\lambda)^2}{2\\sigma}}, \\\\\n    \\label{eq:chap3-f2x}\n    f^x_2(\\lambda, \\sigma) =  \\frac{1}{\\cZ^x} \\int \\dd{x} x^2 \\, p_x(x)e^{-\\frac{(x-\\lambda)^2}{2\\sigma}} - f^x_1(\\lambda, \\sigma)^2.\n\\end{gather}\nThe input update functions can be interpreted as updating the estimation of the mean and variance of the signal $x_i$ based on the information coming from the incoming messages grasped by $\\msg{\\lambda}{i}{\\mu}$ and $\\msg{\\sigma}{i}{\\mu}$ with the information of the prior $p_x$.\n\nTo sum-up, by considering the leading order terms in the thermodynamic limit, the BP equations can be self-consistently re-written as a closed set of equations over mean and variance variables \\eqref{eq:chap3-rbp-om}-\\eqref{eq:chap3-rbp-V}-\\eqref{eq:chap3-rbp-B}-\\eqref{eq:chap3-rbp-A}-\\eqref{eq:chap3-rbp-sig}-\\eqref{eq:chap3-rbp-lbd}-\\eqref{eq:chap3-rbp-xhat-f1}-\\eqref{eq:chap3-rbp-xhat-f2}. Eventually, r-BP can equivalently be thought of as the projection of BP onto the following parametrizations of the messages\n\\begin{gather}\n    \\label{eq:chap3-rbp-1}\n    \\msg{\\tilde{m}^{(t)}}{\\mu}{i}(x_i) \\propto e^{\\msg{B^{(t)}}{\\mu}{i}x_i + \\frac 1 2 \\msg{A^{(t)}}{\\mu}{i}x_i^2} \\propto \\int \\dd{z_\\mu} \\pout(y_\\mu | z_\\mu)e^{-\\frac{(z_\\mu - W_{\\mu i} x_i - \\msg{\\omega^{(t)}}{\\mu}{i})^2}{2 \\msg{V^{(t)}}{\\mu}{i}}} ,\\\\\n    \\label{eq:chap3-rbp-2}\n    \\msg{m^{(t+1)}}{i}{\\mu}(x_i) \\propto e^{-\\frac{(\\msg{\\hat{x}^{(t+1)}}{i}{\\mu}- x_i)^2}{2 \\msg{{C^x}^{(t+1)}}{i}{\\mu}}} \\propto p_x(x_i)e^{-\\frac{(\\msg{\\lambda^{(t)}}{i}{\\mu}- x_i)^2}{2 \\msg{\\sigma^{(t)}}{i}{\\mu}}}. \n\\end{gather}\nNote that, at convergence, an approximation of the marginals is recovered from the projection on the parametrization \\eqref{eq:chap3-rbp-2} of \\eqref{eq:chap3-bp-marginal},\n\\begin{gather}\n    m_i(x_i) = \\frac{1}{\\cZ_i}p_x(x_i)e^{-(\\lambda_i-x_i)^2\/2\\sigma_i} = f^x_1(\\lambda_i, \\sigma_i), \\\\\n    \\msg{\\sigma}{i}{\\mu} = \\left(\\sum_{\\mu}\\msg{A}{\\mu}{i} \\right)^{-1}, \\\\ \n    \\msg{\\lambda}{i}{\\mu} = \\msg{\\sigma}{i}{\\mu} \\left(\\sum_{\\mu} \\msg{B}{\\mu}{i}\\right).\n\\end{gather} \n\nNonetheless, r-BP is scarcely used as such as the computational cost can be readily reduced with little more approximation. Because the parameters in \\eqref{eq:chap3-rbp-1}-\\eqref{eq:chap3-rbp-2} take the form of messages on the edges of the factor graph there are still $O(M \\times N)$ quantities to track to solve the self-consistency equations by iterations. Yet, in the thermodynamic limit, the messages are closely related to the marginals as the contribution of \nthe missing message between \\eqref{eq:chap3-bp2} and \\eqref{eq:chap3-bp-marginal} is to a certain extent negligible. Careful book keeping of the order of contributions of these small differences leads to a set of closed equations on parameters of the marginals, i.e. $O(N)$ variables, corresponding to the GAMP algorithm.\n\nA detailed derivation and developed algorithm of r-BP for the GLM can be found for example in \\cite{Zdeborova2016} (\\citesec~6.3.1). In \\citesec~\\ref{sec:chap3-multivalue} of the present paper, we also present the derivation in a slightly more general setting where the variables $x_i$ and $y_\\mu$ are possibly vectors instead of scalars.\n\n\\paragraph{Generalized approximate message passing}\n\nThe GAMP algorithm with respect to marginal parameters, analogous to the messages parameters introduced above (summarized in  \\eqref{eq:chap3-rbp-1}-\\eqref{eq:chap3-rbp-2}), is given in \\citealg~\\ref{alg:chap3-amp}. \nThe origin of GAMP is again the development of the r-BP message-like equations around marginal quantities. The details of this derivation for the GLM can be found for instance in \\cite{Zdeborova2016} (\\citesec~6.3.1).\nFor a random initialization, the algorithm can be decomposed in 4 steps per iteration which refine the estimate of the signal $\\x$ and the intermediary variable $\\z$ by incorporating the different sources of information.\nSteps 2) and 4) involve the \\emph{update functions} relative to the prior and output channel defined above. \nSteps 1) and 3) are general for any GLM with a random Gaussian weight matrix, as they result from the consistency of the two alternative parametrizations introduced for the same messages in \\eqref{eq:chap3-rbp-1}-\\eqref{eq:chap3-rbp-2}\n\n\n\\input{chap3_amp}\n\n\n\\subparagraph{Relation to TAP equations}\nHistorically the main difference between the AMP algorithm and the TAP equations is that the latter was first derived for binary variables with $2$-body interactions (SK model) while the former was proposed for continuous random variables with $N$-body interactions (Compressed Sensing). The details of the derivation (described in \\cite{Zdeborova2016} or in a more general case in \\citesec~\\ref{sec:chap3-multivalue}), rely on the knowledge of the statistics of the disordered variable $\\W$ but do not require a disorder average, as in the Georges-Yedidia expansion yielding the TAP equations.\nBy\nfocusing on\nthe GLM with a random Gaussian weight matrix scaling as $O(1\/\\sqrt{N})$ (similarly to the couplings of the SK model) we naturally obtained TAP equations at second order, with an Onsager term in the update \\eqref{alg:chap3-amp-om} of $\\omega_\\mu$. \nYet an advantage of the AMP derivation from BP over the high-temperature expansion is that it explicitly provides `correct' time indices in the iteration scheme to solve the self consistent equations \\cite{bolthausen2014iterative}. \n\n\\subparagraph{Reconstruction with AMP}\nAMP is therefore a practical reconstruction algorithm which can be run on a single instance (the disorder is not averaged) to estimate an unknown signal $\\x_0$. Note that the prior $p_x$ and channel $\\pout$ used in the algorithm correspond to the student statistical model and they may be different from the true underlying teacher model that generates $\\x_0$ and $\\y$. In other words, the AMP algorithm may be used either in the Bayes optimal or in the mismatched setting defined in \\citesec~\\ref{sec:chap2-teacher-student}.\nRemarkably, it is also possible to consider a disorder average in the thermodynamic limit to study the average case computational hardness, here of the GLM inference problem, in either of these matched or mismatched configurations.\n\n\\paragraph{State Evolution}\nThe statistical analysis of the AMP equations for Compressed Sensing in the average case and in the thermodynamic limit $N\\to \\infty$ lead to another closed set of equations that was called State Evolution (SE) in \\cite{Donoho2009}. Such an analysis can be generalized to other problems of application of approximate message passing algorithms. The derivation of SE starts from the r-BP equations and relies on the assumption of independent incoming messages to invoke the Central Limit Theorem. It is therefore only necessary to follow the evolution of a set of means and variances parametrizing Gaussian distributions. When the different variables and factors are statistically equivalent, as it is the case of the GLM, SE reduces to a few scalar equations. The interested reader should refer to \\citeapp~\\ref{app:chap6-vect-amp} for a detailed derivation in a more general setting.\n\n\\subparagraph{Mismatched setting} In the general mismatched setting we need to carefully differentiate the teacher and the student. We note $p_{x_0}$ the prior used by the teacher. We also rewrite its channel $\\pouto(y|\\vect{w}\\T\\x)$ as the explicit function $y = g_0(\\vect{w}\\T\\x; \\epsilon)$ assuming the noise $\\epsilon$ to be distributed according to the standard normal distribution.\nThe tracked quantities are the \\emph{overlaps},\n\\begin{gather}\n    q =\\lim_{N\\to \\infty}\\frac{1}{N} \\sum_{i=1}^N \\hat{x}_i^2\\,, \\quad m = \\lim_{N\\to \\infty}\\frac{1}{N} \\sum_{i=1}^N \\hat{x}_i x_{0,i} \\,, \\quad q_0 = \\lim_{N\\to \\infty}\\frac{1}{N} \\sum_{i=1}^N x_{0,i}^2 = \\E_{p_{x_0}}[x_0^2] ,\n\\end{gather}\nalong with the auxiliary $V$, $\\hat{q}$, $\\hat{m}$ and $\\hat{\\chi}$:\n\\begin{align}\n\\label{eq:chap3-se-nonishi-out-q}\n\\hat{q}^{(t)} & = \\int \\D{\\epsilon} \\int \\dd{\\omega} \\dd{z} \\cN(z, \\omega ; 0, \\mat{Q}^{(t)}) \n\t\\gouts(\\omega, g_0\\left( z ; \\epsilon\\right) , V^{(t)})^2 \\, ,\\\\\n\t\\label{eq:chap3-se-nonishi-out-m}\n\\hat{m}^{(t)} & = \\int \\D{\\epsilon} \\int \\dd{\\omega} \\dd{z} \\cN(z, \\omega ; 0, \\mat{Q}^{(t)})\n\t\t\\partial_{z} \\gouts(\\omega, g_0\\left( z ; \\epsilon\\right) , V^{(t)}) \\, ,\\\\\n\\label{eq:chap3-se-nonishi-out-xi}\n\\hat{\\chi}^{(t)} & = - \\int \\D{\\epsilon} \\int \\dd{\\omega}\\dd{z} \\cN(z, \\omega ; 0, \\mat{Q}^{(t)}) \n\t \\partial_{\\omega} \\gouts(\\omega, g_0\\left( z ; \\epsilon_\\mu\\right) , V^{(t)}) \\, ,\n\\end{align}\n\\begin{align}\n\\label{eq:chap3-se-nonishi-in-q}\nq^{(t+1)} & = \\int \\dd{x_0} p_{x_0}(x_0) \\int \\D{\\xi} \n\tf^x_1 \\left( (\\alpha \\hat{\\chi}^{(t)})^{-1}\\left({\\sqrt{\\alpha \\hat{q}^{(t)}} \\xi + \\alpha \\hat{m}^{(t)}\\x_0}\\right); (\\alpha \\hat{\\chi}^{(t)})^{-1}  \\right) ^2  \\, ,\\\\\n\\label{eq:chap3-se-nonishi-in-m}\nm^{(t+1)} & = \\int \\dd{x_0} p_{x_0}(x_0) \\int \\D{\\xi} x_0 \n\tf^x_1 \\left( (\\alpha \\hat{\\chi}^{(t)})^{-1}\\left({\\sqrt{\\alpha \\hat{q}^{(t)}} \\xi + \\alpha \\hat{m}^{(t)}\\x_0}\\right); (\\alpha \\hat{\\chi}^{(t)})^{-1} \\right) \\, ,\\\\\n\\label{eq:chap3-se-nonishi-in-V}\nV^{(t+1)} & = \\int \\dd{x_0} p_{x_0}(x_0) \\int \\D{\\xi} \nf^x_2 \\left( (\\alpha \\hat{\\chi}^{(t)})^{-1}\\left({\\sqrt{\\alpha \\hat{q}^{(t)}} \\xi + \\alpha \\hat{m}^{(t)}x_0}\\right); (\\alpha \\hat{\\chi}^{(t)})^{-1} \\right) \\, ,\n\\end{align}\nwhere we use the notation $\\cN(\\cdot;\\cdot,\\cdot)$ for the normal distribution, $\\D{\\xi}$ for the standard normal measure and the covariance matrix $\\mat{Q}^{(t)}$ is given at each time step by\n\\[\n\\mat{Q}^{(t)} = \n\\begin{bmatrix}\nq_0 & m^{(t)} \\\\\n\\\\\n{m^{(t)}} & q^{(t)} \\\\\n\\end{bmatrix}.\n\\]\n\nDue to the self-averaging property, the performance of the reconstruction by the AMP algorithm on an instance of size $N$ can be tracked along the iterations given \n\\begin{gather}\n    \\label{eq:chap3-se-mse}\n    \\MSE(\\hat{x})= \\frac{1}{N}\\sum_{i=1}^N (\\hat{x}_i - x_{0,i})^2 =  q - 2 m + q_0,\n\\end{gather}\nwith only minor differences coming from finite-size effects.\nState Evolution also provides an efficient procedure to study from the theoretical perspective the AMP fixed points for a generic model, such as the GLM, as a function of some control parameters. It reports the average results for running the complete AMP algorithm on $O(N)$ variables\nusing a few scalar equations. Furthermore, the State Evolution equations simplify further in the Bayes optimal setting.\n\n\\subparagraph{Bayes optimal setting}\nWhen the prior and channel are identical for the student and the teacher, the true unknown signal $\\x_0$ is in some sense statistically equivalent to the estimate $\\xh$ coming from the posterior. More precisely one can prove the Nishimori identities \\cite{Opper1991, Iba1999, Nishimori2001} (or \\cite{Kabashima2016} for a concise demonstration and discussion) implying that $q  = m$, $ V  = q_0 - m$ and $\\hat{q} = \\hat{m} = \\hat{\\chi}$. Only two equations are then necessary to track the performance of the reconstruction:\n\\begin{align}\n    \\label{eq:chap3-se-bo-qh}\n    \\hat{q}^{(t)} & = \\int \\dd{\\epsilon} p_{\\epsilon_0}(\\epsilon) \\int \\dd{\\omega} \\dd{z} \\cN(z, \\omega ; 0, \\mat{Q}^{(t)}) \n    \\gouts(\\omega, g_0\\left( z ; \\epsilon\\right) , V^{(t)})^2 \\\\\n    \\label{eq:chap3-se-bo-q}\n    q^{(t+1)} & = \\int \\dd{x_0} p_{x_0}(x_0) \\int \\D{\\xi} \n\tf^x_1 \\left( (\\alpha \\hat{\\chi}^{(t)})^{-1}\\left({\\sqrt{\\alpha \\hat{q}^{(t)}} \\xi + \\alpha \\hat{m}^{(t)}\\x_0}\\right); (\\alpha \\hat{\\chi}^{(t)})^{-1}  \\right) ^2 \\, .\n\\end{align}\n\n\n\\subsection{Replica method}\n\\label{sec:chap3-replica}\nAnother powerful technique from the statistical physics of disordered systems to examine models with infinite range interactions is the replica method. It enables an analytical computation of the quenched free energy via non-rigorous mathematical manipulations. More developed introductions to the method can be found in \\cite{Mezard1986, Nishimori2001, Castellani2005}. \n\n\\subsubsection{Steps of a replica computation}\nThe basic idea of the replica computation is to compute the average over the disorder of $\\log \\cZ$ by considering the identity $\\log \\cZ = \\lim_{n\\to 0} (\\cZ^n - 1)\/n$. First the expectation of $\\cZ^n$ is evaluated for $n\\in\\mathbb{N}$, then the $n\\to 0$ limit is taken by `analytic continuation'. Thus the method takes advantage of the fact that the average of a power of $\\cZ$ is sometimes easier to compute than the average of a logarithm. We illustrate the key steps of the calculation for the partition function of the GLM \\eqref{eq:chap3-glm-Z}.\n\n\\subparagraph{Disorder average for the replicated system: coupling of the replicas}\nThe average of $\\cZ^n$ for $n\\in \\mathbb{N}$ can be seen as the partition function of a system with $n + 1$ non interacting replicas of $\\x$ indexed by $a \\in \\{0, \\cdots, n\\}$, where the first replica $a=0$ is representative of the teacher and the $n$ other replicas are identically distributed as the student:\n\\begin{align}\n    \\E_{\\W, \\y, \\x_0}\\left[ \\cZ^n \\right] &  = \\E_{\\W}\\left[\n        \\int \\dd{\\y} \\dd{\\x_0} \n        \\pouto(\\y|\\W\\x_0)\n        p_{\\x_0}(\\x_0)\n        \\left( \\int \\dd{\\x} \\pout(\\y|\\W\\x)p_x(\\x)\n        \\right)^n\n    \\right] \\\\\n    & = \\E_{\\W}\\left[ \n        \\int \\dd{\\y} \\prod_{a=0}^{n} \\left( \\dd{\\x_a} \\pouta(\\y|\\W\\x_a)p_{x_a}(\\x_a) \\right)\n        \\right] \\\\\n    & = \\E_{\\W}\\left[ \\int \\dd{\\y} \\prod_{a=0}^{n} \\left( \\dd{\\x_a} \\dd{\\z_a} \\delta(\\z_a - \\W\\x_a)\\pouta(\\y|\\z_a)p_{x_a}(\\x_a) \\right) \n    \\right] \\;.\n\\end{align}\nTo perform the average over the disordered interactions $\\W$ we consider the statistics of $\\z_a = \\W\\x_a$. Recall that $W_{\\mu i} \\sim \\cN(W_{\\mu i} ;0,1\/N)$, independently for all $\\mu$ and $i$. Consequently, \nthe $\\z_a$ are jointly Gaussian in the thermodynamic limit with means and covariances\n\\begin{gather}\n    \\E_{\\W}[z_{a,\\mu}] = \\E_{\\W}\\left[\\sum_{i=1}^N W_{\\mu i}x_{a,i}\\right]  =  0\\, , \\quad E_{\\W}\\left[ z_{a,\\mu} z_{b, \\nu}\\right]  = \\sum_{i=1}^N x_{a, i}x_{b, i} \/ N = q_{ab}.\n\\end{gather}\nThe overlaps, that we already introduced in the SE formalism, naturally re-appear. We introduce the notation $\\q$ for the $(n+1) \\times (n+1)$ overlap matrix. Integrating out the disorder $\\W$ shared by the $n+1$ replicas will therefore leave us with an effective system of now coupled replicas:\n\\begin{align}\n    \\E_{\\W, \\y, \\x_0} \\left[ \\cZ^n \\right] = \n    \\int \\prod_{a,b} \\dd{N q_{ab}}  & \\int \\dd{\\y} \\prod_{a=0}^{n} \\dd{\\z_a} \\pouta(\\y|\\z_a) \\\\\n     \\exp&\\left(-\\frac{1}{2} \\displaystyle \\sum_{\\mu=1}^M \\sum_{a,b}z_{a,\\mu}z_{b,\\mu} (\\q^{-1})_{ab} - M C(\\q,n)\\right)\\notag \\\\\n    & \\int \\prod_{a=1}^n\\dd{\\x_a} p_{x_a}(\\x_a) \\delta(N q_{ab} - \\sum_{i=1}^N x_{a,i}x_{b,i}). \\notag\n\\end{align}\n\n\\subparagraph{Change of variable for the overlaps: decoupling of the variables}\nWe consider the Fourier representation of the Dirac distribution fixing the consistency between overlaps and replicas,\n\\begin{align}\n    \\delta(N q_{ab} - \\sum_{i=1}^N x_{a,i}x_{b,i}) =\n   \\int \\frac{{\\mathrm{d}\\hat{q}_{ab}}}{2 i \\pi } \\, e^{\\hat{q}_{ab}(N q_{ab} - \\sum_{i=1}^N x_{a,i}x_{b,i})},\n\\end{align}\nwhere $\\hat{q}_{ab}$ is purely imaginary, which yields\n\\begin{align}\n    \\E_{\\W, \\y, \\x_0} \\left[ \\cZ^n \\right] = & \\int  \\prod_{a,b} \\dd{Nq_{ab}}  \\int \\prod_{a,b} \\frac{{\\mathrm{d}\\hat{q}_{ab}}}{2 i \\pi } \\, \\exp\\left(N \\hat{q}_{ab}q_{ab}\\right) \\\\\n    & \\int \\dd{\\y} \\prod_{a=0}^{n} \\dd{\\z_a} \\pouta(\\y|\\z_a) \\exp\\left(-\\frac{1}{2} \\displaystyle \\sum_{\\mu=1}^M \\sum_{a,b}z_{a,\\mu}z_{b,\\mu} (\\q^{-1})_{ab} - M C(\\q,n)\\right)\\notag \\\\\n    & \\int \\prod_{a=1}^n\\dd{\\x_a} p_{x_a}(\\x_a) \\exp\\left(- \\hat{q}_{ab} \\displaystyle \\sum_{i=1}^N x_{a,i}x_{b,i}\\right) \\notag\n\\end{align}\nwhere $C(\\q,n)$ is related to the normalization of the Gaussian distributions over the $\\z_a$ variables, and the integrals can be factorized over the $i$-s and $\\mu$-s. Thus we obtain\n\\begin{gather}\n    \\label{eq:chap3-replica-Nscaling}\n    \\E_{\\W, \\y, \\x_0}\\left[ \\cZ^n \\right] = \\int \\prod_{a,b} \\dd{N q_{ab}}  \\int \\prod_{a,b} \\dd{ \\hat{q}_{ab}} e^{N \\hat{q}_{ab}q_{ab}} e^{M \\log \\hat{\\mathcal{I}}_z(\\q)} e^{N \\log \\hat{\\mathcal{I}}_x(\\hat{\\q})} \\, , \n\\end{gather}\nwith\n\\begin{gather}\n    \\hat{\\mathcal{I}}_z(\\q) = \\int \\dd{y} \\prod_{a=0}^{n} \\dd{z_a} \\pouta(y|z_a) \\exp\\left(-\\frac{1}{2} \\displaystyle \\sum_{a,b}z_{a}z_{b} (q_{ab})^{-1} - C(\\q,n)\\right) \\, ,\\\\\n    \\hat{\\mathcal{I}}_x(\\hat{\\q}) = \\int \\prod_{a=1}^n\\dd{x_a} p_{x_a}(x_a) \\exp\\left(- \\hat{q}_{ab} \\displaystyle x_{a}x_{b}\\right) \\, ,\n\\end{gather}\nwhere we introduce the notation $\\hat{\\q}$ for the auxiliary overlap matrix with entries $(\\hat{\\q})_{ab} = \\hat{q}_{ab}$ and we omitted the factor $2i\\pi$ which is eventually subleading as $N\\to + \\infty$.\nThe decoupling of the $x_i$ and the $z_\\mu$ of the infinite range system yields pre-factors $N$ and $M$ in the exponential arguments. In the thermodynamic limit, we recall that both $N$ and $M$ tend to $+\\infty$ while the ratio $\\alpha=M\/N$ remains fixed. Hence, the integral for the replicated average is easily computed in this limit by the saddle point method: \n\\begin{gather}\n   \\log \\E_{\\W, \\y, \\x_0}\\left[ \\cZ^n \\right] \\simeq N \\mathrm{extr}_{\\q \\hat{\\q}}\\left[\\mathcal{\\phi}(\\q, \\hat{\\q})\\right]  \\, , \\quad  \\mathcal{\\phi}(\\q, \\hat{\\q}) = \\sum_{a,b} \\hat{q}_{ab}q_{ab} + \\alpha \\hat{\\mathcal{I}}_z(\\q) + \\hat{\\mathcal{I}}_x(\\hat{\\q}),\n\\end{gather}\nwhere we defined the replica potential $\\mathcal{\\phi}$.\n\n\\subparagraph{Exchange of limits: back to the quenched average}\nThe thermodynamic average of the log-partition is recovered through an a priori risky mathematical manipulation: (i) perform an analytical continuation from $n \\in \\mathbb{N}$ to $n \\to 0$ \n\\begin{gather}\n   \n    \\frac{1}{N}\\E_{\\W, \\y, \\x_0}\\left[ \\log \\cZ \\right] \n    = \n   \n    \\lim_{n\\to 0} \\frac{1}{nN} \\E_{\\W, \\y, \\x_0}\\left[ \\cZ^n -1\\right] \n    = \n   \n    \\lim_{n\\to 0} \\frac{1}{nN} \\log \\E_{\\W, \\y, \\x_0}\\left[ \\cZ^n \\right] \n\\end{gather}\nand (ii) exchange limits \n\\begin{gather}\n    -f   \n    = \\lim_{N\\to \\infty} \\lim_{n\\to 0}\\frac{1}{n}  \\frac{1}{N} \\log \\E_{\\W, \\y, \\x_0}\\left[ \\cZ^n \\right] = \\lim_{n\\to 0} \\frac{1}{n} \\mathrm{extr}_{\\q \\hat{\\q}}\\left[\\mathcal{\\phi}(\\q, \\hat{\\q})\\right].\n\\end{gather}\nDespite the apparent lack of rigour in taking these last steps, the replica method has been proven to yield exact predictions in the thermodynamic limit for different problems and in particular for the GLM \\cite{Reeves2016, Barbier2017a}.\n\n\\subparagraph{Saddle point solution: choice of a replica ansatz}\nAt this point, we are still left with the problem of computing the extrema of $\\mathcal{\\phi}(\\q, \\hat{\\q})$. To solve this optimization problem over $\\q$ and $\\hat{\\q}$, a natural assumption is that replicas, that are a pure artefact of the calculation, are equivalent. This is reflected in a special structure for overlap matrices between replicas that only depend on three parameters each,\n\\begin{align}\n    \\q = \n\\begin{bmatrix}\nq_0 & m & m & m \\\\\nm & q & q_{12} & q_{12}\\\\\nm & q_{12} & q & q_{12}\\\\ \nm & q_{12} & q_{12} & q \\\\\n\\end{bmatrix} \\, , \\quad\n\\hat{\\q} = \n\\begin{bmatrix}\n\\hat{q_0} & \\hat{m} & \\hat{m} & \\hat{m} \\\\\n\\hat{m} & \\hat{q} & \\hat{q}_{12} & \\hat{q}_{12}\\\\\n\\hat{m} & \\hat{q}_{12} & \\hat{q} & \\hat{q}_{12}\\\\ \n\\hat{m} & \\hat{q}_{12} & \\hat{q}_{12} & \\hat{q} \\\\\n\\end{bmatrix},\n\\end{align} \nhere given as an example for $n=3$ replicas. \nPlugging this \\emph{replica symmetric} (RS) ansatz in the expression of $\\mathcal{\\phi}(\\q, \\hat{\\q})$, taking the limit $n\\to0$ and looking for the stationary points as a function of the parameters $q$, $m$, $q_{12}$ and $\\hat{m}$, $\\hat{q}$, $\\hat{q}_{12}$ recovers a set of equations equivalent to SE \\eqref{eq:chap1-dnn-rec1}, albeit without time indices. Hence the two a priori different heuristics of BP and the replica method are remarkably consistent under the RS assumption.\n\nNevertheless, the replica symmetry can be spontaneously broken in the large $N$ limit and the dominating saddle point does not necessarily correspond to the RS overlap matrix. This replica symmetry breaking (RSB) corresponds to substantial changes in the structure of the examined Boltzmann distribution. It is among the great strengths of the replica formalism to naturally capture it. \nYet for inference problems falling under the teacher-student scenario, the correct ansatz is always replica symmetric in the Bayes optimal setting \\cite{Nishimori2001, Castellani2005, Zdeborova2016}, and we will not investigate here this direction further. The interested reader can refer to the classical references for an introduction to replica symmetry breaking \\cite{Mezard1986, Nishimori2001, Castellani2005} in the context of the theory of spin-glasses.\n \n\n\\subparagraph{Bayes optimal setting} As in SE the equations simplify in the matched setting, where the first replica corresponding to the teacher becomes equivalent to all the others. The replica free energy of the GLM is then given as the extremum of a potential over two scalar variables:\n\\begin{gather}\n    \\label{eq:chap3-replica-fe-glm}\n    - f = \\mathrm{extr}_{q \\hat{q}}\\left[ - \\frac{1}{2} q \\hat{q} +  \\mathcal{I}_x(\\hat{q}) + \\alpha \\mathcal{I}_z(q_0, q)\\right]\\\\\n    \\label{eq:chap3-replica-fe-glm_Ix}\n    \\mathcal{I}_x(\\hat{q}) = \\int \\D{\\xi} \\dd{x} p_x(x)e^{-\\hat{q}\\frac{x^2}{2} + \\sqrt{\\hat{q}}\\xi x} \\log\\left( \\int \\dd{x'} p_x(x')e^{-\\hat{q}\\frac{x'^2}{2} + \\sqrt{\\hat{q}}\\xi x'}\\right) \\\\\n    \\mathcal{I}_z(q , q_0) = \\int \\D{\\xi} \\dd{y} \\dd{z} \\pout(y|z)\\cN(z;\\sqrt{q}\\xi,q_0-q) \\notag \\\\\n    \\label{eq:chap3-replica-fe-glm_Iz}\n    \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\log\\left( \\int \\dd{z'} \\pout(y|z')\\cN(z';\\sqrt{q}\\xi,q_0-q)  \\right) .\n\\end{gather}\nThe saddle point equations corresponding to the extremization \\eqref{eq:chap3-replica-fe-glm}, fixing the values of $q$ and $\\hat{q}$, would again be found equivalent to the Bayes optimal SE \\eqref{eq:chap3-se-bo-qh} - \\eqref{eq:chap3-se-bo-q}. This Bayes optimal result is derived in \\cite{Krzakala2012} for the case of a linear channel and Gauss-Bernoulli prior, and can also be recovered as a special case of the low-rank matrix factorization formula (where the measurement matrix is in fact known) \\cite{Kabashima2016}.\n\n\\subsubsection{Assumptions and relation to other mean-field methods}\nA crucial point in the above derivation of the replica formula is the extensivity of the interactions of the infinite range model that allowed the factorization of the $N$ scaling of the argument of the exponential integrand in \\eqref{eq:chap3-replica-Nscaling}. The statistics of the disorder $\\W$ and in particular the independence of all the $W_{\\mu i}$ was also necessary. While this is an important assumption for the technique to go through, it can be possible to relax it for some types of correlation statistics, as we will see in \\citesec~\\ref{sec:chap3-ortho-invariant}.\n\nNote that the replica method directly enforces the disorder averaging and does not provide a prediction at the level of the single instance. Therefore it cannot be turned into a practical algorithm of reconstruction. Nonetheless, we have seen that the saddle point equations of the replica derivation, under the RS assumption, matches the SE equations derived from BP. This is sufficient to theoretically study inference questions under a teacher-student scenario in the Bayes optimal setting, and in particular predict the MSE following \\eqref{eq:chap3-se-mse}.\n\nIn the mismatched setting however, the predictions of the replica method under the RS assumption and the equivalent BP conclusions can be wrong. By introducing the symmetry breaking between replicas, the method can sometimes  be corrected. It is an important endeavor of the replica formalism to grasp the importance of the overlaps and connect the form of the replica ansatz to the properties of the joint probability distribution examined. When BP fails on loopy graphs, correlations between variables are not decaying with distance, which manifests into an RSB phase. Note that there also exists message passing algorithms operating in this regime \\cite{Mezard2001, Mezard2002, Mezard2009, Saglietti2019, Antenucci2019, Antenucci2019a}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Some current directions of research}\n\\label{sec:chapex}\n\nThe great leap forward in the performance of machine learning with neural networks brought by deep learning algorithms, along with the multitude of theoretical and practical challenges it has opened, has re-ignited the interest of physicists for the theory of neural networks. \nIn this \\citesec, far from being exhaustive, we review some current directions of research leveraging mean-field approximations. Another relevant review is \\cite{Carleo}, which provides references both for machine learning research helped by physics methods and conversely research in physics using machine learning.\n\nWorks presented below do not necessarily implement one of the classical inference methods presented in \\citesecs~\\ref{sec:chap3} and \\ref{sec:chap3further}. In some cases, the mean-field limit corresponds to some asymptotic setting where the problem simplifies: typically some correlations weaken, fluctuations are averaged out by concentration effects and, as a result, ad-hoc methods of resolution can be designed. Thus, in the following contributions, different assumptions are considered to serve different objectives. For instance some take an infinite size limit, some assume random (instead of learned) weights or vanishing learning rates. Hence, there is no such a thing as one mean-field theory of deep neural networks. The below cited works are rather complementary pieces of solving a great puzzle. \n\n\n\n\n\n\\subsubsection{Neural networks for unsupervised learning}\n\n\\paragraph{Fundamental study of learning}\nGiven their similarity with the Ising model, Restricted Boltzmann Machines have unsurprisingly attracted a lot of interest.  Studying an ensemble of RBMs with random parameters using the replica method, Tubiana and Monasson \\cite{Tubiana2017} evidenced different regimes of typical pattern of activations in the hidden units and identified control parameters as the sparsity of the weights, their strength (playing the role of an effective temperature) or the type of prior for the hidden layer. Their study contributes to the understanding of the conditions under which the RBMs can represent high-order correlations between input units, albeit without including data and learning in their model. \nBarra and collaborators \\cite{Barra2017,Barra2018}, exploited the connections between the Hopfield model and RBMs to characterize RBM learning understood as an associative memory.  Relying again on replica calculations, they characterize the retrieval phase of RBMs.\nM\u00e9zard \\cite{Mezard2017} also re-examined retrieval in the Hopfield model using its RBM representation and message passing, showing in particular that the addition of correlations between memorized patterns could still allow for a mean-field treatment at the price of a supplementary hidden layer in the Boltzmann Machine representation. This result remarkably draws a theoretical link between correlations in the training data and the necessity of depth in neural network models.\n\nWhile the above results do not include characterization of the learning driven by data, a few others were able to discuss the dynamics of training. Huang \\cite{Huang2017} studied with the replica method and TAP equations the Bayesian leaning of a RBM with a single hidden unit and binary weights.\nBarra and collaborators \\cite{Barra2017} empirically studied a teacher-student scenario of unsupervised learning by maximum likelihood on samples of an Hopfield model which they could compare to their theoretical characterization of the retrieval phase. \nDecelle and collaborators \\cite{Decelle2017,Decelle2018} introduced an ensemble of RBMs characterized by the spectral properties of the weight matrix and derived the typical dynamics of the corresponding order parameters during learning driven by data. Beyond RBMs, analyses of the learning in other generative models are starting to appear \\cite{Wang2018}.\n\n\n\\paragraph{Training algorithm based on mean-field methods}\nBeyond bringing theoretical insights, mean-field methods are also found useful to build tractable estimators of the likelihood in generative models, which in turn serves to design novel training algorithms. \n\nFor Boltzmann machines, this direction was already investigated in the 80s and 90s, \\cite{Peterson1987,Hinton1989,Galland1993,Kappen1998}, albeit in small models with binary units and for artificial data sets very different from modern machine learning benchmarks. More recently, a deterministic training based on naive mean-field was tested on RBMs \\cite{Welling2002,Tieleman2008}. On toy deep learning data sets, the algorithm was found to perform poorly when compared to both CD and PCD, the commonly employed approximate Monte Carlo methods.\nHowever going beyond naive mean-field, considering the second order TAP expansion, allows to bridge the gap in efficiency \\cite{Gabrie2015, Tramel2018} . Additionally, the deterministic mean-field framework offers a tractable way of evaluating the learning success by exploiting the mean-field observables to visualize the representation learned by RBMs. Interestingly, high temperature expansions of estimators different from the maximum likelihood have also been recently proposed as efficient inference method for the inverse Ising problem \\cite{Lokhov2018}. \n\n\n\n\n\nBy construction, variational auto-encoders (VAEs)\nrely on a variational approximation of the likelihood. In practice, the posterior distribution of the latent representation given an input (see \\citesec~\\ref{sec:chap1-unsupervised}) is typically approached by a factorized Gaussian distribution with mean and variance parametrized by neural networks. The factorization assumption relates the method to a naive mean-field approximation. \n\n\\paragraph{Structured Bayesian priors}\nWith the progress of unsupervised learning, the idea of using generative models as expressive priors has emerged. \n\nFor reconstruction tasks, in the event where a set of typical signals is available a priori, the latter can serve as a training set to learn a model of the underlying data distribution with a generative model. Subsequently, in the reconstruction of a new signal of the same type, the generative model can serve as a Bayesian prior.\nIn particular, the idea to exploit RBMs in CS applications was pioneered by \\cite{Dremeau2012} and \\cite{Tramel2015}, who trained binary RBMs using Contrastive Divergence (to locate the support of the non-zero entries of sparse signals) and combined it with an AMP reconstruction. They demonstrated drastic improvements in the reconstruction with structured learned priors compared to the usual sparse unstructured priors. The approach, requiring to combine the AMP reconstruction for CS and the RBM TAP inference, was further generalized in \\cite{Tramel2016, Tramel2018} to real valued distributions. In the line of these applications, several works have also investigated using feed forward generative models for inference tasks.  Using this time multi-layer VAMP inference, Rangan and co-authors \\cite{Pandit2019} showed that VAEs could help for in-painting partially observed images. \nNote also that a different line of works, mainly considering GANs, examined the same type of applications without resorting to mean-field algorithms \\cite{Bora2017, Hand2018, Hand2018a, Mixon2018}. Instead they performed the inference via gradient descent and back-propagation.\n\nAnother application of generative priors is to model synthetic data sets with structure. In \\cite{Gabrie2018, Aubin2019, Goldt2019a}, the authors designed learning problems amenable to a mean-field theoretical treatment  by assuming the inputs to be drawn from a generative prior (albeit with untrained weights so far). This approach goes beyond the vanilla teacher-student scenario where input data is typically unstructured with i.i.d. components. This is a crucial direction of research as the role of structure in data appears as an important component to understand the puzzling efficiency of deep learning. \n\n\\subsubsection{Neural networks for supervised learning}\n\n\\paragraph{New results in the replica analysis of learning}\nThe classical replica analysis of learning with simple architectures, following bases set by Gardner and Derrida 30 years ago, continues to be explored. Among the most prominent results, Kabashima and collaborators \\cite{Kabashima2008, Shinzato2008, Shinzato2009} extended the mean-field treatment of the perceptron from data matrices with i.i.d entries to random orthogonal matrices. It is a much larger class of random matrices where matrix entries can be correlated.\nMore recently, a series of works explored in depth the specific case of the perceptron with binary weight values for classification on random inputs. \nReplica computations showed that the space of solutions is dominated in the thermodynamic limit by isolated solutions \\cite{Huang2013, Huang2014}, but also that subdominant dense clusters of solutions exist with good generalization properties in the teacher-student scenario case \\cite{Baldassi2015, Baldassi2016, Baldassi2018}. This observation inspired a novel training algorithm \\cite{Chaudhari2017a}. \nThe simple two-layer architecture of the committee machine was also reexamined recently \\cite{Aubin2018}. In the teacher-student scenario, a computationally hard phase of learning was evidenced by comparing a message passing algorithm (believed to be optimal) and the replica prediction. In this work, the authors also proposed a strategy of proof of the replica prediction. \n\n\\paragraph{Signal propagation in depth}\nMean-field approximations can also help understand the role and implications of depth by characterizing signal propagation in neural networks. The following papers consider the width of each layer to go to infinity. In this limit, Sompolinsky and collaborators characterized how neural networks manage to progressively separate data manifolds fed as inputs \\cite{Kadmon2016, Chung2018a, Cohen2019}. Another line of works focused on the initialization of neural networks (i.e. with random weights), and found an order-to-chaos transition in the signal propagation as a function of hyperparameters of training \\cite{Poole2016,Schoenholz2017}. As a result, the authors could formulate recommendations for combinations of hyperparameters to practitioners. This type of analysis could furthermore be generalized to convolutional networks \\cite{Novak2019}, recurrent networks \\cite{Gilboa2019} and networks with batch-normalization regularization \\cite{Yang2019}. The space of functions spanned by deep random networks in the infinite-size limit was also studied by \\cite{Li2018,Li2019}, using the different but related approach of the generating functional analysis.\nYet another mean-field argument, this time relying on a replica computation, allowed to compute the mutual information between layers of large non-linear deep neural networks with orthogonally invariant weight matrices \\cite{Gabrie2018}. Using this method, mutual informations can be followed precisely along the learning for an appropriate teacher-student scenario. The strategy offers an experimental test bed to characterize possible links between the generalization ability of deep neural networks and information compression phases in the training (see \\cite{Tishby2015, Shwartz2017, Saxe2018}).\n\n\n\\paragraph{Dynamics of SGD learning in simple networks and generalization}\nA number of different mean-field limits led to interesting analyses of the dynamics of gradient descent learning. In particular, the below mentioned works contribute to shed light on the generalization power of neural networks in the so-called overparametrized regimes, that is where the number of parameters exceeds largely either the number of training points or the underlying degrees of freedom of the teacher rule.\nIn linear networks first, an exact description in the high-dimensional limit was obtained for the teacher-student setup by \\cite{Advani2017} using random matrix theory. The generalization was predicted to improve with the overparametrization of the student. \nNon-linear networks with one infinitely wide hidden layer were considered by \\cite{Mei2018, Rotskoff2018, Chizat2018a, Sirignano2018} who showed that gradient descent converges to a finite generalization error. \nTheir results are related to others obtained in a slightly different limit of infinitely large neural networks \\cite{Jacot2018}. \nFor arbitrarily deep networks, Jacot and collaborators \\cite{Jacot2018} showed that, in a certain setting, gradient descent was effectively performing a kernel regression with a kernel function converging to a fixed value for the entire training as the size of the layers increases. \nIn both related limits, the absence of divergence is accounting for generalization not deteriorating despite of the explosion of the number of parameters. The relationship between the two above limits was discussed in \\cite{Chizat2018, Mei2019, Geiger2019a}. \nSubsequent works, leveraged the formalism introduced in \\cite{Jacot2018}. Scaling for the generalization error as a function of network sizes were derived by \\cite{Geiger2019}. Other authors focused on the characterization of the network output function in this limit, which takes the form of a Gaussian process \\cite{Lee2019}. This fact was probably first noticed by Opper and Winther with one hidden layer \\cite{Opper99}, to whom it inspired a TAP based Bayesian classification method using Gaussian processes. \nFinally, yet another limit was analyzed by \\cite{Goldt2019}, considering a finite number of hidden units with an infinitely wide input. Following classical works on the mean-field analysis of online learning (not covered in the previous sections \\cite{Saad1995, Saad1995a, Biehl1995, Saad1999a}), a closed set of equations can be derived and analyzed for a collection of overlaps. Note that these are the same order parameters as in replica computations. The resulting learning curves evidence the necessity of multi-layer learning to observe the improvement of generalization with overparametrization. An interplay between optimization, architecture and data sets seems necessary to explain the phenomenon.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\n\\label{sec:conclu}\n\nThis review aimed at presenting in a pedagogical way a selection of inference methods coming from statistical physics. In past and current lines of research that were also reviewed, these methods are sometimes turned into practical and efficient inference algorithms, or sometimes the angle stone in theoretical computations.\n\n\\textbf{What is missing}\nThere are more of these methods beyond what was covered here. In particular the cavity method \\cite{Mezard1986}, closely related to message passing algorithms and the replica formalism, played a crucial role in the physics of spin glasses. Note also that we assumed replica symmetry, which is only guaranteed to be correct in the Bayes optimal case. References of introductions to replica symmetry breaking are \\cite{Mezard1986, Castellani2005}, and newly proposed message passing algorithms with RSB are \\cite{Saglietti2019, Antenucci2019, Antenucci2019a}.\nThe methods of analysis of online learning algorithms pioneered by \\cite{Saad1995, Saad1995a, Biehl1995} and reviewed in \\cite{Saad1999a} also deserve the name of classical mean-field analysis. They are currently actively serving research efforts in deep learning theory \\cite{Goldt2019}. Another important method is the off-equilibrium mean-field theory \\cite{Crisanti1988, Crisanti1993,Cugliandolo1993}, recently used for example to characterize a specific type of neural networks called graph neural networks \\cite{Kawamoto2018} or to study properties of gradient flows \\cite{Mannelli2019}. \n\n\n\n\n\\textbf{On the edge of validity}\nWe have also touched upon the limitations of the mean-field approach. To start with, the thermodynamic limit is ignoring finite-size effects. Moreover, different ways of taking the thermodynamic limit for the same problem sometimes lead to different results. Also, necessary assumptions of randomness for weights or data matrices are sometimes in clear contrast with real applications. \n\n\nThus, the temptation to apply abusively results from one field to the other can be a dangerous pitfall of the interdisciplinary approach. We could mention here the characterization of the dynamics of optimization. While physicists have extensively studied Langevin dynamics with Gaussian white noise, the continuous time limit of SGD is unfortunately not an equivalent in the general case. While some works attempt to draw insights from this analogy using strong assumptions (e.g. \\cite{Choromanska2015, Jastrzebski2017}), others seek precisely to understand the differences between the two dynamics in neural networks optimization (e.g. \\cite{Baity-Jesi2018, Simsekli2019}). \nAlternatively, another good reason to consider the power of mean-field methods lies in the observation rooted in the tradition of theoretical physics that one can learn from models a priori far from the exact neural networks desired, but that retain some key properties, while being amenable to theoretical characterization. For example, \\cite{Mannelli2019} studied a high-dimensional non-convex optimization problem inspired by the physics of spin glasses apparently unrelated to neural networks, but gained insights on the dynamics of gradient descent (and Langevin) that is of primal interest. Another example of this surely promising approach is \\cite{Wang2018}, who built and analyzed a minimal model of GANs.\n\nMoreover, the possibility to combine well-studied simple settings to obtain a mean-field theory for more complex models, as recently demonstrated in a series of work \\cite{Tramel2015, Tramel2016,Tramel2018, Manoel2017b, Fletcher2018a, Gabrie2018, Aubin2019}, constitutes an exciting direction of research that should broaden considerably the limit of applications of mean-field methods.\n\n\\textbf{Patching the pieces together and going further}\nThus the mean-field approach alone cannot to this day provide complete answers to the still numerous puzzles on the way towards a deep learning theory. Yet, considering different limits and special cases, combining solutions to approach ever more complex models, the approach should help uncover more and more corners of the big black box. Hopefully, intuition gained at the edge will help revealing the broader picture. \n\n\n\n\n\n\n\n\n\n\n\n    \n\\section{Introduction}\n\nWith the continuous improvement of storage techniques, the amount of available data is currently growing exponentially. While it is not humanly feasible to  treat all the data created, \\emph{machine learning}, as a class of algorithms that allows to automatically infer structure in large data sets, is one possible response.\nIn particular, \\emph{deep learning} methods, based on neural networks, have drastically improved performances in key fields of artificial intelligence such as image processing, speech recognition or text mining. A good review of the first successes of this technology published in 2015 is \\cite{LeCun2015a}. A few years later, the current state-of-the-art of this very active line of research is difficult to envision globally.\nHowever, the complexity of deep neural networks remains an obstacle to the understanding of their great efficiency. Made of many layers, each of which constituted of many neurons, themselves accompanied by a collection of parameters, the set of variables describing completely a typical neural network is impossible to only visualize. Instead, aggregated quantities must be considered to characterize these models and hopefully help and explain the learning process. The first open challenge is therefore to identify the relevant observables to focus on. Often enough, what seems interesting is also what is hard to calculate. In the high-dimensional regime we need to consider, exact analytical forms are unknown most of the time and numerical computations are ruled out.  \n, ways of approximation that are simultaneously simple enough to be tractable and fine enough to retain interesting features are highly needed.\n\nIn the context where dimensionality is an issue, physicists have experimented that macroscopic behaviors are typically well described by the theoretical limit of infinitely large systems. Under this \\emph{thermodynamic} limit, the statistical physics of disordered systems offers powerful frameworks of approximation called \\emph{mean-field theories}.\nInteractions between physics and neural network theory already have a long history as we will discuss in \\citesec~\\ref{sec:chap1-nn-and-mf}. Yet, interconnections have been re-heightened by the recent progress in deep learning, which also brought new theoretical challenges. \n\n\nHere, we wish to provide a concise methodological review of fundamental mean-field inference methods with their application to neural networks in mind. Our aim is also to provide a unified presentation of the different approximations allowing to understand how they relate and differ. \nReaders may also be interested in related review papers. Another methodological review is \\cite{Advani2013}, particularly interested in applications to neurobiology. Methods presented in the latter reference have a significant overlap with what will be covered in the following. Some elements of random matrix theory are there additionally introduced. \nThe approximations and algorithms which will be discussed here are also largely reviewed in \\cite{Zdeborova2016}. \nThis previous paper includes more details on spin glass theory, which originally motivated the development of the classical mean-field methods, and particularly focuses on community detection and linear estimation. \nDespite the significant overlap and beyond their differing motivational applications, the two previous references are also anterior to some recent exciting developments in mean-field inference covered in the present review, in particular extensions towards multi-layer networks. An older, yet very interesting, reference is the workshop proceedings \\cite{opper2001advanced}, which collected both insightful introductory papers and research developments for the applications of mean-field methods in machine learning. Finally, the recent \\cite{Carleo} covers more generally the connections between physical sciences and machine learning yet without detailing the methodologies. This review provides a very good list of references where statistical physics methods were used for learning theory, but also where machine learning helped in turn physics research.  \n\n\nGiven the literature presented below is at the cross-roads of deep learning and disordered systems physics, we include short introductions to the fundamental concepts of both domains. These \\citesecs~\\ref{sec:chap1} and  \\ref{sec:chap2} will help readers with one or the other background, but can be skipped by experts. In \\citesec~\\ref{sec:chap3}, classical mean-field inference approximations are derived on neural network examples. \\citesec~\\ref{sec:chap3further} covers some recent extensions of the classical methods that are of particular interest for applications to neural networks. We review in \\citesec~\\ref{sec:chapex-all} a selection of important historical and current directions of research in neural networks leveraging mean-field methods. As a conclusion, strengths, limitations and perspectives of mean-field methods for neural networks are discussed in \\citesec~\\ref{sec:conclu}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgements}  \nThis paper is based on the introductory chapters of my PhD dissertation, written under the supervision of Florent Krzakala and in collaboration with Lenka Zdeborov\u00e1, to whom both I am very grateful. I would like to thank also Benjamin Aubin, C\u00e9dric Gerbelot, Adrien Laversanne-Finot, and Guilhem Semerjian for their comments on the manuscript. I gratefully acknowledge the support of the `Chaire de recherche sur les mod\u00e8les et sciences des donn\u00e9es' by Fondation CFM pour la Recherche-ENS and of Fondation L'Or\u00e9al For Women In Science. I also thank the Kalvi Institute for Theoretical Physics, where part of this work was written. \n\n\n\n\n\\section{Index of notations and abbreviations}\n\n\\begin{itemize}\n\\item[] {[N]} - Set of integers from $1$ to $N$\n\\item[] {$\\dirac(\\cdot)$} - Dirac distribution\n\\item[] {$\\sigma(x) = (1 + e^{-x})^{-1}$} - Sigmoid\n\\item[] {$\\mathrm{relu}(x)=\\max(0,x)$} - Rectified Linear Unit\n\\item[] {$\\mat{X}$} - Matrix\n\\item[] {$\\vect{x}$} - Vector\n\\item[] {$\\mat{I}_N \\in \\R^{N\\times N}$} - Identity matrix\n\\item[] {$\\langle \\cdot \\rangle$} - Average with respect to the Boltzmann distribution\n\\item[] {$\\mathrm{O}(N) \\subset \\R^{N\\times N}$} - Orthogonal ensemble\n\\item[] {1RSB} - 1 Step Replica Symmetry Breaking\n\\item[] {AMP} - Approximate message passing\n\\item[] {BP} - Belief Propagation\n\\item[] {cal-AMP} - Calibration Approximate Message Passing\n\\item[] {CD} - Contrastive Divergence\n\\item[] {CS} - Compressed Sensing\n\\item[] {CSP} - Constrain Satisfaction Problem\n\\item[] {DAG} - Directed Acyclic Graph\n\\item[] {DBM} - Deep Boltzmann Machine\n\\item[] {EC} - Expectation Consistency\n\\item[] {EP} - Expectation Propagation \n\\item[] {GAMP} - Generalized Approximate message passing\n\\item[] {GAN} - Generative Adversarial Networks \n\\item[] {GD} - Gradient Descent\n\\item[] {GLM} - Generalized Linear Model\n\\item[] {G-VAMP} - Generalized Vector Approximate Message Passing\n\\item[] {i.i.d.} - independent identically distributed\n\\item[] {PCD} - Persistent Contrastive Divergence\n\\item[] {r-BP} - relaxed Belief Propagation\n\\item[] {RS} - Replica Symmetric\n\\item[] {RSB} - Replica Symmetry Breaking\n\\item[] {RBM} - Restricted Boltzmann Machine\n\\item[] {SE} - State Evolution\n\\item[] {SGD} - Stochastic Gradient Descent\n\\item[] {SK} - Sherrington-Kirkpatrick\n\\item[] {TAP} - Thouless Anderson Palmer \n\\item[] {VAE} - Variational Autoencoder\n\\item[] {VAMP} - Vector Approximate message passing\n\\end{itemize}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nMotivated by Platonov's striking work on  the reduced Whitehead group $\\SK(D)$\nof\nvalued division algebras $D$, see \\cite{platonov,platsurvey},\nV. Yanchevski\\u\\i, considered the unitary  analogue, $\\SK(D, \\tau)$, for a\ndivision algebra~$D$ with unitary (i.e., second kind) involution $\\tau$, see\n\\cite{yin,y,yinverse,yy}.\nWorking with division algebras over a field with henselian discrete (rank $1$)\nvaluation whose residue field also contains\na henselian discrete valuation, and carrying out formidable technical\ncalculations, he produced remarkable analogues to Platonov's results. By relating\n$\\SK(D,\\tau)$ to data over the residue algebra, he showed not only that\n  $\\SK(D,\\tau)$ could be nontrivial but that it could be any finite abelian group,\nand he gave a formula in the bicyclic case expressing  $\\SK(D,\\tau)$\nas a quotient of relative Brauer groups.\n Over the years since then   several approaches have been given  to\n understanding and calculating the (nonunitary) group $\\SK$ using different methods,\nnotably by\nErshov~\\cite{ershov}, Suslin~\\cite{sus1,sus2}, Merkurjev and Rost~\\cite{merk}\n(For surveys on the group $\\SK$, see \\cite{platsurvey}, \\cite{gille},\n\\cite{merk} or\n\\cite[\\S6]{wadval}.) However, even after the passage of some  30 years,\nthere does not seem to have\nbeen any improvement in calculating $\\SK$ in  the unitary setting.\nThis may be due in part to the complexity of the formulas in Yanchevski\\u\\i's\nwork, and the difficulty in following some of his arguments.\n\nThis paper is a sequel to \\cite{hazwadsworth} where the reduced Whitehead\ngroup $\\SK$ for a graded division algebra was studied. Here we consider the\nreduced unitary Whitehead group of a graded division algebra\nwith unitary graded involution. As in our\nprevious work, we will see that the graded calculus is much easier and more\ntransparent than the non-graded one. We calculate  the unitary\n$\\SK$ in several important cases. We also show how this enables one to\ncalculate  the unitary $\\SK$ of a tame\ndivision algebra over a henselian field, by passage to the associated\ngraded division algebra.\nThe graded approach allows us not only to recover most of\nYanchevski\\u\\i's results in  \\cite{y,yinverse, yy}, with very substantially\nsimplified proofs, but also extend them\nto  arbitrary value groups and  to calculate the unitary $\\SK$ for\nwider classes of  division algebras.  There is a significant simplification\ngained by considering arbitrary value groups from the outset, rather than\ntowers of discrete valuations. But the greatest gain comes from passage\nto the graded setting, where the reduction  to arithmetic considerations\nin  the degree $0$ division subring is quicker and more transparent.\n\nWe briefly describe our principal results. Let $\\mathsf{E}$ be a graded division\nalgebra, with torsion free abelian grade group $\\Gamma_\\mathsf{E}$,\nand let $\\tau$ be a unitary\ngraded involution on $\\mathsf{E}$.\n\\lq\\lq Unitary\" means that\nthe action of $\\tau$ on the center $\\mathsf{T} = Z(\\mathsf{E})$ is nontrivial\n(see~\\S\\ref{unitsk1}).\n  The {\\it reduced unitary Whitehead group}  for $\\tau$ on~$\\mathsf{E}$ is defined\nas\n$$\n\\SK(\\mathsf{E},\\tau) \\ = \\ \\big\\{a\\in \\mathsf{E}^*\\mid \\Nrd_\\mathsf{E}(a^{1-\\tau})=1\\big \\}\\big\/\n\\big \\langle a\\in \\mathsf{E}^* \\mid a^{1-\\tau}=1 \\big \\rangle,\n$$\nwhere $\\Nrd_\\mathsf{E}$ is the\nreduced norm map $\\Nrd_\\mathsf{E}\\colon\\mathsf{E}^* \\rightarrow \\mathsf{T}^*$\n(see~\\cite[\\S3]{hazwadsworth}). Here, $a^{1-\\tau}$ means $a\\mspace{1mu}\\tau(a)^{-1}$.\nLet $\\mathsf{R}=\\mathsf{T}^\\tau = \\{t\\in \\mathsf{T}\\mid \\tau(t) = t\\}\\subsetneqq\\mathsf{T}$\n(see~\\S\\ref{unitsk1}).\nLet $\\mathsf{E}_0$ be the subring of homogeneous elements of degree~$0$ in $\\mathsf{E}$;\nlikewise for $\\mathsf{T}_0$ and $\\mathsf{R}_0$.\nFor an involution $\\rho$ on $\\mathsf{E}_0$, $S _\\rho(\\mathsf{E}_0)$ denotes\n$\\{a\\in \\mathsf{E}_0 \\mid \\rho(a) = a\\}$ and $\\Sigma_\\rho(\\mathsf{E}_0) = \\langle\nS_\\rho(\\mathsf{E}_0)\\cap \\mathsf{E}_0^*\\rangle$.\nLet $n$ be the index of $\\mathsf{E}$, and $e$  the exponent of the group\n$\\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T}$. Since $[\\mathsf{T}:\\mathsf{R}]=2$, there are just two possible cases:\neither  \\ (i) $\\mathsf{T}$ is unramified over $\\mathsf{R}$, i.e.,\n$\\Gamma_\\mathsf{T}=\\Gamma_\\mathsf{R}$; or \\  (ii)~$\\mathsf{T}$~is~totally~ramified over~$\\mathsf{R}$,\ni.e., $|\\Gamma_\\mathsf{T}:\\Gamma_\\mathsf{R}|=2$ . We will prove the following\nformulas  for the unitary~$\\SK$:\n\n\\begin{itemize}\n\n\\item[(i)] Suppose $\\mathsf{T}\/\\mathsf{R}$ is unramified:\n\n\\medskip\n\n\\begin{itemize}\n\\item [$\\bullet$] If $\\mathsf{E}\/\\mathsf{T}$ is\nunramified,  then $\\SK(\\mathsf{E},\\tau)  \\cong  \\SK(\\mathsf{E}_0, \\tau|_{\\mathsf{E}_0})$\n(Prop.~\\ref{unramified}).\n\n\\medskip\n\n\\item [$\\bullet$] If $\\mathsf{E}\/\\mathsf{T}$ is totally ramified,\nthen (Th.~\\ref{sktotal}):\n\\begin{align*}\n\\SK(\\mathsf{E},\\tau)\\ & \\cong  \\  \\big \\{a\\in \\mathsf{T}_0^*\\mid a^n\\in \\mathsf{R}_0^*\\}\\big \/\n\\{a\\in \\mathsf{T}_0^*\\mid a^e\\in \\mathsf{R}_0^* \\}\n\\\\\n&\\cong  \\  \\big \\{\\omega \\in \\mu_n(\\mathsf{T}_0) \\mid \\tau(\\omega)\n=\\omega^{-1}\\big \\}\\big \/\\mu_e.\n\\end{align*}\n\n\n\\item [$\\bullet$] If $\\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T}$ is cyclic,\nand $\\sigma$ is a generator of $\\operatorname{Gal}(Z(E_0)\/T_0)$,  then (Prop.~\\ref{cyclic}):\n\\smallskip\n\\begin{itemize}\n\\item [$\\circ$] $\\SK(\\mathsf{E},\\tau) \\ \\cong  \\  \\{ a\\in E_0^* \\mid N_{Z(\\mathsf{E}_0)\/\\mathsf{T}_0}\n(\\Nrd_{\\mathsf{E}_0}(a)) \\in \\mathsf{R}_0  \\}\\big \/\\big(\n\\Sigma_\\tau(\\mathsf{E}_0)\\cdot \\Sigma_{\\sigma\\tau}(\\mathsf{E}_0)\\big)$.\n\n\\smallskip\n\n \\item [$\\circ$] If $\\mathsf{E}_0$ is a field, then $\\SK(\\mathsf{E},\\tau)=1$.\n\\end{itemize}\n\n \\medskip\n\n\\item[$\\bullet$]  If $\\mathsf{E}$ has a maximal graded subfield\n$\\mathsf{M}$ unramified over $\\mathsf{T}$ and another maximal graded subfield~$\\mathsf{L}$ totally\nramified over $\\mathsf{T}$, with $\\tau(\\mathsf{L} ) =\\mathsf{L}$, then $\\mathsf{E}$ is semiramified and\n(Cor.~\\ref{seses})\n\\begin{equation*}\n\\SK(\\mathsf{E},\\tau) \\ \\cong \\ \\big\\{a \\in \\mathsf{E}_0 \\mid N_{\\mathsf{E}_0\/\\mathsf{T}_0}(a)\\in \\mathsf{R}_0\\big\\}\n \\, {\\big\/} \\,  \\textstyle{\\prod\\limits_{h\\in \\operatorname{Gal}(\\mathsf{E}_0\/\\mathsf{T}_0)}}\n\\mathsf{E}_0^{*h \\tau}.\n\\end{equation*}\n\\end{itemize}\n\n\n\\item[(ii)] If $\\mathsf{T}\/\\mathsf{R}$ is totally ramified,  then\n$\\SK(\\mathsf{E},\\tau)=1$ (Prop.~\\ref{completely}).\n\\end{itemize}\n\nThe bridge between the graded and the non-graded henselian setting is\nestablished by\nTh.~\\ref{involthm2}, which shows that for a tame  division\nalgebra $D$ over a henselian valued field with a unitary involution $\\tau$,\n${\\SK(D,\\tau)\\cong \\SK(\\operatorname{{\\sf gr}}(D),\\widetilde\\tau)}$ where $\\operatorname{{\\sf gr}}(D)$ is the\ngraded division algebra associated to $D$\nby the valuation,\n  and $\\widetilde\\tau$\nis the graded involution on $\\operatorname{{\\sf gr}}(D)$ induced by $\\tau$ (see~\\S\\ref{unitary}).\nThus, each of the results listed above for graded division  algebras\nyields analogous formulas for valued division algebras over a henselian\nfield, as illustrated in Example~\\ref{toex} and Th.~\\ref{appl}.\nThis recovers existing formulas, which were primarily for the\ncase with value group $\\mathbb Z$ or $\\mathbb Z\\times \\mathbb Z$, but with easier and\nmore transparent proofs than  those in the existing literature.\nAdditionally, our results apply   for any value groups whatever.\nThe especially simple case where $\\mathsf{E}\/\\mathsf{T}$~is totally ramified\n and $\\mathsf{T}\/\\mathsf{R}$~is unramified is entirely new.\n\nIn the sequel to this paper \\cite{II}, the very interesting special\ncase will be treated where $\\mathsf{E}\/\\mathsf{T}$ is semiramified (and $\\mathsf{T}\/\\mathsf{R}$\nis unramified) and $\\operatorname{Gal}(\\mathsf{E}_0\/\\mathsf{T}_0)$ is bicyclic.  This case\nwas the setting  of essentially all of Platonov's specifically computed\nexamples with nontrivial $\\SK(D)$ \\cite{platonov,plat76}, and likewise\nYanchevski\\u\\i's unitary examples in \\cite{yinverse}\nwhere the nontrivial $\\SK(D, \\tau)$ was fully computed.  This case is\nnot pursued here because it requires some more specialized arguments.\nFor such an $\\mathsf{E}$, it is known that $[\\mathsf{E}]$ decomposes\n(nonuniquely) as  $[\\mathsf{I}\\otimes _\\mathsf{T} \\mathsf{N}]$  in the\ngraded Brauer group of $\\mathsf{T}$, where $\\mathsf{I}$ is inertial over $\\mathsf{T}$ and\n$\\mathsf{N}$  is nicely semiramified, i.e., semiramified and containing a\nmaximal  graded subfield totally ramified over $\\mathsf{T}$.  Then  a formula will\nbe given for $\\SK(E)$ as a factor group of the relative Brauer\ngroup $\\operatorname{Br}(\\mathsf{E}_0\/\\mathsf{T}_0)$ modulo other relative Brauer groups and the\nclass of $\\mathsf{I}_0$. An exactly analogous formula will be proved for\n$\\SK(\\mathsf{E}, \\tau)$  in the unitary setting.\n\n\n\\section{Preliminaries}\\label{prel}\n\nThroughout this  paper we will be concerned with involutory division algebras and\ninvolutory graded division algebras. In the non-graded setting, we will denote a\ndivision algebra by $D$ and its  center by $K$;  this~$D$ is  equipped with an\ninvolution\n$\\tau$, and we set  $F=K^\\tau = \\{a \\in K\\mid \\tau(a) = a\\}$.\n In the graded setting, we will write $\\mathsf{E}$ for a graded division algebra with\ncenter $\\mathsf{T}$, and $\\mathsf{R}=\\mathsf{T}^\\tau$ where $\\tau$ is a graded involution on~$\\mathsf{E}$.\n(This is consistent  with the notation used in~\\cite{hazwadsworth}.)\nDepending on\nthe context, we will write $\\tau(a)$~or~$a^\\tau$ for the action of the\ninvolution on an element, and $K^\\tau$ for the set of elements of $K$\ninvariant under $\\tau$. Our convention is that $a^{\\sigma\\tau}$ means\n$\\sigma(\\tau(a))$.\n\n In this section, we  recall  the notion of graded division algebras\nand collect the  facts we need about them  in~\\S\\ref{pregda}. We will then\nintroduce the unitary and graded reduced unitary Whitehead groups\nin~\\S\\ref{grinvols} and~\\S\\ref{unitsk1}.\n\n\\subsection{Graded division algebras}\\label{pregda}\n\n\nIn this subsection we establish  notation and\nrecall some fundamental facts about graded division\nalgebras indexed by a totally ordered abelian group. For an extensive\nstudy of such graded division algebras\nand their relations with valued division algebras, we refer the reader\nto~\\cite{hwcor}. For generalities  on graded rings see~\\cite{vanoy}.\n\n\n\nLet\n$\\mathsf{R} = \\bigoplus_{ \\gamma \\in \\Gamma} \\mathsf{R}_{\\gamma}$ be a\ngraded ring, i.e.,\n  $\\Gamma$ is an abelian group,  and $\\mathsf{R}$ is a\nunital ring such that each $\\mathsf{R}_{\\gamma}$ is a\nsubgroup of $(\\mathsf{R}, +)$ and\n$\\mathsf{R}_{\\gamma} \\cdot \\mathsf{R}_{\\delta} \\subseteq \\mathsf{R}_{\\gamma +\\delta}$\nfor all $\\gamma, \\delta \\in \\Gamma$. Set\n\\begin{itemize}\n\\item[] $\\Gamma_\\mathsf{R}  \\ = \\  \\{\\gamma \\in \\Gamma \\mid \\mathsf{R}_{\\gamma} \\neq 0 \\}$,\n \\  the grade set of $\\mathsf{R}$;\n\\vskip0.05truein\n\\item[] $\\mathsf{R}^h  \\ = \\ \\bigcup_{\\gamma \\in\n\\Gamma_{\\mathsf{R}}} \\mathsf{R}_{\\gamma}$,  \\ the set of homogeneous elements of $\\mathsf{R}$.\n\\end{itemize}\nFor a homogeneous element of $\\mathsf{R}$  of degree $\\gamma$, i.e., an\n$r \\in \\mathsf{R}_\\gamma\\setminus\\{0\\}$, we write $\\deg(r) = \\gamma$.\nRecall  that $\\mathsf{R}_{0}$ is a subring of $\\mathsf{R}$ and that for each $\\gamma \\in\n\\Gamma_{\\mathsf{R}}$, the group $\\mathsf{R}_{\\gamma}$ is a left and right $\\mathsf{R}_{0}$-module.\nA subring $\\mathsf{S}$ of\n$\\mathsf{R}$ is a \\emph{graded subring} if $\\mathsf{S}= \\bigoplus_{ \\gamma \\in\n\\Gamma_{\\mathsf{R}}} (\\mathsf{S} \\cap \\mathsf{R}_{\\gamma})$.  For example, the\ncenter of $\\mathsf{R}$, denoted $Z(\\mathsf{R})$, is a graded subring of\n$\\mathsf{R}$.\nIf $\\mathsf{T} = \\bigoplus_{ \\gamma \\in\n\\Gamma} \\mathsf{T}_\\gamma$ is another graded ring,\na {\\it graded ring homomorphism} is a ring homomorphism\n$f\\colon \\mathsf{R}\\to \\mathsf{T}$ with $f(\\mathsf{R}_\\gamma) \\subseteq \\mathsf{T}_\\gamma$\nfor all $\\gamma \\in \\Gamma$.  If $f$ is also bijective,\nit is called a graded ring isomorphism;  we then write\n$\\mathsf{R}\\cong_{\\mathrm{gr}} \\mathsf{T}$.\n\n\nFor a graded ring $\\mathsf{R}$, a {\\it graded left $\\mathsf{R}$-module} $\\mathsf{M}$ is\na left  $\\mathsf{R}$-module with a grading ${\\mathsf{M}=\\bigoplus_{\\gamma \\in \\Gamma'}\n\\mathsf{M}_{\\gamma}}$,\nwhere the $\\mathsf{M}_{\\gamma}$ are all abelian groups and $\\Gamma'$ is a\nabelian group containing $\\Gamma$, such that $\\mathsf{R}_{\\gamma} \\cdot\n\\mathsf{M}_{\\delta} \\subseteq \\mathsf{M}_{\\gamma + \\delta}$ for all $\\gamma \\in \\Gamma_\\mathsf{R},\n\\delta \\in \\Gamma'$.\nThen, $\\Gamma_\\mathsf{M}$ and $\\mathsf{M}^h$ are\ndefined analogously to $\\Gamma_\\mathsf{R}$ and~$\\mathsf{R}^h$.  We say that $\\mathsf{M}$ is\na {\\it graded free} $\\mathsf{R}$-module if it has a base as a free\n$\\mathsf{R}$-module consisting of homogeneous elements.\n\n\nA graded ring $\\mathsf{E} = \\bigoplus_{ \\gamma \\in \\Gamma} \\mathsf{E}_{\\gamma}$ is\ncalled a \\emph{graded division ring} if $\\Gamma$ is a\ntorsion-free abelian group and\nevery non-zero homogeneous\nelement of $\\mathsf{E}$ has a multiplicative inverse in $\\mathsf{E}$.\nNote that the grade set  $\\Gamma_{\\mathsf{E}}$ is actually a group.\nAlso, $\\mathsf{E}_{0}$ is a division ring,\nand  $\\mathsf{E}_\\gamma$ is a $1$-dimensional\nleft and right $\\mathsf{E}_0$ vector space for every $\\gamma\\in \\Gamma_\\mathsf{E}$.\nSet $\\mathsf{E}_\\gamma^* = \\mathsf{E}_\\gamma \\setminus\\{0\\}$.\nThe requirement that $\\Gamma$ be torsion-free is made\nbecause we are interested in graded division rings arising\nfrom valuations on division rings, and all the grade groups\nappearing there are torsion-free.  Recall that every\ntorsion-free abelian group $\\Gamma$ admits total orderings compatible\nwith the group structure.  (For example, $\\Gamma$ embeds in\n$\\Gamma \\otimes _{\\mathbb Z}\\mathbb Q$ which can be given\na lexicographic total ordering using any base of it as a\n$\\mathbb Q$-vector space.)  By using any total ordering on\n$\\Gamma_\\mathsf{E}$, it is easy to see that $\\mathsf{E}$ has no zero divisors\nand that $\\mathsf{E}^*$, the multiplicative group of units of $\\mathsf{E}$,\ncoincides with\n $\\mathsf{E}^{h} \\setminus \\{0\\}$ (cf. \\cite[p.~78]{hwcor}).\nFurthermore, the degree map\n\\begin{equation}\\label{degmap}\n\\deg\\colon \\mathsf{E}^* \\rightarrow \\Gamma_\\mathsf{E}\n\\end{equation} is a group epimorphism with kernel $\\mathsf{E}_0^*$.\n\n\nBy an easy adaptation of the ungraded arguments, one can see\nthat every graded  module~$\\mathsf{M}$  over a graded division ring\n$\\mathsf{E}$ is graded free, and every two\nhomogenous bases have the same cardinality.\nWe thus call $\\mathsf{M}$ a \\emph{graded vector space} over $\\mathsf{E}$ and\nwrite $\\dim_\\mathsf{E}(\\mathsf{M})$ for the rank of~$\\mathsf{M}$ as a graded free $\\mathsf{E}$-module.\nLet $\\mathsf{S} \\subseteq \\mathsf{E}$ be a graded subring which is also a graded\ndivision ring.  Then we can view $\\mathsf{E}$ as a graded left $\\mathsf{S}$-vector\nspace, and  we write $[\\mathsf{E}:\\mathsf{S}]$ for $\\dim_\\mathsf{S}(\\mathsf{E})$.  It is easy to\ncheck the \\lq\\lq Fundamental Equality,\"\n\\begin{equation}\\label{fundeq}\n[\\mathsf{E}:\\mathsf{S}] \\ = \\ [\\mathsf{E}_0:\\mathsf{S}_0] \\, |\\Gamma_\\mathsf{E}:\\Gamma_\\mathsf{S}|,\n\\end{equation}\nwhere $[\\mathsf{E}_0:\\mathsf{S}_0]$ is the dimension of $\\mathsf{E}_0$ as a left vector space\nover the division ring $\\mathsf{S}_0$ and $|\\Gamma_\\mathsf{E}:\\Gamma_\\mathsf{S}|$ denotes the\nindex in the group $\\Gamma_\\mathsf{E}$ of its subgroup $\\Gamma_\\mathsf{S}$.\n\nA \\emph{graded field} $\\mathsf{T}$ is a commutative graded division ring.\nSuch a $\\mathsf{T}$ is an integral domain (as $\\Gamma_\\mathsf{T}$ is torsion free),\nso it has a quotient field,\nwhich we denote $q(\\mathsf{T})$.  It is known, see \\cite[Cor.~1.3]{hwalg},\nthat $\\mathsf{T}$~is integrally closed in $q(\\mathsf{T})$.  An extensive theory of\ngraded algebraic field extensions of graded fields has been developed in\n\\cite{hwalg}.\n\n\n\n\nIf $\\mathsf{E}$ is a graded division ring, then its center $Z(\\mathsf{E})$ is clearly\na graded field.  {\\it The graded division rings considered in\nthis paper will always be assumed finite-dimensional over their\ncenters.}  The finite-dimensionality assures that $\\mathsf{E}$\nhas a quotient division ring $q(\\mathsf{E})$ obtained by central localization,\ni.e., ${q(\\mathsf{E}) = \\mathsf{E} \\otimes_\\mathsf{T} q(\\mathsf{T})}$,  where $\\mathsf{T} = Z(\\mathsf{E})$. Clearly,\n$Z(q(\\mathsf{E})) = q(\\mathsf{T})$ and $\\operatorname{ind}(\\mathsf{E}) = \\operatorname{ind}(q(\\mathsf{E}))$, where\nthe index of $\\mathsf{E}$ is defined by $\\operatorname{ind}(\\mathsf{E})^2 = [\\mathsf{E}:\\mathsf{T}]$\n(see \\cite[p.~89]{hwcor}).\nIf $\\mathsf{S}$ is a graded field which is a graded subring of $Z(\\mathsf{E})$\nand $[\\mathsf{E}:\\mathsf{S}] <\\infty$,\nthen $\\mathsf{E}$ is said to be a {\\it graded division algebra} over~$\\mathsf{S}$.\nWe recall a fundamental connection between $\\Gamma_\\mathsf{E}$ and $Z(\\mathsf{E}_0)$:\nThe field $Z(\\mathsf{E}_0)$\nis Galois over $\\mathsf{T}_0$,  and there is a well-defined group epimorphism\n\\begin{equation}\\label{surj}\n\\Theta_\\mathsf{E}\\colon\\Gamma_\\mathsf{E} \\rightarrow \\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{T}_0)\n\\text{\\,\\,\\,\\,\\, given by \\,\\,\\,\\,\\, } \\deg(e)\\mapsto (z\\mapsto eze^{-1}),\n\\end{equation}\nfor any $e\\in \\mathsf{E}^*$.\n(See \\cite[Prop.~2.3]{hwcor} for a proof).\n\n\n\n\n\n\nLet $\\mathsf{E} = \\bigoplus_{\\alpha \\in \\Gamma_\\mathsf{E}} \\mathsf{E}_{\\alpha}$ be a graded division\nalgebra with a graded center $\\mathsf{T}$ (with, as always, $\\Gamma_\\mathsf{E}$ a\ntorsion-free abelian group).\nAfter fixing  some total ordering on $\\Gamma_\\mathsf{E}$,  define a function\n$$\n\\lambda\\colon  \\mathsf{E}\\setminus\\{0\\} \\rightarrow \\mathsf{E}^{\\ast} \\quad\\textrm{ by }\n\\quad\\lambda(\\textstyle\\sum c_{\\gamma}) =\nc_{\\delta}, \\text{ where $\\delta$ is minimal among the $ \\gamma \\in \\Gamma_\\mathsf{E}$ with\n$c_{\\gamma} \\neq0$}.\n$$\n Note that $\\lambda(a) = a$ for $a \\in \\mathsf{E}^{\\ast}$, and\n\\begin{equation}\\label{valhomin}\n \\lambda(ab)= \\lambda(a) \\lambda(b) \\textrm{ for all } a,b \\in \\mathsf{E}\\setminus\\{0\\}.\n\\end{equation}\n\n\nLet $Q = q(\\mathsf{E})= \\mathsf{E} \\otimes_\\mathsf{T} q(\\mathsf{T})$, which is a division ring\nas $\\mathsf{E}$ has no zero divisors and is finite-dimensional over $\\mathsf{T}$.\nWe can extend $\\lambda$ to a map defined on all of $Q^{\\ast} = Q\\setminus\\{0\\}$\n as follows: for $q\n\\in Q^{\\ast}$, write $q = ac^{-1}$  with $a \\in \\mathsf{E}\\setminus\\{0\\}$, $c \\in\nZ(\\mathsf{E})\\setminus\\{0\\}$, and set $\\lambda(q) = \\lambda(a) \\lambda(c)^{-1}$. It follows\n from (\\ref{valhomin}) that $\\lambda\\colon Q^{\\ast} \\rightarrow \\mathsf{E}^{\\ast}$ is\nwell-defined and is\na group homomorphism. Since the composition\n\\begin{equation}\\label{inji}\n\\mathsf{E}^{\\ast} \\hookrightarrow Q^{\\ast}\n\\stackrel{\\lambda}{\\longrightarrow} \\mathsf{E}^{\\ast}\n\\end{equation}\n is the identity, $\\lambda$ is a\nsplitting map for the injection $\\mathsf{E}^{\\ast} \\hookrightarrow\nQ^{\\ast}$.\n\nFor a graded division algebra $\\mathsf{E}$ over its center $\\mathsf{T}$, there is\n a reduced norm map $\\Nrd_\\mathsf{E}\\colon\\mathsf{E}^*\\rightarrow \\mathsf{T}^*$\n(see~\\cite[\\S3]{hazwadsworth}) such that  for $a\\in \\mathsf{E}$ one has\n$\\Nrd_\\mathsf{E}(a)=\\Nrd_{q(\\mathsf{E})}(a)$. The {\\it reduced Whitehead group},\n$\\SK(\\mathsf{E})$, is defined as\n$\\mathsf{E}^{(1)}\/\\mathsf{E}'$, where $\\mathsf{E}^{(1)}$ denotes\nthe set of elements of $\\mathsf{E}^*$ with  reduced norm 1, and $\\mathsf{E}'$ is the\ncommutator subgroup $[\\mathsf{E}^*,\\mathsf{E}^*]$ of $\\mathsf{E}^*$. This group was studied in detail\nin~\\cite{hazwadsworth}.\nWe will be using the following facts which were established in that paper:\n\\begin{remarks}\\label{grfacts} Let $n = \\operatorname{ind}(\\mathsf{E})$.\n\\begin{enumerate}[\\upshape(i)]\n\\item For $\\gamma\\in \\Gamma_\\mathsf{E}$, if\n$a\\in \\mathsf{E}_\\gamma$ then $\\Nrd_\\mathsf{E}(a)\\in \\mathsf{E}_{n\\gamma}$.  In particular,\n$\\mathsf{E}^{(1)}$ is a subset of $\\mathsf{E}_0$.\n\n\\smallskip\n\n\\item  If $\\mathsf{S}$ is any graded subfield of $\\mathsf{E}$\ncontaining $\\mathsf{T}$ and $a\\in \\mathsf{S}$, then\n$\\Nrd_\\mathsf{E}(a) =  N_{\\mathsf{S}\/\\mathsf{T}}(a)^{n\/[\\mathsf{S}:\\mathsf{T}]}$.\n\n\\smallskip\n\n\\item \\label{rnrd} Set\n\\begin{equation}\\label{dlambda}\n\\partial  \\ = \\  \\operatorname{ind}(\\mathsf{E})\\big\/\\big(\\operatorname{ind}(\\mathsf{E}_0) \\,\n[Z(\\mathsf{E}_0) \\!: \\!\\mathsf{T}_0]\\big).\n\\end{equation}\n  If $a\\in \\mathsf{E}_0$, then,\n\\begin{equation}\\label{nrddo}\n\\Nrd_\\mathsf{E}(a)  \\ = \\   N_{Z(\\mathsf{E}_0)\/\\mathsf{T}_0}\\Nrd_{\\mathsf{E}_0}(a)^\n {\\,\\partial}\n\\in \\mathsf{T}_0.\n\\end{equation}\n\n\\smallskip\n\n\\item \\label{normal} If $N$ is a normal subgroup of\n$\\mathsf{E}^{\\ast}$, then $N^{n} \\subseteq\n\\Nrd_\\mathsf{E}(N)[\\mathsf{E}^{\\ast}, N]$.\n\nFor  proofs of (i)-(iv) see \\cite[Prop.~3.2 and~3.3]{hazwadsworth}.\n\n\\smallskip\n\n\\item \\label{torik} $\\SK(\\mathsf{E})$ is $n$-torsion.\n\\begin{proof}\nBy taking  $N=\\mathsf{E}^{(1)}$, the assertion follows from (\\ref{normal}).\n\\end{proof}\n\\end{enumerate}\n\\end{remarks}\n\n\nA graded division algebra $\\mathsf{E}$ with center $\\mathsf{T}$ is\nsaid to be {\\it inertial} (or {\\it unramified}) if $\\Gamma_\\mathsf{E}=\\Gamma_\\mathsf{T}$.\nFrom~(\\ref{fundeq}), it then follows  that $[\\mathsf{E}:\\mathsf{T}]=[\\mathsf{E}_0:\\mathsf{T}_0]$;\nindeed, $\\mathsf{E}_0$ is central simple over $\\mathsf{T}_0$ and $\\mathsf{E} \\cong_{\\operatorname{{\\sf gr}}} \\mathsf{E}_0 \\otimes _{\\mathsf{T}_0} \\mathsf{T}$.\nAt the other extreme, $\\mathsf{E}$ is said to be {\\it totally ramified}\nif $\\mathsf{E}_0=\\mathsf{T}_0$. In an intermediate case  $\\mathsf{E}$ is\nsaid to be {\\it semiramified} if\n $\\mathsf{E}_0$ is a field and ${[\\mathsf{E}_0:\\mathsf{T}_0]=|\\Gamma_\\mathsf{E}:\\Gamma_\\mathsf{T}|=\\operatorname{ind}(\\mathsf{E})}$.\nThese definitions are motivated by  analogous  definitions for\nvalued division algebras (\\cite{wadval}).\nIndeed,  if a tame valued division algebra\nis unramified, semiramified, or totally ramified, then so is its\nassociated graded division algebra.\nLikewise, a  graded field extension $\\mathsf{L}$~of~\n$\\mathsf{T}$ is said to be  {\\it inertial} (or {\\it unramified})\nif $\\mathsf{L}\\cong_{\\operatorname{{\\sf gr}}} \\mathsf{L}_0 \\otimes_{\\mathsf{T}_0}\\mathsf{T}$ and the field $\\mathsf{L}_0$\nis separable over $\\mathsf{T}_0$. At the other extreme,\n$\\mathsf{L}$ is {\\it totally ramified} over $\\mathsf{T}$ if\n$[\\mathsf{L}:\\mathsf{T}] = |\\Gamma_\\mathsf{L}:\\Gamma_\\mathsf{T}|$.  A graded division algebra\n$\\mathsf{E}$ is said to be {\\it inertially split} if $\\mathsf{E}$ has a maximal\ngraded subfield $\\mathsf{L}$ with $\\mathsf{L}$ inertial  over $\\mathsf{T}$.\nWhen this occurs, $\\mathsf{E}_0 = \\mathsf{L}_0$,  and $\\operatorname{ind}(\\mathsf{E})= \\operatorname{ind}(\\mathsf{E}_0) \\,\n[Z(\\mathsf{E}_0) \\!: \\!\\mathsf{T}_0]$ by Lemma~\\ref{dlambdafacts} below.\nIn particular, if $\\mathsf{E}$ is  semiramified then $\\mathsf{E}$ is inertially\nsplit,\n$\\mathsf{E}_0$ is\nabelian Galois over $\\mathsf{T}_0$, and the canonical map\n$\\Theta_\\mathsf{E}\\colon \\Gamma_\\mathsf{E}\\rightarrow \\operatorname{Gal}(\\mathsf{E}_0\/\\mathsf{T}_0)$ has kernel\n$\\Gamma_\\mathsf{T}$  (see~\\eqref{surj} above and\n~\\cite[Prop.~2.3]{hwcor}).\n\n\\begin{lemma}\\label{dlambdafacts}\nLet  $\\mathsf{E}$ be a graded division algebra  with center $\\mathsf{T}$.\n For the  $\\partial$ of~\\eqref{dlambda},\n$\\partial^2 = |\\ker(\\Theta_\\mathsf{E})\/\\Gamma_\\mathsf{T}|$.  Also,\n$\\partial = 1$ iff $\\mathsf{E}$ is inertially split.\n\\end{lemma}\n\n\\begin{proof}\nSince $\\Theta_\\mathsf{E}$ is surjective,  $\\Gamma_\\mathsf{T}\\subseteq\\ker(\\Theta_\\mathsf{E})$,\nand $Z(\\mathsf{E}_0)$ is Galois over $\\mathsf{T}_0$, we have\n\\begin{align*}\n\\partial^2  \\ &= \\ \\operatorname{ind}(\\mathsf{E})^2 \\, \\big\/ \\,\n\\big(\\operatorname{ind}(\\mathsf{E}_0)^2 \\, [Z(\\mathsf{E}_0):\\mathsf{T}_0]^2\\big)\n \\ = \\ [\\mathsf{E}:\\mathsf{T}] \\, \\big\/\\big([\\mathsf{E}_0:Z(\\mathsf{T}_0)] \\, [Z(\\mathsf{E}_0):\\mathsf{T}_0] \\,\n|\\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{T}_0)|\\big)\\\\\n& =  \\ [\\mathsf{E}_0:\\mathsf{T}_0] \\, |\\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T}| \\, \\big\/ \\,\n\\big([\\mathsf{E}_0:\\mathsf{T}_0] \\, |\\operatorname{im}(\\Theta_\\mathsf{E})|\\big) \\ = \\ |\\ker(\\Theta_\\mathsf{E})\/\\Gamma_\\mathsf{T}|.\n\\end{align*}\n\nNow, suppose $M$ is a maximal subfield of $\\mathsf{E}_0$\nwith $M$ separable over $\\mathsf{T}_0$.  Then,\n$M\\supseteq Z(\\mathsf{E}_0)$ and $[M:Z(\\mathsf{E}_0)] = \\operatorname{ind}(\\mathsf{E}_0)$.  Let\n$\\mathsf{L} = M\\otimes _{\\mathsf{T}_0} \\mathsf{T}$ which is a graded subfield of $\\mathsf{E}$\ninertial over $\\mathsf{T}$, with $\\mathsf{L}_0 = M$.  Then,\n$$\n[\\mathsf{L}:\\mathsf{T}] \\ = \\ [\\mathsf{L}_0:\\mathsf{T}_0] \\ = \\ [\\mathsf{L}_0:Z(\\mathsf{E}_0)]\\, [Z(\\mathsf{E}_0):\\mathsf{T}_0]\n \\ = \\ \\operatorname{ind}(\\mathsf{E})\/\\partial.\n$$\nThus, if $\\partial = 1$, then $\\mathsf{E}$ is inertially\nsplit, since $\\mathsf{L}$ is a maximal graded subfield of $\\mathsf{E}$ which is\ninertial over $\\mathsf{T}$.  Conversely, suppose $\\mathsf{E}$ is inertially\nsplit, say $\\mathsf{I}$ is a maximal graded subfield of $\\mathsf{E}$ with\n$\\mathsf{I}$  inertial over $\\mathsf{T}$.  So, $[\\mathsf{I}_0:\\mathsf{T}_0] = [\\mathsf{I}:\\mathsf{T}] =\n\\operatorname{ind}(\\mathsf{E})$. Since $\\mathsf{I}_0Z(\\mathsf{E}_0)$ is a subfield of $\\mathsf{E}_0$,\nwe have\n\\begin{align*}\n[\\mathsf{I}_0:\\mathsf{T}_0] \\ &\\le \\ [\\mathsf{I}_0Z(\\mathsf{E}_0) :\\mathsf{T}_0] \\ = \\\n[\\mathsf{I}_0 Z(\\mathsf{E}_0):Z(\\mathsf{E}_0)]\\, [Z(\\mathsf{E}_0):\\mathsf{T}_0] \\\\\n&\\le  \\\n\\operatorname{ind} (\\mathsf{E}_0)\\, [Z(\\mathsf{E}_0):\\mathsf{T}_0] \\ = \\ \\operatorname{ind}(\\mathsf{E})\/\\partial\n \\ = \\ [\\mathsf{I}_0:\\mathsf{T}_0] \/\\partial.\n\\end{align*}\nSo, as $\\partial$ is a positive integer, $\\partial = 1$.\n\\end{proof}\n\n\n\n\n\\subsection{Unitary $\\SK$ of division algebras}\\label{grinvols}\n\nWe begin with a description of unitary $K_1$ and $\\SK$ for a division\nalgebra with an involution. The analogous definitions for graded\ndivision algebras will be given in ~\\S\\ref{unitsk1}.\n\n\nLet $D$ be a division ring finite-dimensional over its center $K$ of\nindex $n$, and let $\\tau$ be an involution on~$D$,  i.e., $\\tau$ is\nan antiautomorphism of $D$ with $\\tau^2 =\\operatorname{id}$.\nLet\n\\begin{itemize}\n\\item[] $\\quad S_\\tau(D)  \\, =  \\, \\{d \\in D\\mid\\tau(d) = d \\};$\n\\vskip0.05truein\n\\item [] $\\quad \\Sigma_{\\tau}(D)  \\, = \\,\n\\left< S_{\\tau}(D) \\cap D^{\\ast} \\right>.$\n\\end{itemize}\n\nNote that $\\Sigma_{\\tau}(D)$ is a normal subgroup of $D^{\\ast}$. For,\nif $a \\in S_{\\tau}(D)$, $a \\neq 0$, and $b \\in D^{\\ast}$, then\n$bab^{-1} = [ba\\tau(b)][b\\tau(b)]^{-1}\\in \\Sigma_\\tau(D)$,\nas $ba\\tau(b), b\\tau(b) \\in S_\\tau(D)$.\n\n\n\nLet $\\varphi$ be an isotropic $m$-dimensional, nondegenerate skew-Hermitian form\nover $D$\nwith respect to an involution $\\tau$ on $D$.  Let\n$\\rho$ be the involution on $M_m(D)$ adjoint to $\\varphi$, let\n$U_m(D) = \\{a \\in M_m(D)\\mid a\\rho(a) = 1\\}$ be the unitary group\nassociated to $\\varphi$,  and let $\\operatorname{EU}_m(D)$ denote\nthe normal subgroup of $U_m(D)$ generated by the unitary transvections.\nFor  $m>2$,\nthe Wall spinor norm map\n$\\Theta\\colon U_m(D) \\rightarrow D^*\/\\Sigma_\\tau(D)D'$ was developed\nin \\cite{wall},  where it was shown that  $\\ker(\\Theta) = \\operatorname{EU}_m(D)$.\nHere, $D'$ denotes the multiplicative commutator group $[D^*,D^*]$.\nCombining this\nwith~\\cite[Cor.~1 of~\\S 22]{draxl} one obtains  the commutative diagram:\n\\begin{equation}\\label{unnrd}\n\\begin{split}\n\\xymatrix{U_m(D)\\big\/\\operatorname{EU}_m(D) \\ar[r]_-{\\cong}^-\\Theta  \\ar[d] &\nD^*\\big\/\\big ( \\Sigma_\\tau(D)D' \\big )\\ar[d]^{1-\\tau}\\\\\n\\operatorname{GL}_m(D)\\big\/E_m(D) \\ar[r]^-{\\det} \\ar[d]_{\\Nrd} & D^*\\big\/D'\n\\ar[d]^{\\Nrd}\\\\\nK^* \\ar[r]^-{\\operatorname{id}} & K^*}\n\\end{split}\n\\end{equation}\nwhere the map $\\det$ is the Dieudonn\\'e determinant and\n$1-\\tau\\colon D^*\/\\big ( \\Sigma_\\tau(D)D' \\big ) \\longrightarrow D^*\/D'$\nis defined as\n$x\\Sigma_\\tau(D)D'\\mapsto x^{1-\\tau}D'$, where $x^{1-\\tau}$ means $x\\mspace{1mu}\\tau(x)^{-1}$\n(see~\\cite[6.4.3]{hahn}).\n\nFrom the diagram, and parallel to the ``absolute'' case, one defines\nthe {\\it unitary Whitehead group},\n$$\nK_1(D,\\tau)\\ = \\ D^*\/\\big (\\Sigma_\\tau(D)D'\\big ).\n$$\n\nFor any  involution $\\tau$ on $D$, recall that\n\\begin{equation}\\label{taunrd}\n\\Nrd_D(\\tau(d))  \\ = \\  \\tau(\\Nrd_D(d)),\n\\end{equation} for any  $d\\in D$. For, if $p \\in K[x]$ is the\nminimal polynomial of $d$ over $K$,\nthen $\\tau(p)$ is the minimal polynomial of $\\tau(d)$ over $K$\n(see also~\\cite[\\S22, Lemma~5]{draxl}).\n\n\n We consider two cases:\n\n\\subsubsection{ Involutions of the first kind}\\label{firstk}\nIn this case the center $K$ of $D$ is elementwise invariant under the\ninvolution, i.e.,\n$K \\subseteq S_\\tau(D)$. Then $S_\\tau(D)$ is a $K$-vector space.  The\n involutions of this kind are further subdivided into two types:\n{\\it orthogonal} and\n{\\it symplectic} involutions ~(see \\cite[Def.~2.5]{kmrt}).\nBy ~(\\cite[Prop.~2.6]{kmrt}), if $\\operatorname{char}(K)\\not = 2$ and $\\tau$ is orthogonal\nthen, $\\dim_K(S_\\tau(D))=n(n+1)\/2$, while if $\\tau$ is symplectic then\n$\\dim_K(S_\\tau(D))=n(n-1)\/2$. However, if $\\operatorname{char}(K)=2$, then\n$\\dim_K(S_\\tau(D))=n(n+1)\/2$ for each type.\n\n\nIf $\\dim_K(S_\\tau(D))=n(n+1)\/2$, then  for any\n$x \\in D^*$, we have $xS_\\tau(D) \\cap S_\\tau(D) \\not = 0$\nby dimension count; it then follows that\n$D^*=\\Sigma_\\tau(D)$, and thus $K_1(D,\\tau)=1$. However,\nin the case $\\dim_K(S_\\tau(D))=n(n-1)\/2$, Platonov showed that $K_1(D,\\tau)$ is not in general\ntrivial, settling  Dieudonn\\'e's conjecture in negative~\\cite{pld}.\nNote that whenever $\\tau$  is of the first kind we have\n$\\Nrd_D(\\tau(d)) = \\Nrd_D(d)$ for all $d\\in D$, by \\eqref{taunrd}.\nThus,  $K_1(D,\\tau)$ is sent to  the identity under\nthe composition   $\\Nrd \\circ (1-\\tau)$. This explains why\n one does not consider the kernel of this map, i.e., the\nunitary $\\SK$, for involutions of the first kind.\nIf $\\operatorname{char}(K) \\not =2$ and\n$\\tau$~is symplectic,\nthen as the $m$-dimensional form $\\varphi$ over $D$ is skew-Hermitian,\nits associated  adjoint involution~$\\rho$  on $M_m(D)$ is\nof orthogonal type, so there is  an associated  spin group\n$\\operatorname{Spin}(M_m(D), \\rho)$.\nFor any $a\\in S(D)$ one then has $\\Nrd_D(a) \\in K^{*2}$\n(\\cite[Lemma~2.9]{kmrt}).\nOne defines  $K_1\\operatorname{Spin}(D,\\tau)=R(D)\/\\big(\\Sigma_\\tau(D)D'\\big)$,\nwhere\n$R(D)=\\{d\\in D^* \\mid \\Nrd_D(d) \\in K^{*2}\\}$. This group\nis  related to $\\operatorname{Spin}(M_m(D), \\rho)$, and has been studied\nin~\\cite{monyan},  parallel to the work on absolute  $\\SK$ groups and\nunitary $\\SK$ groups for unitary involutions.\n\n \\subsubsection{Involutions of the second kind $($unitary involutions$)$} \\label{secondk}\n In this case $K \\not \\subseteq S_\\tau(D)$.\n Then, let \\break ${F=K^{\\tau}\\ (=K\\cap\nS_\\tau(D)\\,)}$, which is a subfield of $K$ with $[K:F] = 2$.\n It was already observed by Dieudonn\\'e that\n$U_m(D) \\not = \\operatorname{EU}_m(D)$.  An important property proved\n by Platonov and Yanchevski\\u\\i, which we will use frequently,\nis that\n\\begin{equation}\\label{primeinsigma}\nD' \\,  \\subseteq  \\, \\Sigma_\\tau(D).\n\\end{equation}\n(For a proof, see~\\cite[Prop.~17.26]{kmrt}.) Thus\n$K_1(D,\\tau)=D^*\/\\Sigma_\\tau(D)$, which  is not trivial in general.\nThe kernel of the\nmap $\\Nrd \\circ (1-\\tau)$ in  diagram~\\eqref{unnrd}, is called the\n{\\it reduced unitary Whitehead group}, and denoted by $\\SK(D,\\tau)$.\nUsing~(\\ref{taunrd}), it is straightforward to see that\n$$\n\\SK(D,\\tau) \\ = \\  \\Sigma'_{\\tau}(D) \/ \\Sigma_{\\tau}(D), \\quad \\text{where}\n\\quad\\Sigma'_{\\tau}(D)  \\ = \\  \\{ a \\in D^{\\ast} \\mid \\Nrd_D(a) \\in F^{\\ast}\\}.\n$$\nNote that  we use the notation $\\SK(D,\\tau)$ for the reduced unitary\n Whitehead group as opposed to Draxl's notation\n$\\operatorname{USK_1}(D,\\tau)$ in~\\cite[p.~172]{draxl} and\nYanchevski\\u\\i's notation\n$\\operatorname{SUK_1}(D,\\tau)$~\\cite{y} and the notation\n$\\operatorname{USK_1}(D)$ in \\cite{kmrt}.\n\n\nBefore we define the corresponding groups in the graded setting, let us\nrecall that all the groups above fit in Tits' framework\n\\cite{tits}\nof the\n{\\it Whitehead group} $W(G,K)=G_K\/G_K^+$ where $G$ is an almost simple,\nsimply connected linear algebraic group defined over an infinite field\n$K$, with $\\operatorname{char}(K)\\not =2$, and $G$ is isotropic over $K$. Here, $G_K$ is the\nset of $K$-rational points of $G$ and $G_K^+$, is the subgroup of $G_K$,\ngenerated by the unipotent radicals of the minimal $K$-parabolic\nsubgroups of $G$. In this setting, for $G_K=\\operatorname{SL}_n(D)$, $n>1$, we have\n$W(G,K)=\\SK(D)$; for $\\tau$ an involution of first or second  kind\non $D$ and $F = K^\\tau$, for  $G_F=\\operatorname{SL}_n(D,\\tau):=\\operatorname{SL}_n(D)\\cap U_n(D)$\nwe have  $W(G,F)=\\SK(D,\\tau)$; and for $\\tau$ a symplectic involution\non~ $D$ and $\\rho$ the adjoint involution of an $m$-dimensional\nisotropic skew-Hermitian\n form over $D$ with $m\\ge 3$,\nfor the spinor group\n$G_K=\\operatorname{Spin}(M_m(D),\\rho)$ we have $W(G,K)$\nis a double cover of $K_1 \\operatorname{Spin}(D,\\tau)$ (see~\\cite{monyan}).\n\n\n\\subsection{Unitary $\\SK$ of graded division algebras}\\label{unitsk1}\nWe will now introduce the unitary $K_1$ and $\\SK$ in the graded setting.\nLet $\\mathsf{E} = \\bigoplus_{\\gamma \\in \\Gamma_\\mathsf{E}} \\mathsf{E}_{\\gamma}$ be a graded division\nring (with $\\Gamma_\\mathsf{E}$ a torsion-free abelian group) such that\n$\\mathsf{E}$ has finite dimension $n^2$ over its center $\\mathsf{T}$, a graded field.\nLet $\\tau$ be a graded involution of~$\\mathsf{E}$, i.e., $\\tau$~is\nan antiautomorphism of $\\mathsf{E}$ with $\\tau^2 =\\operatorname{id}$ and $\\tau(\\mathsf{E}_{\\gamma}) =\n\\mathsf{E}_{\\gamma}$ for each $\\gamma \\in \\Gamma_\\mathsf{E}$. We define $S_{\\tau}(\\mathsf{E})$ and\n$\\Sigma_{\\tau}(\\mathsf{E})$, analogously to the non-graded cases, as the set of\nelements of $\\mathsf{E}$ which are invariant under~$\\tau$, and the\nmultiplicative group\ngenerated by the nonzero homogenous elements of $S_\\tau(\\mathsf{E})$, respectively.\nWe say  the involution of the {\\it first kind} if all the elements of the\ncenter $\\mathsf{T}$ are invariant under $\\tau$; it is of the {\\it second kind}\n(or {\\it unitary})\notherwise. If $\\tau$ is of the first kind then,  parallel to the\nnon-graded case, either\n$\\dim_\\mathsf{T}(S_\\tau(\\mathsf{E}))=n(n+1)\/2$ or\n$\\dim_\\mathsf{T}(S_\\tau(\\mathsf{E}))=n(n-1)\/2$.\nIndeed,  one can show these equalities\nby arguments analogous to the\nnongraded case as in the proof of \\cite [Prop.~2.6(1)]{kmrt}, as $\\mathsf{E}$ is split \nby a graded maximal subfield and the Skolem--Noether theorem is available\nin the graded setting (\\cite[Prop.~1.6]{hwcor}).  (These equalities  can also\nbe obtained\nby passing to the quotient division algebra as is done in\nLemma~\\ref{invfacts}\\eqref{type} below.)\n\n\n\n Define the {\\it\n unitary Whitehead group}\n$$\nK_1(\\mathsf{E},\\tau) \\ = \\ \\mathsf{E}^*\/\\big(\\Sigma_\\tau(\\mathsf{E})\\mathsf{E}'\\big),\n$$\nwhere $\\mathsf{E}' = [\\mathsf{E}^*,\\mathsf{E}^*]$.\nIf $\\tau$ is of the first kind,\n$\\operatorname{char}(\\mathsf{T})\\not =2$, and $\\dim_\\mathsf{T}(S_\\tau(\\mathsf{E}))=n(n-1)\/2$, a proof similar\nto \\cite[Prop.~2.9]{kmrt}, shows that if $a\n\\in S_\\tau(\\mathsf{E})$ is homogeneous, then $\\Nrd_\\mathsf{E}(a) \\in \\mathsf{T}^{*2}$\n(This can also be verified by passing to the quotient division\nalgebra, then using Lemma~\\ref{invfacts}\\eqref{type} below and invoking the\ncorresponding result for ungraded division algebras.)   For\nthis type of involution, define the {\\it spinor Whitehead group}\n$$\nK_1\\operatorname{Spin}(\\mathsf{E},\\tau) \\ = \\ \\{a \\in \\mathsf{E}^*\\mid\n\\Nrd_\\mathsf{E}(a) \\in \\mathsf{T}^{*^2} \\} \\, \/ \\big( \\Sigma_\\tau(\\mathsf{E})\\mathsf{E}'\\big).\n$$\n\nWhen the graded involution $\\tau$ on $\\mathsf{E}$ is unitary, i.e.,  $\\tau|_\\mathsf{T} \\neq\n\\operatorname{id}$, let $\\mathsf{R}= \\mathsf{T}^{\\tau}$, which is a graded subfield of~$\\mathsf{T}$\nwith $[\\mathsf{T}:\\mathsf{R}]=2$. Furthermore, $\\mathsf{T}$ is Galois over $\\mathsf{R}$, with\n$\\operatorname{Gal} (\\mathsf{T}\/\\mathsf{R}) = \\{\\operatorname{id}, \\tau|_\\mathsf{T}\\}$. (See \\cite{hwalg}\nfor Galois theory for graded field extensions.) Define the\n{\\it reduced unitary Whitehead group}\n\\begin{equation}\\label{skdef}\n\\SK (\\mathsf{E}, \\tau)  \\ = \\  \\Sigma'_{\\tau}(\\mathsf{E}) \\, \/ \\,\n(\\Sigma_{\\tau}(\\mathsf{E}) \\, \\mathsf{E}')\n \\ = \\ \\Sigma'_{\\tau}(\\mathsf{E}) \\, \/ \\,\n\\Sigma_{\\tau}(\\mathsf{E}) ,\n\\end{equation}\nwhere\n$$\n\\Sigma'_{\\tau}(\\mathsf{E}) \\ = \\ \\big\\{a\\in \\mathsf{E}^*\\mid\n\\Nrd_\\mathsf{E}(a^{1-\\tau})=1\\big \\}  \\ = \\ \\{a\\in \\mathsf{E}^*\\mid \\Nrd_\\mathsf{E}(a) \\in \\mathsf{R}^*\\}\n$$\nand\n$$\n\\Sigma_{\\tau}(\\mathsf{E}) \\ = \\ \\langle a\\in \\mathsf{E}^* \\mid\na^{1-\\tau}=1 \\big \\rangle \\ = \\ \\langle S_\\tau(\\mathsf{E}) \\cap \\mathsf{E}^*\\rangle.\n$$\nHere, $a^{1-\\tau}$ means $a\\tau(a)^{-1}$. See Lemma~\\ref{invfacts}(iv)\nbelow for the second equality in \\eqref{skdef}.\nThe group $\\SK(\\mathsf{E}, \\tau)$ will be the main focus of the rest of the paper.\n\n\nWe will use the following facts repeatedly:\n\n\n\\begin{lemma}\\label{invfacts}\\hfill\n\\begin{enumerate}[\\upshape(i)]\n\\item \\label{type}\nAny graded involution on $\\mathsf{E}$ extends uniquely to\nan involution of the same kind\n$($and type$)$  on $Q=q(\\mathsf{E})$.\n\n\\smallskip\n\n\\item \\label{sigma} For any graded involution $\\tau$ on $\\mathsf{E}$, and its\nextension to $Q=q(\\mathsf{E})$, we have $\\Sigma_\\tau(Q) \\cap \\mathsf{E}^*\n\\subseteq \\Sigma_\\tau(\\mathsf{E})$.\n\n\\smallskip\n\n\\item \\label{firstsig} If $\\tau$ is a graded involution of\nthe first kind on $\\mathsf{E}$ with $\\dim_\\mathsf{T}(S_\\tau(\\mathsf{E}))=n(n+1)\/2$,\n then $\\Sigma_\\tau(\\mathsf{E})=\\mathsf{E}^*$.\n\n\\smallskip\n\n\\item \\label{yan} If $\\tau$ is a unitary graded involution on $\\mathsf{E}$,\nthen $\\mathsf{E}'\\subseteq \\Sigma_\\tau(\\mathsf{E})$.\n\n\n\\smallskip\n\n\\item \\label{tori} If $\\tau$ is a unitary graded involution on $\\mathsf{E}$,\nthen $\\SK(\\mathsf{E},\\tau)$ is a torsion group of bounded exponent\ndividing  $n=\\operatorname{ind}(\\mathsf{E})$.\n\n\n\\end{enumerate}\\end{lemma}\n\n\\begin{proof}\\hfill\n\n\n(i) Let $\\tau$ be a graded involution on $\\mathsf{E}$.  Then\n$q(\\mathsf{E})=\\mathsf{E}\\otimes_\\mathsf{T} q(\\mathsf{T})=\n\\mathsf{E}\\otimes_\\mathsf{T} (\\mathsf{T}\\otimes_{\\mathsf{T}^\\tau}q(\\mathsf{T}^\\tau))=\n\\mathsf{E}\\otimes_{\\mathsf{T}^\\tau} q(\\mathsf{T}^\\tau)$. The unique extension\nof $\\tau$  to $q(\\mathsf{E})$ is $\\tau  \\otimes \\operatorname{id}_{q(\\mathsf{T}^\\tau)}$,\nwhich we denote simply as $\\tau$.\nIt then follows that\n$S_\\tau(q(\\mathsf{E}))=S_\\tau(\\mathsf{E})\\otimes_{\\mathsf{T}^\\tau} q(\\mathsf{T}^\\tau)$.\nSince $q(\\mathsf{T}^\\tau)=q(\\mathsf{T})^\\tau$, the assertion follows.\n\n(ii) Note that for the\n map $\\lambda$ in the sequence (\\ref{inji}) we  have\n$\\tau(\\lambda(a))=\\lambda(\\tau(a))$ for all $a\\in Q^*$.\nHence, $\\lambda(\\Sigma_\\tau(Q)) \\subseteq\\Sigma_\\tau(\\mathsf{E})$.\nSince $\\lambda|_{\\mathsf{E}^*}$ is the identity,   we have\n$\\Sigma_\\tau(Q) \\cap \\mathsf{E}^* \\subseteq \\Sigma_\\tau(\\mathsf{E})$.\n\n(iii) The extension of the graded  involution $\\tau$  to\n$Q=q(\\mathsf{E})$, also denoted $\\tau$, is of the first kind with\n$\\dim_Q(S_\\tau(Q))=n(n+1)\/2$ by (\\ref{type}). Therefore\n$\\Sigma_\\tau(Q)=Q^*$ (see~\\S\\ref{firstk}). Using\n(\\ref{sigma}) now, the assertion follows.\n\n(iv) Since $\\tau$ is a unitary graded involution,   its extension to\n$Q=q(\\mathsf{E})$ is  also  unitary,  by (\\ref{type}). But\n${Q' \\subseteq \\Sigma_\\tau(Q)}$,\nas noted in \\eqref{primeinsigma}.  From~(\\ref{inji}) it follows that\n$Q' \\cap \\mathsf{E}^* = \\mathsf{E}'$. Hence, using \\eqref{sigma}, \\break\n${\\mathsf{E}' \\subseteq \\mathsf{E}^* \\cap Q' \\subseteq \\mathsf{E}^* \\cap \\Sigma_\\tau(Q)\n\\subseteq \\Sigma_\\tau(\\mathsf{E})}$.\n\n\n\n(v) Setting $N=\\Sigma_\\tau'(\\mathsf{E})$,\nRemark~\\ref{grfacts}(\\ref{normal}) above, coupled with the fact\nthat $\\mathsf{E}' \\subseteq \\Sigma_\\tau(\\mathsf{E})$ (\\ref{yan}), implies that\n$\\SK(\\mathsf{E},\\tau)$ is an $n$-torsion group. This assertion also follows\nby using (\\ref{sigma}) which implies the natural map $\\SK(\\mathsf{E},\\tau)\n\\to \\SK(Q,\\tau)$ is injective and the fact that unitary\n$\\SK$ of a division algebra of index $n$ is\n$n$-torsion~(\\cite[Cor. to~2.5]{y}).\n\\end{proof}\n\n\\subsection{Generalized dihedral groups and field extensions}\n\nThe nontrivial case of  $\\SK(\\mathsf{E},\\tau)$ for $\\tau$ a unitary\ngraded involution turns out to be when\n $\\mathsf{T} = Z(\\mathsf{E})$ is unramified over $\\mathsf{R} = \\mathsf{T}^\\tau$ (see \\S\\ref{unram}).\nWhen that occurs, we will see in\nLemma~\\ref{unramfacts}\\eqref{eight} below that $Z(\\mathsf{E}_0)$ is a so-called\ngeneralized dihedral extension over $\\mathsf{R}_0$.\nWe now give the definition and observe a few easy\n facts about generalized dihedral groups and extensions.\n\n\\begin{deff}\\label{gendi}\\hfill\n\\begin{enumerate}[\\upshape(i)]\n\\item \\label{deffo1} A group $G$ is said to be {\\it generalized dihedral} if $G$ has a\nsubgroup $H$ such that $[G:H]=2$ and every $\\tau \\in G \\backslash H$\nsatisfies $\\tau^2=\\operatorname{id}$.\n\nNote that if $G$ is generalized dihedral and $H$ the distinguished subgroup,\nthen $H$ is abelian and $(h\\tau)^2=\\operatorname{id}$, for all $\\tau \\in G\\backslash H$\nand $h\\in H$. Thus, $\\tau^2=\\operatorname{id}$ and $\\tau h \\tau ^{-1}=h^{-1}$ for all\n$\\tau \\in G\\backslash H, h \\in H$. Furthermore, every subgroup of $H$ is\nnormal in $G$. Clearly every dihedral group is generalized dihedral, as\nis every elementary abelian $2$-group. More generally, if $H$ is any\nabelian group and $\\chi \\in \\operatorname{Aut}(H)$ is the map $h \\mapsto h^{-1}$, then\nthe semi-direct product\n$H \\rtimes_i \\langle \\chi \\rangle$ is a generalized dihedral group,\nwhere $i\\colon\n\\langle \\chi \\rangle \\rightarrow \\operatorname{Aut}(H)$ is the inclusion map.\nIt is easy to check that every generalized dihedral group is isomorphic to\nsuch a semi-direct product.\n\n\\item \\label{deffo2} Let $F \\subseteq K \\subseteq L$ be fields with $[L:F]<\\infty$ and\n$[K:F]=2$. We say that $L$ is {\\it generalized dihedral for $K\/F$} if $L$ is\nGalois over $F$ and every element of $\\operatorname{Gal}(L\/F) \\backslash \\operatorname{Gal}(L\/K)$\nhas order $2$, i.e., $\\operatorname{Gal}(L\/F)$ is a generalized dihedral group.\nNote that when this occurs, $L$ is compositum of fields $L_i$\n containing~$K$ with each\n$L_i$ generalized dihedral for $K\/F$ with $\\operatorname{Gal}(L_i\/K)$ cyclic, i.e.,\n$L_i$ is Galois over $F$ with $\\operatorname{Gal}(L_i\/F)$ dihedral (or a Klein $4$-group).\nConversely, if $L$ and $M$ are generalized dihedral for $K\/F$ then so is\ntheir compositum.\n\n\\end{enumerate}\n\\end{deff}\n\\begin{example} Let $n\\in \\mathbb N$, $n \\geq 3$, and let $F\\subseteq K$ be fields with\n$[K:F]=2$ and $K=F(\\omega)$, where $\\omega$ is a primitive $n$-th root of unity\n(so $\\operatorname{char}(F) \\nmid n$). Suppose the non-identity element of $\\operatorname{Gal}(K\/F)$ maps\n$\\omega$ to~$\\omega^{-1}$. For any $c_1,\\dots,c_k \\in F^*$, if\n$\\omega \\not \\in F(\\sqrt[n]{c_1},\\dots,\\sqrt[n]{c_k})$, then\n$K(\\sqrt[n]{c_1},\\dots,\\sqrt[n]{c_k})$ is generalized dihedral for $K\/F$.\n\\end{example}\n\n\n\n\\section{Henselian to graded reduction}\\label{unitary}\n\nThe main goal of this section is to prove an isomorphism\nbetween  the unitary $\\SK$ of a valued division\nalgebra with involution over a henselian field\n  and the graded $\\SK$ of  its associated graded division algebra.\nWe first recall how to associate a graded division algebra to a valued\ndivision algebra.\n\nLet $D$ be a division algebra finite dimensional over its\ncenter $K$, with a valuation\n$v\\colon  D^{\\ast} \\rightarrow\n\\Gamma$. So, $\\Gamma$ is a totally ordered abelian group,\nand $v$ satisifies the conditions that for all\n$a, b \\in D^{\\ast}$,\n\\begin{enumerate}\n\\item $ \\qquad \\, v(ab)  \\, = \\,  v(a) + v(b)$;\n\n\\item $\\quad v(a+b)  \\, \\geq \\,  \\min \\{v(a),v(b) \\}\\;\\;\\;\\;\\; (b \\neq -a).$\n\\end{enumerate}\nLet\n\\begin{align*}\nV_D  \\ &= \\  \\{ a \\in D^{\\ast} \\mid v(a) \\geq 0 \\}\\cup\\{0\\},\n\\text{ the\nvaluation ring of $v$};\\\\\nM_D  \\ &= \\  \\{ a \\in D^{\\ast} \\mid v(a) > 0\n\\}\\cup\\{0\\}, \\text{ the unique maximal left (and right) ideal\n of $V_D$}; \\\\\n\\overline{D}  \\ &= \\  V_D \/ M_D, \\text{ the residue\ndivision ring of $v$ on $D$; and} \\\\\n\\Gamma_D  \\ &= \\  \\mathrm{im}(v), \\text{ the value\ngroup of the valuation}.\n\\end{align*}\n\nNow let $K$ be a field with a valuation $v$,\nand suppose $v$ is {\\it henselian}; that is,\n  $v$ has a unique extension to every  algebraic\nfield extension of $K$. Recall that a\n field extension $L$ of $K$ of degree~$n<\\infty$ is\nsaid to be {\\it tamely ramified} or {\\it tame} over $K$\nif, with respect to the unique extension of $v$ to $L$,\nthe residue field\n$\\overline L$ is separable over\n$\\overline K$ and $\\operatorname{char}({\\overline K})\\nmid\n\\big ( n\\big\/[\\overline L:\\overline K] \\big )$. Such an $L$~is\nnecessarily {\\it defectless} over $K$,\ni.e., $[L:K] = [\\overline L:\\overline K] \\, |\\Gamma_L:\\Gamma_K|$,\nby \\cite[Th.~3.3.3]{EP} (applied to $N\/K$ and $N\/L$, where\n$N$ is a normal closure of $L$ over $K$).\nAlong the same lines, let $D$ be a\ndivision algebra with center $K$ (so, by convention,\n$[D:K] < \\infty$); then the henselian valuation  $v$ on $K$ extends uniquely\nto a valuation\non $D$ ~(\\cite{wad87}). With respect to this valuation, $D$\nis said to be\n{\\it tamely ramified} or {\\it tame} if the center $Z(\\overline D)$ is\n separable over $\\overline K$ and ${\\operatorname{char}({\\overline K}) \\nmid\n\\big[\\operatorname{ind}(D)\\big\/\\big(\\operatorname{ind}(\\overline D)[Z(\\overline D):\\overline K]\\big)}\\big]$.\nRecall from \\cite[Prop.~1.7]{jw}, that whenever the field extension\n$Z(\\overline D)\/\\overline K$ is separable, it is abelian Galois.\nIt is known  that\n$D$ is tame if and only if $D$~is split by the\nmaximal tamely ramified field extension of~$K$, if and only if\n$\\operatorname{char}(\\overline K) = 0$\nor $\\operatorname{char}(\\overline K) = p\\ne 0$ and the $p$-primary component of~\n$D$ is inertially split, i.e., split by the maximal unramified\nextension of~$K$~(\\cite[Lemma~6.1]{jw}).\nWe say $D$ is \\emph{strongly tame}\nif $\\operatorname{char}(\\overline K)\\nmid\\operatorname{ind}(D)$.\nNote that strong tameness implies tameness.\nThis is clear from the last characterization of tameness,\nor from \\eqref{Ostrowski} below.\n Recall also from \\cite[Th.~3]{M}, that for a valued\ndivision algebra $D$ finite dimensional over its  center $K$ (here not necessarily\nhenselian), we have the \\lq\\lq Ostrowski theorem\"\n\\begin{equation}\\label{Ostrowski}\n[D:K]\n\\ = \\ q^k\\,[\\overline D:\\overline K] \\,|\\Gamma_D:\\Gamma_K|,\n\\end{equation}\nwhere $q=\\operatorname{char}({\\overline D})$ and $k \\in \\mathbb Z$ with $k\\geq 0$\n(and  $q^k = 1$ if $\\operatorname{char}(\\overline D) = 0$).\nIf $q^k = 1$ in equation~   \\eqref{Ostrowski}, then $D$~is\nsaid to be  {\\it defectless} over $K$.\nFor background on  valued division algebras,\nsee~\\cite{jw} or the survey paper~\\cite{wadval}.\n\n\\begin{remark}\\label{shensel}\nIf a field $K$ has a henselian valuation $v$ and $L$ is a subfield of\n$K$ with $[K:L] <\\infty$, then the restriction $w = v|_L$ need not be\nhenselian.  But it is easy to see that $w$ is then \\lq\\lq semihenselian,\"\ni.e., $w$~has more than one but only finitely many different extensions to\na separable closure $L_{\\textrm{sep}}$ of $L$.\nSee \\cite{engler} for a thorough analysis of\nsemihenselian valuations. Notably, Engler shows\nthat  $w$~is semihenselian iff the residue\nfield $\\overline L_w$ is algebraically closed but\nthere is a henselian valuation $u$ on $L$ such that $u$\n is a proper coarsening of $w$ and  the residue  field $\\overline L_u$ is\nreal closed. When this occurs, $\\operatorname{char}(L) = 0$, $L$ is formally real,\n$w$ has exactly two extensions to\n$L_{\\textrm{sep}}$, the value group $\\Gamma_{L,w}$\nhas a nontrivial divisible subgroup, and the henselization\nof $L$ re $w$ is $L(\\sqrt{-1})$, which lies in $K$.\nFor example, if we take any prime number $p$,\nlet $w_p$ be the\n$p$-adic discrete valuation on $\\mathbb Q$, and let $L = \\{r\\in \\mathbb R\\,|\\ r\n\\text{ is algebraic over $\\mathbb Q$}\\}$; then any extension of $w_p$ to\n$L$ is a semihenselian valuation.\nNote that if $v$ on $K$ is discrete, i.e., $\\Gamma_K\\cong \\mathbb Z$,\nthen $w$ on $L$ cannot be semihenselian, since $\\Gamma_L$ has no\nnontrivial divisible subgroup; so, $w$ on $L$ must be henselian.\nThis preservation of the henselian property for discrete valuations\nwas asserted in \\cite[Lemma, p.~195]{y}, but the proof given there is invalid.\n\\end{remark}\n\nOne associates to a valued division algebra $D$ a graded division algebra\nas follows:\nFor each $\\gamma\\in \\Gamma_D$, let\n\\begin{align*}\n D^{\\ge\\gamma}  \\ &=  \\\n\\{ d \\in D^{\\ast} \\mid v(d) \\geq \\gamma \\}\\cup\\{0\\}, \\text{ an additive\nsubgroup of $D$}; \\qquad \\qquad\\qquad\\qquad\\qquad \\ \\\\\nD^{>\\gamma}  \\ &=  \\ \\{ d \\in D^{\\ast} \\mid v(d) > \\gamma \\}\\cup\\{0\\},\n\\text{ a subgroup\nof $D^{\\ge\\gamma}$};   \\text{ and}\\\\\n \\operatorname{{\\sf gr}}(D)_\\gamma \\ &= \\\nD^{\\ge\\gamma}\\big\/D^{>\\gamma}.\n\\end{align*}\nThen define\n$$\n \\operatorname{{\\sf gr}}(D)  \\ = \\  \\textstyle\\bigoplus\\limits_{\\gamma \\in \\Gamma_D}\n\\operatorname{{\\sf gr}}(D)_\\gamma. \\ \\\n$$\nBecause $D^{>\\gamma}D^{\\ge\\delta} \\,+\\, D^{\\ge\\gamma}D^{>\\delta}\n\\subseteq D^{>(\\gamma +\n\\delta)}$ for all $\\gamma , \\delta \\in \\Gamma_D$, the  multiplication on\n$\\operatorname{{\\sf gr}}(D)$ induced by multiplication on $D$ is\nwell-defined, giving that $\\operatorname{{\\sf gr}}(D)$ is a graded  ring, called the\n{\\it associated graded ring} of $D$. The\nmultiplicative property\n(1) of  the valuation $v$ implies that $\\operatorname{{\\sf gr}}(D)$ is a graded\ndivision ring.\nClearly,\nwe have ${\\operatorname{{\\sf gr}}(D)}_0 = \\overline{D}$ and $\\Gamma_{\\operatorname{{\\sf gr}}(D)} = \\Gamma_D$.\nFor $d\\in D^*$, we write $\\widetilde d$ for the image\n$d + D^{>v(d)}$ of $d$ in $\\operatorname{{\\sf gr}}(D)_{v(d)}$.  Thus,\nthe map given by $d\\mapsto \\widetilde d$ is\na group epimorphism $\\rho: D^* \\rightarrow {\\operatorname{{\\sf gr}}(D)^*}$ with\nkernel~$1+M_D$, giving us the short exact sequence\n\\begin{equation}\\label{grmap}\n1 \\ \\longrightarrow 1+M_D  \\ \\longrightarrow D^* \\  \\longrightarrow  \\ \\operatorname{{\\sf gr}}(D)^*\n \\ \\longrightarrow  \\ 1,\n\\end{equation} which will be used throughout. For a detailed\nstudy of the associated graded algebra of a valued\ndivision algebra\nrefer to \\cite[\\S4]{hwcor}.  As shown in \\cite[Cor.~4.4]{hazwadsworth},\nthe  reduced norm maps\nfor $D$ and $\\operatorname{{\\sf gr}}(D)$ are related by\n\\begin{equation}\\label{nrdrel}\n\\widetilde{\\Nrd_D(a)}  \\ = \\ \\Nrd_{\\operatorname{{\\sf gr}}(D)}(\\widetilde a) \\ \\ \\ \\text{for all }\na\\in D^*.\n\\end{equation}\n\n\nNow let $K$ be a field with a henselian valuation $v$ and, as before,\nlet $D$ be a division algebra with center~$K$. Then $v$ extends uniquely\nto a valuation on $D$, also denoted $v$,\n and one obtains associated to~$D$ the graded division algebra $\\,\\operatorname{{\\sf gr}}(D)=\n\\bigoplus_{\\gamma \\in \\Gamma_D} D_{\\gamma}$. Further, suppose $D$ is\ntame  with respect\nto $v$. This implies that $[\\operatorname{{\\sf gr}}(D):\\operatorname{{\\sf gr}}(K)] = [D:K]$, $\\operatorname{{\\sf gr}}(K) = Z(\\operatorname{{\\sf gr}}(D))$ and\n$D$ has a maximal subfield $L$ with $L$ tamely ramified over\n$K$~(\\cite[Prop.~4.3]{hwcor}).\nWe can then  associate to an\ninvolution $\\tau$ on $D$, a graded involution $\\widetilde \\tau$ on $\\operatorname{{\\sf gr}}(D)$.\nFirst, suppose\n$\\tau$ is of the first kind on $D$. Then $v \\circ \\tau$ is\nalso a valuation on $D$ which restricts to $v$ on $K$;\nthen, $v \\circ \\tau = v$\n since $v$ has a\nunique extension to $D$. So,\n$\\tau$ induces a well-defined map $\\widetilde{\\tau}\\colon \\operatorname{{\\sf gr}}(D) \\rightarrow \\operatorname{{\\sf gr}}(D)$,\ndefined on homogeneous elements by $\\widetilde{\\tau}(\\widetilde{a}) =\n\\widetilde{\\tau(a)}$ for all $a \\in D^{\\ast}$. Clearly,\n$\\widetilde{\\tau}$ is a well-defined\ngraded involution on $\\operatorname{{\\sf gr}}(D)$; it is of the first kind, as\nit leaves $Z(\\operatorname{{\\sf gr}}(D))=\\operatorname{{\\sf gr}}(K)$ invariant.\n\nIf $\\tau$ is a unitary involution on $D$, let $F=K^\\tau$. In\nthis case, we need to assume  that the restriction of the valuation $v$\nfrom $K$ to $F$\ninduces a henselian valuation on $F$, and that $K$ is tamely ramified\nover $F$. Since $(v\\circ\\tau)|_F = v|_F$, an argument similar to the\none above\n shows that $v \\circ \\tau$ coincides with $v$ on $K$ and\nthus on~$D$,  and the induced  map $\\widetilde \\tau$  on $\\operatorname{{\\sf gr}}(D)$\nas above is a graded\ninvolution. That $K$ is tamely ramified over $F$\nmeans that $[K:F] = [\\operatorname{{\\sf gr}}(K):\\operatorname{{\\sf gr}}(F)]$, $\\overline{K}$ is separable\nover $\\overline{F}$, and $\\operatorname{char}(\\overline F)\n\\nmid  |\\Gamma_K : \\Gamma_F|$.  Since $[K:F]=2$, $K$ is always tamely\nramified over $F$ if $\\operatorname{char}(\\overline F) \\not = 2$. But if\n$\\operatorname{char}(\\overline{F})=2$, $K$ is tamely ramified over $F$ if and\nonly if $[\\overline{K}: \\overline{F}]= 2$, $\\Gamma_K = \\Gamma_F$, and\n$\\overline{K}$ is separable (so Galois) over $\\overline{F}$.\nSince $K$ is Galois over $F$, the canonical map $\\operatorname{Gal}(K\/F)\\to\n\\operatorname{Gal}(\\overline K\/\\overline F)$ is surjective, by \\cite[pp.~123--124, proof of\nLemma~5.2.6(1)]{EP}.\nHence,  $\\tau$~induces the\nnonidentity $\\overline{F}$-automorphism $\\overline \\tau$ of $\\overline{K}$.\nAlso $\\widetilde \\tau$ is unitary, i.e.,\n$\\widetilde{\\tau}|_{\\operatorname{{\\sf gr}}(K)}\n\\neq \\operatorname{id}$. This is obvious if $\\operatorname{char} (\\overline{F}) \\neq 2$,\nsince then $K = F(\\sqrt c)$ for some $c \\in F^{\\ast}$, and\n$\\widetilde{\\tau}(\\widetilde{\\sqrt c}) =\n\\widetilde{\\tau (\\sqrt c)}=\n-\\widetilde{ \\sqrt c} \\neq \\widetilde{\\sqrt c}$.\nIf $\\operatorname{char}(\\overline{F})= 2$, then $K$ is unramified over $F$ and\n$\\widetilde{\\tau}|_{\\operatorname{{\\sf gr}}(K)_0}= \\overline{\\tau}$ (the automorphism\nof $\\overline{K}$ induced by $\\tau|_K$) which is nontrivial as\n$\\operatorname{Gal}(K\/F)$ maps onto $\\operatorname{Gal}(\\overline{K}\/ \\overline{F})$; so\nagain $\\widetilde{\\tau}|_{\\operatorname{{\\sf gr}}(K)} \\neq \\operatorname{id}$.\nThus, $\\widetilde{\\tau}$ is a  unitary graded involution\nin any characteristic.\nMoreover, for the graded fixed field $\\operatorname{{\\sf gr}}(K)^{\\widetilde{\\tau}}$ we have\n$\\operatorname{{\\sf gr}}(F) \\subseteq \\operatorname{{\\sf gr}}(K)^{\\widetilde{\\tau}} \\subsetneqq \\operatorname{{\\sf gr}}(K)$ and\n$[\\operatorname{{\\sf gr}}(K):\\operatorname{{\\sf gr}}(F)] =2$, so $\\operatorname{{\\sf gr}}(K)^{\\widetilde{\\tau}} = \\operatorname{{\\sf gr}}(F)$.\n\n\n\n\n\n\\begin{thm}\\label{involthm1}\nLet $(D,v)$ be a tame valued division algebra over a\nhenselian field $K$,\nwith $\\operatorname{char}(\\overline K)\\not = 2$.\n If  $\\tau$ is an involution of the first kind on $D$,  then\n$$\nK_1(D,\\tau)  \\ \\cong \\  K_1(\\operatorname{{\\sf gr}}(D),\\widetilde \\tau),\n$$\nand if  $\\tau$ is symplectic, then\n$$\nK_1\\operatorname{Spin}(D,\\tau)  \\ \\cong  \\ K_1\\operatorname{Spin}(\\operatorname{{\\sf gr}}(D),\\widetilde \\tau).\n$$\n\\end{thm}\n\n\n\n\\begin{proof}\n\n Let $\\rho\\colon D^* \\rightarrow \\operatorname{{\\sf gr}}(D)^*$ be the\ngroup epimorphism given in (\\ref{grmap}). Clearly\n$\\rho(S_{\\tau}(D))\n\\subseteq S_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D))$, so\n$\\rho (\\Sigma_{\\tau}(D))\n\\subseteq \\Sigma_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D))$.\nConsider the following diagram:\n\n\\begin{equation} \\label{exactdiagramfirst}\n\\begin{split}\n\\xymatrix{\n1 \\ar[r] &  (1+M_D) \\cap \\Sigma_{\\tau}(D)D' \\ar[r] \\ar[d]&\n\\Sigma_{\\tau}(D)D' \\ar[r]^-{\\rho} \\ar[d] &\n\\Sigma_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D))\\operatorname{{\\sf gr}}(D)' \\ar@{.>}[r] \\ar[d]& 1  \\\\\n1 \\ar[r] & (1+M_D)  \\ar[r] &\nD^*  \\ar[r]^-{\\rho} & \\operatorname{{\\sf gr}}(D)^*\n \\ar[r] & 1. }\n\\end{split}\n\\end{equation}\n\nThe top row of the diagram is exact. To see this, note that\n$\\rho(D')=\\operatorname{{\\sf gr}}(D)'$. Thus, it suffices to show that $\\rho$ maps\n$S_{\\tau}(D) \\cap D^{\\ast}$ onto $S_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D)) \\cap\n\\operatorname{{\\sf gr}}(D)^{\\ast}$. For this, take any $d \\in D^{\\ast}$ with $\\widetilde{d} =\n\\widetilde{\\tau}(\\widetilde{d})$.  Let ${b = \\frac{1}{2} (d+ \\tau(d)) \\in\nS_{\\tau}(D)}$. Since $v(b) = v(\\tau(b))$ and $\\widetilde{d} +\n\\widetilde{\\tau(d)} = 2 \\widetilde{d} \\neq 0$, ${\\widetilde{b} = \\frac{1}{2}\n(\\widetilde{d+ \\tau(d)}) = \\frac{1}{2} (\\widetilde{d}+\n\\widetilde{\\tau(d)}) = \\widetilde{d}}$. Since $\\tau$ on $D$ is an\ninvolution of the first kind, the index of $D$ is a power of $2$\n(\\cite[Th.~1, \\S 16]{draxl}). As ${\\operatorname{char}(\\overline K)\\not = 2}$, it follows\nthat the valuation is strongly tame, and by \\cite[Lemma~2.1]{haz},\n$$\n{1+M_D \\ = \\ (1+M_K)[D^*,1+M_D]   \\ \\subseteq  \\ \\Sigma_{\\tau}(D)D'}.\n$$\nTherefore,\nthe left vertical map is the identity map. It follows (for example\nusing the snake lemma) that $K_1(D,\\tau) \\cong K_1(\\operatorname{{\\sf gr}}(D),\\widetilde\n\\tau)$. The proof for  $K_1\\operatorname{Spin}$  when $\\tau$ is of symplectic type\nis similar.\n\\end{proof}\n\nThe key to proving the corresponding result for unitary\ninvolutions is the Congruence Theorem:\n\n\\begin{thm}[Congruence Theorem]\\label{congthm}\nLet $D$ be a tame  division algebra over a\nfield $K$ with henselian valuation $v$. Let\n$D^{(1)} = \\{ a\\in  D^*\\mid \\Nrd_D(a) = 1\\}$.\nThen,\n$$\nD^{(1)} \\, \\cap  \\, (1+M_D)  \\ \\subseteq  \\ [D^*,D^*].\n$$\n\\end{thm}\n\nThis theorem was proved by Platonov in \\cite{platonov} for\n$v$ a complete discrete valuation, and it was an essential tool\nin all his calculations of $\\SK$ for division rings.\nThe Congruence Theorem  was asserted by\nErshov in \\cite{ershov}  in the generality given here.\nA full proof is given in \\cite[Th.~ B.1]{hazwadsworth}.\n\n\n\n\\begin{prop}[Unitary Congruence Theorem]\\label{unitarycongthm}\nLet $D$ be a tame  division algebra over a\nfield~$K$ with henselian valuation $v$, and let\n $\\tau$ be a unitary involution  on $D$.  Let $F=K^\\tau$.\nIf $F$ is henselian with respect to  $v|_F$\nand $K$ is tamely ramified over $F$, then\n$$\n(1+M_D) \\, \\cap \\,\n\\Sigma'_{\\tau}(D)  \\ \\subseteq \\  \\Sigma_{\\tau}(D).\n$$\n\\end{prop}\n\\begin{proof}  The only published proof of this we know  is\n\\cite[Th.~4.9]{y}, which is just for the case $v$ discrete rank~1;\nthat proof is rather hard to follow, and appears to apply for other\n valuations only if $D$ is inertially split. Here we provide another\nproof, in full generality.\n\nWe use the well-known facts that\n\\begin{equation}\\label{nrd1+m}\n\\Nrd_D(1+M_D) \\ = \\ 1+M_K \\text{\\ \\ \\ and  \\ \\ \\ }\nN_{K\/F}(1+M_K)  \\ = \\  1+M_F.\n\\end{equation}\n(The second equation holds as $K$ is tamely ramified over $F$.)\nSee \\cite[Prop.~2]{ershov} or \\cite[Prop.~4.6, Cor.~4.7]{hazwadsworth}\nfor a proof.\n\nNow, take $m \\in M_D$ with $\\Nrd_D(1 +m) \\in F$. Then $\\Nrd_D(1+m) \\in F\n\\cap(1+M_K) = 1 +M_F$. By \\eqref{nrd1+m} there is $c \\in 1 +M_K$ with\n$\\Nrd_D(1 +m) = N_{K\/F}(c) = c \\tau(c)$, and there is $b \\in 1+M_D$\nwith $\\Nrd_D(b) = c$. Then,\n$$\n\\Nrd_D(b \\tau(b))  \\ = \\  c \\tau(c)  \\ = \\  N_{K\/F}(c)  \\ = \\  \\Nrd_D(1+m).\n$$\nLet $s = (1+m)(b \\tau(b))^{-1} \\in 1+M_D$. Since $\\Nrd_D(s) = 1$, by\nthe Congruence Theorem for $\\SK$, Th.~\\ref{congthm} above,\n$s \\in [D^{\\ast}, D^{\\ast}]\n\\subseteq \\Sigma_{\\tau}(D)$, (recall \\eqref{primeinsigma})\n. Since $b \\tau(b) \\in\nS_{\\tau}(D)$, we have $1+m = s(b \\tau(b)) \\in \\Sigma_{\\tau}(D)$.\n\\end{proof}\n\n\n\n\\begin{thm}\\label{involthm2}\nLet $D$ be a tame  division algebra over a\n field $K$ with henselian valuation $v$.\nLet $\\tau$ be a unitary involution  on $D$, and let $F=K^\\tau$.\nIf $F$ is henselian with respect to $v|_F$\nand $K$ is tamely ramified over $F$, then $\\tau$ induces a\nunitary graded involution $\\widetilde{\\tau}$ of $\\operatorname{{\\sf gr}}(D)$  with\n$\\operatorname{{\\sf gr}}(F)=\\operatorname{{\\sf gr}}(K)^{\\widetilde{\\tau}}$, and\n$$\n\\SK(D, \\tau)  \\ \\cong \\  \\SK(\\operatorname{{\\sf gr}}(D), \\widetilde{\\tau}).\n$$\n\\end{thm}\n\n\\begin{proof}\nThat\n$\\widetilde{\\tau}$ is a  unitary graded involution\non $\\operatorname{{\\sf gr}}(D)$ and $\\operatorname{{\\sf gr}}(F)=\\operatorname{{\\sf gr}}(K)^{\\widetilde \\tau}$  was\nalready observed (see the discussion before  Th.~\\ref{involthm1}).\nFor the canonical\nepimorphism $\\rho\\colon D^{\\ast} \\rightarrow \\operatorname{{\\sf gr}}(D)^{\\ast}$,\n$a \\mapsto \\widetilde{a}$, it follows from \\eqref{nrdrel} that\n$\\rho( \\Sigma'_{\\tau}(D) \\subseteq\n\\Sigma'_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D))$. Also, clearly $\\rho(S_{\\tau}(D))\n\\subseteq S_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D))$, so $\\rho (\\Sigma_{\\tau}(D))\n\\subseteq \\Sigma_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D))$. Thus, there is a\ncommutative diagram\n\\begin{equation} \\label{exactdiagram}\n\\begin{split}\n\\xymatrix{1 \\ar[r] & (1+M_D) \\cap \\Sigma_{\\tau}(D) \\ar[d] \\ar[r] &\n\\Sigma_{\\tau}(D) \\ar[d] \\ar[r]^-{\\rho} &\n\\Sigma_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D))\n\\ar[d] \\ar@{.>}[r] & 1 \\\\\n1 \\ar[r] &  (1+M_D) \\cap \\Sigma'_{\\tau}(D) \\ar[r] &\n\\Sigma'_{\\tau}(D) \\ar[r]^-{\\rho} &\n\\Sigma'_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D)) \\ar@{.>}[r] & 1,}\n\\end{split}\n\\end{equation}\nwhere the vertical maps are inclusions, and the left vertical map is\nbijective, by Prop.~\\ref{unitarycongthm} above.\n\n\nTo see that the bottom row of  diagram~\\eqref{exactdiagram} is exact at\n$\\Sigma'_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D))$, take $b \\in D$ with\n$\\Nrd_{\\operatorname{{\\sf gr}}(D)}(\\widetilde{b}) \\in \\operatorname{{\\sf gr}}(F)$. Let $c = \\Nrd_D(b) \\in\nK^{\\ast}$. Then $\\widetilde{c} = \\Nrd_{\\operatorname{{\\sf gr}}(D)} (\\widetilde{b}) \\in \\operatorname{{\\sf gr}}(F)$, so\n$\\widetilde{c} = \\widetilde{t}$ for some $t \\in F^{\\ast}$. Let $ u = c^{-1}\nt \\in 1 +M_K$. By \\eqref{nrd1+m} above, there is $d \\in 1+M_D$ with $\\Nrd_D(d)\n= u$. So, $\\Nrd_D(bd) = cu = t \\in F^{\\ast}$. Thus, $bd \\in\n\\Sigma'_{\\tau}(D)$ and $\\rho(bd) = \\widetilde{\\mspace{1mu} bd\\mspace{1mu} } = \\widetilde{b}$. This\ngives the claimed exactness, and shows that the bottom row of\ndiagram~\\eqref{exactdiagram} is exact.\n\nTo see that the top row of diagram~\\eqref{exactdiagram} is exact at\n$\\Sigma_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D))$, it suffices to show that $\\rho$ maps\n${S_{\\tau}(D) \\cap D^{\\ast}}$ onto $S_{\\widetilde{\\tau}}(\\operatorname{{\\sf gr}}(D)) \\cap\n\\operatorname{{\\sf gr}}(D)^{\\ast}$. For this, take any $d \\in D^{\\ast}$ with $\\widetilde{d} =\n\\widetilde{\\tau}(\\widetilde{d})$. If $\\operatorname{char}(\\overline{F}) \\neq 2$, as\nin the proof of Th.~\\ref{involthm1}, let $b =\n\\frac{1}{2} (d+ \\tau(d)) \\in S_{\\tau}(D)$. Since $v(b) =\nv(\\tau(b))$ and $\\widetilde{d} + \\widetilde{\\tau(d)} = 2 \\widetilde{d} \\neq\n0$, we have ${\\widetilde{b} = \\frac{1}{2} (\\widetilde{d+ \\tau(d)}) = \\frac{1}{2}\n(\\widetilde{d}+ \\widetilde{\\tau(d)}) = \\widetilde{d}}$. If $\\operatorname{char}(\\overline{F})\n= 2$, then $K$ is unramified over $F$, so $\\overline{K}$ is Galois over~\n$\\overline{F}$ with $[\\overline{K}: \\overline{F}]=2$, and the map\n$\\overline{\\tau}\\colon\n\\overline{K} \\rightarrow \\overline{K}$ induced by $\\tau$ is the nonidentity\n$\\overline{F}$-automorphism of $\\overline{K}$. Of course,\n$\\overline{K} = \\operatorname{{\\sf gr}}(K)_0$ and\n$\\overline{\\tau} = \\widetilde{\\tau}|_{\\operatorname{{\\sf gr}}(K)_0}$. Because $\\overline{K}$\nis separable\nover $\\overline{F}$, the trace $\\textrm{tr}_{\\overline{K}\/\\overline{F}}$ is\nsurjective, so\nthere is $r \\in V_K$ with $\\widetilde{r} + \\widetilde{\\tau}(\\widetilde{r}) = 1\n\\in \\operatorname{{\\sf gr}}(F)_0$. Let $c = rd + \\tau(rd) \\in S_{\\tau}(D)$. We have\n$\\widetilde{\\mspace{1mu} rd\\mspace{1mu} } = \\widetilde{r}\\widetilde{d}$ and\n$$\n\\widetilde{\\tau(rd)} \\, = \\,\n\\widetilde{\\tau}(\\widetilde{\\mspace{1mu} rd\\mspace{1mu}})\n \\, = \\,  \\widetilde{\\tau}(\\widetilde{r}\\widetilde{d})  \\, = \\,\n\\widetilde{\\tau}(\\widetilde{d})\\widetilde{\\tau}(\\widetilde{r})  \\, = \\,\n\\widetilde{\\tau}(\\widetilde{r})\\widetilde{d}.\n$$\n Since $v(rd) = v(\\tau(rd))$ and\n$\\widetilde{\\mspace{1mu} rd\\mspace{1mu} } + \\widetilde{\\tau(rd)} = \\widetilde{r}\\widetilde{d}\n+\\widetilde{\\tau}(\\widetilde{r})\\widetilde{d} = \\widetilde{d} \\neq 0$, we\nhave\n$\\widetilde{c} = \\widetilde{\\mspace{1mu} rd\\mspace{1mu} } + \\widetilde{\\tau(rd)} = \\widetilde{d}$. So,\nin all cases $\\rho(S_{\\tau}(D) \\cap D^{\\ast}) = S_{\\widetilde \\tau}(\\operatorname{{\\sf gr}}(D)) \\cap\n\\operatorname{{\\sf gr}}(D)^{\\ast}$, from which it follows that the bottom row of\ndiagram~\\eqref{exactdiagram}\nis exact. Since each row of~(\\ref{exactdiagram}) is  exact,\nwe have a right exact sequence of\ncokernels of the vertical maps, which yields the isomorphism of the\ntheorem.\n\\end{proof}\n\nHaving established the bridge between the unitary $K$-groups in\nthe graded setting and the non-graded henselian  case (Th.~\\ref{involthm1},\nTh.~\\ref{involthm2}), we\ncan deduce\nknown formulas in the literature for the  unitary Whitehead\ngroup of  certain valued division algebras,  by passing to\nthe graded setting.  The proofs are much easier than those\n previously available.\nWe will do this systematically for unitary  involutions\nin Section~\\ref{grsec}. Before we turn to that,  here is an example\nwith an involution of the first kind:\n\n\\begin{example}  Let $\\mathsf{E}$ be a graded division algebra over\nits center $\\mathsf{T}$ with an involution $\\tau$ of the first kind.\nIf $\\mathsf{E}$~is unramified over $\\mathsf{T}$, then, by using $\\mathsf{E}^*=\\mathsf{E}_0^*\\mathsf{T}^*$, it\nfollows easily  that\n\\begin{equation}\\label{kuni}\nK_1(\\mathsf{E},\\tau) \\ \\cong \\  K_1(\\mathsf{E}_0,\\tau|_{\\mathsf{E}_0}),\n\\end{equation} and, if $\\operatorname{char}(\\mathsf{E}) \\ne 2$ and $\\tau$ is symplectic,\n\\begin{equation}\nK_1\\operatorname{Spin}(\\mathsf{E}, \\tau) \\ \\cong \\  K_1\\operatorname{Spin}(\\mathsf{E}_0,\\tau|_{\\mathsf{E}_0}).\n\\end{equation}\nNow if $D$ is a tame and unramified division algebra\nover a henselian valued field and $D$ has an\n involution $\\tau$ of the first kind, then the associated\ngraded division ring $\\operatorname{{\\sf gr}}(D)$ is also unramified with the\ncorresponding graded involution $\\widetilde \\tau$ of the first kind;\nthen Th.~\\ref{involthm1} and (\\ref{kuni}) above show that\n$$\nK_1(D,\\tau)  \\ \\cong  \\ K_1(\\operatorname{{\\sf gr}}(D),\\widetilde{\\tau})\n \\ \\cong \\  K_1(\\operatorname{{\\sf gr}}(D)_0,\\tau|_{\\operatorname{{\\sf gr}}(D)_0}) \\ = \\ K_1(\\overline{D},\n\\overline{\\tau}),\n$$\nyielding a theorem of Platonov-Yanchevski\\u\\i~\\cite[Th. 5.11]{py85}\n(that $K_1(D,\\tau)\\cong K_1(\\overline{D},\n\\overline{\\tau})$ when $D$ is unramified over $K$ and\nthe valuation is henselian and discrete rank $1$.)\nSimilarly, when $\\operatorname{char}(\\overline D) \\ne 2$ and $\\tau$ is  symplectic,\n$$\nK_1\\operatorname{Spin}(D,\\tau)  \\ \\cong \\  K_1\\operatorname{Spin}(\\operatorname{{\\sf gr}}(D),\\widetilde{\\tau})\n \\ \\cong \\  K_1\\operatorname{Spin}(\\operatorname{{\\sf gr}}(D)_0,\\tau|_{\\operatorname{{\\sf gr}}(D)_0}) \\ = \\ K_1\\operatorname{Spin}(\\overline{D},\n\\overline{\\tau}).\n$$\n\\end{example}\n\n\\begin{remark}\\label{goodrem}\nWe have the following commutative diagram connecting  unitary $\\SK$ to\nnon-unitary $\\SK$,\nwhere $\\SH(D,\\tau)$ and $\\SH(D)$ are the cokernels of $\\Nrd\\circ(1-\\tau)$\nand $\\Nrd$ respectively (see diagram~\\eqref{unnrd}).\n\n\n\\begin{equation}\\label{goodd}\n\\begin{split}\n\\xymatrix{\n1 \\ar[r] & \\SK(D,\\tau) \\ar[r] \\ar[d] & D^*\/\\Sigma(D)\n\\ar[rr]^-{\\Nrd\\circ(1-\\tau)} \\ar[d]^{1-\\tau} &\n& K^* \\ar[r] \\ar[d]^{\\operatorname{id}} & \\SH(D,\\tau) \\ar[r]  \\ar[d]& 1\\\\\n1 \\ar[r] & \\SK(D) \\ar[r]  &    D^*\/D' \\ar[rr]^-{\\Nrd} &\n& K^* \\ar[r] & \\SH(D) \\ar[r] & 1.\n}\n\\end{split}\n\\end{equation}\n\nNow, let $D$ be a tame valued division algebra with  center $K$ and with\na unitary involution $\\tau$, such that the valuation\nrestricts to a henselian valuation on $F=K^\\tau$. By\nTh.~\\ref{involthm2}, $\\SK(D,\\tau)\\cong \\SK(\\operatorname{{\\sf gr}}(D),\\widetilde \\tau)$\nand by \\cite[Th.~4.8, Th.~4.12]{hazwadsworth}, $\\SK(D)\\cong \\SK(\\operatorname{{\\sf gr}}(D))$\nand $\\SH(D)\\cong\\SH(\\operatorname{{\\sf gr}}(D))$. However, $\\SH(D,\\tau)$ is not stable\nunder ``valued filtration'', i.e.,\n$\\SH(D,\\tau)\\not \\cong \\SH(\\operatorname{{\\sf gr}}(D),\\widetilde \\tau)$. In fact\nusing~(\\ref{grmap}), we can build a commutative diagram with exact rows,\n\\begin{equation*}\n\\begin{split}\n\\xymatrix{\n1 \\ar[r] & (1+M_K)\\cap \\Nrd(D^*)^{1-\\tau} \\ar[r] \\ar@{^{(}->}[d] & \\Nrd(D^*)^{1-\\tau} \\ar[r] \\ar@{^{(}->}[d]& \\Nrd\\big(\\operatorname{{\\sf gr}}(D)^*\\big)^{1-\\widetilde \\tau} \\ar[r]\\ar@{^{(}->}[d] & 1\\\\\n1 \\ar[r] & 1+M_K \\ar[r] & K^* \\ar[r]  & \\operatorname{{\\sf gr}}(K)^* \\ar[r]  & 1,\n}\n\\end{split}\n\\end{equation*}\nwhich induces the exact sequence\n$$\n1\\longrightarrow (1+M_K)\\big\/\\big( (1+M_K)\\cap\\Nrd(D^*)^{1-\\tau}\\big)\n\\longrightarrow\n\\SH(D,\\tau) \\longrightarrow \\SH(\\operatorname{{\\sf gr}}(D),\\widetilde \\tau) \\longrightarrow 1.\n$$\nBy considering the norm $N_{K\/F}\\colon K^* \\rightarrow F^*$, we clearly have\n$\\Nrd(D^*)^{1-\\tau}\n\\subseteq \\ker N_{K\/F}$. However, by \\eqref{nrd1+m},\n${N_{K\/F}\\colon 1+M_K\\rightarrow 1+M_F}$ is surjective, which shows that\n$1+M_K\\big\/\\big( (1+M_K)\\cap\\Nrd(D^*)^{1-\\tau}\\big)$ is not trivial and thus\n${\\SH(D,\\tau)\\not \\cong \\SH(\\operatorname{{\\sf gr}}(D),\\widetilde \\tau)}$.\n\\end{remark}\n\n\\section{Graded Unitary $\\SK$ Calculus}\\label{grsec}\n\nLet $\\mathsf{E}$ be a graded division algebra over its center $\\mathsf{T}$\nwith a unitary graded involution $\\tau$,  and\nlet $\\mathsf{R}=\\mathsf{T}^\\tau$.\nSince $[\\mathsf{T}:\\mathsf{R}]=2=[\\mathsf{T}_0:\\mathsf{R}_0]\\,|\\Gamma_\\mathsf{T}:\\Gamma_\\mathsf{R}|$,\nthere are  just two possible cases:\n\n\n\\begin{itemize}\n\\item[$\\bullet$] $\\mathsf{T}$ is totally ramified over\n$\\mathsf{R}$, i.e., $|\\Gamma_\\mathsf{T}:\\Gamma_\\mathsf{R}|=2$\n\n\\item [$\\bullet$] $\\mathsf{T}$ is unramfied over\n$\\mathsf{R}$, i.e., $|\\Gamma_\\mathsf{T}:\\Gamma_\\mathsf{R}|=1$.\n\\end{itemize}\n\nWe will consider $\\SK(\\mathsf{E},\\tau)$ in these two cases separately\nin \\S\\ref{totram} and \\S\\ref{unram}.\n\nThe following notation will be used throughout this section and the next:\nLet $\\tau'$ be another involution on $\\mathsf{E}$.  We write\n$\\tau' \\sim \\tau$ if\n$\\tau'|_{Z(\\mathsf{E})}=\\tau|_{Z(\\mathsf{E})}$.  For $t \\in \\mathsf{E}^*$,\nlet $\\varphi_t$ denote  the map from $E$ to $E$ given by conjugation by $t$,\ni.e., $\\varphi_t(x)=txt^{-1}$.\nLet $ \\Sigma_0=\\Sigma_\\tau \\cap \\mathsf{E}^*_0$ and\n$\\Sigma'_0=\\Sigma_\\tau'\\cap \\mathsf{E}_0^*$.\n\n\nWe first  collect some facts which will be used below.\nThey all follow by easy calculations.\n\n\n\\begin{remarks}\\hfill\n\\begin{enumerate}[\\upshape(i)]\\label{easyob}\n\\item \\label{one} We have $\\tau' \\sim \\tau$ if and only if there is a\n$t\\in \\mathsf{E}^*$ with $\\tau(t)=t$ and $\\tau'=\\tau \\varphi_t$.\n(The proof is analogous to the ungraded version given, e.g. in\n\\cite[ Prop.~2.18]{kmrt}.)\n\n\\smallskip\n\n\\item \\label{two} If $\\tau'\\sim \\tau$, then $\\Sigma_{\\tau'}=\\Sigma_{\\tau}$\nand $\\Sigma'_{\\tau'}=\\Sigma'_{\\tau}$; thus $\\SK(\\mathsf{E},\\tau')=\\SK(\\mathsf{E},\\tau)$.\n(See~\\cite[Lemma~1]{yin} for the analogous ungraded result.)\n\\smallskip\n\\item \\label{three} For any $s \\in \\mathsf{E}^*$, we have\n$\\tau \\varphi_s=\\varphi_{\\tau(s)^{-1}}\\tau$.  Hence, $\\tau \\varphi_s$ is an involution\n(necessarily $\\sim \\tau$) if and only if $\\tau \\varphi_s=\\varphi_{s^{-1}}\\tau$\nif and only if $\\tau(s)\/s \\in \\mathsf{T}$.\n\\smallskip\n\\item \\label{four} If $s \\in \\mathsf{E}^*_\\gamma$ and $\\tau(s)=s$, then\n$\\Sigma_\\tau' \\cap \\mathsf{E}_\\gamma= s \\Sigma'_0$ and\n$S_\\tau\\cap \\mathsf{E}_\\gamma=s(S_{\\tau_s}\\cap \\mathsf{E}_0)$ where $\\tau_s=\\tau \\varphi_s$.\n\\smallskip\n\\end{enumerate}\n\\end{remarks}\n\n\n\\subsection{$\\mathsf{T}\/\\mathsf{R}$ totally ramified} \\label{totram}\nLet $\\mathsf{E}$ be a graded division algebra with a unitary graded involution $\\tau$\n such that $\\mathsf{T}=Z(\\mathsf{E})$ is totally\nramified over $\\mathsf{R}=\\mathsf{T}^\\tau$. In this section we will show\nthat $\\SK(\\mathsf{E},\\tau)=1$. Note that  the assumption that $\\mathsf{T}\/\\mathsf{R}$ is\ntotally ramified implies that $\\operatorname{char}(\\mathsf{T}) \\ne 2$.  For, if\n$\\operatorname{char}(\\mathsf{T}) = 2$ and $\\mathsf{T}$ is totally ramified over a graded\nsubfield $\\mathsf{R}$ with $[\\mathsf{T}:\\mathsf{R}] = 2$, then for any $x\\in \\mathsf{T}^*\\setminus\n\\mathsf{R}^*$, we have $\\deg(x^2) \\in \\Gamma_R$, so $x^2\\in \\mathsf{R}$;  thus, $\\mathsf{T}$ is\npurely inseparable over $\\mathsf{R}$.  That cannot happen here,  as\n$\\tau|_\\mathsf{T}$ is a nontrivial $\\mathsf{R}$-automorphism of~$\\mathsf{T}$.\n\n\\begin{lemma}\\label{six}\nIf $\\mathsf{T}$ is totally ramified over $\\mathsf{R}$, then\n$\\tau\\sim\\tau'$ for some graded involution $\\tau'$, where\n$\\tau'|_{\\mathsf{E}_0}$  is of the first kind.\n\\end{lemma}\n\n\\begin{proof}\nLet $Z_0=Z(\\mathsf{E}_0)$.\nSince $\\mathsf{T}$ is totally ramified over $\\mathsf{R}$, $\\mathsf{T}_0=\\mathsf{R}_0$, so $\\tau|_{Z_0}\n\\in \\operatorname{Gal}(Z_0\/\\mathsf{T}_0)$. Since the map\n$\\Theta_\\mathsf{E}\\colon\\Gamma_\\mathsf{E} \\rightarrow \\operatorname{Gal}(Z_0\/\\mathsf{T}_0)$ is surjective\n(see  \\eqref{surj}), there is $ \\gamma\\in \\Gamma_\\mathsf{E}$\nwith $\\Theta_\\mathsf{E}(\\gamma)=\\tau|_{Z_0} $. Choose $y \\in \\mathsf{E}^*_\\gamma$ with\n$\\tau(y)=\\pm y$. Then set $\\tau'=\\tau\\varphi_{y^{-1}}$.\n\\end{proof}\n\n\\begin{example}\\label{trex} Here is a construction of  examples of\n graded division algebras\n$\\mathsf{E}$ with unitary  graded involution~$\\tau$ with $E$ totally ramified over\n$Z(\\mathsf{E})^\\tau$. We will see below that these are all such examples.\nLet $\\mathsf{R}$ be any  graded field\nwith $\\operatorname{char}(\\mathsf{R})\\ne 2$, and let $\\mathsf{A}$ be a graded division algebra\nwith center $\\mathsf{R}$, such that $\\mathsf{A}$~is totally ramified over $\\mathsf{R}$ with\n$\\exp(\\Gamma_\\mathsf{A}\/\\Gamma_\\mathsf{R}) = 2$.  Let $\\mathsf{T}$ be a graded field\nextension of $\\mathsf{R}$ with $[\\mathsf{T}:\\mathsf{R}] = 2$, $\\mathsf{T}$ totally ramified over $\\mathsf{R}$,\nand $\\Gamma_\\mathsf{T}\\cap \\Gamma_\\mathsf{A} = \\Gamma_\\mathsf{R}$.  Let $\\mathsf{E} = \\mathsf{A}\\otimes_\\mathsf{R}\n\\mathsf{T}$,  which is a graded central simple algebra over $\\mathsf{T}$, as\n$\\mathsf{A}$ is graded central simple over $\\mathsf{R}$, by \\cite[Prop.~1.1]{hwcor}.\nBut because $\\Gamma_\\mathsf{T}\\cap \\Gamma_\\mathsf{A} = \\Gamma_\\mathsf{R}$, we have\n$\\mathsf{E}_0 = \\mathsf{A}_0 \\otimes_{\\mathsf{R}_0} \\mathsf{T}_0 = \\mathsf{R}_0 \\otimes_{\\mathsf{R}_0} \\mathsf{R}_0 = \\mathsf{R}_0$.\nSince $\\mathsf{E}_0$ is a division ring, $\\mathsf{E}$ must be a graded division ring, which is\ntotally ramified over $\\mathsf{R}$, as $\\mathsf{E}_0 = \\mathsf{R}_0$.  Now, because $\\mathsf{A}$ is\ntotally ramified over $\\mathsf{R}$, we  have $\\exp(\\mathsf{A}) = \\exp(\\Gamma_\\mathsf{A}\/\n\\Gamma_\\mathsf{R}) = 2$, and $\\mathsf{A}= \\mathsf{Q}_1\\otimes_\\mathsf{R} \\ldots \\otimes_\\mathsf{R}\\mathsf{Q}_m$, where\neach $\\mathsf{Q}_i$ is a  graded symbol algebra of degree at most $2$, i.e., a\ngraded quaternion algebra.  Let $\\sigma_i$ be a graded involution of the\nfirst kind on $\\mathsf{Q}_i$ (e.g., the canonical symplectic graded involution),\n and let $\\rho$ be the nonidentity $\\mathsf{R}$-automorphism of $\\mathsf{T}$.\nThen, $\\sigma = \\sigma_1\\otimes \\ldots \\otimes \\sigma_m$ is a graded involution\nof the first  kind on $\\mathsf{A}$, so $\\sigma\\otimes \\rho$  is a unitary graded\ninvolution on~$\\mathsf{E}$, with $\\mathsf{T}^\\tau = \\mathsf{R}$.\n\n\n\\end{example}\n\n\\begin{prop}\\label{total}\nIf $\\mathsf{E}$  is totally ramified over $\\mathsf{R}$, and $\\mathsf{E} \\ne  \\mathsf{T}$,\nthen $\\Sigma_{\\tau}=\\mathsf{E}^*$, so\n$\\SK(\\mathsf{E},\\tau)=1$.  Furthermore, $\\mathsf{E}$ and $\\tau$ are as described\nin Ex.~\\ref{trex}.\n\\end{prop}\n\n\\begin{proof}\nWe have $\\mathsf{E}_0=\\mathsf{T}_0=\\mathsf{R}_0$.\nFor any  $\\gamma \\in \\Gamma_\\mathsf{E}$, there is a nonzero $a\\in \\mathsf{E}_\\gamma$\nwith $\\tau(a) = \\epsilon a$ where $\\epsilon = \\pm 1$.  Then,\nfor any $b\\in \\mathsf{E}_\\gamma$, $b = ra$ for some $r\\in  \\mathsf{E}_0 = \\mathsf{R}_0$.\nSince $r$ is  central and symmetric, $\\tau(b) = \\epsilon b$.\nThus,  every element of $\\mathsf{E}^*$ is symmetric or\nskew-symmetric. Indeed, fix any $t\\in \\mathsf{T}^*\\setminus \\mathsf{R}^*$.\nThen $\\tau(t) \\ne t$, as $t\\notin \\mathsf{R}^*$.  Hence, $\\tau(t) =\n-t$.  Since $t$ is central and skew-symmetric, every\n$a\\in \\mathsf{E}^* $ is symmetric  iff $ta$ is  skew-symmetric.\nThus,  $\\mathsf{E}^* = S_\\tau^* \\cup tS_\\tau^*$.  To see that\n$\\Sigma_\\tau = \\mathsf{E}^*$, it suffices to show that $t\\in  \\Sigma_\\tau$.\nTo see this, take any $c,d\\in \\mathsf{E}^*$ with $dc\\ne cd$.  (They exist, as\n$\\mathsf{E} \\ne \\mathsf{T}$.)  By replacing $c$ (resp.~$d$) if necessary by\n$tc$ (resp.~$td$), we may assume that $\\tau(c) = c$ and $\\tau(d) = d$.\nThen, $dc = \\tau(cd) = \\epsilon cd$, where $\\epsilon = \\pm 1$;\nsince $dc\\ne cd$, $\\epsilon = -1$;  hence $\\tau(tcd) = tcd$.\nThus, $t = (tcd) c^{-1}  d^{-1} \\in \\Sigma_\\tau(\\mathsf{E})$, completing the proof\nthat $\\Sigma_\\tau(\\mathsf{E})  = \\mathsf{E}^*$.\n\nFor $\\gamma \\in \\Gamma_\\mathsf{E}$, let $\\overline \\gamma = \\gamma+\\Gamma_\\mathsf{T} \\in\n\\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T}$.\nTo see the structure of $\\mathsf{E}$, recall that as $\\mathsf{E}$ is totally ramified\nover~$\\mathsf{T}$ there is a well-defined nondegenerate $\\mathbb Z$-bilinear symplectic\npairing $\\beta\\colon (\\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T}) \\times\\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T})\n\\to \\mathsf{E}_0^*$ given by $\\beta(\\overline \\gamma, \\overline \\delta) = y_\\gamma y_\\delta\ny_\\gamma^{-1}y_\\delta^{-1}$ for any nonzero $y_\\gamma\\in  \\mathsf{E}_\\gamma$,\n$y_\\delta\\in \\mathsf{E}_\\delta$.  The computation above for $c$ and $d$ shows\nthat $\\operatorname{im} (\\beta) = \\{\\pm 1\\}$.  Since the pairing $\\beta$ is nondegenerate\nby \\cite[Prop.~2.1]{hwcor}  there is a symplectic base  of $\\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T}$, i.e., a subset\n$\\{\\overline \\gamma_1, \\overline \\delta_1, \\ldots , \\overline \\gamma_m, \\overline \\delta_m\\}$   of\n$\\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T}$ such that $\\beta(\\overline \\gamma_i, \\overline \\delta_i) = -1$\nwhile $\\beta(\\overline \\gamma_i, \\overline \\gamma_j)  = \\beta(\\overline \\delta_i, \\overline \\delta_j)\n=1$ for all $i,j$, and $\\beta(\\overline \\gamma_i, \\overline \\delta_j) = 1$ whenever\n$i\\ne j$, and $\\Gamma_\\mathsf{E} = \\langle \\gamma_1, \\delta_1,\\ldots,\n\\gamma_m, \\delta_m\\rangle + \\Gamma_\\mathsf{T}$.  Choose any nonzero $\\mathbf{i}_i\\in\n\\mathsf{E}_{\\gamma_i}$ and $\\mathbf{j}_i\\in \\mathsf{E}_{\\delta_i}$.  The properties of the\n$\\overline\\gamma_i, \\overline \\delta_i$  under $\\beta$ translate  to:\n$\\mathbf{i}_i \\mathbf{j}_i = - \\mathbf{j}_i \\mathbf{i}_i$ while $\\mathbf{i}_i \\mathbf{i}_j =  \\mathbf{i}_j \\mathbf{i}_i$\nand $\\mathbf{j}_i \\mathbf{j}_j =  \\mathbf{j}_j \\mathbf{j}_i$ for all $i,j$, and\n$\\mathbf{i}_i \\mathbf{j}_j =  \\mathbf{j}_j \\mathbf{i}_i$ whenever $i \\ne j$. Since $\\beta(\n\\overline {2\\gamma_i}, \\overline \\eta) = 1$ for all $i$ and all $\\eta\\in \\Gamma_\\mathsf{E}$,\neach~$\\mathbf{i}_i^2$ is central in $\\mathsf{E}$.  But also $\\tau(\\mathbf{i}_i^2) = \\mathbf{i}_i^2$,\nas $\\tau(\\mathbf{i}_i) = \\pm \\mathbf{i}_i$. So, each $\\mathbf{i}_i^2 \\in \\mathsf{R}^*$, and likewise each\n$\\mathbf{j}_i^2\\in \\mathsf{R}^*$.  Let\n$\\mathsf{Q}_i = \\mathsf{R}\\text{-span}\\{1, \\mathbf{i}_i,  \\mathbf{j}_i,  \\mathbf{i}_i\\mathbf{j}_i\\}$ in~$\\mathsf{E}$.\nThe relations on the $\\mathbf{i}_i, \\mathbf{j}_i$ show that each $\\mathsf{Q}_i$ is a\ngraded quaternion algebra over $\\mathsf{R}$, and the distinct~$\\mathsf{Q}_i$ centralize\neach other in $\\mathsf{E}$.    Since each $\\mathsf{Q}_i$ is graded central simple over $\\mathsf{R}$,\n$\\mathsf{Q}_1\\otimes_\\mathsf{R} \\ldots \\otimes_\\mathsf{R}\\mathsf{Q}_m$ is graded  central  simple over\n$\\mathsf{R}$\nby \\cite[Prop.~1.1]{hwcor}.  Let $\\mathsf{A} = \\mathsf{Q}_1\\ldots \\mathsf{Q}_m \\subseteq \\mathsf{E}$.  The graded\n$\\mathsf{R}$-algebra epimorphism $\\mathsf{Q}_1\\otimes_\\mathsf{R} \\ldots \\otimes_\\mathsf{R}\\mathsf{Q}_m \\to\n\\mathsf{A}$ must be an isomorphism, as the domain is graded simple.\nIf $\\Gamma_\\mathsf{T} \\subseteq \\Gamma_\\mathsf{A}$, then $\\mathsf{T} \\subseteq \\mathsf{A}$,\nsince $\\mathsf{E}$ is totally ramified over $\\mathsf{R}$.  But this cannot occur, as\n$\\mathsf{T}$ centralizes $\\mathsf{A}$ but $\\mathsf{T}\\supsetneqq \\mathsf{R} = Z(\\mathsf{A})$.  Hence,\nas $|\\Gamma_\\mathsf{T}:\\Gamma_\\mathsf{R}| = 2$, we must have $\\Gamma_\\mathsf{T} \\cap \\Gamma_\\mathsf{A}\n = \\Gamma_\\mathsf{R}$.\nThe graded $\\mathsf{R}$-algebra homomorphism $\\mathsf{A} \\otimes_\\mathsf{R} \\mathsf{T} \\to\n\\mathsf{E}$ is injective since its domain is graded simple, by \\cite[Prop.~1.1]{hwcor};\nit is also surjective, since $\\mathsf{E}_0 = \\mathsf{R}_0 \\subseteq \\mathsf{A}\\otimes_\\mathsf{R} \\mathsf{T}$\nand $\\Gamma_{\\mathsf{A} \\otimes_\\mathsf{R} \\mathsf{T}} \\supseteq \\langle \\gamma_1, \\delta_1,\\ldots,\n\\gamma_m, \\delta_m\\rangle +\\Gamma_\\mathsf{T} = \\Gamma_\\mathsf{E}$.  Clearly,\n$\\tau = \\tau|_\\mathsf{A} \\otimes \\tau|_\\mathsf{T}$.\n\\end{proof}\n\n\n\n\n\n\n\\begin{prop}\\label{completely}\nIf $\\mathsf{E} \\ne \\mathsf{T}$ and $\\mathsf{T}$  is totally ramified over $\\mathsf{R}$, then\n$\\Sigma_{\\tau}=\\mathsf{E}^*$, so $\\SK(\\mathsf{E},\\tau)=1$.\n\\end{prop}\n\n\n\\begin{proof}\nThe case where $\\mathsf{E}_0 = \\mathsf{T}_0$ was covered by Prop.~\\ref{total}.\nThus, we may assume that $\\mathsf{E}_0 \\supsetneqq\\mathsf{T}_0$.\nBy Lemma~\\ref{six} and Remark~\\ref{easyob}(\\ref{two}), we can assume\nthat $\\tau|_{\\mathsf{E}_0}$ is of the first kind. Further, we can assume that\n$\\mathsf{E}_0^*=\\Sigma_{\\tau|_{\\mathsf{E}_0}}(\\mathsf{E}_0).$\nFor, if $\\tau|_{\\mathsf{E}_0}$\nis symplectic, take any $a\\in \\mathsf{E}_0^*$ with $\\tau(a)=-a$, and let\n$\\tau'=\\tau\\varphi_a$.\nThen, $\\tau' \\sim \\tau$ (see Remark~\\ref{easyob}(\\ref{three})).\n Also, $\\tau'|_{Z(\\mathsf{E}_0)}=\\tau|_{Z(\\mathsf{E}_0)}$,  as $a \\in \\mathsf{E}_0$ and so\n$\\varphi_a |_{Z(\\mathsf{E}_0)}=\\operatorname{id}$. Therefore, $\\tau'|_{\\mathsf{E}_0}$ is of the first kind.\nBut as $\\tau(a)=-a$, $\\tau'|_{\\mathsf{E}_0}$ is orthogonal.\nThus $\\mathsf{E}_0^*=\\Sigma_{\\tau'|_{\\mathsf{E}_0}}(\\mathsf{E}_0)$, as noted at the beginning of\n\\S\\ref{firstk}. Now replace $\\tau$ by $\\tau'$.\n\nWe consider two cases.\n\n{Case I.} Suppose for each $\\gamma\\in \\Gamma_\\mathsf{E}$ there is\n$x_\\gamma \\in \\mathsf{E}^*_\\gamma$ such that $\\tau(x_\\gamma)=x_\\gamma$.\n Then, $\\mathsf{E}^* = \\bigcup_{\\gamma \\in \\Gamma_\\mathsf{E}}\\mathsf{E}_0^*x_\\gamma\n\\subseteq \\Sigma_\\tau(\\mathsf{E})$, as desired.\n\n{Case II.} Suppose there is $\\gamma\\in \\Gamma_\\mathsf{E}$ with\n$\\mathsf{E}_\\gamma \\cap S_\\tau=0$. Then $\\tau(d)=-d$ for each $d\\in \\mathsf{E}_\\gamma$.\nFix $t \\in \\mathsf{E}^*_\\gamma$. For any $a \\in \\mathsf{E}_0$, we have $ta \\in \\mathsf{E}_\\gamma$;\n so,\n$-ta=\\tau(ta)=\\tau(a)\\tau(t)=-\\tau(a)t$.  That is,\n\\begin{equation}\\label{localeq}\n\\tau(a) \\, = \\, \\varphi_t(a)\\quad\\text{for all }a \\in \\mathsf{E}_0.\n\\end{equation}\nLet $\\tau''=\\tau\\varphi_t$, which is a unitary involution on $\\mathsf{E}$ with\n$\\tau''\\sim\\tau$ (see Remark~\\ref{easyob}(\\ref{three})). But,\n$\\tau''(a)=a$ for all\n$a \\in \\mathsf{E}_0$, i.e., $\\tau''|_{\\mathsf{E}_0}=\\operatorname{id}$. This implies that $\\mathsf{E}_0$ is\na field. Replace $\\tau$ by $\\tau''$. The rest of the argument uses this\nnew $\\tau$.\nSo $\\tau|_{\\mathsf{E}_0}=\\operatorname{id}$. If we are now in\nCase I for this $\\tau$, then we are done by Case I. So, assume we are\nin Case II.\nTake any $\\gamma\\in \\Gamma_\\mathsf{E}$ with\n$\\mathsf{E}_\\gamma \\cap S_\\tau=0$.  For any nonzero $t\\in \\mathsf{E}_\\gamma$,\nequation~(\\ref{localeq}) applies to $t$, showing $\\varphi_t(a)=\\tau(a)=a$\nfor all $a \\in \\mathsf{E}_0$; hence for the map $\\Theta_\\mathsf{E}$ of\n\\eqref{surj}, $\\Theta_\\mathsf{E}(\\gamma) = \\operatorname{id}_{\\mathsf{E}_0}$. But recall that\n$\\mathsf{E}_0$~is Galois over $\\mathsf{T}_0$ and $\\Theta_\\mathsf{E}\\colon \\Gamma_\\mathsf{E}\\to\n\\operatorname{Gal}(\\mathsf{E}_0\/\\mathsf{T}_0)$ is surjective.  Since $\\mathsf{E}_0 \\ne \\mathsf{T}_0$,\n  there is $\\delta \\in \\Gamma_\\mathsf{E}$ with\n$\\Theta_\\mathsf{E}(\\delta) \\ne \\operatorname{id}$.  Hence, there must be some\n$s\\in \\mathsf{E}_\\delta^*\\cap S_\\tau$.  Likewise, since $\\Theta_\\mathsf{E}(\\gamma-\\delta)\n= \\Theta_\\mathsf{E}(\\gamma) \\Theta_\\mathsf{E}(\\delta)^{-1} \\ne \\operatorname{id}$, there\nis some $r\\in \\mathsf{E}_{\\gamma-\\delta}^*\\cap S_\\tau$.  Then,\nas $rs\\in \\mathsf{E}_\\gamma^*$, we have $\\mathsf{E}_\\gamma^*  = \\mathsf{E}_0^*rs \\subseteq\n\\Sigma_\\tau$.  This is true for every $\\gamma$  with $\\mathsf{E}_\\gamma\\cap S_\\tau\n= 0$.    But for any other $\\gamma\\in \\Gamma_\\mathsf{E}$, there is an $x_\\gamma$\nin  $\\mathsf{E}_\\gamma^* \\cap S_\\tau$; then $\\mathsf{E}_\\gamma^* = \\mathsf{E}_0^*x_\\gamma \\subseteq\n\\Sigma_\\tau$. Thus, $\\mathsf{E}^* = \\bigcup_{\\gamma\\in \\Gamma_\\mathsf{E}}\\mathsf{E}_\\gamma^*\n\\subseteq \\Sigma_\\tau$.\n \\end{proof}\n\n\n\\subsection{$\\mathsf{T}\/\\mathsf{R}$    unramified} \\label{unram}\n\nLet $\\mathsf{E}$ be a graded division algebra with a unitary involution\n $\\tau$ such that $\\mathsf{T}=Z(\\mathsf{E})$ is unramified over $\\mathsf{R}=\\mathsf{T}^\\tau$.\nIn this subsection, we will give a general formula for $\\SK(\\mathsf{E},\\tau)$ in\nterms of data in $\\mathsf{E}_0$.\n\n\\begin{lemma}\\label{unramfacts}\nSuppose $\\mathsf{T}$ is unramified over $\\mathsf{R}$.  Then,\n\\begin{enumerate}[\\upshape (i)]\n \\item \\label{seven}\nEvery $\\mathsf{E}_\\gamma$ contains both nonzero symmetric\nand skew symmetric elements.\n\n\\smallskip\n\n  \\item  \\label{eight}\n$Z(\\mathsf{E}_0)$ is a generalized dihedral extension for\n$\\mathsf{T}_0$ over\n$\\mathsf{R}_0$ $($see Def.~\\ref{gendi}$)$.\n\n\\smallskip\n\n  \\item  \\label{nine}\nIf $\\mathsf{T}$ is unramified over $\\mathsf{R}$, then\n$\\SK(\\mathsf{E},\\tau)=\\Sigma'_0\/\\Sigma_0$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\\hfill\n\n(i) If $\\operatorname{char}(\\mathsf{E})=2$, it is easy to see that every\n$\\mathsf{E}_\\gamma$ contains a symmetric element (which is also  skew\nsymmetric)  regardless of any assumption on $\\mathsf{T}\/\\mathsf{R}$.\nLet $\\operatorname{char}(\\mathsf{E})\\not=2$.\nSince $[\\mathsf{T}_0:\\mathsf{R}_0]=2$ and $\\mathsf{R}_0=\\mathsf{T}_0^\\tau$,  there is\n$ c\\in \\mathsf{T}_0$ with $\\tau(c)=-c$. Now there is\n$t\\in \\mathsf{E}_\\gamma$, $t\\not =0$, with $\\tau(t)=\\epsilon t$ where\n$\\epsilon=\\pm 1$. Then $\\tau(c t)=-\\epsilon c t $.\n\n(ii) Let $G=\\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{R}_0)$ and $H=\\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{T}_0)$.\nNote that $[G:H]=2$. Since $\\tau$ is unitary,\n$\\tau |_{Z(\\mathsf{E}_0)} \\in G\\setminus H$. We will denote\n$\\tau |_{Z(\\mathsf{E}_0)}$ by $\\overline \\tau$ and will show that for\nany $h\\in H$, $(\\overline \\tau h)^2=1$. By (\\ref{surj}),\n$\\Theta_\\mathsf{E}\\colon \\Gamma_\\mathsf{E} \\rightarrow \\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{T}_0)$ is onto,\nso there is $\\gamma \\in \\Gamma_\\mathsf{E}$,  such that $\\Theta_\\mathsf{E}(\\gamma)=h$.\nAlso by (\\ref{seven}),\nthere is an $x\\in \\mathsf{E}_\\gamma^*$ with $\\tau(x)=x$.\nThen $\\tau\\varphi_x$ is an involution,  where $\\varphi_x$ is  conjugation by $x$;\ntherefore, $\\tau \\varphi_x|_{Z(\\mathsf{E}_0)} \\in G$ has order $2$.\nBut $\\varphi_x|_{Z(\\mathsf{E}_0)}=\\Theta_\\mathsf{E}(\\gamma)=h$. Thus\n$(\\overline \\tau h)^2=1$.\n\n(iii) By~(\\ref{seven}), for each $\\gamma \\in \\Gamma_\\mathsf{E}$, there is\n$s_\\gamma \\in \\mathsf{E}_\\gamma$,  $s_\\gamma\\not =0$, with\n$\\tau(s_\\gamma)=s_\\gamma$. By Remark~\\ref{easyob}\\eqref{four},\n${\\Sigma_\\tau'=\\bigcup_{\\gamma \\in \\Gamma_\\mathsf{E}} s_\\gamma \\Sigma'_0}$.\nSince each $s_\\gamma \\in S_\\tau \\subseteq \\Sigma_\\tau$, the injective map\n$\\Sigma'_0\/\\Sigma_0 \\rightarrow \\Sigma_\\tau'\/\\Sigma_\\tau$ is an isomorphism.\n\\end{proof}\n\nTo simplify notation in the next theorem, let $\\overline \\tau =\n\\tau|_{Z(\\mathsf{E}_0)} \\in \\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{R}_0)$, and\nfor any $h \\in \\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{T}_0)$,\n write $\\Sigma_{h \\overline \\tau}(\\mathsf{E}_0)$ for $\\Sigma_\\rho(\\mathsf{E}_0)$ for any unitary\ninvolution $\\rho$ on $\\mathsf{E}_0$ such that $\\rho|_{Z(\\mathsf{E}_0)}=h\\overline \\tau$.\nThis is well-defined, independent of the choice of $\\rho$, by the ungraded\nanalogue of Remark\\eqref{easyob}\\eqref{two}.\n\n\\begin{theorem}\\label{msem}\nLet $\\mathsf{E}$ be a graded division algebra with center $\\mathsf{T}$, with a unitary\ngraded involution $\\tau$, such that $\\mathsf{T}$~is unramified over $\\mathsf{R}=\\mathsf{T}^\\tau$.\nFor each $\\gamma \\in \\Gamma_\\mathsf{E}$ choose a nonzero\n$x_\\gamma \\in S_\\tau \\cap \\mathsf{E}_\\gamma$. Let $H=\\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{T}_0)$.\nThen,\n$$\n\\SK(\\mathsf{E},\\tau) \\ \\cong  \\ (\\Sigma'_\\tau \\cap \\mathsf{E}_0) \\big \/\n(\\Sigma_\\tau \\cap \\mathsf{E}_0),\n$$\nwith\n\\begin{equation}\\label{ersh}\n\\Sigma'_\\tau \\cap \\mathsf{E}_0  \\ = \\\n\\big \\{ a \\in \\mathsf{E}_0^*\\mid\nN_{Z(\\mathsf{E}_0)\/\\mathsf{T}_0}\\Nrd_{\\mathsf{E}_0}(a)^\\partial\\in \\mathsf{R}_0 \\big \\},\n\\textrm{  \\ \\ \\ where \\ \\ \\ } \\partial \\, = \\\n\\operatorname{ind}(\\mathsf{E})\/\\big(\\operatorname{ind}(\\mathsf{E}_0) \\, [Z(\\mathsf{E}_0):\\mathsf{T}_0]\\big)\n\\end{equation}\nand\n\\begin{equation}\\label{ersh1}\n\\Sigma_\\tau\\cap\\mathsf{E}_0 \\ = \\ P\\cdot X, \\textrm{\\ \\ \\ where\\ \\ \\ }\nP\\, =\\ \\textstyle{\\prod}_{h \\in H}\\Sigma_{h\\overline \\tau}(\\mathsf{E}_0)\n\\textrm{ \\ \\ and \\ \\ }\nX\\, = \\ \\langle x_\\gamma x_\\delta x_{\\gamma+\\delta}^{-1} \\mid\n\\gamma,\\delta \\in \\Gamma_\\mathsf{E} \\rangle \\, \\subseteq \\, \\mathsf{E}_0^*.\n\\end{equation}\nFurthermore, if $H=\\langle h_1,\\dots,h_m \\rangle$, then\n$P \\, = \\ \\prod_{(\\varepsilon_1,\\dots,\\varepsilon_m)\\in \\{0,1\\}^m}\n\\Sigma_{h_1^{\\varepsilon_1}\\dots h_m^{\\varepsilon_m} \\overline \\tau}(\\mathsf{E}_0)$.\n\\end{theorem}\n\nBefore proving the theorem, we record the following:\n\n\\begin{lemma}\\label{lem5}\nLet $A$ be a central simple algebra over a field $K$, with an\ninvolution $\\tau$ and an automorphism or anti-automorphism $\\sigma$.\nThen,\n\n\\begin{enumerate}[\\upshape(i)]\n\\item  \\label{hyh1} $\\sigma \\tau \\sigma^{-1}$ is an involution of $A$ of the same\n kind as $\\tau$, and\n$$\nS_{\\sigma \\tau \\sigma^{-1}}  \\, = \\, \\sigma(S_\\tau),\n\\textrm{ \\ \\  so \\  \\  } \\Sigma_{\\sigma \\tau \\sigma^{-1}} \\, = \\,\n\\sigma(\\Sigma_\\tau).\n$$\n\n\\item \\label{hyh2} Suppose $A$ is a division ring.\nIf $\\sigma$ and $\\tau$ are each unitary involutions, then\n$($writing {${S_\\tau^*=S_\\tau\\cap A^*}$}$)$,\n$$\nS_\\tau^* \\ \\subseteq  \\ S_\\sigma^* \\cdot \\sigma(S_\\tau^*)\n \\ = \\ S_\\sigma^* \\cdot S_{\\sigma \\tau \\sigma^{-1}}^*, \\textrm{  \\  \\ so \\  \\ ~~}\n\\Sigma_\\tau  \\ \\subseteq  \\ \\Sigma_\\sigma\\cdot \\Sigma_{\\sigma \\tau \\sigma^{-1}}.\n$$\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\\hfill\n\n(i) This follows by easy calculations.\n\n(ii) Observe that if $a \\in S_\\tau^*$,\nthen $a=\\big (a\\sigma(a)\\big)\\sigma(a^{-1})$ with ${a\\sigma(a) \\in S_\\sigma^*}$\nand $\\sigma(a^{-1}) \\in \\sigma(S_\\tau^*)=S_{\\sigma \\tau \\sigma^{-1}}^*$ by~(\\ref{hyh1}).\nThus,~(\\ref{hyh2}) follows from~(\\ref{hyh1}) and the fact\nthat ${A' \\subseteq \\Sigma_\\tau \\cap \\Sigma_\\sigma}$\n(see \\eqref{primeinsigma}).\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{msem}]\n\nFirst note that by Lemma~\\ref{unramfacts}\\eqref{nine} the canonical map\n$$\n(\\Sigma'_\\tau \\cap \\mathsf{E}_0) \\, \\big\/ \\, (\\Sigma_\\tau \\cap \\mathsf{E}_0)\n \\ \\longrightarrow  \\ \\Sigma'_\\tau\/\\Sigma_\\tau \\ = \\ \\SK(\\mathsf{E},\\tau)\n$$\nis an isomorphism.\nThe description of $\\Sigma'_\\tau \\cap \\mathsf{E}_0$\nin~(\\ref{ersh}) is immediate from the fact that\nfor $a \\in \\mathsf{E}_0$,\n$\\Nrd_{\\mathsf{E}}(a)=N_{Z(\\mathsf{E}_0)\/\\mathsf{T}_0}\\Nrd_{\\mathsf{E}_0}(a)^\\partial\\in \\mathsf{T}_0$\n(see Remark~\\ref{grfacts}\\eqref{rnrd}).\n\nFor $\\Sigma_\\tau \\cap \\mathsf{E}_0$, note that for each\n$\\gamma \\in \\Gamma_\\mathsf{E}$, if $a\\in \\mathsf{E}_0$, then\n$a x_\\gamma \\in S_\\tau$ if and only if\n $x_\\gamma \\tau(a) x_\\gamma^{-1}=a$. That is,\n$S_\\tau \\cap \\mathsf{E}_\\gamma = S(\\varphi_{x_\\gamma} \\tau;\\mathsf{E}_0)x_\\gamma$,\nwhere $S(\\varphi_{x_\\gamma} \\tau;\\mathsf{E}_0)$ denotes the set of\nsymmetric elements in $\\mathsf{E}_0$ for the unitary involution\n$\\varphi_{x_\\gamma} \\tau|_{\\mathsf{E}_0}$.\nTherefore,\n$$\n\\Sigma_\\tau \\cap \\mathsf{E}_0  \\ = \\\n\\big \\langle S(\\varphi_{x_\\gamma} \\tau;\\mathsf{E}_0)^* x_\\gamma \\mid\n\\gamma \\in \\Gamma_\\mathsf{E} \\big \\rangle \\, \\cap \\,  \\mathsf{E}_0.\n$$\nTake a product $a_1x_1\\ldots a_kx_k$ in $\\Sigma_\\tau \\cap \\mathsf{E}_0$\nwhere each $x_i=x_{\\gamma_i}$ for some $\\gamma_i \\in \\Gamma_\\mathsf{E}$\nand $a_i \\in S(\\varphi_{x_i} \\tau;\\mathsf{E}_0)^*$. Then,\n\\begin{equation}\\label{pet}\na_1x_1\\dots a_k x_k \\ = \\\na_1\\varphi_{x_1}(a_2)\\ldots\\varphi_{x_1\\ldots x_{i-1}}(a_i)\\ldots\n\\varphi_{x_1\\ldots x_{k-1}}(a_k)x_1\\ldots x_k \\, \\in \\,\n\\mathsf{E}_{\\gamma_1+\\ldots+\\gamma_k}.\n\\end{equation}\nSo, $\\gamma_1+\\ldots+\\gamma_k=0$. Now, as $a_i \\in S(\\varphi_{x_i} \\tau;\\mathsf{E}_0)$\nand $\\tau \\varphi_{x_j}^{-1}=\\varphi_{x_j}\\tau$ for all $j$,\nby Lemma~\\ref{lem5}(\\ref{hyh1}) we obtain\n\\begin{equation}\\label{longd}\n\\varphi_{x_1\\ldots x_{i-1}}(a_i)\\, \\in \\,\nS(\\varphi_{x_1}\\ldots \\varphi_{x_{i-1}}(\\varphi_{x_i}\\tau)\n\\varphi_{x_{i-1}}^{-1}\\ldots\\varphi_{x_1}^{-1};\n\\mathsf{E}_0)^* \\ = \\ S(\\varphi_{x_1\\ldots x_{i-1}x_ix_{i-1}\\ldots x_1} \\tau;\\mathsf{E}_0)^*\n \\ \\subseteq  \\ \\Sigma_{h\\overline \\tau}(\\mathsf{E}_0)  \\ \\subseteq  \\, P,\n\\end{equation}\nwhere $h=\\varphi_{x_1\\dots x_{i-1}x_ix_{i-1}\\dots x_1}|_{Z(\\mathsf{E}_0)} \\in H$.\nNote also that if $k=1$, then\n$x_1 \\in S_\\tau \\cap \\mathsf{E}_0^* \\subseteq \\Sigma_{\\overline \\tau}(\\mathsf{E}_0)\n\\subseteq P.$\n\nIf $k>1$, then\n$$\nx_1\\ldots x_k \\ = \\ x_{\\gamma_1}\\ldots x_{\\gamma_k} \\\n= \\ (x_{\\gamma_1} x_{\\gamma_2}\nx_{\\gamma_1+\\gamma_2}^{-1})(x_{\\gamma_1+\\gamma_2}x_{\\gamma_3}\n\\ldots x_{\\gamma_k}),\n$$\nwith\n$(\\gamma_1+\\gamma_2)+\\gamma_3+\\ldots+\\gamma_k=0$. It follows by induction\non $k$ that $x_1 \\dots x_k \\in X$. With this and ~(\\ref{pet})\nand ~(\\ref{longd}), we have $a_1x_1\\dots a_kx_k \\in P\\cdot X$\n(which is a group, as $\\mathsf{E}_0' \\subseteq \\Sigma_{\\overline \\tau}(\\mathsf{E}_0)\n\\subseteq  P$\nby\n\\eqref{primeinsigma}), showing that $\\Sigma_\\tau \\cap \\mathsf{E}_0\n\\subseteq P \\cdot X$. For the reverse inclusion, take any\n$h \\in H$ and choose $\\gamma \\in \\Gamma_\\mathsf{E}$ with\n$\\varphi_{x_\\gamma}|_{Z(\\mathsf{E}_0)}=h$. Then,  $x_\\gamma \\in S_\\tau^*\n\\subseteq \\Sigma_\\tau$ and $S(\\varphi_{x_\\gamma}\\tau;\\mathsf{E}_0)^*x_\\gamma=\nS_\\tau^*\\cap \\mathsf{E}_\\gamma \\subseteq \\Sigma_\\tau$, so\n$\\Sigma_{h\\overline \\tau}(\\mathsf{E}_0)=\\Sigma_{\\varphi_{x_\\gamma}\\tau}(\\mathsf{E}_0)=\n\\langle S(\\varphi_{x_\\gamma}\\tau;\\mathsf{E}_0)^* \\rangle \\subseteq\n\\Sigma_\\tau \\cap \\mathsf{E}_0$. Thus, $P \\subseteq \\Sigma_\\tau \\cap \\mathsf{E}_0$,\nand clearly also $X \\subseteq \\Sigma_\\tau \\cap \\mathsf{E}_0$. Hence,\n$\\Sigma_\\tau \\cap \\mathsf{E}_0 =P\\cdot X$.\n\nThe final equality for $P$ in the Theorem follows from\nLemma~\\ref{lembe} below by taking $U=\\mathsf{E}_0^*$, $A=H$, and\n$W_h=\\Sigma_{h\\overline \\tau}(\\mathsf{E}_0)$ for $h \\in H$. To see that\nthe lemma applies, note that each $\\Sigma_{h\\overline \\tau}(\\mathsf{E}_0)$\ncontains $\\mathsf{E}_0'$ by \\eqref{primeinsigma}.  Furthermore, take any $h,\\ell \\in H$, and choose\n$x,y \\in E^* \\cap S_\\tau$ with $\\varphi_x|_{Z(\\mathsf{E}_0)}=h$ and\n$\\varphi_y|_{Z(\\mathsf{E}_0)}=\\ell$. Then,\n$$\n(\\varphi_y\\tau)(\\varphi_x\\tau)(\\varphi_y\\tau)^{-1} \\ = \\\n\\varphi_y\\tau\\varphi_x\\varphi_y^{-1} \\ = \\ \\varphi_{yx^{-1}y}\\tau,\n$$\nand $\\varphi_{yx^{-1}y}|_{Z(\\mathsf{E}_0)}=\\ell h^{-1}\\ell=\\ell^2h^{-1}$. Hence, by\nLemma~\\ref{lem5}(\\ref{hyh2}),\n$\\Sigma_{h\\overline \\tau}(\\mathsf{E}_0)\\subseteq\n\\Sigma_{\\ell\\overline \\tau}(\\mathsf{E}_0)\\Sigma_{\\ell^2h^{-1}\\overline \\tau}(\\mathsf{E}_0)$.\nThis shows that hypothesis~(\\ref{hyp}) of Lemma~\\ref{lembe} below is\nsatisfied here.\n\\end{proof}\n\n\\begin{lemma}\\label{lembe}\nLet $U$ be a group, $A$ an abelian group, and $\\{W_a\\mid a\\in A\\}$ a family\nof subgroups of $U$ with each $W_a \\supseteq [U,U]$. Suppose\n\\begin{equation}\\label{hyp}\nW_a \\, \\subseteq  \\, W_bW_{2b-a} \\textrm{ \\ \\ for~all  } a,b \\in A.\n\\end{equation}\nIf $A=\\langle a_1,\\dots,a_m \\rangle$, then\n$$\n\\textstyle{\\prod\\limits_{a\\in A}} W_a \\ = \\\n\\textstyle{\\prod\\limits_{(\\varepsilon_1,\\dots,\\varepsilon_m)\\in \\{0,1\\}^m}}\nW_{\\varepsilon_1a_1+\\ldots+\\varepsilon_ma_m}. $$\n\\end{lemma}\n\\begin{proof}\nSince each $W_a \\supseteq [U,U]$, we have $W_aW_b=W_bW_a$, and this is a\nsubgroup of $U$, for all $ a,b \\in A$.\nLet\n$$\nQ \\, = \\textstyle{\\prod\\limits_{(\\varepsilon_1,\\dots,\\varepsilon_m)\\in \\{0,1\\}^m}}\n W_{\\varepsilon_1a_1+\\ldots+\\varepsilon_ma_m}.\n$$\nWe prove by induction on $m$ that each $W_a\\subseteq Q$. The lemma then\nfollows, as $Q$ is a subgroup of $U$.\nNote that condition~(\\ref{hyp}) can be conveniently restated,\n\\begin{equation}\\label{hyp1}\n\\textrm{ if  \\ } a+b \\, = \\, 2d \\in A, \\textrm{  \\ then  \\ }\nW_a  \\, \\subseteq \\,  W_dW_b.\n\\end{equation}\nTake any $c\\in A$. Then, (\\ref{hyp1}) shows that $W_{-c}\\subseteq W_0W_c$.\nTake any $i \\in \\mathbb Z$, and suppose $W_{ic}\\subseteq W_0W_c$. Then by\n(\\ref{hyp1}) $W_{-ic}\\subseteq W_0W_{ic} \\subseteq W_0W_c$. So, by\n(\\ref{hyp1}) again, $W_{(i+2)c}\\subseteq W_cW_{-ic} \\subseteq W_0W_c$ and\n${W_{(i-2)c}\\subseteq W_{-c}W_{-ic} \\subseteq W_0W_c}$. Hence, by induction\n(starting with $j=0$ and $j=1$), $W_{jc}\\subseteq W_0W_c$ for every\n$j \\in \\mathbb Z$. This proves the lemma when $m=1$.\n\nNow assume $m>1$ and let $B=\\langle a_1,\\dots,a_{m-1} \\rangle \\subseteq A$.\nBy induction, for all\n$b\\in B$,\n$$\nW_b \\ \\subseteq\n\\textstyle{\\prod\\limits_{(\\varepsilon_1,\\dots,\\varepsilon_{m-1})\\in \\{0,1\\}^{m-1}}}\nW_{\\varepsilon_1a_1+\\ldots+\\varepsilon_{m-1}a_{m-1}} \\ \\subseteq  \\, Q.\n$$\nAlso, by the cyclic case done above,\n$W_{j a_m} \\subseteq W_0W_{a_m} \\subseteq Q$\nfor all $j \\in \\mathbb Z$. So, for any $b \\in B, j \\in \\mathbb Z$,\nusing~(\\ref{hyp1}),\n\\begin{equation}\\label{thte}\nW_{2b+ja_m}  \\, \\subseteq  \\, W_b W_{-ja_m}  \\, \\subseteq  \\, Q,\n\\end{equation}\nand\n\\begin{equation}\\label{thte2}\nW_{b+2ja_m}  \\, \\subseteq  \\, W_{ja_m} W_{-b} \\, \\subseteq  \\, Q.\n\\end{equation}\nLet $d=a_{i_1}+\\ldots+a_{i_\\ell}$ for any indices\n$1\\leq i_1< i_2 <\\ldots <i_\\ell \\leq m-1$. Since\n$W_{d+a_m} \\subseteq Q$ by hypothesis,\nfrom~(\\ref{hyp1}) and~(\\ref{thte2}) it follows that\n\\begin{equation}\\label{d+3}\nW_{d+3a_m} \\,  \\subseteq  \\ W_{d+2a_m} W_{d+a_m}  \\, \\subseteq  \\, Q.\n\\end{equation}\nNow, take any element of $A$; it has the form $b+ja_m$ for some\n$b\\in B$ and $j \\in \\mathbb Z$. If $b \\in 2B$ or if $j$ is even,\nthen~(\\ref{thte}) and~(\\ref{thte2}) show that\n$W_{b+ja_m} \\subseteq Q$. The remaining case is that $j$ is\nodd and $b \\not \\in 2B$, so $b=2c+d$, where $c\\in B$ and\n$d=a_{i_1}+\\ldots+a_{i_\\ell}$ for some indices with\n$1\\leq i_1<i_2<\\ldots <i_\\ell \\leq m-1$. Set $q=1$ if\n$j \\equiv 3 \\pmod{4}$ and $q=3$ if\n$j \\equiv 1  \\pmod{4}$. Then, $W_{d+qa_m}\\subseteq Q$ by definition\nif $q=1$ or by \\eqref{d+3} if $q=3$.\nHence, by~(\\ref{hyp1}),\n\\begin{equation}\\label{ww}\nW_{b+ja_m} \\, = \\ W_{(2c+d)+ja_m}  \\, \\subseteq  \\\nW_{(c+d)+((j+q)\/2)a_m}W_{d+qa_m} \\,  \\subseteq  \\, Q,\n\\end{equation}\nusing~(\\ref{thte2}) as $c+d \\in B$ and $(j+q)\/2$ is even.\nThus, $W_a \\subseteq Q$, for all $a \\in A$.\n\\end{proof}\n\n\\begin{corollary}\\label{unramified}\nIf $\\mathsf{E}$  is unramified over $\\mathsf{R}$,  then\n$\\SK(\\mathsf{E},\\tau) \\cong \\SK(\\mathsf{E}_0,\\tau|_{\\mathsf{E}_0})$.\n\\end{corollary}\n\n\\begin{proof}\nSince $\\mathsf{E}$ is unramified over $\\mathsf{R}$, we have $\\mathsf{T}$ is unramified over\n$\\mathsf{R}$, $Z(\\mathsf{E}_0) = \\mathsf{T}_0$,  and  $\\Gamma_\\mathsf{E} = \\Gamma_\\mathsf{R}$,\nso we can choose all the $x_\\gamma$'s to lie in $\\mathsf{R}$.\n The assertion thus follows immediately\n from Th.~\\ref{msem}, as $P = \\Sigma_{\\tau|_{\\mathsf{E}_0}}(\\mathsf{E}_0)$\nand $X \\subseteq R_0^* \\subseteq P$.\n(Alternatively, more  directly, one can\nobserve that $\\Sigma'_0=\\Sigma'_{\\tau|_{\\mathsf{E}_0}}(\\mathsf{E}_0)$ and\n$\\Sigma_0=\\Sigma_{\\tau|_{\\mathsf{E}_0}}(\\mathsf{E}_0)$ and so deduce the Corollary by\nLemma~\\ref{unramfacts}(\\ref{nine}).)\n\\end{proof}\n\n\\begin{corollary}\\label{seses}\nIf $\\mathsf{T}$ is unramified over $\\mathsf{R}$ and $\\mathsf{E}$ has a maximal graded subfield\n$\\mathsf{M}$ unramified over $\\mathsf{T}$ and another maximal graded subfield $\\mathsf{L}$ totally\nramified over $\\mathsf{T}$  with $\\tau(\\mathsf{L} ) =\\mathsf{L}$, then $\\mathsf{E}$ is semiramified with\n$\\mathsf{E}_0=\\mathsf{M}_0$ $($a field$)$ and $\\Gamma_\\mathsf{E}=\\Gamma_\\mathsf{L}$, and\n\\begin{equation}\\label{semirsk}\n\\SK(\\mathsf{E},\\tau) \\ \\cong \\ \\big\\{a \\in \\mathsf{E}_0\n\\mid N_{\\mathsf{E}_0\/\\mathsf{T}_0}(a)\\in \\mathsf{R}_0\\big\\}\n \\, {\\big\/} \\,  \\textstyle{\\prod\\limits_{h\\in \\operatorname{Gal}(\\mathsf{E}_0\/\\mathsf{T}_0)}}\n\\mathsf{E}_0^{*h\\overline \\tau}.\n\\end{equation}\n\\end{corollary}\n\\begin{proof}\nLet $n=\\operatorname{ind}(\\mathsf{E})$. Since $[\\mathsf{M}_0:\\mathsf{T}_0]=n$ and\n${|\\Gamma_\\mathsf{L}:\\Gamma_\\mathsf{T}|=n}$, it follows from the Fundamental Equality\n~\\eqref{fundeq}   for $\\mathsf{E}\/\\mathsf{T}$,\n$\\mathsf{M}\/\\mathsf{T}$, and $\\mathsf{L}\/\\mathsf{T}$\nthat\n$[\\mathsf{E}_0:\\mathsf{T}_0]=[\\Gamma_\\mathsf{E}:\\Gamma_\\mathsf{T}]=n$,  $\\mathsf{E}_0=\\mathsf{M}_0$,\nwhich is\nis a field, and $\\Gamma_\\mathsf{E} = \\Gamma_\\mathsf{L}$.\nThus $\\mathsf{E}$ is semiramified, so for the $\\partial$ of\n\\eqref{dlambda},  $\\partial = 1$. Now, $\\mathsf{L}^\\tau$ is a graded\nsubfield of\n$\\mathsf{L}$ with $[\\mathsf{L}:\\mathsf{L}^\\tau] = 2$.  Since $\\mathsf{L}_0 = \\mathsf{T}_0$ while $(\\mathsf{L}^\\tau)_0\n= (\\mathsf{L}_0)^\\tau = \\mathsf{R}_0$, $\\mathsf{L}$~must be unramified over $\\mathsf{L}^\\tau$;\nhence, $\\Gamma_\\mathsf{E} = \\Gamma_\\mathsf{L}  = \\Gamma_{\\mathsf{L}^\\tau}$.\nTherefore, one can choose all the\n$x_\\gamma$'s in Th.~\\ref{msem} to lie in $\\mathsf{L}^\\tau$. Then each\n $x_\\gamma x_\\delta x_{\\gamma+\\delta}^{-1} \\in (\\mathsf{L}^\\tau)_0^* =\n\\mathsf{R}_0^*=\n\\mathsf{E}_0^{*\\overline \\tau}$. Hence, $X \\subseteq P =\n \\prod_{h\\in \\operatorname{Gal}(\\mathsf{E}_0\/\\mathsf{T}_0)}\\mathsf{E}_0^{*h\\overline \\tau}$,\nso\nthe formula for\n $\\SK(\\mathsf{E},\\tau)$ in Th.~\\ref{msem} reduces to (\\ref{semirsk}).\n\\end{proof}\n\n\\begin{remark} In a sequel to this paper~\\cite{II}, the following will be\nshown:  With the hypotheses  of Th.~\\ref{msem}, suppose\n $\\mathsf{E}$ is  semiramified with a graded  maximal subfield\n$\\mathsf{L}$ totally ramified over $\\mathsf{T}$ such that $\\tau(\\mathsf{L}) = \\mathsf{L}$, and\nsuppose\n$\\operatorname{Gal}(\\mathsf{E}_0\/\\mathsf{T}_0)$ is bicyclic,\nsay $\\mathsf{E}_0 = N\\otimes _{T_0} N'$ with\n$N$ and $N'$ each cyclic Galois over $\\mathsf{T}_0$. Then,\n$$\n{\\SK(\\mathsf{E},\\tau) \\ \\cong \\ \\operatorname{Br}(\\mathsf{E}_0\/\\mathsf{T}_0;\\mathsf{R}_0)\\big\/\n\\big(\\operatorname{Br}(N\/\\mathsf{T}_0;\\mathsf{R}_0)+\\operatorname{Br}(N'\/\\mathsf{T}_0;\\mathsf{R}_0)\\big)},\n$$\n where\n$\\operatorname{Br}(\\mathsf{E}_0\/\\mathsf{T}_0)$ is the relative Brauer group\n$\\ker\\big(\\operatorname{Br}(\\mathsf{T}_0)\\to \\operatorname{Br}(\\mathsf{E}_0)\\big)$ and\n$\\operatorname{Br}(\\mathsf{E}_0\/\\mathsf{T}_0;\\mathsf{R}_0)$ is the kernel of\n$\\text{cor}_{\\mathsf{E}_0\\to \\mathsf{R}_0}\\colon \\operatorname{Br}(\\mathsf{E}_0\/\\mathsf{T}_0)\\to \\operatorname{Br}(\\mathsf{E}_0\/\\mathsf{R}_0)$.\n(Compare this with~\\cite[Th.~5.6]{y}.)  A further formula will be given\nassuming only that $\\mathsf{E}$ is semiramified over $\\mathsf{T}$ with\n$\\operatorname{Gal}(\\mathsf{E}_0\/\\mathsf{T}_0)$  bicyclic.\n\\end{remark}\n\n\nIn his construction of division algebras $D$ with nontrivial $\\SK$,\nPlatonov worked\noriginally in~\\cite[\\S4]{platonov} with a division algebra $D$\nwhere $Z(D)$ is a\nLaurent power series field; he gave an exact sequence relating\n$\\SK(D)$ with\n$\\SK(\\overline D)$ and what he called the ``group of projective\nconorms.''\nYanchevski\\u\\i ~gave in~\\cite[4.11]{y} an analogous exact\nsequence in the unitary case.\nTheir results and proofs are valid whenever $Z(D)$ has a\nhenselian discrete\n(rank $1$) valuation. We show here that their results hold more generally\nwhenever $Z(D)$ has a henselian\nvaluation with $\\Gamma_D\/\\Gamma_{Z(D)}$ cyclic. We work in\nthe equivalent graded setting where the arguments are more transparent.\n\nAs before, let $\\mathsf{E}$ be a graded division algebra finite dimensional\nover its center $\\mathsf{T}$ with a unitary graded involution~$\\tau$, and let\n$\\mathsf{R}=\\mathsf{T}^\\tau$. Assume that $\\mathsf{T}$ is unramified over $\\mathsf{R}$ and  that\n$\\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T}$ is cyclic group. (This cyclicity holds, e.g.,\n whenever $\\Gamma_\\mathsf{T}\\cong \\mathbb Z$.)\nIt follows that the surjective map $\\Theta_\\mathsf{E}\\colon\\Gamma_\\mathsf{E}\n\\rightarrow \\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{T}_0)$ has kernel~$\\Gamma_\\mathsf{T}$. (For,\nby \\cite[Prop.~2.1, (2.3), Remark 2.4(i)]{hwcor}$, \\ker(\\Theta_\\mathsf{E})\/\\Gamma_\\mathsf{T}$ has a nondegenerate symplectic\npairing, and hence has even rank as a finite abelian group. But here\n$\\ker(\\Theta_\\mathsf{E})\/\\Gamma_\\mathsf{T}$\nis a cyclic group.) Hence,  $\\partial=1$\nby Lemma~\\ref{dlambdafacts}, so $\\mathsf{E}$ is inertially split.\nInvoking Lemma~\\ref{unramfacts}\\eqref{seven},\nchoose any $s\\in \\mathsf{E}^*$\nwith $\\deg(s)+\\Gamma_\\mathsf{T}$ a generator of $\\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T}$,\nsuch that $\\tau(s)=s$. Let $\\sigma=\\varphi_s\\in \\operatorname{Aut}_\\mathsf{T}(\\mathsf{E})$; so\n$\\sigma\\tau$ is another $\\mathsf{T}\/\\mathsf{R}$-graded involution of $\\mathsf{E}$, and\n$\\tau\\sigma=\\sigma^{-1}\\tau$ (see Remark~\\ref{easyob}(\\ref{three})).\nBy the choice of $s$, $\\sigma|_{Z(\\mathsf{E}_0)}$ is a generator of the\ncyclic group $\\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{T}_0)$. Note that\n$\\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{R}_0)=\\langle \\sigma|_{Z(\\mathsf{E}_0)},\\tau|_{Z(\\mathsf{E}_0)}\\rangle$\nis a dihedral group.\nRecall our convention that $c^{\\sigma\\tau}$ means $\\sigma(\\tau(c))$.\nLet\n\\begin{align*}\n\\mathcal S \\ & = \\  \\big \\{ (\\beta,b) \\in Z(\\mathsf{E}_0)^* \\times \\mathsf{E}_0^* \\mid\n\\beta^{\\sigma-1}=\\Nrd_{\\mathsf{E}_0}(b)\\big \\} ;\\\\\n\\mathcal N \\ & = \\  \\pi_1(\\mathcal S)\n\\textrm{ (projection into the first component) } \\\\\\ & \\quad= \\\n\\big \\{ \\beta \\in Z(\\mathsf{E}_0)^* \\mid  \\beta^{\\sigma-1}=\n\\Nrd_{\\mathsf{E}_0}(b) \\textrm{ for some $b \\in \\mathsf{E}_0^*$}\\big \\};\\\\\n\\mathcal W \\ & = \\ \\mathsf{T}_0^* \\cdot \\Nrd_{\\mathsf{E}_0}(\\mathsf{E}_0^*)  \\ \\subseteq\n \\ Z(\\mathsf{E}_0)^*;\\\\\n\\mathcal P \\ & = \\ \\mathcal N \/\\mathcal W ,\\textrm{ which is Platonov's\ngroup of projective conorms for $\\mathsf{E}$ }\\textrm{\\cite[\\S4]{platonov}}.\\qquad\n\\qquad\\qquad\\quad\n\\end{align*}\n\\begin{align*}\n\\mathcal S_\\tau \\ & = \\ \\big \\{(\\alpha,a) \\in Z(\\mathsf{E}_0)^*\\times\\mathsf{E}_0^* \\mid\n\\alpha^{\\sigma-1}=\\Nrd_{\\mathsf{E}_0}(a)^{1-\\sigma\\tau} \\big \\};\\\\\n\\mathcal N_\\tau \\ & = \\ \\pi_1(\\mathcal S_\\tau)\n\\textrm{ (projection into the first component) } \\\\\n\\ & \\quad = \\ \\big \\{ \\alpha \\in Z(\\mathsf{E}_0)^* \\mid \\alpha^{\\sigma-1}=\n\\Nrd_{\\mathsf{E}_0}(a)^{1-\\sigma\\tau} \\textrm{ for some } a\\in \\mathsf{E}_0^* \\big \\};\\\\\n\\mathcal W_\\tau\\ & = \\ \\mathsf{T}_0^*\\cdot \\Nrd_{\\mathsf{E}_0}(\\Sigma_\\tau(\\mathsf{E}_0)) \\\n\\subseteq \\  Z(\\mathsf{E}_0)^* ;\\\\\n\\mathcal PU_\\tau \\ & = \\ \\mathcal N_\\tau \/\\mathcal W_\\tau, \\textrm{ which is\nYanchevski\\u\\i's group of unitary projective conorms for }\n(\\mathsf{E},\\tau)\\textrm{~\\cite[4.11]{y}}.\n\\end{align*}\n\n\n\\begin{proposition}\\label{cyclic}\nIf $\\mathsf{T}$ is unramified over $\\mathsf{R}$ and $\\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T}$ is cyclic,\nthen for any generator\n$\\sigma$ of the cyclic group $\\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{T}_0)$,\nwe have\n\n\\begin{enumerate}[\\upshape(i)]\n\\item \\label{rh1} $\\SK(\\mathsf{E}) \\ \\cong \\  \\{ a\\in E_0^* \\mid\nN_{Z(\\mathsf{E}_0)\/\\mathsf{T}_0}(\\Nrd_{\\mathsf{E}_0}(a))=1 \\}\\, \\big \/ \\, \\big([\\mathsf{E}_0^*,\\mathsf{E}_0^*]\\cdot\n\\{ c^{\\sigma-1}\\mid c\\in \\mathsf{E}_0^* \\}\\big).$\n\n\\smallskip\n\n\\item \\label{rh2} $\\SK(\\mathsf{E},\\tau) \\ \\cong  \\  \\{ a\\in E_0^* \\mid N_{Z(\\mathsf{E}_0)\/\\mathsf{T}_0}\n(\\Nrd_{\\mathsf{E}_0}(a)) \\in \\mathsf{R}_0  \\} \\, \\big \/ \\, \\big(\n\\Sigma_\\tau(\\mathsf{E}_0)\\cdot \\Sigma_{\\sigma\\tau}(\\mathsf{E}_0)\\big)$.\n\n\\smallskip\n\n\\item \\label{rh3} The following sequence is exact:\n\\begin{equation}\n\\SK(\\mathsf{E}_0,\\sigma\\tau)  \\longrightarrow \\SK(\\mathsf{E},\\tau)\n\\overset {f}{\\longrightarrow}  \\mathcal PU_\\tau \\longrightarrow 1,\n\\end{equation}\nwhere the map $f\\colon\\SK(\\mathsf{E},\\tau)\\rightarrow \\mathcal PU_\\tau$ is\nthe composition of $a\\, \\Sigma_\\tau(\\mathsf{E}_0)\\cdot \\Sigma_{\\sigma\\tau}(\\mathsf{E}_0)\n\\mapsto (\\alpha,a) \\in \\mathcal S_\\tau$ and\n${(\\alpha,a)\\mapsto \\alpha \\mathcal W_\\tau \\in \\mathcal PU_\\tau}$.\n\\smallskip\n\n\\item \\label{rh4} There is a commutative diagram with exact rows:\n\\begin{equation*}\n\\begin{split}\n\\xymatrix{\n& \\SK(\\mathsf{E}_0,\\sigma\\tau) \\ar[r] \\ar[d]_{{}_{\\substack{a\\\\ \\downarrow\\\\ a^{1-\\sigma\\tau}}}}\n & \\SK(\\mathsf{E},\\tau) \\ar[r]^-{f} \\ar[d]_{{}_{\\substack{a\\\\ \\downarrow\\\\ a^{1-\\sigma\\tau}}}}\n & \\mathcal PU_\\tau \\ar[r] \\ar[d]_{{}_{\\substack{\\alpha\\\\ \\downarrow\\\\ \\alpha}}}&  1\\\\\n1 \\ar[r] & \\SK(\\mathsf{E}_0) \\ar[r] \\ar[d]_{{}_{\\substack{b\\\\ \\downarrow\\\\ b}}}\n & \\SK(\\mathsf{E}) \\ar[r]^-{g} \\ar[d]_{{}_{\\substack{b\\\\ \\downarrow\\\\ b}}}\n & \\mathcal P \\ar[r] \\ar[d]_{{}_{\\substack{\\beta\\\\ \\downarrow\\\\ \\beta^{1+\\tau}}}} & 1\\\\\n& \\SK(\\mathsf{E}_0,\\sigma\\tau) \\ar[r] &  \\SK(\\mathsf{E},\\tau) \\ar[r] & \\mathcal PU_\\tau \\ar[r] & 1}\n\\end{split}\n\\end{equation*}\nwhere the\n map $g\\colon\\SK(\\mathsf{E})\\rightarrow \\mathcal P$ is the composition of\n$ \\ b\\, [\\mathsf{E}_0^*,\\mathsf{E}_0^*]\\langle c^{\\sigma-1}\\rangle \\mapsto (\\beta,b)\\in \\mathcal S$\nand ${(\\beta,b)\\mapsto \\beta \\mathcal W \\in \\mathcal P}$.\n\\vskip.06truein\n\\item \\label{unrkl} If  $\\mathsf{E}_0$ is a field, then $\\SK(\\mathsf{E},\\tau)=1$.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}  \\hfill\n\n(i) This formula was given by Suslin~\\cite[Prop.~1.7]{sus1} for a division algebra\nover a field  with a complete discrete valuation. In order to prove it in the\ngraded setting we need  two exact sequences which were given in\n \\cite[Th.~3.4]{hazwadsworth}:\n$$\n\\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T}  \\wedge \\Gamma_\\mathsf{E}  \/\\Gamma_\\mathsf{T}\n\\longrightarrow \\mathsf{E}^{(1)}\\big \/[\\mathsf{E}_0^*,\\mathsf{E}^*]\n\\longrightarrow \\SK(\\mathsf{E}) \\longrightarrow 1,\n$$\n$$\n1 \\longrightarrow \\ker \\widetilde N \/[\\mathsf{E}_0^*,\\mathsf{E}^*] \\longrightarrow\n\\mathsf{E}^{(1)}\\big \/[\\mathsf{E}_0^*,\\mathsf{E}^*] \\longrightarrow\n\\mu_\\partial(\\mathsf{T}_0) \\cap \\widetilde N(\\mathsf{E}_0^*)\\longrightarrow 1,\n$$\nwhere $\\mathsf{E}^{(1)} = \\{a\\in \\mathsf{E}^*\\mid \\Nrd_\\mathsf{E}(a) = 1\\} \\subseteq \\mathsf{E}_0$ and\n$\\widetilde N = N_{Z(\\mathsf{E}_0)\/\\mathsf{T}_0}\\circ \\Nrd_{\\mathsf{E}_0}\\colon\n\\mathsf{E}_0^* \\to \\mathsf{T}_0^*$. Since $\\partial=1$ (see the paragraph prior to the\nProposition) and the wedge product of a cyclic group with itself is trivial,\nthese  exact sequences yield\n$$\n\\SK(\\mathsf{E}) \\ \\cong \\  \\{ a\\in \\mathsf{E}_0^* \\mid\nN_{Z(\\mathsf{E}_0)\/\\mathsf{T}_0}(\\Nrd_{\\mathsf{E}_0}(a))=1 \\}\\big \/[\\mathsf{E}_0^*,\\mathsf{E}^*].\n$$\nWe are left to show that $[\\mathsf{E}_0^*,\\mathsf{E}^*]=[\\mathsf{E}_0^*,\\mathsf{E}_0^*]\\cdot\n\\{ c^{\\sigma-1}\\mid c\\in \\mathsf{E}_0^* \\}$. This follows from  the fact\nthat $\\mathsf{E}^*\/\\mathsf{T}^* \\mathsf{E}_0^*\\cong \\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T}$ is cyclic\ntogether with  the\nfollowing observation, which is easily verified  using the standard commutator\nidentities: If $G$ is a group and $N$ is a normal subgroup of $G$ such that\n$G\/Z(G)N$ is a cyclic group generated by, say,  $xZ(G)N$,\nthen $[N,G]=[N,N][x,N]$ where $[x,N]=\\{[x,n] \\mid n \\in N\\}$.\n(Here, take $G = \\mathsf{E}^*$, $N= \\mathsf{E}_0^*$,  and for $x$ take any $s\\in \\mathsf{E}_\\gamma^*$\nfor any $\\gamma\\in \\Gamma_\\mathsf{E}$ such that $\\Theta_\\mathsf{E}(\\gamma+\\Gamma_\\mathsf{T})\n= \\sigma$.)\n\n\n(ii) By Th.~\\ref{msem}, taking into account that $\\partial=1$ and\n$\\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{T}_0)=\\langle \\sigma \\rangle$, we have,\n\\begin{equation} \\label{cycliccase}\n\\begin{split}\n\\SK(\\mathsf{E},\\tau)\\ &  \\cong  \\ (\\Sigma'_\\tau \\cap \\mathsf{E}_0) \\big \/\n(\\Sigma_\\tau \\cap \\mathsf{E}_0)   \\\\\n\\ & = \\  \\{ a \\in \\mathsf{E}_0^*\\mid\nN_{Z(\\mathsf{E}_0)\/\\mathsf{T}_0}\\Nrd_{\\mathsf{E}_0}(a)\\in \\mathsf{R}_0  \\}\\,\n\\big \/ \\, \\big(\\Sigma_\\tau(\\mathsf{E}_0)\\cdot \\Sigma_{\\sigma\\tau}(\\mathsf{E}_0)\\cdot\n\\langle x_\\gamma x_\\delta x_{\\gamma+\\delta}^{-1}\n\\mid \\gamma, \\delta \\in \\Gamma_\\mathsf{E} \\rangle\\big),\n\\end{split}\n\\end{equation}\nwhere for each $\\gamma \\in \\Gamma_\\mathsf{E}$,\n$x_\\gamma$ is chosen in  $\\mathsf{E}^*_\\gamma$\nwith $x_\\gamma=\\tau(x_\\gamma)$ and $x_\\gamma\\ne 0$,\nusing Lemma~\\ref{unramfacts}\\eqref{seven}.\nLet $\\mathsf{L}=\\mathsf{R}[s]$, where $s$ is chosen in $\\mathsf{E}^*$ with $\\varphi(s)|_{\\mathsf{E}_0}\n= \\sigma$, which is possible as $\\Theta_{\\mathsf{E}}\\colon \\Gamma_\\mathsf{E} \\to\n \\operatorname{Gal}(Z(\\mathsf{E}_0)\/\\mathsf{T}_0)$\nis surjective (see \\eqref{surj}).  Moreover, $s$ can  be chosen  with\n$\\tau(s) = s$. Since $\\ker(\\Theta_\\mathsf{E}) = \\Gamma_\\mathsf{T} = \\Gamma_\\mathsf{R}$,\nwe have $\\Gamma_\\mathsf{E} = \\langle \\deg(s) \\rangle+ \\Gamma_\\mathsf{R}$.  Thus,\n$\\mathsf{L}$  is a graded\nsubfield of $\\mathsf{E}$ with $\\Gamma_\\mathsf{L}=\\Gamma_\\mathsf{E}$ and $\\tau|_\\mathsf{L}=\\operatorname{id}$.\nFor each $\\gamma \\in \\Gamma_\\mathsf{E}$\nwe can choose $x_\\gamma \\in \\mathsf{L}_\\gamma^*$; then for all\n$\\gamma,\\delta \\in \\Gamma_\\mathsf{E}$, we have\n$x_\\gamma x_\\delta x_{\\gamma+\\delta}^{-1} \\in \\mathsf{L}_0^*\n\\subseteq \\Sigma_\\tau(\\mathsf{E}_0)$. Thus,  the\n$\\langle x_\\gamma x_\\delta x_{\\gamma+\\delta}^{-1}  \\rangle$ term in\n\\eqref{cycliccase}  is redundant,\nyielding the formula in (ii).\n\n(iii) We first check that $f$ is well-defined: Take any $a \\in \\mathsf{E}_0^*$\nwith $N_{Z(\\mathsf{E}_0)\/\\mathsf{T}_0}(\\Nrd_{\\mathsf{E}_0}(a))\\in \\mathsf{R}_0^*$. Let\n$c=\\Nrd_{\\mathsf{E}_0}(a)$. Then, as $\\mathsf{R}_0=\\mathsf{T}_0^\\tau$,\n$1=N_{Z(\\mathsf{E}_0)\/\\mathsf{T}_0}(c)^{1-\\tau}=N_{Z(\\mathsf{E}_0)\/\\mathsf{T}_0}(c)^{1-\\sigma\\tau}=\nN_{Z(\\mathsf{E}_0)\/\\mathsf{T}_0}(c^{1-\\sigma\\tau})$. By Hilbert~90, there is\n$\\alpha \\in Z(\\mathsf{E}_0)^*$ with $\\alpha^{\\sigma-1}=c^{1-\\sigma\\tau}=\n\\Nrd_{\\mathsf{E}_0}(a)^{1-\\sigma\\tau}$. Hence $(\\alpha,a)\\in \\mathcal S_\\tau$, so\n$\\alpha \\in \\mathcal N_\\tau$, and the choice of $\\alpha$ is unique up to\n$\\mathsf{T}_0^* \\subseteq \\mathcal W_\\tau$.\nThus, the image of $a$ in $\\mathcal  P U_\\tau$ is independent of the choice of\n$\\alpha$.\nSuppose further that $a=pq$ for some\n$p\\in \\Sigma_\\tau(\\mathsf{E}_0)$, $ q \\in \\Sigma_{\\sigma\\tau}(\\mathsf{E}_0)$, say,\n$p=s_1\\ldots s_k$ with each $s_i \\in S_\\tau(\\mathsf{E}_0)$.\nThen,\n\\begin{equation}\\label{skks}\n\\begin{split}\n\\Nrd_{\\mathsf{E}_0}(p)^\\tau \\ &= \\ \\Nrd_{\\mathsf{E}_0}(s_1)^\\tau \\ldots\n\\Nrd_{\\mathsf{E}_0}(s_k)^\\tau  \\  = \\ \\Nrd_{\\mathsf{E}_0}(s_1^\\tau)\\ldots\n\\Nrd_{\\mathsf{E}_0}(s_k^\\tau)\\\\\n& = \\ \\Nrd_{\\mathsf{E}_0}(s_1)\\ldots \\Nrd_{\\mathsf{E}_0}(s_k) \\ = \\ \\Nrd_{\\mathsf{E}_0}(p);\n\\end{split}\n\\end{equation}\nlikewise, $\\Nrd_{\\mathsf{E}_0}(q)^{\\sigma\\tau}=\\Nrd_{\\mathsf{E}_0}(q)$. So,\n$$\n\\alpha^{\\sigma-1} \\, = \\ \\Nrd_{\\mathsf{E}_0}(pq)^{1-\\sigma\\tau} \\,\n= \\ \\Nrd_{\\mathsf{E}_0}(p)^{1-\\sigma\\tau}\\Nrd_{\\mathsf{E}_0}(q)^{1-\\sigma\\tau} \\,\n= \\ \\Nrd_{\\mathsf{E}_0}(p)^{1-\\sigma}.\n$$\nHence, $\\big (\\alpha\\Nrd_{\\mathsf{E}_0}(p)\\big )^{\\sigma-1}=1$, showing that\n$\\alpha \\Nrd_{\\mathsf{E}_0}(p) \\in \\mathsf{T}_0$; Thus $\\alpha \\in \\mathcal W_\\tau$.\nThis proves that $f$ is well-defined.\n\n\nFor the subjectivity of $f$, take any $\\alpha \\in \\mathcal N_\\tau$.\nThen, there is $a\\in \\mathsf{E}_0^*$ with\n$\\alpha^{\\sigma-1}=\\Nrd_{\\mathsf{E}_0}(a)^{1-\\sigma\\tau}$. So,\n$N_{Z(\\mathsf{E}_0)\/\\mathsf{T}_0}(\\Nrd_{\\mathsf{E}_0}(a))^{1-\\sigma\\tau}\n=N_{Z(\\mathsf{E}_0)\/\\mathsf{T}_0}(\\alpha^{\\sigma-1})=1$, which shows that\n$N_{Z(\\mathsf{E}_0)\/\\mathsf{T}_0}(\\Nrd_{\\mathsf{E}_0}(a))\\in \\mathsf{T}_0^{\\sigma\\tau}\n=\\mathsf{T}_0^\\tau=\\mathsf{R}_0$, and hence\n$a \\in \\Sigma_\\tau'(\\mathsf{E})\\cap \\mathsf{E}_0^*$. Since\n$f\\big(a\\Sigma_\\tau(\\mathsf{E}_0)\\Sigma_{\\sigma\\tau}(\\mathsf{E}_0)\\big)\n=\\alpha\\mathcal W_\\tau$, $f$ is surjective.\n\n\nFinally, we determine $\\ker(f)$: The image of $\\SK(\\mathsf{E}_0,\\sigma\\tau)$ in\n$\\SK(\\mathsf{E},\\tau)$ is $\\Sigma_{\\sigma\\tau}'(\\mathsf{E}_0)\\Sigma_\\tau(\\mathsf{E}_0)\\big \/\n\\Sigma_{\\sigma\\tau}(\\mathsf{E}_0)\\Sigma_\\tau(\\mathsf{E}_0)$. An element in this image\nis represented by some\n$a \\in \\Sigma_{\\sigma\\tau}'(\\mathsf{E}_0)$. For such an $a$,\n$\\Nrd_{\\mathsf{E}_0}(a)^{1-\\sigma\\tau}=1$. Then $(1,a) \\in \\mathcal S_\\tau$, so\nthat $f$ maps the image of $a$ to $1$ in $\\mathcal PU_\\tau$. Conversely,\nsuppose\n$a\\Sigma_\\tau(\\mathsf{E}_0)\\Sigma_{\\sigma\\tau}(\\mathsf{E}_0) \\in \\ker(f)$. That is,\n$\\Nrd_{\\mathsf{E}_0}(a)^{1-\\sigma\\tau}=\\alpha^{\\sigma-1}$, where\n$\\alpha \\in \\mathcal W_\\tau$,  so\n$\\alpha=c\\Nrd_{\\mathsf{E}_0}(d)$ with $c\\in \\mathsf{T}_0^*$ and $d\\in \\Sigma_\\tau(\\mathsf{E}_0)$.\nSo, $\\Nrd_{\\mathsf{E}_0}(d)=\\Nrd_{\\mathsf{E}_0}(d)^{\\tau}$ by the argument\nof~(\\ref{skks}) above, and hence\n$$\n\\Nrd_{\\mathsf{E}_0}(a)^{1-\\sigma\\tau} \\, = \\ \\alpha^{\\sigma-1}\n \\, = \\ \\big (c\\Nrd_{\\mathsf{E}_0}(d)\\big )^{\\sigma-1}\n \\, = \\ \\Nrd_{\\mathsf{E}_0}(d)^{\\sigma-1} \\, = \\ \\Nrd_{\\mathsf{E}_0}(d)^{\\sigma\\tau-1}.\n$$\nThus, $\\Nrd_{\\mathsf{E}_0}(ad)^{1-\\sigma\\tau}=1$, i.e.,\n$ad \\in \\Sigma_{\\sigma\\tau}'(\\mathsf{E}_0)$. Hence,\n$a=(ad)d^{-1} \\in \\Sigma_{\\sigma\\tau}'(\\mathsf{E}_0)\\Sigma_\\tau(\\mathsf{E}_0)$. This\nshows that $\\ker(f)$ coincides with the image of $\\SK(\\mathsf{E}_0,\\sigma\\tau)$\nin $\\SK(\\mathsf{E},\\tau)$, completing the proof of exactness of the  sequence.\n\n(iv) Exactness of the middle row is proved by an analogous but easier\nargument to that for~(\\ref{rh3}).\nCommutativity of the left  rectangles of the diagram is evident.\nCommutativity of\nthe top right rectangle  is clear from the definitions.\nCommutativity of the bottom right rectangle is\neasy to check using the identity\n\\begin{equation}\\label{sigmatau}\n(1-\\sigma\\tau)\\circ(\\sigma-1) \\ = \\ (\\sigma-1)\\circ(1+\\tau),\n\\end{equation}\nwhich follows from $(\\sigma\\tau)^2\n = \\operatorname{id}$. Note that for each column of the diagram,\nthe composition of the two maps is the squaring map.\n\n(v) For this part, the proof follows closely Yanchevski\\u\\i's proof in\n\\cite[4.13]{y}. (But our notational convention for products of functions is\n$fg = f\\circ g$, whereas his appears to be $fg = g\\circ f$.)\nSuppose $\\mathsf{E}_0$ is a field. For simplicity we denote\n$\\overline \\tau=\\tau|_{\\mathsf{E}_0}$ by $\\tau$. Take $a\\in\n\\Sigma_\\tau'(\\mathsf{E})\\cap \\mathsf{E}_0$. So,\n$N_{\\mathsf{E}_0\/\\mathsf{T}_0}(a^{1-\\tau})=1$. We will\nshow that $a \\in \\mathsf{E}_0^\\tau \\mathsf{E}_0^{\\sigma\\tau}$.\nIt then follows by (\\ref{rh2}) above\n that $\\SK(\\mathsf{E},\\tau)=1$. But since $\\mathsf{E}_0$ is cyclic over\n$\\mathsf{T}_0$, by  Hilbert~90 there is a $b \\in \\mathsf{E}_0^*$ such that\n$a^{\\tau-1}=b^{\\sigma-1}$ where $\\langle \\sigma \\rangle  =\\operatorname{Gal}(\\mathsf{E}_0\/\\mathsf{T}_0)$.\nSo, ${1=a^{(\\tau+1)(\\tau-1)}=b^{(\\tau+1)(\\sigma-1)}}$.\nAnalogously to \\eqref{sigmatau}, we have\n $(\\tau+1)(\\sigma-1)=(\\sigma-1)(1-\\tau\\sigma)$.\nSo $b^{(\\sigma-1)(1-\\tau\\sigma)}=1$.\nSetting $c=b^{(1-\\tau\\sigma)}$, we have $c^{\\sigma-1}=1$, so  $c\\in \\mathsf{T}_0$.\nBut, $N_{\\mathsf{T}_0\/\\mathsf{R}_0}(c)=c^{1+\\tau\\sigma}=\nb^{(1+\\tau\\sigma)(1-\\tau\\sigma)}=1$.  By Hilbert 90  we have\n$c=d^{\\tau\\sigma-1}$ for some $d\\in \\mathsf{T}_0^*$. Let $t=bd\\in  \\mathsf{E}_0^*$. Then,\n$t^{1-\\tau\\sigma}=b^{(1-\\tau\\sigma)}d^{(1-\\tau\\sigma)}=\nd^{(\\tau\\sigma-1)}d^{(1-\\tau\\sigma)}=1$, i.e.,  $t \\in \\mathsf{E}_0^{\\tau\\sigma}$.\nSo, $\\sigma(t) = \\tau(t) \\in \\mathsf{E}_0^{\\sigma\\tau}$.\nThus,  $a^{\\tau-1}=b^{\\sigma-1}=(t\/d)^{\\sigma-1}=t^{\\sigma-1}=\nt^{\\tau-1}$, as $d\\in \\mathsf{T}_0$. This shows that\n$(a\\tau(t))^{\\tau-1}=1$, i.e., $a\\tau(t) \\in \\mathsf{E}^\\tau$; hence\n$a=(a\\tau(t))\\tau(t)^{-1}\\in  \\mathsf{E}_0^\\tau\n\\mathsf{E}_0^{\\sigma\\tau}$.\n\\end{proof}\n\n\n\n\\section{Totally ramified algebras}\n\nFor a graded division algebra $\\mathsf{E}$ totally ramified over its center\n$\\mathsf{T}$ with a unitary graded involution $\\tau$, two possible cases can\narise: either $\\mathsf{T}$ is totally ramified over $\\mathsf{R}=\\mathsf{T}^\\tau$, or $\\mathsf{T}$ is\nunramified over $\\mathsf{R}$. In the first case, we showed in Prop.~\\ref{total}\nthat $\\SK(\\mathsf{E},\\tau)$ is trivial. We now obtain an\neasily computable explicit formula\nfor $\\SK(\\mathsf{E},\\tau)$ in the second case.\nFor a field $K$ and for $n\\in \\mathbb N$, we write $\\mu_n$ for the group of\nall $n$-th roots of unity in an algebraic closure of $K$.  Then set\n$\\mu_n(K) = \\mu_n \\cap K^*$.\n\n\n\\begin{theorem} \\label{sktotal} If $\\mathsf{E}$ is totally ramified over $\\mathsf{T}$ of\nindex $n$ and $\\mathsf{T}$ is unramified over $\\mathsf{R}$, then\n\\begin{align}\n\\SK(\\mathsf{E},\\tau)\\ & \\cong  \\  \\big \\{a\\in \\mathsf{T}_0^*\\mid a^n\\in \\mathsf{R}_0^*\\} \\,\\big \/\n \\, \\{a\\in \\mathsf{T}_0^*\\mid a^e\\in \\mathsf{R}_0^* \\} \\label{genesa}\n\\\\\n&\\cong  \\  \\big \\{\\omega \\in \\mu_n(\\mathsf{T}_0) \\mid \\tau(\\omega)\n=\\omega^{-1}\\big \\}\\big \/\\mu_e, \\label{genesa1}\n\\end{align}\nwhere $e$ is the exponent of $\\Gamma_\\mathsf{E}\/\\Gamma_\\mathsf{T}$. In particular,\n\\begin{enumerate}[\\upshape(i)]\n\\item \\label{hjh1} The restriction of the map\n$K_1(\\mathsf{E},\\tau)\\rightarrow K_1(\\mathsf{E})$ given by  $a\\Sigma_\\tau \\mapsto a^{1-\\tau}\\mathsf{E}'$, induces an  injective map\n$$\n\\alpha\\colon \\SK(\\mathsf{E},\\tau)\\longrightarrow \\SK(\\mathsf{E})\\ \\cong \\\n\\mu_n(\\mathsf{T}_0)\/\\mu_e.\n$$\n\\item \\label{hhh} If the exponent $e$ of $\\mathsf{E}$ is odd, then $\\alpha$ is an\nisomorphism.\n\n\\smallskip\n\n\\item  If  $e>2$ then $\\mathsf{T}_0=\\mathsf{R}_0(\\mu_e)$, and $\\tau$ acts on $\\mu_e$ by\n$\\omega\\mapsto \\omega^{-1}$.\n\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nSince $\\mathsf{T}$ is unramified over $\\mathsf{R}$ and $\\mathsf{E}_0=\\mathsf{T}_0$, the formulas of\nTh.~\\ref{msem} for $\\SK(\\mathsf{E},\\tau)$ reduce to $\\partial = n$ and\n\\begin{equation}\\label{gendesb}\n\\SK(\\mathsf{E},\\tau) \\ \\cong \\  \\{a\\in \\mathsf{T}_0^*\\mid a^n\\in \\mathsf{R}_0^* \\} \\,\n\\big \/ \\, \\big(\\mathsf{R}_0^*  \\,  \\langle x_\\gamma x_\\delta x_{\\gamma+\\delta}^{-1}\n\\mid \\gamma, \\delta \\in \\Gamma_\\mathsf{E} \\rangle\\big),\n\\end{equation}\nwhere each $x_\\gamma\\in \\mathsf{E}_\\gamma^*$ with $\\tau(x_\\gamma) = x_\\gamma$.\nRecall that as $\\mathsf{E}\/\\mathsf{T}$ is totally ramified,\nthe canonical pairing $\\mathsf{E}^*\\times \\mathsf{E}^* \\rightarrow \\mu_e(\\mathsf{T}_0)$\ngiven by  $(s,t)\\mapsto [s,t]$ is surjective\n(\\cite[Prop.~2.1]{hwcor}), and   $\\mu_e(\\mathsf{T}_0) = \\mu_e$,\ni.e., $\\mathsf{T}_0$ contains all $e$-th roots of unity.\nSince each $\\mathsf{E}_\\gamma = \\mathsf{T}_0 x_\\gamma$ with $\\mathsf{T}_0$ central,\nit follows  that\n$\\{[x_\\delta,x_\\gamma] \\mid \\gamma, \\delta  \\in \\Gamma_\\mathsf{E}\\} = \\mu_e$.\nNow consider $c=x_\\gamma x_\\delta x_{\\gamma+\\delta}^{-1}$ for any $\\gamma,\n\\delta\\in\\Gamma_\\mathsf{E}$. Then,  $\\tau(c)=x_{\\gamma+\\delta}^{-1}x_\\delta x_\\gamma$.\nNote that $x_\\delta x_\\gamma$ and $x_{\\gamma+\\delta}$\neach lie in $\\mathsf{E}_{\\gamma+\\delta} = \\mathsf{T}_0x_{\\gamma+\\delta}$,\nso they commute.  Hence,\n\\begin{equation}\\label{c}\n\\tau(c) c^{-1} \\, = \\ x_{\\gamma+\\delta}^{-1} (x_\\delta x_\\gamma)\nx_{\\gamma+\\delta}x_\\delta^{-1}x_\\gamma^{-1}\n \\ = \\  [x_\\delta,x_\\gamma].\n\\end{equation}\n  Since  $[x_\\delta,x_\\gamma] \\in \\mu_e$,\nthis shows that\n$c \\in \\big \\{a\\in \\mathsf{T}_0^*\\mid a^e\\in \\mathsf{R}_0^*\\big \\}$. For the reverse\ninclusion, take any\n$d$~in~$\\mathsf{T}_0^*$ such that $d^e\\in \\mathsf{R}_0^*$.\nSo $\\tau(d)d^{-1} \\in \\mu_e$. Thus,\n$\\tau(d) d^{-1}= [x_\\delta,x_\\gamma]$, for some $\\gamma, \\delta\\in \\Gamma_\\mathsf{E}$.\n Taking ${c=x_\\gamma x_\\delta x_{\\gamma+\\delta}^{-1}}$, we have\n$\\tau(d) d^{-1}=\\tau(c) c^{-1}$\nby \\eqref{c},  which implies that $dc^{-1}$ is\n$\\tau$-stable, so lies in $\\mathsf{R}_0^*$;\n thus, ${d \\in \\mathsf{R}_0^*  \\,  \\langle x_\\gamma x_\\delta x_{\\gamma+\\delta}^{-1}\n\\mid \\gamma, \\delta \\in \\Gamma_\\mathsf{E} \\rangle}$. Therefore,\n$\\mathsf{R}_0^*  \\,  \\langle x_\\gamma x_\\delta x_{\\gamma+\\delta}^{-1} \\mid\n\\gamma, \\delta \\in \\Gamma_\\mathsf{E} \\rangle\n=\\{a\\in \\mathsf{T}_0^*\\mid a^e\\in \\mathsf{R}_0^* \\}$. Inserting this in ~(\\ref{gendesb}) we\nobtain (\\ref{genesa}).\n\n(i) Consider the well-defined map\n$\\alpha\\colon\\SK(\\mathsf{E},\\tau)\\rightarrow \\SK(\\mathsf{E})$ given by\n$a\\Sigma_\\tau\\mapsto a^{1-\\tau}\\mathsf{E}'$ (see  diagram~\\eqref{goodd} for the\nnon-graded version).\nBy~\\cite[Cor.~3.6(ii)]{hazwadsworth},\n$\\SK(\\mathsf{E})\\cong \\mu_n(\\mathsf{T}_0)\/\\mu_e$. Taking into account\nformula ~\\eqref{genesa} for $\\SK(\\mathsf{E},\\tau)$,\nit is easy to see that $\\alpha$ is  injective.\n\nWe now verify that\n\\begin{equation}\\label{imalpha}\n\\operatorname{im}(\\alpha) \\ = \\ \\big \\{\\omega \\in \\mu_n(\\mathsf{T}_0) \\mid\n\\tau(\\omega)=\\omega^{-1}\\big \\} \\, \\big \/\\mu_e,\n\\end{equation}\nand thus obtain ~(\\ref{genesa1}).\nIndeed, since $\\mu_e =  \\{[x_\\delta,x_\\gamma] \\mid \\gamma, \\delta\n\\in \\Gamma_\\mathsf{E}\\}$,  by setting\n$c=x_\\gamma x_\\delta x_{\\gamma+\\delta}^{-1}$ we have\n${[x_\\delta,x_\\gamma]=\\tau(c) c^{-1}}$ by \\eqref{c}. This shows that $\\mu_e \\subseteq \\big \\{\\omega \\in \\mu_n(\\mathsf{T}_0) \\mid \\tau(\\omega)=\\omega^{-1}\\big \\}$. Now for\nany $\\omega \\in \\mu_n(\\mathsf{T}_0)$ with $\\tau(\\omega)=\\omega^{-1}$,\nwe have $N_{\\mathsf{T}_0\/\\mathsf{R}_0}(\\omega)=1$, so  Hilbert~90 guarantees that\n$\\omega=c^{1-\\tau}$ for some $c\\in \\mathsf{T}_0^*$. Then,\n$(c^n)^{1-\\tau} = \\omega^n = 1$, so $c^n\\in \\mathsf{R}_0^*$.\nThus,\n$c\\in \\Sigma_\\tau'$, and clearly  $\\alpha(c\\Sigma_\\tau)=\\omega\\mu_e$.\nThis shows $\\supseteq$ in \\eqref{imalpha}; the reverse inclusion is clear\nfrom  the definition of $\\alpha$.\n\n\n(ii)\nSuppose $e$ is odd.  Let $m = |\\mu_n(\\mathsf{T}_0)|$.  So, $\\mu_n(\\mathsf{T}_0) = \\mu_m$,\nwith $m \\,|\\, n$.  Also, $e\\,|\\, m$, as $\\mu_e\\subseteq \\mathsf{T}_0$.  Since $e$ and $n$\nhave the same prime factors, this is also true for $e$ and $m$.  Recall that\n$\\operatorname{Aut}(\\mu_m)\\cong (\\mathbb Z\/m\\mathbb Z)^*$,  the multiplicative group of units of the ring\n$\\mathbb Z\/m\\mathbb Z$; so, $|\\operatorname{Aut}(\\mu_m)| = \\varphi(m)$, where $\\varphi$ is Euler's\n$\\varphi$-function. Since $e\\,|\\, m$ and $e$ and $m$ have the same prime\nfactors (all odd), the canonical map $\\psi\\colon \\operatorname{Aut}(\\mu_m) \\to\n\\operatorname{Aut}(\\mu_e)$ given by restriction is surjective with kernel of order\n$\\varphi(m)\/\\varphi(e) = m\/e$, which is odd.  Therefore,\n$\\psi$ induces an isomorphism on the $2$-torsion subgroups,\n$_2\\!\\operatorname{Aut}(\\mu_m) \\cong \\ _2\\!\\operatorname{Aut}(\\mu_e)$.  Now, $\\tau|_{\\mu_m}\n\\in \\, _2\\!\\operatorname{Aut}(\\mu_m)$ and we saw for~(i) that $\\tau|_{\\mu_e}$\nis the inverse map $\\omega \\mapsto \\omega^{-1}$.  The inverse map\non $\\mu_m$ also lies in $_2\\!\\operatorname{Aut}(\\mu_m)$ and has the same restriction\nto $\\mu_e$ as $\\tau$.  Hence, $\\tau|_{\\mu_m}$ must be the inverse map.\nThat is, $\\{\\omega \\in \\mu_n(\\mathsf{T}_0) \\mid\n\\tau(\\omega)=\\omega^{-1} \\} = \\mu_n(\\mathsf{T}_0)$.  Therefore,\n\\eqref{imalpha} above shows that $\\operatorname{im}(\\alpha) = \\mu_n(\\mathsf{T}_0)\/\\mu_e$,\nwhich we noted above is isomorphic to $\\SK(\\mathsf{E})$.\n\n(iii) We saw in the proof of part (i) that\n$\\tau$ acts on $\\mu_e$ by the inverse map.  So,\nif $e>2$, then $\\mu_e \\not \\subseteq \\mathsf{R}_0$.\nSince $[\\mathsf{T}_0:\\mathsf{R}_0]=2$, it then follows that\n$\\mathsf{T}_0=\\mathsf{R}_0(\\mu_e)$.\n\\end{proof}\n\n\\begin{remark}\nThe isomorphism $\\SK(\\mathsf{E},\\tau)\\cong \\SK(\\mathsf{E})$ of part (\\ref{hhh}) of the\nabove theorem can be obtained under the milder condition that\n$\\mathsf{E}_0=\\mathsf{T}_0\\mathsf{E}'$ provided that the exponent of $\\mathsf{E}$ is a prime power. The\nproof is similar.\n\\end{remark}\n\n\n\n\n\\begin{example} \\label{toex}\nLet $r_1, \\ldots, r_m$ be integers with each\n$r_i \\ge 2$.  Let $e = \\text{lcm}(r_1, \\ldots, r_m)$, and let\n$n = r_1\\ldots r_m$.  Let $C$ be any field such that $\\mu_e\n\\subseteq C$ and $C$ has an automorphism $\\theta$\nof order $2$ such that\n$\\theta(\\omega) = \\omega^{-1}$ for all $\\omega\\in \\mu_e$.\nLet $R$ be the fixed field $C^\\theta$.\nLet  $x_1,\\dots, x_{2m}$ be $2m$ independent indeterminates, and let\n$K$ be the  iterated Laurent power series field $C((x_1))\\dots ((x_{2m}))$.\nThis $K$~is equipped with its standard valuation\n$v\\colon K^* \\rightarrow \\mathbb Z^{2m}$ where $\\mathbb Z^{2m}$ is given\nthe right-to-left lexicographical ordering. With this valuation $K$ is\nhenselian (see~\\cite[p.~397]{wadval}). Consider the tensor product of\nsymbol algebras\n\\begin{equation*}\nD\\ = \\ \\Big (\\frac{x_{1},x_{2}}{K}\\Big )_{\\omega_1} \\otimes_K\n\\ldots \\otimes_K\n\\Big (\\frac{x_{2m-1},x_{2m}}{K}\\Big )_{\\omega_m},\n\\end{equation*}\nwhere  for $1\\leq i \\leq m$, $\\omega_i$ is a primitive $r_i$-th root of unity\nin $C$.\nUsing the valuation theory developed for division algebras, it is known that\n$D$ is a division algebra, the valuation $v$ extends to $D$, and $D$ is\ntotally ramified over $K$ (see \\cite[Ex.~4.4(ii)]{wadval} and\n\\cite[Ex.~3.6]{tw}) with\n$$\n\\Gamma_D\/\\Gamma_K \\ \\cong \\\n\\textstyle{\\prod\\limits_{i=1}^m} (\\mathbb Z\/r_i \\mathbb Z)\\times\n(\\mathbb Z\/r_i\\mathbb Z),\n$$\nand $\\overline D = \\overline K\\cong  C$.\nExtend $\\theta$ to an automorphism  $\\theta'$ of order $2$ on $K$ in the obvious\nway, i.e., acting by~$\\theta$  on the coefficients of a Laurent series, and with\n$\\theta'(x_i) = x_i$ for $1\\le i\\le 2m$.\nOn each of the symbol algebras\n$\\Big (\\frac{x_{2i-1},x_{2i}}{K}\\Big )_{\\omega_i}$ with its generators $\\mathbf{i}_i$\nand $\\mathbf{j}_i$ such that $\\mathbf{i}_i^{r_i}=x_{2i-1}$, $\\mathbf{j}_i^{r_i}=x_{2i}$, and\n${\\mathbf{i}_i\\mathbf{j}_i=\\omega_i\\mathbf{j}_i\\mathbf{i}_i}$, define an involution  $\\tau_i$ as follows:\n$\\tau_i(c \\, \\mathbf{i}_i^k \\mathbf{j}_i^l)= \\theta'(c)  \\, \\mathbf{j}_i^l \\mathbf{i}_i^k$, where $c\\in K$ and $0\\leq l,k < r_i$.\n Clearly\n${K^{\\tau_i}= K^{\\theta'} = R((x_1))\\dots ((x_{2m}))}$, and therefore $\\tau_i$~is a\nunitary involution.  Since the $\\tau_i$ agree\non $K$ for $1\\le i\\le m$, they yield   a unitary involution\n$\\tau=\\otimes_{i=1}^m\\tau_i$\non $D$. Now by Th.~\\ref{involthm2},\n$\\SK(D,\\tau)\\cong\\SK(\\operatorname{{\\sf gr}}(D),\\widetilde \\tau)$. Since\n$D$ is totally ramified over $K$, which is unramified over $K^{\\tau}$,\nwe have correspondingly that $\\operatorname{{\\sf gr}}(D)$ is totally ramified over~$\\operatorname{{\\sf gr}}(K)$,\nwhich is unramified over $\\operatorname{{\\sf gr}}(K)^{\\widetilde \\tau}$.\nAlso, $\\operatorname{{\\sf gr}}(K)_0 \\cong \\overline K \\cong  C$.\nWe have ${\\exp(\\operatorname{{\\sf gr}}(D)) = \\exp(D) = \\exp(\\Gamma_D\/\\Gamma_K) =\n\\text{lcm}(r_1, \\ldots, r_m) = e}$ and\n$ {\\operatorname{ind}}(\\operatorname{{\\sf gr}}(D)) = \\operatorname{ind}(D) = r_1\\ldots r_m = n$.\nBy Th.~\\ref{sktotal},\n$$\n\\SK(D,\\tau) \\ \\cong \\ \\SK(\\operatorname{{\\sf gr}}(D),\\widetilde \\tau)\\ \\cong \\\n\\{ \\omega\\in \\mu_n(C) \\mid \\theta(\\omega) = \\omega^{-1}\\}\\big\/ \\mu_e,\n$$\nwhile by \\cite[Th.~4.8, Cor.~3.6(ii)]{hazwadsworth},\n$$\n\\SK(D) \\ \\cong \\ \\SK(\\operatorname{{\\sf gr}}(D))\\ \\cong \\mu_n(C)\/\\mu_e .\n$$\nHere are some more specific examples:\n\n(i) Let $C = \\mathbb C$, the complex numbers, and let\n$\\theta$ be complex conjugation, which maps every root of\nunity to its inverse. So, $R = C^\\theta = \\mathbb R$.\nThen, $\\SK(D, \\tau) \\cong \\SK(D) \\cong \\mu_n\/\\mu_e \\cong\\mathbb Z\n\/(n\/e)\\mathbb Z$.\n\n(ii) Let $r_1 = r_2 = 4$, so $e = 4$ and $n = 16$.  Let\n$\\omega_{16}$ be a primitive sixteenth root of unity in $\\mathbb C$,\nand let\n${C = \\mathbb Q(\\omega_{16})}$, the sixteenth\ncyclotomic extension of $\\mathbb Q$.  Recall that ${\\operatorname{Gal}(C\/\\mathbb Q)\n \\cong \\operatorname{Aut}(\\mu_{16}) \\cong (\\mathbb Z\/4\\mathbb Z) \\times (\\mathbb Z\/2\\mathbb Z)}$,\nLet $\\theta\\colon C \\to C$ be the automorphism which\nmaps  $\\omega_{16} \\mapsto (\\omega_{16})^7$. Then,\n$\\theta^2 = \\operatorname{id}_C$, as $7^2 \\equiv 1 \\ (\\text{mod }16)$,\nand $\\{\\omega \\in  \\mu_{16}\\mid \\theta(\\omega) = \\omega^{-1}\\}\n = \\mu_8$.  Thus, $\\SK(D, \\tau) \\cong \\mu_8\/\\mu_4 \\cong\n\\mathbb Z\/2\\mathbb Z$, while $\\SK(D) \\cong \\mu_{16}\/\\mu_4 \\cong \\mathbb Z\/4\\mathbb Z$.\nSo, here the injection  $\\SK(D,\\tau) \\to \\SK(D)$ is not surjective.\n\n(iii) Let $r_1 = \\ldots = r_m = 2$, so $e = 2$ and $n = 2^m$.\nHere, $C$ could be any quadratic extension of any field~\n$R$ with $\\operatorname{char}(R) \\ne 2$. Take $\\theta$ to be the unique\nnonidentity $R$-automorphism of $C$.  The resulting\n$D$ is a tensor product of\n$m$ quaternion algebras  over $C((x_1))\\ldots((x_{2m}))$,\nand ${\\SK(D,\\tau) \\cong\n\\{\\omega\\in \\mu_{2^m}(C)\\mid \\theta(\\omega) = \\omega^{-1}\\}\\big\/\n\\mu_2}$, while $\\SK(D) \\cong \\mu_{2^m}(C)\/\\mu_2$.\n\\end{example}\n\nEx.~\\ref{toex} gives an indication how to use the graded approach to\nrecover results in the literature on\nthe unitary $\\SK$ in a unified\nmanner and to extend them from division algebras with discrete valued groups to\n arbitrary valued groups.  While $\\SK(D)$ has long been known for the $D$\nof Ex.~\\ref{toex},\nthe formula for $\\SK(D,\\tau)$ is new.\n\nHere is a more complete statement of what the results in the preceding sections\n yield for\n$\\SK(D, \\tau)$ for valued division algebras $D$.\n\n\n\\begin{theorem}\\label{appl}\nLet $(D,v)$ be a tame valued division algebra over a  field $K$\nwith $v|_K$ henselian,\nwith a unitary involution $\\tau$; let $F=K^\\tau$, and suppose $v|_F$ is\nhenselian  and that $K$ is tamely\nramified over $F$. Let $\\overline \\tau$ be the involution on $\\overline D$\ninduced by $\\tau$.\nThen,\n\\begin{enumerate}\n  \\item[$(1)$] Suppose $K$ is unramified over $F$.\n\\begin{enumerate}[\\upshape(i)]\n  \\item \\label{apun} If $D$ is unramified over $K$, then\n$\\SK(D,\\tau)\\cong \\SK(\\overline D,\\overline \\tau)$.\n\\smallskip\n  \\item \\label{tolast} If $D$ is totally ramified over $K$, let $e = \\exp(D)$\nand $n = \\operatorname{ind}(D)$; then,\n  $$\n\\SK(D,\\tau) \\ \\cong  \\  \\{\\omega \\in \\mu_n(\\overline K) \\mid \\tau(\\omega) =\n\\omega^{-1}\\}\\big \/\\mu_e,\n$$\nwhile $\\SK(D) \\cong \\mu_n(\\overline K)\/\\mu_e$.\n\\smallskip\n  \\item \\label{tirl} If $D$ has a maximal graded subfield\n$M$ unramified over $K$ and another maximal graded subfield $L$ totally\nramified over $K$, with $\\tau(L ) =L$, then $D$ is semiramified  and\n\\begin{equation*}\\label{semirsk3}\n\\SK(D,\\tau) \\ = \\ \\big\\{a \\in \\overline D^* \\mid N_{\\overline D\/\\overline K}(a)\\in \\overline F\\big\\}\n \\, {\\big\/} \\,  \\textstyle{\\prod\\limits_{h\\in \\operatorname{Gal}(\\overline D\/\\overline K)}}\n\\overline F^{*h\\overline \\tau}.\n\\end{equation*}\n  \\item\\label{gamcyclic} Suppose $\\Gamma_D\/\\Gamma_K$ is cyclic.  Let\n$\\sigma$ be a generator of $\\operatorname{Gal}(Z(\\overline D)\/\\overline K)$.  Then,\n$$\n\\SK(D, \\tau) \\ \\cong \\ \\{ a\\in \\overline D^*\\mid N_{Z(\\overline D)\/\\overline K}(\\Nrd_{\\overline D}(a))\n\\in \\overline F\\} \\, \\big\/ \\, \\big(\\Sigma_{\\overline \\tau}(\\overline D) \\cdot\n\\Sigma_{\\sigma\\overline \\tau}(\\overline D)\\big).\n$$\n  \\item \\label{apsem} If $D$ is  inertially split, $\\overline D$ is a field and\n$\\operatorname{Gal}(\\overline D\/\\overline K)$ is cyclic, then $\\SK(D,\\tau)=1$.\n\\end{enumerate}\n\\medskip\n  \\item[$(2)$] \\label{apto} If $K$ is totally ramified over $F$, then\n$\\SK(D,\\tau)=1$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nLet $\\operatorname{{\\sf gr}}(D)$ be the associated graded division algebra  of $D$.\nThe tameness assumptions assure that $\\operatorname{{\\sf gr}}(K)$ is the center of\n$\\operatorname{{\\sf gr}}(D)$ with $[\\operatorname{{\\sf gr}}(D):\\operatorname{{\\sf gr}}(K)] = [D:K]$ and that the graded involution\n$\\widetilde\\tau$ on $\\operatorname{{\\sf gr}}(D)$ induced by $\\tau$ is unitary with\n$\\operatorname{{\\sf gr}}(K)^{\\widetilde \\tau} = \\operatorname{{\\sf gr}}(K^\\tau)$. In each case of Th.~\\ref{appl},\nthe conditions on $D$ yield analogous conditions on $\\operatorname{{\\sf gr}}(D)$.\nSince by Th.~\\ref{involthm2}, $\\SK(D,\\tau)\\cong \\SK(\\operatorname{{\\sf gr}}(D),\\widetilde \\tau)$,\n(2) and (1)(v) follow immediately  from Prop.~\\ref{completely}\nand Prop.~\\ref{cyclic}(\\ref{unrkl}), respectively. Part (1)(i),\nalso follows from\nTh.~\\ref{involthm2}, and Cor.~\\ref{unramified} as follows:\n$$\n\\SK(D,\\tau)  \\ \\cong \\ \\SK(\\operatorname{{\\sf gr}}(D),\\widetilde{\\tau})\n  \\ \\cong \\ \\SK(\\operatorname{{\\sf gr}}(D)_0,\\tau|_{\\operatorname{{\\sf gr}}(D)_0}) \\ = \\\n \\SK(\\overline{D},\\overline{\\tau}).\n$$\nParts (1)(ii), (1)(iii),  and (1)(iv) follow similarly using\nTh.~\\ref{sktotal}, Cor.~\\ref{seses}, and Prop.~\\ref{cyclic}\\eqref{rh2}\nrespectively.\n\\end{proof}\n\nIn the special case  that the henselian valuation\non $K$ is discrete (rank $1$),\nTh.~\\ref{appl}~(1)(i), (iii), (iv), (v)  and (2) were obtained by\nYanchevski\\u\\i~ \\cite{y}.  In this discrete case, the assumption\nthat $v$ on $K$ is henselian already implies that $v|_F$ is henselian\n(see Remark~\\ref{shensel}).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nObservations of high-redshift quasars reveal that supermassive\nblack holes (SMBHs) with mass of $\\ga 10^9~M_\\odot$ already exist\nas early as redshifts $z\\ga 6$\n\\citep{2006NewAR..50..665F,2007AJ....134.2435W,2011Natur.474..616M}.\nGas accretion and mergers of the remnant black holes (BHs) formed by the collapse of\nfirst-generation stars ($\\sim 100~M_\\odot$) have been considered for\nproducing such SMBHs\n\\citep[e.g.][]{2001ApJ...552..459H,2003ApJ...582..559V,2007ApJ...665..187L}.\nHowever, various forms of radiative feedback can prevent efficient BH\ngrowth, making it difficult to reach $\\ga 10^9~M_\\odot$ within the age\nof the high-redshift Universe\n\\citep{2007MNRAS.374.1557J, 2009ApJ...701L.133A, 2009ApJ...696L.146M,2012ApJ...747....9P,2012MNRAS.425.2974T}.\n\nOne possible alternative solution is the rapid formation of\nsupermassive stars (SMSs; $\\ga 10^5~M_\\odot$) and their subsequent\ncollapse directly to massive BHs in the first galaxies\n\\citep[e.g.,][]{1994ApJ...432...52L,2006MNRAS.370..289B,2006MNRAS.371.1813L}.\nThe massive seed BHs, formed by direct collapse, shorten the total\nSMBH growth time sufficiently, even in the presence of subsequent\nradiative feedback (e.g. \\citealt{th2009,2012ApJ...745L..29D}).\n\nSMSs can form out of primordial gas in massive `atomic cooling'\nhaloes with virial temperatures $T_{\\rm vir}\\ga 10^4$K, if H$_2$\nformation and line cooling are prohibited through the pre-stellar\ncollapse.  Possible mechanisms to suppress H$_2$ formation are\nphoto dissociation by far-ultraviolet (FUV) radiation\n\\citep{2001ApJ...546..635O,2003ApJ...596...34B,2008MNRAS.391.1961D,\n2009MNRAS.396..343R,2010MNRAS.402.1249S,\n2011MNRAS.416.2748I,2011MNRAS.418..838W,2013MNRAS.428.1857J} and\ncollisional dissociation\n\\citep{2012MNRAS.422.2539I,2014MNRAS.439.3798F}.  In the absence of\n${\\rm H_2}$, the primordial gas remains warm ($\\sim8,000$K) and may\ncollapse monolithically without strong fragmentation\n\\citep{2003ApJ...596...34B,2010MNRAS.402.1249S,\n  2013MNRAS.433.1607L,2014arXiv1404.4630I}.  The resulting central\nprotostar can grow via accretion at a high rate of $\\ga 0.1~M_\\odot~{\\rm yr}^{-1}$.\nIf the accretion rate maintains such a high value, the embryonic\nprotostar evolves to a SMS with mass of $\\ga 10^5~M_\\odot$.  The SMS\ncollapses as a whole to a single BH either directly\n\\citep{2002ApJ...572L..39S}, or via the intermediate stage of a close\nbinary BH \\citep{2013PhRvL.111o1101R}, through a general relativistic\ninstability \\citep[e.g.,][]{1971reas.book.....Z,1983bhwd.book.....S}.\nThe massive remnant BH is then a promising seed that can grow into one\nof the observed SMBHs at $z\\approx 6-7$.\n\n\nIn the scenario above, a major unresolved question is whether rapid\naccretion will continue unabated through the protostellar evolution.\nRecent high-resolution simulations by \\cite{2014MNRAS.439.1160R}\nsuggest that the compact nuclear accretion disc, surrounding a central\nembryonic protostar, is gravitationally unstable.  Thus,\nself-gravitating clumps are expected to form during the early stages\nof the accretion phase.  Such efficient fragmentation could prevent\nthe rapid growth of the central protostar, and preclude eventual SMS\nformation.  This is analogous to the fragmentation of gravitationally\nunstable discs in the cores of lower mass `minihaloes', which had\nbeen suggested to reduce the characteristic masses of Population III (Pop III)\nstars \\citep{2010MNRAS.403...45S, greif+11}.  Moreover, if the gas is slightly polluted by\nheavy elements ($Z\\la 10^{-4}~Z_\\odot$), dust cooling can decrease the\ntemperature and induce the fragmentation \\citep{2008ApJ...686..801O},\nwhich would be a further obstacle to SMS formation.  Because of this,\nthe existence of the pristine, metal-free gas has often been\nconsidered as a necessary condition of forming a SMS in\nsemi-analytical models \\citep[e.g.,][]{2012MNRAS.425.2854A,2014MNRAS.442.2036D}.\nWe note that even very efficient fragmentation may not prevent accretion \non to a central point source \\citep[e.g.,][]{2007ApJ...671.1264E} and\nthe rapid inward migration of the fragments, on a time-scale comparable \nto the orbital period, may help its growth further \n\\citep[e.g.,][]{2010MNRAS.404.2151C}.\nIn our context, however, even a slow down of the accretion rate could bring it \nbelow the critical value required for SMS formation.\n\nIn this paper, motivated by the above, we discuss the expected properties of a compact,\nmarginally unstable nuclear protogalactic disc, and the fate of the\nclumps formed in the disc by gravitational instability.  Using\nanalytical models, we argue that despite fragmentation, the growth of\na SMS remains the most likely outcome. The reason for\nthis conclusion is the rapid inward migration of the fragments, and\ntheir merger with the central protostar.\n\nThe rest of this paper is organized as follows.  In \\S~\\ref{sec:2}, we\ndescribe the basic model of the fragmenting disc and of the clumps\nformed in the disc.  In \\S~\\ref{sec:fate}, we discuss the fate of the\nclumps, considering various important processes: migration, accretion,\ncontraction, and star-formation by reaching the zero-age main sequence\n(ZAMS).  We also consider the possibility that radiative feedback from\nmassive stars, formed from the clumps, may halt gas accretion on to the\nwhole system.  In \\S~\\ref{sec:metal}, we discuss whether\nmetal pollution and associated dust cooling could prohibit SMS\nformation.  Finally, we discuss our results and summarize our\nconclusions in \\S~\\ref{sec:conc}.\n\n\n\n\n\n\\section{Fragmentation of the accretion disc around a supermassive star}\n\\label{sec:2}\n\n\\subsection{Basic equations}\n\nWe consider the properties of a disc formed after the collapse of\nprimordial gas inside an atomic cooling halo,\nwhen molecular hydrogen formation is suppressed.\nSince the parent gas\ncloud is hot ($T\\sim 8000$ K), the accretion rate on to the disc is\nhigh:\n\\begin{equation}\n\\dot M_{\\rm tot} \\sim \\frac{c_{\\rm s}^3}{G}\\sim 0.1~M_\\odot~{\\rm yr}^{-1} \\left(\\frac{T}{8000~{\\rm K}}\\right)^{3\/2},\n\\end{equation}\nwhere $c_{\\rm s}$ is the sound speed and $G$ is the gravitational\nconstant \\citep{1977ApJ...214..488S, 1986ApJ...302..590S}.  The\naccretion disc around the central protostar can become unstable\nagainst self-gravity.  To understand the stability of the disc,\nToomre's parameter \\citep{1964ApJ...139.1217T} defined by\n\\begin{equation}\nQ=\\frac{c_{\\rm s}\\Omega}{\\pi G\\Sigma}\n\\label{eq:Q}\n\\end{equation}\nis often useful, where $\\Omega$ is the orbital frequency and $\\Sigma$\nis the surface density of the disc.  In the marginal case of $1\\la Q\n\\la 2$, strong spiral arms are formed in the disc.  The spiral arms\nredistribute angular momentum and heat the disc by forming shocks\n\\citep[e.g.,][]{2005ApJ...633L.137V,2006ApJ...650..956V}.\nThe resulting disc is self-regulated to the marginal state. \nThus, we here assume $Q\\simeq 1$.\n\n\nSince the gas temperature in the atomic cooling halo is kept near\n$\\sim 8000$ K, the external accretion rate on to the disc, from larger\nradii, is also nearly constant.  We estimate the surface density of\nthe self-regulated disc by assuming that it is in steady state, and\nthat it has an effective viscosity $\\nu$ (arising from gravitational\ntorques),\n\\begin{equation}\n\\Sigma=\\frac{\\dot M_{\\rm tot}}{3\\pi \\nu}.\n\\label{eq:continue}\n\\end{equation}\nThe scale-height of the disc is estimated from vertical hydrostatic\nbalance,\n\\begin{equation}\nH=\\frac{c_{\\rm s}}{\\Omega}. \n\\label{eq:scaleheight}\n\\end{equation}\nWe define the particle number density as $n\\equiv\\Sigma \/(2m_{\\rm\n  p}H)$, where $m_{\\rm p}$ is the proton mass.\n\n\\begin{figure*}\n\\vspace{1mm}\n\\begin{center}\n\\begin{tabular}{ccc}\n \\begin{minipage}{0.33\\hsize}\n  \\begin{center}\n  \\vspace{-1mm}\n   \\includegraphics[width=54mm]{temp_Hm_1.eps}\n  \\end{center}\n \\end{minipage}\n\\begin{minipage}{0.32\\hsize}\n \\begin{center}\n   \\includegraphics[width=54.5mm]{Sigma_d_Hm_1.eps}\n  \\end{center}\n \\end{minipage}\n\\begin{minipage}{0.32\\hsize}\n \\begin{center}\n   \\includegraphics[width=55mm]{alpha_Hm_1.eps}\n  \\end{center}\n \\end{minipage}\n\\end{tabular}\n\\caption{Radial profiles of disc quantities, shown as a function of\n  the orbital period: (a) temperature, (b) surface density, and (c)\n  viscous parameter.  The solid curves show profiles with H$^-$\n  free-bound emission as the only radiative cooling process.  The\n  dashed curves additionally include Ly$\\alpha$ emission.  The\n  horizontal dot-dashed line in panel (c) indicates the critical value\n  of the viscous parameter for fragmentation, $\\alpha_{\\rm f}=1$.  The\n  shaded bar marks the location of the fragmenting radius at\n  $P_{\\rm f}=2\\pi\/\\Omega_{\\rm f}=10^4$ yr (or $\\Omega_{\\rm f}=2\\times 10^{-11}$\n  s$^{-1}$).  }\n  \\label{fig:profile}\n\\end{center}\n\\end{figure*}\n\n\nTo determine the thermal state of the disc we assume that it is in\nequilibrium, i.e. we balance heating and radiative cooling\n($Q_+=Q_-$).  In particular, we consider viscous heating by turbulence\nand spiral shocks,\n\\begin{align}\nQ_+&=\\frac{9}{4}\\nu \\Sigma \\Omega ^2,\n\\end{align}\nand radiative cooling\n\\begin{align}\nQ_-&=2H\\Lambda,\n\\end{align}\nwhere $\\Lambda$ is the cooling rate in units of erg s$^{-1}$\ncm$^{-3}$.  For the cooling process, we consider H$^-$ free-bound\nemission, which is the dominant channel in the thermal evolution at\n$n\\ga 10^8~{\\rm cm}^{-3}$ for a warm primordial gas in an atomic cooling halo.\nThe form of the cooling rate is given in \\S~\\ref{sec:chem}.\n\nFinally, we adopt the standard $\\alpha$-prescription\n\\citep{1973A&A....24..337S} as a model for the viscosity by\ngravitational torques:\n\\begin{equation}\n\\nu = \\alpha c_{\\rm s}H,\n\\label{eq:viscous}\n\\end{equation}\nwhere $\\alpha$ is the viscous parameter.  From hydrodynamical\nsimulations of a self-gravitating disc, the disc is found to be\nsusceptible to fragmentation if the viscous parameter exceeds a\ncritical threshold value, $\\alpha \\ga \\alpha_{\\rm f}$\n\\citep[e.g.,][]{2001ApJ...553..174G}.  \\cite{2005MNRAS.364L..56R} have\nstudied fragmentation conditions for various specific heat ratios, and\nobtained $\\alpha_{\\rm f}\\sim 0.06$.  However, the critical value\ndepends on both initial conditions and on numerical resolution\n\\citep{2011MNRAS.410..559M, 2011MNRAS.411L...1M}.  Recently,\n\\cite{2012ApJ...746..110Z} have considered mass loading from an\ninfalling envelope, realistic radiative cooling, and radiative\ntrapping of energy inside clumps, and suggested that the critical\nvalue for fragmentation is around $\\alpha_{\\rm f}\\sim 1$. Moreover,\nnumerical simulations of fragmentation of discs with relatively high\naccretion rates ($\\dot M_{\\rm tot}\\sim 10^{-4}-10^{-2}~M_\\odot~{\\rm yr}^{-1}$), in\nthe context of present-day massive star formation \\citep[e.g.,\n][]{2007ApJ...656..959K} and Pop III star formation\n\\citep{2011Sci...331.1040C} have shown that the effective viscous\nparameter arising from gravitational torques is $\\sim 0.1-1$.  Based\non these results, we here set $\\alpha _{\\rm f}=1$ as our fiducial\nvalue, and analyze disc fragmentation using the condition of $\\alpha\n\\geq \\alpha_{\\rm f}$ following previous analytical works\n\\citep[e.g.,][]{2005ApJ...621L..69R, 2007MNRAS.374..515L}.\n\n\n\\subsection{Radiative cooling and chemistry}\n\\label{sec:chem}\n\nIn dense ($n\\ga 10^8~{\\rm cm}^{-3}$) and warm ($3000\\la T\\la 8000$ K) primordial\ngas, the dominant cooling processes is the free-bound emission of\nH$^-$ ions (H + e$^-$ $\\rightarrow$ H$^-$ + $\\gamma$;\n\\citealt{2001ApJ...546..635O,2014arXiv1404.4630I}). The cooling rate,\nin units of erg s$^{-1}$ cm$^{-3}$, is given by\n\\begin{equation}\n\\Lambda =\\lambda(T)n^2x_{\\rm e},\n\\end{equation}\nwhere $x_{\\rm e}(\\ll1)$ is the free-electron fraction and\n$\\lambda(T)\\simeq 10^{-30}~ T$ (with $T$ in units of Kelvin).  \nIn such a warm gas, prior to protostar formation, \nthe electron fraction is determined by the balance between\nrecombination (H$^+$ + e$^-$ $\\rightarrow$ H + $\\gamma$) and\ncollisional ionization (2H $\\rightarrow$ H$_2^+$ + e$^-$).\nThe evolution of the electron fraction follows the equation\n\\begin{equation}\n\\frac{dx_{\\rm e}}{dt}=-\\alpha_{\\rm rec}nx_{\\rm e}^2 + \\alpha_{\\rm ci}n.\n\\end{equation}\nAt the densities of interest here, the electron fraction is in equilibrium,\nand given simply by \n\\begin{equation}\nx_{\\rm e}=\\sqrt{\\frac{\\alpha_{\\rm ci}}{\\alpha_{\\rm rec}}}.\n\\end{equation}\nFrom Appendix \\ref{sec:appA}, we obtain the specific value of\n\\begin{align}\nx_{\\rm e}&\\simeq f(T)~n^{1\/2} \\exp(-\\epsilon\/2T)\\label{eq:x_e}\\\\\\nonumber\n&\\simeq 5.0\\times 10^{-11}~T^{1.2}~n^{1\/2} \\exp(-\\epsilon\/2T).\n\\end{align}\n\n\n\\subsection{Radial profile of the disc}\n\\label{sec:disc}\n\nWe next obtain the properties of the marginally fragmenting disc,\nusing the energy conservation equation ($Q_+=Q_-$).  Using\nEq.~(\\ref{eq:Q}), the particle density can be written by\n\\begin{equation}\nn=\\frac{\\Sigma \\Omega}{2m_{\\rm p}c_{\\rm s}}=\\frac{\\Omega ^2}{2m_{\\rm p}\\pi G}.\n\\end{equation}\nThen, we can solve the energy conservation equation with respect to $\\Omega$:\n\\begin{equation}\n\\Omega ^2=\n\\frac{3\\dot M_{\\rm tot}}{8\\pi}\n\\frac{(2\\pi G m_{\\rm p})^{5\/2}}{c_{\\rm s} \\lambda(T)f(T)}\n\\exp(\\epsilon\/2T).\n\\end{equation}\n\n\nWe present the profiles of the temperature, surface density, and\nviscous parameter as a function of $\\Omega$ for the case of\n$\\dot M_{\\rm tot} =0.1~M_\\odot~{\\rm yr}^{-1}$ in Figure~\\ref{fig:profile}.  As\ndescribed above, we assume that fragmentation occurs efficiently at\nthe radii where $\\alpha \\ga \\alpha_{\\rm f}\\simeq 1$.  From\nFigure~\\ref{fig:profile}(c), we find that fragmentations occur in the\ncentral regions where the orbital period is shorter than $10^4$ yr.\nWithin the fragmenting region, the surface density is approximately\ngiven by $\\Sigma =\\Sigma_{\\rm f}(\\Omega\/\\Omega_{\\rm f})$, where\n$\\Sigma_{\\rm f}=50$ g cm$^{-2}$ and $\\Omega_{\\rm f}=2\\times 10^{-11}$\ns$^{-1}$, respectively.  \nDuring the earliest stages, i.e. prior to the formation of any central\nprotostar embryo, or immediately following it, the disc mass dominates\nthe central protostellar mass.  We can then obtain the radius within\nwhich the disc fragments effectively,\n\\begin{equation}\nR_{\\rm f}=\\frac{2\\pi G\\Sigma_{\\rm f}}{\\Omega_{\\rm f}^2}\\simeq 2\\times 10^{-2}~{\\rm pc}.\n\\label{eq:r_f_early}\n\\end{equation}\nMoreover, during this stage, the profiles of the surface density, disc\nmass, and number density can be written as functions of the radial\ndistance $R$ from the central protostar:\n\\begin{equation}\n\\Sigma = \\Sigma_{\\rm f}\\left(\\frac{R}{R_{\\rm f}}\\right)^{-1},\n\\label{eq:sig0}\n\\end{equation}\n\\begin{equation}\nM_d \\simeq 430~M_\\odot \\left(\\frac{R}{R_{\\rm f}}\\right),\n\\label{eq:md0}\n\\end{equation}\nand\n\\begin{equation}\nn \\simeq 6\\times 10^8~{\\rm cm}^{-3} \\left(\\frac{R}{R_{\\rm f}}\\right)^{-2},\n\\label{eq:numdens}\n\\end{equation}\nrespectively.\nThe typical mass of the clumps formed at $\\simeq R_{\\rm f}$\nis estimated as\n\\begin{equation}\nM_{\\rm c}\\simeq \\Sigma_{\\rm f} H_{\\rm f}^2\\simeq 30~M_\\odot.\n\\label{eq:M_c}\n\\end{equation}\nThese disc profiles and the clump mass are in good agreement with the\nhighest resolution results in the numerical simulations by\n\\cite{2014MNRAS.439.1160R}, lending credence to our simplified `toy'\nmodel.\n\nIn the above, we have assumed that the mass of the central protostar\nis negligible.  However, the central protostar grows via rapid\naccretion and having a central point source can then modify the disc\nstructure.  After $\\ga 10^4$ yr, the gas within $R_{\\rm f}$ accretes\non to the central protostar, and the protostellar mass exceeds the disc\nmass within $R_{\\rm f}$.  The fragmentation radius (defined by\n$\\alpha(R_{\\rm f})=1$)\nremains roughly constant for the first $\\sim {\\rm a~few}~\\times 10^4$ yr, after which it\nbegins to move outward slowly, according to\n\\begin{align}\nR_{\\rm f}&=\\left(\\frac{GM_\\ast}{\\Omega_{\\rm f}^2}\\right)^{1\/3}\\label{eq:rf_late}\n\\\\\\nonumber\n&\\simeq 5\\times 10^{-2}~{\\rm pc}\\left(\\frac{M_\\ast}{10^4~M_\\odot}\\right)^{1\/3}\n\\propto t^{1\/3},\n\\end{align}\nwhere $M_\\ast$ is the mass of the protostellar embryo at the center,\nand we have assumed that the central accretion rate remains constant\nover time.\nIn the regions where $M_\\ast>M_{\\rm d}$, the orbital frequency is\nproportional to $R^{-3\/2}$ and thus the surface density and mass of\nthe disc are given by\n\\begin{equation}\n\\Sigma =\\frac{\\Sigma_{\\rm f}}{\\Omega_{\\rm f}}\\sqrt{\\frac{GM_\\ast}{R^3}}\n\\label{eq:sig1}\n\\end{equation}\nand\n\\begin{equation}\nM_d =4\\pi \\frac{\\Sigma_{\\rm f}}{\\Omega_{\\rm f}}\\sqrt{GM_\\ast R},\n\\label{eq:md1}\n\\end{equation}\nrespectively. The radius where $M_\\ast=M_{\\rm d}$ is given by\n\\begin{equation}\nR_{\\rm g\\ast} =\\frac{M_\\ast \\Omega_{\\rm f}^2}{16\\pi^2 G\\Sigma_{\\rm f}^2}\\propto t.\n\\end{equation}\nThe condition $R_{\\rm f}\\la R_{\\rm g\\ast}$ is satisfied when the\nprotostellar mass exceeds $\\simeq 3\\times 10^3~M_\\odot$, which corresponds to \n$t\\simeq 3\\times 10^4~{\\rm  yr}~(M_\\ast\/3\\times 10^3~M_\\odot )(\\dot M_{\\rm tot}\/0.1~M_\\odot~{\\rm yr}^{-1} )^{-1}$.\nFor convenience in the following section (\\S~\\ref{sec:fate}), we define the transition epoch as $t_{\\rm g\\ast}=10^5$ yr\nneglecting the difference of the factor of 3 because the ratio of $R_{\\rm g\\ast}\/R_{\\rm f}\\propto t^{2\/3}$ \ndoes not change significantly in the range $3\\times 10^4<t<10^5$ yr. \nTherefore, we use Eq.~(\\ref{eq:sig0}) and (\\ref{eq:md0}) for $t<t_{\\rm g\\ast}$\nand Eqs.~(\\ref{eq:sig1}) and (\\ref{eq:md1}) for $t\\geq t_{\\rm g\\ast}$\n(hereafter the `late phase') for the radial profile of the disc,\nrespectively.\n\n\n\n\n\\section{Fate of the central protostar and clumps}\n\\label{sec:fate}\n\nIn the fragmenting disc, the evolution of the clumps is determined by\nseveral physical processes, including migration, accretion, and\nevolution to ZAMS stars.  In this section, we consider these\nprocesses, and discuss the fate of the clumps formed at the\nfragmentation radius.\n\n\n\\subsection{Clump migration}\n\\label{sec:mig}\n\nThe clumps formed at $R_{\\rm f}$ interact with the disc through\ngravitational torques, which results in their orbital evolution, i.e.\n`migration'.  We first discuss the time-scales for limiting cases of\ninward migration applicable to planets embedded in smooth discs; this\nserves as a useful reference.  We then discuss the complications and\nuncertainties for protogalactic discs, arising from the fact that the\ndisc is clumpy, and the nominal migration rate is very rapid.\n\nLimiting cases of idealized migration are well-known in the theory of\ndisc-planet interactions.  One is the case when the planet mass is so\nsmall that the perturbation induced in the disc is in the linear\nregime.  The density waves excited in the disc by the planet exert\ntorques on the planet through resonances, and thus promote the\nmigration (so-called Type~I migration;\ne.g. \\citealt{1980ApJ...241..425G,1986Icar...67..164W}).  The\ncharacteristic time-scale of Type~I migration is\n\\begin{equation}\nt_{\\rm mig,I}=\\frac{1}{4Cq\\mu}\\left(\\frac{H}{R}\\right)^2\\frac{2\\pi}{\\Omega},\n\\label{eq:typeI}\n\\end{equation}\nwhere $C=3.2+1.468~\\xi$, $\\xi$ is the power of the surface density\nprofile $\\Sigma(R)$, $q=M_{\\rm c}\/M_\\ast$, and $\\mu=\\pi \\Sigma\nR^2\/M_\\ast$ \\citep{2002ApJ...565.1257T}.  A second case is when the\nplanet mass is massive enough to clear a gap in the disc (but still\nless massive than the local disc).  Then, the planet migrates inward\non approximately the discs's viscous time-scale (so-called Type~II\nmigration; e.g. \\citealt{1986ApJ...309..846L,1997Icar..126..261W}):\n\\begin{equation}\nt_{\\rm mig,II}\\simeq t_{\\rm vis}\n=\\frac{1}{3\\pi \\alpha}\\left(\\frac{R}{H}\\right)^2 \\frac{2\\pi}{\\Omega}.\n\\label{eq:typeII}\n\\end{equation}\n\nThe theoretical basis for these time-scales is derived from idealized\nsetups, assuming point-like planets and a smooth disc profile.\nWhether these results are applicable for an extended clump, in a\ngravitationally unstable and lumpy disc, as we discuss in this paper\nis poorly known.  Nevertheless, recent numerical simulations have\nprovided some useful information.  In particular, low-mass planets\nembedded in a turbulent disc have been found to migrate inwards,\ndespite the stochastic kicks due to the turbulent density\nfluctuations, essentially on the time-scale of Type I migration\n\\citep{2011MNRAS.416.1971B, 2011ApJ...737L..42M}.  If the clump grows\nvia accretion and opens a gap during its migration in the\ngravitationally unstable disc, a transition between Type~I and Type~II\nmigration occurs and the migration speed slows down\n\\citep{2012ApJ...746..110Z}.\n\n\nFrom the disc properties discussed in \\S~\\ref{sec:disc}, we expect\nthat the characteristic migration time shifts from that of Type~II to\nthat of Type~I, because the mass ratio $q$ decreases as the central\nprotostar (and the mass of the corresponding steady-state disc) grows.\nWe therefore estimate the transition epoch and the migration time for\nthe late phase ($M_\\ast \\ga 10^4~M_\\odot$) using\nEq.~(\\ref{eq:typeI}).  The gap-opening condition has been\ninvestigated by many authors\n\\citep[e.g.,][]{1979MNRAS.186..799L,2002ApJ...572..566R,2006Icar..181..587C,duffellmacfadyen2013},\nbut again, remains poorly known for gravitationally unstable and\nclumpy discs.  To give a conservative estimate, we here do not discuss\nthe gap-opening condition, but compare the time-scales of Type~I and II\nmigration.  The ratio of the two time-scales at $R_{\\rm f}$ (Eq.~\\ref{eq:rf_late}) \nis independent of the central mass, and is given by\n\\begin{equation}\n\\frac{t_{\\rm mig,I}}{t_{\\rm mig,II}}\\simeq \\frac{0.4}{q}\\left(\\frac{H}{R}\\right)^{3}\n\\simeq 2.0\\left(\\frac{M_{\\rm c}}{30~M_\\odot}\\right)^{-1},\n\\end{equation}\nwhere we have used the condition $Q=1$.  Since the clumps grow via\naccretion at a typical rate of at least $\\dot M_{\\rm c}\\sim {\\rm a~few}\n\\times 10^{-3}~M_\\odot~{\\rm yr}^{-1}$ (see \\S~\\ref{sec:kh}), the clump\nmigrates towards the central protostar on a time-scale comparable or\nshorter than that in the Type~II case, independent of the presence or\nabsence of a gap.  In the following, we therefore adopt $t_{\\rm mig,\n  II}$ as a conservative choice for the migration time-scale.\n\nWe first estimate the migration time for the case that the\nself-gravity of the disc within $R_{\\rm f}$ dominates the protostellar\ngravity (i.e. $t<t_{\\rm g\\ast}$).  At this stage, the mass of\nthe central protostar is $\\la 10^4~M_\\odot$.  From\nEqs.~(\\ref{eq:md0}) and (\\ref{eq:M_c}), the clump mass is\ninitially smaller than the local disc mass ($M_{\\rm c}\\simeq\n0.1~M_d$).  In the disc-dominated case, the orbital decay time of the\nclump ($t_{\\rm mig}=-R\/\\dot R$) is approximately given by the viscous\ntime (Eq.~\\ref{eq:typeII}; see also \\citealt{duffell+2014} who\nfind a factor of few faster migration in the case they studied).\nSince $H\/R=Q\/2\\simeq 0.5$ at $R_{\\rm f}$, $t_{\\rm mig}\\sim 0.4\n(2\\pi\/\\Omega_{\\rm f})\\sim 4\\times 10^3$ yr.  Thus, we find that the\nclump orbit decays slightly faster than the orbital period. This\nfollows essentially from our assumption of $\\alpha_{\\rm f}=1$,\nand leads to a complication: Eq.~(\\ref{eq:typeII}) for the\nType~II migration rate is strictly valid only for $t_{\\rm mig}>\n2\\pi\/\\Omega$ because the torques exerted on the clump are assumed to\nbe averaged over an orbit (at a fixed radius).  Despite this\ncomplication, we can still safely conclude $t_{\\rm mig}\\la\n2\\pi\/\\Omega$. This is because assuming a slower migration ($t_{\\rm\n  mig}\\ga 2\\pi\/\\Omega$) would justify the use of orbit-averaged\nType~II torques, which would then yield the contradiction $t_{\\rm\n  mig}< 2\\pi\/\\Omega$.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=58mm,width=82mm]{mig_time.eps}\n\\end{center}\n\\caption{The decay time of the clump's orbit, for the case of $M_\\ast\n  <10^4~M_\\odot$ ($M_\\ast<M_{\\rm d}$; $t<t_{\\rm g\\ast}$), $M_\\ast =10^4$ and $10^5~M_\\odot$\n  ($M_\\ast>M_{\\rm d}$; $t>t_{\\rm g\\ast}$), respectively.  In each case, the initial\n  position of the clump is set to the fragmentation radius $R_{\\rm\n    f}$.  In the first ($M_\\ast<10^4~M_\\odot$) case, we show the\n  slow-down of the migration expected if the clump grows by accretion\n  at a rate of $f\\dot M_{\\rm tot}$ with $f=0$ (solid), $0.2$ (dashed),\n  and $0.5$ (dotted).}\n\\label{fig:mig}\n\\end{figure}\n\nWhen the clump migrates to $\\simeq 0.1~R_{\\rm f}$, its mass becomes\ncomparable to the local disc mass (neglecting for now any growth of\neither the disc or the clump during the migration).\nWithin $\\sim 0.1~R_{\\rm f}$, the migration speed slows down because\nthe disc outside the clump can no longer absorb the orbital angular\nmomentum of the clump \\citep{1995MNRAS.277..758S}.\nIn the clump-dominated case, the migration time is somewhat modified\nas\n\\begin{equation}\nt_{\\rm mig}\\simeq q_{\\rm B}^{-k}t_{\\rm vis},\n\\end{equation}\nwhere\n\\begin{align}\nq_{\\rm B}&=\\frac{(1+q)\\dot M_{\\rm tot}}{M_{\\rm c}}t_{\\rm vis}\\\\\\nonumber\n&\\simeq 1.1~\\left(\\frac{1+q}{1.1}\\right)\\left(\\frac{M_{\\rm c}}{30~M_\\odot}\\right)^{-1}\n\\left(\\frac{\\dot M_{\\rm tot}}{0.1~M_\\odot~{\\rm yr}^{-1}}\\right)\\left(\\frac{t_{\\rm vis}}{300~{\\rm yr}}\\right)\n\\end{align}\nand\n\\begin{align}\nk=1-\\left(1+\\frac{\\partial \\ln \\Sigma}{\\partial \\ln \\dot M_{\\rm tot}}\\right)^{-1}\\simeq 0.4.\n\\end{align}\n\nIn Figure~\\ref{fig:mig}, we show the evolution of the orbital radius\nof the clump (black lines: $M_\\ast<10^4~M_\\odot$).  The horizontal axis\nis the time from the onset of the orbital decay at $R_{\\rm f}$.  To see the\neffect of the slow down, we assume that the clump grows at the\naccretion rate of $f\\dot M_{\\rm tot}$ with $f=0$ (solid curve), $0.2$\n(dashed), and $0.5$ (dotted), respectively.  In each case, the clump's\norbit decays within $10^4$ yr, even if we consider the slow down of\nthe migration.\n\nNext, we consider the case that the protostellar gravity exceeds the\nself-gravity of the disc within $R_{\\rm f}$ (i.e. $t\\geq t_{\\rm g\\ast}$).  \nIn Figure~\\ref{fig:mig}, we show the corresponding\norbital evolution for $M_\\ast =10^4$ (blue middle curve) and\n$10^5~M_\\odot$ (red right most curve) as examples of this stage.\nIn both the cases, we do not consider the clump growth because\nthe initial clump mass is much smaller than the disc mass within \n$R_{\\rm f}$ and thus the slow-down effect works after the orbit\nhas decayed by more than two orders of magnitude.\nFor $M_\\ast =10^5~M_\\odot$, the fragmentation radius moves out to $\\sim 0.1$ pc. \nIn this case, the decay time becomes longer than the orbital\nperiod, and the clump could evolve into a normal massive star, rather\nthan an SMS. However, this scenario is realized only by assuming that\na SMS with $M_\\ast \\sim 10^5~M_\\odot$ has already grown at the center of\nthe disc.\n\n\n\n\\subsection{Clump accretion and evolution}\n\\label{sec:kh}\n\nIn the fragmenting disc, many clumps are formed within $R_{\\rm f}$ and\ntheir orbits decay through interaction with the accretion disc.  Since\nthe typical density of the clumps is $\\sim 10^9~{\\rm cm}^{-3}$ (Eq.~\\ref{eq:numdens})\nand since they are optically thin to the H$^-$ free-bound emission,\ntiny protostars are formed in the clumps within their free-fall time\n$\\sim 10^3$ yr, which is an order of magnitude shorter than the\norbital decay time.  In this section, we simply estimate the accretion\nrate on to the clumps (which we assume quickly evolve to protostars),\nand then discuss the possibility of forming ZAMS stars from the\nclumps.\n\nThe accretion rate on to a point-like clump in a Keplerian disc,\nwhere we assume a rotationally supported disc \\citep{2014MNRAS.439.1160R},\nis estimated as\n\\begin{equation}\n\\dot M_{\\rm c}=\\frac{3}{2}\\Sigma \\Omega (f_{\\rm H}R_{\\rm H})^2,\n\\end{equation}\nwhere $R_{\\rm H}$ is the Hill radius defined by $R(M_{\\rm\n  c}\/3M_\\ast)^{1\/3}$ and $f_{\\rm H}\\sim O(1)$\n\\citep[e.g.,][]{2004ApJ...608..108G}.  We have also assumed that the\nclump accretes gas orbiting in the nearby disc, within an impact\nparameter $f_{\\rm H}R_{\\rm H}$.\nAt the fragmentation radius, the accretion rate is given by\n\\begin{align}\n\\dot M_{\\rm c}|_{R_{\\rm f}}&=\\frac{3}{2}\\Sigma_{\\rm f} \\Omega_{\\rm f} f_{\\rm H}^2 R_{\\rm f}^2\\left(\\frac{M_c}{3M_\\ast}\\right)^{2\/3}\\label{eq:clump_acc}\\\\\\nonumber\n&\\simeq 1.1\\times 10^{-2}~M_\\odot~{\\rm yr}^{-1}~\\left(\\frac{f_{\\rm H}}{1.5}\\right)^2\\left(\\frac{M_{\\rm c}}{30~M_\\odot}\\right)^{2\/3},\n\\end{align}\nwhich is similar to the critical rate ($\\approx4\\times\n10^{-3}~M_\\odot~{\\rm yr}^{-1}$) at which the evolution of a protostar changes\nqualitatively \\citep{2001ApJ...561L..55O, 2003ApJ...589..677O}.  Below\nthe critical rate, the protostar grows to a usual ZAMS star.  Above\nthe critical rate, the protostar evolves instead to a structure with a\nbloated envelope resembling a giant star \\citep{2012ApJ...756...93H, 2013ApJ...778..178H}.\nThe accretion rate given by Eq.~(\\ref{eq:clump_acc}) \nis only a factor of $\\approx 2$ above the critical value. We\ntherefore consider the case in which some of the clumps grow at a\nsub-critical rate and form massive ZAMS stars.\n\nThe protostar embedded in the clump begins to undergo Kelvin-Helmholtz\n(KH) contraction, loosing energy by radiative diffusion.  After the\ncontraction, the star reaches a ZAMS star when the central temperature\nincreases to $\\sim 10^8$K and hydrogen burning begins\n\\citep{2001ApJ...561L..55O, 2003ApJ...589..677O}. \nBelow the critical rate, the time-scale of the KH contraction is estimated as \n\\begin{align}\nt_{\\rm KH}&\\simeq \\frac{M_{\\rm c}}{\\dot M_{\\rm c}}\n\\ga 10^4~{\\rm yr}\\left(\\frac{M_{\\rm c}}{30~M_\\odot }\\right)\n\\left(\\frac{\\dot M_{\\rm c}}{3\\times 10^{-3}~M_\\odot~{\\rm yr}^{-1}}\\right)^{-1}.\n\\end{align}\nThus, it is longer than\nboth the migration time for $M_\\ast \\leq 10^4~M_\\odot$ and the orbital\nperiod at the fragmentation radius.\nTherefore, most clumps formed\npromptly in the early (disc-dominated) phase are expected to migrate\nand merge with the central protostar before reaching the massive ZAMS\nstars ($M_\\ast \\leq 10^4~M_\\odot$). On the other hand, clumps formed in\nthe late (central protostar-dominated) phase ($M_\\ast >10^4~M_\\odot$)\ncould survive and evolve to ZAMS stars.\n\n\n\\subsection{Radiative feedback}\n\nAs we have seen in \\S~\\ref{sec:kh}, the decay time of the clump orbit\nbecomes longer than the KH time during the late stage for\n$M_\\ast>10^4~M_\\odot$.  In this case, the clump can contract by loosing\nenergy and reach the ZAMS star before migrating towards the central\nprotostar.  This, however, requires a central SMS to be already present.\nOn the other hand, even during the early stage for $M_\\ast <10^4~M_\\odot$, \nsome clumps could survive for their KH times, because\nclumps interact with each other, as well as with the disc, within the\nfragmentation radius.  In a clumpy disc, these gravitational\ninteractions will make the orbital decay time longer or shorter\nstochastically, and can also cause clumps to be temporarily ejected\nfrom the disc\n\\citep{2005ApJ...630..152E,2012MNRAS.424..399G,2012ApJ...746..110Z,2013ApJ...777L..14F}.\nSome of these clumps can evolve to ZAMS stars and emit strong UV\nradiation, which could prevent the accretion on to the protostar and\non to the disc as a while.\n\nWe briefly estimate the number of the clumps which exist in the disc at a moment.\nThe corresponding range where the fragmentation occurs is $\\Delta R\\simeq R_{\\rm f}$.  \nThe Hill radius of the clump at $R_{\\rm f}$ is written by\n\\begin{equation}\nR_{\\rm H}=1.5\\times 10^{16}~{\\rm cm}\\left(\\frac{M_{\\rm c}}{30~M_\\odot}\\right)^{1\/3}.\n\\label{eq:hill}\n\\end{equation}\nThus, we roughly estimate the maximum number of the clumps which survives as\n\\begin{equation}\nN_{\\rm c}\\sim \\frac{\\Delta R}{R_{\\rm H}}\\simeq 20\\left(\\frac{M_\\ast}{10^5~M_\\odot}\\right)^{1\/3}\n\\left(\\frac{M_{\\rm c}}{30~M_\\odot}\\right)^{-1\/3}.\n\\label{eq:nc}\n\\end{equation}\nIf some clumps grow at rates higher than $\\sim 10^{-3}~M_\\odot~{\\rm yr}^{-1}$ and\ntheir masses increase within $t_{\\rm KH}$, the maximum number\ndecreases.  \nWe here consider $N_{\\rm c}\\approx10-20$ as a conservative value.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=36mm,width=83mm]{disk.eps}\n\\end{center}\n\\caption{Schematic figure of the accretion disc and a small\n  circum clump disc, showing the distance of the clump from the\n  central star $R$, the Hill radius of the clump $R_{\\rm H}$ (Eq.~\\ref{eq:hill}), \n  the size of the circum clump disc $R_{\\rm d,c}$, the\n  size of the H$_{\\rm II}$ region around the clump $R_{\\rm ion}$\n  (Eq.~\\ref{eq:ion}), the gravitational radius of the clump\n  $R_{\\rm g,c}$ (Eq.~\\ref{eq:grav}), and the size of the whole\n  nuclear disc $R_{\\rm d}$ (Eq.~\\ref{eq:rd}).  }\n\\label{fig:clump_disc}\n\\end{figure}\n\n\nFirst, let us consider radiative feedback on the accretion on to the\nZAMS star (clump) itself.  Figure~\\ref{fig:clump_disc} illustrates the\ngas structure around the clumps and the characteristic radii.  The gas\naccreting on to the ZAMS star makes a small circum-clump disc.  The\ndisc size is roughly estimated as $R_{\\rm d,c}\\simeq R_{\\rm H}\/3$\n\\citep{1998ApJ...508..707Q}, based on numerical simulations\n\\citep{2009MNRAS.397..657A, 2012ApJ...747...47T}.  The ZAMS star emits\nUV radiation and an H$_{\\rm II}$ region is formed above and below the\ndisc.  The size of the H$_{\\rm II}$ region depends on the ionizing\nluminosity of the ZAMS star and on the density profile of the\nin-falling material.  For simplicity, we estimate the size $R_{\\rm\n  ion}$ in the polar direction using Eq.~(36) of\n\\cite{2008ApJ...681..771M} as\n\\begin{equation}\n\\frac{R_{\\rm ion}}{R_{\\rm d,c}}\\simeq 0.16\\left(\\frac{M_{\\rm c}}{30~M_\\odot}\\right)^{1.25}\n\\left(\\frac{\\dot M_{\\rm c}}{3\\times 10^{-3}~M_\\odot~{\\rm yr}^{-1}}\\right)^{-1},\n\\end{equation}\nwhere we assume that the disc mass is equal to the clump mass and we\ntake the temperature of the H$_{\\rm II}$ region to be $3\\times 10^4$ K\n\\citep{2011Sci...334.1250H}.  The sound speed in the H$_{\\rm II}$\nregion is $c_{\\rm s,ion}\\simeq 20$ km s$^{-1}$.  For a ZAMS star\nlocated at $R_{\\rm f}$, we obtain\n\\begin{equation}\nR_{\\rm ion}\\simeq 8.0\\times 10^{14}~{\\rm cm} \\left(\\frac{M_{\\rm c}}{30~M_\\odot}\\right)^{19\/12}\n\\left(\\frac{\\dot M_{\\rm c}}{3\\times 10^{-3}~M_\\odot~{\\rm yr}^{-1}}\\right)^{-1}.\n\\label{eq:ion}\n\\end{equation}\nSince the sound-crossing time within $R_{\\rm ion}$ is much shorter\n($\\sim13$ yr), the ionization front expands rapidly.  When the front\nreaches the gravitational radius\n\\begin{equation}\nR_{\\rm g,c}=\\frac{GM_{\\rm c}}{c_{\\rm s,ion}^2}\n\\simeq 1.0\\times 10^{15}~{\\rm cm} \\left(\\frac{M_{\\rm c}}{30~M_\\odot}\\right),\n\\label{eq:grav}\n\\end{equation}\nthe ionized gas breaks out through the neutral infalling gas.  The\nionizing photons subsequently heat the disc surface, and thus\nphoto-evaporation begins to suppress the accretion rate.  The\nphoto-evaporation rate can be expressed as\n\\begin{align}\n\\dot M_{\\rm PE}\\simeq 3.8\\times 10^{-4}\\left(\\frac{\\Phi_{\\rm EUV}}{10^{50}~{\\rm s}^{-1}}\\right)^{1\/2}\n\\left(\\frac{R_{\\rm d,c}}{10^{16}~{\\rm cm}}\\right)^{1\/2}~M_\\odot~{\\rm yr}^{-1},\n\\end{align}\nwhere $\\Phi_{\\rm EUV}$ is the ionizing photon number flux\n\\citep{2013ApJ...773..155T}.  Using the relation $\\Phi_{\\rm\n  EUV}=3.7\\times 10^{49}N_{\\rm c}$ s$^{-1}(M_{\\rm c}\/60~M_\\odot)^{3\/2}$\nin the mass range $60\\la M_{\\rm c} \\la 300~M_\\odot$, we obtain\n\\begin{align}\n\\dot M_{\\rm PE}\\simeq 7.4\\times 10^{-4}\\left(\\frac{M_\\ast}{10^5~M_\\odot}\\right)^{1\/6}\n\\left(\\frac{M_{\\rm c}}{60~M_\\odot}\\right)^{3\/4}\n~M_\\odot~{\\rm yr}^{-1},\n\\end{align}\nwhich is smaller than $\\dot M_{\\rm c}\\sim {\\rm a~few}\\times\n10^{-3}~M_\\odot~{\\rm yr}^{-1}$ by approximately an order of magnitude.  We therefore\nconclude that accretion on to the clump could not be suppressed by its\nown photoionization heating.  After the H$_{\\rm II}$ region breaks out\nof the disc, however, the accretion proceeds only from the shadow of\nthe disc and thus the accretion rate is reduced.  Moreover,\n\\cite{2011Sci...334.1250H} suggest that the gas behind the disc is\nshocked and accelerated outward by the pressure gradient after the\nexpansion of the H$_{\\rm II}$ region.  This process halts the\naccretion at $M_{\\rm c} \\approx 40~M_\\odot$.  \n\\citet{2013ApJ...773..155T}\nhave shown that photo-evaporation starts to suppress the accretion \nwhen $\\dot M_{\\rm PE}\/\\dot M_{\\rm c}\\ga 0.2$.\nNote that accretion on to a clump inside $R_{\\rm f}$ cannot be\nsuppressed, even if the H$_{\\rm II}$ region expands by a large factor.\nIn particular, a clump growing at a super critical rate, $\\dot M_{\\rm\n  c}\\ga 10^{-2}~M_\\odot~{\\rm yr}^{-1}$, is not affected by the radiative feedback\nbecause $\\dot M_{\\rm PE}\/\\dot M_{\\rm c} \\la 0.08$.\n\nNext, we discuss the possibility that the ionizing photons from the\ncollection of all ZAMS stars together suppress the accretion (with\n$\\dot M_{\\rm tot}\\simeq 0.1~M_\\odot~{\\rm yr}^{-1}$) from the parent cloud on to the\ndisc as a whole.  To make our discussion conservative, we assume that\nthe clumps which emit strong UV radiation grow at the rate just below\nthe critical accretion rate.  For $M_\\ast \\sim 10^5~M_\\odot$, the\norbital decay time is $\\sim 3\\times 10^4$ yr and so each clump then\ngrows to $\\sim 100~M_\\odot$.  The corresponding total ionizing photon\nrate is $\\sim 1.1\\times 10^{51}$~s$^{-1}$.  From numerical simulations\nof the gravitational collapse of an atomic-cooling cloud\n\\citep{2014arXiv1404.4630I}, the rotational velocity is proportional\nto the Keplerian velocity, $v_{\\rm rot}=f_{\\rm Kep}v_{\\rm Kep}$, with\n$f_{\\rm Kep}\\simeq 0.5$.  From this relation, the size of the whole\ndisc around the central protostar is given by\n\\begin{align}\nR_{\\rm d}&=f_{\\rm Kep}^2 R_{\\rm env},\\label{eq:rd}\\\\\\nonumber\n&\\simeq 2.5~{\\rm pc}\\left(\\frac{f_{\\rm Kep}}{0.5}\\right)^{2}\n\\left(\\frac{R_{\\rm env}}{10~{\\rm pc}}\\right),\n\\end{align}\nwhere $R_{\\rm env}$ is the size of the quasi-spherical parent cloud\nfuelling the inner disc. We approximate $R_{\\rm env}$ using a critical\nBonnor-Ebert sphere with a temperature of $8,000$ K and a central\ndensity of $10^4~{\\rm cm}^{-3}$, \nwhich corresponds to that of the gravitationally unstable core \nin the atomic cooling haloes without H$_2$ molecules \\citep[e.g.,][]{2008ApJ...682..745W}.\nTherefore, we obtain $\\dot M_{\\rm PE}=3.5\\times 10^{-2}~M_\\odot~{\\rm yr}^{-1}~(\\Phi_{\\rm EUV}\/1.1\\times 10^{51}~{\\rm s}^{-1})^{1\/2}\n(R_{\\rm d}\/2.5~{\\rm pc})^{1\/2}$ and thus $\\dot M_{\\rm PE}\/\\dot M_{\\rm tot}\\simeq 0.35$,\nwhich is comparable or only slightly above the critical value.\nTherefore, even with conservative assumptions, the strong UV radiation\ncannot deplete the gas supply on to the disc from the parent cloud, and\nthe central protostar can grow to a SMS star with $M_\\ast \\simeq 10^5~M_\\odot$.\n\n\n\n\\section{fragmentation by metals: dust cooling}\n\\label{sec:metal}\n\nThe site envisioned for forming a SMS is atomic cooling gas in a halo\nwith virial temperature $>10^4$ K, in which H$_2$ cooling is\nprohibited prior to and throughout the protostellar collapse.  Recent\nnumerical simulations suggest that the majority of the atomic cooling\nhaloes are polluted by heavy elements due to Pop III supernovae from\nprior star formation \\citep{2010ApJ...716..510G,2012ApJ...745...50W}.\nThe resulting metallicity is $Z\\la 10^{-4}~Z_\\odot$ ($\\la\n10^{-3}~Z_\\odot$) if the Pop III supernovae are core-collapse type\n(pair-instability type).  If the gas is slightly polluted with $Z\\ga\n5\\times 10^{-6}~Z_\\odot$, the temperature decreases below $\\sim 500$ K\nby dust cooling \\citep{2008ApJ...686..801O}, which could promote\nefficient fragmentation.  The details of this fragmentation, and\nwhether dust cooling ultimately prevents SMS formation, are not yet\nunderstood.\n\nThe temperature of the gas with dust grains begins to decrease through\nheat exchange with cool dust grains ($T_{\\rm gr}\\ll T$) when collisions\nbetween gas particles and dust are sufficiently frequent.  Collisional\ncooling of the gas is efficient until $T\\sim 500$ K, where the gas and\ndust is thermally coupled ($T\\simeq T_{\\rm gr}$).  At the cooling\nphase, the compressional heating and collisional cooling are balanced\nas\n\\begin{equation}\n\\frac{\\rho c_{\\rm s}^2}{t_{\\rm ff}}\\simeq H_{\\rm gr},\n\\label{eq:d1}\n\\end{equation}\nwhere $t_{\\rm ff}=\\sqrt{3\\pi \/(32G\\rho)}$ is the free-fall time and\n$H_{\\rm gr}$ is the energy exchange rate (in erg s$^{-1}$ cm$^{-3}$)\nbetween the gas and dust, defined by\n\\begin{equation}\nH_{\\rm gr}\\simeq 2k_{\\rm B}Tn_{\\rm H}n_{\\rm gr}\\sigma_{\\rm gr}c_{\\rm s},\n\\label{eq:d2}\n\\end{equation}\nfor $T\\gg T_{\\rm gr}$ \\citep{2006MNRAS.369.1437S,2012MNRAS.419.1566S}.\nWe adopt the number density and cross-section of the dust particles\nfrom \\cite{2003A&A...410..611S},\n\\begin{equation}\n\\frac{n_{\\rm gr}\\sigma_{\\rm gr}}{\\rho}=4.7\\times 10^{-2}\\left(\\frac{Z}{10^{-4}~Z_\\odot}\\right)~{\\rm cm^2~g^{-1}},\n\\label{eq:d3}\n\\end{equation}\nwhere the depletion factor of metals to dusts is assumed as high as the present-day Galactic value \n$f_{\\rm dep}\\simeq 0.5$.\nFrom Eqs.~(\\ref{eq:d1}), (\\ref{eq:d2}), and (\\ref{eq:d3}), we obtain \n\\begin{equation}\nn\\simeq 6.8\\times 10^7~{\\rm cm}^{-3} \\left(\\frac{T}{6000~{\\rm K}}\\right)^{-1}\\left(\\frac{Z}{10^{-4}~Z_\\odot}\\right)^{-2},\n\\label{eq:d4}\n\\end{equation}\nabove which the temperature begins to decrease compared to the\nzero-metallicity case.  The density given by Eq.~(\\ref{eq:d4}) is\ncomparable to that at the fragmenting radius.  This implies that the\ndiscussion in \\S~\\ref{sec:fate} remains valid in metal-polluted gas,\nas long as the metallicity remains below $Z\\la 10^{-4}~Z_\\odot$.  We\nconclude that disc fragmentation cannot prevent SMS formation, as long\nas the metallicity is below this value. \nOn the other hand, at\nhigher metallicity ($Z> 3\\times10^{-4}~Z_\\odot$),\nthe gas temperature rapidly decreases by metal-line cooling at\ndensities below $\\sim 10^4~{\\rm cm}^{-3}$ \\citep{2008ApJ...686..801O,2012MNRAS.422.2539I}.  \nNumerical simulations of metal-enriched collapse into atomic cooling haloes\nalso capture the character of gas fragmentation for $Z\\ga 10^{-3}~Z_\\odot$\n\\citep{2014MNRAS.438.1669S,2014MNRAS.440L..76S}.\nThe outcome would be\na compact star-cluster which consists of many low-mass stars with\n$\\sim 1~M_\\odot$ \\citep{2014MNRAS.439.1884T}.\n\n\n\n\n\n\n\n\n\\section{Discussion and conclusions}\n\\label{sec:conc}\n\nIn this paper, we discussed the properties of a disc around the embryo\nof a SMS ($\\ga 10^5~M_\\odot$), expected to be present\nin a primordial gas without H$_2$ molecules in massive haloes with\nvirial temperature $\\ga 10^4$ K.  A high accretion rate $\\ga\n0.1~M_\\odot~{\\rm yr}^{-1}$, sustained for $\\ga 10^5$yr, is required to form a SMS.\nThe inner region of such a disc is gravitationally unstable, and\nfragments into $O(10)$ clumps with characteristic mass of $\\sim\n30~M_\\odot$.  We discuss the possibility that this fragmentation\nprevents SMS formation.  We argue that most of the clumps formed in\nthe disc rapidly migrate towards the central protostar and merge with\nit.  The orbital decay time is shorter than or comparable to the\norbital period of the clumps ($\\la 10^4$ yr).  Some of the clumps can\ngrow via accretion and evolve to ZAMS stars within\ntheir KH time, either because they survive longer\ndue to the stochasticity of the migration process, or because they\nform later, further out in the disc.\n\nOur toy model, on which these conclusions are based,\n  can be tested against simulations of Pop III star formation in\n  lower mass minihaloes.  In this case, the dominant cooling process is\n  H$_2$ line emission ($T\\sim 10^3$ K) and the resulting accretion\n  rate is $\\sim 10^{-3}-10^{-2}~M_\\odot~{\\rm yr}^{-1}$, which is smaller than that\n  in the SMS formation case.  According to high-resolution numerical\n  simulations by \\cite{2012MNRAS.424..399G}, clump formation occurs at\n  $\\sim 10$ AU ($\\Sigma \\sim 5\\times 10^3$ g cm$^{-2}$, $\\Omega \\sim\n  0.1$ yr$^{-1}$, $c_{\\rm s}\\sim 2.5$ km s$^{-1}$ and $M_{\\rm d}\\sim\n  1~M_\\odot$ in their Fig.~2) in the gravitationally unstable disc at\n  the early stage $\\la 10$ yr.  From Eq.~(\\ref{eq:r_f_early}), we can\n  estimate the fragmentation radius as $\\sim 20$ AU, which is\n  consistent with the fragmentation radii in the simulations.  At this\n  stage, the average accretion rate is high, $\\dot M_{\\rm tot}\\simeq\n  10^{-2}~M_\\odot~{\\rm yr}^{-1}$ in their Fig.~8.  From Eq.~(\\ref{eq:continue}),\n  (\\ref{eq:scaleheight}), (\\ref{eq:viscous}), and $Q\\simeq 1$,\n\\begin{equation}\n\\dot M_{\\rm tot}\\simeq 3\\alpha \\frac{c_{\\rm s}^3}{G}.\n\\end{equation}\nUsing these equations, the sound speed at the fragmentation radius\n($\\alpha _{\\rm f}=1$) is estimated as $c_{\\rm s}=2.5$ km s$^{-1}$, which\nagrees well with the numerical result.  The viscous time at $\\sim 10$\nAU is roughly estimated as $t_{\\rm vis}\\sim M_{\\rm d}\/\\dot M_{\\rm\n  tot}\\sim 10^2$ yr.  Since some clumps migrate inward within $10$ yr\n($< t_{\\rm vis}$) in simulation, other effects (Type I migration and\ninteraction between clumps) could accelerate the clumps migration.\nTherefore, our results from the toy model are expected to be\nconservative, and appear to capture the essential features of disc\nfragmentation and clump migration.\n\nThe gas in most of the massive haloes is likely to be polluted by heavy\nelements due to prior star formation and Pop III supernovae.  With the\nmetallicity of $5\\times 10^{-6} \\la Z \\la 10^{-4}~Z_\\odot$, the dust\ncooling decreases the temperature rapidly and could promote the\nfragmentation at $n> 10^{8}~{\\rm cm}^{-3}$ \\citep{2008ApJ...686..801O}.  As a\nresult, the existence of dusts has been considered as one of the most\nsevere obstacles to the SMS formation.  However, the fragmentation\ninduced by the dust cooling is expected to occur only within the\nfragmenting radius we estimated. \nThis means that the clumps formed by \nthe dust cooling can migrate towards the center on a time-scale comparable to the \norbital period, in the same way as the primordial case. We conclude\nthat the SMS formation is not prevented, unless the metal-line cooling\ndominates for $Z\\ga 3\\times 10^{-4}~Z_\\odot$.  Our results therefore\nremove a significant obstacle for SMS formation.\n\nOur estimates for the orbital decay time of the clumps are based on\nthe well-known formulae of Type~I and II planetary migration.\nAlthough some numerical simulations find that these formulae remain\napproximately valid in gravitationally unstable and clumpy discs,\nthere are many uncertainties regarding the migration rate in this\ncase.  For example, we neglect the orbital eccentricities of the\nclumps.  The clumps formed by the disc fragmentation are expected to\nhave eccentric orbits and thus the angular momentum transfer between\nthe disc and clumps could change from the case of the circular orbits.\nMoreover, in such a situation, the interaction with eccentric clumps\nretard the orbital decay of the other clumps\n\\citep{2013ApJ...777L..14F}.  To estimate the migration time within\nthe clumpy disc around the SMS star, more sophisticated numerical\nsimulations are necessary.\n\nIt is worth further emphasizing the limitations and uncertainties of\nour disc model.  In Figure~\\ref{fig:profile}, we present the disc\nprofile based on cooling by free-bound emission of H$^-$ and thermal\nequilibrium.  Hydrogen atomic cooling (Ly$\\alpha$ and two-photon\nemission) dominates at $T\\ga 8000$ K ($P>10^4$ yr).  To quantify the\neffect, we include Ly$\\alpha$ cooling ($\\Lambda =\\lambda_{\\rm\n  Ly\\alpha}n^2x_{\\rm e}$; \\citealt{2007ApJ...666....1G}) in the\nlimiting optically thin case (dashed lines in Fig.~\\ref{fig:profile}).\nSince the gas is in fact optically thick to the Ly$\\alpha$ photons at\n$n>10^5~{\\rm cm}^{-3}$, and the cooling efficiency of the two-photon channel is\nsmaller than the Ly$\\alpha$ emission, the dashed lines overestimate\nthe additional atomic cooling rate.  Nevertheless, the profiles\ndeviate from our fiducial disc model only modestly, beginning at\n$P>10^4$ yr and thus the critical period changes at most within the\nshaded region in Figure~\\ref{fig:profile} (c).\n\nThe crucial ingredient of this paper is the critical value of the\neffective viscous parameter $\\alpha_{\\rm f}$ for fragmentation.  In\nour model, the fragmentation radius depends sensitively on the value\nof $\\alpha_{\\rm f}$.  For example, if we chose $\\alpha_{\\rm f}=0.5$,\nwe obtain $\\Omega_{\\rm f}=2\\times 10^{-12}$ s$^{-1}$ ($P=10^5$ yr),\n$\\Sigma_{\\rm f}=6$ g cm$^{-2}$, \n$M_{\\rm c}\\simeq 400~M_\\odot$ and $R_{\\rm f}\\simeq 0.2$ pc.  In this\ncase, the orbital decay time becomes longer than the KH time of the\nclumps by a factor of 4, and most of the clumps may evolve to\nmassive stars and emit strong UV radiation. \nNevertheless, the number\nof clumps decreases by a factor of 2 from the case of $\\alpha_{\\rm f}=1.0$\n(see Eq. \\ref{eq:nc}).\nTherefore, we expect that our main conclusion also does not change. \nHowever, if $\\alpha_{\\rm f}\\la 0.1$, the fragmenting radius moves outside the size of the\nwhole disc given by Eq.~(\\ref{eq:rd}), which means that the disc\ncannot exist, i.e., our model is no longer self-consistent.  To\nexplore the precise value of $\\alpha_{\\rm f}$ is left for future\ninvestigations.  In particular, our results motivate high-resolution\nnumerical simulations similar to \\cite{2014MNRAS.439.1160R}.\nMoreover, more realistic treatments of radiative cooling and chemical\nreactions at densities higher than $\\sim 10^8~{\\rm cm}^{-3}$\n\\citep{2014arXiv1404.4630I} should be included since the fragmentation\nefficiency strongly depends on the equation of state of the gas.\n\nDespite these caveats, we expect that SMS formation in metal-poor gas\nin atomic-cooling haloes is difficult to avoid, as it requires that\nthe high mass inflow rate, $\\ga 0.1~M_\\odot~{\\rm yr}^{-1}$, is democratically\ndistributed among $O(100)$ fragments (so that {\\em none} of them\naccretes at a super critical rate for SMS formation) and that most of\nthese fragments survive rapid migration and avoid coalescence with the\ngrowing central protostar for several orbital times.  The toy models\npresented in this paper disfavor this scenario.\n\n\n\\section*{Acknowledgements} \nWe thank Greg Bryan, Kazuyuki Omukai, Takashi Hosokawa, Kei Tanaka and\nEli Visbal for fruitful discussions.  This work is supported by the\nGrants-in-Aid by the Ministry of Education, Culture, and Science of\nJapan (KI), and by NASA grant NNX11AE05G (ZH).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDuring recent years, there has been a substantial increase of research activity in the field of medical image reconstruction. One particular application area is the acceleration of Magnetic Resonance Imaging (MRI) scans. This is an area of high impact, because MRI is the leading diagnostic modality for a wide range of exams, but the physics of its data acquisition process make it inherently slower than modalities like X-Ray, Computed Tomography or Ultrasound. Therefore, the shortening of scan times has been a major driving factor for routine clinical application of MRI.\n\nOne of the most important and successful technical developments to decrease MRI scan time in the last 20 years was parallel imaging ~\\cite{Sodickson1997,Pruessmann1999,Griswold2005}. Currently, essentially all clinical MRI scanners from all vendors are equipped with parallel imaging technology, and it is the default option for a large number of scan protocols. As a consequence, there is a substantial benefit of using multi-coil data for machine learning based image reconstruction. Not only does it provide a complementary source of acceleration that is unavailable when operating on single channel data, or on the level of image enhancement and post-processing, it also is the scenario that ultimately defines the use-case for accelerated clinical MRI, which makes it a requirement for clinical translation of new reconstruction approaches. The drawback is that working with multi-coil data adds a layer of complexity that creates a gap between cutting edge developments in deep learning~\\cite{LeCun2015} and computer vision, where the default data type are images. The goal of this manuscript is to bridge this gap by providing both a comprehensive review of the properties of parallel MRI, together with an introduction how current machine learning methods can be used for this particular application.\n\t\t\n\\begin{figure*}[!t]\n\t\\centering\n\t\\includegraphics[width = 6.5in]{.\/Picture1_v2}\n \t\\caption{In k-space based parallel imaging methods, missing data is recovered first in k-space, followed by an inverse Fourier transform and combination of the individual coil elements. In image space based parallel imaging, the Fourier transform is performed as the first step, followed by coil sensitivity based removal of the aliasing artifacts from the reconstructed image by solving an inverse problem.}\n \t\\vspace{-.3cm}\n \t\\label{fig:recon_PI_image_kspace}\n\\end{figure*}\t\t\n\t\t\n\\subsection{Background on multi-coil acquisitions in MRI}\nThe original motivation behind phased array receive coils~\\cite{Roemer1990} was to increase the SNR of MR measurements. These arrays consist of $n_c$ multiple small coil elements, where an individual coil element covers only a part of the imaging field of view. These individual signals are then combined to form a single image of the complete field of view. The central idea of all parallel imaging methods is to complement spatial signal encoding of gradient fields with information about the spatial position of these multiple coil elements. For multiple receiver coils, the MR signal equation can be written as follows\n\\begin{equation}\n\\label{eq:signal_multiplecoils}\nf_j(k_x,k_y) = \\int \\limits_{-\\infty}^{\\infty}\t\\int \\limits_{-\\infty}^{\\infty} u(x,y) c_j(x,y) e^{-i (k_x x + k_y y)}\\dd{x} \\dd{y}.\n\\end{equation}\nIn \\Cref{eq:signal_multiplecoils}, $f_j$ is the MR signal of coil $j=1,\\ldots,n_c$, $u$ is the target image  to be reconstructed, and $c_j$ is the corresponding coil sensitivity. Parallel imaging methods use the redundancies in these multi-coil acquisitions to reconstruct undersampled k-space data. After discretization, this undersampling is described in matrix-vector notation by\n \\begin{equation}\n  \\mathbf{f}_j = \\mathbf{F}_{\\Omega}\\mathbf{C}_j\\mathbf{u} + \\mathbf{n}_j,\n \\end{equation}\n where $\\mathbf{u}$ is the image to be reconstructed, $\\mathbf{f}_j$ is the acquired k-space data in the $j^\\textrm{th}$ coil, $\\mathbf{C}_j$ is a diagonal matrix containing \n the sensitivity profile of the receiver coil \\cite{Pruessmann1999}, $\\mathbf{F}_{\\Omega}$ is a partial Fourier sampling operator that samples locations ${\\Omega},$ and $\\mathbf{n}_j$ is measurement noise in the $j^{\\textrm{th}}$ coil.\n\nHistorically, parallel imaging methods were put in two categories: Approaches that operate in image domain, inspired by the sensitivity encoding (SENSE) method~\\cite{Pruessmann1999} and approaches that operated in k-space, inspired by simultaneous acquisition of spatial harmonics (SMASH)~\\cite{Sodickson1997} and generalized autocalibrating partial parallel acquisition (GRAPPA)~\\cite{Griswold2002}. This is conceptually illustrated in \\Cref{fig:recon_PI_image_kspace}. While these two schools of thought are closely related~\\cite{Kholmovski,Uecker2014}, we organized this document according to these classic criteria for historical reasons.\n\n\n\n\n\n\n\n\\section{Classical parallel imaging in image space}\n\\label{sec:imagespace_PI}\nClassical parallel imaging in image space follows the SENSE method~\\cite{Pruessmann1999}, which can be identified by two key features. First, the elimination of the aliasing artifacts is performed in image space after the application of an inverse Fourier transform. Second, information about receive coil sensitivities is obtained via precomputed, explicit coil sensitivity maps from either a separate reference scan or from a fully sampled block of data at the center of k-space (all didactic experiments that are shown in this manuscript follow the latter approach). More recent approaches jointly estimate coil sensitivity profiles during the image reconstruction process~\\cite{Ying2007,Uecker2008}, but for the rest of this manuscript, we assume that sensitivity maps were precomputed. The reconstruction in image domain in \\Cref{fig:recon_PI_image_kspace} shows three example undersampled coil images, corresponding coil sensitivity maps and the final reconstructed images from a brain MRI dataset. The coil sensitivities were estimated using ESPIRiT~\\cite{Uecker2014}.\n\n\n\nMRI reconstruction in general and parallel imaging in particular can be formulated as an inverse problem. This provides a general framework that allows easy integration of the concepts of regularized and constrained image reconstruction as well as machine learning that are discussed in more detail in later sections. \\Cref{eq:signal_multiplecoils} can be discretized and then written in matrix-vector form:\n\\begin{equation}\n\t\\label{eq:forward_problem}\n\t{\\mathbf{f} = \\mathbf{E}\\mathbf{u} +\\mathbf{n}},\n\\end{equation}\nwhere $\\mathbf{f}$ contains all k-space measurement data points and $\\mathbf{E}$ is the forward encoding operator that includes information about the sampling trajectory and the receive coil sensitivities and $\\mathbf{n}$ is measurement noise. The task of image reconstruction is to recover the image $\\mathbf{u}$. In classic parallel imaging one generally operates under the condition that the number of receive elements is larger than the acceleration factor. Therefore, \\Cref{eq:forward_problem} corresponds to an over-determined system of equations. However, the rows of $\\mathbf{E}$ are linearly dependent because individual coil elements do not measure completely independent information. Therefore the inversion of $\\mathbf{E}$ is an ill-posed problem, which can lead to severe noise amplification, described via the g-factor in the original SENSE paper~\\cite{Pruessmann1999}. \\Cref{eq:forward_problem} is usually solved in an iterative manner, which is the topic of the following sections.\n \n\\subsection{Overview of conjugate gradient SENSE (CG-SENSE)}\nThe original SENSE approach is based on equidistant or uniform Cartesian k-space sampling, where the aliasing pattern is defined by a point spread function that has a small number of sharp equidistant peaks. This property leads to a small number of pixels that are folded on top of each other, which allows a very efficient implementation~\\cite{Pruessmann1999}. When using alternative k-space sampling strategies like non-Cartesian acquisitions or random undersampling, this is no longer possible and image reconstruction requires a full inversion of the encoding matrix in \\Cref{eq:forward_problem}. This operation is demanding both in terms of compute and memory requirements (the dimensions of $\\mathbf{E}$ are the total number of acquired k-space points times $N^2$ where $N$ is the size of the image matrix that is to be reconstructed), which lead to the development of iterative methods, in particular the CG-SENSE method introduced by Pruessmann et al. as a follow up of the original SENSE paper~\\cite{Pruessmann2001}. In iterative image reconstruction the goal is to find a $\\hat{\\mathbf{u}}$ that is a minimizer of the following cost function, which corresponds to the quadratic form of the system in \\Cref{eq:forward_problem}:\n\\begin{equation}\n  \\label{eq:inverse_leastSquares}\n  \\hat{\\mathbf{u}} \\in \\argmin\\limits_{\\mathbf{u}} \\frac{1}{2} \\| \\mathbf{E} \\mathbf{u} - \\mathbf{f} \\|_2^2.\n\\end{equation}\nIn standard parallel imaging, $\\mathbf{E}$ is linear and \\Cref{eq:inverse_leastSquares} is a convex optimization problem that can be solved with a large number of numerical algorithms like gradient descent, Landweber iterations~\\cite{Landweber1951}, primal-dual methods~\\cite{Pock_PD_2010} or the alternating direction method of multipliers (ADMM) algorithm~\\cite{Boyd2011} (a detailed review of numerical methods is outside the scope of this article). \nIn the original version of CG-SENSE~\\cite{Pruessmann2001}, the conjugate gradient method~\\cite{Hestenes1952} is employed.  However, since MR k-space data are corrupted by noise, it is common practice stop iterating before theoretical convergence is reached, which can be seen as a form of regularization.\nRegularization can be also incorportated via additional constraints in \\Cref{eq:inverse_leastSquares}, which will be covered in the next section.\n\nAs a didactic example for this manuscript, we will use a single slice of a 2D coronal knee exam to illustrate various reconstruction approaches. This data were acquired on a clinical 3T system (Siemens Skyra) using a 15 channel phased array knee coil. A turbo spin echo sequence was used with the following sequence parameters: TR=2750ms, TE=27m, echo train length=4, field of view 160mm$^2$ in-plane resolution 0.5mm$^2$, slice thickness 3mm.\nReadout oversampling with a factor of 2 was used, and all images were cropped in the frequency encoding direction (superior-inferior) for display purposes. In the spirit of reproducible research, data, sampling masks and coil sensitivity estimations that were used for the numerical results in this manuscript are available online\\footnote{\\url{https:\/\/app.globus.org\/}: Endpoint: NYULH Radiology Reconstruction Data, coronal pd data, subject 17, slice 25.}. \n\\Cref{fig:knee_reconstructions} shows an example of a retrospectively undersampled CG-SENSE reconstruction with an acceleration factor of 4 for the data from \\Cref{fig:knee_reconstructions}. Early stopping was employed by setting the numeric tolerance of the iteration to $5\\cdot10^{-5}$, which resulted in the the algorithm stopping after 14 CG iterations.\n\n\\subsection{Nonlinear regularization and compressed sensing} \\label{sec:2c}\n\n\\Cref{eq:inverse_leastSquares} can be extended by including a-priori knowledge via additional penalty terms, which results in a constrained optimization problem defined in \\Cref{eq:inverse_regularization}, which forms the cornerstone of almost all modern MRI reconstruction methods\n\\begin{equation}\n  \\label{eq:inverse_regularization}\n  \\hat{\\mathbf{u}} \\in \\argmin\\limits_{\\mathbf{u}} \\frac{1}{2}  \\| \\mathbf{E} \\mathbf{u} - \\mathbf{f} \\|_2^2 + \\sum_{i} \\lambda_i \\Psi_i(\\mathbf{u}).\n\\end{equation}\nHere, $\\Psi_i$ are dedicated regularization terms and $\\lambda_i$ are regularization parameters that balance the trade-off between data fidelity and prior. Since the introduction of compressed sensing~\\cite{Candes2006, Donoho2006} and its adoption for MRI~\\cite{Lustig2007,Block2007,Lustig2008a}, nonlinear regularization terms, in particular $\\ell_1$-norm based ones, are popular in image reconstruction and are commonly used in parallel imaging~\\cite{Block2007,Lustig2010,Knoll2011,Knoll2012, Akcakaya2011, Akcakaya2014, Jung2009}. The goal of these regularization terms is to provide a separation between the target image that is to be reconstructed from the aliasing artifacts that are introduced due to an undersampled acquisition. Therefore, they are usually designed in conjunction with a particular data sampling strategy. The classic formulation of compressed sensing in MRI~\\cite{Lustig2007} is based on sparsity of the image in a transform domain (Wavelets are a popular choice for static images) in combination with pseudo-random sampling, which introduces aliasing artifacts that are incoherent in the respective domain. For dynamic acquisitions where periodic motion is encountered, sparsity in the Fourier domain common choice~\\cite{Gamper2008}. Total Variation based methods have been used successfully in combination with radial~\\cite{Block2007} and spiral~\\cite{Valvano2016} acquisitions as well as in dynamic imaging~\\cite{Feng2013}. More advanced regularizers based on low-rank properties have also been utilized \\cite{Lingala2011}.\n\n\\Cref{fig:knee_reconstructions} shows an example of a nonlinear combined parallel imaging and compressed sensing reconstruction with a Total Generalized Variation~\\cite{Knoll2011} constraint. The regularization parameter $\\lambda$ was set to $2.5\\cdot10^{-5}$ \nand the reconstruction was using 1000 primal-dual~\\cite{Pock_PD_2010} iterations. The used equidistant sampling was chosen for consistency with the other reconstruction methods is not optimal for the incoherence condition in compressed sensing. Nevertheless, the nonlinear regularization still provides a superior reduction of aliasing artifacts and noise suppression in comparison to the CG-SENSE reconstruction from the last section. \n\t\t\t\t\t\t\n\\section{Classical parallel imaging in k-space}\n\nParallel imaging reconstruction can also be formulated in k-space as an interpolation procedure. The initial connections between the SENSE-type image-domain inverse problem approach and k-space interpolation has been made more than a decade ago \\cite{Kholmovski}, where it was noted that the forward model in \\Cref{eq:forward_problem} can be restated in terms of the Fourier transform, ${\\bm \\kappa}$ of the combined image, $\\mathbf{u}$ as \n\\begin{equation}\n    \\mathbf{f} = \\mathbf{A}\\mathbf{F}^*{\\bm \\kappa} \\triangleq \\mathbf{G}_{\\text{acq}} {\\bm \\kappa},\n\\end{equation}\nwhere $\\mathbf{f}$ corresponds to the acquired k-space lines across all coils, and $\\mathbf{G}_{\\text{acq}}$ is a linear operator. Similarly, the unacquired k-space lines across all coils can be formulated using \n\\begin{equation}\n    \\mathbf{f}_{\\text{unacq}} = \\mathbf{G}_{\\text{unacq}} {\\bm \\kappa}\n\\end{equation}\nCombining these two equations yield\n\\begin{equation}\n    \\mathbf{f}_{\\text{unacq}} = \\mathbf{G}_{\\text{unacq}} \\mathbf{G}_{\\text{acq}}^{\\dagger} \\mathbf{f}.\n\\end{equation}\nThus, the unaccquired k-space lines across all coils can be interpolated based on the acquired lines across all coils, assuming the pseudo-inverse, $\\mathbf{G}_{\\text{acq}}^{\\dagger}$, of $\\mathbf{G}_{\\text{acq}}$ exists \\cite{Kholmovski}. Thus, the main difference between the k-space parallel imaging methods and the aforementioned image domain parallel imaging techniques is that the former produces k-space data across all coils at the output, whereas the latter typically produces one image that combines the information from all coils.\n\n\\subsection{Linear k-space interpolation in GRAPPA}\nThe most clinically used k-space reconstruction method for parallel imaging is GRAPPA, which uses linear shift-invariant convolutional kernels to interpolate missing k-space lines using uniformly-spaced acquired k-space lines \\cite{Griswold2002}. For the $j^\\textrm{th}$ coil k-space data, ${\\kappa}_j$, we have\n\\begin{align}\n    {\\kappa}_j&(k_x, k_y - m\\Delta k_y)  \\nonumber \\\\\n    &= \\sum_{c=1}^{n_c} \\sum_{b_x = -B_x}^{B_x} \\sum_{b_y = -B_y}^{B_y} g_{j,m} (b_x,b_y,c) \\nonumber \\\\\n    &\\quad\\quad \\quad \\quad \\kappa_c(k_x - b_x \\Delta k_x,k_y - Rb_y \\Delta k_y),\n\\end{align}\nwhere $R$ is the acceleration rate of the uniformly undersamped acquisition; $m \\in \\{1, \\dots, R-1\\}$; $g_{j,m}(b_x,b_y,c)$ are the linear convolutional kernels for estimating the $m^\\textrm{th}$ spacing location in $j^\\textrm{th}$ coil; $n_c$ is the number of coils; and $B_x$, $B_y$ are parameters determined from the convolutional kernel size. A high-level overview of such interpolation is shown in the reconstruction in k-space section of \\Cref{fig:recon_PI_image_kspace}.\n\nSimilar to the coil sensitivity estimation in SENSE-type reconstruction, the convolutional kernels, $g_{j,m}(b_x,b_y,c)$ are estimated for each subject, from either a separate reference scan or from a fully-sampled block of data at the center of k-space, called autocalibrating signal (ACS) \\cite{Griswold2002}. A sliding window approach is used in this calibration region to identify the fully-sampled acquisition locations specified by the kernel size and the corresponding missing entries. The former, taken across all coils, is used as rows of a calibration matrix, $\\mathbf{A}$; while the latter, for a specific coil, yields a single entry in the target vector, $\\mathbf{b}$. Thus for each coil $j$ and missing location $m \\in \\{1, \\dots, R-1\\}$, a set of linear equations are formed, from which the vectorized kernel weights $g_{j,m}(b_x,b_y,c)$, denoted $\\mathbf{g}_{j,m}$, are estimated via least squares, as $\\mathbf{g}_{j,m}\\in \\arg \\min_{\\mathbf{g}} ||{\\mathbf{b}- \\mathbf{Ag}}||_2^2$. GRAPPA has been shown to have several favorable properties compared to SENSE, including lower g-factors, sometimes even less than unity at certain parts of the image \\cite{Robson2008}, and more smoothly varying g-factor maps \\cite{Breuer2009}. Furthermore, k-space interpolation is often less sensitive to motion \\cite{Breuer2005}. Due to these favorable properties, GRAPPA has found utility in multiple large-scale projects, such as the Human Connectome Project \\cite{Ugurbil2013}.\n\n\n\n\\subsection{Advances in k-space interpolation methods}\n\nThough GRAPPA is widely used in clinical practice, it is a linear method that suffers from noise amplification based on the coil geometry and the acceleration rate \\cite{Pruessmann1999}. Therefore, several alternative strategies have been proposed in the literature to reduce the noise in reconstruction.\n\nIterative self-consistent parallel imaging reconstruction (SPIRiT) is a strategy for enforcing self-consistency among the k-space data in multiple receiver coils by exploiting correlations between neighboring k-space points \\cite{Lustig2010}. Similar to GRAPPA, SPIRiT also estimates a linear shift-invariant convolutional kernel from ACS data. In GRAPPA, this convolutional kernel used information from acquired lines in a neighborhood to estimate a missing k-space point. In SPIRiT, the kernel includes contributions from all points, both acquired and missing, across all coils for a neighborhood around a given k-space point. \nThe self-consistency idea suggests that the full k-space data should remain unchanged under this convolution operation. SPIRiT objective function also includes a term that enforces consistency with the acquired data, where the undersampling can be performed with arbitrary patterns, including random patterns that are typically employed in compressed sensing \\cite{Block2007, Lustig2007}.\nAdditionally, this formulation allows incorporation of regularizers, for instance based on transform-domain sparsity, in the objective function \nto reduce reconstruction noise via non-linear processing \\cite{Lustig2010\n. Furthermore, SPIRiT has facilitated the connection between coil sensitivities used in image-domain parallel imaging methods and the convolutional kernels used in k-space methods via a subspace analysis \\cite{Uecker2014}.\n\nAn alternative line of work utilizes non-linear k-space interpolation for estimating missing k-space points for uniformly undersampled parallel imaging acquisitions \\cite{Chang2012}. It was noted that during GRAPPA calibration, both the regressand and the regressor have errors in them due to measurement noise in the acquisition of calibration data, which leads to a non-linear relationship in the estimation. Thus, the reconstruction method, called non-linear GRAPPA, uses a kernel approach to map the data to a higher-dimensional feature space, where linear interpolation is performed, which also corresponds to a non-linear interpolation in the original data space. The interpolation function is estimated from the ACS data, although this approach typically required more ACS data than GRAPPA \\cite{Chang2012}. This method was shown to reduce reconstruction noise compared to GRAPPA. Note that non-linear GRAPPA, through its use of the kernel approach, is a type of machine learning approach, though the non-linear kernel functions were empirically fixed a-priori and not learned from data.\n\n\\subsection{k-space reconstruction via low-rank matrix completion}\nWhile k-space interpolation methods remain the prevalent method for k-space parallel imaging reconstruction, there has been recent efforts on recasting this type of reconstruction as a matrix completion problem. Simultaneous autocalibrating and k-space estimation (SAKE) is an early work in this direction, where local neighborhoods in k-space across all coils are restructured into a matrix with block Hankel form \\cite{Shin2014}. Then low-rank matrix completion is performed on this matrix, subject to consistency with acquired data, enabling k-space parallel imaging reconstruction without additional calibration data acquisition. Low-rank matrix modeling of local k-space neighborhoods (LORAKS) is another method exploiting similar ideas, where the motivation is based on utilizing finite image support and image phase constraints instead of correlations across multiple coils \\cite{Haldar2014}. This method was later extended to parallel imaging to further include the similarities between image supports and phase constraints across coils \\cite{Haldar2016}. A further generalization to LORAKS is annihilating filter-based low rank Hankel matrix approach (ALOHA), which extends the finite support constraint to transform domains \\cite{Jin2016a}. By relating transform domain sparsity to the existence of annihilating filters in a weighted k-space, where the weighting is determined by the choice of transform domain, ALOHA recasts the reconstruction problem as the low-rank recovery of the associated Hankel matrix.\n\n\n\\section{Machine learning methods for parallel imaging in image space}\n\\label{sec:imagespace_learning}\nThe use of machine learning for image-based parallel MR imaging evolves naturally from \\Cref{eq:inverse_regularization} based on the following key insights. First, in classic compressed sensing, $\\Psi$ are a general regularizers like the image gradient or wavelet transforms, which were not designed specifically with undersampled parallel MRI acquisitions in mind. These regularizers can be generalized to models that have a higher computational complexity. $\\Psi$ can be formulated as a convolutional neural network (CNN)~\\cite{LeCun1989}, where the model parameters can be learned from training data inspired by the concepts of deep learning~\\cite{LeCun2015}. This was already demonstrated earlier in the context of computer vision by Roth and Black~\\cite{Roth2009}. They proposed a non-convex regularizer of the following form:\n\\begin{equation}\n    \\label{eq:foe_model}\n    \\Psi{(\\mathbf{u})} = \\sum_{i=1}^{N_k} \\langle \\rho_i(\\mathbf{K}_i\\mathbf{u}),\\mathbf{1} \\rangle.\n\\end{equation}\nThe regularizer in \\Cref{eq:foe_model} consists of $N_k$ terms of non-linear potential functions $\\rho_i$, $\\mathbf{K}_i$ are convolution operators. $\\mathbf{1}$ indicates a vector of ones. \nThe parameters of the convolution operators and the parametrization of the non-linear potential functions, form the free parameters of the model, which are learned from training data.\n\n\\begin{figure}[!t]\n\n\t\\includegraphics[width = 1 \\columnwidth]{.\/figure_learning_image_space_illustration_150dpi}\n \t\\caption{Illustration of machine learning-based image reconstruction. The network architecture consists of $T$ stages that perform the equivalent of gradient descent steps in a classic iterative algorithm. Each stage consists of a regularizer and a data consistency layer. Training the network parameters $\\Theta$ is performed by retrospectively undersampling fully sampled multi-coil raw k-space data and comparing the output of the network $\\mathbf{u}^T_s(\\Theta)$ to a target reference reconstruction $\\mathbf{u}_{\\text{ref}}$ obtained from the fully sampled data.}\n \t\\label{fig:learning_image_space_illustration}\n\\end{figure}\n\n\n\\begin{figure*}[!t]\n\\begin{center}\n\n    \\includegraphics[width = 1 \\textwidth]{.\/figure_knee_reconstructions_150dpi}\n \t\\caption{Comparison of image-domain based parallel imaging reconstructions of a retrospectively accelerated coronal knee acquisition.\n \tThe used sampling pattern, zero-filling, CG-SENSE, combined parallel imaging and compressed sensing with a TGV constrained and a learned reconstruction are shown together with their SSIM values to the fully sampled reference. See text in the respective sections for details on the individual experiments.}\n \t\\label{fig:knee_reconstructions}\n \t\\vspace{-.3cm}\n\\end{center}\n\\end{figure*}\n\nThe second insight is that the iterative algorithm that is used to solve \\Cref{eq:inverse_regularization} naturally maps to the structure of a neural network, where every layer in the network represents an iteration step of a classic algorithm \\cite{Gregor2010}.\nThis follows naturally from gradient descent for the least squares problem in \\Cref{eq:inverse_leastSquares} that leads to the iterative Landweber method~\\cite{Landweber1951}. After choosing an initial $\\mathbf{u}^0$, the iteration scheme is given by \\Cref{eq:landweber}:\n\\begin{equation}\n    \\label{eq:landweber}\n\t\\mathbf{u}^{t} = \\mathbf{u}^{t-1} - \\alpha^t \\mathbf{E}^*(\\mathbf{E}\\mathbf{u}^{t-1} - \\mathbf{f}), \\quad t > 0.\n\\end{equation}\n$\\mathbf{E}^*$ is the adjoint of the encoding operator and $\\alpha^t$ is the step size of iteration $t$. Using this iteration scheme to solve the reconstruction problem in \\Cref{eq:inverse_regularization} with the regularizer defined in \\Cref{eq:foe_model} leads to the update scheme defined in \\Cref{eq:landweber_foe}, which forms the basis of recently proposed image space based machine learning methods for parallel MRI:\n\\begin{equation}\n    \\label{eq:landweber_foe}\n\t\\mathbf{u}^{t} = \\mathbf{u}^{t-1} - \\alpha^t \\left( \\sum_{i=1}^{N_k} (\\mathbf{K}_i)^\\top \\rho_i'(\\mathbf{K}_i \\mathbf{u}^{t-1}) + \\lambda^t \\mathbf{E}^*(\\mathbf{E}\\mathbf{u}^{t-1} - \\mathbf{f}) \\right).\n\\end{equation}\nThis update scheme can then be represented as a neural network with $T$ stages corresponding to $T$ iteration steps \\Cref{eq:landweber_foe}. $\\rho_i'$ are the first derivatives of the nonlinear potential functions $\\rho_i$, which are represented as activation functions in the neural network. The transposed convolution operations ${\\mathbf{K}}_i^\\top$ correspond to convolutions with filter kernels rotated by 180 degrees. Most recently proposed approaches follow this structure, and their difference mainly lies is the used model architecture. The idea of the variational network~\\cite{Hammernik2018} follows the structure of classic variational methods and gradient-based optimization, and the network architecture is designed to mimic a classic iterative image reconstruction. The approach from Aggarwal et al.~\\cite{Aggarwal2019} follows a similar design concept, while using convolutional neural networks (CNNs), but shares the same set of parameters for all stages of the network, thus reducing the number of free parameters. It also uses an unrolled conjugate-gradient step for data consistency instead of the gradient based on in \\Cref{eq:landweber}. \n\nAn illustration of image space based machine learning for parallel MRI along the lines of~\\cite{Hammernik2018,Aggarwal2019} is shown in \\Cref{fig:learning_image_space_illustration}. \nTo determine the model parameters of the network that will perform the parallel imaging reconstruction task, an optimization problem needs to be defined that minimizes a training objective. In general, this can be formulated in a supervised or unsupervised manner. Supervised approaches are predominantly used while unsupervised approaches are still a topic of ongoing research (an approach for low-dose CT reconstruction was presented in~\\cite{Wu2017}). Therefore, we will focus on supervised approaches for the remainder of this section. We define the number of stages, corresponding to gradient steps in the network, as $T$. $s$ is the current training image out of the complete set of training data $S$. The variable $\\Theta$ contains all trainable parameters of the reconstruction model. The training objective then takes the following form:\n\\begin{equation}\n  \\label{eq:image_space_learning_training}\n  L(\\Theta) = \\min_{\\Theta} \\frac{1}{2S} \\sum_{s=1}^{S} \\| \\mathbf{u}^T_s(\\Theta) - \\mathbf{u}_{\\text{ref},s} \\|_2^2.\n\\end{equation}\n\n\n\nAs it is common in deep learning, \\Cref{eq:image_space_learning_training} is a non-convex optimization problem that is solved with standard numerical optimizers like stochastic gradient descent or ADAM~\\cite{Kingma2014}. This requires the computation of the gradient of the training objective with respect to the model parameters $\\Theta$. This gradient can be computed via backpropagation~\\cite{LeCun2012}:\n\\begin{equation}\n\t\\frac{\\partial L(\\Theta)}{\\partial \\Theta^t} = \\frac{\\partial \\mathbf{u}^{t+1}}{\\partial \\Theta^t} \\cdot \\frac{\\partial \\mathbf{u}^{t+2}}{\\partial \\mathbf{u}^{t+1}} \\hdots \\cdot \\frac{\\partial \\mathbf{u}^T}{\\partial \\mathbf{u}^{T-1}} \\cdot \\frac{\\partial L(\\Theta)}{\\partial \\mathbf{u}^T}.\n\\end{equation}\n\n\nThe basis of supervised approaches is the availability of a target reference reconstruction $\\mathbf{u}_{\\text{ref}}$. This requires the availability of a fully-sampled set of raw phased array coil k-space data. This data is then retrospectively undersampled by removing k-space data points as defined by the sampling trajectory in the forward operator $\\mathbf{E}$ and serves as the input of the reconstruction network. The current output of the network $\\mathbf{u}^T_s(\\Theta)$ is then compared to the reference $\\mathbf{u}_{\\text{ref}}$ via an error metric. The choice of this error metric has an influence on the properties of the trained network, which is a topic of currently ongoing work. A popular choice is the mean squared error (MSE), which was also used in \\Cref{eq:image_space_learning_training}. Other choices are the $\\ell_1$ norm of the difference~\\cite{Hammernik2017b} and the structural similarity index (SSIM)~\\cite{Wang2004}. Research on generative adversarial networks~\\cite{Goodfellow2014} and learned content loss functions is currently in progress. The current literature in this area is further noted in the discussion section. \n\n\nAn example reconstruction that compares the variational network learning approach from~\\cite{Hammernik2018} to CG-SENSE and constrained reconstructions from the previous sections is shown in \\Cref{fig:knee_reconstructions} together with the SSIM to the fully sampled reference. It can be observed that the learned reconstruction outperforms the other approaches in terms of artifact removal and preservation of small image features, which is also reflected in the highest SSIM. All source code\\footnote{\\url{https:\/\/github.com\/VLOGroup\/mri-variationalnetwork}} for this method is available online. \n\n\n\n\\section{Machine learning methods for parallel imaging in k-space} \\label{sec:kspace_learning}\n\nThere has been a recent interest in using neural network to improve the k-space interpolation techniques using non-linear approaches in a data-driven manner. These newer approaches can be divided into two groups based on how the interpolation functions are trained. The first group uses scan-specific ACS lines to train neural networks for interpolation, similar to existing interpolation approaches, such as GRAPPA or non-linear GRAPPA. The second group uses training databases, similar to the machine learning methods discussed in image domain parallel imaging. \n\nRobust artificial-neural-networks for k-space interpolation (RAKI) is a scan-specific machine learning approach for improved k-space interpolation \\cite{Akcakaya2019}. This approach trains CNNs on ACS data, and uses these for interpolating missing k-space points from acquired ones. The interpolation function can be represented by\n\\begin{align}\n     {\\kappa}&_j(k_x, k_y - m\\Delta k_y) = f_{j,m}(\\{{\\kappa}_c(k_x - b_x \\Delta_x, \n     \\nonumber \\\\\n     &\\quad k_y - R b_y \\Delta_y)\\}\n     _{b_x \\in [-B_x,B_x], b_y \\in [-B_y, B_y], c \\in [1, n_c]}),\n\\end{align}\nwhere $f_{j,m}$ is the interpolation rule implemented via a multi-layer CNN for outputting the k-space of the $m^\\textrm{th}$ set of uniformly spaced missing lines in the $j^\\textrm{th}$ coil, $R$ is the undersampling rate, $B_x, B_y$ are parameters specified by the receptive field of the CNN, $n_c$ is the number of coils. \nThus, the premise of RAKI is similar to GRAPPA, while the interpolation function is implemented using CNNs, whose parameters are learned from ACS data with an MSE loss function. The scan-specific nature of this method is attractive since it requires no training databases, and can be applied to scenarios where a fully-sampled gold reference cannot be acquired, for instance in perfusion or real-time cardiac MRI, or high-resolution brain imaging. \n\\begin{figure}\n\\centering\n\t\\includegraphics[width = \\columnwidth]{.\/RAKI_7T}\n\t \\caption{A slice from a high-resolution (0.6 mm isotropic) 7T brain acquisition, where all acquisitions were performed with prospective acceleration. It is difficult to acquire fully-sampled reference datasets for training for such acquisitions, thus two scan-specific k-space methods were compared. The CNN-based RAKI method visibly reduced noise amplification compared to the linear GRAPPA reconstruction.}\n \n \t\\label{fig:raki7T}\n \t\\vspace{-.2cm}\n\\end{figure}\nExample RAKI and GRAPPA reconstructions for such high-resolution brain imaging datasets, which were acquired with prospective undersampling are shown in \\Cref{fig:raki7T}. These data were acquired on a 7T system (Siemens Magnex Scientific) with 0.6 mm isotropic resolution. $R = 5,6$ data were acquired with two averages for improved SNR to facilitate visualization of any residual artifacts. Other imaging parameters are available in \\cite{Akcakaya2019}. For these datasets, RAKI leads to a reduction in noise amplification compared to GRAPPA. Note the noise reduction here is based on exploiting properties of the coil geometry, and not on assumptions about image structure, as in traditional regularized inverse problem approaches, as in \\Cref{sec:2c}. \nHowever, the scan-specificity also comes with downsides, such as the computational burden of training for each scan \\cite{Zhang2018a}, as well as the requirement for typically more calibration data. In \\Cref{fig:knee_kspace}, reconstructions of the knee dataset from \\Cref{fig:knee_reconstructions} are shown, where all methods, which rely on subject-specific calibration data, exhibit a degree of artifacts, due to the small size of the ACS region, while RAKI has the highest SSIM among these.\n\nWhile originally designed for uniform undersampling patterns, this method has been extended to arbitrary sampling, building on the self-consistency approach of SPIRiT \\cite{Hosseini2019}. Additionally, recent work has also reformulated this interpolation procedure as a residual CNN, with residual defined based on a GRAPPA interpolation kernel \\cite{Zhang2019}. Thus, in this approach called residual RAKI (rRAKI), the CNN effectively learns to remove the noise amplification and artifacts associated with GRAPPA, giving a physical interpretation to the CNN output, which is similar to the use of residual networks in image denoising \\cite{Zhang2017denoise}. An example application of the rRAKI approach in simultaneous multi-slice imaging \\cite{Zhang2018b} is shown in \\Cref{fig:rraki}.\n\\begin{figure}[!t]\n\\centering\n\t\\includegraphics[width =\\columnwidth]{.\/knee_kspace}\n \t\\caption{Comparison of k-space parallel imaging reconstructions of a retrospectively accelerated coronal knee acquisition, as in \\Cref{fig:knee_reconstructions}. Due to the small size of the ACS data relative to the acceleration rate, the methods, none of which utilizes training databases, exhibit artifacts. GRAPPA has residual aliasing, whereas SPIRiT shows noise amplification. These are reduced in RAKI, though the residual artifacts remain. Respective SSIM values reflect these visual assessment.}\n \t\\label{fig:knee_kspace}\n \t\\vspace{-.2cm}\n\\end{figure}\n\n\\begin{figure*}[!t]\n\\centering\n\t\\includegraphics[width = \\textwidth]{.\/fmri_mid8slice}\n \t\\caption{Reconstruction results of simultaneous multi-slice imaging of 16 slices in fMRI, where the central 8 slices are shown. GRAPPA method exhibits noise amplification at this high acceleration rate. The rRAKI method, whose linear and residual components are depicted by $G^C$ and $F^C$ respectively, exhibits reduced noise. Due to imperfections in the ACS data for this application, the residual component includes both noise amplification and residual artifacts.}\n \t\\label{fig:rraki}\n\\end{figure*}\n\nA different line of work, called DeepSPIRiT, explores using CNNs trained on large databases for k-space interpolation with a SPIRiT-type approach \\cite{Cheng2018}. Since sensitivity profiles and number of coils vary for different anatomies and hardware configurations, k-space data in the database were normalized using coil compression to yield the same number of channels \\cite{Buehrer2007, Huang2008}. Coil compression methods effectively capture most of the energy across coils in a few virtual channels, with the first virtual channel containing most of the energy, the second being the second most dominant, and so on, in a manner reminiscent of principal component analysis. After this normalization of the k-space database, CNNs are trained for different regions of k-space, which are subsequently applied in a multi-resolution approach, successively improving the resolution of the reconstructions, as illustrated in \\Cref{fig:deepspirit}. The method was shown to remove aliasing artifacts, though difficulty with high-resolution content was noted. Since DeepSPIRiT trains interpolation kernels on a database, it does not require calibration data for a given scan, potentially reducing acquisition time further. \n\n\\begin{figure}[!b]\n\\centering\n\n\t\\includegraphics[width =\\columnwidth]{.\/deepspirit_small}\n\t\\caption{The multi-resolution k-space interpolation in DeepSPIRiT uses distinct CNNs for diffrent regions of k-space, successively refining the resolution of the reconstructed k-space.}\n \n \t\\label{fig:deepspirit}\n\\end{figure}\n\nNeural networks have also been applied to the Hankel matrix based approaches in k-space \\cite{Han2018a}. Specifically, the completion of the weighted k-space in ALOHA method has been replaced with a CNN, trained with an MSE loss function. The method was shown to not only improve the computational time, but also the reconstruction quality compared to original ALOHA by exploiting structures beyond low-rankness of Hankel matrices.\n\n\n\n\n\\section{Discussion}\n\\subsection{Issues and open problems}\nSeveral advantages of machine learning approaches over classic constrained reconstruction using predefined regularizers have been proposed in the literature. First, the regularizer is tailored to a specific image reconstruction task, which improves the removal of residual artifacts. This becomes particularly relevant in situations where the used sampling trajectory does not fulfill the incoherence requirements of compressed sensing, which is often the case for clinical parallel imaging protocols. Second, machine learning approaches decouple the compute-heavy training step from a lean inference step. In medical image reconstruction, it is critical to have images available immediately after the scan, while prolonged training procedures that can be done on specialized computing hardware, are generally acceptable. The training for the experiment in \\Cref{fig:knee_reconstructions} took 40 hours for 150 epochs with 200 slices of training data on a single NVIDIA M40 GPU with 12GB of memory. Training data, model and training parameters exactly follow the training from~\\cite{Knoll2019}.  Reconstruction of one slice then took 200ms, in comparison to 10ms for zero filling, 150ms for CG-SENSE and 10000ms for the PI-CS TGV constrained reconstruction.\n\nThe focus in \\Cref{sec:imagespace_learning} and \\Cref{sec:kspace_learning} were on methods that were developed specifically in the context of parallel imaging. Some architectures for image domain machine learning have been designed specifically towards a target application, for example dynamic imaging~\\cite{Schlemper2018,Qin2019}. In their current form, these were not yet demonstrated in the context of multi-coil data. The approach recently proposed by Zhu et al. learns the complete mapping from k-space raw data to the reconstructed image~\\cite{Zhu2018a}. The proposed advantage is that since no information about the acquisition is included in the forward operator $A$, it is more robust against systematic calibration errors during the acquisition. This comes at the price of a significantly higher number of model parameters. The corresponding memory requirements make it challenging use this model for matrix sizes that are currently used in clinical applications. \nWe also note that there are fewer works in k-space machine learning methods for MRI reconstruction. This may be due to the different nature of k-space signal that usually has very different intensity characteristics in the center versus the outer k-space, which makes it difficult to generalize the plethora of techniques developed in computer vision and image processing that exploit properties of natural images.\n\nMachine learning reconstruction approaches also come with a number of drawbacks when compared to classic constrained parallel imaging. First, they require the availability of a curated training data set that is representative so that the trained model generalized towards new unseen test data. While recent approaches from the literature~\\cite{Hammernik2018,Aggarwal2019,Schlemper2018,Qin2019,Chen2018} have either been trained with hundreds of examples rather than millions of examples as it is common in deep learning for computer vision~\\cite{Deng2009,LeCun2015}, or trained on synthetic non-medical data~\\cite{Zhu2018a} that is publicly available from existing databases. However, this is still a challenge that will potentially limit the use of machine learning to certain applications. Several applications in imaging of moving organs, such as the heart, or in imaging of the brain connectivity, such as diffusion MRI, cannot be acquired with fully-sampled data due to constraints on spatio-temporal resolutions. This hinders the use of fully-sampled training labels for such datasets, highlighting applications for scan-specific approaches or unsupervised training strategies.\n\nThese reconstruction methods also require the availability of computing resources during the training stage. This is a less critical issue due to the increased availability and reduced prices of GPUs. The experiments in this paper were made with computing resources that are available for less than 10,000 USD, which are usually available in academic institutions. In addition, the availability of on-demand cloud based machine learning solutions is constantly increasing. \n\nA more severe issue is that in contrast to conventional parallel imaging and compressed sensing, machine learning models are mostly non-convex. Their properties, especially regarding their failure modes and generalization potential for daily clinical use, are understood less well than conventional iterative approaches based on convex optimization. For example, it was recently shown that while reconstructions generalize well with respect to changes in image contrast between training and test data, they are susceptible towards systematic deviations in SNR~\\cite{Knoll2019}. It is also still an open question how specific trained models have to be. \nIs it sufficient to train a single model for all types of MR exams, or are separate models required for scans of different anatomical areas, pulse sequences, acquisition trajectories and acceleration factors as well as scanner manufacturers, field strengths and receive coils? \\cite{icassp2017_open_problems} \nWhile pre-training a large number of separate models for different exams would be feasible in clinical practice, if certain models do not generalize with respect to scan parameter settings that are usually tailored to the specific anatomy of an individual patient by the MR technologist, this will severely impact their translational potential and ultimately their clinical use.\n\nFinally, the choice of the loss function that is used during the training has an impact on the properties of the trained network, and a particular ongoing research direction is the use of GANs~\\cite{Goodfellow2014,Arjovsky2017,Gulrajani2017,Mao2017}. This is an interesting direction because these models have the potential to create images that are visually indistinguishable from fully sampled reference images, where features in the images that are not supported by the amount of acquired data, are hallucinated by the network. This situation obviously must be avoided in any application in medical imaging. Strategies to mitigate this effect are the combination of GANs with conventional error metrics like MSE~\\cite{Isola2016image,Ledig2017c}. Comparable approaches were used in the context of MRI reconstruction~\\cite{Quan2017,Shitrit2017,Yang2018,Hammernik2018a,Kim2018a,Mardani2018}. \n\n\\subsection{Availability of training databases and community challenges}\nAs mentioned in the previous section, one open issue in the field of machine learning reconstruction for parallel imaging is the lack of publicly available databases of multi-channel raw k-space data. This restricts the number of researchers who can work in this field to those who are based at large academic medical centers where this data is available, and for the most part excludes the core machine learning community that has the necessary theoretical and algorithmic background to advance the field. In addition, since the used training data becomes an essential part of the performance of a certain model, it is currently almost impossible to compare new approaches that are proposed in the literature with each other if the training data is not shared when publishing the manuscript. While the momentum in initiatives for public releases of raw k-space data is growing~\\cite{fastMRIcommunity}, the number of available data sets is still on the order of hundreds and limited to very specific types of exams. Examples of publicly available rawdata sets are mridata.org\\footnote{\\url{https:\/\/mridata.org}} and the fastMRI dataset\\footnote{\\url{https:\/\/fastmri.med.nyu.edu\/}}.\n\n\\section{Conclusion}\n\nMachine learning methods have recently been proposed to improve the reconstruction quality in parallel imaging MRI. These techniques include both image domain approaches for better image regularization and k-space approaches for better k-space completion. While the field is still in its development, there are many open problems and high-impact applications, which are likely to be of interest to the broader signal processing community.\n\n\\bibliographystyle{IEEEbib}\n\\input{output.bbl}\n\n\n\n\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\IEEEPARstart{R}{ecent} work on visual recognition focuses on \nthe importance of obtaining large labeled datasets \nsuch as ImageNet~\\cite{deng2009imagenet}. \nLarge-scale datasets when used for \ntraining deep neural network models tend to produce \nstate-of-the-art results on visual recognition~\\cite{krizhevsky2012imagenet}. \nHowever, in some cases, it may be difficult to obtain a large number of samples \nfor certain rare or fine-grained categories. \nHence, recognizing these rare categories become difficult. \nHumans, on the other hand, can easily recognize these rare categories \nby identifying the semantic description of the new category \nand how it is related to the seen categories. \nFor example, a person can identify a new animal zebra by \nidentifying the semantic description of a zebra having \nblack and white stripes and looking like a horse.\nA similar approach is undertaken for \nlearning models to recognize unseen and rare categories. \nThis learning scenario is known as zero-shot learning (ZSL) \nbecause zero labeled samples of the unseen categories \nare available for the training stage. ZSL has promising ramifications in autonomous vehicles, medical imaging, robotics, etc., where it is difficult to annotate images of novel categories but high-level semantic descriptions of classes can be obtained easily.\n\nTo be able to recognize unseen categories, \nwe usually train a learning model using a large collection of labeled samples \nfrom the seen categories and then adapt it to unseen categories. \nFor zero-shot recognition, the seen and the unseen categories \nare related through a high-dimensional vector space \nknown as semantic-descriptor space. \nEach category is assigned a unique semantic descriptor. \nExamples of semantic descriptor can be manually \ndefined attributes~\\cite{lampert2014attribute} \nor automatically extracted word vectors~\\cite{word2vec}. \nFigure~\\ref{fig:zsl} depicts the ZSL problem in terms of \nhow much information is available during training and testing.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=8cm]{img\/zsl.png}\n\\vspace*{-0.1in}\n\\caption{Depiction of the zero-shot learning problem. \nDuring training, we have lots of labeled images from seen classes (cat, dog, elephant) \nbut no labeled images from unseen classes. \nWe do have semantic descriptors of all the classes available. \nUsing all the information, the goal is to recognize the unseen classes.}\n\\label{fig:zsl}\n\\vspace*{-0.2in}\n\\end{figure}\n\nMost ZSL methods involve mapping from the visual feature space \nto the semantic-descriptor space or \nvice versa~\\cite{Zhang_2017_CVPR,akata2016label,frome2013devise,socher2013zero}. \nSometimes, both the visual features and the semantic descriptors \nare mapped to a common feature space~\\cite{zhang2016zero,changpinyo2016synthesized}. \nMost of these mapping-based approaches \nlearn an embedding function for samples and semantic descriptors. \nThe embedding is learned by minimizing a similarity function \nbetween the embedded samples and the corresponding embedded semantic descriptors. \nThus, most ZSL methods differ in the choice of the embedding and similarity functions. \nLampert et. al~\\cite{lampert2014attribute} used linear classifiers, \nidentity function and Euclidean distance for the sample embedding, \nsemantic embedding and similarity metric, respectively. \nRomera-Paredes et al.~\\cite{romera2015embarrassingly} \nused linear projection, identity function and dot product. \nALE~\\cite{akata2016label}, DEVISE~\\cite{frome2013devise}, \nSJE~\\cite{akata2015evaluation} all used a bilinear compatibility framework, \nwhere the projection was linear and the similarity metric was a dot product. \nThey used different variations of pairwise ranking objective to train the model. \nLATEM~\\cite{xian2016latent} was an extension of the above method, \nwhich used piecewise linear projections to account for the non-linearity. \nCMT~\\cite{socher2013zero} used a neural network to map image features \nto semantic descriptors with an additional novelty detection stage to detect unseen categories.\nSAE~\\cite{kodirov2017semantic} used an auto-encoder-based approach, \nwhere the image feature is linearly mapped to a semantic descriptor \nas well as being reconstructed from the semantic-descriptor space. \nDEM~\\cite{Zhang_2017_CVPR} used a neural network to map \nfrom a semantic-descriptor space to an image-feature space. \n \nAfter the embedding is carried out, \nclassification is performed using the nearest-neighbor search. \nAn earlier study~\\cite{radovanovic2010hubs} showed that the nearest-neighbor search \nin such a high-dimensional space suffers from the \\emph{hubness phenomenon} \nbecause only a certain number of data-points becomes nearest neighbor \nor hubs for almost all the query points, resulting in erroneous classification results. \nHowever, Shigeto et al.~\\cite{shigeto2015ridge} showed that \nmapping from a semantic-descriptor space \nto a visual-feature space does not aggravate the hubness problem. \nThus, in this paper, we pursue a semantic-descriptor-space\nto a visual-feature-space mapping approach. \nWe further introduce the concept of relative features \nthat uses pairwise relations between data-points. \nThis not only provides additional structural information about the data \nbut also reduces the dimensionality of the feature space implicitly~\\cite{2stage}, \nthus alleviating the hubness problem.\n\nZero-shot learning further suffers from a projection-domain-shift problem \nbecause the mapping from the semantic-descriptor space \nto the visual-feature space is learned from \nthe data belonging to only the seen categories. \nAs a result, the projected semantic descriptors of \nthe unseen categories are misplaced from the unseen test-data distribution.  \nFu et al.~\\cite{fu2015transductive} identified the domain-shift problem \nand used multiple semantic information sources and label propagation \non unlabeled data from the unseen categories to counter the problem. \nKodirov et al.~\\cite{kodirov2015unsupervised} cast ZSL \nas a dictionary-learning problem and constrained the dictionary \nof the seen and unseen data to be close to each other. \nThis transductive approach is unrealistic \nas it assumes access to the unlabeled test data from unseen categories \nduring the training stage. \nAt the very least, we could carry out the test-time post-processing \nof the semantic descriptors. \nFor test-time adaptation, we propose to find correspondences \nbetween the projected semantic descriptors and the unlabeled test data \nafter which the descriptors are further mapped to the corresponding data-points. \nThis is inspired by recent work on local correspondence-based approach \nto unsupervised domain adaptation~\\cite{das2018sample}, \nwhich produces better results than global domain-adaptation methods.\n\nAnother problem with ZSL is that models \nare generally evaluated only on unseen categories. \nIn a real-world scenario, we expect the seen categories to appear \nmore frequently compared to the unseen categories. \nAs a result, it is appropriate to test our model on both seen and unseen categories. \nThis evaluation setting is known as Generalized Zero-Shot Learning (GZSL) \nand was initially introduced by Chao, et al.~\\cite{chao2016empirical}. \nThey found that the performance of unseen categories \nin the GZSL setting was poor and proposed a shifted-calibration mechanism \nto improve the performance. \nThis shifted-calibration mechanism lowers the classification scores of the seen categories. \nWe propose to develop a scaled-calibration mechanism to study the effect on recognition performance. \nThis has an effect of changing \nthe effective variance of a class and is therefore more interpretable.\n\nOther methods for ZSL include hybrid and synthesized methods. \nHybrid models expressed image features or semantic embeddings \nas a combination\/mixture of existing seen features or semantic embeddings. \nSemantic Similarity Embedding (SSE)~\\cite{zhang2015zero} \nexploits class relationship at both the image-feature and semantic-descriptor spaces \nto map them into a common embedding space. \nOur proposed ZSL method also exploits pairwise relationships \nbetween classes by minimizing the discrepancy between \nthe projected semantic descriptors and the corresponding class prototype \nobtained from the image features. \nCONSE~\\cite{norouzi2013zero} learns the probability of a seen sample \nbelonging to a seen class and uses the probability of an unseen sample \nbelonging to seen classes to relate to the semantic-descriptor space. \nSynthetic Classifiers (SYNC)~\\cite{changpinyo2016synthesized} \nlearn a mapping between the semantic-embedding space \nand the model-parameter space. \nThe model parameters of the classes are represented as a combination of phantom classes, \nthe relationship with which is encoded through a weighted bipartite graph. \nSynthesized methods generally convert ZSL into \na standard supervised-learning problem \nby generating samples for the unseen categories. \nSome of these methods\ninclude~\\cite{verma2018generalized,guo2017synthesizing}. \nThe limitations of these methods lie in not being able \nto generate samples very close to the true distribution. \nA more comprehensive overview of recent work on ZSL can be found in~\\cite{xian2018zero}.\n\nTo summarize, we propose a three-step approach to zero-shot learning. \nFirstly, to prevent aggravating the hubness problem, \na mapping is learned from the semantic-descriptor space to the image-feature space that minimizes both one-to-one and pairwise distances \nbetween semantic embeddings and the image features. \nSecondly, to alleviate the domain-shift problem at test time, \nwe propose a domain-adaptation method that finds\ncorrespondences between the semantic descriptors and the image features of test data.\nThirdly, to reduce biased-ness in the GZSL setting, \nwe propose scaled calibration on the classification scores of the seen classes\nto balance the performance on the seen and unseen categories.\nFinally, we evaluated our proposed approach on four standard ZSL datasets \nand compared our approach against state-of-the art methods \nfollowed by further analyzing the contribution of each component of our approach.\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=16cm]{img\/approach.png}\n\\vspace*{-0.1in}\n\\caption{The semantic descriptors are mapped to the image-feature space through the multi-layer perceptron. Then the semantic embeddings are regressed to the corresponding features through one-to-one and pairwise relations. After that, the semantic embeddings of the unseen classes are adapted to the unseen test data. This is followed by scaled calibration during testing when classification scores of seen classes are modified.}\n\\label{fig:approach}\n\\vspace*{-0.2in}\n\\end{figure*}\n\n\\section{Methodology}\n\\subsection{Problem Description}\nLet the training dataset $\\mathcal{D}_{tr}$ \nconsist of $N_{tr}$ samples such that \n$\\mathcal{D}_{tr}=\\{(\\mathbf{x}_i, \\mathbf{a}_i, y_i), i=1,2,\\ldots,N_{tr}\\}$. \nHere, $\\mathbf{x}_i \\in \\mathbb{R}^{m\\! \\times \\! n \\! \\times \\!c}$ \nis an image sample ($m\\! \\times \\!n$ is the image size and $c$ is the number of channels) \nand $\\mathbf{a}_i \\in \\mathbb{R}^{s}$ is the semantic descriptor \nof the sample's class. \nEach semantic descriptor $\\mathbf{a}_i$ is uniquely associated with \na class label $y_i \\in \\mathcal{Y}_{tr}$. \nThe goal of ZSL is to predict the class label $y_j \\in \\mathcal{Y}_{te}$ \nfor the $j^{th}$ test sample $\\mathbf{x}_j$. \nIn the traditional ZSL setting, \nwe assume that $\\mathcal{Y}_{tr} \\cap \\mathcal{Y}_{te} = \\varnothing$; \nthat is, the seen (training) and the unseen (testing) classes are disjoint. \nHowever, in the GZSL setting, both seen and unseen classes \ncan be used for testing; \nthat is, $\\mathcal{Y}_{tr} \\subset \\mathcal{Y}_{te}$. \nIn the training stage, we have the semantic descriptors \nof both the seen and unseen classes available \nbut no labeled training data of the unseen classes are available. \nThe overall framework of our proposed ZSL approach is shown in Fig.~\\ref{fig:approach}.\n\n\\subsection{Relational Matching}\nOur goal is to learn a mapping $\\mathbf{f}(\\cdot)$ \nthat maps a semantic descriptor $\\mathbf{a}_i$ \nto its corresponding image feature $\\mathbf{\\phi}(\\mathbf{x}_i)$. \nHere, $\\mathbf{x}_i$ is an image and $\\mathbf{\\phi}(\\cdot)$ \nrepresents a CNN architecture that extracts a high-dimensional feature map. \nThe mapping ${\\bf f}(\\cdot)$ is a fully-connected neural network. \nSince our goal is to make the embedded semantic descriptor \nclose to the corresponding image feature, \nwe use a least square loss function to minimize the difference. \nWe also need to regularize the parameters of $\\mathbf{f}(\\cdot)$. \nIncluding these costs and averaging over all the instances, \nour initial objective function $\\mathcal{L}_1$ is as follows:\n\\begin{equation}\n\\mathcal{L}_1 = \\frac{1}{N_{tr}}\\sum_{i=1}^{N_{tr}}||\\mathbf{f}({\\bf a}_i)-\\mathbf{\\phi}(\\mathbf{x}_i)||_2^2 + \\lambda_rg({\\bf f})~,\n\\end{equation}\nwhere $g(\\cdot)$ is the regularization loss for the mapping function. \nThe loss function $\\mathcal{L}_1$ minimizes the point-to-point discrepancy \nbetween the semantic descriptors and the image features. \nTo account for the structural matching between \nthe semantic-descriptor space and the image-feature space, \nwe try to minimize the inter-class pairwise relations in these two spaces. \nThus, we construct relational matrices for both the semantic descriptors and image features. \nThe semantic relational matrix $\\mathbf{D}_a$ is established \nsuch that each element, \n$[\\mathbf{D}_a]_{uv} = ||{\\bf f}({\\bf a}^u)-{\\bf f}({\\bf a}^v)||_2^2$, \nwhere ${\\bf a}^u$ and ${\\bf a}^v$ are semantic descriptors \nof seen categories $u$ and $v$, respectively. \nThe image feature relational matrix $\\mathbf{D}_\\phi$ \nis constructed such that each element, \n$[\\mathbf{D}_\\phi]_{uv} = ||\\overline{\\mathbf{\\phi}}^u-\\overline{\\mathbf{\\phi}}^v||_2^2$, \nwhere $\\overline{\\mathbf{\\phi}}^u$ and $\\overline{\\mathbf{\\phi}}^v$ \nare mean representations of the categories $u$ and $v$, respectively. \n$\\overline{\\mathbf{\\phi}}^u$ can be represented as \n\\begin{equation}\n\\overline{\\mathbf{\\phi}}^u=\\frac{1}{|\\mathcal{Y}^{u}_{tr}|}\\sum_{y_i \\in \\mathcal{Y}^{u}_{tr}}^{}\\mathbf{\\phi}({\\mbox{\\boldmath $x$}}_i)~,\n\\end{equation}\nwhere the summation is over the representations of class $u$, \nand $|\\mathcal{Y}^{u}_{tr}|$ is the cardinality of the training set of class $u$. \nA similar formula holds for class $v$. \nFor structural alignment, we want the two relational matrices, \n$\\mathbf{D}_a$ and $\\mathbf{D}_{\\phi}$, \nto be close to one another. \nHence, we want to minimize the structural alignment loss function $\\mathcal{L}_2$,\n\\begin{equation}\n\\mathcal{L}_2=||\\mathbf{D}_a-\\mathbf{D}_{\\phi}||_F^2~,\n\\end{equation}\nwhere $||\\cdot||_F^2$ stands for the Frobenius norm. \nCombining the loss functions $\\mathcal{L}_1$ and $\\mathcal{L}_2$, \nwe have the total loss $\\mathcal{L}_{total}$,\n\\begin{equation}\n\\mathcal{L}_{total} = \\mathcal{L}_1 + \\rho \\mathcal{L}_2~,\n\\end{equation}\nwhere $\\rho \\geq 0$ weighs the loss contribution of $\\mathcal{L}_2$.\n$\\mathcal{L}_{total}$ is to be optimized with respect to \nthe parameters of the semantic-descriptor-to-visual-feature-space mapping ${\\bf f}(\\cdot)$.\n\n\\subsection{Domain Adaptation}\nAfter the training is carried out, \ndomain discrepancy may be present between \nthe mapped semantic descriptors and the image features of unseen categories. \nThis is because the unseen data has not been used in the training \nand our regularized model does not \ngeneralize well for the unseen categories. \nHence, we need to adapt the mapped semantic descriptors \nfor the unseen categories using the test data from the unseen categories. \nLet the mapped descriptors for the unseen categories be stacked vertically \nin the form of a matrix ${\\bf A} \\in \\mathbb{R}^{n_u\\!\\! \\times \\!\\!d}$, \nwhere $n_u$ is the number of unseen categories \nand $d$ is the dimension of the mapped semantic-descriptor space, \nand therefore it is also the dimension of the image-feature space. \nLet ${\\bf U} \\in \\mathbb{R}^{o_u\\!\\! \\times \\!\\!d}$ be the unseen test dataset, \nwhere $o_u$ is the number of test instances from the unseen categories. \nFor adapting the mapped descriptors, \nwe propose to find the point-to-point correspondence \nbetween the descriptors and the test data. \nLet the correspondence be represented as a matrix \n${\\bf C} \\in \\mathbb{R}^{n_u\\! \\times \\!o_u}$. \nWe want to rearrange the rows of ${\\bf U}$ such that \neach row of the modified matrix corresponds to the row in ${\\bf A}$. \nThis is done by minimizing the following loss function $\\mathcal{L}_3$,\n\\begin{equation}\n\\mathcal{L}_3=||{\\bf C}{\\bf U}-{\\bf A}||_F^2~.\n\\end{equation}\n\nThis loss function enforces that ${\\bf C}{\\bf U}$ produces the adapted semantic descriptors. \nHowever, a problem may exist that an instance in ${\\bf U}$ \ncorresponds to more than one descriptor in ${\\bf A}$. \nThis would essentially result in a test sample corresponding to more than one category. \nTo avoid that, we use an additional group-based regularization function \n$\\mathcal{L}_4$ using Group-Lasso,\n\\begin{equation}\n\\mathcal{L}_4=\\sum_{j}\\sum_{c}||[{\\bf C}]_{I_cj}||_2~,\n\\end{equation}\nwhere $I_c$ corresponds to the indices of those rows in ${\\bf A}$ \nthat belong to the unseen class $c$. \nTherefore, $[{\\bf C}]_{I_cj}$ is the vector consisting of \nthe row indices from $I_c$ and the $j^{th}$ column. \nSince ${\\bf C}$ is a correspondence matrix, \nsome constraints should be enforced such as ${\\bf C} \\geq \\mathbf{0}$, \n${\\bf C}\\mathbf{1}_{o_u}=\\mathbf{1}_{n_u}$ \nand ${\\bf C}^{T}\\mathbf{1}_{n_u}=\\frac{n_u}{o_u}\\mathbf{1}_{o_u}$, \nwhere $\\mathbf{1}_{n}$ is an $n\\! \\times \\!1$ vector of one's. \nThe second equality constraint is scaled by the factor $\\frac{n_u}{o_u}$ \nto account for the difference in the number of instances \nin the mapped semantic-descriptor space and the image-feature space \nfor the unseen categories. \nHence, the domain adaptation optimization problem becomes \n\\begin{equation}\n\\underset{{\\bf C}}{\\text{min}} \\hspace{0.1em} \\left\\{\\mathcal{L}_3 + \\lambda_g \\mathcal{L}_4\\right\\}  \\hspace{0.5em} s.t. \\hspace{0.5em} {\\bf C} \\geq \\mathbf{0}, {\\bf C}\\mathbf{1}_{o_u}=\\mathbf{1}_{n_u},{\\bf C}^{T}\\mathbf{1}_{n_u}=\\frac{n_u}{o_u}\\mathbf{1}_{o_u}~,\n\\label{eq7}\n\\end{equation}\nwhere $\\lambda_g$ weighs the loss function $\\mathcal{L}_4$.\n\nThe above optimization problem is convex and can be efficiently solved \nusing the conditional gradient method~\\cite{frank1956algorithm}. \nThe conditional gradient method requires solving \na linear program as an intermediate step over the constraints \n${\\bf C} \\in \\mathcal{D}=\\{{\\bf C}: {\\bf C} \\geq {\\bf 0}, {\\bf C}\\mathbf{1}_{o_u}=\\mathbf{1}_{n_u}, \n{\\bf C}^{T}\\mathbf{1}_{n_u}=\\frac{n_u}{o_u}\\mathbf{1}_{o_u}\\}$ \nas shown in Algorithm 1. \nThe linear program of finding the intermediate variable \n${\\bf C}_d$ in Algorithm 1 can be easily solved \nusing a network simplex formulation \nof the earth-mover's distance problem~\\cite{bonneel2011displacement}.\n\\begin{algorithm}[]\n\\SetAlgoLined\n \\textbf{Intitialize :} ${\\bf C}_0 = \\frac{1}{(n_uo_u)}\\mathbf{1}_{n_u\\! \\times \\!o_u}$, $t=1$\\\\\n \\textbf{Repeat}\\\\\n \\quad ${\\bf C}_d = \\underset{{\\bf C}}{\\text{argmin}} \\hspace{0.5em} \\text{Tr}(\\nabla_{{\\bf C}={\\bf C}_0} (\\mathcal{L}_3 + \\lambda_g \\mathcal{L}_4)^{T}{\\bf C}),  \\hspace{0.5em} s.t.  \\hspace{0.5em} {\\bf C} \\in \\mathcal{D}$ \\\\\n \\quad ${\\bf C}_1 = {\\bf C}_0 + \\alpha({\\bf C}_d - {\\bf C}_0), \\quad \\text{for} \\quad \\alpha = \\frac{2}{t+2}$ \\\\\n \\quad ${\\bf C}_0={\\bf C}_1$ \\quad \\text{and} \\quad $t=t+1$ \\\\\n \\textbf{Until} Convergence \\\\\n \\textbf{Output :} ${\\bf C}_0 = \\text{arg}\\min \\limits_{{\\bf C}}\\{\\mathcal{L}_3 + \\lambda_g \\mathcal{L}_4\\} \\quad s.t. \\quad {\\bf C} \\in \\mathcal{D}$\n\\caption{Conditional Gradient Method (CG)}\n\\end{algorithm}\n\nOnce the final solution of the correspondence matrix ${\\bf C}_0$ \nin Algorithm 1 is obtained, \nwe inspect ${\\bf C}_0$. \nFor each test instance, \nwe assign the class correspondence to the highest value of the correspondence variable. \nThis is done for all the test instances. \nThe new semantic descriptors are obtained by taking the mean \nof the feature instances belonging to the corresponding class. \nThe adapted semantic descriptors are then stacked vertically in the matrix ${\\bf A}'$.\n\n\\subsection{Scaled Calibration}\nIn the GZSL setting, \nit is known that the classification results are biased \ntowards the seen categories~\\cite{chao2016empirical}. \nTo counteract the bias, \nwe propose the use of multiplicative calibration on the classification scores. \nIn our case, we use 1-Nearest Neighbor (1-NN) \nwith the Euclidean distance metric as the classifier. \nThe classification score for a test point \nis given by the Euclidean distance of the test image feature \nto the mapped semantic descriptor of a category. \nFor a test point ${\\mbox{\\boldmath $x$}}$, \nwe adjust the classification scores on the seen categories as follows\n\\begin{equation}\n\\hat{y}=\\underset{c \\in \\mathcal{T}}{\\text{argmin}}\n\\hspace{0.5em}||{\\mbox{\\boldmath $x$}}-{\\bf f}({\\bf a}^c)||_2\\cdot\\mathbb{I}[c \\in \\mathcal{S}]~,\n\\end{equation} \nwhere $\\mathbb{I}[\\cdot]=\\gamma$ if $c \\in \\mathcal{S}$ \nand 1 if $c \\in \\mathcal{U}$ and $\\mathcal{S} \\cup \\mathcal{U} = \\mathcal{T}$. \nHere, $\\mathcal{S}, \\mathcal{U}$ and $\\mathcal{T}$ \nrepresent the sets of seen, unseen and all categories, respectively. \nThe effect of scaling is to change the effective variance of the seen categories. \nWhen the nearest-neighbor classification is carried out \nwith the Euclidean distance metric, \nit assumes that all classes have equal variance. \nBut since the unseen categories are not used \nfor learning the embedding space, \nthe variance of the unseen-category features is not accounted for. \nThat is why the Euclidean distance metric \nfor the seen categories needs to be adjusted for. \nFor $\\gamma > 1$, if we obtain a balanced performance \nbetween the seen and unseen classes, \nit implies that the variance of the seen classes has been overestimated. \nSimilarly, if we obtain a balanced performance for $\\gamma < 1$, \nit means that the variance of the seen classes has been underestimated. \nThe overall procedure of our proposed zero-shot learning method \nfrom training to testing is given in Algorithm 2.\n\\begin{algorithm}[]\n\\SetAlgoLined\n\\textbf{Input:} Training Dataset $\\{(\\mathbf{x}_i, \\mathbf{a}_i, y_i)\\}_{i=1}^{N_{tr}}$\\\\\n\\textbf{Parameters:} $\\lambda_r$, $\\rho$, $\\lambda_g$, $\\gamma$\\\\\n\\textbf{Repeat} (Training)\\\\\n\\quad Sample Minibatch of $\\{({\\mbox{\\boldmath $x$}}_i,{\\bf a}_i)\\}$ pairs\\\\\n\\quad Gradient descent $\\mathcal{L}_1 + \\rho \\mathcal{L}_2$ w.r.t parameters of ${\\bf f}(\\cdot)$\\\\\n\\textbf{Until} Convergence \\\\\n\\textbf{Input:} Test Dataset $\\{(\\mathbf{x}_i)\\}_{i=1}^{N_{te}}$\\\\\n\\quad Apply Algorithm 1 to obtain adapted descriptors \\\\\n\\quad of unseen classes ${\\bf A}'$ (Adaptation)\\\\\n\\textbf{Repeat} for each test point ${\\mbox{\\boldmath $x$}}$ (Testing)\\\\\n\\quad $\\hat{y}=\\underset{c \\in \\mathcal{T}}{\\text{argmin}}\\hspace{0.5em}||{\\mbox{\\boldmath $x$}}-{\\bf f}({\\bf a}^c)||_2\\cdot\\mathbb{I}[c \\in \\mathcal{S}]$ (Calibration)\\\\\n\\textbf{Until} all test points covered\n\\caption{Proposed Zero-shot Learning Algorithm}\n\\end{algorithm}\n\n\\section{Experimental Results}\nFollowing the previous experimental settings~\\cite{xian2018zero}, \nwe used the following four datasets for evaluation: \n\\textbf{AwA2}~\\cite{lampert2014attribute} (Animal with Attributes) \ncontains 37,322 images of 50 classes of animals. \n40 classes of animals are considered to be the seen categories \nwhile 10 classes of animals are considered to be the unseen categories. \nEach class is associated with a 85-dimensional continuous semantic descriptor. \n\\textbf{aPY}~\\cite{farhadi2009describing} (attribute Pascal and Yahoo) \nconsists of 20 seen categories and 12 unseen categories. \nEach category has an associated 64-dimensional semantic descriptor. \n\\textbf{CUB}~\\cite{welinder2010caltech} (Caltech-UCSD Birds-200-2011) \nis a fine-grained dataset consisting of 11,788 images of birds. \nFor evaluation, all the bird categories are split into 150 seen classes and 50 unseen classes. \nEach class is associated with a 312-dimensional continuous semantic descriptor. \n\\textbf{SUN}~\\cite{patterson2012sun} (Scene UNderstanding database) \nconsists of 14340 scene images. \nAmong these, 645 scene categories are selected as seen categories \nwhile 72 categories are selected as unseen categories \nand it consists of a 102-dimensional semantic descriptor.\n\nFor the purpose of evaluation, \nwe used class-wise accuracy because it prevents dense-sampled classes \nfrom dominating the performance. \nAccordingly, class-wise accuracy is averaged as follows\n\\begin{equation}\nacc=\\frac{1}{|\\mathcal{Y}|}\\sum_{y=1}^{|\\mathcal{Y}|}\\frac{\\text{No. of correct predictions in class}\\hspace{0.5em} y}{\\text{No. of samples in class}\\hspace{0.5em} y},\n\\end{equation}\nwhere $|\\mathcal{Y}|$ is the number of testing classes.\nIn the GZSL case, class-wise accuracy of both seen and unseen classes \nare obtained separately and then averaged using harmonic mean $H$~\\cite{xian2018zero}. \nThis is done so that the performance on seen classes \ndoes not dominate the overall accuracy,\n\\begin{equation}\nH=\\frac{2\\times acc_s\\! \\times \\!acc_u}{acc_s + acc_u},\n\\end{equation}\nwhere $acc_s$ and $acc_u$ are the class-wise accuracy \non seen and unseen categories, respectively. \nIn the GZSL classification setting, \nthe search space of predicted categories consists of both seen and unseen categories. Based on \\cite{xian2018zero} and for fair comparison, a single trial of experimental results on a large batch of training and testing dataset is reported.\n\nFor the experiments, \nwe used a two-layer feedforward neural network \nfor the semantic embedding ${\\bf f}(\\cdot)$. \nThe dimensionality of the hidden layer was chosen as 1600, 1600, 1200 \nand 1600 for the \\textbf{AwA2}, \\textbf{aPY}, \\textbf{CUB} \nand \\textbf{SUN} datasets, respectively. \nThe activation used was ReLU. \nThe image features used were the ResNet-101. \nWe compared different variations of our proposed method with previous approaches. \nOURS-R variation is with the training stage \nincluding the structural loss $\\mathcal{L}_2$. \nOURS-RA includes the structural loss as well as the domain adaptation stage \nincluding the loss functions $\\mathcal{L}_3$ and $\\mathcal{L}_4$. \nOURS-RC includes the structural loss as well as the calibrated testing stage. \nOURS-RAC includes all the components of structural loss, \ndomain adaptation and calibrated testing. \nWithout all these components,  the proposed method reduces to \nthe Deep Embedding Model (DEM)~\\cite{Zhang_2017_CVPR} baseline. \nThe parameters $(\\lambda_r, \\rho, \\lambda_g, \\gamma)$ \nfor the \\textbf{AwA2}, \\textbf{aPY}, \\textbf{CUB} and \\textbf{SUN} datasets \nare set as $(10^{-3}, 10^{-1}, 10^{-1}, 1.1)$, \n$(10^{-4}, 10^{-1}, 10^{-1}, 1.1)$, $(10^{-2}, 0, 10^{-1}, 1.1)$ \nand $(10^{-5}, 10^{-1}, 10^{-1}, 1.1)$, respectively. \nFor the OURS-RAC variation, \nwe used different calibration parameter values of $0.98, 1.1,0.97, 0.999$ \nfor the \\textbf{AwA2}, \\textbf{aPY}, \\textbf{CUB} \nand \\textbf{SUN} datasets, respectively. \n$\\rho$ was set to $0$ for the \\textbf{CUB} dataset \nbecause it is a fine-grained dataset and since the categories \nare very close to each other in the feature space, \nstructural matching does not provide additional information. \nIn Table~\\ref{table:compare}, \nwe reported class-wise accuracy results \nfor the conventional unseen classes setting (\\textbf{tr}), \ngeneralized unseen classes setting (\\textbf{u}), \ngeneralized seen classes setting (\\textbf{s}), \nand the Harmonic mean (\\textbf{H}) of the generalized accuracies.\n\n\\begin{table*}[]\n\\caption{Results of variations of our proposed approach in comparison \nwith previous methods on the \\textbf{AwA2}, \\textbf{aPY}, \\textbf{CUB} \nand \\textbf{SUN} datasets. \nThe best results of each setting in each dataset are shown in boldface.}\n\\label{table:compare}\n\\centering\n\\scalebox{1.0}{\\begin{tabular}{l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}\n\\hline\n       & \\multicolumn{4}{c|}{\\textbf{AwA2}} & \\multicolumn{4}{c|}{\\textbf{aPY}} & \\multicolumn{4}{c|}{\\textbf{CUB}} & \\multicolumn{4}{c}{\\textbf{SUN}} \\\\ \\hline\nMethod & \\textbf{tr}    & \\textbf{u}    & \\textbf{s}    & \\textbf{H}  \n& \\textbf{tr}    & \\textbf{u}    & \\textbf{s}    & \\textbf{H}  \n& \\textbf{tr}    & \\textbf{u}    & \\textbf{s}    & \\textbf{H}  \n& \\textbf{tr}    & \\textbf{u}    & \\textbf{s}    & \\textbf{H}  \n\\\\ \\hline\nDAP~\\cite{lampert2014attribute}    &46.1       &0.0      &84.7      &0.0     &33.8       &4.8      &78.3     &9.0     &40.0       &1.7      &67.9     &3.3     &39.9       &4.2      &25.1     &7.2     \\\\ \nIAP~\\cite{lampert2014attribute}    &35.9       &0.9      &87.6      &1.8     &36.6       &5.7      &65.6     &10.4     &24.0       &0.2      &\\textbf{72.8}     &0.4     &19.4       &1.0      &37.8     &1.8     \\\\ \nCONSE~\\cite{norouzi2013zero}   &44.5      &0.5      &\\textbf{90.6}      &1.0     &26.9       &0.0      &\\textbf{91.2}     &0.0     &34.3       &1.6      &72.2     &3.1     &38.8 &6.8      &39.9     &11.6     \\\\ \nCMT~\\cite{socher2013zero}  &37.9         &0.5      &90.0      &1.0     &28.0       &1.4      &85.2     &2.8     &34.6       &7.2      &49.8     &12.6     &39.9 &8.1      &21.8     &11.8     \\\\ \nSSE~\\cite{zhang2015zero}   &61.0       &8.1      &82.5      &14.8     &34.0       &0.2      &78.9     &0.4     &43.9       &8.5      &46.9     &14.4     &51.5 &2.1      &36.4     &4.0     \\\\ \nLATEM~\\cite{xian2016latent}   &55.8     &11.5      &77.3     &20.0     &35.2       &0.1      &73.0     &0.2     &49.3       &15.2      &57.3     &24.0     &55.3 &14.7      &28.8     &19.5     \\\\ \nALE~\\cite{akata2016label}    &62.5       &14.0     &81.8     &23.9     &39.7       &4.6      &73.7     &8.7     &54.9       &23.7      &62.8     &34.4     &58.1 &21.8      &33.1     &26.3     \\\\ \nDEVISE~\\cite{frome2013devise}    &59.7    &17.1      &74.7    &27.8     &\\textbf{39.8}      &4.9      &76.9     &9.2     &52.0       &23.8      &53.0     &32.8     &56.5 &16.9      &27.4     &20.9     \\\\ \nSJE~\\cite{akata2015evaluation}    &61.9       &8.0      &73.9      &14.4     &32.9       &3.7      &55.7     &6.9     &53.9       &23.5      &59.2     &33.6     &53.7 &14.7      &30.5     &19.8     \\\\ \nESZSL~\\cite{romera2015embarrassingly}    &58.6       &5.9      &77.8      &11.0     &38.3       &2.4      &70.1  &4.6     &53.9       &12.6      &63.8     &21.0     &54.5 &11.0      &27.9     &15.8     \\\\ \nSYNC~\\cite{changpinyo2016synthesized}   &46.6       &10.0      &90.5      &18.0     &23.9       &7.4      &66.3     &13.3     &55.6       &11.5      &70.9     &19.8     &56.3 &7.9      &\\textbf{43.3}     &13.4     \\\\ \nSAE~\\cite{kodirov2017semantic}   &54.1       &1.1      &82.2      &2.2     &8.3       &0.4      &80.9     &0.9     &33.3       &7.8      &54.0     &13.6     &40.3 &8.8      &18.0     &11.8     \\\\ \nGFZSL~\\cite{verma2017simple}   &63.8       &2.5      &80.1     &4.8     &38.4       &0.0      &83.3     &0.0     &49.3       &0.0      &45.7     &0.0     &60.6 &0.0      &39.6     &0.0     \\\\ \nSR~\\cite{annadani2018preserving}   &63.8       &20.7      &73.8      &32.3     &38.4       &13.5      &51.4     &21.4     &\\textbf{56.0}       &24.6      &54.3     &33.9     &61.4 &20.8      &37.2     &26.7     \\\\ \nDEM~\\cite{Zhang_2017_CVPR}   &\\textbf{67.1}       &30.5      &86.4     &45.1     &35.0       &11.1      &75.1     &19.4     &51.7       &19.6      &57.9     &29.2     &40.3       &20.5      &34.3     &25.6     \\\\ \n\\hline\nOURS-R   &63.4    &36.5 &80.6            &50.3     &29.9   &15.3 &71.4          &25.2   &46.6       &20.2 &48.6           &28.6     &59.9      &21.7  &{38.1}          &27.6     \\\\ \nOURS-RA   &64.4      &\\textbf{61.8}   &69.9         &65.6     &35.4        &30.4 &72.9          &42.9     &52.6      &\\textbf{47.6}     &41.0       &44.1     &\\textbf{67.5}      &\\textbf{54.4}  &36.6           &\\textbf{43.7}     \\\\ \nOURS-RC  &63.4      &57.9  &72.0            &64.2     &29.9       &26.4 &53.3    &35.3     &46.6       &27.2      &43.9     &33.6     &59.9        &42.4 &32.6          &36.8     \\\\ \nOURS-RAC  &64.4            &60.6  &72.3     &\\textbf{65.9}     &35.4        &\\textbf{34.1}  &63.5         &\\textbf{44.4}     &52.6       &44.0  &45.1         &\\textbf{44.6}     &\\textbf{67.5}      &54.1 &36.6           &\\textbf{43.7}     \\\\ \\hline\n\\end{tabular}}\n\\vspace*{-0.2in}\n\\end{table*}\n\nFrom the table, we observed that our proposed approach \noutperforms previous methods \nby a large margin in the generalized harmonic mean setting. \nTo be more specific, our proposed method produces \nan improvement of around 20\\%, 23\\%, 10\\% and 16\\% \nharmonic mean accuracy over the previous best approach for the \\textbf{AwA2}, \\textbf{aPY}, \\textbf{CUB} and \\textbf{SUN} datasets, respectively. \nThe large improvement in performance can be attributed \nto our three-step procedure for improvement. \nUsing only the structural matching (OURS-R), \nwe produced better results than previous approaches \nexcept for the \\textbf{CUB} dataset, \nwhere it produces a harmonic mean accuracy of about 28\\%. \nThis is because \\textbf{CUB} requires minute fine-grained feature extraction. \nAdditional usage of domain adaptation (OURS-RA) \nand calibrated testing (OURS-RC) produced \nmuch better results than OURS-R for all the datasets. \nHowever, domain adaptation produced better result than the calibration procedure. \nThis is because our correspondence-based approach \nproduced class-specific adaptation of the unseen class semantic embeddings. \nThe scaled-calibration procedure is not class-specific \nand just differentiates between seen and unseen classes. \nIt also does not adapt to the test data. \n\nIt is to be noted that the difference in performance \nbetween OURS-RA and OURS-RAC is negligible. \nThis is because the domain adaptation step transforms \nthe unseen semantic embeddings away \nfrom the seen categories towards the unseen categories, \nthus reducing the bias towards the seen categories \nand rendering further calibration ineffective. \nThe effect of domain adaptation is visualized in \nFig.~\\ref{fig:tsnezsl1} for the \\textbf{AwA2} dataset \nusing t-SNE~\\cite{maaten2008visualizing}. \nIn Fig.~\\ref{fig:tsnezsl1}(a), \nthe unseen class semantic embeddings (blue) remained very close to \nthe seen class features (maroon). \nHowever, with the domain adaptation step, \nthe unseen class semantic embeddings get transformed to \nnear the centre of unseen class feature clusters (green) \nas shown in Fig.~\\ref{fig:tsnezsl1}(b).\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=8cm]{img\/tsnezsl1.png}\n\\vspace*{-0.1in}\n\\caption{2D t-SNE map of the embedded instances. (a) Without domain adaptation and \n(b) with domain adaptation for the \\textbf{AwA2} dataset. \nHere, the seen and unseen image features are shown in maroon and green, respectively. \nThe embedded semantic descriptors for the seen and unseen classes \nare shown in red and blue color, respectively. }\n\\label{fig:tsnezsl1}\n\\vspace*{-0.2in}\n\\end{figure}\n\nWe also analyzed the effect of the structural matching \nby varying $\\rho \\in \\{10^{-3}, 10^{-2}, 10^{-1}, 10^{0}, 10^{1}, 10^{2}\\}$ \nand observed how the class-wise accuracy changes. \nWe carried out experiments using the \\textbf{AwA2} and \\textbf{SUN} datasets, \nthe results of which are reported in Fig.~\\ref{fig:rhosens}. \nWe also reported the DEM baseline ($\\rho=0$) in dotted lines. \nFrom the plots, the Conventional Unseen and the Generalized Seen accuracies \nare better than or equal to the baseline for only a small range of $\\rho$. \nOn the other hand, the Generalized Unseen accuracy is greater than the baseline \nover a large range of $\\rho$ for the \\textbf{AwA2} dataset \nwhile it oscillated about the baseline for the \\textbf{SUN} dataset. \nFor the \\textbf{SUN} dataset, \nwe do not have a significant gain over the baseline \nbecause \\textbf{SUN} is a fine-grained dataset \nwhere structural matching does not carry additional information. \nThe goal of structural regularization is to exploit \nthe pairwise relations among classes so as to generalize better to novel classes. \nTherefore, we did not see huge difference in performance \nfrom the baseline for the Generalized Seen accuracy. \nSurprisingly, there was a drop in conventional unseen accuracy \nas $\\rho$ was increased. \nThis might be probably because there was no overlap \nbetween the classes used for testing and the classes used for structural matching. \nThis is not the case though in the generalized setting. \n\\begin{figure}[]\n\\centering\n\\includegraphics[width=8cm]{img\/rhosens.png}\n\\vspace*{-0.1in}\n\\caption{Results of class-wise accuracy as $\\rho$ is varied for different settings \non the \\textbf{AwA2} and the \\textbf{SUN} datasets. \nThe baseline used is DEM. The different performance settings \nare Conventional Unseen Accuracy (Left Top), \nGeneralized Seen Accuracy (Right Top), Generalized Unseen Accuracy (Left Bottom) \nand Generalized Harmonic Mean Accuracy (Right Bottom).}\n\\label{fig:rhosens}\n\\vspace*{-0.2in}\n\\end{figure}\n\nWe also studied the effect of varying the calibration parameter $\\gamma$ \non the generalized accuracy for the \\textbf{AwA2} and \\textbf{SUN} datasets. \nThe results are shown in Fig.~\\ref{fig:calib}. \nAs expected, the generalized unseen accuracy increases \nand the generalized seen accuracy decreases with increasing $\\gamma$. \nThe peak of the harmonic mean accuracy was observed close to \nwhen the seen and unseen accuracies became equal. \nThe maximum unseen accuracy is less than the maximum seen accuracy \nfor the \\textbf{AwA2} dataset because the unseen classes \nare less separated and therefore more difficult to classify. \nThe situation is reversed for the \\textbf{SUN} dataset \nwhere the maximum unseen accuracy is more than the maximum seen accuracy.\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=8cm]{img\/calib.png}\n\\vspace*{-0.1in}\n\\caption{Results of Generalized Seen Accuracy (Red), \nGeneralized Unseen Accuracy (Green) \nand Generalized Harmonic Mean Accuracy (Blue) \nas the calibration parameter $\\gamma$ is varied \non the \\textbf{AwA2} and \\textbf{SUN} datasets.}\n\\label{fig:calib}\n\\vspace*{-0.2in}\n\\end{figure}\n\nWe also reported convergence results of the test accuracy \nwith respect to the number of epochs for both the \\textbf{AwA2} \nand the \\textbf{SUN} datasets in Figs.~\\ref{fig:epAwA2} and~\\ref{fig:epSUN}, respectively.\nWe used the OURS-R variation with $\\rho=0.1$ to compare with the DEM baseline. \nThe convergence rate for the baseline and OURS-R variation seems \nto be similar in all the settings for both datasets. \nHowever, our steady-state values were higher \nfor the generalized unseen and generalized harmonic mean setting. \nFor the conventional unseen and generalized seen setting, \nour steady-state value was less than the baseline. \nThe reason is explained previously \nwhile describing performance sensitivity to $\\rho$. \n\\begin{figure}[]\n\\centering\n\\includegraphics[width=8cm]{img\/epAwA2.png}\n\\vspace*{-0.1in}\n\\caption{Convergence results of test accuracy with respect to the number of epochs \nunder different settings for the \\textbf{AwA2} dataset. \nOURS-R results are shown in red color while the DEM baseline is shown in blue color.}\n\\label{fig:epAwA2}\n\\vspace*{-0.2in}\n\\end{figure}\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=8cm]{img\/epSUN.png}\n\\vspace*{-0.1in}\n\\caption{Convergence results of test accuracy \nwith respect to the number of epochs under different settings \nfor the \\textbf{SUN} dataset. \nOURS-R results are shown in red color while the DEM baseline is shown in blue color.}\n\\label{fig:epSUN}\n\\vspace*{-0.2in}\n\\end{figure}\n\nWe also studied the effect of varying the number of test unseen samples per class \non the generalized harmonic mean accuracy. \nWe used OURS-RA variation of our model for this study. \n$\\rho=0.1$ was set for the experiments on the \\textbf{AwA2} (blue color) \nand the \\textbf{SUN} (yellow color) datasets \nand the result was reported in Fig.~\\ref{fig:nsamp}. \nWhen the fraction is 0.01 for the \\textbf{SUN} dataset, \nthe number of samples in some classes becomes zero \nand therefore the performance is not reported. \nFrom the results, it is seen that the test accuracy was stable \nwith change in the fraction of total number of samples used for testing. \nThere is a slight increase in accuracy with decreasing number of samples, \nwhich is surprising because domain adaptation would perform poorly \nwith less number of samples. \nHowever, this effect is nullified since the probability \nof including challenging examples is reduced \nand so we observed a slight improvement in performance.\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=6cm]{img\/nsamp.png}\n\\vspace*{-0.1in}\n\\caption{Generalized Harmonic Mean Accuracy results \nas the number of test samples per class is varied for the \\textbf{AwA2} (blue) \nand \\textbf{SUN} (yellow) datasets.}\n\\label{fig:nsamp}\n\\vspace*{-0.2in}\n\\end{figure}\n\nWe also studied how the test performance varies \nas the number of seen classes for training is reduced \nfor the \\textbf{AwA2} dataset using OURS-R model. \nWe set $\\rho=0.1$ and reported results over 5 trials in Fig.~\\ref{fig:nclass}. \nWe observed that the change in the seen-class accuracy \nis not much because the training and testing distributions are the same. \nThe conventional unseen-class accuracy dips by a large amount \nas the number of training classes decreases because \nthere is less representative information to be transferred to novel categories. \nHowever, we obtained a peak for the generalized unseen accuracy results \nat a fraction of 0.4 of the number of seen classes. \nThis is because as the number of training classes decreases, \nthe amount of representative information decreases, causing decrease in performance. \nOn the other hand, less number of seen classes implies \nless bias towards seen categories and improvement of unseen-class accuracies. \nAlso, there is large performance variation for unseen-class accuracy \nbecause training and testing distributions are different \nand the performance can vary depending on \nhow related are the training classes to the unseen classes in a trial.\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=6cm]{img\/nclass.png}\n\\vspace*{-0.1in}\n\\caption{Test accuracy results as the number of seen classes \nused for training is varied for the \\textbf{AwA2} dataset.}\n\\label{fig:nclass}\n\\vspace*{-0.2in}\n\\end{figure}\n\nWe also performed experiments to find whether the OURS-R variant \nreduces hubness compared to DEM. \nThe hubness of a set of predictions is measured \nusing the skewness of the 1-Nearest-Neighbor histogram ($N_1$). \nThe $N_1$ histogram is a frequency plot for $N_1[i]$ \nof the number of times a search solution $i$ (in our case a class attribute) \nis found as the Nearest Neighbor for the test samples. \nLess skewness of $N_1$ histogram implies less hubness of the predictions. \nWe used the test samples of the unseen classes in the generalized setting \nfor both DEM and OURS-R on the \\textbf{AwA2} and the \\textbf{aPY} datasets. \nWe used $\\rho=0.1$ and reported results averaging over 5 trials \nin Table~\\ref{table:hubness}. \nFrom the results, \nOURS-R method produced less skewness of the $N_1$ histogram on both the datasets. \nThis implies that using the additional structural term reduces \nhubness and therefore the curse of dimensionality is reduced.\n\n\\begin{table}[]\n\\center\n\\caption{Hubness comparison using skewness for DEM and OURS-R methods \non the \\textbf{AwA2} and \\textbf{aPY} datasets}\n\\label{table:hubness}\n\\begin{tabular}{@{}ccc@{}}\n\\toprule\nSkewness & AwA2  & aPY \\\\ \\midrule\nDEM      & 3.39    & 1.85  \\\\\nOURS-R   & 2.41    & 1.33  \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\section{Conclusion}\nThis paper proposed a three-step approach to improve \nthe performance of zero-shot learning for image classification. \nThe three-step approach involved exploiting structural information in data, \ndomain adaptation to unseen test samples and calibration of classification scores. \nWhen the proposed method was applied to \nstandard datasets of zero-shot image classification, \nit outperformed previous methods by a large margin, where\nthe most effective component was the domain adaptation step. \n\n\\iffalse\n\\section*{Acknowledgement}\nThis work was supported in part by the National\nScience Foundation under Grant IIS-1813935.\nAny opinion, findings, and conclusions or recommendations \nexpressed in this material are those of the authors \nand do not necessarily reflect the views of the National Science Foundation. \nWe also gratefully acknowledge the support of NVIDIA Corporation \nwith the donation of an TITAN XP GPU used for this research.\n\\fi\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFrom isotopic studies of meteorites it is known that the solar nebula\ncontained at least a dozen different short-lived radionuclides, or SLRs\n\\citep[see reviews by][]{MckeD03, MeyeZ06, Wadh07}.\nIdentification of the sources of these SLRs could greatly constrain the \nSun's birth environment and processes acting during star formation. \nThe half-lives of some of these isotopes are shorter than the timescales\n$\\sim 10^{6} - 10^{7} \\, {\\rm yr}$ typically associated with star formation, \nso they must have been produced near the time and place of the Sun's formation.\nThe SLRs $\\mbox{${}^{26}{\\rm Al}$}$, $\\mbox{${}^{41}{\\rm Ca}$}$ and $\\mbox{${}^{60}{\\rm Fe}$}$, in particular, cannot have been inherited from\nthe Sun's molecular cloud in abundances consistent with ongoing Galactic nucleosynthesis,\nand must have been late additions \\citep{Jaco05}. \nA leading candidate for the source of these and other SLRs is one or more core-collapse \nsupernovae in the Sun's birth environment, contaminating either its molecular cloud \n\\citep{CameT77, VanhB02, GounK09}, or\nits protoplanetary disk \\citep{Chev00, OuelD05, Ouel07, OuelDH10, LoonT06}.\nAnother leading candidate is production of SLRs by irradiation (by solar cosmic \nrays, essentially), within the solar nebula \\citep{Goun01}.\nBecause this latter mechanism by itself is inadequate to explain the abundance of \n$\\mbox{${}^{60}{\\rm Fe}$}$ in the early solar system \\citep{LeyaH03, Goun06}, it is generally\naccepted that the source of $\\mbox{${}^{60}{\\rm Fe}$}$ is core-collapse supernovae \\citep[e.g.][]{Wadh07}, \nalthough it is not clear whether the source of $\\mbox{${}^{60}{\\rm Fe}$}$ is a single, nearby \nsupernova, or many (possibly distant) supernovae \\citep[as in][]{GounK09}.\n\nThe origins of the other SLRs are also debated.\nA correlation between $\\mbox{${}^{26}{\\rm Al}$}$ and $\\mbox{${}^{41}{\\rm Ca}$}$ has been observed in meteorites, demanding\na common source for these two isotopes {\\it after} the formation of the solar nebula,\nas in the so called ``late-injection\" hypothesis of \\citet{SahiG98}.\nIt is not yet clear whether these two SLRs are correlated with $\\mbox{${}^{60}{\\rm Fe}$}$.\nIf evidence for a corelation could be found, this would strongly suggest that $\\mbox{${}^{26}{\\rm Al}$}$ \nand $\\mbox{${}^{41}{\\rm Ca}$}$ were injected by the same supernova or supernovae that injected $\\mbox{${}^{60}{\\rm Fe}$}$.\nThe lack of such evidence, though, leaves open the possibility that $\\mbox{${}^{26}{\\rm Al}$}$ and $\\mbox{${}^{41}{\\rm Ca}$}$ \nwere created by irradiation within the solar nebula while $\\mbox{${}^{60}{\\rm Fe}$}$ was injected separately\nby one or more supernovae, into the Sun's molecular cloud or protoplanetary disk. \n\nThe abundances of the SLRs alone have not yet enabled a discrimination between \nthese possibilities, but \\citet[][hereafter GM07]{Goun07} have proposed \nthat the oxygen isotopic ratios of early solar system materials may be used to rule \nout certain hypotheses.  \nSpecifically, they argue that if $\\mbox{${}^{26}{\\rm Al}$}$ and $\\mbox{${}^{41}{\\rm Ca}$}$ were injected by a nearby\nsupernova into the Sun's protoplanetary disk, sufficient to produce the observed \nmeteoritic ratio $\\alratio \\approx 5 \\times 10^{-5}$ \\citep{MacPD95}, then \nthe oxygen isotopic ratio of the solar nebula would be considerably altered:\nsolar nebula materials formed before the injection would have oxygen isotopic \nratios {\\it significantly} different from later-formed materials. \nGM07 calculated the shifts in oxygen isotopic ratios accompanying injection of \nsupernova $\\mbox{${}^{26}{\\rm Al}$}$ into the Sun's protoplanetary disk, using the isotopic yields in\nbulk supernova ejecta calculated by \\citet{Raus02}.\nA robust prediction of the GM07 models is that ${}^{17}{\\rm O} \/ {}^{16}{\\rm O}$ \nof pre-injection materials should be significantly higher, by several percent,\nthan post-injection materials.\nExamples of pre-injection materials may exist in meteorites, or especially in the \nsolar wind sample returned by the ${\\it Genesis}$ mission \\citep{Burn03}. \nSince preliminary results from {\\it Genesis} suggest the Sun is {\\it not} \nisotopically heavy in oxygen \\citep{Mcke09}, and because no such  \n${}^{17}{\\rm O}$-rich (or ${}^{16}{\\rm O}$-poor) components have been discovered \nin meteorites, GM07 rule out a supernova origin for the $\\mbox{${}^{26}{\\rm Al}$}$ and $\\mbox{${}^{41}{\\rm Ca}$}$ in meteorites.\n\nThe purpose of this paper is to reproduce and refine the method pioneered by GM07, \nand to test the conclusion that $\\mbox{${}^{26}{\\rm Al}$}$ cannot have a supernova origin.\nGM07 originally considered only bulk ejecta of spherically symmetric supernova \nexplosions.\nWe begin our analysis with this case, but make necessary refinements to the method,\nand use our current nucleosynthesis models to predict the isotopic yields.\nWe then expand on the analysis of GM07, calculating the isotopic yields  by allowing \nthe disk to intercept ejecta from different \nparts of the supernova explosion rather than a uniformly mixed total yield, and by examining \nanisotropic explosions.\nWe also simultaneously consider the injection of ${}^{41}{\\rm Ca}$ into the disk.\n\nThe paper is organized as follows. \nIn \\S 2, we outline the method used to calculate shifts in oxygen isotopic\ncomposition due to supernova injection of $\\mbox{${}^{26}{\\rm Al}$}$ and $\\mbox{${}^{41}{\\rm Ca}$}$, including updates to the method of GM07.\nIn \\S 3 we describe the results of nucleosynthesis simulations we have carried out,\nto determine the isotopic yields in supernova ejecta under various explosion scenarios.\nWe determine the inputs needed to compute the shifts in solar nebula oxygen isotopic \ncomposition.  \nThese shifts in oxygen isotope before and after injection are presented in \\S 4, and \nin \\S 5 we draw conclusions. \n\n\n\n\\section{Method}\n\n\\subsection{Calculation of Isotopic Shifts} \n\nThe method of GM07 is fairly straightforward. \nThey assume that meteoritic components that sample the solar nebula's starting composition,\n{\\it before} the acquisition of ${}^{26}{\\rm Al}$, can be identified and measured. \nLikewise, they assume samples {\\it after} the acquisition of ${}^{26}{\\rm Al}$ can be \nidentified and measured. \nAny difference in the oxygen isotopic content between samples of those two \ngroups would then constitute a shift in oxygen isotopes brought about by the injection of \nsupernova material.\nThe preditced shift in oxygen isotopes due to injection of supernova material into the \nprotoplanetary disk can then be compared to the actual difference in oxygen isotopes\nbefore and after.\nIn practice, because the vast majority of meteoritic components sample the solar nebula\nafter injection, GM07 assumed a ``final\" value for the solar nebula oxygen isotopes, and\nused the isotopic yields in supernova ejecta to predict the initial composition.\nTesting the supernova injection hypothesis thus amounts to finding meteoritic inclusions \nwith this initial oxygen isotopic composition. \nSuch inclusions should have evidence for no live ${}^{26}{\\rm Al}$ at the time of their\nformation, and should be among the oldest meteoritic inclusions. \n\nThe earliest-formed solids in the solar system are widely accepted to be the calcium-rich, \naluminum-rich inclusions (CAIs), both because they contain minerals that are the first \nsolids expected to condense in a cooling solar nebula \\citep{Gros72}, and because their\nPb-Pb ages are the oldest measured, at 4568.6 Myr \\citep{BouvW09}.\nIt is worth noting that because many of the minerals in CAIs are condensates, their\nisotopic composition should reflect that of the solar nebula gas. \nThe vast majority of CAIs have inferred initial ratios $\\alratio \\approx 5 \\times 10^{-5}$\nor appear to have been isotopically reset at a later date \\citep{MacPD95}.\nOnly in a handful of CAIs known as ``FUN\" CAIs (fractionation with unknown nuclear\neffects) has it been possible to set firm upper limits on the initial $\\alratio$ ratio\nand show these CAIs did not contain live $\\mbox{${}^{26}{\\rm Al}$}$ when they formed \\citep{FaheG87, MacPD95}.\nThus, CAIs overall reflect the composition of the solar nebula at an early time, and \nFUN CAIs possibly record the oxygen isotopic abundance before the solar nebula acquired $\\mbox{${}^{26}{\\rm Al}$}$.\n\nTo make more precise statements, it is necessary to quantify the oxygen isotopic composition\nof the nebula and various components. \nThe molar fraction of oxygen in gas and rock can vary, so the relevant quantities are the \nratios of the stable oxygen isotopes, $\\mbox{$^{17}{\\rm O}$} \/ \\mbox{$^{16}{\\rm O}$}$ and $\\mbox{$^{18}{\\rm O}$} \/ \\mbox{$^{16}{\\rm O}$}$.\nIn the field of cosmochemistry, these ratios are commonly expressed as deviations from a \nstandard, in this case Standard Mean Ocean Water (SMOW), which has \n$\\mbox{$^{17}{\\rm O}$} \/ \\mbox{$^{16}{\\rm O}$} = 3.8288 \\times 10^{-4}$ and $\\mbox{$^{18}{\\rm O}$} \/ \\mbox{$^{16}{\\rm O}$} = 2.0052 \\times 10^{-3}$\n\\citep{oneil86}. \nThe fractional deviations of the isotopic ratios from these standard values are\n$\\dxvii$ and $\\mbox{$\\delta^{18}{\\rm O}$}$, and are measured in parts per thousand, or ``permil\" ($\\promille$). \n[That is, $\\dxvii = 1000 \\times \\left( (\\mbox{$^{17}{\\rm O}$} \/ \\mbox{$^{16}{\\rm O}$}) \/ (\\mbox{$^{17}{\\rm O}$} \/ \\mbox{$^{16}{\\rm O}$})_{\\rm SMOW} - 1 \\right)$.]\nIt is also standard to report the quantity $\\Delta^{17}{\\rm O} \\approx \\dxvii - 0.52 \\, \\mbox{$\\delta^{18}{\\rm O}$}$,\nbecause this quantity is conserved during almost all chemical fractionation processes. \n[More precisely, $\\Delta\\mbox{$^{17}{\\rm O}$} \\equiv \\ln(1+\\dxvii) - 0.5247 \\ln(1+\\mbox{$\\delta^{18}{\\rm O}$})$ \\citep{Mill02}.]\n\nIt is clear that the final oxygen isotopic composition of the nebula, $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})'$, \nwill depend on its starting composition $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})_{0}$, the composition of the \nsupernova material, $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})_{\\rm SN}$, and the mass of supernova material injected\n(relative to the mass of the disk). \nIt is straightforward to show that \n\\begin{equation}\n\\delta^{17}{\\rm O}' - \\delta^{17}{\\rm O}_{0} =\n \\frac{ x }{ 1 + x } \\, \\left( \\delta^{17}{\\rm O}_{\\rm SN} - \\delta^{17}{\\rm O}_{0} \\right),\n \\label{eq:shift}\n\\end{equation}\nwhere $\\delta^{17}{\\rm O}_{\\rm SN}$ is the isotopic ratio of the supernova material\ninjected into the disk, and $x \\equiv M({}^{16}{\\rm O})_{\\rm SN} \/  M({}^{16}{\\rm O})_{\\rm disk}$\nmeasures the mass of injected oxygen relative to the oxygen present in the disk (with a similar\nformula applying to $\\mbox{$\\delta^{18}{\\rm O}$}'$).\nIn terms of the masses involved, \n\\begin{equation}\nx = \\frac{ M({}^{16}{\\rm O})_{\\rm SN}   }{ M({}^{26}{\\rm Al})_{\\rm SN}   } \\times\n    \\frac{ M({}^{26}{\\rm Al})_{\\rm SN}  }{ M({}^{26}{\\rm Al})_{\\rm disk} } \\times\n    \\frac{ M({}^{26}{\\rm Al})_{\\rm disk}}{ M({}^{27}{\\rm Al})_{\\rm disk} } \\times\n    \\frac{ M({}^{27}{\\rm Al})_{\\rm disk}}{ M({}^{16}{\\rm O})_{\\rm disk}  }. \n \\label{eq:x}\n\\end{equation}\nMost of these terms are defineable. \nFirst, $M({}^{26}{\\rm Al})_{\\rm SN} \/ M({}^{26}{\\rm Al})_{\\rm disk} \\equiv \\exp( +\\Delta t \/ \\tau)$,\nwhere $\\Delta t$ is the time delay between supernova injection and isotopic closure of the\nmeteoritic materials, and $\\tau = 1.03 \\, {\\rm Myr}$ is the mean lifetime of $\\mbox{${}^{26}{\\rm Al}$}$.\nBy definition, $M({}^{26}{\\rm Al})_{\\rm disk} \/ M({}^{27}{\\rm Al})_{\\rm disk} \\equiv$\n$(26\/27) \\times (5 \\times 10^{-5})$, because sufficient $\\mbox{${}^{26}{\\rm Al}$}$ must be injected to\nyield the meteoritic ratio. \nFinally, the isotopic abundances in the solar nebula are known (the ratio \n${}^{27}{\\rm Al} \/ {}^{16}{\\rm O}$ is taken from \\citet{Lodd03}), so \nwe derive \n\\begin{equation}\nx = \\frac{ 4.846 \\times 10^{-7} }{ [ M({}^{26}{\\rm Al}) \/ M({}^{16}{\\rm O})]_{\\rm SN} } \\, \\exp( +\\Delta t \/ \\tau).\n\\end{equation}\nNote that $x$ is independent of the mass of the disk, but it increases with $\\Delta t$, since \nlarger values of $\\Delta t$ imply that more supernova material had to be injected to yield the\nsame $\\alratio$ ratio, thereby implying larger isotopic shifts in oxygen associated with this\ninjection.\n\nBesides the time delay $\\Delta t$, the major inputs needed to infer $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})_{0}$ are\nthe isotopic composition $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})_{\\rm SN}$ and ratio of ${}^{16}{\\rm O}$ to ${}^{26}{\\rm Al}$ \nin the supernova ejecta, and the oxygen isotopic composition of the post-injection solar nebula.\nGM07 used bulk abundances of supernova ejecta calculated by \\citet{Raus02}\nfor the first set of quantities.\nThey also assumed that the oxygen isotopic ratios of the post-injection solar nebula matched the SMOW \nvalues of the present-day Earth: $(\\dxvii',\\mbox{$\\delta^{18}{\\rm O}$}') = (0\\, \\promille, 0\\, \\promille)$. \nThis assumption is the main reason why they concluded that the pre-injection solar nebula had to be \n${}^{17}{\\rm O}$-rich, as we now demonstrate. \nRearranging equation~\\ref{eq:shift} yields\n\\begin{equation}\n\\delta^{17}{\\rm O}_{0} = \\delta^{17}{\\rm O}' \n +x \\left( \\delta^{17}{\\rm O}' - \\delta^{17}{\\rm O}_{\\rm SN} \\right). \n\\end{equation} \nSupernova ejecta tend to be ${}^{16}{\\rm O}$-rich; in the extreme limit, \n$\\delta^{17}{\\rm O}_{\\rm SN} \\approx -1000\\, \\promille$.  \nIf $\\delta^{17}{\\rm O}' \\approx 0\\, \\promille$ also, then \n$\\delta^{17}{\\rm O}_{0} \\approx +(1000 x)\\, \\promille$.\nThat is, $\\delta^{17}{\\rm O}_{0}$ is inferred to have been positive and potentially\nquite large if $x > 10^{-2}$. \nThe isotopic yields of the supernova ejecta computed by \\citet{Raus02}\nwere consistent with such large values of $x$ and $\\delta^{17}{\\rm O}_{\\rm SN} < 0$, \nleading GM07 to conclude that generally $\\delta^{17}{\\rm O}_{0} > 0\\, \\promille$.\nIndeed, for progenitor masses $15 - 25 \\, M_{\\odot}$, GM07 inferred\n$\\delta^{17}{\\rm O}_{0} \\approx +35$ to $+220\\, \\promille$.\nSince there are no early-formed meteoritic components with $\\delta^{17}{\\rm O}$ this\nhigh, and because the oxygen isotopic composition of the Sun appears to be consistent\nwith $\\delta^{17}{\\rm O} \\approx -60\\, \\promille$ \\citep{Mcke09}, GM07 ruled\nout supernova injection of $\\mbox{${}^{26}{\\rm Al}$}$ and $\\mbox{${}^{41}{\\rm Ca}$}$.\nThis conclusion depends on a few key assumptions that we update below.\nWe consider the starting composition of the solar nebula, and take into account the \nnon-homogeneity of supernova ejecta. \n\n\\subsection{Solar Nebula Oxygen Isotopic Composition}\n\nOxygen isotopic ratios potentially can test or rule out the supernova injection \nhypothesis, but several caveats must be applied to the method of GM07.\nThe first and most important correction involves the oxygen isotopic composition of the \nsolar nebula immediately before and after the injection of supernova material.\nGM07 assumed the post-injection composition was equal to SMOW; \nhowever, SMOW is widely understood {\\it not} to reflect the oxygen \nisotopic ratios of the solar nebula immediately after injection.\nOn a three-isotope diagram of $\\dxvii$ versus $\\mbox{$\\delta^{18}{\\rm O}$}$, the oxygen isotopes of planetary and\nmeteoritic materials are arrayed along a mixing line called the Carbonaceous Chondrite Anhydrous\nMineral (CCAM) line discovered by \\citet{Clay73}.\nAfter correcting for isotopic fractionation by thermal and chemical processes, \\citet{YounR98}\ninferred a mixing line with slope 1.0 in the three-isotope diagram, and so we will\nrefer to this mixing line as the ``slope-1\" line.\nToday the oxygen isotopic composition of the Earth (SMOW) is widely recognized to reflect a mixture\nof an isotopically lighter rocky component (to which CAIs belong), and an isotopically heavy reservoir\n\\citep[e.g.][]{Clay03,YounR98}.\nIt is very likely that this component is isotopically heavy water, with\n$\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$} > +30\\, \\promille$\n\\citep{Clay84,LyonY05,Lyonea09}.\nThe existence of isotopically heavy water is supported by the discovery (in the primitive\ncarbonaceous chondrite Acfer 094) of a poorly characterized product of aqueous alteration, with\n$\\dxvii \\approx \\mbox{$\\delta^{18}{\\rm O}$} \\approx +180\\, \\promille$ \\citep{SakaS07}.\nQuite possibly this heavy water is the result of a mass-dependent photodissociation of CO in the outer\nsolar nebula by an external ultraviolet source \\citep{LyonY05,Lyonea09}.\nThe photodissociation can be isotopically selective because the different isotopologues of CO\nmolecules can self-shield; ${\\rm C}^{17}{\\rm O}$ and ${\\rm C}^{18}{\\rm O}$ are optically thin and \ndissociate more completely, releasing ${}^{17}{\\rm O}$ and ${}^{18}{\\rm O}$ atoms that react with \n${\\rm H}_{2}$ to form isotopically heavy water, while the abundant molecule ${\\rm C}^{16}{\\rm O}$ \nis more optically thick and does not as completely dissociate.\nThe light CO molecule is eventually lost with the nebular gas.\nWhatever the source of the isotopically heavy component, \nSMOW only represents a late stage in nebular evolution, and does not represent\nthe state of the nebula immediately after injection of supernova material. \n\nApplying the same reasoning, it is likely that the starting composition of the solar nebula\nwas lower (more ${}^{16}{\\rm O}$-enriched) on the slope-1 line than most CAIs. \nThe majority of CAIs tend to cluster near $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$}) \\approx$ \n$(-41\\, \\promille,-40\\, \\promille)$, i.e., $\\Delta^{17}{\\rm O} \\approx -20.2\\, \\promille$\n\\citep[see][and references therein]{Clay03};\nbut many of the most primitive and unaltered CAIs cluster near \n$(\\dxvii,\\mbox{$^{18}{\\rm O}$}) \\approx$ $(-50\\, \\promille,-50\\, \\promille)$,\nor $\\Delta\\mbox{$^{17}{\\rm O}$} \\approx -24\\, \\promille$ \\citep{ScotK01}.\nLikewise, \\citet{MakiN09} report $\\Delta\\mbox{$^{17}{\\rm O}$} = -23.3 \\pm 1.9\\, \\promille$ for\n``mineralogically pristine\" CAIs. \nCAIs also contain grains of hibonite, spinel, and corundum, which are among the first \nminerals expected to condense from a cooling gas of solar composition \\citep{EbelG00},\nand which are presumably even more primitive than CAIs themselves. \n\\citet{ScotK01} report that hibonite grains are also found to cluster near\n$(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$}) \\approx (-50\\, \\promille,-50\\, \\promille)$,\nor $\\Delta^{17}{\\rm O} = -24 \\,\\promille$, while \\citet{MakiN09b}\nobserved 4 hibonite grains from Allende and Semarkona to have oxygen isotopes\nin the range $\\Delta^{17}{\\rm O} = -32$ to $-17\\, \\promille$. \nThey also found that spinel grains from the CV chondrite Allende had \n$\\Delta^{17}{\\rm O} = -25 \\pm 5\\,\\promille$, and that corundum grains from the CM chondrite \nSemarkona clustered strongly in the range $\\Delta^{17}{\\rm O} = -24 \\pm 2\\, \\promille$. \n\\citet{KrotN10} likewise report $\\Delta^{17}{\\rm O} = -24 \\pm 2\\, \\promille$ for primitive CAIs \nand amoeboid olivine aggregates, which are also believed to have condensed from solar nebula gas. \nFrom these results we infer that $(\\dxvii',\\mbox{$\\delta^{18}{\\rm O}$}') \\approx (-50\\, \\promille,-50\\, \\promille)$\nin the solar nebula immediately after the injection of supernova material. \n\nMeteoritic and other samples also constrain the initial (pre-injection) oxygen isotopic \ncomposition of the solar nebula, and find it to be very similar. \nAs described above, very firm and low upper limits to initial $\\alratio$ exist for\nFUN CAIs that mark them as having formed before the injection of $\\mbox{${}^{26}{\\rm Al}$}$ and $\\mbox{${}^{41}{\\rm Ca}$}$\n\\citep{SahiG98}. \n\\citet{KrotC08} have identified a fractionation line associated with the FUN CAIs\nwith $\\Delta^{17}{\\rm O} = -24.1\\, \\promille$ that passes through\n$(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$}) \\approx (-51\\, \\promille,-52\\, \\promille)$.\nPresumably the original isotopic composition of the nebula matched that of the Sun,\nwhich might therefore be measured by {\\it Genesis} mission \\citep{Burn03}.\nPreliminary measurements can be interpreted as clustering on a fractionation line\nwith $\\Delta^{17}{\\rm O} \\approx = -26.5 \\pm 5.6\\, \\promille$ \\citep{Mcke09}, \nwhich would intersect the slope-1 line at \n$(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$}) \\approx (-56\\, \\promille, -57\\, \\promille)$, and other analyses\nsuggest $\\Delta^{17}{\\rm O} \\approx -33 \\pm 8 \\, \\promille$ [$2\\sigma$ errors]\n\\citep{McKeea10}. \n\nFrom these results it seems likely that the original solar nebula oxygen isotopic\ncomposition was near the {\\it Genesis} preliminary result of\n$(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$}) \\approx (-54\\, \\promille,-53\\, \\promille)$, or possibly much lower\nalong the slope-1 line. \nSubsequent reaction of rock with a ${}^{16}{\\rm O}$-depleted reservoir \nthen moved material along the slope-1 line to $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$}) \\approx$ \n$(-41\\, \\promille,-40\\, \\promille)$,\nwhere most CAIs are found \\citep[][and references therein]{Clay03}.\nFUN CAIs appear to represent an intermediate stage in this process, only partially \nevolved along the slope-1 line. \nTo fix values, we will simply assume the solar system protoplanetary disk isotopic\nratios started as $(\\delta\\mbox{$^{17}{\\rm O}$},\\delta\\mbox{$^{18}{\\rm O}$}) = (-60\\,\\promille,-60\\,\\promille)$.\n\nThe above discussion changes the criterion by which one can reject the supernova\ninjection hypothesis. \nBecause GM07 assumed an initial solar nebula composition near SMOW, \nthey concluded that pre-injection samples necessarily would have had $\\dxvii > 0$, \nand the lack of such samples in meteorites ruled out the hypothesis.\nBut we assert that the supernova injection hypothesis can be ruled out only if \ninjection of supernova material necessarily shift the oxygen isotopic composition of \nthe solar nebula from a composition near $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$}) \\approx (-60\\, \\promille,-60\\, \\promille)$ \nto one far off the slope-1 line, or one on the slope-1 line but with $\\dxvii > -50 \\, \\promille$.\nIn this way, the GM07 method of using oxygen isotopic constraints might still allow\na test of the supernova injection hypothesis. \n\n\\subsection{Magnitude of Isotopic Shift \\label{magofshifts}}\n\nThere are at least three scenarios wherein the shift in oxygen isotopes following injection of \nsupernova material can be consistent with the above constraints.\nFrom equation~\\ref{eq:shift}, it is seen that even if the supernova ejecta and the protoplanetary \ndisk differ in oxygen isotopic composition by hundreds of permil, the shift in oxygen isotopes \nmay be small ($< 1$ permil) if the injected mass is small, so that $x < 10^{-2}$. \nMore precisely, if $\\mbox{${}^{26}{\\rm Al}$}$ and the other SLRs are injected by a supernova into the solar nebula\ndisk, then the magnitude of the shift in oxygen isotopes will depend on the fraction of \nejecta oxygen that accompanies Al. \nIn this first scenario, O and Al may be significantly fractionated during delivery of the ejecta \nto the solar nebula.\nFor example, \\citet{Ouel07} find that effectively only material condensed from the supernova \nejecta into large ($> 1 \\, \\mu{\\rm m}$ radius)  grains can be injected directly into a protoplanetary \ndisk.\nIn the extreme event that the only grains that entered the protoplanetary disk were corundum\n(${\\rm Al}_{2}{\\rm O}_{3}$) grains, the isotopic shifts in oxygen would be negligible\n($< 0.001\\,\\promille$).\nOr, if only 10\\% of the oxygen in the ejecta condensed into grains, and 90\\% remained in gas\nthat was excluded from the disk, then the isotopic shifts in oxygen isotope (for a given amount\nof injected $\\mbox{${}^{26}{\\rm Al}$}$) would be 10 times smaller than predicted by GM07. \nIt is therefore not possible to determine the shifts in oxygen isotopes following injection into \na disk without quantifying the degree to which O and Al are fractionated between gas and solids. \nIn what follows, we assume no fractionation, as such a calculation is beyond the scope of the \npresent investigation; but we consider dust condensation in supernova ejecta to be a very important \neffect, one that potentially could significantly reduce the predicted isotopic shifts. \n\nIn the second scenario, the shifts in oxygen isotopes could also remain small if the injected \nmaterial was simply higher than expected in $\\mbox{${}^{26}{\\rm Al}$}$ (or lower in O), so that again $x < 10^{-2}$. \nThe calculations of GM07 relied on the {\\it bulk} abundances calculated by \\citet{Raus02}.\nThat is, GM07 assumed that the injected material uniformly sampled the entirety of the supernova\nejecta.\nSuch a uniform sampling is unlikely, as supernovae often do explode in \na clumpy fashion and asymmetrically. It has long been understood that asymmetries or hydrodynamic \ninstabilities may disrupt the stratification of the progenitor star, but they do not result in large scale \ncompositional mixing \\citep[e.g.][]{joggerst08, hftm05, fam91}.\nThe X-ray elemental maps of the Cassiopeia A supernova remnant \\citep{Hwan04} dramatically \ndemonstrate that massive stars are likely to explode as thousands of clumps of material, each sampling\ndifferent burning zones within the progenitor.\n\\citet{OuelDH10} have argued that this may be a near-universal feature of core-collapse\nsupernovae; at the very least, observations do not rule out this possibility. \nSo it is more than possible that the solar nebula received materials from only limited regions \nwithin the ejecta in which the $\\mbox{${}^{26}{\\rm Al}$} \/ {}^{16}{\\rm O}$ ratio could have varied considerably from\nthe average value for the ejecta. \nThe non-uniformity of the $\\mbox{${}^{26}{\\rm Al}$} \/ {}^{16}{\\rm O}$ ratio may be magnified if the star explodes\nasymmetrically, allowing explosive nucleosynthesis to proceed differently even in parcels of \ngas in the same burning zone. \n\nFinally, in the third scenario by which isotopic shifts may conform to measurements, $x$ need not\nbe small, and the isotopic shifts may approach $10\\, \\promille$ in magnitude,\nso long as the injection moved the composition {\\it up} the slope-1 line by $\\approx 10\\, \\promille$ \n(i.e., the change in $\\dxvii$ equalled the change in $\\mbox{$\\delta^{18}{\\rm O}$}$, both being $< 10\\, \\promille$), \nor {\\it down} the slope-1 line by a comparable or even larger amount.\nA shift from an initial composition $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})_{0} \\approx (-60\\, \\promille, -60\\, \\promille)$,\nconsistent with ${\\it Genesis}$ measurements of the Sun's composition, to \n$(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})' \\approx (-50\\, \\promille, -50\\, \\promille)$, consistent with primitive meteoritic\ncomponents, would not conflict with the data.\nAlternatively,  a shift from an initial composition $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})_{0} \\approx (-60\\, \\promille, -60\\, \\promille)$,\nto $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})' \\approx (-70\\, \\promille, -70\\, \\promille)$, \nor even $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})' \\approx (-80\\, \\promille, -80\\, \\promille)$, \nfollowed by mixing with the ${}^{16}{\\rm O}$-poor reservoir that moves solar nebula solids up the \nslope-1 line, would also conform to the data.  \n\nIn the next section we compute the isotopic yields in core-collapse supernovae of various progenitor\nmasses, both in spherically symmetric explosions \\citep[as considered by][]{Raus02}\nand asymmetric explosions.\nThese calculations allow us to predict the oxygen isotopic composition $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})_{\\rm SN}$ \nand the ratio $x$ of the supernova material at various locations within the explosion, to assess the \nrange of possible isotopic shifts under the second and third scenarios. A supernova injection scenario\nwould be ruled out, {\\it unless} either the injection of material results in small overall shifts (i.e. the \ninjected material contains a high \\mbox{${}^{26}{\\rm Al}$}~ abundance relative to oxygen, or vice versa, a low oxygen \nabundance relative to Al), or the oxygen isotopes are shifted along the \n'slope-1 line', in which case shifts of up to $\\sim$10 permil in either direction are allowed.\n\n\n\n\\section{Isotope Production within Supernovae}\n\n\\begin{deluxetable}{lcccc}\n\\tablewidth{0pt}\n\\tablecaption{Explosion Simulations\\label{tab1}}\n\\tablehead{\n  \\colhead{Simulation}\n& \\colhead{progenitor}\n& \\colhead{Energy}\n& \\colhead{M$_{\\rm Remn}$}\n& \\colhead{Delay} \\\\\n  \\colhead{}\n& \\colhead{}\n& \\colhead{$10^{51}$\\,erg}\n& \\colhead{(\\Msol)}\n& \\colhead{ms}\n}\n\\startdata\n23e-0.8 & 23 \\Msol\\, single & 0.8 & 5.7 & 20 \\\\\n23e-1.2 & 23 \\Msol\\, single & 1.2 & 4.1& 20 \\\\\n23e-1.5 & 23 \\Msol\\, single & 1.5 & 3.2 & 20 \\\\\n23e-0.7-0.8 & 23 \\Msol\\, single & 0.8 & 3.2 & 700 \\\\\n23e-0.7-1.5 & 23 \\Msol\\, single & 1.5 & 2.3 & 700 \\\\\n16m-run1 & 16 \\Msol\\, binary &  1.5 &  2.06 & 20 \\\\\n16m-run2 & 16 \\Msol\\, binary &  0.8 &  2.43 & 20 \\\\\n16m-run4 & 16 \\Msol\\, binary &  5.9 &  1.53 & 20 \\\\\n23m-run1 & 23 \\Msol\\, binary &  1.9 &  3.57 & 20 \\\\\n23m-run2 & 23 \\Msol\\, binary &  1.2 &  4.03 & 20 \\\\\n23m-run5 & 23 \\Msol\\, binary &  6.6 &  1.73 & 20 \\\\\n40m-run1 & 40 \\Msol\\, single &  10 &  1.75 & 20 \\\\\n40m-run5 & 40 \\Msol\\, single &  1.8 &  4.51 & 20 \\\\\n40m-run9 & 40 \\Msol\\, single &  2.4 &  6.02 & \\,20\n\\enddata\n\\label{tb:explsim}\n\\end{deluxetable}\n\n\\subsection{Numerical Methods}\n\nWe calculate the yields of oxygen isotopes, \\mbox{${}^{41}{\\rm Ca}$}, and \\mbox{${}^{26}{\\rm Al}$}\\, in several core-collapse\nsupernova scenarios, listed in Table \\ref{tb:explsim}.\nThese calculations explore four different progenitor models, with a range of explosion scenarios\nfor each.\nWe use a large set of thermally driven 1D explosions with varying kinetic energies and delays,\nfor a star of initial mass 23 $\\Msol$ and a more restricted range of explosions for a 16 $\\Msol$, \nand 23 $\\Msol$, with the hydrogen envelope stripped in a case B binary scenario, and a single 40 \n$\\Msol$ progenitor that ends its life as a type WC\/O after extensive mass loss.\nWe also examine a 3D explosion of the 23 $\\Msol$ binary progenitor.\nDetails of the simulations can be found in \\citet{yca09}.\nThe set of progenitor models we selected by no means samples the entire diversity of supernovae, but it\nrepresents a variety of cases across a large range of progenitor masses and explosion parameters.\nIt is sufficiently diverse to make generalizations for behaviors that appear across all models.\n\nProgenitor models were produced with the TYCHO stellar evolution code\n\\citep{YouA05}.\nTo model collapse and explosion, we use a 1-dimensional Lagrangian code developed by \\citet{her94}\nto follow the collapse through core bounce.\nThis code includes 3-flavor neutrino transport using a flux-limited diffusion calculation and a\ncoupled set of equations of state to model the wide range of densities in the collapse phase\n\\citep[see][for details]{her94,fryer99a}.\nIt includes a 14-element nuclear network \\citep{bth89}\nto follow the energy generation.\nTo get a range of explosion energies, we opted to remove the neutron star and drive an explosion by\ninjecting energy in the innermost 15 zones (roughly 0.035 $\\Msol$).\nThe duration and magnitude of energy injection of these artificial explosions were altered to produce the\ndifferent explosion energies.\nOur 3-dimensional simulation uses the output of the 1-dimensional explosion (23 $\\Msol$ binary star, 23m-run5)\nwhen the shock has reached 10$^9$\\,cm.\nWe then map the structure of this explosion into our 3D Smooth Particle Hydrodynamics code SNSPH\n\\citep{frw06}\nby placing shells of particles whose properties are determined by\nthe structure of our 1-dimensional explosion \\citep[see][for details]{hun05, frw06}.\nThe 3D simulation uses 1 million SPH particles and is followed for 800 seconds after collapse.\n\nThe initial intent of the 3D simulation was to create a fully 3-dimensional\ncalculation of an explosion with a moderate bipolar asymmetry \\citep{Younea06}.\nThe interesting behavior of \\mbox{${}^{26}{\\rm Al}$}\\, in the explosion then prompted us to \nconsider the composition of this material in relation to an isotopic enrichment scenario of\nthe solar system.\nBoth observational and theoretical evidence indicate that asymmetry is strong and ubiquitous in supernovae \n\\citep[e.g.][]{grb07,yf07,hun05, Lopeea09}. In the situation of supernova injection the asymmetric \nmodel's primary utility lies not in modeling a specific event, but rather sampling a wide range of \nthermodynamic histories for material capable of producing \\mbox{${}^{26}{\\rm Al}$}. Injected material is likely to sample \nonly a small region of the supernova, meaning we can treat each SPH particle as an isolated trajectory \nwhose further evolution is not dependent upon the progenitor star or or global parameters of the \nexplosion. Any explosion that produces a similar thermodynamic trajectory will end up with similar \nyields. We can thus probe a large variety of explosion condition not accessible to 1D calculations \nwithout a prohibitive investment of computational time. Therefore we consider this asymmetric\nsimulation sufficiently generic to justify its usage for this investigation.\nWe create an asymmetric explosion with a geometric aspect ratio and final kinetic energy axis ratio \ndesigned to be roughly consistent with the degree of asymmetry implied by supernova polarization \nmeasurements. To simulate an asymmetric explosion, we modify the velocities within each shell by \nincreasing those of\nparticles within 30$^\\circ$ of the z-axis by a factor of 6; we will refer to\nthese parts of the supernova as high velocity structures (HVSs). The velocities of the remaining particles were\ndecreased by a factor of 1.2, roughly conserving the explosion energy. This results in a 2:1 morphology\nbetween the semimajor to semiminor axes ratio by the end of the simulation.\nWe did not introduce any angular dependence in the thermal energy.\nAt these early times in the explosion, much of the explosion energy remains in thermal energy, so the total\nasymmetry in the explosion is not as extreme as our velocity modifications suggest.\nFor a detailed discussion on choosing asymmetry parameters for 3D explosions see\n\\citet{hun03,hftm05,fw04}.\n\n\\begin{figure}[tbh]\n\\centering      \n\\subfloat[]{\\label{fig:dens}\\includegraphics[angle=90,width=0.49\\columnwidth]{temperature.eps}}\\\\\n\n\\subfloat[]{\\label{fig:temp}\\includegraphics[angle=90,width=0.49\\columnwidth]{density.eps}}\\\\\n\n\\subfloat[]{\\label{fig:entr}\\includegraphics[angle=90,width=0.49\\columnwidth]{entropy.eps}}\n\\caption{Shown is the time evolution of temperature, density, and entropy \nfor representative examples of particles from\nthe Ring and Bubble region of the 3D calculation. Each line is labelled with the abundance of \n\\mbox{${}^{26}{\\rm Al}$}, \\mbox{$^{18}{\\rm O}$}, and \\mbox{$^{18}{\\rm F}$}~(above each graph) in mass fraction per particle at the end of \nthe simulation.}\n\\label{fig:traj}\n\\end{figure}\n\nAs noted above, although we are only considering one 3D model, the results we get from that \nmodel are representative of a range of nucleosynthetic conditions that may occur in multiple \nexplosion\/progenitor scenarios. Each parcel of gas follows its \nown density and temperature evolution, which is determined by the \\emph{local} velocity of the \nparcel of gas. It is the local conditions of the gas that matter; it is unaware of the global evolution.  \nAn example of the trajectories from the Ring and Bubble regions are shown in Figure \n\\ref{fig:traj}. The Figure shows the temperature, density, and radiation entropy evolution for \nrepresentative $\\mbox{${}^{26}{\\rm Al}$}$-rich particles in the explosion. The lines are labeled with $\\mbox{${}^{26}{\\rm Al}$}$, $^{18}$F, \nand $\\mbox{$^{18}{\\rm O}$}$ abundances at the emd of the simulation (before complete radioactive decay of $^{18}$F). \nWe see three classes of trajectories: high temperature and high entropy, high temperature and \nlow entropy, and low temperature, high entropy. Predictably, the high temperature, low entropy \ntrajectories tend to have low $^{18}$F (and therefore $\\mbox{$^{18}{\\rm O}$}$) abundances due to the \nphotodisintegration of $^{18}$F into $^{14}$N $+\\ \\alpha$. High temperature, high entropy \ntrajectories have a higher reverse rate for that reaction, preserving slightly more $^{18}$F. The low \ntemperature particles have the highest $^{18}$F abundance at the end of burning.\n\nThe asymmetric explosion samples trajectories with a large span \nof velocity evolutions reasonable for plausible asymmetries. As we demonstrate with the 1D models and in \n\\citet{yca09}, the sites of production for \\mbox{${}^{26}{\\rm Al}$}~ are similar across a wide range of stellar \nmasses, so as long as we sample the particle trajectories well in a single asymmetric explosion, the \nresults are robust to very large changes in progenitor mass and explosion asymmetry. \nThis assumption is valid, since we are \nlooking for regions in the explosion that produce plausible abundances, not a bulk yield.\n\nThe network in the explosion code terminates at $^{56}$Ni and cannot follow neutron excess, so to\naccurately calculate the yields from these models we turn to a post-process step.\nNucleosynthesis post-processing was performed with the Burn code \\citep{yf07},\nusing a 524 element network terminating at $^{99}$Tc.\nThe initial abundances in each SPH particle are the 177 nuclei in the initial stellar model.\nThe network machinery is identical to that in TYCHO \\citep[for details of the simulations see][]{yca09}.\n\n\n\\begin{figure}[ptbh]\n\\centering\n\\includegraphics[angle=0,width=0.69\\columnwidth]{figure1.eps}\n\\\\\n\\includegraphics[angle=0,width=0.69\\columnwidth]{figure2.eps}\n\\caption{Shown is a $2.0\\times10^{11}$ cm thick slice in the x-z plane of\nthe 3D simulation. \\mbox{${}^{26}{\\rm Al}$}\\, abundances are in red-tones (amount per particle in \\Msol) and number\nratio of $^{17}$O\/$^{16}$O (top panel) or $^{18}$O\/$^{16}$O (bottom panel)\nper particle in blue-tones. The sizes of the data points are arbitrarily chosen, but\nscale with their values. A color gradient was also used to visualize the different\nabundances per particle. The ligher colors\/bigger data points correspond to\nhigher abundances.\nApparent is a Ring on either side of the center along the axis of symmetry,\nand further out from them the Bubbles, where the highest \\mbox{${}^{26}{\\rm Al}$}\\, abundance is found.}\n\\label{fig:al}\n\\end{figure}\n\n\\begin{figure}[tbh]\n\\centering\n\\includegraphics[angle=0,width=0.7\\columnwidth]{figure3.eps}\n\\caption{Same as Figure \\ref{fig:al} but with $^{41}$Ca shown in green. The highest \\mbox{${}^{41}{\\rm Ca}$}\\, abundance\nis adjacent to the Rings, and only partially overlaps with the Ring- regions.}\n\\label{fig:ca}\n\\end{figure}\n\n\n\\subsection{Production of ${}^{26}{\\rm Al}$ in 1D and 3D explosions}\n\nThere are three primary sites for production of \\mbox{${}^{26}{\\rm Al}$}\\, in a massive star and its accompanying\nsupernova.\nIt can be produced by hydrogen burning at high temperatures in the shell-burning regions of\nmassive stars or evolved AGB stars. But neither of these production sites\nis important to the supernova injection scenario, as discussed in \\citet{yca09}.\nIn the 1D simulations, the two dominant production sites are two peaks in $^{26}{\\rm Al}$\nabundances that coincide with peak temperatures\nin the explosion of  $2.2 \\times 10^9 \\, {\\rm K}$ and $1.5 \\times 10^9 \\, {\\rm K}$, in material\nthat has undergone hydrostatic C burning in the progenitor.\nThe higher of the two temperatures is sufficient for explosive C and Ne burning, the lower of the\ntemperatures is near explosive C burning.\nThe production of \\mbox{${}^{26}{\\rm Al}$}\\, in both regions is due to a significant increase in the\nflux of free p, n, and $\\alpha$-particles.\nAt higher temperatures, characteristic of O burning, $^{26}{\\rm Al}$ is quickly destroyed.\n\nWithin the 3D calculations, $^{26}{\\rm Al}$ is produced in two main regions\n(see Figure \\ref{fig:al}), similar to the 1D results.\nThe first is a ring-like structure and an associated small bubble where the two\nHVSs emerge into a lower density region, and which we denote the ``Ring\".\nMaterial in the Ring has undergone explosive Ne and C burning during the explosion at\ntemperatures slightly above $2 \\times 10^9 \\, {\\rm K}$, and\ncorresponds to the explosive C and Ne region in the 1D simulations identified above.\nThe second region, further out at the terminal end of the HVSs, is denoted the ``Bubble\"\n(see Figure \\ref{fig:al}).\nWithin the Bubble, material has undergone hydrostatic C burning and then experienced\npeak shock temperatures $\\sim 1.5 \\times 10^9 \\, {\\rm K}$ during the explosion,\nand corresponds to the second of the \\mbox{${}^{26}{\\rm Al}$}\\, peaks in the 1D\nsimulations identified above (which we will refer to as sub-explosive C burning region).\nWhile the production sites of \\mbox{${}^{26}{\\rm Al}$}\\, in the simulation in 3D occur in zones of about the same\ntemperatures as in the 1D cases, the peak in \\mbox{${}^{26}{\\rm Al}$}\\, abundance in those regions are reversed\nfrom the corresponding regions in 1D (i.e. the peak that is higher in \\mbox{${}^{26}{\\rm Al}$}\\, in 1D is lower in \\mbox{${}^{26}{\\rm Al}$}\\,\nin 3D, and vice versa). An important aspect of the Bubble is that due to the rapid expansion\nof the HVSs, its density drops rapidly, quenching some of the nuclear reactions.\nThe decrease in density in the 1D simulations occurred at\na slower rate, conversely more of the \\mbox{${}^{26}{\\rm Al}$}\\, was able to be processed into other species.\nThis freezeout of nuclear reactions (suppressing subsequent destruction of \\mbox{${}^{26}{\\rm Al}$})\nin the  3D simulation is the reason for the higher production of $^{26}{\\rm Al}$\nin the Bubble, as compared to the Ring.\n\nIn 3D, the \\mbox{${}^{41}{\\rm Ca}$}\\, production occurs in only one main production site, in a region adjacent to and\npartly overlapping the Ring in the 3D simulation (see Figure \\ref{fig:ca}), and in both the \nexplosive C\/Ne and sub-explosive C burning regions in the 1D calculations. The production of \n\\mbox{${}^{41}{\\rm Ca}$}~ requires a high $^{40}$Ca abundance as the seed \nnucleus, and the main production channel is p- and n-capture onto $^{40}$Ca. \nThe \nslower drop in density and temperature in the 1D calculations tended to favor a low level production of \n\\mbox{${}^{41}{\\rm Ca}$}, which is why its abundance is slightly higher as compared to the 3D calculation. In the 3D \ncalculation, \\mbox{${}^{41}{\\rm Ca}$}~ is produced in the Ring, but the faster expansion of the material there due to the \nvelocity asymmetry shuts off the reactions faster than in the 1D models, and the final \\mbox{${}^{41}{\\rm Ca}$}~ abundance\nis lower than in 1D. In the bubble region, the lower temperature and rapid density falloff preclude \nany significant \\mbox{${}^{41}{\\rm Ca}$}~ production. \n\nWithin the zones where \\mbox{${}^{26}{\\rm Al}$}\\, is produced, the M$({}^{26}{\\rm Al})$\/M(${}^{16}{\\rm O})$\nratios can differ significantly from the bulk abundances.\nFor the 1D models, these ratios can vary by a factor of $\\sim 1$ up to a factor of\n$\\sim 100$ between the\nexplosive C\/Ne burning region and the sub-explosive C burning region, with a typical\nvariation of a factor of $\\sim 2-3$.\nFor example, in model 23e-1.5 the\nM(\\mbox{${}^{26}{\\rm Al}$})\/M(\\mbox{$^{16}{\\rm O}$}) ratio varies from $(6.7-8.4)\\times10^{-6}$, and\nfor model 16m-run2 varies from $1.6\\times10^{-7}$ to $1.7\\times10^{-5}$.\nThese are to be compared to the abundances in the bulk of the ejecta, which are\nM(\\mbox{${}^{26}{\\rm Al}$})\/M(\\mbox{$^{16}{\\rm O}$}) $\\approx 6.7 \\times 10^{-6}$ for model 23e-1.5,\n$\\approx 1.7 \\times 10^{-6}$ for model 16m-run2, and varies between\n$4.0 \\times 10^{-7}$ to $1.9 \\times 10^{-4}$\nacross all 1D explosions. In the 3D model, the ratios are\nM(\\mbox{${}^{26}{\\rm Al}$})\/M(\\mbox{$^{16}{\\rm O}$}) $\\approx 1 - 4 \\times 10^{-4}$ in the SPH particles\nin the Bubble, $\\approx 1 - 4 \\times 10^{-5}$ in the SPH particles\nin the Ring, and\n$2.88 \\times 10^{-5}$ for the bulk supernova abundances.\nThus we see that injection of material from the Bubble brings in an order of\nmagnitude less oxygen (essentially all ${}^{16}{\\rm O}$) per \\mbox{${}^{26}{\\rm Al}$}\\, atom than injection\nof material from the supernova overall or from the Ring.\n\n\n\\subsection{Production of O isotopes in ${}^{26}$Al-producing regions}\n\nThe abundances and isotopic compositions of oxygen within a localized region of the\nsupernova can vary significantly from the bulk values, as their production is sensitive to\nthe density, temperature, and composition, and to their variations with time in that region.\nIn the 1D explosions, density falls off roughly as a power law \\citep{arnett96}.\nBecause this maintains a high density in the region where\n\\mbox{${}^{26}{\\rm Al}$}\\, forms by explosive C and Ne burning,\n\\mbox{$^{18}{\\rm O}$}\\, is effectively synthesized into heavier species.\n\\mbox{$^{17}{\\rm O}$}\\, is also synthesized into heavier species but is also created by neutron\ncaptures onto \\mbox{$^{16}{\\rm O}$}.\nThe net effect is that both \\mbox{$^{17}{\\rm O}$}\\, and \\mbox{$^{18}{\\rm O}$}\\, are reduced relative to \\mbox{$^{16}{\\rm O}$}, and\nthe \\mbox{$^{18}{\\rm O}$}\/\\mbox{$^{17}{\\rm O}$}\\, ratio is reduced.\nIn the 23e-1.5 model, their mass fractions in the \\mbox{${}^{26}{\\rm Al}$}\\, rich zones never exceed\n$\\sim 10^{-5}$, and the isotopic composition in nearly all our 1D cases\napproaches $(-1000\\,\\promille,-1000\\,\\promille)$, effectively pure \\mbox{$^{16}{\\rm O}$}.\n\nThe details of oxygen isotopic abundances in the \\mbox{${}^{26}{\\rm Al}$}\\, rich zones of the 3D explosion\ndiffer.\nThe more rapid expansion of the material in the 3D calculation limits the processing of \n\\mbox{$^{17}{\\rm O}$}~ and other isotopes into heavier species, so the yield of those is higher than in the 1D calculations.\nAs one of the main burning products of explosive Ne burning, $^{16}$O is\nquite abundant in the Ring.\nHowever, some of the free p and n produced during explosive burning capture onto\n\\mbox{$^{16}{\\rm O}$}, producing $^{17}$O, so the Ring (as Figure \\ref{fig:al} shows) is quite enriched\nin \\mbox{$^{17}{\\rm O}$}\\, relative to the rest of the explosion. \nIn the Bubble, \\mbox{$^{16}{\\rm O}$}\\, is not produced explosively, and is mostly\nleft over from the progenitor.\nThe increased flux of free particles also burns some of the $^{16}$O there to $^{17}$O\nand $^{18}$O.\nThe freezeout from the expansion limits the processing of\nthese isotopes into heavier species, so the 3D explosion is richer in these isotopes than\nthe 1D simulation.\n\nPart of the reason for the large variation in ${}^{18}{\\rm O}$ isotopic yields is that\nmost of it is produced by decay of $^{18}{\\rm F}$ ($t_{1\/2} = 110$ minutes), which was\nco-produced with \\mbox{$^{17}{\\rm O}$}\\, and \\mbox{$^{18}{\\rm O}$}. \nThus it depends sensitively on how much \\mbox{$^{18}{\\rm F}$}~ is \npresent once nuclear burning shuts off, which in turn depends sensitively on the \ntrajectories taken by the gas.\nAt low temperatures the classical decay reaction $^{18}{\\rm F} \\, \\rightarrow \\, {}^{18}{\\rm O} + e^{+}$ \ncompletely dominates, but at high temperatures (above $\\sim 10^{9} \\, {\\rm K}$), \nand low proton density, another decay channel opens up for\n${}^{18}{\\rm F}$, and it\ncan decay also via \n$^{18}{\\rm F} \\, \\rightarrow \\, {}^{14}{\\rm N} + \\alpha$ \\citep{ga00}.\nThe branching ratio of these two reactions is very sensitive to temperature at around $1\\times10^9$ K,\nwith higher temperatures overwhelmingly favoring the decay to  ${}^{14}{\\rm N} + \\alpha$.\n\nThe amount of \\mbox{$^{18}{\\rm F}$}~ remaining at the end of burning is highly dependent on the time taken \nto drop below that temperature, and the density evolution, as high entropies favor the destruction over \nthe synthesis. \nBecause of the power law drop off, the density in the 1D calculations stayed higher for a longer period of time,\nas compared to the 3D calculation, thus isotopes had a longer time window in which they could be processed\nto higher species. The density of the 3D calculation dropped faster due to the \nincreased velocities of particles to create the asymmetry, so the nuclear burning shut off earlier, \nand more isotopes like \\mbox{$^{17}{\\rm O}$}, \\mbox{$^{18}{\\rm O}$}, or \\mbox{$^{18}{\\rm F}$}~ survived the nucleosynthesis of the explosion.\nThis results in a substantial variation in the $^{18}{\\rm F}$ abundance\nbetween the 1D and the 3D calculation, and that same effect (i.e. how quickly the density drops) \nis also responsible for the variation in abundance \nof particles in the Ring and the Bubble by the end of the 3D simulation.\n\nIn the 1D simulations the full decay of all \\mbox{$^{18}{\\rm F}$}\\, after the explosion was calculated in the reaction network.\nThe 3D simulation was terminated earlier in its evolution before complete decay of the \\mbox{$^{18}{\\rm F}$}. As the\n temperature at that point in the explosion was well below $10^9$ K,\nwe assumed that any \\mbox{$^{18}{\\rm F}$}\\, still present would decay into \\mbox{$^{18}{\\rm O}$}, as this is the only significant \nchannel at these lower temperatures.\n\n\n\n\\section{Solar system oxygen isotopic shifts accompanying ${}^{26}{\\rm Al}$ delivery}\n\n\\begin{figure}[tbh]\n\\centering\n\\includegraphics[angle=90,width=0.8\\columnwidth]{figure5.eps}\n\\caption{Three isotope plot showing the shifts in the oxygen isotopes we calculate following\ninjection of supernova material for different scenarios. The shifts from he 3D \ncases are plotted in green, those from the 1D bulk cases are plotted in blue, the 1D\nexplosive C\/Ne burning cases are plotted in orange, and the 1D sub-explosive C burning cases \nare plotted in cyan. Indicated by the bigger black dots are SMOW at \n($0\\, \\promille$, $0\\, \\promille$) and our assumed pre-injection composition at \n($-60\\, \\promille$, $-60\\, \\promille$). The very large shift of the 3D bulk scenario was omitted for clarity.}\n\\label{fig:3isob}\n\\end{figure}\n\n\\subsection{Spherically Symmetric Supernova Explosions}\n\nIt is now possible to calculate the shifts in oxygen isotopic abundances before and after\nthe injection of supernova material, using equation~\\ref{eq:shift}.\nAs described in \\S 2, the initial composition of the solar nebula was probably close to \n$(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})_{0} \\approx (-60\\, \\promille,-60\\, \\promille)$, and we adopt this as \nthe starting value.\nBased on the numerical simulations of \\S 3, we have calculated the ratios $x$ and the isotopic \nabundances $(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$})_{\\rm SN}$ within the ejecta overall, and within the regions where \n$\\mbox{${}^{26}{\\rm Al}$}$ is produced.\n\nWe begin with the case of the 1D explosions.\nThe regions where $\\mbox{${}^{26}{\\rm Al}$}$ is produced are those we identified as the C\/Ne explosive burning region,\nand the sub-explosive C burning region.\nFor our purposes, these regions were defined based on the $\\mbox{${}^{26}{\\rm Al}$}$ content. \nThe exact amount of $\\mbox{${}^{26}{\\rm Al}$}$ produced varied among the simulations, but the (radial) abundance \ndistribution of $\\mbox{${}^{26}{\\rm Al}$}$ in each simulation showed two distinct peaks that were at least one order \nof magnitude higher than the average $\\mbox{${}^{26}{\\rm Al}$}$ mass fraction. \nThus the $\\mbox{${}^{26}{\\rm Al}$}$-rich regions in the 1D simulations were defined to be at least one order of \nmagnitude higher in mass fraction than the average distribution.\nThe final isotopic composition of the solar nebula following injection of supernova material\nhas been calculated first assuming the material had the average (bulk) composition of the ejecta for comparison with \\citet{Goun07},\nand then that of one of these $\\mbox{${}^{26}{\\rm Al}$}$-rich regions.\nThe results are presented in Tables~\\ref{tb:1Da}- \\ref{tb:1Dc} and in Figure \\ref{fig:3isob}.\nOur results for injection of bulk ejecta from 1D explosions conforms closely to the findings \nof GM07 using the 1D models of \\citet{Raus02}.\nThe ejecta are generally very ${}^{16}{\\rm O}$-rich, with $\\dxvii$ and $\\mbox{$\\delta^{18}{\\rm O}$}$ that are \nlarge and negative.\n\n\\begin{deluxetable}{lrrrrr}\n \\tablewidth{0pt}\n \\tablecaption{Oxygen isotopic shifts following injection from a 1D supernova}\n \\tablehead{\n \\colhead{$\\,$}  & \\colhead{16m-}    & \\colhead{23m-}    & \\colhead{40m-}    & \\colhead{23e-}    & \\colhead{23e-0.7-} \\\\\n \\colhead{$\\,$}  & \\colhead{average} & \\colhead{average} & \\colhead{average} & \\colhead{average} & \\colhead{average} }\n  \\startdata\n  \\multicolumn{6}{c}{\\underline{Bulk}}\\\\\n  \\\\\n  \\mbox{$^{16}{\\rm O}$} & $1.97$ \\Msol  & $1.48$ \\Msol  & $3.29$ \\Msol  & $2.93$ \\Msol  & $5.44$ \\Msol \\\\\n  \\mbox{$^{17}{\\rm O}$}        & $2.24\\times10^{-4}$ \\Msol     & $7.67\\times10^{-6}$ \\Msol     & $1.80\\times10^{-4}$ \\Msol     & $3.6\\times10^{-4}$ \\Msol      & $2.94\\times10^{-4}$ \\Msol \\\\\n  \\mbox{$^{18}{\\rm O}$}       & $1.85\\times10^{-3}$ \\Msol     & $1.53\\times10^{-5}$ \\Msol     & $3.29\\times10^{-5}$ \\Msol     & $6.63\\times10^{-5}$ \\Msol     & $1.27\\times10^{-4}$ \\Msol \\\\\n  \\mbox{${}^{26}{\\rm Al}$}   & $3.86\\times10^{-6}$ \\Msol     & $1.91\\times10^{-4}$ \\Msol     & $1.52\\times10^{-5}$ \\Msol     & $2.17\\times10^{-5}$ \\Msol     & $2.15\\times10^{-5}$ \\Msol \\\\\n  \\mbox{${}^{41}{\\rm Ca}$}   & $4.06\\times10^{-6}$ \\Msol & $2.63\\times10^{-6}$ \\Msol & $1.21\\times10^{-5}$ \\Msol & $1.06\\times10^{-5}$ \\Msol & $1.49\\times10^{-4}$ \\Msol \\\\\n  \\\\\n  \\dxvii               & $-719\\,\\promille$ & $-987\\,\\promille$ & $-899\\,\\promille$ & $-788\\,\\promille$ & $-854\\,\\promille$ \\\\\n  \\mbox{$\\delta^{18}{\\rm O}$}              & $-581\\,\\promille$ & $-996\\,\\promille$ & $-997\\,\\promille$ & $-993\\,\\promille$ & $-990\\,\\promille$ \\\\\n  $x$                  & 0.247           & 0.00375         & 0.105           & 0.0655          & 0.123           \\\\\n  \\\\\n  Final \\dxvii         & $-191\\,\\promille$ & $-63.5\\,\\promille$ & $-139\\,\\promille$ & $-105\\,\\promille$  & $-147\\,\\promille$ \\\\\n  Final \\mbox{$\\delta^{18}{\\rm O}$}        & $-163\\,\\promille$ & $-63.5\\,\\promille$ & $-149\\,\\promille$ & $-117\\,\\promille$  & $-162\\,\\promille$ \\\\\n  Final $\\Delta\\mbox{$^{17}{\\rm O}$}$ & $-118\\,\\promille$ & $-31.1\\,\\promille$ & $-65.6\\,\\promille$& $-45.1\\,\\promille$ & $-66.2\\,\\promille$ \\\\\n  $\\Delta$t \t\t& 1.41 Myr\t\t& 0.65 Myr\t\t& 1.36 Myr\t\t& 1.31 Myr\t\t& 1.73 Myr \\\\\n\\enddata\n\\label{tb:1Da}\n\\end{deluxetable}\n\n\\begin{deluxetable}{lrrrrr}\n \\tablewidth{0pt}\n \\tablecaption{Oxygen isotopic shifts following injection from a 1D supernova}\n \\tablehead{\n \\colhead{$\\,$}  & \\colhead{16m-}    & \\colhead{23m-}    & \\colhead{40m-}    & \\colhead{23e-}    & \\colhead{23e-0.7-} \\\\\n \\colhead{$\\,$}  & \\colhead{average} & \\colhead{average} & \\colhead{average} & \\colhead{average} & \\colhead{average} }\n  \\startdata\n  \\multicolumn{6}{c}{\\underline{explosive C\/Ne burning}}\\\\\n  \\\\\n  \\mbox{$^{16}{\\rm O}$} & $5.19\\times10^{-1}$ \\Msol     & $5.14\\times10^{-1}$ \\Msol     & $4.83\\times10^{-1}$ \\Msol     & $1.35$ \\Msol  & $1.86$ \\Msol \\\\\n  \\mbox{$^{17}{\\rm O}$}        & $1.47\\times10^{-7}$ \\Msol     & $2.89\\times10^{-7}$ \\Msol     & $2.76\\times10^{-8}$ \\Msol     & $8.79\\times10^{-7}$ \\Msol     & $4.96\\times10^{-6}$ \\Msol \\\\\n  \\mbox{$^{18}{\\rm O}$}       & $4.41\\times10^{-7}$ \\Msol     & $4.02\\times10^{-6}$ \\Msol     & $1.70\\times10^{-8}$ \\Msol     & $1.64\\times10^{-8}$ \\Msol     & $3.21\\times10^{-8}$ \\Msol \\\\\n  \\mbox{${}^{26}{\\rm Al}$}   & $8.98\\times10^{-7}$ \\Msol     & $1.04\\times10^{-4}$ \\Msol     & $4.55\\times10^{-7}$ \\Msol     & $9.37\\times10^{-6}$ \\Msol     & $1.53\\times10^{-5}$ \\Msol \\\\\n  \\mbox{${}^{41}{\\rm Ca}$}   & $1.74\\times10^{-6}$ \\Msol & $2.74\\times10^{-6}$ \\Msol & $5.59\\times10^{-8}$ \\Msol & $6.70\\times10^{-6}$ \\Msol & $1.16\\times10^{-4}$ \\Msol \\\\\n  \\\\\n  \\dxvii               & $-999\\,\\promille$  & $-998\\,\\promille$ & $-999\\,\\promille$  & $-997\\,\\promille$  & $-994\\,\\promille$ \\\\\n  \\mbox{$\\delta^{18}{\\rm O}$}              & $-1000\\,\\promille$ & $-996\\,\\promille$ & $-1000\\,\\promille$ & $-1000\\,\\promille$ & $-1000\\,\\promille$ \\\\\n  $x$                  & 0.280            & 0.00241         & 0.514            & 0.0698           & 0.0587 \\\\\n  \\\\\n  Final \\dxvii         & $-266\\,\\promille$ & $-62.3\\,\\promille$ & $-379\\,\\promille$ & $-121\\,\\promille$  & $-112\\,\\promille$ \\\\\n  Final \\mbox{$\\delta^{18}{\\rm O}$}        & $-266\\,\\promille$ & $-62.2\\,\\promille$ & $-379\\,\\promille$ & $-121\\,\\promille$  & $-112\\,\\promille$ \\\\\n  Final $\\Delta\\mbox{$^{17}{\\rm O}$}$ & $-147\\,\\promille$ & $-30.6\\,\\promille$ & $-227\\,\\promille$ & $-61.3\\,\\promille$ & $-56.1\\,\\promille$ \\\\\n  $\\Delta$t \t\t& 1.51 Myr\t\t& 0.77 Myr\t\t& 1.03 Myr\t\t& 1.34 Myr\t\t& 1.75 Myr \\\\\n\\enddata\n\\label{tb:1Db}\n\\end{deluxetable}\n\n\n  \\begin{deluxetable}{lrrrrr}\n \\tablewidth{0pt}\n \\tablecaption{Oxygen isotopic shifts following injection from a 1D supernova}\n \\tablehead{\n \\colhead{$\\,$}  & \\colhead{16m-}    & \\colhead{23m-}    & \\colhead{40m-}    & \\colhead{23e-}    & \\colhead{23e-0.7-} \\\\\n \\colhead{$\\,$}  & \\colhead{average} & \\colhead{average} & \\colhead{average} & \\colhead{average} & \\colhead{average} }\n  \\startdata\n  \\multicolumn{6}{c}{\\underline{sub-explosive C burning }}\\\\\n  \\\\\n  \\mbox{$^{16}{\\rm O}$} & $1.78\\times10^{-1}$ \\Msol     & $3.85\\times10^{-1}$ \\Msol     & $2.47$ \\Msol  & $1.84$ \\Msol  & $1.91$ \\Msol \\\\\n  \\mbox{$^{17}{\\rm O}$}        & $1.81\\times10^{-4}$ \\Msol     & $1.03\\times10^{-6}$ \\Msol     & $1.88\\times10^{-7}$ \\Msol     & $4.18\\times10^{-5}$ \\Msol     & $4.86\\times10^{-5}$ \\Msol \\\\\n  \\mbox{$^{18}{\\rm O}$}       & $3.12\\times10^{-4}$ \\Msol     & $7.32\\times10^{-6}$ \\Msol     & $6.51\\times10^{-8}$ \\Msol     & $8.31\\times10^{-5}$ \\Msol     & $9.14\\times10^{-5}$ \\Msol \\\\\n  \\mbox{${}^{26}{\\rm Al}$}   & $2.95\\times10^{-6}$ \\Msol     & $6.77\\times10^{-5}$ \\Msol     & $1.54\\times10^{-5}$ \\Msol     & $1.42\\times10^{-5}$ \\Msol     & $8.75\\times10^{-6}$ \\Msol \\\\\n  \\mbox{${}^{41}{\\rm Ca}$}   & $1.65\\times10^{-7}$ \\Msol & $1.35\\times10^{-6}$ \\Msol & $8.42\\times10^{-6}$ \\Msol & $7.78\\times10^{-6}$ \\Msol & $8.44\\times10^{-6}$ \\Msol \\\\\n  \\\\\n  \\dxvii               & $1503\\,\\promille$ & $-990\\,\\promille$ & $-1000\\,\\promille$ & $-944\\,\\promille$ & $-938\\,\\promille$ \\\\\n  \\mbox{$\\delta^{18}{\\rm O}$}              & $-221\\,\\promille$ & $-995\\,\\promille$ & $-1000\\,\\promille$ & $-980\\,\\promille$ & $-979\\,\\promille$ \\\\\n  $x$                  & 0.0292          & 0.00276         & 0.0775           & 0.0628          & 0.106 \\\\\n  \\\\\n  Final \\dxvii         & $-15.7\\,\\promille$ & $-62.6\\,\\promille$ & $-128\\,\\promille$  & $-112\\,\\promille$  & $-144\\,\\promille$ \\\\\n  Final \\mbox{$\\delta^{18}{\\rm O}$}        & $-64.6\\,\\promille$ & $-62.6\\,\\promille$ & $-128\\,\\promille$  & $-114\\,\\promille$  & $-148\\,\\promille$ \\\\\n  Final $\\Delta\\mbox{$^{17}{\\rm O}$}$ & $+19.2\\,\\promille$  & $-30.7\\,\\promille$ & $-64.9\\,\\promille$ & $-55.3\\,\\promille$ & $-71.4\\,\\promille$ \\\\\n $\\Delta$t \t\t& 0.90 Myr\t\t& 0.72 Myr\t\t& 1.29 Myr\t\t& 1.29 Myr\t\t& 1.39 Myr \\\\\n\\enddata\n\\label{tb:1Dc}\n\\end{deluxetable}\n\nGM07 likewise found, using the 1D models of \\citet{Raus02}, that the ejecta are depleted \nin $\\mbox{$^{17}{\\rm O}$}$, and in most cases $\\mbox{$^{18}{\\rm O}$}$ as well. (Their $15 \\, M_{\\odot}$\ncase is enriched in $\\mbox{$^{18}{\\rm O}$}$, in contrast to our $16 \\, M_{\\odot}$ case).\nSimilarly to our $23 \\Msol$ models, both the $21 \\Msol$ and the \n$25 \\Msol$ model of \\citet{Raus02} are very depleted in \\mbox{$^{17}{\\rm O}$}~ and \\mbox{$^{18}{\\rm O}$}, and the composition of \nthose ejecta approach $-1000\\, \\promille$ for both \\dxvii~ and \\mbox{$\\delta^{18}{\\rm O}$}, although the \\citet{Raus02}\nmodels tend to be slightly richer in \\mbox{$^{17}{\\rm O}$}~ and \\mbox{$^{18}{\\rm O}$}~ than ours. The similarity between the \\citet{Raus02} \nbulk abundances and those calculated for this work are unsurprising. Post-He burning stages rapidly \ndestroy  $\\mbox{$^{17}{\\rm O}$}$ and $\\mbox{$^{18}{\\rm O}$}$, resulting in the oxygen in interior zones being nearly pure $\\mbox{$^{16}{\\rm O}$}$. \nA 20-25 \\Msol model has a very large oxygen mantle. The slightly higher fraction of heavy isotopes \nin the \\citet{Raus02} arise in a somewhat larger He shell that results from a less accurate treatment \nof mixing in their earlier stellar models. \n\nThese differences reflect the variability inherent in calculations of nucleosynthesis in \nmassive stars, especially where small shifts in stable isotopes are concerned.\nWe also find, as did GM07, that the isotopic shifts associated with injection from ejecta from \n1D supernova explosions {\\it tend} to be large (tens of permil), but not in all cases.\nIn the 23m runs, $\\mbox{${}^{26}{\\rm Al}$}$ is produced more abundantly, and the ${}^{26}{\\rm Al} \/ {}^{16}{\\rm O}$\nratios yield $x < 10^{-2}$ in these explosions.\nIn the 23m cases, the ejecta are particularly ${}^{16}{\\rm O}$-rich, but relatively less oxygen\nneeds to be injected per $\\mbox{${}^{26}{\\rm Al}$}$ because more $\\mbox{${}^{26}{\\rm Al}$}$ is produced. (It should be remembered that 23m \nis a binary case, where a significant fraction of the $\\mbox{$^{17}{\\rm O}$}$ and $\\mbox{$^{18}{\\rm O}$}$ have been removed by \nmass loss from the He shell, and production of $\\mbox{${}^{26}{\\rm Al}$}$ has been enhanced by higher peak shock \ntemperatures relative to the 23 $\\Msol$ single star models.)\nThe isotopic shifts associated with injection from 23m 1D explosions are typically\n$< 3\\,\\promille$.\n\nFor all the cases the shifts in both $\\dxvii$ and $\\mbox{$\\delta^{18}{\\rm O}$}$ are negative and similar in magnitude; that is, \nthe injection of material from this supernova moves the composition down\nthe slope-1 line.\nLater nebula evolution would produce materials that move back up this line. The most favorable case \nis the 23m supernova.\nThe magnitude and direction of the isotopic shifts associated with the 23m case\nare such that current measurements of solar nebula materials do not rule out this possibility, even \nfor bulk abundances. As we argue below, however, bulk abundances are not the best representation\n of the abundances of injected material.\n\n\nTables~\\ref{tb:1Da}, \\ref{tb:1Db}, and~\\ref{tb:1Dc} and Figure \\ref{fig:3isob} \nalso show the isotopic shifts associated with \ninjection from only $\\mbox{${}^{26}{\\rm Al}$}$-rich regions within the supernova.\nEven when considering injection of $\\mbox{${}^{26}{\\rm Al}$}$-rich regions only, the conclusions are not much changed:\nthe isotopic shifts in oxygen generally are many tens of permil, and make the solar nebula more\n${}^{16}{\\rm O}$-rich. The sub-explosive C burning region of the 16m model is the only case that does \nnot move the nebular composition along the slope 1 line. In general the sub-explosive C burning in the \nhigher mass models provide the best results, as they produce the highest ratio of $\\mbox{${}^{26}{\\rm Al}$}$ to oxygen. \n\n\nTables~\\ref{tb:1Da}, \\ref{tb:1Db}, and~\\ref{tb:1Dc} also give the yields of $\\mbox{${}^{41}{\\rm Ca}$}$ produced in\neach of the 1D explosion scenarios.\nThe post-injection ${}^{41}{\\rm Ca} \/ {}^{40}{\\rm Ca}$ ratio is generally more than sufficient \nto match the meteoritic ratio, and a time delay is implied before isotopic closure, so that \n$\\mbox{${}^{41}{\\rm Ca}$}$ can decay. \nFor the ejecta from the $23 \\, M_{\\odot}$ progenitors, the implied time delay (for $\\mbox{${}^{41}{\\rm Ca}$}$ to \ndecay to a level ${}^{41}{\\rm Ca} \/ {}^{40}{\\rm Ca} = 1.4 \\times 10^{-8}$) for the 23m cases is\n$\\sim$ 0.7 Myr for all three regions considered (i.e. bulk, explosive C\/Ne burning, and sub-explosive \nC burning).\nThe implied time delay for injection from the 23e cases is $\\sim$ 1.3 Myr for all three regions, \nand from the 23e-0.7 cases is 1.4 Myr (for the sub-explosive C burning region) -- 1.7 Myr (for the other \ntwo regions). The time delays for the \nother progenitor cases are all within those ranges. The range of these time delays are very similar to \nthe range of 1.0 -- 1.8 Myr calculated by \\citet{Goun07}.\nThe effect of this time delay is to cause $\\mbox{${}^{26}{\\rm Al}$}$ to decay, too, before isotopic closure, and \nto increase the isotopic shifts in oxygen.\nThe shifts are increased by factors of 2 (for the 23m) at the low end to 5.2 (for the 23e-0.7 cases) \nat the high end.\nIf isotopic closure is to be achieved in a few $\\times (10^{5} - 10^6) \\, {\\rm yr}$ \\citep{MacPD95, KitaH05},\nthen injection from the 23e and 23e-0.7 would seem to introduce too much \n$\\mbox{${}^{41}{\\rm Ca}$}$ to match constraints.\nInjection from the more energetic 23m progenitor cases are consistent with a small shift in oxygen isotopes downward along the slope-1 line,\nas well as the final ${}^{41}{\\rm Ca} \/ {}^{40}{\\rm Ca}$ ratio of the solar nebula. \n\n\n\\subsection{Asymmetric Supernova Explosions}\n\nIn Table~\\ref{tb:3D} and Figure \\ref{fig:3isob}, \nwe present the yields of $\\mbox{${}^{26}{\\rm Al}$}$, $\\mbox{${}^{41}{\\rm Ca}$}$, and oxygen isotopes in various \nregions of the ejecta in our simulation of the 3D explosion.\nWe calculate the isotopic shifts if the injection uniformly samples all of the ejecta (bulk),\nif it samples the Ring material, and if it samples the Bubble material.\nBy design, membership in the Bubble and Ring material is defined by high $\\mbox{${}^{26}{\\rm Al}$}$ content.\nThese $\\mbox{${}^{26}{\\rm Al}$}$-rich regions do not have well-defined edges, instead fading out monotonically\nin $\\mbox{${}^{26}{\\rm Al}$}$-abundance as one moves out into the surrounding ejecta (see Figure \\ref{fig:al}).\nIn order to not impose an arbitrary geometry on these regions we determined\nmembership by $\\mbox{${}^{26}{\\rm Al}$}$ amount per SNSPH particle. \nWe used two different lower limits or thresholds for inclusion -- $1.5 \\times 10^{-13} \\, \\Msol$ \nof $\\mbox{${}^{26}{\\rm Al}$}$ per particle for a maximum extent of the \\mbox{${}^{26}{\\rm Al}$}\\, rich region, and \n$1.5 \\times 10^{-11} \\, \\Msol$ per particle for a minimum extent of the $\\mbox{${}^{26}{\\rm Al}$}$-rich region \n(``high \\mbox{${}^{26}{\\rm Al}$}\" case).\nThe most $\\mbox{${}^{26}{\\rm Al}$}$-rich SPH particles in the Ring and Bubble had $1.5 \\times 10^{-10} \\, \\Msol$\nand $4.8 \\times 10^{-10} \\, \\Msol$ of $\\mbox{${}^{26}{\\rm Al}$}$, respectively. \nEach threshold picked out {\\it all} SPH particles in the respective regions that it identified.\n\n\\begin{deluxetable}{lrrrrr} \n \\tablewidth{0pt}\n \\tablecaption{\\mbox{${}^{26}{\\rm Al}$}\\, and O in the 3D explosion \\label{tb:3D}} \n \\tablehead{\n \\colhead{$\\,$}   & \\colhead{Bulk}      & \\colhead{Ring}       & \\colhead{Ring}      & \\colhead{Bubble}    & \\colhead{Bubble} }\n \\startdata                  \n                  &                     &                      & high Al26           &                     &  \nhigh Al26           \\\\\n \\mbox{${}^{26}{\\rm Al}$}\\,(\\Msol)      & $1.474\\times10^{-6}$ & $3.510\\times10^{-7}$  & $2.420\\times10^{-7}$ & $9.996\\times10^{-7}$ & $9.585\\times10^{-7}$  \\\\\n \\mbox{$^{16}{\\rm O}$}\\,($\\Msol$)  & 0.511               & $2.888\\times10^{-2}$  & $5.949\\times10^{-3}$ & $7.025\\times10^{-3}$ & $2.550\\times10^{-3}$  \\\\\n \\mbox{$^{17}{\\rm O}$}\\,($\\Msol$) & $2.170\\times10^{-4}$ &  $3.400\\times10^{-5}$ & $2.241\\times10^{-5}$ & $1.021\\times10^{-5}$ & $1.738\\times10^{-9}$  \\\\ \n \\mbox{$^{18}{\\rm O}$}\\,($\\Msol$) & $1.509\\times10^{-1}$ &  $3.553\\times10^{-3}$  & $1.018\\times10^{-4}$ & $7.314\\times10^{-3}$ & $5.635\\times10^{-3}$  \\\\ \n \\mbox{${}^{41}{\\rm Ca}$}\\,(\\Msol) & $2.132\\times10^{-8}$ & $2.039\\times10^{-8}$ & $4.312\\times10^{-9}$ & $1.827\\times10^{-11}$ & $1.239\\times10^{-11}$ \\\\\n                      \\\\\n $\\delta\\mbox{$^{17}{\\rm O}$}$   & $+43.9\\,\\promille$    &  $+1894\\,\\promille$    & $+8258\\,\\promille$    & $+2573\\,\\promille$    & $-998.3\\,\\promille$    \\\\\n $\\delta\\mbox{$^{18}{\\rm O}$}$  & $+129887\\,\\promille$  &  $+53546\\,\\promille$   & $+6582\\,\\promille$    & $+460545\\,\\promille$  & $+978646\\,\\promille$   \\\\ \n $x$              & 0.16808             & 0.03987              & 0.01192             & 0.003406            & 0\n.001289             \\\\\n \\\\\n Final $\\delta\\mbox{$^{17}{\\rm O}$}$ & $-45.1\\,\\promille$    &  $+14.8\\,\\promille$    & $+37.9\\,\\promille$    & $-51.1\\,\\promille$    & $-61.2\\,\\promille$     \\\\\n Final $\\delta\\mbox{$^{18}{\\rm O}$}$& $+18638\\,\\promille$   &  $+1995\\,\\promille$    & $+18.2\\,\\promille$    & $+1503\\,\\promille$    & $+1200\\,\\promille$     \\\\\n Final $\\Delta\\mbox{$^{17}{\\rm O}$}$ & $-1608\\,\\promille$    &  $-561\\,\\promille$     & $+27.8\\,\\promille$    & $-534\\,\\promille$     & $-477\\,\\promille$ \\\\\n $\\Delta$t \t\t& 0.66 Myr\t\t& 0.90 Myr\t\t& 0.70 Myr\t\t& --\t\t& -- \\\\\n\n\\enddata\n\\end{deluxetable}\n\nOverall, the ejecta of the 3D simulation are much richer in \\mbox{$^{17}{\\rm O}$}\\, and \\mbox{$^{18}{\\rm O}$}\\, than the 1D\nsimulations, but also contain  two regions (the Ring and the Bubble) in which the $\\mbox{${}^{26}{\\rm Al}$}$\nproduction is increased over the 1D calculations.\nAs we have previously discussed, the production of \\mbox{$^{18}{\\rm O}$}\\, is significantly altered\nfrom the 1D results. \nThe added yield from the decay of \\mbox{$^{18}{\\rm F}$}\\, to \\mbox{$^{18}{\\rm O}$}\\, makes the ejecta significantly richer \nin this isotope, and results in large (tens of permil) to very large\n(hundreds of permil) and positive shifts in \\mbox{$\\delta^{18}{\\rm O}$}\\, for material from both the Ring and\nBubble, and the bulk. \nThis is in stark contrast to the \\mbox{$^{18}{\\rm O}$}\\, poor ejecta produced in the 1D simulations, and emphasizes \nthe sensitive dependence on the prevailing thermodynamic conditions of \\mbox{$^{18}{\\rm O}$}\\, production. \nThe production of \\mbox{$^{17}{\\rm O}$}\\, is much less sensitive to the thermodynamic conditions.\nIn the 3D simulation we also see an increase over the 1D cases in the production of \\mbox{$^{17}{\\rm O}$}\\, in the \n\\mbox{${}^{26}{\\rm Al}$}\\, rich regions and the bulk; however the change is not as drastic as in \\mbox{$^{18}{\\rm O}$}. \nThis again differs from the 1D calculations, and the more $\\mbox{$^{17}{\\rm O}$}$-rich ejecta result in positive \nshifts in \\dxvii, on the order of $-1 \\, \\rm{to }+15\\,\\promille$ for the Bubble and bulk,\nand close to $+100\\,\\promille$ for the Ring.\n\nTable~\\ref{tb:3D} also shows the ${}^{41}{\\rm Ca} \/ {}^{40}{\\rm Ca}$ ratio following injection\nof material from the 3D supernova into the solar nebula.\nIf injection comes from the Bubble region only, the amount of $\\mbox{${}^{41}{\\rm Ca}$}$ injected is too low to \nconform to meteoritic ratios, and injection from the Bubble can be ruled out on these grounds.\nInjection of material from the Ring or bulk regions, in contrast, imply reasonable time delays \n$\\approx 0.66 - 0.90 \\, {\\rm Myr}$. \nThis implies an increase in oxygen isotopic shifts of $< 2.5$ over what is presented in Table~\\ref{tb:3D}. \nThese time delays are just below the ones \\citet{Goun07} calculate, which again is explained by \nthe faster density- drop off in the 3D calculation producing slightly less \\mbox{${}^{41}{\\rm Ca}$}~ than in 1D.\n\n\nWhen an explosion samples a variety of thermodynamic trajectories through asymmetry, including those \nthat result in freeze-out conditions due to rapid expansion, the overriding conclusion to be derived is that a very large range in \noxygen isotopic shifts is allowed.  \nIt would seem extremely unlikely that conditions in an asymmetric explosion would conspire to \nyield a small isotopic shift consistent with the meteoritic constraints, though more``normal\" trajectories that \ndo not experience this freeze-out process are still candidate production sites, as wee see in 1D. \n\n\n\\section{Discussion} \n\nAs \\citet{NichP99} strongly advocated, injection of supernova ejecta can produce measurable\n``collateral damage\" to stable isotope systems in protoplanetary disks.\nGM07 in particular point to the role of oxygen isotopes in constraining this process.\nThe point of that paper was that the injection of $\\mbox{${}^{26}{\\rm Al}$}$ (and $\\mbox{${}^{41}{\\rm Ca}$}$) from a single nearby \nsupernova necessarily would have brought in significant levels of oxygen isotopically distinct\nfrom the pre-injection solar nebula.\nThe solar nebula after injection, they argued, would differ in its oxygen isotopes\nby several tens of permil from the pre-injection values, which they robustly predicted\nwould be more ${}^{17}{\\rm O}$-rich than the solar nebula.\nThey cited the {\\it Genesis} measurements of solar wind oxygen as those most likely to\nsample the pre-injection solar nebula.\nSince preliminary results from {\\it Genesis} \\citep{Mcke09, }(McKeegan et al.\\ 2009, 2010) \nare revealing the Sun to be ${}^{16}{\\rm O}$-rich, GM07 would rule out injection\nof $\\mbox{${}^{26}{\\rm Al}$}$ and $\\mbox{${}^{41}{\\rm Ca}$}$ from a single supernova. \n\nIn this paper, we attempted to reproduce the calculations of GM07, to apply\ntheir method of using oxygen isotopes to test the supernvoa injection hypothesis.\nWe made necessary corrections to their method, mostly in regard to the presumed\noxygen isotopic composition of the (post-injection) solar nebula.\nGM07 assumed this was identical to SMOW, meaning the pre-injection \nsolar nebula had to be more $\\mbox{$^{17}{\\rm O}$}$-rich than almost any known inclusions.\nWe presented considerable evidence that the post-injection composition was in fact \nmuch more ${}^{16}{\\rm O}$-rich than that, closer to $(-60\\, \\promille,-60\\, \\promille)$. \nWe carried out stellar nucleosynthesis calculations, to calculate the isotopic yields\nof $\\mbox{${}^{26}{\\rm Al}$}$, $\\mbox{${}^{41}{\\rm Ca}$}$ and oxygen isotopes in a variety of supernova explosion scenarios, \nincluding the 1D (spherically symmetric) cases \nas well as 3D (asymmetric) explosions. \nBecause $\\mbox{${}^{26}{\\rm Al}$}$ and $\\mbox{${}^{41}{\\rm Ca}$}$ are observed to be correlated \\citep{SahiG98}, \nwe also simultaneously considered injection of $\\mbox{${}^{41}{\\rm Ca}$}$ into the solar nebula. \nWe then computed the shifts in oxygen isotopes and the final ${}^{41}{\\rm Ca} \/ {}^{40}{\\rm Ca}$\nratio in the solar nebula following injection of sufficient supernova material to produce\nthe meteoritic ratio ${}^{26}{\\rm Al} \/ {}^{27}{\\rm Al} = 5 \\times 10^{-5}$. \n\nOur 1D simulations largely confirm the results of GM07, that isotopic shifts are likely\nto be tens of permil and to make the solar nebula more ${}^{16}{\\rm O}$-rich than before\nthe injection.\nWe found that injection of material from either the bulk or the explosive C\/Ne burning and sub-explosive C burning\nregions of supernovae moved the composition of the solar nebula down the slope 1 line. Our \n$23 \\Msol$ progenitors led to isotopic shifts in oxygen\n which moved the composition of the solar nebula down the slope-1 line, with the less energetic \nexplosions producing larger shifts and time delays.\nThe 23m progenitors, which were the most energetic of the $23 \\Msol$ cases and especially effective in\nproducing $\\mbox{${}^{26}{\\rm Al}$}$, generated shifts that amounted to only $< 6\\, \\promille$, including a time delay of \n0.7 Myr for $\\mbox{${}^{41}{\\rm Ca}$}$ to decay to its meteoritic value. \nThis scenario, at least, is consistent with all the applied meteoritic constraints. If less than 100\\% of the\n oxygen penetrated the solar nebula material due to, for example, dust condensation, all but one of the \n 1D cases are consistent with the evidence from the early solar system.\n\nWe note that this conclusion differs from what GM07 infer for injection of bulk ejecta from\n21 and $25 \\, M_{\\odot}$ progenitors. \nGM07 likewise found isotopic shifts downward along the slope-1 line, but with a magnitude\nof 40 to 50 permil.\nIt is worth noting that had GM07 assumed the same starting composition for the solar nebula,\n$(-60\\, \\promille,-60\\, \\promille)$, \nthat we do, \nthen they would have found the solar nebula oxygen isotopic composition to be \n$(-82\\, \\promille,-82\\, \\promille)$ after injection of supernova material from an\n$25 \\, M_{\\odot}$ progenitor, and $(-81\\, \\promille,-74\\, \\promille)$ after injection of supernova material from an\n$21 \\, M_{\\odot}$ progenitor. Although these shifts are moderately large, they are {\\it down} the slope-1 line.\nAs we established in \\S \\ref{magofshifts}, this would not have been {\\it in}compatible with the meteoritic constraints, as some \nvery ${}^{16}{\\rm O}$-rich meteoritic samples in this range are known, including CAIs in \nIsheyevo, at $\\approx (-68\\,\\promille,-66\\,\\promille)$ \\citep{GounK09},\nand a ferromagnesian cryptocrystalline chondrule in the CH chondrite Acfer 214,\nat $\\approx (-75\\,\\promille,-75\\,\\promille)$ \\citep{KobaI03}.\nSubsequently the mass-independent fractionation process would have shifted the nebula upward\nalong the slope-1 line, erasing this isotopic shift and eventually producing the composition\n$(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$}) \\approx (-40\\,\\promille,-40\\,\\promille)$ common to most CAIs\n\\citep[e.g.][]{ItohK04}.\nWe conclude that the supernova injection hypothesis cannot be rejected based\non 1D models. \n\nOur investigation of other parameters suggest that it is even more difficult to be conclusive\nabout supernova injection.\nWe have considered a small number of progenitor masses undergoing spherically symmetric collapse;\nin a few cases we varied other parameters, such as varying the explosion energy, or allowing for loss of a\nhydrogen envelope in a binary scenario, or allowing an asymmetrical explosion.\nIn most of these cases the isotopic shifts in oxygen were large.\nAmong the cases considered here, the final $\\dxvii$ values in the solar nebula varied from\n$-379\\,\\promille$ to $+15\\,\\promille$, and the final $\\mbox{$\\delta^{18}{\\rm O}$}$ values varied from\n$-379\\,\\promille$ to $+18000\\,\\promille$.\nAs GM07 found, most of the cases where meteoritic abundances of \\mbox{${}^{26}{\\rm Al}$}\\, are injected lead to \nlarge ($>10\\,\\promille$) shifts in oxygen isotopes. \nWe also considered the yields in a 3D anisotropic explosion of a $23 \\, M_{\\odot}$\nprogenitor, in the bulk ejecta and two $\\mbox{${}^{26}{\\rm Al}$}$-rich zones analogous to those in the 1D\nexplosions.\nWe find that a wide range of outcomes is possible, with oxygen isotopic shifts as large\nas hundreds of permil, or as low as $< 3\\,\\promille$.\nThe fact that ${}^{18}{\\rm F}$ can decay to ${}^{14}{\\rm N}$ instead of ${}^{18}{\\rm O}$\nat high temperatures makes the yield of ${}^{18}{\\rm O}$ especially sensitive to the\nthermodynamic trajectory of the ejecta, which partially accounts for the spread in the\n$\\mbox{$^{18}{\\rm O}$}$ yields.\nOn the one hand, the wide range of possible outcomes makes it nearly impossible to state \nconclusively that all supernova injection scenarios can be ruled out.\nOn the other hand, the wide range of possible outcomes seems to imply a degree of fine\ntuning so that the oxygen isotopic shifts in the solar nebula were not large, especially\nfor the 3D case.\n\nWe conclude that the hypothesis, that the $\\mbox{${}^{26}{\\rm Al}$}$ in the solar nebula was due to supernova \nmaterial injected into the Sun's protoplanetary disk, can still be made compatible with\nmeteoritic constraints, under two scenarios.\nThe first is that the injected supernova material came from either the bulk ejecta, or\nfrom \na region in a supernova that experienced thermodynamic conditions like the \nsubexplosive C burning zone. The latter is more physically likely.\nWith a $0.7$ Myr time delay, the injection would have moved the solar nebula oxygen\nisotopic composition from \n$(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$}) \\approx (-60\\, \\promille,-60\\, \\promille)$ to a more $\\mbox{$^{16}{\\rm O}$}$-rich value along the slope one \nline. All but one of our explosions produce movement along the slope 1 line. We produce shifts as small as \n$\\approx (-63\\, \\promille,-63\\, \\promille)$, which would have produced an accompanying meteoritic ratio  \n${}^{41}{\\rm Ca} \/ {}^{40}{\\rm Ca} = 1.4 \\times 10^{-8}$.\nSubsequent mixing of rocky material with a ${}^{16}{\\rm O}$-depleted reservoir would have\nthen moved the composition of meteoritic inclusions up the slope-1 line to values \n$(\\dxvii,\\mbox{$\\delta^{18}{\\rm O}$}) \\approx (-50\\, \\promille,-50\\, \\promille)$, consistent with primitive\nCAIs, and further up the slope-1 line with time.\n\nThe second scenario is one in which only dust grains are injected in to the protoplanetary\ndisk, and very little of the supernova oxygen condenses into dust grains. \nIf only the most refractory grains such as corundum were injected, then potentially\n$x \\ll 10^{-5}$, and the isotopic shifts would be negligible ($\\ll 1\\, \\promille$),\nfor nearly all the cases considered here. \nIt is worth noting that Ca is equally refractory to Al and is likely to condense from\nsupernova ejecta under the same conditions that Al condenses, so the meteoritic abundance\nof $\\mbox{${}^{41}{\\rm Ca}$}$ could still be matched following injection of $\\mbox{${}^{26}{\\rm Al}$}$.\n\\citet{Ouel07} have calculated that only $1\\%$ of gas-phase ejecta are injected \ninto a disk.\nIf almost all of the Ca and Al in the ejecta are locked up in large grains \n(radii $> 0.1 \\, \\mu{\\rm m}$) that are efficiently injected \\citep{OuelDH10}, but\nless than a few percent of the oxygen is, then \npotentially all of the isotopic shifts in oxygen calculated here should be reduced by a \nfactor of about 100. \nEssentially all of the 1D cases considered here would then conform with the meteoritic \nconstraints, and even some of the 3D cases as well. \n\n\nTo summarize, we agree with GM07 that oxygen isotopes can be a powerful constraint on \nsupernova injection models.\nOur calculations of oxygen isotopic shifts following injection from the bulk ejecta\nof 1D supernovae broadly match the results of GM07.\nHad GM07 assumed the same starting composition of the solar nebula that we did, and\nconsidered a smaller time delay between injection and isotopic closure, they would\nhave found isotopic shifts for $20 - 25 \\, M_{\\odot}$ progenitors that would not\nbe inconsistent with meteoritic constraints.\nOur own calculations of the same case predict shifts that are similar, although smaller\nin magnitude, and which are also consistent with meteoritic constraints. \nThe existence of an example that is consistent with the oxygen isotopic composition and\nthe ${}^{41}{\\rm Ca} \/ {}^{40}{\\rm Ca}$ ratio of the solar nebula means that the \nsupernova injection hypothesis cannot be ruled out. \nBecause the nucleosynthesis of oxygen differs in asymmetric explosions, a much wider\nrange of oxygen isotopes is possible in 3D explosions.  \nBecause of the contingent nature of the injection it becomes difficult to make any statement \nabout the possibility that the solar nebula acquired\n$\\mbox{${}^{26}{\\rm Al}$}$ from such an asymmetric explosion.\nFinally, all oxygen isotopic shifts are reduced if only large grains are injected\ninto the protoplanetary disk, and only a small fraction of oxygen condenses into\nlarge grains.\nQuantifying the fractionation of Al and O during injection into a protoplanetary disk\nis the focus of ongoing work by this research group.\nIf only a few percent of the total oxygen is injected, then nearly all the 1D explosions \nconsidered here could be consistent with the meteoritic constraints on oxygen isotopes\nand $\\mbox{${}^{41}{\\rm Ca}$}$ abundances. \nWe therefore conclude it is premature to rule out the supernova injection hypothesis\nbased on oxygen isotopes. \n\n\n\n\n\n\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{}\n\n\\section{Introduction}\nElectron impact excitation of nitrogen molecules plays an \nimportant role in atmospheric emission of planets and satellites \nsuch as the Earth, Titan and Triton. \nFor example, excitation of the ${a}^{1} \\Pi_g$ state \nand subsequent transitions to the ground ${X}^{1} \\Sigma_g^+$ \nstate are responsible for the far ultraviolet emissions of the \nLyman-Birge-Hopfield system which is prominent in \nthe airglow of the Earth's atmosphere\\cite{SpaceScienceRev.58.1}.  \nRecently, Khakoo et al.\\cite{2005PhRvA..71f2703K} measured \ndifferential cross sections (DCSs) \nof electron impact excitation of N$_2$ molecule from the ground \n${X}^1 \\Sigma^{+}_{g}$ state to the 8 lowest excited electronic  \nstates of \n$A^{3} \\Sigma_{u}^{+}$,  $B^{3} \\Pi_{g}$, $W^{3} \\Delta_{u}$, \n${B'}^{3} \\Sigma_{u}^{-}$, ${a'}^{1} \\Sigma_{u}^{-}$, \n$a^{1} \\Pi_{g}$, $w^{1} \\Delta_{u}$ \nand $C^{3} \\Pi_{u}$ states. \nBased on their differential cross section data, \nJohnson et al.\\cite{2005JGRA..11011311J} \nderived integral cross sections (ICSs) \nfor these electron impact excitations. \nIn general, their ICSs are smaller than the \nother experimental cross sections at low impact energies below \n30 eV. These deviations may have some significance on \nstudy of atmospheric emissions, because a mean kinetic energy of \nelectron at high altitudes is about 10 eV\\cite{Wayne2000}. \nTo shed light on this situation from a theoretical point of view, \nwe perform the ab initio R-matrix calculations of \nelectron impact excitations of N$_2$ molecule in this work. \n\nMany previous experimental measurements have been focused on \nexcitation to a specific electronic state. \nFor example, Ajello and Shemansky\\cite{JGeophysResSpacePhys.90.9845} and \nMason and Newell\\cite{JPhysB.20.3913} \nmeasured ICSs for electron impact excitation to the ${a}^1 \\Pi_g$ state, \nwhereas Poparic et al.\\cite{ChemPhys.240.283}, \nZubek\\cite{JPhysB.27.573} and \nZubek and King\\cite{JPhysB.27.2613} measured \ncross sections for the ${C}^3 \\Pi_u$ state. \nIn addition to these works, Zetner and Trajmar \\cite{Zetner1987}\nreported excitation cross sections to \nthe ${A}^{3} \\Sigma_{u}^{+}$,  $B^{3} \\Pi_{g}$, \n$W^{3} \\Delta_{u}$ and $a^{1} \\Pi_{g}$ states. \nSo far, comprehensive measurements of the excitation to\nthe 8 lowest electronic states are limited to three groups of \nCartwright et al. \\cite{PhysRevA.16.1013}, \nBrunger and Teubner \\cite{PhysRevA.41.1413} and \nKhakoo et al.\\cite{2005PhRvA..71f2703K}.  \nThe measurements of Brunger and Teubner \\cite{PhysRevA.41.1413} include \nexcitation DCSs for the ${E}^{3} \\Sigma_{g}^{+}$ and \n${a''}^{1} \\Sigma_{g}^{+}$ states in addition to the 8 lowest excited states. \nThe DCSs of Brunger and Teubner\\cite{PhysRevA.41.1413} and \nKhakoo et al.\\cite{2005PhRvA..71f2703K} were later converted \nto ICSs by Campbell et al.\\cite{JPhysB.34.1185} and \nJohnson et al.\\cite{2005JGRA..11011311J}, respectively. \nDetailed reviews on electron N$_2$ collisions \ncan be found in Itikawa\\cite{JPhysChemRefData.35.31} \nand Brunger and Buckman\\cite{Br02}. \n\nSeveral groups have performed theoretical calculation of \nlow energy electron collisions with N$_2$ molecule.  \nFor example, Chung and Lin\\cite{PhysRevA.6.988} employed the Born \napproximation to calculate excitation cross sections for the 11 target states \nincluding the $A^{3} \\Sigma_{u}^{+}$,  $B^{3} \\Pi_{g}$, \n$W^{3} \\Delta_{u}$, $a^{1} \\Pi_{g}$, $w^{1} \\Delta_{u}$ \nand $C^{3} \\Pi_{u}$ states. \nLater, the same group of Holley et al.\\cite{PhysRevA.24.2946}\ncalculated excitation ICSs for the $a^{1} \\Pi_{g}$ state \nusing a two-state-close-coupling method. \nFliflet et al.\\cite{JPhysB.12.3281} and \nMu-Tao and McKoy\\cite{PhysRevA.28.697} reported \ndistorted-wave cross sections for excitation of \nthe $A^{3} \\Sigma_{u}^{+}$, $B^{3} \\Pi_{g}$, \n$W^{3} \\Delta_{u}$, $w^{1} \\Delta_{u}$, $C^{3} \\Pi_{u}$,  \n$E^{3} \\Sigma_{g}^{+}$, ${b'}^{1} \\Sigma_{u}^{+}$ and \n${c'}^{1} \\Sigma_{u}^{+}$ states. \nIn general, these approximate methods are expected to \nbe accurate at high impact energies above 30 eV.  \nHowever, more elaborate method is required for precise comparison \nwith experiment at low energies. \nGillan et al.\\cite{JPhysB.23.L407} calculated excitation ICSs for \nthe $A^{3} \\Sigma_{u}^{+}$, $B^{3} \\Pi_{g}$ and \n$W^{3} \\Delta_{u}$ states using the fixed nuclei R-matrix method. \nThey included the 4 lowest target states in their R-matrix model, \nwith target CI wave functions containing 2-13 CSFs.  \nTheir cross sections for the $A^{3} \\Sigma_{u}^{+}$ and \n$W^{3} \\Delta_{u}$ states agree well with the experimental results \nof Cartwright et al. \\cite{PhysRevA.16.1013}. \nHowever, ICSs for the $B^{3} \\Pi_{g}$ state \ndeviate considerably from the experimental cross sections. \nSubsequently, they extended their R-matrix model to include \nthe 8 lowest valence states\\cite{JPhysB.29.1531}. \nTheir target CI wave functions were much improved from \ntheir previous work by employing valence active space description, \nresulting in 68-120 CSFs per target state. \nIn their paper, the ICSs were shown for the $A^{3} \\Sigma_{u}^{+}$, \n$B^{3} \\Pi_{g}$, $W^{3} \\Delta_{u}$ and ${B'}^{3} \\Sigma_{u}^{-}$ states,   \nwhile the DCSs were presented for only the $A^{3} \\Sigma_{u}^{+}$ state. \nAgreement with the ICSs of Cartwright et al.\\cite{PhysRevA.16.1013} \nis good for these 4 excited states. \nHowever, agreement is marginal at DCS level. \n\nIn this work, we study electron impact excitation of \nN$_2$ molecule by the fixed nuclei R-matrix method \nas in our previous work on electron O$_2$ \nscatterings \\cite{PhysRevA.73.052707, 2006TASHIRO-2}. \nAlthough theoretical treatment is similar to the previous work of \nGillan et al.\\cite{JPhysB.29.1531}, more target states and \npartial waves of a scattering electron are included in the present work. \nMain purpose of this work is comparison of ICSs as well as DCSs \nfor the 8 lowest excited states \nwith the experimental results of Cartwright et al. \\cite{PhysRevA.16.1013}, \nBrunger and Teubner\\cite{PhysRevA.41.1413}, \nCampbell et al.\\cite{JPhysB.34.1185}, Khakoo et al. \\cite{2005PhRvA..71f2703K} \nand Johnson et al.\\cite{2005JGRA..11011311J}. \nThis is because previous theoretical works have covered only a part of \nthese 8 excitations. \n\nIn this paper, details of the calculation are presented in section 2, \nand we discuss the results in section 3 comparing our ICSs and DCSs with \nthe previous theoretical and available experimental data.  \nThen summary is given in section 4. \n\n\\clearpage\n\n\n\\section{Theoretical methods}\n\nThe R-matrix method itself has been described extensively in the literature \n\\cite{Bu05,Go05,Mo98} as well as \nin our previous paper\\cite{PhysRevA.73.052707}. \nThus we do not repeat general explanation of the method here. \nWe used a modified version of the polyatomic programs in the UK molecular \nR-matrix codes \\cite{Mo98}.  \nThese programs utilize the gaussian type orbitals (GTO) to \nrepresent target electronic states as well as a scattering electron. \nAlthough most of the previous R-matrix works in electron N$_2$ collisions \nhave employed Slater type orbitals (STO), we select GTO mainly because of \nsimplicity of the input and availability of basis functions. \nIn the R-matrix calculations, we have included 13 target states; \n${X}^1 \\Sigma^{+}_{g}$, $A^{3} \\Sigma_{u}^{+}$, $B^{3} \\Pi_{g}$, \n$W^{3} \\Delta_{u}$, ${B'}^{3} \\Sigma_{u}^{-}$, ${a'}^{1} \\Sigma_{u}^{-}$, \n$a^{1} \\Pi_{g}$, $w^{1} \\Delta_{u}$, $C^{3} \\Pi_{u}$, \n${E}^{3} \\Sigma_{g}^{+}$, ${a''}^{1} \\Sigma_{g}^{+}$, \n$c^{1} \\Pi_{u}$ and ${c'}^{1} \\Sigma_{u}^{+}$. \nThe potential energy curves of these target electronic states are\nshown in figure \\ref{fig1} for reference. \nThese target states were represented by valence configuration interaction \nwave functions constructed by state averaged complete active space SCF \n(SA-CASSCF) orbitals.  \nNote that some target states, ${E}^{3} \\Sigma_{g}^{+}$, \n${a''}^{1} \\Sigma_{g}^{+}$ and ${c'}^{1} \\Sigma_{u}^{+}$, \nare Rydberg states and cannot be described \nadequately in the present valence active space. \nInclusion of these states are intended to improve quality of \nthe R-matrix calculations by adding more target states in the \nmodel, as in our previous works \\cite{PhysRevA.73.052707,2006TASHIRO-2} \nas well as other R-matrix works \\cite{No92,Hi94}. \nTest calculation was performed with an extra $4 a_g$ orbital in the target \norbital set. However, the target excitation energies as well as the \nexcitation cross sections did not change much compared to the results \nwith valence orbital set described above. \nAlso, removal of $3 b_{1u}$ orbital from target active space \ndid not affect the result much in our calculation. \nIn this study, the SA-CASSCF orbitals were obtained by calculations with \nMOLPRO suites of programs \\cite{molpro}. \nThe target orbitals were constructed from the [5s3p1d] level of \nbasis set taken from Sarpal et al. \\cite{Sa96}. \nIn our fixed-bond R-matrix calculations, the target states were \nevaluated at the equilibrium bond length $R$ = 2.068 a$_0$ \nof the N$_2$ ${X}^1\\Sigma^{+}_{g}$ ground electronic state. \nAlthough we also performed calculations with $R$ = 2.100 a$_0$ as in \nthe previous R-matrix calculation of Gillan et al.\\cite{JPhysB.29.1531}, \nthe cross sections with $R$ = 2.068 a$_0$ and $R$ = 2.100 a$_0$ are \nalmost the same. Thus, we will only show the results with \nthe equilibrium bond length of N$_2$ in the next section. \nThe radius of the R-matrix sphere was chosen to be 10 a$_0$ in our  \ncalculations.\nIn order to represent the scattering electron, we included diffuse\ngaussian functions up to $l$ = 5, with 9 functions for $l$ = 0, 7 functions \nfor $l$ = 1 - 3 and 6 functions for $l$ = 4 and 5.  \nExponents of these diffuse gaussians were fitted using the GTOBAS \nprogram \\cite{Fa02} in the UK R-matrix codes.  \nIn addition to these continuum orbitals, we included 8 extra virtual \norbitals, one for each symmetry. \n\nWe constructed the 15-electron configurations from the orbitals\nlisted in table \\ref{tab0}. \nThe CI target wave functions are composed of the valence orbitals in \ntable \\ref{tab0} with the 1$a_g$ and 1$b_{1u}$ orbitals kept doubly \noccupied. \nWe have included 3 types of configurations in the calculation. \nThe first type of configurations has the form,  \n\\begin{equation}\n1a_g^2 1b_{1u}^2 \\{ 2a_g 3 a_g 1 b_{2u} 1 b_{3u} 2 b_{1u} 3 b_{1u} \n1 b_{3g} 1 b_{2g} \\}^{10}  \n\\left( {}^{1} A_{g} \\right) \\{5a_{g}...39a_{g} \\}^{1} \n\\left( {}^{2}A_g \\right), \n\\end{equation}\nhere we assume that the total symmetry of this 15 electrons system is\n${}^2A_g$. \nThe first 4 electrons are always kept in the 1$a_g$ and 1$b_{1u}$\norbitals, then the next 10 electrons are distributed over the valence\norbitals with restriction of target state symmetry, ${}^{1} A_{g}$\nsymmetry of the N$_2$ ground state in this case. \nThe last electron, the scattering electron, occupies one of the\ndiffuse orbitals, of $a_{g}$ symmetry in this example. \nTo complete the wave function with the total symmetry ${}^2A_g$, \nwe also have to include configurations with the other target states \ncombined with diffuse orbitals having appropriate symmetry in the same\nway as in the example. \nThe second type of configurations has the form,  \n\\begin{equation} \n1a_g^2 1b_{1u}^2 \\{ 2a_g 3 a_g 1 b_{2u} 1 b_{3u} 2 b_{1u} 3 b_{1u} \n1 b_{3g} 1 b_{2g} \\}^{10}  \n\\left( {}^{1} A_{g} \\right) \\{ 4a_{g} \\}^{1} \\left( {}^{2}A_g\n\\right),\n\\end{equation}\nwhere the scattering electron occupies a bound $4a_{g}$ extra virtual \norbital, instead of the diffuse continuum orbitals in the \nexpression (1). \nAs in table \\ref{tab0}, we included one extra virtual orbital for each\nsymmetry. \nThe third type of configurations has the form, \n\\begin{equation}\n1a_g^2 1b_{1u}^2 \\{ 2a_g 3 a_g 1 b_{2u} 1 b_{3u} 2 b_{1u} 3 b_{1u} 1\nb_{3g} 1 b_{2g} \\}^{11} \n\\left( {}^{2}A_g \\right).\n\\end{equation} \nIn this case, the last 11 electrons including the scattering electron \nare distributed over the valence orbitals with the restriction of \n${}^2A_g$ symmetry. \nNote that the third type of configurations are crucial in\ndescription of N$_2^-$ resonance states, which often have dominant\ncontributions to the excitation cross sections. \nIn this way, the number of configurations generated for a specific total \nsymmetry is typically about 60000, though the final dimension of the inner \nregion Hamiltonian \nis reduced to be about 600 by using CI target contraction and \nprototype CI expansion method \\cite{Te95}. \n\nThe R-matrix calculations were performed for all 8 irreducible \nrepresentations of the D$_{2h}$ symmetry, \n$A_g$, $B_{2u}$, $B_{3u}$, $B_{1g}$, $B_{1u}$, $B_{3g}$, $B_{2g}$ \nand $A_u$, in doublet spin multiplicity of the\nelectron plus target system. \nDCSs were evaluated in the same way as \nin our previous paper\\cite{2006TASHIRO-2}. \n\n\\clearpage\n\n\\section{Results and discussion}\n\n\\subsection{Excitation energies}\n\nFigure \\ref{fig1} shows the potential energy curves of all N$_2$ target \nstates included in the present R-matrix model. \nThese curves were obtained by the same SA-CASSCF method employed \nin our R-matrix calculation. \nTable \\ref{tab1} compares the excitation energies of the N$_2$ target states \nfrom the present calculation with the previous R-matrix results of \nGillan et al.\\cite{JPhysB.29.1531}, multi-reference coupled cluster results of \nBen-Shlomo and Kaldor \\cite{JChemPhys.92.3680} as well as experimental values. \nSince these energies are evaluated at different inter-nuclear distance, \n2.068 $a_0$ in our case, \n2.100 $a_0$ in Gillan et al.\\cite{JPhysB.29.1531} and \n2.074 $a_0$ in Ben-Shlomo and Kaldor \\cite{JChemPhys.92.3680}, \nprecise comparison is not so meaningful. \nHowever, deviations of excitation energies from the experimental \nvalues are less than 0.8 eV in our calculation, which \nis good considering the level of calculation. \nIn terms of excitation energies, our calculation and \nthe previous R-matrix calculation of Gillan et al.\\cite{JPhysB.29.1531} \nhave similar quality. \n\nIn addition to this good agreement of target energies with experimental\nresults, N$_2^+$ energies are also well described in our SA-CASSCF\ncalculation. In our calculation, N$_2^+$ $X {}^2 \\Sigma_g^+$ and \n$A {}^2 \\Pi_u$ states are located \nat 15.63 and 17.21 eV above N$_2$ $X {}^1 \\Sigma_g^+$ state, respectively. \nCompared to the experimental values of 15.61 and 17.08 eV, \nour SA-CASSCF calculation gives good results. \nNote that the energy ordering of N$_2^+$ $X {}^2 \\Sigma_g^+$ and \n$A {}^2 \\Pi_u$ states are not well described in \nthe Hartree Fock level calculation, see Ermler and McLean \\cite{JChemPhys.73.2297} for\nexample.   \n\n\n\\subsection{Integral cross sections}\n\nFigure \\ref{fig2} shows integral cross sections for electron impact \nexcitation from the N$_2$ $X^{1} \\Sigma_{g}^{+}$ state to the \n$A^{3} \\Sigma_{u}^{+}$,  $B^{3} \\Pi_{g}$, $W^{3} \\Delta_{u}$ \nand ${B'}^{3} \\Sigma_{u}^{-}$ states. \nIn this figure, present results are compared with \nthe previous R-matrix calculations of Gillan et al.\\cite{JPhysB.29.1531}, \nrecent calculations of da Costa and Lima \\cite{2006IJQC..106.2664D}, \nexperimental results of Cartwright et al. \\cite{PhysRevA.16.1013}, \nCampbell et al.\\cite{JPhysB.34.1185} and recent measurements of \nJohnson et al.\\cite{2005JGRA..11011311J}.  \nRenormalized values of Cartwright et al. \\cite{PhysRevA.16.1013} are \nused as recommended by Trajmar et al. \\cite{PhysRep.97.221}. \nFigure \\ref{fig3} compares the present excitation cross sections \nof the ${a'}^{1} \\Sigma_{u}^{-}$, $a^{1} \\Pi_{g}$, $w^{1} \\Delta_{u}$ \nand $C^{3} \\Pi_{u}$ states with the previous experimental results of \nCartwright et al. \\cite{PhysRevA.16.1013}, \nCampbell et al.\\cite{JPhysB.34.1185} and \nJohnson et al.\\cite{2005JGRA..11011311J}.\nFor the $a^{1} \\Pi_{g}$ state cross sections, \nthe recent calculations of \nda Costa and Lima \\cite{2006IJQC..106.2664D},\nother experimental values \nof Ajello and Shemansky \\cite{JGeophysResSpacePhys.90.9845}, \nZetner and Trajmar \\cite{Zetner1987} and \nMason and Newell \\cite{JPhysB.20.3913} are included. \nFor the $C^{3} \\Pi_{u}$ state cross sections, \nthe experimental results of Zubek \\cite{JPhysB.27.573}, \nZubek and King \\cite{JPhysB.27.2613} and \nPoparic et al. \\cite{ChemPhys.240.283} are included. \n\nOur excitation cross sections for the $A^{3} \\Sigma_{u}^{+}$ state  \nhave a resonance feature at approximately 12 eV \nas in the previous R-matrix results of Gillan et al.\\cite{JPhysB.29.1531}. \nThe N$_2^-$ ${}^2 \\Pi_u$ resonance state is responsible for \nthis peak structure. \nThe main configuration of this resonance state\nis $1\\pi_u^3 1\\pi_g^2$. \nOther than the ${}^2 \\Pi_u$ symmetry partial cross sections, \nThe ${}^2 \\Pi_g$ symmetry contributes to the ICSs as \na smooth background component (not shown in the figure).  \nCompared to the previous R-matrix cross sections, \nthe peak at 12 eV is more pronounced in our case. \nOur results are slightly larger than theirs at 12-17.5 eV.  \nCompared to the recent experimental results of \nJohnson et al.\\cite{2005JGRA..11011311J}, \nour cross sections are about 50\\% larger at 12.5-20 eV, though \n50\\% smaller at 10 eV. Also our calculation overestimates \nthe results of Campbell et al.\\cite{JPhysB.34.1185}, \nhowever, the results of Cartwright et al. \\cite{PhysRevA.16.1013} \nagree well with our results except at 12.5 eV. \nThe position of the resonance peak depends rather strongly on the \ninter-nuclear distance of N$_2$ molecule, which is 12.2 eV \nfor 2.068 $a_0$ and 11.75 eV for 2.100 $a_0$ in our calculations. \nThus, inclusion of vibrational motion may be necessary to resolve \nthis discrepancy of the resonance peak. \n\nOur excitation cross sections for the $B^{3} \\Pi_{g}$ state \nhave a small bump at 12.8 eV, which is not evident in \nthe previous R-matrix cross sections. \nThe origin of this bump is the N$_2^-$ ${1}^2 \\Delta_g$ state, with main\nconfiguration of $3\\sigma_g^1 1\\pi_g^2$.\nOther than this bump, the ICSs are mostly composed of the ${}^2 \\Pi_g$ \nsymmetry contribution and have a shape similar to \nthe previous R-matrix results. \nThe magnitude of our ICSs is about 50\\% larger than the previous \nresults of Gillan et al.\\cite{JPhysB.29.1531}. \nRecently, da Costa and Lima \\cite{2006IJQC..106.2664D} calculated \nICSs for the $B {}^{3} \\Pi_{g}$ state using the Schwinger multichannel\nmethod with the minimal orbital basis for the single configuration \ninteractions (MOB-SCI) approach. \nThere cross sections are much larger than our results above 12 eV. \nAlso, there is a prominent peak around 10 eV in their ICSs, which \ndoes not exist in the R-matrix calculations. \nCompared to the experimental ICSs, our results agree well with the cross\nsections of Cartwright et al. \\cite{PhysRevA.16.1013}, especially above 15 eV.  \nHowever, the results of Campbell et al.\\cite{JPhysB.34.1185} are \nmuch larger than ours. \nRecent measurements of Johnson et al.\\cite{2005JGRA..11011311J} agree \nbetter with the previous R-matrix calculation of \nGillan et al.\\cite{JPhysB.29.1531}.   \n\nFor the excitation cross sections for the $W^{3} \\Delta_{u}$ state, \nour results have a shape and magnitude similar to \nthe previous R-matrix results. \nMost of our ICSs are composed of the ${}^2 \\Pi_g$ symmetry \npartial cross sections. \nAgreement with the experimental cross sections of \nJohnson et al.\\cite{2005JGRA..11011311J} is good in this case. \nThe cross sections of Campbell et al.\\cite{JPhysB.34.1185} \nagree well with our results at 15 and 17.5 eV, \nbut their value is about half as much as our result at 20 eV. \nThe results of Cartwright et al. \\cite{PhysRevA.16.1013} are \nabout two times larger than our cross sections.  \n\nOur excitation cross sections for the ${B'}^{3} \\Sigma_{u}^{-}$ state \nare about half of the previous R-matrix cross sections of \nGillan et al.\\cite{JPhysB.29.1531}. \nApart from this difference in magnitude, the shape of the \ncross sections is similar. \nDominant component in these ICSs is the ${}^2 \\Pi_g$ symmetry partial \ncross sections, \nalthough the ${}^2 \\Pi_u$ symmetry also has certain contribution \naround 18-20 eV. \nAmong 3 different experimental measurements, our results agree \nwell with the results of Johnson et al.\\cite{2005JGRA..11011311J}. \nThe experimental cross sections of other two groups \nare much larger than our results at 15 and 17.5 eV, and have \na different energy dependence compared to the present calculation. \n\nThe situation of the excitation cross sections for \nthe ${a'}^{1} \\Sigma_{u}^{-}$ state is similar to the case \nof the ${B'}^{3} \\Sigma_{u}^{-}$ state. \nThe ${}^2 \\Pi_g$ and ${}^2 \\Pi_u$ symmetry partial cross sections \ncontribute almost equally to the ICSs. \nOur cross sections roughly agree with the results of \nJohnson et al.\\cite{2005JGRA..11011311J}, while the cross sections of \nCartwright et al. \\cite{PhysRevA.16.1013} and \nCampbell et al.\\cite{JPhysB.34.1185} at 15 eV are much larger than our result. \nThe results of Cartwright et al. \\cite{PhysRevA.16.1013} and \nCampbell et al.\\cite{JPhysB.34.1185} \ndecrease as impact energy increases from 15 to 20 eV, however,  \nour cross sections increase mildly in this energy region. \n\nIn case of excitation to the $a^{1} \\Pi_{g}$ state, \nseveral other experimental results are available in addition \nto the measurements of Cartwright et al. \\cite{PhysRevA.16.1013}, \nCampbell et al.\\cite{JPhysB.34.1185}, \nand Johnson et al.\\cite{2005JGRA..11011311J}.  \nThe cross section profiles of Johnson et al.\\cite{2005JGRA..11011311J}, \nAjello and Shemansky \\cite{JGeophysResSpacePhys.90.9845}, \nCartwright et al. \\cite{PhysRevA.16.1013} and \nMason and Newell \\cite{JPhysB.20.3913} are similar to our ICSs. \nHowever, the magnitude of our cross sections is \nlower than the experimental values in most case except \nthe cross sections of Johnson et al.\\cite{2005JGRA..11011311J}.  \nAt 15, 17.5 and 20 eV, agreement of our results with \nthe cross sections of Johnson et al.\\cite{2005JGRA..11011311J} \nis very good, although our cross section at 12.5 eV is \ntwice as large as their value. \nNote that there is no dominant symmetry contribution to the calculated \nICSs. All partial cross sections contribute rather equally to the ICSs. \nRecent ICSs of da Costa and Lima \\cite{2006IJQC..106.2664D} by \nthe Schwinger multichannel method are also shown in the panel (b) of the\nfigure \\ref{fig3}. Their result has a sharp peak at 12 eV as in their \ncalculation for the $B {}^{3} \\Pi_{g}$ state excitation. \nThis difference between our and their results may come from \ndifferent number of target states considered in the scattering calculation. \nOnly the $X {}^{1} \\Sigma_{g}^1$, $a {}^{1} \\Pi_{g}$ and $B {}^{3}\n\\Pi_{g}$ states were included in the calculations of da Costa and Lima.    \nThe other part of the cross section profile is similar to the\nshape of our cross sections, although the magnitude of their cross\nsections are about twice as large as our results at 15-20 eV. \n\n\nOur excitation cross section for the $w^{1} \\Delta_{u}$ state \ngradually increases as a function of energy from the threshold to \nthe broad peak around 17.5 eV, then decreases toward 20 eV. \nIn this case, agreement with the results of \nJohnson et al.\\cite{2005JGRA..11011311J} is not so good compared \nto the excitations of the $a^{1} \\Pi_{g}$ and \n${a'}^{1} \\Sigma_{u}^{-}$ states. \nOur cross sections are about 50\\% larger than their values at 17.5 and 20 eV. \nAt 15 eV, our results agree well with the cross section of \nJohnson et al.\\cite{2005JGRA..11011311J}, however, they are \nabout 50\\% lower than the results of \nCartwright et al. \\cite{PhysRevA.16.1013} and \nCampbell et al.\\cite{JPhysB.34.1185}.  \nIn the calculated ICSs, the ${}^2 \\Pi_u$ symmetry partial cross section is \na major component, with a minor contribution from the ${}^2 \\Pi_g$ symmetry. \n\n\n\nThe calculated excitation cross sections for \nthe $C^{3} \\Pi_{u}$ state has a peak  \nsimilar to the experimental results of Zubek \\cite{JPhysB.27.573} and \nPoparic et al.\\cite{ChemPhys.240.283}.   \nAlthough the shape of the cross sections is similar, \nposition of the cross section peak is different from \nexperimental results.  \nIn our case, it is located at about \n17 eV, whereas corresponding peaks are located at \n14 eV in the experimental cross sections. \nThe height of the peak in our ICSs is lower than the \nexperimental values of Zubek \\cite{JPhysB.27.573} \nand Poparic et al.\\cite{ChemPhys.240.283}.  \nIt is unclear whether there is a cross section peak in  \nthe experimental cross sections of Cartwright et al. \\cite{PhysRevA.16.1013}, \nCampbell et al.\\cite{JPhysB.34.1185} \nand Johnson et al.\\cite{2005JGRA..11011311J}.  \nAt least, it appears that they do not have a peak around 17 eV. \nThe origin of this discrepancy in the cross section peak is uncertain, \nbut may be related to the employment of the fixed-nuclei approximation or \ninsufficiency of higher excited target states in the R-matrix model. \nThe calculated ICSs are composed of the ${}^2 \\Sigma_u^+$ and \n${}^2 \\Sigma_u^-$ symmetry partial cross sections near the peak \nstructure at 17 eV. \nThe contribution of the ${}^2 \\Sigma_u^+$ symmetry is about 50\\% larger \nthan the ${}^2 \\Sigma_u^-$ component. \nOther than these two symmetries, the ${}^2 \\Pi_g$ symmetry partial cross \nsection contributes to the ICSs as a smooth background component. \n\n\n\n\n\n\\subsection{Differential cross sections}\n\nFigure \\ref{fig4} shows calculated DCSs for excitation of \nthe ${A}^{3} \\Sigma_{u}^{+}$ state with the experimental \nresults of Khakoo et al. \\cite{2005PhRvA..71f2703K}, \nBrunger and Teubner \\cite{PhysRevA.41.1413}, \nCartwright et al. \\cite{PhysRevA.16.1013}, \nZetner and Trajmar \\cite{Zetner1987}, \nLeClair and Trajmar \\cite{JPhysB.29.5543} and \nthe previous R-matrix DCSs of Gillan et al.\\cite{JPhysB.29.1531}.   \nOur DCSs at 12.5, 15 and 17.5 eV have similar shape in common.  \nThey are enhanced in backward direction and have a small dimple \nat 120 degrees with a bump at 75 degrees. \nAt 17.5 eV, our cross sections are located between the \nexperimental values of Khakoo et al.\\cite{2005PhRvA..71f2703K} \nand Cartwright et al. \\cite{PhysRevA.16.1013}.   \nThe profile of the experimental DCSs are reproduced well in \nour calculation. \nAt 15 eV, our results agree better with the results of \nKhakoo et al. \\cite{2005PhRvA..71f2703K} compared to the other experiments. \nIn the DCSs of the previous R-matrix calculation of \nGillan et al.\\cite{JPhysB.29.1531},  \na bump is located at 40 degrees and a small dimple \nis located at 100 degrees, which agree better with the \nexperimental results of Brunger and Teubner\\cite{PhysRevA.41.1413}. \nIn our calculation, these dimple and bump are shifted toward \nbackward direction by 20 degrees, and agreement with \nthe results of Brunger and Teubner \\cite{PhysRevA.41.1413} is not so good. \nAt 12.5 eV, our calculation overestimates the experimental results by  \na factor of two. As seen in panel (a) of Fig.\\ref{fig2}, \nthis discrepancy is related to the existence of a resonance \npeak around 12.5 eV. \n\nFigure \\ref{fig5} compares calculated excitation DCSs for the \n$B^{3} \\Pi_{g}$ state with the experimental and recent theoretical results. \nOur DCSs at 12.5, 15 and 17.5 eV have backward-enhanced feature \nwith a broad peak at 130 degrees. \nAt 15 and 17.5 eV, our DCSs agree well with the results of \nKhakoo et al. \\cite{2005PhRvA..71f2703K} \nat forward direction below 80 degrees. \nHowever, their DCSs are smaller than ours by a factor of \ntwo at 80-130 degrees. Agreement with the results of \nCartwright et al. \\cite{PhysRevA.16.1013} \nat 15 eV is good at 20-130 degrees, although their DCSs are twice as \nlarge as our DCSs at 17.5 eV for low scattering angles. \nBecause of a resonance-like feature at 12.5 eV as seen in panel (b) \nof Fig.\\ref{fig2}, our results are larger than the experimental results \nat 12.5 eV. \nRecent Schwinger multi-channel results of da Costa and Lima \n\\cite{2006IJQC..106.2664D} are much larger than our DCSs at 12.5 \nand 15 eV. The deviation is especially large at 12.5 eV, which is\npossibly related to the difference in the excitation energies of \nthe target state. \n\n\nFigure \\ref{fig6} shows the excitation DCSs for \nthe $W^{3} \\Delta_{u}$ state with the experimental cross sections. \nAt 15 and 17.5 eV, our cross section gradually increases \nas a function of scattering angle, without noticeable bump or dip. \nAt 12.5 eV, the shape of DCSs is nearly symmetric around 90 degrees. \nAgreement with the experimental DCSs of \nKhakoo et al. \\cite{2005PhRvA..71f2703K} is good, \nalthough their results at 15 and 17.5 eV have more complex structure \nsuch as a small peak at 80 degrees. \nOur DCSs are generally smaller than the other experimental results \nof Brunger and Teubner\\cite{PhysRevA.41.1413}, \nCartwright et al. \\cite{PhysRevA.16.1013}, \nZetner and Trajmar \\cite{Zetner1987}. \n\nExcitation cross sections for the ${B'}^{3} \\Sigma_{u}^{-}$ state \nare shown in figure \\ref{fig7}. \nCalculated DCSs decrease to be zero toward 0 and 180 degrees, \nbecause of a selection rule associated with \n$\\Sigma^{+}$-$\\Sigma^{-}$ transition \\cite{Go71,Ca71}. \nOur DCSs have a broad single peak near 90 degrees at 12.5 and 15 eV, \nwhereas there are two broad peaks at 17.5 eV. \nThe position of the right peak at 17.5 eV coincides with that of \nthe experimental DCSs of Khakoo et al. \\cite{2005PhRvA..71f2703K} and \nCartwright et al. \\cite{PhysRevA.16.1013}, \nalthough the peak of Cartwright et al. \\cite{PhysRevA.16.1013} is \nmuch higher than ours. \nOur results agree well with the DCSs of \nKhakoo et al. \\cite{2005PhRvA..71f2703K} at 15 and 17.5 eV. \nHowever, their cross sections at 15 eV have a small dip at 100 degrees \nand a small bump 60 degrees, which do not exist in our results. \nAt 12.5 eV, our cross sections are slightly larger than the results \nof Khakoo et al.\\cite{2005PhRvA..71f2703K}. \nOn the whole, agreement with the other experimental results of \nBrunger and Teubner \\cite{PhysRevA.41.1413} and \nCartwright et al. \\cite{PhysRevA.16.1013} is not good. \n\nFigure \\ref{fig8} shows the excitation DCSs for the \n${a'}^{1} \\Sigma_{u}^{-}$ state. \nBecause of $\\Sigma^{+}$-$\\Sigma^{-}$ selection rule, \nDCSs at 0 and 180 degrees become zero as in the case of \nthe ${B'}^{3} \\Sigma_{u}^{-}$ state DCSs. \nCalculated DCSs have a broad single peak near 60 degrees at 12.5 and 15 eV. \nAt 17.5 eV, there are two broad peaks at 50 and 120 degrees. \nAlthough there is slight overestimation of DCSs near 50-60 degrees, \nour DCSs agree marginally with the results of \nKhakoo et al.\\cite{2005PhRvA..71f2703K}. \nAgreement with the other experimental results is \nnot good except low scattering angles at 17.5 eV.  \n\nFigure \\ref{fig9} compares our excitation DCSs for \nthe $a^{1} \\Pi_{g}$ state with the experimental cross sections. \nBecause of large variation of the DCSs, the cross sections are \nshown in logarithmic scale. \nCalculated DCSs are strongly forward-enhanced, \nwhich is consistent with all experimental results shown \nin the figure. \nOur DCSs at 12.5 eV have a small dip around 100 degrees, which \nmoves forward to 85 degrees at 15 eV and 75 degrees at 17.5 eV. \nThis behavior roughly agrees with the results of \nCartwright et al.\\cite{PhysRevA.16.1013} and \nKhakoo et al.\\cite{2005PhRvA..71f2703K}. \nAt 15 eV, our DCSs agree better with \nthe results of Khakoo et al.\\cite{2005PhRvA..71f2703K} than \nthe other experimental DCSs. \nAt 17.5 eV, the results of Cartwright et al.\\cite{PhysRevA.16.1013}  \nare closer to our DCSs at scattering angles above 40 degrees.  \nBelow 40 degrees, our calculation significantly underestimates \nthe experimental DCSs. \nOur results at 12.5 eV are located between the DCSs of \nCartwright et al. \\cite{PhysRevA.16.1013} \nand Khakoo et al.\\cite{2005PhRvA..71f2703K}, \nhowever the shape of the DCSs is similar to their results. \nThe shapes of DCSs calculated by da Costa and Lima \n\\cite{2006IJQC..106.2664D} are similar to our results. \nHowever, their cross sections are larger than our results at\nlow-scattering angles below 80 degrees, where their results agree better \nwith the experimental DCSs of Brunger and Teubner \\cite{PhysRevA.41.1413} and \nZetner and Trajmar \\cite{Zetner1987}.\n\n\nFigure \\ref{fig10} shows calculated excitation DCSs for \nthe $w^{1} \\Delta_{u}$ state with the experimental cross sections. \nOur DCSs are enhanced in forward direction as in the case of \nthe $a^{1} \\Pi_{g}$ state. However, magnitude of the \nenhancement is much smaller than that of the $a^{1} \\Pi_{g}$ state. \nAgreement with the DCSs of Cartwright et al. \\cite{PhysRevA.16.1013} \nis good at 17.5 eV except low scattering angles below 20 degrees. \nAt 12.5 and 15 eV, their results are much larger than our DCSs. \nAt 15 eV, our DCSs agree marginally with the results of \nKhakoo et al. \\cite{2005PhRvA..71f2703K}, \nalthough details of the DCS profile are different. \nTheir results are smaller than ours at 17.5 and 12.5 eV. \nDiscrepancy is especially large for forward scattering at 12.5 eV. \n\nFigure \\ref{fig11} shows excitation DCSs for the $C^{3} \\Pi_{u}$ state \nwith the experimental cross sections of \nKhakoo et al.\\cite{2005PhRvA..71f2703K}, \nBrunger and Teubner\\cite{PhysRevA.41.1413}, \nZubek and King \\cite{JPhysB.27.2613} and \nCartwright et al. \\cite{PhysRevA.16.1013}.  \nCalculated DCS profiles are almost flat at 12.5 and 15 eV, \nwhereas they are enhanced in backward direction at 17.5 eV. \nBelow 90 degrees, slope of the calculated DCSs at 17.5 eV \nis similar to the results of Khakoo et al.\\cite{2005PhRvA..71f2703K}, \nZubek and King \\cite{JPhysB.27.2613} and \nCartwright et al. \\cite{PhysRevA.16.1013}, \nthough our results are about 50\\% larger than their DCSs. \nIn general, our results do not agree well with the experimental DCSs. \nAlthough the ICS of \nKhakoo et al. \\cite{2005PhRvA..71f2703K} at 15 eV agrees well \nour result as shown in panel (d) of Fig.\\ref{fig3}, \nangular dependence of the cross sections appears to be different. \n\n\n\\subsection{Discussion}\n\n\n\n\nThe excitation ICSs of the $B^3 \\Pi_g$ state, shown in panel (b) of figure\n\\ref{fig2}, have a small bump around 13 eV. However, there is no such\nstructure in the previous R-matrix ICSs of \nGillan et al. \\cite{JPhysB.29.1531}. \nThe origin of this bump in our calculation is the N$_2^-$ ${1}^2\n\\Delta_g$ state, with main configuration of $3\\sigma_g^1 1\\pi_g^2$. \nThe existence of the N$_2^-$ ${1}^2 \\Delta_g$ state can also be verified by \nusual CASSCF calculation on N$_2^-$ with valence active space ignoring \ncontinuum orbitals. In molpro calculation, the energy of the \n${1}^2 \\Delta_g$ state is 15.7 eV. Since diffuse continuum orbitals are\nadded in the R-matrix calculation, the energy of the state is stabilized\nto be 12.8 eV in the present scattering calculation. \nIn the same way, the N$_2^-$ ${}^2 \\Pi_u$ ($1\\pi_u^3 1\\pi_g^2$) \nresonance peak in the $A{}^{3} \\Sigma_u^+$ excitation ICSs can be \nverified by usual CASSCF calculation. In molpro calculation, \nit is located at 14.7 eV, whereas the position of the resonance \nis stabilized to be 12.2 eV in our R-matrix scattering calculation.  \nIt is unclear why the bump in the ICSs of the $B^3 \\Pi_g$ state  \nis not evident in the previous R-matrix \ncross sections of Gillan et al.\\cite{JPhysB.29.1531}.  \nSome details of the R-matrix calculations are different in their\ncalculation and ours, e.g., they used hybrid orbitals with Slater type \nfunctions, whereas we employed SA-CASSCF orbitals with Gaussian type functions. \nThese difference may contribute to the difference in magnitude of \nthe ${}^2 \\Delta_g$ partial cross section. \n\n\nIn this study, we employed the fixed-nuclei (FN) approximation. \nAs we can see in figure \\ref{fig1}, equilibrium bond lengths of \nthe excited N$_2$ states are longer than that of the ground state. \nThus, in principle, it would be desirable to include the effect of \nnuclear motion in the R-matrix calculation. \nUse of the FN approximation may be responsible for \nseveral discrepancies between our calculation and experiments, \nincluding bumps in the ICSs of the $A^{3} \\Sigma_{u}^{+}$ and \n$B^{3} \\Pi_{g}$ states, the position \nof the peak in the ICSs of the $C^{3} \\Pi_{u}$ state. \nAlthough the calculated DCSs agree very well with experimental results \nin general, our DCSs of the $A^{3} \\Sigma_{u}^{+}$, $B^{3} \\Pi_{g}$, \n$w^{1} \\Delta_{u}$ and $C^{3} \\Pi_{u}$ states at 12.5 eV are 2-4 times\nlarger than experimental results. These deviations in the near-threshold\nDCSs can also be related to the FN approximation. \nIn spite of these discrepancies, good agreements are observed between \nour calculation and experiments in most ICS and DCS cases as we can see \nin the figures. Agreements with the recent experimental results of \nKhakoo et al.\\cite{2005PhRvA..71f2703K} and Johnson et al.\n\\cite{2005JGRA..11011311J} are especially impressive. \nIt is possible to include nuclear motion in the R-matrix \nformalism through vibrational averaging of T-matrix elements \nor the non-adiabatic R-matrix method, though application of \nthese methods will be a difficult task in the presence of many \ntarget electronic states. In the future, we plan to perform \nthe R-matrix calculation with these methods including nuclear \nmotion. \n\n\n\n\n\n\\clearpage\n\n\\section{summary}\n\nWe have investigated electron impact excitations of \nN$_2$ molecule using the fixed-bond R-matrix method \nwhich includes 13 target electronic states,\n${X}^1 \\Sigma^{+}_{g}$, $A^{3} \\Sigma_{u}^{+}$, $B^{3} \\Pi_{g}$, \n$W^{3} \\Delta_{u}$, ${B'}^{3} \\Sigma_{u}^{-}$, ${a'}^{1} \\Sigma_{u}^{-}$, \n$a^{1} \\Pi_{g}$, $w^{1} \\Delta_{u}$, $C^{3} \\Pi_{u}$, \n${E}^{3} \\Sigma_{g}^{+}$, ${a''}^{1} \\Sigma_{g}^{+}$, \n$c^{1} \\Pi_{u}$ and ${c'}^{1} \\Sigma_{u}^{+}$. \nThese target states are described by CI wave functions in the valence \nCAS space, using SA-CASSCF orbitals. Gaussian type orbitals \nwere used in this work, in contrast to the STOs in \nthe previous R-matrix works. \nWe have obtained integral cross sections as well as \ndifferential cross sections of excitations to the \n$A^{3} \\Sigma_{u}^{+}$,  $B^{3} \\Pi_{g}$, $W^{3} \\Delta_{u}$, \n${B'}^{3} \\Sigma_{u}^{-}$, ${a'}^{1} \\Sigma_{u}^{-}$, \n$a^{1} \\Pi_{g}$, $w^{1} \\Delta_{u}$ \nand $C^{3} \\Pi_{u}$ states, which have been studied a lot experimentally \nbut not enough theoretically before. \nIn general, good agreements are observed both in the \nintegrated and differential cross sections, \nwhich is encouraging for further theoretical and experimental \nstudies in this field.   \nHowever, some discrepancies are seen in the integrated \ncross sections of the $A^{3} \\Sigma_{u}^{+}$ \nand $C^{3} \\Pi_{u}$ states, especially around a peak structure. \nAlso, our DCSs do not agree well with the experimental results at \nlow impact energy of 12.5 eV, compared to the higher energies \nof 15 and 17.5 eV. \nThese discrepancies may be related to the fixed-nuclei approximation or \ninsufficiency of higher excited target states in the R-matrix model. \n\n\n\n\\begin{acknowledgments}\nThe work of M.T. is supported by the Japan Society for the \nPromotion of Science Postdoctoral Fellowships for Research Abroad. \nThe present research is supported in part by the grant from the Air Force \nOffice of Scientific Research: the Advanced High-Energy Closed-Cycle Chemical \nLasers project (PI: Wayne C. Solomon, University of Illinois, \nF49620-02-1-0357). \n\\end{acknowledgments}\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDeep learning has led to breakthroughs in various fields, such as computer vision \\cite{Krizhevsky2012, Kaiming2016} and language processing \\cite{Oord2016}. Despite its success, it is still limited by its vulnerability to adversarial examples \\cite{Szegedy2014}. In image processing, adversarial examples are small, typically imperceptible perturbations to the input that can lead to misclassifications in real-world scenarios. In domains like autonomous driving or healthcare this can potentially have fatal consequences. Since the weakness of neural networks to adversarial examples has been demonstrated, many methods were proposed to make neural networks more robust and reliable \\cite{Goodfellow2015, Madry2018, Tramer19}. In a constant challenge between new adversarial attacks and defenses, most of the proposed defenses have been shown to be rather ineffective \\cite{Guo2018, Kurakin2018, Samangouei2018}. Yet, substantial progress in this field is needed to increase the reliability and trustworthiness of neural networks in our daily lives.\n\n\n\n\\begin{figure*}\n    \\centering\n   \n   \n   \n   \n   \n    \\begin{subfigure}{0.205\\textwidth}\n      \\includegraphics[width=\\textwidth]{Images\/gradient_descent_normal.pdf}\n      \\caption{}\n      \\label{fig:gradient_descent}\n    \\end{subfigure} \n    \\begin{subfigure}{0.205\\textwidth}\n      \\includegraphics[width=\\textwidth]{Images\/gradient_descent_smoothed.pdf}\n      \\caption{}\n      \\label{fig:gradient_descent_smoothed}\n    \\end{subfigure}\n    \\begin{subfigure}{0.210\\textwidth}\n      \\includegraphics[width=\\textwidth]{Images\/surface.pdf}\n      \\caption{}\n      \\label{fig:surface}\n    \\end{subfigure} \n    \\begin{subfigure}{0.210\\textwidth}\n      \\includegraphics[width=\\textwidth]{Images\/surface_smoothed.pdf}\n      \\caption{}\n      \\label{fig:surface_smoothed}\n    \\end{subfigure}\n    \\begin{subfigure}{0.05\\textwidth}\n      \\includegraphics[width=\\textwidth]{Images\/colorbar.pdf}\n      \\label{fig:gradient_descent_cb}\n      \\caption*{}\n    \\end{subfigure}\n  \\caption{Comparison between the standard \\acf{GD} method \\textbf{(A)} and the proposed \\acf{SNGD} \\textbf{(B)}. \\textbf{(C)} shows the side view of the noisy loss surface displayed in (A) and (B). The calculated ascent direction is displayed by a white arrow. For SNGD the ascent direction is obtained by averaging multiple gradients (red arrows) in the vicinity of the current point on the loss surface. \\textbf{(D)} illustrates the effect of SNGD when the amount of sampling operations $N$ tends to infinity. In the limit, using SNGD is equivalent to using GD on a loss surface that is convoluted with a convolution kernel defined by the sampling distribution.}\n  \\label{fig:gradient_descent_comparison}\n\\end{figure*}\n\n\nIn this paper we aim to improve the effectiveness of adversarial attacks on artificial neural networks. For this we adopt an intuitive idea that is known as gradient sampling \\cite{Burke05}. This method is designed to reliably estimate the significant optima of non-smooth non-convex functions with unreliable gradient information. Gradient sampling can be interpreted as a generalized steepest descent method where the gradient is not only calculated at a given point but additionally for points in the direct vicinity. The final gradient direction is subsequently calculated in the convex hull of all sampled gradients and thus incorporates additional information about the local geometry of the loss surface. The central idea behind this approach is that point estimates of gradients can be misleading, especially for noisy and non-convex loss surface. However, this method is only feasible for low-dimensional functions, due to the complexity of computing the convex hull in higher dimensions. To overcome this problem, we propose to simplify the algorithm so that it works with high dimensional data. \nWe discard the calculation of the convex hull and compute the final gradient direction as a weighted average of all sampled gradients. Figure \\ref{fig:gradient_descent_comparison} illustrates how the convergence of the standard \\acf{GD} method can be improved for noisy and non-convex loss surfaces by the proposed \\acf{SNGD}. Standard GD calculates nearly random ascent directions while SNGD approximates the global ascent direction more accurately as it is less susceptible to noise and local optima. \n\nThe contributions of this paper can be summarized as follows. First we relate the proposed gradient sampling method to the empirical mean, which converges towards a nonlocal gradient with respect to an underlying probability distribution for enough sampling operations. Subsequently we show that SNGD-based attacks are more query efficient while achieving higher success rates than prior attacks on popular benchmark datasets. Finally, we demonstrate that SNGD approximates the direction of an adversarial example more accurately during the attack and empirically show the effectiveness of SNGD on non-convex loss surfaces. \n\n\\section{Preliminary}\nIn the following we introduce the necessary mathematical notation to describe adversarial attacks and to review current contributions in this field.\n\n\\subsection{Notation}\n\nLet ($x$, $y$) with $x\\in\\mathbb{R}^d$ and $y\\in\\{1, 2, \\dots, C\\}$ be pairs of samples and labels in a classification task with $C$ different classes, where each sample is represented by a $d$-dimensional feature vector. In the following we assume that the samples are drawn from the $d$-dimensional unit cube, i.e., $x \\in [0,1]^d$, as this is typically the case for image data. Let $\\mathcal{L}$ be the loss function (e.g., categorical cross-entropy) of a neural network $F_{\\theta}$, parameterized by the parameter vector $\\theta \\in \\Theta$. \nConstructing an adversarial perturbation $\\gamma \\in \\mathbb{R}^d$ with maximum effect on the loss value can be stated as the following optimization problem: \n\\begin{align} \\label{eq:adversarial_opt}\n\\underset{\\gamma}\\max ~\\mathcal{L}(F_{\\theta}(x + \\gamma), y)\n\\end{align}\nThe perturbation $\\gamma$ is usually constrained in two ways: 1) The value range of the adversarial example is still valid for the respective domain (e.g, between [$0$, $1$] or [$0$, $255$] for images), 2) The adversarial example $x_{adv} = x + \\gamma$ is within a set of allowed perturbations $S$ that are unlikely to change the class label for human perception. In the following, we focus on \\textit{untargeted gradient-based adversarial attacks} that are constrained by the $L_{\\infty}$ norm such that $||\\gamma||_{\\infty} \\leq \\epsilon$, as done in prior work \\cite{Lin2020}.\n\n\\subsection{Related Work}\n\nA variety of adversarial attacks have been proposed. We give a brief overview for the most successful gradient-based algorithms and also a proven zeroth order attack known as \\ac{SPSA}.\n\n\\subsubsection{Fast Gradient Sign Method (FGSM)}FGSM introduced by \\citeauthor{Goodfellow2015}, is one of the first gradient-based adversarial attacks. FGSM calculates an adversarial example according to the following equation:\n\\begin{align} \\label{eq:fgsm}\n    x_{adv} = \\Pi_{S} \\, (x + \\epsilon \\cdot \\operatorname{sign}(\\nabla_x\\mathcal{L}(F_{\\theta}(x), y))\n\\end{align}\nwhere $\\Pi_{S}(x)$ is a projection operator that keeps $\\gamma$ within the set of valid perturbations $S$ and $\\operatorname{sign}$ is the componentwise signum operator. \n\n\\subsubsection{Basic Iterative Method (BIM)}\\citeauthor{Kurakin2017} proposed an iterative variant of the FGSM attack, in which multiple smaller gradient updates are used to find the adversarial perturbation:\n\\begin{align} \\label{eq:ifgsm}\n    x_{adv}^{t + 1} = \\Pi_{S} \\, (x_{adv}^{t} + \\alpha \\cdot \\operatorname{sign}(\\nabla_x  \\mathcal{L}(F_\\theta(x_{adv}^{t}), y))\n\\end{align}\nwhere $\\epsilon \\geq \\alpha > 0$ and $x_{adv}^{t + 1}$ describes the adversarial example at iteration $t$ and $x_{adv}^{0} = x$. \\citeauthor{Madry2018} introduced a slightly modified version of BIM called \\textbf{Projected Gradient Descent (PGD)}, in which the starting point of the attack is randomly chosen from the set of valid perturbations $S$. \nMore variants of iterative gradient-based attacks have been proposed. This includes several momentum-based methods \\cite{Dong18, Useato2018, Lin2020} MI-FGSM ADAM-PGD, NI-FGSM.\n\n\\subsubsection{Nesterov Iterative Fast Gradient Sign Method (NI-FGSM)}\\citeauthor{Lin2020} incorporated Nesterov momentum into iterative attacks to improve their transferability and success rate. The attack is formally given by:\n\\begin{equation}\n\\begin{split}\n\\label{eq:nifgsm}\n    x_{nes}^{t} &= x_{adv}^{t} + \\alpha \\cdot \\mu \\cdot g^{t} \\\\ \n    g^{t + 1} &= \\mu \\cdot g^{t} + \\frac{\\nabla_{x}\\mathcal{L}(F_{\\theta}(x_{nes}^{t}), y)}{||\\nabla_{x}\\mathcal{L}(F_{\\theta}(x_{nes}^{t}), y)||_{1}} \\\\\n    x_{adv}^{t + 1} &= \\Pi_{S} \\, (x_{adv}^{t} + \\alpha \\cdot \\operatorname{sign}(g^{t + 1})\n\\end{split}\n\\end{equation}\nwhere $g^{t}$ denotes the accumulated gradients at iteration $t$ and $g^{0} = 0$ and $x_{adv}^0=x$. $\\mu>0$ describes the momentum parameter.\n\n\\subsubsection{Output Diversified Initialization (ODI)}\\citeauthor{tashiro2020} introduced ODI, which aims at finding efficient starting points for adversarial attacks. Therefore, they calculate the starting point of an adversarial attack such that it maximizes the difference in the output space compared to a clean sample. They empirically show that this initialization provides more diverse and effective starting points than random initialization. The initialization of an ODI-based attack is given by: \n\\begin{align}\n   \n    x_{ODI}^{t + 1} &= \\Pi_{S} \\, (x_{adv}^{t} + \\alpha \\cdot \\operatorname{sign}(\\frac{\\nabla_{x}\\omega^{\\top}F_{\\theta}(x_{ODI}^t)}{||\\nabla_{x}\\omega^{\\top}F_{\\theta}(x_{ODI}^t)||_{2}})) \n\\end{align}\nwhere $x_{adv}^0 = x$ and $\\omega \\in \\mathbb{R}^{C}$ is sampled from a uniform distribution in a predefined range. This procedure is repeated for a given number of steps. Afterwards, the calculated starting point can be used for initializing an adversarial attack.\n\n\\subsubsection{Brendel \\& Bethge Attack (B\\&B)}\\citeauthor{Brendel19} propose an alternative gradient-based attack. They use gradients to estimate the decision boundary between an adversarial and a benign sample. Subsequently the attack follows the decision boundary towards the clean input to find the smallest perturbation that leads to a misclassification. Simultaneously, the updated perturbation is forced to stay within a box-constraint of valid inputs. The optimization problem can be formulated as follows:\n\\begin{equation}\n\\begin{split}\n    \\underset{\\delta}{\\min} ~& ||x-x_{adv}^{t-1} - \\delta^{t}||_{\\infty}  \\quad \\text{ s.t. } \\\\\n   &x-x_{adv}^{t-1} + \\delta^{t} \\in [0,1]^d, \\ b^{t \\top}\\delta^{t} = c^{t}, \\ ||\\delta^{t}||_{2}^{2} \\leq r\n\\end{split}\n\\end{equation}\nwhere $\\delta^{t} \\in \\mathbb{R}^d$ is the step along the decision boundary in iteration $t\\in\\mathbb{N}$ within a given trust region with radius $r>0$, $b^t$ denotes the current estimate of the normal vector of the local boundary, and $c^t$ describes the constraint that the current perturbation is on the decision boundary. In particular the $L_{\\infty}$-variant of this attack is of interest for our investigations.\n\n\\subsubsection{\\acf{SPSA}}proposed by \\cite{Spall1992, Useato2018} is a zeroth order optimization algorithm which has been successfully used in prior work to break models with obfuscated gradients \\cite{Useato2018}. It uses finite-differences to approximate the optimal descent direction. Zeroth order attacks are usually less efficient than their gradient-based counterparts. Nevertheless, they can be used in situations where the gradient information of the model is either removed by a non-differentiable operation (e.g., JPEG compression \\cite{Guo2018}) or is highly obfuscated, as described in \\cite{Kurakin2018}. \n\n\n\\section{Sampled Nonlocal Gradient Descent} \\label{SampledNonlocal}\n\nWe propose an alternative to the standard \\acf{GD} algorithm, which we name \\acf{SNGD}. Our aim is to use this method to further augment existing gradient-based attacks. SNGD is specifically designed for noisy and non-convex loss surfaces as it calculates the gradient direction as the weighted average over multiple sample points in the vicinity of the current data point. The gradient calculation of SNGD is given by:\n\\begin{equation} \\label{eq:gradient_averaging}\n\\begin{split}\n\\nabla_{\\scriptscriptstyle{SNGD}} &\\mathcal{L}(F_\\theta (x,y)) :=\\\\\n&\\nabla_x \\frac{1}{N} \\sum_{i=1}^{N} w_{i} \\cdot \\mathcal{L}(F_\\theta (\\operatorname{clip_{[0,1]}}\\{x + \\xi_i^\\sigma\\}), y)),\n\\end{split}\n\\end{equation}\nwhere $N\\in\\mathbb{N}$ is the number of sampling operations, $w_i$ is the $i$-th weight of a sampled gradient, and $\\operatorname{clip_{[a,b]}}$ is the component-wise clipping operator with value range $[a, b]$. The clipping operator is needed to ensure that the data stays in the normalized range, e.g., in the case of images. It can be discarded for other applications with unbounded data. The random variables $\\xi_i^\\sigma$ are considered to be drawn i.i.d. from a distribution $P^\\sigma$ parametrized by the standard deviation $\\sigma>0$,\nwhich effectively determines the size of the neighborhood.\nBy the law of large numbers the sampled nonlocal gradient converges (with a rate of order $N^{-1\/2}$ in variance) to the respective expectation value, which is the given by the following nonlocal gradient\n\\begin{align} \n\\nabla_x \\mathbb{E}_{\\xi \\sim P^\\sigma} \\left[ \\mathcal{L}(F_\\theta (\\operatorname{clip_{[0,1]}}\\{x +\\xi\\}), y))  \\right]\n\\end{align}\nThe expectation is effectively a local averaging of the likelihood around $x$. Note that by linearity the proposed SNGD method is equivalent to an averaging of the gradients, i.e., the standard form of a nonlocal gradient, cf. \\cite{du2019nonlocal}.\nThis observation is also useful to efficiently compute the attack, as one can use only a single backward pass for all sampled gradients in each iteration. Note that as the forward-passes can be parallelized, the effective runtime of SNGD is equivalent to GD with sufficient memory.\nA SNGD-based BIM attack is formally given by:\n\\begin{align} \\label{eq:sngdifgsm}\n    x_{adv}^{t + 1} = \\Pi_{S} \\, (x_{adv}^{t} + \\alpha \\cdot \\operatorname{sign}(\\nabla_{\\scriptscriptstyle{SNGD}}  \\mathcal{L}(F_\\theta(x_{adv}^{t}), y))\n\\end{align}\nWe demonstrate the benefit of this approach in terms of efficiency in the results section below.\n\n\\section{Experiments}\n\nWe conduct several experiments to evaluate SNGD. We first analyze if SNGD can improve the success rate of adversarial attacks compared to GD. Secondly, we explore the possibility of combining SNGD with other methods, e.g., ODI. Also we investigate the option to decay the standard deviation $\\sigma$ during an attack and to individually weight the sampled gradients. Furthermore, we inspected the performance difference between several attacks, as efficient attacks play an important role to facilitate the evaluation of model robustness in real-world applications. Lastly, we analyze the ability of SNGD to better approximate the global descent direction and show that SNGD is effective for non-convex loss surfaces. Additional and ineffective experiments are included in the supplementary material.\n\n\\subsection{Setup} \n\nIn the following we give an overview of general hyperparameters used for the experiments, including thread model, training, evaluation, and datasets. We describe dataset-specific hyperparameters such as the model architecture in the corresponding sections.\n\n\\subsubsection{Threat model}In this work we focus our evaluation on the $L_{\\infty}$ norm and untargeted attacks. We combine our proposed method, SNGD with state-of-the art attacks, including PGD \\cite{Madry2018} and PGD with Nesterov Momentum (N-PGD) \\cite{Lin2020}, which we call SN-PGD and SN-N-PGD respectively. We additionally combine all PGD-based attacks with ODI \\cite{tashiro2020}. ODI-based attacks achieve one of the highest Success rates on the Madry MNIST leaderboard \\cite{Madry2018}. Moreover, we compare our approach to the B\\&B attack \\cite{Brendel19}, one of the most recent and effective gradient-based attacks. We additionally evaluated if models are obfuscating their gradients with the zeroth order \\ac{SPSA} attack \\cite{Spall1992, Useato2018}. If not stated otherwise all SNGD-based attacks are performed with $w_{i} = 1,\\, \\forall i \\in \\{1,2,\\dots, N\\}$. \n\n\nWe limited the amount of model evaluations to $2000$ for each gradient-based attack and distributed them over multiple restarts and samples (SNGD), such that $R\\cdot I \\cdot N = 2000$, where $R$ denotes the total number of restarts, $I$ the amount of attack iterations, and $N$ the amount of sampling operations for SNGD. We tried multiple combinations of model evaluations and restarts, as we observed that for a fixed budget of total evaluations this considerably impacts the performance of the attacks. For the standard PGD attack we obtained the highest success rates between $20-400$ iterations and $5-100$ random restarts. For SN-PGD attacks we used $100$ iterations, $4$ sampling operations and $5$ random restarts. The B\\&B attack was performed with $1000$ iterations and two restarts. The SPSA attack was performed with $100$ steps and a sample size of $8192$, as shown to be effective in \\cite{Useato2018}. Additional hyperparameters are included in the supplementary material. We performed all attacks on the same subset of $1000$ ($10\\%$) randomly selected test images as in \\cite{Useato2018, Brendel19}.\n\n\\subsection{Data and architectures}\n\nThree different image classification datasets were used to evaluate the adversarial robustness of the different models (MNIST \\cite{LeCun98}, Fashion-MNIST \\cite{Xiao2017} and CIFAR10 \\cite{Krizhevsky2009}). We split each dataset into the predefined train and test sets and additionally removed $10\\%$ ($5000$ samples) of the training data for validation.\n\n\\subsubsection{Training}We decided to evaluate our attack on two of the strongest empirical defenses to date, adversarial training \\cite{Athalye18, Madry2018, Useato2018} and TRADES \\cite{Zhang19}. For adversarial training we used the fast FGSM-based adversarial training algorithm \\cite{Wong2020}. In preliminary experiments on MNIST we observed that the loss surfaces of these models are not as convex as described in prior work \\cite{Kurakin2018, Chan20} (see Figure \\ref{fig:loss_landscape_fgsm_mnist}) compared to models trained with PGD-based adversarial training. For comparison we additionally trained each model with the typically used PGD-based adversarial training \\cite{Madry2018}. For TRADES we used the pre-trained model provided by the authors of the original paper \\cite{Zhang19} for the MNIST and CIFAR10 datasets.\n\nFor fast FGSM-based training we used the same hyperparameters as proposed in \\cite{Wong2020}. For PGD-based training we used $7$ steps and a step size of $1\/4$ $\\epsilon$ \\cite{Madry2018}. All self-trained networks were trained and evaluated $5$ times using stochastic gradient descent with the Adam optimizer ($\\beta_{1} = 0.9$, $\\beta_{2} = 0.999$) \\cite{Kingma14}. We used a cyclical learning rate schedule \\cite{Smith2017} which has been successfully used for adversarial training in prior work \\cite{Wong2020}. Thereby, the learning rate $\\lambda$ is linearly increased up to its maximum $\\Lambda$ over the first $2\/5$ epochs and then decreased to zero over the remaining epochs. The maximum learning rate $\\Lambda$ was estimated by increasing the learning rate of each individual network for a few epochs until the training loss diverged \\cite{Wong2020}. All models were optimized for $100$ epochs, which was sufficient for convergence, and the checkpoint with the lowest adversarial validation loss was chosen for testing.\n\n\\subsubsection{MNIST}consists of greyscale images of handwritten digits each of size $28\\times28\\times1$ ($60,000$ training and $10,000$ test). We used the same MNIST model that \\citeauthor{Wong2020} used for fast adversarial training. However, we doubled the number of filters for the convolutional layers, as we noticed that the performance of the model sometimes diverged to random guessing during training. The optimal maximum learning rate we found for MNIST was about $0.005$, which is in line with \\cite{Wong2020}. As in prior work, we used a maximum perturbation budget of $\\epsilon = 0.3$.\n\n\n\n\\subsubsection{Fashion-MNIST}consists of greyscale images of $10$ different types of clothing, each of size $28\\times28\\times1$ ($60,000$ training and $10,000$ test). The Fashion-MNIST classification task is slightly more complicated than MNIST, as it contains more intricate patterns. For Fashion-MNIST we used the same architecture as for MNIST. The optimal learning rate we found for Fashion-MNIST was approximately $0.007$. To the best of our knowledge there is no standard perturbation budget $\\epsilon$ commonly used for Fashion-MNIST. Since this dataset contains more complicated patterns than MNIST we used a lower maximum perturbation budget of $\\epsilon = 0.15$.\n\n\n\\subsubsection{CIFAR10}consists of color images, each of size $32\\times32\\times3$, with $10$ different labels ($50,000$ training and $10,000$ test). CIFAR10 is the most challenging classification task out of the three. For CIFAR10 we used the same PreActivationResNet18 \\cite{Kaiming2016} architecture as in \\cite{Wong2020}. All images from the CIFAR10 dataset were standardized and random cropping and horizontal flipping were used for data augmentation during training as in \\cite{Kaiming2016, Madry2018, Wong2020}. We found the optimal learning rate to be around $0.21$. Inline with previous work, we set the maximum perturbation budget to $\\epsilon = 8\/255$.\n\n\\subsection{Experiments on noise distributions and sampling}\n\n\\subsubsection{Noise distribution:}To combine \\ac{SNGD} with an adversarial attack we need to define the distribution $P^\\sigma$ from which we sample data points in the local neighborhood. In preliminary experiments we evaluated the performance for the Uniform, Gaussian and Laplacian distributions on the MNIST dataset. We analyzed the success rate for a wide range of distribution parameters but did not observe any considerable differences between the optimally tuned distributions. Since the Gaussian distribution achieved marginally superior results, we decided to use it for the remaining experiments.\n\nWe constrained the search space for the optimal standard deviation $\\sigma$ to $0< \\sigma <\\epsilon$ since the gradient information outside of the attack radius should be non-relevant for the optimization of the attack.\nTo rapidly find a good estimate for the standard deviation $\\sigma$ for each model and attack, we fine-tuned $\\sigma$ on a single batch of the validation set. We observed that for a wide range of $\\sigma$ values the performance of the PGD attack increases. This is exemplified in Figure \\ref{fig:success} for the MNIST dataset ($\\epsilon = 0.3)$.\n\n\\begin{figure}\n  \\centering\n  \\begin{subfigure}{0.22\\textwidth}\n    \\includegraphics[width=\\textwidth]{Images\/success.pdf}\n    \\caption{Success Rate}\n    \\label{fig:success}\n  \\end{subfigure} \n  \\begin{subfigure}{0.22\\textwidth}\n    \\includegraphics[width=\\textwidth]{Images\/sampling.pdf}\n    \\caption{Sampling}\n    \\label{fig:sampling}\n  \\end{subfigure}\n\\caption{The two plots show the success rate of SNGD-based PGD attacks (SN-PGD) on a single validation batch (512 samples) of the \\textbf{MNIST} dataset ($\\epsilon$ = 0.3).  \\textbf{(A)} SN-PGD success rate for varying standard deviations $\\sigma$. All displayed attacks are performed with $100$ iterations, $5$ restarts and $N = 4$. For noise free attacks ($\\sigma = 0$) we made an exception and used $400$ iterations and $5$ restarts. \\textbf{(B)} SN-PGD success rate for different amounts of sampling operations $N$ and $\\sigma = 0.05$. Every attack was performed with $5$ restarts and the amount of attack iterations $I$ was set to $400\/N$.}\n\\label{fig:results}\n\\end{figure}\n\n\\subsubsection{Sampling:}In an additional experiment we evaluated the number of sampling operations $N$ that are the most effective for \\ac{SNGD}-based attacks. Each sampling operation $N$ increases the computational overhead of SNGD-based attacks but should in turn improve the calculated descent direction. Therefore, we evaluated the success rate for different $\\sigma$ values and number of sampling operations on the MNIST validation set. Figure \\ref{fig:sampling} illustrates that $4$ sampling operations were optimal to increase the success rate for MNIST. We set $N = 4$ for all datasets for the remaining experiments.   \n\n\\section{Results and Discussion} \\label{Sec:Results}\n\n\\subsection{Success Rate} Table \\ref{tab:first_results} demonstrates that the proposed SN-PGD attack surpasses the mean success rate of prior attacks in our experiments. Nevertheless, for the TRADES model trained on the CIFAR10 dataset the B\\&B attack was more effective than our approach. In contrast to the results reported in \\cite{Brendel19} the B\\&B attack did not always outperform the standard PGD attack for the adversarially trained models. We could also not see a consistent increase in performance when combining SN-PGD or PGD and Nesterov momentum denoted by SN-N-PGD and N-PGD. N-PGD outperformed PGD in $3$ out of $8$ cases and SN-N-PGD outperformed SN-PGD in $1$ out of $8$ cases. We additionally analyzed the performance for individual runs as the standard deviation was high in some cases (e.g., F-MNIST). We observed that in general, the individual models were either more robust or vulnerable against all attacks. For all experiments where SN-PGD showed the highest mean success rate it also showed the highest success rate for all individual models. An exception was a single individual run (CIFAR10, PGD training), in which it was surpassed by the B\\&B attack.\n\n\\begin{table*}\n    \\centering\n    \\begin{tabular}{lcccccccc}\n        \\toprule\n        & Clean & FGSM & PGD & N-PGD &B\\&B & SN-PGD & SN-N-PGD & SPSA \\\\\n        \\midrule\n        \\textbf{MNIST} \\\\\n        RFGSM & 99.2 \\textsubscript{\\textpm 0} & 96.4 \\textsubscript{\\textpm 2} & 88.8 \\textsubscript{\\textpm 3} & 88.6 \\textsubscript{\\textpm 4} & 87.0 \\textsubscript{\\textpm 3} & \\textbf{86.4}\\textsubscript{\\textpm 2} & 88.8 \\textsubscript{\\textpm 3} & 92.8\\textsubscript{\\textpm 3}\\\\\n        PGD & 99.0 \\textsubscript{\\textpm 0} & 97.0 \\textsubscript{\\textpm 2} & 92.4 \\textsubscript{\\textpm 2} & 93.6 \\textsubscript{\\textpm 2} &  91.0 \\textsubscript{\\textpm 4} & \\textbf{90.8\\textsubscript{\\textpm 1}} & 91.2 \\textsubscript{\\textpm 2} & 95.0 \\textsubscript{\\textpm 3} \\\\\n        TRADES & 99.5 & 96.2 & 91.2 & 91.6 & 90.6 & \\textbf{90.4}  & 91.3 & 92.2 \\\\\n        \\midrule\n        \\textbf{F-MNIST} \\\\\n        RFGSM & 85.4 \\textsubscript{\\textpm 1} & 74.8 \\textsubscript{\\textpm 4} & 60.4 \\textsubscript{\\textpm 8} & 61.6  \\textsubscript{\\textpm 8}  & 60.6  \\textsubscript{\\textpm 9} & \\textbf{58.8}\\textsubscript{\\textpm 8} & 61.0 \\textsubscript{\\textpm 9} & 66.6\\textsubscript{\\textpm 8}\\\\\n        PGD &  85.7 \\textsubscript{\\textpm 0} &  83.2 \\textsubscript{\\textpm 5} & 70.0 \\textsubscript{\\textpm 7} & 69.2  \\textsubscript{\\textpm 7} & 70.8 \\textsubscript{\\textpm 7} & 68.8 \\textsubscript{\\textpm 6} & \\textbf{68.6} \\textsubscript{\\textpm 6} & 74.2 \\textsubscript{\\textpm 6} \\\\\n         \\midrule\n        \\textbf{CIFAR10} \\\\\n        RFGSM & 83.6 \\textsubscript{\\textpm 0} & 54.0 \\textsubscript{\\textpm 7} & 43.4 \\textsubscript{\\textpm 6} & 44.0 \\textsubscript{\\textpm 4} & 45.8 \\textsubscript{\\textpm 5} & \\textbf{43.0} \\textsubscript{\\textpm 6} & 44.6\\textsubscript{\\textpm 5}  & 49.1\\textsubscript{\\textpm 6}\\\\\n        PGD & 79.7 \\textsubscript{\\textpm 0} & 55.0 \\textsubscript{\\textpm 3} & 48.4 \\textsubscript{\\textpm 2} & 49.6 \\textsubscript{\\textpm 2}  & 49.0 \\textsubscript{\\textpm 3} &\\textbf{48.2} \\textsubscript{\\textpm 2} & 49.2\\textsubscript{\\textpm 3} & 50.5 \\textsubscript{\\textpm 4} \\\\\n        TRADES & 84.9 & 63.2 & 59.3 & 59.2 & \\textbf{58.3} & 58.5 & 59.0 & 60.2 \\\\\n    \\end{tabular}\n     \\caption{Mean accuracy and standard deviation ($\\%$) on \\textbf{MNIST}, \\textbf{Fashion-MNIST} and \\textbf{CIFAR10} for various adversarial attacks. The lower the accuracy the better (higher success rate). The attack with the highest success rate is displayed in bold for each row. RFGSM- and PGD-trained models were trained and evaluated five times.}\n     \\label{tab:first_results}\n\\end{table*}\n\n\\subsection{Additional experiments}\n\nThe following section summarizes additional experiments: 1) combination of SNGD with ODI, 2) combination of SNGD with noise decay, 3) different approaches to weight the gradients sampled with SNGD, 4) the runtime between different attacks, 5) the ability to approximate the global descent direction between GD- and SNGD-based attacks, and 6) the effectiveness of SNGD on increasingly convex loss surfaces. \n\n\\subsubsection{1) Combination with ODI}We evaluated if combining the different PGD-based attacks with \\acf{ODI} increases their success rate. From Table \\ref{tab:odi_results} we can see the accuracy and accuracy difference after combining the attacks with ODI. In $9$ out of $24$ cases the success rate improved while it decreased in $4$ out of $24$. For the PGD trained CIFAR10 model initialization with ODI improved the SN-PGD attack the most by $2.2\\%$. For the remaining experiments the performance was not changed considerably. This is in line with the original paper, where the success rate increased only marginally on the MNIST dataset and more substantially on CIFAR10 \\cite{tashiro2020}. Note that ODI was partly designed to increase the transferability of adversarial attacks to other models which was not tested in this experiment.\n\n\n\\begin{table}[ht]\n    \\centering\n    \\begin{tabular}{lccc}\n        \\toprule\n        ODI & PGD & N-PGD & SN-PGD \\\\\n        \\midrule\n       \\textbf{MNIST} \\\\\n        RFGSM & 88.6 (-0.2) & 88.4(-0.2) & \\textbf{86.8}(+0.4) \\\\\n        PGD & 92.4(+0.0) & 93.3(-0.3) & \\textbf{90.8}(+0.0)  \\\\\n        TRADES & 91.2 (+0.0) & 91.4(-0.2) & \\textbf{90.3}(-0.1) \\\\\n        \\midrule\n        \\textbf{F-MNIST} \\\\\n        RFGSM & 60.4(+0.0)  & 61.4(-0.2) & \\textbf{59.2}(+0.4) \\\\\n        PGD & 70.0(+0.0)  & 69.2(+0.0) & \\textbf{68.8}(+0.0)  \\\\\n        \\midrule\n        \\textbf{CIFAR10} \\\\\n        RFGSM & 43.7(+0.3)  & 44.4(+0.4) & \\textbf{43.2}(+0.2)  \\\\\n        PGD & 48.4(+0.0)  & 49.6(+0.0)  & \\textbf{46.0}(-2.2)  \\\\\n        TRADES & 57.5(-1.8) & 57.7(-1.5) & \\textbf{56.8}(-1.7) \\\\\n    \\end{tabular}\n    \\caption{Mean accuracy ($\\%$) on \\textbf{MNIST}, \\textbf{Fashion-MNIST} and \\textbf{CIFAR10} for various adversarial attacks with ODI. The performance difference to attacks without ODI is given by a subscript (negative subscript values indicate an attack success rate increase). The attack with the highest success rate (With or without ODI) is displayed in bold for each row. RFGSM- and PGD-trained models were trained and evaluated five times.}\n    \\label{tab:odi_results}\n    \n\\end{table}\n\n\\subsubsection{2) Noise decay}\\citeauthor{Madry2018} demonstrate that individual runs of a PGD attack converge to distinctive optima with similar loss values. Based on this observation, they concluded that that PGD might be an optimal first-order adversary. We noticed that SN-PGD converges to different loss values than PGD and further evaluated if different noise levels $\\sigma$ are optimal for different samples, as the characteristics of the loss surface are likely to change between different samples. Therefore, we evaluated if decaying the noise during a SN-PGD attack and between restarts improves the success rate. The idea is that decaying the noise during the attack should make the optimization more stable, as it is less likely that the algorithm alternates around a local optimum. A similar approach has also been proposed for the gradient sampling methodology \\cite{Burke05}. We compared three different decay schedules: 1) decaying the noise in each attack iteration (SN-PGD + ID), 2) decaying the noise at every attack restart (SN-PGD + RD), 3) decaying the noise at every iteration and restart (SN-PGD + IRD). The results are summarized in Table \\ref{tab:noise_decay_results}. Every schedule improved the performance on average. Note that we did not tune the hyperparameters for noise decay and divided the noise by $1.05$ at each iteration and by $2$ at each restart.\n\n\n\\begin{table}[ht]\n    \\centering\n    \\begin{tabular}{lcccc}\n        \\toprule\n         & SN-PGD & +ID & +RD & +IRD \\\\\n        \\midrule\n        \\textbf{MNIST} \\\\\n        RFGSM & 86.4\\textsubscript{\\textpm 2} & \\textbf{85.8} \\textsubscript{\\textpm 2} & 86.0 \\textsubscript{\\textpm 3} & 86.2 \\textsubscript{\\textpm 3} \\\\\n        PGD & 90.8\\textsubscript{\\textpm 1} & \\textbf{90.2} \\textsubscript{\\textpm 2} & \\textbf{90.2} \\textsubscript{\\textpm 2} & 90.4 \\textsubscript{\\textpm 2} \\\\\n        TRADES & 90.4 & \\textbf{90.0} & 90.4 & 90.1 \\\\\n        \\midrule\n       \\textbf{F-MNIST} \\\\\n        RFGSM & 58.8\\textsubscript{\\textpm 8}  & 58.0 \\textsubscript{\\textpm 9} & 58.4 \\textsubscript{\\textpm 9} & \\textbf{57.8} \\textsubscript{\\textpm 9} \\\\\n        PGD & 68.8 \\textsubscript{\\textpm 6} & \\textbf{67.4} \\textsubscript{\\textpm 6} & 68.0 \\textsubscript{\\textpm 6} & 67.6 \\textsubscript{\\textpm 6} \\\\\n         \\midrule\n        \\textbf{CIFAR10} \\\\\n        RFGSM & 43.0 \\textsubscript{\\textpm 6} & \\textbf{42.8} \\textsubscript{\\textpm 3} & 43.0 \\textsubscript{\\textpm 5} & 43.0 \\textsubscript{\\textpm 5} \\\\\n        PGD & 48.2 \\textsubscript{\\textpm 2} & 48.2  \\textsubscript{\\textpm 2} & 48.0 \\textsubscript{\\textpm 1} & \\textbf{47.8} \\textsubscript{\\textpm 2} \\\\\n        TRADES & 58.5 & \\textbf{58.0} & 58.4 & \\textbf{58.0} \\\\\n    \\end{tabular}\n    \\caption{Mean accuracy and standard deviation ($\\%$) on \\textbf{MNIST}, \\textbf{Fashion-MNIST} and \\textbf{CIFAR10} for various \\ac{SNGD}-based adversarial attacks with different noise decay schedules. The lower the accuracy the better (higher success rate). The attack with the highest success rate is displayed in bold for each row. RFGSM- and PGD-trained models were trained and evaluated five times.}\n    \\label{tab:noise_decay_results}\n    \n\\end{table}\n\n\\subsubsection{3) Gradient weighting}The dimensionality of the optimization problem is high compared to the amount of sampling operations $N$. This makes it less likely that the SNGD algorithm behaves like it would in the limit of $N$. Thus, we explored heuristics to improve the convergence rate of the algorithm. Instead of computing the final gradient direction as the weighted average of the gradients of all samples, we weighted the gradients according to their relation to the gradient at the original data point. We tested three different measurements and their reciprocals as weights: 1) Cosine similarity, 2) Euclidean distance, 3) Scalar-product. In our experiment weighting with the cosine similarity was the most effective approach. This reduced the sensitivity of the method to the standard deviation $\\sigma$ without reducing the performance of the attack. However, we must know the individual gradients of each sample in order to weigh them. Thus, we cannot sum up the activation during the forward-passes and calculate only the average gradient, but rather have to do one backward-pass for each forward-pass. This introduces a considerable additional computational overhead. Overall, averaging the gradients was more effective in our experiments with respect to runtime and success rate. \n\n\\subsubsection{4) Runtime comparison}Table \\ref{tab:runtime} shows the runtime average and standard deviation for each attack over all experiments shown in Table \\ref{tab:first_results}. Due to the lack of gradient information, SPSA requires a high amount of model evaluations and takes by far the longest time to find adversarial examples. The B\\&B attack was also considerably slower than PGD for the same amount of model evaluations in our experiments. Note that in the beginning of each B\\&B attack, we need to find the decision boundary which introduces an additional computational overhead. The fastest attack was the proposed SN-PGD where we use the same amount of forward-passes as with the PGD attack but use less backward passes. We did not parallelize the sampling operations of SNGD in our experiments. The runtimes of the attacks were compared on a Nvidia Geforce GTX1080.\n\n\\begin{table}\n    \\centering\n    \\begin{tabular}{lccccccc}\n        \\toprule\n        PGD & SN-PGD & B\\&B & SPSA \\\\\n        \\midrule\n         100\\% \\textsubscript{\\textpm 4.9\\%} & 75\\% \\textsubscript{\\textpm 1\\%}  & 455\\%  \\textsubscript{\\textpm 49\\%}  & 2134\\% \\textsubscript{\\textpm 367\\%}\n    \\end{tabular}\n    \\caption{Mean relative runtime of several attacks compared to standard PGD on all datasets (\\textbf{MNIST}, \\textbf{Fashion-MNIST} and \\textbf{CIFAR10}) for various adversarial attacks.}\n    \\label{tab:runtime}\n    \n\\end{table}\n\n\\subsubsection{5) Approximation of the adversarial direction}To get a better understanding of the effectiveness of SNGD, we inspected if \\ac{SNGD}-based PGD attacks approximate the final direction of a successful adversarial attack more accurately. This was achieved by computing the cosine similarity between subsequent iterations during the attack. Figure \\ref{fig:cosinesim_subsequent} shows that for the correct noise values $\\sigma$ the cosine similarity between subsequent attack iterations increases. Thus, the final adversarial direction is estimated more accurately by the \\ac{SNGD}-based attack. Depending on the characteristics of the loss surface the standard deviation $\\sigma$ of a SNGD-based attack can be adjusted to avoid improper local optima.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth]{Images\/cosine_sub.pdf}\n\\caption{Average cosine similarity between subsequent gradients directions of a SN-PGD attack for varying standard deviations $\\sigma$. For specific $\\sigma$ values the cosine-similarity between subsequent steps increases.}\n\\label{fig:cosinesim_subsequent}\n\\end{figure}\n\n\\subsubsection{6) Loss surface}After observing that SNGD can improve the approximation of the global descent direction we examined if SNGD has a bigger impact on models with non-convex loss surfaces, where the global descent direction is hard to approximate for standard GD. Therefore we calculated an approximate visualization of the loss landscape by calculating the loss value along the direction of a successful adversarial perturbation ($g$) and a random orthogonal direction ($g^\\perp$) originating from a clean sample as exemplified in Figure \\ref{fig:loss_landscape}.\nIn contrast to prior work \\cite{Kurakin2018, Chan20} we found that the loss surface of the adversarial trained models (RFGSM and PGD) is often not increasing most rapidly towards the adversarial direction, which shows the non-convexity of the optimization problem. Furthermore, we noticed that in cases where the loss surface is less convex, the performance difference between PGD and SN-PGD increases. The sub-figures \\textbf{(A)}, \\textbf{(B)}, and \\textbf{(C)} show loss surfaces which are increasingly convex, simultaneously the performance difference between SN-PGD and PGD for these models decreases (\\textbf{(A)}:$4.1\\%$, \\textbf{(B)}: $1.2\\%$, \\textbf{(C)}: $0.1\\%$). Table \\ref{tab:first_results} shows that the difference between SN-PGD and PGD is higher for RFGSM-trained networks, where we generally observed that the loss surface is less convex. Note that this kind of loss surface visualization is limited and provides only an approximation of the true characteristics.\n\n\n\n\\begin{figure}[ht] \n \\centering\n \\begin{subfigure}{0.23\\textwidth}\n      \\includegraphics[width=\\textwidth]{Images\/loss_landscape_fgsm_mnist.pdf}\n      \\caption{FGSM MNIST}\n      \\label{fig:loss_landscape_fgsm_mnist}\n    \\end{subfigure} \n    \\begin{subfigure}{0.23\\textwidth}\n      \\includegraphics[width=\\textwidth]{Images\/loss_landscape_pgd_mnist.pdf}\n      \\caption{PGD MNIST}\n      \\label{fig:loss_landscape_pgd_mnist}\n    \\end{subfigure} \n    \\begin{subfigure}{0.23\\textwidth}\n      \\includegraphics[width=\\textwidth]{Images\/loss_landscape_pgd_cifar.pdf}\n      \\caption{PGD CIFAR10}\n      \\label{fig:loss_landscape_cifar}\n    \\end{subfigure} \n \\caption{Representative loss surfaces around a clean sample $X_i$ (red dot) for a fast FGSM-trained model \\textbf{(A)} and PGD-trained models \\textbf{(B)}, \\textbf{(C)}. We calculate the loss value for sample $x_{i} + \\epsilon_{1} \\cdot \\gamma + \\epsilon_{2} \\cdot \\gamma^\\perp$ where $\\gamma$ is the direction of a successful adversarial attack and $\\gamma^\\perp$ a random orthogonal direction.}\n \\label{fig:loss_landscape}\n\\end{figure}\n\n\\section{Conclusion}\n\nIn this paper, we propose \\acf{SNGD}, an easy to implement modification of gradient descent to improve its convergence for non-convex and noisy loss surfaces. Through our experiments on three different datasets, we demonstrate that this method can be effectively combined with state-of-the-art adversarial attacks to achieve higher success rates. Furthermore, we show that SNGD-based attacks are more query efficient than current state-of-the-art attacks. Although the method proved to be effective in our experiments, larger datasets like ImageNet \\cite{Deng2009} still need to be explored. Additionally, the performance of the attack on alternative defense methods can be tested.  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAfter the observation of GRB~990123 and its afterglow, it was realized\nthat Gamma-Ray Bursts (GRBs) are collimated and not isotropic\nexplosions. GRB~990123 was a most problematic event since it called\nfor more than a solar mass of energy entirely converted into\n$\\gamma$-ray photons (Kulkarni et al. 1999) to explain its fluence\nunder the assumption of isotropic emission. It also had a peculiar\nafterglow, with a steepening of the decay rate approximately one day\nafter explosion (Castro-Tirado et al. 1999). This behavior had been\npredicted to be the signature of a collimated outflow (Rhoads 1997,\n1999). After correcting for beaming the implied energy release was\nreduced to $\\sim5\\times10^{51}$~erg (Bloom, Frail \\& Kulkarni 2003),\nwell below the energy crisis level of $10^{53}$~erg.\n\nGRB jets were initially postulated to be uniform, i.e., with constant\nproperties (energy flux, Lorentz factor, pressure) within a\nwell-defined opening angle $\\theta_j$ and nothing outside. Such jets\nwould produce a steepening afterglow light curve, with the break time\nscaling as $\\theta_j^{8\/3}\\,(E_{\\rm{iso}}\/n)^{1\/3}$ (Sari, Piran \\&\nHalpern 1999), where $E_{\\rm{iso}}$ is the isotropic equivalent energy\nof the outflow and $n$ the density of the ambient medium. Frail et\nal. (2001) and Panaitescu \\& Kumar (2001) discovered a remarkable\nanti-correlation between the jet break time and the isotropic\nequivalent energy release $E_{\\gamma,\\rm{iso}}$ in the prompt\nphase. Such a correlation, initially entirely empirical, can be\naccounted for if it is assumed that the same total energy is given to\nevery GRB but is channeled into jets with different opening angles.  A\nnarrow jet would appear as a very bright GRB followed by an\nearly-breaking afterglow, while a wide jet would produce a weak GRB\nand an afterglow with a late break.\n\nRossi, Lazzati \\& Rees (2002) realized that the relation\n$E_{\\gamma,\\rm{iso}}\\,\\theta_j^2=$constant may reflect a universal\nangular distribution of jet energy rather than a distribution of\nopening angles among different jets (see also Lipunov, Postnov \\&\nProkhorov 2001; Zhang \\& \\mes\\ 2002).  In this case one would write\n\\begin{equation}\n\\frac{dE(\\theta)}{d\\Omega} \\propto \\theta^{-2}.\n\\label{eq:stru}\n\\end{equation}\nRossi et al. also showed that such an energy pattern would produce an\nafterglow that reproduces the Frail et al. correlation and does not\nviolate other observations. Different observed properties would\nreflect different observing geometries rather than different intrinsic\njet properties. Several other jet morphologies have since been\nproposed and studied, specifically general power-law profiles and\nGaussian profiles (Granot \\& Kumar 2003; Salmonson 2003, Rossi et\nal. 2004; Zhang et al. 2004a). Despite this theoretical interest,\nresearch on structured jets has focused on their late-time dynamics\n(Kumar \\& Granot 2003) and observable properties (Perna, Sari \\& Frail\n2003; Nakar, Granot \\& Guetta 2004) and not on the origin of the\nenergy distribution. The only exception is the electromagnetic\nforce-free model, in which a structured jet is predicted naturally\n(Lyutikov, Pariev \\& Blandford 2003).\n\nSome degree of structure in GRB jets is implied by their connection to\nsupernova (SN) explosions (Galama et al. 1998; Stanek et al. 2003;\nHjorth et al. 2003; Malesani et al. 2004). The shear forces and mixing\nof the cold stellar material with the outflowing relativistic plasma\nare expected to create an interface or hollow slower jet around the\nhyper-relativistic flow (MacFadyen \\& Woosley 1999; MacFadyen et\nal. 2001; Proga et al. 2003). The resulting energy distributions of\nthe highly relativistic material are, however, far from a power-law\n(Zhang, Woosley \\& Heger 2004b). If one includes the\ntrans-relativistic outflow energy a power-law profile is obtained, but\nwith the steep index $-3$.\n\nIn this letter we propose a mechanism to create structured jets with\n$dE\/d\\Omega \\propto \\theta^{-2}$ as a result of the time evolution of\nthe opening angle of an hydrodynamic jet. This mechanism relies on the\nnature of the jet propagation through the star but is more robust and\npredictable than the poorly understood shear effects and mixing\ninstabilities. We show that, under a simple ansatz (constant\nluminosity engine) this mechanism yields a beam pattern following the\nprescription of eq.~\\ref{eq:stru} under wide conditions. This letter\nis organized as follows: in \\S~2 we describe the star, cocoon and jet\nconditions at breakout; in \\S~3 we compute the jet evolution during\nthe release of the cocoon material and the resulting jet structure at\nlarge radii. We discuss our results and their implications in \\S~4 and\nwe summarize our findings in \\S~5.\n\n\\newpage\n\n\\section{The jet and its cocoon at breakout}\n\nConsider a high entropy mildly relativistic jet ($\\Gamma_0 \\gtrsim 1$)\ngenerated at some small radius $r_0\\approx10^7-10^8$~cm in the core of\na massive star with an opening angle $\\theta_0$. The jet tries to\naccelerate under the pressure of internal forces. We assume that the\ninitial jet has enough momentum to define a direction of propagation\nand avoid a spherical explosion of the star. Such initial conditions\nare analogous to those adopted in numerical simulations (Zhang et\nal. 2003, 2004b). In the absence of any external material, it would\naccelerate in a broad conical outflow, with its Lorentz factor scaling\nwith radius as $\\Gamma\\propto{}r\/r_0$.\n\nIf the jet is surrounded by dense cold matter, as in the\ncollapsar\/hypernova scenario (Woosley 1993; MacFadyen \\& Woosley\n1999), such acceleration is inhibited. Once the jet reaches supersonic\nspeeds with respect to the stellar material, the ram pressure of the\nshocked gas ahead of the jet drives a reverse shock into the head of\nthe jet, slowing the jet head down to subrelativistic speeds. The\nmixture of shocked jet and shocked stellar material surrounds the\nadvancing jet to form a high pressure cocoon, analogous to the cocoon\nformed by a low-density jet from a radio galaxy advancing into the\nintergalactic medium (Scheuer 1974).  As a result the jet dynamics is\ngoverned by three competing effects. First, the internal pressure\naccelerates the jet material, whose Lorentz factor scales as\n$\\Gamma\\propto(\\Sigma_j\/\\Sigma_0)^{1\/2}$ if the acceleration is\nisentropic, where $\\Sigma_j$ is the jet cross section of a single jet\nand $\\Sigma_0$ its value at $r_0$ (see, e.g., Beloborodov 2003,\n\\S~3.1). If the jet suffers internal dissipation as it propagates,\ne.g., from entrainment of cocoon material or internal shocks, then the\nincrease of $\\Gamma$ with $\\Sigma_j$ is slower. This can be caused,\ne.g., by shear instabilities at the jet-cocoon boundary.  Second, the\nhead of the jet is slowed down by its interaction with the massive\nstar. The larger the jet cross-section, the slower its head\npropagates. Both shocked jet and stellar material flow to the sides\nfeeding the cocoon. Third, the cocoon pressure acts as a collimating\nforce on the jet, which therefore becomes narrower and more\npenetrating, albeit less relativistic.\n\nMatzner (2003) developed a simple analytic treatment to follow this\npropagation.  Using the Kompaneets approximation (Kompaneets 1960) to\ndescribe the cocoon expansion, he was able to compute the jet\npropagation time within the star and the cocoon properties at the\nmoment the jet head reaches the stellar surface --- the breakout\ntime, $t_{\\rm br}$. For our purposes we need merely to compare the\nwidth of the cocoon at breakout with the width of the jet.\n\nConsider a jet with luminosity $L_j$ that reaches the stellar surface\nat $t_{\\rm{br}}$. The energy stored in the cocoon (neglecting\nadiabatic losses) is $E_c = L_j\\,(t_{\\rm{br}}-{r_\\star}\/{c})$, where\n$r_\\star$ is the radius of the star.  About half of the cocoon energy\nis transferred to the nonrelativistic shocked stellar plasma via the\ncocoon shock, with the other half remaining in relativistic, shocked\njet material. If we adopt a relativistic equation of state for the\nentire cocoon, we can write the cocoon pressure as\n\\begin{equation}\np_c \\approx \\frac{E_c}{3\\,V_c}\n\\label{eq:pc}\n\\end{equation}\nwhere $V_c\\approx{}r_\\star\\,r_\\perp^2$ is the volume and\n$r_\\perp=v_{\\rm{sh}}\\,t_{\\rm{br}}$ the transverse radius of the cocoon\n(we assume that the jet occupies a negligible fraction of the cocoon\nvolume.) The velocity of the shock $v_{\\rm{sh}}$ driven by the cocoon\npressure into the star can be computed by balancing the cocoon\npressure against the ram pressure exerted by the stellar material\n(which has negligible internal pressure):\n$v_{\\rm{sh}}=\\sqrt{p_c\/\\rho_\\star}$, where $\\rho_\\star$ is the matter\ndensity of the star. This set of relations can be used with\neq.~\\ref{eq:pc} to obtain an equation for the cocoon pressure that\nreads (see also Begelman and Cioffi 1986 for a similar treatment in\nthe context of extragalactic radio sources):\n\\begin{equation}\np_c=\\left[\\frac{L_j\\rho_\\star}\n{3\\,r_\\star\\,t_{\\rm{br}}^2}\\left(t_{\\rm{br}}-\\frac{r_\\star}{c}\\right)\n\\right]^{1\/2}\n\\simeq\\left(\\frac{L_j\\,\\rho_\\star}{3\\,r_\\star\\,t_{\\rm{br}}}\\right)^{1\/2} .\n\\label{eq:pc2}\n\\end{equation}\nWe assume that the head of the jet propagates subrelativistically\nthrough the star and write $t_{\\rm br} \\equiv \\eta r_\\star\/c$, where\n$\\eta >1$.  Then, adopting the notation $Q=10^x\\,Q_x$ and using cgs\nunits throughout, we have\n\\begin{equation}\np_c\\simeq 3\\times10^{19} \\eta^{-1\/2} \\left(\n\\frac{L_{j,51}\\,\\rho_{\\star}}{r_{\\star,11}^2} \\right)^{1\/2} .\n\\end{equation}\nThe opening angle of the cocoon is given by \n\\begin{equation}\n\\theta_c\\simeq\\frac{r_\\perp}{r_\\star}\\simeq\n\\frac{\\sqrt{p_c\/\\rho_\\star}\\,t_{\\rm{br}}}{r_\\star}\n\\simeq 0.2 \\eta^{3\/4} \n\\left(\\frac{L_{j,51}}{r_{\\star,11}^2\\,\\rho_\\star}\\right)^{1\/4} ,\n\\label{eq:thc}\n\\end{equation}\ni.e., of the order of tens of degrees.\n\nNow let us consider the properties of the jet as it reaches the\nbreakout radius. The jet pressure is given by $p_j = L_j\/(4 c \\Sigma_j\n\\Gamma_j^2)$, where $\\Gamma_j$ is the Lorentz factor of the jet\nmaterial before it is shocked at the jet head. Setting $p_j = p_c$\n(Begelman \\& Cioffi 1989; Kaiser \\& Alexnder 1997) with $\\Sigma_j =\n\\pi r_\\star^2 \\theta_j^2$, we find\n\\begin{equation}\n\\theta_{j,{\\rm br}} \\simeq 0.1 \\eta^{1\/4} \\left(\\frac{L_{j,51}}\n{r_{\\star,11}^2 \\rho_{\\star}}\\right)^{1\/4} \\Gamma_{j,{\\rm br}}^{-1} .\n\\label{eq:thj}\n\\end{equation}\nFor isentropic flow between $r_0$ and $r_\\star$ the confined jet\nreaches a Lorentz factor\n\\begin{equation}\n\\Gamma_{j,{\\rm br}}\\simeq 14 \\eta^{1\/8}\n\\Gamma_0\\left(\\frac{r_{\\star,11}}{r_{0,8}} \\right)^{1\/2}\n\\left(\\frac{\\theta_0}{30^\\circ} \\right)^{-1\/2}\n\\left(\\frac{L_{j,51}}{r_{\\star,11}^2 \\rho_{\\star}}\\right)^{1\/8} ,\n\\end{equation}\nwith somewhat lower values if the jet is dissipative. The\ncorresponding jet opening angle at breakout is $\\la 1^\\circ$; however,\nthe fact that $\\theta_{j,{\\rm br}}\\Gamma_{j,{\\rm br}} < 1$ suggests\nthat the jet should freely expand to a few times $\\theta_{j,{\\rm br}}$\nafter exiting the star. Even so, the jet at breakout is expected to be\nmore than an order of magnitude narrower than the cocoon. A more\ndetailed treatment of the evolution of the cocoon and its interaction\nwith the jet can be obtained by numerical integration of the set of\ndifferential equations that govern the flow. This allows us to take\ninto account the density gradient of the star and the shape of the\ncocoon. Results (Lazzati et al. 2005) do not differ substantially from\nthese analytic estimates.\n\nThese results may be altered if substantial dissipation due to\nrecollimation shocks and\/or shear instabilities takes place. We do not\nattempt to model these effects here. Recollimation shocks are however\nrelatively weak, being mostly oblique to the flow. As a matter of\nfact, these results are in qualitative agreement with results from\nnumerical simulations (Zhang, Woosley \\& MacFadyen 2003). Simulations\nshow, indeed, that isentropic conditions are not respected throughout\nthe propagation, since collimation shocks take place. These shocks\neffectively create a new nozzle at a larger radius $r>r_0$ which\nmodifies the scaling of the Lorentz factor with the jet cross\nsection. What is of relevance here is that no matter how wide the jet\nis initially (and it is likely to be poorly collimated, especially if\ninitial collimation is provided by the accretion disk), the jet that\nemerges from the star is narrow and highly collimated by the cocoon\npressure.  This result is observed in all numerical simulations of\njets in collapsars (MacFadyen \\& Woosley 1999; MacFadyen, Woosley \\&\nHeger 2001; Zhang et al. 2003). The cocoon, on the other hand, covers\na wide solid angle, several tens of degrees across.  In the next\nsection, we explore what happens after jet breakout.\n\n \n\\section{Jet breakout}\n\n\\begin{figure}\n\\psfig{file=unijet_f1.ps,width=\\columnwidth}\n\\caption{{Energy distribution for the afterglow\nphase for three instantaneous beam patterns (see inset). In all three\ncases a well defined $dE\/d\\Omega\\propto\\theta^{-2}$ section is clearly\nvisible. Only the edge of the jet and its core show marginal\ndifferences. The results shown are for a jet\/star with\n$L=10^{51}$~erg~s$^{-1}$, $T_{\\rm{GRB}}=40$~s and\n$r_\\star=10^{11}$~cm. Inset: Instantaneous beam patterns that reach\nthe surface of the star. The solid line shows a uniform jet, dashed\nline shows a Gaussian energy distribution, while the dash-dotted line\nshows an edge brightened (or hollow) jet.}\n\\label{fig:pat}}\n\\end{figure}\n\nWe now consider the jet development after the breakout, with the inner\nengine still active. At this stage, which has not been investigated in\nnumerical simulations so far, it is likely that dissipation will have\na lesser role, since the jet, as we shall see, is de-collimated rather\nthan recollimated.\n\nAs the jet reaches the stellar surface, it clears a channel for the\ncocoon. The cocoon material is therefore now free to expand out of the\nstar and its pressure drops. We assume that from this moment on the\nshock between the cocoon and the cold stellar material stalls, as a\nconsequence of the dropping cocoon pressure. This is equivalent to\nassuming a constant volume of the cocoon cavity inside the star. The\npressure drop for the relativistic cocoon can be derived through\n$dE_c=-\\epsilon_c\\,\\Sigma_c\\,c_s\\,dt$ where $\\Sigma_c$ is the area of\nthe free surface through which the cocoon material expands,\n$\\epsilon_c$ the cocoon energy density and $c_s=c\/\\sqrt{3}$ is the\nsound speed of the relativistic gas.  Writing the cocoon volume as\n$V_c\\sim\\Sigma_c\\,r_\\star$, we can obtain the pressure evolution:\n\\begin{equation}\np_c=p_{c,{\\rm br}}\\,\\exp\\left({-\\frac{ct}{\\sqrt{3}\\,r_\\star}}\\right)\n\\end{equation}\nwhere $p_{c,{\\rm br}}$ is the cocoon pressure at the moment of shock\nbreakout.\n\nAs the cocoon pressure decreases, fresh jet material passing through\nthe cocoon is less tightly collimated. Under isentropic conditions the\njet Lorentz factor increases linearly with the opening angle and\npressure balance yields $\\theta_j \\propto p_c^{-1\/4}$, implying an\nexponentially increasing opening angle of the form\\footnote{Note that,\nif the jet is causally connected at breakout as suggested by\neq.~\\ref{eq:thj}, the jet would freely expand to an angle\n$\\theta_j=1\/\\Gamma_{j,{\\rm{br}}}>\\theta_{j,{\\rm{br}}}$ of the order of\na few degrees. This would result in an initially constant opening\nangle. The only effect on the final energy distribution of\neq.~\\ref{eq:wow} is to increase the size of the jet core from\n$\\theta_{j,{\\rm{br}}}$ to $1\/\\Gamma_{j,{\\rm{br}}}$.}\n\\begin{equation}\n\\theta_j=\\theta_{j, {\\rm br}}\\,\\exp\\left[\\frac{c\\,t}{4\\sqrt{3}\\,r_\\star}\n\\right] .\n\\label{eq:thopen}\n\\end{equation}\nDissipative jet propagation gives similar results (with merely a\ndifferent numerical coefficient $\\sim O(1)$ inside the exponential),\nprovided that $\\Gamma$ varies roughly as a power of $\\Sigma_j$.  The\nangular distribution of integrated energy, as observed in the\nafterglow phase, is computed by integrating the instantaneous\nluminosity per unit solid angle from the moment the jet becomes\nvisible along a given line of sight ($t_{\\rm{l.o.s.}}$) until the end\nof the burst:\n\\begin{equation}\n\\frac{dE}{d\\Omega}=\\int_{t_{\\rm{l.o.s.}}}^{T_{\\rm{GRB}}}\n\\frac{dL}{d\\Omega}\\,dt \\simeq\\int_{t_{\\rm{l.o.s.}}}^{T_{\\rm{GRB}}}\n\\frac{L(t)}{\\pi\\,\\theta_j^2(t)}\\,dt .\n\\label{eq:struj0}\n\\end{equation}\n$t_{\\rm{l.o.s.}}$ is obtained by inverting eq.~\\ref{eq:thopen}. Such\nintegration is valid provided that the jet opening angle at time\n$T_{\\rm{GRB}}$ is smaller than the natural opening angle of the jet:\n$T_{\\rm{GRB}}<4\\sqrt{3}(r_\\star\/c)\\log(\\theta_0\/\\theta_{\\rm{br}})$. For\nthe fiducial numbers assumed ($r_\\star=10^{11}$~cm,\n$\\theta_{j,\\rm{br}}=1\\degr$ and $\\theta_0=30\\degr$) this corresponds to\n$\\sim100$~s comoving burst duration. Interesting effects should be\nexpected for longer bursts, as discussed in \\S~4. Assuming a jet with\nconstant luminosity $L$ and for all the lines of sight that satisfy\n$t_{\\rm br} < t_{\\rm{l.o.s.}} \\ll T_{\\rm GRB}$, eq.~\\ref{eq:struj0}\ngives the jet structure\n\\begin{equation}\n\\frac{dE}{d\\Omega} = \\frac{2\\sqrt{3}\\,L\\,r_\\star}\n{\\pi\\,c}\\,\\theta^{-2}  \\qquad\\theta_{j,{\\rm{br}}}\\le\\theta\\le\\theta_0\n\\label{eq:wow}\n\\end{equation}\nand $dE\/d\\Omega\\sim$constant inside the core radius\n$\\theta_{j,{\\rm{br}}}$.  This angular dependence, which characterizes\na ``structured jet'' or ``universal jet'' (Rossi et al. 2002, 2004;\nZhang \\& Meszaros 2002; Salmonson 2003; Lamb, Donaghy \\& Graziani\n2003), is of high theoretical interest.  Jets with this beam pattern\nreproduce afterglow observations. If the jet is powered by fall-back\nof material from the star to the accretion disk, the mass accretion\nrate would be anti-correlated with the radius of the star, for a given\nstellar mass. This would produce a roughly constant value of\n$L_j{}r_\\star$ that would reproduce the so-called Frail relation\n(Frail et al. 2001; Panaitescu \\& Kumar 2001) as a purely\nviewing-angle phenomenon.  In this case all jets would be alike, but\ndifferent observers would see them from different angles, deriving\ndifferent energetics.\n\nIn the above equations we have assumed for simplicity that the jet\nreaching the surface of the star is uniform. As shown by simulations\n(e.g. Zhang et al. 2003), it is more likely that a Gaussian jet\nemerges from the star. On the contrary, boundary layers may be\nproduced by the interaction of the jet with the collimating star,\nresulting in edge brightened jets (see the inset of\nFig.~\\ref{fig:pat}).  It can be shown easily that the $\\theta^{-2}$\npattern does not depend on the assumption of\nuniformity. Fig.~\\ref{fig:pat} shows the integrated energy\ndistribution for uniform, Gaussian and hollow intrinsic jets. Even\nthough small differences are present at the edges (the jet core and\nthe outskirts), the general behavior is always\n$dE\/d\\Omega\\propto\\theta^{-2}$.\n\n\\section{Discussion}\n\nOur prediction that the jet opening angle evolves with time has two\nmajor, in principle testable, consequences. First, the brightness of a\ntypical GRB should tend to decrease with time, assuming a constant\nefficiency of gamma-ray production, since the photon flux at earth is\nproportional to the energy per unit solid angle. This behavior is\ncertainly seen in FRED (Fast Rise Exponential Decay) single pulsed\nevents, while it is harder to make a quantitative comparison in the\ncase of complex light curves. Some events, like the bright GRB~990123,\nhave multi-peaked lightcurves which show a clear fading at late\ntimes. On the other hand there exist rare events, such as GRB~980923,\nin which an increase of luminosity with time is observed. Lacking\nafterglow observations, it is hard to determine whether such events\nare peculiar.\n\nTo understand the second consequence of the model, consider an\nobserver at an angle $\\theta_{\\rm{obs}}>\\theta_{\\rm{br}}$ where\n$\\theta_{\\rm{br}}$ is the jet opening angle at breakout. Initially,\nthis observer lies outside the beaming cone and does not detect the\nGRB. Later, as the jet spreads beyond $\\theta_{\\rm{obs}}$, the\nobserver detects the burst. The beginning of the GRB emission is\ntherefore observer dependent. This causes a correlation between the\nburst duration and energetics: the longer the burst the larger its\ndetected energy output. A correlation between burst duration and\nfluence is detected in the BATSE sample but whether it is intrinsic or\ndue to systematic effects is still a matter of debate. Finally, if any\nisotropic emission should mark the jet breakout (MacFadyen \\& Woosley\n1999; Ramirez-Ruiz, MacFadyen \\& Lazzati 2002), such emission would not\nbe followed immediately by $\\gamma$-rays for observers lying outside\nof $\\theta_{\\rm{br}}$. This delay may explain the unusually long\ndelays between precursors and main emission detected in several BATSE\nGRBs (Lazzati 2005).\n\nSome complications can arise for very compact stars and\/or very\nlong-lived jets. In computing the energy profile (eq.~\\ref{eq:wow}) we\nhave implicitly assumed that the jet expands exponentially until it\ndies. This approximation obviously fails if the engine is long-lived,\nsince at some point the jet reaches its natural opening angle --\ndetermined, e.g., by the accretion disk geometry -- and the expansion\nstops. Alternatively for some combination of the parameters, the\ncocoon may occupy a small opening angle (eq.~\\ref{eq:thc}) . As the\njet expands, it eventually hits the cold star material and a different\nexpansion law, driven by the jet pressure onto the stellar material,\nsets in. Again, the expansion of the jet is slowed down and as a\nconsequence a shallower distribution of energy with off-axis angle is\nexpected. Such modifications to the $\\theta^{-2}$ law would manifest\nthemselves in the late stages of the afterglow evolution. The higher\nthan expected energy at large angles would flatten the light curve\ndecay or produce a bump in the afterglow. Such behavior has been\nclaimed in several radio afterglows (Frail et al. 2004; Panaitescu \\&\nKumar 2004), and could also explain the two breaks observed in the\nlight curve of GRB~030329\\footnote{Note that a simple two-component\njet cannot explain the fast variability observed in the optical\nlightcurve. It can only reproduce the overall behavior of the optical\nand radio lightcurves.}. If the outflow reaches its natural opening\nangle or crashes into the cold stellar material while still active,\nthe time-integrated jet can be described as the sum of a $\\theta^{-2}$\njet plus a roughly uniform jet with a larger opening angle --- in\nother words, a two-component jet (Berger et al. 2003). This behavior\nis tantalizing also if we consider the interaction from the point of\nview of the star. The expansion of the cocoon drives a shock wave into\nthe cold stellar material where it deposits approximately\n$10^{51}$~erg (Zhang et al. 2003; Lazzati \\& Begelman in\npreparation). If the cocoon is small and the jet comes into contact\nwith the star while still active and expanding, additional energy is\ngiven to the star. Complex afterglows should therefore be associated\nwith particularly energetic hypernov\\ae, such as SN2003dh.\n\n\\section{Summary and conclusions}\n\nIn this Letter we have analyzed the evolution of a GRB jet during its\ntransition from stellar confinement to free expansion. We focused\nmainly on the resulting angular structure of the outflow, and reached\ntwo main conclusions. First, the angular structure of the jet in the\nafterglow phase does not necessarily reflect the way in which the\nenergy is released by the inner engine. If the jet opening angle is\nnot constant during the $\\sim100$~s of the engine activity, the\nangular structure of the jet results from the integration of this\nevolution and has little to do with the pristine beam pattern.\nSecond, we analyzed the most probable evolution in the collapsar\nscenario. We find that the interaction of the jet with the surrounding\ncocoon causes the opening angle to increase exponentially with\ntime. This evolution, for a constant energy release from the inner\nengine and independently of the intrinsic beam pattern, produces an\nangular structure with $dE\/d\\Omega\\propto\\theta^{-2}$.  Such an energy\ndistribution produces a broken power-law afterglow (Rossi et al. 2002,\n2004; Salmonson 2003, Kumar \\& Granot 2003; Granot \\& Kumar 2003). It\ncan also explain the so-called Frail relation (Frail et al. 2001;\nPanaitescu \\& Kumar 2001) if $L_j r_\\star\/c$ is roughly constant for\ndifferent GRBs. The origin of this energy distribution is here\nexplained for the first time in the context of hydrodynamical\nfireballs (see Lyutikov et al. 2003 for a discussion on the context of\nelectromagnetically dominated fireballs).  We briefly discuss in \\S~4\nsome testable prediction of this model.\n\n\\bigskip\n\nWe thank Miguel Aloy, Gabriele Ghisellini and Andrew MacFadyen for\nuseful discussions. This work was supported in part by NSF grant\nAST-0307502 and NASA Astrophysical Theory Grant NAG5-12035.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nLondon Interbank Offered Rate (LIBOR) was established in 1986 by\nthe British Banking Association (BBA), who defines LIBOR as ``the\nrate at which an individual Contributor Panel bank could borrow\nfunds, were it to do so by asking for and then accepting\ninter-bank offers in reasonable market size, just prior to 11:00\n[a.m.] London time''. Every London business day each bank within\nthe Contributor Panel (selected banks from BBA) makes a blind\nsubmission (each banker does not know what the quotes of the other\nBanks are) and a compiler (Thomson Reuters) averages the second\nand third quartiles. In other words, LIBOR is the trimmed average\nof the expected borrowing rates of leading banks. LIBOR rates are\npublished for several maturities and currencies.\n\nOver the time LIBOR became a fundamental  interest rate with three\nmain characteristics: (i) it was viewed as an (intended) measure\nof the borrowing cost in the interbank market, (ii) before the\nfinancial crisis, it was interpreted as a risk free rate and (iii)\nit is a signal of global credit market conditions. Libor is\nenormously influential due to its use for the valuation of\nfinancial products worth trillions of dollars (\\cite{BISstat})\n\nThe way in which LIBOR is fixed is peculiar, because it does not\narise from actual transactions. It is not the result of the competing forces of supply and demand. There is a panel of banks selected\nby the BBA. Each of them should submit their best estimate\naccording to the following question: ``At what rate could you\nborrow funds, were you to do so by asking for and then accepting\ninter-bank offers in a reasonable market size just prior to 11\nam?'' (\\cite{BBATrent}). At some point, individual bank LIBOR\nsubmissions are often regarded as a proxy for the financial health\nof the submitting entity. Usually, an employee or group of\nemployees responsible for cash management in a bank are in charge\nof  the daily submission to BBA. They should base their submission\non the money market conditions for the bank, and should not be\ninfluenced by other bank divisions such as the derivatives trading\ndesks. A fair Libor could signals the state of the interbank money market, and the central banks could act to alleviate frictions in it.\n\nUntil May 29, 2008 LIBOR was presumed a pretty honest estimation\nof the borrowing costs of prime banks. On that day,\n\\cite{MollenkampWhitehouse} published an article on the Wall\nStreet Journal casting doubts on the transparency of LIBOR's\nsetting, implying that published rates were lower than those\nimplied by credit default swaps (CDS). Investigations conducted by\nseveral market authorities such as US Department of Justice, the\nEuropean Commission, and the Financial Services Authority\n(FSA)\\footnote{It is noteworthy that the Financial Services Act\n2012 renamed FSA as Financial Conduct Authority (FCA), raising the\nimportance of ``fair conduct'' in financial markets.} detected\ndata manipulation and imposed severe fines to banks involved in\nsuch illegal procedure.\n\nSeveral leading banks applied for leniency. Jurists use to say\n\\textsl{``confessio est probatio probatissima''}, i.e. confession\nis the best proof. Therefore, we can accept that, at least, there\nwas some kind of unfair individual submissions or even worse, a\ncollusion attempt by a cartel of banks. This manipulation had two\nmain objectives. On the one hand, low submissions were intended to\ngive the market a signal of the  bank's own good financial health.\nIf a bank steadily submits greater rates, this could indicate\nproblems in raising money, generating concerns regarding  a\nunderlying solvency problem. On the other hand, some low\nsubmissions could be aimed to earn money from certain portfolio\npositions, whose assets are valued according to LIBOR.\n\nThe effect of erroneous LIBOR extends beyond the financial\nmarkets. In addition to provide a biased interbank lending cost,\n\\cite{Stenfors} affirms that  it corrupts a  ``key variable in the\nfirst stage of the monetary transmission mechanism''.\n\nThe importance of a good pricing system is based  on its\nusefulness for making decisions. As Hayek \\cite{Hayek45} affirmed ``we\nmust look at the price system as such a mechanism for\ncommunicating information if we want to understand its real\nfunction''. If the price system is contaminated, but perceived as\npure, the effect could reach also the real economy, making it\ndifficult to find a way out the financial crisis.\n\nThis rate-rigging scandal made economists to examine  the\nevolution of LIBOR rates and compare it with other market rates.\n\\cite{TaylorWilliams2009} documented the decoupling of the LIBOR\nrate from other market rates such as the Overnight Interest Swap\n(OIS), Effective Federal Fund (EFF),  Certificate of Deposits\n(CDs), Credit Default Swaps (CDS), and Repo rates. They\nhypothesize that the reasons for the divergent behavior were due\nto expectations of future interest rates and in the accompanying\ncounterpart risk. \\cite{SniderYoule} study individual quotes in\nthe LIBOR bank panel and corroborate the claim by\n\\cite{MollenkampWhitehouse} that LIBOR quotes in the US are not\nstrongly related to other bank borrowing cost proxies. In their\nmodel, the incentive for misreporting borrowing costs is profiting\nfrom a portfolio position. Consequently, the misreporting could\npoint upwards in one currency and downwards in another one,\ndepending on the portfolio exposition. The evidence of such\nbehavior is detected with the formation of a compact cluster of\nthe different panel bank quotes around a given point.\n\\cite{AbrantesMetz2011} track daily LIBOR rates over the period\n1987 to 2008.\n\n\\vskip 3mm\n\nIn particular, this paper analyzes the empirical distribution of\nthe Second Digits (SDs) of the Libor interest rate, and compares\nthem with the uniform and Benford's distributions. Taking into\naccount the whole period, the null hypothesis that the empirical\ndistribution follows either the uniform or the Benford's\ndistribution cannot be rejected. However, if only the period after\nthe sub-prime crisis is taken into account, the null hypothesis is\nrejected. This result puts into question the ``aseptic'' setting\nof LIBOR. In a recent paper  Bariviera \\textit{et al.} \\cite{Bariviera2015} the authors\nuncover strange changes in the information endowment of LIBOR time\nseries, as measured by two information theory quantifiers, namely\npermutation entropy and permutation statistical complexity. Their\nresults allow to infer some degree of manipulation or, at least,\nchanges in the underlying stochastic process that governs interest\nrate's time series.\n\nAntitrust law enforcement is complex,  because manipulation and fraud can be elegantly camouflaged. An\nstatistical procedure could  hardly be accepted as legal proof in\na court of law. However, its use by surveillance authorities makes\nthe attempted manipulation more costly and more difficult to\nbe maintained. Consequently, we view our proposal as a market watch\nmechanism that could make manipulation and\/or collusion attempts\nmore difficult in the future. Additionally, an efficient\noverseeing mechanism could increase the incentives to apply for\nleniency at earlier stages of the manipulation\n(\\cite{AbrantesSokol2012}.)\n\n\nThe aim of this paper is to show that a forecasting method based\non Maximum Entropy Principle (MaxEnt) is very useful not only to\nproduce accurate forecasts, but also to detect some anomalous\nsituations in time series. In particular, we claim that, in\nabsence of data manipulation, forecast accuracy should be\napproximately the same at all times under examination. On the\ncontrary, manipulation would produce more  predictable\nconsequences, increasing the predictive-power of our model, that\nwe apply here to LIBOR and other UK interest rates.\n\n\nThis paper is  organized as follows. Section \\ref{sec:ME}\ndescribes our methodology based on the Maximum Entropy method.\nSection \\ref{sec:data} describes the data used in the paper and\ndeals with the results obtained with the proposed methodology.\nFinally, section \\ref{sec:conclusions} draws the main conclusions.\n\n\n\n\\section{MaxEnt approach for predictions in time-series}\\label{sec:ME}\n\nIn a recent paper, Mart\\'in \\textit{et al. }\\cite{Martin2014} developed an information\ntheory based method  for time series prediction. Given its\noutstanding results in approaching the true dynamics underlying a\ngiven time series, we believe that it is a suitable method to\napply here. In order to make the paper self-contained, we review\nbelow the description of the method, taken from \\cite{Martin2014}.\n\n\\noindent The  behavior of a dynamical  system can be registrated as\na time series i.e.  a sequence of measurements\n$\\{ v(t_n), n=1,\n\\ldots , N\\}$ of an observable of the system\nat discrete times $t_n$, where\n$N$ is the length of the time series.\n\n\n\n\\noindent The Takens theorem of 1981 asserts that, for $T \\in\n\\mathbb{R}$, $T>0$, there exists a functional form of the type,\n\\begin{equation} \\label{mapping}\nv(t+T)=F({\\bf v}(t)),\n\\end{equation}\nwhere\n\\begin{equation} \\label{vector v}\n{\\bf v}(t)=[v_1(t),v_2(t),\\ldots,v_d(t)],\n\\end{equation}\nand  $v_i(t)=v(t-(i-1) \\Delta)$, for $i=1,\\ldots,d$. $\\Delta$ is\nthe time lag and $d$ is the embedding dimension of the\nreconstruction.  $T$ represents the \\emph{anticipation time} and\nit is of fundamental importance for a prediction model.\n\n\\noindent We will consider (as in [\\cite{Martin2014}) and\nreferences therein] a particular representation for the mapping\nfunction of Eq. (\\ref{mapping}), expressing it, using Einstein's\nsummation notation,  as an expansion of the form\n\n\\begin{eqnarray}\nF^*({\\bf v}(t))=&a_{0}+a_{i_1} v_{i_1}(t)~ +a_{i_1 i_2}v_{i_1}(t)~\nv_{i_2}(t)~ + ~a_{i_1 i_2 i_3}v_{i_1}(t)~ v_{i_2}(t)~v_{i_3}(t)~ +\n\\ldots  \\label{Fasterisco}\n  \\\\ \\nonumber & +a_{i_1 i_2... i_{n_p}}v_{i_1}(t)~\nv_{i_2}(t)~\\ldots v_{i_{n_p}}(t)~,\n\\end{eqnarray}\nwhere $1 \\le i_k \\le d $ with $1 \\le k \\le n_p$  and  $n_p$ being\nan  adequately chosen polynomial degree so as to to series-expand\nthe mapping $F^*$. The number of parameters in\nEq.(\\ref{Fasterisco})  corresponding to  the terms of  degree  $k$\ndepends on the embedding dimension and  can be calculated using\ncombination with repetitions,\n\\begin{equation} \\label{variaciok}\n\\left(\n\\begin{array}{c}\n  d \\\\\n  k\\\\\n\\end{array\n\\right)^* = \\frac{(d+k-1)!}{k! (d-1)!}\n\\end{equation}\n\n\n\n\\noindent Accordingly, the length of the vector of parameters\n$\\textbf{a}$, $N_c$, is\n\\begin{equation} \\label{Nc}\nN_c=\n \\sum_{k=1}^{n_p}\\left\n\\begin{array}{c}\n  d \\\\\n  k\\\\\n\\end{array\n\\right)^*\n\\end{equation}\n\n\n\n\n\\noindent The computations are  made on the basis of a specific\ninformation supply,  given by $M$ points of the series\n\\begin{equation}\\label{Ecuacion5}\n\\{{\\bf v}(t_n),v(t_n+T)\\},~~~n=1,\\ldots,M.\n\\end{equation}\n\n\n\n\\noindent Given the data set in Eq. (\\ref{Ecuacion5}), the\nparametric mapping  in Eq. (\\ref{Fasterisco}) will be determined\nby  the following condition,\n\n\\begin{equation}\\label{Ecuacion7}\nF^*({\\bf v}(t_n))=v(t_n+T)~~~~~~n=1,\\ldots,M ,\n\\end{equation}\nwhich can be expressed in matrix form as,\n\\begin{equation}\\label{Ecuacion12}\n W   \\textbf{a} =\\textbf{v}_T,\n\\end{equation}\nwhere   $W$ is a matrix of size $ M \\times N_c$, whose $n$-th row\nis \\\\ $[1,v_{i_1}(t_n), v_{i_1}(t_n) v_{i_2}(t_n),\\ldots ,\nv_{i_1}(t_n) v_{i_2}(t_n) \\ldots v_{i_{n_p}}(t_n)] $ (Cf.\nEq.(\\ref{Fasterisco})) and \\\\ $(\\textbf{v}_T)_n=v(t_n+T)$, for\n$n=1,\\ldots,M$.   Shannon's entropy, defined for a discrete random\nvariable, can be extended to situations for which the random\nvariable under consideration is continuous.\n\n\\noindent In order to infer coefficients which are consistent with\nthe data we shall assume that each set $\\textbf{a}$ is realized\nwith probability $P(\\textbf{a})$. Thus,\n a normalized probability distribution over the possible sets $\\textbf{a}$ is introduced,\n\\begin{equation}\\label{EcuacionP}\n \\int_{I} P(\\textbf{a}) \\  d\\textbf{a} =1 ,\n\\end{equation}\nwhere $d\\textbf{a}=da_1da_2 \\cdots da_ {N_c}$ and $N_c$ is the\nnumber of parameters of the model.\n\n\n\n\\noindent The problem then becomes that of  finding\n$P(\\textbf{a})$ subject to the requirement that the associated\nentropy $H$  be maximized, since this is the best way of avoiding\nany bias. The expectation value of $\\textbf{a}$, is defined by\n\\begin{equation}\\label{EcuacionEa}\n \\left\\langle \\textbf{a} \\right\\rangle = \\int_{I}{P(\\textbf{a}) \\textbf{a}\\  d\\textbf{a} }.\n\\end{equation}\n\\noindent Consider the  continuous random variable $\\textbf{a}$\nwith probability density function $ p(\\textbf{a})$ on $I$ and $\\,\nI=(-\\infty,\\infty)$. The entropy is given by\n\n\\begin{equation}\\label{EcuacionH}\n\\displaystyle H(\\textbf{a})=-\\int_{I}{P(\\textbf{a})\\ \\ln\\\nP(\\textbf{a})\\ d\\textbf{a}},\n\\end{equation}\nwhenever it exists, and the  relative entropy reads\n\n\\begin{equation}\\label{EcuacionHr}\n\\displaystyle H=-\\int_{I} P(\\textbf{a})\\ \\ln\\ \\frac\n{P(\\textbf{a})} {P_0(\\textbf{a})} \\ d\\textbf{a},\n\\end{equation}\nwhere $P_0(\\textbf{a})$ is an appropriately chosen a priori\ndistribution.\n\n\n\n\\noindent This measure exhibits many of the properties of a\ndiscrete entropy but, unlike the entropy of a discrete random\nvariable, that for a continuous random variable may be infinitely\nlarge, negative, or positive (Ash, 1965 [6]). \\noindent We\ncharacterize, via the entropic maximum principle, various\nprobability distributions, subject to the constraints\nEq.(\\ref{EcuacionP}) and Eq.(\\ref{Ecuacion12}) for the expectation\n$\\left\\langle \\textbf{a} \\right\\rangle$ of $\\textbf{a}$. \\noindent\nThe method for solving this constrained optimization problem is to\nuse Lagrange multipliers for each of the operating constraints and\nmaximize the following functional with respect to $P(\\textbf{a})$,\n\n\\begin{equation}\\label{EcuacionHrmultip}\n J=- \\left[\\int_{I} P(\\textbf{a}) \\ln \\frac {P(\\textbf{a})} {P_0(\\textbf{a})} d\\textbf{a} + \\lambda_0 \\left[\\int_{I} P(\\textbf{a}) d\\textbf{a}-1\\right] +  {\\lambda}^t \\int_{I} \\left[W P(\\textbf{a}) \\textbf{a} - \\textbf{v}_T \\right]  d\\textbf{a} \\right],\n\\end{equation}\nwhere $\\lambda_0$  and $\\lambda$  are Lagrange multipliers\nassociated, respectively, with the normalization condition and\nwith the constraints, Eq.(\\ref{EcuacionP}) and\nEq.(\\ref{Ecuacion12}).\n\n\n\n\\noindent Taking the functional derivative with respect to\n${P(\\textbf{a})}$  we get\n\\begin{equation}\\label{Ecuacion20}\n\\frac{ \\partial J}{\\partial P(\\textbf{a})} = \\ln\n\\left(\\frac{P(\\textbf{a})}{P_0(\\textbf{a})}\\right) +1 + \\lambda_0\n+ {\\lambda}^t  W \\textbf{a} =0,\n\\end{equation}\nwhich implies that the maximum entropy distribution must have the\nform\n\\begin{equation}\\label{Ecuacion25}\n P(\\textbf{a})= \\exp -(1 +  \\lambda_0 )  \\exp(\\lambda^t W \\textbf{a}) P_0(\\textbf{a})\n\\end{equation}\n\n\n\\noindent If the a priori probability distribution\n$P_0(\\textbf{a})$  is chosen to be proportional to $\\exp(\n-\\frac{1}{2} \\textbf{a}^t [\\sigma^2]^{-1} \\textbf{a})$,  where\n$\\sigma^2$ is the covariance matrix, a Gaussian form  for the\nprobability distribution $P(\\textbf{a})$ is obtained, with\n\\begin{equation}\\label{Ecuacion30}\n\\left\\langle \\textbf{a} \\right\\rangle = -\\sigma W^t \\lambda\n\\end{equation}\n\n\\noindent Considering  Eq.(\\ref{Ecuacion12}), the Lagrange\nmultipliers $\\lambda$ can be eliminated:\n\n\n\\begin{equation}\\label{Ecuacion40}\n\\lambda= -\\sigma^{-1} (W W^t)^{-1} \\textbf{v}_T,\n\\end{equation}\nand one can thus write\n\n\\begin{equation}\\label{Ecuacion45}\n\\left\\langle \\textbf{a} \\right\\rangle = W^t  (W W^t)^{-1}\n\\textbf{v}_T .\n\\end{equation}\n\\noindent The matrix $ W^t  (W W^t)^{-1}$ is  known as the\nMoore-Penrose pseudo-inverse of the matrix $W$ (see\n\\cite{Martin2014} and references therein). Consequently,\n this result shows that the  maximum entropy principle  coincides with a least square criterion.\n\\noindent Once the pertinent parameter vector  $\\textbf{a}$ is determined, it is\nused to predict  {\\bf new} series' values, $\\widehat v(t_n+T))_{n=1,\\ldots, M_P}$, according to\n\n\n\\begin{equation} \\label{Prediccion}\n(\\widehat v(t_n+T))_{n=1,\\ldots, M_P}= \\widehat{ W} \\textbf{a},\n\\end{equation}\n where $\\widehat W$ is the matrix of size $ M_P \\times N_c $ (see Eq.(\\ref{Ecuacion12})), obtained using  $\\widehat v(t_n)$ values.\n\n\n\n\n\\section{Data and results }\\label{sec:data}\n\nWe analyze the Libor in pound Sterling. The data span is from 01\/01\/1999 until  21\/10\/2008, with a total of 2560 datapoints. All data were retrieved from DataStream.\n\nIn this section we present the results obtained using the  methodology  proposed in Section \\ref{sec:ME}. We consider the embedding dimension $d=4$ and the polynomial degree $n_p=2$. The length of the vector of parameters, according to Eq. \\ref{Nc} is $N_c=15$.\n\nWe fit our model with $M=700$ datapoints, corresponding to approximately two and a half years beginning on  01\/01\/1999. Once the model's parameters were determined, we forecasted the rest of the time series, up to 21\/10\/2008.\n\n\\begin{figure}[!ht]\n\\center\n\\includegraphics[width=16cm,height=10cm]{Figura1_29_mayo.eps}\n\\caption{Original and forecasted time series for different anticipation times}\n\\label{fig:T1}\n\\end{figure}\n\n\n\\begin{figure}[!ht]\n\\center\n\\includegraphics[width=16cm,height=10cm]{Figura2_29_mayo.eps}\n\\caption{Original and forecasted time series for different anticipation times}\n\\label{fig:T2}\n\\end{figure}\n\n\nIn the figures \\ref{fig:T1} and \\ref{fig:T2}  the original time series values and the predicted ones are overlapped (blue and red refer to original and\npredicted values, respectively) for different anticipation time values.  The time-interval between the beginning of the time series and the vertical dashed lines corresponds to the model interval, used to estimate the parameters. The other part corresponds to the out-of-sample forecasts.\n\nIn order to prove the robustness of our proposal we did forecast for different anticipation times (T=\\{7, 10, 13, 16\\} days).\n\n\nWe can observe in figures \\ref{fig:T1} and \\ref{fig:T2} that, as\nexpected, during the model interval period, the original and the\npredicted time series are very close. This is the consequence of\nthe adequate fitting power of the model. As is the case for any\nforecast method, one tries to mimic the behavior of the time\nseries to be estimated. When we move into the (out of the sample)\nprediction interval, we note that during the first months, our\nmethod behaves very well. We expect that, as  economic theory\naffirms, competitive prices should behave randomly\n(\\cite{Samuelson65}). Consequently, if we assume that the time\nseries under study is generated by a memoryless stochastic\nprocess, accurate forecasts are not possible. In spite of the fact\nthat the original time series changes, we can see that the\npredicted time series is rather constant between 2002 and 2007.\nThis is the consequence of the stochastic nature of the original\ntime series. The prediction performance is very poor. In addition,\nthe distance between the original and the predicted series in this\nperiod increases monotonically with the anticipation time, as\nexpected. Surprisingly enough, beginning with 2007, our model\nbegins to fit real data very well. Predicted time series moves\n\\textit{pari passu} with the original one, even during the large\nincreases during 2008. A similar analysis can be done with\nreference to figure \\ref{errorpor}. In that figure, we display the\nrelative mean square error between the original and forecasted\ntime series, year by year.\n\nWhat could make  the same model to change its forecast accuracy in\nsuch  dramatic fashion? According to Wold's theorem\n(\\cite{Wold1954}), a time series can be separated into a\ndeterministic part and an stochastic part. If the stochastic part\ndominates the behavior of the time series, forecast is\nunsuccessful. This is what we can observe between 2002 and the end\nof 2006. On the contrary, beginning in 2007, and until the end of 2008,\nprediction becomes very accurate. Given that the prediction model\nis the same for both periods, we conjecture that the time series\nis dominated by a deterministic process in the last of the two\nperiods. Recalling the literature review of Section\n\\ref{sec:intro}, we can state that this result is an indirect\nproof of LIBOR manipulation. We emphasize that such\n``manipulation'' necessarily comprises the contamination of the\ntime series with a deterministic device, which was detected by the\nMaxEnt model.\n\n\n\\begin{figure}[!ht]\n\\center\n\\includegraphics[scale=.4]{relerror_junio.eps}\n\\caption{Relative mean square errors }\n\\label{errorpor}\n\\end{figure}\n\n\n\\section{Conclusions \\label{sec:conclusions}}\n\nIn this paper we present a novel  prediction method based on the\nMaxEnt principle. Taking into account its previous performance\n(\\cite{Martin2014}), we believe it is suitable for the study of the\n``Libor Case''. We study Libor time series between 1999 until\n2009. Based on the prediction accuracy of our method, we are able\nto detect two distinctive regimes. The first one, extends between\n2002 and the end of 2006. In this period the time series behaves\nas predicted by standard economic theory, reflecting the random\ncharacter of prices in competitive environments. The prediction\npower is, consequently, poor. The second time-period spans  2007 and\n2008. In this period the time series changes its\nregime, moving to a more predictable one. We can safely think that\na deterministic device was introduce into the Libor setting. This\nsituation takes place at the time that what was called by the\nnewspapers as the ``Libor manipulation'' one.  As a consequence,\nour paper is able to detect such manipulation, using exclusively\ndata from Libor time series. We would like to emphasize the\nrelevance of advanced statistical models in market's watch\nmechanisms. Our results could be of interest to surveillance\nauthorities, given the importance of fair market conditions in\nfree market economies.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Large Hadron Collider (LHC) at CERN confirmed the predictions of the Standard Model (SM) of particle physics by discovering the Brout-Englert-Higgs boson~\\cite{Aad:2012tfa,Chatrchyan:2012xdj} in 2012. However, until now, high energy searches did not discover any particles beyond the ones present in the SM. Therefore, great hopes of finding new physics (NP) rest on low energy precision physics where flavor experiments have accumulated intriguing hints for physics beyond the SM within the recent years, most prominently in $b\\to s\\ell^+\\ell^-$ data~\\cite{Aaij:2017vbb,Aaij:2019wad,Aaij:2020nrf}, $b\\to c\\tau\\nu$ transitions~\\cite{Lees:2012xj,Aaij:2017uff,Abdesselam:2019dgh} and the anomalous magnetic moment (AMM) of the muon ($a_\\mu=(g-2)_{\\mu}\/2$)~\\cite{Bennett:2006fi,Mohr:2015ccw,Abi:2021gix}. Interestingly, these hints for NP fall into a common pattern: they can be considered as signs of lepton flavor universality violation (LFUV)~\\footnote{Recently, it has been pointed out that also the Cabibbo Angle Anomaly can be interpreted as a sign of LFUV~\\cite{Coutinho:2019aiy,Crivellin:2020lzu}.}, which is respected by the SM gauge interactions and is only broken by the Higgs Yukawa couplings.\n\nAmong these anomalies, $a_\\mu$, which displays a $4.2\\,\\sigma$ deviation from the SM prediction~\\cite{Aoyama:2020ynm}, is most closely related to Higgs interactions as it is a chirality changing observable. I.e. it involves a chirality flip and therefore a violation of $SU(2)_L$ is required to obtain a non-zero contribution. Furthermore, the required NP effect to explain $a_\\mu$ is of the order of the electroweak (EW) SM contribution and TeV scale solutions need an enhancement mechanism, called chiral enhancement, to be able to account for the deviation (see e.g. Ref.~\\cite{Crivellin:2018qmi} for a recent discussion). Obviously, also $h\\to\\mu^+\\mu^-$ is a chirality changing process and any enhanced effect in $a_\\mu$ should also result in an enhanced effect here~\\footnote{Correlations between $a_\\mu$ and $h\\to\\mu^+\\mu^-$ were considered in the EFT in Ref.~\\cite{Feruglio:2018fxo} and in the context of vector-like leptons (see Ref.~\\cite{Crivellin:2020ebi} for a recent global analysis) in Ref.~\\cite{Kannike:2011ng,Dermisek:2013gta,Dermisek:2014cia,Crivellin:2018qmi}.}. Recently, both ATLAS and CMS measured $h\\to\\mu^+\\mu^-$, finding a signal strength w.r.t. the SM expectation of $1.2\\pm0.6$ \\cite{Aad:2020xfq} and $1.19^{+0.41+0.17}_{-0.39 -0.16}$ \\cite{CMS:2020eni}, respectively. \n\nThe mechanism of chiral enhancement, necessary to explain $a_\\mu$, has been well studied (see Ref.~\\cite{Crivellin:2018qmi} for a recent account). \nHere leptoquarks (LQs) are particularly interesting since they can give rise to an enhancement factor of {$m_t\/m_\\mu \\approx 1700$}~\\cite{Djouadi:1989md, Chakraverty:2001yg,Cheung:2001ip,Bauer:2015knc,Popov:2016fzr,Chen:2016dip,Biggio:2016wyy,Davidson:1993qk,Couture:1995he,Mahanta:2001yc,Queiroz:2014pra,ColuccioLeskow:2016dox,Chen:2017hir,Das:2016vkr,Crivellin:2017zlb,Cai:2017wry,Crivellin:2018qmi,Kowalska:2018ulj,Mandal:2019gff,Dorsner:2019itg,Crivellin:2019dwb,DelleRose:2020qak,Saad:2020ihm,Bigaran:2020jil,Dorsner:2020aaz}, allowing for a TeV scale explanation with perturbative couplings that are not in conflict with direct LHC searches. In fact, there are only two LQs, out of the 10 possible representations~\\cite{Buchmuller:1986zs}, that can yield this enhancement: the scalar LQ $SU(2)_L$ singlet ($S_1$) and the scalar LQ $SU(2)_L$ doublet ($S_2$) with hypercharge $-2\/3$ and $-7\/3$, respectively. In addition, there is the possibility that $S_1$ mixes with the $SU(2)_L$ triplet LQ $S_3$, where $S_1$ only couples to right-handed fermions~\\cite{Dorsner:2019itg}.\n\nFurthermore, LQs are also well motivated by the hints for LFUV in semi-leptonic $B$ decays, both in $b\\to s\\mu^+\\mu^-$~\\cite{Aaij:2017vbb,Aaij:2019wad,Aaij:2020nrf} and $b\\to c\\tau\\nu$ data~\\cite{Lees:2012xj,Aaij:2017uff,Abdesselam:2019dgh}, which deviate from the SM with up to $\\approx 6\\,\\sigma$~\\cite{Alguero:2019ptt,Aebischer:2019mlg,Ciuchini:2019usw,Arbey:2019duh} and $\\approx 3\\,\\sigma$~\\cite{Amhis:2019ckw,Murgui:2019czp,Shi:2019gxi,Blanke:2019qrx,Kumbhakar:2019avh}, respectively. Here possible solutions include again $S_1$~\\cite{Fajfer:2012jt, Deshpande:2012rr, Sakaki:2013bfa, Freytsis:2015qca, Bauer:2015knc, Li:2016vvp, Zhu:2016xdg, Popov:2016fzr, Deshpand:2016cpw, Becirevic:2016oho, Cai:2017wry, Buttazzo:2017ixm, Altmannshofer:2017poe, Kamali:2018fhr, Azatov:2018knx, Wei:2018vmk, Angelescu:2018tyl, Kim:2018oih, Crivellin:2019qnh, Yan:2019hpm}, $S_2$~\\cite{Tanaka:2012nw, Dorsner:2013tla, Sakaki:2013bfa, Sahoo:2015wya, Chen:2016dip, Dey:2017ede, Becirevic:2017jtw, Chauhan:2017ndd, Becirevic:2018afm, Popov:2019tyc} and $S_3$~\\cite{Fajfer:2015ycq, Varzielas:2015iva, Bhattacharya:2016mcc, Buttazzo:2017ixm, Barbieri:2015yvd, Kumar:2018kmr, deMedeirosVarzielas:2019lgb, Bernigaud:2019bfy}, where $S_1$ and $S_3$ together can provide a common explanation of the $B$ anomalies and the AMM of the muon~\\cite{Crivellin:2017zlb,Buttazzo:2017ixm,Marzocca:2018wcf, Bigaran:2019bqv,Crivellin:2019dwb}. We take this as a motivation to study these correlations for the LQs which can generate $m_t\/m_\\mu$ enhanced effects by considering three scenarios: 1) $S_1$ only, 2) $S_2$ only, 3) $S_1+S_3$ where $S_1$ only couples to right-handed fermions. Note that these are the only scenarios which can give rise to the desired $m_t\/m_\\mu$ enhanced effect.\n\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\begin{overpic}[scale=.55,,tics=10]\n\t\t{higgs-decays_higgs.pdf}\n\t\t\\put(5,27){$h$}\n\t\t\\put(90,39){$\\mu$}\n\t\t\\put(90,6){$\\mu$}\n\t\t\\put(42,36){$t^{(c)}$}\n\t\t\\put(42,5){$t^{(c)}$}\n\t\t\\put(70,22){$S_{i}$}\n\t\\end{overpic}\n\t\\\\\n\t\\vspace{5mm}\n\t\\begin{overpic}[scale=.55,,tics=10]\n\t\t{LQ_on_shell_photon.pdf}\n\t\t\\put(5,45){$\\mu$}\n\t\t\\put(90,45){$\\mu$}\n\t\t\\put(57,5){$\\gamma$}\n\t\t\\put(48,48){$S_{i}$}\n\t\t\\put(27,24){$t^{(c)}$}\n\t\t\\put(66,24){$t^{(c)}$}\n\t\\end{overpic}\n\t\\caption{Sample Feynman diagrams which contribute to $h\\to\\mu^{+}\\mu^-$ (top) and the AMM of the muon (bottom). In addition, we have to include the diagrams where the Higgs and photon couple to the LQ, as well as self-energy diagrams.}\n\t\\label{FeynmanDiagrams}\n\\end{figure}\n\n\n\n\n\n\\section{Setup and Observables}\n\nThe most precise measurements of the anomalous magnetic moment (AMM) of the muon ($a_\\mu=(g-2)_{\\mu}\/2$) has been achieved by the E821 experiment at Brookhaven~\\cite{Bennett:2006fi,Mohr:2015ccw} and recently be the g-2 experiment at Fermilab~\\cite{Abi:2021gix}, which differs from the SM prediction by\n\\begin{equation}\n\\label{Delta_amu}\n\\delta a_\\mu=a_\\mu^{\\rm{exp}} - a_\\mu^{\\rm{SM}} = (251 \\pm 59) \\times 10^{-11} \\,,\n\\end{equation}\ncorresponding to a $4.2\\,\\sigma$ deviation~\\cite{Aoyama:2020ynm}\\footnote{This result is based on Refs.~\\cite{Aoyama:2012wk,Aoyama:2019ryr,Czarnecki:2002nt,Gnendiger:2013pva,Davier:2017zfy,Keshavarzi:2018mgv,Colangelo:2018mtw,Hoferichter:2019gzf,Davier:2019can,Keshavarzi:2019abf,Kurz:2014wya,Melnikov:2003xd,Masjuan:2017tvw,Colangelo:2017fiz,Hoferichter:2018kwz,Gerardin:2019vio,Bijnens:2019ghy,Colangelo:2019uex,Blum:2019ugy,Colangelo:2014qya}. The recent lattice result of the Budapest-Marseilles-Wuppertal collaboration (BMWc) for the hadronic vacuum polarization (HVP)~\\cite{Borsanyi:2020mff} on the other hand is not included. This result would render the SM prediction of $a_\\mu$ compatible with experiment. However, the BMWc results are in tension with the HVP determined from $e^+e^-\\to$ hadrons data~\\cite{Davier:2017zfy,Keshavarzi:2018mgv,Colangelo:2018mtw,Hoferichter:2019gzf,Davier:2019can,Keshavarzi:2019abf}. Furthermore, the HVP also enters the global EW fit~\\cite{Passera:2008jk}, whose (indirect) determination is below the BMWc result~\\cite{Haller:2018nnx}. Therefore, the BMWc determination of the HVP would increase tension in EW fit~\\cite{Crivellin:2020zul,Keshavarzi:2020bfy} and we opted for using the community consensus of Ref.~\\cite{Aoyama:2020ynm}.}. Therefore, it is very interesting to investigate if and how this discrepancy can be explained by physics beyond the SM.\n\n\n\\begin{table}\n\\begin{equation}\n\\renewcommand{\\arraystretch}{2}\n\\begin{tabular}{c|c|c}\n& $\\mathcal{G}_{\\text{SM}}$ & $\\mathcal{L}_{q\\ell}$\\\\\n\\hline\n$S_1$ & $\\bigg(3,1,-\\dfrac{2}{3}\\bigg)$ & $\\left(\\lambda_{fj}^{R}\\,\\bar{u}^c_f\\ell_{j}+\\lambda_{fj}^{L}\\,\\bar{Q}_{f}^{\\,c}i\\tau_{2}L_{j}\\right) S_{1}^{\\dagger}+\\text{h.c.}$\\\\\n$S_{2}$ & $\\bigg(3,2,\\dfrac{7}{3}\\bigg)$ & $\\gamma_{fj}^{RL}\\,\\bar{u}_{f}S_{2}^{T}i\\tau_{2}L_{j}+\\gamma_{fj}^{LR}\\,\\bar{Q}_f\\ell_j S_{2}+\\text{h.c.}$\\\\\n$S_{3}$ & $\\bigg(3,3,-\\dfrac{2}{3}\\bigg)$ & $\\kappa_{fj}^{}\\,\\bar{Q}^{\\,c}_{f}i\\tau_{2}\\left(\\tau\\cdot S_{3}\\right)^{\\dagger}L_{j}+\\text{h.c.}$\n\\end{tabular}\\nonumber\n\\end{equation}\n\\caption{Scalar LQ representations together with their couplings to quarks and leptons, generating the desired $m_t\/m_\\mu$ enhanced effect in the AMM of the muon. Here  $\\mathcal{G}_{\\text{SM}}$ refers to the SM gauge group $SU(3)_c\\times SU(2)_L\\times U(1)_Y$,  $L$ ($Q$) is the lepton (quark) $SU(2)_{L}$ doublet, $u$ ($\\ell$) the up-type quark (lepton) singlet and $c$ refers to charge conjugation. Furthermore, $j$ and $f$ are flavor indices and $\\tau_{k}$ the Pauli matrices.}\n\\label{LQrep}\n\\end{table}\n\nAs we motivated in the introduction, we will focus on the three scalar LQs $S_1$, $S_2$ and $S_3$ for explaing $a_\\mu$. These representations couple to fermions as given in Table~\\ref{LQrep}\\footnote{Note that ``pure'' LQs with couplings only to one quark and one lepton do not give rise to proton decays at any perturbative order. The reason for this is that di-quark couplings are necessary in order to break baryon and\/or lepton number which is otherwise an unbroken symmetry forbidding proton decay (see Ref.~\\cite{Dorsner:2012nq} for a recent detailed discussion).}. Since we are in the following only interested in muon couplings to third generation quarks, we define\n$\\lambda _R^{} \\equiv \\lambda _{32}^R$, $\\lambda _L^{} \\equiv \\lambda _{32}^L$, $\\gamma _{LR}^{} \\equiv \\gamma _{32}^{LR},$ $\\gamma _{RL}^{} \\equiv \\gamma _{32}^{RL}$, $\\kappa  = {\\kappa _{32}}$. \n\nIn addition to the gauge interactions, which are determined by the representation under the SM gauge group, LQ can couple to the SM Higgs~\\cite{Hirsch:1996qy}\n\\begin{align}\n{{\\cal L}_H} &= {Y_{13}}S_1^\\dag \\left( {{H^\\dag }\\left( {\\tau \\cdot{S_3}} \\right)H} \\right) + {\\rm{h}}{\\rm{.c}}{\\rm{.}} \\label{eq:LQ_mixing}\\\\\n&- {Y_{22}}{\\left( {Hi{\\tau _2}{S_2}} \\right)^\\dag }\\left( {Hi{\\tau _2}{S_2}} \\right) - \\sum\\limits_{k = 1}^3 ( m_k^2 + {Y_k}{H^\\dag }H)S_k^\\dag {S_k}\n\\nonumber\n\\end{align}\nHere $m_k^2$ are the $SU(2)_L$ invariant bi-linear masses of the LQs. After $SU(2)_L$ breaking, the term $Y_{13}$ generates off-diagonal elements in the LQ mass matrices and one has to diagonalize them through unitary transformations in order to arrive at the physical basis. Therefore, non-zero values of $Y_{13}$ are necessary to generate $m_t\/m_\\mu$ enhanced effects in scenario 3). $Y_1$ and $Y_{2,22}$ are phenomenologically relevant for $h\\to\\mu^+\\mu^-$ in scenario 1) and 2), respectively, but not necessary for an $m_t\/m_\\mu$ enhancement.\n\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.54\\textwidth]{plot6.pdf}\n\t\\includegraphics[width=0.352\\textwidth]{plot7.pdf}\n\t\\caption{Correlations between the ${\\rm Br}[h\\to\\mu^+\\mu^-]$, normalized to its SM value, and the NP contribution in the AMM of the muon $\\delta a_\\mu$ for scenario 1) (left) and scenario 2) (right) with $m_{1,2}=1.5\\,$TeV. The predictions for different values of the LQ couplings to the Higgs are shown, where for scenario 1) $Y=Y_1$ while in scenario 2) $Y=Y_2+Y_{22}$. Even though the current ATLAS and CMS results are not yet constraining for these models, sizeable effects are predicted, which can be tested at future colliders. Furthermore, scenario 1) yields a constructive effect in $h\\to\\mu^+\\mu^-$ while the one in scenario 2) is destructive such that they can be clearly distinguished with increasing experimental precision. }\\label{S1S2}\n\\end{figure*}\n\nNow we can calculate the effects in $a_\\mu$ and $h\\to\\mu^+\\mu^-$~\\footnote{Correlations between the related modes $\\tau\\to\\mu\\gamma$ and $h\\to\\tau\\mu$ were studied in Refs.~\\cite{Dorsner:2015mja,Cheung:2015yga,Baek:2015mea} in the context of LQs} for which sample diagrams are shown in Fig.~\\ref{FeynmanDiagrams}. In both cases we have on-shell kinematics. For $a_\\mu$ the self-energies can simply be taken into account via the Lehmann-Symanzik-Zimmermann formalism and no renormalization is necessary. This is however required for $h\\to\\mu^+\\mu^-$ in order to express the result in terms of the physical muon mass. Here, the effective Yukawa coupling, which enters $h\\to\\mu^+\\mu^-$, is given by\n\\begin{equation}\nY_\\mu ^{{\\rm{eff}}} = \\frac{{m_\\mu ^{} - \\Sigma _{\\mu \\mu }^{LR}}}{v} + \\Lambda _{\\mu \\mu }^{LR}\\,,\n\\end{equation}\nwhere $\\Lambda _{\\mu \\mu }^{LR}$ is the genuine vertex correction shown in Fig.~\\ref{FeynmanDiagrams} and $\\Sigma _{\\mu \\mu }^{LR}$ is the chirality changing part of the muon self-energy. In these conventions $-i\\Sigma _{\\mu \\mu }^{LR}P_R$ equals the expression of the Feynman diagram for the self-energy. Note that $Y_\\mu ^{{\\rm{eff}}}$ is finite without introducing a counter-term. For $a_\\mu$ we expand in the muon mass and external momenta up to the first non-vanishing order, while in $h\\to\\mu^+\\mu^-$ external momenta can be set to zero from the outset but we expand in $m_h^2\/m_{1,2,3}^2$. The resulting amplitudes can be further simplified by expanding the LQ mixing matrices and mass eigenvalues in $v^2\/m_{1,2,3}^2$ and the loop functions in $m_h^2\/m_t^2$, which gives a very precise numerical approximation, resulting in\n\\begin{widetext}\n\t\\begin{align}\n\t\\dfrac{{{\\rm{Br}}\\left[ {h \\to \\mu^{+} \\mu^{-} } \\right]}}{{{\\rm{Br}}{{\\left[ {h \\to \\mu^{+} \\mu^{-} } \\right]}_{{\\rm{SM}}}}}} \\approx \\Bigg| \n\t1 + \\dfrac{m_t}{m_\\mu }\\dfrac{N_c}{8\\pi^2}\\bigg[ \\dfrac{\\lambda _R^*\\lambda_L}{m_1^2}\\left( \\dfrac{m_t^2}{8}\\mathcal{J}\\!\\left( {\\dfrac{{{m_h^2}}}{{m_t^2}},\\dfrac{{m_t^2}}{{m_1^2}}} \\right) + v^{2}Y_1 \\right)+v^{2}\\lambda_R^{*}\\kappa Y_{13} {\\dfrac{{\\log \\left( {m_3^2\/m_1^2} \\right)}}{{m_3^2-m_1^2}}}\\nonumber\\\\\n\t\\qquad\\qquad\\qquad+ \\dfrac{\\gamma_{LR}^*\\gamma _{RL}}{m_{2}^2}\\left( \\dfrac{m_t^2}{8}\\mathcal {J}\\!\\left( {\\dfrac{{{m_h^2}}}{{m_t^2}},\\dfrac{{m_t^2}}{{m_2^2}}} \\right) + v^{2}{(Y_2+ Y_{22})} \\right)\\bigg]\n\t\\Bigg|^2\\,,\\\\\n\t{\\delta a_\\mu }\\approx \\frac{{{m_\\mu }}}{{4{\\pi ^2}}}\\frac{{{N_c}{m_t}}}{{12}}{\\rm{Re}}\\left[ \n\t\\dfrac{\\gamma_{LR}^{}\\gamma _{RL}^*}{m_{2}^2} { {\\cal E}_{1}\\!\\left( {\\frac{{m_t^2}}{{m_2^2}}} \\right)} - \\frac{\\lambda_R}{m_{1}^2} \\left( {\\lambda _L^* {{\\cal E}_{2}\\!\\left( {\\frac{{m_t^2}}{{m_1^2}}} \\right)} + \\kappa {Y_{13}}\\frac{{{v^2}}}{{m_3^2}}{\\cal E}_{3}\\!\\left( {\\frac{{m_1^2}}{{m_3^2}}},\\frac{{m_t^2}}{{m_3^2}} \\right)} \\right)\\right]\\,,\n\\label{hmumuamuFormula}\n\t\\end{align}\n\\end{widetext}\nwith the loop functions given by\n\\begin{align}\n{\\cal J}\\left( x,y \\right) &= 2\\left( {x - 4} \\right)\\log (y) - 8 + \\frac{{13}}{3}x{\\mkern 1mu} \\,,\n\\end{align}\n\\begin{align}\n\\begin{aligned}\n{\\cal E}_{1}(x)&=1+4\\,\\log(x)\\,,\\;\\;\n{\\cal E}_{2}(x)=7+4\\,\\log(x)\\,,\\\\\n{\\cal E}_{3}( x,y ) &= {\\cal E}_{2}(y) + \\frac{4\\,{\\log (x)}}{{x - 1}} \\,.\n\\end{aligned}\n\\end{align}\nWe only considered the $m_t$ enhanced effects and neglected small CKM rotations, which in principle appear after EW symmetry breaking. As anticipated, in \\eq{hmumuamuFormula} one can see that scenario 3) only contributes if $Y_{13}$ is non-zero. Furthermore, since in this scenario $a_\\mu$ has a relative suppression of $v^2\/m_{1,3}^2$ with respect to $h\\to\\mu^+\\mu^-$, one expects here the largest effects in Higgs decays. In principle also $Y_1$, $Y_2$ and $Y_{22}$ enter in \\eq{hmumuamuFormula}. However, their effect is sub-leading as it is suppressed by $v^2\/m_{1,2}^2$.\n\n\\subsection{Effective Field Theory}\n\nIn the SM effective field theory (SMEFT), which is realized above the EW breaking scale and therefore explicitly $SU(2)_L$ invariant, there are only two chirality flipping 4-fermion operators~\\cite{Grzadkowski:2010es} which can give rise to $m_t$ enhanced effects in $a_\\mu$ and $h\\to\\mu^+\\mu^-$ via re-normalization group evolution (RGE) effects:\n\\begin{align}\n\\begin{aligned}\nQ_{{\\rm{\\ell equ }}}^{(1)} &= \\left( {\\bar \\ell _2^a{e_2}} \\right){\\varepsilon _{ab}}\\left( {\\bar q_3^b{u_3}} \\right)\\,,\\\\\nQ_{{\\rm{\\ell  equ }}}^{(3)} &= \\left( {\\bar \\ell _2^a{\\sigma _{\\mu \\nu }}{e_2}} \\right){\\varepsilon _{ab}}\\left( {\\bar q_3^b{\\sigma ^{\\mu \\nu }}{u_3}} \\right)\\,.\n\\end{aligned}\n\\end{align}\nImportantly, while both operators mix at order $\\alpha_{(s)}$ with each other, only the second operators mixes (directly) into the magnetic operator~\\cite{Jenkins:2013zja,Jenkins:2013wua,Alonso:2013hga}\n\\begin{equation}\n\\begin{aligned}\n{Q_{eB}} &= {{\\bar \\ell }_2}{\\sigma^{\\mu \\nu }}{e_2}H{B_{\\mu \\nu }}\\,,\\\\\n{Q_{eW}} &= {{\\bar \\ell }_2}{\\sigma^{\\mu \\nu }}{e_2}{\\tau ^I}HW_{\\mu \\nu }^I\\,,\n\\end{aligned}\n\\end{equation}\ngiving rise to the AMM of the muon after EW symmetry breaking\\footnote{Note that LQs are the only renormalizable extensions of the SM that can generate these operator at tree-level~\\cite{deBlas:2017xtg}.}.  Furthermore, as $Q_{{\\rm{\\ell equ }}}^{(1)}$ mixes into ${Q_{e\\varphi }} =  {{H^\\dag }H}  {{{\\bar \\ell }_2}{e_2}H}$ (generating modified Higgs couplings to muons) it is clear that a UV complete (or at least simplified) model is necessary to correlate $a_\\mu$ to $h\\to\\mu^+\\mu^-$.\n\nThe EFT approach is beneficial in our LQ setup since it allows for the inclusion of RGE effects, as recently done in Ref.~\\cite{Aebischer:2021uvt}. In a first step, the LQ model is matched on the SMEFT (at the LQ scale), giving tree-level effects in $C_{{\\rm{\\ell equ }}}^{(1,3)}$~\\cite{Alonso:2015sja} and a loop effect in ${Q_{eB}}$ and \n${Q_{eW}}$~\\cite{Gherardi:2020det}. Then the SMEFT is used to evolve the Wilson coefficients of these operators to the weak scale where the EW gauge bosons, the Higgs and the top quark are integrated out~\\cite{Crivellin:2013hpa,Dekens:2019ept,Hurth:2019ula}. Next, the magnetic operator of the muon is evolved to the muon scale~\\cite{Crivellin:2017rmk,Aebischer:2017gaw} where the AMM is measured. Ref.~\\cite{Aebischer:2021uvt} finds a reduction of $a_\\mu$ by $\\approx20\\%-30\\%$ compared to the leading order estimate of LQ masses between $1$--$10\\,$TeV. Furthermore, as $C_{{\\rm{lequ }}}^{(1)}$ is enhanced by $\\approx5\\%-10\\%$ by the running from the LQ scale to the EW scale~\\cite{Aebischer:2018bkb}, this leads to an important enhancement of $50\\%-70\\%$ of the prediction for ${\\rm Br}[h\\to\\mu^+\\mu^-]$ w.r.t the leading order calculation. To be conservative, we will use $50\\%$ in our following phenomenological analysis.\n\n\\section{Phenomenology}\n\\label{pheno}\n\nLet us now study the correlations between $a_\\mu$ and $h\\to\\mu^+\\mu^-$ in our three scenarios with $m_t$-enhanced contributions. First, we consider scenario 1) and 2) where $S_1$ and $S_2$ give separately rise to $m_t$-enhanced effects in $a_\\mu$ and $h\\to\\mu^+\\mu^-$. Since both processes involve the same product of couplings to SM fermions, the correlation depends only weakly via a logarithm on $m_t^2\/m_{1,2}^2$. However, there is a dependence on $Y_1$ and $Y_{22}+Y_2$ which breaks the direct correlation but cannot change the sign of the effect for order one couplings. This can be seen in Fig.~\\ref{S1S2}, where the correlations are depicted for $m_{1,2}=1.5$ TeV, respecting LHC bounds~\\cite{Sirunyan:2018ryt,Diaz:2017lit,Aaboud:2016qeg}. The predicted effect is not large enough such that the current ATLAS and CMS measurements are sensitive to it. However, note that it is still sizeable due to the $m_t$ enhancement and therefore detectable at future colliders where the ILC~\\cite{Behnke:2013lya}, the HL-LHC~\\cite{ApollinariG.:2017ojx}, the FCC-ee~\\cite{Abada:2019zxq}, CEPC~\\cite{An:2018dwb} or the FCC-hh~\\cite{Benedikt:2018csr} aim at a precision of approximately 10\\%, 8\\%, 6\\% and below 1\\%, respectively. Furthermore, the effect in ${\\rm Br}[h\\to\\mu^+\\mu^-]$ in scenario 1) is necessarily constructive while in scenario 2) it is destructive, such that in the future a LQ explanation of $a_\\mu$ by $S_1$ could be clearly distinguished from the one involving $S_2$. \n\nIn scenario 3), where $S_1$ only couples to right-handed fermions, the effect in ${\\rm Br}[h\\to\\mu^+\\mu^-]$ is even more pronounced due to the relative suppression of the contribution to $a_\\mu$ by $v^2\/m_{1,3}^2$, see \\eq{hmumuamuFormula}. Furthermore, in this case the correlation between $a_\\mu$ and $h\\to\\mu^+\\mu^-$ depends to a good approximation only on the ratio $m_1\/m_3$. As the effect is symmetric in $m_1$ and $m_3$ we fix one mass to $1.5$ TeV and obtain the band shown in Fig.~\\ref{S1S3Y13} by varying the other mass between $1.5$ and $3$ TeV. The effect in $h\\to\\mu^+\\mu^-$ within the preferred region for $a_\\mu$ is necessarily constructive and large enough that an explanation of the central value of $a_\\mu$ is already disfavored by the ATLAS and CMS measurements of $h\\to\\mu^+\\mu^-$. Clearly, with more data the LHC will be able to support (disprove) this scenario if it finds a (no) significant enhancement of the $h\\to\\mu^+\\mu^-$ decay, assuming $\\delta a_\\mu$ is confirmed. This scenario also leads to sizeable effects in $Z\\mu\\mu$~\\cite{Dorsner:2019itg} which are compatible with LEP data~\\cite{ALEPH:2005ab}, but could be observed at the ILC~\\cite{Behnke:2013lya}, CLIC~\\cite{Aicheler:2012bya} or the FCC-ee~\\cite{Abada:2019zxq}. \n\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.52\\textwidth]{plot5.pdf}\n\t\\caption{Correlations between the NP contribution to the AMM of the muon ($\\delta a_\\mu$) and ${\\rm Br}[h\\to\\mu^+\\mu^-]$, normalized to its SM value in scenario 3). This correlation depends to a good approximation only on the ratio $m_1\/m_3$. As the effect is symmetric in $m_1$ and $m_3$, we fix one mass to $1.5\\,$TeV and obtain the dark-blue band by varying the other mass between $1.5\\,$TeV and $3\\,$TeV. The effect in $h\\to\\mu^+\\mu^-$ within the preferred region for $a_\\mu$ is necessarily constructive and so large that an explanation is already constrained by the ATLAS and CMS measurements of $h\\to\\mu^+\\mu^-$.}\n\t\\label{S1S3Y13}\n\\end{figure*}\n\n\n\\section{Conclusions}\n\\label{conclusions}\n\nLQs are prime candidates for an explanation of the intriguing hints of LFUV. As LFUV within the SM only originates from the Higgs, chirality changing observables as the AMM of the muon and, of course, $h\\to\\mu^+\\mu^-$ are especially interesting. In particular, there are three possible LQ scenarios which can address the discrepancy in the AMM of the muon by an $m_t\/m_\\mu$ enhancement. This also leads to enhanced corrections in $h\\to\\mu^+\\mu^-$, which involve the same coupling structure as the $a_\\mu$ contribution. This leads to interesting correlations between $a_\\mu$ and $h\\to\\mu^+\\mu^-$, which we study in light of the recent ALTAS and CMS measurements. \n\nWe find that scenario 3), in which $S_1$ only couples to right-handed fermions and mixes after EW symmetry breaking with $S_3$, predicts large constructive effects in $h\\to\\mu^+\\mu^-$ such that the current ATLAS and CMS measurements are already excluding part of the parameter space. In case $\\delta a_\\mu$ is solely explained by $S_1$ or $S_2$ the effect in  ${\\rm  Br}[h\\to\\mu^+\\mu^-]$ is of the order of several percent and therefore detectable at future colliders, in particular at the FCC-hh. Furthermore, while the $S_1$ scenario predicts constructive interference in $h\\to\\mu^+\\mu^-$ for the currently preferred range of $a_\\mu$, the $S_2$ scenario predicts destructive interference such that they can be clearly distinguished in the future. \n\\medskip\n\n\\begin{acknowledgments}\nAcknowledgements -- A.C. thanks Martin Hoferichter for useful discussions. The work of A.C. and D.M. supported by a Professorship Grant (PP00P2\\_176884) of the Swiss National Science Foundation and the one of F.S. by the Swiss National Science Foundation grant 200020\\_175449\/1.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nIn the mathematics literature, lowering and rasing operators operators are known as generators of step algebras, which were originally\nintroduced by Mickelsson \\cite{Mick} for reductive pairs of Lie algebras, $\\mathfrak{g}'\\subset \\mathfrak{g}$. These algebras naturally act on  $\\mathfrak{g}'$-singular vectors in $U(\\mathfrak{g})$-modules  and are important in representation theory,\n\\cite{Zh1,Mol}.\n\nThe general theory of step algebras for classical universal enveloping algebras was developed in \\cite{Zh1,Zh2} and\nwas extended to the special liner and orthogonal quantum groups in \\cite{Kek}. They admit a natural description in\nterms of extremal projectors, \\cite{Zh2}, introduced for classical groups in \\cite{AST1,AST2}\nand extended to the quantum group case in \\cite{T}. It is known that the step algebra $Z(\\mathfrak{g},\\mathfrak{g}')$ is generated by\nthe image of the orthogonal complement $\\mathfrak{g}\\ominus \\mathfrak{g}'$ under the extremal projector of the $\\mathfrak{g}'$.\nAnother description of lowering\/rasing operators for classical groups was obtained in\n\\cite{NM,Pei,PH,W} in an explicit form of polynomials in $\\mathfrak{g}$.\n\nA generalization of the results of \\cite{NM,Pei} to quantum $\\mathfrak{g}\\l(n)$ can be found in \\cite{ABHST}. In this special case, the lowering operators can be also conveniently expressed through  \"modified commutators\" in the Chevalley generators of $U(\\mathfrak{g})$\nwith coefficients in the field of fractions of $U(\\mathfrak{h})$. Extending\n\\cite{PH,W} to a general quantum group is not straightforward, since there are no\nimmediate candidates for the nilpotent triangular Lie subalgebras $\\mathfrak{g}_\\pm $ in $U_q(\\mathfrak{g})$. We suggest such a generalization, where the lack of\n$\\mathfrak{g}_\\pm $ is compensated by the entries of the universal R-matrix with one leg  projected to the natural representation.\nThose entries are nicely expressed through  modified commutators in the Chevalley generators turning into elements of $\\mathfrak{g}_\\pm$\nin the quasi-classical limit. Their commutation relation with the Chevalley generators modify the classical commutation\nrelations with $\\mathfrak{g}_\\pm$ in a tractable way.  This enabled us to generalize the results of \\cite{NM,Pei,PH,W} and construct\ngenerators of Mickelsson algebras for the non-exceptional quantum groups.\n\n\\subsection{Quantized universal enveloping algebra}\n\\label{ssecQUEA}\nIn this paper,  $\\mathfrak{g}$ is a complex simple Lie algebra of type $B$, $C$ or $D$.\nThe case of $\\mathfrak{g}\\l(n)$ can be easily derived from here due to the natural inclusion $U_q\\bigl(\\mathfrak{g}\\l(n)\\bigr)\\subset U_q(\\mathfrak{g})$, so we do not pay special attention to it.\nWe choose a Cartan subalgebra $\\mathfrak{h}\\subset \\mathfrak{g}$ with the canonical inner product $(.,.)$ on $\\mathfrak{h}^*$.\nBy  $\\mathrm{R}$ we denote the root system of $\\mathfrak{g}$ with a fixed subsystem of\npositive roots $\\mathrm{R}^+\\subset \\mathrm{R}$ and the basis of simple roots $\\Pi^+\\subset \\mathrm{R}^+$.\nFor every $\\lambda\\in \\mathfrak{h}^*$ we denote by  $h_\\lambda$  its image under the isomorphism $\\mathfrak{h}^*\\simeq \\mathfrak{h}$,\nthat is $(\\lambda,\\beta)=\\beta(h_\\lambda)$ for all $\\beta\\in \\mathfrak{h}^*$.\nWe put $\\rho=\\frac{1}{2}\\sum_{\\alpha\\in \\mathrm{R}^+}\\alpha $ for the Weyl vector.\n\nSuppose that $q\\in \\mathbb{C}$ is not a root of unity. Denote by $U_q(\\mathfrak{g}_\\pm)$ the $\\mathbb{C}$-algebra generated by  $e_{\\pm\\alpha}$, $\\alpha\\in \\Pi^+$, subject to the q-Serre relations\n$$\n\\sum_{k=0}^{1-a_{ij}}(-1)^k\n\\left[\n\\begin{array}{cc}\n1-a_{ij} \\\\\n k\n\\end{array}\n\\right]_{q_{\\alpha_i}}\ne_{\\pm \\alpha_i}^{1-a_{ij}-k}\ne_{\\pm \\alpha_j}e_{\\pm \\alpha_i}^{k}\n=0\n,\n$$\nwhere  $a_{ij}=\\frac{2(\\alpha_i,\\alpha_j)}{(\\alpha_i,\\alpha_i)}$,\n$i,j=1,\\ldots, n=\\mathrm{rk}\\: \\mathfrak{g}$, is the Cartan matrix, $q_{\\alpha}= q^{\\frac{(\\alpha,\\alpha)}{2}}$, and\n$$\n\\left[\n\\begin{array}{cc}\nm  \\\\ k\n\\end{array}\n\\right]_{q}\n=\n\\frac{[m]_q!}{[k]_q![m-k]_q!},\n\\quad\n[m]_q!=[1]_q\\cdot [2]_q\\ldots [m]_q.\n$$\nHere and further on, $[z]_q=\\frac{q^z-q^{-z}}{q-q^{-1}}$ whenever $q^{\\pm z}$ make sense.\n\nDenote by $U_q(\\mathfrak{h})$ the commutative $\\mathbb{C}$-algebra generated by $q^{\\pm h_\\alpha}$, $\\alpha\\in \\Pi^+$. The quantum group $U_q(\\mathfrak{g})$ is a $\\mathbb{C}$-algebra generated by  $U_q(\\mathfrak{g}_\\pm)$ and $U_q(\\mathfrak{h})$ subject\nto the relations\n$$\nq^{ h_\\alpha}e_{\\pm \\beta}q^{-h_\\alpha}=q^{\\pm(\\alpha,\\beta)} e_{\\pm \\beta},\n\\quad\n[e_{\\alpha},e_{-\\beta}]=\\delta_{\\alpha, \\beta}\\frac{q^{h_{\\alpha}}-q^{-h_{\\alpha}}}{q_\\alpha-q^{-1}_\\alpha}.\n$$\nRemark that $\\mathfrak{h}$ is not contained in $U_q(\\mathfrak{g})$, still it is convenient for us to keep reference to $\\mathfrak{h}$.\n\nFix the comultiplication in $U_q(\\mathfrak{g})$ as in \\cite{CP}:\n\\begin{eqnarray}\n&\\Delta(e_{\\alpha})=e_{\\alpha}\\otimes q^{h_{\\alpha}} + 1\\otimes e_{\\alpha},\n\\quad\n\\Delta(e_{-\\alpha})=e_{-\\alpha}\\otimes 1 + q^{-h_{\\alpha}} \\otimes e_{-\\alpha},\n\\nonumber\\\\&\n\\Delta(q^{\\pm h_{\\alpha}})=q^{\\pm h_{\\alpha}}\\otimes q^{\\pm h_{\\alpha}},\n\\nonumber\n\\end{eqnarray}\nfor all $\\alpha \\in \\Pi^+$.\n\nThe subalgebras $U_q(\\b_\\pm)\\subset U_q(\\mathfrak{g})$ generated by $U_q(\\mathfrak{g}_\\pm)$ over $U_q(\\mathfrak{h})$ are quantized universal enveloping algebras of the\nBorel subalgebras $\\b_\\pm=\\mathfrak{h}+\\mathfrak{g}_\\pm\\subset \\mathfrak{g}$.\n\nThe Chevalley  generators $e_{\\alpha}$ can be extended to a set of higher root vectors $e_{\\beta}$ for\nall $\\beta\\in \\mathrm{R}$. A normally ordered set of root vectors generate a Poincar\\'{e}-Birkhoff-Witt (PBW) basis of $U_q(\\mathfrak{g})$\nover $U_q(\\mathfrak{h})$, \\cite{CP}. We will use $\\mathfrak{g}_\\pm$ to denote the vector space spanned by $\\{e_{\\pm \\beta}\\}_{\\beta\\in \\mathrm{R}^+}$.\n\nThe universal R-matrix is an element of a certain extension of $U_q(\\mathfrak{g})\\otimes U_q(\\mathfrak{g})$.\nWe heavily use the intertwining relation\n\\begin{eqnarray}\n\\mathcal{R} \\Delta(x)= \\Delta^{op}(x)\\mathcal{R},\n\\label{intertiner}\n\\end{eqnarray}\nbetween the coproduct and its opposite for all $x\\in U_q(\\mathfrak{g})$.\nLet $\\{\\varepsilon_i\\}_{i=1}^n\\subset \\mathfrak{h}^*$ be the standard orthonormal basis  and $\\{h_{\\varepsilon_i}\\}_{i=1}^n$ the corresponding dual basis in $\\mathfrak{h}$.\nThe exact expression for $\\mathcal{R}$ can be extracted from \\cite{CP}, Theorem 8.3.9, as the ordered product\n\\begin{eqnarray}\n\\mathcal{R}= q^{\\sum_{i=1}^n h_{\\varepsilon_i}\\otimes  h_{\\varepsilon_i}}\\prod_{\\beta} \\exp_{q_\\beta}\\{(1-q_\\beta^{-2})(e_\\beta\\otimes e_{-\\beta} )\\} \\in U_q(\\b_+)\\hat \\otimes U_q(\\b_-),\n\\label{Rmat}\n\\end{eqnarray}\nwhere  $\\exp_{q}(x )=\\sum_{k=0}^\\infty q^{\\frac{1}{2}k(k+1)}\\frac{x^k}{[k]_q!}$.\n\nWe use the notation $e_i=e_{\\alpha_i}$ and $f_{i}=e_{-\\alpha_i}$ for  $\\alpha_i\\in \\Pi^+$, in all cases apart\nfrom $i=n$, $\\mathfrak{g}=\\mathfrak{s}\\o(2n+1)$, where we set $f_n=[\\frac{1}{2}]_q e_{-\\alpha_n}$.\nThe reason for this is two-fold.\nFirstly, the natural representation can be defined through the classical assignment on the generators, as given below.\nSecondly,\nwe get rid of $q_{\\alpha_n}=q^{\\frac{1}{2}}$ and can work over $\\mathbb{C}[q]$, as the relations involved turn into\n$$\n[e_{n},f_{n}]=\\frac{q^{h_{\\alpha_n}}-q^{-h_{\\alpha_n}}}{q-q^{-1}},\n$$\n$$\nf_{n}^3f_{n-1}-(q+1+q^{-1})f_{n}^2f_{n-1}f_{n}+(q+1+q^{-1})f_{n}f_{n-1}f_{n}^2-f_{n-1}f_{n}^3=0.\n$$\nIt is easy to see that the square root of $q$  disappears from the corresponding factor in the presentation\n(\\ref{Rmat}).\n\n\nIn what\nfollows, we regard $\\mathfrak{g}\\l(n)\\subset \\mathfrak{g}$ to be the Lie subalgebra with the simple roots $\\{\\alpha_i\\}_{i=1}^{n-1}$ and $U_q\\bigl(\\mathfrak{g}\\l(n)\\bigr)$ the corresponding\nquantum subgroup in $U_q(\\mathfrak{g})$.\n\nConsider the natural representation of $\\mathfrak{g}$ in the vector space $\\mathbb{C}^N$.\nWe use the notation $i'=N+1-i$ for all integers $i=1,\\ldots,N$. The assignment\n$$\n\\pi(e_{i})=e_{i,i+1}\\pm e_{i'-1,i'}, \\quad \\pi(f_{i})= e_{i+1,i}\\pm e_{i',i'-1}, \\quad \\pi(h_{\\alpha_i})= e_{ii}-e_{i+1,i+1}+e_{i'-1,i'-1}-e_{i'i'},\n$$\nfor $i=1, \\ldots,n-1$, defines a direct sum of two representations of $\\mathfrak{g}\\l(n)$ for each sign.\nIt extends to the natural representation of the whole $\\mathfrak{g}$ by\n$$\n\\pi(e_{n})= e_{n,n+1}\\pm e_{n'-1,n'}, \\quad \\pi(f_{n})=  e_{n+1,n}\\pm e_{n',n'-1}, \\quad \\pi(h_{\\alpha_n})=\ne_{nn}-e_{n'n'},\n$$\n$$\n\\pi(e_{n})= e_{nn'}, \\quad \\pi(f_{n})= e_{n'n}, \\quad \\pi(h_{\\alpha_n})=\n2e_{nn}-2e_{n'n'},\n$$\n$$\n\\pi(e_{n})= e_{n-1,n'}\\pm e_{n,n'+1}, \\quad \\pi(f_{n})= e_{n',n-1}\\pm e_{n'+1,n}, \\quad \\pi(h_{\\alpha_n})=\ne_{n-1,n-1}+e_{nn}-e_{n'n'}-e_{n'+1,n'+1},\n$$\nrespectively, for $\\mathfrak{g}=\\mathfrak{s}\\o(2n+1)$, $\\mathfrak{g}=\\mathfrak{s}\\mathfrak{p}(2n)$, and $\\mathfrak{g}=\\mathfrak{s}\\o(2n)$.\n\nTwo values of the sign give equivalent representations.\nThe choice of minus corresponds to the standard representation that preserves the bilinear form\nwith entries $C_{ij}=\\delta_{i'j}$, for $\\mathfrak{g}=\\mathfrak{s}\\o(N)$, and $C_{ij}=\\mathrm{sign}(i'-i)\\delta_{i'j}$, for $\\mathfrak{g}=\\mathfrak{s}\\mathfrak{p}(N)$.\nHowever, we fix the sign to $+$ in order to simplify calculations. The above assignment also defines representations of $U_q(\\mathfrak{g})$.\n\n\\section{$R$-matrix of non-exceptional quantum groups}\nDefine $\\check{\\mathcal{R}}=q^{-\\sum_{i=1}^n h_{\\varepsilon_i}\\otimes h_{\\varepsilon_i}}\\mathcal{R} $.\nDenote by $\\check{R}^-=(\\pi\\otimes \\mathrm{id})(\\check{\\mathcal{R}}) \\in \\mathrm{End}(\\mathbb{C}^N)\\otimes U_q(\\mathfrak{g}_-)$\nand by $\\check{R}^+=(\\pi\\otimes \\mathrm{id})(\\check{\\mathcal{R}}_{21}) \\in \\mathrm{End}(\\mathbb{C}^N)\\otimes U_q(\\mathfrak{g}_+)$.\nIn this section, we deal only with $\\check{R}^-$ and suppress the label \"$-$\" for simplicity,\n$\\check{R}=\\check{R}^-$.\n\nDenote by $N_+$ the ring of all upper triangular matrices in $\\mathrm{End}(\\mathbb{C}^N)$ and by $N'_+$ its ideal spanned by $e_{ij}$, $i<j+1$.\n\\begin{lemma}\nOne has\n$$\n\\check{R} =1\\otimes 1+(q^{1+\\delta_{1n}}-q^{-1-\\delta_{1n}})\\sum_{i=1}^n\\pi(e_i)\\otimes f_{i}\\mod N'_+\\otimes U_q(\\mathfrak{g}_-),\n$$\nwhere $\\delta_{1n}$ is present only for $\\mathfrak{g}=\\mathfrak{s}\\mathfrak{p}(2n)$.\n\\end{lemma}\n\\begin{proof}\nFor all positive roots $\\alpha, \\beta$ the matrix $\\pi(e_\\alpha e_\\beta)$ belongs to $N'_+$.\nAlso, $\\pi(e_\\beta)\\in N'_+$ for all $\\beta\\in \\mathrm{R}^+\\backslash \\Pi^+$.\n Therefore, the only terms that\ncontribute to $\\mathrm{Span}_{\\varepsilon_i-\\varepsilon_j\\in \\Pi^+}\\{e_{ij}\\otimes U_q(\\mathfrak{g}_-)\\}$\nare those of degree 1 from the  series $\\exp_{q_\\alpha}(1-q_\\alpha^{-2})(e_\\alpha\\otimes e_{-\\alpha} )$ with $\\alpha\\in \\Pi^+$.\n\\end{proof}\n\\noindent\nWrite $\\check{R}=\\sum_{i,j=1}^N e_{ij}\\otimes \\check{R}_{ij}$, where $\\check{R}_{ij}=0$ for $i>j$. Due to the $\\mathfrak{h}$-invariance\nof $\\check R$, the entry $\\check{R}_{ij}\\in U_q(\\mathfrak{g}_-)$ carries weight $\\varepsilon_j-\\varepsilon_i$.\n\n\nFor all $\\mathfrak{g}$, we have $f_{k,k+1}=f_k=f_{k'-1,k'}$ once $k<n$\nand $f_{n,n+1}=f_n=f_{n+1, n'}$ for $\\mathfrak{g}=\\mathfrak{s}\\o(2n+1)$,\n$f_{n-1,n'}=f_n=f_{n, n'+1}$ for $\\mathfrak{g}=\\mathfrak{s}\\o(2n)$, and\n$f_{nn'}=[2]_qf_n$ for $\\mathfrak{g}=\\mathfrak{s}\\mathfrak{p}(2n)$.\nWe present explicit expressions for the  entries $f_{ij}$ in terms of modified commutators in Chevalley generators,\n $[x,y]_a=xy- ayx$, where $a$ is a  scalar; we also put $\\bar q = q^{-1}$.\n\\begin{propn}\n\\label{expl_f}\nSuppose that $\\varepsilon_i-\\varepsilon_j \\in \\mathrm{R}^+\\backslash \\Pi^+$. Then the elements $f_{ij}$ are given by the following formulas:\\\\\nFor all $\\mathfrak{g}$ and  $i+1<j\\leqslant\\frac{N+1}{2}$:\n\\begin{eqnarray}\nf_{ij}=[f_{j-1},\\ldots [f_{i+1},f_i]_{\\bar q^{}}\\ldots ]_{\\bar q},\n\\quad\nf_{j'i'}=[\\ldots [f_i,f_{i+1}]_{\\bar q^{}},\\ldots f_{j-1}]_{\\bar q}.\n\\label{gl(n)}\n\\end{eqnarray}\nFurthermore,\n\\begin{itemize}\n\\item\nFor $\\mathfrak{g}=\\mathfrak{s}\\o(2n+1)$: $f_{nn'}=(q-1)f_n^2$ and\n$$\nf_{i,n+1}=[f_{n},f_{i,n}]_{\\bar q^{}}, \\quad f_{n+1,i'}=[f_{n',i'},f_{n}]_{\\bar q^{}}, \\quad i<n,\n$$\n$$f_{ij'}=q^{\\delta_{ij}}[f_{n+1,j'},f_{i,n+1}]_{\\bar q^{\\delta_{ij}}},  \\quad i,j<n.$$\n\\item\nFor $\\mathfrak{g}=\\mathfrak{s}\\mathfrak{p}(2n)$:  $f_{nn'}=[2]_qf_n$ and\n$$\nf_{in'}=[f_{n},f_{in}]_{\\bar q^{2}}, \\quad f_{ni'}=[f_{n'i'},f_{n}]_{\\bar q^{2}}, \\quad i<n,\n$$\n$$f_{ij'}=q^{\\delta_{ij}}[f_{nj'},f_{in}]_{\\bar q^{1+\\delta_{ij}}}, \\quad i,j<n.\n$$\n\\item\nFor $\\mathfrak{g}=\\mathfrak{s}\\o(2n)$: $f_{nn'}=0$ and\n$$f_{in'}=[f_{n},f_{i,n-1}]_{\\bar q^{}}, \\quad f_{ni'}=[f_{n'+1,i'},f_n]_{\\bar q^{}},\\quad i<n-2,$$\n$$\nf_{ji'}=q^{\\delta_{ij}}[f_{ni'},f_{j,n}]_{\\bar q^{1+\\delta_{ij}}}, \\quad i,j\\leqslant n-1.\n$$\n\\end{itemize}\n\\end{propn}\n\\begin{proof}\nThe proof is a direct calculation with the use of the identity\n$$\n(f_\\alpha\\otimes 1)\\check{\\mathcal{R}}- \\check{\\mathcal{R}}(f_\\alpha\\otimes 1)  =\\check{\\mathcal{R}}(q^{-h_\\alpha}\\otimes f_\\alpha)-(q^{h_\\alpha}\\otimes f_\\alpha)\\check{\\mathcal{R}},\n$$\nwhich follows from\nthe intertwining axiom (\\ref{intertiner}) for $x=f_\\alpha$.\nThis allows us to construct the elements $f_{ij}$ by induction  starting from $f_\\alpha$, $\\alpha\\in \\Pi^+$.\n\\end{proof}\nFor each $\\alpha\\in \\Pi^+$, denote by $P(\\alpha)$ the set of ordered pairs $l,r=1,\\ldots,N$,\nwith $\\varepsilon_l-\\varepsilon_r= \\alpha$. We call such pairs simple.\n\\begin{propn}\n\\label{non-dyn_f_ij}\nThe matrix entries $f_{i,j}\\in U_q(\\mathfrak{g}_-)$ such that  $\\varepsilon_i-\\varepsilon_j \\not \\in \\Pi^+$\nsatisfy the identity\n$$[e_{\\alpha},f_{ij}]=\\sum_{(l,r)\\in P(\\alpha)}\n\\bigl(f_{il}\\delta_{jr}q^{h_{\\alpha}}-q^{-h_{\\alpha}}\\delta_{il}f_{rj}\\bigr),$$\nfor all simple positive  roots  $\\alpha$.\n\\end{propn}\n\\begin{proof}\nThe proof is a straightforward calculation based on the intertwining relation  (\\ref{intertiner}), which\nis equivalent to\n$$\n(1\\otimes e_\\alpha) \\check{\\mathcal{R}}-\\check{\\mathcal{R}}(1\\otimes e_\\alpha)=\\check{\\mathcal{R}}(e_\\alpha\\otimes q^{h_\\alpha})-(e_\\alpha\\otimes q^{-h_\\alpha}) \\check{\\mathcal{R}},\n$$\nfor $x=e_\\alpha$, $\\alpha\\in \\Pi^+$. Alternatively, one can use the  expressions for $f_{ij}$ from Proposition \\ref{expl_f}.\n\\end{proof}\n\\section{Mickelsson algebras}\nConsider the Lie subalgebra $\\mathfrak{g}'\\subset \\mathfrak{g}$ corresponding to the root subsystem $\\mathrm{R}_{\\mathfrak{g}'}\\subset \\mathrm{R}_\\mathfrak{g}$ generated by $\\alpha_i$, $i>1$, and\nlet $\\mathfrak{h}'\\subset \\mathfrak{g}'$ denote its Cartan subalgebra.\nLet the triangular decomposition $\\mathfrak{g}'_-\\oplus \\mathfrak{h}'\\oplus \\mathfrak{g}'_+$ be compatible with the triangular decomposition of $\\mathfrak{g}$. Recall the definition of step algebra\n$Z_q(\\mathfrak{g},\\mathfrak{g}')$ of the pair $(\\mathfrak{g},\\mathfrak{g}')$.\nConsider the left ideal $J=U_q(\\mathfrak{g})\\mathfrak{g}'_+$ and its normalizer  $\\mathcal{N}=\\{x\\in U_q(\\mathfrak{g}): e_\\alpha x \\subset J, \\forall \\alpha \\in \\Pi^+_{\\mathfrak{g}'}\\}$. By construction, $J$ is a two-sided ideal in the algebra $\\mathcal{N}$. Then $Z_q(\\mathfrak{g},\\mathfrak{g}')$ is\nthe quotient $\\mathcal{N}\/J$.\n\nFor all $\\beta_i\\in \\mathrm{R}^+_\\mathfrak{g}\\backslash \\mathrm{R}^+_{\\mathfrak{g}'}$ let $e_{\\beta_i}$ be the corresponding PBW generators\nand let  $Z$ be  the vector space spanned by $e_{-\\beta_l}^{k_l}\\ldots e_{-\\beta_1}^{k_1}e_0^{k_0} e_{\\beta_1}^{m_1}\\ldots e_{\\beta_l}^{m_l}$,\nwere  $e_0=q^{h_{\\alpha_1}}$, $k_i\\in \\mathbb{Z}_+$, and $k_0\\in \\mathbb{Z}$.\nThe PBW factorization\n$\nU_q(\\mathfrak{g})=U_q(\\mathfrak{g}'_-)Z  U_q(\\mathfrak{h}') U_q(\\mathfrak{g}'_+)\n$\ngives rise to the decomposition\n$$\nU_q(\\mathfrak{g})=Z U_q(\\mathfrak{h}') \\oplus (\\mathfrak{g}'_- U_q(\\mathfrak{g})+ U_q(\\mathfrak{g})\\mathfrak{g}'_+).\n$$\n\\begin{propn}[\\cite{Kek}, Theorem 1]\nThe projection $U_q(\\mathfrak{g})\\to Z U_q(\\mathfrak{h}')$ implements an embedding of $Z_q(\\mathfrak{g},\\mathfrak{g}')$ in $Z U_q(\\mathfrak{h}')$.\n\\label{Kekalainen}\n\\end{propn}\n\\begin{proof}\nThe  statement is proved in  \\cite{Kek} for the orthogonal and special linear quantum groups\n but the arguments apply to symplectic groups too.\n\\end{proof}\nIt is proved within the theory of extremal projectors that\ngenerators \nof $Z_q(\\mathfrak{g},\\mathfrak{g}')$ are labeled by\nthe roots $\\beta\\in \\mathrm{R}_\\mathfrak{g}\\backslash \\mathrm{R}_{\\mathfrak{g}'}$ plus $z_0=q^{h_{\\alpha_1}}$.\nWe calculate them in the subsequent sections, cf. Propositions \\ref{negative} and \\ref{positive}.\n\n\\subsection{Lowering operators}\nIn what follows, we extend $U_q(\\mathfrak{g})$ along with its subalgebras containing $U_q(\\mathfrak{h})$ over the field of fractions of $U_q (\\mathfrak{h})$\nand denote such an extension by hat, e.g. $\\hat U_q(\\mathfrak{g})$. In this section we calculate representatives\nof the negative generators of $Z_q(\\mathfrak{g},\\mathfrak{g}')$  in $\\hat U_q(\\b_-)$.\n\nSet $h_i=h_{\\varepsilon_i}\\in \\mathfrak{h}$ for all $i=1,\\ldots,N$ and introduce $\\eta_{ij}\\in \\mathfrak{h}+\\mathbb{C}$ for $i,j=1,\\ldots,N$, by\n\\begin{eqnarray}\n\\eta_{ij}=h_i-h_j+(\\varepsilon_i-\\varepsilon_j, \\rho)-\\frac{1}{2}|\\!|\\varepsilon_i-\\varepsilon_j|\\!|^2.\n\\end{eqnarray}\nHere $|\\!|\\mu|\\!|$ is the Euclidean norm on $\\mathfrak{h}^*$.\n\\begin{lemma}\n\\label{thetas}\nSuppose that $(l,r)\\in P(\\alpha)$ for some $\\alpha\\in \\Pi^+$. Then\n\\begin{itemize}\n\\item[i)] if  $l<r<j$, then $\\eta_{lj}-\\eta_{rj}=h_\\alpha+(\\alpha,\\varepsilon_j-\\varepsilon_r)$,\n\\item[ii)] if  $i<l<r $, then $\\eta_{li}-\\eta_{ri}=h_\\alpha+(\\alpha,\\varepsilon_i-\\varepsilon_r)$,\n\\item[iii)]\n $\\eta_{lr}=h_\\alpha.\n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\nWe have $(\\alpha, \\rho)=\\frac{1}{2}|\\!|\\alpha|\\!|^2$ for all $\\alpha \\in \\Pi^+$. This proves iii). Further, for $\\varepsilon_l-\\varepsilon_r=\\alpha$:\n\\begin{eqnarray}\n\\eta_{lj}-\\eta_{rj}&=&h_\\alpha+\\frac{1}{2}|\\!|\\alpha|\\!|^2+\\frac{1}{2}|\\!|\\varepsilon_j-\\varepsilon_r|\\!|^2-\\frac{1}{2}|\\!|\\varepsilon_j-\\varepsilon_r-\\alpha|\\!|^2=h_\\alpha+(\\alpha,\\varepsilon_j-\\varepsilon_r),\n\\quad r< j,\n\\nonumber\n\\\\\n\\eta_{li}-\\eta_{ri}&=&h_\\alpha+\\frac{1}{2}|\\!|\\alpha|\\!|^2+\\frac{1}{2}|\\!|\\varepsilon_i-\\varepsilon_r|\\!|^2-\\frac{1}{2}|\\!|\\varepsilon_i-\\varepsilon_r-\\alpha|\\!|^2=h_\\alpha+(\\alpha,\\varepsilon_i-\\varepsilon_r),\n\\quad i< l,\n\\nonumber\n\\end{eqnarray}\nwhich proves i) and ii).\n\\end{proof}\n\nWe call a strictly ascending sequence $\\vec m=(m_1,\\ldots,m_s)$ of integers a route from $m_1$ to $m_s$.\nWe write $m<\\vec m$ and $\\vec m< m$ for $m\\in \\mathbb{Z}$ if, respectively, $m< \\min\\vec m$ and $\\max\\vec m< m$. More generally,\nwe write $\\vec m< \\vec k$ if $\\max\\vec m < \\min \\vec k$.\nIn this case, a sequence $(\\vec m, \\vec k)$ is a route from $\\min \\vec m$ to $\\max\\vec k$.\n\nGiven a route $\\vec m=(m_1,\\ldots,m_s)$, define the product\n$\nf_{\\vec m}=f_{m_1,m_2} \\cdots f_{m_{s-1},m_s}\\in U_q(\\mathfrak{g}_-)$.\nConsider a free right $\\hat U_q(\\mathfrak{h})$-module $\\Phi_{1m}$ generated by $f_{\\vec m}$ with $1\\leqslant\\vec m \\leqslant  j$ and\ndefine an operation $\\partial_{lr}\\colon \\Phi_{1j}\\to \\hat U_q(\\b_-)$ for $(l,r)\\in P(\\alpha)$ as follows. Assuming\n$1\\leqslant \\vec\\ell < l< r< \\vec \\rho < j$, set\n$$\n\\begin{array}{rrccc}\n\\partial_{lr}f_{(\\vec\\ell,l)}f_{(l,r)}f_{(r,\\vec \\rho)}&=&f_{(\\vec\\ell,l)}f_{(r,\\vec \\rho)}[\\eta_{lj}-\\eta_{rj}]_q,\\\\\n\\partial_{lr}f_{(\\vec\\ell,l)}f_{(l,\\vec \\rho)}&=&-f_{(\\vec\\ell,l)}f_{(r,\\vec \\rho)}q^{-\\eta_{lj}+\\eta_{rj}},\n\\\\\n\\partial_{lr}f_{(\\vec\\ell,r)}f_{(r,\\vec \\rho)}&=&f_{(\\vec\\ell,l)}f_{(r,\\vec \\rho)}q^{\\eta_{lj}-\\eta_{rj}},\n\\\\\n\\partial_{lr}f_{\\vec m}&=&0, & l\\not\\in \\vec m, &r \\not \\in \\vec m.\\\\\n\\end{array}\n$$\nExtend $\\partial_{lr}$ to entire $\\Phi_{1j}$  by $\\hat U_q(\\mathfrak{h})$-linearity. Let $p\\colon \\Phi_{1j}\\to \\hat U(\\mathfrak{g})$\ndenote the natural homomorphism of $\\hat U_q(\\mathfrak{h})$-modules.\n\\begin{lemma}\n\\label{partial}\nFor all $\\alpha \\in \\Pi^+$   and all $x\\in \\Phi_{1j}$,\n$e_{\\alpha}\\circ p(x)=\\sum_{(l,r)\\in P(\\alpha)}\\partial_{lr}x \\mod \\hat U_q(\\mathfrak{g})e_\\alpha$.\n\\end{lemma}\n\\begin{proof}\nA straightforward analysis based on Proposition \\ref{non-dyn_f_ij} and Lemma \\ref{thetas}.\n\\end{proof}\n\\noindent\nTo simplify the presentation, we suppress the symbol of projection $p$ in what follows.\n\nIntroduce elements\n$A_{r}^{j}\\in  \\hat U_q(\\mathfrak{h})$ by\n\\begin{eqnarray}\nA_{r}^{j}&=&\\frac{q-q^{-1}}{q^{-2\\eta_{rj}}-1},\n\\label{A_r}\n\\end{eqnarray}\nfor all $r,j\\in [1,N]$ subject to $r< j$.\nFor each simple pair $(l,r)$ we define  $(l,r)$-chains as\n\\begin{eqnarray}\nf_{(\\vec \\ell,l)}f_{(l,\\vec\\rho)}A_{l}^{j}+f_{(\\vec\\ell,l)}f_{(l,r)}f_{(r,\\vec\\rho)}A_{l}^{j}A_{r}^{j}+f_{(\\vec\\ell,r)}f_{(r,\\vec\\rho)}A_{r}^{j},\n\\quad f_{(\\vec\\ell,l)}f_{l,j}A_{l}^{j}+f_{(\\vec\\ell,j)},\n\\label{chain}\n\\end{eqnarray}\nwhere $1\\leqslant\\vec \\ell < l$ and $r<\\vec\\rho \\leqslant j$. Remark that $f_{(l,r)}=\\left[\\frac{(\\alpha,\\alpha)}{2}\\right]_q e_{-\\alpha}$, where $\\alpha=\\varepsilon_l-\\varepsilon_r$.\n\\begin{lemma}\n\\label{open-right}\nThe operator $\\partial_{lr}$ annihilates $(l,r)$-chains.\n\\end{lemma}\n\\begin{proof}\nApplying $\\partial_{lr}$ to the 3-chain in (\\ref{chain}), we get\n$$\nf_{(\\vec\\ell,l)}f_{(r,\\vec\\rho)}(-q^{-\\eta_{lj}+\\eta_{rj}}A_{l}^{j}+[\\eta_{lj}-\\eta_{rj}]_qA_{l}^{j}A_{r}^{j}+q^{\\eta_{lj}-\\eta_{rj}}A_{r}^{j}).\n$$\nThe factor in the brackets turns zero on substitution of \\ref{A_r}.\n\nNow apply $\\partial_{lj}$ to the right expression in (\\ref{chain}) and get\n$$\nf_{(\\vec\\ell,l)}([h_\\alpha]_qA_{l}^{j}+q^{h_\\alpha})\n=f_{(\\vec\\ell,l)}(\\frac{q^{h_\\alpha}-q^{-h_\\alpha}}{q^{-2\\eta_{lj}}-1}+q^{h_\\alpha})\n=f_{(\\vec\\ell,l)}\\frac{[h_\\alpha-\\eta_{lj}]_q}{[-\\eta_{lj}]_q}=0,\n$$\nso long as $\\eta_{lj}=h_\\alpha$ by Lemma \\ref{thetas}.\n\\end{proof}\n\nGiven a route $\\vec m=(m_1,\\ldots,m_s)$, put\n $A_{\\vec m}^{j}= A_{m_1}^{j}\\cdots  A_{m_s}^{j}\\in \\hat U_q(\\mathfrak{h})$ (and $A_{\\vec m}^{j}=1$ for the empty route) and define\n\\begin{eqnarray}\nz_{-j+1}=\\sum_{1< \\vec m < j}f_{(1,\\vec m,j)} A_{\\vec m}^{j} \\in \\hat U_q(\\b_-),\\quad j=2,\\ldots,N,\n\\label{dyn_f}\n\\end{eqnarray}\n where the summation is taken over all possible $\\vec m$\nsubject to the specified inequalities  plus the empty route.\n\n\n\\begin{propn}\n\\label{negative}\n$\ne_{\\alpha} z_{-j}=\n0 \\mod\n\\hat U_q(\\mathfrak{g})e_\\alpha\n$\nfor all $\\alpha \\in \\Pi^+_{\\mathfrak{g}'}$ and $j=1,\\ldots,N-1$.\n\\end{propn}\n\\begin{proof}\nThanks to Lemma \\ref{partial}, we can reduce consideration to the action of operators $\\partial_{lr}$,\nwith $(l,r)\\in P(\\alpha)$. According to the definition of $\\partial_{lr}$\nthe summands in (\\ref{dyn_f}) that survive the action of $\\partial_{lr}$ can be organized into\na linear combination of $(l,r)$-chains with coefficients in $\\hat U_q(\\mathfrak{h})$. By Lemma \\ref{open-right} they are killed by $\\partial_{lr}$.\n\\end{proof}\nThe elements  $z_{-i}$, $i=1,\\ldots,N-1$,\nbelong to the normalizer $\\mathcal{N}$ and form the set of negative generators of $Z_q(\\mathfrak{g},\\mathfrak{g}')$\nfor symplectic $\\mathfrak{g}$. In the orthogonal case, the negative part of $Z_q(\\mathfrak{g},\\mathfrak{g}')$ is generated by $z_{-i}$, $i=1,\\ldots,N-2$.\n\\subsection{Raising operators}\nIn this section we construct positive generators of $Z_q(\\mathfrak{g},\\mathfrak{g}')$, which are called raising\noperators.\nConsider an algebra automorphism $\\omega\\colon U_q(\\mathfrak{g})\\to U_q(\\mathfrak{g})$ defined on the generators\nby $f_\\alpha\\leftrightarrow e_\\alpha$, $q^{\\pm h_\\alpha}\\mapsto q^{\\mp h_\\alpha}$. For  $i<j$,  let $g_{ji}$ be the image of $f_{ij}$ under this isomorphism. The natural representation  restricted to $U_q(\\mathfrak{g}_\\pm)$ intertwines $\\omega$ and matrix transposition.\nSince $(\\omega\\otimes \\omega)(\\check{\\mathcal{R}})=\\check\\mathcal{R}_{21}$, the matrix  $\\check{R}^+=(\\pi\\otimes \\mathrm{id})(\\check\\mathcal{R}_{21})$ is equal to $1\\otimes 1+(q-q^{-1})\\sum_{i<j}e_{ji}\\otimes g_{ji}$.\n\\begin{lemma}\nFor all $\\alpha\\in \\Pi^+_{\\mathfrak{g}'}$ and all $i>1$,\n$\ne_\\alpha g_{i1}=\\sum_{(l,r)\\in P(\\alpha)}\\delta_{il} g_{r1} \\mod \\hat  U_q(\\mathfrak{g})e_\\alpha\n$.\n\\label{[e,g]}\n\\end{lemma}\n\\begin{proof}\nFollows from the intertwining property of the R-matrix.\n\\end{proof}\nConsider the right $\\hat U_q(\\mathfrak{h})$-module $\\Psi_{i1}$ freely generated by $f_{(\\vec m,k)}g_{k1}$ with $i\\leqslant\\vec m< k$.\nWe define  operators $\\partial_{lr}\\colon \\Psi_{i1}\\to \\hat U_q(\\mathfrak{g})$ similarly as we did it for $\\Phi_{1j}$.\nFor a simple pair $(l,r)\\in P(\\alpha)$, put\n$$\n\\partial_{l,r}f_{(\\vec m,k)}g_{k1}=\n\\left\\{\n\\begin{array}{rrrrr}\nf_{(\\vec m,l)}g_{r1},&l =k,\\\\\n\\bigl(\\partial_{l,r}f_{(\\vec m,k)}\\bigr)g_{k1},&l\\not =k,\n\\end{array}\n\\right.\n \\quad i\\leqslant \\vec m< r.\n$$\nThe Cartan factors appearing in $\\partial_{lr}f_{(\\vec m,k)}$ depend on $h_\\alpha$. When pushed to the right-most position,\n$h_\\alpha$ is shifted by $(\\alpha,\\varepsilon_1-\\varepsilon_r)$.\nWe extend $\\partial_{lr}$ to an action on $\\Psi_{i1}$ by the requirement that $\\partial_{lr}$ commutes with the right action of $\\hat U_q(\\mathfrak{h})$. Let $p$ denote the natural homomorphism of $\\hat U_q(\\mathfrak{h})$-modules,\n$p\\colon \\Psi_{i1}\\to \\hat U_q(\\mathfrak{g})$.\nOne can prove the following analog of Lemma \\ref{partial}.\n\\begin{lemma}\n\\label{partial_+}\nFor all $\\alpha \\in \\Pi^+_{\\mathfrak{g}'}$  and all $x \\in \\Psi_{i1}$,\n$e_{\\alpha}\\circ p(x)=\\sum_{(l,r)\\in P(\\alpha)} \\partial_{lr} x \\mod \\hat U_q(\\mathfrak{g})e_\\alpha$.\n\\end{lemma}\n\\begin{proof}\nStraightforward.\n\\end{proof}\n\\noindent\nWe suppress the symbol of projection $p$ to simplify the formulas.\n\nDefine $\\sigma_i$ for all $i=1,\\ldots,N$ as follows. For $i<j$ let $|\\!\\!|i-j|\\!\\!|$ be the number of simple positive roots\nentering $\\varepsilon_i-\\varepsilon_j$. For all $i,k= 2,\\ldots, N$, $i< k$, put\n$$\nA_k^i=\\frac{q^{\\eta_{k1}-\\eta_{i1}}}{[\\eta_{i1}-\\eta_{k1}]_q}, \\quad B_k^i=\\frac{(-1)^{|\\!\\!|i-k|\\!\\!|}}{[\\eta_{i1}-\\eta_{k1}]_q},\n$$\nFor each $(l,r)\\in \\P(\\alpha)$, where $\\alpha \\in \\Pi^+_{\\mathfrak{g}'}$, define 3-chains  as\n\\begin{eqnarray}\nf_{(i,\\vec m, l)}g_{l1}B_{l}^i\n+\nf_{(i,\\vec m, l)}f_{(l,r)}g_{r1}A_{l}^iB_{r}^i\n+\nf_{(i,\\vec m, r)}g_{r1}B_{r}^i,\n\\label{3-chain_AB}\n\\end{eqnarray}\nwith $i< \\vec m< l< r \\leqslant N$ and\n\\begin{eqnarray}\nf_{(i,\\vec \\ell, l)}f_{(l,\\vec \\rho, k)}g_{k1}A_{l}^i\n+\nf_{(i,\\vec \\ell, l)}f_{(l,r)}f_{(r,\\vec \\rho, k)}g_{k1}A_{l}^iA_{r}^i\n+\nf_{(i,\\vec \\ell, r)}f_{(r,\\vec \\rho, k)}g_{k1}A_{r}^i\n\\label{3-chain_AA}\n\\end{eqnarray}\nwith\n$i< \\vec \\ell <  l  < r < \\vec \\rho <  k\\leqslant N$.\nThe 2-chains are defined as\n\\begin{eqnarray}\ng_{i1}+f_{(i,r)}g_{r1}B_{r}^i, \\quad\n f_{(i,\\vec m, k)}g_{k1}+f_{(i,r)}f_{(r,\\vec m, k)}g_{k1}A_{r}^i\n \\label{2-chain AB}\n\\end{eqnarray}\nwhere $r$ is such that $\\varepsilon_i-\\varepsilon_r\\in \\Pi^+_{\\mathfrak{g}'}$\nand $i< r< \\vec m< k \\leqslant N$. In all cases the empty routes $\\vec m$ are admissible.\n\\begin{lemma}\n\\label{chains_killed}\nFor all $\\alpha \\in \\Pi^+_{\\mathfrak{g}'}$ and all $(l,r)\\in P(\\alpha)$ the $(l,r)$-chains are\nannihilated by $\\partial_{lr}$.\n\\end{lemma}\n\\begin{proof}\nSuppose that $i=l$ and apply $\\partial_{ir}$ to the left 2-chain in (\\ref{2-chain AB}). The result is\n$$\ng_{r1}+[h_\\alpha]_qg_{r1}B^{i}_r\n=g_{r1}(1+[h_{\\alpha} +(\\alpha,\\varepsilon_1-\\varepsilon_r)]_qB_{r}^i)=g_{r1}(1+[\\eta_{i1}-\\eta_{r1}]_qB_{r}^i)=0,\n$$\nby Lemma \\ref{thetas}.\nApplying $\\partial_{ir}$ to the  right  2-chain  in (\\ref{2-chain AB}) we get\n$$\nf_{(r,\\vec m, k)}g_{k1}(-q^{-\\eta_{i1}+\\eta_{r1}}+[\\eta_{i1}-\\eta_{r1}]_qA_{r}^i)=0.\n$$\nNow consider 3-chains. The  action of $\\partial_{lr}$ on the (\\ref{3-chain_AA}) produces\n\\begin{eqnarray}\n-f_{(i,\\vec \\ell, l)}q^{-h_\\alpha}f_{(r,\\vec \\rho, k)}g_{k,1}A_{l}^i\n+\nf_{(i,\\vec \\ell, l)}[h_\\alpha]_qf_{(r,\\vec \\rho, k)}g_{k,1}A_{l}^iA_{r}^i\n+\nf_{(i,\\vec \\ell, l)}q^{h_\\alpha}f_{(r,\\vec \\rho, k)}g_{k,1}A_{r}^i,\n\\nonumber\n\\end{eqnarray}\nwhich turns zero since\n$\n-q^{\\eta_{r1}-\\eta_{l1}}A_{l}^i\n+\n[\\eta_{l1}-\\eta_{r1}]_qA_{l}^iA_{r}^i\n+\nq^{\\eta_{l1}-\\eta_{r1}}A_{r}^i=0\n$.\nThe action of $\\partial_{lr}$ on (\\ref{3-chain_AB}) yields\n$$\nf_{(i,\\vec m, l)}g_{r1}B_{l}^i\n+\nf_{(i,\\vec m, l)}[h_\\alpha]g_{r1}A_{l}^iB_{r}^i\n+\nf_{(i,\\vec m, l)}q^{h_\\alpha}g_{r1}B_{r}^i.\n$$\nThis is vanishing since\n$\nB_{l}^i\n+\n[\\eta_{l1}-\\eta_{r1}]A_{l}^iB_{r}^i\n+\nq^{\\eta_{l1}-\\eta_{r1}}B_{r}^i\n=\nB_{l}^i\n+\n\\frac{[\\eta_{i1}-\\eta_{r1}]_q}{[\\eta_{i1}-\\eta_{l1}]_q}B_{r}^i=0.\n$\n\\end{proof}\nGiven a route $\\vec m = (m_1,\\ldots, m_k)$ such that $i<\\vec m$ let $A_{\\vec m}^i$ denote the product $A^i_{m_1}\\ldots A^i_{m_k}$.\nIntroduce elements $z_{i}\\in \\hat U_q(\\mathfrak{g}_-)\\mathfrak{g}_+$ of weight $\\varepsilon_1-\\varepsilon_i$ by\n$$\nz_{i-1}=g_{i1}+\\sum_{i< \\vec m< k\\leqslant N}f_{(i,\\vec m,k)}g_{k1}A_{\\vec m}^iB_k^i, \\quad i=2,\\ldots,N.\n$$\nAgain, the summation includes empty $\\vec m$.\n\\begin{propn}\n$\ne_{\\alpha} z_{i}=\n0 \\mod\n\\hat U_q(\\mathfrak{g})e_\\alpha\n$,\nfor all $\\alpha \\in \\Pi^+_{\\mathfrak{g}'}$\nand $i=1,\\ldots,N-1$.\n\\label{positive}\n\\end{propn}\n\\begin{proof}\nBy Lemma \\ref{[e,g]},  the vectors $g_{2'1}$ and $z_{N-1}=g_{1'1}$ are normalizing the left ideal $\\hat U_q(\\mathfrak{g})\\mathfrak{g}'_+$,\nso is $z_{N-2}=g_{2'1}+f_1g_{1'1}B^{1'}_{2'}$.\nOnce the cases $i=2',1'$ are proved, we  further assume $i<2'$.\nIn view of Lemma \\ref{partial_+}, it is sufficient to show that $z_{i-1}$ is killed, modulo $\\hat U_q(\\mathfrak{g})\\mathfrak{g}'_+$,\nby all $\\partial_{lr}$ such that $\\varepsilon_l-\\varepsilon_r\\in \\Pi^+_{\\mathfrak{g}'}$. Observe that $z_{i-1}$ can be arranged into\na linear combination of chains, which are killed by  $\\partial_{lr}$, as in Lemma \\ref{chains_killed}.\n\\end{proof}\nThe elements  $z_{i}$, $i=1,\\ldots,N-1$,\nbelong to the normalizer $\\mathcal{N}$. They form the set of positive generators of $Z_q(\\mathfrak{g},\\mathfrak{g}')$\nfor symplectic $\\mathfrak{g}$. In the orthogonal case, the positive part of $Z_q(\\mathfrak{g},\\mathfrak{g}')$ is generated by $z_{i}$, $i=1,\\ldots,N-2$.\n\n\n\\vspace{20pt}\n{\\bf Acknowledgements}.\nThis research is supported in part by the RFBR grant 12-01-00207-a.\nWe are grateful to the Max-Plank Institute for Mathematics in Bonn for hospitality and\nexcellent research atmosphere.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\nTranslating thoughts into commands understandable by external applications and devices is the basic principle ruling the development of Brain Computer Interfaces (BCIs) \\cite{singh2021comprehensive}. \nThe most appreciated method to collect neural signals is the electroencephalogram (EEG), having that it records data with non-invasive surface sensors called electrodes, it is sufficiently low-cost and with possible high temporal and spatial resolution \\cite{craik2019deep}. Moreover, the EEG signals are characterized by rhythms, whose fluctuations may be exploited to detect specific brain states \\cite{vaid2015eeg}. Among these brain conditions, the imagination of voluntary movements, called Motor Imagery (MI), may be observed over the primary sensorimotor cortex with amplitude variations of the $\\mu$ and $\\beta$ rhythms \\cite{wriessnegger2018frequency} \\cite{dai2019eeg}. These effects can be exploited to create MI EEG-based BCIs that can be used for a variety of applications spanning from rehabilitation procedures to the control of wheelchair movements \\cite{singh2021comprehensive}, and in conjunction with virtual and augmented reality \\cite{kohli2022review}.\\\\\nTherefore, one of the main BCI life-cycle components is represented by feedback, which benefits from the evolution and improvement of Artificial Intelligence (AI) approaches \\cite{cao2020review} in predicting the different brain states to be translated into system commands.\\\\\nConsequently, \\textit{how have the AI techniques evolved in and influenced the field of MI EEG-based BCIs, considering the great number of possibilities offered by these systems?}\\\\ \nThis work aims at providing a brief overview and discussion on this topic by analysing the AI approaches that have been applied to  some representative datasets present in the domain literature. \n\nTherefore, in Section~\\ref{sec:overview} the paper firstly provides an overall view of the timeline related to MI EEG-based BCIs and justifies the choice of specific datasets, that are described in Section~\\ref{sec:datasets}. An overview of the AI techniques applied to these data is provided in Section~\\ref{sec:ai} and observations on the presented AI approaches are discussed in Section~\\ref{sec:discussion}. Finally, conclusions are drawn in Section~\\ref{sec:conclusion}.\n\n\\section{Overview}\\label{sec:overview}\nThe research on MI EEG-based BCIs has become particularly prolific in the last years.\nIn fact, typing a quick query on Scopus\\footnote{\\url{https:\/\/www.scopus.com\/}} title, abstract and keywords of indexed works, i.e., \\textit{TITLE-ABS-KEY((mi OR motor AND imagery OR  motor AND imagination)  AND  (eeg OR electroencephalographic) AND based AND (bci OR  brain AND computer AND interface))}, we obtain the graphic in Figure~\\ref{fig:MI-EEG-based-BCI-query}. Notice that the search was conducted during September 2022.\\\\\nBesides having a clear understanding of a constant increase of publications on these topics starting from 2017 and having for now its apex in 2021, Figure~\\ref{fig:MI-EEG-based-BCI-query} provides a story of the early phases of the MI EEG-based BCIs, which set the foundation for later works.\\\\\nAn initial period (1996-2003) during which the BCI community began to discuss these topics was followed by a discrete boost (2004) of the research production probably due to the insightful work of \\textit{Schalk et al.} \\cite{schalk2004bci2000} and the outcome of the \\textit{BCI Competition 2003} \\cite{blankertz2004bci}.\\\\ \nStarting from the evident direction took by many laboratories concerning the development of systems capable of enabling a mean of communication and control for patients with severe motor disabilities, \\textit{Schalk et al.} presented the BCI2000\\footnote{\\url{https:\/\/www.bci2000.org\/mediawiki\/index.php\/Main_Page}}, which is an open-source platform that allows the management of BCI systems and that remains active and maintained to the present day. \nMoreover, not only some of the authors of \\cite{schalk2004bci2000} were involved in the \\textit{BCI Competition 2003}, but also the names of the researchers of the pioneer work of 1996 \\cite{kalcher1996graz} obtained with the Scopus search, unsurprisingly appear. \nIn fact, they contributed to dataset III titled \\textit{Motor Imagery}, which presented data acquired on the main central cortical electrodes (C\\{3,4,z\\}) during the imagination of left or right hand movements, with the main aim of providing a continuous feedback to the BCI users. The winning strategy to the proposed problem was presented by \\textit{Lemm et al.} \\cite{lemm2004bci}, who tried to disclose motor intention by characterizing the EEG signal rhythmic activity through complex Morlet wavelet \\cite{torrence1998practical} application and using a probabilistic model to predict left or right hand MI.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{MI-EEG-based-BCI-query.png}\n\t\\caption{Number of papers per year obtained by querying Scopus on MI EEG-based BCI related keywords (search conducted during September 2022).}\n\t\\label{fig:MI-EEG-based-BCI-query}\n\\end{figure}\n\nAfterwards, the \\textit{BCI Competitions - Berlin Brain Computer-Interface}\\footnote{\\url{https:\/\/www.bbci.de\/competition\/}} datasets quickly became benchmarks on which test strategies to provide more efficient and reliable BCI systems.\\\\\nBy adding to the first query the string \\textit{(bci AND competition AND  (2003 OR ii OR iii OR iv))}, representing the BCI Competition II (or 2003), III and IV, and limiting the results to year 2021 only, 27 papers are provided in output, of which 23 use the BCI Competition IV datasets (especially, 2a and 2b). Considering that Figure~\\ref{fig:MI-EEG-based-BCI-query} reports 97 works published in 2021 and that the screening through Scopus is done only on the title, abstract, and keywords of the works indexed by this search engine, the number of publications using the \\textit{BCI Competition IV dataset 2a} and \\textit{2b} \\cite{tangermann2012review} seems to be fairly high and justifies a deeper analysis concerning the AI techniques tested on them to better understand their evolution during a long time span.\n\nHowever, these datasets present EEG recordings acquired on a restricted number of subjects, considering a restricted number of electrodes and experimental conditions (as detailed in Section~\\ref{sec:datasets}). Wanting to have a general overview of the evolution of AI approaches in MI EEG-based BCIs and noticing its increased use as a benchmark in the last 10 years, the \\textit{EEG Motor Movement\/Imagery Dataset} \\cite{schalk2004bci2000} \\cite{goldberger2000physiobank} collected from a larger population, using a different montage, and considering diverse MI tasks, but using the BCI2000 system, is also considered. \n\nMoreover, a great attention has been given to wearable technologies, especially in the last few years, since the necessity of moving the use of BCIs from medical and laboratory environments to real-world scenarios is becoming more pressing due to a variety of needs like developing in-home rehabilitation tools \\cite{daly2015brain}, exploiting customer-grade devices \\cite{vasiljevic2020brain}, and providing continuous assistive technologies \\cite{minguillon2017trends} that patients could easily use alone without having to buy expensive equipment.\\\\\nFollowing these principles, \\textit{Peterson et al.} \\cite{peterson2020feasibility} have collected the \\textit{MI-OpenBCI} dataset using wearable low-cost technologies to record MI EEG-based BCI data. Therefore, an overview of this dataset is provided to have a closer look on future developments of these new systems and the changing role of AI when facing them.\n\n\n\\section{Datasets}\\label{sec:datasets}\nConsidering the brief overview presented in Section~\\ref{sec:overview}, the datasets that will be at the center of this paper analysis are the \\textit{BCI Competition IV dataset 2a} (2012) and \\textit{2b} (2012) \\cite{tangermann2012review}, the \\textit{EEG Motor Movement\/Imagery Dataset} (2009) \\cite{schalk2004bci2000} \\cite{goldberger2000physiobank}, and the \\textit{MI-OpenBCI} (2020) one \\cite{peterson2020feasibility}. In this section a concise description of their characteristics is reported for completeness.\n\n\n\\subsection{BCI Competition IV dataset 2a}\nThe \\textit{BCI Competition IV dataset 2a} \\cite{tangermann2012review} is provided under the name \\textit{Continuous Multi-class Motor Imagery}. In fact, it has been collected from 9 subjects executing a cue-based BCI paradigm consisting of left\/right hand, both feet and tongue MI. Each subject participated in different days to 2 experimental sessions containing 6 experimental runs of 48 trials each.\\\\ \nFigure~\\ref{fig:montage-BCI-CompetitionIV-dataset2a} depicts the montage consisting of 22 Ag\/AgCl electrodes for scalp recording, reference and ground electrodes placed on the left and right mastoids, and 3 monopolar electrooculogram channels (positioned to provide reference for artifact removal).\\\\\nThe signals have been acquired with 250Hz sampling rate and the dataset authors provided them bandpass (0.5-100Hz) and notch (50Hz) filtered. The  noisy trials were removed by manual screening of experts.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.7\\linewidth]{montage-BCI-CompetitionIV-dataset2a.png}\n\t\\caption{Electrode positioning used for the recording of BCI Competition IV dataset 2a. Notice that on the right are reported the positions of the electrooculogram channels. This setting is also used for BCI Competition IV dataset 2b, considering electrodes C\\{3,4,z\\} only. Original montage taken from \\cite{tangermann2012review}.}\n\t\\label{fig:montage-BCI-CompetitionIV-dataset2a}\n\\end{figure}\n\n\\subsection{BCI Competition IV dataset 2b}\nThe \\textit{BCI Competition IV dataset 2b} \\cite{tangermann2012review} presents the following short description: \\textit{Session-to-Session Transfer of a Motor Imagery BCI under Presence of Eye Artifacts}.\nIn fact, the motivation driving its presence into the BCI competition was to provide a correct EEG signal classification despite having data affected by ocular noise.\\\\\nThe dataset has been collected from 9 right-handed healthy subjects using only the central cortical electrodes (C\\{3,4,z\\}), the reference and ground channel and the 3 monopolar electrooculogram channels (depicted in Figure~\\ref{fig:montage-BCI-CompetitionIV-dataset2a}), during a cue-based BCI paradigm of left\/right hand MI \\cite{leeb2007brain}. The experiment was designed to have 2 separate sessions consisting of 6 runs of 10 trials each without user feedback and 3 separate sessions consisting of 4 runs of 40 trials each with online feedback.\\\\\nThe EEG signal was sampled at 250Hz and bandpass (0.5-100Hz) and notch (50Hz) filtered.\n\n\\subsection{EEG Motor Movement\/Imagery Dataset}\nThe \\textit{EEG Motor Movement\/Imagery Dataset}\\footnote{Dataset description and recordings available at \\url{https:\/\/physionet.org\/content\/eegmmidb\/1.0.0\/}.} \\cite{schalk2004bci2000} \\cite{goldberger2000physiobank} has been collected by using the BCI2000 system introduced in Section~\\ref{sec:overview} and considering an electrode montage of 64 channels respecting the 10-10 International System (excluding electrodes Nz, F\\{9,10\\}, FT\\{9,10\\}, A\\{1,2\\}, TP\\{9,10\\}, P\\{9,10\\}).\\\\\n109 subjects were asked to perform a cue-based experiment in a single session consisting of 14 experimental runs divided in 2 baseline recordings (eyes open and closed) and 3 runs per experimental condition. The experimental conditions consisted of the motor execution or imagination of left and right hand movements or both hands and feet movements.\\\\\nThe signal was acquired with a sampling rate of 160Hz and no pre-processing was performed on the data.\n\n\\subsection{MI-OpenBCI}\nThe \\textit{MI-OpenBCI} dataset \\cite{peterson2020feasibility} \\cite{peterson2022motor} has been collected and provided to the research community very recently (year 2020).\\\\\nThe authors performed a feasibility study on the use of consumer-grade MI EEG-based BCI system. Therefore, they employed the OpenViBE software platform\\footnote{\\url{http:\/\/openvibe.inria.fr\/}.}, the OpenBCI Cyton sensing board with Daisy Module\\footnote{\\url{https:\/\/docs.openbci.com\/GettingStarted\/Boards\/DaisyGS\/}.} and used Electrocap System II with 19 electrodes to acquire the EEG signal wirelessly. Of these electrodes, 16 were effectively used (F\\{z,3,4,7,8\\}, C\\{z,3,4\\}, T\\{3,4,5,6\\}, P\\{z,3,4\\}). Reference and ground electrodes were placed on the left and right ear lobes.\\\\\nNotice that electromyographic signals were also acquired, but details are not provided here, having that these physiological data are not the focus of the present work.\\\\\n12 healthy right-handed subjects with no prior experience with BCIs performed a cue-based MI of the dominant hand grasping or resting during a single experimental session. 4 runs of 20 trials were executed.\\\\\nThe EEG signal was acquired with sampling rate of 125Hz and filtered with a 3rd order Butterworth bandpass-filter (0.5-45Hz). \n\n\\section{Artificial intelligence approaches on the investigated datasets}\\label{sec:ai}\nHaving provided a brief description of each dataset under scrutiny, it is possible to proceed with the analysis of the AI approaches applied on them during their lifetime. \\\\\nThe analyses will start from the first publication presenting results on the investigated dataset and then proceed by considering some relevant works that allow further discussions on the influence that AI has on MI EEG-based BCIs. \n\n\\subsection{On the BCI Competition IV dataset 2a}\\label{sec:bciiv2a}\nThe winners of the \\textit{BCI Competition IV dataset 2a} presented their approach in a dedicated publication \\cite{ang2012filter}. \\textit{Ang et al.} wanted to enhance the performances of the Common Spatial Pattern (CSP) algorithm, and thus proposed the use of Filter Bank CSP (FBCSP) \\cite{ang2008filter}. FBCSP consists of 4 phases, i.e., the band-pass filtering, the spatial filtering, the feature selection based on mutual information, and the classification phase. Concerning this last step, the authors chose to apply the Na\u00efve Bayesian Parzen Window classifier, having obtained good results from a previous competition \\cite{ang2008filter}. \nMoreover, having that the dataset presented 4 conditions and that the algorithm was initially developed for binary classification, they propose multi-class extensions of FBCSP by using the divide and conquer, pairwise and one versus rest approaches. This last strategy was chosen as the one to present to the competition having that it obtained better or similar average Cohen's kappa value (0.57) with less computational cost, datum that is particularly important when thinking of the real-time nature that BCI responses should have.\n\nIn 2015, \\textit{Nicolas-Alonso et al. }\\cite{nicolas2015adaptive} thought about this issue and proposed an adaptive semi-supervised classification that could also face the non-stationarity of the EEG signals. Therefore, they firstly reduced non-stationarity by using exponentially weighted moving average before performing the classification, which is done by using the newly developed method based on spectral regression kernel discriminant analysis. In fact, the authors train the model with labelled data and then introduce a self-training algorithm using new unlabelled data to sequentially update the model. \nNotice that the feature extraction is done through CSP.\\\\\nThe joint semi-supervised learning and adaptive processing provided better results with a significant computational efficiency compared to previous literature works considering the 4-class problem (0.70 Cohen's kappa coefficient and 77\\% accuracy).\\\\\nThe authors further improved the performances (0.74 maximum average kappa value) by using the stacked regularised linear discriminant analysis for classification and thus integrating temporal, spectral, and spatial information \\cite{nicolas2015adaptiveIEEE}.\n\nAn adaptive learning strategy has been also proposed by \\textit{Raza et al.} \\cite{raza2016adaptive}, in 2016, who exploited the covariate shift detection based on exponentially weighted moving average to monitor the EEG-based BCIs. The authors obtained an average accuracy of 76.70\\% employing the upper bound version of their methodology.\n\nIn 2017, \\textit{Jafarifarmand et al.} \\cite{jafarifarmand2017new} proposed a new framework that could allow the improvement of multi-class BCIs. The process consisted of a feature extraction step using artifact rejected CSP and a self-regulated\nadaptive resonance theory based neuro-fuzzy classifier that is able to model the non-stationarity of the EEG signal as well as uncertainties in the data. The authors obtained better results in respect to the competition winners with an average kappa value of 0.63.\n\nIn 2019, \\textit{Olivas-Padilla} \\cite{olivas2019classification} do not only test their strategy on \\textit{BCI Competition IV dataset 2a} but also acquire a proprietary dataset using OpenBCI with a similar experimental paradigm. The feature extraction was again provided by a novel modification of the CSP algorithm, i.e., the discriminative FBCSP, to model the spatial and spectral characteristics of the EEG signal. A modular network composed of 4 Convolutional Neural Networks (CNNs) specialised in the binary classification of combinations of the 4 experimental conditions, was used to provide a final classification. Notice that Bayesian optimisation was employed for hyperparameter selection. The authors' model achieved 80.03\\% accuracy and a kappa score of 0.61.\n\nIn 2019, deep learning methods were also investigated by \\textit{Majidov \\& Whangbo} \\cite{majidov2019efficient} who highlighted the necessity of these models to use a good number of data. Therefore, they proposed a pipeline consisting of different combinations of the following steps: (i) a data augmentation step based on sliced moving windows, (ii) the extraction of the power spectrum density from 3 EEG rhythms and the application of FBCSP, (iii) information theoretical feature extraction, (iv) tangent space mapping feature extraction and, (v) wrapper-based feature-selection with particle swarm optimization. The classification was performed with a 1D CNN, avoiding deeper networks that could overfit and obtaining an average accuracy of 87.94\\%.\n\nIn the same year (2019), 83\\% accuracy and 0.80 Cohen's kappa value were instead achieved by \\textit{Zhang et al.} \\cite{zhang2019novel}. The authors proposed the use of one-versus-rest FBCSP to extract characterising features for each of the 4 classes and a deep architecture based on the CNN and Long Short Term Memory (LSTM) models. Notice that they performed the classification by training the model on the data merged from all the subjects and then evaluating the data of each subject separately, obtaining a subject-invariant strategy.\n\nA year later (2020), \\textit{Luo et al.} \\cite{luo2020motor} proposed a novel ensemble support vector learning based approach to combine event related synchronisation\/desynchronisation features typical of the MI domain. The authors employed class discrepancy-guided sub-band filter-based common spatial pattern to extract more separable and stable features before the classification step. The proposed approach achieved an average kappa value of 0.60.\n\nFinally, in 2021, a CNN model built on the inception-time network called \\textit{EEG-inception} \\cite{zhang2021eeg} and, in 2022, a transfer learning-based CNN and LSTM hybrid deep learning model \\cite{khademi2022transfer} have been  proposed. \\textit{EEG-inception} is directly fed with the raw EEG data augmented with noise addition, avoiding complex pre-processing steps and decreasing the possibility of overfitting. 88.39\\% average accuracy was obtained on the 4 classes.\\\\\nSimilarly, \\textit{Khademi et al.} \\cite{khademi2022transfer} firstly applied a data augmentation by cropping and secondly provided a time-frequency characterization of the EEG signal through continuous wavelet transform. Afterwards, the authors exploited ResNet-50 and Inception-v3, 2 pre-trained CNNs, to provide a transfer learning strategy supporting their CNN\/LSTM model. The mean kappa values obtained by using the provided approach were 0.86 and 0.88 for the ResNet-50 and Inception-v3 models, respectively. The accuracy values were around 90\\% and 92\\%.\n\nTable~\\ref{tab:bci_civ_2a} summarises the time-line of the presented AI strategies and highlights (bold) the best achieved results.\n\n\\begin{table}\n\t\\caption{\\textit{BCI Competition IV dataset 2a} time-line on AI strategies summary. Best results are highlighted (bold).}\n\t\\label{tab:bci_civ_2a}\n\t\\resizebox{\\textwidth}{!}{%\n\t\\begin{tabular}{ccp{7.5cm}cc}\n\t\t\\hline\n\t\tPaper & Year & Strategy & Cohen's kappa & Accuracy \\\\\n\t\t\\hline\n\t\t\\cite{ang2012filter} & 2012 & FBCSP $+$ Na\u00efve Bayesian Parzen Window classifier & 0.57 & NA \\\\\n\t\t\\cite{nicolas2015adaptive} & 2015 & CSP $+$ adaptive semi-supervised classification & 0.70 & 77.00\\% \\\\\n\t\t\\cite{nicolas2015adaptiveIEEE} & 2015 & Added on \\cite{nicolas2015adaptive} stacked regularised linear discriminant analysis & 0.74 & NA \\\\\n\t\t\\cite{raza2016adaptive} & 2016 & Adaptive learning with covariate shift detection & NA & 76.70\\% \\\\\n\t\t\\cite{jafarifarmand2017new} & 2017 & Artifact rejected CSP $+$ self-regulated neuro-fuzzy framework & 0.63 & NA \\\\\n\t\t\\cite{olivas2019classification} & 2019 & FBCSP $+$ 4 CNNs $+$ Bayesian optimisation & 0.61 & 80.03\\% \\\\\n\t\t\\cite{majidov2019efficient} & 2019 & Data augmentation $+$ feature engineering $+$ 1D CNN & NA & 87.94\\% \\\\ \n\t\t\\cite{zhang2019novel} & 2019 & One-vs-rest FBCSP $+$ CNN and LSTM (subject-invariant) & \\textbf{0.80} & 83.00\\% \\\\\n\t\t\\cite{luo2020motor} & 2020 & CSP variation $+$ ensemble support vector learning & 0.60 & NA \\\\\n\t\t\\cite{zhang2021eeg} & 2021 & \\textit{EEG-inception} & NA & 88.39\\% \\\\\n\t\t\\cite{khademi2022transfer} & 2022 & Data augmentation $+$ Inception-v3 & NA & \\textbf{92.00\\%} \\\\\n\t\t\\hline\n\t\\end{tabular}\n}\n\\end{table}\n\n\n\\subsection{On the BCI Competition IV dataset 2b}\nA good number of works are in common between the two BCI Competition IV datasets under analysis. In fact, the winners using the \\textit{BCI Competition IV dataset 2b} were again \\textit{Ang et al.} \\cite{ang2012filter}, who used the same approach described for \\textit{BCI Competition IV dataset 2a} at the beginning of Section~\\ref{sec:bciiv2a}. They obtained an average accuracy around the 60\\%.\\\\\nSimilarly, the work of \\textit{Raza et al.} \\cite{raza2016adaptive} appears again in the performed search. The average accuracy obtained on dataset 2b was of 73.33\\% using the upper bound version of the proposed strategy.\n\nIn 2019, \\textit{Zhu et al.} \\cite{zhu2019separated} proposed an end-to-end deep learning framework based on the transfer of knowledge obtained from previously analysed subjects with the main aim of removing the training phase from MI BCIs. CSP was used to extract features in the temporal domain and the authors introduced a separated-channel CNN to provide the correct characterisation of the multi-channel EEG signal. Besides testing the proposed strategy on the scrutinised dataset, they performed analyses on a proprietary recorded set of data, considering left\/right hand MI. Notice that a single subject data was used as the test set, while the remaining subject data were exploited for model training. A comparison with widely used traditional machine learning techniques was also performed considering the K-nearest neighbour, logistic regression, linear discriminant analysis and Support Vector Machine (SVM) classifiers. Therefore, close performances were detected between the novel approach without transfer learning and the traditional models, especially considering the information transfer rate on the BCI Competition IV dataset 2b. Instead, better average results were provided when the transfer learning was applied. The proposed approach achieved 0.83 Information Transfer Rate (ITR), while the best result for the traditional techniques was obtained by the SVM with an ITR value of 0.02. The accuracy values were around 64\\% and 50\\% for the two strategies, respectively.\n\nInstead, in the same year, \\textit{Malan \\& Sharma} \\cite{malan2019feature} proposed a paradigm shift by considering a better definition of the feature vector to enhance a SVM classifier performances. Therefore, after extracting features by considering frequency-related and statistical measures, the authors performed feature selection by proposing a regularised neighbourhood component analysis. In fact, the authors wanted to reduce the feature vector by selecting the features providing the maximum accuracy or minimum generalisation error.\\\\\nThe final results show that the proposed algorithm was able to provide better average performances in terms of accuracy (80.70\\%), kappa coefficient (0.62), precision (0.85), recall (0.79), specificity (0.83) and F1-score (0.81) by using a lower number of features (6 on 42) when compared with feature selection techniques based on ReliefF, principal component analysis and genetic algorithm.\n\nAn year later (2020), the ensemble support vector learning strategy of \\textit{Luo et al.} \\cite{luo2020motor} described in Section~\\ref{sec:bciiv2a} provided 0.71 average max kappa value on the analysed dataset.\\\\ \nEven \\textit{Zhang et al.} \\cite{zhang2021eeg} (2021) tested their \\textit{EEG-inception} model on both the BCI Competition IV datasets and obtained the 88.60\\% average accuracy on dataset 2b.\n\nFinally, \\textit{Malan \\& Sharma} \\cite{malan2022motor} presented a new work in 2022, focusing again on the optimisation of the feature vector to improve a SVM classifier performances. Therefore, they proposed a novel methodology consisting of (i) dual tree complex wavelet transform based filter bank, (ii) CSP for spatial feature extraction from the previously obtained EEG sub-bands, and again (iii) a regularised neighbourhood component analysis. Comparing their approach with other CSP variations, they obtained better results in terms of accuracy (84\\%) and kappa coefficient (0.68).\n\nTable~\\ref{tab:bci_civ_2b} summarises the time-line of the presented AI strategies and highlights (bold) the best achieved results.\n\n\\begin{table}\n\t\\caption{\\textit{BCI Competition IV dataset 2b} time-line on AI strategies summary. Best results are highlighted (bold).}\n\t\\label{tab:bci_civ_2b}\n\t\\resizebox{\\textwidth}{!}{%\n\t\\begin{tabular}{ccp{7.5cm}cc}\n\t\t\\hline\n\t\tPaper & Year & Strategy & Cohen's kappa & Accuracy \\\\\n\t\t\\hline\n\t\t\\cite{ang2012filter} & 2012 & FBCSP $+$ Na\u00efve Bayesian Parzen Window classifier & NA & 60.00\\% \\\\\n\t\t\\cite{raza2016adaptive} & 2016 & Adaptive learning with covariate shift detection & NA & 73.33\\% \\\\\n\t\t\\cite{zhu2019separated} & 2019 & CSP $+$ knowledge transfer $+$ CNN & NA & 64.00\\% \\\\\n\t\t\\cite{malan2019feature} & 2019 & Feature selection through regularised neighbourhood component analysis & 0.62 & 80.70\\% \\\\\n\t\t\\cite{luo2020motor} & 2020 & CSP variation $+$ ensemble support vector learning & \\textbf{0.71} & NA \\\\\n\t\t\\cite{zhang2021eeg} & 2021 & \\textit{EEG-inception} & NA & \\textbf{88.60\\%} \\\\\n\t\t\\cite{malan2022motor} & 2022 & Feature vector optimisation $+$ SVM & 0.68 & 84.00\\% \\\\\n\t\t\\hline\n\t\\end{tabular}\n}\n\\end{table}\n\n\n\\subsection{On the EEG Motor Movement\/Imagery Dataset}\nThe \\textit{EEG Motor Movement\/Imagery Dataset} has not been used as much as the previous 2 datasets, but provides a good starting point to perform analyses on a larger pool of data considering both executed and imagined movements.\\\\\nThe first work by \\textit{Sleight et al.} (2009) \\cite{sleight2009classification} reporting results on this dataset performed a particularly interesting analysis having that the authors' aim was not to discriminate different MI tasks, but the imagined from the real movements. Therefore, they firstly considered feature extraction based on independent component analysis on different frequency bands and a channel selection, to provide a well characterised feature vector as input to a SVM model. After performing different analyses, the best average accuracy (69\\%) was obtained by considering a subject-based approach on data normalized per frequency band and avoiding the use of independent component analysis.\n\nAlmost 5 years later, in 2013, \\textit{Park et al.} \\cite{park2013augmented} proposed the augmented complex CSP to deal with non-circular EEG signals. The SVM model (with a Gaussian kernel) was employed to classify data from a synthetic dataset and the one under scrutiny, considering left\/right hand MI. Among the 109 subjects, 56 performed over the 64\\% accuracy.\n\nIn 2017, a LSTM model was proposed \\cite{zhang2017intent} to obtain intent recognition considering all the experimental MI conditions of 10 subjects. A hyper-parameter selection through orthogonal array application was also performed obtaining a final average accuracy of 95.53\\%.\\\\\nAnother deep learning approach has been chosen in 2018 by \\textit{Dose et al.} \\cite{dose2018end}, who considered the combination of different MI tasks for signal classification, i.e., left\/right hand, open eyes with left\/right hand, feet with open eyes and left\/right hand conditions. The authors designed a CNN architecture that could be complemented with a transfer learning strategy. The average accuracy values obtained on the different task combinations without (with) the transfer learning approach were 80.38\\% (86.49\\%), 69.82\\% (79.25\\%) and 58.58\\% (68.51\\%), respectively. A general performance increase was detected when applying the transfer learning strategy.\n\nAn ensemble learning approach for MI data classification on 10 subjects was instead chosen by \\textit{Zhang et al.} \\cite{zhang2018converting} in 2018. Firstly, the authors exploited a recurrent neural network and a CNN model to learn features and then an autoencoder for feature adaption. Afterwards, they applied eXtreme Gradient Boosting for intent recognition, obtaining an average accuracy of 95.53\\%.\n\nThe prediction of different MI conditions (left\/right hand, plus rest, plus feet) was also provided by \\textit{Wang et al.} \\cite{wang2020accurate} (2020), who exploited a model based on a pre-trained CNN, \\textit{EEG-Net}, obtaining the following accuracy values: 82.43\\%, 75.07\\% and 65.07\\%.\n\nFinally, in 2021, \\textit{Varsehi \\& Firoozabadi} \\cite{varsehi2021eeg} focused on a channel selection method, taking as an assumption the causal interaction between channels. Therefore, the authors proposed a novel channel selection algorithm based on Granger causality analysis and after different processing steps, classified the resulting data by using the linear\/quadratic discriminant analysis, kernel Fisher discriminant, SVM, multi-layer perceptron, learning vector quantisation, neural network, k-nearest neighbour and decision tree classifiers. The best results obtained by the proposed strategy and using 8 channels only were of 93.03\\% accuracy, 92.93\\% sensitivity, and 93.12\\% specificity.\n\nTable~\\ref{tab:physionet} summarises the time-line of the presented AI strategies and highlights (bold) the best achieved results.\n\n\\begin{table}\n\t\\caption{\\textit{EEG Motor Movement\/Imagery Dataset} time-line on AI strategies summary. Best results are highlighted (bold).}\n\t\\label{tab:physionet}\n\t\\resizebox{\\textwidth}{!}{%\n\t\\begin{tabular}{ccp{7.5cm}cc}\n\t\t\\hline\n\t\tPaper & Year & Strategy & Cohen's kappa & Accuracy \\\\\n\t\t\\hline\n\t\t\\cite{sleight2009classification} & 2009 & Imagined vs real movements $+$ independent component analysis $+$ SVM  & NA & 69.00\\% \\\\\n\t\t\\cite{park2013augmented} & 2013 & Augmented complex CSP $+$ SVM (56 subjects) & NA & 64.00\\%\\\\\n\t\t\\cite{zhang2017intent} & 2017 & LSTM (all MI conditions) $+$ hyperparameter selection through orthogonal array & NA & \\textbf{95.53\\%} \\\\\n\t\t\\cite{dose2018end} & 2018 & CNN (left\/right hand MI) $+$ transfer learning & NA & 86.49\\% \\\\\n\t\t\\cite{zhang2018converting} & 2018 & Recurrent neural network $+$ CNN for feature extraction $+$ eXtreme Gradient Boosting (all MI conditions) & NA & \\textbf{95.53\\%} \\\\\n\t\t\\cite{wang2020accurate} & 2020 & \\textit{EEG-Net} (left\/right hand MI) & NA & 82.43\\%\\\\\n\t\t\\cite{varsehi2021eeg} & 2021 & channel selection based on Granger causality & NA & 93.03\\% \\\\\n\t\t\\hline\n\t\\end{tabular}\n}\n\\end{table}\n\n\\subsection{On the MI-OpenBCI dataset}\nBeing very recent (2020), the \\textit{MI-OpenBCI} dataset has not yet been deeply analysed by the research community.\\\\\nThe dataset authors, \\textit{Peterson et al.},  \\cite{peterson2020feasibility} proposed data denoising and feature extraction based on CSP. The best results were achieved by the penalised time-frequency band CSP with an average accuracy of 83.30\\%. Notice that generalised sparse discriminant analysis was used for both feature selection and classification considering both online and offline scenarios.\\\\\nTherefore, the proposed methodology is set as a benchmark for future works.\\\\\nNotice that the works citing \\textit{Peterson et al.} paper have exploited it to make a point regarding the necessity of moving to low-cost and consumer-grade technologies to provide more ecological and easy-to-use BCIs.\\\\\nFor example, \\textit{Koo et al.} \\cite{koo2021demonstration} cite it when considering that locked-in patients do not have communication tools based on BCI paradigms that are easy to maintain, low cost and comfortable. Moreover, \\textit{Peterson et al.} work is taken as an example considering the use of open-source and low cost platforms \\cite{butsiy2021comprehensive} \\cite{zambrana2022validation} like the OpenBCI one.\n\n\\section{Discussion}\\label{sec:discussion}\nConsidering the general overview given on the AI techniques applied to the analysed datasets, observations can be made towards the scope of the present paper.\n\\begin{enumerate}\n\t\\item Initially, the BCI Competition IV datasets were considered as the sole test beds for novel MI detection techniques. Afterwards, researchers began to use them as terms of comparison for their proprietary data. The \\textit{EEG Motor Movement\/Imagery Dataset} seemed to immediately take the role of a benchmark.\n\t\\item The CSP and its variations have been widely used and remain between the most applied techniques for feature extraction, meant to provide a correct data characterisation as input to AI models.\n\t\\item From an initial application of statistical measures or SVM classifiers, the AI techniques evolved following this pattern: (i) traditional machine learning approaches, (ii) introduction of transfer or adaptive learning, (iii) focus on the feature vector and channel selection, (iv) deep learning architectures, especially based on CNN and LSTM, or ensemble techniques, and (v) exploitation of pre-trained networks.\n\t\\item The necessity of introducing data augmentation techniques has also appeared when the researches conducted with deep learning approaches became more mature.\n\t\\item The check on performances has shifted from the use of the sole kappa coefficient to an extreme tendency of accessing the system behaviour through accuracy only.\n\t\\item New technologies and demands from the general public have asked a paradigm shift related to the use of low-cost and easy-to-use devices, which should enable a rapid response from BCI systems and thus provide lightweight computation techniques to allow efficient and reliable feedback to the users as part of their requirements.\n\t\\item The acquisition of the \\textit{MI-OpenBCI} dataset has set a good standard to provide systems in line with the requirements reported at the previous point.\n\\end{enumerate}\nTherefore, the evolution of AI techniques is clearly detectable and followed the discoveries presented by the AI community. However, this evolution demanded changes in how the experimental paradigms and devices are made. Especially, the need for more data to feed to the models has become particularly pressing as well as the possibility of providing signals with higher signal-to-noise ratio.\\\\\nInterestingly, a return to simpler and quicker traditional machine learning models in combination with feature engineering strategies, may prove effective for new real-time and low-cost BCI systems. In fact, instead of answering the demands from a computational perspective, the user needs will probably become the main focus when developing new technologies. \n\n\\section{Conclusion}\\label{sec:conclusion}\nIn this paper, a brief over all survey on the AI techniques applied to specific MI EEG-based BCI datasets has been provided. \\\\\nStarting from the development of the BCI2000 platform and the opening of the BCI Competitions, great interest has been given by the field research community towards the generation of new approaches enabling the correct response of such systems.\\\\\nIn the presented overview, the evolution of AI techniques and their influence on MI EEG-based BCIs seemed to have been twofold: the AI approaches have provided better discrimination of experimental conditions and the devices used to collect data and control applications have changed according to the needs of the general public.\\\\\nHowever, many issues remain to be answered, starting from the necessity of finding reliable metrics for feedback procedure evaluations not only based on the sole accuracy but also accessing the usability, comfort, reliability and reproducibility of the new consumer-grade technologies.\n\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe past few years have witnessed a growing interest in an online learning-based Linear Quadratic (LQ) control,\nin which an unknown LTI system is controlled while guaranteeing a suitable scaling of the regret (defined as the average difference between the closed-loop quadratic cost and the best achievable one, with the benefit of knowing the plant's parameters)  over a desired horizon $[0,T]$.\n\nTable 1 summarizes the regret's scaling achieved by several recent works in the literature (\\cite{abbasi2011regret, mania2019certainty, cohen2019learning, lale2020explore}). As can be seen, the bounds scale like $\\sqrt{T}$, but also include an exponential term in $m$ (the dimension of the plant's state \\textit{plus inputs}) when an initial stabilizing controller is unavailable. Recently, \\cite{chen2020black} have shown that this dependency is unavoidable in this setting, at least for an initial exploration time period, by providing a matching lower bound. On closer inspection, this undesirable dependency stems from an exponential growth of the system's state during the initial period of learning, which eventually contributes a $\\mathcal{O}(m^m)$ term like that appearing on the last row of Table 1. Apart from negatively impacting regret, this transient state growth can also be damageable if, e.g., the linear plant to be controlled is in fact the linearization of a nonlinear process around some particular equilibrium, and the state can be driven outside the neighborhood of this equilibrium where the linearization is adequate.\n\n\nA natural idea to try and partially alleviate this effect in over-actuated systems is to reduce the ambient dimension $``m\"$ during the initial period by employing only a subset of all available actuators, before potentially switching to a different mode to simultaneously learn and control the plant. The goal of this paper is to show that this approach yields the desired result (a lower bound on the plant's state and the regret than in the presence of all actuators) and to provide a rigorous proof of the achieved $\\mathcal{O}((n+d_i)^{(n+d_i)})$ bound (where $d_i$ is number of actuators during initial exploration). We note that, while the idea is conceptually simple, obtaining these rigorous guarantees is not completely straightforward, and requires a revisit of the tools of \\cite{lale2020explore} to overcome a potential challenge: ensuring that, at the end of the period when a strict subset of actuators is used, all entries of the B-matrix (including those corresponding to unused actuators) are sufficiently well-learned for the closed-loop to become stable and the regret to remain appropriately bounded. This can only be achieved by adding additional exploratory noise with respect to the approach of \\cite{lale2020explore} and results in an additional linear term in the regret bound.\n\n\\begin{table}[h!]\n\\scriptsize\n\\caption{Overview of Prior Works}\n\\centering\n\\begin{tabular}{c c c}\n\n\\hline \\hline\nAlgorithm  & Regret& *\\\\ [1ex]\n\\hline\n\\cite{abbasi2011regret}&  $\\mathcal{O}(m^m\\sqrt{T})$ & No \\\\\n\\cite{mania2019certainty} & $\\mathcal{O}(poly(m)\\sqrt{T})$  & Yes \\\\\n\\cite{cohen2019learning} &  $\\mathcal{O}(poly(m)\\sqrt{T})$  & Yes  \\\\\n\\cite{lale2020explore}&  $\\mathcal{O}(m^m)+\\mathcal{O}(poly(m)\\sqrt{T})$  & No \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n* requirement for initial stabilizing controller.\n\nWhile this idea of leveraging over-actuation can generally be applied to any model-based type of online learning algorithms presenting the exponential scaling mentioned above, we focus on the special kind of methods based on ``Optimism in the Face of Uncertainty (OFU)''.\n\n\nOFU type of algorithms, which couple estimation and control design procedures, have shown their ability to outperform the naive certainty equivalence algorithm. \\cite{campi1998adaptive} propose an OFU based approach to address the optimal control problem for LQ setting with guaranteed asymptotic optimality. However, this algorithm only guarantees the convergence of average cost to that of the optimal control in the limit but does not provide any bound on the measure of performance loss in finite time. \\cite{abbasi2011regret} propose a learning-based algorithm to address\nthe adaptive control design of LQ control system in finite time with worse case regret bound of $\\mathcal{O}(\\sqrt{T})$ with exponential dependence in dimension. Using $l_2$-regularized least square estimation, they construct a high-probability confidence set around unknown parameters of the system and designed an algorithm that optimistically plays with respect to this set. Along this line, many works attempt to get rid of the exponential dependence with further assumption, e.g. highly sparse dynamics (see \\cite{ibrahimi2012efficient}) or access to a stabilizing controller (see \\cite{cohen2019learning}). Furthermore, \\cite{faradonbeh2017optimism} propose an OFU-based learning algorithm with mild assumptions and $\\mathcal{O}(\\sqrt{T})$ regret. This class of algorithms was extended by  \\cite{lale2020regret,lale2020adaptive} to LQG setting where there is only partial and noisy observations on the state of system. In addition, \\cite{lale2020explore} propose an algorithm with more exploration for both controllable and stabilizable systems. \n\nThe remainder of the paper is organized as follows:\nSection \\ref{sec:prelim} reviews the preliminaries, assumptions and presents the problem statement and formulation. Section\n\\ref{problem statement} overviews the proposed initial exploration (IExp), stabilizing OFU (SOFUA) algorithms and discusses in detail how to choose best actuating mode for the initial exploration purpose.  Furthermore, in Section \\ref{problem statement} the performance analysis (state norm upper-bound and regret bound) is given while leaving the details of the analysis to Section \\ref{sec:analysis} and technical theorems and proofs of analysis to Section\n\\ref{appendix}. Numerical experiments are given in Section \\ref{simulation}. Finally, Section \\ref{conclusion} summarizes the paper's key contributions.\n\n\n\n\\section{Assumptions and Problem Formulation}\\label{sec:prelim}\n\nConsider the following linear time invariant dynamics and the associated cost functional given by:\n\\begin{subequations}\n\t\\begin{align}\n\t\tx_{t+1} &=A_{*}x_{t} + B_{*}u_{t}+\\omega_{t+1}\\label{eq:dyn}\\\\\n\t\tc_{t} &=x_{t}^\\top Q_*x_{t} + u_{t}^\\top R_*u_{t}\\label{eq:obs}\n\t\\end{align}\n\\end{subequations}\nwhere the plant and input matrices $A_{*}\\in  \\mathbb{R}^{n \\times n}$ and  $B_{*}\\in  \\mathbb{R}^{n \\times d}$ are initially unknown and have to be learned and $(A_*,B_*)$ is controllable. $Q_*\\in  \\mathbb{R}^{n \\times n} $ and $R_*\\in  \\mathbb{R}^{d \\times d} $ represent known and positive definite matrices. $\\omega_{t+1}$ denotes process noise, satisfying the following assumption. \n\n\\begin{assum}\n\n\\label{Assumption 1}\nThere exists a filtration $\\mathcal{F}_{t}$ such that\n\n$(1.1)$ $\\mathbb{E}[\\omega_{t+1}\\omega_{t+1}^\\top|\\;\\mathcal{F}_{t}]=\\bar{\\sigma}_{\\omega}^2I_{n}$ for some $\\bar{\\sigma}_{\\omega}^2>0$;\n\n$(1.2)$ $\\omega_{t}$ are component-wise sub-Gaussian, i.e., there exists $\\sigma_{\\omega}>0$ such that for any $\\gamma \\in \\mathbb{R}$ and $j=1,2,...,n$\n\\begin{align*}\n\\mathbb{E}[e^{\\gamma\\omega_{j}(t+1)}|\\;\\mathcal{F}_{t}]\\leq e^{\\gamma^2\\sigma_{\\omega}^2\/2}.\n\\end{align*}\n\\end{assum}\n\n\n\nThe problem is designing a sequence $\\{u_t\\}$ of control inputs such that the regret $\\mathcal{R}_T$ defined by\n\n\\begin{align}\n\t\\mathcal{R}_T =\\sum _{t=1}^{T} \\bigg( x_{t}^\\top Q_*x_{t} + u^\\top_{t} R_* u_{t}-J_*(\\Theta_*, Q_*, R_*)\\bigg)\\label{eq:Reg} \n\\end{align}\n\nachieves a desired specification which scales sublinearly in $T$. The term $J_*(\\Theta_*, Q_*, R_*)$ in (\\ref{eq:Reg}) where $\\Theta_*=(A_*\\; B_*)^\\top$ denotes optimal average expected cost. For LQR setting with controllable pair $(A,\\;B)$ we have $J_*(\\Theta,Q,R)=\\bar{\\sigma}_{\\omega}^2 trace( P(\\Theta,Q,R))$, where $P(\\Theta,Q,R)$ is the unique solution of discrete algebraic riccati equation (DARE) and the average expected cost minimizing policy has feedback gain of \n\\begin{align*}\nK(\\Theta,Q,R)= -(B^\\top P(\\Theta, Q,R)B+R)^{-1}B^\\top P(\\Theta, Q, R)A.\n\\end{align*}\n\n\n\n\n\n\n\n\nWhile the regret's exponential dependency on system dimension appears in the long-run  in \\cite{abbasi2011regret} the recent results of \\cite{mania2019certainty} on the existence of a stabilizing neighborhood, makes it possible to design an algorithm that only exhibits  this dependency during an initial exploration phase (see \\cite{lale2020explore}).  \n\nAfter this period, the controller designed for any estimated value of the parameters is guaranteed to be stabilizing and the exponentially dependent term thus only appears as a constant in overall regret bound. As explained in the introduction, this suggests using only a subset of actuators during initial exploration to even further reduce the guaranteed upperbound on the state.\n\nIn the remainder of the paper, we pick the best actuating mode (i.e. subset of actuators) so as to minimize the state norm upper-bound achieved during initial exploration and characterize the needed duration of this phase for all system parameter estimates to reside in the stabilizing neighborhood. This is necessary to guarantee both closed loop stability and acceptable regret, and makes it possible to switch to the full actuation mode. \n\nLet $\\mathbb{B}$ be the set of all columns, $b^i_*$ ($i\\in \\{1,...,d\\}$) of $B_*$. An element of its power set $2^{\\mathbb{B}}$ is a subset $\\mathcal{B}_{j}$ $j\\in \\{1,..., 2^d\\}$ of columns corresponding to a submatrix $B_*^j$ of $B_*$ and mode $j$. For simplicity, we assume that $B^1_*=B_*$ i.e., that the first mode contains all actuators. Given this definition we write down different actuating modes dynamics with extra exploratory noise as follows\n\n\\begin{align}\n\tx _{t+1} ={\\Theta^i_{*}}^\\top z_{t}+B_*\\nu_t+\\omega_{t+1}, \\quad z_t=\\begin{pmatrix} x_t \\\\ u^i_t \\end{pmatrix}. \\label{eq:dynam_by_theta} \n\\end{align}\nwhere $\\Theta^i_*=(A_*,B^i_*)^\\top$ is controllable. \n\nThe associated cost with this mode is\n\\begin{align}\n\t\tc^i_{t} &=x_{t}^\\top Q_* x_{t} + {u^i_t}^\\top R^i_* u^i_t \\label{eq:obsswitch}\n\\end{align}\nwhere $R_*^i\\in \\mathbb{R}^{d_i\\times d_i}$ is a block of $R_*$ which penalizes the control inputs of the actuators of mode $i$. \n\n\nWe have the following assumption on the modes which assists us in designing proposed strategy.\n\n\n\n\n\n\\begin{assum}(Side Information) \\label{Assumption2}\n\\begin{enumerate}\n\\item  There exists $s^i$ and $\\Upsilon_i$ such that $\\Theta_{*}^i \\in \\mathcal{S}^i_c$ for all modes $i$ where\n\\begin{align*}\n  \\mathcal{S}^i_c =&\\{{\\Theta^i} \\in R^{(n+d_i)\\times n}  \\mid trace({\\Theta^i}^\\top \\Theta^i)\\leq ({s^i})^2,\\\\\n &\\text{$(A,B^i)$ is \n controllable,}\\\\\n &  \n\\|A+B^iK(\\Theta^i,Q_*,R^i_*)\\|\\leq  \\Upsilon_i<1  \\\\\n&\n\\text{and $(A,M)$ is observable,} \\text{where $Q=M^\\top M$} \\}.\n\\end{align*}\n.\n\\item There are known positive constants  $\\eta^i$, $\\vartheta_i$, $\\gamma^i$ such that $\\|B_*^i\\|\\leq \\vartheta_i$,\n\\begin{align}\n&\\sup_{\\Theta^i\\in \\mathcal{S}^i}\\|A_*+B^i_*K({\\Theta}^i,Q_*,R^i_*)\\|\\leq \\eta^i \\label{Assum3_1}\n\\end{align}\nand\n\\begin{align}\nJ_*(\\Theta_*^{i},Q_*,R^i_*)-J_*(\\Theta_*,Q_*,R_*)\\leq \\gamma^i. \\label{Assum3_4}\n\\end{align}\t\nfor every mode $i$.\n\\end{enumerate}\n\\end{assum}\n\nBy slightly abusing notation, we drop the superscript label for the actuating mode 1 (e.g. $\\Upsilon_1=\\Upsilon$, $s^1=s$, and $\\mathcal{S}_c^1=\\mathcal{S}_c$). It is obvious that $s^i\\leq s$ $\\forall i$. \n\nNote that the item (1) in Assumption \\ref{Assumption2} is typical in the literature of OFU-based algorithms (see e.g., \\cite{abbasi2011regret,lale2020explore}) while (2) in fact always holds in the sense that $\\sup_{\\Theta^i\\in \\mathcal{S}^i}\\|A_*+B^i_*K({\\Theta}^i,Q_*,R^i_*)\\|$ and $J_*(\\Theta_*^{i},Q_*,R^i_*)-J_*(\\Theta_*,Q_*,R_*)$ are always bounded (see e.g., \\cite{abbasi2011regret,lale2020explore}). The point of (2), then, is that upper-bounds on their suprema are available which can in turn be used to bound regret explicitly. The knowledge of these bounds does not affect Algorithms 1 and 2 but their value enters Algorithm 3 for determination of the best actuating mode and the corresponding exploration duration. In that sense \"best actuating mode\" should be understood as \"best given the available information\".\n\nBoundedness of $S^i_c$'s implies boundedness of $P(\\Theta^i,Q_*,R_*^i)$ with a finite constant $D_c^i$ (see \\cite{anderson1971linear}), (i.e., $D_c^i=\\sup \\{\\left\\lVert P(\\Theta^i,Q_*,R_*^i)\\right\\rVert \\mid\\Theta^i \\in \\mathcal{S}^i_c \\}$). We define $D=\\max_{i\\in \\mathcal{B}^*} D^i$. Furthermore, there exists $\\kappa^i_c>1$ such that $\\kappa^i_c=\\sup \\{\\left\\lVert K(\\Theta^i, Q_*, R_*^i)\\right\\rVert \\mid\\Theta ^i\\in \\mathcal{S}^i_c \\}$.\n\n\nRecalling that the set of actuators of mode $i$ is $\\mathcal{B}_{i}$, we denote its complement by $\\mathcal{B}_i^c$ (i.e. $\\mathcal{B}_i\\cup \\mathcal{B}_i^c=\\{1,...,d\\}$). Furthermore, we denote the complement of control matrix $B_*^i$ by $\\bar{B}^i_*$.\n\n\nIf some modes fail to satisfy Assumption\n\\ref{Assumption2} they can simply be removed from the set $2^{\\mathbb{B}}$ without affecting algorithm or the derived guarantees.\n\n\\section{Overview of Proposed Strategy}\\label{problem statement}\n\n\n\nIn this section, we propose an  algorithm in the spirit of that first proposed by \\cite{lale2020explore} which leverages actuator redundancy in the \"more exploration\" step to avoid blow up in the state norm while minimizing the regret bound. We break down the strategy into two phases of initial exploration, presented by Algorithm (IExp), and optimism (Opt), given by SOFUA algorithm. \n\nThe IExp algorithm, which leverages exploratory noise, is deployed in the actuating mode $i^*$ for duration $T_c^{i^*}$ to reach a stabilizing neighborhood of the full-actuation mode and alleviate state explosion while minimizing regret. \n\nAfterwards, Algorithm 2 which leverages all the actuators comes into play. This algorithm has the central confidence set, given by the Algorithm 1, as an input. The best actuating mode $i^*$ that guarantees minimum possible state norm upper-bound and initial exploration duration $T^{i^*}_c$ is determined by running Algorithm 3 at the subsection \\ref{deterBest}. \n\n\n\\begin{algorithm} \n\t\\caption{\\small Initial Exploration (IExp) \\normalsize} \\label{alg:IExp}\n\t\t\\begin{algorithmic}[1]\n\t\t\\STATE \\textbf{Inputs:}$T^{i^*}_c$$\\,s^{i^*}>0,$$\\,\\delta>0,$$\\, \\sigma_{\\omega},\\, \\sigma_{\\nu}\\,,\\lambda>0$\n\t\t\\STATE set $V^{i^*}_0=\\lambda I$, $\\hat{\\Theta}^{i^*}=0$ \n\t\t\\STATE $\\tilde{\\Theta}^i_0=\\arg\\min_{\\Theta \\in \\mathcal{C}^{i^*}_0(\\delta)\\cap S^i}\\,\\, J(\\Theta^i,Q_*,R_*^i)$\n\t\t\\FOR {$t = 0,1,..., T^{i^*}_c$}\n\t\t\\IF {$\\det(V^{i^*}_t)> 2 \\det(V^{i^*}_{\\tau})$ or $t=0$} \n\t\t\\STATE Calculate $\\hat{\\Theta}^{i^*}_t$ by (\\ref{eq:LSE_Solf}) and set $\\tau=t$\n\n\t\t\\STATE Find $\\tilde{\\Theta}^{i^*}_t$ by (\\ref{eq:nonconvexOpt}) for $i=i^*$\n\t\t\\ELSE\n\t\t\\STATE $\\tilde{\\Theta}^{i^*}_t=\\tilde{\\Theta}^{i^*}_{t-1}$\n\t\t\\ENDIF \n\t\t\\STATE For the parameter $\\tilde{\\Theta}^{i^*}_t$ solve the Ricatti equation and find $u^{i^*}_t=K(\\tilde{\\Theta}^{i^*}_t,Q_*, R_*^{i^*})x_t$\n\t\t\\STATE Construct $\\bar{u}^{i^*}_t$ using (\\ref{eq:controlAlg}) and apply it on the system $\\Theta_*$ (\\ref{eq:dynam_by_theta2}) and observe new state $x_{t+1}$.\n\t\t\\STATE Using $u^{i^*}_t$ and $x_t$ form $z^{i^*}$ and Save $(z^{i^*}_t,x_{t+1})$ into dataset\n\t\t\\STATE Set $V^{i^*}_{t+1}=V^{i^*}_t+z^{i^*}_t{z^{i^*}_t}^\\top$ and form $\\mathcal{C}^{i^*}_{t+1}$\n\t\t\\STATE using $\\bar{u}^i_t$ and $x_t$ form $\\bar{z}^i_t$\n\t\t\\STATE Form $(\\bar{z}^i_t,x_{t+1})$\n\t\t\\STATE Set $V_{t+1}=V_t+\\bar{z}^i_t {\\bar{z}_t^i}^\\top$ and form $\\mathcal{C}_{t+1}$\\ENDFOR\n\t\t\\STATE Return $V_{T_c+1}$ and corresponding $\\mathcal{C}_{T_c}$ \n\t\\end{algorithmic}\n\t\\end{algorithm}\n\n\\begin{algorithm} \n\t\\caption{Stabilizing OFU Algorithm (SOFUA)} \\label{alg:SOFUA}\n\t\t\\begin{algorithmic}[1]\n\t\t\\STATE \\textbf{Inputs:}$T,$$\\,S>0,$$\\,\\delta>0,$$\\,Q\\,, L,\\, V_{T_c},\\, \\mathcal{C}_{T_c},\\, \\hat{\\Theta}_{T_c}$\n\t\t\\STATE $\\tilde{\\Theta}_{T_c}=argmin_{\\Theta \\in \\mathcal{C}_{T_c}(\\delta)\\cap S}\\,\\, J(\\Theta)$\n\t\t\\FOR {$t = T_c,T_c+1,T_c+2,...$}\n\t\t\\IF {$\\det(V_t)> 2 \\det(V_{\\tau})$ or $t=T_c$} \n\t\t\\STATE Calculate $\\hat{\\Theta}_t$ by (\\ref{eq:LSE_Solf}) and set $\\tau=t$\n\n\t\t\\STATE Find $\\tilde{\\Theta}_t$ by (\\ref{eq:nonconvexOpt}) for $i=1$\n\t\t\\ELSE\n\t\t\\STATE $\\tilde{\\Theta}_t=\\tilde{\\Theta}_{t-1}$\n\t\t\\ENDIF \n\t\t\\STATE For the parameter $\\tilde{\\Theta}_t$ solve Ricatti and calculate $\\bar{u}_t=K(\\tilde{\\Theta}_t, Q_*,R_*)x_t$\n\t\t\\STATE Apply the control on $\\Theta_*$ and observe new state $x_{t+1}$.\n\t\t\\STATE Save $(z_t,x_{t+1}) $ into dataset\n\t\t\\STATE $V_{t+1}=V_t+z_tz_t^\\top$\n\t\t\\ENDFOR\n\t\\end{algorithmic}\n\t\\end{algorithm}\n\\subsection{Main steps of Algorithm 1}\n\n\\subsubsection{Confidence Set Contruction}\n\nIn IExp phase we add an extra exploratory Gaussian noise $\\nu$ to the input of all actuators even those not in actuators set of mode $i$. Assuming that the system actuates in an arbitrary mode $i$, the dynamics of system, used for confidence set construction (i.e. system identification), is written as\n\\begin{align}\n\tx _{t+1} ={\\Theta^i _{*}}^\\top \\underline{z}^i_{t}+\\bar{B}^i_*\\bar{\\nu}^i_t+\\omega_{t+1}, \\quad \\underline{z}^i_{t}=\\begin{pmatrix} x_t \\\\ \\underline{u}^i_t \\end{pmatrix}. \\label{eq:dynam_by_theta3} \n\\end{align}\n\nin which $\\bar{B}^i_*\\in \\mathbb{R}^{d-d_i}$ and $\\underline{u}^i_{t}=u^i_t+\\nu_t(\\mathcal{B}_i)$,  and $\\bar{\\nu}^i_t=\\nu_t(\\mathcal{B}_i^c)$ where, if $\\nu_t \\in \\mathbb{R}^{d}$ and $\\mathcal{N}\\subset\\mathbb{B}$, the vector $\\nu (\\mathcal{N})\\in \\mathbb{R}^{card(\\mathcal{N})}$ is constructed by only keeping the entries of $\\nu_t$ corresponding to the index set of elements in $\\mathcal{N}$. Note that (\\ref{eq:dynam_by_theta3}) is equivalent to (\\ref{eq:dynam_by_theta}) but separates used and unused actuators.\n\nBy applying self-normalized process, the least square estimation error, $e(\\Theta^{i})$ can be obtained as:\n\t\\begin{align}\n\t\\nonumber & e(\\Theta^{i})= \\lambda  \\operatorname{Tr}({\\Theta^{i}}^\\top\\Theta^{i})\\\\\n\t&+\\sum _{s=0}^{t-1} \\operatorname{Tr} \\big((x_{s+1}-{\\Theta^{i}}^\\top \\underline{z}^{i}_{s})(x_{s+1}-{\\Theta^{i}}^\\top \\underline{z}^{i}_{s})^\\top)\\big)  \\label{eq:LSE_op}\n\t\\end{align}\nwith regularization parameter $\\lambda$. This yields the $l^{2}$-regularized least square estimate:\n\n\\begin{align}\n\t\\hat{\\Theta}^i_t &=\\operatorname*{argmin_{\\Theta^{i}}} e(\\Theta^{i})=({\\underline{Z}_t^{i}}^\\top \\underline{Z}_t^{i}+\\lambda I)^{-1}{\\underline{Z}_t^{i}}^\\top X_t\n\t\\label{eq:LSE_Solf}\n\\end{align}\n\nwhere $\\underline{Z}_t^{i}$ and $X_t$ are matrices whose rows are ${\\underline{z}^{i}_{0}}^\\top,..., {\\underline{z}^{i}_{t-1}}^\\top$ and $x_{1}^\\top,...,x_{t}^\\top$, respectively.\nDefining covariance matrix $V^{i^*}_{t}$ as follows:\n\\begin{align*}\nV^{i}_{t}=\\lambda I + \\sum_{s=0}^{t-1} \\underline{z}^{i}_{s}{\\underline{z}^{i}_{s}}^\\top=\\lambda I +{\\underline{Z}_t^{i}}^\\top \\underline{Z}_t^{i},\n\\end{align*}\n\nit can be shown that with probability at least $(1-\\delta)$, where $0<\\delta<1$, the true parameters of system $\\Theta^i_*$ belongs to the confidence set defined by (see  \\ref{thm:Conficence_Set_Attacked}): \n\\begin{align}\n\t\\nonumber\\mathcal{C}^{i}_{t}(\\delta)& =\\{{\\Theta^i}^\\top \\in R^{n \\times (n+d_i)}  \\mid \\\\  & \\nonumber\\operatorname{Tr}((\\hat{\\Theta}^{i}_{t}-\\Theta^{i})^\\top V_{t}^i(\\hat{\\Theta}^{i}_{t}-\\Theta^{i}))\\leq \\beta^{i}_{t}(\\delta)  \\}, \\\\\n\t\\nonumber\\beta^{i}_t(\\delta) &=\\bigg(\\lambda^{1\/2}s^i+\\sigma_{\\omega}\\sqrt{2n\\log(n\\frac{\\det(V^{i}_{t})^{1\/2}\\det(\\lambda I)^{-1\/2}}{\\delta}})\\\\\n\t&+\\|\\bar{B}_*^i\\|\\sigma_{\\nu}\\sqrt{2d_i\\log(d_i\\frac{ \\det(V^{i}_{t})^{1\/2}\\det(\\lambda I)^{-1\/2}}{\\delta}})\\bigg)^{2} \\label{eq:Conf_set_radius_unAtt}\n\\end{align}\nAfter finding high-probability confidence sets for the unknown parameter, the core step is implementing Optimism in the Face of Uncertainty (OFU) principle. At any time $t$, we choose a parameter  $\\tilde{\\Theta}^{i}_t \\in \\mathcal{S}^i_c\\cap \\mathcal{C}^{i}_{t}(\\delta)$ such that:\n\n\\begin{align}\nJ(\\tilde{\\Theta}^{i}_t, Q_*, R_*^i)\\leq \\inf\\limits_{\\Theta^{i} \\in \\mathcal{C}^{i}_t(\\delta)\\cap \\mathcal{S}^i_c }J(\\Theta^{i},Q_*,R^i_*)+\\frac{1}{\\sqrt{t}}. \\label{eq:nonconvexOpt} \\end{align}\n\nThen, by using the chosen parameters as if they were the true parameters, the linear feedback gain $K(\\tilde{\\Theta}^{i}, Q_*, R_*^{i})$ is designed. We synthesized the control $\\underline{u}^{i}_{t}=u^{i}_t+\\nu_t(\\mathcal{B}_{i})$  on (\\ref{eq:dynam_by_theta3}) where $u^{i}_t=K(\\tilde{\\Theta}^{i}, Q_*, R_*^{i})x_t$. The extra exploratory noise $\\nu_t \\sim \\mathcal{N}(\\mu,\\,\\sigma_{\\nu}^{2}I)\\in \\mathbb{R}^{d}$ with $\\sigma_{\\nu}^{2}=2\\kappa^2\\bar{\\sigma}_{\\omega}^2$ is the random ``more exploration\" term. \n\nAs can be seen in the regret bound analysis, recurrent switches in policy may worsen the performance, so a criterion is needed to prevent frequent policy switches. As such, at each time step $t$ the algorithm checks the condition $\\det(V^{i}_{t})>2\\det(V^{i}_{\\tau})$ to determine whether updates to the control policy are needed where $\\tau$ is the last time of policy update.\n\n\\subsubsection{Central Ellipsoid Construction}\nNote that (\\ref{eq:Conf_set_radius_unAtt}) holds regardless of the control signal $\\underline{z}^{i}_t$. The formulation above also holds for any actuation mode, being mindful that the\ndimension of the covariance matrix changes. Even while actuating in the IExp phase, by applying augmentation technique, we can build a confidence set (which we call the central ellipsoid) around the parameters of the full actuation mode  thanks to extra exploratory noise. For $t\\leq T^{i}_c$, this simply can be carried out by rewriting (\\ref{eq:dynam_by_theta3}) as follows:\n\\begin{align}\n\tx _{t+1} =\\Theta _{*}^\\top \\bar{z}^i_{t}+\\omega_{t+1}, \\quad \\bar{z}^i_t=\\begin{pmatrix} x_t \\\\ \\bar{u}^i_t \\end{pmatrix} \\label{eq:dynam_by_theta2} \n\\end{align}\nwhere $\\bar{z}^i_t\\in \\mathbb{R}^{n+d}$ and $\\bar{u}^i_t\\in \\mathbb{R}^d$ is constructed by augmentation as follows\n\n\\begin{align}\n\\bar{u}^i_t(\\mathcal{B}_i)=u^i_t+\\nu(\\mathcal{B}_i),\\quad \\bar{u}^i_t(\\mathcal{B}^c_i)=\\nu(\\mathcal{B}^c_i). \\label{eq:controlAlg}\n\\end{align}\n\nBy this augmentation, we can construct the central ellipsoid \n\n\\begin{align}\n&\t\\nonumber\\mathcal{C}_{t}(\\delta) =\\{\\Theta^\\top \\in R^{n \\times (n+d)}  \\mid \\operatorname{Tr}((\\hat{\\Theta}_{t}-\\Theta)^\\top V_{t}(\\hat{\\Theta}_{t}-\\Theta))\\\\\n& \\nonumber\\leq \\beta_{t}(\\delta)  \\} \\\\\n\t& \\beta_t(\\delta) =(\\sigma_{\\omega}\\sqrt{2n\\log(\\frac{\\det(V_{t})^{1\/2}\\det(\\lambda I)^{-1\/2}}{\\delta}})+\\lambda^{1\/2}s)^{2}. \\label{eq:Conf_set_radius_centralElips} \n\\end{align} \n\nwhich is an input to Algorithm 2 and used to compute IExp duration. \n\n\\subsection{Main steps of Algorithm 2}\n\nThe main steps of Algorithm 2 are quite similar to those of Algorithm 1 with a minor difference in confidence set construction. Algorithm 2 receives $V_{T^{i^*}_c}$, $Z_{T^{i^*}_c}$, and $X_{T^{i^*}_c}$ from Algorithm 1, using which for  $t>T^{i^*}_c$ we have \n\\begin{align*}\nV_{t}&=V_{T^{i^*}_c} + \\sum_{s=T^{i^*}_c+1}^{t-1} {z}_{s}{{z}_{s}}^\\top\\\\\nZ_tX_t&=Z_{T^{i^*}_c}X_{T^{i^*}_c}^\\top + \\sum_{s=T^{i^*}_c+1}^{t-1} {z}_{s}{{x}_{s}}^\\top\n\\end{align*}\nand the confidence set is easily constructed.\n\nThe following theorem summarizes boundedness of state norm when Algorithm 1 and 2 are deployed.\n\n\n\n\n\\begin{thm} \\label{lemma:stabilization}\n\\begin{enumerate}\n\\item  The IExp algorithm keeps the states of the underlying system actuating in any mode $i$ bounded with the probability at least $1-\\delta$ during initial exploration, i.e., \n\n\\begin{align} \\label{eq:upperbound_state}\n\\nonumber \\|x_t\\|&\\leq \\frac{1}{1-\\Upsilon_i}\\big(\\frac{\\eta_i}{\\Upsilon_i}\\big)^{n+d_i}\\bigg[G_iZ_t^{\\frac{n+d_i}{n+d_i+1}}\\beta^i_t(\\delta)^{\\frac{1}{2(n+d_i+1)}}+\\\\\n&\\quad (\\sigma_{\\omega}\\sqrt{2n\\log\\frac{nt}{\\delta}}+\\|s I\\|\\sigma_{\\nu}\\sqrt{2d_i\\log\\frac{d_it}{\\delta}})\\bigg]=:\\alpha_t^i,\n\\end{align}\nfor all modes $i \\in \\{1,..., 2^d\\} $. \n\n\\item For $t>T^{i^*}_c+\\frac{(n+d_{i^*})\\log(n+d_{i^*})+\\log c^{i^*}-\\log \\chi_s}{\\log\\frac{2}{1-\\Upsilon}}:=T_{rc}$ we, with probability at least $1-\\delta$, have $\\|x_t\\|\\leq 2\\chi_s$ where\n\\begin{align}\n\\chi_s:=\\frac{2\\sigma_{\\omega}}{1-\\Upsilon}\\sqrt{2n\\log\\frac{n(T-T^{i^*}_c)}{\\delta}}.\n\\end{align}\n\\end{enumerate}\n\\end{thm}\n\n\n\n\n\n\n\nFrom parts (1) and (2) of Theorem  \\ref{lemma:stabilization} we define the following \\textit{good events}:\n\\begin{align}\nF^i_{t}=\\{\\omega \\in\\Omega  \\mid \\forall s \\leq T^{i}_c, \\left\\lVert x_{s}\\right\\rVert \\leq \\alpha^i_{t} \\}.\\label{eq:GoodEven_state_unat} \n\\end{align}\nand \n\\begin{align}\nF^{op,c}_{t}=\\{\\omega \\in\\Omega  \\mid \\forall\\;\\; T^{i^*}_c\\leq s \\leq t, \\left\\lVert x_{s}\\right\\rVert^2 \\leq X_c^2 \\}.\\label{eq:GoodEven} \n\\end{align}\nin which\n\\begin{align}\nX_c^2=\\frac{32n\\sigma^2_{\\omega}(1+\\kappa^2)}{(1-\\Upsilon)^2}\\log \\frac{n(T-T_c)}{\\delta}.\\label{eq:upperBoundOptimisimphase} \n\\end{align}\nwhere both the events are used for regret bound analysis and the former one specifically is used to obtain best actuating mode for initial exploration.\n\n\\subsection{Determining the Optimal Mode for IExp}\n\\label{deterBest}\nWe still need to specify the best actuating mode $i^*$ for initial exploration along with its corresponding upperbound $X_t^{i^*}$. Theorem \\ref{lemma:estimationerror_timeMinumum22} specifies $i^*$. First we need the following Lemma.\n\n\\begin{lem} \\label{lemma:estimationerror_timeMinumum}\nAt the end of initial exploration, for any mode $\\forall i\\in \\{1,..., 2^d\\}$ the following inequality holds\n\\begin{align}\n||\\hat{\\Theta}_{T^i_{\\omega}}-\\Theta_*||_2\\leq \\frac{\\mu^i_c}{\\sqrt{T^i_{\\omega}}} \\label{eq:stab_neighb}\n\\end{align}\nwhere $\\mu^i_c$ is given as follows\n\n\\begin{align}\n\\nonumber \\mu^i_c:= & \\frac{1}{\\sigma_{\\star}}\n \\bigg(\\sigma_{\\omega}\\sqrt{n(n+d)\n\\log\\big(1+\\frac{\\mathcal{P}_c}{\\lambda (n+d)}\\big)+2n\\log\\frac{1}{\\delta}}+\\\\\n&\\sqrt{\\lambda}s\\bigg)\n \\label{eq:kappaDefControllable}\n\\end{align}\nwith,\n\\begin{align*}\n\\mathcal{P}_c&:={X_{T^i_{\\omega}}^{i}}^2(1+2{\\kappa^i}^2)T^i_{\\omega}+\n4T^i_{\\omega}\\sigma^2_{\\nu}d_i\\log (dT^i_{\\omega}\/\\delta)\n\\end{align*}\n\nin which $T^i_{\\omega}$ stands for initial exploration duration of actuating in mode $i$. Furthermore, if we define\n\\begin{align}\nT^i_c:=\\frac{4(1+\\kappa)^2\\mu^{i2}_c}{(1-\\Upsilon)^2}\\label{eq:T_cDef}\n\\end{align}\nthen for $T^i_{\\omega}>T^i_c$, $||\\hat{\\Theta}_{T^i_{\\omega}}-\\Theta_*||_2\\leq \\frac{1-\\Upsilon}{2(1+\\kappa)}$ holds with probability at least $1-2\\delta$.\n\\end{lem}\nThe proof is provided in Appendix \\ref{appendix}.\n\n\\begin{thm}\\label{lemma:estimationerror_timeMinumum22}\nSuppose Assumptions \\ref{Assumption 1} and \\ref{Assumption2} hold true. Then for a system actuating in the mode $i$ during initial exploration phase, the following results hold true\n\\begin{enumerate}\n\n\\item $I_{F^i_t} \\max_{1\\leq s\\leq t} \\|x_s\\|\\leq x_t$ \nwhere $I_{F^i_t}$ is indicator function of set $F^i_t$ and\n\\begin{align}\n& x_t=Y_{i,t}^{n+d_i+1}\\\\\n& \\nonumber Y_{i,t}:=\\max \\big(e, \\lambda (n+d^i)(e-1), \\frac{-\\bar{L}_i+\\sqrt{\\bar{L}_i^2+4\\bar{K}_i}}{2\\bar{K}_i}\\big),\n\\end{align}\nwith\n\\begin{align*}\n&\\bar{L}^i=(\\mathcal{D}^i_1+\\mathcal{D}^i_2)\\big(2n\\sigma_{\\omega}\\log\\frac{1}{\\delta}+\\sigma_{\\omega}\\sqrt{\\lambda}s^i\\big)\\log t +\\\\\n&\\mathcal{D}^i_3\n\\sqrt{\\log t\/\\delta}+\n(\\mathcal{D}^i_1+\\mathcal{D}^i_2)n\\sigma_{\\omega}(n+d_i)\\times \\\\ &\\bigg(\\log\\frac{(n+d_i)\\lambda+2{\\mathcal{V}^i_t}^2}{(n+d_i)\\lambda}t+\\log\\frac{(1+2{\\kappa^i}^2)}{(n+d_i)\\lambda}t\\bigg)\\log t\\\\\n&\\quad \\bar{K}^i=2(\\mathcal{D}^i_1+\\mathcal{D}^i_2)n\\sigma_{\\omega}(n+d_i)(n+d_i+1)\\log t.\n\\end{align*}\nwhere \n\t\\begin{align*}\n\t &\\mathcal{D}^i_1:=\\frac{4}{1-\\Upsilon_i}\\big(\\frac{\\eta_i}{\\Upsilon_i}\\big)^{n+d_i}\\bar{G}_i (1+2{\\kappa^i}^2)^{\\frac{n+d_i}{2(n+d_i+1)}}\\\\\n\t  &  \\mathcal{D}^i_2:=\\frac{4}{1-\\Upsilon_i}\\big(\\frac{\\eta_i}{\\Upsilon_i}\\big)^{n+d_i}\\bar{G}_i 2^{\\frac{n+d_i}{2(n+d_i+1)}}\\mathcal{V}_T^i,\\\\ &\\mathcal{D}^i_3:=\\frac{n\\sqrt{2}}{1-\\Upsilon_i}\\big(\\frac{\\eta_i}{\\Upsilon_i}\\big)^{n+d_i}\\sigma_{\\omega}\n\t\\end{align*}\n\t\nin which $\\mathcal{V}^i_t=\\sigma_{\\nu}\\sqrt{2d_i\\log d_it\/\\delta}$ holds with probability least $1-\\delta\/2$.\n\n\\item The best actuating mode $i^*$ for initial exploration is,\n\\begin{align}\n\\nonumber i^*&=argmin_{i\\in \\{1,...,2^{\\mathbb{B}}\\}} Y_{i,T^i_{\\omega}}^{n+d_i+1}\\\\\n& s.t\\;\\; \\;\\; T^i_{\\omega}\\geq T^i_c \n\\label{optimiza}\n\\end{align}\n\n\\item The upper-bound of state norm of system actuating in the mode $i^*$ during initial exploration phase can be written as follows:\n\\begin{align}\n\\|x_t\\|\\leq c_c^{i^*}(n+d_{i^*})^{n+d_{i^*}}\\label{simpleBound}\n\\end{align}\nfor some finite system parameter-dependent constant $c_c^{i^{*}}$.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{rem}\\label{Remark:Find Best Actuating mode}\nWhile optimization problem (\\ref{optimiza}) cannot be solved \nanalytically because\n$T^i_c$ itself depends \non $x_{T^i_{\\omega}}$, it can be \ndetermined using Algorithm 3.\n\\end{rem}\n\n\n\n\\begin{algorithm} \n\\label{alg3}\n\n\t\\caption{\\small Find best actuating mode $i^*$ and its corresponding $T_c^{i^*}$\\normalsize} \\label{alg:FindMode}\n\t\t\\begin{algorithmic}[1]\n\t\t\\STATE \\textbf{Inputs:}$\\lambda,\\,\n\t\t\\kappa,\\,S^i>0,$$\\,\\delta>0,$$\\,, \\sigma_{\\omega},\\, \\vartheta_i,\\, \\eta_i,\\, \\Upsilon_i \\, \\forall i$\n\t\t\\FOR {$\\forall i\\in \\{1,...,2^{\\mathbb{B}}\\} $}\n\t\t\\STATE $T^i_{itr}=1$\n \t\t\\FOR {$t=1,2,..., T^i_{itr} $}\n\t\t\\STATE  compute $T_c^i$ by (\\ref{eq:T_cDef}) \n\t\t\\IF {$t<T_c^i$} \n\t\t\\STATE $T^i_{itr}=T^i_{itr}+1$\n\t\t\\ELSE \n\t\t\\STATE $T^i_{\\omega}=t$\n\t\t\\ENDIF\n\t\t\\ENDFOR\n\t\t\\ENDFOR\n\t\t\\STATE Compute $x_{T^i_{\\omega}}=Y_{i,T^i_{\\omega}}^{n+d_i+1}$ $\\forall i\\in \\{1,...,2^{\\mathbb{B}}\\}$\n\t\t\\STATE Solve \n\t\t$i^*=\\arg \\min_{i\\in \\mathcal{B}_*} x_{T^i_{\\omega}}$.\n\t\t\\STATE \\textbf{Outputs:} $i^*$ and $T^{i^*}_{\\omega}$\n\t\t\\end{algorithmic}\n\t\\end{algorithm}\n\n\\subsection{Regret Bound}\nRecall (\\ref{eq:Reg}), the regret for the proposed strategy (IExp+\\\\\nSOFUA) can be defined as follows:\n\n\\begin{align}\n\t\\mathcal{R}_T =\\mathbb {E}\\left[\\sum_{t=1}^{T} ( x_{t}^\\top Q_*x_{t} + {\\bar{u}^{i(t)}_{t}}^\\top R_* \\bar{u}_{t}^{i(t)}-J_*(\\Theta_*,Q_*,R_*))\\right] \\label{eq:RegBB} \n\\end{align}\n\nwhere $i(t)=i^*$ for $t\\leq T^{i^*}_c$ and $i(t)=1$ for $t> T^{i^*}_c$.\n\nAn upper-bound for $\\mathcal{R}_T$ is given by the following theorem which is the next core result of our analysis. \n\n\\begin{thm} (Regret Bound of IExp+SOFUA) \\label{lemma:RegretBoundControllable}\nUnder Assumptions \\ref{Assumption 1} and \\ref{Assumption2}, with probability at least $1-\\delta$ the algorithm SOFUA together with additional exploration algorithm IExp which runs for $T^{i^*}_{c}$ time steps achieves regret of  $\\mathcal{O}\\big((n+d_{i^*})^{(n+d_{i^*})}T^{i^*}_{rc}\\big)$ for $t\\leq T^{i^*}_{c}$ and $\\mathcal{O}\\big(poly(n+d)\\sqrt{T-T^{i^*}_{rc}}\\big)$ for $t>T^{i^*}_{c}$ where $\\mathcal{O}(.)$ absorbs logarithmic terms.\n\\end{thm}\t\n\n\\section{Numerical Experiment}\n\\label{simulation}\nIn this section, we demonstrate on a practical example how the use of our algorithms successfully alleviates state explosion during the initial exploration phase. We consider a control system with drift and control matrices to be set as follows:\n\n\\begin{align*}\n\tA_*=\\begin{pmatrix}\n\t\t1.04 & 0 & -0.27 \\\\\n\t\t0.52 & -0.81 & 0.83\\\\\n\t\t 0 & 0.04 & -0.90\n\t\\end{pmatrix},\\; B_*=\\begin{pmatrix}\n\t0.61 &-0.29 & -0.47\\\\\n\t0.58 & 0.25 &  -0.5\\\\\n\t0 & -0.72 & 0.29\n\\end{pmatrix}.\n\\end{align*}\n\nWe choose the cost matrices as follows:\n\n\\begin{align*}\n\tQ_*=\\begin{pmatrix}\n\t0.65 &-0.08& -0.14 \\\\\n\t\t-0.08 & 0.57 & 0.26\\\\\n\t\t-0.14& 0.26& 2.5\n\t\\end{pmatrix},\\;R_*=\\begin{pmatrix}\n\t\t0.14 &0.04& 0.05 \\\\\n\t\t0.04 &0.24 &0.08\\\\\n\t\t0.05 &0.08 &0.2\n\t\\end{pmatrix}.\n\\end{align*}\n\t\nThe Algorithm 3 outputs the exploration duration $T^{i^*}_c=50s$ and best actuating mode $i^*$ for initial exploration with corresponding control matrix $B_*^{i^*}$ and $R_*^{i^*}$\n\\begin{align*}\nB_*^{i^*}=\\begin{pmatrix}\n\t0.61 &-0.29\\\\\n\t0.58 & 0.25\\\\\n\t0 & -0.72\n\\end{pmatrix},\\;\\;\\;\\;\\;\\; R_*^{i^*}=\\begin{pmatrix}\n\t\t0.14 &0.04  \\\\\n\t\t0.04 &0.24\n\t\\end{pmatrix}.\n\\end{align*}\t\n\n\\begin{figure}\n\t\\centering\n\t\\vspace{5pt}\n\t\t\\hspace{4pt}\n\t\t\\includegraphics[trim=4cm 6cm 6cm 8cm, scale=.4]{MainNormState.pdf}\n\t\\hfill\n\t\n\t\t\\hspace{4pt}\n\t\t\\includegraphics[trim=4cm 7cm 6cm 8cm, scale=.4]{mainRegretBound.pdf}\n\t\n\t\\hfill\n\t\\caption{Top. State norm, Bottom. regret bound}\n\t\\label{fig:naive_attacked_estimateANDregret}\n\\end{figure}\n\nIt has graphically been shown in \\cite{abbasi2013online} that the optimization problem (\\ref{eq:nonconvexOpt}) is generally non-convex for $n,d>1$. Because of this fact, we decided to solve optimization problem (\\ref{eq:nonconvexOpt}) using a projected gradient descent method in Algorithm 1 and 2, with basic step\n\n\\begin{align}\n\\tilde{\\Theta}^i_{t+1}\\leftarrow PROJ_{\\mathcal{C}^i_t(\\delta)} \\bigg(\\tilde{\\Theta}^i_{t}-\\gamma  \\nabla_{\\Theta^i}(L^itr (P(\\Theta^i,Q_*,R^{i}_*))) \\bigg)\n\\end{align}\n\nwhere $L^{i}=\\bar{\\sigma}^2_{\\omega}+\\vartheta^2\\bar{\\sigma}^2_{\\nu}$ for $i=i^*$ and $L^{i}=\\bar{\\sigma}^2_{\\omega}$ for $i=1$. $\\nabla_{\\Theta^i}f$ is the gradient of $f$ with respect to $\\Theta^i$. $\\mathcal{C}^i_t(\\Theta^i)$ is the confidence set, $PROJ_g$ is Euclidean projection on $g$ and finally $\\gamma$ is the step size. Computation of gradient $\\nabla_{\\Theta^i}$ as well as formulation of projection has been explicited in \\cite{abbasi2013online}, similar to which we choose the learning rate as follows:\n\\begin{align*}\n\t\\gamma=\\sqrt{\\frac{0.001}{tr(V^i_t)}}.\n\\end{align*}\n\nWe apply the gradient method for 100 iterations to solve each OFU optimization problem and apply the projection technique until the projected point lies\ninside the confidence ellipsoid. The inputs to the OFU algorithm are $T=10000$, $\\delta=1\/T$, $\\lambda=1$, $\\sigma_{\\omega}=0.1$, $s=1$ and we repeat simulation $10$ times. \n\nAs can be seen in Fig. \\ref{fig:naive_attacked_estimateANDregret}, the maximum value of the state norm (attained during the initial exploration phase) is smaller when using mode $i^*$ than when all actuators are in action.\n\n\nRegret-bound for both cases is linear during initial exploration phase, however SOFUA guarantees $\\mathcal{O}(\\sqrt{T})$ regret for $T>50 s$.\n\n\\section{Conclusion}\\label{conclusion}\nIn this work, we proposed an OFU principle-based controller for over-actuated systems, which combines\na step of \"more-exploration\" (to produce a stabilizing\nneighborhood of the true parameters while guaranteeing a bounded state during exploration) with one of \"optimism\", which efficiently\ncontrols the system. Due to the redundancy, it is possible to further optimize the speed of convergence of the exploration phase to the stabilizing neighborhood by choosing over actuation modes, then to switch to full actuation to guarantee an  $\\mathcal{O}(\\sqrt{T})$ regret in closed-loop, with\npolynomial dependency on the system dimension.\n\nA natural extension of this work is to classes of systems in which some modes are only stabilizable. Speaking more broadly, the theme of this paper also opens the door to more applications of switching as a way to facilitate learning-based control of unknown systems, some of which are the subject of current work.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nTo control magnetic behavior for device applications, it is crucial to engineer magnetic interactions.\nUsually, the interaction between spins has the symmetric form $\\bm S_i \\cdot \\bm S_j$ and, as a result, the spins tend to be (anti)parallel.\nOn the other hand, in magnetic materials with broken inversion symmetry, a qualitatively different interaction, $\\bm S_i \\times \\bm S_j$, called the Dzyaloshinskii--Moriya (DM) interaction, appears as a consequence of spin-orbit interactions\\cite{dzyaloshinskii1958,moriya1960}. \nThe DM interaction is antisymmetric for spin operators and favors twisted spin structures, which induces numerous interesting magnetic behaviors such as chiral soliton lattices in chiral helimagnets\\cite{kishine2015}, skyrmion formation\\cite{bogdanov1989,roszler2006,muhlbauer2009,yu2010,nagaosa2013}, and the enhancement of domain wall mobility\\cite{thiaville2012,chen2013,ryu2013,emori2013,torrejon2014}.\nIn addition, the DM interaction relates magnetic properties and electric polarizations in multiferroic materials\\cite{katsura2005,sergienko2006,tokura2014}.\n\nIn 1960, Moriya first proposed the microscopic derivation of the DM interaction at the first order of the spin-orbit coupling and discussed two different contributions\\cite{moriya1960};\nthe first one is the extension of the superexchange mechanism to multiorbital spin-orbit systems, and the other is the combination of the direct exchange interaction and the spin-orbit coupling.\nAlthough the physical pictures of these mechanisms are clear particularly for insulating systems, these formulations are not suitable for practical calculation.\nFor quantitative analysis, several techniques have been developed to calculate the DM interaction from first-principles calculations\\cite{liechtenstein1987,igor1996,katsnelson2000,katsnelson2010,heide2008,ferriani2008,heide2009,freimuth2014,kikuchi2016,koretsune2015}.\nMost of these approaches consider the energy difference by twisting the magnetic structures in various ways.\nOne of the most direct approaches is to calculate the energies of spirals with the finite vector $\\bm q$ as $E(\\bm q)$ and extract the $q$-linear term\\cite{heide2008,ferriani2008,heide2009},\nalthough this approach is sometimes time-consuming.\nWhen the twisting angle is small, the energy change can be evaluated from the information of the uniform magnetic structures\nby utilizing, for example, the magnetic force theorem\\cite{liechtenstein1987,igor1996,katsnelson2000},\nBerry phase\\cite{freimuth2014}, or\nspin gauge field transformation\\cite{kikuchi2016}.\nOn the other hand, perturbation expansion with respect to the exchange couplings gives a different formulation to evaluate the DM interaction\\cite{fert1980,imamura2004,kundu2015,wakatsuki2015,koretsune2015,shibuya2016}.\n\n\nIn this paper, we overview three approaches to evaluate the DM interaction from first-principles calculations,\nthat is, the methods using the energy of spirals, $E(\\bm q)$, the spin current, and the off-diagonal spin susceptibility.\nBy applying these methods to chiral ferromagnets, namely, Mn$_{1-x}$Fe$_x$Ge\\ and Fe$_{1-x}$Co$_x$Ge\\ systems, we discuss the relationship among these approaches,\nhow the band structures affect the DM interactions, and how each approach explains experimental results.\n\n\\section{Formulation to Compute the DM Interaction}\nIn this section, we describe three approaches to calculate the DM interaction.\nHereafter, we consider the low-energy effective Hamiltonian for the local direction of the magnetic continuum, $\\bm n(\\bm r)$, and neglect the charge degrees of freedom.\nThen, the exchange interaction and the DM interaction are given as\n\\begin{align}\n\tH = \\int d \\bm r \\;\\sum_\\mu\\left[ J_{\\mu} (\\nabla_{\\mu} \\bm n)^2 + \\sum_\\alpha D^{\\alpha}_{\\mu} (\\nabla_{\\mu} \\bm n \\times \\bm n)^{\\alpha}\\right],\n\t\\label{eq:Hamiltonian_continuum}\n\\end{align}\nwhere $J_{\\mu}$ and $D_{\\mu}^{\\alpha}$ denote the spin stiffness and the DM interaction, respectively.\nHere, ${\\mu}=x,y,z$ represents the relative direction of the two spins in the lattice representation, and ${\\alpha}=x,y,z$ represents the spin rotation axis.\nThis Hamiltonian can be derived from the local-spin Hamiltonian\n\\begin{align}\n\tH = \\sum_{ij} \\left[ - J_{ij} \\bm S_i \\cdot \\bm S_j + \\bm D_{ij} \\cdot ( \\bm S_i \\times \\bm S_j ) \\right].\n\\end{align}\nIn fact, when we assume the isotropic system,\n$J_{\\mu}$ and $D_{\\mu}^{\\alpha}$ can be written as\n\\begin{align}\n\tJ &= \\frac{1}{V}\\sum_{j} \\frac{1}{6}| \\bm r_j |^2J_{0j},\\\\\n\t\\bm D_\\mu &= \\frac{1}{V} \\sum_{j} (-r_j)^\\mu \\bm D_{0j},\n\t\\label{eq:JDcontinuum}\n\\end{align}\nwhere $V$ is the volume of one site.\nIn this paper, we consider the total magnetic moment along the $z$-axis as a starting point and the DM interaction with ${\\alpha}=x,y$.\n\n\n\\subsection{Twisting magnetic structures}\nUsing the Hamiltonian in Eq.~\\eqref{eq:Hamiltonian_continuum}, we can easily calculate the energy for a given $\\bm n(\\bm r)$.\nFor example, the exchange term always gives a non-negative value for $J_{\\mu} > 0$ and the uniform magnetic structure, $\\nabla_{\\mu} \\bm n = 0$, is most favorable.\nOn the other hand, the DM interaction term favors the twisted magnetic structure and its chirality depends on the sign of $D_{\\mu}^{\\alpha}$.\nIn fact, when $J_{\\mu} = J$ and $D_{\\mu}^{\\alpha} = D \\delta_{{\\mu} {\\alpha}}$, a helical structure such as $\\bm n = (\\cos q z, \\sin q z, 0)$ has $q$-dependent energy, $E(q) = J q^2 - D q$,\nand the most stable wavevector is given as $q = D\/2J$.\nThis means that once we can calculate the energy of the helical structure, $E(q)$, in actual systems, then we can evaluate the DM interaction as well as the spin stiffness in this effective Hamiltonian.\nBy changing the twisting axis and direction of $\\bm n$, we can discuss each component of the DM interaction, $D_{\\mu}^{\\alpha}$.\n\n\n\nOne way to compute $E(q)$ in the first-principles calculations is to employ the generalized Bloch theorem,\nthat is, apply different boundary conditions for up and down spins to simulate twisted magnetic structures\\cite{heide2008,heide2009}.\nCalculations of energies for actual twisted magnetic structures with supercells have also been performed\\cite{yang2015}.\n\n\\subsection{Perturbation with respect to spin gauge field}\nHere, we describe the spin current approach to the DM interaction using the method of the spin gauge field\\cite{kikuchi2016}.\nIn Eq.~\\eqref{eq:Hamiltonian_continuum}, electron degrees of freedom are integrated out and the Hamiltonian depends only on the magnetic moment $\\bm n(\\bm r)$.\nTo obtain this Hamiltonian from the microscopic model, we consider the following Hamiltonian in the field representation:\n\\begin{align}\n\tH &= \\int d \\bm r \\; \\sum_{l} c^\\dagger_{l} \\left[ -\\frac{\\hbar^2{\\bm \\nabla}^2}{2m} - J_{\\rm ex}^{l} {\\bm n}\\cdot \\bm \\sigma\n  +\\frac{i}{2}\\sum_{\\mu} \\bm \\lambda_{\\mu} \\cdot\\bm \\sigma \\overleftrightarrow{\\nabla}_{\\mu} \\right] c_{l},\n\\label{eq:Hamiltonian}\n\\end{align}\nwhere  $c^\\dagger_{l}$ and $c_{l}$  are electron creation and annihilation operators for orbital ${l}$, respectively, \n$c^\\dagger \\overleftrightarrow{\\nabla}_{\\mu} c\\equiv c^\\dagger \\nabla_{\\mu} c- (\\nabla_{\\mu} c^\\dagger) c$, and $\\bm \\lambda_{\\mu}$ denotes the spin-orbit interaction.\nThe local direction of the magnetization $\\bm n(\\bm r)$, with $\\bm n=(\\sin\\theta\\cos\\phi, \\sin\\theta\\sin\\phi, \\cos\\theta)$, is static, and $J_{\\rm ex}^{l}$ denotes the exchange constant.  \nThe form of $\\bm \\lambda_{\\mu}$ is determined by the symmetries of the system; for example, $\\lambda^\\alpha_\\mu \\propto \\delta^\\alpha_\\mu$ for B20 alloys such as MnSi and FeGe, while $\\lambda^\\alpha_\\mu\\propto \\varepsilon_{\\alpha\\mu z}$ for Rashba systems with $z$ as its perpendicular direction.  \nWe consider here a simplified model with a quadratic dispersion and a spin--orbit interaction linear in the momentum but the extension to general cases is straightforward.\n\nFor this Hamiltonian, we consider the local spin gauge transformation so that the local magnetic moment $\\bm n(\\bm r)$ points to the $z$ direction.\nFor this purpose, we introduce a unitary transformation in spin space as\n$c_{l}(\\bm r)=U(\\bm r)a_{l}(\\bm r)$, where $U$ is a $2\\times2$ unitary matrix satisfying \n$U^\\dagger(\\bm n\\cdot\\bm \\sigma) U=\\sigma^z$ \\cite{TKS_PR08}. \nAn explicit form of $U$ is given as $U=\\bm m\\cdot \\bm \\sigma$ with $\\bm m\\equiv (\\sin\\frac{\\theta}{2}\\cos\\phi, \\sin\\frac{\\theta}{2}\\sin\\phi, \\cos\\frac{\\theta}{2})$. \nGeometrically, $U$ rotates the spin space by $\\pi$ around the $\\bm m$-axis.\nUsing this unitary transformation, derivatives of the electron field become covariant derivatives as  $\\nabla_{\\mu} c_{l} = U(\\nabla_{\\mu} + iA_{{\\rm s},{\\mu}})a_{l}$, where  \n$A_{{\\rm s},{\\mu}} \\equiv \\sum_{\\alpha} A_{{\\rm s},{\\mu}}^{\\alpha} \\frac{\\sigma^{\\alpha}}{2} = -iU^\\dagger \\nabla_{\\mu} U$ is an SU(2) gauge field, called a spin gauge field, given by $A_{{\\rm s},{\\mu}}^{\\alpha}=2(\\bm m\\times \\nabla_{\\mu}\\bm m)^{\\alpha}$.\nThen, the Hamiltonian for the electron in the rotated frame is given as $H=H_0+H_A$ with \n\\begin{align}\n\tH_0\\equiv & \\int d \\bm r \\; \\sum_{{l}} a^\\dagger_{{l}} \\left[- \\frac{\\hbar^2 {\\bm \\nabla}^2}{2m}  - J_{\\rm ex}^{{l}} \\sigma^z + \\frac{i}{2}\\sum_{\\mu} \\tilde{\\bm \\lambda}_{\\mu} \\cdot \\bm \\sigma \\overleftrightarrow{\\nabla}_{\\mu}   \\right]a_{{l}},\n\\label{H0}\\\\\nH_A \\equiv & \\int d\\bm r \\; \\sum_{{l}}\\left[\\sum_{{\\mu} {\\alpha}}\\hat{\\tilde{j}}_{{\\rm s},{l},{\\mu}}^\\alpha A^{\\alpha}_{{\\rm s},{\\mu}} +\\frac{\\hbar^2}{8m}\\hat{n}_{{\\rm el},{l}}(A_{{\\rm s},{\\mu}}^{\\alpha})^2\\right].\n\\label{HA}\n\\end{align}\nHere, \n$\\tilde{\\lambda}^\\sb_{\\mu}\\equiv \\sum_{\\alpha} R_{{\\alpha} \\sb}\\lambda^{\\alpha}_{\\mu}$ is the spin--orbit coupling constant rotated by the SO(3) matrix $R_{{\\alpha} \\sb}\\equiv 2m_{\\alpha} m_\\sb - \\delta^{{\\alpha}\\sb}$ satisfying  $U^\\dagger \\sigma^{\\alpha} U=\\sum_\\sb R_{{\\alpha} \\sb}\\sigma^\\sb$, and $\\hat{n}_{{\\rm el},{l}}\\equiv a^\\dagger_{l} a_{l} $.\nThe spin current density operator $\\hat{\\tilde{j}}_{{\\rm s},{l},{\\mu}}^{\\alpha}$ in the rotated frame is given by $\\hat{\\tilde{j}}_{{\\rm s},{l},{\\mu}}^{\\alpha} \\equiv - \\frac{i\\hbar^2}{4m}a^\\dagger_{l} \\sigma^{\\alpha} \\overleftrightarrow{\\nabla}_{\\mu} a_{l} - \\frac{1}{2}\\tilde{\\lambda}^\\alpha_{\\mu} a_{l}^\\dagger a_{l}$.\n\nThe interaction between electrons and the magnetization structure is originally given by the exchange interaction, $J_{\\rm ex}^{l} c^\\dagger_{l}{\\bm n}(\\bm r)\\cdot \\bm \\sigma c_{l}$ in Eq.~\\eqref{eq:Hamiltonian}.  After the spin gauge transformation, \nthis exchange interaction becomes a trivial one, $J_{\\rm ex}^{l} a^\\dagger_{l} \\sigma^z a_{l}$ in Eq.~\\eqref{H0}.\nThe interaction between electrons and the magnetization structure is instead given by $H_A$ in Eq.~\\eqref{HA}.  \nWe regard $H_0$ as the non-perturbative Hamiltonian and treat $H_A$ as a perturbation.\nSince $A_{{\\rm s},{\\mu}}^{\\alpha} =2(\\bm m\\times \\nabla_{\\mu} \\bm m)^{\\alpha}$ is proportional to the derivative of $\\bm n(\\bm r)$, the perturbative expansion by $H_A$ gives the derivative expansion with respect to the magnetization structure.  \n\n\nLet us derive the effective Hamiltonian for magnetization, $H_{\\rm eff}$, by integrating out the electron degrees of freedom in this rotated frame: \n\\begin{equation}\n\\exp\\left(-\\frac{i}{\\hbar}\\int dt\\; H_{\\rm eff}\\right) \\equiv \\int \\mathcal D a^\\dagger \\mathcal D a \\exp\\left(\\frac{i}{\\hbar}(S_0-\\int dt\\; H_A)\\right),\n\\label{Seff}\n\\end{equation}\nwhere $S_0$ is the action corresponding to $H_0$, that is, $S_0\\equiv \\int dt[\\int d\\bm r\\sum_{l} i\\hbar a_{l}^\\dagger \\partial_t a_{l} - H_0]$.   We expand the right-hand side of Eq.~\\eqref{Seff} by $H_A$, and obtain $H_{\\rm eff}$ as \n\\begin{align}\n\\int dt\\; H_{\\rm eff} =& i\\hbar \\ln Z_0 + \\int dt \\average{H_A} + \\mathcal O((\\partial \\bm n)^2), \n\\label{Seff2} \\\\\nZ_0 \\equiv& \\int \\mathcal D a^\\dagger \\mathcal D a \\exp\\left(\\frac{i}{\\hbar}S_0\\right). \n\\label{Z0}\n\\end{align}\nLet us extract the DM interaction from $H_{\\rm eff}$.  Since $\\tilde{\\lambda}^{\\alpha}_{\\mu}(\\bm r)=\\sum_\\sb R_{\\sb {\\alpha}}(\\bm r)\\lambda^\\sb_{\\mu}$ in $H_0$ depends on the magnetization structure, not only $\\average{H_A}$ but also $\\ln Z_0$ contributes to the DM interaction.  However, the contribution from $\\ln Z_0$ is only higher order in $\\lambda$, while the contribution from $\\average{H_A}$ contains first-order terms in $\\lambda$.  Therefore, we can neglect the contribution from $\\ln Z_0$ when $\\lambda$ is sufficiently small.  \nFor $\\average{H_A}$, since the DM interaction is the first-order derivative term in the effective Hamiltonian of the magnetization, it is sufficient to consider only the terms in $\\average{H_A}$ proportional to $A_{\\rm s}$.  Therefore, the DM interaction is included in   \n\\begin{align}\n\tH_{\\rm eff} = & \\int d\\bm r \\;   \\sum_{{l} {\\mu} {\\alpha}} \\tilde{j}_{{\\rm s},{l},{\\mu}}^\\alpha A^\\alpha_{{\\rm s},{\\mu}} ,\n\\label{Heff}\n\\end{align}\nwhere $\\tilde{j}_{{\\rm s},{l},{\\mu}}^\\alpha\\equiv \\average{ \\hat{\\tilde{j}}_{{\\rm s},{l},{\\mu}}^{\\alpha} }$ is the expectation value of the spin current density in the rotated frame evaluated by $H_0$ in Eq.~\\eqref{H0}. \nThe spin current density $j_{{\\rm s},{l},{\\mu}}^{\\alpha}$ in the laboratory frame is related as ${j}_{{\\rm s},{l},{\\mu}}^{\\alpha}=\\sum_\\sb R_{{\\alpha} \\sb}\\tilde{j}_{{\\rm s},{l},{\\mu}}^\\sb$.\nThen, by using the identity \n$\\sum_\\sb R_{{\\alpha} \\sb}A_{{\\rm s},\\mu}^\\sb = (\\nabla_{\\mu} \\bm n\\times \\bm n)^{\\alpha} + n^{\\alpha} A_{{\\rm s},{\\mu}}^{z}\n$,\nthe effective Hamiltonian reads \n\\begin{align}\n\tH_{\\rm eff} = & \\int d\\bm r \\;  \\left[\\sum_{{\\mu} {\\alpha}} D_{\\mu}^{\\alpha} (\\nabla_{\\mu} \\bm n\\times \\bm n)^{\\alpha} +  \\sum_{\\mu {l}} {j}_{{\\rm s},{l},{\\mu}}^\\parallel A_{{\\rm s},{\\mu}}^{z} \\right],\n\\label{Heff2}\n\\end{align}\nwhere \n${j}_{{\\rm s},{l},{\\mu}}^\\parallel \\equiv \\tilde{j}_{{\\rm s},{l},{\\mu}}^z=\\bm n\\cdot\\bm{j}_{{\\rm s},{l},{\\mu}}$, and \n\\begin{align}\n        D_{\\mu}^{\\alpha} \\equiv\n \\sum_{{l}} {j}_{{\\rm s},{l},{\\mu}}^{\\perp, {\\alpha}}\n\\label{eq:dm_spincurrent}\n\\end{align}\nwith $j_{{\\rm s},{l},{\\mu}}^{\\perp, {\\alpha}}\\equiv {j}_{{\\rm s},{l},{\\mu}}^{\\alpha}-n^{\\alpha} {j}_{{\\rm s},{l},{\\mu}}^\\parallel$.\nThus, the DM interaction is given by the expectation value of the spin current density of electrons.\nMore precisely, the transversely polarized component of the spin current $j_{{\\rm s},{l},{\\mu}}^{\\perp}$ contributes to the DM interaction.        \nOn the other hand, the longitudinally polarized component  $j_{{\\rm s},{l},{\\mu}}^{\\parallel}$ contributes to the spin-transfer torque term, $j_{{\\rm s},{l},{\\mu}}^{\\parallel}A_{{\\rm s},{\\mu}}^z=j_{{\\rm s},{l},{\\mu}}^{\\parallel}(1-\\cos\\theta)\\nabla_{\\mu}\\phi$.  We have checked that, in the case of the simplified model in Eq.~\\eqref{eq:Hamiltonian},  the contribution to the spin-transfer torque term from $j_{\\rm s}^\\parallel$ is cancelled by that from $\\ln Z_0$ in Eq.~\\eqref{Seff2} up to the $\\lambda^3$-order.   This is expected since the spin-transfer torque will not arise spontaneously at equilibrium states.   \n\nIn the practical calculation, we consider a general form of the spin current density as\n\\begin{align}\n\tj_{{\\rm s},{l},{\\mu}}^{\\alpha} = \\sum_{\\bm k}\\frac{1}{4} \\langle  c_{\\bm k {l}}^\\dagger ( v_{\\mu} \\sigma^{\\alpha} + \\sigma^{\\alpha} v_{\\mu} ) c_{\\bm k {l}} \\rangle,\n\t\\label{J_DFT}\n\\end{align}\nwhere the velocity operator is defined as $v_{\\mu} = d H_{\\bm k}\/ d k_{\\mu}$ with \n$H_{\\bm k}=e^{-i\\bm k\\cdot \\bm x}H e^{i\\bm k\\cdot \\bm x}$.   \nThis general form of spin current actually corresponds to the DM interaction when we apply the spin gauge field technique to a generalized Hamiltonian instead of Eq.~\\eqref{eq:Hamiltonian}.\nNote that the DM interaction for the local-spin model derived from the tight-binding Hamiltonian\\cite{katsnelson2010} corresponds to the discrete representation of Eqs.~\\eqref{eq:dm_spincurrent} and \\eqref{J_DFT}, which can be confirmed using Eq.~\\eqref{eq:JDcontinuum}.\n\nThe spin current in Eq.~\\eqref{eq:dm_spincurrent} is the expectation value at equilibrium states.  Spin currents at equilibrium states arise mainly by two mechanisms.\nOne is due to the magnetization structure, which induces a spin current of the form $j_{{\\rm s},{\\mu}}^{\\alpha} \\propto (\\bm n\\times \\nabla_{\\mu} \\bm n)^{\\alpha}$.\nSuch a magnetization-induced spin current is known to be relevant, for example, in multiferroic systems\\cite{katsura2005}.\nSince this spin current is proportional to $\\nabla_{\\mu} \\bm n$, it does not contribute to the first-order derivative terms in Eq.~\\eqref{Heff2}, but contributes to higher-order derivative terms in the effective Hamiltonian.\nThe other mechanism that induces a spin current at equilibrium states is the spin--orbit interaction with broken inversion symmetry, represented by the last term in Eq.~\\eqref{eq:Hamiltonian}.  \nThis interaction tends to lock the relative angle of the spin and the momentum of electrons and yields a finite spin current.  The existence of such a spin--orbit-induced spin current has been noticed and discussed in the literature\\cite{rashba2003,usaj2005,wang2006_2,sonin2007,sonin2007_prb,tokatly2008}. \nThis spin--orbit-induced spin current arises even when $\\nabla_{\\mu} \\bm n=0$ and  contributes to the first-order derivative terms in Eq.~\\eqref{Heff2}.\nThus, to be precise, $j_{\\rm s}^\\perp$ in Eq.~\\eqref{eq:dm_spincurrent} is generally not the total amount of spin current flowing in the system; when we expand a spin current in powers of $\\nabla_\\mu \\bm n$, then $j_{\\rm s}^\\perp$ in Eq.~\\eqref{eq:dm_spincurrent} corresponds to the non-derivative part.  In the practical calculation to evaluate, for example, the $D_\\mu^x$ and $D_\\mu^y$ components of the DM interaction, we set the magnetization direction $\\bm n$ uniformly in the $z$-direction and calculate $j_{{\\rm s},\\mu}^x$ and $j_{{\\rm s},\\mu}^y$, respectively.  \n\n\nOur result, Eq.~\\eqref{eq:dm_spincurrent}, clarifies that a spin current is a direct origin of the DM interaction.  Let us give a physical interpretation of this result.  Generally,  when an interaction is mediated by some medium, the interaction changes with its flow.  This phenomenon is known as the Doppler effect.  In the case of magnets, magnetic interactions are mediated by electron spin hopping among magnetic moments.  Therefore, when the electron spin flows as a spin current, the magnetic interaction changes and an additional interaction will emerge.   Since the spin current makes two adjacent magnetic moments inequivalent in the sense that one is located upstream and the other downstream,  the additional magnetic interaction is antisymmetric with respect to the exchange of the two adjacent magnetic moments.  Thus, an antisymmetric magnetic interaction, that is, the DM interaction, arises as the Doppler effect due to the spin current.  Let us see this Doppler effect more closely in Fig.~\\ref{fig:Doppler}.  Consider two adjacent magnetic moments with directions $\\bm n$ and $\\bm n'=\\bm n+ (\\bm a\\cdot \\bm \\nabla)\\bm n$, where $\\bm a$ is the vector connecting the two sites, and electrons hopping between them.  When an electron with spin $\\bm s$ hops from $\\bm n$ to $\\bm n'$,  its spin precesses by $\\epsilon (\\bm s\\times \\bm n)$, with $\\epsilon$ a small coefficient, due to the torque from the magnetic moments.  For the electron hopping with such a precession, the spatial variation of the magnetic moments will look like not $(\\bm a\\cdot \\bm \\nabla)\\bm n$ but rather $(\\bm a\\cdot \\bm \\nabla)\\bm n - \\epsilon (\\bm s\\times \\bm n)$.  When we express a spin current as $j_{{\\rm s},\\mu}^\\alpha=s^\\alpha v_\\mu$ with $v_\\mu$ the velocity of an electron, the discussion so far suggests that the derivative of the magnetization vector changes from $\\nabla_\\mu \\bm n$ to $\\mathfrak{D}_\\mu\\bm n\\equiv\\nabla_\\mu\\bm n-\\eta(\\bm j_{{\\rm s},\\mu}\\times \\bm n)$, with $\\eta$ some coefficient, due to the presence of the spin current.  Accordingly,  the interaction energy, which was originally  $(\\nabla_\\mu \\bm n)^2$ when there are no spin currents, changes into  \n$(\\mathfrak{D}_\\mu\\bm n)^2\\cong (\\nabla_\\mu\\bm n)^2+2\\sum_\\mu\\eta\\bm j_{{\\rm s},\\mu}\\cdot(\\nabla_\\mu\\bm n\\times \\bm n)$.  Thus, the DM interaction $\\sum_\\mu\\bm D_\\mu\\cdot(\\nabla_\\mu\\bm n\\times \\bm n)$ arises as the spin-current-induced Doppler shift of the exchange energy $(\\nabla_\\mu \\bm n)^2$.  \nNote that only the spin current component $j_{\\rm s}^\\perp$ perpendicular to $\\bm n$ contributes to the Doppler shift $\\mathfrak{D}_\\mu\\bm n\\equiv\\nabla_\\mu\\bm n-\\eta(\\bm j_{{\\rm s},\\mu}\\times \\bm n)$, which is consistent with the result of Eq.~\\eqref{eq:dm_spincurrent}.  \n\n\nFinally, let us mention that the spin current method can be applied to insulating systems.\nIn fact, the intrinsic spin current can be finite even in insulators.\nThe application to insulators such as Cu$_2$OSeO$_3$ will be discussed elsewhere.\n\n\\begin{figure}\n\t\\includegraphics[bb=0 0 1224 738, width=0.45\\textwidth]{81712Fig1.png}\n\t\\caption{\n\t\t(Color online) Schematic picture showing the mechanism of the spin-current-induced Doppler shift.\t\n A spin current $j_{{\\rm s},\\mu}^\\alpha=s^\\alpha v_\\mu$ is flowing with spin polarization $\\bm s$ and velocity $\\bm v$.  Due to the torque from the localized spin $\\bm n$, this spin current flows with its spin precessing.  Such a spin current with precession changes the spatial variation of the localized spins from $\\nabla_\\mu \\bm n$ to $\\nabla_\\mu \\bm n - \\bm j_{{\\rm s},\\mu}\\times \\bm n$, which is the Doppler shift due to the spin current.    \n Adapted with permission.\\cite{kikuchi2016} Copyright 2016, American Physical Society.\n\t}\n\\label{fig:Doppler}\n\\end{figure}\n\n\\subsection{Perturbation with respect to exchange couplings}\nHere, we consider Eq.~\\eqref{eq:Hamiltonian} directly to derive the DM interaction.\nWe assume that $J_{\\rm ex}$ is small and consider the second-order perturbation to integrate out the electron degrees of freedom as in the derivation of the RKKY interaction.\nSince there is a spin-orbit interaction, the effective Hamiltonian for $\\bm n(\\bm r)$ includes the DM interaction term $D_{\\mu}^{\\alpha} (\\nabla_{\\mu} \\bm n\\times \\bm n)^{\\alpha}$ with\\cite{koretsune2015}\n\\begin{align}\n\tD_{\\mu}^\\sb = \\frac{J_{\\rm ex}^2}{2} \n\t\\lim_{q \\to 0} \\frac{\\partial \\chi_0^{{\\alpha} \\gamma}(\\bm q,i\\omega_n=0)}{i \\partial q^{\\mu}}.\n\t\\label{eq:dm_chi}\n\\end{align}\nHere, $({\\alpha},\\sb,\\gamma)=(x,y,z), (y,z,x),$ or $(z,x,y)$, and\n$\\chi_0$ is the non-interacting spin susceptibility defined as\n\\begin{align}\n\t&\\chi_0^{{\\alpha} \\gamma}(\\bm q, i\\omega_l) = -\\frac{T}{V} \\sum_{l,l',s_1,s_2,s_3,s_4}\\sum_{\\bm k, m} \\sigma^{\\alpha}{s_4 s_1}\\nonumber\\\\\n\t&\\times G^0_{ls_1 l' s_2} (\\bm k, i\\omega_m) \\sigma^\\gamma_{s_2 s_3} G^0_{l' s_3 l s_4}(\\bm k + \\bm q, i\\omega_m + i \\omega_l),\n\\end{align}\nwhere $\\sigma$ is the Pauli matrix and $G^0$ is the non-interacting Green's function in the orbital basis.\nUsing this spin susceptibility, we can write the DM interaction as\n\\begin{align}\n\tD_{\\mu}^\\sb &= \\frac{1}{V} \\sum_{\\bm k} D_{\\mu}^\\sb(\\bm k)\\\\\n\tD_{\\mu}^\\sb(\\bm k) &= \n\t\\lim_{\\bm q \\to 0} \\frac{\\partial}{i \\partial q^{\\mu}} \\sum_{n,n'}\n\t\\frac{f(\\varepsilon_{n' \\bm k+ \\bm q}) - f(\\varepsilon_{n \\bm k})}{\\varepsilon_{n' \\bm k+ \\bm q} - \\varepsilon_{n \\bm k}}\\nonumber\\\\\n\t& \\times \\langle n\\bm k| \\sigma^\\alpha | n' \\bm k + \\bm q \\rangle\n\t\\langle n'\\bm k+ \\bm q| \\sigma^\\gamma | n \\bm k \\rangle,\n\\end{align}\t\nwhere $| n \\bm k \\rangle$ is the eigenvector of the Kohn-Sham Hamiltonian with the eigenvalue of $\\varepsilon_{n \\bm k}$.\nNote that this representation uses the expectation value of two Green's functions, which is similar to that using the magnetic force theorem and is in sharp contrast to the spin current approach.\n\n\n\\subsection{Relationship between band structures and DM interactions}\nOne advantage of spin current and spin susceptibility approaches is that we can discuss the relationship between band structures and the DM interaction.\nIn fact, we can easily calculate the contribution from the band anticrossing points, where the nontrivial spin texture arises owing to the spin-orbit couplings.\nHere, let us consider the simple 2$\\times$2 Hamiltonian\n\\begin{align}\n\tH = \\gamma (k_x \\sigma^x + k_y \\sigma^y + m \\sigma^z).\n\t\\label{eq:2x2ham}\n\\end{align}\nThen, using Eq.~\\eqref{eq:dm_spincurrent} or Eq.~\\eqref{eq:dm_chi},\nwe obtain\n\\begin{align}\n\tD_{\\mu}^{\\alpha} = \\gamma D \\begin{pmatrix}\n\t\t1 & 0 \\\\\n\t\t0 & 1\n\t\\end{pmatrix}\n\\end{align}\nwith\n\\begin{align}\n\tD = n_{\\rm el} = \\pi (\\mu^2 - m^2) [ \\theta(\\mu - m) - \\theta(-\\mu - m) ]\n\\end{align}\nfrom Eq.~\\eqref{eq:dm_spincurrent}\nand\n\\begin{align}\n\tD = \\frac{J_{\\rm ex}^2}{16 \\pi} [ \\theta(\\mu - m) - \\theta(-\\mu - m) ]\n\\end{align}\nfrom Eq.~\\eqref{eq:dm_chi}.\nHere we assume that the number of electrons, $n_{\\rm el}$, is zero at $\\mu = 0$ and the row and column correspond to spatial (${\\mu}$) and spin (${\\alpha}$) indices, respectively.\nSchematic pictures of the energy band and the DM interactions are shown in Fig.~\\ref{fig:schematic2x2}.\nIt is interesting to note that the DM interaction is negative for $\\mu < -m$ and positive for $\\mu > m$ for $\\gamma > 0$,\nindicating that this anticrossing point gives positive contributions to the DM interaction.\nThat is, when the chemical potential sweeps across the anticrossing points from below to above, the DM interaction increases.\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[scale=0.35]{81712Fig2.eps}\n\t\t\\caption{\n\t\t\t(Color online) (a) Band structure of the two-band model defined in Eq.~\\eqref{eq:2x2ham} and (b) the chemical potential dependence of the DM interaction corresponding to this band structure calculated by the spin current method (black line) and spin susceptibility method (red line).\n\t\t}\n\t\\label{fig:schematic2x2}\n\t\\end{center}\n\\end{figure}\n\nThe relationship between the spin configurations and the symmetry of the DM coefficient is also clear in these formalisms.\nLet us consider three typical spin configurations of a conduction electron in the momentum space, the Rashba (which arises in polar systems), Dresselhaus, and Weyl (in chiral systems) configurations, represented by the Hamiltonians\n$H_{\\rm R}=\\alpha(k_x\\sigma^y-k_y\\sigma^x)$,\n$H_{\\rm D}=\\beta(k_x\\sigma^x-k_y\\sigma^y)$, and\n$H_{\\rm W}=\\gamma(k_x\\sigma^x+k_y\\sigma^y)$, respectively.\nThe schematic spin textures are shown in Fig.\\ \\ref{fig:spin_textures}.\nThe DM interactions in these cases (denoted by $D_{\\rm R}$, $D_{\\rm D}$, and $D_{\\rm W}$, respectively) are\n\\begin{align}\n\t&D_{{\\rm R},{\\mu}}^{\\alpha} =  \\alpha n_{\\rm el} \\left(\\begin{array}{ccc} 0 &1&0\\\\ -1 & 0 & 0 \\\\ 0&0& 0\\end{array}\\right),\n\\;\nD_{{\\rm D},{\\mu}}^{\\alpha} = \\beta n_{\\rm el} \\left(\\begin{array}{ccc} 1 &0&0\\\\ 0 & -1 & 0 \\\\ 0&0& 0\\end{array}\\right),\n\\;\n\\notag \\\\\n&D_{ {\\rm W},{\\mu}}^{\\alpha} = \\gamma n_{\\rm el} \\left(\\begin{array}{ccc} 1 &0&0\\\\ 0 & 1 & 0 \\\\ 0&0& 0\\end{array}\\right).\n\\label{DR and DD and DW}\n\\end{align}\nThis result clearly relates the symmetry of crystal structures, spin-orbit couplings, and the DM interactions.\nFor example, DM interactions are antisymmetric in polar systems whereas diagonal DM interactions are expected in non-polar systems, as discussed by a different approach\\cite{Kim13,Gungordu16}.\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[bb=0 0 1200 600, scale=0.20]{81712Fig3.png}\n\t\t\\caption{\n\t\t\t(Color online) Spin texture in momentum space for (a) Rashba, (b) Dresselhaus, and (c) Weyl-type Hamiltonians.\n Adapted with permission.\\cite{kikuchi2016} Copyright 2016, American Physical Society.\n\t\t}\n\t\t\\label{fig:spin_textures}\n\t\\end{center}\n\\end{figure}\n\n\n\n\n\n\\section{Application to Chiral Ferromagnets}\n\n\\subsection{Electronic structure of FeGe}\nFeGe is a B20-type chiral ferromagnet and is extensively studied experimentally.\nA skyrmion crystal state due to the DM interaction has been observed near room temperature\\cite{yu2011} and the divergence of the skyrmion size has been pointed out in Mn$_{1-x}$Fe$_x$Ge\\ with $x=0.8$, indicating the sign change of the DM interaction\\cite{shibata2013}.\nNeutron scattering experiments also suggest the sign change of the DM interaction in Mn$_{1-x}$Fe$_x$Ge\\ with $x=0.8$\\cite{grigoriev2013} and Fe$_{1-x}$Co$_x$Ge\\ with $x=0.6$\\cite{grigoriev2014}.\nIn MnGe, a unique three-dimensional spin structure has been observed\\cite{kanazawa2011,kanazawa2012}.\nThe crystal symmetry of the B20 compounds is P2$_1$3, and owing to its symmetry, the DM interaction should be given as $D_\\mu^{\\alpha} = D \\delta_{{\\alpha}\\mu}$.\n\nFigure \\ref{fig:band}(a) shows the DFT band structure of FeGe (black solid lines).\nHere, the spin-orbit couplings are included and the total ferromagnetic moment is parallel to the (001) direction ($z$ axis).\nThe calculated local magnetic moment is 1.18 $\\mu_B$ per Fe atom, which is consistent with the results of experiments\\cite{wappling1968,lundgren1968} and previous calculations\\cite{yamada2003}.\nThe red broken lines are the Wannier-interpolated band structure with Fe 3d and Ge 4p Wannier orbitals.\nAccording to the average energy difference between up and down spins for the Fe 3d orbitals, the exchange splitting of the 3d orbitals is estimated to be $\\Delta = 1.17$ eV.\nFigure \\ref{fig:band}(b) shows the obtained tight-binding band structure around the Fermi level with the color representing the weight of the up spin.\nSince the spin-orbit coupling of FeGe is not strong, each band is basically characterized as either an up-spin or down-spin band, and the complex spin texture emerges only around the band anticrossing region.\nHereafter, to discuss the atomic composition dependences of Mn$_{1-x}$Fe$_x$Ge\\ and Fe$_{1-x}$Co$_x$Ge, we show the results obtained using the electronic structures with self-consistent charge densities for corresponding carrier densities by fixing the atomic geometries and the lattice constant to the experimental values of FeGe\\cite{lebech1989}.\n\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[scale=0.60]{81712Fig4a.eps}\n\t\t\\includegraphics[scale=0.60]{81712Fig4b.eps}\n\t\t\\caption{\n\t\t\t(Color online) (a) Comparison between DFT band structure (black solid lines) and tight-binding band structure (red broken lines).\n\t\t\t(b) Detailed band structure around the Fermi level with colors representing the weight of the up spin;\n\t\t\tthat is, red (blue) lines correspond to up-spin (down-spin) bands.\n\t\t\tThe Fermi level is set to zero.\n\t\t\tAdapted from Ref.\\ \\citen{koretsune2015} under the CC-BY 4.0 license.\n\t\t}\n\t\t\\label{fig:band}\n\t\\end{center}\n\\end{figure}\n\n\\subsection{DM interaction using $E(q)$}\n\n\\begin{figure*}\n\t\\begin{center}\n\t\t\\includegraphics[scale=0.70]{81712Fig5.eps}\n\t\t\\caption{\n\t\t\t(Color online) Spin stiffness $J$ and\n\t\t\tDM interaction $D$ for Mn$_{1-x}$Fe$_x$Ge\\ and Fe$_{1-x}$Co$_x$Ge\\ calculated using the energies of helical spin structures $E(q)$ from Ref.\\ \\citen{gayles2015} (green line) and Ref.\\ \\citen{kikuchi2016} (red lines).\n\t\t\tThe error bars for the red lines indicate the fitting errors of $E(q) = J q^2 - D q$.\n\t\t}\n\t\t\\label{fig:DM_Eq}\n\t\\end{center}\n\\end{figure*}\n\nFigure \\ref{fig:DM_Eq} shows the spin stiffness and the DM interaction calculated using $E(q)$ for Mn$_{1-x}$Fe$_x$Ge\\ and Fe$_{1-x}$Co$_x$Ge$\\;$ in two different studies.\\cite{gayles2015,kikuchi2016} \nAlthough there is a slight difference in the two studies due to the treatment of the alloys, the results well reproduce the sign change of the DM interaction observed in the experiments, that is, at $x=0.8$ in Mn$_{1-x}$Fe$_x$Ge\\ and $x=0.6$ in Fe$_{1-x}$Co$_x$Ge.\nIn addition, a recent spin-wave spectroscopy experiment has shown that the DM interaction for FeGe is $\\sim$ -3.6 meV\\AA, which is also consistent with the calculation.\nIn contrast, around MnGe, the DM interaction is very small while the experiments show a very small skyrmion size, indicating a large DM interaction\\cite{kanazawa2011}.\nThis discrepancy may come from the validity of the evaluation using $E(q)$ since this method assumes that the spatial spin variation is small.\nIn contrast with other skyrmion materials like FeGe and MnSi, MnGe is proposed to have a unique three-dimensional skyrmion lattice structure\\cite{kanazawa2012}, which suggests that not only the DM interaction but also other mechanisms such as the frustration of the exchange coupling may play an important role.\n\nThe values of $J$ do not change much for Mn$_{1-x}$Fe$_x$Ge\\ and are on the order of 1 eV\\AA$^2$.\nFor Fe$_{1-x}$Co$_x$Ge, on the other hand, the values of $J$ decrease with increasing $x$ and almost vanish for $x > 0.6$, indicating that the ferromagnetic state becomes unstable.\nThe wavelength of the helix, $4 \\pi J\/D$, for FeGe is estimated to be 160 nm, which is almost the same order as the experimental value.\n\n\n\n\\subsection{DM interaction using the spin current}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[scale=0.60]{81712Fig6.eps}\n\t\t\\caption{\n\t\t\t(Color online) Spin currents, $j_{{\\rm s},x}^x$ and $j_{{\\rm s}, y}^y$ for FeGe as a function of spin-orbit coupling strength.\n\t\t}\n\t\t\\label{fig:lambda_dep}\n\t\\end{center}\n\\end{figure}\n\n\\begin{figure*}\n\t\\begin{center}\n\t\t\\includegraphics[scale=0.70]{81712Fig7.eps}\n\n\t\t\\caption{\n\t\t\t(Color online) DM interactions $D$ for Mn$_{1-x}$Fe$_x$Ge\\ and Fe$_{1-x}$Co$_x$Ge\\ calculated using the spin current (blue line) and the $\\lambda$-linear contribution in the spin current (red line).\n\t\t\tThe error bars indicate the variances of $D_x^x$ and $D_y^y$.\n\t\t}\n\t\t\\label{fig:DM_js}\n\t\\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\begin{center}\n\t\t\\includegraphics[scale=0.70]{81712Fig8.eps}\n\t\t\\caption{\n\t\t\t(Color online) (a) Contribution of each band to the DM interaction, $D_{n\\bm k}$, with dominant band anticrossing points circled, and (b) the energy distribution of the DM interaction, $D(E)$ (black line), for FeGe. \n\t\t\tThe Fermi energy dependence of the DM interaction, $D\\equiv\\int^{E_F} D(E')f(E')dE'$, within the rigid band approximation is also shown as the red line.\n\t\t\tAdapted with permission.\\cite{kikuchi2016} Copyright 2016, American Physical Society.\n\t\t\t}\n\t\t\\label{fig:Dk}\n\t\\end{center}\n\\end{figure*}\n\n\nNext, let us discuss the DM interaction using Eq.~\\eqref{J_DFT}.\nSince this equation is valid within the first order of the spin-orbit coupling, it is important to check the spin-orbit-coupling-strength dependence of the spin current.\nFor this purpose, an electronic structure calculation without the spin-orbit coupling is performed, and a tight-binding model that reproduces this band structure is constructed.\nBy mixing the two tight-binding Hamiltonians with and without the spin-orbit coupling, the spin current as a function of the spin-orbit coupling strength is calculated as shown in Fig.\\ \\ref{fig:lambda_dep}.\nAs can be seen, the spin current is determined almost within the first order of the spin-orbit coupling in this system, indicating that the spin current is a good approximation for the DM interaction.\nHere, the difference between $j_{{\\rm s},x}^x$ and $j_{{\\rm s},y}^y$ originates from the magnetic moment, which breaks the symmetries that connect the $x$- and $y$-axes.\nIn fact, if we consider the magnetic moment along the (111) direction, the two perpendicular spin currents have the same values due to the C$_3$ symmetry.\nNote that for large spin-orbit-coupling systems such as a Co\/Pt bilayer, the higher-order contributions cannot be neglected\\cite{freimuth2017}.\n\nFigure \\ref{fig:DM_js} shows the DM interaction estimated directly using the spin current and by extracting the first-order contribution from the spin current.\nSince the idea of the spin current approach is the same as the method using $E(q)$, these results give almost the same result as shown in Fig.\\ \\ref{fig:DM_Eq}.\nThe agreement between the first-order contribution and the original spin current supports the validity of this estimation.\nInterestingly, the difference between $j_{{\\rm s},x}^x$ and $j_{{\\rm s},y}^y$ becomes smaller for the first-order contribution.\nThe higher-order contribution of the spin-orbit coupling and the possible anisotropy of the DM interaction are important future issues.\n\nIn the spin current approach, it is possible to discuss the relationship between the band structure and the DM interaction and, as a result, the chemical potential dependence of the DM interaction.\nIn fact, we can rewrite Eq.~\\eqref{J_DFT} as\n\\begin{align}\n\n\tD = \\sum_{n \\bm k} D_{n\\bm k} f(\\epsilon_{n\\bm k}) = \\int D(E) f(E) dE,\n\\end{align}\nwhere $n$, $f(E)$, $D_{n\\bm k}$, and $D(E)$ are the band index, the Fermi distribution function, the contribution of each band to the DM interaction, and the density of the DM interaction, respectively.\nFigure \\ref{fig:Dk}(a) shows the band structure of FeGe with the color representing $D_{n\\bm k}$.\nAs discussed in Sect. 2, we can see that the DM interaction comes from the restricted region of the band structure where the complex spin texture arises due to band anticrossing.\nThe density of the DM interaction, $D(E)$, shown in Fig.\\ \\ref{fig:Dk}(b), also gives useful information for discussing the carrier density dependence of the DM interaction. \nThat is, in this case, $D(E) < 0$ for $E<0$ and $D(E) > 0$ for $E>0$ indicate the dip structure around FeGe ($E=0$) and the resulting two sign changes in Mn$_{1-x}$Fe$_x$Ge\\ and Fe$_{1-x}$Co$_x$Ge.   \n\n\n\\subsection{DM interaction using the spin susceptibility}\n\n\\begin{figure*}\n\t\\begin{center}\n\t\t\\includegraphics[scale=0.70]{81712Fig9.eps}\n\t\t\\caption{\n\t\t\tDM interactions for Mn$_{1-x}$Fe$_x$Ge\\ and Fe$_{1-x}$Co$_x$Ge\\ calculated using the off-diagonal spin susceptibility.\n\t\t\tThe error bars indicate the variances of $D_x^x$ and $D_y^y$.\n\t\t}\n\t\t\\label{fig:DM_chiq}\n\t\\end{center}\n\\end{figure*}\n\nThe DM interaction using Eq.~\\eqref{eq:dm_chi} is shown in Fig.\\ \\ref{fig:DM_chiq}.\nHere, the exchange coupling $J_{\\rm ex}$ is estimated by the exchange splitting of the $d$ orbitals, that is, the average energy difference of $3d$ Wannier orbitals for up and down spins.\nIn this result, the two sign-change points ($x\\sim0.4$ for Mn$_{1-x}$Fe$_x$Ge\\ and $x\\sim0.02$ for Fe$_{1-x}$Co$_x$Ge) are shifted to some extent from the positions in the experiments.\nThis may be because the perturbation expansion with respect to $J_{\\rm ex}$ is not a good approximation in this material.\nIn fact, $J_{\\rm ex}$ is not so small and the effect of exchange splitting on the electronic structure cannot be explained only by $J_{\\rm ex} {\\bm n} \\cdot \\bm \\sigma$.\nOn the other hand, the DM interaction for Mn$_{1-x}$Fe$_x$Ge\\ with small $x$ is larger than the previous two results and is more consistent with the experimental result.\nThis suggests that the pertubation expansion with respect to $J_{\\rm ex}$ reasonably works even for large $J_{\\rm ex}$ systems and can give a better estimation once the derivative expansion with respect to the magnetic structure does not work.\nIn this way, this method can play a complementary role in the evaluation of the DM interaction.\n\n\n\\section{Conclusion}\nIn this paper, we have reviewed three approaches to evaluate the DM interaction.\nThe first one is to evaluate the energy of the helical spin structure, $E(\\bm q)$, directly from first principles and extract the DM interaction.\nThe method is very powerful while the obtained information is limited.\nThe second one is the perturbation expansion with respect to the spin gauge field, which represents the spin twisting.\nThe idea is to extract the DM interaction term by twisting the spin structure, which is almost the same as the first approach, while this method only needs the electronic structure calculation for the uniform magnetic state.\nThis method gives a clear picture that the DM interaction can be expressed in terms of the spin current at the equilibrium within the first order of the spin-orbit couplings.\nFurthermore, the relationship to the band structure and the effect of carrier doping can be easily discussed.\nThe third one is the perturbation expansion with respect to the exchange coupling, which can be understood as the extension of the RKKY mechanism.\nThis method also clarifies the relationship between the band structure and the DM interaction, which is roughly consistent with the spin current approach.\n\nBy applying these methods to chiral ferromagnets, it has been shown that the first two approaches give almost the same results, which agree well with the experiments, while the third one gives a slightly different result.\nSince the starting points of the first two approaches and the third approach are completely different, these approaches will play complementary roles in the evaluation of the DM interaction.\nThe application of these methods to various systems and clarifying their validity are important future issues for designing materials utilizing this unique antisymmetric interaction.\n\n\n\\acknowledgement\nThe authors thank H. Fukuyama, N. Kanazawa, H. Kawaguchi, W. Koshibae, H. Kohno, T. Momoi, D. Morikawa, N. Nagaosa, M. Ogata, S. Seki, K. Shibata, Y. Suzuki, G. Tatara, and Y. Tokura for valuable discussions.\nIn particular, the authors thank G. Tatara for the collaboration in Ref.\\ \\citen{kikuchi2016} and T. Ko. and R. A. thank N. Nagaosa for the collaboration in Ref. \\citen{koretsune2015}.\nThis work was supported by JSPS KAKENHI Grant Numbers JP25400344 and JP26103006 and JST PRESTO Grant Number JPMJPR15N5.  T. Ki. is a Yukawa Research Fellow supported by the Yukawa Memorial Foundation.    \n\n\\bibliographystyle{jpsj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDetecting, interpreting and comparing structures and properties of network data about social interactions and complex physical phenomena is critically important to a variety of problems. However, this is a difficult task because comparisons between two or more networks can involve checking for graph or subgraph isomorphism, for which no tractable solution is known. Instead, various network properties (\\textit{e.g.}, degree distribution, centrality distributions) have been used to describe and compare networks. \n\nAnother approach is to consider a network's global structure as a by-product of a graph's local substructures~\\cite{ugander2013subgraph}. More sophisticated graph statistics are based on counting the number of small motifs \\cite{milo2002network} or graphlets~\\cite{ahmed2015efficient} present in the graph and comparing their distributions \\cite{yaveroglu2014revealing}. Unfortunately, graphlet counting presupposes that all possible graphlets be enumerated ahead of time. Because the number of unique graphlets increases exponentially with the number of nodes in the graphlet, previous work has been limited to graphlets of at most five nodes.\n\nAn alternative to developing sophisticated graph statistics is to learn graph generation models that encode properties of the graph in various ways. Graph generators like the Exponential Random Graph Model (ERGM) \\cite{robins+al}, the Chung-Lu Edge Configuration Model (CL) \\cite{chung+lu}, the Stochastic Kronecker Graph (SKG) \\cite{leskovec2010kronecker}, and the Block Two-Level Erd\\H{o}s-R\\'{e}nyi Model (BTER) \\cite{seshadhri2012community} can be fitted to real-world graphs.\n\nRecent work has found that many social and information networks have a more-or-less tree-like structure, which implies that detailed topological properties can be identified and extracted from a graph's {\\em tree decomposition}~\\cite{adcock2016tree}. \nBased on these findings, Aguinaga~\\textit{et~al.}~described a method to turn a graph's tree decomposition into a \\emph{Hyperedge Replacement Grammar} (HRG)\n~\\cite{aguinaga2016growing}. The HRG model can then generate new graphs with properties similar to the original.\n\nOne limitation of the HRG model is that the instructions for reassembling the building blocks, \\textit{i.e.}, the graph grammar, encode only enough information to ensure that the result is well-formed. HRG production rules are extracted directly from the tree decomposition; some rules from the top of the tree, some from the middle of the tree, and some from the bottom of the tree. Then, to generate an new graph that is similar to the original graph, we would expect that rules from the top of the tree decomposition are applied first, rules from the middle next, and rules from the leaves of the tree are applied last. However, when generating a new graph the HRG model applies rules probabilistically; where a rule's probability relative to its frequency in the grammar.  However, when generating a new graph, the rules in HRG models have no context on when they should fire. HRG models are in need of a mechanism that corrects for this problem by providing context to the rules.\n\nIn the present work, we make three contributions:\n\\begin{enumerate}\n    \\item We improve the HRG model by encoding context in latent variables.\n    \\item We propose a methodology for evaluating our model that enforces a strict separation between training and test data, in order to guard against overfitting.\n    \\item We test our model on 6 train\/test pairs of graphs and find that it discovers a better model, that is, one that generalizes better to the test data, than the original HRG model as well as Kronecker and Chung-Lu models.\n\\end{enumerate} \n\n\n\\section{Background}\n\nBefore we introduce our model, we first provide an overview and examples of the HRG model.\n\n\\subsection{Hyperedge Replacement Grammars}\n\nLike a context free string grammar (CFG), an HRG has a set of production rules $A \\rightarrow R$, where $A$ is called the left-hand side (LHS) and $R$ is called the right-hand side (RHS).\nIn an HRG, a rule's RHS is a graph (or hypergraph) with zero or more \\emph{external} nodes. Applying the rule replaces a hyperedge labeled $A$ with the graph $R$; the nodes formerly joined by the hyperedge are merged with the external nodes of $R$. The HRG generates a graph by starting with the start nonterminal, $S$, and applying rules until no more nonterminal-labeled hyperedges remain.\n\n\\subsection{Tree Decomposition}\n\\begin{figure}\n\\centering\n\\input{.\/figs\/td}\n\\caption{An example graph and its tree decomposition. The width of this tree decomposition is 3, \\textit{i.e.}, the size of the largest bag minus 1. The sepset between each bag and its parent is labeled in blue. Bags are labeled ($\\eta_0$, etc.) for illustration purposes only.\n}\n\\label{fig:td}\n\\end{figure}\n\nGiven a graph $H = (V,E)$, a \\emph{tree decomposition} is a tree whose nodes, called \\emph{bags}, are labeled with subsets of $V$, in such a way that the following properties are satisfied:\n\\begin{itemize}\n\\item For each node $v \\in V$, there is a bag $\\eta$ that contains $v$.\n\\item For each edge $(u,v) \\in E$, there is a bag $\\eta$ that contains $u$ and $v$.\n\\item If bags $\\eta$ and $\\eta^\\prime$ contain $v$, then all the bags on the path from $\\eta$ to $\\eta^\\prime$ also contain $v$.\n\\end{itemize}\n\nIf $\\eta^\\prime$ is the parent of $\\eta$, define $\\bar{\\eta} = \\eta' \\cap \\eta$ (also known as the \\emph{sepset} between $\\eta'$ and $\\eta$). If $\\eta$ is the root, then $\\bar{\\eta} = \\emptyset$.\n\nAll graphs can be decomposed (though not uniquely) into a tree decomposition, as shown in Fig.~\\ref{fig:td}. In simple terms, a tree decomposition of a graph organizes its nodes into overlapping \\emph{bags} that form a tree. The \\emph{width} of the tree decomposition, which is related to the size of the largest bag, measures how tree-like the graph is. Finding optimal tree decompositions is NP-hard, but there is significant interest in finding fast approximations because many computationally difficult problems can be solved efficiently when the data is constrained to be a tree-like structure. \n\n\\subsection{Grammar Extraction}\n\\label{sec:extraction}\n\n\\begin{figure}\n\\centering\n\\input{.\/figs\/rule}\n\\caption{Extraction of an HRG production rule from $\\eta_2$ containing graph vertices \\{2,3,4,5\\}. The LHS of the production rule corresponds to the sepset of the bag and its parent. The RHS of the production rule contains nodes from the bag, terminal edges induced from the original graph, and nonterminal edges from the sepset between the bag and its children.}  \n\\label{fig:rule}\n\\end{figure}\n\n\\iffalse\n\\begin{figure*}\n\\centering\n\\input{.\/figs\/grammar}\n\\caption{The full grammar extracted from the example of Figure~\\ref{fig:td}.}\n\\label{fig:grammar}\n\\end{figure*}\n\\fi\n\nAguinaga et al.~\\cite{aguinaga2016growing} extract HRG rules from a graph, guided by a tree decomposition of the graph. For example, Figure~\\ref{fig:rule} illustrates how one HRG rule is extracted from a tree decomposition. \n\nIf we assume that the tree decomposition is rooted, then every bag $\\eta$ of the tree decomposition corresponds to an edge-induced subgraph, which we write $G_\\eta$, defined as follows: For each edge $(u,v) \\in E$, if every bag $\\eta'$ containing $u,v$ is either equal to $\\eta$ or a descendant of $\\eta$, then $(u,v) \\in H_\\eta$.\nFor example, in Figure~\\ref{fig:td}, the bag $\\eta_2 = \\{2,3,4,5\\}$ corresponds to the subgraph $H_{\\eta_2}$ whose edges are 1--2, 1--5, 2--3, 2--4, and 3--5.\n\nIf $H=(V,E)$ is a graph and $H^\\prime=(V^\\prime,E^\\prime)$ is an edge-induced subgraph of $H$, we define an \\emph{external} node of $H^\\prime$ to be any node of $H^\\prime$ that has a neighbor not in $H^\\prime$. Then, define the operation of \\emph{replacing} $H^\\prime$ with a hyperedge to be:\n\\begin{itemize}\n\\item Remove all edges in $E'$.\n\\item Remove all nodes in $V'$ except for the external nodes.\n\\item Add a hyperedge joining the external nodes.\n\\end{itemize}\n\nEvery bag $\\eta$ also induces a HRG rule $\\text{N}^{|\\bar{\\eta}|} \\rightarrow R$, where $R$ is constructed as follows. \n\\begin{itemize}\n\\item Make a copy of $H_\\eta$.\n\\item Label the nodes in $\\bar{\\eta}$ as external nodes.\n\\item For each child $\\eta_i$ of $\\eta$, replace $H_{\\eta_i}$ with a hyperedge labeled $\\text{N}^{|\\bar{\\eta}_i|}$.\n\\end{itemize}\nFor example, in Figure~\\ref{fig:rule}, the bag $\\eta_2$ induces the rule shown at right. The LHS is $\\text{N}^3$ because the sepset between $\\eta_2$ and its parent has three nodes ($3,4,5$); in the RHS, these three nodes are marked as external. The node numbers are for illustration purposes only; they are not actually stored with the production rules. The RHS has two terminal edges (2--3, 2--4) from the original graph and one nonterminal edge (2--5) corresponding to the sepset between $\\eta_2$ and its one child. \n\n\\iffalse\nThree other HRG production rules are extracted from the example tree decomposition in the same way (Figure~\\ref{fig:grammar}). Leaf bags in the tree decomposition do not produce RHSs with nonterminals, because they do not have children. The root bag has no parent, thus its sepset is empty and its LHS is $\\text{N}^0$, the start nonterminal.\n\\fi\n\nAfter an HRG is extracted from the tree decomposition, its production rules are gathered into a set, merging identical rules and assigning to each unique rule $(A\\rightarrow R)$ a probability $P(A \\rightarrow R) = P(R \\mid A)$ proportional to how many times it was encountered during extraction. This grammar can then be used to randomly generate new graphs, or compute the probability of another graph.\n\n\\section{Latent Variable Probabilistic Hyperedge Replacement Grammars}\n\n\nHere, we improve upon the HRG model by encoding more context into the model via latent variables, in a process that is analogous to how a first-order Markov chain can simulate a higher-order Markov chain by expanding the state space\n\nIn this section, we adopt a notational shortcut. In an HRG production $A \\rightarrow R$, the RHS $R$ is a hypergraph fragment containing zero or more hyperedges with nonterminal labels $Y_1, \\ldots, Y_r$. We suppress the graph structure of $R$ and write the rule simply as $X \\rightarrow Y_1 \\cdots Y_r$.\n\n\\subsection{Nonterminal Splitting}\n\nFollowing previous work on probabilistic CFGs \\cite{matsuzaki+al:2005,petrov+al:2006}, we increase the context-sensitivity of the grammar by splitting each nonterminal symbol $X$ in the grammar into $n$ different \\emph{subsymbols}, $X_i, \\ldots, X_n$, which could potentially represent different contexts that the rule is used in. Thus, each rule in the original grammar is replaced with several \\emph{subrules} that use all possible combinations of the subsymbols of its nonterminal symbols. \n\nFor example, if $n=2$, the rule $\\n2\\rightarrow \\epsilon$ would be split into $\\n2_1\\rightarrow \\epsilon$ and $\\n2_2\\rightarrow \\epsilon$.\n\n\\iffalse\nFor example, if $n=2$, the rules in Fig.~\\ref{fig:grammar} would be split as follows:\n\\begin{align*}\n\\text{(from }\\n0&\\rightarrow \\n2~\\n3\\text{)} & \\n0_1 &\\rightarrow \\n2_1~\\n3_1 & \\n0_2 &\\rightarrow \\n2_1~\\n3_1 \\\\\n&&\\n0_1 &\\rightarrow \\n2_1~\\n3_2 &\\n0_2 &\\rightarrow \\n2_1~\\n3_2 \\\\\n&&\\n0_1 &\\rightarrow \\n2_2~\\n3_1 &\\n0_2 &\\rightarrow \\n2_2~\\n3_1 \\\\\n&&\\n0_1 &\\rightarrow \\n2_2~\\n3_2 &\\n0_2 &\\rightarrow \\n2_2~\\n3_2 \\\\\n\\text{(from }\\n2&\\rightarrow \\epsilon\\text{)} & \\n2_1 &\\rightarrow \\epsilon & \\n2_2 &\\rightarrow \\epsilon \\\\\n\\text{(from }\\n3&\\rightarrow \\n2\\text{)} & \\n3_1 &\\rightarrow \\n2_1 & \\n3_2 &\\rightarrow \\n2_1 \\\\\n&&\\n3_1 &\\rightarrow \\n2_2 & \\n3_2 &\\rightarrow \\n2_2.\n\\end{align*}\n\\fi\n\nIn general, a rule with $r$ nonterminal symbols on its right-hand side is split into $n^{r+1}$ subrules.\n\n\n\\subsection{Learning}\n\n\\newcommand{\\inside}[2]{\\ensuremath{P_{\\textit{in}}( #1,#2)}}\n\\newcommand{\\outside}[2]{\\ensuremath{P_{\\textit{out}}(#1,#2)}}\n\\newcommand{\\ruleprob}[1]{\\ensuremath{P(#1)}}\n\n\n\nAfter obtaining an $n$-split grammar from the training graphs, we want to learn probabilities for the subrules that maximize the likelihood of the training graphs and their tree decompositions.\nHere we use Expectation-Maximization (EM)~\\cite{dempster+al:1977}. In the E (Expectation) step, we use the Inside-Outside algorithm \\cite{lari+young:1990} to compute the expected count of each subrule given the training data, and in the M (Maximization) step, we update the subrule probabilities by normalizing their expected counts.\n\n\n\\begin{figure}\n\\centering\n\\input{.\/figs\/egg_ill}\n\\caption{Inside and outside probabilities. Here, a hyperedge labeled $X_i$ with three external nodes has generated a subgraph $\\eta$. The inside probability (left) of $\\eta,X_i$ is the probability of all the subderivations that generate subgraph $\\eta$ from $X_i$. The outside probability (right) is the probability of all the partial derivations that generate the complement of subgraph $\\eta$, with a hyperedge labeled $X_i$ in place of the subgraph.}\n\\label{fig:egg_ill}\n\\end{figure}\n\n\nThese expected counts can be computed efficiently using dynamic programming. \nGiven a tree decomposition $T$, consider a bag $\\eta$ and its corresponding subgraph $H_\\eta$. The grammar extraction method of  Background Section~\\ref{sec:extraction} assigns $H_\\eta$ a nonterminal symbol, which we write $X$. Let $X_i$ be a subsymbol of $X$.\nThe \\emph{inside} probability of $H_\\eta$ with label $X_i$, written as $\\inside{\\eta}{X_i}$, is the total probability of all derivations starting from $X_i$ and ending in $H_\\eta$. The \\emph{outside} probability of $H_\\eta$ with label $X_i$, written as $\\outside{\\eta}{X_i}$, is the total probability of all derivations starting from $S$ and ending in $H$ with $H_\\eta$ replaced with a hyperedge labeled $X_i$. See Figure~\\ref{fig:egg_ill}.\n\n\\begin{figure}\n\\centering\n\\input{.\/figs\/egg_calc}\n\\caption{Computation of inside and outside probabilities. Here, a hyperedge labeled $X_i$ has been rewritten with a rule rhs with two hyperedges labeled $Y_j$ and $Z_k$. At left, the inside probability of $\\eta,X_i$ is incremented by the product of the rule and the inside probabilities of $\\eta_1,Y_j$ and $\\eta_2,Z_k$. At right, the outside probability of $\\eta_1,Y_j$ is incremented by the product of the outside probability of $\\eta,X_i$, the rule, and the inside probability of $\\eta_2,Z_k$.}\n\\label{fig:egg_calc}\n\\end{figure}\n\nThe inside probabilities can be calculated recursively, from smaller subgraphs to larger subgraphs. \nWe assume that bag $\\eta$ has at most two children, which follows if $T$ is in Chomsky Normal Form~\\cite{chomsky1959certain}. If $\\eta$ has two children, let $\\eta_1$ and $\\eta_2$ be the children, let $Y$ and $Z$ be the labels of $H_{\\eta_1}$ and $H_{\\eta_2}$, and let $Y_j$ be and $Z_k$ be subsymbols of $Y$ and $Z$. Then the inside probability of $H_\\eta$ with subsymbol $X_i$ is defined by:\n\\begin{align*}\n    \\inside{\\eta}{X_i} &= \\sum_{j, k} \\ruleprob{X_i\\rightarrow Y_j Z_k} \\, \\inside{\\eta_1}{Y_j} \\, \\inside{\\eta_2}{Z_k} \\\\\n\\intertext{and similarly if $\\eta$ has only one child:}    \n\\inside{\\eta}{X_i} &= \\sum_{j} \\ruleprob{X_i\\rightarrow Y_j} \\, \\inside{\\eta_1}{Y_j} \\\\\n\\intertext{or no children:}\n\\inside{\\eta}{X_i} &= \\ruleprob{X_i\\rightarrow \\epsilon}.\n\\end{align*}    \n\nThe outside probabilities are calculated top-down. \nIf a bag~$\\eta$ has two children, then the outside probabilities of its children are defined by:\n\\begin{align*}    \n    \\outside{\\eta_1}{Y_j} &= \\sum_{i, k} \\ruleprob{X_i\\rightarrow Y_j Z_k} \\, \\outside{\\eta}{X_i} \\, \\inside{\\eta_2}{Z_k} \\\\\n    \\outside{\\eta_2}{Z_k} &= \\sum_{i, j} \\ruleprob{X_i\\rightarrow Y_j Z_k} \\, \\outside{\\eta}{X_i} \\, \\inside{\\eta_1}{Y_k}. \\\\\n\\intertext{See Figure~\\ref{fig:egg_calc} for an illustration of this computation. Similarly, if $\\eta$ has only one child:}\n    \\outside{\\eta_1}{Y_j} &= \\sum_{i} \\ruleprob{X_i\\rightarrow Y_j} \\, \\outside{\\eta}{X_i}.\n\\end{align*}\n\nIn the Expectation step, we compute the posterior probability of each subrule at each bag of each training tree decomposition~$T$:\n\\begin{align*}\n    P(\\eta, X_i\\rightarrow Y_j  Z_k\\mid T) &= \\frac1{P(T)}\\outside{\\eta}{X_i}  \\ruleprob{X_i\\rightarrow Y_j Z_k} \\cdot{} \\\\ &\\qquad \\inside{\\eta_1}{Y_j}  \\inside{\\eta_2}{Z_k} \\\\\n    P(\\eta, X_i \\rightarrow Y_j \\mid T) &= \\frac1{P(T)}\\outside{\\eta}{X_i}  \\ruleprob{X_i \\rightarrow Y_j}  \\inside{\\eta_1}{Y_j} \\\\\n    P(\\eta, X_i \\rightarrow \\epsilon \\mid T) &= \\frac1{P(T)}\\outside{\\eta}{X_i}  \\ruleprob{X_i \\rightarrow \\epsilon}\n\\end{align*}\nwhere $P(T) = \\inside{\\eta_0}{S}$ and $\\eta_0$ is the root bag of $T$.\nThe expected count of each subrule is calculated by summing over the posterior probability of the rule over all nodes of all training trees:\n\\begin{equation*}\n    E[c(X_i\\rightarrow \\alpha)] = \\sum_{\\text{trees $T$}} \\sum_{\\eta \\in T} P(\\eta, X_i\\rightarrow \\alpha\\mid T) \n\\end{equation*}\nwhere $\\alpha$ is any right-hand side.\n\nIn the Maximization step, we use the expected counts calculated above to update the rule probabilities:\n\\begin{align*}\n    \\ruleprob{X_i\\rightarrow \\alpha} &:= \\frac{E[c(X_i\\rightarrow \\alpha)]}{\\sum_{\\alpha'} E[c(X_i \\rightarrow \\alpha')]}.\n\\end{align*}\n\nThese probabilities are then used to repeat the E step. The method is guaranteed to converge to a local maximum of the likelihood function, but not necessarily to a global maximum.\n\n\\section{Evaluation}\n\nCurrent research in graph modelling and graph generation evaluate their results by comparing the generated graphs with the original graph by aggregate properties like degree distribution, clustering coefficients, or diameter~\\cite{aguinaga2016growing,leskovec2010kronecker,aguinaga2016infinity,seshadhri2012community,leskovec2005graphs,chakrabarti2006graph}. There are two potential problems with such metrics. First, these metrics do not test how well the model generalizes to model other graphs that represent similar data. Second, they are heuristics from which a generated graph's ``goodness'' is difficult to define or standardize. We discuss and address both of these problems in this section.\n\n\\subsection{Train\/Test Data}\n\nComparing generated graphs with the original graph cannot test how well the model generalizes to other graphs that represent similar data or different versions of the same network phenomena. To see why, consider the extreme case, in which a model simply memorizes the entire original graph. Then, the generated graphs are all identical to the original graph and therefore score perfectly according to these metrics. This is akin to overfitting a machine learning classifier on training data and then testing it on the same data, which would not reveal whether the model is able to generalize to unseen instances.\n\nIn standard data mining and machine learning tasks, the overfitting problem is typically addressed through cross-validation or by evaluating on heldout test data sets. In the present work, we adapt the idea of using heldout test data to evaluate graph grammars.\nIn experiments on synthetic graphs, this means that we generate two random graphs using the same model and parameters; we designate one as the training graph and the other as the test graph. In experiments on real world graphs, we identify two graphs that represent the same phenomenon, \\textit{e.g.}, citations or collaborations, and we mark one as the training graph and one as the test graph.\n\nIn reality, we might not be able to find test graphs that have similar properties as the training graph. Fortunately, cross-validation can also be adapted to cases where no test graph is available by using disjoint subgraph samples from a single graph.\n\n\\subsection{Likelihood}\n\nIn addition to the possibility of overfitting, high-level aggregations of graph properties may not always be good comparators of two or more graphs. Indeed, examples abound in related literature showing how vastly different graphs can share similar aggregate statistics~\\cite{yaveroglu2014revealing}. We propose, as an additional metric, to evaluate models by using them to measure the likelihood of a test graph or graphs. Intuitively, this measures how well a model extracted from the training graph generalizes to a test graph. If the model simply memorizes the entire training graph, then it will have zero likelihood (the worst possible) on the test graph. If the model is better able to generalize to new graphs, then it will have higher likelihood on the test graph.\n\nUnfortunately, it is not always computationally feasible to compute the likelihood of graphs under previous models. But with HRGs, it can be computed in linear time given a tree decomposition. (It would also be possible, but slower, to sum the probabilities of \\emph{all} possible tree decompositions \\cite{chiang2013parsing}.)  The likelihood on a test graph is simply $\\inside{\\eta_0}{S}$, where $\\eta_0$ is the root of the tree decomposition. Note that the model probabilities are estimated from the training graphs, even when computing likelihood on test graphs. As this number is usually very small, it's common to take logs and deal with log-likelihoods.\n\n\\subsection{Smoothing}\nA problem arises, however, if the test graph uses a rule that does not exist in the grammar extracted from the training graph. Then the inside probability will be zero (or a log-probability of $-\\infty$). This is because an HRG missing any necessary rules to construct the test graph cannot generate the test graph exactly, and therefore results in a zero probability.\n\nIn this case, we would still like to perform meaningful comparisons between models, if possible. So we apply smoothing as follows. To test an HRG $H$ on a test graph, we first extract an HRG, $H^\\prime$, from the test graph using the latent-variable HRG method. Define an \\emph{unknown rule} to be a rule in $H^\\prime$ but not in $H$. Then for each unknown rule, we add the rule to $H$ with a probability of $\\epsilon$. We can then compute the log likelihood on the test graph under the augmented grammar $H \\cup H^\\prime$. The final test log likelihood is calculated as\n\\begin{align*}\n    L = L_{H \\cup H^\\prime} - c(H^\\prime \\setminus H) \\cdot \\log \\epsilon\n\\end{align*}\nwhere $L_{H \\cup H^\\prime}$ is the log likelihood of the test graph under the augmented grammar, $c(H^\\prime \\setminus H)$ is the number of times that unknown rules are used in the test graph, and $\\epsilon$ is the probability of each unknown rule. Note that as long as $\\epsilon$ is much smaller than the probability of any known rule, its value is irrelevant because $L$ does not depend on it.\n\nIdeally, we would like the number of unknown rules to be zero. In our experiments, we find that increasing the number of training graphs and\/or decreasing the size of the training graphs can bring this number to zero or close to zero. Note that if two HRGs have differing sets of unknown rules, then it is \\emph{not} meaningful to compare their log-likelihood on test graphs. But if two HRGs have identical sets of unknown rules, then their log-likelihoods can be meaningfully compared. We will exploit this fact when evaluating models with latent variables in the next section.\n\n\n\\section{Experiments}\n\nIn this section we test the ability of the latent-variable HRG (laHRG) to generate graphs using the train-test framework described above. We vary $n$, the number of subsymbols that each nonterminal symbol is split into, from 1 to 4. Note that the 1-split laHRG model is identical to the original HRG method. By varying the number of splits, we will be able to find the value that optimizes the test likelihood. We have provided all of the source code and data analysis scripts at \\url{https:\/\/github.com\/cindyxinyiwang\/laHRG}.\n\n\\subsection{Setup}\nGiven a training graph, we extract and train a latent-variable HRG from the graph. Then we evaluate the goodness of the grammar by calculating the log likelihood that the test graph could be generated from the grammar. \n\n\\begin{table}\n\\tabcolsep=1.5pt\\relax\n \\smal\n\\caption{Datasets used in experiments}\n\\label{table: real-world-graphs}\n\\centering\n\\begin{tabular}{lllrr} \n\\toprule\n & Type & Name &  Nodes & Edges \\\\ \n\\midrule\n \\multirow{4}{*}{Synth} &\n \\multirow{2}{*}{Barabasi-Albert}\n & train-ba &  30,000 & 59,996 \\\\ \n & & test-ba &  30,000 & 59,996 \\\\\n \\cmidrule{2-5}\n & \\multirow{2}{*}{Watts-Strogatz}\n & train-ws & 30,000 & 60,000 \\\\\n & & test-ws  & 30,000 & 60,000 \\\\\n\n\\midrule\n\n \\multirow{10}{*}{Real} &\n \\multirow{2}{*}{Citation}\n & train-cit-HepTh &  27,770 & 352,807 \\\\ \n & & test-cit-HepPh &  34,546 & 421,578 \\\\\n \\cmidrule{2-5}\n & \\multirow{2}{*}{Internet}\n & train-as-topo & 34,761 & 171,403 \\\\\n & & test-as-733  & 6,474 & 13,895 \\\\\n \\cmidrule{2-5}\n & \\multirow{2}{*}{Purchase}\n & train-amazon0312  & 400,727 & 3,200,440 \\\\\n && test-amazon0302 & 262,111 & 1,234,877 \\\\ \n \\cmidrule{2-5}\n & \\multirow{2}{*}{Wikipedia}\n & train-wiki-vote  & 7,115 & 103,689 \\\\ \n & & test-wiki-talk & 2,394,385 & 5,021,410 \\\\ \n\\bottomrule\n\\end{tabular}\n\\end{table}\n\nFor evaluation, we need to make sure that the test graph is disjoint from the training graph to guard against overfitting. Here we introduce two techniques to achieve that: 1. we partition a single graph into two disjoint parts so that they do not have overlapping vertices. Then we train laHRG from one part of the graph, and calculate the log likelihood of the other disjoint part; 2. we choose 2 graphs of the same type, and use one for training and the other for evaluation.\n\nWe evaluate laHRG with log likelihood metric for both evaluation methods mentioned above on 6 types of graphs: 2 synthetic graphs (generated from random graph generators), and 4 real world graphs. For the first evaluation method, we use 1 graph, each from 6 different types of graphs, and partition the graph for training and testing purpose. For the second evaluation method, we do not partition the training graph, but choose 1 additional graph from each type of graphs for testing. The graphs were obtained from the SNAP\\footnote{\\url{https:\/\/snap.stanford.edu\/data}} and KONECT\\footnote{\\url{http:\/\/konect.uni-koblenz.de}} graph repositories and are listed in Table~\\ref{table: real-world-graphs}.\n\n\\iffalse\nThe graph types are described below:\n\\begin{itemize}\\setlength\\itemsep{1em}\n    \\item Barabasi-Albert: two random graphs generated using the preferential attachment algorithm with $\\alpha=2$.\n    \\item Watts-Strogatz: two random graphs generated using the Watts-Strogatz ring method with 2 neighbors and a rewiring probability of 0.2.\n    \\item Citation: cit-HepPh and cit-HepTh are the citation graphs of research papers in high energy physics phenomenology and theory, respectively, from January 1993 to April 2003. A edge in the a citation graph represents a citation between two papers.\n    \\item Internet: The as-topo network of the relationships between autonomous systems (controlled by independent network operators) on the Internet collected between Feb. 15, 2010 and Mar 10, 2010. The as-733 is a similar network from Jan. 2, 2000 -- the largest of 733 such graphs.\n    \\item Purchase: The amazon0302 and amazon0312 graphs are co-purchasing networks from Mar. 2, 2003, and Mar. 12, 2003, respectively, as crawled from the ``customers who bought $x$ also bought $y$'' feature on Amazon.\n    \\item Wikipedia: The wiki-talk network has an edge between two users if one user edits the talk page of another user on Wikipedia anytime before Jan. 2008. The wiki-vote network has an edge between two Wikipedia users if one users votes (for or against) the promotion of another user as an administrator.\n   \n\\end{itemize}\n\\fi \n\nMany of these graphs are too large for a tree decomposition to be calculated. Instead, we randomly sampled a set of fixed-size subgraphs from the training graph and a set of fixed-size subgraphs from the test graph. Besides the concern from the calculation of large tree decompositions, sampling multiple graphs is also important for the extraction of a broad set of rules. Recall that if a single rule required to generate the test graph is not found within the training graph, then the likelihood will be 0. Therefore, large test graphs would require many more training graphs in order to reduce (or hopefully eliminate) the need for smoothing.\n\n\nIn all experiments, we extract 500 samples of size-25 subgraphs from the training graph. We extract an HRG from each size-25 subgraph, perform nonterminal splitting and EM training. The 500 HRGs are then combined and their weights are normalized to create the final laHRG model. We also take 4 samples of size-25 subgraphs from the test graph, calculate the log-likelihood of each under the laHRG model, and report the mean log-likelihood and confidence interval. \n\nWe chose these parameters empirically such that there is no need for smoothing.\nIf we were to increase the subgraph size for the test graphs, then we would also need to increase the number of training graph samples or rely on smoothing to ensure non-zero likelihood. \n\nTo compute tree decompositions, we used a reimplementation of the QuickBB algorithm \\cite{gogate-dechter-uai04}, with only the ``simplicial'' and ``almost-simplicial'' heuristics.\n\n\\subsection{Log-Likelihood Results} \\label{sec:loglikelihood}\n\n\\begin{figure}[t]\n\\centering\n\\input{figs\/gen_loglikelihood.tex}\n\\caption{On synthetic graphs, splitting nonterminal symbols ($n \\geq 2$) always improves log-likelihood on the test graph, as compared to no splitting ($n=1$). We did not observe overfitting up to $n=4$. Left\/right column: train on Barabasi-Albert (ba) or Watts-Strogatz (ws). Top\/bottom row: test on ba\/ws.\n}\n\\label{fig:gen_loglikelihood}\n\\end{figure}\n\nThis section explains the performance of laHRG in terms of log-likelihood metric on test graph for two different train-test split methods mentioned in the previous section. We mainly analyze the results for the second method: train on one graph and test on another graph of the same type. The first method has similar results, and we include them here to show that our evaluation method also works for graphs that are difficult to find different test graphs of the same type.\n\n\\subsubsection{Validate on Different Graph of Same Type}\nWe first show the log-likelihood results on synthetic datasets. The two random graph models, the Barabasi-Albert graph and Watts-Strogatz graph, generate very different graph types.\nThe four panels in Fig.~\\ref{fig:gen_loglikelihood} show the log-likelihood results of four combinations of training graphs and test graphs. Higher is better.\n\nAs a sanity check, we also trained an laHRG model on a Barabasi-Albert graph and tested it against a Watts-Strogatz graph and vice versa. We expect to see much lower log-liklihood scores because the laHRG trained on one type of graph should be different than another type of graph. The top-right and bottom-left panels in Fig.~\\ref{fig:gen_loglikelihood} show that this is indeed the case; the log-likelihood measure and the laHRG model pass the sanity check.\n\n\n\n\\begin{figure}\n\\centering\n\\input{figs\/real_loglikelihood.tex}\n\\caption{On real-world graphs, splitting nonterminal symbols ($n \\geq 2$) always improves log-likelihood on the test graph, as compared to no splitting ($n=1$), peaking at 2 or 3 splits and then dropping due to overfitting. Error bars indicate 95\\% confidence intervals. Higher is better.}\n\\label{fig:real_loglikelihood}\n\\end{figure}\n\nNext we extracted and tested the laHRG model on real world graphs. The log-likelihood results are illustrated in Fig.~\\ref{fig:real_loglikelihood} for laHRG models of up to 4-splits.\nWe find that the log-likelihood scores peak at $n$ = 2 or 3 and then decreases when $n$ = 4.  \n\nRecall that laHRG is the same as HRG~\\cite{aguinaga2016growing} when $n$ = 1. Based on the results from Fig.~\\ref{fig:real_loglikelihood}, we find that splitting does indeed increase HRG's ability to generate the test graph. However, as increasing $n$ shows diminishing returns and sometimes decreases performance. The decrease in log likelihood when $n>2$ is caused by model overfitting. The increase in node-splitting allows laHRG to fine-tune the rule probabilities to the training graph, which we find does not always generalize to the test graph.\n\n\\subsubsection{Validate on Disjoint Subgraph}\n\nIf it is difficult to find a test graph of the same type with the training graph, it is still possible to evaluate laHRG with the log likelihood metric. We can partition the graph into two disjoint parts, and use one for training and the other for testing. Fig.~\\ref{fig:single_loglikelihood} shows the log likelihood of the test subgraphs that are disjoint from the training graphs. Again, splitting nonterminal symbols increase the log likelihood on the test graph, but as the number of splits ($n$) increases, log likelihood decreases due to overfitting.\n\\begin{figure}\n\\centering\n\\input{figs\/single_loglikelihood.tex}\n\\caption{Loglikelihood on subgraphs that are disjoint from training graphs. Trends similar to Fig.~\\ref{fig:real_loglikelihood} and Fig.~\\ref{fig:gen_loglikelihood} can be observed with this method. Error bars indicate 95\\% confidence intervals. Higher is better.}\n\\label{fig:single_loglikelihood}\n\\end{figure}\n\n\n\\subsection{Comparing against other Graph Generators}\n\nThe log-likelihood metric is a principled approach to calculating the performance of a graph generator. Unfortunately, other graph generators are not capable of performing this type of likelihood calculation. In order to compare the laHRG graph model to other state of the art graph generators including the Kronecker~\\cite{leskovec2010kronecker} and Chung-Lu~\\cite{chung+lu} models, we revert to traditional graph metrics to compare a generated graph to a test graph(not against the original graph).\n\n\nAmong the many choices for heuristic graph comparison metrics, we chose the degree distribution and graphlet correlation distance (GCD).\n\nRecall that the sampling of 25-node subgraphs was necessary to ensure a non-zero probability for the log likelihood evaluation. No such requirement exists for evaluation on GCD.\nNevertheless, to maintain an apples to apples comparison, we performed similar graph sampling methods for degree distribution distance and GCD: we trained Kronecker and Chung-Lu models on a 25-node subgraph from the training graph, generated a 25-node graph, compared the generated graph against a 25-node subgraph of the test graph, and repeated this process 500 times. \n\nAs a baseline, we also compared the training and test graph directly to get a basic sense of their similarity. So, we directly compared 25-node subgraphs from the training graph to 25-node subgraphs of the test graph without any model. We repeated this direct comparison 500 times and report the mean and 95\\% confidence interval. \nWe call this the ``Direct'' comparison because it does not involve any graph generation.\n\n\\subsubsection{Degree Distribution Distance}\n\nIn the present work we apply the degree distribution distance of Pr{\\v{z}}ulj~\\cite{prvzulj2007biological} to compare two or more degree distributions. Lower degree distribution distance between two graphs means they are more similar. \n\nwhich is defined as follows. Given a graph $H$, we first scale and normalize the degree distribution of $H$:\n\n$$S_H(k) = \\frac{d_H(k)}{k}$$\n$$T_H = \\sum_{k=1}^\\infty{S_H(k)}$$\n$$N_H(k) = \\frac{S_H(k)}{T_H}$$\n\n\\noindent in order to reduce the effect of higher degree nodes, where $d_H(K)$ is the number of nodes in $H$ that have a degree of $k$. Then we calculate the distance between two degree distributions $D\\left(d_H, d_{H^\\prime}\\right)$ as:\n\n$$D\\left(d_H, d_{H^\\prime}\\right) = \\frac{1}{\\sqrt{2}}\\sqrt{ \\sum^\\infty_{k=1}\\left( N_H(k) - N_{H^\\prime}(k) \\right)^2 },$$\n\n\\noindent which is essentially a normalized sum of squares between the two distributions. We call this metric the degree distribution distance. Because this is a ``distance'' metric low values indicate high similarity.\n\n\nFigure~\\ref{fig:degree_distribution} illustrates the results of the degree distribution distance. Recall that the laHRG is identical to HRG~\\cite{aguinaga2016growing} when $n=1$. The Kronecker and Chung-Lu do not have an $n$ parameter, so their plots are flat. All points represent the mean of 500 repetitions; each point contains error bars indicating the 95\\% confidence intervals -- although many error bars are too small to be seen.\n\nThe laHRG model generates graphs that more closely follow the degree distribution of the test graph than graphs generated by Kronecker and Chung-Lu models. Higher nonterminal splitting, \\textit{i.e.}, $n>1$, shows little change on the degree distribution distance.\n\nIt is expected that the Direct baseline outperforms all graph models, because the Direct baseline simply compares two graphs generated from the exact same generation process, which rewards an overfit model.\n\nHere, the HRG models predict the test graph's degree distribution better than the Direct baseline does; whether this is because they generalize better, or due to chance, or some other reason, would need further analysis to determine. In any case, nonterminal splitting ($n \\geq 2$) has only a slight effect on the model, generally attracting the degree distribution toward the training graph's and away from the test graph's. \n\nInterestingly, the Direct baseline has similar or better performance than the Kronecker and Chung-Lu methods. It is unlikely that these results can be completely explained by overfitting. Instead, Kronecker or Chung-Lu methods may perform poorly due to underfitting, wherein these models do not model the training graph well enough. More work is needed to understand these results.\n\n\\begin{figure}[t]\n\\centering\n\\input{figs\/degree_distribution.tex}\n\\input{figs\/legend.tex}\n\\caption{HRG models are shown to generate graphs with lower (= better) degree distribution distance to the test graph when compared to other models. Splitting nonterminals ($n \\geq 2$) sometimes decreases degree distance but sometimes increases it.}\n\\label{fig:degree_distribution}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\centering\n\\input{figs\/gcd.tex}\n\\input{figs\/legend.tex}\n\\caption{HRG models are shown to generate graphs with lower (= better) graphlet correlation difference (GCD) to the test graph, when compared with other models. Splitting nonterminals ($n \\geq 2$) sometimes inproves GCD and sometimes decreases it.}\n\\label{gcd_real}\n\\end{figure}\n\n\n\n\\subsubsection{Graphlet Correlation Distance}\n\nAlthough the degree distribution is the most well known and widely adopted graph comparison metric, it is far from complete. The degree distribution can be easily mimicked by two very large and different networks. \nFor example, previous work has shown that it is easy to construct two or more networks with exactly the same degree distribution but substantially different structure and function~\\cite{prvzulj2004modeling,li2005towards}. \nThere is mounting evidence which argues that the graphlet comparisons are a better way to measure the similarity between two graphs~\\mbox{\\cite{prvzulj2007biological,ugander2013subgraph}}.\nRecent work from systems biology has identified a metric called the Graphlet Correlation Distance (GCD). The GCD computes the distance between two graphlet correlation matrices -- one matrix for each graph~\\cite{yaveroglu2015proper}.\nBecause the GCD is a distance metric, low values indicate high similarity where the GCD is 0 iff the two graphs are isomorphic. \n\nFigure~\\ref{gcd_real} illustrates the results of the GCD. Recall that the laHRG is identical to HRG [10] when n = 1. The Kronecker and Chung-Lu do not have an n parameter, so their plots are flat. All points represent the mean of 500 repetitions; each point contains error bars indicating the 95 confidence intervals \u2013 although many error bars are too small to be seen.\n\nThe Direct baseline illustrates how similar the training and test graphs are. As expected, we find that the Direct comparison is best on the random Watts-Strogatz graphs. But laHRG outperforms it on all of the real-world graphs.\n\n\n\\subsection{Comparison with Log Likelihood Metric}\nGCD and Degree Distribution metrics indicate that laHRG is a better graph generator than other options like Kron and Chung-Lu graph generators, but our experiments seem to suggest that splitting nonterminals in HRG does not have much effect in terms of GCD and Degree Distribution. However, nonterminal splitting does increase log likelihood of the test graph, as explained in Section~\\ref{sec:loglikelihood}. This discrepancy is probably because log likelihood metric is able to capture more general structure and properties of a graph than GCD and Degree Distribution. Both GCD and Degree Distribution only focus on a specific graph property, which might not be perfectly correlated with overall structure of a graph. On the other hand, the log likelihood metric we propose does not overemphasize a particular graph property, but directly measures the ability of the graph generator to generate the test graph.\n\n\n\\begin{table}[t]\n\\caption{Number of rules\/parameters in grammars extracted from training graphs}\n\\label{fig:sizes}\n\\begin{center}\n\\begin{tabular}{lrrrr}\n\\toprule\n& \\multicolumn{4}{c}{$n$} \\\\\n\\addlinespace[1ex]\nTrain & \\multicolumn{1}{c}{1} & \\multicolumn{1}{c}{2} & \\multicolumn{1}{c}{3} & \\multicolumn{1}{c}{4} \\\\\n\\midrule\nCitation & 1,156 & 7,193 & 19,885 & 44,410 \\\\\nInternet & 1,005 & 5,686 & 14,057 & 29,247 \\\\\nPurchase & 969 & 6,196 & 18,237 & 38,186 \\\\\nWikipedia & 1,065 & 6,891 & 20,891 & 42,841 \\\\\n\\midrule\nBarabasi-Albert & 48 & 298 & 930 & 2,126 \\\\\nWatts-Strogatz & 60 & 346 & 1,023 & 2,380 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\section{Grammar Analysis}\n\nRecall that the HRG models merges two production rules if they are identical. Splitting rules produces subrules that have the same structure, but different symbols, so they cannot be merged; splitting nonterminal nodes will therefore increase the size of the grammar. In the worst case, the blowup could be cubic in $n$. Table~\\ref{fig:sizes} shows the sizes of all the grammars used in the experiments. Because rules with probability zero are excluded, the blowup is slightly less than cubic.\n\nHere we see a trade-off between model size and performance. The larger the grammar gets, the better it is able to fit the training graph. On the other hand, we prefer smaller models to mitigate the possibility of overfitting. If we had not used separate training and test graphs, it would not be clear how to manage this trade-off, but our evaluation is able to demonstrate that larger grammars (up to a point) are indeed able to generalize to new data.\n\n\n\\begin{figure}\n\\[\\begin{aligned}\n\\n0 &\\xrightarrow{0.22}\n\\begin{tikzpicture}[baseline=-2pt]\n\\node (n0)[externalnode] at (0,0) {};\n\\node (n1)[externalnode] at (1,0) {};\n\\node (e0)[nonterminal] at (0,-1) {$\\nss11$};\n\\node (e01)[nonterminal] at (0.5,1) {$\\nss22$};\n\\draw (n0)--(n1);\n\\draw (e0)--(n0);\n\\draw (e01)--(n0);\n\\draw (e01)--(n1);\n\\end{tikzpicture} \\\\[3ex]\n\\n1_1 &\\xrightarrow{0.88} \\begin{tikzpicture}[baseline=-2pt]\n\\node (n0)[externalnode] at (1,0) {};\n\\node (na)[internalnode] at (0,0) {};\n\\draw (n0)--(na);\n\\end{tikzpicture} \\\\[3ex]\n\\n1_2 &\\xrightarrow{0.64} \\begin{tikzpicture}[baseline=-2pt]\n\\node (na)[internalnode] at (0,0) {};\n\\end{tikzpicture}\n\\end{aligned}\n\\hspace{0.5in}\n\\begin{aligned}\n\\n2_1 &\\xrightarrow{0.70}\n\\begin{tikzpicture}[baseline=-2pt]\n\\node (na)[internalnode] at (0,0) {};\n\\node (nb)[internalnode] at (1,0) {};\n\\node (n0)[externalnode] at (0.5,1) {};\n\\draw (na)--(n0);\n\\draw (nb)--(n0);\n\\end{tikzpicture} \\\\[1ex]\n\\n2_2 &\\xrightarrow{0.60}\n\\begin{tikzpicture}[baseline=-2pt]\n\\node (na)[internalnode] at (0,0) {};\n\\node (nb)[internalnode] at (1,0) {};\n\\node (eab)[nonterminal] at (0.5,1) {$\\nss22$};\n\\node (ea)[nonterminal] at (0,-1) {$\\nss11$};\n\\draw (eab)--(na);\n\\draw (eab)--(nb);\n\\draw (ea)--(na);\n\\end{tikzpicture} \\\\[1ex]\n\\n2_2 &\\xrightarrow{0.18}\n\\begin{tikzpicture}[baseline=-2pt]\n\\node (na)[internalnode] at (0,0) {};\n\\node (nb)[internalnode] at (1,0) {};\n\\node (ea1)[nonterminal] at (-0.35,-1) {$\\nss11$};\n\\node (ea2)[nonterminal] at (0.35,-1) {$\\nss11$};\n\\draw (ea1)--(na);\n\\draw (ea2)--(na);\n\\end{tikzpicture} \\\\[1ex]\n\\n2_2 &\\xrightarrow{0.14}\n\\begin{tikzpicture}[baseline=-2pt]\n\\node (na)[internalnode] at (0,0) {};\n\\node (nb)[internalnode] at (1,0) {};\n\\node (eb)[nonterminal] at (1,-1) {$\\nss12$};\n\\draw (eb)--(nb);\n\\end{tikzpicture}\n\\end{aligned}\\]\n\\caption{Rules extracted from as-topo, 2-split nonterminals, showing only those with probability at least $0.1$ and with at most two external nodes.}\n\\label{fig:as-grammar}\n\\end{figure}\n\nWhat do the HRG grammars look like? The models learned by unsupervised methods like EM can often be difficult to interpret, especially when the number of splits is high and the grammar is large. Figure~\\ref{fig:as-grammar} shows selected 2-split rules extracted from the as-topo training graph, namely, those with probability at least 0.1 and with at most two external nodes. \n\nWe can see that the subsymbols behave quite differently from each other.\nFor example, the $\\n2_1$ rule adds a connection between its two external nodes (via a third node), whereas none of the $\\n2_2$ rules adds a connection (perhaps because, as can be seen in the RHS of the $\\n0$ rule, they are already neighbors).\n\nWhat can we learn from these graph grammars? This is an open question. If we assume that the tree decomposition provides a meaningful representation of the original graph, then we may be able to interrogate and assign meaning to these rules depending on their context.  But we save this as a matter for future work.\n\n\n\\section{Conclusion}\n\nThis present work identifies and addresses two problems in applying Hyperedge Replacement Grammars (HRGs) to network data~\\cite{aguinaga2016growing} by adding latent variables in order to make production rules more sensitive to context and by introducing a principled evaluation methodology that computes the log likelihood that an HRG model generates a graph.\n\nTo guard against the possibility of the new model overfitting the original graph, we enforced a separation between the original graph from which the model is trained and a different graph on which the model is tested. This methodology should be better at selecting models that generalize well to new data. We confirmed Aguinaga et al.'s original finding that HRGs perform better than the widely-used Kronecker and Chung-Lu models, and showed that adding latent variables usually improves performance further. \n\nFurthermore, we evaluated our method against the original HRG model by directly measuring the log-likelihood of the test graphs under all models. This metric is more principled than aggregation of statistics of select graph properties. Under this metric, our method improves over the original in all cases, peaking at either $n=2$ or $3$ splits.\n\nHRGs extracted from tree decompositions are large.\nSplitting nonterminals grows the model even more. But our finding that 2- or 3-split grammars still generalize better to unseen graphs suggests that these models are not unreasonably large.\n\nIt remains for future work to test this claim by evaluating other generative graph models on test graphs distinct from training graphs. It should also be possible to simplify the HRG and laHRG models by trying to prune low-probability rules while maintaining high performance. Finally, more analysis is needed to provide an interpretation for the patterns automatically discovered by laHRGs.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThere are a great number of applications for differential geometry and\nmathematical physics. This applications can be use in many areas in this\ncentury. One of the most important applications of differential geometry is\non geodesics. A geodesic is the shortest route between two points. Geodesics\ncan be found with the help of the Euler-Lagrange and Hamilton equations.\nAlso, the information about them can be seen in many mechanical and geometry\nbooks. It is well known that differential geometry provides a suitable field\nfor studying Lagrangians and Hamiltonians of classical mechanics\\ and field\ntheory. So, the dynamic equations for moving bodies were obtained for\nLagrangian and Hamiltonian mechanics by many authors and are illustrated as\nfollows:\n\n\\textbf{I.} \\textbf{Lagrange Dynamics Equations} \\cite{klein,deleon,abraham}\n\\textbf{\\ }Let $M$ be an $n$-dimensional manifold and $TM$ its tangent\nbundle with canonical projection $\\tau _{M}:TM\\rightarrow M$. $TM$ is called\nthe phase space of velocities of the base manifold $M$. Let $L:TM\\rightarrow\nR$ be a differentiable function on $TM$\\ and is called the \\textbf\nLagrangian function}. We consider closed 2-form on $TM$\n\\begin{equation}\n\\Phi _{L}=-dd_{J}L.  \\label{0.1}\n\\end{equation\n(if $J^{2}=-I,$ $J$\\ is a complex structure or if $J^{2}=I,$ $J$ is a\npara-complex structure and $Tr(J)=0$). Consider the equatio\n\\begin{equation}\n\\mathbf{i}_{\\xi }\\Phi _{L}=dE_{L}.  \\label{1}\n\\end{equation\nWhere the semispray $\\xi $ is a vector field. We know that $E_{L}=V\\left(\nL\\right) -L$\\ \\ is an energy function and $V=J(\\xi )$\\ a Liouville vector\nfield. Here $dE_{L}$ denotes the differential of $E_{L}$. It is well-known\nthat (\\ref{1}) under a certain condition on $\\xi $ is the intrinsical\nexpression of the Euler-Lagrange equations of motion. This equation is named\n\\ as \\textbf{Lagrange dynamical equation}. The triple $(TM,\\Phi _{L},\\xi )$\nis known as \\textbf{Lagrangian system} on the tangent bundle $TM$. The\noperations run on (\\ref{1}) for\\ any coordinate system $(q^{i}(t),p_{i}(t))\n. Infinite dimension \\textbf{Lagrangian's\\ equation }is obtained the form\nbelow\n\\begin{equation}\n\\begin{array}{l}\n\\frac{dq^{i}}{dt}=\\dot{q}^{i}\\ ,\\ \\frac{d}{dt}\\left( \\frac{\\partial L}\n\\partial \\dot{q}^{i}}\\right) =\\frac{\\partial L}{\\partial q^{i}}\\ ,\\\ni=1,...,n\n\\end{array}\n\\label{2}\n\\end{equation\n\\textbf{II.} \\textbf{Hamilton Dynamics Equations }\\cite{deleon,abraham\n\\textbf{: }Let $M$ be the base manifold and its cotangent manifold $T^{\\ast\n}M$. By a symplectic form we mean a 2-form $\\Phi $ on $T^{\\ast }M$ such that:\n\n\\textbf{(i)} $\\Phi $ is closed , that is, $d\\Phi =0;$ \\textbf{(ii)} for each \n$z\\in T^{\\ast }M$ , $\\Phi _{z}:T_{z}T^{\\ast }M\\times T_{z}T^{\\ast\n}M\\rightarrow \n\\mathbb{R}\n\\ $is weakly non$\\deg $enerate. If $\\Phi _{z}$ in (ii) is non$\\deg $enerate,\nwe speak of a \\textbf{strong symplectic form}. If (ii) is dropped we refer\nto $\\Phi $\\ as a presymplectic form\\textbf{. }Now\\textbf{\\ }let $(T^{\\ast\n}M,\\Phi )$ us take as a symplectic manifold. A vector field $Z_{H}:T^{\\ast\n}M\\rightarrow TT^{\\ast }M$ is called \\textbf{Hamiltonian vector field}\\ if\nthere is a $C^{1}$ \\textbf{Hamiltonian function}\\ $H:T^{\\ast }M\\rightarrow \n\\mathbb{R}\n$ such that \\textbf{Hamilton dynamical equation} is determined b\n\\begin{equation}\n\\mathbf{i}_{Z_{H}}\\Phi =dH.  \\label{3}\n\\end{equation\nWe say that $Z_{H}$ is locally Hamiltonian vector field if $\\Phi $\\ is\nclosed. Where $\\Phi $ shows the canonical symplectic form so that $\\Phi\n=-d\\lambda ,$ $\\lambda =J^{\\ast }(\\omega ),$ such that $J^{\\ast }$ a dual of \n$J,$ $\\omega $ a 1-form on $T^{\\ast }M$. The triple $(T^{\\ast }M,\\Phi\n,Z_{H}) $ is named \\textbf{Hamiltonian system }which it is defined on the\ncotangent bundle $T^{\\ast }M.$ From the local expression of $Z_{H}$ we see\nthat $(q^{i}(t),p_{i}(t))$ is an integral curve of $Z_{H}$ iff \\textbf\nHamilton's equations }are expressed as follows\n\\begin{equation}\n\\dot{q}^{i}=\\frac{\\partial H}{\\partial p_{i}}\\ ,\\ \\dot{p}_{i}=-\\frac\n\\partial H}{\\partial q^{i}}.  \\label{4}\n\\end{equation\n\\textbf{Considering information the above, in a lot of articles and books,\nit is possible to show how differential geometric methods are applied in\nLagrangian's and Hamiltonian's mechanics in the below. Some works in\nparacomplex geometry are used for mathematical models. }\n\n\\textit{Cruceanu}, \\textit{Fortuny} and \\textit{Geada} have presented\nparacomplex geometry which is related to algebra of paracomplex number and\nthe study of the structures on differentiable manifolds called paracomplex\nstructures. Furthermore, they have considered a compatible neutral\npseudo-Riemannian metric, the para-Hermitian and para-Kahler structures, and\ntheir variants \\textbf{\\cite{cruceanu}. }Kaneyuki and Kozai have introduced\na class of affine symmetric spaces, which are called para-Hermitian\nsymmetric spaces, a paracomplex analogue Hermitian symmetric space \\textbf\n\\cite{kaneyuki}. }In the study of para-Kahlerian manifolds, \\textit{Tekkoyun}\nhas introduced paracomplex analogues of the Euler-Lagrange and Hamilton\nequations. Furthermore, the geometric results on the related mechanic\nsystems have been presented\\textbf{\\ \\cite{tekkoyun1}. }\\textit{Etayo} an\n\\textit{\\ Santamaria} studied connections attached to non-integrable almost\nbiparacomplex manifolds. Manifolds endowed with three foliations pairwise\ntransversal are called $3$-webs. Similarly, they can be algebraically\ndefined as biparacomplex or complex product manifolds, i.e., manifolds\nendowed with three tensor fields of type $(1,1),$ $F,$ $P$ and $J=FoP$,\nwhere the two first are product and the third one is complex, and they\nmutually anti-commute. In this case, it is well known that there exists a\nunique torsion-free connection parallelizing the structure. A para-K\\\"{a\nhlerian manifold $M$\\ is said to be endowed with an almost\nbi-para-Lagrangian structure (a bi-para-Lagrangian manifold) if $M$\\ has two\ntransversal Lagrangian distributions (involutive transversal Lagrangian\ndistributions) $D_{1}$\\ and $D_{2}$\\ \\cite{etayo}. \\textit{Carinena} and \n\\textit{Ibort} obtained the Lax equations which are associated with a\ndynamical endowed with a bi-Lagrangian connection and a closed two-form \n\\Omega $ parallel along dynamical field $\\Gamma $. The case of Lagrangian\ndynamical systems is analysed and the nonnoether constants of motion found\nby Hojman and Harleston are recovered as being associated to a reduced Lax\nequation. Completely integrable dynamical systems have been shown to be a\nparticular case of these systems by their \\textbf{\\cite{carinena}. }\\textit\nGordejuela} and \\textit{Santamaria} have proved that the canonical\nconnection of a bi-Lagrangian manifold introduced which was by Hess is a\nLevi-Civita connection by showing that a bi-Lagrangian manifold (i.e. a\nsymplectic manifold endowed with two transversal Lagrangian foliations) is\nendowed with a canonical semi-Riemannian metric \\textbf{\\cite{etayo2}. \n\\textit{Kanai} has been concerned with closed $C^{\\infty }$ Riemannian\nmanifolds of negative curvature whose geodesic flows have $C^{\\infty }$\nstable and unstable foliations. In particular, we have shown that the\ngeodesic flow of such a manifold is isomorphic to that of a certain closed\nRiemannian manifold of constant negative curvature if the dimension of the\nmanifold is greater than two and if the sectional curvature lies between \n\\frac{-9}{4}$ and $-1$ strictly \\textbf{\\cite{kanai}. }Since they have shown\nfundamental physical properties in turbulence (conservation laws, wall laws,\nKolmogorov energy spectrum,...), symmetries are used to analyse common\nturbulence models. A class of symmetry preserving turbulence models has been\nproposed. This class has been refined in such a way that the models respect\nthe second law of thermodynamics. Moreover, an example of model belonging to\nthe class has been numerically tested by \\textit{Razafindralandy} and \n\\textit{Hamdouni} \\textbf{\\ \\cite{dina}. }A base-equation method has been\nimplemented to actualize the hereditary algebra of the Korteweg--de Vries\n(KdV) hierarchy and the N-soliton manifold is reconstructed. The novelty of\nour approach is the fact that it can in a rather natural way, predict other\nnonlinear evolution equations which admit local conservation laws.\nSignificantly enough, base functions themselves are found to provide a basis\nto regard the KdV-like equations as higher order degenerate bi-Lagrangian\nsystems by \\textit{Chakrabarti} and \\textit{Talukdar}\\textbf{\\ \\cit\n{chakrabarti}. }Bi-para-complex analogue of Lagrangian and Hamiltonian\nsystems has been introduced on Lagrangian distributions by \\textit{Tekkoyun}\nand \\textit{Sari}. Additionally, the geometric and physical results related\nto bi-para-dynamical systems have also been presented by them \\textbf{\\cit\n{tekkoyun2010}. }Authors introduced generalized-quaternionic K\\\"{a}hler\nanalogue of Lagrangian and Hamiltonian mechanical systems. Moreover , the\ngeometrical-physical results which are related to generalized-quaternionic \n\\\"{a}hler mechanical systems have also been also given by Tekkoyun and Yayl\n\\textbf{\\ \\cite{tekkoyun2011}.}\n\nIn the above studies; although para-complex mechanical systems were analyzed\nsuccessfully in relatively broad area of science, they have not dealt with\nbi-para-complex conformal mechanical systems on the bi-Lagrangian conformal\nmanifold. In this study, therefore, equations related to bi-para-conformal\nmechanical systems on the bi-Lagrangian conformal manifold used in obtaining\ngeometric quantization have been presented.\n\n\\section{Preliminaries}\n\nIn this study, all the manifolds and geometric objects are $C^{\\infty }$\\\nand the Einstein summation convention $\\left( \\sum x_{i}=x_{i}\\right) $ is\nin use. Also, $A$, $F(TM)$, $\\chi (TM)$\\ and $\\Lambda ^{1}(TM)$\\ denote the\nset of para-complex numbers, the set of para-complex functions on $TM$, the\nset of para-complex vector fields on $TM$\\ and the set of para-complex\n1-forms on $TM$, respectively. The definitions and geometric structures on\nthe differential manifold $M$\\ given in \\cite{cruceanu} may be extended to \nTM$\\ as follows:\n\n\\section{Conformal Geometry}\n\nA conformal map is a function which preserves angles. Conformal maps can be\ndefined between domains in higher dimensional Euclidean spaces, and more\ngenerally on a Riemann or semi-Riemann manifold. A conformal manifold is a\ndifferentiable manifold equipped with an equivalence class of (pseudo)\nRiemann metric tensors, in which two metrics $g^{\\prime }$ and $g$ are\nequivalent if and only i\n\\begin{equation}\ng^{\\prime }=\\lambda ^{2}g  \\label{4.1}\n\\end{equation\nwhere $\\lambda >0$ is a smooth positive function. An equivalence class of\nsuch metrics is known as a conformal metric or conformal class \\cite{wiki}.\nTwo Riemann metrics $g$ and $g^{\\prime }$ on $M$ \\ are said to be equivalent\nif and only i\n\\begin{equation}\ng^{\\prime }=e^{\\lambda }g  \\label{7}\n\\end{equation\nwhere $\\lambda $ is a smooth function on $M$. The equation given by (\\ref{7\n) is called a \\textbf{Conformal Structure }\\cite{folland}.\n\n\\section{Bi-Para-Complex Geometry}\n\nLet $M$ be a differentiable manifold. An almost bi-para-complex structure on \n$M$ is denoted by two tensor fields $F$ and $P$ of type (1,1) giving \nF^{2}=P^{2}=1$, $F\\circ P+P\\circ F=0$ \\cite{etayo}. It is seen that $P\\circ\nF $ is an almost complex structure. If the matrix-structure defined by the\nalmost bi-para-complex structure is integrable then for every point $p\\in M$\nthere exists an open neighbourhood $U$ of $p$ and local coordinates \n(U;x^{_{i}},y^{i})$ such tha\n\\begin{eqnarray}\nF(\\partial \/\\partial x^{i}) &=&\\partial \/\\partial y^{i},F(\\partial \/\\partial\ny^{i})=\\partial \/\\partial x^{i},  \\label{2.0} \\\\\nP(\\partial \/\\partial x^{i}) &=&\\partial \/\\partial x^{i},\\text{ }P(\\partial\n\/\\partial y^{i})=-\\partial \/\\partial y^{i},\\text{ }\\forall i=\\overline{1,n} \n\\notag\n\\end{eqnarray\n\\cite{etayo1}. The existence of these kind of local coordinates on $M$\npermits to construct holomorphic local coordinates, $(U;z^{k}),$ \nz^{k}=x^{k}+\\mathbf{i}y^{k}$, $\\mathbf{i}^{2}=-1,$ $k=\\overline{1,n},$ or\npara-holomorphic local coordinates, $(U;z^{k}),$ $z^{k}=x^{k}+\\mathbf{j\ny^{k},k=\\overline{1,n},$ $\\mathbf{j}^{2}=1$ \\cite{gadea, newlander}. \n(M,g,J) $ is a para-K\\\"{a}hlerian manifold that always has two transversal\ndistributions defined by the eigen-spaces associated to $+1$ and $-1$\neigenvalues of $J$. Besides, the mentioned distributions are involutive\nLagrangian distributions if somebody thinks of the symplectic form $\\Phi $\ndefined by \n\\begin{equation*}\n\\Phi (X,Y)=g(JX,Y),\\forall X,Y\\in \\chi (M).\n\\end{equation*\nConsider $(x^{i},y^{i})$ to be a real coordinate system on a neighborhood $U$\nof any point $p$ of $M.$ Also let $\\{\\frac{\\partial }{\\partial x^{i}},\\frac\n\\partial }{\\partial y^{i}}\\}$ and $\\{dx^{i},dy^{i}\\}$ be natural bases over \nR$ of the tangent space $T_{p}(M)$ and the cotangent space $T_{p}^{\\ast }(M)$\nof $M,$ respectively. Then the below equalities may be written b\n\\begin{equation}\nJ(\\frac{\\partial }{\\partial x^{i}})=\\frac{\\partial }{\\partial y^{i}\n,\\,\\,\\,\\,\\,\\,\\,J(\\frac{\\partial }{\\partial y^{i}})=\\frac{\\partial }{\\partial\nx^{i}}.  \\label{2.1}\n\\end{equation\nLet $\\ z^{i}=\\ x^{i}+\\mathbf{j}\\ y^{i},$ $\\mathbf{j}^{2}=1,$ also be a\npara-complex local coordinate system on $M.$ So the vector fields will be\nshown\n\\begin{equation}\n\\frac{\\partial }{\\partial z^{i}}=\\frac{1}{2}\\{\\frac{\\partial }{\\partial x^{i\n}-\\mathbf{j}\\frac{\\partial }{\\partial y^{i}}\\},\\,\\,\\,\\frac{\\partial }\n\\partial \\overline{z}^{i}}=\\frac{1}{2}\\{\\frac{\\partial }{\\partial x^{i}}\n\\mathbf{j}\\frac{\\partial }{\\partial y^{i}}\\}.  \\label{2.2}\n\\end{equation\nwhich represent the bases of $M$. Also, the dual covector fields are \n\\begin{equation}\ndz^{i}=dx^{i}+\\mathbf{j}dy^{i},\\,\\,\\,\\,\\,d\\overline{z}^{i}=dx^{i}-\\mathbf{j\ndy^{i}  \\label{2.22}\n\\end{equation\nwhich represent the cobases of $M$. Then the following expression can be\nwritte\n\\begin{equation}\nJ(\\frac{\\partial }{\\partial z^{i}})=\\mathbf{-j}\\frac{\\partial }{\\partial \n\\overline{z}^{i}},\\,\\,\\,\\,\\,\\,J(\\frac{\\partial }{\\partial \\overline{z}^{i}})\n\\mathbf{j}\\frac{\\partial }{\\partial z^{i}}.  \\label{2.3}\n\\end{equation\nThe dual endomorphism $J^{\\ast }$ of $T_{p}^{\\ast }(M)$ at any point $p$ of\nthe manifold $M$ satisfies that $J^{\\ast 2}=I,$ and is denoted b\n\\begin{equation}\nJ^{\\ast }(dz^{i})=\\mathbf{-j}d\\overline{z}^{i},\\,\\,\\,\\,\\,\\,J^{\\ast }(\n\\overline{z}^{i})=\\mathbf{j}dz^{i}.  \\label{2.4}\n\\end{equation\nLet $V^{A}$ be a commutative group $(V,+)$ endowed with a structure of\nunitary module over the ring $A.$ Let $V^{R}$ denote the group $(V,+)$\nendowed with the structure of real vector space inherited from the\nrestriction of scalars to $R\\mathbf{.}$ The vector space $V^{R}$ will then\nbe called the real vector space associated to $V^{A}.$ Settin\n\\begin{equation}\nJ(u)=ju,\\,\\,\\,\\,\\,\\,P^{+}(u)=e^{+}u,\\,\\,\\,P^{-}(u)=e^{-}u\\,\\,,\\,\\,\\,u\\in\nV^{A},  \\label{2.5}\n\\end{equation\nthe equalitie\n\\begin{equation}\n\\begin{array}{l}\nJ^{2}=1_{V},\\,\\,\\,\\,P^{+2}=P^{+},\\,\\,\\,\\,P^{-2}=P^{-},\\,\\,\\,\\,\\,\\,P^{+}\\circ\n\\,\\,\\,P^{-}=P^{-}\\circ P^{+}=0 \\\\ \nP^{+}+P^{-}=1_{V},\\,\\,\\,P^{+}-\\,\\,\\,P^{-}=J, \\\\ \nP^{-}=(1\/2)(1_{V}-J),\\,\\,\\,\\,\\,\\,P^{+}=(1\/2)(1_{V}+J), \\\\ \nj^{2}=1,\\,\\,\\,\\,e^{+2}=e^{+},\\,\\,\\,\\,e^{-2}=e^{-},\\,\\,\\,\\,\\,\\,e^{+}\\circ\n\\,\\,\\,e^{-}=e^{-}\\circ e^{+}=0,\\, \\\\ \n\\,\\,e^{+}+e^{-}=1,\\,\\,\\,e^{+}-\\,\\,\\,e^{-}=j, \\\\ \ne^{-}=(1\/2)(1-j),\\,e^{+}=(1\/2)(1+j)\n\\end{array}\n\\label{2.6}\n\\end{equation\ncan be found. Also, we calculated whic\n\\begin{equation}\n\\begin{array}{l}\nP^{\\mp }\\left( \\frac{\\partial }{\\partial z^{i}}\\right) =-e^{\\mp }\\frac\n\\partial }{\\partial \\overline{z}^{i}}\\text{ \\ \\ \\ \\ },\\text{ \\ \\ \\ \\ }P^{\\mp\n}\\left( \\frac{\\partial }{\\partial \\overline{z}^{i}}\\right) =e^{\\mp }\\frac\n\\partial }{\\partial z^{i}}, \\\\ \nP^{\\ast \\mp }\\left( dz^{i}\\right) =-e^{\\mp }d\\overline{z}^{i}\\text{ \\ \\ \\ \\ \n,\\text{ \\ \\ \\ \\ }P^{\\ast \\mp }\\left( d\\overline{z}^{i}\\right) =e^{\\mp\n}dz^{i}\n\\end{array}\n\\label{2.8}\n\\end{equation\nIf the conformal manifold $(M,g,J=P^{+}-P^{-})$ satisfies the following\nconditions simultaneously then the conformal manifold is an almost\npara-conformal\\textbf{\\ }Hermitian manifold. The first expression can be\ngiven as follows\n\\begin{equation}\ng(X,Y)+g(X,Y)=0\\Leftrightarrow g(X,Y)=0,\\,\\,\\,\\,\\forall X,Y\\in \\chi (D_{1}),\n\\label{2.9}\n\\end{equation\nsince $P^{+}$ and $P^{-}$ are the projections over $D_{1}$ and $D_{2}$\nrespectively. Then $(P^{+}-P^{-})(X)=P^{+}X-P^{-}X=P^{+}X=X,$ \n(P^{+}-P^{-})(Y)=P^{+}Y-P^{-}Y=P^{+}Y=Y.$ Similarly the second expression\ncan be shown as follows\n\\begin{equation}\ng(X,Y)+g(X,Y)=0\\Leftrightarrow g(X,Y)=0,\\,\\,\\,\\,\\forall X,Y\\in \\chi (D_{2}).\n\\label{2.10}\n\\end{equation\nLet $X=X_{1}+X_{2},Y=Y_{1}+Y_{2}$ be vector fields on $M$ such that \nX_{1},Y_{1}\\in D_{1}$ and $X_{2},Y_{2}\\in D_{2}.$ The\n\\begin{equation}\n\\begin{array}{c}\ng(JX,Y)=g(JX_{1}+JX_{2},Y_{1}+Y_{2})=g(X_{1}-X_{2},Y_{1}+Y_{2}) \\\\ \n=g(X_{1},Y_{1})-g(X_{2},Y_{1})+g(X_{1},Y_{2})-g(X_{2},Y_{2}) \\\\ \n=-g(X_{2},Y_{1})+g(X_{1},Y_{2}), \\\\ \ng(X,JY)=g(X_{1}+X_{2},JY_{1}+JY_{2})=g(X_{1}+X_{2},Y_{1}-Y_{2}) \\\\ \n=g(X_{1},Y_{1})+g(X_{2},Y_{1})-g(X_{1},Y_{2})-g(X_{2},Y_{2}) \\\\ \n=g(X_{2},Y_{1})-g(X_{1},Y_{2})\n\\end{array}\n\\label{2.101}\n\\end{equation\nand hence \ng(JX,Y)+g(X,JY)=-g(X_{2},Y_{1})+g(X_{1},Y_{2})+g(X_{2},Y_{1})-g(X_{1},Y_{2})=0, \n$ for all vector fields $X,Y$ on $M$. If the conditions (\\ref{2.9}) and (\\re\n{2.10}) are true then $D_{1}$ and $D_{2}$ are Lagrangian distributions in\nterms of the 2- form $\\Phi (X,Y)=g(JX,Y).$ Therefore, if the almost\npara-complex structure $J$\\ is integrable then $(M,g,J)$ is a para-conformal\nK\\\"{a}hlerian manifold, or equivalently, $(M,\\Phi ,D_{1},D_{2})$ is a\nbi-Lagrangian conformal manifold. \\cite{gilkey,weyl,kim}. Where $W_{\\pm }$\nis a conformal para-complex structure to be similar to an integrable almost\n(para)-complex $P^{\\mp }$ given in (\\ref{2.8}). Similarly $W_{\\pm }^{\\ast }$\nare the dual of $W_{\\pm }$ structures. So, we adapt the following equations\nusing (\\ref{7})\n\\begin{equation}\n\\begin{array}{l}\nW^{\\mp }\\left( \\frac{\\partial }{\\partial z^{i}}\\right) =-e^{\\mp }e^{\\lambda \n\\frac{\\partial }{\\partial \\overline{z}^{i}}\\text{ \\ \\ \\ \\ },\\text{ \\ \\ \\ \\ \nW^{\\mp }\\left( \\frac{\\partial }{\\partial \\overline{z}^{i}}\\right) =e^{\\mp\n}e^{-\\lambda }\\frac{\\partial }{\\partial z^{i}}, \\\\ \nW^{\\ast \\mp }\\left( dz_{i}\\right) =-e^{\\mp }e^{\\lambda }d\\overline{z}_{i\n\\text{\\ \\ \\ \\ },\\text{ \\ \\ \\ \\ }W^{\\ast \\mp }\\left( dz_{i}\\right) =e^{\\mp\n}e^{-\\lambda }d\\overline{z}_{i}\n\\end{array}\n\\label{2.11}\n\\end{equation}\n\n\\section{Conformal Bi-Para Euler-Lagrangians}\n\nHere, conformal bi-para-Euler-Lagrange equations and a conformal\nbi-para-mechanical system will be obtained under the consideration of the\nbasis $\\{e^{+},e^{-}\\}$ on the bi-Lagrangian conformal manifold $(M,\\Phi\n,D_{1},D_{2})$. Let $(W^{+},W^{-})$ be an almost bi-para-complex conformal\nstructure on $(M,\\Phi ,D_{1},D_{2})$, and $(z^{i},\\overline{z}^{i})$ be its\npara-complex coordinates. Let the vector field $\\xi $ be a semispray given b\n\\begin{equation}\n\\begin{array}{l}\n\\xi =e^{+}(\\xi ^{i+}\\frac{\\partial }{\\partial z^{i}}+\\overline{\\xi }^{i+\n\\frac{\\partial }{\\partial \\overline{z}^{i}})+e^{-}(\\xi ^{i-}\\frac{\\partial }\n\\partial z^{i}}+\\overline{\\xi }^{i-}\\frac{\\partial }{\\partial \\overline{z\n^{i}}); \\\\ \nz^{i}=\\,z^{i+}e^{+}+z^{i-}e^{-};\\,\\overset{.}{\\,z}^{i}=\\overset{.}{\\,z\n^{i+}e^{+}+\\overset{.}{z}^{i-}e^{-}=\\xi ^{i+}e^{+}+\\xi ^{i-}e^{-}; \\\\ \n\\overline{z}^{i}=\\,\\overline{z}^{i+}e^{+}+\\overline{z}^{i-}e^{-};\\,\\overset{\n}{\\,\\overline{z}}^{i}=\\overset{.}{\\,\\overline{z}}^{i+}e^{+}+\\overset{.}\n\\overline{z}}^{i-}e^{-}=\\overline{\\xi }^{i+}e^{+}+\\overline{\\xi }^{i-}e^{-}\n\\end{array}\n\\label{3.1}\n\\end{equation\nwhere the dot indicates the derivative with respect to time $t$. The vector\nfield denoted by $V=(W^{+}-W^{-})(\\xi )$ and given b\n\\begin{equation}\n(W^{+}-W^{-})(\\xi )=e^{+}(-e^{\\lambda }\\xi ^{i+}\\frac{\\partial }{\\partial\nz^{i}}+e^{-\\lambda }\\overline{\\xi }^{i+}\\frac{\\partial }{\\partial \\overline{\n}^{i}})-e^{-}(-e^{\\lambda }\\xi ^{i-}\\frac{\\partial }{\\partial z^{i}\n+e^{-\\lambda }\\overline{\\xi }^{i-}\\frac{\\partial }{\\partial \\overline{z}^{i}\n)  \\label{3.2}\n\\end{equation\nis called conformal \\textit{bi}-\\textit{para} \\textit{Liouville vector field}\non the bi-Lagrangian conformal manifold. The maps given by $T,$ \nP:M\\rightarrow A$ such that $T=\\frac{1}{2}m_{i}(\\overline{z}^{i})^{2}=\\frac{\n}{2}m_{i}(\\overset{.}{z}^{i})^{2},$ $P=m_{i}gh$ are called \\textit{the\nkinetic energy} and \\textit{the potential energy of the system,}\nrespectively.\\textit{\\ }Here\\textit{\\ }$m_{i},g$ and $h$ stand for mass of a\nmechanical system having $n_{i}$ particle, the gravity acceleration and\ndistance to the origin of a mechanical system on the bi-Lagrangian conformal\nmanifold,\n\nrespectively. Then $L:M\\rightarrow A$ is a map that satisfies the conditions;\n\n\\textbf{i)} $L=T-P$ is a conformal \\textit{bi-para} \\textit{Lagrangian\nfunction,}\n\n\\textbf{ii)} the function given by $E_{L}=V(L)-L$ is \\textit{a conformal\nbi-para energy function}.\n\nThe operator $i_{(W^{+}-W^{-})}$ induced by $W^{+}-W^{-}$ and shown b\n\\begin{equation}\ni_{W^{+}-W^{-}}\\omega (Z_{1},Z_{2},...,Z_{r})=\\sum_{i=1}^{r}\\omega\n(Z_{1},...,(W^{+}-W^{-})Z_{i},...,Z_{r})  \\label{3.3}\n\\end{equation\nis said to be \\textit{vertical derivation, }where $\\omega \\in \\wedge\n^{r}{}M, $ $Z_{i}\\in \\chi (M).$ The \\textit{vertical differentiation} \nd_{(P^{+}-P^{-})}$ is defined b\n\\begin{equation}\nd_{(W^{+}-W^{-})}=[i_{(W^{+}-W^{-})},d]=i_{(W^{+}-W^{-})}d-di_{(W^{+}-W^{-})}\n\\label{3.4}\n\\end{equation\nwhere $d$ is the usual exterior derivation. For an almost para-complex\nstructure $W^{+}-W^{-}$, the closed para-conformal K\\\"{a}hlerian form is the\nclosed 2-form given by $\\Phi _{L}=-dd_{_{(W^{+}-W^{-})}}L$ such tha\n\\begin{equation}\nd_{\\left( W^{+}-W^{-}\\right) }L=\\mathbf{e}^{+}\\left( -e^{\\lambda }\\frac\n\\partial L}{\\partial \\overline{z}^{i}}dz^{i}+e^{-\\lambda }\\frac{\\partial L}\n\\partial z^{i}}d\\overline{z}^{i}\\right) -\\mathbf{e}^{-}\\left( -e^{\\lambda \n\\frac{\\partial L}{\\partial \\overline{z}^{i}}dz^{i}+e^{-\\lambda }\\frac\n\\partial L}{\\partial z^{i}}d\\overline{z}^{i}\\right) :\\mathcal{F\n(M)\\rightarrow \\wedge ^{1}{}M  \\label{3.5}\n\\end{equation\nLet $\\xi $ be the second order differential equations given by equation (\\re\n{3.1}) an\n\\begin{equation}\n\\begin{array}{l}\ni_{\\xi }\\Phi _{L}=\\Phi _{L}(\\xi ) \\\\ \n=-\\mathbf{e}^{+}\\xi ^{i+}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial z^{j}\n\\frac{\\partial L}{\\partial \\overline{z}^{i}}dz^{j}+\\mathbf{e}^{+}\\xi\n^{i+}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial z^{j}}\\frac{\\partial L}\n\\partial \\overline{z}^{i}}dz^{i}-\\mathbf{e}^{+}\\xi ^{i+}e^{\\lambda }\\frac\n\\partial ^{2}L}{\\partial z^{j}\\partial \\overline{z}^{i}}dz^{j}+\\mathbf{e\n^{+}\\xi ^{i+}e^{\\lambda }\\frac{\\partial ^{2}L}{\\partial z^{j}\\partial \n\\overline{z}^{i}}dz^{i} \\\\ \n-\\mathbf{e}^{+}\\overline{\\xi }^{i+}e^{-\\lambda }\\frac{\\partial \\lambda }\n\\partial z^{j}}\\frac{\\partial L}{\\partial z^{i}}dz^{i}+\\mathbf{e}^{+\n\\overline{\\xi }^{i+}e^{-\\lambda }\\frac{\\partial ^{2}L}{\\partial\nz^{j}\\partial z^{i}}dz^{i}+\\mathbf{e}^{+}\\overline{\\xi }^{i+}e^{\\lambda \n\\frac{\\partial \\lambda }{\\partial \\overline{z}^{j}}\\frac{\\partial L}\n\\partial \\overline{z}^{i}}dz^{i}+\\mathbf{e}^{+}\\overline{\\xi \n^{i+}e^{\\lambda }\\frac{\\partial ^{2}L}{\\partial \\overline{z}^{j}\\partial \n\\overline{z}^{i}}dz^{i} \\\\ \n+\\mathbf{e}^{-}\\xi ^{i-}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial z^{j}\n\\frac{\\partial L}{\\partial \\overline{z}^{i}}dz^{j}-\\mathbf{e}^{-}\\xi\n^{i-}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial z^{j}}\\frac{\\partial L}\n\\partial \\overline{z}^{i}}dz^{i}+\\mathbf{e}^{-}\\xi ^{i-}e^{\\lambda }\\frac\n\\partial ^{2}L}{\\partial z^{j}\\partial \\overline{z}^{i}}dz^{j}-\\mathbf{e\n^{-}\\xi ^{i-}e^{\\lambda }\\frac{\\partial ^{2}L}{\\partial z^{j}\\partial \n\\overline{z}^{i}}dz^{i} \\\\ \n+\\mathbf{e}^{-}\\overline{\\xi }^{i-}e^{-\\lambda }\\frac{\\partial \\lambda }\n\\partial z^{j}}\\frac{\\partial L}{\\partial z^{i}}dz^{i}-\\mathbf{e}^{-\n\\overline{\\xi }^{i-}e^{-\\lambda }\\frac{\\partial ^{2}L}{\\partial\nz^{j}\\partial z^{i}}dz^{i}-\\mathbf{e}^{-}\\overline{\\xi }^{i-}e^{\\lambda \n\\frac{\\partial \\lambda }{\\partial \\overline{z}^{j}}\\frac{\\partial L}\n\\partial \\overline{z}^{i}}dz^{i}-\\mathbf{e}^{-}\\overline{\\xi \n^{i-}e^{\\lambda }\\frac{\\partial ^{2}L}{\\partial \\overline{z}^{j}\\partial \n\\overline{z}^{i}}dz^{i} \\\\ \n\\mathbf{e}^{+}\\xi ^{i+}e^{-\\lambda }\\frac{\\partial \\lambda }{\\partial z^{j}\n\\frac{\\partial L}{\\partial z^{i}}d\\overline{z}^{i}-\\mathbf{e}^{+}\\xi\n^{i+}e^{-\\lambda }\\frac{\\partial ^{2}L}{\\partial z^{j}\\partial z^{i}}\n\\overline{z}^{i}-\\mathbf{e}^{+}\\xi ^{i+}e^{\\lambda }\\frac{\\partial \\lambda }\n\\partial \\overline{z}^{j}}\\frac{\\partial L}{\\partial \\overline{z}^{i}}\n\\overline{z}^{j}-\\mathbf{e}^{+}\\xi ^{i+}e^{\\lambda }\\frac{\\partial ^{2}L}\n\\partial \\overline{z}^{j}\\partial \\overline{z}^{i}}d\\overline{z}^{j} \\\\ \n-\\mathbf{e}^{+}\\overline{\\xi }^{i+}e^{-\\lambda }\\frac{\\partial \\lambda }\n\\partial \\overline{z}^{j}}\\frac{\\partial L}{\\partial z^{i}}d\\overline{z}^{i}\n\\mathbf{e}^{+}\\overline{\\xi }^{i+}e^{-\\lambda }\\frac{\\partial \\lambda }\n\\partial \\overline{z}^{j}}\\frac{\\partial L}{\\partial z^{i}}d\\overline{z}^{j}\n\\mathbf{e}^{+}\\overline{\\xi }^{i+}e^{-\\lambda }\\frac{\\partial ^{2}L}\n\\partial \\overline{z}^{j}\\partial z^{i}}d\\overline{z}^{i}-\\mathbf{e}^{+\n\\overline{\\xi }^{i+}e^{-\\lambda }\\frac{\\partial ^{2}L}{\\partial \\overline{z\n^{j}\\partial z^{i}}d\\overline{z}^{j} \\\\ \n-\\mathbf{e}^{-}\\xi ^{i-}e^{-\\lambda }\\frac{\\partial \\lambda }{\\partial z^{j}\n\\frac{\\partial L}{\\partial z^{i}}d\\overline{z}^{i}+\\mathbf{e}^{-}\\xi\n^{i-}e^{-\\lambda }\\frac{\\partial ^{2}L}{\\partial z^{j}\\partial z^{i}}\n\\overline{z}^{i}+\\mathbf{e}^{-}\\xi ^{i-}e^{\\lambda }\\frac{\\partial \\lambda }\n\\partial \\overline{z}^{j}}\\frac{\\partial L}{\\partial \\overline{z}^{i}}\n\\overline{z}^{j}+\\mathbf{e}^{-}\\xi ^{i-}e^{\\lambda }\\frac{\\partial ^{2}L}\n\\partial \\overline{z}^{j}\\partial \\overline{z}^{i}}d\\overline{z}^{j} \\\\ \n+\\mathbf{e}^{-}\\overline{\\xi }^{i-}e^{-\\lambda }\\frac{\\partial \\lambda }\n\\partial \\overline{z}^{j}}\\frac{\\partial L}{\\partial z^{i}}d\\overline{z}^{j}\n\\mathbf{e}^{-}\\overline{\\xi }^{i-}e^{-\\lambda }\\frac{\\partial \\lambda }\n\\partial \\overline{z}^{j}}\\frac{\\partial L}{\\partial z^{i}}d\\overline{z}^{j}\n\\mathbf{e}^{-}\\overline{\\xi }^{i-}e^{-\\lambda }\\frac{\\partial ^{2}L}\n\\partial \\overline{z}^{j}\\partial z^{i}}d\\overline{z}^{j}+\\mathbf{e}^{-\n\\overline{\\xi }^{i-}e^{-\\lambda }\\frac{\\partial ^{2}L}{\\partial \\overline{z\n^{j}\\partial z^{i}}d\\overline{z}^{j}\n\\end{array}\n\\label{3.7}\n\\end{equation\nSince the closed conformal para K\\\"{a}hlerian form $\\Phi _{L}$ on $M$ is in\na para-symplectic\\emph{\\ }structure, it is found tha\n\\begin{equation}\n\\begin{array}{l}\nE_{L}=\\mathbf{e}^{+}\\left( -\\xi ^{i+}e^{\\lambda }\\frac{\\partial L}{\\partial \n\\overline{z}^{i}}+\\overline{\\xi }^{i+}e^{-\\lambda }\\frac{\\partial L}\n\\partial z^{i}}\\right) -\\mathbf{e}^{-}\\left( -\\xi ^{i-}e^{\\lambda }\\frac\n\\partial L}{\\partial \\overline{z}^{i}}+\\overline{\\xi }^{i-}e^{-\\lambda \n\\frac{\\partial L}{\\partial z^{i}}\\right) -\n\\end{array}\n\\label{3.8}\n\\end{equation\nand thu\n\\begin{equation}\n\\begin{array}{l}\ndE_{L}=\\mathbf{e}^{+}\\left[ -\\xi ^{i+}e^{\\lambda }\\frac{\\partial \\lambda }\n\\partial z^{j}}\\frac{\\partial L}{\\partial \\overline{z}^{i}}dz^{j}-\\xi\n^{i+}e^{\\lambda }\\frac{\\partial ^{2}L}{\\partial z^{j}\\overline{z}^{i}}dz^{j}\n\\overline{\\xi }^{i+}e^{-\\lambda }\\frac{\\partial \\lambda }{\\partial z^{j}\n\\frac{\\partial L}{\\partial z^{i}}dz^{j}+\\overline{\\xi }^{i+}e^{-\\lambda \n\\frac{\\partial ^{2}L}{\\partial z^{j}\\partial z^{i}}dz^{j}\\right] \\\\ \n-\\mathbf{e}^{-}\\left[ -\\xi ^{i-}e^{\\lambda }\\frac{\\partial \\lambda }\n\\partial z^{j}}\\frac{\\partial L}{\\partial \\overline{z}^{i}}dz^{j}-\\xi\n^{i-}e^{\\lambda }\\frac{\\partial ^{2}L}{\\partial z^{j}\\partial \\overline{z\n^{i}}dz^{j}-\\overline{\\xi }^{i-}e^{-\\lambda }\\frac{\\partial \\lambda }\n\\partial z^{j}}\\frac{\\partial L}{\\partial z^{i}}dz^{j}+\\overline{\\xi \n^{i-}e^{-\\lambda }\\frac{\\partial ^{2}L}{\\partial z^{j}\\partial z^{i}}dz^{j\n\\right] -\\frac{\\partial L}{\\partial z^{j}}dz^{j} \\\\ \n+\\mathbf{e}^{+}\\left[ -\\xi ^{i+}e^{\\lambda }\\frac{\\partial \\lambda }\n\\partial \\overline{z}^{j}}\\frac{\\partial L}{\\partial \\overline{z}^{i}}\n\\overline{z}^{j}-\\xi ^{i+}e^{\\lambda }\\frac{\\partial ^{2}L}{\\partial \n\\overline{z}^{j}\\partial \\overline{z}^{i}}d\\overline{z}^{j}-\\overline{\\xi \n^{i+}e^{-\\lambda }\\frac{\\partial \\lambda }{\\partial \\overline{z}^{j}}\\frac\n\\partial L}{\\partial z^{i}}d\\overline{z}^{j}+\\overline{\\xi }^{i+}e^{-\\lambda\n}\\frac{\\partial ^{2}L}{\\partial \\overline{z}^{j}\\partial z^{i}}d\\overline{z\n^{j}\\right] \\\\ \n-\\mathbf{e}^{-}\\left[ -\\xi ^{i-}e^{\\lambda }\\frac{\\partial \\lambda }\n\\partial \\overline{z}^{j}}\\frac{\\partial L}{\\partial \\overline{z}^{i}}\n\\overline{z}^{j}-\\xi ^{i-}e^{\\lambda }\\frac{\\partial ^{2}L}{\\partial \n\\overline{z}^{j}\\partial \\overline{z}^{i}}d\\overline{z}^{j}-\\overline{\\xi \n^{i-}e^{-\\lambda }\\frac{\\partial \\lambda }{\\partial \\overline{z}^{j}}\\frac\n\\partial L}{\\partial z^{i}}d\\overline{z}^{j}+\\overline{\\xi }^{i-}e^{-\\lambda\n}\\frac{\\partial ^{2}L}{\\partial \\overline{z}^{j}\\partial z^{i}}d\\overline{z\n^{j}\\right] -\\frac{\\partial L}{\\partial \\overline{z}^{i}}d\\overline{z}^{j}\n\\end{array}\n\\label{3.9}\n\\end{equation\nUse of equation (\\ref{1}) gives\n\\begin{equation}\n\\begin{array}{l}\n\\mathbf{e}^{+}\\xi ^{i+}e^{\\lambda }\\frac{\\partial ^{2}L}{\\partial\nz^{j}\\partial \\overline{z}^{i}}+\\mathbf{e}^{+}\\overline{\\xi }^{i+}e^{\\lambda\n}\\frac{\\partial ^{2}L}{\\partial \\overline{z}^{j}\\partial \\overline{z}^{i}}\n\\mathbf{e}^{-}\\xi ^{i-}e^{\\lambda }\\frac{\\partial ^{2}L}{\\partial\nz^{j}\\partial \\overline{z}^{i}}-\\mathbf{e}^{-}\\overline{\\xi }^{i-}e^{\\lambda\n}\\frac{\\partial ^{2}L}{\\partial \\overline{z}^{j}\\partial \\overline{z}^{i}}\n\\\\ \n+\\mathbf{e}^{+}\\xi ^{i+}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial z^{j}\n\\frac{\\partial L}{\\partial \\overline{z}^{i}}+\\mathbf{e}^{+}\\overline{\\xi \n^{i+}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial \\overline{z}^{j}}\\frac\n\\partial L}{\\partial \\overline{z}^{i}}-\\mathbf{e}^{-}\\xi ^{i-}e^{\\lambda \n\\frac{\\partial \\lambda }{\\partial z^{j}}\\frac{\\partial L}{\\partial \\overline\nz}^{i}}-\\mathbf{e}^{-}\\overline{\\xi }^{i-}e^{\\lambda }\\frac{\\partial \\lambda \n}{\\partial \\overline{z}^{j}}\\frac{\\partial L}{\\partial \\overline{z}^{i}}\n\\frac{\\partial L}{\\partial z^{j}} \\\\ \n-\\mathbf{e}^{+}\\xi ^{i+}e^{-\\lambda }\\frac{\\partial ^{2}L}{\\partial\nz^{j}\\partial z^{i}}-\\mathbf{e}^{+}\\overline{\\xi }^{i+}e^{-\\lambda }\\frac\n\\partial ^{2}L}{\\partial \\overline{z}^{j}\\partial z^{i}}+\\mathbf{e}^{-}\\xi\n^{i-}e^{-\\lambda }\\frac{\\partial ^{2}L}{\\partial z^{j}\\partial z^{i}}\n\\mathbf{e}^{-}\\overline{\\xi }^{i-}e^{-\\lambda }\\frac{\\partial ^{2}L}\n\\partial \\overline{z}^{j}\\partial z^{i}} \\\\ \n\\mathbf{e}^{+}\\xi ^{i+}e^{-\\lambda }\\frac{\\partial \\lambda }{\\partial z^{j}\n\\frac{\\partial L}{\\partial z^{i}}+e^{+}\\overline{\\xi }^{i+}e^{-\\lambda \n\\frac{\\partial \\lambda }{\\partial \\overline{z}^{j}}\\frac{\\partial L}\n\\partial z^{i}}-\\mathbf{e}^{-}\\xi ^{i-}e^{-\\lambda }\\frac{\\partial \\lambda }\n\\partial z^{j}}\\frac{\\partial L}{\\partial z^{i}}-e^{-}\\overline{\\xi \n^{i-}e^{-\\lambda }\\frac{\\partial \\lambda }{\\partial \\overline{z}^{j}}\\frac\n\\partial L}{\\partial z^{i}}+\\frac{\\partial L}{\\partial \\overline{z}^{j}}=0\n\\end{array}\n\\label{3.10}\n\\end{equation\nIf a curve denoted by $\\alpha :A\\rightarrow M$ is considered to be an\nintegral curve of $\\xi ,$ $\\xi (L)=\\frac{\\partial L}{\\partial t}$ $,$ then\nthe following equation is obtained\n\\begin{equation}\n\\begin{array}{l}\n\\left( \\mathbf{e}^{+}-\\mathbf{e}^{-}\\right) e^{\\lambda }\\left[ \\mathbf{e\n^{+}\\xi ^{i+}\\frac{\\partial }{\\partial z^{j}}+\\mathbf{e}^{+}\\overline{\\xi \n^{i+}\\frac{\\partial }{\\partial \\overline{z}^{j}}+\\mathbf{e}^{-}\\xi ^{i-\n\\frac{\\partial }{\\partial z^{j}}+\\mathbf{e}^{-}\\overline{\\xi }^{i-}\\frac\n\\partial }{\\partial \\overline{z}^{j}}\\right] \\left( \\frac{\\partial L}\n\\partial \\overline{z}^{j}}\\right) \\\\ \n+\\left( \\mathbf{e}^{+}-\\mathbf{e}^{-}\\right) e^{\\lambda }\\left[ \\mathbf{e\n^{+}\\xi ^{i+}\\frac{\\partial }{\\partial z^{j}}+\\mathbf{e}^{+}\\overline{\\xi \n^{i+}\\frac{\\partial }{\\partial \\overline{z}^{j}}+\\mathbf{e}^{-}\\xi ^{i-\n\\frac{\\partial }{\\partial z^{j}}+\\mathbf{e}^{-}\\overline{\\xi }^{i-}\\frac\n\\partial }{\\partial \\overline{z}^{j}}\\right] \\left( \\lambda \\right) \\left( \n\\frac{\\partial L}{\\partial \\overline{z}^{j}}\\right) +\\frac{\\partial L}\n\\partial z^{j}} \\\\ \n-\\left( \\mathbf{e}^{+}-\\mathbf{e}^{-}\\right) e^{-\\lambda }\\left[ \\mathbf{e\n^{+}\\xi ^{i+}\\frac{\\partial }{\\partial z^{j}}+\\mathbf{e}^{+}\\overline{\\xi \n^{i+}\\frac{\\partial }{\\partial \\overline{z}^{j}}+\\mathbf{e}^{-}\\xi ^{i-\n\\frac{\\partial }{\\partial z^{j}}+\\mathbf{e}^{-}\\overline{\\xi }^{i-}\\frac\n\\partial }{\\partial \\overline{z}^{j}}\\right] \\frac{\\partial L}{\\partial z^{j\n} \\\\ \n\\left( \\mathbf{e}^{+}-\\mathbf{e}^{-}\\right) e^{-\\lambda }\\left[ \\mathbf{e\n^{+}\\xi ^{i+}\\frac{\\partial }{\\partial z^{j}}+\\mathbf{e}^{+}\\overline{\\xi \n^{i+}\\frac{\\partial }{\\partial \\overline{z}^{j}}+\\mathbf{e}^{-}\\xi ^{i-\n\\frac{\\partial }{\\partial z^{j}}+\\mathbf{e}^{-}\\overline{\\xi }^{i-}\\frac\n\\partial }{\\partial \\overline{z}^{j}}\\right] \\left( \\lambda \\right) \\frac\n\\partial L}{\\partial z^{j}}+\\frac{\\partial L}{\\partial \\overline{z}^{j}}=\n\\end{array}\n\\label{3.11}\n\\end{equation\no\n\\begin{eqnarray}\n\\left( \\mathbf{e}^{+}-\\mathbf{e}^{-}\\right) e^{\\lambda }\\xi \\left( \\frac\n\\partial L}{\\partial \\overline{z}^{i}}\\right) +\\left( \\mathbf{e}^{+}-\\mathbf\ne}^{-}\\right) e^{\\lambda }\\xi \\left( \\lambda \\right) \\left( \\frac{\\partial \n}{\\partial \\overline{z}^{i}}\\right) +\\frac{\\partial L}{\\partial z^{i}} &=&0,\n\\label{3.12} \\\\\n-\\left( \\mathbf{e}^{+}-\\mathbf{e}^{-}\\right) e^{-\\lambda }\\xi \\left( \\frac\n\\partial L}{\\partial z^{i}}\\right) +\\left( \\mathbf{e}^{+}-\\mathbf{e\n^{-}\\right) e^{-\\lambda }\\xi \\left( \\lambda \\right) \\frac{\\partial L}\n\\partial z^{i}}+\\frac{\\partial L}{\\partial \\overline{z}^{i}} &=&0.  \\notag\n\\end{eqnarray\nThen the following equations are found\n\\begin{equation}\n\\left( \\mathbf{e}^{+}-\\mathbf{e}^{-}\\right) \\frac{\\partial }{\\partial t\n\\left( e^{\\lambda }\\frac{\\partial L}{\\partial \\overline{z}^{i}}\\right) \n\\frac{\\partial L}{\\partial z^{i}}=0\\text{ \\ , \\ }\\left( \\mathbf{e}^{+}\n\\mathbf{e}^{-}\\right) \\frac{\\partial }{\\partial t}\\left( e^{-\\lambda }\\frac\n\\partial L}{\\partial z^{i}}\\right) -\\frac{\\partial L}{\\partial \\overline{z\n^{i}}=0.  \\label{3.13}\n\\end{equation\nThus equations\\textbf{\\ }(\\ref{3.13}) are seen to be \\textbf{conformal\nbi-para Euler-Lagrange equations} on the distributions $D_{1}$ and $D_{2},$\nand then the triple $(M,\\Phi _{L},\\xi )$ is seen to be a \\textbf{conformal\nbi-para mechanical system}\\textit{\\ }with taking into account the basis \n\\{e^{+},e^{-}\\}$ on the bi-Lagrangian conformal manifold $(M,\\Phi\n,D_{1},D_{2})$.\n\n\\section{Conformal Bi-Para Hamiltonians}\n\nIn the part, conformal bi-para Hamilton equations and a conformal bi-para\nHamiltonian mechanical system on the bi Lagrangian conformal manifold \n(M,\\Phi ,D_{1},D_{2})$ will be derived. Let $(z_{i},\\overline{z}_{i})$ be\nits para-complex coordinates. Let $\\{\\frac{\\partial }{\\partial z_{i}},\\frac\n\\partial }{\\partial \\overline{z}_{i}}\\}$ and $\\{dz_{i},d\\overline{z}_{i}\\},$\nbe bases and cobases of $T_{p}(M)$ and $T_{p}^{\\ast }(M)$ of $M,$\nrespectively. Let us assume that an almost bi-para-complex conformal\nstructure, a bi-para-conformal Liouville form and a bi-para-complex\nconformal 1-form on the distributions ${}D_{1}$ and $D_{2}$ are shown by \nW^{\\ast +}-W^{\\ast -}$, $\\lambda $ and $\\omega $, respectively. Then, we\nusing \\cite{miron} and (\\ref{2.11}):\n\n\\begin{equation}\n\\begin{array}{c}\n\\omega =\\frac{1}{2}[(z_{i}dz_{i}+\\overline{z}_{i}d\\overline{z\n_{i})e^{+}+(e^{2\\lambda }z_{i}dz_{i}+e^{2\\lambda }\\overline{z}_{i}d\\overline\nz}_{i})e^{-}], \\\\ \n\\lambda =(W^{\\ast +}-W^{\\ast -})(\\omega )=\\frac{1}{2}[(\\mathbf{-\ne^{+}e^{\\lambda }z_{i}d\\overline{z}_{i}+e^{+}e^{\\lambda }\\overline{z\n_{i}dz_{i})] \\\\ \n-\\frac{1}{2}(\\mathbf{-}e^{-}e^{\\lambda }z_{i}d\\overline{z\n_{i}+e^{-}e^{\\lambda }\\overline{z}_{i}dz_{i})]\n\\end{array}\n\\label{4.111}\n\\end{equation\nIt is well known that if $\\Phi $ is a closed para-K\\\"{a}hlerian form on the\nbi-Lagrangian conformal manifold, then $\\Phi $ is also a para-symplectic\nstructure on ${}$the bi-Lagrangian conformal manifold. Given a\nbi-para-conformal Hamiltonian vector field $Z_{H}$ fixed with the\nbi-para-conformal Hamiltonian energy\\textit{\\ }$H$ that is \n\\begin{equation}\nZ_{H}=(Z_{i}\\frac{\\partial }{\\partial z_{i}}+\\overline{Z}_{i}\\frac{\\partial \n}{\\partial \\overline{z}_{i}})e^{+}+(Z_{i}\\frac{\\partial }{\\partial z_{i}}\n\\overline{Z}_{i}\\frac{\\partial }{\\partial \\overline{z}_{i}})e^{-}.\n\\label{4.2}\n\\end{equation\nThen closed 2-form i\n\\begin{eqnarray}\n\\Phi &=&-d\\lambda =e^{+}\\overline{Z}_{i}dz_{i}-e^{-}\\overline{Z}_{i}dz_{i}\n\\frac{1}{2}\\left[ \n\\begin{array}{c}\ne^{+}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial z_{i}}\\bar{Z\n_{i}z_{i}dz_{i}+e^{+}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial \\overline\nz}_{i}}\\overline{Z}_{i}\\overline{z}_{i}dz_{i} \\\\ \n-e^{-}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial z_{i}}\\overline{Z\n_{i}z_{i}dz_{i}-e^{-}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial \\overline\nz}_{i}}\\overline{Z}_{i}\\overline{z}_{i}dz_{i\n\\end{array\n\\right]  \\label{4.3} \\\\\n&&-e^{+}Z_{i}d\\overline{z}_{i}+e^{-}Z_{i}d\\overline{z}_{i}-\\frac{1}{2}\\left[ \n\\begin{array}{c}\n\\mathbf{-}e^{+}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial z_{i}\nZ_{i}z_{i}d\\overline{z}_{i}-e^{+}e^{\\lambda }\\frac{\\partial \\lambda }\n\\partial \\overline{z}_{i}}Z_{i}\\overline{z}_{i}d\\overline{z}_{i} \\\\ \n+e^{-}e^{-}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial z_{i}}Z_{i}z_{i}\n\\overline{z}_{i}+e^{-}e^{\\lambda }\\frac{\\partial \\lambda }{\\partial \n\\overline{z}_{i}}Z_{i}\\overline{z}_{i}d\\overline{z}_{i\n\\end{array\n\\right] .  \\notag\n\\end{eqnarray\nAnd then it follow\n\\begin{equation}\n\\begin{array}{c}\ni_{Z_{H}}\\Phi =\\Phi (Z_{H}) \\\\ \n=\\bar{Z}_{i}e^{+}\\left[ 1-\\frac{1}{2}e^{\\lambda }\\left( z_{i}\\frac{\\partial\n\\lambda }{\\partial z_{i}}+\\overline{z}_{i}\\frac{\\partial \\lambda }{\\partial \n\\overline{z}_{i}}\\right) \\right] dz_{i}-Z_{i}e^{+}\\left[ 1+\\frac{1}{2\ne^{\\lambda }\\left( z_{i}\\frac{\\partial \\lambda }{\\partial z_{i}}+\\overline{z\n_{i}\\frac{\\partial \\lambda }{\\partial \\overline{z}_{i}}\\right) \\right] \n\\overline{z}_{i} \\\\ \n-\\bar{Z}_{i}e^{-}\\left[ 1-\\frac{1}{2}e^{\\lambda }\\left( z_{i}\\frac{\\partial\n\\lambda }{\\partial z_{i}}+\\overline{z}_{i}\\frac{\\partial \\lambda }{\\partial \n\\overline{z}_{i}}\\right) \\right] dz_{i}+Z_{i}e^{-}\\left[ 1+\\frac{1}{2\ne^{\\lambda }\\left( z_{i}\\frac{\\partial \\lambda }{\\partial z_{i}}+\\overline{z\n_{i}\\frac{\\partial \\lambda }{\\partial \\overline{z}_{i}}\\right) \\right] \n\\overline{z}_{i\n\\end{array}\n\\label{4.4}\n\\end{equation\nOn the other hand, the differential of the bi-para-conformal Hamiltonian\nenergy $H$\\textit{\\ }is calculated as follows\n\\begin{equation}\ndH=(\\frac{\\partial H}{\\partial z_{i}}dz_{i}+\\frac{\\partial H}{\\partial \n\\overline{z}_{i}}d\\overline{z}_{i})e^{+}+(\\frac{\\partial H}{\\partial z_{i}\ndz_{i}+\\frac{\\partial H}{\\partial \\overline{z}_{i}}d\\overline{z}_{i})e^{-}.\n\\label{4.5}\n\\end{equation\nBy means of equation\\textbf{\\ }(\\ref{3}), using equation (\\ref{4.4}) and \n\\ref{4.5}), the conformal bi-para Hamiltonian vector field is seen to b\n\\begin{equation}\nZ_{H}=(Z_{i}\\frac{\\partial }{\\partial z_{i}}+\\overline{Z}_{i}\\frac{\\partial \n}{\\partial \\overline{z}_{i}})e^{+}+(Z_{i}\\frac{\\partial }{\\partial z_{i}}\n\\overline{Z}_{i}\\frac{\\partial }{\\partial \\overline{z}_{i}})e^{-}.\n\\label{4.6}\n\\end{equation\nIf a curve $\\alpha :I\\subset A\\rightarrow M$ is an integral curve of the\nconformal bi-para Hamiltonian vector field $Z_{H}$, i.e., $Z_{H}(\\alpha (t))\n\\dot{\\alpha}(t)$ $,\\,\\,t\\in I.$ In the local coordinates, we get $\\alpha\n(t)=(z_{i}(t),\\overline{z}_{i}(t))$ an\n\\begin{equation}\n\\dot{\\alpha}(t)=(\\frac{dz_{i}}{dt}\\frac{\\partial }{\\partial z_{i}}+\\frac{\n\\overline{z}_{i}}{dt}\\frac{\\partial }{\\partial \\overline{z}_{i}})e^{+}+\n\\frac{dz_{i}}{dt}\\frac{\\partial }{\\partial z_{i}}+\\frac{d\\overline{z}_{i}}{d\n}\\frac{\\partial }{\\partial \\overline{z}_{i}})e^{-}.  \\label{4.10}\n\\end{equation\nTaking equations\\textbf{\\ }(\\ref{4.6}),\\textbf{\\ }(\\ref{4.10}), the\nfollowing equations are foun\n\\begin{equation}\n\\frac{dz_{i}}{dt}=\\frac{-(e^{+}-e^{-})}{\\left[ 1+\\frac{1}{2}e^{\\lambda\n}\\left( z_{i}\\frac{\\partial \\lambda }{\\partial z_{i}}+\\overline{z}_{i}\\frac\n\\partial \\lambda }{\\partial \\overline{z}_{i}}\\right) \\right] }\\frac{\\partial\nH}{\\partial \\overline{z}_{i}}\\text{ , }\\frac{d\\overline{z}_{i}}{dt}=\\frac\n(e^{+}-e^{-})}{\\left[ 1-\\frac{1}{2}e^{\\lambda }\\left( z_{i}\\frac{\\partial\n\\lambda }{\\partial z_{i}}+\\overline{z}_{i}\\frac{\\partial \\lambda }{\\partial \n\\overline{z}_{i}}\\right) \\right] }\\frac{\\partial H}{\\partial z_{i}}.\n\\label{4.12}\n\\end{equation\nHence, equations\\textbf{\\ }(\\ref{4.12}) are seen to be \\textbf{conformal\nbi-para Hamilton equations} on the bi-Lagrangian conformal manifold $(M,\\Phi\n,D_{1},D_{2}),$ and then the triple $(M,\\Phi ,Z_{H})$ is seen to be a \n\\textbf{conformal bi-para Hamiltonian mechanical system}\\textit{\\ }with the\nuse of basis $\\{e^{+},e^{-}\\}$ on $(M,\\Phi ,D_{1},D_{2})$.\n\n\\section{Conclusion}\n\nIt is seen in the above, formalisms of Lagrangian and Hamiltonian mechanics\nhad intrinsically been described by taking into account the basis \n\\{e^{+},e^{-}\\}$\\ on the bi-Lagrangian conformal manifold $(M,\\Phi\n,D_{1},D_{2})$. Conformal bi-para Lagrangian and bi-para Hamiltonian models\nhave arisen to be very important tools since they present a simple method to\ndescribe the model for bi-para-conformal mechanical systems. So, the\nequations derived in (\\ref{3.13}) and (\\ref{4.12}) are only considered to be\na first step to realize how bi-para-complex conformal geometry has been used\nin solving problems in different physically spaces. For further research,\nbi-para-complex conformal Lagrangian and Hamiltonian vector fields derived\nhere are suggested to deal with problems in different fields of physics. In\nthe literature, the equations, which explains the linear orbits of the\nobjects, were presented. This study explained the non-linear orbits of the\nobjects in the space by the help of revised equations using Weyl theorem.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{The Context: Tiny Robots}\n\nIn the recent years, there has been a wide interest in\nthe cooperative behavior of tiny robots.\nIn particular, many distributed coordination protocols have been devised\nfor a wide range of models and for a wide range of problems,\nlike convergence, gathering, pattern formation,\nflocking, etc.\nAt the same time, researchers have also started characterizing\nthe scenarios in which such problems cannot be solved, deriving\nimpossibility results.\n\n\\subsection{Our Motivation: Even Simpler Robots}\n\nAn interesting question regards the minimal cognitive\ncapabilities that such tiny robots need to have for completing a particular task.\nIn particular, researchers have initiated the study of ``weak robots''\\cite{FlocchiniPSW99}.\nWeak robots are \\textit{anonymous} (they do not have any identifier),\n\\textit{autonomous} (they work independently),\n\\textit{homogeneous} (they behave the same in the same situation),\nand \\textit{silent} (they also do not communicate with each other).\n\nWeak robots are usually assumed to have their own local view, represented as a Cartesian\ncoordinate system with origin and unit length and axes. The orientation of axes, or the \\textit{chirality} (relative order of the orientation of axes or handedness),\nis not common among the robots.\nThe robots move in a sequence of three consecutive actions, \\textit{Look-Compute-Move}:\nthey observe the positions of other robots in their local coordinate system and the observation step returns a set of points to the observing robot.\nThe robots cannot distinguish if there are multiple robots at the same position,\ni.e., they do not have the capability of \\textit{multiplicity detection}.\nImportantly, the robots are \\textit{oblivious} and cannot maintain state between rounds\n(essentially moving steps).\nThe computation they perform are always based on the data they have collected in the \\emph{current} observation step; in the next round they again collect the data.\nSuch weak robots are therefore interesting\nfrom a self-stabilizing perspective: as robots do not rely on memory,\nan adversary cannot manipulate the memory either.\nIndeed, researchers have demonstrated that weak robots\nare sufficient to solve a wide range of problems.\n\nWe in this paper aim to relax the assumptions on the tiny robots\nfurther. In particular, to the best of our knowledge, all prior literature\nassumes that robots can observe the positions of other robots in their local view. This enables them to calculate the distance between any pair of robots.\nThis seems to be a very strong assumption, and accordingly,\nwe in this paper initiate the study of even weaker robots which\ncannot locate other robots positions in their local view, preventing them from measuring distances. We define these kind of robots as \\textit{monoculus robots}.\n\n\nIn particular, we\ninitiate to explore two naturally weaker models for monoculus robots with less cognitive capabilities\n\\begin{enumerate}\n\\item \\emph{Locality Detection} ($\\mathcal{LD}$): The robots can distinguish whether a neighbor robot is at a distance more than a predefined value $c$ or not.\n\\item \\emph{Orthogonal Line Agreement} ($\\mathcal{OLA}$): The robots agree on a pair of orthogonal lines (but not necessarily the orientation of the lines).\n\\end{enumerate}\n\n\n\\subsection{The Challenge: Convergence}\n\nWe focus on the fundamental convergence problem for monoculus robots\nand show that the problem is already non-trivial in this setting.\n\nIn particular, many naive strategies lead to non-monotonic\nbehaviors. For example, strategies where boundary robots\n(robots located on the convex hull) move toward the\n``median'' robot they see, may actually \\emph{increase}\nthe area of the convex hull in the next round,\ncounteracting convergence as shown in Fig.~\\ref{fig:anglebisector}$(a)$.\nA similar counterexample exists for a strategy where\nrobots move in the direction of the angle bisector as shown in Fig.~\\ref{fig:anglebisector} ($b$).\n\\noindent\n\\begin{figure}[H]\\centering\n\\includegraphics[width=\\linewidth]{figanglebisector.pdf}\n\\caption{ The 4 boundary robots are moving $(a)$ towards the median robot $(b)$ along the angle bisector. The discs are the old positions and circles are the new positions. The old convex hull is drawn in solid line, the new convex hull is dashed. The arrows denote the direction of moving.  }\\label{fig:anglebisector}\n\\end{figure}\nBut not only enforcing convex hull invariants is challenging,\nalso the fact that visibility is restricted and we cannot detect\nmultiplicity:\nWe in this paper assume that robots are not transparent, and\naccordingly, a robot does not see whether and how many robots\nmay be hidden behind a visible robot. As robots are also not able\nto perform multiplicity detection (i.e., determine how many robots\nare collocated at a certain point), strategies such as ``move toward\nthe center of gravity'' (the direction in which most robots are located),\nare not possible.\n\n\n\\subsection{Our Contributions}\n\nThis paper studies distributed convergence problems\nfor anonymous, autonomous, oblivious, non-transparent, monoculus, point robots under a most general asynchronous scheduling model\nand makes the following contributions.\n\\begin{enumerate}\n\\item We initiate the study of a new kind of robot,\n the \\emph{monoculus robot} which cannot measure distances.\n The robot comes in two natural flavors, and we introduce the\n Locality Detection ($\\mathcal{LD}$)\nand the Orthogonal Line Agreement ($\\mathcal{OLA}$) model accordingly.\n\\item We present and formally analyze deterministic and self-stabilizing distributed convergence algorithms for both\n$\\mathcal{LD}$ and $\\mathcal{OLA}$.\n\\item We show our assumptions in $\\mathcal{LD}$ and $\\mathcal{OLA}$ are minimal in the sense that\nrobot convergence is not possible for monoculus robots without any additional capability.\n\\item We report on the performance of our algorithms through simulation.\n\\item We show that our approach can be generalized to higher dimensions and, with a small extension, supports termination.\n\\end{enumerate}\n\n\\subsection{Paper Organization}\n\nThe remainder of this paper is organized as follows.\nSection~\\ref{sec:preli} introduces the necessary\nbackground and preliminaries.\nSection~\\ref{sec:algo} introduces two algorithms for convergence.\nSection~\\ref{sec:impossibilty} presents an impossibility result which shows the minimality of our assumptions.\nWe report on simulation results in\nSection~\\ref{sec:simul} and discuss extensions in\nSection~\\ref{sec:discussion}.\nIn Section~\\ref{sec:relwork}, we review related work,\nbefore we conclude in Section~\\ref{sec:conclusion}.\n\n\\section{Preliminaries}\\label{sec:preli}\n\n\n\\subsection{Model}\n\nWe consider anonymous, autonomous, homogeneous, oblivious, non-transparent robots with unlimited visibility, unless the view is obstructed by another robot:\nSince the robots are non-transparent, any robot can see at most one robot in any direction.\nAs usual, the robots in each round execute a sequence of \\textit{Look-Compute-Move} steps:\nFirst, the robot observes other robots (\\textit{Look} step); second, on the basis of the observed information, it executes an algorithm\nwhich computes a direction which the robot must move towards\n(\\textit{Compute} step);\nthe robot then moves in this direction (\\textit{Move} step),\nfor a fixed distance $b$ (the step size).\nThe robots are silent, cannot detect multiplicity points, and can\npass over each other (no collision occurs).\n\nIn this paper, we introduce monoculus robots:\n\\begin{definition}(Monoculus Robot)\nA robot is called \\emph{monoculus} if it is anonymous, autonomous, oblivious, homogeneous, and silent. We assume the robot is a non-transparent point\nrobot, has unlimited visibility, and can neither determine the position of other robots nor detect multiplicty.\n\\end{definition}\n\nWe consider the most general CORDA or ASYNC scheduling model\nknown from weak robots~\\cite{FlocchiniPSW99} as well as\nthe ATOM or Semi-Synchronous (SSYNC) model~\\cite{suzu}.\nThese models define the activation schedule of the robots:\nthe SSYNC model considers instantaneous computation and movement, i.e.,\nthe robots cannot observe other robots in motion,\nwhile in the ASYNC model any robot can look at any time. In SSYNC the time is divided into global rounds and a subset of the robots are activated in each round\nwhich finish their \\textit{Look-Compute-Move} within that round.\nIn case of ASYNC, there is no global notion of time.\nThe Fully-synchronous (FSYNC) is a special case of SSYNC, in which all the robots are activated in each round.\nThe algorithms presented in this paper work in both the ASYNC and the\nSSYNC setting. For the sake of generality, we present our proofs\nin terms of the ASYNC model.\n\n\\subsection{Notation and Terminology}\n\n A configuration $C$ is a multiset containing all the robot positions in 2D.\n At any time $t$ the configuration (the mapping of robots in the plane)\n is denoted by $C_t$. The convex hull of configuration $C_t$ is denoted as $CH_t$.\n\\emph{Convergence} is achieved when the distance between any pair of robots is less than a predefined value $c$ (and subsequently does not violate this anymore).\nOur multi-robot system is vulnerable to adversarial manipulation,\nhowever the algorithms presented in this paper are self-stabilizing~\\cite{dolev2000self}\n and robust to state manipulations.\nSince the robots are oblivious, they only depend on the \\emph{current state}: if the state is perturbed,  the algorithms are still able to converge in a self-stabilizing manner \\cite{Gilbert2009}.\n\n\n\n\\section{Convergence Algorithms}\\label{sec:algo}\n\nWe now present distributed robot convergence algorithms for both our models,\n$\\mathcal{LD}$ and $\\mathcal{OLA}$.\n\n\\subsection{Convergence for $\\mathcal{LD}$}\n\nIn this section we consider the convergence problem for the monoculus robots in\nthe $\\mathcal{LD}$ model. Our claims hold for any $c\\geq 2b$.\nAlgorithm~\\ref{algo:convergelocality}\ndistinguishes between two cases: (1) If the robot only sees one other robot,\nit infers that the current configuration must be a line (of 2 or more robots),\nand that this robot must be on the border of this line;\nin this case,  the boundary robots always move inside (usual step size $b$).\n(2) Otherwise, a robot moves towards  any visible, non-local robot (distance\nat least $c$),\nfor a $b$ distance (the step size).\n\nOur proof unfolds in a number of lemmas followed by a theorem.\nFirst, Lemma~\\ref{lem:4bdistance} shows that it is impossible to have a pair of robots with distance larger than $2c$ in the converged situation. Lemma~\\ref{lem:chsubset} shows that our algorithm ensures a monotonically decreasing convex hull size.  Lemma~\\ref{lem:finitedecrement} then proves that the decrement in perimeter for each movement is greater than a constant (the convex hull decrement is strictly monotonic). Combining all the three lemmas, we obtain the correctness proof of the algorithm.\nIn the following, we call two robots \\emph{neighboring} if they see each other (line of sight is not obstructed by another robot).\n\n\\begin{algorithm}\n\\DontPrintSemicolon\n\\caption{\\textsc{ConvergeLocality}}\\label{algo:convergelocality}\n\\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n\\Input{Any arbitrary configuration}\n\\Output{All robots are inside a circle of radius $c$}\n\\eIf{only one robot is visible}{Move distance $b$ towards that robot}\n{\\eIf{there is at least one robot farther than $c$}{Move distance $b$ towards any one of the robots with distance more than $c$}{Do not move\\tcp*{All neighbor robots are within a distance $c$}}}\n\\end{algorithm}\n\n\\begin{lemma}\\label{lem:4bdistance}\nIf there exists a pair of robots at distance more than $2c$ in a non-linear configuration, then there exists a pair of neighboring robots at distance more than $c$.\n\\end{lemma}\n\\begin{proof}\nProof by contradiction. If there is a pair of robots with distance more than $2c$, then for them not to move, there are at least two robots on the line joining them positioned such that each pair has a distance less than $c$. Since the robots are non-transparent, the end robots cannot look beyond their neighbors to know that there is a robot at a distance more than $c$.\nIn Fig.~\\ref{fig:4bdistance}, $r_1$ and $r_4$ are $2c$ apart. So $r_2$ and $r_3$ block the view. Since it is a non-linear configuration, say robot $r_5$ is not on the line joining $r_1$ and $r_4$. $l$ is the perpendicular bisector of $\\overline{r_1r_4}$. If $r_5$ is on the left side, then it is more than $c$ distance away from $r_4$ and vice versa. If there is another robot on $\\overline{r_4r_5}$, then consider that as the new robot in a non-linear position, and we can argue similarly. Hence there would at least be a single robot similar to $r_5$ in a non-linear configuration for which the distance is more than $c$.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[height=0.2\\linewidth]{fig4bdistance.pdf}\n\\caption{A non-linear configuration with a pair of robots at a distance $2c$}\\label{fig:4bdistance}\n\\end{figure}\n\\qed\n\\end{proof}\n\n\\begin{lemma}\\label{lem:chsubset}\nFor any time $t' > t$ before convergence, $CH_{t'}\\subseteq CH_{t}$.\n\\end{lemma}\n\\begin{proof}\nThe proof follows from a simple observation. Consider any robot $r_i$. If $r_i$ decides to move towards some robot, say $r_j$, then it is at least $c$ distance away. Even if $r_j$ is on the boundary, $r_i$ cannot cross the boundary. If $r_i$ is already on the boundary, then it always moves on the perimeter or inside the convex hull. Hence the convex hull gradually decreases.\n\n If all the robots are on a straight line, then the boundary robots move monotonically closer in each step. The distance between the end robots is a monotonically decreasing sequence until it reaches $c$.\n\\qed\n\\end{proof}\n\n\\begin{lemma}\\label{lem:finitedecrement}\nIn finite time the decrement in the perimeter of the convex hull is at least $ b \\left(1- \\sqrt{\\frac{1}{2}\\left(1+\\cos\\left(\\frac{2\\pi}{n}\\right)\\right)}\\right)$.\n\\end{lemma}\n\\begin{proof}\nThe sum of internal angles of a $k$-sided convex polygon is $(k-2)\\pi$. So there exists a robot $r$ at a corner $A$ (ref. Fig.~\\ref{fig:decrement}) of the convex hull such that the internal angle is less than $(1-\\frac{2}{n})\\pi$, where $n$ is the total number of robots.\nLet $B$ and $C$ be the points where the  circle centered at $A$ with radius $b\/2$ intersects the convex hull.\nAny robot lying outside the circle will not move inside the circle according to Algorithm~\\ref{algo:convergelocality}.\nAll the robots inside the circle will eventually move out once they are activated.\nAfter all the robots are activated at least once, the decrement in perimeter is at least $AB + AC - BC$. From cosine rule,\n$$AB + AC - BC = \\frac{b}{2}+\\frac{b}{2} -\\sqrt{\\left(\\frac{b}{2}\\right)^2+\\left(\\frac{b}{2}\\right)^2-  2\\frac{b}{2}\\frac{b}{2}\\cos\\left(\\pi -\\frac{2\\pi}{n}\\right)}  $$\n$$ = b \\left(1- \\sqrt{\\frac{1}{2}\\left(1+\\cos\\left(\\frac{2\\pi}{n}\\right)\\right)}\\right)$$\n\\begin{figure}[H]\\centering\n\\includegraphics[height=0.3\\linewidth]{exampleforfinitedecrement.pdf}\n\\caption{Once activated the robots $r$ and $r'$ will move outside the solid circle with radius $b\/2$. The robot $r''$ moves a distance $b$ towards $r'$ because distance between them is more than $2b$ and stops at $D$.}\\label{fig:decrement}\n\\end{figure}\n\\qed\n\\end{proof}\n\n\\begin{theorem}(Correctness)\nAlgorithm~\\ref{algo:convergelocality} terminates when all the robots are within a $c$ radius disc.\n\\end{theorem}\n\\begin{proof}\nFrom Lemmas~\\ref{lem:chsubset}\nand~\\ref{lem:finitedecrement} we know that the convex hull never\nincreases, and eventually a robot will be activated which strictly\ndecreases the hull.\nAccording to Lemma~\\ref{lem:4bdistance}, eventually there will not be a pair of robots with more than $2c$ distance.\nNote that the distance between any two points in a disc of radius $c$ is less than or equal to $2c$. Hence the robots will converge within a disc of radius $c$.\n\\qed\\end{proof}\n\n\n\\subsection{Convergence for $\\mathcal{OLA}$}\n\nIn this section we consider monoculus robots in the $\\mathcal{OLA}$ model.\nOur algorithm will distinguish between\n\\textit{boundary-}, \\textit{corner-} and \\textit{inner-robots}, defined\nin the canonical way. We note that robots can determine their type:\nFrom the Fig.~\\ref{fig:orthogonalline}, we can observe that for $r_2$, all the robots lie below the horizontal line. That means, one side of the horizontal line is empty\nand therefore $r_2$ can figure out that it is a boundary robot. Similarly all $r_i$, $i\\in \\{3,4,5,6,7,8\\}$ are boundary robots. Whereas, for $r_1$, both horizontal and vertical lines have one of the sides empty, hence $r_1$ is a corner robot. Other robots are all inner robots.\n Consequently, we define \\textit{boundary robots} to be those, which have exactly one side of one of the orthogonal lines empty.\n\n  Algorithm~\\ref{algo:convergequadrant} (\\textsc{ConvergeQuadrant}) can be described as follows.\n  A rectangle can be constructed with lines parallel to the orthogonal lines passing through boundary robots such that, all the robots are inside this rectangle.\n  In Fig.~\\ref{fig:orthogonalline}, each boundary robot always moves inside the rectangle perpendicular to the boundary and the inside robots do not move.\n  Note that the corner robot $r_1$ has two possible directions to move. So it moves toward any robot in that common quadrant.\n  Gradually the distance between opposite boundaries becomes smaller and smaller and the robots converge. In case all the robots are on a line which is parallel to either of the orthogonal lines, then the robots will find that both sides of the line are empty.\n  In that case they should not move. But the robots on either end of the line would only see one robot. So they would move along the line towards that robot.\n\n\\begin{figure}[!h]\\centering\n\\includegraphics[height=0.4\\linewidth]{figdirectionlesscompass.pdf}\n\\caption{Movement direction of the boundary robots}\\label{fig:orthogonalline}\n\\end{figure}\n\n\\begin{algorithm}[!h]\n\\DontPrintSemicolon\n\\caption{\\textsc{ConvergeQuadrant}}\\label{algo:convergequadrant}\n\\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n\\Input{Any arbitrary configuration and robot $r$}\n\\Output{All robots are inside a square with side $2b$ }\n\\uIf{only one robot is visible}{Move towards that robot}\n\\uElseIf{$r$ is a boundary robot}{Move perpendicular to the boundary to the side with robots}\n\\uElseIf{$r$ is a corner robot}{Move towards any robot in the non-empty quadrant}\n\\Else{Do not move \\tcp*{It is an inside robot}}\n\\end{algorithm}\n\n\\begin{theorem}(Correctness)\nAlgorithm~\\ref{algo:convergequadrant} moves all the robots inside some $2b$-sided square in finite time.\n\\end{theorem}\n\\begin{proof}\nConsider the distance between the robots on the left and right boundary. The horizontal distance between them decreases each time either of them gets activated. The rightmost robot will move towards the left and the leftmost will move towards the right. The internal robots do not move. So in at most $n$ activation rounds of the boundary robot, the distance between two of the boundary nodes will decrease by at least $b$. Hence the distance is monotonically decreasing until $2b$.\nAfterwards, the total distance will never exceed $2b$ anymore.\n\nGiven there is a corner robot present in the configuration, that robot will move towards any robot in the non-empty quadrant. So, the movement of\nthe corner robot contributes to the decrement in distance in both directions.\nConsider robots inside the quadrant are presently very close to one of the boundaries and the corner robot moves towards that robot, then the decrement in one of the dimensions can be small (an $\\epsilon > 0$).\nConsider for example the configuration of a strip of width $b$,\nthen the corner robot becomes the adjacent corner in the next round;\nthis can happen only finitely many times.\nEach dimension converges within a distance $2b$, so in the converged state the shape of the converged area would be $2b$-sided square.\n\\qed\\end{proof}\n\n\\begin{remark}\nIf the robots have some sense of angular knowledge, the corner robots can always move in a $\\pi\/4$ angle, so the decrement in both dimension is significant, hence convergence time is less on average.\n\\end{remark}\n\n\n\\section{Impossibility and Optimality}\\label{sec:impossibilty}\n\nGiven these positive results, we now show that we cannot make the monoculus robots much weaker, otherwise we lose convergeability.\n\n\\begin{theorem}\nThere is no deterministic convergence algorithm for monoculus robots without any additional capability.\n\\end{theorem}\n\\begin{proof}\nWe prove the theorem using a symmetry argument.\nConsider the two configurations $C_1$ and $C_2$ in Fig.~\\ref{fig:indistinguishableconfig}.\nIn $C_1$, all the robots are equidistant from robot $r$, while in $C_2$, the robots are at different distances, however the relative angle of the robots is the same at $r$. Now considering the local view of robot $r$, it cannot distinguish between $C_1$ and $C_2$.\n Say a deterministic algorithm $\\phi$ decides a direction of movement for robot $r$ in configuration $C_1$.\n Since both $C_1$ and $C_2$ are the same from robot $r$'s perspective, the deterministic algorithm outputs the same direction of movement for both cases.\n \\begin{figure}[H]\n\\centering\n\\includegraphics[height=0.3\\linewidth]{figindistinguishableconfig.pdf}\n\\caption{Locally indistinguishable configurations with respect to $r$}\\label{fig:indistinguishableconfig}\n\\end{figure}\nNow consider the convex hull $CH_1$ and $CH_2$ of $C_1$ and $C_2$ respectively.\nThe robot $r$ moves a distance $b$ in one round.\nThe distance from any point inside $CH_1$ is more than $b$ but we can skew the convex hull in the direction of movement, so to make it like $CH_2$, where if the robot $r$ moves a distance $b$ it exits $CH_2$.\n Therefore there always exists a situation for any algorithm $\\phi$ such that the area of the convex hull increases. Hence it is impossible for the robots to converge.\n\n\\qed\\end{proof}\n\n\n\n\\section{Simulation}\\label{sec:simul}\n\nWe now complement our formal analysis with simulations, studying the average\ncase. We assume that robots are distributed uniformly at random\nin a square initially, that $b=1$\nand $c=2$, and\nwe consider FSYNC scheduling. As a baseline to evaluate performance, we consider the\noptimal convergence distance and time if the robots had capability to observe positions,\ni.e., they are \\emph{not} monoculus. Moreover, as a lower bound,\nwe compare to an algorithm which converges all robots to\nthe centroid, defined as follows:\n$$ \\{\\bar{x}, \\bar{y}\\} = \\left\\{\\cfrac{\\sum_{i=1}^nx_i}{n},\\cfrac{\\sum_{i=1}^ny_i}{n}\\right\\}$$\n\\noindent where $\\{x_i,y_i\\} \\forall i \\in \\{1,2,\\cdots,n\\}$ are the robots' coordinates.\n\nWe calculate distance $d_i$ from each robot to the centroid\nin the initial configuration. The optimal distance we have used as\nconvergence distance is the sum of distances from each robot to the unit disc\ncentered at the centroid. So the sum of the optimal convergence distances $d_{opt}$ is given by\n$$d_{opt} = \\sum_{i=1}^n (d_i -1), \\quad if \\, d_i>1 $$\nIn the simulation of Algorithm~\\ref{algo:convergelocality}, we define $d_{CL}$ as the cumulative number of steps taken by all the robots to converge (sometimes also known as the \\emph{work}). Now we define the performance ratio, $\\rho_{CL}$ as\n $$ \\rho_{CL} = \\cfrac{d_{CL}}{d_{opt}}$$\nSimilarly for Algorithm~\\ref{algo:convergequadrant} we define $d_{CQ}$ and $\\rho_{CQ}$. \\\\\n In Fig.~\\ref{fig:varrobot}, we plot the distribution of 100 iterations of simulation of Algorithm~\\ref{algo:convergelocality}, varying the number of robots for a fixed region of deployment.\nThe median increases if we increase the number of robots deployed in the same region.\n\nIn Fig.~\\ref{fig:varrobotCQ}, we plot the distribution of 100 iterations of simulation of Algorithm~\\ref{algo:convergequadrant} varying the number of robots for a fixed region of deployment.\nThe median increases if we increase the number of robots deployed in the same region.\nIn Fig.~\\ref{fig:varregionCQ}, we plot the distribution of 100 iterations of simulation of Algorithm~\\ref{algo:convergequadrant} for a fixed number of robots deployed in different regions. Here we can observe that the distribution does not vary much even if we change the region of deployment.\n\\begin{minipage}{\\linewidth}\n\\begin{minipage}{0.45\\linewidth}\n\\begin{figure}[H]\n\\includegraphics[width=\\linewidth]{VarRobotFixRange100.pdf}\n\\caption{Different number of robots in same region in $\\mathcal{LD}$}\\label{fig:varrobot}\n\\end{figure}\n\\end{minipage}\\hspace{0.1\\linewidth}\n\\begin{minipage}{0.45\\linewidth}\n\\begin{figure}[H]\n\\includegraphics[width=\\linewidth]{VarRangeFixRobot40.pdf}\n\\caption{Fixed number of robots deployed in different region in $\\mathcal{LD}$}\\label{fig:varrange}\n\\end{figure}\n\\end{minipage}\n\\begin{minipage}{0.45\\linewidth}\n\\begin{figure}[H]\n\\includegraphics[width=\\linewidth]{CQVarRobotFixRange100.pdf}\n\\caption{Different number of robots in same region in $\\mathcal{OLA}$}\\label{fig:varrobotCQ}\n\\end{figure}\n\\end{minipage}\\hspace{0.1\\linewidth}\n\\begin{minipage}{0.45\\linewidth}\n\\begin{figure}[H]\n\\includegraphics[width=\\linewidth]{CQVarRangeFixRobot40.pdf}\n\\caption{Fixed number of robots deployed in different region in $\\mathcal{OLA}$}\\label{fig:varregionCQ}\n\\end{figure}\n\\end{minipage}\n\\end{minipage}\\vspace{5mm}\nIn Fig.~\\ref{fig:varrange}, we plot the distribution of 100 iterations of simulation of Algorithm ~\\ref{algo:convergelocality} for a fixed number of robots deployed in different regions. Here we can observe that the distribution does not vary much even if we change the region of deployment.\n\n\n\\begin{figure}[!h]\\centering\n\\includegraphics[height=0.37\\linewidth]{CLvsCQdistance40robot.pdf}\n\\caption{ $\\rho_{CL}$ VS $\\rho_{CQ}$ for the same number of robots}\\label{fig:compareCLCQrobotdistance}\n\\end{figure}\n\\begin{figure}[!h]\\centering\n\\includegraphics[height=0.37\\linewidth]{CLvsCQdistance100region.pdf}\n\\caption{ $\\rho_{CL}$ VS $\\rho_{CQ}$ for the same size of region}\\label{fig:compareCLCQregiondistance}\n\\end{figure}\n\\begin{figure}[!h]\\centering\n\\includegraphics[height=0.37\\linewidth]{CLvsCQtime40robot.pdf}\n\\caption{ $\\tau_{CL}$ VS $\\tau_{CQ}$ for the same number of robots}\\label{fig:compareCLCQrobot}\n\\end{figure}\n\\begin{figure}[!h]\\centering\n\\includegraphics[height=0.37\\linewidth]{CLvsCQtime100region.pdf}\n\\caption{$\\tau_{CL}$ VS $\\tau_{CQ}$ for the same size of region}\\label{fig:compareCLCQregion}\n\\end{figure}\n\nFig.~\\ref{fig:compareCLCQrobotdistance}, and ~\\ref{fig:compareCLCQregiondistance} show the comparison between the performance ratio (PR) for distance. We can observe that Algorithm~\\ref{algo:convergequadrant} performs better. This is due to the fact that, in Algorithm~\\ref{algo:convergequadrant} only boundary robots move.\n\n Let $d_{max}$ be the distance of farthest robot from the centroid and  $t_{CL} $ be the number of synchronous rounds taken by Algorithm~\\ref{algo:convergelocality} for convergence. We define $\\tau_{CL}$ as follows\n$$\\tau_{CL} =\\cfrac{t_{CL}}{d_{max}}$$\nSimilarly for Algorithm~\\ref{algo:convergequadrant}, we define $t_{CQ}$ and $\\tau_{CQ}$. $\\tau_{CL}$ and $\\tau_{CQ}$ show performance ratio for convergence time of Algorithm~\\ref{algo:convergelocality} and ~\\ref{algo:convergequadrant} respectively.\n In Fig.~\\ref{fig:compareCLCQrobot} and ~\\ref{fig:compareCLCQregion}, we can observe that $\\tau_{CL}$ is very close to 1, so Algorithm~\\ref{algo:convergelocality} converges in almost the same number of synchronous rounds (proportional to distance covered, since step size $b=1$) as the maximum distance.\nWe can observer that Algorithm~\\ref{algo:convergequadrant} takes more time as the number of robots and the side length of square region increases.\n\\section{Discussion}\\label{sec:discussion}\n\nThis section shows that our approach supports some interesting\nextensions.\n\n\\subsection{Termination for $\\mathcal{OLA}$ Model}\n\nWhile we only focused on convergence and not termination so far,\nwe can show that with a small amount of memory,\ntermination is also possible in the $\\mathcal{OLA}$\nmodel.\nTo see this, assume that each robot has a 2-bit persistent memory in the $\\mathcal{OLA}$ model for each dimension, total 4-bits for two dimensions. \n Algorithm~\\ref{algo:convergequadrant} has been modified to Algorithm~\\ref{algo:convergequadranttermination} such that it can accommodate termination.\nAll the bits are initially set to 0. \nEach robot has its local coordinate system, which remains consistent over the execution of the algorithm. The four bits correspond to four boundaries in two dimensions, i.e., left, right, top and bottom.\n If a robot finds itself on one of the boundaries according to its local coordinate system, then it sets the corresponding bit of that boundary to 1. Once both bits corresponding to a dimension become 1, the robot stops moving in that dimension.\nConsider a robot $r$. Initially it was on the left boundary in its local coordinate system. Then it sets the first bit of the pair of bits corresponding to $x$-axis. It moves towards right. Once it reaches the right boundary, then it sets the second bit corresponding to $x$-axis to 1. Once both the bits are set to 1, it stops moving along the $x$-axis. Similar movement termination happens on the $y$-axis also. Once all the 4-bits are set to 1, the robot stops moving.\n\n\\begin{algorithm}[!h]\n\\caption{\\textsc{ConvergeQuadrantTermination}}\\label{algo:convergequadranttermination}\n\\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n\\DontPrintSemicolon\n\\Input{Any arbitrary configuration and robot $r$ with 4-bit memory}\n\\Output{All robots are inside a square with side $2b$ }\n\\eIf{the robot is on a boundary(ies)}{set the corresponding bit(s) to 1}{Do nothing \\tcp*{$r$ is an inside robot}}\n\\uIf{$r$ is a boundary robot and the bits corresponding to that dimension are not 1}{Move perpendicular to the boundary to the side with robots}\n\\uElseIf{$r$ is a corner robot}{\\eIf{Both bits corresponding to a dimension is 1}{Move in other dimension to the side with robots}{Move towards any robot in the non-empty quadrant}}\n\\Else{Do not move \\tcp*{$r$ is not on boundary OR all four bits are 1}}\n\\end{algorithm}\n\n\n\n\n\n\n\\subsection{Extension to $d$-Dimensions}\n\nBoth our algorithms can easily be extended to $d$-dimensions.\nFor the $\\mathcal{LD}$ model, the algorithm remains exactly the same.\nFor the proof of convergence, similar arguments as Lemma~\\ref{lem:finitedecrement} can be used in $d$ dimensions.\nWe can consider the convex hull in $d$-dimensions and the boundary robots of the convex hull always move inside. The size of convex hull reduces gradually and the robots converge.\n\nAnalogously for the $\\mathcal{OLA}$ model, the distance between two robots in the boundary of any dimension gradually decreases and the corner robots always move inside the $d$-dimensional cuboid. Hence it converges.\nHere the robot would require $2d$ number of bits for termination.\n\n\\section{Related Work}\\label{sec:relwork}\n\n The problems of gathering \\cite{suzuki1999distributed}, where all the robots gather at a single point,\n  convergence \\cite{cohen2006convergence}, where robots come very close to each other and\n Pattern formation \\cite{flochini1,suzuki1999distributed} have been studied intensively in the literature.\n\n   Flocchini et al. \\cite{FlocchiniPSW99} introduced the CORDA or Asynchronous (ASYNC) scheduling model for weak robots. Suzuki et al. \\cite{suzu} have introduced the ATOM or Semi-synchronous (SSYNC) model.\n       In \\cite{suzuki1999distributed}, impossibility of gathering for $n=2$ without assumptions on local coordinate system agreement for \\textit{SSYNC} and \\textit{ASYNC} is proved.\nAlso, for $n>2$ it is impossible to solve gathering without assumptions on either coordinate system agreement or multiplicity detection \\cite{Prencipe2007}. Cohen and Peleg \\cite{CohenP04} have proposed a center of gravity algorithm for convergence of two robots in ASYNC and any number of robots in SSYNC.\n\nTo the best of our knowledge in all the previous works, the mathematical models always assume that the robots can find out the location of other robots in their local coordinate system in the Look step. This in turn implies that the robots can measure the distance between any pair of robots albeit in their local coordinates. All the algorithms exploit this location information to create an invariant point or a robot where all the other robots gather. But in this paper we deprive the robots of the capability to determine the location of other robots. This leads to robots incapable of finding any kind of distance or angles.\n\nAny kind of pattern formation requires these robots to move to a particular point of the pattern. Since the monoculus robots cannot figure out locations, they cannot stop at a particular point. Hence any kind  of pattern formation algorithm described in the previous works which requires location information as input are obsolete. Gathering problem is nothing but the point formation problem \\cite{suzuki1999distributed}. Hence gathering is also not possible for the monoculus robots.\n\n\n\n\\section{Conclusion}\\label{sec:conclusion}\n\nThis paper introduced the notion of \\emph{monoculus robots}\nwhich cannot measure distance: a practically relevant generalization of\nexisting robot models. We have proved that the two basic models\nstill allow for convergence (and with a small memory,\neven termination), but with less capabilities,\nthis becomes impossible.\n\nThe $\\mathcal{LD}$ model converges in an almost optimal number of rounds, while the $\\mathcal{OLA}$ model takes more time.\nBut the cumulative number of steps is less for the $\\mathcal{OLA}$ model compared to the $\\mathcal{LD}$ model since only boundary robots move. Although we found in our simulations that the median and angle bisector strategies successfully converge, finding a proof accordingly remains  an\nopen question.\nWe see our work as a first step, and believe that the study of weaker robots opens an interesting field for future research.\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{A Rounding procedure using random walks}\n\\label{sec:algo}\nThe simple random walk based algorithm outlined in the introduction \ndoesn't take into account any of the constraints $V_j \n\\cdot x \\leq 1$ and therefore likely to violate them after some\nrandom walk steps. However, the probability that an $x_i$ reaches\n1 before it reaches 0 is equal to the ratio \n$\\frac{x'_i \/\\gamma}{1- x'_i + x'_i \/\\gamma}\n= x'_i$. \nThis is\na consequence of a more general property of martingales known as Doob's\noptional stopping theorem  (\\cite{feller1}).\n\\begin{theorem}\nLet $(\\Omega , \\Sigma, P)$ be a probability space and $\\{ F_i \\}$ be a \nfilteration of $\\Omega$, and $X = \\{ X_i \\}$ a martingale with respect to\n$\\{ F_i \\}$. Let $T$ be a stopping time such that $\\forall \\omega \\in \\Omega, \n\\ \\forall i , \\ \\ | X_i (\\omega ) | < K$ \nfor some positive integer $K$, and $T$ is almost surely bounded.\nThen $\\mathbb{E} [ X_T ] = \\mathbb{E} [ X_0 ]$.   \n\\label{stopping-theorem}\n\\end{theorem} \nIn the above application, the stopping times are $X_T = \\{-b , a \\}$ whichever\nis earlier. So $-b \\cdot \\Pr [ X = -b ] + a \\Pr [ X =a] = 0 \\cdot \\Pr[ X = 0]$.\nSince $\\Pr [X = -b] + \\Pr [ X=a ] = 1$ from the stopping criteria, \n$\\Pr [ X= a] = \\frac{b}{a+b}$. In our context, we start from $X = x'_i$\nand $b = \\frac{x'_i}{\\gamma}$ and $a = \\frac{1- x'_i}{\\gamma}$, so\nthe probability that $\\hat{x}_i = 1$ equals $\\frac{x'_i \/\\gamma}\n{(1 - x'_i + x'_i )\/\\gamma} = x'_i$. \nSo the distributions of the variables \nbeing absorbed at 0 or 1 are identical to the independent rounding. \nHowever, the random walk process has many intermediate steps that do not\nhave corresponding mappings in the $2^n$ possible configurations of the\nindependent rounding. Whether it makes this framework more powerful compared\nto one-step independent rounding is a difficult question but we feel that \nit could\ngive us a superior understanding of the process of independent rounding.   \n\nIn the algorithm presented in Figure \\ref{algo1}, instead of the simple\nBernoulli random walk step, we use a normal Gaussian random walk on\neach coordinate. \nWe run the basic algorithm for\n$T$ iterations such that the number of un-fixed variables in each \nconstraint of $C^T$ (the active set of constraints after $T$ iterations) \nis bounded by $\\log n$. \nThe value of $T$, more specifically $\\mathbb{E}[ T ]$, will be\ndetermined during the course of our analysis. \n\n Now we run the algorithm of \\cite{MT:10} (c.f. section \\ref{sec5}) \non the constraints in $C^T$ which\nare projections of the original constraints on the unfixed variables after\n$T$ steps.\n\\begin{figure}\n\\fbox{\\parbox{6.0in}{\n\\begin{center}\nAlgorithm {\\bf Iterative Randomized Rounding }\n\\end{center}\n{\\footnotesize\nInput : $x'_i \\ \\ 1 \\leq i \\leq n$ satisfying $Ax \\leq b \\cdot e^m$. $C^0$: set of constraints.\n\\\\\n$0 < \\gamma < \\delta < 1$ - the exact values are discussed in the analysis.\n\\\\\nOutput : $\\hat{x_i} \\in \\{0, 1\\}$ \\\\\n\nInitialize all variables as {\\it un-fixed}\nand set $X_0 = [x'_1 , x'_2 , \\ldots x'_n ] $.\\\\\n{\\bf Repeat} for iterations $i \\ \\  = 1, 2 \\ldots $\n\\begin{quote}\n\\begin{enumerate}\n\\item Generate a random vector $ R^{i} = U_i$ where $U_i$ is\na multidimensional gaussian r.v. restricted to the unfixed variables.\n\\item $X_{i+1} = X_i + \\gamma \\cdot R^i$.\n\\item {\\it Fix} a variable if it is\nless than $\\delta$ or greater than $1-\\delta$.\\\\ \nIf all variables are fixed then exit. \\\\\n\\{ * multiple variables may get fixed in a single iteration. * \\}\n\\item Update the set of constraints $C^t$ that contains at least one\nunabsorbed variable.\n\\end{enumerate}\n\n\\end{quote}\n{\\bf until} stopping condition ${\\cal S}$ (* \n$\\max_j \\{ \\mbox{ number of unfixed variables in } C_j \\}  \\leq \\log n $ *)\n\\\\[0.1in]\nRun Moser-Tardos \\cite{MT:10}  algorithm on $C^T$ on the unfixed \nvariables of $X_T$ according\ngiven in Figure \\ref{algo2} \\\\[0.1in]\nRound the {\\it fixed} variables to 0 or 1 whichever is closer \n and return this vector denoted by $\\hat{x}$.\n}\n}}\n\\caption{An iterative randomized rounding algorithm based on random walks}\n\\label{algo1}\n\\end{figure}\n\\subsection{The framework and some notations }\nIn the rounding algorithm, $X_t$ denotes\nthe random walk vector after $t$ steps. \nLet $U^d$ denote a\n $d$-dimensional Gaussian random variable\nand let $U_t$\ndenote the projection of $U^n$ on the current subspace corresponding to\nthe unfixed variables in the\n$t$th iteration. \nThe normal distribution is\ndenoted by ${\\cal N}(\\mu , \\sigma^2)$ where $\\mu$ is the\nmean and $\\sigma^2$ is the variance. \nUnless otherwise stated, we will\nrefer to the {\\it standard} normal distribution where $\\mu = 0$ and\n$\\sigma^2 = 1$. \n\nThe parameters $\\gamma, \\delta$ are chosen to ensure that the walk stays within\nthe feasible region. It suffices to have $\\gamma \\leq \\frac{\\delta}{\\log n}$\nfrom the pdf of the normal distribution if we are executing $O(\\frac{1}{\n\\gamma^2})$ steps (see \\cite{LM:12} for a rigorous proof). The value of\n$\\delta$ will be determined according to the approximation factor and the\nerror bound that we have in a specific application. In our analysis, \nthe the focus will be on \nvariables absorbed at 0 that will be rounded down. The decrease in\nthe objective value can be at most $n\\delta$ (recall the maximum weight \nof the coefficients is 1). If we choose $\\delta$ such that $n\\delta$ is\n$o(OPT)$ it will suffice. For the main theorems, choosing $\\delta \\leq\n\\frac{1}{polylog n}$ will work.\n\nAfter $t$ iterations, the $j$-th constraint is denoted by\n$C_j^t : < V_j , X_t > \\ \\leq 1 + \\beta_t$ that has many of variables\n{\\it fixed}. We shall denote the set of unfixed variables in iteration $t$ \nby ${\\cal U}(t) \\subset\n\\{1, 2 \\ldots n\\}$ and the corresponding vector by $X^{{\\cal U}}_t$ where the\nthe fixed variables are set to 0. The constraint vector $V^t_j =\nV_j \\cap {\\cal U}(t)$, where only the coefficients in $V^t_j$ corresponding to \n${\\cal U}(t)$ can be non-zero. Then\\\\\n$|< V^t_j , X_t - X_{t+1}>| = |< \\frac{V_j}{\\twonorm{ V_j }} , \nX^{{\\cal U}}_t - X_{t+1}^{{\\cal U}}>| \\cdot\n\\twonorm{ V_j }$ is the change in the value of $C_j$ in\niteration $t$ as the remaining\nvariables do not change. \nNote that while $X^{\\cal U}$ changes in every step, $V^t_j$ changes only when\nsome variable is absorbed. \n\nFor convenience of the analysis, we will club successive iterations \ninto {\\it phases}, where\nwithin a phase $p$, $\\beta_p$ remains unchanged. Equivalently, $\\beta_p \n- \\beta_{p-1}$\nreflects the cumulative effect of a number of random walk steps within the \nphase $p$ referred to as the {\\it accumulated error} or simply {\\it error}.  \nThe phase $p$ corresponds to the $L_2$ norm of any\nconstraint $\\twonorm{V^{p}_j}$ that is bounded by $\\sqrt{n\/2^p}$. \nIntuitively, with additional random walk steps, we are more \nlikely to violate the original constraints and $\\beta_p$ is a measure of\nthe violation. \nIn terms of the above notation, it is obvious that\n$ \\twonorm{V^{p+1}_j} \\leq \\twonorm{V^{p}_j}$. We will use the index $p$ (\nrespectively $t$) to indicate reference to phases (resp. iterations).\n\nWlog, we assume that all constraints have at least 2 \nvariables and at most $n-1$\nvariables. \\footnote{A constraint with $n$ variables has a trivial solution\nwhere any one variable can be set to 1 and a constraint with one\nvariable is redundant.} \nThe successive Brownian motion steps (defined by multidimensional normal\nGaussian) form a martingale sequence.\nThe following result forms the crux of our analysis - see \\cite{bansal:10,LM:12}\nfor more details and proof.\n\\begin{observ}\nIn iteration $t$, let $Y_t = < V_j , U_t \\cdot \\gamma >$, that is the\nmeasure of the change in $< V_j , x >$ in the step $t$. Then $Y_t$\nis a Gaussian random variable with mean 0 and variance $\\gamma\n\\cdot\\twonormsq{ V_j }$.\n\\end{observ}\nNote that the scaled\nrandom variable $Y'_t = \\frac{Y_t}{\\gamma \\twonorm{V^t_j}} $ is a\nGaussian with mean 0 and variance $\\leq 1$. \nWhen $Y'_i$ correspond to standard Gaussian, then\n\\begin{lemma}\nFor any $\\beta > 0$,\n $\\Pr [ | Y'_1 + Y'_2 \\ldots Y'_T | > \\beta ] \\leq 2 \\exp ( - \\beta^2 \/2T\n) $.\n\\label{chernoff-mart}\n\\end{lemma}\n\nThe behavior of random walks starting from an arbitrary initial position\nand subsequently absorbed at 0 can be\nobtained from the {\\it gambler's ruin problem} where the underlying martingale\nis the Brownian motion.\nThere exists a wealth of literature on Brownian motion \\cite{feller1,Ross:2006},\nbut the specific form in which we invoke them for analyzing our algorithm\nis stated below. A proof is presented in the appendix.\n\n\\begin{lemma}\nConsider a random walk starting from position $a$ in an\ninterval of length $a +b$ with absorbing barriers at both end-points.\nThen the expected number of steps for the walk to get absorbed (at any of the\nends) is $a \\cdot b$.\nMoreover, the probability of the random walk being absorbed at 0 is\n$\\frac{b}{a+b} - 1\/k$\nafter $k\\cdot a \\cdot b $ steps for any $k > 1$.\n\\label{boost-prob-absorb}\n\\end{lemma}\n{\\it Remark} By choosing $b > k \\cdot a$,\nthe probability\ncan be made arbitrarily close to 1 for $k \\gg 1$ - conversely,\nthe probability of non-absorption at 0 is $O(\\frac{1}{k})$ after $k^2 a^2$\nsteps.\n\n\\ignore{\nThe success of the rounding algorithm depends on the {\\it race} between the\nhitting of constraints $C^t_j$ and the unfixed \nvariables of $V_j^t$. In particular, we would like to\nhit the variable (hyper)-planes $x_i =1$ or $x_i = 0$ more often than the\nhyperplanes $C^t_j$. \n The sooner the ${(X_t )}_i$ hits the \nboundaries 0 or 1, the smaller is the displacement $\\twonorm{ X_t }$ in \n${\\mathbb R}^n$. \nLovette and Meka \\cite{LM:12} also make use of this observation\nimplicitly and in their case, the discrepancy hyperplanes are much further\ncompared to the variable hyperplanes. In our case, the situation is much \ntighter and much of our technical analysis revolves around that.  \nIf there are $r^2$ {\\it unfixed} \nvariables (a subspace with dimension $r^2$)\nthen the variance of the displacement in each step is proportional to \nthe $L_2$ norm of $V_j$ which is $r$.\nSince the\nsum of the $r^2$ variables is bounded by 1 in any feasible LP solution, at \nleast $r^2 \/2$ variables have LP values bounded by $ x'_i \\leq \n\\frac{2}{r^2}$ which is the starting point of the random walk, ${(X_0 )}_i$. \nThe 1-dimensional projection is also a brownian walk ${(X_t)}_i$\nwhere each step is scaled by \n$\\gamma$, and consequently, the $r^2 \/2$ variables are within $\\frac{2}\n{\\gamma \\cdot r^2}$ from 0. Using a simpler argument based on number\nof $\\gamma$-length step sizes, one needs to move \n$\\frac{\\beta}{r \\gamma}$ steps towards some constraint $C_j$ with \nslack $\\beta$\nversus $\\frac{2}{\\gamma \\cdot r^2}$ steps to hit 0s of\nvariables associated with $C_j$. \nClearly the \nlatter is closer if $\\beta > \\frac{2}{r}$. This {\\it bias} in favor of \nhitting the variables holds the key to the success and efficiency of \nour algorithm and we need to capitalize on this by choosing $T_p$\nappropriately, so that the many variables become fixed at 0 before\n$X_p$ hits the constraints (of the scaled polytope). This in turn slows\ndown the Brownian motion further, as the norm reduces and we keep repeating\nthis process. \n\n\n\\section{Appendix}\n{\\bf Proof of Lemma \\ref{boost-prob-absorb}}\n\\begin{proof}\nThe proof of the gambler's ruin problem actually uses a {\\it quadratic\nmartingale}\\footnote{The proof that it is a martingale can be found in\nstandard books on stochastic process}  $B^2 (t) - t$ where $B(t)$ is the \nBrownian motion random variable.\nUsing the optional stopping theorem on this, we obtain\n\\[ \\mathbb{E} [ B^2 (T) - T ] = \\mathbb{E}[ B^2 (0) - 0 ] = a^2 \\]\nSince $p = \\frac{b}{a+b}$ is the probability that it is at 0, we obtain\n$\\mathbb{E}[T] = a \\cdot b$.\n\nNow, consider the events leading to the failure of being absorbed at 0 -\nthese correspond to the absorption of the random walk at either ends or\nthe non-absorption at either ends. \n\\[ \\Pr [ Failure ] = \\Pr [ Failure \\cap Absorption ] + \\Pr [ Failure \\cap\nNon-absorption ]\\]\n\\[  = \\Pr [ Failure | Absorption ] \\cdot \\Pr [ Absorption] \n+ \\Pr[ Failure | Non-absorption ] \\cdot \\Pr [Non-absorption]  \\]\n\\[\n\\leq \\Pr [ Failure | Absorption ] + \\Pr [Non-absorption] \\] \nFrom the preceding argument ,the probability that the random walk is not \nabsorbed after $k a\\cdot b$ steps \nis less than $1\/k$ using Markov's inequality. \nFrom optional stopping criteria, the probability of\nabsorption at 0 is $\\frac{b}{a+b}$. The probability that the random walk is\nnot absorbed at 0 after $k a  b $ steps is bounded by \n$ 1 - \\frac{b}{a+b} \n+ 1\/k$. Consequently, the probability of being absorbed at 0 is \nat least $\\frac{b}{a+b} - 1\/k$. \n\\end{proof}\n\\subsection{A Lower Bound on error}\n\\begin{lemma}\nFor $A \\in {\\cal A}^{m\\times n}_{\\log n}$ such that $A\\cdot \\bar{x} \\leq e^m$,\nwe cannot simultaneously obtain $\\sum_i \\hat{x}_i = \\Omega ( n\/\\log n )$ and\n${|| A \\hat{x}||}_\\infty  \\leq o(\\frac{\\log\\log n}{\\log\\log\\log n})$\nfor $\\hat{x}_i \\in \\{ 0, 1 \\}$.\n\\label{lbnd}\n\\end{lemma}\n\n\\begin{proof}\n\nAs mentioned before, it is easy to see that $x_i = 1\/k$ is a solution \nfor $\\bar{x}$ with objective\nfunction value $n\/k$. After rounding $\\bar{x}$ we want to have at least \n$\\Omega ( n\/k )$ 1s in the rounded vector, say $\\bar{y}$.\nWe must have $A \\cdot \\bar{y} \\leq t\\cdot e$ for some rounding guarantee\n$t$. Let $b$ be a fixed vector with $n\/k$ 1s. \n\nLet $r$ be a row vector with $k$ 1s in random location - this corresponds to\na row of $A$. The probability that $< r , \\bar{y} > \\ \\ \\geq t$ is given by\n \\[  \\frac{{ (n\/k) \\choose t} \n\\cdot { (n-n\/k) \\choose (k-t) }}{{n \\choose k }} \\]\nUsing $ {(\\frac{n}{k})}^k \\leq {n \\choose k} \\leq {(\\frac{ne}{k})}^k $, the \nabove expression is\n\\begin{eqnarray}\n \\geq & \\frac{ {( \\frac{n}{tk})}^t \\cdot {(\\frac{n -n\/k}{k-t})}^{k-t}}{\n{(\\frac{ne}{k})}^k} \\\\\n = & \\frac{ \\frac{1}{t^t} \\cdot {( \\frac{k-1}{k-t})}^{k-t}} {\n e^k } \\\\ \n = & \\frac{ k^{k-t}}{ {( k-t )}^{k-t} \\cdot e^k \\cdot t^t }\\\\\n = & \\frac{1}{ {( 1 - t\/k )}^{k-t} \\cdot e^k \\cdot t^t } \\\\\n \\geq & \\frac{1}{{( 1 - t\/k )}^{k\/2} \\cdot e^k \\cdot t^t } \\text{ for } k \\gg t\\\\\n = & \\frac{1}{ e^{ k - t\/2} \\cdot t^t }\n\\end{eqnarray}\nSo the probability that for $m$ independently chosen rows, the probability that\n$< r , \\bar{y} > \\ \\ < t$ is \n\\begin{eqnarray}\n  p(m,k,t) &  \\leq & {\\left( 1 - \\frac{1}{ e^{ k - t\/2} \\cdot t^t } \\right)}^m \n\\leq  {\\left( 1 - \\frac{1}{ e^{ k } \\cdot t^t } \\right)}^m =\n{\\left( 1 - \\frac{1}{ e^{ k } \\cdot t^t } \\right)}^{e^k t^t \\frac{m}{e^k t^t}}\n\\\\\n & \\leq & {\\left( \\frac{1}{e} \\right)}^{\\frac{m}{e^k t^t}} \n\\end{eqnarray} \nSince there are no more than ${n \\choose\n(n\/\\log n)}$ choice of columns with $n\/\\log n$ 1s, the probability that all\nthe dot products are less than $t$ is less than ${n \\choose (n\/\\log n)} \\cdot\np(m,k,t)  \\leq p(m,k,t) \\cdot {(\\frac {n e}{n\/\\log n})}^{n\/\\log n} \n \\leq e^n \\cdot p(m,k,t)$. If $e^n \\cdot p(m,k,t) < 1$ then there must exist\nmatrices in ${\\cal A}^{m \\times n}_k$ that do not satisfy the error bound $t$.\nSo,\n\\[    1  > {\\left( \\frac{1}{e} \\right)}^{\\frac{m}{e^k t^t}} \\cdot e^{n} \n    \\Rightarrow {( e ) }^{\\frac{m}{e^k t^t}}  >  e^n \\] \n which is equivalent to the condition \n\\begin{equation}\n \\frac{m}{n}  > e^k t^t \\text{  or  }\n\\log(m\/n) > k + t\\log t \n\\label{lbndreq}\n\\end{equation}\n \n For $k = \\log n$ and $m = n\\cdot polylog(n) $, this holds for some $t \\geq \n\\frac{\\alpha \\log \\log n}{\\log\\log\\log n})$, i.e., for some constant \n$\\alpha > 0$, i.e., \nthe error cannot be\n$o (\\frac{\\log\\log n}{\\log\\log\\log n})$. \n\\end{proof}\n\n\\ignore{\n\\subsection{An alternate proof for the RT bound}\nOur algorithm also provides an alternate proof for the \n$O(\\frac{\\log m}{\\log\\log m})$ error bound in constraints incurred in \ncase of classical randomized rounding by Raghavan and \nThompson \\cite{RT:87}\n\nWe run our algorithm on the above problem(with scaling $S=1$).\nAs shown in the proof of Lemma \\ref{absorp_rate},the expected number of \nunfixed variables in $i^{th}$ constraint $=E(u_i^p) (=\\|V_i^p\\|_2 ^2) \\leq \\frac{n}{2^p}$ (when $B = S = 1$).The above holds with high probability when $p \\leq \\log(n)-\\log(\\log(m))(=p*)$.\\\\\nTo simplify our analysis we define variables indexed by the number of phases after p* phases are over.\nLet q indicate the number of phases after $p*$ phases are over.\nLet $w_i^q=u_i^{q+\\log (n)-\\log (\\log (m))}$ (that is the number of unfixed variables in $i^{th}$ constraint indexed by number of phases after p*)and let $\\overset{\\sim}{T_q}=T_{q+\\log (n)-\\log (\\log (m))}$ (that is the number of iterations as a function of number of phases after p*).\\\\\nNow $E(w_i^q)=\\frac{n}{2^{\\log (n)-\\log (\\log (m))+q}}=\\frac{\\log m}{2^q}$.\nTo get bound with sufficiently high probability,bound we need to run our algorithm for a sufficiently large number($>>\\overset{\\sim}{T_q}$) of steps so that,number of unabsorbed variables is bounded by $\\frac{\\log (m)}{2^q}$ with high probability.\n\n\n\nWe claim the following:-\n\\begin{lemma}\nIf r=$2^q \\cdot 2^{2^q +1}$, then after $\\overset{\\sim}{T_r}=\\frac{2^{r+\\log(n)-\\log(\\log(m))} \\cdot 2^{r+\\log(n)-\\log(\\log(m))}}{n^2 \\gamma^2}$ steps,the number of unfixed variables in $i^{th}$ equation is bounded by $\\frac{\\log(m)}{2^q}$ with probability $\\geq 1-\\frac{1}{m}$.\n\\end{lemma}\n\\begin{proof}\nBy definition $w_i^r$ is the number of unfixed variables in the $i^{th} constraint$ after $\\overset{\\sim}{T_r}$ iterations.\\\\\nWe run it for $\\overset{\\sim}{T_r}=\\frac{2^{\\log (n)-\\log (\\log (m))+r} \n2^{\\log (n)-\\log (\\log (m))+r}}{n^2 \\gamma^2}$ steps, where r is some function of q, so that $E(w_i^r)=\\frac{\\log (m)}{2^r}$\\\\\nThus,$Pr(w_i^r> \\frac{\\log (m)}{2^q})$\\\\\n$=Pr(w_i^r>E(w_i^r)(1+\\delta ))$, where $(1+\\delta)\\frac{\\log (m)}{2^r}=\\frac{\n\\log (m)}{2^q}$ \\\\\n$\\leq (\\frac{e^\\delta}{(1+\\delta)^{1+\\delta}})^{\\frac{\\log (m)}{2^r}}$\\\\\n$=e^{\\delta \\frac{\\log (m)}{2^r}-(1+\\delta )\\ln(1+\\delta )\\frac{\\log (m)}{2^r}}$,  where $(1+\\delta )\\frac{\\log (m)}{2^r}=\\frac{\\log (m)}{2^q}$\\\\\n$=e^{\\frac{\\log (m)}{2^q}-\\frac{\\log (m)}{2^r}-\\frac{\\ln(1+\\delta)\\log (m)}{2^q}}  \\text{  since } (1+\\delta )\\frac{\\log (m)}{2^r}=\\frac{\\log (m)}{2^q} \\text{   and  hence  }  \\delta \\frac{\\log (m)}{2^r}=\\frac{\\log (m)}{2^q}-\\frac{\\log \n(m)}{2^r}$\\\\\n$\\leq e^{\\frac{\\log (m)}{2^q}(1-ln(1+\\delta))}$\\\\\nTo bound the probability inverse polynomial in m,\nwe need to choose$\\frac{\\log (m)}{2^q}(ln(1+\\delta)-1) \\geq O(\\log (m))$\\\\\nor $\\ln(1+\\delta) \\geq 2^q+1$\\\\\nor $1+\\delta  \\geq 2^{2^q+1}$\\\\\nor $ 2^r \\geq 2^q 2^{2^q+1}$\\\\\nThus it suffices to run for $\\overset{~}{T_r}=\\frac{2^{r+\\log (n)-\\log \n(log (m))}.2^{r+\\log (n)-\\log (\\log (m))}}{n^2 \\gamma^2}$ iterations so as to have $w_i^r$ close to O($\\frac{\\log (m)}{2^q}$), with high probability, where r=$2^{2^q}.2^q$\\\\\n\\end{proof}\n\nTo prove a bound on the error, \nwe proceed similar to the analysis in Section \\ref{}. \nConsider a fixed constraint $C_j:V_j.\\bar{x} \\leq 1$.\n\\begin{lemma}For all constraints the error in $q^{th}$ phase after p* is \n$\\overset{\\sim}{\\delta_q} \\leq \\delta_{q+\\log(n)-\\log(\\log(m))}=2^{\\frac{q}{2}} \\cdot 2^{2^q+1}$\n\\begin{align*}\n\\text{Consider } &Pr(\\vert < \\gamma (\\sum_{i=\\overset{\\sim}{T_{r-1}}+1}^{\\overset{\\sim}{T_{r}}} U_i),V_j>\\vert \\geq \\beta_q)\\\\ \n&=Pr(\\vert < \\sum_{i=\\overset{\\sim}{T_{r-1}}+1}^{\\overset{\\sim}{T_{r}}} U_i, \\frac{V_j}{\\| V_j \\|}> \\vert \\geq \\frac{\\beta_q}{\\gamma \\| V_j \\|}).\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nNow $<U_i,\\frac{V_j}{\\vert \\vert V_j \\vert \\vert}> \\sim \\mathcal{N}(0, \\sigma^2)$ where $\\sigma^2 \\leq 1$.\\\\\nThus the above is bounded by $e^{-\\frac{\\beta_q^2}{\\gamma^2 \\vert \\vert V_j \\vert \\vert ^2 2 (\\overset{\\sim}{T_r}-\\overset{\\sim}{T_{r-1}})}}$\nThus for the $q+\\log (n)-\\log(\\log (m))$ phase,we will need to choose $\\beta_q$ satisfying:-\n$\\frac{\\beta_q^2}{\\gamma^2 \\vert \\vert V_j \\vert \\vert ^2 2 (\\overset{\\sim}{T_j}-\\overset{\\sim}{T_{j-1}})} \\geq \\log (m)$\\\\\ni.e. if $\\beta_q \\geq \\gamma \\vert \\vert V_j \\vert \\vert \\sqrt{2.\\log (m).(\\overset{\\sim}{T_j}-\\overset{\\sim}{T_{j-1}})}$\\\\\nSince $\\vert \\vert V_j \\vert \\vert \\leq \\sqrt{\\frac{\\log n}{2^i}}$ and $\\overset{\\sim}{T_j}=\\frac{2^{q+\\log (n)-\\log (\\log (m))} 2^{2^q+1}.2^{q+\\log (n)-\n\\log (\\log (m)) 2^{2^q+1}}}{n^2 \\gamma^2}$,\\\\ \nit suffices to choose \\begin{align*}\n \\beta_q &\\geq 2^{q\/2}.2^{2^q+1}\n\\end{align*}\nFor $q \\leq \\log (\\log (\\log (m)))-2$, the error is bounded by $2^{\\frac{\\log\n(\\log (\\log (m)))-2}{2}}.2^{2^{\\log (\\log (\\log (m)))-2}}=\\sqrt{\\log \n(m).\\log (\\log (m))}$\\\\\nThis leaves the number of unfixed random variables bounded by O($\\frac{\\log \n(m)}{2^q}=\\frac{\\log (m)}{\\log (\\log (m))}$).Increasing $q$ \nbeyond this value will not give any advantage since the error will then exceed the error due obtained by setting variables to 1.\n\\end{proof}\n\n\\subsection{Proof of the  general approximation bound: Proof of Theorem \\ref{mainthm3} }\n\nIf we are interested in maintaining feasibility of constraints, we can\nensure it by sacrificing the value of the objective function according to\nsome trade-offs. The same algorithm works, where we use the known method \nof damping the probabilities of rounding (\\cite{RT:87,AS:95}).\n\nLet $X_1 , X_2 \\ldots X_u$ denote the unabsorbed random \nvariables in a constraint\nafter the Brownian walk phase and let $ U = \\sum_i X_i  = \\frac{B}{\\beta}$\nfor some $\\beta \\geq 1$. Let $\\hat{X_i}$ be the $\\{0, 1\\}$ random variables\nwhere $\\Pr [ \\hat{X_i} =1] = X_i$ and so $\\mathbb{E} [ \\sum_i  \\hat{X_i} ] = \n\\sum_i X_i = \\frac{B}{\\beta}$. To apply LLL, we need\n$\\Pr [ \\sum_i \\hat{X_i} > B ] < \\frac{1}{d}$.\n\nSince $U = \\sum_i \\hat{X_i}$, then from Chernoff bounds, we know that \n\\begin{equation}\n \\Pr [ U \\geq (1+ \\Delta)\\mathbb{E}[ U ] \\leq {\\left[ \\frac{e^\\Delta }{ { (1+ \n\\Delta )}^{ 1+ \\Delta }} \\right]}^{E[U]} \n\\label{chernoff}\n\\end{equation}\nUsing $\\mathbb{E} [ U ] = \\frac{B}{\\beta}$ and $\\beta = 1+ \\Delta $ we obtain\n$\\Pr [ U \\geq B ] \\leq {\\left[ \\frac{e^{\\beta -1}}{ {(\\beta )}^{\\beta}}\\right]\n}^{B\/\\beta} \\leq {\\left( \\frac{e}{\\beta} \\right)}^B \\leq \\frac{1}{\\alpha d}$\nfor some scale parameter $\\alpha \\geq 1$.\nIt follows that \n\\begin{equation}\n {( \\beta \/e )}^{B} > d\\alpha \n\\label{damp_req}\n\\end{equation}\nTherefore $\\beta = e {(d \\alpha)}^{1\/B}$.\nSince our matrices are 0-1, the\nminimum infeasibility happens at $B+1$, so we can substitute $B+1$ instead\nof $B$ in the previous calculations to obtain $\\beta = e {(d \\alpha)}^{1\/B+1}$.  \nUsing $S \\geq 1$ as the initial scaling for Brownian walk,\nwe obtain the following result using Lemma \\ref{err_bnd}\n\n\\begin{lemma}\nFor $B \\geq 1$,\nwe can round the fractional solution to a feasible integral solution\nwith objective function value $\\Omega ( OPT \/S )$ for S=max(${(\\frac{m}{OPT}})^\\frac{1}{B-1},d^\\frac{1}{B-1}$).\nIf $B \\in \\mathbb{Z}$,B can be replaced by B+1 in above exprssion and proof \n\\label{colbndapprox}\n\\end{lemma}\n\\begin{proof}\nIf we start from $\\overset{\\sim}{x}=\\frac{x'}{S}$, the solution produced by brownian walk satisfies the constraints with RHS  \n$\\frac{B}{\\beta}$(as shown in Lemma $\\ref{err_bnd}$).\\\\\nNow we define events for LLL similar to the proof of Lemma \\ref{bnd_col}.\\\\\n$\\forall 1 \\leq i \\leq m$ define $E_i\\equiv (A_i^T.x >B)$.\\\\\nand define $E_{m+1}\\equiv c^T.x <(1-\\epsilon).OPT$.\\\\\nAs shown above, $\\forall 1\\leq i \\leq m$, $Pr(E_i)\\leq (\\frac{e}{\\beta})^B$.\\\\\nAlso by chernoff's bound $Pr(E_{m+1})\\leq e^{{-\\epsilon^2}{OPT\/2}}$.\\\\\nWe apply LLL with weights $y_i=\\frac{1}{\\alpha d}$, for $1\\leq i \\leq m$\\\\\nand $y_{m+1}=\\frac{1}{2}$.\\\\,\nfor some appropriately defined $\\alpha \\geq 1$.\\\\\nNow we need $(\\frac{e}{\\beta})^B \\leq \\frac{1}{\\alpha.d}(1-\\frac{1}{\\alpha d})^d.\\frac{1}{2}$\\\\\nand $e^{\\frac{-\\epsilon^2 OPT}{2 \\beta}} \\leq \\frac{1}{2}.(1-\\frac{1}{\\alpha .d})^m \\leq \\frac{1}{2}.e^{-\\frac{m}{\\alpha.d}}$(From Equation (\\ref{eqnobj}))\\\\\nThe above is satisfied for $\\frac{OPT}{\\beta}\\geq \\frac{m}{\\alpha.d}$\\\\\nThus we need to choose $\\frac{OPT}{(\\alpha.d)^\\frac{1}{B}}\\geq \\frac{m}{\\alpha.d}$\\\\\nOR $\\beta=(\\alpha.d)^{1\/B} \\geq (\\frac{m}{OPT})^{\\frac{1}{B-1}}$.\\\\\nAlso $\\alpha \\geq 1$.For $B \\in \\mathbb{Z}$, it would have sufficed to use $B'=B+1$ as opposed to choosing B which proves the stated lemma.\n\\end{proof}\nTo prove Theorem \\ref{mainthm3}, we can observe that for $c_i \\geq \\frac{1}{p}$, we have $OPT \\geq \\frac{n}{kp}$.\nFor $\\frac{mk}{n}=O(\\log(m))$, we get the scaling as $max((\\log(m))^\\frac{2}{B},(p \\log(m))^\\frac{1}{B})$, which proves Theorem \\ref{mainthm3}\\\\\n\n\n\\color{black}\n\n\n\\section{Applications of our rounding results}\n\\label{sec6}\n\nIn this section we briefly sketch the applications of our rounding theorems.\nAlthough these do not significantly improve prior results,\nour parameterization could simplify further applications.\n\\ignore{For some of the applications, we need a weighted version of Theorem\n\\ref{mainthm2} that we state without proof - which follows from our\nearlier applications of LLL.\n\\begin{lemma}\nFor $A \\in {\\{0,1 \\}}^{m \\times n}$ with a maximum of\n$\\rho$ 1's in each column, we can round\nthe optimum solution $x^{*}$ of the linear program\n$\\max_{x} \\sum_i c_i x_i \\ \\ s.t. A x \\leq 1 \\ \\ , 0 \\leq\nx_i \\leq 1 $ and\n $1 = \\max_i c_i \\geq c_i \\geq 1\/\\alpha \\mbox{ for some } \\alpha \\geq e $\nto $\\hat{x} \\in {\\{ 0, 1 \\}}^n$ such that\n\\[\n{|| A \\hat{x}||}_\\infty \\leq \\left( \\frac{\\log \\rho + \\log \\alpha + \\log\\log n}{\\log\\log\n(\\rho \\alpha \\log n)}\\right) \\ \\ \\text{ and } \\sum_i c_i \\hat{x}_i \\geq\n(1 - \\epsilon ) OPT \\text{ for } OPT \\geq \\Omega ( m\/(\\alpha d ))\n\\label{bnd_col}\n\\]\n\\end{lemma}\n}\n\n\\subsection{Application to Switching circuits}\n\nConsider an $n$-input butterfly network (with $n\\log n$ total nodes) \nwith $( s_i , t_i ) \\ \\ i \\leq n\\log n$ source destination\npairs where each input\/output node has $\\log n$ sources and $\\log n$ \ndestinations. \nFor any arbitrary instance of routing, we can do a two phase\nrouting with a random intermediate destination, say, we choose a random\nintermediate destination $r_i$ for the source-destination pair $(s_i , t_i )$. \nWe want to route a maximum number of pairs subject to some edge capacity\nconstraints. \n\nIf $n = 2^L$, then the the expected\ncongestion $k$ of an edge for a random permutation is $\\log n \\cdot \n\\frac{ 2^{\\ell} \\times\n2^{L - \\ell} }{2^L} = \\log n$ and moreover it can be \nbounded by $c\\log n$ with high probability for some constant $c$. \nSo there exists a fractional solution with flow value $\\frac{1}{\\log n}$\nfor each of the $n\\log n$ paths with objective function value $n$. \n\nLet $A$ be an $m \\times t$ matrix where $m$ is the number of edges\nand $t = n\\log n$ is the number of paths. The edges are denoted \nby $e_1 , e_2 \\ldots e_m$\nand the paths are denoted by $\\Pi_1 , \\Pi_2 \\ldots \\Pi_t$. Then \n$A_{i,j} =1$ iff \nthe flow $\\Pi_j$ passes through edge $e_i$. The number of edges in a path\nis bounded by $L = \\log n$. The value of the flow through \npath $\\Pi_j$ (\ndenoted by $f_j$) is the amount of flow and let $\\bar{f}$ be the\nvector denoting all the flows. Then $A \\cdot f \\leq \\bar{c}$ where $\\bar{c} =\n( c_1 , c_2 \\ldots c_m )$ is the vector corresponding to the congestion\nin edges $( e_1 , e_2 \\ldots e_m )$.\n\nSince the congestion is bounded by $c \\log n $, $A \\in {\\cal A}^{m \\times \nn\\log n}_{c\\log n}$ and there exists a fractional solution \n$\\bar{f} = ( 1\/c\\log n , 1\/c\\log n \\ldots 1\/c\\log n$ with objective value\n$\\Omega (n)$. Using $\\rho = O(\\log n)$ in \nLemma \\ref{bnd_col}, we can round it to\na 0-1 solution that yields the following result. \n\\begin{theorem}\nIn an $n$ input BBN butterfly network having $2(n \\log n)$ edges, \nwe can route the $\\Omega (n)$ source-sink pairs\nwith congestion $O(\\log\\log n\/\\log\\log\\log n )$. From the capacity \nconstraint, it follows that no node contains more than $O(\\log\\log n)$\nsources or destination. \n\\end{theorem}\nNote that we can also handle weighted objective functions. \n\nThe above result matches the previous results of \\cite{MS:99,CMHMRSSV:98} \nwhich have the advantage of being online. Using $B = \n\\log\\log n\/\\log\\log\\log n$, in Lemma \\ref{colbndapprox} we can obtain an\noptimal bound. Note that, in this case we can match asymptotically \nthe optimal fractional flow of $n\\log\\log n\/\\log\\log\\log n)$ with \n$x_i = \\frac{\\log\\log n}{c \\log\\log\\log n\\cdot \\log n}$.\n\\begin{theorem}\nIn an $n$ input multi-butterfly network having $n\\log n$ edges, \nwe can route $\\Omega (\\frac{n\\log\\log n}{\\log\\log\\log n})$\nflows with congestion $O(\\log\\log n\/\\log\\log\\log n )$ \nwhere each node can be the source or sink of at most \n\\\\\n$O(\\log\\log n\/\\log\\log\\log n )$ flows.\n\\end{theorem}\nThe above result marginally extends a similar result of Maggs and Sitaraman\n\\cite{MS:99} where they can route $(n\/\\log^{1\/B} n)$ pairs in the online\ncase. It remains an open problem to find a fast online implementation of\nour rounding algorithms. \n\\subsection{Maximum independent set of rectangles}\n\nConsider a $\\sqrt{n} \\times \\sqrt{n}$ grid and a set of axis-parallel\nrectangles that are aligned with the grid points,i.e., the upper-left\nand the lower-bottom corners are incident on the grid\\footnote{We can\nassume that all the grid-points that are flush with the sides are\nin the interior by slightly enlarging the rectangle}. Each rectangle\ncontains $A$ grid points for some $A \\leq n$ but they can have any\naspect ratio. Trivially, the different types of such rectangles can be\nat most $A$ where $A = a \\cdot b$ for $a = 1, 2 \\ldots A$. Two such\nrectangles $r_1$ and $r_2$ are {\\it overlapping} if $r_i \\cap r_2$\ncontains one or more grid points. \n\nFrom the $A$ possible rectangles whose upper-right corners are anchored\nat a specific grid point $p$, the input consists of one such rectangle.\nTo avoid messy calculations, we assume that the grid is actually embedded on\na torus. Given a set $S$ of $n$ such rectangles, our goal is to select a\nlarge non-overlapping set of rectangles. Although this is a restricted\nversion of the general MISR problem it still captures many applications. \n\nFor any given grid point $p$, let $S(p)$ denote the set of rectangles\ncontaining $p$. This can be bounded by $\\sum_{a = 1}^{A} a \\times A\/a = \nO( A^2 )$. After solving the relevant packing linear program, the\n$n \\times n$ point-rectangle matrix ${\\cal A}$ contains 1 in the $(i,j)$\nposition if the $i$-th grid point is incident on the $j$-th rectangle.\nFrom our previous observation, no row contains more than $A^2$ 1's. \n\nBy setting $k = A^2$, we can apply our rounding results to obtain a wide\nrange of trade-offs for this problem. For example, if $A$ is polylog($n$), \nthen using $\\rho = A$ in Lemma \\ref{bnd_col}, \nwe can choose $\\Omega (OPT)$ rectangles with no more than \n$O(\\log\\log n\/ \\log\\log\\log n )$ overlapping rectangles on any clique, \nwhere $OPT$ is the fractional\noptimal solution of the corresponding packing problem. \n\nFurther, using a result of \\cite{ENO:04}, we\ncan obtain $\\Omega ( \\frac{OPT \\log\\log\\log^2 n}{\\log\\log^2 n })$ \nnon-overlapping rectangles as well as a $\\frac{\\log\\log^2 n}\n{\\log\\log\\log^2 n}$ approximate solution in the weighted case.\n\\subsection{$b$-matching in random hypergraphs}\n\nIn a hypergraph $H$ on $n$ vertices has hyper-edges defined by subsets of \nvertices.\nWe wish to choose the maximum set of hyper-edges such that no more than\n$b$ hyperedges are incident on any vertex. An integer linear program\ncan be written easily for this problem with $n$ constraints and $m$\nvariables corresponding to each of the $m$ edges. Let $A$ be an $n \\times m$\nmatrix where $A_{i,j} = 1$ if vertex $i$ is incident on the $j$-th hyperedge.\nLet $x_i = \\{ 0,1 \\}$ depending on if the $i$-th hyperedge is selected in the\n$b$-matching.\n\\[ \\textbf{Maximize} \\sum_i x_i \\ \\ s.t. \\ A x \\leq b \\cdot e^m \\ \\ x_i \\in\n\\{ 0, 1 \\} \\] \nThe relaxed LP can be solved and the solution may be rounded. The weighted\nversion can be formulated similarly by associating weights $w_i$ for $x_i$.\n\nIf the $m$ edges of $H$ are random \nsubsets of vertices, then we can represent the fractional \nsolution as an $n \\times\nm$ random matrix with each entry of the matrix set to 1 with probability\n$k\/n$. That is, each hyperedge has $k$ vertices and chooses $k$ vertices \nrandomly. So the expected number of edges incident on a vertex is \n$\\frac{mk}{n}$. There exists a feasible fractional solution using\n${x_i} \\ \\ 1 \\leq i \\leq m = \\frac{n}{mk}$ with objective value $OPT \\geq n\/k$.\n\nFor $b = \\Omega ( \\log\\log n\/\\log\\log\\log n )$, we can can obtain a \n$b$ matching of\nsize $\\Omega (OPT)$ for {\\it most} hypergraphs which matches the best\npossible size given by the fractional optimum. In general, we obtain \nan approximation $k^{1\/b}$ for $b \\geq 2$ using the result of Theorem\n\\ref{mainthm3} including the weighted version. \n\nIt is known that $k$-uniform $b$-matching problem cannot be approximated\nbetter than $\\frac{k}{b\\log k}$ for $b \\leq k\/\\log k$ unless \n$P = NP$ \\cite{OFS:11,HSS:06}. So our\nresult shows that the bound can be much better for many $k$-uniform\nhypergraphs, for example $k = \\log n$ and $b \\geq 2$. \n A closely related result for bounded column case was observed by\nSrinivasan \\cite{AS:96}.\n\n\\section{Concluding Remarks}\n\n\n\n\\subsection{A general approximation bound :Proof of theorem \\ref{mainthm3}}\n\nIf we are interested in maintaining feasibility of constraints, we can\nensure it by sacrificing the value of the objective function according to\nsome trade-offs. The same algorithm works, where we use the known method \nof damping the probabilities of rounding (\\cite{RT:87,AS:95}).\n\nLet $X_1 , X_2 \\ldots X_u$ denote the unabsorbed random \nvariables in a constraint\nafter the Brownian walk phase and let $ U = \\sum_i X_i  = \\frac{B}{\\beta}$\nfor some $\\beta \\geq 1$. Let $\\hat{X_i}$ be the $\\{0, 1\\}$ random variables\nwhere $\\Pr [ \\hat{X_i} =1] = X_i$ and so $\\mathbb{E} [ \\sum_i  \\hat{X_i} ] = \n\\sum_i X_i = \\frac{B}{\\beta}$. To apply LLL, we need\n$\\Pr [ \\sum_i \\hat{X_i} > B ] < \\frac{1}{d}$.\n\nSince $U = \\sum_i \\hat{X_i}$, then from Chernoff bounds, we know that \n\\begin{equation}\n \\Pr [ U \\geq (1+ \\Delta)\\mathbb{E}[ U ] \\leq {\\left[ \\frac{e^\\Delta }{ { (1+ \n\\Delta )}^{ 1+ \\Delta }} \\right]}^{E[U]} \n\\label{chernoff}\n\\end{equation}\nUsing $\\mathbb{E} [ U ] = \\frac{B}{\\beta}$ and $\\beta = 1+ \\Delta $ we obtain\n$\\Pr [ U \\geq B ] \\leq {\\left[ \\frac{e^{\\beta -1}}{ {(\\beta )}^{\\beta}}\\right]\n}^{B\/\\beta} \\leq {\\left( \\frac{e}{\\beta} \\right)}^B \\leq \\frac{1}{\\alpha d}$\nfor some scale parameter $\\alpha \\geq 1$.\nIt follows that \n\\begin{equation}\n {( \\beta \/e )}^{B} > d\\alpha \n\\label{damp_req}\n\\end{equation}\nTherefore $\\beta = e {(d \\alpha)}^{1\/B}$.\nSince our matrices are 0-1, the\nminimum infeasibility happens at $B+1$, so we can substitute $B+1$ instead\nof $B$ in the previous calculations to obtain $\\beta = e {(d \\alpha)}^{1\/B+1}$.  \nUsing $S \\geq 1$ as the initial scaling for Brownian walk,\nwe obtain the following result using Lemma \\ref{err_bnd}\n\\begin{lemma}\nFor $S \\geq c'\\cdot {(d\\alpha )}^{1\/(B+1)}$ , and $B \\geq 1$,\nwe can round the fractional solution to a feasible integral solution\nwith objective function value $\\Omega ( OPT \/S )$ for $OPT \\geq \\Omega \n( mS\/(\\alpha d)))$. \n\\label{colbndapprox}\n\\end{lemma}\n The last condition in the Lemma follows since Equation (\\ref{eqnobj}) is \nsatisfied. \n\nIt would be worthwhile to compare and contrast our results and techniques\nwith \\cite{AS:95,AS:96}. For the case of the number of 1's bounded\nby $\\rho$ in a 0-1 matrix\\footnote{Our techniques and proofs must be\nmodified to work for more general matrices having entries from $[0,1]$} $A$, an objective value $OPT\/ {\\rho}^{1\/B}$  \nwas obtained which is marginally superior to our result of\n$OPT\/{(\\rho \\log n)}^{1\/(B+1)}$.\nStill, we believe, that our techniques are \neasier to use and build upon for various applications compared to the FKG\ninequality based techniques. Following the Brownian motion, the number\nof non-zero coefficients in any constraint is at most $O(\\log n)$ but it\nis not clear how this can be exploited by FKG based analysis of \\cite{AS:95,\nBS:00} which is independent of the number of columns.\n\nFor weighted objective function with $c_i \\geq 1\/p$, we can choose\nthe value of the scaling factor $\\alpha$ more carefully so that $p =\n\\alpha^{1 - 1\/B'} \\cdot {(\\log n)}^{1 -2\/B'} $, and $mk\/n \\leq \\log m$,\nusing $S = {(\\alpha \\log^2 n )}^{1\/B'}$ gives an objective value\nof at least $ nB'\/(p Sk)  \\geq n\/(k \\alpha \\log n) $\nwhich satisfies condition \\ref{eqnobj}. Note that $S = {( p \\log n)}^{1\/(B'-1)}\n$ from the above parameter settings.\\\\\nBy subsituting back $B+1$ for $B'$ in the previous calculations,\n$S = {( p \\log n)}^{1\/B}$. So for $B = 1$, the approximation is\n$O(p\\log n)$.\nThis completes the proof of Theorem \\ref{mainthm2}. \\\\[0.1in]\n\n\\section{Introduction}\nMany combinatorial optimization problems can be modeled\nusing a weighted packing integer program.\n$ \\mbox{ Maximize } \\sum_{i=1}^{i=n} c_i \\cdot x_i$\nsubject to\n$ \\sum_{ i \\in S_j } x_i \\leq 1 \\ \\ 1 \\leq j \\leq m \\ \\ \\ \n x_i \\in \\{0, 1 \\}$.\\\\ \nWlog, we can assume that $\\max_i c_i = 1$.\nAlthough the above formulation appears somewhat restrictive\nhaving a fixed right hand side of each inequality, the\nmethods that we develop extend to more general parameters.\nOften the constraints are expressed as \n$ C_j: V_j \\cdot x \\leq 1$ \nwhere $V_j$ is a 0-1 incidence vector corresponding to set $S_j \\subset\n\\{ 1, 2\\ldots n \\}$. We will also use $x_i$ to denote the $i$-th coordinate\nof a vector $x$.\nSince this version is also NP-hard as it captures many intractable\nindependent set problems,\na common strategy is to solve the Linear Program (LP)\n corresponding to the relaxation $0 \\leq x_i \\leq 1$. Then\nuse the LP optimum $OPT$ to obtain a good approximation of the optimal\nintegral solution. \nSuppose the optimum is achieved by the\nvector $x' = [ x'_1 , x'_2 \\ldots x'_n ]$ \nIn the conventional {\\it randomized rounding} \\cite{RT:87}, we \nround each $x_i$ independently in the following manner.\\\\ \n$\\hat{x}_i =\n\\{ 1 \\text{ with probability $x'_i$ and } \n0 \\text{ otherwise} \\}$.\n\\\\\nSince $\\mathbb{E}[ \\hat{x}_i ] = x'_i$, it follows that $\\mathbb{E} [ \\sum_i c_i \\hat{x}_i ] \n=  \\sum_i c_i \\cdot x'_i $ and using Chernoff bounds, we can show that for\nall $j$, w.h.p. $ \\sum_{ i \\in S_j } \\hat{x}_i \n\\leq \\left(\\frac{\\log m}{\\log\\log m}\n\\right) $.\n\nThe primary motivation for improved randomized rounding\nis improved approximation of\ninteger linear packing problems. Historically, the original randomized\nrounding technique was proposed for obtaining better approximation of \nmulticommodity\nflows \\cite{RT:87}. \nSrinivasan \\cite{AS:95,AS:99} presents an extensive survey of many \nsophisticated variations of\nrandomized rounding techniques and applications to approximation algorithms.\nIt was established much later \\cite{CGKT:07}\nthat the Raghavan-Thompson bound cannot be improved as a consequence of the\nfollowing result.\n\\begin{lemma}[\\cite{CGKT:07}]\nThere exists a constant $\\delta > 0$ such that the integrality gap of the \nmulticommodity flow relaxation problem is $\\Omega ( 1\/c' \\cdot \nn^{\\frac{1}{3c' + 13}} )$ for any congestion $c'$, $1 \\leq c' \\leq \n(\\delta \\log n)\/\n\\log\\log n $ where $n$ is the number of vertices in the graph and the \nintegrality gap for $c'$ is with respect to congestion 1. \n\\end{lemma}\n\nTherefore no rounding algorithm can simultaneously achieve error\n$c' = o( \\log n\/\\log\\log n)$, \nand an objective function value of $\\Omega ( OPT )$ for all\npolynomial size input ($m \\leq n^{O(1)}$). \nNote that this does not preclude improvement of the multicommodity flow problem\napproximation by alternate formulation - for this, there are alternate \nhardness bounds given by \\cite{CGKT:07}. \n\nSubsequent to the work of Raghavan-Thompson, Srinivasan \\cite{AS:95,AS:96}\nBaveja and Srinivasan \\cite{BS:00}, Kolliopoulous and Stein \\cite{KS:98}\nobtained results that focussed on circumventing the above bottleneck\nby making use of some special properties of the constraint matrices like\ncolumn-restricted matrices. In \nparticular, the focus shifted to designing approximation algorithms that\nare feasible (unlike the basic independent rounding of RT) by bounding\ndependencies between inequalities and using {\\it Lovasz Local Lemma} (LLL)\nor even more sophisticated corelation inequalites like FKG \\cite{AS:96}.\nLeighton, Rao and Srinivasan \\cite{LRS:98} made very clever use of\nLLL to achieve nearly optimal results in graphs with {\\it short} flow paths.\nThe notion of short flow paths was developed further in the context of \nthe {\\it unsplittable} \nmulticommodity flow problem with an explicit objective function in many\npapers including \\cite{KR:96,BS:00,KS:06,CCGK:07}. Intuitively, short\nflow paths reduce dependencies between conflicting routing\npaths and makes the algorithms more efficient in terms of assigning paths\nwithout exceeding the maximum allowed congestion (for edge-disjoint paths\nit is 1). \n\nIn this paper, we attempt to provide a uniform framework behind many prior\nuses of independent rounding by recasting the process as\na Brownian walk in high dimension and analyzing its convergence. For readers\nfamiliar with this technique, it may appear to be an overkill since the\nfinal outcome of Brownian motion and independent rounding are identical \n(we shall formalize this in the next section). However, we will demonstrate\nthat this framework offers a cleaner exposition and simpler explanations \nof many of the previous clever, yet {\\it ad-hoc} techniques. \nWe have presented\na detailed characterization of the time-dependent behavior of the random variables \nassociated with each inequality executing Brownian motion which was not studied before\nto the best of our knowledge.   \nIt could potentially offer new tools for analysis of more complicated rounding\nmethods where the random variables may not be independent. \n\nWe demonstrate two distinct applications of our new framework\nby addressing the rounding problem for the {\\it average} case of\nan input family that we will define more precisely. \n\\\\\n(i) A simple approximation algorithm for {\\it weighted} objective\nfunction. \n\\\\\n(ii) An improvement of the Raghavan-Thompson rounding error for\na restricted class of weighted objective function.\n\nAs opposed to\nthe one-shot rounding of \\cite{RT:87} , also referred as {\\it independent\nrounding}, our rounding is iterative and based on successive refinements\nof the LP solution that has a lower variance per step, to yield a sparser\nset of constraints. This reduces dependencies between inequalities\nand makes the use of techniques like Lovasz Local Lemma (LLL) more effective. \n\\subsection{Main results and some applications}\n\\begin{theorem}\nGiven an $m \\times n$ 0-1 matrix $A$, such that $A \\cdot x' \\leq b$ for \n$x'_i \\in [0,1]$,\nand an objective function $\\sum_i c_i x_i$, \nin randomized polynomial time $x'$ can be transformed into $y'$ which \nare $[0,1]$ valued random variables such that for all $(\\log n - \\log\\log m \n+ \\log b ) \\geq p \\geq 0$\\\\\n(i) All constraints have $ \\leq \\frac{nB}{2^p}$ non-zero variables from \n$y'$ and\n\\\\\n(ii) For all $i$, $\\sum_i A_i \\cdot y' \\leq O(\\sqrt{\\frac{2^p B \\log m}{n}})$ where \n$A_i$ is the $i$-th row of $A$.\n\\\\\n(iii) $\\mathbb{E} [ c_i y'_i ] = \\sum_i c_i x_i$\n\\label{sparsify}\n\\end{theorem}\nIn other words we have a sparse system of equivalent constraints with the objective\nfunctional value unchanged in the expection.\n\\\\\n\n{\\bf Remark} For $p = \\log n - \\log\\log m + \\log B$, there are $\\log m$ \nvariables in each inequality summing upto $O(1)$. This specific result has\nbeen observed before by using a clever application of independent rounding\nin the following manner. Round each $x'_i < \\frac{1}{\\log m}$ to \n$\\frac{1}{\\log m}$ with probability $x'_i \\log m$. Then $\\mathbb{E} [x_i] =\nx'_i$ and no inequality has more than $\\log m$ variables with high probability.\n\\ignore{\nOur transformation does not use any extra variables unlike the \nprevious methods that makes multiple copies of each variable \n(see for example \\cite{CC:09}). \nThey transform the initial matrix $A$ to an $m\\times n\\log n$ matrix $A'$ and \n$x'$ is transformed into $y'$ that has dimension $n\\log n$, using \nanother application of independent rounding (some papers present it as \na technique to do $\\frac{1}{\\log n}$ rounding) so\nthat the fractional solution remains feasible. Subsequently,\ncare has to be taken to ensure that the rounded solution chooses exactly\none copy of a variable which is an implicit case of dependent rounding. \nThe {\\it path decomposition} strategy for multicommodity flow \nin the original paper of Raghavan and Thompson \\cite{RT:87} is also \nalong similar lines.  While it works for specific instances, it will be\ntechnically challenging to formalize it as a {\\it blackbox} \ntechnique similar to the previous theorem.    \n}\n\nUsing the above sparsification, we obtain a number of interesting \nresults arguably in simpler ways than known previously.  \n\\ignore{\n\\begin{theorem}\nLet $A \\in {\\{0,1\\}}^{m \\times n}$, with no more than $\\rho$\n1's per column.\nSuppose $x' \\in {[0,1]}^n$ maximizes $\\sum_i c_i x_i $ and\nsatisfies $ A \\cdot x \\leq \\cdot e^m $ with $OPT = \\sum_i c_i x'_i$.\nThen $x'$ can be rounded to $\\hat{x} \\in {\\{0,1 }\\}^n$ in polynomial time \nsuch that\nthe following holds with high probability, \n\\begin{eqnarray}\n {|| A \\hat{x}||}_\\infty & \\leq & 1\n\\text{ and } \\\\\n\\sum_i c_i \\hat{x}_i & \\geq & \\Omega ( OPT\/(\\rho \\log m) )\n\\end{eqnarray}\n\\label{mainthm2}\n\\end{theorem}\n\\vspace{-0.3in}\nThe above result follows by applying a simple greedy procedure to the \nsparsified matrix.\n\nFor a more general RHS, $b \\geq 1$, we use LLL-based techniques to obtain the following\nresults.\n}\n\\begin{theorem}\nLet ${\\cal A}^{m \\times n}_k$ denote the family of $m \\times n$ \nmatrices where each row (independently) has $k$ ones in randomly \nchosen columns from $\\{ 1, 2, \\ldots n \\}$ and 0 elsewhere\\footnote{Alternately\nwe can set every entry to be 1 with probability $k\/n$ independently}. \nLet $A \\in \n{\\cal A}^{m \\times n}_k$, then\nfor any point $x' = ( x'_1 , x'_2 \\ldots x'_n ) , \\\n0 \\leq x'_i \\leq 1$ such that\n$ A \\cdot x' \\leq e^m $, where $e^m$ is a vector of $m$ 1's,\nand $OPT = \\sum_i c_i \\cdot x'_i$ where \n$1 = \\max_i \\{ c_i \\}$ and\n$\\forall i\\ c_i \\geq \\frac{1}{p}$. Then,\n$\\bar{x'}$ can be rounded to $\\hat{x} \\in {\\{0,1 }\\}^n$ such that\nthe following holds with probability $\\geq 1 - \\frac{1}{m}$\n\\begin{eqnarray}\n{|| A \\hat{x}||}_\\infty & \\leq & O\\left(\\frac{\\log(mkp\\log m\/n) + \\log\\log m }{\\log\\log (mkp\\log m\/n + \\log m)}\\right)\n\\text{ and } \\label{cond1}\\\\\n\\sum_i c_i \\hat{x}_i & \\geq & \\Omega ( OPT )\n\\end{eqnarray}\nMoreover such an $\\hat{x}$ can be computed in randomized polynomial time.\n\\label{mainthm0-1}\n\\end{theorem} \n{\\bf Remark} \n(i) The result relates the rounding error to the average number of ones\nin a column, i.e., $\\frac{mk}{n}$ and characterizes tradeoffs between the\nparameters $m, n, k$. For example, for $k = \\sqrt{n}$, we can bound the error\nto $O(\\log\\log n)$ for $m \\leq \\sqrt{n} \\cdot polylog(n)$ for\n$p  \\leq \\frac{1}{\\log^{O(1)} n}$.\nThis is a significant improvement over the $O(\\frac{\\log n}{\\log\\log n})$\nbound of the independent rounding. \n\\\\\n(ii) For $k = \\log n$, the bound is indeed tight.\nA proof is given in the appendix (\\cite{V:15}).\n\\\\\n(iii) For the unweighted case, the result holds for arbitrary distribution \nof the $k$ 1's in each row.\\footnote{ \nSet each variable to 1 with\nprobability $1\/k$, and for each violated constraint that has more than $q =\n\\log ((mk)\/n )$\nones, zero all its variables (or zero enough of its variables, chosen\narbitrarily, so that it has only $q$ ones). At most $n\/2$\nof the variables are heavy in the sense that they appear in\nmore than $2mk\/n$ constraints. For a light variable, even if it comes up 1, the\nprobability that it is a member of a violated constraint is small (say, below\n1\/2), and hence the light variables (using linearity of expectation) guarantee\na value of $\\Omega(n\/k)$.\\\\\nThis proof sketch was given by an anonymous reviewer of an earlier version.\n}\n\\begin{theorem}\nLet $A \\in {\\cal A}^{m \\times n}_k$,\nsuch that  $x' \\in {[0,1]}^n$ maximizes $\\sum_i c_i x_i $ and\nsatisfies $ A \\cdot x \\leq b \\cdot e^m $ with $OPT = \\sum_i c_i x'_i$.\nIf $1 = \\max_i c_i \\geq 1\/p ,\\ \\ p \\geq 1$ \nand $m \\leq \\frac{n\\log m}{k}$,\nthen $x'$ can be rounded to $\\hat{x} \\in {\\{0,1 }\\}^n$ in polynomial time \nsuch that the following holds with high probability for $b \\geq 1$\n\\begin{eqnarray}\n {|| A \\hat{x}||}_\\infty & \\leq & b\n\\text{ and } \\\\\n\\sum_i c_i \\hat{x}_i & \\geq & \\Omega ( OPT\/(max(p \\log m ,\\log^2(m))^{1\/b})) \n\\end{eqnarray}\n\\label{mainthm3}\n\\end{theorem}\n\\vspace{-0.3in}\n\\color{black}\nWe observe a trade-off between the\nweights and the approximation factor ${(p \\log m ))}^{1\/b}$. For $b = O(\n\\frac{\\log\\log m}{\\log\\log\\log m})$, we can obtain $O(1)$ approximation for \nfor $c_i \\geq \\frac{1}{\\log^C n}$ for any constant $C \\geq 1$. \n\nThe random matrices provides a natural framework for combinatorial\nresults related to random hypergraphs. We sketch one such application to\n$b$-matching of $k$-regular hypergraphs that yields \nan approximation $k^{1\/b}$ for $b \\geq 2$ using the result of Theorem\n\\ref{mainthm3} and\nextends a result of Srinivasan \\cite{AS:96}. This implies that \nfor $b = \\Omega ( \\log\\log n\/\\log\\log\\log n )$, we can can obtain a\n$b$ matching of\nsize $\\Omega (OPT)$ for {\\it most} hypergraphs which matches the best\npossible size given by the fractional optimum. \nIt is known that $k$-uniform $b$-matching problem cannot be approximated\nbetter than $\\frac{k}{b\\log k}$ for $b \\leq k\/\\log k$ unless\n$P = NP$ \\cite{OFS:11,HSS:06}. \n\n\\ignore{For lack of space, we have sketched details of the other applications\nin the Appendix, section \\ref{sec6}.}\n\\subsection{An overview and related work}\nOur algorithm can be best characterized as a randomized iterated rounding where we begin\nfrom a fractional feasible (specifically optimal) solution $x'$ and iteratively\nconverge to a {\\rm good} integral solution. Our algorithm\nhas two distinct stages - {\\it random walk} stage (more precisely, Brownian walk) and \nsubsequently in the second stage invokes the Moser-Tardos iterative scheme \nfor constructive Lovasz\nLocal Lemma (LLL). In the first stage we effectively \n{\\it slow down} the RT rounding \nprocess. \nOur approach is intuitive  - starting from $x'$, for each\nvariable (dimension), we will roughly increment \n(actually a normal Gaussian increment)\n$x_i$ by $\\pm \\gamma$ for\na suitably chosen $1 > \\gamma > 0$ where the sign (direction) is a\nrandom variable. In each iteration,\nthe values of $x_i$'s are modified and we continue this\nprocess for each $x_i$ until it is in the range $[0, \\delta ] \\cup\n[1- \\delta,1]$ for an appropriate $1 > \\delta > 0$ such that\n$\\delta > \\gamma$. \nAt this point we {\\it fix} the variable\nand we terminate when all inequalities have less than some predetermined \nvalue $u$ of {\\it unfixed} variables . \nThis stage has some similarities with the method \nof Lovette and Meka \\cite{LM:12} but our analysis requires completely \ndifferent techniques.\nThe crux of the method called {\\it partial coloring lemma}\nis a rounding strategy of an arbitrary $ x \\in {[-1 , +1]}^n$ vector within\nthe discrepancy polytope defined by the constraints starting with\n$x = (0,0, \\ldots 0 )$.\nTheir method can be mapped to\n$\\{ 0, 1\\}$ rounding as well, that was observed by Rothvoss\\cite{R:13}.\nCompared to the setting of the discrepacy rounding, we are dealing with\nsmaller error margins and the variable and polytope constraints are\nnot widely separated in terms of distances from the starting point.\nMoreover, one needs\nto also account for the deviation of the objective function which is\nnot required for Spencer's discrepancy result.\n\n\\ignore{\nAt a high level, our framework provides an alternate \ntechnique for limiting dependencies. In the remark following Theorem \n\\ref{mainthm0-1} we noted that the rounding error is related to the average\nnumber of 1's in a column which is related to the path-lengths in switching\ncircuits (c.f. section \\ref{sec6}). \nWe show that $\\Omega (\\frac{n \\log\\log n}{ \\log\\log\\log n})$ \nconnections can be supported in a \n{\\it multi-butterfly} network with congestion\nbounded by $O(\\log\\log n\/\\log\\log\\log n )$, extending a result of \n\\cite{MS:99,CMHMRSSV:98} where they achieve a similar congestion for $n$ flows. \n\nWe also present approximation algorithms for the Maximum Independent Set of\nRectangles (MISR) problem where the rectangles are aligned with grid points.\nThe rectangles can have arbitrary aspect ratios but their areas are bounded.\nThis version can be useful for many applications including map labelling\nwhere the bounding boxes are not arbitrarily large and do not intersect \ntoo many cliques. For the case that the rectangles contain at most\npolylog grid points, we present a\n$O(\\frac{\\log\\log^2 n}{\\log\\log\\log^2 n})$ approximation bound for the \nweighted version. Obtaining an approximation factor $o(\\frac{\\log n}\n{\\log\\log n})$ \nbeen an important open problem (\\cite{CC:09,CH:09}) and has \npartly motivated the recent work in \nquasi-polynomial time approximation algorithms \\cite{AW:13}.   \n\nFurther, the random matrices provides a natural framework for combinatorial\nresults related to random hypergraphs. We sketch one such application to\n$b$-matching of $k$-regular hypergraphs that yields \nan approximation $k^{1\/b}$ for $b \\geq 2$ using the result of Theorem\n\\ref{mainthm2} and\nextends a result of Srinivasan \\cite{AS:96}. This implies that \nfor $b = \\Omega ( \\log\\log n\/\\log\\log\\log n )$, we can can obtain a\n$b$ matching of\nsize $\\Omega (OPT)$ for {\\it most} hypergraphs which matches the best\npossible size given by the fractional optimum. \nIt is known that $k$-uniform $b$-matching problem cannot be approximated\nbetter than $\\frac{k}{b\\log k}$ for $b \\leq k\/\\log k$ unless\n$P = NP$ \\cite{OFS:11,HSS:06}. \n\nFor lack of space, we have sketched details of the above applications\nin the Appendix, section \\ref{sec6}.\n\\subsection{Prior related work in randomized rounding}\n\nSubsequent to the work of Raghavan-Thompson, Srinivasan \\cite{AS:95,AS:96}\nBaveja and Srinivasan \\cite{BS:00}, Kolliopoulous and Stein \\cite{KS:98}\nobtained results that are similar in spirit to this paper. A direct\ncomparison of these results is a complex exercise \nsince they had not addressed this specific framework of random matrices. \nLater, we\nwill be able to compare \nthem with respect to our result on column-restricted \n0-1 matrices that we use as an intermediate step\nto prove the main results. Although, this intermediate\nresult falls a little short of the best known, we feel that our techniques\nare simpler and more general that could open up a new paradigm for rounding \nmore general integer programs.\n\nThere exists a very extensive and rich literature on the use of randomized\nrounding for solving various generalizations \nof the edge-disjoint path problem starting\nwith the seminal paper of Raghavan and Thompson. Leighton and Rao \\cite{LR:88,\nLR:99} formalized many algorithmic an combinatorial \naspects of the multicommodity flow problem. \nThe problem of edge\ndisjoint paths in expander graphs was resolved by Bohman and Frieze\n\\cite{BF:01}. \n\n\n\nThe recent work on constructive discrepancy perhaps comes\nclosest to our technique of randomized iterative rounding. \nBansal's \\cite{bansal:10} seminal\npaper on a constructive proof of Spencer's discrepancy theorem is based\non rounding solutions of successive\na semi-definite program that captures the discrepancy constraints.\nThis method was further refined and simplified in the work of\nLovette and Meka\\cite{LM:12} who\nderived an alternate proof\nbased on a very elegant analysis of a multidimensional random walk.\nThe crux of the method called {\\it partial coloring lemma}\nis a rounding strategy of an arbitrary $ x \\in {[-1 , +1]}^n$ vector within\nthe discrepancy polytope defined by the constraints starting with\n$x = (0,0, \\ldots 0 )$. \nTheir method can be mapped to\n$\\{ 0, 1\\}$ rounding as well, that was observed by Rothvoss\\cite{R:13}.\nCompared to the setting of the discrepacy rounding, we are dealing with\nsmaller error margins and the variable and polytope constraints are \nnot widely separated in terms of distances from the starting point. \nMoreover, one needs\nto also account for the deviation of the objective function which is\nnot required for Spencer's discrepancy result.\n}\n\n\\subsection{A Lower Bound on error}\n\\begin{lemma}\nFor $A \\in {\\cal A}^{m\\times n}_{\\log n}$ such that $A\\cdot \\bar{x} \\leq e^m$,\nwe cannot simultaneously obtain $\\sum_i \\hat{x}_i = \\Omega ( n\/\\log n )$ and\n${|| A \\hat{x}||}_\\infty  \\leq o(\\frac{\\log\\log n}{\\log\\log\\log n})$\nfor $\\hat{x}_i \\in \\{ 0, 1 \\}$.\n\\label{lbnd}\n\\end{lemma}\n\n\\begin{proof}\n\nAs mentioned before, it is easy to see that $x_i = 1\/k$ is a solution \nfor $\\bar{x}$ with objective\nfunction value $n\/k$. After rounding $\\bar{x}$ we want to have at least \n$\\Omega ( n\/k )$ 1s in the rounded vector, say $\\bar{y}$.\nWe must have $A \\cdot \\bar{y} \\leq t\\cdot e$ for some rounding guarantee\n$t$. Let $b$ be a fixed vector with $n\/k$ 1s. \n\nLet $r$ be a row vector with $k$ 1s in random location - this corresponds to\na row of $A$. The probability that $< r , \\bar{y} > \\ \\ \\geq t$ is given by\n \\[  \\frac{{ (n\/k) \\choose t} \n\\cdot { (n-n\/k) \\choose (k-t) }}{{n \\choose k }} \\]\nUsing $ {(\\frac{n}{k})}^k \\leq {n \\choose k} \\leq {(\\frac{ne}{k})}^k $, the \nabove expression is\n\\begin{eqnarray}\n \\geq & \\frac{ {( \\frac{n}{tk})}^t \\cdot {(\\frac{n -n\/k}{k-t})}^{k-t}}{\n{(\\frac{ne}{k})}^k} \\\\\n = & \\frac{ \\frac{1}{t^t} \\cdot {( \\frac{k-1}{k-t})}^{k-t}} {\n e^k } \\\\ \n = & \\frac{ k^{k-t}}{ {( k-t )}^{k-t} \\cdot e^k \\cdot t^t }\\\\\n = & \\frac{1}{ {( 1 - t\/k )}^{k-t} \\cdot e^k \\cdot t^t } \\\\\n \\geq & \\frac{1}{{( 1 - t\/k )}^{k\/2} \\cdot e^k \\cdot t^t } \\text{ for } k \\gg t\\\\\n = & \\frac{1}{ e^{ k - t\/2} \\cdot t^t }\n\\end{eqnarray}\nSo the probability that for $m$ independently chosen rows, the probability that\n$< r , \\bar{y} > \\ \\ < t$ is \n\\begin{eqnarray}\n  p(m,k,t) &  \\leq & {\\left( 1 - \\frac{1}{ e^{ k - t\/2} \\cdot t^t } \\right)}^m \n\\leq  {\\left( 1 - \\frac{1}{ e^{ k } \\cdot t^t } \\right)}^m =\n{\\left( 1 - \\frac{1}{ e^{ k } \\cdot t^t } \\right)}^{e^k t^t \\frac{m}{e^k t^t}}\n\\\\\n & \\leq & {\\left( \\frac{1}{e} \\right)}^{\\frac{m}{e^k t^t}} \n\\end{eqnarray} \nSince there are no more than ${n \\choose\n(n\/\\log n)}$ choice of columns with $n\/\\log n$ 1s, the probability that all\nthe dot products are less than $t$ is less than ${n \\choose (n\/\\log n)} \\cdot\np(m,k,t)  \\leq p(m,k,t) \\cdot {(\\frac {n e}{n\/\\log n})}^{n\/\\log n} \n \\leq e^n \\cdot p(m,k,t)$. If $e^n \\cdot p(m,k,t) < 1$ then there must exist\nmatrices in ${\\cal A}^{m \\times n}_k$ that do not satisfy the error bound $t$.\nSo,\n\\[    1  > {\\left( \\frac{1}{e} \\right)}^{\\frac{m}{e^k t^t}} \\cdot e^{n} \n    \\Rightarrow {( e ) }^{\\frac{m}{e^k t^t}}  >  e^n \\] \n which is equivalent to the condition \n\\begin{equation}\n \\frac{m}{n}  > e^k t^t \\text{  or  }\n\\log(m\/n) > k + t\\log t \n\\label{lbndreq}\n\\end{equation}\n \n For $k = \\log n$ and $m = n\\cdot polylog(n) $, this holds for some $t \\geq \n\\frac{\\alpha \\log \\log n}{\\log\\log\\log n})$, i.e., for some constant \n$\\alpha > 0$, i.e., \nthe error cannot be\n$o (\\frac{\\log\\log n}{\\log\\log\\log n})$. \n\\end{proof}\n\n\n\n\\subsection{A general approximation bound }\n\nIf we are interested in maintaining feasibility of constraints, we can\nensure it by sacrificing the value of the objective function according to\nsome trade-offs. The same algorithm works, where we use the known method \nof damping the probabilities of rounding (\\cite{RT:87,AS:95}).\n\nLet $X_1 , X_2 \\ldots X_u$ denote the unabsorbed random \nvariables in a constraint\nafter the Brownan motion phase and let $ U = \\sum_i X_i  = \\frac{B}{\\beta}$\nfor some $\\beta \\geq 1$. Let $\\hat{X_i}$ be the $\\{0, 1\\}$ random variables\nwhere $\\Pr [ \\hat{X_i} =1] = X_i$ and so $\\mathbb{E} [ \\sum_i [ \\hat{X_i} ] = \n\\sum_i X_i = \\frac{B}{\\beta}$. To apply LLL, we need\n$\\Pr [ \\sum_i \\hat{X_i} > B ] < \\frac{1}{d}$.\n\nSince $U = \\sum_i \\hat{X_i}$, then from Chernoff bounds, we know that \n\\begin{equation}\n \\Pr [ U \\geq (1+ \\Delta)\\mathbb{E}[ U ] \\leq {\\left[ \\frac{e^\\Delta }{ { (1+ \n\\Delta )}^{ 1+ \\Delta }} \\right]}^{E[U]} \n\\label{chernoff}\n\\end{equation}\nUsing $\\mathbb{E} [ U ] = \\frac{B}{\\beta}$ and $\\beta = 1+ \\Delta $ we obtain\n$\\Pr [ U \\geq B ] \\leq {\\left[ \\frac{e^{\\beta -1}}{ {(\\beta )}^{\\beta}}\\right]\n}^{B\/\\beta} \\leq {\\left( \\frac{e}{\\beta} \\right)}^B \\leq \\frac{1}{\\alpha d}$\nfor some scale parameter $\\alpha \\geq 1$.\nIt follows that \n\\begin{equation}\n {( \\beta \/e )}^{B} > d\\alpha \n\\label{damp_req}\n\\end{equation}\nTherefore $\\beta = e {(d \\alpha)}^{1\/B}$.\nUsing $S \\geq 1$ as the initial scaling for Brownian motion,\nwe can obtain the following result using Lemma \\ref{err_bnd}\n\\begin{lemma}\nFor $S \\geq c'\\cdot {(d\\alpha )}^{1\/B}$ , and $B \\geq 1$,\nwe can round the fractional solution to a feasible integral solution\nwith objective function value $\\Omega ( OPT \/S )$ for $OPT \\geq \\Omega \n( mS\/(\\alpha d)))$. \n\\label{colbndapprox}\n\\end{lemma}\n The last condition in the Lemma follows from Equation \\ref{eqnobj} is \nsatisfied. \nThis completes the proof of Theorem \\ref{mainthm2}.\n\nIt would be worthwhile to compare and contrast our results and technqiues\nwith \\cite{AS:95,AS:96}. For the case of the number of 1's bounded\nby $\\rho$ in a 0-1 matrix $A$, an objective value $OPT\/ {\\rho}^{1\/B}$  \nis obtained which is marginally superior to our result of\n$OPT\/{(\\rho \\log n)}^{1\/B}$. Still, we believe, that our technqiues are \neasier to use and build upon for various applications compared to the FKG\ninequality based technqiues. Following the Brownian motion, the number\nof non-zero coefficients in any constraint is at most $O(\\log n)$ but it\nis not clear how this can be capitalized by FKG based analysis.\n\n\n\n\\subsection{Random matrices}\n\\label{sec5}\n\nLet ${\\cal A}^{m\\times n}_k$ denote the family of \n$m \\times n$ 0-1 matrix with exactly $k$ 1s in each of\nthe $m$ rows chosen uniformly at random. \nClearly $x'_i = 1\/k$ is a feasible solution with objective\nfunction value $n\/k$. After rounding $x'$ to $\\hat{x}$, \nwe want to achieve an objective value\n$\\Omega ( n\/k )$ 1s in the rounded vector. \nIn addition, $A \\cdot x' \\leq t\\cdot e^m$ for error guarantee\n$t$. \n\nTo compute the dependency $d$ for $C_r$, we observe that \nanother constraint $C_i$ will contain a 1 in $j_1 (r)$ with \nprobability $\\frac{k}{n}$, \ni.e., if it had one of the $k$ randomly chosen 1's in that column, which\nis $\\frac{k}{n}$. Since all the rows were chosen independently, the expected\nnumber of rows among $m$ rows that have a 1 in column $j_1 (r)$ can\nbe bounded by $\\frac{m k}{n}$ and by $O(\\max\\{ \\frac{m k }{n} , \\log m \\} )$ \nwith probability greater than $1 - 1\/m$. Since this holds for all the positions,\n$j_i (r)$, by the union bound, and \nusing $\\max \\{ a , b \\} \\leq (a+b)$, we can claim the following\n\\begin{claim}\nThe total number of constraints that are \ncorrelated to $C_r$ can be\nbound by $O(\\frac{m k \\log m}{n} +  \\log^2 m )$ with high \nprobability.\n\\label{deg_bnd}\n\\end{claim}\nIn our context, for $k = \\log m$, $d = O( \\frac{m\\log^2 m}{n} + \n\\log m ^2)$. \nWe can verify this by \nexhaustively computing the dependence - if it exceeds this then our algorithm\nis deemed to have failed which is bounded by inverse polynomial probability.\nIf we choose $t$ such that $e \\cdot 2^{-t} \\cdot (d+1) \\leq 1$ or\nequivalently $t = \\Omega ( \\log (\\frac{m\\log^2 m}{n}+ \\log^2 m  ) ) $, \nthen we can apply\nthe previous theorem to obtain a rounding that satisfies an error bound of\n$O( \\log (\\frac{m\\log^2 m}{n}+ \\log^2 m ) )$. \\\\\nFor $m$ bounded by \n$n \\log^{O(1)} m$ this is\n$O(\\log\\log m)$ which is substantially better than the Raghavan-Thompson\nbound.  \\\\However for $m \\geq n^{1 + \\epsilon}$ for any constant $\\epsilon > 0$,\nit is no better.\n\\begin{proof} (Completing the Proof of Theorem \\ref{mainthm0-1})\nNow we formally define our bad events $E_i$ here.\\\\\n$\\forall 1\\leq i \\leq m$, define $E_i\\equiv(A_i^T x > \\delta )$.(where we choose error to be $\\delta$).\nDefine $E_{m+1}\\equiv(c^T.x <(1-\\epsilon)OPT)$.\nWe have $Pr(E_i)\\leq \\frac{e^{\\delta}}{{1+\\delta}^{1+\\delta}}$.\nand $Pr(E_{m+1})<e^{\\frac{-\\epsilon^2.OPT}{2}}$.\nWe choose $y_i=\\frac{1}{\\alpha.d}$,for some $\\alpha\\geq1$.\nand choose $y_{m+1}=\\frac{1}{2}$.\nTo apply LLL we need,\n$Pr(E_i)<\\frac{1}{\\alpha.d}(1-\\frac{1}{\\alpha.d})^d.(1-\\frac{1}{2})$\nand $Pr(E_{m+1})<\\frac{1}{2}(1-\\frac{1}{\\alpha.d})^m$.\nTo satisfy these equations we get error,$\\delta=\\frac{\\log(\\alpha.d)}{\\log(\\log(\\alpha.d))}$.\\\\\nAlso we get $e^{\\frac{-\\epsilon^2.OPT}{2}}<e^{\\frac{-m}{\\alpha.d}}$.\nThe second equation is satisfied for $OPT>\\frac{m}{\\alpha.d}$ or $\\alpha.d >\\frac{m}{OPT}$.\nCombining the 2 results we have error $\\delta$ in constraints as $max\\left\\{\\frac{\\log(d)}{\\log(\\log(d))},\\frac{\\log(\\frac{m}{OPT})}{\\log(\\log(\\frac{m}{OPT}))}\\right\\}$\\\\which is $max\\left\\{\\left( \\frac{\\log (\\frac{mk\\log(m)}{n}) +  \\log\\log m}{\\log\\log \n(\\frac{mk \\log(m)}{n} \\log m)}\\right),\\frac{\\log(\\frac{m}{OPT})}{\\log(\\log(\\frac{m}{OPT}))}\\right\\}$.\nWhen $c_i \\geq \\frac{1}{p}$ we have $OPT \\geq \\frac{n}{k.p}$.\nSubstituting the value of OPT in the equation and using the fact that $\\frac{mk}{n}=O(\\log(m))$, we prove Theorem \\ref{mainthm0-1}. \n\\ignore{\n\\color{red}\n\n Then, using \n$d \\leq \\frac{mk\\log m}{n}$ (Claim \\ref{deg_bnd}), $\\exp (- m\/(d\\alpha ) \\leq\n\\exp (- n\/(\\alpha k\\log n))$ so the condition \\ref{eqnobj} is \nsatisfied for LLL to be applicable. In general, Equation \\ref{eqnobj} can be\nsatisfied when the objective function value $OPT$ \nexceeds $\\frac{n}{\\alpha k\\log n}$ for choosing a suitable $\\alpha \\geq e$. \n\n\\subsection{Putting it together}\nFor a random $m \\times n$ matrix $A \\in {\\cal A}^{m \\times n}_k$ \nfor $k \\geq \\log n$ \nin $A^k$, we first apply the Brownian\nwalk based algorithm to sparsify it to $\\log m$ unfixed variables\nper constraint. \n\n\nFor weighted objective function, observe that Equation \\ref{eqnobj} \ncan be satisfied for value of the objective function at least $\\Omega \n(n\/(k\\log n)$. This can be ensured by the condition $c_i \\geq 1\/\\log n$.\nMore generally, for $c_i \\geq 1\/(\\alpha \\log n)$, \nwe can use a scale factor $\\alpha \\geq 1$ to obtain\nan error bound of $O(\\log (d \\alpha) \/\\log\\log (d \\alpha))$ with\nobjective value $\\Omega ( n\/(k\\alpha \\log n))$.\\\\ \nThe rest follow from LLL and the algorithm of Moser-Tardos.} The expected \nrunning time for the second phase is $O( \\frac{ m^2 }{n} polylog (n))$ in\nour case, due to application of Moser Tardos.\\\\ \nHere the error bound may fail for the additional reason that\nthe dependence may exceed $\\frac{m k polylog (n)}{n}$ after the Brownian \nmotion phase. This can happen with probability $\\frac{1}{m^{\\Omega (1) }}$. \nThis part is related the input distribution of ${\\cal A}^{m\\times n}_k$ and\nrepeating the algorithm may not work.\n\\end{proof}\n\n\n{\\it Remark} A direct application of LLL (without running the Brownian \nmotion) would increase the dependence to $O(\\frac{m k^2 \\log n}{n} +\\log m \n\\log n )$. In order to maintain the same asymptotic error bound, the number\nof rows in the matrix, $m$, could be significantly less. For example, if\n$m,k = n^{1\/2}$, then the difference would be $O(\\log n\/\\log\\log n)$\nversus $O(\\log\\log n\/\\log\\log\\log n )$.  \n\nThe proof of Theorem \\ref{mainthm3} is along similar lines after appropriate \nscaling and is given in the appendix.\n\n\\ignore{\nTo obtain the approximation factor of \nTheorem \\ref{mainthm2} we use the previous scaling technique with\n$d = \\alpha \\max\\{ \\frac{e mk u}{n} , u\\log m\\}$, where $u$ is the\nnumber of unabsorbed variables in any constraint and $\\alpha \\geq 1$ is an\nappropriate scaling factor. Let $B' = B+1$, then \n\\[ \\beta = e \\alpha^{1\/B'} \\cdot \\max\\{ {(\\frac{e mk u}{n})}^{1\/B'} ,\n {(u\\log m)}^{1\/B'} \\} = e \\cdot {(\\alpha u)}^{1\/B'} \n\\cdot \\max\\{ {(\\frac{e mk }{n})}^{1\/B'} ,\n {(\\log m)}^{1\/B'} \\} . \\]\n\n\nFor weighted objective function with $c_i \\geq 1\/p$, we can choose\nthe value of the scaling factor $\\alpha$ more carefully so that $p = \n\\alpha^{1 - 1\/B'} \\cdot {(\\log n)}^{1 -2\/B'} $, and $mk\/n \\leq \\log m$,\nusing $S = {(\\alpha \\log^2 n )}^{1\/B'}$ gives an objective value\nof at least $ nB'\/(p Sk)  \\geq n\/(k \\alpha \\log n) $ \nwhich satisfies condition \\ref{eqnobj}. Note that $S = {( p \\log n)}^{1\/(B'-1)} \n$ from the above parameter settings.\\\\  \nBy subsituting back $B+1$ for $B'$ in the previous calculations,\n$S = {( p \\log n)}^{1\/B}$. So for $B = 1$, the approximation is\n$O(p\\log n)$.\nThis completes the proof of Theorem \\ref{mainthm2}. \\\\[0.1in]\n}\n\\color{black}\n\n\\section{Brownian walk analysis}\n\nWe will divide the analysis into two components - \nFirst we will compute the rate at which the variables are absorbed at 0. \nSecond, we compute the increments of all variables during each iteration that \ncauses an inequality to be violated. \nThis causes the right hand side of any inequality to \nbehave as a martingale and we will refer to it as the {\\it error}. \n Note that the increase in {\\it error} is related to the number of \nunabsorbed variables as they execute random walk. Once all variables are\nabsorbed, then the error doesn't change. For subsequent applications, we\nwill derive the bounds for starting position scaled by $S \\geq 1$, i.e.,\nfrom $\\frac{x'}{S}$. Readers who are familiar with \n\\cite{LM:12} may note that our analysis focuses on variables associated with\neach constraint as opposed to the global number of variables.\n\\begin{lemma}\n\nLet $x' \\in \\mathbb{R}^n$ be a feasible solution to $A\\cdot x \\leq B\\cdot \\vec{1}^m$, \n$A \\in {\\{0,1\\}}^{m\\times n}$, and \n$\\overset{\\sim}{x}=\\frac{x'}{S}$, $S \\geq 1$ be chosen as the starting point \nfor brownian walk. Then,\n after $T_p=\\frac{2^p. 2^p}{n^2 \\gamma^2}$ steps, with probability \n$\\geq 1-\\frac{1}{m^{\\Omega(1)}}$, the number of unfixed variables in $i^{th}$ \nconstraint is $O(\\frac{n B}{S \\cdot 2^p}$) \nfor $2^p \\leq \\frac{n B}{S  (\\log (m)))}$\\footnote{For $m$ polynomial in $n$\nwe need not distinguish between $\\log m$ and $\\log n$}.\n\\footnote{Applying a union bound over all constraints, we satisfy the conditions of the lemma for each of the phases and constraints with high probability}\n\\label{absorp_rate}\n\\end{lemma}\n First, we can\nuse Lemma \\ref{boost-prob-absorb} to get a bound on the probability of \nabsorption.\n\\begin{claim}\nFor $b \\geq r a$,  after $O( \\frac{ r^2 a^2}{\\gamma^2} )$ Brownian motion \nsteps, the probability of non-absorption at 0 is $\\leq O(\\frac{1}{r})$. \n\\label{eq_var_bnd}\n\\end{claim}\nThe above claim is trivially true\nfor $r \\leq 1$.\nIn Lemma \\ref{boost-prob-absorb} , use $k = r $, $b = ra\/\\gamma$ \nthat gives absorption probability $\\leq O(1\/r)$ after $r\\cdot a\/\\gamma \\cdot\na r\/\\gamma = \\frac{ r^2 a^2 }{\\gamma^2 }$ steps.   \n\\begin{proof}(of Lemma \\ref{absorp_rate})\nChoosing $T_p=\\frac{2^p 2^p}{n^2 \\gamma^2}=\\frac{(\\overset{\\sim}{x_i})^2}{\\gamma^2}(\\frac{2^p}{n \\overset{\\sim}{x_i}})^2$,\\\\\nwe can apply Claim \\ref{eq_var_bnd} with $r=\\frac{2^p}{n \\overset{\\sim}{x_i}}$ to obtain that\nthe probability that $x_i$ is not absorbed at 0 is $\\leq \\frac{n \\overset{\\sim}{x_i}}{2^p}$.\nLet $V_i^p\\:x\\leq B$ be the $i^{th}$ constraint after $p$ phases.\n\nIf $u^p_i$\n(= $\\twonormsq{ V_i^p }$) denotes the number of unabsorbed variables in \nconstraint $i$ after $p$ phases then\n$\\mathbb{E} ( u^p_i ) \n\\leq \\sum_{j=1}^{n} \\frac{n \\overset{\\sim}{x_{j}}}{2^p} A_{i,j} \\leq \n\\frac{n B}{S \\cdot 2^p}$ since \n $\\sum_{j=1}^{n} A_{i,j}\\overset{\\sim }{x_{j}} \\leq \\frac{B}{S}$.\n \n Note that the variables are independently executing Brownian walks.\nFor $\\frac{n B}{S \\cdot 2^p} \\geq  \\log{(m)}$ or equivalently, \n$p \\leq \\log ({n})- \\log (\\log (m) )+\\log (B)-\\log (S)$,\n we can apply Chernoff bound to claim that $u_i^p$\nis $O(\\frac{n B}{S \\cdot 2^p})$ with probability $\\geq 1-\\frac{1}{m }$ \n\\end{proof}\n\\begin{cor}\nSince $\\mathbb{E} ( u^p_i ) =  \\mathbb{E} ( \\twonormsq{V_i^p} )$ for $V_i^p = {\\{ 0, 1 \\}}^n$,\nwe can bound $ \\twonormsq{V_i^p} $ by $\\frac{n B}{S \\cdot 2^p}$ with high\nprobability for $p \\leq \\log ({n})- \\log (\\log (m \\log(n)) )+\\log (B)-\\log (S)$.\n\\label{normbnd}\n\\end{cor}\n\\textbf{Remark}\n(i) For $A_{i,j} \\in [0,1]$, the above proof on $\\mathbb{E} ( u_i^p )$ \ndoesn't hold directly but the bound on $ \\twonormsq{V_i^p} $ is still valid.\n\\\\\n(ii) If we proceed upto $p* = \\log ({n})-\\log (\\log (m  ))+\n\\log (B)-\\log (S)-\\log (c)$, all constraints have\nat most $O(\\log m )$ \nunfixed coordinates.\n \n\\subsection{Error Bound}\n\nFrom the previous result, \nfor all $j , \\twonormsq{V_j^p} \\leq  O(\\frac{nB}{S2^p}$) \nwith high probability.\nTo bound the error consider a fixed constraint $C_j:V_j.x \\leq B$,\nwe denote the error accumulated in the $p^{th}$ phase by $\\delta_p$, so\nthat $\\sum_{q=1}^p \\delta_q = \\beta_p$.\n\n\\begin{lemma}\nFor all constraints $C_j$, the error \n $\\delta_p \\leq\n\\sqrt{\\frac{2 B \\log m } {S\\cdot n }} 2^{\\frac{p}{2}} .$\\\\ \nThus the total error $\\beta_p$ upto $\\log n - \\log (\\log m )+ \n\\log (B)- \\log (S)$ is bounded by $c' \\frac{B}{S}$ for some constant $c'$. \n\\label{err_bnd} \n\\end{lemma}\n\\begin{proof}\nConsider $\\Pr(< \\gamma (\\sum_{i=T_{p-1}+1}^{T_{p}}U_i ),V_j>\\vert \\geq \\delta_p )$\\\\ \n=$\\Pr(\\vert < \\sum_{i=T_{p-1}+1} ^{T_p} U_i, \\frac{V_j}{\\vert \\vert V_j \\vert \\vert}> \\vert \\geq \\frac{\\delta_p}{\\gamma \\vert \\vert V_j \\vert \\vert}).$\\\\\n\nNow $<U_i,\\frac{V_j}{\\vert \\vert V_j \\vert \\vert}> \n\\sim \\mathcal{N}(0, \\sigma^2)$ where $\\sigma^2 \\leq 1$. From Lemma \n\\ref{chernoff-mart}, it follows that\nthe above is bounded by $\\exp (-\\frac{(\\delta_p)^2}\n{\\gamma^2 \\vert \\vert V_j \\vert \\vert ^2 2 (T_p-T_{p-1})})$.\nIt will hold simultaneously for all the $m$ constraints in the $p^{th}$ \nphase, if $\\delta_p$ satisfies :-\n\n$\\frac{(\\delta_p)^2}{\\gamma^2 \\vert \\vert V_j \\vert \\vert ^2 \n2 (T_p-T_{p-1})} \\geq \\Omega( \\log m )$\\\\\ni.e. if $\\delta_p \\geq \\gamma \\vert \\vert V_j \\vert \\vert \\sqrt{2 \\cdot\n\\log  m .(T_p-T_{p-1})}$\\\\\nFrom Corollary \\ref{normbnd} $\\twonormsq{V^{p-1}_j } \\leq \\frac{nB}{S\\cdot 2^{p-1}}$ and $T_p=\\frac{2^p 2^p}{n^2 \\gamma^2}$,\\\\ \nit suffices to choose \n \\[ \\delta_p \\geq \\gamma \\sqrt{\\frac{nB}{S\\cdot 2^{p-1}}}\\cdot \n\\sqrt{2 \\log m }\\cdot \\sqrt{\\frac{2^p\\cdot 2^p}{n^2 \\gamma^2}}\n =2 \\sqrt{\\frac{ B \\log m  2^p}{S\\cdot n}}\n\\]\nThe above bound for $\\delta_p$ holds with high probability when $p \\leq \\log n -\n\\log (\\log m )+ \\log (B)- \\log (S)$ since only in this situation \nwe can bound effective value of $V_j$.\\\\\nThus the total error upto $\\log n - \\log (\\log m )+ \\log (B)- \\log (S)$ is \nbounded by \n\\begin{eqnarray}\n\\sum_{p=1}^{\\log n -\\log (\\log m )+\\log (B)-\\log (S)} 2\\sqrt{\\frac{ B\\log m }\n{S\\cdot n}} 2^{\\frac{p}{2}} &\n \\leq & 2\\sqrt{\\frac{ B\\log m }{S\\cdot n}} \\cdot O(2^{\\frac{\\log n - \n\\log (\\log m )+\\ log B - \\log (S)}{2}}) \\\\\n & = & 2\\sqrt{\\frac{ B \\log m }{S\\cdot n}} \\cdot O(\\sqrt{\\frac{nB}\n{S\\cdot \\log m }})=O(\\frac{B}{S})\n\\end{eqnarray}\nTherefore, the total error in every constraint is bounded by\n $<V_j,x^p>=<V_j,\\overset{\\sim}{x}+\\gamma\\sum_{i=1} ^{T_p} U_i>=\nO(\\frac{B}{S})$.\n\\end{proof}\nThus the solution obtained satisfies the constraints \n$A \\cdot x \\leq \\frac{B}{S}$.\n\n The above method can be extended to obtain an alternate proof the \n$O(\\log m\/\\log\\log m)$ error bound of Raghavan-Thompson that we\nhave ommitted from this version. \n\nBased on the above results, we summarize as follows.\n\\begin{cor}\nIn the first stage of Algorithm {\\bf Iterative Randomized Rounding}, we \nrun the algorithm for\n$T = \\frac{B^2}{S^2 \\log^2 m \\gamma^2}$ steps. Then with high \nprobability\\\\ \n(i) all constraints have $ \\leq \\log m$ unfixed variables and\n\\\\\n(ii) the total error in any constraint is bounded by $c'B\/S$ for some\nconstant $c'$. \n\\label{browniansteps}\n\\end{cor}\n\nUsing {\\it multiple copies} of a variable, results similar\nto this section have been used before that are {\\it ad hoc} to specific \napplications \\cite{CC:09,AS:96}.\nHowever, it requires two stages of rounding - once choosing exactly one\ncopy followed by a phase of independent rounding. In comparison, our\ntechnique and analysis are more general. \n \n\n\\section{Applications using LLL}\n\n\\label{short}\n\nAt the end of brownian walk(that is after $\\frac{B^2}{S^2\\log^2m \\gamma^2}$ steps), we have atmost $\\log(m)$ unconverged variables \noccurring(with non zero coefficients) in each equation.\nNow we need to bound the error in independantly rounding the unconverged \nNow we need to bound the error in independantly rounding the unconverged \nvariables.\nAs per the notations setup in section \\ref{sec:algo}, $V_i^{p*}$ represents the \ncoefficient vector of $i^{th}$ constraint after p* phases.\n\\\\\n\nDefine $\\overset{\\wedge}{A}$ as a matrix having rows as $V_i^{p*}$.\n\nNow if the unconverged variables are rounded from $\\overset{\\sim}{x}$ to $\\overset{\\wedge}{x}$,\nthe change in the value of RHS is \n$A\\cdot(\\overset{\\wedge}{x}-\\overset{\\sim}{x})=\\overset{\\wedge}{A} \\cdot (\\overset{\\wedge}{x}-\\overset{\\sim}{x})$.\n(Since only unconverged variables change).\n\nHence to calculate additional error due to independant rounding we only need to consider the \"`sparsified\"' matrix $\\overset{\\wedge}{A}$.\n\\\\\nBefore we prove the main results, we consider the simpler \ncase of columns with bounded number of ones -\nSuppose the matrix $A^{m\\times n}$ has no more than\n$\\rho$ 1's in any column. Let $OPT$ be the optimal fractional \nobjective value for the weighted objective function $\\sum_i c_i \\cdot x_i$. \nConsider a fixed constraint $C_r$, that contains $\\log m$ 1's after the Brownian\nmotion and let $j_1 (r) , j_2 (r) \\ldots $ denote the (at most) $\\log n$\ncolumns that contain 1.\nWe say that a constraint $C_y$ is {\\it dependent } on $C_r$ if they\nshare at least one column where the value is 1.\nSo, the dependency of any single constraint can be bound by $\\rho\\log m$\n(Since there are $\\log m$ 1's per row in the sparsified matrix $\\overset{\\wedge}{A}$).\nIf we use the independent rounding to round the \nfractional solution $x'$, the probability that the value of a constraint\nexceeds $t > 1$ is bounded $\\frac{1}{2^t}$ from Chernoff\nbounds \\footnote{This\nis a slightly weaker version to keep the expression simple}. Let\n$E_i$ denote the event that $C_i$ exceeds $t$ when we use randomized\nrounding.  We are interested to know the\nprobability of the event $\\bigcap_{1 \\leq i \\leq m} \\bar{E_i}$ \nsince this implies the event that all the inequalities are less than $t$.\nThis is tailor-made for Lovasz Local Lemma (LLL).\nWe also want to guarantee a large\nvalue of the objective function. For example, setting all variables equal to\nzero would guarantee feasibility but also return an objective function value 0.\nThus we define an additional event $A_{m+1}$ corresponding to the objective\nfunction value less than $(1- \\epsilon ) \\cdot OPT$ for some suitable $1 >\n\\epsilon > 0$.\nSince $A_{m+1}$ is a function of all the variables, it has dependencies with\nall other $A_i \\ \\ i = 1 \\ldots m$, therefore we have to use the generalized\nversion of LLL in this case. \n\\begin{theorem}[Lovasz Local Lemma \\cite{EL:75}]\nLet $A_i , 1 \\leq i \\leq N$ be events such that $\\Pr [ A_i ] = p$ and each\nevent is dependent on at most $d$ other events. Then if $ep(d+1) < 1$, then\\\\\n$ \\Pr ( \\bar{A_1} \\cap \\bar{ A_2 } \\ldots \\bar{ A_N } ) > 0 $.\n\\\\\nAlternately, in a more general (asymmetric) case, where the dependencies are\ndescribed by a graph $(\\{ 1, 2 \\ldots N \\}, E)$\nwhere an edge between $i, j$ denotes dependency between\n$A_i , A_j$ and $y_i$ are real numbers such that $\\Pr ( A_i ) \\leq\ny_i \\cdot \\prod_{(i,j) \\in E} ( 1 - y_j )$ then\n$ \\Pr \\left( \\bigcap_{i=1}^{N} \\bar{ A_i } \\right) \\geq \\prod_{i=1}^{N}\n( 1 - y_j )$.\\\\\nMoreover, such an event can be computed in randomized polynomial time using\nan algorithm of Moser and Tardos \\cite{MT:10}.\n\\label{lll}\n\\end{theorem}\n\nIf we choose $t$ such that $e \\cdot 2^{-t} \\cdot (d+1) \\leq 1$ or\nequivalently $t = \\log d$, then we can apply\nthe previous theorem to obtain a rounding that satisfies an error bound of\n$O( \\log \\rho + \\log\\log m )$. \nThe error $t$ can be improved to $O(\\frac{\\log d}{\\log\\log d})$\nby using a tighter version of the Chernoff bound (Equation \\ref{chernoff}).\n\n\nWe define $A_{m+1}$\nas the event where the objective value is less than $(1- \\epsilon ) OPT$. From\nChernoff-Hoeffding bounds, we know that $\\Pr ( A_{m+1} ) \\leq \n\\exp ( - \\epsilon^2  OPT\/2 )$.\nWe define $y_i$ for $i = 1, 2 \\ldots m$ as before, corresponding\nto the probability of exceeding $t = \\log d\/\\log\\log d$.\n\\\\ By choosing\n$y_i = 1\/(\\alpha d) \\ i \\leq m$ and $y_{m+1} = \\frac{1}{2}$, \nfor some suitable scaling factor $\\alpha \\geq e$, we\nmust satisfy the following inequalities\n\\begin{eqnarray}\n\\Pr (A_i ) & \\leq & 1\/(\\alpha d) {( 1 - 1\/(\\alpha d) )}^{d} \\cdot \\frac{1}{2} \\ \\  i = 1, 2 \\ldots m \\\\\n\\Pr ( A_{m+1} ) & \\leq & \\frac{1}{2} \n{( 1 - 1\/(\\alpha d) )}^{m} \\leq\n\\frac{1}{2}\\cdot \\exp (- m\/(\\alpha d))\n\\label{eqnobj}\n\\end{eqnarray}\nThe first condition is easily satisfied when $\\Pr ( A_i ) = \\frac{1}{\\Omega \n(\\alpha d)}$.\\\\\nTo satisfy the second inequality, we can choose $\\alpha$\nso that  $OPT \\geq \\Omega ( \\frac{m}{\\alpha d})$. \nThis implies that $\\exp (- m\/(d\\alpha ) \\geq\n\\exp (- \\epsilon^2 OPT\/2))$,  so condition (\\ref{eqnobj}) is\nsatisfied for LLL to be applicable. \\\\\n\n\n\\begin{figure}\n\\fbox{\\parbox{6.0in}{\n\n{\\footnotesize \n\\begin{center}\nAlgorithm {\\bf LLL based Iterative Randomized Rounding }\n\\end{center}\nInput : $x'_i \\ \\ 1 \\leq i \\leq n, t$ (error parameter)\n\\\\\nOutput : $\\hat{x_i} \\in \\{0, 1\\}$ \\\\\n\nDo independent rounding on all the variables having values $x'_i$ to\n$\\hat{x_i}$.\n\\\\\nCompute the value of each constraint $C_i$ as $< V_i , \\hat{x} >$\n\n{\\bf While} any inequality exceeds $t$ {\\bf or} objective value is $< OPT\/2$\n\\begin{quote}\n\\begin{enumerate}\n\\item Pick an arbitrary constraint $C_j$ that exceeds $t$\nand perform independent rounding on all the variables in $V_j$.\n\\item Update the value of the constraints whose variables have changed\n\\end{enumerate}\n\\end{quote}\nReturn the rounded vector $\\hat{x}$.\n}}\n}\n\\caption{An iterative randomized rounding algorithm based on Moser-Tardos}\n\\label{algo2}\n\\end{figure}\n\n\n\nWe summarize our discussion as follows\n\\begin{lemma}\n\nFor $A \\in {\\{0,1 \\}}^{m \\times n}$ with a maximum of \n$\\rho$ 1's in each column, we can round \nthe optimum solution $x^{*}$ of the linear program\n$\\max_{x} \\sum_i c_i x_i \\ \\ s.t. A x \\leq 1 \\ \\ , 0 \\leq \nx_i \\leq 1 $ \nto $\\hat{x} \\in {\\{ 0, 1 \\}}^n$ such that \n\\[\n{|| A \\hat{x}||}_\\infty \\leq max(\\left( \\frac{\\log \\rho +  \\log\\log m}{\\log\\log \n(\\rho \\log m)}\\right),\\frac{\\log(\\frac{m}{OPT})}{\\log(\\log(\\frac{m}{OPT}))}) \\ \\ \\text{ and } \\sum_i c_i \\hat{x}_i \\geq \n(1 - \\epsilon ) OPT   \n\\label{bnd_col}\n\\]\n\\end{lemma}\n\\begin{proof} \nThe error bound follows from the previous discussion by setting $y_i = \\Omega (\n\\frac{1}{\\alpha d})$ and using the stronger form of Chernoff bound in\nEquation \\ref{chernoff}. \nNote that Equation \\ref{eqnobj} can be satisfied by ensuring\n$OPT \\geq \\frac{m}{\\alpha d}$ by choosing an appropriately large $\\alpha$.\\\\\nThe above is satisfied for $\\alpha \\geq \\frac{m}{OPT.d}$ i.e. if  $\\alpha.d=\\frac{m}{OPT}$.\\\\\nIn this case the discrepancy is $\\frac{\\log (\\frac{m}{OPT})}{\\log(\\log(\\frac{m}{OPT}))}$\\\\\nHence the result follows.\n\\end{proof}\n\nThe objective function still follows the martingale property since the\nvariables starting the random walk at $x'_i $ have probability\n$ x'_i $ of being absorbed at 1 which is identical to  the independent\nrounding that we use in the second phase\nfor Moser-Tardos algorithm.\nAlthough, the random walk is short-cut by a single step independent\nrounding, the distribution for absorption at 0\/1 remains unchanged. To\nsee this, consider the {\\it last} time a variable $x_i$ is rounded by the\nMoser-Tardos algorithm - the probability is $x'_i$ since it is done\nindependently every time.\n\n\\ignore{\nThe rest follow from LLL and the algorithm of Moser-Tardos. The expected\nrunning time for the second phase is $O( \\frac{ m^2 }{n} polylog (n))$ in\nour case.\\\\\nThat the error bound may fail if either the\nBrownian walk and the Moser-Tardos algorithm fail\nwith inverse polynomial probability because of the inherent randomization.\nThis probability can be boosted by repetition.\n\\end{proof}\nAs an example, consider $\\rho = \\log m$, $OPT \\geq \\frac{m}{\\rho \n\\log^{c+1} m }$ then for $\\alpha = \\log^{c} m$,\nthe error is $O(\\log\\log m\/\\log\\log\\log m )$.\n\\\\\n{\\it Remark} A direct application of LLL (without running the Brownian\nwalk) would increase the dependence to $O(\\rho \\cdot n)$ resulting in an\nerror bound of $O( \\frac{\\log \\rho + \\log m}{\\log\\log m })$ that is not \nbetter than the RT bound. \n}\n\\input{randmat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA viable Machine Translation (MT) system must have a solution to anaphoric and elliptical ambiguities. It is not viable until it provides mechanism to handle the problem(s) of unidentified words such as names and abbreviations. Earlier, a discourse based approach is used to resolve anaphoric and elliptical ambiguities in Text-based Machine Translation (TBMT) \\cite{1}. To produce high quality translation, the source text is dissected into mono-sentential discourses where complex discourses require further dissection either directly into primitive discourses or first into compound and later into primitive discourses \\cite{2}. The resolution of ambiguities is performed during discourse processing stage \\cite{3}. However, less attention has been given to resolve unknown words in TBMT. This paper improves dissection and discourse processing procedure by providing an algorithm for the resolution of unidentified lexical units. Firstly, we discuss discourse analysis and the dissection of complex and compound discourses into primitive discourses. Secondly, we present our proposed algorithm and analyze its validity by applying it to real world text, i.e., newspaper fragments. Finally, we present conclusion to our work.    \n\n\\begin{table*}[!t]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Primitive discourses and Generalized patterns (First complex discourse)}\n\\label{tab:1}\n\\centering\n\\begin{tabular}{|c|c|}\n\\hline\nPrimitive discourses &\tGeneralized patterns\\\\\n\\hline\nIPEC of ILO &\tA of B\\\\\n\\hline\nIPEC has initiated &\tA has B\\\\\n\\hline\nProject is new &\tA is B \\\\\n\\hline\n(Time\\textunderscore Bound\\textunderscore Program) of (Government\\textunderscore of\\textunderscore Pakistan) is supported &\tA of B is C\\\\\n\\hline\nElimination of WFCL\t& A of B\\\\\n\\hline\nElimination from Pakistan &\tA from C \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}[!t]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Primitive discourses and Generalized patterns (Second complex discourse)}\n\\label{tab:2}\n\\centering\n\\begin{tabular}{|c|c|}\n\\hline\nPrimitive discourses &\tGeneralized patterns\\\\\n\\hline\nProject of IPEC &\tA of B\\\\\n\\hline\nIPEC will provide &\tA will B\\\\\n\\hline\nAssistance is technical &\tA is B\\\\\n\\hline\nProvided to the (Government\\textunderscore of\\textunderscore Pakistan) &\tA to the B\\\\\n\\hline\n(Convention\\textunderscore 182) of ILO &\tA of B\\\\\n\\hline\nImplement on children &\tA on B\\\\\n\\hline\nChildren are working &\tA are B\\\\\n\\hline\nWorking conditions are hazardous &\tA B are C\\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\section{Discourse analysis and discourse unit}\nThe term discourse is introduced by Zellig Harris in 1952. A discourse is a connected piece of text of more than one sentence spoken by one or more speakers \\cite{4}. Bellert defines discourse as a sequence of utterances $S_{1}, S_{2}, S_{3}....S_{n}$ such that the semantic interpretation of each utterance $S_{i}(2\\leq i \\leq n)$ is dependent on the interpretation of the utterances $S_{1}, S_{2}, S_{3}....S_{i-1}$ \\cite{5}. However, interpretation of $S_{i}$ may depend on any subset of the set (say $U$) of previous utterances where $U=S_{1}, S_{2}, S_{3}....S_{n}$ \\cite{2}. We think this definition needs further improvement in the context of cataphora resolution where interpretation of $S_{i}$ may sometimes depend on any subset of the set (say $\\overline{U}$) of the subsequent utterances. A discourse unit is an atomic utterance that has no reference beyond its limitations or boundaries and can be mono-sentential or poly-sentential.     \n\n\\section{Dissection and Discourse processing}  \nThe entire dissection procedure is divided into two phases, i.e., dissection phase and discourse processing phase. In dissection phase, the source text is converted into primitive discourses. These primitive discourses are used to get generalized predicates. In discourse processing phase, various ambiguities are resolved including anaphoric and elliptical ambiguities. Mathematically, the dissection procedure can be represented as\n\n\\begin{equation}\nT = (D_{1}, D_{2}, D_{3}......D_{i}...D_{l})l\\geq1 \n\\end{equation}\n\\begin{equation}\nT=\\sum_{i=1}^l D_{i}\n\\end{equation}\n\nwhere $T$= Source text and $D_{i}$= Poly or Mono sentential Discourse.\n\nThe resolution of the unidentified words is assumed to be processed during the discourse processing stage. The problem of unknown words resulted when we continued to apply the dissection concept to the real world text. For instance, consider the following newspaper fragment\n\n\\emph{[The International Labour Organization ILO's International Programme on the Elimination of Child Labour (IPEC) has initiated a new project to support the Government of Pakistan's Time Bound Programme on the elimination of the Worst Forms of Child Labour (WFCL) from Pakistan. The project will provide technical assistance to the Government of Pakistan to implement ILO Convention 182 on children working in hazardous working conditions.]}.\n\nThe two complex discourses are\n\n1- [The International Labour Organization ILO's International Programme on the Elimination of Child Labour (IPEC) has initiated a new project to support the Government of Pakistan's Time Bound Programme on the elimination of the Worst Forms of Child Labour (WFCL) from Pakistan].\n\n2- [The project will provide technical assistance to the Government of Pakistan to implement ILO Convention 182 on children working in hazardous working conditions].\n\nThe above discourses are dissected into the primitive discourses as given in Table \\ref{tab:1} and Table \\ref{tab:2}.\n\nIt is worth noting that one variable is used for the abbreviations as well as for the compound nouns. For example, Government of Pakistan is treated as a noun. Computer considers it noun by concatenating the text including some special characters (such as \\textunderscore) defined by the programmer. Government of Pakistan is therefore considered as Government\\textunderscore of\\textunderscore Pakistan. Recognizing the correct noun is a challenging task for the computer\/MT scientists. For instance, the Time Bound Program is considered noun from the previous discussion, but it might be treated otherwise. The decision must be based on the context. The concept of dissection is further explained in \\cite{1}.   \n\n\\section{Unidentified lexical units} \n\nCreation of a list or glossary that contains well-known nouns, i.e., abbreviation and names, could be the best solution. The computer should add the new abbreviations or names (as they come along) to the list but sometimes the repetitive uses of nouns create problems for MT system. A pseudo code is written to resolve this problem. The pseudo code could be useful in Question Answering (QA) system, Information Retrieval (IR) system and in MT system, respectively.\n\nThe proposed algorithm updates new names and abbreviations not present in the lexicon. Firstly, the existence of the noun is checked, i.e., whether the noun is correct or not. For instance, if a user enters Paksin Skidn Odind instead of Pakistan State Oil then the computer will subsequently reject it since these words not present in the lexicon. A problem could result if a user enters a word already in the lexicon, but not appropriate for the given abbreviation. For instance, the computer updates wrong abbreviation, if in case Pakistan Supreme Oil is entered instead of Pakistan State Oil. The best solution is that user is allowed to update inappropriate words for the given abbreviation, but these inappropriate words are less likely to be used by another user. Hence, the abbreviations that are less likely to be used are deleted automatically after a month. This could decrease the chances of errors. If these are not automatically deleted, then MT system could give invalid information, i.e., MT system may display Pakistan Supreme Oil instead of Pakistan State Oil.  The flow chart of the pseudo code is given in Fig \\ref{fig:1}. It identifies the existence of noun in a lexicon. It takes primitive discourse as a unit of analysis. If noun doesn't exist, the nature of the noun either abbreviation or name is initially identified. If the noun is abbreviation\/name, it is updated to the lexicon. Otherwise the user is requested to enter the required abbreviation\/name for the updating purposes.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=5in]{fig}\n\\caption{Flow chart of the proposed algorithm}\n\\label{fig:1}\n\\end{figure*\n\n\\section{Evaluation} \n\nThe algorithm is applied to newspaper fragments and substantial numbers of unidentified lexical units are manually resolved. The evaluation of unknown words took place when we applied the dissection procedure to newspaper fragments . During our experiments, 124 names have been updated to the lexicon as compared to 57 abbreviations. Moreover, the number of unknown names appeared more than unknown abbreviations. However, updating new nouns depends on the lexicon. A poor lexicon results in considerable number of unidentified lexical units while a rich lexicon results in fewer unknown words.\n\n\\section{Conclusions} \n\nThis paper explained the resolution of unidentified lexical units in TBMT. The resolution was considered as a part of discourse processing stage where apart from resolving other ambiguities, the resolution of unknown words was also considered. We presented an algorithm, which updates unknown nouns to the lexicon. The presented algorithm takes mono-sentential discourse as an input. The algorithm was manually applied to newspaper fragments and unidentified words were updated to the lexicon. In future this algorithm will be implemented as a part of our dissection model. Additionally, this could also be useful in QA and IR system.\n\\section*{Acknowledgement}\n\nThis research is supported by the MIC (Ministry of Information and Communication) South Korea, under ITRC (Information Technology Research Center) support program supervised by II TA  (Institute of Information Technology Advancement).\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nWhile wave functions of quantum systems may be complex, spectra of their\nenergy eigenvalues must be real, which is usually secured by restricting the\nunderlying Hamiltonian to be Hermitian \\cite{qm}. However, the condition of\nthe reality of the energy spectrum does not necessarily imply that it is\ngenerated by an Hermitian Hamiltonian. Indeed, it is well known that\nnon-Hermitian Hamiltonians obeying the parity-time ($\\mathcal{PT}$) symmetry\nmay also produce entirely real spectra~\\cite%\n{bender1,dorey,bender2,bender3,review,ptqm}. In terms of the single-particle\ncomplex potential,\n\\begin{equation}\nP(\\mathbf{r})\\equiv V(\\mathbf{r})+iW(\\mathbf{r}),  \\label{U}\n\\end{equation}%\nthe $\\mathcal{PT}$ symmetry requires its real and imaginary parts to be even\nand odd functions of coordinates \\cite{bender1}: $V(\\mathbf{r})=V(-\\mathbf{r}%\n),W(-\\mathbf{r})=-W(\\mathbf{r})$,~i.e.,~\n\\begin{equation}\nP(-\\mathbf{r})=P^{\\ast }(\\mathbf{r}),  \\label{minus}\n\\end{equation}%\nwhere the asterisk stands for the complex conjugate. Actually, Hamiltonians\nwhich keep $\\mathcal{PT}$ symmetry may be transformed into Hermitian ones\n\\cite{Mostafazadeh,Barash2,Barash}.\n\nIn the general case, the energy spectrum generated by the $\\mathcal{PT}$%\n-symmetric potential remains real (physically relevant) below a certain\ncritical value of the strength of the imaginary part of the underlying\npotential, $W(\\mathbf{r})$ in Eq. (\\ref{U}), which is a threshold of the $%\n\\mathcal{PT}$ symmetry breaking. Above the critical value, the system is\nmade unstable by emerging imaginary parts of energy eigenvalues. In some\nmodels, the breakup of the $\\mathcal{PT}$ symmetry may follow the onset of\nthe jamming anomaly, which means a transition from increase to decrease of\nthe power flux between the spatially separated gain and loss spots with the\ngrowth of the gain-loss coefficient \\cite{jamming1}. The fragility of the $%\n\\mathcal{PT}$ symmetry essentially limits the use of this property in\napplications, where new effects, such as unidirectional transmissivity \\cite%\n{uni}, enhanced absorption of light \\cite{Longhi}, lasing in microrings \\cite%\n{exp5}, acoustic sensors \\cite{sensors}, as well as the operation of $%\n\\mathcal{PT}$-symmetric metamaterials \\cite{exp4} and microcavities \\cite%\n{exp6} strengthen with the increase of the gain-loss coefficient.\n\nThus far, the $\\mathcal{PT}$ symmetry was not experimentally realized in\nquantum systems, and, moreover, it was argued that, strictly speaking, $%\n\\mathcal{PT}$-symmetric systems do not exist in the framework of the quantum\nfield theory \\cite{Szameit}. On the other hand, a possibility to implement\nthe concept of the $\\mathcal{PT}$ symmetry in terms of classical physics was\npredicted for optical media with symmetrically placed gain and loss elements\n\\cite{theo1}-\\cite{Kominis}, which is based on the similarity between the\nSchr\\\"{o}dinger equation in quantum mechanics and the paraxial-propagation\nequation for optical waveguides. Experimentally, this possibility was\nimplemented in several waveguiding settings \\cite{exp1}-\\cite{exp7}, as well\nas in other photonic media, including exciton-polariton condensates \\cite%\n{exci2,exci3}, and in optomechanical systems \\cite{om}. In these contexts,\nbreaking of the $\\mathcal{PT}$ symmetry was observed. Emulation of the $%\n\\mathcal{PT}$ symmetry was also demonstrated in acoustics \\cite{acoustics}\nand electronic circuits \\cite{electronics}, and predicted in atomic\nBose-Einstein condensates \\cite{Cartarius}, magnetism \\cite{magnetism}, and\nchains of coupled pendula \\cite{Peli}.\n\nThe $\\mathcal{PT}$ symmetry, being a linear feature, is often combined with\nintrinsic nonlinearity of settings in which it is realized. Most typically,\nit is the Kerr nonlinearity of underlying optical media, which gives rise to\nnonlinear Schr\\\"{o}dinger equations (NLSEs) with the cubic term and complex\npotentials, subject to constraint Eq. (\\ref{minus}). Such equations may\ngenerate $\\mathcal{PT}$-symmetric solitons, which were considered in many\ntheoretical works \\cite{soliton}, \\cite{Konotop}-\\cite{Alexeeva} (see also\nreviews \\cite{review1,review2}), and experimentally demonstrated too \\cite%\n{exp7}. Although these works were chiefly dealing with one-dimensional (1D)\nmodels, stable $\\mathcal{PT}$-symmetric solitons were also predicted in some\n2D settings \\cite{Yang}, \\cite{2D-1}-\\cite{2D-3}. A characteristic feature\nof $\\mathcal{PT}$-symmetric solitons is that, although existing in\ndissipative systems, they appear in continuous families, similar to their\ncounterparts in conservative models \\cite{families}, while usual dissipative\nsolitons exist as isolated solutions (\\textit{attractors}, if they are\nstable) \\cite{diss2,diss3}. The realization of the $\\mathcal{PT}$ symmetry\nin 2D geometry may provide essential extension of the above-mentioned\napplications, such as the unidirectional transmission, enhanced absorption,\nand lasing for broad optical beams.\n\nSolitons are also vulnerable to destabilization via the $\\mathcal{PT}$%\n-symmetry breaking at the critical value of the gain-loss coefficient \\cite%\n{breaking}. Nevertheless, it was found that, in some settings, the solitons'\n$\\mathcal{PT}$ symmetry can be made \\emph{unbreakable}, extending to\narbitrarily large values of the strength of the model's imaginary potential\n\\cite{unbreakable}-\\cite{China}, see also a brief review of the\nunbreakability concept in \\cite{book}. The particular property of these\nmodels is that self-trapping of solitons is provided not by the\nself-focusing sign of the nonlinearity, but by the defocusing sign, with the\ncoefficient in front of the cubic term growing fast enough from the center\nto periphery. In the absence of gain and loss, this scheme of stable\nself-trapping was elaborated for 1D, 2D, and 3D bright solitons \\cite%\n{Barcelona1}-\\cite{Barcelona5}. It is essential to stress that such models\nare \\textit{nonlinearizable}, which means that decaying tails of solitons\nare determined by the full nonlinear equation. In other words, the models\nhave no linear spectrum, the spectrum of eigenstates being represented by\nnonlinear self-trapped modes (solitons). Accordingly, the models elaborated\nin Refs. \\cite{unbreakable}-\\cite{China} realize the $\\mathcal{PT}$ symmetry\nin a sense different from that defined in the usual systems---not in terms\nof the linear spectrum, which does not exist in this case, but in the form\nof stable families of complex-valued solitons with real propagation\nconstants (eigenvalues), which exist in the presence of spatially odd\nimaginary potentials.\n\nThe present work introduces 2D models which maintain stable solitons,\nincluding (nearly) unbreakable ones, in the presence of the spatially\ngrowing self-defocusing nonlinearity and antisymmetric imaginary potentials,\n$iW\\left( x,y\\right) $ in Eq. (\\ref{U}). One model, with\n\\begin{equation}\nW(x,y)=\\gamma _{0}x\\exp \\left( -\\beta r^{2}\\right) ,~r^{2}=x^{2}+y^{2},\n\\label{x}\n\\end{equation}%\nwhere $\\gamma _{0}>0$ and $\\beta \\geq 0$ are constants, features the\nunbreakable or nearly unbreakable 2D $\\mathcal{PT}$ symmetry, represented by\nseveral species of families of stable solitons: single- and double-peak\nones, as well as 2D solitons with embedded integer vorticity (topological\ncharge), $m=1,2,3$. The second model, with\n\\begin{equation}\nW\\left( x,y\\right) =\\gamma _{0}xy\\exp \\left( -\\beta r^{2}\\right) ,\n\\label{xy}\n\\end{equation}%\nis not, strictly speaking, a $\\mathcal{PT}$-symmetric one, but it is equally\nrelevant for the realization in optics, and it shares basic manifestations\nof the $\\mathcal{PT}$ symmetry, maintaining families of single- and\nmulti-peak solitons [featuring up to five peaks, in accordance with the\nstructure of $W(x,y)$] and solitary vortices, also with $m=1,2,3$. The\nlatter result is a contribution to the general topic of constructing models\nmore general than the $\\mathcal{PT}$-symmetric ones with similar\nproperties(including the case of the partial $\\mathcal{PT}$ symmetry \\cite%\n{partial}), which has been addressed in various settings \\cite%\n{2D-0,2D-02,2D-03,zezyu,families,Kominis1,Kominis2,Nixon,Kominis3}, see also\nreview \\cite{review1}.\n\nIn both models, universal analytical forms are obtained for tails of\nsolitons, and full exact solutions are produced for particular species of\nsingle-peak solitons, with $\\beta =0$ in Eqs. (\\ref{x}) and (\\ref{xy}). In\nthe former case, the existence of the exact solitons at arbitrarily large\nvalues of $\\gamma _{0}$ in Eq. (\\ref{x}) explicitly demonstrates the\nunbreakability of the $\\mathcal{PT}$ symmetry. In the latter case two\ndifferent families of exact solutions are found, which, however, exist only\nfor $\\gamma _{0}\\leq 2$ in Eq. (\\ref{xy}) with $\\beta =0$. In addition, an\nanisotropic version of the latter model gives rise to particular exact\nsolutions for vortex solitons with topological charge $m=1$. Generic soliton\nfamilies with $m=0,1,2,3$, which include the exact single-peak solutions as\nparticular ones, are constructed in a numerical form in both models, and\ntheir stability is investigated numerically---both through computation of\neigenvalues for small perturbations and by means of direct simulations.\n\n\n\\section*{Results}\n\n\\subsection*{\\textbf{The models and analytical solutions for solitons}}\n\n\\subsubsection*{\\textbf{The underlying equations}}\n\nThe 1D NLSE for the amplitude of the electromagnetic field, $u(x,z)$, with\nthe local strength of the self-defocusing nonlinearity, $\\Sigma (x)$,\ngrowing from $x=0$ towards $x=\\pm \\infty $ faster than $|x|$ (this condition\nis necessary for self-trapping imposed by the self-repulsion \\cite%\n{Barcelona1}), which is capable to maintain bright solitons with unbreakable$%\n\\mathcal{\\ PT}$ symmetry, is \\cite{unbreakable}%\n\\begin{equation}\ni\\frac{\\partial u}{\\partial z}+\\frac{1}{2}\\frac{\\partial ^{2}u}{\\partial\nx^{2}}-\\Sigma (x)|u|^{2}u=iW(x)u.  \\label{qq}\n\\end{equation}%\nHere $z$ and $x$ are scaled propagation coordinate and transverse\ncoordinate, in terms of the planar optical waveguide. In work \\cite%\n{unbreakable}, the analysis was presented for a steep 1D modulation profile,%\n\\begin{equation}\n\\Sigma (x)=\\left( 1+\\sigma x^{2}\\right) \\exp \\left( x^{2}\\right) ,\n\\label{Gauss}\n\\end{equation}%\nwith $\\sigma \\geq 0$, where coefficients equal to $1$ may be fixed to these\nvalues by means of rescaling. The choice of this profile allows one to\nobtain a particular exact solution for solitons \\cite{Barcelona1}. Of\ncourse, in a real physical medium the local strength of the nonlinearity,\ndefined as per Eq. (\\ref{Gauss}), cannot grow to infinitely large values at $%\n|x|\\rightarrow \\infty $. However, in reality it is sufficient that it grows\naccording to Eq. (\\ref{Gauss}) to finite values, that correspond to $|x|$\nwhich is essentially larger than the width of the soliton created by this\nprofile. The growth of $\\Sigma (x)$ may be safely aborted at still larger $%\n|x|$ \\cite{Barcelona1}.\n\nFurther, the spatially-odd imaginary potential, which accounts for the $%\n\\mathcal{PT}$-symmetric gain-loss profile (cf. Eq. (\\ref{U})), was\nintroduced in Ref. \\cite{unbreakable} as%\n\\begin{equation}\nW(x)=\\gamma _{0}x\\exp \\left( -\\beta x^{2}\\right) ,  \\label{gamma}\n\\end{equation}%\nwith $\\gamma _{0}>0$ and $\\beta \\geq 0$. In the case of the spatially\nuniform self-focusing cubic nonlinearity, the 1D imaginary potential in the\nform given by Eq. (\\ref{gamma}) was introduced in Ref. \\cite{extra-China}.\n\nHere, we aim to introduce a 2D extension of the model, as the NLSE for the\npropagation of the electromagnetic field with amplitude $u\\left(\nx,y,z\\right) $ in the bulk waveguide with transverse coordinates $\\left(\nx,y\\right) $:\n\n\\begin{equation}\ni\\frac{\\partial u}{\\partial z}+\\frac{1}{2}\\left( \\frac{\\partial ^{2}u}{%\n\\partial x^{2}}+\\frac{\\partial ^{2}u}{\\partial y^{2}}\\right) -\\Sigma\n(r)|u|^{2}u=iW\\left( x,y\\right) u,  \\label{NLS}\n\\end{equation}%\nwhere $r\\equiv \\sqrt{x^{2}+y^{2}}$ is the radial coordinate, and the\nnonlinearity-modulation profile is chosen similar to its 1D counterpart (\\ref%\n{Gauss}):\n\\begin{equation}\n\\Sigma (r)=\\left( 1+\\sigma r^{2}\\right) \\exp \\left( r^{2}\\right)\n\\label{sigma}\n\\end{equation}%\nwith $\\sigma \\geq 0$. Further, we consider two different versions of the 2D\nimaginary potential. First, it is a $\\mathcal{PT}$-symmetric one given by\nEq. (\\ref{x}). The other imaginary potential, defined as per Eq. (\\ref{xy}),\nis not $\\mathcal{PT}$-symmetric, because the $\\mathcal{P}$ transformation, $%\n\\left( x,y\\right) \\rightarrow \\left( -x,-y\\right) $, does not reverse the\nsign of $W\\left( x,y\\right) $, in this case. However, in terms of the\nimplementation in optics the gain-loss distribution corresponding to Eq. (%\n\\ref{xy}) is as relevant as that defined by Eq. (\\ref{gamma}), and, as\nmentioned above, properties of solitons in models which are akin to $%\n\\mathcal{PT}$-symmetric ones is a subject of considerable interest.\n\nStationary states with a real propagation constant, $k$, are looked for as\nsolutions to Eq. (\\ref{NLS}) in the form of\n\\begin{equation}\nu\\left( x,y\\right) =\\exp \\left( ikz\\right) U\\left( x,y\\right) ,  \\label{uU}\n\\end{equation}%\nwith complex function $U\\left( x,y\\right) $ satisfying the following\nequation:%\n\\begin{equation}\nkU=\\frac{1}{2}\\left( \\frac{\\partial ^{2}U}{\\partial x^{2}}+\\frac{\\partial\n^{2}U}{\\partial y^{2}}\\right) -\\Sigma (r)|U|^{2}U-iW\\left( x,y\\right) U.\n\\label{UU}\n\\end{equation}\n\n\\subsubsection*{\\textbf{Asymptotic solutions}}\n\nAs mentioned above, Eqs. (\\ref{NLS}) and (\\ref{UU}) are \\textit{%\nnonlinearizable}, i.e., they cannot be characterized by a linear spectrum.\nIndeed, straightforward analysis of Eq. (\\ref{UU}) demonstrates that it may\nproduce localized solutions (solitons), with tails decaying at $r\\rightarrow\n\\infty $ according to an asymptotic expression which is determined by the\nfull nonlinear equation, rather than by its linearization. For the $\\mathcal{%\nPT}$-symmetric imaginary potential (\\ref{x}) with $\\beta =0$, it is%\n\\begin{equation}\nU_{\\mathrm{asympt}}\\left( x,y\\right) =\\frac{1}{\\sqrt{2\\sigma }}\\exp \\left( -%\n\\frac{1}{2}r^{2}-i\\gamma _{0}x\\right) ,  \\label{asympt1}\n\\end{equation}%\nprovided that $\\sigma \\neq 0$. In the case case of $\\sigma =0$, this\nasymptotic solution is replaced by%\n\\begin{equation}\nU_{\\mathrm{asympt}}\\left( x,y\\right) =\\frac{r}{\\sqrt{2}}\\exp \\left( -\\frac{1%\n}{2}r^{2}-i\\gamma _{0}x\\right) .  \\label{asympt2}\n\\end{equation}\nNote that asymptotic solutions given by Eqs. (\\ref{asympt1}) and (\\ref%\n{asympt2}) exist at \\emph{arbitrarily large} $\\gamma _{0}$, suggesting the\n\\textit{unbreakability} of the $\\mathcal{PT}$ symmetry in this case, as\ncorroborated by exact solution (\\ref{exact1}) produced below.\n\nThe imaginary potential defined by Eq. (\\ref{xy}) with $\\beta =0$ produces\nthe following result:%\n\\begin{equation}\nU_{\\mathrm{asympt}}\\left( x,y\\right) =\\sqrt{\\frac{1-\\left( \\gamma\n_{0}\/2\\right) ^{2}}{2\\sigma }}\\exp \\left( -\\frac{1}{2}r^{2}-\\frac{1}{2}%\ni\\gamma _{0}xy\\right) ,  \\label{asympt3}\n\\end{equation}%\nfor $\\sigma \\neq 0$, and if $\\sigma =0$, the result is%\n\\begin{equation}\nU_{\\mathrm{asympt}}\\left( x,y\\right) =\\sqrt{\\frac{1-\\left( \\gamma\n_{0}\/2\\right) ^{2}}{2}}r\\exp \\left( -\\frac{1}{2}r^{2}-\\frac{1}{2}i\\gamma\n_{0}xy\\right) .  \\label{asympt4}\n\\end{equation}%\nOn the contrary to the the above asymptotic solutions, given by Eqs. (\\ref%\n{asympt1}) and (\\ref{asympt2}), which are available for arbitrarily large $%\n\\gamma _{0}$, their counterparts produced by Eqs. (\\ref{asympt3}) and (\\ref%\n{asympt4}) exist only at $\\gamma _{0}<2$, i.e., if the gain-loss coefficient\nis not too large.\n\nIt is relevant to stress the \\textit{universal character} of all asymptotic\napproximations given by Eqs. (\\ref{asympt1}) - (\\ref{asympt4}): they depend\nsolely on coefficients $\\sigma $ and $\\gamma _{0}$ of the underlying model,\nand, unlike the commonly known asymptotic forms of solitons in usual\nsystems, do not depend on the propagation constant, $k$. The single\nexception is presented by exact solution Eq. (\\ref{exact0}) given below,\nwhose asymptotic form (actually coinciding with the exact soliton solution,\nin that case) explicitly depends on $k$, but this happens solely for\nspecially chosen parameters given by Eq. (\\ref{special}). In the generic\ncase, a dependence on $k$ appears in the next-order correction to the shape\nof the asymptotic tail. In particular, the correction to the tails given by\nEqs. (\\ref{asympt1}) and (\\ref{asympt2}) are%\n\\begin{equation}\n\\delta U_{\\mathrm{asympt}}\\left( x,y\\right) =-\\left( k\/r^{2}\\right) U_{%\n\\mathrm{asympt}}\\left( x,y\\right) .  \\label{delta}\n\\end{equation}%\nFurthermore, for more complex solutions, such as multi-peak solitons and\nsolitary vortices, as well as for higher-order radial states of the\nsingle-peak solitons, which are produced below in the numerical form, the\nasymptotic form at large $r$ is exactly the same as given by Eqs. (\\ref%\n{asympt1})-(\\ref{asympt4}).\n\n\\subsubsection*{\\textbf{Exact solutions for single-peak solitons}}\n\nPrecisely at the above-mentioned critical value $\\gamma _{0}=2$, the\nasymptotic solutions (\\ref{asympt3}) and (\\ref{asympt4}) vanish. However, in\nthe special case,\n\\begin{equation}\n\\sigma =0,\\gamma _{0}=2,\\beta =0,  \\label{special}\n\\end{equation}%\nthe vanishing asymptotic solution Eq. (\\ref{asympt4}) is replaced by a\ndifferent one, which, as can be easily checked, is an \\emph{exact solution}\nto Eq. (\\ref{UU}) (not just an asymptotic approximation valid at large $r$),%\n\\begin{equation}\n\\left( U_{\\mathrm{exact}}^{\\left( xy\\right) }\\right) _{\\gamma _{0}=2}=\\sqrt{%\n-\\left( 1+k\\right) }\\exp \\left( -\\frac{1}{2}r^{2}-\\frac{1}{2}i\\gamma\n_{0}xy\\right) .  \\label{exact0}\n\\end{equation}%\nIt exists, as the continuous family, at all values of $k<-1$.\n\nFurther, Eq. (\\ref{UU}) which includes the $\\mathcal{PT}$-symmetric\nimaginary potential Eq. (\\ref{x}), with $\\beta =0$, gives rise to an exact\nsolution at a special value $k_{0}^{(x)}$ of the propagation constant:%\n\\begin{equation}\nU_{\\mathrm{exact}}^{(x)}=\\frac{1}{\\sqrt{2\\sigma }}\\exp \\left( -\\frac{1}{2}%\nr^{2}-i\\gamma _{0}x\\right) ,  \\label{exact1}\n\\end{equation}%\n\\begin{equation}\n~k_{0}^{(x)}=-\\left( 1+\\frac{\\gamma _{0}^{2}}{2}+\\frac{1}{2\\sigma }\\right) ,\n\\label{exact1parameters}\n\\end{equation}%\nwhich exists at all values of coefficients $\\gamma _{0}$ and $\\sigma $,\nexcept for $\\sigma =0$. In other words, at $k=k_{0}^{(x)}$ the asymptotic\napproximation Eq. (\\ref{asympt1}) is tantamount to the exact solution. This\nsolution features the \\emph{unbreakable} $\\mathcal{PT}$ symmetry, as it\npersists at arbitrarily large values of the gain-loss coefficient, $\\gamma\n_{0}$. Moreover, although Eq. (\\ref{exact1}) yields the exact solution at\nthe single value of the propagation constant, given by Eq. (\\ref%\n{exact1parameters}), which is embedded in a generic family of numerically\nfound fundamental solitons, as demonstrated below in Figs. \\ref{fig1}-\\ref%\n{fig3}, the entire family asymptotically shrinks to the exact solution in\nthe limit of large $\\gamma _{0}$. Indeed, it is easy to find that, for $%\n\\gamma _{0}^{2}\\gg 1$ and a relatively small deviation of the propagation\nconstant from the special value (\\ref{exact1parameters}), $\\left\\vert \\delta\nk\\right\\vert \\equiv \\left\\vert k-k_{0}^{(x)}\\right\\vert \\ll \\gamma _{0}^{2}$%\n, the fundamental soliton is%\n\\begin{equation}\nU_{\\mathrm{approx}}^{(x)}\\approx \\frac{1}{\\sqrt{2\\sigma }}\\exp \\left[ -\\frac{%\n1}{2}\\left( r^{2}+\\frac{\\delta k}{\\gamma _{0}^{2}}x^{2}\\right) -i\\left(\n\\gamma _{0}-\\frac{\\delta k}{\\gamma _{0}}\\right) x\\right] ,  \\label{approx}\n\\end{equation}%\nfeaturing weak anisotropy of the shape, $\\left\\vert U_{\\mathrm{approx}%\n}^{(x)}\\left( x,y\\right) \\right\\vert $.\n\nNext, Eq. (\\ref{UU}) with the imaginary potential taken as per Eq. (\\ref{xy}%\n) with $\\beta =0$, and with $\\sigma \\neq 0$ in the nonlinearity-modulation\nprofile (\\ref{sigma}), gives rise to the following exact solution, at the\nrespective single value of $k$:%\n\\begin{equation}\n\\left( U_{\\mathrm{exact}}^{\\left( xy\\right) }\\right) _{\\gamma _{0}<2}=\\sqrt{%\n\\frac{1-\\left( \\gamma _{0}\/2\\right) ^{2}}{2\\sigma }}\\exp \\left( -\\frac{1}{2}%\nr^{2}-\\frac{1}{2}i\\gamma _{0}xy\\right) ,  \\label{exact2}\n\\end{equation}%\n\\begin{equation}\nk_{0}^{(xy)}=-\\left[ 1+\\frac{1}{2\\sigma }\\left( 1-\\left( \\frac{\\gamma _{0}}{2%\n}\\right) ^{2}\\right) \\right] .  \\label{exact2parameters}\n\\end{equation}%\nIn this case too, the asymptotic approximation Eq. (\\ref{asympt3}) becomes\nidentical to the exact solution at $k=k_{0}^{(xy)}$, both existing at $%\n\\gamma _{0}<2$, on the contrary to exact solution (\\ref{exact1}), which\nexists at all values of $\\gamma _{0}$.\n\nThus, the models considered here do not have the linear spectrum. Instead of\nit, they are characterized by spectra (families) of self-trapped nonlinear\nsolutions (solitons). The radical change of the concept of the system's\nspectrum implies a respective change in the concept of the $\\mathcal{PT}$\nsymmetry, which now applies not to the set of eigenvalues of the linearized\nsystem, but directly to the existence of families of nonlinear states.\nLastly, it is worthy to note that all the asymptotic and exact solutions\nproduced above, including the first correction (\\ref{delta}) to the\nasymptotic tails, feature isotropic shapes of $\\left\\vert U(x,y\\right\\vert $%\n, although the imaginary potentials Eqs. (\\ref{x}) and (\\ref{xy}) are\nobviously anisotropic.\n\n\\subsubsection*{\\textbf{Exact solutions for elliptic vortices in an\nanisotropic model}}\n\nIn addition to 2D fundamental solitons, similar to the exact ones presented\nhere, we also address below, by means of numerical methods, solitons with\nembedded vorticities, $m=1,2,3...$ . A challenging issue is to seek for\nexact solutions for vortex solitons. Such solutions can be found in the case\nof imaginary potential Eq. (\\ref{xy}) with $\\beta =0$, but for a more\ngeneral \\textit{anisotropic} version of the nonlinearity-modulation profile\nin Eq. (\\ref{NLS}) with $\\sigma =0$, namely,%\n\\begin{equation}\n\\Sigma \\left( x,y\\right) =\\exp \\left( x^{2}+gy^{2}\\right) ,  \\label{g}\n\\end{equation}%\nwhere positive $g\\neq 1$ accounts for the ellipticity of the modulation\nprofile. Then, an exact solution for elliptically deformed vortex solitons\nwith $m=1$ is given by the following ansatz [cf. Eq. (\\ref{exact2})]:\n\\begin{equation}\nU\\left( x,y\\right) =U_{0}\\left( x+iby\\right) \\exp \\left( -\\frac{1}{2}\\left(\nx^{2}+gy^{2}\\right) -iaxy\\right) ,  \\label{vortex}\n\\end{equation}%\nwhere real $b\\neq 1$ accounts for the ellipticity of the soliton's phase\nfield, and $a$ is another real constant. The substitution of this ansatz and\nexpressions Eq. (\\ref{g}) and Eq. (\\ref{xy}) (with $\\beta =0$) in the\naccordingly modified equation (\\ref{UU}) leads to the following relations\nbetween parameters of the ansatz:%\n\\begin{gather}\n\\left( 1+g\\right) a=-\\gamma _{0},  \\notag \\\\\n(g-1)b-\\left( 1+b^{2}\\right) a=0,  \\label{conditions} \\\\\nb^{2}\\left( 1-a^{2}\\right) +a^{2}=g^{2},  \\notag\n\\end{gather}%\nsupplemented by expressions for the propagation constant and soliton's\namplitude:%\n\\begin{equation}\nk=-\\left( 3\/2+g\/2+ab\\right) ,~U_{0}^{2}=\\left( 1-a^{2}\\right) \/2.  \\label{kU}\n\\end{equation}%\nThe system of three equations (\\ref{conditions}) for two free parameters $a$\nand $b$ demonstrates that the exact vortex solution is a nongeneric one, as\nit may exist only if an additional constraint, which can be derived by\neliminating $a$ and $b$ in Eq. (\\ref{kU}), is imposed on parameters $g$ and $%\n\\gamma _{0}$:%\n\\begin{equation}\n\\left( g^{2}-1\\right) ^{2}\\left[ g^{2}\\left( g+1\\right) ^{2}-\\gamma _{0}^{2}%\n\\right] \\left[ \\left( g+1\\right) ^{2}-\\gamma _{0}^{2}\\right] =\\gamma _{0}^{2}%\n\\left[ \\left( g^{2}+1\\right) \\left( g+1\\right) ^{2}-2\\gamma _{0}^{2}\\right]\n^{2}.\n\\end{equation}\n\nIn the isotropic model, with $g=1$, Eq. (\\ref{conditions}) has no nontrivial\nsolutions. However, they can be found for $g\\neq 1$. A particular example is%\n\\begin{gather}\nb=1\/\\sqrt{2}\\approx 0.707\\,1,a=-\\left( 3-\\sqrt{5}\\right) \/\\left( 4\\sqrt{2}%\n\\right) \\approx -0.1351, \\\\\nU_{0}=\\sqrt{3\\left( 3+\\sqrt{5}\\right) }\/\\left( 4\\sqrt{2}\\right) \\approx\n0.7006,\n\\end{gather}%\nwhich is a valid solution at $g=\\left( 3\\sqrt{5}-1\\right) \/8\\approx\n\\allowbreak 0.713\\,5\\ $and $\\gamma _{0}=\\left( 3+\\sqrt{5}\\right) \/\\left( 16%\n\\sqrt{2}\\right) \\approx \\allowbreak 0.231\\,4$. This value of $g$ corresponds\nto eccentricity $e\\equiv \\sqrt{1-g}=\\sqrt{(9-3\\sqrt{5})\/8}\\approx\n\\allowbreak 0.535\\,2$ of the elliptic profile in Eq. (\\ref{g}).\n\nNumerical results are reported below for the isotropic model, while the\nanisotropic one should be a subject for separate consideration.\n\n\\subsection*{\\textbf{Numerical results for zero-vorticity solitons}}\n\n\\subsubsection*{The\\textbf{\\ }$\\mathcal{PT}$\\textbf{-symmetric imaginary\npotential (\\protect\\ref{x}): single- and double-peak solitons}}\n\nThe isolated exact solution of the model with the $\\mathcal{PT}$-symmetric%\n\\textbf{\\ }gain-loss distribution, given by Eqs. (\\ref{exact1}) and (\\ref%\n{exact1parameters}), can be embedded in a continuous family of solitons,\nproduced by a numerical solution of Eq. (\\ref{UU}), with $\\Sigma (r)$ and $%\n\\gamma \\left( x\\right) $ taken as per Eqs. (\\ref{sigma}) and (\\ref{x}). The\nappropriate numerical algorithm is the Newton conjugate gradient method \\cite%\n{Yang-book}, which is briefly outlined in section Method below. The\nstability of the stationary states was identified by numerical computation\nof eigenvalues of small perturbations, using linearized equations (\\ref%\n{eigen}) for perturbations around the stationary solitons. Finally, the\nstability predictions, produced by the eigenvalues, were verified by\nsimulations of the perturbed evolution of the solitons (some technical\ndetails are reported elsewhere \\cite{book}).\n\nIt is relevant to stress that the convergence of the algorithm which\nproduces stationary states depends on appropriate choice of the initial\nguess. While stationary modes were not found in \\textquotedblleft holes\"\nappearing in stability charts which are displayed below in Figs. \\ref{fig2}, %\n\\ref{fig3}, \\ref{fig7}, \\ref{fig10}-\\ref{fig12} and \\ref{fig16}, \\ref{fig17}%\n, it is plausible that stationary solutions exist in the holes too, being,\nhowever, especially sensitive to the choice of the input. On the other hand,\nthe intricate alternation of stability and instability spots, which is also\nobserved in the charts, is a true peculiarity of the present model.\nMoreover, genuine structure of the stability charts may be fractal, but\nanalysis of this possibility is beyond the scope of the present work.\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig1_PT.pdf}\n\\caption{Typical examples of stable solitons produced by the model with the $%\n\\mathcal{PT}$-symmetric imaginary potential defined by Eq. (\\protect\\ref{x}%\n). (a) A fundamental single-peak soliton for $\\protect\\gamma _{0}=1.2$ in\nEq. (\\protect\\ref{x}) and propagation constant $k=-3.2$ in Eq. (\\protect\\ref%\n{uU}). (b) A higher-order radial state of the single-peak soliton for $%\n\\protect\\gamma _{0}=0.2$ and $k=-4$. (c) A double-peak soliton for $\\protect%\n\\gamma _{0}=1.4$ and $k=-4$. In all the cases, $\\protect\\sigma =1$ and $%\n\\protect\\beta =0$ are fixed in Eqs. (\\protect\\ref{sigma}) and (\\protect\\ref%\n{x}).}\n\\label{fig1}\n\\end{figure}\n\nGeneric examples of numerically found \\emph{stable} solitons with single-\nand double-peak shapes are displayed in Fig. \\ref{fig1}. Note that the\ndouble-peak modes have their two maxima separated in the direction of $x$,\nin accordance with the anisotropic shape of the imaginary potential in Eq. (%\n\\ref{x}). As concerns single-peak modes, two different varieties of stable\nones were found: fundamental solitons, with the shape similar to that of the\nexact solution given by Eqs. (\\ref{exact1}) and (\\ref{exact1parameters})\n[see Fig. \\ref{fig1}(a)], and higher-order states with a radial ring\nsurrounding the central peak, see Fig. \\ref{fig1}(b). It is worthy to note\nthat, unlike many other models, where higher-order radial states are\nunstable \\cite{Atai1}-\\cite{Atai7}, they are stable in the present case.\nNote also that shapes of both species of the single-peak solitons,\nfundamental and higher-order ones, seem isotropic in terms of $\\left\\vert\nU\\left( x,y\\right) \\right\\vert $, similar to exact solution (\\ref{exact1}).\nThe isotropy is obviously broken by double-peak modes, see Fig. \\ref{fig1}%\n(c).\n\nResults of the stability analysis, based on the computation of perturbation\neigenvalues, are summarized in the stability map in the plane of $\\left(\nk,\\gamma _{0}\\right) $ [the soliton's propagation constant and strength of\nthe gain-loss term in Eq. (\\ref{x})], for $\\beta =0$ and $\\beta =0.2$ in\nFigs. \\ref{fig2} and \\ref{fig3}, respectively. Several noteworthy features\nare revealed by these plots. First, it is worthy to note significant\nstability areas for both the double-peak and higher-order single-peak $%\n\\mathcal{PT}$-symmetric solitons in Figs. \\ref{fig2} and \\ref{fig3}.\nFurther, bistability is observed at many points, in the form of coexisting\nstable fundamental and double-peak solitons, or fundamental and higher-order\nsingle-peak ones. As concerns the possibility of maintaining the unbreakable\n$\\mathcal{PT}$ symmetry, Fig. \\ref{fig2} demonstrates shrinkage of the\nexistence and stability regions of the modes with the increase of $\\gamma\n_{0}$ at $\\beta =0$ to the exact soliton solution given by Eqs. (\\ref{exact1}%\n) and (\\ref{exact1parameters}), in agreement with the trend represented by\napproximate solution (\\ref{approx}). Eventually, the exact solution loses\nits stability at $\\gamma _{0}\\geq 2$. On the other hand, the introduction of\na relatively weak confinement of the gain-loss term, with $\\beta =0.2$ in\nEq. (\\ref{x}), demonstrates that the $\\mathcal{PT}$ symmetry remains\nunbreakable in \\ref{fig3}, where both the existence and stability regions\nextend in the direction of large values of $-k$ and $\\gamma _{0}$, without\nfeaturing any boundary.\n\nAs concerns unstable solitons, they typically blow up in the course of the\nevolution, see an example below in Fig. \\ref{fig18}. Although it shows the\nblowup of a vortex soliton, the instability development of zero-vorticity\nones is quite similar.\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.6\\textwidth]{Fig2_PT.jpg}\n\\caption{The stability map for the $\\mathcal{PT}$-symmetric solitons\nmaintained by imaginary potential (\\protect\\ref{x}), in the case of $\\protect%\n\\sigma =1$ and $\\protect\\beta =0$ in Eqs. (\\protect\\ref{sigma}) and (\\protect\n\\ref{x}). Stable fundamental single-peak solitons are marked by green dots.\nAll unstable solitons are marked by red crosses, irrespective of their\nstructure. Exact soliton solutions, given by Eqs. (\\protect\\ref{exact1}) and\n(\\protect\\ref{exact1parameters}), are indicated by green stars (except for\none at $\\protect\\gamma _{0}=2$, which is designated by the red cross, as the\nexact solutions are unstable at $\\protect\\gamma _{0}\\geq 2$). Green numbers $%\n\\geq 2$ in this figure and below denote stable solitons with the same number\nof peaks. Further, green numbers $1$ label stable single-peak solitons with\nthe higher-order radial structure, as in Fig. \\protect\\ref{fig1}(b). Green\nnumbers $1$ or $2$, placed close to green dots, imply bistability, i.e.,\ncoexistence of stable fundamental single-peak solitons and stable\nhigher-order or double-peak ones. Red crosses placed on top of green dots\nimply coexistence of fundamental single-peak solitons with some unstable\nmode. Soliton solutions were not found in white areas.}\n\\label{fig2}\n\\end{figure}\n\nThe stability charts, drawn in Figs. \\ref{fig2} and \\ref{fig3} for $\\sigma\n=1 $ in Eq. (\\ref{sigma}), are quite similar to their counterparts produced\nat other values of $\\sigma $, including $\\sigma =0$, when the exact solution\ngiven by Eqs. (\\ref{exact1}) and (\\ref{exact1parameters}) does not exist,\nwhile the asymptotic form of the solitons' tails is given by Eq. (\\ref%\n{asympt2}).\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.6\\textwidth]{Fig3_PT.pdf}\n\\caption{The same as in Fig. \\protect\\ref{fig2}, but for $\\protect\\beta =0.2$\nin Eq. ( \\protect\\ref{x}), i.e., with the gain-loss term subject to weak\nspatial confinement. In this case, there are no exact solitons solutions,\nwhile the asymptotic solution for the tails is given by Eq. (\\protect\\ref%\n{asympt1}) with $\\protect\\gamma _{0}=0$ (the confinement eliminates $\\protect%\n\\gamma _{0}$ from the asymptotic solution).}\n\\label{fig3}\n\\end{figure}\n\n\\subsubsection*{\\textbf{The\\ imaginary potential (\\protect\\ref{xy}): single-\nand multi-peak solitons}}\n\nA drastic difference revealed by the stability analysis of the model based\non Eqs. (\\ref{NLS}), (\\ref{sigma}) and (\\ref{xy}) is that the respective\nexact solutions, given by Eq. (\\ref{exact0}) for the special case (\\ref%\n{special}), and by Eqs.\\ (\\ref{exact2}) and (\\ref{exact2parameters}) for $%\n\\sigma >0$, $\\beta =0$ and arbitrary $\\gamma _{0}$, are completely unstable,\non the contrary to the stability of the exact solutions in the case of the $%\n\\mathcal{PT}$-symmetric imaginary potential Eq. (\\ref{x}) (at $\\gamma _{0}<2$%\n). Furthermore, all numerical solutions found in the full 2D model with $%\n\\beta =0$ in Eq. (\\ref{xy}) are unstable too. The stabilization in this\nmodel is provided by $\\beta >0$, i.e., by imposing the spatial confinement\non the gain-loss term in Eq. (\\ref{xy}). For fixed $\\sigma $, there is a\nminimum value $\\beta _{\\min }$ of $\\beta $ which secures the stabilization.\nFor instance, we have concluded that the solitons may be stable in the model\nwith $\\sigma =1$ in Eq. (\\ref{sigma}) at $\\beta \\geq \\beta _{\\min }\\approx\n0.2$ in Eq. (\\ref{xy}), still being completely unstable, e.g., at $\\beta\n=0.1 $.\n\nAs mentioned above, the steep growth of $\\Sigma \\left( r\\right) $ in Eq. (%\n\\ref{sigma}) cannot extend to infinity, it being sufficient to maintain the\nadopted profile of $\\Sigma (r)$ on a scale which is essentially larger than\na characteristic size of solitons supported by this profile. The same\npertains to the linear growth of the imaginary potential at large $|x|$ in\nEq. (\\ref{x}): in reality, it should not continue at distances much larger\nthan the size of the stable solitons considered in the previous section.\nHowever, the presence of $\\beta _{\\min }$ implies that the corresponding\n\\textquotedblleft tacit\" confinement of $\\gamma \\left( x,y\\right) $ in Eq. (%\n\\ref{xy}) is not sufficient to produce stable 2D solitons. At $\\beta >\\beta\n_{\\min }$, the numerical solution generates stable fundamental single-peak\nsolitons and their higher-order radial counterparts with isotropic shapes of\n$\\left\\vert U\\left( x,y\\right) \\right\\vert $, as shown in Fig. \\ref{fig4}%\n(a,b). Further, stable multi-peak solitons are found too. Due to the 2D\nstructure of the imaginary potential (\\ref{xy}), they feature a four- or\nfive-peak structure, built along both the $x$ and $y$ axes, as shown in Fig. %\n\\ref{fig4}(c,d), instead of the uniaxial double-peak modes supported by the\nquasi-1D imaginary potential (\\ref{x}), cf. \\ref{fig1}(c).\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig4_PT.pdf}\n\\caption{Examples of stable single- and multi-peak $\\mathcal{PT}$-symmetric\nsolitons, found in the model based on Eqs. (\\protect\\ref{sigma}) and (%\n\\protect\\ref{xy}), with $\\protect\\sigma =1$ and (a) $\\protect\\beta =0.5$, $%\n\\protect\\gamma _{0}=1$, $k=-1$; (b) $\\protect\\beta =0.5$, $\\protect\\gamma %\n_{0}=0.2$, $k=-4$; (c) $\\protect\\beta =0.2$, $\\protect\\gamma _{0}=1.4$, $%\nk=-2.8$; (d) $\\protect\\beta =0.5$, $\\protect\\gamma _{0}=0.4$, $k=-1.8$. }\n\\label{fig4}\n\\end{figure}\n\nA typical stability chart for the 2D solitons generated by the model with $%\n\\beta >\\beta _{\\min }$ is displayed in Fig. \\ref{fig5}. It features\nbistability between the fundamental single-peak solitons and the\nhigher-order ones, or four- and five-peak complexes, in a relatively small\nregion of the $\\left( k,\\gamma _{0}\\right) $ plane, at sufficiently small\nvalues of $\\gamma _{0}$. Figure \\ref{fig5} clearly shows that no solitons\nwere found at $\\gamma _{0}\\geq 2$, this restriction coinciding with that for\nthe exact solution given by Eqs. (\\ref{exact2}) and (\\ref{exact2parameters}%\n). Thus, unlike the $\\mathcal{PT}$-symmetric imaginary potential (\\ref{x}),\nthe model based on potential (\\ref{xy}) does not produce unbreakable soliton\nfamilies.\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.6\\textwidth]{Fig5_PT.jpg}\n\\caption{The stability chart, defined as in Figs. \\protect\\ref{fig2} and\n\\protect\\ref{fig3}, but for the model including imaginary potential (\\protect\n\\ref{xy}), with $\\protect\\sigma =1$ and $\\protect\\beta =0.5$ in Eqs. (%\n\\protect\\ref{sigma}) and (\\protect\\ref{xy}). As indicated by the upper\ndashed red curve, no solitons were found at $\\protect\\gamma _{0}\\geq 2$,\nwhere the exact solution given by Eqs. (\\protect\\ref{exact2}) does not exist\neither.}\n\\label{fig5}\n\\end{figure}\n\n\\subsection{\\textbf{Vortex solitons}}\n\nSoliton solutions of Eq. (\\ref{UU}) with embedded vorticity were found\nnumerically by means of the above-mentioned Newton conjugate gradient\nmethod, initialized by the ansatz with integer vorticity $m\\geq 1$ added to\nthe previously found 2D stationary solutions of Eq. (\\ref{UU}):\n\n\\begin{equation}\nU\\left( x,y\\right) \\rightarrow U\\left( x,y\\right) r^{m}\\exp (im\\theta\n)\\equiv U\\left( x,y\\right) \\left( x+iy\\right) ^{m},  \\label{AngularPush}\n\\end{equation}%\nwhere $\\left( r,\\theta \\right) $ are the polar coordinates. The stability of\nresulting vortex solitons was again analyzed through the computation of\neigenvalues for modes of small perturbations around the vortex states, see\nEqs. (\\ref{eigen}), and then verified by direct simulations.\n\n\\subsubsection*{\\textbf{Vortex solitons in the case of the }$\\mathcal{PT}$%\n\\textbf{-symmetric imaginary potential}}\n\nIn the framework of the model with imaginary potential (\\ref{x}), stable\nvortex solitons were found in the case of $\\beta =0$ (no gain-loss\nconfinement) with $m=1$, while vortices with $m\\geq 2$ do not exist or are\nunstable. An example of stable vortices is shown in Fig. \\ref{fig6}, and the\nrespective stability charts for different values of $\\sigma $ in Eq. (\\ref%\n{sigma}) are presented in Fig. \\ref{fig7}. The strongly anisotropic shape of\nthe vortex is a consequence of the anisotropy of the underlying imaginary\npotential (\\ref{x}).\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig6_PT.pdf}\n\\caption{Three-dimensional (a) and top-view (b) shapes of $\\left\\vert\nU\\left( x,y\\right) \\right\\vert $ for a typical stable vortex soliton with $%\nm=1$, supported by the $\\mathcal{PT}$-symmetric imaginary potential (\\protect\n\\ref{x}) with $\\protect\\gamma _{0}=0.6$, $\\protect\\beta =0$, and $\\protect%\n\\sigma =0$ in Eq. (\\protect\\ref{sigma}), the propagation constant being $%\nk=-3 $. Panel (c) displays the phase structure of the vortex.}\n\\label{fig6}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig7_PT.pdf}\n\\caption{Stability charts for vortex solitons with topological charge $m=1$\nin the model including the $\\mathcal{PT}$-symmetric imaginary potential (%\n\\protect\\ref{x}) with $\\protect\\beta =0$, and $\\protect\\sigma =0$ or $1$ in\nEq. (\\protect\\ref{sigma}), in panels (a) and (b) panels, respectively. Green\ncircles and red crosses denote stable and unstable vortex solitons,\nrespectively. The same notation is used below in other stability charts for\nvortex solitons.}\n\\label{fig7}\n\\end{figure}\n\nThe introduction of the confinement of the gain and loss in Eq. (\\ref{x})\n(in particular, setting $\\beta =0.5$) makes it possible to construct stable\nvortex solitons with higher vorticities, corresponding to $m>1$ in Eq. (\\ref%\n{AngularPush}). An example of a stable vortex with $m=3$ is shown in Fig. %\n\\ref{fig8}.\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig8_PT.pdf}\n\\caption{The same as in Fig. \\protect\\ref{fig6}, but for stable vortex\nsoliton with $m=3$ and parameters $\\protect\\gamma _{0}=0.8$, $\\protect\\beta %\n=0.5$, $\\protect\\sigma =0$, $k=-4$.}\n\\label{fig8}\n\\end{figure}\n\nIn most cases, stable vortices generated by input (\\ref{AngularPush}) from\ndouble-peak stationary solutions have the same shape as those originating\nfrom their single-peak counterparts. However, in few cases the application\nof the lowest vorticity, with $m=1$ in Eq. (\\ref{AngularPush}), to the\ndouble-peak input leads to the creation of stable vortex solitons with a\ncomplex shape, see an example in Fig. \\ref{fig9}.\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig9_PT.pdf}\n\\caption{The same as in Fig. \\protect\\ref{fig6}, but for a case when the\nstable vortex soliton with $m=1$ and a complex shape is created, the\nparameters in Eqs. (\\protect\\ref{x}) and (\\protect\\ref{sigma}) being $%\n\\protect\\gamma _{0}=0.4$, $\\protect\\beta =0$, and $\\protect\\sigma =1$. The\npropagation constant is $k=-3.6$.}\n\\label{fig9}\n\\end{figure}\n\nStability charts for the vortex solitons with $m=1,2$, and $3$, supported by\nthe $\\mathcal{PT}$-symmetric imaginary potential which is subject to the\nspatial confinement, with $\\beta =0.5$ in Eq. (\\ref{x}), are shown in Figs. %\n\\ref{fig10} - \\ref{fig12}. While the stability area shrinks with the\nincrease of $m$, a few stable isolated modes were found even for $m=4$ (not\nshown here). The comparison of Figs. \\ref{fig10} and \\ref{fig7} shows that\nthe introduction of the spatial confinement of the gain-loss profile helps\nto expand the stability area for $m=1$ towards larger values of $\\gamma _{0}$%\n, thus upholding the trend to observe the unbreakable $\\mathcal{PT}$\nsymmetry in this 2D model. In direct simulations, the evolution of unstable\nvortex modes leads towards the blowup, via their fusion into a single peak,\nsimilar to what is displayed below in Fig. \\ref{fig18}.\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig10_PT.pdf}\n\\caption{Stability charts for solitons with vorticity $m=1$ in the case of\nthe $\\mathcal{PT}$-symmetric imaginary potential (\\protect\\ref{x}) with $%\n\\protect\\beta =0.5$, and $\\protect\\sigma =0$ or $1$ in Eq. (\\protect\\ref%\n{sigma}), in panels (a) and (b), respectively.}\n\\label{fig10}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig11_PT.pdf}\n\\caption{The same as in Fig.(\\protect\\ref{fig10}) (stability charts) but for\nvortex solitons with $m=2$.}\n\\label{fig11}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig12_PT.pdf}\n\\caption{The same as in Fig. (\\protect\\ref{fig10}), but for vorticity $m=3$.}\n\\label{fig12}\n\\end{figure}\n\n\\subsubsection*{\\textbf{Vortex solitons in the model with imaginary\npotential (\\protect\\ref{xy})}}\n\nStarting from input Eq. (\\ref{AngularPush}), stable vortices can be\nconstructed in the model with the gain-loss profile Eq. (\\ref{xy}) only if\nit is subject to the spatial confinement (recall the same is reported above\nfor zero-vorticity solitons). Examples of stable solitons with vorticities $%\nm=1,2$ and $3$ found in this model are shown in Figs. \\ref{fig13} - \\ref%\n{fig15}. Note that higher-order states with $m\\geq 2$ are actually compound\nstates built of $m$ unitary vortices, whose pivots do not merge into a\nsingle one, remaining separated, although with a small distance between\nthem, as can be seen for $m=2$ in Fig. \\ref{fig14}. The pivots form arrays\nalong axes $x$ or $y$, the particular direction being randomly chosen by the\ninitial conditions. Nevertheless, the overall shapes of the unitary and\nhigher-order vortices are nearly isotropic, due to the structure of the\ngain-loss term in Eq. (\\ref{xy}) (cf. strongly anisotropic shapes of\nvortices in Figs. \\ref{fig6}, \\ref{fig8}, and \\ref{fig9}, supported by the\nimaginary potential (\\ref{x})).\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig13_PT.pdf}\n\\caption{The same as in Fig. \\protect\\ref{fig6}, but for the stable vortex\nsoliton with $m=1$ in the case of imaginary potential Eq. (\\protect\\ref{xy}%\n), with $\\protect\\gamma _{0}=0.4$, $\\protect\\beta =0.5$, $\\protect\\sigma =0$%\n, and propagation constant $k=-3.4$.}\n\\label{fig13}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig14_PT.pdf}\n\\caption{The same as in Fig. \\protect\\ref{fig13}, but for stable vortex\nsolitons with $m=2$ and $k=-3.6$.}\n\\label{fig14}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig15_PT.pdf}\n\\caption{The same as in Fig. \\protect\\ref{fig13}, but for stable vortex\nsolitons with $m=3$ and parameters $\\protect\\gamma _{0}=0.2$, $\\protect\\beta %\n=0.5$, $\\protect\\sigma =0$, $k=-2.2$.}\n\\label{fig15}\n\\end{figure}\n\nStability charts obtained in this model for the solitons with embedded\nvorticities $m=1$ and $2$ are shown in Figs. \\ref{fig16} and \\ref{fig17}.\nOnly few examples of stable vortices with $m=3$, not shown here, have been\nfound in this case (for instance, at $\\sigma =0$, $\\gamma _{0}=0.4$, $k=-1.2$%\n).\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig16_PT.pdf}\n\\caption{Stability charts for vortex solitons with $m=1$ in the model\nincluding imaginary potential (\\protect\\ref{xy}), with $\\protect\\beta =0.5$\nand $\\protect\\sigma =0$ in (a) or $\\protect\\sigma =1$ in (b).}\n\\label{fig16}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig17_PT.pdf}\n\\caption{The same as in Fig. (\\protect\\ref{fig16}), but for vorticity $m=2$.}\n\\label{fig17}\n\\end{figure}\n\nFinally, a generic example of the evolution of an unstable vortex soliton is\nshown in Fig. \\ref{fig18}. The strong difference between vertical scales in\ndifferent panels of the figure clearly suggests that the instability leads\nto the blowup of the unstable mode, in the course of which the original\nvortex tends to fuse into a single peak. In fact, all unstable solitons\nconsidered in this work tend to develop the blowup in direct simulations.\n\n\\begin{figure}[tbp]\n\\centering\\includegraphics[width=0.75\\textwidth]{Fig18_PT.pdf}\n\\caption{The blowup of an unstable vortex soliton with $m=2$ and $\\protect%\n\\gamma _{0}=1.2$, $\\protect\\beta =0.5$, $\\protect\\sigma =1$, $k=-2.4$, in\nthe model with imaginary potential (\\protect\\ref{xy}). Panels display the\nfield at $z=60$ (a), $z=200$ (b) and $z=300$ (c). Note the difference in\nvertical scales between them.}\n\\label{fig18}\n\\end{figure}\n\n\\section*{Discussion}\n\nThe objective of this work is to elaborate 2D models with the spatially\nmodulated self-defocusing nonlinearity and gain-loss distributions\n[imaginary potentials, $iW\\left( x,y\\right) $] which give rise to families\nof stable single-peak, multi-peak, and vortical solitons, including ones\nwhich may persist and remain stable (\\textquotedblleft unbreakable\") at\narbitrarily large values of strengths $\\gamma _{0}$ of the imaginary\npotential. The unbreakability is possible in the case of the $\\mathcal{PT}$%\n-symmetric imaginary potential, which is given by Eq. (\\ref{x}). An asset of\nthe models, which can be implemented in bulk nonlinear optical waveguides\nwith embedded gain and loss elements, is that they produce universal\nasymptotic solutions for solitons' tails, along with full exact solutions\nfor selected species of 2D fundamental and vortex solitons (the latter one\nis available in the elliptically deformed version of the model). In\nparticular, in the limit of large $\\gamma _{0}$, the unbreakable family of\nfundamental solitons tends to shrink towards the exact solution. Generic\nfamilies of zero-vorticity solitons, including single- and multi-peak ones\nand higher-order radial states of single-peak solitons, as well as families\nof self-trapped modes with embedded vorticity $m=1,2$, and $3$, are\nconstructed in the numerical form, and their stability is identified by\nmeans of the numerical computation of eigenvalues for small perturbations,\nand verified by direct simulations. In the case of the $\\mathcal{PT}$%\n-symmetric imaginary potential (\\ref{x}) the solitons are stable in vast\nparameter regions, and feature a trend towards maintaining the unbreakable $%\n\\mathcal{PT}$ symmetry. Under the action of the imaginary potential (\\ref{xy}%\n), families of stable fundamental and vortex solitons exist too, provided\nthat the imaginary potential is subject to spatial confinement.\n\nA relevant extension of the analysis may be to address the elliptically\ndeformed model, which is considered in the present work in a brief form. A\nchallenging problem is the possibility of the fractal structure of the\nstability patterns in the models' parameter planes.\n\n\\section*{Methods}\n\n\\subsection*{\\textbf{The Newton conjugate gradient method for 2D robust }$PT$%\n\\textbf{-symmetry model}}\n\nSolutions of the stationary equation (\\ref{UU}) were constructed by means of\nthe Newton conjugate gradient method, which is presented in detail in book\n\\cite{Yang-book}. In terms of this method, the stationary-solution operator $%\n\\mathbf{L}_{0}$ is defined by Eq. (\\ref{UU}), while the respective\nlinearization operator $\\mathbf{L}_{1}$ is defined as\n\n\\begin{equation}\n\\mathbf{L}_{1}=%\n\\begin{bmatrix}\nA & B \\\\\nC & D%\n\\end{bmatrix}%\n,  \\label{L1_linearization_operator}\n\\end{equation}%\nwith matrix elements\n\\begin{eqnarray}\nA &=&-k+\\frac{1}{2}\\nabla ^{2}-\\Sigma (r)(\\left[ 3(\\mathrm{Re}U)^{2}+(%\n\\mathrm{Im}U)^{2}\\right] , \\\\\nB &=&-2\\Sigma (r)\\mathrm{Re}U\\cdot \\mathrm{Im}U+W(x,y), \\\\\nC &=&-2\\Sigma (r)\\mathrm{Re}U\\cdot \\mathrm{Im}U-W(x,y), \\\\\nD &=&-k+\\frac{1}{2}\\nabla ^{2}-\\Sigma (r)\\left[ 3(\\mathrm{Im}U)^{2}+(\\mathrm{%\nRe}U)^{2}\\right] ,\n\\end{eqnarray}%\nwhere the nonlinearity coefficient, $\\Sigma (r)$, and imaginary potential, $%\nW\\left( x,y\\right) $ are defined, respectively, by Eq. (\\ref{sigma}) and\nEqs. (\\ref{x}) or (\\ref{xy}).\n\n\\subsection*{\\textbf{Simulations of the evolution of the wave fields}}\n\nDirect simulations of the evolution equation (\\ref{NLS}), written as\n\n\\begin{equation}\ni\\frac{\\partial U}{\\partial z}=-\\frac{1}{2}\\left( \\frac{\\partial ^{2}U}{%\n\\partial x^{2}}+\\frac{\\partial ^{2}U}{\\partial y^{2}}\\right) +\\left[\nk+\\Sigma (r)|U|^{2}+i\\gamma \\left( x,y\\right) \\right] U,  \\label{EqOnU}\n\\end{equation}%\ncf. Eq. (\\ref{UU}), have been performed by means of the commonly known\nsplit-step method. Marching forward in $z$ at each step was split in two\nparts, according to the following equations:\n\n\\begin{eqnarray}\n\\mathbf{I} &\\mathbf{:~}&i\\frac{\\partial U}{\\partial z}=\\left[ k+\\Sigma\n(r)|U|^{2}+i\\gamma \\left( x,y\\right) \\right] U, \\\\\n\\mathbf{II} &\\mathbf{:~}&i\\frac{\\partial U}{\\partial z}=-\\frac{1}{2}\\left(\n\\frac{\\partial ^{2}U}{\\partial x^{2}}+\\frac{\\partial ^{2}U}{\\partial y^{2}}%\n\\right) .\n\\end{eqnarray}\n\nThe solutions were numerically constructed in the 2D spatial domain, $%\n\\left\\vert x,y\\right\\vert \\leq 9$, which was covered by a discrete grid of\nsize $N_{x}\\times N_{y}=512\\times 512$. The direct simulations were carried\nout with step $\\Delta z=10^{-5}$. This small step was selected to provide\nsufficient accuracy of the numerical solutions obtained in the presence of\nthe \\textquotedblleft exotic\" nonlinearity-modulation and gain-loss profiles\n(\\ref{sigma}) and (\\ref{x}) or (\\ref{xy}).\n\n\\subsection*{\\textbf{The stability analysis}}\n\nThe stability of the stationary states against small perturbations were\nbased, as usual, on the general expression for a perturbed solution,%\n\\begin{equation}\nu\\left( x,y,z\\right) =e^{ikz}\\left\\{ U\\left( x,y\\right) +\\varepsilon \\left[\ne^{\\Gamma z}v\\left( x,y\\right) +e^{\\Gamma ^{\\ast }z}w^{\\ast }\\left(\nx,y\\right) \\right] \\right\\} ,  \\label{pert}\n\\end{equation}%\nwhere $\\varepsilon $ is an infinitesimal perturbation amplitude, with\neigenmodes $\\left\\{ v\\left( x,y\\right) ,w\\left( x,y\\right) \\right\\} $ and\n(complex) eigenvalue $\\Gamma $, which should be found from the numerical\nsolution of the respective linearized equations,%\n\\begin{equation}\n\\begin{array}{c}\n\\left( -k+i\\Gamma \\right) v+\\frac{1}{2}\\left( \\frac{\\partial ^{2}}{\\partial\nx^{2}}+\\frac{\\partial ^{2}}{\\partial y^{2}}\\right) v-2\\Sigma\n(r)|U|^{2}v-\\Sigma U^{2}w=i\\gamma \\left( x,y\\right) v, \\\\\n\\left( -k-i\\Gamma \\right) w+\\frac{1}{2}\\left( \\frac{\\partial ^{2}}{\\partial\nx^{2}}+\\frac{\\partial ^{2}}{\\partial y^{2}}\\right) w-2\\Sigma\n(r)|U|^{2}w-\\Sigma U^{2}v=-i\\gamma \\left( x,y\\right) w,%\n\\end{array}\n\\label{eigen}\n\\end{equation}%\nsubject to zero boundary conditions at $\\left\\vert x,y\\right\\vert\n\\rightarrow \\infty $ (in fact, at borders of the solution domain). These\nequations were solved by means of the known spectral collocation method \\cite%\n{Yang-book}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe evolution of the structure and stellar populations of massive galaxies at high redshifts entails some of the key puzzles in galaxy evolution. The mean size of quiescent galaxies increases by a factor of about four since $z = 2.5$ \\citep[e.g.,][]{Buitrago08,vanDokkum08,Toft09,Williams11,Damjanov11,Newman12}, as reflected in a progressive buildup of light in their outer envelopes \\citep{vanDokkum10}, while the typical morphologies of the massive examples appear to become more spheroid-dominated \\citep{vanderWel11,Chang13b,Chang13}. At the same time, star formation is being truncated in many galaxies as they transition onto the red sequence. Both the rates of structural growth and the increase in number density of quenched galaxies appear to accelerate at $z \\gtrsim 1.5$ \\citep[][hereafter N12]{Newman12}.\n\nThe physical mechanisms driving these changes are only partially understood. Accretion of material in low-mass, gas-poor satellites has emerged as a popular explanation for the structural changes, since this adds stars at large radii while increasing the overall mass comparatively little \\citep[e.g.,][]{Naab09,Bezanson09,Hopkins09}. However, the observed (N12) and theoretical \\citep{Nipoti12} rates of such minor mergers appear too low to account fully for the rate of size growth, suggesting that additional processes may be at play. The continual arrival of new galaxies onto the red sequence whose sizes may differ from those of the older population already in place further complicates the interpretation; this could lead to a type of progenitor bias whose significance is still debated observationally (e.g., N12; \\citealt{Whitaker12b,Carollo13,Poggianti13}). Whether a stochastic merger history can lead to the tight scaling relations seen locally has also been questioned by some authors \\citep[e.g.,][]{Nipoti09,Nair10}. \n\nAdditional insight into the growth mechanisms arises from lookback studies that compare the rates of structural evolution as a function of environment. It is expected that the merger history of a galaxy depends on the local density or halo mass \\citep{McIntosh08,Fakhouri10,Lin10,Jian12,Kampczyk13}. If size growth is primarily merger driven, it is natural to expect that it will proceed at a rate that depends on past merger activity. Internally driven growth processes such as expansion via mass loss \\citep{Fan08,Fan10,Damjanov09}, on the other hand, should be less sensitive to environment. At the same time as gradual size growth proceeds, morphological transformations occur through a variety of processes that are both environmentally related (e.g., mergers, galaxy harassment, tidal interactions, and gas deprivation; see \\citealt{Treu03,Moran07}, and references therein) and internally driven (e.g., secular bulge growth; \\citealt{Kormendy04}). Lookback studies to $z \\sim 0.5 - 1$ have been essential to determine the history of morphological change in clusters \\citep[e.g.,][]{Dressler97,Postman05,Poggianti09}.\n\nSimilarly, while the cessation of star formation is clearly influenced by both environmentally related processes ---  e.g., ram-pressure stripping, gas starvation, galaxy--galaxy interactions --- as well as internal mechanisms, such as feedback from supernovae or an active galactic nucleus (AGN), the underlying physical processes and their relative importance as a function of mass and cosmic time remain uncertain. Understanding the history of star formation quenching in different environments aids in disentangling the influence of these processes. Observationally, this is constrained by the evolution of the fraction of quenched galaxies and their star formation histories in clusters, groups, and the field \\citep[e.g.,][]{Finn10,Tran10,Quadri12,Muzzin12,Raichoor12b,Dressler13,Brodwin13,Bedregal13,Alberts14}.\n\nHigh-redshift galaxy clusters represent excellent laboratories in which to address these questions, since they probe extreme overdensities at the epoch when quiescent galaxy growth and also the buildup of the red sequence appear most rapid. The expected decline in the number of clusters at high redshifts, coupled with the increasing difficultly of the observations necessary to locate and confirm them, has limited our knowledge of these systems. To date, only a handful of $z>1.6$ clusters hosting red galaxies are known \\citep[e.g.,][see Section~\\protect\\ref{sec:discussion}]{Papovich10,Gobat13,Stanford12,Zeimann12,Muzzin13,Tanaka13,Galametz13}. Spectroscopic data is required not only to confirm a putative cluster and isolate its members but also to precisely constrain the stellar populations and past star formation activity. Very few red cluster galaxies have been spectroscopically studied thus far, which has been the prime limiting factor in undertaking a study of the role of the environment in their evolution.\n\nIn this paper we present imaging and grism spectroscopy for the cluster candidate JKCS~041~using the Wide Field Camera 3 (WFC3) on board the \\emph{Hubble Space Telescope} (\\emph{HST}). JKCS~041~was originally discovered as an overdensity of galaxies with similar colors \\citep{Andreon09} in images from the UKIRT Infrared Deep Sky Survey \\citep{Lawrence07}. It exhibits a tight red sequence coincident with diffuse X-ray emission \\citep{Andreon11,Andreon11b} detected securely in a 75~ks \\emph{Chandra} observation. The X-ray observations and the galaxy richness indicate a relatively high halo mass of $\\log M_{200}\/\\msol \\simeq 14.2-14.5$ \\citep{Andreon13}. JKCS~041~was not detected in a Sunyaev--Zel'dovich (SZ) survey conducted by \\citet{Culverhouse10}, but the present upper limit on the mass is consistent with these X-ray-- and richness-based mass estimates (Section~\\ref{sec:discussion}). Estimates of the redshift of JKCS~041~based on different photometric techniques and data sets ranged from $z = 1.9 - 2.2$.  However, earlier attempts to confirm the reality of the cluster and to secure its spectroscopic redshift were unsuccessful.\n\nHere we use the WFC3 grisms to show that JKCS~041~is a genuine $z = 1.80$ rich cluster, confirmed via the spectroscopic confirmation of 19 member galaxies, of which 15 are quiescent. This is by far the largest number of quiescent cluster members beyond $z\\simeq1.5$ with spectroscopic data, making JKCS~041~a unique probe of early evolution in a dense environment. Our observations provide an ideal complement recent \\emph{HST} field surveys based on similar WFC3 data, such as CANDELS \\citep{Grogin11,Koekemoer11} and 3D-HST \\citep{Brammer12}.\n\nAfter describing our observations and methods in Sections 2 and 3, we introduce the cluster members and their basic properties in Section 4. In Section 5, we construct composite spectra of the quiescent cluster members. The stacking technique has been successfully employed in many cluster studies at lower redshifts \\citep[e.g.,][]{Dressler04,Gobat08,Poggianti09,Muzzin12} to discern variations in galaxy populations and star formation histories with mass and environment. For the first time in such a distant cluster, the quality of our spectra is sufficient to measure age-sensitive stellar absorption features and derive mean stellar ages as a function of galaxy mass. Additionally, through a comparison with composite spectra assembled by \\citet{Whitaker13} based on 3D-HST data, we are able to compare the stellar ages of quenched galaxies in JKCS~041~and the field near the same epoch. We demonstrate that although the fraction of quiescent systems in the cluster is elevated, the mean ages of these galaxies do not differ appreciably from the field sample.\n\nTo investigate the role of the environment in structural evolution, in Section 6 we compare the shapes, sizes, and radial mass profiles of members of JKCS~041~to a large sample of coeval field galaxies drawn from the CANDELS survey. By comparing the distribution of axis ratios, we find some evidence that a lower fraction of quiescent galaxies in the cluster contain a significant disk-like component. We consider the effect that variations in the morphological mixture of quenched galaxies in different environments may have on comparisons of the mass--radius relation, and conclude that there is no significant difference in the sizes of the JKCS~041~members compared to the field sample, particularly when these are better matched in morphology. In Section~7 we compare to results derived in other $z > 1.6$ clusters. We discuss the physical significance of our findings in Section~8, and finally summarize them in Section~9.\n\nThroughout we adopt a $\\Lambda$CDM cosmology with $\\Omega_m = 0.3$, $\\Omega_{\\Lambda} = 0.7$, and $H_0 = 70$~km~s${}^{-1}$~Mpc${}^{-1}$. All magnitudes are in the AB system, and stellar masses refer to a \\citet{Salpeter55} initial mass function (IMF).\n\n\\section{\\emph{HST} Observations and Data Reduction}\n\nWe observed JKCS~041~with the infrared channel of WFC3 (GO 12927, Cycle 20, P.I.~Newman) in four visits with a common pointing center but various spacecraft orientations. One two-orbit visit was devoted to imaging in the F160W and F105W filters, and the remaining 14 orbits were divided among 3 visits comprising G141 and G102 grism observations. In addition to our new \\emph{HST}~data, JKCS~041~benefits from an array of earlier ground- and space-based photometry. In this section we describe our reduction of the \\emph{HST}~observations and construction of a multi-wavelength catalog.\n\n\\subsection{HST Imaging}\n\nJKCS~041~was imaged through the F160W and F105W filters for approximately 4\/3 and 2\/3 orbits, respectively, using a four-point dither pattern identical to that adopted by the CANDELS survey \\citep{Koekemoer11}. After combining these deeper exposures with the grism pre-images, described below, the total exposure times were 4.5~ks in F160W and 2.7~ks in F105W. Although the calibrated frames produced using \\texttt{calwfc3} by the archive on-the-fly pipeline were mostly sufficient, we found it necessary to expand the pixel mask to include additional warm and hot pixels. The exposures were then registered and combined using \\texttt{multidrizzle} with a pixel scale of $0\\farcs06$.\n\n\\subsection{Photometric Catalog}\n\\label{sec:catalog} \n\nIn addition to the new \\emph{HST} imaging, JKCS~041~has been observed in the $ugrizJHKs$ filters by the MegaCam and WIRCam instruments at the Canada--France--Hawaii Telescope (CFHT) as part of the CFHT Legacy Survey (Deep Field 1) and the WIRCam Deep Survey \\citep{Bielby12}. We also made use of \\emph{Spitzer} Infrared Array Camera (IRAC) observations in the $3.6\\mu$m and $4.5\\mu$m channels taken as part of the \\emph{Spitzer} Wide-Area Infrared Extragalactic Survey (SWIRE; P.I.~Lonsdale).\n\nA multi-wavelength catalog was created using \\texttt{SExtractor} \\citep{SExtractor} with F160W as the detection band. The procedures followed those detailed in N12. All images were first aligned and drizzled onto the F160W pixel scale. Colors were then measured in apertures on images of matched resolution. To account for systematic uncertainties in zeropoints and PSF matching, we added a 3\\% uncertainty (10\\% for IRAC) in quadrature to the random flux errors. For a few of the galaxies that we confirm to be members of JKCS~041~(IDs 359, 375, 376, and 281; see Section~\\ref{sec:specconfmembers}), the aperture photometry was affected by neighboring sources. In order to measure accurate colors in these cases, we used \\texttt{Galfit} \\citep{Peng02} to fit S\\'{e}rsic profiles to all nearby sources simultaneously in each observed band.\n\nPhotometric redshifts $z_{\\rm phot}$ were computed using the $z_p$ estimator provided by \\texttt{EAZY} \\citep{Brammer08}. Stellar population parameters were derived using a custom code for the sample of bright galaxies with strong continuum signal in the grism spectra (see Section~\\ref{sec:contfit}). For fainter galaxies, we used \\texttt{FAST} \\citep{Kriek09b} to fit \\citet[][BC03]{BC03} models with exponentially declining star formation histories, dust attenuation, and a Salpeter IMF to the photometry; details of the grid can be found in N12. Finally, we use \\texttt{InterRest} \\citep{Taylor09} to interpolate to rest-frame colors in the \\citet{Bessell90} $UBV$ and 2MASS $J$ filters.\n\n\\subsection{HST Grism Spectroscopy}\n\\label{sec:hstgrism}\n\nA total of 14 orbits, split among 3 visits, was devoted to spectroscopy using the G102 and G141 grisms. The spacecraft orientations were spaced by 26~deg and 72~deg from the initial visit to facilitate the deblending of spectral traces. At the beginning of each sequence of grism exposures, a short undispersed exposure through the F160W (for G141) or F105W (for G102) filter was taken to register the grism images, which were then taken following the same dither pattern used for the imaging. The total integration time was 17.0~ks for each grism. In three exposures we noticed a rapidly increasing background in the final few reads; we successfully recovered data with the normal background level by masking the final reads and reprocessing the up-the-ramp readouts using \\texttt{calwfc3}. G102 covers the wavelength range $\\simeq850-1140$~nm with a dispersion of 2.4~nm per pixel, whereas G141 spans $\\simeq1110-1670$~nm at 4.6~nm per pixel. The wide wavelength range provided by the combination of grisms proved essential to locating the Balmer\/4000~\\AA~continuum break at the redshift of JKCS~041. \n\nThe grism data were reduced using the \\texttt{aXe} package \\citep{Kummel09}. For each object in the catalogs described in Section~\\ref{sec:catalog} and for each visit, \\texttt{aXe} generates a calibrated two-dimensional (2D) spectrum and an extracted spectrum, along with estimates of the noise and the flux contamination from other objects. A vertical extraction was used, with the wavelength constant perpendicular to the grism trace. Contamination from overlapping spectra was taken into account using the Gaussian emission model, which estimates the spectrum of each object by linearly interpolating the fluxes in the $i$, $z$, F105W, $J$, and F160W filters and distributes the flux spatially according to the Gaussian shape parameters estimated by \\texttt{SExtractor}. We found the extracted spectra generated by \\texttt{aXe} sufficient for deriving emission line redshifts (Section~\\ref{sec:emlines}); however, we made several improvements to the extraction of the brighter sources whose continuum emission we have modeled (Section~\\ref{sec:contfit}). \n\nFirst, the global background subtraction performed by \\texttt{aXe} often left significant residual trends, especially for the G102 grism. We improved upon this by fitting and subtracting a linear trend in wavelength to the background pixels in each 2D spectrum, omitting pixels in the extraction aperture and those with significant contaminating flux from other objects. The 2D spectra were created with larger dimensions than the \\texttt{aXe} default in order to ensure they contain a significant number of blank pixels. With this improvement, the G102 and G141 spectra generally join together smoothly.\n\nSecond, \\texttt{aXe} relies on a Gaussian approximation to the object light profile when it performs optimally-weighted extraction of spectra. While adequate for many objects, this is a poor representation of the extended light profiles of large spheroidal galaxies, which include many of our primary targets. Thus, for each galaxy for which we extract a continuum spectrum, we use the F160W image to measure the light profile in the cross-dispersion direction appropriate to each visit. This profile was then used to extract a one-dimensional (1D) spectrum, including error and contamination estimates, from the 2D spectrum with improved weighting. At the same time we measure the light profile of each galaxy in the dispersion direction. In grism spectroscopy this sets the line spread function (the LSF, i.e., the spectral resolution) and so is essential for the modeling we perform in Section~\\ref{sec:contfit}.\n\n\\section{Redshift Measurements and Spectral Fitting}\n\\label{sec:z}\n\nThe WFC3 G102 and G141 grisms represent a powerful combination, particularly for faint continuum spectroscopy: they cover a wide wavelength range continuously with uniform sensitivity, reach magnitudes that remain difficult from the ground, and sample all objects in the field of view with no pre-selection of targets. In this section we describe the measurement of 98 redshifts in the field of JKCS~041, which form the basis for our identification of the cluster members and the study of their properties in the remainder of the paper. The full catalog of redshift measurements is tabulated in Appendix~A. Our single WFC3 pointing covers the region within 1~arcmin, or 0.51 Mpc, of the X-ray centroid of JKCS~041. This is well-matched to the virial radius $R_{500} = 0.52$~Mpc estimated by \\citet{Andreon09} based on the X-ray temperature.\n\nThe galaxies included in our redshift survey consists of two distinct samples with very different selection properties: an \\emph{emission line sample} of galaxies showing one or more spectral lines, and a \\emph{continuum sample} of brighter galaxies for which we extract and model the continuum emission. The former is approximately limited by line flux, whereas the latter is limited by broadband flux. In Section~\\ref{sec:completeness} we estimate how these selections correspond to physical galaxy properties at the cluster redshift.\n\n\\subsection{The Emission Line Sample}\n\\label{sec:emlines}\n\nWe searched for emission lines in the 1D and 2D spectra of all galaxies having $H_{160} < 25.5$ using the plots generated by \\texttt{aXe2web}. These include contamination estimates, which are very useful for distinguishing true emission lines from overlapping zero order images of other galaxies. We additionally verified the reality of the emission lines by comparing the three independent spectra obtained for each object at the various orientations. In total we identified 63 emission line sources. An example spectrum is shown in the left panel of Figure~\\ref{fig:zexamples}.\n\nWavelengths of emission lines were measured by fitting Gaussian profiles in \\texttt{IRAF}. We averaged the wavelengths that were measured separately in each valid spectrum (i.e., each orientation at which the spectrum fell in the field of view and was not strongly contaminated). In 35 of 63 sources unambiguous redshifts were derived through the identification of multiple lines, primarily H$\\alpha$, [\\ion{O}{2}], and [\\ion{O}{3}]. When only a single line was identified (28 sources), it was interpreted as H$\\alpha$ (22 sources) or [\\ion{O}{3}] (6 sources) depending on which was more consistent with the photometric redshift.\n\nThe rms redshift uncertainty was estimated internally from the scatter in independent measurements as $\\sigma_z = 0.003$. For nine galaxies we can compare with redshifts measured at higher spectral resolution in the VIMOS VLT Deep Survey \\citep[VVDS;][]{LeFevre13}. After excluding one outlier with $\\Delta z = 0.07$, the rms scatter is $\\sigma_{\\Delta z} = 0.005$ with no detectable systematic bias. This is 20 times smaller than the median uncertainty in the photometric redshifts of the these galaxies.\n\nWe estimate a typical $5\\sigma$ line flux limit of $5 \\times 10^{-17}$ erg cm${}^{-2}$ s${}^{-1}$ in G141 data over $\\lambda \\approx 1.2 - 1.6 \\mu{\\rm m}$ and in G102 over $\\lambda \\approx 0.9 - 1.1 \\mu{\\rm m}$. By simulating artificial emission lines in the extracted spectra, we verified that we would visually identify $\\sim 80\\%$ of lines exceeding this flux limit. This limit applies to the spectra from each visit, which are the basis of our line search. These have 2--3 orbit depth, which is comparable to the 3D-HST \\citep{Brammer12} and WISP \\citep{Atek10} surveys, and these programs have estimated similar limits.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.49\\linewidth]{117example}\n\\includegraphics[width=0.49\\linewidth]{64example}\n\\caption{{\\bf Left:} example spectrum in which three emission lines are identified to yield an unambiguous redshift. {\\bf Right:} example of a luminous ($H_{160} = 21.2$) continuum-selected galaxy at $z = 2.414$ showing a prominent continuum break and several absorption lines. Blue and red lines show the coadded G102 and G141 spectra, respectively, binned to 48~\\AA~pixels with associated $1\\sigma$ errors shaded. The black line shows the best-fitting model (Section~\\ref{sec:contfit}), and broadband photometry is shown in green. The inset shows the full set of photometry on an expanded wavelength scale. The upper panels show the 2D spectra, displayed without applying a flux calibration.\n\\label{fig:zexamples}}\n\\end{figure*}\n\n\\subsection{The Continuum Sample}\n\\label{sec:contfit}\n\nFor all galaxies in the \\emph{HST} field of view brighter than $H_{160} < 23.3$ with photometric redshifts $1.4 < z_{\\rm phot} < 3$, we model the continuum emission in order to derive precise redshifts and stellar population properties. This flux limit corresponds to a typical signal-to-noise ratio of 5 per spectral pixel in the coadded spectra, suitable for continuum fitting, while the redshift range restricts the sample to galaxies for which the Balmer\/4000 \\AA~break is expected to fall well within the grism spectral coverage.\n\nFor each galaxy, we visually examined the spectra obtained during each of the three visits extracted using the improved weighting described in Section~\\ref{sec:hstgrism}. The contamination model was subtracted from each spectrum. Heavily contaminated wavelength regions, often comprising an entire visit, were identified and masked. The spectra were then coadded using inverse variance weighting to produce a combined spectrum for each grism. The galaxy light profiles, measured for each visit along the dispersion direction (Section~\\ref{sec:hstgrism}), were averaged with the same weights to estimate the LSF. The exposure times of the spectra vary significantly, since the number of visits that contribute to the stack ranges from 1 to 3. Of the 59 galaxies in the continuum sample, we were able to extract G102 and G141 spectra for 40 objects (68\\%). The remaining 19 sources were either heavily contaminated by neighboring sources or dispersed off the detector.\n\nTo make optimal use of the extensive data we gathered for JKCS~041, we developed a code designed to fit stellar population  models jointly to spectroscopic and photometric data with flexible models and arbitrary LSFs. \\texttt{pyspecfit} is written in Python. It is Bayesian in nature and uses \\texttt{MultiNest} \\citep{Feroz09}, a Markov Chain Monte Carlo (MCMC) engine, to explore the parameter space and properly estimate uncertainties and degeneracies.  The details of the code are described in Appendix~B. An example fit is shown in the right panel of Figure~\\ref{fig:zexamples} for a luminous red galaxy at $z = 2.414$. While our fits are based on the BC03 models, we note in passing that we experimented with using the 2007 models instead, but decided against this due to their uniformly poorer fits to the spectrophotometry. The poorer fits arise from excess light in the rest-frame near-infrared, which is consistent with other studies indicating that the contribution of the TP-AGB stars in these more recent models is overstated \\citep[e.g.,][]{Kriek10,Zibetti13}.\n\nA potential source of error in deriving redshifts from the continuum shape arises from joining spectra from the two grisms. This is of particular concern in the present sample since, as we show in Section~\\ref{sec:specconf}, the 4000~\\AA~break at the redshift of JKCS~041~falls near the division between the grisms. We tested for errors arising from this possible confusion by reanalyzing the spectra of the 17 continuum-selected cluster members (detailed in Section~\\ref{sec:specconf} below) after explicitly forcing the G102 and G141 flux levels to agree, on average, in the small wavelength range where they overlap. This process may introduce some additional noise, but it eliminates the possibility of a spurious spectral break. We found that only 2 of 17 redshifts shift by a significant amount ($> 2 \\sigma$).\\footnote{These are IDs 286, where there is some confusion about the location of the break (see the $P(z)$ distribution in Figure~\\ref{fig:memberspectra}), and ID 375, which is likely a satellite of a nearby, luminous red cluster members whose spectrum is difficult to clearly separate.} Both galaxies are on the red sequence and are very likely cluster members.\n\nOnly five galaxies in the continuum sample show strong emission features; in these cases, we adopt the emission redshift. We slightly increased the noise estimates for the spectral data by 20\\% to obtain a median $\\chi^2_{\\rm spec} \/ n_{\\rm pixels} = 1.0$, while for the photometry we find a median $\\chi^2_{\\rm phot} \/ n_{\\rm filters} = 1.1$. This indicates that the models provide good fits and that the noise estimates are reasonable. The median random uncertainty in $z_{\\rm grism}$ is $\\sigma_{z_{\\rm grism}}\/(1+z) = 0.0025$ for the continuum-selected galaxies, which is a factor 15 improvement over their photometric redshift errors.\n\n\\section{Spectroscopic Confirmation of JKCS~041~and Identification of Member Galaxies}\n\\label{sec:specconf}\n\nIn this section we use our grism redshift survey to identify JKCS~041~spectroscopically. Thanks to the excellent precision of the grism redshifts, which are typically $\\sim 15-20$ times more precise than the photometric estimates, we will show that JKCS~041~stands out as a strong overdensity of massive galaxies at $z = 1.80$ which are spatially coincident with diffuse X-ray emission, thus supporting the identification of JKCS~041~as a galaxy cluster with a hot intracluster medium (ICM). We then isolate a sample of spectroscopically confirmed member galaxies and discuss its likely completeness, before turning to the color distribution and star formation activity of these cluster members.\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{redshift_dist}\n\\caption{{\\bf Top:} distribution of grism redshifts at $z_{\\rm grism} > 0.7$ derived from emission lines and continuum fitting. Red and blue colors refer to the $UVJ$-based quiescent and star-forming classifications, respectively (see Figure~\\ref{fig:uvj}). {\\bf Middle:} stellar mass and redshift distribution for the same galaxies as in the top panel. Circles and crosses denote continuum and emission line redshifts, respectively. Vertical lines encompass the 19 identified cluster members. The green dashed line approximates the mass completeness of the continuum sample ($z_{\\rm phot} > 1.4$, $H_{160} < 23.3$) for a solar metallicity galaxy formed in a burst at $z_f = 5$. {\\bf Bottom:} redshift distribution of a mass-limited sample of galaxies found within the WFC3 field of view, divided into those located inside and outside of the outermost contour of detected X-ray emission (Figure~\\ref{fig:clusterimage}). The histograms are normalized by the area of these regions. A spectroscopic redshift is available for 83\\% of sources from one of the sources described in the text; for the remainder we rely on $z_{\\rm phot}$. JKCS~041~is the clear excess evident at $z=1.8$.\\label{fig:zdist}}\n\\end{figure}\n\n\\subsection{Spectroscopic Identification of JKCS~041~and Alignment with X-ray Emission}\n\nThe redshift distribution of the emission line and continuum-selected samples in our grism survey is shown in the top panel of Figure~\\ref{fig:zdist}. JKCS~041~is the richest structure in the field, comprising 19 galaxies, and is located at $z = 1.80$. The prominence of this peak is more remarkable when one considers that many of the members are red and massive systems with $M_* > 10^{11} \\msol$. Since the uncertainties in the grism redshifts are $\\sigma_z \\lesssim 0.01$, this shows that the $6.5\\sigma$ overdensity of red galaxies discovered by \\citet{Andreon09} identified a dominant structure and not a blend of several poorer ones.\n\nFigure~\\ref{fig:clusterimage} shows the that the distribution of galaxies in the $z=1.80$ cluster is clearly centered upon the diffuse X-ray emission \\citep{Andreon09}. Similar to some other high-redshift clusters \\citep[e.g.,][]{Zeimann12}, JKCS~041~does not have a single dominant galaxy located at the cluster center, which presumably reflects a lack of dynamical relaxation compared to lower-redshift systems. Nonetheless, the centroid of the spectroscopic cluster members is R.A.~=~02:26:44.0 $\\pm$ 6 arcsec, Decl.~=~--04:41:36 $\\pm$ 4 arcsec (red box in Figure~\\ref{fig:clusterimage}), which coincides with the X-ray centroid determined by Andreon et al.~to within $1\\sigma$. The cluster members are not distributed uniformly over the field; instead, all lie within $R_{500}$ of the X-ray center, and the majority are confined to a much smaller, elongated structure overlapping the X-ray emission. By considering a larger sample of red sequence candidate members extending to fainter magnitudes than our spectroscopic sample, \\citet{Andreon13} show that the red sequence galaxies follow a smoothly declining radial profile with parameters resembling those of lower-redshift clusters.\n\nJKCS~041~is therefore a natural identification for the source of the X-ray emission. Based on our grism data, we can now assemble a highly complete redshift catalog in the zone of the X-ray emission and verify that JKCS~041~is indeed the most likely origin. A stellar mass-selected sample is ideal for thus purpose, since it allows us to compare similar galaxy populations uniformly at different redshifts, and massive galaxies are better tracers of a deep gravitational potential. The green line in Figure~\\ref{fig:zdist} (middle panel) shows the limiting stellar mass for our continuum flux-selected sample, estimated as described in the caption, and demonstrates that this sample is fairly complete at masses $M_* > 10^{10.8} \\msol$ and redshifts $z = 1.4 - 2$. At lower redshifts, since the 4000~\\AA~break lies outside our spectral coverage, we combine our grism catalog with redshifts from the VVDS \\citep{LeFevre13} and the Carnegie--\\emph{Spitzer}--IMACS Survey \\citep{Kelson14}. This yields a spectroscopic redshift for 83\\% of the mass-limited sample; for the remainder we use $z_{\\rm phot}$. To assess the association of galaxies with the X-ray emission, we consider systems that are located within the outermost contour of the X-ray emission shown in Figure~\\ref{fig:clusterimage}. The bottom panel of Figure~\\ref{fig:zdist} shows their redshift distribution and clearly demonstrates that the $z = 1.80$ peak is dominant and concentrated within the X-ray emission.\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\linewidth]{clusterimage}\n\\caption{\\emph{HST}\/WFC3 image of JKCS~041~in the F160W and F105W filters. Confirmed cluster members are indicated by yellow (quiescent galaxies) and light blue (star-forming) circles.  The smoothed \\emph{Chandra} X-ray emission \\citep{Andreon09} is overlaid as contours. The centroid of the spectroscopically confirmed members and its $1\\sigma$ uncertainty is shown by the red rectangle, which is well-aligned with the X-ray centroid. Similarly, the dashed rectangle shows the mass-weighted centroid of the quiescent members, including the three likely members listed in Table~1 whose positions are indicated by dashed circles (contaminated spectra preclude a spectroscopic determination for these galaxies). White squares show spectroscopically confirmed non-members that are on the cluster red sequence (Section~\\ref{sec:completeness}). \\label{fig:clusterimage}}\n\\end{figure*}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{groups}\n\\caption{Peaks of the redshift distribution in the field of JKCS~041. The positions of galaxies with $M_* > 10^{10.8} \\msol$ in four redshift peaks, as described in the text, are plotted with symbol area proportional to stellar mass. Stars and circles distinguish star-forming and quiescent galaxies, respectively, as classified by their $UVJ$ colors. The outer isophote of the X-ray emission (Figure~\\ref{fig:clusterimage}) is approximated by the ellipse, and the dotted region outlines the field of the \\emph{HST} imaging. The centroids of the quiescent galaxies in each peak, weighted by stellar mass, are indicated by crosses.\\label{fig:groups}}\n\\end{figure}\n\nSecondary peaks of the redshift distribution in the field of JKCS~041~are expected and are seen in front of other high-redshift clusters \\citep[e.g.,][]{Zeimann12,Mantz14}. The next strongest peaks are located at $z=0.96, 1.13$, and 1.48; each contains 1 or 2 massive galaxies within the zone of X-ray emission, compared to 11 in JKCS~041~(Figure~\\ref{fig:zdist}, lower panel). Figure~\\ref{fig:groups} shows the positions of galaxies in these foreground structures, with crosses marking their centroids.\\footnote{For galaxies with spectroscopic redshifts, we plot those within $|z_{\\rm spec} - z| < 0.03$ in Figure~\\ref{fig:groups}, whereas for those with only photometric redshifts, we allow $|z_{\\rm phot} - z| < 0.08$.} In addition to being more sparsely populated, the $z = 0.96$ and 1.13 structures are not concentrated within the X-ray emission: the $z=0.96$ structure is very diffuse, and most galaxies in the $z=1.13$ peak lie outside of the X-ray--emitting region. The sparse $z = 1.48$ structure is better aligned with the X-ray emission than the other foreground peaks, but it seems far too poor to contribute a significant fraction of the flux. Only a single galaxy is massive enough to be included in Figure~\\ref{fig:groups}. For comparison, the $z=1.62$ group discovered by \\citet{Tanaka13b} in ultra-deep \\emph{Chandra} data appears to be richer, yet it exhibits diffuse X-ray flux that is still $\\sim 15\\times$ fainter than that observed around JKCS~041. \n\nWhile \\citet{Bielby10} considered these foreground structures as possible sources of the X-ray emission, they were unable to locate the dominant $z = 1.80$ cluster in a ground-based optical redshift survey. With the less biased selection and dense sampling afforded by the WFC3 grisms, we have shown that JKCS~041~is the most likely origin and is a genuine high-redshift cluster: it exhibits a spectroscopically confirmed population of massive, red galaxies that are concentrated within diffuse X-ray emission, and the observed X-ray properties are fairly consistent with expectations for a cluster with the observed richness of JKCS~041~\\citep{Andreon13}. After making a small correction to the luminosity distance, the bolometric X-ray luminosity estimated by \\citet{Andreon09} is $L_X = (6.5 \\pm 1.5) \\times 10^{44}$~erg~s${}^{-1}$ within $R_{500}$.\n\n\n\\subsection{Spectroscopically Confirmed Cluster Members}\n\\label{sec:specconfmembers}\n\nWith the redshift of JKCS~041~established, we now construct a sample of spectroscopically confirmed member galaxies that will form the basis of the remainder of the paper. The identification of cluster members is relatively unambiguous due to the high precision of the grism redshifts. We selected as cluster members those galaxies for which $>50\\%$ of the integrated probability density $P(z)$ is located within $z_{\\rm clus} \\pm 3 \\sigma_z$. Here $P(z)$ is derived from the MCMC chains for the continuum sample and is approximated as a Gaussian for the emission line sample (Section~\\ref{sec:emlines}). We estimate the cluster velocity dispersion $\\sigma_v = c \\sigma_z \/ (1+z) = 800$~km~s${}^{-1}$ based on the X-ray luminosity presented by \\citet{Andreon09} and the scaling relation derived by \\citet{Zhang10} for nearby clusters, which is consistent with the $z \\sim 1$ relation determined by \\citet{Andreon08}. We began with an initial estimate of $z_{\\rm clus}$ and iterate by updating $z_{\\rm clus}$ with the mean redshifts of the selected members.\n\nThis procedure converged in only one iteration to yield 19 members with a mean redshift of $z_{\\rm clus} = 1.803 \\pm 0.003$. The selected members are precisely those in the interval $z_{\\rm grism} = 1.803 \\pm 0.022$, which is indicated by the vertical lines in the lower panel of Figure~\\ref{fig:zdist}. We note that adopting the velocity window of $\\pm 2000 (1+z_{\\rm clus})$~km~s${}^{-1}$ advocated by \\citet{Eisenhardt08} would remove only one galaxy from this sample. Among the several previously published estimates of the redshift of JKCS~041, the {\\tt EAZY} photometric redshifts with no corrections applied gave the true $z_{\\rm clus}$ \\citep{Raichoor12b}. Spectra, images, and $P(z)$ distributions for the 19 confirmed members are shown in Figure~\\ref{fig:memberspectra}, and their coordinates and photometric properties are listed in Table~\\ref{tab:memberdata}. \n\n\\begin{figure*}\n\\includegraphics[width=0.49\\linewidth]{272} \n\\includegraphics[width=0.49\\linewidth]{355} \\\\\n\\includegraphics[width=0.49\\linewidth]{376} \n\\includegraphics[width=0.49\\linewidth]{356} \\\\\n\\includegraphics[width=0.49\\linewidth]{657} \n\\includegraphics[width=0.49\\linewidth]{286} \\\\\n\\includegraphics[width=0.49\\linewidth]{352} \n\\includegraphics[width=0.49\\linewidth]{411}\n\\caption{Spectroscopically confirmed cluster members. For each object, the main panel shows the grism spectra (blue is G102, red is G141, $1\\sigma$ errors are shaded) binned to 48~\\AA~(red) and 96~\\AA~(blue) pixels for display purposes. Photometry (green circles) and the best-fitting model (black) are overlaid. The top and bottom axes shows the rest- and observed-frame wavelength in nm, and the units of $F_{\\lambda}$ are $10^{-18}$ erg cm${}^{-2}$ s${}^{-1}$ \\AA${}^{-1}$. The inset shows the complete photometry on an expanded scale in the same units. Cutouts show F105W\/F160W images, displayed on a linear scale, with a side length of $5\\arcsec$. The $P(z)$ subpanels show the redshift probability density derived from the broadband photometry only using \\texttt{EAZY} (black curves) and from our joint fits to the spectra and photometry (filled histograms). Galaxies are ordered by decreasing F160W flux. For the two galaxies in the emission line sample (IDs 332 and 531) no continuum fit is plotted.\\label{fig:memberspectra}}\n\\end{figure*}\n\n\\addtocounter{figure}{-1}\n\\begin{figure*}\n\\includegraphics[width=0.49\\linewidth]{447} \n\\includegraphics[width=0.49\\linewidth]{289} \\\\\n\\includegraphics[width=0.49\\linewidth]{387} \n\\includegraphics[width=0.49\\linewidth]{375} \\\\\n\\includegraphics[width=0.49\\linewidth]{317} \n\\includegraphics[width=0.49\\linewidth]{359} \\\\\n\\includegraphics[width=0.49\\linewidth]{281} \n\\includegraphics[width=0.49\\linewidth]{693}\n\\caption{Continued}\n\\end{figure*}\n\n\\addtocounter{figure}{-1}\n\\begin{figure*}\n\\includegraphics[width=0.49\\linewidth]{531} \n\\includegraphics[width=0.49\\linewidth]{255} \\\\\n\\includegraphics[width=0.49\\linewidth]{332}\n\\caption{Continued}\n\\end{figure*}\n\n\\begin{deluxetable*}{lllcccccccccc}\n\\tablewidth{\\linewidth}\n\\tablecaption{Redshifts and Photometric Data for Spectroscopically Confirmed Cluster Members and Red Sequence Members}\n\\tablehead{\\colhead{ID} & \\colhead{R.A.} & \\colhead{Dec.} & \\colhead{$H_{160}$} & \\colhead{$z$} & \\colhead{Type} & \\colhead{$\\log M_*^{\\rm auto}\/\\textrm{M}_{\\odot}$} & \\colhead{$z-J$} & \\colhead{$(U-B)_{\\rm r}$} & \\colhead{$(U-V)_{\\rm r}$} & \\colhead{$(V-J)_{\\rm r}$} & \\colhead{$UVJ$} & \\colhead{Quality}}\n\\startdata\n\\cutinhead{\\emph{Spectroscopically confirmed cluster members}}\n272 & 36.68173 & -4.68934 & 20.63 & $1.798_{-0.003}^{+0.002}$ & C & $11.71 \\pm 0.03$ & $2.02 \\pm 0.04$ & 1.20 & 1.84 & 1.15 & Q & A\\\\\n355 & 36.68644 & -4.69239 & 20.80 & $1.798_{-0.002}^{+0.002}$ & C & $11.52 \\pm 0.02$ & $2.01 \\pm 0.03$ & 1.15 & 1.63 & 1.05 & Q & A\\\\\n376 & 36.67501 & -4.69286 & 21.20 & $1.811_{-0.008}^{+0.004}$ & C & $11.56 \\pm 0.03$ & $2.07 \\pm 0.05$ & 1.34 & 1.90 & 1.14 & Q & A\\\\\n356 & 36.69423 & -4.69235 & 21.35 & $1.805_{-0.004}^{+0.003}$ & C & $11.36 \\pm 0.04$ & $1.97 \\pm 0.07$ & 1.16 & 1.81 & 1.13 & Q & A\\\\\n657 & 36.67557 & -4.70257 & 21.61 & $1.812_{-0.002}^{+0.002}$ & C & $11.11 \\pm 0.02$ & $2.02 \\pm 0.05$ & 1.20 & 1.77 & 0.92 & Q & A\\\\\n286 & 36.68790 & -4.68994 & 21.69 & $1.798_{-0.013}^{+0.068}$ & C & $11.47 \\pm 0.03$ & $1.94 \\pm 0.08$ & 1.16 & 1.88 & 1.37 & Q & B\\\\\n352 & 36.69051 & -4.69215 & 21.88 & $1.797_{-0.004}^{+0.006}$ & C & $11.22 \\pm 0.05$ & $2.05 \\pm 0.08$ & 1.23 & 1.87 & 1.08 & Q & A\\\\\n411 & 36.67382 & -4.69384 & 22.11 & $1.821_{-0.004}^{+0.004}$ & C & $11.15 \\pm 0.04$ & $1.84 \\pm 0.09$ & 1.11 & 1.84 & 1.19 & Q & A\\\\\n447 & 36.69121 & -4.69487 & 22.12 & $1.797_{-0.009}^{+0.011}$ & C & $10.81 \\pm 0.03$ & $1.42 \\pm 0.09$ & 0.82 & 1.34 & 0.64 & Q & A\\\\\n289 & 36.68965 & -4.68994 & 22.17 & $1.802_{-0.004}^{+0.003}$ & C & $10.89 \\pm 0.03$ & $1.97 \\pm 0.08$ & 1.18 & 1.74 & 0.70 & Q & A\\\\\n387 & 36.68231 & -4.69296 & 22.36 & $1.801_{-0.009}^{+0.009}$ & C & $11.00 \\pm 0.04$ & $1.50 \\pm 0.11$ & 0.94 & 1.51 & 1.49 & SF & B\\\\\n375 & 36.67488 & -4.69278 & 22.43 & $1.819_{-0.008}^{+0.008}$ & C & $10.88 \\pm 0.02$ & $1.91 \\pm 0.09$ & 1.09 & 1.64 & 1.05 & Q & B\\\\\n317 & 36.69911 & -4.69091 & 22.45 & $1.787_{-0.003}^{+0.003}$ & C & $10.75 \\pm 0.04$ & $2.00 \\pm 0.11$ & 1.14 & 1.61 & 1.11 & Q & A\\\\\n359 & 36.67696 & -4.69228 & 22.54 & $1.792_{-0.005}^{+0.004}$ & C & $10.67 \\pm 0.03$ & $1.90 \\pm 0.11$ & 1.10 & 1.56 & 0.61 & Q & B\\\\\n281 & 36.69061 & -4.68944 & 22.77 & $1.806_{-0.004}^{+0.004}$ & C & $10.73 \\pm 0.06$ & $2.06 \\pm 0.17$ & 1.12 & 1.75 & 0.98 & Q & B\\\\\n693 & 36.67771 & -4.70379 & 22.86 & $1.820_{-0.010}^{+0.019}$ & C & $10.51 \\pm 0.05$ & $1.14 \\pm 0.09$ & 0.75 & 1.11 & 0.78 & SF & C\\\\\n531 & 36.67919 & -4.69839 & 23.12 & $1.818_{-0.002}^{+0.002}$ & E & $9.73 \\pm 0.06$ & $0.49 \\pm 0.11$ & 0.27 & 0.46 & 0.16 & SF & A\\\\\n255 & 36.68793 & -4.68838 & 23.30 & $1.795_{-0.075}^{+0.004}$ & C & $10.53 \\pm 0.04$ & $1.35 \\pm 0.24$ & 0.85 & 1.70 & 0.76 & Q & C\\\\\n332 & 36.67165 & -4.69125 & 23.83 & $1.785_{-0.003}^{+0.003}$ & E & $9.35 \\pm 0.28$ & $0.22 \\pm 0.21$ & 0.11 & 0.21 & 0.82 & SF & B\\\\\n\\cutinhead{\\emph{Candidate cluster members on red sequence (not spectroscopically confirmed), $H_{160} < 23.3$ and $R < R_{500}$}}\n772 & 36.67527 & -4.70738 & 22.26 & $1.81_{-0.11}^{+0.08}$ & P & $10.91 \\pm 0.28$ & $2.00 \\pm 0.09$ & 1.20 & 1.72 & 1.00 & Q & \\ldots\\\\\n275 & 36.68274 & -4.68931 & 22.68 & $1.81_{-0.19}^{+0.12}$ & P & $10.78 \\pm 0.28$ & $1.87 \\pm 0.17$ & 1.00 & 1.66 & 1.02 & Q & \\ldots\\\\\n404 & 36.68949 & -4.69338 & 22.89 & $1.59_{-0.09}^{+0.17}$ & P & $10.71 \\pm 0.28$ & $1.86 \\pm 0.16$ & 1.22 & 1.91 & 1.33 & Q & \\ldots\n\\enddata\n\\tablecomments{The ``r'' subscript denotes colors in the rest frame. C and E types indicate continuum and emission line redshifts, whereas P denotes photometric redshifts. Q and SF refer to galaxies in the quiescent and star-forming regions of the $UVJ$ color--color plane. For type C, $M_*$ is derived from fits to the full spectrophotometry (Section~\\ref{sec:contfit}); for types E and P, $M_*$ is based on \\texttt{FAST} fits to the photometry. Median random uncertainties in the rest-frame $U-B$, $U-V$, and $V-J$ colors are 0.07, 0.03, and 0.08 mag, respectively. $H_{160}$ is F160W magnitude in the \\texttt{MAG\\_AUTO} aperture, and $M_*^{\\rm auto}$ is scaled here to this total flux. See Appendix~A for notes on the redshift quality flags.\\label{tab:memberdata}}\n\\end{deluxetable*}\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{uvj}\n\\caption{Rest-frame colors of the spectroscopically-confirmed cluster members. Circles and crosses denote galaxies with continuum and emission line redshifts, respectively, while filled and open symbols denote massive ($M_* > 10^{11} \\msol$) and less massive ($M_* < 10^{11} \\msol$) systems, respectively. The grayscale shows the field distribution for galaxies drawn from the NMBS survey (see Section~\\ref{sec:fq}) that have $z = 1.8 \\pm 0.2$ and $M_* > 10^{10.6} \\msol$. The solid line divides the quiescent and star-forming selection regions, while the dashed line shows the partition between bluer and redder quiescent galaxies used by \\citet{Whitaker13}. Median color uncertainties are illustrated by the error bar. \n\\label{fig:uvj}}\n\\end{figure}\n\n\\subsection{Completeness}\n\\label{sec:completeness}\n\nAlthough the continuum sample is strictly flux-limited ($H_{160} < 23.3$), it forms a nearly mass-limited sample at $z = 1.80$. Based on the catalog of N12~that covers a much wider area, we expect 88\\% of galaxies at $z=1.8$ with $M_* > 10^{10.6} \\msol$ to be brighter than $H_{160} = 23.3$. Within the WFC3 field of view surrounding JKCS~041, all galaxies above this mass threshold that are photometric candidate members ($z_{\\rm phot} = 1.8 \\pm 0.2$) are brighter than $H_{160} = 22.8$, even though the imaging depth is $\\sim 3$~mag fainter. Independently, we estimate nearly the same limiting mass using the BC03 model for a solar-metallicity galaxy formed in a burst at $z_f = 5$ (see green line in Figure~\\ref{fig:zdist}, middle panel). Conversely, all confirmed cluster members in the continuum sample have $M_* > 10^{10.5} \\msol$.\n\nWe thus expect the parent continuum-limited sample to be reasonably complete for stellar masses $M_* > 10^{10.6} M_{\\odot}$. Additional incompleteness arises from those spectra that could not be extracted due to contamination from nearby sources. This affects 19 of the 59 galaxies in the continuum sample (Section~\\ref{sec:contfit}). Three of these lie on the red sequence and are located at $R < R_{500}$. These are likely cluster members whose properties we list in Table~\\ref{tab:memberdata}. Three additional bluer systems located within $R_{500}$ have $z_{\\rm phot}$ consistent with JKCS~041~within their 68\\% confidence intervals; however, the redshift uncertainties are too large to associate them with the cluster with any confidence. None of the candidate members discussed above has a stellar mass $M_* > 10^{11} \\msol$. Therefore, most likely we have spectroscopically confirmed all members with $M_* > 10^{11} \\msol$ and $R < R_{500}$. At lower masses $M_* = 10^{10.6-11} \\msol$, considering the three most likely photometric candidates, our estimated spectroscopic completeness is $\\sim 75\\%$. Given this high completeness, for the rest of the paper we focus our analysis on the spectroscopically-confirmed cluster members.\n\nCompleteness for the emission line sample is less straightforward to interpret. For this reason, we confine our quantitative analysis in Sections~\\ref{sec:stellarpops} and onward to the better-defined continuum-selected sample and classify galaxies based on their colors, not on the presence of emission lines. Nonetheless, it is useful to have a rough idea of the star formation rate (SFR) corresponding to the limiting line luminosity of $3 \\times 10^8 L_{\\odot}$ (Section~\\ref{sec:emlines}). [\\ion{O}{2}] and [\\ion{O}{3}] lie within our spectral coverage for JKCS~041~members. For [\\ion{O}{2}] emission, this limit corresponds to a SFR of $\\gtrsim 30$~$M_{\\odot}$~yr${}^{-1}$ according to the \\citet{Kewley04} calibration with dust attenuation of $A_V = 1$. For galaxies with significantly subsolar metallicity, the [\\ion{O}{2}] emission will be weaker, but [\\ion{O}{3}] will be more visible. Limits will also be weaker for galaxies with higher dust content $A_V > 1$, which is expected for massive systems.\n\n\\subsection{Colors and Star Formation Properties of the Cluster Members}\n\\label{sec:galaxyprops}\n\nFigure~\\ref{fig:uvj} shows the distribution of the confirmed cluster members in the rest-frame $UVJ$ color--color diagram. This plane is frequently used to distinguish quiescent and star-forming systems \\citep{Williams09}, and for the remainder of the paper we refer to quiescent and star-forming galaxies based on this criterion, using the specific form proposed by \\citet{Whitaker11}.\n\nOf the 19 confirmed members, 17 arise from the continuum sample, and 15 of of these fall in the quiescent region of the $UVJ$ plane. This large number of quiescent members with spectroscopic data makes JKCS~041~an invaluable laboratory for studying environmental processes at high redshifts. None of the quiescent members shows unambiguous ($>3\\sigma$) residual line emission above the continuum models, although there is a hint of [\\ion{O}{2}] in IDs 657 and 447. Galaxy 447 is a borderline case: it falls near the edge of the quiescent selection box. It has a specific SFR of $10^{-10.2}$ Gyr${}^{-1}$ inferred from the spectrophotometric fitting, which is intermediate between the other 14 $UVJ$-quiescent members (all $<10^{-11}$ Gyr${}^{-1}$) and the star-forming members ($\\sim 10^{-9}$ Gyr${}^{-1}$). Of the cluster members in the star-forming region of the $UVJ$ plane, two show emission lines (IDs 531 and 332) and have low stellar stellar masses $M_* = 10^{9.4-9.8} \\msol$, while two more massive examples having $M_* = 10^{10.5-11} \\msol$ were identified through continuum fitting (IDs 387 and 693). Note that we are able to secure redshifts of these bright blue galaxies even though they lack detectable emission lines.\n\nMorphologically, virtually all of the quiescent confirmed members appear spheroid-dominated (see Figure~\\ref{fig:memberspectra}). This visual impression is supported by a quantitative analysis of the galaxy shapes in Section~\\ref{sec:structure}. Of the four star-forming members, two appear compact (IDs 693 and 531), ID 332 appears diffuse and irregular, and ID 387 (located near the cluster center) appears to be an inclined disk with a red bulge.\n\nOnly one spectroscopic member is detected as an X-ray point source in the 75~ks \\emph{Chandra} data  \\citep{Andreon09}: ID 352, a $UVJ$-quiescent galaxy with $L_{X, 0.5-2~{\\rm keV}}= 6 \\times 10^{42}$~erg~s${}^{-1}$. To investigate the presence of obscured star formation or AGN activity in other cluster members, particularly those classified as quiescent by their $UVJ$ colors, we measured 24 $\\mu$m fluxes in the \\emph{Spitzer} Multiband Imaging Photometer (MIPS) data taken for the SWIRE survey.\\footnote{We used a simple $7''$ diameter aperture and applied an aperture correction factor of 2.56. The X-ray source (ID 352) has a detected close neighbor whose flux was subtracted using a PSF model.} None of the quiescent members is detected at $2\\sigma$ significance ($> 0.13$~mJy), and there is no detection in a mean stack to a $2\\sigma$ limit of 32~$\\mu$Jy. \n\nThe 2 more massive star-forming members (IDs 387 and 693) are detected with fluxes of $0.20 \\pm 0.06$ mJy each. Based on the \\citet{Wuyts08} templates, this corresponds to a total infrared luminosity of $L_{\\rm IR} = (1.3 \\pm 0.4) \\times 10^{12} L_{\\odot}$ for each source and SFRs of $140 \\pm 44$ $M_{\\odot}$ yr${}^{-1}$ for a \\citet{Chabrier03} IMF \\citep{Bell05}. These are typical for star-forming galaxies in this mass and redshift range \\citep[e.g.,][]{Reddy06}. Thus, among the galaxies in our continuum-selected sample, we see a one-to-one correspondence between those which lie in the quiescent region of the $UVJ$ plane and those which lack detectable 24$\\mu$m emission, albeit in fairly shallow MIPS imaging. \\citet{Papovich12} also found a good correspondence between these diagnostics in a $z=1.62$ proto-cluster using deeper MIPS data, and \\citet{Fumagalli13} recently showed that $UVJ$-quiescent galaxies at high redshift generally lack mid-infrared emission to very deep limits. We conclude that the $UVJ$ diagram provides reasonable classifications of cluster and field galaxies and is suitable for making differential comparisons, as we do in Sections 5 and thereafter.\n\n\n\n\\subsection{The Red Sequence}\n\\label{sec:candidates}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{redsequence}\n\\caption{Red sequence of JKCS~041. \\emph{Red circles:} spectroscopically-confirmed quiescent cluster members. \\emph{Blue circles}: Confirmed star-forming members. \\emph{Black crosses:} confirmed non-members. \\emph{Green squares:} candidate cluster members on the red sequence (dashed region) that lack a grism redshift due to contamination of their spectra. \\emph{Gray circles:} remaining galaxies with no grism redshift. Only galaxies within $R_{500}$ of the cluster center are plotted; this includes all confirmed members.\\label{fig:redseq}}\n\\end{figure}\n\nIn the absence of spectroscopic data, members of high-redshift clusters are frequently identified based on the red sequence. With our grism observations we can assess the purity and completeness of this method. Figure~\\ref{fig:redseq} shows the color--magnitude diagram for galaxies with $R < R_{500}$, where $R$ is the distance from the X-ray centroid. \n\nJKCS~041~shows a clear red sequence with a mean observed color $\\langle z-J \\rangle = 1.98 \\pm 0.02$ and a measured scatter of $\\sigma_{z-J} = 0.07$. This is comparable to the rms measurement error of $\\delta_{z-J} = 0.09$, indicating that the intrinsic scatter is low \\citep{Andreon11b}. We define red sequence galaxies as those within $\\pm 2\\sigma$ of the mean color to a limiting magnitude of $H_{160} < 23.3$ (dashed in Figure~\\ref{fig:redseq}).\n\nThe majority of the spectroscopically confirmed members in the continuum sample (13 of 17) are on the red sequence. However, in addition to the two star-forming members, two galaxies that are classified as quiescent according to their $UVJ$ colors are bluer than the $z-J$ red sequence (IDs 255 and 447). These are likely systems where star formation has been most recently truncated. Naturally, some galaxies located on the $z-J$ red sequence will not be associated with the cluster. Using the grism redshifts, we identified five interlopers over the full field of view, which are indicated by boxes in Figure~\\ref{fig:clusterimage}. Only two of these are located at $R < R_{500}$. Thus, a red sequence selection yields a fairly pure and complete sample (13 of 15, or 87\\%) of quiescent members within $R < R_{500}$, as anticipated from the high overdensity of red sequence galaxies compared to the field \\citep{Andreon11}. At larger radii contamination is more severe.\n\n\n\n\\section{Stellar Populations of Quiescent Galaxies: JKCS~041~Compared to the Field}\n\\label{sec:stellarpops}\n\nHaving identified a well-defined set of cluster members based on grism spectroscopy, we now turn to the effect of the cluster environment on their stellar populations. We first consider the fraction of quenched systems in JKCS~041~relative to coeval field galaxies of matched stellar mass. Additional insight can then be gained from the ages of the quiescent cluster members. We construct composite spectra that reveal age-sensitive stellar absorption lines at high signal-to-noise for the first time in such a distant cluster. Using these, we investigate the mean stellar age both as a function of mass within the cluster, and relative to similar field galaxies whose composite spectrum was constructed by \\citet{Whitaker13} using 3D-HST grism data.\nThe 17 spectroscopically confirmed cluster members in the continuum-selected sample ($H_{160} < 23.3$) , which is approximately mass-limited ($M_* \\gtrsim 10^{10.6} \\msol$, Section~\\ref{sec:completeness}) and confined to $R < R_{500} \\approx 500$~kpc, form the basis for the following comparisons.\n\n\\subsection{The Quiescent Fraction}\n\\label{sec:fq}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{quiescentfrac}\n\\caption{Fraction of galaxies classified as quiescent by their $UVJ$ colors in several stellar mass bins. Spectroscopic members of JKCS~041~(black) are compared to coeval field galaxies drawn from the NMBS survey (green). Horizontal error bars show the range of masses in each bin, with points placed at the median mass, while vertical $1\\sigma$ errors are based on binomial statistics.\\label{fig:fq}}\n\\end{figure}\n\nFigure~\\ref{fig:fq} compares the fraction $f_Q$ of galaxies in JKCS~041~with quiescent $UVJ$ colors to that of field galaxies in the same range of stellar mass and redshift. The comparison sample is drawn from the NEWFIRM Medium Band Survey catalogs in the AEGIS and COSMOS fields \\citep{Whitaker11}, selected from $z_{\\rm phot} = 1.8 \\pm 0.2$ and converted to a Salpeter IMF. Although this ``field'' sample includes galaxies that inhabit a range of environments, a differential comparison is still informative because JKCS~041~is a strong overdensity.\\footnote{For example, 9 members having $M_* > 10^{11} \\msol$ lie within 1~arcmin of the cluster center, whereas only 1.8 are expected from the mean surface density in the field.}\n\nClearly, the cluster environment has had a powerful role in determining the number of quenched systems: 88\\% (15 of 17) of the cluster members in the continuum sample are quiescent, whereas this fraction is less than half in the field. Roughly half of the quiescent cluster members were thus quenched by environmentally-related processes. \nRecalling that our spectroscopic sample may be missing some cluster members with masses $M_* = 10^{10.6-11} \\msol$ due to contamination of their spectra, we have tested the effects of adding in the six unconfirmed candidate members described in Section~\\ref{sec:candidates}. This would move $f_Q$ in the lowest-mass bin only with the plotted $1\\sigma$ uncertainty, resulting in a fraction that would still be elevated above the field. Using a photometric redshift selection and a statistical background subtraction, \\citet{Raichoor12b} also estimated a high quiescent fraction $f_Q \\gtrsim 85\\%$ ($1\\sigma$ limit) among massive galaxies ($M_* \\gtrsim 10^{11} \\msol$) in the core of JKCS~041~($R < 0.5 R_{200}$), consistent with our spectroscopic sample.\n\n\\subsection{Composite Spectra of Quiescent Cluster Members}\n\\label{sec:stackspec}\n\nHaving determined that the efficiency of quenching in JKCS~041~is high, we now consider the ages of its quiescent members by constructing composite spectra of these galaxies. \nStacking increases the signal-to-noise ratio and averages over residual contamination or background subtraction errors that may affect individual spectra. Rather than stacking the flux-calibrated spectra and photometry, we average continuum-normalized spectra covering $\\simeq 4000-5900$~\\AA~redward of continuum break. This technique has several advantages. First, we are able to measure the age-sensitive Balmer (H$\\beta$,$\\gamma$,$\\delta$) and Mg~\\emph{b} absorption lines; since these are narrowband features, they are more robust against errors in the continuum shape and uncertainties in dust attenuation. Second, we avoid the rest-frame near-infrared where model uncertainties related to the TP-AGB phase can influence the derived ages around 1~Gyr. Third, we are able to make a homogeneous comparison to coeval, quiescent galaxies in the field, whose composite continuum-normalized spectrum was measured by \\citet{Whitaker13} using 3D-HST survey data.\n\nIn order to investigate mass-dependent trends, we split the sample of 15 confirmed quiescent members into a higher-mass subsample consisting of 8 galaxies with $M_* > 10^{11} \\msol$, and a lower mass subsample whose 7 members span $M_* = 10^{10.5-11} \\msol$. The continuum of each spectrum was first determined by fitting a third order polynomial to the models shown in Figure~\\ref{fig:memberspectra}, excluding the strong absorption lines. Each spectrum was then divided by the continuum, shifted to the mean redshift of the cluster, and interpolated onto a grid with 48~\\AA~pixels (17~\\AA~in the rest frame), which is close to the native dispersion. The spectra were then combined by averaging each spectral pixel, excluding the highest and lowest measures. Uncertainties were estimated by bootstrapping. The LSFs of the galaxies entering the stack (Section~\\ref{sec:hstgrism}) were also averaged to construct a mean LSF.\n\nWe fit the stacked spectra to simple stellar population (single-burst) models using \\texttt{pyspecfit}, taking the redshift, age, and metallicity as free parameters. Although the actual star-formation histories are possibly more complex, using the burst models enables us to make a direct comparison with other work, particular that of \\citet[][Section \\ref{sec:ages}]{Whitaker13}. The model spectra were continuum-normalized using the same method that was applied to the data. A broad, log-uniform prior was placed on the age. We allow the metallicity to vary to quantify the degeneracy with age. Since these galaxies are expected to evolve into the cores of present-day massive ellipticals \\citep[e.g.,][]{Bezanson09,Hopkins09}, which are metal-enriched to ${\\rm [Z\/H]} \\approx 0.1 - 0.3$ \\citep[e.g.,][]{Thomas10, Conroy14}, we place a broad uniform prior on [Z\/H] over the range 0--0.3.\n\nThe top left panel of Figure~\\ref{fig:stacks} shows the spectrum of the more massive ($M_* > 10^{11} \\msol$) quiescent members of JKCS~041. The quality of the spectrum is remarkably high, with a signal-to-noise ratio of 55 per pixel, and it clearly shows several absorption lines as indicated in the figure. The model (black curve) fits the data well with an age of $1.45^{+0.24}_{-0.18}$~Gyr, marginalized over metallicity, which corresponds to a formation redshift $z_f = 3.0^{+0.4}_{-0.2}$. \n\nThe lower left panel displays the mean spectrum of the lower-mass ($M_* = 10^{10.5-11} \\msol$) quiescent members. Although the spectrum is necessarily noisier, with a signal-to-noise ratio of 22, it is clearly different from that of the higher-mass galaxies. The clearest difference is the enhanced strength of the Balmer absorption lines: H$\\beta$, H$\\gamma$, H$\\delta$ are all markedly deeper in the lower mass sample. We derive a younger luminosity-weighted mean age of $0.90^{+0.19}_{-0.10}$ Gyr, corresponding to a formation redshift $z_f = 2.4^{+0.2}_{-0.1}$. The Mg~\\emph{b} absorption in this spectrum is too deep to be matched even by a maximally old, metal-rich model; this may be due to residual non-Gaussian noise in the stack. In any case, masking Mg~\\emph{b} shifts our age inference by only $\\sim 1\\sigma$ to $0.79 \\pm 0.19$~Gyr (dashed lines in Figure~\\ref{fig:stacks}).\n\nThe quiescent galaxies in JKCS~041~thus have a range of ages that follow the well-known mass-dependent trends seen in the field, in which lower-mass early type galaxies typically have younger luminosity-weighted ages \\citep[e.g.,][]{Treu05L,Thomas10}. Although the absolute ages depend somewhat on metallicity, the right panel of Figure~\\ref{fig:stacks} shows that the age difference of $0.52 \\pm 0.26$~Gyr between the two subsamples is more robust, provided that they have broadly similar metallicity. We indeed expect the mean metallicities of our mass-selected subsamples to differ by $\\lesssim 0.1$~dex, based on abundance studies at low redshift.\\footnote{Given the ratio of the median stellar masses entering our two bins, we estimate a velocity dispersion ratio of $\\Delta \\log \\sigma \\approx 0.2$, which corresponds to abundance variations of $\\Delta {\\rm [Fe\/H]} \\approx 0.03$ and $\\Delta {\\rm [Mg\/Fe]} \\approx 0.08$ in $z \\sim 0$ ellipticals \\citep{Conroy14}.}\n\nTwo additional pieces of data support this conclusion. First, the ages of the individual galaxies as measured from fits to their grism spectra and photometry (Section~\\ref{sec:contfit}) show the same trend: the median age is 1.6~Gyr and 0.96~Gyr for the high- and low-mass subsamples, respectively, which is consistent with the ages derived from their mean continuum-normalized spectra. Second, the lower-mass galaxies have bluer colors, as shown in Figure~\\ref{fig:uvj}. We can predict the mean color differences between the mass-selected subsamples that should arise purely from the difference in ages inferred from their absorption lines. The predicted $\\Delta \\langle U-V \\rangle = 0.14 \\pm 0.08$ and $\\Delta \\langle V-J \\rangle = 0.26 \\pm 0.12$ are consistent with the measured values of $\\Delta \\langle U-V \\rangle = 0.20$ and $\\Delta \\langle V-J \\rangle = 0.29$. Thus, the color trend can be explained by a mass-dependent trend in age, rather than metallicity or dust content.\n\nThese results should be interpreted with the usual understanding the ages are luminosity-weighted and so skew toward the most recent star formation episode. Our focus is robustly constraining the mean age as a function of mass, and some cluster members at given mass may of course be older or younger. (For example, the spectrum of  ID 355, shown in Figure~\\ref{fig:memberspectra}, is clearly younger than that of the first-rank cluster member.) The tightness of the red sequence led \\citet{Andreon11b} to infer that the spread in ages at a fixed mass is quite small. Their analysis, however, is sensitive to the assumed cluster redshift, which we have now revised to $z = 1.80$. For further details and a revised estimate of the age scatter based on our spectroscopic data, we refer to \\citet{Andreon13}.\n\n\\subsection{Age and Line Emission in Quiescent Galaxies as a Function of Environment}\n\\label{sec:ages}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.63\\linewidth]{contnorm}\n\\includegraphics[width=0.36\\linewidth]{contour_sps}\n\\caption{{\\bf Left:} composite spectra of confirmed quiescent members of JKCS~041~in two bins of stellar mass. Red curves show the data and $1\\sigma$ uncertainties, and black lines show the model fit. Dashed blue curves show composite spectra of quiescent field galaxies from \\citet{Whitaker13}: the upper and lower panels show their stacks of redder and bluer quiescent galaxies, respectively. {\\bf Right:} constraints on the simple stellar population model derived for the two mass-selected subsamples. Contours show $1\\sigma$ and $2\\sigma$ constraints; dashed contours show results for the lower-mass subsample when Mg~\\emph{b} is masked. The upper panel shows the marginalized posterior distribution for the age and compares to field constraints derived by \\citet{Whitaker13} for their bluer and redder quiescent galaxy subsamples ($1\\sigma$ error bars).\\label{fig:stacks}}\n\\end{figure*}\n\n\\citet{Whitaker13} recently constructed composite spectra of 171 quiescent field galaxy observed in the 3D-HST grism survey. This presents an interesting opportunity to compare quenched field and clusters galaxies at the same early epoch. The Whitaker et al.~data are very well suited for this comparison. In addition to being observed with the same instrument, they selected quiescent galaxies using the same $UVJ$ color selection, and their limiting magnitude of $H_{140} < 22.8$ (measured in the F140W filter) is similar to our limit of $H_{160} < 23.3$. Their median stellar mass $10^{11.08} \\msol$, converted to a Salpeter IMF, matches the $10^{11.11} \\msol$ of our sample. The main difference is that the Whitaker et al.~stacks combine field galaxies spanning a wide range in redshift, $z = 1.4 - 2.2$,  whereas the members of JKCS~041~are obviously at a single redshift. Nonetheless, the median redshift of the galaxies in their stacks is $\\langle z \\rangle \\simeq 1.6 - 1.7$, close to JKCS~041. \n\n\nRather than subdividing their sample by stellar mass, Whitaker et al.~split the quiescent selection region of the $UVJ$ plane into two regions indicated by the dashed line in Figure~\\ref{fig:uvj}. Among the quiescent JKCS~041~members, such a color division is very similar to a division in stellar mass: the eight quiescent members with $M_* > 10^{11} \\msol$ would all fall in the redder subsample of Whitaker et al., and the seven less massive members fall in or near their bluer region. The mean color difference between the galaxies in their blue and red subsamples ($\\Delta \\langle U-V \\rangle = 0.2$, $\\Delta \\langle V-J \\rangle = 0.3$) is consistent with that described above for our mass-selected subsamples.\n\nWith this in mind, in the top left panel of Figure~\\ref{fig:stacks} we compare our composite spectrum of massive JKCS~041~members to the composite field spectrum of redder quiescent galaxies investigated by Whitaker et al. First, we note that the Mg~\\emph{b} lines are nearly identical. Correspondingly, Whitaker et al.~derived an age of $1.6^{+0.5}_{-0.4}$~Gyr for their redder field sample, consistent with our measurement (see right panel). Interestingly, the field stacks show faint line emission in [\\ion{O}{3}] $\\lambda\\lambda 4959, 5007$ and in filling of H$\\beta$, whereas the spectrum of the JKCS~041~members clearly lacks this emission and instead follows the stellar population model closely.\\footnote{We note that the \\citet{Whitaker13} stacks are median spectra and so should be relatively immune from strong line emission in a small fraction of the field sample.}\n\nIn the lower left panel of Figure~\\ref{fig:stacks} we compare our composite spectrum of lower-mass JKCS~041~members to the composite spectrum of bluer quiescent field galaxies. The strong Balmer lines seen in the cluster members are also evident in the field. Whitaker et al.~derived a reduced age of $0.9^{+0.2}_{-0.1}$ Gyr, again consistent with our measurement for the lower-mass ($M_* = 10^{10.5-11} \\msol$) quiescent cluster members. Whitaker et al.~infer [\\ion{O}{3}] emission in their bluer subsample as well, although the signal there is more ambiguous. Our stack of lower-mass members shows no clear evidence of emission, but the lower signal-to-noise ratio makes this distinction marginal.\n\nComparing the ages derived in our two stacks to the Whitaker et al.~measurements in the upper right panel of Figure~\\ref{fig:stacks}, we find that the cluster and field samples span a very similar range. Quantitatively, the differences in luminosity-weighted mean stellar ages are $\\Delta t = {\\rm age}_{\\rm JKCS041} - {\\rm age}_{\\rm field} = -0.2 \\pm 0.5$~Gyr and $0.0^{+0.3}_{-0.1}$~Gyr for the more-massive\/redder and less-massive\/bluer subsamples, respectively. These results are marginalized over a range of metallicity, whereas Whitaker et al.~instead fixed the metallicity to solar abundance in their analysis. If we do the same, these age differences shift to $\\Delta t = 0.2 \\pm 0.5$~Gyr and $0.3^{+0.3}_{-0.2}$~Gyr, respectively. In this solar metallicity case, however, the age of the lower-mass cluster members is strongly influenced by the Mg~\\emph{b} region, where we noted that the fit is poor. Masking Mg~\\emph{b} and relying on Balmer line indictors yields $\\Delta t = 0.0^{+0.2}_{-0.1}$~Gyr for the lower-mass subsample.\n\nIn each of these comparisons, we do not detect a difference between the field and cluster mean ages at the $\\sim 1\\sigma$ level, or about 0.5~Gyr and 0.3~Gyr for the more- and less-massive subsamples, respectively. Because the median redshift of the galaxies entering the Whitaker et al.~stacks is slightly lower than that of JKCS~041, comparing ages is not precisely the same as comparing formation times. However, the difference in median lookback time is $\\sim0.3$~Gyr for the massive\/redder subsample and only 0.1~Gyr for less-massive\/bluer examples; both are less than the statistical uncertainties. We also note that the mean ages derived above will not include any galaxies that were very recently truncated and are in transition to the quiescent region of the $UVJ$ plane.\n\nIn summary, the mean luminosity-weighted ages of the quiescent members of JKCS~041~varies with mass, with lower-mass galaxies having younger ages. The cluster members span a remarkably similar range of ages to that seen in quiescent field galaxies near the same redshift. Intriguingly, however, the line emission seen in quiescent field samples is absent in JKCS~041, at least among its more massive members where the high quality of the spectrum permits a comparison. We discuss the physical significance of these findings in Section~\\ref{sec:discussion}.\n\n\\section{Structure of Quiescent Galaxies: JKCS~041~Compared to the Field}\n\\label{sec:structure}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\linewidth]{galfits}\n\\caption{F160W\/F105W images (left panels) of the 15 confirmed quiescent members of JKCS~041~ordered by F160W flux, displayed with a logarithmic scaling. Center panels show logarithmically spaced F160W isophotes. Right panels show residuals of the S\\'{e}rsic fits to each F160W image, scaled linearly over $\\pm 23$~mag~arcsec${}^{-2}$. Pixels masked in the fits are set to zero. The cutout side length is $4\\arcsec \\approx 34$~kpc.\\label{fig:galfits}}\n\\end{figure*}\n\nTo gain insight into the role of the environment in the rapid structural evolution of quiescent galaxies at $z \\sim 2$, we now compare the structural properties of the members of JKCS~041~to their field counterparts. In addition to our \\emph{HST} imaging of the cluster, this comparison requires a large field sample. Furthermore, in order to minimize systematic differences, the structural measurements should be conducted following the same procedures in the cluster and field. The CANDELS data provide an excellent basis for such a comparison, since the survey has imaged a large area using \\emph{HST}\/WFC3 to a depth similar to our F160W observations. Here we assemble a sample of 225 galaxies spanning $z = 1.8 \\pm 0.3$ drawn from the CANDELS fields. Using this large sample, we are able to make a precise and homogeneous comparison between galaxy structure in JKCS~041~and the field.\n\n\\subsection{Structural measurements and field sample}\n\\label{sec:sizemethod}\n\nWe used \\texttt{Galfit} to fit 2D S\\'{e}rsic profiles to the F160W images of all spectroscopically confirmed quiescent cluster members (Figure~\\ref{fig:galfits}). The detailed procedures for PSF construction and masking or simultaneous fitting of nearby galaxies follow those described in N12. The only procedural difference is that we estimate the sky in a larger rectangular annulus around the object, with a width of 80 pixels, and mask objects more aggressively when the sky level is estimated. The derived structural parameters are listed in Table~\\ref{tab:sersic}. Throughout this section, we refer sizes using the semi-major axis $a = R_e^{\\rm maj}$ of the ellipse enclosing half of the light, and \\emph{not} a ``circularized'' effective radius $\\sqrt{ab}$ that is also frequently quoted in the literature. We prefer $R_e^{\\rm maj}$ because it is independent of inclination for oblate objects, which form one focus of our analysis, whereas the circularized radius is very sensitive to viewing angle for flattened systems. For the lowest-mass confirmed quiescent member (ID 255), we were unable to secure a reliable size measurement, since this galaxy is essentially unresolved. Based on our simulations, its size is likely $R_h \\lesssim 1~{\\rm pixel} \\approx 0.5$~kpc. Our comparison to the field is limited to galaxies having $M_* > 10^{10.7} \\msol$, so this low-mass galaxy does not enter our analysis.\n\nIn this section we refer to stellar masses $M_*^{\\rm tot}$ that are scaled to the total flux in the S\\'{e}rsic profile fit. This is preferable when constructing the mass--radius relation, since the size and luminosity are derived  consistently from the same light profile. For the largest galaxies, we note that $M_*^{\\rm tot}$ can exceed the {\\tt MAG\\_AUTO}-scaled masses $M_*^{\\rm AUTO}$ (Table~1) by up to 0.25 dex.\n\n\\begin{deluxetable}{cccccc}\n\\tablewidth{\\linewidth}\n\\tablecaption{S\\'{e}rsic Fits to Confirmed Quiescent Cluster Members} \n\\tablehead{\\colhead{ID} & \\colhead{$R_e^{\\rm maj}$ (kpc)} & \\colhead{$q$} & \\colhead{$n$} & \\colhead{$H_{160}^{\\rm tot}$} & \\colhead{$\\log M_*^{\\rm tot}\/M_{\\odot}$}}\n\\startdata\n272 & 14.7 & 0.71 & 6.8 & 20.03 & 11.96 \\\\\n376 & 5.00 & 0.70 & 6.5 & 20.84 & 11.70 \\\\\n286 & 5.27 & 0.83 & 8.0 & 21.17 & 11.68 \\\\\n356 & 10.6 & 0.97 & 7.7 & 20.71 & 11.62 \\\\\n355 & 4.72 & 0.56 & 2.7 & 20.65 & 11.58 \\\\\n352 & 2.45 & 0.74 & 5.2 & 21.56 & 11.34 \\\\\n411 & 0.85 & 0.57 & 4.1 & 21.93 & 11.22 \\\\\n657 & 1.56 & 0.91 & 3.2 & 21.45 & 11.18 \\\\\n289 & 0.83 & 0.65 & 3.8 & 21.97 & 10.97 \\\\\n447 & 3.13 & 0.81 & 3.3 & 21.95 & 10.88 \\\\\n317 & 1.43 & 0.47 & 1.9 & 22.30 & 10.81 \\\\\n281 & 0.89 & 0.75 & 3.0 & 22.61 & 10.80 \\\\\n375 & 0.62 & 0.95 & 3.4 & 22.64 & 10.79 \\\\\n359 & 1.47 & 0.86 & 6.9 & 22.29 & 10.77 \\\\\n255 & \\multicolumn{5}{l}{(unresolved --- see text)}\n\\enddata\n\\tablecomments{Stellar masses in the final column are scaled to the total S\\'{e}rsic magnitude and so differ from the \\texttt{MAG\\_AUTO}-scaled masses in Table~\\ref{tab:memberdata}. See Section~\\ref{sec:sizemethod} for estimates of uncertainties.\\label{tab:sersic}}\n\\end{deluxetable}\n\nOur field comparison sample is drawn from four of the CANDELS survey fields. We have augmented the UDS and GOODS-S catalogs in N12~by adding data in COSMOS and GOODS-N, where we make use of the NMBS and MOIRCS Deep Survey \\citep{Kajisawa11} photometry. In each field, photometric redshifts, stellar masses, and rest-frame colors were estimated using the same procedures described in Section~\\ref{sec:catalog}, based throughout on the BC03 models and a Salpeter IMF. S\\'{e}rsic profiles were fit to the CANDELS F160W images using the same methods applied to JKCS~041. Our field comparison sample consists of 225 galaxies with $M_* > 10^{10.7} \\msol$ in the redshift interval $z = 1.8 \\pm 0.3$ that are classified as quiescent according to their $UVJ$ colors. Galaxies within 1 Mpc of the known $z=1.62$ cluster at the edge of the UDS field (\\citealt{Papovich10,Tanaka10}; see Section~\\ref{sec:discussion}) were removed. For 17 galaxies in the field sample (7.5\\%) and 1 of the cluster members, the S\\'{e}rsic index reached the maximum value $n = 8$ allowed in our fits. Since the radii derived in such cases are often unreliable (see N12, \\citealt{Raichoor12}), we indicate these galaxies separately in our plots and omit the $n=8$ field galaxies when fitting the mass--radius relation.\n\nTo validate our fitting method, we inserted hundreds of simulated galaxies with S\\'{e}rsic profiles into the UDS and JKCS~041~images with a distribution of parameters similar to that in our sample. We found that $n$, $R_h$, and the total flux are recovered with negligible biases, i.e., less than a few percent. The typical $1\\sigma$ uncertainties in $R_h$ are $\\sigma_{R_h} = 10\\%$ for the majority of systems having $R_h < 0\\farcs5$, increasing to 17\\% for larger galaxies. In about 7\\% of cases, $R_h$ differs from the true value by more than factor of 1.5. The S\\'{e}rsic index $n$ is recovered with errors of $\\sigma_n = 0.4$ when $n < 5$, increasing to $\\sigma_n = 0.9$ for more extended profiles having $n = 5-7$. Total fluxes are recovered with a scatter of $\\sigma_{\\rm mag} \\simeq 0.1$~mag. These estimates can be applied to the measurements in Table~\\ref{tab:sersic}.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.8\\linewidth]{qplot_full}\n\\caption{Projected axis ratios $q$ as a function of stellar mass for the quiescent galaxies in our field sample (top panel) and in JKCS~041~(bottom). In the top panel, a grid of randomly selected cutouts having the corresponding $M_*$ and $q$ is shown, with the blue points denoting the actual parameters of the field galaxies and the blue line indicating the running mean. A representative error bar in shown in the lower panel, which includes only random uncertainties in $M_*$. Histograms in the right panels show the $q$ distributions with Poisson error bars. Red curves show the best-fitting two-component model described in the text: dotted and dashed curves denote the disk-like, oblate population and the spheroid population, respectively, while solid curves show their sum. The JKCS~041~members are best fit by a pure spheroid population, whereas about half of the field sample belongs to the oblate population in this model (see Figure~\\ref{fig:fobl}).\\label{fig:qplot}}\n\\end{figure*}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{fobl} \\hfill\n\\caption{Posterior probability density for the fraction $f_{\\rm obl}$ of quiescent galaxies that belong to the disk-like, oblate population, based on the model proposed by \\citet{Chang13}.\\label{fig:fobl}}\n\\end{figure}\n\n\\subsection{Shapes of Quiescent JKCS~041~Members versus the Field}\n\\label{sec:shapes}\n\nWe begin our structural comparison of quiescent field and cluster galaxies by considering their shapes. Figure~\\ref{fig:qplot} compares the projected axis ratios $q = b\/a$ of the two samples. The top panel shows that the field sample spans a wide range of shapes that extends to highly flattened systems with low $q$. This suggests that many quiescent field galaxies at $z \\sim 1.8$ harbor a significant disk component. A visual inspection of images of the systems having $q \\lesssim 0.5$ supports this conclusion. Other authors have noted evidence of significant disk-like structures in quiescent galaxies at $z > 1$, even at the highest stellar masses, based on both their projected axis ratio distribution \\citep{vanderWel11,Weinzirl11,Buitrago13,Chang13,Chang13b} and on results from two-component bulge\/disk decompositions \\citep{Stockton08,McGrath08,Bruce12,Papovich12}\n\nTurning to the JKCS~041~members in the lower panel of Figure~\\ref{fig:qplot}, there appear to be fewer flattened galaxies: only one, for example, has $q < 0.5$. Quantitatively, the difference in mean projected axis ratios is $\\langle q_{\\rm JKCS} \\rangle - \\langle q_{\\rm field} \\rangle = 0.11 \\pm 0.04$, and we derive a $p$-value of 0.03 from a permutation test that indicates this difference is moderately significant.\\footnote{The $p$-value is the fraction of random permutations of the field and cluster identifications for which $\\langle q_{\\rm JKCS} \\rangle - \\langle q_{\\rm field} \\rangle$ exceeds that which is observed in absolute value (i.e., a two-sided test).} This suggests a probable difference in the underlying morphological composition of the cluster and field galaxies.\n\nMore physical insight can be gained from the $q$ distribution using a model for the distribution of intrinsic galaxy shapes. \\citet{Chang13} have shown that the $q$ distribution of quiescent galaxies can be understood as arising from a two-component population viewed at random angles. One component consists of mildly triaxial galaxies that are nearly spherical, and the other consists of a highly flattened, oblate population. In the following, we refer to these as the spheroid and disk-like components, respectively, although it should be kept in mind that the quiescent disk-like galaxies are likely composite objects containing significant bulges \\citep{Bruce12} and may be related to the lenticular population at lower redshift; we note that these passive disk-like galaxies appear to span a range of S\\'{e}rsic indices $n \\approx 1 - 5$. This decomposition of the $q$ distribution is not unique, but it is motivated by more detailed photometric and kinematic classifications at lower redshift and serves as a useful starting point for understanding the $z > 1$ population. Chang et al.~showed that the fraction $f_{\\rm obl}$ of quiescent systems belonging to the disk-like population appears to be roughly independent of mass over the range of masses and redshifts relevant for the present paper. In support of this, we see no trend in $\\langle q \\rangle$ with mass in Figure~\\ref{fig:qplot}.\n\nOverlaid on the histograms in Figure~\\ref{fig:qplot} are fits based on this two population model.\\footnote{We use the distribution of intrinsic axis ratios within the oblate and triaxial populations from the first entry in Table 3 of \\citet{Chang13}.} Figure~\\ref{fig:fobl} shows the inferred fraction $f_{\\rm obl}$ of disk-like galaxies. We find that about half ($f_{\\rm obl} = 0.52 \\pm 0.08$) of the $z \\sim 1.8$ field sample belongs to the disk-like population, consistent with Chang et al., whereas in JKCS~041~the $q$ distribution is best fit with a pure spheroid population ($f_{\\rm obl} = 0$), with $f_{\\rm obl} < 0.28$ at 68\\% confidence. Comparing the two samples, we find that $f_{\\rm obl}$ is lower in the cluster at 90\\% confidence. \n\n\\subsection{Sizes and Radial Profiles of Quiescent JKCS~041~Members versus the Field}\n\\label{sec:sizes}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.7\\linewidth]{simple_mass_size} \\\\\n\\includegraphics[width=0.7\\linewidth]{mass_size_qtrend}\n\\caption{{\\bf Top:} stellar mass--$R_e^{\\rm maj}$ relation for quiescent galaxies in JKCS~041~(black symbols) and in our $z = 1.8 \\pm 0.3$ field sample (blue). The solid line shows the field relation at $z = 1.8$ (Equation~\\ref{eqn:fit_full_field}), and the dashed red line shows the $z \\sim 0$ relation for early type galaxies from \\citet{Shen03}, where we have converted their circularized radii to $R_e^{\\rm maj}$ estimates by assuming a mean axis ratio of $\\langle q \\rangle \\approx 0.75$ \\citep[e.g.,][]{Padilla08}. Open symbols denote field galaxies best fit with $n = 8$, whose sizes may be unreliable. {\\bf Bottom:} stellar mass--$R_e^{\\rm maj}$ relation for our color-selected sample of quiescent field galaxies (black symbols with error bars) is compared to that defined by the subset of flattened galaxies with $q < 0.4$ (green) and to our inferred relation for the spheroid population (blue). Bands indicate $1\\sigma$ uncertainties, and gray circles show the JKCS~041~members as in the top panel.\\label{fig:simpleMSR}}\n\\end{figure*}\n\nThe stellar mass--radius relations for the quiescent field galaxies and the quiescent JKCS~041~members are shown in Figure~\\ref{fig:simpleMSR}. As a first step toward comparing the two, we fit a linear relation with Gaussian scatter $\\mathcal{N}(\\sigma)$ to the field sample\n\\begin{align}\n\\log R_e^{\\rm maj} \/ {\\rm kpc}  &= \\alpha + \\beta \\log {\\rm M}_*^{\\rm tot} \/ 10^{11} {\\rm M}_{\\odot} \\nonumber\\\\\n & - 0.26(z-1.8) + \\mathcal{N}(\\sigma),\\label{eqn:fit_full_field}\n\\end{align}\nwhere $\\beta = 0.61 \\pm 0.07$, $\\alpha = 0.22 \\pm 0.02$, and $\\sigma = 0.23 \\pm 0.01$. Here we have taken into account the mild redshift evolution $\\partial \\log R \/ \\partial z = -0.26$ expected within field sample based on the results by N12. This fit is shown by the blue line. Comparing the JKCS~041~members to the mean field relation, there is no evidence for a systematic difference between the two: $\\langle \\Delta \\log R_e^{\\rm maj} \\rangle = 0.01 \\pm 0.09$.\\footnote{Throughout, the uncertainty in the mean $\\langle \\Delta \\log R_e^{\\rm maj} \\rangle$ is estimated as $\\sqrt{\\sigma_{\\rm clus}^2 + \\sigma_{\\rm field}^2}$. Here the uncertainty $\\sigma_{\\rm clus} = 0.23 \/ \\sqrt{N_{\\rm clus}}$ in the mean cluster galaxy offset is based on the scatter seen in the field relation (Equation~\\ref{eqn:fit_full_field}), and the uncertainty $\\sigma_{\\rm field}$ in the mean field relation is derived from the fit parameters.} There is a hint, however, of a mass-dependent trend: the five most massive galaxies are all displaced above the mean field relation, by an average $\\langle \\Delta \\log R_e^{\\rm maj} \\rangle = 0.21 \\pm 0.12$. \n\nSince the axis ratio distribution suggests that the morphological mix of quiescent galaxies may be different in JKCS~041~and the field (Section~\\ref{sec:shapes}), it is important to consider what effect this may have on a comparison of sizes. If the morphological compositions indeed differ, then a simple comparison of radii --- such as that performed above --- will conflate the sizes of spheroids and disks, rather than isolating the effect of the environment on galaxies of comparable morphologies. While nearly edge-on disk-like galaxies are easily identified, it is not easy to locate the same systems viewed at lower inclination. A division in S\\'{e}rsic index is not very effective, since flatted ($q \\lesssim 0.4$) quiescent galaxies are seen in the field over a wide range of $n \\approx 1-5$. Therefore, rather than attempting to morphologically classify the individual galaxies in the distant field sample, we proceed from the model of the underlying shape distribution discussed in Section~\\ref{sec:shapes} and follow its implications for the mass--radius relation.\n\nThe lower panel of Figure~\\ref{fig:simpleMSR} demonstrates that the flattened quiescent field galaxies having $q < 0.4$ (green symbols) appear to follow a different mass--radius relation: they have smaller $R_e^{\\rm maj}$ than the bulk field sample (black symbols), and increasingly so at higher masses.\\footnote{For a single population of triaxial objects, the smallest $q$ is seen when longest and shortest axes are in the plane of the sky, and the projected $R_e^{\\rm maj}$ is maximal. The fact that small-$q$ galaxies have \\emph{smaller} $R_e^{\\rm maj}$ thus supports the notion that they are a distinct population with a different size distribution. We also emphasize that our discussion is confined to \\emph{quiescent} galaxies, and star-forming disks are well known to have larger sizes (e.g., \\citealt{Williams10}, N12, and references therein). \\citet{Chang13} present evidence that highly-inclined galaxies with quiescent $UVJ$ colors are transparent and are not preferentially affected by obscured star formation (excluding the small fraction of MIPS sources does not alter the $q$ distribution).} We expect the $q < 0.4$ galaxies to be a fairly pure ($f_{\\rm obl} = 0.89$, according to the decomposition in Section~\\ref{sec:shapes}) but incomplete sample of the disk-like population. Since $R_e^{\\rm maj}$ is independent of inclination for transparent, oblate objects, those galaxies in the disk-like population that are viewed more nearly face-on, i.e., with higher $q$, should follow the same mass--radius relation. Assuming that a fraction $f_{\\rm obl} = 0.52 \\pm 0.08$ of quiescent field galaxies --- of all inclinations --- belong to this disk-like population, it is then straightforward to estimate the mass--radius relation for the spheroids. Specifically, at each mass we consider the mean $\\langle \\log R_e^{\\rm maj} \\rangle$ as a weighted average: $f_{\\rm obl} \\langle \\log R_{e,{\\rm obl}}^{\\rm maj} \\rangle + (1 - f_{\\rm obl}) \\langle \\log R_{e,{\\rm sph}}^{\\rm maj} \\rangle$. \n\nThe blue band in Figure~\\ref{fig:simpleMSR} shows the resulting constraint on the relation for quiescent field spheroids. If the cluster galaxies are indeed dominated by spheroids, as suggested by their axis ratio distribution, it is clear that any difference between the field and cluster relations at high masses is much reduced. Quantitatively, \nthe sizes of the five most massive cluster members do not differ systematically ($\\langle \\Delta \\log R_e^{\\rm maj} \\rangle = -0.06 \\pm 0.19$) from the field spheroid relation, although the uncertainties are necessarily increased, and when considering the full range of masses, the cluster members are slightly smaller but still consistent with the field spheroids ($\\langle \\Delta \\log R_e^{\\rm maj} \\rangle = -0.14 \\pm 0.10$). We regard our morphological separation of the mass--radius relation of quiescent galaxies as a first approximation, since it relies on a very simple model for the underlying distribution of shapes (Section~\\ref{sec:shapes}; \\citealt{Chang13}) and its apparent invariance with mass at $z \\sim 2$. More data is needed to test this model and its implication that the fraction of massive, quiescent galaxies with significant disk components increases with redshift. However, it is clear that a difference in the morphological mixtures of the field and cluster samples could significantly affect comparisons of their mass--radius relations.\n\nIn summary, there is no significant difference overall between the mass--radius relation defined by the quiescent JKCS~041~members and that defined by our coeval field sample. There is a weak hint of a mass-dependent trend in which the most massive cluster members are offset to larger radii, if all color-selected quiescent galaxies are considered irrespective of morphology. However, a closer inspection reveals that this may arise because the cluster population is richer in spheroids, and spheroids are ``larger'' than quiescent disk-like galaxies. Figure~\\ref{fig:light_profiles} supports this conclusion via a direct comparison the surface mass density profiles of the JKCS~041~members to the field galaxies. Here we consider only field galaxies with $q > 0.45$ to better match the cluster sample. The \\emph{HST} PSF was deconvolved from the observed F160W light profile using the technique proposed by \\citet{Szomoru10}, and the resulting light profile was  converted to a stellar mass profile using a constant $M_*\/L$ for each galaxy. There is no clear difference in the mass profiles in the field and JKCS~041~samples. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{light_profiles}\n\\caption{Azimuthally averaged surface mass density $\\Sigma_*$ profiles of JKCS~041~members (red lines), plotted down to a limiting surface brightness of $H_{160} = 26$~mag~arcsec${}^{-2}$ and PSF-deconvolved as described in the text. In each of three stellar mass bins, we compare to the population of quiescent field galaxies at $z\\sim1.8$ that have $q > 0.45$, excluding highly flattened galaxies that are absent in the cluster sample. The thick dashed line shows median surface density profile of the field sample derived from our S\\'{e}rsic fits, and the gray region encloses 68\\% of the field profiles at each radius.\\label{fig:light_profiles}}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{othersizestudies_ev}\n\\caption{Comparison of published results on the environmental dependence of the mass--radius relation of quiescent galaxies. Each point represents the mean offset $\\Delta \\log R_e$ from the field relation. For studies of individual clusters, listed in the upper-left legend, we compare to Equation~\\ref{eqn:fit_full_field}. For ensembles of clusters \\citep{Delaye14} and studies of group-scale overdensities \\citep[][dashed error bars]{Cooper12,Lani13}, the published offsets from the authors' field relation are quoted directly. The shaded band denotes the weighted mean of the $z > 1.6$ clusters and its $1\\sigma$ uncertainty. Appendix~C describes our method for compiling and harmonizing these diverse data sets and describes systematic uncertainties inherent in such a comparison.\\label{fig:othersizestudies}}\n\\end{figure}\n\n\\subsection{Comparison to other studies of the environmental dependence of the mass--radius relation}\n\\label{sec:literatureMSR}\n\nSeveral recent studies of the environmental dependence of galaxy sizes at high redshifts are compared in Figure~\\ref{fig:othersizestudies}. The references in the upper-left legend refer to individual clusters, for which we have compiled published structural measurements of their quiescent or early type members. In Section~\\ref{sec:discussion} we review the bulk physical properties of the $z > 1.6$ clusters themselves; our focus here on the mass--radius relation. To synthesize these published results into a quantity that can be compared as directly as possible, given the diversity of samples and methods (see Appendix~C for details), we compute the mean offset $\\langle \\Delta \\log R_e^{\\rm maj} \\rangle$ between the quiescent members of each cluster and the field relation in Equation~\\ref{eqn:fit_full_field}.  We regard Figure~\\ref{fig:othersizestudies} as a first step toward synthesizing results from various high-$z$ studies, but caution that systematic differences in measurement techniques may affect a comparison of our field sample with other authors' cluster data; some of these are discussed in Appendix~C.\n\nConsidering the $z > 1.6$ clusters first, \\citet[][see also \\citealt{Bassett13}]{Papovich12}, \\citet{Zirm12} and \\citet{Strazzullo13} have all remarked on evidence for larger sizes among the quiescent members of the clusters they studied. (We note that many of these members are actually photometric candidates, whereas the members of JKCS~041~are confirmed by grism redshifts.)  Based on Figure~\\ref{fig:othersizestudies}, we regard the present evidence for a variation in the mass--size relation in the cores of these most distant clusters and proto-clusters as very marginal. On the other hand, present sample sizes are too small to rule out a modest size enhancement of $\\sim 0.05$~dex. \nMoving to lower redshifts, \\citet{Delaye14} studied 9 clusters at $z = 0.8-1.4$ along with a field sample selected and analyzed in a homogeneous way. They found significant evidence for an offset in the mass--radius relation by $\\Delta \\log R_e \\simeq 0.1$~dex. In the two $z \\sim 1.2$ clusters studied by \\citet{Rettura10} and \\citet{Raichoor12}, however, we find no significant offset.\n\n\\citet{Bassett13} noticed that the slight trend for the quiescent candidate members of the cluster they studied (IRC-0218A, $z=1.62$) to have larger $R_e$ and smaller $n$ was mostly driven by a population of disk galaxies located at large cluster-centric radii $R \\approx 1-1.5$~Mpc.\\footnote{As described in Appendix~C, we include only the members of this cluster within $R < 1$~Mpc in Figure~\\ref{fig:othersizestudies} for a better comparison with other data sets.} Although their remark that differences in morphology can influence comparisons of the mass--radius relation is similar to our findings, we note that that the nearly pure disks they discuss ($n \\sim 1$) have \\emph{larger} $R_e$ than the mean quiescent galaxy --- consistent with faded spirals that have been starved of gas during infall --- whereas the disk-like quiescent field population discussed in Section~\\ref{sec:sizes} is offset to \\emph{smaller} $R_e$ and exhibits a wide range of $n$ indicating a significant build-up of bulges (see a similar trend in \\citealt{HuertasCompany13}). Altogether, this points to a complex mixture of morphologies varying from the field to the cluster outskirts and core. \n\nIn addition to these cluster studies, two recent studies have examined the dependence of the mass--radius relation on local density in blank field surveys, where the densest regions are typically groups or low-mass clusters. These results are distinguished with dashed error bars in Figure~\\ref{fig:othersizestudies}. \\citet{Cooper12} found a size enhancement of $\\Delta \\log R_e \\simeq 0.1$~dex among early type galaxies in the densest regions in the DEEP3 survey fields. In the UDS field, \\citet{Lani13} detected a similar enhancement that was dominated by the most massive and highest-redshift galaxies. This is the regime where we found that differences in the morphological mix could affect our interpretation of JKCS~041. Lani et al.~considered such a possibility and tested it by cutting their sample in S\\'{e}rsic index $n$. Although this is a reasonable first approach, we find the connection between the oblate, disk-like quiescent population and S\\'{e}rsic index to be loose (Section~\\ref{sec:sizes}). Additionally, while the $M_*$--$R_e^{\\rm maj}$ relation likely varies with $q$ (Figure~\\ref{fig:simpleMSR}), we find no such dependence on $n$ for quiescent galaxies. In future work, it would be useful to consider the $q$ distributions of samples whose mass--radius relations are being compared.\\footnote{Interestingly, further testing by C.~Lani et al. (2013, private communication) following the submission of this paper has shown that their results are not affected by an axis ratio cut of $q > 0.4$.}\n\nIn contrast to these $z \\gtrsim 1$ studies, there appears to be \\emph{no} dependence at $z \\sim 0$ of the size of early type galaxies on local density, halo mass, or position within the halo \\citep{Weinmann09,Guo09,Nair10,HuertasCompany13local}. These $z \\sim 0$ results, however, have been challenged by \\citet{Valentinuzzi10}, who claim an excess of \\emph{compact} massive galaxies in local clusters; interestingly, these compact galaxies show a tendency to have S0 morphologies. The only clear point of agreement is that the BCGs in very massive clusters are exceptionally large \\citep[e.g.,][]{Bernardi07}. \n\nIn summary, the evidence for environmental variation in the mass--radius relation in the most distant $z > 1.6$ clusters is still limited by small samples. At $z \\sim 1$ there is good evidence for an offset to larger sizes in the cluster sample studied by \\citet{Delaye14}, as well as in group-scale overdensities \\citep{Cooper12,Lani13}. At $z \\sim 0$, most evidence points toward a remarkable independence of early type galaxy structure on environment. There are contrary indications for many secondary trends that might shed light on an underlying physical picture: are galaxy sizes enhanced primarily in distant clusters' cores (Delaye et al.) or their outskirts \\citep{Bassett13}? Is the enhancement stronger for higher (Lani et al.) or lower mass (Delaye et al.) galaxies? Furthermore, the evolutionary connection between $z \\gtrsim 1$ results and the precise constraints available at $z \\sim 0$ remains unclear.\n\n\\section{Discussion}\n\\label{sec:discussion}\n \n\n\n\n\n\n\n\\begin{deluxetable*}{lccccl}\n\\tablewidth{\\linewidth}\n\\tablecaption{JKCS~041~Compared to Other Spectroscopically Verified $z > 1.6$ Proto-clusters and Clusters} \n\\tablehead{\\colhead{Cluster} & \\colhead{$z$} & \\colhead{Mass $M_{200}$} & \\colhead{Diffuse X-ray flux} & \\colhead{$N_{\\rm spec}$ \/} & \\colhead{References} \\\\\n\\colhead{} & \\colhead{} & \\colhead{$(\\msol)$} & \\colhead{(erg cm${}^{-2}$ s${}^{-1}$)} & \\colhead{$N_{\\rm spec~Q}$} & \\colhead{}}\n\\startdata\nJKCS~041${}^{\\dagger}$ & 1.80 & $(2-3) \\times 10^{14}$ & $2 \\times 10^{-14}$ & 19 \/ 15 & This work, \\citet{Andreon13} \\\\\nIRC-0218A${}^*$ & 1.62 & $(2-7) \\times 10^{13}$ & $\\sim 3 \\times 10^{-15}$ & 11 \/ 3 & P10, T10, P12, B13, L13, Pi12 \\\\\nSpARCS J022427-032354 & 1.63 & \\ldots & \\ldots & 12 \/ 3 & \\citet{Muzzin13} \\\\\nIDCS J1426+3508${}^{\\dagger}$ & 1.75 & $4 \\times 10^{14}$ & $3 \\times 10^{-14}$ & 7 \/ 2 & S12, B12 \\\\\nIDCS J1433.2+3306${}^{\\dagger}$ & 1.89 & $\\sim 10^{14}$ & \\ldots & 7 \/ 2 & B07, Z12 \\\\\nCl J1449+0856${}^{\\dagger}$ & 2.00 & $5 \\times 10^{13}$ & $9 \\times 10^{-16}$ & 22 \/ 7 & G11, G13, S13 \\\\\nMRC 0156-252 & 2.02 & \\ldots & $\\sim 2 \\times 10^{-15\\ddag}$ & 10 \/ 1 & O05, Ga13 \\\\\nMRC 1138-262 & 2.16 & \\ldots & \\ldots & 11 \/ 4 & Zi12, T13, and references therein\n\\enddata\n\\tablecomments{$N_{\\rm spec}$ is the number of spectroscopic members, of which $N_{\\rm spec~Q}$ are quiescent. Masses and X-ray fluxes are only indicative, since various energy bands, apertures, and scaling relations are used. References: P10, P12: \\citet{Papovich10,Papovich12}, Pi12: \\citet{Pierre12}, S12: \\citet{Stanford12}, B12: \\citet{Brodwin12}, B07: \\citet{Brodwin07}, Z12: \\citet{Zeimann12}, G11, G13: \\citet{Gobat11,Gobat13}, S13: \\citet{Strazzullo13}, T10, T13: \\citet{Tanaka10,Tanaka13}, B13: \\citet{Bassett13}, L13: \\citet{Lotz13}, Zi12: \\citet{Zirm12}, O05: \\citet{Overzier05}, Ga13: \\citet{Galametz13}. ${}^{\\dagger}$Based on WFC3 grism data.\\label{tab:otherclusters} ${}^*$Also called XMM-LSS J02182-05102. ${}^{\\ddag}$The X-ray emission is suspected to be associated with the radio galaxy rather than thermal ICM emission.}\n\\end{deluxetable*}\n \nIn addition to JKCS~041, seven overdensities containing a red galaxy population have been identified at $z>1.6$ and confirmed spectroscopically.\\footnote{In addition to these, we note that \\citet{Spitler13} recently discovered a $z = 2.2$ cluster candidate containing a red galaxy population using medium-band photometric redshifts.} Although all have been labeled ``clusters'' or ``proto-clusters,'' these are in fact a diverse set of structures that span a wide range of masses and evolutionary states. The properties of these systems are summarized in Table~\\ref{tab:otherclusters}. \n\nJKCS~041~is remarkable in several ways. First, it appears to be a fairly massive cluster for its redshift. As \\citet{Andreon13} describe, the X-ray luminosity, X-ray temperature, and galaxy richness give mass estimates of $\\log M_{200} \\simeq 14.2 - 14.5$ that are reasonably consistent given the uncertainties in the evolution of the relevant scaling relations. \\citet{Culverhouse10} report the non-detection of an SZ signal in the direction of JKCS~041, corresponding to an upper limit of $\\log M_{200} \\lesssim 14.5$. Therefore, given the depth of the observation, this non-detection is still consistent with the range of independent X-ray-- and richness--based estimates.\\footnote{Here we use the \\citet{Bonamente08} scaling relation between $Y_{2500}$ and $M_{2500}$ to estimate $\\log M_{2500} < 13.7$, which corresponds to $\\log M_{200} < 14.5$ assuming the  \\citet{Duffy08} mass--concentration relation.} While deeper SZ observations of JKCS~041~will be very valuable, we conclude that all present data are consistent with a mass in the range $M_{200} \\simeq (2-3) \\times 10^{14} \\msol$. Compared to the other $z > 1.6$ clusters in Table~\\ref{tab:otherclusters} with estimated masses, JKCS~041~appears to be the most massive other than IDCS~J1426+3508, which is possibly more massive by a factor $\\sim 1.5 - 2$.\n\nSecond, we have been able to confirm a large number of member galaxies via grism redshifts (see Table~\\ref{tab:otherclusters}), especially those that are quiescent. This has allowed us to construct a spectroscopic sample that is fairly complete at radii $R < R_{500}$ and masses $M_* > 10^{10.6} \\msol$, and is thus suitable for studying environmental effects on the member galaxies. We emphasize that comparing numbers of spectroscopic members is not the same as comparing the underlying galaxy populations, given the diversity of observations and analysis methods used in Table~\\ref{tab:otherclusters}. However, the bright end of the red sequence is quite rich in JKCS~041.  For a detailed comparison of its red sequence morphology with those of other high-redshift structures, we refer to \\citet{Andreon13}.\n\nMotivated by our unique data on the quiescent population in JKCS~041, we have compared their structural and stellar population properties to coeval field samples. Considering first the structure and morphology of the cluster members, we found some evidence for a lack of quiescent disk-like galaxies relative to the field population. In the context of cluster studies at lower redshift, this is consistent with the idea that the cluster ellipticals are formed early ($z > 2$) in dissipative mergers, probably continuing to evolve via dry mergers, whereas many S0's are formed much later at $z \\lesssim 0.5$ and decline in numbers toward higher redshifts \\citep[e.g.,][]{Dressler97,Andreon97,Smith05,Postman05,Poggianti09}. \nAn interesting related development is the observation that the fraction of quiescent galaxies in the field with disk-like components appears to \\emph{increase} at $z > 1$, particular among massive ($M_* > 10^{11} \\msol$) systems (see references in Section~\\ref{sec:shapes}). The relative lack of these compact, disk-like quiescent galaxies in JKCS~041~suggests that the cluster environment either inhibits their formation or else is effective in destroying the more loosely bound disk material, perhaps through tidal stripping or galaxy--galaxy encounters that build up the bulge. Larger samples of distant clusters in a range of evolutionary stages are needed to verify this trend.\n\nComparing the radial profiles of the cluster members to their field counterparts, we detect no statistically significant differences overall, but found a hint of a trend for larger effective radii among the most massive cluster members. One interpretation, which has been promoted in studies of other $z > 1.6$ clusters and proto-clusters \\citep{Zirm12,Papovich12,Bassett13,Lotz13}, is that size growth proceeds at an accelerated rate in the cluster environment, perhaps due to a higher rate of mergers or a higher fraction that are dry. We cannot rule out this possibility, but we note that present constraints in these most distant clusters remain statistically weak (Section~\\ref{sec:literatureMSR}). Furthermore, in the case of JKCS~041, we found that a difference in the morphological mixture of color-selected quiescent galaxies relative to the field may account for our observations just as well. Although this explanation also points toward environment-dependent evolution, it suggests a more nuanced picture in which bulge growth and morphological transformation may play a role in shaping the mass--radius relation in clusters, and not only a pure acceleration of ``inside-out'' spheroid growth.\n\nA weak environmental dependence of size among quiescent galaxies of the same mass and morphology would  indicate that either the galaxy merger rate does not vary substantially among the environments sampled, or that the rate of size growth is decoupled from the merger activity. This would be surprising given that mergers are thought to be the prime driver of spheroid growth (see \\S1). Presently, however, it is not clear how to connect observations of the mass--radius relation in clusters at different redshifts into an evolutionary sequence. As discussed in Section~\\ref{sec:literatureMSR}, results at $z \\gtrsim 1.5$ are not conclusive, the $z \\sim 1$ study with the most statistical power \\citep{Delaye14} indicates that cluster members are enlarged by $\\Delta \\log R_e \\approx 0.1$~dex, while at $z \\sim 0$ there seems to be no relation between the structure of early type galaxies and their local environment or halo mass. One possibility is that cluster members experience an initially enhanced rate of galaxy--galaxy encounters and mergers during infall, as the cluster is forming, while the virialization of the cluster and the resulting high velocity dispersion then inhibits future merging (see, e.g., \\citealt{Lotz13} and \\citealt{Delaye14}). In this picture, the mass--radius relation of cluster members is offset to larger $R_e$ at high redshift, while at later times the field galaxies ``catch up'' and this offset declines. It will be interesting to test this hypothesis as larger samples of distant clusters and richer data sets become available.\n\nWhile high-$z$ studies have used local density or cluster membership to quantify the environment, a galaxy's status as central galaxy in its dark matter halo may be more physically relevant. Central galaxies are expected grow more rapidly than satellites in some models, and they benefit from the accretion of stars that are tidally stripped from disrupted sinking satellites \\citep[e.g.,][]{Shankar13}. This process of ``cannibalism'' becomes increasingly important in higher halo mass, with the giant BCGs being the most extreme examples. The BCG of JKCS~041~indeed has the most extended light profile of all the cluster members, and it is the third nearest of the spectroscopic members to the cluster center. The BCG appears similar to that of the massive \\citet{Stanford12} cluster at $z=1.75$, which is also exceptionally luminous and extended ($R_e = 18$~kpc).\n\nA complementary approach is to quantify the rate of galaxy interactions and mergers more directly. \\citet{Lotz13} indeed inferred a high ongoing merger rate --- exceeding that in the field by a factor of 3--10 --- in IRC-0218A at $z=1.62$, based on their estimation that $57^{+13}_{-14}\\%$ of the massive proto-cluster members have double nuclei or a close satellite galaxy. By visual inspection of the 17 spectroscopic members of JKCS~041~in our continuum-selected sample (Figure~\\ref{fig:galfits}), we find that 3, i.e., $18^{+12}_{-6}\\%$, have close companions within the same search radius used by Lotz et al.~(20 kpc comoving).\\footnote{These are IDs 376 and 375, which are paired with one another and a faint, diffuse blue system (see Figure~\\ref{fig:galfits}), and ID 286.} Although a full analysis would require accounting for projected pairs in the cluster, this suggests a lower rate of ongoing mergers in JKCS~041, consistent with the latter being in a more dynamically evolved state.\n\nTurning to the stellar populations of the galaxies in JKCS~041, we found a high fraction of quenched systems compared to coeval field galaxies of the same mass (Figure~\\ref{fig:fq}). Elevated quiescent fractions $f_Q$, indicating the early onset of a star-formation--density relation, have been reported in the cores of other $z > 1.6$ clusters  \\citep{Quadri12,Strazzullo13}. When comparing our results with others, it is important to bear in mind several factors. First, some studies have emphasized the presence of galaxies in $z \\gtrsim 1.4$ cluster cores that have unusually high levels of star formation compared with cluster galaxies at lower redshift \\citep{Hilton10,Fassbender11}. While we also have located two massive galaxies with SFRs~$\\sim 140 \\msol$~yr${}^{-1}$ (Section~\\ref{sec:galaxyprops}) in the core of JKCS~041, we emphasize that they still represent a lower fraction of the galaxy population than in the field. Second, our grism-based study is confined to relatively massive galaxies in the cluster core ($M_* > 10^{10.6} \\msol$, $R < R_{500} \\approx 500$~kpc). Measurements of $f_Q$ that extend to lower stellar masses and larger cluster-centric radii are expected to be lower. Finally, there is likely a significant variation in $f_Q$ from cluster to cluster \\citep[e.g.,][]{Brodwin13}, and the color-based selection method used to discover JKCS~041~may prefer higher-$f_Q$ clusters relative to a cluster mass-limited sample. What we have clearly shown is that the cluster core environment does affect the fraction of massive galaxies that are quenched by $z=1.8$ in at least some clusters.\n\nWhen considering the physical processes responsible for truncating star formation, it is common to distinguish internal quenching mechanisms (often referred to as mass- or self-quenching) from environmentally related processes that correlate with the local density or the position of a galaxy within its halo \\citep[e.g.,][]{Peng10}. The clear signature of the environment on star formation activity in JKCS~041~at $z = 1.8$ implies that truncation by cluster processes has been fairly rapid, since the galaxies must have fallen into the cluster fairly recently (see also \\citealt{Quadri12}). Some semi-analytic models in fact predict the disappearance of environmental quenching beyond $z \\gtrsim 1.5$ \\citep{McGee09}, when the $\\sim 2$~Gyr timescale for stripping of hot halo gas (``strangulation'') exceeds the time for which the necessary dense ICM has existed. Observations of a star formation--density relation at earlier epochs suggests that more rapid quenching mechanisms may be at work, such as ram-pressure stripping.\n\nAlthough roughly half of the spectroscopic members of JKCS~041~have been quenched by environmentally related processes (Section~\\ref{sec:fq}), we nonetheless found that the mean ages of these galaxies does not differ greatly from similarly selected samples in the field. This indicates that the quenching mechanism had no large effect on \\emph{when} truncation occurred.\nThis finding is consistent with the idea that the environment modulates the \\emph{fraction} of quiescent systems without much affecting their ages. Evidence at lower redshift for a null or weak ($\\lesssim 0.4$~Gyr) environmental dependence of age among quiescent systems comes from studies of spectroscopic age diagnostics \\citep{Thomas10,Moresco10,Muzzin12} and spectral energy distributions \\citep{Andreon96,Raichoor11} at $z \\simeq 0 - 1.2$, as well as from the evolution of the fundamental plane in clusters and the field at $z < 1.3$ \\citep{vanDokkum07}. Our study extends earlier work by probing cluster galaxy ages through spectral diagnostics close to the epoch of their star formation and comparing these to similar observations of coeval field systems.\n\nThere are no AGN members with bright optical line emission in the core of JKCS~041, as are present in several other $z > 1.6$ clusters \\citep[e.g.,][]{Stanford12,Zeimann12,Gobat13}. Much fainter line emission can be reached in our composite spectra. Interestingly, there is no sign of the centrally concentrated, faint emission in H$\\beta$ and [\\ion{O}{3}] that was seen by \\citet{Whitaker13} in their composite spectra of quiescent field galaxies. Equally strong line emission would have been detected in our stack of $M_* > 10^{11} \\msol$ cluster members. If the field emission traces star formation, this finding would indicate that the dead cluster members lack the residual nuclear star formation present in field samples. Whitaker et al.~suggest that a LINER-type spectrum is more likely, given their estimate of the [\\ion{O}{3}]\/H$\\beta$ line ratio and the line luminosity. At $z \\sim 0$ the prevalence of faint [\\ion{O}{3}] emission does not decrease in denser environments \\citep{Kauffmann04}, so such a trend at $z \\sim 2$ would be intriguing if verified in other clusters.\n\n\\section{Summary}\nBased on our \\emph{HST} WFC3 imaging and grism observations of JKCS~041, along with associated multi-wavelength data, we conclude:\n\\begin{enumerate}\n\\item JKCS~041~is a genuine rich, X-ray luminous cluster at $z = 1.80$, confirmed through the spectroscopic identification of 19 members that are spatially aligned with diffuse X-ray emission. The spectroscopic members include 15 quiescent galaxies, the largest number yet confirmed in any $z > 1.6$ cluster. Five of these are very massive galaxies having $M_*^{\\rm tot} = 10^{11.6-12} \\msol$.\n\n\\item High-quality composite grism spectra of the quiescent cluster members allow us to measure their stellar ages via the strengths of the H$\\delta$, H$\\gamma$, H$\\beta$ and Mg~\\emph{b} absorption lines. Less massive quiescent members with $M_* < 10^{11} \\msol$ have mean luminosity-weighted ages of $0.9^{+0.2}_{-0.1}$ Gyr, while more massive galaxies are older ($1.4^{+0.3}_{-0.2}$ Gyr).\n\n\\item Comparing the spectra of the quiescent cluster members to those of similarly-selected field galaxies studied by \\citet{Whitaker13}, we find that the field and cluster samples span a very similar range of ages. At the same time, the fraction of quenched galaxies at fixed stellar mass is much higher in JKCS~041. This implies that the cluster environment is responsible for quenching of a substantial fraction of massive galaxies in JKCS~041, but that the mode of quenching (environmental versus internal) does not have a large effect on \\emph{when} star formation is truncated within the $\\sim 0.3-0.5$~Gyr uncertainties in our comparison.\n\n\\item The centrally concentrated H$\\beta$ and [\\ion{O}{3}] emission seen by Whitaker et al.~in median spectra of quiescent field galaxies is absent in the JKCS~041~members, at least among the more massive galaxies ($M_* > 10^{11} \\msol$) where the high quality of the grism spectra permit a detailed comparison.\n\n\\item Comparing the quiescent members of JKCS~041~to a large sample of coeval field galaxies, we find that the distribution of projected axis ratios suggests a lower fraction of disk-like systems among quiescent galaxies in the cluster.\n\n\\vspace{-0.2cm}\n\\item We find no statistically significant difference in the mass--radius relation or in the radial mass profiles of the quiescent cluster members compared to their field counterparts. While the most massive cluster members ($M_* > 10^{11.5} {\\rm M}_{\\odot}$) are marginally offset from the field mass--radius relation when considering all quiescent systems together, this apparent difference is weakened when the samples are better matched in morphology. Larger samples are still needed to clarify the structure of galaxies in distant, forming clusters, as well as to connect these results to studies at lower redshift.\n\n\\end{enumerate}\n\n\\acknowledgments\nWe thank the referee for a detailed report. It is a pleasure to acknowledge insightful conversations with Marc Huertas-Company, John Mulchaey, and Sirio Belli. We also thank Kate Whitaker, Rik Williams and the CSI team, Alessandro Rettura, Andrew Zirm, and Casey Papovich for sharing their data in an electronic format, as well as Nor Pirzkal and Beth Perriello for their assistance in planning and executing the \\emph{HST} observations.  \nBased on observations made with the NASA\/ESA \\emph{Hubble Space Telescope}, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with program number GO-12927, which was supported under NASA contract NAS 5-26555. A.R.~acknowledges financial contribution from the agreement ASI-INAF I\/009\/10\/0 and from Osservatorio Astronomico di Brera. Based on observations obtained with MegaPrime\/MegaCam, a joint project of CFHT and CEA\/IRFU, at the Canada--France--Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at Terapix available at the Canadian Astronomy Data Centre as part of the Canada--France--Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS.\n\n\\begin{appendix}\n\n\\section{Grism Redshift Catalog}\n\nTable~4 lists the redshifts derived for the 98 galaxies described in Section~\\ref{sec:z}. For emission line sources, we assign a quality flag `A' when more than one line is visible and `B' otherwise. For continuum sources, we qualitatively assign a quality flag based on the appearance of the spectrum and the posterior probability distribution $P(z)$. Spectra with a weak or absent continuum break, often with a multimodal $P(z)$, carry a `C' flag. The `B' flag corresponds to a more clearly detected continuum break; we expect the vast majority of these redshifts to be reliable. The `A' flag is reserved for the highest signal-to-noise objects with unambiguous continuum breaks and, in some cases, absorption lines.\n\n\\LongTables\n\\begin{deluxetable*}{lllccccc}\n\\tablewidth{0.6\\linewidth}\n\\tablecaption{Grism Redshifts}\n\\tablehead{\\colhead{ID} & \\colhead{R.A.} & \\colhead{Dec.} & \\colhead{$H_{160}$} & \\colhead{$z_{\\rm grism}$} & \\colhead{Type} & \\colhead{Quality}}\n\\startdata\n220 & 36.695309 & $-4.687007$ & 19.08 & $0.285 \\pm 0.005$ & E & B \\\\\n167 & 36.694981 & $-4.685004$ & 20.15 & $1.064 \\pm 0.005$ & E & B \\\\\n698 & 36.673634 & $-4.704338$ & 20.18 & $1.127 \\pm 0.005$ & E & B \\\\\n13 & 36.683534 & $-4.672097$ & 20.39 & $0.609 \\pm 0.005$ & E & B \\\\\n516 & 36.683296 & $-4.698129$ & 20.59 & $0.963 \\pm 0.005$ & E & B \\\\\n272 & 36.681727 & $-4.689340$ & 20.63 & $1.798 \\pm 0.002$ & C & A \\\\\n355 & 36.686442 & $-4.692394$ & 20.80 & $1.798 \\pm 0.002$ & C & A \\\\\n409 & 36.692244 & $-4.693913$ & 20.85 & $0.692 \\pm 0.005$ & E & A \\\\\n60 & 36.687740 & $-4.677383$ & 20.86 & $0.608 \\pm 0.005$ & E & B \\\\\n448 & 36.691822 & $-4.694914$ & 21.05 & $0.797 \\pm 0.005$ & E & B \\\\\n376 & 36.675006 & $-4.692865$ & 21.20 & $1.811 \\pm 0.006$ & C & A \\\\\n64 & 36.675602 & $-4.677701$ & 21.24 & $2.415 \\pm 0.001$ & C & A \\\\\n628 & 36.678489 & $-4.701768$ & 21.26 & $1.592 \\pm 0.010$ & C & B \\\\\n499 & 36.681694 & $-4.697093$ & 21.27 & $1.127 \\pm 0.005$ & E & B \\\\\n445 & 36.673416 & $-4.694926$ & 21.33 & $0.893 \\pm 0.005$ & E & B \\\\\n356 & 36.694233 & $-4.692351$ & 21.35 & $1.805 \\pm 0.004$ & C & A \\\\\n546 & 36.665075 & $-4.699060$ & 21.36 & $2.187 \\pm 0.054$ & C & C \\\\\n485 & 36.670279 & $-4.696597$ & 21.41 & $1.131 \\pm 0.005$ & E & B \\\\\n164 & 36.661774 & $-4.684718$ & 21.45 & $1.325 \\pm 0.005$ & E & B \\\\\n743 & 36.697722 & $-4.705844$ & 21.47 & $1.324 \\pm 0.003$ & E & A \\\\\n657 & 36.675567 & $-4.702566$ & 21.61 & $1.812 \\pm 0.002$ & C & A \\\\\n48 & 36.678011 & $-4.676309$ & 21.67 & $0.962 \\pm 0.005$ & E & B \\\\\n165 & 36.661849 & $-4.684869$ & 21.68 & $1.302 \\pm 0.005$ & E & B \\\\\n286 & 36.687899 & $-4.689939$ & 21.69 & $1.798 \\pm 0.041$ & C & B \\\\\n342 & 36.696650 & $-4.691744$ & 21.74 & $1.323 \\pm 0.005$ & E & A \\\\\n519 & 36.702752 & $-4.697865$ & 21.76 & $1.055 \\pm 0.005$ & E & B \\\\\n352 & 36.690511 & $-4.692148$ & 21.88 & $1.797 \\pm 0.005$ & C & A \\\\\n601 & 36.689218 & $-4.700765$ & 21.89 & $1.339 \\pm 0.018$ & C & C \\\\\n451 & 36.680181 & $-4.695045$ & 21.90 & $1.470 \\pm 0.047$ & E & B \\\\\n556 & 36.675557 & $-4.699295$ & 21.91 & $1.591 \\pm 0.006$ & C & A \\\\\n249 & 36.702231 & $-4.688053$ & 21.98 & $1.935 \\pm 0.003$ & C & A \\\\\n410 & 36.673327 & $-4.693843$ & 22.00 & $2.406 \\pm 0.009$ & C & A \\\\\n107 & 36.676193 & $-4.681298$ & 22.01 & $1.623 \\pm 0.004$ & E & A \\\\\n452 & 36.683320 & $-4.695092$ & 22.02 & $1.464 \\pm 0.004$ & C & A \\\\\n779 & 36.695368 & $-4.707747$ & 22.03 & $1.713 \\pm 0.009$ & C & B \\\\\n320 & 36.668857 & $-4.691090$ & 22.04 & $1.125 \\pm 0.005$ & E & A \\\\\n411 & 36.673819 & $-4.693840$ & 22.11 & $1.821 \\pm 0.004$ & C & A \\\\\n447 & 36.691213 & $-4.694868$ & 22.12 & $1.797 \\pm 0.010$ & C & A \\\\\n197 & 36.699141 & $-4.685847$ & 22.13 & $1.704 \\pm 0.007$ & C & B \\\\\n166 & 36.695278 & $-4.685600$ & 22.16 & $0.484 \\pm 0.005$ & E & B \\\\\n289 & 36.689652 & $-4.689939$ & 22.17 & $1.802 \\pm 0.003$ & C & A \\\\\n589 & 36.693715 & $-4.698247$ & 22.21 & $0.702 \\pm 0.005$ & E & B \\\\\n392 & 36.685294 & $-4.693101$ & 22.33 & $2.065 \\pm 0.012$ & E & A \\\\\n85 & 36.689254 & $-4.679838$ & 22.35 & $1.519 \\pm 0.005$ & E & A \\\\\n387 & 36.682313 & $-4.692964$ & 22.36 & $1.801 \\pm 0.009$ & C & B \\\\\n655 & 36.682254 & $-4.702452$ & 22.40 & $0.795 \\pm 0.005$ & E & B \\\\\n375 & 36.674884 & $-4.692784$ & 22.43 & $1.819 \\pm 0.008$ & C & B \\\\\n317 & 36.699109 & $-4.690911$ & 22.45 & $1.787 \\pm 0.003$ & C & A \\\\\n80 & 36.690513 & $-4.679514$ & 22.51 & $1.174 \\pm 0.005$ & E & A \\\\\n798 & 36.667559 & $-4.708978$ & 22.51 & $1.065 \\pm 0.005$ & E & B \\\\\n105 & 36.676666 & $-4.681000$ & 22.54 & $1.623 \\pm 0.004$ & E & B \\\\\n359 & 36.676956 & $-4.692278$ & 22.54 & $1.792 \\pm 0.004$ & C & B \\\\\n365 & 36.691019 & $-4.692373$ & 22.54 & $1.511 \\pm 0.005$ & E & A \\\\\n569 & 36.681467 & $-4.699630$ & 22.61 & $1.834 \\pm 0.022$ & C & C \\\\\n637 & 36.679943 & $-4.701682$ & 22.70 & $1.490 \\pm 0.094$ & C & C \\\\\n385 & 36.702109 & $-4.692868$ & 22.71 & $1.257 \\pm 0.005$ & E & B \\\\\n281 & 36.690609 & $-4.689444$ & 22.77 & $1.806 \\pm 0.004$ & C & B \\\\\n334 & 36.690954 & $-4.691279$ & 22.79 & $1.133 \\pm 0.005$ & E & B \\\\\n674 & 36.687376 & $-4.703028$ & 22.85 & $1.302 \\pm 0.005$ & E & A \\\\\n693 & 36.677710 & $-4.703786$ & 22.86 & $1.820 \\pm 0.014$ & C & C \\\\\n323 & 36.674250 & $-4.691128$ & 22.99 & $1.369 \\pm 0.009$ & C & C \\\\\n224 & 36.684922 & $-4.686954$ & 23.04 & $0.966 \\pm 0.005$ & E & A \\\\\n201 & 36.676671 & $-4.686139$ & 23.04 & $0.924 \\pm 0.005$ & E & A \\\\\n8 & 36.680094 & $-4.670625$ & 23.07 & $0.968 \\pm 0.005$ & E & A \\\\\n16 & 36.692232 & $-4.672568$ & 23.12 & $1.474 \\pm 0.005$ & E & A \\\\\n531 & 36.679186 & $-4.698393$ & 23.12 & $1.818 \\pm 0.005$ & E & A \\\\\n414 & 36.696719 & $-4.693920$ & 23.16 & $1.334 \\pm 0.005$ & E & A \\\\\n459 & 36.675068 & $-4.695578$ & 23.25 & $1.599 \\pm 0.053$ & C & C \\\\\n653 & 36.676695 & $-4.702391$ & 23.25 & $1.611 \\pm 0.041$ & C & C \\\\\n368 & 36.679813 & $-4.692496$ & 23.26 & $1.951 \\pm 0.033$ & C & C \\\\\n587 & 36.665178 & $-4.700129$ & 23.27 & $1.917 \\pm 0.015$ & C & C \\\\\n446 & 36.679765 & $-4.694762$ & 23.29 & $1.485 \\pm 0.054$ & C & C \\\\\n255 & 36.687932 & $-4.688383$ & 23.30 & $1.795 \\pm 0.040$ & C & C \\\\\n77 & 36.681823 & $-4.678888$ & 23.35 & $0.902 \\pm 0.005$ & E & A \\\\\n300 & 36.696786 & $-4.690403$ & 23.36 & $0.693 \\pm 0.005$ & E & A \\\\\n582 & 36.691929 & $-4.700078$ & 23.42 & $1.132 \\pm 0.005$ & E & A \\\\\n161 & 36.684522 & $-4.684455$ & 23.48 & $1.137 \\pm 0.005$ & E & B \\\\\n61 & 36.687598 & $-4.677597$ & 23.49 & $2.049 \\pm 0.005$ & E & A \\\\\n593 & 36.698937 & $-4.700375$ & 23.51 & $2.164 \\pm 0.005$ & E & A \\\\\n117 & 36.689081 & $-4.681802$ & 23.52 & $1.474 \\pm 0.005$ & E & A \\\\\n177 & 36.672820 & $-4.685007$ & 23.65 & $0.798 \\pm 0.005$ & E & A \\\\\n156 & 36.694895 & $-4.684186$ & 23.66 & $1.965 \\pm 0.005$ & E & A \\\\\n504 & 36.690690 & $-4.697156$ & 23.79 & $1.064 \\pm 0.005$ & E & B \\\\\n477 & 36.700287 & $-4.696110$ & 23.82 & $1.833 \\pm 0.005$ & E & B \\\\\n332 & 36.671646 & $-4.691251$ & 23.83 & $1.785 \\pm 0.005$ & E & B \\\\\n21 & 36.681790 & $-4.673701$ & 23.88 & $1.489 \\pm 0.005$ & E & A \\\\\n39 & 36.689767 & $-4.675155$ & 23.94 & $2.047 \\pm 0.005$ & E & A \\\\\n282 & 36.669934 & $-4.689446$ & 23.98 & $1.940 \\pm 0.005$ & E & B \\\\\n145 & 36.665669 & $-4.683570$ & 24.00 & $1.631 \\pm 0.005$ & E & A \\\\\n149 & 36.679709 & $-4.683808$ & 24.01 & $1.173 \\pm 0.005$ & E & A \\\\\n175 & 36.698451 & $-4.684924$ & 24.03 & $1.520 \\pm 0.005$ & E & A \\\\\n538 & 36.683767 & $-4.698538$ & 24.22 & $1.111 \\pm 0.005$ & E & B \\\\\n427 & 36.671179 & $-4.694407$ & 24.29 & $1.000 \\pm 0.005$ & E & A \\\\\n677 & 36.670007 & $-4.703092$ & 24.29 & $0.665 \\pm 0.005$ & E & A \\\\\n742 & 36.678980 & $-4.705792$ & 24.43 & $1.135 \\pm 0.005$ & E & A \\\\\n581 & 36.691821 & $-4.699903$ & 24.55 & $1.170 \\pm 0.005$ & E & A \\\\\n598 & 36.688125 & $-4.700496$ & 24.77 & $1.470 \\pm 0.005$ & E & A \\\\\n87 & 36.687851 & $-4.679866$ & 24.94 & $2.154 \\pm 0.005$ & E & B \n\\enddata\n\\tablecomments{Type `E' and `C' denote emission line and continuum-based grism redshifts, respectively. Uncertainties on emission line redshifts are listed as 0.005, based on our external comparison with higher-resolution data in Section~\\ref{sec:emlines}; errors on the continuum-derived redshifts are based on the MCMC chains. Quality flags are explained in the text.\\label{tab:z}}\n\\end{deluxetable*}\n\n\\section{Method for Fitting of Grism Spectra and Photometry}\n\\texttt{pyspecfit} is based on the MCMC sampler \\texttt{MultiNest} \\citep{Feroz09}. It accepts as input one or more spectra, with associated LSFs, along with broadband photometric data. For a given set of model parameters proposed by the sampler, the likelihood $\\mathcal{L}$ is computed as follows. We begin with a grid of \\citet{BC03} simple stellar population (SSP, or ``burst'') models with a \\citet{Salpeter55} IMF. First, the SSPs are interpolated to the desired metallicity and integrated over the star formation history. We adopt an exponentially declining model with $\\textrm{SFR} \\propto e^{-t\/\\tau}$ for $t > -t_0$, where SFR is the star formation rate at time $t$, $\\tau$ is the $e$-folding time, $t = 0$ at the epoch of observation, and  $t_0$ is the age. Gas lost during stellar evolution is not recycled. Next, dust attenuation is applied using the \\citet{Calzetti00} law, parameterized by the attenuation $A_V$ at 5500~\\AA. Finally, the spectrum is redshifted, and attenuation by the intergalactic medium blueward of Ly$\\alpha$ is taken into account following \\citet{Madau95}. \n\nThe model is then binned to the wavelength grid of each observed spectrum and convolved by the LSF (i.e., the galaxy light profile) to produce model spectra, i.e., $M^{\\rm G102}$ and $M^{\\rm G141}$ for the fits described in Section~\\ref{sec:contfit}. The model is also integrated over the filter transmission curves to obtain the model flux density $M^{\\rm phot}_k$ through each observed filter. The likelihood is $\\mathcal{L} = \\exp(-\\frac{1}{2} \\chi^2)$, where\n\\begin{align}\n\\chi^2 = &\\sum_i \\left(\\frac{D^{\\rm G102}_i - P(\\lambda_i) M^{\\rm G102}_i}{\\sigma^{\\rm G102}_i}\\right)^2 + \n\\sum_j \\left(\\frac{D^{\\rm G141}_j - P(\\lambda_j) M^{\\rm G141}_j}{\\sigma^{\\rm G141}_j}\\right)^2 + \n\\sum_k \\left(\\frac{D^{\\rm phot}_k - M^{\\rm phot}_k}{\\sigma^{\\rm phot}_k}\\right)^2.\n\\end{align}\nHere $D^{\\rm G102}$ and $D^{\\rm G141}$ are the observed spectra with associated uncertainties $\\sigma^{\\rm G102}$ and $\\sigma^{\\rm G141}$, $i$ and $j$ run over the pixels in each spectrum, and $D^{\\rm phot}_k$ is flux density measured in filter $k$ with uncertainty $\\sigma^{\\rm phot}_k$. $P(\\lambda)$ is a polynomial that scales and modulates the shape of the spectra. At minimum a constant is necessary to scale the spectra to the total flux, but it is also desirable to allow for some variation in the broadband spectral shape (see also \\citealt{Brammer12}). We use a linear $P(\\lambda)$, which is continuous across the entire wavelength range spanned by both grisms, and determine the coefficients that minimize $\\chi^2$ for a given set of model parameters using a linear least-squares approach. Essentially, this procedure allows for a mild deformation of the spectral shape to match the photometric data, but the low polynomial order prevents the introduction of a spectral break. \n\nFor our fits to the spectra and photometry of the individual galaxies in our continuum sample (Section~\\ref{sec:contfit}), we chose uniform priors over $1 < z < 3$, $7 < \\log \\tau \/ {\\rm yr} < 10$, $8 < \\log t_0 \/ {\\rm yr} < \\log a(z) \/ {\\rm yr}$, and $0 < A_V < 2$, and where $a(z)$ is the age of the universe at redshift $z$. The metallicity was fixed to solar. For our analysis of the continuum-normalized stacked spectra, we allow the metallicity to vary and fit simple stellar populations as described in Section~\\ref{sec:stackspec}. \\texttt{pyspecfit} produces samples from the posterior distribution for these parameters, as well as the stellar mass $M_*$ (including remnants) and SFR at the observation epoch. In this paper we primarily make use of the redshift and stellar mass estimates and report the median, with $1\\sigma$ errors representing the 16th and 84th percentiles. We have compared our stellar mass estimates for the continuum sample of 40 galaxies to the estimates produced by \\texttt{FAST}, which fits only the broadband photometry. The redshift was fixed to $z_{\\rm grism}$ in \\texttt{FAST}. We find that the median difference between the two mass estimates is consistent with zero, and there is no systematic trend with mass.\n\n\\section{Literature Compilation in Figure~16}\n\nHere we describe our compilation of literature measurements of the variation of the stellar mass--size relation with environment used in Figure~\\ref{fig:othersizestudies}. For the six individual clusters plotted, including JKCS~041, we use the masses and radii of individual quiescent galaxies and compare these to the mean relation that we measured in the field (Equation~\\ref{eqn:fit_full_field}). We take this field relation as a uniform basis of comparison for every cluster, since it is based on a much larger sample of field galaxies than those used in the following studies, but we note that this may introduce some systematic errors, which are estimated below. In each case, stellar masses are converted to a Salpeter IMF and a cut of $M_* > 10^{10.7} \\msol$ is applied to ensure that similar mass ranges are probed. From \\citet{Papovich12}, we take the 10 $UVJ$--quiescent galaxies above this limit with $P_z > 0.5$ and $R_{\\rm proj} < 1$~Mpc. From \\citet{Zirm12}, we take the eight photometrically-selected candidates in their Table 1; since their masses are derived using the \\citet{Maraston05} models, we divide them by 0.69 to account for the typical offset from BC03-based masses found by \\citet{Muzzin09}. From \\citet{Strazzullo13} we take the four ``passive early type'' galaxies above our mass limit listed in their Figure~12. From \\citet{Rettura10} we take the 18 galaxies in RDCS1252.9--2927 in their Table 1. From \\citet{Raichoor11,Raichoor12} we take the sizes and BC03-based masses of 23 galaxies in the Lynx cluster E and W. For the Rettura et al.~and Raichoor et al.~data, we apply a mean offsets determined by \\citet{Raichoor11} of $\\Delta \\log M_* = -0.05$, which includes an aperture correction ($+0.06$~dex) to total S\\'{e}rsic magnitudes and the mean effect of including dust attenuation ($-0.11$~dex), which should better match our procedure.\n\nFor each cluster we compute the mean offset $\\Delta \\log R_e^{\\rm maj}$ from Equation~\\ref{eqn:fit_full_field} and estimate its uncertainty as $0.23 \/ \\sqrt{N}$~dex, where $N$ is the number of cluster members, based on the scatter in the field relation. Several sources of systematic uncertainty may affect this comparison between our field relation and independently-measured masses and sizes of cluster galaxies. First, different authors use different photometric apertures. Using {\\tt MAG\\_AUTO}-scaled masses for our field galaxy sample, rather scaling to the total S\\'{e}rsic  magnitude, produces a shift of only $\\Delta \\log R_e^{\\rm maj} = -0.01$~dex in Equation~\\ref{eqn:fit_full_field}, but larger offsets could apply to other data sets. Second, the inclusion with galaxies having questionable S\\'{e}rsic fits can lead to shifts of $\\sim0.02$~dex. Third, although we have tried to harmonize stellar mass to first order by applying offsets based on the IMF and the set of stellar population models used, other differences in the priors and fitting procedure remain. Since the Papovich et al.~sample overlaps our UDS data, we are able in this case to directly compare stellar mass estimates. For the 20 overlapping $UVJ$-quiescent galaxies with $M_* > 10^{10.7} \\msol$, we find that our $M_*^{\\rm tot}$ are offset from the Papovich et al.~measures by $-0.05$~dex, which corresponds to a shift in $\\Delta \\log R_e^{\\rm maj}$ of $0.63 \\times (-0.05) = -0.03$~dex. These uncertainties should be kept in mind pending future studies that homogeneously analyze data from an ensemble of high-$z$ clusters.\n\nIn addition to these studies of individual clusters, we also directly quote results from 3 studies of larger samples. From \\citet{Delaye14}, we take the mean mass-normalized radii in their field and cluster samples in three redshift bins from their Table~9. From \\citet{Cooper12} we take the difference in median sizes of matched galaxy samples in high- and low-density regions of the DEEP3 survey from their Figure~3. \\citet{Lani13} publish relative sizes of red galaxies in high- and low-density regions in the UDS field, broken down by mass (their Figures 5 and 6). To better compare with the above works, we average these mass-dependent measurements in each of their redshift bins, weighting by the number of galaxies in each mass bin. Only mass bins with $M_* > 10^{10.7} \\msol$ were used, after converting to a Salpeter IMF.\n\n\\end{appendix}\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Introduction}\n\nThe kernel trick provides an elegant and natural technique to extend linear models to non-linear models with a great representation power. In the past decade, numerous works have studied bandit and reinforcement learning problems under the assumption that the reward function conforms to a kernel-based model~\\citep[]{srinivas2010gaussian,Krause11Contexual, wang2014theoreticalGPEI,   nguyen2017regretGPEI,  Scarlett2017Lower, Chowdhury2017bandit,wang2018meta,kandasamy2018parallelised,   Javidi, yang2020provably,shekhar2020multi,bogunovic2020corruption, zhou2020neuralUCB,Vakili2020Scalable,vakili2020information,  cai2020lower, zhang2020neuralTS}.\n\n\nThe analysis of online learning problems with a kernel-based model typically utilizes confidence intervals applicable to the elements of a reproducing kernel Hilbert space (RKHS). However, the state-of-the-art confidence intervals in this setting~\\citep{Chowdhury2017bandit} do not appear to be tight, resulting in suboptimal regret bounds.\nThe main challenge seems to stem from the online (sequential) nature of the observation points, in contrast to an offline (fixed in advance) design. \nWe first overview the existing results, and then formalize the open problem of tight confidence intervals for the RKHS elements under the online setting. We also discuss the consequences of these bounds on the regret performance. \nFor clarity of exposition, we focus on bandit problems and the GP-UCB algorithm~\\citep{srinivas2010gaussian, Chowdhury2017bandit}, but the problem is equally relevant to reinforcement learning problems and other algorithms such as GP-TS. \n\n\\vspace*{-1ex}\n\\section{Problem Setup}\n\nConsider a positive definite kernel $k:\\mathcal{X} \\times \\mathcal{X}\\rightarrow \\mathbb{R}$ with respect to a finite Borel measure,\nwhere $\\mathcal{X}\\subset \\mathbb{R}^d$ is a compact\nset. Let $\\mathcal{H}_k$ denote the RKHS corresponding to $k$, defined as a Hilbert space equipped with an inner product $\\langle.,.\\rangle_{\\mathcal{H}_k}$ satisfying the following: $k(.,x)\\in \\mathcal{H}_k$, $\\forall x\\in \\mathcal{X}$, and $\\langle f,k(.,x)\\rangle_{\\mathcal{H}_k}=f(x)$, $\\forall x\\in\\mathcal{X}, \\forall f \\in \\mathcal{H}_k$ (reproducing property). The typical assumption in kernel-based models is that the \\emph{objective function} $f$ satisfies $f\\in\\mathcal{H}_k$ for a known kernel $k$. Let $\\{\\lambda_m\\}_{m=1}^\\infty$ and $\\{\\phi_m\\}_{m=1}^\\infty$ denote the Mercer eigenvalues and eigenfeatures of $k$, respectively~\\citep[see, e.g.,][Theorem~$4.1$]{Kanagawa2018}. \nUsing Mercer's representation theorem~\\citep[see, e.g.,][Theorem~$4.2$]{Kanagawa2018}, an alternative representation for $f\\in\\mathcal{H}_k$ is given by\n\\begin{eqnarray}\\label{kernelmodel}\nf(x) = \\mathbf{w}^{\\top} \\mathbf{\\Lambda}^{\\frac{1}{2}}\\bm{\\phi}(x),\n\\end{eqnarray}\nwhere $\\mathbf{w} = [w_1,w_2,...]^\\top$ and $\\bm{\\phi}(x) = [\\phi_2(x), \\phi_2(x),...]^{\\top}$ are the (possibly infinite-dimensional) \\emph{weight} and feature vectors, and $\\mathbf{\\Lambda}$ is a (possibly infinite dimensional) diagonal matrix with $\\mathbf{\\Lambda}_{i,j} = \\lambda_i$, if $i=j$. The RKHS norm of $f$ satisfies $\\|f\\|_{\\mathcal{H}_k} = \\|\\mathbf{w}\\|_{\\ell^2}$.\n\n\n\n\\paragraph{Kernelized Bandits:}\nConsider an online learning setting where a learning algorithm is allowed to collect a sequence of noisy observations $\\{(x_i, y_i)\\}_{i=1}^\\infty$, where $y_i=f(x_i)+\\epsilon_i$ with $\\epsilon_i$ being well-behaved noise terms. The objective is to get as close as possible to the maximum of $f$. The performance of the algorithm is measured in terms of regret, defined as the cumulative loss in the values of the objective function at observation points, compared to a global maximum:\n\\begin{eqnarray}\n\\mathcal{R}(N) = \\sum_{i=1}^N \\left(f(x^*) - f(x_i)\\right), \n\\end{eqnarray}\nwhere $x^*\\in \\text{argmax}_{x\\in \\mathcal{X}} f(x)$ is a global maximum.  Under the assumption $f \\in \\mathcal{H}_k$, this setting is often referred to as that of kernelized bandits, Gaussian process (GP) bandits, or Bayesian optimization. The latter two terms are motivated by the algorithm design which often employs a GP surrogate model. Throughout this paper, we make the following assumptions.\n\\begin{assumption}\\label{ass1}\nThe RKHS norm of $f$ is bounded as $\\|f\\|_{\\mathcal{H}_k}\\le B$, for some $B>0$. Moreover, the noise terms are i.i.d. sub-Gaussian random variables, i.e., for some $R>0$, $\\mathbb{E}[\\exp(\\eta\\epsilon_i)]\\le \\exp(\\frac{\\eta^2R^2}{2})$, $\\forall \\eta\\in \\mathbb{R}, \\forall i \\in \\mathbb{N}$. \n\\end{assumption}\n\n\nIn online learning problems, the observation points are collected sequentially. In particular, the observation point $x_{i+1}$ is determined after all the values $\\{(x_j, y_j)\\}_{j=1}^i$ are revealed. This is in contrast to an offline design, where the data points are fixed in advance. We next formalize this distinction.\n\\begin{definition}\ni) In the \\textbf{online setting}, for the sigma algebras $\\mathcal{F}_i = \\sigma(x_1,x_2,\\dots, x_{i+1}, \\epsilon_1, \\epsilon_2, \\dots, \\epsilon_i)$, $i\\ge 1$, it holds that $x_i$ and $\\epsilon_i$ are $\\mathcal{F}_{i-1}$ and $\\mathcal{F}_i$ measurable, respectively.\nii) In the \\textbf{offline setting}, for all $i\\ge 1$, it holds that $x_i$ is independent of all $\\epsilon_j$, $j \\ge 1$.\n\\end{definition}\n\n\n\\paragraph{Surrogate GP Model:}\n\nIt is useful for algorithm design to employ a zero-mean surrogate GP model $\\hat{f}$ with kernel $k$ which provides a surrogate posterior mean (regressor) and a surrogate posterior variance (uncertainty estimate) for the kernel-based model.  Defining $\\mu_n(x) = \\mathbb{E}\\big[\\hat{f}(x)|\\{(x_i,y_i)\\}_{i=1}^{n}\\big]$ and $\\sigma_n^2(x) = \\mathbb{E}\\big[(\\hat{f}(x) - \\mu_n(x))^2|\\{(x_i,y_i)\\}_{i=1}^{n}\\big]$, it is well known that\n$\n\\mu_n(x) =  \\mathbf{z}^{\\top}_n(x)\\mathbf{y}_n$ and \n$\\sigma_n^2(x) = k(x,x) - \\mathbf{k}_n^{\\top}(x)  (\\lambda^2 \\mathbf{I}_n+\\mathbf{K}_n)^{-1}\\mathbf{k}_n(x), \n$\nwhere $\\mathbf{k}_n(x) = [k(x,x_1), k(x,x_2), \\dots, k(x,x_n)]^{\\top}$, $\\mathbf{K}_n$ is the positive definite kernel matrix $[\\mathbf{K}_n]_{i,j} = k(x_i, x_j)$, $\\mathbf{z}_n(x) = (\\lambda^2\\mathbf{I}_n + \\mathbf{K}_n)^{-1}\\mathbf{k}_n(x)$, $\\mathbf{I}_n$ is the identity matrix of dimension $n$, and $\\lambda>0$ is a regularization parameter. \n\n\n\n\n\n\\vspace*{-1ex}\n\\section{Confidence Intervals Applicable to RKHS Elements}\n\nDeriving confidence intervals applicable to RKHS elements is significantly more challenging in the online setting compared to the offline setting. In the latter case, for any fixed $x\\in \\mathcal{X}$, we have with probability at least $1-\\delta$ that\n$\n|f(x) - \\mu_n(x)| \\le \\rho_0(\\delta)\\sigma_n(x),\n$\nwhere $\\rho_0(\\delta)=  B+ \\frac{R}{\\lambda} \\sqrt{2\\log(\\frac{2}{\\delta})}$, $B$ and $R$ are the parameters specified in Assumption~\\ref{ass1}, and $\\lambda$ is the regularization parameter of the surrogate GP model. Moreover, when $f$ is Lipschitz (or H\\\"older) continuous~\\citep[that is true with typical kernels; see,][]{shekhar2020multi}, this easily extends to a uniform guarantee:  With probability at least $1-\\delta$, we have uniformly in $x$ that $|f(x)-\\mu_n(x)| = \\mathcal{O}\\big(\n\\big(B+\\frac{R}{\\lambda}\\sqrt{d\\log(n)+\\log(\\frac{1}{\\delta})}\\big)\\sigma_n(x)\n\\big)$, where the implied constants in $\\mathcal{O}(.)$ depend on the Lipschitz (or H\\\"older) continuity parameters.\n\nIn the online setting, strong uniform bounds are also well-known in the case of a linear model $f(x) = \\mathbf{w}^{\\top}x$: \\cite{Abbasi2011} proved that, with probability $1-\\delta$, uniformly over $x$,\n\\begin{eqnarray}\\label{conf:lin}\n|f(x) - \\mu_n(x)| \\le \\rho_n(\\delta)\\sigma_n(x),\n\\end{eqnarray}\nwhere $\\rho_n(\\delta)= B + \\frac{R}{\\lambda}\\sqrt{d\\log(\\frac{1+n\\bar{x}^2\/\\lambda^2}{\\delta})} $ and $\\bar{x}=\\max_{x\\in \\mathcal{X}}\\|x\\|_{\\ell^2}$.\nThe crux of the proof is a \\emph{self-normalized bound for vector valued martingales} $S_n = \\sum_{i=1}^n \\epsilon_i x_i$~\\citep[][Theorem $1$]{Abbasi2011},  which yields the following \\emph{confidence ellipsoid} for $\\mathbf{w}$~\\citep[][Theorem $2$]{Abbasi2011}: \n$\n\\|\\mathbf{w}-\\hat{\\mathbf{w}}_n\\|_{V_n} \\le \\lambda\\rho_n(\\delta), ~\\text{with probability at least}~1-\\delta\n$, where $V_n= \\lambda^2\\mathbf{I}_d+\\sum_{i=1}^nx_ix_i^{\\top}$.\nThis confidence ellipsoid for $\\mathbf{w}$ can then be represented in terms of the confidence interval for $f(x)$ given in~\\eqref{conf:lin}.\nNotice that the linear model is a special case of~\\eqref{kernelmodel} with $\\mathbf{w} = [w_1,w_2,\\dots, w_d]^{\\top}$ and $\\bm{\\phi}(x) = x$ being $d$ dimensional weight and feature vectors respectively, and $\\Lambda = \\mathbf{I}_d$ being the square identity matrix of dimension $d$.\n\n\\cite{Chowdhury2017bandit} built on the self-normalized bound for the vector valued martingales to prove the following theorem for the kernel-based models. \n\\begin{theorem}\\label{TheoremOnK}\nUnder Assumption~\\ref{ass1}, in the online setting, with probability at least $1-\\delta$, we have for all $x\\in \\mathcal{X}$ that\n\\begin{eqnarray}\n|f(x)-\\mu_n(x)|\\le \\rho_n(\\delta)\\sigma_n(x),\n\\end{eqnarray}\nwhere $\\rho_n(\\delta) = B+R\\sqrt{2\\left(\\gamma_{n-1}+1+\\log(\\frac{1}{\\delta})\\right)}$, and $\\gamma_n=\\sup_{\\{\\mathbf{x}_i\\}_{i=1}^n\\subset\\mathcal{X}}\\log\\det(\\lambda^2\\mathbf{I}_n+\\mathbf{K}_n)$ is the maximal information gain at time $n$, which is closely related to the effective dimension associated with the kernel (e.g., see~\\cite{srinivas2010gaussian, Valko2013kernelbandit}). \n\\end{theorem}\n\nOur open problem is concerned with improving this confidence interval. \n\n\\paragraph{Open Problem.} Under Assumption~\\ref{ass1}, in the online setting, consider the general problem of proving a confidence interval of the following form uniformly in $x\\in \\mathcal{X}$:\n\\begin{eqnarray}\n|f(x)-\\mu_n(x)|\\le \\rho_n(\\delta)\\sigma_n(x),~\\text{with probability at least}~1-\\delta.\n\\end{eqnarray}\nWhat is the lowest growth rate of $\\rho_n(\\delta)$ with $n$? In particular, is it possible to reduce the confidence interval width in Theorem~\\ref{TheoremOnK} by an $\\tilde{\\mathcal{O}}(\\sqrt{\\gamma_n})$ factor?\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\vspace*{-1ex}\n\\section{Discussion}\n\n\nFollowing standard UCB-based bandit algorithm techniques, it can be shown that the GP-UCB algorithm (namely, repeatedly choosing $x$ to maximize the current upper confidence bound) attains\n\\begin{eqnarray}\n\\mathcal{R}(N) = \\tilde{\\mathcal{O}}(\\rho_N(\\delta)\\sqrt{N\\gamma_N}), ~\\text{with probability at least}~1-\\delta.\n\\end{eqnarray}\nSubstituting $\\rho_N(\\delta)$ from Theorem~\\ref{TheoremOnK}, we have $\\mathcal{R}(N) = \\tilde{\\mathcal{O}}(\\gamma_N\\sqrt{N})$. Unfortunately, this is not always sublinear in $N$, since $\\gamma_N$ can grow faster than $\\sqrt{N}$, e.g., in the case of the Mat{\\'e}rn family of kernels.  Hence, the regret bound can be trivial in many cases of interest. It is unknown whether this suboptimal regret bound is a fundamental shortcoming of GP-UCB or a result of suboptimal confidence intervals, but the latter appears likely to be the most significant factor. The same question can be asked about the analysis of many other bandit algorithms including GP-TS~\\citep{Chowdhury2017bandit} and GP-EI~\\citep{nguyen2017regretGPEI}, as well as KOVI in the reinforcement learning setting~\\citep{yang2020provably}.  \n\n\nComparing the results under the online and offline settings, we see a stark contrast of an $\\mathcal{O}(\\sqrt{\\gamma_n})$ factor in the width of confidence intervals. We expect that the $\\mathcal{O}(\\sqrt{\\gamma_n})$ factor in the confidence interval width in the online setting can be replaced by an $\\tilde{\\mathcal{O}}(d\\log(n))$ term, resulting in an \n$\\tilde{\\mathcal{O}}(\\sqrt{dN\\gamma_N })$ regret bound. Roughly speaking, we are suggesting that a square root of the effective dimension of the kernel in the regret bound can be traded off for a square root of the input dimension. \n\n\n\nOf significant theoretical importance is a less practical algorithm \\emph{SupKernelUCB}~\\citep{Valko2013kernelbandit}, which achieves an $\\tilde{\\mathcal{O}}(\\sqrt{N\\gamma_N})$ regret bound for the kernelized bandit problem with a finite action set ($|\\mathcal{X}|<\\infty$). The finite action set assumption can be relaxed to compact domains using a discretization argument contributing only an $\\tilde{\\mathcal{O}}(\\sqrt{d\\log(N)})$ factor to the regret bound~\\citep[see,][Appendix A.4]{cai2020lower}. This bound is tight for the cases where a lower bound on regret is known, namely for commonly used squared exponential and Mat{\\'e}rn kernels~\\citep{Scarlett2017Lower, vakili2020information}.  In view of this discussion, the above-mentioned improvement is information-theoretically feasible, but it remains to determine whether GP-UCB can achieve it.\n\n\n\n\n\n\n\\section{Introduction}\n\nThe kernel trick provides an elegant and natural technique to extend linear models to non-linear models with a great representation power. In the past decade, numerous works have studied bandit and reinforcement learning problems under the assumption that the reward function conforms to a kernel-based model~\\citep[]{srinivas2010gaussian,Krause11Contexual, wang2014theoreticalGPEI,   nguyen2017regretGPEI,  Scarlett2017Lower, Chowdhury2017bandit,wang2018meta,kandasamy2018parallelised,   Javidi, yang2020provably,shekhar2020multi,bogunovic2020corruption, zhou2020neuralUCB,Vakili2020Scalable,vakili2020information, Janz2020SlightImprov, cai2020lower, zhang2020neuralTS}.\n\n\nThe analysis of online learning problems with a kernel-based model typically utilizes confidence intervals applicable to the elements of a reproducing kernel Hilbert space (RKHS). However, the state-of-the-art confidence intervals in this setting~\\citep{Chowdhury2017bandit} do not appear to be tight, resulting in suboptimal regret bounds.\nThe main challenge seems to stem from the online (sequential) nature of the observation points, in contrast to an offline (fixed in advance) design. \nWe first overview the existing results, and then formalize the open problem of tight confidence intervals for the RKHS elements under the online setting. We also discuss the consequences of these bounds on the regret performance. \nFor clarity of exposition, we focus on bandit problems and the GP-UCB algorithm~\\citep{srinivas2010gaussian, Chowdhury2017bandit}, but the problem is equally relevant to reinforcement learning problems and other algorithms such as GP-TS. \n\n\\vspace*{-1ex}\n\\section{Problem Setup}\n\nConsider a positive definite kernel $k:\\mathcal{X} \\times \\mathcal{X}\\rightarrow \\mathbb{R}$ with respect to a finite Borel measure,\nwhere $\\mathcal{X}\\subset \\mathbb{R}^d$ is a compact\nset. Let $\\mathcal{H}_k$ denote the RKHS corresponding to $k$, defined as a Hilbert space equipped with an inner product $\\langle.,.\\rangle_{\\mathcal{H}_k}$ satisfying the following: $k(.,x)\\in \\mathcal{H}_k$, $\\forall x\\in \\mathcal{X}$, and $\\langle f,k(.,x)\\rangle_{\\mathcal{H}_k}=f(x)$, $\\forall x\\in\\mathcal{X}, \\forall f \\in \\mathcal{H}_k$ (reproducing property). The typical assumption in kernel-based models is that the \\emph{objective function} $f$ satisfies $f\\in\\mathcal{H}_k$ for a known kernel $k$. Let $\\{\\lambda_m\\}_{m=1}^\\infty$ and $\\{\\phi_m\\}_{m=1}^\\infty$ denote the Mercer eigenvalues and eigenfeatures of $k$, respectively~\\citep[see, e.g.,][Theorem~$4.1$]{Kanagawa2018}. \nUsing Mercer's representation theorem~\\citep[see, e.g.,][Theorem~$4.2$]{Kanagawa2018}, an alternative representation for $f\\in\\mathcal{H}_k$ is given by\n\\begin{eqnarray}\\label{kernelmodel}\nf(x) = \\mathbf{w}^{\\top} \\mathbf{\\Lambda}^{\\frac{1}{2}}\\bm{\\phi}(x),\n\\end{eqnarray}\nwhere $\\mathbf{w} = [w_1,w_2,...]^\\top$ and $\\bm{\\phi}(x) = [\\phi_2(x), \\phi_2(x),...]^{\\top}$ are the (possibly infinite-dimensional) \\emph{weight} and feature vectors, and $\\mathbf{\\Lambda}$ is a (possibly infinite dimensional) diagonal matrix with $\\mathbf{\\Lambda}_{i,j} = \\lambda_i$, if $i=j$. The RKHS norm of $f$ satisfies $\\|f\\|_{\\mathcal{H}_k} = \\|\\mathbf{w}\\|_{\\ell^2}$.\n\n\n\n\\paragraph{Kernelized Bandits:}\nConsider an online learning setting where a learning algorithm is allowed to collect a sequence of noisy observations $\\{(x_i, y_i)\\}_{i=1}^\\infty$, where $y_i=f(x_i)+\\epsilon_i$ with $\\epsilon_i$ being well-behaved noise terms. The objective is to get as close as possible to the maximum of $f$. The performance of the algorithm is measured in terms of regret, defined as the cumulative loss in the values of the objective function at observation points, compared to a global maximum:\n\\begin{eqnarray}\n\\mathcal{R}(N) = \\sum_{i=1}^N \\left(f(x^*) - f(x_i)\\right), \n\\end{eqnarray}\nwhere $x^*\\in \\text{argmax}_{x\\in \\mathcal{X}} f(x)$ is a global maximum.  Under the assumption $f \\in \\mathcal{H}_k$, this setting is often referred to as that of kernelized bandits, Gaussian process (GP) bandits, or Bayesian optimization. The latter two terms are motivated by the algorithm design which often employs a GP surrogate model. Throughout this paper, we make the following assumptions.\n\\begin{assumption}\\label{ass1}\nThe RKHS norm of $f$ is bounded as $\\|f\\|_{\\mathcal{H}_k}\\le B$, for some $B>0$. Moreover, the noise terms are i.i.d. sub-Gaussian random variables, i.e., for some $R>0$, $\\mathbb{E}[\\exp(\\eta\\epsilon_i)]\\le \\exp(\\frac{\\eta^2R^2}{2})$, $\\forall \\eta\\in \\mathbb{R}, \\forall i \\in \\mathbb{N}$. \n\\end{assumption}\n\n\nIn online learning problems, the observation points are collected sequentially. In particular, the observation point $x_{i+1}$ is determined after all the values $\\{(x_j, y_j)\\}_{j=1}^i$ are revealed. This is in contrast to an offline design, where the data points are fixed in advance. We next formalize this distinction.\n\\begin{definition}\ni) In the \\textbf{online setting}, for the sigma algebras $\\mathcal{F}_i = \\sigma(x_1,x_2,\\dots, x_{i+1}, \\epsilon_1, \\epsilon_2, \\dots, \\epsilon_i)$, $i\\ge 1$, it holds that $x_i$ and $\\epsilon_i$ are $\\mathcal{F}_{i-1}$ and $\\mathcal{F}_i$ measurable, respectively.\nii) In the \\textbf{offline setting}, for all $i\\ge 1$, it holds that $x_i$ is independent of all $\\epsilon_j$, $j \\ge 1$.\n\\end{definition}\n\n\n\\paragraph{Surrogate GP Model:}\n\nIt is useful for algorithm design to employ a zero-mean surrogate GP model $\\hat{f}$ with kernel $k$ which provides a surrogate posterior mean (regressor) and a surrogate posterior variance (uncertainty estimate) for the kernel-based model.  Defining $\\mu_n(x) = \\mathbb{E}\\big[\\hat{f}(x)|\\{(x_i,y_i)\\}_{i=1}^{n}\\big]$ and $\\sigma_n^2(x) = \\mathbb{E}\\big[(\\hat{f}(x) - \\mu_n(x))^2|\\{(x_i,y_i)\\}_{i=1}^{n}\\big]$, it is well known that\n$\n\\mu_n(x) =  \\mathbf{z}^{\\top}_n(x)\\mathbf{y}_n$ and \n$\\sigma_n^2(x) = k(x,x) - \\mathbf{k}_n^{\\top}(x)  (\\lambda^2 \\mathbf{I}_n+\\mathbf{K}_n)^{-1}\\mathbf{k}_n(x), \n$\nwhere $\\mathbf{k}_n(x) = [k(x,x_1), k(x,x_2), \\dots, k(x,x_n)]^{\\top}$, $\\mathbf{K}_n$ is the positive definite kernel matrix $[\\mathbf{K}_n]_{i,j} = k(x_i, x_j)$, $\\mathbf{z}_n(x) = (\\lambda^2\\mathbf{I}_n + \\mathbf{K}_n)^{-1}\\mathbf{k}_n(x)$, $\\mathbf{I}_n$ is the identity matrix of dimension $n$, and $\\lambda>0$ is a regularization parameter. \n\n\n\n\n\n\\vspace*{-1ex}\n\\section{Confidence Intervals Applicable to RKHS Elements}\n\nDeriving confidence intervals applicable to RKHS elements is significantly more challenging in the online setting compared to the offline setting. In the latter case, \\citep{vakili21} showed that for any fixed $x\\in \\mathcal{X}$, we have with probability at least $1-\\delta$ that\n$\n|f(x) - \\mu_n(x)| \\le \\rho_0(\\delta)\\sigma_n(x),\n$\nwhere $\\rho_0(\\delta)=  B+ \\frac{R}{\\lambda} \\sqrt{2\\log(\\frac{2}{\\delta})}$, $B$ and $R$ are the parameters specified in Assumption~\\ref{ass1}, and $\\lambda$ is the regularization parameter of the surrogate GP model. Moreover, when $f$ is Lipschitz (or H\\\"older) continuous~\\citep[that is true with typical kernels; see,][]{shekhar2020multi}, this easily extends to a uniform guarantee:  With probability at least $1-\\delta$, we have uniformly in $x$ that $|f(x)-\\mu_n(x)| = \\mathcal{O}\\big(\n\\big(B+\\frac{R}{\\lambda}\\sqrt{d\\log(n)+\\log(\\frac{1}{\\delta})}\\big)\\sigma_n(x)\n\\big)$, where the implied constants in $\\mathcal{O}(.)$ depend on the Lipschitz (or H\\\"older) continuity parameters.\n\nIn the online setting, strong uniform bounds are also well-known in the case of a linear model $f(x) = \\mathbf{w}^{\\top}x$: \\cite{Abbasi2011} proved that, with probability $1-\\delta$, uniformly over $x$,\n\\begin{eqnarray}\\label{conf:lin}\n|f(x) - \\mu_n(x)| \\le \\rho_n(\\delta)\\sigma_n(x),\n\\end{eqnarray}\nwhere $\\rho_n(\\delta)= B + \\frac{R}{\\lambda}\\sqrt{d\\log(\\frac{1+n\\bar{x}^2\/\\lambda^2}{\\delta})} $ and $\\bar{x}=\\max_{x\\in \\mathcal{X}}\\|x\\|_{\\ell^2}$.\nThe crux of the proof is a \\emph{self-normalized bound for vector valued martingales} $S_n = \\sum_{i=1}^n \\epsilon_i x_i$~\\citep[][Theorem $1$]{Abbasi2011},  which yields the following \\emph{confidence ellipsoid} for $\\mathbf{w}$~\\citep[][Theorem $2$]{Abbasi2011}: \n$\n\\|\\mathbf{w}-\\hat{\\mathbf{w}}_n\\|_{V_n} \\le \\lambda\\rho_n(\\delta), ~\\text{with probability at least}~1-\\delta\n$, where $V_n= \\lambda^2\\mathbf{I}_d+\\sum_{i=1}^nx_ix_i^{\\top}$.\nThis confidence ellipsoid for $\\mathbf{w}$ can then be represented in terms of the confidence interval for $f(x)$ given in~\\eqref{conf:lin}.\nNotice that the linear model is a special case of~\\eqref{kernelmodel} with $\\mathbf{w} = [w_1,w_2,\\dots, w_d]^{\\top}$ and $\\bm{\\phi}(x) = x$ being $d$ dimensional weight and feature vectors respectively, and $\\Lambda = \\mathbf{I}_d$ being the square identity matrix of dimension $d$.\n\n\\cite{Chowdhury2017bandit} built on the self-normalized bound for the vector valued martingales to prove the following theorem for the kernel-based models. \n\\begin{theorem}\\label{TheoremOnK}\nUnder Assumption~\\ref{ass1}, in the online setting, with probability at least $1-\\delta$, we have for all $x\\in \\mathcal{X}$ that\n\\begin{eqnarray}\n|f(x)-\\mu_n(x)|\\le \\rho_n(\\delta)\\sigma_n(x),\n\\end{eqnarray}\nwhere $\\rho_n(\\delta) = B+R\\sqrt{2\\left(\\gamma_{n-1}+1+\\log(\\frac{1}{\\delta})\\right)}$, and $\\gamma_n=\\sup_{\\{\\mathbf{x}_i\\}_{i=1}^n\\subset\\mathcal{X}}\\log\\det(\\lambda^2\\mathbf{I}_n+\\mathbf{K}_n)$ is the maximal information gain at time $n$, which is closely related to the effective dimension associated with the kernel (e.g., see~\\cite{srinivas2010gaussian, Valko2013kernelbandit}). \n\\end{theorem}\n\nOur open problem is concerned with improving this confidence interval. \n\n\\paragraph{Open Problem.} Under Assumption~\\ref{ass1}, in the online setting, consider the general problem of proving a confidence interval of the following form uniformly in $x\\in \\mathcal{X}$:\n\\begin{eqnarray}\n|f(x)-\\mu_n(x)|\\le \\rho_n(\\delta)\\sigma_n(x),~\\text{with probability at least}~1-\\delta.\n\\end{eqnarray}\nWhat is the lowest growth rate of $\\rho_n(\\delta)$ with $n$? In particular, is it possible to reduce the confidence interval width in Theorem~\\ref{TheoremOnK} by an $\\tilde{\\mathcal{O}}(\\sqrt{\\gamma_n})$ factor?\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\vspace*{-1ex}\n\\section{Discussion}\n\n\nFollowing standard UCB-based bandit algorithm techniques, it can be shown that the GP-UCB algorithm (namely, repeatedly choosing $x$ to maximize the current upper confidence bound) attains\n\\begin{eqnarray}\n\\mathcal{R}(N) = \\tilde{\\mathcal{O}}(\\rho_N(\\delta)\\sqrt{N\\gamma_N}), ~\\text{with probability at least}~1-\\delta.\n\\end{eqnarray}\nSubstituting $\\rho_N(\\delta)$ from Theorem~\\ref{TheoremOnK}, we have $\\mathcal{R}(N) = \\tilde{\\mathcal{O}}(\\gamma_N\\sqrt{N})$. Unfortunately, this is not always sublinear in $N$, since $\\gamma_N$ can grow faster that $\\sqrt{N}$, e.g., in the case of the Mat{\\'e}rn family of kernels.  Hence, the regret bound can be trivial in many cases of interest. It is unknown whether this suboptimal regret bound is a fundamental shortcoming of GP-UCB or a result of suboptimal confidence intervals, but the latter appears likely to be the most significant factor. The same question can be asked about the analysis of many other bandit algorithms including GP-TS~\\citep{Chowdhury2017bandit} and GP-EI~\\citep{nguyen2017regretGPEI}, as well as KOVI in the reinforcement learning setting~\\citep{yang2020provably}.  \n\n\nComparing the results under the online and offline settings, we see a stark contrast of an $\\mathcal{O}(\\sqrt{\\gamma_n})$ factor in the width of confidence intervals. We expect that the $\\mathcal{O}(\\sqrt{\\gamma_n})$ factor in the confidence interval width in the online setting can be replaced by an $\\tilde{\\mathcal{O}}(d\\log(n))$ term, resulting in an \n$\\tilde{\\mathcal{O}}(\\sqrt{dN\\gamma_N })$ regret bound. Roughly speaking, we are suggesting that a square root of the effective dimension of the kernel in the regret bound can be traded off for a square root of the input dimension. \n\nIn a recent work addressing the above challenges and limitations, \\cite{Janz2020SlightImprov} showed that partitioning the domain to many subdomains and fitting an independent GP to each one of them leads to an improved online confidence interval, based on local observations (see their Lemma~$5$). They leveraged this method to prove sublinear regret under Mat{\\'e}rn kernels whenever the smoothness parameter $\\nu$ exceeds one.  This is a significant improvement over the typical $\\tilde{\\mathcal{O}}(\\gamma_N\\sqrt{N})$ bound, which turns out to be sublinear only when $\\nu > d\/2$.\n\nOf significant theoretical importance is a less practical algorithm \\emph{SupKernelUCB}~\\citep{Valko2013kernelbandit}, which achieves an $\\tilde{\\mathcal{O}}(\\sqrt{N\\gamma_N})$ regret bound for the kernelized bandit problem with a finite action set ($|\\mathcal{X}|<\\infty$). As noted in \\cite{Janz2020SlightImprov}, the finite action set assumption can be relaxed to compact domains using a discretization argument; this turns out to contribute only an $\\tilde{\\mathcal{O}}(\\sqrt{d\\log(N)})$ factor to the regret bound~\\citep[e.g., see,][for the details]{li2021gaussian}\n. This bound is tight for the cases where a lower bound on regret is known, namely for commonly used squared exponential and Mat{\\'e}rn kernels~\\citep{Scarlett2017Lower, vakili2020information}.  \n\nVery recently, a line of works has attained $\\tilde{\\mathcal{O}}(\\sqrt{N\\gamma_N})$ regret bounds for various other algorithms beyond SupKernelUCB.  However, all of them rely on more sophisticated techniques rather than a standard combination of GP-UCB and an online confidence bound.  Briefly, \\cite{salgia2020} uses a tree-based partitioning on the domain along with localized search techniques, \\cite{jamieson21} uses experimental design techniques to perform batch sampling and eliminate suboptimal inputs, and \\cite{li2021gaussian} observes that the above-mentioned offline confidence bounds of \\cite{vakili21} can be used within batches as long as the GP is reset at the start of each batch.\n\nIn view of this discussion, the improvement from $\\tilde{\\mathcal{O}}(\\gamma_N\\sqrt{N})$ to $\\tilde{\\mathcal{O}}(\\sqrt{N\\gamma_N})$ has now been obtained using several different algorithms, but it remains to determine whether GP-UCB (or a simple variation thereof) can achieve it.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThere are various approaches to calculate the $B\\to\\pi$ transition\nform factor, such as the lattice QCD\ntechnique\\cite{lattice,lattice2,lattice3}, the QCD light-cone sum\nrules (LCSRs)\\cite{sumhuang,sumrule,sumrule2,pball} and the\nperturbative QCD (PQCD)\napproach\\cite{wirbel,huangl,lihn,lihn1,weiy2,lucai}. The PQCD\ncalculation is reliable only when the involved energy scale is\nhard enough, i.e. in the large recoil regions. Due to the\nrestriction to the $\\pi$ energies smaller than the inverse lattice\nspacing, the lattice QCD calculation becomes more difficult in the\nlarge recoil regions and at the present, the lattice QCD results\nof the $B\\to\\pi$ transition form factor are available only for\nsoft regions, i.e. $q^2>15GeV^2$. The lattice QCD results can be\nextrapolated to small $q^2$ regions, and the different\nextrapolation methods might cause uncertainties about\n$5\\%$\\cite{lattice2}. While, the QCD LCSRs can involve both the\nhard and the soft contributions below $q^2<18GeV^2$\\cite{sumhuang}\nand can be extrapolated to higher $q^2$\nregions\\cite{sumrule,sumrule2,pball}. Therefore, the results from\nthe PQCD approach, the lattice QCD approach and the QCD LCSRs are\ncomplementary to each other, and by combining the results from\nthese three methods, one may obtain a full understanding of the\n$B\\to\\pi$ transition form factor in its physical region, $0\\leq\nq^2 \\leq (M_B-M_\\pi)^2\\simeq 25GeV^2$.\n\nCertain exclusive process involving hadrons can be described by\nPQCD if the momentum transfer is sufficiently large. The amplitude\ncan be factorized into the convolution of the non-perturbative\nwavefunction for each of the hadrons with a PQCD calculable\nhard-scattering amplitude. The PQCD factorization theorem has been\nworked out in Refs.\\cite{li,liyu} based on the earlier works on\nthe applications of PQCD to hard exclusive processes \\cite{hard}.\nIn the present paper, we shall use the PQCD approach to calculate\nthe $B\\to\\pi$ transition form factor in the large recoil regions.\n\nIn the PQCD approach based on collinear factorization theorem, a\ndirect calculation of the one-gluon-exchange diagram for the $B$\nmeson transition form factor suffers singularities from the\nendpoint region of a momentum fraction $x\\to 0$. Because of these\nsingularities, it was claimed that $B\\to\\pi$ transition form\nfactor is dominated by soft dynamics and not calculable in\nPQCD\\cite{pqcdno}. In fact, in the endpoint region the parton\ntransverse momenta $\\mathbf{k}_\\perp$ are not negligible. After\nincluding the parton transverse momenta, large double logarithmic\ncorrections $\\alpha_s \\ln^2 k_\\perp$ appear in higher order\nradiative corrections and must be summed to all orders. In\naddition, there are also large logarithms $\\alpha_s \\ln^2x$ which\nshould also be summed (threshold resummation\\cite{threshold1}).\nThe relevant Sudakov form factors from both $k_\\perp$ and the\nthreshold resummation can cure the endpoint singularity which\nmakes the calculation of the hard amplitudes infrared safe, and\nthen the main contribution comes from the perturbative regions.\n\nAn important issue for calculating the $B\\to\\pi$ transition form\nfactor is whether we need to take both the two wavefunctions\n$\\Psi_B$ and $\\bar\\Psi_B$ into consideration or simply $\\Psi_B$ is\nenough? In literature, many authors (see\nRefs.\\cite{huangl,lihn,lihn1}) did the phenomenological analysis\nwith only $\\Psi_B$, setting $\\bar\\Psi_B=0$ (or strictly speaking,\nignoring the contributions from $\\bar\\Psi_B$). However, As has\nbeen argued in Refs.\\cite{wu1,descotes}, one may observe that the\ndistribution amplitudes (DAs) of those two wavefunctions have a\nquite different endpoint behavior, such difference may be strongly\nenhanced by the hard scattering kernel. Even though $\\bar\\Psi_B$\n(with the definition in Ref.\\cite{lucai}) is of subleading order\ncontribution, there is no convincing motivation for setting\n$\\bar\\Psi_B=0$. In the present paper, we shall keep both the two\nwavefunctions $\\Psi_B$ and $\\bar\\Psi_B$ to do our calculations and\nshow to what extent the $\\bar\\Psi_B$ can affect the final results.\nAnother issue we need to be more careful is about the pionic\ntwist-3 contributions. Based on the asymptotic behavior of the\ntwist-3 DAs, especially $\\phi^{as}_p(x)\\equiv 1$, most of the\npeople pointed out a large twist-3 contribution\\cite{li1,weiy2} to\nthe $B\\to\\pi$ transition form factor, i.e. bigger than that of the\nleading twist in almost all of the energy regions. In\nRef.\\cite{huangwu}, the authors have made a detailed analysis on\nthe model dependence of the twist-3 contributions to the pion\nelectro-magnetic form factor, and have raised a new twist-3\nwavefunction with a better endpoint behavior for $\\Psi_p$, which\nis derived from the QCD sum rule moment calculation\\cite{huang3}.\nAnd their results show that with such new form for $\\Psi_p$, the\ntwist-3 contributions to the pion electro-magnetic form factor are\npower suppressed in comparison to the leading twist contributions.\nAccording to the power counting rules in Ref.\\cite{li1}, the\npionic twist-2 and twist-3 contributions should be of the same\norder for the case of the B meson decays. With the new form for\n$\\Psi_p$\\cite{huangwu}, we show that for the case of the $B\\to\\pi$\ntransition form factor, even though the twist-3 contributions are\nof the same order of the leading twist contributions, its values\nare less than the leading twist contribution.\n\nThe purpose of the paper is to examine the $B\\to\\pi$ transition\nform factor in the PQCD approach, and to show how the PQCD results\ncan match with the QCD LCSR results and the extrapolated lattice\nQCD results. In the PQCD approach, the full transverse momentum\ndependence ($k_T$-dependence) for both the hard scattering part\nand the non-perturbative wavefunction, the Sudakov effects and the\nthreshold effects are included to cure the endpoint singularity.\nIn section II, based on the $k_T$ factorization formulism, we give\nthe PQCD formulae for the $B\\to\\pi$ transition form factor in the\nlarge recoil regions. In section III, we give our numerical\nresults and carefully study the contributions from $\\Psi_B$ and\n$\\bar\\Psi_B$, and those from the different pionic twist\nstructures. The slope of the obtained form factors\n$F_{+,\\;0}^{B\\pi}(q^2)$ in the large recoil regions can match with\nthose obtained from other approaches. Conclusion and a brief\nsummary are presented in the final section.\n\n\\section{$B\\to\\pi$ transition form factor in the large recoil regions}\n\nFirst, we give our convention on the kinematics. For convenience,\nall the momenta are described in terms of the light cone (LC)\nvariables. In the LC coordinate, the momentum is described in the\nform, $k=(\\frac{k^+}{\\sqrt 2}, \\frac{k^-}{\\sqrt 2},\n\\mathbf{k}_\\bot)$, with $k^{\\pm}=k^0\\pm k^3$ and\n$\\mathbf{k}_\\bot=(k^1, k^2)$. The scalar product of two arbitrary\nvectors $A$ and $B$ is, $A\\cdot B= \\frac{A^+B^- + A^-B^+}{ 2} -\n\\mathbf{A}_\\bot\\cdot\\mathbf{B}_\\bot$. The pion mass is neglected\nand its momentum is chosen to be in the minus direction. Under the\nabove convention, we have $P_B =\\frac{M_B}{\\sqrt\n2}(1,1,\\mathbf{0}_\\perp)$, $P_{\\pi}=\\frac{M_B}{\\sqrt 2}(0,\\eta\n,\\mathbf{0}_\\bot)$ and $\\bar{P}_{\\pi}=\\frac{M_B}{\\sqrt\n2}(\\eta,0,\\mathbf{0}_\\bot)$, with $\\eta=1-\\frac{q^2}{M_B^2}$ and\n$q=P_B-P_{\\pi}$.\n\nThe two $B\\to \\pi$ transition form factors $F_+^{B\\pi}(q^2)$ and\n$F_0^{B\\pi}(q^2)$ are defined as follows:\n\\begin{equation}\\label{eq:bpi1}\n\\langle \\pi(P_{\\pi})|\\bar u \\gamma_{\\mu}b|\\bar\nB(P_B)\\rangle=\\left((P_B+P_{\\pi})_{\\mu}-\n\\frac{M_B^2-m_{\\pi}^2}{q^2}q_{\\mu}\\right)F_+^{B\\pi}(q^2)+\n \\frac{M_B^2-m_{\\pi}^2}{q^2} q_{\\mu}F_0^{B\\pi}(q^2),\n\\end{equation}\nwhere $F_+^{B\\pi}(0)$ should be equal to $F_0^{B\\pi}(0)$ so as to\ncancel the poles at $q^2=0$.\n\nThe amplitude for the $B\\to \\pi$ transition form factor can be\nfactorized into the convolution of the wavefunctions for the\nrespective hadrons with the hard-scattering amplitude. The\nwavefunctions are non-perturbative and universal. The momentum\nprojection for the matrix element of the pion has the following\nform,\n\\begin{equation}\nM_{\\alpha\\beta}^{\\pi} = \\frac{i f_{\\pi}}{4} \\Bigg\\{\n\\slash\\!\\!\\!p\\,\\gamma_5\\,\\Psi_{\\pi}(x, \\mathbf{k_\\perp})-\nm^p_0\\gamma_5 \\left(\\Psi_p(x, \\mathbf{k_\\perp})\n-i\\sigma_{\\mu\\nu}\\left(n^{\\mu}\\bar{n}^{\\nu}\\,\\frac{\\Psi_{\\sigma}'(x,\n\\mathbf{k_\\perp})}{6}-p^\\mu\\,\\frac{\\Psi_\\sigma(x,\n\\mathbf{k_\\perp})}{6}\\, \\frac{\\partial}{\\partial\n\\mathbf{k}_{\\perp\\nu}} \\right)\\right)\n\\Bigg\\}_{\\alpha\\beta},\\label{benek}\n\\end{equation}\nwhere $f_{\\pi}$ is the pion decay constant and $m^p_0$ is the\nparameter that can be determined by QCD sum rules\\cite{huang3}.\n$\\Psi_{\\pi}(x,\\mathbf{k_{\\perp}})$ is the leading twist (twist-2)\nwave function, $\\Psi_p(x,\\mathbf{k_{\\perp}})$ and\n$\\Psi_{\\sigma}(x,\\mathbf{k_{\\perp}})$ are sub-leading twist\n(twist-3) wave functions, respectively. $\\Psi_{\\sigma}'(x,\n\\mathbf{k_\\perp})=\\partial \\Psi_{\\sigma} (x,\n\\mathbf{k_\\perp})\/\\partial x$, $n=(\\sqrt{2},0,\\mathbf{0}_\\bot)$\nand $\\bar{n}=(0,\\sqrt{2},\\mathbf{0}_\\bot)$ are two null vectors\nthat point to the plus and the minus directions, respectively. The\nmomentum projection for the matrix element of the B meson can be\nwritten as \\cite{weiy2,BenekeFeldmann}:\n\\begin{equation}\\label{projectorB}\nM^B_{\\alpha\\beta}=-\\frac{if_B}{4}\n\\left\\{\\frac{\\dirac{p}_B+M_B}{2}\\left[\\dirac{n}\n\\Psi^+_B(\\xi,\\mathbf{l_\\bot}) +\n\\bar\\dirac{n}\\Psi^-_B(\\xi,\\mathbf{l_\\bot})-\\Delta(\\xi,\n\\mathbf{l}_{\\bot}) \\gamma^\\mu \\frac{\\partial} {\\partial\nl_\\perp^\\mu}\\right] \\gamma_5\\right\\}_{\\alpha\\beta}\\ ,\n\\end{equation}\nwhere $\\xi=\\frac{l^+}{M_B}$ is the momentum fraction for the light\nspectator quark in the B meson and $\\Delta(\\xi, \\mathbf{l}_{\\bot})\n=M_B \\int_0^{\\xi} d\\xi' (\\Psi^-_B(\\xi',\\mathbf{l}_{\\bot})\n-\\Psi^+_B(\\xi',\\mathbf{l}_{\\bot}))$. Note the four-component\n$l_\\perp^\\mu$ in Eq.(\\ref{projectorB}) is defined through,\n$l^{\\mu}_\\perp=l^\\mu-\\frac{(l^+ n^\\mu +l^-\\bar{n}^\\mu)}{2}$ with\n$l=(\\frac{l^+}{\\sqrt{2}},\\frac{l^-}{\\sqrt{2}},\\mathbf{l}_\\perp)$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\textwidth]{fig1.eps}\n\\caption{Lowest order hard-scattering kernel for $B\\to\\pi$ form\nfactor, where the cross denotes an appropriate gamma matrix.}\n\\label{figbpi}\n\\end{figure}\n\nIn the large recoil regions, the $B\\to\\pi$ transition form factor\nis dominated by a single gluon exchange in the lowest order as\ndepicted in Fig.(\\ref{figbpi}). In the hard scattering kernel, the\ntransverse momentum in the denominators are retained to regulate\nthe endpoint singularity. The masses of the light quarks and the\nmass difference ($\\bar\\Lambda$) between the b quark and the B\nmeson are neglected. The terms proportional to $\\mathbf{k}_\\bot^2$\nor $\\mathbf{l}_\\bot^2$ in the numerator are dropped, which are\npower suppressed compared to other ${\\cal O}(M_B^2)$ terms. Under\nthese treatment, the Sudakov form factor from $k_T$ resummation\ncan be introduced into the PQCD factorization theorem without\nbreaking the gauge invariance\\cite{li1}. In the transverse\nconfiguration $b$-space and by including the Sudakov form factors\nand the threshold resummation effects, we obtain the formulae for\n$F_+^{B\\pi}(q^2)$ and $F_0^{B\\pi}(q^2)$ as following,\n\\begin{eqnarray}\nF_+^{B\\pi}(q^2) &=& \\frac{\\pi C_F}{N_c} f_{\\pi}f_BM_B^2\\int d\\xi\ndx\\int b_Bdb_B~ b_\\pi db_\\pi~ \\alpha_s(t)\n\\times\\exp(-S(x,\\xi,b_\\pi,b_B;t)) \\nonumber\\\\\n&\\times& S_t(x)S_t(\\xi)\\Bigg \\{ \\Bigg [ \\Psi_\\pi(x, b_\\pi)\\left (\n(x\\eta+1)\\Psi_B(\\xi, b_B)+  (x\\eta-1)\\bar\\Psi_B(\\xi, b_B) \\right\n)\\nonumber \\\\\n&+& \\frac{m_0^p}{M_B}\\Psi_p(x, b_\\pi)\\cdot\\left(\n(1-2x)\\Psi_B(\\xi,b_B)+\\left(\\frac{2}{\\eta}-1\\right)\\bar\\Psi_B(\\xi,b_B)\\right)\n-\\frac{m_0^p}{M_B}\\frac{\\Psi'_\\sigma(x,b_\\pi)}{6} \\cdot\\nonumber\\\\\n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\left(\\left(1+2x-\\frac{2}\n{\\eta}\\right)\\Psi_B(\\xi,b_B)-\\bar\\Psi_B(\\xi,b_B) \\right)\n+6\\frac{m_0^p}{M_B}\\frac{\\Psi_\\sigma(x,b_\\pi)}\n{6}\\Psi_B(\\xi,b_B)\\Bigg ]h_1(x,\\xi,b_{\\pi},b_B)\\nonumber \\\\\n&-& (1+\\eta+x\\eta) \\frac{m_0^p}{M_B}\n\\frac{\\Psi_\\sigma(x,b_\\pi)}{6}\n[M_B\\Delta(\\xi,b_B)]h_2(x,\\xi,b_{\\pi},b_B)\\nonumber \\\\\n&+& \\Bigg [ \\Psi_\\pi(x, b_\\pi)\\left ( -\\xi\\bar\\eta\n[\\Psi_B(\\xi,b_B)+\\bar\\Psi_B(\\xi,b_B)]+\\frac{\\Delta(\\xi,b_B)}{M_B}\n\\right )+2\\frac{m_0^p}{M_B}\\Psi_p(x,b_\\pi)\\cdot\\nonumber \\\\\n&& \\left( (1-\\xi)\\Psi_B(\\xi,b_B)+(1+\\xi-\\frac{2\\xi}\n{\\eta})\\bar\\Psi_B(\\xi,b_B) +2\\frac{\\Delta(\\xi,b_B)} {M_B}\\right )\n\\Bigg ] h_1(\\xi,x,b_B,b_\\pi) \\Bigg \\}, \\label{fbc+}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nF_0^{B\\pi}(q^2) &=& \\frac{\\pi C_F}{N_c} f_{\\pi}f_BM_B^2\\int d\\xi\ndx\\int b_Bdb_B~ b_\\pi db_\\pi~ \\alpha_s(t)\n\\times \\exp(-S(x,\\xi,b_\\pi,b_B;t))\\nonumber\\\\\n&\\times& S_t(x)S_t(\\xi) \\Bigg \\{ \\Bigg [ \\Psi_\\pi(x,\nb_\\pi)\\eta\\left ( (x\\eta+1)\\Psi_B(\\xi,\nb_B)+(x\\eta-1)\\bar\\Psi_B(\\xi, b_B) \\right )\n\\nonumber \\\\\n&+&\\frac{m_0^p}{M_B}\\Psi_p(x, b_\\pi) \\big((2-\\eta-2x\\eta)\n\\Psi_B(\\xi,b_B)+\\eta\\bar\\Psi_B(\\xi,b_B)\\big) \\nonumber \\\\\n&-&\\frac{m_0^p}{M_B}\\frac{\\Psi'_\\sigma(x,b_\\pi)}{6}\\cdot\\big(\n\\eta(2x-1)\\Psi_B(\\xi,b_B)-(2-\\eta)\\bar\\Psi_B(\\xi,b_B)\n\\big) \\nonumber\\\\\n&+&6\\frac{m_0^p}{M_B}\n\\eta\\frac{\\Psi_\\sigma(x,b_\\pi)}{6}\\Psi_B(\\xi,b_B)\\Bigg ]\nh_1(x,\\xi,b_\\pi,b_B) \\nonumber \\\\\n&-& [3-\\eta-x\\eta]\\frac{m_0^p}{M_B} \\frac{\\Psi_\\sigma(x,b_\\pi)}{6}\n[M_B\\Delta(\\xi,b_B)]h_2(x,\\xi,b_{\\pi},b_B)\\nonumber \\\\\n&+& \\Bigg [ \\Psi_\\pi(x, b_\\pi)\\eta\\left ( \\xi\\bar\\eta\n(\\Psi_B(\\xi,b_B)+\\bar\\Psi_B(\\xi,b_B))+\\frac{\\Delta(\\xi,b_B)}{M_B}\n\\right )\\nonumber \\\\\n&+& 2\\frac{m_0^p}{M_B}\\Psi_p(x,b_\\pi)\\cdot\\Big(\n(\\eta(1+\\xi)-2\\xi)\\Psi_B(\\xi,b_B) +\\eta(1-\\xi)\\bar\\Psi_B(\\xi,b_B)\n\\nonumber\\\\\n&+& 2(2-\\eta) \\frac{\\Delta(\\xi,b_B)}{M_B}\\Big) \\Bigg ]\nh_1(\\xi,x,b_B,b_\\pi) \\Bigg \\},\\label{fbc0}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nh_1(x,\\xi,b_\\pi,b_B)&=&K_0(\\sqrt{\\xi x\\eta}~M_B b_B)\n    \\Bigg [ \\theta(b_B-b_\\pi)I_0(\\sqrt{x\\eta}~M_Bb_\\pi)\n      K_0(\\sqrt{x\\eta}~M_B b_B) \\nonumber \\\\\n&&+\\theta(b_\\pi-b_B)I_0(\\sqrt{x\\eta}~M_Bb_B)\n    K_0(\\sqrt{x\\eta}~M_B b_\\pi) \\Bigg ], \\\\\nh_2(x,\\xi,b_\\pi,b_B)&=&\\frac{b_B}{2\\sqrt{\\xi\nxy}M_B}K_{1}(\\sqrt{\\xi x\\eta}~M_B b_B)\n    \\Bigg [ \\theta(b_B-b_\\pi)I_0(\\sqrt{x\\eta}~M_Bb_\\pi)\n      K_0(\\sqrt{x\\eta}~M_B b_B) \\nonumber \\\\\n&&+\\theta(b_\\pi-b_B)I_0(\\sqrt{x\\eta}~M_Bb_B)\n    K_0(\\sqrt{x\\eta}~M_B b_\\pi) \\Bigg ],\n\\end{eqnarray}\nand we have set,\n\\begin{equation}\\label{oldpsi}\n\\Psi_B=\\frac{\\Psi_B^{+}+\\Psi_B^-}{2}\\ ,\\;\\;\\;\n\\bar{\\Psi}_B=\\frac{\\Psi_B^{+}-\\Psi_B^-}{2}\\ .\n\\end{equation}\nThe functions $I_i$ ($K_i$) are the modified Bessel functions of\nthe first (second) kind with the $i$-{\\it th} order. The angular\nintegrations in the transverse plane have been performed. The\nfactor $\\exp(-S(x,\\xi,b_\\pi,b_B;t))$ contains the Sudakov\nlogarithmic corrections and the renormalization group evolution\neffects of both the wave functions and the hard scattering\namplitude,\n\\begin{equation}\nS(x,\\xi,b_\\pi,b_B;t)=\n\\left[s(x,b_\\pi,M_b)+s(\\bar{x},b_\\pi,M_b)+s(\\xi,b_B,M_b)\n-\\frac{1}{\\beta_{1}}\\ln\\frac{\\hat{t}}{\\hat{b}_\\pi}\n-\\frac{1}{\\beta_{1}}\\ln\\frac{\\hat{t}}{\\hat{b}_B} \\right],\n\\end{equation}\nwhere ${\\hat t}={\\rm ln}(t\/\\Lambda_{QCD})$, ${\\hat b}_B ={\\rm\nln}(1\/b_B\\Lambda_{QCD})$, ${\\hat b}_\\pi ={\\rm\nln}(1\/b_\\pi\\Lambda_{QCD}) $ and $s(x,b,Q)$ is the Sudakov exponent\nfactor, whose explicit form up to next-to-leading log\napproximation can be found in Ref.\\cite{liyu}. $S_t(x)$ and\n$S_t(\\xi)$ come from the threshold resummation effects and here we\ntake a simple parametrization proposed in Refs.\\cite{li1,kls},\n\\begin{equation}\nS_t(x)=\\frac{2^{1+2c}\\Gamma(3\/2+c)}{\\sqrt{\\pi}\\Gamma(1+c)}\n[x(1-x)]^c\\;,\n\\end{equation}\nwhere the parameter $c$ is determined around $0.3$ for the present\ncase.\n\nThe hard scale $t$ in $\\alpha_s(t)$ and the Sudakov form factor\nmight be varied for the different hard scattering parts and here\nwe need two $t_i$\\cite{li1,lucai}, whose values are chose as the\nlargest scale of the virtualities of internal particles, i.e.\n\\begin{equation}\nt_1={\\rm MAX}\\left(\\sqrt{x\\eta}M_B,1\/b_\\pi,1\/b_B\\right),\\;\nt_2={\\rm MAX} \\left(\\sqrt{\\xi\\eta}M_B,1\/b_\\pi,1\/b_B\\right).\n\\end{equation}\nThe Fourier transformation for the transverse part of the wave\nfunction is defined as\n\\begin{equation}\\label{fourier}\n\\Psi(x,\\mathbf{b})=\\int_{|\\mathbf{\\mathbf{k}}|<1\/b}\nd^2\\mathbf{k}_\\perp\\exp\\left(-i\\mathbf{k}_\\perp\n\\cdot\\mathbf{b}\\right)\\Psi(x,\\mathbf{k}_\\perp),\n\\end{equation}\nwhere $\\Psi$ stands for $\\Psi_\\pi$, $\\Psi_p$, $\\Psi_\\sigma$,\n$\\Psi_B$, $\\bar\\Psi_B$ and $\\Delta$, respectively. The upper edge\nof the integration $|\\mathbf{k}_\\perp|<1\/b$ is necessary to ensure\nthat the wave function is soft enough\\cite{huang2}.\n\nIn summary, we compare the results in Eqs.(\\ref{fbc+},\\ref{fbc0})\nwith those in Refs.\\cite{descotes,li1,weiy2,lucai}. In\nRef.\\cite{descotes}, only leading twist ($\\Psi_\\pi$) of the pion\nis discussed. Setting the twist-3 terms to zero, the two formulae\nin Eqs.(\\ref{fbc+},\\ref{fbc0}) and Ref.\\cite{descotes} are in\nagreement. In Ref.\\cite{li1}, the single B meson wave function\n$\\Psi_B$ is assumed and the terms of $\\bar\\Psi_B$ and $\\Delta$ are\nneglected. And in Ref.\\cite{lucai}, with a new definition for\n$\\Psi_B$ and $\\bar\\Psi_B$, i.e.\n\\begin{equation}\n\\label{newpsi} \\Psi_B=\\Psi_B^{+}\\ ,\\;\\;\\;\n\\bar{\\Psi}_B=(\\Psi_B^{+}-\\Psi_B^-),\n\\end{equation}\nboth contributions from $\\Psi_B$ and $\\bar\\Psi_B$ are taken into\nconsideration, with only the terms of $\\Delta$ are neglected. The\nmomentum projector used in \\cite{li1,lucai} for the pion is\ndifferent from the present projector in Eq.(\\ref{benek}), i.e.\nthere is no term proportional to $\\Psi_\\sigma$ in\nRefs.\\cite{li1,lucai}. Except for these\ndifferences\\footnote{According to the power counting rules in\nRef.\\cite{li1}, the terms that do not existent in Ref.\\cite{li1}\nare defined as sub-leading terms in $1\/M_B$ and are neglected\naccordingly. And here, we keep all the terms with care.}, the\nformulae in \\cite{lucai,li1} are consistent with ours. Our results\nagree with Ref.\\cite{weiy2}, except for several minus errors that\nshould be corrected there.\n\n\\section{numerical calculations}\n\nIn the numerical calculations, we use\n\\begin{eqnarray}\n\\Lambda^{(n_f=4)}_{\\over{MS}}=250MeV,\\;\\; f_\\pi=131MeV,\\;\\;\nf_B=190MeV,\\;\\;m^p_{0}=1.30GeV.\n\\end{eqnarray}\nThe wavefunctions in the compact parameter $b$-space,\n$\\Psi^B_+(\\xi,b_B)$, $\\Psi^B_-(\\xi,b_B)$, $\\Psi_\\pi(x,b_\\pi)$,\n$\\Psi_p(x,b_\\pi)$ and $\\Psi_\\sigma(x,b_\\pi)$ can be found in the\nappendix. The $k_T$-dependence has been kept in both the B meson\nand the pion wavefunctions. As has been argued in several\npapers\\cite{huangww,weiy,jk,huangwu}, the intrinsic\n$k_T$-dependence of the wave function is important and the results\nwill be overestimated without including this effect, so it is\nnecessary to include the transverse momentum dependence into the\nwave functions not only for the B meson but also for the pion. As\nhas been argued in Ref.\\cite{huangwu}, we take $m^p_0=1.30GeV$ for\nlatter discussions, which is a little below the value given by the\nchiral perturbation theory\\cite{chiral}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{afpilamnew.eps}%\n\\includegraphics[width=0.5\\textwidth]{pfpilamnew.eps}\n\\caption{PQCD results for the $B\\to \\pi$ transition form factors\n$F_+^{B\\pi}(q^2)$ (Left) and $F_0^{B\\pi}(q^2)$ (Right) with\ndifferent values for $\\bar\\Lambda$. The dashed line stands for\n$\\bar\\Lambda=0.5GeV$, the dotted line stands for\n$\\bar\\Lambda=0.6GeV$, the upper edge of the shaded band\ncorresponds to $\\bar\\Lambda=0.40GeV$ and the lower edge of the\nband corresponds to $\\bar\\Lambda=0.70GeV$. For comparison, the\nsolid line comes from the QCD LCSR\\cite{sumhuang,sumrule} and the\nfuscous shaded band shows its theoretical error $\\pm 10\\%$.}\n\\label{lambda}\n\\end{figure}\n\nThe two wavefunctions $\\Psi_B$ and $\\bar\\Psi_B$ of the B meson\nshown in the appendix depend only on the effective mass\n($\\bar\\Lambda=M_B-m_b$) of the B meson. An estimate of\n$\\bar\\Lambda$ using QCD sum rule approach gives\n$\\bar\\Lambda=0.57\\pm0.07GeV$\\cite{lambdavalue}. In\nFig.(\\ref{lambda}), we show the $B\\to\\pi$ transition form factor\nwith different value of $\\bar\\Lambda$, where the shaded band is\ndrawn with a broader range for $\\bar\\Lambda$, i.e.\n$\\bar\\Lambda\\in(0.4GeV,0.7GeV)$. And for comparison, we show the\nQCD LCSR result \\cite{sumrule} in solid line and its theoretical\nerror $(\\pm 10\\%)$ by a fuscous shaded band in Fig.(\\ref{lambda}).\nThe results show that the $B\\to\\pi$ transition form factor will\ndecrease with the increment of $\\bar\\Lambda$. When $\\bar\\Lambda\\in\n(0.5GeV,0.6GeV)$, one may observe that the present results agree\nwell with the QCD LCSR results\\cite{sumhuang,sumrule} up to\n$q^2\\sim 14GeV^2$. In Fig.(\\ref{lambda}), for simplicity, only the\nQCD LCSR results of Ref.\\cite{sumrule} are shown. The LCSR results\nin Refs.\\cite{sumhuang,sumrule} are in agreement with each other\neven though they have taken different ways to improve the QCD LCSR\ncalculation precision, i.e. in Ref.\\cite{sumhuang}, an alternative\nway to do the QCD LCSR calculation is adopted in which the pionic\ntwist-3 contributions are avoided by calculating the correlator\nwith a proper chiral current and then the leading twist\ncontributions are calculated up to next-to-leading order; while in\nRef.\\cite{sumrule}, the usual QCD LCSR approach is adopted and\nboth the twist-2 and twist-3 contributions are calculated up to\nnext-to-leading order. In Ref.\\cite{lucai}, $\\bar\\Lambda$ is\ntreated as a free parameter and a bigger value is adopted there,\ni.e. $\\bar\\Lambda=(0.70\\pm0.05)GeV$. The main reason is that in\nthe present paper, we have used an improved form (with better\nendpoint behavior than that of the asymptotic one) for the pionic\ntwist-3 wavefunction $\\Psi_p$, while in Ref.\\cite{lucai}, they\ntook $\\phi_p$ in Ref.\\cite{pball} (with an endpoint behavior even\nworse than the asymptotic one) other than $\\Psi_p$ to do the\ncalculations, so the value of $\\bar\\Lambda$ in Ref.\\cite{lucai}\nmust be big enough to suppress the endpoint singularity coming\nfrom the hard kernel. For clarity, if not specially stated, we\nshall fix $\\bar\\Lambda$ to be $0.5GeV$ in the following\ndiscussions.\n\n\\begin{figure}\n\\centering\n\\begin{minipage}[c]{0.48\\textwidth}\n\\centering\n\\includegraphics[width=2.9in]{afpitwist.eps}\n(a)\n\\end{minipage}%\n\\begin{minipage}[c]{0.48\\textwidth}\n\\centering\n\\includegraphics[width=2.9in]{afpibwave.eps}\n(b)\n\\end{minipage}\n\\caption{PQCD results for the $B\\to \\pi$ transition form factor\n$F_+^{B\\pi}(q^2)$ with fixed $\\bar\\Lambda=0.5GeV$. The left\ndiagram is for the different pion twist structures, $\\Psi_\\pi$,\n$\\Psi_p$ and $\\Psi_\\sigma$. The right diagram is for the different\nB meson structures, $\\Psi_B$, $\\bar\\Psi_B$ and $\\Delta$, where\n$\\Psi_B$ and $\\bar\\Psi_B$ are defined in Eq.(\\ref{oldpsi}).}\n\\label{pionwave}\n\\end{figure}\n\nSecond, to get a deep understanding of the $B\\to\\pi$ transition\nform factor, we discuss the contributions from different parts of\nthe B meson wavefunction or the pion wave function,\ncorrespondingly. Here we take $F_+^{B\\pi}(q^2)$ to do our\ndiscussions and the case of $F_0^{B\\pi}(q^2)$ can be done in a\nsimilar way. In Fig.(\\ref{pionwave}a), we show the contributions\nfrom the different twist structures of the pion wave function,\ni.e. $\\Psi_\\pi$, $\\Psi_p$ and $\\Psi_\\sigma$ (the contributions\nfrom the terms involving $\\Psi'_\\sigma$ are included in\n$\\Psi_\\sigma$), respectively. From Fig.(\\ref{pionwave}a), one may\nobserve that the contribution from $\\Psi_\\pi$ is the biggest, then\ncomes that of $\\Psi_p$ and $\\Psi_\\sigma$. And the ratio between\nall the twist-3 contributions and the leading twist contribution\nis $\\sim 70\\%$ in the large recoil regions. This behavior is quite\ndifferent from the conclusion that has been drawn in\nRefs.\\cite{weiy2,li1}, in which they concluded that the twist-3\ncontribution is bigger than that of twist-2 contribution,\nespecially in Ref.\\cite{weiy2}, it claimed that the twist-3\ncontribution is about three times bigger than that of twist-2 at\n$q^2=0$. Such kind of big twist-3 contributions are due to the\nfact that they only took the pion distribution amplitudes into\nconsideration (or simply adding a harmonic transverse momentum\ndependence for the pion wavefunctions), and then the endpoint\nsingularity coming from the hard kernel can not be effectively\nsuppressed, especially for $\\Psi_p$ whose DA's asymptotic behavior\nis $\\phi_p\\equiv 1$. In Ref.\\cite{huangwu}, the authors have made\na detailed analysis on the model dependence of the twist-3\ncontributions to the pion electro-magnetic form factor, and have\nraised a new twist-3 wavefunction (as is shown in the appendix)\nwith a better endpoint behavior for $\\Psi_p$, which is inspired\nfrom QCD sum rule moment calculation. With this model wave\nfunction for $\\Psi_p$, Ref.\\cite{huangwu} shows that the twist-3\ncontributions of the pion electro-magnetic form factor agree well\nwith the power counting rule, i.e. the twist-3 contribution drops\nfast and it becomes less than the twist-2 contribution at $Q^2\\sim\n10GeV^2$. For the present B meson case, according to the power\ncounting rules in Ref.\\cite{li1}, the twist-3 contribution and the\ntwist-2 contribution are of the same order, however one may find\nfrom Fig.(\\ref{pionwave}a) that with a new form with better\nendpoint behavior for $\\Psi_p$, the twist-3 contribution can be\neffectively suppressed and then its contribution is less than the\nleading twist contribution.\n\n\\begin{figure}\n\\centering\n\\begin{minipage}[c]{0.48\\textwidth}\n\\centering\n\\includegraphics[width=2.9in]{afpibwave2.eps}\n(a)\n\\end{minipage}%\n\\begin{minipage}[c]{0.48\\textwidth}\n\\centering\n\\includegraphics[width=2.9in]{comparebbar.eps}\n(b)\n\\end{minipage}\n\\caption{PQCD results for the $B\\to \\pi$ form factor\n$F_+^{B\\pi}(q^2)$ with fixed $\\bar\\Lambda=0.5GeV$, where $\\Psi_B$\nand $\\bar\\Psi_{B}$ are defined in Eq.(\\ref{newpsi}). The left\ndiagram shows the contributions from different B meson\nwavefunctions, $\\Psi_B$, $\\bar\\Psi_{B}$ and $\\Delta$,\nrespectively. The right diagram is the distribution of the ratio\n$R=\\left(\\frac{F^{B\\pi}_{+}|_{\\bar\\Psi_B}}{F^{B\\pi}_{+}|_{All}}\\right)$\nversus $q^2$. } \\label{bwaves2}\n\\end{figure}\n\nNow, we show to what extent, $\\bar\\Psi_B$ will affect the final\nresults. Fig.(\\ref{pionwave}b) presents the contributions from\n$\\Psi_B$, $\\bar\\Psi_B$ and $\\Delta$ respectively, where $\\Psi_B$\nand $\\bar\\Psi_{B}$ are defined in Eq.(\\ref{oldpsi}). From\nFig.(\\ref{pionwave}b), one may observe that the contribution from\n$\\Delta$ is quite small and can be safely neglected as has been\ndone in most of the calculations. However the contribution from\n$\\bar\\Psi_B$ is quite large, i.e. at $q^2=0$, the ratio between\nthe contributions of $\\bar\\Psi_B$ and $\\Psi_B$ is about $(-70\\%)$,\nwhich roughly agrees with the observation in Ref.\\cite{weiy2}. So\nthe negative contribution from $\\bar\\Psi_B$ can not be neglected,\nand it is necessary to suppress the big positive contribution from\n$\\Psi_B$ so as to get a more reasonable total contributions from\nboth $\\Psi_B$ and $\\bar\\Psi_B$. The above results of\nFig.(\\ref{pionwave}b) is obtained by using the definition\nEq.(\\ref{oldpsi}). A new definition (\\ref{newpsi}) for $\\Psi_B$\nand $\\bar\\Psi_B$ has been raised in Ref.\\cite{lucai} and the\ncontributions from the $\\Psi_B$, $\\bar\\Psi_B$ and $\\Delta$ with\nsuch a new definition (\\ref{newpsi}) are shown in\nFig.(\\ref{bwaves2}a). We draw the distribution of the\ncorresponding ratio $R=\\left(\\frac{F^{B\\pi}_{+}\n|_{\\bar\\Psi_B}}{F^{B\\pi}_{+}|_{All}}\\right)$ versus $q^2$ in\nFig.(\\ref{bwaves2}b), where $\\left(F^{B\\pi}_{+}\n|_{\\bar\\Psi_B}\\right)$ means that only the contributions from\n$\\bar\\Psi_B$ are considered and $\\left(F^{B\\pi}_{+}|_{All}\\right)$\nmeans that all the contributions from the B meson wavefunctions\nare taken into consideration. One may observe from\nFig.(\\ref{bwaves2}b) that even with the new definition\n(\\ref{newpsi}) for $\\Psi_B$ and $\\bar\\Psi_B$, the contribution\nfrom $\\bar\\Psi_B$ is not small ($\\sim 25-40\\%$) and it can not be\nsafely neglected. Thus both $\\Psi_B$ and $\\bar\\Psi_B$ should be\nkept in the calculation for giving a better understanding of the B\ndecays.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{comparefpia.eps}%\n\\includegraphics[width=0.5\\textwidth]{comparefpip.eps}\n\\caption{Comparison of different PQCD results for the $B\\to \\pi$\ntransition form factors $F_+^{B\\pi}(q^2)$ (Left) and\n$F_0^{B\\pi}(q^2)$ (Right). The solid, dashed and dotted lines are\nthe results obtained in Ref.\\cite{li1} and are for\n$\\omega_B=0.36$GeV, $0.40$GeV and $0.44$GeV respectively. The\nshaded band are our present results with the upper edge for\n$\\bar\\Lambda=0.50GeV$ and the lower edge for\n$\\bar\\Lambda=0.60GeV$, respectively. For comparison, the dash-dot\nline stands from the QCD LCSR result\\cite{sumhuang,sumrule}.}\n\\label{lambda2}\n\\end{figure}\n\nFinally, we make a comparison of the present results for\n$F_{+,0}^{B\\pi}(q^2)$ with those obtained in Ref.\\cite{li1} in\nFig.(\\ref{lambda2}). In Ref.\\cite{li1}, $\\bar\\Psi_B$ has been\nneglected and $\\Psi_B$ takes the form\n\\begin{equation}\n\\Psi_B(x,b_B)=N_Bx^2(1-x)^2\\exp\\left[-\\frac{1}{2}\\left(\n\\frac{xM_B}{\\omega_B}\\right)^2-\\frac{\\omega_B^2 b_B^2}{2}\\right],\n\\end{equation}\nwhere $N_B$ is the normalization factor and $\\omega_B$ is taken to\nbe $(0.40\\pm 0.04)GeV$. In Fig.(\\ref{lambda2}), we show their\nresults for $\\omega_B=0.36$GeV, $0.40$GeV and $0.44$GeV and our\npresent results with $\\bar\\Lambda\\in (0.5GeV, 0.6GeV)$,\nrespectively. The two results in the large recoil regions $q^2\\sim\n0$ are consistent with each other, however one may observe that\nthe fast rise in Ref.\\cite{li1} has been suppressed in our present\nresults and the slope of the present obtained form factors\n$F_{+,\\;0}^{B\\pi}(q^2)$ are more consistent with the QCD LCSR\nresults in Ref.\\cite{sumhuang,sumrule}. The main reason for the\ndifferences between our present results and those in\nRef.\\cite{li1} is that we have used a better endpoint behavior\nwavefunction for $\\Psi_p$\\cite{huangwu}. With this new form for\n$\\Psi_p$, we find that the total twist-3 contributions are in fact\nless than ($\\sim 70\\%$) the leading twist contribution in the\nlarge recoil regions. While in Ref.\\cite{li1}, the twist-3\ncontributions are about two times bigger than that of the leading\ntwist, especially for the bigger $q^2$ regions, and then the total\ncontributions will give a fast rise in shape.\n\n\\section{discussion and summary}\n\nIn the present paper, we have examined the $B\\to\\pi$ transition\nform factor in the PQCD approach, where the transverse momentum\ndependence for the wavefunction, the Sudakov effects and the\nthreshold effects are included to regulate the endpoint\nsingularity and to derive a more reasonable result. We emphasize\nthat the transverse momentum dependence for both the B meson and\nthe pion is important to give a better understanding of the\n$B\\to\\pi$ transition form factor. The pionic twist-3 contributions\nto the $B\\to\\pi$ transition form factor are carefully studied with\na better endpoint behavior wavefunction for $\\Psi_p$, and\nFig.(\\ref{pionwave}) shows that the twist-3 contributions are of\nthe same order of the leading twist contribution, however its\nvalues are less than that of the leading twist. This observation\nimproves the results obtained in Refs.\\cite{weiy2,li1}, in which\nthe asymptotic behavior for $\\phi_p$ was used and they claimed a\nlarge twist-3 contributions to the $B\\to\\pi$ transition form\nfactor, i.e. bigger than that of the leading twist.\nFig.(\\ref{pionwave}b) and Fig.(\\ref{bwaves2}) show that both\n$\\Psi_B$ and $\\bar\\Psi_B$ are important, no matter what definition\n(Eq.(\\ref{oldpsi}) or Eq.(\\ref{newpsi})) is chosen. Under the\ndefinition (\\ref{oldpsi}), the negative contribution from\n$\\bar\\Psi_B$ is necessary to suppress the big contribution from\n$\\Psi_B$ and to obtain a reasonable total contributions. While\nunder the definition Eq.(\\ref{newpsi}), the contribution from\n$\\bar\\Psi_B$ is power suppressed to that of $\\Psi_B$, however it\nstill can contribute $25-40\\%$ to the total contributions. As is\nshown in Fig.(\\ref{lambda2}), a comparison of our present results\nfor $F_{+,\\;0}^{B\\pi}(q^2)$ with those in Ref.\\cite{li1} shows\nthat a better PQCD result (with its slope closes to the QCD LCSR\nresults) can be obtained by carefully considering both the pionic\ntwist-3 contributions and the contributions from the two\nwavefunctions $\\Psi_B$ and $\\bar\\Psi_B$ of the B meson.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{acombined.eps}%\n\\includegraphics[width=0.5\\textwidth]{pcombined.eps}\n\\caption{PQCD results for the $B\\to \\pi$ form factors\n$F_+^{B\\pi}(q^2)$ (Left) and $F_0^{B\\pi}(q^2)$ (Right). The shaded\nband are our present results with the upper edge for\n$\\bar\\Lambda=0.50GeV$ and the lower edge for\n$\\bar\\Lambda=0.60GeV$, respectively. The dashed and dotted lines\nstand for the QCD LCSR result Eq.(\\ref{sumruleapp}) and the fits\nto the lattice QCD results with errors\\cite{lattice3},\nrespectively. } \\label{combined}\n\\end{figure}\n\nIn the literature, the values of the $B\\to\\pi$ transition form\nfactors $F^{B\\pi}_{+}(0)$ and $F^{B\\pi}_{0}(0)$ are determined\naround $0.3$. With $\\bar\\Lambda\\in (0.50GeV,0.60GeV)$, we obtain\n$F^{B\\pi}_{+,0}(0)=0.265\\pm 0.032$. This result is consistent with\nthe extrapolated lattice QCD result $F^{B\\pi}_{+,0}(0)=0.27\\pm\n0.11$\\cite{lattice} and the newly obtained QCD LCSR result\n$F^{B\\pi}_{+,0}(0)=0.258\\pm0.031$\\cite{sumrule}. The PQCD\ncalculation are reliable only when the involved energy scale is\nhard enough. The lattice QCD calculations which presently are\navailable only for the soft regions, i.e. $q^2>15GeV^2$. The QCD\nLCSR can treat both hard and soft contributions with $q^2\\precsim\n18GeV^2$\\cite{sumhuang,sumrule} on the same footing. Therefore,\nthe results from the PQCD approach, the lattice QCD approach and\nthe QCD LCSRs are complementary to each other and by combining the\nresults of those three approaches, one may obtain an understanding\nof the $B\\to\\pi$ transition form factor in the whole physical\nregions. The $B\\to\\pi$ transition form factors $F^{B\\pi}_+(q^2)$\nand $F^{B\\pi}_0(q^2)$ derived from QCD LCSRs can be written in the\nfollowing parameterization \\cite{sumrule}:\n\\begin{equation}\\label{sumruleapp}\nF^{B\\pi}_+(q^2)=\\frac{r_1}{1-q^2\/m_1^2} + \\frac{r_2}{1-q^2\/m_{\\rm\nfit}^2},\\;\\;\\;F^{B\\pi}_0(q^2)=\\frac{r_3}{1-q^2\/m_{\\rm 0fit}^2},\n\\end{equation}\nwhere $r_1$, $r_2$, $r_3$, $m_1$, $m_{\\rm fit}$ and $m_{\\rm 0fit}$\nare fitted parameters and can be taken as\\cite{sumrule},\n$r_1=0.744$, $r_2=-0.486$, $r_3=0.258$, $m_1=5.32GeV$, $m_{\\rm\nfit}^2=40.73GeV^2$ and $m_{\\rm 0fit}^2=33.81GeV^2$. With the\nparameterization Eq.(\\ref{sumruleapp}), the QCD LCSR results can\nbe extrapolated up to the upper limit of $q^2$, i.e. $q^2\\sim\n25GeV^2$, and then it can be treated as a bridge to connect both\nthe PQCD results and the lattice QCD results. In\nFig.(\\ref{combined}), we show the results of the PQCD approach,\nthe lattice QCD approach and the extrapolated QCD LCSR results\ndefined in Eq.(\\ref{sumruleapp}), respectively. Our present PQCD\nresults with $\\bar\\Lambda\\in (0.5GeV, 0.6GeV)$ are in agreement\nand can match with the QCD LCSR results and the lattice QCD\ncalculations, which are shown in Fig.(\\ref{combined}).\n\nIn summary, we have shown that the PQCD approach can be applied to\ncalculate the $B\\to\\pi$ transition form factor in the large recoil\nregions. The twist-3 contributions are less than the leading twist\ncontribution with a better endpoint behavior twist-3 wavefunctions\nand both of the two wavefunctions $\\Psi_B$ and $\\bar\\Psi_B$ of the\nB meson are necessary to give a deep understanding of the B\ndecays, e.g. $B\\to\\pi$ transition form factor. Combining the PQCD\nresults with the QCD LCSR and the lattice QCD calculations, the\n$B\\to\\pi$ transition form factor can be determined in the whole\nkinematic regions.\n\n\n\\begin{center}\n\\section*{Acknowledgements}\n\\end{center}\n\nThe authors would like to thank H.N. Li for some useful\ndiscussions. This work was supported in part by the Natural\nScience Foundation of China (NSFC). \\\\\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section*{Introduction}\nAt leading twist 3 quark distribution functions (DF) in the nucleon are present; the experimentally well measured unpolarized quark DF, the experimentally less  known quark helicity DF and the so far undetermined transversity DF. The latter cannot be measured in inclusive DIS due to its chiral-odd nature, since all possible interactions are chiral-even for nearly massless quarks. Therefore one needs an additional chiral-odd function in the cross section to access transversity. This can be achieved either by an anti quark transversity DF in double transversely polarized Drell-Yan processes or, alternatively, one can have a chiral-odd fragmentation function in semi-inclusive deep inelastic scattering (SIDIS) or hadroproduction.\n\\section{The Belle experiment}\nThe Belle \\cite{belledetector} experiment at the asymmetric $e^+e^-$ collider KEKB at Tsukuba, Japan, is mainly dedicated to the study of CP violation in B meson decays. Its center of mass energy is tuned to the $\\Upsilon(4S)$ resonance at $\\sqrt{s} = 10.58$ GeV. Part of the data was also recorded 60 MeV below the resonance. These off-resonance events are studied in order to measure spin dependent fragmentation functions (FF). For the present analysis an integrated luminosity of 29.0 fb$^{-1}$ has been analyzed. The aerogel \\v{C}erenkov counter, time-of-flight detector and the central drift chamber enable a good particle identification and tracking, which is crucial for these measurements. Using the information from the silicon vertex detector, one selects tracks originating from the interaction region and thus reducing the contribution of hadrons from heavy meson decays. To reduce the amount of hard gluon radiative events a cut on the kinematic variable thrust of $T> 0.8$ is applied. This enhances the typical 2-jet topology and the thrust axis is used as approximation of the original quark direction. We also require that the fractional energy $z \\stackrel{CMS}{=} 2E_h\/Q > 0.2$ in order to significantly reduce the contribution of pions arising from the vector meson decays or of those assigned to a wrong hemisphere.\n\\section*{Collins FF}\nThe Collins effect occurs in the fragmentation of a transversely polarized quark with polarization $S_q$ and 3-momentum k into an unpolarized hadron of transverse momentum $P_{h\\perp}$ with respect to the original quark direction. According to the Trento convention \\cite{trento} the number density for finding an unpolarized hadron h produced from a transversely polarized quark q is defined as:\n\\begin{equation}\nD_{h\\\/q^\\uparrow}(z,P_{h\\perp})  = D_1^q(z,P_{h\\perp}^2) + H_1^{\\perp q}(z,P_{h\\perp}^2)\\frac{(\\hat{\\mathbf{k}} \\times \\mathbf{P}_{h\\perp})\\cdot \\mathbf{S}_q}{zM_h} ,\n\\label{eq:cdef}\n\\end{equation}\nwhere the first term describes the unpolarized FF $D^q_1(z,P_{h\\perp})$, with $z\\stackrel{CMS}{=} \\frac{2E_h}{Q}$ being the fractional energy the hadron carries relative to half of the center of mass system (CMS) energy Q. The second term, containing the Collins function $H_1^{q\\perp} (z,P^2_{h\\perp})$, depends on the spin of the quark and thus leads to an asymmetry as it changes sign under flipping the quark spin. The vector product can accordingly be described by a $\\sin(\\phi)$ modulation, where $\\phi$ is the azimuthal angle spanned by the transverse momentum and the plane defined by the quark spin and its momentum. In $e^+e^-$ hadron production the Collins effect can be observed by a combined measurement of a quark and an anti quark fragmentation. Combining two hadrons from different hemispheres in jetlike events, with azimuthal angles $\\phi_1$ and $\\phi_2$ as defined in Fig. 1, would result in a $\\cos(\\phi_1 +\\phi_2)$ modulation. In the CMS these azimuthal angles are defined between the transverse component of the hadron momenta with regard to the thrust axis $\\hat{n}$ and the plane spanned by the lepton momenta and $\\hat{ n}$. The comparison of the thrust axis calculations using reconstructed and generated tracks in the MC sample shows an average angular separation between the two of 75 mrad with a a root mean square of 74 mrad. Due to that small biases in one of the reconstruction methods used could arise and were studied as discussed later. Following reference \\cite{daniel} one either computes the azimuthal angles of each pion relative to the thrust axis which results in a $\\cos(\\phi_1 +\\phi_2)$ modulation or one calculates the azimuthal angle relative to the axis defined by the $2^{nd}$ pion which results in a $\\cos(2\\phi_0)$ modulation. While the first method directly accesses moments of the Collins functions, the second method contains a convolution integral of the Collins FF over possible transverse momenta of the hadrons.\n\\begin{figure}\n\n\\includegraphics[width=4.8cm]{plotangle.eps}\n\\includegraphics[width=4cm]{plotangle1.eps}\n\\caption{Description of the azimuthal angles $\\phi_0$, $\\phi_1$ and $\\phi_2$ relative to the scattering plane defined by the lepton axis and either the thrust axis  $\\hat{n}$ or the momentum of the $2^{nd}$ hadron $P_{h2}$. \\label{angle}}\n\n\\end{figure}\n\\subsection{Measured asymmetries}\nWe measure the azimuthal asymmetries $N(2\\phi )\/N_0$, where $N(2\\phi )$ denotes the number of hadron pairs in bins of either $2\\phi_0$ or $\\phi_1 +\\phi_2$ and $N_0$ is the average number of hadron pairs in the whole angle interval. The main backgrounds, producing similar azimuthal asymmetries as the Collins effect, are the radiation of soft gluons and possible acceptance effects. The gluonic contribution is proportional to the unpolarized FF and is independent of the charge of the hadrons. Consequently taking the ratio of the normalized distributions for unlike-sign over like-sign pairs the gluonic distributions cancel in the leading order:\n\\begin{eqnarray}\nA_\\alpha & := &\\frac{\\frac{N(2\\phi )}{N_0} |_{unlikesign}}\n{\\frac{N(2\\phi )}{N_0} |_{likesign}}  \\nonumber \\\\\n& \\approx &  1+\n\\frac{\\sin^2\\theta}{1+cos^2\\theta} \\left( F\\left(\\frac{H_1^{\\perp, fav}}{D_1^{fav}}\n,\\frac{H_1^{\\perp,dis}}{D_1^{dis}}\\right)+\\mathcal{O} f(Q_T ,\\alpha_S)^2\\right)cos(2\\phi ) \\quad,\n\\end{eqnarray}\nwhere $\\theta$ is either the angle between the colliding leptons and the produced hadron or the thrust axis for methods $\\alpha = 0, 12$, respectively. Favored and disfavored FF (fav,dis) describe the fragmentation of a light quark into a pion of same or opposite charge sign. Obviously also acceptance effects cancel in the double ratios. The latter are fit by the sum of a constant term and a $\\cos(2\\phi_0)$ or $\\cos(\\phi_1 +\\phi_2)$ modulation. The double ratios of unlike sign over like sign pairs showed the existence of the Collins effect and gave a hint about the overall magnitude \\cite{prl}. As suggested in \\cite{peter}, measuring in addition double ratios containing any combination of charged pion pairs reveals additional information on the ratio of the favored and disfavored Collins functions. Preliminary results for the double ratios of unlike-sign (UL) over all charged (C) pion pairs can be seen in Fig.2 together with the final results of the unlike sign (UL) over like sign (L) pion pairs. The combined zbins are obtained by adding the symmetric bins of the $4\\times 4$ $z \\in$ [0.2,0.3,0.5,0.7,1.0] bins. The data has been corrected for the contribution of charmed hadron decays. A nonzero asymmetry is visible for both double ratios, while the UL\/C are about 40\\% of the UL\/L results (the average values are: $A^{UL\/C}_0 = (1.27 \\pm 0.49 \\pm 0.35)\\%$ and $A^{UL\/C}_{12} = (1.752 \\pm 0.59 \\pm 0.41)\\%$ compared to $A^{UL\/L}_0 = (3.06 \\pm 0.57 \\pm 0.55)\\%$ and $A^{UL\/L}_{12} = (4.26 \\pm 0.68 \\pm 0.68)\\%$). The data shows a rising behavior with rising fractional energy z for both results. Several systematic cross-checks of the analysis method were performed and the differences in the results are quoted as systematic uncertainties: Instead of double ratios we used the subtraction method for the unlike from the like sign or charged pion asymmetries; the constant fit to the double ratios obtained in MC (without a Collins contribution) together with its statistical error and a similar fit to double ratios of positively charged over negatively charged pion pair data were assigned as systematic error. The differences to the results when fitting the double ratios also with higher order azimuthal modulations were added to the systematic errors. All contributions to the systematic errors were added in quadrature. Studies using introduced asymmetries in the MC data revealed that the $\\cos(\\phi_1+\\phi_2)$ method undersetimates the generated asymmetries due to the discrepancies between the thrust axis calculated for generated and reconstructed tracks. These results have therefore been rescaled by a factor 1.21.\nThe presented measurement represents the first evidence of the Collins effect and will help to disentangle the favored to disfavored Collins function ratio.\n\\begin{figure}[ht]\n\\includegraphics[width=8cm]{final5pipi.eps}\n\\caption{Results for the $\\cos(2\\phi_0)$ and the $\\cos(\\phi_1+\\phi_2)$ method for the UL\/C (squares, preliminary) and UL\/L (triangles, final) double ratios. The upper error band correspond to systematic errors of the UL\/L double ratios, the lower one to those of the UL\/C ratios.\\label{fig:pipi}}\n\\end{figure}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOur universe as we know is an expanding one. So to describe a consistent quantum theory of gravity at early times i.e. near the big bang singularity one requires the solutions describing quantum gravity in time dependent backgrounds. String theory in Linear Dilaton backgrounds provide a way to tackle this problem in a somewhat systematic manner. In \\cite{Antoniadis:1988aa},\\cite{Polchinski:1989fn},\\cite{Antoniadis:1988vi},\\cite{Antoniadis:1990uu} it was noticed that time-like linear dilaton solutions of supercritical string theories are strongly coupled at assymptotically early times and hence difficult to solve. Although at assymptotically late times one has much more control since there, the theories become weakly coupled and one can solve them perturbatively. In this regard we should mention that we take the following conventions for the dilaton $\\Phi$ and the string coupling $g_{st}$,\n$$\\Phi=-QX^0\\ ,\\quad g_{st}=e^{\\langle\\Phi\\rangle}=e^{-QX^0}\\ ,\\ \\ \\text{with}\\ \\ Q>0\\ .$$\nIn \\cite{Aharony:2006ra}, the authors found that in such cases the spectrum of relevant deformations of the ``matter CFT\" has modes which grow in time although their backreaction on the geometry is small at assymptotically late times due to the small string coupling. The condition for the backreaction being small is stated below but one should note that these are not the usual tachyons we are familiar with. These modes were termed ``\\textit{Pseudo-tachyons}\" in the literature. Tachyonic solutions also grow in time but they have large back reactions on the geometry causing instabilities in the background. If we consider the deformations,\n$$\\int d^2z\\ \\sum_i\\mu_i\\mathcal{O}_ie^{\\kappa_iX^0}$$\nwith $\\mathcal{O}_i$ having dimension $\\Delta_i$. This is marginal if,\n$$\\Delta_i+\\frac{1}{4}\\kappa_i(\\kappa_i+2Q)=2\\ \\Rightarrow\\ \\kappa_i=-Q\\pm\\sqrt{Q^2-4(\\Delta_i-2)}\\ .$$\nFor $2<\\Delta_i<2+Q^2\/4$ we see that the properly normalised relevant mode goes like,\n$$\\tilde{\\lambda}=\\frac{\\lambda}{g_{st}}\\sim e^{\\pm QX^0\\sqrt{1-4(\\Delta_i-2)\/Q^2}}\\ .$$\nSo we see that one of the modes grow in time and these are the ones identified as \\textit{Pseudo-tachyons}.\n\nThe condition that the pseudo-tachyons have small backreaction and hence provide a well defined i.e. stable background, is that we begin with an appropriate initial state. This is sometimes called a ``capping\" state. As per current understanding, unambiguous description of such states is still an open problem. Although some work has been done in this regard, where smooth evolution from ``nothing\" was considered \\cite{McGreevy:2005ci}. Basically the authors studied the effect of closed string tachyon condensation on the spacetime with a space-like singularity at the level of general relativity. \n\nAnother way to identify the ``capping\" state is to embed the Supercritical Linear Dilaton(SCLD) phase in a strong coupling completion of string theory. The goal of the current paper is to take this route for identifying the capping state. First let us look at the obstacles we face in achieving this goal, then we mention cases where there is a systematic way to circumvent these problems to obtain a reasonably accurate result.\n\n\\subsubsection*{Obstacles}\n\\begin{enumerate}\n\\item{The most obvious obstactle is that we do not know whether there exists a consistent strong coupling completion for supercritical string theories.}\n\\item{Even if we are able find certain examples where such completion is possible, computing physical quantites relevant for the target spacetime is considerably difficult since spacetime supersymmetry is broken due to the linear dilaton which couples to the world-sheet curvature.\\\\\nFor example suppose we want to compute the renormalised mass of different operators at a given mass level. In presence of supersymmetry it is sufficient to compute the renormalised mass of one component of the supermultiplet or superfield, the rest are fixed by supersymmetry. This will obviously not be true if supersymmetry is broken.}\n\\item{Full non-perturbative treatment of string theory is yet to be as well understood as perturbative string theory which is why even if we resolve the first obstacle, the resolution may be true only in some special cases.}\n\\end{enumerate}\nWe argue that if we have two supercritical theories which are S-dual to each other describing the weak and strong coupling regime then we can attempt to construct the capping state within reasonable accuracy using the technique of interpolating functions. Although we fail to find a specific example and produce numbers, it may help to know the strategy of constructing the capping state in case we suceed in finding an example. \n\nThe technique of interpolating function involves computing the renormalized mass function $F(g)$ as a perturbative expansion in string coupling. If we have two theories at the weak and strong coupling end which are S-dual to each other, then we have two functions $F_W$ and $F_S$ which have perturbative expansions in the weak and strong coupling respectively at the two ends. We will closely follow the approach described in \\cite{Sen:2013oza} to find an interpolating function $F_{m,n}(g)$ which matches $F_W$ upto $m$-th power in the weak coupling expansion and $F_S$ upto the $n$-th power in the strong coupling expansion. This interpolating function is supposed to give the renormalized mass of the theory for the whole range of coupling to a reasonable accuracy. Other approaches are also available in the literatre \\cite{Kleinert:2001ax} and the Pad\\'e approximant approach.\\\\\n\nThe rest of the paper is organised as follows. In Section \\ref{sec:LD} we discuss briefly the known results and the conventions we use for linear dilaton backgrounds and mass renormalization in string perturbation theory. We will also derive the partition function for SCLD backgrounds and show that modular invarince dictates that only certain spacetime dimensions are allowed, which is well known. In Section \\ref{sec:Q1} we compute the two point amplitude of a pseudo-tachyonic mode on the torus, which gives the renormalized mass at one loop for this mode. In Section \\ref{sec:IF} we describe strategy for determining the ``capping\" state with the help of interpolating functions. And finally we conclude with some discussions in Section \\ref{disc}.\n\n\\section{Known results and conventions to be used}\n\\label{sec:LD}\n\\subsection{Linear dilaton backgrounds}\nIn this section we give a brief review of known results in the case when a dilaton which is linear in the string fields, couples to the world-sheet curvature \\cite{Polchinski:101b}. So,\n\\begin{equation}\n\\phi=V_{\\mu}X^{\\mu}\\ ,\n\\end{equation} \nand the world-sheet Eucledian action for the strings is given by,\n\\begin{equation}\nS_P=\\frac{1}{4\\pi\\alpha^{\\prime}}\\int d^2x\\sqrt{\\gamma}\\gamma^{ab}\\partial_aX^{\\mu}\\partial_bX_{\\mu}\\ +\\ \\frac{1}{4\\pi}\\int d^2x\\sqrt{\\gamma}\\phi R\n\\end{equation}\nVarrying this action w.r.t the world-sheet metric $\\gamma$ we obtain the energy momentum tensor for this theory. We give the result here for the holomorphic and anti-holomorphic part of the E.M. tensor,\n\\begin{eqnarray}\nT(z)&=& -\\frac{1}{\\alpha^{\\prime}}:\\partial X^{\\mu}\\partial X_{\\mu}:\\ +\\ V_{\\mu}\\partial^2X^{\\mu} \\nonumber\\\\\n\\bar{T}(\\bar{z})&=& -\\frac{1}{\\alpha^{\\prime}}:\\bar{\\partial} X^{\\mu}\\bar{\\partial} X_{\\mu}:\\ +\\ V_{\\mu}\\bar{\\partial}^2X^{\\mu}\\ .\\label{eq:emLD}\n\\end{eqnarray}\nWith this energy momentum tensor we want to find the modified central charge as well as the modifed conformal weights of the vertex operators. Now we want to work with time-like dilaton so we choose $V_{\\mu}\\equiv(Q,\\vec{0})$   and hence we end up with (space-time metric is the minkowski metric)\n\\begin{equation}\nT(z)= -\\frac{1}{\\alpha^{\\prime}}:\\partial X^{\\mu}\\partial X_{\\mu}:\\ -\\ Q\\partial^2X^{0}\n\\end{equation}\nand similarly for $\\bar{T}(\\bar{z})$.\n\nWe want to compute the the $T(z)T(0)$ as well as the $T(z)e^{ik.X}$ OPE's. We know that now $\\partial X^0$ is no longer a primary while $e^{-ik^0X^0}$ is still a primary. Looking only at the leading order divergences, we get the following results\\footnote{Here $D$ is the dimensionality of the target space-time.},\n\\begin{eqnarray}\nT(z)T(0)&\\sim & \\frac{D-6\\alpha^{\\prime}Q^2}{2z^4}\\ +\\ ....\\\\\nT(z)e^{ik.X}(0) &\\sim & \\left(\\frac{\\alpha^{\\prime}k^2}{4}-\\frac{iQk^0\\alpha^{\\prime}}{2}\\right)\\frac{e^{ik.X}}{z^2}\\ +\\ ....\n\\end{eqnarray}\nThus we have $c=\\bar{c}=D-6\\alpha^{\\prime}Q^2$. And the conformal dimension of the vertex operator, $$h_v=\\frac{\\alpha^{\\prime}k^2}{4}-\\frac{iQk^0\\alpha^{\\prime}}{2}=\\frac{\\alpha^{\\prime}}{4}\\left[\\vec{k}^2-(k^0)^2-2ik^0Q)\\right]$$\n\n\\subsection{The one loop partition function}\nBefore computing the partition function for such theories we include world-sheet supersymmetry so that in the case of $Q = 0$ we have a GSO projected spectrum of critical superstrings in 10 dimension where the tachyon is projected out.\n\nWe use the known results for fermionic traces \\cite{Polchinski:102b},\n\\begin{eqnarray}\nZ^{\\alpha}_{\\ \\beta}=\\text{Tr}_{\\alpha}\\left[q^{H}\\text{exp}(i\\pi\\beta F)\\right]=\\frac{1}{\\eta(\\tau)}\\vartheta\\begin{bmatrix}\n\\alpha\/2\\\\\n\\beta\/2\n\\end{bmatrix}(0|\\tau)\\ .\n\\end{eqnarray}\nHere $F$ is the world-sheet fermion number, $\\alpha$ and $\\beta$ take integer values modulo 2. For $\\alpha$ even we have trace over the NS and for odd we have trace over R states. \n\nThe total central charge due to the \\textit{bc} and $\\beta\\gamma$ ghosts is given by, $c_g=-26+11=-15$ and the matter CFT has central charge $c_m=D-6\\alpha^{\\prime}Q^2+\\frac{D}{2}$. So for the central charge for the matter + ghost CFT to vanish we need,\n\\begin{equation}\n\\frac{3D}{2}-6\\alpha^{\\prime}Q^2=15\\ \\ \\Rightarrow\\ D-4\\alpha^{\\prime}Q^2=10\\ .\\label{eq:D_Q}\n\\end{equation}\nFor specificity we assume $D\/2$ to be odd and we will GSO project onto even world-sheet fermion number to remove the tachyon.\n\nNow we have the following well results on the torus,\n\\begin{equation}\n\\langle c(w_1)b(w_2)\\bar{c}(\\bar{w}_3)\\bar{b}(\\bar{w}_4)\\rangle_{T^2}=|\\eta(\\tau)|^4\\ .\n\\end{equation}\nThe scalar partition function ($q=e^{2\\pi i\\tau}\\ \\Rightarrow\\ q\\bar{q}=e^{-4\\pi\\tau_2}$),\n\\begin{eqnarray}\nZ_s(\\tau)&=&(q\\bar{q})^{-\\frac{D-6\\alpha^{\\prime}Q^2}{24}}\\text{Tr}\\left(q^{L_0}\\bar{q}^{\\bar{L}_0}\\right) \\\\\\nonumber\\\\\nZ_s(\\tau)&=&V_D(q\\bar{q})^{-\\frac{D-6\\alpha^{\\prime}Q^2}{24}}\\int\\frac{d^{D-1}k}{(2\\pi)^{D-1}}\\frac{dk^0}{2\\pi}\\text{exp}\\left[-\\pi\\alpha^{\\prime}\\tau_2\\big(\\vec{k}^2-(k^0)^2-2ik^0Q\\big)\\right]\\nonumber\\\\\n&&\\hspace*{6cm}\\times\\prod_{\\mu,n}\\sum_{N_{\\mu n},\\bar{N}_{\\mu n}}q^{nN_{\\mu n}}\\bar{q}^{n\\bar{N}_{\\mu n}}\\nonumber\\\\\nZ_s(\\tau)&=&iV_D(q\\bar{q})^{-\\frac{D}{24}}(q\\bar{q})^{\\frac{\\alpha^{\\prime}Q^2}{4}}\\int\\frac{d^{D-1}k}{(2\\pi)^{D-1}}\\frac{d\\omega}{2\\pi}\\text{exp}\\left[-\\pi\\alpha^{\\prime}\\tau_2\\big(\\vec{k}^2+\\omega^2 + 2\\omega Q\\big)\\right]\\nonumber\\\\\n&&\\hspace*{6cm}\\times\\prod_{n}\\frac{1}{(1-q^n)^D(1-\\bar{q}^n)^D}\\nonumber\\\\\nZ_s(\\tau)&=& iV_D|\\eta(\\tau)|^{-2D}\\int\\frac{d^{D-1}k}{(2\\pi)^{D-1}}\\frac{d\\omega}{2\\pi}\\text{exp}\\left[-\\pi\\alpha^{\\prime}\\tau_2\\big(\\vec{k}^2+\\omega^2 + 2\\omega Q + Q^2\\big)\\right]\n\\end{eqnarray}\nNow just redifing the variable $\\omega\\rightarrow\\omega+Q$ and integrating over the momenta variables we get,\n\\begin{equation}\nZ_s(\\tau)=iV_D\\left((4\\pi^2\\alpha^{\\prime}\\tau_2)^{-\\frac{1}{2}}|\\eta(\\tau)|^{-2}\\right)^D=iV_D(Z_X(\\tau))^D\n\\end{equation}\nFrom the fermionic traces we obtain after projecting onto even world-sheet fermion number we get,\n\\begin{equation}\nZ_{\\psi}^+=\\frac{1}{2}\\left(Z^0_{\\ 0}(\\tau)^{\\frac{D}{2}-1}-Z^0_{\\ 1}(\\tau)^{\\frac{D}{2}-1}-Z^1_{\\ 0}(\\tau)^{\\frac{D}{2}-1}-Z^1_{\\ 1}(\\tau)^{\\frac{D}{2}-1}\\right)\\ .\n\\end{equation}\nNow we know that $Z^1_{\\ 1}(\\tau)=0$ so,\n\\begin{equation}\nZ_{\\psi}^+=\\frac{1}{2}\\left(Z^0_{\\ 0}(\\tau)^{\\frac{D}{2}-1}-Z^0_{\\ 1}(\\tau)^{\\frac{D}{2}-1}-Z^1_{\\ 0}(\\tau)^{\\frac{D}{2}-1}\\right)\\ .\n\\end{equation}\nThus the full partition function is given by\\footnote{Here we consider the case of Type IIB theories although generalization to other cases is simple.},\n\\begin{equation}\nZ_{LD}=\\int \\frac{d\\tau d\\bar{\\tau}}{4\\tau_2}iV_D\\left((4\\pi^2\\alpha^{\\prime}\\tau_2)^{-\\frac{1}{2}}|\\eta(\\tau)|^{-2}\\right)^D|\\eta(\\tau)|^4Z_{\\psi}^+(\\tau)Z_{\\bar{\\psi}}^{+}(\\bar{\\tau})\\nonumber\n\\end{equation}\n\\begin{equation}\n\\Rightarrow Z_{LD}=iV_D\\int \\frac{d\\tau d\\bar{\\tau}}{16\\pi^2\\alpha^{\\prime}\\tau_2^2}\\left((4\\pi^2\\alpha^{\\prime}\\tau_2)^{-\\frac{1}{2}}|\\eta(\\tau)|^{-2}\\right)^{D-2}Z_{\\psi}^+(\\tau)Z_{\\bar{\\psi}}^{+}(\\bar{\\tau})\n\\end{equation}\n\n\\subsection{Modular Invariance}\nWhile testing the modular invariance of the above partition function one immediately notices that for arbitrary values of $D>10$ we do not have modular invariance i.e. even though the measure $d\\tau d\\bar{\\tau}\/\\tau_2^2$ and $Z_X(\\tau)$ are modular invariant by themselves $Z^+_{\\psi}Z^+_{\\bar{\\psi}}$ break this invarince. But one can show explicitly that for $D=10+16n$ with integer values of $n$,\\footnote{In \\cite{Aharony:2006ra} this result was stated and also this result was argued by K. Narayan and collaborators in an unpublished notes.} $Z^+_{\\psi}Z^+_{\\bar{\\psi}}$ is modular invariant owing to the following transformation laws,\n\\begin{eqnarray}\nZ^+_{\\psi}(-1\/\\tau) = Z^+_{\\psi}(\\tau) &;& Z^+_{\\psi}(\\tau + 1) = \\exp\\left(\\frac{2\\pi(2n+1)i}{3}\\right)Z^+_{\\psi}(\\tau) \\\\\nZ^+_{\\bar{\\psi}}(-1\/\\bar{\\tau}) = Z^+_{\\bar{\\psi}}(\\bar{\\tau}) &\\text{and}& Z^+_{\\bar{\\psi}}(\\bar{\\tau} + 1) = \\exp\\left(-\\frac{2\\pi(2n+1)i}{3}\\right)Z^+_{\\bar{\\psi}}(\\bar{\\tau})\n\\end{eqnarray}\nHence for the rest of this paper we take $D=10+16n$. Thus we have including \\eqref{eq:D_Q}, \n\\begin{equation}\n4\\alpha^{\\prime}Q^2=16n\\ ,\\quad\\Rightarrow\\quad Q=2\\sqrt{\\frac{n}{\\alpha^{\\prime}}} \\ .\\label{eq:largeQ}\n\\end{equation} \n\nSo we see that due to the modular invarince we have,\n\\begin{equation}\nZ_{LD}=iV_D\\int_{\\mathbb{F}_0} \\frac{d\\tau d\\bar{\\tau}}{16\\pi^2\\alpha^{\\prime}\\tau_2^2}\\left(Z_X(\\tau)\\right)^{8+16n}Z_{\\psi}^+(\\tau)Z_{\\bar{\\psi}}^{+}(\\bar{\\tau})\n\\end{equation}\nwhere, $\\mathbb{F}_0$ denotes the fundamental domain in the complex $\\tau$ upper half plane. Notice that only for $n=0$ i.e. the critical case $Z_{LD}$ vanishes by the use of Riemann Identity. For $n\\neq 0$ the partition function is non vanishing which implies that linear dilaton backgrounds break spacetime supersymmetry.\n\n\\subsection{Mass renormalization in string pertubation theory}\nWe know that one has to start with the BV master action when attempting an off-shell formulation of string theory \\cite{Zwiebach:1992ie},\\cite{Hata:1993gf},\\cite{Okawa:2004ii},\\cite{Sen:2016bvm},\\cite{Berkovits:1995ab},\\cite{Witten:1986qs},\\cite{Sen:2015hha}. It is then clear that if one is able to solve the resulting equation of motion pertubatively, then there exists a well defined procedure to compute several physical quantities such as mass renormalization \\cite{Pius:2013sca},\\cite{Pius:2014iaa}, vacuum shift \\cite{Pius:2014gza} etc. order by order in perturbation theory. From \\cite{Pius:2014iaa} we learn that using the 1PI effective action to some given order in pertubation theory, we have to sum over all the 1PR diagrams for the two point function to get the renormalised mass at that order in the pertubation expansion. The full propagator is given, in accordance with the above statement by,\n\\begin{equation}\n\\Pi=\\Delta+\\Delta\\hat{\\mathcal{F}}\\Delta+\\Delta\\hat{\\mathcal{F}}\\Delta\\hat{\\mathcal{F}}\\Delta+....=\\Delta(1-\\hat{\\mathcal{F}}\\Delta)^{-1}=(1-\\Delta\\hat{\\mathcal{F}})^{-1}\\Delta\\ .\n\\end{equation}\nHere $\\hat{\\mathcal{F}}$ denotes the contribution from the Riemann surfaces that are 1PI in the external momenta. And,\n$$\\Delta=\\frac{1}{2(L_0+\\bar{L}_0)}\\delta_{L_0,\\bar{L}_0}$$\ngives the tree level propagator. If $\\mathcal{F}$ is the full off-shell two point funtion then,\n\\begin{equation}\n\\Pi=\\Delta+\\Delta\\mathcal{F}\\Delta.\n\\end{equation}\n\nNow suppose that we project on to a given mass level $m$, if the full off-shell two point function restricted to this level be denoted by $F_T$ then we have,\n\\begin{equation}\n\\tilde{F}_T=F_T(1+(k^2+m^2)^{-1}\\mathcal{I}F_T)^{-1}\n\\end{equation}\nwith,\n$$\\mathcal{I}=\n\\begin{pmatrix}\n & I & \\\\\n I& &  \\\\\n & & I \n\\end{pmatrix}\\ .$$\nThe propagator restricted to the modified physical sector is given by,\n$$\\mathcal{V}=U(k)(k^2+m^2-\\tilde{F}_d(k))^{-1}U(k)^{\\dagger},\\ \\text{with}\\ \\tilde{F}(k)=U(k)\\tilde{F}_d(k)U(k)^{\\dagger}\\ \\text{and}\\ \\tilde{F}_{\\alpha\\beta}(k)=\\langle\\bar{\\alpha}|\\tilde{F}_T|\\bar{\\beta}\\rangle_p.$$\nSo we see that $\\tilde{F}(k)$ is the renormalized mass matrix for the physical sector. On diagonalizing we will get the renormalized masses for each physical state. As an exercise we work out the mass renormalization of the massless states like the graviton, in critical superstring theory i.e. $Q=0$ case in Appendix \\ref{app:B}. For the critical case, the two point graviton amplitude should vanish since we expect general co-ordinate invarince in the quantum theory of superstrings as well. So the massless states remain massless under quantum corrections. \n\n\\section{Two point amplitude at one loop with $Q\\neq 0$}\n\\label{sec:Q1}\nIn case of $Q\\neq 0$ i.e. the linear dilaton background with world sheet supersymmetry, instead of \\eqref{eq:emLD} we get the EM tensor and the world-sheet supercurrent given by,\n\\begin{eqnarray}\nT_B(z)&=&-\\frac{1}{\\alpha^{\\prime}}:\\partial X^{\\mu}\\partial X_{\\mu}:\\ +\\ V_{\\mu}\\partial^2X^{\\mu}\\ -\\ \\frac{1}{2}\\psi^{\\mu}\\partial\\psi_{\\mu} \\\\\nT_F(z)&=& i(2\/\\alpha^{\\prime})^{1\/2}\\psi^{\\mu}\\partial X_{\\mu}\\ -\\ i(2\\alpha^{\\prime})^{1\/2}V_{\\mu}\\partial\\psi^{\\mu}\n\\end{eqnarray}\nWith these results we can straight forwardly compute the BRST current,\n\\begin{equation}\nQ_B=\\frac{1}{2\\pi i}\\oint (dz j_B(z)\\ -\\ d\\bar{z} \\bar{j}_B(\\bar{z}) )\n\\end{equation}\nwhere,\n\\begin{equation}\nj_B=cT_B+\\gamma T_F+bc\\partial c+\\frac{3}{4}\\partial c\\beta\\gamma+\\frac{1}{4}c\\partial(\\beta\\gamma)-c\\beta\\partial\\gamma-b\\gamma^2 \\label{eq:BRST}\n\\end{equation}\nand the same expression with antiholomorphic fields for the antiholomorphic part. \n\nLet us first find the equation of motion for the first excited level with this BRST current. The on shell condition for an operator $\\Phi$, as always is given by, $Q_B\\Phi=0$. Since the tachyon has been projected out the first excited level gives the lowest energy i.e. the ground state. The vertex operator for the NS-NS state in the -1 picture number is given by,\n\\begin{equation}\n\\bar{c}cV_{-1}(k,z,\\bar{z})=\\bar{c}c\\ e^{-(\\phi+\\bar{\\phi})}\\zeta_{\\mu\\nu}\\psi^{\\mu}\\bar{\\psi}^{\\nu}e^{ik.X}(z,\\bar{z})\\label{eq:pic-1}\n\\end{equation}\nUntil the next subsection, we concentrate on the holomorphic part only since the anti-holomorphic part works out in a similar fashion. So applying the BRST charge  we get,\n\\begin{eqnarray}\nQ_B (c\\ e^{-\\phi}\\zeta_{\\mu\\nu}\\psi^{\\mu}e^{ik.X}(0,0)) &=& \\frac{1}{2\\pi i}\\oint dz\\ j_B(z)c\\ e^{-\\phi}\\zeta_{\\mu\\nu}\\psi^{\\mu}e^{ik.X}(0,0) \\\\\n&=&\\frac{1}{2\\pi i}\\oint dz\\ \\frac{(\\partial c)c(0)e^{-\\phi}\\zeta_{\\mu\\nu}\\psi^{\\mu}e^{ik.X}(0,0)}{z}\\left(\\frac{\\alpha^{\\prime}k^2}{4}+\\frac{iV.k\\alpha^{\\prime}}{2}+\\frac{1}{2}\\right) \\nonumber\\\\\n&+& \\frac{1}{2\\pi i}\\oint dz\\ \\left(\\frac{\\alpha^{\\prime}}{2}\\right)^{1\/2}\\frac{\\eta(0)c(0)e^{ik.X}(0,0)}{z}\\zeta_{\\mu\\nu}(k^{\\mu}+2iV^{\\mu})\\nonumber\\\\\n&+&\\frac{1}{2\\pi i}\\oint dz\\ \\left(-\\frac{1}{2}\\right)\\frac{(\\partial c)c(0)e^{-\\phi}\\zeta_{\\mu\\nu}\\psi^{\\mu}e^{ik.X}(0,0)}{z}\\ +\\ ....\\ .\n\\end{eqnarray}\nThe $....$ denote terms which are regular in $z$ and hence they vanish under the closed contour integral. Just to be precise, the first two terms are the contributions due to $cT_B$ and $\\gamma T_F$ while the third term is the single pole contribution due to the ghost part. Adding up all the contribution we get the on-shell conditions,\n\\begin{equation}\nk^2+2iV.k=0\\ ,\\quad\\text{and}\\quad \\zeta_{\\mu\\nu}(k^{\\mu}+2iV^{\\mu})=0\\ .\\label{eq:onshell}\n\\end{equation}\nNotice that if we take $V_{\\mu}\\rightarrow 0$ we get back the expected on-shell conditions for the massless states in critical superstring theory. For time like $V$ we have, \n\\begin{equation}\nk^2-2iQk^0=0\\ \\Rightarrow\\ k^2=2iQk^0\\ ,\\quad\\text{and}\\quad \\zeta_{\\mu\\nu}k^{\\mu}=2iQ\\zeta_{0\\nu}\\ .\\label{eq:onshell_tl}\n\\end{equation}\nFrom the above equation \\eqref{eq:onshell_tl} we see that,\n\\begin{equation}\nk^2-2iQk^0=0\\ \\Rightarrow\\ -(k^0+iQ)^2+|\\vec{k}|^2=Q^2\\ \\Rightarrow\\ -M^2=Q^2 \\label{eq:treeM}\n\\end{equation}\ni.e. the tree level mass squared go as $-Q^2$.\n\nIn what follows we take $Q$ to be large. We will work with the convention of $\\alpha^{\\prime}=2$. Thus \\eqref{eq:largeQ} becomes\n\\begin{equation}\nQ=\\sqrt{2n}\\quad\\Rightarrow\\ \\text{$n\\rightarrow\\infty$ as $Q\\rightarrow\\infty$.} \\label{eq:largeQ2}\n\\end{equation}\nWhile computing the two point amplitude on the torus we will focus on the terms that have the highest power of $n$ or equivalently $Q$. \nWe will first determine the 0-picture vertex operators and determine first the real part and then the imaginary part of the two-point amplitude.\n\nFrom the -1 picture vertex in \\eqref{eq:pic-1} and the PCO,\n\\begin{equation}\n\\chi(z)=e^{\\phi}T_F(z)+c\\partial\\xi(z)+\\frac{1}{4}\\partial b\\eta e^{2\\phi}(z)+\\frac{1}{4}b(2\\partial\\eta e^{2\\phi}+\\eta\\partial(e^{2\\phi}))(z)\\ ,\n\\end{equation}\nwe get for the holomorphic part,\n\\begin{eqnarray}\n\\chi(z)cV_{-1}(k,0,0)&=& zT_F(z)c\\zeta_{\\mu}\\psi^{\\mu}e^{ik.X}(0,0)\\ +\\ z\\partial\\xi(z)e^{-\\phi}(0)c\\zeta_{\\mu}\\psi^{\\mu}e^{ik.X}(0,0)\\nonumber\\\\\n&-&\\frac{1}{4}z^2.\\frac{1}{z^2}\\eta(z)e^{\\phi}(0)c\\zeta_{\\mu}\\psi^{\\mu}e^{ik.X}(0,0)\\ +\\ \\frac{1}{4}2z.\\frac{1}{z}\\eta(z)e^{\\phi}(0)c\\zeta_{\\mu}\\psi^{\\mu}e^{ik.X}(0,0)\\nonumber\\\\\n&+&\\frac{1}{2}z^2.\\frac{1}{z}\\partial\\eta(z)e^{\\phi}(0)c\\zeta_{\\mu}\\psi^{\\mu}e^{ik.X}(0,0)\\ .\n\\end{eqnarray}\nFor the coefficient of $z$ we need the single pole contribution for a finite answer in the $z\\rightarrow0$ limit. While for the $z^0$ coefficient we need the constant regular piece. Thus in the $z\\rightarrow0$ limit we have,\n$$V_0(k,0,0)=\\zeta_{\\mu}\\left(i\\partial X^{\\mu}-\\psi^{\\mu}(k.\\psi)\\right)e^{ik.X}(0,0)+\\frac{1}{4}\\eta e^{\\phi}(0)\\zeta_{\\mu}\\psi^{\\mu}e^{ik.X}(0,0)\\ .$$\nThe last term above does not conserve the picture number inside the two point one loop amplitude and hence we can drop it. We see that we reach at the same result as in the $Q=0$ case, but now with on shell conditons \\eqref{eq:onshell_tl}. And similarly for the anti-holomorphic part.\n\nNow we can compute the two-point amplitide. As in the case with $Q=0$, we have the full amplitude given by,\n\\begin{equation}\n\\mathcal{M}_g=g_s^2\\int_{\\mathbb{F}_0}\\frac{d\\tau d\\bar{\\tau}}{4\\tau_2}\\int d^2z\\ \\langle b\\bar{b}\\bar{c}c(0)V_0(k_1,0,0)V_0(k_2,z,\\bar{z})\\rangle\\ .\n\\end{equation}\n\n\\subsection{The real part}\nLet us first turn our attention to the real part of the correlator.\n\\begin{eqnarray}\n\\left[\\int d^2z\\ \\zeta^{(1)}_{\\mu\\nu}\\zeta^{(2)}_{\\rho\\sigma}\\left\\langle\\left\\lbrace\\partial X^{\\mu}(0)\\bar{\\partial}X^{\\nu}(0)-\\psi^{\\mu}(0)(k_1.\\psi(0))\\bar{\\psi}^{\\nu}(0)(k_1.\\bar{\\psi}(0))\\right\\rbrace\\right.\\right. \\nonumber\\\\\n\\left.\\left.\\times\\lbrace\\partial X^{\\rho}(z)\\bar{\\partial}X^{\\sigma}(\\bar{z})-\\psi^{\\rho}(z)(k_2.\\psi(z))\\bar{\\psi}^{\\sigma}(\\bar{z})(k_2.\\bar{\\psi}(\\bar{z}))\\rbrace e^{ik_1.X}(0,0)e^{ik_2.X}(z,\\bar{z})\\right\\rangle\\right. \\nonumber\\\\\n\\left. -\\int d^2z\\ \\zeta^{(1)}_{\\mu\\nu}\\zeta^{(2)}_{\\rho\\sigma}\\left\\langle\\left\\lbrace \\psi^{\\mu}(0)(k_1.\\psi(0))\\bar{\\partial}X^{\\nu}(0)+\\partial X^{\\mu}(0)\\bar{\\psi}^{\\nu}(0)(k_1.\\bar{\\psi}(0))\\right\\rbrace\\right.\\right. \\nonumber\\\\\n\\left.\\left.\\times\\lbrace \\psi^{\\rho}(z)(k_2.\\psi(z))\\bar{\\partial}X^{\\sigma}(\\bar{z})+\\partial X^{\\rho}(z)\\bar{\\psi}^{\\sigma}(\\bar{z})(k_2.\\bar{\\psi}(\\bar{z}))\\rbrace e^{ik_1.X}(0,0)e^{ik_2.X}(z,\\bar{z})\\right\\rangle\\right] \\ . \\label{eq:RELD}\n\\end{eqnarray}\n\nWe will look at the different terms that contribute to this correlator and focus our attention to the terms that have the highest power of $n$.\n\nWe will be using the result,\n\\begin{align}\n\\langle e^{ik_1.X}(0,0)e^{ik_2.X}(z,\\bar{z})\\rangle &=& iV_D(Z_X(\\tau))^D(2\\pi)^D\\delta^D(k_1+k_2)\\left|\\vartheta_1(z,\\tau)\\exp\\left(-\\frac{\\pi(\\text{Im}\\ z)^2}{\\tau_2}\\right)\\right|^{\\alpha^{\\prime}k_1.k_2}\\nonumber\\\\\n&&\\times\\prod_{i=1}^{2}\\left|\\frac{2\\pi}{\\vartheta_1^{\\prime}(0,\\tau)}\\right|^{-\\frac{\\alpha^{\\prime}k_i^2}{2}}\\ .\n\\end{align}\n\\textbf{Note}: For linear dilaton backgrounds with charge $Q$, the correlation function of tachyon vertices yields, $\\delta^D(k_1+k_2+2Q(1-h))$ with $h$ being the genus of the Riemann surface we consider \\cite{Ishibashi:2016bno}. In our case we have the torus with $h=1$. Hence the factor of $\\delta^D(k_1+k_2)$, which forces $k_2=-k_1$.\n\nThus applying \\eqref{eq:onshell_tl}, \\eqref{eq:largeQ2} and $\\alpha^{\\prime}=2$ we have,\n\\begin{eqnarray}\n\\langle e^{ik_1.X}(0,0)e^{ik_2.X}(z,\\bar{z})\\rangle &=& i(2\\pi)^D\\delta^D(k_1+k_2)\\mathcal{C}_X^{T^2}\\left|\\frac{2\\pi\\vartheta_1(z,\\tau)}{\\vartheta^{\\prime}_1(0,\\tau)}\\exp\\left(-\\frac{\\pi(\\text{Im}\\ z)^2}{\\tau_2}\\right)\\right|^{-4iQk_1^0}. \\nonumber\\\\ \n&=& i(2\\pi)^D\\delta^D(k_1+k_2)\\mathcal{C}_X^{T^2}\\left|\\frac{2\\pi\\vartheta_1(z,\\tau)}{\\vartheta^{\\prime}_1(0,\\tau)}\\exp\\left(-\\frac{\\pi(\\text{Im}\\ z)^2}{\\tau_2}\\right)\\right|^{8n} \\label{eq:Tachvert}\n\\end{eqnarray}\nWith $\\mathcal{C}_X^{T^2}=V_D(Z_X(\\tau))^D$. Since we are computing everything in a Lorentz's invariant manner we can go to the rest frame of the pseudo-tachyonic mode for which,\n$$\n(k_1^0)^2=M^2=-Q^2\\quad\\Rightarrow\\quad k_1^0=iQ=i\\sqrt{2n}\n$$ \nIn the rest of this section we will directly state the results for the different correlators we need. The details on how to obtain these results are provided in the Appendix \\ref{app:A}.\n\\begin{itemize}\n\\item{First let us focus on the piece,\n\\begin{equation}\n\\left\\langle\\partial X^{\\mu}(0)\\partial X^{\\rho}(z)\\bar{\\partial}X^{\\nu}(0)\\bar{\\partial}X^{\\sigma}(\\bar{z}) e^{ik_1.X}(0,0)e^{ik_2.X}(z,\\bar{z})\\right\\rangle .\n\\end{equation}\n\\begin{align}\n=\\left[\\delta^{\\mu\\rho}\\delta^{\\nu\\sigma}\\left(\\frac{(\\vartheta_{11}\\partial^2_{z}\\vartheta_{11}-\\partial_z\\vartheta_{11}\\partial_z\\vartheta_{11})(z)}{\\vartheta_{11}(z)^2}+\\frac{1}{4\\pi\\tau_2}\\right)\\left(\\frac{(\\vartheta_{11}\\bar{\\partial}^2_{z}\\vartheta_{11}-\\bar{\\partial}_z\\vartheta_{11}\\bar{\\partial}_z\\vartheta_{11})(\\bar{z})}{\\vartheta_{11}(\\bar{z})^2}+\\frac{1}{4\\pi\\tau_2}\\right)\\right. \\nonumber\\\\\n\\left. +\\ \\frac{\\pi^2}{\\tau_2^2}\\delta^{\\mu\\sigma}\\delta^{\\rho\\nu}+\\big(\\delta^{\\mu\\rho}k_1^{\\sigma}k_2^{\\nu}(..)+k_1^{\\sigma}k_2^{\\mu}\\delta^{\\rho\\nu}(..)\\big)+k_1^{\\sigma}k_1^{\\rho}k_2^{\\mu}k_2^{\\nu}(..)+....\\right]\\left\\langle\\prod_i e^{ik_i.X}\\right\\rangle.\\label{eq:boson}\n\\end{align}\nWhen we now contract the indeces with $\\zeta^{(1)}_{\\mu\\nu}\\zeta^{(2)}_{\\rho\\sigma}$ and use \\eqref{eq:onshell} along with $k_1=-k_2$, we see that the first two terms are $\\sim O(1)$, the third term which is quadratic in the momenta contribute at $\\sim O(2n)$ and the term which is quartic in momenta contributes at $\\sim O(4n^2)$. So we focus on the last piece and it is given by,\n\\begin{align}\n64n^2\\zeta^{(1)}_{00}\\zeta^{(2)}_{00}\\left[\\left|\\left(\\frac{\\vartheta^{\\prime}_{11}(z)}{\\vartheta_{11}(z)}\\right)^2-\\frac{4\\pi^2}{\\tau_2^2}(\\text{Im}(-z))^2\\right|^2+\\frac{16\\pi^2}{\\tau_2^2}\\left|\\frac{\\vartheta^{\\prime}_{11}(z)}{\\vartheta_{11}(z)}\\text{Im}(-z)\\right|^2\\right.\\nonumber\\\\\n\\left.+\\frac{8\\pi}{\\tau_2}\\text{Im}\\left\\lbrace\\frac{\\vartheta^{\\prime}_{11}(z)(\\bar{\\vartheta}^{\\prime}_{11}(\\bar{z}))^2}{\\vartheta_{11}(z)(\\bar{\\vartheta}_{11}(\\bar{z}))^2}\\text{Im}(-z)\\right\\rbrace\\right]\\left\\langle\\prod_{i=1,2} e^{ik_i.X}\\right\\rangle.\n\\end{align}\nThis contribution is positive as we can see from the above expression.}\n\\item{Next we consider the cross terms of the type,\n\\begin{eqnarray}\n\\zeta^{(1)}_{\\mu\\nu}\\zeta^{(2)}_{\\rho\\sigma}\\left\\langle\\partial X^{\\mu}(0)\\bar{\\partial}X^{\\nu}(0)\\psi^{\\rho}(z)(k_2.\\psi(z))\\bar{\\psi}^{\\sigma}(\\bar{z})(k_2.\\bar{\\psi}(\\bar{z})) e^{ik_1.X}(0,0)e^{ik_2.X}(z,\\bar{z})\\right\\rangle \\nonumber\\\\\n=-\\zeta^{(1)}_{\\mu\\nu}\\zeta^{(2)}_{\\rho\\sigma}k_2^{\\mu}k_2^{\\nu}k_2^{\\rho}k_2^{\\sigma}\\left(\\left|\\frac{\\vartheta_{11}^{\\prime}(z)}{\\vartheta_{11}(z)}\\right|^2+\\frac{4\\pi^2}{\\tau_2^2}(\\text{Im}(-z))^2+\\frac{4\\pi}{\\tau_2}\\text{Im}\\left\\lbrace\\frac{\\vartheta_{11}^{\\prime}(z)}{\\vartheta_{11}(z)}\\text{Im}(-z)\\right\\rbrace\\right)\\ \\nonumber\\\\\n=\\ -64n^2\\zeta^{(1)}_{00}\\zeta^{(2)}_{00}\\left(\\left|\\frac{\\vartheta_{11}^{\\prime}(z)}{\\vartheta_{11}(z)}\\right|^2+\\frac{4\\pi^2}{\\tau_2^2}(\\text{Im}(-z))^2+\\frac{4\\pi}{\\tau_2}\\text{Im}\\left\\lbrace\\frac{\\vartheta_{11}^{\\prime}(z)}{\\vartheta_{11}(z)}\\text{Im}(-z)\\right\\rbrace\\right)\\ .\n\\end{eqnarray}\nHence we see that these terms go as $\\sim\\ O(Q^4)\\ \\sim\\ O(4n^2)$. And all the cross terms in \\eqref{eq:RELD}} come with a minus sign hence these terms also give positive contributions to the amplitude.\n\\item{The term with four holomorphic and four anti-holomorphic fermions is given by,\\\\ \n\\begin{eqnarray}\nk_{1\\alpha}k_{2\\beta}k_{1\\gamma}k_{2\\delta}\\langle\\psi^{\\mu}\\psi^{\\alpha}(0)\\psi^{\\rho}\\psi^{\\beta}(z)\\bar{\\psi}^{\\nu}\\bar{\\psi}^{\\gamma}(0)\\bar{\\psi}^{\\sigma}\\bar{\\psi}^{\\delta}(z)e^{ik_1.X}(0,0)e^{ik_2.X}(z,\\bar{z})\\rangle\n\\end{eqnarray}\nNow for the fermionic correlator we have,\n\\begin{equation}\nk_{1\\alpha}k_{2\\beta}\\langle\\psi^{\\mu}\\psi^{\\alpha}(0)\\psi^{\\rho}\\psi^{\\beta}(z)\\rangle\\ =\\ k_{1\\alpha}k_{2\\beta}\\left(A_z\\delta^{\\mu\\alpha}\\delta^{\\rho\\beta}+B_z\\delta^{\\mu\\rho}\\delta^{\\alpha\\beta}+C_z\\delta^{\\mu\\beta}\\delta^{\\alpha\\rho}\\right)\\ . \\label{eq:Fcorreln}\n\\end{equation}\nThe coefficients can be computed easily and are given by \\cite{Sen:2013oza}, \\cite{Atick:1986rs},\n\\begin{eqnarray}\nA_z &=& \\langle\\psi^{1}(0)\\psi^{1}(0)\\psi^{2}(z)\\psi^{2}(z)\\rangle\\ =\\ 1\\ ,\\\\\nB_z &=& \\langle\\psi^{1}(0)\\psi^{2}(0)\\psi^{1}(z)\\psi^{2}(z)\\rangle \\nonumber\\\\\n&=& -\\frac{1}{8(\\eta(\\tau)^4)^{2n+1}}\\left(\\frac{\\vartheta_{11}^{\\prime}(0)}{\\vartheta_{11}(z)}\\right)^2\\sum_{\\nu}\\delta_{\\nu}\\left(\\vartheta_{\\nu}(z)^2+2\\vartheta_{\\nu}(z)\\vartheta_{\\nu}(-z)+\\vartheta_{\\nu}(-z)^2\\right)\\big(\\vartheta_{\\nu}(0)^2\\big)^{4n+1}\\ ,\\nonumber\\\\\n\\\\\nC_z &=& \\langle\\psi^{1}(0)\\psi^{2}(0)\\psi^{2}(z)\\psi^{1}(z)\\rangle\\ =\\ -B_z\\ .\n\\end{eqnarray}\nSo the full correlator on using \\eqref{eq:largeQ} behaves like,\n\\begin{eqnarray}\n\\zeta^{(1)}_{\\mu\\nu}\\zeta^{(2)}_{\\rho\\sigma}\\big(A_zk_1^{\\mu}k_{2}^{\\rho}-B_z(k_1)^2\\delta^{\\mu\\rho}+C_zk_1^{\\rho}k_2^{\\mu}\\big)\\big(A_{\\bar{z}}k_1^{\\nu}k_{2}^{\\sigma}-B_{\\bar{z}}(k_1)^2\\delta^{\\nu\\sigma}+C_{\\bar{z}}k_1^{\\sigma}k_2^{\\nu}\\big)\\nonumber\\\\\n= 16n^2\\left(4\\zeta^{(1)}_{00}\\zeta^{(2)}_{00}|A_z+C_z|^2+\\zeta^{(1)}_{\\mu\\nu}\\zeta^{(2)\\mu\\nu}|B_z|^2+4Re[B_z(A_z+C_z)]\\zeta^{(1)}_{0\\mu}\\zeta^{(2)\\mu}_{\\ 0}\\right)\\\\\n\\end{eqnarray}\nThis contribution as we can see gives positive contribution as well and appears in \\eqref{eq:RELD} with a $`+$' sign.}\n\\item{Finally from the last two lines of \\eqref{eq:RELD} also, we have negative contributions but since they appear with a $`-$' sign in the equation. So their full contribution to the amplitude is also positive when we keep the highest order terms $n$ i.e. prortional $4n^2$ like in the above cases. The explicit details are given in Appendix \\ref{app:A}.}\n\\end{itemize}\n\n\\subsection{The imaginary part}\nLet us now look at the contribution from the imaginary part which is given by,\n\\begin{equation}\n\\left[\\int d^2z\\ \\zeta^{(1)}_{\\mu\\nu}\\zeta^{(2)}_{\\rho\\sigma}\\left\\langle \\left\\lbrace \\psi^{\\mu}(0)(k_1.\\psi(0))\\bar{\\partial}X^{\\nu}(0)+\\partial X^{\\mu}(0)\\bar{\\psi}^{\\nu}(0)(k_1.\\bar{\\psi}(0))\\right\\rbrace\\right.\\right. \\label{eq:IMLD}\n\\end{equation}\n$$\\left.\\left.\\times\\left\\lbrace\\partial X^{\\rho}(z)\\bar{\\partial}X^{\\sigma}(\\bar{z})-\\psi^{\\rho}(z)(k_2.\\psi(z))\\bar{\\psi}^{\\sigma}(\\bar{z})(k_2.\\bar{\\psi}(\\bar{z}))\\right\\rbrace e^{ik_1.X}(0,0)e^{ik_2.X}(z,\\bar{z})\\right\\rangle\\right.$$\n$$\\left. +\\int d^2z\\ \\zeta^{(1)}_{\\mu\\nu}\\zeta^{(2)}_{\\rho\\sigma}\\langle (\\mu\\leftrightarrow\\rho,\\ \\nu\\leftrightarrow\\sigma,\\ k_1\\leftrightarrow k_2,\\ z,\\bar{z}\\leftrightarrow 0,0)e^{ik_1.X}(0,0)e^{ik_2.X}(z,\\bar{z})\\rangle\\right]\\ .$$\n\nFrom the first two lines of \\eqref{eq:IMLD} we have four terms. We first look at,\n\\begin{eqnarray}\n&& \\langle\\bar{\\partial}X^{\\nu}(0)\\partial X^{\\rho}(z)\\bar{\\partial}X^{\\sigma}(\\bar{z})\\psi^{\\mu}(0)(k_1.\\psi(0))e^{ik_1.X}(0,0)e^{ik_2.X}(z,\\bar{z})\\rangle \\\\\n&& +\\ \\langle\\partial X^{\\mu}(0)\\partial X^{\\rho}(z)\\bar{\\partial}X^{\\sigma}(\\bar{z})\\bar{\\psi}^{\\mu}(0)(k_1.\\bar{\\psi}(0))e^{ik_1.X}(0,0)e^{ik_2.X}(z,\\bar{z})\\rangle . \\nonumber\n\\end{eqnarray}\nNow using \\eqref{eq:Bcorreln}, \\eqref{eq:Fcorreln} and the onshell condition \\eqref{eq:onshell_tl} one can easily check the following facts\n\\begin{itemize}\n\\item{The first term has two contributions, one at $\\sim O(2n)$ which we will neglect with respect to the contribution at $\\sim O(4n^2)$,\n$$-64n^2\\zeta^{(1)}_{00}\\zeta^{(2)}_{00}\\left(-i\\frac{\\vartheta^{\\prime}_{11}(z)}{\\vartheta_{11}(z)}+\\frac{2\\pi}{\\tau_2}\\text{Im}(z)\\right)\\left(-i\\frac{\\bar{\\vartheta}^{\\prime}_{11}(\\bar{z})}{\\bar{\\vartheta}_{11}(\\bar{z})}-\\frac{2\\pi}{\\tau_2}\\text{Im}(z)\\right)\\left(i\\frac{\\bar{\\vartheta}^{\\prime}_{11}(\\bar{z})}{\\bar{\\vartheta}_{11}(\\bar{z})}+\\frac{2\\pi}{\\tau_2}\\text{Im}(z)\\right)$$\n\\begin{equation}\n\\Rightarrow 64n^2\\zeta^{(1)}_{00}\\zeta^{(2)}_{00}\\left(\\left|\\frac{\\vartheta^{\\prime}_{11}(z)}{\\vartheta_{11}(z)}\\right|^2+\\frac{4\\pi^2}{\\tau_2^2}(\\text{Im}(z))^2+\\frac{4\\pi}{\\tau_2}\\text{Im}\\left\\lbrace\\frac{\\vartheta^{\\prime}_{11}(z)}{\\vartheta_{11}(z)}\\text{Im}(z)\\right\\rbrace\\right)\\left(i\\frac{\\bar{\\vartheta}^{\\prime}_{11}(\\bar{z})}{\\bar{\\vartheta}_{11}(\\bar{z})}+\\frac{2\\pi}{\\tau_2}\\text{Im}(z)\\right)\n\\end{equation}\nSame argument are true for the second term and the contribution at $\\sim Q^4$ or $4n^2$ is given by,\n$$-64n^2\\zeta^{(1)}_{00}\\zeta^{(2)}_{00}\\left(-i\\frac{\\vartheta^{\\prime}_{11}(z)}{\\vartheta_{11}(z)}+\\frac{2\\pi}{\\tau_2}\\text{Im}(z)\\right)\\left(-i\\frac{\\bar{\\vartheta}^{\\prime}_{11}(\\bar{z})}{\\bar{\\vartheta}_{11}(\\bar{z})}-\\frac{2\\pi}{\\tau_2}\\text{Im}(z)\\right)\\left(i\\frac{\\vartheta^{\\prime}_{11}(z)}{\\vartheta_{11}(z)}-\\frac{2\\pi}{\\tau_2}\\text{Im}(z)\\right).$$\nThus adding them up we get,\n\\begin{equation}\ni64n^2\\zeta^{(1)}_{00}\\zeta^{(2)}_{00}\\left(\\left|\\frac{\\vartheta^{\\prime}_{11}(z)}{\\vartheta_{11}(z)}\\right|^2+\\frac{4\\pi^2}{\\tau_2^2}(\\text{Im}(z))^2+\\frac{4\\pi}{\\tau_2}\\text{Im}\\left\\lbrace\\frac{\\vartheta^{\\prime}_{11}(z)}{\\vartheta_{11}(z)}\\text{Im}(z)\\right\\rbrace\\right)\\left(\\frac{\\bar{\\vartheta}^{\\prime}_{11}(\\bar{z})}{\\bar{\\vartheta}_{11}(\\bar{z})}+\\frac{\\vartheta^{\\prime}_{11}(z)}{\\vartheta_{11}(z)}\\right). \\label{eq:x3psi}\n\\end{equation}}\n\\item{And one can check that from the last line in \\eqref{eq:IMLD} these contribution which is conjugate to the one described above comes out to be \n$$-i64n^2\\zeta^{(1)}_{00}\\zeta^{(2)}_{00}\\left(\\left|\\frac{\\vartheta^{\\prime}_{11}(z)}{\\vartheta_{11}(z)}\\right|^2+\\frac{4\\pi^2}{\\tau_2^2}(\\text{Im}(z))^2+\\frac{4\\pi}{\\tau_2}\\text{Im}\\left\\lbrace\\frac{\\vartheta^{\\prime}_{11}(z)}{\\vartheta_{11}(z)}\\text{Im}(z)\\right\\rbrace\\right)\\left(\\frac{\\bar{\\vartheta}^{\\prime}_{11}(\\bar{z})}{\\bar{\\vartheta}_{11}(\\bar{z})}+\\frac{\\vartheta^{\\prime}_{11}(z)}{\\vartheta_{11}(z)}\\right).$$\nSo when we add this piece to \\eqref{eq:x3psi} the total contribution vanishes.}\n\\item{One can check that similar arguments hold for the other type of term as well i.e. terms with one bosonic field and rest all fermionic fields. When we take the contribution from all possible terms of this type the total contribution vanishes.}\n\\end{itemize}\n\n\nOne can choose a particular co-ordinate patch i.e. fix some gauge to carry out the integral over the moduli space, but we will not do it in this note.  There is one point we should mention; for the bosonic piece we see explicitly that there are no other divergences other than $z\\rightarrow 0$ i.e. they are free from spurious divergences. But for a function like $B_z$ or $C_z$ since we do not completely know zeros and poles for the spin structure sum it is hard to conclude the absence of spurious poles. This can be an interesting direction to check that the perturbative amplitude is indeed completely well defined.\n\nNow from \\cite{Pius:2014iaa} we have to get the renomalized mass one has to solve the following equation iteratively,\n\\begin{equation}\nE^2=m^2-\\tilde{F}(E)\\ ,\n\\end{equation}  \nto a given order in perturbation theory. Here $m^2$ is the tree level mass squared and $\\tilde{F}(E)$ is the quantum correction to the propagator. in the current case,\n$$m^2=-Q^2\\ ,\\quad \\text{and}\\quad \\text{Im}(\\tilde{F}(E))=0\\ ,\\quad \\tilde{F}(E)>0\\ .$$\nfrom the one loop correction. Also since the imaginary part is zero at the order we considered, we have zero decay width so the modes are stable. Hence atleast to this order we can confirm the following statement unambiguously.\n\\begin{itemize}\n\\item{The \\textit{Pseudotachyon} modes remain pseudotachyonic under quantum corrections and the system stabilizes towards condensation of such modes.}\n\\end{itemize} \n\n\n\\section{Interpolating functions due to Sen}\n\\label{sec:IF}\nWe now attempt to construct an interpolating function which gives the renormalized mass for all coupling and matches the pertubative expansion to a given order at both the weak and strong coupling end if such expansions indeed exists. We will follow the strategy described in the seminal work \\cite{Sen:2013oza} for this purpose. We will first briefly discuss the know example where this technique succeeds to within $10\\%$ accuracy, and then try to implement it for the Linear Dilaton backgrounds.\n\n\\subsection{Example of the heterotic\/type I duality}\nIn case of the Heterotic and type I, S-dual theories the couplings at the two ends are related by,\n$$g_Hg_I=2^5\\pi^7\\ .$$\nThus we can parametrize them as follows,\n$$g_H=(2\\pi)^{7\/2}g,\\quad g_I=2^{3\/2}\\pi^{7\/2}g^{-1},$$\nso that the two formulae meet at $g=1$. The renormalized mass function of the second excited state\\footnote{In our convention we always state the tachyons to be the ground state or zero excitation state, even though in superstring theories they are projected out by GSO projection. } (i.e. the first massive level) was considered. We define, the renormalized mass function $F(g)$ by,\n$$M(g)=2^{15\/8}\\pi^{7\/8}F(g)$$ \nFor weak coupling end we have the Heterotic theory whereas for the strong coupling end we have the type I theory with,\n\\begin{equation}\nF^W_2(g)=g^{1\/4}\\left(1+K_wg^2+O(g^4)\\right),\\ \\ F^S_1(g)=g^{3\/4}\\left(1+K_sg^{-1}+O(g^{-2})\\right);\\ K_w\\simeq 0.23,\\ K_s\\simeq 0.351\n\\end{equation}\nwhere one loop or the subleading corrections at both ends where considered.\n\nWe now look at the interpolating function which has to agree with the coupling expansions to order $m$ in the weak coupling and order $n$ in strong coupling end.\n\\begin{equation}\nF_{m,n}(g)=g^{1\/4}\\left[1+a_1g+...+a_mg^m+b_ng^{m+1}+b_{n-1}g^{m+2}+...+b_1g^{m+n}+g^{m+n+1}\\right]^{\\frac{1}{2(m+n+1)}}\\label{eq:Fg}\n\\end{equation} \nIn the present example it is known that for the Heterotic theory being a closed superstring theory has the expansion in $g^2$ so $a_m=0$ for $m$ odd.\nThe interpolating function which is within $10\\%$ accuracy of the actual function was calculated to be,\n$$F_{3,1}=g^{1\/4}\\left(1+10K_wg^2+10K_sg^4+g^5\\right)^{1\/10}$$\n\n\\subsection{Application to the linear dilaton backgrounds}\nIn our case we would like to apply the same technique to the first excited state of strings which are the \\textit{Pseudotachyonic} modes. It is fairly obvious that now all the coefficients for the interpolating function may depend on the background charge $Q$.\n\n\\textbf{Note:} It should be understood that this technique is useful only in the cases where it is possible to determine the physical quantities of the theory at both ends perturbatively, with the use of S-duality. In cases where this is not true, this process cannot yield sensible results and one indeed has to carry out a full non-perturbative computation.\\\\\\\\\n\nSo we will begin with a theory where the weak coupling ($g_w$) regime is S-dual to another theory at the strong coupling ($g_s$) regime. Hence we will have,\n\\begin{equation}\ng_wg_s=\\mathcal{C}_0\\ ,\\quad\\text{where $\\mathcal{C}_0$ is a constant.} \\label{eq:coupling}\n\\end{equation}\nWe do the following parametrization,\n$$g_w=c_0g,\\ \\text{and}\\ g_s=(\\mathcal{C}_0\/c_0)g^{-1}$$\nsuch that the formulae meet at $g=1$ i.e. both the strong and weak couplings become of the same order ($\\sim\\ O(1)$) while satisfying \\eqref{eq:coupling}. This parameter now can be treated as,\n$$g\\equiv g_{st}=e^{-QX^0}\\ \\ \\text{i.e. the coupling varrying with time.}$$\nAs in the earlier example we define the renormalized mass function via,\n$$M(g)=\\text{(const.)}\\times F(g)\\ .$$\nLet us first try to see the leading order coefficient of the interpolating function for the case at hand. From \\eqref{eq:treeM} we have,\n\\begin{equation}\nk^2-2iQk^0=0\\ \\Rightarrow\\ M^2=-Q^2\n\\end{equation}\nand the leading order coupling behavior at the strong and weak coupling end are assumed to be,\n\\begin{equation}\nF_W(g)\\sim g^{\\kappa},\\quad F_S(g)\\sim g^{\\delta}\\ .\n\\end{equation}\nNote that in case of the example in previous subsection $\\kappa=1\/4$ and $\\delta=3\/4$. Here we don't fix the values since they are theory dependent.\n\nWith these details, now we can venture to make an ansatz for the interpolating function that agrees the expansions at both ends upto some given order.\n\\begin{align}\nF_{m,n}(g)&=& -Q^2g^{\\kappa}\\left[1\\ +\\ a_1(Q)g\\ +\\ a_2(Q)g^2\\ +\\ ...+\\ a_m(Q)g^m\\ +\\ b_n(Q)g^{m+1}\\ +\\ ...\\right. \\nonumber\\\\\n&&\\left. +\\ b_1(Q)g^{m+n}\\ +\\ g^{m+n+1}\\right]^{\\frac{\\delta-\\kappa}{m+n+1}}.\n\\end{align}\nWe saw that the weak coupling expansion is in powers of $g^2$ since it is a cosed superstring perturbation theory. So $a_m(Q)=0$ for odd $m$. And the coefficient $a_2(Q)\\sim O(Q^2)$ in the large $Q$ limit. Similarly at the strong coupling end if the theory is an open superstring theory then $b_1(Q)$ starts at $\\sim O(Q^2)$ otherwise, $b_2(Q)$ starts at $\\sim O(Q^2)$ and $b_n(Q)=0$ for odd $n$. The above interpolating function matches the leading order expansion at both ends by construction\\footnote{In the large $Q$ limit we saw that the corrections go as $\\sim\\ Q^4$. So if an overall factor of $-Q^2$ is pulled out we are left with a correction factor of $Q^2$ within the parenthesis.}. \n\\begin{eqnarray}\n\\text{Expanding around $g=0$}&:& F_{m,n}(g)\\approx -Q^2g^{\\kappa}\\left(1+\\frac{a_2(Q)(\\delta - \\kappa)}{m+n+1}g^2+O(g^4)\\right)\\\\\n\\text{Expanding around $g=\\infty$}&:& F_{m,n}(g)\\approx -Q^2g^{\\delta}\\left(1+\\frac{b_1(Q)(\\delta - \\kappa)}{m+n+1}g^{-1}+O(g^{-2})\\right)\n\\end{eqnarray}\nFor the one loop corrected mass at both ends one can compute the interpolating function $F_{3,1}$ (like in the previous example) we have, \n\\begin{equation}\nF_{3,1}(g)=-Q^2g^{\\kappa}\\left(1\\ +\\ a_2Q^2g^2\\ +\\ b_1Q^2g^4\\ +\\ g^5\\right)^{(\\delta-\\kappa)\/5}.\n\\end{equation}\nWe have written, \n$$a_2(Q)\\equiv a_2Q^2,\\quad b_1(Q)\\equiv b_1Q^2,\\ \\text{with}\\ a_2>0,\\ b_1>0$$\n$$\\text{in accordance with their large $Q$ expansion.}$$\\\\\nNow that we have a function which gives the renormalized mass for the lowest energy state we can take the limit $g\\rightarrow 1$ i.e. at time $X^0=0$ to find the lowest energy state to which the universe tunnels to from early time strongly coupled region in case of the linear dilaton (time dependent) background. Hence after tunneling we reach a capping state configuration with the lowest energy state given by,\n\\begin{equation}\nM_{cs}= -Q^2\\left[2\\ +\\ Q^2(a_2\\ +\\ b_1)\\right]^{(\\delta-\\kappa)\/5}\\times (\\text{const.}).\n\\end{equation} \nThe over all constant is also theory specific like $\\kappa$ and $\\delta$. The accuracy of the result obviously depends on the accuracy of the interpolating function and it increases or stabilizes as we go to higher order in pertubation theory at both ends.\n\n\\section{Discussions}\n\\label{disc}\nIn this note we have presented a strategy of constructing the so called ``capping\" state in SCLD theories by embedding the SCLD phase in a strong coupling completion. Although we are unable to produce exact numbers by working with some specific theories, we do give consistency arguments to support that the strategy employed may give sensible results. Let us list the results we obtained for summarising this work,\n\\begin{itemize}\n\\item{We show for the two point function on the torus, that in the large $Q$ limit the imaginary part vanishes to the highest order in $Q$ implyng a vanishing decay width to this order. From the real part we get a positive contribution so that the energy of the Pseudo-tachyonic mode lowers under the one loop correction of the mass.}\n\\item{We find the interpolating function which is supposed to match at both the weak and strong coupling end in case of theories which are S-dual to each other.}\n\\end{itemize}\n\nAn interesting future direction is to actually compute the fermionic correlation function using some mathematical tools to figure out whether there are any spurious poles for this amplitude. Another open problem is to find specific theories satisfying the relevant conditions so that this procedure can be checked explicitly. But these are currently at the level of speculation. If we do infact find specific theories then we can produce numbers which will render a concrete example where this strategy can be tested for determining the initial capping state within some reasonable accuracy. \n\n\\section*{Acknowledgement}\nI would like to thank Ashoke Sen, Harold Erbin and K. Narayan for their useful comments on the earlier versions of the draft which improved my understanding significantly.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Acknowledgement}\nWe thank Junjie Chen and Guancheng Wang for computing the number of distinct bugs and helping with the experiments related to HiCOND.\n\n\\section{Approach}\n\\label{sec:approach}\n\nFigure~\\ref{fig:workflow} depicts the overall workflow of our approach that constitutes the following three main steps: \n(1) extracting configurable test features from the initial test suite, \n(2) using the test features to cluster test program\\xspace{}s into similar groups, and\n(3) generating configurations based on the centroid of clusters.\nWe describe each step in the rest of this section.\n\n\\subsection{Extracting configurable test features}\n\n\\Part{Test features.}\nWe use the term \\emph{test features}, or \\emph{features} for short, similar to Groce et al.~\\cite{Groce:ISSRE:2013}: ``Test features are basic \\emph{compositional units} used to describe what a test does (and does not) involve.'' Test features are essentially building blocks of test cases. For example, in testing libraries, the API calls are the functions; in grammar-based testing, the production rules in the grammar can be features; and, in testing compilers or interpreters, programming structures can be test features. In Csmith\\xspace, test features are the same as C language features, therefore, in the rest of the paper, we use terms test feature, language feature, and feature interchangeably.\n\n\\Part{Configurable test generator.}\nA test generator essentially composes new test programs\\xspace with or based on the test features. \nSome test generators allow users to use configuration files or command-line options to customize test generation by modifying the test generation parameters. \nWe call such test generators \\emph{configurable}.\nCsmith\\xspace{}~\\cite{Regehr:TestGenGrammar:Csmith:PLDI:2011} is a configurable test generator for C compilers. \nCsmith\\xspace{} then uses these parameters to decide which features to include in the test programs.\n\n\\Part{Controllable test features in Csmith\\xspace.}\nCsmith\\xspace{} allows choosing the C programming constructs to be included in the test programs\\xspace through command-line options.\nThe order and number of the constructs are, however, chosen randomly and developers cannot control them---mainly because Csmith\\xspace{} is a random generator that uses grammar to generate test program\\xspace{}s. \nAs of Csmith\\xspace{} version $2.3.0$ \\footnote{\\label{csmith}\\url{https:\/\/embed.cs.utah.edu\/csmith\/}}, the Csmith\\xspace{} provides $32$ configurable features in the form of \\texttt{--ft} and \\texttt{--no-ft} command-line options, where $ft$ denotes a feature, and \\texttt{--ft} indicates inclusion and \\texttt{--no-ft} indicates exclusion of $ft$ in the generation of the test programs\\xspace. \nFigure ~\\ref{fig:feature_count_testinputs} shows the list of available controllable features in Csmith\\xspace. \nThese features describe a wide range of language features from programming constructs in C language, e.g., \\texttt{compound\\_assignment} for compound assignments, and \\texttt{float} for floating-point data types.\nDifferent features may impact the behavior of the compiler under test in different ways. \nFor example, Groce et al.~\\cite{Groce:ISSRE:2013} observed that pointer manipulation in a test program\\xspace can suppress certain class of bugs in the C compilers, as most compilers would conservatively avoid certain optimizations for pointers in the input program, consequently, masking potential bugs in those optimizations.\n\n\\Part{Extracting features.}\nIn the context of this paper, we only consider test programs\\xspace that have been reported in the GCC bug reports.\nFigure ~\\ref{fig:bug_report} shows an example of bug report (\\#11492) in GCC Bugzilla \\footnote{\\url{https:\/\/gcc.gnu.org\/bugzilla\/show_bug.cgi?id=11492}}.\nWe use the number of occurrences of features in each test program\\xspace to create a feature vector for the test program\\xspace.\nFor this, we use ClangAST\\xspace{} \\footnote{\\url{https:\/\/clang.llvm.org\/docs\/IntroductionToTheClangAST.html}} to extract the abstract syntax tree (AST) of the programs in the initial failing test suite.\nFeature vectors are normalized to the range of $[0,1]$ using min-max normalization~\\cite{Pedregosa:ScikitLearn:2011}, i.e.,\n$ z_{ij} = \\frac { x_{ij} \\, - \\, \\min(x_{.j}) } { \\max(x_{.j}) \\, - \\, \\min(x_{.j}) } $, where $x_{ij}$ is the value of the $j^{th}$ feature of the $i^{th}$ test program\\xspace $x_{i}=(x_{i1},...,x_{in})$, $x_{.j}$ is the list of $j^{th}$ feature from all test programs\\xspace, and $z_{ij}$ is the corresponding min-max normalized value of $x_{ij}$.\n\n\\subsection{Clustering test programs\\xspace}\nTo identify the clusters, we use X-Means\\xspace{} clustering algorithm \\cite{Pelleg:Cluster:xmeans:ICML:2000} that estimates the optimal number of clusters in the underlying distribution of data.\nThe X-Means\\xspace{} clustering is an unsupervised machine learning algorithm that performs clustering of unlabeled data without the need for presetting the number of clusters.\nEach cluster is represented by a \\emph{centroid} that has a minimum distance to the data points of the cluster.\nSince the feature vectors contain the value in the range of $[0,1]$, the values in the centroids would be a number between $0$ and $1$ as well.\n\n\\subsection{Generating test programs\\xspace}\nOur implementation of \\textsc{K-Config}\\xspace uses the centroid in X-Means\\xspace{} clustering \\cite{Pelleg:Cluster:xmeans:ICML:2000} to suggest configurations for the test generator.\nTo generate configurations, we use the corresponding value of a feature in the cluster as the probability of including the feature in a test program\\xspace{}.\nThe feature is more dominant in the test program\\xspace{}s if the corresponding value of a feature is closer to $1$, and conversely, if the corresponding value is closer to $0$, it denotes that the feature is less prevalent in the test program\\xspace{}s.\nTherefore, in this approach, for each cluster, we create a configuration wherein the probability of inclusion of a feature in the test program\\xspace is equal to its corresponding value in the centroid of that cluster.\nFor instance, suppose the centroid of a cluster is $(0.1, 0.7)$ for features \\texttt{f1} and \\texttt{f2}, respectively, \nthe configuration generation algorithm in \\textsc{K-Config}\\xspace, whenever called, it includes \\texttt{--f1} with probability $0.1$ and \\texttt{--no-f1} with probability $0.9$, similarly it would \\texttt{--f2} with $0.7$ probability and \\texttt{--no-f2} with $0.3$ probability.\nIt is in contrast with the swarm testing~\\cite{Groce:TestGenRnd:Swarm:ISSTA:2012} that uses the simplest form of fair coin-toss probability (i.e., $0.5$) for the inclusion of a feature.\n\nAlgorithm~\\ref{alg:generator} describes the process of generating new test program\\xspace{}s using \\textsc{K-Config}\\xspace in more detail.\nGiven a testing budget $totalBudget$ and a set of centroids $CS$, the algorithm calls $ConfigGen$ in round-robin fashion until the test budget expires.  \nProcedure $ConfigGen$ takes a centroid $C\\in CS$ and generates a new configuration.\nIn generating a new configuration, $ConfigGen$ chooses to include feature $f_i$ with a random probability $c_i$ where $f_i$ is represented by the element $c_i$ in the centroid $C$. \nFinally, $TS$ will have all the generated failure-inducing test programs.\n\n\\begin{algorithm} [ht]\n\\caption{\\textsc{K-Config}\\xspace}\n\\label{alg:generator}\n $totalBudget$  $\\leftarrow$ Testing budget (i.e. 10,000 test count)\\;\n $CS$ $\\leftarrow$ Set of centroids\\;\n $TS \\leftarrow \\{\\}$\\; \n \\DontPrintSemicolon\\;\n \n \\While{$spentBudget \\leq totalBudget$}{\n \\ForAll{centroid $C$ $\\in$ $CS$}{\n     $config \\leftarrow ConfigGen(C)$\\;\n     $test \\leftarrow Csmith(config)$\\;\n     \\If{doesFail(test,GCC)}{\n      $TS \\leftarrow TS \\cup test$\\;\n     }\n    }\n}\n     \n\\DontPrintSemicolon\\;\n\\SetKwProg{Fn}{Function}{:}{\\KwRet}\n\\Fn{ConfigGen($C$)}{\n    $features \\leftarrow \\emptyset$\\;\n    \\ForAll{value $c$ $\\in$ $C$}{\n      $randProb \\leftarrow [0:1]$\\;\n    \\eIf{randProb $\\leq$ c}{\n        $features.put(1)$\\;\n    }{\n        $features.put(0)$\\;\n    }\n }\n}\n\\end{algorithm}\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nThis paper proposed \\textsc{K-Config}\\xspace, a test configuration generation approach, that uses the code snippets in the bug reports to guide the test generation.\nGiven a \\textsc{Seed}\\xspace test suite and a configurable test generator, \\textsc{K-Config}\\xspace computes configurations for the test generator.\n\nWe extensively evaluated \\textsc{K-Config}\\xspace with GCC regression test suite and Csmith\\xspace{} on eight GCC versions. \nThe results suggest that the configurations computed by \\textsc{K-Config}\\xspace approach can help Csmith\\xspace{} to trigger more miscompilation failures than the state-of-the-art approaches. \nThe results signify the benefits of analyzing bug reports in generation of new test programs\\xspace.\nOur source code for this work is publicly available at \\url{https:\/\/github.com\/mdrafiqulrabin\/kconfig\/}.\n\n\n\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\n\n\\KC-Round-Robin\\xspace and \\textsc{Swarm}{} are complementary approaches in proposing configurations: \\textsc{Swarm}{} chooses the configurations randomly, while \\KC-Round-Robin\\xspace takes into account the bug reports in determining the configurations.\n\\KC-Round-Robin\\xspace generally finds more failures and provides higher code coverage than \\textsc{Swarm}. Moreover, the number of distinct bugs detected by the techniques are: $4$ by both, $3$ only by \\KC-Round-Robin\\xspace, and $3$ only by \\textsc{Swarm}.\n\nThe particle swarm optimization technique used in HiCOND\\xspace requires a considerable amount of failing and passing test programs\\xspace generated by Csmith\\xspace to find an optimal configuration. Unfortunately, as GCC matures, it becomes harder for Csmith\\xspace to trigger a failure in it---e.g., in Table \\ref{table:result_10k}, compare the number of failures triggered by Default Csmith\\xspace in GCC 4.3.0 and 4.8.2. Therefore, HiCOND\\xspace loses its applicability to newer, more stable versions of GCC. Notice that in \\cite{Junjie:PSO:ASE:2019}, HiCOND\\xspace is only trained on an older version (i.e. GCC-4.3.0), and in the cross-version experiment is tried in other versions (Section IV-F in \\cite{Junjie:PSO:ASE:2019}).\n\n\n\\Part{Searching around the lamp posts}.\n\\textsc{K-Config}\\xspace analyzes features of failing test program\\xspace{}s to create new \\emph{configurations} for a test generator.\nThe result of our experiment shows that \\textsc{K-Config}\\xspace could find up to \\KMC{} miscompilations in GCC, and outperforms \\textsc{Swarm}{} and HiCOND\\xspace in triggering miscompilation failures.\n\nIt suggests that analysis of the regression test suite can enable designing techniques to guide random test generators such as Csmith\\xspace. \nThe result indicates that processing failing test programs\\xspace can provide insights into the regions of code that are susceptible to bugs.\nIt might suggest that many bug fixes are incomplete~\\cite{Yin:IncompleteBugFix:ESEC:FSE:2011}, but we are unable to verify the similarity between root causes of the bugs that \\textsc{K-Config}\\xspace triggers and the test programs\\xspace of regression test suite.\n\n\n\\Part{\\textsc{Seed}\\xspace vs. generated test suite}.\nThe coverage of GCC for the test programs\\xspace in the regression test suite shows the power of small, directed test programs\\xspace.\nAlthough the size of the regression test suite was smaller than the test suites generated by Csmith\\xspace, they significantly outperformed those test suites in the coverage of GCC in terms of statements, branches, and functions.\nThe coverage of GCC for the regression test suite can serve as a benchmark to measure the shortcomings of the generated test suites.\nFrom this experiment, the gap still is very wide. \nNote that we only could compile the test programs\\xspace in the regression test suite without linking them because they did not accompany a \\texttt{main} function.\n\n\n\\Part{Difference with deep learning-based approaches}.\nThis approach is different from (deep) learning-based approaches such as DeepSmith~\\cite{Cummins:TestGenLearn:DeepSmith:ISSTA:2018} that build a generative model for the test programs\\xspace of the programs. \nMoreover, learning-based techniques face two challenges. \nFirst, learning-based approaches require many test programs\\xspace with millions of tokens to train a model. \nSecond, learning-based approaches tend to converge to a restrictive language model of test program\\xspace{} that overly restricts the type of test program\\xspace{}s that can be produced ~\\cite{Godefroid:FeatureBased:LearnFuzz:ASE:2017}. \n\\textsc{K-Config}\\xspace instead uses the configuration of test generators to guide testing which is less constrained than the generation of test program\\xspace{}s in learning-based approaches. \nIn particular, \\textsc{K-Config}\\xspace only specifies the programming constructs that should be present in the generated test program\\xspace{}s, and the order or number of those constructs are determined by the test generator. \n\n\n\\Part{Limiting requirements}.\nThere are two main limitations to the application of \\textsc{K-Config}\\xspace. First, it assumes that a stable test generator exists. Second, it requires a set of failing test programs\\xspace. GCC compiler has been under development for decades and the bug reports are available.\nMoreover, GCC has Csmith\\xspace \\cite{Regehr:TestGenGrammar:Csmith:PLDI:2011}, a mature and well-engineered test generator, that allowed us to evaluate the effectiveness of \\textsc{K-Config}\\xspace in testing GCC compilers.\n\n\\section{Evaluation Setup}\n\\label{sec:evaluation}\n\n\\input{data\/possible_failures.tex}\n\\input{img\/difftest}\n\nThis section discusses the evaluation setup we used for the \\textsc{K-Config}\\xspace approach.\n\n\\Part{Test oracles}.\nThe common practice for testing compilers is differential testing \\cite{mckeeman1998differential, Regehr:TestGenGrammar:Csmith:PLDI:2011}.\nThat is, a test program\\xspace{} is compiled and executed by two or more versions of compilers, or two or more optimization levels, and the results are compared.\nThe differential test oracle specifies the result of the output of the compiled programs by all compilers and optimization levels must be the same.\nUndefined behaviors \\cite{Regehr:TestGenGrammar:Csmith:PLDI:2011} in the C language can complicate this process, however, Csmith\\xspace{} does the best effort to avoid the undefined behaviors in C.\n\nWe use differential testing to evaluate the behavior of the compilers on the test programs\\xspace.\nFigure ~\\ref{fig:diff_test} shows the differential testing in optimization level.\nMore specifically, we compile a program with different levels of optimization and compare the output of the executed programs. \nDue to the large scale of the experiment, and to optimize the experimentation time, we only use the lowest \\texttt{-O0} and highest \\texttt{-O3} optimization levels. We do not include the intermediate optimization levels \\texttt{-O1} and \\texttt{-O2}, as most optimization passes that a compilation of a program can invoke can be captured in \\texttt{-O3}.\nNote that although almost all optimization passes are enabled in \\texttt{-O3}, there may be cases that the interactions of viable, competing optimizations may put compiler in a position to choose ones over another one, hence not exercising some of the passes that would have been otherwise exercised in \\texttt{-O1} or \\texttt{-O2} levels. Our experiment does not capture such cases, and we speculate that such cases would be rare and hence negligible.\n\n\n\\Part{Failure types}.\nWe expect three types of compiler failures in the compilation of a program: (1) Crash when compiler terminates the compilation abruptly with a crash report on screen, (2) Timeout when compiler gets into non-termination on the compilation of a program, and (3) Miscompilation when the executable generated by the compiler produced different results on different optimizations.\nWe check the behavior of the compiler (i.e., exit code) to identify the crashes.\nWe use $10$ seconds timeout in compiling test programs\\xspace.\nWe identify miscompilation by comparing the output of executables generated by the compiler on different optimization levels.\n\n\n\\Part{Experiment parameters}.\nOverall we evaluate five configuration generation for Csmith\\xspace{}: \\textsc{Default}\\xspace, \\textsc{Swarm}, HiCOND\\xspace, \\KC-Round-Robin\\xspace and \\KC-Weighted\\xspace.\n\\textsc{Default}\\xspace{} denotes the default configuration that includes all the features by default.\n\\textsc{Swarm}{} represents the swarm testing configuration where the probability of inclusion of a feature in a test program\\xspace is $0.5$, same as the fair coin-toss approach \\cite{Groce:TestGenRnd:Swarm:ISSTA:2012}.\nHiCOND\\xspace denotes the test configurations computed from the history-guided approach proposed by Chen et al.~\\cite{Junjie:PSO:ASE:2019}.\nIn \\textsc{K-Config}\\xspace, this probability is equal to the corresponding value in the centroid of the cluster.\nFor example, if the corresponding value for \\texttt{volatile} in the centroid of a cluster is $0.2$, there is $20\\%$ chance that \\textsc{K-Config}\\xspace to use \\texttt{--volatile} and $80\\%$ chance that it includes \\texttt{--no-volatile} in the configuration.\n\\KC-Round-Robin\\xspace{} adopts the round-robin strategy with uniform weights for all clusters, while \\KC-Weighted\\xspace{} uses a weighted strategy based on the size of clusters.\nNote that we do not evaluate the regression test suite, as only a few of its test programs\\xspace have \\texttt{main} functions that can generate an executable.\n\n\n\\Part{Clustering parameters}.\nWe use the implementation of X-Means\\xspace{} algorithm in \\texttt{pyclustring} data mining library ~\\cite{Novikov:PyClustering:2019} with default hyperparameters.\nWe use the default Bayesian information criterion (BIC)~\\cite{BIC} for the splitting type to approximate the number of clusters.\nThe stop condition for each iteration is $0.025$;\nthe algorithm stops processing whenever the maximum value of change in centers of clusters is less than this value.\nMoreover, the default distance metric used is the sum of squared errors (SSE) \\cite{kwedlo2011clustering} which is the distance between data points and its centroid.\n\n\n\\Part{Test budget}.\nWe create $10,000$ test programs\\xspace for each approach and evaluate the effectiveness of the test (i.e., bug finding, coverage, and the number of distinct bugs in each approach).\nTo account for the random effects in the approaches, we run each experiment three times, average them, and round the values to the closest integer.\nWe use $10$ seconds as the timeout for GCC to compile a test program\\xspace{}.\nA small experiment with 6 hours test budget and 30 seconds timeout yielded similar observations.\nOverall, we evaluated the techniques for over $1,500$ hours (approximately $62$ days).\n\n\\section{Experimental Setting}\n\\label{sec:experimental}\nThis section discusses the experimental setting we used for the \\textsc{K-Config}\\xspace approach.\n\n\n\\Part{Reference test suite}.\nA reference test suite is a collection of test programs\\xspace that exhibit interesting behaviors, e.g., high code coverage or triggering failures.\nWe use the GCC regression test suite that has \\N{} parsable C code snippets.\nThese code snippets have been collected from the confirmed bug reports in GCC Bugzilla $^{\\ref{bugzilla}}$ and Testsuite $^{\\ref{testsuite}}$.\nThe code snippets in the bug reports are small and usually do not include \\texttt{main} functions.\nWe call this test suite \\textsc{Seed}\\xspace henceforth.\nWe, hereby, use terms regression test suite and \\textsc{Seed}\\xspace test suite interchangeably.\n\nFigure ~\\ref{fig:feature_count_testinputs} shows the number of test program\\xspace{}s in \\textsc{Seed}\\xspace test suite for each feature.\nIt shows that the distribution of test features in failing test program\\xspace{}s is not uniform. \nFeatures such as comma operators, global variables, and pointers occurred more frequently than features such as unary plus operator (\\texttt{+}), safe math, or int8. \nWe observe that all Csmith\\xspace{} configurable test features appear in one or more failing test programs\\xspace in the regression test suite. Each feature has been present in $6$ to $3871$ test programs\\xspace.\nNote that it is difficult to identify the role of individual features in the failures in the large, complex systems such as GCC, without substantial simplification of the test~\\cite{DD} and close inspection of the program execution. \n\n\n\\Part{Test generation tool}.\nWe use Csmith\\xspace{} \\cite{Regehr:TestGenGrammar:Csmith:PLDI:2011} that is an open-source automatic test generation tool for C compilers. \nGiven a set of C language features as options, Csmith\\xspace{} can generate a random C program that contains those features. \nThe programs generated by Csmith\\xspace{} are \\emph{closed}; that is, all variables are initialized in the source code by Csmith\\xspace and they do not require any external inputs for the execution. \nWe use Csmith\\xspace{} $2.3.0$ $^{\\ref{csmith}}$ in our experiments.\n\n\n\\Part{Test generation techniques}.\nWe compare the effectiveness of configurations proposed by \\textsc{K-Config}\\xspace with the effectiveness of configurations in three comparable techniques: (1) Csmith\\xspace default configuration, (2) swarm configurations, and (3) HiCOND\\xspace configurations. \nIn the \\emph{default configuration}, we use the default configuration of Csmith\\xspace wherein all C language features are enabled by default.\n\\emph{Swarm configurations} are configurations that the Csmith\\xspace features are enabled by a fair coin-toss as in ~\\cite{Groce:TestGenRnd:Swarm:ISSTA:2012}.\n\\emph{HiCOND\\xspace configurations} are created by particle swarm optimization as in ~\\cite{Junjie:PSO:ASE:2019} that attempts to optimize the effectiveness of Csmith\\xspace by systematically exploring the bug-finding capability of configurations in Csmith\\xspace.\n\n\n\\Part{Configuration generation strategies}.\nWe evaluate \\textsc{K-Config}\\xspace with two configuration generation strategies: round-robin, and weighted.\nIn the round-robin strategy, centroids of the clusters are used in a round-robin fashion to generate test programs\\xspace.\nRound-robin strategy ignores the size of clusters.\nThe weighted \\textsc{K-Config}\\xspace strategy is similar to the round-robin, except the number of test programs\\xspace generated by each centroid is proportional to the size of the cluster that the centroid represents. \nIn weighted \\textsc{K-Config}\\xspace, larger clusters will have more test programs\\xspace generated. \n\n\n\\Part{GCC versions under test}.\nWe use eight versions of GCC to evaluate the effectiveness of the \\textsc{K-Config}\\xspace approach: GCC $4.3.0$, $4.8.2$, $5.4.0$, $6.1.0$, $7.1.0$, $8.1.0$, $9.1.0$, and GCC trunk (as of $09\/03\/2019$).\nThe official releases are mature and have been widely in use for building various programs and operating systems,\nwhile the trunk version contains the newest experimental features and, as a result, is not as stable as the official releases. \n\n\n\n\n\\section{Introduction}\n\nCompilers are key parts of software development infrastructure that translate high-level programs understandable by developers to low-level programs that machines can execute.\nDevelopers \\emph{rely} on compilers to build, profile, and debug the programs; therefore any bugs in compilers threaten the integrity of software development process.\nFor this reason, researchers have been focused on testing compilers to uncover bugs, \\cite{Alipour:FocusedTesting:ISSTA:2016,Cummins:TestGenLearn:DeepSmith:ISSTA:2018,Regehr:TestGenGrammar:Csmith:PLDI:2011}. \n\nTesting compilers is particularly difficult due to their sheer size and complexity. \nTesting such massive, sophisticated systems is a non-trivial task, and researchers and developers still can find bugs in them\n\\cite{CompilerSurvey:2020}.\nRandom testing, also know as fuzzing, is a common, lightweight approach for generating test programs\\xspace for compilers \\cite{miller1990empirical}.\nMany programming languages have specialized random test generators that use the language grammar and language-specific heuristics to produce test programs in those languages; Csmith\\xspace~\\cite{Regehr:TestGenGrammar:Csmith:PLDI:2011} for C, jsFunFuzz~\\cite{Link:jsFunFuzz} for JavaScript, and Go-Fuzz~\\cite{Link:goFuzz} for Go are good examples of such test generators that have been successful in helping developers to find hundreds of bugs in various compilers and interpreters. \n\nMature test generators, like Csmith\\xspace, are configurable and allow developers to guide the test generation through configurations. \nPrior studies have proposed different techniques to exploit configurations to generate more effective test programs.\nFor example, swarm testing~\\cite{Groce:TestGenRnd:Swarm:ISSTA:2012} configures Csmith\\xspace such that test programs contain a random subset of the language features, focused random testing ~\\cite{Alipour:FocusedTesting:ISSTA:2016} uses statistical analysis of the programs generated by Csmith\\xspace and their coverage to suggest Csmith\\xspace configurations that target specific blocks in GCC, or\nHiCOND\\xspace~\\cite{Junjie:PSO:ASE:2019} analyzes on the historical test programs to create Csmith\\xspace configurations that are more likely to find bugs.\n\nIn this paper, we propose \\textsc{K-Config}\\xspace, an approach that uses the bugs reported by users to propose configurations for Csmith\\xspace.\nMore specifically, \\textsc{K-Config}\\xspace clusters the programs in the bug reports by X-Means\\xspace{} algorithm \\cite{Pelleg:Cluster:xmeans:ICML:2000} and uses the centroids of clusters as a basis for proposing configurations for Csmith\\xspace.\nWe implemented \\textsc{K-Config}\\xspace for GCC C compiler and Csmith\\xspace test generator. \nWe collected \\N{} failing test program\\xspace{}s from the GCC Bugzilla~\\footnote{\\label{bugzilla}\\url{https:\/\/gcc.gnu.org\/bugzilla\/}} and Testsuite~\\footnote{\\label{testsuite}\\url{https:\/\/github.com\/gcc-mirror\/gcc\/blob\/master\/gcc\/testsuite\/}}.\nWe performed an extensive experiment to evaluate the effectiveness of \\textsc{K-Config}\\xspace on eight versions of GCC. We compared it with swarm testing~\\cite{Groce:TestGenRnd:Swarm:ISSTA:2012}, HiCOND\\xspace~\\cite{Junjie:PSO:ASE:2019} and the default configuration of Csmith\\xspace~\\cite{Regehr:TestGenGrammar:Csmith:PLDI:2011}.\n\nTo the best of our knowledge, the other comparable techniques do not use the information available in the bug reports in the generation of new test programs\\xspace. \\textsc{K-Config}\\xspace is the first attempt in that direction \\cite{rabin2019kconfig}. The reasoning it uses is that the code snippets in the bug reports that triggered bugs earlier are more likely to be of interest to developers. We believe bug reports contain insights that can be extracted to improve the test generation for compilers.\n\nThe results of our experiments suggest that \\textsc{K-Config}\\xspace could find up to \\KMC{} miscompilations, while swarm finds up to \\SMC{} miscompilations, and the default configurations couldn't find any miscompilations. Moreover, the coverage of \\textsc{K-Config}\\xspace is higher than other techniques in all versions of GCC compilers in our experiment.\nOur results also show that the coverage of regression test programs\\xspace is still higher than any automatically generated test suite. It suggests that despite the advances in generating test programs for compilers, there is still a wide gap between the effectiveness of the generated test programs, and the small, regression test suites.\n\nThe intuition behind \\textsc{K-Config}\\xspace is that the language features that have participated in previous bugs are likely to participate in new bugs as well. \nSimilar intuition has been used in LangFuzz~\\cite{Zeller:TestGenMutation:LangFuzz:Security:2012} and HiCOND\\xspace~\\cite{Junjie:PSO:ASE:2019}.\nLangFuzz transplants fragments of failing test program\\xspace{}s to generate new test programs\\xspace for JavaScript.\nAlthough similar in the intuition, instead of manipulating existing test programs, \\textsc{K-Config}\\xspace uses configurations to guide the test generator to generate new test programs\\xspace.\nHiCOND\\xspace uses the configuration of Csmith\\xspace \\emph{generated} test programs to optimize and propose diverse test configurations.\nWhile \\textsc{K-Config}\\xspace and HiCOND\\xspace both are similar in the sense that they use an initial test suite to infer effective configurations for Csmith\\xspace, they differ in two main ways. \nFirst, HiCOND\\xspace requires the configuration of the Csmith\\xspace in the generation of each test program. In other words, all initial test programs must be generated by Csmith\\xspace. Therefore, the test programs that are not created by Csmith\\xspace, like the ones available in the bug reports, cannot be used by HiCOND\\xspace.\nSecond, the particle swarm optimization technique used in HiCOND\\xspace requires a large initial test suite of failing and passing test programs\\xspace generated by Csmith\\xspace. Unfortunately, as GCC matures, it becomes harder for Csmith\\xspace to trigger failures in GCC hence, which negatively impacts HiCOND\\xspace applicability on newer, more stable versions of GCC. \n\n\\Part{Contributions}.\nThis paper makes the following contributions.\n\\begin{itemize}\n    \\item We propose an approach for computing configuration of the Csmith\\xspace test generator by processing the code snippets in the bug reports.\n    \\item We perform a large-scale case study on the effectiveness of our approach on eight versions of GCC---a large, complex C compiler.\n\\end{itemize}\n\n\\Part{Paper Organization}.\nThe paper is organized as follows.\nSection~\\ref{sec:approach} describes the proposed \\textsc{K-Config}\\xspace approach. \nSection~\\ref{sec:experimental} and Section~\\ref{sec:evaluation} describe the experimental setting and evaluation setup.\nSection~\\ref{sec:results} presents the results of the experiments. \nSection~\\ref{sec:discussion} discusses the results. \nSection~\\ref{sec:related} presents the related works.\nSection~\\ref{sec:threat} describes the main threats to validity. \nFinally, section~\\ref{sec:conclusion} concludes the paper.\n\n\n\\section{Related Work}\n\\label{sec:related}\n\n\nSeveral approaches for testing compilers have been proposed; for example, \ncode mutation \\cite{Zeller:TestGenMutation:LangFuzz:Security:2012}, \nrandom test generation \\cite{Regehr:TestSelection:Rank:PLDI:2013}, \nmetamorphic testing \\cite{Zhendong:TestOracle:classfuzz:PLDI:2016, Zhendong:TestGenMutation:Orion:PLDI:2014, Zhendong:TestGenMutation:Athena:OOPSLA:2015,Zhendong:TestGenMutation:Hermes:OOPSLA:2016},\nlearning-based testing \\cite{Cummins:TestGenLearn:DeepSmith:ISSTA:2018}, \nand swarm testing \\cite{Groce:TestGenRnd:Swarm:ISSTA:2012}, to name few. \nThese approaches either generate test programs\\xspace from scratch by grammar \\cite{Regehr:TestGenGrammar:Csmith:PLDI:2011} and learning \\cite{Cummins:TestGenLearn:DeepSmith:ISSTA:2018}, \nor they create new test programs\\xspace by manipulating \\cite{Zeller:TestGenMutation:LangFuzz:Security:2012} or transforming the existing test programs\\xspace, e.g., \\cite{Zhendong:TestGenMutation:Orion:PLDI:2014}.\n\nSwarm testing \\cite{Groce:TestGenRnd:Swarm:ISSTA:2012} randomly chooses a subset of features available to generate new test cases. \nThe generated test cases are very diverse and the evaluation result shows that this approach outperforms Csmith\\xspace{}'s default configuration in both code coverage and crash bug finding. \nSPE \\cite{Zhendong:TestSelection:SPE:PLDI:2017} where authors enumerate a set of programs with different variable usage patterns. \nThe generated diverse test cases exploit different optimization and the evaluation result shows that the skeletal program enumeration has confirmed bugs in all tested compilers. \nTwo more related studies in this area are LangFuzz \\cite{Zeller:TestGenMutation:LangFuzz:Security:2012} and Learn\\&Fuzz \\cite{Godefroid:FeatureBased:LearnFuzz:ASE:2017}. \nThe LangFuzz approach extracts code fragments from a given code sample that triggered past bugs and then apply random mutation within a pool of fragments to generate test inputs. \nOn the other hand, the Learn\\&Fuzz approach uses the learnt seq2seq model to automate the generation of an input grammar suitable for PDF objects using different sampling strategies. \nBoth approaches have revealed several previously unknown bugs in popular compilers and interpreters.\n\nSeveral approaches to accelerate the speed of test selection and triage have been proposed; for example, \\cite{Regehr:TestSelection:Rank:PLDI:2013}, \\cite{Chen:TestSelection:LET:ICSE:2017}, \\cite{Chen:TestSelection:4steps:ICSE:2018}, \\cite{Chen:TestSelection:COP:IEEE-TSE:2018}.\nChen et al. \\cite{Regehr:TestSelection:Rank:PLDI:2013} evaluate the impact of several distance metrics on test case selection and prioritization.\nChen et al. \\cite{Chen:TestSelection:LET:ICSE:2017} proposed LET where authors use machine learning to schedule the test inputs. This learning-to-test approach has two steps: learning and scheduling. In learning steps, LET extracts a set of features from the past bug triggering test cases and then trains a capability model to predict the bug triggering probability of the test programs, and trains another time model to predict the execution time of the test programs. In scheduling steps, LET ranks the target test programs based on the probability of bug triggering in unit time. The evaluation result shows that the scheduled test inputs significantly accelerate compiler testing.\nAnother example in this area is COP \\cite{Chen:TestSelection:COP:IEEE-TSE:2018} where authors predict the coverage information of compilers for test inputs and prioritize test inputs by clustering them according to the predicted coverage information. The result shows that COP significantly outperforms state-of-the-art acceleration approaches in test acceleration.\n\n\\section{Results}\n\\label{sec:results}\n\n\n\\input{data\/result_10k.tex}\n\\subsection{Comparison with \\textsc{Default}\\xspace and \\textsc{Swarm}}\n\nIn Table \\ref{table:result_10k}, $10,000$ test program\\xspace{}s of \\KC-Round-Robin\\xspace{}, on average, triggered up to $96$ miscompilations, while \\textsc{Swarm}{} triggered up to $70$ miscompilations, on average. The default configuration of Csmith\\xspace{} (\\textsc{Default}\\xspace) triggered $16$ miscompilations in GCC-4.3.0 but did not find any miscompilations in other versions of GCC.\n\nThe \\textsc{Default}\\xspace found $1708$ crashes in GCC-4.3.0 compared to $755$ by \\KC-Round-Robin\\xspace{} and $666$ by \\textsc{Swarm}{}. However, the \\textsc{Default}\\xspace did not find any crashes in GCC-4.8.2, where the \\KC-Round-Robin\\xspace{} found $111$ crashes and the \\textsc{Swarm}{} found $125$ crashes.\n\nIn all cases, \\KC-Round-Robin\\xspace{} triggered the highest number of miscompilations compared to \\textsc{Default}\\xspace and \\textsc{Swarm}.\n\n\n\\subsection{Comparison with HiCOND\\xspace}\n\nHiCOND\\xspace uses particle swarm optimization to search for configurations that can find more bugs. \nWe received the configurations of HiCOND\\xspace for GCC-4.3.0 from the authors.\nIn GCC-4.3.0, HiCOND\\xspace finds the highest number of crashes on \\texttt{-O3}, while swarm followed by \\KC-Round-Robin\\xspace{} discover the highest number of crashes under \\texttt{-O1}. \\KC-Round-Robin\\xspace{} finds the highest number of miscompilations.  \nConfigurations proposed by HiCOND\\xspace do not find any crashes in GCC-4.8.2, as it is highly optimized for bugs in GCC-4.3.0. \nMoreover, HiCOND\\xspace relies on a set of bugs generated by \\textsc{Default}\\xspace to search for the optimal configurations. \n\\textsc{Default}\\xspace does not trigger any failures on GCC-4.8.2, we, therefore, could not compute new configurations using the approach in \\cite{Junjie:PSO:ASE:2019} for GCC-4.8.2 and beyond.\n\n\n\\subsection{Number of distinct bugs}\n\n\nTo identify the number of distinct bugs we used the correcting commits heuristic that has been used in previous studies~\\cite{Regehr:TestSelection:Rank:PLDI:2013, Chen:TestSelection:LET:ICSE:2017}.\nThis heuristic uses the commit that a test program switched from failing to passing as the proxy for the minimum number of distinct bugs.\nNote that in our results, there are test programs that still exhibit miscompilation characteristics on the latest versions of the GCC, therefore, this heuristic can not be used for them.\nDue to the submission deadline, we only compute the distinct bugs for \\textsc{Swarm}{} and \\KC-Round-Robin\\xspace in GCC-4.8.2.\nFor the crash and miscompilation of failing test programs\\xspace, the numbers of distinct bugs detected by the techniques are: 3\\xspace only by \\KC-Round-Robin\\xspace, 3\\xspace only by \\textsc{Swarm}, and 4\\xspace by both.\n\n\n\\subsection{Effectiveness of cluster weighting strategies}\n\nWe evaluated the effectiveness of \\textsc{K-Config}\\xspace for two cluster weighting strategies: \\KC-Round-Robin\\xspace{} that ignores the size of clusters, and \\KC-Weighted\\xspace{} which generates test programs proportional to the size of clusters.\nThe results in Table \\ref{table:result_10k} show that \\KC-Round-Robin\\xspace{} was more effective than \\KC-Weighted\\xspace{} in finding bugs. \n\\KC-Round-Robin\\xspace{} could trigger twice as more miscompilations, and thrice as more crashes than \\KC-Weighted\\xspace{}. \nThe potential reason can be that larger clusters represent prominent combinations of features in the bug reports, it is likely that developers already noticed them and addressed them. Therefore, test programs\\xspace generated based on \\KC-Weighted\\xspace{} would not lead to new failures.\nDue to the poor performance of \\KC-Weighted\\xspace, we excluded the \\KC-Weighted\\xspace{} from subsequent experiments to save computing time.\n\n\n\\subsection{Effectiveness of individual clusters}\n\nTo measure the effectiveness of individual clusters in \\KC-Round-Robin\\xspace, we count the number of failures triggered by the test programs\\xspace generated based on their centroid configuration.\nFigure ~\\ref{fig:nbug} shows the number of crashes and miscompilations triggered by each cluster. We excluded the timeouts from the figure due to their sheer numbers.\nThe $x$-axis in this figure denotes the clusters, and the $y$-axis denotes the number of miscompilation and crash failures triggered by each cluster.\n\nAmong $134$ clusters, $50$ did not contribute to finding any failures, $29$ triggered only one crash, $18$ triggered only one miscompilation, while there are configurations with $20$ to $26$ (crash or miscompilation) failures.\nTable~\\ref{table:top5_nbug} shows the top-$5$ most effective configurations in triggering failures (crash and miscompilation combined).\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{img\/nbug.png}\n    \\caption{Number of failures in each configuration.}\n    \\label{fig:nbug}\n\\end{figure}\n\n\\input{data\/top5_centroids.tex}\n\n\n\\subsection{Coverage across test suites}\n\nTable~\\ref{table:coverage_10k} shows the number of covered statements, branches, and functions on GCC versions 4.3.0, 4.8.2, 5.4.0, 6.1.0, and 7.1.0 for the \\textsc{Seed}\\xspace{}, \\textsc{Default}\\xspace{}, \\textsc{Swarm}{}, and \\KC-Round-Robin\\xspace{} test suites.\nWe computed the coverage with -O3 optimization. \nAt the time of writing, we did not extract the coverage information for most recent versions of GCC, i.e., 8.1.0, 9.1.0, and trunk, due to difficulties stemming from recent changes in the file structure in GCC project.\nThe coverage shows that, in the measured GCC versions, the regression test suite has higher coverage than the generated test suites. \nIn the generated test suite, the \\KC-Round-Robin\\xspace{} has higher coverage than others (\\textsc{Default}\\xspace{}, \\textsc{Swarm}{}, and HiCOND\\xspace{}).\n\n\\input{data\/coverage_10k.tex}\n\n\n\\subsection{The probability of features}\n\nIn the swarm testing~\\cite{Groce:TestGenRnd:Swarm:ISSTA:2012}, the probability of inclusion of a test feature in the test program depends on the fair coin-toss for all test features. Therefore, the probability is the same for all test features, it is 0.5. \nFigure~\\ref{fig:centroid:probabilities}, on the other hand, shows the distributions of probabilities per test feature in the \\KC-Round-Robin\\xspace approach.\nThe mean and median of the probabilities of all features are less than 0.5. However, 28 out of 32 features have been in one or more configurations with probability 1.\nIn more than $25\\%$ of configurations, $75\\%$ or more of features are excluded.\nIt highlights the importance of a few test features that dominate the test programs\\xspace generated by those clusters.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{img\/centroids_boxplots.png}\n    \\caption{Distribution of probability of inclusion for individual GCC features in \\KC-Round-Robin\\xspace.}\n    \\label{fig:centroid:probabilities}\n\\end{figure}\n\n\n\n\\section{Threats to Validity}\n\\label{sec:threat}\n\nIn this section, we describe several threats to validity for our study.\n\n\\Part{Internal Validity.} \nDespite our best effort, some bugs may exist in the tools and scripts used in this paper.\nHowever, we used the well-tested programs in the implementation and evaluation of \\textsc{K-Config}\\xspace to reduce the chance of mistakes.\nWe have taken care to ensure that our results are unbiased, and have tried to eliminate the effects of variability by repeating the experiments multiple times.\n\n\\Part{External Validity.}\nWe evaluated the \\textsc{K-Config}\\xspace approach on Csmith\\xspace and several GCC versions, therefore, the extent to which it generalizes to other compilers and test generators is quite unknown.\nIn this approach, the feature set is restricted to the ones that can be translated to Csmith\\xspace configuration options, therefore the feature space is also limited.\nAdditionally, \\textsc{K-Config}\\xspace relies on the features that have triggered compiler bugs in the past, therefore the effectiveness may decrease in future GCC versions for the bug fixes.\nAnother potential issue can be that the bug reports to build the \\textsc{Seed}\\xspace test suite may not be representative of all GCC bugs and can impact the clusters.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\n\\paragraph{A factor of two.}\n\nThe early universe from at least big bang nucleosynthesis onwards\nis well described by a model where the geometry is locally\nspatially homogeneous and isotropic up to linear perturbations,\nthe matter consists of a gas of particles with\npositive pressure, and the relation between geometry\nand matter is given by the Einstein equation\nbased on the four-dimensional Einstein-Hilbert action.\nHowever, at late times such a model underpredicts the\ndistance to far-away sources and the expansion rate.\nCompared to the simplest possibility, the spatially flat\nmatter-dominated model, the discrepancy is a factor\nof about two in both the distance\n(for a fixed Hubble constant) and the expansion rate\n(for a fixed energy density or age of the universe).\nTherefore at least one of the three assumptions\n--homogeneity and isotropy, standard matter content\nand standard gravity-- is wrong, assuming that light\npropagation is correctly modeled by null geodesics.\nNo deviations from standard gravity have been observed in\nlocal physics, not in the solar system\n(apart from the Pioneer anomaly and the flyby anomaly, where\nthe possibility of systematics is not ruled out) nor\nin pulsars \\cite{solar, Will:2005}.\nNeither is there any detection of an effect of exotic\nmatter with negative pressure on local physics.\nThe factor of two discrepancy only appears in\nobservations of distance and expansion rate\nwhich involve quantities integrated over\nlarge scales\\footnote{It has been argued that locally\nrepulsive gravity has been observed in the motions\nof galaxies near the Local Group \\cite{localde}.\nThis is an interesting possibility, but the present data\nis not precise enough for such a detection.}.\nThis situation is quite different from that of dark matter,\nfor which there is evidence from various different systems\non several scales.\n\nWhile there is no evidence against standard general relativity\nor standard matter apart from the increased distance and\nexpansion rate,\nthe universe is known to be locally far from homogeneity and\nisotropy due to the formation of non-linear structures at late\ntimes. It is possible that the breakdown of the homogeneous\nand isotropic approximation could explain the failure of\nthe prediction of homogeneous and isotropic models with\nordinary matter and gravity\n\\cite{Buchert:2000, Tatekawa:2001, Wetterich:2001, Schwarz, Rasanen, Kolb:2004}.\nThe effect of inhomogeneity and\/or anisotropy on the\nevolution of the universe was first discussed in detail\nin \\cite{fitting} under the name ``the fitting problem'',\nand the effect on the average expansion rate is known as\nbackreaction\n\\cite{Buchert:1999, Buchert:2001, Ellis:2005, Rasanen:2006b, Buchert:2007}.\nIt has been shown with toy models that inhomogeneities\ncan lead to accelerating expansion\n\\cite{Rasanen:2006b, Chuang:2005, Paranjape:2006a, Kai:2006, Rasanen:2006a},\nbut whether this happens for the distribution of structures\npresent in the real universe is not yet clear.\nThe order of magnitude of the observed change in\nthe expansion rate and the correct timescale\nof around 10 billion years do emerge from the physics of\nstructure formation in a semi-realistic model\n\\cite{Rasanen:2008a, peakrevs}, but there is no fully\nrealistic calculation yet.\n\n\\paragraph{Light propagation and statistical homogeneity and isotropy.}\n\nMost cosmological observations probe quantities\nrelated to light propagation, such as the redshift,\nthe angular diameter distance (or equivalently the\nluminosity distance) and image distortion.\nIn linearly perturbed homogeneous and isotropic\nFriedmann-Robertson-Walker (FRW) models, the redshift\nand the distance are to leading order determined by the\nexpansion rate and the spatial curvature.\nThe corrections due to the perturbations are small for\ntypical light rays, and are important only\nfor the cosmic microwave background (CMB), whose redshift\nanisotropies are very accurately measured, and image\ndistortion, which is zero for the background and remains\nsmall when perturbations are included. (At least this\nis the case when the perturbations are statistically\nhomogeneous and isotropic and the homogeneity scale is small;\nsee \\cite{Enqvist:2009} for a counterexample with a large\nspherically symmetric structure.)\n\nThe fact that the optical properties of a FRW universe can be\nexpressed in terms of the expansion rate and the spatial curvature\nis rather obvious, because light propagation is, for small\nwavelengths, purely geometrical, and these are the only\ndegrees of freedom in the FRW geometry.\nIn a general spacetime, the situation is more involved.\nNevertheless, if the distribution of the geometry is statistically\nhomogeneous and isotropic, it could be expected that light\npropagation over distances longer than the homogeneity scale\ncan to a first approximation be similarly described with a few\nquantities related to the overall geometry, regardless of complicated\nlocal details \\cite{Rasanen:2008a}.\nLight propagation in statistically homogeneous and isotropic\nuniverses with irrotational dust as the matter content was studied\nin \\cite{Rasanen:2008b}, where it was argued that if the distribution\nevolves slowly compared to the time it takes for light\nto cross the homogeneity scale, then the redshift and the\nangular diameter distance are determined by the\nexpansion rate as a function of redshift and the\nmatter density today.\nThe study \\cite{Rasanen:2008b} had three shortcomings.\n\nFirst, it was assumed that the variation in the spatial\ndirection of the null geodesics (i.e. light deflection) is small.\nThe magnitude of the null shear was also left undetermined.\nObservationally, both light deflection and image distortion\nare known to be small for typical light rays \\cite{Munshi:2006},\nand it should be established that this follows from the\nsymmetry properties of the spacetime.\nSecond, the treatment of matter as irrotational dust\nis not locally valid \\cite{Buchert:2005, Pueblas:2008},\nbecause effects such as rotation and velocity dispersion are\nimportant for stabilising structures on small scales.\nThe vorticity and non-dust nature of the matter content\nmay be expected to be unimportant for the overall cosmological\nevolution in the real universe at late times.\nNevertheless, such effects should be included to establish under\nwhich conditions they can be neglected, and put the dust\napproximation on better footing. Including matter\nother than dust is also necessary for treating backreaction\nin the early universe such as during inflation\n\\cite{Buchert:2001, Woodard, Unruh, Geshnizjani, Brandenberger:2002}\nor preheating.\nThird, the arguments were qualitative and the corrections\nto the mean behaviour were not determined.\n\nWe now remedy the first and second problems.\nWe derive results for light propagation, including\ndeflection and shear, using only assumptions\nabout the symmetry of the spacetime geometry and matter content.\nWe consider general matter content and include rotation.\nConcentrating on observable quantities related to light propagation,\nwe show how the relevant averaging hypersurface is\ngiven by the statistical symmetry of the spacetime.\nHowever, our analysis is not more quantitative\nthan \\cite{Rasanen:2008b}, and the arguments should\nbe followed up with a more rigorous study.\n\nIn section 2 we go through our assumptions, set up\nthe covariant formalism and derive\nresults for the redshift, the deflection,\nthe null shear and the angular diameter distance.\nIn section 3 we derive the evolution equations\nfor the scale factor, which generalise the\nBuchert equations of the irrotational\ndust case \\cite{Buchert:1999}, and consider\nthe validity of the dust approximation.\nIn section 4 we discuss the possible effect\nof the discreteness of the matter content,\nthe relevance of average quantities and the FRW\ndescription, and summarise the situation.\n\n\\section{Light propagation} \\label{sec:light}\n\n\\subsection{Spacetime geometry} \\label{sec:geom}\n\n\\paragraph{Statistical symmetry.}\n\nWe assume that there exists a foliation of the\nspacetime into spatial hypersurfaces of statistical\nhomogeneity and isotropy, which we denote by $\\mathcal{N}$.\nThe time which is constant on such a hypersurface\nis denoted $t$, and when referring to a particular\nhypersurface, we use the notation $\\mathcal{N}(t)$.\nBy this we mean that when we consider\nany region larger than the homogeneity scale, the average\nquantities within the region do not depend on its\nlocation, orientation or size.\nIn other words, over large scales, there are no\npreferred locations or directions, and no correlations.\nLocally the dynamics can be complex, as the assumption\nof statistical homogeneity and isotropy only concerns\naverage quantities evaluated over large scales.\nThe frame of statistical homogeneity and isotropy\nmay not locally coincide with either\nthe Eckart frame (where there particle number flux is zero) or\nthe Landau-Lifshitz frame (where the energy flux is zero)\n\\cite{Maartens:1998}.\nHowever, statistical homogeneity and isotropy does imply\nthat the integrated flux of any quantity through the\nboundary of a volume larger than the homogeneity scale\nvanishes.\n\nIn this view, the universe consists of identical (up\nto statistical fluctuations) boxes stacked next\nto each other.\nIn the real universe, there are correlations even\nover scales longer than the Hubble scale,\ndue to inflation (or some other process in the early universe)\nwhich produces a large region that is exactly homogeneous\nand isotropic except for linear perturbations.\nThe distribution of the perturbations\nis statistically homogeneous and isotropic.\nWhen the perturbations become non-linear at late\ntimes (in typical supersymmetric dark matter models,\nthe first structures form around a redshift of 40--60 \\cite{SUSYCDM}),\nlocal homogeneity and isotropy are lost, but the distribution\nof non-linear structures remains statistically homogeneous\nand isotropic, and the amplitude of\ncorrelations is small beyond the homogeneity scale.\nWhat one finds as the homogeneity scale depends on\nthe limit that one sets for this amplitude.\nBased on the fractal dimension of the point set of galaxies,\nit has been argued that the distribution becomes homogeneous on a\nscale of around 100 Mpc to an accuracy of about 10\\% \\cite{hom}.\nHowever, there are still large fluctuations on 100 Mpc scales,\nand it has been argued that the sample size is not large\nenough to establish that the distribution is self-averaging,\nwhich is a necessary condition for statistical homogeneity\n\\cite{inhom, SylosLabini:2009}.\nStudies of morphology also suggest that the homogeneity\nscale could be 300 Mpc or more \\cite{morphology}.\n\nStatistical homogeneity and isotropy is\nformulated in terms of spatial hypersurfaces, but\nlight travels along null geodesics, not in a spacelike\ndirection. Therefore we also need information about the\nevolution which relates one hypersurface to the next.\nWe assume that the evolution is slow in the sense\nthat the timescale of change in the spatial distribution\nis much larger than the homogeneity scale.\nPhrased differently, the variation of the geometry\nalong a null geodesic is rapid compared to the\nscale over which the mean varies significantly.\nIn the real universe, the timescale for change\nin the distribution of matter and geometry\nis the Hubble time $H^{-1}$.\nToday $H_0^{-1}=3000$\\mbox{$h^{-1}$Mpc}{} (with $h$ somewhat below unity\n\\cite{Hubble}), much larger than 100--300 Mpc.\nIn the past, the homogeneity scale was even smaller\nrelative to the Hubble scale, as structure formation\nwas less advanced.\n\nThe combination of statistical homogeneity and\nisotropy on spatial hypersurfaces and slow evolution\nfrom one hypersurface to the next can be heuristically\nthought of as a distribution that is statistically\napproximately homogeneous and isotropic in four dimensions\nwhen considering scales larger than the homogeneity scale,\nbut smaller than the timescale of change in the distribution.\nThe notion of statistical homogeneity and isotropy\nin general spacetimes should be made more rigorous,\nand the role of slow evolution in the arguments we\nmake below on light propagation should be quantified.\n\n\\paragraph{The two frames.}\n\nWe denote the vector normal to $\\mathcal{N}$ by $n^\\a$ and\nthe velocity of the observers by $u^\\a$.\nBoth are normalised to unity, $n_\\a n^\\a=u_\\a u^\\a=-1$.\nThe observer velocity is completely general, it\nis not assumed to be geodesic or irrotational.\nFor reviews of the covariant approach we use, see\n\\cite{Ehlers:1961, Ellis:1971, Ellis:1998c, Clarkson:2000, Tsagas:2007};\nfor the relation to the ADM formalism \\cite{Arnowitt:1962},\nsee \\cite{Jantzen:2001}.\nThe tensors which project on the hypersurface\northogonal to $n^\\a$ and the rest space\northogonal to $u^\\a$ are, respectively,\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{h}\n  h_{\\a\\b} &\\equiv& g_{\\a\\b} + n_\\a n_\\b \\nonumber \\\\\n  h_{\\a\\b}^{(u)} &\\equiv& g_{\\a\\b} + u_\\a u_\\b \\ ,\n\\eea\n\n\\noindent where $g_{\\a\\b}$ is the spacetime metric.\nThe restriction of the projection tensor\n$h_{\\a\\b}$ to $\\mathcal{N}$ is the metric on $\\mathcal{N}$.\nThe spatial derivative of a scalar is defined as\n$\\hat{\\nabla}_\\a f\\equiv h_{\\a}^{\\ \\b} \\nabla_\\b f$, for vectors we have\n$\\hat{\\nabla}_\\b f_\\a \\equiv h_{\\b}^{\\ \\d} h_{\\a}^{\\ \\c} \\nabla_\\d f_\\c$,\nand similarly for higher order tensors.\nThe spatially projected traceless part of a tensor is\n$f_{\\langle\\a\\b\\rangle}\\equiv h_{(\\a}^{\\ \\ \\c} h_{\\b)}^{\\ \\ \\d} f_{\\c\\d} - \\frac{1}{3} h_{\\a\\b} h^{\\c\\d} f_{\\c\\d}$.\nThe volume element on $\\mathcal{N}$ is $\\epsilon_{\\a\\b\\c}\\equiv\\eta_{\\a\\b\\c\\d} n^\\d$, \nwhere $\\eta_{\\a\\b\\c\\d}$ is the spacetime volume element.\nThe derivative with respect to the proper time $s$\nof the frame of statistical homogeneity and isotropy\nis $n^\\a\\nabla_\\a$, and it is denoted by an overdot.\nUnless $\\dot{n}^\\a=0$, the proper time $s$ does not coincide\nwith the time $t$ which is constant on $\\mathcal{N}$.\nWe can write $n_\\a=-\\dot{t}^{-1}\\partial_\\a t$.\nThe derivative with respect to $t$ is\n$m^\\a \\nabla_\\a$, with $m^\\a=\\dot{t}^{-1} n^\\a$.\nWe define $\\Gamma\\equiv-n_\\a m^\\a$, so\n$\\Gamma=\\dot{t}^{-1}=\\partial_t s$ and $m^\\a=\\Gamma n^\\a$.\nPhysically, $\\Gamma$ describes the time dilation due to\nthe non-geodesic motion of the $n^\\a$ frame; we have\n$\\dot{n}_\\a=\\hat{\\nabla}_\\a\\ln\\Gamma$.\n(Note that a $\\Gamma$ which depends only on $t$\ncorresponds to a different time coordinate, not\ndifferent physics.)\nOnly if $\\dot{n}^\\a=0$ can we choose $\\Gamma=1$ and $s=t$,\nwhich is equivalent to the statement that $\\mathcal{N}$ is a\nhypersurface of constant proper time.\nIn addition to $s$ and $t$, we also have the\nproper time of the observers, defined by $u^\\a$.\n\nBecause the timescale for the evolution of structures\nis determined by their proper time, the hypersurface of statistical\nhomogeneity and isotropy could be expected to coincide with the\nhypersurface of constant proper time of observers\ncomoving with the structures, as argued in\n\\cite{Rasanen:2006b, Rasanen:2008a, Rasanen:2008b}.\nHowever, if the matter consists of several components\nwhich form structures differently, the situation is not\nso simple. For example, in the real universe,\ndark matter and baryons cluster differently (though the differences\nare not expected to be important on scales larger\nthan the homogeneity scale).\nAnd on small scales, dark matter is multistreaming,\nso there is more than one proper time\nassociated with the matter flow at a single point.\nWe keep the hypersurface of statistical homogeneity\nand isotropy arbitrary.\n\nWithout loss of generality, we write the observer\nvelocity $u^\\a$ in terms of $n^\\a$ and a component\northogonal to $n^\\a$,\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{n}\n  u^\\a = \\gamma ( n^\\a + v^\\a ) \\ ,\n\\eea\n\n\\noindent where $\\gamma\\equiv -n_\\a u^\\a=(1-v^2)^{-1\/2}$, with\n$v^2\\equiv v_\\a v^\\a$ and $v_\\a n^\\a=0$.\nNote that $v^\\a$ is not the peculiar velocity, either\nin the perturbation theory sense of a velocity with\nrespect to a fictitious background, or in the\nphysical sense of deviation from\na shearfree velocity field \\cite{peculiar}.\nThe quantity $v^\\a$ measures the deviation of the local\nobserver velocity from the time direction set by the\nframe of statistical homogeneity and isotropy.\nEven if $v^\\a$ is zero, there can be arbitrarily large\nspatial variations in the expansion rate.\nWe will see that a large $v$ implies significant anisotropy\nin the CMB. We therefore often take $v$ to be small,\nand expand to first order in $v$. We use $\\simeq$ to indicate\nequality up to and including terms first order in $v^\\a$.\n(We do not assume that derivatives of $v^\\a$ are small.)\nPhysically, this means that the motion of the observers\nwith respect to the frame of homogeneity and isotropy is\nnon-relativistic.\n\n\\paragraph{Fluid kinematics.}\n\nThe covariant derivative of $n^\\a$ can be decomposed as\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{gradn}\n  \\nabla_\\b n_\\a\n  &=& \\frac{1}{3} h_{\\a\\b} \\theta + \\sigma_{\\a\\b} - \\dot{n}_\\a n_\\b \\ ,\n\\eea\n\n\\noindent where\n$\\theta\\equiv\\nabla_\\a n^\\a=\\hat{\\nabla}_\\a n^\\a$\nis the volume expansion rate and\n$\\sigma_{\\a\\b}\\equiv\\nabla_{\\langle\\b} n_{\\a\\rangle}=\\hat{\\nabla}_{\\langle\\b} n_{\\a\\rangle}$\nis the shear tensor.\nThe tensor $\\sigma_{\\a\\b}$ and the acceleration vector\n$\\dot{n}^\\a$ are spatial in the sense that they are orthogonal\nto $n^\\a$, $\\sigma_{\\a\\b} n^\\b=0$, $\\dot{n}_\\a n^\\a=0$.\nThe shear scalar is defined as\n$\\sigma^2\\equiv\\frac{1}{2}\\sigma_{\\a\\b}\\sigma^{\\a\\b}$.\nBecause $n^\\a$ is hypersurface-orthogonal, it follows\nfrom Frobenius' theorem that the vorticity \n$\\omega_{\\a\\b} \\equiv \\nabla_{[\\b} n_{\\a]} + \\dot{n}_{[\\a} n_{\\b]}=\\hat{\\nabla}_{[\\b} n_{\\a]}$\nis zero\n\\cite{Ehlers:1961, Ellis:1971, Ellis:1998c}, \\cite{Wald:1984} (page 434).\n\nThe covariant derivative of the observer velocity $u^\\a$ can be\nanalogously decomposed with respect to itself,\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{gradu}\n  \\nabla_\\b u_\\a\n  &=& \\frac{1}{3} h^{(u)}_{\\a\\b} \\theta^{(u)} + \\sigma^{(u)}_{\\a\\b} + \\omega^{(u)}_{\\a\\b} - A_\\a u_\\b \\ ,\n\\eea\n\n\\noindent where $\\theta^{(u)}\\equiv\\nabla_\\a u^\\a$,\n$\\sigma_{\\a\\b}^{(u)}\\equiv h^{(u)}_{\\a\\c} h^{(u)}_{\\b\\d} \\nabla^\\d u^\\c-\\frac{1}{3}\\theta^{(u)} h^{(u)}_{\\a\\b}$,\n$\\omega^{(u)}_{\\a\\b} \\equiv \\nabla_{[\\b} u_{\\a]} + A_{[\\a}u_{\\b]}$\nand $A^\\a\\equiv u^\\b\\nabla_\\b u^\\a$.\n\nGiven \\re{n}, the expansion rates in the two frames \nare related as (see \\cite{Tsagas:2007} for the expressions\nfor the acceleration, shear and vorticity)\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n  \\label{reltheta} \\theta^{(u)} &=& \\gamma \\theta + \\gamma ( \\hat{\\nabla}_\\a v^\\a + \\dot{n}_\\a v^\\a ) + \\gamma^3 ( \\dot{v}_\\a v^\\a + v^\\a v^\\b \\hat{\\nabla}_\\a v_\\b ) \\nonumber \\\\\n  &\\simeq& \\theta + \\hat{\\nabla}_\\a v^\\a + \\dot{n}_\\a v^\\a + \\dot{v}_\\a v^\\a \\ .\n\\eea\n\n\\paragraph{The energy-momentum tensor.}\n\nIn the geometrical optics approximation, light propagation\nis kinematical, and independent of the laws which determine\nthe evolution of the geometry.\nHowever, we prefer to replace the Einstein tensor\nwith the energy-momentum tensor which describes the\nmatter content, and to do that we assume that the geometry\nis related to the matter by the Einstein equation,\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{Einstein}\n  G_{\\a\\b} &=& 8 \\piG_{\\mathrm{N}} T_{\\a\\b} \\ ,\n\\eea\n\n\\noindent where $G_{\\a\\b}$ is the Einstein tensor, $G_{\\mathrm{N}}$\nis Newton's constant, and $T_{\\a\\b}$ is the energy-momentum tensor.\n\nWithout loss of generality, the energy-momentum tensor can\nbe decomposed with respect to $n^\\a$ as\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{emdecn}\n  T_{\\a\\b} = \\rho^{(n)} n_\\a n_\\b + p^{(n)} h_{\\a\\b} + 2 q^{(n)}_{(\\a} n_{\\b)} + \\pi^{(n)}_{\\a\\b} \\ ,\n\\eea\n\n\\noindent where $\\rho^{(n)}\\equiv n^\\a n^\\b T_{\\a\\b}$ is the energy density,\n$p^{(n)}\\equiv\\frac{1}{3} h^{\\a\\b} T_{\\a\\b}$ is the pressure,\n$q^{(n)}_\\a\\equiv -h_\\a^{\\ \\b} n^\\c T_{\\b\\c}$ is the energy\nflux and\n$\\pi^{(n)}_{\\a\\b}\\equiv h_{\\a}^{\\ \\c} h_{\\b}^{\\ \\d} T_{\\c\\d} - \\frac{1}{3} h_{\\a\\b} h^{\\c\\d} T_{\\c\\d}=T_{\\langle\\a\\b\\rangle}$\nis the anisotropic stress.\nBoth $q^{(n)}_\\a$ and $\\pi^{(n)}_{\\a\\b}$ are spatial in\nthe sense that $q^{(n)}_\\a n^\\a=0, \\pi^{(n)}_{\\a\\b} n^\\b=0$.\nThe quantities measured by the observers\nare given by the decomposition with respect to $u^\\a$,\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{emdecu}\n  T_{\\a\\b} = \\rho^{(u)} u_\\a u_\\b + p^{(u)} h^{(u)}_{\\a\\b} + 2 q^{(u)}_{(\\a} u_{\\b)} + \\pi^{(u)}_{\\a\\b} \\ ,\n\\eea\n\n\\noindent where $\\rho^{(u)}, p^{(u)}, q^{(u)}_\\a$ and $\\pi^{(u)}_{\\a\\b}$\nare defined analogously to the $n^\\a$ frame quantities.\nLocally, dust is defined as matter for which\n$p^{(u)}$, $q^{(u)}_\\a$ and $\\pi^{(u)}_{\\a\\b}$ are zero;\nit then follows from the equations of motion that $A^\\a$\nis also zero.\nIn the $u^\\a$ frame, the non-dust terms have a clear physical\ninterpretation in terms of what the observers measure.\nSuch terms can arise from the properties of matter\n(it may be that the matter cannot be treated as dust in any frame)\nand from the fact that an ideal fluid looks non-ideal to a\nnon-comoving observer. We discuss treating the matter\napproximately as dust in \\sec{sec:evo}.\n\nWe could equally take \\re{emdecn} and \\re{emdecu}\nas decompositions of the Einstein tensor rather than the\nenergy-momentum tensor.\nWe use assumed symmetry properties of \\re{emdecn}\nsuch as the absence of preferred directions over large\ndistances, and these could be equally phrased in terms of the\ngeometry expressed in $G_{\\a\\b}$.\nHowever, $T_{\\a\\b}$ is more transparent because it can be\nunderstood in terms of a matter model.\n\n\\subsection{Photon energy and redshift}\n\n\\paragraph{The photon momentum.}\n\nWe want to relate quantities integrated along null\ngeodesics to average quantities which characterise\nthe spatial geometry.\nWe use assumptions about the symmetry properties\nof the spacetime, so averages are most meaningfully\ndiscussed in terms of quantities on $\\mathcal{N}$\nand the vector $n^\\a$.\nIn contrast, the observable redshift and light\ndeflection are defined by the observer velocity $u^\\a$.\n(The angular diameter distance and the null shear\nscalar are independent of the velocity field \\cite{Sachs:1961}.)\n\nIn the geometrical optics approximation light travels on null geodesics\n\\cite{Misner:1973} (page 570), \\cite{Schneider:1992} (page 93).\nWe do not consider caustics, which are not expected to be\nimportant for typical light rays in cosmology (though see \\cite{caustic}).\nFor treatment of the CMB in the covariant formalism, see\n\\cite{Maartens:1998, Dunsby:1997, Zibin:2008a}.\nThe null geodesic tangent vector\nis given by the gradient of the phase of the wave,\nidentified with the photon momentum, and denoted by $k^\\a$.\nIt satisfies $k_\\a k^\\a=0$ and $k^\\a \\nabla_\\a k^\\b=0$.\nThe redshift plus one is proportional to the energy\nmeasured by the observer, $1+z\\propto E^{(u)}$, which in turn is\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{Eu}\n  E^{(u)} = - u_\\a k^\\a \\ .\n\\eea\n\nThe photon momentum can be decomposed into an\namplitude and the direction, and the direction\ncan be split into components parallel and orthogonal to $u^\\a$,\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{kdecu}\n  k^\\a = E^{(u)} ( u^\\a + r^\\a ) \\ ,\n\\eea\n\n\\noindent with $u_\\a r^\\a=0$, $r_\\a r^\\a=1$.\n\nBecause the vector $n^\\a$ is adapted to the\nsymmetry of the spacetime, it is more convenient\nto calculate quantities in the $n^\\a$ frame and then transform\nto the $u^\\a$ frame.\nThe decomposition of $k^\\a$ with respect to $n^\\a$ reads\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{kdecn}\n  k^\\a = E^{(n)} ( n^\\a + e^\\a ) \\ ,\n\\eea\n\n\\noindent with $E^{(n)} \\equiv - n_\\a k^\\a, n_\\a e^\\a=0$, $e_\\a e^\\a=1$.\nThe quantities $E^{(n)}$ and $e^\\a$ do not have a\nstraightforward observational interpretation, unlike $E^{(u)}$ and $r^\\a$.\n\nThe observed energy $E^{(u)}$ is given in terms of $E^{(n)}$ by\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{EuEn}\n  E^{(u)} &=& \\gamma ( 1 - v_\\a e^\\a ) E^{(n)} \\ ,\n\\eea\n\n\\noindent and the observed direction $r^\\a$ is related to $e^\\a$ by\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{re}\n  r^\\a &=& \\frac{1}{\\gamma ( 1 - v_\\b e^\\b )} ( n^\\a + e^\\a ) - \\gamma ( n^\\a + v^\\a ) \\nonumber \\\\\n  &\\simeq& ( 1 + v_\\b e^\\b ) e^\\a + v_\\b e^\\b n^\\a - v^\\a \\ .\n\\eea\n\n\\noindent The inverse relation is\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{er}\n  e^\\a &=& \\frac{1}{ \\gamma + v_\\b r^\\b } ( u^\\a + r^\\a ) - \\gamma^{-1} u^\\a + v^\\a \\nonumber \\\\\n  &\\simeq& ( 1 - v_\\b r^\\b ) r^\\a - v_\\b r^\\b u^\\a + v^\\a \\ .\n\\eea\n\n\\paragraph{Statistical homogeneity and isotropy.}\n\nWe obtain the evolution of $E^{(n)}$ by operating with the derivative\nalong the null geodesic, $\\patl{}\\equiv k^\\a\\nabla_\\a$.\nDenoting $\\pate{} \\equiv (n^\\a + e^\\a) \\partial_\\a$ and using\n\\re{gradn} and \\re{kdecn}, we have\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{Eder}\n  E^{(n)} \\pate{E^{(n)}} &=& k^\\b \\nabla_\\b E^{(n)} \\nonumber \\\\\n  &=& - k^\\a k^\\b \\nabla_\\b n_\\a \\nonumber \\\\\n  &=& - {E^{(n)}}^2 \\left( \\frac{1}{3} \\theta + \\dot{n}_\\a e^\\a + \\sigma_{\\a\\b} e^\\a e^\\b \\right) \\ ,\n\\eea\n\n\\noindent which integrates into\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{Eint}\n  E^{(n)}(\\eta) &=& E^{(n)}(\\eta_0) \\exp\\left( \\int_{\\eta}^{\\eta_0} \\mathrm{d} \\eta \\left[ \\frac{1}{3} \\theta + \\dot{n}_\\a e^\\a + \\sigma_{\\a\\b} e^\\a e^\\b \\right] \\right) \\nonumber \\\\\n  &=& E^{(n)}(t_0,\\bi{x}_0) \\exp\\left( \\int_{t}^{t_0} \\mathrm{d} t \\Gamma \\left[ \\frac{1}{3} \\theta + \\dot{n}_\\a e^\\a + \\sigma_{\\a\\b} e^\\a e^\\b \\right] \\right) \\nonumber \\\\\n  &\\approx& E^{(n)}(t_0,\\bi{x}_0) \\exp\\left( \\int_{t}^{t_0}\\mathrm{d} t \\frac{1}{3} \\av{ \\Gamma \\theta } \\right) \\ ,\n\\eea\n\n\\noindent where the integral is along the null geodesic and\nthe subscript $0$ refers to the observer's position and time.\nOn the second line we have taken the time $t$ as the integration\nvariable; the spatial coordinates $\\bi{x}$ on $\\mathcal{N}$ are understood\nas functions of $t$ on the null geodesic.\nWe have then taken into account\nthat if there are no preferred directions in the\ngeometry of $\\mathcal{N}$ over long distances, and the\ndirection $e^\\a$ changes only little or\nevolves much more slowly than the distribution\nof the geometry (we discuss this in \\sec{sec:def}),\nthe dominant contribution is given by the average\nexpansion rate. (We use the symbol $\\approx$\nto indicate dropping terms which are suppressed due\nto statistical homogeneity and isotropy, in contrast\nto $\\simeq$, which indicates dropping terms which\nare small because $v\\ll1$.)\nThe argument for this is the following \\cite{Rasanen:2008b}.\nIf $\\Gamma\\dot{n}^\\a$ has no preferred orientation,\nit points equally in the directions along\nand opposite to $e^\\a$, so its contribution vanishes.\nSimilarly, $\\Gamma\\sigma_{\\a\\b}$ contributes only via\nits trace, which is zero.\nThe term $\\frac{1}{3}\\Gamma\\theta$ then gives the dominant contribution.\nUnder the assumption that the timescale for the evolution of\nthe distribution of the geometry is much larger than the time\nit takes for light to cross the homogeneity scale, the\nintegral is dominated by the average value of $\\Gamma\\theta$,\nas the contributions of the variation around the average cancel.\n(See \\sec{sec:av} for details of the averaging.)\nIn reality, there is some evolution of the quantities\nalong the null geodesic, and the cancellations are not\nperfect, so the contributions of $\\Gamma\\dot{n}_\\a e^\\a$,\n$\\Gamma\\sigma_{\\a\\b} e^\\a e^\\b$ and of the variation of\n$\\Gamma\\theta$ are only suppressed instead of zero.\n\nThe observed energy $E^{(u)}$ is, using \\re{EuEn},\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n  E^{(u)} &\\approx& E^{(n)}(t_0,\\bi{x}_0) \\gamma ( 1 - v_\\a e^\\a ) \\exp\\left( \\int_{t}^{t_0}\\mathrm{d} t \\frac{1}{3} \\av{ \\Gamma \\theta } \\right) \\ ,\n\\eea\n\n\\noindent and the redshift $1+z=E^{(u)}(\\eta)\/E^{(u)}(\\eta_0)$ is\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{z}\n  1+z &\\approx& \\frac{ \\gamma ( 1 - v_\\a e^\\a ) }{ \\gamma_0 ( 1 - v_\\a e^\\a )|_0 } \\exp\\left( \\int_{t}^{t_0}\\mathrm{d} t \\frac{1}{3} \\av{ \\Gamma \\theta } \\right) \\ .\n\\eea\n\n\\noindent Expressing $e^\\a$ in terms of the observed direction\n$r^\\a$ with \\re{er}, it is transparent that there are\nlarge observed anisotropies in the redshift of isotropically\ndistributed sources unless $v$ is small or constant or there\nis a conspiracy of cancellations.\nConversely, if $v$ is small, the anisotropy is small,\neven though the variations in the geometry can be large.\nIn particular, the near-isotropy of the CMB does\nnot imply that the universe would be nearly FRW\n\\cite{Rasanen:2008b, Rasanen:2009}.\nAssuming $v\\ll1$, the correction due to $v$ reduces to \n$v_\\a e^\\a|_0-v_\\a e^\\a\\simeq v_\\a r^\\a|_0-v_\\a r^\\a$.\nThe first term is the dipole due to the motion of the observer\nwith respect to the frame of statistical homogeneity and isotropy,\nand the second, which can have arbitrary angular dependence,\nis the corresponding term at the source. These are in addition\nto the usual dipole due to the difference between the velocity\nof the observer and the source.\nAs long as the difference between $u^\\a$ and $n^\\a$ is small,\nthe difference in the redshift between the two frames is small,\neven though the expansion rates $\\theta$ and $\\theta^{(u)}$\ncan be very different, as the gradient of $v^\\a$ can be\nlarge even when $v$ is small.\n\n\\paragraph{The local environment.}\n\nWhen we argue for the cancellation of terms other than\n$\\av{\\Gamma\\theta}$ in the integral \\re{Eint} due to symmetry,\nthis only applies to propagation over distances longer than the\nhomogeneity scale, and deviations due to the local environment\nare not accounted for.\nFor example, in linearly perturbed FRW spacetimes, the shear term\n$\\sigma_{\\a\\b} e^\\a e^\\b$ contains the usual local dipole,\nwhich we have neglected.\nTo be consistent in our approximation of concentrating on\npropagation over long distances and neglecting the\neffect of the local environments near the source and the observer,\nwe should approximate $1 - e^\\a v_\\a + v_\\a e^\\a|_0 \\simeq 1$.\nIn the real universe, this approximation seems to hold well.\nThe velocity difference between the CMB frame and our\nrest frame is of the order $10^{-3}$, and the rest frame\nof local large-scale structures is also near the CMB frame \\cite{dipole}.\nThe effect of the local environment is likely to be small\nas long as structures are small compared to the\ndistance the light travels and the observer is not in a\nspecial location \\cite{Rasanen:2008a}.\nThis is true for the structures which are known to exist\nand which are expected in usual models of structure formation,\nbut may not be valid for speculative large spherical structures,\noften described with the Lema\\^{\\i}tre-Tolman-Bondi (LTB)\nmodel \\cite{LTB, February:2009}.\n\nFor the CMB anisotropies, the corrections\ndue to the local environment and the deviations around the mean\ncannot be neglected. They are important\nfor the low multipoles, as in the Integrated Sachs-Wolfe\neffect and the Rees-Sciama effect, and could be\nrelated \\cite{asymmodels} to observed violations of\nstatistical isotropy of the CMB \\cite{asymobs}.\nFormalism for the CMB in the case when the geometry is\nnot perturbatively near FRW has been developed in \\cite{Maartens:1998}.\n\n\\paragraph{The mean redshift and the scale factor.}\n\nThe redshift characterises a single geodesic (or more accurately,\ntwo points and two frames along a single geodesic), so its spatial\naverage is not well defined. However, it is useful to introduce the \n``mean redshift'' $\\bar{z}$ by\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n  1 + \\bar{z} \\equiv \\exp\\left( \\int_{t}^{t_0}\\mathrm{d} t \\frac{1}{3} \\av{ \\Gamma \\theta } \\right) \\ .\n\\eea\n\n\\noindent The physical interpretation of $1+\\bar{z}$ is that\nif we take any two points on $\\mathcal{N}(t)$ and $\\mathcal{N}(t_0)$ which are\nconnected by a null geodesic (or several), the redshift\nalong the null geodesic(s) is $\\bar{z}$ plus small corrections\n(assuming that the rest frames of the source and the observer\nare close to the frame of statistical homogeneity and isotropy).\nFrom the arguments above and the observational fact that the CMB\ndeviations from isotropy and from the blackbody shape of the\nspectrum are small \\cite{Fixsen:1996} we know that temperature\ndifferences between different spatial locations are small,\nand the mean value of the redshift gives the dominant contribution.\n\nWe define the scale factor $a$ as (setting $a(t_0)=1$)\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{a}\n  a(t) &\\equiv& (1 + \\bar{z})^{-1} = \\exp\\left( - \\int_{t}^{t_0}\\mathrm{d} t \\frac{1}{3} \\av{ \\Gamma \\theta } \\right) \\ .\n\\eea\n\n\\noindent The quantity $\\theta$ gives the change of rate of the\nlocal volume element with respect to the proper time $s$, so\n$\\Gamma\\theta$ gives the rate of change with respect to $t$.\nTherefore $a(t)^3$ is proportional to the\nvolume of $\\mathcal{N}$: the mean redshift is determined by the\nchange of the overall volume of space.\n\nThe change of the redshift of a given source with time,\ncalled redshift drift \\cite{Sandage:1962}, has been suggested\nas a test of the FRW metric and LTB\nmodels \\cite{Uzan:2008, Quartin:2009}. Essentially, the change of\nredshift with time tests the relationship $1+z=a(t)^{-1}$ between\nthe redshift and the scale factor, with the scale factor\nassociated with an average expansion rate. In the present case,\nunlike in LTB models, the relationship between\nthe mean redshift and the average expansion rate is the\nsame as in FRW models. (See \\sec{sec:avexp} for discussion\nof the average expansion rate.)\nHowever, because redshift drift is a small effect,\nthe variations around the mean would have to be considered\ncarefully to make a prediction.\n\n\\subsection{Deflection} \\label{sec:def}\n\n\\paragraph{Picard's proof.}\n\nIn deriving \\re{Eint} it was assumed that the\nspatial direction of the null geodesic does not change\nrapidly along the geodesic.\nThe direction $e^\\a$ is the direction of the null geodesic\nprojected on $\\mathcal{N}$, so it enters into the arguments\nabout cancellation due to symmetry.\nThe direction $r^\\a$ in turn describes the apparent position\nof the source as seen by the observer, so it is an observable,\nand the change in $r^\\a$ is called the deflection.\nAs we do not have information about the 'original'\nposition of the source, i.e. the position in the hypothetical\nsituation that the spacetime would be flat along the photon path,\nthe deflection can only be measured statistically, unless the\napparent position changes on the timescale of the observation,\nas in microlensing.\n\nLet us look at the change of $e^\\a$ and $r^\\a$ along the\nnull geodesic. As with the redshift, it is simpler to\nconsider $e^\\a$ and then relate it to $r^\\a$.\nIt is convenient to choose a non-coordinate basis and introduce\ntetrads adapted to the 1+3 decomposition\n\\cite{Ellis:1967, Ellis:1998c, vanElst:1996}.\nWe denote the tetrad basis by ${t^\\a_{\\ A}}$, with\n$\\eta^{AB} t_{\\a A} t_{\\b B}=g_{\\a\\b}, t^\\a_{\\ A} t_{\\a B}=\\eta_{AB}$\nas usual, where $\\eta_{AB}=\\mathrm{diag}(-1,1,1,1)$.\nWe use capital Latin letters to denote components in the tetrad\nbasis, e.g. $e^A\\equiv e^\\a t_{\\a}^{\\ A}$.\n\nThe change of $e^A$ along the null geodesic is given by\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{eeq1}\n  {E^{(n)}}^{-1} \\patl{e^A} &=& ( n^B + e^B ) \\nabla_B e^A \\nonumber \\\\\n  &=& n^A \\left( \\frac{1}{3} \\theta + \\dot{n}_B e^B + \\sigma_{BC} e^B e^C \\right) - \\dot{n}^{A} \\nonumber \\\\\n  && + e^A ( \\dot{n}_B e^B + \\sigma_{BC} e^B e^C ) - \\sigma^A_{\\ \\, B} e^B \\ .\n\\eea\n\n\\noindent where we have on the second line inserted\n$e^A= {E^{(n)}}^{-1} k^A - n^A$ inside the covariant derivative\nand used \\re{gradn} and \\re{Eder}.\nWe specialise the choice of basis by taking $t_\\a^{\\ 0}=n_\\a$,\nso that $n^A=\\delta^{A0}, n_A=-\\delta_{A0}$.\n(In a coordinate basis, this choice is not\npossible in general \\cite{Ehlers:1961, Ellis:1971}.)\nThen $e^A$ is zero for $A=0$, while for the\nspatial components (which we denote by small Latin letters from\nthe middle of the alphabet, $e^i\\equiv h^i_{\\ A} e^A$)\nwe obtain, using the definition of the covariant derivative,\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{grade}\n  {E^{(n)}}^{-1} \\patl{e^i} &=& ( n^B + e^B ) \\nabla_B e^i \\nonumber \\\\\n  &=& \\pate{e^i} + ( n^B + e^B ) \\Gamma^i_{\\ CB} e^C \\nonumber \\\\\n  &=& \\pate{e^i} + a^i + \\Omega^i_{\\ j} e^j - a_j e^j e^i - \\epsilon^{i}_{\\ j k} N^j_{\\ l} e^k e^l \\ ,\n\\eea\n\n\\noindent where the connection components $\\Gamma^A_{\\ BC}$\nhave been expressed in terms of the decomposition \\re{gradn}\nof $\\nabla_B n_A$ as well as an object $a_i$, a symmetric object $N_{ij}$\nand an antisymmetric object $\\Omega_{ij}$; see\n\\cite{vanElst:1996, Ellis:1998c} for details\\footnote{In the\nnotation of \\cite{vanElst:1996, Ellis:1998c},\n$\\Omega_{ij}\\equiv\\epsilon_{ijk}\\Omega^k$.}.\nTogether with the 9 degrees of freedom\nin $\\{ \\theta, \\dot{n}^i, \\sigma^i_{\\ j} \\}$, the 12 degrees\nof freedom in $\\{ a^i, N^i_{\\ j}, \\Omega^i_{\\ j} \\}$ completely\ncharacterise the spacetime geometry.\n\nPutting \\re{eeq1} and \\re{grade} together, we have a system\nof ordinary differential equations for the components $e^i$\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{eeq}\n  \\partial_\\eta e^i &=& - \\dot{n}^i - a^i - ( \\sigma^i_{\\ j} + \\Omega^i_{\\ j} ) e^j + ( \\dot{n}_j e^j + a_j e^j + \\sigma_{jk} e^j e^k ) e^i + \\epsilon^{i}_{\\ j k} N^j_{\\ l} e^k e^l \\nonumber \\\\\n  &\\equiv& f^i(\\eta, e^j) \\ .\n\\eea\n\n\\noindent If the geometry is statistically homogeneous and\nisotropic and its distribution evolves slowly, the change in\n$e^i$ remains small due to the lack of preferred directions in\n$\\{\\dot{n}^i, a^i, \\sigma^i_{\\ j}, N^i_{\\ j}, \\Omega^i_{\\ j} \\}$.\nThis can be expressed more formally as follows.\nAccording to Picard's theorem, the system of equations \\re{eeq}\nhas a unique solution given by an iteration\n(see e.g. \\cite{Duff:1966}, page 19). Let us define\n$e^i_{(N+1)}(\\eta)\\equiv e^i_{(0)} + \\int_{\\eta_0}^\\eta \\mathrm{d}\\eta' f^i [\\eta',e^i_{(N)}(\\eta')]$,\nwith $e^i_{(0)}\\equiv e^i(\\eta_0)$. The solution\nto \\re{eeq} is given by the $N\\rightarrow\\infty$ limit.\nAt first step, we have\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{esol1}\n  e^i_{(1)} &=& e^i_{(0)} - \\int_{\\eta_0}^\\eta \\mathrm{d}\\eta' ( \\dot{n}^i + a^i ) -  e_{(0)}^j \\int_{\\eta_0}^\\eta \\mathrm{d}\\eta'( \\sigma^i_{\\ j} + \\Omega^i_{\\ j} ) + e_{(0)}^i e_{(0)}^j \\int_{\\eta_0}^\\eta \\mathrm{d}\\eta' ( \\dot{n}_j + a_j ) \\nonumber \\\\\n  && + e^i_{(0)} e_{(0)}^j e_{(0)}^k \\int_{\\eta_0}^\\eta \\mathrm{d}\\eta' \\sigma_{jk}  + e_{(0)}^k e_{(0)}^l \\int_{\\eta_0}^\\eta \\mathrm{d}\\eta' \\epsilon^{i}_{\\  j k} N^j_{\\ l} \\ .\n\\eea\n\n\\noindent As with \\re{Eint}, we could write $\\mathrm{d}\\eta=\\Gamma\\mathrm{d} t$,\nwith the understanding that the spatial coordinates are functions\nof $t$ along the null geodesic.\nIf $\\dot{n}^i$ and $a^i$ have no preferred direction\nand evolve slowly, their integral vanishes,\nprovided the distance over which the integral is taken\nis longer than the homogeneity scale.\nIn practice, the cancellation is not perfect, because\nthere is evolution and statistical fluctuations,\nso the integral is simply suppressed.\nSimilarly, all diagonal components of any tensor\nshould contribute almost equally to the integrals\nand the contribution from non-diagonal components should\nbe suppressed.\nBecause $\\sigma^i_{\\ j}, \\Omega^i_{\\ j}$\nand $\\epsilon^{i}_{\\ j k} N^j_{\\ l}$ are traceless, their\ncontributions are suppressed.\nOne can repeat the argument at every iteration to conclude\nthat the solution $e^i$ is $e^i_{(0)}$ plus a small deviation.\n(For the linear term in \\re{eeq} this is obvious,\nas it simply gives the $\\eta$-ordered exponential of the\nintegral of $\\sigma^i_{\\ j} + \\Omega^i_{\\ j}$.)\nThe change in the observed position of the source\n$r^A$ is then also small as long as $v$ is small, as\nwe see from \\re{re}.\nThis qualitative understanding should be made more\nrigorous, and the amplitude and the distribution of\nthe deflection should be evaluated.\n\n\\subsection{Null shear and angular diameter distance} \\label{sec:da}\n\n\\paragraph{The null shear.}\n\nThe distortion of the size and shape of the source\nimage are described by the null expansion rate $\\tilde{\\theta}$\n(or equivalently the angular diameter distance $D_A$)\nand the null shear tensor $\\tilde{\\sigma}_{\\a\\b}$.\nTo find these quantities, we need\nto introduce a tensor $\\tilde{h}_{\\a\\b}$ which projects onto\na two-space orthogonal to the null geodesic, $\\tilde{h}_{\\a\\b} k^\\b=0$.\nAnalogously to \\re{gradn} and \\re{gradu}, the covariant\nderivative of $k^\\a$ can be decomposed as follows:\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{kgrad}\n  \\nabla_\\b k_\\a &=& \\tilde{\\theta}_{\\a\\b} \\nonumber \\\\\n  &=& \\frac{1}{2} \\tilde{h}_{\\a\\b} \\tilde{\\theta} + \\tilde{\\sigma}_{\\a\\b} + k_{(\\a} P_{\\b)} \\ ,\n\\eea\n\n\\noindent where the trace\n$\\tilde{\\theta}=\\tilde{h}^{\\a}_{\\ \\b} \\nabla_\\a k^\\b=\\nabla_\\a k^\\a$\nis the expansion rate of the area of the null geodesic bundle,\n$\\tilde{\\sigma}_{\\a\\b}= \\tilde{h}_{\\a}^{\\ \\d} \\tilde{h}_{\\b}^{\\ \\c} \\nabla_\\c k_\\d - \\frac{1}{2} \\tilde{h}_{\\a\\b} \\tilde{\\theta}$\nis the null shear and $P_\\a$ is a vector which depends on\nthe choice of $\\tilde{h}_{\\a\\b}$ and plays no role in what follows. \nWe have $\\tilde{\\sigma}_{\\a\\b} k^\\b=0$, $P_\\a k^\\a=0$.\nThe null geodesic vorticity is zero,\nbecause $k^\\a$ is a gradient.\nThe null shear scalar is defined as\n$\\tilde{\\sigma}^2\\equiv\\frac{1}{2}\\tilde{\\sigma}_{AB}\\tilde{\\sigma}^{AB}$.\nThe area expansion rate $\\tilde{\\theta}$ is related to the\nangular diameter distance by (see e.g. \\cite{Schneider:1992, Sasaki:1993})\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{DA}\n  D_A \\propto \\exp \\left( {\\frac{1}{2} \\int \\mathrm{d}\\l \\tilde{\\theta}} \\right) \\ .\n\\eea\n\nThe angular diameter distance $D_A$ and the shear amplitude\n$\\tilde{\\sigma}_{\\a\\b}\\tilde{\\sigma}^{\\a\\b}$ do not depend on $\\tilde{h}_{\\a\\b}$.\nThey also do not depend on the observer velocity, so it is\nenough to look at the $n^\\a$ frame without having to transform\nto the $u^\\a$ frame at the end.\nIt is again convenient to use tetrads instead of sticking to\na coordinate basis. We choose $\\tilde{h}_{AB}$ to be orthogonal\nto both $n^A$ and $e^A$,\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{htt}\n  \\tilde{h}_{AB} &=& g_{AB} + n_{A} n_{B} - e_{A} e_{B} \\nonumber \\\\\n  &=& g_{AB} - {E^{(n)}}^{-2} k_{A} k_{B} + 2 {E^{(n)}}^{-1} k_{(A} n_{B)} \\ ,\n\\eea\n\n\\noindent We proceed with $\\tilde{\\sigma}_{AB}$ the same way\nas we did with $e^A$ in \\sec{sec:def}. Taking the derivative\nalong the null geodesic and projecting, we obtain\n(see e.g. \\cite{Schneider:1992, Sasaki:1993})\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{shear1}\n  \\tilde{h}_{(A}^{\\ \\ C} \\tilde{h}_{B)}^{\\ \\ D} \\patl{\\tilde{\\sigma}_{CD}} = - \\tilde{\\theta} \\tilde{\\sigma}_{AB} + C_{AB} \\ ,\n\\eea\n\n\\noindent where \n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n  C_{AB} &\\equiv& k^M k^N \\tilde{h}_{(A}^{\\ \\ C} \\tilde{h}_{B)}^{\\ \\ D} C_{MCND} \\nonumber \\\\\n  &=& 2 {E^{(n)}}^2 \\tilde{h}_{(A}^{\\ \\ C} \\tilde{h}_{B)}^{\\ \\ D} \\left( E_{CD} + \\frac{1}{2} \\tilde{h}_{CD} e^M e^N E_{MN} - \\tilde\\epsilon_{CM} H^{M} _{\\ \\ D} \\right) \\,\n\\eea\n\n\\noindent where $C_{AB}$ has been expressed in terms\nof the electric and magnetic components of the Weyl tensor,\n$E_{AB} \\equiv C_{ACBD} n^C n^D=C_{A0B0}$,\n$H_{AB} \\equiv \\frac{1}{2} \\epsilon_A^{\\ \\ CD} C_{CDBE} n^E=\\frac{1}{2} \\epsilon_A^{\\ \\ CD} C_{CDB0}$, and $\\tilde\\epsilon_{AB}\\equiv\\epsilon_{ABC} e^C$.\n\nOn the other hand, from the definition of the covariant derivative\nwe obtain (see \\cite{Ellis:1998c, vanElst:1996} for the\ndecomposition of $\\Gamma^A_{\\ BC}$)\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{shear2}\n  {E^{(n)}}^{-1} \\tilde{h}_{(A}^{\\ \\ C} \\tilde{h}_{B)}^{\\ \\ D} \\patl{\\tilde{\\sigma}_{CD}} &=& \\pate \\tilde{\\sigma}_{AB} - 2 \\tilde{h}_{(A}^{\\ \\ C} \\tilde{h}_{B)}^{\\ \\ D} ( n^E + e^E ) \\Gamma^F_{\\ C E} \\tilde{\\sigma}_{FD} \\nonumber \\\\\n  &=& \\pate \\tilde{\\sigma}_{AB} + \\left( 2 \\Omega_{ij} + 2 \\tilde\\epsilon_{k[i} N_{j]}^{\\ k} + \\epsilon_{ijk} N^{k}_{\\ l} e^l \\right) \\tilde{h}_{(A}^{\\ \\ \\, i} \\tilde{\\sigma}_{B)}^{\\ \\ \\, j} \\ .\n\\eea\n\n\\noindent Note that in a tetrad basis contracting with $\\tilde{h}_{AB}$\ncommutes with $\\pate{}$, but not with $\\patl{}$.\nWe denote indices on the space orthogonal to $n^A$\nand $e^A$ with small Latin letters from the beginning of the\nalphabet, $e^a\\equiv \\tilde{h}^a_{\\ B} e^B$, and specialise the choice\nof basis by taking $t_{\\a}^{\\ 3}=e_\\a$, so that $e^A=\\delta^{A3}, e_A=\\delta_{A3}$.\nPutting together \\re{shear1} and \\re{shear2}, we have\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{patsigma2}\n  \\pate \\tilde{\\sigma}_{ab} &=&  - {E^{(n)}}^{-1} \\tilde{\\theta} \\tilde{\\sigma}_{ab} + {E^{(n)}}^{-1} C_{ab} + ( 2 \\Omega_{c(a} +  2 \\tilde\\epsilon_{d[c} N^{d}_{\\ (a]} + \\tilde\\epsilon_{c(a} N^{3}_{\\ 3} ) \\tilde{\\sigma}_{b)}^{\\ \\, c} \\ .\n\\eea\n\nLike equation \\re{eeq} for the components of the deflection,\n\\re{patsigma2} is an ordinary differential equation for the two\nindependent components of $\\tilde{\\sigma}_{ab}$. However, we cannot\nstraightforwardly apply Picard's theorem. First, \\re{patsigma2}\ncontains the unknown $\\tilde{\\theta}$.\nEven if we include the equation \\re{Raynull} given below\nto obtain a closed first order system of equations, Picard's theorem does\nnot apply, because it assumes that the variables remain bounded,\nwhereas the initial condition for the area expansion rate is\n$\\tilde{\\theta}(\\eta_0)=-\\infty$ at the observer.\n\nNevertheless, the reasoning about cancellations due the lack\nof preferred directions still holds, because the solution\ndepends on the source term $C_{ab}$ only via an integral.\nThis is transparent with the change of variable\n$\\tilde{\\sigma}_{ab}\\equiv\\tilde\\Sigma_{ab}+\\int\\mathrm{d}\\eta {E^{(n)}}^{-1} C_{ab}$.\nWe now argue as before that due to statistical homogeneity\nand isotropy $C_{ab}$ contributes dominantly via its trace,\nwhich is zero, so $\\tilde{\\sigma}_{ab}\\approx\\tilde\\Sigma_{ab}$.\nThis eliminates the source term in \\re{patsigma2}.\nGiven the initial condition $\\tilde\\Sigma_{ab}(\\eta_0)=0$,\nwe obtain $\\tilde{\\sigma}_{ab}\\approx\\tilde\\Sigma_{ab}\\approx0$.\nAs with the deflection, this argumentation needs to\nbe made more rigorous, and the amplitude of the small\nshear that is generated should be calculated.\n\n\\paragraph{The angular diameter distance.}\n\nApplying the derivative $\\patl{}$ to $\\tilde{\\theta}$, we\nobtain the evolution equation for the area expansion rate\n(see e.g. \\cite{Schneider:1992, Sasaki:1993})\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{Raynull}\n  \\patl\\tilde{\\theta} &=& - G_{AB} k^A k^B - 2 \\tilde{\\sigma}^2 - \\frac{1}{2} \\tilde{\\theta}^2 \\nonumber \\\\\n  &=& - 8\\piG_{\\mathrm{N}} T_{AB} k^A k^B - 2 \\tilde{\\sigma}^2 - \\frac{1}{2} \\tilde{\\theta}^2  \\nonumber \\\\\n  &=& - 8\\piG_{\\mathrm{N}} \\big( \\rho^{(n)} + p^{(n)} - 2 q^{(n)}_A e^A + \\pi^{(n)}_{AB} e^A e^B \\big) {E^{(n)}}^2 - 2 \\tilde{\\sigma}^2 - \\frac{1}{2} \\tilde{\\theta}^2 \\ ,\n\\eea\n\n\\noindent where we have used the Einstein equation\n\\re{Einstein} and applied the decomposition \\re{emdecn}\nof the energy-momentum tensor.\nAs discussed in \\sec{sec:geom}, we could equally\nregard \\re{emdecn} as the decomposition of the\nEinstein tensor (or, in the present context, the\nRicci tensor, as the trace does not contribute to \\re{Raynull}).\n\nUsing \\re{DA}, we obtain from \\re{Raynull} the evolution\nequation for the angular diameter distance:\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{DAeq}\n  \\frac{\\mathrm{d}^2 D_A}{\\mathrm{d}\\l^2} &=& - \\left[ 4 \\piG_{\\mathrm{N}} \\big( \\rho^{(n)} + p^{(n)} - 2 q^{(n)}_A e^A + \\pi^{(n)}_{AB} e^A e^B \\big) {E^{(n)}}^2 + \\tilde{\\sigma}^2 \\right] D_A \\ .\n\\eea\n\nThe solution again depends on the source functions only via\nan integral. This is transparent with the change of variable\n$\\tilde{\\theta}\\equiv\\tilde\\Theta - 8\\piG_{\\mathrm{N}} \\int\\mathrm{d}\\l ( \\rho^{(n)} + p^{(n)} - 2 q^{(n)}_A e^A + \\pi^{(n)}_{AB} e^A e^B ) {E^{(n)}}^2$.\nWe can write\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{muint}\n  && \\int\\mathrm{d}\\l ( \\rho^{(n)} + p^{(n)} - 2 q^{(n)}_A e^A + \\pi^{(n)}_{AB} e^A e^B ) {E^{(n)}}^2 \\nonumber \\\\\n  && \\approx E^{(n)}(\\eta_0) \\int\\mathrm{d} t a^{-1} \\av{ \\Gamma ( \\rho^{(n)} + p^{(n)} ) } \\nonumber \\\\\n  && = E^{(n)}(\\eta_0) \\int\\mathrm{d} t a^{-1} \\av{ \\Gamma ( \\rho^{(u)} + p^{(u)} )+ \\frac{4}{3} \\Gamma F } \\ ,\n\\eea\n\n\\noindent where \n$F\\equiv v^2 (\\rho^{(u)} + p^{(u)}) + 2 \\gamma^{-1} q^{(u)}_A v^A + \\pi^{(u)}_{AB} v^A v^B\\simeq 2 q^{(u)}_A v^A $.\nOn the second line we have taken into account\n$\\mathrm{d}\\l={E^{(n)}}^{-1}\\mathrm{d}\\eta$ and the approximate scaling\n$E^{(n)}\\approx E^{(n)}(\\eta_0) a^{-1}$.\nWe have also again applied the reasoning that statistical\nhomogeneity and isotropy together with slow evolution implies\nthat the contributions of $q^{(n)}_A e^A$ and $\\pi^{(n)}_{AB} e^A e^B$\nare suppressed, and that the dominant\ncontribution of $\\rho^{(n)} + p^{(n)}$ comes from the average,\nFinally, we have written $\\rho^{(n)} + p^{(n)}$ in terms of $u^\\a$\nframe quantities using \\re{h}, \\re{n}, \\re{emdecn} and \\re{emdecu}.\nDropping the null shear, the solution $\\tilde{\\theta}$ to \\re{Raynull}\ndepends only on the quantity \\re{muint} and\n$\\l=\\int\\mathrm{d}\\eta {E^{(n)}}^{-1} \\approx E^{(n)}(\\eta_0)^{-1}\\int\\mathrm{d} t a\\av{\\Gamma}$,\nwhere we have assumed that $\\Gamma$ has a statistically\nhomogeneous and isotropic distribution and varies slowly, so\nthat the integral is dominated by the average value.\nBecause both \\re{muint} and $\\l$ depend approximately only\non $t$ (and $E^{(n)}(\\eta_0)$), so does $\\tilde{\\theta}$.\nWriting \\re{Raynull} as an integral equation, dropping\nsubdominant parts which depend on position, taking the\ntime derivative and expressing the equation in terms\nof the angular diameter distance, we obtain\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{DAbareq}\n  H \\partial_{\\bar{z}} \\left[ \\av{\\Gamma}^{-1} (1+\\bar{z})^2 H \\partial_{\\bar{z}} \\bar{D}_A \\right] &=& - 4\\piG_{\\mathrm{N}} \\av{ \\Gamma (\\rho^{(u)} + p^{(u)}) + \\frac{4}{3} \\Gamma F } \\bar{D}_A \\nonumber \\\\\n  &\\simeq& - 4\\piG_{\\mathrm{N}} \\av{ \\Gamma (\\rho^{(u)} + p^{(u)}) + \\frac{8}{3} \\Gamma q^{(u)}_A v^A } \\bar{D}_A \\ ,\n\\eea\n\n\\noindent where the notation $\\bar{D}_A(t)$ refers to the dominant\npart of the angular diameter distance with the corrections to the\nmean dropped, and we have used the relation\n$\\partial_t\\bar{D}_A=-(1+\\bar{z}) H \\partial_{\\bar{z}}\\bar{D}_A$, with $H\\equiv\\pat_t{a}\/a$.\nThe quantity $\\bar{D}_A$ has a similar physical interpretation\nas $\\bar{z}$: if we take any two points on $\\mathcal{N}(t)$ and $\\mathcal{N}(t_0)$\nconnected by a null geodesic (or several of them), the angular\ndiameter distance along the geodesic(s) is $\\bar{D}_A$\nplus small corrections.\nAs noted in \\cite{Rasanen:2008b}, while $\\bar{D}_A(\\bar{z})$\nis well-defined, there does not exist a function $D_A(z)$ even along\na single geodesic, because the redshift is in general\nnot monotonic along the null geodesic\\footnote{In\n\\cite{Rasanen:2008b} it was incorrectly claimed that when\nthe factor in the parenthesis on the right-hand side of\n\\re{DAeq} is positive, $D_A$ would be monotonic, because\nthe initial condition (at the observer) for $\\patl{D_A}$\nis negative. However, because $\\l$ decreases\nalong the null geodesic away from the observer, this\nonly implies that $\\patl{D_A}$ has at most one zero,\nso $D_A$ is separately monotonic on at most two sections\nof the null geodesic, as is well known from FRW spacetimes.\nIn the present case, the sign of the factor in the parenthesis\nis not determined, and the number of zeros of $\\patl{D_A}$ is not limited.}.\n\nApart from $\\Gamma$ and $F$, the equation \\re{DAbareq}\nfor the mean distance in terms of $\\bar{z}$ is the same as in\nFRW spacetimes.\nHowever, the relation between $\\av{\\rho^{(u)} + p^{(u)}}$ and\n$\\bar{z}$ is different than in the FRW case, as we discuss\nin the next section (see also \\cite{Rasanen:2008b}).\nAs with the redshift and the\ndeflection, it is important to make the arguments\nabout the null shear and the angular diameter distance\nmore rigorous by evaluating the variation around the\nmean. Observationally, the angular diameter distance\nis known not to vary much with direction \\cite{peakvar}.\n\n\\section{The average expansion rate and the scale factor} \\label{sec:av}\n\n\\subsection{The average expansion rate} \\label{sec:avexp}\n\n\\paragraph{Defining the average.}\n\nWe have started with light propagation, which involves\nthe observed quantities directly, and have been led to\nconsider averages. The average of a scalar $f$ on $\\mathcal{N}$ is\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{av}\n  \\av{f}(t) \\equiv \\frac{ \\int \\epsilon f }{ \\int \\epsilon } \\ ,\n\\eea\n\n\\noindent where $t$ is constant on $\\mathcal{N}$.\nRecall that in general, $t$ is not a proper time.\nIn particular, it is neither the proper time\nassociated with $n^\\a$ nor the proper time measured\nby the observers. The commutation rule between averaging\nand taking a derivative with respect to $t$ is\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{comm}\n  \\partial_t\\av{f} = \\av{\\partial_t f} + \\av{\\Gamma\\theta f} - \\av{f} \\av{\\Gamma\\theta}\n\\eea\n\nThe scale factor $a$ was defined in \\re{a} to give the mean\nredshift. From the definition it follows that $a^3$ is\nproportional to the volume of $\\mathcal{N}$. The average expansion\nrate of interest is\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{H}\n  3 \\H &=& \\av{\\Gamma\\theta} \\nonumber \\\\\n  &=& \\av{ \\Gamma \\gamma^{-1} \\theta^{(u)} - \\hat{\\nabla}_\\a ( \\Gamma v^\\a ) - \\Gamma \\gamma^2 (\\dot{v}_\\a v^\\a + v^\\a v^\\b \\hat{\\nabla}_\\a v_\\b) } \\nonumber \\\\\n  &\\simeq& \\av{ \\Gamma \\theta^{(u)} - \\hat{\\nabla}_\\a ( \\Gamma v^\\a ) - \\Gamma \\dot{v}_\\a v^\\a } \\ ,\n\\eea\n\n\\noindent where we have used \\re{reltheta} and the relation\n$\\dot{n}_\\a=\\Gamma^{-1}\\hat{\\nabla}_\\a\\Gamma$.\nIn the irrotational ideal fluid case, the scale factor\nwas originally defined as the volume of the hypersurface\northogonal to $u^\\a$, or equivalently by using the\naverage of $\\theta^{(u)}$, with the lapse function\nincluded \\cite{Buchert:1999, Buchert:2001}.\nWe start from light propagation, and while $u^\\a$\nis the relevant velocity field for the redshift, the\nsymmetry of the spacetime selects $\\Gamma \\nabla_\\a n^\\a=\\Gamma\\theta$\nas the relevant local expansion rate.\nLocally $\\theta$ can be very different from $\\theta^{(u)}$, even for\nsmall $v$, because the derivatives of $v^\\a$ can be large.\nHowever, using Gauss' theorem \\cite{Wald:1984} (page 433)\nthe total derivative in \\re{H} can be converted into a surface\nintegral which describes the flux of $\\Gamma v^\\a$ through the boundary.\nIf the distribution is statistically homogeneous and isotropic,\nthere should be an equal flux through the surface in both\ndirections, so the integral vanishes (up to statistical\nfluctuations).\nTherefore, the difference between the average quantities\n$\\av{\\Gamma\\theta}$ and $\\av{\\Gamma\\theta^{(u)}}$ is suppressed\nby $v$.\n\n\\subsection{Evolution equations for the scale factor} \\label{sec:evo}\n\n\\paragraph{The average equations.}\n\nWe now write down the evolution equations for\nthe scale factor $a$, or equivalently for the\naverage expansion rate.\nThese generalise the Buchert equations derived\nfor irrotational dust \\cite{Buchert:1999}.\nIn \\cite{Buchert:2001} the average equations were\nwritten down in the irrotational ideal fluid case\nusing the ADM formalism, assuming that the\naveraging hypersurface is orthogonal to the fluid flow\\footnote{These\naverage equations, apart from the conservation law\nof the energy-momentum tensor, were written down in the context of\ngeneral irrotational matter content in \\cite{Behrend}.\nNote that the perturbative calculation in \\cite{Behrend} is incorrect,\nbecause the averages of both first order terms and intrinsic second order\nterms taken in the perturbed spacetime are neglected in comparison\nwith the averages of squares of first order terms. In fact,\nall these terms are of the same order, and the distinction between\nthem is gauge-dependent \\cite{Kolb:2004}.}.\nIn \\cite{Larena:2009} the average equations were derived\n(also in the ADM formalism) for an ideal fluid\nincluding rotation, taking the expansion rate to be\n$h^{\\a\\b}\\nabla_\\b u_\\a=\\theta^{(u)}+n_\\a \\dot u^\\a$,\nand keeping the hypersurface of averaging arbitrary.\n(The formalism was applied to second order perturbation\ntheory in \\cite{Clarkson:2009a}, with the hypersurface\nfixed by the condition $H_{\\a\\b}=0$.\nAveraging in second order perturbation theory was also\nconsidered in \\cite{Brown:2009}, with different hypersurfaces\nfixed by coordinate conditions.)\nIn \\cite{Gasperini:2009}, the average equations were\nderived for an ideal fluid in the covariant formalism,\nwith an arbitrary averaging hypersurface.\nWe consider general matter content, an arbitrary\nhypersurface of averaging and use the covariant formalism.\n\nCombining the Einstein equation \\re{Einstein}\nwith the Bianchi and Ricci identities for $n^\\a$,\nthe evolution equations can be conveniently written in\nterms of the decompositions \\re{gradn} and \\re{emdecn}\nand the electric and magnetic components of the Weyl tensor\n\\cite{Ellis:1998c, Clarkson:2000, Tsagas:2007}.\nWe are only interested in the three scalar equations\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n   \\label{Rayloc} \\dot{\\theta} + \\frac{1}{3} \\theta^2 &=& - 4\\piG_{\\mathrm{N}} ( \\rho^{(n)} + 3 p^{(n)} ) - 2 \\sigma^2 + \\dot{n}_\\a \\dot{n}^\\a + \\hat{\\nabla}_\\a \\dot{n}^\\a \\\\\n  \\label{Hamloc} \\frac{1}{3} \\theta^2 &=& 8 \\pi G_{\\mathrm{N}} \\rho^{(n)} - \\frac{1}{2} {^{(3)}R} + \\sigma^2 \\\\\n  \\label{consloc} \\dot{\\rho}^{(n)} + \\theta ( \\rho^{(n)} + p^{(n)} ) &=& - \\hat{\\nabla}_\\a q^{(n)\\a} - 2 \\dot{n}_\\a q^{(n)\\a} - \\sigma_{\\a\\b} \\pi^{(n)\\a\\b} \\ .\n\\eea\n\n\\noindent where ${^{(3)}R}$ is the spatial curvature of $\\mathcal{N}$.\nIf $\\mathcal{N}$ is a hypersurface of constant proper time,\nthen $\\dot{n}^\\a=0$, and the equations differ from\nthe irrotational dust case only by the pressure term in the\nRaychaudhuri equation \\re{Rayloc} and the non-dust terms in\nthe conservation law \\re{consloc}. In terms of the Hamiltonian\nconstraint \\re{Hamloc}, the only difference is the different\nevolution of the energy density. We keep $\\dot{n}^\\a$ arbitrary.\nChanging to derivatives with respect to $t$, averaging,\napplying the commutation rule \\re{comm} and using\nthe relations $\\av{\\Gamma\\theta}=3\\pat_t{a}\/a$ and\n$\\dot{n}_\\a=\\Gamma^{-1}\\hat{\\nabla}_\\a\\Gamma$, we obtain\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n  \\label{Rayavn} && 3 \\frac{\\pat_t^2{a}}{a} = - 4\\piG_{\\mathrm{N}} \\av{\\rho^{(n)} + 3 p^{(n)}} + \\av{\\dot{n}_\\a \\dot{n}^\\a} + \\av{\\hat{\\nabla}_\\a \\dot{n}^{\\a}} + \\mathcal{Q} \\nonumber \\\\\n  && \\quad + \\av{ \\frac{1}{3} (\\Gamma^2-1) \\theta^2 + (1-\\Gamma^{-2})\\Gamma \\partial_t\\theta + \\theta \\partial_t\\Gamma } \\\\\n  \\label{Hamavn} && 3 \\frac{(\\pat_t a)^2}{a^2} = 8 \\pi G_{\\mathrm{N}} \\av{\\rho^{(n)}} - \\frac{1}{2} \\av{{^{(3)}R}} - \\frac{1}{2} \\mathcal{Q} + \\frac{1}{3} \\av{(\\Gamma^2-1)\\theta^2} \\\\\n  \\label{consavn} && \\partial_t \\av{\\rho^{(n)}} + 3 \\H \\av{\\rho^{(n)} + p^{(n)}} = - \\av{\\Gamma\\theta p^{(n)}} + \\av{\\Gamma\\theta} \\av{p^{(n)}} - \\av{\\Gamma \\dot{n}_\\a q^{(n)\\a} + \\Gamma \\sigma_{\\a\\b} \\pi^{(n)\\a\\b}} \\nonumber \\\\\n  && \\quad - \\av{\\hat{\\nabla}_\\a (\\Gamma q^{(n)\\a})} \\ ,\n\\eea\n\n\\noindent where\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n  \\mathcal{Q} &\\equiv& \\frac{2}{3}\\left( \\av{{\\Gamma\\theta}^2} - \\av{\\Gamma\\theta}^2 \\right) - 2 \\av{\\sigma^2} \\ .\n\\eea\n\n\\noindent In \\cite{Buchert:2001, Behrend, Larena:2009},\nthe equivalent of the factors of $\\Gamma$ were put inside the\naverages of the terms which appear on the right-hand side of\n\\re{Rayloc} and \\re{Hamloc}, rather than the left-hand side.\nInserting \\re{Rayloc} into the last term of \\re{Rayavn} and\n\\re{Hamloc} into the last term of \\re{Hamavn} would recover\nthat form of the equations.\nHowever, the present convention keeps the $\\hat{\\nabla}_\\a\\dot{n}^\\a$\nterm and the last term of \\re{consavn} as total derivatives,\nwhich we can neglect.\n\nBecause the backreaction variable $\\mathcal{Q}$ is a statistical\nquantity which characterises the distribution of the\nspatial geometry, it is appropriate to give it in\nterms of $\\theta$ and $\\sigma$, which are related to $n^\\a$.\nHowever, it seems more appropriate to express the energy-momentum tensor\nin terms of the decomposition with respect to $u^\\a$ from the point of\nview of estimating the magnitude of the different terms.\nUsing \\re{h}, \\re{n}, \\re{emdecn} and \\re{emdecu}, we obtain\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n  \\label{Rayavu} && 3 \\frac{\\pat_t^2{a}}{a} = - 4 \\piG_{\\mathrm{N}} \\av{\\rho^{(u)} + 3 p^{(u)}} + \\av{\\dot{n}_\\a \\dot{n}^\\a} + \\av{\\hat{\\nabla}_\\a \\dot{n}^{\\a}} + \\mathcal{Q} \\nonumber \\\\\n  && \\quad + \\av{ \\frac{1}{3} (\\Gamma^2-1) \\theta^2 + (1-\\Gamma^{-2})\\Gamma \\partial_t\\theta + \\theta \\partial_t\\Gamma } - 8 \\piG_{\\mathrm{N}} \\av{F} \\\\\n  \\label{Hamavu} && 3 \\frac{(\\pat_t a)^2}{a^2} = 8 \\pi G_{\\mathrm{N}} \\av{\\rho^{(u)}} - \\frac{1}{2} \\av{{^{(3)}R}} - \\frac{1}{2} \\mathcal{Q} + \\frac{1}{3} \\av{(\\Gamma^2-1)\\theta^2} + 8 \\piG_{\\mathrm{N}} \\av{F} \\\\\n  \\label{consavu} && \\partial_t \\av{\\rho^{(u)}} + 3 \\H \\av{\\rho^{(u)} + p^{(u)}} = - \\av{\\Gamma\\theta ( p^{(u)} + \\frac{1}{3} F ) } + \\av{\\Gamma\\theta} \\av{p^{(u)} + \\frac{1}{3} F} \\nonumber \\\\\n  && \\quad - \\av{\\gamma \\Gamma \\dot{n}_\\a q^{(u)\\a} + \\gamma^2 \\Gamma\n    (\\rho^{(u)}+p^{(u)}) \\dot{n}_\\a v^\\a + \\gamma \\Gamma \\dot{n}_\\a\n    q^{(u)}_\\b v^\\a v^\\b + \\Gamma \\dot{n}_\\a \\pi^{(u)\\a\\b} v_\\b } \\nonumber \\\\ \n  && \\quad - \\av{ \\Gamma \\sigma_{\\a\\b} \\pi^{(u)\\a\\b} + \\gamma^2 \\Gamma (\\rho^{(u)}+p^{(u)}) \\sigma_{\\a\\b} v^\\a v^\\b + 2 \\gamma \\Gamma \\sigma_{\\a\\b} q^{(u)\\a} v^\\b } \\nonumber \\\\\n  && \\quad - \\partial_t \\av{F} - 4 \\H \\av{F} - \\av{\\hat{\\nabla}_\\a (\\Gamma q^{(n)\\a})} \\ ,\n\\eea\n\n\\noindent with $F=v^2 (\\rho^{(u)} + p^{(u)}) + 2 \\gamma^{-1}\nq^{(u)}_\\a v^\\a + \\pi^{(u)}_{\\a\\b} v^\\a v^\\b$ as before.\nWe have not written $\\hat{\\nabla}_\\a (\\Gamma q^{(n)\\a})$ in terms of\n$u^\\a$ frame quantities, because it is suppressed due to\nstatistical homogeneity and isotropy, like $\\hat{\\nabla}_\\a \\dot{n}^\\a$.\nVorticity does not appear in the above equations, because\n$n^\\a$ is hypersurface-orthogonal by construction.\nWere we to decompose $\\nabla_\\b n_\\a$ with respect\nto the $u^\\a$ frame, the vorticity of $u^\\a$ would\n(to leading order in $v$) emerge from\n$\\hat{\\nabla}_\\a \\dot{n}^\\a$ and ${^{(3)}R}$ \\re{Rayloc} and \\re{Hamloc}.\n(For the definition of ${^{(3)}R}$ for velocity fields which are not\nhypersurface-orthogonal, see \\cite{Ellis:1990}.)\nIn particular, the leading order contribution of the $u^\\a$\nframe vorticity to the average Raychaudhuri equation \\re{Rayavn}\nvanishes due to statistical homogeneity and isotropy, because it\nis contained in the boundary term $\\av{\\hat{\\nabla}_\\a \\dot{n}^\\a}$.\n(In Newtonian gravity, the vorticity combines with $\\mathcal{Q}$ to give\na boundary term, so backreaction vanishes for periodic\nboundary conditions and for statistical homogeneity and\nisotropy \\cite{Buchert:1995}.)\n\nDropping the boundary terms $\\hat{\\nabla}_\\a (\\Gamma q^{(n)\\a})$ and\n$\\hat{\\nabla}_\\a \\dot{n}^\\a$ as well as all terms higher than first\norder in $v$, we have\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n  \\label{Rayapp} && 3 \\frac{\\pat_t^2{a}}{a} \\simeq - 4 \\piG_{\\mathrm{N}} \\av{\\rho^{(u)} + 3 p^{(u)}} + \\av{\\dot{n}_\\a \\dot{n}^\\a} + \\mathcal{Q} \\nonumber \\\\\n  && \\quad + \\av{ \\frac{1}{3} (\\Gamma^2-1) \\theta^2 + (1-\\Gamma^{-2})\\Gamma \\partial_t\\theta + \\theta \\partial_t\\Gamma } - 16 \\piG_{\\mathrm{N}} \\av{q^{(u)}_\\a v^\\a} \\\\\n  \\label{Hamapp} && 3 \\frac{(\\pat_t a)^2}{a^2} \\simeq 8 \\pi G_{\\mathrm{N}} \\av{\\rho^{(u)}} - \\frac{1}{2} \\av{{^{(3)}R}} - \\frac{1}{2} \\mathcal{Q} + \\frac{1}{3} \\av{(\\Gamma^2-1)\\theta^2} + 16 \\piG_{\\mathrm{N}} \\av{q^{(u)}_\\a v^\\a} \\\\\n  \\label{consapp} && \\partial_t \\av{\\rho^{(u)}} + 3 \\H \\av{\\rho^{(u)} + p^{(u)}} \\simeq - \\av{\\Gamma\\theta \\big( p^{(u)}+\\frac{2}{3}q^{(u)}_\\a v^\\a \\big)} + \\av{\\Gamma\\theta} \\av{ p^{(u)} + \\frac{2}{3} q^{(u)}_\\a v^\\a } \\nonumber \\\\\n  && \\quad - \\av{ \\Gamma \\dot{n}_\\a q^{(u)\\a} + \\Gamma (\\rho^{(u)}+p^{(u)}) \\dot{n}_\\a v^\\a + \\Gamma \\dot{n}_\\a v_\\b \\pi^{(u)\\a\\b} } - \\av{ \\Gamma\\sigma_{\\a\\b} \\pi^{(u)\\a\\b} + 2 \\Gamma \\sigma_{\\a\\b} q^{(u)\\a} v^{\\b} } \\nonumber \\\\\n  && \\quad - 2 \\partial_t \\av{q^{(u)}_\\a v^\\a} - 8 \\H \\av{q^{(u)}_\\a v^\\a} \\ .\n\\eea\n\n\\paragraph{The dust approximation.}\n\nOne reason for deriving the general equations\n\\re{Rayavu}--\\re{consavu} is to take into account\ndeviations from the approximation of treating\nthe matter as dust in the late universe.\nThe importance of the different terms depends\non the matter model, and cannot be determined from\ngeneral arguments. However, it is possible\nto say what would would be necessary for the\nnon-dust terms to have a significant effect.\nFor the $\\dot{n}^\\a$ terms to be important in \\re{Rayapp}--\\re{consapp},\n$\\dot{n}_\\a \\dot{n}^\\a$ would have to be of the order of the\nsquare of the expansion rate in a large fraction of space (contrary\nto what was argued in \\cite{Tsagas:2009}; see also \\cite{Coley}),\nor the contraction of the energy flux and $\\dot{n}^\\a$ would\nhave to be of the order of the product of the average energy\ndensity and the average expansion rate.\nIn order for the pressure or the anisotropic stress\nto be important, they would have to be on average of the same order\nof magnitude as the average energy density.\nFor the time dilation to be important, the spatially\nvarying part of $\\Gamma$ would have to be of order one\nin a fraction of space which is of order one.\nIf the matter content is a gas of Standard Model particles\nplus cold or warm dark matter, and structures evolve from\nsmall adiabatic perturbations with a nearly scale-invariant\nspectrum, it seems unlikely that any of these conditions would\nbe satisfied in the late universe when radiation pressure can be neglected.\n\nLet us assume that the matter is approximately dust in\nthe $u^\\a$ frame, i.e. that $p^{(u)}, q^{(u)\\a}, \\pi^{(u)}_{\\a\\b}$\nand $A^\\a$ are small\\footnote{From the equations\nof motion it follows that $A^\\a$ is zero if\n$p^{(u)}, q^{(u)\\a}$ and $\\pi^{(u)}_{\\a\\b}$ are zero, and $A^\\a$ is small if\n$p^{(u)}, q^{(u)\\a}, \\pi^{(u)}_{\\a\\b}$ as well as $\\hat{\\nabla}_\\a p^{(u)}$ and\n$\\hat{\\nabla}^\\b \\pi^{(u)}_{\\a\\b}$ are small.},\nand the deviation of $\\Gamma$ from unity (and the\ntime derivative of the deviation) is small,\n$\\Gamma\\equiv1-\\delta\\Gamma$, with $|\\delta\\Gamma|\\ll1$.\nWhen we drop all squares of small terms (whether\nthey are non-dust terms or $v^\\a$), the\nequations \\re{Rayapp}--\\re{consapp} simplify to\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n  \\label{Rayav} 3 \\frac{\\pat_t^2{a}}{a} &\\simeq& - 4 \\piG_{\\mathrm{N}} \\av{\\rho^{(u)} + 3 p^{(u)}} + \\mathcal{Q} \\nonumber \\\\\n  && + \\av{ \\dot{v}_\\a \\dot{v}^\\a + \\frac{2}{3} \\delta\\Gamma \\theta^2 + 2 \\delta\\Gamma\\partial_t\\theta + \\theta \\partial_t\\delta\\Gamma } \\\\\n  \\label{Hamav} 3 \\frac{(\\pat_t a)^2}{a^2} &\\simeq& 8 \\pi G_{\\mathrm{N}} \\av{\\rho^{(u)}} - \\frac{1}{2} \\av{{^{(3)}R}} - \\frac{1}{2} \\mathcal{Q} + \\frac{2}{3} \\av{ \\delta\\Gamma \\theta^2} \\\\\n  \\label{consav} \\partial_t \\av{\\rho^{(u)}} + 3 \\H \\av{\\rho^{(u)}} &\\simeq& - \\av{\\theta p^{(u)}} + \\av{ \\dot{v}_\\a q^{(u)\\a} + \\rho^{(u)} \\dot{v}_\\a v^\\a } - \\av{\\sigma^{(n)}_{\\a\\b} \\pi^{(u)\\a\\b} } \\ ,\n\\eea\n\n\\noindent where we have used the fact that\n$\\dot{n}^\\a\\simeq A^\\a-\\dot{v}^\\a$ plus corrections of order $v$.\nEquations \\re{Rayav}--\\re{consav} give the leading\ncorrections to the treatment of matter as irrotational dust,\ncompared to the original Buchert equations \\cite{Buchert:1999}.\nIn particular, they cover the case when the matter can be\nlocally treated as dust, but has rotation, and $n^\\a$ is orthogonal\nto the hypersurface of constant proper time of observers\ncomoving with the dust. Then $\\dot{v}^\\a$ is of order $v$,\nand we can choose $\\Gamma=1$, so the difference between\n$n^\\a$ and $u^\\a$ arises only from vorticity.\nWe see that vorticity alone has (to leading order in $v$)\nno effect on the averages, because the dominant contribution\ncomes from the total derivative $\\hat{\\nabla}_\\a \\dot{n}^\\a$. \nIt does change the relation of the scale factor $a$\nto the geometry, because $a$ will be defined with a\ndifferent vector field, but not the relation of $a$ to the redshift.\n\nFor irrotational dust, the Raychaudhuri equation \\re{Rayloc}\ncan be integrated as an inequality before averaging to obtain\nthe bound $Ht\\leq1$ \\cite{Wald:1984} (page 220),\n\\cite{Nakamura:1995, Rasanen:2005}.\nWhen rotation or non-dust terms are important,\nthere is no such local inequality.\nIndeed, having $\\dot{\\theta}+\\frac{1}{3}\\theta^2>0$ locally\nis required in order for collapsing regions to stabilise.\nFrom a physical point of view, we would still expect\nto recover $Ht\\leq1$ unless there is sustained acceleration\nin a significant fraction of space, but the conditions for this\nderived from \\re{Rayloc} involve combinations of spatial\naverages and integrals over time, and are not entirely transparent.\n\nNote that in order for the approximation of treating the matter\nas dust on average to hold, it is only necessary that the contribution of\nthe non-dust terms to the averages is smaller than that of the average\nenergy density. It is not required that the energy-momentum\ntensor of matter could be locally approximated as dust everywhere.\nIn fact, deviations from the irrotational dust behaviour\nare necessarily important on small scales.\nAs the local Raychaudhuri equation \\re{Rayloc}\nshows, in order to stabilise structures, a large\n$\\dot{n}^\\a$ or its gradient is needed.\nThis can correspond to $u^\\a$ frame vorticity\nas with rotating baryonic structures,\nor the acceleration can be generated by\nanisotropic stress (or a pressure gradient or energy flux)\nas in the case of dark matter \\cite{Pueblas:2008}.\nFor dark matter, the dust approximation is\nlocally invalid in structure formation due to\nmultistreaming \\cite{Buchert:2005}.\nNevertheless, as long as the volume occupied\nby regions where such terms are important is small, their\ncontribution to the average is not important.\nIn Newtonian calculations, this is certainly the case \\cite{Pueblas:2008}.\n\nApproximating the matter content as dust on average does not\nimply  viewing the matter as infinitesimal grains.\nFor example, the issue of what the ``particles''\nof the dust fluid are is sometimes raised, and whether one\ncan consistently ``renormalise'' the scale of the\ndescription as larger stable structures form\n\\cite{Rasanen:2006b, Rasanen:2008a}.\nHowever, the dust approximation is properly understood\nas the statement that when considered on large scales,\nthe energy density dominates over the pressure, the energy\nflux, the anisotropic stress, and their gradients.\n\nIt has been argued that because of gradients of\nspatial curvature, clocks in different regions\nrun at different rates, and that this effect is important\nfor cosmology but neglected in the dust approximation \\cite{Wiltshire}.\nAny such effects are accounted for in the present analysis,\nto the extent they are part of general relativity, and\nnot outside of the geometrical optics approximation.\nIf the $n^\\a$ frame is non-geodesic, different points on the\nhypersurface of statistical homogeneity and isotropy indeed\nhave different values of the $n^\\a$ frame proper time\n(though this cannot be understood as being due to spatial curvature\ngradients), which in turn is close to the proper time\nmeasured by the observers if $v$ is small.\nFor this time dilation to be significant, the acceleration\n$\\dot{n}^\\a$ would have to be of the order of the average\nexpansion rate in a significant fraction of space.\nThis in turn requires that either the motion of the observers\nis very non-geodesic (large $A^\\a$) or the acceleration\nbetween the two frames is significant (large $\\dot{v}^\\a$);\nthe latter possibility however has to contend with the fact\nthat the velocity between the frames cannot become large,\nas this would violate the small anisotropy observed in the CMB.\nIn order to generate such large accelerations, the non-dust terms\nin the energy-momentum tensor would have to be significant in a large\nfraction of space. This would likely have important cosmological\neffects apart from the time dilation.\nNote that it follows from statistical homogeneity and isotropy\nthat the spatial difference in the CMB temperature\nbetween different regions is small, in contrast to\nthe arguments made in \\cite{Wiltshire}.\nThis issue can be observationally probed with the\nblackbody shape of the CMB spectrum \\cite{Fixsen:1996, blackbody}.\n\n\\section{Discussion}\n\n\\subsection{Modelling issues}\n\n\\paragraph{Discreteness.}\n\nWhile we have kept the energy-momentum tensor generic,\nthe arguments about light propagation in \\sec{sec:light}\ncontain the implicit assumption that matter is so\nfinely distributed in space that it can be treated\nas a continuous distribution which light rays sample.\nThe redshift, the angular diameter distance and other quantities\nrelated to light propagation depend on the spacetime geometry\nonly via an integral along the null geodesic. If the matter\nconsists of discrete clumps whose size and number density is\nso small that a typical light ray will never encounter matter,\nthe energy-momentum tensor along the light ray is zero, regardless\nof the average energy density (or pressure or other components).\nFor example, the integrand in \\re{muint} vanishes identically,\nand our arguments about cancellation between regions of low\nand high density do not apply.\n\nThe approximation of discrete matter as a continuous fluid\nhas been studied from first principles for the dynamics of\nmatter in Newtonian gravity \\cite{discrete}.\nHowever, the effect on light propagation has been looked\nat mostly from the perspective of adding perturbations to\na FRW metric. With a statistically homogeneous and isotropic\ndistribution and small structures, it is then not surprising\nthat the deviation of the expansion rate from the FRW case\nis small \\cite{Rasanen:2008a}.\nOne exception is \\cite{Clifton}, where the effect of discreteness\nlight propagation was considered in a lattice model without any FRW\napproximation.\n\nIf the light travels in vacuum, and we assume statistical\nhomogeneity and isotropy, the mean angular diameter distance\nis given by \\re{DAbareq} with zero on the right-hand side.\nThe equation can be integrated to yield\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray} \\label{DAempty}\n  \\bar{D}_A = \\int_0^{\\bar{z}} \\mathrm{d} z \\frac{\\bar{\\Gamma(z)}}{(1+z)^2 \\bar{H}(z)} \\ ,\n\\eea\n\n\\noindent where $\\bar{H}$ and $\\bar{\\Gamma}$ are the mean expansion rate\nand the mean time dilation along the null geodesic,\nwhich in general do not coincide with the spatial averages.\nBecause the null geodesic samples only vacuum, the expansion rate along\nthe null geodesic is larger than the average\nexpansion rate (assuming that the matter satisfies the\nstrong energy condition). The equations \\re{Rayloc} and \\re{Hamloc}\nshow that if the acceleration $\\dot{n}^\\a$ and the shear\n$\\sigma_{\\a\\b}$ could be neglected, the expansion rate sampled\nby the light ray would be the same as in an empty FRW universe.\nIf the time dilation could be neglected, the angular diameter\ndistance would then correspond to the 'coasting universe'\n(or Minkowski space, for non-expanding regions).\nHowever, it is probably not reasonable to neglect\nthe shear $\\sigma_{\\a\\b}$. For example, the existence of both expanding\nand non-expanding regions means that there is a gradient\nin the expansion rate, which implies non-zero shear.\nEvaluating the expansion rate along a null geodesic,\nand the angular diameter distance, is thus reduced to the\nquestion of realistically modelling the shear scalar (and the\nacceleration $\\dot{n}^\\a$) along the geodesic.\n\nDiscussing light propagation in terms of null geodesics assumes\nthe validity of the geometrical optics approximation.\nGeometrical optics is in turn based on modelling\nlight as local plane waves, which requires the wavelength\nof the light to be much smaller than both the curvature scale\nand the scale over which the amplitude, wavelength and\npolarisation of the light change significantly. If the fraction of the\nvolume occupied by matter is so small that light rays\nnever come close to the matter particles, this is satisfied.\nHowever, if the light passes through small discrete\nregions where the energy-momentum tensor is non-zero,\nthe situation is very different from the geometrical optics limit.\n(We are here concerned only with gravitational\ninteractions, and are not taking into account gauge\ninteractions between photons and matter.)\nIn evaluating the validity of the continuous fluid\napproximation, the extension of the wave-packet of the\nmatter particles should therefore be taken into account.\nThe treatment of the photon waves should also be more\ndetailed, instead of simply treating their transverse\nwidth as zero (as in the null geodesic picture) or\ninfinite (as implicit in the plane wave approximation).\nThe effect of discreteness on light propagation is not\nobvious, and should be studied in a realistic model.\n\n\\paragraph{The relevance of averages.}\n\nThe equations \\re{Rayavn}--\\re{consavn} or\n\\re{Rayavu}--\\re{consavu} generalise\nthe average equations for the irrotational\ndust case derived by Buchert \\cite{Buchert:1999}\nto arbitrary matter content and rotation,\nand an arbitrary averaging hypersurface.\nThe scale factor has been defined to give the mean\nredshift in the case that the difference between the\nobserver frame and the frame of statistical homogeneity\nand isotropy is small, and we have assumed statistical\nhomogeneity and isotropy and slow evolution, which are\nrequired to have a meaningful notion of mean redshift.\nThe equations \\re{Rayavn}--\\re{consavn} or\n\\re{Rayavu}--\\re{consavu} are of course valid\nwithout any symmetry assumptions. However,\nthe quantity $a^3$ is of limited use in interpreting observations\nunless the space is statistically homogeneous and isotropic,\nso that the change in the total volume of the spatial hypersurface\nis the dominant effect for light propagation.\n\nThe system of equations \\re{Rayavn}--\\re{consavn}\nor \\re{Rayavu}--\\re{consavu} is derived from the\nscalar part of the full Einstein equation, which is\nnot closed, because it is coupled to the vector and tensor parts.\nAs the sum of two tensors at different spacetime\npoints is not a tensor, the average of a tensor (or vector)\nin curved spacetime is not well-defined, so it is sometimes\nsaid that the rest of the evolution equations cannot be averaged.\nHowever, we can write the evolution equations\nin terms of components, and average these.\nThe problem is not lack of covariance\\footnote{One can even average\nthe vector and tensor part of the equations covariantly\nby first projecting with a vector field such as $v^\\a$\nor $\\hat{\\nabla}_\\a\\theta$. The Einstein equation can be\nexpressed in full generality in scalar form\nby using a projection \\cite{Padmanabhan:2004}.},\nbut the fact that products of variables become\nindependent correlation terms, so the number of unknowns\nincreases when taking the average, and the set of average\nequations does not close.\n\nMethods for covariantly averaging tensors on curved spacetime\nhave been suggested, including the macroscopic gravity\nformalism \\cite{Zalaletdinov, Paranjape}, the Ricci flow \\cite{Carfora},\na statistical averaging formalism \\cite{Debbasch},\na procedure which relies on a specific choice of tetrads \\cite{Behrend:2008},\nand the proposal of \\cite{Korzynski:2009} which is more\na way to rewrite the tensors than average them.\nHowever, the relevant issue is not the mathematical definition\nof averages in some covariant formalism which is an extension\nof general relativity or in a statistical ensemble of spacetimes.\nRather, we want to determine the impact of structures in\nthe spacetime which actually describes the universe we\nobserve, with the dynamics determined by the Einstein equation\nand the local equations for light propagation.\nAverages are useful insofar as they provide an approximate\ndescription of observed quantities in this complex system.\nThe relevant averaging procedure, and the hypersurface\nof averaging, emerges from considering observations,\nand cannot be determined on abstract mathematical grounds. \n\nIt bears emphasising that taking the average on a different\nhypersurface would correspond to considering a different velocity\nfield, and this is a physical choice. This issue is\nseparate from the question of gauge-invariance, i.e. dependence of\nunphysical quantities on the chosen coordinate system.\nThe Buchert equations were originally derived using the ADM\nformalism \\cite{Buchert:1999}, where the distinction between\nchoice of velocity field and choice of coordinates is\nnot entirely transparent. However, the problem can be considered\ncompletely covariantly, without introducing coordinates\n\\cite{Tsagas:2007, Rasanen:2008a, Rasanen:2008b}.\nThe averages depend on the choice of the averaging hypersurface\n\\cite{Geshnizjani, Rasanen:2004}, but not on the coordinate\nsystem \\cite{Kolb:2004}\\footnote{In \\cite{Brown:2009},\nchoice of the averaging hypersurface and the coordinate system\nwas conflated. This mixes up defining a quantity of interest\nusing physical criteria and using different coordinates to\ndescribe the same physics.}.\nThe relevant velocity field is singled out as that of the\nobserver by the redshift, and the relevant averaging hypersurface\nis given by the symmetry properties of the spacetime.\n\n\\paragraph{Deviation from the FRW universe.}\n\nIf backreaction is important for the average\nexpansion rate, i.e. if $\\mathcal{Q}$ contributes significantly\nto \\re{Rayavn} and \\re{Rayavu}, there is no ``FRW background''\nthat would emerge on large scales \\cite{Rasanen:2006b, Rasanen:2008a}.\n(Note that $\\mathcal{Q}$ being small does not guarantee that the spacetime\ncan be described by the FRW metric.)\nThe FRW metric describes a universe that is exactly\nhomogeneous and isotropic, not a universe where there\nare large non-linearities with a statistically homogeneous\nand isotropic distribution.\n\nWhile the deviation of the average expansion rate from the\nFRW equations could be attributed to an effective matter\ncomponent in a FRW universe, this is not the case for\nother observables. For the shear scalar, this is obvious,\nbecause it is zero in FRW models, but generally positive.\nAs a less trivial example, the spatial curvature \nin the FRW case is fixed to be proportional to $a^{-2}$, while the\naverage spatial curvature in an inhomogeneous and\/or\nanisotropic space can evolve non-trivially \\cite{Rasanen:2007}.\nIn fact, if the\nmatter can be treated as dust, the effect of backreaction\nis encoded in the non-trivial evolution of the spatial curvature\n\\cite{Buchert:1999, Rasanen:2005, Rasanen:2006b}. (The\ndifference between backreaction in general relativity\nand Newtonian gravity can also be understood in terms\nof the spatial curvature \\cite{Rasanen:2008a}.)\nFor this reason, calling the effect of backreaction a\nchange of background as in\n\\cite{Kolb:2009, Clarkson:2009a, Clarkson:2009b}\nis misleading. A FRW model which reproduces the\naverage expansion rate of a clumpy model is better called\na ``fitting model'' or something similar. The metric associated\nwith it cannot be used to calculate quantities other than\nthe one specifically fitted for, and usual perturbation theory\naround it does not make sense.\n\nIn particular, if backreaction is important,\nthe relation between the expansion rate\nand the angular diameter distance given by \\re{DAbareq}\n(assuming that discreteness is not important) is\ndifferent from the FRW case \\cite{Rasanen:2008b}\\footnote{In contrast,\nthe relation $D_L=(1+z)^2 D_A$ for the luminosity distance $D_L$\nis universal \\cite{Ellis:1971}, \\cite{Schneider:1992} (page 111),\n\\cite{Etherington:1933}.}.\nEven though either the change of the expansion rate or the change\nof the distance due to backreaction can be reproduced in a FRW\nmodel by introducing extra sources of matter or changing the\nEinstein equation, it is not possible to do both\nat the same time, since FRW models cannot mimic\nthe correlation \\re{DAbareq}.\nIn a clumpy space, if the dust approximation holds, the distance\nis uniquely determined by the function $H(\\bar{z})$ and\nthe matter density today \\cite{Rasanen:2008b}.\n(Note that fitting the distance observations may not\nnecessarily require accelerating expansion, because the relation\nbetween $H$ and $D_A$ is different from the FRW case.)\nAnalogously, in a FRW universe with general matter content,\nthe distance is determined by $H(z)$ and the spatial curvature\ntoday \\cite{Clarkson:2007b}.\nIn LTB models, the relationship is different from\neither of these cases \\cite{February:2009}.\n\nThis prediction for both FRW models and backreaction\ncan be tested with independent observations of distance\nand the expansion rate \\cite{Avgoustidis:2009}.\nThe deviation of the backreaction case from the FRW\nconsistency condition is related to the\ndifference of the average expansion rate from the FRW\ncase with vacuum energy and dust \\cite{Rasanen:2008b}.\nThere are relatively good constraints on the distance scale as\na function of redshift from type Ia supernova observations\n\\cite{FRWexp}, but measurements of the expansion rate\nusing the ages of passively evolving galaxies are less\nprecise \\cite{ages}. The expansion rate at different\nredshifts also enters into the radial mode of the baryon\nacoustic oscillations, and a measurement\nwas reported in \\cite{BAO2} (see also\n\\cite{SylosLabini:2009, MiraldaEscude:2009}).\nWith better observations of the expansion rate, it\nwill be possible to more tightly test the statement that\nthe universe is described by a FRW metric,\nindependent of the possible existence of exotic matter\nor modified gravity \\cite{Clarkson:2007b, Shafieloo:2009}\\footnote{Note\nthat this is not a test of the Copernican principle.\nThe statement that our position in the universe is\nnot untypical is different from the statement that the\nmetric is FRW. In fact, the Copernican principle says\nnothing about the metric.}.\nSimilarly, backreaction can be tested without having a\nprediction for the average expansion rate.\nIn the case of backreaction, it helps that there are\nindependent observational constraints on the matter density today,\nwhile the only way to determine the spatial curvature of a\nFRW universe is to make independent measurements\nof the expansion rate and distance, and use the FRW relation\nbetween them. (In particular, the CMB has no model-independent\nsensitivity to the spatial curvature.)\n\n\\subsection{Conclusions} \\label{sec:disc}\n\n\\paragraph{Summary.}\n\nIn \\cite{Rasanen:2008b} it was argued that light\npropagation can be approximately described in terms\nof the overall geometry (meaning the average expansion rate\nand other average scalar quantities) in statistically homogeneous\nand isotropic dust universes where the distribution\nevolves slowly compared to the time it takes for\nlight to cross the homogeneity scale.\nThe calculation was incomplete because it\nwas simply assumed that the light deflection is small,\nand there was no result for the amplitude of the image shear.\nIt was also assumed that the matter is dust and irrotational,\nwhile it is known that such a description does not locally\nhold everywhere.\n\nWe have now considered general matter content,\nwith a general observer velocity and a general\nhypersurface of statistical homogeneity and isotropy,\nwith a slowly evolving distribution.\nFrom these assumptions about the spacetime symmetry,\nwe find that the propagation of typical light rays\nover distances longer than the homogeneity scale\ncan to leading order be treated in terms of a\nfew average quantities, as in the irrotational dust case.\nThe redshift is given by the average volume\nexpansion rate of the hypersurface of statistical\nhomogeneity and isotropy, assuming that the velocity difference\nbetween the frame of statistical homogeneity and\nisotropy and the observer frame is non-relativistic.\nThe relevant averaging hypersurface is selected by the\nsymmetry of the situation as the one of statistical\nhomogeneity and isotropy, while the relevant\nvelocity is that of the observers, because it\ngives the observed redshift.\nThe angular diameter distance is to leading order determined\nby the averages of the expansion rate, time dilation,\nenergy density and pressure.\nThe light deflection and the image shear are small.\n\nWe have also written down the generalisation of the\nBuchert equations \\cite{Buchert:1999} for the\nevolution of the average expansion rate to general\nmatter content and averaging hypersurface.\nProvided that the difference between frame\nof statistical homogeneity and isotropy and the\nobserver frame is small, and that pressure,\nenergy flux and anisotropic stress are not significant\nin a large fraction of space, we recover the Buchert\nequations for dust.\n\n\\paragraph{Outlook.}\n\nIf backreaction has a large effect on the average\nexpansion rate, the relation between\nthe expansion rate and the angular diameter distance\nis different from the FRW case, and the difference\ngrows with the deviation of the expansion rate\nfrom that of the FRW model with dust and vacuum energy.\nThis is a distinct prediction of backreaction, which\ncannot be mimicked by any FRW model \\cite{Rasanen:2008b}.\n\nOur arguments are qualitative, and \nshould be followed up by a more rigorous quantitative study.\nIn particular, evaluating the deviations from the mean is\nnecessary to study lensing and the low multipoles of the CMB.\nThe equations we have written in the covariant formalism\ncontain all general relativistic effects and are exact, except\nfor the geometrical optics approximation.\nThe study of light propagation in a general\nspacetime is reduced to the system of coupled ordinary differential\nequations \\re{Eder}, \\re{eeq}, \\re{patsigma2} and \\re{Raynull}.\nTo obtain a solution, we do not have to know\nthe global geometry, it is only necessary to specify\nthe distribution along the null geodesic. This approach was\nused in the perturbative case in \\cite{Kainulainen:2009}.\nThis formulation is well-suited to slowly evolving statistically\nhomogeneous and isotropic universes, where the solution is\nexpected to depend only on the statistical properties of the\ndistribution. In particular, because in the real universe the\ndistribution originates from small almost Gaussian fluctuations, its\nstatistics are even at late times determined by the\ninitial power spectrum, processed by gravity.\nThe effect of the discreteness of the matter content of the\nuniverse on light propagation should be clarified.\nFinally, even though the relation between the expansion rate\nand the distance is already a prediction which can be checked,\nit remains of central importance to derive the average expansion\nrate from the statistics of structure formation \\cite{Rasanen:2008a}\nin a reliable manner to allow easier and more comprehensive\ncomparison with observations.\n\n\\acknowledgments\n\nI thank Thomas Buchert, Timothy Clifton, Ruth Durrer, Pedro\nFerreira and David Wiltshire for discussions and disagreements, \nand Thomas Buchert also for comments about the manuscript,\nand the Helsinki Institute of Physics and the Tuorla Observatory\nfor hospitality.\n\nThis paper is dedicated to Heidi Hopeamets\\\"{a}.\\\\\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sect:1}\n\nLet $\\Omega$ and $D$ be bounded Lipschitz domains in $\\mathbb{R}^n$, $n\\geq 2$, such that $D\\Subset\\Omega$ and $\\Omega\\backslash\\overline{D}$ is connected. Let $\\eta\\in L^\\infty (D)$ be such that $\\eta(\\mathbf{x})>\\eta_0\\in\\mathbb{R}_+$, $\\mathbf{x}=(x_j)_{j=1}^n\\in \\Omega$, and $|\\eta(\\mathbf{x})-1|>0$, $\\mathbf{x}\\in\\mathrm{neigh}(\\partial D)\\cap D$, where $\\mathrm{neigh}(\\partial D)$ denotes an open neighbourhood of $\\partial D$. Set\n \\begin{equation}\\label{form}\n \\gamma(\\mathbf{x})=1+(\\eta-1)\\chi_D(\\mathbf{x}),\\: \\mathbf{x}\\in \\Omega.\n \\end{equation} \n Introduce \n \\[\n H_0^{-1\/2}(\\partial \\Omega):=\\{f\\in H^{-1\/2}(\\partial\\Omega); \\int_{\\partial\\Omega} f\\, ds=0\\},\n \\]\nand consider the following elliptic PDE problem for $u\\in H^1(\\Omega)$,\n\\begin{equation}\\label{EQ_Calderon}\n\\begin{cases}\n\\displaystyle{\\mathrm{div}(\\gamma\\nabla u)=0}\\qquad & \\mbox{in}\\ \\ \\Omega,\\medskip\\\\\n\\displaystyle{\\frac{\\partial u}{\\partial\\nu}=g\\in H_0^{-1\/2}(\\partial\\Omega)} & \\mbox{on}\\ \\ \\partial\\Omega,\n\\end{cases}\n\\end{equation} \nwhere $\\nu\\in\\mathbb{S}^{n-1}$ is the exterior unit normal vector to $\\partial\\Omega$.\nAssociated with \\eqref{EQ_Calderon}, we introduce the following Neumann-to-Dirichlet (DtN) map, $\\Lambda_\\gamma: H_0^{-1\/2}(\\partial\\Omega)\\mapsto H^{1\/2}(\\partial\\Omega)$, \n\\begin{equation}\\label{eq:NtD}\n\\Lambda_\\gamma(g)=u|_{\\partial\\Omega},\n\\end{equation}\nwhere $u\\in H^1(\\Omega)$ is the solution to \\eqref{EQ_Calderon}. For a given $g\\in H_0^{-1\/2}(\\partial\\Omega)$, we are concerned with the inverse inclusion problem of determining $D$ by the pair of boundary data $(g, \\Lambda_\\gamma(g))$, independent of $\\eta$. That is, \n\\begin{equation}\\label{eq:ic1}\n(g, \\Lambda_\\gamma(g))\\rightarrow \\partial D,\\quad \\mbox{independent of $\\eta$}. \n\\end{equation}\n\n\nThe inverse inclusion problem constitutes a particular set of sub-problems for the celebrated Calder\\'on's inverse problem. The Calder\\'on problem is concerned with the recovery of the conductivity of a body by the associated boundary electric current and flux measurements. Mathematically, it can formulated as determining $\\gamma$ by knowledge of $\\Lambda_\\gamma$ associated with the PDE system \\eqref{EQ_Calderon}, where $\\gamma\\in L^\\infty(\\Omega)$ is not necessarily of the specific form \\eqref{form}. Calder\\'on's inverse problem arises from industrial applications of practical importance including electrical impedance tomography (EIT) in medical imaging and geophysical exploration. The problem was first posed and studied by A. P. Calder\\'on \\cite{CALDERON2006} and has a profound impact on the mathematical developments of inverse problems. On the other hand, in many physical and engineering applications such as detecting defects in a composite medium, one is concerned more about recovering the geometric shape of an inhomogeneous inclusion embedded in a homogeneous conductive body. This naturally leads to the inverse inclusion problem \\eqref{eq:ic1}. In the practical point of view, many reconstruction methods have been developed in the literature for the inverse inclusion problem. They include the monotonicity-based method \\cite{HU}, the factorization method \\cite{GH}, the enclosure method \\cite{I}, and the method of using the generalized polarization tensors deduced from the layer potential techniques \\cite{ammari2004reconstruction}.\n\nIt is noted that knowing $\\Lambda_\\gamma$ is equivalent to knowing $(g, \\Lambda_\\gamma(g))\\in H_0^{-1\/2}(\\partial\\Omega)$ $\\times H^{1\/2}(\\partial\\Omega)$ for any $g\\in H_0^{-1\/2}(\\partial\\Omega)$, which corresponds to infinitely many boundary measurements in the practical scenario. In order to recover a generic $\\gamma(x)$, it is necessary to make use of infinitely \nmany boundary measurements. In fact, one can see that the cardinality of the unknown $\\gamma(x)$ is $n$, whereas the cardinality of the given data encoded into $\\Lambda_\\gamma$ is $2(n-1)$. Here, by cardinality, we mean the number of independent variables in a given quantity. In such a case, one has $2(n-1)\\geq n$ for $n\\geq 2$, which means that the inverse problem in recovering a generic $\\gamma(x)$ is at least formally determined (indeed, it is formally determined when $n=2$, and over-determined when $n\\geq 3$). Nevertheless, for the inverse inclusion problem \\eqref{eq:ic1}, the cardinality of the unknown $\\partial D$ is $n-1$, and for a fixed $g$, the cardinality of the boundary measurement data $(g, \\Lambda_\\gamma(g))$ is also $n-1$. That is, the inverse inclusion problem \\eqref{eq:ic1} is formally determined with a single measurement. Hence, it is unobjectionable to expect that one can establish the unique identifiability result for \\eqref{eq:ic1} in a generic scenario.  However, this problem constitutes a long-standing open problem in the literature. \n\nIn \\cite{beretta2019lipschitz,AMR,ammari2017identification,moi2,moi3}, the uniqueness and reconstruction issues for \\eqref{eq:ic1} were studied under multiple measurements. There are also several works for \\eqref{eq:ic1} by a single boundary measurement, provided that the inclusion $D$ is a-priori known to belong to certain specific geometrical classes. If $D$ is within the radial geometry, uniqueness and stability results were established in \\cite{fabes1999inverse,moi1,Kang_ball_R3} by a single measurement. In the case that $D$ is a convex polygon in $\\mathbb{R}^2$ or a convex polyhedron in $\\mathbb{R}^3$, the uniqueness results are respectively derived in \\cite{friedman1989uniqueness} and \\cite{barcelo1994inverse}. The reconstruction of an insulating curvilinear polygonal inclusion was considered in \\cite{CHL}. In a recent paper \\cite{StabilityPolygon}, we proved a logarithmic type stability in determining polygonal inclusions. In those studies mentioned above by a single measurement, it is a technical requirement that the content of the inclusion has to be uniform; that is, the conductivity $\\eta$ of the inclusion in \\eqref{eq:ic1} is a positive constant. Moreover, in all of the aforementioned literature except \\cite{StabilityPolygon}, full boundary measurements are required. Here, by full boundary measurement, we mean that the measurement dataset $(\\psi, \\Lambda_\\gamma(\\psi))$ is given over the whole boundary $\\partial\\Omega$. For comparison, the following partial boundary measurement was used in \\cite{StabilityPolygon}: $(\\psi, \\gamma\\partial_\\nu u|_{\\Gamma_0})$ with $\\mathrm{supp}(\\psi)\\subset\\Gamma_0$, where $\\psi=u|_{\\partial\\Omega}$ and $\\Gamma_0\\Subset\\partial\\Omega$ is a proper subset. We would like to mention the partial-data Calder\\'on problem constitutes another challenging topic in the field of inverse problems (cf. \\cite{CSU,IUY}). Nevertheless, it is pointed out that a mild condition was imposed for the study in \\cite{StabilityPolygon} which depends on the a-priori knowledge of the underlying inclusion as well as the corresponding boundary input. \n\nThe mathematical argument in \\cite{StabilityPolygon} is of a localized feature, which is based on carefully studying the singular behaviors (in the phase space) of the solution to the conductivity problem \\eqref{EQ_Calderon} around a corner point on the polygonal inclusion. In this paper, we show that the corner singularity in \\cite{StabilityPolygon} can be relaxed to be a certain high-curvature condition. Indeed, the corner singularity can be regarded as having an extrinsic curvature being infinity. Our argument in tackling the singular behaviors of the solution to \\eqref{EQ_Calderon} around an admissible high-curvature point on the boundary of the conductive inclusion is mainly motivated by a recent article \\cite{Liu_curve}. However, the study in \\cite{Liu_curve} mainly deals with high-curvatures occurring on the support of a parameter $q$, which is the coefficient for the lower-order term of an elliptic partial differential operator, namely $-\\Delta+q$. In the current study, the high-curvatures enter into the coefficient of the leading-order term, namely $\\gamma$ associated with $\\nabla(\\gamma\\nabla u)$. It is pointed out that in \\cite{StabilityPolygon}, quantitative stability estimates are established in determining polygonal inclusions in two dimensions, whereas in this paper, we are mainly concerned with the qualitative unique identifiability issue in any dimension $n\\geq 2$. Finally, we would also like to mention in passing some recent related works \\cite{ACL,B1,B2,BLa,BLb,BL17,BL172,BLX,CX,CDL,CDL2,LX} on characterizing the geometric singularities in the coefficients of certain partial differential operators and their implications to the related inverse inclusion problems. \n\nThe rest of the paper is organized as follows. In Section 2, we derive some auxiliary results. Section 3 is devoted to the main results for the inverse inclusion problem. In the Appendix, we present some discussion about a generic condition in the main theorem. \n\n\\section{Some auxiliary results}\n\nIn this section, we derive several auxiliary results that shall be of critical importance for establishing the unique determination results in Section 3. By following the treatment in \\cite{Liu_curve}, we first introduce an important geometric notion for our study. \n\n\\begin{definition}\\label{def:k}\nLet $K,L,M,\\delta$ be positive constants and $D$ be a bounded domain in $\\mathbb{R}^n$, $n\\geq 2$. A point $\\mathbf{p}\\in \\partial D$ is said to be an admissible $K$-curvature point with parameters $L,M,\\delta$ if the following conditions are fulfilled.\n\\begin{enumerate}\n\\item Up to a rigid motion, the point $\\mathbf{p}$ is the origin $\\mathbf{x}=\\mathbf{0}$ and $\\mathbf{e}_n:=(0,\\cdots,0,1)$ is the interior unit normal vector to $\\partial D$ at $\\mathbf{0}$.\n\\item Set $b=\\sqrt{M}\/K$ and $h=1\/K$. There is a $\\mathcal{C}^{2,1}$ function $w: B(\\mathbf{0},b)\\rightarrow \\mathbb{R}_+\\cup \\{0\\}$ with $B(\\mathbf{0},b)\\subset \\mathbb{R}^{n-1}$ such that if \n\\begin{equation}\nD_{b,h}=B(\\mathbf{0},b)\\times(-h,h)\\cap D,\n\\end{equation}\nthen\n\\begin{equation}\nD_{b,h}=\\{\\mathbf{x}\\in \\mathbb{R}^n; |\\mathbf{x}'|<b, -h<x_n<h, w(\\mathbf{x}')<x_n<h\\},\n\\end{equation}\nwhere $\\mathbf{x}=(\\mathbf{x}',x_n)$ with $\\mathbf{x}'\\in \\mathbb{R}^{n-1}$. Here and also in what follows, $B(\\mathbf{x}, b)$ signifies a ball centered at $\\mathbf{x}$ and of radius $b$. \n\\item The function $w$ in (2) satisfies\n\\begin{equation}\nw(\\mathbf{x}')=K|\\mathbf{x}'|^2+\\mathcal{O}(|\\mathbf{x}'|^3),\\qquad \\mathbf{x}'\\in B(\\mathbf{0}',b).\n\\end{equation}\n\\item $M\\geq 1$ and there are $0<K_-\\leq K\\leq K_+<\\infty$ such that \n\\[ K_-|\\mathbf{x}'|^2\\leq w(\\mathbf{x}')\\leq K_+|\\mathbf{x}'|^2,\\qquad |\\mathbf{x}'|<b, \\]\n\\[M^{-1}\\leq \\frac{K_\\pm}{K}\\leq M, \\qquad K_+-K_-\\leq LK^{1-\\delta}.\\]\n\\item The intersection $V=\\overline{D_{b,h}}\\cap \\mathbb{R}^{n-1}\\times\\{h\\}$ is a Lipschitz domain.\n\\end{enumerate}\n\\end{definition}\n\nIn what follows, we suppose that $\\partial D$ possesses an admissible $K$-curvature point $\\mathbf{p}$. We shall work within an Euclidean system of coordinates, which is transformed from the cardinal one after a rigid motion such that $\\mathbf{p}$ is the origin. We assume that in this coordinate system, the boundary $\\partial D$ in a neighborhood of $\\mathbf{p}$ can be represented by the equation $x_n=w(\\mathbf{x}')$, where $w(\\mathbf{x}')$ is the function in Definition~\\ref{def:k} satisfying $w(\\mathbf{0}')=0, \\nabla_{\\mathbf{x}'} w(\\mathbf{0}')=\\mathbf{0}'$. We denote by $U_{b,h}:=B(\\mathbf{0}',b)\\times (-h, h)$ the cylinder centered at $\\mathbf{p}$, and by $D_{b,h}:=U_{b,h}\\cap D$ the neighborhood of $\\mathbf{p}$, and by $\\partial S :=\\partial D_{b,h}\\setminus \\partial D$ the complementary part of its surface.\n\nWe next study the local behaviors of the solution $u$ to \\eqref{EQ_Calderon}. We recall that $\\eta$ in \\eqref{form} is a positive constant and $\\eta\\neq 1$. Nevertheless, we shall remark the case that $\\eta$ is a variable function in Section 3. It is straightforward to show that \\eqref{EQ_Calderon} is equivalent to the following transmission problem with $u_i:=u|_{D}$ and $u_e:=u|_{\\Omega\\backslash\\overline{D}}$: \n\\begin{equation}\\label{EQ2}\n\\begin{cases}\n     \\triangle u_e =0 & \\text{in}\\ \\ \\Omega\\setminus\\overline{D}, \\\\\n     \\triangle u_i =0 & \\text{in}\\ \\ D,\\\\\n     u_i=u_e & \\text{on}\\ \\ \\partial D,\\\\\n     \\eta\\partial_\\nu u_i=\\partial_\\nu u_e & \\text{on}\\ \\ \\partial D,\\\\\n     \\partial_\\nu u_e=g & \\text{on}\\ \\ \\partial \\Omega. \n\\end{cases}  \n\\end{equation}\nIn principle, we shall show that $u_e$ in \\eqref{EQ2} cannot be harmonically extended across the admissible $K$-curvature point $\\mathbf{p}\\in\\partial D$, provided $K$ is sufficiently large. It is easily seen that the harmonic extension across $\\mathbf{p}$ is equivalent to the analytic extension of $u_e$ across $\\mathbf{p}$. Then using this non-extension result together with an absurdity argument, we can establish the unique identifiability results for the inverse problem \\eqref{eq:ic1}. To that end, throughout the rest of this section, we assume on the contrary that the function $u_e$ admits a harmonic extension into $D_{b, h}$, which is still denoted by $u_e$. That is, when $u_e|_{D_{b,h}}$ is involved in what follows, it is actually referred to the harmonic extension mentioned above. Under such a situation, we shall derive several key properties of the solution $u\\in H^1(U_{b,h})$ locally around the point $\\mathbf{p}\\in\\partial D$, which shall be used for the absurdity argument in Section~\\ref{sect:3} for the inverse inclusion problem. \n\n\\begin{proposition}\\label{TIntId}\nLet $u_0\\in H^1_{loc}(U_{b,h})$ be harmonic in $U_{b,h}$. Then,\n\\begin{equation}\\label{IntId}\n(\\eta-1)\\int_{D_{b,h}}\\nabla u\\cdot \\nabla u_0 \\, d\\mathbf{x}=\\int_{\\partial S}(\\eta\\partial_\\nu u_i-\\partial_\\nu u_e)u_0-(u_i-u_e)\\partial_\\nu u_0\\, ds,\n\\end{equation}\nwhere $u=u_i\\chi_D+u_e\\chi_{\\Omega\\backslash\\overline{D}}\\in H^1(U_{b,h})$ is the solution to (\\ref{EQ2}). \n\\end{proposition}\n\\begin{proof}\nThe integral identity \\eqref{IntId} can be directly verified by using Green's formula and the transmission conditions across $\\partial D$ of $u$.\n\\end{proof}\n\\begin{proposition}\\label{prop:regularity}\nUnder the assumption that $u_e$ can be harmonically extended into $D_{b,h}$, one has $u_i \\in \\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})$ and $u_e\\in \\mathcal{C}^{1,\\alpha}(\\overline{U_{b,h}}\\backslash D_{b,h})$ for some $0<\\alpha \\leq 1$ in two and three dimensions. \n\\end{proposition}\n\\begin{proof}\n Clearly, from \\eqref{EQ2}, one sees that $u_i\\in H^1(D_{b, h})$ satisfies $\\Delta u_i=0$ in $D_{b, h}$, and \n $$\n u_i=u_e\\quad \\mbox{and}\\quad \\partial_\\nu u_i=\\eta^{-1}\\partial_\\nu u_e\\quad\\mbox{on}\\ \\ {\\partial D_{b, h}\\cap \\partial D}. \n $$   \n According to Definition~\\ref{def:k}, we know that ${\\partial D_{b, h}\\cap \\partial D}$ is $\\mathcal{C}^{2,1}$. Hence, by the regularity estimate up the boundary for elliptic PDEs (cf. \\cite[Theorem 4.18]{McL}), we have that $u_i\\in H^3(D_{b, h})$. Finally, by the Sobolev embedding, we readily see $u_i \\in \\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})$ for some $0<\\alpha \\leq 1$ in two and three dimensions. $u_i\\in \\mathcal{C}^{1,\\alpha}(\\overline{U_{b,h}}\\backslash D_{b,h})$ is obvious since we assume that $u_e$ allows a harmonic extension into $D_{b,h}$. \n\\end{proof}\n\n\\begin{remark}\nThe H\\\"older regularity of $u^i$ up to interface $\\partial\\Omega$ around $\\mathbf{p}$ is an important ingredient in our subsequent argument. In Proposition~\\ref{prop:regularity}, we derive the $H^3$-smoothness, which yields the desired H\\\"older regularity by the Sobolev embedding in two and three dimensions. For general dimensions greater than 3, one can follow a completely similar argument to derive the $\\mathcal{C}^{1,\\alpha}$ regularity of $u^i$ in $\\overline{D_{b,h}}$, but subject to requiring that ${\\partial D_{b, h}\\cap \\partial D}$ is $\\mathcal{C}^{r+1,1}$, where $r:=\\lceil{\\frac n 2-1+\\alpha}\\rceil$. That is, one needs to modify $\\mathcal{C}^{2,1}$ in item (2) in Definition~\\ref{def:k} to be $\\mathcal{C}^{r+1,1}$ for the general dimension case in order to have Proposition~\\ref{prop:regularity}. However, only two and three dimensions are physically meaningful, and hence we mainly focus on the two and three dimensional cases. Nevertheless, it is emphasized that all our subsequent arguments can be extended to higher dimensions after some necessary but slight modifications.  \n\\end{remark}\n\n\n\\begin{theorem}\\label{estimationT}\nLet $u_0\\in H^1_{loc}(U_{b,h})$ be a harmonic function, known as the CGO (Complex Geometric Optics) solution, constructed in the from\n\\begin{equation}\\label{CGO}\nu_0=\\exp(\\mathbf{\\xi}\\cdot \\mathbf{x}),\n\\end{equation}\n with the parameter \n \\begin{equation}\\label{eq:rho}\n \\xi=\\mathrm{i}\\tau \\hat{\\mathbf{v}}-\\tau \\mathbf{e}_n\\in\\mathbb{C}^n, \\ \\tau\\in\\mathbb{R}_+,\n \\end{equation}\n where\n \\begin{equation}\\label{eq:pv1}\n \\hat{\\mathbf{v}}:=\\begin{cases}\n \\displaystyle{\\frac{\\nabla u_i(\\mathbf{p})-(\\nabla u_i(\\mathbf{p})\\cdot \\mathbf{e}_n)\\mathbf{e}_n}{\\big|\\nabla u_i(\\mathbf{p})-(\\nabla u_i(\\mathbf{p})\\cdot \\mathbf{e}_n)\\mathbf{e}_n\\big|}} \\ \\ \\ &\\mbox{if\\ \\ $\\nabla u_i(\\mathbf{p})\\nparallel \\mathbf{e}_n$,}\\medskip\\\\\n \\mathbf{e}_1:=(1,0,\\ldots,0)\\ \\ &\\mbox{if\\ \\ $\\nabla u_i(\\mathbf{p})\\parallel \\mathbf{e}_n$}. \n \\end{cases}\n \\end{equation}\n Then it holds that\n \\begin{equation}\\label{estimation}\n \\begin{split}\n &C_{n,\\eta,\\alpha}|\\nabla u_i(\\mathbf{p})|\\\\\n \\leq&  \\Vert u_i \\Vert_{\\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})}(1+(\\tau h)^{(n-1)\/2})e^{\\tau(\\frac{1}{4K}-h)}\\\\\n &+\\Vert u_i \\Vert_{\\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})}\\left((\\frac{K}{K_-})^{\\frac{n-1}{2}}-(\\frac{K}{K_+})^{\\frac{n-1}{2}}\\right)e^{\\frac{\\tau}{4K}}\\\\\n&+\\Vert u_i \\Vert_{\\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})}(h+K^{-1}_-)^{\\alpha\/2}h^{(n+1+\\alpha)\/2}(K\/K_-)^{(n-1)\/2}\\tau^{3\/2}e^{\\frac{\\tau}{4K}}\\\\\n&+(\\Vert u_i \\Vert_{\\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})}+\\Vert u_e \\Vert_{\\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})})h^{\\alpha+(n-1)\/2}(K\/K_-)^{(n-1)\/2}\\\\\n&\\quad \\times (1+\\tau h)\\tau^{(n-1)\/2}e^{\\tau(\\frac{1}{4K}-h)},\n \\end{split}\n \\end{equation}\nwhere $C_{n,\\eta,\\alpha}$ is a positive constant depending on $n, \\eta$ and $\\alpha$. \n\\end{theorem}\n\\begin{proof}\nIt is directly verified that $u_0$ constructed in \\eqref{CGO}--\\eqref{eq:pv1} is a harmonic function. Next, we apply the constructed $u_0$ to the integral identity \\eqref{IntId}. \nBy straightforward calculations, we split the integral at the left hand side of (\\ref{IntId}) into the following identity,\n\\begin{eqnarray}\\label{IntId2}\n\\nabla u_i(\\mathbf{p})\\cdot \\xi \\int_{x_n>K|\\mathbf{x}'|^2}e^{\\xi\\cdot \\mathbf{x}} dx=\\nabla u_i(\\mathbf{p})\\cdot \\xi \\int_{x_n>\\max(h, K|\\mathbf{x}'|^2)}e^{\\xi\\cdot \\mathbf{x}}\\, d\\sigma_\\mathbf{x} \\nonumber\\\\\n+\\nabla u_i(\\mathbf{p})\\cdot \\xi \\left( \\int_{K|\\mathbf{x}'|^2<x_n<h}e^{\\xi\\cdot \\mathbf{x}}\\, d\\sigma_{\\mathbf{x}}-\\int_{D_{b,h}} e^{\\xi\\cdot \\mathbf{x}}\\, d\\sigma_\\mathbf{x} \\right) \\nonumber\\\\\n+\\int_{D_{b,h}} \\xi\\cdot(\\nabla u_i(\\mathbf{x})-\\nabla u_i(\\mathbf{p})) e^{\\xi\\cdot \\mathbf{x}}\\, d\\sigma_\\mathbf{x}\\nonumber \\\\\n+\\frac{1}{\\eta-1}\\int_{\\partial S}(\\eta\\partial_\\nu u_i-\\partial_\\nu u_e)e^{\\xi\\cdot \\mathbf{x}}-(u_i-u_e)\\partial_\\nu e^{\\xi\\cdot \\mathbf{x}}\\, ds.\n\\end{eqnarray}\nFor notational convenience, we rewrite the integral identity (\\ref{IntId2}) in the following form,\n\\begin{equation}\\label{eq:eee1}\n\\nabla u_i(\\mathbf{p})\\cdot \\xi I_0=\\nabla u_i(\\mathbf{p})\\cdot \\xi I_1+\\nabla u_i(\\mathbf{p})\\cdot \\xi I_2+ I_3+\\frac{1}{\\eta-1}I_4.\n\\end{equation}\nwhere $I_j$, $j=1,\\ldots,4$, are respectively defined as\n\\begin{equation}\\label{eq:I}\n\\begin{split}\nI_0=& \\int_{x_n>K|\\mathbf{x}'|^2}e^{\\xi\\cdot \\mathbf{x}}\\, d\\sigma_\\mathbf{x},\\\\\nI_1=& \\int_{x_n>\\max(h,K|\\mathbf{x}'|^2)}e^{\\xi\\cdot \\mathbf{x}}\\, d\\sigma_\\mathbf{x},\\\\\nI_2=& \\int_{K|\\mathbf{x}'|^2<x_n<h}e^{\\xi\\cdot \\mathbf{x}}\\, d\\sigma_\\mathbf{x}-\\int_{D_{b,h}} e^{\\xi\\cdot \\mathbf{x}}\\, d\\sigma_\\mathbf{x},\\\\\nI_3=& \\int_{D_{b,h}} \\xi\\cdot(\\nabla u_i(\\mathbf{x})-\\nabla u_i(\\mathbf{p})) e^{\\xi\\cdot \\mathbf{x}}\\, d\\sigma_\\mathbf{x}, \\\\\nI_4=& \\int_{\\partial S}(\\eta\\partial_\\nu u_i-\\partial_\\nu u_e)e^{\\xi\\cdot \\mathbf{x}}-(u_i-u_e)\\partial_\\nu e^{\\xi\\cdot \\mathbf{x}}\\, ds.\n\\end{split}\n\\end{equation}\nUsing Lemmas 2.17--2.20 in \\cite{Liu_curve} we have the following estimates of the integrals $I_0,I_1,I_2,I_3,$:\n\\begin{equation}\\label{eq:E1}\n\\begin{split}\nI_0=& \\frac{1}{-\\xi_n}\\left(\\frac{\\pi}{-\\xi_n K}\\right)^{(n-1)\/2}\\exp\\left( -\\frac{\\xi'\\cdot\\xi'}{4\\xi_n K} \\right),\\\\\n|I_1|\\leq & C_n \\frac{1+(\\tau h)^{\\frac{n-1}{2}}}{\\tau^{\\frac{n+1}{2}}K^{\\frac{n-1}{2}}}e^{-\\tau h},\\\\ \n|I_2|\\leq & C_n \\left(K_-^{-\\frac{n-1}{2}}-K_+^{-\\frac{n-1}{2}}\\right)\\tau^{-\\frac{n+1}{2}},\\\\ \n|I_3|\\leq & C_{n,\\alpha}\\Vert u_i \\Vert_{\\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})}(h+K_-^{-1})^{\\alpha\/2}h^{(n+\\alpha+1)\/2}K_-^{-(n-1)\/2},\n\\end{split}\n\\end{equation}\nwhere $\\xi=(\\xi_j)_{j=1}^n$ and $\\xi'=(\\xi_j)_{j=1}^{n-1}$. \nBy following a similar argument to the proof of Proposition 2.21 in \\cite{Liu_curve}, the last integral $I_4$ can be estimated as follows,\n\\begin{equation}\\label{eq:I4}\n|I_4|\\leq C_{n,\\alpha}h^{\\alpha+(n-1)\/2}K_-^{-(n-1)\/2}(1+\\tau h)e^{-\\tau h}(\\eta\\Vert u_i\\Vert_{\\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})}+\\Vert u_e\\Vert_{\\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})}).\n\\end{equation}\nFinally, by applying the estimates in \\eqref{eq:E1} and \\eqref{eq:I4} to \\eqref{eq:eee1}, together with grouping similar terms, one can arrive at the estimate (\\ref{estimation}).\n\nThe proof is complete. \n\\end{proof}\n\n\\section{Unique determination results for Calder\\'on's inverse inclusion problem}\\label{sect:3}\n\nIn this section, we consider the inverse problem \\eqref{eq:ic1}. In principle, we aim to establish a local unique determination result as follows. Let $(D, \\eta)$ and $(\\widetilde D, \\widetilde \\eta)$ be two conductive inclusions supported in $\\Omega$ as described in \\eqref{form}. Let $\\Gamma_0\\Subset\\partial\\Omega$ be an open subset and let $g\\in H_0^{-1\/2}(\\partial\\Omega)$ with $\\mathrm{supp}(g)\\subset\\Gamma_0$. If\n\\begin{equation}\\label{eq:u1}\n\\Lambda_{\\gamma}(g)=\\Lambda_{\\widetilde{\\gamma}}(g)\\quad\\mbox{on}\\ \\ \\Gamma_0, \n\\end{equation}\nwhere $\\gamma$ and $\\widetilde\\gamma$ are given in \\eqref{form}, respectively associated with $\\eta$ and $\\widetilde\\eta$, then in a certain generic scenario as shall be detailed in what follows, $D\\Delta\\widetilde D:=(D\\backslash\\overline{\\widetilde{D}})\\cup(\\widetilde{D}\\backslash\\overline{D})$ cannot possess a $K$-curvature point as described in Definition~\\ref{def:k} with $K$ sufficiently large, which lies on $\\partial \\Sigma$. Here, $\\Sigma$ is the connected component of $\\Omega\\backslash\\overline{D\\cup\\widetilde{D}}$ that connects to $\\partial\\Omega$. In the simpler geometric setup with $D$ and $\\widetilde{D}$ both being convex, this means $D\\Delta\\widetilde{D}$ cannot possess a $K$-curvature point with $K$ sufficiently large. To establish the local uniqueness results, we shall make use of the contradiction argument by assuming that $D\\Delta\\widetilde D$ possesses an admissible $K$-curvature point. Let $\\mathbf{p}$ signify the aforementioned admissible $K$-curvature point. Without loss of generality, we assume that $\\mathbf{p}\\in \\partial D\\cap \\partial \\Sigma$; see Fig.\\ref{dispo_inverse} for a schematic illustration. Let $u$ and $\\widetilde u$ be the electric potential fields in \\eqref{EQ2} associated with $\\gamma$ and $\\widetilde\\gamma$, respectively. By virtue of \\eqref{eq:u1}, we readily see that \n\\begin{equation}\\label{eq:u2}\nu=\\widetilde u\\quad\\mbox{and}\\quad \\partial_\\nu u=\\partial_\\nu \\widetilde u\\quad\\mbox{on}\\ \\ \\Gamma_0. \n\\end{equation}\nNoting that $\\Delta u=\\Delta \\widetilde u=0$ in $\\Sigma$, we thus have from the unique continuation principle that $u=\\widetilde u$ in $\\Sigma$. The contradiction shall be established from the quantitative properties of the solution $u\\in H^1(U_{b,h})$ locally around the point $\\mathbf{p}\\in\\partial D$. It is recalled that according to our discussion below \\eqref{EQ2}, $u=u_i\\chi_{D_{b,h}}+u_e\\chi_{U_{b,h}\\backslash\\overline{D_{b,h}}}$ in $U_{b,h}$. By taking $b$ and $h$ sufficiently small, we can assume that $U_{b,h}$ stays away from $\\widetilde D$ with a positive distance.  Since $u_e=\\widetilde{u}$ in ${U_{b,h}\\backslash\\overline{D_{b,h}}}$, and $\\widetilde{u}$ is obviously analytic in $U_{b,h}$, we clearly have that $u_e$ can be harmonically extended into $D_{b,h}$, which is exactly $\\widetilde u$. \n\nTo establish the uniqueness result mentioned above, we start with introducing an admissibility condition on the input $g\\in H_0^{-1\/2}(\\partial\\Omega)$ in \\eqref{EQ2} for our inverse problem study.  \n\n\\begin{figure}\n\\includegraphics[scale=0.6]{dispo_inverse} \n\\caption{A $K$-curvature point and its neighborhood}\n\\label{dispo_inverse}\n\\end{figure}\n\\begin{proposition}\\label{inf_grad}\nLet $u$ be the solution to \\eqref{EQ_Calderon}. Then there exists $g\\in H_0^{-1\/2}(\\partial\\Omega)$ such that for all $\\mathbf{x}\\in \\Omega$,\n\\begin{equation}\\label{eq:inf_grad}\n\\lim_{r\\rightarrow +0}\\frac{\\int_{B_r(\\mathbf{x})} |\\nabla u(\\mathbf{z})|d\\mathbf{z}}{|B_r(\\mathbf{x})|}\\geq m_g,\n\\end{equation}\nwhere $m_g>0$ is a positive constant independent of the conductivity function $\\gamma$. Here and also in what follows, $B_r(\\mathbf{x})$ signifies a ball centered at $\\mathbf{x}$ with radius $r$.\n\\end{proposition}\n\n\\begin{remark}\\label{rem:input}\nIn fact, the statement of Proposition \\ref{inf_grad} in the 2-dimensional case is true if the input data $g$ has only one local maximum and only one local minimum on $\\partial \\Omega$ (see e.g. \\cite{gradu1,gradu2}). Through out the rest of the paper, we assume that the condition \\eqref{eq:inf_grad} holds true in our study. In the context of the inverse inclusion problem, it means that the boundary input should be properly chosen. \n\\end{remark}\n\n\n\n\\subsection{Local behaviours of the solution $u$ near the high curvature point $\\mathbf{p}$}\nIn Theorem \\ref{estimationT}, we obtain an estimate of $|\\nabla u_i (\\mathbf{p})|$ in terms of the $K$-curvature parameters and the $\\mathcal{C}^{1,\\alpha}$-norm of $u_{i}$. In this subsection, we further refine the estimate \\eqref{estimation}. Specifically, we shall need an estimates of $\\Vert u_i \\Vert_{\\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})}$ and $\\Vert u_e \\Vert_{\\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})}$ appeared in \\eqref{estimation} in terms of the geometric parameters of the a-priori parameters. It is recalled that according to our discussion at the beginning of this section, $u_e=\\widetilde u$ in $U_{b,h}$, with $\\widetilde u$ satisfying \\eqref{EQ2} associated with $(\\widetilde D, \\widetilde\\eta)$ that arises from the contradiction argument in what follows. It is noted again that $\\widetilde u$ is harmonic in $U_{b,h}$. \n\nBy Proposition~\\ref{prop:regularity}, we know that $u_i \\in \\mathcal{C}^{1,\\alpha}(\\overline{D_{b,h}})$ and $u_e\\in \\mathcal{C}^{1,\\alpha}(\\overline{U_{b,h}}\\backslash D_{b,h})$ for some $0<\\alpha\\leq 1$. However, the H\\\"{o}lder norms of the solution $u$ on each side intricately depend on the geometric shape, and particularly depend on the local geometric parameter $K$ of the admissible $K$-curvature point $\\mathbf{p}$. The following lemmas give an estimate of these H\\\"{o}lder norms in terms of the interface's local curvature \\cite{DEF}.\n\n\\begin{lemma}\\label{lemma_DEF}\nLet $Q_R$ be a cube in $\\mathbb{R}^n$ of side length $R$, centered at $\\mathbf{p}\\in\\partial\\Omega$. We denote the two sub-domains of $Q_R$ lying on the two sides of $\\partial D$ by $Q^+_R:=Q_R\\cap \\{x_n>w(\\mathbf{x}')\\}$ and $Q^-_R:=Q_R\\cap \\{x_n<w(\\mathbf{x}')\\}$, respectively. We consider the conductivity equation,\n\\begin{equation}\\label{eq_DEF}\n\\mathrm{div} [(1+(\\eta-1)\\chi_\\pm)\\nabla u]=0 \\ \\ \\text{in}\\ \\ Q_R,\n\\end{equation}\nwhere $\\chi_\\pm(\\mathbf{x})=1$ in $Q^+_R$ and $\\chi_\\pm(\\mathbf{x})=0$ in $Q^-_R$. Then there exist positive constants $\\alpha\\in (0, 1),\\mu,C$ independent of $u$ and $w$ such that\n\\begin{equation}\\label{holder_estimate}\n\\Vert \\nabla u\\Vert_{\\mathcal{C}^\\alpha(Q^\\pm_{R\/4})}\\leq C(1+\\Vert \\nabla w \\Vert_{\\mathcal{C}^1(Q_R)})^\\mu \\Vert \\nabla u\\Vert_{L^2(Q_R)}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{remark}\\label{rem:n1}\nIn the context of our study, we let $R_0>0$ be such that $U_{b,h} \\Subset Q_{R_0}\\Subset \\Omega$, then Lemma~\\ref{lemma_DEF} implies that there exist $C_{n,\\eta,R_0}$ and $\\mu$ which depend only on the a-priori data such that\n\\begin{equation}\\label{holder_estimate_K}\n\\Vert \\nabla u\\Vert_{\\mathcal{C}^\\alpha(Q^\\pm_{R_0\/4})}\\leq C_{n,\\eta,R_0} K^\\mu \\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}.\n\\end{equation}\n\\end{remark}\n\nClearly, \\eqref{holder_estimate_K} gives the estimates of $\\|\\nabla u_i\\|_{C^\\alpha(\\overline{D_{b,h}})}$ and $\\|\\nabla u_e\\|_{C^\\alpha(\\overline{U_{b,h}}\\backslash{D_{b,h}})}$. We next further derive the estimates of $\\|u_i\\|_{C(\\overline{D_{b,h}})}$ and $\\|u_e\\|_{C(\\overline{U_{b,h}}\\backslash {D_{b,h}})}$ in terms of the local curvature of the admissible $K$-curvature point, which then yield the desired estimates of $\\|u_i\\|_{C^{1,\\alpha}(\\overline{D_{b,h}})}$ and $\\|u_e\\|_{C^{1,\\alpha}(\\overline{U_{b,h}}\\backslash {D_{b,h}})}$.\n\n\\begin{lemma}\\label{lem:holder}\nLet $u=u_i\\chi_D+u_e\\chi_{\\Omega\\backslash\\overline{D}}\\in H^1(\\Omega)$ be the solution to (\\ref{EQ2}). Suppose that $D\\Subset B(\\mathbf{x}_0, r_0)\\Subset\\Omega$, where $B(\\mathbf{x}_0, r_0)$ is a given ball. Furthermore, it is assumed that for any point $\\mathbf{x}\\in B(\\mathbf{x}_0, r_0)\\backslash {D}$, there exists a line segment within $\\Omega$ which connects $\\mathbf{x}$ to a point $\\mathbf{x}'\\in\\Omega\\backslash\\overline{B(\\mathbf{x}_0,r_0)}$. Then one has that \n\\begin{equation}\\label{eq:estn1}\n\\|u_i\\|_{C^{1,\\alpha}(\\overline{D_{b,h}})}\\leq C K^\\mu\\|g\\|_{H^{-1\/2}(\\partial\\Omega)},\n\\end{equation}\nand\n\\begin{equation}\\label{eq:estn2}\n\\|u_e\\|_{C^{1,\\alpha}(\\overline{U_{b,h}}\\backslash {D_{b,h}} )}\\leq C K^\\mu\\|g\\|_{H^{-1\/2}(\\partial\\Omega)},\n\\end{equation}\nwhere the positive constant $C$ depends on generic constant in the estimate \\eqref{holder_estimate_K} as well as $\\Omega$ and $B(\\mathbf{x}_0, r_0)$, but independent of $K$. \n\\end{lemma}\n\n\\begin{proof}\nWe first prove \\eqref{eq:estn2}, and by virtue of \\eqref{holder_estimate_K}, we see that\n\\begin{equation}\\label{eq:estn25}\n\\|\\nabla u_e\\|_{C^\\alpha(U_{b,h}\\backslash\\overline{D_{b,h}})}\\leq C K^\\mu\\|g\\|_{H^{-1\/2}(\\partial\\Omega)}. \n\\end{equation}\nand hence it suffices for us to show that \n\\begin{equation}\\label{eq:estn3}\n\\|u_e\\|_{C(U_{b,h}\\backslash\\overline{D_{b,h}})}\\leq C K^\\mu\\|g\\|_{H^{-1\/2}(\\partial\\Omega)}. \n\\end{equation}\n\nBy the standard elliptic PDE estimate, we know $u_e\\in C(\\Omega\\backslash\\overline{B(\\mathbf{x}_0, r_0)})$, and for any $\\mathbf{x}'\\in\\Omega\\backslash\\overline{B(\\mathbf{x}_0, r_0)}$, \n\\begin{equation}\\label{eq:estn4}\n|u_e(\\mathbf{x}')|\\leq C_1\\|g\\|_{H^{-1\/2}(\\partial\\Omega)},\n\\end{equation}\nwhere $C_1$ is a positive constant depending only on $\\eta, \\Omega$ and $B(\\mathbf{x}_0, r_0)$, but independent of $K$. For any $\\mathbf{x}\\in \\overline{U_{b, h}}\\backslash{D_{b,h}}$, we let $\\mathbf{x}'\\in \\Omega\\backslash\\overline{B(\\mathbf{x}_0, r_0)}$ and ${l}(\\mathbf{x},\\mathbf{x}')$ be the line segment connecting $\\mathbf{x}$ and $\\mathbf{x}'$ such that $l(\\mathbf{x},\\mathbf{x}')\\Subset\\Omega$. Then by the intermediate value theorem, we have\n\\begin{equation}\\label{eq:estn5}\nu_e(\\mathbf{x})-u_e(\\mathbf{x}')=\\nabla u_e(\\xi)\\cdot(\\mathbf{x}-\\mathbf{x}'),\n\\end{equation}\nwhere $\\xi\\in l(\\mathbf{x}, \\mathbf{x}')$. By combining \\eqref{eq:estn25} and \\eqref{eq:estn4}, we readily have from \\eqref{eq:estn5} that\n\\begin{equation}\\label{eq:est6}\n|u_e(\\mathbf{x})|\\leq C_2(1+K^\\mu)\\|g\\|_{H^{-1\/2}(\\partial\\Omega)}\\leq 2C_2 K^\\mu\\|g\\|_{H^{-1\/2}(\\partial\\Omega)},\n\\end{equation}\nwhere we have made use of the facts that $|\\mathbf{x}-\\mathbf{x}'|\\leq \\mathrm{diam}(\\Omega)$, and without loss of generality that $K^\\mu\\geq 1$. \\eqref{eq:est6} clearly implies \\eqref{eq:estn3}, which in combination with \\eqref{eq:estn25} immediately yields \\eqref{eq:estn2}. \n\n\\eqref{eq:estn1} can be proved by following a completely similar argument. Indeed, for any $\\mathbf{x}\\in D_{b, h}$, one can take $\\mathbf{x}'\\in \\partial D\\cap U_{b,h}$, and then it holds that\n\\begin{equation}\\label{eq:estn7}\nu_i(\\mathbf{x})-u_e(\\mathbf{x}')=u_i(\\mathbf{x})-u_i(\\mathbf{x}')=\\nabla u_i(\\xi)\\cdot(\\mathbf{x}-\\mathbf{x}'),\n\\end{equation}\nwhere $\\xi\\in l(\\mathbf{x}, \\mathbf{x}')\\subset\\overline{D_{b, h}}$. Finally, by combining Remark~\\ref{rem:n1}, \\eqref{eq:est6} and \\eqref{eq:estn7}, one can show \\eqref{eq:estn1}. \n\nThe proof is complete.  \n\n\\end{proof}\n\n\\begin{remark}\\label{rem:n2}\nThe geometric condition in Lemma~\\ref{lem:holder}, namely for any point $\\mathbf{x}\\in B(\\mathbf{x}_0, r_0)\\backslash {D}$, there exists a line segment within $\\Omega$ which connects $\\mathbf{x}$ to a point $\\mathbf{x}'\\in\\Omega\\backslash\\overline{B(\\mathbf{x}_0,r_0)}$, can be easily fulfilled if $D$ is convex or star-shaped. \n\\end{remark}\n\nAs mentioned earlier, we assume that $u_e$ can be harmonically extended into $D_{b,h}$, which is still denoted by $u_e$. That is, $u_e$ is harmonic in $U_{b,h}$ and hence is real analytic in $U_{b,h}$. Then for $b,h\\in\\mathbb{R}_+$ sufficiently small, we can have from \\eqref{eq:estn2} that\n\\begin{equation}\\label{eq:estn8}\n\\|u_e\\|_{C^{1,\\alpha}(\\overline{U_{b,h}})}\\leq C K^\\mu\\|g\\|_{H^{-1\/2}(\\partial\\Omega)},\n\\end{equation}\nwhere $C$ depends on the same a-priori data as those in \\eqref{eq:estn2}. Since throughout the paper, our argument is localized around an admission $K$-curvature point $\\mathbf{p}$ (cf. Theorems~\\ref{estimationT} and \\ref{decayT}). Hence, in what follows, we shall always assume that \\eqref{eq:estn8} holds true. By combining \\eqref{estimation} of Theorem~\\ref{estimationT} and Lemma~\\ref{lem:holder}, one can derive the following theorem.\n\n\\begin{theorem}\\label{decayT}\nLet $u\\in H^1(\\Omega)$ be the solution to \\eqref{EQ2} and $\\mathbf{p}\\in\\partial D$ be an admissible $K$-curvature point. Suppose that \\eqref{eq:estn1} and \\eqref{eq:estn2} hold. We further suppose that the exponent $\\mu$ in \\eqref{eq:estn1} and \\eqref{eq:estn2} (or equivalently in (\\ref{holder_estimate_K})) satisfies\n\\begin{equation}\\label{eq:cond1}\n\\mu <\\frac{\\min(1,\\delta)}{2}, \n\\end{equation}\nwhere $\\delta$ is the a-priori parameter associated to $\\mathbf{p}$ (cf. Definition~\\ref{def:k}). \nThen it holds that\n\\begin{equation}\\label{decay}\n|\\nabla u_i(\\mathbf{p})|\\leq \\mathcal{E}  \\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}(\\ln K)^{(n+1)\/2}K^{\\mu-\\min(\\alpha,\\delta)\/2},\n\\end{equation}\nwhere $\\mathcal{E}$ depends on the same a-priori data as those in \\eqref{eq:estn1}--\\eqref{eq:estn2} as well as $\\alpha$ and $L, M$ in Definition~\\ref{def:k} , but independent of $g$ and $K$. \n\\end{theorem}\n\\begin{proof}\nBy plugging the estimates \\eqref{eq:estn1} and \\eqref{eq:estn8} into \\eqref{estimation} of Theorem \\ref{estimationT}, we can obtain\n\\begin{equation}\\label{estimation_2}\n\\begin{split}\n& C |\\nabla u_i(\\mathbf{p})|\\leq K^\\mu \\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}(1+(\\tau h)^{(n-1)\/2})e^{\\tau(\\frac{1}{4K}-h)} \\\\\n& +K^\\mu \\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}\\left((\\frac{K}{K_-})^{\\frac{n-1}{2}}-(\\frac{K}{K_+})^{\\frac{n-1}{2}}\\right)e^{\\frac{\\tau}{4K}}\\\\\n& +K^\\mu \\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}(h+K^{-1}_-)^{\\alpha\/2}h^{(n+1+\\alpha)\/2}(K\/K_-)^{(n-1)\/2}\\tau^{3\/2}e^{\\frac{\\tau}{4K}} \\\\\n& +K^\\mu \\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}h^{\\alpha+(n-1)\/2}(K\/K_-)^{(n-1)\/2}\\\\\n& \\quad \\times (1+\\tau h)\\tau^{(n-1)\/2}e^{\\tau(\\frac{1}{4K}-h)}.\n \\end{split}\n \\end{equation}\n It is noted that in \\eqref{estimation_2}, we have absorbed the generic constant involved into the right-hand side term, and it depends on the a-priori data as stated in the theorem, which should be clear in the context. \n \nNext, by following a similar argument to the proof of Proposition 2.22 in \\cite{Liu_curve}, one can show that there exists $C_{n,L,M}>0$ such that \n \\begin{equation}\n \\left|\\left(\\frac{K}{K_-}\\right)^{\\frac{n-1}{2}}-\\left(\\frac{K}{K_+}\\right)^{\\frac{n-1}{2}}\\right|\\leq C_{n,L,M}K^{-\\delta}.\n \\end{equation}\n Using $h=1\/K$ and $b=\\sqrt{M}\/K$ in the definition of the $K$-curvature point in Definition~\\ref{def:k}, the estimate (\\ref{estimation_2}) further yields\n \\begin{eqnarray}\\label{estimation_3}\n& &C|\\nabla u_i(\\mathbf{p})|\\leq (1+(\\tau \/K)^{(n-1)\/2})K^\\mu e^{-\\frac{3\\tau}{4K}}\\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}\\nonumber \\\\\n& &+K^{\\mu-\\delta}e^{\\frac{\\tau}{4K}}\\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}+K^{\\mu-(n+2\\alpha+1)\/2} \\tau^{3\/2}e^{\\frac{\\tau}{4K}}\\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}\\nonumber \\\\\n& &+K^{\\mu-\\alpha-(n-1)\/2} (1+\\tau \/K)\\tau^{(n-1)\/2}e^{-\\frac{3\\tau}{4K}}\\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}.\n \\end{eqnarray}\nChoosing $\\tau=4K\\ln K^\\rho$ for some $\\rho>0$ and dividing by $\\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}$, the left-hand side of (\\ref{estimation_3}) can be estimated by\n\\begin{equation}\\label{estimation_4}\n(\\ln K)^{(n-1)\/2}K^{\\mu-3\\rho}+K^{\\mu-\\delta+\\rho}+(\\ln K)^{3\/2}K^{\\mu+1-n\/2-\\alpha+\\rho}+(\\ln K)^{(n+1)\/2}K^{\\mu-\\alpha-3\\rho}.\n\\end{equation}\nBy setting $\\rho=\\min(\\alpha,\\delta)\/2$, each of the terms in (\\ref{estimation_4}) can be estimated by\n\\begin{equation}\nC(\\ln K)^{(n+1)\/2}K^{\\mu-\\min(\\alpha,\\delta)\/2},\n\\end{equation}\nand thus the claim of this theorem follows.\n\nThe proof is complete. \n\\end{proof}\n\n\\begin{remark}\\label{rem:nn2}\nIn Appendix, we shall present two examples to numerically verify that the condition~\\eqref{eq:cond1} can be fulfilled in generic scenarios of practical interest.\n\\end{remark}\n\n\\subsection{Local uniqueness result}\n\nWe are in a position to present the main local uniqueness result for the inverse inclusion problem \\eqref{eq:ic1}. \n\n\\begin{theorem}\\label{mainT}\nLet $(D, \\eta)$ and $(\\widetilde D, \\widetilde\\eta)$ be two conductive inclusions in $\\Omega$, and $u, \\widetilde u$ be the solutions to \\eqref{EQ2} associated respectively to $(D, \\eta)$ and $(D, \\widetilde\\eta)$. Suppose that  \n\\begin{enumerate}\n\\item $\\partial D$ and $\\partial\\widetilde D$ are of class $C^{2, 1}$; \n\n\\item $D$ and $\\widetilde D$ satisfy the geometric condition in Lemma~\\ref{lem:holder};\n\n\\item $G:=\\Omega\\setminus \\overline{(D\\cup \\widetilde{D})}$ is connected;\n\n\\item the condition~\\eqref{eq:cond1} is fulfilled for $u\/\\widetilde u$; \n\n\\item Proposition \\ref{inf_grad} holds for $u$ and $\\widetilde{u}$. \n\\end{enumerate}\nLet $d_0\\in\\mathbb{R}_+$ and $\\Gamma_0\\subset\\partial\\Omega$. If $u=\\widetilde u$ on $\\Gamma_0$, then $D\\Delta\\widetilde{D}=(D\\setminus \\widetilde{D})\\cup (\\widetilde{D}\\setminus D)$ cannot possess an admissible $K$-curvature point $\\mathbf{p}$ such that \n\\begin{equation}\\label{eq:cc1}\n\\max\\{\\mathrm{dist}(\\mathbf{p}, \\partial D), \\mathrm{dist}(\\mathbf{p}, \\partial\\widetilde D)> d_0, \n\\end{equation}\nand $K\\geq K_0$, where $K_0\\in\\mathbb{R}_+$ is sufficiently large and depends on the a-priori parameters of $\\mathbf{p}$ in Definition~\\ref{def:k} as well as $\\Omega, d_0, g, \\eta, \\widetilde\\eta$ and $B(\\mathbf{x}_0, r_0)$ in Lemma~\\ref{lem:holder}. \n\\end{theorem}\n\n\\begin{proof}\nBy absurdity, we assume without loss of generality that there exists an admissible $K$-curvature point $\\mathbf{p}\\in \\partial D \\cap \\partial G$, such that \n$B(\\mathbf{p},d_0)\\Subset\\Omega\\backslash\\overline{\\widetilde{D}}$. We next show that as $K\\rightarrow+\\infty$, $|\\nabla u(\\mathbf{p})|\\rightarrow 0$, which yields a contraction the assumption (5) in the statement of the theorem. \n\n\nFirst, let us consider the function $u-\\widetilde{u}$. This function is harmonic in $G$. Moreover, $u-\\widetilde{u}$ and $\\partial_\\nu (u-\\widetilde{u})$ vanish on $\\Gamma_0$. It follows from the unique continuation property that $u=\\widetilde{u}$ in $G$.\n\nIt is recalled that $u=u_i\\chi_D+u_e\\chi_{\\Omega\\backslash\\overline{D}}$. Since $u=\\widetilde u$ in $G$ and both $u$ and $\\widetilde u$ are harmonic in $G$, we see that $u_e$ admits a harmonic extension $D\\cap B(\\mathbf{p}, d_0)$, which is actually $\\widetilde u$. Hence, Theorem~\\ref{decayT} applies and we immediately obtain from \\eqref{decay}\n that\n\\begin{equation}\\label{eq:nn2}\n|\\nabla u(\\mathbf{p})|\\leq \\mathcal{ET}  \\Vert g \\Vert_{H^{-1\/2}(\\partial \\Omega)}(\\ln K)^{(n+1)\/2}K^{\\mu-\\min(1,\\delta)\/2}.\n\\end{equation}\nClearly, the right-hand side of the above estimate tends to zero as $K\\rightarrow +\\infty$. Therefore, we can choose $K_0$ such that when $K>K_0$, $|\\nabla u(\\mathbf{p})| < m_g$ which contradicts to the assumption (5) stated in the theorem. \n\nThe proof is complete. \n\\end{proof}\n\n\\begin{remark}\nIt is remarked that the assumptions (2) and (3) in Theorem~\\ref{mainT} can be fulfilled if $D$ and $\\widetilde D$ are convex; see Remark~\\ref{rem:n2}. Nevertheless, they might be more general than being convex in Theorem~\\ref{mainT}. Moreover, since our argument is localized around the admission $K$-curvature point, one may consider a even more general geometric situation where there are multiple conductive inclusions. In such a case, one may also relax the requirement that $G:=\\Omega\\setminus \\overline{(D\\cup \\widetilde{D})}$ is connected and replace $G$ to be the connected component of $\\Omega\\backslash\\overline{D\\cup\\widetilde D}$ that connects to $\\partial\\Omega$. \n\\end{remark}\n\n\\begin{remark}\nAs discussion in Remark~\\ref{rem:input}, the condition~(5) in Theorem~\\ref{mainT} can be fulfilled by choosing a suitable input $g$. As mentioned in Remark~\\ref{rem:nn2}, we shall show in the Appendix that the condition~(4) can be fulfilled in generic scenarios. \n\\end{remark}\n\n\n\\begin{remark}\nSince our argument is localized around an admissible $K$-curvature point, it is sufficient for us to require that the conductivity parameter $\\eta$ is constant in a small neighbourhood of the admissible $K$-curvature, and it can be an $L^\\infty$ variable function in the rest part of the inclusion $D$. Our unique recovery result in Theorem~\\ref{mainT} can be extended to such a case by straightforwardly modifying the relevant arguments. Finally, we would like to point out if sufficient a-priori information is available about the underlying inclusion, the local uniqueness result in Theorem~\\ref{mainT} also implies a certain global uniqueness result, and we shall explore more along this direction in our future study. \n\\end{remark}\n\n\n\n\\section*{Acknowledgment}\nThe work of H Liu was supported by the startup grant from City University of Hong Kong, Hong Kong RGC General Research Funds, 12302017, 12301218 and 12302919.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe aim of this paper is to find a part of a John decomposition\non which a given nontrivial operator is invertible in a certain sense, \nand to apply this to the study of contact points of convex bodies. \n\nJohn decomposition of identity is by now a classical tool in the local \ntheory of Banach spaces.\nSuppose $X  =  ({\\bf R}^n, \\| \\cdot \\|)$ is a Banach space whose \nellipsoid of maximal volume contained in $B(X)$ \ncoincides with the unit euclidean ball. \nThe John decomposition of the identity operator on $X$ is\n\\begin{equation}                                                             \\label{Johndecomp}\n  {\\it id}  =  \\sum_{j = 1}^m  x_j \\otimes x_j,\n\\end{equation}\nwhere $x_j \/ \\|x_j\\|_X$ are some contact points of the surfaces\nof $B(X)$ and the unit euclidean ball.\nThis celebrated theorem of F.~John has been used extensively \nover past 30 years. Recently it was interpreted as an isotropic\ncondition \\cite{G-M}, and was generalized to a non-convex case\nin \\cite{G-P-T}.\n\nIt is important to know good parts of John decomposition, \nas it provides a good set of contact points of $X$, \nwhich can be useful for understanding the geometrical structure of $X$,\nsee \\cite{R2}.\nIn the present paper we find a part of a decomposition (\\ref{Johndecomp})\nwhich preserves the orthogonal structure under action of a given \nlinear operator $T$.\nMore precisely, if $\\|T\\|_{2 \\rightarrow 2}  =  1$\nthen there exists a subset of indices $\\sigma$\nof cardinality $|\\sigma|  \\ge  (1 - \\varepsilon) \\|T\\|_{\\rm HS}^2$ such that\nthe system $(T x_j)_{j \\in \\sigma}$ is $C(\\varepsilon)$-equivalent\nto an orthogonal basis in Hilbert space.\nThis result is non-trivial even for the coordinate decomposition\n${\\it id}  =  \\sum e_j \\otimes e_j$.\nIn this case it generalizes the principle of restricted invertibility\nproved by J.~Bourgain and L.~Tzafriri \\cite{B-Tz}. \nThey considered only operators $T$ for which all norms $\\|T e_j\\|_2$ \nare well bounded below and proved the principle with some fixed $0 < \\varepsilon < 1$.\n\n\n$T$ being an orthogonal projection, we derive a new lemma of \nDvoretzky-Rogers type. \nSuppose $P$ is an orthogonal projection in $X$ with ${\\rm rank} P = k$.\nThen for any $\\kappa  <  k$\nthere are contact points $x_1, \\ldots, x_\\kappa$\nsuch that setting $z_j  =  P x_j \/ \\|P x_j\\|_2$\nwe have\n\n\\begin{itemize}\n\\item{the system $(z_j)$ is $C(\\kappa \/ k)$-equivalent in $l_2$-norm \n    to the canonical basis of $l_2^\\kappa$;}\n\\item{$\\|z_j\\|_X  \\ge  c \\sqrt{\\frac{k - \\kappa}{n}}$  for all $j$.}\n\\end{itemize}\n\n\\noindent To put the result in other words, \nthe orthonormal system in $Z$ guaranteed by the classical \nDvoretzky-Rogers Lemma is essentially the normalized projections \nof contact points of $X$.\nMoreover, the result holds for selfadjoint operators as well \nas for projections, the Hilbert-Schmidt norm substituting ${\\rm rank} P$.\nFor a general operator $T$ the best lower bound for $\\|T x_j\\|_X$ \nis equivalent to $\\frac{1}{n} |{\\rm trace} T|$.\n\n$T$ being the identity operator, we obtain a set of \n$k  >  (1 - \\varepsilon) n$ contact points of $X$\nwhich is $C(\\varepsilon)$-equivalent in $l_2$-norm to the canonical \nbasis in $l_2^k$.\nThis settles an isomorphic version of a problem of \nN.~Tomczak-Jaegermann (\\cite{T-J}, p.127),\nand confirms the feeling that contact points are always \ndistributed fairly uniformly on the surface of the maximal volume ellipsoid\n(see \\cite{B1}).\nBesides, this yields the proportional Dvoretzky-Rogers factorization\n(with constant $C(\\varepsilon)  =  \\varepsilon^{c \\log \\varepsilon}$, \nwhich is however not the best known estimate).\n\nThe Dvoretzky-Rogers Lemma is proved useful in study \nof subspaces of $X$ well isomorphic to $l_\\infty^k$.\nThe use of the refined Dvoretzky-Rogers lemma above\nimproves a \"Gaussian\" version of Alon-Milman-Talagrand Theorem \nabout $l_\\infty^k$-subspaces of $X$.\nLet $P$ be an orthogonal projection in $X$ with ${\\rm rank} P = k$.\nThen there exists a subspace $Z  \\subset X$ \nwhich is $M$-isomorphic to $l_\\infty^m$ \nfor $m  \\ge  c k \/ \\sqrt{n}$\nand $M  =  c \\sqrt{\\frac{n}{k}} \\ell(P)$.\nThe subspace $Z$ is canonically spanned by the projections\nof $m$ contact points $x_j$.\nMoreover, the norm on $Z$ is $M$-equivalent to \n$|||z|||  =  \\max_{j \\le m} | \\< z , x_j \\> |$.\nThis improves the estimates obtained by \nM.~Rudelson in \\cite{R1}, and also provides information\nabout the position of $Z$ in $X$.\nBesides, this yields a refinement of M.~Rudelson's \nresult about $l_\\infty^k$-subspaces in spaces\nwith large volume ratio.\nIf ${\\rm vr}(X)  \\ge  a \\sqrt{n}$ then $X$ has a subspace $Z$ \nof dimension $m  \\ge  C_1(a) \\sqrt{n}$\nwhich is $C_2(a) \\log{n}$-isomorphic to $l_\\infty^m$.\n\nThe extraction results about John decompositions\ncan be reformulated in the language of frames. \nSuppose we are given a tight frame $(x_j)$ in Hilbert space $H$,\nand a norm-one linear operator $T : H  \\rightarrow  H$. \nThen there is a subsequence \n$(T x_j)_{j \\in \\sigma}$ with $|\\sigma|  \\ge  (1 - \\varepsilon) \\|T\\|_{\\rm HS}^2$ \nwhich is $C(\\varepsilon)$-equivalent to an orthogonal basis in Hilbert space.\nThis theorem again can be interpreted as an extension of\nthe invertibility principle. \nIt also generalizes results of P.~Casazza \\cite{C2} and \nthe author \\cite{V}, who worked with the identity operator \n$T = {\\it id}$.\n\nThe rest of the paper is organized as follows. \nIn \\S \\ref{SecPrelim} we recall some basic tools used later. \nThe extraction result about John decompositions, as\nwell as some modifications, is proved in \\S \\ref{SecMain}.\nIts relation to the principle of restricted invertibility\nand infinite-dimensional analogs are discussed in \\S \\ref{SecInvert}.\nDvoretzky-Rogers type lemmas are derived from these results\nin \\S \\ref{SecDR}.\nThey help to understand structure of the set of contact points. \nApplications to $l_\\infty^k$-subspaces of a finite dimensional \nspace are given in \\S \\ref{SecCube}.\nFinally, in \\S \\ref{SecFrames} we discuss a relation of \nthese results to the theory of frames in Hilbert space. \n\nI am grateful to M.~Rudelson for many important discussions.\nThe research would not be accomplished without help and \nencouragement of my wife Lilya.\n\n\n\n\n\n\n\n\n\\section{Preliminaries}                                                \\label{SecPrelim}\n\n\nWe denote by $c, c_1, c_2$ absolute constants, \nand by $C(t)$, $C_1(t)$, $C_2(t)$ constants which depend on the\nparameter $t$ only. The values of these constants may differ\nfrom line to line.\nThe canonical vectors in ${\\bf R}^n$ are denoted by $e_j$.\n\nA sequence of vectors $(x_j)$ in a Banach space is called \n{\\em $K$-Hilbertian} if \n$\\| \\sum a_j x_j \\|  \\le  K ( \\sum |a_j|^2 )^{1\/2}$\nfor any finite set of scalars $(a_j)$.\nSimilarly, $(x_j)$ is called \n{\\em $K$-Besselian} if \n$K \\| \\sum a_j x_j \\|  \\ge  ( \\sum |a_j|^2 )^{1\/2}$\nfor any finite set of scalars $(a_j)$.\nSuppose we are given two sequences $(x_j)$ and $(y_j)$ \nin Banach spaces $X$ and $Y$ respectively.\nThe sequences $(x_j)$ and $(y_j)$ are called {\\em $K$-equivalent}\nif there exist constants $K_1$ and $K_2$ with $K_1 K_2  \\le  K$\nsuch that for any finite sequence of scalars $(a_j)$\n$$\nK_1^{-1} \\| \\sum a_j y_j \\|_Y\n\\le  \\| \\sum a_j x_j \\|_X\n\\le  K_2 \\| \\sum a_j y_j \\|_Y.\n$$\nIn other words, the linear operator \n$T : {\\rm span}(x_j)  \\rightarrow  {\\rm span}(y_j)$ defined as\n$T x_j  =  y_j$ for all $j$\nis a $K$-isomorphism: $\\|T\\| \\|T^{-1}\\|  \\le  K$.\n\nHere and in the next section we work in a Hilbert space $H$\nwhose scalar product is denoted by $\\< \\cdot, \\cdot \\> $, \nand the norm by $\\| \\cdot \\|$.\nFirst we observe that the Hilbert-Schmidt norm of an operator on $H$\ncan be computed on the elements\nof certain decompositions of identity.\n\n\\begin{lemma}                                                             \\label{hs}\n  Let ${\\it id}  =  \\sum x_j \\otimes x_j$\n  be a decomposition on a Hilbert space $H$, \n  and $T : H \\rightarrow H$ be a linear operator. \n  Then \n  $$\n  \\|T\\|_{\\rm HS}^2  =  \\sum \\|T x_j\\|^2.\n  $$\n\\end{lemma}\n\n\\noindent {\\bf Proof.}\\ \\  \nIt is enough to write \n$$\n\\sum T x_j  \\otimes  T x_j\n=  T T^*\n= \\sum T e_j  \\otimes  T e_j\n$$\nand to take traces.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\n\\noindent As an immediate consequence we have \n\n\\begin{corollary}                                                             \\label{squares}\n  Let ${\\it id}  =  \\sum x_j \\otimes x_j$\n  be a decomposition on a Hilbert space $H$. \n  Then \n  $$\n  \\sum \\|x_j\\|^2  =  \\dim H.\n  $$\n\\end{corollary}\n\n\n\\begin{lemma}\n  Let ${\\it id}  =  \\sum x_j \\otimes x_j$\n  be a decomposition on a Hilbert space.\n  Then the system $(x_j)$ is $1$-Hilbertian. \n\\end{lemma}\n\n\\noindent {\\bf Proof.}\\ \\ \nNotice that for every vector $x$\n$$\n\\|x\\|^2  =  \\langle x, x \\rangle  \n  =  \\Big\\langle \\sum \\langle x_j, x \\rangle x_j, x \\Big\\rangle\n  =  \\sum | \\langle x_j, x \\rangle |^2.\n$$\nThus $\\| \\sum x_j \\otimes e_j \\| = 1$, \nand by duality $\\| \\sum e_j \\otimes x_j \\| = 1$.\nThis yields that $(x_j)$ is $1$-Hilbertian.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\n\nThe starting point of this paper is the \nprinciple of restricted invertibility proved \nby J.~Bourgain and L.~Tzafriri \\cite{B-Tz}.\n\n\\begin{theorem} (J.~Bourgain, L.~Tzafriri).                          \\label{BTz}  \n  Let $T$ be a linear operator in $l_2^n$\n  for which $\\|T e_j\\| = 1$, $j = 1, \\ldots, n$. \n  Then there exists a subset $\\sigma  \\subset  \\{1, \\ldots, n\\}$ \n  of cardinality $|\\sigma|  \\ge  c_1 n \/ \\|T\\|^2$ such that\n  $$\n  \\Big\\| \\sum_{j \\in \\sigma} a_j T e_j \\Big\\|\n    \\ge  c \\Big( \\sum_{j \\in \\sigma} |a_j|^2 \\Big)^{1\/2}\n  $$\n  for any choice of scalars $(a_j)$.\n\\end{theorem}\n\n\nThe invertibility principle will be used together with\nthe following restriction theorem. It can easily be recovered \nfrom A.~Kashin's and L.~Tzafriri's paper \\cite{K-Tz}, \nsee the proofs of Theorem 1 and Corollary 2 there. \n\n\\begin{theorem} (A.~Kashin, L.~Tzafriri).                                  \\label{Lunin}\n  Let $A$ be a norm-one linear operator in $l_2^m$.\n  Fix a number $\\lambda$ with $1\/m  \\le  \\lambda  \\le 1$.\n  Then there exists a subset $\\nu  \\subset \\{1, \\ldots, m\\}$\n  of cardinality $|\\nu|  \\ge  \\lambda m \/ 4$\n  such that \n  $$\n  \\| P_\\nu A \\|  \\le  c \\left( \\sqrt{\\lambda} + \\frac{\\|A\\|_{\\rm HS}}{\\sqrt{m}} \\right).\n  $$\nHere $P_\\nu$ denotes the coordinate projection onto ${\\bf R}^\\nu$.\n\\end{theorem}\n\n\nNow we introduce an elementary procedure of splitting a sequence.\nGiven a sequence $(x_j)$ in $H$, let $(y_k)$ be any sequence of vectors in $H$ \nsuch that for every $j = 1, 2, \\ldots$\n\n\\begin{itemize}\n\\item{the vectors $y_k$, $k \\in \\sigma_j$, are multiples of the vector $x_j$;}\n\\item{$\\sum_{k \\in \\sigma_j} \\|y_k\\|^2  =  \\|x_j\\|^2$.}\n\\end{itemize}\n\n\\noindent Then we say that $(y_k)$ is the {\\em splitted sequence} $(x_j)$.\nSplitting allows us to make the norms of the vectors almost equal. Still the key \nproperty of a sequence, being $h$-Hilbertian, is preserved by splitting.\n\n\n\n\n\\section{The main result}                                               \\label{SecMain}\n\nIn this section we prove an extraction theorem which is a core of the paper.\n\n\\begin{theorem}                                                          \\label{main}\n  Let ${\\it id}  =  \\sum x_j \\otimes x_j$\n  be a decomposition on $l_2^n$,\n  and $T$ be a norm-one linear operator. \n  Then for any $\\varepsilon > 0$ there exists a set of indices $\\sigma$\n  of cardinality $|\\sigma|  \\ge  (1 - \\varepsilon) \\|T\\|_{\\rm HS}^2$ such that\n  \n  (i) the system $(T x_j)_{j \\in \\sigma}$ is $C(\\varepsilon)$-equivalent\n      to an orthogonal basis in $l_2^\\sigma$;\n\n  (ii) $\\|T x_j\\|  \\ge  c \\sqrt{\\varepsilon} \\frac{\\|T\\|_{\\rm HS}}{\\sqrt{n}} \\|x_j\\|$\n      for all $j \\in \\sigma$.\n\\end{theorem} \n\n\\noindent {\\bf Proof.}\\ \\ \nPut $h  =  \\|T\\|_{\\rm HS}^2$.\nBy an approximation one can assume that the system $(x_j)$ is finite,\nso we enumerate it as $(x_j)_{j \\le m}$.\nDenote $y_j  =  T x_j$ for all $j$. \nSplitting the system $(x_j)_{j \\le m}$ we can assume that \n$$\n0.9 \\sqrt{ \\frac{h}{m} }  \\le  \\|y_j\\|  \\le  1.1 \\sqrt{ \\frac{h}{m} }\n\\ \\ \\ \\ \\mbox{for all $j \\le m$.}\n$$\nLet $\\delta  =  \\varepsilon \/ 3$.\nConsider the set of indices satisfying (ii), i.e.\n$$\n\\tau  =  \\Big\\{ j \\le m :  \n          \\|y_j\\|  \\ge  0.9 \\sqrt{\\delta} \\sqrt{\\frac{h}{n}} \\|x_j\\|  \\Big\\}.\n$$\nWe claim that \n\\begin{equation}                                            \\label{sizeoftau}\n  |\\tau|  \\ge  (1 - \\delta) m.\n\\end{equation}\nIndeed, by Corollary \\ref{squares}\n\\begin{eqnarray*}\nn  =  \\sum_{j \\le m} \\|x_j\\|^2  \n    &\\ge&  \\sum_{j \\in \\tau^c} \\|x_j\\|^2                                     \\\\\n    &\\ge&  \\sum_{j \\in \\tau^c} \n                  \\left( \\frac{1}{0.9 \\sqrt{\\delta}} \\sqrt{\\frac{n}{h}} \n                  \\|y_j\\| \\right)^2                                        \\\\\n    &\\ge&  |\\tau^c| \\left( \\frac{1}{0.9 \\sqrt{\\delta}} \\sqrt{\\frac{n}{h}} \n                           0.9 \\sqrt{\\frac{h}{m}} \\right)^2                \n      =    |\\tau^c| \\cdot \\frac{n}{\\delta m}.\n\\end{eqnarray*}\nThus $|\\tau^c|  \\le  \\delta m$, which proves (\\ref{sizeoftau}). \n\nNow we have to find a further subset $\\sigma  \\subset \\tau$ \nof cardinality $|\\sigma|  \\ge  (1 - \\varepsilon) h$,\nsuch that the system $(y_j)_{j \\in \\sigma}$ is $C(\\varepsilon)$-equivalent\nto an orthogonal basis in $l_2^\\sigma$.\nThe set $\\sigma$ will be constructed by successive iterations. \nOn the first step $\\sigma = \\emptyset$.\nOn each successive step, the remainder $h - |\\sigma|$ \nwill be reduced in a fixed proportion.\nSo, it is enough to prove the following.\n\n\\begin{lemma}\n  Let $\\sigma  \\subset \\tau$ with $|\\sigma|  <  (1 - \\varepsilon)h$ be given, \n  and suppose the system $(y_j)_{j \\in \\sigma}$ is $K$-equivalent to an \n  orthogonal basis in Hilbert space.\n  Then $\\sigma$ can be extended in $\\tau$ to a subset $\\sigma_1$ so that \n  \n  (a) the system $(y_j)_{j \\in \\sigma_1}$ is $C(K, \\varepsilon)$-equivalent to an\n      orthogonal basis in Hilbert space;\n\n  (b) for some absolute constant $\\alpha < 1$ \n      $$\n      h - |\\sigma_1|  \\le  \\alpha (h - |\\sigma|).                          \\label{remainder}\n      $$\n\\end{lemma}\n\n\\noindent {\\bf Proof.}\\ \\ \nLet $P$ be the orthonormal projection in $l_2^n$\nonto $l_2^n  \\ominus  {\\rm span}(y_j)_{j \\in \\sigma}$\n(at the first step $P = {\\it id}$).\nFirst observe that by Lemma \\ref{hs}\n\\begin{eqnarray*}\n\\sum_{j \\le m} \\|P y_j\\|^2\n  & = &  \\sum_{j \\le m} \\|P T x_j\\|^2\n    =    \\|PT\\|_{\\rm HS}^2                                          \\\\\n  & = &  h  -  \\| ({\\it id} - P) T \\|_{\\rm HS}^2                         \\\\\n  &\\ge&  h  -  \\|{\\it id} - P\\|_{\\rm HS}^2 \\|T\\|^2                       \\\\\n  & = &  h  -  |\\sigma|.\n\\end{eqnarray*}\nUsing (\\ref{sizeoftau}), we get\n\\begin{eqnarray*}\n\\sum_{j  \\in  \\tau \\setminus \\sigma} \\|P y_j\\|^2  \n  & = &  \\sum_{j \\in \\tau} \\|P y_j\\|^2                                       \\\\\n  &\\ge&  \\sum_{j \\le m}  \\|P y_j\\|^2  \n            -  |\\tau^c| \\cdot \\Big( 1.1 \\sqrt{h \/ m} \\Big)^2                 \\\\\n  &\\ge&  h - |\\sigma| - 1.21 \\delta h                                              \\\\\n  &\\ge&  (1 - 2 \\delta) h - |\\sigma|   =:  h_0.\n\\end{eqnarray*}\nNote that $h_0$ is comparable with $h$. \nIndeed, since $|\\sigma|  <  (1 - \\varepsilon) h  =  (1 - 3 \\delta) h$, we have\n\\begin{equation}\n  h_0  \\ge  \\delta h.                                              \\label{hzeroh}\n\\end{equation}\nNow we can split the system $(y_j)_{j  \\in  \\tau \\setminus \\sigma}$\nso that the resulting system $(y'_j)_{j \\le M}$ satisfies\n\\begin{equation}\n  \\|P y'_j\\|  \\ge  0.9 \\sqrt{\\frac{h_0}{M}}                   \n     \\ \\ \\ \\ \\mbox{for all $j \\le M$}.                                \\label{Pydash}\n\\end{equation}\n\nWe are going to apply Kashin-Tzafriri's extraction result, Theorem \\ref{Lunin}.\nIn the dual setting it can be reformulated as follows.\n\n\\begin{itemize}\n\\item{Let $(x_j)_{j \\le m}$ be a $1$-Hilbertian system in a Hilbert space, \n  and put $\\sum\\|x_j\\|^2  =  h$. \n  Fix a number $\\lambda$ with $1\/m  \\le  \\lambda  \\le 1$.\n  Then there exists a subset $\\nu  \\subset \\{1, \\ldots, m\\}$\n  of cardinality $|\\nu|  \\ge  \\lambda m \/ 4$\n  such that setting \n  $K  =  \\Big( \\sqrt{\\lambda} + \\sqrt{h \/ m} \\Big)^{-1}$,\n  $$\n  \\mbox{the system $(K x_j)_{j \\in \\sigma}$ is $c$-Hilbertian}.\n  $$}\n\\end{itemize}\n\n\\noindent Apply this to the system $(y'_j)_{j \\le M}$ \nwhich is $1$-Hilbertian and \n$$\n\\sum_{j \\le M} \\|y'_j\\|^2  \n  \\le  \\sum_{j \\le m} \\|y_j\\|^2  \n   =   h.\n$$\nWith $\\lambda  =  4 h \/ M$, we obtain a subset \n$\\nu  \\subset \\{1, \\ldots, M\\}$\nof cardinality $|\\nu|  \\ge  h$\nsuch that \n\\begin{equation}                                                    \\label{ishilb}\n\\mbox{the system $\\Big( \\sqrt{\\frac{M}{h}} y'_j \\Big)_{j \\in \\nu}$ \n      is $c$-Hilbertian}\n\\end{equation}\n(notice that we could make $M$ large enough to have $\\lambda  \\le  1$\nas required in Kashin-Tzafriri's Theorem).\nTherefore, the system \n$\\Big( \\sqrt{\\frac{M}{h}} P y'_j \\Big)_{j \\in \\nu}$ \nis $c$-Hilbertian, too.\nMoreover, by (\\ref{Pydash}) and (\\ref{hzeroh})\n$$\n\\Big\\| \\sqrt{\\frac{M}{h}} P y'_j \\Big\\|   \n  \\ge  0.9 \\sqrt{\\frac{h_0}{h}}\n  \\ge  0.9 \\sqrt{\\delta}  \n  \\ \\ \\ \\ \\mbox{for all $j \\in \\nu$}.\n$$\n\nAt this point we use the original invertibility principle\nof J.~Bourgain and L.~Tzafriri \\cite{B-Tz}, \nwhich can be reformulated as follows.\n\n\\begin{itemize}\n\\item{Let $(x_j)_{j \\le n}$ be an $H$-Hilbertian system in $l_2$ \n      and $\\|x_j\\|  \\ge  \\alpha$ for all $j$.\n      Then there exists a subset $\\rho  \\subset \\{1, \\ldots, n\\}$\n      of cardinality $|\\rho|  \\ge  c (\\alpha \/ H)^2 n$ such that\n      the system $(x_j)_{j \\in \\rho}$ is $(c_1 \/ \\alpha)$-Besselian.}\n\\end{itemize}\n\n\\noindent Apply this to the system \n$\\Big( \\sqrt{\\frac{M}{h}} P y'_j \\Big)_{j \\in \\nu}$.\nThere exists a subset $\\rho'  \\subset  \\nu$ of cardinality \n$|\\rho'|  \\ge  c (0.9 \\sqrt{\\delta})^2 h  =  c \\delta h$\nsuch that\n\\begin{equation}                                               \\label{isbess}\n\\mbox{the system $(\\sqrt{\\frac{M}{h}} P y'_j)_{j \\in \\rho'}$\n      is $c_1 \/ \\sqrt{\\delta}$-Besselian.}\n\\end{equation}\nRecall that each vector $y'_j$ with $j \\in \\rho'$ \nis a multiple of some vector $y_{k(j)}$ \nwith $k(j)  \\in  \\tau \\setminus \\sigma$.\nBy (\\ref{isbess}), these vectors $y_{k(j)}$ must be linearly independent. \nIn particular, the correspondence $j \\mapsto k(j)$ is one-to-one.\nConsider the subset $\\rho  \\subset  \\tau \\setminus \\sigma$\nconsisting of vectors\n$$\n\\rho  =  \\{ k(j) : j \\in \\rho' \\}.\n$$\nNow put $\\sigma_1  =  \\sigma \\cup \\rho$.\n\nWe see that (b) is satisfied with $\\alpha  =  1 - c \\delta \/ 2$:\n$$\nh - |\\sigma_1|  \\le  h - |\\sigma| - c \\delta h  \\le  \\alpha (h - |\\sigma|).\n$$\n\nThe reason why (a) holds is that \nthe system $(y_j)_{j \\in \\rho}$ is well equivalent to an\northogonal basis, \nand the spans of $(y_j)_{j \\in \\sigma}$ and $(y_j)_{j \\in \\rho}$\nare well disjointed.\nTo implement this idea, note that by the preseding observations\n(\\ref{ishilb}) and (\\ref{isbess}) yield that there exist constants \n$(\\lambda_j)_{j \\in \\rho}$ such that\n\n\\begin{itemize}\n\\item{the system $(\\lambda_j   y_j)_{j \\in \\rho}$ is $c$-Hilbertian,}\n\\item{the system $(\\lambda_j P y_j)_{j \\in \\rho}$ \n                               is $c_1 \/ \\sqrt{\\delta}$-Besselian.}\n\\end{itemize}\n\n\\noindent Now it is easy to complete the proof.\nFix any scalars $(a_j)_{j \\in \\rho}$.\nThen defining $\\lambda_j  =  y_j \/ \\|y_j\\|$ for $j \\in \\sigma$\nwe have\n\\begin{eqnarray*}\n\\Big( \\sum_{j \\in \\sigma \\cup \\rho} |a_j|^2 \\Big)^{1\/2} \n  & = &  \\Big( \\sum_{j \\in \\sigma}  |a_j|^2 \n             + \\sum_{j \\in \\rho}|a_j|^2 \\Big)^{1\/2}                   \\\\\n  &\\le&  (K + c_1 \/ \\sqrt{\\delta}) \\left( \n         \\Big\\| \\sum_{j \\in \\sigma}   a_j \\lambda_j   y_j \\Big\\|^2\n       + \\Big\\| \\sum_{j \\in \\rho} a_j \\lambda_j P y_j \\Big\\|^2\n                               \\right)^{1\/2}                            \\\\\n  & = &  (K + c_1 \/ \\sqrt{\\delta}) \n         \\Big\\| \\sum_{j \\in \\sigma}   a_j \\lambda_j   y_j \n             +  \\sum_{j \\in \\rho} a_j \\lambda_j P y_j \\Big\\|       \n            \\\\ \\mbox{by orthogonality}                       \\\\\n  &\\le&  (K + c_1 \/ \\sqrt{\\delta}) \n         \\Big\\| \\sum_{j \\in \\sigma \\cup \\rho} a_j \\lambda_j y_j \\Big\\|,\n\\end{eqnarray*}\nbecause $P y_j = y_j$ for $j \\in \\sigma$.\nThis shows that the system $(\\lambda_j y_j)_{j \\in \\sigma_1}$ \nis $(K + c_1 \/ \\sqrt{\\delta})$-Besselian.\nNext, \n\\begin{eqnarray*}\n\\Big\\| \\sum_{j \\in \\sigma_1} a_j \\lambda_j y_j \\Big\\|\n  &\\le&  \\Big\\| \\sum_{j \\in \\sigma}   a_j \\lambda_j y_j \\Big\\|\n       + \\Big\\| \\sum_{j \\in \\rho} a_j \\lambda_j y_j \\Big\\|               \\\\\n  &\\le&  K \\Big( \\sum_{j \\in \\sigma}   |a_j|^2 \\Big)^{1\/2}        \n       + c \\Big( \\sum_{j \\in \\rho} |a_j|^2 \\Big)^{1\/2}              \\\\\n  &\\le&  \\sqrt{2} (K + c) \n           \\Big( \\sum_{j \\in \\sigma \\cup \\rho} |a_j|^2 \\Big)^{1\/2},\n\\end{eqnarray*}\nso the system $(\\lambda_j y_j)_{j \\in \\sigma_1}$ is $\\sqrt{2}(K + c)$-Hilbertian as well. \nThis establishes (a) of the Lemma\nand completes the proof of Theorem \\ref{main}.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\nLet us rewrite Theorem \\ref{main} in a different form,\nwhich is useful in applications like Dvoretzky-Rogers type lemmas.\n\n\\begin{theorem}                                                            \\label{mainc}\n  Let ${\\it id}  =  \\sum x_j \\otimes x_j$\n  be a decomposition on $l_2^n$,\n  and $T$ be a norm-one linear operator, $\\|T\\|_{\\rm HS}^2  =  h$.\n  Then for any integer $\\kappa  <  h$ there exists a set of indices $\\sigma$\n  with $|\\sigma| = \\kappa$ such that\n  \n  (i) the system $(T x_j)_{j \\in \\sigma}$ is $C(\\kappa \/ h)$-equivalent\n      to an orthogonal basis in $l_2^\\sigma$;\n\n  (ii) $\\|T x_j\\|  \\ge  c \\sqrt{\\frac{h - \\kappa}{n}} \\|x_j\\|$\n      for all $j \\in \\sigma$.\n\\end{theorem} \n\n\n\\noindent It is sometimes more useful to get a lower bound \nfor $\\< x_j, T x_j \\> $ rather than for $\\|T x_j\\|$.\n\n\\begin{proposition}                                                   \\label{mainscalar}\n  In Theorem \\ref{main} statement (ii) can be replaced by \n  \n  (ii') $| \\< x_j, T x_j \\> |  \n          \\ge  c \\varepsilon \\frac{|{\\rm trace} T|}{n} \\|x_j\\|^2$\n          for $j \\in \\sigma$.\n\\end{proposition}\n\n\\noindent {\\bf Proof.}\\ \\ \nNotice that $\\sum  \\< x_j, T x_j \\>  =  {\\rm trace} T$.\nTherefore, splitting our system $(x_j)_{j \\le m}$\nwe can assume that\n\\begin{equation}                                                       \\label{tracem}\n  | \\< x_j, T x_j \\> |  \n    \\ge  0.9 \\frac{|{\\rm trace} T|}{m}\n    \\ \\ \\ \\ \\mbox{for all $j$}.\n\\end{equation}\n\nLet us examine the proof of Theorem \\ref{main}.\nThe set $\\tau$ was responsible for the lower bound of $\\|T x_j\\|$.\nSo, we replace $\\tau$ by \n$$\n\\tau'  =  \\Big\\{ j \\le m :\n       | \\< x_j, T x_j \\> |  \n       \\ge  (\\varepsilon \/ 5) \\frac{|{\\rm trace} T|}{n} \\|x_j\\|^2\n        \\Big\\},\n$$\nand all we have to check is\n\\begin{equation}                                                   \\label{tdash}\n  |\\tau'|  \\ge  (1 - \\varepsilon\/3) m.\n\\end{equation}\nBy (\\ref{tracem})\n$$\n\\tau'  \\supset  \\rho  \n  =  \\Big\\{ j \\le m : \\|x_j\\|^2  \\le  (3 \/ \\varepsilon) \\frac{n}{m} \\Big\\}.\n$$\nSince $\\sum_{j \\le m} \\|x_j\\|^2  =  n$, \nwe have $|\\rho|  \\ge  (1 - \\varepsilon\/3) m$.\nThis verifies (\\ref{tdash}) and allows us \nto finish the proof as in Theorem \\ref{main}.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\n\n\n\n\n\\section{Principle of restricted invertibility}                    \\label{SecInvert}\n\nOur first application stems from the look at Theorem \\ref{main}\nas an extension of the \"principle of restricted invertibility\",\nTheorem \\ref{BTz}, proved by J.~Bourgain and L.~Tzafriri.\nIndeed, for the coordinate decomposition \n$id = \\sum e_j \\otimes e_j$ we get\n\n\\begin{corollary}                                                \\label{BTzgeneralized}\n  Let $T$ be a norm-one linear operator in $l_2$.\n  Then for any $\\varepsilon > 0$ there exists a subset $\\sigma \\subset \\{1, 2, \\ldots\\}$\n  of cardinality $|\\sigma|  \\ge  (1 - \\varepsilon) \\|T\\|_{\\rm HS}^2$ such that\n  the sequence $(T e_j)_{j \\in \\sigma}$ is $C(\\varepsilon)$-equivalent to an orthogonal\n  basis in Hilbert space.\n\\end{corollary}\n\nThis theorem generalizes the invertibility principle in two ways. \nFirst, instead of requiring that {\\em all} norms $\\|T e_j\\|$ \nbe large, we can assume only largeness of their average, \nwhich is the Hilbert-Schmidt norm of $T$.\nThis makes the result independent of the dimension $n$ of the space,\n$\\|T\\|_{\\rm HS}^2$ being a natural substitute for the dimension $n$.\n\nThe second improvement is that we obtain the subset $\\sigma$ with \nthe largest possible cardinality.\nCorollary \\ref{BTzgeneralized} allows us to get \n$|\\sigma|  \\ge  (1 - \\varepsilon) n \/ \\|T\\|^2$ in the original invertibility\nprinciple for any $0 < \\varepsilon < 1$\n(while the Bourgain-Tzafriri's argument proves only the \nexistence of such $\\varepsilon$).\nIn some applications one really needs almost full percentage;\nparticularly, this is important in estimating the distance to the \ncube, see \\cite{Sz-T}.\n\nNotice that the infinite-dimensional analogs of Theorem \\ref{main}\nand Corollary~\\ref{BTzgeneralized} hold, too.\n\n\\begin{proposition}\n  Let ${\\it id}  =  \\sum x_j \\otimes x_j$ be a decomposition \n  on a Hilbert space, and $T$ be a linear operator which is \n  not Hilbert-Schmidt. \n  Then for any $\\varepsilon > 0$ there exists an infinite subset $\\sigma$\n  such that the sequence $(T x_j)_{j \\in \\sigma}$ \n  is $(1 + \\varepsilon)$-equivalent to an orthogonal basis in Hilbert space. \n\\end{proposition}\n  \n\\noindent {\\bf Proof.}\\ \\ \nThe subset $\\sigma$ is constructed by a standard induction argument, \nmodulo the following claim:\n\n\\begin{itemize}\n\\item{For any finite dimensional subspace $E$\n$$\n\\sup_j {\\rm dist} (T x_j \/ \\|T x_j\\|, E)  =  1.\n$$\n}\n\\end{itemize}\n\nAssume the contrary. \nDenoting the orthogonal projection in $l_2$ onto $E$ by $P$, we would have \n$$\n\\inf_j  \\Big\\| P (T x_j \/ \\|T x_j\\|) \\Big\\|  =  \\delta  >  0,\n$$\nthat is \n$$\n\\|P T x_j\\|  \\ge  \\delta \\|T x_j\\| \n  \\ \\ \\ \\ \\mbox{for all $j$.}\n$$\nBy Lemma \\ref{hs}, this yields\n$$\n\\|PT\\|_{\\rm HS}  \\ge  \\delta \\|T\\|_{\\rm HS}  =  \\infty.\n$$\nBut the operator $PT$ has finite rank, \nthus $\\|PT\\|_{\\rm HS}$ must be finite. \nThis contradiction completes the proof. \n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\nAn infinite dimensional analogue of Bourgain-Tzafriri's Theorem \\ref{BTz}\nsays that, given a linear operator $T$ in $l_2$ with \n$\\|T e_j\\|  =  1$, $j = 1, 2, \\ldots$, \nthere exists a subset $\\sigma$ of $\\{1, 2, \\ldots \\}$\nwith upper density $\\overline{\\rm dens}\\; \\sigma  \\ge  c \/ \\|T\\|^2$, \nsuch that the sequence $(T e_j)_{j \\in \\sigma}$\nis $c$-Besselian \\cite{B-Tz}.\n\nAs we loose the normalizing condition $\\|T e_j\\| = 1$, nothing can be said\nin general about the density of $\\sigma$.\nIndeed, let $(y_k)$ be the result of a splitting of a canonical basis\nin $l_2$, so that the sets $\\sigma_j$ from the definition of splitting\nsatisfy $|\\sigma_j|  \\rightarrow  \\infty$ as $j  \\rightarrow  \\infty$.\nThere exist a norm-one operator $T$ in $l_2$ such that \n$\\|T\\|_{\\rm HS}  =  \\infty$ and\n$T e_k  =  y_k$, $k = 1, 2, \\ldots$.\nHowever, each term $y_k$ in the sequence $(y_k)$ is repeated $|\\sigma_j|$\ntimes. Therefore, the upper density of any subset $\\sigma$ satisfying\nthe conclusion of Corollary \\ref{BTzgeneralized} must be zero.\n\nSimilarly, in some cases $\\sigma$ must be a sparse set with respect \nto the dimension.\nMore precisely, in general the sequence $(T x_j)_{j \\in \\sigma}$ \nspans a subspace of infinite codimension.\nThis follows easily from a result of P.~Casazza and O.~Christensen\ndiscussed in Section \\ref{SecFrames}.\n\n\n\n\n\n\n\n\\section{Contact points and Dvoretzky-Rogers lemmas}                       \\label{SecDR}\n\nHere we apply Theorem \\ref{main} to John decompositions. \nLet $X = ({\\bf R}^n, \\| \\cdot\\|)$ be a Banach space. \nThe ellipsoid of maximal volume contained in $B(X)$ is unique \nand it is called {\\em maximal volume ellipsoid} of $X$.\nSuppose the euclidean structure on $X$ is chosen so that\nthe maximal volume ellipsoid coincides with the unit euclidean ball $D_n$.\nLet us write a John decomposition on $X$:\n\\begin{equation}                                               \\label{Johndecomposition}\n  {\\it id}  =  \\sum_{j = 1}^m  x_j \\otimes x_j,\n\\end{equation}\nwhere $x_j \/ \\|x_j\\|_X$ are some contact points of $B(X)$ with the John's \nellipsoid (see \\cite{T-J}, \\S 15.3).\nJohn decompositions can be considered as a subclass\nin the class of all decompositions of type ${\\it id}  =  \\sum x_j \\otimes x_j$.\nConversely, each decomposition (\\ref{Johndecomposition})\nis a John decomposition for a suitable Banach space \n$X = ({\\bf R}^n, \\| \\cdot\\|)$, whose maximal volume ellipsoid is the unit\neuclidean ball $D_n$.\nActually, the norm on $X$ is given by \n$\\|x\\|_X  =  \\max_{j \\le m} | \\< x, \\frac{x_j}{\\|x_j\\|_X} \\> |$.\nThis result goes back to F.~John \\cite{J}, \nalthough other proofs were found recently by \nK.~Ball \\cite{B2} and A.~Giannopoulos and V.~Milman \\cite{G-M}.\nTherefore, working with contact points instead of decompositions\n${\\it id}  =  \\sum x_j \\otimes x_j$ we do not lose generality.\n\nRecasting Theorem \\ref{main} in this light, we have\n\n\\begin{corollary} \n  Let $X$ be an $n$-dimensional Banach space whose maximal volume ellipsoid\n  is the unit euclidean ball. \n  Let $T$ be a linear operator with $\\|T\\|_{2 \\rightarrow 2}  \\le  1$.\n  Then for any $\\varepsilon > 0$ there are contact points $x_1, \\ldots, x_k$\n  with $k  \\ge  (1 - \\varepsilon) \\|T\\|_{\\rm HS}^2$\n  such that the system $(Tx_j)_{j \\le k}$ is $C(\\varepsilon)$-equivalent \n  in $l_2$-norm to the canonical basis of $l_2^k$.\n\\end{corollary}\n\n\\noindent Moreover, the norms $\\|T x_j\\|_2$ are well bounded below:\n$$\n\\|T x_j\\|  \\ge  C_1(\\varepsilon) \\frac{\\|T\\|_{\\rm HS}}{\\sqrt{n}}\n\\ \\ \\ \\ \\mbox{for all $j \\le k$.}\n$$\n\n$T$ being an orthogonal projection, this result leads us to \na new Dvoretzky-Rogers type lemma , which we will discuss now. \nSuppose $X$ is an $n$-dimensional Banach space whose John's \nellipsoid is the unit euclidean ball. \nLet $Z$ be a $k$-dimensional subspace of $X$.\nDvoretzky-Rogers Lemma states that, given a positive integer\n$\\kappa < k$,\nthere is an orthonormal system $(z_j)_{j  \\le  \\kappa}$ in $Z$ such that\n$$\n\\|z_j\\|_X  \\ge  \\sqrt{\\frac{k - \\kappa + 1}{n}}\n  \\ \\ \\ \\ \\mbox{for all $j  \\le  \\kappa$}.\n$$\nLet us sketch the proof. By induction, it is enough to find {\\em one}\nvector $z$ in $Z$ such that $\\|z\\|_2 = 1$ and \n$\\|z\\|_X  \\ge  \\sqrt{\\frac{k}{n}}$\n(then substitute $Z$ by $Z  \\ominus  {\\rm span}(z)$ and repeat the argument).\nBy duality, this is equivalent to finding a functional \n$x^*  \\in  B(X^*)$ with $\\|P x^*\\|_2  \\ge  \\sqrt{\\frac{k}{n}}$,\nwhere $P$ is the orthogonal projection onto $Z$.\nLet ${\\it id}_{X^*}  =  \\sum \\lambda_j x_j^* \\otimes x_j^*$\nbe a John decomposition in $X^*$, that is $\\sum \\lambda_j  =  n$\nand $x_j^*$ are contact points of $B(X^*)$.\nThen $P  =  \\sum \\lambda_j P x_j^* \\otimes P x_j^*$.\nTaking the trace, we get $k  =  \\sum \\lambda_j \\|P x_j^*\\|_2^2$.\nSo, there is a $j$ such that $\\|P x_j^*\\|_2^2  \\ge  \\sqrt{\\frac{k}{n}}$.\nThis completes the proof.\n\nHowever, this argument, as well as other known proofs of the \nDvoretzky-Rogers Lemma, only establish the existence of the vectors $z_j$.\nIn contrast to that, the argument based on Theorem \\ref{main}\nprovides information about their position.\n\n\\begin{theorem}                                                           \\label{DR}\n  Let $X$ be an $n$-dimensional Banach space \n  whose maximal volume ellipsoid is the unit euclidean ball.\n  Let $P$ be an orthogonal projection, ${\\rm rank} P = k$.\n  Then for any positive integer $\\kappa  <  k$\n  there are contact points $x_1, \\ldots, x_\\kappa$\n  such that setting $z_j  =  P x_j \/ \\|P x_j\\|_2$\n  we have\n  \n  (i)  the system $(z_j)$ is $C(\\kappa \/ k)$-equivalent in $l_2$-norm \n       to the canonical basis of $l_2^\\kappa$;\n\n  (ii) $\\|z_j\\|_X  \\ge  c \\sqrt{\\frac{k - \\kappa}{n}}$\n       for all $j$.\n\\end{theorem}\n\n\n\\noindent {\\bf Proof.}\\ \\ \nNote that $\\|P\\|_{\\rm HS}  =  \\sqrt{k}$ and apply Theorem \\ref{mainc}\nto a John decomposition on $X$. \nThis gives us contact points $x_1, \\ldots, x_\\kappa$ such that\n(i) is satisfied, and\n$$\n\\|P x_j\\|_2  \\ge  c \\sqrt{\\frac{k - \\kappa}{n}}\n  \\ \\ \\ \\ \\mbox{for every $j \\le m$}.\n$$\nIt remains to note that for every $j \\le m$\n$$\n\\|P x_j\\|_X  \\ge  \\frac{\\< P x_j, x_j \\> }{\\|x_j\\|_{X^*}}\n  =  \\< P x_j, x_j \\>  =  \\|P x_j\\|_2^2,\n$$\nhence\n$$\n\\|z_j\\|_X  \\ge  \\|P x_j\\|_2\n  \\ \\ \\ \\ \\mbox{for every $j \\le m$}.\n$$\nThis completes the proof.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\nA natural question is whether Theorem \\ref{DR} \ncan be extended for arbitrary operator $T$, \n$\\|T\\|_{2 \\rightarrow 2} = 1$, \nwith $k$ substituted by $\\|T\\|_{\\rm HS}^2$. \nThe answer is negative. \nIndeed, let $n  =  2^m$ for a positive integer $m$,\nand denote by $W_m$ the Walsh $n \\times n$ matrix.\nConsider the operator $T  =  n^{-1\/2} W_m$\nacting in the space $X  =  l_\\infty^n$.\nAll contact points $(x_j)$ of $X$ are simply the\ncoordinate vectors. However, denoting $z_j = T x_j \/ \\|T x_j\\|_2$\nwe have for any $j$\n$$\n\\|z_j\\|_X  =  \\|T x_j\\|_X \/ \\|T x_j\\|_2  =  n^{-1\/2}.\n$$\nThis shows that (ii) in Theorem \\ref{DR} would fail \nfor the operator $T$.\n\nStill, a lower bound for $\\| \\cdot \\|_X$-norm exists\nand is equivalent to $\\frac{1}{n} |{\\rm trace}{T}|$.\n\n\\begin{proposition}\n  Let $X$ be an $n$-dimensional Banach space\n  whose maximal volume ellipsoid is the unit euclidean ball.\n  Let $T$ be an operator with $\\|T\\|_{2 \\rightarrow 2} \\le 1$.\n  Then for any $\\varepsilon > 0$\n  there are $k  >  (1 - \\varepsilon) n$ contact points $x_1, \\ldots, x_\\kappa$\n  such that \n  \n  (i)  the system $(T x_j)$ is $C(\\varepsilon)$-equivalent in $l_2$-norm \n       to the canonical basis of $l_2^\\kappa$;\n\n  (ii) $\\|T x_j\\|_X  \\ge  c \\varepsilon \\frac{|{\\rm trace} T|}{n}$.\n\\end{proposition}\n\n\\noindent {\\bf Proof.}\\ \\ \nThe argument is similar to that of Theorem \\ref{DR};  \none only uses Proposition \\ref{mainscalar}\ninstead of Theorem \\ref{main}.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\nThe estimate in (ii) is essentially sharp. \nIndeed, for any positive integers $k \\le n$\none can construct an orthogonal projection $P$\nin ${\\bf R}^n$ with ${\\rm rank} P = k$, and such that\n$$\n\\|P e_j\\|_2  =  \\sqrt{\\frac{k}{n}}\n\\ \\ \\ \\ \\mbox{for all $j \\le n$}.\n$$\nNotice that \n$\\|P\\|_{1 \\rightarrow 2}       =  \\sqrt{\\frac{k}{n}}$, \ntherefore \n$\\|P\\|_{2 \\rightarrow \\infty}  =  \\sqrt{\\frac{k}{n}}$.\nThus \n$\\|P\\|_{1 \\rightarrow \\infty}  \\le  \\frac{k}{n}$.\nNow consider the space $X  =  l_\\infty^n$\nand the operator $T = P$ on it.\nThe only contact points $(x_j)$ of $X$ are the coordinate \nvectors $(e_j)$. Then for all $j$\n$$\n\\|T x_j\\|_X  =  \\|P e_j\\|_\\infty  \n  \\le  \\frac{k}{n}  \n   =  \\frac{|{\\rm trace} T|}{n}.\n$$\nThis shows that the lower bound in (ii) is essentially sharp.\n\nFinally, there is a class of operators for which \nTheorem \\ref{DR} itself can be extended: selfadjoint operators.\nSo, the desired general result for arbitrary operator can be obtained\nif we allow a suitable rotation of $(z_j)$.\nThere is always a unitary operator (coming from the polar\ndecomposition of $T$) which sends the vectors $(z_j)$\nto vectors of a good $\\| \\cdot \\|_X$-norm.\n\n\\begin{theorem}                                                    \n  Let $X$ be an $n$-dimensional Banach space \n  whose maximal volume ellipsoid is the unit euclidean ball.\n  Let $T$ be an operator with $\\|T\\|_{2 \\rightarrow 2} \\le 1$, \n  and put $\\|T\\|_{\\rm HS}^2  =  h$.\n  Then for any positive integer $\\kappa  <  h$\n  there are contact points $x_1, \\ldots, x_\\kappa$\n  such that setting $z_j  =  |T| x_j \/ \\|T x_j\\|_2$\n  we have\n  \n  (i)  the system $(z_j)$ is $C(\\kappa \/ h)$-equivalent in $l_2$-norm \n       to the canonical basis of $l_2^\\kappa$;\n\n  (ii) $\\|z_j\\|_X  \\ge  c \\sqrt{\\frac{h - \\kappa}{n}}$\n       for all $j$.\n\\end{theorem}\n\n\\noindent {\\bf Proof.}\\ \\ \nLet $T = U |T|$ be the polar decomposition of $T$, \nwhere $|T| = (T^* T)^{1\/2}$ is a positive selfadjoint operator\nand $U$ is a partial isometry on $l_2^n$.\nApply Theorem \\ref{mainc} to the operator $|T|$\nand a John decomposition on $X$. \nAs before, this gives us contact points $x_1, \\ldots, x_\\kappa$ \nsuch that (i) is satisfied, and\n$$\n\\Big\\| |T| x_j \\Big\\|_2  \\ge  c \\sqrt{\\frac{h - \\kappa}{n}}\n  \\ \\ \\ \\ \\mbox{for every $j \\le m$}.\n$$\nFrom diagonalization of $|T|$ it follows that\n$\\Big\\| |T|^{1\/2} \\Big\\|_{2 \\rightarrow 2}  =  \\|T\\|_{2 \\rightarrow 2}^{1\/2}  \\le  1$.  \nTherefore we can bound for every $j \\le m$\n\\begin{eqnarray*}\n\\Big\\| |T| x_j \\Big\\|_X  \n &\\ge&  \\frac{\\< |T| x_j, x_j \\> }{\\|x_j\\|_{X^*}}\n   =    \\Big\\| |T|^{1\/2} x_j \\Big\\|_2^2                \\\\\n &\\ge&  \\Big\\| |T| x_j \\Big\\|_2^2\n   =    \\Big\\| T x_j \\Big\\|_2^2.\n\\end{eqnarray*}\nHence\n$$\n\\|z_j\\|_X  \\ge  \\|T x_j\\|_2\n  \\ \\ \\ \\ \\mbox{for every $j \\le m$}.\n$$\nThis completes the proof.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\n\n\n\n\n\nNow we turn to a particular case when $T$ is the identity operator, \nwhich also happens to be interesting. We clearly have \n\n\\begin{corollary}                                            \\label{contact}\n  Let $X$ be an $n$-dimensional Banach space \n  whose maximal volume ellipsoid is the unit euclidean ball.\n  Then for any $\\varepsilon > 0$ there is a set of $k  >  (1 - \\varepsilon) n$\n  contact points which is $C(\\varepsilon)$-equivalent to \n  the canonical vector basis of $l_2^k$.\n\\end{corollary}\n\n\\noindent This result is related to another variant of the \nclassical Dvoretzky-Rogers Lemma, which establishes the existence\nof contact points whose distance to a certain orthonormal basis\nis controlled (\\cite{T-J}, Theorem 15.7).\nMore precisely, there exist contact points $x_1, \\ldots, x_n$\nand an orthonormal basis $h_1, \\ldots, h_n$ such that\n$$\n\\|x_j - h_j\\|_2  \\le  2 \\Big(1 - \\sqrt{\\frac{n - i + 1}{n}} \\Big)\n\\ \\ \\ \\ \\mbox{for $i  \\le  n$}.\n$$\nHowever, this estimate is to crude to assure that a fixed \nproportion of the system $(x_j)_{j \\le n}$ is equivalent \nin $l_2$-norm to an orthonormal system. Such an isomorphism \ncan be established only for $c \\sqrt{n}$ contact points. \n\nUsing this argument, it is proved that for $k = [\\sqrt{n} \/ 4]$\nthere exist orthonormal vectors \n$x_1, \\ldots, x_k$ in $(E, \\|\\cdot\\|_2)$ on which all three norms \n$\\|\\cdot\\|$, $\\|\\cdot\\|_2$, and $\\|\\cdot\\|_*$ \ndiffer by the factor $2$ at most (\\cite{T-J}, p.127). \nIt has been an open problem to make $k$ proportional to $n$.\nBy Corollary \\ref{contact} we actually have $k  \\ge  (1 - \\varepsilon)n$\nand make all three norms $\\|\\cdot\\|$, $\\|\\cdot\\|_2$, and $\\|\\cdot\\|_*$\nequal to $1$ on our sequence (paying however in exact orthogonality).\n\nBy duality, Corollary \\ref{contact} holds also for the ellipsoid of minimal \nvolume containing $B(X)$ instead of the maximal volume ellipsoid. \nThis variant of Corollary \\ref{contact} yields also a proportional \nDvoretzky-Rogers factorization result from \\cite{B-Sz}. \nNamely, given an $n$-dimensional Banach space $X$ and $\\varepsilon > 0$, \nthere is a $k > (1 - \\varepsilon) n$ such that the formal identity \n${\\it id} : l_2^k \\rightarrow l_\\infty^k$\ncan be written as ${\\it id} = \\alpha \\cdot \\beta$ for some \n$\\beta : l_2^k \\rightarrow X$, $\\alpha : X \\rightarrow l_\\infty^k$, \nwith $\\|\\alpha\\| \\|\\beta\\|  \\le  C(\\varepsilon)$.\nThis can be obtained by duality from the factorization of the identity\non the contact points, ${\\it id} : l_1^k  \\rightarrow  X  \\rightarrow  l_2^k$, \nguaranteed by Corollary \\ref{contact}.\n\nA few comments about the dependence $C(\\varepsilon)$ in Theorem \\ref{main}.\nIt is a challenge to find the correct asympthotics. \nIndeed, by an argument of S.~Szarek and M.~Talagrand \\cite{Sz-T}\nthe proportional Dvoretzky-Rogers factorization above\nyields a non-trivial estimate on the distance from $X$ to $l_\\infty^n$ --\na well known and hard problem in the local theory of normed spaces.\nThe factorization constant $C(\\varepsilon)$ lies in the heart of the computation\nof this distance.\n\nThe proof of Theorem \\ref{main} guarantees that\n$C(\\varepsilon)  \\le  \\varepsilon^{c \\log \\varepsilon}$.\nHowever, for the Dvoretzky-Rogers factorization constant \n$C_{\\rm DR} (\\varepsilon)$ much better bounds are found \\cite{G}: \n$C_{\\rm DR} (\\varepsilon)  \\le  c \\varepsilon^{-1}$. \nSince $C_{\\rm DR} (\\varepsilon)  \\le  C(\\varepsilon)$ and \n$C_{\\rm DR} (\\varepsilon)  \\rightarrow  \\infty$ as $\\varepsilon \\rightarrow 0$  (see \\cite{Sz-T}), \nwe necessarily have $C(\\varepsilon)  \\rightarrow  \\infty$ as $\\varepsilon \\rightarrow 0$.\n\nHowever, $C(\\varepsilon)  \\rightarrow  1$ as $\\varepsilon \\rightarrow 1$.\nThis follows directly from \n\n\\begin{lemma}\n  Let a normalized sequence $(x_j)_{j \\le n}$ in Hilbert space \n  be $M$-Hilbertian, and $\\varepsilon > 0$.\n  Then there is a subset $\\sigma  \\subset  \\{1, \\ldots, n\\}$\n  of cardinality $|\\sigma|  \\ge  C(M, \\varepsilon) n$ such that\n  the system $(x_j)_{j \\in \\sigma}$ is $(1 + \\varepsilon)$-equivalent\n  to the canonical basis of $l_2^\\sigma$.\n\\end{lemma}\n\n\\noindent {\\bf Proof.}\\ \\ \nWe can assume that the given Hilbert space is $l_2^n$.\nDefine a linear operator $T : l_2^n  \\rightarrow  l_2^n$ by \n$$\nT e_j  =  x_j  \n\\ \\ \\ \\ \\mbox{for $j \\le n$.}\n$$\nLet $A = T^*T - {\\it id}$.\nThen the matrix of $A$ has zeros on the diagonal and \n$\\|A\\|  \\le  M^2 + 1$.\nNow, by a theorem of J.~Bourgain and L.~Tzafriri\n(\\cite{B-Tz} Theorem 1.6, see also \\cite{K-Tz})\nthere is a subset $\\sigma  \\subset  \\{1, \\ldots, n\\}$ \nof cardinality $|\\sigma|  \\ge  C(M, \\varepsilon)$ such that\n$$\n\\| P_\\sigma A P_\\sigma \\|  <  \\varepsilon.\n$$\nThis shows that for any sequence of scalars $(a_j)_{j \\le n}$\n$$\n\\Big| \\Big\\langle\n(T^*T - {\\it id}) \\sum_{j \\in \\sigma} a_j e_j , \\sum_{j \\in \\sigma} a_j e_j\n\\Big\\rangle \\Big|\n<  \\varepsilon,\n$$\nhence\n$$\n\\Big| \\Big\\langle\n\\sum_{j \\in \\sigma} a_j x_j , \\sum_{j \\in \\sigma} a_j x_j\n\\Big\\rangle \n  -  \\sum_{j \\in s} |a_j|^2\n\\Big|\n<  \\varepsilon.\n$$\nThis clearly finishes the proof.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\n\n\n\n\n\n\n\n\n\\section{Embeddings of the cube}                                     \\label{SecCube}\n\nIn this section we apply Theorem \\ref{main} to the study \nof embeddings of $l_\\infty^k$ into finite dimensional spaces. \nN.~Alon and V.~Milman proved that if a given normalized sequence $(x_j)$ \nin a Banach space $X$ has small Rademacher average ${\\bf E} \\| \\sum \\varepsilon_j x_j\\|_X$, \nthen it must contain a large subsequence well equivalent \nto the canonical basis of $l_\\infty^k$.\nLater on, M.~Talagrand \\cite{T} improved this result \nand simplified the argument.\n\n\\begin{theorem} (M.~Talagrand).  \\                                   \\label{Talagrand}\n  Suppose we are given vectors $(x_j)_{j \\le n}$ in a Banach space $X$\n  with $\\|x_j\\|_X  \\ge  1$.\n  Set $M = {\\bf E} \\| \\sum \\varepsilon_j x_j \\|_X$ \n  and $\\omega = \\sup \\Big\\{ \\sum |x^*(x_j)| : x^* \\in B(X^*) \\Big\\}$.\n  Then there exists a subset $\\sigma  \\subset  \\{1, \\ldots, n\\}$ \n  of cardinality $|\\sigma|  \\ge  c n \/ \\omega$ such that\n  $$\n  \\frac{1}{2} \\max_{j \\in \\sigma} |a_j|\n    \\le  \\Big\\| \\sum_{j \\in \\sigma} a_j x_j \\Big\\|_X\n    \\le   4 M \\max_{j \\in \\sigma} |a_j|\n  $$  \n  for any choice of scalars $(a_j)$.\n\\end{theorem}\n\nA few years earlier, M.~Rudelson obtained in \\cite{R1} \na \"Gaussian\" version of this theorem.\nRecall that the $\\ell$-norm of an operator $u : l_2^n  \\rightarrow  X$\nis defined as \n$$\n\\ell(u)^2  =  \\int \\|ux\\|_X^2 \\, d \\gamma_n(x),\n$$\nwhere $\\gamma_n$ is the canonical Gaussian measure on ${\\bf R}^n$.\nWe will sometimes write $\\ell(X)$ instead of $\\ell({\\it id}_X)$.\n\nSuppose $X$ is an $n$-dimensional Banach space whose maximal volume ellipsoid\nis the unit euclidean ball.\nLet $P$ be an orthogonal projection in $X$ \nand set $k  =  {\\rm rank} P$, $a  =  k \/ n$.\nThe result of M.~Rudelson states that there is a subspace \n$Z  \\subset  P(X)$  of dimension $m  \\ge  C_1(a) \\frac{\\sqrt{n}}{\\ell(P)}$\nwhich is $C_2(a) \\ell(P)$-isomorphic to $l_\\infty^m$.\n\nWe will remove $\\ell(P)$ from the estimate on the dimension.\nFurther, it will be shown that $Z$ is canonically spanned by the projections\nof some contact points of $X$.\nIn particular, the norm on $Z$ is well equivalent to \n$\\max_{j \\le m} | \\< z , x_j \\> |$,\nwhere $x_j$ are contact points.\nThis yields automatically that $Z$ is well complemented \nby the orthogonal projection. \nMoreover, the dependence on $a$ will be improved.\n\n\\begin{theorem}                                                    \\label{GT}\n  Let $X$ be an $n$-dimensional Banach space \n  whose maximal volume ellipsoid is $B(l_2^n)$.\n  Let $P$ be an orthogonal projection, ${\\rm rank} P = k$.\n  Then there are contact points $(x_j)_{j  \\le  m}$\n  with $m  \\ge  c_1 k \/ \\sqrt{n}$ such that \n  $$\n  \\max_{j \\le m} | \\< x , x_j \\> |    \\le   \\|x\\|_X    \n  \\le   c \\sqrt{\\frac{n}{k}} \\ell(P) \\max_{j \\le m} | \\< x , x_j \\> |\n  $$\n  for every $x$ in $Z  =  {\\rm span}(P x_j)_{j \\le m}$.\n\\end{theorem}\n\nA particular case $k = n$ is also interesting: \nwe get a sequence of $m  \\ge  c_1 \\sqrt{n}$ contact points \nwhich is $c \\ell(X)$-equivalent to the canonical vector basis \nof $l_\\infty^m$.\nLet us prove this latter fact separately. \nBy Corollary \\ref{contact}, there exists a set of contact points \n$(x_j)_{j \\le m'}$, $m'  \\ge  n \/ 2$, \nwhich is $c$-equivalent in $l_2$-norm \nto the canonical basis of $l_2^{m'}$.\nLet $(g_j)$ be independent standard Gaussian random variables.\nTo apply Talagrand's Theorem \\ref{Talagrand}, we bound\n$$\nM   =   {\\bf E} \\| \\sum \\varepsilon_j x_j \\|_X\n   \\le  c {\\bf E} \\| \\sum g_j x_j \\|_X\n   \\le  c c_2 {\\bf E} \\| \\sum g_j e_j \\|_X\n$$\nby the ideal property of the $\\ell$-norm, see Lemma \\ref{Slepian} below. \nFurther, for any finite system of scalars $(a_j)$\n\\begin{eqnarray*}\n\\| \\sum a_j x_j \\|_X   \n  &\\le&   \\| \\sum a_j x_j \\|_2                     \\\\\n  &\\le& c_2 \\Big( \\sum |a_j|^2 \\Big)^{1\/2}          \n   \\le  c_2 \\sqrt{n} \\max_j |a_j|.\n\\end{eqnarray*}\nThus\n$$\n\\omega  =  \\sup \\Big\\{ \\sum |x^*(x_j)| : x^* \\in B(X^*) \\Big\\}\n   \\le c_2 \\sqrt{n}.\n$$\nNow Talagrand's Theorem \\ref{Talagrand} finishes the proof. \n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\nThe proof of Theorem \\ref{GT} is longer, but the main idea remains \nto combine results of Section \\ref{SecDR} with Talagrand's theorem. \nFirst, we need to know what vectors canonically span a large\nsubspace $Z$ well isomorphic to $l_\\infty^m$.\nThey happen to be multiples of some contact points $(x_j)_{j \\le m}$.\nMore precisely, we have\n\n\\begin{proposition}                                                           \\label{spancube}\n  Under the assumptions of Theorem \\ref{GT}, the system \n  $\\Big( \\frac{Px_j}{\\|Px_j\\|_2} \\Big)_{j \\le m}$\n  is $c \\sqrt{\\frac{n}{k}} \\ell(P)$-equivalent \n  to the canonical basis of $l_\\infty^m$.\n\\end{proposition}\n\nTo prove this, let $(x_j)_{j \\le k'}$, $k'  \\ge  k \/ 2$,\nbe the contact points provided by Theorem \\ref{DR}.\nPut \n$$\nz_j  =  \\alpha \\sqrt{\\frac{n}{k}} \\frac{Px_j}{\\|Px_j\\|_2}\n$$\nfor an appropriate $\\alpha > 0$ and all $j$.\nThen \n\n(i)  the system $(z_j)_{j \\le k'}$ is $(c \\sqrt{\\frac{n}{k}})$-equivalent\n     in $l_2$-norm to the canonical basis of $l_2^{k'}$;\n\n(ii) $\\|z_j\\|  \\ge  1$ for all $j$.\n\n\\noindent For a future reference note that the proof of \nTheorem \\ref{DR} gives also \n\\begin{equation}                                                                 \\label{Pxj}\n  \\|Px_j\\|_2  \\ge  c_1 \\sqrt{\\frac{n}{k}}.\n\\end{equation}\n\nTo apply Talagrand's Theorem \\ref{Talagrand} to the system $(z_j)_{j \\le k'}$,\nrecall the ideal property of the $\\ell$-norm (cf. \\cite{T-J}, \\S 12)\n\n\\begin{lemma}                                                                   \\label{Slepian}\n  For any two linear operators \n  $A : l_2^n  \\rightarrow  X$ and $B : l_2^n  \\rightarrow  l_2^n$\n  $$\n  \\ell(AB)  \\le  \\|B\\| \\ell(A).\n  $$\n\\end{lemma}\nLet $(h_j)_{j \\le k'}$ be an orthonormal basis in ${\\rm span}(z_j)$.\nNow we bound \n\\begin{eqnarray*}\nM  & = &    {\\bf E} \\| \\sum \\varepsilon_j z_j \\|_X  \n    \\le   c {\\bf E} \\| \\sum  g_j z_j \\|_X                          \\\\\n   &\\le&  c \\sqrt{\\frac{n}{k}} {\\bf E} \\| \\sum g_j h_j \\|_X         \n          \\ \\ \\ \\ \\mbox{by (i) and Lemma \\ref{Slepian}}       \\\\\n   & = &  c \\sqrt{\\frac{n}{k}} \\ell(P(X))\n     =    c \\sqrt{\\frac{n}{k}} \\ell(P).\n\\end{eqnarray*}\nNoting (ii) above, we apply Talagrand's Theorem \\ref{Talagrand}.\nThis finishes the proof. \n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\n\nTo obtain Theorem \\ref{GT}, Talagrand's Theorem will be used more delicately.\nIts proof in \\cite{T} gives the following additional property.\n\n\\begin{lemma}                                                                 \\label{Talagrandplus}\n  In the situation of Theorem \\ref{GT}, suppose $(x_j^*)_{j \\le n}$\n  are functionals in $X^*$ such that \n  $$\n  x_j^* (x_j)  \\ge  1  \\ \\ \\ \\ \\mbox{and} \\ \\ \\ \\ \n  \\|x_j^*\\|_{X^*}  =  1      \\ \\ \\ \\ \\mbox{for all $j \\le n$.}\n  $$\n  Then the subset $\\sigma$ can be found so that \n  $$\n  \\sum_{j  \\in  \\sigma \\setminus \\{i\\}} |x_i^* (x_j)|  \\le  1\/2\n  \\ \\ \\ \\ \\mbox{for all $i \\in \\sigma$}.\n  $$\n\\end{lemma}\n\n\nLet us turn again to the proof of Proposition \\ref{spancube}.\nWe applied Talagrand's Theorem to the system $(x_j)_{j \\le k'}$.\nNote that by (\\ref{Pxj})\n\\begin{eqnarray*}\n\\< x_j, z_j \\>  & = &  \\alpha \\sqrt{\\frac{n}{k}} \\frac{\\< x_j, Px_j \\> }{\\|Px_j\\|_2}  \\\\\n  & = &   \\alpha \\sqrt{\\frac{n}{k}} \\|Px_j\\|_2\n   \\ge  1   \\ \\ \\ \\ \\mbox{for all $j$.}\n\\end{eqnarray*}\nTherefore we obtain a subset $\\sigma  \\subset \\{1, \\ldots, k'\\}$\nof cardinality $|\\sigma|  \\ge  c_1 k \/ \\sqrt{n}$ such that\n\\begin{itemize}\n  \\item{the system $(z_j)_{j \\in \\sigma}$ \n        is $c \\sqrt{\\frac{n}{k}} \\ell(P)$-equivalent to the \n        canonical basis of $l_\\infty^\\sigma$;}\n  \\item{$\\sum_{j  \\in  \\sigma \\setminus \\{i\\}} |\\< x_i , z_j\\> |  \\le  1\/2$\n        for all $j \\in \\sigma$.} \n\\end{itemize}\nNow fix an $x  =  \\sum_{j \\in \\sigma} a_j z_j$,\nand let $i \\in \\sigma$ be such that $|a_i|  =  \\max_{j \\in \\sigma} |a_j|$.\nThen\n\\begin{eqnarray*}\n\\max_{j \\in \\sigma} |\\< x, x_j \\> |  \n  &\\ge&  |\\< x_i, x \\> |\n    =    \\Big| \\sum_{j \\in \\sigma} a_j \\< x_i, z_j \\> \\Big|          \\\\\n  &\\ge&  |a_j| \\Big( |\\< x_i, z_i \\> |  \n           -  \\sum_{j  \\in  \\sigma \\setminus \\{i\\}} | \\< x_i, z_j \\> | \\Big) \\\\\n  &\\ge&  \\frac{1}{2} |a_i|\n    =    \\frac{1}{2} \\max_{j \\in \\sigma} |a_j|                    \\\\\n  &\\ge&  \\Big( c \\sqrt{\\frac{n}{k}} \\ell(P) \\Big)^{-1} \\|x\\|_X.\n\\end{eqnarray*}\nThis proves the second inequality in Theorem \\ref{GT}, \nwhile the first one is obvious.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\nTheorem \\ref{GT} yields also that $Z$ is well complemented by \nthe orthogonal projection. \n\n\\begin{proposition}\n  In the situation of Theorem \\ref{GT}, \n  let $P_Z$ be the orthogonal projection in $X$ onto $Z$. Then \n  $$\n  \\|P_Z\\|  \\le  c \\sqrt{\\frac{n}{k}} \\ell(P).\n  $$\n\\end{proposition}\n\n\\noindent {\\bf Proof.}\\ \\  \nFor any $x \\in X$\n\\begin{eqnarray*}\n\\|P_Z x\\|_X  \n  &\\le&  c \\sqrt{\\frac{n}{k}} \\ell(P) \\max_{j \\le m} | \\< P_Z x, x_j \\> | \\\\\n  & = &  c \\sqrt{\\frac{n}{k}} \\ell(P) \\max_{j \\le m} | \\< x, x_j \\> |     \\\\\n  &\\le&  c \\sqrt{\\frac{n}{k}} \\ell(P) \\|x\\|_X.\n\\end{eqnarray*}\nThis completes the proof.\n{\\mbox{}\\nolinebreak\\hfill\\rule{2mm}{2mm}\\par\\medbreak} \n\n\nAnother consequence of Theorem \\ref{GT} is a refined isomorphic \ncharacterization of spaces with large volume ratio.\nRecall that the volume ratio of an $n$-dimensional Banach space $X$\nis defined as\n$$\n{\\rm vr}(X)  =  \\min \\left( \\frac{{\\rm Vol}(B_X)}{{\\rm Vol}(E)} \\right)^{1\/n}\n$$\nover all ellipsoids $E$ contained in $B_X$, see \\cite{L-M}, \\cite{P}.\nThe maximal value of ${\\rm vr}(X)$ among all $n$-dimensional spaces \nis of order $\\sqrt{n}$, \nand the only space with maximal volume ratio is $l_\\infty^n$\n(K.~Ball \\cite{B1}).\nLater on, M.~Rudelson proved in \\cite{R1} that\nif ${\\rm vr}(X)$ is proportional to the maximal volume ratio \nthen $X$ has a subspace isomorphic to $l_\\infty^m$ with \nthe isomorphism constant of order $\\log n$, \nand such that \n$m  \\sim  \\frac{\\sqrt{n}}{\\log n}$.\nUsing Theorem \\ref{GT} in the M.~Rudelson's proof of this\nresult removes the parasitic factor $\\log n$ from the estimate \non the dimension. \n\n\\begin{theorem}\n  Let $a > 0$, and $X$ be an $n$-dimensional Banach space. \n  If ${\\rm vr}(X)  \\ge  a \\sqrt{n}$ \n  then there exists a subspace $Z$ of dimension \n  $m  \\ge  C_1(a) \\sqrt{n}$ which is $C_2(a) \\log n$-isomorphic \n  to $l_\\infty^m$.\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n\\section{Frames}                                                     \\label{SecFrames}\n\nThe notion of frame goes back to the work of R.~Duffin and A.~Schaeffer\non nonharmonic Fourier series \\cite{D-S}.\nA sequence $(x_j)$ in a Hilbert space $H$ is called a {\\em frame} \nif there exist positive numbers $A$ and $B$ such that\n$$\nA \\|x\\|^2  \\le  \\sum_j | \\langle x, x_j \\rangle |^2  \\le  B \\|x\\|^2\n\\ \\ \\ \\ \\mbox{for $x \\in H$.}\n$$\nThe number $(B\/A)^{1\/2}$ is called a {\\em constant} of the frame.\nWe call $(x_j)$ a {\\em tight frame} if $A = B = 1$.\nFor introduction to the theory of frames, its relation to wavelets and signal \nprocessing, see \\cite{B-W}.\nGeometric structure of frames is studied extensively in recent years, see\n\\cite{Ho}, \\cite{A}, \\cite{C-C1}, \\cite{C-C2}, \\cite{C2}, \\cite{V}.\n\nIt is known by now that finite dimensional frames are essentially the \nsame object as John decompositions. \nIn the equivalent theory, it is sufficient to work only with tight frames,\nbecause every frame with constant $M$ is $M$-equivalent to a tight frame\n(cf. e.g. \\cite{Ho}).\nFurther, one has the following equivalence between frames and \nJohn decompositions:\n$$\n\\mbox{ $(x_j)$ is a tight frame in $H$ }   \n  \\iff   {\\it id}_H  =  \\sum_j x_j \\otimes x_j.\n$$\nThis observation allows to interpret the results of \nSections \\S \\ref{SecMain} and \\S \\ref{SecInvert} as statements about frames.\nTheorem \\ref{main} yields:\n\n\\begin{corollary}                                                    \\label{frame}\n  Let $(x_j)$ be a tight frame in Hilbert space $H$,\n  and $T$ be a norm-one linear operator in $H$. \n  Then for any $\\varepsilon > 0$ there exists a subset of indices $\\sigma$\n  of cardinality $|\\sigma|  \\ge  (1 - \\varepsilon) \\|T\\|_{\\rm HS}^2$ such that\n  the system $(T x_j)_{j \\in \\sigma}$ is $C(\\varepsilon)$-equivalent\n  to an orthogonal basis in Hilbert space.\n\\end{corollary}\nFor operators $T$ which are not Hilbert-Schmidt this means \nthat the subset $\\sigma$ is infinite.\n\nClearly, Corollary \\ref{frame} itself generalizes the invertibility\nprinciple of J.~Bourgain and L.~Tzafriri.\nWhen applied to the identity operator, it yields that every tight \nframe in $l_2^n$ has a subset of length $(1 - \\varepsilon) n$ \nwhich is $C(\\varepsilon)$-equivalent to an orthogonal basis in Hilbert space.\nThis result was proved in \\cite{V} as a generalization of \nP.~Casazza's theorem \\cite{C2}.\n\nNotice that one necessarily has $C(\\varepsilon)  \\rightarrow  \\infty$ as $\\varepsilon  \\rightarrow  1$,\nas explained in Section \\S \\ref{SecDR}.\nAn infinite dimensional analog of this phenomenon holds, too.\nA frame may not in general contain a complete subsequence equivalent\nto an orthogonal basis. The counterexample was found \nby P.~Casazza and O.~Christensen in \\cite{C-C2}, see also \\cite{V}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n{\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThere is now substantial evidence that environments of intermediate density,\nsuch as galaxy groups, play an important role in the transformation of field\ngalaxies into the passively evolving populations found in galaxy clusters.\nSeveral studies have found that group populations already exhibit reduced\nstar formation rates (SFR; Lewis et al.~2002; Gomez et al.~2003) and high\nearly-type fractions similar to those observed in denser environments\n(Zabludoff \\& Mulchaey 1998; Jeltema et al.~2007).  While the physical\nmechanisms responsible for the preprocessing of galaxies in the group regime\nare still heavily debated, several recent studies have reported an\noverdensity of X-ray luminous Active Galactic Nuclei (AGN) on the outskirts\nof clusters and within the substructure surrounding unrelaxed systems (D'Elia\net al.~2004; Cappelluti et al. 2005; Kocevski et al.~2009a,2009b; Gilmour et\nal.~2009). These observations suggest that increased nuclear activity may be\ntriggered in such environments and that AGN-driven outflows may play a role\nin suppressing star formation within galaxies during cluster assembly.\nIndeed the increased dynamical friction within groups and their low relative\nvelocity dispersions make them conducive to galaxy interactions which can\ntrigger such activity (Hickson 1997; Canalizo \\& Stockton 2001) and recent\nhydrodynamical simulations suggest that merger-triggered AGN feedback can\nhave a profound effect on the gas content and star formation activity of\ntheir host galaxies (Hopkins et al.~2007; Somerville et al.~2008). \n\nSince the mass density of virialized structures increases with redshift,\nmergers are expected to play an even greater role in the group environment in\nthe past.  Therefore, if galaxy interactions and subsequent AGN feedback are\ndriving a significant portion of the preprocessing found in intermediate\ndensity environments, we may expect to find an overdensity of AGN in high\nredshift groups in the early stages of hierarchical cluster formation.  In\nthis Letter we report on \\emph{Chandra} observations of one such system, the\nCl0023+0423 (hereafter Cl0023) four-way group merger at $z=0.84$ (Lubin et\nal.~2009a).  The Cl0023 structure consists of four interacting galaxy groups\nwhich simulations suggest are the direct progenitors of a future massive\ncluster.  As such, the system provides a unique look at the level\nof processing and evolution already underway in the group environment prior\nto cluster assembly. \n\nTo search for evidence of triggered nuclear activity within the Cl0023\nstructure, we present the number counts of X-ray point sources detected in a\nfield covering the entire system, as well as a cross-correlation of these\nsources with our extensive spectroscopic database.  Surprisingly, we find\nno evidence for an overdensity of X-ray detected point sources in the\ndirection of the Cl0023 groups.  We discuss the implications of this finding\non the role of AGN feedback in regulating galaxy evolution in such\nstructures. We also examine possible explanations for the lack of increased\nnuclear activity in the system.  Throughout this Letter we assume a\n$\\Lambda$CDM cosmology with $\\Omega_{m} = 0.3$, $\\Omega_{\\Lambda} = 0.7$, and\n$H_{0} = 70$ $h_{70}$ km s$^{-1}$ Mpc$^{-1}$. \n\n\n\\begin{figure*}\n\\epsscale{1.1}\n\\plotone{.\/f1.astroph.eps}\n\\caption{(\\emph{left}) Adaptively smoothed density map of\n  color-selected red galaxies in the Cl0023 field.  Three density peaks that\n  correspond with four spectroscopically confirmed galaxy groups are marked.\n  Adapted from Lubin et al.~(2009a).  (\\emph{right}) Adaptively smoothed, ACIS-I image of the Cl0023 field\n  in the soft X-ray band (0.5-2 keV). \\label{fig-densmap}}\n\\end{figure*}\n\n\\vspace{0.5in}\n\\section{The Cl0023+0423 System}\n\nOriginally detected in the cluster survey of Gunn, Hoessel \\& Oke (1986),\nthe Cl0023 system consists of four galaxy groups separated by roughly 3000\nkm s$^{-1}$ in radial velocity (Lubin et al.~2009a).  Two of the constituent\ngroups have measured velocity dispersions of 428 and 497 km s$^{-1}$, while\nthe second, poorer pair, have dispersions of 206 and 293 km s$^{-1}$ (Lubin et\nal.~2009a).  $N$-body simulations suggest that the groups are likely bound and in\nthe process of forming a massive cluster within the next $\\sim1$ Gyr, which,\nbased on virial mass estimates of the individual groups, will have a final\nmass of $\\sim5\\times10^{14}$ M$_{\\odot}$ (Lubin, Postman \\& Oke 1998). \n\nDetails of our optical observations of the Cl0023 field are presented in\nLubin et al.~(2009a).  In short, the system was imaged with the Sloan Digital\nSky Survey (SDSS) $r'i'z'$ filters using the Large Format Camera (LFC; Simcoe\net al.~2000) on the Palomar 5-m telescope and follow-up spectroscopy carried\nout with the Deep Imaging Multi-object Spectrograph (DEIMOS; Faber et\nal.~2003) on the Keck 10-m telescopes.  Our spectroscopic observations\nyielded 423 extragalactic redshifts in the Cl0023 field and an additional 73\ngalaxy redshifts were incorporated from the spectroscopic survey of Oke,\nPostman \\& Lubin (1998). The combined catalog contains redshifts for 134\ngalaxies in the Cl0023 structure with $0.820<z<0.856$.   An adaptively smoothed\ndensity map of color-selected red galaxies in the Cl0023 field constructed\nfrom our ground-based optical imaging of the system is shown in Figure\n\\ref{fig-densmap}, while the redshift distribution of the system is shown in\nFigure \\ref{fig-zhist}.  Of the three density peaks visible in Figure\n\\ref{fig-densmap}, concentrations A and C consist of galaxies belonging to the\nredshift peaks at $z=0.839$ and $z=0.845$, respectively, while concentration\nB is a superposition of two groups along the line of sight at $z=0.828$ and\n$z=0.845$.  The galaxies within the highest redshift peak at $z\\sim0.864$ are\nnot centrally concentrated but rather extend across the entire region,\nsuggesting a sheet of galaxies in the near background.\n\n\n\\section{X-ray Observations} \n\\label{sect-data}\n\nObservations of the Cl0023 group system were carried out with {\\it Chandra's}\nAdvanced CCD Imaging Spectrometer (ACIS; Garmire et al.~2003) on 2007 August\n30 (obsID 7914).  The observation consists of a single 49.4 ks pointing of\nthe $16\\farcm9\\times16\\farcm9$ ACIS-I array, with the aimpoint located at\n$\\alpha_{2000} = 00^{\\rm h}23^{\\rm m}50.9^{\\rm s}$, $\\delta_{2000} =\n+04^{\\circ}22^{\\prime}55^{\\prime\\prime}$.  The dataset was reprocessed and\nanalyzed using standard CIAO 3.3 software tools\\footnote{Available through\n  the {\\it Chandra} X-ray Center at http:\/\/cxc.harvard.edu\/} in an identical\nmanner to the procedure detailed in Kocevski et al.~(2009a).  Images for use\nin object detection were created from the level 2 event lists with a 0.492\n$^{\\prime\\prime}$ pixel$^{-1}$ binning and corresponding spectrally-weighted\nexposure maps were constructed to account for vignetting.  An adaptively\nsmoothed, exposure corrected image of the Cl0023 field in the soft band is\nshown in Figure \\ref{fig-densmap}.\n\n\\begin{figure}\n\\epsscale{1.125}\n\\plotone{.\/f2.astroph.eps}\n\\caption{Redshift distribution in the Cl0023 field.  Shaded regions\n  correspond to galaxies within the Cl0023 structure.\\label{fig-zhist}}\n\\end{figure}\n\n\\begin{figure*}\n\\epsscale{1.125}\n\\plotone{.\/f3.astroph.eps}\n\\caption{Combined cumulative source number counts versus flux for the\n  Cl0023 group merger system in the soft (0.5-2 keV;\n  \\emph{left}) and hard band (2-10 keV; \\emph{right}).  The shaded region denotes a\n  $1\\sigma$ variance in the number counts.  Only sources detected above the\n  $3\\sigma$ level are included.  The results of an \\emph{XMM-Newton} survey of the COSMOS\n  field (Cappelluti et al.~2007) and that of a 130 ks \\emph{Chandra} observation\n  of the CDFN and CDFS are shown for comparison (solid, dashed and\n  dashed-dotted lines, respectively). The vertical dotted line represents the flux at which our\n  sky coverage dropped to 20\\% of the full ACIS-I field of view. \\label{fig-lognlogs}}\n\\end{figure*}\n\nWe searched for point sources using the wavelet-based {\\tt wavdetect}\nprocedure in CIAO, employing the standard $\\sqrt2$$^i$ series of wavelet\npixel scales, with $i=0-16$.   We adopted a minimum exposure threshold of\n20\\% relative to the exposure at the aimpoint of the observation and a\nthreshold significance for spurious detections of $10^{-6}$.  Object\ndetection was carried out on the unvignetting-corrected images and source\nproperties, including count rates and detection significances, were\ndetermined with follow-up aperture photometry on the vignetting-corrected\nimages.  A total of 151 sources were found by {\\tt  wavdetect}, of which 91\nhad detection significances greater than $3\\sigma$ in at least one of the\n0.5-2 keV (soft), 2-8 keV (hard) and 0.5-8 keV (full) bands. \n\nSource fluxes in the soft and hard bands were determined by normalizing a\npower-law spectral model to the net count rate measured for each source.\nWe assumed a photon index of $\\gamma = 1.4$ for the power-law\nmodel\\footnote{As this is the slope of the X-ray background (Tozzi et\n  al.~2001; Kushino et al.~2002)} and a\nGalactic neutral hydrogen column density of $2.66\\times10^{20}$ cm$^{-2}$\n(Dickey \\& Lockman 1990). Full-band fluxes were determined by summing the\nflux in the soft and hard bands.   Finally, rest frame X-ray luminosities\nwere calculated for sources matched to galaxies with measured redshifts (see\n\\S\\ref{sect-opt-matching}) using the luminosity distance equation and a\n$(1+z)^{\\gamma - 2}$ $k$-correction appropriate for a power-law spectrum:\n\n\\begin{equation}\n   L_{\\rm X} = 4\\pi d^{2}_{L} f_{\\rm X} (1+z)^{\\gamma - 2},\n\\end{equation}\n\n\\noindent where $d_{L}$ is the luminosity distance and $f_{\\rm X}$ is the 0.5-8 keV X-ray\nflux.\n\n\n\n\n\\section{Point Source Number Counts}\n\nTo calculate the cumulative number counts, $N(>S)$, of X-ray sources in\nthe field of the Cl0023 system, we employed the method described by Gioia et\nal.~(1990):\n\n\\begin{equation}\n   N(>S) = \\sum_{i=1}^{N} \\frac{1}{\\Omega_{i}} {\\rm deg}^{-2}.\n\\end{equation}\n\n\\noindent Here $N$ is the total number of detected point sources and\n$\\Omega_{i}$ is the sky area in square degrees sampled by the detector down\nto the flux of the $i$th source.  The variance of the number counts was in turn calculated as\n\n\\begin{equation}\n   \\sigma_{i}^{2} = \\sum_{i=1}^{N} \\left(\\frac{1}{\\Omega_{i}}\\right)^{2}.\n\\end{equation}\n\nIn order to determine $\\Omega_{i}$ we constructed a flux limit map using the\nmethod employed by Kocevski et al.~(2009a). First, all point sources detected\nby {\\tt wavdetect} were replaced with an estimate of the local background\nwith the CIAO tool {\\tt dmfilth} and the resulting images binned to a pixel\nscale of $32^{\\prime\\prime}$ pixel$^{-1}$ to produce a coarse background map.  This\nmap is used to determine the flux limit, $S_{\\rm lim}$, for a $3\\sigma$ point\nsource detection in any one pixel.  We can then calculate $\\Omega_{i}$ by\nsumming the sky area covered by all pixels with $S_{\\rm lim}$ equal to or\ngreater than the flux of the $i$th source.   \n\nThe resulting cumulative source number counts for the Cl0023 group system in\nthe 0.5-2 keV (left panel) and 2-10 keV (right panel) bands are shown in\nFigure \\ref{fig-lognlogs}.  The latter was chosen to ease comparison with\nprevious studies and obtained by extrapolating our 2-8 keV fluxes to 10 keV.\nAlso shown are the cumulative number counts measured in the COSMOS field\n(Scoville et al.~2007) and the {\\it Chandra} Deep Field South and North (CDFS\nand CDFN; Rosati et al.~2002; Brandt et al.~2001).   The COSMOS results are\nthose of Cappelluti et al.~(2007) converted to a spectral index of $\\gamma = 1.4$, while the CDFS and CDFN counts are the\nresults of our own reanalysis of single ACIS-I pointings in each\nfield\\footnote{Observation ID numbers 581 and 2232.}. \n\n\nIn both the soft and hard bands, we find that the number of sources detected\nin the Cl0023 field is statistically consistent ($<1\\sigma$ deviation) with\nthe source counts observed in the reference blank fields.  In the hard band,\nthis agreement extends over roughly the entire sampled flux range, while in\nthe soft band, we find an underdensity of bright sources relative to the\nCOSMOS field, in agreement with the underdensity of soft sources previously\nreported at these fluxes in the CDFS (Yang et al.~2003).  Our measured source\ncounts suggest that there is no increased nuclear activity in the Cl0023 system\ndetectable at X-ray wavelengths above our $3\\sigma$ flux limit, which amounts\nto $2.9\\times10^{-15}$ erg s$^{-1}$ cm$^{-2}$ (0.5-8 keV).  At the median\nredshift of the Cl0023 system, this corresponds to a rest-frame 0.5-8 keV\nluminosity of $6.9\\times10^{42}$ $h_{70}^{-2}$ erg s$^{-1}$, making our\nobservations sensitive to moderate luminosity Seyferts and QSOs in the\ncomplex. \n\nTo parameterize the number counts, we fit the unbinned soft- and hard-band\ncounts in the Cl0023 field with a power-law model of the form $N(>S) =\nk(S\/S_{0})^{-\\alpha}$ using the maximum likelihood method of Murdoch et\nal.~(1973).  Our best-fit slopes, $\\alpha$, to the faint- and bright-end\nnumber counts in the soft band are $\\alpha_{\\rm faint} = 0.57\\pm0.12$ and\n$\\alpha_{\\rm bright} = 1.61\\pm0.30$, with a break in the distribution at\nroughly $S\\simeq3\\times10^{-15}$ erg s$^{-1}$ cm$^{-2}$.  In the hard band we\nonly fit to the bright-end counts as we do not sample the faint-end\npopulation sufficiently to obtain a separate fit.  Our best-fit slope above\n$S\\simeq8\\times10^{-15}$ erg s$^{-1}$ cm$^{-2}$ is $\\alpha_{\\rm bright} =\n1.69\\pm0.31$.  These slopes are in good agreement (within the errors) with\nprevious studies of the \\emph{Chandra} Deep Fields, which found Euclidean\nslopes at the bright-end and $\\alpha_{\\rm faint} = 0.63\\pm0.13$,\n$0.67\\pm0.14$ in the soft band for the CDFS and CDFN, respectively (Brandt et\nal.~2001; Rosati et al.~2002).  \n\n\n\n\\section{Optical Source Matching }\n\\label{sect-opt-matching}\n\n\\begin{center}\n\\tabletypesize{\\scriptsize}\n\\begin{deluxetable}{cccrrr}\n\\tablewidth{0pt}\n\\tablecaption{Properties of X-ray Detected Galaxies in the Cl0023 Field with\n  Measured Redshifts\\label{tab-prop}}\n\\tablecolumns{6}\n\\tablehead{\\colhead{RA} & \\colhead{Dec} &\n           \\colhead{} &  \\colhead{Net} &  \\colhead{$F_{\\rm x}$$^{\\dagger}$} & \\colhead{$L_{\\rm x}$$^{\\ddagger}$}  \\\\  \n           \\colhead{(J2000)} &  \\colhead{(J2000)} &  \n           \\colhead{$z$} & \\colhead{Counts} & \\colhead{($\\times10^{-15}$)} & \\colhead{($\\times10^{42}$)}} \n\\startdata\n00:23:47.5   &  04:21:17.6  &  1.487  &  7.7  &  1.49  & 12.02  \\\\   \n00:23:43.3   &  04:18:06.3  &  0.169  &  6.4  &  2.94  &  0.21  \\\\    \n00:23:56.3   &  04:17:60.0  &  0.683  &  3.8  &  1.14  &  1.71  \\\\   \n00:23:55.2   &  04:25:20.2  &  1.091  & 13.5  &  4.77  & 19.86  \\\\   \n00:24:02.5   &  04:22:12.9  &  0.442  & 39.5  & 11.29  &  6.46  \\\\    \n00:23:51.1   &  04:27:19.8  &  0.113  & 19.7  &  5.34  &  0.17  \\\\   \n00:23:52.2   &  04:25:53.7  &  0.682  &  5.7  &  0.69  &  1.03  \\\\   \n00:23:48.9   &  04:21:23.7  &  0.745  & 43.0  &  9.53  & 17.23  \\\\   \n00:23:58.5   &  04:24:51.1  &  1.336  & 10.5  &  3.76  & 24.13  \\\\   \n\\vspace*{-0.075in}\n\\enddata\n\\tablecomments{All X-ray properties measured in the 0.5-8 keV band; $^{\\dagger}$ In units of erg\n  s$^{-1}$ cm$^{-2}$;  $^{\\ddagger}$ In units of $h_{70}^{-2}$ erg s$^{-1}$.} \n\\end{deluxetable}\n\\end{center}\n\\vspace{-0.2in}\n\nDespite the absence of a clear overdensity of X-ray sources in the Cl0023\nfield, we searched for AGN within the Cl0023 structure by matching our X-ray\nsource list to our preexisting spectroscopic catalog.  To perform this cross-correlation,\nwe determined the positional uncertainty associated with each X-ray source\nusing the empirical relationship of Kim et al.~(2007), who find that\ncentroiding errors increase exponentially with off-axis angle from the\naimpoint of the observation and decrease as the source counts increase with a\npower-law form.  To determine the reliability of a given match, we employed a\nmaximum likelihood technique described by Sutherland \\& Saunders (1992) and\nmore recently implemented by Kocevski et al.~(2009a).  The method gauges the\nlikelihood that a given optical object is matched to an X-ray source by\ncomparing the probability of finding a genuine counterpart with the\npositional offset and magnitude of the optical candidate relative to that of\nfinding a similar object by chance.  We refer the reader to Kocevski et\nal.~(2009a) for details.  \n\nUsing this technique we have matched a total of nine X-ray sources to\ngalaxies with measured redshifts in our spectroscopic catalog.  These\ngalaxies cover a broad range in redshift ($0.442 < z < 1.487$) and there is no\nevidence for a concentration near the redshift of the Cl0023 system.  In fact,\nwe find no X-ray point sources matched to the 134 galaxies spectroscopically\nassociated with the four groups in the Cl0023 structure.  \nHowever, we should note that we did not specifically target X-ray\nsources with our spectroscopic observations, but instead simply cross-correlated\ntheir positions with out existing spectroscopic database.  In future DEIMOS\nobservations of the system we plan to have dedicated masks for X-ray and\nradio detected AGN in order to both increase our spectroscopic completeness\nof X-ray sources and to determine if the lack of AGN currently observed in the Cl0023 groups holds.\nThe coordinates, redshifts, and X-ray properties of the nine galaxies\ncurrently matched to X-ray point sources in the Cl0023 field are listed in Table 1.  \n\n\n\n\n\\section{Discussion and Conclusions}\n\nUsing \\emph{Chandra} imaging of the Cl0023 complex, we have searched for\nevidence of triggered nuclear activity within a dynamically active system of\nfour galaxy groups in the early stages of cluster formation.   Both the\nredshift distribution and cumulative number counts of X-ray point sources in\nthe Cl0023 field reveal little evidence to suggest that the system contains X-ray\nluminous AGN in excess to what is observed in the field population.  \nThese results are at odds with previous reports of source excesses on the outskirts\nof dynamically unrelaxed clusters at high redshift.  They also appear to challange \nthe notion that AGN-driven outflows play a significant role in the preprocessing \nobserved in galaxy groups and environments of moderate overdensity relative to the field.\nIf preprocessing is underway in the Cl0023 system, our\nobservations suggest that powerful (quasar mode) nuclear activity is not the\npredominant mechanism quenching star formation and driving the evolution of\nCl0023 galaxies.   Of course we cannot rule out a population of\nlow-luminosity AGN powering ``radio mode'' feedback (Croton et al.~2006) in\nthe Cl0023 complex as our observations are only sensitive to moderate\nluminosity Seyferts and QSOs.  We are currently analyzing \\emph{Very Large Array} (\\emph{VLA}) 20-cm\nobservations of Cl0023 to search for such a population and expect to present\na full radio study of the system in a forthcoming paper (L.~M.~Lubin et al.~2009b,\nin preparation). \n\nOur current findings are in stark contrast to the overdensity of AGN recently detected in\nsimilar \\emph{Chandra} observations of the Cl1604 supercluster at $z=0.9$,\nwhere we find a population of Seyferts associated with an unrelaxed cluster and two rich groups \n(Kocevski et al.~2009a, 2009b).  However, the galaxy populations of these groups\ndiffer in significant ways from those of the Cl0023 system.  The Cl1604\ngroups tend to have higher velocity dispersions and more evolved galaxy\npopulations than the Cl0023 groups, as indicated by their average SFRs and\nmorphological fractions (Gal et al.~2008; Lubin et al.~2009a).  Previous\nobservations of Cl0023 galaxies found them to be predominately late-type\nsystems (75\\%; Lubin et al.~1998) with substantial amounts of ongoing star\nformation\\footnote{This is consistent with the galaxy properties of high\n  redshift groups with similar velocity dispersions (e.g.~Poggianti et\n  al.~2006)} (Postman, Lubin, Oke 1998; Lubin et al.~2009a), whereas the hosts\nof the Cl1604 AGN tend be bulge-dominated, post-starburst galaxies which show\nsigns of recent or ongoing galaxy interactions.   Therefore, while Cl0023\ncontains galaxies which have the gas necessary to fuel nuclear activity, it apparently\nlacks the bulge-dominated and massive early-type hosts in which powerful AGN\nhave been shown to reside (Kauffmann et al.~2003). \n\nA likely explanation for the absence of luminous AGN in the Cl0023 groups is\nthat the system lacks galaxies with sufficiently massive nuclear black holes\nrequired to power such activity.  It has previously been shown that the\nbulge-dominated S0 population in clusters and groups builds up over time at\nthe expense of the spiral population and that this morphological evolution is\nmore pronounced in lower mass systems (Poggianti et al.~2009).  There is also\nevidence that these galaxies are typically more massive than their suspected\nprogenitors (Dressler et al.~2009), suggesting they experience growth in\ntheir stellar bulges while in overdense environments, possibly via a\ncentrally concentrated burst of star formation (Dressler et al.~1999). \nGiven the correlation between bulge mass and central black hole mass\n(Gebhardt et al.~2000), we would expect similar growth in galactic nuclei\nover the same period.  Therefore, if disruptive AGN-driven\noutflows play a role in quenching star formation in groups, as has been\nsuggested, it may only become an important factor in the preprocessing of\ngalaxy populations during a later stage in the evolution of such groups and\nstructures, when sufficiently massive galaxies (and nuclear black holes) have\nbuilt up, but prior to hydrodynamical processes within clusters stripping\nthem of their gas reservoirs.   \n\nFurther observations of a larger sample of systems in the early\nstages of cluster formation, with a variety of velocity dispersions and\nmorphological fractions, will be required to test this scenario. \nIn the mean time, we are planning additional spectroscopic follow-up of the\nCl0023 groups targeting the radio bright population as well as the\nremaining X-ray point sources that currently lack redshifts.  \nThis will give us a greater spectroscopic completeness of X-ray luminous AGN in\nthe Cl0023 field, which will enable us to test our current findings and should allow us to better discern\nthe prevalence of powerful nuclear activity during cluster formation.\n\n\n\n\n\\acknowledgments\nThis work is supported by the Chandra General Observing Program under award\nnumber G07-8126X.  The spectrographic data used herein were obtained at the\nW.M. Keck Observatory, which is operated as a scientific partnership among\nthe California Institute of Technology, the University of California and the\nNational Aeronautics and Space Administration. The Observatory was made\npossible by the generous financial support of the W.M. Keck Foundation.\n\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\n_This book is dedicated to Gary Gregor whose knowledge about Edgar Evans and whose enthusiasm for the work has been a constant source of encouragement_\n\n_Front cover photograph_ : Edgar Evans dressed for exploration. (Courtesy of Scott Polar Research Institute \u2013 SPRI)\n\n## Contents\n\n_Acknowledgements_\n\n_Introduction_\n\n 1 The Gower Peninsula: Early Life\n\n 2 The Boy Sailor: Naval Training\n\n 3 The _Discovery_ Expedition\n\n 4 From England to South Africa\n\n 5 The Southern Ocean to Antarctica\n\n 6 Early Months in Antarctica: February to September 1902\n\n 7 The Antarctic Spring: September to October 1902\n\n 8 The Antarctic Summer: October 1902 to January 1903\n\n 9 The End of the _Discovery_ Expedition, 1903\u201304\n\n10 Return from Antarctica, then Home Again, 1904\u201310\n\n11 _Terra Nova_\n\n12 The First Western Party\n\n13 The Winter Months, 1911\n\n14 The Polar Assault\n\n15 The Aftermath\n\n16 Why Did Edgar Die First?\n\n Epilogue\n\n## Acknowledgements\n\nDr David Wilson, the great-nephew of Dr Edward Wilson, Scott's confidant and friend, has been a remarkable source of friendship, encouragement and advice throughout the work. My colleagues, Dr John Millard, Dr Howell Lloyd and Dr Aileen Adams, have diligently read the work and offered helpful comments, as has Mrs Jackie McDowell.\n\nProfessor Stuart Malin has patiently guided me through the intricacies of longitude and latitude. Lieutenant Commander Brian Witts, Curator of HMS _Excellent_ Museum, Portsmouth, assisted me with details of the field gun run competition at Olympia. I am greatly indebted to these colleagues and friends. I accept responsibility for any misunderstandings or omissions.\n\nThe assistance of the staff at the Scott Polar Research Institute, The Naval Museum at Portsmouth and the Swansea Library has been greatly appreciated; all have been unfailingly courteous, helpful and enthusiastic.\n\nAlison Stockton and James Oram, a Classics Undergraduate at Durham University, have patiently proofread the work and I am grateful for their help.\n\nI have to thank Professor Julian Dowdeswell, Director of the Scott Polar Institute, for permission to publish extracts from those manuscripts over which the Institute has rights, also the Debenham, Shackleton, Skelton and Scott families for kind permission to quote from family papers. The Auckland Institute and Museum of New Zealand have allowed me to quote from Charles Ford's journals as have the Dundee Art Galleries and Museums in relation to James Duncan's papers. Mr John Evans, Edgar's grandson, has allowed me to quote from Edgar's sledging journal. Every reasonable effort has been made to trace copyright holders; any omissions or mistakes will be inserted into subsequent editions of this work.\n\nFinally, my thanks must go to my husband, Dr David Williams, whose assistance and help made this work possible.\n\n## Introduction\n\nSaturday 17 February 1912, Antarctica.\n\nA man crawls helplessly on the icy snow, his clothes are torn open, his skis are off, his gloves and boots lie discarded on the snow, bandages trail from his frostbitten fingers.\n\nHe dimly sees four images coming towards him, but in his confusion he cannot work out what is happening. When his companions arrive he can hardly speak. He can barely stand and after a hopeless attempt at walking he falls back onto the snow. Three of his exhausted rescuers plod back wearily to the camp for a sledge.\n\nThey lay him on the sledge and struggle to pull him over the snowy waste. On the way he loses consciousness. He is never to be aware of his surroundings again. In the tent, as his companions watch, his breathing becomes irregular and shallow; he dies quietly at 10 p.m.1\n\nSo ended the life of Edgar Evans, the 'Welsh Giant' from Middleton in South Wales, a man who contributed hugely to Antarctic exploration, a Petty Officer who had built a relationship with his leader, Robert Falcon Scott, that transcended the barriers of class, rank and education. Theirs was a loyalty that had been built over long periods of interdependence as they endured the horrors of prolonged man-hauling at sub-zero temperatures in Antarctica.\n\nScott wrote that Edgar was a 'giant worker with a truly remarkable head piece',2 that Edgar was 'hard and sound' on a trek and had an 'inexhaustible supply of anecdotes'.3 He chose Edgar as one of the five to go to the Pole.\n\nEdgar died on the return, overcome by circumstances so awful that his four companions were soon to join him in icy tombs in Antarctica.\n\nThis is Edgar Evans' story.\n\n### Notes\n\n1 Ed. King, H.G.R., Edward Wilson _Diary of the Terra Nova Expedition to the Antarctic 1910\u20131912_ , Blandford Press, London, 1972, p. 243.\n\n2 Ed. Jones, M. _Robert Falcon Scott Journals Scott's Last Expedition_ , Oxford University Press, Oxford, 2005, p. 369.\n\n3 Ibid., p. 303.\n\n## 1\n\n## The Gower Peninsula: Early Life\n\nEdgar Evans came from the Gower Peninsula in South Wales. Jutting into the Bristol Channel and open to the Atlantic gales, Gower is a place of outstanding natural beauty, a location that attracts visitors to its shores year after year. It boasts other famous attractions: in one of its coastal caves, the Paviland Cave, the oldest human skeleton in the British Isles was discovered \u2013 the 'Red Lady of Paviland'1 (actually male remains) is tens of thousands of years old.\n\nIt would have been a remarkable astrologer who foretold fame for Edgar Evans when he was born on 7 March 1876 in Middleton Hall Cottage, Middleton \u2013 a village in Rhossili and one of the remote parishes on the peninsula. Edgar's mother, Sarah, had moved to Middleton Cottage, her sister's home, for her confinement.\n\nThis was a modest family. Their roots were firmly in Gower. Evans' paternal grandfather, Thomas, and three previous generations of his family, came from the peninsula. Thomas was employed in a local limestone quarry (limestone was shipped across the Bristol Channel to fertilise the fields of north Devon). Thomas' son, Charles (1839\u20131907), the father of Edgar Evans, was one of the famous 'Cape Horners', hardy seamen who sailed from Europe around Cape Horn to the west coast of America, a journey that could last six months.\n\nThe 'Cape Horn' trade grew because Swansea was then the world centre for smelting copper, essential in industry, construction and ship-building (the copper covering on ships' hulls prevented the wood from rotting and made the vessels faster).\n\nThere were copper works in Swansea from as early as 1717. Approximately 2 tons of coal was necessary to smelt 1 ton of copper, and since South Wales was rich in coal, copper was brought to Swansea rather than coal being taken to the copper sources. When British ore was worked out, copper mines further afield were sought and Cuba and South American countries, particularly Chile, were used. These voyages to South America, in coal-carrying sailing ships, were hazardous undertakings. Life at sea was brutal and unforgiving. Off the Horn, with 'winds at full-gale strength, waves as high as the maintops, sometimes hail and then snow coming down thick, clouds so low they enfold the mastheads, spume and sky indistinguishable',2 forward progress was often impossible, some days the ship was set back by miles. Sometimes the voyage lasted four months, often much longer and many men died on the 'widow-making' passage. Added to the physical horrors of the crossing was the ever-lurking possibility of spontaneous combustion of the coal, likely to be disastrous in wooden ships and more probable if the coal was damp. After managing to survive the voyage, the sailors still had to contend with the perils of disease in South American ports. And then, having endured all that, the sailors faced the daunting prospect of the return journey. Years later, one of Edgar's companions in the Antarctic wrote that only those who had had the experience could realise what it meant: handling frozen sails in the dark, short handed... and 'so cold that the chocks (fittings for securing the ropes) have to be thawed with hot water before a rope will run through them'.3\n\nBut Charles Evans pursued this trade until he was in his mid-30s, well after the time he married and had children. In 1862, when he was 23, Charles, described as 'Mariner, son of Thomas Evans, Quarryman', married Sarah Beynon in St Mary's church, in the village of Rhossili. Rhossili is one of the many villages dotted over Gower. It had 294 residents4 and was connected to its closest neighbour by just a muddy lane. Sarah was a local girl, the daughter of William Beynon, the licensee of the Ship Inn in Middleton, and his wife, Ann. She was 22 at the time of her marriage and her family had held the licence for the Ship Inn for most of the nineteenth century.\n\nThe ceremony was performed by the Reverend John Ponsonby Lucas BA, MA, an Oxford graduate, who ministered to several of the local villages. St Mary's, with its beautiful Norman doorway, remains an active, functioning church. For many years a plaque in the aisle wall has proudly commemorated the life of Charles and Sarah's famous son, Edgar.\n\nAs was usual in Victorian households, the couple produced a large family \u2013 there were eight known children. Birth control was unknown in working class communities in the late 1800s, and a high birth rate was a type of insurance policy against an unsupported old age. Four of the children are listed in the 1871 census: Charles, 7; John, 4 (both described as scholars); Mary-Ann, 2; and Annie Jane aged 1. The gap of three years between Charles and John suggests an infant death. In 1874, another son, Arthur, was born, followed, in 1876, by Edgar. A seventh child, George, was born in 1878 and a sister, Eliza Jane, in 1879. In fact Sarah Evans gave birth to more than the eight children; in 1913, after Edgar's death, she was interviewed by a local reporter, and exhibiting stoicism difficult to imagine nowadays, she said that she had buried nine of her twelve children, three having died from consumption.5\n\nMrs Sarah Evans registered the birth of her fourth son in the sub-district of Gower Western on 13 April 1876. The 7 March was recorded as the birth date and Mrs Evans, unable to write (as was common, even six years after compulsory education was introduced),6 recorded her mark with a cross.\n\nInterestingly, when Edgar entered the navy in April 1891, his Certificate of Service states that his date of birth was 9 March. Probably, once the error was officially recorded, the Boy, 2nd Class, then aged 15 years and 37 (or 39) days, thought it more prudent to go along with the official record than to challenge it, and he never corrected the date, although he is likely to have been aware of his registered birth date. Years later, in 1911, he wrote in his diary on 9 March, when he was on a sortie, that it was the 'first time he has spent his birthday sledging'.7\n\nEdgar was born into a small, tight-knit community. In the pre-First World War era, many people stayed within a few miles of their birthplace for the whole of their lives, and there was a huge interconnection of families through marriage. Sarah Evans had family links with many people in her village as well as brothers and sisters, some of whom were still living at the Ship Inn. So Edgar was born into a ready-made network of uncles, aunts, grandparents and cousins, as well as the immediate family crowded into his family's cottage, which housed up to four of his older brothers and sisters as well as the babies, George and Eliza Jane. In addition, his father added to the crush on his intermittent visits home.\n\nHe learnt to speak in English as Welsh was hardly ever heard on that part of the peninsula. There must have been some incomers in Gower over the years because the villagers spoke in the 'Gower Dialect'. This dialect, now virtually forgotten, had evolved through the influence of settlers from south-west England.\n\nAs a little lad he kept well out of the way of the local dignitaries; when the doctor visited on horseback or, occasionally, driving his horse and trap which carried the brightly coloured bottles of medicine that could be prescribed for virtually every ailment (since many of the residents could not read they were thought to be particularly impressed by the colours), Edgar took care to avoid him. Likewise, the Rector was an important local man. The Evans family were definitely 'Church' rather than 'Chapel' (the place of worship for the local Methodists). It is surprising nowadays to read of the chasm that existed between the two in some parts of the country (reminiscent of the Catholic\/Protestant divide in Northern Ireland), but in Gower the division was only a pale and peaceful reflection of those clashes. When the Reverend John Ponsonby Lucas had married Edgar's parents, he gave a girl a lift in his pony and trap, a journey of about an hour, but he was forbidden to speak to her because he was of the established Church whilst she was a Nonconformist.\n\nBy the 1881 census the family had moved to Pitton, the next small hamlet east of Rhossili. As was usual, only the people actually in the house on the day the census was recorded were counted, and Mrs Evans, 'Mariners Wife', registered her four younger children, now including Edgar, aged 5, as 'scholar' at the village school at Middleton. Forster's Elementary Education Act of 1870 had stipulated that all children between the ages of 5 and 12 were obliged to attend school. The thrust behind this act was the fear that Britain's status in the world could be threatened by the lack of an efficient national education system. It was a move by no means universally welcomed; there were fears that education would make members of the labouring classes, such as the Evans family, 'think' and so become dissatisfied with their lot. The Church also had doubts; its support for the biblical story of creation (which implies, amongst other things, that we are born to the station that we are meant to remain in) resulted in reservations. Also, the Church was already the recipient of state money for educating the poor and was reluctant to relinquish this. But once the Act was law, children were educated perforce. Edgar would be at Standard 1. He learnt his letters from an elementary reading book by copying a line of writing, in 'good, round, upward writing', and later wrote a few common words from dictation. He did simple addition and subtraction (of not more than four figures), as well as learning his multiplication tables (up to 6). Strict instruction was given on how to hold a pen \u2013 in the right hand with the thumb nearly underneath and three fingers flat out on the top; if his teacher saw him with one of his fingers bent he would have been rewarded by a rap on the knuckles.8 Edgar certainly benefited from the education he was given before he left school at the age of 12. His writing in later years was clear and his prose concise. The only thing that seems to have escaped his attention is punctuation; sentences flow effortlessly and sometimes confusingly, one into another.\n\nIn 1883, when his father Charles was 44, the family moved their home again. By now Charles had left long-haul shipping and was employed on a boat _The Sunlight_ , which was involved in local coastal work based in Swansea, so the Evans family moved to the town. Swansea was important; it was part of the nation's 'workshop of the world' and also known as 'Copperopolis' because of the prominence of the copper trade. The family moved to Hoskin's Place, Swansea. They would have lived in one of the thousands of identical 'two up, two down' little terraced houses, with a communal back yard and 'privy'. It is not clear just how many of the family made the move to Swansea; only the four youngest are recorded as being in the house on the 1881 census, but it is unlikely that Edgar's 13-year-old sister Mary Anne or 11-year-old Annie would have left home by 1883, so it is probable that seven or eight Evans members (at least) shared the overcrowded facilities. Life was not easy. When coal could be afforded, the downstairs room was warmed by a coal fire, which was an integral part of an iron oven. Food was scarce: homemade bread and pies, meat once a week if possible, and potatoes. Water was heated by the stove and a tin bath (decorously concealed behind a clotheshorse decked with washing for privacy), was used for the weekly or fortnightly baths. Three or four members of the family used the same water.\n\nWith its population of over 50,000, busy streets, horse traffic, pollution from the copper works and noise, the town must have come as a shock to the country children; a huge contrast to sparsely populated Rhossili. Young Edgar was enrolled at St Helen's School, Vincent Street, Swansea and remained there until he was 13. The school had just been enlarged when 7-year-old Edgar enrolled as a pupil and, with its 250 pupils, it too must have seemed huge. The life of the school and the education it offered is described by N.L. Thomas in a centenary booklet, _A Hundred Years in School, St Helen's 1874\u20131974_ ,9 which shows how very fortunate Edgar and his fellow pupils were to fall under the influence of an enlightened, humane headmaster, Mr Lewis Schleswick. This was an opportunity certainly not enjoyed by all Victorian children. Mr Schleswick's service was stretched; he had a small staff, certified assistants, uncertified assistants and pupil teachers10 and, to keep the teaching standards as high as possible, he taught the pupil teachers each morning before school began. The school remains. It is proud of its famous old boy and has Edgar's picture prominently displayed.\n\nAs the school year progressed, Edgar, no doubt with the other pupils, was tempted by those infrequent but exciting diversions which lightened the drab routine of the Victorian school room, and often (as the daily school roll recorded) cut school attendance dramatically. Many of the children had to work for their parents before and after school. By a young age they were accustomed to a life of repetitive monotony and any glamorous excitement must have been a glorious break. Such delights were visits by circuses to the St Helen's area, the occasional fair, regattas at Mumbles and Swansea, Saint Patrick's Day11 celebrations (when school attendance was noticeably small) and, on occasion, processions. Once, after a Sunday school outing, over one hundred boys were absent. The reason given by the miscreants, according to Mr Schleswick, was that they were too tired to get to school.12 Occasionally, however, absences were official. When General William Booth, the founder of the Salvation Army, visited Swansea in 1883, the school was given a half-day holiday and when there was a large public procession in relation to the Blue Ribbon Movement,13 the boys were allowed time off to watch it. The headmaster wrote that not only the pupils but also the pupil teachers were given an official day's holiday as a reward for their 'unremitting zeal and energy'.14\n\nAttendance could fall for more serious reasons. Mr Schleswick recorded that the summer of 1885 was exceptionally hot and it was difficult to keep the boys at their work as they were in a 'state of exhaustion'. The area around St Helen's was overcrowded, poor and susceptible to disease. Typhoid fever, that curse of unsanitary water supplies, attacked the school in 1896 'in spite of the drains being regularly disinfected by the Urban Sanitary Authority'. When Mr Schleswick inspected the drinking-water cistern, he found that it was filled with a deposit, to the depth of an inch and had a dirty filter.15 Other infectious diseases extracted a heavy toll: an outbreak of measles would close the school for three weeks,16 scarlet fever, that harbinger of rheumatic fever, sinus and ear infections, also visited regularly. Boys from houses where infection lurked were sent home to reduce the risk of cross contamination in this pre-antibiotic era, but death was a common caller. In 1887, there was a drought which had an impact both by causing dehydration and because the boys drank infected water. On this occasion wily local entrepreneurs profited by collecting barrels of water from springs in the countryside and transporting and selling the water to whichever urbanite could afford to buy it.\n\nThe end of the school day was the signal for those boys, who did not have to work for their parents, to escape to freedom. They made their own entertainment; since there were no cars or buses, but only slowly moving horse drawn vehicles, they could play on the street: trundling hoops, whipping tops or just standing in the middle of the street and gossiping. St Helen's was close to Swansea Bay, famous for oysters, but probably of more interest to the boys as a glorious beach playground for football and swimming. Sundays were rest days. Edgar went with his family to Sunday school during the day and church in the evening. They all wore their 'Sunday best' clothes for the church visit.\n\nSoon after his tenth birthday Edgar became a 'half timer'. This exploitative use of cheap child labour meant that school time was cut, so that Edgar spent half the day at school, half at work. For his work he earned about a shilling (approximately \u00a35 in current value) a week. He was relatively lucky. Fourteen years previously he could have been working as a 'half timer' from the age of 8.17 So Edgar's total education was five years full-time (from 5 to 10), thereafter three years of half-time education. Nothing highlights the difference between the privileged and working classes of Victorian England better than their educational opportunities. By the time he was 10, Edgar, an intelligent child, would have been competent in the basic subjects: able to read, write to dictation and do arithmetic. Later he would have been introduced to a smattering of more interesting topics: geometry and geography.18 He would have been used to the idea of homework or 'home lessons'. He would have sung \u2013 the Welsh are natural singers \u2013 and St Helen's had a tradition for music and singing and the pupils were examined on their prowess.19 But his formal education was virtually at its end. By contrast, Dr Edward Wilson, who served with Edgar on both Scott's expeditions as an officer and who came from a privileged background, was (although an average student) immersed in Latin, Greek, English, arithmetic and spelling by the age of 10.20 And Wilson's education would continue for many more years. Education for the upper classes provided shibboleths to enter into a society that was virtually closed for people of Edgar's education. His was the class that sailed the ships, worked the mines, smelted the ore and so underpinned for Britain those social and economic foundations that maintained her pre-eminence in the world. But the country was (in the main) proud of the Empire and proud to serve Queen and Country, and Edgar would have imbibed this pride.\n\nFrom 1886, Edgar worked as a telegraph messenger boy in Swansea's head post office. He carried his bag around Swansea delivering telegrams. His hours were long and tiring, and a fellow pupil from the 1880s recalls working till 10.30 at night.21 After Edgar's death, his photograph, taken after his first Antarctic sortie, was displayed in the Swansea Head Post Office for many years. The photograph shows a good-looking young man. He was described as having blue eyes and a 'fresh' complexion.22 He was clean-shaven with brown hair, a straight nose, a strong jaw and a generous mouth.\n\nThe Head Post Master of 1886 decreed that messenger boys began their day with musket-duty and Edgar, between the ages of 10 to 13, was drilled and marched in procession, carrying his musket on his shoulders. A big event of 1887 was the visit of W.E. Gladstone23 (Victoria's former Prime Minister) to Swansea to open a local Public Free Library. By now the boys were sufficiently drilled to march in procession to the library. How much they appreciated Gladstone's speech on Irish Home Rule is not recorded.\n\nBehind the post office was the North Dock. Here ships from exotic destinations would tie up and the boys were sometimes allowed on board. They would badger the sailors with questions, their imagination soaring along with stories of lands and adventures far, far beyond the confines of Swansea. Edgar had never been out of Gower. These visiting seamen and his father's stories nurtured his determination to see the world, to become a sailor. In his early teenage years he decided that he would join the Navy as soon as they would have him.\n\nHowever, he had to curb his impatience for a few years. In the meantime his mother took him to visit her family in Middleton. The journey was a step back in time for the newly sophisticated urbanite. To get there they had to travel by road to Pitton Cross and then brave the rigours of a high-banked, muddy, narrow lane, just wide enough for a horse and cart. But the Gower Coast had attractions other than family visits for a young boy. It was littered with wrecks: schooners, paddle ships, barques, oyster boats and ketches.24 Over fifty vessels \u2013 from a French vessel in 1557, to the Norwegian Barque _Helvetia_ in 1887 \u2013 were known to have foundered in its treacherous waters. Edgar was 11 when the _Helvetia_ ran aground in the southern part of the bay. He was enthralled at the story of her battle against the elements. On this occasion there were no fatalities, but her cargo of timber floated onto the beach and every available man, boy, horse and cart spent days loading the wood. _Helvetia_ 's bare wooden ribs can be clearly seen today, sticking out of the sand in Rhossili Bay.\n\nWhen he was 13, half-time work finished and Edgar left school for full-time employment in the Castle Hotel. Many of the captains of those copper ore barques berthed at North Dock actually frequented the hotel25 and their stories must have strengthened Edgar's resolve to join the Navy. He read the _Boys' Own Paper_ (a relentless recruiting agent for the Navy), too.26 By now he was so keen to see the world that he actually tried to join up when he was 14. He was refused but returned to the Castle Hotel announcing, 'I am coming back to you for another year and then I am going to join the Navy.'27\n\nHis parents were dismayed. Sarah Evans had known the hardship of bringing up (and probably already burying) her children with a husband away for months at a time. Charles Evans also tried to dissuade his son; he had had to have a leg amputated after it was damaged in an accident on his ship. But Edgar was determined. As soon as he could, at the age of 15, he applied to join the Navy.\n\nThe die was cast.\n\n### Notes\n\n1 Investigated in 1823 by the Reverend William Buckland (1784\u20131856), Professor of Geology at Oxford, an eminent palaeontologist, who, because he was a Creationist and thought that no human remains could be older than the Biblical Great Flood, hugely underestimated the age of his find.\n\n2 Lundy, D., _The Way Of A Ship_ , Jonathan Cape, London, 2002, p. 15.\n\n3 Wild, J.R.F., Letter to Mrs Bostock. SPRI MS 1078\/3\/1; D.\n\n4 Lee, S., _The Population of Rhossili_ Gower, IV. Swansea, 1951, p. 27.\n\n5 _South Wales Daily Post_ , Tuesday 18 February: 'Consumption' is tuberculosis, then endemic and causing death in over half its victims.\n\n6 Forster's Education Act. Drafted by William Forster, a Liberal Member of Parliament and introduced on 17 February 1870. The act provided elementary education for children aged 5\u201312. Parents were still expected to pay fees, though if they were poor, the board of each school would pay.\n\n7 _Edgar Evans' Journal, 27\/1\/11\u201312\/3\/11_ , SPRI: Ms 1487: BJ 9\/3\/11.\n\n8 Thomas, N.L., _A Hundred Years in School, St Helen's, 1874\u20131974_ , Souvenir Centenary Booklet, held at Swansea Library, 1974, p. 22.\n\n9 Thomas, N.L., _A Hundred Years in School, St Helen's 1874\u20131974_ , Souvenir Centenary Booklet, held at Swansea Library, 1974.\n\n10 Pupil Teachers. Students who also taught.\n\n11 The Patron Saint of Ireland.\n\n12 Thomas, N.L., _A Hundred Years in School, St Helens 1874\u20131974_ , Souvenir Centenary Booklet, held at Swansea Library, 1974, p. 2.\n\n13 A Temperance Union.\n\n14 Thomas, N.L., _A Hundred Years in School, St Helen's 1874\u20131974_ , Souvenir Centenary Booklet, held at Swansea Library, 1974, p. 15.\n\n15 Ibid., p. 17.\n\n16 Ibid., p. 15.\n\n17 Factory Act of 1874.\n\n18 Thomas, N.L., _A Hundred Years in School_ , St Helen's 1874\u20131974, Souvenir Centenary Booklet, 1974, held at Swansea Library, p. 22.\n\n19 Ibid., p. 19.\n\n20 Williams, I., _With Scott in the Antarctic Edward Wilson, Explorer, Naturalist, Artist_ , The History Press, Gloucestershire, 2009, p. 25.\n\n21 Thomas, N.L., _A Hundred Years in School, St Helen's 1874\u20131974_ , Souvenir Centenary Booklet, held at Swansea Library, 1974. p. 22.\n\n22 The National Archives, Service Certificate (No.160225) Ref. ADM 188\/235.\n\n23 William Ewart Gladstone (1809\u20131898). Liberal politician and repeatedly Victoria's Prime Minister. At the time of his visit to Swansea he was out of office, but was later to serve his final, fourth term.\n\n24 _Gower, The Treacherous Coast_. Map based on the original idea and research by Mike Downie. \u00a9 Mike Downie, 1985.\n\n25 A three-masted ship.\n\n26 Winton, J., _Hurrah For The Life Of A Sailor, Life on the lower deck of the Victorian Navy_ , Michael Joseph Limited, London, 1977, p. 288.\n\n27 Gregor, G.C., _Swansea's Antarctic Explorer, Edgar Evans, 1876\u20131912_. Swansea City Council, Swansea, 1995, p. 9.\n\n## 2\n\n## The Boy Sailor: Naval Training\n\nHe did not have to wait long. The 5 April 1891, soon after his 15th birthday, saw Edgar attending his medical examination. Rules for medical fitness to enter the navy were laxer than today. A boy had to be without a physical deformity and to be able to speak clearly; he had to have good eyesight, colour vision and good hearing in both ears. There should be no obvious signs of injury to the head and he should not be of 'weak intellect'.1 Boys who could read and write clearly were favoured.2\n\nThe navy wanted boys with 'good heart and lungs' and without any hernias or 'tendency thereto'. There should be no disease or malformation of the genital organs.3\n\nOf particular interest to Edgar were the regulations concerning teeth. These stipulated that boys below the age of 17 could have _seven_ defective (decayed) teeth. Entrants over 17 could have _ten_ problematic teeth, the only proviso being that all ages had to have four sound, opposing molars (two in each jaw) and the same number of incisors similarly placed.4 Dental hygiene was little practised in the 1890s and tooth decay was commonplace. Young people in the United Kingdom frequently had all their teeth removed as a 21st birthday present (an option clearly not open to would-be sailors), to avoid the infection, pain and expense of dental work. Edgar had eight decayed teeth.5 He presumably had to have these (or at least some of them) attended to before he was finally accepted after a special application.6\n\nHis career began on the training establishment for Royal Navy Boy Seamen, HMS _Impregnable_ , on the 15 April 1891 for three days.7 After this Edgar Evans, Boy 2nd Class, official number 160225, was transferred to the wooden training ship HMS _Ganges_.8 Edgar's Certificate of Service continued until 17 February 1912, when, as Chief Petty Officer, he was discharged, 'lost in British Antarctic Expedition'.9 Roland Huntford denigrates Edgar Evans in his book _Scott and Amundsen_ , by writing that over the years he turned into a 'beery womaniser', exposed to the risk of venereal disease.10 This might imply absences during his training, but Edgar's naval Certificate of Service records no evidence of this, rather a seamless progress through the ranks.11 His 'Character and Efficiency' throughout this time is described as being 'Very Good', except for 1897 and 1899 when it was 'Good'.12\n\nEdgar started his new life at a time when Britain's commercial and imperial power was at its zenith and the navy an important guardian of that power. But as there had been no major sea battle since the Battle of Trafalgar, Britain faced no obvious rivals and the service was perceived to be stagnating and becoming hidebound by tradition.13 It was also becoming a subject for national debate. The Naval Defence Act (1889) authorised the expenditure of \u00a321,000,000 on the navy and the building of seventy-two new warships. The navy was to be on a scale 'at least equal to the naval strength of any other two countries'.14\n\nThe navy was becoming fashionable, too. The Royal Naval Exhibition of 1891 was hugely popular; visited by over 2.5 million people, it aimed to draw attention to important aspects of naval life and history. Attractions in the exhibition included a life-size model of the lower deck of HMS _Victory_ at Trafalgar (showing the death of Nelson), an area for field gun drill and manoeuvres, a lake with two miniature battleships fighting out naval engagements, a 167ft model of the Eddystone lighthouse,15 relics from Arctic expeditions and a fleet of fifty model silver ships.16 In 1893 the Navy Records Society was first published. This publication featured historical documents that illustrated the prestigious history of the Royal Navy and in 1894 the Navy League17 was established. This aimed to underline Britain's status as a world peace power, to promote public awareness of the country's dependence on the sea and to emphasize the fact that a powerful navy was necessary to maintain that power. The League stated that the primary aim of national policy was the command of the sea.18\n\nBluejackets (enlisted men with Edgar amongst them) were becoming the sentimentalised nation's darlings and nautical dramas, such as _Black Eyed Susan, True Blue and HMS Pinafore_ , became popular.19 Even Edgar, a 15-year-old 'boy', could share in this nationalistic pride. Indeed, in the popular imagination the British 'tar' was the envy of the world. Led properly, 'he would go anywhere, do anything and do it with a will'.20 When Queen Victoria's Golden Jubilee Naval Review took place in 1887, the _Daily Telegraph_ wrote that the people loved their navy and believed in it.\n\nThe reality of training was very different from this imperialistic, jingoistic attitude. Many a fond hope for adventure and excitement must have been irredeemably crushed within hours of entering the service. _Ganges_ , which Edgar joined in 1891 with his parents' consent, was an old hulk in Falmouth, Cornwall, which served as a training establishment for Boy Cadets. Later, at 18, he signed for a further twelve years (in his case from 9 March 1894 to 9 March 1906) and then at the age of 30 he signed on for a further ten years. This second signing was essential because twenty-two years actual service (gaps due to illness or for other reasons were not counted) was the minimum required for a sailor to be eligible for a pension.21\n\nReading the accounts of life on the _Ganges_ as recorded in extracts from the _Falmouth Packet Newspaper 1866\u20131899_ , and reading personal accounts of the life endured by the Boys, is like looking through the two ends of a telescope. Both are undoubtedly true, but the training, aimed at toughening the boys, was harsh and often cruel. It must have often seemed overwhelming to the 15-year-old boy, now classified as Boy 2nd Class, and to the thousands of other Boys who went through the system.\n\nThere was no soft introduction. From the moment he was on board Edgar was caught up in the everyday routine. First he was told how and where to sling his hammock, then issued with his kit (for which he had an allowance: \u00a36 in 1891, lesser amounts in 1893 and 1906).22 His civilian clothes were sold and he was introduced to the overcrowded, under-ventilated, unsanitary ship that was to be his base for the next year. The kit issued was quite extensive; over sixty items are listed in the _Navy List_ for 1891, including jacket, jerseys, trousers, hats and caps, boots, bed and covers, a knife and two lanyards (ropes worn around the neck for securing whistles or knives). Intriguingly, two 'cholera belts' are listed; these are bands of flannel, sometimes with strips of copper in them, to be worn around the waist and thought to increase 'bodily resistance'.23 It is not clear whether these were issued routinely against the possible perils of the training ships or held back for use in the east.\n\nThe Boys all had a 'Housewife' containing needles, buttons, thread and cotton, so that they could keep their kit in the condition demanded by the service.\n\nIn their day-to-day existence the Boys were entirely at the mercy of their Instructors. Lionel Yexley, a Boy on HMS _Impregnable_ just a few years before Edgar joined HMS _Ganges_ , recorded an existence that would have been similar to Edgar's experience. The day began at 5.30 a.m. when, wakened by the shrill notes of the Bosun's pipes and with his hammock safely slung and his kit in place, he was given a ship's biscuit, plus a basin of hot cocoa with a little sugar. These biscuits were staple naval diet; they were routinely and famously full of weevils \u2013 little beetles, which swarmed out and floated in the cocoa when the biscuit was dunked in it. Breakfast came only after the Boys had worked for several hours scrubbing the decks. For this activity they had to pull their trousers above their knees and were not allowed to wear shoes or socks, even on the coldest days.24 Breakfast was a hulk of bread with a scraping of butter or dripping (on another ship, HMS _Vincent_ , at about this time, no Boy was allowed his bread ration until he had collected two hundred cockroaches to exchange for the food). The meal was followed by sail drill and mast and yard drill, considered important in spite of the fact that there were few sail-driven war ships by the 1890s, although, as it happens, this training was to be of particular relevance to Edgar. He was instructed in all aspects of sail maintenance: shortening and setting, loosing and furling sails. Later he would learn about the rigging, climb the mast and gain knowledge of the hull.\n\nAfter this came the prayers, read by the Captain or Chaplain, followed by gunnery training, with the Instructors concentrating on muzzle loading cannons similar to those used at Trafalgar.25 The Boys still had Cutlass drill.26\n\nAt 11.30 a.m. the instruction finished and the Boys fell in to witness the daily punishments. Flogging was abolished in Britain in 1891, partially due to the long-term efforts of a man with indirect connections to Edgar, Sir Clements Markham, 'father' of Scott's first Antarctic expedition, but the cane, the birch and the rope's-end (the stonnicky), were feared symbols of discipline. 'Miscreants' were whipped with a cane bound at both ends with waxed twine to prevent splitting. They were punished for offences that seem minor today: no chinstrap sewn on the cap, a button off the trousers or being 'slack' at falling in. For this ordeal, hammocks were lashed into a cross shape, the Boy to be caned had his shirt drawn up around his waist, leaving only his duck (heavy cotton) trousers to protect his buttocks from the vicious cane. Usually the punishment was six to nine cuts and the weals took about ten days to heal. Serious offences (theft or desertion) were punished by the birch, up to twenty-four strokes being permitted.27 This was an appalling punishment. In 1892 on the training ship HMS _Boscawen_ in Portland, the birch was pickled in brine; this made it tougher, so that it caused more tearing and laceration of the skin (on this ship, the Corporals apparently took it in turns to administer alternate strokes, laughing as they did so).\n\nAfter this terrifying experience, those with any appetite had their dinner. The Boys prepared this themselves and took it to the overworked galley cooks to be put in the oven. They ate meat and potatoes with occasional helpings of cabbage or doughboys.\n\nThe final meal of the day was at 3.30 p.m., thus giving a literal interpretation to the word 'breakfast' the following morning. Tea consisted of bread and treacle (the treacle often being taken by the strongest bully, it was considered that actually getting hold of the food was a wonderful way of encouraging initiative), washed down by tea.\n\nEven this meagre provision of foodstuffs depended on the weather being good enough for the supply boat to get to _Ganges_. If the weather was bad, the diet reverted to salt beef, canned pork and the weevily ship's biscuit. Although officially vegetables were provided every day, the food given to the Boys seems like a subsistence allowance and it is difficult to see how some of them did not succumb to deficiency diseases such as scurvy. However there was probably was no evidence of full-blown nutritional disorders; those who survived the system and wrote their memoirs do not record deficiency illnesses. No questions were asked in Parliament about malnourished Boys.\n\nThe afternoon was spent in further instruction; tying knots and splicing, more sail and arms drill, boat handling. There was also a little instruction in reading, writing, arithmetic, geography, religious instruction and also naval history, though Edgar's formal education had been virtually completed by the time he entered the service. After 4 p.m. there was recreation: drill, dumb bells, Indian clubs, football on the shore in the summer (though the time available was limited in Edgar's time as British Summertime was not introduced until many years later). When he went back on board _Ganges_ he had to mend his own clothes and then he could pass the evening with draughts, ludo (a board game with dice) and reading. Remarkably Edgar remained a keen reader \u2013 his schoolmaster at St Helen's had done well.\n\nLife on _Ganges_ was endured without heating and with candles and oil lamps to light the evening hours. The Boys had a bath each week. _Ganges_ had six baths; everyone went through them in turn with no change in the cold water and the Boys had to wash their clothes in the same water too, which quickly became black from their dirt and from the dye in blue serge uniforms. After the bath the Boys were lined up for inspection by their instructor. They were only too keen to look clean; the instructor was aided in his inspection by his stonnicky, the fearsome symbol of authority.\n\nIt was considered that sailors should be able to swim. Though this is entirely reasonable, the methods employed seem horrendous to the modern reader. A sort of bathing tray was lowered by the side of the ship, a large crate with the boards at the bottom set apart so that water got in. Barnacles (small organisms with sharp shells) were put on the bottom, so that once a Boy was in the water he was committed to try and swim if he wanted to avoid his feet being cut by the barnacles. In this respect Edgar was lucky; he was used to the sea whereas many unlucky Boys came from the countryside and had never been in water. Indeed, the authorities felt very strongly about swimming; in one training ship those Boys who still could not swim after thirteen months were flogged. A Parliamentary Question revealed that, after this, only six unfortunates still failed the test.28\n\nIt was not all repetitive training. There were other activities. Efforts were obviously made to enliven the Boy's lives; for example singing was thought to be good for the boys and Edgar was a tuneful singer. It was written that 'the songs of home never sounded sweeter than when heard far away from home'29 and the Chief Instructor for Singing (under the School Board for London) visited Boys' training ships to recommend a suitable selection of songs; _A Life on the Ocean Wave, Britannia, the Pride of the Oceans_ or _We'll soon sight the Isle of Wight my Boys_ were popular. In 1890 the Bandmaster of _Ganges_ edited the _Royal Naval Song Book_ , a collection of fifty national melodies and a naval song book has been in print ever since. In the _Falmouth Packet_ , newspaper reports on _Ganges_ ' activities, singing and recitations are mentioned frequently. Also there were concerts. Remarkably, Offenbach's one-act operetta, _The Two blind Men (Les deux Aveugles_ ), was performed (in English)30 and theatricals and lantern slides were put on, in addition to cricket matches, boat races, football matches and sports days (the name of 'Evans' appears in the report on the prize-giving of 27 May 1893 as coming second in the sack race,31 but he did not shine athletically when he was training on _Ganges_ ). The Boys could have breaks from the ship, if they had anywhere to go to: half days on Sundays and, if their parents could afford it, two weeks in the summer and four weeks at Christmas.\n\nThis was the life that Edgar entered to. It is no wonder that years later he said that although he practically ran away to join the navy, he was sorry he had done so for the first two years were so arduous, despite getting used to it after that.32 He was on _Ganges_ for twelve months. By 1892, newly promoted to Boy 1st Class on 21 April, he would have enough practical knowledge to be sent to sailing brigs which cruised along the south coast of England and down to Spain for six weeks. His thin uniform offered poor protection against the conditions at sea. He had not yet reached his full height of 5ft 10in, and at 5ft 6in33 he probably would have just been able to stand upright between the decks.\n\nIn spite of the modernisation to be introduced by that naval innovator, Admiral Sir Jackie Fisher,34 for the likes of Edgar the discipline and routine remained horribly traditional. One contemporary, a lad named Fred Parsons, who was 15 in 1893 when Edgar was 17, described how he and his mates were fed with pork from a cask stamped 1805 and how, when he cried out when a marine stepped on his bare toe, he was disciplined to six cuts of the cane, for 'talking rather than hoisting'. He wrote that the cane was not struck straight down, but rotated in a half circle so that the exposed flesh was struck as the cane travelled upwards, which was more painful. Fred wrote that he held out till the third stroke when he let out a scream that brought the upper deck to a standstill.35\n\nIn January 1893 Edgar was transferred to HMS _Trafalgar_ , a battleship of 11,940 tons, carrying twenty-nine guns of various descriptions and six torpedo tubes. _Trafalgar_ was the second flagship of the Mediterranean Fleet and based in Malta. At 17 Edgar was at last beginning to see the world. A brief spell on HMS _Cruiser_36 was followed by a return to _Trafalgar_ where, in March 1894, he was promoted to Ordinary Seaman. This is the lowest rank in the navy and marks the beginning of his official naval career. He was 18 (and now described in official records as having 'a device' on his right forearm and a stabbed heart on his left forearm).37\n\nThe next stage up the promotion ladder from Ordinary Seaman is Able Seaman and Edgar, having presented himself to _Trafalgar_ 's Captain for an examination of his skills in seamanship, sail reefing, knotting and his familiarity and ability to work on any part of the ship, was duly promoted in July 1895. He remained on _Trafalgar_ for a further ten months, until May 1896, when he was transferred back to shore barracks in Portsmouth, firstly at HMS _Vivid. Vivid_ was a new establishment, thought by some in the naval hierarchy to be so expensively and lavishly built that it was a complete waste of money (it had gas lighting, electric bells, good washing facilities and an immense drill hall where Edgar exercised). Edgar then had a barrack transfer to HMS _Victory_ , whose most famous engagement, he learnt, had been the Battle of Trafalgar in 1805.\n\nIn September 1896 he was transferred to HMS _Excellent_ , another shore establishment and the Royal Naval Gunnery School at Whale Island, where he was taught the principles and practice of firing and maintaining naval guns. Whale Island still values its connection with Edgar. An accommodation block for Warrant Officers and Senior Ratings was named after him in the 1960s, the first ever to be named after a Petty Officer (rather than an Admiral). The buildings were replaced in the 1990s; Edgar's name was honoured by a plaque commemorating his service. He remained at the shore base _Excellent_ and another 'stone frigate' HMS _Vernon_ until May 1898. _Vernon_ was the home of the Royal Navy's torpedo branch, based independently but near to HMS _Excellent_. In his three months at _Vernon_ , Edgar learnt about torpedoes and the art and purpose of firing these missiles from battleships. So by the time of his next posting, Able Seaman Edgar Evans had received training in gunnery and torpedoes.\n\nUpward progression was continued when Edgar became a Leading Seaman. The Queen's Regulations and Admiralty instructions state that 'Men who are thorough seamen, good helms-men, able to assist in repairing sails and practical riggers capable of doing duty in any part of the Ship, may be examined for the rating of Leading Seamen. If found qualified, the Captain may rate them as such, as vacancies occur for that description of men in the complement'.38 Edgar passed this milestone in June 1898 when he was on the shore establishment HMS _Pembroke_. The only blemish in his exemplary record occurred during a three-month stint on HMS _The Duke of Wellington_ , a gun ship, from 1 April to 26 June 1899. He was demoted back to Able Seaman by his Captain for twenty days (6\u201326 June) and for a further three days when he was transferred to the battleship HMS _Majestic_.39 After this he is re-registered as Leading Seaman again. As the time is of short duration, the misdemeanour must have been a minor one.\n\n_Majestic_ was awe-inspiring. She was the largest battleship in existence. She had forty-four guns of different sizes, five torpedo tubes and a crew of 700 and led the Channel and Atlantic Fleet. The Commander of the Fleet was Vice Admiral, Prince Louis of Battenberg, GCB, GCVO, KCMG, PC,40 probably called something less respectful by the lower deck. The torpedo officer on this tremendous warship was the man who was to shape Edgar's destiny, Lieutenant Robert Falcon Scott. Edgar's basic training in gunnery and torpedo work immediately put him into contact with Scott as he served on _Majestic_ in the Channel Fleet for the next two years.\n\nWhen Scott was given command of the British Antarctic Expedition and of _Discovery_ , the Admiralty gave permission for naval men to apply to join the expedition, which aimed to penetrate the unknown mysteries of the Antarctic and amass as much scientific and geographic information as possible, about that part of Antarctica that had been seen in the 1840s by James Clark Ross. Scott always felt that he wanted a Royal Naval crew; men in the Merchant Navy were less drilled in the rigid hierarchy of command than the Royal Navy. He knew that there would be volunteers and when Edgar, as well as Petty Officer David Silver Allan and Stoker Arthur Lester Quartley from _Majestic_ , volunteered, all three were appointed. Edgar's long association with Scott and the Antarctic had begun.\n\nScott knew that the bluejackets would bring the sense of naval discipline with which he was familiar. In fact, as _Discovery_ was not in government employment, her crew was not legally subject to the Naval Discipline Act, but they signed on voluntarily under Scott's command. Everyone on board must have been aware of the real position and the men, as well as the officers, deserve credit for observing this 'fiction'.\n\nEdgar had one more step up the promotion ladder before he sailed in _Discovery_ ; he was promoted to Petty Officer 2nd Class on 18 November 1900. (Later, on Scott's recommendation on 2 April 1904, he was promoted to Petty Officer 1st Class and allowed to qualify as a gunner. To complete this advance he had to pass a further 'Education Certificate' in 1908).41 Petty Officers were, and are, the Sergeants of the navy. They are in charge of the seamen and the daily working routine of the ship. They are selected from Leading Seamen, preference being given to those who had signed on for continuous service (as Edgar had done), and they may have special skills (for example a stoker or carpenter). Edgar was classified as a Seaman Petty Officer, Gunnery Class. For his promotion Edgar had to take the examination laid down by the Admiralty; he had to show knowledge of the King's Regulations, he had to have an understanding of the principles of seamanship and be considered capable of enforcing regulations on board.\n\nEdgar's service record shows a man of ability, intelligence and determination. Scott valued him from the start. He valued Edgar's intelligence, physical strength and practical ability, but also his kind spirit and good nature.42 Edgar repaid him with a loyalty that lasted to their deaths on the ill-fated return from the South Pole in 1912.\n\n### Notes\n\n1 The Queen's Regulations and Admiralty Instruction for the Government of Her Majesty's Naval Service. Her Majesty's Stationery Office, London, Regulation 1153: 1899, p. 514.\n\n2 Ibid., 1861, Chapter 1X, Para.2.\n\n3 Ibid., Regulation 1154, section 6, p. 514.\n\n4 Ibid., Regulation 1154, section h, p. 516.\n\n5 Gregor, G.C. _Swansea's Antarctic Explorer, Edgar Evans, 1876\u20131912_ , Swansea City Council, 1995, p.11.\n\n6 _Cambria Daily Leader_ , 13 February 1913.\n\n7 HMS _Impregnable_ was a training establishment started in Davenport in 1862. As ships were added to the establishment each was renamed _Impregnable_. The ship in Edgar's day had been previously named HMS _Howe_.\n\n8 _Ganges_ was commissioned in 1821. She had seen a good deal of action and was the last sailing ship to sail as a Flagship.\n\n9 The National Archives, Service Record (No. 160225), Ref. ADM188\/235.\n\n10 Huntford, R., _Scott and Amundsen_ , Hodder and Stoughton, London, Sydney, Auckland, Toronto, 1979, p. 328.\n\n11 Edgar was kept on the ledger of HMS _President_ when he was on Scott's second expedition of 1910\u201312. National Archives, Service Record (No. 160225), Ref. ADM 188\/235.\n\n12 Certificate of Service. D.N.A. 3A\/S.R. Official number. 160225. Portsmouth Division.\n\n13 Carew, A., _The Lower Deck of the Royal Navy 1900_ \u2013 _1939,_ Manchester University Press, 1981, p. xiv.\n\n14 Winton, J., _Hurrah For The Life Of A Sailor, Life on the lower deck of the Victorian Navy_ , Michael Joseph, London, 1977, p. 287.\n\n15 Eddystone Lighthouse. Lighthouse built to protect mariners from the treacherous Eddystone rocks in Cornwall. This was the fourth lighthouse here. It was designed by James Douglas in 1880.\n\n16 Winton, J., _Hurrah For The Life Of A Sailor, Life on the lower deck of the Victorian Navy_ , Michael Joseph, London, 1977, p. 292.\n\n17 The Navy League was established in 1894.\n\n18 Winton, J., _Hurrah For The Life Of A Sailor, Life on the lower deck of the Victorian Navy_ , Michael Joseph, London, 1977, p. 287.\n\n19 Ibid., p. 288.\n\n20 Ibid., p. 301.\n\n21 Carew, A., _The Lower Deck of the Royal Navy 1900\u20131939,_ Manchester University Press, Manchester, 1981, p. xvii.\n\n22 The National Archives, Service Record (No. 160225), Ref. ADM188\/235.\n\n23 _Navy List, corrected to March 1891_ , Her Majesty's Stationery Office, London, 1891, p. 576.\n\n24 Boys only wore shoes and socks with their best uniform or when they were ashore or visiting the sickbay. Summers, D.L., _One hundred years of training Boys for the Royal Navy_. HMS _Ganges_ , Shotty Gate, Suffolk, 1966, p. 34.\n\n25 Ibid., p. 35.\n\n26 Phillipson, D., _Band of Brothers: Boy Seamen in the Royal Navy 1800_ \u2013 _1956_ , Sutton Publishing, Gloucestershire, 1966, p. 18.\n\n27 Summers, D.L., _One hundred years of training Boys for the Royal Navy_. HMS _Ganges_ , Shotty Gate, Suffolk, 1966, p. 35.\n\n28 Ibid., p.40.\n\n29 Winton, J., _Hurrah For The Life Of A Sailor, Life on the lower deck of the Victorian Navy_ , Michael Joseph Limited, London, 1977, p. 297.\n\n30 HMS _Ganges_ , Mylor. Extracts from the Falmouth Packet Newspaper, 1866\u20131899. Compiled by Harwood, B. HMS Ganges Association, Cornwall Division. 05\/03\/1891.\n\n31 Ibid., 27\/05\/1893.\n\n32 Debenham, F., _Journal_ , 19\/01\/1911\u201308\/03\/1911 MS 279\/2: BJp, p. 15.\n\n33 The National Archives, Service Record (No. 160225), Ref.ADM188\/235.\n\n34 Admiral Sir Jackie Fisher. When Edgar entered the navy Fisher was in charge of the Portsmouth Naval Dockyard.\n\n35 Winton, J., _Hurrah For The Life Of A Sailor, Life on the lower deck of the Victorian Navy_ , Michael Joseph, London, 1977, p. 297.\n\n36 HMS _Cruiser_ was an Osprey-class sloop that Edgar served on from May until August 1893.\n\n37 The National Archives, Service Record (No. 160225), Ref. ADM188\/235.\n\n38 _Queen's Regulations and Admiralty Instructions 1861_ Chapter V11. Regulation 13.\n\n39 The National Archives, Service Record (No. 160225), Ref. ADM188\/235.\n\n40 Louis of Battenberg, GCB (Knight Grand Cross); GCVO (Grand Cross of the Victorian Order); KCMG (Knight Commander of St Michael and St George); PC (Privy Councellor); (1854\u20131921), Ist Marquess of Milford Haven and Grandfather of Prince Phillip, Duke of Edinburgh, the husband of Queen Elizabeth the Second.\n\n41 The National Archives, Education Certificate number 1025\/08.\n\n42 Bernacchi, L., Saga, London, p. 96.\n\n## 3\n\n## The _Discovery_ Expedition\n\nMuch has been written about the Scott's journeys in the early 1900s yet relatively little about the contribution of the seamen to their achievements. Yet the expeditions' successes self-evidently depended on the men's huge contributions, amongst which Edgar Evans' input can be ranked with the highest.\n\nEdgar was following Scott, his commander, rather than a geographical target when he joined the British National Antarctic Expedition, the _Discovery_ expedition. Other destinations would have been as acceptable. Almost certainly he knew where the Antarctic was \u2013 that large white mass on the bottom of the globe \u2013 but if he knew little more than this, he was on par with the majority of British citizens. Antarctica had been described by Captain James Cook, on his second voyage of 1772\u201375, as lands doomed by nature to everlasting frigidness, never to feel the warmth of the sun's rays and the land mass had been identified by subsequent explorers. It had been visited briefly by a number of expeditions: the Norwegian Borchgrevink claimed to have planted the first footstep on its icy surface when he leapt out of his rowing boat in 1895. Later, in 1899, he led an expedition, funded by the British newspaper magnate George Newnes, which landed and overwintered on Cape Adare. But sorties to the interior had been limited to a short trip on the Barrier; no expedition had penetrated into the heart of Antarctica. Its geography was obscure; was it the mythical seventh continent, or a series of islands? Was the South Pole on land or water? Edgar would probably have read about previous expeditions and would have known about these uncertainties; he would also have heard that more was known about the moon than about Antarctica.\n\nBut 1901 was planned as the year for exploration and discovery in Antarctica. There was to be scientific cooperation between nations: Great Britain, Germany, Scotland and Sweden planned explorations and aimed to pool their scientific findings. Of these expeditions _Discovery_ was the most successful in making inroads into Antarctica, but the continent would take decades more to reveal her secrets.\n\nToday we know that Antarctica is a continent; the coldest, windiest and most remote place on earth, it occupies a tenth of the world's landmass and is covered by an icecap that (as Scott's team discovered), flows slowly towards the surrounding oceans. It has very little rain (less than the Sahara Desert) and has few indigenous inhabitants, though visited regularly by varieties of penguins, seals, birds and whales. These facts would, of course, have become of interest to Edgar and his fellow crew-members, but in a peripheral way. Their focus was more immediate and practical: to obey orders, to maintain discipline, to be cheerful, and to rise to any crisis that might befall the ship or the expedition.\n\nIn Britain, plans for the expedition aroused national interest. Apart from grants from the government and The Royal Geographical Society (including a huge private donation from one of its Fellows), large and small donations were received from all over the country. Thousands applied to become part of the voyage; there were 3,000 applications from the navy alone1 and Edgar was elated to be part of the patriotic voyage. When, on Tuesday 6 August at 11.30 a.m. precisely, _Discovery_ slipped her moorings and sailed for Madeira, the young Edgar was an obscure, 25-year-old Petty Officer (2nd Class). By the time _Discovery_ returned to England three years later, he had become a national hero, a local lad made good, written about glowingly in the South Wales newspapers,2 a modest young man who had sailed to South Africa, New Zealand and Antarctica and remained on the Antarctic continent for over two years, enduring the worst conditions that it could throw at him. By 1904, he had contributed to seven sledge journeys totalling 174 days (a record only beaten by Scott) and experienced the horrors of man-hauling at sub-zero temperatures. He had become a man of new knowledge and experience.\n\nHowever, in August 1901, none of the crew could be certain that _Discovery_ would even reach Antarctica (in 1897 the Belgian vessel _Belgica_ had been trapped in Antarctic ice for thirteen months). They did know, however, that in _Discovery_ they had a ship that had been especially built for the Antarctic conditions. The ship was the brainchild of one man, Sir Clements Markham (1830\u20131916), who for years had pursued his vision of Antarctic exploration with determination, tenacity and astuteness. By 1900 Sir Clements had managed to get enough support from the government and other sources to commission his purpose-designed, 1,600-ton _Discovery_ \u2013 a coal-fired ship, rigged as a barque.3 Her name was deliberately chosen to 'continue the spirit of maritime enterprise' which Sir Clements felt had always been a distinguishing feature of the British nation.4 She was painted black, her profile lightened by yellow masts and funnel and white boats painted with a 'D' in black and gold. She was constructed of wood (which would flex when ice pressed against it); her bow was essentially 11ft of oak with sheathing of ironbark, which allowed her to function as an icebreaker, and along her sides were two outer layers of wood, about 26in thick, to protect from glancing ice blows. Water could be drained from the engine to prevent freezing, and both rudder and the screw could be detached and brought into the ship if need be in icy conditions, as was done later in the voyage.\n\nOne reason that the expedition finally received government subsidies related to the need to advance work on terrestrial magnetism. The Magnetic Pole is not in a static position \u2013 it moves each year \u2013 yet an accurate assessment of its position was needed to calculate longitude. In the early 1900s movement of the Magnetic Pole caused significant problems (and therefore concerns for commercial shipping) in the Southern Ocean. Ships sometimes went miles out of their optimal routes and one of _Discovery_ 's briefs was to investigate the location of the Magnetic Pole. Edgar was well aware of this difficulty; his father's Cape Horn voyages via the Southern Ocean to South America had all too much experience of the problem. So _Discovery_ was equipped with a magnetic observatory, and to ensure accurate recordings no iron or steel was allowed within 30ft (fore, aft, either side, above and below) of the magnetic instruments. Inside the 30ft radius, copper was used and, instead of wire in the rigging, ropes were made of hemp and in the cabins the beds were made of rolled brass with wooden battens. Officers within the circle were threatened with having to shave with brass razors. The zoological and biological laboratories on either side of the meteorological observatory were not allowed any iron tools or even steel-wire bottlebrushes.\n\nIn 1900 Robert Falcon Scott, then a relatively unknown Lieutenant, was appointed to lead the British National Antarctic Expedition. There was never any question that Scott's expedition was to have scientific as well as exploratory ambitions; the support of the Royal Geographic Society and the Royal Society (who were interested in the behaviour of tides, currents, glaciers and southern weather), had been an important factor in Sir Clements Markham obtaining government finance. The expedition was instructed to investigate whether or not an Antarctic Continent actually existed, thereafter to investigate the position of the South Magnetic Pole, record those problematic southern winds and currents and study the geography, meteorology and physics of the region. A determination to claim as much land as possible for Great Britain would have been heartily endorsed by Edgar and his comrades.\n\nScott knew that bluejackets5 would bring naval discipline to the venture. In total, thirty-two personnel were appointed to _Discovery_ from the Royal Navy and two from the Royal Naval Reserve: Officers, an Officer\/Engineer, a Bosun, Petty Officers, Able Seamen, Royal Marines, Stokers, a Steward and a Carpenter.6 Edgar, as well as David Silver Allan (Petty Officer), Arthur Lester Quartley (a Leading Stoker) and James Dellbridge, the 2nd engineer, had served with Scott on _Majestic_ and were quickly appointed when they volunteered. Although _Discovery_ was run as if she sailed under the Naval Discipline Act, she was not in fact in Government employment. This was not a problem for Edgar; his appointment was a simple continuation of his naval appointment (as was the case for the other naval men, a substantial form of government support). During the _Discovery_ expedition his records show he was on the pay roll of HMS _President_ , seconded to the National Antarctic Expedition. All the crew signed voluntarily under Scott's command. Scott wrote that the success of the expedition was not due to a single individual, but to the loyal cooperation of all the members. He paid tribute to the petty officers and men who had worked so cheerfully and loyally for the general good.\n\n_Discovery_ was launched in Dundee on 3 June 1901. The imagination of the nation was caught as she progressed around the coast towards the East India Dock in London, her progress cheered by thousands on land. Edgar joined his ship on 27 July 1901. He was immediately precipitated into the bustle and hustle of preparing the ship and storing the huge amount of provisions and equipment needed for at least a year in Antarctica. The work of the crew was interrupted and constantly delayed by dignitaries and members of the public eager to inspect the ship. Finally, the Bishop of London came on board to address and bless the officers and men. The Bishop spoke of the difficulties and dangers of the voyage but also impressed with his comments concerning the need to remember that God was with them always. Quoting from the psalms, he stated the necessity of good comradeship and sympathy for each other \u2013 'Behold how good and how pleasant it is for brethren to dwell together in unity.'\n\n_Discovery_ sailed from London to Cowes in the Isle of Wight, an event of sufficient importance to be reported in _The Times_.7 Cowes was to be her final port of call in Britain before she sailed away for three years.\n\n### Notes\n\n1 Wild, R.F.J., Notes related to the British National Antarctic Expedition 1901\u201304, SPRI, MS 944\/3: D.\n\n2 _South Wales Daily Post_ , 20 September 1904.\n\n3 Three masts: two square-rigged masts, the third, (the aft 'mizzen' mast), rigged fore and aft.\n\n4 Skelton, J.V., & Wilson, D.M. _Discovery Illustrated_ , Reardon Publishing, Cheltenham, 2001, p. 10.\n\n5 Seamen in the British Navy.\n\n6 Markham, C., _Antarctic Obsession: The British National Antarctic Expedition 1901\u20131904_ , Bluntisham Books and the Erskine Press, Norfolk, England, 1986, p. 18.\n\n7 _Times_ , Thursday, 1 August 1901, p. 6. Issue 36522.\n\n## 4\n\n## From England to South Africa\n\nIn Cowes, the newly crowned King Edward VII and his beautiful queen, Alexandra, came on board to inspect the ship and her complement. Edward's sister, Empress Frederick, had just died,1 but he managed to speak well and impressed the lower deck with his comments. The King said that he was glad of the opportunity of saying goodbye to an expedition that was (for once), bound on peaceful aims for the increase of scientific knowledge. Seaman Thomas Williamson wrote that the King said 'we were a fine lot of fellows and before going he wished us God speed, a pleasant voyage and a safe return. We manned the rigging when he left, the Captain calling for three cheers for His Majesty which was heartily responded to'.2 The men were, for the most part, patriotically enthusiastic about the venture. Marine Gilbert Scott said that the expedition was 'to remove any lack of the unknown from the map of the world and to press forward the frontiers of human knowledge a little further'.3\n\nAlthough Edgar's local newspapers4 do not report on _Discovery_ 's departure, it was a national event. Government and official donations were complemented by small gifts and contributions from schools and individuals throughout the country. There were fundraising events and dinners. Williamson included a ditty in his diary, written for a dinner in a London club and sung to the strains of _Auld Lang Syne_ , which went:\n\nTo night, my brother Savages we bid a warm adieu\n\nTo Captain Scott and all the lot of his good ship's gallant crew\n\nThey sail, as sailors sailed of yore, like Britain's brave and bold\n\nFor a British crew is as stout and true as in the days of old\n\nSo here's to Scott and all his lot\n\nWith all his might and main\n\nAnd we'll meet them in the Savage Club\n\nWhen they come home again\n\nIt continues in typical populist fashion:\n\nThey sail to find the Southern Pole and bring it safely home.5\n\nThe 6 August 1901 was a brilliant sunny day when Edgar left England. Thousands cheered and waved from the shore. Hundreds, on little boats, shouted and tooted. Bands blared. The noise and the sight of yacht racing at Cowes faded gradually into the distance as _Discovery_ slowly sailed away.\n\nScott's planned route was from England to Madeira, then Cape Town and afterwards to Melbourne, Lyttelton and Antarctica. In the event several problems were to slow _Discovery_ 's progress; it was obvious from early in the voyage that her speed was slower than expected, the sail area was small \u2013 'it would take nearly a gale of wind to produce a respectable speed'6 \u2013 and she was a glutton for consuming coal. In spite of continual attempts at repair en route, she also leaked throughout the voyage (probably due to unequal contraction of poorly seasoned oak in the main frames which shrunk and bent the rivet-like bolts7 and, conversely, widening of the planking in the ship's sides close to the scorching engine room). In addition, she was 'a poor sailer', she had difficulty tacking into the wind and 'sagged leeward' (away from the wind).8 These problems meant that Scott could not keep to his schedule.\n\nFrom the start _Discovery_ was run on the naval 'tier system' with a line of command from Officers (including scientists), Warrant Officers, Petty Officers (POs) and Ratings. Sir Clements Markham lists three Warrant Officers and six Petty Officers, including Edgar, in the ship's complement.9 Each rank had its own quarters. POs were berthed in a separate, starboard-side area and had their own Mess. The Ratings lived and ate in their own quarters (which, being relatively near the stove, had an advantage in that it was warm in winter).10 They slept on hammocks that were stowed away during the day; they used their lockers as seats and slung their tables between the beams when they did not need them. Plans of _Discovery_ show six mess tables and total accommodation for forty-five crew (more than eventually sailed, which was thirty-five).11 The average age of the Petty Officers was 27 (Edgar was 25); the crew members' 23.12\n\nAt sea, a routine was quickly established on naval lines with watches of four hours. In between duties Edgar read, mended his kit, got to know his new shipmates and slept. As a Petty Officer of the watch he gave orders to the seamen for the smooth running of the ship. When he was on watch he instructed the bluejackets to climb the rigging and climb out on the yards to unfurl and furl the sails. The diaries are full of details of working on the topsail yards, shortening cables, cleaning, painting, pumping out the main hold and clearing up in general.\n\nThroughout the week duties followed the same routine, but on Saturdays the ship was given a thorough cleaning. This was for medical reasons as well as naval standards. In 1901, the cause of scurvy, the dreaded disease that had caused so many deaths in previous years, was still unknown. The value of citrus fruit juice, which had provided protection to earlier travellers, had become questioned as an effective defence,13 and one of the many theories as to scurvy's cause was that it was exacerbated by dark, damp conditions; Scott was determined to attend to every precaution that might keep this horror from his ship. On Sunday, Scott did a full inspection of the men and the ship. A religious service followed, Scott reading the prayers; the men, with _Discovery_ 's First Lieutenant Charles Royds (1876\u20131931) accompanying them on the piano, sang heartily. Scott used the time after the service to address the crew on specific problems or general matters, and, after this, the crew were (officially) at leisure.\n\nOn the first part of the voyage _Discovery_ proved to be a 'remarkably comfortable ship'.14 Scott described the voyage as 'most delightful'15 and the officers were very satisfied with the morale and general bonhomie \u2013 a satisfaction that persisted throughout the voyage. As in many organisations, however, the views of the 'managers' were not automatically reflected by those in the lower echelons. Some of the crew were unhappy with their conditions and grumbles surfaced early. After only twodays at sea, Thomas Williamson complained that he had 'six hours sleep in twenty-four hours of hard work'16 and that the routine consisted of 'all day at work and keeping watch at night'. Later, he complained that the officers must have thought that the sailors were 'automobiles or traction engines, which only needed to be oiled to keep on the go, rather than human beings'.17 But although Williamson commented freely in his logbook, his observations are by no means all negative. He liked the ship and thought she sailed well; he appreciated the novelty, the interest and (sometimes) the beauty of the places he was visiting. His day's log entry frequently ends with 'All's well'.\n\nAnother Able Seaman, Frank Wild (later to become a well-known Antarctic hero when he was on Shackleton's famous expeditions), also recorded his reservations. Wild wrote that the voyage from England was neither eventful nor happy because the crew were worked very hard, day and night, and often unnecessarily so. He wrote, puzzlingly, since he himself was a naval seaman, that this way of working was the naval way and 'as the Captain and most of the officers were naval men they could not get out of these ways'. Only when _Discovery_ reached Antarctica did things 'work round to a reasonable state', and they had 'occasional glimpses of sunshine through the clouds of discontent'.18\n\nDifferences between ranks permeated many aspects of life on board. Although the same food was provided for officers and men, the men served themselves, whilst the officers were served at a table in the mess refined by oak furniture, linen on the table, silver cutlery and monogrammed china. Marine Gilbert Scott, one of the stewards, has left an account of his duties in the wardroom; he found the hours long and weary. There were differences in some supplies for officers and crew; for example, jam, marmalade, tea, coffee, tobacco and alcohol.19 In relation to tobacco, _Discovery_ carried 1,800 pounds, 1 and a half pounds per officer per month, 1 pound per man (the men smoked Navy Leaf, stronger tobacco than that smoked by the officers, and they rolled the leaf tobacco into 'pricks' with spun yarn, the traditional naval way).20 In relation to alcohol, the provision was different too; the 30 gallons of brandy, 60 of port and 36 of sherry were presumably for officer consumption, as were the 28 cases of champagne (listed under 'medical comforts'). Cases of these had been brought on board _Discovery_ from the king's yacht when Edward inspected the ship and had, most conveniently, been forgotten. The 800 gallons of rum were provided for the crew. Ratings were issued with grog each day \u2013 grog being a 'tot' of rum (an eighth of a pint), diluted with water, and the rum was a potent mixture of Demerara, Trinidad and other rums.21 The addition of water made it virtually undrinkable after an hour, but Dr Wilson wrote that somehow the tots were occasionally saved and drunk in bulk, causing a comfortable fuddledness. Non-drinking seamen, such as William Lashly, a Leading Stoker, could opt for lime juice but probably rarely did. Lashly claimed his ration, which he either distributed to friends22 or, more likely, traded for benefits such as a change of duties.\n\nBut the most important divides between men and their officers, present not only in the navy but in virtually every section of Victorian society, were education and pay. This combination of relative disadvantages created an almost impassable chasm for a man such as Edgar to cross. For most of the crew, their formal education had finished in their early teens. In relation to pay, there was little hope of Edgar ever achieving the remuneration received by the officers. In this instance, in comparison to Scott's annual pay of \u00a3865 (expedition pay \u00a3500, plus \u00a3365 naval pay),23 Edgar received a total annual pay of \u00a382 4 _s_ (\u00a341 and 1 _s_ from the expedition and \u00a341 3 _s_ from the navy).24 Seamen received just over \u00a35525 (for comparison, General Labourers received \u00a362, Porters \u00a389, Surgeons and Medical Officers \u00a3475).26 One man who did cross the divide was Able Seaman Frank Wild, who years after Scott's expedition and after heroic expeditions with Ernest Shackleton in the Antarctic, rejoined the naval forces in 1916, and in 1917 was given a commission as Temporary Lieutenant Royal Naval Voluntary Reserve (RNVR) and sent to the North Russian Front.27 Another seaman, Garrett, became a Lieutenant in the RNVR.28\n\nHowever, some links did exist. In the Victorian era, literature and poetry were important interests. There were hundreds of books on _Discovery_ ; publishers had given volumes, a 'well wisher' had given fifty novels and some authors had given copies of their works. There were also quantities of magazines,29 and Edgar and his companions had ample opportunities to read the authors and poets. Edgar was well read and Scott found him superior to the other seamen in this respect. Edgar had firm opinions on the works he read; he did not like Kipling's poems and Dickens did not appeal, but he greatly liked Dumas (whose name he anglicised to Dumm-ass), because the works had 'more plot'.30 Also, when they were on later expeditions, there was no differentiation between men and officers as they struggled equally in their quest to explore new ground. In one of the longest sorties of the _Discovery_ expedition, Scott spent sixty-six days in close proximity and complete equality with Evans and William Lashly, as they battled against Antarctica's awesome conditions. Scott felt he got to know his co-explorers well and had nothing but praise for them. Edgar, for his part, developed a deep loyalty to Scott that lasted until their deaths in 1912.\n\nAlthough on HMS _Majestic_ Edgar had served with Scott, he had to get to know the officers and scientists who would be making the long journey with him. He felt that the officers would be of importance to him, the scientists less so. Alongside Commander Scott, there were seven officers (plus the two doctors), who all took on specialist work in addition to their naval duties. The hierarchy was as follows: Second in Command was Albert Armitage, a veteran of polar work who had already been in the Arctic for three years. Armitage was Scott's navigator and ice master. The First Lieutenant was Charles Royds, who was in charge of the day-to-day running of the ship, allocating officers and men to the duties of the watch. His specialist area was meteorological observations and he had had training in magnetism. Royds had good contact with the men and was credited with achieving much of the relative harmony that existed on _Discovery_. He was a talented piano player and organised concerts for the men's entertainment. The Second Lieutenant was Michael Barne, a shipmate of Scott's on _Majestic_. He helped with magnetic observations and was in charge of the deep-sea sounding apparatus. The Third Lieutenant was a man later to become famous in the annals of Antarctic history \u2013 Ernest Shackleton. For the _Discovery_ expedition he was trained up in seawater density and salinity studies. Reginald Skelton, an ex-shipmate of Scott's from his _Majestic_ days, was the Engineer. Doctor Reginald Koettlitz, 6ft tall with a droopy moustache, who had already survived winters in the Arctic, took on the role of botanist. Dr Koettlitz had his problems with Edgar; although Edgar was always respectful in the presence of officers, his language, when he was working, was colourful. When Koettlitz heard some of Edgar's more florid expressions he reported him to Scott for bad language. Dr Koettlitz's junior was the recently qualified Edward Wilson, who was the zoologist and artist. The geologist was Hartley Travers Ferrar, aged 22 and also recently qualified.\n\nAmong the lower deck, Edgar's new colleagues were four Warrant Officers. The Boatswain (Bosun) was Thomas Alfred Feather, aged 31, who was in charge of seamen's duties. Scott was full of praise for his Bosun, writing that no rope or sail was lost on the three-year voyage under Feather's expert supervision.31 The other Warrant Officers were James Dellbridge (29), the 2nd engineer who came from _Majestic_ ; Fred Dailey (28), the carpenter who possessed an 'eye' for defects,32 and Charles Reginald Ford (23), the ships steward, whose duties were to keep an exact account of the stores. Feather's deputy was David Silver Allan from _Majestic_ , a Scotsman aged 31, who, with Thomas Kennar (25) and William MacFarlane (27), carried out the duties of Quartermaster \u2013 which means they had some responsibility for navigation and signals. Other POs for _Discovery'_ s journey to Antarctica were William Smythe (24), Jacob Cross (26) and William Lashly (33), the Acting Chief Stoker (re-rated by Scott as Chief Stoker). An American, Stoker Arthur Lester Quartley was described by Dr Wilson as 'quite the finest figure of a man I have ever set eyes on, standing just over six feet and a perfect giant in strength and sinew'.33\n\nEdgar himself was tall and big for a Welshman. He was nearly 6ft and weighed 12 stone 10lbs. He was strong, competitive and in 'hard condition'. Along with Cross, Kennar, MacFarlane and Smythe he was appointed as a supernumerary rating for the thirty-one-strong shore party.34 The shore explorations were to include a balloon ascent. Lashly and Engineer Skelton had been sent to the Army Balloon Factory at Aldershot to be given instruction on balloon ascents, one of Scott's more unusual briefs for the Antarctic.\n\nEight days out of England the ship approached Madeira, and Seaman Williamson wrote that the scientists were hard at work doing their depth soundings (difficulties were encountered with the apparatus which frequently broke). In Funchal, Madeira's capital, _Discovery_ 's crew took on 54 tons of coal. Coal was already being consumed more quickly than expected and this was to remain a concern throughout the voyage.\n\nAll the men took their turn with coaling \u2013 a filthy, backbreaking task. All the ventilators and any cracks had to be papered over before work began, otherwise the coal dust got everywhere, being most particularly unwelcome in the living quarters. In Madeira, a new worry relating to the ship's construction at Dundee surfaced, one that concerned Edgar. Various metal attachments for the sails had broken, and although these could obviously be repaired, the defect clearly made the crew anxious about the ship's performance in a severe gale.35 Gilbert Scott, the Marine, wrote that some of the men got drunk and started rowing and fighting, 'which was caused by the same thing that causes most of the trouble'.36\n\nOfficial plans for cooperation between _Discovery_ and other European explorations are underlined by the fact that at the same time that _Discovery_ was in Madeira, the German ship _Gauss_ was departing from Germany for Cape Town and the Antarctic. In the case of _Gauss_ , plans for collaboration were not as comprehensive as was hoped; she became trapped in the Antarctic ice.37\n\n_Discovery_ left Madeira bound for Cape Town with a stop planned in South Trinidad, a small island in the Atlantic. A few days out and Skelton's grumbling tooth abscess gave him so much pain that extraction became essential. As a sensible precaution Scott had arranged dental treatment for all the crew before _Discovery_ sailed, an attention that Skelton had presumably missed. This was much in the way that prophylactic appendectomies may be performed nowadays on men and women who will be cut off from medical attention for months. In the check-up over ninety teeth were removed, and nearly 200 fillings put in (at a cost of just over \u00a362).\n\nEdgar's teeth remained far from perfect; he had two extractions and three fillings38 (one unfortunate had ten fillings and five extractions).39 In fact, very little dental trouble was recorded on the expedition and the doctors must have reflected with relief on Scott's forethought.\n\nHowever, a problem was discovered when it was found that a serious new leak had developed.40 Black, stinking, slimy water had seeped into the hold and covered and rotted the bottom layer of provisions. All hands were called on to help. They were not pleased, and were particularly furious with the men who had stowed the hold in London. One hand wrote, 'and by all the saints above, the man or rather the men who were responsible for the leakage ought to be strung up, for the greater part of the provisions are in a dilapidated state and those at the bottom are utterly destroyed'.41 It was very hot (140\u00b0F in the engine room), but the gruelling work went on round the clock under a relentless tropical sun. Tins of cheddar cheese had gone putrid, eggs in unsoldered tins had gone rotten, and sugar had fermented. The holds had to be cleared and disinfected and the provisions stored above until a new platform was built in the hold.\n\nEdgar was involved in this unenviable task, made particularly obnoxious by a stench that was so awful that they could only go on with the work when a ventilation hole had been made. In spite of repeated attempts to solve the problem of leaks the problem continued to plague _Discovery_ as she sailed south, but this occasion gave the most trouble.\n\n_Discovery_ headed west to take advantage of the winds and currents sailing almost as far as South America before heading south-east towards Cape Town. On 31 August she 'crossed the line', the equator, a milestone associated with a rather violent initiation for those crew members crossing the equator for the first time42 \u2013 although Williamson described it as 'good sport'43 and likewise Lashly as a 'very good afternoon's sport'.44 It may have been good sport to the onlookers, but hardly for the initiates. The oldest seaman (in this case PO David Allan, Bosun Feather's deputy), was 'Neptune' who 'visited' the ship. Initiates were introduced into his court 'in the most thorough manner'. The ritual could be humiliating and probably settled a few scores.\n\nPO David Allan was resplendent in his crown and his oilskins were chalked all over with fish. His 'wife', merchant seaman John Mardon, was wearing a flowered silk outfit; he had pink cheeks, a mass of thin rope twist down his back and a hat with paper roses. The two were accompanied by attendants (tritons): Leading Seaman Arthur Pilbeam, Able Seaman Frank Wild and seven other crew members arrived to sit the initiates on a platform 12ft above a canvas bath, shave their heads roughly, wash their mouths out with soap, and finally to pull the chair from under them so that the victims dropped backwards into the water bath. The officers went first, uninitiated lower-deck men followed. Edgar had not 'crossed the line' and had to endure the ritual. Unfortunately he did not record his feelings or, more probably, his fury. The 'court's' activities were lubricated by two bottles of whisky, which were passed quickly around. The carnival atmosphere soon turned sour. Neptune's queen, Mardon, fell into the sail bath, the entire court became covered in soot, flour and soap, and when the men returned to the mess, aggressive complaining followed against the Officers and Quartermasters with Mardon becoming particularly obnoxious. The next morning one of the stokers collapsed in the boiler room and when Dr Wilson was called to the scene he found that the man's thumb had been bitten right through to the bone in a drunken brawl by his fellow Dundee citizen, James Duncan, seaman.45\n\nNaval Regulations gave clear guidelines to cover excessive drinking and Scott had to respond with formal discipline. He discharged Mardon to be handed over to the authorities at the next port. He also, optimistically, ordered the lower deck to clean up their language. At least some of the lower deck agreed with him; Gilbert Scott thought that some of the men's behaviour was disgraceful.46\n\nDrink was both a regular feature of sailors' routine as well as a curse. Events such as the 'crossing of the line' and shore leave were characterised regularly by drunken bouts, though drunkenness was unusual at sea. Although most sailors drank whenever the opportunity presented itself, and in the later expedition Edgar drew particular attention to himself with his drinking, there is no mention in the _Discovery_ diaries, or in Edgar's naval record of this expedition, that he was prone to noteworthy overindulgence.\n\nLife at sea was not all work. Scott understood the importance of trying to keep the crew entertained. In the early part of the voyage the heat was tropical and as they sailed away from Madeira, the men had time for deck games: boxing, tug-of-war, deck cricket. They sang, accompanied by the mandolin. They played with the animals on board: Scott's Aberdeen Terrier, Scamp; Armitage's dog, Vinca, a Samoyed (whose mother was an Arctic Veteran); and also with the cats and the rabbits. Lectures, such as 'Vegetable and Animal Marine Life' and 'The Causes of Phosphorescence', were organised. The opportunity to look at specimens under the microscope (magnified 500 times) was appreciated. Remarkably, a form of hockey was played on the upper deck; officers against scientists, keenly watched by the crew.\n\nOn 13 September, _Discovery_ stopped at the small, uninhabited island of South Trinidad in the South Atlantic, approximately 500 miles east of Brazil and described by Wild as 'a very pleasant visit'.47 Edgar saw a land covered with yellow-grey craggy rock that rose steeply from the shore to peaks of 1,000ft, covered with fern and scrub. Frigate birds wheeled overhead as some of the crew landed. A bluejacket was assigned to each of the scientists to aid in the collection of specimens. They found a shore alive with crabs, some large and red and green, with eyes that stared bulbously at their visitors; others were pale and globular, like black-eyed apples on legs. The stop was short but important for getting samples of the wildlife to send back to London, and sixteen types of bird were found. Not all the crew landed; Williamson, for example, was detailed to look after the transport boats. He passed the time by fishing, adding sixteen fish and one young shark to the haul of new treasures.\n\nMany different birds followed the ship and Edgar became adroit at identifying them. Catching different types of albatrosses and the Cape pigeons became a popular pastime.\n\nThe run to Cape Town was made by sail as much as possible to reserve the coal supply. With a good wind, reasonable progress was made, though not at the speed that was hoped for (174 nautical miles on 26 September, 180 and 165 on 27th and 28th). _Discovery_ weathered her first gale on 26 September; the topgallant sheet was carried away and the sails had to be furled 'after a hard struggle'.48 The wind continued to blow very heavily till 2 a.m. in the middle watch, when it began to drop, and by 5.30 a.m. it had gone, leaving behind it a big heavy swell which made _Discovery_ roll unmercifully. But Edgar and the whole crew were satisfied with the way their ship had stood up to her first gale, she had proved herself a good sea ship.\n\nApart from battling the difficult conditions, the crew were occupied with preparing for their arrival in South Africa. They painted the ship \u2013 that unending naval duty \u2013 and prepared the coal chutes. Scott told them what he expected in terms of dress and behaviour, and later he met them to help them make their wills. Edgar, dutiful son that he was, left everything to his mother.49\n\nOn the afternoon of the 3 October, _Discovery_ entered Table Bay eight days behind schedule, in spite of the effort of the crew. The remarkable sight of Table Mountain impressed Edgar. The mountain was covered in a cloth of cloud, which moved continuously, so that the mountain tops showed variably in shade and in sunlight. Because of the difficulty in keeping to schedule, Scott decided that he would have to bypass Melbourne, a decision that meant that dogs and supplies waiting in Melbourne had to be transported to New Zealand. These were problems beyond Edgar's sphere as he had more pressing things to think about. Hundreds of visitors came to the ship in spite of the fact that the Boer War between the British and the Dutch settlers was continuing. Although Afrikaner guerrillas were close by and martial law was in place, _Discovery_ and her crew were welcomed with open arms by the British community.\n\nEdgar attended a garden party hosted by the Admiral in Cape Town and a picnic hosted by the Governor.50 He visited a Boer prisoners' camp but, sadly, did not record his thoughts or impressions about the war, which was, by now, in its third year. His Warrant Officer, Ford, and others were hosted at a dinner given by the Chief Warrant and Dockyard Officers of the port. On this occasion Ford gave a speech saying that although doubtless many dangers would be met in the Antarctic, he dreaded more the last half hour knowing that he was about to speak. Ford said that _Discovery_ was manned with volunteers motivated with that love of adventure and carelessness of danger, which was the birthright of every English man,51 a view endorsed by Edgar.\n\nMeanwhile Scott's disciplinary duties continued: Mardon, the merchant seaman who had caused so much trouble 'crossing the line' and the only one of the crew whom Scott thought was completely unsatisfactory, was finally dismissed. But in spite of this warning, drink-related concerns surfaced again. The relaxed discipline in Cape Town meant that alcohol was freely available to the crew and their new friends plied them with drink. This time two other crewmen, Donkeyman52 William Hubert and Stoker William Page, were drunk and incapable. Worse, Page was grossly insubordinate to a superior. The two offenders were paraded before Scott. Such insubordination should have resulted in discharge and naval detention on shore, but at this stage Scott needed his crew. He merely stopped the men's pay, shore leave and rum allowance.\n\nWork continued alongside the social activities. In spite of the numerous visitors on board, which naturally held things up, Edgar was busily involved in supervising the resetting of the rigging and the recaulking of _Discovery_ 's deck and sides above the waterline. A crew member wrote that it 'proved a very painful job but of course it must be done'.53\n\nOther important duties continued: _Discovery_ 's magnetic observatory had to be recalibrated with the Cape Town observatory. This could not be done in the town because of the Cape Town trams, which distorted readings, so the observations were made on a plateau behind the town on the Cape of Good Hope peninsula. Guerrilla activity meant that the work could only be done in the day and the work extended to ten days, rather than the seven that had been planned.54\n\nAfter taking on coal, _Discovery_ sailed to the naval base of Simon's Town on the other side of the Cape on 5 October. As she made the short journey the ship again demonstrated her remarkable ability to roll, sometimes through nearly forty degrees, so that everything not carefully fastened down just flew about. When they arrived in Simon's Town, the crew received the customary friendly welcome; Edgar was allowed ashore in the evening even though he had plenty to do during the day. _Discovery_ was refitted and supplied at no cost to the expedition.\n\nOn Monday 14 October 1901, _Discovery_ left Simon's Town and steamed round the fleet. It was the grandest send-off. Some of the sentiments expressed by the crew as they left South Africa seem dated today. 'The ships in harbour gave us a splendid send off with all good wishes for a successful voyage and a safe return to dear old England'.55 Williamson was positive, too. _Discovery_ received 'three glorious cheers from each of the ships as we steamed past them, I think that this was the best send off ever I saw a ship get'.56\n\n### Notes\n\n1 _South Wales Daily Post_ , Tuesday 6 August 1901, p. 3.\n\n2 Williamson, T.S., _Log_ 1901\u20131904, SPRI, MS 774\/1\/1; BJ, p. 2.\n\n3 Scott, G., _Journal during the British National Antarctic Expedition_ SPRI, MS 1485: D.\n\n4 _Cabbrian, Herald of Wales_ and _South Wales Daily Post_.\n\n5 Williamson, T.S., _Log_ 1901\u20131904, SPRI, MS 774\/1\/1; BJ, p. 4.\n\n6 Markham, C., _Antarctic Obsession: The British National Antarctic Expedition 1901\u20131904_ , Bluntisham Books and the Erskine Press, Norfolk, England, 1986, p. 107.\n\n7 Yelverton, D., _Antarctica Unveiled, Scott's First Expedition and the Quest for the Unknown Continent_ , University Press of Colorado, USA, 2000, p. 94.\n\n8 Markham, C. _Antarctic Obsession: The British National Antarctic Expedition 1901\u20131904_ , Bluntisham Books and the Erskine Press, Norfolk, England, 1986, p. 113.\n\n9 Ibid., p. 66.\n\n10 _Times_ , 20\/03\/1901.\n\n11 Yelverton, D., _Antarctica Unveiled, Scott's First Expedition and the Quest for the Unknown Continent_ , University Press of Colorado, USA, 2000, p. 378.\n\n12 Markham, C., _Antarctic Obsession: The British National Antarctic Expedition 1901\u20131904_ , Bluntisham Books and the Erskine Press, Norfolk, England, 1986, p. 67.\n\n13 Williams, I., _With Scott in the Antarctic, Edward Wilson, Explorer, Naturalist, Artist_ , The History Press, Gloucestershire, 2008, p. 86\n\n14 Markham, C. _Antarctic Obsession: The British National Antarctic Expedition 1901\u20131904_ , Bluntisham Books and the Erskine Press, Norfolk, England, 1986, p. 109.\n\n15 Ibid., p. 106.\n\n16 Williamson, T.S., _Log 1901\u20131904_ , SPRI, MS 774\/1\/1; BJ. p. 3.\n\n17 Ibid., p. 4.\n\n18 Wild, J.R.F., _Letter to Mrs A.C. Bostock_ (his cousin), SPRI, MS 1078\/3\/1; D.\n\n19 Baughman, T.H., _Pilgrims on the Ice, Robert Falcon Scott's First Antarctic Expedition_ , University of Nebraska Press, USA, 1999, p. 77\u201378.\n\n20 Ellis, A.R., _Under Scott's Command, Lashly's Antarctic Diaries_ , Victor Gollancz, London, 1969, p. 37.\n\n21 Pack, J. _Nelson's Blood: The Story of Naval Rum_ , Mason, Havant, Hampshire, 1982, p. 85.\n\n22 Ellis, A.R., _Under Scott's Command: Lashly's Antarctic Diaries_ , Baylis, London, 1969, p. 37.\n\n23 Markham, C., _Antarctic Obsession: The British National Antarctic Expedition 1901\u20131904_ , Bluntisham Books and the Erskine Press, Norfolk, England, 1986, p. 70.\n\n24 Ibid., p. 90.\n\n25 Ibid., p. 93.\n\n26 Williamson, J.G., _The Structure of Pay in Britain 1710\u20131911, Research in Economic History_ , 7, 1982, p. 1\u201354,\n\n27 Afterwards Wild sailed with Shackleton on _Quest_ in 1921 as second-in-command. When Shackleton died in January 1922, Wild took over command.\n\n28 Abbott, G.P., _Letters to Cherry-Garrard_ , SPRI MS 559\/22\/1-3; D, 13\/04\/1916.\n\n29 Armitage, A.B., _Two Years in the Antarctic_ , Paradigm Press, Bungay, Suffolk, 1984, p. 116.\n\n30 Gregor, G.C., _Swansea's Antarctic Explorer, Edgar Evans, 1876\u20131912_. City of Swansea Publication, 1995 (Appendix 3, _The Martyred Hero_. Richards, H.S.) p.96.\n\n31 Scott, R.F., _Voyage of Discovery_ , Vol. 1, London, p. 54.\n\n32 Ibid., p. 54.\n\n33 Ed. Savours, A., _Edward Wilson, Diary of the Discovery Expedition to the Antarctic Regions 1901\u20131904_ , Blandford Press, London, 1966, p. 286.\n\n34 Yelverton, D., _Antarctica Unveiled, Scott's First Expedition and the Quest for the Unknown Continent_ , University Press of Colorado, USA, 2000, Appendix 2.\n\n35 Markham, C., _Antarctic Obsession: The British National Antarctic Expedition 1901\u20131904_ , Bluntisham Books and the Erskine Press, Norfolk, England, 1986, p. 106.\n\n36 Scott, G., _Journal during BNA Expedition_ SPRI, MS 1485; D.\n\n37 _Gauss_ was trapped in sea ice for twelve months, many miles from the region the Germans had intended to explore.\n\n38 Markham, C., _Antarctic Obsession: The British National Antarctic Expedition 1901\u20131904_ , Bluntisham Books and the Erskine Press, Norfolk, England, 1986, p. 89.\n\n39 Ibid., p. 97.\n\n40 Yelverton, D., _Antarctica Unveiled, Scott's First Expedition and the Quest for the Unknown Continent_. University Press of Colorado, Colorado, USA, 2000, p. 76.\n\n41 Williamson, T.S., _Log 1901\u20131904_ , SPRI, MS 774\/1\/1: BJ, p. 8.\n\n42 A traditional 'hazing' ceremony, in which initiates are subjected to a strenuous, humiliating and sometimes, dangerous ritual.\n\n43 Williamson, T.S. _Log 1901\u20131904_ , SPRI, MS 774\/1\/1: BJ, p. 11.\n\n44 Ellis, A.R., _Under Scott's Command, Lashly's Antarctic Diaries_ , Victor Gollancz, London, 1969, p. 17.\n\n45 Ed. Savours, A., _Diary of the Discovery Expedition to the Antarctic 1901\u20131904_ , Blandford Press. London. 1966. p. 73.\n\n46 Scott, G., _Journal during BNA Expedition_ , SPRI, MS 1485; D.\n\n47 Wild, J.R.F., _Letter to Mrs A.C. Bostock_ (his cousin), SPRI, MS 1078\/3\/1; D.\n\n48 Ibid., p. 22.\n\n49 Markham, C., _Antarctic Obsession: The British National Antarctic Expedition 1901\u20131904_ , Bluntisham Books and the Erskine Press, Norfolk, England, 1986, p. 90.\n\n50 Baughman, T.H., _Pilgrims on the Ice, Robert Falcon Scott's First Antarctic Expedition_ , University of Nebraska Press. USA, 1999, p. 75.\n\n51 Ibid., p. 74.\n\n52 A sailor working in the engine room.\n\n53 Williamson, T.S., _Log 1901\u20131904_ , SPRI, MS 774\/1\/1; BJ. p. 26.\n\n54 Yelverton, D., _Antarctica Unveiled, Scott's First Expedition and the Quest for the Unknown Continent_ , University Press of Colorado, USA, 2000, p. 83.\n\n55 Ellis, A.R., _Under Scott's Command: Lashly's Antarctic Diaries_ , Victor Gollancz, London, 1969, p. 19.\n\n56 Williamson, T.S., _Log 1901\u20131904_ , SPRI, MS 774\/1\/1; BJ. p. 28.\n\n## 5\n\n## The Southern Ocean to Antarctica\n\nWhen _Discovery_ departed from Simon's Town on 14 October 1901, she was twelve days behind schedule, with Scott's route taking him away from the main shipping routes. He needed to investigate the magnetic fields in the area south of 40\u00b0 latitude (his scientists were to record the intensity and dip of the magnetic forces), to comply with an important part of his instructions from the Admiralty. After making the scientific observations, _Discovery_ would make for Lyttelton, New Zealand.\n\n_Discovery_ soon caught a westerly breeze and made way by sail alone. The atmosphere amongst the sailors was cheerful; they caught a good haul of fish in the trawling net but subsequently managed to tangle the net and lose their haul in the process. Edgar enjoyed the evening singsongs.\n\nBut the weather soon deteriorated and the run became cold, rough and wet. From late October to early November _Discovery_ went through a series of storms of wind, snow and hail. These made for unpleasant conditions but impressive daily distances, 'nipping along' through the thousands of miles still to be covered: 207 miles on 22 October and 200 on the 28th, for example. The storms were a real test of the ship's sea worthiness; Gilbert Scott wrote that 'even the most experienced crewmen had never seen the like',1 and the crew agreed that _Discovery_ did well despite waves of 30 to 40ft rising as high as the Upper Topsail.2 The ship lifted over them, though Williamson wrote that she was reeling 'like a drunken man'. He was concerned that she would 'roll her masts out of her'.3 On the 28th a vast wave flooded on board, deluging the forecastle4, mess deck and laboratories. The men had just brought their winter clothes up from the hold. They were soaked through, and the crew had a lively time bailing water out of the wardroom. To make matters worse, the men were wearing sea boots, and Williamson wrote 'it's a wonder to me that some of us did not break our necks, once during the day I thought my checks were in. I was flung from one side of the ship to the other, with such force that I thought I had stove my ribs in... All's well'.5 But this second baptism confirmed to Edgar that he could have confidence in his ship.\n\n_Discovery_ continued south until, on 12 November, she was at a focus of magnetic interest at 52\u00b0S 131\u00b0E. Scott aimed to get even further south, as far as 65\u00b0, so that changes in the magnetic force and dip closer to the Magnetic Pole could be recorded. _Discovery_ was now about 1,000 miles from land and well away from the shipping tracks, so the cry of 'ship afire' on the night of the 14th caused considerable alarm. A sailor's oilskin had been left too close to a lamp; the ship was rolling and a fire started close to some paraffin oil tanks. Tales from his father's 'Cape Horn' years of fires on wooden ships must have come flooding back to Edgar as the alarm was raised. However, the crew were well rehearsed and the fire was put out quickly, but it was a cautionary experience nevertheless.\n\nOn 16 November, Edgar saw his first piece of ice bobbing around on the sea \u2013 only the size of a soup plate, but indisputably ice. Soon the ship was surrounded by bigger pieces succeeded by strange weathered blocks. The crew hooked up some ice to taste. It was fresh.6 Edgar was on deck when _Discovery_ first 'charged' an ice block as she entered the pack (the band of ice that surrounds Antarctica). By the evening she was enclosed by ice. The pack's shrouded grey-white mystery could not fail to impress; shafts of light reflected as a ghostly shimmer, the occasional bird flitted silently through the gloom. But Edgar did not have much time for introspection. As Petty Officer he was busy as he supervised the removal of the topgallant sails at night (to reduce strains on the masts should there be collisions with ice) and their replacement in the morning.\n\n_Discovery_ punched through as far south as 62\u00b0 50'S 139\u00b0E on steam. Then, after the magnetic observations had been made, she had to turn northwards; neither time nor coal reserves permitted further southern ventures. A glass of rum was issued all round; the boundaries of science had been pushed a little further.\n\nOn 22 November the crew sighted Macquarie Island. Here, in only a few hours, more seals, birds and penguins were added to the haul on board. The skins were to be of interest to the Natural History Museum in London, while the flesh was to be eaten. This visit was made lively by an encounter with a very large seal which, reluctant to give up its life for the advancement of scientific knowledge, charged at Skelton's camera and stand while Skelton himself attacked it with the boat's foot spur. The crew had seen penguins on the sail to the island, but here there were thousands of birds: handsome King penguins and smaller, orange-crested Royal penguins.\n\nAs the ship approached New Zealand Scott wrote that no sail, rope yarn or supplies on the deck had been lost (though on the final part of the run, in a 56\u00b0 roll, big waves damaged several cabins and a huge amount of china). Finally, on 28 November, _Discovery_ was off Lyttelton, the port of Christchurch. She had covered over 1,500 miles under a whole variety of conditions: heat, cold and wind. The crew had performed well.\n\nThe New Zealanders were fascinated by their visitors and made a fuss of them. They viewed them as young heroes sailing away to unravel the unsolved mysteries of Antarctica. Thus hundreds of locals turned out to greet them, and the crew were impressed and appreciative, writing that there were always willing hands ready to do the least little thing to make things easier and that the New Zealanders would not be forgotten. One wrote, 'when we are in Antarctica there will be as many thinking of us in New Zealand as at home, perhaps more'.7 Edgar attended a 'smoking concert' in a local Working Mans Club. There were other smoking concerts, on the ships and in Lyttelton. The mayor and people of Christchurch hosted a banquet for the crew,8 and the Sydenham Working Men's Club invited the sailors to dinner. Railway journeys to Christchurch were provided free of charge. Indeed, the time in New Zealand passed quickly.\n\nFor Scott, though, the time was too long. He would have liked to bypass some of the hospitality and sail straight on to the Antarctic, but he had no choice \u2013 he had to remain (for nearly four weeks) because of some possible damage to the hull, which meant that _Discovery_ had to go into dry dock for inspection. Visitors swarmed over the ship. Meanwhile, everything in the hull had to be unloaded, every item in every box recorded, and, after restowing, every box's exact location in the hull accounted. Stores shipped from England, including the unassembled magnetic huts, had to be squeezed in somehow. The Third Officer, Ernest Shackleton, was assigned this unenviable task and needed all the help he could get, but the crew, exhausted by their journey, keen to sample generous local hospitality and allowed to spend their evenings on shore,9 soon reached the stage where many were incapable of doing anything practical. But although Lieutenant Royds wrote that there wasn't a sober man on board by early evening and Engineer Skelton recorded that, 'there has been a great deal of fighting and drunkenness and I hope two of the seamen will be discharged',10 Edgar Evans does not rate a mention in relation to poor behaviour.\n\nBut poor behaviour was rife amongst other members of the crew. An Able Seaman (Baker) struck the Quartermaster Kenner (an arrestable offence), for some unknown reason, and deserted with Seaman Robert Sinclair and one of Edgar's fellow Petty Officers, William Smythe. Royds was sure that all three had deserted permanently but Smythe and Sinclair eventually reappeared; Smythe was reported to the Admiralty to be disrated from Petty Officer to Able Seaman.\n\nApart from these problems, New Zealand was a success for the lower deck. The city of Christchurch was greatly admired; it was beautifully laid out with wide streets and a river running through it, which enabled much boating and fishing. Maori ladies in local costume visited the ship and were given lunch. New Zealand sheep farmers offered as many sheep as the ship could carry (fifty); the farmers drew lots to see which farmer would have the honour of presenting them.11 When questions were asked about the reason for the expedition, the reply was that the question missed an important point \u2013 'how could they expect to know anything of the mighty universe of which the world is but an atom, if they didn't explore to the uttermost recesses of their own little globe?'12\n\nFinally, on 21 December, Julius, Bishop of Christchurch, came on board to bless the explorers. The men in their working clothes sang their final land-based hymn, _O God our help in ages past_ , lustily. The send off was a rousing event; cheer after cheer followed the ship as she sailed, bands played _Say Au Revoir But Not Good-bye_ as _Discovery_ , with her forty-four crew members, made her way to Port Chalmers \u2013 her last port of call before Antarctica. But the happy, noisy, band-blaring atmosphere was instantly destroyed when one of the young seamen, Charles Bonner, who had climbed to the top of the main mast, fell 100ft to his death from the mast truck.13 He seems to have stood up, holding on to just a wind vane, and lost his balance at the first sea swell. There was a 'wild cry' as he hurtled headfirst onto the corner of an iron deckhouse,14 spilling his brains over the deck. Skelton thought he was the worse for liquor when he went up and he hoped the death would be an object lesson to the men who had been drinking too much,15 but records from the crew do not mention alcohol. They were dumbfounded; the accident cast a gloom over the ship's company \u2013 death had visited them in seconds, depriving them of a popular crew member who, they knew, had left a fianc\u00e9e in England. Able Seaman Duncan wrote, 'Every one went about his work silently and quietly; they were afraid that the least sound would disturb the dead... Myself and another of his messmates washed poor Charlie and put a Union Jack over him and put him quietly to rest on the poop ready for internment at Port Chalmers.'16 Bonner was buried with full military honours; his body was carried on a gun carriage.17 Bluejackets from _Ringarooma_ , another British ship in port, formed a funeral escort and firing party.\n\nWhen _Discovery_ sailed into the unknown, weighed down by personnel and scientific equipment, provisions, coal, terrified sheep, snarling dogs and livestock, _Ringarooma_ (whose officers and crew had given a benefit concert for _Discovery_ ) gave the crew three cheers and crewman Williamson wrote that 'as Englishmen and brothers in arms, we could not hold it any longer, so we bucked up in spirit and gave them something of a return. The best possible under the circumstances. Good-bye civilisation.'18\n\nChristmas Day passed without celebration; the crew had no heart for it. Just a religious service as they sailed over the Southern Ocean. New Year's Day also started with melancholy memories of those loved ones 14,000 miles away. A feeling of isolation permeated all ranks. Scott wrote of this, so did Seaman Duncan. But as the day progressed and the crew sailed under clear skies, they were diverted to a degree by a new phenomenon \u2013 icebergs. These came as a revelation to the men, most of whom could never even have imagined such extraordinary icy sculptures. Soon six or more icebergs were in view, each one a miracle of blue and green and white. Some bergs were tabular, flat topped and with perpendicular sides.\n\nOn 3 January 1902, _Discovery_ crossed the Antarctic Circle at 66\u00b0 33'S and was now within the Antarctic Circle, the second major British exploration to be there since Captain James Clark Ross in the 1840s.19 (Bernacchi recorded the custom, which allows seamen to drink a toast with both feet on a table.)20 On that day the ship re-entered the pack ice. This belt of ice, which consists of the sea-ice from previous seasons and which is sprinkled liberally with icebergs, surrounds Antarctica. It is separated from the mainland by a rim of water in the Antarctic summer; in winter the whole sea freezes northwards for hundreds of miles. Edgar had aimed at adventure and excitement when he joined the navy, he must have felt fulfilled as he looked across the remarkable sight; hummocks of ice piled up in endless confusion, the surface white intensified by the greens and blues in the hollows and showing starkly against pools of dark water. The later calm of the sea was a big relief after the blustery Southern Ocean.\n\nTime spent going through the pack was not wasted. As _Discovery_ pushed her way through her crew took every opportunity to catch seals, and soon the upper deck looked more like a butcher's shop than His Majesty's Ship, as gory carcases of sheep and seal meat were hung on the rigging to freeze. Scott was determined that all the crew should eat fresh meat. He hoped this might be an added protection against scurvy, and the liver was often served at breakfast, the meat at dinner. Some thought the meat 'very good, better than beef, especially Bombay beef'21, but Edgar could never get used to it and didn't enjoy it. The biggest seal caught on the pack was a huge crabeater seal that crew members Cross, Heald and Joyce and two officers bagged. It was nearly 8ft and over 1,200lbs. It certainly gave the crew enough food for a few days. Edgar saw Emperor penguins for the first time; the birds looked large, more like small seals, but they were indisputably penguins. One was standing on his hind legs, his characteristic beak and yellow throat clearly visible.\n\n_Discovery_ sailed close to the 170\u00b0E meridian in the direction of the hut where the explorer Borchgrevink had overwintered in 1899. As she battled through the pack the crew 'watered the ship'. They swarmed over the side armed with picks and shovels to cut blocks of snow from the top of the floe.22 The ice was melted in long tanks fitted with steam coils. In this way tons of water could be collected in a few hours.23 In 1901 Scott had no way of knowing how long _Discovery_ would be in the pack (he knew that one expedition had been held in the pack for fifteen months and some of her crew had died before she was released without reaching her objective). In the event, she got through the pack in five days in spite of a few collisions with heavy pieces of ice. The crew picked a path through open patches of water whenever they could find them.\n\nThe 5 January 1902 was celebrated as the Christmas holiday. It was a Sunday and Edgar joined the morning service celebrating with Christmas and New Year hymns. On that day three memorable things happened: firstly, the sun 'forgot to set' (and did not remember how to do so again till shortly before midnight on 15 February, to be followed by long days which gradually grew shorter until 'Old Sol' disappeared on the 24 April for four months).24 Secondly, the skis that were to be used in Antarctica were tried for the first time. In 1901 skis were much longer than those used today and only one 150cm bamboo pole was used as a ski stick. The pole had an iron ferrule and a point at the end. This unwieldy combination must have been difficult to master but Edgar took to it well. He was competitive, athletic and strong, and by the end of the afternoon, in spite of collisions and falls, he felt confident enough to take part in races. Thirdly, the crew saw their first Ad\u00e9lie penguins. These birds, named after the French explorer Dumont d'Urville's wife, are quarrelsome and noisy, but the crew were instantly captivated by the way the Ad\u00e9lies ran eagerly and fearlessly up to them. Later, Dr Edward Wilson was embarrassed to record how a male Ad\u00e9lie made overtures to him by offering him a stone for a nest. In his collection of First World War writings, _Goodbye to All That and Other Great War Writings_ , Robert Graves included the story of how a male Ad\u00e9lie passed Wilson, looked at him admiringly and returned to deposit his gift at Wilson's feet.25\n\nAfter these new experiences, everyone congregated on the mess deck for a sing-along and a barrel of beer, presented before _Discovery_ left England.26 Cards were given to the men by Dr Wilson (a gift from his wife), and he gave the Petty Officers a box of crackers. They also received gifts from Royd's mother. The Mess desk was dressed and Scott wrote that it 'looked very nice'.27 Celebrations ended with the men singing and cheering, and an extra tot of rum was served.\n\nSoon open sea was visible beyond the pack, backed in the distance by the Antarctic mountains \u2013 mountains hitherto unexplored. For the first time Edgar could appreciate the awesome, austere beauty of Antarctica. Early on 8 January, basked in beautiful sunshine and surrounded by calm blue water, _Discovery_ steamed into the open sea. A day later Edgar, with his companions, landed at Cape Adare. Crew member Williamson wrote, 'Soon all was excitement, we lowered the boats and were soon scrambling up the beach where the great man Borchgrevink spent that lonely and tiresome winter.'28 Some local inhabitants were also excited; another colony of Ad\u00e9lie penguins gave the men their close attention. The birds jumped in and out of the water, squawking to their friends and managing to look both earnest and comical. Borchgrevink's hut impressed; Seaman Duncan wrote that it was constructed on a log cabin principle and could be assembled quickly. Duncan thought it was better than the one they had brought.29 Some useful provisions and coal remained from 1899.\n\nThe sun shone all night and the distant mountains gleamed as the crew left a tin cylinder on the Cape's stony shore. This contained records and private letters and was the first in the series of the hopeful paper chase of messages left so that the relief vessel could follow _Discovery_ 's progress. Lashly left a letter for his wife; 'she may get it some day if the postman should happen to come this way'.30\n\n_Discovery_ sailed down the west coast of the Ross Sea looking for likely landing places and winter quarters. All too soon Edgar and his colleagues understood the power of the elements ranged against them. The crew kept a watchful eye on proceedings as the ship escaped from a chain of icebergs, then had to shelter from a raging snowstorm. Williamson later said it was, 'more like two gales lashed together'.31 On the 15 January, once the wind had subsided, men were able to land to leave another of those hopeful messages, marked by a red pole, for the relief ship. These small containers were literally their sole method of communication with the outside world. As she sailed along the coastline _Discovery_ passed close by icebergs and the crew captured a variety of seal that completed their collection of the 'set' of pack-ice seals: Weddell, Crabeater, Ross, Leopard.32\n\nWhen ice blocked further exploration south on 22 January, _Discovery_ turned eastwards, charting the coastline. Amongst the confusion of mountains that they passed, some named, others not, the crew saw the two volcanoes recorded by Ross, Mount Erebus and Mount Terror, at over 100 miles away. Mount Erebus was, and remains, an active volcano (the world's most southern). Mount Terror was thought to be dormant, but the crew decided immediately that during the expedition they would climb it and settle any doubts. _Discovery_ passed Cape Crozier. This cape, home to thousands of Emperor penguins was a place that was to linger permanently in Edgar's mind after he was involved in an abortive expedition to reach it. But now it was merely the recipient of a third record of instructions and letters for the relief ship. This time two red cylinders were placed conspicuously in the centre of the penguin rookery.\n\nFrom Cape Crozier, Edgar had his first view of Ross' famous Barrier, then one of the unsolved mysteries of Antarctica. The Barrier stretched as far as the eye could see in an irregular coastline. The crew were a bit let down \u2013 Gilbert Scott was 'greatly disappointed' at its appearance,33 as they had heard so much about it from Ross' descriptions. Perhaps its contrast with the huge mountains they had just passed made it look smaller than they had thought, a bit like a flat sea broken up by wind furrows.34 _Discovery_ sailed east along the Barrier for seven days. New findings were made; the Barrier is 400 miles long and between 200\u2013400ft high. Soundings were taken along the whole length, and on good days, with the sun shining, it was like a pleasure trip.\n\nBy 28 January the height of the Barrier was getting lower. Sea soundings recorded a depth of around 90 fathoms (at the other end of the Barrier the sea soundings were 640 fathoms) and on the 30th, two high peaks of land were seen. Ross had believed that there was land to the east of the Barrier but had been unable to prove it, so this was a big discovery. Lashly was unmoved, commenting laconically, 'so we have passed the ice barrier at last'.35 But Gilbert Scott wrote that it was the first finding of land in the twentieth century 'which human eye had never seen before'.36 It was named after King Edward VII, _Discovery_ 's patron.\n\nFinding winter quarters was becoming increasingly urgent. It was time to turn back. However, the return was difficult and there was a close shave with icebergs.37 On 1 February it was feared that _Discovery_ would be trapped in rapidly forming ice, as she steamed round and round trying to find an opening. Still, discipline was maintained. Williamson grumbled; 'this monstrous idea of scrubbing decks every morning... in below freezing temperature... it seems as though they cannot forget that navy idea or commandment of thou shalt not miss scrubbing decks no matter under what circumstances, if it did any good I would not mind but as soon as you turn the water on it is frozen and then you have to come along with shovels to pick the ice up'.38\n\nAfter this danger the emphasis to get back to secure winter quarters increased, but on the sail along the Barrier a stop was made to make a balloon ascent. _Discovery_ was secured in a little bay called Balloon Bight (later the Bay of Whales), and this was the furthest south any ship had been; it was also the site from which Roald Amundsen was to make his successful sortie to the Pole in 1911. In 1902 all hands were occupied in filling the balloon with hydrogen. It was hoped that seeing the Antarctic from on high would yield information about her interior. Scott made the first ascent, he forgot the camera so, when Shackleton went up next, he became the first aerial photographer of the Antarctic. None of the hands were invited to ascend, not even those who had undergone special training in Aldershot. In the event the only finding was a view of miles and miles of snow, and the realisation that ballooning in Antarctica, in inexperienced hands, is a very chancy business.\n\nBy 8 February _Discovery_ had reached the top of McMurdo Sound.39 Scott decided to make his winter base in the shallow bay close to a tongue of land jutting out from the slopes of Mount Erebus (which steamed intermittently throughout the expedition). It was relatively close to the Barrier and gave access to the interior. The crew blasted the ice away with gun-cotton to get as far inland as possible and the ship was snug in snow and ice from 12\u201330ft deep. Here they landed. The shore party, with Edgar as Petty Officer, landed to erect a hut on a rocky promontory looking across to Mount Erebus \u2013 the base was known as 'Hut Point'. Edgar must have been relieved. Along with many of the crew he must have wondered if he would ever get this far.\n\n### Notes\n\n1 Scott, G., _Journal during the BNA Expedition_ , SPRI. MS 1485: D.\n\n2 Duncan, J., _Journal kept during the British National Antarctic Expedition 1901\u20131904, 4\/10\/01\u20138\/11\/02_. SPRI MS.1415; D, p. 3.\n\n3 Williamson, T.S., _Logs 1901\u20131904_ , MS, 774\/1\/1: BJ, p. 47. p. 31.\n\n4 Forecastle. The space at the front of the ship below the main deck where the crew's quarters were.\n\n5 Williamson, T.S., _Logs 1901\u20131904_ , MS, 774\/1\/1: BJ, p. 31.\n\n6 Sea ice loses its salinity after about three years.\n\n7 Williamson, T.S., _Logs 1901\u20131904_ , MS, 774\/1\/1: BJ, p. 47.\n\n8 Scott, G., _Journal during BNA Expedition_ SPRI MS 1485: D.\n\n9 Ibid.\n\n10 Skelton, R., _The Antarctic Journals of Reginald Skelton_ , Reardon Publishing, Cheltenham, England, 2004, p. 31.\n\n11 James Duncan. _Journal kept during the British National Antarctic Expedition 1901\u20131904, 4\/10\/01\u20138\/11\/02_. SPRI MS.1415; D, p. 3.\n\n12 Baughman, T.H., _Pilgrims on the Ice_ , University of Nebraska Press, Lincoln and London, 1999, p. 85.\n\n13 A truck \u2013 a guide for ship's ropes in the form of a disk with holes, fitted to the top of the mast.\n\n14 Scott, R.F., _Scott's Voyage of the Discovery_ , John Murray, London, 1929, p.84.\n\n15 Skelton, R., _The Antarctic Journals of Reginald Skelton_ , Reardon Publishing, Cheltenham, England, 2004, p. 33.\n\n16 James Duncan. _Journal kept during the British National Antarctic Expedition_ , 19011904. 4\/10\/01\u20138\/11\/02. SPRI MS.1415; D .\n\n17 Ibid.\n\n18 Williamson, T.S., _Logs 1901\u20131904_ , MS, 774\/1\/1: BJ, p. 51.\n\n19 Borchgrevink's British Antarctic Expedition of 1898\u20131890, which sailed in _Southern Cross_ was British, but in name only. Only three of the thirty-one men on board were not Norwegian.\n\n20 Bernacchi, L.C., _Saga of the Discovery_ , Blackie, London, 1938, p. 23.\n\n21 Ellis, A.R., _Under Scott's Command: Lashly's Antarctic Diaries_ , Victor Gollancz, London, 1969, p. 21.\n\n22 Scott, R.F. _The Voyage of the Discovery_ , John Murray, London, 1929, p. 94.\n\n23 Ibid., p. 95.\n\n24 Wild, J.R.F., _Letters to Mrs Bostock_ , SPRI MS 1078\/3\/1: D, p. 3.\n\n25 Robert Graves, _Goodbye to All That and Other Great War writings_ , Carcanet, London, Postscript, p. 280.\n\n26 Williamson, T.S., _Logs 1901\u20131904_ , MS, 774\/1\/1: BJ, p. 55.\n\n27 Scott, G. _Journal during BNA Expedition_ SPRI MS 1485: D.\n\n28 Williamson, T.S., _Logs 1901\u20131904_ , MS, 774\/1\/1: BJ, p. 59.\n\n29 James Duncan, _Journal kept during the British National Antarctic Expedition 1901\u20131904_ , 4\/10\/01\u20138\/11\/02. SPRI MS.1415; D.\n\n30 Ellis, A.R., _Under Scott's Command: Lashly's Antarctic Diaries_ , Victor Gollancz, London, 1969, p. 21.\n\n31 Williamson, T.S., _Logs 1901\u20131904_ , MS, 774\/1\/1: BJ, p. 63.\n\n32 Skelton, R., _The Antarctic Journals of Reginald Skelton_ , Reardon Publishing, Cheltenham, England, 2004, p. 43.\n\n33 Scott, G., _Journal during BNA Expedition_ SPRI MS 1485: D\n\n34 Ed. Savours, A., _Edward Wilson, Diary of the Discovery Expedition to the Antarctic 1901\u20131904_ , Blandford Press, London, 1966, p. 105.\n\n35 Ellis, A.R., _Under Scott's Command, Lashly's Antarctic Diaries_ , Victor Gollancz, London, 1969. p. 24.\n\n36 Scott, G., _Journal during BNA Expedition_ SPRI MS 1485: D.\n\n37 Ibid.\n\n38 Williamson, T.S., _Logs 1901\u20131904_ , MS, 774\/1\/1: BJ. p. 79.\n\n39 Ed. Savours, A., _Edward Wilson, Diary of the Discovery Expedition to the Antarctic 1901\u20131904_. Blandford Press, London, 1966, p. 112.\n\n## 6\n\n## Early Months in Antarctica: February to September 1902\n\nWork started on Hut Point immediately. Speed was essential in the construction; the magnetic huts, store huts, kennels, provisions, coal and water all needed to be on land quickly. Scott couldn't be certain that _Discovery_ would not be torn away from her moorings, so any shore party had to be self-sufficient.\n\nIt was a hard job: the supports had to be dug deep into the frozen ground and it was difficult to locate all the pieces for making up the sides.1 When the hut was finished its roof was felted and covered in heavy double wood and, surprisingly, it had a veranda around its walls (the men thought it was better suited to a colonial shooting lodge than a polar base). In fact, the original idea, that it should be the permanent base for the expedition, became unnecessary when _Discovery_ stayed in Antarctica over the winter rather than returning to New Zealand. Thus the men lived on the ship, mainly using the hut for storage, drying clothes and putting on entertainments. Problems with the hut's construction became apparent; when the stove was lit, snow on the roof melted and leaked through, icing up clothes stored there for weeks. Some of the men were not too overwhelmed by Antarctica; Seaman Duncan wrote 'this seems like a dreary place to be spending twelve months in'.2 Ross Island and the hut were to be their base for the next two years. Behind the land rose gradually towards hills of about 500ft. The crew named these 'Arrival Heights' and 'Harbour Heights', and they called the mass of rock north of these hills, which rose over 1000ft above sea level, 'Castle Rock'.\n\nScott was aware that the problem of keeping over thirty men occupied might become acute throughout the freezing, dark Antarctic winter, when poor visibility precluded any sorties and when small irritations could easily become magnified. Recreational activities were not neglected. With the continuing light in the Antarctic autumn (February\u2013March), football and hockey were regularly played. Teams were imaginatively varied to cross ranks: married or engaged against singles, older men against the youngsters. Sometimes, Officers against Men (Officers 3, Men 2 on 13 February was a particular highlight). The rules were lax; there was no such thing as 'off-side'. Hockey matches lasted an hour and were played in temperatures of \u221230\u00b0C, or even lower. Improvising, the crew used light sticks and a homemade hardball. This was just as well with men such as Edgar on the attack. Big (just under 12 stone) and athletic, he dashed about, a misty vapour steaming round his head helmet, his stick high in the air, eager to attack the opposition with deadly intent; a fearsome opponent indeed. Skiing was encouraged also whilst the light held: in spite of the cumbersome equipment there was only one serious injury \u2013 a broken leg.\n\nEdgar made himself useful immediately. Scott said he was a man of 'Herculean strength' and 'well muscled'. Equally important in a party isolated from the rest of the world, he had a cheerful nature and was reliable and practical, outgoing and gregarious but not boastful or pushy. On this expedition during the next twenty-three months, he was to take part in some of _Discovery_ 's most important sorties. These would make him a hardened veteran of Polar exploration and one of Scott's most trusted colleagues.\n\nMost of the outings would take him towards, or on to, the Western Mountains, that enormous range of mountains to the west of McMurdo Sound. On three expeditions he was with Scott on journeys (totalling ninety-four days), the last being a stupendous 950 statute mile journey into the mountains. This expedition was a sledging feat that conclusively proved the immensity of the continent. It was a feat equal to the huge journeys of the Arctic explorers half a century earlier.\n\nThe crew understood that Scott retained absolute authority and realised that he was prepared to use this authority. Two days after _Discovery_ berthed he ordered that the cook, Charles Brett, should be put in irons for insubordination. One of the crew, Duncan, simply recorded 'Mr Brett getting troublesome'.3 (Brett refused to report for duty; he was fed up with working in the galley all day.) The lively scene, as Brett fought his captors, before he was finally clapped in irons, was recorded in the diaries. But this was the only time that physical discipline was used on the expedition.\n\nNaval routine continued on land. Whilst the light lasted, the crew worked from 7 a.m. until 8 a.m. when they had breakfast: porridge, followed by curry with rice\/salmon\/mince\/sardines\/rissoles\/cold tongue, washed down with cocoa or coffee.4 Work continued until midday when there was a break for an hour and a half for dinner (soup followed by ham\/beef pie\/seal\/tinned meat with vegetables then a pudding).5 Virtually all the food came from tins. Soon after they arrived the men continued work until 5 p.m., but as the light faded they finished for the day by early afternoon.6 The provision of artificial light was a necessity as there were twenty-four hours of darkness during the Antarctic winter. Scott was very conscious of the necessity for light, both for psychological reasons and for health (darkness was thought to contribute to Polar Madness). An attempt to manufacture a windmill dynamo to produce electricity failed repeatedly (the windmill blades snapped in the wind) so an oxyacetylene lamp was used to give light for about eight hours, otherwise candles (provided or homemade) were used and gave enough light for the men to write up their logs, play cards, draughts or chess. Royds showed magic lantern slides.7\n\nEdgar's first sledging expedition was a catastrophe. It highlighted total British inexperience in coping with Antarctic conditions and was notable for its lack of proper preparation and for its incompetence. It was a hard lesson in survival. On this expedition Edgar was one of the inadvertent but blameless causes of a domino-type series of disasters that left one crewman dead and that could have easily resulted in the deaths of seven other crewmen and the officer in charge of them, Lieutenant Michael Barne. None of the equipment had been tested before the expedition set out. It was to be, as Scott wrote, 'one of our blackest days in the Antarctic'.8\n\nThe sortie was to Cape Crozier, and its practical purpose was to update the messages for the relief ship on the position of _Discovery_ 's winter quarters. When the expedition set out no one understood how weather in Antarctica could change in a very short time from reasonable conditions to howling blizzards. The expedition lasted a memorable fifteen days, from 4\u201319 March (autumn in Antarctica).\n\nThe party was under the command of Lieutenant Charles Royds. There were three other officers, including Lieutenant Barne, and also nine crew members. Scott wrote later that the packed sledges 'presented an appearance of which we should afterwards have been wholly ashamed'.9 The party took two sledges and four tents, with Edgar sharing a tent with Stoker Frank Plumley and Seaman William Heald. They slept in a three-man sleeping bag for extra warmth (but extra discomfort) and quickly understood the horrors of Antarctic journeys in the autumn.\n\nThey took pyramidal tents, big enough for three people to lie down in but for only one person to stand in comfortably. To erect the tent they firstly had to spread and stabilize bamboo poles on the ice and then cover them with the tent cloth: when a high wind was blowing, this would involve a long struggle with flapping canvas and collapsing poles. With the tent erected, snow was piled around its base to stabilize it. Edgar and his companions kept their woollen underwear on throughout the sorties but changed into their reindeer fur suits for sleeping (a prolonged business as the furs got as stiff as boards) and slept on a canvas cloth. Anything they took off had to be put in the shape it needed to be in the following morning, since garments froze (in a few minutes) into whatever shape they were left in. Unlacing the leggings was another ordeal. The men had to use their bare fingers which meant putting their hands back into their mitts every few minutes. One of the most important aspects of the night routine was foot care: frostbite could progress to gangrene so the utmost care had to be taken. The night socks were carried next to their skin during the daytime march; at night the innermost pair of day socks was changed with this pair and in turn put next to the skin. Grass was also used to absorb sweat and was often put next to the skin at night. Long fur boots reaching to the knee were worn with the fur nightwear. When they eventually got into their sleeping bags, they shivered and shook for hours, rime inside the tent from the evening cooking dripping onto their faces. Morning was always a welcome relief.\n\nOn the Cape Crozier Expedition, following three exhausting days and 21 miles progress, the work of finding the trail and keeping to the course was passed from the officers to Able Seaman William Heald and Edgar. But, after a short time, as conditions remained so difficult, the officers changed the plan again, unwisely deciding that those party members with skis \u2013 Royds, Skelton and Koettlitz (who had previous experience of Polar conditions in the Arctic) \u2013 should go on, leaving the inexperienced Barne to lead the nine men home (the Royds trio did not reach Cape Crozier).\n\nOn 9 March, Lieutenant Barne set off to return to the ship. The party made good progress initially, but on the 11th they ran into such atrocious flying drift that they stopped and pitched their tent for protection. Miserably they ate biscuits and solid, cold pemmican (the stove was broken). Two of the seamen, Vince and Hare, had such problems with frozen feet in the leather ski boots that they changed them for fur finnesko.\n\nThey were only about 4 miles from the ship, but the wrong decision was made when the group decided to abandon the tent, let the dogs loose and go on by foot in the direction which they thought led to _Discovery_. This was unwise because blizzards only last for two to three days in Antarctica and they would probably have been safe if they had stuck in their tent. But they had no way of knowing this; they were exhausted, freezing, frightened, hungry and thirsty with no means of heating ice for drinking water. As the wind thundered against the tent, they thought they could be blown away with every gust and they decided that it would be safe to make a dash for the ship, just a few miles away. In fact Lieutenant Barne, in the negligible visibility, miscalculated their position; they were actually close to cliffs on the north side of Ross Island and close to the sea. The group of nine progressed cautiously along a steep icy slope with snow swirling around them. Vince and Hare in finnesko (no grip) slipped and slithered on the icy surface. When Hare decided to return for his boots he disappeared almost immediately. Lieutenant Barne and the remaining eight spread out to try to find him, and as they did so Edgar lost his footing and disappeared down a steep slope. Showing more courage than reason, Barne threw himself down the slope after him. He found himself accelerating down an ever-increasing gradient until he was, miraculously, stopped by the same snow bank that had halted Edgar's descent.\n\nWhen the American seaman Arthur Quartley made the same decision to go down the slope to find Barne and Edgar, he left behind him, on that exposed hillside, six Polar novices in a blizzard that reduced vision to a few feet and with no idea of where they were. As they attempted a diagonal descent down the slope, they too began to slip, Wild said, 'at a fearful rate with no idea where they were going and no hope of stopping'.10 Vince was first, Seaman Wild second, the other three close behind. Suddenly Wild stopped with a jerk on a snow ledge. The men behind all slid down helplessly and landed on his ledge except for George Vince, who was unable to stop, probably because he was in finnesko, who shot on and disappeared. The men could see a ledge 20ft from where they had landed. As they crawled cautiously towards it, the drift lifted and they saw, to their horror, the drop of 90ft into the sea that Vince had spiralled down. Shouting had no effect in that howling hurricane. They had to leave Vince to his dreadful fate.11 The young seaman was to be Scott's only Antarctic fatality on the _Discovery_ expedition.\n\nEdgar, Lieutenant Barne and Arthur Quartley had no hope of getting up the slope to rejoin their companions. They cautiously made their way in what they hoped was the direction of the ship. At one point Barne was just saved from joining Vince in the sea by the quick thinking of Quartley who pulled him back from the edge of the cliff. Barne's compass was not working and the three men walked and crawled along until they finally saw a landmark they knew. Then they progressed more confidently, aided by the blaring of the ship's siren.\n\nWhen they stumbled across a search party, the three were virtually unrecognisable. They could not speak intelligibly (Dr Wilson writes that one of them was talking rubbish),12 and they were all dazed. Barne was so affected by the cold that he could hardly speak at all, intelligibly or otherwise, and the trio had to be identified by Quartley. Edgar's ear was badly frostbitten. It looked like an apple, but he said it felt like a cabbage.13 He had frostbite of his fingers and feet, and his nose was swollen to a prodigious size. He was to continue to have problems with his nose on later expeditions; he would refer to it as his 'Old Blossom'. All three were put in the sick bay and looked after by Dr Wilson. The experience had no long-term physical effect on Edgar, though by now he undoubtedly understood the necessity for thorough preparation.\n\nBut what of Hare, the man who had first left the party to get his ski boots? He had an extraordinary adventure, reappearing uninjured, even without frostbite, after being caught in below-freezing temperatures for two days with no protection. Hare was put in the Magnetic Laboratory at 17\u00b0F and the temperature was gradually raised to freezing point. Remarkably he suffered no long term effects and could only have survived because he stopped near to the shelter of a pile of rocks, pulled his arms into his woollen blouse (under the gabardine wind jacket) and covered the opening to his helmet. He must have been covered in snow, which left enough space for him to breathe, a sort of primitive snow hole.14 When he got back to the ship Scott was so relieved he looked 'as if he thought the dead were walking in'.15\n\nBy 19 March _Discovery_ was firmly frozen into her winter quarters. Good Friday, 28 March, was celebrated with hot cross buns 'or bricks, could hardly tell which'.16 By 3 April the ship was in darkness. Throughout the winter the doctors examined the men carefully with blood tests, chest measurements, waist, biceps, forearm and calf measurements, weight and a record of blowing power.\n\nThere was one further expedition in the autumn of 1902. Edgar was not on it. It was also unsuccessful because the conditions were simply too bad for expedition work. Called the Southern Depot Attempt, the expedition from 31 March to 4 April 1902 was to lay a series of depots to the south in preparation for exploration the following year. The expedition took eighteen dogs and four sledges, but the horrors of sledging in late autumn were again all too apparent. The temperature was between \u221226\u00b0 and \u221240\u00b0F. If the men touched any metal it stung like a hot iron and left a white mark; the dogs would hardly pull; the snow surface was awful and the wind made progress terrible. The trip lasted three days with the men enduring miserable nights. Scott decided to depot the provisions early and turn back. On the way home the dogs pulled for all they were worth.\n\nThe 23 April was the official beginning of the long winter. Although the moon's movements (disappearing and returning every month), the rotation of the stars and the shortening of each day told them that time was actually passing, the sun, a vital force, had disappeared from their lives. On the 25th, the first edition of the _South Polar Times_ was produced \u2013 'The paper is a very good one quite interesting and amusing'.17 Editions were to continue throughout the expedition. A single copy of an alternative paper, _The Blizzard_ , was produced for those submissions not quite up to the standard of the _South Polar Times_. Unfortunately entries were anonymous, and if Edgar made an attempt it is not recorded. Scrap albums were popular, as was making metal models, but it was cold in the workshop and the men had to wear mittens to avoid a skin burn when the cold metal was touched.18 The winter blizzards raged with gusts of over 100 miles an hour.19 However, Edgar never tired of looking at the auroras, those brilliant undulating lights which lit up the pitch-black sky.20\n\nBy mid May, there was only half an hour of twilight in the whole day. A good moon meant that they could get about a bit, and the crew played football by moonlight when they could (though the doctor said it was injurious to health at such low temperatures). They played until 14 June, only a few days before Midwinter's Day.\n\nAs the winter months progressed some resentment creeps into the diaries; things were allegedly 'slack' with the officers who had intended to work wonders to keep the men entertained during the winter, but had not even started when the winter was nearly half gone (only one concert having been performed). The crew wanted the promised classes and information about the scientific work. The Lower Deck felt hard done by.\n\nBut the bleak months passed without any serious problems and the men were soon to hear Scott's plans for the expedition. He may have been aware of the rumbles. On Sunday 22 June, Midwinter's Day, he spoke to the men. He said firstly that he was pleased with the general health of the ship's company and he emphasised how each man must continue to look after his own health so as to be fit for sledging. He went on to say that he wanted all the men to have their chance to take part in expeditions and explained how different parties would go south, west and towards the Magnetic Pole.\n\nMidwinter's Day was indeed a welcome diversion. The mess was beautifully decorated with coloured paper and draped in flags, garlands, wreathes and the men's photographs.21 The stokers had carved an ice block into the shape of a frost king with his crown. More photos were taken. The crew had bloaters (soaked, smoked herrings) for breakfast (they needed a lot of water afterwards), and little toys and puzzles had been sent by Mrs Royds, which helped to pass the time, as did cards from Mrs Wilson. Dinner was real turtle soup; ham, kidney beans, potatoes and plum pudding with brandy sauce washed down with bottles of Bass ale, this showing that the crew were served a good and varied diet. After this largesse and a post prandial nap, there were ices made of condensed milk and chocolate vanilla, cakes and sweets, before the day was rounded off with grog and, finally, a concert in the Royal Terror Theatre. Some of the crew went to the Captain asking for more lubrication. Unsurprisingly, Scott had no difficulty in refusing.22\n\nOn 6 August, the anniversary of _Discovery_ sailing from England, a minstrel show was held in the Royal Terror Theatre (The Hut); the singing was 'very fair indeed',23 and in the later part of the winter months there were lectures on geology, wireless telegraphy (very interesting), 'Sledging To-day', the 'Wonders of the Deep' and magic lantern shows.24 These probably received a varied reception, but in general the bleak months passed without any serious problems. Lashly wrote that he had plenty of work to do, and that everyone was preparing for the sledging journeys and weeks passed very well.25 Alterations and improvements were made to the furs, their sleeping bags and the sledging equipment in general.\n\nOn Friday 22 August 1902, the sun returned at last. Most of the men climbed the hills to get a glimpse of it. Its first appearance for four long months, a brilliant red and the sky all around it, looked, as Williamson said, 'something beautiful.'26 As daylight increased so everyone's mood improved. The sledges and automatic sledgemeters were tested. Edgar became, unsurprisingly, an excellent sledge hauler; he was soon considered one of the best.27\n\n### Notes\n\n1 James Duncan, _Journal kept during the British National Antarctic Expedition 1901\u20131904. 4\/10\/01\u20138\/11\/02_ , SPRI MS.1415; D, p. 45.\n\n2 James Duncan, _Journal kept during the British National Antarctic Expedition 1901\u20131904. 4\/10\/01\u20138\/11\/02_ , SPRI MS.1415; D, 10 February.\n\n3 Ibid.,10 February.\n\n4 Yelverton, D., _Antarctica Unveiled_ , University Press of Colorado, 2000, p. 154.\n\n5 Ibid., p.154.\n\n6 Williamson, T.S., _Log 13\/07\/01\u201323\/06\/02_ SPRI MS 744\/1\/1: BJ, 22\u201330 April 1902.\n\n7 Ibid., 1 May 1902.\n\n8 Scott, R.F., _The Voyage of the Discovery_ , John Murray, London, 1929, p. 173.\n\n9 Ibid., p. 170.\n\n10 Wild, J.R.F., Letter to Mrs A.C. Bostock (his cousin), SPRI, MS 1078\/3\/1; D\n\n11 Ibid.\n\n12 Ed. Savours, A., _Edward Wilson, Diary of the Discovery Expedition to the Antarctic 1901\u20131904_ , Blandford Press, London, 1966, p. 123.\n\n13 Ibid., p. 124.\n\n14 Williams, I., _With Scott in the Antarctic: Edward Wilson, Explorer Naturalist, Artist_ , The History Press, Stroud, Gloucestershire, England, 2009.\n\n15 Ed. Savours, A., _Edward Wilson, Diary of the Discovery Expedition to the Antarctic 1901\u20131904_ , Blandford Press, London, 1966, p. 125.\n\n16 Ellis, A.R., _Under Scott's Command, Lashly's Antarctic Diaries_ , Victor Gollancz, London, 1969, p. 35.\n\n17 Ibid., p. 40.\n\n18 James Duncan, _Journal kept during the British National Antarctic Expedition 1901\u20131904. 4\/10\/01\u20138\/11\/02._ SPRI MS.1415; D, p. 45.\n\n19 Priestley, R., Lecture, _The Psychology of Polar Exploration_ , SPRI, MS 1097\/16\/1; D.\n\n20 Auroras are phenomena related to the sun because of its emission of electric particles. Protons and electrons originating in the sun are caught by the terrestrial magnetic fields. When these electrical particles meet the ionised gases in the higher layers of the atmosphere a light is produced in the sky.\n\n21 Ed. Savours, A., _Edward Wilson, Diary of the Discovery Expedition to the Antarctic 1901\u20131904_. Blandford Press, London, 1966, p.155.\n\n22 James Duncan, _Journal kept during the British National Antarctic Expedition 1901\u20131904. 4\/10\/01\u20138\/11\/02_. SPRI MS.1415; D. 23 June.\n\n23 Ed. Savours, A., _Edward Wilson, Diary of the Discovery Expedition to the Antarctic 1901\u20131904_. Blandford Press, London, 1966, p. 168.\n\n24 James Duncan, _Journal kept during the British National Antarctic Expedition 1901\u20131904. 4\/10\/01\u20138\/11\/02_. SPRI MS.1415; D. 23rd, p. 41.\n\n25 Ibid., p. 42.\n\n26 Williamson, T.S., _Log 13\/07\/01\u201323\/06\/02_ SPRI MS 744\/1\/1: BJ 22 August 1902.\n\n27 Ed. Skelton, J., _The Antarctic Journals of Reginald Skelton_ , Reardon Publishing, Cheltenham, England, 2004, p. 109.\n\n## 7\n\n## The Antarctic Spring: September to October 1902\n\nEdgar's first spring expedition was south-west, towards the mountains bordering the west coast of McMurdo Sound, which were such an important feature of _Discovery_ 's exploratory and scientific itinerary. In fact the Western Mountains were to be the main destination for Edgar's expeditions throughout the _Discovery_ expedition. The aim of this first sortie was to find a southerly route into the interior of Victoria Land; in addition the team were to study ice formation and geological features. But the expedition soon ran into trouble. Six men set out \u2013 two officers, Lieutenant Royds and Dr Koettlitz, with four from the lower deck, Edgar, William Lashly, Frank Wild and Arthur Quartley. Quartley and Lashly were Stokers and Wild was an Able Seaman. The four were recognized as an impressive combination. As the party set out with two sledges, one behind the other, Skelton wrote that the four were the 'best possible you could pick out of the ship. I wish I was going with them',1 a comment echoed by the mess deck who already called them the Guarantee Party.\n\nEven with the Guarantee Party, the team soon encountered problems as they aimed towards Black and Brown Islands, two islands engulfed by the great ice Barrier to the east of Mount Discovery and just over 20 miles from Hut Point. Snow ridges split the sledge runners, the light was awful, the temperature dropped to \u221256\u00b0F and they were engulfed in a violent blizzard. Although the party did manage some survey work in the area around the two islands, they found the going terrible. Wild, in particular, was unimpressed: the party, he wrote, 'did little except getting frost bitten and bad tempered and had a miserable time for ten days, for two of which we were confined to the tents whilst a glorious blizzard played Old Harry with things in general'.2\n\nAs the blizzard whirled around them and they started to shut the camp down, Lashly left his sleeping bag beside his tent and went back to collect more gear. He saw his bag being swept away into the swirling whiteness. On this sortie Lashly was sleeping in the officers' tent and had a single sleeping bag. In 1902, opinions varied about the relative benefits of one-man and three-man sleeping bags. Some thought the three-man bag warmer (though more uncomfortable than the single bag). Also, one-man sleeping bags for all would be excessively heavy, so whilst officers and scientists had single bags, the men usually shared a three-man bag. Lashly, since he shared a tent with the officers, had been issued with a single bag, but when it disappeared he had to crowd in with Edgar, Wild, and Quartley. When one of them said 'turn', they all had to turn. Lashly, in a phlegmatic understatement, said that it 'was rather crowded'.3 As the blizzard raged the men had to sit in the tent with their backs against the canvas to stop it tearing to pieces. The following day the bag was still missing and the party decided to retreat \u2013 the men were fed up and frostbitten. They decided that they were every type of fool to have ever come to Antarctica and swore that nothing would bring them back... and yet, the glamour of the vastness of the place, the scenery, the anticipation of finding something new, the satisfaction of winning against nature 'remain with one and call men again and again to, \" _that stark and sullen solitudes, that sentinel, The Pole_ \".'4\n\nBefore Edgar's next sledging party a problem surfaced that was to surprise and worry the entire crew. Second-in-Command Armitage had led an expedition to the interior of the south-western mountains. His party was also searching for a route that would lead into the interior of Antarctica. Six days into his trip and on a glacier high in the mountains, the team were held up by storms and deteriorating health (one man, Ferrar, had aches in his legs, another a troublesome ankle and another sore gums). Armitage suspected scurvy. He left two men behind in the camp and continued to explore the glacier valley before leading the team back to Hut Point. Confirmation of the disease caused considerable anxiety. The belief that scurvy was due to putrefaction in tinned food had been proved wrong and despite all the efforts that had been taken to avoid the problem (cleaning the ship, taking as much exercise as possible,5 providing as much light as could be managed and carefully examining every tin of food before it was eaten for possible contamination)6 scurvy was 'an unwelcome surprise'.7 The advice Scott had been given was wrong.\n\nAs we now know scurvy is a deficiency disease due to a lack of vitamin C. It appears when the vitamin has been absent from the diet for about three months. Vitamin C is mostly present in fruits, green vegetables and potatoes. Although 'vitamins' were an unknown concept in the early 1900s, _Discovery_ in fact carried many tinned vegetables on board: artichokes, Brussels sprouts, carrots, cauliflower, haricots verts, petit pois, tomatoes and many dried and preserved fruits (for example, apples, apricots, peaches, pears, rhubarb).8 According to Hawk and Bergeim's _Practical Physiological Chemistry_ of 1924, some canned vegetables remained rich in vitamin C9 (the canning methods used were apparently less destructive to the vitamin than boiling).10 But even if this is true and the men ingested some vitamin C with the tinned vegetables and fruit, Armitage had only been away from the ship for a short time so the diet clearly provided less than the 125mg now thought to be required in a male smoker (less in the non-smoker).11 When, as a result of this outbreak, fresh, rather than tinned meat was provided, this too was to prove ineffective. Vitamin C is present in offal (liver and kidney), but meat itself has little of the essential vitamin.12 But in 1902 this information was tantalizingly unknown, though the symptoms, of lassitude, fatigue, spongy gums, swollen joints, aching muscles and swollen, red spots (which could later break down and rot), were all familiar. Armitage's party's symptoms were characteristic.\n\nThis diagnosis could clearly have dreadful consequences and radical action was called for. The ship was thoroughly cleaned, the bilges disinfected, overcrowding was reduced by some men sleeping in the hut and exercise was insisted on. Tinned meat was given up except for Tuesdays when the cook, Brett, had a day off ('Scurvy Tuesdays'), otherwise some form of fresh meat or, importantly, liver and kidney, was served daily, plus porridge each morning, 'liberal' jam and extra portions of bottled fruit.13 Since it was thought that the problem was made worse by Brett's cooking, Armitage informed him that his (Brett's) bonus depended on an improvement. Suddenly, palatable, even tasty food appeared; and thereafter the men mostly enjoyed the meals and some thought the seal liver was delicious. An exception was Edgar, who never took to seal meat. But he must have eaten some offal and may have made up his vitamin intake with the tinned foodstuffs. Certainly he never developed scurvy on _Discovery_ , in spite of all his sledging miles.\n\nOn 4 October, Edgar was off again, this time a return to Cape Crozier, the Cape where messages for the relief ship were left. In addition the party aimed to survey the land in general and, if possible, to climb Mount Terror to confirm whether it really was, or was not, an extinct volcano. This was a smaller party than the expedition in March, which had started out with thirteen members. It had two officers and the Guarantee Party, Edgar, Lashly, Quartley and Wild, who all went on skis pulling two sledges. It took a week's hard pulling for them to reach Cape Crozier, but on 11 October, Edgar and Engineer Skelton climbed down to the mail post on Cape Crozier and fixed their tin below the one already there, writing on the binding: 'planted 2.45 PM, 11th October 1902'. Then they both added their signatures.\n\nThe 11 October was productive. The two men were to make an important natural history discovery. They saw Emperor penguins on the shore and noted a track they thought was worth investigating. Emperors were a fascinating mystery in 1902; their life cycle obscure. It was assumed that no animal could possibly have evolved to breed in the caterwauling gloom of Antarctica and that the birds probably migrated north in the winter. When Skelton returned the following day he and his companions saw 300 Emperors squawking together on the sea ice. Skelton was convinced that he had made a significant discovery, but it would take another six days (18 October) before he could prove this, when with Edgar and Quartley he reached the rookery and captured two young birds. Dead chicks of various ages were strewn over the colony. This proved conclusively that Emperors did, indeed, breed in Antarctica.\n\nThose five days delay between the visits to the shore was due to another blizzard. Wild wrote that the wind was 100mph. He said that the weight of snow bent the tent poles alarmingly. It also reduced the space inside the tents so much that the men could not lie straight in their bags; instead they had to take turns to keep the 'doorway' open and they could only cook one hot meal a day and that was with difficulty. Wild wrote that his companions' tent got completely snowed up and had to be dug out, its occupants having existed on biscuits and a little sugar for three days (another version is that they all lived on biscuits and sugar for five days and had nothing to drink in that time).14 But Wild writes that the 'other' tent was revived by brandy and a good hot meal!\n\nWhen the six men reached home base on 24 October, proudly carrying their young birds, all the men were suffering so badly from snow blindness that they did not recognise Dr Wilson who went out to meet them. They recorded their lowest temperature on the sortie had been \u221258.6\u00b0F. Unsurprisingly they had not attempted to climb Mount Terror.\n\nThey had a wonderful meal: seal liver and bacon. On the same night the sun started its 'shameful routine of forgetting to go to bed and staying out all night'.15\n\n### Notes\n\n1 Ed. Skelton. J & Wilson, D., _The Antarctic Journals of Reginald Skelton_ Reardon Publishing, 2004, p. 109.\n\n2 Wild, J.R.F., Letter to Mrs Bostock. SPRI MS 1078\/3\/1; D.\n\n3 Ellis, A.R., _Under Scott's Command, Lashly's Antarctic Diaries_ , Victor Gollancz, London, 1969, p. 53.\n\n4 _The Lure of Little Voices_ , by Robert W Service (1874\u20131958) published 1907, Songs of a sourdough, Quotation written by Wilde, J.R.F., in Letter to Mrs Bostock. SPRI MS 1078\/3\/1;\n\n5 Carpenter, K.J., _The History of Scurvy and Vitamin C,_ Cambridge University Press, Cambridge, 1986, p. 138.\n\n6 Scott had been strongly advised that the main cause of scurvy was ptomaine poisoning, putrefaction in tins. Dr Wilson's duty every morning was to sniff and taste the contents of all the tins to be eaten that day and discard any that were 'tainted'.\n\n7 Wilson, E.A., _The Medical Aspect of the Discovery's Voyage to the Antarctic_ , British Medical Journal, 1905 2 p. 77\u201380.\n\n8 Information supplied by the Discovery Centre, Dundee, (Julie Millerick).\n\n9 Hawk, P.B.; Bergeim, O., _Practical Physiological Chemistry_ , Blackiston's Son. Philadelphia, 1926, p. 817.\n\n10 Ibid., p. 818.\n\n11 Food and Nutritional Board. Institute of Medicine, Vitamin C, Dietary Reference Intakes, National Academy Press: 2000: 95\u2013185.\n\n12 Personal communication. Professor Jeffrey Wood. Professor of Food and Animal Science, Bristol University, 2006.\n\n13 Armitage, A.B., _Two Years in the Antarctic_ , Paradigm Press, Bungay, Suffolk, 1984, p. 138.\n\n14 Ed. Savers, A., _Diary of the Discovery Expedition to the Antarctic 1901\u20131904_ , Blandford Press, London, 1966, p. 205.\n\n15 Ibid.\n\n## 8\n\n## The Antarctic Summer: October 1902 to January 1903\n\nCommanded by Lieutenant Armitage, and with Chief Engineer Skelton and nine lower-deck companions, Edgar was on the Main Western Party from 29 November 1902 to 18 January 1903. The aim of the journey was to get as close as possible to the Magnetic Pole and to make recordings.\n\nIn 1902 the exact location of the South Magnetic Pole had not yet been identified,1 and when the Main Western Party set off Edgar was well aware of the importance of its location to navigation. This related to the fact that magnetic compass readings differed significantly from the true north and by different amounts in different parts of the world. Charts showing magnetic declination, the difference between magnetic north and true north, were first produced in the 1770s. These charts had to be updated regularly (as is still done today), because the magnetic field pattern and the Magnetic Pole position change continuously, whereas the Geographical Pole remains fixed.\n\nHistorically, in spite of the charts, navigation errors had led to calamitous losses. Failure to correct a ship's compass accurately for the magnetic declination could result in the ship navigating a course miles from its intended land destination. This frequently led to disaster. Recordings made at sea were not as accurate as land-based recordings so the _Discovery_ expedition was a unique opportunity to make many land-based recordings close to the Magnetic Pole, so that accurate adjustments could be made to the charts. The work, which was to be coordinated with the German and Swedish records, fulfilled an important part of _Discovery_ 's brief.\n\nIt was a well-manned party: the Main Western Party comprised two officers (Skelton and Armitage), ten men (three from the Guarantee Party, Edgar, Wild and Quartley) and seven others. There was a Supporting Party of nine (two officers, seven men), who were to return after an anticipated three weeks, leaving the Main Western Party to continue. The ten sledges and four teams started off under a cloudless sky, pulling a total of nearly 5,000lbs, which included just under 2.5lbs of food per man per day and 112lbs of seal, mostly liver, which unfortunately had been cooked in fat and then oven dried in order to reduce weight. This reduced its vitamin C content to a minimum, a problem that was completely unsuspected at the time.\n\nOn any expedition enough food has to be carried for safe survival but not an ounce more. The basic sledge ration included: Pemmican, a concentrated rich mixture of fat and protein, soup squares and 'Red Ration' (a mixture of bacon and pea-flour to thicken the food), sugar, Bovril (a beef and yeast extract), rations, small amounts of chocolate, plasmon (a concentrated powder milk preparation), cheese and cocoa. The food was all individually marked; 'R' (Red Ration), 'Choc', etc, and was carried in weekly bags. Edgar shared his bag with Quartley and Horace Buckridge, a laboratory assistant.2 Tea and matches were carried in a tin.\n\nAs they set off, the ship gave three cheers. Photographs were taken. The twenty-one-man party started off in good fettle with sails flying. Armitage's aim was to get to the south side of the Ferrar Glacier, the glacier he had been forced to return from a few months earlier, and thence onto the Polar Plateau. This glacier is enormous, bigger than all the European glaciers,3 and Armitage thought it would be impossible to get straight onto it from the sea ice. He aimed to make the early part of the ascent via another glacier (later named the Blue Glacier) and then cross onto the Ferrar Glacier via a pass running between the two.\n\nThis expedition was to try everyone's stamina, including Edgar's, but his good nature and quick-wittedness survived intact. The teams started well; a stiff wind allowed them to use sails initially, but all too soon the wind dropped. On the hard, irregular sea ice, the teams found that wooden runners pulled badly in comparison to the German Silver (an alloy of copper and nickel) runners and had to be removed. By the end of the first day, 8 miles had been covered. When they camped Edgar did the cooking in his three-man tent; he was always practical and resourceful.\n\nVictoria Land had never been charted, and it was hoped initially that if it was narrow the expedition might even reach as far as its western coast. But the Main Western Party that Edgar was on only had provisions for eight weeks and progress was slow. Reaching the Magnetic Pole quickly became an impossible goal, so repeated magnetic observations became the priority.4 A dipping compass will incline downwards to 90\u00b0 when directly over the Magnetic Pole. On 1 December, Skelton recorded a set of 'dips' of 86\u00b032 (almost 2\u00b0 higher than the recording on the ship),5 i.e. they were closer to the Magnetic Pole.\n\nIt had taken them three days to get off the sea ice and onto the Blue Glacier. Pulling the sledges up the slope was heavy work, but with steady dragging they were 600ft above sea level in a day. But the glacier itself was steep and variably snow-covered or icy. It became so difficult to pull the sledges that the men had to relay; this happens when the surface or the incline makes man-hauling impossible with a full load. The men had to take half a load from the sledge, travel a certain distance, unload it and return with the empty sledge to load the remainder, thereby covering three times the distance they would cover normally. This was a time consuming and exhausting business, made worse on this occasion by their boots continually slipping on the treacherous surface so they had to keep changing from crampons to skis, and back again. But Edgar remained stoic and practical. He had a good memory and an excellent grasp of detail; when the teams stopped to make a camp, Edgar and three others were detailed to make a survey of the terrain before the others climbed out of their pulling gear.6\n\nAfter six hard days the team had climbed 2,480ft and at last the surface levelled out in a big open valley filled with ice and snow. Mountains gleamed in the distance. Armitage's plan was to push through the valley and find a pass through the mountains and towards the Ferrar Glacier. After one more day (6 December) they had reached 3060ft. The scenery was breathtaking.\n\nWhen Armitage skied off to investigate the pass to the Ferrar Glacier, Edgar was reorganising the sledges for the Support Party's departure. Two of his party's 11ft sledges were changed over with two of the Support Party's 9ft sledges, and he unpacked and repacked them. For once he was despondent; the loads the Main Western Party were to pull were too large for the 9ft sledges; the straps wouldn't go around them. Skelton told him he just had to make the best of it.7 They named the site Separation Camp and left a depot of provisions there. The ongoing party separated from the nine men of the support group on 10 December.\n\nArmitage found his Descent Glacier. It looked perilously steep. Its lower reaches out of sight, its upper part a steep icy slope. He, along with Dr Koettlitz, decided that it would be madness to try and get their heavily loaded sledges down that way, so five precious days were wasted in trying to find a better route through the mountains and down to the Ferrar Glacier. In this abortive search the sledges had to be hauled one by one, and Edgar spent two days easing six sledges up a slope by block and tackle over a rise of about 800ft in half a mile.8 When the attempt was abandoned, Edgar and his companion had to lower the sledges down that almost 40\u00b0 slope.9\n\nRather than give up, the team decided finally to try the Descent Glacier. Although it looked impassable from its upper parts, it is known now that the glacier is made up of a series of 'steps', which slope down to the Ferrar Glacier over a distance of approximately 3 miles. The upper step, which is very steep, slopes down for 400 yards and ends in a small shelf. The lower slopes are less extreme. So the glacier is actually a series of decreasing but significant gradients, interspersed with shelves.\n\nThe trial descent, on 16 December, was hair-raising. Edgar, Wild, Armitage and Skelton set off with ice axes, a long rope, and an empty 9ft sledge. They found that the upper slope was initially relatively easy, with no crevasses or ridges, but then it became very steep and dangerous. Edgar, Wild and Skelton put on crampons and tied themselves together to act as an anchor for Armitage (a heavy man), who was attached to the rope and lowered, with the sledge, down the slope. When the rope had been let out to its limit, Armitage secured himself and the sledge with an ice axe and the three went down cautiously to join him. They continued this way for about 700ft. Some of the stretches were very steep and they eventually ended up in a fog of clouds, which blocked out any further vision. But they had shown the initial descent was possible. To climb up the slope to the top again pushed them to their limits.\n\nBut, amazingly, on the very same day, all twelve men got down that fearful descent. The sledges were tied together in pairs; one 11ft\u20137ft pair with Edgar, Quartley and two others, and a 12ft\u20139ft combination. The first four fell and slipped down the slope for yards before bringing the slithering mass to a halt, thus the second group, with Edgar in it, started cautiously. They also lost control and slipped down helplessly onto the ledge (Armitage wrote that the run was more exhilarating than the water-chute at Earl's Court in London).10 The next slope took three hours by which time they were over a third of the way down. A series of less horrendous gradients followed.\n\nThey got down the Descent Glacier in a day and a half. They had conquered the first great obstacle by reaching the Ferrar Glacier.\n\nThe remainder of the exploration of the Ferrar Glacier reads like the trials of Job. The team were trapped by blizzards, sticky snow made pulling awful, freezing fog engulfed them, snow blocked visibility, but still they kept going with stoicism and determination. On Christmas Eve, the men asked Armitage if they could have a holiday over Christmas Day: 'definitely not' was the response. But they were determined to mark the occasion and hid the small gifts they had carried in their kit under the snow. When the wind scattered the snow, exposing the little offerings, Wild wrote they were rewarded by 'a thorough wigging' for carrying extra weight!11\n\nThe Ferrar Glacier continued unendingly. Whenever they thought they had reached the summit they found they were still on another of the series of icefalls. By New Year's Day they only had enough provisions to last a few more days, but at last, on 2 January 1903, a final icefall saw them on the summit, the Polar Plateau. The party was the first to set eyes on that vast ice plain and comprehend the enormity of the Polar Plateau icecap. This was a significant advance in Antarctic exploration.\n\nThey went on for a few miles taking further magnetic readings. But food supplies and illness determined their return. One man had collapsed with breathlessness and chest pains. He may have been suffering from altitude sickness, secondary to the relatively low oxygen levels at high altitude, which can cause problems at a height of 10,000ft. Even today men regularly need to be evacuated down to sea level because of problems at this height. In 1903 others felt unwell too, but there is no record of Edgar suffering.\n\nThe descent meant going down the Ferrar Glacier, ascending the daunting Descent Glacier and returning via the Blue Glacier. As he went down the Blue Glacier, Edgar fell 20ft into a crevasse, the lower parts of which were so deep as to be out of sight. The rope attached to his harness was not thick. He did not panic. Another rope was lowered round him and he was hauled up, though with some difficulty.\n\nThe team got back to the ship on 19 January. They had broken through the Western Mountain chain to the immense Antarctic Plateau, reaching nearly 9,000ft, and in doing so had made important magnetic recordings.\n\nThey had achieved a remarkable exploration: they were to find that others also had achieved remarkable feats. On one of the most important expeditions of the _Discovery_ years, Scott, Wilson and Shackleton had made a pioneering sledge journey, towards the Geographic South Pole. They were away from November 1902 until February 1903. The three hoped to get as close as possible to the Pole. Although they failed to reach it, in fact they did not get off the Barrier, they did reach a notable 82\u00b0 11' S, by far the furthest south achieved, they formed an impression of the nature of the Barrier (one of _Discovery_ 's briefs) and they made a record of the mountains fringing the Barrier on Victoria Land. All three suffered with scurvy, with swollen spongy gums, knees that would not bend and swelling of their legs. Shackleton was the worst afflicted and became very breathless. Much against his will, he was sent home on the relief ship SS _Morning_.12\n\nIndeed, these two expeditions yielded significant new knowledge about the mysteries of Antarctica.\n\n### Notes\n\n1 The location of the South Magnetic Pole was to be found in 1909, by Douglas Mawson, Edgeworth David and Alistair Mackay on Shackleton's _Nimrod_ expedition.\n\n2 Ed. Skelton, J., _The Antarctic Journals of Reginald Skelton_ , Reardon Publishing, Cheltenham, England, 2004, p. 136.\n\n3 Cherry-Garrard, A., _Journal_ SPRI, MS 559\/18\/1\u20134; BJ VOL 2, 24\/08\/1911.\n\n4 Yelverton, D., _Antarctica Unveiled_ , University Press of Colorado, 2000, p. 203\n\n5 Ed. Skelton, J., _The Antarctic Journals of Reginald Skelton_ , Reardon Publishing, Cheltenham, England, 2004, p. 138.\n\n6 Ibid., p. 138.\n\n7 Ibid., p. 140.\n\n8 Yelverton, D., _Antarctica Unveiled_ , University Press of Colorado, 2000, p. 204.\n\n9 Ibid., p. 205.\n\n10 Armitage, A.B., _Two Years in Antarctica_ , Paradigm Press, Bungay, Suffolk, 1984 p. 171.\n\n11 Wild, J.R.F., Notes related to the BNAE, 1901\u20131904, SPRI, MS 944\/3:D\n\n12 Wilson, E.A., _The Medical Aspect of Discovery's voyage to the Antarctic_. British Medical Journal, 1905, 2, p. 77\u201380.\n\n## 9\n\n## The End of the _Discovery_ Expedition, 1903\u201304\n\nEdgar was on three expeditions to the mountains that led to the interior of Antarctica: the Western Depot Party, 9\u201320 September, the Western Attempt, 12\u201321 October, and the Western Summit, 26 October to 24 December 1903.\n\nThe ship's company were now experienced Antarcticans, and the Antarctic winter of March to September had lost its power to frighten or astonish. The men endured the months of freezing darkness and howling blizzards with grim determination. Even the _South Polar Times_ had become a routine. Edgar played cards, wrote letters, looked forward to the return of the sun. The monotony was broken when the King's birthday was celebrated with a sports day, despite, the _South Polar Times_ wrote, 'that ever constant friend the wind, squalling with its constant shrillness'. A large silk Union Jack was hoisted on Hut Point and a Royal Salute fired. Edgar led a tug of war team, generously allowing the opposition to take the best ground (his team lost two to one), but he won the 2 mile flat ski race easily. His team lost the sledge-dragging competition by 20 seconds; here, rival teams on skis, pulled sledges loaded with 900lbs of iron. The day was a big success. It was rounded off by a concert after a magic lantern show, where views of New Zealand and Maoris were shown and Engineer Skelton showed views of the ships in which he had served, followed by a concert.1\n\nScott spent the long winter planning a second sledging campaign. He aimed to have all the sledging parties back by Christmas (the Antarctic summer). He wanted to avoid a further year in Antarctica and needed to have the men free to concentrate on _Discovery_ 's release from her icy manacles. Since Antarctica's low temperatures made major sledging expeditions impractical before October, this only left about ten weeks for exploration before the teams needed to return to base and release _Discovery_ for her return to England.\n\nScott's object was to find a new road to the Ferrar Glacier and lay a depot on it; then he planned to push on from Armitage's furthest west, over the inland Plateau, and, hopefully, find the western shores of Victoria Land.\n\nThe Western Depot Party was important, both for laying a depot for subsequent exploration and for its success in finding a direct route to the Ferrar Glacier. The glacier descends gradually to an inlet called New Harbour; an inlet that Armitage had thought would be impassable for sledges and so had reached the glacier via the Blue Glacier and the awful connection between the two, the Descent Glacier. Scott hoped that a route could be found via New Harbour that would make exploration to the west easier and quicker.\n\nThe inlet of New Harbour was indeed an awesome sight when Edgar and the team first saw it on 14 September. Gigantic ice blocks and high masses of earth and rock debris blocked the entrance. Boulders looking like giant tabletops rested on ice columns. But when Scott, Edgar and Skelton reconnoitred cautiously into the maze they did find a route, which led gradually from the inlet to the lower part of the glacier. As they followed this course the team had to carry their sledges and loads (430lbs of provisions) across the jagged ground until they reached a trail on the north side of the glacier,2 which twisted and turned until it finally arrived on the Ferrar Glacier itself. The glacier looked like 'a smooth polished road \u2013 a ribbon of blue, down the centre of which ran a dark streak caused by a double line of boulders'.3 The team pushed on up its smooth icy surface until they came to an area below Cathedral Rocks4 where, surrounded by glorious pinnacles reaching to over 3,000ft, they established their depot of three weeks provisions for six people, by a big boulder.5 They turned for home on 17 September. Scott's hopes for greater speed had been fulfilled. The 140 miles had been completed in twelve days rather than the twenty-one days that had been allowed,6 and it was possible that it could be done in even quicker time. Armitage's party had taken three weeks to get to the depot spot the previous year.7\n\nThe Western Depot Party was followed about a month later by the Western Attempt, an abortive effort to actually get through the Western Mountains and onto the Plateau. Twelve men in three sledge groups left base on 12 October. Edgar's group was led by Scott and included Engineer Skelton, Boatswain Feather, Stoker Lashly and Able Seaman Handsley. There was a second group, which planned to make a geological survey of the region. A third group of three men was a support group. Scott's party was to be away for nine weeks, the other groups were to return earlier. The four 11ft sledges, carrying 200lbs per man, were cheered off from _Discovery_. The party left with high hopes, but they were to be back at base in nine days.\n\nScott set a cracking pace which Edgar kept up with well. In spite of their heavy loads the party covered the 45 miles to the cape of New Harbour in two days. They called the cape Butter Point because Scott thought this would be the highest point that the group would get fresh meat on their return \u2013 and so left butter there to cook the meat.8\n\nIt was probably on this sortie that Edgar caused Scott an irritation that he told his skiing companions about years later. As the temperature was \u221247\u00b0F, Scott told the party to put on three pairs of socks. Edgar put on two. His feet were soon frostbitten and the party had to stop. Scott asked him, 'How many socks have you got on?' Edgar replied respectfully, 'Locks, sir. How many pairs did you say, sir?' To which Scott replied wearily, 'Why three.' He told Edgar to take his boots off and get the circulation back. Edgar dropped out of sight, but Scott came up to hurry him up. Trying to avoid Scott's observant eye Edgar tried to take his two pairs of socks off as if they were three. He fumbled away but without success. Scott accused him of disobedience. 'Can you count?' 'Yes, sir, fairly well.' 'Did you think you had put three socks on?' 'Well sir, I was a bit sleepy when I put them on.' Finally Edgar had to own up and was told off sternly. When he came to put his boots on they were as stiff as iron and he had to bend them with a geological hammer. The ice at the bottom of the boot was impossible to get out.9\n\nThey continued at such a pace that they reached their depot on 16 October (four days) and camped on the glacier in the valley under Cathedral Rocks. If Edgar, in his imagination, had ever considered Antarctica's majestic, severe beauty, he must have been impressed by the view in front of him that evening. The sunlight pinnacles of Cathedral Rocks showed a rich dark brown; lower down the rock became greyish-black, splashed with lighter areas. There were patches of snow, and here and there a glacier gleamed, sparkling white and contrasting markedly with the bare rock.10 The glacier itself could be seen curving down towards the sea. Beyond this was the sea itself, pearly grey in the distance. There was complete stillness.\n\nContinuing the punishing pace they reached the enormous boulders below the Knobhead Moraine, reaching this landmark in six days rather than the twenty-seven days of the previous year.11 But, at 6,000ft, disaster struck. The German silver covering the wooden sledge runners had split to shreds on two of the four sledges; one was less damaged, the fourth intact. The men knew that without metal protection, the wooden runners underneath would disintegrate on the hard sharp ice. There was nothing for it but to leave the sound sledge and stores and return to base for repairs. Their return was as near to flying as was possible. They covered nearly 90 miles in three days.\n\nThe definitive expedition, the Western Summit, left on 26 October. It was a smaller party than the Western Attempt as there were just nine members: Scott's Advance Party and Ferrar's Geological Party. Again, rapid progress was made. On this run they reached the Knobhead Moraine in three days (rather than six), in spite of the runners still giving problems and having to be repeatedly mended by Lashly, Skelton and Edgar. Unexpectedly they were careless about leaving things outside the tents to dry and on 27 October they only narrowly escaped a serious problem when a sudden blast of wind scattered sleeping bags, socks and finnesko left out on the ground to dry. But at the depot on 1 November, they found 'a loss, the gravity of which could scarcely be exaggerated'.12 The lid of the instrument box had not been fastened securely and had blown open. The wind had whirled away Scott's copy of _Hints to Travellers_.13 This was hugely important. Skelton wrote 'it contains all the data for the skipper to work out his sights for time and position'.14 The explorers had no landmarks to rely on accurately and made observations of the sun to work out their latitude and longitude. Edgar, familiar with his father's stories of the problems of navigation around the Horn, appreciated the importance of the loss all too well.\n\nThe team had intended to work out their latitude by observing the sun's height above the horizon at noon and adding a calculation for the sun's declination.15 This calculation involved the tables in _Hints for Travellers_. The calculation of longitude involved time. We all know that as we travel around the globe the local time changes (for example there are five hours time difference between London and New York). To calculate their longitude, the explorers made a comparison between the time at whatever position they were at \u2013 usually at midday, when the sun was at its highest and, at the exact same moment, the time on their chronometer was standardised to Greenwich Mean Time.16 The difference between the two was used to calculate their longitude position. Since the earth takes twenty-four hours to revolve through 360\u00b0, one hour is one twenty-fourth of that spin, or 15\u00b0, so if, for example, the difference in the readings was three hours, then the explorers were 45\u00b0 away from the north-south meridian, i.e. 45\u00b0 east or west. Logarithmic tables were required here also; at the equator, where the girth of the earth is at its biggest, those fifteen degrees equal 1,000 miles, but as the lines of longitude converge, the distance each degree represents shrinks until it is nothing at the Geographic Pole. If the explorers' course deviated by even a few degrees of longitude, they needed to be able to convert those degrees into miles. This, too, needed the tables in _Hints for Travellers_.\n\nSo the loss was a most serious blow, but Scott could not consider returning again. He thought that he could measure the sun's altitude at noon and use this to work out latitude. He planned to keep the party on a due west course on the ascent. He asked his group if they agreed to go on, and Edgar and his companions agreed, though Skelton wrote 'now we shall never know exactly where we are'.17 They were marching away into the unknown without having any idea of their precise position or how to get back.\n\nThey had food for over six weeks, pulling 230lbs per man.18 They struggled to climb towards the summit, repeatedly needing to repair the metal on the sledges, avoid ice falls and move from hard ice to snow (which made pulling harder but was kinder on the runners). They were frostbitten and engulfed by driving snow and icy blasts of wind sweeping down from the summit. Edgar lost all sensation in one foot, a known precursor of frostbite, which could become gangrenous. Progress was halted whilst his companions rubbed and warmed the injured foot until he could feel it again. The weather deteriorated, and on Wednesday 4 November, at 7,000ft, they were imprisoned by a violent gale and thick suffocating snow for seven days. 'Desolation Camp' was their name for this base. Edgar endured the gale but found inactivity even worse than hauling. Apart from managing to cook two meals a day he spent most of the day in his sleeping bag looking up at the fluttering green canvas, unable to sleep because of the gale. He spent his time reading William le Queux and _The Red Magazine_. Communication between the tents was only possible when there was a lull in the storm. Edgar's usual cheerfulness deserted him and he merely endured.\n\nOn 11 November they escaped. Ferrar's Geological Party split off (Ferrar's group was to make startling and important finds of fossil plant remains that would prove that Antarctica was once a part of a temperate climate, the first confirmatory find of this nature from this part of Antarctica). The remaining nine plodded westwards and upwards. Comments such as 'dragging heavy' and 'surface bad' occur with depressing frequency in the records.\n\nBy 15 November, the twenty-first day out, Skelton wrote that the surface was 'practically on the level all day'.19 On 16 he wrote they were 'on practically level surface at 900 feet'.20 As the vast level snow plain stretched out before them at 8,900ft they could congratulate themselves. They had achieved a remarkable 'first' and they still had five weeks of rations, enough to cover a good many miles over the Plateau before returning. Scott wanted to discover with certainty if the high land was just the plateau of an island (the island of Victoria Land) or part of a vast continent. But plans to locate the Magnetic Pole had been abandoned, despite magnetic records being made regularly.\n\nScott decided to pull the sledges separately; Edgar pulled with Scott and Boatswain Feather (whose back was giving him trouble). Skelton led the other two men. But by the end of the morning's march on the 16th, Skelton's team was three quarters of an hour behind the other team and he wrote on the following day, 'the work was really too hard for us'.21 On the 19th this observation was confirmed; one of the men, Handsley, felt ill, and subsequently collapsed, unable to breathe (and probably not helped by the brandy that was administered).22 Three days later Scott decided to split the party, changing Feather with Stoker Lashly and progressing forward as a threesome, leaving the others to return. It was their twenty-ninth day out; the start of what Scott described as 'three weeks of the hardest physical work that I have ever experienced, and yet three weeks on which I can look with unmixed satisfaction, for I do not think it would have been possible to have accomplished more in the time'.23\n\nLashley, Edgar and Scott shared a single sleeping bag. They endured the hardships, the dangers and the hunger equally. Scott was the leader but always consulted his two companions, whom he grew to like and admire, over big decisions. He always said 'we' not 'I', when referring to the journey. He wrote, 'with these two men behind me, our sledge became a living thing and the days of slow progress were numbered.'24 But in spite of this apparent equality it might be questioned whether Scott, having lost his instruction manual, was not foolhardy to proceed, risking his own and his companions' lives. It would be interesting to know what Edgar and Lashly, who knew the risks, really thought. But being lower deck they would have been unlikely to question their leader.\n\nEverything the three men recorded was new information; the surface of the mostly smooth, though variably broken up by sastrugi, sharp ice edges like waves whipped up by the wind. On this surface their progress resembled a small boat at sea, climbing up a wave and then diving down into the hollow. They recorded that the wind blew from west to east across the Plateau during the winter.\n\nThat wind was terrible. It blew continuously and cut them to pieces. Things were worse in the mornings but got slightly better as they warmed up on the march. Edgar had a deep cut on the side of a fingernail and his finnesko wore out. His face was cracked; his cheeks and lips were sore and raw. Eating was difficult. Laughing, if he had wanted to, was impossible.\n\nThey turned back on 30 November. They had kept going simply because they wanted to last out to the end of the month. They were 300 miles from the ship and had nineteen days' provisions and sixteen days' oil to get them back to their depot where they had cached another ten days' food and oil. Going through the fields of sastrugi was exhausting; they fell often and as they fell the harnesses jerked them. But they found no sign of the west coast of Victoria Land in spite of the fact that, as they reflected, they could have crossed Greenland in many places on a trek of the length they had covered.25 They had shown the immensity of the Plateau and recorded its conditions. Antarctica was clearly not a series of islands but truly continental in size. Man had now penetrated its silent snow-topped plain.\n\nThe month they took to return was grim. The sledge capsized repeatedly, they fell time and again and Scott was worried that they had overestimated their marching abilities. But the phlegmatic characters of his two companions shone throughout. When poor visibility marooned them in their tent at a time when an hour's delay was critical because of their limited food and fuel supplies, it was a blessing to have Edgar stick his head out of the tent and announce in his usual matter-of-fact tone that the sun was now shining;26 the calm, solid reassurance of the British lower deck representatives was a relief to the overwrought mind. Scott thought they were undefeatable and wrote, 'however tiresome our day's march or however gloomy the outlook they always find something to jest about. In the evenings we have long arguments about naval matters and generally agree that we could rule the service a great deal better than any Board of Admiralty. Incidentally I learn a great deal about lower deck life \u2013 more than I could hope to have done under ordinary conditions.'27\n\nBy 6 December, concern about the oil supply got worse. They were getting more and more hungry. They were gaunt. Edgar looked wild with sunken cheeks, frostbitten face and a bulbous frostbitten nose. They were so eager not to miss the smallest scrap of food that they used Shackleton's 'noble game of shut eye' to allocate each man his share;28 one man turned his head when the food was divided and he decided who would eat which portion. They dreamed of food; Edgar's idea of happiness was roast pork, Lashly's apples and vegetables, while Scott thought continuously about bowls of Devonshire cream.\n\nThe snow surface varied but was often abysmal, and on 9 December it was like sand; the sledge felt like a log and they could hardly cover a mile in an hour. Edgar had never done such hard pulling. It took all his energy; for once he could not talk. Skis were hopeless to move the sledge in these conditions. They were worried about food but more worried about oil which was now down to one can, also Scott and Edgar's tobacco supply was at an end (Edgar had been on half a pipe per day). Although they were certain that they were near the edge of the Plateau they were uncertain of their precise location on this never ending plain. Scott suggested that they increased the marches by half an hour each day and he halved the oil allowance, (which meant a cold lunch). Edgar and Lashly would not disagree with a suggestion from Scott and indeed, under the circumstances, there was no alternative. They were travelling by the rule of thumb, but Scott wrote that Edgar's face fell dismally \u2013 he only believed that food was beneficial when it was warm and had 'a chance of sticking to the ribs'.29\n\nOn 10 December they slogged away for five hours, had a cold lunch and started again. But in the afternoon, as he peered through the unending snow, Edgar's sharp eye spotted land. They knew the end of the Plateau was within reach, but where were they precisely? There were innumerable glaciers falling down from the mountains but which was the Ferrar? Time and food made the right choice vital.\n\nOn 13 December Edgar's nose was badly frostbitten. The threesome had to stop to massage it back to life. It had given trouble for weeks and by now looked like a large, swollen potato. He always talked about this member as if it was something that was not actually his, but something he had to look after, 'my poor old nose again; well there, it's chronic'.30 As they held to an easterly direction the land began to slope downwards. They really had no idea where they were.\n\nThe 14 December was a date that Edgar would remember for the rest of his life. They were heading east and, although they were lost, they decided to keep going; a snowstorm was brewing and incarceration in a tent, in a blizzard, might well mean death from starvation. So they advanced into the unknown, steering their sledge through great ice hummocks and crevasses. As they progressed, Scott in front, Edgar and Lashly behind, the slope grew steeper and smoother. Lashly lost his footing, Edgar was pulled over and the three men and the sledge careered into the unknown like an unstoppable express train. The surface became rougher and, as they bounced onwards and downwards, they must have thought that death, or at the very least serious injury, was on its way. But as they eventually came to a halt on hard, rough, windswept snow, amazingly all three were able to struggle to their feet. They had careered down 300ft. The first question Edgar and Lashly asked Scott was if he was all right (sir).\n\nLuck seemed to be with them. As they looked around they recognised their own glacier and other familiar landmarks. In the distance they could see their friend, the volcano Erebus, smoking away. Their food depot was within reach. After all their tribulations they thought they were safe. But malicious fate had not given up yet. Soon after setting off, Edgar and Scott fell into a crevasse. Lashly, left on the surface to mastermind the rescue, used one hand to hang on to the broken sledge, straddling the crevasse and his other to slide skis under the sledge to support it. As Scott and Edgar hung in their traces, surrounded by blue ice walls and with an unfathomable gulf below, Scott asked Edgar how he was doing. The reply, from a man facing death for the second time in a day, was characteristically calm. He was 'good enough'.31\n\nScott had to climb up a rope to escape first. This was hugely difficult. His fingers were frostbitten, his clothes were thick and cumbersome, and he hadn't climbed a rope for years. But he took his gloves off, and in the subzero temperatures he hoisted himself up slowly. A harness was swung down to Edgar and he was hauled up, frostbitten but unbowed, 'Well I'm blowed' was the (unlikely) official report of what he said.32\n\nThat night, black and blue with bruises, frostbitten, sore and exhausted, he mused over his extraordinary day; Scott wrote that Edgar ruminated continually on the day's experience, 'My word but that was a close call. My word, but that WAS a close call'.33\n\nThey picked up supplies from the depots, and on the 16th they were at their old quarters at Knobhead Moraine in the large glacier basin. Instead of rushing back to base Scott decided to investigate the direction of ice streams from the basin. He wanted to follow a glacier tributary that sloped north and then eastwards. The team were about to make one of the great geographical finds of the expedition.\n\nThey negotiated the steep ice slope roped together and continued until they found a shallow frozen lake resting on deep layers of mud; 'what a splendid place for growing spuds'.34 The glacier, it seemed, did not end by pouring icebergs into the sea, as had been assumed. Instead it ended in a lake high in the mountains. The three could not find any moss or lichen, though it is now known that far beneath the icy surface these lakes remain unfrozen and support colonies of bacteria and phytoplankton.35 Microbes, bacteria and pollen have been found in the ice. This was information well beyond the explorers' wildest imagination and would take about another century to be discovered.36\n\nBut there was more. Progressing further down, through the valley of startling beauty and ruggedness, they came across stretches of undulating sand, then areas covered by rocks of different colours and sizes, more sandy stretches and boulder heaps. Extraordinarily there was no snow or ice, though the surface seemed to be the result of ice and water action. As they advanced, Edgar, always mindful of his stomach, asked if there was any point in carrying the lunch any further. The three had their lunch, sitting at a place that gave them memorable views up and down the valley. Except for the mountain summits there was no ice or snow in view. They ran their fingers through the sand and drank from the streams. It was amazing to think that they were less than 100 miles from their terrible experiences on the summit. They saw nothing alive, just the skeleton of a Weddell seal. Scott named this dry area a 'valley of the dead'.37 The trio had discovered one of the extraordinary Dry Valleys of Victoria Land38 \u2013 they were in what was to be later named the Taylor Valley. The find was one of the major geographical discoveries of the entire expedition.\n\nThe trio returned. The Western Summit expedition was now over. Having managed to fry up some meat at Butter Point they got back to _Discovery_ on Christmas Eve 1903. They were very thin and exhausted. The steward made a celebratory meal of steak and tomatoes, after which the three seemed to put on weight with every meal. They could congratulate themselves that they had achieved a remarkable sledging feat of nearly 1,000 miles. They had travelled by far the furthest into Victoria Land.\n\nThese expeditions, which opened up the Western Mountains, made Edgar a true veteran of Antarctic exploration. He was away from the ship for nearly sixty days on sorties that would tax the most experienced modern-day explorer, confirming (if this was needed), his strength, his alert intelligence and his composure.\n\nWhen _Discovery_ returned to England a number of the men were singled out for special mention. One of these was Edgar, who was promoted to PO 1st Class. Another was Stoker Lashly. Scott described them as men of magnificent physique. He wrote that the journey to the interior of Victoria Land reached the limit of possible performance under such awful conditions and that it could not have been accomplished had either man failed in the slightest. Scott recorded that Edgar's and Lashly's determination, courage and patience were often taxed to the utmost yet the two men were always cheerful and respectful. The outings cemented the loyalty and mutual respect between Scott and Edgar.\n\n### Notes\n\n1 _South Polar Times_ , vol. 2, VI, April 1903, p. 28.\n\n2 Scott, R.F., _Scott's Voyage of the Discovery_ , James Murray, London, 1929, p. 563\n\n3 Ibid., p. 563.\n\n4 The four abrupt cliffs surmounted by sharp peaks thought to resemble a cathedral. Named by Armitage in 1902.\n\n5 Ed. Skelton, J., _The Antarctic Journals of Reginald Skelton_ , Reardon Publishing, Cheltenham, England, 2004, p.184.\n\n6 Ibid., p. 184.\n\n7 Scott, R.F., _Scott's Voyage of the Discovery_ , James Murray, London, 1929, p. 564.\n\n8 Ibid., p. 574.\n\n9 Debenham, F., _Journal_ , 19\/01\/1911\u201308\/03\/1911 MS 279\/2:BJp, p.78.\n\n10 Ibid., p. 575.\n\n11 Ibid., p. 576.\n\n12 Scott, R.F., _Scott's Voyage of the Discovery_ , James Murray, London, 1929, p. 587.\n\n13 _Hints for Travellers_ , a publication issued by the Royal Geographical Society of London which supplied the data to locate altitude and longitude accurately.\n\n14 Skelton, R., _Sledging Diary_ SPRI MS 342\/2\/6;BJ, 01\/11\/1903\n\n15 The angle between the magnetic north and true north at a particular point on the Earth's surface.\n\n16 Greenwich is at 0\u00b0 longitude. This cut Harrison's Clock.\n\n17 Skelton, R., _Sledging Diary_ , SPRI, MS 342\/2\/6;BJ,01\/11\/1903.\n\n18 Ellis, A.R., _Under Scott's Command: Lashly's Antarctic Journals_ , Victor Gollancz, London, 1969, p. 71.\n\n19 Skelton, R., _Sledging Diary_ , SPRI, MS, 342\/2\/4;BJ, 15\/11\/03.\n\n20 Ibid., 16\/11\/1903.\n\n21 Ibid., 17\/11\/1903.\n\n22 Ibid., 20\/11\/1903.\n\n23 Scott, R.F., _Scott's Voyage of the Discovery_ , James Murray, London, 1929, p. 601.\n\n24 Ibid., p 602.\n\n25 Ibid., p. 605.\n\n26 Ibid., p.609.\n\n27 Ibid., p. 609.\n\n28 Devised by Shackleton on the Southern Journey 1902\/03 which was undertaken by Shackleton with Scott and Wilson.\n\n29 Scott, R.F., _Scott's Voyage of the Discovery_ , James Murray, London, 1929, p. 613.\n\n30 Ibid., p. 615.\n\n31 Ibid., p. 621.\n\n32 Ibid., p. 621.\n\n33 Ibid., p 622.\n\n34 Ellis, A.R., _Under Scott's Command: Lashly's Antarctic Journals_ , Victor Gollancz, London, 1969, p. 83.\n\n35 Subglacial Lake, a lake under an ice cap or ice sheet. There are over 120 subglacial lakes in Antarctica; Lake Vostok is the largest known at the current time.\n\n36 Small free floating aquatic plants have now been identified.\n\n37 Scott, R.F., _Scott's Voyage of the Discovery_ , James Murray, London, 1929, p. 627.\n\n38 A row of valleys in Victoria Land with low humidity and without ice or snow.\n\n## 10\n\n## Return from Antarctica, then Home Again, 1904\u201310\n\n_Discovery_ was still incarcerated in miles of ice when the explorers returned and open water was yet 20 miles away. Second-in-Command Armitage had ordered highly organised teams to blast and saw through the ice to try and speed up its dissolution. It was hoped that _Discovery_ could escape in January 1904, and 'Saw Camp' was well established by the time Edgar got back to the ship on Christmas Eve. The aim was to saw and blast a path through the ice (in the event a hopeless task, the ice was 7ft thick and any crack froze up almost immediately) and three shifts worked round the clock. When Edgar arrived at 'Saw Camp' with Scott he was happy not to have to joined the band of 'unwashed, unshaven, sleepless, swearing, grumbling, laughing, joking reprobates',1 because Scott, seeing that the attempt was futile, stopped the work. Two days later all hands returned to the ship except for Scott, Dr Wilson, Edgar and Lashly, the cook Charles Clark and Seaman William Heald. They were to kill penguins. It seemed likely that _Discovery_ would remain stuck for another year and her larder needed stocking.\n\nOn 5 January, bewilderment was rife. Not one but two ships came into sight. One was _Morning_ , the relief ship, but Edgar joined in the wild guessing about the second: was it _Gauss_ (the German vessel); a private yacht; a man o' war?2\n\n_Morning_ brought mail with much missed news of the family at home. Edgar had no personal bad news, but _Morning_ carried orders that were dumbfounding to the hard-pressed crew of _Discovery_. Scott was ordered to abandon _Discovery_ if she could not be freed from the ice (the government did not want the expense of yet another rescue attempt on a third year) and to facilitate the repatriation of _Discovery_ 's crew and equipment, the government had sent a whaling ship, the _Terra Nova_ , to accompany _Morning_.\n\nScott read orders to the crew that they could hardly believe. Although the royal societies and the Admiralty were apparently satisfied with the achievements of the expedition, the government had now taken over complete control. This was because the expedition had run out of money; there were no funds for a further year in Antarctica if _Discovery_ could not be extricated from the ice. _Discovery_ was now public property. Scott confirmed that if _Discovery_ could not be freed, she was to be abandoned to the mercies of Antarctica. The men saw that Scott had tears in his eyes and Edgar, deeply loyal to Scott, was outraged. The crew all thought that there had never been orders framed to give an officer less option; they were full of 'you are on no account' and 'you are to distinctly understand'. Scott said that he had not fully taken in the idea of abandoning the ship but he thanked everyone for the way they had stood by him loyally. The men responded with three cheers. Williamson wrote, 'and there's no doubt that he deserves it if ever a man did',3 and Ford agreed, 'I think it is very hard times that he should be given no option at all after proving himself such a capable Polar Commander and it is a pity this work should be spoilt by quarrelling at home'.4\n\nWork to abandon _Discovery_ began on 14 January; supplies and equipment were hauled over the ice to load into the relief ships, a miserable task. The crews met midway to transfer the loads and Edgar's strength was put to good use. But nature is capricious. In 1904, the ice, which had remained solid the previous year, started breaking up. On 3 February 7 miles separated _Discovery_ from _Morning_ and the _Terra Nova_. Hopes for a release started to rise. The men wondered if the ice's break up was helped by the weight of heavy snow that pushed it under water.5 Edgar was involved in further blasting, the idea being to make a crack in the ice between Hut Point and the relief ships and then create a 'lake' that _Discovery_ could escape to. By the 13th, the sea was only 2.5 miles away and on the 14th, the ice finally split as the swell broke up the ice and _Terra Nova_ butted it again and again, her crew using the old whalers dodge of rolling the ship by running backwards and forwards across her deck to widen the crack. Finally _Terra Nova_ sailed through triumphantly with _Morning_ 'following meekly behind'.6 The men hoisted the Union Jack on Hut Point and rushed onto the floe and onto the ships. Scott sent a case of whisky for the two ships.7\n\nBut still nothing was easy. When _Discovery_ finally escaped into the water she was caught immediately by irresistibly strong winds, which pushed her back onto the shoreline at Hut Point where she lay stranded, pounded by wind, waves and tide for over eight hours. Edgar thought that there would be permanent damage to his ship; that having finally escaped the ice the sea would break her up. It must have been awful to see the way her decks buckled under the strain and to see planking from the ship's bottom floating to the surface. As heavy seas broke and washed over the ship, there was frozen ice all over the deck. But when at last the storm subsided and _Discovery_ got off the sandbank, mercifully no permanent damage had been done.\n\nEdgar gave a lingering look to those familiar landmarks that he thought he would never see again: Mount Erebus (still smoking), Castle Rock and Observation Hill. Then _Discovery_ retraced her route up the coast of Victoria Land. The voyage to New Zealand was notable for gales and _Discovery_ 's penchant for heavy rolling, sometimes through 50\u00b0, as she tossed about like a cork. In storms she leaked 'everywhere', the bedding was soaked, the ship was shaken, and, wrote Williamson, the top gallants were left on her until it was too late to take them in and the 'jib blew out of bolt ropes with a loud report'. Several of the men were seasick. They all 'stood by to turn out'.8 In a lull the topgallants were furled. This took two hours work and then the gale returned and continued all day, rolling the ship abominably. The men were utterly fed up, but they were, Williamson wrote, 'doing it for King and Empire and we are taking it joyfully!!... Pumping all day. How we are looking forward to Lyttelton and civilisation again.'9\n\nThey had some days of relative rest in Auckland Islands preparing for the grand arrival. All the ships were freshly painted as _Discovery_ , with _Morning_ and _Terra Nova_ , entered Lyttelton harbour on 1 April, serenaded by the strains of _Home Sweet Home_. No one was sorry to be back in civilisation with hospitable friends. Gifts flowed on board, boxes filled with mutton, potatoes and greens. The ship overflowed with well-wishers. After two months in New Zealand _Discovery_ (with a further twenty sheep on board and gifts from New Zealand farmers), sailed towards South America. She went through the Straits of Magellan10 and called at Chile, Port Stanley in the Falklands and the Azores, where the Prince of Monaco, a distinguished scientist, came on board on 2 September. Passing ships signalled to the crew, congratulating them on their achievements and wishing them a safe return home. The scientific programme continued throughout.\n\nOn 10 September 1904, _Discovery_ docked in Portsmouth. She had been away for thirty-seven months. _The Times_ wrote that there need be no reserve in the welcome extended to the _Discovery_ when she arrived in Portsmouth from an expedition 'full of hazard. The scanty knowledge which we possessed of an area of several million miles, Captain Scott and his companions have succeeded in dispelling our ignorance'.11 The Mayor of Portsmouth welcomed the new knowledge of the character and interior of the continent, the geological collections, the penguins, the meteorology and terrestrial magnetism (the last two of practical importance). There was national unanimity (at least initially) about the value of Polar research to the nation.12 Sir Clements Markham said that the _Discovery_ expedition was 'the best-conducted and most successful expedition that had ever entered the Polar regions, Arctic or Antarctic'.13 The sailors, interviewed by the _Daily News_ , were enthusiastic: they had reached further south than any other expedition, they were glad to be back but 'didn't have a bad time of it' \u2013 they had got their mail every twelve months.14\n\nThe public were fascinated to meet those who had experienced the actuality of this place of mystery. People wanted to shake the heroes' hands; they wanted to hear them speak. There was support for the medals granted to the crew. Edgar, on Scott's recommendation, received the highly prized 'Silver Medal with Clasp' (a duplicate medal was issued to his widow in 1914)15 and a small silver medal for success in an Antarctic sports competition (the ski race). Many years later a posthumous Royal Geographical Medal was added to this distinguished collection.16\n\nHe was given two months leave and was delighted to be home. Friendly and outgoing and with a great supply of new anecdotes, he was happy to return to Swansea, his family and friends. When _Gower Church Magazine_ interviewed him, the reporter wrote that he was, 'robust and courageous to a degree and has during his voyage added much to his previous knowledge and attainments'.17\n\nIn the _South Wales Daily Post_ he reflected on his Western Journey:\n\nIt's an uncanny feeling standing there, surrounded by everlasting snow, gigantic nunataks18 all around you and dead silence, which is almost deafening. Not a sign of life, no birds to speak of, only a melancholy seal to look at, and his blessed hide not worth a cent in the European market. Six of us were chosen to do this trip, which was 300 miles from the ship, and lasted nine weeks and three days: but three went back. We saw absolutely nothing. We were 9,200 feet on the ice cap and away towards the Pole was a range of unclimbable mountains. Nobody knows what lies beyond it.19\n\nHe went to London to be 'Paid Off' from the expedition. The Certificate of Discharge states his date of discharge as 30 September 1904; importantly the certificate grades both his 'Character for Conduct' and 'Character for Ability' as 'Very Good'. He continued in full-time naval employment as a Petty Officer. His Certificate of Service records the date of his promotion to 1st Class Petty Officer as 2 April 1904 (when _Discovery_ was in New Zealand) and this provided a modest financial boost; pay for a 1st Class Petty Officer was \u00a339 10 _s_ 10 _d_.20 Although he had assured the reporter of the _South Wales Daily Post_ that he had no marriage plans, he was in fact going back to Middleton regularly to pursue his friendship with his pretty cousin, Lois Beynon. He had been away from any domestic comfort for three years. Lois was a girl from a similar background (her father William, was Edgar's uncle, his mother's brother), who had been brought up in the public house in Middleton. She was someone who understood him. She was spirited, attractive, and musical. Edgar was a well-known young man, intelligent, good looking, affectionate and musical. A marriage was arranged in a very short time and was solemnised on 13 December 1904, just a little less than fourteen weeks from Edgar's arrival back in England.\n\nWelsh interest in the marriage was not only directed towards the now famous Swansea boy but also to Lois. As well as being familiar to patrons of the Ship, Lois was known in Rhossili for her lovely singing voice. Evening entertainment then was often in the form of musical soir\u00e9es and Lois frequently contributed to these; duets (often with the daughter of the Rector who had married Edgar's parents), _Blow Gentle Wind_ , and solos, _O'er Life's Dark Sea_ , were regularly performed and, more cheerfully, Arthur Sullivan's _Three Little Maids_. When she married her explorer on 13 December, 'Rhossili was agog with excitement'.21 Lois was described as the youngest daughter of Mr and Mrs Beynon of the popular Ship Inn. Edgar, as a man who had 'sprung to prominence by reason of the fact that he was one of the crew of _Discovery_ sent out for the purpose of Antarctic exploration'.22\n\nHe described himself modestly, according to the clearly impressed _Gower Church Magazine_ , 'as a simple mariner'.23 The Rector, The Reverend Lewis Hughes, performed the ceremony and the celebrations were noisy. A 'feu de joie', a rifle salute with rifles fired in quick succession so that the sound is continuous, added to the boisterous festivities.24 Lois looked delightful. She wore white silk, trimmed with cream chiffon with a matching picture hat. She carried the ivory-bound prayer book that Edgar had given her and was attended by two bridesmaids whom Edgar presented with dress rings. One of Lois' brothers, Enoch, was best man and the full choir gave vocal support. A wedding breakfast in the Ship Inn followed and the happy pair departed for London, their departure serenaded by coastguards and farmers who vied with each other by firing more gun salutes.25\n\nThe _Gower Church Magazine_ reported inaccurately on Edgar's perilous Western Journey saying that he had been '270 miles further south than all the rest of the crew',26 but enthusiastically describing his modesty; 'like every truly brave man he is far from being boastful and requires considerable persuasion to make himself relate anything about himself'. The article went on to say that Lois' contributions to local concerts were much appreciated 'as evidenced in the numerous and costly wedding presents given her by a large number of friends and relatives... the singing was worthy of the bright occasion, the voices in chanting the special psalm and the two well-known hymns were wonderfully sweet and one could feel that the hearts of all were full... May every blessing follow them in their new home'.27\n\nA few days in London was exciting beyond Lois' wildest imaginings: the noise, the traffic, the people, the entertainment and Edgar; Edgar who was used to large cities, who knew about traffic, omnibuses, theatres, who could navigate confidently through the confusing whirl. Edgar was her hero.\n\nAfter London came Portsmouth, the naval town on the south coast of England. The family were to be here until Edgar left again for the Antarctic. Married life began at 12 Walden Road, in the Tipner district of Portsmouth. Walden Road is a long terrace built over a number of years from the 1890s. Number 12 was an Edwardian house, first listed in the early 1900s. It had a bay window at the front and was 'modern', in that it probably had its own 'privy'.28 There were shops: a fruiterer, a general store, a grocer (3lbs of sugar, just over 5 pence; 1.5lbs butter, 1 shilling 6 pence; a pint of beer, 2 pence; 2oz tobacco, 6 pence) and a confectioner conveniently close, also the local doctors,29 but the change from rural Gower must have been a tremendous challenge for Lois. The only point of similarity was the sea; the dock was only about half a mile away, but for the first time she was away from her family and lifelong friends and thrown into a life of a very different pace. She had to manage long days on her own, to act independently, to cope with pregnancy and to look after babies. The couple had three children: Norman Edgar (18 August 1905) and their daughter Muriel (9 November 1906) were born when the family was at Walden Road. Edgar then moved his family to 52 Chapel Street, in the Portsmouth suburb of Buckland, where their second son, Ralph, was born on 4 December 1908. Both boys were to be baptised at home in Rhossili by the Reverend Lewis Hughes.\n\nEdgar was fully occupied. After postings for a total of nine months on HMS _President_ and HMS _Firequeen_ , Petty Officer 1st Class Edgar Evans was posted to HMS _Excellent_ , the Royal Naval Gunnery School in Portsmouth, on 15 January 1905 to train as a gunnery instructor. Apart from a three-month period in 1906, he remained on _Excellent_ for two years and then stayed in other shore bases in the Portsmouth area until he left for the Antarctic again in 1910, having qualified also as a Torpedo Instructor. He was based at HMS _Barfleur_ from April to July 1906 ( _Barfleur_ was the Flagship of the Rear Admiral of the Portsmouth Division of the Reserve Fleet), _Excellent_ , again, from July 1906 to January 1907 and HMS _Victory_ from February 1907 to April 1910.\n\nIn 1906 the Royal Naval and Military Tournament opened in Olympia in London. This tournament was a national entertainment with competitions, massed bands, musical rides and historic battles, but undoubtedly its centrepiece was the field gun run. This was a competition that had evolved after Royal Naval involvement in the relief of the South African town of Ladysmith30 during the Boer War.31 At that time naval personnel hauled guns from HMS _Powerful_ up to Ladysmith to defend the vulnerable town against the Boer attack. Edgar knew of this notable achievement; he had been in South Africa before the Boer War ended. In 1906, in Olympia, as HMS _Excellent_ 's Gunnery Trainer, he led his eighteen-man gun crew in a display of dismantling and reconstructing the heavy guns as they were hoisted over walls and bridges. This was no easy task. The guns used were 12-pounder 6cwt,32 the weight of the gun barrel was 896lbs, the carriage 350lbs and the wheels 120lbs, so the total weight of the equipment to be taken over the course was approximately 1 ton (1,016kg, over 2,000lbs).\n\nIn the following year, Edgar led his crew to victory in the field gun competition (as apposed to display).33 This took place during each afternoon performance throughout the tournament. It involved two crews vying with each other to initially haul the field gun and its carriage across the arena towards 'a wall', dismantling the gun to get it over the obstacle, reassembling the gun, and then getting it across a 28ft 'chasm'. For this the gun had to be dismantled again, some of the team swinging across the chasm carrying the 120lb wheels, the others sweating to get the carriage and gun barrel over the gap before they swung across themselves. The gun carriage was then put together again and raced towards a gap in the 'enemy wall' \u2013 too narrow for the gun to get through with its wheels on, so they had to be removed and put on again before a stiff contest to get to the enemy line, fire a round of ammunition and repeat the process in reverse to get back to the starting line. The team that crossed the original starting line first was awarded a point, and whichever team achieved the biggest number of points during the entire tournament was deemed victorious. The competition, accompanied by the virtually delirious shouts of support from loyal devotees, was hugely demanding and required relentless rehearsal, and a strong coordinated team. Here Edgar clearly excelled. The Royal Tournament, including the field gun exercise, continued until 1999 when it was axed, but currently there are plans to revive it.\n\nYears later he told another story about the gun run. Whilst the training was in progress he was stationed in Corfu for a short while. With six friends he hired a ramshackle four-wheeled cart for trip into the country. Having somehow deposited the driver, some brilliant 'idjit' thought that it was a good time to carry out a gun run practice. They broke up the cart, reasonably scientifically, into about twenty pieces and then charged back with all the bits. An Officer appeared, 'What is this tomfoolery?', 'Gun Practice, sir, dismounting and retiring with gear.' And then, Edgar concluded to his spellbound audience, 'We couldn't make the bleeding fool see how important it was and he sent us back to the ship without letting us set the gun up again.'34 The driver's reactions are not recorded.\n\nEdgar's five-and-a-half years of married life were full of action and excitement. In relation to the navy, in February 1906, King Edward VII launched HMS _Dreadnought_ in Portsmouth as part of the modernisation reforms instigated by Admiral Sir 'Jacky' Fisher. This was thought to be the most powerful warship in the world, a ship that would make all others obsolete. Later that year there was rioting in Portsmouth Barracks.35 This extraordinary and much publicised event was precipitated when young recruits were kept on the parade ground for an inordinate length of time in appalling weather. They decamped, without permission, to the gymnasium where the Duty Officer (a gunner) dismissed the lesser offenders but kept junior stokers back for a reprimand. Unfortunately he started with a short, curt order of 'on the knee', apparently a common command in the gunnery division and given so that men at the back could hear what was being said. But the order and the way it was given caused indignation amongst the stokers who responded with a resounding 'No'. The stokers' arrest and subsequent disquiet continued for days, with stokers ransacking the canteen before finally mutinying. The gates of the barracks were barred. There was no movement in or out. The event finally culminated in over 200 stokers being arrested and the Duty Officer and Senior Officers being relieved of their commands. After this, 'on the knee' was used for drill only. The mutiny would not have affected Edgar directly (except for him being unable to get into barracks), but the tension, uncertainty and excitement must have been intense and spread throughout the ranks.\n\nBut Portsmouth offered attractions beyond naval and domestic duties. Apart from enjoying a drink in the pubs, for the first time Edgar sampled the novelty of cabaret shows, sometimes going to the 'artistes' door to see and talk to the performers as they left the theatre.36 There were several theatres to choose from: The Hippodrome, The Royal, The People's Palace where performances were often twice nightly (so giving time to be home at a reasonable hour) and presenting 'all the latest novelties from America and the Continent... no expense is spared to present a bright and exhilarating entertainment'.37 There were entertainments on the pier, too, such as marine bands and vocal concerts.38 Lois wrote later that Edgar had been a good husband. But although he was an affectionate husband and father who worked hard, it was difficult to chain him to domesticity exclusively. It was still less than two years after his battles against Antarctica's worst excesses. Gregarious and outgoing, he needed new experiences. The field gun run was only one of these.\n\nHe may have been the father of two other children, twins, born a few months before his daughter Muriel. Beatrice Louise Pharoah was a teacher who, at the time of the twins' birth, lived in Sultan Road, about five roads along from Chapel Street, the Evans' home. Following Beatrice's history through several changes of name and address is a challenge. She was born in 1873, the daughter of a farmer, James Enos Pharoah. Her father died in 1877 and by the 1881 census Beatrice Pharoah is listed as the stepdaughter of Edward James Anderson and his wife Martha (although in fact the actual marriage of Edward and Martha does not appear to have taken place until 1883).39 Beatrice seems to have taken her stepfather's name and the 1891 census lists her as Beatrice Anderson, living with her now widowed mother in 298 Commercial Road, Portsmouth. Ten years later, in the 1901 census, Beatrice, still known as Anderson, was living with her mother in Lake Road, Portsmouth.\n\nBeatrice married Thomas Henry Glazier in 1902 (giving her name as Beatrice Louise Pharoah). Her husband died after just over two years of marriage. She was to marry again. In 1914, after Edgar's death, she married Charles James Amsden, recording Anderson as her father's name.40 When Beatrice's daughter married in 1928, she recorded her maiden name as Amsden, but withheld details regarding her father's name and profession (although her new husband supplied details of his own father).41\n\nBeatrice gave birth to twins, confusingly named Kathleen Lillian42 and Lillian Kathleen,43 on 31 July 1906 at Albert House, Albert Road, Cosham. Beatrice Louise Evans (formally Anderson) is recorded as the mother, the address as 39 Sultan Road. Edgar Evans, Petty Officer RN, is listed as father. This is possibly true. Beatrice was between marriages and Edgar was a virile, active young man. Lois, preoccupied with domestic commitments, her pregnancy and baby Norman, may well have been unable to satisfy him sexually. Whatever the truth Beatrice undoubtedly had a tough time with her twins. Lillian Kathleen (aged seventeen months) died at Sultan Road on the 23 January 1908. The father was registered as Edgar Evans, 1st Class Petty Officer Royal Navy, HMS _Victory_. The certified cause of death was 'Dentition' (presumably problems with teething) and bronchopneumonia.44 Beatrice was with her daughter when she died, but there is no mention of Edgar being present.\n\nIn 1907, the erstwhile Third Officer of _Discovery_ , Ernest Henry Shackleton, returned with his own expedition to Antarctica on _Nimrod_. Shackleton reached to within 97 geographic miles of the South Pole, having ascended onto the plateau via a momentous, 150-mile haul up a glacier, which he called the Beardmore after his principal financial backer. A group from his expedition reached the Magnetic South Pole and he was hailed as a hero. When Scott had first heard of Shackleton's intentions, he was back in the navy and marooned in the Atlantic. He wrote to Shackleton saying that he felt he had 'cut right across my plans' (for a return) and that he had the 'right to my own field of work in the way that Peary claimed Smith's Sound and many African travellers their particular locality'.45 He wrote that foreigners had 'conseeded [sic] that the sphere of the Ross Sea was English', surely therefore 'the English must admit the same argument to apply amongst themselves'.46\n\nWhen Shackleton failed to reach the Pole, Scott wrote that a sportsman is not jealous of his record or slow to praise those who surpass it, but there was surely _Schadenfreude_ behind these comments. Shackleton's failure left the prize open for him. His new expedition, the British Antarctic Expedition, was announced on 13 September 1909, and aroused immense public interest; 8,000 people applied to join. Preference was given to those with Antarctic experience and to members of the Royal Navy. Scott wrote to Edgar in March 1910, saying that he (Scott) had applied for Edgar's services on the proposed expedition and that he expected that Edgar would be appointed in two weeks. He wanted Edgar to be at the ship to help in fitting her out. Edgar decided to accept the offer and started working for the expedition in its headquarters in Victoria Street, London. Funding for the expedition was a huge task. The British Government, weary of requests for polar exploration, initially refused financial support. Also, Lloyd George's 'Peoples Budget' had significantly increased taxation on the wealthy, making them less likely to donate money to philanthropic causes. _Discovery_ , now owned by the Hudson's Bay Company, could not be got for a reasonable price so Scott bought the whaling ship _Terra Nova_ for his expedition paying \u00a312,500. Refitting and a thorough cleaning were needed (her hull was full of seal blubber and the stench was overpowering). Edgar was also fully occupied sorting the gear for the sledges. During the refit he shared lodgings in London near the dock.47 By now he was a big, burly man at 5ft 10in, and nearly 14 stone.48 He certainly made an impact on Griffith Taylor, the geologist of the expedition, when he visited expedition headquarters in 1910. Taylor said that Edgar, who was sorting the gear for the sledges, almost filled the room and that he (Taylor) looked at Edgar with considerable respect. Taylor's friend Charles Wright, a physicist, had just had his application to join the expedition turned down, so Taylor, fired up competitively by the sight of Edgar's sturdy proportions, decided to show that some of the scientists could at least 'walk' up to naval standards. The two walked 50 miles in twenty-four hours from Cambridge, to resubmit the application.49\n\nShould Edgar have accepted Scott's offer and left Lois and their three children? When Admiralty released him from the navy to join _Terra Nova_ , he and his colleagues were taken off the naval payroll, though they kept their place on the promotion list. Edgar's Certificate of Service records that his 'home' base, from 20 April 1910 until 17 February 1912, was HMS _President_ , when it finishes with the words, 'Lost in British Antarctic Expedition'. But the loss of naval pay meant that, unlike his _Discovery_ days, Edgar was dependent on expeditionary funds.\n\nThere were obvious reasons against him going. He knew there was no guarantee that he would ever get back to England, having had all too vivid experiences of the dangers of Antarctica. He knew his wife would need him, having seen his mother bringing up her children single-handedly when his own father was at sea. But the lure of Antarctica remained strong; he was not the only married man or father to go with Scott (who had only recently got married himself). He knew that the _Discovery_ expedition had brought him recognition well beyond the dreams of a lad from Gower, and he knew there would be a Pole attempt and thought he would be chosen for that party. Further recognition and fame beckoned. It was a chance worth taking. In the words of one of the _Terra Nova_ explorers:\n\nI hear the white wastes calling\n\nAcross the restless seas\n\nCivilization's palling\n\nThe wanderer's disease\n\nI wish that I could once again\n\nAround the cooker sit\n\nAnd hearken to its soft refrain\n\nAnd feel so jolly fit50\n\nThe die was cast.\n\nHe certainly did not foresee that Lois was to face difficult financial circumstances. He had signed on for extra years to secure a naval pension. And his family supported him. After he had died his father-in-law (whose loyalty might be expected to be directly focused on his daughter and her struggles and conversely stretched towards his son-in-law) said, 'He was a fine boy. He was a good husband and a good son to his old mother.'51 Indeed, Lois did not falter in her loyalty.\n\nEdgar's father Charles had died in 1907 and before Edgar left, he and Lois returned to Gower. Scott had decided that Cardiff would be the place of departure for the _Terra Nova_. He visited his mother, now living in the village of Pitton with her sister, and then walked 16 miles to visit his older brother, Charles, at Cwm Farm, Sketty. This visit was made memorable to one of his nieces when Edgar swept her up in his arms and promised to visit her again on his return. When he left, Lois and the children stayed on with her sister in Cardiff.52\n\nHe told Lois he would be back with her and the children in a year. But the parting was sad and was to be final. She would never see him again.\n\n### Notes\n\n1 Ed. Savours, A., _Edward Wilson, Diary of the Discovery Expedition 1901\u20131904_ , Blandford Press, London, 1966, p. 331.\n\n2 Ford, C.E., Journal 10\/12\/03\u201314\/03\/04, SPRI, MS 1174; D, 02\/02\/1904.\n\n3 Williamson, T.S., Journal 1901\u20131904, SPRI, MS, 774\/1\/2; BJ. 10\/01\/1904.\n\n4 Ford, C.E., Journal 10\/12\/03\u201314\/03\/04, SPRI, MS 1174; D, 10\/01\/1904.\n\n5 Ibid., 02\/02\/1904.\n\n6 Ibid., 14\/02\/1904.\n\n7 Williamson, T.S., Journal 1901\u20131904, SPRI, MS, 774\/1\/2;BJ. 13\/02\/1904.\n\n8 Ibid., 07\/03\/1904.\n\n9 Ibid., 07\/03\/1904.\n\n10 Ibid., 02\/07\/1904.\n\n11 _Times_ , Editorial, Saturday 10\/09\/904, Issue 37496, p. 9.\n\n12 Unknown newspaper clipping. SPRI, 24\/04\/1906.\n\n13 Unknown newspaper clipping. SPRI, 12\/09\/1904.\n\n14 Unknown newspaper clipping. SPRI, 12\/09\/1904.\n\n15 Yelverton, D., _Antarctica Unveiled, Scott's First Expedition and the Quest for the Unknown Continent_ , University Press of Colorado, Colorado USA, 2000, Appendix 8.\n\n16 Ibid., Appendix 8. The Royal Geographical Society Medal was awarded in 1913.\n\n17 _Gower Church Magazine_ , January 1905.\n\n18 A mountain or peak sticking up through an ice sheet.\n\n19 _South Wales Daily Post_ , 20 September 1904.\n\n20 Information from Royal Naval Library, Portsmouth.\n\n21 _South Wales Daily Post_ , 14 December 1904.\n\n22 Ibid., 14 December 1904.\n\n23 _Gower Church Magazine_ , January 1905.\n\n24 _South Wales Daily Post_ , 14 December 1904.\n\n25 Ibid., 14 December 1904.\n\n26 _Gower Church Magazine_ , January 1905.\n\n27 Ibid., January 1905.\n\n28 Multimap.com. Postal code PA2 8PJ.\n\n29 Information from the Local History Section of the Portsmouth Library.\n\n30 A town in Natal besieged by the Boers between 2 November 1899 and 28 February 1900. Relieved by Sir Henry Buller. Named after the Spanish wife of the Governor of Cape Town.\n\n31 Boer War, 1899\u20131902.\n\n32 Cwt is a hundredweight, 50.80kg.\n\n33 Personal communication, 2010, Lieutenant Commander Brian Witts, Curator, HMS _Excellent_ Museum, Portsmouth.\n\n34 Debenham, F., _Journal_ , 19\/01\/1911\u201308\/03\/1911 MS 279\/2: BJp p. 56.\n\n35 04\/11\/1906, the 'On the Knee Mutiny'. The most serious and widely publicised breakdown in naval discipline. It was confined to the Royal Naval Barracks, Portsmouth.\n\n36 John Evans, grandson, personal communication, 2010.\n\n37 _Portsmouth Guide_ , The Hippodrome, Commercial Road, Portsmouth, 1907. (Prices 3 _d_ , 6 _d_ , 1\/-, 1\/6, 2\/6, 10\/6, 15\/-).\n\n38 Ibid., p. 93.\n\n39 Information from the Museums and Records Service, Portsmouth City Council, 2010.\n\n40 Certified Copy of an Entry of Marriage, Registration District Portsmouth, 23 September 1914, TE 159326.\n\n41 Certified Copy of an Entry of Marriage, Registration District Portsmouth, 3 May 1928, TE 159292.\n\n42 Certified Copy of an Entry of Birth, Registration District Fareham, 31 July 1906, No. 269, CJ 735173.\n\n43 Certified Copy of an Entry of Birth, Registration District Fareham, 31 July 1906.\n\n44 Certified Copy of an Entry of Death, Registration District Portsmouth, 23 January 1908. HC 326019.\n\n45 Scott, R.F., _Letter to Ernest Henry Shackleton_ ,18\/02\/1907, SPRI, MS 1456\/23: D.\n\n46 Ibid., Undated letter, but soon after 18\/02\/1907.\n\n47 Cheetham, A.B., _Letter to Mr and Mrs Brewer_ , 14\/02\/1913, SPRI, MS 1365\/1\/1\u20132: D.\n\n48 Debenham, F., _Journal_ , 19\/01\/1911\u201308\/03\/1911 MS 279\/2: BJp p. 17.\n\n49 Taylor, G., _Letter to H.S. Richards_ 11\/06\/1962, Swansea Museum. Wright was successful.\n\n50 Cherry-Garrard, A., _Diary, The Barrier Blight by One Who Has Not Had It_ , SPRI. MS 559\/4; BJ.\n\n51 _South Wales Daily Post_ , 11 February 1913.\n\n52 Johnson, A.M., _Scott of the Antarctic and Cardiff_ , The Captain Scott Society, Cardiff, 1995, p. 23.\n\n## 11\n\n## Terra Nova\n\nThe _Terra Nova_ expedition lasted from 1910\u201313. The ship was to return to Cardiff without Edgar Evans and his four companions, Robert Falcon Scott, Edward Wilson, 'Birdie' Bowers and 'Titus' Oates. The five men died in Antarctica in 1912.\n\nThe journey to Antarctica was full of incident for Edgar. He came near to dismissal in New Zealand and _Terra Nova_ came near to disaster soon after leaving her last port of call. If Edgar had retired from Antarctic challenges after the _Discovery_ expedition he would have disappeared gently into the quiet backwaters of historical oblivion. But he chose to follow Scott again and so became, after his death, nationally famous and, to an extent, nationally defamed.\n\nHe had an unfortunate start to the expedition. Scott had chosen Cardiff, the capital of Wales, as the point of departure simply because the Welsh had offered generous financial and practical support to the expedition. For their part, the dignitaries in Cardiff were quick to appreciate the publicity and commercial advantages offered to their city by the expedition, Cardiff had only been granted city status by Edward VII in 1905, and national exposure was a bonus. In Cardiff, Scott and his officers were invited by the Cardiff Chamber of Commerce for a farewell banquet at the Royal Hotel (fillets of beef _Terra Nova_ , souffl\u00e9 Captain Scott, South Pole ice pudding).1 The crew were entertained at a nearby hotel (unfortunately no record of that menu remains), and after the meal the men were invited to join the officers. Scott requested that Edgar, the local South Wales hero, who had been lionised in Cardiff,2 should sit between him and the Lord Mayor of Cardiff. No doubt Edgar's glass had been well filled throughout the evening when he rose to his feet to give an impromptu, but effective speech. The event was reported fully by _The Cambrian:_\n\nEdgar Evans' Cardiff Speech\n\nAbertawe Boy who is Southward Bound\n\nBreezy Speech at Cardiff Banquet\n\nCaptain Scott C.V.O and the officers of the British Antarctic Expedition vessel, Terra Nova, were entertained to dinner on Monday evening by the commercial community, the President of the Cardiff Chamber of Commerce in the chair. The crew were also entertained to dinner and \u00a31,000 for the funds of the expedition were collected at the former ceremony, at which an event of special interest to Swansea also occurred, when the Lord Mayor presented to the Expedition a banner emblazoned with the arms of Cardiff.\n\nChief Seaman Edgar Evans of Swansea, one of the biggest and burliest members of the crew, was received with three times three as he rose from his seat between the Lord Mayor and Captain Scott. With the typical modesty of a Jack Tar and an unmistakable West Wales accent, he said;\n\nI think it's out of place for me to sit up here with Captain Scott, but like Lord Charles Beresford, whatever I have to say I'll say it in as few words as possible. (cheers)\n\nEvery man in the ship has confidence in Captain Scott. I know him well and he knows me very well\u2014(laughter)\u2014and I know Lieutenant Evans very well. (cheers)\n\nEvery man in the Expedition is heart and soul in the business, and it has got to be a success this time\u2014 (cheers)\u2014every man will do his best.\n\nAs a representative of Wales I am pleased to meet you all, but whether Wales or Ireland if Captain Scott had only said he was going again I would go too. (cheers) No one else would have induced me to go again, but if there is one man in the world who will bring this to a successful issue, Captain Scott is the man, (renewed cheers)\n\nAs regards the flag, if Captain Scott wants to know the English translation of the Welsh mottoes, here it is: 'Awake, it is day' and 'The Welsh dragon leads the van'.\n\nThe crew appreciate what you have done for them; I hope we shall meet again\u2014and we shall. (Cheers) Of course that depends on Captain Scott bringing back the Pole, (loud laughter). We cannot put it in the museum, but if we do bring it back I hope you will let it go to Swansea. (loud cheers and laughter)\n\nEveryone has great confidence in Captain Scott and Lieut. Evans, and if we do ever come back we hope to meet you again in Cardiff again. (loud cheers)3\n\nIt was a big occasion for Edgar; the dark, oak-panelled room glowed in the candlelight. The great and the good of Cardiff paid him attention, their badges of office shining. His officers, their medals gleaming on their chests, listened. He was bedazzled. He was excited. He was plied with drink. No wonder he got more than well lubricated.\n\nIt took six men to get Edgar back onto the _Terra Nova_.4 He was not aggressive,5 just incapable. His niece said later that he did tend to drink too much on occasions; she thought it was understandable considering his hazardous career.6 He was also, almost certainly, not the only man worse for wear on this occasion. Sixteen months later, Lieutenant Henry Bowers wrote to Kathleen Scott when the British team were leaving for the Pole, saying that he was glad that by then the men would have had a certain amount of experience which would be of help in the approach to the job, with \u2013 in the case of some of them \u2013 'a little less of that spirit that did not do us credit on our departure from Cardiff.'7 Yet this story is repeated whenever Edgar Evans' story is told.\n\nThe dinner was actually a great success financially. The \u00a31,000 collected towards expedition funds was of great importance, because it went some way to relieving Scott of the embarrassment of departing from the United Kingdom still unable to guarantee the wages of some of his officers and men.\n\nThe _Terra Nova_ was the trusty whaling ship, the unexpected relief vessel that had accompanied _Morning_ in Antarctica in 1904. Now she sailed under the white ensign rather than a merchant flag because Scott had been elected as a member of the Royal Yacht Squadron and she was registered as a yacht under Scott's name.8 The expedition was expensive, about \u00a350,000, mainly because of the personnel numbers, sixty-five men, eleven of whom were scientists or doctors. There were three motorised sledges that Scott hoped would be of use in pulling supplies across the Barrier, sixteen Siberian ponies, dogs, equipment and supplies for several years in Antarctica.\n\nAlthough after his death the _Western Mail_ printed a photograph of Edgar and Lois aboard _Terra Nova_ in Cardiff,9 Lois did not go to see Edgar off on 15 June. She had three children under five, the youngest only eighteen months, and she may have thought that the pressure of seeing him off from the dock would be too much. It was probably a wise decision; a truckload of coffins clearly visible to the crew hardly encouraged cheerfulness.10 But his niece, Sarah Evans, travelled from Swansea for the departure which was viewed by thousands. Scott was clearly not fazed about Edgar's performance the night before. Sarah recalled that he called Edgar into his cabin to receive the Mayor of Cardiff's final good wishes and that he (Scott) gave her a sledging biscuit as a memento.11 Edgar was clearly back to his normal self; described as 'burley Chief Seaman', he hoisted Cardiff's flag to the foremast. As the breeze shook out the flag to show its Welsh dragon 'rampant and confident', the crowds 'burst into a mighty roar', particularly when a Welsh leek fastened onto the masthead joined the Cardiff flag.12 Sarah was on a steamer following _Terra Nova_ as, surrounded with tugs and pleasure steamers, the ship started down the Bristol Channel. When the last tugboat departed, Edgar is reported to have said in a 'thunderous' whisper; 'Goodbye we shall always remember you'.13 Members of Edgar's family waved from the Gower cliffs at Rhossili as _Terra Nova_ made her way towards the open sea. They would never see their man again.\n\n_Terra Nova_ followed the same route as _Discovery_ had taken but she did berth in Melbourne rather than bypassing Australia: she sailed via Madeira, South Africa, Australia and New Zealand to Antarctica. She left England without her Captain; Scott remained behind to continue fundraising engagements and the journey to South Africa was commanded by _Discovery_ 's Second-in-Command, Lieutenant Edward 'Teddy' Evans. Edgar knew several of the people on board: Chief of Scientific Staff was the erstwhile Junior Doctor and Zoologist on _Discovery_ , Dr Edward Wilson. There were others who had been on that expedition: Petty Officers Thomas Crean, Thomas Williamson and William Heald, and Chief Stoker Lashly. Other crew members had journeyed south before: Lieutenant Evans had been second-in-command of _Morning_ , a trip that irrevocably whetted his appetite for Antarctic exploration. Engineer Bernard Day had been on the _Nimrod_ expedition with Shackleton in 1907. Bosun Alf Cheetham was the veteran of Antarctic travel; he had also been on _Morning_ 's relief trip in 1903 and with _Nimrod_ on Shackleton's 1907\u201309 expedition.14\n\nThe _Terra Nova_ carried an ambitious team. Although the South Pole was high on the list of priorities, Scott was also determined to undertake a big scientific programme that would add academic status to the venture. Edgar's officers, apart from Scott and Lieutenant Evans were: Henry Robertson Bowers, a Lieutenant in the Royal Indian Marines, a man whose beaky nose quickly earned him the nickname 'Birdie'. He was short, 5ft 4in, stocky and full of energy \u2013 'the hardiest traveller that ever undertook a polar journey'. Lieutenant Wilfred Montague Bruce, Royal Naval Reserve and Kathleen Scott's brother, travelled to Vladivostok to meet Cecil Meares, the man appointed to choose the sledge dogs and give Meares assistance in getting the dogs and the ponies, (which Meares also selected), from Russia and across land and sea to join up with _Terra Nova_. Scott was determined to get the best use possible from skis on this expedition, so Tryggve Gran, a Sub-Lieutenant in the Norwegian Navy, was appointed as ski expert. The First Officer was Victor Campbell, 'The Mate' often 'The Wicked Mate'. Stories about him were many; Gran said he never liked being on watch with Campbell because he (Campbell) turned him into a drumstick (a domestic).15 Laurence Titus Oates, a Captain in the Inniskilling Dragoons, was in charge of the ponies. He was destined for Antarctic heroism and contributed \u00a31,000 to Scott's coffers. The Commander was Harry Pennell (Pennylope). He was the navigator and was also in charge of magnetic work. He was considered one of the most competent members of the expedition. Finally there was Lieutenant Rennick, who looked after the hydrographical work and deep-sea sounding.\n\nThe scientific complement was the biggest ever to travel to Antarctica. Dr Wilson's prot\u00e9g\u00e9 Apsley Cherry-Garrard was a biologist. Cherry-Garrard was an immensely wealthy young man who also gave \u00a31,000 to the expedition and who would later write _The Worst Journey in the World_ about his experiences. Many other members of the scientific staff would go on to international fame: Raymond Priestley and Australians Thomas Griffith Taylor and Frank Debenham were the geologists and Canadian Charles Wright the physicist. George Simpson was the meteorologist who was to make pivotal observations and conclusions on wind and weather conditions of Antarctica. Dennis Lillie and Edward Nelson were biologists and there were two surgeons, Edward Atkinson and George Levick.\n\nHerbert Ponting was the famous 'Camera Artist' who photographed the crew, including Edgar. His photographs and film, which were to prove invaluable for advertising and providing funds for the expedition, remain prized collectable items to this day. The motor expert was Bernard Day. He had had experience with the motors used on Shackleton's expedition.\n\nThirty-three of the company were assigned to the shore parties16 and Edgar was in this group. Edgar's letter to his mother in the early part of the voyage said that he was well, that he had seen Sarah and two of her sisters in Cardiff, but that other cousins had not turned up. He asked his mother to write to him in Cape Town. He sent his love to his aunts and uncles 'and anyone that enquires'. He was her 'Ever loving Son'.17\n\nOn board, the men, in cramped accommodation and working conditions, getting to know each other slowly. They worked at the bilge pumps (in spite of all efforts _Terra Nova_ leaked and the men had to spend about half an hour each watch pumping out seawater). Her rolling was enough to make some of the men seasick. Edgar did not start off well with his temporary Captain, Lieutenant Teddy Evans. He had apparently spotted an error over the ordering of some ski bindings, a subject he had good experience in, and he reported this error to Scott. Teddy Evans, who had made the order, was not pleased, particularly when Edgar was put in charge of the ski equipment rather than himself.\n\n'Crossing the line' was on 15 July and Edgar was now experienced at the ritual. He was Neptune. The show began on the 14th, when Neptune's messenger, Triton, arrived to announce that Neptune and his Queen, Amphitrite, would be visiting the ship. Resplendent in his royal robes Edgar appeared, supported by Queen Amphitrite, the strapping Petty Officer Frank Browning. Their followers were the doctor (Seaman James Paton), a barber (Bosun Alfred Cheetham), a barrister (another Petty Officer, George Abbott), all ably helped by Captain Oates and Lieutenant Atkinson as bears. Edgar's friend, Tom Crean, and Petty Officer Thomas Williamson were the policemen.\n\nThe Clerk (Abbott), read Neptune's address to the Ship's company, which showed Edgar's quick wit. After welcoming his audience to Father Neptune's domain and stating that unwilling initiates would be attended to by Neptune's stalwart policemen, the clerk went on to wish the expedition every success. But he commented on the ship's bad leak (perhaps due to cargo shifting so often, or perhaps the tuneless singing of Shanties at the pumps?) and said that it was tough that the stiff breezes he had sent, had veered round so that they travelled straight up the hawse pipes (the area on the ship's bow that the cables go through), causing the sails to have to be furled in the middle watch (the watch between midnight to 4 a.m.). He asked if the ship was towing a sea anchor or whether the rudder was athwartships (lying across) and checking progress, because the ship moved so remarkably slowly. He presumed, 'she would go a long way in a long time'.18\n\nThe actual initiation, which followed Edgar's advice that the 'Main Brace' should be spliced (drink should be issued), was predictably violent. The doctor inevitably 'prescribed' a pill, a gobstopper of soap and tallow, washed down by a mixture of vinegar and cayenne to every initiate. This was followed by lathering from a bucket of whitewash and another of soot, a shave with a 3ft wooden razor and then the drop into the bath 12ft below. The first to go (Nelson) pulled the 'barber' into the bath with him. Others tried to fight Neptune's assistants in spite of the warning they had been given and most were overcome by the 'police' Crean and Williamson. But Gran (Norwegian, so perhaps unfairly included) chucked the 'doctor' over his shoulder into the bath and was 'lathered very gingerly after that'.19 The captain decorated Father Neptune with the Grand Cross of the Victorian Order. The Main Brace was promptly spliced with port wine (drinks all round). A concert followed, but to some of the disappointed crewmen the atmosphere was still 'somewhat dry';20 they wanted more in the alcoholic line.\n\nFollowing _Discovery_ 's trail, _Terra Nova_ called on South Trinidad, one of the fabled Treasure Islands,21 great rocks and corals, many sharks swimming around and with one accessible shore. Edgar recorded the birds again: Terns, Petrels, Gannets. The stretch down to South Africa was helped by strong westerly winds, which allowed _Terra Nova_ to fly along under full sail and arrive on 15 August, (though still fifteen days overdue), in the Naval Base of Simon's Bay. She had travelled over 7,000 miles. In South Africa good will abounded, the locals were as hospitable as ever. The ship was repainted in the dockyard. Other ships sent gifts of bread, eggs, and fresh meat. Edgar had reason to be proud of _Terra Nova_ when she left Simon's Bay after eighteen days with Scott now in command. He wrote to his mother cheerfully, saying that the plans were for them to leave for the ice in December and saying he had received a letter from his brother, Charlie, 'quite a spasm for him wasn't it?' Again he signed himself her 'ever-loving son' and he sent his regards to all the relations.\n\nIllustrating a perennial worry of all expeditions, Stoker Lashly reported problems with his teeth. He asked Dr Atkinson to pull some out and, in a way that makes the modern reader blanch, Drs Atkinson and Wilson attacked the problem. They had six goes at the first tooth, a tooth that remained as tightly in its socket when they gave up as when they started. Then they pulled out three others, breaking two during the process. Afterwards Lashly wanted them to have another go at the first.22\n\nAs they sailed onto Melbourne, the voyage was characterised by the usual delights and discomforts of seaboard life: _Terra Nova_ rolled vigorously, sometimes the lee rail was well under water and the sea flooded along the starboard side and into the laboratory and sleeping quarters. By 2 October, the wind had fallen. The crew postulated that they were travelling in front of a storm system that was moving at about 150 miles a day. They hypothesised that if they had been in a sailing ship without steam, the cyclone would have caught them and they would have been in continually bad weather. The fact that _Terra Nova_ had auxiliary steam meant that they could keep ahead of the storm. They speculated that this could explain the reports of particularly bad weather encountered by sailing ships on that latitude.23 Lieutenant Evans wrote that he and Scott had definitely selected Edgar to be one of the seaman selected for the shore party (with three Irish companions: Robert Forde, Pat Keohane and Tom Crean).24\n\nOn 12 October Melbourne was reached. A telegram was waiting for Scott, one that outraged the loyal Edgar. The telegram read: 'Beg leave to inform you, _Fram_ [Amundsen's ship] proceeding Antarctic. Amundsen.' The race to the Pole was about to begin though the British did not at first appreciate this; their initial reaction was muted, they did not think that Amundsen would go to the same part of Antarctica as themselves. Roald Amundsen, (1872\u20131928), the famous Norwegian explorer, had left Norway with the avowed intent of travelling to the Arctic basin and the North Pole. But this was a ruse and he kept the truth from everyone except his brother. Amundsen needed a coup and his South Pole ambitions were hatched when two American explorers, Frederick Cook and Robert Peary, both claimed independently to have reached the North Pole. Geologist Raymond Priestley later remembered Amundsen's conquest of the South Pole as 'the greatest geographical impertinence that history records'25 (Priestley's opinion was that Amundsen's expedition was to make money). If Scott had known earlier that there would be rivalry in the Antarctic he might well have modified his plans which were not based on speed, but he only fully appreciated the competition risks months later when Amundsens's ship was found by chance, just miles along the Barrier from Scott's base. By contrast, Amundsen, fully aware of Scott's plans, knew that swift progress was essential for the Norwegians to get to the Pole first.\n\nThe British expedition plans continued. The Australian Government contributed \u00a32,500 to the expedition. Marconi, the radio pioneer, offered wireless assistance (refused, because the equipment was too bulky),26 and on 28 October, _Terra Nova_ sailed on to another warm welcome at Lyttelton, the port of Christchurch, New Zealand, nine years after Edgar's visit on _Discovery_.\n\n_Terra Nova_ was in New Zealand for a month. Her persistent leak was attacked again and reduced to a degree that the hand pump could control it in two daily sessions of a quarter of an hour. The stores were unpacked and repacked with 'Birdie' Bowers in capable charge. Each item was marked with a red or green band depending on whether it was designated for Scott's Main Party or an Eastern Party that, it was planned, would investigate the land east of the Barrier. The men practised assembling the prefabricated huts. Their living space on the main deck was horribly overcrowded, but Edgar, as the men's spokesman, requested that their comfort should not be considered. He said that because there was such a need to squeeze in extra supplies, 'they were prepared to pig it anyhow'.27 The ponies and dogs were quarantined on Quail Island before being taken on board. The New Zealand press was enthusiastic, writing that 'Our American cousins have discovered one of the Poles and the record of British exploration will be fittingly crowned if the expedition succeeds in planting the Union Jack on the other'.28\n\nThe sailors enjoyed Christchurch; some of them enjoyed themselves in the traditional naval way; drink and women. On this occasion Edgar definitely disgraced himself by going on a drinking spree. Before the ship departed for Port Chalmers for coaling on 26 November, and after the Bishop of Christchurch had blessed the ship, Edgar, drunk, fell into the harbour whilst getting on board. Although Scott seemed to have taken the Cardiff episode in a matter-of-fact way, this second episode was different. The expedition had been disgraced publicly. He dismissed Edgar.\n\nWhy did Edgar do it? He may simply have wanted the satisfaction of a last, good and prolonged, drinking bout. It may equally have been a guilt reaction. New Zealand was his last link with civilization to be followed by irrevocable separation from his dependent wife and family. He may well have wanted to blot out these and other disagreeable thoughts. Either way dismissal would have been a disaster. He had come off the naval payroll to join _Terra Nova_ , so the loss of expeditionary pay would have been a big financial blow for his family. Probably the financial loss would have not been permanent. Edgar was still officially on the naval list and it is likely that after such an episode, he would have faced disciplinary action in Portsmouth and probably disrated. But he would have been on reduced naval pay and the shame to his family would have been considerable\n\nWhen he was sober Edgar went to Scott, who was still in Lyttelton, to apologise and to ask Scott to reconsider his decision. After initial resistance Scott relented and the two men travelled in the same express train to Port Chalmers, Edgar acting as if nothing had happened.29 Scott's decision annoyed Lieutenant Evans. Teddy, who was still unhappy about Edgar's promotion to ski master.30 Teddy also thought that the reinstatement was bad for discipline. But Scott's loyalty and affection for Edgar was genuine. Edgar was a member of the Guarantee Party. They had covered miles of Antarctic wasteland together and gone through conditions that Lieutenant Evans could only imagine; Edgar had been tried and trusted in the worst Antarctic circumstances. Scott did not want to lose his talisman.\n\n_Terra Nova_ left Lyttelton on 29 November. There was the usual excited send off; special trains were put on so that people could watch the departure \u2013 all the ships in the harbour were decorated. Cherry-Garrard wrote there was 'a general hullabaloo'.31 The ship was dangerously overloaded, her deck like a floating farmyard: there were nineteen ponies, all swaying continuously as the ship lifted up and down (Scott has specified that he wanted white ponies only because on Shackleton's expedition, the dark ponies had died before white ones), thirty-three dogs (presented by schools from all over the country, which barked and snarled and strained at their chains), two cats, two rabbits, a pigeon, squirrels and a guinea pig. In addition the deck groaned with 'thirty tons of coal, 2,500 gallons of petrol, some tons of pony fodder and petroleum'. _Terra Nova_ also carried '162 frozen sheep and three bullocks'.32\n\nIn addition there were three caterpillar-track motor sledges. These were potentially a huge innovation in Antarctica. Shackleton had taken a motorcar on the 1907 expedition and thought that motorised transport was feasible in Antarctica.33 Scott went further; he was the first to pioneer motor sledges in Polar conditions, a possible development that worried Amundsen. There were innumerable sacks of coal and stacks of petrol cases. Mutton from the New Zealand farmers found a place in the icehouse along with three carcasses of beef and boxes of sweetbreads and kidneys.\n\nThe seas through which they passed to reach the pack ice are amongst the stormiest in the world. Dante wrote that those who have committed carnal sin are tossed about ceaselessly by the most furious winds in the second circle of Hell, and this is how it appeared to one of the officers as _Terra Nova_ pitched and plunged about in a force 10 gale for thirty-six hours.34 Edgar understood, all too well, the implications for the overloaded ship. As the waves broke with increasing fury over the deck, the ponies began to fall over, the coal loosened petrol cases and the chained dogs were thrown to and fro by each successive wave.\n\nThe hatches were battened down, but by 4 December the ship had slowly filled with water. The crew tried unsuccessfully to stop the mountainous waves that washed all over the deck by pouring oil overboard. Coal sacks became battering rams and loosened the petrol drums (150 gallons were lost).35 The ship's violent tossing opened the deck seams and allowed coal dust to pour into the bilges (the part of the ship below water level where the sides curve towards the keel) and, in spite of the clean up, the dust mixed with blubber from _Terra Nova_ 's previous occupation as a whaler. Geologist Priestly wrote later that, in addition to the blubber, one of the sailors must have spilt a barrel of oil in the mainhold which also got mixed with the coal and formed into coal balls 'about the size of composition cricket-balls and these had blocked the pipes leading from the pump'.36 Lashly worked for hours, unsuccessfully, to try to clear the pipes. The boiler fires had to be closed down; if water got into contact with the boilerplates the boiler would buckle and become useless for further steaming. The engine driven pump was shut down. The ship was at the mercy of the sea as the men worked on furiously, clinging to the rails and up to their waists in water. Officers, scientists and men formed a chain gang and bailed for their lives for twenty-four hours, as the wind raged up to 72mph. Though the men knew that they were dependent on each ship's plank staying firm, they still sang sea shanties (that helped the rhythm of heavy bucket passing) that could be heard above the roar of the waves. When allowed a rest period, Edgar threw himself into his hammock and slept, oblivious of the pitching and rolling of the tortured ship.\n\nFinally the engineers managed to cut a hole in the bulkhead,37 so that Lieutenants Evans and Bowers could crawl to the hand pumps and pull out those lumps of oily coal dust. Often working under water, the two finally managed to get the hand pumps working. At last the storm subsided, the water level gradually receded, the fires were relit and the ship pumped dry. It had been a near miss. Two ponies and one dog died, tons of coal had been lost overboard along with 100 gallons of petrol.\n\nThe ship reached the pack ice, ominously further north than expected, on 9 December. Although Scott had thought that _Terra Nova_ was large enough to make an attempt at getting through the pack early,38 and though she could and did butt away at the heavy ice, she had nothing of the power of modern icebreakers and eventually took a month to get through. Sometimes progress was limited to one knot at full power, so to conserve coal the fire was put out. But every day spent pushing through ice had an effect on the timetable, 'Truly getting into our winter quarters is no light task; at first the gales and heavy seas and now this continuous fight with the pack ice.'39 By 23 December the coal supply was down to 300 tons.40 The 25 December, in the pack, was altogether too Christmassy for Scott, but the day was celebrated with a church service, Christmas hymns and lusty singing. The Men had mutton for the celebration lunch (they thought that penguin was not good enough for Christmas),41 plus beer and whisky. Crean's rabbit gave birth to seventeen babies.\n\nFinally, on 30 December, _Terra Nova_ escaped from the ice pack. Victoria Land in all its mysterious, pristine, majestic beauty could be seen about 60 miles ahead. Mount Sabine and the Admiralty Range looked glorious. The high snow peaks were lit by the sun and looked as if they lay over the clouds, like a layer of white satin. On New Year's Day the watch sighted Mount Erebus. Scott headed for his intended base camp, Cape Crozier, but found it impossible to land because of a heavy northerly swell, so _Terra Nova_ steamed directly to the Skuary, a rocky cape just north of the ice edge, renamed Cape Evans in honour of Lieutenant Teddy Evans, Scott's second in command. The base was 12 geological miles north of Hut Point,42 meaning that later on in the expedition the men would have an additional 12 miles to sledge. Ice anchors were let down and unloading began.\n\nThe first task was to build the accommodation in which the men would spend the winter. The ship was moored 1.5 miles from the landing place and all the stores had to be ferried by sledge across the pack ice. In the disembarkation one of the motor sledges disappeared through the ice. These expensive experiments had cost \u00a31,000 each; the loss was equivalent to Edgar's wages for thirty years.\n\nEdgar worked tirelessly; he helped unload the ship, build the hut, 10ft above sea level,43 check the sledges and assist Dr Wilson in the bloody occupation of killing and preparing seal carcasses for the larder. Scott thought he was impressively competent. He had no doubt that the (non-motorised) sledges that Edgar had fitted would work well. The hut was triple-walled and heavily insulated, with seaweed quilting on the roof. It was divided by a wall of packing cases, with scientists and officers on one side, men on the other. The arrangement has been criticised, with detractors saying that the division demonstrated Scott's over reliance on naval hierarchy and his discomfort in being with men of a different social class. This view can be challenged. Scott had already shown on the _Discovery_ expedition how he could exist easily with the seamen; in fact one of his attractive characteristics was that he was comfortable with all classes. But what of the sailors? Their whole upbringing and education had schooled them against social integration. Their cultural connections and framework were different. They would have found enforced intimacy with the officers and scientists an unwelcome constraint on their behaviour. They needed a safety valve, a separate unit to let off steam. This was shown later; when Edgar was on a sortie with three officers, he never swore in front of them. But when they returned to base he reverted to his normal vocabulary. When Debenham heard him through the partition he said; 'that sounds like Taff but it can't be \u2013 he never talked like that with us'.44 Edgar would not have wanted to be curbing his tongue full-time.\n\nHe wrote to his mother on 3 January 1911. Headed \"' _Terra Nova_ \" Cape Crozier, Victoria Land', he mentioned the bad weather and the pack ice, but only in passing. He was certainly no moaner. He said he expected to be in Antarctica for about fifteen months. He stamped the letter with an Antarctic stamp, marked 'Victoria Land', a unique curiosity, he thought. He asked his mother to keep it. He mentioned his wife; she probably had a job getting the children back to school. He sent his love to all.\n\nThey had brought a farmyard with them, but now it was even bigger. In addition to the ponies, dogs, rabbits and cats, Skua gulls nested and fought over seals and squawking penguins.45 They all wondered where _Fram_ was. Probably Meares was the only one to voice the horrid suggestion that if Amundsen was on the ice near them, he could go straight for the Pole.46\n\nAfter a week's hard sledging and the cargo over the ice, the equipment and supplies were well stored at Cape Evans. Scott planned a series of sorties before the winter: a depot-laying party in preparation for the attempt on the Pole the following year and two other expeditions \u2013 an Eastern Party that would carry out scientific and surveying work in King Edward VII Land and a Western Party which would carry out a similar mission in South Victoria Land. Edgar was a member of the second party. Scott's depot-laying party was to have far-reaching effects; there was near disaster as some members avoided death by a hair's breadth, and of the eight fittest ponies that had been taken on the trip five died, a misfortune that Scott was ultimately to claim contributed to the deaths of the Polar Party.47\n\nThe plans were in action.\n\n### Notes\n\n1 The oak panelled room where the dinner took place remains today and a seven-course dinner is recreated accurately and enjoyably by the Captain Scott Society of Cardiff on the 13 June every year.\n\n2 Richards, S., _Letter to Mr Pound_ relating his discussion with Sarah Owen (Evans) about Edgar getting tight. 18\/06\/1965, Swansea Museum, Box 210, PO Edgar Evans.\n\n3 _The Cambrian 17_ \/06\/1910.\n\n4 Copy of letter from Stanley Richards dated 18\/06\/1965 concerning his conversation with Edgar's niece, Sarah Owen who recalled Edgar's condition after the reception. Royal Institution Swansea.\n\n5 Richards, S., _Letter to Mr Pound_ relating to Edgar's drunken episode, 10\/06\/1965, Swansea Museum, Box 210, PO Edgar Evans.\n\n6 Ibid.\n\n7 Gwynn, S., _Captain Scott_ , The Golden Hind Series, London, 1930, p. 204.\n\n8 Ibid., p. 165.\n\n9 _Western Mail_ , 12\/02\/1913.\n\n10 Bowers, H.R., _Letter to Edith Bowers 07\/\/06\/1910_ , SPRI, MS 1505: D.\n\n11 Gregor, G., _Swansea's Antarctic Explorer, Edgar Evans, 1876\u20131912_ , Swansea City Council, 1995, p. 33.\n\n12 _The South Wales Times_ , ?\/06\/1910. Newspaper Clipping, SPRI.\n\n13 Ibid.\n\n14 Having served on _Morning_ and _Nimrod_ , Cheetham continued his Antarctic service when he went with Shackleton on _Endurance_ , (1914\u20131916). He was drowned when his ship was torpedoed in 1918.\n\n15 Cherry-Garrard, A., _Diary No. 1_ SPRI MS 559\/18\/1\u20134: BJ, 10\/09\/1910.\n\n16 Ed. Jones, M., _Robert Falcon Scott, Journals_ , Oxford University Press, Oxford 2005, p. 5.\n\n17 Evans, Edward, _Letter to Lois Evans 21\/06\/1910_ , Swansea Museum.\n\n18 Abbott, G.P., _Journal 01\/06\/1910\u201317\/10\/1911_. SPRI, MS 1754\/1D.\n\n19 Cherry-Garrard, A., _Diary No. 1_ SPRI MS 559\/18\/1\u20134: BJ, 15\/07\/1910.\n\n20 Abbott, G.P., _Journal 01\/06\/1910_ \u2013 _17\/10\/1911_. SPRI, MS 1754\/1D.\n\n21 Cherry-Garrard, A., _Diary No. 1_ SPRI MS 559\/18\/1\u20134: BJ, 26\/07\/1910.\n\n22 Ibid., 30\/08\/1910.\n\n23 Ibid., 10\/10\/1910.\n\n24 Evans, Edward, _Letter to Daniel Radcliffe_ , SPRI, MS 1013\/2\/3.\n\n25 Priestley, R., Lecture; _The Antarctic Past and Present_ , SPRI, MS 1097\/15:D.\n\n26 Cherry-Garrard, A., _Diary No. 1_ SPRI MS 559\/18\/1-4:BJ, 26\/07\/1910.\n\n27 Ed. Jones, M., _Robert Falcon Scott Journals_ , Oxford University Press, Oxford 2005, p. 10.\n\n28 New Zealand _Evening Post_. 10\/06\/1911.\n\n29 Gregor, G., _Swansea's Antarctic Hero Edgar Evans, 1876_ \u2013 _1912_ , Swansea City Council. Swansea. 1995, p. 37.\n\n30 Richards, S., _Letter to Reginald Pound_ , 14\/06\/1965. Edgar Evans Swansea Museum Box 210.\n\n31 Cherry-Garrard, A., _Diary No. 1_ SPRI MS 559\/18\/1\u20134:BJ, 29\/11\/1910.\n\n32 Evans, Edward. _Letter to Daniel Radcliffe_ , SPRI, MS 1013\/2\/6.\n\n33 SPRI, Unknown newspaper clipping, 02\/1907. Shackleton was described as FRGS, FRAS and Silver Medallist of the Royal Geographical Society.\n\n34 Cherry-Garrard, A., _The Worst Journey in the World_ , Picador, London, 2001, p. 49.\n\n35 Evans, Edward. _Letter to Daniel Radcliffe_ , SPRI, MS 1013\/2\/6.\n\n36 Priestly, R., _The Polar Expedition As A Psychological Study_ , SPRI, MS 1097\/16.\n\n37 A division that creates watertight compartments in the hull of a ship, so that leaking in one compartment will not flood the whole ship.\n\n38 Unknown newspaper clipping, 28\/08\/1910, SPRI.\n\n39 Scott R.F., _Scott's Last Expedition Vol. 1_ , John Murray, London 1935, p. 29.\n\n40 Cherry-Garrard, A., _Diary No. 1_ SPRI MS 559\/18\/1\u20134:BJ, 23\/12\/1910.\n\n41 Ibid. 25\/12\/1910.\n\n42 Hut Point was avoided because of the fear that _Terra Nova_ would be iced in as _Discovery_ had been.\n\n43 Evans Edward, _Letter to Daniel Radcliffe_ , SPRI MS 1013\/2\/4.\n\n44 Debenham, F., _Letter to Stanley Richards_ , 25\/05\/1962 Swansea Museum, box 210 PO Edgar Evans (Red File).\n\n45 Cherry-Garrard, A., _Diary No. 1_ SPRI MS 559\/18\/1\u20134:BJ 10\/01\/1910.\n\n46 Ibid., 10\/01\/1911.\n\n47 Scott, R.F., _Scott's Last Expedition, Vol.1_ , John Murray. London 1935 p. 472.\n\n## 12\n\n## The First Western Party\n\nThe 1911 autumn sledging trip: 27 January to 15 March. Edgar, with scientists Griffith Taylor, Charles Wright and Frank Debenham, spent over six weeks exploring and studying the geology of the Dry Valleys, the Ferrar Glacier, the Koettlitz Glacier and the Taylor Glacier.\n\nThe Dry Valleys in Victoria Land are one of the few areas in Antarctica where geologists can easily study the rocks because the valleys are perennially free from ice and snow. Scott, Edgar and Lashly were the first to discover these oases and briefly explored them, on their way back from their Western Journey of 1903. The 1911 expedition was to examine them in more detail.\n\nEdgar returned to the area as sledge master and cook of the Western Party. He was particularly suited for the expedition; he had more experience of Antarctic sledging than practically anyone and the three scientists were tyros; they knew they were lucky to have him. Taylor wrote later that Edgar was 'at ease with the officers',1 good in emergencies, unfailingly cheerful, amusing and he kept everyone's spirits up. He was a compulsive, funny, storyteller: he told them how at home he trimmed ducks' bills so that fowls could get a fair amount of food.2 He kept the scientists 'in stitches' with stories about his school; stories, which Debenham thought unexpectedly, were as good as _Stalky and Co._ (a popular book by Kipling about a badly run boarding school). He told them about the _Discovery_ expedition. Once, when he, Dr Koettlitz and Lieutenant Armitage had killed a seal and he was called in for supper, he asked where the sweetbreads were.3 The officers told him they had eaten them; an acquired taste they said, he would not have liked them. When it was his turn to cook, he fried and ate the remaining sweetbreads. When Koettlitz and Armitage enquired where they were he replied, respectfully, that he had eaten them; 'What?' 'Yes, I acquired the taste for them in the night, sir.'4\n\nLater in the expedition he announced his prospective method of proposing to a girl: 'Kin you keep yourself and help me a bit too? If so, then _you're_ the \"pizened critter\" for me.' 'If she doesn't \"bite\" then you're better off without her, if she does then you're richer instead of poorer.'5 He offered to teach the others a one-handed clove hitch and bet them the price of a dinner that they would not be able to do it after he had demonstrated it six times. He exaggerated when he told his companions that he had run away to join the navy at the age of 13. He said that he had been very sorry for it for two years, but had gradually grown to like the service.\n\nHe was a keen reader and his conversation was full of literary allusions. On this expedition the team carried volumes of Browning, Tennyson, a dictionary and novels as well as scientific volumes. Edgar had _The Red Magazine_ (a monthly publication) and a thriller by William Le Queux.6 He liked thrillers. He did not like Kipling whose stories about the navy were, in his opinion, much too concentrated. He extolled the writings of his favourite author Alexandre 'Dummass'. The scientists could not understand whom he meant until he described the plot of _The Three Musketeers_.7\n\nHis practical skills were invaluable. He taught his companions how to sledge and camp; how to put up and take down the tent, how to cook for four men in that tent (ice melted in the stove quicker than snow and so used less fuel), 'the hiss of the primus stove was a particularly welcome sound',8 and how to cobble ski boots (sewing from inside the boot with a sewing awl, a curved implement that could be manoeuvred inside the boot). He could advise them on the first signs of frostbite or scurvy. He was always ready to do the hardest jobs and, importantly, he was tactful and respectful with his advice; he gave the tips without making the novices feel inferior.9 But he was always ready to give his opinion; when they argued about their scientific finds he would break in with the most 'fearful and wonderful suggestions'.10\n\nAfter Edgar had died, Taylor remembered with affection the Canary Island hat (a large floppy creation) that he had worn on sunny days,11 'but it soon turned into the official balaclava'.12 He remembered also the bets that he lost trying to do the one-handed clove hitch. Debenham clearly liked him. In the accounts there is no sense of a class barrier; instead there is a sense that Edgar had qualities that the more educated men admired.\n\n_Terra Nova_ carried both the Western and Eastern Parties to their drop off point. The Western Party was left at Butter Point, across McMurdo Sound, about 30 miles from their base. There were cheers and goodbyes from the whole ship's company as the ship sailed out of sight.\n\nAustralian Griffith Taylor was the principal geologist; he was to investigate the effects of wind, water and ice on the land. His fellow Australian, Frank Debenham, also a geologist, was to help by collecting specimens. Charles Wright was the 'iceologist', the physicist\/glaciologist, who was to examine and photograph ice crystals. The men set out with two sledges, a 12ft and a 9ft, but were able to leave one at a depot, so pulled just one sledge for much of the time.\n\nTheir aim was to make a geological exploration of the region between the Dry Valleys and the Koettlitz Glacier, to find how the land had been affected by glacier movement, wind, frost and water. Taylor, particularly, wanted to ascertain how their findings compared with observations in warmer climates. Their orders were to climb the Ferrar Glacier to the junction with the Dry Valley Glacier, to go down the glacier and investigate the Dry Valleys, then to make a geological exploration of the Koettlitz Glacier, returning via Hut Point (Scott's base of 1902), to Cape Evans. Scott suggested that when they had investigated the Dry Valleys they could move east to get to the Koettlitz either by climbing up that feared 'Descent Pass' that Edgar had navigated with such difficulty in 1902, or, if this proved impossible, return to the sea ice and progress to the east around Butter Point. In the event, the expedition developed the geological discoveries made in _Discovery_ days, made maps of the Lower Ferrar and Dry Valleys and then went on to the Taylor Glacier (named for Griffith Taylor after the expedition), exploring the lower part of that valley. Edgar gave the name 'Wales Glacier' to one of its tributaries.13 They returned via the Koettlitz Glacier and made detailed maps of its tributary valleys.\n\nThe point about this expedition was that its primary focus was scientific. The team could take any time they wanted to examine features of interest.\n\nThe expedition was tough; no dogs could be spared so they man\u2013hauled throughout. Debenham wrote later; 'we got into one or two tight spots on the journey but when we did, he (Edgar) never showed any alarm and usually made a joke in the middle of what looked like being a very risky job'.14 His main problem throughout the expedition was his nose, which regularly got frostbitten. When his sledge-mates told him the trouble was flaring again, he talked about the offending member as if it was a difficult dependent, not directly connected to him and he told his 'old Blossom' off severely. Later in the expedition he also got a frostbitten ear when unbelievably, he was just wearing a tam-o'-shanter (no ear protection). He admitted that the problem was pure carelessness on his part.\n\nThey left the drop off point on 27 February taking provisions for eight weeks. They went off at a cracking pace, not even saying goodbye to those friends who remained on the ship. Taylor had geological hammers, notebooks, binoculars and specimen bags hanging out of every pocket. Edgar was cook for the first week, Debenham, cook's mate, to take over after a week. The pots were aluminium, a good conductor of heat or cold. They had to be careful not to touch the pots with their bare fingers or their skin would stick to the surface. Each man hauled about 270lbs.15 On their first tramp they noted unusual features for Antarctica \u2013 extensive patches of moss.16 By the end of the first day's sledging they were in a wonderful position; they could look up the Ferrar Glacier, where straight lines of dark hills ran upwards on each side of the ice, the mountains behind showed clearly against the western sky. When they looked back they could see the ship, now a tiny speck on the horizon, their last link with (relative) civilisation for two months. When Edgar cooked their first meal they could hardly manage the pemmican, which is often far too rich for the beginning of a sortie. The temperature was 13\u00b0F.17\n\nOn the lower part of the glacier they found, surprisingly, Emperor penguins in their moult phase. By 1911 it was known that the Emperors breed in Antarctica, also that they went through a moulting phase of two or three weeks before they returned to the sea, but it was not known previously that these lordly birds chose the Ferrar Glacier as one of their moulting spots.18 Throughout the moult the Emperors go without food; their old feathers would get waterlogged if they went into the sea and Edgar watched the birds as they wandered around, old patches of feathers hanging from them unattractively, as they bad-temperedly batted each other with their flippers.\n\nHauling was heavy work; the scientists were in a 'somewhat flabby condition'19 and Taylor in his account of the journey gives credit to Edgar for his 'mighty strength' and his care of the sledging equipment.20 After three days, they were well advanced on the glacier, the sledging became 'damnable' as they pulled through snow of up to 10in deep.21 By 31 January they camped below Cathedral Rocks, near where the Ferrar Glacier divides into two. Cathedral Rocks were known to Edgar, but new to the scientists who decided that they were well named. They thought that the high ridges and sharply cut ends looked like the transepts of a cathedral;22 they could also see Descent Pass that Edgar had navigated in 1902. It looked formidable.\n\nFrom their vantage point they could still look back on Ross Island with Mount Erebus still smoking and the sea. The glacier stretched above. Edgar wrote in his journal about how impressed he was with the rugged surroundings. In early February they descended the same steep glacier leading to the Dry Valleys that Edgar had gone down in 1902. But the 1911 expedition went further beyond the glacier and deeper into the valley than they had done in 1902; the men spent a week studying the geology of the valley which is 25 miles long and 4 miles wide, encased by mountains of over 5,000ft and completely free of ice and snow at a latitude of over 77\u00b0S. It was a remarkable and beautiful spot. Thaw streams ran down the glacier, and Edgar wrote that he 'did not expect to see scenery like this'.23 Taylor studied the glacial landscape24 and Edgar became interested in collecting rock samples and later fossils in the moraine rocks (Taylor offered him cash if he found any).25\n\nThe Dry Valleys deserve their name insofar as there is no snow, but there is plenty of water, due to the thaws. After Scott, Edgar and Seaman Lashly had discovered the upper part of the Dry Valleys in 1903 the scientists of 1911 were understandably keen to add useful knowledge about these phenomena. We now know that the dry valleys are a row of valleys in Victoria Land named because of their low humidity and lack of snow and ice cover. The floors of the valleys are covered by grey, loose gravelly material. They were formed when katabatic winds,26 reaching 200mph, swept through, evaporating any moisture in their path. Scott originally named the valley 'Death Valley', because there was nothing obviously alive there, but in fact bacteria proliferate in the summer melt water and provide nutrients for the soil. The Americans in their preparation for the Mars probe used the area, as the conditions are the nearest earth equivalent to that planet.\n\nThe valley's lack of snow meant that they could not use their sledge and they set off for a few days exploration with a tent, sleeping bags and dry provisions; Edgar carried his sleeping bag, the tent, the tent poles and his provisions slung over his shoulder. This was not a problem for him. The others made do, carrying their sleeping bags, collecting bags, camera and biscuits.27 Importantly, at least from Edgar's point of view, they did not take the cooker. The meals were all cold, 'make-believe' meals.28 Each day they had ten biscuits, a stick of chocolate, 2oz cheese and 1.5oz butter.29 Edgar felt the lack of a hot meal keenly and he believed that food could only be of benefit when it was warm; a diet of cheese, biscuits and chocolate was simply not enough. His journal over these days is full of complaints about the unsatisfactory nature of his rations. One day he complained he only had biscuits, butter and icy water for one meal, not even cheese and chocolate. Sucking ice or snow did not help. Their thirst was only quenched for a few minutes. They thought that the biscuits were similar to porridge in that their comforting effect wore off in a short time leaving a horrid vacuum; in addition, Debenham wrote, they were so hard that they sometimes had to be broken up with a hammer.30 But he thought the scenery was lovely.\n\nThe glacier they had descended ended in a drainage lake two miles long surrounded by mountains. The lake was partially frozen; its edge was covered with four inches of smooth, clear ice \u2013 ideal for skating. A rock bar, a 'reigel', projected into it giving the lake an hourglass appearance. Taylor named the 'reigel' and the lake after his friend Professor Bonney of Cambridge. At the far side of Lake Bonney, reached with much slipping and sliding, was an area that Edgar instantly dubbed 'the football fields'. It was full of holes filled up with gravel and sand, obviously the sort of football field he was used to. This was where Scott, Edgar and Lashly had turned back in 1903, after this everything was new. Edgar wrote that now, he had had the satisfaction of seeing the whole of it.31 They found about twenty seal skeletons and wondered how they came to be there. Edgar was amazed that they had managed to climb so far up and thought that they had probably died of starvation, because having got up, they could not get down. The alternative theory, that they had come up purposely to die, did not appeal. The men explored south till they were almost back to the coast again. When Taylor and Edgar climbed a reigel near the end of the valley they could see the sea, just about 13 miles away32 and they travelled further south towards the water, climbing over moraine heaps.33\n\nEdgar described the lower part of the valley; 'the more one sees of this place the more one is impressed with the rugged scenery, there are mountains all around with glaciers coming down the sides of them, then the valley is extremely interesting from a geological point of view there are six inland lakes, of course at present they are frozen over but in summer they are not \u2013they are made from the thaw of the glaciers and melting of snow, some are quite two miles square and there is any quantity of rocks of all descriptions... The last time I was here I only came a third of the way through the Dry Valley now I have the satisfaction of seeing the whole if it'.34\n\nAfter four days the end was in sight and about time too; 'my belly fairly rattles. We hope to get back to Glacier Camp tomorrow, a feed of Pemmican will be very welcome or anything hot in fact. Four days of dry biscuits is enough for a while.'35 He complained that it was the first time that a sledging party had tried to go without hot food in the Antarctic. The venture into the Dry Valleys had been a new departure; explorers did not usually separate themselves from their sledges for more than a few hours.\n\nBy 10 February, they were below Cathedral Rocks again. They could congratulate themselves. They had found a treasure trove of geological and biological specimens in the Dry Valleys. Some of the fossils that Edgar collected contained primitive flora and microorganisms. It was very cold and they celebrated their return to hot food eagerly. They ate, a lot, double hoosh and double cocoa, at last liquid food. This was followed predictably by stomach-ache. Edgar's solution was a 'massage with an ice axe' or 'an operation for appendicitis with the same'.36\n\nOn 11 February, Edgar and Taylor, roped together, went, as instructed by Scott, to explore the possibilities of the Descent Pass, 7 miles away. If they could have negotiated it they would have been saved the journey back to the coast. The going was very heavy. They got into a maze of crevasses. As they progressed there was a noise like an earthquake as they stepped on a crust of snow, this was followed by a peculiar shudder, lasting for seconds. They found that their axes went far too easily through the ice. Suddenly they found they had sunk up to their thighs in the snow. The surface started caving in. They were on the edge of a 'profound' crevasse and retreated cautiously.37 They thought that they could never have got their sledge over the gaping void and prudently decided to go down to Butter Point and along the coast to the Koettlitz Glacier.\n\nThey were soon back from where they started, at the base of the Ferrar, at the junction of glacier and sea ice, though the rapid descent was accompanied by 'plenty of cusses'38 as they fell on the ice and ridges. At the base they could still see their old sledge tracks. The interesting discussions continued: for example, did they get enough sugar? Edgar thought they did not. He wrote that they discussed several things but 'did not settle them'.39 One of these queries was that he did not think that New Harbour was at the mouth of the DryValley. He opined that this was not what he had seen from the mouth of the Dry Valleys the week before. Wright supported his argument. Debenham bet him he was wrong.40 Edgar was incorrect.\n\nAt the base of the glacier the sea ice was on the move. It had gone out 8 miles in the two weeks they had been in the mountains. Killer whales circled in the sea, below cracks in the ice that got bigger as the men looked at them. Debenham wrote that Edgar normally kept his diary with 'much pain and tribulation', but on 13 February he was excited to write about the sea ice moving. It was clear that it would not withstand any weight. They retreated to the fast ice close to the land and then started climbing towards the mountains again. The aim was to get to the Blue Glacier in a couple of days and then onto the Koettlitz Glacier. Progress was slow as they pulled through snow up to their knees on a steep upward slope. Taylor wondered if anyone had ever adopted a worse route with a laden sledge.41 They could only make strides of a few inches and could not get a good pull on the traces. They fell repeatedly and the sledges capsized, altogether an exhausting business. Edgar delivered his strongest curse in the presence of officers: 'May the curse of the seven blind beggars of Egypt be upon you.'42 This was delivered with emphasis at every halt. Their breath froze as soon as it reached the air. They had icicles hanging onto their moustaches and beards that made them look like walruses.\n\nThey did 5 miles in eight hours, climbing 600ft up the long snow slope that runs along the coast from Butter Point to the Blue Glacier. Their footgear gave them trouble. It took much tugging, shoving and chafing to get their feet into boots that were as stiff as iron, then having to do them up with frozen fingers.43 The nails in the soles transmitted the cold into the boots, and Debenham wrote that sometimes they had to hit them with a geological hammer to get them into shape, especially if snow had got in and frozen. Edgar wore puttees to guard against this and found them remarkably successful. Debenham thought that the worst part of the ordeal was when the boot actually thawed; he said that then there was a pitched battle between the owner of the boot and the boot itself as to which gave in to the others temperature. He wrote that he had never guessed that cold feet could give such excruciating pain.44\n\nBetween 17 and 25 February they struggled through truly ter r ible surfaces up the middle of the 'desiccated [Koettlitz] glacier, now weathered into pie-crusts, bastions and pinnacles of every conceivable shape'.45 They reached the north side of the glacier and explored the moraines, hanging valleys and 'ice slabs' in the foothills of Mount Hooker. There was a lake with seals swimming in it. A stream flowed from this lake; over 20 miles long it reached the sea near the Blue Glacier. They named the stream 'Alph', from Coleridge's poem _Kubla Khan_ in which a sacred river runs in a pure stream into the Mediterranean Sea:\n\nWhere Alph, the sacred river ran\n\nThrough caverns measureless to man,\n\nDown to a sunless sea.\n\nThere are repeated comments about card games: 24 February, the two Australians (Debenham and Taylor) versus Canada and Wales (Wright and Edgar). Australia lost handsomely. In March, Edgar won a dinner from Debenham and Wright. He admitted he had lost, at least once, to Taylor (when they were safely back Edgar lost a game of cribbage to Taylor 'to the astonishment of the seamen').46\n\nThe return began on 2 March. Travelling via the north-west side of the Koettlitz, they found, for a change, that the ice was smooth and comparatively easy and assumed they were on the frozen surface of the Alph. This was confirmed when one day water suddenly rose up through the snow flooding the floor of the tent. It seemed that tidal water had come surging into the Alph. On the 9th Edgar wrote that it was the first time he had spent his birthday sledging. He wrote that pulling one of the sledges that day was 'a bugger'; they pulled hard enough to 'break the heart of the sledge, never mind the party pulling'.47 To celebrate he had two cups of tea and an extra biscuit.\n\nThe Western Party reached Hut Point on 14 March, laden with sacks of geological and fossil samples. They had made maps of the Lower Ferrar and Taylor Glacier and explored the lower part of the Dry Valleys for the first time. They had added many new features to the map and named them.48 Scott wrote that the party 'gave Edgar a very high character'.49\n\nSo Edgar ended another exciting and productive Antarctic exploration.\n\n### Notes\n\n1 Taylor, G., _Letter to Stanley Richard_ , 11\/06\/1962, Swansea Museum, Box 210, (Edgar Evans).\n\n2 Ibid.\n\n3 The pancreas or thymus of a calf, lamb, or other young animal soaked, fried and eaten as food. They were considered a delicacy.\n\n4 Debenham, F., _Journal_ , 19\/01\/1911\u201308\/03\/1911 MS 279\/2: BJp.\n\n5 Ibid., p. 78.\n\n6 William Le Queux, 02\/17\/1864\u201313\/10\/1927. An Anglo-French journalist and writer who wrote 150 novels dealing with international intrigue, also books warning of Britain's vulnerability to European invasion before the First World War.\n\n7 Taylor, G., _Journeyman Taylor, The Education of a Scientist_ , Robert Hale, London, 1958, p. 99.\n\n8 Speak, P., _DEB, A Biography of Frank Debenham_ , Polar Publishing, Guildford England, 2008, p. 31.\n\n9 Debenham, F., _Journal_ , 19\/01\/1911\u201308\/03\/1911 MS 279\/2: BJ p. 40.\n\n10 Debenham, F., _Letter to Stanley Richard_ , 25\/05\/1962, Swansea Museum, Box 210 (Edgar Evans).\n\n11 Taylor, G., _Letter to Stanley Richard_ , 11\/06\/1962, Swansea Museum, Box 210 (Edgar Evans).\n\n12 Ibid.\n\n13 Ibid.\n\n14 Debenham, F., _Letter to Stanley Richard_ , 25\/05\/1962, Swansea Museum, Box 210 (Edgar Evans).\n\n15 Debenham, F., _Journal_ , 19\/01\/1911\u201308\/03\/1911 MS 279\/2: BJ p. 40.\n\n16 Taylor, G., _Journeyman Taylor, The Education of a Scientist_ , Robert Hale, London, 1958, p. 98.\n\n17 Evans, E., _Journal, 27\/01\/1911\u201312\/05\/1911_ , SPRI, MS 1487: BJ, 27\/01\/1911.\n\n18 Debenham, F., _Journal_ , 19\/01\/1911\u201308\/03\/1911 MS 279\/2: BJ p. 45.\n\n19 Ibid., p. 46.\n\n20 Taylor, G., _Journeyman Taylor, The Education of a Scientist_ , Robert Hale, London, 1958, p. 98.\n\n21 Debenham, F., _Journal_ , 19\/01\/1911\u201308\/03\/1911 MS 279\/2: BJ, p. 48.\n\n22 Ibid., p. 49.\n\n23 Ibid., 04\/02\/1911.\n\n24 Taylor, G., _Journeyman Taylor, The Education of a Scientist_ , Robert Hale, London, 1958, p. 98.\n\n25 Debenham, F., _Journal_ , 19\/01\/1911\u201308\/03\/1911 MS 279\/2: BJp, p. 56.\n\n26 Derived from the Greek 'going down'. They occur when cold dense winds are pulled down by the force of gravity.\n\n27 Debenham, F., _Journal_ , 19\/01\/1911\u201308\/03\/1911 MS 279\/2: BJp, p. 60.\n\n28 Evans, E., _Journal 27\/01\/1911\u201312\/03\/1911_ , SPRI, MS 1487: BJ, 03\/02\/1911.\n\n29 Debenham, F., _Journal_ , 19\/01\/1911\u201308\/03\/1911 MS 279\/2: BJp, p. 57.\n\n30 Ibid., p. 64.\n\n31 Evans, E., _Journal 27\/01\/1911\u201312\/03\/1911_ , SPRI, MS 1487: BJ, 06\/02\/1911.\n\n32 Ibid., 05\/02\/1911.\n\n33 Debenham, F., _Journal_ , 19\/01\/1911\u201308\/03\/1911 SPRI, MS 279\/2: BJp, p. 61.\n\n34 Evans, E., _Journal 27\/01\/1911\u201312\/03\/1911_ , SPRI, MS 1487: BJ, 06\/02\/1911.\n\n35 Ibid., 06\/02\/1911.\n\n36 Debenham, F., _Journal_ , 19\/01\/1911\u201308\/03\/1911 MS 279\/2: BJp, p. 70.\n\n37 Taylor, G., _Journeyman Taylor, The Education of a Scientist_ , Robert Hale, London, 1958, p. 100.\n\n38 Evans, E., _Journal 27\/01\/1911\u201312\/03\/1911_ , SPRI MS 1487: BJ, 12\/02\/1911.\n\n39 Ibid., 12\/02\/1911.\n\n40 Debenham, F., _Journal_ , 19\/01\/1911\u201308\/03\/1911 MS 279\/2: BJp, p. 77.\n\n41 Taylor, G., _Journeyman Taylor, The Education of a Scientist_ , Robert Hale, London, 1958, p. 100.\n\n42 Debenham, F., _Journal_ , 19\/01\/1911\u201308\/03\/1911 MS 279\/2: BJp, p. 79.\n\n43 Ibid., p. 78.\n\n44 Ibid., p. 78.\n\n45 Taylor, G., _Journeyman Taylor, The Education of a Scientist_ , Robert Hale, London, 1958, p. 100.\n\n46 Taylor, G., _Letter to Stanley Richard_ , 11\/06\/1962, Swansea Museum, Box 210, (Edgar Evans).\n\n47 Evans, E., _Journal 27\/01\/1911\u201312\/03\/1911_ , SPRI MS 1487;BJ, 09\/03\/1911.\n\n48 The naming was difficult to agree on. Years later Debenham wrote that Edgar's suggestions had been too naval and too mess-deck. He (Debenham) decided against female names, but when he named a glacier the Kitticarrara Glacier, this was resented by Edgar who opined that the rule against female names was being broken.\n\n49 Ed. Jones, M., _Robert Falcon Scott Journals_ , Oxford University Press, Oxford, 2005, p. 146.\n\n## 13\n\n## The Winter Months, 1911\n\nAlthough they had reached Hut Point ( _Discovery_ 's headquarters) they were still some 15 miles south of Cape Evans. The home base could not be reached until the sea ice had frozen over sufficiently for safe transport. Hut Point was already crowded; Scott and his party, having returned from laying depots on the Barrier, had been in the hut for two weeks, and the scientific party increased the cramped community to sixteen. There was ominous news for Taylor's party. After _Terra Nova_ had deposited Edgar's group, she had carr ied the Eastern Party to the far end of the Barr ier. Unable to get ashore on King Edward VII Land, she steamed back along the Barrier and into a bay, the Bay of Whales, some 60 miles closer to the Pole than the English base. Here, the crew were astounded to find a ship. It was Amundsen's _Fram_. She was carrying just eight men but over 100 dogs (Amundsen magnanimously offered some dogs to Scott). There was now absolutely no doubt that the Norwegian was single-minded in his determination to get to the Pole first. Lieutenant Evans wrote that he hoped that the 'best man may win'.1\n\nIn the face of this unexpected challenge Scott decided that he would not change his plans; the scientific aims of the expedition could not be sacrificed in order to win a race. But the Pole remained a priority, both in terms of national pride and because the achievement would be sure to attract funds to help the expedition's big financial outlay. Edgar thought Amundsen's change of plan deceitful; he knew, none better, of Scott's months of careful planning and was angry and indignant. The scientists were upset too. But they all understood that in Amundsen they faced formidable competition. He was experienced both in Arctic travel, having navigated through the North-West Passage, and he had been on an expedition to the Antarctic (though not to its interior) in 1897. His display of dog driving was formidable when the British met him in the Bay of Whales. But Amundsen, too, had his worries. He was worried about Scott's motor sledges. He feared they could rob him of success.\n\nThe sixteen men were in Hut Point for a month waiting for the sea to freeze. Evenings, lit by the dim glow of candles and blubber lamps, were spent in long discussions.2 The hut had a central room, 'Villa Virtue',3 where Dr Wilson, Cherry-Garrard, Lieutenant Bowers, Captain Oates and the dog handler, Meares, slept. The other officers, including Scott, slept in shared accommodation around the hut with the men.4 Hardly a class-conscious division. Seals were killed and the food supplemented by the remains of Shackleton's 1908 visit; his biscuits were thought to be particularly good, especially when they were toasted and smeared with blubber. Debenham was a cook and had a particular talent for making chapatis.\n\nThey left Hut Point on 11 April as a large party. Edgar was with Scott, Lieutenant Evans, 'Birdie' Bowers, Taylor, Wr ight, Debenham, Petty Officer Crean and the Norwegian ski expert, Gran. They took enough food for twenty-four hours. The aim was to keep on land initially and then to go over the sea-ice to Cape Evans. When the group stopped for lunch Edgar and Taylor prospected the ice to make sure it was safe to cross (Taylor had previously fallen through a weak patch). Eventually they all attempted to reach Cape Evans in a night's march. The journey was difficult; they were caught by a blizzard (when they finished their food) and it eventually took them two days to make the journey to Cape Evans. They arrived, ravenous and exhausted, on 13 April.\n\nOn 23 April the sun disappeared. There were a few excitements over the winter months: a bitch had six puppies and killed them all.5 The men played football on the ice when there was still enough light. Edgar was always Captain Oates' first choice for his team: 'Go on Taff, break them up', 'Right-o, sir'. Dr Atkinson left the hut and got lost; his hand got badly frostbitten. Edgar was in charge of a search party and, always in the forefront of activities, was photographed by Ponting bandaging Atkinson's hand.\n\nThere were also lectures. The Men attended the first few, but a lecture on parasitology was too much and Edgar did not attend any after this, so missing Dr Atkinson's lecture on scurvy (which Atkinson interestingly thought was catching and in which he reiterated the theory that the disease was caused by bacterial acid poisoning and correctly stated that eating fresh vegetables was a way to halt it). Edgar also missed Scott's suggestion for building igloos on the Southern Attempt, a suggestion turned down because of the two hours' labour that would be needed after a day's hauling.\n\nMidwinter was celebrated with the hut decorated with flags, an enormous cake, an extravagant meal and alcohol. Edgar, according to Scott, enjoyed himself by 'imparting confidences in heavy whispers'.6\n\nPhotographer Ponting wrote that Edgar was the dominant personality of the mess-deck over the winter months. Ponting said that Edgar's previous Polar experience, his build, his stentorian voice and manner of using it, all compelled 'the respect due to one who would have been conspicuous in any company'. Ponting thought that Edgar was one of Scott's towers of strength; he had heard Scott telling Edgar that he did not know what the expedition would do without him on more than one occasion.7\n\nThe focus of the winter was preparation for the Pole attempt. A first at the Pole would add a huge kudos to the expedition; failure would diminish its achievements. Scott's plans were explained. The sledges were overhauled and Edgar worked hard on them. The dogs and the ponies needed attention; the dogs, in particular, were a source of irritation. As well as providing interest and amusement, they repeatedly escaped their traces and rushed off after seals and penguins, or got their tongues stuck to frozen tins, a problem only remedied by catching the dog and warming the tin.\n\nScott was worried about ski boots and bindings. He knew that the 2lb boots could chafe the men's Achilles tendons. Edgar's practical skills resolved the problem (one which had concerned Amundsen for years). He made a ski shoe with a double sole of sealskin, stiffened with wood, into which the men could fit their soft fur finnesko. The shoe was held in place by a strap and the modification was stronger, allowing more flexibility than the standard boot. He sewed the boots with waxed thread.8 The finnesko\/shoe combination weighed less than a ski boot9 and was an undoubted success.\n\nAnticipation for the Pole attempt was increasing.\n\n### Notes\n\n1 Evans, Edward. _Letter to Daniel Radcliffe_ , SPRI, MS1013\/2\/4.\n\n2 Ibid., p. 102.\n\n3 Taylor, G., _Journeyman Taylor, The Education of a Scientist_ , Robert Hale, London, 1958, p.101.\n\n4 Ibid., p. 101.\n\n5 Cherry-Garrard, A., _Journal_ , MS 559\/18\/1\u20134;BJ Vol. 2, 19\/08\/1911.\n\n6 Ibid., p. 232.\n\n7 Ponting, H.G., _The Great White South_ , Duckworth, London, 1932, p. 162.\n\n8 Personal communication. September 2010, Jean Scholar, granddaughter of PO Fred Parsons.\n\n9 Ed. Jones, M., _Robert Falcon Scott's Last Expedition_ , Oxford University Press, Oxford, 2005, p. 239.\n\n## 14\n\n## The Polar Assault\n\nScott took Edgar on one sortie before the assault on the Pole. Along with Edgar, scientist George Simpson and 'Birdie' Bowers, he set out to check the rate of flow of the Ferrar Glacier by recording how far ice stakes, positioned seven months previously, had moved. This was the first observation on the movements of a coastal glacier and they found that the stakes had moved variable distances; between 24 and 32ft. They thought that the Ferrar Glacier was 'lively'. The group were away for thirteen days, on the last day they covered 21 miles, man-hauling into the teeth of a freezing headwind. They 'captured many frostbites'.1 Edgar was 'a treasure'.2\n\nEdgar had been well briefed on Scott's plans for the Polar Assault. There were to be three stages \u2013 the Barrier, the Beardmore Glacier and the Plateau.3 It was planned to get across the Barrier with motorised sledges (which, he hoped might significantly improve progress over the Barrier by comparison with Shackleton), ten horses and the dogs. Scott thought that the ponies would be reliable for the Barrier stage, after which they were to be killed.4 He calculated that each pony could pull 550lbs (perhaps more), between them a total of 5,500lbs. The dogs were to return to base when they had pulled their loads over the Barrier. Scott did not think that animals would be able to survive the Beardmore Glacier's fearsome conditions and crevasses.5 Here he was influenced by Shackleton, who had consulted Frederick Jackson, the leader of the Jackson Harsworth Expedition of 1894\u201397. Jackson had recommended Russian ponies for Polar exploration6 and Shackleton had become critical of dogs.7 Scott himself had seen one dog team disappear down a crevasse in February 1911 and he knew that Shackleton's last pony had fallen into an endless abyss on the Beardmore on the 1908 Pole attempt. The British position was that if the plateau was as bad as Shackleton described, no beast could stand the trials of getting onto it, but that 'man could do what beasts would not'.8\n\nScott's plans were made before he had any idea that Amundsen's presence would make the lack of dogs crucial, but he determined to stick to the plans. He did take dogs, but only thirty-four, compared to Amundsen, who took over a hundred. But by this stage he was realistic about his chances; he had heard how well the Norwegian team controlled their dogs, he knew he would have to start later than his rival because ponies suffered so badly in Antarctic conditions and he wrote that if Amundsen achieved the Pole it would be before the British, because he (Amundsen) could travel fast with dogs and was certain to start early. Scott warned that the British venture might be belittled and he wrote: 'After all it's the work that counts not the applause that follows.'9 But he worried also that Amundsen would get the news back first in 1912.10\n\nScott wrote to Edgar's wife Lois just before the team departed. He said that Edgar had told him a great deal about her and that he could imagine that she and the children wanted to see him come home. He assured Lois that Edgar was very well, very strong and in good condition. He knew the family would be disappointed, but he thought that it was likely that Edgar would stay in the South for an extra year. If so, he asked Lois to remember that Edgar was certain to be in the best of health and that it would be all the better when he did come home. Scott wrote that he hoped that Edgar would get a good billet on his return, which would make it unnecessary for him to leave her again. 'He is such an old friend of mine that no one deserves so well all I can do for him.' She must not be anxious or Worried.11 12\n\nThe attempt was critically dependent on dates. Scott recorded that the distance was 1,530 geographical miles; if they could keep up with Shackleton's daily distances they would return near the end of March, in the early part of winter when temperatures would be very low. In Antarctica there is a 'Coreless Winter'; temperatures drop immediately at the end of summer and remain low till October (lower for longer than the Arctic). Meteorologist George Simpson estimated that the temperatures at the last stage of the Barrier journey would be around \u221220\u00b0F, very challenging, after a long plateau trek, but endurable.13 In the event, at the end of their Barrier journey, the British were to suffer temperatures that were up to \u221220\u00b0F \u2013 colder than the typical horrors.14\n\nEdgar was full of optimism at the start; he thought that the motor sledges had real potential. When he watched the four men, Lieutenant Teddy Evans, Stoker Lashly, Engineer Bernard Day and the Steward Henry Hooper, set off on 24 October with the caterpillar-track sledges he was enthusiastic: 'Lord, Sir, I reckon if them things can go like that you wouldn't want nothing else.'15 The motor sledges pulled at about 1 mile per hour,16 but did not get far; the caterpillar tracks were reasonably successful but the cylinders got too hot, whilst wind on the carburettors made them too cold; one motor failed after 14 miles, the other after 50. Their loads were repacked, 740lbs onto a 10ft sledge, which the four men pulled furiously to a pre-arranged rendezvous. The extra physical work of hauling would take its toll and affect Lieutenant Evans and Lashly later on the attempt at the Pole.\n\nOn the same day (1 November) that the tractors were abandoned the pony caravan set out. There were ten pony leaders in three groups, Edgar was one of them, leading Snatcher, and when the cavalcade halted he shared a tent with Scott, Oates and Dr Wilson.17 As the team got ready to set out Scott again praised Edgar, writing: 'Edgar Evans has proved a useful member of our party; he looks after the sledges and sledge equipment with a care of management and a fertility of resource which is truly astonishing \u2013 on \"track\" he is just as sound and hard as ever and has an inexhaustible supply of anecdotes.'18 Edgar's 'fitness to travel' was self-evident; any suggestion that he was below par seems misplaced. The dog teams with two dog-drivers, Cecil Mears and Dmitri Gerof, had left earlier and when the ponies set out, Edgar's Snatcher romped away, leading the party.19\n\nPonting filmed the pony parties' departure. The weakest went first, followed by the stronger. Some ponies struggled from the start and wide gaps opened between them. They were still in their summer coats and got seriously chilled as they plodded over the featureless, monotonous white surface; the teams saw no sign of land for days, sky and snow merging into a white pall.\n\nAfter five days they were delayed by a blizzard. The ponies suffered particularly badly. Scott had organised for snow walls to be built to protect them from wind and snow, but the fine snow still got in their eyes, noses, ears and under their coats, where it turned to ice. Rugs were little help as they quickly became soaked.20 The ponies' progress was in marked contrast to the dogs, which pulled their loads with little difficulty \u2013 their nictitating membranes protecting their eyes from the snow.21 22 The teams built depots and snow cairns at roughly 70 mile intervals. Each was provisioned with enough fuel and food (buried deeply to prevent fat in the pemmican from deteriorating) for a week, so that the returning parties would be well supplied. Five depots were built on the Barrier: Corner Camp, Bluff, One Ton, Mid, and Southern Depots.\n\nOne Ton Camp was reached on 15 November. Scott had left provisions there earlier in the year and the ponies' loads were lightened by leaving bundles of seal meat for food supplies for the return journey. The party covered about 13 geographical miles a day, but the snow surface was awful. Scott thought that a worse set of conditions for the ponies could not be imagined as their hooves sank deep into the snow; snow-shoes were not worn regularly, and Scott wrote that he wished the animals would wear them.23 On 21 November they caught up with Lieutenant Evans' motor party and camped together. This motor party foursome, having man-hauled, were fit but already ominously hungry.24 It was becoming clear that rations that satisfied men leading ponies (or men sitting on dog sledges), were not enough for those actually pulling the sledges. Hunger was to bedevil man-hauling parties. After a few days on a sortie, men began to feel an overwhelming craving for food that was only eased for a few hours by a meal. Days were occupied with thoughts of food and nights were plagued by food dreams; when they woke the craving was almost unbearable, and every hair-covered morsel was 'watched over with the eager solicitude of a dog for a bone'.25\n\nScott followed Shackleton's daily progress chart. He reasoned that if he could keep going at a pace equal to, or even ahead of Shackleton, he would have a chance of success. The ponies were all to be killed at, or before, the Beardmore Glacier and the meat used to feed both dogs and men. The dogs would return from the Beardmore and after the pony slaughter, the men would start to man-haul. Dog handler Mears eyed the ponies with anticipation, and when Oates shot the first pony (Jehu), on the night of 23 November, Jehu made glorious feeds for four days for twenty dogs.26 Over the next eleven days four more ponies (but not Edgar's Snatcher) were killed. The pony handlers watched Mears suspiciously. They were fond and protective of their charges; they also knew that the longer their particular pony survived, the less time they would have to man-haul. One pony 'cut up well' and the man-hauling team enjoyed a nice piece of undercut27 (the meat was boiled and added to the pemmican). After a delay due to poor conditions the teams pushed on and reached their second depot, Lower Depot.\n\nThe first two of Scott's planned returnees, Hooper and Day (from the motor party), turned back on 24 November. As the fourteen remaining men continued slowly south, they began to see mountains fringing the Barrier in the distance. They progressed in three groups according to speed: man-haulers, ponies, dogs, with Scott always comparing his progress to Shackleton's. But his advance was slower: visibility was poor and they were soon several days behind schedule. By comparison, Amundsen's men either skied beside the sledges or rode on them, covering over 20 miles a day.\n\nWhen, on 29 November, the mists rolled away, the surface continued to be bad and the ponies' hooves sank deeply into the snow. But on that day, in glorious sunshine, they passed the 82\u00b0 21S 'furthest south landmark' of Scott's, Shackleton's and Wilson's 1903 expedition. The huge twin-peaks of Mount Markham, discovered by that party, could be seen in the distance. They could congratulate themselves that in contrast to the fifty-eight days it had taken the three to get there in 1903, the 1911 party reached it in twenty-nine.\n\nIn 1908, Shackleton had found a route from the Barrier onto the plateau via the Beardmore Glacier and Scott aimed at this tributary, the Gateway. This path, leading onto the main glacier, avoided the most awful of the pressure ridges that piled up at the glacier\/Barrier junction. But as the teams approached the Gateway a gale struck and snow piled in drifts, burying the sledges. The men steered by compass in 'simply horrible conditions'.28 Bowers mused about Amundsen; if Amundsen had not had any problems, he should have reached the Pole. But Bowers' opinion was that the Norwegian was a sneaking ruffian: 'Old England may be a long way off, but we will do our best for her honour down here at the limit of our globe.'29\n\nAny hope of immediate British progress was lost when, on 5 December, the gale developed into a howling, white, thick, blizzard, which raged 'such as one might expect to be driven at us by all the powers of darkness'30 and which trapped the men at the bottom of the Beardmore for four days. The peculiar feature of this blizzard was that it was warm, making things even more unpleasant. Streams of water ran down the door flap and into the tent. The men lay in sopping sleeping bags. The temperature was 27\u00b0F.\n\nThe snow is melting and everything's afloat\n\nIf this goes on much longer we shall have to turn the tent upside down and use it as a boat\n\nThey could not see as far as the next tent, let alone the nearby mountains. No one understood what it meant: was it exceptional local circumstances? Scott wrote that no foresight could have prepared for this state of affairs.31 Because of the unanticipated halt, Scott had to break into summit rations, the more generous allowance meant for glacier work: 16oz biscuit (made by Huntley and Palmer), 12oz pemmican, 2oz butter, 0.57oz cocoa, 3oz sugar and 0.86oz tea per day (no oatmeal, which would be too difficult to cook).32\n\nThe delay was serious. It was to make them late on the glacier, late at the Pole and late in the season on the return journey. This was to expose them on their return from the Pole to the lowest Barrier temperatures recorded for over seventy years.33\n\nThose ponies that remained were in a pitiable state; Edgar had to dig Snatcher out of his snowy covering every few hours. But they had to keep the ponies going somehow; they were needed for that last haul onto the glacier as the loads were too heavy for the men to pull. At last the weather improved a little and on 9 December the team started out again. The ponies, stiff from days in the blizzard, floundered on. Sometimes they could only pull for a few yards. Edgar had to flog Snatcher to keep him going. The dogs followed with the remainder of the load. When the ponies had finally got their loads onto the glacier they were shot; in a way it was a relief not to see them suffer any more, but Edgar reluctantly led Snatcher to his execution and the men called the camp, 'Shambles Camp'. By now Amundsen was half way up his glacier.\n\nThe expedition made three depots on the Beardmore Glacier: Lower, Mid and Upper. At the Lower Depot, Mears turned back with his dogs on 11 December. He took letters from all the men; Edgar sent messages to his family.34 The dogs had performed well but there was no more food for them and Scott, still believing that the glacier would be too tough for them, wanted to preserve them in good condition in case they were needed for a possible attempt the following year. Twelve men were now left to man-haul up the awesome 120 mile, crevasse-ridden glacier as it rose from Barrier level to over 9,000ft. They hauled about 200lbs each. Bowers wrote that 'he had never pulled so hard, or so nearly crushed my inside into my backbone by the everlasting jerking with all my strength on the canvas band round my unfortunate tummy'.35 Edgar was the same.\n\nHe pulled with Scott, Edward Wilson and Titus Oates. It could take them eleven jerks to get the 800lb sledges started and then the men had to strain every muscle and fibre to keep the thing moving onwards and upwards. They had to relay on the soft snow of the lower slopes, taking half the load and then going back for the remainder, so doubling the distance covered. As they pulled their legs were buried up to calf level, the sledges were covered in snow and if a sledge stopped, they had to jerk again on their harnesses (often up to fifteen times) to get it going again. Their breath fogged their goggles and snow blindness caused agonies. They stumbled over and into crevasses; the sledges had to be continually turned over to scrape frozen snow off the runners. The surface changed to shiny blue ice with an irregular surface resembling a series of combs. The men ate the pony meat which they thought was beneficial, but they were in fact already suffering from serious malnutrition. Each day their body fat, essential for insulation against the cold, diminished and they were noticeably thinner. Just as important was the loss of muscle bulk which meant pulling became more exhausting. Although they were taking in about 4,500 calories they needed over 7,000 calories to man-haul up the glacier36 so they were already in negative balance. In addition, their diet contained no vitamin C and virtually no other vitamins.\n\nOn 14 December, Dr Wilson wrote that his team, with Scott, Edgar and Oates, were the strongest pullers, although the weights they were pulling were the same as the other two teams. They experimented by changing sledges; Edgar's team still pulled the best.37 This must have influenced Scott when he made his final decision on the men to haul to the Pole.\n\nOn that day, as the British teams struggled up the glacier, Amundsen and his four companions reached the Pole.\n\nBy 17 December the going was better. The British advanced 11 miles, climbing up a series of pressure ridges and tobogganing as fast as possible down the other side. On the 18th they made over 12 miles and on the 19th over 14. They were nearly 6,000ft above the Barrier. They all wore the crampons, and the ski shoes that Edgar had made during the winter which were a great success. Scott was delighted and Edgar very pleased. Scott wrote that the team owed Edgar much.38\n\nBy 21 December they had found a good place for the Upper Glacier Depot at 85\u00b0S, over 6,000ft above the Barrier and close to the steep slope that ascends to the plateau. The march that followed was long and hot, over blue rugged ice with crevasses everywhere. Scott managed to get the party through the crevasses but it took a huge amount of energy to pull the sledges up the steep slopes and to stop them overrunning on downward slopes. But Edgar thought the glaciers and mountains were stupendous. As they pulled, Scott watched the individual performances of the men, and on 21 December, at 85\u00b03', he decided on the Support Party that was to be sent home. This left just eight men to pull two sledges: Scott, Edward Wilson, Edgar and Titus Oates were in one team,'Birdie' Bowers, Stoker Lashly, PO Crean and Lieutenant Teddy Evans in the other. Scott forced a fast pace, marching for over nine hours, and he wrote of the delightful feeling of security he experienced when they finally reached the summit proper on 23 December. Though the surface was covered with sastrugi, the horizon levelled off in every direction and it was a wonderful feeling to have reached a horizontal surface at last. There was a vast silence around them, only broken by the sounds of the sledges. The teams made three depots on the plateau: Three Degrees, One and a Half Degrees and Last Depot.\n\nBut the strain was telling on all the men and must have been particularly bad for Lashly and Lieutenant Evans who, in addition to man-hauling up the Beardmore, had pulled their loaded sledge for 400 miles across almost the whole length of the Barrier. Dehydration was a problem that affected them all. At this altitude (where the oxygen level was lower) to cope with their extreme exertion they had to hyperventilate and needed about 6L of fluid (over 10 pints) each day;39 they were actually taking 6 pints per day. Even the indomitable Bowers wrote on the 23rd that he was getting exhausted and all his muscles 'have had their turn at being stiffened up'.40 Christmas Day was made memorable when, perhaps because of their fatigue, Lieutenant Evans' team nearly came to grief. Lashly fell into deep crevasse and nearly pulled the rest of his crew in with him. Although Scott wrote that even the fall had not disturbed Lashly's equanimity, his (Lashly's) comments were considered afterwards to be unsuitable to record. He was hauled out with difficulty. It was his 44th birthday.\n\nEdgar loved his food, especially a good, hot meal and he appreciated the celebration Christmas food; four courses: pemmican, horse meat flavoured with onion and curry powder and thickened with biscuit, then a sweet arrowroot cocoa and biscuit hoosh, plum pudding, followed by _Captain Scott's Invaluable Assistant: Edgar Evans_ cocoa with raisins. Finally a dessert of caramels and ginger41 enhanced the general sense of well-being.\n\nThe men hoped that the worst was over \u2013 it was not.\n\nOn the last day of the year Edgar had an accident that did much to imperil the whole expedition. Scott had decided to strip down the 12ft sledges, remove the worn runners, put on fresh 10ft runners and so convert the sledges into lighter ones. Edgar, Crean and Lashly did the work in sub-zero temperatures, a task that took them till 11 p.m. Scott wrote (again) that Edgar was the most valuable asset to the party and that to build a sledge under those conditions was a fact worth special record.42 But Edgar cut his hand, it was probably not a big cut and he hid the injury. Certainly Lashly does not mention the incident in his diary. Edgar may have thought that the cut was not serious; in any event he wanted desperately to be in the Pole Party. If his _Discovery_ explorations had brought him fame, how much more would a 'First to the Pole' achieve? He may well have planned to run a pub in South Wales when he left the navy. It was an occupation he was familiar with. His father-in-law had been the long-established licensee of the Ship Inn in Middleton. His wife had been brought up in the pub and Edgar, with his practical mind, his out-going personality and his new fund of Antarctic stories, would have been a natural as a publican. A first to the Pole would have guarenteed the pub's success.\n\nBy 30 December, the expedition finally caught up with the dates in Shackleton's journal. Two important things happened on 31 December: Scott laid the first Polar cache, Three Degree Depot, and he also took the remarkable decision to order the 'other' team, Lieutenants Evans and Bowers, Crean and Lashly, to leave their skis, sticks, ropes and axes at the Depot, possibly to save the 80lbs of weight. This is a decision that has been much cr iticised, as it was eventually to leave 'Birdie' Bowers to march 300 miles to the Pole and back. But for now the four men plodded through the snow whilst their companions continued with skis and sticks for a further three days until the final decision was made. Up to this point it was assumed that Scott had planned for four men to go to the Pole, and probably for days he had decided in favour of his own team of Wilson, Oates and Edgar. But on 3 January, 170 miles from the Pole, he went to Lieutenant Evans, Lashly, Crean and Bowers' tent and announced that he was sure he could reach the Pole if they would give one man up and make the homeward journey shorthanded. Evans said that 'of course we consented'43 and 'Birdie' joined Scott's team. In the event the three who returned were to have huge problems; Lieutenant Evans suffered terribly from scurvy and eventually had to be left with Lashly, whilst Crean made an amazingly brave solo journey to get help for his leader.44 But on that final day of 1911, supplies were redistributed; 'Birdie' transferred his share of the food and, uncomplainingly, trudged to and from the Pole. But it meant that extra time and fuel had to be spent cooking meals for five, a point that Scott had not considered.\n\nOn 8 January Scott reiterated his praise of Edgar saying that it was only at that time that he realised how much was due to Edgar, commenting on the indispensable ski shoes and crampons \u2013 the product of Edgar's manufacture, design and good workmanship. Edgar was also responsible for every sledge fitting, tent, sleeping bag and harness. Scott said there had not been a single expression of dissatisfaction relating to any of them.45\n\nScott, Bowers, Edgar, Wilson and Oates journeyed on, man-hauling on difficult surfaces with minimum temperatures averaging \u221223\u00b0F. They had a month's supply of food for the five men to get to the Pole and back to the last depot. Scott and Wilson pulled in front, Oates and Edgar behind; Bowers, on foot, was between the four. Although the pulling was fear-some they were buoyed up because they thought they were ahead of the Norwegians; on 9 January 1912 they passed Shackleton's furthest south-ernmost point and since they assumed that Amundsen would get onto the Plateau via the Beardmore Glacier and they could see no signs of dogs or sledges, they thought that they were in the lead; 'All is new ahead'.46 But every mile was at a tremendous cost. Scott wrote on 12 January 'With the surface as it is one gets horribly sick of the monotony and can easily imagine oneself getting played out.... It is going to be a close thing'.47 They started to descend to the Pole and made their final depot on 15 January. There was sunshine at last, not a cloud in the sky. 'Only twenty-seven miles from the Pole. We _ought_ to do it now.'48\n\nBut on 16 January 'the worst has happened or nearly the worst'.49 Bowers' sharp eyes detected what he thought was a cairn.50 Half an hour later he made out a black speck, which clearly was not snow. As the five marched towards it, the speck became a black flag fluttering in the wind. They had been beaten. Nearby were sledge tracks and ski tracks and many dog prints in the snow, underlining one of the reasons for Amundsen's success. The disappointment was intense and Scott wrote that they had 'many (bitter) thoughts' and 'much discussion'.51 The prize had been snatched from them, but at least one of the party was glad that they had got there by good British man-haulage.52 Edgar's thoughts are not recorded. The five thought that they had been beaten by two or three weeks and now they had to face the return home. Scott wondered if they had the strength. Bowers wrote to his mother that they were all fit and well and should, with luck, catch the ship in time for the news. He thought that he could not have better companions and that five was a pleasant little crowd when he was so far from home. To his sister, however, 'Birdie' admitted that they were losing strength and that he felt very weary at the end of a long busy march.53\n\nOn the 17th, they marched in the coldest conditions that Wilson could ever remember.54 There was a force five gale and 54\u00b0 of frost. They had, in fact, endured colder temperatures, but now loss of their body fat cruelly reduced their resistance to cold. They made their own exact British calculation for the Pole (about half a mile from Amundsen's flag)55 and 'Birdie' wrote to his mother from 'the apex of the earth'.56 The conditions were so awful that after five hours on the march, Edgar's hands (in spite of double woollen and fur mitts) were so cold that the team stopped and treated themselves to a good 'week-end' lunch with pieces of chocolate. Edgar smoked a cigarette brought by Wilson, a queer taste after weeks without tobacco.\n\nAt the point that the British judged to be the Pole, they flew the Queen Mother's (Queen Alexandra) Union Jack and their own flags and took photos. In these images the men are so engulfed by clothes that it is difficult to comment on their appearance, but Edgar's nose looks white, his face sunken. In the afternoon they passed the Norwegian's most southerly camp 'Polheim' and found a small tent with equipment and a letter for Scott to forward to the Norwegian King Haakon. Then they started north. 'Well we have turned our back now on the goal of our ambition with sore feelings and must face our 800 miles of solid dragging \u2013 and goodbye to most of our day-dreams!'57 On the day the British reached the Pole, Amundsen and his companions were only a week away from their quarters in the Bay of Whales having done the journey in ninety-nine days. They had taken fifty-two dogs and killed twenty-four of them in comparison to Scott's lack of dogs on the Plateau.58\n\nIn the weeks between the Norwegian and British arrival at the Pole the Antarctic winter had begun. The temperature was already dropping below zero. The wind blew from the south, whistling around the tent at night.59 Their successful return depended both on speed and their ability to pick up their food cairns, and both these were to cause problems. The return was doomed \u2013 all five men were to die.\n\nThey had to begin by pulling up a rise. The Pole is lower than the highest part of the plateau, which had to be climbed before the descent to the Beardmore Glacier. The return journey began well;60 wind from the south allowed them to use their floor cloth as a makeshift sail to help with the sledge, but as the wind strengthened, the snow blew in drifts and this made it difficult to pick up the tracks from their outward journey and they sometimes had to unharness and search for the tracks. Poor visibility made picking up the cairns difficult too. Edgar's fingertips were badly blistered and the snow surface became like sand. Sledge hauling was exhausting and appallingly monotonous. On 20 January Oates recorded that one of his toes had turned black.\n\nEdgar was deteriorating too. As well as the problems that beset them all \u2013 malnutrition, loss of body fat and muscle (as the heaviest man in the party he was the most affected by the deficiency in calories), dehydration, lack of vitamins, low body temperature and problems with altitude \u2013 he suffered from specific problems; his cut hand (the injury on 31 December) and a postulated brain injury that was caused, it is suggested, by relatively minor falls into crevasses.61\n\nHis hand trauma is well recorded. On 7 January, well before the Pole and seven days after the initial injury, the cut had 'a lot of pus in it'.62 Scott commented on the same day that the cut was nasty and secondary to sledge making.63 On 17 January, as the British team searched for their exact South Pole position, Wilson recorded that they stopped and camped for lunch because of Edgar's cold hands (Oates and Bowers, as well as Edgar, had bad frostbite of their noses and cheeks too).64 On the 23rd Scott wrote that Edgar was far from well; 'There is no doubt that (Edgar) Evans is a good deal run down, his fingers are badly blistered and his nose is rather seriously congested with frequent frostbites. He is very much annoyed with himself which is not a good sign.'65 On the 25th, Edgar's fingers and nose were in a bad state (and Oates was suffering from a very cold foot). On 28 January, as they pulled north in the biting wind, Wilson commented again on Edgar's badly blistered fingertips.66 By the end of the month, when they were still 600 miles from base, Edgar's nails were falling off, the fingers were raw and oozing;67 it was agonising for Edgar to remove his mitts and gloves. Scott wrote that Edgar's hands were really bad and to his (Scott's) surprise, he was showing signs of losing heart.68 To this, Scott added that he was disappointed in Edgar.\n\nWilson dressed Edgar's fingers every day with an antiseptic solution, melting snow with a spir it lamp. Although the fingers were still 'quite sweet' (not apparently infected) on 4 February, by the following day they were suppurating and his nose was 'bad and rotten looking'.69 Wilson wrote that Edgar was feeling the cold a lot and always getting frostbitten. He was visibly thinner.\n\nEdgar's 'head injury' is not recorded as a major event. On 4 February, Wilson wrote that Scott and Edgar had fallen in a crevasse to their waists70; Scott wrote that it was the second fall for Edgar.71\n\nThe return was a race between the season, the conditions, the men's fitness and their food supplies. By 31 January they reached Three Degree depot, the last depot on the plateau, 180 miles from the Pole. They picked up Bowers' skis, a week's food supply and a note from Lieutenant Evans before progressing north, where the Beardmore, with its chaos of crevasses, awaited them. They reached the rim of the plateau on 4 February. Not only did Edgar fall into a glacier that day, but Scott described him as becoming rather dull and incapable.72 From the rim of the plateau they could see those rocks, which were the upper markers of the Beardmore Glacier. It was a relief to see rocky outcrops after days and days of featureless whiteness. The party's initial progress down the glacier was adequate, but after three days spent threading their way through crevasse fields to finally reach the Upper Glacier Depot, Edgar deteriorated significantly. Daily distances were halved. He was now incapable of helping with the camp work and was holding the party up. His hands were festering, his nose looked awful, he became withdrawn and unlike himself, and was feeling the cold terribly (in spite of the fact that they were now taking extra food, seven days rations in six days).73 Oates, too, was deteriorating, his toes were black and his nose and cheeks were 'dead yellow'.74\n\nIn spite of these anxieties Wilson took some time on 7 February to collect rock and fossil specimens. On the 8th, they spent half a day collecting more. Wilson was a passionate investigator. The Polar expedition gave a unique opportunity for study of the Beardmore Glacier and collection from this fascinating area had always been part of the scientific programme. Also Scott may have thought that to give Edgar (and Oates) a rest would be beneficial. In the event, the specimens that were collected were later found to have embedded fern-like fossil leaves and stems of Glosopteris, a plant that flourishes in warm temperate climates. These would give incontestable proof of profound changes in the earth's climate and show that Antarctica had once formed part of a great, warm, southern continent.\n\nOn 11 February, in hazy and distorting light, they got lost in a maze of ridges, getting more and more despondent. Food became the dominant anxiety. Could they find the next depot? Breakfast was one biscuit, then they had a single meal left.To their exquisite relief they stumbled on their depot, but, because they were not covering the necessary daily distances to get them safely to the next depot, they had to reduce their rations making three meals of pemmican stretch to four. When they left the mid-glacier depot on 13 February they had only enough food for just over three days. On the 14th, Scott wrote that Edgar showed them a huge blister on his foot; this delayed the march whilst his crampons were adjusted. Scott felt that Edgar was getting worse continually. His slowness added to the over-whelming anxieties about food, now reduced to an evening meal of biscuit with a thin hoosh of pemmican. By 15 February they were approximately 18 miles from the Lower Glacier Depot.\n\nEdgar collapsed on 16 February. He was giddy and could not even walk beside the sledge. The party camped, but by this time they had only enough food for a day, so no further delay could be considered. The 17 February started without a premonition of its eventual tragic outcome, although by now Edgar was a shambling caricature of his former self. His foot worked out of his ski shoe (the shoe that he had manufactured) and he stayed behind to readjust the shoe. The remainder of the party went on, but when they saw he was not coming behind them, they stopped and cooked a meal. When still he did not arrive, they went back to look for him and found he had collapsed again, his clothes were dishevelled and he was crawling over the snow. He talked slowly and said, when asked what had happened, that he did not know, but he thought he must have fainted. When he tried to walk, he collapsed again. Oates stayed with him whilst Wilson, Scott and Bowers went back for the sledge. He lapsed into unconsciousness and died two hours after reaching the tent. His companions made a prayer over his body, covered it and left it to be engulfed by snow. At the time of his death he had been 109 days on a diet low in all vitamins and wholly lacking in vitamin C. He died three weeks before news of Amundsen's victory was blazoned around the world.\n\nHis companions obviously went over the events. But this was no time for prolonged reflection; they were in a stark struggle for survival themselves. They decided, however, that he had begun to get weak before the Pole and that the downward path was caused by his fingers, his falls and his loss of confidence in himself.\n\nThey left the foot of the glacier on 19 February. Bowers' diary finished on the 25th, Wilson's, with no warning, on the 27th. At the Southern Barrier depot there was no paraffin.75 In the Middle Barrier depot, oil was short again. On 16 or 17 March (Scott had lost track of dates), Scott said that Oates had announced that he could not continue. He had gan-grene (needing treatment by amputation), his leg was unbearably painful, his hands were frostbitten and he could not feed himself. Months before, at Cape Evans, Oates had said that it would be a sick man's duty to eliminate himself. Now he held out until he could see that there was no possibility of surviving, and then he ended his life by crawling out of the tent to die on the snow in the freezing temperatures.Wilson had a supply of opium and morphine and Scott had ordered him to hand these out. Whether Oates could have taken the tablets before crawling out of the tent is debatable, his hands were so bad that Wilson was feeding him. It seems reasonable to surmise, however, that if he had come to his decision earlier, the remaining three might have been able to progress faster.\n\nScott,Wilson and Bowers made their final camp 11 miles south of One Ton Depot. There they per ished slowly. On 18 November 1912, eight months later, a search party found the tent along with their bodies and graphic accounts of their doomed return.\n\n### Notes\n\n1 Ed. Jones, M., _Robert Falcon Scott Journals_ , Oxford University Press, 2005, p. 287.\n\n2 Ibid., p. 287.\n\n3 Scott, R.F., _Preliminary lecture on Southern Journey_ , 1911, SPRI, MS 1453\/28: D.\n\n4 Scott, R.F., _Preliminary lecture on Southern Journey_ , 1911, SPRI, MS 1453\/28: D.\n\n5 _Scott's Last Expedition The Journals_ , Carroll and Graff, New York, 1996, p. 196.\n\n6 Riffenburg, B., _Nimrod_ , Bloomsbury, London, 2004, p. 120.\n\n7 Fisher, M & J., _Shackleton and the Antarctic_ , Houghton Mifflin Company, Boston, 1958, p. 107.\n\n8 Bowers, H., _Journals Relating to the British Antarctic Expedition, 1910\u20131912_ MS 1505\/3\/5\/9; BJ.\n\n9 Ed. Jones, M., _Robert Falcon Scott, Journals_ , Oxford University Press, Oxford, p. 302.\n\n10 Wilson, E.A., _Letter to Mr and Mrs Reginald Smith_ , SPRI, MS.599\/142\/9\/D.\n\n11 _Evening Post_ , Vol. LXXXV, Issue 38, 14\/02\/1913.\n\n12 _South Wales Daily Post_ , 13\/02\/1913.\n\n13 Soloman, S., _The Coldest March_ , Yale University Press, London, 2001, p. 165.\n\n14 Ibid., p. 192.\n\n15 _Scott's Last Expedition The Journals_ , Carroll and Graff, New York, 1996, p.323.\n\n16 Cherry-Garrard, A. _Diary_ , SPRI, MS 559\/4; BJ, 24\/10\/1911.\n\n17 Ed. King, H.G.R., _Diary of the Terra Nova Expedition to the Antarctic_ , Blandford Press, London, 1972, p. 183.\n\n18 Ed. Jones, M., _Robert Falcon Scott Journals_. Oxford University Press, Oxford, 2005, p. 303.\n\n19 Ibid., p. 312.\n\n20 Bowers, H., _Journals relating to the British Antarctic Expedition 1910\u20131912_ , SPRI, MS 1505\/3\/5\/9; BJ.\n\n21 A transparent protective layer under the eyelid that can cover the eye surface, protecting and moistening it.\n\n22 Although both dogs and ponies have nictitating membranes the exposed corneal surface is much bigger in the pony than the dog.The membrane has no blood supply and the cold sustained for long periods of movement may have resulted in corneal damage more easily in the ponies. Personal communication, Professor Peter Bedford, 2011.\n\n23 Ed. Jones, M., _Robert Falcon Scott Journals_ , Oxford University Press, 2005, p. 334.\n\n24 Cherry-Garrard, A., _Diary 03\/11\/1911\u201328\/01\/1912_. SPRI, MS 559\/5: BJ.\n\n25 Priestley, R., Lecture, _The Psychology of Exploration_ , SPRI, MS 1097\/16\/1; D.\n\n26 Ed. King, H.G.R., _Diary of the Terra Nova Expedition to the Antarctic_ , Blandford Press, London, 1972, p. 206.\n\n27 Ellis, A.R., _Under Scott's Command Lashly's Antarctic Diaries_ ,Victor Gollancz, London, 1969, p. 126.\n\n28 Bowers, H., _Journals relating to the British Antarctic Expedition 1910_ \u2013 _1912_ , SPRI, MS 1505\/3\/5\/9; BJ.\n\n29 Ibid.\n\n30 Seaver, G., _'Birdie' Bowers of the Antarctic_ , John Murray, London, 1947, p. 239.\n\n31 Ed. Jones, M. _Scott's Last Expedition The Journals_ , Carroll and Graff, New York, 1996, p. 339.\n\n32 Bowers, H., _Journals relating to the British Antarctic expedition, 1910\u20131912_ , SPRI, MS 1505\/3\/5\/9; BJ.\n\n33 Soloman, S., _The Coldest March_ , Yale University Press, London, 2001, p. 293.\n\n34 The letters were not preserved.\n\n35 Bowers, H., _Journals relating to the British Antarctic Expedition, 1911_ , SPRI MS 1505\/3\/5\/9; BJ.\n\n36 Fiennes, R., _Captain Scott_ , Hodder and Stoughton, London, 2003, p. 285.\n\n37 Ed. King, H.G.R., _Diary of the Terra Nova Expedition to the Antarctic_ , Blandford Press, London, 1972, p. 217.\n\n38 Soloman, S., _The Coldest March_ ,Yale University Press, New Haven and London, 2001, p. 190.\n\n39 Personal communication, Dr Edward Coats, 2010. (From the Omega Challenge Race to the pole 2009.)\n\n40 Seaver, G., _'Birdie' Bowers of the Antarctic_ , John Murray, London, 1947, p. 245.\n\n41 Ellis, A.R., _Under Scott's Command: Lashly's Antarctic Diaries_ ,Victor Gollancz, London, 1969, p. 132.\n\n42 Ed. Jones, M. _Scott's Last Expedition: The Journals_ , Carroll and Graff, New York, 1996, p. 363.\n\n43 _Daily Mirror_ , 22\/05\/1913.\n\n44 Lieutenant Evans returned to England with the relief ship.\n\n45 Ed. Jones, M. _Scott's Last Expedition: The Journals_ , Carroll and Graff, New York, 1996, p. 369.\n\n46 Ibid., p. 370.\n\n47 Ibid., p. 373.\n\n48 Ibid., p. 374.\n\n49 Ibid., p. 375.\n\n50 Ibid., p. 376.\n\n51 Ibid., _Appendix 111, 'Significant changes to Scott's original Base and Sledging Journals'_ , p. 470.\n\n52 Bowers, H., _Journals relating to the British Antarctic Expedition, 1910_ \u2013 _12_ , SPRI MS 1505\/3\/5\/9: BJ.\n\n53 Ibid.\n\n54 Ed. King, H.G.R., _Diary of the Terra Nova Expedition to the Antarctic_ , Blandford Press, London, 1972, p. 232.\n\n55 _Daily Mirror_ , 23\/05\/1913, (Commander Evans' Lecture in the Albert Hall).\n\n56 Bowers, H., _Journals relating to the British Antarctic Expedition, 1911_ , SPRI MS 1505\/3\/5\/9.\n\n57 Ibid., p. 378.\n\n58 At Cape Evans, some skeletons of Scott's dogs remain with collars and chains still attached.\n\n59 Bowers, H., _Journals relating to the British Antarctic Expedition, 1911_ , SPRI MS 1505\/3\/5\/9.\n\n60 Evening Post,Vol. XXXV Issue 38 14\/02\/1913.\n\n61 Ed. Jones, M., _Scott's Last Expedition: The Journals_ , Carroll and Graff, New York,1996, p. 397.\n\n62 Ed. King, H.G.R., _Diary of the Terra Nova Expedition to the Antarctic_ , Blandford Press, London, 1972, p. 230.\n\n63 Ed. Jones, M., _Scott's Last Expedition:The Journals_ , Carroll and Graff, New York, 1996, p. 368.\n\n64 Ibid., p. 376.\n\n65 Ibid., p. 383.\n\n66 Ed. King, H.G.R., _Diary of the Terra Nova Expedition to the Antarctic_ , Blandford Press, London, 1972, p. 238.\n\n67 Ibid., p. 289.\n\n68 Ed. Jones, M., _Scott's Last Expedition The Journals_ , Carroll and Graff, New York, 1996, p. 387.\n\n69 Ed. King, H.G.R., _Diary of the Terra Nova Expedition to the Antarctic_ , Blandford Press, London, 1972, p. 240.\n\n70 Ibid, p. 239.\n\n71 Ed. Jones, M., _Scott's Last Expedition : The Journals_ , Carroll and Graff, New York, 1996, p. 389.\n\n72 Ibid., p. 390.\n\n73 Ed. King, H.G.R., _Diary of the Terra Nova Expedition to the Antarctic_ , Blandford Press, London, 1972, p. 239.\n\n74 Ibid., p. 240.\n\n75 The paraffin had leaked out because of inadequate seals on the tins that had allowed evaporation.\n\n## 15\n\n## The Aftermath\n\nThe news was cabled to Britain on 11 February 1913 and the reality of the disaster flew immediately around the world. Amongst the many letters and messages that had been found in the tent was Scott's 'Message to the Public', which (though this may not have been Scott's intention) forcibly focussed attention on Edgar's deterioration as a most significant contribution to the failure and death of the whole party. Over the next months in some quarters, Edgar was to be stigmatised as being the primary cause of the tragedy, not only being the first to die, but also as the only member of the party who was not of officer status.\n\nScott wrote that 'the advance party would have returned to the glacier in fine form and with a surplus of food, but for the astonishing failure of the man whom we had least expected to fail. Edgar Evans was thought the strongest man of the party'.1 After this Edgar became labelled as the 'Strong Man', with the implication that physical strength was his only asset, an implication often made by those with no knowledge of his contributions to Scott's expeditions. When the official report was telegraphed to London, two paragraphs from Scott's sledging journals were quoted in full; one relating to Edgar's death, the other, Scott's account of Titus Oates' death. The contrast in the modes of death, the self-sacrificing Captain and the Petty Officer (who held them up before he finally died a natural death), was gripping.\n\nSome of the early reports that made him the scapegoat were refuted; the early rumour that the fate of the Southern Party was sealed when Edgar's four companions had to drag him hundreds of miles on a sledge was denied,2 and Lieutenant Teddy Evans stated that reports that Edgar had gone mad were cruel, scandalous and without foundation.3 However, his four companions, notably Oates, who also seriously slowed the party's progress, were hailed as heroes.\n\nThree days after the official report, one and a half million children in elementary schools throughout the country gathered to hear 'The Immortal Story of Captain Scott's Expedition'. The 'Message' and Scott's account of Oates' death were quoted in full.4 The scene was set. Many Edwardians saw a tenuous relationship between physical strength, mental capacity and social status,5 and there are photographs and descriptions of Edgar as the 'strong man' which emphasised how mere strength did not necessarily imply a good character. Self-control and self-discipline were viewed as the epitome of masculinity, particularly for the middle or upper classes. It was the core of the heroic character and Edgar appeared to have lost control of his rational thought. He definitely did not represent the heroic ideal. _The Times Weekly Edition_ quoted a report from Christchurch, New Zealand, that said 'It would seem from what escaped some of the survivors that (Edgar) Evans lost his reason for the time, being under the great stress of fatigue and privation and was incapable of obeying orders, or assisting his hard pushed companions in the weary work of pulling the sledge. Indeed it became necessary in the end to lay him on it.'6 Edgar, the 'strong man', failed first and contributed significantly to the fate of his companions. Throughout February 1913, newspapers projected the view that Edgar had had serious psychological problems.\n\nRemarkably, none of the articles suggests a physical cause for Edgar's deterioration. Class and education were promoted as the important issues. Edgar's demise was hastened by his relative lack of education. He was not a gentleman and, therefore, apparently less able to withstand the strain of the return. The _Daily Express_ ' front-page article on 12 February 1913 quoted an 'eminent mental specialist' who stated that it was the uneducated man who would feel 'the mental strain and the dreary monotonous life amid eter nal snows' more acutely than the educated man.\n\nThe specialist wrote that:\n\n... experiments have proved that the brain is only kept active and healthy by the stimulus it is constantly receiving from the senses. The limitless white of the still snows would provide little stimulus for the eye to transmit. The deadly silence would deprive the ear of work. The monotony of the food would prevent the brain receiving the stimulus of a new taste. To an educated man this strain would be bad enough but he would be able to stimulate his brain with his store of learning... The absence of such a stimulus in an uneducated man such as presumably Seaman Evans would have been, might have been succeeded by a kind of self-mesmerism followed by mania and the delusion that he was being kept from food and home, both close at hand.7\n\nIn short, Edgar's education had not given him the psychological reserve to cope with the conditions, though it seems implausible nowadays that a knowledge of Virgil was considered helpful to a man battling against Antarctica's furies. Edgar's behaviour was contrasted unfavourably with the heroism of Oates and his comrades Scott, Wilson and Lieutenant Bowers.\n\nHistorian Professor Max Jones states that the extent of the criticism should not be exaggerated and that most commentators did not single out Edgar particularly as the cause of the disaster.8 This may be so, but the effect of the articles that were published must have been devastating to his wife, children and relatives in South Wales who were faced with the intimation, made by clever, educated men, that Edgar a seaman, had not only caused the death of the man he sincerely admired, but other important members of the 'gentleman class'. His oldest child, Norman, would have had taunts at school; his father had let the side down.The family's hero had been demoted to anti-hero status.\n\nHis children must have been devastated when they saw the Player's cigarette card series. Player's produced small coloured cards to go in their cigarette packets, a collection avidly collected by children. In the series on Polar exploration the first four are: Captain Scott, Commander Evans, Captain Oates and Dr Wilson. Number six is Lieutenant Bowers, but number five \u2013 'Taking Sea Temperatures' \u2013 was presumably the proposed illustration of Edgar, deliberately omitted. Other cards in this British series include a sledge party crossing a crevasse, sledge flags, Ad\u00e9lie penguins, the victorious Amundsen and his compatriot, Oscar Wisting, at the South Pole. Edgar's omission is marked and extraordinary.\n\n_Like English Gentlemen: To Peter Scott from the Author of Where's Master?_ (actually H.D. Rawnsley) was a children's book published in 1913. One can only hope that neither Edgar's children nor their friends read it: 'There were four men with the hero at the South Pole and their names are worth remembering, One was Petty Officer Edgar Evans, a great big brawny seaman who had been with him for many years.'9 The book recounts how they reached the Pole and started their return, 'but Evans, the man of mighty muscles, seemed to have lost his strength. He was always a little behind the others, found it harder than they to pull himself out of the snow drifts... He stopped dead \"I'm done\", said he10... But they were English gentlemen these four, the hero and Dr Wilson and Captain Oates and Lieutenant Bowers and so, such a thing as leaving Evans behind never came into their heads.'11 The diatribe continues: 'they counted the cost. They were willing to pay the price. Even if the price was to be their own lives... they were English gentlemen'.12 In contrast to Edgar, when Oates failed, his whole thought was for his comrades to save themselves. He knew, when the time came, 'how English Gentlemen die'.13\n\nRawnsley also wrote a Sonnet Sequence dedicated to the Antarctic Heroes. The Seventh Sonnet, dedicated to Edgar, contains the lines:\n\nAh, well for him he died, nor ever knew\n\nHow his o'er wearied stumbling forward drew\n\nDeath's snare about his friends to hold them fast14\n\nThe insidious damage was done. The feeling that Edgar had somehow failed because of his class persisted for years. The scientist and explorer Brian Roberts15 wrote in the 1930s: 'Science is no inspiration to those who have only done manual labour... This was surely the explanation of Seaman Evans' unaccountable breakdown when Scott's party realised they were not first at the Pole. The others knew that the true value of their journey had not been lost but to him it must have seemed that all their effort had been in vain.'16\n\nIt took years for Edgar to regain his rightful status as one of the Antarctic heroes. Gary Gregor, the librarian at Swansea Museum, wrote the first biography of his life as late as 1995.\n\n### Notes\n\n1 Arranged Huxley, L., _Scott's Last Expedition Vol.1_ , Smith Elder, London, 1913, p. 605.\n\n2 Pictures Past, The National Library of New Zealand, Evening Post, Vol. LXXXV, Issue 40, 17\/02\/1913, p. 7.\n\n3 _Daily Mail_ , 15\/02\/1913.\n\n4 Ed. Cubitt, G.; Warren, A., _The King Upon His Knees_ (Chap. 6 by Jones, M.) Manchester University Press, 200, p. 110.\n\n5 Jones, M., _The Last Great Quest_ , Oxford University Press, Oxford, 2003, p. 111.\n\n6 _Times Weekly Edition_ , 21\/02\/1913.\n\n7 _Daily Express_ 12\/02\/1913, No. 4009.\n\n8 Jones, M., _The Last Great Quest_ , Oxford University Press, Oxford, 2003, p. 112.\n\n9 Anon, _Like English Gentlemen To Peter Scott_ , Hodder and Stoughton, London, [1913], p. 31.\n\n10 Ibid., p. 37.\n\n11 Ibid., p 41.\n\n12 Ibid., p. 43\n\n13 Ibid., p 48.\n\n14 Rawnsley, H. D., _To the Heroes of the Antarctic, A Sonnet Sequence_ British Review April 1913 80\u201384.\n\n15 Brian Birley Roberts 1912\u20131978. Member of British Graham Land Expedition 1934\u201337. Researched into cold climate clothing. Involved in drafting Antarctic Treaty of 1959.\n\n16 Brian Roberts, Personal Journals 1934\u201337 Vol. 1 British Graham Land Expedition Vol.1 01\/12\/1934\u201309\/12\/1935, p. 168.\n\n## 16\n\n## Why Did Edgar Die First?\n\nAmundsen wrote later that the main reason for the final tragedy was that the British had endured the terrific strain of man-hauling, whereas the Norwegians 'had dogs all the way to the Pole and all the way back to our base'.1 He said he never pulled anything at all.2 He was probably right in his conclusion insofar that if Scott had actually managed to get dogs on the plateau and kept them healthy, the dogs would have been able to pull the enfeebled travellers across the plateau and down to the Barrier, saving the explorers precious calories and time by allowing the injured to ride on the sledges. Certainly by the time the explorers died their bodies had been ravaged by a multiplicity of problems that were interdependent in contributing to their deaths. Conditions such as vitamin deficiency, malnutrition, dehydration and hypothermia all combined inextricably to accelerate the downward spiral, and Edgar, the heaviest man in the party, suffered the most. In addition, he and Captain Oates stand out with specific problems. Oates had an old war wound and gangrene. Edgar suffered from a cut hand that festered and needed constant attention; he had a fall that could have caused an intracerebral bleed and, as the only non-officer, he could have suffered from a feeling of cultural isolation.\n\nMalnutrition plagued the whole party; they were unaware of this but records of dreams of delicacies that vanished, food-shops that were shut and obsessive ruminations about food are commonplace. Edgar and his companions had been man-hauling since early December 1911. The summit rations of approximately 4,571 kcal per man per day had been started in those bleak days at the base of the Beardmore Glacier. These rations consisted of 1,054 kcal of protein, 1,953 kcal of fat and 1,564 kcal of carbohydrate.3 Amundsen's assault team's calorie intake was similar, but the two expeditions had totally different calorific demands. The British were man-hauling whilst the Norwegians travelled on sledges pulled by the dogs, or on skis without loads. Recent studies have shown that Scott's men would have used over 7,000 calories per day for the exhausting occupation of man-hauling.4 So although the diet contained the correct percentage of carbohydrate according to modern information,5 the total intake was too low. Taken from 7 December 1911, until Edgar died on 17 February 1912 (seventy-two days), each member of the team had built up a staggering calorie shortfall of approximately 175,000 calories. To fully comprehend what this deficit means, an average man eats 2,500 calories a day when performing his daily activities, so it is easy to see how this astounding calorie deficit, approximately equal to a single man's intake for seventy days in normal life, would have affected the explorers.\n\nWhen Sir Ranulph Fiennes and Dr Mike Stroud man-hauled across the Antarctic for sixty-eight days (fewer days than Edgar), they each lost 44lbs in weight.6 But, in addition, Edgar was the biggest man of the party. An individual's food requirements depends on his weight as well as his physical activity and Edgar's resting and hauling metabolic rate would have been higher than his companions; he would have needed a greater calorie intake to support his size. His rate of weight loss must have been considerably faster than his companions, and Scott wrote, 'we are pretty thin, especially Evans (he meant Edgar), but none of us are feeling worked out'.7 As Edgar lost weight he lost his insulating body fat.8 He would have lost muscle also,9 and this would have made man-hauling increasingly challenging. Cherry-Garrard wrote that Edgar must have had 'a most terrible time'. He went on, 'I think it is clear from the diaries that he had suffered very greatly without complaint. At home he would have been nursed in bed; here he must march'.10\n\nIntertwined with malnutrition is hypothermia. This is defined as a core body temperature of below 35\u00b0C or 95\u00b0F. The men were often pulling into the wind and sometimes into snow \u2013 a dangerous combination. The cold and his sweat reduced the insulation properties of Edgar's clothes to practically nothing.11 His temperature dropped, he shuddered in the wind and the only time he would have been reasonably comfortable was when he was actually pulling the sledge.12 The cold struck as soon as he stopped. Shivering and shuddering increases heat production,13 but as the concomitant disadvantage of increasing calorie requirements and the deadly combination of malnutrition, exhaustion, insufficient insulation and insufficient fluids decreased his ability to maintain body heat. Dr Wilson wrote that Edgar was 'feeling the cold a lot always getting frost bitten'.14 He had always had this tendency. It has been shown that a previous cold injury is a significant risk factor for frostbite,15 and this was the case with Edgar. As his body temperature fell he would fumble with accustomed tasks, he may have become forgetful16 and he may have felt sick. As his struggle continued he was increasingly at risk of irregular heart rhythms.17\n\nOn the high elevation of the Antarctic Plateau, there is a lower atmospheric pressure than at sea level. This means there is less oxygen to breathe as the air is 'thinner'. To maintain a sufficient level of oxygen in the blood, the explorers had to breathe rapidly. Each breath contains water vapour, as can be seen even at land level, when breath exhaled on cold dry days 'steams', and these droplets, which freeze instantly, are sometimes called 'the shower of life'. The explorers had to overbreathe constantly and, as a result, lost significant water vapour. The fluid lost in their breathing contributed to their dehydration.\n\nIn addition to the sweat and water vapour lost when they were man hauling, the problems continued when they got into their sleeping bags. When Edgar toggled down to rest in his bag, he knew he had no hope of an undisturbed sleep until the moisture from his breath and the perspiration from his clothes (which added to the icy layer already coating up the inside of the bag to a degree that it took over half an hour to wriggle into), warmed to something approaching body temperature.18 Until then the men shivered (and probably cursed) until something approaching comfort overtook them.\n\nOn 17 December Scott wrote that the team got thirsty and chipped up ice on the march as well as drinking 'a great deal' of water at the halts.19 In fact 'they were always thirsty'.20 In 1911, the importance of keeping well hydrated was not understood and the men drank about 6 pints (3,400ml) of fluid each day21 (mugs of tea at breakfast, lunch and dinner, plus fluid in the pemmican). This is completely inadequate. It is now known that enough fluid should be taken to keep the urine fairly dilute (i. e. a pale colour) and flowing freely. When, in 2009, the British team man-hauled across the Plateau to the Pole in the Omega 3 race (at about the same time of year as Scott, though ninety-seven years later), they covered more ground on a daily basis but drank between 6\u20138L, per day, virtually double Scott's intake. They drank 4\u20135L when they were pulling and the remainder in the tent. On this intake their urine was darker when they drank 6L, pale when they drank 8.22 In Antarctica the atmosphere is dry and sweat evaporates quickly from the skin, and in 1911 the men may not have actually realised that they were sweating, though they would probably have had headaches and felt short-tempered. Edgar may have been conscious of his heart racing. But even if the team had been aware of the need for the increased fluid intake, this would have been difficult to manage logistically. Enough fuel to heat ice or snow was a cause for concern on the return from the Pole when the fuel (paraffin) was found to have evaporated from the tins,23 and dehydration was to be a crucial factor in their fate.\n\nOf the general problems endured by all the party altitude sickness is an unlikely cause of Edgar's death. He had been on the Plateau for weeks and had acclimatised to the altitude during his exhausting climb up the Beardmore Glacier. Although people are sent back to sea level from the Plateau every year,24 the problem most frequently arises in those who have not had the time to acclimatise and nowadays the majority who work in the Antarctic have arrived by plane. Edgar and the remainder of the team had had time to acclimatise.\n\nVitamin deficiency unites the party's general problems with a specific problem in Edgar's case. Vitamin C deficiency has been suggested as the primary cause of Edgar's death on 17 February.25 This is unlikely, although the explorers had been on a diet deficient in vitamin C (on the Barrier, the Beardmore Glacier and the Plateau) for more than fifteen weeks. Summit rations also contained less of the other vitamins than the Barrier rations26 and had insufficient folic acid and vitamin B12. But clearly Edgar did not have overt scurvy; Dr Wilson, a careful reporter and not afraid to chart the men's medical problems, makes no mention of scurvy, a condition he was all too familiar with. However, Edgar must have had sub-clinical vitamin C deficiency, a condition that can exist without any clinical signs of scurvy whatsoever. We now know a good deal about scurvy, its causation and its signs. It is known that when a person previously saturated with vitamin C goes on a diet without fruit or vegetables, signs of scurvy do not appear for about sixteen weeks, after which time skin dryness and thickening occurs, followed by small petichiae \u2013 small purplish skin spots due to the release of a tiny amount of blood from very small blood vessels, the capillaries. Problems with the healing of cuts come later. However, blood levels of vitamin C reach low levels well before clinical signs develop.27\n\nEdgar always hated eating seal meat and its offal (which contains the vitamin), so his blood levels are very likely to have been low. But since he had no clinical signs of early scurvy, it is less likely that the fall into a crevasse up to his waist on 4 February would have caused a significant intracerebral bleed secondary to vitamin C deficiency. Scott would have been most unlikely to have allowed time to be spent looking for specimens on 8 February, however important this was for their scientific programme, if he had thought that Edgar was on an irreversible downward course. On 13 February, Edgar was able to cooperate in looking for a depot and raised hopes with the shout of 'depot ahead', (although his 'depot' was, in fact, just a shadow).28 He had no obvious localised weakness on one side of the body (which commonly happens in an intracerebral bleed) and there is no record of headaches. He was able to get into his skis and start in the traces on the day that he died, no doubt attempting to keep up till the end. The problems that were reported: 'no power to assist with camping work'29 and 'giving us serious anxiety'30 can equally well be ascribed to a number of conditions. His difficulty with his boots was caused, not by paralysis, but because of frostbitten hands. Scott's comment that Edgar was 'becoming rather dull and incapable'31 were made on the very day that Edgar had his fall, suggesting (as Scott had written previously), a problem that had been going on for far longer than a few hours.\n\nOther factors that could have contributed to Edgar's deterioration are: psychological problems and infection. Could the whole deterioration be secondary to psychological troubles? Dr Wilson recorded in his diary that he wrote for his family that Edgar's collapse was 'much to do with the fact that he had never been sick in his life and is now helpless with his hands frost-bitten'.32 But psychological problems are an unlikely suggestion. Edgar was an intelligent extrovert. He had good social skills and could talk equally well to officers and non-officers.33 34Although the concept of 'depression' did not exist in 1910, there are no positive clues to suggest this diagnosis, which is characteristically a recurrent condition. Edgar's behaviour before and between expeditions carries absolutely no hint of a 'personality wobble'. He had good family links and bonds to sustain him. He wanted to return to Antarctica and volunteered to do so. He contributed well until the final few weeks and on the day he died he was alert enough to ask for some string to tie up his boots. He may have been low, but his symptoms do not suggest serious, incapacitating depression. Debenham interpreted Edgar's low mood as being due to the fact that he (Edgar) thought that he had let Scott down, firstly by cutting his hand and then by delaying the party, i. e. that his psyche was responding to external factors, rather than being the primary cause of his problems. This was a comment supported by Scott's writing on 30 January, that Edgar had not been cheerful since the accident. If Edgar's plans for opening a pub in South Wales35 were dashed by him not getting to the South Pole first (and there is no obvious reason why they should have been; his friend PO Tom Crean ran a successful public house in Ireland until his death in 1938),36 Edgar's resilient, steely, gregarious mind would have readily turned to other enterprises. He was not a man to suffer from cultural isolation. He was too confident of himself.\n\nInfection may well have contributed to Edgar's death. Bacteria can exist in Antarctica's sub-zero temperatures.37 One such bacterium is Staphylococcus aureus, a bacterium commonly carried in the nose38 and sleeping bags,39 and the likely cause of Edgar's hand infection. Abscess formation typically occurs seven days after a wound becoming infected, as happened to Edgar. By February, his frostbitten fingers were suppurating and his nose was very bad (almost as bad as his fingers) and rotten looking. Infection would have increased his resting metabolic rate, so exacerbating his calorie requirements and further increasing his weight loss. A possible scenario is nasal carriage of staphylococci resulting in a wound infection in his hand and, following this, invasion of the blood stream, probably repeated.40 There need not have been any signs in the arm \u2013 the bacterium could have silently gained ascendancy and the final picture can be interpreted as low blood pressure and collapse related to infection.\n\nAnother infection that had been put forward as the cause of Edgar's death is anthrax: a bacterial infection particularly found in animals (in this case the ponies), that can be transmitted to humans.41 Typically, when anthrax affects the skin, an ulcer develops surrounded by fluid-filled blisters. It seems a less likely diagnosis here. It was Edgar's fingers and nails that persistently gave the greatest trouble. Anthrax was a condition that Dr Wilson would certainly have been familiar with from his training in St George's Hospital and he would have recorded his findings, particularly the characteristic black, necrotic (dead) tissue at the point where infection had started. Furthermore, pus is not a feature of anthrax and pus was a marked feature in Edgar's case. Also, no other members of the expedition, who had also been in close contact with the animals, showed evidence of the disease. Interestingly when floors in the Base were renovated recently and examined for anthrax, none was identified.42\n\nRoland Huntford writes that Edgar had been 'exposed to the risk of venereal disease',43 despite providing no supporting evidence. Whatever disease is implied by this statement, it would not have contributed to Edgar's death. The legal definition of venereal disease includes syphilis, gonorrhoea and chancroid (which can be ignored in this account).44 The genital ulcers of primary syphilis would have been noted on Edgar's medical examination, as would a rash related to secondary syphilis. In addition, none of the symptoms described in Edgar's case suggest the long-term complications of syphilis as a contributing factor to his death. However, it is impossible to disprove (or prove) whether Edgar had the post-acute phase of gonorrhoea. This could have caused discomfort in passing urine but would not interfere with his ability to man-haul and would have had no effect on his final days.\n\nAn intriguing suggestion was made by a Welsh television company that Scott, pushed beyond his limits by Edgar's incapacities and convinced that the team would do better without him, shot Edgar as he lay confused and helpless on the snow.45 This can be dismissed. Guns were not taken onto the plateau (every consideration was given to weight). When his companions found Edgar crawling around on the snow, Scott went back with Wilson and Bowers to get the sledge, leaving Oates with Edgar. Wilson, deeply religious and conscious of the sanctity of life, could never have written his untroubled farewell letters to his wife, family and friends, if there had been any suggestion of murder.\n\nAn interesting condition called paradoxical undressing could have affected Edgar in his last few hours. This forms a part of many hypothermic deaths. The term 'paradoxical' is used because the victim's temperature is already too low to sustain life, but as hypothermia tightens its insidious grip, the victim begins taking off his clothes. Scott wrote that Edgar's clothes were disarranged, his hands uncovered and frostbitten.46 The cause of paradoxical undressing, which even today causes police to assume some victims of hypothermia have been assaulted, is not clear. A possible explanation is that the hypothalamus, that part of the brain that regulates body temperature, finally fails and releases its control of the small blood vessels in the skin, allowing them to dilate so that the victim suddenly feels hot and throws his clothes off. This condition clearly was not the cause of Edgar's downward spiral, but may have contributed to his demise.\n\nWhen all these diverse medical conditions have been considered, it has to be asked if Edgar should have included in the Pole Party in the first place. The answer must be that he was a natural choice. Scott considered that he was strong, fit, a good companion and a representative of the lower deck. Of the other two lower deck representatives Lashly and Crean, Chief Stoker Lashly had already man-hauled the 400 miles from Corner Camp and was in no fit state for a gruelling further advance, so the choice was between Edgar and Crean. The choice in favour of Edgar was probably because Edgar was already a member of Scott's sledge team, a team that had proved how well it could pull. Scott had no reason to doubt Edgar's fitness so there was no reason to change the team's membership.47 Debenham wrote later supporting the choice of Edgar and saying that he 'did not think for a minute that \"Taff\" was chosen as the last of the five men'. Debenham thought that Scott had intended to take Edgar from the first, as the 'most skilful sledge master and rigger, the strongest man and unendingly cheerful'. Debenham understood the significance of Edgar's cut hand mentioned in Scott's diary and he (Debenham) wrote that after Edgar had cut his hand he was no longer cheerful, but he interpreted Edgar's slowing down as being firstly, 'because he thought he had let his Captain down by having the accident and secondly, later, because he knew he was delaying the party'.48 Debenham added that he 'thought that Scott added Oates as the last of the five as a reward for his management of the ponies and because he wanted the Army to be represented'.49 He wrote that if one could talk of an 'odd man out', it would be Oates; he was not an accomplished sledger and he had a leg wound from the Boer War. This was a comment supported by a note in Cherry-Garrard's journals in which he writes that Drs Atkinson and Wilson had discussed the final choice and agreed that Oates was very 'done up'. Dr Atkinson had said that he did not think that Oates wanted to go.50 Debenham reiterated that Edgar always got on well with officers (with the exception of Lieutenant Evans), and that he personally thought that his subsequent death was related to the cut hand.51 In Edgar, Scott chose a tried and true companion, an experienced and intelligent sledger, a cheerful, good-natured man. When it came to the return, any other man would have suffered comparably. Edgar's unique problems, his larger size and his hand infection, could not have been anticipated.\n\nWhen Edgar was clearly failing should his companions have given him the morphine tablets that Dr Wilson carried and allowed him to die painlessly before abandoning him to the snowy wastes? Although his death released them from this appalling ethical dilemma, two of those companions, at least, would not have left him if he had become incapable. The moral issue of what to do if one of them failed had obviously been discussed: Dr Wilson would never abandon a comrade. His whole life had been governed by a love of God and a desire to serve others. He had no fear of death. He would have stayed with Edgar; his creed, the essential condition that bound all dangerous enterprises, was that men must stand by each other in distress, even beyond the bounds of reason. Bowers, too, was deeply religious; to abandon Edgar would have been unthinkable. Furthermore, although Edgar was slowing their progress hugely, they did not consider that he was actually dying.\n\nIn relation to the remainder of the doomed party, after Edgar's death Scott said that the delays the party had suffered on the return had greatly weakened them by firstly making inroads into their surplus provisions and, secondarily, by making them later in the season than had been planned. This delay meant that the snow surface resembled sand \u2013 impossible to pull over. This well-known phenomenon is due to the fact that when the temperature falls to 30\u00b0 below zero, sledges cease to glide. The low temperature of between \u221230 and \u221240\u00b0 does much to explain the slowness of the British party on the Barrier. With a distance between depots of 70 miles and only enough food and fuel in each depot to cover that distance, the party had to average over 9 miles per day. For a week, however, the best march on the Barrier approximated 9 miles and, in the later stages, progress deteriorated to as low as 3. Their failure to maintain the required speed was undoubtedly due to Oates' breakdown, which became a tax on the party's energies. When they met persistent winds and frequent blizzards, they must have known that the outlook was hopeless. The _Times Weekly Edition_ wrote, however, that they 'never relinquished their gallant struggle and fought on to the bitter end'.52\n\nEdgar did not fail because he had not got to the Pole first. No one, neither his companions on that fateful journey nor the newspaper articles afterwards, seem to have grasped the basic fact that he was ill. The effects of the insufficient calories affected Edgar greatly as the largest and strongest of the party and his extra needs were exacerbated by his hand infection. These problems accelerated on the return journey, causing weakness and a greater susceptibility to the cold. The deterioration was so gradual that it was not understood by his companions, but in the light of modern understanding of body physiology it is possible to say that the collapse was due to a failure of the survival measures to maintain core body temperature and that his acquired infection is likely to be a pivotal cause of his premature demise. He probably had no idea what was happening in the toxic, confused state of his last few hours as he played out a twentieth-century Greek tragedy in which nature and malevolent fate combined to defeat him. But, to the last, he tried to obey and support Scott.\n\n### Notes\n\n1 _The Cambrian_ , 01\/02\/1913.\n\n2 Pound, R., _Evans of the Brook_ , Oxford University Press, Oxford, 1963, p. 119.\n\n3 Rogers, A.F., _The Death of Chief Petty Officer Evans_ , The Practitioner,Vol. 212, 1974, p. 576. Butter in the diet gave over half the vitamin A and Carotene requirements for work at 4,500 kcal per day, plus a little vitamin D. Milk powder in the biscuits provided insufficient quantities of thiamine, nicotinic acid, riboflavin, folic acid and vitamin B complex. There was no vitamin C.\n\n4 Fiennes, R., _Captain Scott_ , Hodder and Stoughton, London, 2003, p. 283.\n\n5 Stroud, M., _Nutrition Across Antarctica_ , BNF Nutrition Bulletin, 19, 1994, p. 150.\n\n6 Fiennes, R., _Captain Scott_ , Hodder and Stoughton, London, 2003, p. 285.\n\n7 Scott, R.F., _Scott's Last Expedition vol 1_ , Murray, London, 1935, p. 434.\n\n8 Ward, M.P.; Milledge, J.S.; West, J.B., _High Altitude Medicine and Physiology_ , Arnold, London, 2000, p. 171.\n\n9 Ibid., p. 168.\n\n10 Cherry-Garrard, A., _The Worst Journey in the World_ , Picador, London, 1994, p. 544.\n\n11 Ward, M.P., Milledge, J.S. West, J.B. _High Altitude Medicine and Physiology_ , Arnold, London, 2000, p. 296.\n\n12 Priestley, R.E., _The Psychology of Polar Exploration_ , SPRI, MS 1097\/16\/1; D.\n\n13 Ward, M.P.; Milledge, J.S.; West, J.B., _High Altitude Medicine and Physiology_ , Arnold, London, 2000, p. 296.\n\n14 Ed. King, H.G.R., _Edward Wilson, Diary of the Terra Nova Expedition to the Antarctic_ , Blandford Press, London, 1972, p. 240.\n\n15 Cattermole, T.J., _The epidemiology of cold injury in Antarctica_ , Aviation & Space Environmental Medicine, 1999, Alexandria,Virginia, USA. p. 135\u2013140.\n\n16 Ward, M.P.; Milledge, J.S.; West, J.B., _High Altitude Medicine and Physiology_ , Arnold, London, 2000, p. 297.\n\n17 Ibid., p. 298.\n\n18 Priestley, R.E., _The Psychology of Polar Exploration_ , MS 1097\/16\/1; D.\n\n19 Scott, R.F., _Scott's Last Expedition vol 1_ , Murray, London, 1935, p. 396.\n\n20 Ibid., p. 397.\n\n21 Bowers, H., Miscellaneous stores list (compiled with Robert Falcon Scott British Antarctic Expedition 1910\u20131913), SPRI, MS 1453\/30; D.\n\n22 Personal communication, Dr Edward Coats, member of the British team, 2010.\n\n23 This evaporation was due to insufficient seals on the tins.\n\n24 Fiennes, R., _Captain Scott_ , Hodder and Stoughton, London, 2003, p. 322.\n\n25 Rogers, A.F. _The Death of Chief Petty Officer Evans_ The Practitioner Vol. 212, 1974, p. 580.\n\n26 Ibid., p. 576. The rations contained little Thiamine, Riboflavine, Pyridoxine.\n\n27 Crandon, J.H.; Lund, C.C.; Dill, D.B., _Experimental Human Scurvy_ , New England Journal of Medicine, 1940, vol. 233, p. 353\u2013369.\n\n28 Scott, R.F. _Scott's Last Expedition vol 1_ , Murray, London, 1935, p. 444.\n\n29 Ibid., p.444.\n\n30 Ibid., p. 445.\n\n31 Ibid., p. 437.\n\n32 Ed. King, H.G.R., _Edward Wilson, Diary of the Terra Nova Expedition to the Antarctic_ , Blandford Press, London 1972, p. 243.\n\n33 Taylor, G., _Letter to Stanley Richard_ , 11\/06\/1962, Swansea Museum, Box 210, (Edgar Evans).\n\n34 Debenham, F., _Journal_ , 19\/01\/1911\u201308\/03\/1911 MS 279\/2: BJp, p. 54.\n\n35 Soloman, S., _The Coldest March_ ,Yale University Press, New Haven and London, 2001, p. 283.\n\n36 Smith, M., _An Unsung Hero,The Remarkable Story of Tom Crean_ , Headline Book Publishing, 2001, p. 309.\n\n37 Hadley, M.D., _Nasal carriage of staphylococci in an Antarctic Community_. The Staphylococci, Proceedings of the Alexander Ogston Centennial Conference, Aberdeen University Press, 1981, p. 241\u2013253.\n\n38 Ibid., p. 239.\n\n39 Ibid., p. 245.\n\n40 Personal communication, Professor T.H. Pennington, University of Aberdeen, 2006.\n\n41 Falckh, R.C.F., _The Death of Chief Petty Officer Evans_ , Polar Record, 1987, 23 (145) p. 397.\n\n42 Personal Communication, Robert Headland, Emeritus Associate, SPRI.\n\n43 Huntford, R., _Scott & Amundsen_, Hodder and Stoughton, London, 1979, p. 328.\n\n44 Chancroid is a sexually transmitted disease that causes a ragged ulcer at the site of the infection and is treated with antibiotics.\n\n45 Davies, J., _The Last Journey_ , Drama-documentary by Cardiff television and film company, Fflie. HTV Wales 31\/07\/2002, previously screened on Welsh TV company S4C, as _Y Daith Olaf_.\n\n46 Scott, R.F., _Scott's Last Expedition vol 1_ , Murray, London, 1935, p. 446.\n\n47 Crean came to the fore when on the return journey to base camp, Lieutenant Evans became dangerously ill with scurvy. Crean undertook an amazing 18 hour solo walk to Hut Point. He and Lashly were awarded the Albert medal for saving Lieutenant Evans' life.\n\n48 Debenham, F., Letter to Stanley Richards 25\/05\/1962, Swansea Museum, Box 201, Edgar Evans Red File.\n\n49 Ibid.\n\n50 Cherry-Garrard, A., _Diary_ , SPRI, MS 559\/4; BJ,.Vol 3 19\/02\/11\u201311\/10\/11 24\/10\/1911.\n\n51 Debenham, F., Letter to Stanley Richards 25\/05\/1962, Swansea Museum, Box 201, Edgar Evans Red File.\n\n52 _Times Weekly Edition_ , 21\/02\/1913.\n\n## Epilogue\n\n_Terra Nova_ (carrying the Cardiff flag)1 reached Cape Evans on 18 January 1913. The ship had been thoroughly cleaned and a celebratory meal prepared for the five who, it was assumed, had reached the Pole and come back safely. Their letters were in individual pillowcases with each man's name printed on it; _Terra Nova_ carried supplies of chocolate, cigars and champagne \u2013 festive luxuries for all the men who had been marooned in Antarctica for months.\n\nTeddy Evans had recovered from his scurvy. He had been promoted to Commander while he was in England and now he returned with the _Terra Nova_. He shouted across the water through a megaphone,'Are you all well?'2 The ominous pause that greeted his enquiry spoke volumes. The silence was followed eventually by Lieutenant Campbell's reply,'the Southern Party reached the South Pole on January 18th last year, but were all lost on the return journey \u2013 we have their records'.3 The news of the heroes' deaths was greeted with silent shock, flags were lowered to half mast, celebrations shelved and the letters re-stored, for return to those wives, mothers, family and friends who had written with such eager anticipation.\n\nA 9ft cross was erected on Observation Hill, the hill that overlooked Hut Point, Edgar's home on the _Discovery_ days. The cross recorded the names of the dead men and the final line of Tennyson's poem _Ulysses_ , a tribute chosen by Cherry-Garrard:\n\nTo strive, to seek, to find and not to yield.\n\nWhen _Terra Nova_ reached New Zealand, press contracts required her to lay out to sea for twenty-four hours after the sad news had been cabled to Britain (on 11 February) and the next of kin informed. It was 12 February 1913 when she eventually entered Lyttelton harbour, her white ensign flying at half-mast. Here the crew found 'the Empire \u2013 almost the civilised world \u2013 in mourning. It was as though they had lost great friends'.4\n\nAt home, Lois heard the news on 11 February. She was in Gower. When the expedition funds had become seriously depleted after a year in Antarctica, volunteers in the crew, including Edgar, had offered to forgo their pay for twelve months. The decision had had a domino effect on the dependents and Lois and her children had moved back to Gower to live with her parents. She was on the beach with her youngest son, Ralph, when the telegram arrived. It was from New Zealand, sent by Commander Teddy Evans and forwarded from Portsmouth. It read simply: 'Members wish to express deepest sympathy in your sad loss.'5\n\nThe older children heard rumours of the disaster at school, but the full extent of their loss was only clarified when a journalist from the _South Wales Daily Post_ arrived to interview Lois. The reporter, describing her condescendingly as 'quite a superior and refined little woman', wrote that she had said: 6\n\nI received a bundle of letters last May, which had been brought to New Zealand by Commander Evans when he left the party.7 They were about fifty in number and covered the period of a whole year. The last one which though undated appeared either to have been written in December 1911 or January of last year was written in pencil. It stated that he was only 150 miles from the Pole and that the party were in good health and very confident of success. Since then I have heard nothing until this morning.8\n\nFrom the first she stoically supported her man: she told the press, 'I have this consolation; my husband died bravely and it seems he did not have to undergo such suffering as the other members of the party went through.'9 Her widowed mother-in-law, Mrs Sarah Evans, also rallied to the cause: 'I was always proud of my boy and am prouder than ever to know that he died a hero's death.'10 But just a few days later a reporter from the _South Wales Daily Post_ found Mrs Evans distressed at the rumour that it was 'through Edgar that the other members of the party had lost their lives'; Mrs Evans said, 'I'm worried because I feel that if he hadn't broken down they \u2013 Captain Scott and the rest of them \u2013 would have been alive to-day.' She went on, 'I can't help thinking about it all the time ever since I read about them being \u2013 forced to wait for him... Perhaps it would have been better if they had left him behind.'11\n\nThe reporter wrote that despite all efforts, the 'worthy old dame refused to be comforted'.12\n\nA local memorial service was speedily arranged on Sunday 16 February in St Mary's church, Rhossili. Following this, mention was made in the _Gower Church Magazine_ of that damaging phrase in Captain Scott's last message about, the 'strong man of the party \u2013 the man whom we had least expected to fail'. But the _Gower Church Magazine_ went on: 'A thrill of sympathy was felt all over the world on receiving the tragic news of the death of the five heroes on their return journey after the discovery of the South Pole. Rich and poor have sent messages of heartfelt sympathy with those who have been stricken with grief and have suffered such a loss.'13\n\nMemorial Services were held for Edgar in Cardiff and in chapels and churches throughout Gower. In Swansea's Albert Hall, prayers were offered for the bereaved of Gower,14 and Lois did have valued support from many sources; Commander Teddy Evans had suffered his own bereavement when his wife, Hilda, died from peritonitis on the return from New Zealand. He had moved on from the days of his antagonism to Edgar and clearly empathised with Lois to whom he wrote a gracious and consoling letter:\n\nDear Mrs Evans,\n\nI am writing to sympathise with you on your terrible bereavement.\n\nYour husband died a gallant death on the return march from the Pole after faithfully serving his leader, Capt. Scott, through a most trying time\n\nHe lost his life for the honour of his country, and the British Navy will be proud of having possessed such a brave man. His 'grit' will for ever be an example to the lower deck, his ability was remarkable and I wish to convey to you from the whole expedition our sorrow.\n\nI also write to tell you of the admiration we felt for your dead husband.\n\nI shall soon be in England, and I will see that you and yours will never want. If you are in immediate need write at once to:\n\nMr Wilkinson Green\n\nSecretary to Sir Edgar Speyer, Bart,\n\n7 Lothbury, London EC,\n\nI cannot tell you how sorry I am for you.\n\nBelieve me,\n\nYour sincere friend\n\nEdward G.R. Evans\n\nCommander R.N.15\n\nThis generous and supportive letter must have been a talisman to Lois that she could treasure. A letter from one of her husband's officers to cherish, words that would sustain and guard her and the children against all slights. Teddy Evans was as good as his word. He travelled to meet Lois to hand over Edgar's pocket book (sealed with two government seals and only to be opened by her). Yet another Evans, this time the Acting Secretary of the British Antarctic Expedition Committee, visited her and guaranteed that she and her children would receive enough money to cover their needs, until the relief fund, organised by his committee was in actual operation.\n\nLois said afterwards that, in relation to finance, she was pleased with the arrangements made for her future. The Admiralty had announced that the two naval representatives on the expedition, Scott and Edgar, would be treated as if they had been killed in action, so Lois received Admiralty and government pensions of approximately \u00a391 p.a. and a lump sum of \u00a396. The Admiralty pension included money for her children (2 shillings per week for each child until the boys were 14 and Muriel was 16),16 though Lois had to prove her children were still alive each year before receiving the government pension.17 Additionally she received income of \u00a31,250 from a trust fund raised by voluntary subscription. International as well as national organisations had contributed to the fund; it was said that the widows and orphans were wards not only of England but of the Empire.18 As well as these monies she had Edgar's British Antarctic Expedition salary of \u00a344 per year.\n\nAll this was a relief to Lois; the total she received was a very significant figure to her, far more than a Petty Officer's salary and probably a larger sum than she had ever dreamed of (though the monies recluded her taking up an offer from the London Orphanage Asylum, to board and lodge and educate one of her children until the age of 15). The sum was, in fact, considerably less than the money allocated to Scott's widow Kathleen, who received a lump sum of \u00a32,676, Admiralty and government pensions of \u00a3325 p.a. and the income from a combined trust fund of \u00a312,000 in addition to Scott's British Antarctic Expedition salary and income from books and articles.19 But the settlements granted to Lois, relatively modest as they were, gave her independence and dignity.\n\nThere were some national tributes: The Royal Humane Society's tribute of 11 February recorded 'its deepest sympathy and condolence to the relatives of Seaman (Petty Officer) Edgar Evans R.N'.20 The British Schools and University Club of New York sent Lois a three-page address in careful calligraphy (saying that 'overcome in the Polar desert, they died for the honour of England').21 _The Scotsman_ of 17 February 1913 wrote that Edgar had been a splendid seaman and his Commander's faithful sledge comrade in the memorable inland expeditions. The _Daily Express_ started a fund by selling Scott Memorial Booklets (6 _d_ for twelve).\n\nLois was not present at the memorial service in St Paul's cathedral, on 14 February, which was reported in _The Times_ as a 'National Homage to the Dead'.22 The mourners were led by King George and attended by representatives of the Scott, Oates, Bowers and Wilson families, but not Lois. It is not clear if she was invited. If so, perhaps she simply had not got the funds, or the inclination to travel to London so soon after receiving the news that she was a widow. It was a big occasion; attending were government ministers, Ambassadors and 'ministers of foreign states', national services, scientific societies and 'representatives of official life'. The _Dead March_ from _Saul_ was introduced by a roll of drums and was one of the most moving parts of the ceremony, which concluded with the National Anthem.23 The Cathedral was full; an estimated 10,000 people were unable to get in. But Lois was pleased when, after the ceremony, Frank and Eliza Evans, the parents of Commander Evans, wrote to her saying that it was only the Will of Providence that had taken one Evans instead of another. 'We trust that you, your children and your husband's mother will in time recover from the effects of the terrible misfortune which has fallen upon you.' 24\n\nOn 26 July King George V, with Prince Louis of Battenberg in attendance (a Lord Commissioner of the Admiralty), received the widows of those who had lost their lives in the expedition at an investiture in Buckingham Palace. During the private ceremony the King presented the wives with the medals and clasps that had been awarded to their late husbands in an event that must have been overwhelming for Lois. These Polar Medals were octagonal and hung on ribbons on which engraved clasps were suspended, each clasp recording the dates of the expedition that the recipient had been on. Edgar had been awarded his Polar Medal and Clasp after the _Discovery_ expedition, so Lois was presented with his 'Antarctic 1910\u20131913' clasp by the King in recognition of her husband's role.25\n\nKing George then went on to decorate surviving members of the expedition with their Antarctic Medals and Clasps (plus additional Albert Medals26 to PO Crean and Chief Stoker William Lashly for their bravery in saving the life of Commander Evans). 'Birdie' Bowers' medal was then presented to his mother and Commander Evans accepted Oates' medal on behalf of the family.27 There was a problem, however, over Edgar's actual 1902\u201304 medal; Lois had sold it,28 presumably when she was in financial difficulties in the later years of the _Terra Nova_ expedition. Subsequently a duplicate medal was made and given Lois in 1914.29 Robert Swan carried this medal and the clasps to the South Pole on his 1984\u201386 expedition that followed Scott's 1911 route.30\n\nIn contrast to the other explorers, Edgar's death was not followed by an outpouring of Welsh (or British) patriotic grief. This can be explained in several ways. Firstly, those national newspapers which had expressed their reservations about Edgar's role, that he had 'let the side down', that his companions, by staying with him had imperilled their own lives,31 had resulted in unease and, consequently, a reluctance to bang the nationalistic drum with too much fervour. If it was true that Edgar had been the weak link that brought the party down, especially if he had not faced death with the dignity of Captain Oates, then this was an outcome that could not be eulogised. Secondly, there was local suspicion that Edgar had neglected his family.32 But Edgar had never meant to ignore his family. When he signed on he was due to retire from the Navy after a few years and he hoped that the years in Antarctica would actually secure their future. The expedition's finances were well out of his sphere. Other crew members left their wives and children. To criticise him for leaving Lois and the children in need is being wise after the event. Importantly, his family never doubted this; his father-in law and his wife praised him as a good father, son and husband. The third reason for reserve was the nationally deeply engrained Edwardian prejudices concerning rank and education.\n\nThere were a few attempts to raise support for a memorial in Wales. The Swansea newspaper _The Cambrian_ called for there to be... 'at Swansea or in Gower some permanent memorial to the honour of Petty Officer Evans, who thus links this locality with one of the most heroic exploits of the British race'.33 The _South Wales Daily Post_ reported the mayor of Swansea as saying, 'this is an occasion when the whole country will take the matter up. But there is also a local aspect and in movements of this kind Swansea has never been behind'.34 But virtually nothing happened. Calls for a permanent memorial in South Wales fell on deaf ears. No specific memorial for Edgar appeared in Cardiff or Swansea, though a clock tower in the shape of a lighthouse, with the names of the Polar Party above the door, was presented to Cardiff city and sited by the lake in a city park.\n\nThe only memorial erected in the early years was put up in 1914, by Edgar's loyal, determined, widow.\n\nThe _Gower Church Magazine_ reads:\n\nA beautiful memorial tablet has been erected in Rhossili Church to the memory of the late Mr Edgar Evans Chief Petty Officer, who accompanied Captain Scott to the utmost point of the South Pole, and who perished on the return journey, to be much lamented by his widow and widowed mother, both at the time the news reached them residing at Rhossili Parish.35\n\nThe tablet is inscribed:\n\n_To the Glory of God_\n\n_In the memory of_\n\n_EDGAR EVANS_36\n\nThe Rector, who had married Lois and Edgar nearly ten years earlier, told the congregation that Edgar would go down in history as 'one who was deemed worthy to be chosen among the few last out of the band of heroes to accompany Captain Scott to the South Pole'.37 The Reverend Lewis Hughes said that Scott had valued Edgar as 'the strong man of the party, one with a wonderful head, equal to any emergency and brave in the face of difficulty'.38 He added (optimistically) that Edgar would never be 'forgotten by his country or fade from its annals' and that the expedition had 're-taught a world growing more luxurious and effeminate, the glory of a  soldier's endurance and capacity for stern duty'. He concluded (including Edgar in his comments, in spite of the pejorative comments concerning Edgar's education) that the 'expedition encouraged the possession of scientific courage to the last'.39\n\nEdgar would not have cast himself in the heroic mould. He was a sailor who had made the most of his opportunities. Like many people he probably hoped to benefit from the remarkably successful ventures to which he had contributed. But Antarctica had become a fascination to him. Personal gain had not been the main motive for him taking part in sorties that had so significantly increased knowledge about the unknown continent for the benefit of Britain and the world. But equally, he did nothing to deserve the comments about his 'astonishing failure', which cast a long shadow. Mud sticks and Edgar was temporarily, if lightly, airbrushed out of the record of the Antarctic heroes. He was not mentioned in the 1920 or 1933 _Who's Who_ of Wales. He is still not in the _Dictionary of Eminent Welshmen_ or the _Dictionary of Welsh Biography_. Indeed, national recognition took years.\n\nEven after the Second World War, appreciation came in small, slow steps. In 1948, Lois, by now the only wife remaining of the Polar Party, was an honoured guest at the premiere of the Royal Command film _Scott of the Antarctic_ with her sons, Ralph and Norman. Edgar was played by the acclaimed actor, James Robertson Justice. The film gave belated recognition to Edgar's contribution to Scott's expeditions.\n\nIn 1954, Sarah Evans (the niece who had gone to wave goodbye to Edgar in Cardiff in 1910) recorded her reminiscences which were the basis of an article, _Edgar Evans: A Gower Hero_40 written by Dr Gwent Jones, a founder member of the Gower Society. Later, a piece appeared in the _South Wales Evening Post_.41 This reignited the enthusiasm of the curator of Swansea Museum, Mr Stanley Richards, who canvassed vigorously for greater recognition, at least in Wales, of Edgar's contributions. Mr Richards wrote about Edgar as _The Martyred Hero of Antarctica_ , and mounted a vigorous crusade on Edgar's behalf. Specifically he campaigned for a local memorial in Swansea.\n\nMajor recognition came initially from outside Wales. In 1964 HMS _Excellent_ , the Royal Naval Gunnery School at Whale Island, Portsmouth, built a new accommodation block for the petty officers; the building was named 'The Edgar Evans Building'. It had a plaque that commemorated Edgar and displayed a pair of original skis used during the expedition. This was an important piece of Naval social history; the building was the first to be named for a Petty Officer (rather than a famous admiral) and showed that, fifty years after his death, Edgar's contribution was being recognised \u2013 at least by the Navy. Members of Edgar's family attended the opening and the Second Sea Lord; Sir Royston Wright, spoke of Edgar's strength of character, devotion, loyalty and bravery.42 The buildings were replaced in 2009 by a bigger, better accommodation block with a conference room and ballroom, but which retained Edgar's name. Opened by The Princess Royal in the presence of Edgar's grandson, John, and his great-grandson, Joshua, the buildings were named after two naval war heroes: Chief Gunner Israel Harding VC,43 44 Sergeant Norman Fitch VC45 and Edgar. The plaque and skis remain in a prominent position.\n\nIn 1974, Dr A.F. Rogers, an Antarctic veteran, produced an article on Edgar's death in _The Practitioner_.46 The article was of extreme importance because it was the first to consider in an objective manner, _medical_ causes that could have contributed to Edgar's death. But critical assessments of Edgar's perceived disadvantages47 and the inference he was not wholly loyal to Scott,48 belittled his reputation in spite of other objective articles, which considered his medical problems.49 50\n\nFinally, in 1994, Swansea Council hosted a Civic Ceremony with Edgar's daughter Muriel, aged 87, as guest of honour. Camera artist Herbert Ponting's film of the _Terra Nova_ expedition was shown and the Lord Lieutenant of West Glamorgan presented a bust of Edgar to the city. Here at last was the recognition that Edgar deserved. Sir Michael Llewellyn, the Lord Lieutenant, described Edgar as a very courageous man and the Lord Mayor of Swansea, reminding his audience that Edgar had his roots firmly in south-west Gower and suggesting that perhaps his recognition had been 'much too long in coming'.51 Gary Gregor's appreciation of _Swansea's Antarctic Explorer_52 was published in 1995.\n\nAppreciation continued outside Wales. Books such as _Captain Scott_ by Ranulph Fiennes,53 _Antarctic Destinies_ by Stephanie Barczewski,54 and _A First Rate Tragedy_ by Diana Preston55 gave an unbiased assessment of Edgar's contribution. New Zealand survey expeditions preserved his name by naming two geographical features in the Ross Dependency after him: the Evans Piedmont Glacier, a low coastal ice sheet offVictoria Land named in the 1950s by a New Zealand survey party and the Evans N\u00e9v\u00e9, named by a New Zealand geological survey in the 1960s.\n\nEdgar's status and character is now being re-established after years of disparagement. Reappraisal of Scott's expedition, in relationship to current scientific knowledge, has produced physical reasons for his untimely death.\n\nThose perceived defects, Edgar's rank and relative lack of education (factors completely beyond his control), are now understood to have played an insignificant role. Any doctor today, giving as his first diagnosis that Edgar's deterioration was due primarily to non-physical causes, could reasonably be accused of negligence. The implication that the lack of rational thought reflects inferiority and that rational thought is evidence of man at his highest level (and can protect a man by supplying him with greater reserves in adversity), is surely wrong. To suggest that Edgar could have avoided death, or faced it with greater composure, grit and courage if he had had a better education, is fanciful. The consolations of education and philosophy can only go so far. Any man facing death is more likely, if he can think at all, to be thinking of his immediate necessities, rather than of Virgil or other thinkers' philosophies.\n\nIn relation to the contrasting ways that Edgar and Captain Oates met their deaths, the situations are completely different. Oates was credited with meeting death in a magnificent, gentlemanly way because he remained in control of himself. He was able to articulate his last thoughts for his mother and his regiment, whilst Edgar was not able to make decisions or leave messages. But the reason that Oates remained in control of his actions was that, in contrast to Edgar, he was fully conscious throughout, only too horribly aware of his agonising gangrenous foot (which it is reasonable to assume obliterated the comforting solaces of philosophy or education). Although Oates may have wished for the relief of diminished consciousness, he remained, perforce, fully cognisant. He was able to make a decision, therefore, on when he thought that all hope for recovery or survival was finished and when his condition was materially damaging his companion's chances. Then, he ended his life for the benefit of his comrades. If Edgar had been aware of what was happening to him, he might well have done the same, but in his toxic and confused state he had no control over the way he died.\n\nEdgar is now mentioned in the _Oxford Companion to the Literature of Wales_56 and the _Oxford Dictionary of National Biography_.57 He fully deserves his heroic place in Antarctic history: for his contributions to the _Discovery_ expedition when with Scott and Lashly, he was in the first group to travel on the plateau and, on their return, to make the striking geographical discoveries of the Dry Valleys in the Western Mountains; for his contributions to the _Terra Nova_ expedition, his expertise and practicality as sledging expert, his willingness to impart this knowledge to his colleagues, his humour and his self-control, right up until the last days of the final Polar Assault.\n\nScott appreciated his numerous gifts and Edgar reciprocated with a loyalty that endured despite the divisions of class, rank and education. He died as he had lived \u2013 doing his best.\n\n### Notes\n\n1 _Western Mail_ , 13\/02\/1913.\n\n2 _Nottingham Guardian_ , 13\/02\/1913.\n\n3 Pound, R., _Evans of the Brook_ , Oxford University Press, Oxford, 1963, p. 123.\n\n4 Cherry-Garrard, A., _The Worst Journey in the World_ , Picador, London, 2001, p. 593.\n\n5 _Morning Post_ , 12\/02\/1913.\n\n6 _Morning Post_ , 12\/02\/1913.\n\n7 Lieutenant Teddy Evans suffered very badly, almost fatally, from scurvy when he led the final Supporting Party back from the plateau in 1912. He returned to England on the _Terra Nova_.\n\n8 All these letters appear to have been destroyed.\n\n9 _Morning Post_ , 12\/02\/1913.\n\n10 _South Wales Daily Post_ , 12\/02\/1913.\n\n11 _Cambrian_ , 21\/02\/1913.\n\n12 Ibid.\n\n13 _Gower Church Magazine_ , 17\/02\/1913.\n\n14 Gregor, G.C., _Swansea's Antarctic Explorer: Edgar Evans 1876\u20131912_ , Swansea City Council, 1995, p. 73.\n\n15 _South Wales Daily Post_ , 13\/02\/1913.\n\n16 Jones, M., _The Last Great Quest_ , Oxford University Press, Oxford, 2003, p. 108.\n\n17 Ibid., p.107.\n\n18 _Times_ , 19\/02\/1913.\n\n19 Jones, M., _The Last Great Quest_ , Oxford University Press, Oxford, 2003, p. 108.\n\n20 Sotheby's catalogue, SPRI 07\/10\/1984.\n\n21 Ibid.\n\n22 _Times_ , Saturday, 15 February 1913, p. 8, issue 40136; Col A.\n\n23 _Western Mail_ , 15\/02\/1913.\n\n24 _Cambrian_ , 21\/02\/1913.\n\n25 Ibid.\n\n26 The Albert Medal was first issued in 1866 and discontinued in 1971. Originally issued for saving life at sea, it was extended in 1877 to cover saving life on land.\n\n27 _Times_ , Monday 28 July 1913, p. 9, Issue 40275, Col A. Court Circular.\n\n28 Personal communication, John Evans, Edgar's grandson, 2010.\n\n29 Yelverton, D., _Antarctic Unveiled_ , University Press of Colorado, Boulder, Appendix 8.\n\n30 Mear, R.; Swan, R., _In the Footsteps of Scott_ , Jonathan Cape, London, 1987. p. 142.\n\n31 _Times_ , 06\/11\/1913.\n\n32 Gregor, G., _Swansea's Antarctic Explorer Edgar Evans 1876\u20131912_ , Swansea City Council, Swansea, 1995, p. 82.\n\n33 _Cambrian_ , 14\/02\/1913.\n\n34 _South Wales Daily Post_ , 14\/02\/1913.\n\n35 _Gower Church Magazine_ , March 1914.\n\n36 There were two mistakes on this commemorative tablet. The words 'to seek, to strive, to find and not to yield' were written on it rather than, 'to strive, to seek, to find' and in the relief above the tablet the explorers were depicted with one ski-stick (as had been used in _Discovery_ days) rather than two.\n\n37 _Gower Church Magazine_ , March 1914.\n\n38 Ibid.\n\n39 Ibid.\n\n40 Jones, G., _Edgar Evans: A Gower Hero_ , Gower, 1954, vol. vii.\n\n41 _South Wales Evening Post_ , 05\/02\/1954.\n\n42 _Portsmouth Evening News_ , 18\/12\/1964.\n\n43 Victoria Cross. This is the highest military decoration. It is awarded for valour in the face of the enemy to members of the armed forces of the Commonwealth countries.\n\n44 Israel Harding, 1833\u20131917.Victoria Cross awarded for Harding's bravery in defusing a live shell that landed on HMS _Alexandra_ in Alexandria in1882. His action saved many lives.\n\n45 Norman Finch 1890\u20131966.Victoria Cross awarded for his defence of HMS _Vindictive_ at Zeebrugge in 1918, when, severely wounded, he continued to defend the ship against enemy fire, firing from an exposed position and saving many lives.\n\n46 Rogers, A.F., _The Death of Chief Petty Officer Evans_ , The Practitioner, _212_ , 1974.\n\n47 Huntford, R., _Scott and Amundsen_ , Hodder and Stoughton, London 1979, p. 522.\n\n48 Ibid., p. 520.\n\n49 Falckh, R.C.F., _The Death Of Chief Petty Officer Evans_ , Polar Record, 1987, 23(145).\n\n50 Williams, I., _Edward Wilson, medical aspects of his life and career_ , Polar Record, 2008, _44_ (228).\n\n51 Gregor, G., _Swansea's Antarctic Explorer Edgar Evans 1876\u20131912_ , Swansea City Council, Swansea, 1995, p. 82.\n\n52 Ibid.\n\n53 Fiennes, R., _Captain Scott_ , Hodder and Stoughton, London, 2003.\n\n54 Barczewski, S., _Antarctic Destinies_ , Hambledon Continuum, London, 2007.\n\n55 Preston, D., _A First Rate Tragedy_ , Constable, London, 1997.\n\n56 Ed. Stephens, M., _Oxford Companion to the Literature Of Wales_ , Oxford University Press, 1986, p. 187.\n\n57 _Oxford Dictionary of National Biography,_ Oxford University Press, 2004, p. 680\u2013681.\n\n## Plate Section\n\n1. Edgar as a young adult showing his proficiency badges. (Courtesy of Swansea Museum)\n\n2. Edgar's father in 1893. By this time he was a Quartermaster, sailing up the west coast of England from Swansea to Glasgow. (Courtesy of Keith Roberts)\n\n3. St Mary's church, Rhossili. (Courtesy of G.C. Gregor)\n\n4. Rhossili Bay. (Courtesy of G.C. Gregor)\n\n5. HMS _Ganges_ , a Training Hulk in Cornwall. (Wikipedia)\n\n6. SS _Discovery_. (By kind permission of Dundee Heritage Trust)\n\n7. The pack ice. (Courtesy of D.J. Williams)\n\n8. The ice Barrier. (Courtesy of D.J. Williams)\n\n9. Map of Antarctica showing Transantarctic mountain range. (Courtesy of D.J. Williams)\n\n10. A general view of the hut and Discovery at the bayside of Hut Point. (By kind permission of Dundee Heritage Trust)\n\n11. Southern depot parties preparing to start, 1902. (By kind permission of Dundee Heritage Trust)\n\n12. Sastrugi formed by the wind on Crater Hill. (By kind permission of Dundee Heritage Trust)\n\n13. Scott's hut at Cape Evans (named after Lieutenant 'Teddy' Evans), with Mount Erebus in the background. (Courtesy of Scott Polar Research Institute \u2013 SPRI)\n\n14. The Emperor penguin colony in Cape Crozier. (Image by Reginald Skelton. Courtesy of the Skelton Bequest to SPRI)\n\n15. _Discovery_ and the Aurora Australis. (Edward A. Wilson painting. Courtesy of Dundee Heritage Trust)\n\n16. The foot of the Ferrar Glacier. (NASA image)\n\n17. Modern picture of the field gun run which Edgar's team won for Portsmouth in 1907. (Courtesy of HMS _Excellent_ , Portsmouth Museum)\n\n18. The Western Depot Party, 1911. _From left to right_ : Griffith Taylor, Charles Wright, Lieutenant 'Teddy' Evans, Lieutenant 'Birdie' Bowers, Captain Scott, Frank Debenham, Sub-Lieutenant Tryggve Gran, PO Edgar Evans, PO Thomas Crean. (Courtesy of SPRI)\n\n19. Edgar on tour. (Courtesy of Adrian Raeside)\n\n20. The Ferrar and Koettlitz Glaciers, Cape Evans, Hut Point and the Dry Valley Map. (Courtesy of Peter Fretwell)\n\n21. Satellite image of the winter ice around Ross Island. Note that two large icebergs, A and B, have calved off the Ross Ice Shelf. A is nearly as big as Ross Island. B is approximately 178 miles long. (NASA image annotated by D.J. Williams)\n\n22. Edgar dressed for exploration. (Courtesy of SPRI)\n\n23. A dry valley. (NASA image)\n\n24. The journey to the Pole, 1911. (Map courtesy of D.J. Williams)\n\n25. The motor party led by Lieutenant 'Teddy' Evans, October 1911. (Courtesy of SPRI)\n\n26. Edgar Evans, 'Birdie' Bowers, Edward Wilson and Robert Scott in the tent. (Courtesy of SPRI)\n\n27. Tom Crean (left) repairing sleeping bags with Edgar Evans, during the winter months 1911. (Courtesy of SPRI)\n\n28. Edgar's modification to enable finneskoes to be securely fitted to skis, Antarctic winter 1911. (Courtesy of SPRI)\n\n29. Dr Atkinson's frostbitten fingers. (Courtesy of SPRI)\n\n30. Edgar Evans dressing Dr Atkinson's fingers. (Courtesy of SPRI)\n\n31. Edgar's naval memorial plaque fixed to an accommodation block named after Edgar in 1964 at HMS _Excellent_ , an important piece of Royal Naval social history. (Image courtesy of Jane Gregor)\n\n32. Lois' memorial for Edgar. In the memorial the men are shown using one ski stick rather than the two that were used in the _Terra Nova_ expedition. The quotation from Tennyson's _Ulysses_ should read: 'to strive, to seek, to find and not to yield.' (Courtesy of Jane Gregor)\n\n33. Edgar's widow Lois with Norman, one of their sons, at the premier of _Scott of the Antarctic_ , in 1948. (Courtesy of John Evans)\n\n34. Edgar's Polar medals taken to the South Pole by Swann in 1987. (Courtesy of G.C. Gregor)\n\n## Copyright\n\nFirst published in 2012\n\nThe History Press\n\nThe Mill, Brimscombe Port\n\nStroud, Gloucestershire, GL5 2QG\n\nwww.thehistorypress.co.uk\n\nThis ebook edition first published in 2012\n\nAll rights reserved\n\n\u00a9 Isobel Williams, 2012\n\nThe right of Isobel Williams, to be identified as the Author of this work has been asserted in accordance with the Copyrights, Designs and Patents Act 1988.\n\nThis ebook is copyright material and must not be copied, reproduced, transferred, distributed, leased, licensed or publicly performed or used in any way except as specifically permitted in writing by the publishers, as allowed under the terms and conditions under which it was purchased or as strictly permitted by applicable copyright law. Any unauthorised distribution or use of this text may be a direct infringement of the author's and publisher's rights, and those responsible may be liable in law accordingly.\n\nEPUB ISBN 978 0 7524 7760 2\n\nMOBI ISBN 978 0 7524 7759 6\n\nOriginal typesetting by The History Press\n\nEbook compilation by RefineCatch Limited, Bungay, Suffolk\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n_The Bad Wife Handbook_\n_The Bad Wife Handbook_\n\nRachel Zucker\n\nWESLEYAN POETRY\n\nWesleyan University Press  \nMIDDLETOWN, CONNECTICUT\nPublished by Wesleyan University Press, Middletown, CT 06459\n\nwww.wesleyan.edu\/wespress\n\n\u00a9 2007 by Rachel Zucker\n\nAll rights reserved\n\nPrinted in the United States of America\n\n5 4 3 2 1\n\nLibrary of Congress Cataloging-in-Publication Data\n\nZucker, Rachel.\n\nThe bad wife handbook \/ Rachel Zucker.\n\np. cm. -- (Wesleyan poetry)\n\nISBN-13: 978-0-8195-6846-5 (alk. paper)\n\nISBN-10: 0-8195-6846-5 (alk. paper)\n\nI. Title.\n\nPS3626.U26B33 2007\n\n811'.6--dc222007019778\n\n\"Cover photograph by Celeste Fichter.\"\n_In spite of_  \n _& because,  \nJoshua Goren_\nContents\n\nMonogamist 3\n\nThe Museum of Accidents 4\n\nCodary 5\n\nThe Secret Room 6\n\nFirmament 7\n\nMonogamist 8\n\nGalaxies Rushing Away 9\n\nAxon, Dendrite, Rain 10\n\nRhyme, Lascivious Matchmaker 11\n\nHermeneutic 12\n\nThe Tell 13\n\nWhere I Went Instead of Paris 14\n\nWife, Wife, Duck 15\n\nIt Took 24 Hours to Make the Moon 16\n\nAlluvial 17\n\nMonogamist 18\n\nMy Beautiful Wickedness 19\n\nFloating Wick in Petrol 20\n\nBridle 21\n\nThought, Antithoughts 22\n\nSex 23\n\nWhat Is Not Science is Art is Nature 24\n\nFreud Had Sex but Jung Had God 26\n\nSquirrel in a Palm Tree 31\n\nAnnunciation 51\n\nThe Rise and Fall of the Central Dogma 71\n\nAutographies 89\n\n_Acknowledgments, Dedications, and Notes_ 113\n\n... synonyms do not exist.\n\n\u2014 _Donald Hall_\n_Monogamist_\n\nA human being can't compare  \nsize and brightness\n\non two occasions. So we say  \n _the moon has a dark side_.\n\nWe say _the tide twice a day_.  \nI say _that man there, so unlike_\n\n_my husband_.\n_The Museum of Accidents_\n\nThe school girl's tights speckle  \nin the rain. In the city\n\nthe sparrow on sparrow feet skips  \nacross my path, legs invisible.\n\nWe are bound. Similar,\n\nindistinct forms called bodies,\n\nour Milky Way's spiral arms\u2014\n\nstars, nebulae, matter\u2014\n\nbound\n\nto great disaster.\n_Codary_\n\nOnce he was a type, kind, tide,  \nbut became a singularity.\n\nI stopped breathing.\n\nWhere the husband's orbit overlaps: darkness.  \nNo light can be shed on what lies beyond this\n\ngravitational sheer,  \nharsh polarity\n\nof wanting.\n_The Secret Room_\n\nIsn't hidden. Nor filled with goods  \nor bodies. This feeling\u2014\n\n[strip the wallpaper,  \nknock for panels]\n\nI can't explain it\u2014is always,  \nI think his gaze made it. I say\n\nwhat I don't intend  \nso as to say something of\n\nthis tending, tendency, tender  \nunsayable place I mean to take him.\n_Firmament_\n\nBelow his clean shadow:  \na sunlit prairie. A wheat field\n\nfrom the air: plush and temperate.\n\nThe breeze is a brave caress. There is  \nsomething I see in him: tip, edge, hint\n\n\u2014the skin of it. Shifting wheat\n\nover soil over cavern over water  \nover igneous over molten.\n_Monogamist_\n\nRiding a bike down a flight  \nof steps misnames them,\n\nreveals their lusty gravity.\n\nHave you heard that Brontosaurus  \nis a Camarasaurus head on\n\nan Apatosaurus body?\u2014my  \nlove's like that: shaped,\n\nnamed beast did, did not exist.\n\nThey should be called falls, this  \nplummet.\n_Galaxies Rushing Away_\n\nI'm trying not to try to  \nget him into bed. Instead I try\n\nbut the husband flinches when I  \nand flinches when I say\n\n_I love you_ and I do  \nlove you but say\n\n_I'm meeting a woman named Kate_.  \nThen, off to the winebar, order\n\n_sancerre_ , nice summery white at $7\/glass;  \nhe, me, and vast millions are fast,\n\n\u2014red shift getting redder, every galaxy  \nfrom every galaxy, vow, promise, primordial\n\natom\u2014rushing faster, all on our way  \nto greater disorder.\n_Axon, Dendrite, Rain_\n\nWhen he speaks I am allowed to look at him.  \nLet this perfect conjure slide over (all over)  \nthe thought reaching out to my loud now\u2014\n\nI want to\u2014\n\nbut find no way to make my hands  \nnatural, accidental. I try to make his skin  \na chaste idea. But even his gloves, made from slaughtered  \ngoats, their pliable kid leather become a bias-cut  \nslip, myelin sheath, the impulse _jumps_  \nnode-to-node, too fast for capture.\n\nThe body.\n\nLess, less real. I am aware of wanting  \nto look at him. In the long space  \nin which others speak I cannot look at him.\n\n_take your clothes off_\n\nAnd I do. In dream after dream, except  \nlast night when I'm running a long way  \nin the rain and, basketball in one hand, he  \nstands watching. And when he watches\u2014  \nI run and run, do not wake up  \nbut that\u2014(there,) that, that, that: rain  \nat my window, husband in my bed.\n_Rhyme, Lascivious Matchmaker_\n\nEach time I try to\u2014  \nhere comes my husband again and\n\nmy mind, I'm describing; context.\n\nForgive me, anemone, my green clearing.  \nHe is no still pool, but actual.\n\nIf I showed him my skull below the skin  \nthen threw out the skin, would he wipe clean\n\nthe bone? A thin gold wire  \nprevents my jaw from metaphor or...\n\nHis v-neck suggests\u2014  \nThe bruised way he sits\u2014\n\nWhat to do with his lips\u2014\n_Hermeneutic_\n\nThe sea is supposed to be something  \nmore than a saline menagerie.\n\nI thought to be full of feeling  \nrather than _with child_ was\n\nmutable, could stay small, but now I'm  \ndesolate, fleeting, pierced with this blunt\n\nfissure. My babies left a narrow passage  \nwhere longing festers. And here he entered.\n\nBrutal shunt, my heart fills  \nwith sea water. Involuntary muscles\n\nseize, shudder, refuse to scar.\n_The Tell_\n\nThe basketball makes him not my husband  \nand saying so in poems makes me\n\nthe bad wife. Where is the private, i.e., impassive  \nmask I purchased for my wedding\n\nbut then forgot to wear?\n\nMy mind wrote me a letter requesting to be  \nleft out of it. My body sent flowers\n\nand a note: \"sorry for your loss.\"  \nBut both paid to see the flop and stayed in 'til the river.\n\n_Better to fold the winning hand than fall in love_  \n _with your cards_ , says the husband.\n_Where I Went Instead of Paris_\n\nIn the city, out windows, I fit his face  \nonto the faces of other men and boys\n\nand look away before it fades.  \nI have learned to fly by running fast,\n\nthough the waking body won't comply.  \nHis face is the face of all men\n\nnot my husband; I see him everywhere.\n\nIn the next dream I shave my head  \nand find my skull misshapen. In the next dream\n\nI am raped in the elevator. The doorman  \nsteps over my body. He has your face.\n_Wife, Wife, Duck_\n\nI'm not sure what this could be called \"doubt\"  \nbut that's too simple these clouds: grayer than white\n\n(the white sky behind) like the sky at evening.  \nTo wish the best for someone\n\nI love might mean leaving  \nor leaving him alone. To wish for\n\nhim. Wish for him to\u2014\n\n_It looks like rain_ means  \nit's not raining.\n_It Took 24 Hours to Make the Moon_\n\nI forgot to think of him today.\n\nMade of carbon, oxygen, calcium: you, him, I, stars.\n\nWhen a Mars-like body and Earth collided  \nwithin hours was a protoplanet named Moon\n\nand a planet moved away.  \nFor days\n\nI forget.\n\nMantle, core, ocean, air, I  \nam made of our\n\n\u2014air, air, air and air\u2014  \ncarved-out crater of impact.\n_Alluvial_\n\nThey say God's voice in the city  \nsounds like a man but in the desert\n\nsounds like a woman. His voice, the spine  \nof nighttime, sounds like water.\n\nRock grazed by streamlets long enough  \nwill sunder. One word against my sternum and\n\nI unzip.\n_Monogamist_\n\nI've fallen ________ with him, stupid  \nclich\u00e9, with his dark blue\n\nofficewear. Maybe\n\nI just love my little boy too much\u2014he  \nlooks like him\u2014itself a grievous treason.\n\nJust ask my older son. Ask  \nthe husband. Ask anyone. Ask\n\nthe language for one decent synonym  \nand watch it stutter: perseveration,\n\nobsession, attention to detail  \naren't love exactly nor is\n\nchastity enough punishment.\n_My Beautiful Wickedness_\n\nSomeone dropped a house on me  \nand stole my blood shoes.\n\nThe girl with her skipping and singing  \ncomes to kill me. What then will become\n\nof my spells, sole treasure I possess?\n\nWhat I see when what I see  \nis not there\u2014I know he feels it.\n\nLooking at him like this\n\nisn't a spell to make him  \nlove anyone\n\nbut might. All the good wife  \nwants is to go home.\n\nWhen no one watches  \nI teach the dog to fly.\n_Floating Wick in Petrol_\n\nI am too happy to see him.\n\nSomeone must be blamed. Perhaps  \nthe therapist or my marrying young.\n\n_Say_ , _are you really this beautiful?_\n\nI dream a woman puts a gun in my mouth  \nto make me choose\u2014lustrous, sleek, sexed.\n\nNext a jade green sandal from a bottom  \ndrawer. Suede wedge with straps\n\nthat wind around my shin. My foot  \nin the smooth cradle is lavish, ignitable.\n\nPlease, say you are a dress I can put on for tonight,  \nsay you are a gun or untouched leather\n\npurse, a beaded belt or denim  \npatch or felt-bottomed box or basted hem, say\n\nyou are a spiral binding or photo of a forest  \nframed in beeswax, say a hat pin, say a buckle\n\nsay a gun or polished knob, say anything\n_Bridle_\n\nI promised to stay steady,  \nbut who knew the rage\n\nof arbors?\n\nForests, groves, flagpoles,  \n _Stand_ , we told them. _Stay_.\n\nWhen we set up the blocking,  \nmarked my toe-stops with tape,\n\nI can't describe it\u2014  \nhow my shoes abrade,\n\nfit, like casket.\n_Thought, Antithoughts_\n\nI've nothing to hold him,  \nsuspect I've been dreaming\u2014\n\na woman awake, her  \nhusband breathing\u2014she wants\n\nto be anywhere.\n\nHe's a man  \nwho happened to notice\n\nI made him want  \nto play guitar\n\nbut he didn't. This is the winter  \nthe husband started snoring\n\nand science said free will  \nis a feeling we believe in.\n\nPost hoc confabulation.\n\nI must get up and attend  \nthe microorganisms.\n_Sex_\n\nWane, wax, wobble.  \nMy mind is a map of hunger.\n\nThey say Abulafia could stop his heart  \nwith one letter. _Alef_\n\nlodged in his semi-lunar valve.\n\nSmall _e_ after breath  \nis what I do to keep living.\n_What Is Not Science Is Art Is Nature_\n\nI am dreaming a hole right into the voice of God.  \nStraight into the dark place where my children were made\n\nbut can't follow me back to. Right into the room  \nwhose windows are too high up to see out,\n\nthough the sloped roof is too low for me to stand up.\n\nIn New York snow is unusual, arrives like childhood  \nmemories that might not have happened, disappears\n\nwithout changing anything. But do we say,  \nwhen it snows, because some countries\n\ndon't believe in snow, _I dreamed_  \n _of snow_? No, we say the news was right or wrong.\n\nWe say this strong desire for a window\u2014huge square  \nglass through which a child standing up in a crib\n\nat night alone in a room at the bottom of a flight  \nof stairs far from the mother in winter sees:\n\na Greenwich Village garden cast in urban glow,\n\nquiet, because snow in the '70s was enough  \nto make the city slow and mute\u2014is real.\n\nSo, say it really happened. That doesn't mean\n\nit will again or did. Or that the dream  \ndoesn't make you ordinary.\n_Freud Had Sex but Jung Had God_\n\nI take water  \ninto my lungs\n\nin lieu of him, want for air,  \nhave none and not\n\nbecause a good wife rose up in me  \nor a sharp right turn, bright\n\ndiscipline befell me: I wanted  \nsugar and salt in equal measure\n\none making the other desperate  \nthe now tasteless by turns desperate\n\nthis was this wanting of course  \nit was the kind of snow that never\n\nsticks\u2014O blizzard! wild sky at wit's end\u2014\n\nbut when I look again  \nthe street is barely stained\n\n(sugar, flurry, salt, drift)\n\nand the flat, clean air swears  \nsnow never fell here.\n\n_Squirrel in a Palm Tree_\n\nup, out of the sentry box over the parapet, bastion, rampart, breastwork\n\n[don't think \"I have left them...\"]\n\ndraw and look, lift\u2014erase, draw and lift and lift and lift\n\nan erasable slate  \nthe velum top sheet takes away\n\n[\"left them\"]\n\nup, over the country\n\nthe edge of coast and further out the clouds like stones in deep waters\n\na river delving the lush green\n\nmarsh an amorphous rum babba, soaked and spongy\n\ngrasses and cattails misstate the surface\n\nthe cabin has the sharp inhale of opening a gift\n\n\u00d8\n\nhigh ceiling (blue) and pink and gray striped walls shape me\n\nmake a naked Alice in the bath\n\nbig and tiny\n\nhere and far away\n\na wonder the body fits\n\nso mythic is the mother-absentia\n\ntundra of abandon\n\nI suffer the gift, silence,\n\nfor once, nothing happening\n\nnone using my name to mean anything\n\n\u00d8\n\nbed as wide as it is long\n\nthe night inhuman calm\n\nthe outlets and picture frames and decorative plates are safe\n\nthe bathtub and mirror and doors and linens\n\nI am as light as negligee\n\nhave not my army's entourage\n\n\u00d8\n\non Sunday I will step back into the living room littered with toys  \nthe two boys happy\/shy\/mad to see me  \nbut like I dawdled in the shower\n\nlike I never was anywhere but ready to answer\n\n_where is my?_\n\n_can't find the..._\n\nlook here, the light through the sycamores and dense magnolias  \nlive oaks tasked with spanish moss\n\na veranda you reach through a twelve-foot window\n\n_be_ real\n\n\u00d8\n\nunnaturally light  \nlike a various gravity exhibit at a science museum  \nmy mother has a necklace made of severed reeds  \nthat seem to weigh  \nless than air\n\nthey look like bird teeth or shell splinters\n\n\u2014Haiti? Australia? Peru?\n\nshe can't remember where she got it\n\nbut the stones\u2014emerald rough from Sri Lanka  \non the bookshelf near the kitchen; square, flat rock from Arizona  \non the mantle in the bedroom\u2014those she knows by heart\n\na life of picking things up and bringing them elsewhere\n\n\u00d8\n\nhere is the tree of my thirtieth birthday:  \na palm between two sycamores\n\nthe pineapple-totem trunk is a woven present\n\nfronds rustling to offset loneliness\n\nsquirrel feasts on hope\n\n\u00d8\n\nalone, the room gets smaller despite there being fewer people  \nthe TV approaches like a hopeful lover\n\nlet us, I say to myself, consider the children objectively, which is impossible:\n\nthe boys who are babies create a slavish planet. this means I am bending and lifting and every each moment listening for disaster which is silence where his \"dadaka\" and \"teka-te _kah_ \" pause is surely climbing or choking or considering mischief\n\non guard, keeper! be lively!\n\nanything which requires concentration is danger\u2014\n\nso drag myself to watchfulness with a stab of catastrophic thinking so tired, delighted  \nI've half a mind to leave them and no mind left to do it and nothing to spare of this utter love\n\nmother in a foreign make this real\n\n\u00d8\n\nsome day they will leave _you_\n\nand you will visit the kingdom of adult concerns and never leave\n\nand will want to and will dream of night wakings and tiny spoons of temperate cereal on hands and knees for spilled cumin seeds you will remember the every night of tiny things back in boxes and on shelves and under and in and the ache ache ache of your back as he learns to walk or the relief of finally squatting in a parking lot to nurse him stop that wail\n\na woman with young children is not a woman but a mammal, salve, croon, water carrier\n\nshe has a prize they all desire\n\nlift, lift, life\n\n\u00d8\n\nif there are nests discarded on the sidewalk  \nI step around them knowing what it costs to weave one\n\nonce my shadow was the shape of a bear or egg with arms and legs  \nnow slim and bony the boys sucked the melon-sweet milk right out of\n\na letting, flesh mongers\n\nand if the nest, a relic, outlasts the wind, rain, marauders,  \nit is always the cupped halo of ambivalence\n\ndesire won over by desire is not the same as satisfaction nor lust nor yet resolve  \nI don't believe in happiness\n\n\u00d8\n\nI am equally and at once estranged from the person I knew as I  \nand from the mossy being made so carefully\n\nthe child becomes a wedge between actions and self like a cyclone of gauze wraps himself around my mothering  \nand makes a hollow form\n\nshape: human\n\ncocoon around a maelstrom\n\n\u00d8\n\nin New York the apartment's windows face south and my son knows little of the sunset only that at night it's on the other side of the world\n\nsleep with me, he says  \nI like the other sheets, he says  \nlime in my sippy cup?  \nanything to keep me\n\nobject of desires, I never satisfy because my body is impractical,  \nboundaried, impermanent\n\nhere, on the balcony, dusk draws the bundled leaves on winter trees  \nlike hanging planters or Christmas ornaments\n\ntwenty minutes later the leaves are hedgehogs,  \nthe branches: flaws and fractures in the skin of twilight\n\nnow on the other side of the world, the sun's fiery descent means little when witnessed less when missed\n\nperception or staying is a mighty effort\n\n\u00d8\n\nif the language would slip I could see what limber chance remains me\n\nbut is always my chaperone\n\nthe moon is so full it must recline  \nthe hip is the location the child claims\n\nand aches\n\nfrom use, from absence, the whole pelvis an isosceles arrowhead  \nbarely a ledge the arm comes to scaffold him, the elbow buffer from gaudy onlookers  \nthe breast becomes the shy boy's brow-rest\u2014does he remember the Cyclops wonders?\n\nI remember his greedy squeezing kneading tiny nail edge  \nmy love a tinderboxinflamed, viral\n\nobese, inhale\n\nI miss...\n\nwhen the child falls forward and catches himself with his hands, stands carefully, bunches his face: _fine, fine_ , I get to hold him now and kiss his palms and put my nose against his cheek\n\n\u00d8\n\nwhen away from my tree I want to brag the treasure\n\nsmooth green globe  \nbefore the husk mars innocence\n\nmonumental nut  \nwhere can I bury such bounty?\n\nthe sugar-milk is too much at once and must be dealt with  \nbut there is no dividing beauty, no rationing\n\nI must escape my reputation for hoarding\u2014so in love with the heft of the Asian pear,  \nthe lusty hue of the persimmon, I keep and keep until they spoil\n\n\u2014 _Crack it!_\n\nthese edibles, not memories,  \nthe fontanel bones of his skull about to close\n\n\u00d8\n\nsome women cherish the fathomless want of infants\n\nas it is all around me I cannot muster judgment and having been stayed from my sentence these three days by a stutter of double dashes -- I --\n\nam still, I\n\non either side of the long spine lie two shallow ditches walk your finger tips along these furrows but never pressure the raised column holy like the horizon it is the going and the getting and the lifting and the carrying the bending and listening kneeling and squatting it is my fingers' careful sweeping the alphabet floor mat at four am for the rubber binky it is the way my body in his twin bed tricks him to sleep the way I tucked his baby face against my belly and stood and stooped and swung like a mechanical gadget and set him down and made the back stay hunched so the hand could stroke his hair and sleep him and stayed when the back protested and when the mind tried to make sense the body stopped it\n\nmy love is the bent body, the mastered spine\n\nthe coast is a sure painter's mark  \nbut the horizon is nothing human\n\nfrom this height the flat expanse of farms and plots and houses, speckled towns  \nlike the oyster, lime, and sand sidewalk they call \"tabby\"\n\nthen like flecks of polished shell the tilted glance makes the settled patches rise  \nlike lily pads on tree-green ponds, the roads lascivious zippers\n\nand the sun through the horizon crawl-space may be the moon  \nfor it is everywhere a glowing ring and cannot be a star rather some bulb  \njust this width greater than the earth's diameter\n\nwhether my body is in conflict with the plane's intention is irrelevant\n\nmy children at this remove are figures, figments\n\nthe difference between here and there\n\nplanes always wanted to\n\nlighthouse, cliffside, pride\n\nthey carry but do not mother\n\n20 miles from LaGuardia the houses are little studs punched into denim, no longer in fashion\n\nthe tugs make snags in the nylon surface of the ocean, the houses and trucks  \nalong the capillaries, when,\n\n_oh_ my\u2014\n\nthe fortress edges of Battery Park, airshaft depressions of empty lots  \nand unbought air rights and\n\nthe Empire State buildingamazing  \nwith her glinting, ramrod posture, suddenly\n\nalone above her waist-high charges\n\n_Annunciation_\n\n_traveling or intussusception (an introduction)_\n\nthere is an inside as in my child as I see him or, earlier, felt him\u2014\n\nmyself a stricture outside the real\u2014 there is an outside as in his real face, fake\n\nlaugh, mimic cry and, before, the body bigger: what it looked like\u2014\n\nthere is an inside: my mind at night a plan the word escape, and outside:\n\nI in bed the body in place, in my place I put myself, the body, in, a stand-in, harbor\u2014\n\nthere is inside a bric-a-brac system: faith or habit, what turns my head,\n\nfires along the ocular chiasm\u2014 pinpoint this\/that while the hand goes thoughtless\n\nafter: grab and pull, peel and core, locate and obtain\u2014\n\n_balcony (florence)_\n\nthe fact (Italy) being around us\u2014 we go-betweens\u2014 the sky an impossible duomo\n\nor drop cloth over the cage\u2014 traverse and return\u2014 we traverse and rest\u2014 one thought\n\nat the edges transgresses\u2014 what is disillusionment?\u2014 a young man whistles by\n\non a bike\u2014 Catholics believe in impure thoughts\u2014 even saying is happen\u2014 but I want\n\nto believe in interiors: spaces others don't conceive\u2014\n\nthe bicycle is overtaken by a moped and that by car and that by bus and everywhere\n\nthe crowd teeming, seeming from here human, a given, a context, granted\u2014\n\neach particular (woman with two small dogs stops, searches for key in purse) is a point\n\non a tangent (man on moped) of some veiny larger\u2014 you are out, I am here\u2014\n\nwithin the city wall\u2014 _portes_ \u2014 breaks in the wall, notional membrane on the map\u2014\n\n_santa maria del carmine_\n\nyou become is\u2014 and suddenly are\u2014 [we] not just an emblem, my other\u2014\n\nyou tried to avoid saying always avoiding I tell you to say, say _it_ but I also avoid\n\nbecause emblem and becoming symbol but suddenly between Masolino\n\nand Masaccio, symbol and symbol, you suddenly become and are\u2014 at home the baby\n\navoids sleep, crying, wanting to say but not knowing how, let's avoid him\n\nfor the while occupied, crying his own space\u2014 meanwhile my intestines and internal\n\nhave returned to normal although the package somewhat slack my mind sets\n\nto tighten and look around, look for\u2014 then at night, last night, you suddenly are\n\nand not this time the obstacle but rather good company, a voice in the room\u2014\n\nribs heavy on the bed my body a cage pushing both ways voice in my head open mouth slips\n\nout\u2014 suddenly is\u2014 _is_ \u2014 and edged, mountainous\u2014\n\n_bed_\n\ncannot call it forgiving this thin pad over wood but when you knock on the cage\n\nof my body I do not break no less forgiving I do not crush or falter this is what it is to be\n\nknocked-up? though mostly the motion is down and I am 'under' and then\n\ndon't think just am until that woman breathing isn't me, baby at the foot of the bed not mine\n\nbody not around or inside but me and rather than situation I feel you are a problem\n\nforce to reckon with I am you are breath, then an occasional motor bike, long gasping night\n\nthen nothing my body inside-out and in the after notice the quiet as it closes in\n\n_avoid_\n\nin the morning gone again and only body, presence but not is\u2014 is this\n\nwhat I married? you: here, a given\u2014 _is_ , as in: situation, situated (choice made before)\u2014\n\nsome search for the soul or mind in the body: I think it is a membrane around\n\nthe void, scrim between skin and bones\u2014 I feel it lingering, clinging\u2014 not deep but\n\nrooted to itself\u2014 not leaking (contained) not touching\u2014 one cannot touch it\u2014\n\ndoctors open but do not find, open but not reveal\u2014\n\nsituation: we are two within a larger\u2014 each around an empty inside a pulled skin\u2014\n\nsometimes in contact we have no choice but choose proximity to make a smaller\n\nsystem\u2014 parcel, context\u2014 block out the bright or heavy rain verdant\u2014 what I am trying\n\nto describe is disappearance: what happened and how you can be and invisible,\n\nbe but not is\u2014 even the language knows it, fact without gerund\u2014 voice with edges\n\nbut not edgy\u2014\n\n_balcony_\n\nthere are two sides two centers therefore an interior or, instead, between\u2014\n\na space that is neither, a passage, a pause\u2014 of moment is the thing exposed but not within\u2014\n\nhere we walk as if together as if\u2014 scent of testosterone, then gone\u2014 another's\n\nMadeleine, another epic\u2014 I am outside and in and up and on\u2014 who knows perhaps I am also\n\nin your thoughts as you, your stand-in, are in mine?\n\nlater the terrace is chilly, exposed and the bells make the air softer the light pinkish\u2014 there is\n\nan advantage to self-possession but I can't remember what\u2014 the clouds watercolor\n\nambivalent, a misplaced seasky far from the sea\u2014 not at home but having brought it\n\nwith us\u2014 who snuck it into the luggage, what pocket? under whose saddle?\u2014\n\nthe silver cup of longing displaces me\u2014 I, almost unseated, wanted to make not an emblem but\n\nwhat, what is there that is real?\u2014 if you are a situation it is what you are\u2014 I tell\n\nthe story and we are still here despite being _d\u00e9pays\u00e9s_ or out of country, dizzy as if trying to find\n\nmy way with only a mirror and not eyes\u2014\n\n_santo spirito_\n\nthe angel humbles before her and she with her book raises her other hand\n\nas if to a man offering stolen watches, \"not interested\"\u2014 she looks comfortable,\n\nperhaps the chair was made specially\u2014 between them the showy tiles, a vase\n\nwith three dark flowers, her hand stays as it is\u2014\n\nwhen the child is born she becomes younger, more innocent, na\u00efve, she wears\n\nher blue cloak constantly\u2014 becomes younger until, in the end, the _piet\u00e0_ , who knows\n\nbut she _is_ younger than the son\u2014\n\n_sacristy_\n\nwhat if you only had one scene to work with, say the nativity or the annunciation\n\nand it had to say everything and was the only story you had\u2014 the angel over and over,\n\nslightly different tiles, flat or this time perspective\u2014 it has will have the same ending\n\nbut the curve of her hand, no thank you, slope of her gaze say something new\u2014 today\n\nshe hides behind the scene\u2014 the angel's arms crossed at the wrists like a double bow\n\nbut also, look again, like an embrace\u2014\n\nhow dare he with his golden curls the angel presumptuous his down-turned eyes and\n\nfolded wings\u2014 he has already removed his shoes and has beautiful feet\u2014 she alone, not yet\n\nof child, in some grand palazzo says shyly, no thank you, one hand poised in half prayer,\n\n(open, possible) not altogether proper\u2014 the other stubborn on the book\u2014 not yet a virgin\n\nher blue is French almost gray\u2014\n\nthen, when time flattens, the big fish comes to swallow Noah, ark and all, and Mary\n\nin her blue kimono goes to Holofernes\u2014 someone has taken her desire and used it to fashion\n\na child\u2014 deep, deep into the colonnade one vertebrae at a time someone has taken her\n\nand left a little changeling in exchange, now a virgin except that in the flimsy of her negligee\n\na knife and map of the jugular\n\n_portrait of unknown_\n\na pomegranate, calla lily, sprig of rosemary, meanwhile we do not say but rather tell\n\nand tell around the jelly-centered heart\u2014 come away! soft sugary evidence\u2014 what is the form\n\nof woman besides body?\u2014 there were flowers a given and a garden\u2014 one must\n\npay to enter but may leave as one desires\u2014 unencumbered by regret or so they said but the story\n\nrefuses to unfold peacefully\u2014 the past a torn ticket or smeared receipt\u2014 the story refuses\n\nto reveal or adhere but is dropped behind, a savory trail, birds circle wearily\u2014 I should have\n\nbrought pearls or marbles or pen caps\u2014 instead or _in her place an anemone_ \u2014 where I\n\ngo down springs up a delicate poison to mark descent: flower between my lips\u2014\n\n_cloister or \"vietato toccare le rose\"_\n\nthe church no matter how big is the idea of space but not space\u2014 they have painted\n\nthe windows which are sharp and pedantic, they have hung paintings in all the alcoves\n\nand in the paintings are windows and in the windows people look out at pictures\n\nof the world\u2014 they are therefore never going out and when pictured the child pale\n\nwhite and shrunken, the mother young and small in her dark blue, there is no escaping\n\nthe painting the world not real as it is allegory this place a huge tomb\u2014 turns out there is a price\n\nfor leaving\u2014 now we have found other places small places banal to inhabit\n\nand it is not so easy to break a gash in the wall and look out\u2014\n\noutside, the cloister is: breath, sinus, the habitable\u2014 here day comes into its own\n\nand takes me with it\u2014 into itself, a light blue with folds and tiny flaws\u2014 who knows\n\nif the column reaches up or down or even reaches as much as is and is again\n\nbut this time not a cage\u2014 the roof surrounds what is not roof, the walls create an inside-out:\n\ncloister\u2014 a woman could breathe here, someone celibate\u2014 building outside,\n\nair in\u2014 a woman could live among other women exposed and enveloped\u2014 but this is\n\nnever what I choose\u2014 a boy drawing, long sleeved thermal under a short sleeved T,\n\nlike a fallen Mormon looks up as though to sketch what is not there\u2014 don't touch the roses,\n\ndon't trample the grass\u2014 it's _interdit_ , _vietato_ \u2014 and the past meanwhile that shabby habit\n\ninsinuates and even the light blue day cannot protect me\u2014\n\n_ministries of grace (kansas city)_\n\nthe wings are not attached\u2014 background\u2014 he wears a scarf, almost a bib or like\n\na young boy planning to rob a train in his backyard, bandana\u2014 the sperm of God\n\nin straight gold rays streams in through the window\u2014 later Christ, a naked homely\n\nchild, unlike anything in the natural world\u2014\n\nGabriel always enters from the left or from inside the painting the right\u2014 which one\n\nor where are you?\u2014 I wasn't waiting just reading on my own writing letters\u2014 I wasn't waiting\n\nfor the natural world to perform or people to take their places I wasn't waiting\n\nwas occupied otherwise when came this new work some holy imposition\u2014\n\nhotel room when I finally sleep I have that dream there is a room in my apartment\n\nI've never seen it is my favorite dream this time the room is tiled: floor, walls and ceiling\u2014\n\nsquare white tiles the walls at odd angles like corners of an old house\u2014 the dream says\n\nI will spend a year in there making light stick to film and paper in low or no light then out\n\nthrough the light-safe door to look at what I made, will hit my head and feel lucky\u2014\n\n_annunciation (new york)_\n\nwhen the situation became upside down everything changed: perspective,\n\ncounting, there was none of it\u2014 I cannot even say unprepared the mother father angry\n\nnot wanting and not able to say solution or consequences when all I meant to say is,\n\n\"this happened\"\u2014 one could gloss perhaps a history but truth is who knows\u2014 who was\n\nthat masked man his wings folded back in the night no candle to read by\n\nthe baby sleeping, quiet\u2014\n\nwhen I woke up it occurred to me then at night and at morning and afternoon the calendar\n\nstaring its hard boxes\u2014 go in pull up the blanket around the baby he is too sweet\n\nsleeping the blanket will not kill him despite the doctor\u2014 I must sleep also but afraid, this way\n\nfear this way fear leads two ways is not reliable\u2014 how did this? who did we?\u2014\n\nI must think, sharpen my mind even on a stray arrowhead it is worth remembering\u2014 I saw\n\nthe angel it is a sick joke I was not I said interested in this or in that fucking angel\u2014\n\nbut no one will believe me\u2014\n\nMary was reading (an anachronism) probably religious texts (improbable) when\n\nGabriel interrupts\u2014 humbled? he thinks himself his message more important must insist\u2014\n\nwho is this woman of leisure in grand palazzo\u2014 what father taught her letters what\n\nsire built the chair, suitor gave her leather bound volumes\u2014 I write faster until progesterone\n\npulls me under not time left\u2014 haze of motherhood\u2014 she's never seen again reading\n\nor in repose, only holding: He alive, He dead, she holding, watching, surrounded\u2014 there is no\n\ngrace in this but work though I can't say I put up much resistance\u2014 mine a small\n\npalazzo: marriage, port-a-crib at the foot of the bed blocking the exit\u2014 one hand feeble\n\nbefore me, window already open, gold rays inside\u2014 I was am tired\u2014 there is nothing\n\nto say, it is not about language\u2014 so get up, get up now what we do the only matter\u2014\n\n_central park_\n\nJesus in the water does not look clean or fish-like or marine at all but only unafraid\n\nas if he knew he had gills as if the oxygen in the sea were the same as air\u2014 I will not\n\nget clean but will drown I have one name only\u2014 a spider rests in his lazy cross of web\n\nhigh up above the park bench\u2014 one branch to another\u2014 why in the world did the dove's\n\ntiny sprig comfort Noah, he had no gills\u2014\n\nI was not the subject but location\u2014\n\nand these first buds, some plot within my very death, I hardly mind\n\nhave eyes to see: this ravishing, there may not be another like it\u2014\n\n_sighting (nova scotia)_\n\nyou say it is a cloud formation not land but when two mornings later true mist\n\nrolls in we know we are missing something\u2014 was it just a strip of darker water over\n\nthe horizon\u2014 a craggy one line list of the missing: sex, romance, tourism, humidity\n\nobscured by the new baby, the baby, discipline, weather, will\u2014 our work, our work,\n\nthe distance to town, the meager produce, slippery rocks and what was for a moment\n\na whale on second look just driftwood behind a buoy\u2014\n\nthe sea today has got a grudge against something\u2014 look! I say, waves! and the baby\n\nwaves\u2014 a gull on the low-tide rocks tries to lift but is blown back and forced to land\u2014\n\nagain and again\u2014 wanting, wanting in spite of\u2014\n\n_maritimes_\n\nit was not where bodies washed up, were fished out or brought ashore but where\n\nwe landed, marine and rocky\u2014 the sea calm, a bay, but the tides in and out with fierce\n\nambition\u2014 two gray chairs in the sun almost touching\u2014 dry rocks, mossy rocks,\n\nseaweed covered and glossy close to the edge\u2014\n\ninside, the celibacy of pages turning, a computer-hum in the short utopia of a sleeping child\u2014\n\nthe I-mama growing and looking longingly out of doors through the big wall windows\u2014\n\nthe town has no church, library or playground just pale lettuce or fiddleheads for a short\n\nseason\u2014 the top rocks shiver their tidal pools, a striped-back bird like a skunk\u2014\n\nand on thick mist days I think of Cordelia, imprisoned, wishing\u2014 and not for Lear and not\n\nfor Burgundy and not the King of France but no one bothers asking\u2014\n\n_the shore_\n\nonce a virgin, one can change and change back\u2014 but once a mother\n\nalways\u2014 she was not a conduit or vessel but holy messenger, chosen\u2014 they were all\n\ntaking and wanting and swore his leaving made him stay, would make him\n\ndeep in the heart of the people\u2014 but what mother wants a child deep in the heart\n\nof the people\u2014 how then should she guard him, watch, gather?\n\nhis body a seed within her, a tadpole or parasite\u2014 docile, lazy child: animal born\n\nin captivity\u2014 what was new about him? only that he made her a mother,\n\nround with waiting\u2014 patient like all famous women, her blue like the part of the sea\n\nnone survive\u2014 when she looks again he is a trilliant-shaped glimmer, bitter\n\njewel\u2014 what is real? house, body, tide?\u2014 the water is too cold for anything, the house\n\nvelvety, badly decorated\u2014 it is a sin to stay inside\u2014 be rather by the edge\u2014\n\nshorelined\u2014\n\nin cities elevators never fail to sicken\u2014 too many things inside others\u2014\n\nbut here, the sea itself and its ambivalent rushing\u2014 back and forth, towards\n\nand fast away does not unsettle\u2014 largely a house myself, I have, when seated,\n\na kind of balance and from my gray chair are everywhere windows and color\n\nand in the distance the thin blue promise of what I know must be another coast\u2014\n\nChildren- (if it Please God)\u2014Constant companion, (& friend in\n\nold age) who will feel interested in one,\u2014object to be beloved\n\nand played with.\u2014better than a dog anyhow.\u2014Home, & someone\n\nto take care of house-charms of music and female chit-\n\nchat.\u2014These things good for one's health.\u2014but terrible loss of\n\ntime.\u2014\n\nMarry, Marry, Marry Q.E.D.\n\n\u2014 _Charles Darwin_\n\n_The Rise and Fall of the Central Dogma_\n\nI. REPLICATION\n\nIs the Soul just a notion, a drug?\n\n\u2014 _Alice Notley_\n\nThat it was politically impossible.\n\nThat there was an alternative.\n\nThat I would stay.\n\nThat the lighthouse was useful.\n\nThat I would leave.\n\nThat germs infiltrate.\n\nThat babies before conception.\n\nThat the white collar means concern; the long beard, the shaved head, intensely dyed garment.\n\nThat sex is an effective way of generating warmth.\n\nThat a bay mare and wild iris are unalike.\n\nThat the gaze is not chemical or electricity requires a conductor.\n\nThat a saucepan and spoon are better toys.\n\nThat the office is not a tundra and not a mirage and not.\n\nThat traits are inherited.\n\nThat ideas will save us.\n\nThat a mother's face is not her mindset\u2014\n\neven the infant knows it. And yet we put her\n\non television, donate particle scopes.\n\nI tell you there is a secret world.\n\nThat children too young to walk must be carried.\n\nThat weight can be assessed or described.\n\nThat speech is directional.\n\nThat God knows genomes.\n\nThat the phone in and of itself is not a husband.\n\nThat a dream is not reason to evacuate.\n\nThat the man's death was made of 53 wing bones and the well-planned curve of highway.\n\nThat walking one way was, or this way is, or that the curved back of old bones as seen through\n\na low slit of window is anything but interference: the spiking lights on the mute-stereo or turned-down\n\ncardiogram.\n\nThe city is less \"planned\" than _exposed_\n\nand emerges, one bodega, tenement, row house,\n\npark, highway, until they've pushed out ambiguity.\n\nI believed greatly in the grabby valence of molecules,\n\ntricky protons and funny quarks, my son's invisible puppies:\n\nMuffin, Apple, Muffin, and Lemon.\n\nThe map's coincidence; the body's freak resemblance.\n\nThe _They never told me_ of women in labor or\n\n_Here we are_ under the chuppah.\n\nA flat bed excavator builds its own scaffold.\n\nSecret: to put away opposing evidence.\n\nII. TRANSCRIPTION\n\nwe could have been happy sooner\n\n\u2014 _Brenda Hillman_\n\n_this just in from biology:_\n\nbreasts signal fitness (lateral symmetry) to potential mates\n\n_this just in from sociobiology:_\n\nthe ape mother insures social standing for her daughters\n\n<in the language the real way I feel\n\nbut not _separable_ or _explain_ , and if I\n\nfelt otherwise, outside articulate,\n\nI would not, like a bad wife,\n\ngo tearing around in constant _where is the o o o of my belief ?_\n\n[we are fabric, of cloth, a plucked-at knot]\n\n(are you trying to read her poems and find out if she was sleeping\n\nwith someone or to reason the clitoris placement, whether just dreams\n\nor where she'd been going\u2014)\n\neven science believes in modesty>\n\n_this just in:_\n\nwe are happier than the poems describe\n\nI more I, you completely\n\nbut we came from a code like this with themes, similar names\n\nin the beginning the poem can be anything\n\nbut later not so many, likewise the idea:\n\nhuman marriage: where human means fallible\n\n<and despite knowing long ago that gender doesn't explain\n\nthe exploited and less-fun mother, the wrestling father,\n\nphysically removed husband or why\u2014\u2014\n\nit so appealed to us a system a code something to count on new god much more organic and when we saw it in models we said ah yes and I always knew it and started up with metaphor and talked about language and rhyme and adherence and valence and when they made a sheep with one mother it was proof and when the poem seemed to have some internal logic or visceral impact and when the dogs' eyes were so human and the other culture so like us and when a man savagely murdered his wife we looked for him all over her severed body because we so needed every part of our bodies to be us because it was unfashionable to say spirit this obviously valuable organism as something outside the body god must be the ultimate protein and everywhere was suddenly proof like a seventh grade girl knows, knows _He must notice\u2014_ >\n\nonly after, when we are in it\/ incontrovertible whatever it is (this) [marriage]\/ do we think to question\/\n\nhow did we, did we come to be here? what, other than the simple, well-used markers, made us\n\nmade us want to?\n\n_this:_\n\nprotein alone doesn't make a body or alone a body make\n\nIII. PROCESSING\n\nDarwin looked at the green disc of water surrounded by blue and knew.\n\nThe corals and coconut trees just above sea-level.\n\nSome made sense and other pieces, he made them.\n\nZoophytes, polyps, and actinozoa alive in their stony shelters.\n\nOnce, trying to keep three beetles, he put one under his tongue.\n\nWhen it released some vile liquid, he almost lost all.\n\nWrite it, decipher. Write it: desire.\n\nThe others believed in floods, want of floods, and famine.\n\nThe vegetable kingdom as God's green shadow.\n\nWere we born believing?\n\nOr, like the good giraffe, grow to seek some higher?\n\nAdults of the same species, we use and disuse,\n\ntalk and tool-make, carve a valley.\n\nWe blast, burn, marry, reason, and with signs and gestures,\n\ndance and drawing, make the earth subdue.\n\nIf we are animals we are not orphans.\n\nBut have not God.\n\nIf we are animals, need companions, love offspring.\n\nAlso tobacco, coffee, liquor.\n\nIf we're of animals then have not words for, nor coiled regret,\n\nnor cunning morals.\n\nVelum and air and in the water various cross-current similes for smother.\n\nThe actual time a heart packed in ice survives.\n\nWhat the skeleton wife comes home to make for dinner.\n\nHer fierce metabolism, narrowed profile: body, body-away.\n\nWhat two men witnessed under the four-polled canopy is,\n\non greater magnification, how the old hag in the new mother,\n\nfourth protein on the strand, sustains the huge placenta.\n\nHubris of scalpels: they never made a sheep\n\nso lonesome as a wife.\n\nThe world is full of feeble creatures in changed circumstances.\n\nAs the duck flies less, his wings diminish.\n\nThe human baby with its too big head, a homunculus addiction.\n\n_How amazing_ , wrote Darwin, _that the gelatinous bodies of polyps can conquer ocean waves_.\n\nHe was crazy for orchids, corals, doomed species.\n\nI go on loving you like water but\n\n[That a moment can stretch and fracture, etched with fault-lines,\n\ntactile, tensile, but taut, like the skated-over surface of ice\u2014]\n\n[When the sound of the life with small children, insatiable\n\nsalt cravings, or wanting to be fucked to sleep\u2014]\n\nsay: _mitochondria, the powerhouse of the cell_.\n\nRename misery and we have knowledge.\n\nYoung marriage-lust for zygotes makes bassinets\n\nout of organelle cross-sections.\n\n_Mitochondria, the powerhouse of the cell_.\n\nIV. TRANSLATION\n\nTwist sense and anti-sense strands together; cut one;\n\nwe unwind.\n\nIt takes energy to nick the supercoiled structure hard enough\u2014\n\n(they _want_ to bind)\n\nbut it can be broken.\n\nWe are wound, not knotted.\n\n_a mother, all function, has no morphology_\n\nThat we could look at this and decide the past.\n\nThat we could, that day under the chuppah, know.\n\nThat a single butterfly under the bough-lined structure, the rabbi with a glass and napkin, the\n\nmuffled pre-sound in the crowd's minds, the groom's trembling, the bride's gold frenum.\n\nThat hydrogen binds with a polar molecule.\n\nThat the vow could move two ways in time and the characters in the drama and the audience's\n\ncomplicity and the camera's perspective and the two men watching and the scene folding up inside the album and\n\nthe two boys on the bed wondering where they were before they were born and the mind of their mother and the\n\nsmooth-bottomed shoe of their father and if the glass so wrapped and swaddled broke or shattered or muffled was\n\nthe inorganic portent: sand, water, air, the compressed doctrine: an object's permanence and the social fabric the\n\nmind's usefulness the body's faithlessness the two-chambered heart the pornographic memory jealousy profit\n\naltruism patents partnerships postulates corporate metaphysics and history progress synonyms paradigms human\n\nmind the child's welfare.\n\nThat regret is not biological or an obstacle.\n\n_somewhere a heavy isotope named N-15 reveals my whereabouts_\n\nPhotography can't account for the edge of vision, how I,\n\nnaked, make fear brilliant. The eye is not\n\na camera\n\ntelescope\n\nlooking glass\n\nbut all collude.\n\nOrgans don't arise from needs. Nor does cardiomegaly mean\n\nwanderlust.\n\nAn ancient prototype of which we know nothing\n\npassed us organs of extreme perfection\n\nand organs of little apparent importance.\n\nUnstoppable eye in the cranial orbit.\n\nThe giraffe's small fly-flapper.\n\nProblems with the Central Dogma:\n\nIt doesn't allow for the promiscuity of proteins: how they seek\n\nothers.\n\nThat DNA made protein and protein made us is an attractive notion,\n\nlovely\u2014theorem, sermon, anodyne\u2014and resists evidence:\n\nchaotic cytoplasm,\n\nclandestine mechanisms.\n\nWe understand nothing\n\nfully, have eaten only\n\nthe ripe periphery, without\n\na palate for buds or boughlettes.\n\nFeral, _more at_ Fierce\n\nThe first wife was a hard-working molecule.\n\nA ribosome without membranes making\n\nand making unsheathed to every loving master.\n\nBy the millions she colonized the endoplasmic\n\nreticulum, the enfolding and crenellated mitochondria.\n\nInside the cell: the flap and footfalls of birds and nymphs,\n\na dulcet hymenaeus, rushing piety.\n\nPeppered-moths never rest on tree trunks; the textbook\n\nphotos are of dead specimens glued to sooty,\n\nwhite lichen-covered bark. Nocturnal creatures,\n\nlive moths hide in the high-up canopy\u2014\n\nWant to know everything?\n\nA woman's rope-like hair.\n\nA man's clavicle, how it forms an x-axis plateau of sternum-scapula.\n\nOur beautiful theories, their press of fealty.\n\nThe platonic tongue of wedlock.\n\nThe hush-hush palaver of optics.\n\nNote the raised perimeter and then deduce what fell:\n\nan edged islet I once believed bore life.\n\n_Autography_\n\nI want to change your mind. Not\n\nyou.\n\nYou're, as you are, what I want, even his\n\nblinking neon: [no] indecision\n\nvacancy sign. I have room\n\nfor you and these untrue\n\nI mean disloyal\n\naffections. I'm\n\na penny. Hardly\n\nsomething. One\n\nin a history of immodest\n\nwomen: want, wants, wonton, I.\n\n_Autography 2_\n\nLied. Said I'd be\n\nsatisfied with______.\n\nTruth is:\n\nI want to ruin your life.\n\nThrow them over, some\n\noverture, ignominious\n\nruin, some proven\u2014\n\nthe rest is marriage\n\nby which we bear up\n\nand better ourselves.\n\n_Autography 3_\n\nShall we discuss married sex?\n\nYes, let's take our clothes off and talk of pros and cons, the lag and lapse. The body carries on and there's no other\n\nless-revealed. Real: the husband over or under and going forward while the mind\u2014do you believe in it?\u2014why speak\n\nof it?\u2014flint-streak, it sparks and wanders, will not tame. I'm talking clearly, sincerely, when I say I saw a man and he\n\nwas not my married. This he'd you. He made you he. I made you husband; it was so. We chose to.\n\nAnd suppose I stayed shut-up in the always? Suppose I could have stayed shut-up like so, but o, the bad girl breeze\n\nblows in everywhere, finding the cracks and torments.\n\n_Autography 4_\n\nToday the pigeons tack\n\nin flocks above the city.\n\nThe air is crisp, forgetful.\n\nA plume became a cloud\n\nbecame a plume became a fog,\n\n_No!_ I said to the TV and tried\n\nto hold the whole thing up\n\nbefore I could say stop\n\nit\u2014did not.\n\n_Autography 5_\n\nDuring this time people protested. I didn't, though I never for one moment was for it. And people bought supplies and became political but I didn't though I never for one moment doubted these necessities. A poet acquaintance had a baby. I saw her and the baby\u2014they'd just been at a protest\u2014and felt like I'd never had a baby despite my two boys. I stopped reading newspapers except about science and stopped the TV news though poets were at protests and writing blogs and someone asked me how I could write such abstract lyrics at a time like this and I looked at him and wondered what it felt like to write a poem. Pregnant women looked freakish to me, like costumes or experiments. On my way to the day care I looked at the big bellied women or new mothers with strollers and wondered what was it like to push a baby out of your body. Last night as I gathered my little son out of the bath into a green towel\u2014clean, smooth, slippery, sleepy\u2014I wondered what that was like.\n\n_Autography 6_\n\nA poet friend said I was writing about the shifting you in marriage. I tried to set him straight by being clearer next time. Later he said my poetry \"uses slippage in point of view (between addressing a 'you' and narrating a 'he') to emphasize ambivalence about marriage, as though the speaker were struggling to keep her distance from the 'problem' and avoid using an intimate tone.\" So I saw then how little clear I'd been. All these problems with syntax. I'd tried to strip it, but the smooth under-bark, without its bumpy elephant skin, confused them into thinking I'd found something clean and solid. And they said I'd been hiding in science and in jargon and in metaphor and they said they liked my searing honesty but what did my husband think and they said we like this found language we like this information we like the idea of a young woman writing about her marriage but what did my husband think and when I'd written \"let's discuss married sex\" then I didn't, and they wanted more married sex, for the husband and wife to actually _have_ sex and then someone said, but they are, right? they do, right? it's just pretty subtle and they said we want more _blatant_ sex and someone said does anyone else see another _man_ in these poems? And then they said science is off-putting.\n\n_Autography 7_\n\nThat, between episodes, the husband,\n\nfacing the TV says _hey_... in a sweet way,\n\nshould not be so surprising.\n\n\"Clean me!\" cries my son in his spilled water.\n\n\"Dry you,\" I say with my exposed shirt hem.\n\n\"Clean!\" he begs, but there is no substance\n\nto clean water.\n\nThe budding trees misstate the season.\n\nI am astonished by my screaming offspring.\n\n_Autography 8_\n\nWhat the mother will not\n\n[myriad] say. Many\n\nto secret. This is not just\n\nabout being a woman. No one\n\nbelieves mothers are, anyway.\n\nThey want to know how many\n\nher love is\n\nand want more.\n\nSilence keeps them\n\nsafe so she\n\ngives it\n\naway: mute.\n\nMute, mute, mutter (her\n\nmouth's a busted clasp).\n\n_Autography 9_\n\nMy father with hardly hair on arms or legs and husband in his plush hide. I'm some other creature they look at say, _who knows what you're about?_ I had a body once, remember?\n\nMy boys are two porcelain bell-sounds inside expensive, ornate eggs. Their skin is almost transparent and still soft as a breast. O ache, how the boy already starts his crossing over. Only four years old when a little boat pushed off from shore. I thought he was on it but saw then it was me, the baby-mother, floating away. We lit the craft on fire by shooting torched arrows at that mother, and told the husband in his pelts don't stand too close. Watch her burn and flicker off the edge of boyland where I've long since interloped and dug my heels in. But see here, these were only the idea of feet and I fast become an indistinct sound with indeterminate source. Murmur. Rumor.\n\nOnly a hot, palm-shaped stigma where my littler boy touches my thigh keeps me alive, staves off the specter-mother's residence.\n\n_Autography 10_\n\nIn the sandbox my son pretends\n\nto be dead. On the wooden edge I try\n\nto look alive. I am\n\na wife, in other words\n\n\"woman acting\n\nin a specified capacity\" as in\n\n\"fishwife\" or \"mother.\"\n\nWhat I would rather\n\nis irrelevant other than\n\nafterhours otherwising.\n\nNo one begged me to become\n\nthis fiery virago, ceaselessly\n\nscraping and gutting, searching\n\nout places to bury the wasted\n\nentrails. One night I dug\n\ntoo close and struck a root,\n\nlooked up and saw him:\n\n\u2014plaque nailed\n\nto his trunk read \"this one?\"\n\n\u2014there. I must look for\n\nsomething else to look for\n\nor look at, like my boy\n\non his back in a box. Eyes\n\nclosed, palms up and open.\n\n_Autography 11_\n\nThe woman opening to speak, to say, what can this [this]\n\nif this is [real] then what other way of saying other are there?\n\n_It is like_ this. _This, everyday_.\n\nOpened her and cracked her chest and clamped it.\n\nQuick, we must pack it with ice, _I_\n\n_went to a wedding where two people loved one another_\n\n_they rang bells and I cried because the dancing_\n\nin time everything dies except furniture _my son said_\n\n_my intention was a little two step thinking the remedy was something similar_\n\n_like treating fever with marigolds or love with love but can die like that_\n\nThe spine tries to protect.\n\nBut the real this that she knows the this that the then this.\n\nEvery day. Every day. Every day.\n\n_Autography 12_\n\nWhat is likewise hard can cleave.\n\nAnd so my roving eye sought a sharp punishment wanting the underlying shape to realize and when a man returned my wanting I made this punishment and when I wanted more punishment I looked straight at him like a blade like a punishing cleaver needing sharpening the shrill tones of the knifeman's traveling business pierce every writing through with wanting business he seeks to punish the blade I sought to punish meant to cut away the excess thereby polishing but punished the dust of the knife the cleaver the glance the gaze\n\n_Autography 13_\n\nHere, take this.\n\nIf you die it is not a good remedy. If you are healthy and develop symptoms it is a good remedy for someone else who had a sickness with these symptoms. This is the law of similars. It applies.\n\nWriting is a way to cultivate illness. In other words torture. In other words pervade. If you die you might have had a sickness or rendezvous.\n\n_Autography 14_\n\nAt night I become exquisitely pretty.\n\nI don't want to. Seriously, I don't. Won't be convinced. Not. This time. I really, this time, I seriously. Not. Tonight.\n\nThen when you fall asleep I suddenly, well, want to. Is that what makes me beautiful?\n\nBetter pull the shades down against this blinking want-to, wasted body, the short-long-short signal I'm sending out across the city.\n\n_Autography 15_\n\nThe smoke is from the falling down\n\nI told my son who is afraid\n\nof fire. Forgive me\n\nfor lying. One day he will find out:\n\nthe building's mangled corset,\n\ncracked femurs and blown out lungs\u2014\n\nthere was no one, not one,\n\nto give our blood to. And birds\n\nclogged the gutters. And rats. And paper.\n\nThe paper.\n\nMy son dreams of fire. He dreams\n\nof a mouse with claws.\n\nMeanwhile a father I know buys potassium\n\nchloride, cipro, soup, duct tape.\n\nDreams of ghosts under the kitchen table.\n\nOf his brother. Of falling.\n\nAnother man says he'd have stepped over\n\nanyone to find his wife.\n\n_Autography 16_\n\nThings I've been asked not to write about include: the death of a young child, money, group therapy. What's the harm in an affair that never happened asks N? Therapist: when something's gone this far in the mind it means something's wrong in the marriage. We pay her and I tell them about my husband's cousin's daughter who, at two years, eight months and a matter of days\n\n_Autography 17_\n\nMy son almost made but missed the toilet: that was real.\n\nAnd when the night and its many beasts breathing brought him up again I said, _get back_\n\n_with your sharp bite, gorgeous fangs_.\n\n_Stop writing_.\n\nIt makes me visible, I meant to write invisible. Damn that night with her pester\n\nwhispering _lean_...\n\n_against him_.\n\nCotton dress shirt.\n\nI see what the stupid phrase means: \"mind like a steel trap,\" as I gnaw my leg off to escape mine.\n\nA reader, anonymous, suggested my poems would be better\n\nif the marriage\/motherhood stuff wasn't so literal.\n\nLife too, I'd say.\n\n_Autography 18_\n\nI've fallen in love with everyone\n\nto unlove him by comparison.\n\nAll men. All women.\n\nThe cross-eyed gym guy on the subway.\n\nThe woman with blond braids under her pink hat.\n\nI have banished and exalted humor. Was\n\nyoung. Old. Showed my sadness\n\nas a corpse shows the surgeon:\n\nsee my facts of living?\n\nFor my next act I must jump off a ferry for more\n\ngood material. Right?\n\nSomewhere, this snowy night, Spalding Gray's body\n\nfloats the Hudson, I'm certain.\n\nGathering material.\n\nIt's impossible to unlove some ideas.\n\nNo matter how stupid.\n\nThere he is again:\n\nthe unloved only.\n\n_Autography 19_\n\nI wanted to write a tiny poem,\n\nbut as soon as I built it, it lied.\n\nWas the closed-mouth kiss\n\nof marriage and children. I wanted\n\nto leave something radiant on the pillow,\n\nbut was needed elsewhere\n\nto proffer idioms; be stern and soothing, subtly\n\nclairvoyant, familiar, original, over and over,\n\na pot boiling over puts out its fire but is still\n\nnot safe. Bough so laden\n\nit wastes bounty. The lyric\n\nwas meant to contain desire;\n\nI want some precious toxin.\n\n_Autography 20_\n\nOne night a woman showed up.\n\nWe talked about poetry. She\n\nwas full-grown and had lived a life.\n\nHer poems were dreams and animals,\n\nflying objects and little wails or wafts of witchery.\n\nShe brought black bean soup from the place\n\ndownstairs and ate it at my table.\n\nA man showed up, a younger man, another woman, but\n\nI knew them: where they lived, if they had children.\n\nEven her name didn't make sense and her rusted voice\n\nmade us all be quiet. She drank tea, water, seemed to be\n\nAmerican, but her poems moved like the clean-plucked\n\nwing of a chicken when you run it under water: too\n\nhuman. Who invited her? Later I dreamed she was riding\n\nthe express subway uptown, studying flash cards.\n\n_Don't wake up until you see the flip sides_\n\nsaid the dream but that\n\nwoke me up.\nAcknowledgments, Dedications, and Notes\n\nMany thanks to the editors of the following journals for publishing poems from this book: _Barrow Street, Black_ _Warrior Review, Black Clock, Bridge, Chicago Review, Court Green, Crowd, Five Fingers Review, Gulf Coast, Lyric, Maggid_ , _New Orleans Review, Now Culture, Tango, Xantippe_ , and _Zeek_.\n\nSeven sections from \"Annunciation\" were published in _Barrow Street_ and won the Barrow Street prize. The poem, as a whole, was later awarded The Center for Book Arts Prize by Lynn Emanuel and was printed in a limited edition chapbook designed by Roni Grosz.\n\nI am deeply grateful to Catherine Barnett, Arielle Greenberg, Joy Katz, Wayne Koestenbaum, D. A. Powell and Suzanna Tamminen for their generous and insightful readings of these poems.\n\nLove and boundless appreciation for my \"sisters\": Joan, Arielle, Stacy, Erin, Miriam, and Dana.\n\n\u2022\n\n\"Squirrel in a Palm Tree\" takes place en route to and from and in Savannah, Georgia, at the end of December 2001. The poem was made possible by the teachers\u2014Eric, Rajihah, Joy, Aury, Jennifer, Rafiyah, Amanda, Martina\u2014and Peggy at Basic Trust Day Care Center and by Lynn Heitler and Philip Levy. Thank you.\n\n\"Annunciation\" is for Abram, child of light, and for Josh, Moses, and Stacy, who made the journey with me.\n\n\"Autography 6\" is for Jeff Enke.\n\n\"Autography 20\" is for Hermine Meinhard, John O'Connor, and Patricia Carlin.\n\nSome of the poems are for Alex Wright.\n\n\u2022\n\n\"... synonyms do not exist\" is from Donald Hall's essay \"The Unsayable Said,\" published by Copper Canyon Press.\n\nThe title, \"The Museum of Accidents\" refers to a proposal by French philosopher Paul Virilio for a new museum that would expose and exhibit \"the accident,\" an inevitable consequence of our accelerated, highly technological society.\n\n\"Is the Soul just a notion, a drug?\" is from Alice Notley's poem \"Sun is Very Near Hot and Buttockslike\" in _Disobedience_ , Penguin, 2001.\n\n\"We could have been happy sooner\" is from Brenda Hillman's poem \"Cascadia\" in _Cascadia_ , Wesleyan University Press, 2001.\n\nOn page 80, \"I go on loving you like water but\" is from John Ashbery's poem \"The Tennis Court Oath.\"\n\nOther sources for \"The Rise and Fall of the Central Dogma\":\n\nAmerican Museum of Natural History, \"Darwin\" show, November 2005\u2013August 2006.\n\nBarry Commoner, \"Unraveling the DNA Myth,\" _Harper's_ , February 2002.\n\nBenjamin Farrington, _What Darwin Really Said_ , Shocken, 1966.\n\nSarah Blaffer Hrdy, _Mother Nature: Maternal Instincts and How They Shape the Human Species_ , Ballantine Books, 1999.\n\nJonathan Wells, Ph.D., \"Second Thoughts about Peppered Moths,\" <http:\/\/www.arn.org\/docs\/wells\/jw_pepmoth.htm>\nAbout the author\n\nRachel Zucker is the author of two previous books of poetry, _Eating in the_ _Underworld_ and _The Last Clear Narrative_ , both published by Wesleyan University Press. Zucker is the co-editor of the anthology _Efforts and_ _Affections: Women Poets on Mentorship_ , published by the University of Iowa Press. She has taught at NYU and Yale and was the poet in residence at Fordham University. She lives in New York City.\n\nwww.rachelzucker.net\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n**EARLY BIRD BOOKS**\n\n**FRESH EBOOK DEALS, DELIVERED DAILY**\n\nLOVE TO READ?\n\nLOVE GREAT SALES?\n\nGET FANTASTIC DEALS ON BESTSELLING EBOOKS\n\nDELIVERED TO YOUR INBOX EVERY DAY!\n\n**The Web's Creepiest Newsletter**\n\n**Delivered to Your Inbox**\n\nGet chilling stories of\n\ntrue crime, mystery, horror,\n\nand the paranormal,\n\ntwice a week.\n\n# Felicia\n\nGeorge Alec Effinger\n\n**Acknowledgments**\n\nA great deal of research had to be done before the writing of _Felicia_ could begin. I would like to express my debt to the people who made this task simpler.\n\nI would like to thank the staff of the New Orleans Public Library for its generous help.\n\nI am also indebted to Justin Wilson, the well-known Cajun humorist, whose books and records of Cajun anecdotes first introduced me to these people and their way of life.\n\nI must thank Bobby Conner of the Cameron Parish Sheriff's Office for his help, and also Lieutenant Robert K. Lindsey of the Jefferson Parish Sheriff's Office, who answered many questions and supplied me with details concerning the duties of law enforcement organizations during hurricane emergencies.\n\nFinally, I would like to thank Philip J. Decker of International Harvester for much information concerning trucks; C. W. McCall for his album _Wolf Creek Pass_ ; Dr. Shirley Van Ferney, for her friendship and support.\n\nMost especially, I have to thank Beverly Kandrac, without whose help _Felicia_ might have remained only a chapter and an outline.\nThe way of the superior man is threefold, but I am not equal to it. Virtuous, he is free from anxieties; wise, he is free from perplexities; bold, he is free from fear.\n\n\\- Confucius\n\n_Analects_\n\nWhat have I gained, that I no longer immolate a bull to Jove or to Neptune, or a mouse to Hecate; that I do not tremble before the Eumenides, or the Catholic Purgatory, or the Calvinistic Judgment Day\u2014if I quake at opinion, the public opinion, as we call it; or at the threat of assault, or contumely, or bad neighbours, or poverty, or mutilation, or at the rumour of revolution, or of murder? If I quake, what matters what I quake at?\n\n\\- Ralph Waldo Emerson\n\n_Character_\n**ONE**\n\n**The Calm**\n\n****\n****\n\n**1**\n\n****\n\nThe Louisiana Town was named Arbier, after a French priest who had ministered to the spiritual needs of the area's Indian population, back when the area's population had been only Indian. The present inhabitants of Arbier were proud of their heritage; only a very few of them could claim Indian blood, but that did not stop others from trying.\n\nThese local people pronounced the name of their town _Arber._ Outsiders always tried to pronounce it as Ar- _byay'_. The foreigners, as the locals thought of all outsiders, could be picked out easily. There were no tourists, only foreigners. Even other Louisianans were foreigners, from up north in New Orleans or Baton Rouge. Shreveport was practically Yankee country. The pride and clannishness of the townsfolk was such that when a visitor made the pronunciation error, no one would correct him. He would go on making the same mistake until he left the town, and the residents of Arbier would smile in a way that no foreigner could understand.\n\nIn the late 1950's, the town in St. Didier Parish had numbered only twenty-five hundred people. Now, some two decades later, the population of Arbier was hovering around three thousand. Nothing changed in the town except the names of some of those people, and the seasonal price of shrimp.\n\nOutside Paul Pierson's apartment there were two small balconies. They were shaded by the roof of the neighboring house in the morning, but as noon approached more of the black iron railings and bases absorbed the August heat. It was past eleven o'clock in the morning, and the topmost railings were already beginning to broil in the nearly direct rays of the summer sun. A mockingbird flew from the crepe myrtle tree in the backyard and landed on the railing outside Pierson's living room. The bird hopped sideways uncomfortably until it sat on the shaded part of the iron bar. Once it had settled down, the mockingbird began to run through its extensive catalogue of songs.\n\nPierson had sliding glass doors leading from his bedroom and living room onto his two balconies. The doors were open for ventilation, and the mockingbird's loud singing pierced the stillness of the apartment. Pierson was asleep, but his large gray Persian cat was not. The cat watched the bird intently from across the living room; almost every morning the bird sat on the balcony and taunted the cat. Almost every morning the Persian ran full speed across the living room carpet and jumped at the bird, hitting the sliding screen door. The mockingbird didn't fly away in fear; it did what its name suggested, it mocked. It sat on the iron railing and twittered at the furious cat.\n\nThe Persian may have failed to catch the mockingbird, but the noise succeeded in waking Pierson up. He rubbed a hand through his hair and stared blearily at the wall opposite him. Then he walked into the living room. \"Knock it off, Cy,\" he muttered. He picked up the cat and tossed him halfway across the floor. Then Pierson slid the glass door shut so the bird's singing wouldn't bother the cat any longer. Pierson closed the drapes in front of the door. Then he went back into the bedroom and lay down. After ten minutes he realized that he wouldn't be able to fall asleep again. He shrugged. It was going on noon, and he decided that he might as well get up. Still, though, it was a Saturday morning and he wasn't in any hurry. He pictured to himself his first few movements of the day: get up, pick his clothes up off the floor, go into the bathroom, wash up, get dressed. Maybe eat. Then the rest of the day was his. He thought about these moves from beneath the sheet that covered him. He made no motion to get up yet.\n\nInstead, Pierson rolled over and faced the young woman who was still asleep beside him. Her name was Maddie Gargotier, and Pierson thought that she was a little bit strange. It seemed to him that every few days she would make a crazy decision concerning her future or their life together. He was always amazed to learn how she arrived at these decisions. He guessed that her thought processes rarely had anything to do with real life. Her plans always seemed to her to be carefully made and irrevocable; fortunately, Pierson thought, she forgot them in a matter of hours.\n\nMaddie was a pretty young woman, twenty-two years old, with long dark hair that set off her pale, delicate face. She had high cheekbones, a small, freckled nose, and greenish eyes that changed color depending on the clothing she wore. Now, as she opened them, while she wore nothing, they were just greenish.\n\nPierson reached out fondly to touch her sleepy face.\n\n\"I have something to tell you,\" she said.\n\nPierson pulled his hand back, his gesture incomplete. This morning her words hit him very hard. They were precisely the same words his ex-wife had used on the day she had left him. Coming from Maddie, they were an unpleasant reminder. Here we go, thought Pierson. Either she's leaving me too, or she's pregnant.\n\nHe said nothing. He stared at her, blinking, waiting for her to say what she had to say.\n\n\"I'm joining the Navy,\" she said.\n\nThere wasn't anything for Pierson to do. He was relieved, in a way. It was another of Maddie's quirky inspirations. But he had learned from experience that at the moment she was determined to do what she said. Pierson lay in bed, waiting quietly, like someone who was sitting through a dull joke in the wan hope of a killer punch line. Anything Pierson could have said would have sounded mean and self-serving.\n\nThere was a long, uncomfortable pause.\n\n\"Well?\" asked Maddie.\n\n\"Well what?\" said Pierson, knowing precisely what she was waiting for.\n\n\"What do you think?\"\n\n\"Maddie,\" he said, \"I think that you'll see a lot of the world, get to wear fashionable uniforms, and mostly not have to make many decisions for a few years.\"\n\n\"Oh, Paul,\" said Maddie with a disgusted expression. \"I knew you'd say that.\"\n\n\"How?\"\n\n\"How what?\"\n\nPierson stared up at the ceiling. \"How did you know what I'd say? And what _did_ I say?\"\n\n\"Selfish,\" said Maddie. Pierson did not reply. \"I shouldn't even have asked you,\" she said. \"Me and Shelley and Betsy decided we'd do it together. We're all going down to take the tests the day after tomorrow. I want to see if I can get into the submarine service.\"\n\n\"I don't think they'll put you on one,\" said Pierson. He rubbed his eyes, which were itching and burning. He hadn't tried to touch Maddie since his first attempt. Now he didn't feel much like trying. He'd feel strange; in a way, she had become naval personnel, almost.\n\n\"I don't actually expect to be put on a submarine,\" said Maddie, \"but there's still lots that I can do.\"\n\nPierson had an answer to that, and he knew that he shouldn't give it to her. He did anyway. \"You don't like your job with Krieger-LaChapiet, right? And so you'll join the Navy. I give you five-to-two odds that you'll get the same job, except then you won't be able to quit.\"\n\n\"You're selfish, you're just plain selfish,\" said Maddie, rolling even farther away from him. \"You don't want me to try anything. You don't want me to make something out of my life.\" Pierson's previous reply had been just what he had thought it would be. A mistake.\n\n\"Sure,\" he said, \"I want you to make something out of your life.\" He sighed. Maddie Gargotier had little enough to work with. Her father owned a popular bar in Arbier, and Maddie had practically grown up in it. The St. Didier Parish School Board didn't expend itself very hard making certain that the children of the electorate attended classes. Maddie's education had been intermittent at best. When she did go to the regional high school, she took mostly typing and secretarial courses. Her guidance counselor advised Maddie that these courses were the most practical; in Arbier they were certainly better than a life of crime, but little more. Most of the employees in the town and the surrounding areas needed farmhands or crewmen for fishing boats. There wasn't much demand for clerks or typists. Maddie had vague plans of going to New Orleans or Lake Charles or Lafayette to look for a job. Pierson couldn't understand how those hypothetical jobs might be better than her present position, in which she was virtually the entire main office of a small fishing supply house.\n\nPierson sat up and tried to untangle the sheet on his side of the bed. \"I don't want you to make a mess of your life, is all,\" he said.\n\n\"You let me worry about that, Paul,\" she said.\n\n\"Since when?\"\n\n\"You go to hell, Pierson,\" she cried. \"You come down here from Ohio all full of superiority. You think you can run anybody's life.\"\n\n\"I came down here to stay. I'm almost as southern as you are.\"\n\n\"You are not,\" said Maddie, still angry. \"You got to be born to it. You can't even remember if you say 'crick' or 'creek.' You can't remember for here, or for Ohio neither.\" That was true.\n\n\"I'm lost without you, Maddie,\" said Pierson. He smiled.\n\n\"I'll tell you what,\" she said. \"Forget about what I said.\n\nCome on back, and go to hell _again._ \" She started sorting through the pile of clothing on the floor. \"I want the bathroom first,\" she said.\n\n\"You got it,\" said Pierson. \"And tonight, Navy person, you sleep in a hammock. See how you like it.\"\n\n\"Yeah,\" said Maddie, turning around and facing Pierson belligerently, \"who says I'll even come back here tonight?\"\n\n\"I didn't think you would,\" he said. \"If you really want to sleep in a hammock, you'll have to go somewhere else.\"\n\nMaddie didn't know how to interpret his remark. This was natural, because Pierson didn't know, either. He wasn't sure if he was playing or genuinely angry. Maddie just stared. Pierson stared back, and after a few seconds Maddie turned and went into the bathroom. Pierson decided to go into the living room and wait; by the time she was finished in the bathroom he would know what he was feeling. He got dressed and brushed his hair. Then he went into the kitchen and took a bottle of Dr. Pepper from the refrigerator for breakfast. He opened it and went into the living room, where he put on the television.\n\nThe Channel Five weatherman was giving his noon report. \"There's good news and bad news in the weather today,\" he said. \"You can see all of it in this morning's satellite photograph from the New Orleans Weather Service. This white mass here is Hurricane Dinah. She's the good news. This high-pressure ridge extends down through Florida and out over the ocean. Dinah spun into it late last night, and it looks like she's rebounding to the north-northeast, away from the Florida coast and definitely no threat to the Gulf region. The latest predictions are that Dinah will rain herself out over the Atlantic, gradually filling in, presenting no threat to any populated areas. But we all know what hurricanes like to do. They like to make fools of weathermen. So we'll still keep an eye on Dinah until she's officially downgraded off the map.\n\n\"That's the good news. The bad news is this tight spiral of clouds over the Virgin Islands. Yesterday that was Tropical Storm Elsie. Today she's Hurricane Elsie. You can see on the satellite loop the definite cyclonic rotation. Her winds are clocked at eighty-five miles per hour, with some winds up to a hundred and twenty miles per hour. Her course has been generally to the west, but we're hoping that the same high-pressure system that diverted Dinah will push Elsie aside. We'll have a better idea of what's going on with Elsie on the Six O'clock Report tonight.\"\n\n* * *\n\nThe weatherman's words were listened to in silence all over Arbier; the people in the town knew what a passing run from a hurricane meant. They had lived through them before. They had no desire at all to do it again. But every year, from about the first of June through October, the Hurricane \"season,\" the residents of Arbier watched the weather reports closely. They needed every hour of warning, in case a hurricane turned toward the Gulf coast. Arbier was hit early in such a case; after the small town was mauled the hurricane moved inland, losing some of its force and terror. But Arbier always took the full strength of the storm's power.\n\nWalter Boshardt, the sheriff of St. Didier Parish, watched the Channel Five news program from his usual lunchtime booth in Mrs. Perkins' diner, the Crisis Caf\u00e9, on Ridge Street. He shook his head when he heard what Strahan the meteorologist had to report. As the weatherman said, it was good news and bad news. But to Boshardt's mind, the good news hardly made up for the bad. It meant that the sheriff had to live with an imaginary killer named Elsie for a few days. Elsie, the personality engendered by the Weather Service, was imaginary; Elsie the murderous storm was very real, already taking lives and destroying property in the Caribbean.\n\nSheriff Boshardt had been elected to his office four times in sixteen years. The reason was simply that the people of St. Didier Parish liked him. He was a tall, trim man, darkly tanned by the Gulf coast sun, with deep-set, intelligent eyes and short blond hair which he described as \"rapidly failing.\" He liked to sit in the diner and laugh with his friends, he liked to sit in Mike's and drink a couple of beers and watch football or basketball on the color television, and he liked his job. Maybe that was the reason his constituents were so pleased with him.\n\nBoshardt divided his duties into two categories. First, he enforced the laws of the parish and the state. This part of his job was necessary, even vital, but Boshardt didn't care for it very much. He was sometimes embarrassed to arrest someone for a misdemeanor, particularly a person whom Boshardt knew well. He never let that interfere with his work, though; he felt uncomfortable, but he did it.\n\nThe second category of duties was more in the line of public service. He enjoyed this a great deal more. He spoke to children in the schools about safety and avoiding accidents. He delivered talks to groups of parents on what to do when their children ignored the safety speeches. He presented information on teenage drinking and drug use. He gave short courses in disaster relief. He was aided in these duties by several deputies, but he performed most of his tasks himself. He hated to delegate jobs; he wanted to be certain that things would be done right, and the best way to do that was to do the work himself.\n\nHe was eating a plate of red beans and rice and washing it down with a glass of cold beer. He mopped up the gravy with chunks of French bread, almost obsessively, every three forkfuls. While he ate, he stared at the television screen.\n\nSkip Strahan, the Channel Five weatherman, was finishing his segment of the noon news. \"Present temperature on the square in downtown Linhart is eighty-nine degrees. Humidity at sixty-seven percent. Winds from the south-southeast at eight miles per hour. Outlook for the Linhart viewing area: clear today and tonight, high in the low nineties, low tonight in the upper seventies. Partly cloudy tomorrow afternoon, with a fifty percent chance of rain late in the day. High tomorrow in the low nineties. That's all for now. Sheila Downing will be back at six o'clock with the weather update.\" The image of Strahan's young face was replaced by a bank commercial. Sheriff Boshardt turned his attention back to his lunch.\n\n\"Glad that storm moved north instead of heading on into the Gulf,\" said the waitress, a seventeen-year-old girl named Lauren.\n\n\"Yeah,\" said Boshardt.\n\n\"I don't know,\" said Lauren. \"Seems this is the time of year for hurricanes to make a run for us.\"\n\n\"Yeah,\" said Boshardt, mopping up the gravy on a piece of bread.\n\n\"And I don't know why they have to name hurricanes after women.\"\n\n\"Neither do I, honey,\" said the sheriff. He looked at her thoughtfully.\n\n\"So Dinah's gone,\" said the waitress. \"It's sad, sort of. If we were going to have a hurricane, I like the name Dinah better than\u2014what is it? Elsie?\"\n\n\"I had a grandmother name of Elsie,\" said Boshardt quietly. That wasn't true; he hadn't, really. He just wanted to take a quiet shot at the girl. He swallowed a long mouthful of beer, staring at the wall beyond the row of booths.\n\n\"Oh, I'm sorry,\" said Lauren, flustered. \"I didn't mean anything by it.\"\n\nNo, you didn't, thought Boshardt. And neither did I. He laughed softly. So we're all even, and that's the way I like it to be.\n\nHe smiled at the waitress, and she started to say something, but didn't. She turned and went into the kitchen. Boshardt shook his head sadly. At the time, he didn't know why.\n\nIt was almost twelve-thirty. The sheriff stood up and ran a hand through his hair. He yawned.\n\n\"Had enough?\" called Mrs. Perkins from the kitchen.\n\n\"Yeah, Ma,\" said Boshardt. \"Real good today. I'll leave you a couple of bucks on the counter.\"\n\n\"You don't have to, Sheriff,\" said the old woman.\n\n\"I know,\" said Boshardt. \"If I had to, I'd eat somewheres else.\" It was about the same thing she said to him every day, and the same thing he said to her. He left two dollars beside the cash register, then walked back out into the hot air that covered Ridge Street like clear fire.\n\nThe patrol car was parked about forty yards from the Crisis Caf\u00e9. Boshardt hoped that he wouldn't meet anyone he knew on the way to the car; he was a very friendly person, and he valued his close relationship with the people of Arbier, but after a heavy meal on a hot day he didn't want to have to smile and make idle conversation. He wanted to get in the car and drive for about fifteen miles and let the garlic and hot sausage fade away into his tissues. He decided to drive up to Linhart and check on the office.\n\nRidge Street, one of Arbier's four main streets, ran straight from the northern corporation limit south to the Gulf of Mexico. The street ended with a driveway paved poorly with asphalt and covered with small white shells; the drive narrowed into a rutted dirt track, which led directly to a very old wooden pier that visibly swayed with the slapping of the waves. Beside the pier on the narrow beach was a bait shop, long since closed down and abandoned. Sheriff Boshardt had to go down to that old shack about two or three times a year, to chase away high school kids or drunks or noisy couples; these people could easily find other places to have their disorderly times, and Boshardt encouraged them to do just that.\n\nHe thought about that bait shop as he passed by the Sea-Ray Motel, which was usually the second stop for the unmarried couples, some of the high school kids, and a surprising number of the drunks, not all of whom were unemployed vagrants. The shack made a southern border for Arbier, a kind of outpost. The Sea-Ray Motel guarded Arbier's northern frontier. Beyond the Sea-Ray, Ridge Street became a parish road, as bad as the asphalt and shell driveway. The noise grew irritating as Boshardt drove along; the car bumped and swayed from one deep chuckhole to another. The shells popped and cracked beneath the tires. A thick cloud of dust and dirt fanned up around the car. Boshardt rolled up the windows as the fine dirt sifted in on him.\n\nThe parish of St. Didier had three main population centers. They were Arbier; Linhart, the parish seat, with about six thousand people; and, northwest of Linhart, the town of Delochitaches, which the people there pronounced De- _lock'_ -i-tus. Boshardt thought that it was just plain unfriendly for a town of eighteen hundred to have a name like that. He had to be careful to give Delochitaches as much attention as he gave Arbier and the rest of the parish; he really didn't like that community.\n\nArbier, where Boshardt had been born and where he grew up, received the greatest share of his time. Many of the people in the parish believed that he should live in Linhart; but in the first place, Boshardt didn't want to leave the house he had lived in all of his life, the same house his parents had moved into after their wedding; in the second place, Boshardt could cite statistics to prove that the crime rate in St. Didier Parish did not justify his moving. It really didn't make any difference where he lived. He could do almost as good a job from Seattle, Washington.\n\nMost of the people of Arbier earned their livings either on the sea or in the fields. On the sea were shrimp boats and fishing trawlers; there were also off-shore oil rigs. In the fields to the north and west of the town was sugar cane. Boshardt looked at the fields of green, waist-high cane as he drove by. Boshardt thought that St. Didier Parish was one of the most beautiful in the state, although by national standards the parish was impoverished.\n\nArbier's residents had a characteristic shrug that they presented to inquisitive foreigners. Like the unwillingness to correct a visitor's pronunciation, reserving that for local folk, the shrug was meaningful to the people of Arbier. Tourists and reporters interpreted the shrug as part of the Cajun spirit; only another native would have understood just how hostile the gesture was.\n\nThe sheriff left Arbier behind, and the cane fields. Soon the farming communities began to flash by on the sides of the road. They were each nothing more than a couple dozen pine board shacks, roofed with tarpaper or sheets of galvanized metal. There was always a gas station, a grocery store that might also contain a post office, and three or four lounges. That's what they called themselves. He just wished that they wouldn't call _him._ Sometimes there was trouble, and sometimes he had to go tend to it. Out here, in tiny villages of one or two hundred people, there was very little concern or respect for the law. People didn't feel part of the parish or the state; they felt separate and virtually exempt from the rules of distant lawmakers. Store owners and lounge operators often had reason to disagree.\n\nLounges. Hell. Boshardt took a quick glance into the open door of Marie & Pal's Lounge, in Capita, Louisiana, as he drove by at thirty-five miles per hour. The inside of the bar seemed as black as death, with only a fuzzy red rectangle glowing faintly in the back. That was all Boshardt could see. No people, no life. Blackness. He drove on.\n**2**\n\nThere was a man registered under the name of Robert Branford in Room 8 of the Sea-Ray Motel. The day manager suspected that this was not the man's real name; the manager didn't especially care. Most of the business he did was with people using false names. What made Robert Branford unusual was that he was alone.\n\nA light went on on the switchboard in the office. The manager saw that it was the man in Room 8. \"Yes?\" he said.\n\n\"I'd like to make a long-distance call, please,\" said Robert Branford.\n\n\"Sure,\" said the manager. He dialed the long-distance operator. \"Operator? This is the Sea-Ray Motel. A guest in Room 8 would like to make a long-distance call. Thank you.\" He pulled the patch cord and went back to his reading. He didn't need to eavesdrop to be entertained. Not at the Sea-Ray.\n\n\"Yes,\" said Robert Branford. \"I'd like a number in Miami Beach, please.\" He gave the operator the number and waited for the connection to go through. He heard the phone in Florida ringing.\n\n\"Hello?\" said a man's voice on the other end.\n\n\"Hello, Tom?\" said Robert Branford. \"This is Chuck in Louisiana. In Arbier.\" He pronounced the town's name wrong.\n\n\"Right,\" said Tom. \"I was expecting you half an hour ago.\"\n\n\"I'm sorry about that. It took me a while just finding a motel around here. They don't get much tourist trade, you know.\"\n\n\"As a matter of some interest,\" said Tom, \"I don't know. But that's fine. We don't want a lot of strange people getting in our way. Who's at the motel with you?\"\n\n\"With me? Nobody,\" said Chuck.\n\n\"Don't be stupid. In the other rooms.\"\n\n\"Oh. A couple of guys that work on the off-shore rigs. A black hooker turning tricks in Room 13\u2014\"\n\n\"It figures you'd find that pretty fast.\"\n\nThere was a pause. Chuck was annoyed, but he kept his anger under control. Tom kept Chuck and the rest of the crew under control; that was what Tom was for. \"Yeah,\" said Chuck at last. \"The rest of the rooms go by the hour instead of the day. It's that kind of place.\"\n\n\"That's good,\" said Tom. \"You must fit in real nice. What do they think you're doing there?\"\n\n\"I dropped kind of a hint that I'm doing research for a magazine article. That covers all the questions I might ask, and also driving around casing the town.\"\n\n\"Terrific. You're using your head. What do you think of the place? What was it? Arbier?\"\n\n\"Yeah,\" said Chuck. \"I think it's just what we want. Maybe about three thousand people, but you never see more than a dozen at a time, even downtown, even during business hours. Everything's on four main streets. You could drive right through the town and miss everything if you took your eyes off the road to change radio stations. Of course, then you'd be in the Gulf of Mexico.\"\n\n\"Of course,\" said Tom. He sounded weary.\n\n\"That's the thing, too. It's right on the water, just the way you wanted. The highest point in the town is about seven feet above sea level. Any kind of heavy weather off the Gulf, and the whole place could be wringing itself out for days.\"\n\n\"So the people there probably know what to do during heavy weather.\"\n\n\"Right,\" said Chuck. \"That's what we want, isn't it?\"\n\n\"That's what we want.\"\n\n\"How about, uh, Bill?\"\n\n\"Who?\" asked Tom.\n\n\"Nelson. You know. He wants everybody to call him Bill.\"\n\n\"He called in from New Orleans. When he was supposed to. He says that's a piece of cake, too. No sweat.\"\n\n\"Everything's going our way\u2014so far,\" said Chuck.\n\n\"I might remind you that so far isn't very far at all yet,\" said Tom.\n\n\"No,\" said Chuck, \"but we're getting there.\"\n\nThere was a brief pause. Chuck could hear the sound of liquid being poured into a glass. The sound made him thirsty. Over the telephone he heard Tom sigh. \"Look,\" said Tom, \"Denny and his crew will be there in two days then,\" he said. \"If he can find the goddamn place.\"\n\n\"If he gets lost,\" said Chuck, \"tell him to call me.\"\n\n\"Yeah,\" said Tom, sounding even more tired. \"That's just what I'll have him do.\"\n\n\"Look, if you don't think I can handle this, just say so. I'll be happy to watch. Just so I get my share.\"\n\n\"Chuck, I have a few things I want you to understand. This is the kind of operation that requires careful planning. You want to know why? Because the parts of this operation will be working at great distances from each other. I will be coordinating things, but to a certain extent you will all be on your own. So you have to understand your individual roles completely, and perform them perfectly. Your individual role, Chuck, is not being cute. Please don't be cute, Chuck. Because if you keep it up, there are Curt and Allen, neither of whom you've ever met, and they'll be looking through your dresser drawers in a matter of hours. And you won't like that at all. So don't be cute.\"\n\n\"All right,\" said Chuck bitterly. \"You plan everything. You tell us what to do. Okay, we'll do it. But it all better work out, because there are more of us than there are of you, even with whatever their names were.\"\n\n\"Goddamn it, there you go again. You're being cute. Last time, Chuck. Positively the last time.\"\n\nThere was silence. Chuck glared at the wall in his motel room. Finally he spoke. \"What about Miami? Is Ed ready there?\"\n\n\"Ed has been redeployed. We came to the conclusion that the operation could be better handled by Nelson in New Orleans. It will be less expensive and involve fewer people that way. The simpler the better.\"\n\n\"Simple! This thing's getting to be like the Normandy invasion.\"\n\n\"True,\" said Tom, \"cute, but true. When you're taking advantage of natural disasters and the panic of simple people, you have to do it with quickness and finesse. And style, too. I like to do everything with style.\"\n\n\"Right,\" said Chuck, \"style will see us through.\"\n\n\"It will,\" said Tom, \"better than cuteness.\"\n\n\"I'll call again tomorrow.\"\n\n\"You do that, Chuck. I'll be waiting anxiously.\" He hung up. In Arbier, Chuck held the receiver for a few seconds, frowning. Then he slammed the phone down angrily and went out to find a liquor store. He returned a short while later; he flicked the television set on and sat on the edge of the bed. He listened to Skip Strahan's noon weather report while he mixed Seagram's and 7-Up in a plastic tumbler. The liquor and the 7-Up had been easy enough to obtain in a chain drug store, but the plastic tumbler had cost Chuck\u2014Robert Branford\u2014a quarter in the motel office.\n\nChuck heard little of the weather report, but he did pay attention to the news of Hurricane Dinah and Hurricane Elsie. Hurricanes were going to play a large part in his immediate future. He raised his tumbler toward the television screen, in silent toast Jo the weatherman and his storms. Then he stretched out on the bed and drank until he lost the bottle.\n\n* * *\n\nSkip Strahan turned from his weather map and looked into the number two camera. \"We'll have a better idea of what's going on with Elsie on the Six O'clock Report tonight,\" he said. Then he gave a quick summary of the forecast for the Linhart viewing area, smiled, and waited for the red light on the camera to go off. When it did, he heard the intro to the bank commercial; he held his pose, smiling determinedly, until the director signaled for him to stop. \"Okay,\" said Strahan.\n\n\"Good job, Skip,\" said the director.\n\n\"Okay.\" Strahan walked off the news show set and headed for his small office and dressing room. He wanted to get the makeup off as quickly as possible. He hated it. He had never grown used to wearing it. It made him feel filthy and greasy.\n\nAfter he finished cleaning it off and changing from the station's uniform tie and blazer into his own more casual clothes, he sat for a minute looking into the mirror. He was feeling kind of rocky. He had taken three Valium before the show began. He didn't feel them at all. He felt terrible, and as he sat looking at his reflection he realized that he was feeling steadily worse. His ears were buzzing loudly and his throat was very dry. He felt a panic coming on. He opened the drawer in his dressing table and took out the bottle of tranquilizers. He swallowed two more with a gulp of warm Coca-Cola. He waited, pretending that he could feel the Valium attacking his anxiety, returning him to a normal frame of mind. His head still buzzed, though, and his hands were sweaty. He thought about taking two more pills.\n\nThe telephone on the dressing table interrupted him. He picked up the receiver. It had to be important, because the receptionist wouldn't let viewers get through to the station personnel otherwise. \"Hello?\" he said.\n\n\"Skip?\"\n\n\"Yes.\"\n\n\"This is Darlaine.\"\n\nStrahan had to think for a few seconds. Darlaine? A weird name. Who was it? Then he remembered, and he knew how she had gotten past the woman at the switchboard, and he didn't like it. Darlaine was the wife of Walter Boshardt, the sheriff of St. Didier Parish. When Skip Strahan made a romantic conquest, he made it count. He wondered angrily why he never learned from past mistakes; he knew that this mistake was not one of his ordinary lapses. If this one blew up, he'd be lucky to get a job broadcasting weather reports to penguins in Antarctica. \"How are you, Darlaine?\" he said, wishing that someone would call him away, or that some fire or emergency would happen.\n\n\"Fine, Skip.\"\n\nThe conversation sat there for a long, uncomfortable moment.\n\nStrahan almost asked how her husband was, but that was a bad question. He said nothing instead.\n\n\"I was wondering,\" said Darlaine Boshardt, \"maybe you'd be interested in talking to the Arbier Boys' Club about what to do in a hurricane.\"\n\n\"I did that two months ago,\" he said. \"To the Boys' Club. They didn't pay much attention.\"\n\n\"Oh.\" She thought for a moment. \"How about the P.T.A.? Have you talked to them yet?\"\n\n\"Not recently,\" said Strahan.\n\n\"I'm sure I could get you scheduled. And you could get your regular fee. I'll bet the Boys' Club didn't give you your regular fee.\"\n\n\"I'll be happy to talk to the parents,\" said Strahan. \"When you have it all worked out, call the station and leave word with one of the secretaries.\" He hoped that she would take the hint and hang up.\n\nShe didn't. \"I was also hoping to see you soon, myself,\" she said. \"Nothing about the weather or anything.\" She tried to sound coy, but the result was so unbearable that Strahan wanted to throw down the receiver.\n\n\"I'll try, Darlaine,\" he said, \"but you know what it's like during hurricane season.\" He left it at that, because she didn't know what it was like. She wouldn't realize that his job was just about the same as it was all during the year. He got most of his information, forecasts, and satellite photographs from the Weather Service in New Orleans. All he did was organize it, draw his lines and numbers on the maps, and read the official conclusions to the folks at home.\n\n\"I guess so,\" said Darlaine.\n\n\"Yeah. Well. I'll try to get down to Arbier for one reason or another soon. I'll see you then.\"\n\n\"Oh, yes, please.\" She almost cooed; Strahan shook his head.\n\n\"I have to go now,\" he said. \"I have to draw the maps for Sheila's evening report.\" That was a lie.\n\n\"All right, Skip. Don't forget. Goodbye.\"\n\n\"'Bye, Darlaine.\" He hung up, wondering what she didn't want him to forget. He unclenched the fist that hadn't been holding the telephone; he looked at the bottle of Valium, then decided not to take any more. He was beginning to feel a little less edgy now. It was all beginning to overwhelm him; the tranquilizers helped him cope with the pressure and the fear he felt during the day and at night. The powerful sleeping pills got him to sleep. The mood elevators and amphetamines got him going in the morning. It was one long cycle. He got the Valium and the Quaaludes from a doctor in Linhart, who was treating him for excessive anxiety. He got the mood pills and the pep pills from a doctor in Arbier, who was treating him for acute depression. He didn't dare tell either doctor about the other. Now he was locked into the cycle, and he couldn't break it. Without the uppers or downers he would be lost; he hated even to think about what he would feel like. And he had to feel good, he had to be ready, every day, in front of television cameras, in front of many thousands of people. He had to smile.\n\nHe didn't want to. He just wanted to lie down. He just wanted to forget about Darlaine Boshardt.\n\nThe telephone rang again. He didn't answer it this time. The sound of the ringing disturbed him, but he didn't move.\n\nAcross town, Corinne Strahan listened to the ringing of the phone in her receiver. Her husband wasn't answering the call; she guessed that he was in his dressing room, because just a few moments before she had tried to call and his line was busy. She hung up and dialed again; the switchboard operator put her through again, and once more she listened to the sound of the telephone ringing. After ten rings, she gave up.\n\nCorinne could easily have gotten into her car and driven to the studio to see her husband. In fact, that was something she used to do very often, after the noon broadcast. They had had lunch together a few times every week. Now, Skip didn't seem to mind missing their lunches together; she understood that he needed all the rest he could get between broadcasts.\n\nOn this day, there was very little for her to do. She had taken two loads of washing to the laundromat\u2014the washateria, as they called it in Linhart. Corinne thought the word was funny. All that work had been finished by ten-thirty. She still had to go shopping for dinner, but there wasn't any hurry about that. The market was right across the street. Skip didn't get home until seven-thirty, anyway, even though he didn't have to do the late report on Sunday. They didn't eat dinner until nine o'clock. . . .\n\nCorinne sighed. She had a lot of trouble getting through the day, too. She knew about the pills and things that Skip was taking. She sure didn't need anything to get her going in the morning, though. Her problem was filling up the time she had. She could watch television, but that meant hearing Skip's voice or seeing him at noon, and that just made her feel worse. She had thought about getting a job.\n\nWhat would Skip say if she told him she was going to work in the cane fields? The idea made her laugh. It was a brief interruption in her boredom. Too brief. Quickly, the tedium re-formed itself around her.\n\nIt was after one o'clock. To Corinne, it seemed as though it ought to be the middle of next week, so slowly had the morning gone. She picked up the phone again, letting a telephone number materialize in her mind. She didn't care whom she called. It turned out to be her father; that didn't surprise her very much. Corrine had talked Skip into renting an apartment for her father, an aging, sick widower. Skip had insisted that Corinne's father not live near them. He didn't want to have to go over to his father-in-law's every time a faucet started dripping. They had agreed to find a place for the elderly man in Arbier; Corinne's father had lived his entire life in Baltimore, and he resisted moving south. Faced with the prospect of living the rest of his life entirely alone, however, he did not resist very strongly. Now Corinne kept in close touch with him and visited him whenever she could talk Skip into driving down.\n\n\"Hello, Dad?\"\n\n\"Corinne?\"\n\n\"How many other daughters do you have?\" she asked. \"Some I don't know about? You been stepping out that I don't know about?\"\n\nHer father laughed. The sound seemed hoarse and frightening to Corinne. \"I'm glad you called,\" he said. \"I was hoping the two of you would visit me this weekend. I know how Skip feels but, damn it, I don't like to be alone here. I'm not going to pretend that there are things that need fixing. I just want to see you. You're my only child, Corinne.\"\n\n\"I know, Dad. But you have to understand how Skip thinks. He believes that this way is the best for all of us.\"\n\n\"Corinne, I'm not going to be here forever. One of these days soon, something's going to come along and carry me away. And before that happens, I want to see as much of you as I can. You're the only thing in the whole world that gives me any happiness. It's selfish, I know, but I think I've earned the right to a little of that.\"\n\n\"Dad,\" said Corinne sadly, \"you're trying to make me feel guilty again. You're trying to blackmail me emotionally. That's just the very thing that Skip objects to. And so do I. You know that it would take a team of angels to carry you off; you're just too stubborn for anything less.\"\n\n\"All right. Well. Remember what I said.\"\n\n\"Okay, Dad. I just wanted to make sure you were all right.\"\n\n\"I'm fine.\"\n\n\"Goodbye, Dad.\"\n\n\"Goodbye, Corinne.\"\n\nShe hung up the phone. The conversation lasted less than ten minutes. It wasn't even quarter after one yet.\n\nShe thought about calling Skip again, but she pictured him resting in his office. She wondered if that's all people did in small towns in places like St. Didier Parish\u2014try to think up people to call on the telephone. Maybe they had to do it to reassure themselves that they were part of a genuine community after all. In places like Linhart and the parish's other towns, it was very difficult to prove that any other way. The telephones were the only link between people. That and the weather, which was the favorite topic of conversation. Maybe that was why Skip was so popular with the viewers, why he was always in demand as a speaker at functions. The weather governed peoples' lives, interfered with them, to a greater extent than in many other parts of the country.\n\nNone of it made the least damn bit of difference to Corinne Strahan. Suddenly she hated everything. Everything. Without exception.\n**3**\n\n****\n\nWalter Boshardt checked the Linhart office and found that his deputy there was doing an acceptable job. The deputy in charge, Captain Brierrer, had little enough to do. Sergeant Marty Theriot and several other officers were there to help him, and they had to make work for themselves in order to justify being on the parish payroll. Theriot particularly enjoyed setting up speed traps, a practice that Boshardt disapproved of but did not forbid. Today the sheriff just ducked into the office, told one of the officers to attend to a few minor tasks, and ducked out again. The drive up to Linhart had been enjoyable. Now Boshardt could drive back to Arbier; he would enjoy that, too. Law and order were thriving in his parish, and everywhere he looked there were peace and open bottles of Dr. Pepper. Except here, by his office. He frowned at the machine, dug in his pocket for some change, and bought a bottle to drink on the way back. Then he walked across the shell-paved parking lot to his car.\n\nWalter Boshardt was comfortably in his middle years. He had lived through many hurricane warnings, and he had lived through four of the actual monsters themselves. It never felt exciting until afterward.\n\nThe sheriff had several special fears, things that had shadowed him since childhood. He had an almost paralyzing fear of dangling power lines, which could fry him as he walked the streets of Arbier or rode in his patrol car. He had nightmares during Hurricane Watches; he saw the black living wires snaking toward him from snapped poles. He could feel the prickling touch of the wires, just before he awoke. He was not afraid of flooding or the invisible dangers of disease that accompanied the rising water. He did wait anxiously for a tile from a nearby roof to come smashing through the taped and boarded windows of his office, driven by winds to a speed of nearly two hundred miles an hour. He had unpleasantly clear notions of what debris traveling at that speed could do to an unprotected human head.\n\nThe people of Arbier, for the most part, did not share the sheriff's fears. They had their own. Most of these worries were over matters of finance: would the hurricane ruin the new stock of merchandise; would the flood waters destroy a house full of furniture; would the winds rip the half-built home apart, or heave a family's house trailer across the lot, or flatten this season's half-grown money crop. . . . Their own personal safety came in a poor second. It was often risked in defense of a locked box of securities or a new automobile.\n\nLittle thought was given to the future, other than the necessities of next year's farming, or the stocking of merchandise for a holiday sale. One natural disaster might make people think temporarily of securing themselves against whatever blows the future might hold; but these thoughts were quickly forgotten, and the next natural disaster would catch them just as ill-prepared, just as surprised, just as frantic. The sale of plywood would be phenomenal for a couple of days, because in between storms no one had thought to install any more permanent safeguards. Flashlight batteries would disappear from stores, because families had neglected to set anything aside. Stores would be stripped of canned goods. Sheriff Boshardt and his deputies had to step in and stifle that fear before it grew beyond containable limits.\n\nBoshardt disliked that, more than any other part of an emergency. A frightened crowd could do more damage to a town, and to itself, than any number of loose house trailers or untethered boats crashing in the hurricane winds.\n\nDarlaine Boshardt, the sheriff's wife, had watched her husband in action for nearly two decades. She shared none of the pride which the people of St. Didier Parish felt toward him. She did not respect his capable manner. She thought that everything he did, he did for his own personal glory. She had little to say to him, and she avoided him at home. This situation made him unhappy, but over the years he had become accustomed to it. He had to; it wouldn't change.\n\nHer real name wasn't Darlaine. She had been born Dorothy Sauk Micheton. She had decided that Dorothy was a plain name, and had required everyone of her acquaintance to call her Darlaine, shortly after her marriage to Boshardt. He still found it difficult to call her by that name; he avoided it as much as possible, rarely calling her by any name other than \"Honey.\"\n\nDarlaine Boshardt had her own medicine for her quirky tantrums and grudges. She usually found a young man she had never met before, and she took him out to the Sea-Ray Motel, in full view of the entire town of Arbier. She always registered under her married name. Everyone in the town, almost everyone in the parish, knew what she did. Instead of laughing at the sheriff, though, the people of St. Didier Parish just thought that his wife was still a little unbaked in the middle.\n\nThe sheriff was well-liked throughout the parish. He was not a descendant of an old Acadian family, that was true. But his parents had lived in Arbier for many years, and Boshardt's mother had grown up in St. Didier Parish. His wife's flagrant adulteries were as well-known to him as to the whole town; both the sheriff and the town chose to ignore her, for different reasons. Walter Boshardt had achieved a kind of uneasy truce with his wife. The town of Arbier passed her behavior as the actions of a foreigner. Walter Boshardt had been too popular in Arbier for Darlaine's antics to undermine that acceptance. In private, perhaps, the residents of the town clucked their tongues and talked about her. But none of that tainted their feelings toward the sheriff.\n\nSkip Strahan, a recent selection of Darlaine Boshardt's, tended to agree with the parish's estimate of her. He was sorry that he had gone along with her. It seemed that he had gotten himself into a situation that might be difficult to work out of. She was more interested in him than she had been with previous men, possibly because he was a television personality. Darlaine understood all of this; she knew of Strahan's reluctance to continue their ugly affair. It was his reluctance that excited her. She had no intention of letting him get away now. She would do anything to watch him wriggle uncomfortably; calling him up and behaving foolishly was nothing compared to what she was fully ready to do.\n\nWhatever she had to do, it was probably something she had already done some time in the past. She had done quite a lot in the past.\n**4**\n\n****\n\n_It was in the town of Kepton, Ohio, twenty years before, near the end of President Eisenhower's second term._\n\n_Paul Pierson went home from school early, His third grade teacher had told her class that there was a bad weather warning, and they were all dismissed to go straight home. Paul and his classmates celebrated, because it was something different, something that broke up the daily pattern. The teacher cut their cheers short, making sure the children understood that they were to go directly home._\n\n_Outside, the air was chilly and damp. It was late March, and little patches of dirty, unmelted snow still remained in protected areas among bushes and in shaded recesses. Paul walked home feeling strange and dreamlike. The wind was oddly quiet. Nothing rustled the stark, black branches of the trees around the schoolyard. The sky was dark. It was obvious to Paul that a storm was approaching. But Paul knew that the school wouldn't send them home almost half an hour early just because of a storm. This sky was different than any he'd ever seen; there was a greenish cast to the heavy clouds. Paul didn't like the way the light made familiar things look. The peculiar color and the stillness exaggerated the frightening sensation. When he got to the end of his street, Paul ran the rest of the way home._\n\n_Another ominous thing was that his father's car was parked in the driveway. Never before had Paul's father come home from work before school let out. All of these unusual things made Paul wonder what was happening. They made him afraid._\n\n_\"I'm home, Ma,\" he called as he went in the side door of the house._\n\n_\"Is that you, Paul?\" called his mother from the living room._\n\n_\"Of course it is, Connie,\" said his father. \"Who did you think it was?\"_\n\n_\"I'm just wondering why the school sent him home, is all,\" said Paul's mother._\n\n_\"Because there's a tornado, that's why,\" said his father._\n\n_\"A tornado?\" said Paul, startled._\n\n_\"Don't scare him, Michael.\"_\n\n_\"I'm not,\" said Paul's father. \"Are you scared, Paul?\"_\n\n_\"No,\" said Paul. He was lying._\n\n_\"We ought to go down to the basement,\" said his mother nervously._\n\n_\"I don't want to go down there,\" said Paul. \"There's bugs.\"_\n\n_\"There's a Tornado Watch for this part of Ohio,\" said his mother. \"The radio said we should plan to seek shelter in case a tornado is seen. That means going down to the basement.\"_\n\n_\"We don't have to go down yet,\" said Paul's father. \"Just if we hear that a tornado's been spotted. We'll have plenty of time.\"_\n\n_\"What if we don't hear?\" cried his wife. \"What then? Are you going to take a chance like that?\"_\n\n_\"The monster lives down there,\" said Paul. No one paid any attention to his objection._\n\n_Paul's father began to get annoyed. \"All right,\" he said. \"Bring the radio and the flashlight.\"_\n\n_\"What about food and water and blankets?\" asked Paul's mother._\n\n_Her husband stared for several seconds. \"How long do you think we'll be down there?\" he shouted at last._\n\n_\"I don't want to go down there,\" said Paul._\n\n_They went down anyway. When they were all in the damp cellar, Paul's mother began worrying again. \"Which corner?\" she asked._\n\n_\"What?\" asked Paul's father._\n\n_\"The tornado corner,\" she said, \"which corner is it?\"_\n\n_\"How should I know?\" Paul's father was so irritated that the boy thought his dad might prefer sitting upstairs in the path of a tornado to any more of this nonsense._\n\n_\"They said on the radio which corner was the safe one for waiting for a tornado,\" said Paul's mother. \"I think it was the north corner. Then the tornado would blow everything over you, instead of on top of you.\"_\n\n_\"I don't think it was the north corner,\" said Paul. \"We studied it in school last semester. I don't think it was the north corner.\"_\n\n_\"This house doesn't have a north comer,\" said Paul's father in a quiet, restrained voice. It was a voice of fury only barely held back. \"I found that out last spring when they put in the sewers.\"_\n\n_\"Maybe it was the west corner, then,\" said Paul's mother._\n\n_\"It doesn't have a west corner, either,\" said her husband. \"That's logical.\"_\n\n_\"What?\" she asked._\n\n_\"This house has four corners,\" said Paul's father, sighing. \"It has a north-northeast corner, a east-southeast corner, a south-southwest corner, and a west-northwest corner. Take your pick.\"_\n\n_\"I don't know,\" said his wife, her face gray and frowning in the dim light. \"If we go into the wrong corner, we'll be killed.\"_\n\n_\"In that case,\" said Paul's father, \"I'd rather be killed upstairs on the couch.\"_\n\n_\"Shut up, Michael.\"_\n\n\"You _shut up.\" It was beginning to sound like one of their usual arguments. Paul's mother started to cry. \"Stop crying,\" said the boy's father._\n\n_Fortunately for the Pierson family, on that occasion there were no tornadoes sighted in the entire state of Ohio. But the argument did not die; over the years, whenever there was a Tornado Watch, the family would go down into the basement and wait. The correct comer remained a mystery; in between tornadoes the matter was forgotten, and no one ever learned which corner was the safe one. Again, luckily, Paul Pierson grew to maturity without ever seeing the evil funnel cloud of a tornado. But tornadoes played an important part in his growing up, and they figured in his nightmares and anxiety dreams for the rest of his life. So did the dark, damp cellar._\n**TWO**\n\n****\n\n**The Tropical Storm**\n**5**\n\nMonday Morning, an August Monday morning, began in Arbier. Cyrus, the Persian cat, flung himself futilely against the screen door of the balcony. Paul Pierson woke up, noticed that Maddie Gargotier was already missing from her side of the bed, and made a gesture that he almost believed was amused. He went into the living room, tossed the cat into the hallway then went back into the bedroom to pick up a set of clothes for the day's work. This task was not too difficult; in the summer heat and humidity of the southern Louisiana climate, Pierson wore a light short-sleeve shirt, a pair of dungarees, and a pair of sneakers every day. The shirt was changed daily, and the blue jeans once a week. On this particular day, Pierson grabbed a blue tee shirt from a laundry bag on the floor, found the blue jeans on the rug by the bed, and searched until he found where he had kicked the sneakers the night before. He didn't worry about socks. In the town of Arbier, the wearing of sneakers made Pierson eccentric. There were many sayings about the behavior of people who didn't conform to the Acadian fashion, and they all had to do with _le Bon Dieu:_ \"If God had wanted man to wear shoes. . . .\" Pierson had an appropriate reply which he never had the courage to use. It went something like, \"If God had wanted people to dance, he would have built the world with polished hardwood floors.\" Knowing the Acadian love of festivals and weekly fais-do-does\u2014community dances\u2014Pierson risked social banishment for such a remark. Such a thing was not unheard of. Why, to call a native a Cajun to his face, unless the situation was particularly friendly, was enough to fuel a life-long feud. It was never the person an outsider spoke to who was the Cajun. It was always someone else who was the down-the-bayou Cajun.\n\nPierson washed, dressed, made himself a breakfast of Dr. Pepper and stale doughnuts, put some dry food in a dish for Cy along with a dish of water, and left for work. At the time, Pierson was employed by a Monsieur de Crout. The job was as unattractive as it sounded. Pierson was a fish sorter. After turning his back on his unsatisfying home life, college life, married life, he had looked for more rewarding experiences among the \"real Americans,\" the ones he had read about but never seen. If the general mixtry\u2014as the local phrase had it\u2014represented real America, forty-nine other states and a large portion of Louisiana had never been so informed. It satisfied Pierson though. He had never seen or even suspected that such a foreign enclave of people existed right in the heart of rural America.\n\nIt wasn't his duty to bring that information to me ignorant millions. It was his job to put the red snappers in one ice-filled box, the flounder in another, the pompano in a third, and so on, all day. It was a very unglamorous way to earn a living; Maddie had made that point several times in their stormy relationship. Pierson had to agree; he came home every day in the most pungent of states. Pierson had no intention of making the fish-sorting profession his life-long vocation, but according to theory it paid the rent and kept Cy in fish-flavored foodlike substances.\n\nThat was the theory. The reason that Pierson's stride was a bit more determined than usual on this Monday morning was that the actual practice in the matter did not come close to its platonic ideal. Pierson had worked the entire summer for Monsieur de Crout and had not received any pay for three weeks. Old Monsieur de Crout was a good man, Pierson thought. He wouldn't have called his employer lovable, but that isn't a quality essential in someone whose main function in your life is the paying out of money. Three weeks is a long time to work for someone on the basis of likableness. De Crout was an old-time Arbierian; he had lived in the town for decades, before the influx of newer, non-Gallic families shortly after the Second World War. For that reason, de Crout enjoyed certain privileges which Pierson would never know. De Crout's shop had no sign outside. Its display windows were forever empty\u2014why not? Everyone in Arbier knew that old Monsieur de Crout ran a fish shop. Members of the old Acadian families treated each other even more specially than the town as a whole treated foreigners. And as far as Pierson could see, it was too late for anyone in the world to enter into that confraternity.\n\nStill, money was money. The rent was due. There was a stack of bills on Pierson's nightstand that was already falling onto the floor and getting kicked inadvertently beneath the bed. The good-time cash that Pierson liked to keep in his pockets was slowly dwindling also. Pierson had decided that he couldn't take any more of Monsieur de Crout's muttered excuses or obscure gestures. The situation was embarrassingly simple: either Pierson got his back pay, or he quit his job. That idea wasn't so bad, as the young man thought about it. After a few days, he might even stop smelling like a fish store himself. Then he could learn what his cat's real feelings for him were.\n\nEvery day, old man de Crout would buy fresh fish caught in the nearby waters of the Gulf, or in the brackish marshes. During crawfish season, the little, ugly mudbugs were de Crout's main source of income. But late summer meant the end of crawfish season, and Pierson had to work with the fish he had learned to identify. These, plus case after case of fresh oysters, were shipped by truck to restaurants within the parish and as far away as New Orleans and Baton Rouge. It wasn't a demanding job. Pierson had as one of his main objections that the job wasn't demanding enough. Line after line of argument went through his mind as he walked along Ridge Street toward Monsieur de Crout's fish store.\n\nThe town of Arbier was waking up and beginning its day at the same time. Pierson recognized several people on his way to work, and he greeted them all. The greetings weren't always returned, because to some of the residents he wasn't much more than the advance party of an invasion, determined to crush the Acadian way of life and replace it with the spiritless and detestable lifestyle they witnessed every day on television commercials. They could swear they would never let that happen. So it was that from these culturally proud people Pierson received never as much as a nod, a recognition that he existed. That would be an invitation to the very kind of change they feared the most. To these stern Acadians, \"change\" did not always mean \"progress.\" Indeed, \"progress\" was rarely a positive word. A common attitude was \"it was good enough for my _grand-p\u00e8re_ , it's good enough for me, for true.\" Anywhere else, this kind of reasoning might be foolish, but here in the odd cultural climate of southwestern Louisiana, it was perfectly natural. So every day Pierson greeted people he knew would ignore him, because he wanted to belong to the community, and because he had no one else to greet.\n\nIt was still early, because Pierson's job required that he have the fish sorted and boxed, ready for shipment, when the truckers were ready. They had deadlines to meet at their destinations as well. The restaurants couldn't wait, and customers couldn't be put off with a shrug and a muttered explanation that the fish packer in St. Didier Parish had taken his time getting to work. If he had stopped to consider it, Pierson would have realized that he was an important cog in one of the chief industries of the state. But on this Monday morning, he didn't feel exceptionally important. He felt angry, angrier by the step, angrier the closer he got to old man de Crout's shop. He was building his anger within him, because otherwise he would be helpless against the odd ways of the old man.\n\nThere was no light inside the shop. There never was. De Crout didn't think that it was necessary. He didn't need a bank of fluorescent lights to help him tell a pound of crawfish from a flounder, and neither did his patrons. As for quality, that was the job of Pierson and his co-worker, Ti' Jacques Barditon. They sorted and threw away the poor quality fish. The restaurant clients to the north and the local housewives who did their shopping in Monsieur de Crout's shop rarely had cause for complaint.\n\nPierson opened the front door. There was complete silence. It was a disquieting feeling. Pierson always felt that there should be a bell tinkling over the door to announce the entry of a patron, or the faint sounds of activity from the back room. But whenever Pierson arrived at the shop\u2014and no matter how early he arrived, old man de Crout was already there\u2014he was always met by the same unpleasant stillness.\n\nThe noise of the door shutting brought old Monsieur de Crout out from the back of the shop. \"You ready to do a day's work today, _hein_?\" he asked. \"We got us a load of shrimp to pack, the Lord himself would get tired just thinking about how hard it was making those little fish.\"\n\nPierson didn't even think about telling his employer that the shrimp weren't fish; he had given that up long ago. Besides, he was still in the grip of the anger he had carefully built up on his walk along Ridge Street. \"Monsieur de Crout,\" he said, and his voice was much less forceful than he wanted it to be, \"I have a couple of things I have to say to you.\"\n\n\"And after the shrimp, you and Ti' Jacques can start on those snappers. Save two or three good ones, too.\"\n\n\"What I meant was, I haven't been paid in over three weeks, you know. If you can't pay me today, I'm going to have to quit this job and find me another. It's not that I don't like working here. It's just that my landlord won't understand my excuse if I can't give him his rent.\"\n\nDe Crout stared silently for a moment. _\"Le sacr\u00e9 am\u00e9ricain,\"_ he muttered finally. Pierson had lived in Arbier long enough to know how derisive the expression was. \"I do my best, me,\" said de Crout. He took a sip of coffee from the demitasse he held. \"If that's not good enough, _eh bien_ , I don't know what to say.\" He turned and went into the back room of the shop. He had left the decision up to Pierson: either the young man could follow de Crout and get to work, or he could walk out the door, quitting his job, and making sure that almost the entire three thousand people of Arbier would hear about it by the end of the day.\n\nPierson glared at the empty doorway to the back room. He took a deep breath, hating the fishy smell of it. \"If God had wanted people to eat fish,\" he said quietly, \"he wouldn't have put them in the ocean and us on the land.\" He turned and started for the door, but before he touched the doorknob he heard de Crout's voice from the back room. \"There are some people, and some times, when the name of _le bon Dieu_ may be used, but this was not the time, and this was not the person.\" Pierson was amazed that de Crout had heard his remark, and even more frustrated because he knew that it would just aggravate the situation.\n\n\"All right,\" said Pierson, \"we'll see what the sheriff's office has to say about this.\" As he left the store, he met Ti' Jacques coming in.\n\n\"Allo, Paul,\" said the tall, skinny boy who had been his helper for the last few weeks.\n\n\"Morning, Jacques,\" said Pierson.\n\n\"I see you leaving by the door.\"\n\n\"Yeah,\" said Pierson.\n\n\"Well,\" said Ti' Jacques, \"you could be leaving, you, for some coffee or some rope or something. Or you could be leaving-leaving.\" The doubling of the word intensified it much more to the Acadian ear than any other more rigorously correct grammar might.\n\n\"I'm leaving for sure,\" said Pierson. \"I quit.\"\n\nJacques squinted his eyes a little. \"You fooling me with fun, boy?\" he asked.\n\n\"No,\" said Pierson. \"I just told old man de Crout to shove his job. He ain't no charity case, and I'm not no volunteer.\"\n\n\"Just let me warn you. These old Frenchmens, they a little different than me and you. He'll be mad in there, I tell you, mad like goddamn. And he won't let it sit either. By the end of the afternoon you'll be lucky to find a job as a stake in the ground. These old Frenchmens, I tell you, they been here a long time, they been here forever. I only been here seventeen years. So when it comes to putting up against one of them, me, I'd just rather not, _hein_?\"\n\n\"Right,\" said Pierson, \"but what does a person do in this town if he wants to get paid for the work he does?\"\n\nTi' Jacques's squint changed into a smile. \"He gets born here,\" he said. Then he pushed past Pierson, into De Crout's fish store.\n\nPierson was back on the sidewalk, in the bright sunlight. Ridge Street was busier now, as more people were on their way to work. He couldn't think of anything to do, so he headed back to his apartment building, hoping that Maddie might have returned. He supposed that she had made good on her promise to join the Navy, or at least gone up to Linhart with her friends to find out all the wonderful things the recruiter there could promise them. But, knowing Maddie as well as he did, Pierson thought that it was just as likely that she had forgotten the entire project already and had just gone off to catch a few catfish for supper. His threat concerning the sheriff could wait until later.\n\nAs he walked slowly back to his apartment, it was simple for Pierson to pick out the members of old Acadian families and people who had moved into the area in more recent times. The Acadians were the descendants of French settlers of Nova Scotia, some of the earliest European settlers in the New World. After a time, during which they followed their old lifestyle, brought over from Brittany, they were forced out of their homes by a political upheaval that saw Great Britain, not France, master of Canada. The Acadians' ancestors were conscripted, and sent back to England; they returned to the fishing life along the coast of Brittany; they settled along the Atlantic coast wherever the local settlers permitted them. But most of all, the Acadians traveled to south Louisiana, which at that time belonged to France. This dispersion is still spoken of as _le grand d'erangement_ , and it is the beginning of Cajun history. The Cajuns\u2014the descendants of the original Acadians\u2014were justifiably proud of their heritage and bitter about the harsh treatment they had received. They developed their own culture out of whatever material came to hand. If they found something as ugly and unappetizing as a crawfish, they found ways to turn the thing into one of the most versatile and delicious of local ingredients.\n\nIt is a matter of pride to the Cajun housewives and chefs that they will not attempt substitutions; nothing is made that cannot be completed with purely local items. The meat, the fowl, the seafood, the spices, the vegetables, everything is plucked from the earth where _le bon Dieu_ allowed them to settle.\n\nPerhaps this is one of the things that separates the Acadian spirit from the Creole. To most people, the two words mean the same thing. Nothing could be more incorrect. A Creole, technically, is a purebred descendant of the original French or Spanish settlers of the Louisiana territory. Of course, there are incredibly few of these people around the Gulf coast parishes today, although many an old _grand-p\u00e8re_ will try to tell you that he can trace his lineage back to the pirate Jean Lafitte himself. Creole today is reserved for the more metropolitan areas of the Louisiana area. One dines on Creole cooking in New Orleans or St. Martinville\u2014 _le petit Paris_ \u2014but everywhere the Cajun influence makes itself known.\n\nThere is a kind of insolence and pride in the genuine Cajun attitude that outsiders can't help but admire, as much as they puzzle over it. Why did that fellow insist that he was not a Cajun, but that everyone else for miles around was? Why is the word sometimes a cause of clannish spirit and other times a curse filthy enough to bring out guns and knives? Most visitors to the area\u2014and there are a few, a little-little, as a Cajun would say\u2014never catch the distinction as they go around looking for the typical Cajun town or the typical Cajun restaurant.\n\nThere is a perfect parallel in America, thought Pierson as he waited for a truck to pass on Ridge Street. He used to play basketball in Ohio. On those playgrounds, which often came close to being battlegrounds as well, he heard the black ballplayers call each other \"nigger\" with a kind of blithe humor. But Pierson knew goddamn well that he could _never_ use that word in that way, no matter how long he played in pick-up basketball games nor how well the blacks in the area got to know him. The same rule applied to \"Cajun.\" Maddie had made that clear early in their relationship. She wasn't no coonass Cajun, her. Maybe them stumpjumpers up to Delochitaches, them, they were Cajuns for goddamn. But Maddie Gargotier and her father, well, it didn't even bear talking about.\n\nThe building that Pierson lived in fit into the neighborhood about as well as he did. It had been built in the mid-sixties as an investment. The typical house in Arbier was short and squat, white, with a porch to entertain neighbors and friends. On one side of the house, outside the kitchen, was a shelf called a _tablette_ where the housewife could work without having to put up with the intolerable heat of the kitchen itself. Above the house proper, reached by a narrow flight of stairs outside the house, was the _gar\u00e7onni\u00e8re_ , a small attic where the young boys slept. The shutters over the windows were painted on the inside only, as were the doors. The Acadian logic said that when the shutters and doors were open, as they were most times, the painted sides showed to the world. When they needed to be closed, there was no one around to see the unpainted sides.\n\nPierson's landlord had made a bad mistake. He had built a brick, modern apartment house in the middle of a row of necessarily identical Acadian homes. Of the eight apartments in the building, only three were occupied\u2014one by Pierson, one by a woman downstairs, the third by the landlord's caretaker. Actually, Pierson himself would have preferred living in the attic of a legitimate Cajun household, but the treatment he got on the first two days after his arrival in Arbier persuaded him otherwise.\n\nHe stopped for a moment and opened his mailbox. It was empty. There was an elevator, but Pierson walked up to the third floor where his apartment was. Walking was his only exercise, and he was vaguely proud of his refusal to give in to such luxuries as two-story elevator rides.\n\nPierson opened the front door, to be met by Cyrus. \"Nothing, Cy,\" said Pierson. The young man sighed. \"Nothing,\" he said again. He walked into the living room. \"Maddie?\" he called. There was no answer. He looked into the bedroom and the bathroom; they, too, were empty. No Maddie, no mail, no job. What about friends? Pierson passed on that one, too. Health? That was a little better, except for his allergy to one of the billion-odd local growing things. He wasn't that young anymore; he couldn't just go down and get into a basketball game with the other kids. Besides, the other kids would all be with their _p\u00e8res,_ netting crabs, shrimping, even trapping m'sieu muskrat. There was no such thing as a summer vacation for Cajun children, not the way Pierson had known vacation in Ohio. He didn't even bother to wonder which way was better; he just sat down on the couch, let Cy jump into his lap, and began to feel sorry for himself.\n\n* * *\n\nChuck sat in a chair in his room at the Sea-Ray. He was bored. He was more bored than he could ever remember. He got up and went into the bathroom. He looked at the toilet, but he felt no urgent need. He looked on the wall to the left. A rectangular piece of cardboard hung there. The words \"Your _Personal_ Bathmat\" were printed on it. \"All for me,\" muttered Chuck. He looked at his reflection in the mirror over the sink. He wasn't feeling well. \"Damn it,\" he said. Then he went back to his chair.\n\nAfter watching a few minutes of television, Chuck turned off the set and picked up the telephone.\n\n\"Yes?\" said the desk man.\n\n\"Long distance,\" said Chuck. He put through a call to Tom in Miami Beach.\n\n\"What?\" said Tom, by way of greeting.\n\n\"It's me,\" said Chuck.\n\n\"Who?\"\n\n\"Chuck.\"\n\n\"All right,\" said Tom. \"What?\"\n\n\"I just wanted to be sure that everything was going okay.\"\n\nChuck heard Tom sigh. \"I really hate to say this, Chuck,\" said Tom, \"but I guess I have to. Do you remember the Boston job a couple of years ago?\"\n\nChuck frowned. \"Sure,\" he said.\n\n\"You'll recall how you almost blew that one for us.\"\n\n\"Yeah,\" said Chuck, \"thanks for reminding me.\"\n\n\"Any time,\" said Tom. \"And, naturally, you remember last year in Detroit.\"\n\nChuck didn't answer. There was an uncomfortable pause.\n\n\"Well,\" said Tom, \"I assume you do. See, Chuck, it's just that I feel that of everyone involved in this enterprise, this masterpiece of planning, you might be the best choice for the person most likely to foul up. No personal feelings involved. Please understand that. I'm just going on past experiences.\"\n\nChuck took a deep breath. \"Tom,\" he said, \"I've always done\u2014\"\n\nTom interrupted. \"I know, I know.\"\n\n\"You really hate me, don't you, Tom?\"\n\n\"I said there were no personal feelings involved. In a job this complex, there can't be.\"\n\n\"Sometimes I wonder if I really understand you,\" said Chuck. He realized that his hands were sweating and his throat was sore.\n\nTom laughed loudly. \"You don't,\" he said. \"I'll bet you don't.\"\n\n\"I don't have to, right?\" said Chuck.\n\n\"You don't have to,\" said Tom, \"right.\"\n\n\"Okay,\" said Chuck, \"I'll call again.\"\n\n\"I know,\" said Tom. \"By the way, I really do hate your guts a lot.\" He hung up.\n\nChuck hung up his receiver. He felt terrible. He stood up, then sat down again. All that he could think to do was go into the bathroom. He thought that he was spending too much time walking back and forth from one room to the other. He just sat and stared. His stomach hurt.\n**6**\n\n****\n\nThe Monday August morning in Linhart, Louisiana broke in almost the identical way in which it did further south, in Arbier. Corinne Strahan opened her eyes but did not move her head from the embroidered pillow. She listened for a moment to the scream of a bluejay and the more melodious call of a bird she couldn't identify. She took a deep breath and let it out slowly. There wasn't much to do, there wasn't any hurry to do it, and she wasn't in a mood to finish her few chores so that she could enjoy hour after hour of solitary leisure.\n\nIt was still early. Skip would be at the station, getting ready his Monday noon weather report. She picked up the telephone and called the station. The switchboard operator recognized her voice and connected her with her husband. \"Hello, Skip?\" she said when he answered.\n\n\"Uh huh,\" he said. He didn't sound pleased at being interrupted.\n\n\"Listen, I was just sitting here and I can't think of any reason why we couldn't have some kind of party for the people there at the station. I mean, we could all pack lunches and go down to Arbier when they bless the shrimp fleet or something.\"\n\n\"Did you just wake up?\" asked Strahan.\n\n\"Yes.\"\n\n\"Well, give yourself a few minutes, and then think about how ridiculous the whole idea is. We're not one big, tight-knit family up here. We don't have a station bowling team, and we don't have a station softball team, and the idea of everybody going down to Arbier to watch some guy in white robes throw holy water at a bunch of fishing boats\u2014\"\n\n\"All right, Skip,\" said Corinne, already exhausted by the defeat, \"it was just an idea. I just wanted to talk to you. I woke up, and you were gone, and I just wanted to talk to you.\"\n\n\"You wake up every morning, and I'm gone most of them.\"\n\n\"Yes, Skip,\" said Corinne, \"I know.\"\n\n\"Everything okay there?\"\n\n\"What's to go wrong?\"\n\n\"Then I'll see you when I get home.\"\n\n\"Sure, Skip. I'll watch you at noon and six.\"\n\n\"Great. Tell me how I do. Well, I have to go.\"\n\n\"I love you, Skip.\"\n\n\"I love you, Corinne. Goodbye.\" There was a click, and then Corinne Strahan was listening to dead air. She hung up her phone and put her head back on the pillow. She stared at the ceiling.\n\nHalf an hour later, showered and dressed, she was taking care of the tiny troubles that were her responsibility. She went shopping first. She got into her car and drove to the supermarket. That took ten minutes. She went into the store and picked out a shopping cart. That took another minute. The day was flying by.\n\nIn the supermarket she met a woman who called her by name. Corinne tried, but she couldn't recall the woman's name, or even where they knew each other from. All Corinne could do was act friendly and hope that the other woman would give her a clue.\n\n\"Corinne!\" said the other woman. She was dressed in blue corduroy jeans, brown loafers, and a faded sweatshirt that said TKE on it. It was clear that this woman didn't care at all about what the Creole and Cajun families thought of her. After all, she was beyond salvation, beyond acceptance, so why bother? Corinne envied that attitude for a moment. She struggled to identify the woman.\n\n\"Hello,\" said Corinne, tossing a box of pretzels into her shopping cart.\n\n\"Listen, I don't have time to stop and chat,\" said the woman. \"I really wish I did, but I have to pick up the kids in about five minutes. But I really do want to talk to you about that charity drive. Skip could be so helpful.\"\n\n\"Sure,\" said Corinne. She didn't remember a charity drive, either. \"Give me a call anytime. I'm home most of the day, anyway.\"\n\n\"Great,\" said the woman. \"I'll talk to you then.\" She pushed her own cart up the aisle. Corinne noticed that her blond hair looked about as real as a passport photograph. She shook her head and finished her shopping. It didn't take long. She was home again before half-past eleven. It took her ten minutes to put the groceries away. She carefully folded the bags and stored them under the sink. Then she went into the living room, turned on the television to Channel Five, and waited for the news.\n\nShe didn't listen very hard to the newscasters' reports. She knew vaguely that the President was somewhere, that some European country that she thought was a friend of America suddenly was angry with us, and that a once-wealthy Hollywood star had been found dead in a sleazy hotel in New York. She couldn't remember where the President was, which country had a grievance with us, or who had died. But her attention increased when Skip came on with the weather report.\n\n\"Let's run the satellite film,\" he said. Corinne felt a little thrill as she watched him. He was carefully made up, wearing a tie and the station's news team blazer. He looked very authoritative, which was probably the most important quality in a weatherman. The satellite film filled the television screen with fuzzy areas of gray and white. Corinne had a difficult time deciding where the Gulf coast was.\n\nSkip held a pointer and indicated a fuzzy white area. \"This is all that remains of Hurricane Dinah. She's almost completely filled in and is busily raining herself out over the mid-Atlantic. She shouldn't even be a threat to shipping routes. And Hurricane Elsie bumped into the same high pressure ridge and followed almost in Dinah's tracks. Elsie, too, has been downgraded. She's no longer a hurricane. She's officially a tropical storm, with gusty winds reaching a maximum of about fifty miles per hour. She's heading out to sea in a northeasterly direction, and as the surface temperature cools down, she ought to dissipate just the way Dinah has.\"\n\nThat was when Corinne's interest failed. She was proud of the way Skip did his job; she just didn't care anything about the job itself. She listened as he finished his report, watched a few commercials, then turned off the television. She waited a quarter of an hour to let Skip get the makeup off and change clothes. Then she called the station again.\n\n\"I'm sorry, Mrs. Strahan,\" said the switchboard operator, \"but Mr. Strahan's not here.\"\n\n\"He's not there?\" Corinne said, feeling a little stupid. \"He was just on, doing his weather report.\"\n\n\"I know,\" said the operator, \"but as soon as he finished, he hurried out of here. He told me to tell anyone who called that he had an important appointment.\"\n\n\"He didn't mention an appointment to me this morning,\" said Corinne.\n\n\"I'm sorry, ma'am. That's all I know.\"\n\n\"Yes, well, thanks a lot,\" said Corinne. She hung up, wondering where her husband had gone. She knew that he often gave lectures to schools or social groups, but those were almost always in the evenings. He should still be at the station, monitoring the Weather Service reports coming in from New Orleans.\n\nIt was twelve-thirty. Corinne Strahan couldn't think of a thing to do. She wandered around her house. In the bedroom she went to her husband's dresser. She knew he kept his medicines in the sock drawer, in the back. They were supposed to be secret, but it was almost impossible to keep secrets in marriage unless one partner was horribly naive. Corinne pushed the socks away and saw six large vials of pills. She recognized the names of two of them. One was a stimulant, and Corinne knew that she definitely didn't need excess chemical energy. She couldn't find uses for her own natural vitality. The other vial contained a mild tranquilizer. Corinne thought that it might be pleasant to spend a few hours in a nice, calm state, like the time she had the accident and they gave her that injection in the hospital. She didn't have anything else to do. She counted out six of the pills, put the vials back behind the socks, went into the bathroom, and swallowed the tranquilizers with a glass of water.\n\n* * *\n\nThere was a high bamboo hedge mixed with wide-leaved banana plants around the home of Walter and Darlaine Boshardt in Arbier. Nevertheless, when Darlaine got out of bed and got dressed, she could see the town she hated so much clearly enough. Unlike her husband, who had been in Arbier all his life and who loved the town, Darlaine wanted to leave with a passion that went past rationality. She dreamed of the most incredible excuses, but she could never find anything plausible enough. Her husband had lived his whole life not only in Arbier, but in this same damn house. It would take something very drastic to make him decide to move. Darlaine Boshardt knew that her happiness was certainly not drastic enough of a reason.\n\nShe recognized his way of thinking, and she knew that his feelings were every bit as strong as hers. She had mentioned on several occasions that she might feel better living in Linhart, where she wouldn't feel so stifled. His reply was that Linhart was exactly the same as Arbier, except that there were a few more churches, a few more \"lounges,\" and the majority of the black population in St. Didier Parish.\n\nThe sheriff's family had settled in Arbier long enough ago so that he was granted almost the same acceptance as any of the native Cajun families. Almost, but not quite. The difference didn't seem to bother Walter. The difference was much greater in Darlaine's case. That she was tolerated at all was only because of her marriage to the sheriff. Whenever one of the local people came up to her and spoke in that crazy dialect, calling her \"Madame Bozar,\" she wanted to strangle him. And because Darlaine Boshardt of Arbier had started out as Dorothy Micheton of New Orleans, she had the suspicion that most of the old-timers of Arbier spat at her shadow as she passed.\n\nIt was early yet, and she had the whole day to kill. That was the way she thought of her life: killing one day after another. Corinne Strahan had her husband's drugs stuffed behind his socks in the dresser. Darlaine Boshardt had something hidden also, in an unused chest of drawers. It was an old black book for listing telephone numbers. She thought for a few minutes, then went to find the notebook.\n\n* * *\n\nPaul Pierson decided that feeling sorry for himself was getting him nowhere. It wasn't solving any of his problems, and it wasn't getting him any closer to finding solutions to emergencies that needed immediate attention. So, using a line of reasoning that made him proud of himself, he stopped feeling sorry for himself. He also tossed Cy off his lap. The cat landed with his own peculiar gracefulness in the middle of the living room floor, looked around, blinked, then jumped up to the plastic dust cover on Pierson's stereo set. Cy was asleep again in a matter of seconds. Pierson wished that he could summon up that same kind of nonchalance. He tried to brush off the long gray hairs that Cy had left on his clothes, but that, as always, was impossible. Pierson made a few halfhearted attempts, then went to the telephone. First he called the Landry School of Secretarial Skills in Linhart, where Maddie sometimes attended various classes. No one there had seen her among the two dozen people in the small building. He called her father's bar, but old M'sieu Gargotier had not seen or heard from his daughter in a few days. Pierson shrugged. He had to let Maddie have her way. Actually, the situation was a little different. Maddie _took_ her way, and Pierson had to adjust to it. He guessed that she had gone up to Linhart after all, to see the Navy recruiter. He was kind of curious about what the man would tell her, particularly when she said that she was interested in the submarine service. He wished that he could be in the recruiter's office when the Navy man asked her if she could type.\n\nPierson went to the refrigerator and found a half-finished bottle of Dr. Pepper. It was a little flat, but that never bothered him. He swallowed what was in the bottle in two gulps. Then he went downstairs and out to Ridge Street, where he picked up a copy of the _St. Didier Dispatch_ , a weekly newspaper printed in Linhart. The local paper, the weekly _Arbier Observateur_ , was printed mostly in French and had little useful information for Pierson, in any event. The feeling in Arbier was that if anything was worth knowing, the news would travel faster by the network of women working at their _tablettes._\n\nPierson looked through the Linhart weekly and found three possible job opportunities. Each was followed by a telephone number but no name. Pierson called the first number. A man's voice answered. \"Allo,\" it said in thick Cajun accent.\n\n\"Hello,\" said Pierson. \"I'm calling about your advertisement in the _Dispatch._ \"\n\n\"Oh, _dommage_ ,\" said the man. \"She is filled, that one, for true.\"\n\n\"Oh,\" said Pierson. \"I'm sorry to have bothered you.\"\n\n_\"C'est rien,_ \" said the man. There was a click as he hung up.\n\nPierson tried the second job offer, and got the same response. As soon as his voice made it clear that he was not a genuine native, and when he had to declare that he wasn't related to some family in Delochitaches, that job, too, was filled. The third job interview took even less time.\n\n\"Damn it,\" said Pierson, throwing the newspaper down. He thought that it wouldn't be any problem learning French; after all, school children did it all over the United States at an early age. But he knew he would never learn Cajun French, in speech, manner, or style. That was a severe handicap in the immediate neighborhood.\n\nPierson, in his restlessness, wandered down Ridge Street toward the center of town. He noticed that a few shops were already putting up two-by-four frames on which sheets of plywood would later be nailed in the event of a Hurricane Watch to protect plate glass windows. An old wives' rhyme that Maddie had chanted recurred in Pierson's memory: \"June, too soon. July, stand by. August, get ready you must. September, remember. October, it's over.\" Already, early in August, some of the weather-wise shopkeepers were getting ready. He had noticed advertisements in the Linhart newspaper for plywood sheets. The ads covered almost two full pages. August and September were big months for the local lumberyards.\n\nThe Gulf coast residents, from Florida, along the Alabama, Mississippi, and Louisiana coasts, to Texas, could be divided into three classes. First, there were the people who scoffed at the damage a hurricane might do. \"It's a storm,\" said these people. \"The winds are higher, the water comes up, maybe you get a few inches on your new carpet, but, what the hell, it's just a storm.\" These people, it could be determined, had never lived through the full fury of a hurricane. Then there were the store-strippers, the ones who knew that all facilities and utilities might be out for days on end. They descended on every grocery store in the neighborhood and bought canned goods to last them four or five times as long as necessary. When these easily panicked people finished, the storekeepers had full cash registers, empty shelves, and weary, dazed expressions. The third category knew well what the turbulent winds of a hurricane could do, especially close to the ocean's edge, as Arbier was. They knew about the hurricane surge, the tidal wave that arrived about the same time as the calm eye of the storm. They knew that water that swirled around one's feet could rise to chest level in a matter of seconds. As soon as a Hurricane Warning was issued, meaning that there was the possibility of a hurricane within the next twenty-four hours, these people packed everything they could into their cars and drove inland. Hotels and motels miles from the coast were booked solid. Families went to stay with relatives and friends until the storm passed. Then they returned to their homes, to see what destruction the hurricane had caused.\n\nPierson viewed the nailing of the wooden frames as farsighted and wise, but he knew that he didn't have much to worry about in his third-story apartment. Maddie had often laughed and called him a fool, almost hoping a hurricane would come and knock some of his Yankee wise-assness out of him. But she took back those wishes, remembering things she had seen on the sand and mud-filled streets of Arbier in years past.\n\n* * *\n\nCorinne felt very nice, and then she felt a little drowsy, so she stretched out on the couch. She dozed for a little while, all the time feeling that there was nothing else in the whole world beside the cushion she rested on, her head, and a little portion of her body down to her shoulders. Her arms and legs were something she remembered pleasantly. After a while, the numbness wore off, and she discovered that she was in the same house in the same Linhart, Louisiana, with the same nothing around her. She only had one useful sensory channel: the telephone. She dialed the television station.\n\n\"Hello, Mrs. Strahan,\" said the operator at the station.\n\n\"Hi,\" she said. \"Could you give me Skip?\"\n\n\"Gee, I'm sorry, Mrs. Strahan,\" said the operator in a voice that conveyed absolutely no regret. \"You know when you called before that he was away on some appointment.\"\n\n\"Didn't he leave a message for me?\"\n\n\"I don't know. Let me check.\" There was a brief silence while the switchboard operator made a quick search of her tiny realm. \"I'm really sorry, Mrs. Strahan, but there ain't no message, neither.\"\n\n\"All right, all right,\" said Corinne. \"Have him call me when he gets back.\"\n\n\"That's just fine,\" said the operator. \"I have your other message waiting here for him, too. He'll give you a call when he gets in. He'll have to be back pretty soon for the evening show.\"\n\n\"Uh huh,\" said Corinne Strahan and slammed the receiver down into its cradle, cutting off the outside world more than ever before. She took out a cigarette, lit it, took two puffs, put the cigarette in an ashtray from some Florida motel, and forgot about it. It burned down without attracting any attention.\n\nCorinne Strahan stalked her domain, her dominion, her pitiable meaningless territory. She lit another cigarette. She sat down in an armchair in the living room. The chair was a color Skip always called \"off-green.\" Corinne had picked out the chair, and every time Skip made his little remark she cringed. He would go on about the chair. \"Why doesn't it commit itself?\" he would ask, as she became angrier. \"It could either be part of the room or definitely decide against it. It's the chair's goddamn decision.\" Then Skip would look very tired, and he would collapse on another chair, a sort of beige armchair, but Corinne refused to grant him any pity on the green chair's account. His tiredness might be legitimate, but he had spoiled the whole effect with that constant round of dull humor.\n\nToday, though, Skip was somewhere else. Somewhere. Corinne would have traded her neighbor's children\u2014she had none of her own\u2014to be somewhere else. Baltimore, for one, where she grew up. Tahiti was nice, she heard.\n\nThe phone rang. She listened to it for a while, still a little groggy from the pills she had taken. She was thinking about Tahiti. She realized that she didn't have the slightest idea where Tahiti was, or what it looked like, or what the people there looked like. But she knew enough to bet her life that it was a much more interesting place than Linhart. The telephone stopped ringing. Corinne was glad of that. The ringing noise just didn't belong, not in the peace she had built for herself in the last hour or so.\n\nThe peace was fading fast. That was why the prescriptions had to be refilled. Damn it, she thought, why isn't there something you can take to. . . . She caught herself before she finished the childish wish. All right. Life wasn't what she dreamed it would be. But whose was? She watched the smoke curl from the second cigarette, breathing its last in a Howard Johnson's Motor Lodge ashtray. She sneered at it. She wondered if their whole life was going to be put together with artifacts from a world that didn't seem to care very much about them. Every year they received a letter from some mission in Arizona, pleading for money for the starving children. There was a picture of a starving child, his eyes big and round, his stomach distended, his arms skinny and frail. Every year Corinne sent the mission five dollars, but every year they got the same appeal, with the same picture. The poor Indian kid didn't seem to get any better on their five dollars. Was the mission hinting that all that kept the boy from health and full development was an increase in their donation? Corinne lit another cigarette and put it in the Howard Johnson's ashtray, and another in the Florida ashtray, and a third in an ashtray from a hotel in New Orleans, and a fourth in an ashtray shaped like a bird with a yawning bill. She put the cigarette in the bird's beak. No worm today, bird, but with luck you could develop a harmful illness. The cigarettes around the room sent up vague curls of smoke. It was like burning incense, back when she had posters of the Jefferson Airplane on the wall and she listened to the Doors and sang along with Joan Baez, hoping that the world's problems would disappear if only someone listened.\n\nThe room was quiet. No one listened. The cigarettes burned.\n\nThere was a mild cloud of smoke drifting toward the ceiling. Corinne moved through it soundlessly. She put on the television set and changed the channel. They were playing _The Maltese Falcon_. That pleased Corinne. She always loved that movie. She could never figure if Humphrey Bogart, as Sam Spade, was a very good guy, or just a mildly good guy who missed his big chance. She had fantasies, sometimes, and she remembered that she once had a fantasy with Humphrey Bogart playing the male lead. She was playing the female lead, except that she was Lauren Bacall. They were at a beautiful nightclub. There were lots of potted palms, and tables with candles sunk deep in round glass jars, and a shiny dance floor. There was an orchestra, but it was playing Simon and Garfunkel songs. Corinne's husband appeared. Bogart, with a sneer and a bit of a lisp, said, \"You spend so much time learning the tricks of the trade, why don't you learn the trade?\" Skip had been speechless. Corinne had loved that fantasy; she still loved it even now, as she resurrected it from her memories.\n\nThe cigarettes had all burned down and left the air with an odor Corinne hated. She regretted lighting those cigarettes. Now she not only felt lonely, she felt stifled.\n\nHumphrey Bogart, as Sam Spade, was just meeting Sidney Greenstreet, as Kaspar Gutman. Gutman was pouring Spade a drink, obviously waiting for Spade to tell him to stop. \"You begin well, sir,\" said Greenstreet as Gutman. \"I distrust a man who says 'when.'\" Gutman continues pouring. \"He's got to be careful not to drink too much. It's because he's not to be trusted when he does.\" The fat man hands Bogart the drink. \"Well, sir,\" says Gutman, \"here's to plain speaking and clear understanding.\"\n\n\"Don't do it, Spade,\" said Corinne. \"Never trust a man who's twice as big as you are.\" There was a silence in the room, as the two actors drank their drinks and Corinne wondered if Skip were twice as big as she was. No, she thought, she was about a hundred and ten pounds, depending on whether or not the bathroom scale was in a good mood, and her husband was an inch over six feet tall and well-built. He was not too big to be untrustworthy.\n\nShe wondered where he was. She went to a closet and got out a hobby kit Skip had given her for their anniversary. He was thirty-five, she was five years younger, and already he felt that she was ready for the cheap, easy craft industry. This particular kit was a board of pine, stained to look almost like walnut, with small white dots all over it. The kit builder was supposed to hammer many identical brass nails into the places marked by the white dots. Then, following a set of printed instructions, the bored fool would wind white string from nail one to nail two to nail three until the kit was finished. It would take about as long as the rest of _The Maltese Falcon,_ a happy coincidence. And then Corinne would have a beautiful string image of a sailboat against a walnut-stained sea. There was even a frame for the finished likeness. She could have the whole thing finished before Skip came home after the six o'clock show. He would be pleased. He would be tired and he would say something about the green chair, but he would be glad that she had put the kit to use. She planned what she would say. \"Where will we put it, Skip?\"\n\nOn the television, Sam Spade was making another call on Kaspar Gutman. He was taking another drink. \"Oh, no,\" said Corinne. She knew what was going to happen. Skip would say, \"Anywhere you think it would look good.\" That's what would happen.\n\nHitting all the brass nails with the small hammer the kit supplied hurt her fingers after a while. Then she started stringing the white cord. She felt like a fool.\n\nThe telephone rang. She dropped the kit to the floor as she jumped up. \"Hello?\" she said.\n\n\"Hello,\" said a man's voice. It was a little squeaky and high-pitched. It isn't Sam Spade, thought Corinne. \"Is Skip homer?\"\n\n\"No,\" said Corinne. \"I was wondering where he was myself.\"\n\n\"I'm calling on behalf of the St. Didier Parish Baby Town. We were hoping that Mr. Strahan would donate some time to our annual fund-raising bazaar. When I talked with him last week, he seemed a little enthusiastic.\"\n\n\"He gets that way sometimes an hour or two after work,\" said Corinne. She immediately regretted her words.\n\n\"Eh?\" said the man.\n\n\"Nothing,\" said Corinne, \"I'm just very edgy right now, so just don't pay any attention to anything I say.\"\n\nThe man's voice changed. It slowed down and dropped in pitch. \"I've had experience handling situations like yours,\" he said. On the television screen, Humphrey Bogart was falling over in a drugged stupor. \"Maybe you would permit me . . . ?\" He left the question dangling, but it was such an open question that Corinne had no idea what to say.\n\nThe St. Didier Parish Baby Town, she thought. It sounds like a used-car dealer. A couple of dozen babies on this shell-paved lot, with strings of light bulbs above them, some of them burned out, and spinners in the breeze, and pennants of different colors. And the babies. They'd have OK stickers on them. This baby with 495 written across its forehead, and on its chest you could have it's a steal! A used-baby lot.\n\n\"Look, Mrs. Strahan, my name is Carl Steinbrenner. You can get in touch with me here, or your husband can. Or. . . .\" Again, he left the possibilities open.\n\nCorinne knew what the possibilities were. One, there was finishing the stupid string boat. Two, there was watching _The Maltese Falcon_ , even though she'd seen it at least ten times before. And there was the third possibility. \"My husband lets me handle all his social and business engagements,\" she said, which was not the truth at all.\n\n\"Ah, I see,\" said Steinbrenner. \"Then perhaps we could discuss this over dinner.\"\n\n\"Oh, I'm sorry,\" said Corinne, \"it's kind of a ritual when Skip gets home. I couldn't be away at dinnertime.\"\n\n\"A drink then,\" said Steinbrenner, \"just a drink or two, and we can have the whole thing settled.\"\n\n\"That's fine,\" said Corinne, her mouth and throat suddenly dry, her hands shaking, the feeling of contentment she had been encouraging disappearing as quickly as the smoke from the long-dead cigarettes. On the television, Sam Spade's face looked very bad. He was talking to his secretary on the telephone. Every expression showed pain. \"Yeah,\" said Spade, \"let's do something right for a change.\" Corinne thought for a few seconds.\n\n\"Are you still there, Mrs. Strahan?\" asked Steinbrenner.\n\n\"Corinne,\" she said slowly. \"Call me Corinne.\" She was still frightened, but she was oddly excited, too.\n\n\"Can I come by, or shall I meet you somewhere?\"\n\n\"Where are you now?\" she asked.\n\n\"In my office, here in Linhart,\" He said.\n\n\"Do you know Bar's Mike and Grill? You go out Hanson Highway south, make a right on Couletain Boulevard. It ends on a small parish road. Take it left, past the causeway over Bayou Chien Mort.\"\n\n\"No, I don't know the place at all,\"\n\n\"That's fine,\" said Corinne, \"because I've never been in there either.\"\n\nThere was silence from both Steinbrenner and Corinne. On the television, a man with bullet holes in him was dropping a black bird wrapped in old newspapers at Sam Spade's feet.\n\n\"It would be a terrific opportunity,\" said Steinbrenner.\n\n\"Yes,\" said Corinne, in what she hoped was a low, breathy voice, one like Mary Astor used on Humphrey Bogart sometimes in _The Maltese Falcon._\n\n\"No,\" said Steinbrenner. \"I meant\u2014\"\n\n\"Never mind,\" said Corinne. \"Keep on that goddamn asphalt road. Stay on it until you hit the causeway over Bayou Chien Mort. Bar's Mike and Grill will be another mile or two, on your left. That road turns into Ridge Street at Arbier. If you get that far, you know you've passed the place.\"\n\n\"Fine,\" said Steinbrenner, his voice even more shrill, giving away his expectations. \"I'll be there in, oh, an hour.\"\n\n\"I'll be at least a drink ahead of you then,\" said Corinne.\n\n\"Don't worry,\" said Steinbrenner, \"I'll catch up.\"\n\n\"Okay. We have a lot to talk about.\"\n\n\"Yes,\" said the man.\n\nOn Corinne's television screen, Spade had checked the black falcon at a San Francisco baggage terminal and mailed the claim check to his post office box. Very shrewd, thought Corinne.\n\n\"We have a lot to talk about,\" she said. She hoped that she sounded enigmatic. \"I'll see you then.\" Without waiting for any further word from Steinbrenner, Corinne hung up the phone. She paused to look in a decorative mirror in the dining room. She was not displeased.\n**7**\n\nAbout noon, Darlaine Boshardt pulled up to the office of the Sea-Ray Motel. She got out of the car, walked across a narrow strip of grass, crossed the sidewalk that led to the rooms, and walked up the sidewalk to the motel's office. She walked in a way that indicated that she was sorry there weren't more people to see her. As it was, there was just the day manager. She opened the door and approached her solitary audience. He was reading a book with a cover that was tastelessly pink, a book he was quick to hide when Darlaine came in.\n\n\"Oh,\" said the day manager, \"hello, Mrs. Boshardt. We're pretty filled up, I'm afraid. I'll have to give you Room 9. There are permanents in 8 and 13, and every other room is occupied right now. Maybe you'd rather wait. I mean, some of those rooms will be vacated not too long from now. If you know what I mean.\"\n\n\"Of course I know what you mean,\" said Darlaine sharply. She crossed her legs, pulled her skirt down in a habitual gesture, and stared.\n\n\"I, uh, I see you're by yourself,\" said the day manager. \"Do you want the key to Number 9?\"\n\nDarlaine gave the man a malicious look that easily conveyed the notion that he had asked the wrong question. \"Yeah,\" she said, standing up and crossing the distance between the chair and the desk in two long strides, \"give me the key and let me register. When Mr. Right gets here, he won't have to go through the trouble. Tell him I'm in\u2014what was it?\u2014Number 9, waiting eagerly, my skin almost tingling with the anticipation of his gentle caress. Hell. Give me the key and the book.\"\n\nThe day manager didn't dare say anything. He merely handed her the key to Room 9 and turned a registration book to face her.\n\n\"You know,\" she said, her voice still a little bitter, \"this place has a quaint old charm. I mean, you don't fill out small white cards when you check in. You still sign a book. There aren't many places that still operate that way.\"\n\n\"It depends on the parish ordinances,\" said the manager. He allowed himself a little smile. Darlaine's promiscuities were well-known around Arbier. The male partners' identities were protected by the manager. Some of those male partners were very influential in the immediate area, in the south, or, on occasion, in the whole country. It didn't make any difference though. All of those men signed themselves in as John Smith, sometimes from New York, sometimes from Los Angeles. It didn't make any difference at all, thought the manager. Darlaine Boshardt and this guy John Smith were having a regular and torrid romance. The manager looked at the register: _Darlaine Boshardt, 8 W. 3 Street, Arbier, Louisiana._ The manager made a notation of the room number and the time. Before he could say anything, Darlaine was walking out of the office, slamming the screen door. The manager hated it when people slammed the door.\n\nAbout ten minutes later a man came in and sat down in the same chair that Darlaine Boshardt had used. The manager looked up questioningly but said nothing. It wasn't his job to ask anything. The man could sit there all day if he wanted. Sometimes they spent a long time in the chair and then silently left. The manager assumed that this was Darlaine's current lover. Ha, thought the motel employee, he could suggest to Mrs. Boshardt that she should see a psychiatrist, but as soon as she got herself down on the doctor's couch, her problems would start all over again.\n\n\"Uh,\" said the man.\n\nThe day manager looked up. His expression was as blank as he could make it. He had worked very hard at that.\n\n\"Uh, I'm, uh, supposed to meet someone at this motel.\"\n\n\"Let's see,\" said the clerk, pretending to give the matter thought. \"Female?\"\n\n\"Yes,\" said the man. He was obviously almost ready to panic.\n\n\"Oh, about forty-five real years old, making a very good try at looking ten years younger. Hair very coarse, like straw. It used to be a nice auburn but now it's so mousy you could bait a trap with it. The wrinkles are making their presence felt. Yeah, felt and _seen._ A thin woman on a large frame, dedicated to every fad diet that comes along. She was pretty once, you can see it, but now she's. . . .\" The manager paused, unable to finish the sentence with just the right word. All the time he had been staring down at the Formica top barrier. Now, though, he looked up at the man. \"Now,\" said the manager, \"she's what she is.\" He showed the register to the man. The stranger looked at the book and saw Darlaine's name on it. He reacted with shock, with a visible jerk. Then he took the pen and wrote in the register. The desk clerk turned the book around again and made his notations. He saw that the man had signed himself in as John Smith, of New York, New York. At least Darlaine was faithful to the men she wasn't being faithful with. The desk man smiled a little. Good old John Smith.\n\n\"Can I have the key, please?\" asked John Smith.\n\n\"Key?\" said the manager. \"Oh, don't worry about it. She has it, and I'll give you one-to-ten odds she'll answer if you knock. I'll give you one-to-eight odds that she'd answer if any man knocked.\"\n\n\"You're a real son of a bitch,\" said John Smith. \"You know that?\"\n\n\"Listen, friend,\" said the manager, \"you're the one going down to Number 9. I'm just going to stay here where I belong and fantasize about all the grotesque and ugly things the two of you are going to do, that the maid is going to have to clean up in an hour.\" He gave John Smith a big grin. He knew that if John Smith were Humphrey Bogart or Robert Mitchum or even, for God's sake, Robert Redford, he would punch the desk clerk in the mouth. But none of those three would be skulking down to Room 9 for a quickie with the wife of the parish sheriff.\n\nJohn Smith just turned around and walked out. He slammed the screen door, harder than most people. The manager winced. The door slamming was just payment for what he had said to John Smith. A few seconds later he had gotten over the entire affair and was back with his pink-spined pornographic novel.\n\nJohn Smith was as mad as he had ever been before. He was so angry, he had passed the stage of trying to think up clever replies to the manager's insolent, smirky behavior. He thought that he ought to have said something, at least, or even hit the man. But John Smith had never hit a man in his entire life. He reminded himself that he had never met a woman at a motel before either. That thought made him feel better. It made the manager seem insignificant. After all, he was just being the kind of man John Smith expected all motel desk clerks to be. He couldn't know from previous experience. And John Smith was hoping that the woman, Darlaine, in Room 9, would be the kind of woman he expected. Of course she was, he reasoned. Otherwise she wouldn't be there. That made him feel better. The manager, who was already shrunken to a maggot with human tendencies, disappeared entirely from John Smith's thoughts. He went over one idea, again and again, as he walked down the narrow sidewalk to Room 9: Darlaine was the kind of woman he thought she was. She would probably be dressed in the kind of strangely erotic things that he had seen in pornographic magazines he had confiscated from his students when he had taught in Shreveport. She would be standing by the desk, in front of the mirror, wearing a black corset, black garter belt, and black stockings. Maybe she had high-heeled shoes. John Smith had never decided whether or not that was an erotic effect. With a flush of excitement, he realized that he would find out in a moment or two.\n\nHe knocked on the door of Room 9. Darlaine Boshardt's voice answered from the other side. \"Come on in,\" she said. \"I left it unlocked.\" John Smith hesitated and took a deep breath. There were two thoughts that he should have considered, but didn't. First, if the manager was as low as John Smith thought him to be, a human worm in fact, what did that make John Smith? Second, if Darlaine Boshardt was as cheap as John Smith thought her to be, what did that make John Smith? These two thoughts remained buried as he turned the knob and went into the room. It didn't occur to him that she had left the door unlocked and, as the manager had said, any man at all could have gotten there first.\n\nDarlaine Boshardt was dressed in a light print dress and brown panty hose. She had kicked off her shoes and was lying on the bed. She was smoking a cigarette. There was a large number of butts in the ashtray on the bedside table. \"I'm glad you got here. _The Maltese Falcon_ gets on in a little while, and I figured I'd end up spending the room rent watching that old thing.\"\n\n\"You already took care of the room rent?\" asked John Smith.\n\nDarlaine sighed wearily. \"Yeah, yeah, don't worry about it.\"\n\nJohn Smith sat down in a red imitation leather chair in a corner of the room. He was too intimidated to lie down beside her on the bed. \"Well,\" he said nervously, \"I guess we're here.\"\n\nDarlaine raised her eyes to the ceiling. \"I got sent a good one this time.\"\n\nJohn Smith got up from the chair and sat on the edge of the bed. \"I just sort of meant that we didn't have much time to get to know each other, that's all.\"\n\n\"What do you want to know?\" She wondered why they always asked that same series of questions: what do you do, are you lonely, are you glad you met me, have I helped to make you forget?\n\n\"Nothing, really,\" said John Smith. His fingers rested on the zipper down the back of Darlaine's dress.\n\n\"Well,\" she said, \"I'm terribly fascinated by you. What do you do?\"\n\nJohn Smith pulled down the zipper and answered the question at the same time. \"I'm a teacher. Social Studies. They hired me because I learned French during the war, I guess.\"\n\n\"You know French?\" she asked, pulling the dress up over her head. Now she wore nothing but the panty hose and a bra. \"What's a _pirogue_?\"\n\n\" _Pirogue_? I never heard the word before.\"\n\n\"Yeah,\" she said, pulling off the panty hose. John Smith was aroused and exasperated by this woman at the same time. \"Almost every kid you teach will have one, and I'm not going to tell you what the word means. What's the French word for sidewalk?\"\n\n_\"Le trottoir,\"_ said John Smith.\n\n\"Not around here, it isn't. It's _banquette._ You pick up a little if you live here as long as I have.\" She sighed loudly. \"But you don't pick up enough, and you won't pick it up fast enough.\"\n\nJohn Smith was struggling with Darlaine's bra. \"They warned me about this distrust of outsiders,\" he said, \"but they said I would do better, because I can speak French.\"\n\n\"You talk French French, not Cajun French. By the way, those are clasps, not snaps. Here, let me do it.\" She took off the bra quickly. Her breasts were large and sagging. Now she wore nothing but her bikini underpants. \"Oh,\" she said, as though the undressing had been completely without erotic effect, which in her case was true, \"you'll do better than most strangers. You give some people a _bonsoir_ or a _bonjour_ and you might get a little better service in a store. You give your class a _comment \u00e7a va, classe?_ when you enter the classroom, and you won't have dirty tricks played on you. But the first time a kid tries showing off his English and says something like, 'My father say I got to told you that you got to brought yourself around for some fine eating like you never ate before,' and you say 'Huh?' you can be sure you'll be marked. How long have you been in Arbier?\"\n\nJohn Smith was a little angered by Darlaine's attitude. \"A couple of days,\" he said.\n\n\"Well, then, don't worry. You're probably marked already. So, anyway, the tits are here, the underpants and all the rest of the good stuff is down there. I suppose they do it the same in Shreveport as they do it here.\"\n\n\"Yes,\" said John Smith. He took off his clothes and piled them on the chair, unlike Darlaine, who had thrown hers randomly to the floor.\n\n\"That's too bad,\" said Darlaine, \"I was hoping for something different.\"\n\n\"Look,\" said John Smith, \"if you want something different, I can give you different.\"\n\n\"No, sir,\" she said, \"I've been through all of that. No, you're going to put that thing in me, and we're going to move around, and you're going to come but I won't, and then we get dressed, I turn the key in to the office and go home in my car. You go home in your car.\"\n\n\"You make it sound so exciting,\" said John Smith.\n\n\"I never promised you excitement,\" she said, crushing the cigarette in the ashtray. \"I just gave you the idea that you were going to get laid. And you are. So start with the foreplay.\"\n\nJohn Smith was angry and a little hurt. He started sucking on one of her nipples while he fondled the other between his thumb and index finger. Both nipples grew hard, but there wasn't a single noise from Darlaine. He reached down and started to slide her underpants off. She raised up a little to make it easier for him, but she wouldn't help any further. When his hand moved through her pubic hair and started rubbing, faster and faster, just a little bit away from where he could give her some genuine pleasure, she grabbed his erection and started stroking it. He immediately groaned with pleasure and increased the tempo of his rubbing. Darlaine wished that John Smith would stop making such ecstatic noises. She imagined with distaste how he would sound when he actually entered her. He rubbed harder, in the wrong place. It wasn't actually bad, she thought; he was making her feel better than she had felt before he arrived. But that wasn't saying very much.\n\nApparently John Smith decided that he had accomplished the goals that foreplay was designed to achieve. He grabbed both of her shoulders and tried to shove his erection into her. He missed his mark and prodded against her. Oh, Lord, he thought.\n\nDarlaine made an impatient noise, grabbed him, and guided him into her. His moans and cries grew louder. She wanted to stuff the sheet into his mouth. She moved rhythmically but without enthusiasm. In a few minutes he came. He stayed on top of her because one time, years before, a woman had told him that she hated to have a man pull out immediately after he came. John Smith waited for what he thought was a reasonable length of time, then rolled over to the other side of the bed. He was sweaty and exhausted.\n\nDarlaine was not. \"If you ask me 'How was it?' I'll get up and get dressed, and then I'll leave and take your clothes with me.\" She lit a cigarette and smoked it. Then she got up and got dressed.\n\n\"I think I'll take a shower,\" said John Smith. He waited for a sarcastic remark from Darlaine. He didn't have to wait long.\n\nShe picked up her purse from the floor under the nightstand and walked to the door. She touched the doorknob and then turned around to face him. \"Well,\" she said, \"I guess sex isn't everything.\" Then she opened the door and left.\n**8**\n\nAt the time that Corinne Strahan was waiting anxiously for her husband to appear on her television set, at the same time that Darlaine Boshardt was making plans to meet John Smith at the Sea-Ray Motel, Paul Pierson was trying to work off some of his frustration. The first thing that he did only made the situation worse. When he was younger, in Ohio, he used to skip stones across Tinker's Creek. He could skip stones for an hour and then, somehow, he would feel better. On this Monday, however, Pierson went down to the end of Ridge Street. He stood on the edge of the narrow beach and felt the anger in him grow. He couldn't even skip stones in this lousy town. They didn't even have stones. All they had was small white shells. Pierson knew that he couldn't skip a small shell. He looked out at the pier, which seemed at any moment ready to topple into the waves. Waves, thought Pierson. Even if he had a nice supply of flat rocks, it wouldn't be as much fun trying to skip them against the waves. He kicked the sand, kicked some shells, turned around, and headed back toward town.\n\nThe weather report was on the television as Pierson entered the Crisis Caf\u00e9 on Ridge Street. He saw that Sheriff Boshardt was in the same booth that he usually took. Another booth was occupied, and one table. Pierson slid into a third booth. While he waited for Lauren, the seventeen-year-old waitress, to bring him a menu and a glass of water, he listened to what the Channel Five weatherman had to say. It seemed that there was another tropical storm, moving south of Cuba, heading for the Yucat\u00e1n Peninsula. The other two storms were dying quickly.\n\n\"Hello, Mr. Pierson,\" said Lauren. She stood beside the booth, an order pad in her hand, waiting patiently. There was a menu on the table top. Pierson wondered how it got there without his knowing it.\n\n\"Oh, hi,\" he said. Lauren just stood there, waiting. Pierson didn't want to make the situation worse by picking up the menu and looking at it. \"How about a small bowl of gumbo and the stuffed crab?\" he said.\n\n\"Fine,\" said Lauren. \"Anything to drink with that?\"\n\n\"A Coke, I guess.\"\n\n\"Okay,\" she said. \"I hear you quit your job over at Monsieur de Crout's. He can be awful stubborn when he wants, and he wants to be most of the time.\"\n\nPierson was amazed. \"How did you know that?\" He thought that he ought to talk to the sheriff, to demand his money. He would, too, later.\n\nLauren was surprised at Pierson's reaction. \"In a town like this, everybody knows everything. Sometimes that's good. Sometimes that's bad.\"\n\n\"For crying out loud,\" said Pierson bitterly, \"maybe that's why I haven't had any luck getting another job today.\"\n\n\"Maybe,\" said Lauren, with the typical deprecating shrug of a genuine Acadian resident of Arbier, \"but I don't think so.\" She turned and went into the kitchen.\n\nPierson stared after her. She was a lovely girl, and he wished that she didn't treat him like a foreigner. After all, he thought, he came into the Crisis Caf\u00e9 almost every day. He thought of himself as a regular. He watched as she came out of the kitchen, carrying a tray of food to the party at the table. She's really all right, thought Pierson. His light brown hair was cut medium length, half covering his ears. His mustache was carefully trimmed. He looked like he would fit into any social situation with ease, and he would, too, in New York, or Los Angeles, or Chicago. He got the first inklings that what were normal characteristics and advantages in the big cities were definite drawbacks in Arbier, Louisiana. It wasn't just that he was a foreigner. He was a damned _odd_ foreigner.\n\nLauren's long, dark hair hung down her back and moved only very little as she walked. Suddenly Pierson had an upsetting thought. He was twenty-nine years old. Lauren was seventeen. If he had been going out with her in his senior year in high school, she would have been just getting ready for kindergarten. On top of feeling in the wrong place, Pierson was beginning to feel old.\n\nA short time later, Lauren brought a bowl of gumbo. The steaming fragrance made Pierson forget his troubles for a few seconds. But the sight of Lauren helped him to remember. \"Say, Lauren,\" he said, \"twenty-nine isn't really old, is it?\"\n\n\"You can't say,\" she said, setting the stuffed crab on the table beside the hot gumbo. \"Old for what? It would be awful old for a muskrat, I think.\"\n\n\"You know what I mean,\" said Pierson.\n\n\"And you probably know my answer, Monsieur Pierson,\" she said, smiling. She turned and walked away again. Pierson felt even more frustrated. She had given him a perfect answer. The trouble was that he didn't know what her answer was. He devoted his attention to his lunch, which was a much more satisfying way of passing time.\n\nHe finished lunch and left Lauren a large tip. The sun was bright and he had to squint as he stepped out on the sidewalk bordering Ridge Street. He took a deep breath. He smelled the ocean, which made him think of fish, which made him think of old man de Crout. For a tiny, flashing moment, he had the idea of going back, apologizing, and asking for his job back. But the thought passed like a mild attack of heartburn, and Pierson was faced with what he feared the most: what to do with the rest of his life. Maybe coming south had been a mistake. Or, if the south had not been a mistake, maybe a town like Arbier was. He could go to New Orleans, but that would be admitting his weakness. New Orleans, for all its French influence and continental grace, was just another big city in some respects.\n\nHe walked up Ridge Street toward Monsieur Gargotier's bar. He took one of the side streets before he got to the bar. The houses were all the same, white frame buildings, one main story and an attic, with an outside flight of stairs running up to it. Around the houses there were miniature jungles of flowers, shrubs, and trees. Each house seemed to have its own pecan tree, its own magnolia, and its own palm tree. The magnolias had already blossomed, and the creamy white flowers had disappeared or were a dead, shriveled brown. There were spiky bushes of Spanish dagger, reaching up three feet or more. The crepe myrtles were still in bloom, and the rose bushes. There were dozens of other flowers and shrubs, all with their own special blossoms, but Pierson couldn't name any of them. Overhead flew gulls and terns. At night came the nighthawks, birds no larger than mockingbirds, which flew in erratic paths, chasing insects for food.\n\nPierson passed a row of houses on this street, stopped, and thought. He didn't have much money\u2014he wasn't the kind of person to put part of his paycheck away for emergencies. All right, he decided, if they didn't want him in their Acadian kingdom, it wouldn't hurt his pride too much to leave. He was sure he could find a job in New Orleans, where they didn't care if your mother's maiden name was French. If it came to the worst, he could always stay with his parents in Ohio for a while. That was a terrible thought. He didn't like the idea of going back at the age of twenty-nine, just because some hick town had driven him out with its clannishness. He was getting a bit old to go running back to his parents. If he gave the matter more thought, he could probably come up with a few better alternatives. He finally decided to do that, but not for a while yet. It was only a little after lunchtime. He had the rest of the day to do serious thinking. He made a right turn at the first cross street and in a few minutes he was back at his apartment. He took the elevator up to his floor. Maybe every Cajun in the parish would laugh at him for using the elevator, instead of the stairs, as God intended. At the moment, however, that didn't make any difference at all to Pierson. The Cajuns would most likely call the thing an elligator and a big-big foolishment. Pierson thought that the Cajuns could take their elligator and shove it.\n\nThe apartment was empty, except for Cyrus. The cat rubbed himself against Pierson's legs, but Pierson was not in the mood to pet the cat. \"Not now,\" he said, pushing Cy aside. The cat came back. Pierson picked him up and tossed him into the living room. For a few seconds, Cy stared at Pierson. Then the cat blinked a few times and jumped up to his place on the dust cover of Pierson's stereo. Pierson gave the apartment a quick check, but Maddie wasn't home yet. Pierson sat down on the couch in the living room, feeling more frustrated and angrier than he had ever before. He had to do something to relieve the tension, but the town of Arbier offered very little for him to choose from. For the moment, Pierson just sat and stared.\n\n* * *\n\nCorinne Strahan was going to meet Carl Steinbrenner at Bar's Mike and Grill. She arrived first. That made her more anxious. She parked her car, a small Japanese station wagon, and went inside. It was dark. There was a bar that ran the length of the room, a glowing juke box at the far end of the room, and two pinball machines against the wall opposite the bar. There were three small tables breaking up the space between the bar and the pinball machines. There were three men sitting at the bar. None of them looked like he could possibly be Steinbrenner. Corinne hesitated.\n\n\"What'll it be, Ma'mselle?\" asked the bartender.\n\nCorinne felt panic rising up in her. It didn't seem like she could go anywhere or do anything without making a fool of herself. She wished first that Steinbrenner hadn't called, and then she wished that she hadn't agreed to meet him. Finally, she wished that she had a couple of her husband's pills. That thought shocked her. She sat down at the bar, thinking that it might be more conspicuous to take a table. Then she decided that a single woman sitting alone at a bar looked pretty bad, too. The panic went away, replaced by the feeling of being evil that she had enjoyed so much earlier. She enjoyed it more now. \"I'm sorry,\" she said to the bartender, \"a whiskey sour, please. I'm waiting for someone.\"\n\n\"I know, I know,\" said the bartender, raising both hands to indicate that he didn't want to hear her story. He made the drink and set it in front of her, along with a cocktail napkin with very bad, very unfunny jokes on it. \"I just make you three or two drink, when your friend come. I just aks you, 'How you are?' but you know I don't care. I just make drinks, me, and I hear all kinds of stories. Now I got to brought myself down to where them three coonass Cajuns is, and listen to their stories like I never before hear them. And I got to go, me, because they three of them and only one of you.\" He gave her a shrug and went back to the three men sitting at the other end of the bar. Corinne was glad, actually. She didn't feel like pouring out her heartaches to a local bartender. That was sinking too low. It was bad enough that she would end up doing that with Steinbrenner.\n\nShe sipped her drink slowly, stretching it out, hoping that Steinbrenner would arrive before she finished it. He didn't. She sat with the empty glass in front of her for a while. Then she read the cocktail napkin. She couldn't believe that someone had actually been paid to write the jokes on the napkin and someone else got money for the cartoon drawings. She wondered how much they got. She wondered what the competition in the cocktail napkin business was like.\n\nAfter about ten minutes, the woman she had met in the supermarket came in, wearing the same loafers and corduroy jeans, but she had changed from the sweatshirt to a light blouse. Her coarse, dyed blond hair was puffed up in a style that Corinne thought had been dead since the Johnson administration. \"Hi,\" said the woman. Corinne still couldn't remember who she was.\n\n\"Hello,\" said Corinne. Then she felt obliged to say, \"I'm waiting for someone.\"\n\n\"So am I,\" said the woman. Corinne noticed that the bartender had already mixed the woman's drink. She must be one of the regulars. The woman took the drink and sat at one of the tables.\n\nThe bartender came over to Corinne. \"Listen,\" he said, \"it don't make some different to me if you drink another drink or you just sat there.\"\n\nCorinne smiled. \"Thanks,\" she said, \"I think I'll wait until my friend comes.\" Again, the bartender shrugged. He muttered something as he turned to walk back to the three men.\n\n* * *\n\nSt. Didier Parish bordered the Gulf of Mexico, which made shrimping and fishing an important industry. The parish was shaped roughly like a triangle, with its broad base along the Gulf. It came to a rounded point a few miles south of Interstate 10, the main east-west route leading to Houston and beyond on the west, and Mobile and beyond on the east. In the southern part of the parish, west of Arbier, there were marshes, incredible mazes of water and dry knolls, cypress trees and water hyacinths. It wouldn't take a stranger ten minutes to get lost in those marshes, and it wouldn't take much longer for him to drown. Still, there were people living in there, in ancient shacks built up on stilts, wherever a tiny piece of dry land permitted it. These were the most isolated Cajuns of all, people who knew no English, who spoke French from the day they were born to the day they died. They called the marshes _prairie tremblante._ Close to the ocean the water became so brackish that it killed the trees. The Gulf's brine lightened the color of the water; further inland the water in the marshes sometimes was dark brown or purple.\n\nThere were little canals cut through the marsh, alleyways for hunters, fishermen, or trappers. Sometimes bridges covered stretches of marsh where there was absolutely no firm footing. A native of the area could walk through the marshes with complete confidence. Alligator grass, salt cane, oyster grass, cattails, and Spanish dagger plants all grew wherever they could find firm ground for their roots. A native could tell by the color of these plants whether it was safe to walk there or not. Tiny variations in color conveyed important information, knowledge that meant the difference between a safe journey and an unmarked grave among the cattails.\n\nSometimes water hyacinths and alligator grass floated on the surface yet were attached to the shore. When they died and sank, new plants would occupy their old space. Watery marsh area was filled in from both directions at once. The water hyacinths were considered a menace, because they grew so rapidly and covered deep stretches of water, choking off bayous that were necessary for transportation. Many studies have been done and suggestions made about how to get rid of them. Still, they grow and cluster and trap the unwary traveler. There is even an argument about where they came from. Some experts claim that they were introduced at the International Cotton Exposition in New Orleans in 1884. But even these experts are divided as to which nation is responsible. The leading contenders are Japan and Venezuela.\n\nBut it makes no difference to the Cajuns who live in the marsh area. There are little hummocks of ground, called _ch\u00eani\u00e8res_ , from the French word for oak. Great, gnarled oaks, heavy with gray-green Spanish moss grow on these isolated islands. The local Cajuns earn their meager livings in several ways. A visitor might be shocked to learn that the principal agricultural concern of these Cajuns was cattle; he might be even more startled to see a cow swimming from one _ch\u00eani\u00e8re_ to another. Sugar cane cannot be cultivated so close to the Gulf, because the salt spray kills the plant. But further back in the marshes, there is a large industry in rice farming.\n\nAll of this was within fifteen miles of Arbier. It's very possible that a Cajun from the marshes, a crab netter or a muskrat trapper, would have the same disdain for the old families of Arbier as the latter do for \"foreigners.\" It is difficult to say, because it is difficult to mingle the two groups. On road maps, these marshy regions are indicated by large, unblemished expanses of white. There are no roads. Yet there may be as many people living among the _ch\u00eani\u00e8res_ as in Arbier itself.\n\nThere are many bayous through St. Didier Parish, some of them unnavigable. The main bayou, Bayou Chien Mort, or Dead Dog Bayou, runs from the marsh land east to a brackish swamp, a _cypri\u00e8re._ There are two causeways across the bayou, one on the old parish road and the other on Hanson Highway, the two main routes between Arbier and Linhart. Just before the bayou, the sugar cane fields begin to occupy all the available land. There are little cuts in Bayou Chien Mort, coulees, leading back to more shacks on untrustworthy-looking stilts.\n\nThe old parish road and Hanson Highway. Paul Pierson made his choice. The highway would get him to Linhart quicker, but he didn't especially want to get to Linhart in a hurry. So he took the old asphalt road. As he drove, he could see houses that had been ripped apart by hurricanes in previous years. He could see that with most of the land only slightly above sea level, most of St. Didier Parish was particularly vulnerable. He drove along the old blacktop road, listening to a radio station playing Cajun music.\n\n* * *\n\nCorinne Strahan had another whiskey sour at Bar's Mike and Grill on the old parish road. She sat, nervously tapping one foot and hating the taste of the cigarettes she was smoking. The woman who seemed to know her seemed also to know the three men at the end of the bar. In any event, she joined them, and after a little while, the woman left the bar with one of the men. Corinne said nothing, and the woman didn't say goodbye. Corinne waited.\n\nAfter a while Carl Steinbrenner arrived. He was completely unremarkable, Corinne thought. He wore a light suit and he didn't seem to sweat. He came up to her and said, \"Mrs. Strahan?\" Corinne was a little annoyed. There wasn't another woman in the place.\n\n\"Call me Corinne,\" she said.\n\n\"Fine,\" said Steinbrenner, \"and you call me Carl.\"\n\n\"All right,\" said Corinne, and there the conversation died until the bartender took his order for a gin fizz. A gin fizz, thought Corinne. Would Sam Spade have ordered a gin fizz?\n\nHalf an hour later, Corinne was in her station wagon, driving south on the parish road. Behind her was Carl Steinbrenner, driving a battered Ford. His car was about the best thing about him, she thought. It was the only thing with personality.\n\nWhen they arrived at the Sea-Ray Motel, Corinne's pulse rate jumped. She felt sick. She felt unreal, as though she were in a dream, and someone else's dream at that. She parked the car but sat behind the wheel. Both hands clutched the steering wheel.\n\nCarl Steinbrenner parked his car next to hers. He got out and walked over to the station wagon. \"I'll go check us in,\" he said, smiling. She said nothing. She didn't even want to nod. Steinbrenner went into the office. He was very careful not to slam the screen door. Corinne could hear everything that was said in the office.\n\n\"I'd like a room, please,\" said Steinbrenner.\n\n\"All right,\" said the desk clerk. \"Sign the book.\"\n\nThere was a moment of silence. Corinne relaxed enough to look around the place. It was long and low, a single row of rooms. How many private dramas were enacted every day in this place? she wondered. Then she saw one of the private dramas coming down the sidewalk, out of Room 6. It was the bleached blonde and the man from Bar's Mike and Grill. \"My God!\" murmured Corinne.\n\n\"'Mr. and Mrs. John Smith,' eh?\" said the desk clerk as he made his notations in the register. \"If you take a glance at this page, you can see that you're the fourth Mr. and Mrs. John Smith since I came on duty. I don't expect you'll be with us long?\"\n\n\"Just give me the key, please,\" said Steinbrenner.\n\nThe woman in the blue corduroy jeans got into a car with the man. Before the man could start his engine, Corinne had turned her key and driven away from the motel. In her rearview mirror she could see the strange man's car behind her and Carl Steinbrenner running out of the office, waving one arm and shouting something. Corinne was almost sick with disgust. She couldn't imagine how she had let herself get into the situation. She was glad that she had the nerve to leave, but she loathed the person who had agreed to go to the Sea-Ray Motel with Carl Steinbrenner.\n\nNot far from the motel was the turn-off for Hanson Highway. She took it and did fifty-five miles per hour until she reached Linhart. She was shaking and her head was buzzing. She wanted to cry, but somehow she couldn't. She wanted to get out, but she didn't know what she wanted to get out of. Her body. She hated her body. She hated what it made itself do. She hated what her body wanted, how it acted when it got those things. And her mind was worse yet. It made her do things even when her body didn't want to.\n\nShe parked the station wagon in the garage and hurried inside her house. It was getting late. She should think about dinner. She washed her hands first. Then she went into the kitchen. Everything looked hateful. She forced herself to take a few deep breaths and calm down. She acknowledged that she felt guilty, but she reminded herself that she really hadn't done anything to feel guilty about. If anything, she proved that she was a different kind of woman than the bleached blonde. She should be feeling better, but she wasn't.\n\nThe telephone started to ring. Corinne's eyes widened. Who? Skip? No, most likely it was a frustrated Carl Steinbrenner. She listened to the noise of the telephone. It kept ringing. It was almost enough to make her pull the cord from the wall. Then it stopped. The silence was frightening, too. She was very alone. She went into the living room and turned on the television. She had a choice of three shows. There was a talk show with a celebrity host that Corinne despised, a soap opera that had been running on the radio when Corinne was a baby, and a movie with John Payne and Virginia Mayo. She turned off the television and went into the bedroom. She pulled open the drawer where her husband's drugs were hidden behind his socks. She took out a vial that said _One capsule at bedtime._ Corinne poured two of the blue capsules into the palm of one hand, closed the vial, and put it back behind Skip's socks. She went into the bathroom and swallowed the blue capsules with a cup of water. About fifteen minutes later she started to feel a tingling sensation in her hands and feet. She felt that everything was, after all, all right; and if it was the blue capsules that made her feel that way, then she thanked the blue capsules. After another few minutes she had a difficult time walking from the living room to the kitchen. She stumbled over something. Her head was filled with a buzzing noise. She went back into the bedroom, knocking against the wall repeatedly along the way. She fell on the bed and was asleep. That was the way Skip found her, when he came home at seven-thirty.\n\n* * *\n\nIt was early afternoon when Darlaine Boshardt left Room 9 after her assignation with her own John Smith. As she was leaving, a man came out of the room next door. \"Hi,\" he said.\n\n\"I suppose,\" said Darlaine. She was in a foul mood. Sometimes sex did that to her. Sometimes it put her to sleep, and those were the times she looked forward to.\n\n\"I guess you and your husband are on vacation,\" said Chuck. \"My name is Robert Branford.\"\n\n\"I suppose we all have our crosses to bear,\" said Darlaine. She didn't like talking with this man, and she wanted to be far away from the motel by the time John Smith had finished showering and dressing.\n\n\"It's really a shame,\" said Chuck. \"It's about two o'clock now, isn't it? And checkout time was noon. You could have saved some money.\"\n\n\"It's all right,\" said Darlaine. \"The guy at the desk gives me special rates.\" She turned, cutting off the conversation as cleanly as if she had used a knife. Behind her, Chuck shrugged. Darlaine walked to her car, unlocked the door, and got in. She pulled out of the motel's driveway and headed toward the turn-off onto Hanson Highway. She pushed the gas pedal as far as it would go, and she watched the needle swing from left to right, from ten to thirty to fifty to seventy. She held a steady seventy miles per hour as she drove north toward Linhart. She felt terrible, and pushing the car around helped a little. Besides, being the wife of the parish's sheriff, all the police knew her. She knew them, too. She wouldn't be stopped.\n\nShe crossed the new causeway over Bayou Chien Mort. She was passing small farm communities. There were fields of sugar cane, soybeans, and sweet potatoes. There were many birds, of many colors, but Darlaine paid them no attention. After a while, the anger or the frustration, whatever she felt, went away. She took a shell-paved side road, turned around, and headed back home, to Arbier.\n\nThe drive home was at a slower speed than the drive north. She drove with her teeth clenched, as if she was forcing herself to do something she hated. She didn't ask herself whether or not that was true. It was enough that she had her nice home in Arbier, and she had her Sea-Ray Motel ready whenever she needed it, and she had her car to go back and forth, from Arbier sometimes as far as Linhart, and back.\n\nDarlaine tried to imagine what it would be like to be married to John Smith. She gave one derisive snort. Then how about that idiot in the room next door? She gave another derisive snort. She passed the motel and followed Ridge Street to W. 3rd. What would it be like to be married to the sheriff of St. Didier Parish? That was an important question, one deserving more consideration than the other two. She thought for a moment before she gave her final derisive snort, stopped the car in the driveway, and got out.\n\n* * *\n\nPaul Pierson was determined to do something positive, forceful, and decisive. There was nothing he could do about squaring his chin or broadening his shoulders, but he knew that he could still play some of the social games with the best of them. He drove up the parish road. He passed Bar's Mike and Grill and the rickety causeway over Bayou Chien Mort. He passed two more little bars and a gas station. Then he saw a solitary building by the side of the road. A sign on one side of the building said _Gussie's Lounge._ There were two cars parked beside the bar. Pierson pulled into the small parking lot and got out. The sweat had made his shirt stick to the seat cover. He felt dirty. He went into the bar.\n\nGussie's Lounge was dimly lit. There was a bar along one side of the single room. There were a number of tables, each with a tablecloth and a deep, round candleholder. The candles were not lit. There were three pinball machines at the back of the bar and a juke box in a corner. Pierson waited for a minute until his eyes adjusted to the darkness. He saw a bartender, a man on a stool, and two girls sitting at a table. All four people were looking at him. He tried to smile coolly, the way Philip Marlowe would have. For an instant, Pierson wondered how that character of Raymond Chandler would succeed in the Greater Arbier area. The answer came very quickly: better, much better, than Paul Pierson.\n\nPierson walked to the back of Gussie's Lounge. He put a quarter into one of the pinball machines. Before he pushed the re-set button, he noticed the score the previous player had registered. Fifty-two thousand. Pierson needed sixty-five thousand to win a free game, according to a little card on the machine. He pushed the re-set button, and the previous score cleared. A silver ball popped in front of a spring-operated plunger. He pulled back on the plunger and let go. The silver ball shot forward, bumped against a few obstacles, made a few bells ring, then sped directly between his useless flippers. Again the ball popped up in front of the plunger. He pulled it back and let go. The ball bounced around, hitting things, raising his score. This time he got to use a flipper to keep the ball in play, but not for very long. His third and last ball was much like the second. His final score was a little over seven thousand. Today, thought Pierson, pinball playing is not the decisive thing I am looking for. He went to the bar and ordered a beer. He took the glass and went to the table where the two girls were sitting.\n\n_\"Bonjour,\"_ he said.\n\nThe girls did not answer. Pierson thought it was odd that two young girls would be sitting in a bar so early in the day. He pulled out one of the other chairs at the table, but before he could sit down, the man at the bar said something in French.\n\n\"What was that?\" asked Pierson.\n\n\"He say he don't like you some at all,\" said one of the girls. They both laughed. Pierson looked at them for a moment, took a sip from the glass of beer, and walked back out into the bright sunlight. It was a beautiful day for skipping stones.\n**9**\n\n****\n\n_It was in New Orleans, Louisiana, almost thirty years before, and the President was Harry S. Truman._\n\n_Dorothy Sauk Micheton was excited. After dinner, Michael Grey was going to call on her. They were going to see_ High Noon, _with Gary Cooper and Grace Kelly. Dorothy was excited because this was the first time that her parents had let her go out on a date without another couple. Michael Grey went to the Boisogn\u00e9 School, a very exclusive boys' school in New Orleans. Dorothy went to the St. Teresa School for Young Women. Everyone knew that was probably the most exclusive school in the city._\n\n_Along with the excitement went a good deal of nervousness. There were all kinds of stories about the boys over at Boisogn\u00e9, and how she would have to fight Michael Grey off all evening. She just laughed when her friends teased her, and she just nodded politely when her mother gave her a rather strict set of instructions. But the truth of the matter was that Dorothy Sauk Micheton did not want to fight anyone off. She rather hoped that Michael Grey would prove to be less of a gentleman than the picture she had of him in her mind. These thoughts made her feel guilty, but she thought them just the same, again and again until she heard the doorbell ring._\n\n_She was so nervous she didn't know what to do. Suddenly she seemed awkward and completely without any of the social graces that girls who attended St. Teresa's were supposed to have. She answered the door, said hello to Michael, and invited him into the living room. She introduced him to her parents and her younger sister. He seemed very nice, almost as shy as she was. That surprised her._\n\n_The two teenagers stood in the living room, trying to make pleasant conversation with Mr. and Mrs. Micheton, but it was almost impossible to do. Dorothy's mother rescued the two youngsters from the situation by saying, \"I suppose you had better be getting along, now. They're not going to hold up the start of the movie for you.\"_\n\n_\"All right, Mother,\" said Dorothy, kissing her mother on the cheek._\n\n_\"And you remember what I said,\" said Mrs. Micheton._\n\n_\"Yes, yes, of course,\" said Dorothy. She put on a light coat, with Michael's aid, and the two of them went to the door. They were followed by Dorothy's father._\n\n_\"It was very nice meeting you, Mr. Grey,\" said Mr. Micheton. \"I'd like to remind you that Dorothy has to be home by midnight.\"_\n\n_\"Yes, sir,\" said Michael Grey. He shook hands with Dorothy's father, opened the front door, and guided Dorothy down the front stairs. They walked up to a blue Packard. He opened the door for her._\n\n_She was astounded. \"Your car?\" she asked. She had thought that they were going to take two streetcars to get to the theater._\n\n_\"Well,\" said Grey, \"it's my father's. He lets me use it sometimes. Get in.\"_\n\n_The nervousness increased as they drove downtown to Canal Street. Dorothy had never been alone with a boy in a car before. That in itself was a thrilling, provocative thing. She got a great deal of pleasure from merely thinking about the possibilities. But those thoughts brought a burden of guilt. The important factor for Dorothy Sauk Micheton was that the guilt was not enough to prevent her from making new fantasies._\n\n_\"You don't have to be so cold, you know,\" said Grey._\n\n_\"Excuse me?\" said Dorothy, her voice sounding oddly shrill to her._\n\n_\"I mean, you could sit a little closer.\"_\n\n_\"Oh.\" She slid across the seat. The pleasurable feelings grew, and the guilt grew with them. What if her parents knew? What if the sisters at St. Teresa's knew?_\n\n_Grey put one arm around her shoulders as he drove. \"I hope we can find someplace to dump this machine downtown,\" he said. He didn't notice that Dorothy had almost cried out when his hand touched her shoulder. Oh, there was something she wanted. . . ._\n\n_They parked the car and walked a few blocks to the theater. It was a Friday evening, and there was a large crowd already forming a line in front of the ticket window._ High Noon _was supposed to be a good movie. A girlfriend of Dorothy's said that it was swell, and Dorothy had planned to see it herself some Saturday afternoon. But seeing it with Michael Grey was so much better._\n\n_Inside the theater he put his arm around her again. Sometimes the feeling was wonderful, and sometimes she was afraid. Michael Grey's hand moved slowly down her arm, and his thumb touched, just barely touched, her breast. She took a sudden breath. He moved the thumb up and down, an inch at a time, slowly caressing her breast. She felt the nipple inside her brassiere grow hard, and she knew that Michael Grey would know that too. She pushed his hand away, and it retreated to her shoulder. Her pulse throbbed in her ears. She felt giddy. Michael Grey sat like a statue through the rest of the movie, and it took that long for Dorothy to recover. She never noticed enough of_ High Noon _to tell her parents when she came home. She just said that it was terrific._\n\n_After the movie, Michael Grey ushered Dorothy out of the theater and across the French Quarter to the_ Caf\u00e9 du Monde _for beignets and coffee. Dorothy was feeling very adult. She loosened up a little and her conversation began to consist of more than one- or two-word answers to his questions. It was obvious that he was impatient about something, but Dorothy just didn't know what to do._\n\n_After they left the_ Caf\u00e9 du Monde, _Michael and Dorothy walked back to where he had left the car. They drove to Audubon Park. In the moonlight the huge live oaks draped with Spanish moss were like things out of a dream. Dorothy was happy, but she was still excited, and she knew, suddenly, what she wanted. Oh, how she wanted. But she knew neither her parents nor the sisters would teach her how to get it. Dorothy hoped that Michael knew. She hoped that he would grab her, force her, take the full responsibility and she could have the pleasure, but not the guilt. . . ._\n\n_He touched her breasts, and she shivered. The pleasure, and the guilt. . . . He put one hand on her leg and she gasped. He moved the hand higher, to her knee. Please, she thought, don't. Please, she thought. The pleasure grew, and she felt herself wanting to be controlled by him. The guilt grew just as rapidly. If she could only let him know what she wanted . . . But guilt prevented that._\n\n_With a sudden stroke of insight, Dorothy realized that she had no idea what_ he _wanted. Should_ she _be doing something? Responding to his increasingly vigorous caresses with some kind of lovemaking of her own? All she knew to do was fall back helplessly, or seemingly helplessly, and let him have his way. Somehow she knew that if he did have his way, her own wants would be satisfied._\n\n_\"Don't be such a dead weight,\" he said, and the words and the panting, growling voice shocked her. For the moment, the guilt overrode any other feelings. She pushed his hand away. It was obvious that he was not going to force her or overwhelm her. He was being just enough of a gentleman to frustrate them both. She hated him suddenly, for the pleasure he was causing, and the guilt that the pleasure caused._\n\n_\"You know,\" he said, sitting up and running a hand through his tousled hair, \"if I had to choose between you and my right hand, I don't know who'd win. Come on. Loosen up.\"_\n\n_But Dorothy was crushed for that evening. That great longing, the aching need, was not going to be filled. She didn't know what to do with him, to encourage him. She wanted to be taken, and he would never do that unless he received the proper responses from the girl. What responses? she asked herself. She wished she could loosen up, like he suggested. There was too much ignorance and too much guilt. She would lose the ignorance eventually, but the cycle of guilt and pleasure would never be broken. Never._\n\n_Dorothy Sauk Micheton might be frustrated again some time in the future, but she swore that she would never be helpless again._\n**THREE**\n\n****\n\n**The Hurricane**\n**10**\n\nA full-scale hurricane may be the most powerful and terrifying thing in the world. Some people might argue that a tornado was worse, that the winds in a tornado reached velocities several times greater than the worst hurricane. That may be true. But a tornado is a short-lived phenomenon, compared to the hurricane which has an average life span of eight or nine days. Some hurricanes die quickly, others defy all Weather Service logic and continue on with their devastation for two weeks or more. Sometimes hundreds of thousands of people are left homeless, and gigantic areas of ruin mark the storm's passing.\n\nOn Saturday, there had been a small area of low pressure in the south Atlantic. This was nothing to be alarmed about. It was a common enough occurrence. Skip Strahan had noted its existence. He saw that it had grown on Sunday, and began to suspect that it might develop into another tropical storm. It seemed that there were more hurricanes every year. Some people blamed the testing of nuclear devices for all the changes in the weather. Other people blamed other things, some so ridiculous that the speakers themselves had trouble believing them. The truth was that there probably weren't more hurricanes than in earlier times; it was only that the satellite monitors spotted storms that might otherwise have gone undetected. Before the first weather satellite was launched, the Weather Service might go through an entire hurricane season and not get past D or E. Now, however, it was only early August, and already Skip was following the sixth tropical storm of the young season. That was the classification of all storms with winds in the fifty-six to seventy-four miles per hour range. Skip showed satellite films of the tropical storm, which had been named Felicia by the Weather Service. On Tuesday, Felicia became a hurricane about one hundred miles southwest of Cuba. Her winds had been clocked at ninety miles per hour, with some gusting up to one hundred and twenty miles per hour.\n\nSkip Strahan looked at the information which he received from New Orleans and the data from the National Weather Service there, whose duty it was to issue Hurricane Warnings. He took a deep breath. Unlike her predecessors, Felicia was moving into the Gulf of Mexico. There was no known method of predicting the path of a hurricane. They followed whimsical, twisting routes, sometimes defying all that the science of meteorology had learned about storms.\n\nThere was a hurricane in the Gulf. Her name was Felicia. Her position was about fifty miles east of the Yucat\u00e1n Peninsula. From now on, Skip would have to stay at his post, relaying the information he received from New Orleans to the people of St. Didier Parish. He hoped that Felicia wouldn't cause him too much trouble.\n\n* * *\n\nSheriff Walter Boshardt woke up. He yawned and rubbed his temples. He let his head fall back on the pillow. He felt warm and drowsy. Then he shook his head and forced himself to sit up again. For some reason, it was more difficult to get out of bed this morning. But Boshardt did it. He was the sheriff of St. Didier Parish, the top law enforcement officer in the parish. There were no local police in any of the communities in the parish, so the sheriff and his thirty men and their ten patrol cars had to guard the safety and well-being of a lot of Cajun coonass bastards who were probably still asleep.\n\nBoshardt shook his head again, to get the idea out of it. It was more likely that the muskrat trappers in the marshes and the rice field workers and the cane workers and the cattlemen and the farmers were already awake and at work. It was time for Boshardt to check in.\n\nHe looked across the bedroom, at the other twin bed where his wife was asleep. Dorothy. That was her name when he married her. Darlaine. Boshardt frowned and went into the bathroom. He thought about the life he had with her, the unpleasant tone of that life. He thought about her flagrant affairs around the parish. He thought about how everyone he spoke to carefully avoided that subject, out of respect for him, for the competent, reliable job he had done as sheriff. No one spoke of Darlaine's activities within his hearing, but Boshardt knew what they were thinking. Maybe there was no way out of the situation. Maybe there was. Boshardt had to get to work.\n\nHe shaved and brushed his teeth and took a quick shower. Then he got dressed, putting on the light brown uniform of the St. Didier Parish Sheriff's Office. There was a flat-brimmed hat that went with the uniform, but Walter Boshardt usually just carried it and left it in his patrol car. He made certain, however, that the rest of his staff wore their uniforms, including their hats, properly at all times while on duty. Boshardt smiled when he thought about that. Why did he do it? Why did he make his men so conscious of their appearance, when his own example showed that the hat wasn't important? Well, he answered himself as he drank a cup of coffee, he made the deputies wear their hats because it seemed the thing to do. He smiled again. The coffee was brewed with chicory, and it was hot and strong and dark. It was good southern Louisiana coffee. They couldn't drink the stuff up to Shreveport. Those Protestants up there were too solemn and quiet. How he'd like to get the sheriff from one of the northern parishes, along the Arkansas border, to come to a Saturday night fais-do-do. It would be like a preacher in a whorehouse.\n\nBoshardt finished the coffee, checked his appearance in a full-length mirror inside the bedroom closet door, and left the house. He got into his patrol car, which he had left in the driveway behind Darlaine's car, and drove to the Arbier office. When he opened the door, the air-conditioned coolness hit him. It felt good, and he took a deep breath. His brown uniform was already darkened with sweat.\n\n\"Morning, Sheriff,\" said a uniformed man behind a desk.\n\n\"Morning, Sergeant,\" said Boshardt. \"What do we have?\"\n\n\"We had a complaint last night,\" said the sergeant. \"You won't believe it, but we had a complaint from the Sea-Ray Motel.\"\n\n\"Don't tell me Sparkle's back,\" said Boshardt. He sat in a chair beside the desk.\n\n\"No,\" said the sergeant, \"the last I heard of him or her or whatever it is, he or she was in New Orleans.\"\n\n\"So what about last night?\"\n\n\"Some customer complained to the night manager about too much noise from the next room, and when the night manager didn't do anything, he called us. Somebody, let me see, it was Auguste who went, but by the time he got there, the people in the next room had checked out and everybody was happy.\"\n\n\"Everybody's always happy at the Sea-Ray,\" said Boshardt, shaking his head.\n\n\"It's the biggest amusement park in the parish,\" said the sergeant.\n\n\"Anything else?\"\n\n\"Yeah, this.\" The sergeant handed Boshardt a sheet of paper.\n\n\"Hurricane, huh? Well, you know what to do. Captain Shaeffer will be in charge. I'll have to go up to Linhart and have a meeting with the department heads. We'll have to figure out the deployment of the patrol cars and the men. Then I'll have to stop by Delochitaches, too, I suppose, though you could evacuate the whole place with a phone call to the right person. God, I hate this.\" Boshardt crumpled the paper and tossed it onto the desk. Then he reached over, picked it up, and straightened it out as best he could.\n\n\"It's an early one, isn't it?\" said the deputy.\n\n\"Yeah, well, you usually think of late August, more likely September for hurricanes hitting Louisiana. But right now I'm thinking about Audrey, back in 1957, and I'm thinking how she hit us and Cameron Parish. And she was June, I think, if I remember right. I'm thinking of the streets of this town filled with the goddamndest stinking mud you'd ever want to smell, and it was covered with sand from the beach, like snow. There wasn't any Arbier after that for weeks. Most of the parish was wiped out. That storm blew sea water in a huge wave that rolled almost twenty miles inland. Nobody had any idea of what to do or where to start. That storm killed a lot of people. I was home from college then, and I was working with my father, shrimping. We thought we had the boat safely moored. We took her up Black Run Bayou and secured her. After Audrey, we went to get the boat, and she plain wasn't there. Not a sign. To this day, I don't know what happened to that boat. And our house. . . .\" Boshardt's voice trailed off. He stared for a while. The sergeant said nothing.\n\nSuddenly Boshardt roused himself. \"All right,\" he said, \"this time will be like the last few years. Linhart gets the news from New Orleans, and everybody in the parish holes up safe and sound. _If_ the hurricane comes here.\"\n\n\"It's still down there off Mexico.\"\n\n\"You've got to be ready,\" said Boshardt. \"I'll see you later. You get the message to Captain Shaeffer when he comes in.\"\n\n\"Check. Anything else?\"\n\n\"Isn't it enough?\" asked Boshardt, as he walked out the door.\n\nSheriff Boshardt didn't know it yet, but there was a large game being set around him, with himself as one of the principal players. If he had been told all the details, he might not have called it a game; still, it had all the necessary elements. There was a playing area\u2014the town of Arbier. There were players\u2014how many? It was too early to tell exactly how many, or even into how many teams they could be divided. There were goals and rewards and maybe penalties. There was a requirement for skill. There was a large element of luck.\n\nWhenever one of the major factors in a situation is a hurricane, there is a goddamn huge element of luck.\n\n* * *\n\nHurricanes need two things to stay alive: heat and moisture. The storm starts as a small area of low pressure over water at least eighty degrees Fahrenheit. Warm, moist, light air is drawn into the area. Because warm air is lighter than cool air, it begins to circulate upward. Water vapor in the air condenses as it cools at higher altitudes. The condensation releases the energy to keep the cycle going. The spinning air mass grows, and clouds start to form. More warm, moist air is drawn into the system. More water vapor condenses, more energy is released. The spinning clouds begin to take on a characteristic spiral formation. The air that enters at the bottom and rises to lose its moisture content is blown away by winds in the upper atmosphere. The hurricane grows and grows, a self-perpetuating pump that has all the heat and moisture it needs in the places where hurricanes are born. These places are the south Atlantic, the Caribbean, and the Gulf of Mexico.\n\nIt is only when the hurricane, pushed by some vague influence of barometric pressure or other atmospheric conditions, comes over land that the pump stops working. The friction of the air mass moving across the land slows the hurricane. No longer is there moist air to be drawn up and around. The land mass rises above sea level. The hurricane rapidly fills in and becomes a scattered mass of squalls.\n\nBut meteorologists know that there is absolutely no way to predict where a hurricane is going to move. Sometimes, against all scientific theory, they maintain their tight spiral formations and hurricane-force winds across huge stretches of land.\n\nThe only thing to do with a hurricane is hide from it.\n\n* * *\n\nChuck woke up in Room 8 of the Sea-Ray Motel. He had had a very bad night's sleep, and he was in a foul mood. The sheet and blanket were twisted around his legs. He kicked free of them and sat up. He had a terrible taste in his mouth. He went into the bathroom and looked in the mirror over the sink. His tongue was coated and swollen. \"Goddamn this place,\" he muttered. He thought he was probably being poisoned by the drinking water, which most likely came out of some stagnant bayou. Or the coffee, which he could barely drink without a good quantity of milk and sugar. Or the bugs. God, he knew he'd been bitten by at least a dozen different kinds of disease-carrying bugs. Chuck just wanted to go home. He almost picked up the telephone to call Tom, to tell Tom that he was going to quit. Arbier wasn't Chuck's kind of place at all, although the Sea-Ray approached his idea of civilized living.\n\nHe got dressed in a tee shirt and a pair of wash-and-wear slacks. He put a pair of sandals on his feet and a pair of sunglasses over his eyes. Then he went outside. He went into the office. The screen door slammed, and the desk clerk jumped a little. He put down the book he was reading.\n\n\"Haven't you finished that book yet?\" asked Chuck.\n\n\"Listen, friend,\" said the desk clerk, \"I read it once for the story line, and now I'm going back over it for style.\"\n\n\"How's the style?\"\n\n\"Style ain't so bad. How's yours?\"\n\nChuck wanted to hit the desk clerk, but he stopped himself. \"I registered a complaint in here last night,\" he said.\n\n\"So I heard.\"\n\n\"Well, your friend the night manager isn't very effective in crisis situations, is he?\"\n\n\"That's not his job,\" said the clerk. \"It's not mine, either.\"\n\n\"Then what are you here for?\"\n\nThe desk clerk smiled. He held up the porno book, but he didn't say a word.\n\n\"Wonderful,\" said Chuck. \"And the cops were really terrific too. I told those morons in Room 9 that I was calling the sheriff's office, and they were gone like a flash. Then I just about get to sleep, and this goddamn defender of law and order wakes me up.\"\n\n\"Seems to me that some of that was your own fault,\" said the clerk. \"We got thirty officers to patrol the whole parish. So a complaint about noisy neighbors isn't going to get very high priority.\"\n\n\"Especially in this motel,\" said Chuck.\n\nThe desk clerk gave Chuck a smile that Chuck wanted to break into a million pieces.\n\n\"You're in a good mood today, aren't you?\" asked Chuck.\n\nThere was a brief silence from the desk clerk. \"Yeah,\" he said at last, \"on the whole, I think it's not a bad day.\"\n\n\"Well, if I get any more trouble out of you, you'll start to have a bad day all around.\"\n\n\"I don't see how. I really don't see how.\" The clerk went back to his reading, ignoring Chuck.\n\n\"Look,\" said Chuck, \"you know why I'm here.\"\n\n\"Yeah,\" said the clerk, not looking up. \"You're complaining about that rock band that checked in at three this morning.\"\n\n\"No, that's not what I meant. I mean, you know why I'm in Arbier.\"\n\n\"You said you were doing a magazine article, I think,\" said the clerk. \"But so far, all I've seen go into your room were a couple of bottles of liquor, and all I've seen come out is that black bitch in Room 13.\"\n\nChuck fought to keep himself under control. It was one of the most difficult things he ever had to do. Finally, he said, \"Well, that business about the magazine article wasn't the truth.\"\n\n\"You know, I half suspected that,\" said the clerk.\n\n\"I work for a film producer. Sort of an advance man. I'm finding locations and noting typical local people.\" He paused, expecting that the desk clerk would be surprised and interested. He was disappointed. \"This film we're working on,\" he said, \"will probably have some scenes shot in Arbier. The rest of my crew was supposed to get in last night, but they haven't got here yet. We're going to need two rooms, each with twin beds.\"\n\n\"We got them,\" said the desk clerk, still not looking up. \"If you want them, put down a deposit.\"\n\n\"How much do you want?\"\n\n\"Give me, oh, twenty-eight bucks and you can have Room 7 and Room 9 until noon tomorrow.\"\n\nChuck took the money out of his wallet and put it on the counter.\n\n\"If I didn't know you better,\" said the clerk, \"I'd think you were renting those rooms on either side of you for the quiet. But that's stupid.\"\n\n\"Right,\" said Chuck. \"It's stupid.\"\n\nThe desk clerk continued his reading. He hadn't looked at the bills Chuck had put on the counter. \"How much you got there?\" asked the clerk.\n\n\"Twenty-eight, like you said.\"\n\n\"Terrific.\" There was a long silence. Chuck soon realized that the clerk wasn't going to move from his chair until Chuck had left the office. Chuck turned and left the office, muttering under his breath. He gave the door a loud slam on the way out.\n\n\"Wait a minute,\" shouted the clerk. Chuck stopped on the sidewalk. He went back into the office. \"Here,\" said the clerk, handing Chuck the keys to Rooms 7 and 9.\n\nChuck nodded and took the keys. \"Go to hell,\" he said.\n**11**\n\nThere was an extra noise that woke Paul Pierson up. There was the sound of the mockingbird, and the sound of Cy's frantic scrabbling at the screen. But there was something else. It was the sound of the lock opening and the front door swinging and the hinges making their croaking, protesting sound. Pierson sat up in bed quickly. Sometimes the landlord came in, uninvited, to spray for roaches. Once the landlord had come into the bathroom, while Pierson was taking a shower, because the woman downstairs had some trouble with her plumbing. Very often, Pierson couldn't figure his landlord out.\n\nIn any event, Pierson grabbed his blue jeans and pulled them on. He also pulled on an Ohio State sweatshirt. As he freed his head and jerked the sweatshirt down, he saw Maddie, standing in the doorway of the bedroom, smiling.\n\n_\"Bonjour,\"_ she said happily.\n\n\"Well,\" said Pierson, \"look who it is. The mistress of the oceans herself.\" He was in a very bad mood, but Maddie didn't seem to notice.\n\n_\"Allo, mon cher cousin,\"_ she said. _\"Je suis Mathilde Gargotier, la plus belle\u2014\"_\n\nPierson interrupted her. \"Maddie,\" he said, \"you know goddamn well that I don't understand your French. And you know just as well that I'd probably be burned up because you never came home last night.\"\n\n_\"Mon Dieu!\"_ she cried in false outrage. \" _Quel_ shock! _Quel_ surprise! What am I, your little _b\u00e9b\u00e9,_ that you get mad as goddamn if I don't come home when you think I should? Since when do I have to check in and out, like this was a prison? I am here because I wanted to be here. Last night, I didn't want to be here, so I wasn't. It is very simple.\"\n\n\"It may be simple,\" said Pierson, \"but it's also not plain, common courtesy. I never said that I owned you, or that I had any right to keep you here if you wanted to leave, or the other way around. I just think I deserve a little more consideration.\"\n\n\"You know, I'm beginning to think that I don't want to be here now, either. I think I may go visit _mon p\u00e8re,_ because I haven't seen him in many-many days. Or I will go to the caf\u00e9 for a demitasse of _caf\u00e9 noir._ But standing here, looking at your idiotic Ohio State sweatshirt, is beginning to make me angry.\"\n\n\"All right,\" said Pierson, \"I'll bite. Why are you getting angry?\"\n\nMaddie grunted and kicked one of Pierson's sneakers across the room. She turned around and walked out of the room. Pierson stared after her for a few seconds, controlling his own temper. He couldn't understand what she had to be angry about. It seemed to him that he was the injured party. After a while he went out and found her sitting on the couch in the living room.\n\n\"Well?\" she said.\n\n\"Well what?\"\n\n\"Well, what are you so mad about?\"\n\nPierson took a deep breath. He had to choose his words carefully, he knew, or Maddie would blow up all over again. \"I was worried about you,\" he said. \"I thought you'd get home late, but when you didn't get home at all, and you didn't call, I got worried.\"\n\n\"You sound like a parent,\" said Maddie.\n\n\"Well, why didn't you call?\"\n\nMaddie smiled. \"Well, you know me and Shelley and Betsy went up to Linhart yesterday, to join the Navy. So, along the way we argued, and we decided it would be better to join the Air Force instead. But then I changed my mind again, and I went back to the Navy.\"\n\n\"Come on already.\"\n\nMaddie smiled, remembering the day before. \"When we got to Linhart, we went to the post office. Shelley and Betsy went to see the Air Force man, and I went to see the Navy man.\" She paused for a long while, and her expression changed. She looked as if she had just tasted something that she wanted to spit out.\n\n\"What happened?\"\n\n\"They wanted to know if I could type,\" she said.\n\nPierson laughed, and Maddie threw a cushion at him. \"No,\" she said, \"they had all kinds of interesting things I could do, but at the last minute I told them I would think it over for three or two days. And I waited, and Shelley and Betsy came out, and you know what? They told the Air Force man the same thing. So we started home, and we took the old road, and we stopped at Henriette's for _un petit,_ maybe two, and we stopped at Marie & Pal's, for _un petit,_ and we stopped at Medoux's for _un petit,_ and then Shelley told Betsy to drive, and we stopped at Beaumont's who used to be Schoenberg for _un petit,_ and then I drove, and we stopped at Gussie's for _un petit,_ and there may have been one or two more in there, I don't remember. But we got to Betsy's house, and we laughed a lot, I remember that all right, and I remember Shelley passing out, and Betsy asked me if I wanted her to drive me here, and all of a sudden, like that, I couldn't remember any English. And I answered her in Cajun, and she couldn't understand me, but it didn't make any difference because in a little while I passed out. And today I feel fine.\" She smiled.\n\n\"You're proud of yourself.\"\n\n\"I am,\" she said.\n\n\"And what are you going to do now?\" he asked. \"You've quit your job and you're not going into the Navy.\"\n\n\"Who says I'm not?\" she said with a frown.\n\n\"You're not, are you?\"\n\n\"No, I'm not. This morning, when the three of us woke up, we decided what we're going to do.\"\n\nPierson gave a short laugh. \"I can hardly wait to hear it. What are the three of you going to do?\"\n\nMaddie's expression became very serious. She put on a very affected attitude. \"We're going to Las Vegas,\" she said. \"We're going to be show girls.\"\n\n* * *\n\nDarlaine Boshardt got up at half-past ten. She lay in bed for a while, wondering how she was going to kill the day. She didn't feel up to taking someone out to the Sea-Ray. She could drive up to Linhart, that thriving metropolis, and do some shopping. She might even get on I-10 all the way into New Orleans. She hadn't been to New Orleans in a long while, she thought. There were too many bad memories. Since her parents had died, there didn't seem to be any reason for going to New Orleans. All that the city had were places for which she had a lingering hatred. She sighed and got out of bed. She washed and dressed and brushed her hair. God, how she hated her hair. It had been beautiful once. She looked in the mirror. _She_ had been beautiful once. Sometimes she felt so frustrated that she came close to doing something violent. Feeling helpless was the most painful thing of all. There were times when she was helpless, like now. She wanted to be her younger self again, and she was helpless to do anything about it. She wanted to be somewhere where her presence was important, but, thanks to her husband, that was impossible. She was helpless to do anything about it. And as powerful as she was at controlling most situations and people, she had to admit that now and then she was totally helpless.\n\nShe tried to shrug the feeling off. She went to the telephone and dialed the television station in Linhart. The switchboard operator there didn't recognize Darlaine's voice. \"I'm sorry, ma'am,\" said the operator, \"but Mr. Strahan is busy because of the hurricane in the Gulf.\"\n\n\"Hurricane?\" said Darlaine. \"Listen, miss, I'm Darlaine Boshardt, the wife of the sheriff. I'm calling because of an emergency. This is official business.\"\n\n\"Yes, Mrs. Boshardt,\" said the operator. Her voice was shaky. There was a click, a few seconds of line noise, and then Darlaine heard Skip Strahan answer the phone.\n\n\"Yes?\" he said. He sounded angry.\n\n\"Hello, Skip. It's me, Darlaine. I wanted to call about confirming your engagement to speak at\u2014\"\n\n\"Look, Mrs. Boshardt\u2014\"\n\n\"Darlaine.\"\n\n\"Yeah. Anyway, right against the eastern coast of the Yucat\u00e1n Peninsula we have this hurricane. A real live one, getting bigger and bigger all the time. Now this hurricane could rain itself out over the Yucat\u00e1n, and then we'd all be happy. But there is the very real possibility that the hurricane will recurve to the north and east. That means us. That's why there's a Hurricane Warning out. Okay?\"\n\n\"Sure, Skip. I know you must be working hard, but\u2014\"\n\n\"I'm working my ass off, Mrs. Boshardt. I don't have any time to talk to you. Maybe give me a call come wintertime.\" He hung up, and Darlaine was listening to static. She hung up too.\n\n\"If you work your ass off, honey,\" she said to the phone, \"you're going to miss it some day, and that ain't no lie.\" She walked into the living room and switched on the television set. Channel Five was showing re-runs of \"Leave it to Beaver\" with a white caption running along the bottom of the picture. The caption said, _A Hurricane Warning has been issued for the entire Gulf coast, from Brownsville, Texas to Key West, Florida. Hurricane Felicia is located just off the eastern coast of the Yucat\u00e1n Peninsula, and is described as \"very dangerous.\" There is a possibility that Hurricane Felicia will turn and strike somewhere along the Gulf coast during the next twenty-four hours. Stay tuned to this channel for further advisories._\n\nDarlaine turned the channel selector. She had a choice of a soap opera, where someone evidently had amnesia but was a key witness in a court case involving an important man in the community and a woman with a shady past, or a game show where contestants were going out of their minds with rapture by winning washer-drier combinations. These two stations were running virtually the same caption at the bottom of the screen. Darlaine turned off the television. \"Well,\" she said aloud, \"that's how I spend my day. I wait for the hurricane.\"\n\nThe telephone rang. She wondered who would be calling her. Perhaps the day wasn't a total waste after all. \"Hello?\" she said into the phone, after running from the living room.\n\n\"Hello, Mrs. Boshardt?\"\n\nDarlaine was disappointed. \"Yes,\" she said, \"speaking.\"\n\n\"Uh, this is Sergeant Hebert at the Arbier sheriff's office. Sheriff Boshardt gave me instructions to call you and tell you that there is a Hurricane Warning out and that you should get ready to take the proper precautions.\"\n\n\"Right,\" said Darlaine, \"thanks.\"\n\n\"Do you need any help, ma'am?\"\n\n\"Thanks for calling, Sergeant.\"\n\n\"Sure, ma'am,\" said the officer. \"Any time.\"\n\n\"Any time is right.\" Darlaine hung up the telephone.\n\n* * *\n\nOn a normal day, Skip Strahan received his day's weather predictions from the office of the United States Weather Service in New Orleans. He also took his own readings with a small battery of sophisticated equipment that the television station operated. Now, however, he was on twenty-four-hour alert. His desk was covered with maps and charts, barometer readings along the path of the hurricane, and dozens of pages of related data. He had only one assistant at the station, Sheila Downing, and her desk was just as littered with material as his. The trouble was that, at the moment, there was little for them to do. The hurricane was stalled against the Yucat\u00e1n. It hadn't moved for hours, although it was growing in size. Perhaps there was a small area of high pressure there holding off the oncoming hurricane. Strahan knew that hurricanes often stalled when they reached land. This was a particularly dangerous time for the coastal communities. They could feel the full strength of the hurricane's winds for hour after hour. Then, for some reason, perhaps a change in pressure inland, the hurricane would move on. There was no way of foretelling the movements of the storm.\n\nSilently Skip prayed that the hurricane would rain itself out, fill in, and die over the Yucat\u00e1n. Many lives would be lost there though.\n\nSuddenly Skip shoved the maps away from him. He felt as if he was going to scream if he had to chart any more reports from New Orleans. He went to his jacket, which hung on a coat rack. He took out a small vial and poured two yellow pills into the palm of one hand. He paused for a moment, then added a third. He got a paper cup of water from the water cooler and took the three pills. In less than fifteen minutes he was feeling better. In half an hour he realized that he was whistling while he plotted the latest data from New Orleans.\n**12**\n\nAbout eleven o'clock in the morning, Sheriff Walter Boshardt completed his tour of the shops along Ridge Street. He was putting off the drive to Linhart, for reasons which weren't clear. For a moment, he stood on the sidewalk. The sky was bright, clear blue. There was a slight breeze from the Gulf, and the fresh smell of it roused so many memories in Boshardt's mind that they mingled and coalesced simply into pleasure. He looked down W. 3rd Street, his street. He wondered if his wife were awake yet. He stared down that street for a while too, thinking of all the years he had walked along, kicking the little white shells that were everywhere. And the great oaks with their weight of Spanish moss covered it all, like a canopy. The cry of a killdeer broke his train of thought; he looked up to see the bird flying back to the mud flats where it made its home. Boshardt wondered if that bird, as well as the town he loved so well, was already marked for destruction by Hurricane Felicia. It was too early to panic, but it was just the right time to prepare. Now Sheriff Boshardt knew that all the shopkeepers in Arbier were aware of the Hurricane Warning. What preparations they made were their concern from then on.\n\nThere was nothing left to do but make the drive to Linhart, and then to Delochitaches. He wanted personally to see that his meager forces were well-prepared for whatever might happen.\n\nBoshardt thought that a cup of coffee before the drive would do him some good. He walked the short distance along Ridge Street to the Crisis Caf\u00e9. He opened the door and was welcomed by the air-conditioned coolness. He tried to remember when he was a boy, before every shop in the town could boast air-conditioning. He tried to recall the heat and the humidity, the constant heaviness of the air, the whirring of the small electric fans that seemed to do no good at all. He laughed. That was one memory that came easily enough.\n\n\"Now this sheriff what we got here,\" said a man at a table, noticing Boshardt's laugh, \"he ain't got the worry, _mais non,_ I tell you for true. Hey, _cousin,_ you look how he brought himself here and he laugh, that one. _Certainement,_ if that how you call Felicia brought herself here, there don't be no laugh. Other hurr'canes, I never will forgot, whoo man! Don't they swat us like _les maringouins!_ \" He slapped his arm as though he were swatting a mosquito.\n\n\"You don't pay no attention to this in the back Cajun,\" said another man. \"He lives so far out in the marshes, he was surprised when I told him Hoover wasn't President no more. He picks moss, this one, a moss cutter. And he talks like that to the sheriff.\"\n\n\"It's all right, boys,\" said Boshardt. He sat in his usual booth, and Lauren came right over to take his order. \"How about the catfish?\" he asked.\n\n\"Have you ever know when we had to say 'no' to the catfish?\" she asked smiling. \"The catfish, special for the sheriff. And coffee. Black?\"\n\n\"Black-black,\" he said. She smiled again and went into the kitchen. Boshardt watched her go. She reminded him. . . .\n\n\"We bring you a special weather advisory from the Channel Five weather station,\" said the announcer on the television. All at once, all conversation in the diner stopped. Lauren and the owner, old Mrs. Perkins, came out of the kitchen to listen.\n\nOn the television, the picture changed to Skip Strahan, standing before a large map of the Gulf region. \"There is little more to report,\" he said. \"Hurricane Felicia has remained stationary off the Yucat\u00e1n Peninsula for several hours. Her winds are now measured at over one hundred and twenty-five miles per hour, with strong gusts of over two hundred miles per hour. She is about two hundred miles in diameter, a giant storm system that is throwing gale-force winds a great distance from the central eye of the hurricane. This information has been compiled from data gathered by the National Hurricane Center in Miami and supplied to the National Weather Service Hurricane Warning Bureau in New Orleans. New advisories will be put out every six hours by the office in New Orleans. If Felicia makes any movement, we will get the information to you as soon as we can. Stay tuned to this channel for further information.\"\n\n\"Oh, Lord,\" whispered Mrs. Perkins.\n\n\"What?\" asked Lauren, shrugging. She walked over to the sheriff's booth. \"I can't wait to see the look on your face when you put that catfish in your mouth,\" she said.\n\n\"Why?\" asked the sheriff. \"Is it special catfish or something?\"\n\n\"Well,\" said Lauren, \"I was watching Mrs. Perkins because I always think I'm going to learn some of her secrets. I never do, but I always try. Anyway, she skinned this here catfish, and then she gutted it, and she cut the head off, and she cut the tail off. Then she goes to fry the thing, and I say, 'Aren't you going to wash it, Mrs. Perkins?' And she says, The damn thing's been swimming in water all its life. If it ain't clean now, it ain't _never_ going to be clean.' So I learned one of her secrets, after all.\"\n\n\"You can take the old catfish out of the bayou,\" said Boshardt.\n\n\"Yes, sir, I know the rest,\" said Lauren, laughing. She went into the kitchen and came back out with a tray for another booth. Then she came back to Boshardt's booth. \"Say,\" she said, \"I wish you could say something to Mrs. Perkins. She's a nervous wreck on account of this hurricane business.\"\n\n\"She has a right to be,\" said the sheriff.\n\n\"I suppose business will drop like an egg off a table,\" said Lauren.\n\n\"That's a bad thought,\" said the sheriff. He pictured the egg spattered all over the floor, an image of how the town of Arbier might be blown apart, hopelessly battered beyond repair.\n\n\"It's a dumb thought,\" said Lauren. \"This whole thing is starting to get me down, too.\"\n\n\"If that hurricane starts to move, and if it starts to swing toward us, you're going to get the strangest feeling up your backside, and then you won't feel so much like laughing.\"\n\n\"How about you, Sheriff?\" asked the pretty seventeen-year-old.\n\n\"Well, to tell you the truth, I got that feeling right now, listening to Strahan on the television. It'll go away, the feeling, I mean, but it'll come back.\"\n\n\"Then come in here for some coffee,\" said Lauren. \"Maybe we'll have some more back-of-the-bayou Cajuns. They're always fun.\" She disappeared once again into the kitchen.\n\nWalter Boshardt stared after her. A little girl from a tiny town in a very small parish in Louisiana. She laughed at the \"back-of-the-bayou Cajuns,\" but what would she do in a city, like New Orleans? How would she react to New York City? Sheriff Boshardt relaxed in the booth. He sighed. He realized that Lauren would most likely think that the people in New York were just as funny as the Cajuns. He wished that he could have her simple outlook. It would take a lot of the sting out of the next day or two.\n\n* * *\n\nCorinne Strahan woke up, got out of bed, washed, dressed, did the laundry, and sat down on the couch in the living room. She was finished for the day. The modern conveniences in her wonderful home had given her hours and hours of leisure time. What was she going to do? she asked herself. There was volunteer work, except that she hated being with old people or crippled people or mentally retarded people. She could take a course or two at the extension school of Hebert College. She could learn conversational Portuguese, so she could talk with some of the shrimpers.\n\nCorinne went to the closet, and took out another kit that Skip had given her. She couldn't even remember when it was that she had gotten it. There was a layer of dust on the box lid. It had probably sat in the closet for a long time. It was a decoupage kit. Corinne looked at it for a while. That made her a decoupageuse, she decided. She was going to decorate a lunch pail for Skip. She didn't have a lunch pail, and Skip didn't need one, and would look absolutely foolish going into work with one.\n\nCorinne thought for a few seconds, trying to decide what else she could do with the kit. She could decorate a wastebasket. That was a wonderful idea.\n\nCorinne Strahan realized that she had just decided to spend the whole day decorating a wastebasket. Foreign leaders she had never even heard of were dying or being assassinated, people walked on the moon\u2014or they used to\u2014and she couldn't do anything more meaningful than decoupage a metal wastebasket, one she would have to go out and buy first anyway.\n\nShe put the kit on the coffee table and turned on the television. To her surprise, Skip's face filled the screen. She listened to his words, and she felt the first shiver of fear. In Linhart she was safe enough. But her father in Arbier was in an awfully vulnerable place. Maybe she should go down and get him. She listened to Skip. Evidently there was plenty of time. The hurricane was stuck against Mexico. Sort of decoupaged against the Yucat\u00e1n, she thought. She turned off the television when Skip finished his message. There was nothing good on until _Casablanca_ at one o'clock. She might as well go out and buy the metal wastebasket and get to work. She could have it finished by the time Skip came home.\n\nGod, she wished that she could do something.\n\n* * *\n\nThe worst effects of a hurricane come from the sea. The hurricane surge which accompanies the eye of the storm can be as high as twenty-five feet. Waves pushed by the hurricane winds can reach thirty feet or more. The waves on top of the surge are often enough to wash inland for great distances. Cities such as Galveston, Texas have learned this horrible truth through experience. Thinking themselves protected by a large beach area and sand dunes, people awaited the approach of a hurricane with curiosity rather than fear. Only when the giant wall of seawater roared upon them, crashing and swirling mile after mile inland, did they realize how incautious they had been. This realization did not last long for many people. For many people, it was the final realization of their lives.\n\nToday the United States Weather Service provides accurate information and estimates of the size and potential danger of any hurricane likely to hit the country. The main center in Miami cooperates with other departments and meteorological groups, with radar stations all across the hemisphere, to keep an accurate picture of the hurricane activity in the Atlantic, the Caribbean, and the Gulf of Mexico. This information is funneled down to local weather bureaus, which relay the data to even smaller communities. In modern times, when nearly every household has at least one television set, the news of an approaching hurricane can be circulated many hours ahead of time, giving plenty of time for preparations or evacuation. With the advances in meteorological science, with the aid of weather satellites and the benefits of the communications media, there are no longer disasters on the scale of the Galveston hurricane in 1900, when over six thousand people died and the city was so utterly ravaged that it could be said that there was no more Galveston. The survivors of that disaster showed great courage, and with help from the government rebuilt the city. Arbier had been hit by hurricanes in the past.\n\nArbier had been virtually destroyed and rebuilt. The residents of the town knew that no hurricane could ever wipe Arbier off the map totally and forever. Still, like the old proprietor of the Crisis Caf\u00e9, the people of Arbier were already beginning to feel a heaviness in the stomach as fear began to take hold.\n\nTo someone like Sheriff Boshardt, who could view the situation of the entire parish as a whole, the situation was a great deal worse. He thought of the people living to the west of Arbier, in the marsh country. Many of those people would fail to be warned, or refuse to leave their homes for safer shelter. Those people would die a particularly horrible death.\n\n* * *\n\nIn Linhart, Deputy Sergeant Marty Theriot was just waiting for the day's routine work to progress from the In box to the Out box. Every day he had a huge amount of paperwork, and every day he put off facing it. On top of everything, there was a hurricane in the Gulf. Felicia did not impress Deputy Theriot very much. Linhart was far enough inland to keep it safe from the dangers of the ocean. Every year, there was at least one hurricane in the Gulf, but it was only infrequently that they came near enough to the Louisiana coastline to do more than drop a little rain. Deputy Theriot decided that there was plenty of time to worry later, when the hurricane showed that it intended to land in St. Didier Parish.\n\nMeanwhile, there was at least three hours of paperwork to be done. He looked at it for a long time; for some reason, today it was worse than usual. He fiddled with the reports for fifteen minutes more, accomplishing nothing. Then he called one of the other officers over and assigned that man to do the day's paperwork. \"Sheriff Boshardt will be here in a little while,\" said Theriot. \"He's going to want to meet with us about this hurricane thing. We're all going to be on standby status in uniform on twelve-hour shifts. So before he gets here, I'm going to secure the people along the old parish road. A lot of those people probably don't know about this thing.\"\n\n\"Right,\" said the officer, frowning. He knew well enough that Theriot was just dumping extra work on him.\n\n\"I'll cover the road down to the causeway. I expect the Arbier office will take care of the rest of the route. In any event, I'll be back before the sheriff gets here.\"\n\n\"You damn well better,\" said the officer.\n\n\"You talk like that to me again,\" said Theriot, \"and I'll have you out in the swamp helping alligators cross the road.\"\n\nHe turned and walked toward the door. He didn't see the gesture the officer made. That was all right, though, because he imagined it well enough. It made him smile.\n\nHe took a patrol car and started out along the old parish road. His first stop was going to be Henriette's Lounge, about a half mile before the community of Capita. He pulled into the strip of parking lot and got out of the car. Just as he slammed the door shut, he heard a call come over the radio. \"R Six to R, can you give me a 10-42?\"\n\nTheriot reached in and picked up his mike. He answered before anyone at headquarters in Linhart could reply. He wanted to show them that he was really at work. \"Robert Six, this is Robert Eight,\" he said. \"Where are you?\"\n\n\"On the north fork to Delochitaches,\" was the answer from R Six.\n\n\"Try Pichon's,\" said Theriot. \"It's about halfway between Delochitaches and Linhart.\" A 10-42 was a query about restroom facilities.\n\n\"Pichon's, got it. Know the place. Thanks.\"\n\n\"10-4,\" said Theriot. He replaced the mike and went into Henriette's Lounge.\n\n\"Allo, Monsieur Theriot,\" said the short, fat woman behind the bar. \"I hope you don't brought yourself here for no troublement.\"\n\n\"Not this time, Henriette,\" said Theriot, taking off his hat and putting it beside him on the bar. \"Give me a Screwturkey.\"\n\n\"I don't fix that drink some at all,\" she said. \"What that is?\"\n\n\"You call yourself a barmaid?\" said Theriot in mock astonishment. \"It's a drink I invented myself. Wild Turkey and orange juice.\"\n\n\"I think I be sick, me, for true,\" said Henriette. She made the drink and gave it to Theriot. He started to pay for it, but she wouldn't accept the money.\n\n\"Thanks, Henriette,\" he said. \"I'm not drinking on duty this way.\"\n\n\"How you figure that, _hein_?\"\n\n\"I don't know,\" said the deputy. \"I'll figure it some other time. You know about the hurricane?\"\n\nHenriette shrugged. \"Don't they another thing on the radio all day?\"\n\nTheriot took the drink and went to the single pinball machine at the back of the bar. There were two things that Marty Theriot loved more than anything else. One was busting someone's ass for speeding, and the other was playing pinball. Theriot thought that he was probably the best pinball player in Louisiana.\n\nHe put a quarter in the machine. On the back glass were the words \"Hi-Lo Express\" in bright colors. Theriot played his first ball. He would have three, and he needed seventy-three thousand points to win a free game. He got sixty-one thousand with the first ball. He turned the machine over, past one hundred thousand points, on the second ball. He had won two free games. With his third ball he ran his score to one hundred sixty-four thousand points. He won four free games, and another when the last two digits of his score matched the numbers that appeared on the back glass when the game was over. Theriot grinned. \"You owe me a buck and a quarter, Henriette.\"\n\nThe fat woman shook her head. \"I give you them drink\u2014\"\n\n\"All right, all right,\" said Theriot, \"forget about it. Thanks for the drink.\"\n\n\"Sometime, Monsieur Theriot, you put me mad. Mad-mad.\"\n\n\"Just so you know about the hurricane. Keep listening to your radio.\"\n\n\" _Bonjour,_ Monsieur Theriot.\"\n\nHe put his hat on and left the bar. The bright sunlight made him squint, even after he put on his sunglasses. He got in the car, and the seat covers were hot. He swore softly. He started the car and headed toward Marie & Pal's.\n\n* * *\n\nAt eleven-thirty, Channel Five started running an old episode of \"My Little Margie.\" Darlaine Boshardt got up quickly and changed channels. She found another game show where people were subjected to the grossest humiliations in order to win plastic bowls to keep leftovers in, and another soap opera where a good wife was falsely accused of having an affair with the lawyer husband of a woman having an abortion while their slightly retarded son was being enticed into the automobile of a strange man. Darlaine sighed heavily. She turned the television off and began pacing the living room. \"My God,\" she muttered, \"I'm pacing.\" She lit a cigarette. She gave a short, humorless laugh. She thought, Now I'm lighting a cigarette. What comes next? The hidden bottle of liquor in the closet? She stubbed out the cigarette and went outside. At least there was a world out there.\n\nIt was a hot, humid, bright world. The crepe myrtles on the tree lawn were kind of pretty, much nicer than the bamboo and banana plant stockade she had around her house. She walked a short distance and was met by a neighbor.\n\n\"Allo, Madame Boxar,\" said the woman.\n\n\"Say, Mrs. Lefort, how's it going?\"\n\nMrs. Lefort was confused. _\"Pardon?\"_ she said.\n\n_\"C'est rien,\"_ said Darlaine, shaking her head. She lit another cigarette, from a pack she carried in a pocket of her blouse. She didn't offer one to Mrs. Lefort.\n\n\"Don't that a shame about how you call hurr'cane?\" asked Mrs. Lefort.\n\n\"I'm not worried yet,\" said Darlaine. \"When I see them dragging the shrimp boats up Ridge Street, then I'll start to worry.\"\n\nMrs. Lefort laughed. \"You fooling with me, _hein_?\"\n\nDarlaine said nothing.\n\n\"I got to brought myself home now,\" said Mrs. Lefort. \"I got the jambalaya for Fran\u00e7ois, and then I make _m\u00e9nage,_ I. . . .\" Her voice trailed off and she looked frustrated as she searched for the English word \"housework.\"\n\n\"You what?\" asked Darlaine.\n\n_\"Merde alors!\"_ muttered Mrs. Lefort, angered by Darlaine's attitude. According to Madame Lefort's way of thinking, Arbier was a town of French-speaking people, and she shouldn't have to struggle to get an idea across to a foreigner. She turned and walked hurriedly into her house.\n\nDarlaine was left standing alone on the sidewalk. She noticed the roses growing outside Mrs. Lefort's house. They were a deep, beautiful red. Darlaine looked the other way, toward Ridge Street. There was nothing exciting to see there. She flicked the cigarette into the street and turned back to her own house. So choke on it, Madame Lefort, she thought.\n\n* * *\n\nPaul Pierson tried to imagine Maddie Gargotier as a Las Vegas show girl. Even with the visual aid of Maddie walking very slowly and very stiffly around the living room, he couldn't quite make the image. \"No,\" he said.\n\n\"No, and why not?\" she asked.\n\n\"What happens when some bigshot gambler, some guy who's just lost thirty thousand to the house, tells the pit boss he wants you for the night?\"\n\nMaddie turned to look haughtily at Pierson. \"Then,\" she said coldly, \"I give myself to him. For a great deal of money.\" Her eyes grew wider. \"And I would bring Betsy and Shelley with me, and we would become the most famous, the most desired, the most\u2014\"\n\n\"Worn out,\" said Pierson.\n\n\"A Gargotier never wears out,\" she said. There was a silence in the room while both people considered the lunacy of that remark.\n\nCyrus ran across the room and hurled himself against the screen door. \"Hey, _minou,_ \" said Maddie, walking over and picking up a thrashing, squirming Persian cat, \"did you miss your Maddie?\"\n\nCy twisted from her grasp and threw himself against the screen again. \"Apparently not,\" said Pierson.\n\n\"You don't know everything, _Monsieur_ Pierson,\" she said. \"Try and act that way with me when I am the most beautiful and most glorious show girl in the whole world. People will come from miles around just to see me step outside my house. I will have men, you don't know, I will have men, they will come all the way from Lafayette, from _New Orleans_!\"\n\n\"If you get to be really famous,\" said Pierson, \"maybe you'll have fans as far away as Biloxi.\"\n\n\"Don't laugh at me, Paul. If I decide to do a thing, I shall do it.\"\n\n\"We're all aware of that,\" said Pierson. He clicked on the television and slumped down on the couch. The program was a game show, but what attracted his attention was the caption running across the bottom of the screen. \"Oh, boy,\" he said.\n\n\"What is it?\" asked Maddie, turning to look at the television. She read the news of Hurricane Felicia. She raised both hands to her face, covering her eyes. \"Oh, Paul!\" she cried.\n\nPaul got up and went over to her. He put his arm around her. \"It's all right, Maddie,\" he said. \"A Gargotier doesn't go to pieces, just because of a thing like a hurricane. Besides, it's a long way away, and it might not even affect us.\"\n\n\"No, no,\" she said, \"I'm thinking about Mexico, where Felicia is. Those poor people! The dogs, the monkeys!\"\n\n\"It's all right, Maddie,\" said Pierson, a little bewildered. \"We'll get Mexico some more monkeys.\"\n\n* * *\n\nIn the Crisis Caf\u00e9, Sheriff Boshardt had finished his fried catfish and was drinking his coffee. For some reason he was putting off the drive to Linhart. Right after the coffee, though, he was going to leave and get in the car and get on Hanson Highway north. . . .\n\nHis thoughts trailed off and were cut off sharply by the voice of the Channel Five weatherman. He stood before a large map of the Gulf area. There was a little marker shaped like a circle with two spiral arms; the marker was placed against the Yucat\u00e1n Peninsula.\n\n\"The latest report we have from the New Orleans office is that Hurricane Felicia is still right here.\" He pointed to the marker. \"She may move at any time, but she appears to be stalled against the Mexican coast. Those coastal communities are being inundated with rain and sea water pushed by the hurricane's winds. So far, Hurricane Felicia has posed no threat to the coast of the United States. Updated records of her position will be made regularly, and we will break into our regularly scheduled programming to tell you if and when Felicia moves. Keep tuned to this station for all the available news of the storm.\" Skip Strahan's face and voice disappeared, and were replaced with the regularly scheduled game show.\n\n\"I don't know why they have to make everybody worry so much,\" said Lauren.\n\n\"You're too young to remember Hurricane Audrey, in 1957,\" said the sheriff. \"When it was done, that hurricane probably destroyed the livelihood of every person in the parish. It destroyed the entire cane crop, just about. It ruined many shrimp boats. The farmland as far north as Linhart was covered with seawater. People died, girl. I could tell you stories that are so ugly, you'd go hide now and not wait for the hurricane.\"\n\n\"That was in 1957,\" she said lightly. \"We've got satellite pictures of her now. We'll have hours of warning. That's just what I was saying. There's no need to get everybody all anxious and wrought up over a hurricane that may not come within hundreds of miles of here.\"\n\n\"We'll feel it,\" said the sheriff with finality. \"One way or the other, we'll feel it.\"\n\n\"You feel it in your bones, eh?\" said Lauren.\n\nThe sheriff looked up at her sadly. He shook his head. \"You don't know what it feels like to be responsible for the well-being of twenty thousand people. If even one person, one crazy old country Cajun, dies, it will be my responsibility. Me and my men have to see that everyone is safe, and most people don't realize how unsafe they are.\"\n\n\"A hurricane is just a storm,\" said Lauren insistently. \"It's just a big storm.\"\n\n\"That's exactly the kind of thinking that kills people,\" he said. \"That's the kind of thinking that causes people to die for their stubbornness. If we have to go into a Hurricane Watch, which means that the storm is coming right on us, the sheriff's office is going to have a tough job on its hands.\"\n\n\"You're up to it, Sheriff Boshardt,\" said Lauren. \"That's why you've been sheriff almost as long as I've been alive.\"\n\n\"I don't know,\" said the sheriff. \"Every time there's a Hurricane Watch, we go out and try to persuade people to move to safer places. But those people who live in the marshes are sure that they're safe enough where they are. The bulletins will say, 'People living in low-lying areas should seek safer shelter.' How do those people know that most of the parish is a low-lying area? They think those poor old shacks built up on stilts, on those damned little mounds of earth, are high enough. You can't force people to leave their homes. You can't force people living in a trailer park to believe that a hurricane can pick up a house trailer and throw it a hundred yards. People just don't realize how terrible a hurricane is until it's too late, and then they're dead.\"\n\n\"It's your job to worry,\" said Lauren. \"It's my job to wait on customers. I'll do that, and if you tell me to run, I'll run. But I won't get worked up for nothing.\"\n\n\"All right, Lauren,\" said Boshardt wearily. He went to finish his cup of coffee and discovered that he already had.\n\n\"Want some more?\" asked Lauren.\n\n\"No,\" said the sheriff, \"I have work to do.\"\n**13**\n\nAt noon, Skip Strahan received news from New Orleans that Hurricane Felicia was moving. He plotted the new coordinates and frowned. He knew that a hurricane might make any number of twists and turns and loops before it died. But there were several paths which had recurred often enough so that Strahan, as a meteorologist, had cause for concern. Felicia had crossed part of the Yucat\u00e1n and was now over the Gulf again, north and west of her previous position.\n\nInstead of weakening when she crossed the land, she seemed to be growing in strength now that she was once more over water. Strahan hoped that Felicia didn't recurve as many hurricanes did in similar circumstances. Strahan foresaw Felicia's path: the hurricane might move in a great arc north and west, and turn east, hitting the coast of the United States in Texas or Louisiana. He hoped he was wrong. The trouble was, it was almost impossible for the hurricane to get out of the Gulf without doing immense damage somewhere. It was almost absurd to think that Felicia would thread her way eastward, among the islands of the Caribbean, out safely into the Atlantic, and there die. No, somebody was going to have to take it on the chin from Felicia. As a meteorologist, Strahan knew that they could only wait and see. Privately, he had a growing feeling of fear and vulnerability. They were safe enough in Linhart, safe from some of the storm's effects. But nowhere was completely safe. Weathermen, trying to impress audiences with statistics, liked to say that the energy released by a hurricane was equivalent to thousands of nuclear bombs, that the amount of water lifted and dropped as rain measured in the billions of gallons. Skip avoided that kind of hysterical reporting. The hurricane was enough drama as it was. He didn't have to pump excitement into the situation. He had a captive audience, as long as the hurricane was a danger.\n\nAlmost everyone in the parish was depending on him to relay the information he received from New Orleans. He didn't like to be in so prominent a position. His stomach hurt, and sometimes he felt like he was going to vomit. He took a couple of tranquilizers. Then he told the station manager that he had prepared a new bulletin. Skip would go on the air with it in a few minutes. Skip took another couple tranquilizers. He straightened his tie and checked his hair. Someone touched up his makeup. He was ready to give the news to the people of St. Didier Parish.\n\n* * *\n\nChuck met Sheriff Boshardt outside the Crisis Caf\u00e9. \"Sheriff Boshardt?\" asked Chuck. \"I was wondering if I could talk to you for a minute.\"\n\n\"Not right now,\" said Boshardt. He had just heard Skip Strahan say that Hurricane Felicia had rebounded off the Yucat\u00e1n Peninsula and was heading north and west. \"I've got to get up to Linhart.\"\n\n\"I just want a minute, Sheriff.\"\n\nBoshardt gritted his teeth. He took a deep breath. \"All right,\" he said at last.\n\n\"I'm new in Arbier,\" said Chuck. He pronounced it wrong. \"I'm staying at the Sea-Ray Motel.\"\n\n\"I hear they have a real nice place there,\" said Boshardt. He really wanted to get rid of this man and drive up to Linhart. He had work to do.\n\n\"What the thing is, is that I'm an advance man for a major Hollywood film company. We're planning to do quite a bit of filming around Arbier and St. Didier Parish, and it's my job to clear everything with you.\"\n\nSheriff Boshardt waved at the man impatiently. \"Come on by the office. There's someone there who can help you.\"\n\n\"Well,\" said Chuck, in a confidential manner, \"I've told people that I'm here to do a magazine article. The fewer people who know why I'm really here, the better. You've never seen anything like a town with a film company shooting there. Everybody wants to be in the picture. Suddenly we're swamped with Hollywood hopefuls. You can save us a lot of time and money if you'd just let me know what sort of official permits we might need. I'm expecting four large trucks full of equipment, props, costumes, along with the crew. We're going to need a place to park all of that.\"\n\n\"Well, I can understand your problem, of course,\" said the sheriff. \"There are a couple of small formalities you'll have to go through, but there won't be any trouble. I'm very proud of our town here, and the parish. I'm glad that we've been chosen as a location for a film.\"\n\n\"Well,\" said Chuck, \"Arbier best fits what I've been looking for.\"\n\n\"Thanks,\" said the sheriff. \"Now I've got to get going. There's a hurricane in the Gulf, and we probably won't be hit, but we have to take precautions just the same.\"\n\n\"Sure,\" said Chuck, \"thanks a lot.\"\n\n\"I'll talk to you later,\" said Boshardt. \"Meanwhile, go on over to the office. There will be a deputy there who can take care of the paperwork for you. And don't worry about the deputies trying to get into your movie. Right now, they all have more important things to worry about.\"\n\n\"I hope I'm not making any trouble for you,\" said Chuck.\n\nBoshardt just waved and headed for his car. Chuck followed him as though to continue the conversation, but it was obvious that the sheriff didn't want to be bothered any longer. Chuck just shrugged. It was a little after noon. He thought he would go into the caf\u00e9 for lunch.\n\n* * *\n\nBoshardt had hoped that Felicia would spend herself on the Yucat\u00e1n peninsula, perhaps heading straight down, away from the Texas-Louisiana coast. Sure, those poor Mexicans would suffer. They didn't have adequate shelters. They wouldn't have adequate warning for the most part. Perhaps thousands would die, hundreds of thousands would be left homeless.\n\nBoshardt made a clucking sound with his tongue. When it came right down to it, he'd rather that hundreds of thousands of strangers be left destitute than a smaller number in his own parish. He wished it happened to the other guy.\n\nBut wishes don't always come true, he thought. And now Felicia had left the land and was back over water. She would grow now, she would become more dangerous yet. She might speed up. She could cross the Gulf in as short a time as ten or twelve hours.\n\nWalter Boshardt got into his patrol car and headed north on Hanson Highway to Linhart, thirty miles inland from Arbier. The people in Linhart wouldn't have to fear the storm surge, most likely, the giant tidal wave that came along with the hurricane. But if the hurricane crossed into St. Didier Parish, or anywhere within a couple hundred miles to either side, the winds would devastate the city. Little Lauren, at the Crisis Caf\u00e9, shrugged and wondered what was so frightening about a \"big storm.\" Boshardt had seen enough to answer her, but he didn't. He had seen dead women holding dead babies, the corpses flayed and stripped of their skin by sand blown by two hundred mile per hour winds. Sheriff Boshardt had seen plenty, and he didn't ever, ever want to see it again.\n\nThe cane fields north of Arbier, on either side of Bayou Chien Mort, would be ruined by Hurricane Felicia, if that storm determined to turn her attention to St. Didier Parish. There he was again, giving an unhuman thing, a storm, human attributes. Well, he thought, it was true. The spray carried by the winds would ruin the crop, the winds themselves would flatten the growing sugar cane. The bayou would overflow the levees it had built for itself, depositing more silt. But, in the process, it would kill many people, the marsh dwellers, the moss cutters, the trappers who went after muskrat and nutria, the cattle would die, the rice crop would be ruined. Farther inland, where the ground was less marshy, better drained, the winds would still flatten everything growing. There was such a potential for total catastrophe. . . .\n\nThe sheriff drove at a constant fifty-five miles per hour to Linhart. He put the car in his reserved parking place at the sheriff's office, and went into the building. He was greeted by everyone there, but the greetings came from faces that wore anxious expressions. Boshardt just nodded in return.\n\nHe went to the desk where an officer was doing the day's routine paperwork. \"Where's Theriot?\" he asked angrily.\n\nThe officer looked up with a wry expression. \"He said he was going out to warn the people along the old parish road.\"\n\n\"Like hell he is,\" said Boshardt. He went to another desk. There was a radio transmitter and receiver on it. He sat down in front of the microphone. \"Robert One to Robert Eight,\" he said, \"what is your 20?\"\n\nThere was no reply.\n\nBoshardt waited a minute, then tried again. \"Robert One to Robert Eight, what is your 20?\"\n\n\"Robert Eight to Robert One, I'm on the parish road, not quite to the old causeway.\"\n\n\"Well, listen up, Robert Eight. I'm giving you the goddamndest biggest 10-19 you've ever had. You haul your ass back here super-quick. Got it?\"\n\n\"Check, Roger One,\" came the amplified voice of Marty Theriot.\n\n\"All right, then. We got work to do, Theriot.\"\n\n\"10-4,\" said Theriot. He sounded slightly disgusted. There was a click as he turned off his microphone. Boshardt got up from the radio unit. He walked around the offices for a few minutes, overseeing the activities of his deputies. It didn't take very long.\n\n\"Where's Brierrer?\" he asked.\n\n\"He's gone to set up an emergency station at the high school,\" said one of the deputies.\n\n\"Goddamn it,\" said Boshardt, \"he's supposed to be in charge here. He could have sent somebody else. Goddamn it, we got to get together here.\" There was complete silence in the offices as Boshardt stalked around. He stood behind one deputy and watched the man filling out forms.\n\nAfter a few minutes, Deputy Sergeant Marty Theriot came in. He was met by an icy, threatening glare from Boshardt. \"Good afternoon, Deputy,\" said the sheriff.\n\n\"Say, Sheriff,\" said Theriot. He walked up to the desk where the officer was doing Theriot's paperwork. \"Why don't you let me do that?\" he asked. The officer looked up at him. His expression was vaguely contemptuous. He got up from the chair. Theriot sat down and took up where the officer had left off.\n\n\"All right now,\" said Walter Boshardt. \"When Brierrer gets back, tell him he's in charge here. That means he _stays_ here. The rest of you are on twelve-hour shifts. You can work out the schedule any way you want to. We've been through this before. Let's stop acting like a bunch of old ladies in a thunderstorm.\" He turned, still angry, and left the office.\n\n* * *\n\nThere was a moment's silence in the sheriff's office in Linhart after Walter Boshardt went out the door. Theriot was swearing under his breath as he filled in the official reports. The other deputies went about their routine duties, almost as though there was no threat of a hurricane. The normal maintenance of the parish had to continue, unbroken, even through a time of tension. After about ten minutes, though, Theriot threw down his pencil. \"Hey,\" he said to the officer who had done most of the work that morning, \"finish this up.\" The officer gave Theriot a disgusted look, but said nothing. Theriot took a paperback book out of one drawer and went to the men's room. He sat in a stall and read a few pages. The book he was reading was _The Maltese Falcon,_ which his wife had told him about. She had seen the movie on television the day before. Theriot's expression was sour as he read on. If it were up to him, he would have slapped that Sam Spade character in the parish prison before half the books had gone by. He couldn't believe what the local cops were letting him get away with. He ticked off the charges on his fingers. Withholding evidence. Conspiracy to commit theft. Theft. Aiding and abetting a known felon. Accessory before and after the fact to murder. The list grew longer and longer. Theriot shut the book in disgust. He really hated that guy Spade. He was thankful that there wasn't anyone like him in the parish. Oh, there was that idiot Bordinaro right there in Linhart, who dealt mostly in divorce cases, peeping and trailing stuff. But he rarely got in the way of the parish police. He knew that Boshardt wouldn't stand for it. If anything important turned up, Bordinaro usually turned it over to the sheriff's office.\n\nTheriot left the lavatory and went back to his desk, where the officer was doing the paperwork. Theriot tried to open the drawer, to put the paperback away, but the officer wouldn't move. \"Would you mind?\" asked Theriot. The officer reluctantly shoved his chair back. Theriot opened the drawer and tossed the paperback in. \"Thanks a lot,\" he said.\n\n\"Any time,\" said the officer.\n\nTheriot was feeling angry. He didn't like the way the sheriff had talked to him. He thought it was a little early for Boshardt to go into a full-blown panic reaction to this hurricane. The sheriff had embarrassed him in front of the entire department.\n\nTheriot went to the side window and peered through the blinds. To his disgust, he saw that Boshardt hadn't left yet. The sheriff was leaning against his car, drinking a bottle of soda pop. \"Come on already,\" murmured Theriot. At last, Boshardt finished drinking, put the empty bottle in a case beside the machine, and got into his patrol car. In a few seconds, he was pulling out of the driveway and heading north and west toward Delochitaches.\n\n\"Okay,\" said Theriot. He had been put in a bad mood. He had already played enough pinball for the day. There was only one thing left that might restore his equilibrium. \"I'm going out to catch lawbreakers and evildoers,\" he told the office in general. He put on his flat-brimmed hat and went outside. After the air-conditioning in the office, the heat of the day almost made Theriot gasp. He got in his patrol car and drove south along Hanson Highway.\n\n* * *\n\nCorinne Strahan was more than bored. She was frightened. She wished that Skip were home. It wasn't that he was a protective father-figure to her. He wasn't. It was just unfair that she had to face this whole day alone, with a hurricane wandering God only knew where around the Gulf of Mexico. She thought of their house. A chunk of debris, hurled by a wind of a hundred fifty or two hundred miles per hour, could do horrible things. She had heard stories, when she first came to Louisiana, about the awful things that hurricanes had done in the past.\n\nShe was still thinking these thoughts when she became aware that she was dialing the telephone. \"Hello,\" said the operator on the other end. \"Channel Five, your alive station for entertainment, news, weather, and sports.\"\n\n\"Hello, Eileen,\" said Corinne. \"Can I talk to Skip?\"\n\n\"Well, he gave orders that he wasn't taking any calls today. The hurricane, you know.\"\n\n\"I guess so. I just wanted a little reassurance I guess.\"\n\n\"I know,\" said the switchboard operator. \"Let me see if I can get him on the line. Hold on.\" Corinne heard static for a few seconds. Then she heard Skip's voice. He sounded impatient.\n\n\"Okay, Corinne,\" he said, \"everything's okay. Now let me go. I've got to do the latest plot.\"\n\n\"Skip, what about my father?\"\n\n\"Your father will be okay. You can't start panicking yet. We'll have plenty of warning if Felicia heads our way. Let me go. I have work to do.\"\n\n\"Skip, don't go away yet.\"\n\n\"Corinne,\" he said mildly, \"I'm getting all of this information from New Orleans. They're getting it from Miami and Puerto Rico and radar installations all over. So by the time it gets to me, it may be an hour old. I can't take time off to have a nice conversation with you. You know that.\"\n\n\"I know that, Skip,\" said Corinne. She knew that the situation had become serious, because the captions on the television were now being broadcast in both English and French.\n\n\"Okay, I'm going.\"\n\n\"Skip.\"\n\nThere was silence.\n\n* * *\n\nDarlaine Boshardt walked up her street and turned on Ridge Street. She was going to the Crisis Caf\u00e9. The small restaurant was a meeting place of many of the people in town, many of the shrimpers who had just returned from a long time at sea, and oil men from the offshore rigs. She met many of her men in the Crisis Caf\u00e9. She thought it was a shame there wasn't a better way of checking them out first. She was thinking of John Smith. What an idiot he had turned out to be. But Darlaine had to take what she could get, and the selection around Arbier was sometimes very limited.\n\nThere were a few people in the diner having lunch. A man at a table told her that her husband had been in just a little while before. \"Don't that a shame you miss him?\" said the man.\n\n\"Uh huh,\" said Darlaine. She looked around. There were a few men at the tables or in the booths, but no one she would consider picking up. She took a table by herself. She smoked a cigarette almost without realizing that she was doing it. She waited for the damn waitress to come over. Darlaine felt impatient. She knew her husband was out somewhere; she hardly ever saw him anymore. They both kept such odd schedules that they hardly ever met to exchange even brief series of remarks.\n\nLauren came over to the table. \"Good afternoon, Mrs. Boshardt,\" she said. \"Do you want to see a menu?\"\n\nThe door opened and Chuck came in. Darlaine appraised him quickly. Well, she thought, that's this afternoon.\n\n\"Would you like to see a menu, Mrs. Boshardt?\" asked the young waitress again.\n\n\"What?\" said Darlaine. \"No. Yes. Give me a menu.\" Lauren left to get a menu. Darlaine watched Chuck closely. He said hello to the two Cajun men sitting at the table. They nodded and shrugged. Darlaine smiled, knowing what they were thinking. It made Chuck seem so innocent, as he walked among these people who resented his very obvious air of superiority. He took a table not far from Darlaine. Lauren returned with two menus, one for Darlaine and one for Chuck.\n\nLauren stood by Darlaine's table, patiently waiting for her to decide. \"How about the broiled trout amandine,\" Darlaine said.\n\n\"Anything to drink?\"\n\n\"A bottle of Dixie.\"\n\n\"All right, Mrs. Boshardt,\" said Lauren, making a quick notation on her order pad. Then the young waitress went to Chuck's table and stood beside him. \"Ready to order, sir?\"\n\n\"Uh, not just yet,\" he said.\n\n\"Okay,\" said Lauren, \"I'll come back in a minute.\" She went into the kitchen.\n\nDarlaine looked at Chuck. He wasn't a bad-looking man. She remembered the rather acidic conversation she had had with him the day before, at the Sea-Ray. That made her smile again. She realized that Chuck was looking at her, so she held her smile. He smiled in return.\n\nLauren came back and took his order. \"I'll have the chopped sirloin, medium rare, fried onions, and mashed potatoes,\" he said. \"And a bottle of beer.\"\n\n\"Dixie or Falstaff?\"\n\n\"Surprise me,\" he said, and laughed. Lauren looked at him with an expression of mild distaste, and the other people in the restaurant were very aware of his loud, obnoxious behavior.\n\nDarlaine got up and sat at his table. \"Do you mind if I join you?\" she asked. \"Some of these local coonass types can be very unfriendly to strangers.\"\n\n\"Whatever happened to southern hospitality?\" he asked.\n\n\"Appomattox,\" she said. They both laughed. \"No, really, around here they're a little stiff-necked about strangers. They take a while before they admit they like you. But once you're in, you're in for good. You've never seen anything like the way they help each other. It can be a nice town.\"\n\n\"Oh?\" said Chuck. \"I saw you at the motel yesterday. I figured you were just passing through.\"\n\nTerrific, thought Darlaine. This guy is a real winner. \"It's like this,\" she said. \"Some people pass through the town and live at the motel. I live in the town and pass through the motel.\"\n\n\"I see,\" said Chuck, smiling broadly. \"Then maybe you can help me.\"\n\nDarlaine gave him one of her practiced looks and said, \"I'll certainly give it my damndest.\"\n\nChuck didn't seem to notice. \"You see,\" he said, \"I'm the advance man of a production company. We're making a film, and part of it is going to be shot in Arbier. I need someone who can help me find the right locations and the right people.\"\n\n\"I can certainly try,\" said Darlaine.\n\nLauren came out of the kitchen some time later, with both Darlaine's and Chuck's orders. Their conversation had been tentative and suggestive, but neither had made any strong moves. \"Mrs. Boshardt,\" said Lauren, \"are you sitting at this table now?\"\n\n\"Yes, Lauren,\" said Darlaine. Lauren put the plates of food in front of the two people, and poured two bottles of beer into two glasses.\n\n\"Mrs. Boshardt?\" asked Chuck. \"Are you the wife of the\u2014\"\n\n\"Yes,\" she said, waving a hand impatiently. \"But don't let that stop you.\"\n\nChuck finally got the idea and smiled. \"Maybe after lunch we can go back to the Sea-Ray. I can kind of outline what I'm looking for.\"\n\n\"Say, what's it like,\" she said, \"staying there more than an hour at a time?\"\n\n\"It's not great, let me tell you,\" said Chuck with a sour expression. \"I'm trying to sleep, see, and along about three in the morning this rock band from New Jersey, of all places, checks into the motel. So naturally the manager puts them right next door to me. You wouldn't believe the noise. They were throwing beer cans against the wall and shouting. Those walls must be the legal limit for thinness, too. So I go over to the night manager to complain, but he doesn't want to do anything. So I knock on this group's door, and I tell them to cut it down. They give me this story about how they wanted to travel around the country and discover America and all that. And they were smoking grass in their van, getting really stoned, and for some reason got off I-10 from New Orleans and got good and lost. I told them again to please turn it down. They thought they were really big. A rock band. But I shot them down with a real good line.\"\n\n\"What was it?\" asked Darlaine, bored already.\n\n\"I said, 'Yeah, and just about everybody else from New Jersey.' Then I went back to my room, and the noise got louder. I could hear them mocking me out, tossing more beer cans against the wall. About four o'clock, I hear them playing the radio. Some station at some black college. I hear this line, 'If all the women in Texas is as ugly as Sapphire's mother, well, the Lone Ranger goin' to be alone for a long time.' An old 'Amos and Andy' recording.\"\n\n\"What did you do then?\" asked Darlaine, without any interest at all.\n\n\"I went back there, and I knocked on the door, and I told those kids that I was calling the cops. They were gone in twenty minutes. Then the cops gave me a bad time.\"\n\n\"I'll have to talk to my husband about that,\" said Darlaine, \"if I ever see him.\"\n\n\"Well, how's your trout?\" he asked.\n\n\"That sounds gross.\"\n\n\"Anyway,\" he said, waving for the waitress, \"I would really appreciate any help you could give me.\" When Lauren came to the table, he asked for the ketchup. \"By the way, young lady,\" he said to Lauren, \"you might be just right for one of the local walk-ons we'll cast here in town.\" Lauren looked excited by the possibility.\n\n\"We'll have to see, of course. I can't promise you anything. That's up to the director and the producer and the woman in charge of casting.\" Lauren smiled. The chance of being in a movie was the biggest thing that had ever happened to her, even bigger than being Crab Queen the year before. She went back into the kitchen, and Chuck could see that for a long time she would be thinking of nothing but that movie role. He smiled.\n\n\"By the way,\" he said, \"I haven't introduced myself. My name is Robert Branford.\"\n\n\"What?\" asked Darlaine. \"Not 'John Smith'?\"\n\nChuck looked puzzled. \"What?\" he said.\n\n\"Yeah, yeah,\" said Darlaine, lighting a cigarette. \"I forgot. You're already staying at the Sea-Ray. It's been a while since I've strayed from my John Smith.\"\n\n\"I hope it doesn't cause you too much anxiety,\" he said.\n\n\"If you can stand it, honey,\" she said, \"I figure I can, too.\" She drank half the glass of beer in one long gulp.\n**14**\n\nFrom Linhart there was a single blacktop road to Delochitaches. Sheriff Walter Boshardt drove along, suddenly wishing that it was a week later, that everything was over with, that whatever would happen had happened, that there was only the work of dealing with the effects of Hurricane Felicia, whatever they might be. If the hurricane moved west and hit Mexico, or northwest and hit Texas, all St. Didier Parish would experience would be some heavy showers, some winds in the gale range. Or Felicia might loop all the way over to Florida, and the effects on St. Didier Parish would be about the same. But if Felicia made straight for the vulnerable, low-lying coast of Louisiana, he would just as soon get it over with. That was why he felt a sense of relief when Felicia began to move. The suspense of having her stalled against the Yucat\u00e1n was growing almost unbearable. Now, once more over water, Felicia would become more dangerous than ever, and more prone to movement. But at least there wasn't the terrible waiting, hour after hour, and the same report: no movement, stay tuned, wait for further information.\n\nDelochitaches as a community was barely discernible from the part of Louisiana around it that wasn't Delochitaches. It made a town like Arbier seem like a major urban center. In population, Delochitaches was roughly half the size of Arbier. Almost every person in that number was a farmer. The area was rich with alluvial soil, and where the conditions were right, one might see a variety of crops.\n\nThe incorporated town of Delochitaches itself was very, very unimpressive. There was no way to get to the town except by the very bad road Boshardt had taken from Linhart and, from the other direction, a similar road branching off from Hanson Highway. The downtown area of Delochitaches consisted of four shops, one of which doubled as the town's post office, and a gas station. Along the road Boshardt counted at least a dozen lounges in about eight miles. Delochitaches had three of its own, mingled among the four shops and the gas station.\n\nThe people in Delochitaches were different, too. There was a greater mixture of national heritages. There was a good number of Germans and Scandinavians, fewer genuine Cajun French. The attitude and manner of Delochitaches was completely different than Arbier. Linhart seemed to Walter Boshardt to be working at being a compromise between the two smaller communities.\n\nBoshardt pulled into the sheriff's office in Delochitaches. He rarely came in person. More often, he would send someone else. But today the situation was grave enough for Boshardt to deal with it himself.\n\n\"Afternoon, Sheriff,\" said the deputy that worked at a desk at the front of the office.\n\n\"Yeah,\" said Boshardt, pushing through a swinging gate. \"Where's Mike Miller?\"\n\n\"He's here someplace,\" said the deputy.\n\n\"Right here,\" said Miller, coming out of a room in the back of the office.\n\n\"All right,\" said Boshardt, \"we've been through the hurricane routine before. The seven of you are responsible for Delochitaches. You think you can handle it?\"\n\n\"We're not counting on a lot of looting, if that's what you mean,\" said Miller.\n\n\"Just be careful, Captain,\" said Boshardt. \"Some people will take any chance to loot anything. And if the worst happens, if we have to take the full force of that damn hurricane, you have a lot of people spread around a lot of ground. Getting them all to a shelter could take a lot of time and trouble. You won't get any help from Linhart. They'll have enough of their own problems. You're on your own. But if I get word that the hurricane is heading for us, I'll call for the National Guard as soon as I can. Still, you might have to go a while on your own. Get the people as safely sheltered as you can. If the hurricane makes a move for us, pull your cars in some protected place, I don't know, maybe under a railroad trestle or something. Don't endanger yourselves without damn good reason. I don't want any dead heroes. I can do without state funerals. Got it?\"\n\n\"Yeah,\" said Miller, \"I got it.\"\n\n\"Stay in touch with Linhart,\" said the sheriff, \"and, through them, with me.\"\n\n\"Right.\"\n\n\"Okay,\" said Boshardt. \"The next day or two might get real bad, or it might not. Let's just do our best.\"\n\n\"We always do that,\" said one of the deputies.\n\n\"Do you?\" he asked, with a weary expression. \"Do you really?\" Then the sheriff walked quickly out of the office, back into the tiny collection of peeling, cracked single-story buildings that made up downtown Delochitaches. He walked to his car. His first duties in the situation were finished. The next move was up to Felicia.\n\nSheriff Walter Boshardt leaned against his car. He felt like having a drink. It would be easy enough to get one. But he wouldn't go into one of the lounges, in uniform, in an emergency situation. That wouldn't look right. Still, Boshardt kind of wanted that drink. He looked up at the sky. It was bright blue. There wasn't the faintest trace of clouds anywhere. But the sky didn't fool Walter Boshardt. One of the main reasons that he had been elected sheriff for the past sixteen years was that very little fooled him. He didn't take chances, he didn't take bribes, he didn't take drinks while on duty.\n\nFor a very brief moment, Boshardt wondered if it was too late to change.\n\n* * *\n\n\"Skip?\"\n\n\"Look, baby,\" said Strahan, \"if you keep calling me like this, I'm just plain not going to answer.\"\n\n\"Skip,\" said Corinne, \"don't you understand? I'm really afraid.\"\n\nStrahan paused and took a deep breath. His stomach muscles felt tight, and there seemed to be a band around his head, getting tighter and tighter. When he finished talking with Corinne, he planned to take about six Valiums. The palms of his hands were sweating and he felt his arms and legs making tiny convulsive jerks. \"Corinne, how many times do I have to tell you that there isn't a thing in the world to be afraid of? That's right here in Linhart you're probably in the safest place in the entire parish?\"\n\n\"What about Dad?\"\n\n\"Your father lives in a furnished room in a large old house in Arbier. They built those houses years ago, and those houses were built to take a lot of punishment. You can't worry about your father. Look. There's nothing you can do anyway. Besides, the hurricane hasn't shown any sign that it's headed toward us.\"\n\n\"Then it's okay?\"\n\n\"It's okay, baby.\"\n\n\"That's all I wanted to hear, Skip. I'm sorry.\"\n\n\"You've called me about once an hour, and I've had to go through the same routine with you. This is the last time, understand?\"\n\n\"Yes, Skip.\" She sounded chastised. She sounded slightly embarrassed.\n\n\"Then I can get back to work?\"\n\n\"Yes, Skip. I love you.\"\n\n\"I love you, too, baby, but you've got to buck up now.\"\n\n\"All right, Skip. Call me later?\"\n\n\"If I get a chance. See you when my shift ends.\"\n\nCorinne said goodbye and hung up the telephone. She knew that Skip's \"shift\" wouldn't end until the hurricane emergency was over.\n\nShe went back to her reading. She had the television on, it was one o'clock, and _Casablanca_ had just started. She sat in her usual spot on the couch, a paperback copy of _The Day of the Locust_ in her hands. A neighbor had told her about the movie she had seen, made from Nathanael West's novel. Corinne was reading the book, and it was a great emotional experience for her. She identified with the character of Homer Simpson, a poor, wretched, futile man, impotent in every sense of the word. Like Homer Simpson, Corinne encouraged the sadness she felt in her, because the sadness brought a strange kind of pleasure. Homer Simpson's paralyzing sickness was not apparent to her. She only knew that his sadness was a lovely thing; she missed West's point completely. She wished that she could feel that same aching of the soul. It would be like art.\n\nThe narrator of the movie said, \"But the others wait in Casablanca, and wait . . . and wait . . . and wait.\" At the bottom of the screen, the same weather advisory was making its circuit in English and French. Corinne read _The Day of the Locust,_ and something about the book increased her anxiety. She couldn't quite put her finger on the precise quality of the book that made her feel that way. She kept reading. The movie played on the television. Every once in a while she would glance up. Sometimes she would see Humphrey Bogart. Once she saw Peter Lorre, firing a pistol and running, she didn't know why, and then being dragged away. She didn't see him again for the remainder of the movie. She saw Sidney Greenstreet. It was the whole gang from yesterday's movie. The only exception was that there was Ingrid Bergman instead of Mary Astor.\n\nReading the book, she nearly cried over Homer Simpson. How could a writer treat his own creations so cruelly? And Homer Simpson lived through everything that West threw at him, sad, perhaps, but human. Corinne found so little in books and movies that was human. Like the movie that was on now. Humphrey Bogart's character was so cool, so resourceful. He probably didn't even sweat, even though he was on the upper lip of Africa. She read the book more, and watched the movie less.\n\nFor one thing, the book didn't have a bi-lingual warning running across its bottom section.\n\n* * *\n\nIt was a little after one o'clock. A teletype in the news room at Channel Five began chattering out information, weather data from New Orleans. Skip Strahan read it and nodded. The storm had moved a little further northward. Her size was about three hundred miles in diameter, according to the radar measurements. Her winds were up to two hundred fifty miles per hour, at the eye, where they would be strongest. Gale force winds surrounded the hurricane for hundreds of miles. The eye itself was about fourteen miles in diameter. Hurricane Felicia was a tight, circular system. There was absolutely no sign of dissipation.\n\nStrahan tore the paper off the teletype and took it to his desk. There was a discrepancy between the barometric pressure in the New Orleans data and his own readings. Strahan sat back in his chair, wanting like hell to give up. He didn't have the equipment, the staff, or the time to do what he had to do. He had to get the news to some twenty thousand people who were sitting by radios and televisions all across the parish.\n\nStrahan thought about taking another couple of tranquilizers, but he knew that if he did that, he might just pass out. Go right to sleep at his desk. And it would take a hurricane to wake him up.\n\n* * *\n\n\"What are you doing?\" asked Pierson, amazed at the frantic activity of Maddie Gargotier.\n\n\"Don't you listen?\" she asked. \"Don't you read? There's a hurricane, and you're just standing there.\"\n\n\"What am I supposed to do?\" he asked.\n\nShe stared at him. \"What are you supposed to do? See those big glass sliding doors? What happens when a big old garbage can comes smashing through them? _Hein?_ We got to tape them up. You have, uh, what is it?\"\n\n\"Masking tape.\"\n\n\"Yes. Make Xs on the doors, on the glass.\"\n\n\"What's that going to do?\" he asked.\n\n\"It keeps the glass from shattering,\" she said.\n\n\"A garbage can at two hundred miles an hour,\" he said, \"all the masking tape in the world isn't going to keep the glass from breaking.\"\n\n\"No,\" she said, \"but you won't have little pieces flying around the house like a bomb.\"\n\n\"All right, okay. I'll put masking tape on the windows. Then what?\"\n\n\"Then we take many-many pins and pin the drapes together tight.\"\n\n\"So the shattered glass won't go flying around the room, huh?\"\n\n_\"Mais oui,\"_ she said.\n\n\"Two hundred miles per hour, something heavy like lawn furniture or something, and we're protected by masking tape and pins.\"\n\n\"Well,\" she cried, \"what were you planning on doing?\"\n\n\"I was going to take Cy into the dining area, far away from any windows, and listen to a battery-powered radio until the whole thing was over.\"\n\n\"Oh, you Americans!\" she cried in anger. \"Clean out the bathtub.\"\n\n\"What?\"\n\n\"Clean out the bathtub with Lysol or something. Boiling water. Then fill the tub.\"\n\n\"Why?\"\n\n\"Because, you _imb\u00e9cile,_ the water system breaks, and the electric system, she also breaks, and for days we don't have water to drink or wash or flush the toilet.\"\n\n\"I'm sorry, Maddie,\" said Pierson. \"I really don't know what to do. You just tell me, and I'll do it.\"\n\nShe was furious. _\"Idiot!\"_ she screamed. \"That's what I'm doing!\"\n\n\"And I really appreciate it, Maddie, I really do. I can't possibly tell you how much I appreciate this. Really.\"\n\n\"Are you being sarcastic with me, Pierson?\" she asked.\n\n\"Sarcastic-sarcastic,\" said Pierson. Maddie threw a book at him.\n\n* * *\n\nDarlaine finished her meal, drank another bottle of beer, and lit a cigarette. She tapped the ashes onto her dinner plate, even though there was an ashtray provided.\n\nChuck was going on and on about the movie they were going to film in Arbier. \"It's going to be kind of an _On the Waterfront,_ but about the competition between the traditional old shrimpers and the others with their more modern and expensive boats.\"\n\n\"Dagos, we call them. They're Portuguese, mostly. From Florida. They fish further out in the Gulf and take most of the catch.\"\n\n\"Right, right,\" said Chuck excitedly, \"it's an area that's been completely untapped by the film media.\"\n\n\"Medium,\" said Darlaine. She was watching Lauren. She wondered if her husband had ever taken sweet Lauren\u2014where? The Sea-Ray? She laughed out loud. But that Lauren. That's how I used to look, thought Darlaine. \"Crummy waitress,\" she muttered.\n\n\"What?\" asked Chuck.\n\n\"I said, she's a crummy waitress.\"\n\nChuck laughed. \"Want to see a great trick you can play on crummy waiters and waitresses?\" Before Darlaine could get in a frantic \"No!\" he had done it. He pushed back the tablecloth in front of him. Then he filled his water glass to the brim with hers. He dropped a nickel into the full glass, and took a cardboard sign on the table advertising the Crisis Caf\u00e9's Friday Fish Fare. He put the cardboard over the top of the water glass, and turned the whole thing upside-down on the hard surface of the table. Then he slid out the piece of cardboard.\n\n\"See?\" he said proudly. \"There ain't no way in hell she can get that nickel or take that glass away without spilling the water all over everything. Mostly herself, if she's as clumsy as most hicks are.\"\n\n\"Did you really say 'hicks'?\" asked Darlaine.\n\n\"I didn't mean that,\" said Chuck.\n\n\"Yeah,\" said Darlaine, looking at Chuck unpleasantly. \"We're all just good old boys down here, sitting on our cabin steps, playing the banjo and singing spirituals, going to gumbo festivals, living close to the earth.\"\n\n\"Well,\" said Chuck, slightly offended, \"I didn't mean it like that.\"\n\nDarlaine didn't answer. She was watching Lauren as she walked toward them with a tray.\n\n* * *\n\nCorinne Strahan was an hour into _Casablanca._ Humphrey Bogart was getting drunk with Dooley Wilson. \"I bet they're asleep in New York,\" he said. \"I bet they're asleep all over America.\" He showed his drunken frustration and said, \"Of all the gin joints in all the towns all over the world, she walks into mine.\"\n\n\"Of course,\" said Corinne. \"What kind of a movie would it be if Ingrid Bergman had walked into Marie & Pal's? For God's sake!\" She had put her book aside, and was now devoting her whole attention to the movie. It seemed that Ingrid Bergman was in love with both Paul Henreid and Humphrey Bogart. \"I bet she goes off with Claude Rains,\" she said, as she got up and turned off the television.\n\nShe sat back down and lit a cigarette. The decoupage idea had died. The book was making her depressed. The movie was too political or something for her. She sat in her lime green living room and smoked the cigarette. She thought about Skip. She thought about her father. She thought about lime green being a really rotten color for a living room. When she stubbed out the cigarette, she lit another. When they were choosing colors, she had preferred Sahara beige. Skip had said that it would show fingerprints and dirt. Now they had a lime green living room.\n\nCorinne thought about knocking herself out again on Skip's sleeping pills. He had carefully not mentioned anything about her taking them the day before. After all, she was under a lot of stress too.\n\nShe thought about the drugs while she smoked the cigarette. She wished there was something comforting on television. Art Linkletter's \"House Party.\" \"Strike it Rich.\" She remembered those programs from her childhood. She thought about the drugs and decided they were probably a bad idea, although she really liked the tingling in her hands and feet just before she got wobbly. She had cracked her head against something the day before, and it was bruised now. She couldn't remember doing it. She just sat there and smoked.\n\n* * *\n\nSkip Strahan got the news from New Orleans, the news he had half-expected to get all afternoon. Hurricane Felicia had begun to pick up speed and had begun to curve eastward. Her projected path would have her enter the Texas-Louisiana coast somewhere between Galveston, Texas, and Grand Isle, Louisiana. Of course, there was a time factor involved. It was three in the afternoon, Tuesday, and at Felicia's current rate of progress, landfall might not occur until Wednesday evening, around six o'clock. That is, thought Strahan, if the damned storm kept to the same path, the same curve, the same speed. And when had he ever known a hurricane to do that? He prepared a new advisory based on the data from the National Weather Service Hurricane Warning Bureau. This time, however, the Hurricane Warning would become a Hurricane Watch. The warning lights would be lit to alert the fishermen: a vertical column of three lights, red, white, red. That meant hurricane. The warnings flags would go up: two square flags, red with smaller black squares in their centers. Hurricane. It was coming. Skip stood up from his desk, and he suddenly felt so weak that he almost fainted. Maybe some Dexedrine. Maybe not, considering all the downers he'd been taking.\n\nHe wanted to go home, to go away. But he couldn't.\n\nHurricane Felicia was coming.\n**15**\n\n****\n\n_It was in Arbier, Louisiana, not quite forty years before, and it was near the end of President Franklin D. Roosevelt's second term._\n\n_The rain never stopped. Walter Boshardt was five years old. He never would have believed that something like rain would terrify him so much. But the rain never stopped._\n\n_The first day was like a storm, a bad storm. It was such a bad storm that his mother and father ran around the house doing things. Their fine house. The house that Walter Boshardt's father had lived in for some years before he married the woman who was to become Walter's mother._\n\n_At first, Walter enjoyed the rain. It was a hard storm but, living on the coast in semitropical Louisiana, he was used to rain. It seemed that there was a regular schedule that the weather followed, especially during the summer months. Then the temperature would be about ninety degrees, and the variance from this figure was no more than five degrees either way, day or night. The large house that the Boshardts lived in had rooms with very high ceilings, so that the warmer air would rise up and the cooler air sink down. The humidity was also constantly fierce, day after day ranging through eighty percent to ninety percent and even higher. And, of course, there was the daily shower, about one o'clock. Walter, at the age of five, was still curious about this happening. The days began almost identically. In the morning, when he went out to play, the skies would be bright blue and completely free of clouds. The air would smell fresh, with a small bite in it from the waters in the Gulf, only a matter of a mile or so away. Walter would play with his friends until lunchtime, when his mother would call him back to the house. He would eat\u2014almost always a bowl of tomato soup, a sandwich, and a few cookies, all washed down with milk laced with a small amount of coffee. After lunch, the bright blue sky would begin to collect cumulus clouds. With alarming speed, for the young boy, black thunderheads would pile up above the town. They looked like bad things, sent by a malevolent force, a monster perhaps. In any event, after about fifteen minutes or half an hour, the black thunderheads would begin to drop their burden of rain. There would be a brief summer shower. If Walter went outside, as he often did, he could look across the flatness of the land out into the Gulf itself; the rain would look like a curtain. His mother would sometimes call him inside again, because all around them glows of lightning made immediate, almost artificial displays that were at once attractive and threatening._\n\n_On this particular day, when Walter was five years old, the rain began early in the day. His parents were worried, and he didn't know why. There was a peculiar and incessant rushing sound coming from the palm trees around the house and the broad-leaved banana plants. The sound itself was enough to make Walter uncomfortable. He wished that it would go away. But, like the rain, the constant roar of the wind through the wide leaves of the pliable plants didn't seem ever to stop._\n\n_It rained, and the wind blew. At first it seemed interesting to Walter. He had never experienced anything like this before. The rain was so hard that when he looked out of one of the windows, he couldn't see across the street. There was a mask of rain._\n\n_\"Can I go out in it, Mommy?\" he asked._\n\n_His mother was behaving in a nervous, frantic way that Walter had never seen in her before. \"What?\" she asked. \"For God's sake, no!\" Walter was disappointed._\n\n_\"You know something, Francy?\" asked his father. \"I better go get the car filled up. If the power goes off, there won't be any way of gassing up the car.\"_\n\n_Walter's mother seemed distracted. \"I guess,\" she said, \"but I hate to see you go out in that. The streets are already flooded.\"_\n\n_\"Oh, it won't be bad,\" said Walter's father. \"I have to do it.\"_\n\n_\"All right,\" said Walter's mother. \"But be careful and come straight back.\"_\n\n_\"You sound like you're talking to the kid.\" Walter's mother was too busy with other things to reply._\n\n_\"Can I come, Daddy?\" asked Walter._\n\n_\"No, son, you stay here.\"_\n\n_\"But why?\"_\n\n_\"Because it's safer.\"_\n\n_Walter pouted and started to scream. \"It's only rain,\" he shouted. \"It's just a lot of rain.\"_\n\n_\"I don't know about that,\" said Walter's father, but Walter didn't know what he meant._\n\n_\"Walter,\" said the boy's mother, addressing his father, \"You know, I think perhaps we ought to move away from here.\"_\n\n_\"Oh, you're just thinking up nightmares again,\" said Walter's father._\n\n_\"No, I'm not,\" said Walter's mother. \"When I was about, oh, five years old, there was this hurricane that swamped Cameron Parish so bad, it looked like the people there had all been killed and that it would take a miracle to put the place to rights again. And then, when I was thirteen, there was another hurricane east of us that killed a couple of dozen people or more and did millions of dollars of damage. I'll be afraid if you go.\"_\n\n_Walter's father was getting impatient. \"I told you why I have to go. We'll need supplies. The car needs gas. I'll have to go put the lawn furniture and the kid's toys in the garage. You're starting to panic. That's the worst thing that can happen to you now. Just keeping saying to yourself, 'It's just raining hard.'\"_\n\n_\"Then why are you taking all of these hurricane precautions,\" asked Walter's mother._\n\n_\"Fran\u00e7oise,\" he said, \"there's such a thing as good, constructive behavior. We might as well prepare for the worst.\"_\n\n_There was a silence for several seconds. Then Walter's mother spoke. \"I can assume, then,\" she said, \"you've decided to ride this thing out in this house.\"_\n\n_\"Of course, of course,\" said Walter, Sr. \"What else is there to do? Take little parish roads all the way up to your folks' place in Lafayette? Those roads are probably tiny rivers by now.\"_\n\n_\"That's just what I mean, Walter. You can't go out.\"_\n\n_\"I_ have _to go out,\" said Walter's father. Without a further word, Walter's father began putting on his heavy rain gear._\n\n_\"Can I come, too, Daddy?\"_\n\n_The man was silent for a bit, then said, \"Put on your raincoat.\"_\n\n_\"Oh,_ mon Dieu, _no!\" cried the boy's mother. \"There's no reason to take him along with you. It's dangerous out there.\"_\n\n_\"No, it isn't,\" said the man. \"It's just raining hard.\"_\n\n_\"The radio said that the barometer has dropped a lot,\" she said. \"It said we should prepare for a possible hurricane.\"_\n\n_\"There's nothing we can do except what we've already done, and a few odd precautions. I'll take the boy just down to get gas. I don't want him frightened by this heavy rain.\"_\n\n_\"I'm not scared, Daddy,\" said young Walter Boshardt. No one paid him any attention._\n\n_\"All right,\" said his mother, heaving a great sigh. \"I'll finish putting up water, just in case. While you're out, see if you can get some canned food. This might really be a hurricane coming.\"_\n\n_\"A bad storm,\" said the elder Walter Boshardt._\n\n_Walter and his father went out to the garage. It was difficult fighting against the wind and the rain. The water stung his face. He had never experienced anything like it. He held his father's hand all the way from the house to the garage. It was a short distance, but the water was coming down so hard the garage was almost invisible. In later days they would learn that they had, in fact, been touched by the edge of a hurricane. The rain was particularly hard in southern Louisiana, so the storm must have stalled for quite a long time against the coast. The rain did not stop for four days. In that time, the city of Lafayette reported a total of almost twenty-seven inches of rain, almost twenty inches in one day._\n\n_The journey to the gasoline station was slow and careful, because visibility was almost non-existent. Walter's father cursed, something the boy rarely heard. When they got to the gasoline station, there was no one there. There was a sign that said \"Sorry. No gas.\" Walter's father cursed some more. They stopped at a small grocery store and bought enough canned food to last a few days. There wasn't much left to choose from._\n\n_On the way home, an overhead powerline had broken, and was swinging on the ground, spitting sparks. It looked to Walter like a huge, awful, black snake that would kill anyone who came near it. He knew nothing of electricity. He just knew that the powerline was something to stay away from._\n\n_\"I hope they get that fixed soon,\" muttered Walter's father. \"Some damn fool is going to try to touch it, and he's going to get the last surprise of his life.\"_\n\n_\"I want to go home, Daddy,\" said Walter._\n\n_\"All right, son,\" said his father. \"There's nothing more we can do out here. We close the shutters, bring out the candles, and open a window on the lee side of the house, just in case. Hysterics aren't going to get us anywhere. I wish your mother understood that. Then I suppose I have to go down and secure the boat. I'll run it up one of the bayous as far as I can and anchor it and tie it down. Then I put the car in the garage, come inside, and wait for the rain to go away.\"_\n\n_Walter's father had been musing to himself. Walter was turned around in the seat, looking through the back window, through an almost opaque wall of rain, at the snapping, fizzling powerline. He would dream of it that night. He would dream of it whenever he got into trouble, or whenever his parents had minor arguments that disturbed him._\n\n_After a block or so, he could no longer see the black snake of a cable. He turned around and was quiet. His father had stopped speaking aloud to himself, concentrating on driving through the fierce rain. They arrived safely at home, and neither of them had said another word._\n\n_Walter went inside the house. His mother was still doing frantic things. His father immediately left again to go to his shrimp boat. Walter sat in his room, unable to see anything through the other side of the window. But he could hear the most terrifying sound, a howl, a rumble, a constant and powerful din caused by the wind through the sturdy plants around the house and by the horrible quantity of rain that fell in so brief a time and by the thunder that accompanied it all._\n\n_It was something he would never forget. It was something he never learned to get used to._\n**FOUR**\n\n****\n\n**Felicia**\n**16**\n\nDeputy Sergeant Marty Theriot was sitting in his patrol car, on Couletain Boulevard facing Hanson Highway. He was a little disappointed. There wasn't as much traffic as he had expected, and his speed trap hadn't netted a thing. But, like any fisher or trapper knows, the most important quality is patience. Patience was something that Marty Theriot had a huge supply of. He could sit and watch traffic on Hanson Highway for hours, if he had to. He had been known to spend the entire day doing it.\n\nHe was surrounded by farmland. To his right there was a stretch of land leading down to Bayou Chien Mort. It had once been swampland, but now it had been partially drained and was cultivated with soybeans. Across from Theriot and to his left were great rectangular patches where, according to the soil conditions, sugar cane, corn, or yams were growing.\n\nMarty Theriot put down the book he was reading. It was titled _Pet Lovers,_ but one look at the illustration on the cover or a glance at the opening pages would prove that it wasn't a book for people who raised tropical fish or went to cat shows. Theriot put the book on the seat beside him, and stared. Across Hanson Highway\u2014the \"Coonass Boulevard\" as it was known throughout the parish\u2014was a field of yams. \"Go yams!\" muttered Theriot. He remembered how he used to take girls out to watch \"the yam races.\" Always late at night, always in out-of-the-way places. There never was much yam-watching done, even though Marty promised the girls that such famous yams as Citation, Man o' War, and Secretariat were running. \"Hey, yams,\" said Theriot. \"What's that, yam?\" He now saw the yam as a dog. \"I think the yam is trying to tell me something.\" And the yam would bark several times, skipping about impatiently. \"Yeah,\" said Theriot, \"'Fury, the story of a yam and the boy who loved him.'\" The yam was a horse again.\n\nAt that moment a semi passed him at about seventy miles per hour. \"Holy jump up and sit down,\" said Theriot, \"I got me one.\" He started after the eighteen-wheeler with siren screaming and the blue lights on top of his patrol car flashing. He overtook the truck and signaled for it to pull over to the side of the road. He waited until the satisfied grin left his face before he got out of the car. Then he adjusted his flat-brimmed hat and his sunglasses and walked up to the cab of the truck.\n\n* * *\n\nThe telephone rang. The noise in the quiet room startled Corinne. She was a little frightened, for no reason that she could think of. She picked up the receiver and said, \"Hello?\"\n\n\"Hello, Corinne?\"\n\n\"Yes.\"\n\n\"Uh, Corinne, this is Carl Steinbrenner.\"\n\n\"Oh, the used-baby man,\" said Corinne.\n\n\"Ha, ha,\" came the reply. There wasn't a touch of humor in Steinbrenner's voice. \"I'm just calling about yesterday. I was very sorry when you left. I was very disappointed.\"\n\n\"Oh, I'm sorry,\" said Corinne. She had hoped that she would never hear from the man again. \"I guess I owe you an apology,\" she said.\n\n\"Well,\" said Steinbrenner's voice, \"I can understand your position. It's just that, like I said, I was a little disappointed.\"\n\nCorinne paused. She knew what went on down at the Sea-Ray. If Steinbrenner had been so disappointed, no doubt the desk clerk advised him of some young woman at the motel who was receiving visitors.\n\n\"Well, I'm sorry,\" she said.\n\n\"Okay, let's forget about Monday. Let's talk about today.\"\n\n\"There isn't a whole lot to talk about,\" said Corinne.\n\n\"I'd like to see you again,\" said Steinbrenner.\n\nCorinne thought that he must be really stupid not to pick up the gigantic hints she was dropping. \"I'm very busy today,\" she said. \"Being the wife of a television personality, I have a lot of important things to do. You know.\"\n\n\"Yes, I'm sure you do. But I was thinking that you could give me just an hour or two of your time today. I didn't call you earlier because I thought you might need time to think things over.\"\n\nWhat things? wondered Corinne. Whether or not she would go to bed with Carl Steinbrenner, who dealt in unwanted babies. Corinne had never been to bed with anyone but Skip. That was what she had to think over. She did it very quickly. The book, the television, the craft kits, Skip's distance, the hours. . . . \"I'll meet you there,\" she said. Her voice was a hoarse whisper.\n\n\"What time?\"\n\n\"An hour from now. Is that all right?\" she asked.\n\n\"Fine,\" he said.\n\n\"You know my car?\"\n\n\"Yes,\" said Steinbrenner.\n\n\"I'll wait in it. You get the room.\"\n\n\"Fine,\" he said.\n\n* * *\n\nA telephone rang in the bedroom of an old house in Mobile, Alabama. A man hurried to answer. \"Hello?\" he said.\n\n\"Sure,\" said another man.\n\n\"Tom?\"\n\n\"Yeah,\" said Tom. \"I've been trying to get you for an hour or so.\"\n\n\"Sorry,\" said the man in Alabama.\n\n\"That's all I hear from you guys, Denny, Sorry. Yeah, you're the sorriest lot I've ever worked with.\"\n\n\"What's up?\" asked Denny. \"I wanted to go already. The other guys are getting restless. Truckers, you know. They can spend only so much time drinking beer and playing pinball.\"\n\n\"So go.\"\n\n\"To Arbier?\" asked Denny.\n\n\"I don't understand,\" said Tom. \"Why is it that the guys I have in the gang are all such losers? I give an order, and everybody obeys. But, God, I really get sick thinking about the way you guys obey. It's like improv crime. It's like something out of the 1960s. You turn every job into a happening. It's not fun for me, believe it.\"\n\n\"I really sympathize,\" said Denny. \"I can just picture what a hellish nightmare it must be for you. You hire us, you tell us where to go, who to meet, what to do. Then you say don't go. Then you say go.\"\n\n\"Leave all that to me. I have today's New Orleans _Times-Picayune._ They're giving Hurricane Felicia big play. The idiot in Louisiana seems to be getting his act together.\"\n\n\"Chuck?\" asked Denny. \"I don't believe it. I'd have thought he'd fallen into the Gulf by now.\"\n\n\"Knowing Chuck,\" said Tom, \"he will. But he knows the job comes first. Any accidental suicides will have to come later. Good man, that Chuck.\"\n\n\"Here's to Chuck,\" said Denny. \"Right. So we can go now?\"\n\n\"You heard me,\" said Tom. \"I've been following the Louisiana situation in my usual manner, and I say go. I've got this timed exactly, and only God or Chuck could screw this up now.\"\n\n\"All right, Tom,\" said Denny. \"I'll get the guys together, we'll mount up and strike the pavement. Hit the road. On to Louisiana and a life of luxury. Juleps on the verandah. Darky women clinging to me, because I'se so good. We'll get there as soon as we can.\"\n\n\"Make it sooner,\" said Tom. \"According to the New Orleans paper, Felicia is curving on a path for the Louisiana coast, and I don't want you too late, or you'll get caught up in the evacuation traffic. Get there soon.\"\n\n\"That's what the boys like to hear,\" said Denny. \"The hammer will be down all the way.\"\n\n\"What?\" said Tom.\n\n\"We're going to get there, don't worry,\" said Denny.\n\n\"Everybody says don't worry. When I hear that, I start to worry.\"\n\n\"So worry,\" said Denny.\n\n\"No,\" said Tom, \"you age too fast in this business. Either that, or you don't age at all. I like what the newspapers are saying. Everything's starting to fall in place.\"\n\n* * *\n\nSheriff Walter Boshardt was a little bewildered. He had been reluctant to leave Arbier, to make the trip to Linhart and Delochitaches. Now that he had done it, he was reluctant to go home. Why? He couldn't answer. It was getting to be late afternoon, Tuesday. The hurricane might not make any kind of disturbance for twenty-four hours. Perhaps his mind was trying to tell him to relax, that his resources would be needed later, that it was unwise to waste them now.\n\nWhatever the reason, Boshardt sat in his patrol car and thought. His first thoughts were about his wife. That made him sadder. Dorothy\u2014he never thought of her as Darlaine\u2014didn't belong in a town like Arbier. She was right about that. She belonged in a city, a place where she could be among society people. He understood her frustration, her hostility. She had said, when they were married, that she was willing to sacrifice that social life to be his wife. But it was a rash promise, and one she first regretted and then forgot.\n\nThere didn't seem to be a solution. Boshardt shook his head. Even the idea of discussing the matter was impossible. There were only two choices. Either they left Arbier for a large city like New Orleans or Atlanta, so Darlaine could immerse herself in things that the sheriff could barely imagine, or their life went on as it was. The latter seemed to be the more probable, even though it made the future look a trifle bleak.\n\nA call came through on the sheriff's two-way radio. \"Robert Two, this is Robert One.\" Boshardt picked up his microphone and flicked the switch.\n\n\"Robert One, this is Robert Two,\" he said.\n\n\"Great to hear from you, Sheriff. Just thought you might want to know that the Hurricane Warning is now a Hurricane Watch for the area between Galveston and Grand Isle for the next twenty-four hours.\"\n\nBoshardt felt the same shiver of fear that he had tried to describe to Lauren in the diner. \"Thanks for the good news, Robert One. Terrific. 10-4 and out.\" He switched his mike off and put it back. He headed back to Arbier. He ran a hand through his short, blond hair. He had a very unpleasant feeling, a sense of something waiting for him, coming to meet him. And her name was Felicia.\n\n* * *\n\n\"You want to know how bad things are?\" asked Deputy Sergeant Marty Theriot.\n\n\"Tell me,\" said the trucker, who had stepped down from his International cabover tractor at the order of the deputy.\n\n\"Yeah, I'll tell you, boy,\" said Theriot. \"I had you at seventy.\"\n\n\"Maybe you don't know it, but there's this hurricane, see,\" said the trucker. \"And I have to drive into Arbier to pick up this load of fish and crabs and who knows what the hell else. And I'd dearly love to get in there and get out before this here hurricane hits me.\"\n\n\"Well,\" said the deputy, \"I can sympathize with you, but that don't mean I can just let you get away with breaking the law.\"\n\n\"Yeah.\"\n\n\"So what do you have to give me?\" asked Theriot.\n\nThe trucker dug in his pocket and came out with a ten dollar bill.\n\n\"Hey, boy,\" said Theriot, \"what do you call that?\"\n\n\"It's a bribe,\" said the trucker.\n\n\"Not that little thing it ain't.\"\n\nThe trucker added another ten. \"That's a whole lot better,\" said Theriot.\n\n\"Can I go now, smokey?\" he asked.\n\nTheriot's expression changed quickly. He looked as angry as he could pretend. \"What did you call me, boy?\" he asked in a threatening voice.\n\n\"I just wanted to know if I could go now,\" said the trucker.\n\n\"Not that,\" said Theriot.\n\n\"You mean 'smokey,'\" said the trucker.\n\n\"Yeah,\" said Theriot. \"You calling me some kind of nigger? You're about to get your face twisted on sideways.\"\n\n\"'Smokey' means a cop like you,\" said the trucker. \"Because of your hat. Smokey the bear. Get it?\"\n\n\"Yeah, boy, I get it, but I'm still mad. Get it?\"\n\n\"Yeah,\" said the trucker, pulling out another ten dollars. \"I thought all you guys understood that talk. You must hear it all the time.\"\n\n\"You better hurry up before you get caught by this here hurricane,\" said Theriot. As the trucker turned to climb back into his rig, Theriot said, \"Put the hammer down to Arbier boy. I'll catch you on the flip-flop.\"\n\nThe sudden use of trucker slang stopped the driver. He looked down at Theriot. \"Negatory, buddy,\" he said, \"not if I can help it.\" He got into his cab, slammed the door, and started off again toward Arbier.\n\nTheriot was grinning as he went back to his patrol car. His receiver could pick up all twenty-three channels of Citizen Band radio transmission. He switched from the police channel to the one commonly used by truckers. He could hear a transmission by the trucker he had just stopped.\n\n\"Break one-oh. This is the Ragin' Cajun,\" said the trucker, asking for the use of the channel. \"We got a smokey on the side, on the Coonass Boulevard, about ten miles southbound out of Linhart town. Just fed the bear what I was savin' for lunch. Any of you good buddies out there, come on?\"\n\nTheriot couldn't resist. He flicked the switch on his mike and answered. \"Break one-oh. This is Deputy Sergeant Martin Theriot of the St. Didier Parish Sheriff's Office. Just wanted you to know it wasn't anything personal, Ragin' Cajun. Keep the rubber side down and the shiny side up, boy.\"\n\n\"What do you know, the smokey's got ears,\" said the Ragin' Cajun. \"Mercy sakes, good buddies, you got a copy on me, come on? You better watch for this bear 'cause he's Fine and _Superfine_.\"\n\n\"A big 10-4,\" said Theriot. \"I'm going to pull the big switch now, so you're on your own.\"\n\n\"Thanks a lot, smokey,\" said the trucker. \"All the good numbers to you.\" He sounded bitter. Theriot smiled and switched back to the police channel. He picked up his porno book and read for a while.\n\n* * *\n\n\"I told you not to call, Corinne.\"\n\n\"Skip, I have to talk to you,\" she said.\n\n\"You've said that every time you've called.\"\n\n\"But look. The situation is worse now, isn't it?\"\n\nSkip looked at his charts. He was feeling pretty well, because of the Elavil he had taken. \"It's not much worse,\" he said.\n\n\"But you put out a Hurricane Watch, didn't you?\" she asked. \"Doesn't that mean that the hurricane is coming toward us?\"\n\n\"It means that at the last plotting. New Orleans decided that it was more likely that Felicia is curving toward us rather than south and away from us. There's still almost twenty-four hours leeway, and the hurricane could do almost anything in that time.\"\n\n\"Skip, you're always trying to soothe me like that. I want to know the truth. Are we going to get hit?\"\n\nIt was the same question that everyone in the parish was asking, and it was the one question he couldn't answer. He had his own readings, the data from New Orleans, the satellite pictures from New Orleans, and their estimates. Still, hurricanes hardly ever followed predicted paths. It was unreasonable to expect Skip to know where the damn thing would be hours from now. \"I don't know, Corinne,\" he said. \"Now, let me go. I have a lot of work to do.\"\n\nHe hung up. That decides it, thought Corinne. She had used the telephone conversation with Skip to decide for her whether or not she would meet Carl Steinbrenner. Apparently she was. She sighed. Going to bed with Steinbrenner. Was that any better than decoupaging a wastebasket? It was all the same. Anyway, there was nothing on television now. All the children's shows had started, and she didn't feel up to watching old \"Quickdraw McGraw\" cartoons. She smoked a cigarette and paced around the house. She turned on the television. One of the Little Rascals was saying, \"Aw, gee, Miss Crabtree, we ain't every going to play hooky again.\" Or something like that; Corinne wasn't paying close attention. The Little Rascals or Our Gang or something. She turned off the television when she became aware of the Hurricane Watch message at the bottom of the screen. She felt very frustrated. At last she went into the bathroom, washed, brushed her dark brown hair, and put on a different blouse. Then she went outside, got in her little station wagon, and drove south on Hanson Highway, toward the Sea-Ray Motel.\n\n* * *\n\nAt four o'clock, Darlaine had finished putting on her clothes. She lit a cigarette, but she didn't say anything to Chuck. She checked her appearance in the mirror and went outside.\n\nChuck was still in bed, covered by the sheet. They had hardly said a complete sentence between them. He had found her not terribly exciting, not wildly erotic. \"She's been pushing forty for so long, it's starting to push back,\" he said to himself.\n\nAfter a little while, he sighed and got out of bed. Neglecting his socks and underwear, Chuck put on a pair of slacks and a sports shirt. He thought about Darlaine. He laughed. She had come with him when he left the Crisis Caf\u00e9, and together they went to the sheriff's office. Chuck was a little embarrassed when everyone in the office said hello to Darlaine. The deputies in the Arbier office gave Chuck some strange looks. At least, Chuck thought they did. He felt very conspicuous with Darlaine obviously impatient. A deputy filled in the forms and gave Chuck the appropriate permits so that his film company could operate in the Arbier area.\n\n\"What about the trucks?\" he had asked then.\n\n\"What kind and how many?\" asked the deputy.\n\n\"Four eighteen-wheelers,\" Chuck had said. \"All the props, costumes, lights, that sort of thing. One is a kind of on-location dressing room.\"\n\n\"You want to park them somewhere, is that it?\"\n\n\"Yes,\" Chuck had said.\n\n\"I don't know if you need a permit or not,\" the deputy had said. \"What the hell, I'll write you out one anyway. Then if anyone complains, you'll have something to show them.\"\n\n\"I really appreciate this,\" Chuck had said.\n\n\"No, no,\" the deputy had said. \"We're always glad to cooperate with film crews and documentary people. It's good business for us. Expands the tourist trade.\"\n\nChuck had thought that the only place to house tourists was the Sea-Ray Motel. The idea at the time had seemed ridiculous, and now it still seemed crazy. Chuck had been given official permission to park his four tractor-trailers just inside the town's corporation limits, in a vacant lot opposite the motel.\n\nChuck picked up the telephone and placed a long distance call to Miami Beach. \"Hello,\" said Tom.\n\n\"This is Chuck. Everything here is fine. The only thing I'm wondering about is your trucks. They were supposed to be here yesterday, weren't they?\"\n\n\"Don't worry,\" said Tom. \"That's my business. Give me a call if they're not there tomorrow morning. I heard from Denny, and it sounds like they're taking their time moseying across the south. All of you guys are getting a free ride on this thing, while I have to sit here and listen to you bitch.\"\n\nChuck pictured Tom in Miami Beach. He must be really suffering, thought Chuck. Like hell. \"All right,\" he said into the telephone, \"How's everything else?\"\n\n\"All right,\" said Tom. \"Nelson called in and said the situation in New Orleans was stable. Stan and Ed are just waiting for Denny and his goddamn truckers to get to you.\"\n\n\"Sounds like we're ready to play.\"\n\n\"Yeah,\" said Tom. \"Except for one thing. One of Denny's men never made the appointment. Anyway, you can go on three trucks. But remember our deal. I think the best thing for you to do is get some local to drive a truck. Rent one. I don't want you coming out short on your end of this thing.\"\n\n\"I won't,\" said Chuck, \"don't worry about me.\"\n\nTom sounded tired. \"I always worry about you, Chuck. You can't imagine how I worry about you.\"\n\n\"How do I find another driver?\" asked Chuck.\n\n\"That's your problem,\" said Tom. \"The whole matter is in your hands. This is your chance to show how great you are in a desperate situation.\"\n\n\"Everything's settled,\" said Chuck. He heard Tom hang up his telephone. Chuck got out a telephone book for the St. Didier Parish area and found the truck rental listings. After he reserved a twenty-foot truck, he would have to find someone to drive it. He thought for a while. Then he decided that the best place to find someone was the Crisis Caf\u00e9. Everyone in town seemed to pass through there at some time in the day. But that could wait until morning. In the morning, things would begin to happen very quickly indeed. Chuck took out a bottle of liquor he had bought that morning, opened it, and drank enough so that he didn't wake up until after midnight. At that time he felt sick. He heard pounding on his door. He got up, staggered to the door, and opened it. His eyes were bleary and his mouth felt like it had been used as a container for unpleasant things.\n\n\"Hey, Chuck.\" It was Denny. With him were two other men whom Chuck didn't recognize.\n\n\"Here,\" said Chuck. He felt miserable. He stumbled around in his dark room. He got the keys for the other two rooms and went back to the door. \"Here,\" he said. \"I'll see you in the morning.\" Then he waved Denny and his truckers away.\n\nChuck went into the bathroom and looked at his tongue. It was coated and swollen. The corners of his mouth were cracked and sore. He ran some cold water and splashed it on his face. He had no clear idea of why he was doing that. \"I'll die of some strange tropical disease before all of this is over with, I know it,\" he said. Then he left the bathroom and stretched out on the bed. In the room to his left, Room 7, there was the sound of beer cans being thrown around. \"They're worse than the rock band,\" muttered Chuck. He drank some more of the liquor and was soon asleep again.\n\n* * *\n\nWhile Chuck slept, Denny, in Room 9, was watching television. The Hurricane Watch was broadcast continuously. A map showing the present position of Hurricane Felicia was always on the screen, and Skip Strahan and Sheila Downing took turns with the other Channel Five newscasters. There wasn't much for any of them to say, however, except report on Felicia's present position, speed, and strength. The hurricane was making a curve that would send it into the coast of Louisiana somewhere near Arbier. Denny got bored with the weathermen and turned off the television. Besides, he had a date in fifteen minutes in Room 13.\n**17**\n\nIt was half past eight on Wednesday morning when Sheriff Walter Boshardt walked into the Arbier office. He was greeted by the four deputies who were on duty. They would be relieved in half an hour by the other four deputies of the Arbier office.\n\n\"All right,\" said Boshardt, \"what's happening?\"\n\n\"Quiet night,\" said one of the deputies.\n\n\"Felicia?\" asked Boshardt.\n\n\"Latest reports are that she's moving northwest, that she'll hit Texas around Corpus Christi. We can expect gale force winds. The gale warnings have gone out, but we're still paying close attention to the hurricane.\"\n\n\"Best thing to do,\" said Boshardt. \"We'll have to go on the assumption that she'll hit us anyway. Better safe than sorry.\"\n\n\"Right, Sheriff.\"\n\nBoshardt nodded. He imagined the activity around the town. Even without the hurricane, the gale force winds were enough to do quite a bit of damage. Every shop in the town must be covered with plywood sheets over the plate glass windows. Gale force winds range as high as seventy-three miles per hour. That was a whole gale. It was something to be cautious about.\n\n\"Any changes, let me know as soon as you can,\" said Boshardt. He thought he could afford to go over to the Crisis Caf\u00e9 for a little breakfast. He had left the house too quickly to grab anything to eat.\n\nAs he sat in his booth in the caf\u00e9, he wondered why he still had the same feeling he had experienced the day before. That hurricane wasn't finished with St. Didier Parish and with Arbier. But the first demitasse of strong, black coffee began to wash that feeling away, and Mrs. Perkins' _pain perdu_ completed the job, and the second small cup of coffee left Boshardt feeling better than he had felt in days.\n\n* * *\n\nAbout fifteen minutes after Sheriff Boshardt left the Crisis Caf\u00e9, Chuck came in. He got ugly glances from some of the people sitting in the booths and tables in the diner. Chuck was aware of the looks. He was aware that the people of Arbier had not taken to him with the kind of friendliness he had hoped for. Chuck was a little slow to realize that his manner had made him worse than just a foreigner to the Cajun residents. When the Cajuns spoke of Chuck, which was not very often, the word that came up often was _cagou_ \u2014disgusted.\n\nChuck walked to the back of the diner. He watched as the natives of the town followed him with their eyes. Chuck sat in a booth. The attitude of the people of Arbier made Chuck nervous.\n\nLauren came over with a menu after a few minutes. He ordered bacon and eggs and a cup of coffee. Then he said, \"Say, you wouldn't know anyone who's looking for some work, do you?\"\n\nLauren didn't say anything, but she pointed to a young man sitting by himself at a table. It was Paul Pierson. Chuck got up and sat at Pierson's table. \"Listen,\" said Chuck, almost in a whisper, \"I understand you could use a quick hundred bucks.\"\n\nPierson looked at Chuck. \"Yeah,\" he said hesitantly.\n\n\"Can you drive a truck? One of those twenty-foot rental jobs?\"\n\n\"Sure,\" said Pierson.\n\n\"Some of the local merchants have hired us to remove some of their more valuable merchandise from the town, on account of the hurricane. We'll be going to Linhart with it.\"\n\n\"Fine,\" said Pierson.\n\n\"When you're finished eating, wait for me, and we'll go back to the Sea-Ray together. I'm staying at the Sea-Ray.\"\n\nPierson didn't say anything. He just sipped his coffee.\n\n* * *\n\nSkip Strahan had been running on energy supplied by his small, private pharmacy. He was beginning to show the effects of all the uppers and downers and mood elevators he was taking. When he was on camera he was fine. He was strong enough to control his actions then. But as soon as his shift ended and he stretched out on a small cot in the dressing room, he began to show nervous symptoms. He was highly irritable. His hands shook and his legs seemed to throb with muscular spasms.\n\nThe news from New Orleans was better on Wednesday morning. It looked like Hurricane Felicia would bypass Louisiana completely. Texas would have to prepare for the worst, while Linhart would experience only the gusty winds and heavy rain associated with the outer fringes of the spiral storm.\n\nSkip sat in the dressing room. It was ten o'clock in the morning. The day was warmer than the last two or three had been. The sky was bright, but covered with a feathery tracing of cirrus clouds. There had been no shower activity the day before, and the winds were barely noticeable. That was the kind of weather he had learned to associate with the approach of a hurricane.\n\nSkip got a telephone call from some moron of a college professor who wanted to take his single-engine plane into the hurricane for some dumbass experiment. Skip tried to talk the man out of the idea, but the professor wouldn't listen to reason. Over the years, fliers had learned through bitter experience never to fly into a hurricane on a single engine. The incredible winds, the sudden updrafts and downdrafts, the squalls threw the airplane around like a toy balloon. Those airplanes were the large airplanes used by the Navy and Air Force. A small one-engine job wouldn't have much of a chance against Felicia. At last Skip felt that he had persuaded the professor not to try the experiment. If the man did, Skip would hear about it later. The professor might be the first casualty of Hurricane Felicia.\n\n* * *\n\nAbout eleven-thirty, the Sheriff went for a walk down to the end of Ridge Street. He wanted to look at the Gulf. He noticed that the waves were rolling in at a much slower rate than usual. Normally, waves break about fourteen a minute. Now, Boshardt counted six a minute. The waves were high swells crashing against the small sandy beach. The spray flew around the pier. He could hear the groaning of the wooden piles as the pier took the impact of the huge waves. The wind had picked up a little from earlier in the morning. It was from the northeast, gusty, maybe ten to twenty miles per hour. Looking over the Gulf, the Sheriff could see a shower falling over the ocean. It seemed to be getting closer. Boshardt turned and went back to his office.\n\nThe four deputies who had gone on duty at nine o'clock didn't even look up as he came in. He went to a desk and called the Linhart station, then the Delochitaches station. Everyone was relaxed but ready. That was the way things should be, thought Boshardt.\n\nHe sat at the desk, fiddling with a pen. Then, impatiently, he went outside again. The cirrus clouds overhead had changed to billowy cumulus clouds which were building and piling high over Arbier. The afternoon shower, thought Boshardt. Maybe a little stronger than usual. He could see the cumulus clouds forming huge thunderheads in the distance.\n\nIt was very quiet outside. There were no people about, no traffic. It made Boshardt uncomfortable. He went back into his office. He wished that something would happen to relieve the tension. Then he wished that nothing would happen. Tension was better than disaster.\n\n* * *\n\nWhen the telephone rang, Corinne knew who it was before she answered. She was right. It was Carl Steinbrenner. \"Hello, Corinne,\" he said.\n\n\"Hello,\" she answered. She was feeling the tremendous burden of guilt that she had accepted after meeting Steinbrenner the day before.\n\n\"Guess what,\" said Steinbrenner.\n\n\"I don't know,\" said Corinne. She felt worried, anxious, and excited, all at the same time. She was having an affair. It was like something on one of the soap operas. An affair. Something that was doomed to failure, something that would bring her only tears at the end, but something that would give her life meaning until then. And later, when the affair was over, she would look back. She would always have the memories of those passionate embraces, the longing, the ecstatic moments.\n\n\"I got you a present,\" said Steinbrenner.\n\nThere hadn't been any passionate embraces, thought Corinne. There had been absolutely no longing, and not one single ecstatic moment.\n\n\"Do you want to know what it is?\" he asked.\n\n\"Sure,\" she said. The guilt was heavier than she had expected. It was almost paralyzing.\n\n\"Meet me at the Sea-Ray,\" he said. \"An hour from now, and I'll show you.\"\n\nCorinne didn't know what to say. She suddenly realized that for Steinbrenner, she was probably just a lunch-hour quickie. A present. What? Some perfume? A cute little pin or bracelet? Maybe one of the better of his used babies? \"All right,\" she said.\n\n\"I'll see you then. Same as yesterday,\" he said.\n\n\"Same as yesterday,\" she said, and hung up the telephone. She went into her bedroom, opened Skip's sock drawer, and took out a vial of tranquilizers. She took three, just to calm herself down. She didn't even check her appearance in a mirror. She went outside, got into her station wagon, and drove south on Hanson Highway. She passed a patrol car from the sheriff's office parked just off the highway, on Couletain Boulevard. She slowed down as she went by. Half a mile later, she sped up again. She got to the Sea-Ray long before Steinbrenner. She smoked a cigarette and listened to the radio.\n\n* * *\n\nSkip went on the air. He was very tense as he read the bulletin he had written from data he had just received from New Orleans. \"The eleven-thirty report from the National Weather Service Hurricane Warning Bureau in New Orleans reports that Hurricane Felicia has turned again. She is on a heading that will bring her to hit the Louisiana coast somewhere between Cameron and Terrebonne Parishes. I repeat, Hurricane Felicia has changed course, and is on a heading that will bring her to hit the Louisiana coast somewhere between Cameron and Terrebonne Parishes.\" He paused. His mouth was suddenly very dry.\n\n\"Hurricane Felicia's winds are now measured at an average of one hundred twenty-five miles per hour. It is estimated that Felicia will be a storm of extremely dangerous intensity. The eye of the hurricane should reach the Louisiana coast about three o'clock this afternoon. People living in low-lying districts should seek other shelter as quickly as possible.\" Skip repeated the last part of the bulletin, and the director changed cameras, turning to a newscaster. Skip went back to his dressing room. He took a couple of pills. He noticed that he was running low. He might have to ride out a hurricane without them. He didn't like that thought at all.\n\n* * *\n\nBoshardt was furious. He had heard Strahan's bulletin.\n\n\"That damn fool!\" he shouted. \"Don't those weather people realize they could be costing lives with those bulletins? He says the eye of the storm will hit us at three. So people will say, 'Well, we've got hours to get ready.' But, goddamn it, when the eye gets here, we'll already have gone through half the hurricane. The front edge of the thing should hit the coast about one o'clock.\" Boshardt did some thinking. \"We have to evacuate now,\" he said. \"Arbier is going to go down again. We've got to get everybody up to the shelters in Linhart.\"\n\n\"We only have an hour and a half,\" said one of the deputies.\n\n\"So let's get our asses in gear,\" said Boshardt. \"We have four men here, and three patrol cars. Get everybody out of here. One of you guys go and warn the Sea-Ray. That place will be under water in a couple of hours. We've got to get everyone out of Arbier and up to Linhart.\"\n\n\"What if they won't go?\" asked a deputy.\n\n\"You always have a few who won't leave,\" said the sheriff. \"Try your damndest to make them go, but remember we don't have much time. 'Low-lying areas!' The whole goddamn coast of Louisiana is a low-lying area. Those people out in the marshes think they're all right because they built a shack on stilts on some damn hump of dry ground.\"\n\n\"Do you want me to go out there?\" asked one of the deputies.\n\n\"No,\" said Boshardt, \"I'll do it. I can speak enough of their Cajun to persuade them. Maybe.\"\n\n\"What about manning the office here?\" asked a deputy.\n\n\"Screw the office,\" said Boshardt. \"You have an hour and a half. I don't want any of you to take any chances. When that storm hits, I expect all four of you to be near Linhart. I want the cars in protected areas. If you do your job right, Arbier will be secure. It's just a matter of moving everybody inland. This town won't be good for anything but mosquitoes and mud pies by tonight. Got it?\"\n\n\"Yes, sir,\" said one of the deputies.\n\n\"Then let's get going.\" Boshardt went to a telephone and called his home. There was no answer. Well, thought Boshardt, she's out for the day already. They'll find her at the Sea-Ray. They'll get her to a shelter. He went outside, got in his patrol car, and headed west, toward the marshes.\n\n* * *\n\nWhen Maddie woke up, she was alone. Pierson was gone. She turned on the television and watched for a while. Then she heard of the Hurricane Watch as Felicia turned toward St. Didier Parish. At first she was nervous and frightened. She wished that Pierson were home. Maddie paced the apartment. Finally she called her father and spoke to him. He didn't seem to be worried. He said that he was driving up to Linhart and would take her along with him. He would be by in fifteen minutes. Maddie said goodbye and hung up the telephone.\n\n\"Hey, _minou,_ \" she said to Cyrus. \"I'm sorry, I can't take you along. I'll put you in the bathroom and shut the door. You'll be safe there.\" She went into the bathroom and saw the tub full of water. \"You won't like that,\" she said. She let the water out, thinking that maybe she was making a mistake. But if the hurricane did hit Arbier, they wouldn't be able to return to the apartment for days, very likely. She let the cold water tap on the sink run in a gentle rivulet, in case Cy got thirsty while he was alone. She put a large bowl of dry cat food in the bathroom, picked up the gray Persian, and tossed him into the bathroom. Then she shut the door. She sighed. \"Now I must go be of service,\" she said.\n\nShe went downstairs by the elevator, and waited for her father. The day didn't look too bad. A heaviness to the air that made her more frightened. Dark, scudding clouds above. In a few minutes Monsieur Gargotier arrived in his car. Maddie got in. The back seat was filled with supplies and canned food, enough to last ten people a week or more. They would be all right.\n\n* * *\n\nDarlaine Boshardt was at the Sea-Ray, watching television in bed with John Smith. This was a different John Smith, one she had never tried before. It was in the middle of their unusually quiet lovemaking that Skip Strahan read the bulletin.\n\n\"Wait a minute,\" said Darlaine.\n\n\"What?\" asked John Smith. He was about to give the woman a couple of powerful thrusts that would make her respect him. So far, nothing else seemed to work. She never made a sound.\n\n\"I said, wait a minute,\" said Darlaine. She listened to the news. \"We're going to get the hurricane, after all.\"\n\nJohn Smith looked worried. \"We better get out of here.\"\n\n\"Don't panic,\" said Darlaine. \"This motel is built to stand heavy weather. Why don't we just stay here and have a party?\"\n\n\"A party?\"\n\n\"Yeah,\" said Darlaine. \"A hurricane party. We used to have them all the time, in New Orleans. We could get those four guys next door, and the black bitch. They probably have some bottles with them. And we all sit around and drink.\"\n\nJohn Smith thought about what they would all do. Sit around and drink, sure. But his limited knowledge of Darlaine was enough to persuade him that there would be more going on. With himself and the truckers and a black whore. He smiled. It sounded like a good way to pass the day. All that would be on the television were weather reports, anyway. \"All right,\" he said, \"a hurricane party.\"\n\n\"And I get to send out the invitations,\" said Darlaine.\n\n\"Let's finish here, first,\" said John Smith, giving Darlaine a couple of quick thrusts.\n\n\"Aren't we finished here?\" she asked. \"I thought we were.\"\n\nYou're a mean broad, thought John Smith. He grabbed Darlaine's shoulders and pushed her firmly against the bed, then he brought her legs up, over his shoulders, and pounded her as hard as he could. After a few seconds, he felt his orgasm building. He cried out when he came.\n\n\"Now we're finished here,\" said Darlaine. She got out of bed and was putting on her clothes. \"I'm going over to invite those truckers.\"\n\nYou bitch, thought John Smith.\n\n* * *\n\nDeputy Sergeant Marty Theriot was sitting in his favorite speed trap, just off Hanson Highway on Couletain Boulevard when he got a call from the Linhart office. Captain Brierrer told him of the Hurricane Watch, and that the leading edge of Felicia could be expected to hit around St. Didier Parish in less than an hour and a half.\n\n\"Roger dodger, Robert One,\" said Theriot. Doggone it, he thought, now I have to go warn all those roadhouses again.\n\nHe backed the car in a driveway and turned, heading toward the old parish road. He would have to stop in every one of those goddamn bars and warn the people of Hurricane Felicia. He smiled.\n\nOverhead, a helicopter chuttered by, airlifting men from the off-shore oil rigs to safety in Linhart. They're really serious about this, thought Theriot. For the first time since the first Hurricane Warning, he felt the smallest bit uncomfortable. He didn't like hurricanes, not at all. They were a genuine pain in the ass. They were a hell of a lot of work. Ugly work. Finding buried corpses with only a hand sticking up out of the mud. Crawling through collapsed buildings, looking for more corpses.\n\nThe corpses would be easier to find in a couple of days, he thought. All you had to do was look for about a million flies.\n\n* * *\n\nThe traffic on Hanson Highway north was dense with people from Arbier fleeing to the shelters of Linhart. The small crew that worked out of the sheriff's office in Arbier had done its job. Of course, they had met some resistance, but on the whole most people reacted sensibly. The only intelligent thing to do was run from the storm, run north, where the hurricane surge and the abnormally high tide and the long, slow, huge waves couldn't reach them. Some of the people panicked, and there were minor accidents, but the deputies from the Arbier office helped the traffic flow as quickly as possible.\n\nThe one place where a deputy met resistance was at the Sea-Ray Motel. The desk clerk told the deputy that he couldn't just leave the place. Some people had chosen to ride out the storm in the motel. \"For crying out loud,\" said the desk clerk, \"it must be a good mile to the Gulf. And it would take a pretty good wind to wobble these walls.\"\n\nThe deputy tried to explain that a good mile wasn't very good, not when the land was as flat and low as St. Didier Parish. Besides, they were expecting a pretty good wind, too. Nevertheless, the desk clerk wouldn't leave.\n\n\"I have guests here,\" he said. \"They don't want to leave. What if they want ice or something?\"\n\nThe deputy didn't say anything more. He turned and walked out the screen door. It slammed behind him. The deputy noticed that Mrs. Boshardt's car wasn't in the lot.\n\nBy half-past twelve, the evacuation of Arbier was almost completed. The deputies joined the tail-end of the procession northward. They would all be safe in Linhart when the storm struck. One of the deputies had expressed pity for the people who wouldn't leave. All he got from his partner was a silent but meaningful shrug.\n\n* * *\n\nCorinne left the Sea-Ray when she heard Strahan's announcement. She was only a little way from the house where her father lived. She started the engine of her car and backed out of the motel's parking lot. Carl Steinbrenner wasn't even in her thoughts. She drove down Ridge Street, then made a right turn onto W. 2nd. She pulled into a driveway, got out of the car, and ran to the stairs up to the furnished room where her father was passing the end of his lonely life. She pressed the doorbell and knocked. The elderly man didn't answer for a short while. For a moment, Corinne thought that perhaps he had taken a ride with someone else to Linhart. But finally he came to the door.\n\n\"Corinne,\" he said, surprised. \"What are you doing here? You should be in Linhart.\"\n\n\"Come on, Dad,\" she said. \"I'm taking you home with me now.\"\n\n\"No, no, Corinne,\" said the old man. \"I'm just going to stay here. It's just a storm. I've been through a lot of storms.\"\n\n\"This is a hurricane, Dad,\" said Corinne. \"You're too close to the Gulf here. You've got to come with me.\"\n\n\"Corinne,\" said her father sternly, \"how much self-respect do I have left? Eh? One thing I want, I want to sit here and wait out the storm. That would be a thing to do. I don't want to be dragged to an emergency shelter. That smells too much like putting me away in an old-folks' home.\"\n\n\"Dad, listen. It's not a matter of self-respect. It's a matter of self-preservation. You're in a lot of danger here.\"\n\nThe old man smiled, for the first time in a very long time. \"Yes,\" he said, \"I kind of like it.\"\n\n* * *\n\nThere was a gawky, strange-looking bird wading through the marshes as Sheriff Boshardt drove west. A roseate spoonbill, he told himself. It'll be dead, soon.\n\nA lot of things would be dead, soon, thought Boshardt. The marsh birds, the Louisiana heron, the oyster catchers, the ibis. The muskrat and nutria, the alligator, the snakes, all dead, all killed by the storm. The people of the marshes. All drowned.\n\nBoshardt looked at his wristwatch. He had an hour before Felicia would make travel impossible. Half an hour out, half an hour back. He felt like Paul Revere. Right, he thought. Sure.\n\nHe stopped at many little communities among the marshes, tiny groups of people who lived together on an island of solid ground in the middle of the marsh, the trembling prairie. He urged them all to get to better shelter, to warn others. He watched his time. There were probably as many people living in the marshes as there were in the towns. Boshardt knew that he couldn't reach them all.\n\nBoshardt kept an eye on the Gulf as he rode back. It was getting close to one o'clock. He made a call on his two-way radio to Linhart, to find out if the situation had changed. No. Hurricane Felicia was due at one o'clock. Boshardt pushed the gas pedal to the floor. At one o'clock he was back in Arbier. The town was virtually deserted; the sheriff suspected that some of the three thousand people in town had chosen to remain. Boshardt felt sick. He passed the Sea-Ray, and there were cars parked there. His wife's wasn't among them. He pulled into the motel and went into the office. No one was there. He knocked on doors, but no one answered except at Room 6. A man, dressed only in a pair of trousers, opened the door a little. \"Yeah?\" said the man.\n\n\"Look,\" said Boshardt. \"There's a hurricane due in just a little while. This isn't a safe place. You'd better evacuate as soon as you can.\"\n\n\"Wait a minute,\" said the man. He shut the door. Boshardt could hear murmurings through the door, but the words were indistinct. The man opened the door a bit again. \"We took a vote,\" he said. \"We decided to stay here.\"\n\nBoshardt's hands clenched. Like the old Cajuns in the marshes, these people thought they were safe. At last, the sheriff admitted to himself that he couldn't be responsible for them. There was no way to force such a large number of people to leave their homes. He had to be satisfied with the fact that he had tried.\n\nThere was no satisfaction. There was only the pain that he would feel after the hurricane, when the casualty figures for his parish were published.\n\nBoshardt got back into his patrol car and headed north, to Linhart. Except for a few stragglers in Arbier, and the people who wouldn't leave the marshes, Arbier was secure. Every shop window had been boarded up. Private houses had been protected in a similar fashion.\n\nA mental image of what Arbier would look like after Felicia had finished with the town entered Boshardt's mind. He shuddered.\n\nIt was one o'clock. Boshardt was halfway between Arbier and Linhart. The hurricane was due very soon.\n\n* * *\n\nBoth the day manager and the night manager had efficiency apartments at the Sea-Ray. The night manager had taken the warning of the deputy and gone to a shelter in Linhart. The day manager, however, was going to ride out the storm in his own apartment. He had been through hurricanes before. He knew the constant boredom of watching weathermen on television, plotting the storm's movements. He knew that the rain would be heavy, but he had already done his best to seal the openings around windows and the door. He had filled his bathtub with water, and he had a battery-powered radio, in case the power went off. He had a supply of candles. Now, come hell or high water, he was ready.\n\nSkip Strahan was telling people that was just what to expect. Hell. And high water.\n\n* * *\n\nAt half-past one the storm was late. Pierson was getting nervous. He didn't like the idea of driving a truck right into the bloody arms of a hurricane. But Chuck didn't seem to be too concerned. The one time that Pierson had mentioned to Chuck that they ought to be moving, Chuck had waved the notion aside. \"These big mothers aren't going to be bothered by a little wind and rain,\" said Chuck.\n\n\"What about mine?\" he asked, referring to the twenty-footer they had rented earlier in the day.\n\n\"You're all right, kid,\" said Chuck. \"Just relax.\"\n\nPierson tried, but he couldn't relax. Denny and the two other truckers seemed relaxed. He had been introduced to them. They gave him their names; one was Marsh Rabbit and the other was Old Mole. The nicknames were their Citizen Band radio handles. Denny used the handle Cracker Smacker. The truckers didn't include Pierson in their conversation very often. Pierson wasn't a trucker, he was outside. Pierson was getting very tired of always being outside.\n\nThe truckers were proud of their rigs. They related to their trucks the same way a cowboy related to his horse, at least on television. The tractors were painted and decorated. Old Mole had his name painted in decorative script on the doors of his cab. He was driving a Kenworth cabover tractor. Marsh Rabbit had a longnose Diamond Reo. Denny had a Peterbilt cabover. The three men were discussing the pros and cons of longnoses as opposed to cabovers. A longnose was a tractor that had the engine out in front, while a cabover had the engine beneath the driver. The cabover didn't ride as smoothly as the longnose, but the driver had a clearer view of the stretch of road immediately ahead of him. The vans these cabs pulled were simple, undecorated forty-footers. There were no company names or emblems on them. The only thing that gave them personality were the bumper stickers. Old Mole had one that said, \"Jesus was a trucker.\" Denny's said, \"I speed up to run down little animals.\"\n\n\"Well,\" said Chuck. \"Let's get going. Mount up. Let's ride into town.\"\n\nAs Pierson was walking toward his rented twenty-footer, he heard one of the truckers behind him say, \"Anything with less than thirteen gears is a bicycle.\"\n\nChuck walked from one truck to another, giving assignments. \"Denny,\" he said, \"you're frontdoor. Mole, you're backdoor. And the Rabbit and the kid are rocking-chair.\" Rockingchair were all the trucks between frontdoor and backdoor.\n\nThe trucks moved out of the vacant lot in the order Chuck had assigned. The drive into Arbier was only a matter of a few minutes. The front truck, with Chuck riding with Denny, stopped almost at the end of Ridge Street. Marsh Rabbit stopped his truck a few hundred yards behind him. Chuck came back to tell Pierson to stop and park his truck the same distance behind the Rabbit, and Pierson saw that Old Mole was a good distance behind him.\n\n\"Open the tailgate on your truck,\" said Chuck.\n\nPierson climbed out of his truck and did as he was told. He was still nervous about being in Arbier with Felicia so near, but the element of risk was why Chuck was willing to pay him a hundred dollars.\n\n* * *\n\nIt takes about three days for a tropical depression to grow to hurricane strength. Felicia was already an old hurricane, judging by the satellite photographs that New Orleans received at regular intervals. These films were made available to local weather bureaus and television and radio stations. The constant work of plotting the storm's path was wearying but vital. The southwestern coast of Louisiana was particularly vulnerable to the furious destruction of a hurricane. In that part of the state, it was sometimes difficult to decide which was water and which was land. The storm surge would wipe out everything in its path. Because the coast remained low and marshy for so many miles inland, that path would be a long one.\n\nSometimes hurricanes behaved in ways that were strange, even for those notoriously erratic storms. Sometimes a hurricane will attack a land area and drop virtually no rain. Sometimes the land area is so inundated with the hurricane's rain that floods become a major hazard. In Louisiana, where the water table is relatively close to the surface, swimming pools had been known to pop up out of the ground, or buckle and crack. The water in the pool might become contaminated. After a hurricane, the danger of the spreading of disease was another peril the local inhabitants had to face. Hurricanes had more than one way to strike down human lives in their paths.\n\nThe storms were hundreds of miles across. The gale force winds that announced the arrival of the hurricane might exist four hundred miles from the eye of the storm.\n\nIn Arbier, there were no gale force winds. It was one thirty, and Felicia had not made her presence known. Perhaps she had stalled, as she had stalled against the Yucat\u00e1n. In New Orleans and Linhart, data came in from radar installations. It seemed that the southwestern parishes had been given a little extra time, a short reprieve. Those in charge of evacuation and those running the emergency shelters used the extra time and were grateful. Things would get bad soon enough.\n\n* * *\n\nAt Bar's Mike and Grill on the old parish road, Deputy Sergeant Marty Theriot was sitting on a stool. He was drinking straight whiskey. He was unaccountably nervous. He thought a drink would help him settle down to his job.\n\nHe had strayed across the bayou on the old causeway, into Ward Two, which was technically the responsibility of the Arbier office. He had come to the bar because one of its owners, Michael Bonneaux, was a very close friend. It was half-past one. The hurricane was half an hour late already. Theriot swallowed a shot of whiskey and clenched his teeth to prevent a grimace from showing.\n\n\"They sometimes do this, don't they?\" asked Bonneaux.\n\n\"Sure,\" said Theriot. \"I wish it would come already. I hate this waiting.\"\n\n\"Me, too.\"\n\n\"You ought to get out of here,\" said Theriot. \"The bayou is going to get awful big awful fast. One second everything will be all right, and the next second you'll reach down to scratch yourself and you'll be ass-deep in bayou water.\"\n\n\"I got time, I figure. I've got everything in the car, all ready to go. But I have the radio on, and when it tells me that Felicia has hit Arbier, I'm going to jump in that car and go so fast up to Linhart\u2014\"\n\n\"Yeah,\" said Theriot. \"Give me one more, and then I have to get back into my own ward.\" Bonneaux poured another shot of whiskey for Theriot. The deputy tossed it down again trying to hide the grimace. He put some money on the bar, adjusted his hat and sunglasses, and walked toward the door.\n\n\"Lots of luck,\" said Bonneaux.\n\n\"Yeah. I got to be out in this mess. Lots of luck to you, too.\"\n\nBonneaux paused for a moment. \"I wonder how much the government would give me if this place drowned.\"\n\n\"I don't know,\" said Theriot, \"maybe nothing. Just to be on the safe side, why don't you set it on fire? In the middle of the flood, there's no way for fire equipment to get to you.\"\n\n\"No fire insurance,\" said Bonneaux.\n\n\"No?\" said Theriot. \"Well, you have time to think of something.\"\n\n\"Yeah. See you.\"\n\n\"Check,\" said Theriot. \"10-4.\" He stepped outside and looked at the sky. It didn't look especially threatening.\n\n* * *\n\nChuck rode with Denny. When they arrived in Arbier, he told Denny to go down almost to the end of Ridge Street. Then he picked up the microphone of Denny's CB radio. \"Break one-oh for the Marsh Rabbit. Good buddy, you pull your rig up behind, say a couple of hundred yards. I'll get that local in the four-wheeler to hang back another couple of hundred. And you, good old Mole, you do the same, come on?\"\n\n\"A big 10-4, right from Old Mole's heart to yours.\"\n\n\"Break one-oh for Cracker Smacker. I'm getting you wall to wall and treetop tall, come on?\"\n\n\"Let's get to work,\" said Chuck. He turned off the microphone and put it back. Then he and Denny got out of their truck. They took a look around the area at the end of Ridge Street.\n\n\"Not a whole hell of a lot to choose from, is there?\" asked Denny.\n\n\"No,\" said Chuck. \"Whatever there is though, we got to get it. Old Tom is sitting in the Miami Beach wonderfulness right now, and he wouldn't like it for sure if we screwed this up.\"\n\n\"Tom does have a way about him, don't he?\" said Denny, with a brief smile.\n\n\"You don't know him as well as I do,\" said Chuck. \"He'd just as soon trade in his wife if his girl friend wanted a new car.\"\n\n\"Let's move,\" said Denny. \"I want to get out of here. There ain't anybody left here. This empty town is giving me the creeps.\"\n\n\"It gives me the creeps when it's full of people,\" said Chuck. \"Good old Hurricane Felicia couldn't pick a better spot to land on. There isn't a town in the whole country I'd rather see wiped out.\"\n\n\"I wish we had something to toast good old Hurricane Felicia,\" said Denny.\n\n\"Good old Hurricane Felicia,\" said Chuck, laughing. He stood beside the truck for a while and thought. Finally, he said, \"You think you can get back in the rig, swing it around, and run the rear end through that boarded-up window?\"\n\n\"No sweat,\" said Denny. He climbed back into the cab and drove the Strick trailer through the plywood and the plate glass behind. Then he pulled the truck out a little, and got out of the cab. He and Chuck went into the store and started loading all the merchandise they could into the truck. Televisions, stereo sets, all sorts of appliances. They worked hard to fill the truck. It took some time. Behind them, Old Mole and Marsh Rabbit were doing the same thing to two other stores.\n\n\"You think you can finish up on your own?\" asked Chuck.\n\n\"Sure,\" said Denny. \"The jewelry store is next. Can I blow the safe?\"\n\n\"I don't know,\" said Chuck, chewing his lower lip. \"Yeah, go ahead. I was just thinking that there were some private homes that might be interesting.\"\n\n\"Screw that,\" said Denny. \"You might run into somebody.\"\n\n\"Hell, there ain't anybody here,\" said Chuck.\n\n\"There always is a bunch of diehard types.\"\n\n\"Then don't blow the safe. We can manage without it.\"\n\n\"All right,\" said Denny.\n\n\"I'm going down to help that kid in the rented truck.\"\n\n\"Hurry back,\" said Denny. \"We want to be out of here as soon as we can.\"\n\nChuck gave him a thumbs-up sign and went to where Pierson was watching what was happening.\n\n* * *\n\nAt two o'clock, Skip Strahan was sitting behind a desk in the Channel Five news set. With him on the broadcast was one of the regular newscasters.\n\n\"Maybe you can explain what's happening,\" said the newscaster.\n\n\"There are a number of factors involved,\" said Skip. \"I couldn't really give a definite answer to that, Pete. It's two o'clock, and New Orleans said that the hurricane was due an hour ago. They're getting regular radar scans from the station in Lake Charles, as well as the satellite photos. I have to keep repeating that the path of a hurricane is a very strange thing in a world that is seemingly governed by strict natural laws. The trouble here is that Felicia is governed by those same laws, but we as meteorologists have not as yet completely understood what they are.\"\n\n\"So what you're saying is that Hurricane Felicia is liable to go anywhere at any time,\" said the newscaster.\n\n\"Right, Pete. I wish I could be more exact about this, but I'll just give the latest bulletin, which was compiled here in the Channel Five Weather Center from information relayed to us from the various hurricane warning organizations. 'Hurricane Felicia,'\" he read, \"'is stalled some distance off the coast of Louisiana, due south of St. Didier Parish. The storm has winds that average one hundred twenty-five miles per hour. The eye of the storm is estimated at roughly fourteen miles in diameter.'\" He added, \"I'd also like to repeat the warnings I gave before. People living in low-lying areas should move away north to safer shelters.\"\n\n\"I guess we can't do anything but wait for the storm to hit,\" said the newscaster.\n\n\"Uh huh,\" said Skip. Suddenly, right then, while he was on the air, Skip began to feel an anxiety attack coming on. His hands were sweating and shaking, and he made fists and hid them below the desk. He felt dreamlike, unreal. His head seemed to be spinning off into space. He wished that the director would cut to a commercial soon. A drop of sweat ran down his forehead, and he flicked it away with one hand; the hand immediately went to his lap again. He was aware that the newscaster was speaking, but he didn't know about what. After a short while, the panic eased, and Skip realized that the newscaster had asked him a question. \"I'm sorry,\" he said, \"what was that?\"\n\n\"I asked if the hurricane was likely to lessen in strength while she's stalled off the coast.\"\n\n\"No, Pete, actually the conditions favor just the opposite. The warm, wet air of the Gulf is perfect fuel for a hurricane. Unless the winds in the upper atmosphere start failing to carry the spent air that's risen up the spiral chimney of the hurricane, there's no reason at all to suspect that Felicia will lessen in intensity.\" With gratitude he saw the director make the hand signal that indicated that they were going to a commercial. Skip wanted to hurry during the ads and take a few pills. He was trying to decide which.\n\n* * *\n\nIn Henriette's place, the television was on Channel Five. They had just heard Skip's explanation\u2014or lack of explanation\u2014of why Hurricane Felicia hadn't arrived on time.\n\nOne of the men sitting at the bar raised his glass at the television. \"That guy, what's his name, Strahan, he doesn't know much, but I'll be goddamned if he can't know it at you for the rest of the afternoon.\"\n\n\"Yeah?\" said one of the other customers. \"If you know so much about hurricanes, why don't you go down and apply for his job?\"\n\n\"I've waited for more than one of these things,\" said the first customer. \"The trouble with Strahan is, he works so hard because he's just too plain lazy to steal.\"\n\n\"And your problem,\" said the second customer, \"is that you'd rather be drunk than here.\"\n\n\"I'm working on it,\" said the first customer proudly.\n\n\"You'll be a lot of help, when that hurricane starts piling water up around our noses.\"\n\n\"Me,\" said the first customer, \"I'm going to be plenty safe.\"\n\n\"Going to a shelter?\"\n\n\"Naw,\" said the first man, \"I'm going to get as much liquor and beer in me as I can first, and then I'm just going to float around on old Henriette.\" He laughed. No one else did. The first customer was already too far gone to notice the look he got from Henriette. He was very close to being bounced out the door.\n\n* * *\n\nThe sky didn't look very ominous, but the thunderheads above let loose a shower, much like the regular afternoon showers, that lasted fifteen minutes or more. Then the rain stopped.\n\nIt was half-past two. The hurricane was an hour and a half late. An hour and a half more for the people trying to get to shelter, thought Boshardt. He was in the Linhart office, getting reports from the Linhart deputies and the Delochitaches office. Everything seemed to be moving smoothly. Now if the hurricane would only make up its mind. . . .\n\nBoshardt sat back in his chair. He was tired. He thought that sixteen years was long enough to be sheriff. He was forty-five years old. He might think about quitting at next election time. He didn't know what he'd do instead. Maybe move to New Orleans, to please Dorothy. There were a lot of things he could do. He could become the head of some large company's security force. He could teach at a police training school. He could fish and hunt and do all those things that he put off every year. He loved duck hunting. He loved fishing in the Gulf.\n\nBoshardt caught himself daydreaming. All right, Felicia, he thought, make your move. We're as ready as we'll ever be. That's not too ready, but its the best we can do. He looked at the report. Hurricane Felicia, size, winds, position. It didn't make any sense for a moment. Walter Boshardt rubbed his eyes. He was tired. He wanted to go home.\n\nFor a second time, the vision of what Arbier might look like after Hurricane Felicia had gone through it flashed through Boshardt's mind. No, he thought, he couldn't go home. A lot of people couldn't go home. For a while the roads would be impassable. For a while the flood waters would make it impossible to get to the homes. For some, there would be no homes, not any longer.\n\nBoshardt was just plain tired of waiting.\n\n****\n****\n\n**18**\n\nIt was two-thirty in Arbier, and Hurricane Felicia had not sent the faintest sign of her intentions. Pierson kept looking up at the sky. There was just a flat layer of clouds. There had been a shower, but nothing unusual, nothing to make him worry.\n\nNothing except the activity of the men he was with. As he watched, the four men smashed into almost every shop on Ridge Street and emptied it of its valuable contents.\n\n\"Come on, kid,\" said Chuck at one point. \"I hired you to help.\"\n\n\"You hired me to drive,\" said Pierson.\n\n\"I hired you to do what I tell you to do,\" said Chuck in a threatening voice.\n\n\"Sure,\" said Pierson. He was suddenly afraid.\n\n\"Now,\" said Chuck, \"what I want you to do is this. You back your truck up to that plywood covering that store's window, see? You've seen the Rabbit do it. Then you kind of nudge the plywood. It takes a lot of skill, you see. You kind of give the plywood this kind of push, and the plywood will break. That's good, that's what we want. But you don't stop there. You keep nudging and you keep pushing, and the glass behind the plywood will break too. That's our main objective, or at least the first important part. Do you think you can handle that?\"\n\n\"I can handle it,\" said Pierson. \"I don't think I want to, though.\"\n\n\"Why not?\" asked Chuck. \"You can see all the fun we're having all up and down this street.\"\n\n\"Well,\" said Pierson, \"it looks to me like you're looting the town.\"\n\n\"That's swell,\" said Chuck, throwing up his hands in disgust. \"'Looting the town,' the kid says. Just because we're breaking into every shop and maybe later a few homes, he thinks we're looting the town. What kind of kid is this? Where are you from, boy?\"\n\n\"Ohio.\"\n\n\"That explains it,\" said Chuck, nodding. \"'Looting the town.' I like the way you put it. Very, uh, very\u2014\"\n\n\"Succinct,\" said Pierson.\n\n\"Shut up and get to work.\"\n\n\"I don't know if I want to be a part of this,\" said Pierson.\n\n\"You _are_ a part of it,\" said Chuck. \"You've seen enough movies and enough television to know what would likely happen to you if you tried splitting now.\"\n\n\"I'd die an agonizing death.\"\n\n\"And Tom thinks I'm a wise-ass,\" said Chuck. \"So, what are you going to do? The decision is yours. But hurry. You're running out of time.\"\n\n\"I think maybe I'll try nudging this here store.\"\n\n\"Nudge away with a clean conscience, kid,\" said Chuck. \"You were compelled to do it.\" He turned away and went back to his own work.\n\nPierson climbed into his truck and drew a deep breath. Then he swung the truck out into the street, steered it around so that it was perpendicular to the sidewalk, and put the truck into reverse. He climbed the curb with a jolt. Then he felt and heard the plywood covering the store window break. He heard the bell-like tinkling of many fragments of glass as they fell. Pierson shifted gears and let the truck roll forward a little. Then he got out. Chuck was watching him. Pierson climbed through the hole he had made and began transferring the contents of the store into his truck.\n\nPierson looked up the street, after about half an hour. It was almost three o'clock. There was still no sign of the hurricane. Chuck looked back at him and gave him a nod. \"Well,\" muttered Pierson, \"I'm pillaging a town.\" He wondered what came next. Rape was always fashionable, and it made a great team with pillaging. Pierson shook his head. He wasn't the rape type. He had enough trouble with a woman, even when he had her full cooperation.\n\n* * *\n\nCorinne was having a difficult time talking her father into going with her to safer shelter in Linhart. He wouldn't hear of the idea. He wanted to stay in the house. It was a matter of great importance to him, and Corinne couldn't understand why.\n\n\"Dad,\" she said, \"you've told me that I'm your only child. That's why you moved down here. To be near me. But it works the same way for me. You're the only father I have.\"\n\n\"Corinne,\" said the old man, \"I've told you that I've made a decision. Now, do you think I'm senile?\"\n\n\"Of course not,\" she said.\n\n\"Do you think your father is naturally stupid?\"\n\n\"Of course not, Dad.\"\n\n\"Then I have arrived at my decision through normal, reasonable ways, and I'm sticking to my decision. I'm staying here, in Arbier, in this house, in this room, because it's the only thing left that I can hang onto.\"\n\n\"You're not senile, Dad,\" she said, \"and you're not stupid. But you are stubborn, to the point that you're endangering both our lives.\"\n\n\"I'm not endangering your life, Corinne,\" said her father. \"You're supposed to get in your car and drive away, back to Linhart. I'm fine. I've got canned food here and pots of water. I'm fine.\"\n\n\"Dad, just listen to me. It's too dangerous.\"\n\n\"Then go.\"\n\n\"Oh, for God's sake, Dad, I'm starting to get angry.\"\n\nCorinne's father was sitting in an old, faded armchair. He sank back further into the cushioned wings of the chair and sighed. \"You're treating me like a child,\" he said. \"Is that what happens when you get old?\"\n\n\"Yes, Dad.\"\n\n\"Well, Corinne, I'm still your father. I can still tell you to leave me alone and go home. Would you deliberately disobey me?\"\n\n\"Yes, Dad.\"\n\n\"Then put on the television. We'll watch it together until Felicia comes.\"\n\n* * *\n\nThe party in Room 6 of the Sea-Ray Motel wasn't yet going at full swing. The truckers Darlaine had hoped would come were off doing something.\" They had made some kind of nebulous excuse and gone away, promising to come back later. Now there was just Darlaine, the black hooker, John Smith, and the day manager. They had a couple of bottles of liquor, and they were already progressing nicely through them. The sex part was over, as far as Darlaine was concerned, until the truckers got back. John Smith had had his audition, and the day manager didn't rate one.\n\n\"You think this place is really safe?\" asked John Smith.\n\n\"There have been hurricanes before,\" said the day manager. \"You're going to get a lot of water coming through under that door. We're going to have to seal it off. But, other than that, we're set. This motel isn't made out of cardboard, you know.\"\n\n\"Yeah,\" said Darlaine, \"it's mostly dried mud built up around a framework of sticks.\"\n\n\"If you're getting worried\u2014\" said the day manager.\n\n\"Never mind her,\" said John Smith. \"Mrs. Boshardt has her little jokes.\"\n\n\"I know,\" said the day manager, \"I see most of them as they stop in for the key.\"\n\nJohn Smith flushed with anger, but he said nothing. He went over to the black hooker. \"How come you hang around a place like this?\" he asked.\n\n\"You mean, 'What's a nice girl like me doing in a place like this?' Is that what you asked me?\" said the black woman.\n\n\"No,\" said John Smith. \"I didn't mean it that way. I was just wondering what was so special about Arbier. I wouldn't think you'd have much business.\"\n\n\"I'd have a hell of a lot more if it weren't for that mother,\" she said, pointing to Darlaine. John Smith laughed. Darlaine turned around and gave them both a cool smile.\n\n\"You want to hit me again with a whiskey and water?\" said the day manager.\n\nDarlaine got up and made the drink. She thought about sort of spilling it on the day manager, but she decided that was just too bitchy. She gave him the glass without a word.\n\n\"Thanks,\" he said. Darlaine grunted a reply. She stretched herself out on one of the beds.\n\n\"Well,\" said John Smith, \"any time now.\"\n\n\"And then what?\" asked Darlaine.\n\n\"The hurricane,\" said John Smith.\n\n\"And then what?\" asked Darlaine.\n\n\"That's when the fun starts.\"\n\n\"Yeah?\" asked Darlaine. \"And then what?\"\n\n\"Then I suppose you get it from fifteen directions at once by the truckers, me, and that guy,\" said John Smith.\n\nDarlaine stared at him for a few seconds. \"And then what?\" she asked at last. No one said anything for a while after that.\n\n* * *\n\nAt three o'clock, Chuck gave his crew the signal that they were finished in Arbier. They closed the tailgates on their trucks and got ready to leave. Chuck came back to talk to Pierson. \"Listen,\" said Chuck, \"the plan is to drive out the old parish road. We're meeting the rest of the gang at this place called, uh, wait a minute, it's a place\u2014\"\n\n\"Bar's Mike and Grill?\" asked Pierson.\n\n\"No, no,\" said Chuck, annoyed, \"shut up and let me think.\" There were several seconds of silence. \"Yeah, I got it. This place is run by this German guy, name of Weiss, but he changed it to Blanc down here. We stop there, and Stan and Ed will meet us there.\"\n\n\"All right with me,\" said Pierson.\n\n\"I hope so,\" said Chuck. \"You hang in there behind Old Mole, and you'll have Marsh Rabbit and me and Denny behind you.\"\n\nPierson said nothing. He climbed up into his truck and started the engine. He waited for Old Mole to swing his eighteen-wheeler out and around. Pierson followed slowly as they drove out Ridge Street. He could see in the mirror that Marsh Rabbit and Denny\u2014Cracker Smacker on the CB, which Pierson didn't have in his rental truck\u2014were following closely behind.\n\n\"Oh, Lord,\" muttered Pierson, \"if you get me out of this, I promise to be good. I won't even masturbate again, and I won't sleep late on Sundays, I may even go to church, and maybe I'll marry Maddie.\" He thought for a few seconds. \"That last part, Lord,\" he said, \"uh, don't hold me to that. Maybe all the rest, though. You hanging in there, God?\"\n\nThe small convoy of trucks passed the Sea-Ray Motel, and Pierson looked at it. \"Goodbye, old motel,\" he whispered. He wondered if he would ever see it again.\n\nIt was right then that fear hit him.\n\nIt was right then that he realized that he was in a very dangerous situation. He was in a situation that could lead to some very terrifying possibilities. He was party to a massive felony already. He was stuck in the middle of a bunch of trucks filled with stolen goods, things that had been looted from the empty town of Arbier. If they were caught. . . .\n\nIf they _weren't_ caught. . . .\n\n\"Oh, Lord,\" whispered Pierson.\n\n* * *\n\nThe trucks made their way up the old parish road, about seven miles from Arbier, to Blanc's. They parked their trucks by the side of the road and went into the roadhouse. Denny and the Rabbit went straight to the bar and ordered drinks. Old Mole went to the pinball machine in the back of the place. Pierson stood just inside the door. Chuck watched him. He knew that he'd have to keep a sharp eye on that kid. The kid was a weak link in their carefully planned operation.\n\nChuck paced around the barroom. He ordered a gimlet. Old man Weiss, or Blanc, had to look the drink up in a book to find out how to make it. It didn't come out right, but Chuck drank it anyway.\n\nPierson came into the bar and stood by Old Mole, watching the trucker play pinball. The trucker was good. Chuck watched Pierson from the bar. He knew that the kid was probably trying to figure out how to get away. Chuck couldn't let that happen. It was a little after three o'clock. Stan and Ed should be coming pretty soon. Then they could all get the hell out of this hayseed parish.\n\nChuck wished that Stan and Ed would get there. He went to the door and looked out. There was nothing moving anywhere. There were no birds, no airplanes, no cars, no people. Nothing but sugar cane.\n\nChuck sat down at the bar and had another drink. He thought that he ought to be careful. He didn't want to be half-crocked when Stan and Ed arrived.\n\n\"How come you're still open when there's a hurricane coming?\" asked Pierson.\n\nBlanc shrugged. It was that same old shrug that Pierson had seen so many times since he had come to Arbier.\n\n\"I'll tell you,\" said Blanc, in a German-tinged accent, \"I'm ready to go if it gets bad. But it might not get bad. You know how much money it costs to shut down the oil things out there in the Gulf and fly the men away and lose all that time and work? You know how much it costs for big companies to shut down for a couple of days because of a lousy storm? Millions of dollars. Me, I don't have millions of dollars. So when they show me a hurricane, I go to Linhart; meanwhile, I stay here and sell you drinks.\"\n\n\"Oh,\" said Pierson, \"I see.\" He went to the back of the room and played the pinball machine. His score was so bad, he put another quarter in, hit the reset button just to clear the score off, and left the game free for whoever wanted to play it. Chuck watched all of that carefully. The kid didn't seem to be much of a problem after all. To Chuck, Pierson didn't seem very bright.\n\n\"You stay open here because a guy in Miami Beach paid you to, right?\" asked Chuck.\n\n\"That was a consideration,\" said Blanc. \"But think. When the hurricane comes, how much is my life worth? What I received from the gentleman in Miami Beach? No.\"\n\nNo is right, thought Chuck. Tom probably overpaid the guy.\n\n* * *\n\nAt four o'clock, there was still no sign of Hurricane Felicia. \"What's the latest status?\" asked Sheriff Walter Boshardt, sitting in the office in Linhart.\n\n\"The same,\" said Deputy Auguste. \"Felicia is still stalled, same size, same wind velocity, no forward progress.\"\n\n\"How long are we going to have to wait for the thing?\" asked Marty Theriot.\n\n\"Until the thing isn't a danger,\" said Boshardt. \"That could take hours. A hurricane stalled like that, I don't know. But it's starting to get on my nerves.\"\n\n\"You think it's getting on your nerves,\" said another deputy. \"You should see what it's like over at one of the emergency shelters.\"\n\n\"No, thanks,\" said Boshardt. He pushed his chair back and put his feet up on the desk. \"I wouldn't want to set foot inside one of those shelters. Like at the high school. You got all those families and kids and crying babies, and everyone has about eight square inches to sit in, and there's nothing to do, hour after hour, and everybody gets mad at everybody. I'll stay here and play sheriff. You deputies can go round up strays and take them to the shelters.\"\n\n\"If only the damned thing would move,\" said Theriot.\n\n\"You just want something to do,\" said Boshardt.\n\n\"Sure, I want something to do.\"\n\n\"Don't let the tension get to you,\" said the sheriff. \"It reduces your efficiency.\"\n\nTheriot made an expression of disgust. \"I'll take care of my efficiency,\" he said.\n\n\"Why don't you go out and rouse the countryside?\" said Boshardt.\n\n\"I did that all morning,\" said Theriot.\n\n\"Do it some more. If the storm starts to move, get back here for orders. This office will be the command center for the whole parish now. Keep moving, direct anyone you see to the nearest shelter, stay in your car and listen to the radio. We'll let you know when that hurricane makes its play.\"\n\n\"All right,\" said Theriot. No pinball, no speed traps. It was a bad day. Marty Theriot had a personal grudge against Hurricane Felicia. He didn't know how he could even the score, but he was going to give it some thought. He went out to his patrol car and drove through the streets of Linhart.\n\n* * *\n\nAt four o'clock, Chuck was very impatient. He was pacing around in front of the door to Blanc's. Stan and Ed hadn't shown up yet. Chuck was anxious to get moving. The longer the trucks waited by the side of the road\u2014the longer Chuck and his men had to spend in a little roadhouse on an old parish road in St. Didier Parish\u2014the greater were the chances that they would be caught making the break for safety.\n\nChuck was feeling a little sick. He didn't like this part of the operation. In the four trucks out there were thousands of dollars worth of stolen goods. Hundreds of thousands of dollars. There was even a Jaguar convertible that Old Mole had gotten into his van. Old Mole was very proud of that.\n\nChuck wasn't feeling proud. The first part of the plan had gone off without a hitch. Into the town, loot it, and out. Clean, perfect, not a soul around. But Stan and Ed had a more difficult assignment, and they might take longer to reach this rendezvous. In any event, each passing minute made Chuck's stomach feel worse. His head hurt, too. He decided that it would be best if he didn't drink any more. Old Mole, Marsh Rabbit, and Denny didn't seem to share his anxiety. Pierson, of course, was ready to break from the strain, too. That kid from Ohio was probably ruining his mind, trying to come up with some way of escaping. Chuck was keeping close watch on Pierson, but that was just another source of aggravation.\n\n\"When were they supposed to be here?\" asked Denny.\n\n\"They were supposed to meet us here at three,\" said Chuck.\n\n\"Well, I wish they'd hurry up. I'm getting hungry.\"\n\nHungry, thought Chuck. That's all he has to worry about.\n\nMarsh Rabbit came up to the bar and got a Coke. \"How was that nigger lady last night, boy?\" he asked.\n\n\"Nothing special,\" said Denny. \"About what you'd expect in a town like that. Nothing spectacular. But she got the job done.\" His smile broadened. \"Yeah, she sure did get the job done.\"\n\n\"I want to get out of here,\" said Chuck in a low voice.\n\n\"That town was a pretty poor excuse for a place for people to live in,\" said Marsh Rabbit.\n\n\"Go see if Stan and Ed are coming,\" said Denny.\n\nChuck got up from his stool and went to the door again. Once more, there was nothing to see. The trucks, that was all. Nothing else.\n**19**\n\n****\n\nAt five o'clock, four hours after the initial prediction of Hurricane Felicia's arrival, there was still no change. Sheriff Boshardt realized that the situation was serious, even without the hurricane. What he and his deputies had talked about so light-heartedly an hour before was now becoming true. The people in the shelters might be growing into an uncontrollable mob. The tension was building. There had to be some way of releasing it. There had to be a way, but in all the years of Boshardt's experience, he had never learned what to do in this particular situation.\n\nAbout every half hour the office received calls from the emergency shelters. There was nothing for the sheriff to say beyond what the people on television were saying. The condition was stable. There was no movement evident in the storm.\n\nBoshardt picked up a telephone and dialed his home number. He didn't expect to get an answer, and he didn't. Then he called the emergency shelters in Linhart and tried to have his wife paged. He had no success with that either. He called the shelter in Delochitaches. Once more, there was no word of his wife. Either she wasn't responding to the paging, or she wasn't at any of the shelters. Boshardt was worried. Neither he nor the deputy had seen her car at the Sea-Ray. Still, it was just like her to do something foolhardy in an emergency like this. Boshardt hung up the phone, frustrated. There was nothing more he could do. He returned his attention to the matters at hand.\n\n* * *\n\nDeputy Sergeant Marty Theriot cruised the streets of Ward One, which included Linhart. He was surprised to see that a large number of people in that town were still out. He stopped whenever he saw a pedestrian, warned the person to go to the nearest emergency shelter, and gave directions. Then Theriot would drive on. He listened to the police call channel of his two-way radio, but there was little new information being broadcast.\n\nTheriot, like the people in the shelters, like Boshardt himself, was growing impatient. The storm was four hours overdue. There was always the possibility that the hurricane had turned and was going for Texas or somewhere farther east along the Louisiana coast. But then, he reminded himself, radar fixes on the storm would show that, and the sheriff would have the news from New Orleans. There was nothing anyone could do but wait and prepare.\n\nTheriot gave some thought about driving along the old parish road south to the old causeway over Bayou Chien Mort. The small communities along the road were composed of houses and shacks that had no chance for survival. He was sure there were many people still in those communities, people who thought the masking tape on the windows and the canned food and the tub full of water would see them through the hurricane. But Theriot knew that Hurricane Felicia would knock those houses into splinters of wood fallen down around the tub full of water and into a lot of small pieces of glass with masking tape on them. The people who lived in those houses and shacks would be counted in the casualty list.\n\nHe drove through the town of Capita and was dismayed to see how many people were planning to ride out the hurricane in tiny shacks made of planks and boards, sheets of galvanized metal, and tar paper. The small dwellings looked so weak that Theriot imagined that he could topple them himself, if he gave himself a good running start.\n\nHe stopped in front of one of the shacks. A woman stood in the doorway. There were three young children hanging on to her. The woman was curious but not worried. \"Afternoon, ma'am,\" said Theriot, getting out of his car.\n\n\"Hi,\" said the woman.\n\n\"How many people you got living in this house?\" he asked.\n\n\"We got, uh, let's see.\" The woman paused for a moment, her brows drawn together in concentration. \"We got me and my husband, the kids, there's five of them, and we got my mother living with us.\"\n\n\"Well, you'd best get to a shelter in Linhart,\" said Theriot.\n\n\"How we goin' to do that?\" asked the woman. \"We don't have no car.\"\n\n\"You don't have a car,\" said Theriot absently. He was thinking about communities like this, scattered all over the parish, and people like this woman living with seven others in a thrown-together shanty. \"You know about the hurricane?\" he asked.\n\n\"Sure,\" said the woman.\n\n\"You think you'll be safe here?\"\n\n\"I wouldn't be here if I didn't.\"\n\nTheriot frowned and shook his head. \"I hate to say this, ma'am, but, uh, I don't think you're safe here.\"\n\n\"I don't see why not,\" said the woman. \"We been living here for a while, now, and the place has held up pretty good.\"\n\n\"Has this place ever been hit by a hurricane while you were here?\" asked Theriot.\n\n\"No, but we're far enough from the Gulf, my Larry says, and we figure just to ride out the storm here.\"\n\n\"Ma'am,\" said Theriot, \"if that hurricane comes this way, or even if it misses this town so you get the slow edge of the storm, this whole town is going to be flattened like something big stepped on it. You get me? These houses are going to be lying around in pieces, and underneath the pieces are going to be you and your Larry and your kids and your mother and everyone else all along this road.\"\n\nThe woman shook her head. \"What are you trying to do, scare me? All right, if we could get to a shelter, maybe we'd go. But these people here can't get to a shelter. Larry, he works at the sugar refinery. But right now he doesn't have a job. We get checks from the government, but between you and me, they're awful small. And we just don't have money for no car.\"\n\nFrom inside the shack came the sound of a baby crying. \"I got to go, mister.\"\n\nTheriot just stood there, wondering what to do. \"Right, ma'am,\" he said, \"thanks.\" He turned and went back to his car. He switched on his microphone and called the office in Linhart. \"Robert One, this is Robert Eight.\"\n\n\"Go ahead, Robert Eight.\"\n\n\"Look. I'm sitting by a small row of shacks. The things are so pitiful, they look like they'd fall over if a dog lifted its leg and went on them. Can't we get a bus or something down here?\"\n\n\"We'll look into it on this end, Robert Eight.\"\n\n\"Check. 10-4,\" said Theriot. He put the microphone back. He had done his duty, and now he was going to find a bar open in Ward One. Most of the lounges along the old parish road were closed and boarded up. But there had to be a place open in the town itself.\n\n* * *\n\nAt five o'clock, Chuck was almost on the verge of nervous collapse. Denny and the two truckers were also getting a little concerned. Pierson was long past that stage.\n\n\"You have any idea where they are?\" asked Denny.\n\n\"How would I know?\" said Chuck. He was practically shouting.\n\n\"You know what I'd do?\" asked Denny.\n\n\"Uh, no, I don't know what you'd do,\" said Chuck.\n\n\"What I'd do, see, I'd call Tom in Miami Beach and find out if he knows what's going on.\"\n\nChuck stared for a moment. Then he slowly nodded his head. \"Yeah,\" he said, \"that's a good idea.\" He took out a couple of dollars and stepped up to the bar. \"Hey, Monsieur Weiss, or Herr Blanc, or whatever, give me some change for this.\"\n\nThe bartender gave Chuck a nasty look. \"Maybe I'll just close up here and go up to Linhart,\" said Blanc.\n\n\"Just give me the change.\"\n\nBlanc went to the cash register and made change for Chuck.\n\n\"Thanks,\" said Chuck. He went to the pay phone and put through a call to Miami Beach. When the switchboard operator answered, he said, \"Room 566, please.\"\n\n\"Thank you,\" said the operator. Chuck listened to the telephone ringing. \"Your party doesn't answer, sir,\" said the operator.\n\n\"Let me have the front desk, then,\" said Chuck.\n\n\"Thank you,\" said the operator.\n\nChuck heard the phone ring again, and a male voice said, \"Desk, Can I help you?\"\n\n\"Yes,\" said Chuck. \"I have a friend staying at your hotel. I just called his room, and there was no answer. I just wanted to check and see if I had the right room number.\"\n\n\"What was the name?\"\n\n\"Tom Smith,\" said Chuck. \"I thought it was Room 566.\"\n\n\"Just a second,\" said the desk man. \"Here. Tom Smith, Room 566. He checked out of the hotel about two hours ago.\"\n\n\"I see,\" said Chuck, not understanding at all. \"Did he leave any kind of message or forwarding address.\"\n\n\"No, sir.\"\n\n\"I see. Sorry to bother you.\" The desk man started to tell Chuck how it wasn't any bother at all, but about two syllables into the sentence Chuck hung up the phone. He walked to the bar and stood between Denny and Old Mole.\n\n\"Well?\" asked Denny.\n\n\"Tom checked out two hours ago. No forwarding address.\"\n\nDenny frowned. \"I don't like that.\"\n\n\"Me neither,\" said Chuck.\n\n\"I think we're going to have to do us some big thinking,\" said Denny.\n\n* * *\n\nA telephone rang in the Linhart branch of the parish sheriff's office. One of the deputies answered. He listened for a moment. Then he said, \"I think you'd better talk to the sheriff himself.\" The deputy looked around. Boshardt was just coming out of the men's room. \"Sheriff,\" said the deputy, \"I think you'd better take this call.\"\n\n\"I don't have time,\" said Boshardt. \"Let Captain Brierrer take it.\"\n\n\"I really think you'd better talk to this man,\" said the deputy.\n\nBoshardt gave the deputy a strange look, but took the telephone. \"Hello?\" he said.\n\n\"Is this Sheriff Boshardt?\"\n\n\"Yes.\"\n\n\"This is Isaiah Columbier.\" He was one of the few blacks in Arbier. The sheriff knew him well. \"I just wanted to let you know that the whole town of Arbier done been trashed.\"\n\n\"What was that?\" asked Boshardt.\n\n\"Ridge Street look like it been hit with an A-bomb. They's broken wood and broken glass all over. And the stores and shops, they's all empty now.\"\n\n\"Isaiah,\" said Boshardt, \"why aren't you in Linhart, in an emergency shelter?\"\n\n\"Well,\" said Columbier, \"I was down in the basement of the kids' school, you know, way down, 'cause I figured no hurricane is going to get me down there. And I wait and I wait. And I wait, and there ain't no hurricane. So up I come, and I look around, and I see that every store in town done been cleaned out.\"\n\nBoshardt chewed a knuckle while he thought. \"Isaiah,\" he said absently, \"you're going to drown down there when the hurricane comes.\"\n\n\"I'll be all right.\"\n\n\"You come on up here, fast,\" said Boshardt. \"I don't want to have to go and drag your body out of the muck.\"\n\n\"I think maybe you'd best come down here,\" said Columbier.\n\n\"Yeah,\" said Boshardt. \"That's what I'm thinking about.\"\n\n* * *\n\n\"I do believe that it's time to start hitting the road,\" said Marsh Rabbit.\n\n\"It's five-thirty,\" said Denny. \"Let's roll 'em.\"\n\n\"Wait a minute,\" said Chuck, \"I'm in charge here.\"\n\n\"Right, right,\" said Old Mole, \"give us the order, and we'll follow you into the yawning mouth of Hell.\"\n\n\"Old Mole likes to talk like that,\" said Marsh Rabbit. \"Before he was a trucker, he was an instructor of freshman English at some eastern college.\"\n\n\"Let's get going,\" said Chuck. \"Mole up front. Denny behind. The kid and me. Rabbit, you got backdoor.\"\n\n\"Groovy,\" said Denny.\n\n\"What?\" said Chuck.\n\n\"I said, 'Groovy,'\" said Denny.\n\n\"That's what I thought you said,\" said Chuck. \"I don't believe it, though. Come on.\"\n\nThe truckers and Pierson left Blanc's and climbed into their rigs. Old Mole eased his eighteen-wheeler onto the old parish road. Denny followed. \"All right, kid,\" said Chuck, \"you just act nice.\"\n\n\"And nobody will get hurt,\" said Pierson. \"Isn't that what desperate men always say?\"\n\n\"I'm not especially desperate,\" said Chuck.\n\n\"The way you were acting in the bar, I thought maybe you were waiting for something to hatch.\"\n\nChuck said nothing for a while. He was thinking. The plan seemed to have developed a hitch. It looked like Stan and Ed weren't coming to the rendezvous at Blanc's, and Chuck and his crew couldn't wait any longer. They wanted to get out of St. Didier Parish, to follow the old parish road north, take the road to Linhart, then Hanson Highway to Route I-10. Once they were on the interstate, they were in the clear. But Chuck really wanted to get out of the parish.\n\nWhile Chuck was planning, Pierson was assessing his own predicament. Chuck would probably not want to let Pierson go, because Pierson could go straight to the cops. But Chuck had to get rid of Pierson sooner or later. Pierson drove and thought. Maybe Chuck would say, \"Hey, kid, you're part of the gang. Welcome to the wonderful world of crime.\" No, really no.\n\n\"What time is it?\" asked Chuck.\n\nPierson took a quick glance at his watch. \"Going on six o'clock.\"\n\n\"Wonderful,\" said Chuck sourly.\n\n\"That hurricane is five hours late,\" said Pierson. \"Maybe it turned or something.\"\n\n\"You're dumb, kid,\" said Chuck.\n\n\"Yeah, well, I try to do the best with what my parents gave me.\"\n\n\"You're still real dumb.\" Chuck didn't say anything for a while.\n\nPierson thought about trying to make an escape. Once they got into Linhart, on the way to Hanson Highway, he could open the door and jump to safety. At maybe forty miles an hour, onto the pavement. He'd be real safe. He scratched that plan and started another.\n\n\"There's a few things I don't understand,\" said Pierson after a while.\n\n\"I'll believe that,\" said Chuck.\n\n\"No, I mean, why go to all this trouble, just for four trucks of television sets and toaster-ovens? You're taking an awful big risk, just doing it. I suppose you'd know that the town would be empty, but the hurricane could have hit us anytime. We could have been killed.\"\n\n\"I already told you, kid,\" said Chuck, \"you're dumb. You are awful dumb. You haven't figured it yet. Anybody else would, but you, kid\u2014\"\n\n\"I know,\" said Pierson, \"I'm dumb.\"\n\n\"Yeah,\" said Chuck. \"Let me tell you a little secret. There ain't no hurricane.\"\n\n\"What?\" said Pierson, bewildered.\n\nChuck smiled. \"I love that look on your face. You look like a doctor's just told you that you were pregnant.\"\n\n\"What do you mean about the hurricane?\"\n\n\"We're all part of a gigantic scheme, a monstrous plot against the safety and well-being of hundreds of thousands, maybe millions of people,\" said Chuck.\n\n\"You're kidding.\"\n\n\"Yeah,\" said Chuck.\n\n\"Is there a hurricane?\" asked Pierson.\n\n\"No, goddamn it, no!\"\n\n\"Then what about\u2014\"\n\nChuck cut him off. \"Look. Tom's got a guy and a gang in New Orleans, see? Since Saturday, they've been feeding all this phony information about this phony Hurricane Felicia.\"\n\n\"What about the radar units all over, and the Air Force and Navy planes, and all that?\"\n\n\"Doesn't make any difference. Our guy in New Orleans would take that information and send out to Linhart the rigged stuff. This parish is such a nuthouse, nobody would notice the difference. Not unless they had gone to some other place, like Texas or somewhere, and realized that the television weathermen weren't mentioning any hurricane. And then these people would have to come back here today and try to convince someone in charge that the Linhart station was wrong. No one would listen. We timed it just like this afternoon\u2014zip in, zip out. The plan wouldn't work anywhere but someplace like this. Nobody goes anywhere. Nobody talks to you. Hell, their idea of a good time is to go down to the department store and try on gloves.\"\n\n\"There isn't a department store in Arbier.\"\n\n\"That's what I mean, kid. You're really dumb.\"\n\n\"And all the cops\u2014\"\n\n\"They're getting people to shelter. All the cops will be in Linhart. That's something we have to watch when we drive through there, but we have phony log books, in case we're stopped. There's only one little thing wrong.\"\n\n\"What's that?\"\n\n\"You, kid,\" said Chuck. \"What am I going to do with you?\"\n\n\"Nothing,\" said Pierson, \"I won't tell.\"\n\nChuck looked up, as if to heaven. What he saw was the ceiling of the cab. He threw up his hands in disgust. \"He won't tell, he says. Look, kid, do you promise? Oh, kid, are you dumb.\"\n\n\"So you put this gang of thugs\u2014\"\n\n\"Not 'thugs,'\" said Chuck. \"I don't like that.\"\n\n\"All right, you put this gang of soldiers of fortune in New Orleans, right? They take over the weather service there. Now, to tell you the truth, it's not very likely and I don't believe it. But, okay, they feed Linhart and St. Didier Parish all this phony stuff. But just to get four trucks of junk? It just doesn't seem worth it.\"\n\n\"There's more, kid,\" said Chuck. \"In a job planned by Tom, there's always more.\"\n\n* * *\n\nSheriff Boshardt was speeding down Hanson Highway, from Linhart to Arbier. There was a growing, complaining doubt in his mind. If he saw what he expected to see in Arbier, there was going to be a lot of trouble. Trouble for him, the kind that could end his professional life. Trouble for the town of Arbier. Trouble for the parish.\n\nAs he crossed the northern corporation limits, he saw that the trucks that had been parked across from the Sea-Ray were gone. It wasn't more than a minute later that he saw that what Isaiah Columbier had told him was true. The town had been looted. Looted of everything. Boshardt drove slowly down Ridge Street. Every shop window had been smashed open, the protective wood broken into kindling. Even old man de Crout's fish place. Why would anyone loot a fish store? Boshardt turned down W. 4th Street, where the nicer homes were. Some of their front doors had been broken open, too. Boshardt sat in his patrol car and frowned. For a moment, he was paralyzed by the sight the town of Arbier presented him with. Then he knew that he had to do something about it. He picked up his microphone. \"Robert One, this is Robert Two. Do you copy?\"\n\n\"Go ahead, Robert Two.\"\n\n\"I want you to call the weather service in New Orleans.\"\n\n\"10-4, Sheriff.\"\n\n\"Right,\" said Boshardt. He waited until he got a reply.\n\n\"Robert One to Robert Two. Can't get through, Sheriff. There's just a recorded announcement giving the latest statistics on the hurricane.\"\n\nBoshardt frowned. He took a notebook from the glove compartment of his car. \"Listen up, Robert One. I have an unlisted number I want you to try.\" He gave the deputy the number, sat back again, and waited. He had a sick feeling in his stomach, and his mouth and throat were dry.\n\n\"Robert Two, the same recorded announcement.\"\n\n\"10-4, Robert One. Thanks. Call Channel Five and find out where Strahan gets his information.\"\n\nThe wait was a little longer this time. It seemed like hours to the sheriff. While he was waiting, he called the Delochitaches office. \"Robert Four, this is Robert Two.\"\n\n\"Go ahead, Robert Two.\"\n\n\"I want roadblocks on Hanson Highway north of Linhart. I want roadblocks on the road leading northwest from Delochitaches. I don't care if it takes every man and car in the parish. If you need help, let me know. Get those deputies off their asses in Linhart.\"\n\n\"Check, Robert Two.\"\n\nAfter he finished talking to Delochitaches, Boshardt got a call from Linhart. He was told that Strahan took a lot of his own readings, but his main source of information was fed to him through New Orleans.\n\n\"Thanks,\" said Boshardt. \"I got to put two and two together.\"\n\n\"What's up, Sheriff?\" said the deputy in Linhart.\n\n\"We just got suckered into one of the bigger con jobs I've ever seen. I don't think there _is_ a hurricane.\"\n\nThe Delochitaches station called back. \"The roadblocks aren't a problem, Sheriff. What are we looking for?\"\n\nBoshardt stared straight ahead. \"We're looking for three tractor-trailer rigs with the town of Arbier inside.\"\n\n* * *\n\n\"Simple plan,\" said Chuck. \"Tom likes to keep things simple. The organization is split into three parts. You got me and Denny here. You got Nelson in New Orleans. And there's Stan and Ed in Stiles Creek.\"\n\n\"Stiles Creek?\"\n\n\"Louisiana Power and Light nuclear generator. An inside job. Stan knows the place inside out. We're all going off somewhere with this stuff that Stan and Ed are taking. They have about, I don't know, fifteen guys with them. There will be only a skeleton crew at the generator, because of the hurricane. A piece of cake for Stan and Ed.\"\n\n\"Except they didn't show.\"\n\n\"Forget it,\" said Chuck. \"We're going to build ourselves a bomb, the way I get it from Tom, and then we can make ourselves a whole pile of money.\"\n\n\"Clever, nice,\" said Pierson. \"And these trucks?\"\n\n\"The sheriff sees the town. He figures the town's been looted. All of his attention goes there. Stan and Ed do the job at Stiles Creek and meet us at Blanc's. We all get away, and the future looks bright.\"\n\n\"Except they didn't show.\"\n\n\"Forget it, I said. Maybe they got hung up somewhere.\"\n\n\"I just had a thought,\" said Pierson. \"Try this one on. You said in Blanc's that Tom checked out of his room. Where is he now?\"\n\n\"I don't know, you dumb bastard,\" said Chuck.\n\n\"Where are Stan and Ed now?\"\n\n\"I don't know.\" Chuck was beginning to sound worried.\n\n\"You're a little slow, but you'll pick it up,\" said Pierson. \"Chuck, _mon ami,_ you've been thrown to the wolves.\"\n\n\"Naw,\" said Chuck. His eyes were wide.\n\n\"Where are we going?\" asked Pierson.\n\n\"Linhart, and then I-10.\"\n\n\"And then where?\" asked Pierson.\n\n\"I don't know.\"\n\nThere was a short, painful silence. Finally, Pierson looked over at Chuck. \"Boy are you dumb,\" he said.\n\nJust then, it started to rain.\n\n* * *\n\nIn a hotel in Miami Beach, a telephone rang at the main desk in the lobby. A man working behind the desk picked up the phone. \"Desk,\" he said.\n\n\"Right,\" said Tom. \"I was out for a while. Were there any messages?\"\n\n\"I'll look,\" said the desk clerk. There was silence for a while. \"Yes,\" said the desk man, when he returned to the phone. \"That man from Louisiana called.\"\n\n\"Somehow I just knew he would,\" said Tom.\n\n\"I gave him your message,\" said the desk clerk. \"I told him that you had checked out already. If he calls again, shall I tell him the same thing?\"\n\n\"I wish you would,\" said Tom. \"You don't know how aggravating it can be sometimes.\"\n\n\"I understand,\" said the desk clerk. \"Are you planning to check out soon, then?\"\n\n\"Oh, no,\" said Tom. \"Another few days, I think. Just tell whoever calls that I'm gone. I don't think anyone but Chuck Smith will call, though.\"\n\n\"Thank you, sir,\" said the desk man.\n\n\"What about that storm?\" asked Tom.\n\n\"The hurricane? Felicia? Should be no trouble to the Miami area. I wouldn't worry about it. In fact, from what I recall hearing on the radio at lunch, it should be hitting the Louisiana coast just about now.\"\n\n\"Just about now?\" said Tom. \"How about that. I sure feel sorry for whoever gets caught by it.\"\n\n\"Yes,\" said the desk man, \"I've seen hurricanes. They're no fun at all.\"\n\n\"Oh, I don't know,\" said Tom cheerfully. \"They can be fun to watch. From a distance. I'll read all about it in the paper tomorrow.\"\n\n\"Will there be anything else, Mr. Smith?\"\n\n\"Yes,\" said Tom. \"Could I have some ice sent up?\"\n\n\"You'll have to call Room Service, sir.\"\n\n\"I'll do that. And thanks for delivering my message to Chuck Smith.\"\n\n\"Quite all right, sir,\" said the desk clerk. \"We've dealt with that kind of thing before. We've had many celebrities here.\"\n\n\"Celebrity,\" said Tom. \"Sure, I like that. Thanks.\" He hung up the phone.\n\n* * *\n\nThe rain got worse, and the wind was rapidly getting stronger. Pierson looked at Chuck. Chuck chewed his lower lip, but didn't say anything for a while. Finally, he said, \"Come on. Stop the truck and come with me. I'm going to try to get Stan and Ed on the CB.\"\n\n\"Are they driving a truck?\" asked Pierson.\n\n\"Ed is,\" said Chuck. \"He has a CB in it. Stan has a CB in his car.\"\n\n\"Nice,\" said Pierson, anxiously looking at the weather as it worsened, making driving difficult.\n\n\"Tom's like that,\" said Chuck. \"He always says that good communications make for better, uh, what you call. . . .\"\n\n\"Communications?\" asked Pierson.\n\n\"No, no, I mean, uh. . . .\"\n\n\"Teamwork?\"\n\n\"Yeah, sort of,\" said Chuck. Pierson stopped the rented truck. He saw in his mirror that Marsh Rabbit had stopped also. \"You're coming with me,\" said Chuck.\n\n\"I'd love to,\" said Pierson. Together they climbed out of the truck. The wind was very strong, and the rain was coming down in almost solid sheets of water. In a few seconds, they were drenched. They ran back to the Rabbit's tractor-trailer rig. Chuck opened the door on the passenger side and climbed in. He slammed the door. Pierson opened it and climbed in beside Chuck.\n\n\"You could wait outside,\" said Chuck.\n\n\"I might run away,\" said Pierson.\n\n\"Not likely,\" said Chuck. \"But thanks.\"\n\n\"Sure,\" said Pierson.\n\nChuck picked up Marsh Rabbit's microphone. He used channel ten on the CB, calling for Stan and Ed. There was no response.\n\n\"You're wasting our time,\" said Pierson.\n\n\"Maybe they're on another channel,\" said the Rabbit.\n\nChuck turned to channel eleven. He tried again. No response.\n\n\"Come on,\" said Pierson nervously, \"this storm is getting worse.\"\n\n\"Let me have that,\" said Marsh Rabbit. He took the microphone away from Chuck and switched to channel nine, which is used only in case of emergency. \"Break nine,\" said the Rabbit, \"what's the weather situation, come on? This is the Marsh Rabbit, good buddies, and the water's coming down like all hell.\"\n\n\"Break nine for the Marsh Rabbit,\" came the reply. \"This is Jackson Fenmore. It's a hurricane, good buddy. A hurricane. Felicia.\"\n\n\"10-4,\" said the Rabbit. \"Mercy sakes and 10-4.\"\n\n\"We'll see you,\" said Fenmore.\n\n\"Hope so,\" said the Rabbit. \"Hope to do that. We gone.\" He hung up the microphone. He looked at Chuck. The expression was unpleasant.\n\n\"A hurricane,\" said Chuck. \"What a coincidence.\"\n\n\"Stop it,\" shouted Pierson. \"Can't you understand? This is a big job, you idiot, bigger than even you. That's why the trucks were late. Tom wanted the hurricane. You're supposed to get caught in it. You're just wasting our time here. We got to keep moving.\"\n\n\"What a coincidence,\" murmured Chuck, stunned.\n\n\"Let's get back, let's get moving,\" said Pierson.\n\n\"Sure,\" said Chuck. \"Better take it easy on the hills in this weather.\"\n\n\"Hills?\" asked Pierson. \"What hills?\"\n\n\"Let's get back to our truck. Don't try no funny stuff,\" said Chuck.\n\n\"I don't have to,\" said Pierson. \"Tom's done enough for all of us for at least the next ten years.\"\n\n\"I can't get over it,\" said Chuck. \"I mean, Tom watching for the storm to grow and sending me right into the damn thing. It's got to blow the whole job. We'll never make it away from here.\"\n\n\"Maybe it's some kind of initiation,\" said Pierson, climbing down out of Marsh Rabbit's cab. He had to shout to be heard.\n\n\"No,\" said Chuck, \"I already been through that. It wasn't as bad as I thought.\"\n\n\"Is this?\"\n\n\"Worse,\" said Chuck, still a little dazed. \"A lot worse.\"\n\n* * *\n\nIt was about six-thirty when the rain started to come down so hard that Boshardt could barely see the road ahead of him. He pulled over onto the shoulder and waited until the squall had passed. Boshardt was on his way back to Linhart. He wanted to catch those trucks and he personally wanted to put the cuffs on that guy Chuck. \"A film company!\"\n\nThe rain subsided a little, but didn't stop completely. The wind through the cane made it look like an ocean, rolling with waves. The wind was from the northeast, and it got stronger and stronger. The rain came down harder. Boshardt tried driving slowly along Hanson Highway, but he found it difficult, against the wind and rain.\n\n\"Robert Two, this is Robert One.\"\n\n\"Go ahead, Robert One,\" said Boshardt, peering forward through the wall of rain.\n\n\"The hurricane is moving.\"\n\n\"I got you, Robert One.\" What would he do now? He had thirty men, covering the whole parish. Should he keep up the roadblocks or use his men to rescue stranded people? Well, thought the sheriff, if it came to a choice between human lives and a bunch of electric popcorn poppers. . . .\n\nThe wind grew. Water forced its way into Boshardt's car, even though the windows were shut as tight as he could make them. In a matter of minutes, the winds had grown to gale force. As he tried to drive against them, he felt the push of the wind trying to blow him off the road. Before Boshardt had covered half the distance to Linhart, he was trying to drive against full hurricane strength winds. He gave up. It was impossible.\n\nThe two-way radio still worked, but the static caused by the storm made communication difficult. Boshardt got through a message that the deputies manning the roadblocks should instead be used in search and rescue operations. The sheriff knew that things were going to get a lot worse, very soon. \"Be careful,\" said Boshardt. \"Use your judgment. We want to save as many lives as we can, but not at the expense of our own.\" Boshardt could barely hear the acknowledgment from Linhart, because of the static interference.\n\n* * *\n\nThe howl of the wind was a terrible experience. The shrieking soon became unbearable. Darlaine and her party at the Sea-Ray cheered when the first hard rain started to fall, but it was only a matter of minutes before the black woman said that the constant noise was making her nervous. John Smith smiled. The rain beat against the window and door. It began to come in, driven by the full force of the wind. The palm trees in the courtyard of the Sea-Ray were bending with their tops almost to the ground.\n\n\"This is it,\" said the day manager.\n\n\"Yeah,\" said Darlaine. \"Somebody give me a cigarette.\"\n\n\"Here,\" said John Smith.\n\nDarlaine took the cigarette and lit it. The noise from the wind was so bad that no one spoke. They sat and watched television. Skip Strahan was explaining that Hurricane Felicia had begun to move inland, that the town of Arbier could expect hurricane force winds for about two hours.\n\n\"Two hours of that racket,\" said the black woman. \"It's driving me nuts.\"\n\n\"You should see the rain,\" said the day manager, peering through the window. \"It's all dark out there, and the rain is coming down like hell.\"\n\n\"This isn't as much fun as I thought it would be,\" said Darlaine.\n\n\"It isn't much fun at all,\" said John Smith.\n\n* * *\n\nIn the marshes to the west of Arbier, the bayous swelled and grew. Livestock drowned, whole herds of animals trying to fight winds of one hundred miles per hour or more perished. People who were foolish enough to go out were stung by sand and pebbles, shells, debris. The sand stung and drew blood. The wind made it almost impossible to move. A person couldn't stand against winds that strong; one could barely crawl to safety.\n\nThere was a strange kind of beauty about the hurricane, too. Along the small beach south of Arbier, the sand was blown away, and the static electricity generated by the wind-driven sand made a myriad firefly lights. But in order to see the beauty of the storm, one had to risk its dangers. Most people thought it wasn't worth it. Others did, and many of them died.\n\n* * *\n\nThe water moved inland. The sea marched forward, like an army fighting house-to-house battles. Slowly one part of the town would be surrounded by the water. A few minutes later, the water had reached halfway up Ridge Street. The water moved on and grew deeper.\n\n* * *\n\nThe spiral shape of the hurricane allowed the sheriff moments when he could see a little of the road ahead. The rain never stopped completely, but in between the spiral bands, between squalls, he drove carefully onward. The water was rising, and the road was almost covered. He tried to call Linhart, but he had to give up, because the static on the radio and the gigantic roar of the wind made communication impossible.\n\n* * *\n\n\"How long is this going to last?\" asked Darlaine. She was already tired of the storm, and the endless pounding of the rain and wind were getting on her nerves.\n\n\"He'll tell us,\" said John Smith, pointing to the television.\n\n\"Maybe we should have gone to Linhart,\" said the black woman.\n\n\"We could go now,\" said Darlaine.\n\n\"In that?\" asked John Smith. \"You're kidding.\"\n\nDarlaine wasn't kidding. She was afraid. But she wasn't going to let the others know it.\n\n* * *\n\nCorinne and her father sat in the room and tried to block the noise of the wind. She turned up the television as loud as she could, but the noise from outside overpowered everything else.\n\n\"It's all right, Corinne,\" said the old man, \"sit down.\"\n\n\"I'm sorry, Dad, but the storm is making me nervous.\"\n\n\"Why? Because it's a hurricane? Just think of it as a bad thunderstorm. That's all.\"\n\nIn Arbier, the water crept in farther, grew deeper.\n**20**\n\nIt was about seven o'clock, and Hurricane Felicia was still growing. She was dropping her burden of rain on the Louisiana coastline. The winds along the outside of the storm were not as strong as the winds close to the eye, but they were strong enough to make driving almost impossible.\n\n\"Let's go,\" said Chuck.\n\n\"Right,\" said Pierson. \"If you don't like the way I'm driving, then why don't you take the wheel? It's all I can do to keep us on the road. I can't see where I'm going, and the wind has the broad side of this truck to blow against.\"\n\n\"All right,\" said Chuck. \"This storm is our way to get out of this mess.\"\n\n\"You think so?\"\n\n\"Yeah,\" said Chuck, \"everybody will be too busy with the hurricane to worry about us. We can drive the trucks somewhere, dump them, and go home.\"\n\n\"A swell day's work, I think,\" said Pierson grimly. \"Anyway, you determined that this was a hurricane. How did you get that notion?\"\n\n\"I'm figuring that maybe you were right. For all I know, maybe Nelson was never in New Orleans, and Stan and Ed. . . .\" His voice trailed off.\n\n\"Aw, come on,\" said Pierson. \"It's not so bad. Everybody gets used once in a while. That's the way the human race works. That's evolution. You got a user and a usee, and the fitter of the two survives.\"\n\n\"Shut up,\" said Chuck.\n\n\"You don't know anything, you get set up to take the rap for something you don't even understand, and you still think you're the crime king of the Deep South.\"\n\n\"Shut up,\" said Chuck.\n\n\"Can you see anything out there?\" asked Pierson.\n\n\"Road. There's got to be road.\"\n\n\"I know there's road,\" said Pierson, \"but there's also marshy land that's getting marshier, and bayou that's going to be the mighty Colorado any minute now. I just wanted you to help me tell them apart.\"\n\n\"Well,\" said Chuck, \"what do you see?\"\n\n\"I think I can see the taillights of Cracker Smacker's rig.\"\n\n\"Denny?\"\n\n\"Yeah,\" said Pierson, \"Denny. I thought he was Cracker Smacker.\"\n\n\"He is,\" said Chuck absently. \"Follow his lights, then.\"\n\n\"Right into the bayou, probably. Your mind is starting to fuzz up on you, you know that?\" asked Pierson.\n\n\"My mind is fine.\"\n\n\"I think, under the circumstances, that I can allow myself a biting retort. How about this one: If your mind was so fine, why are we driving through a hurricane in a bunch of trucks full of looted merchandise, without any definite destination in mind?\"\n\n\"We're just going to play it by ear from here on.\"\n\n\"'Play it by ear,' he says,\" said Pierson. \"By the way, where's my hundred bucks?\"\n\n\"We'll talk about your hundred bucks when this merchandise is delivered.\"\n\nThey rode in silence for a while, each concerned with his private thoughts. Pierson knew that he was still in grave danger, although he tried to make the situation seem less like a captor-prisoner affair. Chuck was distracted by the failure of the scheme to conclude as he had expected. He grew angrier as he realized that he had absolutely no way to get in touch with anyone else in the supposed operation. Now he wondered how much of what Tom had planned and confided to Chuck was the truth. It seemed to Chuck that it was very little. He had been made a scapegoat, and he didn't like it.\n\nThe trucks crept on, fighting against the wind and the rain. It took all of Pierson's strength to keep the twenty-footer on the road. \"We have to get clear of the low part of the parish,\" he said.\n\n\"What?\" said Chuck, roused from his angry musings.\n\n\"I said, we have to get clear. If this is a hurricane, you can expect a lot of water to come rushing across the bottom part of the parish. I don't want to see if this truck floats.\"\n\n\"Yeah,\" said Chuck. \"Then push the pedal down and let's get moving.\"\n\n\"There is still the problem of visibility,\" said Pierson. \"I can't see a damn thing out there.\"\n\n\"Well,\" said Chuck, \"if you can't see anything, it's probably the road. If you _can_ see something, it's probably a building or a tree, and you don't want to hit it.\"\n\n\"What a great rule of thumb,\" said Pierson. \"Just drive along into whatever you can't see.\"\n\n\"Can you come up with anything better?\" asked Chuck.\n\n\"I got us this far, didn't I?\"\n\nChuck just turned his head and looked out the window. There was a constant stream of water; it came in the top of the window, where the seal was not perfect. The water ran along the roof of the cab and dropped, every few seconds, into Chuck's crotch. He didn't notice until a sizable puddle had formed. \"Jesus Christ,\" he muttered, trying to slap the water away.\n\n\"Don't worry about him,\" said Pierson. \"He's probably with Tom in Miami Beach.\"\n\n* * *\n\nSkip Strahan was smiling. He held the smile, and held it, and held it, until the director ran a finger across his throat. The station cut to a commercial, and Strahan got up. His shift was over. He could get some rest.\n\nThe first thing he wanted to do, however, was call Corinne. He dialed the number. The phone on the other end rang and rang ten times before Strahan hung up. Maybe Corinne had gone to a shelter.\n\nStrahan went to the dressing room and sat on the cot. He took a couple of Valiums, just in case. He wasn't feeling edgy, but he had put in a long shift and he knew that as soon as he relaxed, the nervousness and anxiety would return. He put his feet up on the cot and stretched out. He was almost asleep when a thought occurred to him. He got up again and tried calling Corinne's father in Arbier. There was dead silence. The telephone lines were down already.\n\nSkip took a deep breath. He didn't think that his wife would be foolish enough to spend the hurricane with her father. More likely she went to get him earlier in the day and bring him up to Linhart. Skip cheered himself with that thought. That was really the most plausible explanation. She had not called him at the station because the facilities in the emergency shelters were limited, and she may not have had access to a telephone.\n\nThat was probably the explanation, he thought, as he stretched out again on the cot. He closed his eyes. He was very tired, and his back and neck muscles hurt. He tried to relax, but he couldn't get the picture of Corinne out of his mind. He saw her clinging to something, with a huge tidal wave raised above her, ready to crush her as it fell.\n\nSkip took another deep breath, and another tranquilizer. He didn't even get up for water. He just swallowed the pill. Everything was fine. Everything would work out fine except for some cowbirds and some Cajuns and some muskrats that would die during Hurricane Felicia's stay in Louisiana.\n\nEverything was fine.\n\n* * *\n\nBayou Chien Mort had grown into a mean, rushing, ravaging flood. The bayou had spawned lesser rivers, and these were hard at work eroding the highway, eating up the ground on both sides of the roadbed, and creating new streams, all running into the swamps of the east.\n\nThe winds were so strong that the sheriff was sure his car was going to be blown over, but he made steady progress whenever a break in the storm would let him see.\n\nAs he passed Couletain Boulevard, he could see a dark shape in front of him. The curtain of water that ran in rivulets down his windshield wouldn't let him see clearly what the object was. He crept forward carefully. The wind was bellowing so loud that the sheriff found himself holding the steering wheel in clenched fists. His knuckles were white. Relax, he told himself. Don't get into a panic.\n\nHe drew next to the dark shape. It was an old, battered car. It was stopped on the side of the road. Hours before, it would have had its left wheels on the edge of a cultivated field. Now a dark, swirling river ran parallel to the highway, joining up with the bayou further south. The river had already cut away several feet of earth.\n\nThe sheriff tried to determine if there were anyone in the car who might need help. He couldn't see anything, looking through his windshield, through the falling torrent, and into the car.\n\n\"Doggone it to hell and back,\" muttered the sheriff. He couldn't leave the car on the side of the road without investigating. That probably meant getting out of his own car and bucking against hurricane winds that most likely could blow him away.\n\nHe pulled his car up next to the other. It was leaning over to the left, where the driver had driven off the road. Yesterday, it wouldn't have been a problem. Today, thanks to Felicia, the car was balanced precariously over a vicious stream of black water. Boshardt slid across his seat. He could peer into the other car. He saw an old woman. He would have to get the woman and whoever was driving the car to shelter.\n\nBoshardt took a deep breath. He opened the door of his patrol car and got out. Immediately, the wind hit him and nearly sent him sprawling. He clung to the door and rose up, using every bit of strength he had. The rain pelted him so hard that he almost cried out. He tasted blood. He didn't know where it came from.\n\nThe old woman in the other car was signaling to Boshardt. He couldn't understand her. Then Lauren, the waitress from the diner, pressed against the window and motioned that the sheriff should transfer the old woman to his car.\n\nBoshardt was having trouble breathing. The rain was so dense that it was hard to take a breath without inhaling a quantity of water. Boshardt had heard stories of people who, for lack of better shelter, tied themselves in trees above the racing floods of water. These people were usually found some time later, drowned by the falling rain.\n\nBoshardt had a better view of the situation than he had in his car, although he had to shade his eyes from the rain in order to see at all. He turned away from the wind. The rain pelted the back of his head. He saw that the arm of the bayou that ran along the roadside was at least six feet deep, judging by the amount of exposed roadbed and earth. The car was teetering over this new outlet of Bayou Chien Mort.\n\nThe sheriff signaled to Lauren to open the passenger's door of her car. Lauren did so. With her help, the sheriff managed to get the old woman in both arms, and to support her as she tried to get out of the old car. The rain hurt her. Boshardt felt sympathy for her; he couldn't tell if the old woman were crying or not. She gasped for breath. Slowly, Boshardt managed to get the old woman into his patrol car. She moved over on the seat and put her head on the dashboard. Now Boshardt could tell. She was crying, in loud, racking sobs. The sheriff wondered what had happened.\n\nHe went back for Lauren. As he turned he noticed that the car had slid forward and to the left. He tried to shout through the storm, but he couldn't make himself understood. Lauren reached out to him, and Boshardt almost had hold of her hand when the car toppled over the embankment and into the black water below. It sank beneath the surface with a brief cluster of bubbles.\n\nBoshardt stared down at the rushing bayou water. He knew that as long as he lived, he would remember the feeling of Lauren's hand slipping out of his and the look on her face as the car carried her down to a quick but ugly death.\n\nBoshardt stood beside his patrol car. He fell to his knees. The wind almost pushed him flat. He hurt. His whole body hurt. But he forgot all of that while he vomited.\n\nIn a little while he stood and made his way back to the car. It wasn't a great distance, but Boshardt's strength had been all but used up. He got into his car. The old woman was muttering, senile, nattering at him, sobbing.\n\nThe sheriff looked at her with loathing. This woman had cost Lauren her life. No, thought Boshardt, he couldn't think that way. But the only alternative was that _he_ had caused Lauren to die. The old, old, wrinkled woman prattled constantly. Saliva ran down her chin. She turned to Boshardt and said, with an unreadable expression, _\"Les grenouilles magiques! Les grenouilles magiques!\"_\n\nHe didn't have any idea what she was trying to say. He only knew that the old woman, who had turned again and was crying softly, would not have been the person he would have chosen to save, if he had been given a choice. And that knowledge, for some reason, made him feel filthy.\n\n* * *\n\nHurricane Felicia moved across the parish like a giant hunting animal, a creature digging out prey wherever a living thing had hidden itself. It crushed its prey, or drowned it, or blasted it with a vast arsenal of debris.\n\nThe strength of Felicia's winds did strange things. Sheets of metal were ripped from roofs and wrapped around poles. A blunt two-by-four was thrown through a tree like a spear. In Arbier, the streets were filling with water. On the water floated broken poles, limbs from trees, wires, chunks and bits and pieces of houses. There were a large number of animal corpses in the water, and some human dead. All these things bumped their way inland as the hurricane pushed the water along.\n\n* * *\n\n\"Are you still having trouble?\" asked Chuck.\n\n\"You mean seeing?\" asked Pierson. \"Yeah.\" They were proceeding cautiously, still on the old parish road, not yet to the causeway. Pierson was thinking about that causeway. He was wondering if it were still there.\n\n\"You think we have a good chance, don't you?\" asked Chuck.\n\n\"A good chance of what?\" asked Pierson.\n\n\"Of getting out of this,\" said Chuck.\n\n\"Sure,\" said Pierson. Both men knew that meant nothing. They were silent again.\n\nThe truck ahead of Pierson swung sharply. Pierson put on the brakes and slowed down while he watched what Denny would do. The Cracker Smacker up ahead brought his rig under control again, and they all continued.\n\n\"That's something, ain't it?\" asked Chuck.\n\n\"The wind?\"\n\n\"Yeah,\" said Chuck, \"It's driving me crazy, listening to it. I wish it would stop. It feels like it's never going to stop.\"\n\n\"It'll stop,\" said Pierson. \"It's coming out of the northeast from right ahead of us. Then the eye comes. Then the storm comes again, from the other direction, right?\"\n\n\"How should I know?\" asked Chuck. \"I'm from Chicago.\"\n\nPierson was about to say something, but he kept quiet. For the moment at least, it seemed to Pierson that Chuck had dropped his pose of dangerous gangster. He was just a frightened guy in a truck. With a hurricane around him and a flood coming behind him. That put him in exactly the same situation as Pierson, and as long as things stayed that way, Pierson felt that he was safe.\n\n\"You know,\" said Chuck after a while, \"I can't stand any more of this.\"\n\n\"Really?\" said Pierson. \"What are you planning to do?\"\n\n\"I don't know,\" said Chuck.\n\n\"Well, I wish somebody had some plans,\" said Pierson. \"The winds are getting stronger, and this old truck, man, I mean, I don't know if I can keep it on the road much longer.\"\n\n\"You'd better keep it on the road\u2014\"\n\nChuck stopped abruptly. Ahead of them Denny's truck was hit by a strong gust, and the truck jackknifed and flipped over. The wheels spun in the rain. There was no sign that Denny had escaped.\n\n\"What do I do?\" asked Pierson.\n\n\"You keep going,\" said Chuck.\n\n\"What about Denny?\"\n\n\"What do I care, what about Denny? Nobody has been thinking too much of what about Chuck.\"\n\nPierson drove the twenty-footer carefully by the tractor-trailer rig. There was still no sign of the driver. In his mirror, Pierson saw that it was impossible for the back door rig, Marsh Rabbit, to get by the jackknifed rig. He saw Marsh Rabbit climb out of his cab and fall to his knees. Then the rain obscured everything between the jackknifed truck and Pierson.\n\n\"You did a good job, there,\" said Chuck, \"squeezing by.\"\n\n\"A job,\" said Pierson wryly, \"just a job, like any other job. In a hurricane, in a convoy of looted wealth and riches.\"\n\n\"What happened to the Rabbit?\"\n\n\"Well, I don't think we have the Rabbit with us any longer,\" said Pierson. \"I saw him climb out of his cab, and I saw him fall.\"\n\n\"Hell,\" said Chuck, \"I hope he's all right.\"\n\n\"Here's an interesting thought,\" said Pierson. \"Supposing you were driving this tractor-trailer, see, and you were way out in the middle of no man's land, and you were in the middle of an out-and-out hurricane, and your truck blows all over the highway, and you're carrying a load of looted stuff, what would you do?\"\n\n\"I don't know,\" said Chuck. \"I'm hoping I won't have to make that decision.\"\n\nMe, too, thought Pierson.\n\nAnother heavy squall broke, and the winds pulled the truck out of Pierson's control. He was almost dragged into the left margin of the road. He knew that if he ever got the truck on that marshy shoulder, there wasn't anything on God's earth that he could do to get it out. He strained on the wheel, and the wind pulled against him, and he pulled, and swore, and kept the truck on the road.\n\n\"I don't know if it's you or the wind, but one or the other is making this ride scary as all hell,\" said Chuck.\n\n\"I think I'll let the storm take all the honors on this one,\" said Pierson. \"Me, I'd prefer just a plain old summer's drive up to Linhart.\"\n\nPierson looked at Chuck. Chuck looked like he wanted to throw up.\n\n* * *\n\nBoshardt was still feeling sick. He felt sick all through his body. He felt sick in places that medicine wouldn't touch but that several jolts of good whiskey liquor might. Beside him crouched the small old woman. She didn't seem to have any knowledge of what was happening around her. She muttered words that Boshardt could make out, now and then, in French. She never said Lauren's name. When Boshardt asked her who she was, the old woman looked at him in a puzzled way. Boshardt repeated the question, in French and English. All he got was that bewildered stare.\n\nThe hurricane was blowing almost directly head-on, but every once in a while a heavy gust would catch the car broadside, and then it took all of Boshardt's skill to keep from going out of control. He was moving forward at about fifteen miles per hour. The sheriff guessed that it was about the same forward speed as Felicia was making. Great, he thought to himself, I'm staying in the same place in the storm. He thought that if he could only go faster, he could outrace the hurricane to Linhart. He knew that he was probably not in the worst part of the storm, the winds at the center, surrounding the eye. If he were there, he would be dead. Those winds could top two-fifty, and his trusty patrol car wasn't about to stand up to that.\n\n\"No, old patrol car,\" said Boshardt to himself, \"you and me, well, we been through a lot together, ain't we, girl? And I think maybe we got one more hard ride left in us. Anyway, we'll sure give it all we got, right, old girl?\" He waited for an answer, and all he could hear was the constant thunderous roar of the wind, and the rain on the roof, which sounded like hammer blows. \"Well, old girl,\" he said after a while, \"maybe we don't have one more ride left in us. Why don't we just lie down right here, open up a couple of cans of Dixie, and pass into history?\" He drove on, talking to himself, keeping his spirits up. The noise that he was driving through reminded him of his childhood, his first hurricane. He shuddered.\n\n\"Well, it's me again, old girl,\" he said nervously. Then he looked over at the old woman tucked into the far corner of the front seat. He realized that he was mumbling to himself in almost exactly the same way as she was. That made him more frightened than the hurricane. And the hurricane made him pretty damn scared.\n\nThere was water standing on the field to the right and left of Hanson Highway. There was water as far as the eye could see. There was at least a foot of standing water everywhere, on the fields as well as on the highway ahead of him. Boshardt grimaced. He hated to go out in that wind again, but he had an idea that removing the fan belt might keep the engine dry as he drove through the water. A lot of wet stuff was going to be kicked back, and the carburetor and distributor could be soaked. He argued, and he finally persuaded himself that while the storm was coming from the northeast, getting out and opening the hood of the car would amount to the same thing. He'd be left with a soaked engine that wouldn't feel up to burning anything.\n\nBoshardt clenched his teeth and drove slowly through the water on Hanson Highway. He drove as quickly as he could without creating waves that would pile up in front of the car. Against all odds, and to his surprise, he made it into Linhart. He arrived at one of the emergency shelters, ordered someone to see to the old woman, and then drank a couple of cups of black coffee. The warmth of the building, the sudden quiet, lulled him. He stood up and stretched. He looked at his watch. In about forty-five minutes, the eye of the storm would reach the coast of St. Didier Parish.\n\nWith the eye came the hurricane surge.\n\nThe surge. A wall of water. Everyone always described it as \"a wall of water.\" But there is no better way of describing it, thought Boshardt. A cubic yard of water, a simple three feet by three feet by three feet, weighed fifteen hundred pounds. And if the surge plus the waves were mounting to twenty-five or thirty feet high, how many thousands and millions of tons of water was that? Smack. Down on Arbier.\n\nEven the town of Arbier wouldn't stop the surge. That land was too low and marshy. With an uncomfortable feeling, Boshardt thought that the surge might well reach Linhart.\n\nForty-five minutes. Then the _real_ trouble would start.\n**21**\n\nThe winds grew stronger as the hurricane moved over the drowned coastline of St. Didier Parish. As the eye of the hurricane approached, the gusts became increasingly dangerous. It was difficult to tell just how fierce Felicia really was. Already, in Linhart, Skip Strahan's anemometer had been ripped loose by the very winds it was measuring.\n\nThe wind played evil tricks as it moved inland, almost as though Felicia knew that she didn't have long to live once she forced her way across country. The wind whipped around houses and buildings and, following the same aerodynamic laws that kept aircraft aloft, pulled off roofs and hurled them away, to crash and break apart.\n\nThe wind tugged at the wood frame building where Corinne and her father were waiting out the storm. The power had gone out some time before. There were no more television newsmen to tell them how the storm was doing. They didn't need to be told. The pressure of the air outside the house was lower than the air pressure inside; the windows popped out with a frightening clatter of broken glass. The hurricane struck the house, again and again. Corinne huddled close to her father. She held on to him as if she were a small child again. They had a battery-powered radio, but even the local station in Linhart had difficulty broadcasting through the storm. All Corinne could hear was static. She turned the radio off. The static made her nervous. The howling of the winds was growing steadily louder. She had to shout to make herself heard to her father.\n\nFelicia pushed water from the Gulf of Mexico ahead of her, mingled with the huge amount of rain she dropped. The water in Arbier rose higher.\n\nCorinne clung to her father. He tried to soothe her, but it did no good. Corinne thought about Skip's pills and how she would like to take a few, fall asleep, and wake up when the whole hurricane was over.\n\nIt was very dark in the house. The dark clouds of the hurricane covered the evening sky. Corinne's father began gently rocking her. She was crying.\n\nThe winds and the force of the water began to tear the house apart. The groaning of the wood frame house as it began to break up was almost swallowed up by the sound of the wind. But Corinne's father heard it. He knew that the house would become no shelter at all in a little while.\n\n* * *\n\nThe huge tidal wave associated with the hurricane was building in the Gulf. It was not a tidal wave precisely; it had nothing at all to do with the tides. It was completely Felicia's doing. It was a long ridge of water almost twenty feet high, and it was moving along with the rest of the water pushed by Felicia. The wave hit at half-past eight. At that time, Corinne was aware that the house was breaking apart. The rush of the wave, the hurricane surge, completed the job. One moment, Corinne was wrapped in her father's arms. The next, the house was destroyed by a massive wave, and Corinne found herself thrashing about, trying to keep her head above the water. She couldn't find her father for a while. The house had been reduced to floating fragments on the black pool that had been the town of Arbier. After several seconds of floundering, Corinne saw her father. He wasn't far away. She struggled to him and supported him. She held onto him with all her strength and soothed him just as he had soothed her.\n\nShe found a large chunk of floating wreckage and climbed on, pulling her father. He gasped for breath. He tried to talk. Corinne hugged him closer. The wind and the rain made it hard to breathe. On a raft made from a part of someone's house, they floated wherever Felicia sent them.\n\n* * *\n\n\"You know what's happening?\" asked Chuck.\n\nPierson's jaws were clenched tightly. He was fighting a deadly battle against the storm. He didn't answer.\n\n\"I said, do you know what's happened? The winds seem to be getting stronger.\"\n\nPierson nodded. He knew that. He could feel Felicia trying to pull the truck out of his control. The muscles of his arms were beginning to feel weak from the strain. Pierson knew that in the personal competition between Hurricane Felicia and Paul Pierson, Felicia would be the victor, in straight sets.\n\n\"I don't like this,\" said Chuck. \"Shouldn't it start to get better by now? How long is this going to last?\"\n\nPierson was tired of hearing Chuck ask that question. \"I don't know,\" he muttered.\n\n\"Yeah,\" said Chuck, \"I know you don't know. You haven't been much of a plus to this operation.\"\n\nPierson turned for a moment to look at Chuck. \"All right,\" said Pierson, \"tell me where I screwed up.\"\n\n\"We ought to be in Linhart by now. We ought to be further than this. We're not even to the causeway yet.\"\n\n\"I'm paying so much attention to the left-to-right movement that I can't spare much time for the forward movement. But we're making progress.\"\n\nChuck stared out the side window. He couldn't see anything but water. Water on the window, blurring vision. Water on the road. Water piling up against the fronts of the lounges along the parish road. \"God, I want to get out of this,\" said Chuck.\n\n\"I'm tired of hearing about it,\" said Pierson.\n\n\"And I'm tired of hearing you complain.\"\n\n\" _Me_ complain!\" The wind had grown in intensity. They had to shout to make themselves heard.\n\n\"Never mind,\" said Chuck. \"How about if we stop at one of these roadhouses, break in, and wait for the hurricane to go away?\"\n\n\"Look,\" said Pierson, pointing to one of the lounges beside the road. He couldn't recognize which one it was, because the sign along with the roof, two walls, and part of a third had caved in and were starting to float about on the water. \"I have to drive through this.\"\n\n\"Just get me out of it,\" said Chuck.\n\nThe hurricane hit them broadside again, and once more Pierson fought against the immense force that was trying to topple him over into the drenched field on the left side of the road. The truck started to skid, and Pierson steered into it, bringing the truck around safely.\n\n\"Oh, God,\" cried Chuck. He was panic-stricken.\n\n\"Never mind that,\" said Pierson. He was watching what was happening ahead of him. His windshield wipers were almost of no value at all, but he could see clearly enough to know that Old Mole was in serious trouble. The tractor was headed almost directly toward the field to the left, while the trailer was pointed along the road. Old Mole brought the truck under control, but the wind pushed the trailer toward the field. This time, the Mole couldn't stop it. The wind won the contest. The trailer was shoved by the wind. As Pierson watched, the trailer began to tip slowly. For a second it seemed balanced on the wheels on the left side. Then, slowly, the trailer fell. It must have made a tremendous noise when it crashed to the ground, but Pierson, only a short distance behind the Mole, heard nothing but the roaring winds. The back of the trailer buckled, and the doors sprang open. The loot that the Mole had so carefully loaded into his van spilled out, all over the parish road. Some of the smaller items were carried along on the sheet of water that covered the road. Television sets broke, and there were displays of silent explosions. Loot was spread in all directions. The Mole had difficulty opening the door of his cab, which now pointed up to the black sky. Eventually, though, the Mole pulled himself out of the cab. The winds pushed him against his truck. He made motions with his hands to Pierson.\n\n\"Do we pick him up?\" asked Pierson.\n\n\"Are you kidding?\" asked Chuck. \"No, we don't pick him up. If he can't keep his damn truck on the road, then to hell with him.\"\n\nPierson didn't like the tone of Chuck's voice. It seemed to him that Chuck was getting a little crazy from the storm. He had reason, of course, thought Pierson. After all, the way he had been played for a sucker, and then having to ride through a hurricane without any goal in mind, and watching the other members of his gang falling like dinosaurs into the tar pits. Chuck was a little way around the bend, Pierson decided. It was an unpleasant conclusion.\n\n\"You want to see my pistol?\" asked Chuck.\n\n\"What?\" asked Pierson.\n\n\"I have this gun, see, and I was wondering if you'd like to see it.\"\n\n\"Not particularly. Maybe tomorrow, when we have more leisure.\" Pierson was driving slowly by the frantically gesturing Old Mole. He left the trucker and his fallen rig behind and continued down the old parish road. They were about two miles from the causeway.\n\n\"The thing is,\" said Chuck, in a low voice, \"I already have the pistol out. It's a .38 caliber Smith & Wesson. It's a nice piece.\"\n\n\"Could I ask a question?\" said Pierson. \"Because, I mean, we've been through a lot today, and I was just wondering why you have the gun out.\"\n\n\"It's the right time,\" said Chuck. \"That was Tom's old line, the bastard. He was always saying that there was a right time for everything.\"\n\n\"I'd like to meet this Tom some day,\" said Pierson. He was straining his eyes to see ahead.\n\n\"Maybe you will,\" said Chuck. \"Maybe you won't. Stop the truck.\"\n\n\"Stop the truck?\"\n\n\"Yeah, right,\" said Chuck, \"stop the truck.\"\n\n\"Why?\"\n\n\"I've got this pistol in my hand is why.\"\n\nPierson stopped the truck.\n\n\"Now get out,\" said Chuck.\n\n\"You're kidding,\" said Pierson.\n\n\"Out.\"\n\nPierson opened the door. It was very difficult. He had to push against the force of the wind. As he got out, the wind slammed the door shut again. Pierson was thrown to the side, like he was a sheet of old newspaper. He fell, sprawled awkwardly in the thick mud of a field. He saw the truck start up again and move slowly forward along the road. \"You're not getting far,\" said Pierson.\n\n* * *\n\nDarlaine sat in the dark motel room. There was no electricity, and there was no party.\n\n\"We should have known the lights would go out,\" said John Smith. He was lying alone on the floor, soaking wet.\n\n\"Look,\" said the day manager, \"it's just something I overlooked.\"\n\n\"Well,\" said Darlaine, \"it seems to me that a hurricane is a damn fine time to start overlooking things.\"\n\n\"What are we going to do now?\" asked the black woman.\n\n\"I got a great idea,\" said Darlaine. \"We send somebody outside, and the rest of us pick an object in the room. Then we call the person back inside, and if he's still alive, he wins.\"\n\n\"Terrific,\" said John Smith.\n\nThe rain water was flowing in smooth streams along the window and from beneath the door, although they had stuffed towels there to soak the water up. The carpet in the room was drenched.\n\n\"How long before the eye?\" asked the day manager. \"I really want to see that.\"\n\n\"The guy on TV said the eye was due around eight or eight-thirty,\" said the black woman, \"but this whole storm was late, so I don't know what to think.\"\n\nThe winds never stopped their roaring. As the eye approached, the winds grew stronger. The walls of the motel started to shake.\n\n\"Oh boy,\" said John Smith, \"the motel's going to fall down.\"\n\n\"It ain't going to fall down,\" said the day manager. \"When they build down here, they build to stand up to these things.\"\n\nThe walls vibrated. With a tremendous crash, the window broke. A small rock had been flung through it by the wind. The rock smacked against a wall and fell to the floor.\n\n\"You know,\" said the day manager in a quiet voice, \"if that rock had hit\u2014\"\n\n\"Sure, sure,\" said Darlaine.\n\nThey had all become very frightened. The walls shook harder. The bellowing of the hurricane was louder, now that the window had broken. Rain came through the opening and quickly began to flood the room.\n\n\"Do we have anything to plug that window with?\" asked John Smith. No one answered.\n\nThere was a soft whimper from part of the darkness. Someone had been injured, cut by the flying glass. \"What's wrong?\" asked John Smith.\n\n\"I hurt,\" said the black woman.\n\nDarlaine stood up. The water was almost an inch deep in the room. It squished under her feet as she walked to the bathroom. She turned on the cold water tap, hat no water came out. Wonderful, she thought, just what we needed. She said nothing. She went back into the room and sat on the bed farthest from the window. She had thought to wet a washcloth for the black woman, but she couldn't. No water. She thought that it might be ironic. After a while she didn't care any more.\n\n\"You know,\" said John Smith in an anxious voice, \"this wall is starting to crack.\"\n\n\"You're crazy,\" said the day manager.\n\n\"I hurt,\" said the black woman. \"I hurt a lot.\"\n\n* * *\n\nPierson remained where he was, lying in the field. He had tried to stand up twice and been blown down both times. Across the road was one of the lounges. He thought about Chuck's idea. He thought about breaking into the bar. He gave it a lot of consideration, but he knew that if he had trouble standing up, he'd have a lot more trouble trying to cross the road, walking almost at a right angle to the force of the wind. The mud was just fine, for now. There was a lot of water on top of the mud, but Pierson could breathe if he turned his head away from the wind and raised it a little.\n\nAfter a couple of minutes, though, he realized that he was in a lot of danger. The rain fell so hard that the exposed parts of his body were red and painful. \"Well,\" he said to himself, \"let's get going.\" He tried to stand again. The mud sucked him back, and the wind pushed him flat. He tried to get to his knees; he succeeded, but the effort cost him a great deal of strength. He knelt in the field, his head hanging down, for another minute, while he tried to get air into his lungs. There seemed to be more water than air in the atmosphere around him, he thought.\n\nWith another effort, he got to his feet. The wind tried again to knock him down, but this time it failed. Step by step, Pierson crossed the road. He was wading through water that was getting deeper every minute.\n\nThe door to the barroom was boarded up. Pierson tried to rip the boards away. He couldn't. The rain stung him like millions of wasps. He was gasping and choking. He thought that he might very well die on that old parish road, unless he could find shelter soon.\n\nHe went around to the side of the bar. The building gave him some protection from the wind, but not much. He saw a small rectangular window set into the wall about a foot over his head. He looked around for a rock. There were only the white shells. He took off his shoe and broke the window. Then he reached up and tried to grab onto the bottom of the window frame. The rough edges of the broken glass ripped his hands. He couldn't pull himself up.\n\nPierson staggered blindly back to the road. Maybe someone would come by, he thought. Maybe the Marsh Rabbit had managed to maneuver his rig around Denny's, and would pick him up.\n\nPierson headed north, toward Linhart. He wanted to get away from the storm. He knew that the farther away from the Gulf he went, the safer he would be.\n\nPierson discovered that the rain, driven by the wind at such tremendous speed, made seeing impossible. He couldn't see. He was walking forward into God only knew what.\n\nHe remembered what Chuck had said. Drive into what you can't see. Pierson wondered if that held true for walking into a hurricane, too.\n\nThe winds had played with Pierson for a little while. Now, as the eye approached, so did the stronger winds. Pierson found it impossible to move. He fell to the ground and curled up. He was bleeding from many little cuts, his hands were torn by the glass. He could barely breathe. The water was rising to cover him like a blanket.\n\n* * *\n\nChuck was not a particularly good driver, even under the most perfect conditions. Now, he found it almost impossible to push the truck through the water that was building up in front of it. He was moving forward at about five miles an hour.\n\nHe was thinking about what would happen if he got safely to Linhart.\n\nSuddenly he felt a cold, shivery feeling. He had already admitted to himself that he might not make it to Linhart. He had to force himself into more positive thoughts.\n\nHe tried one. _When_ he got safely to Linhart, he could ditch the truck, go to an emergency shelter, and wait out the rest of the storm. Chuck looked through the windshield. There was a fraction of a second, as the windshield wiper made its arc, when he could see the road ahead. He had to use those bits of time wisely, or he would drive blindly into the muck and be stranded. It could not be long before the sheriff's office was alerted to the fact that Arbier had been sacked. The county mounties would be out, looking for the trucks. They would find Marsh Rabbit and Denny. A little further along they would find Old Mole. He wanted to be far enough away, possibly safe in a shelter, when that happened. As soon as the hurricane ended, and as soon as regular services began again, he wanted to get out of the parish, out of the state, out of the South altogether.\n\nHe gave no thought to Pierson.\n\nChuck wondered where Tom was. He had a fantasy about meeting Tom at some future time, somewhere, maybe Vegas or Los Angeles or a bar in New York. What would they have to say to each other? He practiced that for a little while. \"Tom!\" he said, pretending. \"Hey, where'd you go?\"\n\n\"To our rendezvous,\" Tom would say.\n\n\"What rendezvous?\" he would ask.\n\n\"Oh,\" Tom would say, \"did I forget to tell you? I'm terribly sorry.\"\n\nThe fantasy ended as the truck stalled out. The water had drowned the engine. Nothing happened when Chuck tried to start the truck up again. He had a very limited notion of what made cars or trucks run. If he went outside, something he really didn't want to do, he couldn't accomplish anything. He couldn't diagnose the problem, he couldn't fix it.\n\nChuck's eyes filled with tears. He was in a dead truck. The wind pelted the windshield of the vehicle with shells that made loud smacking noises. Chuck was very frightened. He sat quietly, his hands in his lap, and stared.\n\nThe idea came to him that if the sheriff's deputies found the three tractor-trailers, someone was bound to tell them about the twenty-footer. People in Arbier knew there were four trucks. A deputy himself had made out the permits for the trucks. He couldn't stay with the truck. As much as he hated to, as frightened as he was, he opened the door. The monstrous noise of the wind terrified him. He was shaking. He was crying.\n\nHe stepped out into the storm. He was knocked down. He cringed by the side of the truck, trying to breathe.\n**22**\n\nCorinne clung to her father as they swept along on the raft of wreckage. It seemed crazy to Corinne. They were sailing by houses, or parts of houses, that only a few hours ago were filled with people. It was like taking a drive or a pleasant walk, except they were borne on the crest of a deepening flood. The water swirled them out onto Ridge Street, and the water became even more cluttered with debris. Every once in a while, Corinne saw a human corpse, and then she would gasp and turn her head away. Her father was weak, she knew, from the experience of the house breaking up around him. She held him and sheltered his head from the rain. She said soothing things to him, things which she herself didn't believe.\n\nCorinne worried about Skip. He would still be at the station. He would be safe in Linhart. But what if he tried to find her? What if he went to all the emergency shelters and she wasn't there? What would he do? He'd call home, of course, but there was no one there to answer. Then he would get the idea that she had come to Arbier, to get her father, and that she might be in some sort of danger.\n\nWhat would Skip do then?\n\nWould he come down himself, against the hurricane, bulling his way through the winds and the rain and the flood waters? Would Skip do that for her? She smiled. She rocked her father's head in her lap, and told him not to talk. She just wanted to hold him and protect him. She felt surprisingly good. The thought of Skip coming to rescue her made her feel good. The act of comforting her father made her feel good. It was like having a child.\n\nWhen the wreckage turned, sometimes Corinne was exposed to the full intensity of the storm. Then the salt spray carried by the winds made it difficult to breathe. She was stung by the airborne objects that the hurricane threw at her. A tear fell down her cheek, not because she was frightened, but because of what the storm had done to Arbier. There was no more Arbier.\n\nSuddenly Corinne wondered if there would be a Linhart after she and her father were rescued. If not, where would they go? What would Skip do? What would her father do?\n\nCorinne was overwhelmed by the uncertainty. After a while the wreckage turned and the high part, the part Corinne was leaning against, protected her from the wind a little. She saw that the water was not very deep as she passed the ruined shops along Ridge Street. She was floating on a few feet of water. She knew that it would get deeper before the storm ended.\n\nShe floated by the Sea-Ray Motel. She stared at it. Some of the roof had been torn off. Some of the front wall had caved in.\n\nThe Sea-Ray. Carl Steinbrenner. The hurricane.\n\n* * *\n\nThe eye of the hurricane arrived at half-past eight, exactly on schedule. To Pierson, it was a strange feeling. Over a period of about ten minutes, the winds died down until there was hardly a breeze. He looked up, and the sky was dusky and clear. The air felt stifling, almost dead. There was complete silence for a while. After the constant shrieking of the hurricane, the silence hit Pierson. He found it hard to breathe.\n\nHe started down the road to Linhart. He had about a mile to the old causeway, and about fifteen miles after that to Linhart. How long would the calm last? He didn't know. Maybe an hour. He'd never make it.\n\nHe stopped running. He looked instead for shelter, any kind of shelter. There wasn't any. About a hundred yards farther down the old parish road was Bar's Mike and Grill. He ran for it.\n\nThe front of the bar was boarded up. He pulled at the boards with his bloody hands, but he couldn't work them loose. He went around to the side of the bar. There was a window in the front, boarded over long ago, judging by the warped wood and rusty nails. There was a window in the back, taped over. Pierson went to the back of the bar, and there was a wooden plank, left over from the boarding of the front door perhaps. Pierson carried the plank to the side of the building. He wondered if it was too wet to support him, even for the short time he needed. He broke the back window, this time making sure that he had smashed all the glass from the bottom. He leaned the plank against the side of the building. He stepped back a little, took a step and then vaulted onto the plank and up into the window. The plank cracked beneath him, but his head, shoulders, and arms were inside. He struggled for a few moments, and then he fell painfully into the dark men's room of Bar's Mike and Grill. There was a splash of water when he landed.\n\nPierson stood up. His arms, shoulders, and chest were lacerated and bleeding. He hurt worse than he had ever hurt before. He cleaned his wounds as best he could. He didn't know what else to do. There was no running water from the sink.\n\nPierson went out into the dark bar. There were a few inches of water on the floor. He went to the pay phone. He picked up the receiver and listened. There was silence. He dropped in two nickels. The silence continued. Pierson hung up the phone.\n\nThe bar was very dark and very quiet. Pierson thought about washing his wounds with alcohol, whiskey. Then he thought that maybe that could wait a while. He pulled a chair out from a table and sat down. His wounds ached and throbbed, but for the first time in several hours he felt at peace. He took a deep breath, one with no water in it, and smiled. He'd wait out the storm in Bar's Mike and Grill.\n\n* * *\n\n\"You should see it out here!\" said the day manager. The party inside was not excited by the sudden calm. \"Come on! Storm's over!\"\n\n\"Storm ain't over,\" said the black woman. Her voice was very low.\n\n\"It's clear up there,\" said the day manager. \"I can see stars starting to come out.\"\n\n\"My God,\" said Darlaine, \"what a way to spend an afternoon. I should have gone to a movie instead. And to think that I could be in Linhart, right now, listening to Skip Strahan, instead of lying here in the dark. I'm soaked.\"\n\n\"Take your clothes off,\" said John Smith, \"you'll catch cold.\"\n\n\"You come near me one more time, baby,\" said Darlaine, \"and you'll wish your father had stuck to jerking off.\"\n\nJohn Smith didn't say anything. There was absolute silence in the motel room.\n\n\"Why don't you come out?\" said the day manager. \"There's not too much damage really. We got a couple of feet of water and some of the motel is down, but we came out of it all right.\"\n\n* * *\n\nThere was no one to see it, but a pile of water twenty feet high was swelling and rolling toward the beach. It crashed on the shore. The pier disappeared, and the old bait shop. They disappeared as though they had never existed. The water raced into Arbier, smashing buildings, tearing down poles and trees. The flood spent its fury quickly, but the water still moved on, like a fast-running river, down the channels of the streets. Corinne's raft was picked up and carried at a fierce speed and deposited in the cane field between the old parish road and Hanson Highway.\n\nThe water flooded the motel. There was no warning. One moment the water was not quite knee-deep, the next moment the water was chest-high. The party in the motel screamed in fear and horror as the water swept in on them. They sputtered as the water filled the room. Then the water settled. It even receded a bit. The day manager had been pushed against the outside wall of the motel. When he had caught his breath, he called through the window, \"Everybody all right?\"\n\n\"I'm okay,\" said Darlaine, \"it just took me by surprise.\"\n\n\"I'm all right,\" said the black woman.\n\n\"What about John Smith?\" asked the day manager.\n\nDarlaine waded slowly through the water. It reached to her armpits. She went over to where John Smith had been sitting. Her foot bumped against him. \"He's, uh, he's still under the water,\" she said.\n\n* * *\n\nCorinne took a deep breath after the flood spun her all the way out of the town, halfway to the bayou. She cradled her father's head in her lap. She prayed and was grateful that they had come through it all right. She wanted to be rescued. She had a disquieting thought. She was between the two main roads. Perhaps, in the growing darkness, rescue teams might not see her. She didn't know what to do. The worst was over, and she was glad of that. She rocked her father and whispered to him.\n\n* * *\n\n\"You better come in here,\" said Darlaine.\n\nThere was a distant roar from the south. The wind picked up again. What light there had been in the dusky sky disappeared as black clouds moved in. The wind grew stronger as quickly as it had diminished during the approach of the eye. Before the day manager could get inside, the winds were already hurricane strength. There were many objects floating on the surface of the water, and the renewed winds picked them up and hurled them around like shells from a mortar. A pink plastic lawn flamingo that had once decorated the courtyard of the Sea-Ray hit the day manager on the back of the head. He fell, stunned, beneath the water. No one noticed.\n\nThe hurricane freshened and grew stronger. Besides the force of the winds, the storm now had the added impetus of its forward motion. The winds were coming in the opposite direction from the first part of the storm. They would last just as long and do just as much damage. They might cost more lives though, because despite all the warnings on television, radio, and in the newspapers, people insisted on going outside when the eye of the storm passed. Like the day manager, many of these people were caught unsheltered when the storm took up again.\n\n* * *\n\nThe hurricane surge reached as far as Bar's Mike and Grill, but by then it had traveled nearly fifteen miles inland. The water level in the bar raised a few inches. Pierson barely noticed. He was in a lot of pain.\n\nHe heard the increasing howl of the winds and nodded. \"Yeah,\" he said, \"here it comes. Second half kick-off.\" He wondered how Maddie was. She must be in a shelter, he told himself. But then he had the frightening idea that she would have one of her insane notions and decide to stay behind and defend Arbier. Would she really do that? he wondered. No. No. She was in a shelter, probably with her father. She was probably running the place. He smiled. His whole body hurt, but he still smiled. He wished the second half of the storm would end. He wanted to see Maddie. Thinking of her, he lost consciousness and his head fell forward onto the table.\n\n* * *\n\nChuck was in a lot of trouble. He couldn't stand pain, and that's all the hurricane offered him: pain. Physical pain and mental pain. He walked along the old parish road until it came to an end, where the old causeway used to be. Now there was no causeway. Bayou Chien Mort had swollen and torn the causeway from its moorings. There was no way to get across the bayou except by the newer, higher, concrete causeway along Hanson Highway. Chuck thought for a moment. He considered his position. He knew that the hurricane would begin again and attack from behind him. But his goal was still Linhart, and to get to Linhart, he had to cross the bayou. That meant getting to Hanson Highway.\n\nChuck frowned. How could he get to the highway? One, he said to himself, I can walk all the way back down the old parish road, passing as I go Denny, Old Mole, and Marsh Rabbit. Or I could walk cross-country, through the muck of the cane fields, to the highway. That seems to be the only thing to do.\n\nHe started off across the field. He was perhaps a hundred yards from the old parish road when the hurricane winds began again. He shouted and cried. The stinging of the rain was making him insane with the torment. The storm would last another two hours.\n\nIf Chuck had been a Louisiana native, if he had lived in Arbier all his life, he might have recognized the water hyacinths when he saw them. But he didn't. To him, through the heavy veil of rain, the plants looked like some kind of flowering thing growing in the earth. He stepped right off the mucky ground into a weed-choked bayou cut-off. The water was up to Chuck's chin. He struggled forward, but the water got deeper. He stopped. The water was old, stagnant, and foul-smelling. The hurricane's roar and the pelting of the raindrops tortured him. He walked back a step, and then another, and he felt the bank of the cut-off at his back. He turned, hardly able to breathe, and tried to clamber back up onto the field. It took a great deal of effort. He pushed himself up, but the muck slid away as he tried to hoist himself out of the cut-off. His feet were sinking in the muddy bottom. He knew that he had to get out of the water or he would be dead in a few minutes. He raised one foot from the mire and kicked a foothold in the cut-off's bank. Then he pulled the other foot free and tried to hoist himself out. He got both elbows onto the ground, but his foot was stuck fast in the cutoff's bank. He wrenched it and screamed with the pain, but he was able to wrestle his way out of the water. He stopped, prone in the field, with water pooling around his head. He was exhausted. He could barely move his arms and legs.\n\nSomething told him that unless he found shelter, he would die of exposure. He was already worn out just from the work of trying to breathe. The hurricane was throwing things at him, and a rock or stick hurled at the speed of the winds would smash his head to an unrecognizable mass. He should have stayed with the truck, he decided. It was safer inside the truck. He stood up painfully. The wind pushed him down again. He decided to crawl through the deep mud, through the water that was standing on it. The mud wouldn't let him go without a struggle. Every yard forward on his hands and knees was a struggle. It was the hardest struggle of his life, and he was certain that it could be his last. Of that, he had no doubts at all.\n\nHe thought this just before he tumbled down the embankment and into the midnight dark waters of Bayou Chien Mort.\n**23**\n\nIt was almost midnight, and the hurricane had moved on. Felicia would continue to make her presence felt however. Behind her trailed a mass of rain squalls and huge thunderstorms. But the wind had dropped to forty miles an hour, and Sheriff Boshardt made the decision that rescue operations should begin.\n\n\"Wait until morning,\" said the head of emergency operations in Linhart.\n\n\"We can't wait until morning,\" said the sheriff. He was thinking about Arbier, people trapped there, people who had climbed to rooftops to escape the rising water. He was thinking about the people of the marshes, the real Cajuns. These people needed rescuing, and a matter of hours might make all the difference between life and death.\n\nSheriff Boshardt was also thinking about his wife. For the first time in his career, he let a personal consideration influence him in making a decision. But, he told himself, there were enough other reasons to rationalize the decision.\n\nAn emergency rescue vehicle started out from Linhart, pulling a trailer with six boats. Following the vehicle were six cars of the parish sheriff's office. They drove down Hanson Highway. The darkness hid the destruction that Felicia had caused. Because it was near midnight, the rescue workers did not notice that the cane had been flattened and ruined, that the cultivated crops were now lifeless stalks in rows of muck that stretched across the parish. These things would be seen at first light, and they would be seen in silence, because the significance of the destruction was too evident to be spoken of. The governor of the state of Louisiana would fly over St. Didier Parish in a helicopter. He would make a speech about the courageous Cajuns and their heritage. He would ask for aid. The President in Washington would declare St. Didier and surrounding parishes disaster areas, and federal funds would be used to rebuild part of the parish. But there wouldn't be enough money, it wouldn't come soon enough, and it couldn't replace what had been lost.\n\nThe emergency rescue vehicle drove as far south as the fork with the old parish road. From there on the water was too deep. The six boats were unloaded. Each carried one man with a searchlight and another man who rowed and steered the boat.\n\n\"I'd like to go in the first one,\" said Boshardt.\n\n\"Certainly,\" said the deputy who was climbing in. He gave Boshardt the searchlight.\n\n\"Come on,\" the sheriff said. He waited until another boat had been unloaded and manned, and he ordered it to follow him. \"The Sea-Ray is just a couple of hundreds yards from here, right? There were people there. Let's pick them up.\"\n\n\"Check,\" said the man in Boshardt's boat. They moved across the water, with only the slight noise of the oars splashing. The rain continued to fall, and the sky southward, over the Gulf, was lit almost continuously by lightning. Every few minutes there was a loud crash of thunder. Sometimes the rain began to fall especially hard, but the rescue workers tried to ignore it.\n\nIn the lead boat, Boshardt swung the searchlight around until he saw what he was looking for. The Sea-Ray Motel. It was demolished. It would be a long time before the Friday-night couples checked into it again. There was virtually no part of the motel that didn't show some damage. Not a room was left intact. Boshardt played the light along the motel, from one end to the other. The light picked out two people, who had climbed the wreckage of the building and were clinging to a part of the roof. \"Move in closer,\" said the sheriff.\n\nThe boat made its way as close to the motel as it could. Boshardt looked up at his wife and the black woman. \"You climb down,\" he called. \"We'll get you in the boats, and we'll get you out of here.\"\n\n\"Goddamn you!\" shouted Darlaine at her husband. \"It had to be you, didn't it? It had to be you, the hero, right? It had to be you who came and got me!\"\n\n\"Come on down, honey,\" he said.\n\n\"You take this black bitch away,\" said Darlaine in a shrill voice, \"but me, goddamn it, I'll wait up here until Christmas before I let you take me away.\"\n\nBoshardt turned around carefully in the boat. He called back to the boat that had followed him. \"Hey,\" he said, \"you just take Mrs. Boshardt and get her back to the cars. We'll handle the other woman.\"\n\nDarlaine was still screeching at her husband, but Boshardt had stopped listening. The black woman climbed carefully down, and the sheriff helped her into the boat. She was badly cut and her blouse was soaked with blood. \"Let's get her back fast, buddy. She's going to a hospital.\" Boshardt's boat turned and went back to the cars on Hanson Highway. The black woman was transferred to a patrol car, which turned and sped to Linhart. The sheriff ignored Darlaine's frantic, hate-filled screaming.\n\n\"Let's see what the town looks like,\" he said, and his boat started down the channel that had been Ridge Street. Boshardt had seen this picture before, but it still sickened him. How long would it take for the water to drain away? How long before the streets were no longer filled with the vilest smelling mud? Before the flies and corpses hidden away and dead carcasses were removed? How long would it be before there was an Arbier again?\n\nThe rain fell, and the forty mile an hour winds made the rowing difficult. Two of the boats had outboard motors, but the rain, and the chance that the rain would worsen, made the rowboats a better choice.\n\nBoshardt traveled up Ridge Street. The damage done by the looters had been hidden by the water. But the water would go away, and the people would come back, and the repairs would be made, and there would be another hurricane some day.\n\n\"Let's go back,\" he said to his rower. \"Take me back to the cars.\"\n\n\"Yes, sir.\" They turned in the middle of Ridge Street and headed north.\n\n* * *\n\nA patrol car, blue lights flashing and siren sounding, made its way up the old parish road. The siren woke Pierson, and he came to with a start. It was completely dark in the bar, darker than before. But it was quiet. The only sounds he heard were a gentle tapping of raindrops and the siren. The hurricane was over, he decided. He had slept through the second half of the storm. He stood up, and his body, cut and slashed by the glass, hurt. He found that he couldn't stand up completely straight. He could hobble a little, hunched over like an old man.\n\nHe took a chair with him into the men's room. He stood on the chair and carefully slid through the broken window. He cut himself again, but not as badly. He hobbled as quickly as he could around the side of Bar's Mike and Grill. The patrol car had passed the bar and gone farther up the old parish road. Then it stopped, backed up, and turned around. The causeway was out. There was nowhere else for the patrol car to go. Pierson stood in the middle of the road. The patrol car slowed down on the return trip. It stopped just in front of Pierson. A deputy inside slid across the front seat and opened the door. \"Get in,\" said the deputy.\n\n\"Am I under arrest?\" asked Pierson.\n\n\"No,\" said the deputy, \"you're all cut up and bloody. We'll get you up to Linhart and get those cuts taken care of.\"\n\n\"Thanks,\" said Pierson, as the patrol car went around the tractor-trailer rigs that looked like huge sea creatures lying dead on some beach.\n\n\"There were these truckers\u2014\" said Pierson.\n\n\"We know, we know. You'd better believe we know.\"\n\nPierson was anxious. Maybe the truckers would tell the deputies that Pierson had been part of their gang.\n\n\"We're looking for a fourth guy,\" said the deputy.\n\n\"Oh,\" said Pierson. Oh, God, hey, he thought.\n\n\"Drove a twenty-foot rental truck. Chuck Smith. I made out the damn permits myself. I feel great, I can tell you.\"\n\n\"One good thing,\" said Pierson, \"he can't have gone too far in the hurricane.\"\n\n\"Yeah,\" said the deputy, \"and it's still raining and it's still heavy winds out there.\"\n\n\"Yeah,\" said Pierson. He suddenly realized that the three trucks held most of the contents of most of the stores in Arbier. The looting of the town had actually saved a lot of things that would otherwise have been ruined.\n\n\"We'll get him,\" said the deputy. \"If the storm didn't get him first.\"\n\nPierson thought about Chuck. That stupid lerp, he thought. Set up by Tom. Couldn't even figure out that he was the fall guy, set up to be eliminated in a phony operation he'd never understand. No gang in New Orleans. Maybe no gang at the nuclear generator either. Pierson laughed out loud, but there was no humor in it. Chuck, the eternal fall guy. Pierson wondered if he would ever know exactly what Tom's real scheme was, if the news would ever trickle down to St. Didier Parish. Something, maybe nearby, maybe far away. Good old Chuck; just a fall guy, just a diversion? Pierson remembered how proud and confident Chuck had sounded. Where was he now? Dead? About to be captured by the sheriff's men? Dead, a fall guy; captured, a fall guy. And Tom had rid himself of his most expendable colleague. Pierson wondered what kind of man Tom was. For one thing, he didn't mind flinging old partners into the garbage if he needed or wanted to. No, thought Pierson, as he shifted painfully on the seat, I can do without meeting Tom.\n\nBut good old Chuck. Boy, was he dumb.\n\nThe deputy stopped where the old parish road met Hanson Highway. There were three other people who had been rescued. The deputy took all four, including Pierson, to an emergency shelter in Linhart.\n\n\"What do you think those looters were trying to do?\" asked Pierson, his expression displaying curious innocence.\n\n\"I don't know,\" said the deputy as they arrived in Linhart. \"The way I see it, it's the old thing about the good guys against the bad guys. Get out here. This school is one of the shelters. You'll get coffee and hot soup and like that.\"\n\n\"Thanks a lot,\" said Pierson. \"And if it comes to the good guys against the bad guys, I think I'll take the good guys less three points. That's giving them the home field advantage.\"\n\nThe deputy smiled. \"See you later,\" he said, and he headed back down Hanson Highway. Pierson and the three other refugees went into the shelter.\n\n* * *\n\nMost of the people who had remained in Arbier, and who were still alive, had been moved up to Linhart. After some time, a rescue worker caught Corinne in his beam of light. He called to her.\n\n\"I'm fine,\" she called back. \"You'll have to come and get me, though. I don't know if I can walk through all that mud, and my father is a little weak too. We've been sitting here for hours.\"\n\n\"Be right there,\" said the man. He walked across the field to Corinne's little shelter. Each step was difficult. The mud clung to the man's boots, and with each step he pulled his foot up with a loud sucking noise.\n\n\"I don't think I can make it through that,\" said Corinne.\n\n\"Don't worry,\" said the rescue worker, who was now only a small distance away. \"I'll help you, then I'll come back and help your father.\"\n\n\"Take my father first, please?\" asked Corinne. \"The hurricane was hard on him.\"\n\nThe rescue worker bent down to help the old man up. Corinne's father didn't move. His arms were locked. The rescue worker felt how cold the old man was. \"Ma'am,\" he said, \"your father's been dead for a long time. I'm really sorry.\"\n\nCorinne looked up at the rescue worker. She felt very strange inside. She remembered the feeling she had experienced when she floated by the Sea-Ray, the feeling that the hurricane was a punishment for her meeting with Carl Steinbrenner. But this. . . .\n\nCorinne screamed. She scrambled away from the corpse of her father. It wasn't so bad that Corinne had been forced to go through the hurricane; but her father's life. . . .\n\nSurely Carl Steinbrenner wasn't worth _that_ much.\n\n* * *\n\nBoshardt sat in an emergency shelter, drinking coffee and eating a sandwich. He felt disgusted. Ever since the hurricane had started, there had been nothing but frustration and failure for the sheriff. He wanted to quit. He wanted to quit so badly that if there had been a way, he would have gone off in the night and left all this behind. But he had always lived in Arbier. Forty-five years. And he knew that he was capable of standing up to the emergency. But the real truth was that he didn't want to.\n\nHe wanted to go home. The mental image of Arbier under water, with a searchlight playing on the boarded-up shops, the ruined homes, made Boshardt cringe. He felt responsible. He wasn't, of course, but that didn't take the feeling away.\n\n\"All right,\" he muttered, \"let's get the show on the road.\" He had to go question the truckers who had looted the town. He had to go out to the marshes, to check the people in the back. But the minimal roads through there would be under water for days.\n\n\"Who do we have here?\" he asked.\n\n\"Theriot, Auguste, and me,\" said the deputy.\n\n\"Right. Tell Theriot to get his car and follow me.\"\n\nSheriff Boshardt put on his flat-brimmed hat and went out of the emergency shelter in the school. He was chagrined to see that it was still raining and that the winds were still so high. Theriot came running out of the building. As soon as he saw the sheriff, he put on the hat he was carrying.\n\n\"Come on, Marty,\" said the sheriff. \"They're holding the truckers where they caught them, and I want to get to them before those dumb-ass deputies spoil everything.\"\n\n\"All right, boss,\" said Theriot. He got in his car and followed the sheriff.\n\nAs they drove, the sheriff thought about how the hurricane had created so many problems that had nothing to do with a freak of weather. Like Darlaine\u2014Dorothy. What was their relationship going to be like in the morning? What about the owners of the ransacked stores? Could they sort through the stuff in the trucks?\n\nBoshardt thought. The night was dark, the rain made the driving hazardous. Boshardt was being particularly careful. For the most part, the storm was over. But it was foolish to think that it was completely over. Felicia still had some kick to her.\n\nThey slowed and came to a stop about halfway between the causeway on Hanson Highway and the town limits of Arbier. A gust of wind hit the sheriff as he got out of the car. Yeah, he thought, Felicia hasn't finished. It may rain for days. He walked toward the deputies who were holding the three truckers. He took about three steps and then heard a loud, crackling sound. One of the deputies pointed behind Boshardt. The sheriff turned and saw that a powerline had broken, and that it was sparking and spitting on top of Theriot's patrol car.\n\n\"Oh, God!\" cried Boshardt. He ran toward Theriot, gesturing, shouting that Theriot shouldn't get out of the car. He was fine, sitting up on four rubber tires, as long as he stayed in the car.\n\nTheriot didn't understand what the sheriff was trying to say. He grabbed the door handle. The door of the patrol car opened, and Theriot put one foot out onto the wet pavement. Theriot's body jerked. The door swung farther, and the deputy's corpse fell to the ground. Rain ran along his body.\n\nThe sheriff turned away. He had seen too much. He had felt too much. He went to one of the deputies to tell him to call Linhart, because another powerline was down and because they had a deputy dead in the line of duty.\n\n\"Can you handle that, deputy?\" asked Boshardt.\n\n\"Sure,\" said the young man. His voice was very shaky.\n\n\"Fine,\" said Boshardt, \"because I believe I'm about through for the evening.\" He walked back to his car. It was difficult, and he leaned into the winds which tried to knock him down. Not this time, he thought. You've had your chance at me. Not this time.\n\nMaybe next time.\n\n* * *\n\nSkip Strahan woke up. It was dark in the room. He stood up and walked back toward the studio. He met a newsroom director on the way.\n\n\"Fifteen minutes, Skip,\" said the director.\n\n\"Fifteen minutes, what?\" asked Strahan. He was feeling very groggy.\n\n\"Fifteen minutes, you go on. You tell people what the storm's doing, where it's gone. You tell them not to do stupid things, like that. You know.\"\n\n\"Storm?\" asked Strahan.\n\n\"You slept through most of it. Sheila covered for you. We couldn't wake you up. Like you were drugged or something.\"\n\nStrahan felt his face flush. \"My God,\" he said. \"Look, Hal, I\u2014\"\n\n\"Forget it,\" said the director. \"We covered, like I said. But look, Skip, uh, I think you might be in some trouble when this is all over.\"\n\n\"Yeah,\" said Strahan. He was worrying about Corinne. \"I have to go,\" he said.\n\n\"Fifteen minutes,\" said the director. \"Less, now.\"\n\n\"Then I go,\" said Strahan. \"My wife.\"\n\n\"Sure, Skip. After you get off.\"\n\nSkip's stomach muscles tightened as he thought about Corinne. He went back to get his blazer. His hands shook as he poured out a few tranquilizers. He stared at them. He slowly poured them back into the vial.\n\n\"Ten minutes, Skip,\" called the director.\n\n* * *\n\nPierson was resting on a cot. His wounds had been treated and dressed. He was resting. He felt warm and good.\n\n\"Hey, Pierson, you!\"\n\nPierson looked up. It was Maddie.\n\n\"I was waiting to see how long it took for you to brought yourself here,\" she said.\n\n\"You're talking funny,\" said Pierson. He had been given an injection to ease his pain, and everything seemed a little vague. A very good vague.\n\n\"That's because with all these people, sometimes I speak English, sometimes I speak French, sometimes in between,\" she said. She was smiling. She was very happy.\n\n\"Did you have a nice hurricane?\" he asked.\n\n\"Nice,\" she said. \"Not nice-nice, but nice.\"\n\n\"That's nice,\" said Pierson.\n\n\"You fooling me? I don't want to be fooled with.\"\n\n\"No,\" said Pierson. \"What are you doing?\"\n\n\"I am aiding my fellow man,\" she said.\n\n\"How are you doing that?\"\n\n\"I don't give orders, I follow them,\" she said, and she was very proud.\n\n\"Will you take care of me?\" he asked.\n\nMaddie frowned. \"Why should I?\" she asked.\n\n\"Look at me,\" he said. \"I'm bandaged and hurt and weak from loss of blood and everything.\"\n\n\"How did you lose your blood?\" asked Maddie.\n\n\"I crawled through the window of Bar's Mike and Grill.\"\n\n\"That will teach you to stay out of barrooms,\" she said.\n\n\"Even your father's?\"\n\n\"My father doesn't have a bar,\" she said. \"He runs an establishment.\"\n\n\"Right, I forgot.\"\n\n\"You'll never guess what I'm going to be,\" she said.\n\n\"What? I won't even guess. Just tell me.\"\n\nMaddie drew herself up and said softly, \"A nun. Or a nurse. A nurse-nun.\"\n\n\"I want to go to sleep, Maddie.\"\n\n\"You go to sleep, then. There are plenty of other people here.\"\n\n\"You still like me?\" he asked.\n\nShe frowned, then nodded. \"Go to sleep,\" she said quietly. \"By the way, you missed the tornado.\"\n\nPierson's eyes grew wide. \"Tornado?\"\n\n\"Oh, yes,\" said Maddie. \"They sometimes come with hurricanes. Not big tornadoes, though. I wish I'd seen it.\"\n\nThe deputy who brought Pierson into the station some time before stopped by. \"We got those truckers, all right,\" he said. \"They tried to walk across the cane fields between the old parish road and Hanson Highway. Still looking for the fourth guy, though. I think he's lost for good.\"\n\n\"Yeah,\" said Pierson, thinking of Chuck.\n\n\"We just lost a deputy now, too.\"\n\n\"The constant battle of good guys against the bad guys,\" said Pierson. \"The good guys won, all right, but they didn't cover the point spread.\"\n\nThe deputy nodded seriously. He was remembering Theriot. \"You take it easy,\" he said.\n\n\"I've got a new nurse-nun to watch over me,\" said Pierson. \"Maddie, what's a marsh rabbit?\"\n\nMaddie made a face. \"You know m'sieu muskrat? You know they skin him for fur? The part that's not fur, they eat. That's marsh rabbit.\"\n\nPierson shuddered. \"Blech,\" he said.\n\n\"You go to sleep,\" said Maddie.\n\n* * *\n\nChuck scrabbled and fought his way to the surface of the churning river that had been Bayou Chien Mort. He was being carried along, and it was all he could do to keep his head above water. After some time, Chuck grabbed the roots of a large live oak tree and hung there. There was something big floating toward him. It was long and square. When he lost his grip on the roots, he made a grab for the floating thing. It bumped up against the bank of the bayou. Chuck struggled onto the thing. Then the strength of the winds made it difficult to breathe, and the constant pelting of the rain hurt so much. . . .\n\nChuck's arms clung to the floating object. He awoke, and the hurricane was over. It was still raining, and still windy, but it was more like a regular shower. The sky was lighter. He wondered if it was morning.\n\nChuck's fingers found depressions carved into the surface of the thing he was riding. They were letters. They spelled \"Gaudier.\" There were dates. \"My God,\" said Chuck, his voice low and hoarse. \"Oh Lord, I spent the night on a tomb.\" That fact frightened him and amused him at the same time. A tomb. He had been rescued by a tomb. He owed a debt to whoever this Gaudier was. Chuck tried to find humor in the situation. After a short while, after he realized that his arms were too tired to hang on much longer, Chuck admitted that there wasn't much humor.\n\nHe had no idea how a tomb came to be floating in the bayou, but he wasn't going to question his good fortune. The floating tomb came to rest against the bank of the bayou, where the gnarled roots of a tree made an obstacle. Chuck took a couple of deep breaths. He was glad that the hurricane had ended. He was glad that he was still alive\u2014thanks to old Gaudier\u2014and, with luck, he might even get away from the cops.\n\nOther things had sought shelter from the storm, and now one crept slowly toward him. Chuck clung to the tomb. The snake crawled slowly, slowly onto the tomb. It stopped just at the name on the tomb's occupant. The snake raised its head. There was a single row of dull yellow scales on its throat and belly. Chuck had seen a snake like that before.\n\nIt crawled slowly toward him.\n**24**\n\n****\n\n_It was in Chicago, Illinois, twenty-two years before, and it was during President Dwight D. Eisenhower's second term._\n\n_Chuck was ten years old, in the fifth grade. On this particular day, his class and two of the other fifth grade classes were taking a field trip. Chuck liked that a lot. It meant getting out of class, not having to turn in assignments\u2014assignments he hadn't done the night before, because he knew they were all going on a field trip._\n\n_The class went to the zoo. They had a lot of fun, but the teachers looked a little weary from trying to hold their classes together. Some of the boys wanted to watch the sea lions, long after the rest of the class had moved on to the large cats. Chuck liked the cats. Lion, tiger, panther, leopard. They paced their cages restlessly. It made Chuck wonder what it would be like if they ever got loose. All that stored-up energy let loose. Chuck followed his class, but as they left the cats, he turned around and shuddered._\n\n_Mrs. Fry moved her class as though they were a flock of sheep. She picked three or four of the good kids to stand in the front of the mob, and the bad kids followed behind. That was because the bad kids knew that if they didn't follow along, they might not have any more field trips._\n\n_Chuck was bored with their guide, who gave the class all kinds of information about all the animals they were seeing. Mrs. Fry hadn't said that they were going to have a test on it, so Chuck wasn't listening. The guide went on about where this antelope lived, and what this sort of furry thing ate, and what that bird did during mating season. Chuck didn't care at all. If Mrs. Fry did give them a test, he could always complain that they didn't have a warning and it wasn't fair. That was the way Chuck was._\n\n_When they came to the Reptile House, however, Chuck was fascinated. He loved looking at the alligators and crocodiles. The guide told the classes how to tell them apart. One had a pointed snout, one had a blunt snout. For the rest of his life, Chuck would remember that, but he never remembered which animal had which snout._\n\n_The girls hated the snakes, but the boys loved them, probably because the girls hated them. The guide laughed. It sounded like a practiced laugh, because he must have guided grade school classes like this for years, and the boys and girls always reacted the same way._\n\n_The guide told an assistant to remove a snake from its glass case._\n\n_\"Here, boys and girls,\" said the guide. \"I'm going to pass this snake around. Don't be afraid. See? It's not slimy. It's not dangerous at all.\" The girls shied away from touching the snake, but the boys were anxious to see what the thing felt like. It was a disappointment. Chuck held the snake for a few seconds; the snake stuck its tongue out at him._\n\n_After the snake was returned to its case, the guide showed them other snakes. He showed them dangerous ones, like the boa constrictor, which didn't look like it could crush much of anything. The coral snake was small but deadly. The cobra was interesting._\n\n_\"What's that one?\" asked Chuck._\n\n_\"That's a water moccasin, son,\" said the guide. \"Don't worry. You'll probably never see one. Not as long as you live.\"_\nAll rights reserved, including without limitation the right to reproduce this ebook or any portion thereof in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of the publisher.\n\nThis is a work of fiction. Names, characters, places, events, and incidents either are the product of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, businesses, companies, events, or locales is entirely coincidental.\n\nCopyright \u00a9 1976 by George Alec Effinger\n\nCover design by Open Road Integrated Media\n\nISBN: 978-1-4976-0570-1\n\nThis edition published in 2014 by Open Road Integrated Media, Inc.  \n180 Maiden Lane  \nNew York, NY 10038  \nwww.openroadmedia.com\n\n**GEORGE ALEC EFFINGER**\n\nFROM OPEN ROAD MEDIA\n\nFind a full list of our authors and\n\ntitles at www.openroadmedia.com\n\nFOLLOW US\n\n@OpenRoadMedia\n\n# Table of Contents\n\nAcknowledgments\n\nEpigraphs\n\nPART ONE: The Calm\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nPART TWO: The Tropical Storm\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nPART THREE: The Hurricane\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nPART FOUR: Felicia\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nChapter 22\n\nChapter 23\n\nChapter 24\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nTable of Contents\n\nTitle Page\n\nDedication\n\nPraise\n\nPrologue\n\nOne\n\nTwo\n\nThree\n\nFour\n\nFive\n\nSix\n\nSeven\n\nEight\n\nNine\n\nTen\n\nEleven\n\nTwelve\n\nThirteen\n\nFourteen\n\nFifteen\n\nSixteen\n\nSeventeen\n\nEighteen\n\nNineteen\n\nTwenty\n\nTwenty-one\n\nTwenty-two\n\nTwenty-three\n\nTwenty-four\n\nTwenty-five\n\nTwenty-six\n\nTwenty-seven\n\nTwenty-eight\n\nTwenty-nine\n\nThirty\n\nThirty-one\n\nAbbreviations in Notes\n\nNotes\n\nBibliography\n\nList of Plates\n\nAcknowledgements\n\nIndex\n\nCopyright Page\n_by the same author_\n\nIT MUST BE BEAUTIFUL: GREAT EQUATIONS OF MODERN SCIENCE (editor)\n\nTo my mother and the memory of my late father\n[T]he amount of eccentricity in a society has generally been proportional to the amount of genius, mental vigour, and moral courage which it contained. That so few now dare to be eccentric, marks the chief danger of the time.\n\nJOHN STUART MILL, _On Liberty_ , 1869\n\nWe are nothing without the work of others our predecessors,  \nothers our teachers, others our contemporaries. Even when, in  \nthe measure of our inadequacy and our fullness, new insight  \nand new order are created, we are still nothing without others.  \nYet we are more.\n\nJ. ROBERT OPPENHEIMER, Reith Lecture, 20 December 1953\nPrologue\n\n[A] good deal of unkindness and selfishness on the part of parents towards children is not generally followed by ill consequences to the parents themselves. They may cast a gloom over their children's lives for many years.\n\nSAMUEL BUTLER, _The Way of All Flesh,_ 1903\n\nAll it took was a single glass of orange juice laced with hydrochloric acid. A few minutes later, it was clear that his digestive problems were due to a chronic deficiency of stomach acid. For months, he had been admitted to hospital every few weeks to be fed vitamins intravenously, but the doctors had no idea why his digestion was so poor. Now, following the orange-juice experiment, a laboratory test on the chemical contents of his stomach confirmed the conclusion that his stomach contained far too little acid. The simple prescription of a pill to be taken after every meal ended almost eight decades of digestive problems. As a result, Kurt Hofer, the friend who suggested the experiment and made the correct diagnosis, became the reluctant health guru to Paul Dirac, one of the most revered - and strangest - figures in the history of science.\n\nHofer and Dirac both worked at Florida State University but otherwise appeared to have little in common. Hofer - just over forty years of age - was a top-drawer cell biologist, a spirited raconteur who told all comers of his early family life among Austrian mountain farmers and his moment of cinematic glory as a well-paid extra in _The Sound of Music._ Hofer's eyes glittered when he told his stories, his thickly accented voice swooped and surged for emphasis, his hands chopped and shaped the air as if it were dough. Even in this lively company, Dirac was unresponsive, speaking only when he had a pressing question to ask or, less often, a comment to make. One of his favourite phrases was: 'There are always more people who prefer to speak than to listen.'\n\nDirac was one of the pre-eminent pioneers of quantum mechanics, the modern theory of atoms, molecules and their constituents. Arguably the most revolutionary scientific breakthrough of the  twentieth century, quantum mechanics uprooted centuries-old prejudices about the nature of reality and what can, in principle, be known for certain about the universe. The theory also proved to be of enormous utility: it underpins the whole of modern microelectronics and has answered many basic questions that had long defied straightforward answers, such as why electricity flows easily through wire but not through wood. Yet Dirac's eyes glazed over during talk of the practical and philosophical consequences of quantum physics: he was concerned only with the search for the fundamental laws that describe the longest strands in the universe's fabric. Convinced that these laws must be mathematically beautiful, he once - uncharacteristically - hazarded the unverifiable conjecture that 'God is a mathematician of a very high order.'\n\nThe ambitions of Kurt Hofer were more modest than Dirac's. Hofer had made his name in cancer and radiation research by carefully carrying out experiments and then trying to find theories to explain the results. This was the conventional, bottom-up technique of the English naturalist Charles Darwin, who saw his mind 'as a machine for grinding general laws out of large collections of facts'. Dirac, a classic example of a top-down thinker, took the opposite approach, viewing his mind as a device for conjuring laws that explained experimental observations. In one of his greatest achievements, Dirac used this method to arrange what had seemed an unlikely marriage - between quantum mechanics and Einstein's theory of relativity - in the form of an exquisitely beautiful equation to describe the electron. Soon afterwards, with no experimental clues to prompt him, he used his equation to predict the existence of antimatter, previously unknown particles with the same mass as the corresponding particles of matter but with the opposite charge. The success of this prediction is, by wide agreement, one of most outstanding triumphs of theoretical physics. Today, according to the cosmologists' standard theory of the early universe - supported by a wealth of observational evidence - antimatter made up half the material generated at the beginning of the Big Bang; from this perspective, Dirac was the first person to glimpse the other half of the early universe, entirely through the power of reason.\n\nHofer liked to compare Dirac with Darwin: both English, both uncomfortable in the public eye, both responsible for changing the way scientists think about the universe. A decade before, Hofer was  amazed when he heard that Dirac was to move from one of the world's leading physics departments, at the University of Cambridge in England, to take up a position at Florida State University, whose physics department was ranked only eighty-third in the USA. When the possibility of his appointment was first mooted, there were murmurings among the professors that it was unwise to offer a post to an old man. The objections ended only after the Head of Department declared at a faculty meeting: 'To have Dirac here would be like the English faculty recruiting Shakespeare.'\n\nAround 1978, Hofer and his wife Ridy began to pay visits to the Diracs on most Friday afternoons, to wind down for a couple of hours after the week's work. The Hofers set off from their home near the campus in Tallahassee at about 4.30 p.m. and took the two-minute walk to 223 Chapel Drive, where the Diracs lived in a modest, single-storey house, a few paces from the quiet residential street. At the front of the house was a flat, English-style lawn, planted with a few shrubs and a Pindo palm tree. The Hofers were always welcomed warmly by Dirac's smartly dressed wife Manci, who laughed and joked as she dispensed sherry, nuts and the latest faculty gossip. Dirac was painfully spare and round-shouldered, dressed casually in an open-necked shirt and an old pair of trousers, content to sit and listen to the conversation around him, pausing occasionally to sip his glass of water or ginger ale. The chatter ranged widely from family matters to local politics at the university, and from the earnest utterances of Mrs Thatcher on the steps of Downing Street to the most recent sermon from Jimmy Carter in the White House garden. Although Dirac was benign and receptive during these conversations, he was so reserved that Hofer often found himself trying to elicit a response from him - a nod or a shake of the head, a few words, anything to make the conversation less one-sided. Just occasionally, Dirac would be moved to contribute a few words about one of his private enthusiasms - Chopin's waltzes, Mickey Mouse and any television special featuring Cher, the brassy chanteuse.\n\nDuring the first two years or so of these visits, Dirac showed no sign of wanting to talk about himself or of having any deep feelings, so Hofer was ill prepared when, one Friday evening in the spring of 1980, Dirac's vacuum-packed emotions burst into the open. 'I remember it well. It was pretty much like all my other visits except that I was  alone,' Hofer says. 'My wife decided not to come as she was tired, heavily pregnant with our first child.' At the beginning of the visit, Dirac behaved normally and looked alert and ready to absorb the conversations around him. After the customary pleasantries, the Diracs took Hofer by surprise when they ushered him through the formal front room - where they always talked during their Friday chats - to the less formal family room at the rear of the house, adjoining the kitchen and overlooking the garden. The Diracs' pre-war taste was reflected in the decor of this room, dominated by the wood of the floorboards, the panelling on all four walls, and the huge 1920s sideboard covered with framed photographs of Dirac in his prime. A mock-Baroque chandelier hung from the ceiling and, on most of the walls, there were paintings with no trace of modernity.\n\nAs usual, Manci and Hofer chatted convivially while the frail Dirac sat motionless in his favourite old chair, occasionally looking through the glass sliding doors to the garden. For the first half an hour or so of the conversation, he was, as usual, mute but came vibrantly to life when Manci happened to mention his distant French ancestors. Dirac corrected one of Manci's historical facts and began to speak about his family origins and his childhood in Bristol, talking fluently in his quiet, clear voice. Like a well-rehearsed actor, he spoke confidently, in carefully articulated sentences, without pausing or correcting himself. 'I was startled - for some reason, he had decided to take me into his confidence,' Hofer says. 'I'd never seen him talk so eloquently in private.'\n\nDirac described his roots in the rural villages of Bordeaux, in western France, and how his family migrated to the Swiss canton of Valais at the end of the eighteenth century. It was in Monthey, one of the region's industrial towns, that his father was born. As soon as Dirac began to talk about his father, he became agitated, and he turned away from his wife and Hofer, adjusting his pose so that he was staring straight into the fireplace. Hofer was now looking directly at the profile of the top half of Dirac's body: his hunched shoulders, his high forehead, his straight and upward-pointing nose, his white smudge of a moustache. The air conditioning and television were switched off, so the room was silent except for the occasional rumblings of traffic, the barking of neighbourhood dogs, the rattling of the lid on the simmering casserole in the kitchen. After spelling out his ancestry with the precision of a genealogist, Dirac reached the  part of his story where his father arrived in Bristol, married Dirac's mother and started a family. His language remained simple and direct, but, as he began to talk about his childhood, his voice tightened. Hofer, watching Dirac's silhouette sharpen with the fading of the early evening light, was transfixed.\n\n'I never knew love or affection when I was a child,' Dirac said, the normally neutral tone of his voice perceptibly tinged with sorrow. One of his main regrets was that he, his brother and younger sister had no social life but spent most of their time indoors: 'we never had any visitors'. The family was dominated, Dirac recalled, by his father, a tyrant who bullied his wife, day in, day out, and insisted that their three children speak to him in his native French, never in English. At mealtimes, the family split into two: his mother and siblings would eat in the kitchen and speak in English, while Dirac sat in the dining room with his father, speaking only in French. This made every meal an ordeal for Dirac: he had no talent for languages, and his father was an unforgiving teacher. Whenever Dirac made a slip - a mispronunciation, a wrongly gendered noun, a botched subjunctive - his father made it a rule to refuse his next request. This caused the young Dirac terrible distress. Even at that time, he had digestive problems and often felt sick when he was eating, but his father would refuse him permission to leave the table if he had made a linguistic error. Dirac would then have no option but to sit still and vomit. This did not happen just occasionally, but over and over again, for years.\n\nHofer was aghast, scarcely able to believe his ears. 'I felt extremely embarrassed, like I was witnessing a friend pouring out his most terrible secrets to his psychiatrist,' he recalls. 'Here he was, a man famous for equability and his almost pathological reticence, openly talking of the demons that had haunted him for nearly seventy years. And he was as angry as if these awful events had happened yesterday.'\n\nManci barely stirred, except once to bring nibbles and alcohol, and to slow down the preparations for dinner. She knew that on the very rare occasions her husband chose to tell his story, it was best to keep well out of his way and to let him get it off his chest. As the evening turned colder, she brought him a blanket and draped it over his legs, covering him from his lap down to his ankles. Hofer braced himself as Dirac resumed and explained why he was so quiet, so ill at ease with normal conversation: 'Since I found that I couldn't express myself in French, it was better for me to stay silent.'\n\nDirac then moved on to talk about other members of his family: 'I was not the only one to suffer,' he said, still agitated. For thirty-seven years, his mother was locked in a disastrous marriage to a man who treated her like a doormat. But it was Dirac's brother who felt the brunt of their father's insensitivity: 'It was a tragedy. My father bullied him and frustrated his ambitions at every turn.' In what appeared to be a change of tack, Dirac mentioned that his father always appreciated the importance of a good education and that he was respected by his colleagues as a conscientious, hard worker. But this was only a brief respite. Seconds later, Dirac was struggling to control his rage when he spelt out the conclusion he eventually reached about the extent of his debt to his father: 'I owe him absolutely nothing.' That final rasp made Hofer flinch; he could not help but grimace. Dirac hardly ever spoke an unkind word about anyone, but here he was, denouncing his own father with a vehemence most people reserve for the cruellest abusers.\n\nDirac stopped talking abruptly, just after nightfall. His monologue had lasted over two hours. Hofer knew that any words from him would be inappropriate, so he said his subdued goodbyes and walked home, numb and drained. Soon to be a father himself, he reflected on his own youth as part of a close and loving family: 'I simply could not conceive of any childhood as dreadful as Dirac's.' Time tends to embellish, distort and even create childhood memories: could it be that Dirac - usually as literal-minded as a computer - was exaggerating? Hofer could not help asking himself, over and again: 'Why was Paul so bitter, so obsessed with his father?'\n\nLater that night, after talking with his wife Ridy about Dirac's account of his young life, Hofer made up his mind to find out more about it. 'I thought he might open up again during our later gettogethers. ' But Dirac never mentioned the subject again.\n**One**\n\nEnglish home life to-day is neither honorable, virtuous, wholesome, sweet, clean, nor in any creditable way distinctively English. It is in many respects conspicuously the reverse [. . .].\n\nGEORGE BERNARD SHAW, Preface to _Getting Married_ , 1908\n\nAs Kurt Hofer had seen, the elderly Paul Dirac was fixated on his father Charles. But most of Dirac's acquaintances knew nothing of this: at home, he allowed no photographs of his father to be displayed, and he kept his father's papers locked in his desk. Dirac examined them from time to time and talked with distant relatives about his father's origins, apparently still trying to understand the man he believed had blighted his life.\n\nDirac knew that his father had endured a childhood no less miserable than his own. By the time Charles Dirac was twenty, in 1888, he had done three stints of national service in the Swiss army, dropped out of university in Geneva and left home, without telling his family where he was heading. He became an itinerant teacher of modern languages - the subject he had studied at university - and held posts in Zurich, Munich and Paris, before he fetched up two years later in London. English was one language that he did not speak well, so it is not clear why he chose to live in Britain; perhaps it was because it was the world's wealthiest economy, with plenty of teaching jobs at relatively high salaries.\n\nSix years later, Charles Dirac had acquired a sheaf of complimentary references. One, written by the headmaster of a school in Stafford, stated that Monsieur Dirac 'is possessed of very great patience combined with firmness [. . .] I believe he is much liked both by his colleagues and pupils.' His employer in Paris had praised 'his capacity to analyze and generalize, which enabled him to point out my mistakes and help me to ascertain scientifically why they were mistakes'. Charles settled in Bristol, a city famous for the high quality of its schools, and he became Head of Modern Languages at the rapidly expanding Merchant Venturers' School on 8 September 1896, contracted to teach thirty-four hours a week for an annual  salary of one hundred and eighty pounds. He stood out among the teachers because of his conscientiousness, his thick Swiss-French accent and his appearance: a short, stocky, slow-moving man with a drooping moustache, a receding hairline and a face dominated by a huge forehead.\n\nMellowest of British industrial cities, Bristol was known for the friendliness of its people, its mild and wet climate and the hilly roads that wend their way down to the moorings on the river Avon, eight miles from the coast. Bristol was then a thriving manufacturing centre, producing Fry's chocolate, Wills's cigarettes, Douglas motorcycles and many other commodities. Together, these industries had eclipsed the declining trade in shipping, which had been the city's main source of wealth for centuries, some of it based on the slave trade. Most of the city's wealthiest maritime figures were members of the Merchant Venturers' Society, a secretive group of industrialists with a strong philanthropic tradition. It was the generosity of the Society that had made possible the founding of Charles's school together with the high standard of its workshop and laboratory facilities.\n\nDuring a visit to the Central Library a few months after his arrival in Bristol, Charles met Florence Holten, the guileless nineteen-year-old librarian who would become his wife. Though no beauty, she was attractive and possessed features that she would later pass on to her most famous child: her oval face was framed by dark, curly hair, and a firm nose darted out from between her dark eyes. Born into a family of Cornish Methodists, she was brought up to believe that Sunday should be a day of rest, that gambling was sinful and that the theatre was decadent and best avoided. She had been named after the nurse Florence Nightingale, whom her father Richard met during the Crimean War, where he served as a young soldier before becoming a seaman. He was often away for months at a time, leaving behind his wife and six children, of whom Flo was second eldest.\n\nFlo Holten and Charles Dirac were an odd couple. She was twelve years younger than him, a daydreamer uninterested in pursuing a career, whereas Charles was strong-minded and industrious, devoted to his job. The couple had been raised in different, scarcely compatible religions. She was from a family of devout Methodists and so had been raised to frown on alcohol, whereas Charles had been brought up in a Roman Catholic home and liked a glass of wine with his meals. Catholicism had been the cause of riots in Bristol and other  English cities, so Charles may at first have kept his religious beliefs to himself. If he did disclose them, his relationship with the young Flo would have raised eyebrows in her circle.\n\nDespite the possible sectarian tensions, by August 1897 Charles and Flo were engaged, though Flo was feeling sore. Charles had chosen to 'break the spell' of their relationship to visit his mother Walla, a dressmaker in Geneva, leaving his fianc\u00e9e to sulk in Bristol's incessant rain. His father had died the year before. He had been a highly strung junior schoolteacher and later a stationmaster at Monthey station in south-west Switzerland but was dismissed for repeatedly being drunk on duty, leaving him plenty of time to pursue his interest in writing romantic poetry. The Swiss stretch of the Rh\u00f4ne valley had been home to the Dirac family since the eighteenth century, when - according to family lore - they moved from the Bordeaux area in western France. The names of many of the towns in this region and its vicinity end in _-ac_ , such as Cognac, Cadillac and the little-known village, about ten kilometres south of the Angoul\u00eame, called Dirac. Charles believed his family had originated there, but there is no evidence for this among the family records, now stored in the town hall of Saint Maurice (near Monthey), where the colourful Dirac coat of arms - featuring a red leopard with a three-leaf clover in its right paw, below three downward-pointing pine cones - is one of many painted on the walls.\n\nUneven postal delays caused Charles's letters from Switzerland to arrive out of order, infuriating Flo, who wished that 'letters went by electricity like tram cars'; a century would elapse before long-distance lovers benefited from the type of communication she was vaguely envisioning - electronic mail. Lonely and disconsolate, she repeatedly read Charles's notes and, when her family was not looking over her shoulder, replied with newsy letters of how they could not resist teasing her about her pining for 'my own boy'. Struggling to put her longing into words, she sent him a poem full of ardour; in return, he sent a posy of Alpine flowers which she hung round his photograph.\n\nAlmost two years later, Flo and Charles were married 'according to the rites and ceremonies of the Wesleyan Methodists' in Portland Street Chapel, one of the oldest and grandest of Bristol's Methodist churches. The couple moved into Charles's residence in 42 Cotham Road - probably in rented rooms - a short walk from Flo's family  home in Bishopston, in the north of the city. Following custom and practice, Flo stopped doing paid work and stayed at home to do the housework and read about the first skirmishes of Britain's latest imperial venture, the Boer War in South Africa. Soon, she had other things on her mind: the Diracs' first son Felix was born on the first Easter Sunday of the new century. Nine months later, the country mourned the passing of an era when Queen Victoria, having reigned for an unprecedented sixty-three years, died in the arms of her grandson, Kaiser Wilhelm II. Soon after a period of national grief, mitigated only by relief at the ending of the war, the family prepared for a new beginning of its own. In July 1902, they moved into a slot in one of the new terraces on Monk Road, to a roomier, two-storey home that Charles named after his native town of Monthey. The Diracs would soon need extra space as Flo was again pregnant, with only a few weeks to go before the birth.\n\nOn Friday, 8 August 1902, Bristol's eyes were on London, where King Edward VII was to be crowned on the following day. Thousands took the train from Bristol to the capital to see the coronation procession, but the celebrations were a sideshow in the Dirac household. On that Friday morning, Flo gave birth at home to a healthy six-pound boy, Paul Adrien Maurice Dirac. He was, as his mother later recalled, a 'rather small', brown-eyed baby, who slept contentedly for hours in his pram in the patch of the front garden. His mother worried that he ate less food than most children, but the family doctor reassured her that Paul 'was OK, perfectly proportioned'. His parents nicknamed him 'Tiny'.\n\nWhen Felix and Paul were young, they resembled each other, each a quiet, round-faced cherub with a thick bonnet of black, curly hair. Flo dressed them stylishly in thick woollen waistcoats topped with stiff, white-lace Eton collars that reached out to their shoulders, like the wings of a huge butterfly. From family letters and Flo's later testimony, it appears that the boys were close and liked to be with their father, whose top priority was to encourage them to learn. With the virtual absence of visitors and opportunities to mix outside their immediate family, Paul and Felix probably did not appreciate they were being brought up in a singularly unusual environment, a hot-house of private education overseen by a father who would speak to them only in French and a mother who would talk only in English.  According to one witness, the young Paul Dirac believed that men and women spoke different languages.\n\nBut Paul and Felix were let off the leash occasionally. Their mother sometimes took them to the Bristol Downs so that they could play on the vast expanse of grassy parkland stretching from the cliffs of the Avon Gorge to the edges of the city's suburbs. From their favourite spot on the Downs, the Dirac boys had an excellent view of the Clifton Suspension Bridge, one of the most famous creations of Isambard Kingdom Brunel, the charismatic engineer who also left Bristol with its Floating Harbour and Temple Meads railway station, two of the city's finest monuments.\n\nIn the summer, the family would take a bus trip to the beach at nearby Portishead, where the boys learned to swim. Like most families of their modest means, the Diracs rarely took vacations, but, in 1905, they went to Geneva to visit Charles's mother, who had an apartment a stone's throw from the lake and ten minutes' stroll from the railway station. The brothers spent hours by the lakeside statue of the philosopher Jean-Jacques Rousseau, playing together and watching the artificial geyser shoot its jet of water ninety metres towards the sky. When the seventy-year-old Dirac told this story, one of his earliest memories, he liked to point out that his first trip to Switzerland took place at the same time as Einstein was having his most successful spurt of creativity in Berne, only a short train journey from Geneva. That year, Einstein wrote four papers that changed the way people think about space, time, energy, light and matter, laying the foundations of quantum theory and relativity. Twenty-three years later, Dirac would be the first to combine the theories successfully.\n\nThere exist two vivid snapshots of life in the Dirac household in the summer of 1907, shortly before Paul started school, a year after the birth of his sister Betty. The first is the correspondence between Charles Dirac and his family when he was in Trinity College, Cambridge, attending the International Esperanto Congress. Earlier in the year, Charles had qualified to teach the language, which he championed in Bristol for the rest of his life. When Charles was away, his family showered him with loving notes. Flo's affectionate gusto was almost as intense as it had been in the heat of their passion, ten years before. Up to her ears in the chaos of having to look after the three children - taking them for walks, feeding the pet mice,  cooking Paul his favourite jam tarts - she had the undivided attention of her boys: 'It is very quiet without you, the boys are sticking to me for a change.' She assured her husband that his family at home 'all had a nice dinner, mutton, peas, junkets [a sweet dessert]'. The boys missed Charles terribly, Flo told him, just as she did: 'I shall miss you in the bye-bye [i.e. bed] tonight.' Flo enclosed in her letters to Charles notes from Felix and from Paul, who wrote in stick-letter capitals of the welfare of the mice and, most importantly, his love for him: 'Tiny hopes Daddie has not forgotten little Tiny' and 'I love you very much. Come home soon to your own Tiny Dirac xxxxx.' Charles replied with a postcard, written mainly in English but with a little French, promising to bring home some Esperanto chocolate and concluding, 'I would not go out if I did not have to.'\n\nNothing in this loving correspondence bears any sign of the terrible home life that Dirac described to Kurt Hofer. Charles's use of English words appears to be inconsistent with the French-only linguistic regime that Paul claimed his father practised, and his father's tone bears no sign of the heartlessness that Paul remembered.\n\nIt is clear that Charles was as keen as any other father to keep a photographic record of his children. At about this time, he purchased a camera - probably one of the fashionable Kodak box Brownies - to take pictures of his children, many of them showing Felix, Paul and Betty reading avidly. Charles also wanted a portrait of his family to be taken by a professional and for the result to be printed on postcards for family and friends. The photograph, the only surviving image of the entire family, was taken on 3 September and gives us the second impression of the Diracs in 1907. Flo looks demure and serious, her long hair tied up at the back, baby Betty on her lap. Felix is leaning towards her, smiling broadly and looking directly into the camera like Paul, whose left arm rests on his father's right leg, apparently seeking reassurance. Charles leans forward to the camera, eagerly, his alert eyes shining. He steals the picture.\n\nThis photograph of a happy family is subverted by Dirac's later memories of trauma and unhappiness. In one stinging memory, his parents bawled at each other in the kitchen while he and his siblings stood in the garden, frightened and uncomprehending. He once remarked in an interview that his parents 'usually ate separately', though twenty years later friends wrote that he told them he 'never'  saw his parents have a meal together - apparently a rare example of his being caught exaggerating. The rift between his parents was, according to Dirac, responsible for his dining-table ordeals. Three times every day, the tinkling of cutlery, the clatter of saucepans on the gas stove, the waft of cooking smells through the house presaged the ritual that he loathed. In none of the surviving accounts of the dining arrangements did he explain why he alone sat with his father, while his brother and sister ate with their mother in the kitchen. The only partial explanation that Dirac ever gave was that he could not sit in the kitchen because there were insufficient chairs. But this says nothing about the mystery of why Charles singled out him, not Felix or Betty, for special treatment.\n\nThe dining ritual was particularly harrowing on winter mornings, Dirac remembered. He would sit at the table with his father in the silent room, warmed by the burning coal in the fireplace and lit by a few oil lamps. Charles would be dressed in his three-piece suit, ready to cycle to the Merchant Venturers' School, always anxious not to be late for Assembly. His wife, scrambling and disorganised in the kitchen, made his anxieties worse by serving breakfast - usually large portions of piping-hot porridge - much too late for comfort. While he was waiting for his breakfast, Charles gave his first French lesson of the day to his younger son. Quite apart from Dirac's hatred of these arrangements, he grew to dislike eating mainly because his parents insisted, even when his appetite had been sated and he felt sick, that he must eat every morsel of food on his plate.\n\nFor the young Dirac, this was normality. In his early thirties, he wrote to a close friend of the sourness of his home life: 'I did not know of anyone who liked someone else - I thought it did not happen outside novels.' In another letter, he wrote: 'I found it to be the best policy as a child [. . .] to make my happiness depend only on myself and not on other people.' According to Dirac, his best defence against the unpleasantness and hostility he perceived all around him was to retreat into the bunker of his imagination.\n\nDirac first experienced the company of children outside his family shortly after his fifth birthday, when he started at the small and intimate Bishop Road Junior School. This was his first opportunity to socialise, to get a sense of other children's lives, of other domestic customs and practices. But he apparently made no attempt to talk to  other children: he remained silent and continued to live in his own private world.\n\nThe school was round the corner from his home, so close that he could hear its bell ringing at the start of the day. Despite the daily hurry of the breakfast routine, he and his brother always arrived on time. Dirac's class typically consisted of about fifty children crammed into a room about twenty-five feet square, the pupils sitting in rows of identical wooden desks, learning in an atmosphere that was, by today's standards, extremely disciplined and competitive. At the end of their time at school, children had to compete for scholarships that would help to pay for their senior education. Success meant that the child's parents would have to pay little or nothing; failure often meant that the child would be sent out to work.\n\nPaul and Felix were recognisably brothers, but Felix had a rounder face, was a few inches taller and was more heavily built. He was placid and well behaved, though given to lapses of concentration, as his headmaster pointed out when he wrote across his school report: 'The boy appears to me to be a perpetual dreamer. He must wake up!' Felix appears to have taken the advice, as he soon improved and did well in most subjects, especially drawing.\n\nFrom Dirac's later descriptions of his early life, we might expect him to have been an unhappy child, but there are no signs of this in the extant descriptions of him at the time. Twenty-seven years later, when his mother wrote a short poem about him for her own amusement, she described him as 'a cheerful little schoolboy', and added that he was 'contented' and 'happy'. In official reports written when he was eight, teachers at Bishop Road do not comment on his demeanour, saying only that he was 'well behaved', 'an intelligent boy' and 'a very steady worker'. But there are indications that Dirac was not performing to his potential. A few teachers allude to this, most notably the Headmaster, who, on seeing that Dirac had only just managed to be ranked in the top third of the class, wrote on his report in November 1910, 'I expected to find you higher.'\n\nAmong the boys Dirac did not get to know at Bishop Road School was Cary Grant, then known as Archie Leach and living in poverty about half a mile from Monk Road. In the classrooms and playground of the Bishop Road School, Dirac acquired the distinctively warm Bristol accent, which sounds slightly hickish to other native English speakers, evocative of farmers in the south-west of the  country. Like other young natives of Bristol, Dirac and Grant added an L to the pronunciation of most words that end in the letter A, a practice that is now dying out, though many English people still recognise Bristol as the only city in Britain to be able to turn ideas into ideals, areas into aerials. Cary Grant shed this accent when he emigrated to the United States, but Dirac kept it all his life. He spoke with a gentle intonation and an unassuming directness that would surprise the many people who expected him to talk like the plummy-voiced English intellectual of popular caricature.\n\nLike his brother, Dirac's ranking in the class gradually improved. He was good though not exceptional at arithmetic, and he did well in most subjects that did not involve his meagre practical skills. Soon after his eighth birthday, his teacher described him as 'An intelligent boy, but must try hard with his hand-work', drawing attention to his poor marks for handwriting (45 per cent) and drawing (48 per cent). His disappointed teacher commented that he should have done better than thirteenth in the class. Two years later, Dirac was consistently at or near the top of his class, his overall grade occasionally lowered by his relatively weak performance in history and brush-work.  At home, he pursued his extra-curricular hobby of astronomy, standing in his back garden at night to check the positions of the visible planets and constellations and, occasionally, to follow the track of a meteor hurtling across the sky.\n\nThe school did not teach science but did give classes in freehand drawing and also technical drawing, a subject that provided Dirac with one of the foundations for his unique way of thinking about science. His mother later drew attention to his 'most beautiful hands', suggesting that his long and bony fingers equipped him well to be an artist. Technical drawing, used by engineers to render three-dimensional objects on a flat piece of paper, is now taught at very few English junior schools, and rarely at senior level. Yet, in the early twentieth century, it was a compulsory subject for half the pupils: for a few lessons each week, the class would split into two: the girls studied needlework, while the boys were taught technical drawing. In these classes, Dirac learned to make idealised visualisations of various manufactured products by showing them from three orthogonal points of view, making no allowance for the distortions of perspective.\n\nBritain was among the slowest of the wealthier European countries to introduce technical drawing into its schools and did so only in the  wake of the Great Exhibition in 1851. Although the Exhibition was a great popular success, the most perceptive of its 6.2 million visitors saw evidence that mass technical education in Britain would have to improve substantially if the country were to retain its economic hegemony against growing competition from the USA and Germany. The Government agreed, enabling the Great Exhibition's prime mover Sir Henry 'King' Cole to change the technical curriculum of English schools so that boys were taught technical drawing and given an appreciation of the beauty of manufactured objects as well as natural forms. There was, however, a backlash to this practical notion of beauty in the form of the Aesthetic Movement, which flourished in England from the mid-1850s. The movement's leader in France was the flamboyant poet and critic Th\u00e9ophile Gautier, a weight-lifting habitu\u00e9 of the Louvre's Greek galleries. His phrase 'Art for art's sake' became the motto of the English aesthetes, including Oscar Wilde, who shared Gautier's belief that formal, aesthetic beauty is the sole purpose of a work of art. This view would later be distantly echoed in Dirac's philosophy of science.\n\nSir Henry Cole's reforms endured: the guidelines set out by him and his associates were being used in Bishop Road School when Dirac began his formal schooling. In 1909, the educationist F. H. Hayward summarised the prevailing philosophy that underlies the contemporary teaching of art: 'drawing aims at truth of conception and expression, love of beauty, facility in invention, and training in dexterity [. . .] nature study and science lessons cannot proceed far without it.' Hayward urged that students should practise their drawing skills by trying to represent accurately both natural and manufactured objects, including flowers, insects, tables, garden sheds and penknives. In autumn 1912, Dirac was asked to draw a penknife, and he did it competently enough - like all his other drawings, it includes not a line of embellishment.\n\nThe school took pains to teach its pupils how to write legibly, according to textbook rules that Dirac and his brother apparently studied closely. They developed a similar style of handwriting - consistent with the rules set out in the books they studied - neat, easy to read and virtually devoid of flourishes, except for the unusual forming of D, with a characteristic curl at the top left. Dirac did not change this calligraphy one iota for the rest of his life.\n\nIn the early summer of 1911, school inspectors noted that 'the  boys who are particularly bright and responsive are being carefully trained in habits of self-reliance and industry.' Nearly three years later, when Dirac was in his final year at the school, the inspectors visited Bishop Road again and wrote warmly of this 'progressive' school and the practical education it offered: 'a keen, vigorous and thoughtful head [teacher]. Staff [are] earnest, painstaking [. . .] Drawing is well taught and handwork is resourceful, the boys make a number of useful models and are allowed considerable freedom in their choice while the work is so taken as to train them in habits of self reliance, observation and careful calculation and measurement. '\n\nBishop Road School wanted to give its pupils the skills they needed to get good jobs. But, for Dirac, the most important consequence of this practical approach was that it helped to shape his thinking about how the universe works. As he was sitting at his desk in his tiny Bristol classroom, producing an image of a simple wooden object, he had to think geometrically about the relationships between the points and lines that lie in a flat plane. In his mathematics classes, he also learnt about this type of Euclidean geometry, named after the ancient Greek mathematician who reputedly discovered it. So, Dirac studied geometry using both visual images and abstract mathematical symbols. Within a decade, he would transfer this geometric approach from concrete technological applications to the abstractions of theoretical physics - from an idealised, visual representation of a wooden fountain-pen stand to an idealised, mathematical description of the atom.\n\nLater in life, Dirac would say that he never had a childhood. He knew nothing of the rites of passage of most other young boys - long weekend afternoons spent stealing eggs from birds' nests, scrumping from nearby orchards, dashing out in front of trams. In many ways, as a child he seems to have behaved much as Newton had done. 'A sober, silent, thinking lad [. . .] never was known scarce to play with the boys abroad' was how one of Newton's friends described him: the description applies equally well to Dirac as an infant.\n\nDirac was not interested in sport, with the exception of ice-skating, which he learned with Betty and Felix at the nearby Coliseum rink, the talk of Bristol when it opened in 1910. Decades later, his mother recalled that he would sit quietly, reading books that he had  placed neatly around him and learning long poems that he would recite to his family. She shed some light on his sheltered childhood when she spoke to reporters in 1933: '[his father's] motto has always been to work, work, work, and if the boy had showed any other tendencies, then they would have been stifled. But that was not necessary. The boy was not interested in anything else.' There is little doubt that Charles Dirac impressed his sedulous work ethic on his younger son, who later wrote admiringly of his father's conscientiousness:\n\nOne day while cycling [to school, my father fell off his bike], trying to avoid a child who ran out in front of him, and broke his arm. He was very conscientious, so he continued to the school and continued with his teaching, in spite of the broken arm. Eventually, the head master found out about it and sent him home, and told him not to come back until he was better.\n\nPaul was also aware that his father was exceptionally careful with money. In April 1913, Charles took the biggest financial decision of his life by purchasing a more expensive and more spacious home. The family moved from the cramped terrace of Monk Road to a neat semi-detached residence a few minutes' walk away in a slightly more salubrious part of Bristol, at 6 Julius Road. The Diracs now had a home befitting Charles's status in the community, with separate rooms for their two boys so that Dirac now had a place to escape, a private place where he could work alone. The family still kept themselves to themselves, inviting no visitors into their home, apart from Flo's family, her guests - all female - at a monthly afternoon tea party and the steady stream of pupils who took private language lessons from her husband.\n\nLike many parents, Charles entered all his children for scholarship exams. When Felix was nine years old, he failed one of these exams, leading his father to demand an explanation from his teachers; Betty also failed the exam a few years later. Paul had no such problems: he passed every scholarship exam with flying colours and, thus, unlike Felix and Betty, ensured he was educated at minimal expense to his parents.\n\nDirac could see new technology making its imprint on Bristol. The city centre was a patchwork of centuries-old buildings and brand-new ones, many of them emblazoned with advertisements for new  services and products. Open-topped motor cars vied for space on the roads with horse-drawn carriages, bone-shaking bicycles and the trams that made their jerky way round the city. When a programme of road construction began, in the early years of the century, cars began to dominate the city. In late 1910, Dirac had witnessed the beginnings of the Bristol aviation industry, one of the first and largest in Britain. The leading figure in this new Bristol industry was the local entrepreneur Sir George White, who founded the British and Colonial Aeroplane Company and supervised the building of some of the earliest aircraft in a tram shed in Filton, a few miles north of the Diracs' home. Long afterwards, Dirac told his children that he would rush out into the back garden to see aeroplanes precariously taking off from the new airfield less than a mile away. It seems that he wanted to find out more about this new technology: among the papers he kept from his youth were details of a programme at a local technical college, beginning in December 1917: 'Ten Educational Lectures on Aeronautics'.\n\nDirac and his brother stood out among the boys in Bishopston as they both spoke good French even before they started school. According to one report, local boys would stop the Dirac brothers on the streets and ask them to speak a few sentences of French. This knowledge of French was also obvious to the students at their next school, where the language was taught by the school's most feared disciplinarian - their father.\n**Two**\n\nIn the world of commerce,  \nIn the crafts and arts,  \nSons of her are honour'd  \nNobly bear their parts;  \nWhile in sports and pastimes  \nThey have made a name,  \nTrain'd to wield the willow,  \nLearn'd to 'play the game'.\n\nVerse of the Merchant Venturers' School song\n\nOn 4 August 1914, when Dirac was preparing to start at senior school, he heard that Britain was at war - the first conflict to involve every industrialised country in Europe. 'The European War', which would claim more British lives than any other, was to be the backdrop to the whole of his secondary education at the Merchant Venturers' School.\n\nLike most other British cities in the UK, Bristol quickly prepared for the war, the urgency of the preparations heightened by the statement by the Boer War hero Lord Kitchener that the conflict would be decided by Britain's last million men. On the last day of August, in his capacity as Secretary of State for War, Kitchener sent a telegram to the Bristol Citizens' Recruiting Committee asking them to form a battalion of 'better class young men', and within a fortnight some 500 professional men had volunteered for the 'Twelfth Gloucesters', part of 'Kitchener's Army'. Within a few weeks, the focus of the city's industries had changed from making money to supplying the military with everything from boots and clothes to cars and aircraft. Even the Coliseum ice-rink was commandeered as a site to assemble warplanes.\n\nThe first casualty lists were published barely a month after the declaration of war. The Bristol newspapers reported that the Allies had contained the initial German onslaught and that the battle lines had hardened to form a series of linked fortifications that stretched from the Franco-Belgian border on the coast right through to the Franco-Swiss border, close to where Charles Dirac had been brought up. After Parliament passed the Aliens Registration Act, Bristol was one of the UK cities to be declared a 'prohibited area'. Charles had to register with the authorities as a foreigner, although he was hardly a threat to British security. By the time his elder son arrived at the all-boys Merchant Venturers' Secondary School, Charles had spent almost a third of his forty-eight years as its Head of French, doing more than any other teacher to extend the school's reputation for excellence beyond its established forte of technical subjects to modern languages.\n\nIt took Charles about fifteen minutes to cycle from his home to the school in Unity Street, in the heart of the city. The building was round the corner from the Hippodrome, Bristol's newest and swankiest music hall, where the young Cary Grant secured his first job, as a trainee electrician helping to operate the lighting rigs - soon after Paul started at the school. The school's Edwardian-Gothic building had been opened in April 1909, after the previous school on the site had burnt down. Everyone in the vicinity of the new school heard the clatter and rumblings from the basement workshops. The vibrations were so violent that the school's near-neighbour, Harvey's wine merchants, complained of the incessant disturbance to their cellars.\n\nThe behaviour of Charles Dirac, whose pupils nicknamed him 'Dedder', emerges clearly in the testimonies of several of his fellow teachers and his students obtained by the Oxford University physicist Dick Dalitz in the mid-1980s. One of Dirac's fellow students, Leslie Phillips, gave a sense of the reputation of Monsieur Dirac:\n\nHe was _the_ disciplinarian in the school, precise, unwinking, with a meticulous, unyielding system of correction and punishments. His registers, in which he recorded all that went on in the class were neat and cabalastic; no scholar could possibly understand their significance. Later, as a senior, I began to realize the humanity and kindness of the man, the twinkle in the eyes. But to us in the junior school, he was a scourge and a terror.\n\nDedder was well known for his old-fashioned, strictly methodical approach to teaching and for springing random tests on his students, so that they always had to be prepared. If he caught them cheating in these tests or in their homework, he punished them with four half-hour periods of detention on Saturday afternoons. 'You never wrote this. Saturday at four for cribbing,' he told Cyril Hebblethwaite, later  Lord Mayor of Bristol. Most teachers routinely meted out corporal punishment by whacking errant boys across their backsides with a slipper or cane with an enthusiasm that bordered on the sadistic. But there is no record that Charles was fond of this form of chastisement, either at school or at home.\n\nIt is easy to imagine Monsieur Dirac's terrified pupils looking at Paul and Felix and wondering, probably out loud, 'What's he like at home?' Their father's strict classroom regime did, however, bring the benefit of a supply of comics that he had confiscated and brought home for his children. The young Dirac read these cheap 'penny dreadfuls', black-and-white comics full of slapstick cartoons, juvenile jokes, detective stories, sensational tales of soldierly adventure and even the occasional topical reference to the build-up of the German military. This one concession to popular culture in the Dirac home gave the young Paul an enduring taste for comics and cartoons.\n\nThe boys' mother also inflicted her share of pain on them by keeping their hair in tight curls and making them wear knickerbockers long after they were fashionable. They wore short breeches and garters so tight that, when they were removed, they each left an angry red line around the boys' legs. Dirac long remembered the taunts of his fellow pupils for being what nowadays would be damned as 'uncool'. Such was his induction into that most characteristic of English anxieties, embarrassment.\n\nLike all parents at that time, Charles and Flo worried that their children would catch tuberculosis, then responsible for one in every eight deaths in the UK. It was particularly brutal in culling adult males: it accounted for more than one death in three among men aged fifteen to forty-four. The Dirac children were all born during the first decade of a government-funded anti-tuberculosis campaign that urged all citizens to get out into the open air, to take plenty of outdoor exercise and thus to get plenty of fresh air into their lungs. This philosophy may have encouraged Charles to decline to pay for his sons' tram fares to and from school and therefore to force them to walk there and back twice a day (they had lunch at home). Paul later resented what he believed was his father's meanness, though it probably led him to acquire a taste for taking long walks, soon to become one of his obsessions.\n\nIt took only weeks for Dirac to establish himself as a stellar pupil at the Merchant Venturers' School. Except for history and German, he shone at every academic subject and so was usually ranked as the top student of his class. The curriculum was wholly practical, with no room for music nor - to Dirac's relief - Latin and Greek. Instead, the school focused on subjects that would equip its boys to take up a trade, including English, mathematics, science (though not biology), some geography and history. What made the education at this school special was the high quality of the teaching of technical skills such as bricklaying, plasterwork, shoemaking, metalwork and technical drawing. For the previous fifty years, government inspectors had praised the school for giving one of the best technical educations available to any child in the country.\n\nIn the school's laboratories, Dirac learned how to fashion pieces of metal into simple products, how to operate a lathe, how to cut and saw, how to turn a screw thread. Away from the clatter of machinery, the puddles of oil and the coils of swarf, he learned more of the art of technical drawing. These lessons built on the introductory classes at Bishop Road and showed Dirac how to produce plans for more complicated objects, developing his ability to visualise them from different angles. In his 'geometric drawing' classes, Dirac considered cylinders and cones, and he learned how to see in his mind's eye what happens when they are sliced at different angles and then viewed from various perspectives. He was also taught to think geometrically about objects that are not static but moving, and he learned how to draw the path of, for example, a point on the outside of a perfect circle as it rolls along a straight line, like a speck of dust on the outside of a wheel rolling along a road. To students who first encounter these shapes - curved, symmetrical and often intricate - they are a source of delight. If, as is likely, Dirac wondered how to describe these curves mathematically, his technical-drawing teachers would probably have been unable to enlighten him as they were usually former craftsmen with little or no mathematical expertise.\n\nAlthough Dirac focused intensely on his college work, he was well aware of the scale of the war. All day long, convoys of trucks passed through Bristol with their supplies for the soldiers at the front, and huge guns were towed through the streets, shaking nearby buildings. At night, the streetlamps were extinguished to make the city a difficult target for the expected convoys of German airships, although  they never arrived. The city's rapidly expanding aviation industry was on a war footing, so the threat of aerial bombing was clear to Dirac, who passed a busy aircraft factory every time he walked to and from school.\n\nUnreliable news of the conflict trickled back from the battlefronts through newspapers and by word of mouth. The Government's censorship policy prevented journalists from reporting on the full extent of the carnage, but readers could form a broad picture of the conflict and its ramifications. In February 1916, the Germans began their campaign to try to wear down the French Army at Verdun, and in July the British Army attacked on the Somme. Casualty figures soared, although the battle lines changed only slowly. In April 1917, the Germans introduced unrestricted U-boat warfare, aiming to cut supplies of food and other resources to the UK and thereby to force the enemy to the conference table. This brought the United States into the war, and Bristol celebrated by giving its schoolchildren a half-day holiday on 4 July, Independence Day. Meanwhile, Russia was in turmoil, with the fall of the monarchy in February followed nine months later by Lenin's Bolshevik revolution.\n\nEvery day, the Dirac family read about these events in the local and national newspapers. The inside pages of the _Bristol Evening News_ showed head shots of uniformed teenage soldiers, with a few lines that listed their regiment, when they fell and whom they left behind. Despite the depressing regularity of these reports, the recruitment campaigners maintained a constant flow of army volunteers, many of them younger than the minimum legal age of eighteen. Some of the boys shipped out to the killing fields were only a year older than Dirac. The nearest he came to military service was a brief stint in the Cadet Corps in 1917, but around him there was plenty of evidence of the experiences of less fortunate young men. He would certainly have seen legions of wounded and maimed soldiers hobbling around the city, having returned from France for treatment.\n\nBut the war was a boon for Dirac's education. The exodus of the school's older boys depleted the higher classes and enabled Dirac and other bright children to fill the gaps and therefore make quick progress. He excelled at science, including chemistry, which he studied in a silence that he broke on one occasion, a fellow student later remembered, when the teacher made an error, which  Dirac gently corrected. In the foul-smelling laboratories, Dirac learned how to investigate systematically how chemicals behave and learned that all matter is made of atoms. The famous Cambridge scientist Sir Ernest Rutherford gave an idea of the smallness of atoms by pointing out that if everyone in the world spent twelve hours a day placing individual atoms into a thimble, a century would elapse before it was filled. Although no one knew what atoms were made of or how they were built, chemists treated them as if they were as palpable as stones. Dirac learned how to interpret the reactions he saw in the laboratory test tubes simply as rearrangements of the chemicals' constituent atoms - his first glimpse of the idea that the way matter behaves can be understood by studying its most basic constituents.\n\nIn his physics lessons, he saw how the material world could be studied by concentrating, for example, on heat, light and sound. But the mind of young Dirac was now venturing far beyond the school curriculum. He was beginning to realise that underneath all the messy phenomena he was studying were fundamental questions that needed to be addressed. While the other boys in his class were struggling to get their homework done on time, Dirac was sitting at home, reflecting for hours on the nature of space and time. It occurred to him that 'perhaps there was some connection between space and time, and that we ought to consider them from a general four-dimensional point of view'.  He appears to have shared much the same opinion as the Time Traveller in the 1895 novel _The Time Machine_ by H. G. Wells, whose science-fiction novels he read: 'There is no difference between Time and any of the three dimensions of Space except that our consciousness moves along it.' Such an opinion had wide currency at the end of the nineteenth century, and Dirac may have read the Traveller's words when he was a child. In any case, the young Dirac was mulling over the nature of space and time before he had even heard of Einstein's theory of relativity.\n\nDirac's teacher, Arthur Pickering, gave up on teaching him with the rest of the boys and sent him to the school library with a book list. Pickering once set the prodigy a set of tough calculations to keep him busy at home that evening, only to hear from Dirac on his way home that afternoon that he had already done them. And Pickering opened up another new vista to Dirac when he suggested that he look  beyond simple geometry to the theories of the German mathematician Bernhard Riemann, who had proposed that the angles of a triangle do not always add up to exactly 180 degrees. Just a few years later, Dirac would hear how Riemann's geometric ideas - superficially without relevance to science - could shed new light on gravity.\n\nCharles Dirac understood as well as anyone that his younger son had an exceptionally fine mind coupled with formidable powers of concentration. By imposing a rigorous educational regime at home, Charles had produced a workaholic son in his own image, as he presumably intended. What Charles did not apparently appreciate as acutely as other people was Paul's odd behaviour. The young Dirac's fellow students certainly regarded him as strange. In testimonies given sixty years later, several of them described him as a very quiet boy; two accounts speak of 'a slim, tall, un-English-looking boy in knickerbockers with curly hair', and 'a serious-minded, somewhat lonely boy [who] haunted the library'. Even at that time, he had a monomaniacal focus on science and mathematics. Games did not appeal to him and, when he was obliged to play, his participation seems to have been superfluous: one of his fellow schoolboys later remembered that Dirac's style of holding a cricket bat was 'peculiarly inept'. As an old man, Dirac attributed his dislike of team games to his having to play soccer and cricket with the older and bigger boys on the Merchant Venturers' playing fields.\n\nHis appreciation of literature was also extremely limited. He never understood the appeal of poetry, though he did read novels written to appeal to young boys, including adventure stories and tales of great battles, scrutinising each text with the care of a literary critic. As a nine-year-old, Bishop Road School had awarded him a prize of Daniel Defoe's _Robinson Crusoe_ , a novel that always strikes a chord with those who are happy to be away from the crowd - almost, but not quite, alone.\n\nIt was the mathematics and science lessons that did most to shape Dirac's way of thinking. Decades later, when his history teacher Edith Williams renewed contact with him, she told him that, when he was a student in her class, she 'always felt you were thinking in another medium of form and figures'. By every account of Dirac's behaviour in his mid-teens, he had the same personality characteristics as today's pasty-faced technophiles who prefer using the latest software and gadgets to mixing with other people and who are happiest sitting  alone at their computer screens. From a modern perspective, the young Dirac was an Edwardian geek.\n\nAt the Merchant Venturers' School, the class sizes shrunk and the range of lessons narrowed. When Dirac began at the school in September 1914, there were thirty-seven boys in his class; by the time he left in July 1918, four months before the end of the war, there were eleven. At the Speech Day, July 1918, he received a prize - as he had done every year - and heard the Headmaster announce that ninety-six boys had been killed and fifty-six wounded in the year 1916-17. For the rest of his life, he would remember these litanies of death.\n\nNor was there any respite at home from the gloom. In Dirac's eyes, when his father returned home from school, his persona changed from the school's fair-minded and respected disciplinarian to bullying tyrant. He still imposed his linguistic regime at the dinner table, where wartime shortages and rationing had made Flo's meals simpler and less abundant. By the beginning of 1918, there were long, morale-sapping queues for bread, margarine, fruit and meat. The price of a chicken rose to a guinea, a week's wages for a manual labourer. The shortages encouraged many families, including the Diracs, to cultivate fruit and vegetables, and it was mainly for this reason that Paul Dirac took up gardening, though the hobby would also have given him another reason to escape the atmosphere inside the house.\n\nAnother source of unhappiness in the Dirac family was that Charles and Flo each had a favourite child: Paul was his mother's, Betty her father's, with Felix left out in the cold. As a student, Felix had done almost as well as his younger brother at Bishop Road, but the gap between their abilities at senior school became so wide that it began to cause serious friction between them. The two brothers no longer walked around together but were continually bickering. In his later life, Dirac was uncharacteristically forthright about the reason for the rift: 'having a younger brother who was brighter than he was must have depressed him quite a lot'. This is a telling remark. Dirac was never socially sensitive and, as an old man, was exceptionally modest and given to understatement, so he was probably making light of how painful Felix found the experience of being academically outclassed by his younger brother.\n\nAs he came to the end of his studies at the school, Felix had set his heart on becoming a medical doctor. His father, however, had other ideas: he wanted Felix to study engineering. This subject was popular among young people, just as Bernard Shaw had foreseen in his novel _The Irrational Knot_ : a new class of engineer-inventors would go 'like a steam roller' through the effete boobies of the aristocracy. The future appeared to be in the hands of H. G. Wells's 'scientific samurai'. It certainly seemed sensible for Felix to use his practical skills to take a course that would virtually guarantee him employment. As Charles probably realised, for Felix to train to be a doctor would entail six expensive years of training, with little prospect of the costs being offset by Felix winning one of the scarce scholarships to medical school. Felix tried to stand firm, but Charles forced him to climb down, doing more harm to their relationship than he probably realised.\n\nThe cheapest and most convenient place for Felix to study was at the university's Faculty of Engineering, housed in the Merchant Venturers' Technical College, which shared the same premises and facilities as the Merchant Venturers' School. Probably with a good deal of resentment, Felix began his course in mechanical engineering there in September 1916, his studies funded by a City of Bristol University Scholarship.\n\nPaul never contemplated studying anything other than a technical subject. He could have taken his pick from dozens of science courses, and seriously considered taking a degree in mathematics, but decided against it after he learned that the likely outcome would be a career in teaching, a prospect that held no appeal for him. In the end, in the absence of a strong preference of his own, he decided to follow his brother - and, apparently, their father's advice - by studying engineering at the Merchant Venturers' College, supported by a generous scholarship.\n\nIn September 1918, Felix was preparing to begin the final year of his engineering course, which he had been finding hard going - throughout, he had languished near the bottom of his class. At the same time, Paul, aged only sixteen, was about to join the ranks of the engineering students - two years younger than the other students in his class. Felix must have known that others were comparing his talent with his brother's and that he would not emerge well from the comparison.\n**Three**\n\nA report by the Bristol Advisory Committee, working in conjunction with the Employment Exchange, issued early in 1916, threw light on the effect of the war on the labour of young people in the preceding year. It stated that boys were almost generally fired by the ambition to become engineers [. . .]\n\nGEORGE STONE and CHARLES WELLS (eds), _Bristol and the Great War_ , 1920\n\nOn the overcast morning of Monday, 11 November 1918, Dirac set off from his home as usual to walk to the Merchant Venturers' College. It was the beginning of his seventh week at the college, and appeared to be like any other day. But when he arrived, he found that all lectures had been cancelled. He soon heard the reason: suddenly and unexpectedly, the war had ended.\n\nBy midday, the centre of Bristol had become the site of a vast, anarchic carnival. During a day of noisy jubilation not seen before in living memory, English reserve was abandoned. Church bells rang out, businesses shut down, everyone felt licensed to drape themselves in the national flag, to march the streets, to bash empty biscuit tins and dustbin lids and anything that would make a lot of noise. All over the city, Union Jacks hung from windows, lamp posts and from the hundreds of trams and motor vehicles that had been commandeered for the day without demur from the police. Among the groups of marchers repeatedly singing 'Rule Britannia' was a group of American soldiers on the way to war, each of them holding a corner of the Union Jack. Nearby, a group of grammar-school students carried an effigy of the Kaiser, once a resident of Bristol. Dirac's fellow Merchant Venturers' students caroused around the city, singing the song they had composed for the occasion. Dirac long remembered the chorus they sang at the top of their voices: 'We are the boys who make no noise,' followed even more loudly by 'Oo-ah, oo-ah-ah.'\n\nThe Prime Minister David Lloyd George spoke that day in the House of Commons of the curious mixture of regret and optimism in the country after 'the cruellest and most terrible War that has ever  scourged mankind. I hope we may say that thus, this fateful morning, came an end to all wars.' Fate, however, had yet more cruelty in store: the Spanish Flu pandemic that broke out towards the end of the conflict cost even more lives than the war. To try to slow the spread of the virus, Bristol's schools had been closed, leaving thousands of children wanting to spend the afternoons laughing at new film comedians such as Fatty Arbuckle, but they were thwarted by the closing of the cinemas during school hours by the local Council's Malvolios.\n\nThe novelist and poet Robert Graves remarked perceptively that before August 1914, the country was divided into the governing and governed; afterwards, although there were still two classes, they had changed into 'the Fighting Forces [. . .] and the Rest, including the Government'. The new divisions were clear at the Merchant Venturers' College after the war: Dirac saw young men returning from the battlefront suddenly outnumber the original intake of students, whose closest brush with the enemy had been through reading newspaper reports. The soldiers had returned to a brief welcome, but they had to settle down quickly to normal life, encumbered by disfigurement and by shell shock and other psychological damage. These men, most of them still in uniform, brought a new grittiness and pragmatism to the lecture rooms. Dirac later observed: 'the new students had a more mature outlook on life, and in the Engineering Faculty they were especially eager to learn results of practical importance and [they] did not have much patience with theory.'\n\nThe returning soldiers were among the thousands who flocked to that year's Christmas treat in Bristol: the opportunity to see and take a tour around the inside of a captured German submarine U86. It was moored in the docks, the Union Jack flag fluttering on one of its masts above the German naval ensign. Everyone knew the significance of the display: the tank, the machine gun, the aircraft, radio and poison gas had all played their part in the war, but none had seemed more menacing than the submarine. Now this most feared weapon was impotently on show, like a dead shark.\n\nEngineering was evidently not the subject best suited to the talents of the young Dirac. The course at the Merchant Venturers' College was more practical than theoretical and therefore exposed his limited  manual skills while not making the most of his mathematical gifts. True to form, Dirac strode ahead in mathematics and was 'a student who got all the answers exactly right, but who had not the faintest idea of how to deal with apparatus'. Not only was he maladroit, his mind was on other things: he spent much of his time in the physics library, reflecting on the fundamentals of science. With no money and nothing else to do during the day, Dirac would walk down from his home in Julius Road to the college and work in the libraries six days a week. He did, however, make his first friend among the other thirty-one students in the class: Charlie Wiltshire, another solitary young man with a mathematical bent.\n\nThey were taught mathematics by Edmund Boulton, nicknamed 'Bandy', as his gait gave the impression that he had just dismounted a mare. Not a strong academic, Bandy showed his class how to tackle textbook mathematical problems in orthodox ways, only for Dirac repeatedly to proffer simpler and more elegant solutions. Soon Dirac and Wiltshire were segregated so that they could work at a pace that would not shame everyone else. Poor Wiltshire may have felt better if he had stayed behind, as he found the task of keeping up with his friend's mathematical progress 'utterly hopeless'. Within a year, they had completed the mathematical content of their degree, but Wiltshire was permanently scarred. Over thirty years later, he wrote that the experience of trying to stay abreast of Dirac had left him with a 'pronounced inferiority complex'.\n\nMathematics was only a small part of Dirac's curriculum: he spent most of the time fumbling in the laboratories with Wiltshire or trying to stay alert during lectures. Unlike most students, he did not like to be spoon-fed and preferred to learn in private, ideally alone in the library, where he would flit back and forth between passages in books and journals, making his own links and associations. One course of lectures that did keep Dirac on his toes was given by the hard-driving head of the electrical-engineering department, David Robertson, a theoretically minded engineer who had been confined to a wheelchair after contracting polio. Dirac admired Robertson for arranging his life methodically and for the way he used clever labour-saving initiatives to help overcome his disability. It was difficult for Robertson to deliver standard chalk-and-blackboard presentations, so he used a precursor of digital presentation software: a continuous series of lantern slides lit - none too reliably - by a flickering carbon arc lamp. Robertson rushed through his commentary, giving no quarter to the intellectual limitations of his audience or to their need to write legible notes. Dirac's favourable opinion of him was not shared by the great majority of his students, who were left trailing in frustration and despair.\n\nRobertson ensured that the electrical-engineering course was built on solid theoretical foundations. Dirac and his colleagues specialised in electrical engineering only in their final year, after they had been given a grounding in physics, chemistry, technical drawing and other types of engineering - civil, mechanical and automotive. No one could reasonably accuse the course of being out of touch with business: Dirac was taught the elements of management, contract law, patents, bookkeeping and accountancy. He even learnt about income tax.\n\nThe course was based in the engineering laboratories. Dirac spent many hours every week there, working with Wiltshire, learning about the mechanical structures and machinery that underpinned industry, including bridges, pulleys, pumps, internal combustion engines, hydraulic cranes and steam turbines. He measured the strength of materials by stretching them until they snapped and by observing how much they bent under stress. The course on electrical engineering was extremely thorough, and Dirac learned about the subject from its roots - simple experiments in electricity and magnetism - through to the minutiae of the design and operation of the latest hardware of the electricity-supply industry. H. G. Wells could not have asked for a more thorough training for a future leader in his technocratic utopia.\n\nThe university Engineering Society organised trips to local factories, partly to give the students a sense of the noise and grime in which most of them would soon be working. A posed photograph taken on one of these trips in March 1919 shows the physical appearance of Dirac and his fellow students, all of them male. Each of them is wearing a tie, a hat and an overcoat, several of them have a stick, and a few are still in military uniform. The sixteen-year-old Dirac is standing at the front, hands in his pockets, looking blankly at the camera with a hint of adolescent rebelliousness. It is the first of many photographs of him as a young man to show confidence and resolve shining out of his eyes.\n\nSix Julius Road was a cold and unloving refuge to Dirac, but for many local people he seemed to be part of an admirable home. The reputation of Charles Dirac was still on the rise: he had become one of the 'Big Four' housemasters at the Merchant Venturers' School, and his private language classes were thriving at home. A few minutes after the beginning of each tutorial, in the small study overlooking the front garden, Flo knocked on the door to bring Charles and his student a pot of tea and a plate of biscuits - part of the attentive service students took for granted at that address. She spent most of her time running the house but liked to while away afternoons reading romantic novels and the poetry of Robert Browning, Robert Burns and Rudyard Kipling. In an exercise book, she wrote out some of her favourite verse and a collection of aphorisms that indicated her penchant for traditional virtues: 'Control, give, sympathise: these things must be learnt and practised: self-control, charity and sympathy.'\n\nThe Diracs' daughter Betty was as timid as her brothers. Most such girls of her generation began a menial job straight after leaving junior school, but Charles and Flo wanted her to continue her education at the nearby Redlands Girls' School, where she studied without special enthusiasm or achievement. It was convenient for her father to accompany her to school after 1919, when his school relocated to Cotham Lawn Road, ten minutes' walk from the Diracs' home. The move was unpopular with its teachers, though it was made palatable for Charles by a sweetener - promotion to the more lucrative post of Associate University Lecturer. His colleagues in the staffroom respected him as one of the most effective teachers in Bristol, though many regarded him as odd. He did nothing to shed this reputation when he told one of them that he had been trepanned: presumably a surgeon had drilled a tiny hole into his head, intending to let out evil spirits.\n\nTo some of Charles's fellow teachers, there was a whiff of fraudulence about him: they found out that the letters B. \u00e8s. L. ( _Baccalaur\u00e9at-\u00e8s-Lettres_ ) that he almost always put after his name signified only that the University of Geneva had pronounced him able to embark on higher education. He had spent only a year at the university, as an _auditeur_ , taking notes but not a degree. One of his colleagues later chuckled as he recounted the minor staffroom scandal involving Charles: as he was not eligible to wear the full academic  dress, he bought a gown and asked his wife to make him a hood in red, white and blue. She knew nothing of the deception and only found out about it several years later.\n\nIn the spring of 1919, for reasons that are not clear, Charles Dirac sought British nationality for the first time. He wrote urgently to the Swiss authorities, saying that after teaching in the UK for thirty years, 'professional reasons' made it essential that he renounce his Swiss nationality. When he submitted his application to the British authorities, he said he wanted the right to vote after the government had withdrawn it, following the recent amendment to the Aliens Registration Act, which also denied Flo - as the wife of a 'foreign national' - the right to vote in future general elections (she had voted for the first time six months before, in common with other British women over thirty years of age). Perhaps, too, he wanted his daughter and elder son to be eligible for the scholarships available only to British citizens? Whatever his motivation, Charles swore allegiance to George V in front of a justice of the peace in Bristol on 22 October 1919. On that day, his children also became Britons, having previously been classed as Swiss, a status that, according to Betty's later recollections, caused her to be teased in the playground for being 'one of those Europeans'. Paul Dirac was no longer a foreigner, but, to many British eyes, he would always have the air of one.\n\nIn the early summer of 1919, when Paul's first-year results confirmed his potential as a top-flight student, Felix became the first person in his extended family to be awarded a degree, though only with third-class honours. The disparity between the brothers' academic talents had never been so stark, so it is probably no coincidence that the relationship between them became seriously troubled at about this time. In the pained and elliptical comments Dirac made later about Felix, he remarked they would often 'get into a row', though he gives no details of the arguments. One possibility is that they were seeded by Felix's jealousy and sense of inferiority, nourished by Paul's lack of empathy with his brother and by his inability to muster tactful words that were sorely needed to preserve Felix's sense of self-worth. Among his colleagues in his later career, Paul Dirac was famous for not understanding the feelings of others and for his lack of tact. It is unlikely that he was any different when he was a young man.\n\nAfter Felix had taken his degree, he left home and moved about  a hundred miles away to Rugby, which was rapidly changing from one of the West Midlands' sleepy market towns into a booming centre of the new electrical technology. Felix took a three-year student apprenticeship at the British Thomson-Houston Company, on a starting wage of a pound a week, giving him a measure of financial independence. Meanwhile, his penniless brother continued to study engineering - while moonlighting in physics - at the Merchant Venturers' College. As he had already chomped his way through the mathematics part of the course, he seemed destined to spend the remaining two years of his engineering degree fumbling his way through his laboratory exercises and listening to his lecturers drone their way through the syllabus. When especially bored, he amused himself in the library by hunting down the longest German words in the technical dictionaries (hyphens barred) and reading about the subject that most interested him, physics. His scientific imagination was ripe for a challenge, and, a few weeks after he began his second year at university, it arrived.\n\nNo event in Dirac's working life ever affected him as deeply as the moment when relativity 'burst upon the world, with a tremendous impact', as he remembered nearly sixty years later. Einstein became a media figure on Friday, 7 November 1919, when _The Times_ in London published what appeared to be just another post-war edition, including the news that the King supported the proposal of an Australian journalist for two minutes' commemorative silence on the anniversary of Armistice Day. On page 12, the sixth column featured a 900-word article that most readers probably passed over, unless the headline, 'Revolution in Science', captured their attention. Yet this was a momentous piece of journalism, and it helped to propel Einstein from relative obscurity in Berlin to international celebrity; soon, his moustachioed face and frizzled mane of black hair were familiar to newspaper readers all over the world. The unsigned article reported the apparent verification of a theory by Einstein that 'would completely revolutionize the accepted fundamental physics' and thereby overturn the ideas of Isaac Newton that had held sway for over two centuries. The observations were made by two teams of British astronomers who had found that the deflection by the Sun of distant starlight during the recent solar eclipse was consistent with Einstein's theory but not Newton's. When he was an old man, Dirac  remembered this as a time of special excitement: 'Suddenly Einstein was on everyone's lips [. . .] [E]veryone was sick and tired of the war. Everyone wanted to forget it. And then relativity came along as a wonderful idea leading to a new domain of thought.'\n\nDirac, Charlie Wiltshire and their fellow students were fascinated by Einstein's new theory and tried to find out what the fuss was about. This was not an easy task. Their teachers, like most academics in the UK, were no more knowledgeable than their students about this alleged scientific revolution. Apart from occasional articles in scientific journals such as _Nature_ , the primary sources of knowledge about the new theory of relativity were newspapers and magazines, whose editors gave commentators thousands of column inches to speculate - usually facetiously - about the new theory and its apparent defiance of common sense. On 20 January 1920, _Punch_ featured an anti-Semitic poem that exemplified popular puzzlement with the theory that had originated behind the lines of the UK's bitter enemy:\n\nEuclid is gone, dethroned,   \nBy dominies disowned,   \nAnd modern physicists, Judaeo-Teuton,   \nFinding strange kinks in space,   \nSwerves in light's arrowy race,   \nMake havoc of the theories of Newton.\n\nThe pages of the newspapers and magazines were replete with advertisements for scores of half-baked accounts of Einstein's work churned out only months after the theory came to public attention. At that time, there were no science journalists, so Dirac and his friend Wiltshire had to rely on popular articles written by scientists, notably Arthur Eddington, the Quaker astronomer and mathematician at the University of Cambridge and the only person in Britain to have mastered the theory. He had even got his hands dirty in one of the eclipse expeditions that produced crucial support for the theory.\n\nIn a stream of entertaining articles and books, Eddington deployed witty, down-to-earth analogies that made even the most complex abstract ideas accessible and arresting. His skill is exemplified in the account he gave in 1918 of Einstein's famous equation _E_ = _mc_ 2. Other authors could only crank out a dreary and barely comprehensible explanation of the equation's neat connection between the  energy _E_ equivalent to a mass _m_ , and the speed of light in a vacuum (symbolised by the letter _c_ ). Eddington knew better. In his explanation, he used the equation to do a calculation that he knew would interest his readers: he worked out the total mass of the light that the Sun shines onto the Earth and then used the result to comment on the controversial question of whether to keep daylight-saving time:\n\nthe cost of light supplied by gas and electricity companies works out at something like \u00a310,000,000 an ounce. This points the moral of Daylight Saving: the Sun showers down on us 160 tons of this valuable stuff every day; and yet we often neglect this free gift and prefer to pay \u00a310,000,000 an ounce for [light of] a much inferior quality.\n\nEddington and other writers fuelled Dirac's interest in understanding how the material universe works. But he spent most of his time studying for his engineering degree, struggling to concentrate in lectures, mastering the theoretical concepts, doing experiments and writing them up in immaculate accounts that feature scarcely a single crossing-out. To the modern eye, they almost look as if they had been printed by machine in a special typeface that successfully mimics ordinary human handwriting, with every repeated letter reproduced identically.\n\nCharlie Wiltshire was one of the very few people who glimpsed the human side of Dirac. To most people, he looked like a cold-hearted solipsist, uninterested in human contact, engaged only by mathematics, physics and engineering. Even in those repressed times, Dirac appeared to be exceptionally narrow-minded and inhibited.\n\nSoon after his eighteenth birthday, Dirac had to spend time away from his sheltered environment for the first time. He travelled to Rugby, where his brother Felix was one of the small army of young apprentices in the local factories, to spend the summer as a trainee engineer, and, perhaps, to see whether he was suited to factory work. By the end of his month-long stay, the answer was clear.\n\nDirac worked in the British Thomson-Houston electrical goods factory, located on a ninety-acre site next to the railway station. The factory dominated the town. It was said that everyone who lived in Rugby either worked there or knew someone who did. Certainly, everyone in the town was familiar with the saw-tooth profile of the factory's roofs, one of them bearing the sign 'Electrical Machinery'.  And everyone, wherever they stood, could see the smoke billowing from two chimneys that pointed to the sky like a pair of smouldering lances.\n\nDirac arrived in Rugby sporting a new wristwatch, a device that had a decade before been regarded as effeminate for men (and outr\u00e9 for women) but had become respectable after soldiers in the war had found them useful. He lodged above a draper's shop on a street corner, precisely midway between the factory's two entrances, a few minutes' walk away. Dirac was one of about a hundred vacation students who provided menial labour, mainly in the relatively quiet testing laboratories well away from the turbine-construction area, when many of the workers were on holiday. It was a slow-news summer, enlivened only by the dramatic lockout of the Electrical Trades Union and by a local polo match in which one of the players was the Secretary of State for War, Winston Churchill.\n\nFlo regularly wrote to Paul, the first of several hundred letters that she sent him between then and her death. It seems that he kept all of them. These first letters were warm and newsy, telling him of Betty's new dog, how 'Daddy missed you when he had all the grass to cut' and of the new overcoat she was going to have done up for him ('I showed it to Pa & he wants it for himself'). Flo repeatedly complained that he was not telling the family enough about what he was doing. 'Do you ever come across Felix?' she asked. The answer was that the two brothers did pass each other on the streets of Rugby, but they did not exchange a word. Their relationship had deteriorated into a state of cold hostility; Paul apparently offered his brother the same expressionless stare that he gave almost everyone else. Either their mother did not know of her sons' falling out, it seems, or she was too blinkered to notice.\n\nDirac's employers in Rugby gave him the only poor report he would receive in his entire life. David Robertson later showed him the damning comments and disclosed that he was the only vacation student from Bristol ever to receive an unfavourable report. It judged Dirac to be 'a positive menace in the Electrical Test Department', to 'lack keenness' and to be 'slovenly', making it clear between the lines that Dirac would be unwise to seek a future on the factory floor.\n\nIn late September 1920, Dirac returned to Bristol to prepare for his final undergraduate year, when he specialised in electrical engineering. His passion, however, was the theory of relativity. One of  his frustrations was that he could not find an accessible technical account of the theory that would explain, step by step, how Einstein had developed his ideas. Of the academic disciplines that contributed the reams of piffle Dirac read about relativity, none was more prolific than philosophy. One commentator wrote: 'A philosopher who regards ignorance of a scientific theory as insufficient reason for not writing about it cannot be accused of complete lack of originality.' The writer of those words was one of the most talented young philosophers working in Britain, Charlie Broad. Having originally wanted to be an engineer, he trained in both philosophy and science at Cambridge and acquired more expertise in relativity theory than the great majority of physicists, many of whom knew next to nothing about Einstein and his work. In the autumn of 1920, soon after Broad was appointed as the Professor of Philosophy at the University of Bristol, he gave a series of lectures for final-year science students on scientific thought, billed to include a description of Einstein's theory. Dirac and several other engineering students sat in on these lectures, though few of them were sitting alongside Dirac to the end, as the going quickly became tough and the material had little to do with engineering. For Dirac, the course was a memorable experience, as it was for Broad, who wrote thirty years later in his autobiography:\n\nthere came to these lectures one whose shoe-laces I was not worthy to unloose. This was Dirac, then a very young student, whose budding genius had been recognized by the department of engineering and was in the process of being fostered by the department of mathematics.\n\nBroad was a wonderfully idiosyncratic lecturer. He always appeared with a carefully prepared script, and he read every sentence twice, except for the jokes, which he delivered three times. Although he spoke drearily, his content was compelling, jargon-free and spiked with witty references to Charles Dickens, Conan Doyle, Oscar Wilde and other literary figures. Trenchancy was one of his strongest suits. During a warning about the snake oil of most popular accounts of relativity, he counselled that 'popular expositions of the Theory are either definitely wrong, or so loosely expressed as to be dangerously misleading; and all pamphlets against it - even when issued by eminent Oxford tutors - are based on elementary misunderstandings.'\n\nBroad's treatment of relativity in his course was unconventional to  the point of quirkiness. He taught Einstein's first theory and his more general version together, taking a unified approach and concentrating on the basic ideas rather than on the mathematics. Broad's aim was to make it clear that the theories give 'a radically new way of looking at Nature'. The first of Einstein's theories is usually dubbed the 'special theory' because it deals only with observers who move in straight lines at constant speeds with respect to one another; for example, passengers on two trains moving smoothly on parallel tracks. Einstein based his theory on just two simple assumptions: first, that when each of the observers measures the speed of light in a vacuum, they will always find the same value, regardless of their speed; and, second, that measurements made by the observers will lead them to agree on all the laws of physics. Einstein's great insight was to see that if these assumptions were followed to their logical conclusion, a new understanding of space, time, energy and matter emerged.\n\nA casualty of Einstein's theory was the widely accepted belief that the universe is pervaded by an ether, which Broad argued had become superfluous:\n\nthere was supposed to be a peculiar kind of matter, called Ether, that filled all Space. On these theories the Ether was supposed to produce all kinds of effects on ordinary matter, and it became a sort of family pet with certain physicists. As physics has advanced, less and less has been found for the Ether to do.\n\nContrary to the theory, the existence of such a substance would imply that there is a uniquely privileged frame of reference, so relativity implies that the ether is an unnecessary assumption and may well not exist, unless experiments say otherwise. Einstein also noted that measurements of space and time are not, as almost everyone else thought, independent but are inextricably linked, leading to the idea of a unified space-time, a concept introduced by his former teacher Hermann Minkowski, a German mathematician. Finally, Einstein showed that an inevitable consequence of this new way of thinking was his equation _E_ = _mc_ 2, implying that the mass of a small coin is equivalent to the vast energy needed to run a city for days or indeed to raze it. An apocalyptic vision of this power had already been presented by H. G. Wells, shortly before the outbreak of the First World War, in his novel _The World Set Free._\n\nFor most purposes, the predictions of Einstein's special theory were extremely similar to the corresponding ones made by Newton's theory. The two sets of predictions, however, were noticeably different at speeds approaching the speed of light in a vacuum: Einstein claimed that, under these conditions, his theory was more accurate, though it would be several decades before the superiority was convincingly demonstrated by experimenters. In the meantime, Einstein's reasoning made it possible to amend the description of anything given by Newton's theory and produce a 'relativistic' version - one that agreed with the principles of the special theory of relativity. Two years later, Dirac took up a new hobby, aiming to produce relativistic versions of Newtonian theories - an activity he pursued like an engineer upgrading tried-and-tested designs to ones that perform to a higher specification: 'There was a sort of general problem one could take, whenever one saw a bit of physics expressed in a non-relativistic form, to transcribe it to make it fit in with special relativity. It was rather like a game, which I indulged in at every opportunity.'\n\nEinstein's second theory of relativity applied to _all_ observers, including ones who are accelerating; for example, observers who fall freely under the action of gravity. In this 'general theory of relativity', Einstein proposed a geometric picture of gravity, replacing Newton's concept that an apple and every other mass is subject to a force of gravity by a radically new way of describing the situation. According to Einstein, every mass exists in a curved space-time - roughly analogous to a curved sheet of rubber - and the motion of the mass at every point in space-time is determined by the curvature of space-time at that point. Because the theory is relativistic, information cannot be transmitted faster than light, and all energies contribute to mass (via _E_ = _mc_ 2) and therefore to gravity. It turns out that, in the Solar System, where almost all matter has comparatively low density and travels much more slowly than light, the predictions of Einstein's theory of gravity are in extremely good agreement with Newton's. But, in some situations, they can be distinguished, and one of the most straightforward ways of doing so involved measuring the bending of starlight by its gravitational attraction to the Sun during a solar eclipse: Einstein's theory predicted that this deflection would be twice Newton's value. This was the prediction that Eddington and his colleagues believed they had verified in their solar-eclipse experiments.\n\nIt was during one of the early lectures in Broad's course that Dirac had a revelation about the nature of space and time. Broad was talking about how to calculate the distance between two points. If they lie at the sharpest corners of a right-angled triangle, then every schoolchild knows that the distance between the points (the hypotenuse) is given by Pythagoras's Theorem: the square of this distance is equal to the _sum_ of the squares of the lengths of the other two sides. In the space-time of the special theory of relativity, things are different: the square of the distance between two points in space-time is equal to the sum of the squares of the spatial lengths _minus_ the square of the time. Dirac later recalled 'the tremendous impact' on him of Broad's writing down that minus sign. This dash of chalk on Broad's blackboard told Dirac that his schoolboy ideas about space and time were wrong. He had assumed that the relationship between space and time could be described using the familiar Euclidean plane geometry, but if that had been true, every sign in the formula for the distance between two points would have been positive. Space and time must be related by a different kind of geometry. Pickering, Dirac's mathematics teacher at the Merchant Venturers' School, had already introduced him to the Riemannian geometry that Einstein had used to describe curved space-time. In this way of looking at space and time, the angles of a triangle may not add up to 180 degrees as they do in ordinary Euclidean space. In Einstein's general theory of relativity, matter and energy are linked with the space and time in which they exist: matter and energy determine how much space-time is curved, and the curvature of space-time dictates how matter and energy move. Thus, Einstein offered a new explanation of why the apple in the tree in Newton's garden fell: it was not the gravitational pull of the Earth that was responsible but the planet's curvature of space-time in the region of the apple.\n\nInspired by Broad's lectures, and by Eddington's semi-popular book _Space, Time and Gravitation_ , Dirac soon taught himself the special and general theories, another early sign of his special talent as a theoretician. The mathematical complexities of Einstein's general theory so terrified most physicists that they found excuses not to bother with it, whereas Dirac - an engineering undergraduate, not a registered student of physics - studied it voraciously. While other nineteen-year-olds were seeking beauty in the flesh, he sought it in equations.\n\nBroad was sceptical of the contribution philosophy can make to advance the understanding of the natural world (he called it 'aimless wandering in a circle'), but his lectures persuaded Dirac that the subject was worth pursuing. One text he took out of the library was John Stuart Mill's _A System of Logic_ , which the young Einstein had studied some fifteen years before. Mill had been the nineteenth century's pre-eminent British philosopher, the most cogent voice of empiricism, the belief that human beings should ground every concept in verifiable experience. His approach to ethics was largely utilitarian, believing that the ultimate good is one that brings the most happiness to the greatest number of people and that the right-ness of any human action should be judged according to its contribution to public happiness. Mill was influenced by other empiricists, notably by his friend Auguste Comte, the French pioneer of the positivist belief that all true knowledge is scientific, including knowledge about 'sociology', a word that Comte coined. Mill had no time for the Kantian 'intuitionist' view that some truths are so exalted that they transcend experience: he dismissed as meaningless many unverifiable statements made by bishops, politicians and others he regarded as airy-fairy moralists. Mill's views and his feet-on-the-ground public spiritedness were enormously influential among Victorians and have become the essence of the liberal English consensus. He influenced Dirac, and many others, more than they knew.\n\n_A System of Logic_ , published in 1843, is a plain-spoken if laborious account of how empiricism can shape every aspect of human life. The book features Mill's agenda for science, which assumes that there is an underlying 'uniformity of nature'. The aim of scientists should be to explain more and more observations in terms of fewer and fewer laws, every one of them grounded in experience and induced from it. For Mill, the agreement between an experimental measurement and a corresponding theoretical prediction does not imply that the theory is correct, as there may well be many other theories that give equally good agreement. He argued that scientists have the never-ending task of finding theories that are in ever-better agreement with empirical observations.\n\nIn a memoir he wrote in his seventies, Dirac said he gave 'a lot of thought' to philosophy, trying to understand what it could contribute to physics. He recalled that he read _A System of Logic_ 'all through',  which we can safely interpret to mean that he read and pondered almost every word of it, his usual practice. Although he found it 'pretty dull', it introduced to him the important idea that the disparate scientific observations and theories he had learned about had an underlying unity. Furthermore, science should seek to describe this unity using the fewest possible laws of nature, each of them formulated in the simplest possible way. Although this probably influenced the thinking of the young Dirac, he concluded that philosophy was not an effective way of finding out what makes nature tick. Rather, as he put it in an interview in 1963, 'it's just a way of talking about discoveries which have already been made'.\n\nThe best way of understanding nature's regularities, he was coming to believe, was through mathematics. Dirac's lecturers in the engineering classes had drummed into him that mathematical rigour is unimportant; mathematics is simply a tool to obtain useful answers that are correct or, at least, accurate enough for the purpose in hand. One exponent of this pragmatic approach to the mathematics of engineering was Oliver Heaviside, an acid-tongued recluse who had invented a battery of powerful techniques that made it easy to study the effects of passing pulses of electric current through electrical circuits. No one quite understood why these methods worked, but he didn't care: what mattered to him was that they gave correct results, with a speed more rigorous methods could not match and without generating inconsistencies with other parts of mathematics. Engineers prized Heaviside's methods for their usefulness, but mathematicians mocked them for their lack of rigour. Heaviside had no time for pedantry ('Shall I refuse my dinner because I do not understand digestion?') and rejected the attacks of his detested opponents. He even entitled his autobiography after them: _Wicked People I Have Known._ 52\n\nDirac studied Heaviside's techniques and later remarked that there was 'some sort of magic' about them. Another of the engineers' clever tricks that impressed Dirac concerned the calculation of the stresses exerted on materials; for example, by a gymnast balancing on a beam. Engineers routinely calculate these stresses using special diagrams that generate correct answers much more quickly than the mathematicians' rigorous techniques. In his classes, Dirac used this method to represent stresses in this way and saw its power; within a few years, he would use similar techniques in a different context, to understand atoms.\n\nOne of the lessons he learned in his engineering classes was the value of approximate theories. In order to describe how something works, it is essential to take into account the quantities that do most to affect its behaviour and to single out the quantities unimportant enough to be ignored. David Robertson taught Dirac a lesson he later regarded as crucial: even approximate theories can have mathematical beauty. So, when Dirac studied electrical circuits, the stresses on revolving shafts in engines and the windings of the rotors in electric dynamos, he was aware that the underlying theories had, like Einstein's general theory of relativity, a mathematical beauty.\n\nIt was probably Dirac's reflections on Einstein's theory that first led him to believe that the goal of theoretical physicists should be to find equations that describe the natural world, but his studies of engineering were the source of a proviso: that the fundamental equations of Nature are only approximations. It was the job of scientists to find ever-better approximations to the truth, which always lies tantalisingly beyond their reach.\n\nApart from the embarrassing report Dirac had been given in Rugby, his record during his degree was almost flawless: only once in three years did he fail to top his class in every subject (the spoilsport was the assessor of a Strength of Materials course who ranked him second).  But it was clear that his real talents were in theoretical subjects and mathematics. Early in 1921, within a few months of completing the degree, his father suggested that he set his sights on studying at Cambridge. Early in February, Charles wrote to St John's College, almost certainly acting on the advice of Ronald Hass\u00e9, head of Bristol University's mathematics department and a member of Cambridge University's network of talent-spotters. Hass\u00e9 was a graduate and research student of the college, notable as the first person in Cambridge to speak of Einstein's 'theory of relativity'.\n\nCharles enquired whether the college would let him have details of 'any open scholarship in mechanical science or mathematics' that his son could apply for. The college responded swiftly and arranged for Dirac to make his trip to Cambridge in June 1921, to sit the college's entrance examination. Dirac's application to the college, made when he had just turned nineteen, is the earliest extant example of his adult handwriting. It shows that he wrote with the precision and  clarity of a calligrapher, each letter standing upright with some of the capitals decorated unobtrusively with a tiny curlicue.\n\nDirac passed the entrance examination handsomely, winning an annual exhibition (a minor exhibition) of \u00a370, which was disappointingly short of the minimum of \u00a3200 a year that he needed to live in Cambridge. Charles argued that it was 'out of the question' to give his son the additional money as he earned only \u00a3420 a year and had no other income, neglecting to mention his lucrative private tuition. Bristol council refused to help because Charles and Paul had become British citizens only two years before and were therefore ineligible for financial assistance. Disappointed, Charles later wrote to Cambridge asking to be kept informed if any other opportunities should arise for his son. He concluded, 'I am sorry to trouble you, but I believe the boy has an exceptionable [ _sic_ ] head for mathematics and I am trying to do my best for him.' When an official at St John's College offered tactfully to advise him further if he would provide more information about his family's finances, Charles did not reply.\n\nAlthough Paul's Cambridge application had stalled, by July he had a first-class honours degree in engineering, a qualification that he and his father hoped would all but guarantee him employment. However, his graduation coincided with the worst depression in the UK since the industrial revolution: unemployment soared to two million. To every job application, Dirac drew a blank. Thus, the most talented graduate Bristol had ever produced found himself unemployed. But this turned out to be a stroke of luck.\n**Four**\n\nMathematics [. . .] does furnish the power for deliberate thought and accurate statement, and to speak the truth is one of the most social qualities a person can possess. Gossip, flattery, slander, deceit all spring from a slovenly mind that has not been trained in the power of truthful statement.\n\nS. T. DUTTON, _Social Phases of Education in the School and the_  \n_Home_ , London, 1900\n\nWhat might have happened to Dirac if he had got one of the jobs he applied for, perhaps in the burgeoning aviation industry? Might the loss to physics have been offset by a commensurate gain for aeronautics? That these are questions of virtual history is due to the mathematician Ronald Hass\u00e9, who deftly steered Dirac's career from engineering to science. Things could easily have worked out quite differently. In September 1921, when Dirac was at a loose end and looking for jobs, David Robertson suggested to Dirac that, rather than hang around doing nothing, he should do an electrical-engineering project. Dirac dabbled in some experiments, but, after a few weeks, Hass\u00e9 wooed him back to the lecture theatres in the mathematics department, having arranged for him to do a full mathematics degree free of charge and for him to skip the first year's work so he could complete it in two years.\n\nDirac's fellow mathematics students were struck by his punctuality. For the first lectures of the day, beginning at 9 a.m., he was always the first to arrive, silently occupying a seat in the front row and showing no interest whatever in his fellow students. He spoke only when spoken to and talked only in clipped, matter-of-fact sentences that bore no trace of emotion. One of the students later recalled that no one even knew the name of the 'tall, pallid youth' or showed much interest in him until the results of the Christmas examination results revealed that the new student 'P. A. M. Dirac' was top of the class.\n\nSome of the students resolved to make some enquiries about their mysterious colleague. They were surprised to learn that although he  was eighteen months younger than anyone else in the class, he already had a degree in engineering. One of his characteristics was that although he was preternaturally silent, he did stir if he spotted a serious scientific error. In one such incident, after a lecturer had filled two and a half blackboards with symbols and left almost all the students frantically scribbling as they tried to keep up with him, he realised that he had made a mistake. He stood back from the blackboard and turned to Dirac: 'I have gone wrong, can you spot it?' After Dirac identified the error and explained how to put it right, the lecturer thanked him and resumed his exposition.\n\nIn Dirac's first year of his new course, he studied pure mathematics - the branch of mathematics pursued with no concern for its applications - and applied mathematics, employed to solve practical problems. One of his lecturers was Peter Fraser, a farmer's son from the Scottish Highlands, a bachelor who lived much of his life in a reverie and liked to tramp the countryside while contemplating the higher truths of mathematics. He did no original research and never wrote a research paper but channelled all his intellectual energy into his teaching. Dirac believed he was the best teacher he ever had.\n\nShortly before 9 a.m. on Mondays, Wednesdays and Fridays, Dirac was in his seat, awaiting the next episode of Fraser's teaching of a special type of mathematics, known as projective geometry, largely a French invention derived from studies of perspective, shadows and engineering drawing. One of its founders was Gaspard Monge, a draughtsman and mathematician who much preferred to solve mathematical problems using geometric ideas rather than complicated algebra. In 1795, Monge founded the descriptive geometry that Dirac had used in the first technical drawings he made in Bishop Road School, representing objects in three orthogonal points of view. Jean-Victor Poncelet, an engineer in Napoleon's army, built on Monge's ideas to set out the principles of projective geometry when he was a prisoner in Russia in 1812. His ideas and their consequences were to become the mathematical love of Dirac's life.\n\nWhen most students come across projective geometry, they find it an unusual branch of mathematics because it primarily taxes their powers of visualisation and does not feature complicated mathematical formulae. What matters in projective geometry is not the familiar concept of the distance between two points but the _relationships_ between the points on different lines and on different planes. Dirac became  intrigued by the techniques of projective geometry and by their ability to solve problems much more quickly than algebraic methods. For example, the techniques allow geometers to conjure theorems about lines from theorems about points, and vice versa - 'that appealed to me very much', Dirac stressed forty years later. To him, an impressionable young mathematician, this was a powerful demonstration of the power of reasoning to probe the nature of space.\n\nFraser also persuaded Dirac of the value of mathematical rigour - an uncompromising respect for logic, consistency and completeness - something he had, as an engineering student, been taught to wink at. In Dirac's studies of applied mathematics, he learned how to describe electricity, magnetism and the flows of fluids using powerful equations that yielded neat solutions, all consistent with experimental observations. He also used Newton's laws of mechanics to study the contrived examples that inform the education of every applied mathematician: rigid ladders resting against walls, spheres rolling down inclined planes, and beads sliding around circular hoops. Dirac filled several exercise books with his answers, most of them flawless. He did most of this work in his bedroom, his escape from the family he perceived to be unloving and a refuge from Betty's yapping dog. Betty was developing into an unambitious, self-deprecating young woman, in awe of her brother Paul's intelligence, content to while away hours doing nothing. Her father doted on her, as Bishopston local Norman Jones remembered sixty years later when he said that his main recollection of Charles Dirac was 'seeing him always carrying an umbrella, struggling up the hill [. . .] often with his daughter, of whom he was very fond'.\n\nDirac saw Felix only occasionally, at weekends, when he returned from his lodgings in the Black Country of the Midlands, near Wolverhampton. The brothers were still not on speaking terms.\n\nIn the final year of his course, Dirac should have been given the choice of specialising in either pure or applied mathematics. He wanted to take the pure option but did not get his way. His fellow student on the honours mathematics degree programme, Beryl Dent - the strong-minded daughter of a headmaster - had the upper hand because she was paying for her tuition, unlike Dirac. She expressed a firm preference for studying applied mathematics, and her wishes carried the day, perhaps partly because it was easiest for the lecturers  to teach the same courses to the two students. So, for the first time since he began senior school, Dirac had to work alongside a young woman, but his relations with her were strictly formal; they seldom spoke.\n\nDirac spent the 1922-3 academic year with his head down, building on the applied mathematics that he had learned the year before. One bonus for him was that his course included a few lectures on the special theory of relativity, though he probably knew more about the subject than his lecturer. By the time he had finished, he had acquired considerable expertise in Newtonian mechanics. Although he knew that Einstein had found fault with Newton's laws of mechanics, they worked extremely well for all real-world applications, so it made good sense to master them, as tens of thousands of other students - including Einstein himself - had done before.\n\nDuring his mathematics degree, Dirac encountered the ideas of William Hamilton, the nineteenth-century Irish mathematician and amateur poet. He was a friend and correspondent of William Wordsworth, who served science well by helping to persuade Hamilton that he would do better to spend his time on mathematics rather than on poetry. Among his discoveries, Hamilton was most enamoured with his invention of quaternions, mathematical objects that behave peculiarly when they are multiplied together. If two ordinary numbers are multiplied, the same result emerges regardless of their order of multiplication (for example 6 \u00d7 9 has the same value as 9 3 6). Mathematicians say that such numbers 'commute'. But quaternions are different: if one quaternion is multiplied by a second, the result is _different_ from the result obtained if the second is multiplied by the first. In modern language, quaternions are said to be 'non-commuting'. Hamilton believed that quaternions have many practical applications, but the consensus was that they are mathematically interesting but scientifically infertile.\n\nDirac also heard about Hamilton's reformulation of Newton's laws of mechanics. Hamilton's approach largely dispensed with the idea of force and, in principle, enabled scientists to study any material thing - from a simple pendulum to cosmic matter in outer space - much more easily than was possible using Newton's methods. The key to Hamilton's technique was a special type of mathematical object that comprehensively describes the behaviour of the thing under study, the Hamiltonian, as it became known. Hamilton's  methods became another of Dirac's fixations and were to become his favourite way of setting out the fundamental laws of physics.\n\nThe mathematics degree did not present a sufficient challenge to keep Dirac occupied, so Hass\u00e9 encouraged him to take as many of the undergraduate physics courses as his timetable allowed. Once again, Dirac chose to study fundamental subjects which were not covered in his syllabus. In one course, he studied the electron, the particle discovered twenty-five years before in the Cavendish Laboratory in Cambridge by J. J. Thomson, a man equally adept at investigating nature theoretically and - despite his ham-fistedness - experimentally. Several of Thomson's colleagues thought he was joking when he argued that the electron was smaller than the atom and was a constituent of every atom; to many scientists, the idea that there could exist matter smaller than the atom was inconceivable. Yet he was proved right, and, by the time Dirac first became acquainted with the electron, textbooks routinely ascribed electric current to the flow of Thomson's electrons.\n\nDirac also attended lectures in atomic physics given by Arthur Tyndall, a kindly and articulate man with a keen eye for scientific talent. Tyndall introduced Dirac to what was to prove one of the central insights of twentieth-century physics: the idea that the laws of 'quantum theory', which describe nature on the smallest scale, are not the same as the scientific laws that describe everyday matter. Tyndall illustrated this by describing how the energy of light arrives not in continuous waves but in separate, tiny amounts called quanta. At first, this idea was not taken seriously, as virtually all scientists were convinced that light behaves as waves. Their faith rested on the unarguable success of the theory of light published several decades before by the Scottish physicist James Clerk Maxwell, the Cavendish Laboratory's first professor. According to this theory, checked by many experiments, the energy of light and all other types of electromagnetic radiation is delivered not in lumps but continuously, like water waves lashing against a harbour wall.\n\nQuantum theory had been discovered - largely by accident - by Max Planck, the Berlin-based doyen of German physics. He happened on the idea of quanta when he was analysing the results of some apparently obscure desktop experiments that investigated the radiation bouncing around inside the reflecting walls of ovens at steady temperatures (the experiments aimed to help German industry  improve the efficiency of lighting devices). The quantum emerged stealthily from the darkness of those ovens through the ingenuity of Planck, who brilliantly guessed a formula for the variation in the intensity of the radiation with its wavelength, at every temperature setting of the oven. In the closing weeks of 1900, Planck found he could explain the formula for the 'blackbody radiation spectrum' only if he introduced a concept that seemed completely contrary to Maxwell's theory: the energy of light (and every other type of radiation) can be transferred to atoms _only_ in quanta.\n\nThe conservative Planck did not view this quantisation as a revolutionary discovery about radiation but as 'a purely formal assumption' needed to make his calculations work. Einstein first recognised the true importance of the idea in 1905, when he took the concept of radiation quanta literally and demonstrated that the reasoning Planck had used to derive his black-body radiation spectrum formula was hopelessly flawed. The challenge was to do better than Planck by finding a logical derivation of the formula.\n\nWhen Planck discovered the quantum of energy, he also realised that its size is directly determined by a new fundamental constant, which he denoted _h_ and others dubbed Planck's constant. It figures in almost every equation of quantum theory, but nowhere in the previously successful theories of light and matter, retrospectively labelled 'classical theories'. The minuscule size of the constant means that the energy of a typical quantum of light is tiny; for example, a single quantum of visible light has only about a trillionth of the energy of the beat of a fly's wing.\n\nIn these lectures, Tyndall introduced Dirac to a new way of thinking about light, to new physics. But although Tyndall was admired for his clear presentations, quantum physics was then vague, provisional and messy, so it was impossible for him to present to Dirac the kind of tidy, well-reasoned course that he preferred, underpinned by clear principles and concise equations. This may explain why, if Dirac's later recollections are correct, his first course in quantum theory made virtually no impact on him. His main interest remained relativity.\n\nDespite his earlier setback, Charles Dirac had not lost hope of sending Paul to Cambridge. Late in March, Ronald Hass\u00e9 wrote to the applied mathematician Ebenezer Cunningham, one of the Fellows of St John's  College, reminding him of Dirac's failure to win a local scholarship that would have enabled him to take up the place that he had won two years earlier. Hass\u00e9 pointed out that he was 'certain to get first class honours in June', and that he was 'an exceedingly good mathematician', interested mainly in 'general questions - relativity, quantum theory etc., rather than in particular details, and is, I think, very keen on the logical side of the subject'. Among his perceptive comments, Hass\u00e9 did include some provisos about the young Dirac's character: 'He is a bit uncouth, and wants some sitting on hard, is rather a recluse, plays no games, is very badly off financially.' Those minor points aside, Hass\u00e9 warmly recommended that the college should accept Dirac if he could find the funds to eke out a living.\n\nThis time, Paul Dirac was successful. In August, after he heard that he had won a place at Cambridge, he asked to study relativity with Eddington's Congregationalist colleague Cunningham, who had introduced an unusual version of Einstein's special theory of relativity to the UK shortly before the Great War. At that time, Cunningham and Eddington were streets ahead of the majority of their Cambridge colleagues, who dismissed Einstein's work, ignored it or denied its significance.  But Cunningham was not available: he had given up supervising graduate students after the war, when he had been pilloried as a conscientious objector, most woundingly by authorities who prevented him from working in schools on the grounds that he 'was not a fit person to teach children'. The supervisor chosen for Dirac was another mathematical physicist, Ralph Fowler, a generous-spirited man with the build of Henry VIII and the voice of a drill sergeant. He was not a master of relativity but the foremost quantum theorist in the country and an expert in linking the way materials behave to the enmasse behaviour of their atoms. For Dirac, wanting above all to study relativity, this was not encouraging news.\n\nTwo scholarships - one of \u00a370 per year from St John's College, the other from the Government's Department of Scientific and Industrial Research for \u00a3140 per year - were sufficient to fund Dirac's first year in Cambridge, provided he lived frugally, as was his wont. The arrangements seemed to have fallen into place, but, in September, he received bitter news: the university required students to settle their bills at the beginning of term, but his government grant was going to arrive too late. He feared that he would again have to forgo his place, all for the sake of \u00a35.\n\nBut his father came to the rescue by handing him the money he desperately needed to be sure of solvency in Cambridge. Dirac was touched. This was a crucial act of compassion, he later said, and it minded him to forgive his father for the browbeatings round the dinner table and all the other earlier miseries. Charles Dirac did not seem so bad after all.\n**Five**\n\n[. . .] I could behold  \nThe antechapel where the statue stood  \nOf Newton with his prism and silent face,  \nThe marble index of a mind for ever  \nVoyaging through strange seas of Thought, alone.\n\nWILLIAM WORDSWORTH, _The Prelude_ ,  \nBook III, 'Residence at Cambridge', 1805\n\nCambridge has never been the most welcoming place. Visitors who first arrive by rail are often surprised when they realise that the station is almost a mile from the town centre. This rebuffing nudge was quite intentional. Four decades before the station opened in 1845, the authorities had helped to fight off proposals to link the town to London with a canal, but pressure to make Cambridge part of the emerging railway network was irresistible. They did, however, ensure that the station was about twenty minutes' walk from the nearest college so that students would be less tempted to flit off to London and that outsiders would think twice about invading the town's privacy. In 1851, the Vice Chancellor of the university complained to the directors of the railway company that 'they had made arrangements for conveying foreigners and others to Cambridge at such fares as might be likely to tempt persons who, having no regard for Sunday themselves, would inflict their presence on the University on that day of rest'.\n\nAs soon as Dirac - and every other new, luggage-laden student - emerged from the station, he had to trek to the city centre or join the queue for one of the few buses that took passengers to Senate House Hill. On Monday, 1 October 1923, when he walked into St John's College through the Tudor Great Gate, he entered an unfamiliar world of tradition, camaraderie and privilege. He would have been greeted by college porters - resplendent in their liveries and silk hats - each of them charged with keeping an eye on the students and with an obligation to report any errant behaviour. The college admitted only men, many of them in jodhpurs and flat caps  and talking in voices that advertised their breeding. Dirac's social standing was given away by his cheap suit - purchased from the Bristol Co-Op - his gauche manners and, on the odd occasion when he spoke, his accent. There was also something out of the ordinary about his appearance. A small and well-tended black moustache lay above his snaggled top teeth, his wan face topped with a thatch of black curly hair and dominated by his assertively pointed nose. Not quite six feet tall and recognisably his father's son, Dirac had bright eyes, a large forehead that revealed a receding hairline and, already, the slightest of stoops.\n\nThe sense of tradition in the college is most powerfully expressed in its architecture. Some of it was four centuries old, its construction funded by the posthumous largesse of Henry VIII's bookish paternal grandmother, Lady Margaret Beaufort. The enduring presence of these buildings reminds students that their academic home will remain long after all but the most talented of them have been forgotten. Dirac arrived there with no great ambition, and he was unaware of his academic standing relative to his fellow science students, though he had already decided to do only the most challenging fundamental research. This tradition dates back to Galileo, the founder of modern physics, who took the first steps to cast what he called 'the book of nature' in the language of mathematics. He did this at the turn of the seventeenth century, almost a hundred years after the completion of the first buildings of the college. In this sense, St John's is older than physics.\n\nCollege life reflected the origins of British academia. The earliest scholars had been monks, all wearing the same clothes, and all going about their contemplative lives within an agreed set of timetables and rules. In 1923, all the official students of the college and the rest of the university were male, each of them required to wear a gown and mortarboard in public. Any student who went into town incorrectly attired knew he ran the risk of being nabbed by one of the university's private policemen (proctors or 'progs') or their assistants ('bulldogs'), who roamed the streets after dusk. A transgression of the dress code was punished by a fine of 6s 8d, no laughing matter for any young man keen to preserve his spare money, though not nearly as serious as the penalty for being caught with a woman in his room.\n\nThe students were waited on hand and foot. By 6 a.m., the invariably female bed-makers ('bedders') were hanging around the stone staircases, ready to begin their morning's work. The gyps - man-servants - were available all day to clean, wash up and run errands for the students and for the Fellows (also known as 'dons'). Such service was not, however, available to young Dirac in his first year. He spent it in a cold and damp shoebox of a room in a four-storey Victorian house, a fifteen-minute walk from St John's, sharing with two other lodgers. At a cost of almost \u00a315 a term, the landlady Miss Josephine Brown delivered coals and wood for their fires, supplied gas for the lamps that lit their musty little rooms, provided them with crockery and cleaned their boots. Like all the other landladies approved by the university, Miss Brown was obliged to keep a record of any failure of Dirac's to return home by 10 p.m. Always early to bed, he would not have given her any trouble.\n\nDirac had his first experience of grand dining in Hall, where he took his meals. The room is magnificently appointed, with an elaborately decorated wooden ceiling, Gothic stained-glass windows and dark-wood panels hung with portraits of some of the college's most distinguished alumni, including William Wordsworth. The formalities began at 7.30 p.m. with the arrival of the procession of Fellows and other senior members of college at their long table, under the calm gaze of Lady Margaret, whose portrait in oils hung above them. The students were already seated in their gowns along the six rows of benches, either side of three long rows of tables, each of them set with crisp white linen tablecloths, the college coat of arms worked into the damask.\n\nIt was expected that every head should be dutifully cocked, every pair of hands solemnly crossed in silence as one of the students read the Latin grace from a tablet. The moment he finished, a hundred conversations surged to fill the hall.\n\nThe menus, written by hand in French, described the three courses in a style that would meet the approval of a Paris gourmet. The meal might begin with scalloped cod or lentil soup, move on to a main course of jugged hare or boiled tongue and end with gooseberry pie and cream or a plate of cheese with cress and radishes, or even sardines on toast. Much of this rich food was wasted on Dirac, whose poor digestion made him favour more basic fare, which he ate slowly and in only modest quantities.\n\nDirac's fellow diners consisted mainly of the young men of the  Brideshead generation (in Evelyn Waugh's novel, Charles Ryder and Sebastian Flyte were then beginning their final year over in Oxford). Many of them had been privately educated at schools such as Eton, Harrow and Rugby, where they had learned Latin and Greek and the art of discoursing easily about the fashionable topics of the day, such as T. S. Eliot's modernist poetry, or of passing supercilious judgement on Shaw's latest provocation. Dirac was ill equipped to join them.\n\nEvery night, alcohol circulated up and down the dinner table in Hall, loosening the students' tongues, freeing them to shout ever more loudly to make themselves heard over the din. Amid the cacophony, Dirac sat impassively, a teetotaller in the Methodist tradition, silently sipping water from his glass. He had left Bristol never having consumed a cup of tea or coffee, so his first sampling of these drinks was an event for him. Neither much appealed to him, though he did have the occasional weak and milky tea, its caffeine dose scarcely exceeding homoeopathic levels. Decades later, he told one of his children that he drank coffee only to give himself courage before giving a presentation.\n\nDirac's manner at the dinner table became the stuff of legend. He had no interest in small talk, and it was common for him to sit through several courses without saying a word or even acknowledging the students sitting next to him. Too diffident even to ask someone to pass the salt and pepper, he made no demands at all on his fellow diners and felt no obligation to maintain the momentum of any dialogue. Every opening conversational gambit would be met with silence or with a simple yes or no. According to one story still in circulation in St John's College, Dirac once responded to the comment 'It's a bit rainy, isn't it?' by walking to the window, returning to his seat, and then stating 'It is not now raining.' Such behaviour quickly persuaded his colleagues that further questioning was both unwelcome and pointless. Yet he did prefer to eat in company and to hear intelligent people talking about serious matters, and it was by listening to such conversations that Dirac slowly learned about life outside science.\n\nHe was fortunate to go up to Cambridge at this time. The colleges had just seen the departure of the last students in military uniform, which took precedence over academic dress until the students were officially demobilised. Now that Britain was under no threat of another international conflict, this was an optimistic time, and the  next generation of students was anxious to get back to academic work. Dirac was studying in the university's largest department, mathematics, famous for its high standards and its competitiveness. Among the students, the highest cachet was reserved for those who both excelled in their studies and who competed successfully in sport, which is why Hass\u00e9 had thought it relevant to remark in his reference for Dirac that he 'played no games'. Most students took at least some part in the social life in Cambridge - chatting in the new coffee bars, singing in choirs, slipping out in the evening to the cinema or to see an ancient Greek play. None of this interested Dirac. Even by the standards of the most ambitious swot, he was exceptionally focused on his work, though dedication is no guarantee of success, as thousands of students find out every year. He had been consistently top of the class in the academic backwater of Bristol, but he had no idea whether he would be able to compete with the best students in Cambridge. From the moment Dirac and his colleagues arrived, the dons were watching every one of them, always on the lookout for a student of truly exceptional calibre - in Cambridge parlance, 'a first-rate man'.\n\nIt did not take long for the extent of Dirac's talent to become clear to his supervisor, Fowler, who took a brisk interest in his progress, giving him carefully chosen problems to tackle, constantly encouraging him to hone his mathematics. Students who brought Fowler a good piece of work were rewarded with his favourite exclamation, 'Splendid!', and, more often than not, a pat on the back. He was an inspirational presence in the department, but sometimes unpopular: by spending much of his time working at home or on trips to the Continental centres of physics, he often frustrated the students who yearned for the succour of his advice. But Dirac was not so dependent; he was content to be lightly supervised, to work alone and to generate many of his own projects. Soon, he realised that he had been lucky to have been allocated the most effective supervisor of theoretical physics in Cambridge.\n\nFowler's manner was unique in the mathematics department. The prevailing culture was intensely formal, and the academics - every one of them male and dressed like a banker - kept their heads down in their offices and college rooms. The use of first names was all but forbidden: even the friendliest of colleagues referred to each other by their surnames and, outside the common room, conversations rarely  lasted longer than politeness deemed necessary. Opportunities for them to meet outside the college were minimal as there was no tradition of communal tea and coffee breaks and no programme of seminars. Nor was there any of the staff-student socialising now almost de rigueur in modern university life. Apart from Fowler's guidance, Dirac was left to his own devices. He soon settled into a private routine that would have rendered him invisible among the thousands of his fellow students. With no room of his own in the department, he worked on problems that Fowler set him, read recommended books and the latest journals and reviewed the notes he had made during the lectures. He relaxed only on Sundays. If the weather was fine, he set off in the morning for a few hours' walk, dressed in the suit he wore all week, his hands joined behind his back, both feet pointing outwards as he made his way around the countryside in his metronomic stride. One of his colleagues said he looked like 'the bride-groom in an Italian wedding photograph'.\n\nDirac would put his calculations firmly at the back of his mind, aiming to clear his head so that he could approach his work fresh on Monday morning. Pausing only to eat his packed lunch, he looked every inch the city gent inspecting the local terrain: to the north, there was the winding valley of the river Great Ouse and to the east, the geometrical network of fenland drains and Tudor-style buildings with their Dutch gables. He would return in time for dinner at St John's and then walk back to his digs through the foggy backstreets of Cambridge, most of them unlit. On Monday morning, he was ready for another six days' uninterrupted study.\n\nDirac's reserve did not prevent him from meeting many of the country's most famous scientists soon after he arrived. Among them was the man who had introduced him to the technicalities of relativity theory, Arthur Eddington. He was a young-looking forty-year-old, always neatly dressed in his three-piece suit, the knot of his dark tie poised just below the top button of his starched shirt. For someone so eminent, he was surprisingly lacking in confidence - he often sat with his arms crossed defensively, weighing his words carefully. His unique strength as a scientist lay in his hybrid skills as a mathematician and astronomer, giving him the ideal qualifications to play a leading role in tests of the general theory of relativity. He was one of the few scientists who could work on the experiments because, as a Quaker, he was registered as a conscientious objector. Unknown to  most of his colleagues, Eddington had used his reputation to contrive the media hullabaloo that followed the announcement in November 1919 that the solar eclipse results supported the prediction of Einstein's theory rather than Newton's.\n\nDirac attended his lectures and, like most people who first encountered him through his dazzling prose, was disappointed to find that he was an incoherent public speaker who had the habit of abandoning a sentence, as if losing interest, before moving on to the next one. But Dirac admired Eddington's mathematical approach to science, which would become one of the most powerful influences on him. There was no love lost between Eddington and the other great figure of Cambridge science, the New Zealand-born Ernest Rutherford. The two men had sharply contrasting personalities and diametrically opposed approaches to physics. Whereas Eddington was introspective, mild-mannered and fond of mathematical abstraction, Rutherford was outgoing, down to earth, given to volcanic temper tantrums and dismissive of grandiose theorising. 'Don't let me catch anyone talking about the universe in my department,' he growled.\n\nUnlike Eddington, Rutherford did not look in the least like an intellectual.  By the time Dirac first felt his surprisingly limp handshake, Rutherford was a burly fifty-two-year-old, with a walrus moustache, staring blue eyes and given to filling his pipe with a tobacco so dry that it went off like a volcano when he lit it. Everyone knew when he was in a room as he spoke more loudly than anyone else. To the people who saw him waddling down Trumpington Street, he had the brash, confident air of a man who had done well out of life by running a chain of betting shops. But his appearance was deceptive: he was the most accomplished experimental scientist alive, as he was the first to confirm. His most famous discovery, the atomic nucleus, followed after he suggested to two of his students that they should investigate what happens when they fired subatomic particles at a thin piece of gold foil. After he heard that a few of the particles were deflected backwards, Rutherford imagined his way into the heart of the atom and concluded that the core of every atom is positively charged and occupies only a tiny fraction of its space, 'like a gnat in the Albert Hall', as he put it. He first identified the existence of atomic nuclei in the summer of 1912, when he was working at the University of Manchester, eight years before he moved to Cambridge  to become J. J. Thomson's successor as Director of the Cavendish Laboratory. Soon after he arrived there, he made one of his bold predictions about atomic nuclei by proposing that most of them are made not only of protons, each positively charged, but also of hitherto-unidentified particles with about the same mass but no electrical charge. Rutherford encouraged his colleagues to hunt for these 'neutrons', but their desultory experiments drew a blank.\n\nThe mid-1920s were not a productive time for Rutherford as he was no longer making ground-breaking discoveries but was devoting his prodigious energy to directing the Cavendish Laboratory, which he ruled like an absolute but benevolent monarch. The laboratory was tucked away in Free School Lane, a side street that was a few minutes' walk from the mathematicians' offices, but a world apart. Built in 1871, the Victorian Gothic fa\u00e7ade of the laboratory was much the most impressive part of the building. After walking through the front door, visitors found themselves in a dingy corridor next to a hall half-filled with haphazardly parked bicycles. To the modern eye, the laboratories look like the kind of functional workshops Heath Robinson might have set up in his garage: bare brick walls and wooden floors, pedal-operated lathes, hand-operated vacuum pumps, glass-blowing equipment, sturdy benches covered with greasy tools and some pieces of equipment so primative that they would be hard to sell from a junk shop. The authorities in Cambridge had worried whether an environment like this was worthy of a university for gentlemen, but they acknowledged that it had established itself as an exceptionally productive centre for physics research, and at only modest cost. In 1925, the total budget of the laboratory, including all salaries and equipment, was \u00a39,628.\n\nAlthough Rutherford was disdainful of mathematical physicists - or pretended to be - he welcomed tame theorists who would do difficult calculations for him, such as his son-in-law and golfing partner Fowler, the only theorist to have his own office in the Cavendish. Visiting theoreticians had nowhere to sit except in the squalid, unheated library, a shabby tearoom that reeked of congealed milk and stale biscuits. Many of the older theoreticians reciprocated Rutherford's disdain by having nothing to do with activities at the Cavendish, but some of the younger students accepted Rutherford's invitations to attend the laboratory's regular Wednesday afternoon seminars, preceded by tea - often poured by Lady Rutherford - and,  sometimes, Chelsea buns. At the Cavendish, Dirac came to know two of Rutherford's 'boys', who were to become his closest friends: the Englishman Patrick Blackett and Russian Peter Kapitza. Both had been trained as engineers, but their personalities were quite different, exemplifying the two extremes that Dirac liked most: shy introverts like himself (Blackett) and boisterous extroverts (Kapitza). In their different ways, these two men would powerfully influence Dirac, drawing him out of his shell in his early years at Cambridge, keeping him at the hub of experimental activity, introducing him to dozens of new acquaintances he would not otherwise have made and to a field that had previously been of no interest to him: politics.\n\nBlackett and Kapitza had recently turned up at the Cavendish, like jetsam thrown up by the war. Blackett had arrived first, in January 1919, when he was twenty-one years old and still in his navy uniform. He had been given a first-rate technical education at a naval college and, days after graduating, went to war, aged sixteen. On 31 May 1916, the first day of the battle of Jutland, the most violent naval conflict of the war, he was at one of the twin fifteen-inch turrets of HMS _Barham_ , relentlessly bombarded by German warships too distant to see. By the end of the day, he was walking on the deck - the air thick with TNT fumes and disinfectant - among the charred corpses, some with their limbs blown off.\n\nThree weeks after arriving in the Cavendish, he resigned his commission and took a degree in natural sciences to prepare himself for a life in experimental physics. He cut a suave, romantic figure: six feet two inches tall, slim, handsome as a movie star, yet with the haunted demeanour of a midshipman who had seen his mates die in agony in front of his eyes. In the laboratory, he quickly proved to be an ingenious experimenter, with the scientific virtues of imagination and scepticism. One colleague noted that he was 'not easily convinced even by his own ideas'.\n\nIn almost any other laboratory, Blackett would have stood out as the finest student of his generation. However, in that exceptional phase in the history of the Cavendish, he had plenty of competition, especially in the chunky form of Kapitza, who had earlier beaten Blackett to the scholarship for the university's best laboratory student, one of several small victories that helped to fuel Blackett's resentment of him. Kapitza had settled in the UK in 1921 looking - as one of his Trinity colleagues observed - 'like a tragic Russian  prince', insecure and depressed after the deaths of four members of his close family within a few months at the end of 1919: scarlet fever took the life of his infant son, shortly before his father, wife and baby daughter fell victim to Spanish Flu. In the summer of 1921, after braving an initial rejection, he persuaded Rutherford to take him on as a student in the Cavendish. Kapitza idolised Rutherford for his straightforwardness, his energy and his uncanny ability to ask nature the right questions to make it yield its deepest secrets. When Rutherford was out of earshot, Kapitza referred to him as 'the Crocodile', the young Russian's favourite creature: Kapitza collected poems about crocodiles and even welded a metal model of one to the radiator of his open-topped Lagonda. Kapitza's name for his boss may have been an unconscious reference to the reptile that appeared prominently in books by the Soviet Union's most popular children's writer, Korney Chukovsky. Like most parents in Russia, Kapitza had probably read his children the famous stories of the crocodile who swallows people and dogs but who good-naturedly disgorges them unharmed. Chukovsky encouraged his readers to regard the crocodile with a mixture of fear and admiration, just as Kapitza saw Rutherford.\n\nBy the time Dirac arrived in Cambridge, Kapitza was one of the town's most colourful characters. Although he did not speak any language well - even, it was said, his own - he loved to talk, words tumbling incessantly out of one side of his mouth. He chatted merrily in his high-pitched voice, delighting his colleagues with his card tricks and the amusing stories he told in 'Kapitzarene', a language that seemed to consist of Russian, French and English in roughly equal parts. He returned to the Soviet Union every year to see his family and to advise on the programme of industrialisation being pushed by Lenin's successor, Joseph Stalin. He was playing a dangerous game, as the economist John Maynard Keynes told his wife in October 1925 after Kapitza mentioned that he was planning to visit Russia to advise Trotsky on their country's electrification programme, having secured a firm promise that he could return to Cambridge: 'I believe that they will catch him sooner or later [. . .] he is a wild, disinterested, vain, and absolutely uncivilized creature, perfectly suited by nature to be a Bolshie.'\n\nDirac had no such reservations. Near the end of his life, in a nostalgic account of his early days with Kapitza, Dirac wrote that he was  immediately taken with his boldness and self-confidence. They shared a passion for science and engineering, but much divided them: Kapitza delighted in chit-chat, whereas Dirac ignored it; Kapitza loved literature and theatre, whereas Dirac had little time for either; and Kapitza was sceptical of the abstractions of theoretical physics, which were meat and drink to Dirac.\n\nOn Kapitza's first day in the Cavendish, he was surprised by one of Rutherford's first instructions, forbidding him to spread Communist propaganda in the laboratory. Kapitza worked sedulously at his bench but in his spare time never made any secret of his support of Lenin's politics and pleasure at the defenestration of Russia's land-owning aristocracy during the 1917 revolution. As he wrote later, although he never joined the Communist Party, he always supported its goals: 'I am in complete sympathy with the socialist reconstruction directed by the working class and with the broad internationalism of the Soviet Government under the guidance of the Communist Party.'\n\nIn the early 1920s, the British Government was worrying about the stability of the country's institutions, concerned that Communists would infiltrate and subvert them. It is hardly surprising that, only two years after he arrived in Cambridge, an anonymous informer had tipped off the Government's Security Service MI5 with a report 'to the effect that Kapitza is a Russian Bolshevist'. In collaboration with the Metropolitan Police Special Branch, they kept him under surveillance, anxious that he did not suspect for a moment that he was being watched.\n\nIt was probably Kapitza who introduced Dirac to Soviet ideology, a subject that would later become a crucial ingredient of their friendship. In the mid- to late 1920s, such beliefs were not in vogue in Cambridge, as the great majority of students and dons were not seriously interested in politics. The only prominent Marxist don was the economist Maurice Dobb, who, like Kapitza, was based at Trinity College. The tenor of political conversations in its senior common room was the soul of moderation, equilibrium being guaranteed by moderates such as Rutherford and by a bevy of conservatives that included the poet and classicist A. E. Housman and Charlie Broad, who had moved to Cambridge and was living in the rooms once occupied by Newton.\n\nKapitza liked to compare himself to Dickens's Mr Pickwick, and it was an apposite comparison: each, with winning brio, had founded a club whose members had elected him to be their permanent president. In setting up the Kapitza Club in October 1922, he had shaken his postgraduate colleagues out of their lethargy and persuaded them to attend a weekly seminar on a topical subject in physics. The talks usually took place in Trinity College on Tuesday evenings, after a good dinner. The speakers, normally volunteers from the club's members, spoke with the aid only of a piece of chalk and a blackboard mounted on an easel and had to be prepared for a series of interruptions, mediated by Kapitza with the quick wit and \u00e9lan of a modern-day game-show host.\n\nThe rules of the club were that a student could become a member only by giving a talk and that his membership would be withdrawn if he missed a few meetings. Soon after Dirac's arrival in Cambridge, he started going to the club and joined the less frequent, more theoretically inclined \u2207 V Club, named after a common symbol in mathematical physics. This club - the nearest the theoreticians came to having a seminar programme - was attended by dons as well as students, so its proceedings were more in keeping with the stiff ambience of the mathematics department. Rutherford attended them only rarely, scoffing that theorists 'play games with their symbols, but we in the Cavendish turn out the real facts of nature'.\n\nDespite all these new experiences, the postcards Dirac sent home did little more than confirm he was still alive:\n\nDear Father and Mother\n\nI am coming home next Thursday. I expect I shall arrive by a late train.\n\nLove to all\n\nPaul\n\nAll his postcards were like this. They each bore a sepia photograph of a Cambridge scene and about a dozen sterile words, consisting entirely of facts and brief summaries of the weather. His mother set the pace of the correspondence by writing to him almost weekly letters that continued until the middle of Dirac's career, giving her view of life in 6 Julius Road and her relationship with Charles. At this stage, the letters give no sign that the family was unusual: chatty and steeped in maternal affection, they continually stress how  much he was missed - an emotion that Dirac never reciprocated. Charles Dirac apparently did not write to him, though Flo went out of her way to underline that his father was 'very anxious' to know how he was getting on.\n\nFlo told her son how excited the family was by its new toy, a radio. The Diracs were in the first generation of families to buy a receiver, scarcely a year after they first became available in 1922. Their home did not yet have a mains supply of gas or electricity, so Charles had to walk down to the local tram station to charge up the radio's accumulator (its battery). It was worth the inconvenience: the new device livened up 6 Julius Road, replacing the day-long silence with a soundtrack of programmes from the new British Broadcasting Corporation, including talks, concerts and news. The Diracs would gather round the radio each night to hear the newsreader orate as if he were addressing a funeral. On 22 January 1924, they heard that Ramsay MacDonald had been appointed Britain's first Labour Prime Minister. The party that had begun as the creature of the trade unions was in Downing Street, its agenda and rhetoric moderate enough to avoid panicking the British public, always wary of rapid change. Flo reported to Dirac that his father was 'pleased that the Labour government have got in at last. It is the best for teachers' salaries.'\n\nIn Flo's letters, she hardly mentions Felix. In the spring of 1924, still based near Wolverhampton, he was earning a modest wage as a draughtsman and was cycling home to Bristol during his short vacations.  Stooped over his drawing board, his rimless eyeglasses perched on his nose, he spent his days making technical drawings for a manufacturer of heavy machinery and advising engineers in the workshops. A steady worker, he was admired for his politeness and reliability by his colleagues, who knew - as he must have done - that he could look forward to nothing more in his professional life than mediocrity. In private, he began to pursue interests that set him apart from his parents and brother: he became a Buddhist and dabbled in astrology, seeking help from a guru, the Revd. Sapasvee Anagami Inyom, based in south-west London. To judge from his communications to Felix, this counsellor was a theosophist, someone who sought knowledge of God through a mixture of Hindu and Buddhist teachings. His letters - long on generalities, short on specifics - each began with a florid salvo ('Greetings in the Glorious Love, Joy  and Peace in the Three Gems') and continued with pages of windy reassurance. By embarking on this spiritual path, Felix was abandoning both the Methodism of his mother's family and his father's Catholicism, and by following astrology he was perhaps goading his brother, who, like every other scientist, will have dismissed the notion that local stars and planets influence human fortunes as fatuous.\n\nUnlike his brother, Felix showed an interest in the opposite sex. He acquired a girlfriend, and the relationship became serious enough for his father to suggest that Felix and his girlfriend should visit the family home when Paul was present so that the whole family could meet her. He may well have been disappointed by his mother's rejection of the idea, and it appears that his brother was miffed. In the first public interview Paul gave about his family life, almost forty-five years later, he laughed when he quoted the words his mother used to veto the request - 'Oh no, she mustn't, she might go after Paul' - and, unusually, gave his description of the incident a dab of colour by commenting on his mother's protectiveness: 'I rather resented it.' He said nothing about whether he would have accepted the invitation to meet the young woman but implied that - in this isolated case - his father behaved much more reasonably than his mother. Paul's account of her behaviour appears to be the only criticism he ever made of her in public or private, perhaps a sign of the anger she caused him by her possessiveness towards him and the insensitivity she showed to his brother. This is a rare example of his recalling empathy with his brother or anyone else.\n\nAfter his arrival in Cambridge, Dirac realised that if he was to work on truly fundamental research, he had some catching up to do. The University of Bristol had given him an excellent technical training and a basic grounding in mathematics, but there were several gaps in his education. Among the most serious was his ignorance of the unified theory of electricity and magnetism set out fifty years before by James Clerk Maxwell. This theory, with Darwin's theory of evolution, was the most important scientific advance of the Victorian era and did for electricity and magnetism what Einstein's general theory of relativity would later do for gravity. Maxwell described electricity and magnetism in a handful of equations and used them to predict successfully that visible light consists of electromagnetic waves (or  'electromagnetic radiation'). Such light waves fall within the small range of wavelengths that human eyes can see. Electromagnetic waves with shorter wavelengths than visible light include ultraviolet radiation and X-rays; waves with longer wavelengths include infrared radiation and microwaves.\n\nDirac first learned about Maxwell's equations in lectures given by Ebenezer Cunningham, who found the precocious Bristol engineer-mathematician to be assertive and quick to ask questions about physics that he did not understand. Maxwell's equations must have been thrilling to Dirac: in just a few lines of mathematics, they could explain the results of every experiment on electricity, magnetism and light that he had ever done in Bristol, and much else besides. When he heard about the equations, he saw why Einstein's light quanta had, until a few years before, been so widely ridiculed: the idea flatly contradicted the accepted Maxwellian view that light consisted of waves, not particles. However, nine months before Dirac arrived in Cambridge, news from Chicago suggested that Einstein might be right: the American experimenter Arthur Compton had found that, in some circumstances, electromagnetic radiation - including, presumably, visible light - really can behave not as waves but as discrete particles. He had scattered X-rays from free electrons and found that he could explain his measurements only if each scattering is due to a collision between two particles, like a pair of snooker balls striking one another. This is just as Einstein had suggested - the radiation and the electrons were both behaving as particles - in contradiction to the wave picture. Many physicists refused to believe these results, but Dirac was one of the few who took them in his stride, unencumbered by years of familiarity with the deceptive success of Maxwell's theory.\n\nOne of the scientists who dismissed the new photon picture of light as nonsense was the Danish theoretician Niels Bohr. He had made his name in 1913, when he built on Rutherford's suggestion that every atom contains a tiny nucleus. Rutherford's picture could not explain the experimental discovery that atoms emit and absorb light with certain definite wavelengths (each type of atom that gives out visible light, for example, emits only light with a particular set of colours). It is as if each atom has its own 'song', composed of light, not sound - instead of musical notes, each played with a characteristic loudness, every atom can give out light with its own set of colours, each  colour with a characteristic brightness. Scientists had, somehow, to understand the composition of every atomic melody. Bohr came up with his idea soon after he heard that the colours of the light emitted by hydrogen - the simplest atom, containing only one electron - had an extremely simple pattern, first spotted in 1885 by Johannes Balmer, a Swiss schoolteacher. He happened on a simple but mysterious formula that accounted for the colours of the light given out by these atoms, a mathematical encapsulation of hydrogen's signature tune. Every other atom was more complicated and much harder to understand. Bohr's achievement was to take the cue from the hints in this pattern, to build a theory of the hydrogen atom and then to generalise it to every other kind of atom.\n\nBohr's atom had a positively charged nucleus, which has most of the atom's mass, orbited by negatively charged electrons which are tethered by the attractive force between the opposite charges. In much the same way, the planets are held in their orbits around the Sun by the attractive force of gravity. He imagined that the electron in a hydrogen atom could move around in its nucleus in only certain circular orbits - called by others 'Bohr orbits' - each of them associated with a particular value of energy, 'an energy level'. Each of these orbits had its own whole number, known as a quantum number: the orbit closest to the nucleus was labelled by the number one, the next orbit by the number two, the next orbit by three, and so on. Bohr's innovation was to imagine that the atom gives out light when it jumps (or, in other words, makes a transition) from one energy level to another of a lower energy, simultaneously emitting a quantum of radiation that has an energy equal to the difference between the energies of the two levels. Bohr was saying, in effect, that matter at the atomic level behaves very differently from everyday matter: if the apple that fell in Newton's garden were able to lose energy by descending down a set of allowed energy values, it would not have fallen smoothly but would have made its way jerkily to the ground, as if bumping its way down an energy staircase. But the energy values of the apple are so close together that their separation is negligible and the fruit appears to slide smoothly down the staircase. Only in the atomic domain are the differences between energy values significant enough for the transitions to be jerky.\n\nBohr's theory offered a simple understanding of Balmer's mysterious formula. In just a few lines of undemanding high-school algebra,  any physicist could derive the formula using Bohr's assumptions, leaving the satisfying impression that the pattern of hydrogen's colours was comprehensible. Yet Bohr's theory was only a qualified success: according to the laws of electromagnetism, it was absurd. Maxwell's theory said that the orbiting electron would shine - continuously give out electromagnetic radiation - and thus gradually radiate its energy away. So it would not take long before the orbiting electron would spiral to its doom in the nucleus, with the result that the atom would not exist at all. The only way Bohr could counter this was to assert, by fiat, that orbiting electrons do not give off such radiation, that Maxwell's theory did not work on the subatomic scale.\n\nWith a remarkable sureness of intuition, Bohr extended his ideas to all other atoms. He suggested that each atom has energy levels and that this helped to explain why the different chemical elements behaved so differently - why, for example, argon is so inert but potassium is so reactive. Einstein admired the way Bohr's ideas explained Balmer's formula and the insights they gave into the differences between each type of atom, hinting at an understanding of the very foundations of chemistry. As Einstein remarked in his autobiographical notes, Bohr's theory exemplified 'the highest form of musicality in the sphere of thought'.\n\nBut no one properly understood the relationship of Bohr's atom to the great theories of Newton and Maxwell. These theories came to be described as 'classical', to distinguish them from their quantum successors. A fundamental question was, how, precisely, does the theory of the very small merge into the theory of the comparatively large? To answer this, Bohr developed what he called the correspondence principle: the quantum description of a particle resembles the classical theory more and more closely as the particle's quantum number becomes larger. Similarly, if a particle vibrates rapidly and therefore has a very small quantum number, quantum theory must be used to describe it; classical theory will almost certainly fail.\n\nThis principle was too vague for Dirac: he preferred theoretical statements to be expressed in an equation with a single, lapidary meaning, not to be set out in words that philosophers could dispute. But he was fascinated by Bohr's theory of the atom. He had not heard of it in Bristol, so Fowler's lectures on the theory were an eye-opener. Dirac was impressed that Bohr had come up with the first tractable  theory of what was going on inside atoms. Dirac spent long afternoons in the libraries studying his notes from Fowler's lectures and poring over the classic textbook _Atomic Structure and Spectral Lines_ , by the Munich theoretician Arnold Sommerfeld. Required reading for every student of quantum theory, the book set out Bohr's picture of the atom and showed how it could be refined and improved. Sommerfeld gave a more detailed description in which the possible orbits of the electron are not circular (as Bohr had assumed) but elliptical, like the path of a planet round the Sun. He also improved on Bohr's work by describing the motion of the orbiting electron not using Newton's laws but using Einstein's special theory of relativity. The result of Sommerfeld's calculation was that the measured energy levels should differ slightly from the levels predicted by Bohr, a conclusion supported by the most sensitive experiments. Bohr knew as well as everyone else in atomic physics that his theory was fatally flawed and therefore only provisional; what was unclear was whether the theory that succeeded it would be based on a few tweaks to Bohr's ideas or on a radically new approach.\n\nAt the same time as he was learning and applying Bohr's theory, Dirac was immersed in geometry, which he studied privately and at weekly tea parties held on Saturdays by the mathematician Henry Baker, a close friend of Hass\u00e9's. Now approaching his retirement, Baker was an intimidating man with the thick moustache which was, in those days, almost mandatory. His parties took place at four o'clock on Saturday afternoons in the Arts school, a grim Edwardian building only a short walk from the Cavendish. Apart from the porter and a few cleaners, the School was as lifeless as a museum at midnight until Dirac and fifteen or so other aspiring scholars of geometry arrived and knocked on the front door. Baker regarded these meetings as his opportunity to promote his love of geometry to his most able students. The subject needed him: for almost a century, it had been the most fashionable branch of mathematics in Britain, but its popularity was waning as fashion began to favour mathematical analysis and the study of numbers.\n\nThe parties - better described as after-hours classes for devotees - were friendly but tense with formality and protocol. Each gathering began promptly at 4.15 p.m., and, in the time-honoured way at English universities, could not begin until everyone had been served  a cup of tea and a biscuit. The only students allowed to be late were the sportsmen - rowers, rugby players and athletes who would arrive red-faced and settle down hurriedly after depositing their knapsacks full of sweaty kit. Each week, Baker arranged in advance for one of the students to give a talk to the party before submitting to a grilling by the audience, most of them writing with one hand and smoking with the other. Baker was a spirited teacher, a no-nonsense mediator but a stern host - he had no compunction about berating any student whose attention showed the slightest sign of wandering. For several of the young men, the parties were a chore, but they were a highlight of Dirac's week: '[they] did much to stimulate my interest in the beauty of mathematics'. He learned that it was incumbent on mathematicians to express their ideas neatly and concisely: 'the all important thing there was to strive to express the relationships in beautiful form'.\n\nIt was at one of these parties that Dirac gave his very first seminar, about projective geometry. From his fellow students and Baker, he also became acquainted with a branch of mathematics known as Grassmann algebra, named after a nineteenth-century German mathematician. This type of algebra resembled Hamilton's quaternions, as they are both non-commuting: one element multiplied by another gives a different result if the two are multiplied in a different order. Some applied mathematicians jeered that Grassmann's ideas were of little practical use, but such concerns did not trouble Baker. He warned his students to expect no public recognition for anything they achieved in pure mathematics, whereas 'if you discover a comet you can go and write a letter to \"The Times\" about it'.\n\nBaker was the type of don Cambridge academics called 'deeply civilised' - a subject specialist whose enthusiasms were grounded in high culture. One of his hobbies was the culture of ancient Greece, and he was fascinated by the Greeks' love of beauty, which he believed was as good a stimulus to a scientific life as any. This may be one reason why Dirac drew attention to the aesthetic appeal of Einstein's theory of gravity in a talk he gave at one of Baker's gatherings, having pointed out that its predecessor, Newton's law of gravity, 'is of no more interest - (beauty?) - to the pure mathematician than any other inverse power of distance'. This is Dirac's first recorded mention of 'beauty'. In Bristol, he had been encouraged to  take an aesthetic view of mathematics; now, in Cambridge, he had found again that the concept of beauty was in vogue. The popularity of the concept was at least partly due to the enduring success of _Principia ethica_ , published in 1903 by the philosopher George Moore, one of Charlie Broad's colleagues in Trinity College. Writing with a refreshing absence of jargon, Moore made the incisive suggestion that 'the beautiful should be defined as that of which the admiring contemplation is good in itself'. Soon the talk of intellectuals, _Principia ethica_ was admired by Virginia Woolf and her colleagues in the Bloomsbury Group and declared by Maynard Keynes to be 'better than Plato'. Over a century before, Immanuel Kant had rendered the subject of beauty too complex and intimidating for most philosophers, but Moore made it accessible again in a way that commanded respect. Although _Principia ethica_ did not consider the aesthetics of science, Moore's common-sense approach to beauty probably influenced his scientific colleagues at Trinity, including Rutherford and the college's most eminent pure mathematician, G. H. Hardy: both often talked about the beauties of their subject. Kapitza, too, looked on experimental physics not as 'business', as it was to several of his colleagues, but as a kind of 'aesthetic enjoyment'.\n\nAlthough Dirac was not interested in philosophy, this fascination with the nature of beauty had powerful resonances for him. Like many theoreticians, he had been moved by the sheer sensual pleasure of working with Einstein's theories of relativity and Maxwell's theory. For him and his colleagues, the theories were just as beautiful as Mozart's _Jupiter Symphony_ , a Rembrandt self-portrait or a Milton sonnet. The beauty of a fundamental theory in physics has several characteristics in common with a great work of art: fundamental simplicity, inevitability, power and grandeur. Like every great work of art, a beautiful theory in physics is always ambitious, never trifling. Einstein's general theory of relativity, for example, seeks to describe all matter in the universe, throughout all time, past and present. From a few clearly stated principles, Einstein had built a mathematical structure whose explanatory power would be ruined if any of its principles were changed. Abandoning his usual modesty, he described his theory as 'incomparably beautiful'.\n\nDirac was extremely hard to read. Usually, he looked blank or wore a thin smile, whether he was making headway with one of his scientific problems or depressed by his lack of progress. He seemed to live in a world in which there was no need to emote, no need to share experiences - it was as if he believed he was put on Earth just to do science.\n\nHis belief that he was working solely for himself led to one of his rare spats with Fowler. Soon after Dirac began in Cambridge, Fowler gauged the ability of his new student by asking him to tackle a non-trivial but tractable problem: to find a theoretical description of the breaking up of the molecules of gas in a closed tube whose temperature gradually changes from one end to the other. Some five months later, when Dirac found the solution, he wanted to file it away and forget it, a suggestion that dismayed Fowler: 'if you're not going to write your work up, you might as well shut up shop!' Dirac succumbed and forced himself to learn the art of writing academic articles. Words did not come easily to him, but he gradually developed the style for which he was to become famous, a style characterised by directness, confident reasoning, powerful mathematics, and plain English. All his life, Dirac had the same attitude to the written word as his contemporary George Orwell: 'Good prose is like a window pane.'\n\nThat first paper was a piece of academic throat-clearing, of little consequence and unrelated to the fundamental theories of physics that Dirac loved. In his next three papers, however, he was on the more congenial ground of relativity. In his first paper on the subject, he clarified a point in Eddington's mathematical textbook on Einstein's general theory of relativity, and in the next two applied the special version of the theory first to atoms jumping between energy levels and then to soups of atoms, electrons and radiation. It was not until the end of 1924 that he produced an outstanding piece of work, an exploration - using Bohr's atomic theory - of what happens to the energy levels of an atom when the forces acting on it change slowly. Although Dirac came to no startling conclusions, his paper attested to his mastery of Bohr's theory and of Hamilton's mathematical methods. Yet Dirac was starting to believe that such exercises were hollow. The more he thought about the Bohr theory, the more dissatisfied he was with its weaknesses. Others shared this dissatisfaction: physicists all over Europe feared that a logical theory of the atom might simply be beyond the human mind.\n**Six**\n\nMy grief lies all within,  \nAnd these external manners of lament  \nAre merely shadows to the unseen grief  \nThat swells with silence in the tortured soul.\n\nWILLIAM SHAKESPEARE, _Richard II_ ,  \nAct IV, Scene 1\n\nTowards the end of Dirac's graduate research, Ebenezer Cunningham described him as 'quite the most original student I have met in the subject of mathematical physics' and 'a natural researcher'. By the time he returned to Bristol for Christmas in 1924, he had every reason to be pleased with himself: he had written five good papers - well above the average for even a strong graduate student - with little help from Fowler or any other senior colleague. He was certain to get his Ph.D. But Dirac knew that his work had so far involved mainly tidying up loose ends in other people's projects and that he had not done nearly enough to deserve a place with Bohr and Einstein at the forefront of theoretical physics. For the moment, Dirac was biding his time in the green room, awaiting inspiration, before he could step out on the international stage.\n\nThroughout the preceding year, Dirac may have noticed that his mother's letters indicated her deepening unhappiness and that she was manoeuvring him into the position of a confidant. Early in the summer, she had complained of having little money of her own, a theme that was to become a leitmotif of her correspondence with him. Charles earned a respectable salary and supplemented it by giving private tuition but was always worried about money and had - like many a husband at that time - no compunction about giving his wife only enough to run the house. Too proud to turn to her friends or siblings, she was reduced to asking Paul for money: '[Pa] is grousing about the bills just now especially the grocer's, so I am wondering if you will be able to spare a few shillings a week next time you are home?' Though Dirac does not appear to have responded in writing, it is reasonable to suppose that he was disturbed by it as he was living frugally on his grant and had no additional income from teaching. To give his mother money would reduce him to penury.\n\nIn June, he had moved out of his digs into one of the grandest buildings in the college, the neo-classical New Court, built in the early nineteenth century. In his rooms in the west wing of the building, he had for the first time the benefits of being able to work in complete private, disturbed only by the cleaner and bed-maker. Many well-off students put their individual stamp on their own patch of the college by bringing their own furniture, oriental rugs, paintings and trinkets. Dirac's room was as bare as a jail cell, but the accommodation gave him all he needed: peace and quiet, regular meals and warmth. The only irritation for him was the regular ringing of the chapel bell: a few years later, he told a friend that it 'gets on my nerves sometimes' - so much so that 'I am a little afraid of [it]'. But his mother knew that he was happier in Cambridge than he was in Bristol, and she feared that he would no longer be content in the modest and ill-kept family home now that he had gone up in the world. Shortly before he returned to Bristol for the Christmas vacation, she prepared his bedroom, beating the carpet and scrubbing the floor, 'the best I can do to such a shabby room'.\n\nFelix had settled in Birmingham, living in lodgings in the south-west of the city and working in the machine-testing laboratory of a factory. With no sign that his career was about to move up a gear, it may have been hard for him to hear his parents talk about the successes of his younger brother in Cambridge. Felix had good reason to be envious: he was still tethered to a stool in a drawing office, plying a trade that brought him little money and, it seems, little satisfaction. Still regretting that his father had refused to let him study medicine, Felix volunteered for the Ambulance Corps, evening work that gave him glimpses of the doctor's life he had longed for. He was sharing none of this with his brother - they lived separate lives, all fraternal affection spent.\n\nEarly in the cold and dreary January of 1925, Felix snapped. He left his job, though he took care to remain on good terms with his employer, the technical manager in the Testing Machine Department, who certified that he always found Felix 'to be obliging, courteous, and painstaking in his work'. He stopped writing to his parents and sister and did not tell either them or his landlady what he had done  or that he was living off his savings. He pretended still to be at work, leaving his digs in the morning and returning for his evening meal, sometimes attending classes at the nearby Midland Institute.\n\nBy the end of winter, his savings ran out. His landlady did not suspect that anything was wrong until the first Thursday evening in March, when he did not return for dinner.\n\nThe chilly, overcast morning of 10 March began like any other term-time Tuesday for Paul Dirac. There was a hint of spring in the air. As usual, before beginning his day's work, he walked across the stone courts of St John's to the Porter's Lodge to see if there was any mail in his pigeonhole. He found a tiny envelope - small enough to fit in the palm of his hand - postmarked in Bristol late on the previous night, though it was not the weekly note from his mother. He opened the folded letter and saw that it was from his mother's sister Nell. She began uneasily, asking him to bear up for the news that she was about to convey because his 'parents are so greatly upset'. Felix was dead.\n\nHis body had been discovered four days before under a holly bush on the edge of a field two miles south of the Shropshire village of Much Wenlock. Smartly dressed in a suit and bow tie, Felix had a spanner in one of his pockets and was still wearing his bicycle clips, though his cycle was nowhere to be seen. The people who found him assumed that he had killed himself by taking poison, as an empty glass bottle lay next to his corpse. He carried no identifying papers and left no final message; the only clue to his identity was the case of his glasses, which bore the name of an optician in Wolverhampton.\n\nNot so long ago, Dirac had loved his brother and looked up to him, shared the same bedroom and the same handed-down comics, ran with him on the Bristol Downs and followed him to university. They had been split by arguments, resentments and jealousies, all of them now rendered pathetically insignificant by grief. Now, the act of suicide had made reconciliation impossible.\n\nDirac's feelings about all this are not known, as there is no documentary evidence of his reactions. If he behaved according to type, he will have received the news with the calm of a statue and told no one in Cambridge about it, apart, perhaps, from Fowler. But it is possible to speculate on his emotions from the testimonies of the few close family members with whom he shared his pain decades later, if only  for a few moments. If we extrapolate the feelings he showed then back to 1925, it is reasonable to conclude that the passing of Felix left his brother with a tapeworm of anger, sadness and guilt gnawing inside him.\n\nThe news of Felix's death had been all over Bristol late on the Monday afternoon: the _Evening News_ announced the death in a front-page article under the headline 'Dead in a Field'. A report on the following day noted that Felix's death had caused 'a profoundly painful sensation in the city', hinting that the tragedy was all the more incomprehensible because the deceased was 'the son of one of the most respected gentlemen connected with education in this city'. Charles and Flo did not read the report when it was published as they were in Shropshire to identify their son's body and attend the first stage of the inquest. Dirac had just received his aunt's letter and may have wondered why his parents had not wired him as soon as they heard the news. Did they really believe that he would not want to be among the first to hear of his brother's death? Four decades later, Dirac told friends that he was shocked by his parents' distress. The death of his brother was 'a turning point' for him: 'My parents were terribly distressed. I didn't know they cared so much. [. . .] I never knew that parents ought to care for their children, but from then on I knew.'\n\nIf these and his other recollections of his early family life are accurate, they indicate the extent of his emotional detachment. He appears to have been unaware of many of the experiences that do most to shape the lives of children - the fondness of their parents, the importance of family rituals, the day-to-day entanglements of family life. Nor does he ever even allude to the possibility that the coldness of the Dirac household could have been due at least in part to his own insensitivity. These are among the strongest clues that he suffered from what amounted to a kind of emotional blindness.\n\nFrom Dirac's portrayals of his father's cold-hearted tyranny and his mother's overweening maternalism, it would be natural to expect that the suicide of Felix would have hurt his mother much more than his father. But it was the other way round. Charles was poleaxed. This was no ordinary grief: his doctor advised him to rest for a year; his family feared for his sanity and even worried that he might take his own life. Flo, by contrast, took it all in her stride, though she was distressed that she had misunderstood Felix and  had not seen the disaster as it approached. In a memorial poem to him she wrote thirteen years later, she wrote, 'He had dropped the mask.'\n\nOn a bitterly cold Sunday, two weeks after Charles and Flo first heard of their son's death, they attended a memorial service for him at a nearby church. When Flo returned home, she wrote to Dirac with a mother's firmness: 'Mind you meet Pa on Thursday & stick to him all the time after the inquest, there's a dear boy, & bring him home safely whatever he may hear.' Dirac did as she requested: a few days later, he travelled to the enquiry, held within a mile of the hills where Felix had been found, a part of the country finely etched into the English imagination by Housman's bitter, nostalgic poetry. At the enquiry, Dirac and his heartbroken father sat next to each other when they listened to the coroner read his report. He began by noting that the body had been found on Friday 6 March. The corpse was of a man about twenty-five years old, five feet nine inches tall, with thin features, dark hair, a slight moustache and good teeth. Felix had taken his life, the coroner concluded, by 'taking cyanide of potassium whilst of unsound mind'.\n\nWitnessing Charles Dirac's grief taught his son a lesson: no matter how painful life might become, he would never commit suicide, because the price paid by his family would be too great. Betty was no less affected: in her later life, she never spoke about the circumstances of Felix's suicide, though she once remarked to her children that he had been killed in a car accident.\n\nIt appears that Dirac kept working to his usual routine. Fowler had gone on sabbatical in Copenhagen to work with Bohr, leaving Dirac in the care of the young astrophysicist Edward Milne. He set Dirac the task of investigating the processes going on at surfaces of stars such as the Sun, a problem that Dirac solved efficiently, though once again he did not come up with any eye-catching conclusions. For several months, Dirac's productivity plummeted. He never explained why, but it is reasonable to speculate that he was slowed down by grief and, possibly, that he was turning his attention from tackling readily solvable problems to looking for a truly fundamental research problem. Dirac had yet to show that he had the ability to identify such a challenge, the hallmark of a great scientist. But it is clear that he was developing the talent: he returned to the unexplained question of understanding black-body radiation, which had first led Planck to the idea of energy quanta.\n\nDirac investigated a daring new idea first introduced by a twenty-six-year-old French student, Louis de Broglie, in his Ph.D. thesis. De Broglie used special relativity to argue with startling boldness and originality that every subatomic particle - including electrons - should have an associated wave of a nature yet to be understood. Dirac was inured to thinking of the electron as a particle, for example, in orbit around an atomic nucleus, so de Broglie's notion of a wave-like electron seemed to be a mathematical fiction of no importance to physicists. He carried out some initial calculations but put the work aside after concluding that he had done nothing worth publishing. Having sniffed the scent of an important problem, he had then lost it; but he would soon return.\n\nIn early May, almost two months after the death of Felix, Dirac was looking forward to the visit of Niels Bohr, widely regarded as the world's leading atomic scientist (he had won the Nobel Prize for physics two years before). Then approaching his fortieth birthday, he was an imposing figure: tall, noble and good-natured, with a huge head and a heavily built body that still bore traces of youthful athleticism.  His sprawling hands had once helped him to become a top Danish goalkeeper, narrowly missing selection for his country's soccer team in the 1908 Olympics. Those hands now spent much of the time relighting his pipe or cigarettes; like his fellow chain-smoker Rutherford, Bohr was a serial cadger of matches. The two men had worked together in Manchester for three months in the early summer of 1912, and Bohr had come to regard Rutherford as a 'fatherly presence'. It was an improbable friendship. Both were profound, intuitive thinkers and impatient with mathematical thinking, but their modes of expression were entirely different: Rutherford was a straight talker whose bluntness could make a navvy blush, whereas Bohr - an inveterate mumbler - was almost always polite and struggled to articulate the tortuous debate going on inside his head. His words were well worth hearing, however, and his audiences sat in silence, straining to hear his every word.\n\nBohr gave his talk, 'Problems of Quantum Theory', on 13 May and spoke again at the Kapitza Club three days later. He underlined his view that the current atomic theory was only provisional and that a better-founded one was sorely needed. Bohr was also  unhappy with the need to describe light sometimes as particles and at other times as waves. Shortly before, he had failed to resolve the dichotomy, and he was now gloomy about the state of quantum physics. Such confusion intimidates mediocre thinkers, but for the most able ones it signals an opportunity to make their name. One student who was bright enough, in Bohr's estimation, to solve the problems of quantum theory was the German prodigy Werner Heisenberg, based in G\u00f6ttingen but soon to visit Cambridge. He was very different to Dirac: widely cultured and with a fondness for conversation and patriotic songs which had been nurtured around campfires during his years in the German Youth Movement. Heisenberg would declare over a glass of beer that 'physics is fun', a phrase that would not have entered the heads of the serious men who had founded the subject eighty years before.\n\nOn the cool Tuesday evening of 28 July, the sweet summer air calm and damp after a day of wind and light showers of rain, Heisenberg addressed the Kapitza Club, his first presentation in Cambridge. He expected to be met with the university's famous formality but, instead, found himself talking in a makeshift college room, with several members of his audience having to sit on the floor. It is not clear whether Dirac was awake throughout Heisenberg's seminar or even if he attended it. Some of the physicists who attended vaguely remembered that Heisenberg spoke about the light emitted and absorbed by atoms and that he remarked in a coda that he had written an article about a new approach to atomic physics. Later, Heisenberg could be sure only that he did mention this article to his host Fowler, but no one in Cambridge - or even Heisenberg himself - appears to have realised that they had been part of history in the making.\n\nDirac returned home for the summer break having secured funding for another three years' research from the Royal Commission for the Exhibition of 1851, which dispensed scholarships funded by the Exhibition's unexpected profits. Dirac's application had been recommended by Maynard Keynes and included encomia from Cunningham, Fowler and the physicist and astronomer James Jeans, who affirmed that Dirac had 'ability of the highest order in mathematical physics'. Much was expected of the young Dirac, though he had published nothing of consequence since his brother's suicide.\n\nDirac probably had to fend off his mourning parents' requests for  him to return to Bristol. His father had already tried to persuade him to apply for the post of Assistant Lecturer in Mathematics at the university, but there can never have been any question that Dirac would accept such a post - he was starting to become aware of his academic worth. And he was still awaiting a challenge equal to his talent.\n\nEarly in September 1925, a postman walked up the steep path to the front door of 6 Julius Road and delivered an envelope that changed Dirac's life. The package, sent by Fowler, contained fifteen pages of the proofs of a paper sent to him by its author, Werner Heisenberg, who had made several corrections to it in his slanting handwriting. This article, written in German, contained the first glimpse of a completely new approach to understanding atoms. Most supervisors would have kept the proofs to themselves, to get a head start on their fellow researchers. Fowler, however, sent the proofs to Dirac with a few words scribbled on the top right-hand corner of the front page: 'What do you think of this? I shall be glad to hear.'\n\nThe paper, technical and complex, would not have been easy reading for Dirac, whose training at the Merchant Venturers' had given him only a modest command of German. He could, however, see that this was not just another run-of-the-mill exercise in the mathematics of quantum theory. Bohr's theory featured quantities such as the position of the electron and the time it takes to orbit its nucleus, but Heisenberg believed that this was a mistake, as no experimenter would ever be able to measure them. He made this point when he summarised the aim of his theory in the article's introductory sentence: 'The present paper seeks to establish a basis for theoretical quantum mechanics founded exclusively upon relationships between quantities which in principle are observable.' Heisenberg knew that it would be extremely difficult to come up with a complete atomic theory built along the lines he envisaged in a single flourish. That would have been too big a task. Instead, he attempted something simpler, by trying to set out a theory of an electron moving not in three dimensions of ordinary space but in just _one_ dimension, that is, in a straight line. Such an electron exists only in the mind of the theoretical physicist, but if this prototype theory worked, then maybe it would be possible to extend it and produce a more realistic version of the theory, one that could be applied to atoms.\n\nHeisenberg considered how classical theory describes his electron,  moving back and forth, and how quantum theory might account for it, bearing in mind that the two theories must merge smoothly, according to the correspondence principle. The new theory looked completely different from its classical counterpart. For example, there is no mention in the quantum theory of single numbers to represent the electron's position; instead, position is replaced by numbers in a square array, an example of what mathematicians call a matrix. Each number in this array is a property of a _pair_ of the electron's energy levels and represents the likelihood that the electron will jump between that pair of energy levels. So, each number can be deduced from observations of the light given out by the electron when it jumps between them. In this way, Heisenberg demonstrated how to build an entirely new atomic theory solely in terms of _measurable_ quantities.\n\nThis picture looks bizarre to anyone coming to it for the first time. With astonishing boldness, Heisenberg had abandoned the assumption that electrons can be visualised in orbit around a nucleus - an assumption no one had previously thought to question - and replaced it by a purely mathematical description of the electron. Nor was this description easy to accept: for example, if it were to apply to ordinary matter, an object's precise location would not be measured with a ruler but would be given in terms of an array of numbers that give the chances of its making transitions to other energy states. This was no one's idea of common sense. In making an imaginative leap like this, Heisenberg was behaving rather like a painter who had switched from Vermeer's classically descriptive style to one based on the abstractions of Mondrian. But whereas painters can use abstraction simply as a technique for producing an attractive image that may or may not refer to real things, abstraction for physicists is a way of representing things en route to the most accurate possible account of material reality.\n\nDirac initially found Heisenberg's approach too complicated and artificial, so he put the paper aside, dismissing it as being 'of no interest'. About ten days later, however, Dirac returned to it and was struck by a point that Heisenberg made in passing, almost halfway through the paper. Heisenberg wrote that some of the quantities in the theory have a peculiar property: if one quantity is multiplied by another, the result is sometimes different from the one obtained if the sequence of multiplication is reversed. This was  exemplified by the quantities he used to represent position and momentum of a piece of matter (its mass multiplied by its velocity): position multiplied by momentum was, strangely, not the same as momentum multiplied by position. The sequence of multiplication appeared to be crucial. Heisenberg later remarked that he mentioned this point as an embarrassing aside, hoping that it would not put off the paper's reviewers and encourage them to think the theory was too far-fetched to be worth publishing. Far from being disconcerted, Dirac saw that these strange quantities were the key to a new approach to quantum physics. Several years later, his mother told an interviewer that Dirac was so excited that he broke his rule of saying nothing about his work to his parents and did his best to explain non-commutation. He did not try again.\n\nUnlike Heisenberg, who had never come across non-commuting quantities before, Dirac was well acquainted with them - from his studies of quaternions, from the Grassmann algebra he had heard about at Baker's tea parties, and from his extensive studies of projective geometry, which also features such relationships. So, Dirac was not only comfortable with the appearance of such quantities in the theory, he was excited by them, although at first he did not understand their significance, nor did he know how to build on Heisenberg's ideas. What Dirac did notice was that Heisenberg had not constructed his theory to be consistent with special relativity so, true to form, Dirac played his favourite game of trying to produce a version of Heisenberg's theory that was consistent with relativity, but he soon gave up. At the end of September, Dirac prepared to return to Cambridge, convinced that the non-commuting quantities in the theory were the key to the mystery. To make progress, he needed to find the lock - a way of interpreting these quantities, a way of linking them to experimentally observed reality.\n\nOne person who, unknown to Dirac, shared his excitement about the theory was Albert Einstein, who wrote to a friend: 'Heisenberg has laid a big quantum egg.'\n\nAt the beginning of October, Dirac began his final year as a postgraduate student. With Fowler's encouragement, he set aside his books of intricate calculations based on the Bohr theory, well aware that - if Heisenberg's theory was right - those calculations were all but worthless.\n\nIt was during one of his Sunday walks, soon after term began, that Dirac had his first great epiphany. Long afterwards, he could not recall the exact date, though he clearly remembered those first exciting hours of discovery. He was, as usual, trying to forget about his work and let his mind wander in the tranquillity of the flat Cambridgeshire countryside. But on that day, the non-commuting quantities in Heisenberg's theory kept intruding into his conscious mind. The crucial point was that two of these quantities, say A and B, give different results according to the order in which they are multiplied: AB is different from BA. What is the significance of the difference AB - BA?\n\nOut of the blue, it occurred to Dirac that he had come across a special mathematical construction, known as a Poisson bracket, that looked vaguely like AB - BA. He had only a faint visual recollection of the construction, but he knew that it was somehow related to the Hamiltonian method of describing motion. This was characteristic of Dirac, as he was much more comfortable with images than with algebraic symbols. He suspected that the bracket might provide the connection he was seeking between the new quantum theory and the classical theory of the atom - between the non-commuting quantities in Heisenberg's theory and the ordinary numerical quantities in classical theory. Fifty-two years later, he remembered, 'The idea first came in a flash, I suppose, and provided of course some excitement, and then of course came the reaction \"No, this is probably wrong\". [. . .] It was really a very disturbing situation, and it became imperative for me to brush up my knowledge of Poisson brackets.'\n\nHe hurried home to see if he could find anything about the Poisson bracket from his lecture notes and textbooks, but he drew a blank. So he had a problem:\n\nThere was just nothing I could do, because it was a Sunday evening then and the libraries were all closed. I just had to wait impatiently through that night without knowing whether this idea was any good or not, but still I think that my confidence grew during the course of the night. The next morning I hurried along to one of the libraries as soon as it was open [. . .].\n\nA few minutes after Dirac entered the library, he pulled from one of the shelves the tome that he knew would provide the answer to his question: _A Treatise on the Analytical Dynamics of Particles and Rigid Bodies_ by the Edinburgh University mathematics professor  Edmund Whittaker. The index directed him first to page 299, where Whittaker set out the mathematical formula for the bracket. Sure enough, as Dirac had surmised, the Poisson bracket, which first appeared over a century before in the writings of French mathematician Sim\u00e9on-Denis Poisson, had the form of two mathematical quantities multiplied together minus two related quantities multiplied together, the multiplication and minus signs making it appear similar to the expression AB - BA. In one of his greatest insights, Dirac saw that he could weave an entire carpet from this thread - within a few weeks of uninterrupted work he had set out the mathematical basis of quantum theory in analogy to the classical theory. Like Heisenberg, he believed that mental pictures of the tiniest particles of matter were bound to be misleading. Such particles cannot be visualised, nor is it possible to describe them using quantities that behave like ordinary numbers, such as position, speed and momentum. The solution is to use abstract mathematical quantities that _correspond_ to the familiar classical quantities: it was these relationships that Dirac pictured, not the particles that they described. Using the analogy with the Poisson bracket, together with the correspondence principle, Dirac found connections between the abstract mathematical quantities in his theory, including the crucial equation connecting the symbols associated with the position and momentum of a particle of matter:\n\nposition symbol \u00d7 momentum symbol - momentum symbol \u00d7 position symbol = _h_ \u00d7 (square root of -1)\/(2 \u00d7 \u03c0)\n\nwhere _h_ is Planck's constant and \u03c0 is the ratio of the circumference to the diameter of every circle (its value is about 3.142). The square root of minus one - the number that, when multiplied by itself gives minus one - plays no role in everyday life but is common in mathematical physics. So there was nothing new on the right-hand side of the equation. The most mysterious part of the equation was on the left-hand side, especially for those unwise enough to think of the position and momentum symbols as anything other than abstractions: they are not numbers or measurable quantities but _symbols_ , purely mathematical objects.\n\nTo all but mathematical physicists of the most austere disposition, Dirac's description looked remote from reality, but, in the right hands, it was possible to manipulate his abstract symbols to make  concrete predictions. In Eddington's words, 'The fascinating point is that as the development process proceeds, actual numbers are _exuded_ from the symbols.' By this, Eddington meant that the underlying symbolic language yielded, after mathematical manipulation, numbers that experimenters could check. The value of the theory depended on whether these predictions agreed with the readings on counters, dials and detecting screens. If the theory did that successfully and was logically consistent, it must be judged a success, according to Dirac, no matter how peculiar it looked.\n\nFowler appreciated that his student had done something special. Dirac's theory, much more ambitious than Heisenberg's prototype description of the artificial case of an electron jiggling about in a straight line, sought to describe the behaviour of _all_ quantum particles in _all_ circumstances throughout _all_ time. He knew, however, that the most important priority was to demonstrate that his theory could account for the most important general observations that experimenters had made about atoms. In a few lines of algebra, Dirac demonstrated that energy is conserved in his theory - as it is in the everyday world - and that when an atomic electron jumps from one energy level to another, it gives out a quantum of light whose energy is equal to the difference between the two levels. This indicated that the theory was able to reproduce Bohr's successes, without having to assume that electrons are in orbit, like planets round a star, doomed to cascade into the nucleus. For Dirac, it was meaningless to use such graphic images - quantum particles can be described only using the precise, rarefied language of symbolic mathematics.\n\nAlthough Dirac had been inspired by Heisenberg's paper, the two men had sharply different approaches to their subject. Heisenberg proudly referred to his paper as 'the great saw', a tool to cut off the limb on which the old Bohr theory rested. Dirac, on the other hand, sought to build a bridge between Newtonian mechanics and the new theory. His dream was that all the mathematics that Hamilton and others had used to recast Newton's theory of mechanics would have exact counterparts in the new theory. If Dirac was right, physicists would be able to use the infrastructure of 'classical mechanics' - the stuff of hundreds of textbooks - in the construction of the new theory, which had been named the year before by Heisenberg's senior colleague, Max Born: 'quantum mechanics'.\n\nBy early November, Dirac had written his paper and had given it an  ambitious title that would catch the attention of even the most casual browser: 'The Fundamental Equations of Quantum Mechanics'. Fowler was delighted. Only a few months before, he had described his student's ability to 'push forward the mathematical development of his ideas' and to 'view old problems in a fresh and simpler way'. Now he could alter the focus of his praise of Dirac from his potential to his achievement. Fowler's highest priority now was to ensure that the paper was published as quickly as printing schedules allowed; if one of Dirac's competitors managed to submit a similar paper before him, then, according to the unwritten rules of the scientific community, Dirac would be regarded as an 'also ran'. Like sport, science is supposed to be an activity in which the winner takes all. Fowler had recently been elected a Fellow of the UK's academy of science, the Royal Society, qualifying him to send manuscripts for publication in its proceedings in the confident expectation that they would be accepted without delay.\n\nFor most physicists in Cambridge, the discovery of quantum mechanics was a non-event. Apart from his discussions with Fowler, Dirac made no effort to draw his colleagues into the new revolution in physics that he knew was afoot. Word was beginning to spread, however, that he was a 'first-rate man' in the making, though his wispy, almost wordless presence gave no clue to the depth and subtlety of his thinking. It appears to have been at about this time that his colleagues invented a new unit for the smallest imaginable number of words that someone with the power of speech could utter in company - an average of one word an hour, 'a Dirac'. On the rare occasions when he was provoked into saying more than yes or no, he said precisely what he thought, apparently with no understanding of other people's feelings or the conventions of polite conversation.\n\nDuring a meal in St John's Hall, he crushed a fellow student who was devoting his time to workaday problems in classical physics: 'You ought to tackle fundamental problems, not peripheral ones.' This was Rutherford's credo, too, though his approach was more down to earth. Rutherford was wary of the theorists' effusions about their latest hieroglyphics until the results were useful to experimenters. Quantum mechanics had yet to do that. Most physicists found it implausible that nature could be so perverse as to favour a theory that required thirty pages of algebra to explain the simplest  atom's energy levels, rather than Bohr's theory, which explained them in a few lines. For Rutherford and his boys, the real sensation that autumn was not the revelations about quantum mechanics but the discovery that electrons have spin. Made at the University of Leiden by two Dutchmen, this discovery took everyone by surprise. In terms of the Bohr picture of the atom, it was easy to envisage crudely what was going on: the orbiting electron is spinning, just as the Earth spins like a top around its north-south axis. Though soon to be taken for granted, many leading physicists thought the idea that the electron has spin was ridiculous.\n\nOne of the postgraduate students who first heard in Cambridge that term about the discovery of spin was Robert Oppenheimer, a dapper, well-to-do American Jew just arrived from Harvard, then riddled with anti-Semitism. He was emotionally fragile, unsure of what he wanted to do with his life but outwardly confident and always keen to display the breadth and depth of his cultural interests. After Rutherford refused to accept him as a student, he spent a few unproductive weeks working with J. J. Thomson, then well over the hill. Oppenheimer disliked Cambridge life - the 'rather pallid science clubs', the 'vile' lectures, having to live in 'a miserable hole'. He saw fellow American students 'literally dying off under the rigors of disregard, climate, and Yorkshire pudding'. By the end of his first term in Cambridge, Oppenheimer was judged by a close American friend to have 'a first class case of depression'.\n\nDirac mentioned none of his new student acquaintances in his postcards home, and virtually nothing about his work. His frustrated parents had to wait six weeks for him even to confirm that his lodgings were comfortable. Flo, having seen her son ratchet up his work rate after tumbling to the importance of Heisenberg's first paper, began what was to become her ineffectual refrain: 'Don't work too hard; have some fun if it comes your way.' Dirac's father was still a broken man, suffering in the cold weather and - in his wife's words - shuffling around 'so slowly that he is like a block of ice'.\n\nOne of Flo's favourite subjects was national and local politics, but that autumn she wrote little about them, probably because there was not much to write about: Britain was stable and quietly prospering. As the country entered the second half of the 1920s, it seemed at last to be coming to terms with its memories of the war, encouraged by the growing international consensus that disagreements should never  again be resolved on the battlefield. This understanding was manifest in the hailed Treaty of Locarno, a non-aggression pact between France, Germany and Belgium, guaranteed by the two supposedly impartial powers, Italy and the UK. Some English schools celebrated by giving their pupils a day off when the treaty was signed in London on 1 December, the day the Royal Society published Dirac's first paper on quantum mechanics. Fowler had managed to cut the time between the submission of the paper and its publication from the usual three months to three weeks.\n\nWord passed around the cognoscenti of quantum theory that a star had been born. Dirac's earlier work had gone largely unnoticed, but here was a paper that appeared to have been written by a mature mathematician and physicist. One of those who had not heard of Dirac before his first work on the new theory was Heisenberg's boss in G\u00f6ttingen, Max Born. Though given to understatement rather than hyperbole, in his memoir he described his first reading of Dirac's early work on quantum mechanics as 'one of the greatest surprises of my life [. . .] the author appeared to be a youngster, yet everything was perfect in its way and admirable'.\n\nHeisenberg, too, was jolted by the paper. On 23 November, a few days after he received the proof copy Dirac sent him, Heisenberg replied in a two-page letter (in German) that began a fifty-year friendship. He began graciously by telling Dirac that he had read his 'beautiful work with great interest', adding that 'There can be no doubt that all your results are correct, insofar as one believes in the new theory.' The discoverer of the new theory was unsure of whether he had hit on ideas of lasting value.\n\nWhat followed must have made Dirac's heart sink: 'I hope you are not disturbed by the fact that part of your results have already been found here some time ago.' Born had independently found the relationship between the position and momentum symbols, a connection that Dirac probably thought he had been first to make. Also, Heisenberg's theory accounted for the Balmer formula for hydrogen atoms, according to a virtuoso calculation by Heisenberg's slightly older friend Wolfgang Pauli, an Austrian theoretician known for his brilliance, his unsparing intellectual aggression and for drinking a glass of wine too many in the nightspots of Hamburg. Heisenberg's note bore the disappointing message that other European theoreticians were on the same track and the deflating prospect that they would repeatedly beat him into print.\n\nIn the ten days following his first letter, Heisenberg wrote Dirac three more warm and complimentary notes, pointing out technical difficulties and minor errors in Dirac's first paper and seeking to clarify details. He concluded his letter of 1 December: 'Please do not take these questions that I write to you as criticisms of your wonderful work. I must now write an article on the state of the theory [. . .] and still wonder at the mathematical simplicity with which you have overcome this problem.' Dirac knew that he was facing some of the toughest competition theoretical physics had to offer. Heisenberg was working in G\u00f6ttingen not only with Born and his student Pascual Jordan but also in association with some of the world's leading mathematicians. The trio of Born, Heisenberg and Jordan were working in the G\u00f6ttingen tradition of a close relationship between the theoretical physicists, mathematicians and experimenters, in sharp contrast with the virtual separation of the communities in Cambridge, where individuality was prized. So, in the undeclared contest to be the first to develop quantum mechanics into a complete theory, the combined might of the mathematicians and physicists in G\u00f6ttingen was pitted against the loner Dirac. He knew that Heisenberg had given his German competitors a head start of two months.\n\nIt would take several years before quantum mechanics crystallised into a complete theory. During that time, it was a work in progress by about fifty physicists. In retrospect, they resembled a group of construction workers who had agreed on a common project - to build a new theory of the behaviour of matter - though not on how to accomplish it. In this case, the construction site was dispersed across north-western Europe, and virtually all the builders were male, under thirty, intensely competitive and craving the respect of their peers as well as the blessing of posterity. There was no official leader, so the workers were free to concentrate on any part of the project they liked. In this quasi-anarchy, some tasks were sure to be done by several people at the same time so, when useful results emerged, there would be quarrels about who most deserved the credit for them. All the workers had their favourite tools and their own preferred way of solving the problems in hand. Some approached  it philosophically, some mathematically and some with their eyes on what experiment could teach them. Some concentrated on the project's grand plans and others on its details. Most of them liked to collaborate and to bounce ideas off their colleagues, while a few others - notably Dirac - had no wish to be in anyone's team. It was rarely easy to see which of the new ideas were duds and which were gems, nor was it obvious whose approaches to the problem were the most promising. Not that any physicist felt bound by a need to take an entirely consistent approach; all that mattered was getting the job done, by whatever means were available. In the end, prizes for a new scientific theory tend to be awarded as they are in architecture for a new building - not to the people who talked most eloquently during the construction but to those who set out its vision and who did most to realise it.\n\nDirac knew that he and his colleagues had taken only the first step towards the building of a complete theory of quantum mechanics. There was much to do.\n**Seven**\n\nA door like this has cracked open five or six times since we got up on our hind legs. It's the best possible time to be alive, when almost everything you knew is wrong.\n\nTOM STOPPARD, _Arcadia,_ 1993, Act 1, Scene 4\n\nEinstein admired the new quantum mechanics, but he was suspicious of it. On Christmas Day 1925 in Berlin, he wrote to a close friend that it seemed implausible to him that something so simple as a number representing a quantum particle's position should have to be replaced by an array of numbers, 'a genuine witches' multiplication table'. Seven weeks later, he was coming to the conclusion that the theory was wrong.\n\nDirac had no such qualms - he was sure that Heisenberg had pointed the best way ahead. Yet although Dirac was working with Heisenberg's theory, their approaches to it were quite different: whereas Heisenberg thought the theory was revolutionary, for Dirac it was an extension of classical theory. While Heisenberg and his G\u00f6ttingen colleagues strove constantly to account for experimental results, Dirac's priority was to lay the theory's 'substrata', following a favourite term of Eddington's. Dirac was following Einstein in taking a top-down approach, beginning with mathematically precise formulations of fundamental principles and only afterwards using the theory to make predictions.\n\nA few weeks after Christmas - the first the Dirac family had spent without Felix - Dirac gave a talk at the Kapitza Club about his just-published paper on quantum mechanics. Two days later, he sent off for publication the proof that his theory reproduced Balmer's formula, the first of three papers on the new theory that he wrote in the first four months of the year. In these first papers on quantum mechanics, Dirac was trying both to understand the theory and to apply it. Puzzled by the symbols in Heisenberg's theory, he spent months unsuccessfully trying to relate them to projective geometry; none of his ideas worked. He was using mathematics that was unknown or at least unfamiliar to most of his colleagues, yet he  rarely gave details of the mathematical techniques he was using or the experimental observations he was trying to explain. He thus managed to perplex both physicists and mathematicians. Nearly fifty years later, Dirac admitted that his attitude to mathematics was cavalier:\n\nI did not bother at all about finding a precise mathematical nature for [some of my symbols] or about any kind of precision in dealing with them. I think you can see here the effects of an engineering training. I just wanted to get results quickly, results which I felt one could have some confidence in, even though they did not follow from strict logic, and I was using the mathematics of engineers, rather than the rigorous mathematics which had been taught to me by Fraser.\n\nThose words would have puzzled Dirac's peers in the spring of 1925. Most of them would have been hard pressed to identify in his papers any remnants of an engineer's training, nor did his writings flaunt the quick-and-dirty approach to calculations favoured by engineers. Rather, Dirac's papers appeared to be impenetrable to all but the mathematically adept. One reason why Dirac's approach was so puzzling was that he was an unusual hybrid - part theoretical physicist, part pure mathematician, part engineer. He had the physicist's passion to know the underlying laws of nature, the mathematician's love of abstraction for its own sake and the engineer's insistence that theories give useful results.\n\nWearing the hat of the physicist, Dirac knew that, for all the mathematical elegance of quantum mechanics, it had yet to make a single prediction whose confirmation would demonstrate its superiority over Bohr's theory. Such a test of the new theory was not easy to find. The best that Dirac could do was to use the theory to describe the most-investigated example of subatomic collision - the scattering of a photon (a particle of light) by a single electron. This process always involves particles travelling at extremely high speeds, close to the speed of light, so any theory that seeks to describe it must be relativistic - consistent with Einstein's special theory of relativity. The problem was that Heisenberg and Dirac's theory of quantum mechanics was not relativistic, and it was unclear how to incorporate relativity into the theory. Dirac made a start on this by tweaking the theory to improve its consistency with relativity and then used it to make testable predictions, using the ideas he had developed at home  in Bristol soon after he received Heisenberg's original paper. The theory was rough and ready, but it enabled Dirac to make the first prediction of quantum mechanics: using a graph, he compared observations of electron scattering with his 'new quantum theory' and showed that it was in better agreement than the classical theory.\n\nQuantum mechanics was still only a rudimentary theory. Much remained to be clarified about the interpretation of its mathematical symbols: what did they really mean? And was it possible to say any more about the motion of subatomic particles? How could the theory be applied to atoms more complicated than hydrogen, containing more than one electron? In later life, Dirac liked to point out that quantum mechanics was the first physical theory to be discovered before anyone knew what it meant. He spent months on the problem of interpreting its symbols and came to see that the theory was mathematically less complicated than he had first thought. Born pointed out to Heisenberg that each array of numbers in his quantum theory was a matrix, which consists of numbers arranged in horizontal rows and vertical columns that behave according to simple rules spelt out in textbooks. Heisenberg had never heard of matrices when he discovered the theory, as Born often reminded his colleagues, adding that he was the one who had ensured that Heisenberg's egg was properly hatched and that its contents were nurtured into infancy.\n\nIt seemed to many physicists that Dirac was working in a private language, and this inaccessibility made his work unpopular. In Berlin, long the global capital of theoretical physics, the consensus was that the approach of the G\u00f6ttingen group - Heisenberg, Born and Jordan - was the most effective. In the United States, then way behind Europe in developing quantum mechanics, the practically minded theoretician John Slater later recalled his frustration with Dirac's writings. In Slater's view, there are two types of theoretical physicist. The first consists of people like himself, 'the prosaic, pragmatic, matter-of-fact sort, who [. . .] tries to write or speak in the most comprehensible manner possible'. The second was 'the magical, or hand-waving type, who like a magician, waves his hands as if he were drawing a rabbit out of a hat, and who is not satisfied unless he can mystify his readers or hearers'. For Slater and many others, Dirac was a magician.\n\nDirac's academic stock rose further in the spring of 1926, during his final term as a postgraduate. He was no longer just another of Cambridge's many brilliant but unfulfilled loners but was recognised as an extraordinary talent. Fowler arranged for him to give two series of lectures on quantum theory for his fellow students. Fowler was also in the audience, aware that his most brilliant prot\u00e9g\u00e9 had overtaken him.\n\nAlthough Rutherford affected to scorn highfalutin theory, he kept abreast of the latest news about quantum physics. At his request, Dirac gave a presentation at the Cavendish about the welter of quantum discoveries that had been made at G\u00f6ttingen, but it was a poor, hastily prepared talk. His audience almost certainly included Oppenheimer and also Kapitza and Blackett, who were - beneath a veneer of amity - increasingly at odds. The tensions were rooted in their relationships with Rutherford. Kapitza shamelessly flattered and courted him, who in return gave favours and even friendship, to the extent that Kapitza was sometimes described as the son Rutherford never had. None of this went down well with Blackett, who admired Rutherford's creative running of the laboratory but had no time for his authoritarianism. Blackett, too, was an object of envy. In the early autumn of 1925, he tutored Oppenheimer at the laboratory bench, teaching him the craft of experimental physics, for which Oppenheimer had little aptitude, as he well knew. With the peculiar logic of neurosis, Oppenheimer decided to get his own back by anonymously leaving on Blackett's desk an apple poisoned with chemicals from the laboratory. Blackett survived but the authorities were outraged and Oppenheimer avoided expulsion from the university only after his parents persuaded the university not to press charges but to put him on probation, on the understanding that he would have regular sessions with a psychiatrist. A few months later, he switched to theoretical physics - a much more congenial field for him - and worked in the same circle as Dirac, who was busy hammering out his vision of quantum mechanics. Oppenheimer recalled that 'Dirac was not easily understood, not concerned with being understood. I thought he was absolutely grand.'\n\nDirac probably did not notice the intrigues among his friends and acquaintances or their personal problems; even if he did, he would probably have ignored them. He worked all day long and took time off only for his Sunday walk and to play chess, a game he played well  enough to beat most students in the college chess club, sometimes several at the same time. Nor did Dirac take much interest in politics. He was an onlooker during the General Strike that almost brought the UK to a halt for nine days in early May 1926 and led many to fear that a Bolshevik revolution was imminent. King George V urged moderation, while in the Government, Churchill demanded 'unconditional surrender' from the workers ('the enemy') who were supporting the demands of the Miners' Union. Some students thought the strike was a national crisis, but to others it was an opportunity to drive a tram or to play at being a docker or a policeman. Almost half the university's students took part in strike-breaking activities, so the authorities had no choice but to postpone the end-of-year examinations, prolonging the merriment. Dirac heard from his mother that trams and buses in Bristol were still running, a relief to his father, so weakened by grief that he could not walk the mile between his home and the Merchant Venturers' School. Fate was about to bring Charles even more sorrow: he heard from Geneva in early March that his mother had died.\n\nThe collapse of the General Strike was important in the development of political thought in Cambridge. The strength of opposition to the strike in the university demonstrated the unwillingness of its dons to disrupt the political status quo; even some of its socialist academics had been strike-breakers. The humiliation of May 1926 was one of the main motivations of a few Marxist scientists who were determined to establish radical politics in Cambridge and then to spread the word across the country. The most effective of the proselytisers was the young crystallographer Desmond Bernal, an energetic and charismatic polymath, who had joined the Communist Party after he graduated in 1923. He had a vision of a just and well-informed collectivist society, with all policy decisions taken according to scientific principles and with the benefit of expert technological knowledge. Scientists were his ideal society's elite, to the extent that he suggested that they might be granted the freedom to form 'almost independent states and be enabled to undertake their largest experiments without consulting the outside world'. The theoretical basis for Bernal's thinking was supplied by Marxism, which seemed to him and his friends to provide a framework for the solution of every social, political and economic problem.\n\nBernal and his colleagues at first made slow progress in converting colleagues to Marxist thinking, partly because of resistance by moderates such as Rutherford, who despised Bernal more than anyone else in Cambridge for his activism and, apparently, for his open sexual promiscuity. The suspicion of card-carrying Communists was so intense that Bernal apparently decided in 1927, when he began a period of working full-time in the Cavendish, that it would be better to let his membership of the party drop. After that, it appears that none of his colleagues officially joined the party.\n\nKapitza did not make the error of alienating senior colleagues: although he shared many of Bernal's political views, he was careful not to offend Rutherford by talking politics in the laboratory. However, Kapitza will have shared his vision of society with Dirac, who had arrived in Cambridge a political innocent and so heard for the first time the claim that Marxism offered an all-embracing scientific theory that could do for society what Newton had done for science. According to this vision, every economy could be the test bed for a theory that promised a brighter future, with intelligent planning taking the place of the sometimes cruel, invisible hand of market forces. Dirac may have noted the strong support Marxists gave to education and industrialisation and the contempt they poured on religion - themes that emerged soon afterwards in his perspective on aspects of life he was discovering outside physics.\n\nDuring the General Strike, Dirac was absorbed in writing his Ph.D. thesis, a compact presentation of his vision of quantum mechanics. Confident though he was of his understanding of the theory, he knew as he wrote his thesis that it was not the whole story, for he had recently heard that an alternative version of quantum theory had appeared, one that looked completely different from Heisenberg's. The author of the new version was the Austrian theoretician Erwin Schr\u00f6dinger, working in Zurich. He was thirty-eight years old, a generation older than Heisenberg and Dirac, with a formidable reputation in Europe as a brilliant polymath.\n\nSchr\u00f6dinger had discovered his quantum theory independently of Heisenberg and a few weeks later, by building on de Broglie's wave theory of matter, which Dirac had admired but had not taken seriously. In the Christmas vacation of 1925, during an illicit weekend with a girlfriend in the Swiss mountains, Schr\u00f6dinger discovered an  equation that described the behaviour of quanta of matter in terms of their associated waves, and then applied the theory in a series of dazzling papers. His achievement was to generalise de Broglie's idea: the young Frenchman's theory applied only to the special case of matter with no overall force acting on it, but Schr\u00f6dinger's theory applied to all matter, in any circumstances.\n\nThe great virtue of Schr\u00f6dinger's theory was that it was easy to use. For the many scientists intimidated by the abstract mathematics in Heisenberg's approach, Schr\u00f6dinger offered the balm of familiarity: his theory was based on an equation that closely resembled those most physicists had mastered as undergraduates, when they were studying water and sound waves. Better still, in Schr\u00f6dinger's theory, the atom could be, at least to some extent, visualised. Roughly speaking, the energy levels of an atom correspond to the waves that can be set up on a piece of rope, held fixed at one end and shaken up and down at the other. The shaker can set up a single half-wavelength (like a crest, moving up and down) on the rope, or, by shaking more vigorously, two half-wavelengths, or three half-wavelengths, or four, or five, and so on. Each of these wave patterns corresponds to a definite energy of the rope, just as each possible Schr\u00f6dinger wave of an atom corresponds to an atomic energy level. The meaning of these Schr\u00f6dinger waves was unclear: their discoverer suggested unconvincingly that they were a measure of the spread of the electron's charge around the nucleus. Whatever the true nature of these waves, they were more intuitively appealing than Heisenberg's matrices to those who lacked mathematical confidence. They, along with everyone else, were relieved when Schr\u00f6dinger gave a preliminary proof (completed two years later by others) that his theory gave the same results as Heisenberg's. The frightened sceptics could then ignore those intimidating matrices.\n\nAt first, Dirac was annoyed by Schr\u00f6dinger's theory, as he resented even the thought of suspending work on the new quantum mechanics and starting afresh. But in late May, as he was finishing the writing of his Ph.D. thesis, he received a persuasive letter from Heisenberg urging him to take Schr\u00f6dinger's work seriously. This wise advice was ironic coming from Heisenberg, an opponent of the rival theory, who had written to Wolfgang Pauli in early June, 'The more I reflect on the physical portion of Schr\u00f6dinger's theory the more disgusting I find it. What Schr\u00f6dinger writes on the visualizability of his theory is probably not quite right. In other words, it's crap.' Schr\u00f6dinger gave as good as he got, dismissing the mathematical arcana of Heisenberg's theory and the idea of quantum jumps. The two theorists clashed unpleasantly when they first met a month later at a packed seminar in Munich, the first skirmish in what was to be a long and acrimonious dispute.\n\nDirac ignored Schr\u00f6dinger's theory in his Ph.D. thesis, 'Quantum Mechanics', the first to be submitted anywhere on the subject. The thesis was a great success with his examiners, including Eddington, who took the unusual step on 19 June of sending him a short handwritten letter on behalf of the Degree Committee of the Mathematical Board, congratulating him on 'the exceptional distinction' of his work. Dirac disliked celebrations and formality, so he was almost certainly not looking forward to the ceremony. He could have taken the degree without attending it but decided to be there in person for the sake of his proud parents, especially his father, who had given him the money that enabled him to begin his Cambridge studies.\n\nDirac's parents and his sister Betty set off at four in the morning, in good time to take the train to Cambridge via Paddington to see Paul be awarded his degree in the setting of the university's grand Senate House. Every detail of the proceedings harked back to the University's monastic origins. The ermine-collared Vice Chancellor presided and, like the other officials, spoke only in Latin, ensuring that Dirac understood scarcely a word. Wearing evening dress with a white bow tie, a small black cap and black silk gown with a scarlet-lined hood, he knelt on a velvet cushion, placed his hands together and held them out to be grasped by the Vice Chancellor, who delivered a prayer-like oration. Dirac arose, a doctor.\n\nIt was the wettest June in Cambridge for five years, but on that day the rain held off. The town was at its most relaxed, teeming with students and their families. Dirac had not learned the local practice of punting, so he and his family could only watch as others steered their flat-bottomed boats along the Cam, through the lawns and fields, past the gorgeous colleges and chapels.\n\nThe Dirac family arrived home at 4 a.m. on Sunday. It had been a happy trip, though its cost had upset Charles. Flo wrote to her son: 'Pa said it cost him \u00a38, so that will be our summer holiday.' It was to be the highlight of her summer, though she was worried that her son was looking drawn and emaciated: 'I wish you would have a nice  rest & feed up & get strong. Do try!' As usual, he took no notice. Like his father, he had no need of holidays - the long vacations were not for relaxing but for hard work. The university was about to hibernate for the summer and would be virtually devoid of social distractions for the few scholars who remained. It was the perfect environment for Dirac to concentrate even more intensively on his work. Heisenberg and Schr\u00f6dinger had knifed a sack of gemstones, and the race was on to pick out the diamonds.\n\nDirac moved out of his lodgings and into a college room, where he worked at his desk through a sweltering July, producing what would prove to be one of his most enduring insights into nature. He realised that he had been wrong to be wary of Schr\u00f6dinger's work. Dirac saw that he could have derived Schr\u00f6dinger's equation using his theory if only he had not been quite so fixated on the links between classical and quantum mechanics. Now, having set aside his prejudice, he could proceed with new gusto. He explained how to generalise Schr\u00f6dinger's first version of his equation, which applied only to cases that stayed the same as time progressed, to situations that _did_ change with time, such as an atom in a fluctuating magnetic field. Quite independently, Schr\u00f6dinger wrote down the same general equation, which is now named - not entirely fairly - only after him.\n\nWithin a few weeks of mastering Schr\u00f6dinger's equation, Dirac used it to make one of his most famous contributions to science. It concerned the most basic particles that exist in nature, usually described as 'fundamental' because they are believed to have no constituents at all. Classic examples are photons and electrons. Today, two established experimental facts form the bedrock of studies about fundamental particles. First, for each type of fundamental particle, every single one of them in the universe is the same and identical to all other particles of the same type - every electron in every atom on Earth is indistinguishable from every electron in galaxies millions of light years away, just as all the trillions of photons given out each second from a light bulb are the same as the photons given out by the most distant star. For electrons and photons, if you have seen one, you have seen them all. Second, the types of fundamental particles fall into one of two classes, much as almost all human beings can be classified as males or females. The first class is exemplified by the  photon, the second by the electron. In 1926, no one knew that there were two such classes.\n\nThe differences between the behaviours of electrons and photons exemplify the sharp contrast in behaviour between the two known classes of particle. For a collection of electrons, say in an atom, each available energy state can usually accommodate no more than _two_ electrons. The situation is quite different for photons: each energy state can host _any number_ of them. One way to visualise this difference is to imagine a pair of bookcases with horizontal shelves arranged vertically above one another in ascending order of energy - the higher the shelf, the higher the energy to which it corresponds. The shelves of the 'electron bookcase' represent the energy states available to electrons, while the shelves of the 'photon bookcase' correspond to the states available to photons. For the 'electron bookcase', each shelf can accommodate at most two books: once the shelf is occupied, it is full and no others can join it. The 'photon bookcase' is different because its shelves can each house any number of books. It is as if electrons are unsociable, whereas photons are gregarious.\n\nPauli first realised the aversion of electrons to their own company in 1925 when he suggested his exclusion principle. This explained the puzzle of why all the electrons in an atom do not all orbit the nucleus in the same, lowest-energy orbit: it is because the electrons simply are not allowed to fit into the same state - they are forced by the exclusion principle to occupy higher-energy states. This is why the different types of atom - manifest as different chemical elements - behave so differently. In common experience, neon is a gas and sodium is a metal, yet the atoms of neon gas are very similar to the sodium atoms: outside their nuclei, they differ only in that a sodium atom contains one more electron than a neon atom. That additional electron determines the differences between the two elements, and the Pauli exclusion principle explains why sodium's extra electron does not simply join the others and form an almost identical type of atom; rather, it occupies a higher-energy quantum state that is responsible for the differences between the behaviour of the two elements. For the same reason, if there were no exclusion principle, the world around us would have none of the huge variety of forms, textures and colours that we take for granted. Not only would our senses have nothing to perceive, they would not exist. Nor, indeed, would human beings or even life itself.\n\nDirac was aware of the exclusion principle's power. But he knew that there was much more to do before theorists could understand, at an atomic level, what was going on in the chemistry experiments that he had done at Bishop Road School. There, chemistry was about describing how the elements and other substances behaved: the prize was to move beyond these descriptions to explanations in terms of universal laws. Quantum mechanics promised to do just this, but in 1926 it was not even possible to apply it to atoms that contain more than just one electron, the so-called 'heavy atoms'.\n\nIn his college room, Dirac reflected on how Schr\u00f6dinger waves might describe heavy atoms and the importance of the Pauli exclusion principle. At the back of Dirac's mind was Heisenberg's tenet that theories should be set up only in terms of quantities that experimenters can measure. He thought about the Schr\u00f6dinger waves that describe two electrons in an atom and wondered whether each wave would be any different if the electrons swapped places. No experimenter could tell the difference, he concluded, because the light given out by the atom would be the same in each case. The way to describe the electrons was, he realised, in terms of waves with the property that they change sign (that is, are multiplied by minus one) when any two electrons are switched. In a few pages of algebra, he used this idea to work out how energy is shared out by groups of electrons as they fill the available energy states. The formulae Dirac derived that summer are now used every day by researchers who study metals and semiconductors; the flows of heat and electricity in them are determined by their electrons, collectively dancing to the tunes of his formulae.\n\nYet the practical applications were of no interest to Dirac. He was concerned only with understanding how nature ticks at the most fundamental level and why there is such a sharp contrast between the waves that describe electrons and those that describe photons. He concluded that, while the wave describing a group of electrons changes sign if two electrons swap places, the corresponding wave describing a group of photons behaves in the opposite way - if two photons swap places, the wave remains the same.\n\nThis tied in neatly with the abortive work he had done on blackbody radiation and led him to explain one of the most puzzling problems of quantum mechanics, a problem that was beyond the ken of Einstein. As Dirac had first heard in Tyndall's lectures in Bristol, quantum theory had begun in the closing weeks of 1900 when Max  Planck suggested that energy is delivered in quanta. The problem was that no one understood how the new theory of quantum mechanics explained Planck's formula. In the months of grief after Felix's death, Dirac had lost the scent of the solution because his theoretical tools were inadequate. Now he had discovered the tool he needed to explain the black-body radiation spectrum: the waves that describe the photons remain unchanged when any two photons are switched. Two pages of calculations in Dirac's notebooks had brought to an end a research project that had been going on for twenty-five years. He must have known he had done something special, but he did not intend to share it with his parents. On 27 July, the message he wrote on his weekly postcard was 'There is not much to say now.'\n\nAt the end of August, Dirac sent off an account of his new theory to the Royal Society. He had every reason to be pleased with himself, but disappointment was in store, as he had again been beaten into print. At the end of October, a month after his paper was published, he received a short, typewritten letter from a physicist in Rome who had published a quantum theory of groups of electrons eight months before. The letter was from Enrico Fermi, an Italian physicist a year older than Dirac. In a short note, written in Berlitz-enhanced English, Fermi drew attention to his paper, which he presumed that Dirac had not seen, and concluded without rancour: 'I beg to attract your attention to it.' But Dirac _had_ seen Fermi's paper several months before and thought it was unimportant; it had slipped his mind. Although Dirac's paper was very different in approach to Fermi's, their predictions for energies of groups of electrons were identical.\n\nIt later turned out that another physicist had also done work similar to Fermi's. In G\u00f6ttingen, Pascual Jordan had independently derived the same results, had written them up in a manuscript and had given it to his adviser Max Born to read during a trip to the USA. Born put the paper at the bottom of his suitcase and forgot all about it until he returned to Germany several months later, but it was too late. Today, physicists associate the quantum description of groups of electrons only with Fermi and Dirac - in this project, Jordan was, unjustly, a loser.\n\nIn September 1926, Dirac was preparing to leave Cambridge to spend a year in Europe funded by his scholarship from the 1851 Commission. His preference was to spend his first year as 'an 1851  man' with Heisenberg and his colleagues in G\u00f6ttingen, but Fowler wanted him to go to Bohr's Institute for Theoretical Physics in Copenhagen. They agreed on a compromise: Dirac would spend half the time in each, beginning with six months in Denmark.\n\nDirac arrived in Copenhagen exhausted, having spent much of the sixteen-hour voyage across the North Sea vomiting. The experience led him to a surprising resolution: he would keep sailing in stormy seas until he had cured himself of the weakness of seasickness. His colleague Nevill Mott was flabbergasted: 'he is quite indifferent to cold, discomfort, food etc. [. . .] Dirac is rather like one's idea of Gandhi.'\n**Eight**\n\nMR PRALINE: [. . .] I wish to complain about this parrot what I purchased  \nnot half an hour ago from this very boutique.  \nPET SHOP OWNER: Oh yes, the, uh, the Norwegian Blue . . . What's,  \nuh . . . What's wrong with it?  \nMR PRALINE: I'll tell you what's wrong with it, my lad. 'E's dead,  \nthat's what's wrong with it!\n\n_Monty Python's Flying Circus_ , script by JOHN CLEESE and  \nGRAHAM CHAPMAN, 1970\n\nMonty Python's famous sketch uncannily resembles a parable Rutherford told Bohr soon after Dirac had arrived in Copenhagen. 'This Dirac,' Bohr grumbled, 'he seems to know a lot of physics, but he never says anything.' This will not have been news to Rutherford, who decided that the best way of answering Bohr's implied criticism was to tell a story about a man who went to a pet store, bought a parrot and tried to teach it to talk, but without success. The man took the bird back to the store and asked for another, explaining to the store manager that he wanted a parrot that talked. The manager obliged, and the man took another parrot home, but this one also said nothing. So, Rutherford continued, the man went back angrily to the store manager: 'You promised me a parrot that talks, but this one doesn't say anything.' The store manager paused for a moment, then struck his head with his hand, and said, 'Oh, that's right! You wanted a parrot that talks. Please forgive me. I gave you the parrot that thinks.'\n\nDirac did a lot of thinking in Copenhagen, mostly alone. No one at Bohr's institute had ever seen anyone quite like him - even by the standards of theoretical physicists he was profoundly eccentric, a retiring figure, happiest when he was alone or listening in silence. His predisposition to answer questions with either yes or no reminded Bohr of Lewis Carroll's description, in _Alice through the Looking Glass_ , of the frustration involved in talking to cats: 'If they would only purr for \"yes\" and mew for \"no\", or any rule of that sort, so that one could keep up a conversation! But how can one deal with a  person if they always say the same thing?' Once in a while, however, Dirac did extend his binary vocabulary of response. When Bohr or one of his friends fussed over him or pressed him to state his preference about something or other, he would bring the interrogation to an end with a curt 'I don't mind.'\n\nPerhaps surprisingly, Dirac thrived in the friendliness and informality of the institute, a world apart from the chilly formalities of Cambridge. Bohr had taken great care to nurture this congeniality since the opening of the building in 1921. Located on the Blegdamsvej, a wide straight road on the north-western edge of the city, from the outside the institute looked anonymous, much like every other new building in the city. But inside, the institute's atmosphere was unique: for most of the day, it hummed with high-minded debate, most of it free of pomposity; individuality was prized, but collaboration was supported; the administration was efficient, free of asinine bureaucracy. Bohr encouraged his colleagues to relax together - to play silly games, to commandeer library tables for ping-pong tournaments, to spend the occasional evening at the cinema, followed by boozy discussions late into the night. Quantum physics was being forged by this generation of physicists, and they knew it. Every researcher was seeking to put their own stamp on the emerging quantum mechanics, nervous of producing trivialities, hopeful that they would come up with insights that would be of lasting value. Their research articles were news that aspired to be history.\n\nBohr was a national hero in Denmark, though he scarcely looked the part. An unassuming but commanding presence, he looked as if he had absconded from the captaincy of a herring trawler. His depth and versatility enormously impressed Dirac, proving to him it was possible to be a premier-division physicist while taking an active interest in the arts, the stock market, psychology and just about any other subject. Like his mentor Rutherford, Bohr had both an eerily sound intuition about the workings of nature and a real talent for getting the best out of his young colleagues. When a special visitor arrived, Bohr would take him or her on a walk among the beech trees of the Klampenborg Forest, just outside the city, to take the measure of his new colleague and give a sense of his non-mathematical approach to physics. Most of the young physicists came under the spell of Bohr, as he had come under Rutherford's.\n\nBohr and his queenly wife Margrethe oversaw life at the institute  like the manager and manageress of a hostel, doing their best to make their guests feel at home. Bohr spent most of the day practising the art of talking and lighting his pipe at the same time, conversing with his colleagues alone or in groups, encouraging them and putting their ideas through the mill. Polite to a fault, his refrain when he cross-examined his young charges was 'Not to criticize, just to learn.' Bohr was the Socrates of atomic physics and he made Copenhagen its Athens.\n\nDirac was billeted in a boarding house in the heart of the city. As he had done in Bristol and Cambridge, he lived life according to a strict routine: every day except Sunday, he took the thirty-minute walk to the institute, past the ducks and swans on the row of artificial lakes on the north-western rim of the city, returning to his lodgings for lunch. On Sundays, he went on long strolls through the local woods or along the coast to the north of the city, usually alone but sometimes accompanied by some colleagues or just with Bohr. Among the new acquaintances he made there, he got on well with Heisenberg - as likeable in person as he was as a correspondent - but apparently not with Pauli. Although prodigiously talented, Pauli was not the most endearing character in physics: he liked the sound of his own voice and routinely meted out casual verbal violence even to his friends, though he was widely admired for his candour, even by his victims. 'You are a complete fool,' Pauli would repeatedly tell his friend Heisenberg, who later said this joshing helped him to raise his game. But Dirac had no taste for it, and Pauli repeatedly broke through the firewall of his self-confidence. However, Dirac showed no sign of discomfort: whether being praised or condemned, he looked straight ahead with his thousand-yard stare, his entire bearing powerfully radiating his unwillingness to speak or even to be approached.\n\nDirac's behaviour was apparently not a complete surprise to Bohr. A few years later, when describing Dirac's first visit to a journalist, Bohr echoed the gravedigger in _Hamlet_ : 'in Copenhagen [we] expect anything of an Englishman'.\n\nThe most pressing problem for quantum theorists remained: what did the symbols in their equations mean? During the summer, Max Born in G\u00f6ttingen had interpreted Schr\u00f6dinger's waves by abandoning the classical principle that the future state of any particle can  always, in principle, be predicted. Born had pictured an electron being scattered by a target. He argued that it is impossible to predict precisely how much the electron will be deflected and that it is possible to know only the _probability_ that the electron will be scattered around any given angle. This led him to suggest that when a particular wave describes an electron, the probability of detecting it in any tiny region follows from a simple calculation that involves, loosely speaking, multiplying the 'size' of the wave in that region by itself. According to Born, the wave is a fictitious, mathematical quantity that enables the likelihood of future behaviour to be predicted. This was a dramatic break with the mechanistic certainties of Newton's picture of the universe, apparently putting an end to the centuries-old notion that the future is contained in the past. Others had the same idea, including Dirac, but it was Born who first published it, though at first even he does not seem to have fully recognised its importance: in the paper where he introduced the concept, he mentions it only in a footnote.\n\nBorn's quantum probabilities seem to have been news to no one at the institute, least of all Bohr, who remarked, 'We had never dreamt it could be otherwise,' though it is unclear why neither he nor any of his colleagues saw fit to publish the idea. Whatever the origins of the probability-based interpretation of quantum mechanics, everyone in the physics community was talking about it in the autumn of 1926, and it was one of the themes of the first Bohr-Dirac 'dialogue'. Only weeks before Dirac's arrival, Schr\u00f6dinger had been a visitor to the institute and made it clear that he found Born's interpretation of quantum waves and the concept of quantum jumps repugnant. On one occasion, after being grilled to a crisp by Bohr, Schr\u00f6dinger retired sick to his bed, but there was to be no escape. Bohr appeared at his bedside and resumed the interrogation.\n\nDirac would not have responded well to such intense questioning, but he made an effective sounding board for Bohr during their autumnal walks. Dirac hardly said a word while Bohr struggled to articulate one point after another, resolution always lying like a phantom, just beyond his grasp. It was on a Sunday hike in October that Bohr, perhaps speculating that Dirac might be interested in classic English literature, took him to the setting of _Hamlet_ , the royal castle of Kronborg, overlooking the stretch of water between Denmark and Sweden. The Bard would have made comic hay from  their verbal exchange, both from the clash of their conversational styles and their contrasting approaches to science and every other subject. Philosophy was an important, compulsory part of Bohr's education, and he took it seriously. Whereas Bohr sought understanding through words, Dirac thought they were treacherous and believed that true clarity could be achieved only in mathematical symbols. As Oppenheimer would later remark, Bohr 'regarded mathematics as Dirac regards words, namely as a way to make himself intelligible to other people, which he hardly needs'.\n\nThere was never any hope that the two would collaborate, as became plain early in Dirac's stay when Bohr called him into his office to help him write a paper. This was Bohr's usual practice: he often dragooned one of his young colleagues into spending a few days as his scribe. The only reward was the honour of being asked and a daily lunch with the Bohrs in their apartment. But the process was not without its frustrations: no sooner would a sentence escape Bohr's lips than he would qualify, amend or delete it in favour of another form of words that might, or might not, be a closer approximation to his intended meaning. So, the tortuous process of dictation continued, never quite reaching a coherent conclusion. Dirac had better things to do than to spend hours disentangling Bohr's fractured locutions and rendering them into prose of exemplary clarity. 'At school', Dirac announced soon after the first session with Bohr began, 'I was always taught not to start a sentence until I knew how to finish it.' His employment as Bohr's amanuensis lasted about half an hour.\n\nIn the evenings, most of the young physicists at the institute liked to relax in the cinema or in their lodgings with a plate of hot dogs and a few beers. But Dirac preferred to spend his nights taking long, solitary walks around the city. He would set out from his lodgings after dinner, take a tram to its terminus and walk the Copenhagen streets back to his digs, thinking about the problems of quantum physics. He probably did not know that he was following in the footsteps of the nineteenth-century philosopher S\u00f8ren Kierkegaard, pioneer of Christian existentialism and almost as famous among his fellow Danes for his eccentricities as his ideas. Kierkegaard chewed over his ideas in his apartment, pacing back and forth for hours, and during the 'people bath' he took each day in the streets of his native city. For two decades from the mid-1830s, the people of Copenhagen saw  the hunch-backed aristocrat walking around in his broad-rimmed hat, his umbrella folded under his arm. 'I have walked myself into my best thoughts,' he said, a remark precisely echoed by the elderly Dirac. But they reacted differently to the people they passed in the street. Dirac said nothing to his fellow pedestrians, but Kierkegaard would startle some of them by interrogating them about some subject on his mind, following in the tradition of Socrates, whom he called 'the virtuoso of the casual encounter'.\n\nDuring the day, Dirac spent most of his time working in the library, occasionally pausing to read the latest publications in the adjoining 'journal room' and to attend a seminar. To Christian M\u00f8ller, one of the young Danish physicists at the institute, Dirac appeared distracted and aloof:\n\nOften he sat alone in the innermost room of the library in the most uncomfortable position and was so absorbed in his thoughts that we hardly dared to creep into the room, afraid as we were to disturb him. He could spend the whole day in the same position, writing an entire article, slowly and without ever crossing anything out.\n\nIn the library, Dirac was cooking up what would turn out to be one of his most famous insights, the connection between the Heisenberg and Schr\u00f6dinger versions of quantum theory. Everyone knew that the theories seemed to give the same results, yet they looked as different as Japanese and English. Dirac found the rules that allow the two languages to be translated into each other, laying bare the relationship between them and giving new clarity to the Schr\u00f6dinger equation. It turned out that the Schr\u00f6dinger waves were not quite as mysterious as they seemed but were simply the mathematical quantities involved in transforming a description of a quantum - an electron, or any other tiny particle - based on its energy values to one based on possible values of its position. Dirac's theory also accommodated Born's interpretation of Schr\u00f6dinger's waves and showed how to calculate the probability of detecting a quantum. He began to realise that the knowledge an experimenter can have about the behaviour of a quantum is also limited. He wrote that 'one cannot answer any question on the quantum theory which refers to the numerical values for both [the initial position and momentum values of a quantum]', and he pointed out cryptically that one would expect to be able to answer questions in which only one of those initial values is known. He was within a split whisker of what would become the most famous principle in quantum mechanics, the uncertainty principle, soon to be snatched from under his nose by Heisenberg.\n\nIn the course of working out his theory, Dirac introduced a new mathematical construction that made no sense within conventional mathematics. The object, which he called the delta function, resembles the outer edge of the finest of needles, pointing vertically upwards from its base. Away from that base, the numerical value of the delta function is zero, but its height is such that the area enclosed between the perimeter and the base is exactly one unit. Dirac knew but did not care that pure mathematicians would regard the function as preposterous as it did not behave according to the usual rules of mathematical logic. He conceded that the function was not 'proper' but added blithely that one can use it 'as though it were a proper function for practically all purposes in quantum mechanics without getting incorrect results'. It was not until the late 1940s that mathematicians accepted the function as a concept of unimpeachable respectability.\n\nIn an interview in 1963, he remarked that it was his study of engineering that led him to his new function:\n\nI think it was probably that sort of training that first gave me the idea of the delta function because when you think of load in engineering structures, sometimes you have a distributed load and sometimes you have a concentrated load at the point. Well, it is essentially the same whether you have a concentrated load or a distributed one but you use somewhat different equations in the two cases. Essentially, it's only to unify these two things which sort of led to the delta function.\n\nBut Dirac's recollections may have been wrong. It may well be that he first read about the delta function from Heaviside, who introduced the function with his customary belligerence in one of the books Dirac read as an engineering student in Bristol. Today, the function is associated with Dirac's name, but he had not been the first to invent it - that appears to have been done in 1822 by Heaviside's favourite mathematician, the Frenchman Joseph Fourier, though several others later discovered it independently.\n\nBohr was indifferent to mathematical rigour, so he would not have been perturbed by the delta function when he read about it in the draft Dirac submitted to him, following the understanding that Bohr had to approve each paper submitted from the institute. However,  Bohr and Dirac were soon in disagreement, like two poets in dispute over the syntax of a stanza. Bohr cared about every word and repeatedly requested detailed changes. For Dirac, the words were there to give the clearest possible expression to his thoughts, and, once he had found the right words, he saw no need to change them. He would have agreed with T. S. Eliot: 'It means what it says and if I had wanted to say it any other way, I should have done so.'\n\nDirac was usually quick to attribute his success to luck, but not in this case - he referred to the paper as 'my darling'. He later remarked that he was pleased to have solved the particular problem he set out to tackle, of laying bare the relationship between Heisenberg's theory and Schr\u00f6dinger's. The main quality needed in its solution was technical skill and application; in his view, no special inspiration was involved. Another reason why Dirac was so fond of his 'darling' was probably that it was a success for his method of developing quantum mechanics by analogy with classical mechanics. During his reading about Hamilton's approach to classical mechanics, he had read how 'transformation theory' related different descriptions of the same phenomenon - by using this idea to find the connection between Heisenberg's theory and Schr\u00f6dinger's, Dirac had shed light on both.\n\nIf he hoped that the paper would establish him as the leader in the field, he was soon to be disappointed. In the late autumn, before he had the proofs of his paper, he heard that Pascual Jordan had solved exactly the same problem. Although Dirac's approach and presentation were more elegant and easier to use, the two papers covered substantially the same ground and featured much the same conclusions. So although Dirac had made another distinguished contribution to quantum mechanics - his second within a year - he had yet to beat all his colleagues to a key innovation in the theory. He had, however, acquired some distinguished admirers, though most of them were struggling to understand his peculiar combination of logic and intuition. One of them was Albert Einstein, who told a friend: 'I have trouble with Dirac. This balancing on the dizzying path between genius and madness is awful.'\n\nOne evening in Dirac's lodgings shortly before Christmas, the telephone rang. It was Professor Bohr, Dirac's landlady told him, as she passed the receiver to him. This was a new experience for him - he  had never before used a telephone. Knowing that Dirac was about to spend the holiday alone, Bohr was calling to ask if he would like to spend Christmas with him and his family. Dirac accepted, though he did not tell his parents. They had been shivering in an unseasonably cold autumn and recovering from the upheaval of having mains electricity installed. Dirac's mother persisted with her doomed campaign to persuade him to do less work and to eat more ('I hope you will take it easy & get nice and plump like Shakespeare's Hamlet') and, for the first time, confided in her son that she was unhappy and tired of the domestic routine. Desperate for a measure of independence, when Charles was out, she and the unemployed Betty sneaked out together to evening classes in French.\n\nThe Dirac family was also preparing itself for its saddest Christmas: a year before, they had had three children at home for the holiday; now they would have only one. On 22 December, the ailing Charles wrote his son a letter, one of only two that Dirac kept from his father, possibly the only letters Dirac received from him in adult life. No longer communicating with Dirac only in French, Charles wrote the four-page letter entirely in English and on black-bordered notepaper that signalled his continuing mourning for Felix.\n\nMy dear Paul\n\nIt will be a lonely time here without you - the first time since you came to us - not so very long ago it seems, but my thoughts are with you to wish you all the happiness a father can wish his only son.\n\nIf you can any time spare a few moments to give me some details of your life there and your work - nothing could please me more, except seeing you again. I should like to feel sure you take sufficient care of yourself - and do not let your studies make you forget your health.\n\nCharles goes on to say that he would like to buy his son a Christmas present, perhaps 'a set of chessmen', and he offers to do 'anything at all' he can to help him. He signs off 'Many kisses from your loving Father'. The note is a window on his grief, his loneliness, his desperation to be closer to his unresponsive 'only son'.\n\nAt midnight on Christmas Eve, Charles and Betty went to a service at a local church, where Felix's death had first been marked. Later, on Christmas Day, Dirac's mother wrote Dirac a fragmentary letter showing that she was as lonely as the man she was living with:\n\nAll we do, as you know, is work & then more work. [. . .] I am trying to get Pa to have [the front room] re-papered. He ought to after 13 years [. . .] He and Betty went up to Horfield Church at 12-midnight for a Service [. . .] This is the first Xmas Day you have been away from home. It is lonely without you.\n\nShe then asked him an unusual favour:\n\nWould you like to send me a few pounds for a diamond ring? I want one so very much. I could wear it in the evenings & think what a darling you are. It is so monotonous doing housework all day long. I get so fed up with it.\n\nPa has pupils all the year round & gives me \u00a38 a year for clothes and everything. It is worse than a servant.\n\nFor the first time in her correspondence, she showed Dirac that he was not just her favourite son but her most intimate confidant and even a substitute for a gift-bearing lover. As her subsequent letters showed, she was in desperate straits, trapped in an unfulfilling marriage to a man who was highly regarded in the community but whom she regarded as an unsympathetic and insensitive brute. In the coming years, her life would unfold like an Ibsen tragedy.\n\nAnother of the out-of-the-blue ideas that Dirac apparently conceived in Copenhagen is now the basis of all modern descriptions of the fundamental constituents of the universe. Such descriptions are based on the nineteenth-century concept of a 'field', which had superseded Newton's vision that nature's basic particles move under the influence of forces exerted by other such particles, often over long distances. Physicists replaced the notion that the Sun and the Earth exert gravitational forces on each other by the more effective picture that the Sun, the Earth and all the other matter in the universe collectively give rise to a gravitational field which pervades the entire universe and exerts a force on each particle, wherever it is located. Likewise, an all-pervasive electromagnetic field exerts a force on every electrically charged particle. Maxwell's theory of electromagnetism and Einstein's theory of gravity are examples of classical 'field theory', each featuring a field that varies smoothly throughout space and time, not mentioning individual quanta. Such classical theories describe the universe in terms of a smooth, underlying fabric. Yet, according to quantum theory, the universe is fundamentally granular: it is ultimately made of tiny particles such as electrons and photons. Loosely speaking, the texture of the underlying fields should, according to classical ideas, be rather like a smooth liquid, whereas quantum theory suggests that it would be like a vast collection of separate grains of sand. To find a quantum version of Maxwell's classical electromagnetism was one of the theoreticians' most pressing problems, and Dirac's next innovation was to solve it.\n\nQuite what put him on to the solution is something of a mystery. Although he was probably aware of the first steps taken a few months before by Jordan, Dirac later said that he first hit on the idea when he was playing with Schr\u00f6dinger waves as if they were mathematical toys, wondering what would happen if they behaved not as ordinary numbers but as _non-commuting_ quantities. The answer began a new way of describing the quantum world.\n\nDirac found a way of mathematically describing the creation and destruction of photons, both commonplace processes. Particles of light are continually created in vast numbers all over the universe in stars and also here on Earth, when an electric light is switched on, a match is struck, a candle is lit. Likewise, photons are continually destroyed - annihilated - for example, when they disappear into human retinas and when leaves convert sunlight to life-giving energy. Neither of these processes of creation and annihilation can be understood using Maxwell's classical theory, which has no way of describing things that appear out of nowhere or disappear into oblivion. Nor did ordinary quantum mechanics have anything to say in detail about the processes of emission or absorption. Yet Dirac showed that this wizardry can be described in a new type of theory, a compact mathematical description of the creation and destruction of photons. He associated each creation with a mathematical object, a creation operator, which is closely related to but quite distinct from another object associated with annihilation, an annihilation operator.\n\nIn this picture, at the heart of modern quantum field theory, the electromagnetic field pervades the entire universe _._ The appearance of every photon is simply an excitation of this field at a particular place and time, described by the action of a creation operator. By a similar token, the disappearance of a photon is the de-excitation of the field, described by an annihilation operator.\n\nDirac had begun to set out a quantum version of Maxwell's unified field theory of electricity and magnetism. He had learned about that theory only three years before, in Cunningham's lectures in  Cambridge, and was now standing on Maxwell's shoulders. So far as Dirac was concerned, his theory put an end to the hand-wringing about the apparent conflict between two theories of light: a wave theory seemed to account for propagation, while a particle theory was needed to explain the interactions with matter. The new theory avoided the embarrassment of having to choose between the wave and particle descriptions and replaced the two sharply contrasting pictures with a single, unified theory. Evidently pleased with himself, Dirac wrote that the pictures were in 'complete harmony'. But he was not interested in sharing the good news with his parents, who read on their weekly postcard their son's familiar message: 'There is not much to say now.'\n\nIn his paper, Dirac applied his theory and compared his results with the successful predictions Einstein had made a decade before, in 1916. Einstein had used old quantum ideas to calculate the rate at which atoms can emit and absorb light, producing formulae that appeared to describe these processes successfully. The question Dirac had to answer was: does the new theory compare favourably with Einstein's?\n\nEinstein's theory had accounted for the interaction of light and matter in terms of three fundamental processes. Two of them were familiar enough: the emission and absorption of a photon by an atom. But Einstein also predicted a previously unknown way of 'persuading' an atom to jump from one energy level to a lower one, by stimulating it with another photon whose energy is exactly equal to the difference between the two energy levels. The result of this process of 'stimulated emission' is that two photons emerge from the atom: the original one and another one given out when the atom jumps to the lower energy level. This process takes place in the ubiquitous laser - there is at least one in every CD and DVD player and in every bar-code reader - and so is the most common technological application of Einstein's science. Dirac's theory produced exactly the same formulae as Einstein's and had the other advantages that it was more general and mathematically more coherent. As he probably realised, he had gone one better than Einstein.\n\nAt the end of January, as he was preparing to leave Copenhagen, Dirac posted his paper to the Royal Society. It turned out that he was the first to introduce the mathematics of creation and annihilation into quantum theory, though his results had been reached independently by John Slater, studying in Cambridge with Fowler. Slater was one of the many who admired Dirac's paper for its content but found its presentation perversely complicated: 'his paper was a typical example of what I very much distrusted, namely one in which a great deal of seemingly unnecessary mathematical formalism is introduced'.\n\nDirac's time in Copenhagen was an unqualified success. The two theories that he had nurtured there had underlined his status as a leading player on the international stage of science. Although he was still the archetypal individualist, he had come to see the value of taking different approaches to his subject and of having his views cross-examined. Apart from Bohr, the interrogator who most fascinated him was Paul Ehrenfest, an intense and disturbed theoretician based at the University of Leiden in the Netherlands. Ehrenfest got on well with Dirac, who was almost half his age, the two no doubt especially comfortable in each other's company because - unusually among the Institute's members - they disliked both alcohol and smoking. Ehrenfest's aversion to smoking was in part due to his extremely sensitive sense of smell. One victim of this was the amiable Dutch graduate student Hendrik Casimir. Soon after he arrived in Leiden, Casimir had his hair cut before a meeting with Ehrenfest, who soon sniffed the perfume of the barber's dressing. Ehrenfest quickly became angry and shouted, 'I will not tolerate perfume here. Get out. Go home, get out. Get out. Get out.' A few days later, Casimir was dismissed.\n\nEhrenfest was at his best during seminars. Unafraid of ridicule, he would politely but persistently interrupt speakers, seeking clarification of every unclear point. When he first met Dirac, Ehrenfest was uncomfortable with quantum mechanics and was worried that his close friend Einstein was unhappy about the central role played in the theory by probability. Einstein had been the first to identify that when an atom spontaneously jumps to a lower energy level, quantum theory cannot predict either the direction of the emergent photon or the precise time of its ejection. This was also true of ordinary quantum mechanics and of Dirac's new quantum field theory. Einstein was sure that a satisfactory theory had to do better than just predict probabilities: 'God is not playing dice,' he wrote to Max Born. Dirac thought his hero worried too much about the philosophical  issues of quantum mechanics. All that mattered to Dirac - true to his mathematical and engineering training - was that the theory was logical and accurately accounted for the results of experiments.\n\nAt the end of January 1927, Dirac was preparing to travel to G\u00f6ttingen. Soon he would be leaving the company of Niels Bohr, whom Dirac would later describe as 'the Newton of the atom' and 'the deepest thinker that I ever met'. But it was Bohr's warmth and humanity that most impressed Dirac. At Christmas - while Charles, Flo and Betty Dirac were going through the family rituals - Dirac had been welcomed into the Bohrs' loving fold and witnessed familial joy for the first time. Dirac had seen that it was possible to be both a great physicist and a dedicated family man and that perhaps - just perhaps - there might be more to life than science.\n\nFor Bohr, Dirac was 'probably the most remarkable scientific mind which has appeared for a very long time' and 'a complete logical genius'. Also intrigued by Dirac's personality, Bohr never forgot one incident, during a visit to an art gallery in Copenhagen, of his young visitor's eccentricity. When they were looking at a French impressionist painting showing a boat sketched by just a few lines, Dirac observed, 'This boat looks as if it was not finished.' Of another picture, Dirac remarked, 'I like that because the degree of inaccuracy is the same all over.' Such anecdotes became part of scientific lore, and physicists vied with one another to relate the most amusing instances of Dirac's verbal economy, his literal-mindedness, mathematical precision and otherworldliness. With no psychological framework available to help understand him, his personality became an object of collective amusement, through a myriad of 'Dirac stories'.\n\nNo one relished telling the stories more than Bohr, who entertained visitors with them over afternoon tea in his office. Four years before he died, he told a colleague that, of all the people who had visited his institute, Dirac was 'the strangest man'.\n**Nine**\n\n[For young Germans after the great inflation they experienced in 1923] their aims were to live from day to day; and to enjoy to the utmost everything that was free: sun, water, friendship, their bodies.\n\nSTEPHEN SPENDER, _World Within World_ , 1951\n\nIn G\u00f6ttingen, Dirac made another of his unlikely friendships. This one was with Robert Oppenheimer, who had fled Cambridge and was flourishing in Max Born's Department of Theoretical Physics as a Ph.D. student of rare ability, self-confidence and superciliousness. Ever the intellectual peacock, Oppenheimer ensured that his colleagues knew he was thinking about more than physics: his eclectic reading list included F. Scott Fitzgerald's collection of short stories _Winter Dreams_ , Chekhov's play _Ivanov_ and the works of the German lyric poet Johann H\u00f6lderlin. He was also composing verse, a hobby that puzzled Dirac. 'I don't see how you can work on physics and write poetry at the same time,' he remarked during one of their walks. 'In science, you want to say something nobody knew before, in words everyone can understand. In poetry, you are bound to say something that everybody knows already in words that nobody can understand.' For decades to come, Oppenheimer liked to recount this anecdote over cocktails, no doubt having polished Dirac's original phrasing to give it the bite of one of Wilde's paradoxes.\n\nDirac kept normal working hours, while Oppenheimer was nocturnal, so the two young men could not have seen much of each other. They boarded with the Cario family in a spacious granite villa on Giesmarlandstrasse, which led from the town centre out to the local countryside. From the outside, the home appeared to be just another of the town's many lavish residences, but there was a bitterness and penury inside. During the unstable early years of the Weimar Republic, the Carios had been victims of the precipitate fall of the German currency: the number of deutschmarks that could be purchased with an American dollar rose from 64.8 in January 1920 to 4.2 trillion in November 1923. Worse, the family's breadwinner, a doctor, had been disqualified for malpractice. Now that the  Republic had stabilised, the Carios made a living by turning their home into a guesthouse for the stream of foreign visitors, many of them American students visiting the Georgia Augusta University, one of the most prestigious academic addresses in Europe. With his fellow boarders, Dirac sat down every evening to a meal based on the local fare of potatoes, smoked meats, sausages, cabbages and apples.\n\nIt took Dirac and Oppenheimer only five minutes to stroll from their lodgings to Born's department in the Second Physics Institute, located in an ugly red-brick building with all the charm of a Prussian cavalry barracks. Born - a handsome, clean-shaven man, who looked younger than his forty-four years - was reserved but warmer than most of his professorial colleagues. He cultivated a competitive environment but was sensitive to the needs of the brightest students and tolerant of their peccadilloes. Dirac and Oppenheimer were among the many students Born invited to his villa on the Planckstrasse, a quiet road on the outskirts of the town. To be invited there was always a pleasure: dinner would be followed by good-humoured conversation and a concert in the huge front room, which contained two grand pianos. Heisenberg, a close friend of the family, took every opportunity to display his pianistic skills in flamboyant renditions of Beethoven, Mozart and Haydn.\n\nDirac lived just a few steps away from the historic centre of G\u00f6ttingen, one of the best-preserved medieval towns in Lower Saxony: its half-timbered houses and shops, its churches and cobbled backstreets had remained virtually unchanged for centuries. Nor was it yet overrun by the motor car. Most people got around on foot or by bicycle, many of the cyclists sporting garishly coloured caps to show their affiliation to one of the clubs and societies. Like Cambridge, G\u00f6ttingen was a tranquil academic town, dominated by the needs and whims of its academics and students. Seniority and intellectual distinction were at a premium there. Its most revered citizens were the most venerable of its distinguished professors, including the gruff David Hilbert, sixty-three years old and the most celebrated mathematician alive.\n\nAlso like Cambridge, many of G\u00f6ttingen's (mainly male) students were there not so much to be well educated as to spend a few hedonistic years in the fug and cacophony of the town's taverns and coffee bars. No doubt having left Dirac to get his sleep, Oppenheimer and his friends spent many a night on the razzle; he happily picked  up the tab after downing a few pints of _frisches Bier_ in the Black Bear pub or dining on _Wienerschnitzel_ at the four-hundred-year-old Junker Hall. The atmosphere in the pub had hardly changed in generations: most evenings, the din of the students would often dissolve into bibulous choruses of favourite folk songs, while virile young men sloped off to put on their chain mail, don their swords and do some 'academic fencing'. When the combatants returned, their faces were 'decorated' with scars, each a bloody badge of honour.\n\nAt weekends, Oppenheimer and other affluent students often took the two-and-a-half-hour train journey to Berlin, the city of Bertolt Brecht, Arnold Sch\u00f6nberg and Kurt Weill. But Dirac had no interest in broadening his horizons much beyond the towns and villages of Lower Saxony, where he went on long Sunday walks, if he was not snowbound. Within twenty minutes of leaving his lodgings, he was walking in the gently rolling countryside, following the fast-flowing rivers and pausing at the scattered monuments to Bismarck. By early spring, the walking conditions were perfect: almost all the winter snow had melted, and the linden trees, shrubs and flowers were scenting the air. He passed occasional groups of young men in the German Youth Movement but otherwise saw scarcely another person, which was just as he preferred - his empathies lay more with uncommunicative forms of nature than with human beings.\n\nSo G\u00f6ttingen gave Dirac everything he wanted in a town - a great university with a world-leading physics department and comfortable lodgings close to walking country, where he could escape from other people. G\u00f6ttingen was a German Cambridge, with hills.\n\nIn early February 1927, within days of Dirac's arrival in G\u00f6ttingen, he had set Oppenheimer's imagination alight. Oppenheimer was completing his Ph.D., on the quantum mechanics of molecules, and looking to the future which appeared to lay in the direction that Dirac had opened up. Near the end of Oppenheimer's life, when he looked back on his career, he remarked that 'perhaps the most exciting time of my life was when Dirac arrived [in G\u00f6ttingen] and gave me the proofs of his paper on the quantum theory of radiation'. While others found Dirac's field theory mystifying, to Oppenheimer it was 'extraordinarily beautiful'.\n\nOppenheimer had been an outsider at Cambridge and Harvard and so he was pleased at last to feel part of the small community of  G\u00f6ttingen physicists, gradually recovering from his clinical depression. Among his colleagues was Pascual Jordan, born a few weeks after Dirac and the youngest of the quantum innovators. Intense, haunted and private, his eyes stared out from behind elliptical glasses with lenses as thick as jam jars. Oppenheimer later remarked that Jordan's peculiarities may have led him to be underestimated: 'it was in part because he was really an unbelievably queer duck with tics and mannerisms and [. . .] apparent brutalities, which put people off very much.' According to Oppenheimer, Jordan had a stutter so crippling that 'it was difficult to get through', though Oppenheimer may have to some extent admired it - he began to affect a stutter, muttering 'njum-njum-njum' before some of his finely crafted declamations.\n\nAlthough Jordan and his colleagues admired Oppenheimer's quick-fire intelligence - one of them likened him to 'an inhabitant of Olympus who had strayed among humans' - they found his arrogance irritating, to the point that it became unacceptable. One morning, Born found on his desk a letter from several of his colleagues threatening to boycott seminars unless he stopped Oppenheimer from disrupting them with his continual interruptions. Always fearful of showdowns, Born chose to leave the letter - a large sheet of parchment lettered in ornamental script - on his desk for Oppenheimer to see. It did the trick. Relations between Born and Oppenheimer were superficially cordial, but Oppenheimer regarded Born as a 'terrible egotist' who continually complained that he had not been given enough credit for pioneering quantum mechanics. Born had good reason to feel slighted. He had been one of the creators of quantum mechanics, having used his battery of mathematical skills to develop Heisenberg's initial idea. Most physicists gave the lion's share of the credit to Heisenberg, but Born believed that it was he who first fully appreciated the idea's potential and he who led its development in G\u00f6ttingen.\n\nBy the time Dirac arrived there, Born was confident that he had found the right way to develop quantum mechanics, using Heisenberg's ideas, not Schr\u00f6dinger's. Although Born knew of Dirac's reputation, he was not expecting his young visitor to be so adept and knowledgeable. The American physicist Raymond Birge, then visiting G\u00f6ttingen, observed that 'Dirac is the real master of the situation [. . .] when he talks, Born just sits and listens to him open-mouthed.'\n\nAnother colleague, the German theoretician Walter Elsasser, later wrote his impressions of Dirac: 'tall, gaunt, awkward and extremely taciturn. [. . .] of towering magnitude in one field, but with little interest and competence left for other human activities'. Elsasser remembered that although Dirac was always polite, his conversations were almost always stilted: 'one was never sure that he would say something intelligible.' Another of Dirac's traits was his inability to comprehend anyone else's point of view if it didn't fit into his way of looking at things: colleagues would spend hours presenting their perspective on a physics problem, only for him to walk away after making a brief comment, apparently apathetic or bored. Oppenheimer was quite different: he would listen to a colleague's ramblings for a few minutes but would then interject with an eloquent summary of what he was probably trying to say.\n\nWhereas Oppenheimer mixed freely with his colleagues, Dirac spent most of his time working in the library or in one of the empty classrooms. But he was not a complete loner: in Copenhagen, he had come to appreciate being with other physicists, provided they didn't put pressure on him to speak. Most mornings, he walked with fellow boarders at the Carios' to the Mathematics Institute, where he attended lectures that kept them abreast of the latest experimental findings. He also took the time to go to the often-combative afternoon seminars. When Ehrenfest was in town, he was their undisputed inquisitor-in-chief, deflating egos and revealing the crux of every new argument, having cut away the underbrush. In the previous June, he had brought along a Ceylonese parrot trained to say 'But, gentlemen, that's not physics' and recommended that it should chair all forthcoming seminars on quantum mechanics.\n\nMax Delbr\u00fcck, one of the young G\u00f6ttingen physicists, was probably not exaggerating when he later described the experience of walking into one of their seminars: 'you could well imagine that you were in a madhouse.'\n\nWord spread to Berlin that Dirac was a difficult man and that his work was impenetrable and overrated. The Hungarian theoretician Jen\u020d (later Eugene) Wigner later said that, in the mid-1920s, his German colleagues were suspicious of 'the queer young Englishman who resolves [questions of physics] in his own language'. Many Germans were put off by Dirac's manner. The English were known  for their reserve - they acted as if everyone else was either an enemy or a bore, as John Stuart Mill had pointed out - but Dirac's frigidity was unlike anything they had ever seen.\n\nBorn was one of the few Germans who warmed to Dirac, but even he had trouble understanding his new field theory and apparently thought it unimportant. His lack of foresight frustrated Jordan, who had begun to develop ideas on field theory very similar to Dirac's, only to be met with indifference. It would have been fascinating to see what Dirac and Jordan could have achieved in quantum field theory, but Dirac had no interest in collaboration. He turned his attention to using field theory to understand what happens when light is scattered by an atom, normally visualised as being rather like a basketball bouncing off the hard rim of the basket. But, in the new field theory, things are not so straightforward. Dirac showed that, in the fleeting moment of a photon's scattering, it appears to pass through some strange, unobserved energy states. What makes these intermediate processes so odd is that they appear to flout the sacred law of conservation of energy. Although these subatomic 'virtual states' cannot be seen directly, experimenters were later able to detect their subtle influences on fundamental particles.\n\nDirac's calculations also threw up a more troubling artefact. He found that his new theory kept generating bizarre predictions: for example, when he calculated the probability that a photon had been emitted after a given interval, the answer was not an ordinary number but was infinitely large. This made no sense. The probability that an atom would emit a photon must surely be a number between zero (no chance) and one (complete certainty), so it seemed obvious that the prediction of infinity was wrong. But Dirac chose to be pragmatic. 'This difficulty is not due to any fundamental mistake in the theory,' he wrote with more confidence than was warranted. The root of the problem, he speculated, was a simplistic assumption he had made in applying the theory; when he had identified his error and tweaked the theory, he implied, the problem would disappear. In the meantime, he dodged the difficulties using clever mathematical tricks, enabling him to use the theory to make sensible, finite predictions. But it would not be long before he saw that his optimism was misplaced: the lamb had caught its first sight of the wolf's tail.\n\nMeanwhile, the debates about the interpretation of quantum theory had not abated, least of all in Copenhagen, where Heisenberg was struggling to understand the theoretical limits of what can be known about a quantum. He achieved this brilliantly with his uncertainty principle, which made him into the nearest the quantum fraternity had to a household name.\n\nThe principle emerged only after anguished and protracted gestation, which apparently began with a letter from Pauli during the previous October. Heisenberg believed that the correct way to think about the quantum world was in terms of particles, and that the more popular wave-based ideas were merely useful supplementaries. Somehow, Heisenberg wanted to find a way of making definite statements about the measurements that could be made on quantum particles, especially about the limitations on what experimenters can know about them. Heisenberg had talked with Einstein about this, and, when Dirac was in Copenhagen developing transformation theory, he had also discussed it with him.\n\nThe nub of what became known as Heisenberg's uncertainty principle is that the knowledge experimenters have of a quantum's position limits what they can know about its speed, at the same instant. The more they know about a quantum's position, the less they can know about its speed. So, for example, if experimenters know an electron's location with perfect precision, then it follows that they can know nothing whatsoever about its speed at the same moment; on the other hand, if they know the exact value of the electron's speed, they will be totally ignorant of its position. There is, Heisenberg argued, no way round this: regardless of the accuracy of the measuring apparatus or the extent of the experimenters' ingenuity, the principle puts fundamental limitations on knowledge. It turns out that even the most accurate knowledge imaginable of the location of an ordinary object puts only negligible constraints on knowledge of its speed (likewise with the location and speed reversed), so the principle is unimportant in everyday life. This is the root of the physicists' joke about the motorist who tries to con the traffic police by pleading not guilty of speeding on the grounds 'I knew exactly where I was, so I had no idea how fast I was travelling': the plea would be perfectly admissible if it were made by a sentient electron.\n\nIn his paper, Heisenberg explained his principle by picturing what  happens when an experimenter uses a photon of light to probe the behaviour of an electron, demonstrating that the very act of probing disturbs the electron. An analysis of this thought experiment led Heisenberg to a mathematical expression that encapsulated the principle. He also derived the expression mathematically, using two of Dirac's innovations: transformation theory and the relationship between the non-commuting position and momentum.\n\nAs spring set in, Dirac will probably have thought about the principle during his constitutional walks along the tree-lined path following the contours of what was once G\u00f6ttingen's outer wall. He was not especially impressed with Heisenberg's discovery, as he noted later: 'People often take [the uncertainty principle] to be the cornerstone of quantum mechanics. But it is not really so, because it is not a precise equation, but only a statement about indeterminacies. ' Dirac was similarly lukewarm a few months later when Bohr announced his principle of complementarity, apparently related to Heisenberg's principle. According to Bohr's idea, quantum physicists have to accept that a complete picture of subatomic events always involves descriptions that appear incompatible but that are actually complementary - both the wave and particle pictures are needed. In Bohr's view, this idea was part of an ancient philosophical tradition, in which truth cannot be pinned down using only one approach but needs complementary concepts: for example, a mixture of reason and feeling, analysis and intuition, innovation and tradition.\n\nThis principle was fundamental to Bohr's thinking, to the extent that he chose it in 1947 as the basis of the design of his coat of arms. The design features the Chinese yin-yang symbol, which represents the two opposing but inseparable elements of nature, and the Latin motto below reads 'Opposites are complementary'. Many physicists thought that Bohr had uncovered a great truth, but Dirac was again unimpressed: the principle 'always seemed to me a bit vague', he later said. 'It wasn't something which you could formulate by an equation.'\n\nDirac's opinion of Heisenberg's uncertainty principle was not shared by most scientists, including Eddington. In his acclaimed book _The Nature of the Physical World_ , published in November 1928, he gave a sparkling account of 'the principle of indeterminacy', describing it  as 'a fundamental general principle that seems to rank in importance with the principle of relativity'. Writing with his usual panache, Eddington introduced tens of thousands of lay readers to the new principle as one of the cornerstones of quantum mechanics.\n\nEddington writes that he is giving an outline of the theory only against his better judgement: 'It would probably be wiser to nail up over the door of the new quantum theory a notice \"Structural alterations in progress - No admittance except on business\", and particularly to warn the doorkeeper to keep out prying philosophers.' Eddington's account of the theory was the clearest account of quantum mechanics for English-speaking lay readers and was the first widespread publicity for the new theory. If Bohr or another influential figure had taken a leaf out of Eddington's book and been savvy enough to provide a dramatic presentation of the uncertainty principle's discovery to well-briefed journalists, then quantum mechanics may well have become much better known, along with its creators.\n\nWith a hint of nostalgia, Eddington pointed out that modern physicists no longer thought about the universe as a giant mechanism, as Victorian physicists such as James Clerk Maxwell had done, but framed their accounts of the fundamental nature of things in the language of mathematics. The images of cogs and gearwheels were now pass\u00e9, but Eddington believed there were dangers inherent in the new, mathematical way of thinking of fundamental physics:\n\nDoubtless the mathematician is a loftier being than the engineer, but perhaps even he ought not to be entrusted with the Creation unreservedly. We are dealing in physics with a symbolic world, and we can scarcely avoid employing the mathematician who is a professional wielder of symbols; but he must rise to the full opportunities of the responsible task entrusted to him and not indulge too freely his own bias for symbols with arithmetical interpretations.\n\nEddington had put his finger on the central conceptual challenge that made quantum mechanics so difficult for most professional physicists. The great majority of them still thought like engineers and were mathematically weak by the standards of Dirac and his peers. So, most physicists were still trying to visualise the atom as if it were a mechanical device.\n\nThe metaphor of nature as a colossal clockwork mechanism, popular since Newton's day, had long been apt for most purposes. But no longer. Quantum mechanics was based fundamentally on mathematical abstractions and could not be visualised using concrete images - that is why Dirac refused to discuss quantum mechanics in everyday terms, except in later life, when he began to use analogies between the behaviour of quanta and the way ordinary matter behaves. Yet Dirac often remarked that he did not think about nature in terms of algebra, but by using visual images. Since he was a boy, he had been encouraged to develop visual imagination in his art and technical-drawing classes, which were an ideal grounding for his studies of projective geometry. None of the other pioneers of quantum mechanics had been given an education in which geometric visualisation played such a prominent part. Five decades later, when he looked back on his early work in quantum mechanics, Dirac declared that he had used the ideas of projective geometry, unfamiliar to most of his physicist colleagues:\n\n[Projective geometry] was most useful for research, but I did not mention it in my published work [. . .] because I felt that most physicists were not familiar with it. When I had obtained a particular result, I translated it into an analytic form and put down the argument in terms of equations.\n\nDirac had a perfect opportunity to explain the influence of projective geometry on his early thinking about quantum mechanics at a talk he gave in the autumn of 1972 at Boston University. Its philosophy department had invited him to give the talk to clarify this influence and had recruited the urbane Roger Penrose, an eminent mathematician and scientist who knew Dirac well, to chair the seminar. If anyone could prise the story out of Dirac, it was he. In the event, Dirac gave a short, clear presentation on basic projective geometry but stopped short of connecting it to quantum behaviour. After Dirac had batted away a few simple questions, the disappointed Penrose gently turned to him and asked him point-blank how this geometry had influenced his early quantum work. Dirac firmly shook his head and declined to speak. Realising that it was pointless to continue, Penrose filled in the time by extemporising a short talk on a different subject. For those who wanted to demystify Dirac's magic, his silence had never been so exasperating.\n**Ten**\n\nHitler is our F\u00fchrer, he doesn't take the golden fee  \nThat rolls before his feet from the Jew's throne  \nThe day of revenge is coming, one day we will be free [. . .]\n\nFrom an early Nazi marching song, c. 1927\n\nAs a Jew, Max Born had every reason to be alarmed and frightened by the rise of anti-Semitism in G\u00f6ttingen. The atmosphere was 'bitter, sullen [. . .] discontent[ed] and angry and loaded with all those ingredients which were later to produce a major disaster', Oppenheimer remembered, a few years before he died. The Nazis had set up one of their first branches in the town in May 1922. Three years later, the chemistry student Achim Gercke secretly began to compile a list of Jewish-born professors, to provide 'a weapon in hand that should enable the German Reich to exclude the last Hebrew and all mixed race from the German population in the future and expel them from the country'.\n\nLife among the G\u00f6ttingen researchers did have its lighter side, however. Many of them gloated that their profession was for the young, and they mocked the sclerotic imaginations of their elderly professors, paid and revered much more for doing much less. As his later comments confirm, Dirac shared this dismissiveness, and, if an improbable G\u00f6ttingen legend is to be believed, he wrote a quatrain about this for a student review:\n\nAge is of course a fever chill   \nThat every physicist must fear   \nHe's better dead than living still   \nWhen he's past his thirtieth year\n\nG\u00f6ttingen students had a penchant for silly songs and for choral renditions of American tunes, which were sung with special enthusiasm at Thanksgiving. The cosmologist Howard Robertson, who introduced Dirac to ways of describing the curvature of space-time across the universe, had brought to the taverns of G\u00f6ttingen one of their most popular new songs, 'Oh My Darling Clementine'. Dirac  probably did not join in, but he took part in the infantile games that helped to sublimate the physicists' intense competitiveness. One of the games was 'bobbing for apples', when professors and students - often woozy, after a few glasses of beer - would try to sink their teeth into an apple floating on water or beer. Another activity involved running a race while trying to balance a large potato on a tiny spoon. After one of these races in Born's home, a student saw Dirac practising surreptitiously - a sight that would have stunned his colleagues in Cambridge, including the theologian John Boys Smith, who described Dirac as being 'childlike but never childish'.\n\nDirac's stay in G\u00f6ttingen ended in early June 1927. St John's wanted him back and had been wooing him to apply for a fellowship, an honour well worth pursuing. If successful, he would benefit from free board and lodging in college, as well as a modest income to supplement the continuing funds from his 1851 scholarship, which would run out in 1928. A tenured academic post in the university's mathematics department would almost certainly follow, and he would be set up for the rest of his working life. In his letters, Dirac was even less forthcoming about his personal life than he had been when he wrote from Copenhagen. In a letter to the college official James Wordie, Dirac wrote just a single sentence about his activities in G\u00f6ttingen: 'The surrounding country is very beautiful.' Although he preferred Bohr's pullulating institute to Born's comparatively cool department, he told his mother that he preferred G\u00f6ttingen, as it gave him the best opportunities for solitary walks.\n\nIn his research, Dirac appeared to be showing signs of running out of steam. In early May 1927, he used quantum mechanics to predict what happens when light is scattered by an atom - a problem that led to no exciting conclusions. Oppenheimer later said that he was disappointed by Dirac's work in G\u00f6ttingen and could not understand why he did not press on with the development of quantum field theory. Dirac wanted to take a long rest over the summer, he told Oppenheimer, and would then turn his attention to the spin of the electron, still not understood.\n\nDirac intended to begin his break from quantum theory when he returned to England, after he had visited Ehrenfest in Leiden, a small university town in the Netherlands. Dirac stayed in the room at the top of Ehrenfest's large Russian-style house, where he signed his name on the bedroom wall that already bore the signatures of Einstein,  Blackett, Kapitza and dozens of others. The house served as a local hostel for the cream of the world's physicists, who traded anecdotes of their lively conversations with Ehrenfest's wife - a Russian mathematician - and their three children, two daughters and a son who had Down's syndrome.\n\nOppenheimer was planning to join Dirac in Leiden and began to learn Dutch so that he could give a seminar in the language of his host. But first he had to defend his Ph.D. thesis in an oral examination held by James Franck, the distinguished experimenter, and Max Born. Franck took only twenty minutes to question Oppenheimer, but that was enough. On leaving the exam room, Franck sighed, 'I'm glad that is over. He was on the point of questioning _me._ ' Born was relieved that his brilliant but troublesome student was off his hands. At the end of a typewritten letter to Ehrenfest, Born wrote a postscript:\n\nI should like you to know what I think of [Oppenheimer]. Your judgement will not be influenced by the fact I openly admit that I have never suffered as much with anybody as him. He is doubtless very gifted but without mental discipline. He's outwardly modest but inwardly very arrogant. [. . .] he has paralyzed all of us for three quarters of a year. I can breathe again since he's gone and start to find the courage to work.\n\nDirac had not been part of this departmental paralysis, nor does he appear to have been aware of it. Oppenheimer was awed by him and showed him a diffidence he granted to few of his other colleagues. Their days in G\u00f6ttingen were the beginning of a forty-year friendship.\n\nG\u00f6ttingen was too far away for Dirac's family to visit. 'Thank goodness, you are saying, I expect,' his mother wrote in a pained aside. She made it clear to her son how much she envied him: 'You are a lucky fellow to be away from home. [Here,] it is all work, work.' When her husband was out, she wore her new ring - seven diamonds set in platinum - which she had furtively bought with \u00a310 of the money Dirac had sent her, considerably more than Charles allowed her to spend on herself in a year. That piece of jewellery was a private symbol of her most important relationship. She wrote to her son: 'Don't tell Pa [. . .] I expect he would tell me to put the money in the housekeeping, but it is giving me such a lot of pleasure to look at it  and think what a darling you are.' In the evenings, she would sit in the front room with photos of her son, re-reading his postcards, trying to imagine what he would be doing at every time of day.\n\nThe twelve-year age difference between Charles and Flo had never been more plain. She still had an upright posture, smooth skin and scarcely a grey hair; he was hunch-backed, white-haired and wizened. In public, she put on the traditional show as the loyal, uncomplaining wife; in private, she was resentful of being an unpaid servant, as she often wrote to her son. At the beginning of 1927, she was surprised when her husband went on a spending spree, probably funded by his mother's legacy. Dirac often condemned the tattiness of the family home, which had not been decorated for thirteen years, so it may well have been that Charles paid for the extensive wallpapering and the installation of a gas fire in every room, with the aim of making 6 Julius Road more attractive to his son. Charles did not entirely neglect his wife - he bought her one of the new vacuum cleaners to help with the housework: 'Pa likes to see them at work on our carpets giving free demonstrations.'\n\nStill in poor health, Charles consulted a herbalist who advised him to become vegetarian, presenting endless catering problems for his wife, who worried incessantly about his nutrition. She wrote to Dirac: 'Pa is getting ever so many pupils he has scarcely time for meals. I am sure he is working his brain too hard and now he is a vegetarian, there are so many little things to cook which are not substantial enough for him.' Although she thought he was mean and ungrateful, she devoted herself to taking care of him, and her letters to Dirac betrayed no sign that the state of affairs was anything less than she should expect or deserve. But her patience was beginning to run out.\n\nCharles Dirac's work ethic had been the making of one of his sons and possibly the death of the other, but it did not have much influence on his daughter. Betty had left school and was, according to her mother, 'too shy or perhaps too lazy [. . .] to want to do anything to earn her own living & she is not fond of housework either'. Without a job, she lolled around the house mourning the death of her dog and went out with her mother to evening classes in elocution and French. In early July, the family chased out the decorators and made sure everything in their house was spick and span, ready for the return of the itinerant son. The family had not spoken to him for nine  months, but in that time had sent him weekly family bulletins, showering him with affection and pleas for news at his end. In return, he had sent his parents fewer than seven hundred words. He had not once asked after his family on his postcards, which each had the warmth of a stone.\n\nWhen Dirac arrived at the door of 6 Julius Road at lunchtime on 13 July - a dull and overcast afternoon - it is easy to imagine the tearful flutterings of his mother and sister as they hugged his unresponsive frame and the stiff handshake with his father, who was probably no less pleased to see him, even if he was unable to show it. He was soon back in his routine, shutting out his family, working alone in his room. One of Charles's students, D. C. Willis, left an anecdote that offers an insight into the domestic environment at the Diracs' that summer. Willis was sent by Monsieur Dirac 'on his errands to his home during the dinner hour [. . .] as he was concerned about his son Paul who, rumour had it, was working in his bedroom, and would not come out, except to collect his food and use the toilet'.\n\nDirac knew he had a filial duty to be with his parents but felt wretched whenever he was with them. 'When I go back to my home in Bristol I lose all initiative,' he sighed in a letter to a friend, a few years later. He felt oppressed by both his parents - by his father's high-handedness and by his mother's suffocating affection. Although Dirac was twenty-five years old and internationally successful, he still felt himself to be writhing under his father's thumb. And he saw no imminent prospect of escape.\n\nIn October 1927, Dirac returned to Cambridge to reacquaint himself with his friends in St John's and Trinity. He now had even fewer social distractions, as Kapitza had recently married. His new wife was the \u00e9migr\u00e9 Russian artist Anna Krylova, a dark-haired beauty whom Kapitza unaccountably called 'Rat', a nickname that nonplussed audiences in Cambridge theatres for years, whenever they heard him holler it across the stalls. She and Kapitza contributed to the design of the detached house that was being built for them on Huntingdon Road, near the city centre, complete with a huge back garden and a studio for her in the loft. Later, this house would become almost Dirac's second home in Cambridge but, in the early autumn of 1927, he was working hard on his project, first mooted to Oppenheimer, aiming to combine quantum theory and Einstein's  special theory of relativity in the simplest practical case: to describe the behaviour of a single, isolated electron. The quantum theories of Heisenberg and Schr\u00f6dinger were deficient because they did not conform to the special theory of relativity: observers moving at different speeds relative to one another would disagree on the theories' equations. At stake here was the prestige of being the first to find the theory; would he be the sole winner of a scientific prize or would he, yet again, have to share it?\n\nDirac worked on the problem for the first six weeks of the term but without success. He took a break in late October to sit, for the first time, at the top table of international physicists at the Solvay Conference in Brussels. The aim of these invitation-only conferences, funded by the Belgian industrialist Ernest Solvay, was to bring together about twenty of the world's finest physicists every few years to ponder the problems of quantum theory. The youngest star of the first conference in 1911 had been Albert Einstein, then emerging from obscurity and quick to point out the prejudices of older, more conservative minds. In 1927, Einstein was the uncrowned king of physics and entering middle age, still a popular and unassuming figure but showing signs of crustiness and disillusion. He was ploughing his own furrow, seeking a unified theory of gravity and electromagnetism without assuming that quantum mechanics was correct. Now it was Einstein who seemed inflexible and backward-looking.\n\nThe conference was to become a landmark in physics - the place where Einstein first publicly articulated his unease with quantum mechanics but failed to dent the confidence of Bohr and his younger colleagues. There is no sign of the lively conference atmosphere in the famous photograph taken outside the building where the sessions took place: the twenty-nine conference delegates all look expressionless, as though they are posing for a communal passport photograph. Einstein sits at the centre of the front row, with Dirac standing behind his right shoulder. Dirac was so proud of this photograph that, for once succumbing to vanity, he prompted the University of Bristol's physics department to have it framed and mounted on one of their walls. This portrait, a dismal memento, was for decades the best visual evidence available of the meeting, but in 2005 more clues about the atmosphere of the meeting appeared, with the release of a home movie of the delegates during a break between the lectures. What is  most striking about this two-minute clip is the delegates' cheerfulness. Marie Curie, the only woman in the group, does a fetching pirouette; the beaming Paul Ehrenfest waggishly pokes out his tongue at the camera. Dirac, the youngest delegate, looks relaxed and happy as he talks with Max Born.\n\nHeisenberg later remembered that the most intense discussions took place not during the conference sessions but over meals at the delegates' nearby Hotel Britannique, near the site of today's European Parliament. At the epicentre of the debates about quantum theory were Bohr and Einstein's disagreements about Heisenberg's uncertainty principle, which Bohr defended successfully against Einstein's repeated onslaughts. Most of their colleagues were fascinated to hear the two men lock horns, but Dirac was an indifferent bystander:\n\nI listened to their arguments, but I did not join in them, essentially because I was not very much interested [. . .] It seemed to me that the foundation of the work of a mathematical physicist is to get the correct equations, that the interpretation of those equations was only of secondary importance.\n\nDirac and Einstein were poles apart, and neither was comfortable speaking the other's language. Dirac was twenty-three years younger, and his awe rendered him even more shy than usual. But probably the main reason why they did not engage was that their approaches to science contrasted so sharply, partly because they responded so differently to philosophical matters. They agreed that science was fundamentally about explaining more and more phenomena in terms of fewer and fewer theories, a view they had read in Mill's _A System of Logic._ Yet, whereas Einstein remained interested in philosophy, for Dirac it was a waste of time. What Dirac had retained from his reading of Mill, bolstered by his studies of engineering, was a utilitarian approach to science: the salient question to ask about a theory is not 'Does it appeal to my beliefs about how the world behaves?' but 'Does it work?'\n\nAt the conference, Dirac made his first recorded outburst on topics outside physics - religion and politics. Some four decades later, Heisenberg described the event, which took place one evening in the hotel's smoky lounge, where some of the younger physicists were lying around on the chairs and sofas. Dirac's youthful outspokenness needed to be indulged, the elderly Heisenberg said: 'Dirac was a very  young man and in some way was interested in Communistic ideas, which of course was perfectly all right at that time.' Most vivid in Heisenberg's memory was a rant from Dirac about religion, triggered by a comment about Einstein's habit of referring to God during discussions about fundamental physics. Like many of Heisenberg's accounts of incidents in the 1920s, this one is implausibly detailed - it consists of two speeches of several hundred words, quoted as if his memory were word perfect - but it is consistent with other accounts of Dirac's views. According to Heisenberg, Dirac thought religion was just 'a jumble of false assertions, with no basis in reality. The very idea of God is a product of the human imagination.' For Dirac, 'the postulate of an Almighty God' is unhelpful and unnecessary, taught only 'because some of us want to keep the lower classes quiet'. Heisenberg wrote that he objected to Dirac's judgement of religion because 'most things in this world can be abused - even the Communist ideology which you recently propounded'. Dirac was not to be deflected. He disliked 'religious myths on principle' and believed that the way to decide what was right was 'to deduce it by reason alone from the situation in which I find myself: I live in a society with others, to whom, on principle, I must grant the same rights I claim for myself. I must simply try to strike a fair balance.' Mill would have approved.\n\nDuring Dirac's assault on religion, Pauli had been uncharacteristically silent. When asked what he thought, he replied, 'Well our friend Dirac, too, has a religion, and its guiding principle is \"There is no God and Dirac is his prophet\".' It was an old joke, but everyone laughed, including Dirac. The opinions he expressed here, with uncharacteristic forwardness, were entirely in keeping with Kapitza's views and would not have drawn comment from any of the intellectuals who were flirting with Bolshevism. Although Dirac never put any of his political views on paper, it was clear from his actions in the coming decade where his sympathies lay.\n\nDuring the Solvay Conference, Dirac gave a talk on his new field theory of light. He annotated his draft script with rewordings and other changes in every paragraph - more than any other talk he gave in his entire life - indicating that he was on edge. Afterwards, he heard that his idea had been taken up and extended in a way he could have easily foreseen. Pascual Jordan, working with Eugene Wigner,  had produced a field theory of the electron to complement Dirac's theory of the photon. Although Jordan and Wigner's mathematics was similar to Dirac's, their theory did not appeal to Dirac, who could not see how their symbols corresponded to things going on in nature. Their work looked to him like an exercise in algebra, though later he realised he was wrong; his mistake stemmed from his approach to theoretical physics, which was 'essentially a geometrical one and not an algebraic one' - if he could not visualise a theory, he tended to ignore it.\n\nThat was not the only surprise Dirac received in the lecture hall. Shortly before the beginning of a lecture, Bohr asked Dirac what he was working on. He replied that he was trying to find a relativistic quantum theory of the electron. Bohr was baffled: 'But Klein has already solved this problem,' he said, referring to the Swedish theoretician Oskar Klein. The lecture began before Dirac could reply, so the question hung in the air, where it remained: Bohr and Dirac did not have the chance to talk further about it before the conference dispersed. Another three months would elapse before Bohr appreciated his error when he read Dirac's wondrous solution to the problem.\n**Eleven**\n\n[T]he true and the beautiful are akin. Truth is beheld by the intellect which is appeased by the most satisfying relations of the intelligible: beauty is beheld by the imagination which is appeased by the most satisfying relations of the sensible.\n\nJAMES JOYCE, _A Portrait of the Artist as a Young Man_ , 1915,  \nChapter 5\n\nDirac always felt out of place at fancy college dinners. Rich food, vintage wines, antiquated formalities, florid speeches, the fetid smoke of after-dinner cigars - all were anathema to him. So he was probably not looking forward to the evening of Wednesday, 9 November 1927, when he was to be one of the toasts of a dinner to celebrate the annual election of new Fellows to St John's College. He was now certifiably a 'first-rate man', with a permanent seat at the college's high table and the freedom to gather after dinner with his colleagues in their grand, candle-lit Combination Room, completed in 1602. In Hall, beneath the portrait of Lady Margaret Beaufort, Dirac celebrated his election to the fellowship in the traditional way, by consuming an eight-course meal that included oysters, a consomm\u00e9, cream of chicken soup, sole, veal escalope and spinach, pheasant with five vegetables and side salad, and three desserts. For him, the meal was not so much a celebration as a penance.\n\nAfter the dinner, Dirac walked to his rooms, close to the Bridge of Sighs, a Gothic stone structure that crosses the river Cam in a brief undulation, leaving just enough room underneath for the punters. He probably went straight to bed, as his aim was always to be fresh for the morning, when he did his best work. His study was devoid of decoration, with only a folding desk of the sort used by schoolchildren, a simple chair, a coal fire and 'a very ancient settee', as one visitor described it. He worked at his little desk like a schoolboy in an empty classroom, writing in pencil on scraps of paper, sometimes pausing to erase an error or to consult one of his books. Now that he was a Fellow, he had a manservant (a 'gyp') on hand during the day.\n\nIn these austere but comfortable surroundings, Dirac made his most famous contribution to science. St John's had created the best environment imaginable for him. He could work all day, taking breaks only to fulfil his modest lecturing duties, give the occasional seminar and visit the library.\n\nHe was now preoccupied with a single challenge: to find the relativistic equation that describes the electron. Dirac was pretty sure that the electron was 'a point particle' but, like other theoreticians, could not understand why it had not one but two states of spin. Several other physicists had suggested candidate equations - all of them contrived and ungainly - and Dirac was not satisfied with any of them, including the one by Klein that Bohr believed had solved the problem. Dirac was sure Klein's theory was wrong, as it predicted, absurdly, that the chance of detecting an electron in a tiny region of space-time is sometimes _less_ than zero.\n\nDirac knew that it was impossible to deduce the equation from first principles and that he would find it only through a happy guess. But what he could do was to narrow the options, by setting out the characteristics the equation _must_ have and the characteristics it _ought_ to have. Rather than tinker with existing equations, he took the top-down approach, trying to identify the most general principles of the theory he was seeking, before going on to express his ideas mathematically. The first requirement was that the equation conformed to Einstein's special theory of relativity, treating space and time on an equal footing. Second, the equation must be consistent with his beloved transformation theory. Finally, when the equation describes an electron moving slowly compared with the speed of light, its predictions must resemble extremely closely ones made by ordinary quantum mechanics, which had already proved its worth.\n\nThose were useful constraints, but there was still too much room for manoeuvre. If he stuck to them, Dirac could still have written down any number of equations for the electron, so he needed to use his intuition to narrow the possibilities. Believing that the relativistic equation would be fundamentally simple, he thought it most likely that the equation would feature the electron's energy and momentum just as themselves, not in complicated expressions such as the square root of energy or momentum squared. Another clue came from the way he and Pauli had independently found to describe the spin of the  electron, using matrices that each consisted of four numbers arranged in two rows and two columns. Might these matrices feature in the equation he was seeking?\n\nDirac tried out one equation after another, discarding each one as soon as it failed to conform to his theoretical principles or to experimental facts. It was not until late November or early December 1927 that he hit on a promising equation, consistent with both special relativity and quantum mechanics. The equation looked like nothing theorists had ever seen before, as it described the electron not using a Schr\u00f6dinger wave but using a new kind of wave with _four_ interconnected parts, all of them essential.\n\nAlthough the equation had an appealing elegance, that would count for nothing if it did not relate to real electrons. What did the equation have to say, for example, about the spin of the electron and its magnetic field? If his equation contradicted the experimenters' observations, he would have had no choice but to abandon it and start all over again. But there was no need for that. In a few pages of calculations, Dirac showed that he had conjured something miraculous: his equation described a particle not only with the mass of an electron but with precisely the spin and magnetic field measured by experimenters. His equation really did describe the electron so familiar to experimenters. Even better, the very existence of the equation made it clear that it was no longer necessary to tack on the electron's spin and magnetism to the standard description of the particle given by quantum theory. The equation demonstrated that if experimenters had not previously discovered the spin and magnetism of the electron, then these properties could have been _predicted_ using the special theory of relativity and quantum mechanics.\n\nAlthough Dirac apparently showed his usual Trappist calm, he was jubilant. In a few squiggles of his pen, he had described the behaviour of every single electron that had ever existed in the universe. The equation was 'achingly beautiful', as theoretical physicist Frank Wilczek later described it: like Einstein's equations of general relativity, the Dirac equation was universal yet fundamentally simple; nothing in it could be changed without destroying its power. Nearly seventy years later, stonemasons carved a succinct version of the Dirac equation on his commemorative stone in Westminster Abbey: _i_ \u03b3.\u2202\u03c8 = _m_ \u03c8. When set out in full, in the form he originally used, the equation looked intimidating even to many theoreticians simply  because it was so unusual, not that this would have disturbed Dirac: all that mattered to him was that it was based on sound principles and that it worked. It might even have crossed his mind that he had done something that John Stuart Mill had articulated as one of the aims of science - to unify disparate theories to explain the widest possible range of observations.\n\nWhen Dirac was an old man, younger physicists often asked him how he felt when he discovered the equation. From his replies, it seems that he alternated between ecstasy and fear: although elated to have solved his problem so neatly, he worried that he would be the latest victim of the 'great tragedy of science' described in 1870 by Thomas Huxley: 'the slaying of a beautiful theory by an ugly fact'. Dirac later confessed that his dread of such an outcome was so intense that he was 'too scared' to use it to make detailed predictions of the energy levels of atomic hydrogen - a test that he knew it had to pass. He did an approximate version of the calculation and showed that there was acceptable agreement but did not go on to risk failure by subjecting his theory to a more rigorous examination.\n\nDuring November and December, he shared with no one the pleasure he took in his discovery or his occasional panic attacks. Not a single significant letter or record of a conversation with anyone exists from those months. He broke his silence only before he set off to Bristol for the Christmas vacation when he bumped into his friend Charles Darwin, a grandson of the great naturalist and one of Britain's leading theoretical physicists. On Boxing Day, in a long letter to Bohr, Darwin wrote: '[Dirac] has now got a completely new system of equations for the electron which does the spin right in all cases and seems to be \"the thing\".' That was how Bohr learned that the remark he had made to Dirac at the Solvay Conference - that the problem of finding a relativistic equation for the electron had already been solved - was completely wrong.\n\nFowler sent Dirac's paper 'The Quantum Theory of the Electron' to the Royal Society on New Year's Day 1928, and a month later sent off a second paper that cleared up a few details. While the first paper was in press, Dirac wrote to Max Born in G\u00f6ttingen, not mentioning his new equation except in a ten-line postscript, where he spelt out the reasoning that had led to it. Born showed these words to his colleagues, who regarded the equation as 'an absolute wonder'. Jordan and Wigner, who were working on the problem that Dirac had solved, were flabbergasted. Jordan, seeing his rival walk off with the prize, sank into depression.\n\nWhen the equation appeared in print at the beginning of February, it was a sensation. Though most physicists struggled to understand the equation in all its mathematical complexities, the consensus was that Dirac had done something remarkable, the theorist's equivalent of a hole in one. For the first time in his career, he had shown that he was capable of tackling one of the toughest problems of the day and beating his competitors to the solution, hands down. The American theoretician John Van Vleck later likened Dirac's explanation of electron spin to 'a magician's extraction of rabbits from a silk hat'. John Slater, soon to be a colleague of Van Vleck's at Harvard, was even more effusive: 'we can hardly conceive of anyone else having thought of [the equation]. It shows the peculiar power of the sort of intuitive genius which he has possessed more than perhaps any of the other scientists of the period.'\n\nEven Heisenberg, more confident than ever after his recent appointment to a full professorship in Leipzig, was taken aback by Dirac's coup. One physicist later recalled Heisenberg speaking of an English physicist - unquestionably Dirac - who was so clever that it was not worth competing with him. Heisenberg was, however, concerned that despite the equation's beguiling beauty, it might be wrong: he was one of many who underlined a problem that Dirac had pointed out in his first paper on the equation - it made a strange prediction about the values of energy that an electron can have.\n\nThe background to the problem with the equation was that, like time, energy is a relative quantity, not an absolute one. The energy of motion of a free electron - one that has no net force acting on it - can be defined as zero when the particle is stationary; when the particle gathers speed, its energy of motion is always positive. Dirac's problem was that his equation predicted that, in addition to perfectly sensible positive energy levels, a free electron has _negative_ energy levels, too. This arose because his theory agreed with Einstein's special theory of relativity, which said that the most general equation for a particle's energy specifies the _square_ of the energy, _E_ 2. So if one knows that _E_ 2 is, say, 25 (using some chosen unit of energy), then it follows that the energy _E_ could be either +5 or -5 (each of them, when multiplied by itself, equals 25). So, Dirac's formula for the energy of a free electron  predicted that there were _two_ sets of energy values - one positive, the other negative. In classical physics, the negative-energy ones could be ruled out, simply because they are meaningless, but this cannot be done in quantum mechanics as it predicts that a positive-energy electron could always jump into one of them.\n\nNo one had observed such a jump, so the Dirac equation was in serious trouble. Despite this unsightly canker, however, the consensus was that his theory of the electron was a triumph. Yet Dirac seemed to take no pleasure from his success and showed none of the relief and elation that Einstein had demonstrated after he published his equation of general relativity. Dirac's younger colleague Nevill Mott later described the extent of Dirac's detachment from his fellow physicists in Cambridge. Mott was - like hundreds of other theorists - concentrating not on extending quantum mechanics but on applying it.\n\nAccording to Mott, no one in the Cambridge mathematics department knew anything about Dirac's equation until they read his paper in the library. Dirac was, Mott said, passive and forbidding, the kind of expert no one quite dares to consult. Dirac did not seem to appreciate the narrowness of his understanding of companionship: he liked to be among fellow physicists, when they were friendly - as they were in Bohr's Institute - but felt no obligation to talk to them about his work or even to disclose his first name. Charles Darwin had known him for six years before writing him a postcard asking him about his signature: 'What does P. A. M. stand for?'\n\nWhereas at Copenhagen and G\u00f6ttingen there were many premier-league quantum physicists, Fowler and Darwin were the only ones in Cambridge, so Dirac believed that it was his duty to deliver his seminars and lectures on the basics of quantum mechanics. But that, in his view, was where his departmental teaching obligations ended. But, surprisingly for a young research scientist, he did agree to write a textbook on quantum mechanics, scheduled to be the first publication in the 'International Series of Monographs on Physics', edited by Kapitza and Fowler. The series was the brainchild of Jim Crowther, the science reporter of the _Manchester Guardian_ , the unofficial writer-in-residence at the Cavendish Laboratory and the only journalist Dirac regarded as a friend. A passionate Marxist, Crowther had joined the Communist Party in 1923 and managed to be close to both Bernal and Rutherford - sworn enemies - making the most of  the talents and influence of each of them. By subtly cultivating relationships with all the finest young scientists in the Cavendish, including Dirac, Crowther became an influential bit-part player in the emerging group of radical scientists in Cambridge. One of his strengths was his sensitivity: he will have realised quickly that, to make friends with the great young theoretician, he had to overcome Dirac's reluctance to have anything to do with importuning journalists. Dirac just wanted to be left in peace.\n\nDirac's family knew nothing of his equation. For Charles, always keen to find out about Dirac's work, his son's unwillingness to share his science was cruel. In April 1928, when he read an anonymous article in _The Times_ about quantum physics, Charles may have been discouraged by the conclusion: 'Far past is the day when the scientist could talk to the layman as man to man [. . .] the world loses much when science has got into such deep waters that only a Channel swimmer can follow it.' When Charles pressed his son to explain something of his new physics - as he surely did - Dirac almost certainly gave his usual response of shaking his head or remarking unhelpfully that the new quantum theories 'are built up from physical concepts which cannot be explained in words at all'. Although Dirac used his visual imagination to think about quantum mechanics, he declined every request to describe images of the quantum world. As he would later remark: 'To draw its picture is like a blind man sensing a snowflake. One touch and it's gone.'\n\nTo judge from the letters Dirac received from his mother, relations between her and Charles had settled down now she was spending more time out of the house. She went to talks on Tennyson's poetry, saw shows at the Hippodrome theatre with Charles and Betty and visited the cinema, including a trip to see one of the last great silent films, _Ben Hur._ But the Dirac family's favourite novelty was the motor car, the most exciting of the new mass-produced technological innovations. One of Charles's private tutees owned a car and treated the Dirac family to afternoon joyrides to the coast and to countryside teashops, keeping to the speed limit of 20 mph. Images of trips like these - carefree families, cutting loose from worldly concerns for a day - symbolised the prosperity of Britain in the third quarter of the 1920s. For the majority, life had never been better.\n\nBut when Dirac was not at home, his mother's life was empty.  Always in search of a plausible excuse to visit him, she invited herself to Cambridge in mid-February to see the Lent boat races, sheepishly asking if he had the time to see her when she was in town ('I shall be dressed quite nicely & shall not be any trouble'). He often ignored such requests, but this time he agreed, and she arrived in a foggy Cambridge at lunchtime to spend a few hours talking with her son, who apparently gave no sign that he was living through one of the most exciting times of his life and that some of his peers were beginning to talk of him as the heir to Newton.\n\nDirac appeared also to resemble Newton in having no interest in forming romantic relationships with women. Many of Dirac's colleagues had the impression that he was frightened of women of his own age and they could scarcely imagine that he would ever marry. But he did have a close friendship with one woman, the fifty-six-year-old mother of his friend Henry Whitehead, a promising mathematician at Oxford University. Isabel Whitehead, a tall, solidly-built Scot, was the wife of the Right Reverend Henry Whitehead, nineteen years her senior and formerly the Bishop of Madras in India. The couple had spent almost twenty years living there, before retiring to the UK in 1923. Among her fellow expatriates, Mrs Whitehead was notorious: according to an authoritative account of the Christian community in India, she was imperious 'even by the domineering standards of the many British memsahibs'.\n\nThe Whiteheads lived in a half-wood, half-brick cottage in Pincent's Hill, near Reading, about three hours' drive from Cambridge. Always accompanied by their dogs, they led a leisurely life, taking just an hour or two each day to run a small farm with pedigree Guernsey cattle and a few chickens. Both Isabel and Henry were Oxford-educated mathematicians, but it seems from Mrs Whitehead's letters that the two of them talked less about science with Dirac than about other matters, especially Henry's enthusiasm for cricket and their adventures in India, including the week they spent in their home entertaining Gandhi. In the coming years, Mrs Whitehead's correspondence with Dirac also makes it clear that she robustly challenged his atheism and that he trusted her with his most private thoughts about his family. Pincent's Hill became a favourite weekend retreat for him and Mrs Whitehead became his second mother, giving him not only support and affection but also something his own mother could not provide - intellectual stimulation.\n\nDuring the early spring of 1928, Dirac was planning his next journey. His six-month itinerary would begin in April and take him back to Bohr's Copenhagen and Ehrenfest's Leiden, on to Heisenberg's Leipzig and Born's G\u00f6ttingen, and finally his first visit to Stalin's Union of Soviet Socialist Republics. Dirac had heard much about this country; now he would be able to judge for himself.\n**Twelve**\n\nSee how physical science, which is Reason's trade  \nAnd high profession, booketh ever and docketeth  \nAll things in order and pattern.\n\nROBERT BRIDGES, _Testament of Beauty_ , 1929\n\nPaul Ehrenfest could be a moody and demanding colleague, but he was a charming and generous host. In April 1928, when he realised that he would not be able to greet Dirac at Leiden railway station at the beginning of his visit, Ehrenfest arranged for a phalanx of his assistants to be waiting for him on the platform when his train steamed in shortly after 10 p.m. The problem was that none of them knew what Dirac looked like. Ehrenfest's solution was to ensure that, outside every train door facing the platform, there was a student waving a reprint of 'The Quantum Theory of the Electron'. The plan worked.\n\nOne member of the welcoming party was Igor Tamm, a thirty-two-year-old Soviet theoretician, soon to become one of Dirac's closest companions. Tamm was famously restless: in group photographs, while others appeared in sharp definition, he would be a blur. A Marxist even before he went to university, he joined the Social-Democratic Workers' Party in 1915 and, during the subsequent years in Moscow, Kiev, Odessa and Elizavetgrad, studied science while being a part-time activist for the Bolsheviks. He tired of their fanaticism and, when they declared all other political parties illegal in the summer of 1918, was concentrating on science. He became the first Soviet theoretician to use quantum mechanics. In January 1927, he arrived in Leiden and, a year later, electrified by the Dirac equation, was looking forward to meeting its discoverer. Tamm wrote to his wife in Moscow that he wanted to see if there was any truth in rumours that 'it costs a tremendous effort to get a word from [Dirac], and that he talks only to children under ten'.\n\nThe two men soon clicked. In Tamm, Dirac had found another intellegent and entertaining Russian extrovert; in Dirac, Tamm found a companion who was surprisingly agreeable, provided he was under  no pressure to speak. The two men spent the spring afternoons strolling around the town's cobbled streets, watching the traffic on the interlocking network of canals and occasionally walking out to the nearby tulip fields. Tamm taught Dirac to ride a bicycle, Dirac taught Tamm physics, and they talked about matters outside science, probably including politics and Tamm's favourite hobby of mountain climbing. Tamm was humbled by Dirac's erudition: 'I feel like a little child next to him,' he wrote to his wife.\n\nAs was customary for visitors to Leiden, Dirac gave a series of lectures. He had much improved his technique as a public speaker: when he strode towards the blackboard, he seemed to change from being a pitiful wallflower to the Demosthenes of quantum mechanics. Standing quite still, he looked into the eyes of his audience and talked plainly and articulately, with the force of an advocate, not letting a pause or hesitation break his rhythm. He did not read from a prepared text but knew exactly what he wanted to say; once he had decided on the clearest way of expressing an idea, he would not deviate from it, from one lecture to another. When Ehrenfest asked for further explanation, Dirac would respond by repeating what he said, almost word for word.\n\nIn mid-June 1928, Dirac moved on with Tamm to Leipzig to spend a week at a conference co-organised by Heisenberg, who was agonising about the Dirac equation. Darwin and others had demonstrated that it perfectly reproduced previously successful formulae for atomic hydrogen's energy levels, but this news cut no ice with Heisenberg. He was troubled by the equation's absurd prediction that a free electron can have negative energy - and it had become clear that no subtle tinkering with the equation could change it. For Dirac, this was simply the next problem to be addressed. For Heisenberg, it was evidence that the equation was sick. A month after Dirac departed from Leipzig, Heisenberg wrote to Bohr: 'I find the present situation quite absurd and on that account, almost out of despair, I have taken up another field, [trying to understand magnetism]. ' A month later, Heisenberg was even more depressed when he wrote to Pauli: 'The saddest chapter of modern physics is and remains the Dirac theory.' Dirac knew Heisenberg's criticisms were well founded and that the onus was on him to demonstrate that the theory was more than a beautiful mirage.\n\nAmong the scientists Dirac met for the first time in Leipzig was  Heisenberg's student Rudolf Peierls, just turned twenty-one. Wiry, bespectacled and with a pronounced overbite, Peierls oozed vitality and ambition. His professors asked him to take Dirac to the opera, a challenge that his guest's Cambridge colleagues regarded as all but impossible. They could scarcely imagine him sitting through any kind of drama: the artifice, the focus on speech or lyrics and the often contorted plotting would surely have no appeal to his literal mind. Decades later, Peierls could not remember the play or his guest's reaction to it but squirmed at the thought of Dirac's insistence on following the English custom of taking his hat with him to the performance, pointedly refusing to follow the German practice of leaving headwear in the theatre cloakroom. Peierls, whose formal Prussian education had given him a strong sense of politesse, found Dirac's behaviour mortifyingly crude. Dirac, probably oblivious of his colleague's discomfiture, often behaved like this: he was a stickler for English conventions of courtesy and saw no reason to deviate from them in other countries. Flexibility was not his forte.\n\nAfter the conference, Dirac travelled with Tamm to G\u00f6ttingen. Its theoretical physics department was losing its edge as its leader, Max Born, struggled to maintain his momentum. Overworked, worried that younger and fresher minds were leaving him behind, depressed by marital problems and the Nazis' 'blood and soil' anti-Semitism, he slid into a nervous breakdown. His colleague Jordan was openly a conservative nationalist but in private was writing reactionary articles in the journal _Deutsches Volkstrum_ ('German Heritage'), under the cover of a pseudonym.\n\nG\u00f6ttingen was, however, still on the itinerary of every young theoretician. During this visit, Dirac began his long friendship with two other visitors, who embodied his taste for the company of both introverts and extroverts and who were to lead him to his first close relationships with women of his own age. At the flamboyant extreme was George Gamow, a Russian theoretician two years Dirac's junior, destined to be the court jester of quantum physics. Variously nicknamed Johnny, Gee-Gee and (by Bohr) Joe, he was a six-foot three-inch, 220-pound giant and close to being Dirac's polar opposite: loquacious, a passionate smoker and drinker, relentlessly jocular. Shortly before his visit to G\u00f6ttingen, he had made  his name by being one of the first to use quantum mechanics to explain the type of radioactive decay in which an alpha particle can be ejected from types of atomic nuclei (impossible, according to classical mechanics). Dirac, probably to Rutherford's frustration, had attended many Cavendish seminars about new findings in nuclear physics but showed no interest in trying to understand them. As theoreticians, Gamow and Dirac were entirely different: Gee-Gee did not try to come up with fundamental new ideas but preferred to apply ones discovered by others. Yet the two men got along well and often dined together, Dirac listening expressionlessly as his new friend told of how he had learned Euclidean geometry under artillery bombardment and other such stories, most of them more impressive for their colour than their accuracy.\n\nAt the other end of the personality spectrum was Eugene Wigner, who had recently arrived in G\u00f6ttingen after spending years with Einstein in Berlin, having switched to physics after being trained as an engineer. The scion of a wealthy Jewish family, Wigner and his two sisters had been raised by a governess in a grand apartment in one of the most exclusive residential areas of Budapest, overlooking the Danube. He loved to reminisce about his boyhood home: the formal family dinners, the scurryings of the two uniformed servant girls, the scent of freshly cut roses. Unlike Dirac, the young Wigner was politically alert and acutely aware of the instability of his country. Since the break-up of the Austro-Hungarian Empire in 1918, Hungary had been through a bloody Bolshevik revolution led by B\u00e9la Kun and the White Terror organised by nationalist and anti-Semitic forces. Wigner was fearful of the future of the country, then under Admiral Horthy's authoritarian regime.\n\nDespite all the political upheavals, Wigner had an exceptionally fine school education in mathematics and science, even more thorough than Dirac's. Historians still debate why Budapest in the early twentieth century produced so many intellectual innovators, including John von Neumann, whom Dirac would later rate as the world's finest mathematician, and Wigner's friends Le\u00f3 Szil\u00e1rd and Edward Teller, both to do important research into the first nuclear weapons. The success of this cohort of Hungarians is partly due to their education, shortly after the war, in Budapest's excellent high schools and partly to the vibrancy and ambition of the city's Western-focused culture.\n\nWigner was one of the shyest and most uncommunicative of the quantum physicists but, compared with Dirac, he was gregariousness itself, so conversation during their evening meals together was probably strained. They had to find a common language - Dirac did not know Hungarian, hated to speak French and spoke fractured German with a bitumen-thick accent, while Wigner's English was weak, and he liked to converse in German or French. They probably settled on German. No record remains of the details of their early conversations, but it is likely that Wigner mentioned his politics and youthful experiences of anti-Semitism: since he was sixteen, he had followed his father in ideologically opposing Communism, and his views had hardened a year later during Kun's regime, in which his father was thrown out of his job as director of a tannery. For a few months, the Wigners had fled to Austria but returned after the Communists were overthrown.\n\nDirac would have been content to listen to as much of Wigner's life story as he was willing to tell. But when Wigner turned his attention to physics, he quickly saw that Dirac had no interest in sharing his thoughts and ideas. The moment Wigner began to probe, Dirac withdrew into himself like a frightened hedgehog. Igor Tamm knew how to avoid this kind of defensiveness: keep conversation to a functional minimum, avoid personal questions and never risk wasting breath on trivialities. Tamm and Dirac's relationship flourished partly because they had complementary talents: intellectual leadership was provided by Dirac, while the social impetus came from Tamm. It was he who introduced Dirac to what would be one of the greatest pleasures of his young life: mountain climbing. In one long trip east, the two journeyed out to the wooded Harz - ablaze with fireflies in the evenings - and they climbed the challenging peak of Mount Brocken (1,142 metres). Dirac was smitten: apart from equations, nothing did more to stir his sense of beauty than mountains.\n\nAt the end of July 1928, Dirac was preparing for his first visit to Russia, a two-month stay that combined the chores of lecturing with the pleasure of relaxing with Kapitza. Dirac's mother was fearful: 'If you go to Russia, do take care of yourself. We hear such dreadful accounts of the Bolshevists in the papers. There seems to be no law and order anywhere. I expect you know more about the facts than  we do, though, as you are so much nearer.' Since 1918, the British press had reported on the Soviet regime's growing repressiveness, which increased with the rise of Stalin to absolute power in 1926. The British Government did not officially recognise the Soviet Union, but profitable trade between the countries was easing relations between them, culminating in the Labour Prime Minister Ramsay MacDonald's restoration in 1929 of full diplomatic relations.\n\nAfter his arrival in Leningrad on 5 August, Dirac's hosts introduced him to caviar, one of the few luxury foods for which he had a taste. Dirac blossomed in Russia - the scenery, the architecture, the museums and the art galleries - as he wrote in a long and chatty letter to Tamm:\n\nI spent the first two days in Leningrad with Born and his [G\u00f6ttingen colleague] Pohl and we saw the sights and visited the Hermitage and the Museum of Russian Art and the Natural History Museum and also the Roentgen Institute [for physics research] [. . .] I found Leningrad a very beautiful place, and was more impressed by it than by any other town during the journey, particularly as I came up the river in the steamer and first saw the large number of churches, with their gilded domes, quite different from anything I had ever seen [. . .].\n\nMoscow still resembled the city of Anna Karenina, with its squat wooden houses, multicoloured cupolas, horse-drawn cabs driven around the sprawl of zigzag streets by peasants in blue robes, bearded traders sipping vodka and eating cucumbers in the Slovenski Bazaar. Dirac was there to attend the no-expense-spared Congress of Russian Physicists, at his hosts' expense. Physicists in the Soviet Union had been quick to realise the importance of quantum mechanics and wanted to learn from the innovators in western Europe. Of the one hundred and twenty physicists who attended the Congress, about twenty were foreign. Dirac was the star of the occasion, but he arrived in Moscow too late to give his talk, scheduled for the opening session. When he should have been giving his presentation, he was walking around one of the royal palaces on the outskirts of the city; in the evening, he went to a performance of Japanese theatre. The next day, Dirac went with the conference delegates to the Kremlin before setting off alone to walk the streets until sundown.\n\nThe venue for the second part of the Congress was a steamer that  sailed down the Volga to Stalingrad. During the week-long cruise, Dirac gave a talk on his theory of the electron and met the leaders of Soviet physics, including his admirer Lev Landau, a twenty-year-old graduate student, soon to be his country's greatest theoretician - the most accomplished but least mature. Mangy and undernourished, he was so tall that in most company people could see his long, thin face standing out, topped with dark wavy hair that was piled on the right of his head like a burnt crest of meringue. As a critic, he was so aggressive that he made Pauli look demure; as a colleague, so socially inept that he made Dirac look suave.\n\nAfter the Congress, Dirac took a two-day train journey to the Caucasus. He stayed with Kapitza and joined a party of sightseers for a six-hour hike up a glacier near Vladikavkas. Dirac described his adventures in a letter to Tamm but did not mention that, during his time with Kapitza, he experienced an incident that was, in some way, his sexual awakening.  Forty-five years later, he remembered that he first saw a naked young woman in the Caucasus: '[she was] a child, an adolescent. I was taken to a girls' swimming pool, and they bathed without swimming suits. I thought they looked nice.' He was twenty-six years old.\n\nDirac was in no hurry to return to Bristol: the journey took him almost a month. The disparity between the excitement of his work and the dreariness of his home life had never been so stark. He was lionised by many of his colleagues, he was financially independent, and he was benefiting from international travel at a time when it was a luxury. Charles, Flo and Betty, on the other hand, were locked in their routine and left their hometown only rarely. Betty was happy to do nothing at all when she was not looking after her new dog; Charles was overworked and run down; Flo was trying to make the most of every opportunity to leave the house. At her elocution classes, she wrote and practised giving speeches, including one opposing the notion that there might one day be a woman prime minister. She rehearsed her speech on the Bristol Downs, beginning with the flourish 'I rise to oppose the motion of a woman prime minister - to oppose most decidedly and definitely.' For one thing, Flo argued, women do not have sufficiently strong constitutions to take on such a responsibility: 'As regards physique - women today are wonderful: but none can say when a woman may faint! None when  she may scream! Is it becoming for a Prime Minister to suddenly fall to the ground, or to burst into hysterics at a crucial moment?'\n\nAlthough Flo was not in the vanguard of feminism, Dirac knew that underneath his mother's apparent submissiveness lay stoicism and an independence of spirit. These qualities would, over the next three years, be tested to breaking point.\n\nWhen Dirac returned to Cambridge in October 1928, he knew that the onus was on him to cure the sickness of his theory of the electron. Somehow, he needed to find a rational explanation for the negative-energy states which were undermining confidence in the Dirac equation; some of his colleagues were becoming worried that the equation might not be right after all.\n\nThat autumn, he was, unusually, working on several projects at the same time: his hole theory, his textbook and a brief paper on one of his favourite subjects - the relationship between classical mechanics and quantum mechanics. The paper was based on the ultra-rigorous work of von Neumann, who had derived one result that caught Dirac's eye. Von Neumann had found a way of describing the overall behaviour of an enormously large number of non-interacting quantum particles, when nothing is known about their individual behaviour. It turned out, surprisingly, that the statistical description given by quantum mechanics is just as simple as the account given by classical mechanics; in both, the behaviour of the individual particles averages out to a smooth overall pattern, just as the behaviour of a swarming crowd can be described without referring to any of its individuals. In this bijou paper, Dirac developed von Neumann's ideas and laid bare the precise analogy between the classical and quantum understandings of vast numbers of particles. This was a divertimento composed during a holiday from fixing his troublesome symphony.\n\nIn those politically tranquil times, the favourite topic of conversation in Cambridge was poetry. The eighty-five-year-old poet laureate Robert Bridges had written the most talked-about poem of the year, _A Testament to Beauty_ , 5,600 lines about the nature of beauty. It is now read only rarely, but then it struck a chord with tens of thousands of lay readers and some literary critics, including one in the _Cambridge Review_ who described it as 'a high philosophical explanation of Keats's \"Beauty is truth, Truth beauty\"'. To some extent,  Bridges was reacting against modernist art - such as Arnold Sch\u00f6nberg's atonal music, Picasso's cubism, Eliot's fragmented poetry. Bridges sought beauty and found it not only in music, art and nature but also in science, food and even in football matches. Dirac knew, too, that beauty was about much more than art and nature. He had seen it in Einstein's equation for the general theory of relativity and he now had an equation of his own that was no less of a contribution to aesthetics. But aesthetic judgements like that count for nothing in science if a theory fails to agree with experiment. Unless someone could explain the meaning of the negative-energy solutions to the Dirac equation, it was doomed to be remembered only as just another scientific fad.\n\nA few of Dirac's colleagues in Cambridge would not have been distraught if fortune had clipped his wings: his ascending reputation had led, inevitably, to envy. No longer were the two leading lights of the university's experimental and theoretical physics cited as Rutherford and Eddington, but as Rutherford and Dirac. Eddington's star was waning, and he knew it. Meanwhile, the old guard of Cambridge physics looked pitifully out of touch. The proud Irishman Sir Joseph Larmor, holder of the most prestigious chair in Cambridge, the Lucasian Professorship of Mathematics, once held by Newton, was living in the past, unable to understand relativity theory and disdainful of quantum mechanics. He and his friend J. J. Thomson wandered the streets of Cambridge, each of them wearing a bowler hat, a black three-piece suit and an immaculate white shirt, and each wagging a stick behind his back. When they peered into one of the shop windows on Trinity Street, the two superannuated professors looked like a pair of penguins.\n\nThe two men knew that their views counted for nothing among physicists who were once their admiring students and who were now running physics. No one symbolised the new generation's ascendancy more powerfully than Dirac, but he still did not have a permanent job. He had turned down Arthur Compton's offer of a post in Chicago and had later declined an offer of a professorship in applied mathematics at Manchester University, commenting that 'my knowledge of and interest in mathematics outside my own special branch are too small for me to be competent [in such a post].' If the spurned mathematicians in Manchester found his modesty hilarious, Dirac would have been uncomprehending, as he was simply being candid. As Mott said:  'He is quite incapable of pretending to think anything that he did not really think.'\n\nIf Dirac and Fowler were away, Cambridge University would struggle to teach quantum mechanics, as Harold Jeffreys virtually admitted when he wrote to Dirac in March 1929, pleading with him to set the questions on quantum mechanics for the summer examinations. Jeffreys and his fellow 'ignorant and philistine' faculty colleagues were in the embarrassing position of having to admit that 'the candidates know more than we [do]'. Fowler led the campaign to ensure that Dirac remained in Cambridge, and he soon had some success: in June 1929, St John's College awarded Dirac a special lectureship, though it was funded for only three years. Dirac's loyalty to Cambridge was to be tested, repeatedly.\n\nAs Dirac was getting nowhere with his top priority of sorting out the difficulties with his equation, he decided to devote himself to other things. In late 1929, he spent most of his time drafting his book and working on another research project, the theory of heavy atoms. This was by no means his favourite branch of physics, but it was closer to the work of the great majority of quantum theorists, who were applying the theory to complicated atoms and molecules. Dirac was, however, in no doubt that quantum mechanics would be successful:\n\nThe underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.\n\nThose words became one of the clarion calls of reductionists, who believe that complex things can be explained in terms of their components, right down to the level of atoms and their constituents. Extreme reductionism implies, for example, that quantum mechanics lies at the bottom of an inverted pyramid of questions that begins, for example, with 'Why does a dog bark?' A reductionist seeks to answer the question by understanding the chemical reactions going on inside the dog's brain, and those reactions are ultimately understood by the interactions of the chemicals' electrons, whose behaviour is ultimately described by quantum mechanics. Although popular with many scientists, the approach does not describe _how_ to make the links between the layers of explanation.\n\nIn his paper, Dirac applied quantum mechanics to atoms that contain more than one electron, such as carbon atoms. Such atoms are much harder to describe than hydrogen atoms because, in every multi-electron atom, the complicated and unwieldy interactions between all the electrons have to be taken into account. Dirac found a way of describing these interactions approximately and investigated the consequences of the fact that it is impossible to detect experimentally if two of the electrons swapped places. As usual, Dirac left it to others to work out the theory's consequences: the American theoretician John Van Vleck, based in Minneapolis, quickly saw the potential of Dirac's ideas and spent years using them to explain the origin of magnetism, the various ways that atoms can bond to form molecules and the patterns of light emitted by multi-electron atoms. This was to be the main legacy of Dirac's excursion into atomic physics - his first paper on the subject, and his last.\n\nAt the end of term, he visited his family briefly and then, in what was becoming a ritual, set off on another long journey. At Southampton, on the freezing Wednesday morning of 13 March, he boarded the liner _Aquitania_ with his travelling companion, Isabel Whitehead's son Henry. In the crowd at quayside was Florence Dirac, who by then had got the message: her only son wanted to spend as little time at home as he could. Just as she must have dreaded, he would be away for as long as his teaching obligations in Cambridge allowed, on his first visit to the United States of America. His reputation had preceded him.\n**Thirteen**\n\n[I]n England there is something very like a cult of eccentricity. [. . .] With us [Americans], as more than one European has said, the trait is more distinguishable nationally than individually.\n\nGARDNER L. HARDING, _New York Times_ , 17 March 1929\n\nIn every branch of science, theorists vie with experimenters to set the agenda. Since Heisenberg's publication of his path-breaking paper in the autumn of 1925, theoreticians had been pointing the way ahead in physics. Yet the foundations of some of the new theoretical ideas had not even been checked experimentally: according to Schr\u00f6dinger's quantum theory, for example, every material particle has an associated wave, but no experimenter had been able to prove the idea or to refute it. So there was an almost palpable sigh of relief among quantum physicists back in early 1927 when news reached Europe that the American experimenters Clinton Davisson and his student Lester Germer had shown that the electron could indeed behave like a wave. Dirac, often believed to regard experiments with a high-minded insouciance, belied his reputation by arranging to visit Davisson's laboratory on West Street in south Manhattan, a few blocks from the meatpacking district, the first stop on his itinerary.\n\nThis was Dirac's first sight of New York, then booming with wealth and new technology. The Jazz Age was, according to the man who named it, F. Scott Fitzgerald, past its 'heady middle age', though Americans were still enjoying 'the most expensive orgy in history'. The hurried pace of American life was not at first to Dirac's taste: it was somehow fitting that during the first night Dirac spent in his hotel on Seventh Avenue, he was kept awake until the small hours by revellers in an adjacent room. As soon as he awoke the next day, shortly before four o'clock in the afternoon, he realised he had missed his appointment with Davisson. Rather than waste the late afternoon, he spent it strolling around rush-hour midtown Manhattan, teeming with four-square black automobiles navigating around the skyscrapers, each of them a powerful symbol of America's soaring prosperity.\n\nIn Davisson's laboratory the next day, Dirac saw the ingenious apparatus that first persuaded the electron to reveal its wave nature. Davisson and Germer had fired beams of electrons towards a nickel crystal and found that the number of electrons they detected at different angles had alternating peaks and troughs. These variations were impossible to understand if the electron is simply a particle: the only explanation was that the electrons behave as waves which are bent ('diffracted') by the crystal, like two waves combining on the surface of pond, forming peaks when the waves reinforce one another and troughs when they cancel each other out. Physicists had no choice but to conclude that the electron behaved sometimes like a particle and sometimes as a wave - a 'wavicle', as Eddington had dubbed it - precisely as quantum theory had supposed.\n\nDirac quickly headed off on his five-month journey across North America, travelling mainly on the railroad. He kept a record of his trip in terms of numbers, not words: his diary contains no descriptions of his experiences, just a cumulative record of the number of nights he had spent on a train and on board ship.\n\nAfter paying brief visits to Princeton and Chicago, Dirac travelled to Madison, capital of the Midwestern state Wisconsin. Like G\u00f6ttingen, Madison was his sort of town, with a good university and surrounded by countryside offering plenty of opportunities for walks. He was the first foreign guest of John Van Vleck, newly appointed to the university faculty. Slightly older than Dirac, Van Vleck excelled at applying quantum physics and had no interest in its mathematical foundations. The two men spent hours together walking in the vast fields overlooking Lake Mendota, one of the four lakes around the town. For Dirac, Van Vleck was the perfect walking companion - fit, uninterested in small talk and content to say nothing for hours. Perhaps Van Vleck mentioned his passion for railroads and his feat of memorising the passenger railway timetable for the whole of Europe and the United States. Like Dirac, Van Vleck was fascinated by technology, numbers and order.\n\nDirac's hosts were aware of his reputation for eccentricity, and they soon saw that it was well justified and that his sangfroid was extreme even by the standards of the English. He left them several Dirac stories, including a classic that appears to have been first spread around by a tickled Niels Bohr. The story begins during one of Dirac's lectures, moments after he has finished talking, when the moderator asks if anyone has any questions. Someone in the audience says, 'I don't understand the equation on the top-right-hand corner of the blackboard. ' Dirac says nothing. The audience shuffles nervously, but he remains silent, whiling away the time of day, looking unconcerned. The moderator, feeling obliged to break the silence, asks for a reply, whereupon Dirac says, 'That was not a question, it was a comment.'\n\nMadison was also the venue of what would become the most widely quoted interview that Dirac ever gave, to the journalist Joseph Coughlin, known to everyone as Roundy owing to his substantial girth. Well known in the town, he was one of Wisconsin's most popular columnists, delivering regular doses of homespun wisdom on sport and other topics in language that was often ungrammatical but always alive with quirky humour. Dirac kept a typed transcript of the four-page article, in which Roundy recounts verbatim his attempts to persuade his interviewee to utter more than one syllable at a time:\n\nROUNDY: Professor, I notice you have quite a few letters in front of your last name. Do they stand for anything in particular?\n\nDIRAC: No.\n\nROUNDY: You mean I can write my own ticket?\n\nDIRAC: Yes.\n\nROUNDY: Will it be all right if I say that P. A. M. stands for Poincar\u00e9 Aloysius Mussolini?\n\nDIRAC: Yes.\n\nROUNDY: Fine! We are getting along great! Now doctor will you give me in a few words the low-down on all your investigations?\n\nDIRAC: No.\n\nROUNDY: Good. Will it be all right if I put it this way: 'Professor Dirac solves all the problems of mathematical physics, but is unable to find a better way of figuring out Babe Ruth's batting average?'\n\nDIRAC: Yes.\n\nThe dialogue continues for another page. According to the transcript, Roundy's interview was published in the 'P. A. M. issue' of the _Wisconsin Journal_ on 31 April ( _sic_ ). However, the records of the newspaper show that no such edition was published, so it appears that this much-anthologised article is a spoof. One possibility is that the typed document was a pastiche presented to Dirac by his Madison colleagues during his farewell dinner at the University  Club, where - as Van Vleck later wrote - they played an elaborate game to tease out of Dirac the names designated by his initials P. A. M. Whatever the origins of the Roundy interview, it is an example of a probably apocryphal Dirac story that captures his behaviour so accurately that it somehow ought to be true.\n\nDirac left Madison with a cheque for $1,800, more than enough to cover his costs for the remainder of his trip. In June, he combined business and pleasure, giving a series of lectures on quantum mechanics in Iowa and Michigan, also walking down and up the Grand Canyon and hiking in Yosemite National Park and the Canadian Rockies - his introductions to grand North American scenery, which he explored on foot during several trips in the coming decades. He again demonstrated his interest in the latest experimental tools when, during a stay at the California Institute of Technology, he visited the Mount Wilson Observatory, near Pasadena, whose telescope was the largest in the world and by far the most productive source of new information about the universe.\n\nA few months before, Heisenberg had proposed to Dirac that they should travel together to 'bring European life into the American hurry'. When they met in early August at their hotel near the Old Faithful geyser, Heisenberg was surprised to find that Dirac had planned a route that would enable them to see the maximum number of geysers erupt. Even his scenic walks were informed by mathematical analysis. Heisenberg had arranged for them to travel first class to Japan on the steamer _Shinyo Maru_ , sharing a roomy cabin with a sea view. Two leading theoreticians were about to spend weeks together, with every opportunity to talk and perhaps to crack the gnawing problem of how to interpret the negative-energy solutions to Dirac's equation. The clubbable Heisenberg would probably have been game for a collaboration, but not Dirac. Although he admired Heisenberg and regarded him as a friend, Dirac felt no obligation to share any of his thoughts about physics with him. His motto was: 'People should work on their own problems.'\n\nIn the middle of August, after they had each given a series of lectures in Oppenheimer's department at the University of California at Berkeley, they set off from San Francisco on their two-week cruise to Japan. On board, Heisenberg was a conventionally hedonistic  tourist, honing his technique at ping-pong and dancing with the flapper girls. Dirac looked on, probably bemused. It is easy to imagine Dirac at one of the evening balls, sitting at a table and gazing quizzically at Heisenberg as he jived on the dance floor. Heisenberg long remembered being asked by Dirac, 'Why do you dance?' After Heisenberg replied, reasonably enough, 'When there are nice girls it is a pleasure to dance,' Dirac looked thoughtful. After about five minutes of silence, he said, 'Heisenberg, how do you know _beforehand_ that the girls are nice?'\n\nAs their steamer approached Yokohama, a reporter sought an interview with the two famous theoreticians. Unfamiliar with Dirac's appearance but not with Heisenberg's, the reporter said to Heisenberg, 'I have searched all over the ship for Dirac, but I cannot find him.' Heisenberg knew how to handle this: he talked affably to the journalist, no doubt giving him the story he wanted and not mentioning that Dirac was standing next to him, looking in another direction.\n\nIn Japan, the two physicists were greeted as heroes. Leading scientists in Japan knew that their science lagged well behind that of Europe and the USA, and physicists flocked from all over the country to see and hear two of the young founders of quantum mechanics. Dirac and Heisenberg were given round-the-clock obeisance and the full VIP treatment, their first taste of international celebrity. From the official photographs, it is clear that Heisenberg slipped easily into the role of the touring dignitary, looking poised and relaxed in the light summer suit he wore to stay cool in the searing heat. Looking less comfortable than his friend, Dirac made no such changes to his wardrobe: he wore the same three-piece suit and boots that he wore in the depths of the Cambridge winter.\n\nThe itinerary was the usual one for academics making a short trip to the country: a stay in Tokyo followed by a visit to the old imperial city of Kyoto, lecturing to packed, hushed audiences of respectful men wearing Western suits splashed with _jako_ perfume, scenting the auditorium with the fragrance of geraniums. The texts of the lectures were swiftly translated into Japanese and published as the Orient's first authoritative book on quantum mechanics, a bible for Japan's next generation of physicists, destined to make a huge impact. Dirac and Heisenberg, each of them only twenty-seven, were already training their successors.\n\nAt the end of their stay in Japan, Dirac and Heisenberg parted company. Dirac wanted to return by the fastest practicable route, by traversing Russia on the Trans-Siberian Railway. The construction of the 5,785-mile railway in Siberia - with brutal extremes of climate, little local labour available and dreadfully primitive supply routes - had been an engineering project that would have daunted even Brunel. It took twenty-five years to complete. Dirac boarded the train on 24 September at Vladivostock on the eastern coast and, nine days later, arrived in Moscow. He met up with Tamm, and they went on a long walk to see the sights of the city, including the sixteenth-century St Basil's cathedral, later converted into one of the country's many anti-religion museums. Dirac then headed back to England after taking what seems to have been his first flight, from Leningrad to Berlin. This was probably not the most agreeable of experiences: for the next few decades, he preferred to admire aviation technology from a secure vantage point on the ground.\n\nWhile he was away, his family were 'plodding along as usual', as his mother put it. The highlight of the year had been the General Election in June. For Flo, new technology had taken much of the thrill out of politics: 'The Election is being conducted mainly by \"Wireless\",' she wrote to Dirac, 'so I don't get any fun out of meetings. ' She and Charles supported Lloyd George's Liberal Party, which was trounced in Bristol by the Labour Party, consistent with the national swing that put Ramsay MacDonald back into 10 Downing Street.\n\nDirac's father, in better health than he had been for some years, was drifting further away from his wife and ever closer to Betty. While Charles and his favourite child played with the family dog in the garden, Flo was left inside, dreaming of _her_ favourite child thousands of miles away. She imagined him touring the Hollywood studios and riding a donkey down the Grand Canyon in a Panama hat, though she was disappointed to hear that he had done neither. Flo and Charles, having not seen their son for six months, were hoping to see him before the beginning of term and prepared the house for his visit. But in early October, Dirac perfunctorily informed them that he was back in Cambridge and mentioned no plans to visit Bristol.\n\nHe and other theoreticians had made virtually no progress with the problem of negative-energy electrons. Although most physicists  wanted to be rid of them, the Swedish physicist Ivar Waller had shown a few months before that they were indispensable to the theory. Waller had found a strange result when he analysed what happens when a photon is scattered by a stationary electron: Dirac's theory could reproduce the successful classical prediction at low energies only if the electron had access to negative-energy states. There could be only one conclusion for Dirac: his equation would survive only if someone could understand these negative-energy electrons.\n\nAs he settled down for the new term, Dirac was aware that the critical chorus had swelled from a whisper to a roar. In the opinion of its most dominant soloist, Pauli, the equation's sickness was incurable and its agreement with experiment was a fluke. The onus was on the equation's discoverer, refreshed after almost six months' vacation, to rescue it. So he set about the problem again.\n\nAt the end of October, news broke from New York of the event that ended the calm of late-1920s politics and began the descent into global economic catastrophe. The Dow Jones index had reached its historic peak a month before. Then panic struck when the bubble burst. On Friday, 25 October, the newspapers in the St John's common room all featured reports that made clear the scale of the crisis: the _Manchester Guardian_ wrote of 'Wild selling in record turnover of 13,000,000 shares'; _The Times_ wrote, 'a Niagara of liquidation took place on the American stock market today'. Four days later, on 'Black Tuesday', Wall Street all but melted down, and, as F. Scott Fitzgerald later noted, the decade of unparalleled prosperity had 'leapt to its spectacular death [. . .] as if reluctant to die outmoded in its bed'.\n\nBritain braced itself for the aftershock. Dirac kept abreast of the news, but he was focusing mainly on solving the mystery of the negative-energy electrons. Why had no one observed jumps of the familiar, positive-energy electrons into negative-energy states? After a few weeks, he had found an answer. He imagined all the electrons in the universe gradually filling up the energy states: the states with negative energy will be populated first, because they have the lower energies. Only when they are full will electrons occupy positive energy states. Because the negative-energy states are full, there are no vacancies into which these positive-energy electrons can jump. It is ironic that the crucial idea that underpinned the theory was supplied by Dirac's harshest critic, Pauli: according to his exclusion principle, every negative-energy state can be occupied by only _one_ electron.  This prevents each negative-energy state from being filled ad infinitum with electrons.\n\nThe bizarre upshot of the theory is that the entire universe is pervaded by an infinite number of negative-energy electrons - what might be thought of as a 'sea'. Dirac argued that this sea has a constant density everywhere, so that experimenters can observe only departures from this perfect uniformity. If this view is correct, experimenters are in rather the same position as a tribe that has spent its entire life hearing the unchanging background sound of a single musical note: this would not seem like torture because people are aware only of _changes_ to their environment.\n\nOnly a disturbance in Dirac's sea - a bursting bubble, for example - would be observable. He envisaged just this when he foresaw that there would be some vacant states in the sea of negative-energy electrons, causing tiny departures from the otherwise perfect uniformity. Dirac called these unoccupied states 'holes'. They would be observed, he reasoned, only when they are filled by an ordinary electron, which would then emit radiation as it makes the transition. It should therefore be possible to detect a hole in the sea when an ordinary positive-energy electron jumps into it. But what characteristics do the holes have? They mark the absence of a negative-energy electron. Within the general scheme of the 'electron sea', the _absence_ of negative energy amounts to the _presence_ of positive energy (two negatives make a positive: when _debt decreases_ by \u00a35, _wealth increases_ by the same amount). Furthermore, a negative-energy electron is negatively charged, so its absence is equivalent to the presence of a positive charge.\n\nIt follows that each hole has positive energy and positive charge - the properties of the proton, the only other subatomic particle known at that time. So Dirac made the simplest possible assumption by suggesting that a hole _is_ a proton. What he could not explain was why the proton is almost two thousand times as heavy as the electron. That was a problem for the theory, he conceded, a 'serious deficiency'.\n\nThe provenance of the hole theory is not entirely clear. The mathematician Hermann Weyl and others suggested that protons were related in some way to the negative-energy electrons, but their thinking was too woolly for Dirac. He later remarked that 'it was not really so hard to get this idea [of the hole theory]' as he was simply  drawing an analogy with the theory of how atoms emit X-rays (high-energy light). This theory says that an electron close to the nucleus can be knocked out of the atom, leaving a gap into which another electron falls, accompanied by the emission of an X-ray. It is also possible that Dirac had acquired the germ of his idea when he was sailing down the river Volga fifteen months before. At the Russian Congress, he met the Soviet theoretician Yakov Frenkel: someone snapped a photograph of them lying on the deck of the steamer, in their dress suits. In 1926, Frenkel had produced a theory of crystals in which 'empty spaces' in the regular lattice structure of the crystal would behave like particles - again, precisely analogous to Dirac's hole theory. Frenkel may have mentioned this theory to Dirac only for him to forget it and retrieve it later from his subconscious. But Dirac had no such recollection.\n\nWhatever the origins of the theory, there is no doubting the boldness of Dirac's application of the idea. Nowhere in the paper does he pause to comment on the theory's credibility. The crucial point for him was that he now had the beginnings of a viable theory of matter, based on an appealing equation and solid principles. Who was going to accept that the universe was full of unseen negative-energy electrons, an infinite sea of negative electrical charge? Yet his short paper 'A Theory of Electrons and Protons' bears no sign that he was expecting his idea to be greeted with incredulity. He wrote the article in his uncluttered style but with fewer equations than usual, free of the windiness that would have been excusable in the first presentation of a theory that suggested a new way of looking at the material universe.\n\nAlthough Dirac never admitted to being nervous about the reception of his hole theory, he often talked of anxiety as the handmaiden of scientific daring. So it is likely that he feared his theory contained a humiliating fallacy, a concern stoked by a letter he received in late November from Bohr, who had heard about the hole idea on the grapevine. For Bohr, the existence of negative energy levels in Dirac's theory of the electron undermined confidence in the entire concept of energy, a problem that - Bohr observed - also occurred in explanations of why some types of atomic nucleus can sometimes spontaneously eject a high-energy electron, a process known as radioactive beta decay. It seemed that energy was not conserved in this process - there was less energy before the decay than there was afterwards - so  energy appeared to emerge out of nowhere. This was serious: Bohr was questioning quantum mechanics and even the law of conservation of energy. Dirac thought his mentor was overreacting and, in a roundabout way, recommended him to calm down. Dirac had already told Bohr that he believed that the law of conservation of energy had to be preserved at all costs and that, to keep it, he would be prepared to abandon the idea that matter consists of separate atoms and electrons. And Dirac thought it premature to be pessimistic about quantum mechanics, which had only just passed its fourth birthday:\n\nI am afraid I do not completely agree with your views. Although I believe that quantum mechanics has its limitations and will ultimately be replaced by something better (and this applies to all physical theories) I cannot see any reason for thinking that quantum mechanics has already reached the limit of its development. I think it will undergo a number of small changes, mainly with regard to its method of application, and by these means most of the difficulties now confronting the theory will be removed.\n\nDirac concluded by reiterating - almost word for word - his reasons for believing in his hole theory. Although his defence could be regarded as stubborn, he does make it clear that he expected his theory to be superseded; the task in hand was to develop the theory as far as it could be taken. Bohr's criticisms do not seem to have shaken him in the least - he would need this thick skin during the coming barrage of scepticism and derision.\n\nA week after he wrote to Bohr, Dirac gave his first public presentation of the hole theory to an audience in Paris, at the Henri Poincar\u00e9 Institute. He will not have taken much pleasure from giving the lecture, as he reluctantly agreed to give it in French, bringing back abhorrent memories of meals with his father. When he returned to Bristol for Christmas, he had no choice but to speak French again. After his absence for nine months, his family was desperate to see him and to show him their latest plaything - the 'Gramaphone' ( _sic_ ). But Dirac was, as always, downhearted even at the thought of returning to his enervating Bristol routine, his mother endlessly fussing over him, his father still intimidating him simply by his presence. Although Dirac appears to have told none of his physicist friends, he believed that his home life had stultified him as a child and was still grinding him down. He appears to have first shared the full extent of  his pain only a few years later with a friend who was not one of his academic associates. In a letter, he wrote, 'going to see my parents will change me very much, I am afraid, and makes me feel like a child again and unable to do anything for myself'. For now, like all his other emotions, his suffering was hidden.\n**Fourteen**\n\nO hear the sad petition we electrons make to you  \nTo free us from the dominion of the hated quantum view  \nFor we are all abandoned to its dread uncertainty.  \nExcept by you, our champion. O we pray you, set us free!  \nOnce in a pleasant order our smooth-flowing time was spent  \nAs the classical equations told us where to go, we went.  \nWe vibrated in the atom, and a beam of light was freed;  \nAnd we hadn't any structure - only mass and charge and speed.  \nWe know not if we're particles, or a jelly sort of phi,  \nOr waves, or if we're real at all, or where we are, or why,  \nTo protons - holes in ether - according to Dirac.\n\nANON.\n\nThose anonymous lines are from an ode to the electron, pinned to a noticeboard in the Cavendish Laboratory around 1930. Only the most hard-headed theorist could fail to sympathise with the poet's nostalgia. A decade before, atomic physics had been a matter of common sense: electrons were just tiny particles, and they behaved predictably, according to straightforward laws of nature - the same ones that described everything else in the universe. How quaint those ideas now seemed: the classical laws that had held sway for a quarter of a millennium were now, in the atomic domain, obsolete, as Dirac liked to point out, the idea Jonathan Swift explored in _Gulliver's Travels_ \\- that no one would notice if naturally occurring things expanded or contracted in the same proportion - was wrong. The laws of the everyday world cannot be scaled down to the atomic domain: things are different there. Theorists could now reject every attempt to picture the electron as meaningless and therefore fraudulent. The particle did not even behave predictably: physicists were calculating odds like croupiers at nature's gambling table, using waves that no one believed were real. To cap it all, Dirac had the temerity to argue that common-or-garden electrons, with positive energy, are outnumbered by negative-energy ones that cannot even be observed.\n\nIt was probably a Cavendish experimenter, one of many who were suspicious of hole theory, who wrote the anonymous poem. Only a  few theoreticians, including Tamm and Oppenheimer, took the theory seriously, and even they soon found it wanting. In February 1930, Oppenheimer showed that the average lifetime of an atom was about a billionth of a second according to Dirac's hole theory, because the atomic electron would quickly fall to its death in the negative-energy sea. Soon afterwards, Tamm and Dirac independently arrived at the same conclusion. Pauli suggested what became known as his Second Principle: whenever a physicist proposes a new theory, he should apply it to the atoms in his own body. Dirac would be the first victim.\n\nPauli's jest appealed to Gamow, who was staying in Cambridge in the first academic term of 1930, mainly to work with Rutherford and his colleagues. Dirac was charmed by Gamow's non-stop good humour and sense of fun: no one did more to show Dirac what he had missed in his youth. Gamow taught Dirac how to ride a motorcycle (and filmed him doing it), gave him a taste for Conan Doyle's detective novels and apparently introduced him to the high jinks of Mickey Mouse, who first appeared on the screen two years before, in _Steamboat Willy._ 4 Dirac adored Mickey Mouse films, the animated successors of the cartoons he had seen as a boy in the penny weeklies. A few years later, he made a point of attending a day-long festival of the films in Boston, though it seems that he kept this innocent pleasure secret from his highbrow Cambridge colleagues. He was self-aware enough to know that his standing in the St John's common room would not be increased if he were too enthusiastic in his praise of Peg-Leg Pete or Horace Horsecollar.\n\nMore respectable at High Table was Dirac's appetite for mathematical games and puzzles that served no purpose at all beyond entertainment. Once, he gave a devastating performance in a game that had been introduced at G\u00f6ttingen in 1929. The challenge was to express any whole number using the number 2 precisely four times, and using only well-known mathematical symbols. The first few numbers are easy:\n\nSoon, the game becomes much more difficult, even for G\u00f6ttingen's finest mathematical minds. They spent hundreds of hours playing the  game with ever-higher numbers - until Dirac found a simple and general formula enabling _any_ number to be expressed using four 2s, entirely within the rules. He had rendered the game pointless.\n\nOn 20 February 1930, Dirac sent his parents the usual newsless weekly postcard, consisting of a ten-word summary of the Cambridge weather. The day after his mother received it, she visited the library and was astonished to read in a newspaper that her son had been elected a Fellow of the Royal Society, one of the highest honours in British science. Excited and flushed with pride, she dashed out to the post office and sent him a congratulatory telegram, keeping in check her annoyance that he had not mentioned the news on the card. Dirac was a 'naughty boy', she told him two days later in a letter, enquiring whether the society was organising a ceremony of induction. 'Do tell me,' she wrote, stressing each word in frustration.\n\nDirac could now put the initials FRS after his name, letters that render all other academic qualifications redundant. The Society, which then had 447 Fellows, usually gave the honour to scientists in their forties and fifties, after they had been nominated and passed over several times, so it was extraordinary for Dirac to be appointed the first time he had been put up for election, when he was only twenty-seven. As the news spread among the High Tables and common rooms of Cambridge, it would not have escaped the dons' notice that he had been elected a Fellow at a younger age than any of his senior colleagues.\n\nThe announcement appears to have made Dirac's parents realise how rapidly the reputation of their son had risen. 'How hard you must have worked to get to the top of the tree like that,' his mother wrote. 'No wonder you didn't take any interest in the Boat Racing.' The news was a welcome fillip for Flo, whose morale was low. Now that her husband was about to retire, her prospects were pitiable: only fifty-two years old, all she had to look forward to were years cooped up at home with a sick man whom she regarded as a browbeating ingrate and who, she knew, saw her as an inadequate nurse and servant. At school, Charles Dirac's colleagues queued up to offer their congratulations, and he received several letters to congratulate him on raising such a successful son. Paul's engineering teacher Andrew Robertson pointed out that he believed Dirac was the first  Bristol graduate to have been elected an FRS; Ronald Hass\u00e9, who first steered Dirac towards a career in theoretical physics, wrote to say how much he was looking forward to Dirac's first public speech in Bristol in September. The city was to host the annual meeting of the British Association for the Advancement of Science, where scientists and members of the public got together to hear a week of lectures on the latest science. At the Cotham Road School - formerly the Merchant Venturers' School - they celebrated by taking a day off. Charles never quite knew when to expect the next plaudit: once, during a lesson, two complete strangers knocked on his classroom door, entered, complimented him on his son's great achievement, and left.\n\nPerhaps to celebrate his latest success, Dirac took his mother's advice and splashed out almost \u00a3200 on his first car, a Morris Oxford Tourer, capable of a then-impressive 50 mph. There was no driving test: after completing the sale, the garage owner gave him a short demonstration drive around Cambridge and then handed him the keys. He was then free to take his chances on the roads. With the scrapping of the 20 mph speed limit that year, the highways became even more dangerous, not least because of Dirac's presence. A colleague laughed that 'Dirac's car has two gears, reverse and top.' Only Mott left an account of being driven by Dirac, to London on an icy March day when 'Dirac ran - very gently - into the back of a lorry and smashed a headlamp.' Like Kapitza, Dirac was a wild driver, and this appears to have been due both to his poor handling of the vehicle - his appreciation of machines always exceeded his competence at using them - and to the virtual absence of a highway code. Dirac was a stickler for obeying rules that he believed were rational and obviously for the common good, so, in the absence of regulations, he was free to drive as he wished.\n\nDirac was, at last, showing signs of mellowing. Leisure was not reserved only for Sundays: at lunchtimes, the bulk of his day's work done, he would often motor out of Cambridge to the Gog Magog Hills, park his car near a tall tree and climb it, still wearing his three-piece suit. He wore it whatever the weather, whatever the occasion, and took it off only during his drives out to secluded sites by the river Cam and in the fens north-east of the city, where he bathed, as Lord Byron had done 125 years before. Later, when he returned to college or to his desk, he would do only the lightest of tasks. He was taking  a leaf out of the book of G. H. Hardy, who believed that the longest a mathematician can profitably spend doing serious work is four hours.\n\nOf all the months in the Cambridge academic calendar, June was the most relaxed. The examinations over, it was time for the students to leave the university, but only after the catharsis of the summer ball. The intoxicating mix of music and dancing, free-flowing champagne, gorgeous frocks and sharply cut dinner suits could cheer up the most abject examinee. Dons could put on their summer suits and wind down to the 'long vac', when they had no administrative duties and were free to spend the long, languid afternoons doing nothing except sit in a deckchair and watch a game of cricket. Dirac was nonplussed by the appeal of an activity that involved twenty-two men spending hours - sometimes days - playing a game that often ended in a draw, which devoted spectators would often deem exciting. The game had no more ardent admirer than G. H. Hardy, for whom it was akin to pure mathematics: all the more beautiful for its lack of useful purpose. A few years later, he gave pride of place in his study to a photograph of the Australian batsman Donald Bradman, one of Hardy's three greatest heroes (the others were Einstein and Lenin). Hardy was probably looking forward to Bradman's first Test appearances on English soil, but the prospect will have left Dirac unmoved; he was busy preparing to spend the summer climbing and hillwalking with friends. He needed a break and some fresh inspiration if he was to sort out the problems with his hole theory and so answer his critics, including the mocking Pauli and the privately scornful Bohr. Several of Dirac's colleagues would be lining up to attend his public lecture at the Bristol meeting at the end of the summer, he knew, to see if he had cracked the problem of negative-energy electrons.\n\nPreparing for his second trip to the Soviet Union, Dirac read in the British press that Stalin was tightening his grip, forcing through his programme of collective farming, squeezing the peasants in order to pay for a crash programme of industrialisation and persecuting political opponents and religious minorities. Some newspapers were in no doubt of Stalin's malevolence - the _Daily Telegraph_ wrote regularly of his 'Reign of Blood' and his 'war on religion' - but others, including the _Manchester Guardian_ , gave him the benefit of the doubt. The _New Statesman_ \\- the house journal of leftist intellectuals in Britain and favourite reading of Kapitza's in the Trinity common room - insisted that Stalin should be given a fair hearing. Dirac agreed: one of the few things that would draw him into conversation were comments that he perceived to be unfairly hostile to the Soviet Union. Rudolf Peierls later recalled: 'At a time when everything Russian was anathema, he questioned why each particular item was wrong, and this often caused raised eyebrows.' Wanting to see life there for himself, he again ignored the fears of his mother: 'I do hope it is safe in Russia. One hears dreadful stories about it.'\n\nDuring his trip, Dirac felt the arm of the Soviet military on his shoulder: en route to Kharkov, when he attempted to cross the Soviet border at a place not mentioned in the visa that Tamm had obtained for him, border guards held him at the crossing point for three days before releasing him. By early July, he had heard that Soviet law forbade foreigners who stayed in the country for more than a month to take out either Soviet money or foreign currency. So he left the USSR in late July, within a month of his arrival, having cancelled his plans to hike in the Caucasus. His vacation foreshortened, he soon returned to England, to what most scientists would regard as the media highlight of their life.\n\nIn September, Hardy was praising Bradman's devastating performances in the Ashes, and Bristol was preparing to host the British Association meeting. Almost three thousand delegates - including George Bernard Shaw - attended, each of them having paid a pound for the privilege. Jim Crowther told readers of the _Manchester Guardian_ that the public delegates were young and dressed informally, many of the women in sleeveless and flowered voile frocks, the men in alpaca jackets and grey flannels. The ticket price had not changed since the meetings began almost a century before, when the Association's leaders were choosing the most appropriate word to describe the participants. They considered 'savants', 'nature peepers' and 'nature pokers', but finally settled on 'scientists', coined in 1834 by William Whewell, one of John Stuart Mill's philosophical adversaries. Though many hated the new word - Michael Faraday disliked it almost as much as the triply sibilant 'physicist' - it had caught on by the time Dirac was in junior school.\n\nThe organisers, probably fearing that Dirac would give a technical talk of limited public appeal, scheduled him to speak in a modest room in one of the university's new physics laboratories, funded by the tobacco manufacturer H. H. Wills. At 11 a.m. on Monday, 8 September, Dirac stood up without fanfare to address a crowded room on the subject of 'The Proton'. Never confident when he spoke at public meetings, he may have been particularly apprehensive at this one: this was the first time he had agreed to address a lay audience and the first time he had spoken to many of the teachers who had seen him flower. If Charles was there, as is likely, he will have had a full heart as he had not heard his son speak in public before: Paul Dirac would now have no choice but to talk about his science to his father.\n\nDirac entered into the spirit of the British Association. Speaking with his usual directness, in lilting Bristol tones, he talked about his research in a way that might almost have passed as colloquial, though with none of Eddington's flair. To ensure that he was intelligible to people with no science training, he began with the statement that 'matter is made from atoms', and quickly went up the gears, ending with his idea that the proton is a hole in the negative-energy sea of electrons. This implied, he pointed out, that there is only one fundamental particle, the electron, adding that such an economy in nature was 'the dream of philosophers'. For many in his audience, this will have been an exciting revelation, but not for Gamow and Landau, who were at the back of the room, sitting on wooden benches. The two of them had roared down to Bristol on Gamow's motorbike, Landau perched behind him on the luggage carrier. They travelled to the meeting, partly as Bohr's unofficial emissaries, specifically to see if Dirac had anything new to say about his theory. During the talk, Gamow and Landau craned their necks to see the speaker, hanging on his every word, Landau, as usual, unable to resist making snide asides.  After twenty minutes of reiterating arguments he had already published, often using the same words as he had used in his papers, Dirac drew to a close, and they realised that he had said nothing new. Their trip to Bristol had been a wild goose chase.\n\nDirac's theory of negative-energy electrons nevertheless captured the imagination of journalists, and the British newspaper reports gave him more publicity than he had ever known. After his presentation, the representative from the American Science News Service wired Washington: 'This new theory may prove to be as important and interesting to the public as Einstein's theories have been.' The _New York Times_ picked up the story and reported that Dirac's 'acclaimed' theory 'upset all present conceptions of space and matter', adding that 'These physical scientists have a more exciting life than Columbus.' But Dirac's peers were unimpressed. On the way back to Cambridge, Landau and Gamow stopped at a post office. Landau sent Bohr a telegram consisting of a single word: 'Crap'.\n\nThe telegram reached Bohr soon after he received from Dirac a copy of his textbook, _The Principles of Quantum Mechanics._ Even if the author's name were not on the cover, his identity would have been obvious to Bohr from a quick flick through: the unadorned presentation, the logical construction of the subject from first principles and the complete absence of historical perspective, philosophical niceties and illustrative calculations. This was the vision of a mathematically minded physicist, not an engineer. Dirac's peers marvelled at its elegance and at the deceptively plain language, which somehow seemed to reveal new insights on each reading, like a great poem. Many of the students - especially the less able ones - were bemused, dissatisfied and sometimes even dispirited. The book had been written with no regard for his readers' intellectual shortcomings, without the slightest sign of emotion, with not a single leavening metaphor or simile. For Dirac, the quantum world was not like anything else people experience, so it would have been misleading to include comparisons with everyday behaviour. He scarcely mentioned empirical observations except at the beginning, where he described an experiment that demonstrates the failure of classical theory to account for matter on the atomic scale and, hence, motivates the need for quantum mechanics. In its 357 pages, _The Principles of Quantum Mechanics_ featured neither a single diagram, nor an index, nor a list of references, nor suggestions for further reading. This was, above all, a personal view of quantum mechanics, which is why Dirac - usually someone who abjured personal pronouns - always referred to it as 'my book'.\n\nPhysicists immediately hailed it a classic. _Nature_ published a rhapsodic review by an anonymous reviewer who - to judge by the eloquence and sharp turn of phrase - may well have been Eddington.  The author made it clear that this was no ordinary account of quantum mechanics:\n\n[Dirac] bids us throw aside preconceived ideas regarding the nature of phenomena and admit the existence of a substratum of which it is impossible to form a picture. We may describe this as the application of 'pure thought' to physics, and it is this which makes Dirac's method more profound than that of other writers.\n\nThe book eclipsed all the other texts on quantum mechanics written at about the same time - one by Born, another by Jordan - and became the canonical text on the subject in the 1930s. Pauli warmly praised it as a triumph and, although he worried that its abstractions rendered the theory too distant from experiment, described the book as 'an indispensable standard work'. Einstein was another admirer, writing that the book was 'the most logically perfect presentation of quantum theory'. _The Principles of Quantum Mechanics_ later became Einstein's constant companion: he often took it on vacation for leisure reading and, when he came across a difficult quantum problem, would mutter to himself, 'Where's my Dirac?'\n\nBut some of Dirac's undergraduate students were not pleased to find that the book was largely a transcript of his lectures: why, these students wondered, was it worth bothering to go and listen to him? Yet others found the course uniquely compelling. He would enter the lecture theatre punctually and in full academic garb, wearing the traditional uniform of gown and mortarboard. Otherwise, there was nothing else theatrical about him. He would clear his throat, wait for silence, then begin. For most of the lecture, he would stand still and erect, enunciating each word, addressing what one of his students described as his 'personal unseen world'. At the blackboard, he was an artist, writing calmly and clearly, beginning at the top left-hand corner, then methodically working downwards, writing every letter and symbol so that someone at the back of the room could see it clearly. The audience was usually quiescent. If a student asked a question, he would dispatch it with the economy of a great batsman and then move on, as if nothing had disturbed his flow. After precisely fifty-five minutes he would draw his presentation to a close and then, unceremoniously, gather his papers together and walk out.\n\nOne of the new students who were impressed by Dirac's course in the autumn of 1930 was Subrahmanyan Chandrasekhar, later a leading astrophysicist but then a wide-eyed student just arrived from Bombay. For him, the course was 'just like a piece of music you want to hear over and over again'. During his time in Cambridge, he attended the entire course four times.\n\nDirac probably knew he had disappointed his colleagues at the British Association meeting by failing to say anything new. He was about to go to his second Solvay Conference, aware that few of the physicists took seriously his unified theory of electrons and protons; his proposal that protons were holes in the negative-energy sea was beginning to look not just implausible but untenable. One of the blows he suffered came shortly after the Bristol meeting when Tamm wrote to tell him that Pauli had proved that the holes have the same mass as the electron. Experimenters had not detected such a particle, which is probably why Tamm added a sympathetic comment: 'I would be very much pleased to hear that Pauli is wrong.'\n\nThis Solvay meeting was later remembered for being the one where leadership of the community of theoreticians passed from Einstein to Bohr. Einstein was looking out of touch, downcast after Bohr had bested him in one of their tussles about quantum mechanics and its meaning. For Einstein, the theory was fundamentally unsatisfactory as it did not even claim to describe physical reality, only the probabilities for the appearance of a particular physical reality on which an observing experimenter's attention is fixed. Such a theory may be good at explaining experimental results, but it is certainly not complete, Einstein argued. Disillusioned, and uninterested in much of what his colleagues had to say, he consoled himself by playing after-dinner violin duets with the Queen of Belgium, one of his new friends.\n\nUnlike the previous Solvay Conference in 1927, the atmosphere at this one was heavy with forebodings about the world outside physics, where the recession was ravaging most industrialised nations and providing fertile ground for political extremists. A month before the conference, Hitler's National Socialists had taken second place in Germany's election, followed by the Communists. G\u00f6ttingen was now bedecked with Nazi flags, many of its shops displaying trinkets decorated with swastikas. Einstein was sick of the anti-Semitism in Berlin and despised Germany's emerging leader: 'If  the stomach of Germany was not empty, Hitler would not be where he is.'\n\nAs Dirac kept his politics almost exclusively to himself, most of his Cambridge colleagues mistakenly believed he had no interests at all, that he was as one-dimensional as the lines in his projective geometry. He was privately alarmed by the rise of Hitler and broadly supportive of Stalin's project in the USSR, especially its commitment to mass literacy and education. Aware of Dirac's interest, Tamm wrote to him about the radical experiment in 'brigade education', in which students studied intensively, alone and in groups, with no lectures, but with a professor on standby for consultation:\n\nI never thought it possible for a large body of students to work as hard as our students do now. Our [brigades, each of five students, work and study together] 9 days out of 10 [. . .] from 9am to 9pm with a 2-hour interruption for a meal (research work included, which is of course conducted individually by each student). Yesterday, speaking with a brigade, I found them troubled by the fact that they have 'lost without cause' six out of 270 working hours of the last month\n\nAlthough Dirac was interested in the Soviet experiment, it was of only marginal interest to him compared with theoretical physics. By late autumn, he had every reason to be dissatisfied with his progress as his hole theory was in deep trouble. Oppenheimer and Weyl had independently come to the same conclusion as Pauli - that Dirac had no theoretical justification for believing that his holes were protons. The implication was that the theory was incorrect; something was amiss with the Dirac equation. But he was convinced that it was correct - what was needed was the correct interpretation of its mathematics. The American theoretician Edwin Kemble later put his finger on the kind of faith Dirac had in his equation: '[He] has always seemed to me a good deal of a mystic [. . .] he thinks every formula has a meaning if properly understood.'\n\nTowards the end of term, Dirac went through his annual chore of refusing most invitations to Christmas parties, though he did occasionally attend the annual dinner of the Cavendish Physical Society, a boisterous evening of eating, drinking and singing. After Kapitza attended the dinner for the first time in December 1921, he wrote incredulously to his mother, observing how quickly even a moderate amount of alcohol freed the inhibitions of his English colleagues and  made their faces 'lose their stiffness and become lively and animated'.  By the end of the meal, after the cheeseboard and port had been passed round, the air was thick with cigar smoke and everyone was shouting to be heard above the din. The ritual was not yet over: the next stage was a series of facetious toasts (one had been 'To the electron: may it never be of any use to anybody') alternating with off-key renditions of popular tunes such as 'I Love a Lassie', their lyrics rewritten as a jokey commentary on the past year at the laboratory.  At the climax, the portly Rutherford, Thomson and everyone else stood on their chairs, linked together with arms crossed and belted out 'Auld Lang Syne' and then, finally, the National Anthem 'God Save the King'. After the bacchanalia ended, usually well after midnight, it was up to those left standing to take their drunken colleagues to their homes.\n\nIn 1930, Dirac did not attend the dinner but will probably have heard later that Kapitza was the focus of attention that night. Rutherford, then President of the Royal Society, had secured a professorship for his favourite colleague and funding for the construction of a new building to accommodate him and his laboratories. At the end of the seven-course dinner, while the sixty guests were chewing their mince pies, Darwin reminded them of the experience of entering Kapitza's laboratory: 'you had to ring to be admitted by a \"flunkey\" and became confronted not with men working in their shirt sleeves, but with Prof Kapitza seated at a table, like the arch criminal in a detective story, only having to press a button to do a gigantic experiment'.\n\nThe laughter at this image of Kapitza, apparently a forerunner of a James Bond villain, will have been hearty, and it is safe to guess that knowing glances will have passed among his colleagues, many of them envious of his relationship with their laboratory's director. Blackett was not there. Rutherford had no time for petty jealousy but was not above making a thinly disguised attack on his recently retired colleague Sir James Jeans, whose _The Mysterious Universe_ had been a best-seller since it first appeared in the bookstores the month before. Rutherford was as down to earth and, at the same time, as snobbish as anyone in science. As the recorder of the dinner wrote: Sir Ernest Rutherford 'deplored the writing of popular books by men who had been serious scientists, to satisfy the craving for the mysterious exhibited by the public'. This was a common opinion  in Cambridge. A few months later, his idoliser C. P. Snow - a scientist about to become a writer - sneered at science popularisers for doing a job that was just too easy: 'there is no argument and no appeal, just worshipper and worshipped'. The result was, Snow declared, a 'great evil'. Within three years, Snow published his semi-autobiographical novel _The Search_ , the first fiction to bring to a wide audience the atmosphere of Rutherford's laboratory, and to feature Paul Dirac.\n\nA week after Christmas, Rutherford was ennobled at the end of his five-year stint as President of the Royal Society. But the pleasure the honour gave him was eclipsed by a family tragedy: his daughter and only child, Fowler's wife, died in childbirth two days before Christmas. Lord Rutherford, grieving as he approached his sixtieth birthday, must have thought his years of glory were over. He was not doing much research of his own, so his remaining hopes of being involved in more of the ground-breaking discoveries that he longed for were in the hands of his 'boys'.\n\nDirac showed none of the confidence that might be expected of a young man at the top of his game. Chandrasekhar wrote home to his father that he was disappointed that Dirac did not show a bit more swagger: '[Dirac is a] lean, meek shy young \"Fellow\" (FRS) who goes slyly along the streets. He walks quite close to the walls (like a thief!), and is not at all healthy. A contrast to Mr Fowler [. . .] Dirac is pale, thin, and looks terribly overworked.'\n\nWork was not Dirac's only concern. Having read his mother's letters, he may have sensed that his parents' relationship, tense and unstable, was fast approaching a flashpoint. Charles Dirac, dreading retirement, was pleading with the Bristol education authorities to be allowed to stay on in his job, but they were resisting. Betty, now with a car of her own, was doing little except chauffeur him three times a day to and from Cotham Road School. Dirac was watching his sister become another of his father's servants.\n\nMeanwhile, Flo knew that, in only a few months, she would be spending most of her life at home alone with her husband: 'It simply won't bear thinking about.'\n**Fifteen**\n\nRussian politics like opium seems infallibly to provoke the most fantastic dreams and imaginings on the part of the people who study them.\n\nE. A. WALKER, British Embassy, Moscow, 1931\n\nIn Cambridge, during the spring of 1931, Dirac happened upon a rich new seam of ideas that would crystallise into one of his most famous contributions to science. In the thick of this project, he received a letter from his mother, beginning:\n\n27 April 1931\n\nMy dear Paul\n\nPa and I had quite a row yesterday all about some wine upset on some cheap stamps. He got in the most awful rage for a few minutes & then said he had had enough of me & should _go_ if I did anything more to upset him.\n\nI apologised most humbly as usual but on thinking it over, I am pretty certain he meant it.\n\nIn three pages of brief, matter-of-fact sentences, she described to Dirac - apparently for the first time - the charade of her marriage. She told him of a young woman who had visited the family when he was a baby, stayed to supper and had been escorted home by Charles to Bedminster. Flo had written to her that she 'wouldn't have it any more and thought it was all finished'. But she was deluding herself, as she realised when she visited Charles's Esperanto exhibition at Bishop Road School and saw that the woman who was presenting it with him, wearing a huge pair of tortoise-shell glasses, was the young woman who had visited them decades before. 'Fancy if they have kept up the acquaintance for 29 years,' Flo wrote. By this account, his father had been cheating on the woman who had spent most of her life looking after him. Her conclusion was: 'She has nothing to do but humour him, I have to keep the house clean, dress him, bath him & worst of all find something to feed him on.'\n\nAs usual, Dirac appears to have said nothing of this to anyone, even to his close friends. In the early months of 1931, a quiet time for  his fellow theoreticians, he was working on the most promising new theory he had conceived for years. The theory broke new ground in magnetism. For centuries, it had been a commonplace of science that magnetic poles come only in pairs, labelled north and south: if one pole is spotted, then the opposite one will be close by. Dirac had found that quantum theory is compatible with the existence of _single_ magnetic poles. During a talk at the Kapitza Club, he dubbed them magnons, but the name never caught on in this context; the particles became known as magnetic monopoles.\n\nThe idea arose accidentally, he later said, when he was playing with equations, seeking to understand not magnetism but electrical charge. The American experimenter Robert Millikan had demonstrated that this charge exists only in discrete amounts, each of them exactly equal to a whole number multiplied by the size of the electron's charge, usually denoted by _e._ So the electrical charge of a piece of matter can be, for example, five times the charge of the electron (5 _e_ ) or minus six times its charge (-6 _e_ ), but _never_ two and a half times its charge (2.5 _e_ ). The question Dirac wanted to answer was: _why_ does electric charge come only in discrete amounts?\n\nAt first, Dirac worked in traditional ways, with quantum mechanics and Maxwell's equations of electromagnetism. Then, like a jazz musician working with two intertwining melodies, he began the riff that led to the monopole. Dirac pictured the magnetic lines of force that end on a quantum particle, much like the ones that terminate on the pole of a bar magnet, usually displayed by patterns of iron filings, each of them obediently aligned to the magnetic force acting on it. He asked: if quantum mechanics and Maxwell's equations of electromagnetism are assumed to be true, what can be said about the magnetic field associated with a quantum particle? To answer the question, he used an innovative combination of geometric thinking - picturing the possible waves in space and time - with powerful algebraic reasoning. He found a way of building on the existing structure of quantum theory, without changing any of its essential foundations and preserving all the rules that governed the interpretation of the theory. If quantum mechanics can be likened to a house of playing cards - with a fragile balance between its interconnected parts - then Dirac can be said to have added a few more cards, preserving the structure's balance, while extending its  range to include a new type of particle. The theory furnished a new connection between electricity and magnetism, an equation that relates the smallest-possible electrical charge with the weakest-possible magnetic charge.\n\nThe equation enabled him to draw some startling conclusions. First, the strength of the magnetic field of a monopole is quantised - it can have only certain allowed values, whole-number multiples of the minimum quantity, whose value he could easily calculate. It turned out that two monopoles of opposite sign are hard to separate: the force pulling them together is almost five thousand times the force that attracts an electron to a proton. This, Dirac suggested, might be why magnetic poles of opposite sign have never been separated and therefore appear in pairs.\n\nHis second conclusion was still more striking: the observation of just one monopole anywhere in the universe would explain why _electrical_ charge is quantised - the very thing Dirac had set out to understand. Having checked his final calculations and having found no errors, he came to a bold conclusion: if an experimenter happens on a single monopole anywhere in the universe, the new theory can explain why nature had chosen to apportion electric charge _only_ in discrete amounts.\n\nDirac's theory did not guarantee the existence of monopoles but did show that quantum mechanics can describe such particles _if_ they occur in nature. Centuries earlier, other scientists had speculated that monopoles might exist, but those ideas were just hunches, with no logical underpinning. Dirac was the first to give clear reasons _why_ such particles might be observed. He may well have thought that the idea was too beautiful to be wrong, but he followed the convention of presenting his conclusion as an understatement: 'one would be surprised if Nature made no use of it'. And he chose not to go the whole hog by trumpeting the magnetic monopole as a prediction of his theory. Like all physicists at that time, he accepted that experimenters had found the need for only two fundamental particles - the electron and the proton - and that it was not the job of theorists to complicate matters by proposing new ones. Ironically, the first physicist to buck the trend was an experimenter, Rutherford, when he proposed in 1920 that most atomic nuclei contain a hitherto undetected particle, roughly as heavy as the proton. He called the new particle 'the neutron'.\n\nYet, in his paper on the monopole, Dirac implied for the first time that he no longer believed there are only two fundamental particles. In the introduction, he declared that he had suggested that a proton is a hole in the negative-energy sea of electrons: Oppenheimer and Weyl had convinced him that the hole must have the same mass as the electron (he did not mention Pauli, who had also come to the same conclusion). So Dirac followed the logic of Sherlock Holmes: 'When you have eliminated all which is impossible, then whatever remains, however improbable, must be the truth.' The conclusion was that each hole corresponded to a new, hitherto undetected type of particle with exactly the same mass as the electron:\n\nA hole, if there were one, would be a new kind of particle, unknown to experimental physics, having the same mass and opposite charge to an electron. We may call such a particle an anti-electron. We should not expect to find any of them in nature, on account of their rapid rate of recombination with electrons, but if they could be produced experimentally in high vacuum they would be quite stable and amenable to observation.\n\nAgain, Dirac is surprisingly circumspect. Although he states the properties of his new particle and even names it, he seems less keen to stress the inevitability of its existence than the difficulty of detecting it. If Dirac had been confident, he would have included a plain-spoken sentence such as 'According to this version of hole theory, the anti-electron should be detectable,' but he held back. Paradoxically, he did underline a radically new interpretation of protons: they were nothing to do with electrons, he suggested, but have their own negative-energy states, 'an unoccupied one appearing as an antiproton'. Within twenty lines of prose, he had foreseen the existence of the anti-electron and the anti-proton.\n\nThough chary about predicting new particles, Dirac showed no timidity at all when he introduced what amounted to a new way of doing theoretical physics. In two paragraphs, consisting of 350 words and no equations, he argued that the best way to make progress was to seek ever-more-powerful mathematical foundations for fundamental theories, not to tinker with existing theories or look to experiment for inspiration. He envisaged the future of physical science as an unending series of revolutions, driven by mathematical imagination, not by opportunistic responses to the latest announcements from experimenters. This was tantamount to a new style of  scientific investigation: seeking laws of ever-greater generality - as Descartes, John Stuart Mill and others had recommended - but relying on mathematical inspiration to find them, rather than taking their cues mainly from observations.\n\nHe began by pointing out that before Einstein used non-Euclidean geometry as the basis of the general theory of relativity and before Heisenberg used non-commutative algebra in quantum mechanics, these branches of mathematics were 'considered to be purely fictions of the mind and pastimes for logical thinkers'. The solution to the hardest problems in fundamental physics, Dirac inferred, will 'presumably require a more drastic revision of our fundamental concepts than any that have gone before'. He set out his manifesto with the blazing confidence of a young scientist at the height of his powers:\n\nQuite likely these changes [to our fundamental concepts] will be so great that it will be beyond the power of human intelligence to get the necessary new ideas by direct attempts to formulate the experimental data in mathematical terms. The theoretical worker will therefore have to proceed in a more indirect way. The most powerful method of advance that can be suggested at present is to employ all the resources of pure mathematics in attempts to perfect and generalise the mathematical formalism that forms the existing basis of theoretical physics, and after each success in this direction, to try to interpret the new mathematical features in terms of physical entities . . .\n\nHis message was clear: theorists should concentrate much more on the mathematical foundations of their subject and much less on the latest bulletins from the laboratories - to abandon centuries of tradition. No wonder Dirac became known as 'the theorist's theorist'.\n\nEarly in May 1931, when Dirac was writing his paper, Tamm arrived in Cambridge to spend a few months in St John's College, having left his wife and children in Moscow. He had no trouble securing permission to work in the UK, as Dirac was officially a favoured scientist in the Soviet Union, having been elected a corresponding member of the USSR Academy of Sciences three months before.\n\nFor once, Dirac was willing to share his ideas and briefed Tamm on his magnetic monopole theory, suggesting that he use the new theory to calculate the energy values and quantum waves that describe an electron in the vicinity of a monopole. Apart from when  he was asleep, Tamm worked non-stop for three and a half days and finished just in time for Dirac to include his results - less exciting than Dirac had hoped - in the paper. In college, Tamm fraternised easily with the dons, including a few who had become friends with Dirac, having broken through his crust of reserve. Among them were the mathematician Max Newman and the Cavendish experimentalist John Cockcroft, both five years older than Dirac. The Yorkshire-born Cockcroft was a trained engineer and a natural manager, intensely focused to the point of near silence and with a flair for helping Kapitza and his other colleagues to solve technical problems. He was 'a sort of scientific dogsbody of genius', Crowther said.\n\nOnly four days after Tamm arrived, Dirac organised a breakfast in his room to talk about Russia with Tamm and the classicist Martin Charlesworth. Dirac's gyp will have delivered the food, probably plates of bacon, eggs and fried bread, served with a pot of tea, toast and marmalade. The three men talked for four and a half hours. Dirac wanted to learn about the Soviet economy, but he was uneasy when there was any sign that Tamm might present his Marxist views in public, as he showed when Tamm told him that he had been invited to speak on 'Higher Education in the Soviet Union' in London. Dirac remarked pointedly to him that he hoped the talk would be on education, not politics.\n\nFrom the tone of the letters he wrote to his wife in Moscow, Tamm was surprised that so many Cambridge dons were interested in the Soviet experiment. When he had lived in Britain eighteen years before, the university was known for its conservatism, but around the time he arrived there this time, the Marxist Bernal and his colleagues had established a nucleus of left-wing thought and activity among the academics. As Dirac will have heard, it was standard Marxist practice to praise the successes of the Soviet Union and not to dwell on its failures, but to draw attention to the millions of victims of unemployment and imperialist wars and the economic waste that could allegedly be prevented by a properly planned cooperation.  The comments Tamm makes in his letters give the impression that Dirac was then no more than an interested observer of the Marxist proselytisers; his passion was physics, though he was now more relaxed about taking time off to pursue other interests. After lunch, Dirac would often drive Tamm out into the countryside,  sometimes pausing by a roadside tree so that Tamm could teach Dirac the elements of rock climbing and help him to overcome his fear of heights; in return, Dirac taught Tamm to drive and even helped him pass the recently introduced driving test.\n\nIn late June, near the end of Tamm's visit, he and Dirac headed north to the more challenging terrain of Scotland, where they spent a week in the mountains of the Isle of Skye with the industrial chemist James Bell. An expert climber, he had been a friend of Tamm's since their student days in Edinburgh and was a close follower and sceptical supporter of the Soviet experiment, steering a moderate course between Soviet propaganda and the anti-Soviet articles in the British press. Skye provided just the kind of scenery and company Dirac loved, and his vacation gave him an excuse to delay his return to Bristol.\n\nThat year, the summer days of Cambridge did not have their usual languor. They were rudely interrupted by a political frisson whose unlikely source was the Science Museum in London, the location of the second International Congress on the History of Science and Technology. For a few days in early July 1931, a red flag flew over South Kensington. Such gatherings usually attracted no attention, but this one was special: it was attended by a high-powered Soviet delegation that included Nikolai Bukharin - formerly one of Lenin's closest associates, now a colleague of Stalin's - and by several leaders of the Soviet scientific community, notably Boris Hessen. A few weeks before, Stalin had announced the end to almost eighteen months of political warfare between the Soviet state and its intelligentsia, so this conference offered an opportunity to present the Soviet outlook on science and technology in a favourable light. Bukharin had been the darling of the Bolshevik Party but had been pilloried in 1929 when he opposed forced collectivisation of farming and the crash industrialisation of the economy. A year later, he was sacked as the editor of _Pravda_ , but remained loyal to Stalin and gave a full-throated presentation of the Marxist view of science to his audience in the museum. Bukharin stressed the historical context of science and the influence of social and economic conditions on scientific development, dismissing the traditional emphasis on the achievements of outstanding individuals, such as Newton and Darwin. The Soviets knew the right way forward, Bukharin concluded \\- by developing science as part of a unified plan for the whole of society:\n\nThe building of science in the U.S.S.R. is proceeding as the conscious construction of the scientific 'superstructures': the plan of scientific works is determined in the first instance by the technical and economic plan, the perspectives of technical and economic development. But this means that thereby we are arriving _not only at a synthesis of science, but at a social synthesis of science and practice._ 18\n\nAt the end of Bukharin's lecture, there was silence, followed by coughs and shufflings. But the talk was a success: it was reported in several British newspapers and magazines and made an indelible impression on many of the delegates. Desmond Bernal called the gathering 'the most important meeting of ideas [. . .] since the [Bolshevik] Revolution'. Dirac did not participate in the meeting but will have heard about it from Tamm, who accompanied the Soviet party to visit Marx's grave in Highgate Cemetery, and from Kapitza, who organised a lunch in their honour at Trinity College.\n\nThat MI5 was carefully monitoring Bukharin's activities during his visit to Britain would not have surprised Kapitza, but he would surely have been taken aback if he had known that, since January, Special Branch had been opening, checking and sometimes copying mail sent to him from Moscow and Berlin. Armed with folders bulging with vaguely incriminating reports - all of them scientifically inaccurate, sometimes to the point of illiteracy - MI5 were concerned that he had access to sensitive military information and suspected 'that he may be sending [it] abroad'. The search revealed nothing and the government warrant to intercept his mail was suspended on 3 June. But MI5 kept its tabs on him.\n\nDirac was shortly to travel to the United States for another hiking vacation and a sabbatical term in Princeton, but he was duty-bound to visit Bristol first. He disliked confrontations, so he must have been steeling himself in late July as he prepared to spend a week in 6 Julius Road. Everyone was even more unhappy than they had been when he had last seen them, as Dirac knew from his mother's letters. Betty, unable to afford to run her car, sold it for a knockdown price. Charles, bitter that he was being forced to retire, consoled himself by spending the evenings with his friends Mr and Mrs Fisher at their  bungalow in Portishead. Flo, suspicious that Mrs Fisher was one of his mistresses, was hoping he would leave to set up home with her or his girlfriend in the Esperanto group: 'I can't help it anyhow, he is tired of me and likes someone younger.'\n\nDirac thought his family home was a disgrace - it was in a state of seedy disrepair, as his father refused to have maintenance work done and his mother disliked housework more every year. According to Flo, the atmosphere inside was toxic, thick with resentment. She despised Charles, and it would not be surprising if he were upset that she had exploited their marital problems by thickening the wedge between him and his son. It would have been out of character for Dirac to do anything other than to keep his head down and to depart after putting in a token appearance. He did just that, driving back to Cambridge after a few days to give a talk. But he could not escape quite so easily: on the day before the seminar, another harrowing letter from his mother arrived:\n\n19 July 1931\n\nMy dear Paul,\n\nI don't know if this will surprise you but your father & I are going to part (as his own father & mother did.)\n\nIt is his own idea; he says he has hated me for 30 years. I know I could never please him but didn't know it was quite so bad as that.\n\nHe will give me \u00a31 a week or more (it will have to be more) & I am to clear out.\n\nI don't mind, if I have never pleased him. I sent one of his lady friends away when you were just born because she came in every night & he took her home to Bedminster & returned nearly 12 P.M. She has kept in with him ever since & he says he wishes he had married her. She is a nurse now & I suppose will come & look after him.\n\nOtherwise, he sits in the waiting room at Zetland Road with Mrs Fisher from Portishead & she comes up here pretty often, or he is always out. Betty says she will stay with him as they are both after his money.\n\nI am going to see a lawyer Fred [my brother] knows, to-morrow morning & will get it settled before he leaves school on Friday or he may clear out.\n\nDo you know of a tiny cottage or bungalow near the sea up your way? It would be a complete change & I love the sea. I expect Louie or Nell would come along occasionally & I should not meet anyone I know.\n\nIf you could find me a tiny place anywhere I should be so grateful. I wouldn't interfere with you in the least but you could come & see me in your car whenever you had time.\n\nWe are not having any row about it - it is not dignified so you need not stay away if you care to come along earlier. I'll post this while they are at Church.\n\nWith love from Mother\n\nDirac could now understand a scene that had haunted him since he was a child: his parents bawling at each other in the kitchen while he, Felix and Betty were locked outside in the garden. The phrase 'he has hated me for 30 years' probably struck home in Dirac's mind, constantly in search of numbers to process: as he was only twenty-nine years old, she had, in effect, told him that he had not been conceived in a loving relationship, let alone raised in one.\n\nFlo did not wait for her son's advice. She went straight to her lawyer, who advised that Charles could not legally throw her out unless she was with another man, otherwise he would lose his pension. As soon as she was alone, she wrote to Dirac: '[Charles and I] don't speak, but never did much, but I guess it better to stick to Betty. Two of us ought to manage him.'\n\nTen days after he received his mother's most recent letter, on 31 July 1931, Dirac sailed from Liverpool to North America, then in the tightening grip of economic depression. He took his mother with him for the first part of the journey, apparently to give her a short break from the acrimony in 6 Julius Road (she appears to have returned home immediately). After another long hiking vacation with Van Vleck, in the Glacier National Park, Dirac arrived in Princeton - a little over an hour's drive from both New York and Philadelphia - then stirring after the long torpor of the summer vacation. The mathematician Malcolm Robertson, who arrived there at the same time, later remembered being overwhelmed when he drove through the town for the first time at dusk:\n\nThis was my first glimpse of the charming college town that was to play such a large part in my life, and a joyful and exhilarating experience it was indeed. I have never forgotten that first encounter, and my feeling of excitement and awe at the lovely stately homes among the old trees, the magnificent university campus with both new and old stone buildings, acres of well-kept lawns, and even a lake and a peaceful golf course.\n\nSoon after Dirac arrived there at the end of August, he was given a handsomely appointed office in Fine Hall, home of the university's  mathematics department, the newest building on the campus. It was largely the initiative of the tweed-suited Princeton mathematician Oswald Veblen, who oversaw every detail of the building's opulent design, right down to the locations of the electrical sockets. Almost a third of its budget for internal decorations had been allocated to rugs woven from seamless Scottish chenille. Throughout the new building, there was other evidence of his Anglophile tastes, with a firm nod to the ambience in G\u00f6ttingen: the hall's faux Oxbridge architecture and furnishings, its freshly varnished oak-panelled walls, even the ritual of taking afternoon tea. In the common room used for special occasions, Veblen had arranged for Einstein's aphorism _Raffiniert ist der Herr Gott, aber boshaft ist Er nicht_ (God is cunning, but He is not malicious) to be engraved in German on the rim of the huge stone fireplace.\n\nOn the morning of Wednesday 1 October, Dirac walked to Fine Hall from his lodgings near the town centre through the blaze of red and orange foliage, dried-out leaves crackling underfoot. A few hours later, for the first time in his career, he was to co-present a seminar, and with the least likely of his colleagues, Wolfgang Pauli. For Princeton University's physicists, walking to the hall through the connecting corridor, and other faculty members, crossing the campus in the biting chill of the late afternoon, this was an exciting start to the new academic term, an opportunity to see two of the subject's luminaries talking about some of their freshest ideas. The occasion was, Pauli wrote to Rudolf Peierls, 'a first national attraction'.\n\nEach speaker was going to present what amounted to a prediction of a new particle: Dirac presented the monopole, Pauli another hypothetical particle, later called the neutrino. The event marked the dawn of a new culture in physics, in which theory could pre-empt experiment. The figures and demeanours of the two speakers contrasted comically. Dirac was thin as a reed, distant and serene, with the smooth and unblemished skin of a young man but, incongruously, with a pronounced stoop. The overweight Pauli was two years Dirac's senior but his waistline made him look older. When sitting, he looked like a judge deep in reflection, his arms folded over his belly, his bulbous torso rocking rhythmically back and forth. At the seminar, he probably looked troubled and in some pain, having broken his left shoulder when he fell downstairs a few months before, the worse for drink.\n\nMany in the audience will have read about Dirac's prediction, but Pauli's had not appeared in an academic journal, though attentive readers of the _New York Times_ read about it in an article published a few months before. Pauli had first proposed the existence of his new particle in a private letter to a meeting of experts on radioactivity.  There, he tentatively suggested that the existence of the particle could explain the problem that Bohr had identified with energy conservation when a radioactive nucleus ejects an electron. The essence of the problem was that electrons from these nuclei did not all have the same energy; rather, the electrons had a continuous range of energies. Pauli put forward a 'desperate' explanation for this spectrum of energies: the electron in each radioactive decay was ejected with another particle - hitherto undetected - so that the two particles shared their total energy in proportions that varied from one decay to the next. According to Pauli's theory, the new particle should have no electrical charge, the same spin as the electron and only a tiny mass. Few of Pauli's peers liked the idea: for Wigner it was 'crazy', for Bohr it was implausible and Dirac thought it was simply wrong. Pauli later described the neutrino as 'that foolish child of the crisis of my life', referring to his troubled psychological state. His problems had begun earlier in the year, following a series of tragedies - the suicide of his mother three years before, the remarriage of his father to a woman Pauli loathed, and the ending of his brief first marriage, when his wife had the impertinence to leave him for a scientific mediocrity ('such an average chemist').\n\nThe next day, Pauli left Princeton to return to Europe, but Dirac stayed to give a six-lecture course on quantum mechanics, ending with a presentation of his hole theory. In the closing few minutes, he affirmed more clearly than ever in public that anti-electrons should be detectable because:\n\n[they] are not to be considered as a mathematical fiction; it should be possible to detect them by experimental means.\n\nDirac repeated his suggestion that the idea could be tested experimentally by arranging for pairs of ultra-energy photons to collide: if the theory were correct, in some of these collisions the photons would disappear and an electron would appear with an anti-electron. But he was pessimistic. So far as he could see, it would not be feasible for experimenters to test the idea in the next few years.\n\nHe did not realise that the solution to his problem lay in the columns of the _New York Times._ Dirac read it regularly and must have seen the articles on the investigations of cosmic rays being carried out by Millikan, who had given them their catchy name in 1925. The rays had been discovered in 1912 but were still a mystery: all that was known for sure was that they had extremely high energy, typically thousands of times higher than particles ejected from atomic nuclei on Earth. Millikan developed a religion-based theory of the cosmic rays and, by 1928, regarded it as 'fairly definite' that they were the 'signals broadcast throughout the heavens [. . .] the birth cries of infant atoms', clear evidence for divine benison.\n\nDirac must have known that high-energy cosmic rays could produce anti-electrons if the rays collided with other particles on Earth. Yet it seems that he was never much interested in these particles, perhaps because he was influenced by modish opinion in the Cavendish Laboratory in the mid-1920s, when no one there studied the rays. Rutherford's deputy James Chadwick had sighed when he came across another of Millikan's research articles on cosmic rays: 'Another cackle. Will there ever be an egg?' But that was six years before, and by the autumn of 1931 the attitude to the rays at the Cavendish was changing. The first of its scientists to latch on to their importance was Blackett, who was at a crossroads in his career, casting around for a new research topic. This subject must have had a special appeal to the independent-minded Blackett as it would distance him from Rutherford, whose ego was becoming overweening.\n\nBlackett was in the audience at a special Cavendish seminar on Monday, 23 November, when Millikan presented the latest photographs of cosmic rays to be taken at the California Institute of Technology (Caltech). The photographer was Carl Anderson, until recently Millikan's Ph.D. student, only twenty-six years old and already touted as one of the brightest experimenters in the United States. Three weeks earlier, he had pointed out to his boss that the new photographs showed 'Very frequent occurrence of simultaneous ejection of electron and positive particle'. Anderson was trying to take images of the charged particles produced by cosmic rays using a cloud chamber, which enables the tracks of electrically charged particles to be photographed as they travel through a cloud of water vapour. Anderson had built his own cloud chamber and, at Millikan's suggestion, arranged for the entire chamber to be bathed  in a strong and uniform magnetic field, which would deflect the paths of the charged particles as they hurtled through it. Each track contained crucial information: from the density of droplets along each track, Anderson could determine the particle's electric charge, and he could calculate the particle's momentum from the deflection caused by the magnetic field.\n\nIt required great skill for Anderson to take any photographs at all. Most of his images were blank, but by early November he had obtained some 'dramatic and completely unexpected' images, which he sent to Millikan in Europe. The photographs made no sense in terms of the theory they were using. In a puzzled letter to Millikan, Anderson remarked that many of the photographs featured the track of a negatively charged electron with a positively charged particle, two particles appearing at the same time, presumably when a cosmic ray strikes an atomic nucleus in the chamber.\n\nWhen Millikan presented Anderson's inexplicable subatomic images in his seminar at the Cavendish, Blackett was fascinated. Here was a cloud-chamber expert with a talent that everyone knew was great but unfulfilled. Here was a new field in a mess. And here was the perfect opportunity for him to make his name.\n\nMillikan's audience in the Cavendish seminar did not include Dirac, who was still in Princeton. Many of his colleagues, including Martin Charlesworth in St John's, feared they were about to lose him to one of the higher-paying American universities. Charlesworth wrote to Dirac saying how much he missed his 'kindly irony', imploring him 'Don't let them persuade you to stay in the USA _._ Here is your home.' Charlesworth was right to be concerned, for Veblen was energetically wooing Dirac. Even before the carpenters and decorators had put the finishing touches to Fine Hall, Veblen had begun to work with the educator Abraham Flexner, who was trying to set up an institute for advanced study, where world-class thinkers could study in peace, free of all distractions. Einstein was at the top of their wish list, but they were competing with others, including the wily Millikan, at Caltech.\n\nCharlesworth may have worried, too, that Dirac might not be looking forward to returning home. From newspaper and radio reports, Dirac knew that his homeland was plunging into difficult times. On 21 September, the Government removed the pound from  the gold standard and allowed the currency to settle down to whatever price the money-market dealers were prepared to pay for it. It was a national humiliation. The economy plunged deeper into crisis: unemployment continued to escalate, and soon the pound had been devalued by 30 per cent, making Dirac's $5,000 fee for his single-term stay look even more generous. The inevitable General Election returned a stabilising coalition government, but the economic privations continued: that year, one in every two British industrial workers had been unemployed for over four months.\n\nYet the depression was still more serious in the United States, even in affluent Princeton. At the university, many students struggled to pay their fees. Around the town, young vagrants were walking the streets, some of the two million roaming the country in search of work. About thirty million Americans, a quarter of the population, had no income at all. Many people who had money were so frightened of losing it that they hoarded their dollars under mattresses or buried it in the garden. Even President Hoover - long in denial about the extent of the depression - realised that ordinary people were losing faith in the American way of life.\n\nAs Dirac will have been aware, unemployment was said to be zero in the USSR. The admirers of Stalin's Five Year Plan in the press included the _New York Times_ 's Moscow correspondent Walter Duranty, who called the plan a 'stroke of genius' and won the Pulitzer Prize the next year for his reports. Yet Dirac's friends in the Soviet Union suffered terribly when Stalin's attitude towards science changed abruptly, from a subject worthy of study for its own sake to a weapon for fighting capitalism. Tamm and Kapitza supported the new Soviet line, at least in public, but Dirac heard the other side of the story from Gamow, who had been exasperated by the change in the Government's attitude when he returned to Russia in the spring of 1931. The Communist Academy had declared Heisenberg's version of quantum mechanics anti-materialistic, incompatible with the state's increasingly rigid version of Marxist philosophy. During a public lecture at the university on the uncertainty principle, Gamow experienced the full force of state censorship when a commissar, responsible for supervising moral standards, interrupted him and told the audience to leave. A week later, Gamow was forbidden to speak again about the principle in public.\n\nSince the mid-1920s, Gamow and Landau had been two leaders of the informal group of young Soviet theorists nicknamed the 'Jazz Band'. In its seminars, the group discussed new physics, the Bolshoi Ballet, Kipling's poetry, Freudian psychology and any other subject that took their fancy. The Jazz Band was mastering the new quantum physics much more quickly than their professors - 'the bisons' - whom they teased unmercifully, while taking care to remain within the bounds of decorum. The Band overstepped the mark in 1931, however, when they ridiculed a new encyclopedia article on relativity theory, edited to conform to the Party's views on the subject. The butt of the Jazz Band's barbs was the Director of the Physics Institute in Moscow, Boris Hessen, a thoughtful Marxist who had fended off several of the Government's attempts to make orthodox theories of physics conform to 'dialectical materialist' principles, the philosophical basis of Stalinist Marxism, which accords much higher priority to concrete matters than to abstractions. Hessen had only a meagre knowledge of quantum mechanics and general relativity, so he was ill equipped to defend them against ideological interference from Stalin's officials. This ignorance led him to write a ludicrous article in the _Greater Soviet Encyclopedia_ about the ether, declaring it to be 'an objective reality together with other material bodies', contrary to Einstein's teaching. Gamow, Landau and three colleagues sent a mocking note to Comrade Hessen and were put on trial as saboteurs of Soviet science. Landau was temporarily banned from teaching at the Moscow Polytechnic, and the miscreants were banned from living in the five largest cities of the USSR, though the ban was not enforced. According to Gamow, the offending physicists had been found guilty by a jury of machine-shop workers.\n\nEven Dirac fell foul of the censors when the Russian translation of his book was being edited, when his publishers objected that his quantum mechanics was in conflict with dialectical materialism. The book eventually appeared in bookstores after an uneasy deal between the publisher and the editor, Dmitry 'Dimus' Ivanenko, a Jazz Band leader and another of Dirac's effervescent Russian friends. In the awkward opening to the book, it is easy to see reflections of the delicacy of the deal: Ivanenko's preface is conventionally laudatory, but it is preceded by an apologetic note from the 'Publishing House', arguing feebly that although the material in the book is ideologically unsound, Soviet scientists need to use its methods to  advance dialectical materialism. A 'counterflow' of ideologically correct science will then follow, the publishers hoped. In a simpering conclusion, Ivanenko thanked Dirac, 'a sincere friend of Soviet science'.\n\nCensors were also scrutinising science in Germany, where the Depression was wreaking economic mayhem. Scruffy buskers, match-sellers and bootlace salesmen walked the streets in the hope of being paid a few _pfennig_ to buy a loaf; tens of thousands of the unemployed queued outside Nazi offices, waiting for the storm troopers to reward them with a mug of hot soup. The once-peaceful G\u00f6ttingen, where Born was Dean of his faculty, was now seething with political tensions: in the physics library he saw Communist leaflets, while outside the Nazis greeted each other ostentatiously with a click of their heels and a 'Heil Hitler' salute. The Nazis, the majority party in the local government and student congress, were insisting that Einstein's 'Jewish physics' was wrong and pernicious. Born was beginning to think that he had no alternative but to emigrate.\n\nTo most people who came across Dirac, he seemed to be no more engaged with world affairs than an automaton. With no need to share his thoughts with others, unless they were close friends, he gave the impression that he was indifferent to the fate of others. He appeared to have none of the usual need to be warmed by the good opinion of other human beings.\n\nAt work in his office in the new Fine Hall, he was putting into practice the philosophy that he had preached earlier in the year, learning advanced topics in pure mathematics in the hope that they would find application in theoretical physics. He had also returned to field theory, a subject he had co-founded four years before. The theory seemed fated to generate predictions that were not ordinary numbers but infinitely large. While Dirac was preoccupied with his ideas, Heisenberg and Pauli had been developing a full-blown theory of how electrons and photons interact with one another, a quantum theory that accounted for the spontaneous creation and destruction of particles, consistent with the special theory of relativity. Heisenberg and Pauli's theory was also consistent with both quantum theory and experiment, but it was ugly and unwieldy. Oppenheimer later described it as 'a monstrous boo-boo'. Unconvinced that this was the right way to describe nature at a fundamental level, Dirac sought a superior description, one that was logically sound and not plagued with infinities. The more Dirac looked into the Heisenberg-Pauli theory, the more he disliked it. In his view, it was not even consistent with the special theory of relativity because it describes processes throughout space using time measured by a single observer, whereas Einstein had taught that no single time could suffice for all observers, as they make different measurements of time. Dirac spent hours in Fine Hall examining the Heisenberg-Pauli theory and coming to terms with the problem of curing the sickness of field theory. The challenge would obsess him for the rest of his life.\n\nBy the end of the autumn, as Dirac's sabbatical was ending, it was clear that the industrialised world was sliding into its worst-ever economic crisis, and there was a disturbing new militarism in Germany, Japan, Italy and throughout much of east-central Europe. In Britain, everyone was talking about the possibility of another war. The spirit of the age was no longer caught in the freewheeling, life-affirming bravura of _Rhapsody in Blue_ but in the headlong, ominous prelude to _Die Walk\u00fcre._\n\nIn Bristol, it had been a sombre autumn at 6 Julius Road. In her letters, Dirac's mother told him that she and his father had recovered from their climactic row and were back to their routine: she waited on him almost full-time, feeding him his vegetarian meals, washing his clothes and spending hours helping him dress. Each Sunday, she would give him - in silence - the 'ninety-degree' bath that he insisted was good for his rheumatism. After one of them, he had a heart attack. The family doctor told her soon afterwards that her husband 'is a man accustomed to his own way & will not take advice [. . .] He may live 20 years or he may go suddenly.'\n\nBy September, the family were feeling the pinch of the economic crisis: Charles cut his tuition fees and insisted that they could no longer afford to run the car. When Betty told the family's bank manager this, he laughed, Flo told her son. She believed Charles had plenty of money stashed away, although he was spending virtually nothing. Earlier, when Flo tried to claim the small amount of money Felix had left six years before, the authorities sent her a form for her husband to sign as the law specified that the funds must be paid to him. She told Dirac: 'I tore up the form.'\n\nDirac did not return in time for Christmas. Three days before the holiday, his mother wrote to him: 'I am always so grateful that you broke away from our narrow little life.'\n\nDirac was about to have one of his most exhilarating years. The word on the physicists' street was that Chadwick was on to something important at the Cavendish Laboratory. Chadwick - a lean, severe figure - was usually busy overseeing his colleagues' work, dispensing the paltry annual budget for equipment. But he had temporarily put administration to one side. Soon after the Christmas vacation, Chadwick had read an article that he suspected might lead to the neutron, a particle whose existence Rutherford had predicted.  In the article, two French experimenters - Fr\u00e9d\u00e9ric Joliot and Madame Curie's daughter Ir\u00e8ne - reported from their Paris laboratory that they had fired helium nuclei at a target made of the chemical element beryllium and found that particles with no electrical charge were ejected. They argued that these particles were photons, but Chadwick believed they were wrong and that the particles were Rutherford's elusive neutrons. Rutherford agreed. Having just turned forty, Chadwick may have sensed that this could be the last chance for him to make his name, to emerge from the shadow of his imperious leader. He hungrily grabbed the opportunity, working alone night and day, borrowing apparatus and radioactive samples from colleagues all over the laboratory, making new equipment, filling his notebook with data and calculations. Oblivious of the freezing Cambridge midwinter, he was in a world of his own, as his colleagues saw. After three exhausting weeks, he had nailed the neutron. He proved to his satisfaction, and Rutherford's, that his results made sense only if a particle with no charge and about the same mass as a proton is ejected in the nuclear collisions he observed. But when he wrote a report on his work for the journal _Nature_ , he gave it the cautious title 'Possible Existence of the Neutron'.\n\nOn 17 February, Chadwick sent off his paper to _Nature_ , which rushed it into print. Six days later, after a good dinner in Trinity College with Kapitza, he presented his results to his colleagues at the Kapitza Club. Relaxed and emboldened by a few glasses of wine, Chadwick confidently described his experiments, giving appropriate credit to his colleagues, and finally set out the powerful arguments for the existence of the neutron. It was a coup for Chadwick and for  the Cavendish Laboratory, which had at last come up with the kind of ground-breaking result that Rutherford longed for - one that put nature into fresh focus, clarifying the very nature of matter. The audience gave him the unusual accolade of a spontaneous ovation. After the meeting, he asked 'to be chloroformed and put to bed for a fortnight'.\n\nThe discovery gave fresh impetus to the notion that new types of subatomic particle might be predicted before they were detected. The ability to foresee the different types of grain in nature's fabric was a challenge to even the greatest scientists: Einstein had, in effect, predicted the existence of the photon but occasionally lost confidence in his idea before he was proved right; Rutherford - the experimenter's experimenter - had actually been more consistent, never wavering in his belief in the reality of neutrons. Perhaps Dirac's anti-electron and Pauli's neutrino were worth taking seriously, after all?\n**Sixteen**\n\nI hope it will not shock experimental physicists too much if I say that we do not accept their observations unless they are confirmed by theory.\n\nSIR ARTHUR EDDINGTON, 11 September 1933\n\nThe character of Paul Dirac first appeared on stage in a special version of _Faust_ , the _Hamlet_ of German literature. Goethe's drama is the literary antithesis of Agatha Christie's penny-plain narratives that Dirac wolfed down in the evenings. He had no taste for epic plays, but he will have been absorbed in this _Faust_ , a forty-minute musical parody of the twenty-one-hour play, written as a physicists' entertainment.\n\nThe authors, the cast and the audience were the physicists at Bohr's spring meeting in April 1932, and Dirac was there. In the oasis of the institute, physics had not looked more exciting for years, in hideous contrast to the world outside. Chadwick's discovery had revitalised interest in the atomic nucleus, whose detailed structure was a mystery to theoreticians. They had a wealth of other problems to solve, too, including the status of quantum field theory and of the predicted anti-electron, monopole and neutrino - each controversial, none yet detected. As Bohr liked to point out, science often flourishes quickest when it faces problems and contradictions; the Princeton physicist John Wheeler once went so far as to spell out the central idea of the institute as 'No progress without paradox'.\n\nThe version of _Faust_ performed at the Institute was in the tradition of office Christmas parties, with their licensed burlesque and private jokes that stay close to the boundaries of good taste but carefully avoid crossing them. The journalist Jim Crowther was among the audience of twenty-odd conference delegates who entered into the spirit of the occasion, happily indulging the manifold crimes against artistic taste. Bohr, represented in the play by the Lord Almighty, sat in the middle of the front row of the audience, convulsed with laughter as one of his colleagues mimicked his tortured oratory.\n\nIn Goethe's original play, the sharp-tongued Mephistopheles seduces Faust, discontented with his limited wisdom, into a bargain  that grants him universal insight and the love of the beguiling virgin Gretchen. The main theme of the Copenhagen version is the story of the neutrino and of Pauli's attempts to persuade Ehrenfest of its existence. Pauli (not at the meeting) was represented by Mephistopheles, Ehrenfest by Faust, and the neutrino by Gretchen, whose songs Heisenberg accompanied at the piano. The original version of the play opens with speeches from three archangels, and the Copenhagen version began in the same way, except that the trio was represented by the English astrophysicists Eddington, Jeans and Milne, who stood on the almost room-wide desk of the main lecture theatre, declaiming in rhyming doggerel about the latest theories of the universe.\n\nEhrenfest's leg was pulled unmercifully. He was played as a character who lay on the couch with his trousers in disarray, meditating on the vanity of science and life. This probably struck some participants, including Dirac, as being too close to home: Ehrenfest was morose, deeply uneasy about the state of physics and losing his spark. At the meeting, when Darwin approached him with a question, he rebuffed him, saying only, 'I'm bored with physics.'\n\nIn the second half of the playlet, Dirac comes under the spotlight. His monopole is a singing character, treated with respectful curiosity, in contrast to his hole theory, portrayed as bizarre and not wholly serious. In a few revealing lines, the character of Dirac describes the state of his subject:\n\nA strange bird croaks. It croaks of what? Bad luck!   \nOur theories, gentlemen, have run amuck.   \nTo 1926 we must return;   \nOur work since then is only fit to burn.\n\nThese few words accurately capture Dirac's despondency about the state of quantum field theory. He had tried to produce an improved version of Heisenberg and Pauli's relativistic version of quantum field theory but had found out during the meeting that his theory was no improvement at all: both field theories were shot through with infinities. The root of the problem appeared to lie in 'singularities', particular points in the theory where the mathematics become ill defined or even incomprehensible. It was a deft decision of the authors of the Copenhagen _Faust_ , headed by Max Delbr\u00fcck, to arrange for Dirac to exit the stage chased by the actor playing a bit part, Singularity.\n\nThe jibes about hole theory were not confined to the entertainment; throughout the meeting, Dirac had to put up with Bohr's hostile questioning and the taunts of other colleagues. Dirac appeared to take it all on the chin; according to one colleague, during the meetings that week he did not utter a word. In the final session of the meeting, Bohr lost patience and put him on the spot: 'Tell us, Dirac, do you really believe in that stuff?' The room went silent, and Dirac stood briefly to intone his twelve-word reply: 'I don't think anybody has put forward any conclusive argument against it.' Although outwardly loyal to his interpretation of hole theory and to his proposal of the anti-electron, the absence of the particle was sapping his morale. Soon, even he stopped believing in his hole theory, he later told Heisenberg.\n\nJust less than three weeks after the Copenhagen meeting, news broke from the Cavendish of another experimental sensation: the atom had been split. It was the work of John Cockcroft and the dishevelled Irishman Ernest Walton, an expert in engineering hardware. Together, the two men had built the largest machine ever constructed in the Cavendish, capable of accelerating protons through 125,000 volts and smashing them into a metal target. Quantum mechanics predicted that the accelerated protons should have enough energy to break up the nuclei at the heart of the lithium atoms, but it was a challenge to prove it. Cockcroft and Walton increased the intensity of their beam until it was high enough to stand a chance of splitting some of the atoms in their lithium target. After eight months of work, when the beam was delivering a hundred trillion protons per second, telltale flashes on the detector in Cockcroft and Walton's darkened laboratory told them that they had split lithium nuclei into two nuclei of a different element, helium. Here, on the nuclear scale, Cockcroft and Walton realised the dream of alchemists by transforming one type of element into another. For the second time in three months, Rutherford was overseeing the announcement of a great experiment. He was not best pleased when Crowther's news-management skills faltered and the story leaked to the press and broke in the popular Sunday newspaper _Reynolds's Illustrated News_ , which trumpeted the latest Cavendish finding as 'Science's Greatest Discovery'. Other newspapers soon followed, including a nervous _Daily Mirror:_ 'Let it be split, so long as it does not explode.'\n\nWhen the discovery was announced, Einstein happened to be in Cambridge to give a lecture. On 4 May, at the height of public interest in the experiment, an intrigued Einstein paid a private visit to the Cavendish Laboratory for a demonstration. He must have been gratified to see that Cockcroft and Walton's results were consistent with his most famous equation: the total energy of the particles involved in the nuclear reaction is conserved only if energy and mass are related by _E_ = _mc_ 2. Cockcroft and Walton had been the first to verify the equation.\n\nEddington - ready, as ever, with a down-to-earth analogy - linked Cockcroft and Walton's fragmentation of the nucleus to what appeared to be the fissuring of society. He observed that splitting the once-indivisible atom had become the ordinary occupation of the physicist since 1932 and that the social unsettlement of the age seemed to have extended to atoms. By 1932, Cambridge University's political centre of gravity had moved sharply to the left. Only six years before, the great majority of students worked to break the General Strike; by May 1932, the Cambridge Union - bellwether of student opinion - supported the motion that they saw more hope in Moscow than in Detroit. The students were fearful of another war, angry that the spirit of the Locarno Treaty was being mocked by events. Another war was beginning to look all but inevitable.\n\nThe Cavendish triumphs demonstrated the quality of Rutherford's leadership of experimental physicists in Cambridge. By comparison, the university's theoreticians were embarrassingly unproductive - their titular head was the Lucasian Professor Sir Joseph Larmor, then seventy-five and about to retire, not before time. To no one's surprise, the authorities announced in July that his successor was Dirac, who was not quite thirty and just a few months older than Newton's age in 1669 when he took the Chair. As soon as the authorities announced his appointment, he left Cambridge for a while to escape the clamour of congratulations.\n\nDirac knew that the Chair was more than an accolade: it was a vote of confidence but also a challenge. He was expected to continue to be a leader, to set the pace in his field, to leave a legacy that scientists would talk about for centuries. By no means all the holders of the Lucasian Chair had justified their promise: William Whiston, John Colson and Isaac Milner are in no one's list of great mathematicians or scientists. Dirac still had more to prove. He was confident in the durability of his early work on quantum mechanics, though he had good reason to fear that his later ideas - field theory, hole theory, the monopole - might one day be regarded as honourable failures. Worse, he worried that he was becoming too old to come up with original theoretical ideas: earlier in the year, soon after Heisenberg's thirtieth birthday, Dirac told him: 'You are now past 30 and you are no longer a physicist.'\n\nRutherford wrote to congratulate Dirac, hoping that he 'will still continue to be a frequent visitor to the Cavendish', probably an allusion to Larmor, who rarely set foot in the Laboratory. One of Dirac's colleagues summed up the mood when he told the new professor: 'I don't think any recent election to a professorship can have been more popular.' Only Larmor was sniffy about his successor's appointment, later cattishly remarking that Dirac was 'an ornament of the German school [. . .] though a minor one.'\n\nDirac did not look the part of the distinguished Cambridge professor. Shy as a mouse, he had so little gravitas outside the lecture theatre that in the streets of Cambridge he passed for a tyro graduate student. He was nervous in the company of women of his own age, so many of his colleagues assumed he was gay, that he would die a bachelor and had no interest in having children. Yet Kapitza knew better. He came to know Dirac well during their relaxed conversations in the Kapitzas' house, a noisy den that always seemed to be teetering on the edge of familial anarchy. Dirac was at ease there, talking with Kapitza and Rat over a Russian-style meal, playing chess and larking about with their two rumbustious sons. The contrast between the dysfunctional household of 6 Julius Road and the happiness he saw in the Kapitzas' home could scarcely have been plainer. Perhaps Dirac was already longing for the vibrant family life that Kapitza and Bohr had shown him, an environment in which sourness and unkindness were rare, not the norm.\n\nBy the standards of British academics, Dirac was wealthy. When he took up the Lucasian Chair, his annual salary rose sharply, from \u00a3150 to \u00a31,200, supplemented by his annual college 'dividend' of \u00a3300. The modern value of his salary at the end of 1932 is \u00a3256,000. He had seen the last of penury, though for him frugality was too ingrained to be anything other than a way of life. So far as he was concerned, a suit and a tie were all he needed, and he wore them  indoors and outdoors, rain or shine, until most men would regard them as being fit only for the bin. His mother, perpetually chivvying him to smarten up, thought it was high time she bought some new clothes for herself and asked him to pay for them: 'If you have a really substantial salary in the autumn you may be able to treat your mother to a winter coat.'\n\nCharles and Flo were the toast of the city for producing its most famous scientist, but the old quarrels continued. Worried that Charles was planning to convert their daughter into a nun, Dirac's mother suggested that he pay for Betty to take a degree in French at the university. There was not much chance that Charles would pay for it as he believed that higher education should be a male preserve. Betty sensed this, as she told her brother in a letter: 'I haven't actually asked Pa for financial assistance, but he takes no interest in it and doesn't seem willing to help in any way.' But Betty was not resentful: she accepted it as part of her father's character and, besides, most other men felt the same way.\n\nIn Betty's letters to Paul around this time, she seems conventionally affectionate to him, but nothing of substance is known about their relationship. It seems safe to conclude that he thought well of her, however, because in July 1932 he generously offered to pay for his sister's fees and expenses for the next four years. Although she struggled before successfully crossing the first hurdle of gaining a mandatory pass in Latin, she was a contented student. In a touching letter to her brother she assured him, 'I will do my best to give you value for your money, and I am honestly working, for the first time in my life, I believe.' Her educational liberation seems to have disheartened Charles, now a stooped and tottering invalid. He was slowly losing his grip on his family, Flo reported to her son: during a routine domestic stand-off about the use of their car, he huffily agreed to give in to her and Betty, but only after an hour's sullen reflection. It was a momentous moment, the first time in thirty-two years of marriage that she could remember him backing down. He may well have wondered how his life had come to such a pass. Perhaps he would have sympathised with Fatty Bowling, the narrator of _Coming Up for Air_ , George Orwell's satire on 1930s suburbia. Like Charles, Bowling was a hostage to his ungrateful family, tied by convention and financial convenience to a slattern he despised. Unlike Bowling, however, Charles took pleasure from his friends and  his work: language students still traipsed up to 6 Julius Road for his tutorials, and he was still active in the local Esperanto Society.\n\nBy early August, Charles was planning to visit his family in Geneva. As usual, he did not tell his wife about his travel plans but disclosed them to his son, in a letter written almost entirely in French (only the final line was in English). He trod carefully:\n\n7 August 1932\n\nMy dear Paul\n\nI suppose that you are very busy so I will only take a few minutes of your time to tell you how happy and proud I am of your great success. All the newspapers have given us the details. Several friends and acquaintances have asked me to congratulate you on their behalf.\n\nWill this new position change your plans to go to Russia? I would like to know the date when you have decided because as soon as I am strong enough to undertake the journey I should go to Switzerland to sort out some family matters and I do not want to be away from Bristol when you are here.\n\nObviously if you could come with me that would please me more.\n\nMy fond good wishes and may God prosper you.\n\nFather\n\nBut Charles was to be disappointed. His son was planning another vacation in the Soviet Union, this time with Kapitza in Gaspra, a mountainous coastal resort in the Crimea. In Stalin's time, it was a place for the scientific elite to take breaks, away from the forced migrations of peasant farmers, the food shortages and rationings and all the other disasters of the Five Year Plan and collectivisation.\n\nDirac had begun his trip at a conference in Leningrad, where he spoke about his field theory of electrons and photons. After Boris Podolsky - an American of Russian-Jewish blood - and Vladimir Fock told him that they were studying the same problem, Dirac agreed to work with them. During his stay in Kharkhov, Dirac collaborated with his Russian colleagues, and, after a long exchange of technical correspondence, they produced a surprisingly simple proof that Dirac's field theory is equivalent to Heisenberg and Pauli's and more transparently consistent with the special theory of relativity.  This project was another sign that Dirac was no longer quite so insular: early in the year, he had written a modest paper on atomic physics with one of Rutherford's students and now here he was, working on quantum fields in equal harness with Soviet  theoreticians. But Dirac remained wary of collaboration: visiting theoreticians who were not previously acquainted with him found him distant, utterly uninterested in sharing his ideas. When Dirac was visited by one of them, Leopold Infeld, the young Pole found him friendly and smiling but unwilling to respond to any statement that was not a direct question. After twice receiving a reply of just 'No', Infeld managed to phrase a technical query that drew from Dirac an answer consisting of five words. They took Infeld two days to digest.\n\nWhen Dirac was relaxing on the Crimean coast, he was unaware that the story of the anti-electron was approaching its conclusion more speedily than he had dared to believe possible. Many of the characters in this strange denouement, including Dirac, behaved in ways that are now barely comprehensible, even bearing in mind that hardly any physicists in 1932 took Dirac's hole theory seriously and few were even vaguely aware of his prediction of the anti-electron.\n\nThe end of the story began shortly before Dirac's vacation, at the end of July 1932 in Pasadena, not far from the Hollywood Bowl, where the Los Angeles Olympic Games were just beginning. It would be a welcome opportunity for the people of the city and millions of radio listeners to have some respite from the economic gloom and political manoeuvrings in advance of the coming presidential election.  At Caltech, many of the scientists were on vacation. But in a comfortably warm room on the third floor of the aeronautics laboratory, Carl Anderson was hard at work on the effects of cosmic rays within his cloud chamber. By the end of the first day of August, a Monday, all he had to show for his latest experiments were blank photographs, but, on the following day, he struck lucky.\n\nAnderson managed to take a photograph of a single track, just five centimetres long. It looked rather like a hair. The density of bubbles around the track seemed to indicate that it had been left by an electron, but the curvature of the path suggested otherwise - it had been left by a _positively_ charged particle, so it could not possibly have been an electron. Still not quite believing his eyes, Anderson spent an hour or two checking that the poles of his magnet were correct and that they had not been switched by jokesters.  Convinced he was not the victim of a prank, he was elated, though his euphoria was cooled by an icy trickle of panic: was this really a discovery or some  stupid mistake? To clinch the existence of the positive electron Anderson needed more evidence, but by the end of the month he had found only two more examples of his unusual tracks, neither as cut and dried as the first. Millikan was not persuaded.\n\nAfter the Olympic pageant had folded and the Caltech staff had returned after the summer break, Anderson wrote a short description of his experiment for the journal _Science._ Like Chadwick's presentation of his apparent discovery of the neutron, Anderson's account was cautious: he examined every conceivable reason why the track might not be a new particle. Even more circumspect than Chadwick had been, Anderson couched his claim to a discovery in a paper that he entitled 'The Apparent Existence of Easily Deflectable Positives', hardly an eye-catching phrase. Readers who reached the end of the article were rewarded with a sentence that qualifies as a masterpiece of scientific conservatism: 'It seems necessary to call upon a positively charged particle having a mass comparable with that of an electron.' According to one report, Anderson was so worried by his failure to find more good examples of the track that he thought of writing to _Science_ to withdraw his paper. But it was too late: the article was at the printers.\n\nHere, under Anderson's nose, was clear evidence for Dirac's anti-electron - a particle with the same mass as the electron but with the opposite charge. Anderson had earlier spent several evenings a week struggling through Oppenheimer's evening lectures on Dirac's hole theory, so it is practically certain that he knew about the part played by the anti-electron within it. Yet he did not make the connection, probably because he was directing his attention almost exclusively to the cosmic-ray theory of his boss.\n\nAnderson sent off his paper on 1 September, and it appeared in the libraries of American physics departments about eight days later, to be greeted with indifference and disbelief. His finding was 'nonsense', one of his Caltech friends told him. Millikan still believed that something was wrong with Anderson's experiment and so did almost nothing to promote it. Anderson, worried that he had not found another track like the one he detected in early August, spoke publicly about the need to be cautious. Oppenheimer was almost certainly among the thousands of physicists who read the article, and he wrote soon after to his brother that he 'was worrying about [. . .] Anderson's positive electrons'. But Oppenheimer failed to put two  and two together. Perhaps he was blinkered by a narrow interpretation of Dirac's sea of negative-energy electrons: Dirac had always believed that this sea would contain some holes, whereas Oppenheimer assumed that the electron sea was always completely full, so that the concept of the hole was redundant. It beggars belief that Oppenheimer never pointed out the connection between Dirac's theory and Anderson's experiment to Dirac, to Anderson or to anyone else. Yet that appears to be what happened.\n\nOne of Anderson's colleagues did, however, take his result seriously. Rudolph Langer - a Harvard-trained mathematician, talented but not noteworthy - had read Dirac's work on the anti-electron and talked with Anderson and Millikan about the new cosmic-ray photographs. The day after _Science_ published Anderson's paper, Langer sent a short paper to the journal, making connections between the new observations and Dirac's theories. Showing none of Anderson's restraint, Langer concluded that Anderson had observed Dirac's anti-electron. He did not stop there; he went on to build an imaginative new picture of matter, suggesting that the photon is a combination of an ordinary electron and a negative-energy electron, that the monopole is built from a positive and negative monopole and that the proton 'of course' comprises a neutron and a positive electron. The paper looks impressively imaginative today, but it made no impact in 1932, probably because Langer was not sufficiently respected to command attention and because it was simply not done to speculate with such abandon. His insight left no trace in Anderson's memory and was soon forgotten.\n\nBy early autumn, Anderson's 'easily deflected positive' appears to have been a minor query in the minds of most Caltech physicists, a rogue result to be refuted or possibly a puzzle to be solved. In Cambridge, no one seems to have been aware of Anderson's experiment or of Langer's article. The journal _Science_ arrived in the Cambridge libraries by early November, but neither Dirac nor any of his colleagues appear to have read it. But, by then, Blackett was hot on Anderson's trail.\n\nRutherford had agreed that Blackett could begin a new programme of research into cosmic rays. But Blackett's patience with his boss's despotic style had worn thin, as a graduate student saw when Blackett returned from Rutherford's office white-faced with rage and  said, 'If physics laboratories have to be run dictatorially [. . .] I would rather be my own dictator.' Blackett carved out a niche in the Cavendish, working with an Italian visitor, Giuseppe Occhialini, a light-hearted Bohemian commonly known by his nickname 'Beppo'. Ten years younger than Blackett, Occhialini was an expert experimenter who tended to rely on his intuition, rarely pausing to write down an equation, preferring to spell out the steps in his reasoning with an impressive range of accompanying gesticulations. When Occhialini arrived in Cambridge the year before, in July 1931, he had already been involved in experiments to detect cosmic rays and brought to the Cavendish years of experience working with Geiger counters, only recently introduced to Cambridge. These counters were delicate and unreliable, Blackett later remembered: 'In order to make it work you had to spit on the wire on some Friday evening in Lent.' For Occhialini, Blackett was a jack of all trades in the laboratory:\n\nI remember his hands, skilfully designing the cloud chamber, drawing each piece in the smallest detail, without an error, lovingly shaping some delicate parts on his schoolboy's lathe. They were the sensitive yet powerful hands of an artisan, of an artist, and what he built had beauty. Some of my efforts produced what he called 'very ugly bits'.\n\nOcchialini often visited Blackett at home in the evening. The two of them would relax in the front room and review their day's work over glasses of lemonade and a plate of biscuits, while Blackett fondled the ears of his sheepdog. During their conversations at home and in the Cavendish, they came up with a clever way of getting cosmic rays to take photographs of themselves: the trick was to place one Geiger counter above their cloud chamber and another counter below it, so that the chamber was triggered when a burst of cosmic rays entered both the upper and lower counters. By the autumn of 1932, Blackett and Occhialini had used this technique to take the art of photographing cosmic rays from a time-wasting matter of pot luck to a new era of automation. Soon, word circulated round the Cavendish corridors that something special was emerging from the Anglo-Italian duo. Even the reserved Blackett, the quintessence of the upper-crust Englishman, was excited.\n\nSoon Blackett and Occhialini were ready to treat their colleagues to the clearest batch of cosmic-ray photographs ever taken. At their seminar, Dirac was in the audience. This was surely his moment: he could quite reasonably have suggested that Blackett and Occhialini had discovered the anti-electron and, therefore, vindicated his hole theory. But he stayed silent. The mention of the possible presence of positive electrons drew Kapitza to turn to the new Lucasian Professor, sitting in the front row, exclaiming, 'Now, Dirac, put that into your theory! Positive electrons, eh! Positive electrons!' Kapitza had spent hours talking with Dirac but had evidently not even heard of the anti-electron. Dirac replied, 'Oh, but positive electrons have been in the theory for a very long time.' Here, unless electrons really were shooting upwards from the Cavendish basement, the anti-electron seemed to be showing its face. Yet Dirac's colleagues so mistrusted his theory that none of them was prepared to believe that it could predict new particles. Nor, it seems, did Dirac try hard to persuade them, perhaps because he believed that there was still a chance that every positive electron in his colleagues' photographs was in some way a mirage. This was reticence taken to the point of perversity.\n\nAt that time, Dirac was not concentrating on his hole theory but on one of his favourite subjects: how quantum mechanics can be developed by analogy with classical mechanics. In the autumn of 1932, he found another way of doing this, by generalising the property of classical physics that enables the path of any object to be calculated, regardless of the nature of the forces acting on it. Newton's laws could also do this job, and gave the same answer, but this technique - named after the French-Italian mathematician Joseph Louis Lagrange - was more convenient in practice. Dirac had first heard about this method when he was a graduate student, from lectures given by Fowler: it had taken some six years for the penny to drop.\n\nAlthough the technique is usually easy to use, it sounds complicated. At its heart are two quantities. The first, known as the Lagrangian, is the difference between an object's energy of motion and the energy it has by virtue of its location. The second, the so-called 'action' associated with the object's path, is calculated by adding the values of the Lagrangian from the beginning of the path to its end. In classical physics, the path taken by any object between two points in any specified time interval turns out, regardless of the forces acting on it, to be the one corresponding to the smallest value of the 'action' - in other words, nature takes the path of least action. The method enables physicists to calculate the path taken by any  object - a football kicked across the park, a moon in orbit around Saturn, a dust particle ascending a chimney - and, in every case, the result is exactly the same as the one predicted by Newton's laws.\n\nDirac thought that the concept of 'action' might be just as important in the quantum world of electrons and atomic nuclei as it is in the large-scale domain. When he generalised the idea to quantum mechanics, he found that a quantum particle has not just one path available to it but an infinite number, and they are - loosely speaking - centred around the path predicted by classical mechanics. He also found a way of taking into account all the paths available to the particle to calculate the probability that the quantum particle moves from one place to another. This approach should be useful in relativistic theories of quantum mechanics, he noticed, because it treats space and time on an equal footing, just as relativity demands. He sketched out applications of the idea in field theory but, as usual, gave no specific examples; his concern was principles, not calculations.\n\nNormally, he would submit a paper like this to a British journal, such as the _Proceedings of the Royal Society_ , but this time he chose to demonstrate his support for Soviet physics by sending the paper to a new Soviet journal about to publish his collaborative paper on his field theory. Dirac was quietly pleased with his 'little paper' and wrote in early November to one of his colleagues in Russia: 'It appears that all the important things in the classical [. . .] treatment can be taken over, perhaps in a rather disguised form, into the quantum theory.'\n\nEven if Crowther had wanted to publicise this idea, he would have found it hard to get his article published in the _Manchester Guardian_ : it was too technical, too abstract. The 'little paper' appears to have been too abstruse even for most physicists and so remained on library shelves for years, a rarely read curiosity. It was not until almost a decade later that a few young theoreticians in the next generation cottoned on to the significance of the paper and realised that it contained one of Dirac's most enduring insights into nature.\n\nIn the closing months of 1932, the news from Germany was that Hitler stood a fair chance of being elected chancellor in the impending elections: if Dirac's later comments on the F\u00fchrer are anything to go  by, he will have been uneasy at the prospect. Einstein, sick of the political climate and the violent anti-Semitism, fled to the USA and agreed to join Abraham Flexner's Institute for Advanced Study in Princeton, while Born hung on in G\u00f6ttingen, where the Nazis were the largest single party: half its voters now supported them. In the USSR, Stalin was showing ever-greater intolerance of academic freedom. In the USA, Franklin D. Roosevelt had been elected by a landslide, but the country remained in desperate economic straits. In the UK, unemployment rose to unprecedented levels, and there were mass demonstrations about unemployment benefits all over the country.\n\nIn the normally calm centre of Bristol, near the Merchant Venturers' College, hundreds of protestors were baton-charged by the police. A mile away, the Dirac household was again a battlefield. With Betty spending most of her time at university, her parents were left to explore every crevasse of their fractured marriage. Flo told Dirac that his father, becoming more aggressive, was still trying to throw her out of the house. Charles was incensed when he heard that she had given a pupil wrong information about his tuition fees and threw a glass of hot cocoa at her, she reported to Dirac. Yet, to most of the people he knew, Charles looked like a model of the contented retiree. At the Cotham School prize-giving, the Headmaster praised him for his son's success, and they talked over tea and cakes about Dirac's recent trip to Russia. Flo wrote to her son, 'Really, he is quite a gossip outside his own home, where he only condescends to scold.'\n\nThe Dirac family was together for what promised to be a torrid Christmas. But Charles and Flo ceased hostilities, and the family had what Flo described as 'quite the best Xmas we have had for years'. Part of the reason for this may have been that Dirac was in a good mood, as news he had wanted to hear for eighteen months had just arrived.\n**Seventeen**\n\nEinstein says that he considers Dirac the best possible choice for  \nanother chair in the Institute [for Advanced Study]. He would like to  \nsee us try for D[irac] even if the chance of getting him is very small.  \nHe rates him ahead of everyone else in their field. He places Pauli of  \nZurich second, apparently.\n\nLetter from OSWALD VEBLEN to ABRAHAM FLEXNER, 17 March 1933\n\nIt seems that it was not until mid-December 1932 that Dirac was confident that the anti-electron exists. Later, memories were too hazy for the date to be made precise: Dirac recalled that he 'probably' heard the news from Blackett, who never said publicly when he was sure of the new particle's existence. It may be that he discovered it independently of Anderson, though Blackett was always careful to give credit to his American rival for being the first to put his observation into print. Blackett and Occhialini probably learned of Anderson's photographs in the autumn through the grapevine, but they read his article on 'easily deflectable positives' only in January, three months after its publication, when they were taking cosmic-ray photographs by the dozen every day. In this bitterly cold Cambridge winter, Blackett and Occhialini had to trudge each morning to the entrance of the Cavendish through snow, slush and ice; inside, the laboratory was buzzing with the thrill of the new cosmic-ray photographs. It seemed that another success was in the offing, but there was a problem: no one was sure precisely what the images were showing.\n\nThe photographs featured a 'shower' of cosmic rays, with tracks that curved both to the left and to the right, emanating from a single location. In several of the snaps it was plain that Blackett and Occhialini had observed positively and negatively charged particles of about the same mass as they zipped through the cloud chamber: these appeared to be electrons and anti-electrons. Blackett asked Dirac to help interpret the data, and soon he was in the laboratory, doing detailed calculations using his hole theory. The most likely explanation was, they concluded, that incoming cosmic rays were  breaking up nuclei and that in the vicinity of some of these breakups, pairs of positive and negative electrons were being created. It was a classic application of Einstein's equation _E_ = _mc_ 2: the energy of the collision was converted into the masses of the particles. Dirac's calculations persuaded the hyper-cautious Blackett that the photographs were strong evidence for anti-electrons that behaved just as the Dirac equation predicted.\n\nWhen Blackett and Occhialini were preparing to make their results public, Dirac was also reading about events in Berlin. In the November election, the Nazis had lost over two million votes and had seen their representation in the Reichstag fall, but on 30 January, after weeks of chicanery by Hitler and his supporters, he was appointed Chancellor. The following night, G\u00f6ttingen was ablaze with torchlight as a procession of uniformed Nazis wended its way through the streets of the old town, singing patriotic songs at the tops of their voices, waving their swastikas and making anti-Semitic jokes. Hitler dashed naive hopes that he would moderate his policies on coming to power, swiftly implementing a dictatorship. On 6 May, the Nazis announced a purge of non-Aryan academics from universities, and, four days later, book-burning ceremonies were held all over Germany, including G\u00f6ttingen and Berlin. Even before Hitler rose to power, Einstein had left Germany, and he quickly announced that he would not return.\n\nHundreds of other Jewish scientists were desperate to emigrate. Dozens were rescued by Frederick Lindemann, Rutherford's counterpart at Oxford University, a prickly and sarcastic snob who had toured universities in Germany in his chauffeur-driven Rolls Royce offering threatened academics a safe haven in his laboratory. Cambridge University did not openly recruit potential refugees but waited for them to apply: from scientists, it received thirty such applications every day. One of them was Max Born, who was given a short-term academic appointment and - partly as a result of Dirac's support - an honorary position at St John's. In November, his colleague Pascual Jordan became one of three million storm troopers and proudly wore his brown uniform, his jackboots and his swastika armband.\n\nAlthough Heisenberg never joined the Party, he remained in Germany and was pleased that Hitler had come to power, if an anecdote related by Bohr's Belgian student L\u00e9on Rosenfeld is correct. Soon after Hitler became Chancellor, Bohr commented to Rosenfeld  that the events in Germany might bring peace and tranquillity, insisting that the situation 'with those Communists' was 'untenable'. When pressed by Rosenfeld, Bohr remarked: 'I have just seen Heisenberg and you should have seen how happy [he] was. Now we have at least order, an end is put to the unrest, and we have a strong hand governing Germany which will be to the good of Europe.'\n\nAlthough Dirac was privately appalled by Hitler's appointment, his outward response was so discreet as to pass unnoticed except by a few colleagues, including Heisenberg: Dirac vowed never again to talk in German. He had learned two foreign languages but now wanted to speak neither of them.\n\nInternational politics were not Dirac's only distraction. He was also turning his attention to moral philosophy, probably as a result of talking with the formidable Isabel Whitehead. 'Don't despise philosophers too much,' she had counselled him after one of his visits, 'a great deal that they say may be useless, but they are after something which matters.' Mrs Whitehead had been on the receiving end of one of Dirac's tirades against the only academic discipline he openly disdained. One of his _b\u00eates noires_ was the internationally admired Trinity College philosopher Ludwig Wittgenstein, regarded by many as one of the cleverest academics in Cambridge. Several decades later, Dirac remarked that he was an 'Awful fellow. Never stopped talking.'\n\nDirac's disenchantment with philosophers had degenerated into hostility when he read the ignorant comments several of them made on quantum mechanics; in a book review, he had already noted that it had taken the Heisenberg uncertainty principle to awaken the dozy philosophers to the revolutionary implications of quantum mechanics.  The philosophers who least offended Dirac and other theoretical physicists were the logical positivists, who held that a statement had meaning only if it could be verified by observation. There are traces of this philosophy in three pages of notes Dirac wrote out by hand in mid-January 1933, the raw and unpretentious jottings of a young man who wants to take stock and clarify his thinking about religion, belief and faith. He had recently told Isabel Whitehead, 'I am mainly guided in my philosophical belief by Niels Bohr', but these notes indicate that mainstream philosophers influenced Dirac more than he knew.\n\nDirac begins by considering belief. Some of the things a person believes in, he remarks, are not based on evidence but simply because they promote happiness, peace of mind or moral welfare. Such things constitute a person's faith or religion. In the only example he gives to illustrate this, he considers suicide, pointing out that most people believe that it 'is not a good thing, although there is no logical reason against it'. He was still haunted by Felix's demise and by the feeble purchase of logic on grief.\n\nWhen Dirac focuses on the transience of life, he is driven to an important moral conclusion: 'A termination of one's life is necessary in the scheme of things to provide a logical reason for unselfishness _._ [. . .] The fact that there is an end to one's life compels one to take an interest in things that will continue to live after one is dead.'\n\nThis, he says, is quite different from the unselfishness preached by orthodox religion, which he characterises as sacrificing one's interests in this life for one's interests in the next. Although he regards such a sacrifice as wrong-headed, he concedes - with uncharacteristic condescension - the argument made by many an imperial missionary that 'Orthodox religion would be very suitable for a primitive community whose members are not sufficiently developed normally to be taught true unselfishness.'\n\nAlthough Dirac rejects religious faith, he accepts that another faith is needed to replace it, something to make human life, effort and perseverance worthwhile. This leads him to his credo, one that would later influence his thinking on cosmology:\n\nIn my case this article of faith is that the human race will continue to live for ever and will develop and progress without limit _._ This is an assumption that I must make for my peace of mind. Living is worthwhile if one can contribute in some small way to this endless chain of progress.\n\nAt the end of his notes, Dirac turns to belief in God. This notion is so vague and ill defined, he says, that it is hard to discuss with any rigour. He first gave his views on the subject in his diatribe at the 1927 Solvay Conference, and is no less scathing here: 'The object of this belief is to cheer one up and give one courage to face the future after a misfortune or catastrophe. It does this by leading one to think that the catastrophe is necessary for the ultimate good of the people.'\n\nPerhaps Dirac had at least partly in mind his father's rediscovery of his childhood Catholicism after the death of Felix. Dirac himself had  no such solace and had to try to cope with the tragedy entirely without a spiritual crutch. Unable to fathom what he takes to be the religious justification for how a benevolent deity could condone natural disasters - they are part of God's plan, ultimately to the good of humanity - Dirac concludes by dismissing the idea that religion has any place in modern life: 'Any further assumption implied by belief in a God which one may have in one's faith is inadmissible from the point of view of modern science, and should not be needed in a well-organized society.'\n\nThe entire document reveals that Dirac's thinking about morality and religion is suffused with two principal concerns: how these types of knowledge square with scientific observations and how they can be used as a guide to living. This is consistent with the approach of John Stuart Mill, who would have applauded Dirac's suggestion that a personally rewarding faith was sometimes needed to replace the untenable belief in eternal life and for everyone to feel that they are contributing in some way to human progress. Some of Dirac's turns of phrase - his reference to 'a well-organized society' in particular - might be a result of the influence of Mill's French colleague and friend Auguste Comte, the founder of positivism. More likely, Dirac was taking the Marxist line that religion is 'the opium of the people'.\n\nOn Thursday 16 February, dozens of scientists made their way through the London fog in the fast-fading light of the late afternoon. They were heading for the grand Piccadilly home of the Royal Society, in the East Wing of Burlington House, on the site of today's Royal Academy of Arts. This was the headquarters of British science, a stone's throw from many of the city's finest shops and restaurants, a few minutes' walk from the West End theatres.  The audience, including Cockcroft and Walton, probably hoped that the first of the five talks that they would hear would be more exciting than its title: 'Some Results of the Photography of the Tracks of Penetrating Radiation'. Unusually for formal presentations like this, the audience included a posse of journalists - no doubt tipped off by Crowther - most of them probably wondering whether they were wasting their time. If there really was a good story here, why announce it so close to their deadline? It is likely that the newshounds hoped, too, that the handsome speaker at the front of the room was more excited than he looked. Shortly after four-thirty, Blackett rose.\n\nHis talk was sensational. He described his experiment and showed vivid photographs of the showers of charged particles that continually rain down on the planet and yet, until these experiments, had never been recorded on film. Blackett had almost no sense of theatre, but when he projected the photographs of cosmic-ray showers - revealing the hitherto unnoticed showers of particles bombarding the planet from outer space - mouths fell open in disbelief. Although cautious in his interpretation of his pairs of positive and negative particles, Blackett said that they fitted 'extraordinarily well' with the Dirac hole theory. Here, in front of the audience's eyes, was plain evidence for particles emerging out of nothing and for the opposite process, in which electrons and anti-electrons annihilate one another as soon as they meet. Blackett described this as their 'death compact'.\n\nAfter the talk, when the applause had faded, Blackett agreed to give interviews to journalists. Always the perfect gentleman, he stressed that the discoverer of the positive electron was Carl Anderson and that the best theoretical interpretation of the photographs had been given by Dirac. Where, then, was Dirac? He was giving a seminar in another part of Burlington House, unavailable for comment.\n\nThe newspaper reports reflected the excitement of the briefing. Of all the London newspapers, the _Daily Herald_ featured the story most prominently: the headline 'Science Shaken by Young Man's Researches' and 'Greatest Atom Discovery of the Century' was followed by a breathless account of the experiment. It made no mention of Dirac's theory. The anonymous writer excised Occhialini from the story, as did Crowther in the same morning's _Manchester Guardian_ , where he interpreted the discovery using Dirac's theory and used Millikan's colourful term 'cosmic rays'. The _New York Times_ also featured the story on the Friday morning and included a wary quote from Rutherford: 'there seems to be strong evidence of the existence of a light positive particle corresponding to the electron. But the whole phenomenon is exceedingly complex and a great deal of work will have to be done on it.' The reporter did well to extract this quote, as Rutherford did not attend the meeting, having made clear that he mistrusted Blackett and Occhialini's use of Dirac's ideas, which Rutherford believed were nonsense.\n\nNot since Eddington's solar-eclipse announcement thirteen years  before had a talk at the Society made such a splash in the international press. Eddington's shrewd handling of the press had made Einstein an international star, but Blackett's presentation was never going to do the same for Dirac. He had no wish at all to be a celebrity; the very thought of it would have revolted him. And, after Rutherford's guarded comments, few journalists will have been motivated to draw Dirac out of his carapace.\n\nAfter the press reported Blackett's announcement, Anderson was on edge. Most physicists had not read or even heard of his paper on the 'easily deflectable positives', and he had not yet published his photographs in a professional journal. He had not even given the new particle a name. For several months, he and his Caltech colleagues had considered contracting the term 'positive electron' to 'positron' and, at the same time, suggested that the ordinary, negatively charged electron might be renamed the negatron. Other names were forthcoming, too: the astrophysicist Herbert Dingle in London recalled that Electra in Greek mythology had a brother Orestes and so suggested that the positive electron should be called the oreston. It was Anderson, hurriedly completing a long paper on his discovery, who chose the name that stuck: the positron.\n\nThe debate about the positron rumbled on for months. Bohr thought the particle might not be real but caused 'by air current drift' in the cloud chamber. Only after Heisenberg and colleagues went on a skiing vacation in Bavaria with Bohr and took one of Anderson's cloud-chamber photographs did Bohr begin to believe that the positron existed. In California, Anderson wavered and Millikan refused to believe that electrons and positrons were produced in pairs, because the observations did not agree with his theory of cosmic rays. Even in Cambridge, the question was controversial for several months. Rutherford, uncomfortable with the idea that abstract theory could predict a new particle, liked his physics done bottom-up: 'I would have liked it better if the theory had arrived after the experimental facts had been established.'\n\nAlthough few theoreticians accepted Dirac's hole theory, many interpreted the positron's detection as another personal triumph, some once again wearily despairing that it was impossible to compete with him. Tamm, writing to Dirac from Moscow, was unstinting in his praise and even implied that Dirac had given up hope that his prediction would be verified: 'your prediction of the existence of  the [positron] [. . .] seemed so extravagant and totally new that you yourself dared not cling to it and preferred to abandon the theory.' Dirac, privately pleased that his controversial theory had been vindicated by experiments, showed no emotion. He remarked thirty years later, with a detachment that went beyond the Olympian, that he derived his greatest satisfaction not from the discovery of the positrons but from getting the original equations right. In case Dirac should be in the least pleased with himself, Pauli was as ready as ever to bring him down to earth: 'I do not believe in your perception of \"holes\" even if the anti-electron is proved.'\n\nIt was only by the end of 1933 that the majority of quantum physicists accepted that the positron existed, that electron-positron pairs could be created out of the vacuum that the positron had figured in Dirac's hole theory before its detection. Only Millikan, almost alone in standing by his 'birth cry' theory of cosmic rays, held out against the pair-creation idea. But by early 1934, the evidence for the new particle was incontrovertible: the number of positrons detected annually had risen, owing mainly to Blackett and Occhialini's technique, from about four in the previous year to a new annual total of thirty thousand. More importantly, experimenters at the Cavendish and at other laboratories had demonstrated that positrons could be produced at will using radioactive sources on the laboratory bench rather than only as a consequence of showers of cosmic rays bombarding the Earth. Again, Dirac monitored the experimenters' results to see if they agreed with his theory's predictions.\n\nIn hindsight, it was clear that if physicists had taken the Dirac hole theory seriously, the positron would have been detected several months earlier. Anderson later remarked that any experimenter who took the theory at face value and who was working in a well-equipped laboratory 'could have discovered the positron in a single afternoon' using radioactive sources. Blackett agreed. As Dirac appeared to realise later, he must shoulder most of the responsibility for this, as he never advocated strongly that experimenters should hunt for the anti-electron or suggested how they might detect it using apparatus readily available to them. Thirty-three years later, when asked why he did not speak out plainly and predict the anti-electron, Dirac replied: 'Pure cowardice.'\n\nAlthough Dirac believed he had predicted the positron, and talked about it publicly from 1933 onwards, some commentators have  objected that 'prediction' is too strong a word. Even Blackett wrote in 1969 that 'Dirac nearly but not quite predicted the positron,' words that will probably have stung Dirac if he read them. The consensus among today's scientists, however, is that Dirac's role in foreseeing the existence of the positron is one of the greatest achievements in science. In 2002, shortly after the centenary of Dirac's birth, the theoretical physicist Kurt Gottfried went further: 'Physics has produced other far-fetched predictions that have subsequently been confirmed by experiment. But Dirac's prediction of anti-matter stands alone in being motivated solely by faith in pure theory, without any hint from data, and yet revealing a deep and universal property of nature.'\n\nDuring the past seven years, theoreticians had driven most of the progress in physics, but there were now clear signs - particularly from the Cavendish and Caltech discoveries - that experimenters were in the driving seat. Disillusioned with quantum field theory, and having worked for two years without coming up with what he regarded as a strong new idea, Dirac joined Kapitza in his laboratory. It was another unlikely pairing: the most reserved, cerebral theoretician working with the most outgoing, practically minded experimenter. Yet they were like brothers at play.\n\nThey were among the first users of the state-of-the-art facilities in the Mond Laboratory, which Rutherford had arranged to be built for Kapitza in the courtyard of the Cavendish, with funds from the Royal Society. Its opening in early February 1933 was a grand occasion, dozens of trilby-hatted journalists scribbling on their notepads as the procession passed, adding flashes of colour to the grey midwinter afternoon. Dirac was there, in his scarlet gown, watching the proceedings led by Stanley Baldwin, the university's Chancellor and Deputy to the Prime Minister Ramsay MacDonald. During one of the ceremonies, Kapitza pointed to the body of a crocodile carved into the brickwork of the laboratory's main entrance by the modernist sculptor and typographer Eric Gill. Inside the laboratory foyer there was another Gill commission, a bas-relief of Rutherford, a carving that exaggerated the size of Rutherford's nose, making him look like a brother of Einstein. Some artistically conservative authorities in Cambridge were so upset by Gill's depiction that they spent three months trying to have it removed; their anger was diffused only  after Bohr declared the carving to be 'most excellent, being at the same time thoughtful and powerful'. During the furore, Rutherford remained indifferent, claiming that he did 'not understand anything about art'.\n\nDirac and Kapitza conceived a new and potentially revealing experiment to look at how electrons and light interact with each other. As Dirac had seen for himself in Davisson's Manhattan laboratory, when a crystal is struck by a beam of electrons, their paths are bent, demonstrating that electrons can behave as waves. Thus, electrons and light resemble one another in that both behave sometimes as waves, sometimes as particles. Dirac and Kapitza hit on the idea of replacing the crystal with light. Their idea was to reflect light back and forth between two mirrors so that only a whole number of half-wavelengths of light can exist between the mirrors, analogous to the number of half-wavelengths on a rope that is held down at one end and swung at the other. Just as the crystal consists of a regular three-dimensional arrangement of atoms, the reflected light has a regular pattern of allowed wavelengths, so both should be able to bend the path of a beam of electrons. Such an experiment should be a unique probe of the wave-like and particle-like behaviour of both electrons and light. Dirac's calculations showed that it should be possible to detect the electron beam's bending but only if the reflected light is extremely bright, brighter than the best-available lamps. So the state of lighting technology had thwarted the first plans of Dirac and Kapitza to do experiments together. It would not be long, however, before they were back in the laboratory.\n\nIn spring 1933, the _Cambridge Review_ , sober chronicler of the university's affairs, published an anonymous article pointing out that 'the young are now more concerned [with politics] than they have been for a long time past'. The hedonism of the late 1920s had all but disappeared, giving way to alarm about the national economic malaise and the threat of war. Hitler, Mussolini and Stalin were shaking the English out of their indifference to political extremes. Winston Churchill, in the political wilderness, repeatedly warned of the need to rearm, but he was ignored.\n\nAt the Cambridge Union in late February, despite a barnstorming performance from the Fascist Sir Oswald Moseley, the motion 'This House Prefers Fascism to Socialism' was heavily defeated, another  sign that the students favoured Stalin over Hitler. The dons were also turning left, many of them dissatisfied with the unscientific approach taken by politicians to social issues and revolted by the harsh treatment meted out to the unemployed. A few political leaders emerged among the academics, egged on by Jim Crowther, who cleverly promoted his Marxist views without ruffling the feathers of the many scientists who were wary of political commitment. The ones who emerged as the socialist leaders were all workaholic males, able to combine high-flying academic careers with an energetic commitment to politics and, in some cases, effective popularisation. Quietest among them was Blackett, not a Communist but a firm supporter of the Labour Party. He was horrified to see that 'the whole structure of liberalism and free trade is collapsing all over the world', and was struck by 'the paradoxical situation in which so many starve in the midst of so much plenty'. Scientists and engineers had, in Blackett's view, 'produced the technical revolution which has led to this situation', and so ' _must_ therefore be directly concerned with the great political struggles of the day'.\n\nMost influential of all was Bernal, 'the Saint Paul of the science and society movement of the thirties', as one of his colleagues later described him. He later remembered how he was inspired by the Soviet experiment:\n\n[T]here was no mistaking the sense of purpose and achievement in the Soviet Union in those days of trial. It was grim but great. Our hardships in England were less; theirs were deliberate and undergone in an assurance of building a better future. Their hardships were compensated by a reasonable hope.\n\nAlthough Dirac talked politics with Kapitza and Blackett, he seems to have been one of the fellow travellers with the socialist and Communist scientists, never in the vanguard. The political activists were becoming impatient with Dirac's indifference to sharing new knowledge with people outside science: in a short article 'Quantum Mechanics and Bolshevism' in the _Cambridge Review_ , the anonymous author reported on Soviet displeasure with the 'completely non-political character of his work, and its detached tone, divorced from problems and questions of the present day'. In the summer, Bernal included Dirac in his list of intellectual 'culprits' - including Joyce, Picasso and Eliot - who were 'tending to a private dream world', indifferent to the popular accessibility of their work. Dirac  would have pleaded guilty as charged as he regarded it as his job to seek better theories of fundamental particles, not to inform the public about the search. Although he did not attend the annual meeting of the British Association for the Advancement of Science in September 1933, he agreed with its conclusion: scientists have a duty to contribute to public debate and should promote the importance of science and technology in getting the country back on its feet. The community was leaning on Dirac and other scientists of his soloist ilk to speak out.\n\nDirac appears not to have bothered to tell his parents about his success with the positron. Their first excitement that year was a spring visit to Paris, where Betty was studying for her degree. She did not write to her mother but sent regular letters to her father, who was so thrilled when he heard that she might be heading for Geneva that he decided to drop everything and join her. Soon after 5 a.m., on the day after Betty's letter arrived, Charles and Flo headed down to the railway station via the tram, Flo carrying her husband's laden suitcase. She returned home to receive a letter from Dirac inviting her to spend a day with him in Cambridge, and he later paid for her to take a ten-day cruise round the Mediterranean. 'Won't it be funny', she wrote to him from her cabin like a truant schoolgirl, 'if I get home and Pa doesn't know anything about it?' So it turned out: Charles and Betty arrived back at 6 Julius Road in the middle of September, having cabled her in advance, the first communication Flo had received from her husband in eight weeks. This act of abandonment seems to have annoyed Dirac. For at least eight years, he had addressed his postcards home to both parents but, from then on, he addressed them only to his mother.\n\nDirac had spent the summer in Cambridge, trying to understand the infinities that plagued his field theory of photons and electrons and reflecting on the work he had done during the previous year. He had proved the equivalence of his theory to Heisenberg and Pauli's, had discovered the action principle in quantum mechanics, had seen his prediction of the positron verified and had begun a promising laboratory project with Kapitza. This was one of the most distinguished years of work by any scientist in modern times, but Dirac was disappointed. He wrote to Tamm, who had complained that he was going through lean times: 'I am like you in feeling dissatisfied with my  research work during the past year, but unlike you in having no external reasons to blame it on.' He needed a vacation.\n\nAfter hiking and climbing in Norway, Dirac was to attend a conference at Bohr's institute before moving on to Leningrad for the first Soviet Conference on Nuclear Physics, where he was sure to be feted as a star. But it turned out that he would be in no mood to savour the acclaim.\n\nThe atmosphere at Bohr's annual meeting in 1933 was tense and uneasy. It hardly felt right to enjoy a spirited debate about the positron or a cathartic game of ping-pong while Jewish colleagues in Germany were being hounded out of the country. But, with most physicists now convinced of the existence of the positron, Dirac could feel that his confidence in hole theory had been rewarded. Pauli, not wanting to be there to see it, skipped the meeting and went on vacation to the south of France.\n\nBohr organised the usual week-long programme, combining talks at the institute and gatherings at his new home, a mid-nineteenth-century mansion in the south-west of Copenhagen, in the grounds of the local Carlsberg brewery. Set in hundreds of acres of immaculate gardens, this was a grace-and-favour residence, a gift of the Government, who offered it, whenever it became vacant, to the person considered the most distinguished living Dane.\n\nThe physicists at the meeting were in buoyant mood, though Ehrenfest was in poor spirits. Pudgy-faced and overweight, he was losing his grip on physics; for him, the succession of research reports were now a dispiriting agglomeration of detail. Convinced that his own work was worthless, he was looking for a new, less prominent academic position where he could motor in the slow lane. But he had not given up completely: during the discussions, he was still the unselfconscious inquisitor, pressing every speaker towards complete clarity, helping to draw attention away from irrelevancies and towards the saliencies of the new ideas. He was especially close to Dirac at this meeting, and they spent hours talking, keeping a few breaths away from the smokers' fug.\n\nAfter the closing speeches in Bohr's home, the physicists put their luggage in the entrance hall and said their goodbyes. It was the usual bitter-sweet parting, but one delegate seemed especially out of sorts: Ehrenfest, about to catch a waiting taxi, looked flustered and  unhappy. When Dirac thanked him for his contributions to the meeting, he was speechless and, apparently to avoid responding, hurried over to Bohr to say farewell. When he returned, Ehrenfest was bowing and sobbing: 'What you have said, coming from a young man like you, means very much to me because, maybe, a man such as I feels he has no force to live.' Ehrenfest should not be allowed to travel home alone, Dirac thought, but he changed his mind. Abandoning his usual assumption that people mean exactly what they say, he concluded that Ehrenfest meant to say not 'maybe' but 'sometimes' - he _sometimes_ felt that life is not worth living. Trying to say the right thing, Dirac stressed that his compliment was sincere. Still weeping, Ehrenfest held on to Dirac's arm, struggling for words. But none came. He climbed into the taxi, which speedily made its way round the small grassy roundabout in front of the mansion, through the gardens, under the arch of the Carlsberg building and on towards the railway station.\n\nA few days later, Dirac was sailing to Helsinki, playing deck games and relaxing in the sun, en route to the Soviet Union. Since Hitler came to power, the attitude of the USSR towards scientists from other countries had changed: Stalin no longer encouraged his own scientists to mix with foreign colleagues, and such liaisons became a crime, except for Dirac and a small number of other friends of the Soviet Union. Dirac was keen to make light of this when he wrote an ambassadorial letter to Bohr a month before, assuring him of a 'warm welcome from Russian physicists' and noting that the economy there was not depressed: 'the economic situation there is completely different from everywhere else'. Like many other gullible guests, Dirac had virtually no idea of the extent of the starvation and economic tribulations in the Soviet Union since the beginning of the Five Year Plan and the adoption of the collectivisation programme: people went round with string bags in their pockets on the off chance that they should come across a queue. In 1933, the privations were at their worst: the Soviet diet included little milk and fruit, and only a fifth of the meat and fish consumed thirty years before. Almost the only people to eat well were state officials and visiting dignitaries, such as Dirac, who was almost certainly unaware of the cost of the collectivisation programme: about 14.5 million lives during the previous four years, a higher death toll than the Great War. But Dirac knew that times were hard and that even basic items of clothing were  not in the shops: when Tamm said that he would not be able to buy a heavy coat he needed for the coming months of freezing cold, Dirac gave his own coat to him and spent the next winter in England without one.\n\nThis conference was shaping up to be a highlight in Dirac's career, until he heard some appalling news from Amsterdam. Lunchtime in the city's Vondelpark on the last Monday in September had been like any other on an early autumn weekday: the mothers teaching their little children to feed the ducks, the cyclists whooshing past the strolling pedestrians, a few picnickers in the last of the bright afternoon light. But suddenly the calm was shattered by gunshots. A few onlookers gathered round a horrifyingly violent scene: a young boy with Down's syndrome, fatally wounded but still breathing, lying next to a man in his fifties, dead, part of his head blown away. The man was Paul Ehrenfest. Moments before, he had shot his son Wassik but had not quite summoned the will to kill him. Two hours later, the boy died.\n\nIn countless confused seminars on the new quantum ideas, he had done more than anyone else to pick out the diamonds from the mud. He had now been drowned by the wave he had helped to create. Dirac, needing to clarify his own thoughts and feelings, wrote Bohr a four-page letter, describing his last moments with Ehrenfest . Of all Dirac's surviving letters, this is among the longest and most emotionally direct. With the fluency of a novelist, he recalls every detail of his last meeting with Ehrenfest, more sensitive to emotional nuance than most of his colleagues would have believed. He lamented to Mrs Bohr that he should have taken Ehrenfest's last words to him more literally - a shortcoming of which no one thought Dirac capable - and that he should have advised her husband to keep Ehrenfest in Copenhagen. Dirac concluded that he 'could not help blaming himself for what happened'. Mrs Bohr replied with consoling words, thanking him for doing 'so much to make Ehrenfest's last days here as happy as his sad mood allowed'. She added, 'he loved you very much.'\n\nEhrenfest had written a suicide note a month before the Copenhagen meeting - to Bohr, Einstein and a few other close colleagues, though not to Dirac. After declaring that his life had become 'unbearable', he concluded:\n\nIn recent years it has become ever more difficult for me to follow developments [in physics] with understanding. After trying, ever more enervated and torn, I have finally given up in DESPERATION [. . .] This made me completely 'weary of life' [. . .] I did feel 'condemned to live on' mainly because of economic cares for the children [. . .] Therefore I concentrated more and more on ever more precise details of suicide [. . .] I have no other 'practical' possibility than suicide, and that after having killed Wassik. Forgive me.\n\nEhrenfest never sent this terrible note. It was tragic that he did not live to take his place a few weeks later at the Solvay Conference, the climax of almost a decade of research into matter at its most elementary level. Originally scheduled to be about the applications of quantum mechanics to chemistry, the organisers had decided in July 1932 - in the wake of the Cavendish discoveries that year - to switch the theme to the atomic nucleus. It was probably expected that Rutherford would be the cock of the walk at the meeting, but by autumn 1933 nuclear physics had moved on and was aflame with new discoveries, new ideas, new techniques. Rutherford, never one to avoid the limelight, may well have felt eclipsed as he saw the focus of attention turn to others: to America's most flamboyant young experimenter, Ernest Lawrence, and his invention of a high-energy particle accelerator so compact that it fitted on a desktop; to Enrico Fermi and his discovery that slow neutrons could induce some nuclei to undergo radioactive decay artificially; to Heisenberg and his new picture of the typical atomic nucleus as a combination of protons and neutrons, but no electrons.\n\nDirac's intuition was not as sure-footed in this subatomic realm: he disagreed with Heisenberg's view of the nucleus - soon to be in textbooks - just as he did not believe in the existence of Pauli's neutrino. Dirac was most at home when he was teasing out the implications of quantum mechanics, and he was able to do so at the conference, but only after the organisers had been pressed to give him a slot by Pauli.\n\nThis was to be another of Dirac's seminal talks. Having pointed out that the discovery of the positron had renewed interest in the existence of a sea of negative-energy electrons, he argued that the presence of these background particles forces physicists to rethink the concepts of the vacuum and of electrical charge. As Oppenheimer and one of his students had independently suggested, the vacuum  was not completely empty but was seething with activity, vast numbers of particle-antiparticle pairs continually bubbling up out of nothing and then annihilating each other, in fractions of a billionth of a second. These processes of creation and destruction are so brief that there is no hope of detecting them directly, but their existence should cause measurable changes in the energies of atomic electrons. Likewise, Dirac suggested that the charge of an ordinary positive-energy electron should be affected by the presence of the negative-energy sea: the electrical charge of an ordinary electron should be slightly less than the value it would have if the background were absent.\n\nBut the theory was still replete with infinities. Dirac suggested ways of coping with this, using special mathematical techniques to make testable predictions. The audience could see that this was the work of a master, if one who was too clever by half. Pauli despaired of the theory ('so artificial'), while for Heisenberg it was 'erudite trash'.\n\nDirac probably agreed with Pauli and Heisenberg more than he let on, for he knew as well as anyone that his techniques involved the sort of procedures results-hungry engineers would be happy to use but that would make any self-respecting mathematician blanch. Convinced that any fundamental theory worth its salt must make perfect mathematical sense, he was becoming seriously disenchanted with quantum field theory. This Solvay talk would be the last time he used the theory to probe the inner workings of the atom: he would go on to make other fundamental contributions to science, but this presentation marks the end of his golden creative streak, which he had sustained for eight years.\n\nMidway through the autumn term in Cambridge, on Thursday, 9 November, Dirac received the telephone call that most first-rate physicists hope for, if only in secret. A voice from Stockholm told him that he was to share the 1933 Nobel Prize for physics with Schr\u00f6dinger for 'the discovery of new and productive forms of atomic theory'; the deferred 1932 prize went to Heisenberg. Dirac was surprised by his own award but not by the other two, certainly not by the one given to Heisenberg - the principal discoverer of quantum mechanics, in Dirac's opinion. Nervous of the inevitable press attention, Dirac considered refusing the prize, but he soon took  Rutherford's advice: 'A refusal will get you more publicity.' The Dirac family first heard the news on the day of the announcement, soon after ten at night, when a note was slipped through their letter-box by Charles's friend Mrs Fisher.\n\nThe Nobel Prize for physics had been instituted in 1901, when it was awarded to the German experimenter Wilhelm R\u00f6ntgen for his discovery of X-rays. The institution of the prize for physics - and also for chemistry, literature and physiology - was the idea of the Swedish inventor, Alfred Nobel, whose legacy funded the prize in perpetuity. Since the first year, the status of the prizes had grown, and, by 1933, the annual announcements of the winners were featured in newspapers all over the world. As some of the reports noted, Dirac was a special winner: at thirty-one, he was the youngest theoretician ever to win the prize for physics.\n\nMost English national newspapers mentioned Dirac's prize on the day after it was announced. The _Daily Mail_ squeezed in a short report about the award to the 'silent celebrity' next to a long article on 'Hitler's homage to fallen Nazis'. Readers of _The Times_ also read of Dirac's award alongside a report from Germany, where Hitler's deputy, Rudolf Hess, had issued regulations to ensure that electioneering is 'conducted in a dignified manner'. None of the hurriedly prepared articles mentioned the discovery of the positron or captured Dirac's personality; it was left to the _Sunday Dispatch_ later in the month to publish an overheated but insightful description of Britain's newest Nobel laureate. The anonymous author noted that 'more than publicity, [Dirac] fears women. He has no interest in them, and even after being introduced to them, cannot remember whether they are pretty or plain.' Dirac was 'as shy as a gazelle and modest as a Victorian maid'.\n\nThe first congratulatory note to arrive in Dirac's pigeonhole was a telegram from Bohr. Dirac replied with forgivable sentimentality:\n\nI feel that all my deepest ideas have been very greatly and favourably influenced by the talks I have had with you, more than with anyone else. Even if this influence does not show itself very clearly in my writings, it governs the plan of all my attempts at research.\n\nIn the Cavendish, the announcement of the prizes was welcomed by everyone except Max Born, bitter that he had been passed over in favour of Dirac. Others in Cambridge were preoccupied with the  most dramatic event to take place in the town for years: on Armistice Day, three days after Dirac heard from Stockholm, the Socialist Society organised a march of hundreds of students through the centre of Cambridge, seeking 'to provoke clashes, to make a stir [. . .] to put politics on the map and into university conversation; to bounce, startle, or shock people into being interested'. In a normal Armistice Day march, a carnival of several hundred undergraduates walked through the city centre, selling blood-red paper poppies to passers-by in order to raise money for survivors of recent wars and to commemorate the lives of soldiers who had fallen in battle. The tragic aspect of the proceedings was often lost in hilarity, making the occasion ripe for subversion. On that grey Sunday afternoon, the pavements of Cambridge were lined with crowds, jeering as they were passed by marchers, some of them holding the banner pole of the Socialist Society, others bearing a wreath inscribed 'To the victims of the Great War, from those who are determined to prevent similar crimes of imperialism'. The second phrase should be removed, the police escorts insisted, as it might provoke a breach of the peace. By the time the marchers reached the entrance to Peterhouse College, an eruption was inevitable. Onlookers threw flour and white feathers over the students and pelted them with rotten eggs, tomatoes and fish; the marchers retaliated by using a car as a battering ram to push back their tormentors.\n\nThe university authorities panicked. Away from the public posturing, students and dons debated round college firesides whether the marchers had desecrated the day of remembrance or had restored seriousness to what had become a maudlin carnival. The event had marked the beginning of a militant student socialist movement in Cambridge.\n\nIn his rooms in St John's, the Lucasian Professor probably watched the events carefully and pondered how he could make his feelings heard.\n**Eighteen**\n\nFew misfortunes can befall a boy which bring worse consequences than to have a really affectionate mother.\n\nW. SOMERSET MAUGHAM, _A Writer's Notebook_ , 1896\n\nIt has often been said that Dirac hated his father so much that he denied him an invitation to attend the Nobel ceremony. Plausible though the story sounds, it is probably untrue. The Nobel Foundation invited the laureates each to bring only one guest, but they could bring others if the prize-winner paid for their travel and accommodation. Heisenberg took his mother, and Schr\u00f6dinger brought his wife, having left behind his pregnant mistress (the wife of his assistant). So it did not look odd that Dirac went with only his mother. She gave her husband a dose of his own medicine by not telling him about her trip until a few days before she set off, determined to make the most of her time away. She knew that, in only eleven days, she would back at the kitchen sink, the Cinderella of 6 Julius Road.\n\nEarly on the Friday evening of 8 December 1933, Dirac and his mother were in the Swedish port of Malm\u00f6, waiting for the night train that would take them to Stockholm in time for breakfast. A few reporters spent several hours hunting for them all over Malm\u00f6 and eventually tracked them down to a station caf\u00e9, which became the unlikely scene of a press conference. The journalists' persistence was rewarded with a newsworthy interview with two prize eccentrics, 'a very shy and timid boy' and 'a lively and talkative lady'.\n\n'Did the Nobel Prize come as a surprise?' asked one journalist. 'Oh no, not particularly,' Dirac's mother butted in, adding, 'I have been waiting for him to receive the Prize as hard as he has been working.' She was so curious about Sweden that one reporter found himself answering her questions rather than asking his own - here was a woman who revelled in the attentions of the press. Dirac did not stay silent but was unusually forthcoming when the journalist from _Svenska Dagbladet_ asked him how quantum mechanics applies to everyday life and was rewarded with a stream of insights into his unapologetic philistinism:\n\nDIRAC: My work has no practical significance.\n\nJOURNALIST: But might it have?\n\nDIRAC: That I do not know. I don't think so. In any case, I have been working on my theory for eight years and now I have started developing a theory that deals with the positive electrons. I am not interested in literature, I do not go to the theatre, and I do not listen to music. I am occupied only with atomic theories.\n\nJOURNALIST: The scientific world that you have built during the past eight years, does it influence the way you look at everyday occurrences? DIRAC: I am not that mad. Or rather, if it did [have such an influence] then I would go mad. When I rest - that is when I am at sleep of course also when I am taking a walk or when I am travelling - then I make a complete break with my work and my experiments. That is necessary so that there is no explosion here. ( _Dirac points to his head_ ).\n\nThe story of the interview was on the news-stands in Stockholm station when the Diracs arrived shortly before eight o'clock in the morning. A quarter of an hour later, Heisenberg, Schr\u00f6dinger and their guests stepped off the train and were met by a posse of dignitaries, all of them concerned that Dirac and his mother were nowhere to be seen. But when the photographers asked for the laureates and guests to pose, Dirac and his mother stepped forward into the flashes of the awaiting cameras. The welcoming committee was apparently too stunned to ask where they had been and only later heard what had happened: after Dirac's absent-minded mother had failed to wake up when the train reached the station, she had been ejected by a guard, who had thrown her clothes, hairbrush and comb out of the carriage window. After the kerfuffle, the Diracs had made their way to the warm waiting room and had sat apart from the party of officials. When the group left the room, the Diracs followed them like a pair of ducks, without saying a word.\n\nHeisenberg and Schr\u00f6dinger obliged the press with interviews, but Dirac wanted to escape to the hotel as quickly as politeness allowed. He and his mother were accompanied on the short chauffeur-driven journey to their hotel by the Nobel Foundation's attach\u00e9 Count Tolstoy, a grandson of the novelist and a polished diplomat. His first challenge was to sort out the Diracs' accommodation in the 500-room Grand Hotel, overlooking the harbour. The staff must have thought they had done Dirac a favour by putting him and his mother in the bridal suite, but Flo was having none of that and demanded a room of  her own. Dirac - about to pocket his prize money, approximately \u00a3200,000 in today's money - took the cost on the chin.\n\nWhile Heisenberg and Schr\u00f6dinger were relaxing in their baths, Dirac escaped the gaggle of journalists by leaving the hotel surreptitiously, taking his mother with him. They were then free to walk anonymously around the chilly city, in its best suit for the Nobel celebrations, a pre-Christmas festival unique to Stockholm. It looked like fairyland when darkness fell, the firs and Christmas trees lit up with coloured electric lights, the murmurings of the crowd accompanied by the tinkling of lounge pianists and the occasional cry of a seagull overhead.\n\nFlo was not going to be deprived of press attention for much longer. While Dirac was resting, she held court with four journalists, inviting them separately to her suite to talk about her son and to show them the frocks, furs and jewellery he had bought her. The reporters already knew she was a colourful character, but they were not prepared for her torrent of maternal ardour, delivered in words that resembled 'shattering beads of quicksilver', as the _Svenska Dagbladet_ put it. In the interviews, her eyes darted around as she delivered a disjointed, stream-of-consciousness lecture, as if she had been given two minutes to convince them that her son was Superman. One of her targets was the Nobel authorities, who had shamefully credited her son only as 'Dr Dirac' when he is 'the top professor in the world!'\n\nAsked about life at home, Mrs Dirac laid into his father, 'the domestic tyrant', a man who hated wasting time and whose motto was 'work, work, work'. Not mentioning Felix, she described how Charles leant heavily, and unnecessarily, on the young Paul to study, not allowing him to play with other boys: 'If the boy had shown any other tendencies they would have been stifled. But that stifling was not necessary. The boy was not interested in anything else.'\n\nAs a result, Dirac had never known what it was to be a child. None of the journalists appears to have asked her if she took any responsibility for this; it was all the fault of her husband, she thought. When a reporter enquired whether Dirac's father was happy about his son's success, Flo replied disingenuously: 'I would not say so. The father has been surpassed and he doesn't like it.' What of her son's interest in the opposite sex? 'He is not interested in young women [. . .] despite the fact that the most beautiful women of England are in  Cambridge.' The only women he cares for are his mother, his sister and 'perhaps ladies with white hair' (she may have been referring to Isabel Whitehead). Since Flo had vetoed the visit of Felix's girlfriend a decade earlier, possibly before, Dirac had known that his mother feared that young women would be attracted to him, and her attitude had not changed.\n\nOn the following day, the Stockholm news-vendors sold newspapers with headlines that included 'Thirty-One-Year-Old Professor Dirac Never Looks at Girls'.\n\nEarly on Sunday evening, hundreds of coiffed men and women packed the galleries at the Stockholm Concert Hall to witness the King's presentation of the prizes. At 5 p.m. sharp, a blazing chorus of trumpets silenced the crowd before the opening of the two huge doors into the room where the prizes would be awarded. Each of the laureates, escorted by one of the Swedish hosts, marched to their separate armchairs by the platform, covered in red velvet and decorated with banks of pink cyclamen, maidenhair ferns and palms. The national flags of the new laureates hung overhead alongside Sweden's. The prize-winners were in the customary starched white shirt and bow tie, and all of them wore dinner suits, except Dirac, who won the sartorial booby prize by wearing a pitifully old-fashioned dress suit. He bowed low to the King before accepting his medal and certificate and then bowed several times to the crowd amid tumultuous applause. Compared with Heisenberg, Dirac looked pallid and sickly: he looked 'far too thin and stooping', one reporter worried, adding that 'All the motherly ladies warmly hoped that he should feed up and get the time to exercise and enjoy himself a bit.'\n\nAfter the ceremony, the laureates were driven back to the Grand Hotel to attend the Nordic midwinter feast of the Nobel Banquet, in the winter garden of the Royal Salon. Even by the standards of Cambridge this was a spectacular setting for a dinner: the tables, lit with hundreds of bright-red candles in silver holders, were arranged in a horseshoe shape around the water fountain in the centre of the room. There were three hundred guests, every woman in her most scintillating gown, every man in a dinner jacket, except Dirac. At the top table, men were seated alternately with women. On a balcony above, liveried musicians played, in competition with canaries chirruping in their cages near the glass roof.\n\nAfter the speeches, a silent toast to the memory of Alfred Nobel and the singing of the Swedish national anthem, a fleet of waiters began to deliver the first course from a menu that featured game consomm\u00e9, sole fillets with clams and shrimps and fried chicken with vegetable-stuffed artichokes. The climax was the chef's _pi\u00e8ce de r\u00e9sistance_ dessert: ice-cream bombes that shone in the dark after they had been doused in alcohol and set alight. Afterwards, each laureate was expected to make a short speech, customarily a few pieties of gratitude and reflection, laced with self-deprecating wit. After the first speech - given by Ivan Bunin, winner of the prize for literature - Dirac rose from his seat and walked to the rostrum, where, as usual, he shed his shyness. After paying his compliments to the hosts, he declared that he was not going to speak about physics but, instead, wanted to outline how a theoretical physicist would approach the problems of modern economics. This was just the kind of applied thinking that Bernal and his colleagues had been urging Dirac to do, but they might have expected him to choose a different venue for his first public comment on social and economic affairs. Nervous glances were exchanged round the great hall as he leaned over the rostrum and presented an argument that all the economic troubles of the industrialised world stemmed from a fundamental error:\n\n[W]e have an economic system which tries to maintain an equality of value between two things, which it would be better to recognise from the beginning as of unequal value. These two things are the receipt of a certain single payment (say 100 crowns) and the receipt of a regular income (say 3 crowns a year) through all eternity. The course of events is continually showing that the second of these is more highly valued than the first. The shortage of buyers, which the world is suffering from, is readily understood, not as due to people not wishing to obtain possession of goods, but as people being unwilling to part with something which might earn a regular income in exchange for those goods. May I ask you to trace out for yourselves how all the obscurities become clear, if one assumes from the beginning that a regular income is worth incomparably more, in fact infinitely more, in the mathematical sense, than any single payment?\n\nWithout bothering to suggest how his explanation could be tested, he concluded with a Rutherfordian swipe at science popularisers, informing the diners that once they had done their homework, they will have 'a better insight into the way in which a physical theory is fitted in with the facts than you could get from studying popular  books on physics'. After thanking the audience for its patience, he returned to his seat. A spatter of clapping gradually gathered into firm applause, many of the diners laughing nervously and apparently wondering what to make of Dirac's speech. Heisenberg and Schr\u00f6dinger did not follow suit by talking about economics and politics; speaking in German, they gave speeches that followed the convention of steering clear of anything that might be politically controversial.\n\nDirac's reasoning puzzled Schr\u00f6dinger and his wife, and Anny described it as a 'tirade of communist propaganda'. But if the written record of Dirac's speech is accurate, she was being unfair: Dirac was addressing a topic of theoretical economics that transcended politics. He was also wrong: his theory is approximately correct only when interest rates are always low, but he had not taken into account that it makes good sense to take the lump sum if interest rates are high and remain so. If Dirac had bothered to consult a professional economist, such as his Cambridge colleague John Maynard Keynes, he would have been spared posterity's judgement that in his first foray outside his own field he had talked nonsense. And he had done so in the glare of the Nobel spotlight.\n\nDirac's fallacy seems to have gone unnoticed or, at least, unremarked in the after-dinner levity. Flo watched Heisenberg and Schr\u00f6dinger closely as they laughed and joked with the other guests, while Dirac strained to make conversation and occasionally disappeared from gatherings, as if vanishing into thin air. Flo kept a sharp eye on Schr\u00f6dinger, not caring much for his braggadocio: by far the oldest of the trio of physics prize-winners, he kept trying to assert himself as their leader, though Heisenberg and Dirac declined to follow him. She also noticed that Schr\u00f6dinger and his wife 'terribly resent' that he had to share his prize with her son. More to her liking was the genial Heisenberg and his mother, dressed like a Dresden shepherdess. Flo admired Heisenberg for having 'no swank at all', although she thought him a 'terrible flirt', like her son, and she complained that both of them cruised the circles of adoring ladies before they ran 'back to [their] poor, tired mother[s] whenever they have had enough'. She had not previously seen Dirac in the company of admiring young women, and she did not like it: whether or not she noticed, he was drifting away from her.\n\nThe lavish hospitality continued for four days, unabated. Dirac's  only task was to give his Nobel lecture on the Tuesday afternoon, traditionally an opportunity for the laureates to present their work to other academics. Dirac spent most of his twenty-minute presentation on 'The Theory of Electrons and Positrons', describing how quantum mechanics and relativity made possible 'the prediction of the positron'. This was the first time he had referred to his speculation about the positron as a prediction, and he went on to repeat another of his speculations, with more confidence than usual: 'It is probable that negative protons can exist.' Finally, after pointing out the apparent symmetry between positive and negative charge, he hinted that the universe might consist of equal amounts of matter and anti-matter:\n\n[W]e must regard it as an accident that the Earth (and presumably the whole solar system), contains a preponderance of negative electrons and positive protons. It is quite possible that for some of the stars it is the other way about, these stars being built up mainly of positrons and negative protons. In fact, there may be half the stars of each kind.\n\nHe had glimpsed a universe made from equal amounts of matter and anti-matter in which, for some unknown reason, human experience is confined almost entirely to matter. But was this a speculation or a prediction? The audience had good reason to be unsure.\n\nDirac appears to have been unaware that he was not the first to imagine a universe made of both matter and anti-matter. In the high summer of 1898, soon after J. J. Thomson had discovered the electron, the Manchester University physicist Arthur Schuster had hatched a similar idea. In a light-hearted article in a summer edition of _Nature_ , he conceived a universe made of equal amounts of 'matter and anti-matter', based on the bizarre idea that atoms are sources of invisible fluid matter that flow into sinks of anti-atoms.  But Schuster's whimsy lacked substantial underpinnings from reason or observation and so remained a 'holiday dream', as he termed it. Within a decade, it was forgotten.\n\nAfter the Nobel festivities, most of the prize-winners usually return home. But Dirac, Heisenberg and their mothers moved on to yet more celebrations, in Copenhagen. Bohr, probably wanting a piece of the action, threw a grand party in their honour on the Saturday evening at his mansion. Schr\u00f6dinger, not a member of Bohr's inner  circle, declined his invitation and returned to Oxford, where he was living, having fled Germany a few months before. His colleagues in England looked askance at his personal life - he lived with his wife and his mistress - and he, in return, despised the colleges as 'academies of homosexuality'.\n\nDirac's mother had heard many stories about the agreeable life at the court of Bohr, and she was not disappointed. Bohr's was a 'commanding' presence, Flo observed, and she was charmed by his wife Margrethe, whose donnish air was lightened by her daring dress, a green morning frock trimmed with leopard skin and yellow beads. The Bohr residence was looking resplendent: the sprays of winter flowers and ferns, the statues, the cubist painting hanging above the grand piano, the huge windows overlooking acres of garden and woodland. For Flo, this opulence had done nothing to spoil the family, least of all the Bohrs' five playful but well-behaved boys.\n\nBohr was out during the guests' first evening at the house and returned to find that Dirac had been the first to retire to bed. Unwilling to lose precious time, Bohr bounded up to Dirac's room and brought him downstairs for a discussion that lasted into the small hours. She could now see why Dirac held Bohr in such affection: here was an older man, authoritative but not authoritarian, forceful but not intimidating, able to bring out the best in everyone. It may well have crossed Flo's mind that Bohr would have been the perfect father for her son.\n\nThe Bohrs' party would not have disgraced one of the Nobel Foundation's receptions. In the mansion's main hall, three hundred guests sat at tables under the huge glass roof, drinking the endless supplies of champagne, beer and wine and eating the food from the generous buffet. When everyone had eaten, Bohr stood in the centre of the hall and gave a speech in English, subtly ensuring that no one overlooked his contribution to the achievements of his 'young pupils'. Heisenberg replied, in German, but Dirac said nothing; throughout the speeches, he stood behind a pillar. After the toasts, Bohr steered the party into the drawing room for a cabaret from a pink-frocked American singer accompanied by the Danish virtuoso Gertrude Stockman and, inevitably, by Heisenberg at the piano.\n\nDirac will probably have found the celebrations a chore and will have been relieved to spend the next day only with people he knew, a relaxed family Sunday. The many in Cambridge who saw Dirac as a shadow of a man, with no sense of fun, would have been surprised to  see him at ease in the Bohrs' nest, playfully squirting water from an indoor fountain over his mother and Margrethe, both of them laughing and protesting as they tried hopelessly to shield themselves from the dousing. Dirac's Cambridge acquaintances would not have expected, either, that he would happily spend a day larking around with Bohr, his boys and Heisenberg, playing badminton and sleighing on the hills near Copenhagen. In the evening, Dirac reverted to his usual stand-offishness: he sloped off to bed early, not bothering to wish anyone goodnight. But Bohr wanted Dirac to talk shop and so yanked him back downstairs.\n\nOn her return to Bristol late on Monday, Flo was met at the railway station by Betty, who was up until the small hours listening to her mother's account of her 'great and wonderful adventure'. Charles was nowhere to be seen.\n\nFor the rest of his life, Dirac was curious about how he came to win the prize with Heisenberg and Schr\u00f6dinger. The Nobel Foundation, always the essence of discretion, releases the papers concerning each year's prize only after keeping them under lock and key for fifty years. Dirac never did find out about the political machinations that led to the first prizes for quantum mechanics; he eventually learned only that the English crystallographer William Bragg had nominated him and that Einstein had not. Only after Dirac died did it come to light that he had been fortunate to win the prize so young.\n\nIn the first three decades of the prize, the committee that decided the Nobel Prize for physics was biased against theoretical contributions, probably because of Alfred Nobel's wish that his prizes should reward practical inventions and discoveries. The committee, not always well informed about theoretical physics, issued a statement in 1929 that the theories of Heisenberg and Schr\u00f6dinger 'have not yet given rise to any discovery of a more fundamental nature'. Behind the scenes in Stockholm, a long and involved battle was being fought about when to award a prize for the new theory and who should receive it. The Foundation was still arguing about this in 1932, when nominations for Heisenberg and Schr\u00f6dinger were accumulating by the month. By early 1933, the pressure to award a prize for the theory was overwhelming, but there were still disagreements about how to share it. Dirac's name had barely registered with the committee.\n\nBy the time the committee met in September 1933, after the discovery of the positron had become widely accepted, his name was much more prominent. The Swedish physicist Carl Oseen, the most influential member of the committee, had heard from his student Ivar Waller of the quality of Dirac's work. More important, the positron's discovery was viewed as 'an actual fact', an observation that illustrated the utility of Dirac's theory. At the end of the meeting, the consensus was that Heisenberg, Schr\u00f6dinger and Dirac were head and shoulders above the other candidates, including Pauli and Born, and that Heisenberg deserved special recognition for being the first to publish the new theory.\n\nToday, the committee's judgements appear capricious. It would, perhaps, have been fairer to award Heisenberg and Schr\u00f6dinger individual prizes in 1932 and 1933, leaving Dirac to win his own prize a year later, an outcome that Dirac himself would almost certainly have regarded as just. None of this really matters; today, no one doubts that the three physicists honoured in Stockholm in December 1933 deserved their Nobel status. Dirac, Heisenberg and Schr\u00f6dinger are now among the select group of winners that give all Nobel Prizes their special lustre.\n**Nineteen**\n\nTo fast, to study, and to see no woman -  \nFlat treason 'gainst the kingly state of youth.\n\nWILLIAM SHAKESPEARE, _Love's Labour's Lost_ ,  \nAct IV, Scene III\n\nAt the age of thirty-two, Dirac appeared to have everything he could wish for. He was in excellent health, was recognised as one of the best theoretical physicists in the world, had plenty of money and could not have been in a more agreeable job. Apart from worries about his home life, his only problem was that all his friends were men. Most people seemed to take it for granted that Dirac would spend the rest of his life being cosseted in the all-male bastion of St John's College and would die a bachelor. Over the next three years, he would surprise them all.\n\nAs several theoretical physicists guessed, their subject was coming to the end of a golden age. The toolkit of quantum mechanics was now available to solve almost all the practical problems encountered by scientists studying atoms and nuclei. In that domain, the theory worked wonderfully well. But for Dirac and others at the forefront of research, the subject was far from finished: most pressing was the need to find a field theory of electrons, positrons and photons - a theory known as quantum electrodynamics - that is free of infinities.\n\nBased in California, Oppenheimer was an international leader in the field, which he studied when he was not immersed in the _Bhagavad Gita_ and a dozen other books. Early in 1934, Oppenheimer and one of his students had dealt a heavy blow to Dirac's hole theory when they proved that quantum field theory accommodates the existence of anti-electrons without assuming the existence of a negative-energy sea. Oppenheimer sent Dirac a copy of his paper, but heard nothing in reply. In Europe, Pauli and his young student Vicki Weisskopf proved that particles with no spin also have anti-particles, flatly contradicting Dirac's theory, which implied that spinless particles should not have anti-particles because they do not obey the Pauli exclusion principle. Pauli was proud of what he called his 'anti-Dirac  paper' and pleased that he was 'able again to stick one on my old enemy - Dirac's theory of the spinning electron'. Pauli and Weisskopf rendered the concept of the negative-energy sea redundant, and it gradually fell into disuse, as physicists became inured to the idea that each positron was just as real as the electron - there was no need to treat the positron as the absence of anything. But Dirac did not accept this - there are no spinless fundamental particles, he noted unconvincingly, so Pauli and Weisskopf's arguments were academic. For this reason, he continued to use the hole theory, which yielded precisely the same results as theories that dispensed with the sea. His authority ensured that many other physicists followed him, and the hole theory continued to be used, if only as a heuristic device.\n\nWhichever version of quantum electrodynamics physicists used, it was plain that the theory was in trouble. However hard Dirac and his fellow physicists tried, they could not find a way of removing the infinities in the theory, to make rigorous calculations possible. Theoretical physics was 'in a hell of a way', Oppenheimer groaned, though he remained optimistic that either Pauli or Dirac would find a way of rescuing the theory by the following summer. If not, they would have to agree with many others that the theory was beyond salvation.\n\nVisitors to Cambridge, including Heisenberg and Wigner, found that Dirac was not working on quantum field theory but doing experiments with Kapitza in his new laboratory. Dirac was trying to solve a practical problem for some Cavendish colleagues, who needed pure samples of chemical elements. Each atom of every element contains the same number of electrons and protons, but the nuclei do not all have the same number of neutrons: the different varieties of nuclei, each with a characteristic number of neutrons, are known as the element's isotopes. There are, for example, three isotopes of hydrogen: most hydrogen nuclei contain no neutrons at all, but there exist others with one and two neutrons. Rutherford's colleagues needed pure samples of some isotopes for their experiments, but this was difficult, as atoms of naturally occurring samples of elements are a mixture of isotopes, extremely difficult to separate because they behave almost identically in chemical reactions. Dirac thought of a neat way of separating a mixture of two isotopes in a gas, using apparatus with no moving parts. His idea was to force a high-pressure jet of gas to follow a spiral path: the heavier, more sluggish molecules should tend to aggregate on the outside of the  rotating mass of gas, while the lighter ones should hog the inside track. Dirac designed his apparatus for this 'jet stream method of isotope separation', then rolled up his sleeves and built it, having borrowed one of the compressors in Kapitza's store. Once again, he was trying his hand at being an engineer.\n\nHe was surprised by the results. The apparatus did not separate the isotopes efficiently but produced what he later described as 'something like a conjuring trick'. When he pumped gas at six times ordinary atmospheric pressure into a small copper pipe, he found that, after the gas had undergone its spiral motion, it separated into two streams with very different temperatures - one stream was hotter than the other by about one hundred degrees Celsius. During a visit to Cambridge in May 1934, Wigner saw the apparatus and asked Dirac questions about it, but Dirac's replies were terse and unhelpful, causing the mannerly Wigner to take umbrage. Wigner understood that Dirac did not want to speak about the apparatus until he knew what he was talking about and that Dirac was unaware of the convention of parrying ignorance with a polite remark. Dirac thought the temperature difference was caused by the differences in the resistance to flow of the two gases, though it is more likely that the rotational motion tends to separate the faster gas molecules from the slower ones. Dirac spent months collaborating with Kapitza under the approving eye of Rutherford, who thought it augured well for theoretical physics that the Lucasian Professor was soiling his hands in the laboratory.\n\nDuring his discussions with Dirac, Kapitza will have talked a good deal about his friends at Trinity College High Table and the interdisciplinary wanderings of their conversations. What Kapitza did not know was that, from March 1934, one of his acquaintances, whom he and Anna often welcomed to their home, was an MI5 informant. Codenamed 'VSO', the colleague was convinced that 'it would be impossible for a Soviet citizen to go backwards and forwards to Russia unless his value to the Soviet authorities in this country were greater than his value in Russia'. The reports submitted by VSO, flecked with jealous asides about Kapitza's scientific reputation, contained no proof that he was a spy but enough circumstantial evidence to worry the security services. Why was Kapitza so sheepish about admitting, even to a friend, that he held a Soviet passport? The rest home for scientists in the Crimea was open only to Communist Party members, so why was Kapitza allowed to stay there if, as he claimed,  he was not a member? Most suspicious were the clandestine meetings Kapitza had near Cambridge with the new Soviet Ambassador in London, Ivan Maysky. So far as MI5 were concerned, Kapitza was now one of their top suspects.\n\nYet Dirac seems to have aroused no suspicion at all, probably because - to most people - he seemed to be a perfect embodiment of the apolitical, head-in-the-clouds don. If VSO had been as diligent as he was suspicious, he might have wondered why Dirac was able to join Kapitza in the exclusive rest home in the Crimea. But Dirac appears to have entirely escaped the attention of MI5; if they kept a file on him, there remains no public record of it.\n\nThe brutality of Hitler's regime was now clear from press reports, though it seems that Heisenberg made light of them when he visited Cambridge in the spring of 1934 for what turned out to be a fruitless attempt to engage Dirac on the future of quantum electrodynamics. Heisenberg stayed in Born's home and tried to persuade him to return to his homeland. During an afternoon walk in the garden with his host, he mentioned that the Nazi Government had agreed that Born could return to Germany to continue his research but not to teach. His family would not be allowed to go with him. Born, indignant that a close family friend could even contemplate conveying such a message, was furious and broke off the conversation. Only much later could Born bear to listen to Heisenberg describing the privations of trying to be a decent citizen amid the Nazi barbarities.\n\nConditions were no better in the USSR for scientists unwilling to toe the Stalinist line. George Gamow, worried that his support of orthodox quantum mechanics would result in his deportation to a Siberian concentration camp, used his invitation to the 1933 Solvay Conference as a way to escape. He persuaded the Soviet Prime Minister Vyacheslav Molotov to grant him and his wife Rho exit visas and then fled, leaving the Soviet authorities livid. The Gamows arrived in Cambridge in early 1934 and were soon a popular couple, delighting all comers with their friendly vivacity. Rho was a strikingly attractive brunette, with a Garboesque presence that could light up a roomful of the dourest dons. Stylishly dressed with smart accessories, all colour-coordinated with her lipstick, she sometimes looked as if she had walked off a photo shoot for _Vogue._ 9 She smoked one cigarette after another, but this did not put Dirac off; he adored her. The  feeling was mutual, and they soon found ways of doing things together that entailed being alone with each other: she would teach him Russian in exchange for his teaching her to drive. Dirac made steady progress with learning his fourth language, as Rho recorded in the coming months by plotting a graph showing a gradual fall in his 'error index', an undefined concept, Dirac could not help noting. After spending just a few weeks in Cambridge, the Gamows departed for Copenhagen, leaving Dirac bereft.\n\nAccording to private comments Dirac made a few years later, he was not in love with Rho. Nonetheless, their affectionate notes bounced back and forth across the North Sea for months, in a rally of infatuation. 'Please read my letters alone,' she pleaded. She returned the letters he had written in Russian, each one marked with a grade and with his errors neatly corrected in red ink. Hoping that he would approve of her cutting down on her smoking, she asked how many times each day he would like her to think of him; he worried that her memories of him were even slightly harmful to her. They were like cooing teenagers, each desperate not to offend the other and constantly seeking forgiveness. When Rho apologised if she had appeared to be insolent, Dirac reassured her that he was not in the least upset and that, in any case, he 'was not expecting Russian women to be as boring as English ones'. Impatient to see each other again, it would not be long before their wish was fulfilled.\n\nIn the meantime, Dirac continued to learn Russian with a woman teacher who gave him hour-long lessons on Saturday mornings in Cambridge. Her name was Lydia Jackson, a Russian \u00e9migr\u00e9 poet known as Elisaveta Fen before her ill-fated marriage to Meredith Jackson, a Fellow at St John's. Romantic and strong-willed, she felt out of place in Cambridge - no place for assertive women, she thought - and made a living by teaching the language of her homeland. At a gathering of one of London's literary circles, she introduced George Orwell - probably one of her lovers - to the woman who became his first wife.  Jackson liked to talk about the Soviet Union with Dirac, and, by her tantalisingly vague account, it seems that she was more sceptical than he was about Stalin's regime. He rarely spoke about science but did once exchange a few words with her about mathematics: she thought it was a human invention, while Dirac maintained it had 'always existed' and had been 'discovered' by humans. 'Doesn't that mean that it was created by God?' she  asked. He smiled and conceded, 'Perhaps animals knew a little mathematics. '\n\nHer familiarity with Dirac is clear from her letters to him. In one, she commends him for being down to earth, not one of his most lauded qualities: 'I know that you are not as absent-minded as all professors and mathematicians are supposed to be: there must be quite a large chunk of an engineer still in you.' After referring teasingly to a spot of nude bathing she had done in a pond on Hampstead Heath, she gives him some stout advice for the sabbatical he was about to take in Princeton:\n\nBy the way, will you try and _not_ forget all your Russian in the barbarous United States. Please try and read a little from time to time. [. . .] And do remember what I told you about not marrying an American: it would be a fatal mistake! An English girl, of firm but tactful disposition will be most suitable for you. As for a Russian - they are a handful under any circumstances [. . .].\n\nDetermined that no one else would read Dirac's letters to her, she routinely burned them. Their opinions about the Soviet Union, as well as the evidence of whether the relationship became physically intimate, were probably destroyed in those flames.\n\nDirac arrived in Princeton at the end of September, after another hiking vacation with John Van Vleck, this time in the mountains of Colorado. Once again, Dirac provided his friend with more stories of his strangeness, including one in Durango where he was wandering around the town at night, probably wearing what might be kindly described as functional clothing, and was mistaken for a tramp. This would not be the last time Americans would mistake the Lucasian Professor for a vagrant.\n\nIn Princeton, Dirac was working at the Institute for Advanced Study, then a suite of offices in Fine Hall. He and his colleagues in Fine Hall liked to eat at one of the modest restaurants in Nassau Street, the rod-straight road that separates the university buildings on one side from the shops on the other. A faculty favourite was the Baltimore Dairy Lunch, known locally as the Balt, which served wholesome food at low prices, though only to white customers.\n\nOne of Dirac's preferred dining companions was his new colleague Eugene Wigner, the courtly Hungarian who was on a mission to bring  modern quantum mechanics into Princeton. Inexplicably parsimonious, he declared proudly to visitors to his two-bedroom apartment that its furnishings had set him back less than $25, as if it were not obvious. On the day after Dirac arrived in Princeton, neither Wigner nor any other Fine Hall colleague was free for lunch, so Dirac set off alone on the five-minute walk into the town centre. When he entered the restaurant, probably the Balt, he saw Wigner sitting with a woman. Well-groomed and slightly younger than Wigner, and with an infectious cackle of a laugh, she looked rather like him, her face similarly long and angular. She spoke faltering English with the same thick accent, though with none of his reserve, and smoked her cigarettes using a long black holder.\n\nThe woman was Wigner's sister Margit, known as Manci to her friends and family. She was struck by the sight of the slender, vulnerable-looking young man who walked into the restaurant, later remembering that he looked lost, sad and disconcerted. 'Who is that?' she asked her brother. Wigner told her that he was one of the town's most distinguished visitors, one of the previous year's Nobel laureates. When he added that Dirac did not like to eat alone, she asked, 'So why don't you ask him to join us?' Thus began a lunch that changed Dirac's life. His personality could scarcely have contrasted more sharply with hers: to the same extent that he was reticent, measured, objective and cold, she was talkative, impulsive, subjective and passionate - she was the kind of extrovert Dirac liked. They occasionally had dinner together but were not officially dating, perhaps partly because he was distracted by Rho Gamow, who was staying in Princeton, having been left in the care of Dirac by her evidently trusting husband. But these social matters were a sideline: he spent most of his time hard at work in his office in Fine Hall and in the rooms he rented in a grand house on one of the leafy avenues close to Nassau Street. So far as his colleagues could see, for all the interest he showed in women, he could have been a eunuch.\n\nIn Fine Hall, Dirac was accommodated on the same corridor as Einstein, their offices separated only by Wigner's. Einstein was the town's most famous celebrity, after Veblen the first faculty member of the institute. He and his wife had arrived in October 1933 and lived in an apartment before settling in a modest detached house in Mercer Street, about five minutes' walk from the centre of the town, which he  described as a 'quaint ceremonious village of puny demigods on stilts'. Although grateful to be in a safe haven and 'almost ashamed to be living in such peace while all the rest struggle and suffer', he could see his new home town was not free of racism and may have discussed this in his meetings with Paul Robeson, the town's most famous son.\n\nThen fifty-four, Einstein looked older: he shambled around the town in his plain raincoat and woolly hat, avoiding eye contact with fellow pedestrians, especially ones who recognised him. On the day he arrived in Fine Hall, newspaper photographers and a crowd of hundreds gathered to catch a glimpse of him through an open library window. The authorities had to smuggle him in and out of the hall through a back entrance.\n\nVeblen and his colleagues were licking their lips at the thought of Einstein and Dirac working together, but it soon became clear that this was only a dream. The two men respected each other, but there was no special warmth between them, no spark to ignite collaboration. They were studying the same subject, but their approaches were quite different: Dirac was developing quantum theory and was deaf to its alleged philosophical weaknesses; Einstein admired the success of the theory but mistrusted it (during the spring of 1935 he completed his collaboration with his younger research associates Boris Podolsky and Nathan Rosen on a paper that cast serious doubts on the conventional interpretation of the theory). Whereas Einstein was a conservative scientist, Dirac was always ready to discard well-established theories, even ones he had helped to create. Language was another barrier: with only weak English, Einstein preferred to talk in his native tongue, which Dirac spoke only with difficulty (in the company of refugees from Hitler's regime, Dirac relaxed his rule of not speaking German). And Dirac tended to avoid smokers, although Einstein temporarily removed that barrier in late November when he gave up his pipe for a few weeks, to demonstrate his willpower to his wife, who disapproved of the habit. 'You see,' he complained to a neighbour, 'I am no longer a slave to my pipe, I am a slave to dat vooman!'\n\nDirac spent much of this sabbatical writing the second edition of _The Principles of Quantum Mechanics_ , making it less mathematical and less intimidating. The completed version preserved the structure of the original and was more accessible than the first edition, though for all but the most gifted students it was aspirational reading. Most students who wanted to use quantum mechanics to do actual calculations used more practically minded texts, secure in the knowledge that the underlying beauty of the subject was nowhere clearer than in this book, sometimes described as 'the bible of modern physics'.\n\nStill believing that mathematics offered the royal road to the truth about the fundamental workings of nature, Dirac spent much of his time in Princeton learning more mathematics. This led him to find a new way of writing his equation for the electron, by describing its behaviour in a space-time whose geometry is not the standard Euclidean type (in which the sum of the angles of a triangle is one hundred and eighty degrees) but is of a more exotic variety developed by the Dutch mathematician Wilhelm de Sitter. Perhaps this would enable the quantum theory of the electron to be harmonised with the general theory of relativity? The result was a sumptuous piece of mathematics, though one that failed to yield new insights into nature. Dirac had yet to show that his idea - that fundamental physics could be gleaned from promising mathematics - was fertile. No other leading theoreticians had taken much notice of it: they remained pragmatic, taking cues from experiment and trying to learn from the weaknesses and loose ends of the best-available theories.\n\nOne of the most intriguing topics for theorists was radioactive beta decay, in which an unstable nucleus spontaneously ejects a high-energy electron. Early in 1934, Fermi underlined his talent as a theoretician once again, this time by setting out the first quantum field theory of beta decay and giving a clearer understanding of the role of the neutrino. He gave a clear mathematical description of how an atomic nucleus undergoes beta decay, one of its neutrons transmuting into a proton, which remains in the nucleus, while two other particles - an electron and a massless neutrino - are simultaneously created and ejected. This decay was caused by the weak force, a previously unidentified type of force that acts only over extremely short distances, unlike the familiar forces of gravity and electromagnetism. Although Dirac admired Fermi's theory, he did not follow him into the nucleus and its complexities. Dirac was adamant that the best way of making progress was to focus on nature's simplest particles, taking inspiration from the most beautiful mathematics. Time would decide whether such purism was wise.\n\nDirac's colleagues in Fine Hall saw that his fanatical dedication to work was on the wane. He spent most afternoons playing games in the two common rooms, each of them furnished in the style of the best-appointed Oxford University common rooms - plush curtains framing every window, deep-pile carpets on the floor, capacious leather armchairs and imitation-antique tables. During the ritual of afternoon tea, he fruitlessly searched for a way that a king could pass eight opposing pawns and got thrashed by his colleagues in their favourite game, Wei Chi (also known as Go), which he had introduced into Fine Hall a few years before. He was relaxed enough to channel some of his intellectual energy away from the toughest problems in science to games that had no point beyond personal pleasure. The impasse in quantum electrodynamics appears to have sapped his morale: he may have feared that he had fallen victim to the alleged 'Nobel disease', said to prevent prize-winners from repeating the quality of their best work after their return from Stockholm.\n\nOver ice-cream sodas and lobster dinners, Dirac's friendship with Manci deepened. She was a lively, big-hearted conversationalist, and, although she often struggled to find the right words in English, she had the rare ability to make him thaw. Between the long - but gradually shortening - silences, he told her of the pain of his youth, of his brother's suicide, of the father whom he believed had tyrannised him into his defensive silence. Manci also had plenty of private unhappiness to share, telling him that she was an unwanted child, less attractive than her sister, intellectually worthless compared with her brother. Mainly to get out of her parents' house, she married when she was only nineteen. Her Hungarian husband, Richard Bal\u00e1zs, turned out to be a playboy and philanderer, and the marriage was an eight-year calamity mitigated only by the birth of her son Gabriel and daughter Judy. She took the bold step of instigating divorce proceedings and had finally become single again two years before she set sail for Princeton. There had been other men after Bal\u00e1zs, but none of them were around for long, and she was lonely and unfulfilled. She was staying with Eugene for a change of scenery, having promised her children - in Budapest with their governess - that she would be home for Christmas. At thirty years old, she had never felt so free in her life.\n\nAlthough a self-declared 'scientific zero', Manci took a lively interest in international ethics, morals and politics, often impressing  experts with her knowledge but at the same time affronting them with her shameless lack of objectivity. Once she had made up her mind, facts alone were rarely enough to budge it; she seemed to think not just with her brain but with her heart. Religion caused her special anguish. Until 1915, when she was eleven, her family had subscribed half-heartedly to the Jewish faith, visiting the synagogue twice a year, but then had become Lutherans. By the time she met Dirac, she was no longer devoutly religious but appears to have somehow yearned to believe in some kind of deity and did not like to hear religion slighted. She would probably not have welcomed Dirac's view that his religion was simply that 'the world has to improve'.\n\nManci was a keen follower of the arts, and she chivvied Dirac into taking more interest in music, literary novels and ballet. In the evenings, like many people during the Depression, they joined the long cinema queues ready to pay their quarters for a few hours' harmless escapism. They may well have seen some films featuring one of Hollywood's new stars, Cary Grant, rapidly establishing himself as a versatile actor with a gift for playing both comedy and - having thoroughly suppressed his Bristol vowels - the charming, all-American gentleman.\n\nAbout ten days before Christmas 1934, during a journey on the New York subway, Dirac read an unexpected and chilling piece of news. He was in the city to buy an overcoat, to replace the one he had given Tamm fifteen months before. Dreading the Christmas throng of Manhattan and its noisy, bullying traffic, he did not hesitate when Manci offered to go along to keep him company. They agreed to meet in Fine Hall, before driving to Princeton Junction, where they would catch the train to Penn Station. After arriving first at the hall, she took a moment to look in his mailbox and found an airmail letter, which she hurriedly put in her handbag and forgot in the excitement of what was her first trip to the shopping capital of America. When she was sitting next to Dirac in a subway car, clattering and squealing its way towards the Midtown stores, she opened her bag to look for a handkerchief and saw the envelope, which she handed to Dirac. It was from Anna Kapitza in Cambridge, he saw, but it was not just another family chronicle. Manci watched Dirac as he read the typewritten letter, a little over a page long. He turned to her with alarming news - the Soviet Government had detained Peter Kapitza in Moscow.\n\nAnna was desperate. She wrote that her husband's detention was 'a terrible blow to him, almost the severest he ever had in his life', and she pleaded with Dirac for help:\n\nI am writing to you as a friend of K and of Russia and you will understand the impossible situation [. . .] People will talk and the last thing I want is the press to get hold of it. [. . .] I wonder if you could write a letter to the Russian Ambassador in Washington, I feel that is the only way to do anything [. . .].\n\nEarlier, Kapitza had boasted that he was the only Soviet citizen who had unrestricted passage across his country's borders. He had scoffed at his colleagues' warnings that he was courting disaster by returning home each summer for his vacation. Irritated by the defection of Gamow and other Soviet scientists, Stalin's authorities were determined to secure the country's best brains to help build its future. During a trip to the USSR in late September with his wife and children, officials in Leningrad told Kapitza that he must stay in the Soviet Union for the foreseeable future, though his family was free to return to Cambridge. Furious, Kapitza tried to talk his way out of it, pleading unsuccessfully that he could not break faith with his colleagues in England, and was dispatched to Moscow, where he lived in a sparsely furnished room at the Hotel Metropole, with little to do except read, write desperate letters to Anna and go for walks - always under the surveillance of the security police. Rutherford and the Foreign Office had kept the matter secret, in the hope that his detention could be resolved diplomatically. No one, certainly none of the officials in the security services, had expected this: not for want of trying, MI5 had not found any hard evidence that he was a spy.\n\nDirac was still digesting the news when he was trying on overcoats in Lord and Taylor, one of the exclusive stores on Fifth Avenue. Manci had an uphill struggle to persuade him, devoid of dress sense, to take the purchase of the coat seriously. No doubt seeing an opportunity to refurbish his entire wardrobe, the salesman asked Manci discreetly whether Sir would also like a new suit, but Manci smiled and shook her head: to press him to buy more than he needed would be futile. The coat he bought there turned out to be a good investment - it lasted him to his death, a memento of the day he heard about Kapitza's plight and was moved to take political action for the first time in his life. Though he knew that he had none of the interpersonal skills and tact needed to be an effective diplomat, he became the de facto coordinator of the American-based campaign for Kapitza's release.\n\nIn Princeton the next day, Dirac urgently sought advice from the well-connected Abraham Flexner and from Einstein, who promptly agreed to help. Dirac was confident enough to write to Anna Kapitza in Cambridge to assure her that matters would 'all come right in the end'. After the Christmas vacation, he would begin his campaign for Kapitza's release, but first he wanted to take a vacation in Florida. He was planning to go on his own, but Manci had other ideas: seeing an opportunity to spend some time alone with her new friend, she postponed her return to Hungary until after Christmas, breaking the promise she had made to her children.\n\nDirac and Manci motored down in early January from freezing Princeton to the warmth of St Augustine, a resort on the north-east coast of Florida. No one - except, possibly, Wigner - knew that they were together. The vacation appears to have been platonic. Their letters before and after the trip show that they were not yet close and still viewed each other differently - he regarded her only as an agreeable companion, but she saw him as a potential husband. They spent their week dodging the rainstorms and taking trips to the local tourist destinations, including a farm where Dirac spent a few dollars buying a baby alligator that he mailed anonymously to the Gamows in Washington, DC. As Rho opened the package in their hotel room, the alligator jumped out and bit her hand - one of her husband's less amusing practical jokes, she thought. Gamow protested that he had nothing to do with the prank; he thought it was a crocodile, a symbol of his favourite experimenter, sent by someone with more playfulness than common sense. A month later, Dirac owned up, and the poor alligator languished, and a few months later died, in the Gamows' bath.\n\nBy the spring of 1935, the campaign for Kapitza's release was not going well. In Cambridge, Anna could see the vultures circling: several of her husband's colleagues in the town privately wanted to see Kapitza get his comeuppance after the years he had spent shamelessly fawning on the Crocodile. There were whispers that Kapitza was merely an engineer, that his experiments were leading nowhere and that he had received financial rewards in return for spying for the  USSR. Anna's reports drew from Dirac some uncharacteristically direct advice: 'You should not pay attention to stupid stories that no one believes in.'\n\nKapitza's Marxist friends sat on their hands, while Rutherford led a discreet campaign for his release. Seeking advice from colleagues all over Europe and working closely with Soviet officials and with the British Foreign Office, Rutherford wanted a face-saving solution. He sought to give Kapitza the option of working wherever he liked, though he confided in a letter to Bohr that he was certain Kapitza wanted to return to Cambridge, adding that he found the Soviet authorities particularly mendacious. The first Cambridge scientist to visit Kapitza was Bernal, accompanied by his lover Margaret Gardiner, and they spent long afternoons trying to cheer him up over pancakes with caviar and soured cream, washed down with wine. 'I feel like a woman who has been raped when she would have given herself for love,' Kapitza sulked. He used the phrase repeatedly.\n\nGardiner had mixed feelings about Moscow, disturbed by the giant posters of Stalin all over the city and the quarter-mile queues that formed outside the shops the moment new supplies arrived. The Moscow hotels were just as bad as their reputation had led her to believe: rooms heated to a tropical swelter, shabbily dressed waiters pretending to be in a hurry, many of them cadging illegal gratuities. The Muscovites walked around their grey, freezing city wrapped in their padded jackets and fur coats, wearing their derigueur galoshes. Gardiner believed that the country's hopes lay in mass education, always an attractive vision for the English left. Decades later, she recalled seeing a platoon of young soldiers marching towards the Military Academy with exercise books under their arms. Her tour guide explained: 'They are having their illiteracy liquidated.'\n\nAfter Manci's departure in mid-January 1935, Dirac's routine in Princeton was unchanged. Each morning, he trudged through the snow from his rented home near Nassau Street to his room in Fine Hall, worked alone all morning, and had lunch at Newlin's restaurant with Wigner and with one of Princeton's most unusual visitors, the Belgian theoretician Abb\u00e9 Georges Lema\u00eetre. He was an amateur scholar of the playwright Moli\u00e8re, an accomplished interpreter of Chopin and the only member of the physics department to wear a  dog collar. Dirac had first seen him, but had apparently not met him, in October 1923, when he began his studies and when Lema\u00eetre was one of Eddington's postgraduate students. Four years later, Lema\u00eetre had introduced into science the idea that the universe had begun when a tiny egg, a 'primeval atom', suddenly exploded into the matter of the universe. Quite independently, the Russian mathematician Alexander Friedmann had applied Einstein's general theory of relativity to the universe as a whole and demonstrated that some mathematical solutions of the equations correspond to an expanding universe, though his work was published only in Russian and at first went unnoticed.\n\nThe Friedmann-Lema\u00eetre picture of the universe's birth seemed to be at odds with the account of creation in Genesis, but this did not bother Lema\u00eetre, who believed that the Bible teaches not science but the way to salvation. The science-religion controversy 'is really a joke on the scientists', he said: 'They are a literal-minded lot.' Dirac found Lema\u00eetre 'quite a pleasant man to speak with - not strictly religious as one might expect from an Abb\u00e9'. It was probably during these conversations in Princeton's diners that Lema\u00eetre reawakened Dirac's interest in cosmology, the study of the entire universe and its workings, soon to become one of his main interests. For now, he focused on mathematics and quantum physics, which he studied during the day, and he took it easy in the evening. After dinner, he would read one of the books Manci had recommended to him (including _Winnie the Pooh_ ) or go out, perhaps to a movie with the von Neumanns.  Probably as a result of Manci's encouragement, he had become much more interested in music: a highlight of the term for him was a university concert, where he heard a searching performance of Beethoven's last piano sonata by the Austrian virtuoso Artur Schnabel, another Jewish refugee from Hitler's Germany.\n\nManci was with her children in Budapest. About once a week, in her spidery hand, she wrote several pages of news and gossip for Dirac, urging him to keep in close contact. Unaccustomed to receiving warm and attentive letters, he struggled to respond: 'I am afraid I cannot write such nice letters to you - perhaps because my feelings are so weak and my life is mainly concerned with facts and not feelings.'\n\nManci, 'very much upset' by this statement, knew that she would have to take the initiative if she were to stir in him the first quantum of romance. Always wearing her heart on her sleeve, she wrote to  Dirac about her family and bombarded him with questions about his life in Princeton in all its minutiae. His reply was chilling: 'You ought to think less about me and take more interest in your own life and the people around you. I am very different from you. I find I can very quickly get used to living alone and seeing very few people.'\n\nHe sent her lists of corrections to her English and answered her queries as tersely as a speak-your-weight machine. When she sent him photographs of herself, he was grateful but critical: 'I do not like this picture of you very much. The eyes look very sad and do not go well with the smiling mouth.' After she complained that he did not answer all her questions, he re-read her letters, numbered them and sent her tabulated responses to every question he had ignored, including:\n\nLetter number| Question| Answer  \n---|---|---  \n5| What makes me (Manci) so sad?| you have not enough interests.  \n5| Whom else could I love?| You should not expect me to answer this question. You would say I was cruel if I tried.  \n5| You know that I would like to see you very much?| Yes, but I cannot help it.  \n6| Do you know how I feel like?| Not very well. You change so quickly.  \n6| Were there any feelings for me?| Yes, some.59\n\nWhen Manci received the list, she thought Dirac was jeering at her but eventually decided that it was 'quite funny'. Beginning to realise that Dirac did not understand rhetorical questions, she seethed: 'Most of them were not meant to be answered.' It is easy to imagine her tearing out her hair in frustration. But his answers gave her an opportunity to engage with his feelings, and she did not hold back: for his statement that she changed so quickly, she told him he should get 'a second Nobel Prize, in cruelty'. Manci was tough, but she made sure that Dirac was aware of her vulnerable and sensitive side: 'I am only a stupid little girl.' With each letter, she flirted more  audaciously, but Dirac made no comment until he realised that he was being targeted. He snapped: 'You should know that I am not in love with you. It would be wrong for me to pretend that I am. As I have never been in love I cannot understand fine feelings.'\n\nBut Manci was not to be deflected. Although Dirac parried her repeated requests to join him during his forthcoming trip to Russia, she was determined to see him before the summer was over.\n\nThe news of Kapitza's detention first appeared in the British _News Chronicle_ on 24 April 1935, after a leak. Soon, Kapitza's case was well known in the British media, and the newspapers featured long reports on the experiments he had been doing in Cambridge. In interviews with journalists, Anna Kapitza was distraught. 'The whole affair has caused great mental pain to both my husband and myself,' she complained, adding that she was concerned about the effect of the upheaval on her highly strung husband: 'in his present state of mind he is not in a position to do any serious work'. Yet she was underplaying his distress: 'Sometimes I rage and want to tear out my hair and scream,' he had written to her. Life in the Moscow science community was dismal for him as most of his former friends there were shunning him until they knew officially, from Stalin's office, whether Kapitza was one of the 'enemies of the people'. His country's reward for his scientific success and for not making a fuss was, he wrote to Anna, to treat him 'like dog's excrement, which they try to mould in their own way'. He knew his letters would be intercepted and read by the police, so he lambasted the agents of his captivity, not the Soviet system that employed them:\n\n'Not only am I sincerely loyal, but I have deep faith in the success of the [plans for] new construction [in the Soviet Union] [. . .] But even in spite of my cursing, I do believe that the country will come out of all these difficulties victorious. I believe it will prove that the socialist economy is not only the most rational one, but will create a State answering to the world's spiritual and ethical demands. But, for me as a scientist, it is difficult to find a place during the birth pangs.\n\nBut the Soviet Government had plans to keep Kapitza busy and to give him all the material goods he could wish for. It decided to set up a new Institute for Physical Problems, to make him the founding director, to give him a salary most academics would envy and then to  throw in some generous perks, including an apartment in Moscow, a summer house in the Crimea for his family and a brand new Buick. From the vantage point of the sofa in his hotel room, however, the future looked so bleak to Kapitza that he considered suicide. His depression was relieved only by trips to the theatre and the opera and by colour reproductions of his favourite modern art pinned to the blank walls. But C\u00e9zanne, Gogol and Shostakovich offered only meagre consolation: he longed to return to his experiments in the Mond Laboratory, to be with his family and friends in Trinity College.\n\nOn the day news of Kapitza's detention broke in the UK, Dirac was relaxing with the Gamows in Washington, DC. On a fine warm day, the three of them took a forty-minute trip in an airship over the city and looked down on the cherry blossoms in the fullness of their second bloom and on Capitol Hill, where FDR was pushing through his controversial New Deal. Dirac was about to tread the streets of the capital as an unlikely lobbyist, having accepted Anna's suggestion that he should approach the first Soviet Ambassador to the USA, Stalin's friend Aleksandr Troyanovsky.\n\nDirac was officially in Washington to attend three consecutive conferences, where he spent most of his time publicising Kapitza's difficulties and collecting signatures to petition for his release. Every delegate approached by Dirac agreed to sign, including L\u00e9o Szil\u00e1rd, who hatched a ludicrous plan to smuggle Kapitza out of Russia by submarine.\n\nBefore Dirac could present the petition, some groundwork had to be done. He arranged for a letter to be written to the Ambassador from Karl Compton, brother of the famous experimenter and President of the Massachusetts Institute of Technology. Compton declared that Kapitza's absence from Cambridge 'is universally considered by scientists to be a major catastrophe' but suggested that his return 'would be universally acclaimed in the scientific world'. The letter did its job: Troyanovsky quickly agreed to receive both Dirac and Millikan. Dirac later explained to Anna Kapitza why he wanted to be accompanied by Millikan: '[he] is known to be rather opposed to the Soviets but that would be counterbalanced by my being known to be rather in favour.'\n\nThus, on the last Friday afternoon of April 1935, Dirac - for a decade regarded as an asocial misfit, out of touch with world affairs  \\- found himself walking to the Soviet Embassy with America's pre-eminent scientist-diplomat. The embassy, just north of the White House, was looking magnificent: Moscow museums had supplied antique furniture, paintings and rugs as contributions to its renovation.  After waiting in the reception room, dominated by a statue of Lenin, Dirac and Millikan shook hands with the lantern-jawed Troyanovsky, whose charm and accommodating manner had made him popular on the city's social circuit. The half-hour meeting was cordial and relaxed. Over a cup of tea, the Ambassador admitted that he had heard of Kapitza's case only when he read Compton's letter and described the Soviets' hurt when some of its most eminent citizens had failed to return home after travelling abroad. Millikan told him that Kapitza's health was deteriorating and suggested that the Soviet Union should bear in mind public opinion in other countries as well as its own. The continued detention of Kapitza would seriously damage relations between Soviet and American scientists, Millikan concluded. As the meeting drew to a close, Dirac spoke up and pleaded for Kapitza's release, in words he recalled the next day in a letter to Anna Kapitza: 'I have known Kapitza very well for a long time and I know him to be thoroughly reliable and honest [. . .] If he were let out under a promise to return he could be depended on to keep that promise.' The Ambassador ended with an assurance that he would raise their concerns with the Soviet Government, so, Dirac told Anna, he left the meeting feeling hopeful.\n\nYet there was more to do. After the meeting, Millikan wrote to the Ambassador to reiterate the points he and Dirac had made, ratcheting up the diplomatic pressure. Dirac collected the last of the petition's sixty signatures, which included those of almost all the leading physicists in the USA, including Einstein. Flexner had agreed to send another petition, addressed to the American Ambassador in Moscow, who would be asked to present it to the Government. Dirac concluded his letter to Anna: 'I feel sure the Soviet Government will do something about it when they see how widespread is the feeling against them. If they don't, you may rely on me to do all I can when I am in Russia to get Kapitza out in any way.'\n\nA few days later, at the beginning of June, Dirac left Princeton. Compared with his most successful stays in Copenhagen and G\u00f6ttingen, this sabbatical had been largely a scientific washout, but for good reasons. He had invested some time in his relationship with  Manci, but it was small compared with his commitment to secure Kapitza's release. Even at the cost of stalling his work, Dirac was not going to abandon his surrogate brother.\n**Twenty**\n\nSTALIN: You, Mr Wells, evidently start out with the assumption that all men are good. I, however, do not forget that there are many wicked men.\n\n'A Conversation between Stalin and [H. G.] Wells', _New Statesman_ , 27 October 1934\n\nMoscow was beckoning again. For the following four months, Dirac's diary was empty, and he was determined to spend most of that time with Kapitza. Dirac knew that the secret police read his letters to Anna Kapitza and that he would probably be followed when he was in Moscow. He told her, 'If anyone follows me around in Moscow he will get some long walks.'\n\nDirac and Tamm had intended to spend the summer hiking and climbing together in the Caucasus, and Dirac hoped to see one of the allegedly productive factories and the new Dneproges hydroelectric power station, one of the proudest achievements of Soviet engineering. But when Anna Kapitza asked Dirac to cancel the trip in order to support her husband, Dirac shelved his plans and declared himself to be at the service of her and her husband: 'I am ready for anything. ' He travelled to Moscow via Berkeley, where Oppenheimer found that Dirac was as tight-lipped as ever about physics. Two of Oppenheimer's students were elated when he told them that their British guest was prepared to hear their ideas about quantum field theory, which built on his work. During their fifteen-minute presentation, Dirac said nothing. Afterwards, the students braced themselves for his perceptive comments, but there was an agonisingly long silence, eventually broken by Dirac when he asked them, 'Where is the post office?' The students offered to take him there and suggested that he could tell them what he thought of their presentation. Dirac told them, 'I can't do two things at once.'\n\nOn the afternoon of 3 June 1935, Dirac waved goodbye to Oppenheimer and boarded the Japanese MS _Asuma Bura._ 4 He settled into his private cabin and prepared to sail through the mist to San Francisco - catching sight of the half-constructed Golden Gate Bridge - and then on to Japan, China and the USSR. Manci,  meanwhile, was lounging around in Budapest, awaiting the arrival of her first car, a six-cylinder Mercedes Benz bought for her by her father. She had persuaded Dirac to visit her in Budapest at the end of his trip. Her complaints that he didn't respond to her questions drew another tabulated response:\n\nHaveyou played ping-pong with pretty girls? ping-pong| With one pretty girl. Most of the passengers were Japanese, and Japanese girls do not play ping-pong.  \n---|---  \nHave you flirted?| No. She was too young (15 years old). But you ought not to mind if I did. Should I not make the most of what you taught me?  \nWhy were you so derisive?| I am sorry, but I cannot help it at times.\n\nSix weeks after he had set sail from the USA, Dirac arrived in Moscow railway station. Even he, with his Gandhian indifference to his surroundings, must have been struck by the contrast between the fresh, early summer air of Princeton and the stench of rotten eggs that hung over the Soviet capital. It was no longer the city that he had seen four years before but a reeking, overcrowded metropolis. The playwright Eugene Lyons described the 'viscous ooze of [Moscow's] dung-coloured people, not ugly but incredibly soiled, patched, drab; the odour and colour of ingrained poverty, fetid bundles, stale clothes'. Dirac stayed there only briefly: he had arranged to spend most of his time in the more agreeable ambience of the Kapitzas' _dacha_ (summer home) in the village of Bolshevo, thirty-five miles south of the city. Kapitza was looking forward to seeing his English friend, though the tone of his comments to his wife indicates that he did not fully reciprocate the intensity of Dirac's affection. But a day after Dirac's arrival, Kapitza appeared to have changed his mind. He wrote to her:\n\n[We] came here with Tamm and have been walking, boating and talking ever since. I haven't had such a pleasant time with anyone up to now. Dirac treats me so simply and so well that I can feel what a good and loyal friend he is. We talk about all sorts of things and this has been very refreshing. [. . .] Dirac's arrival has revived my memories of the respect and reputation I enjoyed in Cambridge [. . .]\n\nThe two friends relaxed together for almost three weeks. Kapitza's abject morale had not improved when he heard that the Soviet authorities had, for unknown reasons, sent 'Dimus' Ivanenko into exile. It was a familiar story, though no one dared to question Stalin's policy in public. Kapitza was considering giving up physics and changing the subject of his research to physiology so that he could work with Russia's most senior scientist, the elderly but still active Ivan Pavlov. Within the modest compass of Dirac's verbal skills, he tried to lift Kapitza's spirits, and in return Kapitza - evidently knowing nothing of Dirac's friendship with Manci - tried to fix him up with a young girl they met, a good-looking, English-speaking language student. Dirac did not respond.\n\nDuring his stay, he met the Trinity College physiologist Edgar Adrian and other British colleagues asked by Rutherford to assess Kapitza's situation and his psychological state. The Soviet Government supported this visit, presumably to demonstrate their flexibility. But, by the time Adrian and his colleagues met with Kapitza, the die was cast: Kapitza had been forbidden to return to Cambridge, and it remained only to secure the best terms for him to work in his new institute. When Dirac left Moscow at the beginning of September, he knew that he had lost his first diplomatic battle; he would have to become accustomed to living in Cambridge without the man he thought of as his closest friend.\n\nThe final stage of his trip was an antidote to his disappointment: he was to visit Manci in Budapest. She was living with her children in an apartment in what had been Archduke Frederick's house, a short stroll from her parents' sumptuous residence opposite Count Batthy\u00e1ny Park. This was a world of plenty - fine food, exquisitely cut clothes, attentive servants and private concerts in the living room. Dirac's modest origins in Bishopston were part of another world. Manci took her material comforts for granted, but she was unhappy and longed to get away from her parents, who must have been taken aback by the arrival on their doorstep of an unkempt Englishman who knew hardly a word of Hungarian. They knew next to nothing about him and surely cannot have expected that their feisty, outspoken daughter would choose such a diffident man. But they liked him and could see that Manci and Dirac clicked during their nine days together, driving around the city in her new car, sightseeing and soaking in the famous indoor public  baths. When he returned to Cambridge, he wrote to Manci: 'I felt very sad when leaving you and still feel that I miss you very much. I do not understand why this should be, as I do not usually miss people when I leave them. I expect you spoil me too much when I am with you.'\n\nManci was making progress. But three weeks later, her heart sank when she read the final entry in Dirac's latest table of unanswered questions: to her query 'Do you miss me a little?', he responded, 'Sometimes.'\n\nWhen Dirac returned to England in the early autumn of 1935, the country was still disfigured by unemployment and worried by Hitler's aggressive rearmament, Mussolini's sabre-rattling in East Africa and Japan's occupation of Manchuria. 'I would like to kill the politicians of middle Europe,' Manci fumed. Dirac was soon back in his Cambridge routine, but the thrill had gone. Although he had not given up on quantum electrodynamics, he seemed to be getting nowhere. Dirac thought a revolution was needed and probably wondered whether he, now thirty-three, might be too old to be one of its leaders.\n\nRutherford had negotiated a deal that involved moving almost every item of Kapitza's equipment to the Institute for Physical Problems, enabling him to resume all his experiments there. Anna had made Dirac a guardian of the Kapitzas' sons, and he took his duties seriously, taking the boys out at the weekends for rides in his crumbling car and organising his first fireworks display for them on 5 November.  These were good times for Dirac, but he was preparing for yet more loneliness: the Blacketts had left for London, Chadwick for Liverpool, Walton for Dublin, and now the Kapitzas were about to depart for good. Dirac was not the self-sufficient eremite that many people believed him to be: he needed new companionship, and he knew it. Manci was eager to oblige, but he was wary of her forwardness, as he showed when she telephoned him one night late in November as he was preparing to go to bed. She thought he would be delighted to receive an unexpected call from her but he was angry and shaken. The college telephone system was arranged so that the porters heard their stilted conversation, as he explained to her in a brusque note. Surely it was sufficient to communicate only by letter, he wrote, with all the warmth of a tax  inspector. She swiftly replied, making clear what she thought of his secretiveness: 'ridiculous'.\n\nIncidents like that rattled him: could he live with someone who had so little sympathy with his need for privacy? He will have had no wish to be party to a disastrous marriage, like his parents', which he had seen in all its unpleasantness two months before, during another rain-soaked visit to Bristol. Charles and Flo were living out their marriage contract in an unwinnable endgame of squabbles and recriminations. Divorce was out of the question for the born-again Catholic Charles, but when he read his copy of George Bernard Shaw's _Getting Married_ , he may well have sympathised with the author's recommendation: 'Make divorce as easy, as cheap, and as private as marriage.' Flo would probably have welcomed a divorce, but the shame would have been too much for her. So they both remained unhappily shackled to each other, with nothing to look forward to except more arguments. Flo told her son that her pleasures were limited to taking long walks on the Downs, sitting alone in the parks and attending meetings of the new Bristol Shiplovers' Society. 'I have made an awful mess of my life somehow,' she wrote, adding that she blamed herself: 'What we sow, we reap.'\n\nDirac's mother appears to have had no more than a passing interest in his work, but his father struggled hard to understand it. Charles looked through the journals in the library, searching for readable accounts of quantum theory, hoping to absorb some of their content by writing out paragraphs of difficult technical prose, verbatim. He kept a record of his findings in a small, red notebook, on whose front cover he had written a two-inch-high letter P. The desultory references and notes inside are heart-rending records of a keen but confused amateur, unable to make any headway in a subject he longed to understand. Charles had written, in his rheumatic hand, some of the most complimentary comments about his son, highlighting some of the most generous ones: 'Dirac stands out amongst his contemporaries in this field for his originality.' Apart from a summary of one of Crowther's articles on 'New Particles', Charles had not tracked down any of the lively and accessible accounts of quantum mechanics by Eddington or any of the other accomplished popularisers. It seems that his son was giving him no help at all.\n\nWith Bristol's long tradition of adult education, it was easy for the city's citizens to find out about new science. Arnold Tyndall, who  gave Dirac his first introduction to quantum theory, was a popular performer at the night classes on science organised by the university. During one of his courses, a male student caught the eye of the genial Tyndall. Much older than the other students, he always sat at the front, taking careful notes. At the end of the final lecture, he shuffled up to Tyndall to thank him. 'I am glad to have heard all this. My son does physics but he never tells me anything about it.' The student was Charles Dirac.\n\nIn the early summer of 1935, Betty had finished her French course and had come bottom of her class, taking a third-class honours degree, as Felix had done. She wanted to be a secretary and to get out of Bristol as quickly as she could. Charles was now open about his relationship with Mrs Fisher, Flo told Dirac: 'I wish he would go and live with her, folks are always seeing them about together and tell me [. . .] He has always had someone ever since I've been married: Betty says it is French.'\n\nDirac's mother, preparing to go on another Mediterranean cruise alone, sensed that her daughter was growing away from her. In a few weeks, she would temporarily move to London, not leaving her mother a forwarding address. But first Betty went on an August vacation with her father, keeping their destination secret. They were travelling with a group of Catholic priests on a pilgrimage to Lourdes, in the French Pyrenees, where Charles may, to try to rid himself of his ailments, have bathed in its reputedly miraculous waters. He knew that his daughter would pray for him but that his wife and son were, at best, indifferent to his fate.\n\nDirac would probably have been happiest if, like Einstein, he had never supervised a graduate student. It was not until the 1935-6 academic year that Dirac first officially became a research supervisor, taking on two students Born left behind when he moved to Edinburgh to take up a professorship. Dirac had almost none of the skills that he had seen in Fowler: the ability to set problems pitched at the right level for his students, to motivate them in lean times and to support them in the early stages of their career. Dirac believed his only obligation was to point his students towards an interesting theoretical concept and then to look over any work they produced in consequence; it was up to the student to take almost all the initiative. Only the cleverest and most independent-minded students could  flourish under such a regime, as the Cambridge authorities knew. Dirac knew it, too, and showed no interest in recruiting apprentices. But several of the finest young minds sought his guidance, including the Indian mathematician Harish-Chandra and the Pakistani theoretician Abdus Salam, both part of a pattern - the great majority of Dirac's successful students were foreign.\n\nDirac encouraged his students to keep abreast of the latest publications in theoretical physics and also to keep an eye on the experimenters' latest findings. But his faith in the veracity of new experimental results was badly shaken by an incident that began in the autumn of 1935. Dirac heard that the Chicago experimenter Robert Shankland had found evidence that sometimes energy is not conserved, contrary to one of the fundamentals of science: when photons are scattered by other particles, he found the particles' total energy before the collision is not the same as it is afterwards. Setting aside his preference to be led by mathematics rather than experiment, Dirac smelt an impending revolution and in December wrote to Tamm, spelling out the consequences of Shankland's findings. First, the neutrino would no longer be needed, as Pauli had based his entire argument for its existence on the energy-conservation law. Second, and more important, as Shankland's experiment involved light, his results might be a hint that energy is not conserved whenever particles collide at speeds close to the speed of light. If so, Dirac pointed out, it would be reasonable to retain the basic theory of quantum mechanics, which applies to comparatively slow-moving particles, though the relativistic extensions of the theory, such as quantum electrodynamics, would have to be abandoned. A few days later at the Kapitza Club - still meeting despite its founder's absence - Dirac gave a talk on the implications of Shankland's results. To most physicists, the experiments looked unreliable, and it seemed wise to wait for the results to be checked independently. But Dirac could not wait: in January 1936, he set out the implications of Shankland's results in a short, equationless article in the journal _Nature_ , addressing his comments to the entire scientific community. If Shankland was right, Dirac said, quantum electrodynamics would have to be abandoned, adding, 'most physicists will be very glad to see the end of it'. Coming from one of the discoverers of relativistic quantum mechanics and field theory, these were striking words. Heisenberg privately dismissed Dirac's thoughts as 'nonsense'. Einstein did not  conceal his glee: 'I am very happy that one of the real experts now argues for the abandonment of the awful \"quantum electrodynamics\". ' Schr\u00f6dinger, disillusioned with the conventional interpretation of quantum theory, was pleased that Dirac had apparently joined the malcontents. Bohr, who in 1924 had been among the first to suggest that energy might not be conserved in every atomic process, was publicly less critical, though he took Shankland's results with a pinch of salt.\n\nExperimenters, including Blackett in London, downed tools, changed their plans and began programmes of experiments to investigate Shankland's claims. A few months later, however, it became clear that he had been wrong and that energy was indeed conserved. The false alarm made a deep impression on Dirac. A year later, he wrote ruefully to Blackett: 'After Shankland, I feel very sceptical of all unexpected experimental results. I think one should wait a year or so to see that further experiments do not contradict the previous results, before getting worried about them.' Dirac's inclination to believe exciting new observations had been irreversibly undermined.\n\nAfter another secret Christmas vacation with Manci and her children in Austria and Hungary, marriage was now on the cards. But Dirac could not bring himself to commit. No one knew of his inner turmoil; all they saw was the familiar meditative Dirac, the prince of asceticism, going wordlessly about his business. But in private he was not quite as cold and detached as he seemed to be. On his mantel-piece, he kept a photograph of Manci in a swimsuit, but no one saw it: when there was a knock on the door of his college rooms, he took the photograph down and hid it in a drawer. When many of his associates thought he was working, he was sloping off to see Mickey Mouse films, taking the Kapitza boys out for runs in his new car and reading T. E. Lawrence's _Seven Pillars of Wisdom._ In a bid to make Dirac more self-aware, Manci recommended that he read Aldous Huxley's _Point Counterpoint_ as she thought Dirac resembled the novel's character Philip Quarles: brilliant, solitary, emotionally 'a foreigner' and entrenched 'in that calm, remote, frigid silence'. Not seeing the likeness, he wrote to Manci: 'I doubt whether I am really like Philip Quarles, because his parents are not really like mine,' underlining - perhaps unconsciously - the importance of his mother and father to his sense of identity.\n\nDirac wrote his letters to Manci before he went to bed, 'the best time for thinking about you'. He never mentioned his work, nor did she enquire about it, and he rarely referred to his colleagues, but he did make an exception in February, shortly before he was due to meet Bohr and his wife in London. It was not long before Manci tired of the praise Dirac heaped on his elderly friend in one letter after another; 'Bohr, Bohr, Bohr,' she yawned. Dirac was surprisingly sensitive to these complaints and showed that he appreciated that her hair-trigger jealousy needed to be handled with care by toning down the complimentary references to colleagues he admired. His tact was tested again shortly before the Easter vacation, when Manci was hoping to see him. He explained to her that he felt duty-bound to visit his parents, as he had not seen them for several months; the problem was that after his visit to Bristol he would be in no fit psychological state to meet her:\n\nIt really will change me very much when I go home; it will make me afraid to do anything for my own pleasure. I shall probably be afraid to think of you [. . .] I find it satisfies me to be able to think of you whenever I wish. Why cannot you be satisfied in the same way? You should cultivate your imagination [. . .] It would be no use for me to see you for one or two days because, as you know, I am never kind to you the first day or two when I meet you.\n\nDirac pleaded with her to understand the paralysis that overcame him whenever he set foot in 6 Julius Road: 'If you cannot understand this, you will never understand me.' But Manci showed no sympathy; he was selfish, she told him. She had no interest in cultivating her imagination - she was not asking the Earth; all she wanted was to see her man in the flesh:\n\nYou do not consider anything but from your point of view. We are very different in [that] you never think to help people or to make them happy in spite [of the fact] that you are in the lucky situation where it would be easy to do so . . . I like you less.\n\nShe got her way. Shortly before Easter, Dirac returned to Bristol for a few days and, after taking a few days to recover, organised a vacation with Manci in Budapest. 'I cannot imagine being happier than I was with you,' she wrote to him. Finding it hard to express his joy, he assured her that the vacation left him 'not wanting feminine society at all'.\n\nAfter Easter, Dirac's colleagues in St John's were surprised to see him so sunburnt, and when they asked him where he had been, he replied, 'Yugoslavia.' The first casualty of Dirac's secret love was his commitment to literal truth.\n\nDuring the first week of June 1936, Dirac was gathering together his rucksack, sleeping bag, ice axe, rope and crampons, preparing for his next climbing vacation in the USSR with Tamm. Besides visiting Kapitza, he wanted to be in the Caucasus on 19 June to see a solar eclipse, the first he will have seen. Before leaving, he wrote to Manci, asking her not to write to him because if Tamm and Kapitza 'notice [that] you and I write very much to each other, then very quickly the news would spread to physicists all over the world and they would all gossip about us'.\n\nKapitza was in better spirits, reading his subscription copies of the _New Statesman_ and supervising the building of his new institute. Many of its rooms were replicas of ones in the Mond Laboratory, though Kapitza ensured that his new director's office was even grander, with an even larger footprint. After he demanded that every item of his laboratory equipment should be transferred, Rutherford complained that it seemed Kapitza would not be happy until the paint of the Mond Laboratory had been scraped off the walls. The Soviet Union was still the talk of the Cambridge common rooms, and the _Cambridge Review_ abounded with articles about it, including a sceptical review of Crowther's _Soviet Science_ , a whitewash that declared the Stalinist state's interference in science to be minimal. The Trinity College scholar Anthony Blunt, later a distinguished art historian, wrote an article on how a gentleman traveller might make the most of Russian hospitality - the champagne and the caviar, if not the bed bugs. Unknown to his colleagues, Blunt had recently become a Soviet spy.\n\nShortly before Dirac set off for Russia, he heard from his mother that his father was severely ill with pleurisy: every breath was painful and liable to be accompanied by a stabbing pain in his midriff. Flo wrote that the family doctor had ordered her husband to stay in bed for ten days but assured her that 'I'm not to worry as Pa is the kind of man to make the worse of anything just to keep me busy.' From the tone of his mother's letter, Dirac sensed that his father was not seriously ill, and he knew his parents were supported by Betty, about  to move permanently to London to become a secretary. So he decided to set off on vacation and arrived in Moscow on Saturday, but within hours received a telegram from his mother, telling him that his father was dying. He decided to head home, perhaps hoping to make one last effort to make his peace with his father, to achieve a reconciliation that had not been possible with Felix. Having left his hiking gear with Tamm, he caught the 7 a.m. flight from Moscow: he had twenty-two hours to find the right parting words.\n\nCharles grumbled that he did not want to be confined to bed at home because his wife was not taking proper care of him. So his doctor arranged for a professional nurse to take up residence in 6 Julius Road at night and to supervise Charles's care during the day. But that was not enough: after a few days, he demanded to be moved to a nursing home on the perimeter of the nearby St Andrew's Park, where he chose a comfortable room whose bay window looked out on the beds of early summer flowers. The staff soon realised that they had an awkward customer on their hands: the matron told Flo that Charles 'was an awful fidget so restless and fussy', and the nurses were instructed to leave him alone and to look into his room every half an hour. Struggling against pleurisy and the onset of pneumonia, he suddenly decided that he wanted to go home, but his doctor forbade it. Flo stopped visiting him, leaving him alone with his stabbing chest pains, his quarrels with the nurses and his reflections on the past sixty-nine years. One of his bitterest regrets must surely have been his estrangement from his son, 'Einstein the second', as the _Daily Mirror_ had described him three months before. This adulatory article, which Charles is almost certain to have read, concluded by telling its readers that their greatgrandchildren might one day talk about him, having forgotten No\u00ebl Coward, Henry Ford and Charlie Chaplin. One sentence in the piece will have taken Charles by surprise: the anonymous author wrote that Paul Dirac is only happy when he is in the lecture room, at the wheel of his sports car and 'in his home in Bristol, where he can talk with his father'.\n\nAt the end, the only member of Charles Dirac's family to be standing by him was his daughter, and she was about to break his heart by moving to London. On the day she was due to start work, Monday 15 June, he died. The end came a few hours before his son  arrived in Bristol: any hope of a deathbed reconciliation had been extinguished.\n\nTwo days later, on a warm and cloudy summer afternoon, Dirac was among the mourners at the funeral. It was a civic occasion, held in St Bonaventure's, the handsome Catholic church at the end of Egerton Road, near the family home. A few hours before, at eight in the morning, the choir had sung a requiem mass over Charles's open coffin near the altar. The funeral was scheduled to begin at 3 p.m. Shortly before, dozens of mourners made their way through the Bishopston streets - representatives from the Esperanto Society, the Merchant Venturers' Technical College, the French Circle and Cotham Road School, including several schoolchildren. Also there was elderly Arthur Pickering, the man who had introduced Dirac to Riemannian geometry, still telling stories of how he had struggled to find challenges for the most precocious student he ever had.\n\nThe eulogy, the weeping, the sacred music, the lowering of Charles's coffin into the grave - together, they may have stirred Dirac to reflect on the good things his father had done for him. Charles had ensured that his younger son had an excellent education and had encouraged him to study mathematics. And it was Charles who had given him the money he desperately needed in order to begin his studies in Cambridge.\n\nStraight after the funeral, Dirac gave vent to his feelings in a single-page letter to Manci. In the most expansive handwriting he ever used in his life, he wrote that he would return to Moscow after he had spent a week with his mother: 'I think that in Russia I can best get used to my new situation.' He wanted to see Manci again, he told her, but gave her firm instructions not to contact him: 'I would rather you did not wire me while I am in Bristol because my mother would probably open it.' Dirac concluded with some simple words of relief: 'I feel much more free now; I feel I am my own master.'\n\nCharles Dirac had left no will - he probably did not want to leave much to his wife and possibly could not face the thought that his true wishes would be known to all the people who revered him as a family man. Flo had long suspected that he had been squirreling his money away, but even she was stunned by the amount he had hoarded: the net value of his estate was worth \u00a37,590 9s 6d, about fifteen times his final annual salary. Half of the legacy was shared by Paul and Betty, and the rest went to Flo, who quickly headed off on a restorative holiday in the Channel Islands, where she wrote to her son: 'I've won my liberty and shall keep it.' Betty, apparently finding her mother's relief unseemly, departed for London and never lived in Bristol again but occasionally corresponded with her mother. Betty was piqued when she read that Flo had destroyed most of her father's papers in a bonfire in the back garden; the remainder of the papers she gave to Paul. From them we know that, somehow, several of his parents' love letters survived.\n\nDirac family, 3 September 1907\n\nPaul Dirac, 17 August 1907\n\nLeft to right: Felix, Betty and Paul Dirac _c_.1909. A French grammar book rests on Paul's lap.\n\nTechnical drawing by Paul Dirac at Bishop Road School, Bristol, 9 December 1913\n\nBristol University Engineering Society's visit to Messrs Douglas' Works, Kingswood, 11 March 1919. Dirac is in the front row, fourth from the right.\n\nCharles Dirac, _c_.1933\n\nFelix Dirac, 1921\n\n6 Julius Road, Bristol, where Dirac lived with his family from April 1913 until he left for Cambridge in 1923. He regularly returned home and began his work on quantum mechanics in his bedroom here.\n\nMax Born (seated, central) with several younger colleagues at his home in G\u00f6ttingen, spring 1926. Dirac is, as usual, diverted. Oppenheimer is in the back row, fourth from the left.\n\nSome members of the Kapitza Club, after a meeting _c_.1925, in the room of Peter Kapitza, Trinity College, Cambridge. Kapitza is directly beneath the drawing of a crocodile on the easel.\n\nPatrick Blackett and Paul Ehrenfest, _c_.1925\n\nIsabel Whitehead with her husband Henry, and their son Henry, 1922.\n\nDirac (standing close to the doorway) at a meeting in Kazan, Russia, 12 October 1928\n\nLeft to right: Heisenberg's mother, Schr\u00f6dinger's wife, Flo Dirac, Dirac, Heisenberg and Schr\u00f6dinger. They have just arrived at Stockholm railway station, 9 December 1933, for the Nobel celebrations.\n\nExtract from a letter from Dirac to his friend Manci Balazs, 9 May 1935\n\nDirac and Manci on their honeymoon, Brighton, January 1937\n\nThe Dirac family in the garden of their Cambridge home, _c_.1946. Left to right: Dirac, Monica, Manci, Gabriel, Mary and Judy.\n\nDirac and Manci (on the far left) with a party during a crossing of the Atlantic on the SS _America_ , 2 April 1963\n\nDirac and Richard Feynman at a conference on relativity, Warsaw, July 1962\n\nDirac at the Institute for Advanced Study, Princeton, _c_.1958\n\nThe Diracs' home in Tallahasse, 223 Chapel Drive\n\nKapitza and Dirac at the Hotel Bad Schachen, Lindau, summer 1982\n\nOne of the last photographs taken of Dirac, Tallahassee, _c_.1983\n\nWhen Flo returned to Bristol, she arranged for Charles's gravestone in Canford Cemetery to be engraved with the words Paul had written for her:\n\nIn loving memory of   \nOur dear son   \nReginald Charles Felix Dirac, B.Sc.   \n\u2605 Easter Sunday 1900   \n= March 5th 1925   \nAnd of my dear husband   \nCharles Adrien Ladislas Dirac, B.\u00e8s.L   \nFather of the above   \n\u2605 July 31st 1866   \n= June 15th 1936\n\nDirac was obviously determined that the tone of family memories of his father should owe more to propriety than honesty. His mother wrote to him: 'One doesn't mind after a few months.'\n\nWhen Dirac resumed his visit to Russia, he celebrated by attempting to climb Mount Elbrus, 5,640 metres above sea level, the highest peak in the Caucasus, a near wilderness. With Tamm and a small party of his Russian colleagues, Dirac hiked through the forest to reach a base camp and then scaled the eastern side of the mountain, fearful of injury, sweat dribbling down his back and sunburned face during the day, shivering in the tent at night. Mount Elbrus yielded its rewards only grudgingly, as hundreds of defeated mountaineers had found, some as they fell to their deaths. After several days, Dirac and his fellow climbers saw Russia's most majestic glacial scenery, sights all the sweeter for the pain that must be suffered to win them. He only just made it; after reaching the  top, he was spent and had to rest for a day before he could begin the journey back to base. Never again would he attempt such an ambitious climb.\n\nAfter recuperating, Dirac joined Kapitza, who was back to his buoyant best. The building of the institute was progressing well, and the first consignments of his equipment were about to arrive from the Cavendish. The authorities were taking care of him: although most Soviets suffered food shortages, Rutherford heard from Kapitza that he was eating oysters, caviar and smoked sturgeon of a quality that would make even the Trinity College 'gourmands at the high table dribble'. In under three years, the Soviet authorities had won him round.\n\nIn the next stage of Dirac's hedonistic trip, he visited the two people he most wanted to see: Manci and Bohr. Having contemplated his bereavement for a few weeks, when Dirac saw Manci in Budapest, he confided his worries that he and his father were so similar: both devoted to work, both extremely methodical, both lacking in empathy. Apparently for the first time, he described how his father had treated his family so unspeakably. After he left Budapest, she urged him to put his resentments behind him: 'One has to try to understand and forgive.' He will have been mulling over Manci's advice towards the end of September when staying with Bohr and his wife in their country retreat. The Bohrs were also recovering from grief, less complicated and probably much more painful than Dirac's: their eldest son Christian had died two years before, at the age of seventeen, in a freak yachting accident. Bohr had been on the deck with him and had been helpless as he watched him drown.\n\nAt Bohr's suggestion, Dirac stayed in Denmark longer than he originally intended, to attend a special conference at the institute about a branch of science that Dirac knew almost nothing about: genetics. He learned, he wrote in a letter to Manci, that this 'is the most fundamental part of biology' and that there are 'laws governing the way in which one inherits characters from one's parents'. There was no escape from his father's genetic legacy - it was in Dirac's blood .\n\nWhen Dirac returned to Cambridge, his adventurous spirit was intact, and he changed his research topic from quantum physics to cosmology, refocusing his imagination from scales of a billionth of a  centimetre to thousands of light years. Einstein's general theory of relativity provided the sturdy theoretical foundations of modern cosmology, but the subject was handicapped by a dearth of reliable data. As a result, theoretical cosmologists had more room for manoeuvre than was good for them and had to rely heavily on intuition.\n\nWithout question, the most successful observational astronomer was the former lawyer Edwin Hubble, an Anglophile American in his mid-forties, given to declaiming on conference platforms in a strangely affected English accent, akin to Oppenheimer's. Hubble had created a public sensation in 1929 when he suggested that galaxies (aggregates of stars and other matter) do not stay still with respect to one another but are always rushing apart. In what became known as Hubble's law, he used the data in his charts and tables to propose that the further a galaxy is away from the Earth, the faster the galaxy is moving away from it. This picture of galaxies dashing away from each other was consistent with Lema\u00eetre's 'primeval atom' theory of the origin of the universe, a precursor of the modern theory of the Big Bang.\n\nDirac's perspective on the subject emerged after a few months' gestation, when he was also contemplating one of the most important decisions of his life: should he marry Manci? Here was a warm, caring and cultured woman, the kind of extrovert he liked, one of the few with the patience to draw out his humanity. On the other hand, she was impulsive, hot-headed and overbearing. Could he be happy with a woman who had something of the controlling personality of his father? He knew it would be pointless to ask his mother, who wanted no competition for his loyalty. It would not be wise to seek the counsel of Wigner, as his loyalties would be divided; besides, he had problems of his own. Having felt undervalued at Princeton, Wigner had moved to the University of Madison, Wisconsin, and was contemplating marriage to his colleague Amelia Frank, one of the few female quantum physicists. When Wigner asked Manci to visit him and to size up his girlfriend, she jumped at the opportunity to sail from Southampton on the _Queen Mary_ , the world's most luxurious liner, whose maiden voyage had taken place five months before. When Manci asked Dirac if she could visit him in Cambridge before she sailed, he fobbed her off but quickly relented. Still unsure whether he should commit to the relationship, he drove Manci over to see Isabel Whitehead for what Manci knew  was an informal grilling. When he returned to Cambridge, he felt confident enough to forward some of Mrs Whitehead's views to Manci, excising points that might upset her:\n\nMrs Whitehead said she liked you. You are very unusual and have the simplicity of a child. I think this is what she meant by your being charming. [. . .] she said that I ought to make up my mind quickly, also that you and I would find it very difficult to get on together because we are so different.\n\nYet Mrs Whitehead had second thoughts. Worried that Dirac was contemplating marriage without the spiritual commitment she believed was essential, she wrote Dirac a long and anguished letter, thundering like Lady Bracknell:\n\nWould it be useful to go and talk to Prof. Eddington about spiritual things? I feel sad that you should have this limitation that you do not seem [?] to believe in God; and I am always afraid that I have failed to help you, how and when you need help.\n\nMrs Whitehead pleaded with him not to make his decision when he was 'in a mood', a phrase he had used when they last met. This stung him into a rare candour about his state of mind. On 6 December, when Manci was preparing to sail from New York, he replied to Mrs Whitehead that he did not believe his decision depended on whether or not he believed in God. She had misunderstood his reference to his state of mind when he took his decision:\n\n[By 'in a mood'] I meant only that I would need to be in a courageous mood to take an irrevocable step, after I had made up my mind what I ought to do. I think I err on the side of trying to be guided too much by reason and too little by feeling, and this makes me feel helpless when it comes to problems that cannot be solved by the clear-cut reasoning that one has in science [. . .] I have felt very favourably inclined to [Manci] for several months, with occasional relapses, which get less and less as time goes on.\n\nBut Mrs Whitehead was not to be deflected; she wrote straight back to Dirac, insisting that 'married love comes to its highest perfection between people who know and love God'. But these words were wasted on Dirac, for whom the concept of God had no precise meaning.\n\nBy the time he was among the dockside crowds at Southampton, waiting for Manci to arrive, he had made up his mind. During the  drive to London in his sporty drop-head Triumph coup\u00e9, he steered his car to the kerbside and asked Manci, 'Will you marry me?' She accepted immediately. When he told his mother the news, she was predictably shocked but summoned the grace to wish him and Manci well, offering to travel to London on the day before Christmas Eve to meet her future daughter-in-law. Dirac accepted, perhaps inadvertently giving his mother one last chance to persuade him to stay single.\n\nManci was staying in the smart Imperial Hotel in Bloomsbury, overlooking Russell Square. During their few hours together, Flo and Manci found a few moments to talk privately, leaving Manci puzzled.  As soon as Flo arrived home, she wrote to Dirac with a detailed account of the conversation:\n\nFLO: You will be having twin-beds soon.\n\nMANCI: Oh no, I must have a room to myself. I cannot allow Dirac to come in my bedroom.\n\nFLO: What are you marrying him for?\n\nMANCI: I like him very much and want a home.\n\nFlo was astute enough to avoid outright condemnation. 'Manci was very nice indeed,' she wrote, before the inevitable qualification: 'I suppose you know she is only contracting a \"marriage of convenience\". ' His mother knew how to unsettle him. She had just seven days to make him reconsider the balance he had struck between reason and feeling.\n**Twenty-one**\n\nPythagoras says that number is the origin of all things; certainly, the law of number is the key that unlocks the secrets of the universe.\n\nPAUL CARUS, _Reflections on Magic Squares_ , 1906\n\nOn the morning of Saturday 2 January 1937, Dirac and Manci married in Holborn Registry Office in central London. He had wed his anti-particle, a woman almost opposite to him in character and temperament, as his father had done thirty-eight years before. That had proved disastrous, resulting in something akin to mutual annihilation, so Dirac may have feared - at least at the back of his mind - that history would repeat itself.\n\nIt was an overcast day, the crowds in London going about their business after the Christmas holiday, girding themselves for the harshness of winter. The wedding was a simple civil ceremony, with only a few guests, including Dirac's mother and sister, the Blacketts, Isabel Whitehead and her husband. After lunching with them in a restaurant near by, the couple returned to their hotel and drove to Brighton. Dirac could not have picked a more conventional place for his honeymoon: for decades, it had been the most popular seaside venue in Britain for romantic trysts. It was a peculiarly raffish town, famous for its two Victorian piers jutting imperiously out to sea, for the pale green domes of its faux-oriental pavilion, its future-telling robot and a host of other tacky attractions.\n\nIt appears that no photographs were taken of the wedding, but Dirac took reels of them during the vacation, the best of them showing the newlyweds on a pebbled beach, smiling broadly, looking coy and love-struck. Dirac looks comfortable lying on the beach in his ill-fitting three-piece suit, pencils still protruding from the pocket of his jacket. In some of the snaps, it is possible to see a string-operated device that he devised to enable him and Manci to photograph themselves with no one else present.\n\nAfter the honeymoon, while Manci was in Budapest with Betty, Dirac looked around Cambridge for a permanent home and discharged his duties as Lucasian Professor. Three weeks after Manci's  departure, rain lashing against the windows of his rooms in St John's, he was overcome with loneliness, sheltering from the wind and drizzle of the Cambridge winter. He wrote to his wife 'the first love letter I have ever written [. . .] Rather late to begin is it not?' In the two passionate letters he wrote in as many days, he revealed an almost Byronic expressiveness:\n\nI realize more and more as time goes on that you are the only girl for me. Before we were married, I was afraid that getting married would cause a reaction, but now I feel that I will go on loving you more and more as I get to know you better and see what a dear, sweet girl you are. Do you think you will go on loving me more and more, or is it now as much as it can be?\n\nHe had, at last, fallen in love. In the evenings, he read Bernard Shaw's _Getting Married_ \\- retrieved from his father's library - and some books recommended by Manci, including John Galsworthy's sprawling _Forsyte Saga._ 3 But Dirac was spending most of his time in a Manci-obsessed reverie, counting the days to when she was due to return, dreaming of embracing her in bed under a new moon. It was now Manci's turn to be sensitive about what others might think: brushing aside her worries that the censors in Hungary might be intercepting their mail, Dirac was uninhibited: 'You have a very beautiful figure, my darling, so round and charming - and to think that it all belongs to me. Is my love too physical, do you think?' Struggling to find words equal to his passion, he continued:\n\nManci, my darling, you are very dear to me. You have made a wonderful alteration to my life. You have made me human. I shall be able to live happily with you, even if I have no more success in my work. [. . .] I feel that life for me is worth living if I just make you happy and do nothing else.\n\nManci appears to have been no less intoxicated: 'If by any reason a war or anything would prevent me to see you again, I could never love anybody else.' She and Betty were getting on well in Budapest, at the Moulin Rouge, skating on the rinks and doing the Charleston on the dance floor after a few glasses of champagne. 'I am very very happy and being thoroughly spoiled,' Betty wrote to Dirac. But she was depressed and mourning her father: 'he was the finest man I ever knew', she wailed. In Betty's view, her parents had each been the victim of an unfortunate marriage, and she gave Manci a reason why her parents disliked each other, though this was too personal for Manci to spell it out explicitly in a letter to her husband.\n\nManci decided to take Betty in hand and to find her a husband: '[Despite] her little faults, a bit of untidiness and unpunctuality, I shall try to [. . .] improve her and she will be a very good wife.' Within days, Manci had decided that her Hungarian friend Joe Teszler was just the man for her sister-in-law: kind, gentle and - an essential requirement for Betty - a Roman Catholic. This was one of Manci's most effective pieces of social engineering: after a brief courtship, Betty married Joe - six years her senior - in London on 1 April 1937. In Bristol, Flo was now quite alone.\n\n'Some say that I got married rather suddenly,' Dirac wrote to his wife. One of the dons who were surprised by Dirac's marriage was Rutherford, who wrote to Kapitza: 'Our latest news is that Dirac has succumbed to the charms of a Hungarian widow with two children,' adding cryptically, 'I think it will require the ability of an experienced widow to look after him.' A few days later, Dirac wrote to tell Kapitza the news: 'Have you heard that I was married during the vacation [. . .]?' Kapitza was probably surprised as he thought he knew Dirac well but had not even known he was seeing a woman. Anna Kapitza quickly wrote to Manci, though she too had not met her:\n\nDear Mrs Dirac (it sounds very official but he did not even write us your name!)\n\nI hope you will be very happy with that strange man, but he is a wonderful creature and we all love him very much. Do come to see us in the summer.\n\nYours, Anna K\n\nAfter a second honeymoon in Brighton - only a month after the first - Dirac returned to Cambridge with Manci, who had left her children in Budapest. By late April 1937, they were still looking for a permanent home and living in a rented house in Huntingdon Road, a short stroll from the Kapitzas' former home. It is not recorded how Dirac referred to her when he introduced her to his university colleagues, but it is quite possible that he described her not as 'my wife' but by his favourite appellation as 'Wigner's sister' (this was a surprising choice of words for Dirac, usually fastidious in his choice of words to the point of pedantry: Manci was Wigner's _younger_ sister). She quickly established herself as one of the most colourful women in the university, holding dons spellbound as she passed on outrageous gossip about life in Princeton. Dirac looked on, adoringly.\n\nFor all her assertiveness, Manci was happy to be part of what she liked to call 'a very old-fashioned Victorian marriage'. She regarded it as her duty to ensure that her husband's meals were ready on time, to put her husband's used clothes in the laundry basket every night, before laying out freshly ironed clothes for the next day. She allowed Dirac to set out a few ground rules of the relationship, including an understanding that French must never be spoken conversationally in their home - he wanted to put to rest all memories of his father's linguistic regime. Perhaps surprisingly, she accepted that nothing in their domestic routine should ever interfere with Dirac's work. This apparently caused no friction when they were alone but it did, on at least one occasion early in their relationship, lead to an embarrassing tiff: Dirac had agreed to go with her to visit friends for afternoon tea but refused to leave his study because he had not finished thinking. Manci went alone, made excuses for her husband and had to put on a brave face when her host took offence.\n\nThe wary British welcome given to Manci was made no more congenial by the inclement weather. The first few months of 1937 were one of the wettest periods Cambridge had seen for years. She felt unwelcome in the university, which seemed to be a place for men; spouses were meant to be agreeable ornaments - decorative but not obtrusive. Colleges did not allow wives to attend dinners, except on special occasions, so she had to sit alone with her novels and magazines while Dirac fulfilled his duty of dining in college at least once a week. Some of his colleagues thought that his marriage had lightened his character, though he was still as uncommunicative as ever, as the archaeologist Glyn Daniel found when he sat next to him at dinner in St John's:\n\nThe soup came and went in silence; halfway through the Sole V\u00e9ronique I decided the effort must be made - the silence must be broken. But how? Not the weather. Not politics. Not the simple approach, 'My name is Daniel. I study megalithic monuments. Have you any views on Stonehenge?' I turned to Dirac, who was examining the grapes on his sole. 'Have you been to the theatre or the cinema this week?' I asked, innocently. He paused, turned to me with what I supposed was meant to be a kindly smile and said 'Why do you wish to know?' The rest of the meal was eaten in silence.\n\nBy early September, the Diracs had moved into their grand new home, 7 Cavendish Avenue, a detached red-brick house south of the town, built sixty years before. It was in a quiet district - he had  checked carefully that they would not be disturbed by the ringing of church bells - was a twenty-minute cycle ride from St John's College and had 'a beautiful garden' of almost two thirds of an acre. In May, Dirac had written out a cheque for \u00a31,902 10s. 0d., which paid for the property in a single transaction; unlike most newly married couples, they were unencumbered by a mortgage. The interior decor of the house reflected Hungarian tastes in the late 1920s. Manci imported much of the furniture from her Budapest apartment - heavy, dark wood sideboards and cabinets, capacious living-room chairs, gaudy side tables - though Dirac vetoed her most ornate items. Patterned, deep-pile carpeting and conventional landscape paintings helped to set the sober decorative tone.\n\nManci's children joined them in Cambridge and began to study at local schools, where they - with their uncertain, thickly accented English - had to work hard to integrate with other pupils. Although Dirac never legally adopted Judy and Gabriel, he raised them as if they were his own children and never referred to them as his stepchildren. But he also wanted biological children of his own.\n\nA few days after Dirac returned from his honeymoon, he completed his first contribution to cosmology. Had physicists known that he was working on this subject, they would probably have predicted a surprising new insight into the structure of the universe, or perhaps a fresh perspective on Einstein's theory of gravity. But he did neither. In a 650-word letter to _Nature_ that included almost no mathematics, he set out a simple idea about the numbers that describe the universe on the largest scale. As soon as Bohr finished reading the letter for the first time, he walked into Gamow's room in the Copenhagen Institute and said, 'Look what happens to people when they get married.'\n\nDirac's cosmological idea was not completely original, as it bore signs of having been strongly influenced by Eddington. Now perceived by many of his peers as a cocksure eccentric, Eddington had largely abandoned research in conventional cosmology and was spending his time trying to derive some of the most important numbers in science - such as the number of electrons in the universe - not by systematic reasoning but by pure thought. Most theoreticians, including Einstein, thought this was hokum: theoretical physics was  about finding general principles, not about explaining numbers that arise in the search. In Rutherford's scabrous words, Eddington was 'like a religious mystic and [. . .] not all there.'\n\nIn his _Nature_ article, Dirac pointed out that the universe is characterised by several numbers that seem to be connected in a simple way. He focused on three numbers, each of them estimates:\n\n1. The number of protons in the observable universe. Experimentally, this number is roughly 1078 (that is, 10 multiplied by itself 77 times).\n\n2. The strength of the electrical force between an electron and a proton divided by the strength of the gravitational force between them. This turns out to be about 10.\n\n3. The distance across the observable universe divided by the distance across an electron (according to a simple classical picture of the electron). Its value is approximately 10.\n\nThe first striking point about these numbers is that they are so much larger than any number that occurs anywhere else in science: 10, for example, exceeds the number of atoms in a human body by a factor of a hundred billion. The second point is that the largest estimated number, 1078, is the square of the smaller one. This, Dirac believed, may not be a coincidence and suggested that these numbers might be related by extremely simple equations such as\n\nHaving noted that in both of these cases the linking number is about one, Dirac proposed a generalisation: this is always the case - _any_ two of the huge numbers occurring in nature are connected by very simple relationships and linking numbers close to one. This is Dirac's large numbers hypothesis, a consequence of his faith that the laws underlying the workings of the universe are simple.\n\nThe suggestion has an intriguing consequence: because the size of the observable universe continuously increases as it expands, it follows that the ratio of this size to the radius of an electron cannot have always had its present value, 10, but has been increasing throughout time. If Dirac was correct to surmise that this number is connected to the ratio of the electrical force and the gravitational force between an electron and a proton, it followed that the relative strengths of these forces must have been changing as time progressed, as Milne had suggested a few years before. Dirac argued that one consequence of this is that the strength of the gravitational force withers proportionately as the universe ages: when the age doubles, the strength of gravity halves.\n\nDirac's decision to introduce his idea in such a short paper suggests that he believed he had hit on an important new principle and did not want to be beaten into print. If he was expecting the reception that greeted most of his papers, he will have been disappointed: this one was given a frosty reception. Yet none of the sceptics went public with their criticisms, with one prominent exception, the eccentric philosopher-astrophysicist Herbert Dingle. For him, the job of the theorist was to find laws based on experimental measurements, just as Dirac had done in quantum mechanics. Dingle spoke for many a more timid colleague when he wrote an article in _Nature_ that condemned 'the pseudo-science of invertebrate cosmythology', and regretted that Dirac was the latest 'victim of the great Universe mania'. Stung into a quick reply, Dirac repeated his earlier reasoning almost word for word, after prefacing his remarks with an uncontroversial comment about the nature of science: 'The successful development of science requires a proper balance to be maintained between the method of building up from observations and the method of deducing by pure reasoning from speculative assumptions.'\n\nIn the same issue of _Nature_ , Dingle resumed his offensive, stressing that he was not attacking Dirac personally: 'I cited Prof. Dirac's letter not as a source of infection but as an example of the bacteria that can flourish in a poisoned atmosphere; in a pure environment it would not have come to birth, and we should still have the old, incomparable Dirac.'\n\nDirac was not deterred. However, after he had written at length about the implications of his hypothesis in a long paper - completed shortly after Christmas 1937 - he returned to quantum mechanics and did not revisit the hypothesis for another thirty-five years.  Although his idea influenced astronomers in the late 1930s, many of Dirac's peers regarded it as an aberration, joining Bohr in believing that Dirac had made a wrong move towards Eddington and Milne's quasi-mystical cosmology. But his status did not suffer significantly. In October, the Institute for Advanced Study in Princeton, still seeking to recruit the world's best theoretical physicists, put Dirac at the top of the list of the scientists they wanted to recruit, just above Pauli.\n\nBack in Bristol, Charles Dirac had left a surprise for his family: solicitors found, after months of delving through his accounts, that he had been a serial tax evader. The authorities required Flo to pay six years of Charles's tax debt, the maximum they were allowed to reclaim, after making her swear affidavits that she knew nothing of his deception. 'No one knows how Pa managed to elude income tax on so many items,' she wrote to Dirac, who heard that his father had claimed \u00a350 a year tax relief for educating Betty at university, while his son paid the bills. But the nastiest revelation for Dirac was still to come, when he learned that the funds that enabled him to begin his studies at Cambridge had been provided not by his father but by the local education authority. Charles had pretended that he had stumped up the money. This petty and unpleasant deception was, for Dirac, the final straw. It negated everything that his father had done to nurture his career and revealed Charles in his true colours. This was why Paul Dirac told his closest friends, including Kurt Hofer, that he owed his father 'absolutely nothing'.  It was an understandable, if harsh, judgement.\n\nAfter her marriage, Betty left England to live with her husband Joe, who owned and ran a flourishing camera shop in Amsterdam. Within a year they had a son, but their happiness was soon blighted by the news from Berlin, where Hitler was seeking 'living space' outside Germany and was thirsty for Jewish blood. It would not be long before the Teszlers would feel the full force of Hitler's ambitions.\n\nAt the High Table in St John's, everyone was talking about the German Chancellor and the pell-mell rush towards another global conflict. The only European country then openly at war was Spain, where Hitler supported Franco's fascist army; the British Government refused to take sides, outraging socialist opinion, particularly in  Cambridge, from where many idealists journeyed to support Franco's opponents. Dirac's eyes were, as usual, focused on the Soviet Union. That the country was suffering from an unconscionably bloody purge was clear to newspaper readers in Britain, but it appears that Dirac - like many others on the left - thought the reports were exaggerated. In Moscow, Kapitza was not aware of the extent of Stalin's murderous rampage; even so, he knew that several of his colleagues were being harassed and that he risked deportation to a labour camp if he complained, though the censors did not allow him to mention this in his letters.\n\nIn the early summer of 1937, when the Diracs were in Budapest to see her family, Manci wrote to Oswald Veblen and his wife. 'Paul would like very much to go to Russia, but everybody advises him not to.' Dirac insisted on making the visit and wanted to take his family, but Hungarian regulations allowed only Manci to accompany him. Kapitza confirmed the arrangements in a telegram intercepted by MI5, still checking mail he was sending to Cambridge.\n\nAt the end of July, during an oppressively hot summer, the Diracs arrived at the Kapitzas' summer home days before Stalin authorised the torture of suspected enemies of the people. Only a short drive away, his henchmen were gouging out the eyes of their victims, kicking their testicles and forcing them to eat excrement. On the roads around Bolshevo, some of the trucks marked 'Meat' and 'Vegetables' hid prisoners on their way to be shot and buried in the forests to the north of the city which Dirac admired through his binoculars. For many years, Soviet people would refer darkly to 'the year 1937', the height of the Great Purge, Stalin's chaotic and brutal campaign of mass intimidation, imprisonment and murder. By the end of the year, the purge had claimed about four million lives. As Kapitza knew, one of the victims was Boris Hessen, a member of the delegation that had visited London and Trinity College six years before. Five of his fellow visitors would also soon be executed. Now confined to the Soviet Union at Stalin's behest, Kapitza had received all his equipment from the Cavendish Laboratory and had resumed his research.\n\nThe Diracs spent three idyllic weeks in Bolshevo with the Kapitzas in their modest summer house in the heart of a pine forest, with wild strawberries ripe for gathering and a fast-flowing river close by. They spent one languorous day after another lounging around on the covered veranda, telling off-colour jokes, the Diracs bringing the latest  news on the Crocodile and his departing 'boys', the Kapitzas gossiping about life under Stalin. The two men took advantage of the cool mornings to do some manual labour - chopping down trees and clearing shrubs close to the house - and messing around with the boys. Manci, always as _soign\u00e9e_ as a duchess, wanted nothing to do with physical exercise and avoiding cooking anything more complicated than a boiled egg. Dismayed by the _dacha_ 's lack of creature comforts, including toilet paper, she could scarcely believe that, for the first time in her life, she had to sleep outside, in a tent. But she was too polite to gripe: she shone in conversation and won over Kapitza, who saw that she had opened Dirac up. He wrote to Rutherford: 'It is great fun to see Dirac married, it makes him much more human.'\n\nKapitza will almost certainly have enthused about the new institute being built for him. He was dealing adroitly with the authorities, bombarding them with complaints but always avoiding confrontation and keeping on the right side of the power brokers. In return, he was given unusual leeway to employ the staff he wanted and to allocate funds as he saw fit, with a minimum of bureaucracy. In the following year, he was even able to hire Lev Landau as the institute's resident theoretician after he had been arrested in Moscow, having fled the Kharkov police, in fear of his life.  Kapitza had resumed the experiments he had begun in the Mond Laboratory and had successfully liquefied helium the previous February. Exciting new results were afoot.\n\nKapitza persuaded Dirac to demonstrate his support of the Russian experiment by sending his next paper to the _Bulletin of the Soviet Academy of Sciences_ , in commemoration of the twentieth anniversary of the Bolshevik revolution. In the article, he investigated the symmetries underlying classical and quantum descriptions of matter, following the lead given by his brother-in-law Wigner. It was another elegant piece of work, though it produced no useful results and appeared to be more evidence that Dirac was losing his touch.\n\nThe Diracs and Kapitzas knew they were in uncertain times but could scarcely have guessed that they would not sit around the same dinner table again for another twenty-nine years.\n\nAt noon on 25 October 1937, Dirac stood among two thousand mourners in Westminster Abbey, probably wondering whether to  join in the prayers and hymns or stay silent. He was at the memorial service for Rutherford. Nine days before, two weeks after the beginning of the autumn term, he had died after complications arising from surgery on his umbilical hernia: Cambridge was rife with rumours of a botched operation. Within days, government officials agreed that he was eligible to be commemorated in the 'science corner' of Westminster Abbey, alongside Newton, Darwin and Faraday. The funeral service was a national event, attended by a representative of the King, members of the cabinet, the former prime minister Ramsay MacDonald, eighty Cambridge scientists, and several foreign guests. Bohr stayed with the Diracs and joined the Rutherford family party for the event, which ended when an official placed a small urn of the great experimenter's ashes a few inches from Newton's grave.\n\nTwo days after the service, Dirac wrote a consoling note to Kapitza, also grieving from the recent death of his mother. In his reply, Kapitza did not mention that the Crocodile's death occurred just as he was making his most exciting discovery - at sufficiently low temperatures, liquid helium could flow entirely without resistance to its motion. Such 'superfluid' helium could climb spontaneously up the walls of its container and behave in other strange ways that were beyond classical mechanics but which later were explained by applying quantum mechanics to the constituents of the fluid. _Nature_ published Kapitza's results in a December issue, alongside a paper by two Mond experimenters who also announced the discovery of superfluidity: although Kapitza had spent two years without laboratory equipment, he had already caught up with the leaders in his field. It was no longer so easy for his detractors to sneer that he was really just a self-promoting lightweight.\n\nWorried that the future of the Cavendish was in danger, Kapitza wrote to Dirac to enjoin him to take an active interest in securing the laboratory's future: 'I think that you who are now the leading personality in physics in Cambridge, you must take some serious interest in upkeeping the great traditions of the Cavendish Laboratory, so important for all the world.'\n\nBut such a role was beyond Dirac - and, besides, he had no interest in it. The directorship of the Cavendish passed to the crystallographer Sir Lawrence Bragg, who steered the laboratory away from studies of the innermost structure of matter, partly because it could  no longer keep up with the competition from the United States. With Rutherford's passing, the Cavendish had seen the last of its glory days as a place where experimenters probed atoms with the finest possible probes, though Bragg steered the laboratory's agenda into productive territory, culminating in Watson and Crick's discovery of the double-helix structure of DNA in 1953.\n\nBy the end of 1937, Dirac was bereft of the company of experimenters with similar interests in physics, and some of his most valued colleagues among the Cambridge theoreticians were in decline. Following a debilitating stroke, Fowler's health was failing, and, by early 1939, he had 'faded out', as he told Eddington. In the sometimes gory seminars in the mathematics department, Eddington was timorous and unable to defend himself against pillory by his younger colleagues. Dirac looked on, unmoved and dissatisfied with his own research. Quantum field theory was virtually at a standstill, and even the best minds were finding it hard to make progress. Dirac often reflected on the contrast with only a decade before, when quantum mechanics had just been discovered: 'It was very easy in those days for any second-rate physicist to do first-rate work; it is very difficult now for a first-rate physicist to do second-rate work.' These words resonated with the theoretician Fred Hoyle, an independent-minded Yorkshire man who had attended Dirac's undergraduate lectures and who had struggled in the late 1930s to find a subject ripe for development. Hoyle's bottom-up approach to physics was the antithesis of Dirac's style, but they got on well: the trick was, Hoyle said, to ask Dirac fewer questions than he asked you. Hoyle was amused by Dirac's conversational eccentricities, though even he was stunned when he called Dirac to ask him a straightforward administrative question, only for Dirac to reply, 'I will put the telephone down for a minute and think, and then speak again.' A few months later, Hoyle was told that he needed to find a supervisor, and Dirac took him on, partly because he was amused by the prospect of a relationship between a supervisor who did not want a student and a student who did not want a supervisor.\n\nCompared with many of the new ideas in quantum physics, the energy of an electron sounds a simple concept, but it was anything but simple to understand. This was because the energy that an electron has purely by virtue of its existence - its self-energy - turns out  to be infinite. According to classical physics, the source of this embarrassment is the electric field of the electron (in some ways analogous to the gravitational field of a planet): the smaller the size of the particle, the stronger its field near by and the higher its energy. So if the electron were an infinitely small point, as it is usually assumed to be, its self-energy must be infinite. This makes no sense: how can a completely natural quantity have such an immeasurably huge value?\n\nThe theory of quantum electrodynamics, based on hole theory, had the same weakness: the self-energy of the electron was infinitely large. The most likely reason for this failure, Dirac believed, was that there was a fault in the classical theory on which his quantum theory was based: Maxwell's classical theory of electromagnetism. Dirac hoped that if he could remove the errors in the classical theory, he would be able to deduce a quantum theory of the electron that did not suffer from the disease of infinite self-energy. This was an unpopular view: most of his colleagues thought the classical theory was fine and that the challenge was to solve the problems with quantum theory. But Dirac, as usual, was unperturbed by popular opinion and spent several months in late 1937 and early 1938 working out a new classical theory and finding equations to describe an electron with a tiny but non-zero size. It was an immaculate theory but failed at its first hurdle: when Dirac tried to use it to find an infinities-free quantum version of the theory, he failed.\n\nHe may have wondered whether he had lost his edge. Besides his work, he was now a family man with other priorities: a wife and two bickering children, the employment of a cook and several domestic helpers, and his dependent mother, now sixty, living a hundred and twenty-five miles away and with no telephone. Flo was, however, in good spirits: she was pottering around in her house, writing verse in bed, occasionally packing her suitcase and taking a Mediterranean vacation funded by her now healthy bank account.\n\nManci still found it hard to settle and never felt completely comfortable in 7 Cavendish Avenue, a damp house that somehow always seemed cold, even in high summer. Disappointed that Dirac had turned down Princeton University's offer of a well-paid professorship, she thought Cambridge had nothing to commend it except its academic status and was beginning to dread the prospect of spending her life there. She resented the snobbery of the Cambridge academics who patronised her from the moment they heard she did not have  a degree. The Kapitzas were her sort of people - respectful, plain-spoken, full of life - but they were fifteen hundred miles away and in touch only irregularly. Always a thoughtful and generous friend, Manci inundated them with supplies to help them overcome shortages; Anna tactfully requested her to send only English books, coffee beans and good-quality pipe tobacco for her husband. She also encouraged Manci to be more positive about Cambridge: 'do you still feel lonely without your gay Budapest? If so, you are naughty and must not feel like this any more, because it worries people who like you and live with you (I mean Paul of course!)'\n\nIncessantly gloomy news bulletins on BBC radio about Hitler's increasingly transparent intentions did nothing to improve Manci's mood. In the spring of 1938, he had annexed Austria, where soldiers were welcomed with flowers and swastikas as they goose-stepped into towns. In late May, Dirac read an item in _Nature_ that will probably have disturbed him: his friend Schr\u00f6dinger was in Austria and appeared to be on Hitler's side. The article reported that Schr\u00f6dinger had written to a local newspaper in March 1938, 'readily and joyfully' affirming his loyalty to the new regime, having 'misjudged up to the last the real will and true destiny of my land'.\n\nDirac wanted to take his summer vacation in the Soviet Union, but this time the embassy in London refused his application and all others, in response to the British Government's denial of visas to Soviet citizens. So Dirac made more modest plans: in August 1938, he travelled to the Lake District in the north-west of England and went walking and climbing with his friend James Bell and with Wigner, still recovering from the tragically early death of his wife almost a year before, barely eight months after their marriage. From their correspondence, it seems that Bell agreed with Wigner that the recent trials in the Soviet Union were frame-ups, though Bell thought they were no worse than ones organised by the English in their colony of India. Meanwhile, Manci took her children and Dirac's mother to Budapest, where anti-Semitism was making her parents' life intolerable: they were beginning to see that they had no future in Hungary.\n\nSoon, the Diracs' home became a popular hostel for physicists and their families fleeing Nazism. Among the first to arrive were the Schr\u00f6dingers, who later settled in Dublin, after Schr\u00f6dinger accepted a post at the newly created Institute for Advanced Studies. During the stay, Schr\u00f6dinger will have explained to the Diracs why he had  earlier declared his support for the Nazis - he had been forced to make public his approval of the Nazi regime, he said, and had done this as ambiguously as he could. Dirac appears to have accepted this explanation and not to have questioned that his friend's integrity had wavered for a minute.\n\nThe house guest whose courtesy Manci most admired was Wolfgang Pauli, en route to the Institute for Advanced Study in Princeton, where he spent most of the war. Dirac told Kapitza: '[Pauli] has got much milder after his second marriage.'\n\nDirac agreed with the political left that the British Government had been weak and negligent in failing to tackle Hitler after his armies had invaded the Rhineland in March 1936. The left also, however, opposed rearmament and defence expenditure, a policy it would later regret. When Neville Chamberlain became British Prime Minister in 1937, he tried to mollify Hitler and waved away the warnings of his despised colleague Winston Churchill from the back-benches that the ambitions of the F\u00fchrer would have to be opposed by force. The mood in Cambridge alternated from hope that a war could be avoided to fear that a conflict was inevitable. Chamberlain brought about the most famous of these swings on 30 September 1938 when he returned from talks in Munich with Hitler, Mussolini and the French Prime Minister \u00c9douard Daladier to declare 'peace for our time', having agreed that Hitler's troops would be free to enter Czechoslovakia. Crowds cheered Chamberlain's return until they were hoarse; the entire country was euphoric even after it became clear that Czechoslovakia had been betrayed. But Churchill thought the agreement was a travesty: '[The] German dictator, instead of snatching his victuals from the table, had been content to have them served course by course.'\n\nAs he spoke those words, two German chemists, Otto Hahn and Fritz Strassman, were making a discovery that would change the course of history. The experiment they had done superficially looked recondite: when neutrons were fired at compounds of uranium, the new chemical elements that were formed were much lighter than had previously been thought. Within a few weeks, by the beginning of January 1939, it was clear that Hahn and Strassman had observed individual uranium nuclei breaking apart into two other nuclei, each with roughly half the mass of the original nucleus, as if a stone had  split into two parts of about the same size. Analogous to cell division in biology, the process came to be called 'nuclear fission'. The key point was that the amount of energy released in the fission of a nucleus exceeds the energy produced when atoms change partners during the burning of gas, coal and other fossil fuels by a factor of about a million - this is energy release on a huge scale.\n\nEddington had long foreseen the possibility of harnessing nuclear energy and in 1930 looked forward to the time when there would be no need to fuel a power station with 'load after load of fuel' but that 'instead of pampering the appetite of our engine with delicacies like coal or oil we shall induce it to work on a plain diet of subatomic energy'. Just over three years later, at the 1933 annual meeting of the British Association, Rutherford had ridiculed his colleague's vision as 'moonshine'. On the following day, after Le\u00f3 Szil\u00e1rd read about the prediction in _The Times_ , it occurred to him as he traversed a pedestrian crossing in Bloomsbury that it might be possible to capture nuclear energy more easily than Rutherford had imagined: 'If we could find an element which is split by neutrons and which would emit _two_ neutrons when it absorbs _one_ neutron, such an element, if assembled in sufficiently large mass, could sustain a nuclear chain reaction.'\n\nWhen Szil\u00e1rd heard about the discovery of fission, he realised that the chemical element he had in mind could be uranium. If more than one neutron was emitted when the uranium nucleus fissioned, those neutrons could go on to fission other uranium nuclei, which would emit more neutrons, and so on. Szil\u00e1rd later recalled that 'All the things which H. G. Wells predicted appeared suddenly real to me.'\n\nThe discovery of nuclear fission on the eve of a catastrophic conflict is one of history's most tragic coincidences. What made the prospect of nuclear weapons worrying for Dirac and other scientists who understood the implications of the discovery was that it had been made in Berlin, Hitler's capital.\n\nPhysicists and chemists were about to be drawn from the tranquillity of their offices and laboratories into a world of warfare, secrecy and power politics. The stakes could not have been higher, nor could the new work have been more troubling to their consciences. Scientists who regarded it as their duty to be open about their findings found themselves worrying that their results were too sensitive to be made  public. Szil\u00e1rd believed that if uranium was in principle capable of sustaining a nuclear chain reaction, then the results should be kept secret from Hitler's scientists, including Heisenberg and Jordan.\n\nThe sometimes bad-tempered exchanges about whether to keep the fission properties of uranium secret involved most of the leading nuclear scientists, including Bohr, Blackett, Fermi, Joliot-Curie, Szil\u00e1rd, Teller and Wigner. By early summer 1939, the campaign to keep the new science secret had failed. It was now public knowledge that uranium should be able to sustain a nuclear chain reaction: nuclear weapons were a practical possibility.\n\nDirac was only peripherally concerned with these discussions, having been asked by Wigner to support Blackett in the campaign to keep sensitive results confidential. In Cambridge, the euphoria of Chamberlain's Munich agreement had faded into despair by the spring of 1939, when Hitler contemptuously absorbed previously unoccupied parts of Czechoslovakia into Nazi protectorates and client states. War now looked inevitable. During those grim early weeks of 1939, Dirac prepared his first lecture as a self-styled philosopher of science who professed no interest in philosophy. Although the two living scientists he most admired - Einstein and Bohr - were both accomplished at talking about science to wide audiences, Dirac had shown no interest in following their lead until the Royal Society of Edinburgh awarded him their Scott Prize and invited him to give the Scott lecture on their favoured theme of the philosophy of science to an audience that included many who knew little or no science. Late on a Monday afternoon early in February 1939, he spoke for an hour on the relationship between the mathematician, who 'plays a game in which he invents the rules', and the physicist, 'who plays a game in which the rules are provided by Nature'.\n\nDirac's themes were the unity and beauty of nature. He identified three revolutions in modern physics - relativity, quantum mechanics and cosmology - and hinted that he expected them one day to be understood within a unified framework. Although he did not mention John Stuart Mill, Dirac was seeking to answer the same question posed in _A System of Logic:_ 'What are the fewest general propositions from which all the uniformities existing in nature could be deduced?' Whereas Mill never used the beauty of a theory as a criterion of its success, an appreciation for the value of aesthetics had  been part of Dirac's education. He now gave vent to his feelings by proposing the principle of mathematical beauty, which says that researchers who seek the truly fundamental laws of nature in mathematical form should strive mainly for mathematical beauty. Ignoring centuries of philosophical analysis about the nature of aesthetics, he declared that mathematical beauty was a private matter for mathematicians: it is '[a quality that] cannot be defined, any more than beauty in art can be defined, but which people who study mathematics usually have no difficulty in appreciating'.\n\nThe success of relativity and quantum mechanics illustrates the value of the principle of mathematical beauty, Dirac said. In each case, the mathematics involved in the theory is more beautiful than the mathematics of the theory it superseded. He even speculated that mathematics and physics will eventually become one, 'every branch of pure mathematics having its physical application, its importance in physics being proportional to interest in mathematics'. So he urged theoreticians to take beauty as their principal guide, even though this way of coming up with new theories 'has not yet been applied successfully'.\n\nThe physicists in the Edinburgh audience heard Dirac's enthusiasm for the discovery that the universe is expanding, which he said 'will probably turn out to be philosophically even more revolutionary than relativity or the quantum theory'. Focusing on how the universe developed from its birth, he suggested that classical mechanics will never be able to explain the present state of the universe because the conditions at the very beginning of the universe would be too simple to seed the complexity we now observe. Quantum mechanics might provide the answer, he believed: unpredictable quantum jumps early in the universe should be the origin of the complexity and 'now form the uncalculable part of natural phenomena'. Cosmologists rediscovered this idea forty years later, when it became one of the foundations of the quantum origins of the universe. While the world was heading into the gutter of war, Dirac was looking up at the stars.\n\nIn Cambridge, the students could not bring themselves to face the consequences of the expected war. In April, the students' sixpenny magazine _Granta_ looked forward to another summer of croquet on the lawns, cucumber sandwiches, paprika salad and cr\u00e8me br\u00fbl\u00e9es washed down with chilled Bollinger. For students wanting to wind  down after the examinations, there were performances of Mozart's _Idomeneo_ and more opportunities to see Disney's _Snow White and the Seven Dwarfs._ 68 The captain of the university cricket team knew that the party was soon to be over, though he said that he hoped to God that Hitler would not start a war before the end of the cricket season. But he was disappointed: after Hitler's invasion of Poland, Chamberlain declared war on 3 September, before the final overs had been bowled.\n\nTen days before, Dirac - on holiday with his family on the French Riviera - read that Stalin had signed a non-aggression pact with Hitler, a moment that George Orwell called 'the midnight of the century'. Stalin's opportunism was incomprehensible to Dirac. He still tended to expect politicians to practise with the consistency of mathematicians, and it is probably no coincidence that Dirac's disillusion with politics and politicians began that summer. From then on, he turned away from public affairs and concentrated on his family, which was about to expand - Manci was pregnant.\n**Twenty-two**\n\nAs I write, highly civilized human beings are flying overhead, trying to kill me. They do not feel any enmity against me as an individual, nor I against them. They are only 'doing their duty' [. . .].\n\nGEORGE ORWELL, _The Lion and the Unicorn_ , 1941\n\nAdvances in aviation technology had made the aerial bombing of Britain inevitable, though some people in Cambridge could not believe the Germans would ever bomb a town of such beauty. Nuclear weapons were being discussed, too, in newspapers and popular magazines, but most of the public and national leaders seem not to have noticed. Dirac, aware of the potential of nuclear fission, had an inkling of what might be in store: like many scientists, he would soon have to decide whether to drop his research and participate in the largest military programme the world had ever seen.\n\nSoon the conflict would disperse Dirac's extended family across two continents. He waited every day for news of Betty in the Netherlands. Manci was worried about her Jewish relatives, especially her parents and sister, who had left Budapest and settled in New York State, assisted by Wigner and his new wife Mary. Although she strongly supported the war, Manci knew the pain of being suspected as an alien and smarted at the subtle signs of disapproval from strangers when she revealed her thick accent, which many took to be German. In her adopted country, she felt like a 'bloody foreigner'.\n\nWhen the Diracs ventured into the centre of Cambridge on the freezing nights of January 1940, they saw that much of the town looked just as it did in Newton's day. Under the moonlight, the architecture of the city - the College buildings, King's Parade, Senate House - had never looked more sublime. The mood of the town was, however, becoming more apprehensive: thousands were bracing themselves for an attack, ready to flee to the new bomb shelters. Dirac and his family stayed indoors, carefully observing the 'blackout', preventing every shard of light from escaping into the night by covering their windows with black paper. By six o'clock each night,  the town was as quiet as a village on a Sunday morning; by ten, it was almost deserted. The church bells had been silenced, the streetlamps switched off.\n\nAt the beginning of the war, the population of the town had swelled by almost a tenth, to about eighty thousand. At the beginning of September 1939, trainloads of children had arrived from London and other towns that were expected to be the targets of enemy bombers. The evacuees, many with their home addresses written on luggage labels tied around their necks, were billeted with local families, many of which received them less warmly than sentiment now recalls. The Diracs did not take in any of these children, though in the coming months they saw them virtually overrun the town.\n\nEveryone, including the dons, carried around a foul-smelling rubber gas mask. For the time being at least, academics in their gowns had lost their special status and were no more important than the thousands of volunteers and part-time workers who were preparing for war. The texture of day-to-day conversations changed: people talked more loudly, endlessly repeating catchphrases such as 'I'm doing my bit' and 'Don't you know there's a war on?' All over the town, posters warned that 'Careless talk costs lives', words that looked comically alarmist, as there were no signs of an imminent conflict: by March 1940, nothing much had happened since the collapse of Poland, and the restless public called it the Phoney War or, sometimes, the Bore War. Most of the evacuated children drifted home.\n\nThe university ticked over, though there were fewer dons as many of them had left to take up posts in government, the armed forces and war research establishments. There were fewer students, too, but a skeleton programme of teaching continued, and Dirac gave his lectures on quantum mechanics as usual. A regular visitor to the college, he saw how much its atmosphere had changed: it now accommodated not only its staff and students but also uniformed members of the Army, Navy and Royal Air Force, who worked in the new buildings completed shortly after the outbreak of war. The college was one of the national centres of the Air Force, and hundreds of its cadets were trained there, mixing uneasily with the undergraduates, who had different catering facilities. The menus for college members were now much more modest: at High Table, about all the Fellows could expect was a ladleful of mutton stew and vegetables grown on college land. Gardeners had dug up the lawns to grow onions and potatoes.\n\nAt home, the Diracs lived like most others in Britain. They queued for their ration books and food coupons and took pots and pans to local collecting points to be melted down and turned into weapons. Dirac had chopped down a tree in the garden for firewood, cultivated potatoes and carrots in a nearby allotment, and grew giant mushrooms in his cellar. But Manci, well into her pregnancy, wanted support. She would not dispense with her servants, and she fretted at the thought of losing even one of them. Dirac's mother in Bristol was counting the days to the birth of her second grandchild, hoping that the child would be a boy and that his parents would name him Paul. But she was to be disappointed: the child was a girl, Mary, born on 9 February 1940, at London's Great Ormond Street Hospital. As Manci wrote in her notebook, Mary was a 'daddy's girl', as she would remain. Dirac was a doting father, in his reserved way, dandling her on his knees, trying to entice her to play with a new doll sent by her godmother, Schr\u00f6dinger's wife Anny.\n\nDesperate to see her first granddaughter, Flo made a flying visit to see the baby and her mother. Flo's manner with the baby did not impress Manci, who complained to Dirac the next day:\n\nIt is awful of me to write about her, you never criticize my parents. But I never felt as much that she has neither heart nor feelings . . . She has no notion of how to handle a tiny thing as a baby but she picked her up. It was quite terrible to me.\n\nDirac may have sensed that this would not be the last clash between the two women closest to him, each jealous of the other's place in his affections. But their disputes appear not to have spoiled his first few months of paternity. He now had the domesticity he craved, but it was soon disrupted by an urgent request to do something he had hoped to avoid: to join the scientists' war effort.\n\nRudolf Peierls was now in Birmingham, a professor of physics by day and volunteer fireman by night, equipped with a uniform, a helmet and an axe. Peierls had settled in England after fleeing Nazi Germany in 1933 with his Russian wife Genia, a former member of the Jazz Band of Soviet physicists. Like most scientists who had lived under Hitler, Peierls wanted him crushed, but the British authorities were slow to accept his offers of assistance: in early February 1940, Peierls and his wife were officially classified as 'enemy aliens'. The couple's  naturalisation papers came through later that month so he was eligible to work on secret projects, though the authorities still looked at him with suspicion and denied his request to work on the new radar technology.\n\nIn early February 1940, when Dirac was cradling his newborn daughter in his arms, Peierls was thinking about nuclear weapons. Like most scientists who were following the debate, he believed that such a weapon would not be possible after all. Niels Bohr and John Wheeler had apparently provided the clinching argument by proving that the fission of uranium by slow neutrons was due entirely to the rare isotope of uranium 235U, containing a total of 235 nuclear particles, not to the much more common uranium isotope 238U, which contains 238 particles. A little less than one part in a hundred of a typical sample of natural uranium is 235U, and the rest is almost entirely 238U. It followed that if a nuclear bomb were made using naturally occurring uranium, very few nuclei would undergo fission, so any chain reaction that started would soon fizzle out. But a loophole was spotted by one of Peierls' Birmingham colleagues, Otto Frisch, the scientist who had given fission its name and been the first to explain it, in collaboration with his aunt, Lise Meitner. Frisch was one of an almost unbroken string of bachelors who lodged with Rudolf and Genia Peierls and became part of the household, helping with the washing up and keeping their children amused during the blackouts.\n\nThe crucial question Frisch asked was: 'Suppose someone gave you a quantity of the pure 235 isotope of uranium - what would happen? ' When Frisch and Peierls did the calculations, they found the amount of 235U needed was about a pound, about the volume of a golf ball. Although it would be difficult and expensive to produce much of this rare isotope, the resources required, compared with the costs of running the war, would be chickenfeed. Frisch later recalled that when he and Peierls tumbled that the purification process could, in principle, be completed in weeks, 'we stared at each other and realized that an atomic bomb might after all be possible'. Even more terrifying was the thought that the Germans might already have done their calculation and Hitler might be the first to have the bomb.\n\nFrisch and Peierls secretly typed up two memos on the properties of a 'Super-Bomb' and the implications of building one, setting out  their conclusions in a total of six foolscap pages, which they sent to the British Government, keeping just one carbon copy. The authorities were grateful but asked them to understand that, as Peierls later recalled, 'henceforth the work would be continued by others; as actual or former \"enemy aliens\", we would not be told any more about it'. If the Government wanted scientists to build a nuclear weapon, they would need to find a way to distil pure 235U from mined uranium ore, which contains the mixture of 238U and 235U. Several groups were set up in the UK to investigate ways of separating the uranium isotopes, including ones at the universities of Liverpool and Oxford. Scientists in these groups knew that Dirac had invented one method of doing it: the centrifugal jet stream method of isotope separation, which he had investigated in the spring of 1934 but abandoned after the Soviets had detained his collaborator, Kapitza. By the late autumn of 1940, Dirac had heard that his long-discarded experiment might, after all, have important applications in developing material to make a nuclear bomb. Soon he would be under pressure to resume his studies of the technique.\n\nIn the United States, Le\u00f3 Szil\u00e1rd - a close friend of Manci's brother Eugene Wigner - was trying frantically to persuade the Government to develop a nuclear bomb before the Germans. He was working at Columbia University in New York with his fellow refugee Enrico Fermi, the experimentalist best qualified to build a nuclear weapon if it were feasible. Progress was slow and funds were short, partly because few government officials took Szil\u00e1rd's hectoring seriously. In the summer of 1939, Wigner, Szil\u00e1rd and Teller persuaded Einstein to write to President Roosevelt, drawing his attention to the possibility of nuclear weapons and the danger that the Germans might produce one first. After a long delay, Roosevelt invited Einstein to join a committee of government advisers but he brusquely declined and sat out the war at the Institute for Advanced Study in Princeton, where word spread that the Nazis were indeed working on a bomb. In the spring of 1940, Dirac's friends Oswald Veblen and John von Neumann wrote to the director Frank Aydelotte, urgently seeking his assistance to fund investigations into the chain reaction. In their letter, they mentioned a recent conversation with the Dutch physical chemist Peter Debye, who had led one of Berlin's largest research institutes until  the German authorities sent him abroad in order to free his laboratories for secret war work.\n\n[H]e made no secret of the fact that this work is essentially a study of the fission of uranium. This is an explosive nuclear process which is theoretically capable of generating 10,000 to 20,000 times more energy than the same weight of any known fuel or explosive [. . .] It is clear that the Nazi authorities hope to produce either a terrible explosive or a very compact and efficient source of power. We gather from Debye's remarks that they have brought together in this Institute the best German nuclear and theoretical physicists, including Heisenberg, for this research - this in spite of the fact that nuclear and theoretical physics in general and Heisenberg in particular were under a cloud, nuclear physics being considered to be 'Jewish physics' and Heisenberg a 'White Jew'.\n\nThere is a difference of opinion among theoretical physicists about the probability of reaching practical results at an early date. This, however, is a well-known stage in the pre-history of every great invention. The tremendous importance of the utilisation of atomic energy, even if only partially successful, suggests that the matter should not be left in the hands of the European gangsters, especially at the present juncture of world history.\n\nAydelotte responded by helping Szil\u00e1rd with his search for funding. The prime responsibility of Aydelotte and Veblen, however, was the Institute for Advanced Study, and they dreamt of setting up a wartime haven for the most eminent quantum physicists, including Bohr, Pauli, Schr\u00f6dinger, Dirac and even Heisenberg. But when the war intensified, it became unthinkable for most of them to concentrate on anything other than the war. The pursuit of the fundamental laws of physics was set aside.\n\nIn April 1940, the Nazis overwhelmed Norway and Denmark and launched a blitzkrieg on Belgium, Luxembourg and the Netherlands a few weeks later: the Phoney War was over. Dirac's sister Betty and her family were now living in an occupied country. Joe, like all the other Jews, lost much of his freedom: he was subjected to a curfew, forbidden to ride in trams or cars and forced to wear a yellow star when outside his house. A month before, the German forces had conquered Denmark unopposed and had invaded Norway, swatting aside the British Government's naval campaign to repel them. Chamberlain was forced out of office and replaced by Churchill - the man regarded by many as a belligerent class warrior soon became the  saviour of his country and the embodiment of bulldog spirit, a national hero.  The Diracs gathered round their radio to listen to his broadcasts and to reports of his speeches. Three days after he entered 10 Downing Street he told the House of Commons in his first speech as Prime Minister that the aim was 'Victory - victory at all costs, victory in spite of all terror; victory, however long and hard the road may be; for without victory, there is no survival.' Manci was star-struck: she sent Churchill a note consisting of just two words - 'God's blessings' - after a broadcast he had made a few days after the Luftwaffe dropped its first bombs on Cambridge on 18 June 1940.\n\nAt 11.30 p.m. on that night, the air-raid sirens began to wail, and the Diracs scurried down to the shelter of their cellar. Moments before midnight, they heard a Heinkel bomber dive low overhead and, after a piercing whistle, a huge explosion when the plane dropped two high-explosive bombs about a mile away. Ten people were killed, a dozen were injured, and a row of Victorian houses was laid waste. The following night, the bombers struck Bristol for the first time, targeting the British Aeroplane Company's factory in Filton. Dirac's mother was desperate to speak to her son but, with no telephone, the best she could do was to write to him:\n\nThe awful raiders pay a midnight call every night. The first was a downright shock on Monday. I flew down with all my dressing gowns, collected all the green cushions from the big chairs & made myself warm & comfortable propped against the kitchen door [. . .] To my surprise I got intensely angry at their cheek & impudence in disturbing my night's rest & daring to visit our Island in such a manner.\n\nChoosing not to take drams of whisky and play poker with her neighbours in their cellars, Flo spent most nights alone, crouched in the cupboard under the stairs with cotton wool in her ears, trying to sleep during the hours of 'fireworks'. At five in the morning, when the sirens and steamers in the docks roared their 'all clear', she went up to Betty's room to catch up on her sleep. Flo was lonely, sick with rheumatism and gout, anxious about her family and disappointed that her son was such a poor correspondent: 'I am sure you can spare five minutes for a few lines if you try very hard.'\n\nBy August 1940, the 'Battle of Britain' was underway. The Luftwaffe was pummelling London and fighting over the skies of England with  the Royal Air Force, helped by the early warnings made possible by the new radar technology. Despite the widespread fear of an imminent Nazi invasion, daily life in Britain continued normally. Food and everyday supplies were in the shops, the trains and buses were running, and there were queues outside cinemas showing _Gone with the Wind ._ 26 It was a summer of almost uninterrupted glorious weather, and the more prosperous Britons, including Dirac, saw no need to forgo their annual vacation. Dirac and Gabriel took a four-week break in the Lake District, renting a cottage in Ullswater with Max Born and his family - his wife, their nineteen-year-old son Gustav, their daughter Gritli and her new husband Maurice Pryce, a theoretical physicist at the University of Liverpool. The outdoor life, primitive facilities and the prospect of communal cooking were not for Manci, who remained in Cambridge with Judy, baby Mary and her nurse, after Dirac had assured her that the danger of air raids in Cambridge had been exaggerated ('you should not let the air raid warnings worry you, dear').\n\nWhile Gabriel stayed in the cottage, his head buried in a book, Dirac and Pryce headed off early to the mountains with a vacuum flask of hot tea and a packed lunch. With Pryce and Gustav Born, Dirac climbed the highest peak in England, Scafell Pike, rowed on the lakes, climbed up several rock faces and followed some of the paths trodden by Wordsworth, who had lived in nearby Grasmere. At night, the party dined on the balcony, overlooking a lake as still as a pond: it scarcely seemed possible that they were in a country fighting for its life until they switched on their radio and heard the news from London.\n\nBarely four days after Dirac's vacation began, Manci was in the cellar with Mary and Judy, following the first of several air raids. 'I am very sorry to be away during these air raids,' Dirac wrote to his wife, though he was not worried enough to return home. Feeling abandoned and dejected, Manci dropped her usual affectionate tone when she wrote to him:\n\nI know very well that you never do or did what people happened to ask you for. So I am not asking you anything; it is but a question. Would you return to Cambridge if I was not here? Because if you would not, then do not come home please.\n\nAs usual, her wrath soon abated. Dirac was habituated to her outbursts and fended them off by remaining silent. It was a singular marriage, not one most people could endure, but it was working.\n\nDirac's climbing partner Maurice Pryce - formerly a colleague of Dirac and Born in Cambridge - was studying isotope separation with the Liverpool team and had recently asked Dirac's advice about his centrifugal jet method. But it seems that Dirac did not think seriously about developing the method until several months later. This delay is surprising, as many of his peers were talking urgently of the need to develop a nuclear weapon ahead of the Nazis. Perhaps part of the explanation for his tardiness is that he was preoccupied with his stepchildren, constantly quarrelling and consuming more of his energy than he would have liked. Gabriel, then an introverted fifteen-year-old, was developing into a talented mathematician. Encouraged by Manci, he revered his stepfather as a hero, looked to him for advice and even copied his handwriting, down the last detail of the curl on the capital D. Judy, two years his junior, was growing into an attractive young woman and quite different from her brother: she was lazy, headstrong and not at all frightened of provoking her mother. Manci's high-handedness sometimes alarmed Dirac, who privately warned Gabriel that he should not take too much notice of her tantrums.\n\nDirac agonised about his sister and her family, behind enemy lines. She had written to him from Amsterdam via the Red Cross mail service on 3 July to report that she was safe, and the letter took three months to arrive. Shortly after he read it, Dirac heard that Dutch citizens would be fined \u00a315,000 if they were caught listening to British radio transmissions. He was also concerned about his mother, who occasionally visited Cambridge but spent most of her time alone in 6 Julius Road, going out only occasionally to the shops, the cinema and to volunteer for the emergency canteen service. Bristol was the fourth most heavily bombed city in the UK (after London, Liverpool and Birmingham): almost every night, the planes attacked the city and, though Julius Road was two miles from the worst of the attacks, Flo was in fear of her life. She went to bed early and tried to sleep through the seven-hour barrages, until the sirens blasted the 'all clear' signal into the dawn.\n\nThese were among the darkest days of the war. Peierls in Birmingham was one of many who believed that the fight against Hitler was then 'hopeless', as he recalled fourteen years later. Although Germany had failed to win the Battle of Britain, the war was going its way, as Hitler well knew: he told his ally Mussolini in October 1940 that the war had been won.\n\nIn mid-December, Dirac's mother was admitted to a nursing home, suffering from concussion, after a stone had fallen on her when she was out walking. Dirac rushed to Bristol and, between visits to Julius Road, walked around the bombed-out heart of the city. At the Merchant Venturers' College, he saw that many of the buildings he had known since he was a child had been pulverised into smouldering piles of rubble. Several of the homes on his route had been bombed out, their once-private spaces now embarrassingly on show for all to see. 'The middle of Bristol is terribly damaged [. . .] most of the best shopping areas are in ruins [. . .] and many beautiful churches have gone,' he wrote to Manci. She was too angered by being left alone to feel much sympathy:\n\nYou know that envy is not in me but I am a little revolted that you had to go, and have to stay. After all 60 years ought to have been enough for anybody to make friends [. . .] she is only interested in people as far as what she will be able to talk about them.\n\nUnmoved, Dirac helped his mother to return home and stayed with her until she could resume her routine, returning to Cambridge shortly before the year's end. All over the UK, the New Year celebrations were subdued, for the country was pinned to the wall.\n\nMost scientists in Britain had put themselves at the service of their country but, as usual, Dirac did not swim with the shoal. In peace-time, he was part of the mainstream of physics but always one step from it, so that his individuality was not constrained. He now had the same relationship with the scientists working for the military: he supported them but only to an extent that neither his daily routine nor his intellectual independence was compromised. One of the first invitations to participate in war work that Dirac received had come, surprisingly, from the mathematician G. H. Hardy, who was contemptuous of the applied mathematics involved in war work as unworthy of 'a first-rate man with proper personal ambitions'. He wrote to Dirac in May 1940, asking him to join a team of twelve mathematicians to code and decode messages at the Civil Defence offices in St Regis, in the event of a Nazi invasion. Dirac appears to  have declined, probably because he would not consider moving from Cambridge and because teams, to him, were anathema.\n\nThe journalist Jim Crowther did not, however, stop trying to involve his retiring friend in public affairs: in mid-November 1940, he tried to persuade Dirac to attend a meeting of the Tots and Quots dining club, an informal gathering of academics who were interested in exploring how their expertise might be useful to society (the name of the club is a reference to the Latin _quot homines, tot sententiae:_ 'so many men, so many opinions'). Its twenty-three members in 1940 - including Bernal, Cockroft and Crowther - were often joined by guests, such as Frederick Lindemann, H. G. Wells, the philosopher A. J. Ayer and the art historian Sir Kenneth Clark. The location of the club's political centre of gravity, well to the left, was reflected in the outcome of their debates, most of them held over a few bottles of wine and an indifferent meal in London's Soho. The meeting Crowther wanted Dirac to attend, on Saturday, 23 November 1940, was scheduled to discuss Anglo-American scientific cooperation and was to take place in Christ's College, Cambridge. Crowther knew the best way to encourage Dirac to attend: 'It would be quite unnecessary for you to join in the discussion if you did not wish to.' Crowther succeeded, and Dirac listened to a wide-ranging discussion about ways of promoting scientific cooperation with American scientists, until shortly after midnight. Bernal opposed the suggestion that British research projects should be transferred to the United States, arguing that the best way forward was to promote personal contacts between British and American scientists. It was important, he stressed, not to give up too easily on preserving the independence of British science.\n\nThe record of this special Tots and Quots meeting makes no mention of any contribution from Dirac. So far as records show, he attended no other social gathering of scientists during the war.\n\nAt about the time of the meeting, Dirac began to think again about his method of separating mixtures of isotopes.  Seven years earlier, he had demonstrated that the technique might work; he now turned to a theoretical analysis of the process, to help engineers investigate ways of separating a mixture of 235U and 238U. His original idea was to deflect a gaseous jet of the mixture through a large angle, so that the heavier and therefore slower-moving isotopes would be deflected less than the lighter ones, and the two components would separate.  He tried to find a general theory of all processes that might separate isotopic mixtures in this way, aiming to deduce the conditions that would most effectively separate them. To solve the problem, he had to use all his talents: the mathematician's analytical skills, the theoretician's penchant for generalisation and the engineer's insistence on producing useful results.\n\nHe gave his first account of the theory in a confidential, three-page memorandum. Dirac wrote it for Peierls and his colleagues, probably in early 1941, between the incessant bombing raids, and typed it at home. He wrote the paper in his usual spare style but taking care to highlight the most important conclusions so that they would be clear even to engineers allergic to complicated mathematics. The memo does not focus on his own jet separation method but concerns every conceivable way of separating isotopes in a liquid or gaseous mixture by causing a variation in the isotopes' concentration. The separation might be achieved, for example, by subjecting the mixture to a centrifugal force or by carefully arranging for the temperature to change across the container. To make the calculations tractable, he made the reasonable assumptions that the fluid mixture contains only two isotopes (each made of simple atoms) and that the concentration of the lighter one is small compared with the concentration of the other. In a short calculation, he derived a formula for what he called the 'separation power' of the apparatus, a measure of the minimum effort needed to cream off a given amount of the lighter isotope. He found that every part of such an apparatus, irrespective of how it is built, has its own maximum separation power, and he showed how to calculate it.\n\nDirac often drove to Oxford to talk with the experimenters who were developing ways of separating isotopes, under the impish Francis Simon, another German refugee physicist. Dirac surprised many of the experimenters by participating vigorously in their meetings and by making practical suggestions about the design of their apparatus. During these discussions, he conceived several other ways of separating isotopes, each of them based on his original centrifugal jet stream method.\n\nThe Oxford group built one of Dirac's designs, and it worked, but his method was less efficient than the competing technique of gaseous diffusion, which exploits the fact that the atoms of two isotopes in equilibrium and with the same energy have different average speeds: the lighter, swifter atoms are more likely to diffuse through a membrane than heavier ones, enabling the mixture to be separated. Consequently, at this stage in the development of nuclear energy, resources were diverted to gaseous diffusion, and Dirac's idea was set aside.\n\nLate at night on 9 May 1941, a bomb fell opposite the Diracs' home, damaging two houses and causing small fires that Judy helped the fire fighters to extinguish. This was the most frightening moment for the Diracs in the worst year of bombing in Cambridge, and it was relentless where they lived, close to the strategic target of the railway station. But the Diracs' everyday life was much the same as it was before the war. Part of this routine involved welcoming visitors; Dirac was determined not to follow his father's example of virtually barring the family home from others, apart from paying students. One of the most frequent visitors to 7 Cavendish Avenue was Jim Crowther, 'the newspaper man'. A one-man clearing house of information about the activities of leftist scientists, he was a favourite of Manci's, who entertained him and his wife Franciska as royally as rationing allowed: she could stretch to a cup or two of tea, but biscuits and cakes were luxuries. After one get-together, Crowther lent her Somerset Maugham's _On Human Bondage_ to help her improve her English and her understanding of British foibles. Still worried that people in Cambridge thought of her as an outsider, she even sensed disquiet that she might be an enemy agent. Suspicions of aliens intensified in the town in the spring of 1941, when an innocent-looking Dutch seller of second-hand books in Sidney Street was unmasked as a spy. When he heard that military intelligence was on to him, he broke into an air-raid shelter on Jesus Green and shot himself.\n\nDuring the Diracs' conversations with the Crowthers, Dirac heard Crowther's bulletins on the scientists' war work, delivered with his subtle political colouring, though almost certainly without the political edge that he reserved for conversations with more committed colleagues. Crowther knew that this was time well spent: Dirac would never commit himself to the cause of the left, but he was a powerful ally, if only because no other British physicist came close to his intellectual prestige.\n\nAlthough Dirac spent most of his time on war work, he was still thinking about quantum mechanics. In one project, he collaborated with Peierls and Pryce to refute accusations made by Eddington that  experts in relativistic quantum mechanics, including Dirac, were persistently misusing the special theory of relativity. This disagreement had been rumbling for years: in the summer of 1939, Sir Joseph Larmor had heard that 'Eddington has lately come to blows with Dirac.' Dirac, Pryce and Peierls tried to make Eddington see reason but, by the early summer of 1941, their patience had run out, and they prepared what Pryce dubbed 'the anti-Eddington manuscript'. The paper appeared a year later, and Eddington's arguments were crushed to the satisfaction of everyone except Eddington himself, who never accepted defeat.\n\nWhen the Royal Society conferred on Dirac the honour of giving its annual Baker Lecture, he took the opportunity to present his latest thinking about quantum physics. In the early afternoon of 19 June 1941, when Dirac arrived at Burlington House, he saw that central London had suffered surprisingly little in the Blitz; most of the damage had been done in the City and the East End. Giving the lecture was in keeping with the spirit of the hour - Londoners were going about their business as usual, and that included attending lectures about matters of no practical importance.\n\nDirac rose to the podium at 4.30 p.m. to describe why he was so unhappy with the current state of quantum mechanics: why is it, he wondered, that the first version - set out by Heisenberg and Schr\u00f6dinger - is so beautiful whereas the relativistic version is so diseased?  It might be possible, he showed, to remove one of the pathologies of the relativistic theory - negative-energy photons - using a technical device later dubbed the 'indefinite metric'. Although not a panacea, the technique demonstrated to the standing army of quantum physicists that Dirac was still one of their generals. Even Pauli was impressed and wrote to Dirac to say so.\n\nDirac's conclusion to the lecture was that the 'present mathematical methods are not final' and that 'very drastic' improvements were needed. He knew, however, that they were unlikely to be made at a time when most of the best scientific brains were working on top-priority projects for the military. Only rarely did the scientists on opposing sides communicate. One such encounter took place in late September 1941, when Heisenberg travelled to Nazi-occupied Denmark to see Bohr (who knew nothing of the Anglo-American project to build a nuclear bomb) in a fraught meeting that was remembered and interpreted quite differently by the two men. The  playwright Michael Frayn dramatised their discussions six decades later in _Copenhagen_ , a metaphor for the uncertainty principle: the more the intentions of the participants at the meeting are probed, the murkier they appear to be. Although it will never be possible to know precisely what the two men said, one consequence of their meeting is now clear: their friendship was damaged beyond repair.\n\nDirac, in touch with neither Bohr nor Heisenberg, knew nothing of the meeting. When it took place, he was in Cambridge, preparing for the new term, no doubt anxiously reading the news of the Nazis' invasion of the USSR, which had begun when Hitler unilaterally broke the pact with Stalin three months before. Kapitza was now in Hitler's sights. On 3 July, a few days after the pact collapsed and Stalin joined the Allies, Kapitza sent Dirac a telegram, one of the few communications that Dirac received from him during the war:\n\nIn this hour of stress when our two countries fight against a common enemy I want [ _sic_ ] send you a friendly word. The united strength of all men of science will help the victory over the treacherous enemy who by brutal force destroyed the liberty and crushed the freedom of scientific thought in Germany and is trying to do the same in all the world. My greetings to all friends united in their will for fighting to complete victory for the freedom of all people for the freedom of scientific thought so dear to our two countries.\n\nLater during the conflict, Dirac was moved to similarly grand words in a rare letter to Kapitza. After offering his 'hearty congratulations' to Kapitza on his second Stalin Prize, Dirac wrote that he hoped 'that the great Hitler menace which now darkens this world will soon be obliterated'.\n\nFlo was also thinking about Kapitza and his compatriots: 'Those plucky Russians are putting up such a grand fight!', she wrote to her son. By the summer of 1941, Bristol appeared to have seen the worst of the bombing; about 1,200 people had been killed.  She was ailing and desperate to stay at 7 Cavendish Avenue, where Manci was struggling to cope after her maid and cook had departed. In early October, Flo arrived with her luggage and hatbox, having declared that she wanted to help with the housework, though her doctor wrote privately to Dirac: 'I want you to see that she does not do extra work' as 'her heart is overstrained and she is rather run down'. She stayed longer than the month she had planned, working under  Manci's direction as a kitchen maid and house cleaner, helping the servants and Mary's nurse. Soon after the Americans entered the war, following the bombing of Pearl Harbor on 7 December 1941, Flo wrote to one of her neighbours: 'Paul says it will take two years to conquer the Japs.' But she was homesick and tired of being Manci's charlady: 'I really am afraid I will be quite ill if I stay on. Manci imposes on me too much.'\n\nFlo never sent the note as, four days before Christmas, she had a fatal stroke. Dirac seems to have taken her death with his usual almost-inhuman stoicism: his sliver-thin vocabulary of emotions did not include conventional expressions of grief. Manci saw no tears. Yet he knew better than anyone the tragedy of her unfulfilled life: the suicide of her first-born; her servitude during a sham marriage and its horrible final years, when she was like a rabbit domiciled with a bear. Dirac knew that his mother had her flaws: she was absent-minded and disorganised, selfishly determined to keep her younger son to herself. But Dirac knew that life had not been generous to his mother and that he had been her greatest love.\n\nHer funeral took place two days after Christmas. Dirac threw away most of her belongings but not the Christmas card on which she had written her feelings about Manci. He kept that among his papers.\n**Twenty-three**\n\nThere is no room now for the dilettante, the weakling, for the shirker, or the sluggard. The mine, the factory, the dockyard, the salt sea waves, the fields to till, the home, the hospital, the chair of the scientist, the pulpit of the preacher - from the highest to the humblest tasks, all are of equal honour; all have their part to play.\n\nWINSTON CHURCHILL, speech to the Canadian Parliament, 30 December 1941, later broadcast on the BBC\n\nTo Dirac's neighbours, it appeared that the war had little impact on his life: he remained another professor going quietly about his business, his civic duties involving nothing more than an occasional night on fire watch at the Cavendish. But none of his neighbours knew that he spent most of 1942 and 1943 working on nuclear weapons. Even Manci had only a vague idea of what he was doing: she told the people she knew in Cambridge that he was working on 'decoding'.\n\nMost leading scientists did more to support the military than Dirac. Patrick Blackett was one of several of Dirac's friends who took his place at the top table of the Government's scientific advisers and attended dozens of interminable policy meetings. He joined his former Cavendish colleagues Cockcroft and Chadwick on a special committee set up to consider the implications of Frisch and Peierls' prediction of the small amount of uranium needed to make a bomb. They consulted Dirac, but he had no wish to be part of the proceedings.\n\nBy August 1941, Churchill authorised the manufacture of a nuclear weapon, following the advice of the committee and approving comments from his friend and chief scientific adviser, Frederick Lindemann. The British Government allocated the resources its scientists requested to begin to build the bomb and set up the 'Tube Alloys' project, a name chosen to be dull enough to escape the attention of prying eyes and ears. Blackett, the one dissenting voice on the committee, believed that the British could not build the bomb alone: the project would be successful only if it were pursued in  collaboration with the Americans. He would soon be proved right. Blackett was no happier in his other dealings with the Government. He was one of the pioneers in the use of science to inform decisions about the management of the war; for example, in weighing the risks and benefits of different military strategies. The hard-headed application of this new discipline of 'operational research' brought Blackett and his colleagues, including Bernal, into disagreements with the military and the politicians, who both preferred to take decisions with their hearts as well as their heads. Blackett insisted that Churchill's policy of aerial bombing enemy civilians - supported by the military and the public - was ineffective, the misguided result of a failure to identify the enemy's key industrial and military targets. It would be better to bomb the enemy's fleet of U-boats, he told an unmoved Lindemann. Churchill persevered with his policy and kept his scientific committees at a distance: for him, 'Scientists should be on tap, not on top.'\n\nLike many mathematicians, Dirac was invited to work at the Government's research station in Bletchley Park. In late May 1942, he was approached by the ancient-history scholar Frank Adcock, who had been charged with recruiting the best Cambridge brains. Adcock wrote to Dirac, 'There is some work concerned with the war which is itself important and would, I believe, be of interest to you. I am not free to say just what the work is.' When Dirac asked to know more, a Foreign Office official wrote to clarify: 'The work would be a full-time job [nominally nine hours a day] and would require you to leave Cambridge.' With Manci four months pregnant, this was too much disruption for Dirac to contemplate, so he never did work in the huts of Bletchley Park with Max Newman and Newman's former student Alan Turing. This would have been one of the most intriguing collaborations of the war.\n\nIn Cambridge, Dirac supervised graduate students and gave his quantum-mechanics lectures to about fifteen students on Tuesday, Thursday and Saturday mornings. In 1942, his audience included Freeman Dyson, an exceptionally talented student, then nineteen years old. Dyson was disappointed: in his view, the course lacked all sense of historical perspective and made no attempt to help students tackle practical calculations. Not one to suffer in silence, Dyson amused his fellow students by bombarding Dirac with questions, sometimes catching him off-guard and once causing Dirac to end a lecture early so that he could prepare a proper response. Almost twenty years before, the young Dirac had pressurised Ebenezer Cunningham in one of his lecture courses; now it was Dirac's turn to be shown the drawn sword of youth.\n\nBy early 1942, Dirac was thinking more about technology than quantum mechanics. He was a consultant to the Tube Alloys project and worked closely with Rudolf Peierls. One of the first reports that Dirac wrote for him concerned another way of separating a mixture of isotopes, using a simple method that involves injecting the mixture into the base of a hollow cylinder spinning rapidly about its long axis. The centrifugal force generated by the rotation causes the heavier isotope to move towards the outer rim and the lighter one to accumulate closer to the central axis, thus effecting a separation. When Dirac sent his report to Peierls in May 1942, he wrote that he had 'written up [his] old work' and did not mention its provenance.  It is clear from the manuscript that Dirac wanted to investigate the motion of the gases in the tube, to find how far up the spinning cylinder the injected gas will reach. Using classical mechanics, he found that the device would be a stable source of separated isotopes and calculated that, if the cylinder had a radius of one centimetre and rotated almost five thousand times a second, its length should be about eighty centimetres. This confidential report, declassified in 1946, proved to be seminal for the designers of centrifuges. Dirac's calculations provided the theoretical underpinning of the counter-current centrifuge, invented three years earlier by the American scientist Harold Urey. This technique was not used during the manufacture of the first nuclear bombs - other methods made less onerous engineering demands - but later became the nuclear engineer's preferred choice as it gives a particularly efficient way of separating uranium isotopes.\n\nDirac's other work for Peierls and his group in Birmingham consisted of theoretical investigations into the behaviour of a block of 235U if a nuclear chain reaction took place inside it. These calculations probed in detail the energy changes going on inside such a block of material and investigated whether the growth of neutrons would change if the uranium were enclosed in a container. Dirac was happy for his results to be shared with the American scientists who were working on the bomb, including Oppenheimer, who by the end of 1942 had been appointed the Scientific Director of what became known as the Manhattan Project. Oppenheimer excelled at nurturing  young theoreticians in Berkeley, but most of his colleagues were surprised when General Leslie Groves - the Project Director, appointed by Roosevelt - asked him to take on responsibility for building the bomb. One of Oppenheimer's Berkeley colleagues chortled, 'He couldn't run a hamburger stand.' Just as surprising was the authorities' decision to appoint someone who, although a brilliant researcher and teacher, was well known to be a fellow traveller of the Communist Party.\n\nDirac worked mainly in his study, the one room in 7 Cavendish Avenue for which only he had the key, allowing in cleaners on the strict condition that they did not move any of his papers. If he saw any sign at all that his desk had been disturbed, he flew into a wordless rage.\n\nThe children were proving to be a handful. Dirac and Manci may well have been alarmed when Gabriel, soon after he began his mathematics degree in Cambridge, joined the Communist Party, though he kept up his membership for only six months. Judy was less academic and more rebellious: when she was sixteen, in 1943, Manci furiously ordered her out of the house and threw her clothes out of her bedroom window.  Although she was allowed home a few days later, relations with her parents did not improve. Manci, always trying to enforce strict discipline, was frustrated by the feeble support she was given by Dirac - when she needed him to back her up in some altercation with one of the children, he retired sheepishly to his study or escaped to his garden. He spent hours tending his rhododendrons and gardenias, pruning his apple trees, sewing seeds and digging up asparagus, carrots and potatoes to help fill the larder. In the summer, he would shield his balding head from the sun by wearing a handkerchief knotted at each of its four corners. Friends noticed that he practised horticulture using the same top-down methods that he used in theoretical physics, trying to base every decision on a few fundamental principles. He stressed that the best way of ripening apples was to place them in linear rows, each item of fruit separated from its neighbour by precisely the same distance. In one project, he coated pea seeds with dripping and rolled them in red lead oxide powder to discourage birds from eating the newly emerged seedlings, a practice that would today induce palpitations in any self-respecting health and safety inspector.\n\nDirac's heart remained in quantum mechanics. In July 1942, he took time off from war work, left his family at home and travelled with Eddington to attend a conference in Dublin organised by Schr\u00f6dinger, who tried to tempt Dirac to accept a job alongside him. 'There is plenty of food here - ham, butter, eggs, cakes, as much as one wants,' he wrote in one of his fond letters to Manci. The Irish Prime Minister \u00c9amon de Valera, a trained mathematician who had helped bring Schr\u00f6dinger to Ireland, took the two guests on a joyride around the local countryside, having met them during the conference. Dirac had been amazed to see him there, attending lectures and taking detailed notes.\n\nOn 29 September, six weeks after his return to Cambridge - still under attack from Nazi bombers - Manci gave birth to a daughter, Florence, named after Dirac's mother, though she was always called by her second name, Monica. Two days after her birth, Dirac received a letter from Peierls gently enquiring, at the request of the project directorate, if he would be prepared to move from Cambridge to work full-time on the war effort. Predictably, Dirac refused.\n\nHis family was now complete. He never had a son of his own, a disappointment Manci later described as one of the saddest of his life.\n\nDirac saw in Cambridge evidence of the prominent role the USA was now taking in the war. Every day, hundreds of uniformed American servicemen - on leave from the nearby airbases - walked the streets of Cambridge, with plenty of money to spend. They organised baseball games and, in November 1942, were visited by the stately Eleanor Roosevelt. At home, Dirac received intelligence reports of the American-led experiments to build a nuclear bomb and, towards the end of the year, heard that a key experiment in the programme had been completed. In a makeshift laboratory built in a disused squash court in Chicago, Enrico Fermi and his team had built a nuclear reactor, and, in the mid-afternoon of 2 December 1942, they got it working for the first time. They had arranged a self-sustaining nuclear chain reaction, releasing energy at a rate of half a watt. Wigner presented Fermi with a bottle of Chianti, which he shared in silence with his team, who had good cause to celebrate but also to be nervous: for all they knew, Hitler's scientists were ahead of them. A member of Fermi's team, Al Wattenberg, later recalled: 'The thought  that the Nazis might get the bomb before us was too terrifying to contemplate.'\n\nShortly before, Peierls asked Dirac to study a sheaf of technical papers written by Oppenheimer and his Manhattan colleagues describing the explosion of a sample of uranium undergoing fission. Early in January, Dirac pointed out inconsistencies in the papers and discussed how a nuclear bomb might be constructed, including the optimal shapes of the two masses of uranium that could be propelled together to make the bomb. During the next six months of 1943, Dirac investigated theoretically the passage of neutrons in a fissioning block of uranium and presented his results in two reports, one of them in collaboration with Peierls and two of his younger Birmingham colleagues. One of them was Peierls' lodger, Klaus Fuchs, a Bristol-educated refugee from Nazi Germany, an inept but courteous young man in his early twenties. When he and Peierls visited 7 Cavendish Avenue to talk about their secret research with Dirac, they all adjourned to the middle of the lawn in the back garden to ensure that they were out of earshot of everyone near by. Manci, asked to stay inside the house, resented what she knew was the implication: she was a potential eavesdropper. During some of these al fresco discussions, Dirac and Peierls noticed that Fuchs sometimes behaved oddly, complaining that he was unwell and leaving them for surprisingly long periods before returning. It would be another seven years before Dirac and Peierls understood Fuchs' behaviour.\n\nThe collaboration between the scientists working on the bomb in the USA and their counterparts in Britain was tense and difficult, but the problems were apparently resolved in the late summer of 1943, after peace-making conversations between Roosevelt and Churchill. It was obvious to most of the British scientists that they should join the Manhattan Project, and about two dozen of them - including Peierls, Chadwick, Frisch and Cockcroft - joined Oppenheimer and his team in their Los Alamos headquarters in the New Mexico desert. Through Chadwick, Oppenheimer asked Dirac to join the Manhattan team, but he declined. About a year later, he stopped working on the project, but never fully explained why. Peierls later suggested, probably correctly: 'I believe this was because he was beginning to feel that atom bombs were not a matter he wanted to be associated with, and who could blame him?'\n\nDirac may have come to believe that the Nazis could be defeated without nuclear weapons. Or perhaps Dirac was influenced by Blackett, who protested that American scientists on the Manhattan Project were given access to all the research done by their British colleagues but did not reciprocate, except with Chadwick, the only Briton to be given full security clearance. Blackett felt so strongly about this that he tried to persuade his British colleagues to take no part in the Manhattan Project.\n\nOn the night of 5 November 1943, the Luftwaffe dropped their bombs on Cambridge for what turned out to be the last time. Since the outbreak of the war, the sirens had sounded 424 times to warn of the bombings that had killed thirty people and destroyed fifty-one homes. As the nights closed in, Dirac and his family were hoping that the blackout would end soon, but the authorities did not lift it until September in the following year. By then, he was worrying constantly about his sister Betty and her family. At Dirac's request, Heisenberg had attested to the occupying Nazis that she was not Jewish, but Joe and their son were still in grave danger. When Dirac last heard from them, in early September 1943, they had recently fled their home in Amsterdam - a short tram ride from Anne Frank's secret annex - after the Nazis told Joe that he could either be sterilised or interned in Poland. He probably knew that internment was tantamount to a death sentence, so the family headed for Budapest, hoping that it would quickly be liberated by the Allies.\n\nPowerless to help Betty, Dirac sat out the end of the war at home. Several of the family photographs taken around this time show him in his back garden, sitting in a deckchair, teaching Mary to read from _The Wizard of Oz._ One of her earliest memories was of her father spelling out the letters D-o-r-o-t-h-y. She and Monica were given a disciplined upbringing, following the motto of English family life, 'Children should be seen and not heard,' but without any exposure to religious ideas.  Yet Dirac appears to have had at least some regard for religion as he and Manci followed the convention of having both their daughters christened. Probably as a result of his wife's influence, the hard-line atheist had softened his line.\n\nTry as Dirac might to concentrate on quantum physics when he was in college, the continuing presence of the military reminded him that although victory over Hitler was in sight, it could not be taken  for granted. Royal Air Force officials still occupied much of the college, and the military had taken over the Combination Room for purposes they kept secret. Only much later did the Fellows of St John's find out that the room contained a huge plaster model of the stretch of the Normandy coastline on which Allied troops landed on 6 June 1944. Churchill's leading general, Montgomery, believed the end of the war was in sight and didn't believe the Germans could go on much longer. Yet still Dirac could not walk over the Bridge of Sighs without being challenged. When the sentry asked, 'Who goes there?', he was satisfied with only one reply: 'Friend.' Dirac knew the threat still posed by the enemy better than most. Even when victory looked inevitable, from June 1944, Dirac was aware that German scientists, including Heisenberg, might already have developed a nuclear weapon. About a year before, he had heard from the refugee Norwegian chemist Victor Goldschmidt that Heisenberg was working on the Germans' counterpart of the Allies' Tube Alloys project. Dirac knew that the fate of hundreds of potential victims could depend on the scientific success of his closest German friend.\n\nWhile he waited for the war to end, Dirac began work on another edition of his book. His main innovation this time was to introduce a new notation he had first invented shortly before the war broke out. This system of symbols enabled the formulae of quantum mechanics to be written with a special neatness and concision: just the sort of scheme that Dirac had learned to appreciate in Baker's tea parties.\n\nThe centrepiece of the notation was the symbol <q for a quantum state labelled q and the complementary q>; together they can be combined to form mathematical constructions such as <q | q>, a bracket. With his rectilinear logic, Dirac named each part of the 'bracket' after its first and last three letters, _bra_ and _ket_ , new words that took several years to reach the dictionaries, leaving thousands of non-English-speaking physicists wondering why a mathematical symbol in quantum mechanics had been named after an item of lingerie. They were not the only ones to be flummoxed. A decade later, after an evening meal in St John's, Dirac was listening to dons reflecting on the pleasures of coining a new word, and, during a lull in the conversation, piped up with four words: 'I invented the bra.' There was not a flicker of a smile on his face. The dons looked at one  another anxiously, only just managing to suppress a fit of giggling, and one of them asked him to elaborate. But he shook his head and returned to his habitual silence, leaving his colleagues mystified.\n\nThe war in Europe ended in anti-climax on 8 May 1945. The relief felt like a national exhalation. In the centre of Cambridge, thousands gathered in Market Square in the blazing heat of the afternoon, dozens of Union Jacks fluttering limply in the breeze. After the Lord Mayor's speech, two bands marched separately round the town, each followed by hundreds of people, with dozens of couples dancing cheek-to-cheek in the streets. The authorities in St John's College abandoned all formalities for the day: the Combination Room swelled not only with Fellows but with dozens of normally excluded undergraduates raising their glasses to the new peace. Dirac and his family celebrated with neighbours at an impromptu tea party in a local street, munching on scones and spam sandwiches served from trestle tables.\n\nIf Dirac believed that science would quickly return to normal, he was mistaken. In the spring of 1945, he and seven colleagues - including Blackett and Bernal - applied for visas to enable them to attend the June celebrations of the 220th anniversary of the USSR Academy of Sciences; for Dirac, the trip would give him the opportunity to see Kapitza and other Russian friends again. But Churchill refused to allow visas to be issued on the grounds, it was later revealed, that Dirac and his colleagues might share with Stalin's scientists some of the nuclear secrets kept from the Soviets during the war. During a discussion about the matter at the Admiralty in London, Blackett lost his temper and strutted magnificently out of the building, furious that the Government had dared to impugn his integrity. Dirac was angry, too, but showed his emotion only by withdrawing into complete silence and taking a long, solitary walk.\n\nFor several weeks after the end of the war in Europe, news had been seeping out about the Nazi concentration camps. Manci was outraged not only with the Germans but also with 'these dirty Poles' - she was sure they had connived in the atrocities. She wrote to Crowther that she had one of her rare rows with Dirac, apparently because his reaction to the revelations of unconscionable cruelty was too restrained for her taste. The Diracs knew that several of  Manci's relatives had probably been murdered in the camps and that Betty's husband Joe might also be dead. News of him arrived in a telegram delivered to the Diracs' home at the beginning of July, when they were preparing to visit the Schr\u00f6dingers in Dublin. Joe was alive. In Budapest, he had fallen into the hands of the Nazis, who dispatched him to the Mauthausen-Gusen concentration camp in Austria, where he was one of thousands forced to work in the Wiener Graben quarry, mining granite with a pickaxe and carrying the slabs up the hundred and eighty-six steps to the top. Many of his fellow prisoners perished from the freezing cold, were worked to death or were summarily shot through the neck by SS guards after being injured or collapsing from exhaustion. After the camp was liberated in the summer of 1945, he emerged looking close to death - desperate for a morsel of food and with a broken wrist, a seriously infected kidney and missing a finger. While recuperating in an American military hostel in France, desperate for news of Betty and their son Roger, he wrote to Manci to suggest that Kapitza might help to find her, as the Russians had taken over Hungary. He did not have to wait long to hear the denouement: in early September, he heard from Manci that Betty and Roger were safe.\n\nOn 6 August, Dirac heard the news he had been dreading: with the tacit agreement of the British Government, the Americans had dropped a nuclear bomb on Hiroshima, killing about forty thousand Japanese civilians. At nine o'clock that evening, Dirac was in his front room listening to the radio news bulletin: 'Here is the news: it's dominated by the tremendous achievement of Allied scientists - the production of the atomic bomb. One has already been dropped on a Japanese army base. It alone contained as much explosive power as two thousand of our great ten-tonners.'\n\nAfter reading official statements, including one from Churchill and President Truman, the BBC announcer ended with almost comic bathos: 'At home, it's been a Bank Holiday of sunshine and thunder-storms; a record crowd at Lords has seen Australia make 273 for five wickets.' All was well again - cricket had resumed. The national press rushed to praise the achievement of the leading British scientists, including Cockcroft and Darwin, who had helped to design the bomb. None mentioned Dirac, probably to his relief. One of the few civilians who were not shocked by the destructiveness of 'the atomic bomb' was the seventy-nine-year-old H. G. Wells, who first coined  the term in 1914. On 9 August, just as President Truman ordered the dropping of another nuclear bomb on Nagasaki, the _Daily Express_ published a weary personal perspective on the age he had foreseen. He died a year later.\n\nOn 14 August, when news reached Britain of Japan's surrender, public euphoria resurged, and, in Cambridge, Market Hill swelled with an encore of the VE Day celebrations . In the USA, the press showered Oppenheimer with praise and likened him to Zeus. He was the triumph of physics personified.\n\nDirac had no idea that, only fifteen miles from Cambridge, Heisenberg had been interned by the British Secret Service with nine other German scientists in Farm Hall, a red-brick Georgian House on the outskirts of the village of Godmanchester.  They were treated well - given the run of the house, provided with daily newspapers and allowed to walk freely around the grounds, though they were warned that their liberties would be curtailed if any of them tried to escape. A few days after their arrival, Heisenberg wondered why the authorities were keeping him and his colleagues interned without making it public: 'It may be that the British Government is frightened of the communist professors, Dirac and so on. They say \"If we tell Dirac or Blackett where they are, they will report it immediately to their Russian friends, [like] Kapitza\".'\n\nWhen Heisenberg and his colleagues heard about the dropping of the first nuclear bomb, soon after the news was broadcast on BBC radio, they were both perplexed and incredulous. One detainee, Otto Hahn, observed sourly: 'If the Americans have a uranium bomb then you're all second raters. Poor old Heisenberg.' Not knowing that the British were recording their conversations - it was unthinkable, Heisenberg chuckled - the Germans talked freely about their feelings. The British authorities declassified their conversations only in 1992; ever since, historians have pored over the transcripts and have come to a variety of conclusions. Some experts believe that Heisenberg never came close to an understanding of how to make a nuclear bomb; others that he could have made one but slow-pedalled his research in order to prevent the Nazis from acquiring the device. It is, however, indisputable that, during the conversations recorded at Farm Hall, neither Heisenberg nor any of his colleagues expressed any serious qualms about working for the Nazi regime.\n\nBy October 1945, Dirac's life in Cambridge had almost returned to normal. A few weeks before, he had been surprised by the high number of students attending his quantum-mechanics course, several of them still in uniform. At the beginning of the first lecture he announced to the audience, 'This is a lecture on quantum mechanics, ' evidently believing that many of the students were in the wrong room. When none of them got up to leave, he repeated his announcement, this time more loudly. But still no student left.\n\nA few weeks later, Betty and her son Roger - both hungry, traumatised and anxious - returned to stay in 7 Cavendish Avenue before they were reunited with Joe. Betty and her son had almost starved to death in Budapest, and she had seen that the liberation was not as joyous as many journalists reported; in her opinion, the Russian troops who liberated the city were far more brutal than the Nazi army they had ousted. In Betty's later years, her memories of the conflict were too painful to share, though she often remarked that she regarded the survival of her family as a miracle: 'Everything afterwards was a bonus.' Best of all was the birth of her daughter, Christine, just over nine months after Betty and Joe were reunited.\n\nFor the sake of tact, Betty may not have mentioned during her stay in Cambridge that she despised most of the Hungarian acquaintances she had met. Her memories of the double-dealing and inhospitable citizens of Budapest were to become a running sore in her relationship with Manci, with Dirac the embarrassed and ineffectual peace-maker.\n\nThe university and St John's College were settling back into their clockwork routine. Dirac preferred this way of life, free of distractions, but he had a few other duties to discharge: during the war, Crowther had persuaded him to support their French colleagues behind Nazi lines by taking on the undemanding role of the British presidency of the Anglo-French Society of Sciences, working with an informal committee whose members included Blackett, Cockcroft and Bernal. After the war, Crowther decided to relaunch the Society with a prestigious series of talks about scientific developments during the conflict, and he persuaded Dirac to give the first presentation, on 'Developments in Atomic Theory'. The venue for the occasion - a red-letter day in French science - was Le Palais de la D\u00e9couverte, a public science centre that stands like a Greek temple  on a dark side road in the seventh arrondissement. Soon after sundown on Tuesday 6 December, hundreds of the city's leading scientists made their way to Le Palais to hear Dirac talk. Two thousand people clamoured for a seat in the lecture theatre, expecting to hear the secrets of the atomic bomb.\n\nMinutes after Dirac began to speak, the audience realised that it was not going to hear about the latest in nuclear technology but a presentation on the state of quantum mechanics. Dozens tried to leave, but there was no escape: the exit was jammed with the overflow crowd of hundreds, listening to the lecture via loudspeakers. For the physicists who were interested, a treat was in store: they heard Dirac coin two of the best-known technical terms that he introduced: 'fermions', quantum particles that obey the laws that he and Fermi had set out in 1926, and 'bosons', the other type of quantum particles, which obey laws set out by Einstein and the Indian theoretician Satyendra Bose. For most of the audience, this was not much consolation for a wasted evening: at the end of the lecture, several of them bolted for the door.\n\nAt the dinner party afterwards, embarrassment was no doubt still in the air, but Dirac was probably oblivious to it. During six bleak years for science, in which he had contributed more to engineering than to quantum physics, he was relieved that life was returning to normal. But he was now well past thirty, the age he once believed marked the end of the theoretician's productive career: was he now too old to have radically new ideas?\n**Twenty-four**\n\nIn America, the young are always ready to give those who are older than themselves the full benefits of their inexperience.\n\nOSCAR WILDE, _The American Invasion_ , 1887\n\nIn September 1946, Dirac was scratched again by the next generation's talons. He was at a conference on 'The Future of Nuclear Science' at Princeton's Graduate College, half a mile from the campus. Nestled among trees at the top of a grassy hill, the college looked like a Gothic abbey, its majestic tower dominating the surrounding countryside - a picture of English arcadia. Many visitors thought the college had been a landmark in Princeton for centuries, but it had stood there for only thirty-three years.\n\nThe conference was the first of a series of international events during the university's bicentennial celebrations - months of ceremonial glad-handing, sybaritic dinners and colourful parades. The conference organiser Eugene Wigner, fresh from the Manhattan Project, had put together an impressive guest list, including Blackett, Fermi, Oppenheimer, Van Vleck and the Joliot-Curies, all ready to put the war behind them and begin the next chapter of physics.\n\nAt 9.30 a.m., at the beginning of the conference's second day, Dirac was introduced by one of the most exciting scientific talents in America, Dick Feynman (he called himself Dick rather than Richard). Brought up in the New York suburb of Far Rockaway, he was a clean-cut twenty-eight-year-old, brimming over with ideas and sophomoric humour but still grieving after the death of his first wife fourteen months before, from tuberculosis. He was afraid he was already burnt out, he later admitted. When he introduced Dirac, Feynman seemed unburdened by self-doubt but felt 'like a ward-heeler [machine politician] in the 53rd district introducing the President of the United States'. Feynman was not expecting to be impressed: a few weeks before, he had been disappointed by his hero's handwritten script, which Feynman thought was backward looking, stale and 'unimportant'.\n\nDirac discussed how elementary particles could be described using his favourite mathematical device, the Hamiltonian: for Dirac, this  was the only way to proceed, and he did not spare his audience - many of them non-specialists - the technical details. As Feynman feared, the talk fell flat. Worse, Dirac was bereft of new ideas. After the applause, Feynman tried to give lay members of the audience a sense of what Dirac was saying, not hiding his disappointment and remarking that Dirac was 'on the wrong track'. He cracked even more than his usual quota of jokes, prompting Bohr to stand up and ask Feynman to take the proceedings more seriously.\n\nA few hours later, Feynman looked out of the window of the lecture room and saw that Dirac had excused himself from the conference programme and was 'paying no attention to anybody', lying on a patch of grass, leaning on an elbow, gazing lackadaisically at the early-autumn sky. Here was Feynman's opportunity to talk informally with Dirac about a matter that had intrigued him for the past four years. When Feynman was a graduate student, he had studied Dirac's 'little paper' on how the classical least-action principle can be applied in quantum mechanics, demonstrating that it could be used to build another version of quantum mechanics, different from Heisenberg's and Schr\u00f6dinger's but giving the same results. In his paper, Dirac had cryptically remarked that a critical quantum quantity is 'analogous' to its classical counterpart, but Feynman believed that the correct phrase was 'proportional to' (that is, if the quantum quantity changes, the classical one always changes proportionately). Here, at last, was Feynman's chance to find out what Dirac meant.\n\nFeynman described his problem to Dirac and came to the crunch:\n\nFEYNMAN: Did you know that they were proportional?\n\nDIRAC: Are they?\n\nFEYNMAN: Yes they are.\n\nDIRAC: That's interesting.\n\nDirac then got up and walked away. Feynman subsequently became famous for his new version of quantum mechanics but thought the credit was undeserved. The more closely he looked at the 'little paper', the more he realised that he had done nothing new. He later said, repeatedly, 'I don't know what all the fuss is about - Dirac did it all before me.'\n\nFeynman knew he had much to do if he was to prove himself a great physicist. When the conference photograph was taken, he appeared to hint at the extent of his ambition by standing behind  Dirac, just as Dirac had done in the 1927 Solvay Conference photograph, when he stood directly behind Einstein. Within a few years, Feynman's power as an analyst and intuitionist made him, in the eyes of many, the finest theoretician in America. Wigner agreed with that judgement: 'Feynman is a second Dirac, only this time human.'\n\nThe next five years saw the emergence of a new theory of electrons and photons, in some ways the climax of fifty years of theoretical physics. This was largely an American success, the accomplishment of hungry young scientists who had suspended their academic careers during the war to work on nuclear weapons, radar and other projects.  Physicists had worked in lavishly funded, goal-driven international teams, having set aside the elitist traditions of European academia and collaborated in the less formal, can-do social environment of the United States. Now it was time for payback.\n\nOn Capitol Hill, the physicists argued that they deserved the support of the government's tax dollars to pursue curiosity-driven research. It is a fair bet Willy Loman and the other struggling bread-winners of middle America would have baulked at the physicists' case if they had been aware of it, but the politicians were persuaded and gave unheard-of levels of federal support for basic physics research and training. The US Government and private institutions funded theoretical physics. At much greater expense, Uncle Sam equipped experimenters with machines that could probe the structure of matter even more finely, using beams of subatomic particles accelerated to within a whisker of the speed of light in a vacuum. The pursuit of 'high-energy physics' had flourished in Europe in similar ways, though there was no doubt that in this branch of science - and many others - America led the world.\n\nThe first conference of leading subatomic physicists to take place in the USA after the war, at the beginning of June 1947, set their subject's agenda for the next thirty years. Twenty-three carefully selected scientists - all of them men - gathered at an inn on Shelter Island, a small and secluded spot near the eastern tip of Long Island, to review their subject. The gathering could scarcely have had a more spectacular opening: in the first two presentations, experimenters announced that the Dirac equation made predictions that disagreed with new experimental results. The first speaker, Willis Lamb, had the air of a cowboy who had strayed into a physics laboratory.  But his appearance was deceptive: he was a deep thinker, an accomplished experimentalist who could hold his own with the best theorists. He got the meeting off to a flying start by announcing a serious flaw in Dirac's theory: two energy levels of atomic hydrogen that, according to the theory, should have the same energy turn out to be slightly different. Photons emitted by hydrogen atoms when they jump between the two energy levels had been detected by Lamb and his student Robert Retherford, at the Columbia Radiation Laboratory. In a masterly experiment using microwave technology developed during the war, they studied these photons and showed that each of them has only about a millionth of the energy of a quantum of visible light.\n\nIn the next presentation, given by the experimenter Isidor Rabi, of Columbia University in New York, the audience heard yet more unexpected news: the strength of the electron's magnetism appeared to be weaker than the Dirac theory had predicted. The audience was euphoric: here were two observations that heralded the end of the reign of Dirac's beautiful theory and provided crucial tests for any theory that presumed to succeed it. Oppenheimer steered the conference, incisively cross-examining the speakers and interspersing the proceedings with his elegant, if ostentatious, editorial arias. By the end of the meeting, it was clear that the main challenge was to explain Lamb's result. But Dirac knew nothing of all this: he had declined an invitation to attend and read about the wounding of his theory on an autumn Sunday in Princeton, on the front page of the _New York Times._ 10\n\nWithin two years of the Shelter Island Conference, Lamb and Retherford's results had been explained by two of the youngest theorists in the audience. One of them was Feynman, the other was a fellow New Yorker, Julian Schwinger, a loner with the manners of a prince and the self-belief of a boxer. Feynman and Schwinger were both the same age and had read Dirac's book when they were precocious teenagers, and both based their theories on Dirac's 'little paper'. Yet the two versions appeared to be quite different: Schwinger's mathematical approach was hard to understand, but Feynman's approach was intuitive and involved special diagrams that made the underlying science easy to visualise, at least superficially. The two methods gave the same results, and everyone except Schwinger agreed that Feynman's methods were quicker and easier.\n\nIt turned out that the same results had been obtained several years earlier by the Japanese theoretician Sin-Itiro Tomonaga, who had based his ideas on Dirac's version of quantum field theory. As a student, Tomonaga had been a fanatical student of Dirac's book and was in the Tokyo audience when Dirac and Heisenberg gave their lectures during their tour of Japan in 1929. This pioneering work had been completed in Tokyo, where Tomonaga was one of the tens of thousands of starving citizens who were trying to rebuild the city after American bombers had flattened it towards the end of the war.\n\nSo there were now three versions of quantum electrodynamics that looked quite different and yet seemed to give the same results. It was Freeman Dyson, the student who had snapped at Dirac's heels during his wartime lectures, who first demonstrated that the three theories were versions of the same underlying theory. Now, at last, physicists could claim they understood the interactions of the photon and the electron in terms of a theory that agreed with observation to within a few parts in ten thousand - roughly a human hair's breadth compared with the width of a door. Four decades later, when much more accurate measurements were still in excellent agreement with the theory, Feynman referred to it as 'the jewel of physics'. As he often stressed, its fundamental concepts had been set out by Dirac in his 1927 theory: Feynman, Schwinger, Tomonaga and Dyson had, in essence, introduced a collection of ingenious mathematical tricks and techniques that made the theory viable and showed how to remove the embarrassing infinities.\n\nThoroughly pleased with himself for becoming 'a big shot with a vengeance' after his triumph, Dyson was keen to hear Dirac's opinion on the new theory. He was expecting a few words of congratulation from his former teacher, but was disappointed:\n\nDYSON: Well, Professor Dirac, what do you think of these new developments in quantum electrodynamics? DIRAC: I might have thought that the new ideas were correct if they had not been so ugly.\n\nThe feature of the new theory that Dirac most loathed was the technique of renormalisation.  According to this theory, the observed energy of an electron is the sum of its self-energy - resulting from the interaction between the electron and its field - and the bare  energy, defined to be the energy the electron is supposed to have when completely separate from its electromagnetic field. But the bare energy is a meaningless concept because it is actually impossible to switch off the interaction between the electron and its field; only the _observed_ energy can be measured.\n\nThe virtue of renormalisation is that it enables every mention of bare energies in the theory to be removed and replaced with quantities that depend only on observed energies. Using this technique, theorists could use quantum electrodynamics to calculate - to any degree of accuracy - the value of any quantity the experimenters cared to measure. Despite the success of the technique, Dirac abominated it, partly because he could see no way of visualising its mathematics but mainly because he felt that the process of renormalisation was artificial, an inelegant way of sweeping the fundamental problems of theory under the carpet. In his opinion, a fundamental theory of nature must be beautiful, whereas renormalisation seemed to Dirac's taste to be as devoid of beauty as the dissonances of Arnold Sch\u00f6nberg.\n\nEngineers, schooled to worry more about the reliability of their results and less about the rigour of their mathematics, might be expected to be happy with renormalisation, as the process gives answers that always tally with observations to extremely high accuracy. But, paradoxically, Dirac believed his engineering training was at the root cause of his hostility to the technique. At the Merchant Venturers' College, he had learned the engineer's art of using well-chosen approximations to simplify complicated, real-life problems so that they can be analysed mathematically. Dirac made this the theme of his 1980 lecture 'The Engineer and the Physicist': 'The main problem of the engineer is to decide which approximations to make.' Good engineers make wise choices, often based on physical intuition, about the mathematical terms they can ignore in their equations: 'The terms neglected must be small and their neglect must not have a big influence on the result. He must not neglect terms that are not small.'\n\nRenormalisation entails a practice that no self-respecting engineer would countenance, Dirac pointed out: the neglect of large terms in an equation. To neglect infinitely large quantities in an equation was, for an engineer, anathema. Most physicists had no such compunctions, and leading theorists paid little heed to Dirac's objections. As  Dyson pointed out, although the infinities in the theory had not been eliminated, they were isolated in mathematical expressions that were quite separate from formulae representing the effects experimenters actually observe. Dirac was unconvinced. He, Schr\u00f6dinger, Heisenberg, Pauli, Born and Bohr - the 'old gang', as Dyson dubbed them - had now joined Einstein in the wings of theoretical physics, while the next generation took centre stage. Of the _ancien r\u00e9gime_ , only Pauli kept closely abreast of new developments in their subject; the rest withdrew into their own private worlds. Dyson and his friends were contemptuous of their elder colleagues:\n\nIn the history of science there is always a tension between revolutionaries and conservatives, between those who build grand castles in the air and those who prefer to lay one brick at a time on solid ground. The normal state of tension is between young revolutionaries and old conservatives [. . .] in the late 1940s and early 1950s, the revolutionaries were old and the conservatives were young.\n\nIn a sense, Dirac was the Trotsky of theoretical physics: he envisioned his subject progressing through one revolution after another, each an improvement on its predecessor. But new quantum electrodynamics did not constitute progress so far as Dirac was concerned: the theory offended the aesthetic sensibilities he had first developed in Bristol, when he was an Eton-collared cherub at junior school, a greasy-aproned engineering student - moonlighting in general relativity - at college, and a budding mathematician at university. Whether this unique aestheticism would be a dependable guide remained to be seen.\n\nWhen Dirac was a young man, he had been uninterested in human companionship, but he had come to value it. The result was that, after the war, Cambridge seemed to him like a ghost town - Fowler and Eddington had died, and all of Rutherford's former 'boys' had left. Manci also felt the pain of the exodus, complaining to her brother Wigner in Princeton that 'Life here is utterly and completely different.'\n\nWith the ascendancy of American physics, Cambridge looked to Dirac to give leadership in the new era, but to no avail. Concerned only with his own research and in doing a modicum of teaching, he did nothing to improve the primitive facilities for students of theoretical physics in Cambridge: there were no offices for them in the department, and they even had to organise the programme of seminars.  Dirac now preferred to work at home, as he had done during the war. Manci ensured that the children did not disturb him: woe betide them if they tried to attract his attention by banging on his study door.\n\nBy late 1950, Gabriel and Judy had left home. Gabriel was pursuing his career, and Judy - apparently settling down after her tempestuous adolescence - had married, leaving the Diracs to bring up their two youngest daughters. According to Manci, Dirac 'kept himself too aloof' from them, and she had to encourage him to kiss them. Neither Mary nor Monica recalled having any sense that their father was a famous or distinguished man - only that he was exceptionally quiet and good-natured, although unemotional and extremely slow to anger. Monica cannot recall seeing him laugh. But in many ways Dirac was a typical father, taking an interest in their hobbies, helping them do their homework and encouraging them to have pets, though he forbade them to bring dogs into the house because, as Monica recalls, 'he did not like being startled when they barked'. Animal welfare was one of his concerns: when designing a flap for the girls' cat, he measured the span of its whiskers to ensure that the animal would not be incommoded as it passed through the hole.\n\nAmong the visitors to the Diracs' home were Esther and Myer Salaman. Esther, born and raised in the Ukraine, had been a student of Einstein's in the early 1920s, joined the Cavendish in 1925 and married Myer, a physiologist, a year later. She was the kind of fine-looking, self-assured woman Dirac admired. He listened carefully to her effusions on the leading nineteenth-century Russian novelists, including her favourite, Tolstoy, whose _War and Peace_ took Dirac two years to complete, having digested every word of it. He brought this same attention to detail to Dostoevsky's _Crime and Punishment_ , which he thought was 'nice', though he pointed out that 'In one of the chapters the author makes a mistake: he describes the sun as rising twice on the same day.'\n\nManci was still feeling out of place in Cambridge, contemptuous of its drab provincialism and despondent at the thought that she might have to spend the rest of her life in colourless England. Every day, newsreaders delivered discouraging news of the sluggish economy, continued rationing and product shortages; there was no sign of an end  to the austerities of wartime. Manci, feeling the pinch, complained to Monica that 'Uncle Eugene pays his cleaner more every week than your father gives me in housekeeping.' These were grim times, accurately summarised by the worldly-wise senior civil servant Bob Morris as 'a right, tight, screwed-down society walled in in every way'.\n\nThe treatment of the dons' wives by the colleges and university was still a sore point with Manci, though she saw a few hopeful signs. In 1948, the authorities symbolically enrolled Queen Elizabeth (later the Queen Mother) as the first woman to take a bona-fide degree, albeit an honorary one. A year later, under this legislation, women students at Cambridge first graduated. Slowly, much more slowly than Manci wanted, women in Cambridge University were making progress towards equality.\n\nTo the emerging generation of physicists, Dirac was a cool and wary stranger, but for Heisenberg and other fellow pioneers of quantum mechanics, he was an attentive friend. After the war, Heisenberg knew he had to justify the work he had done for the Nazis, but this was an enervating struggle - several of his former colleagues, including his former friend and student Peierls, wanted nothing to do with him, and Einstein treated him with contempt. In 1948, when Heisenberg returned to Cambridge - at a time when Dirac was absent - he looked haggard and anxious but was excellent company, delighting his hosts one evening with an unrehearsed performance of Beethoven's _Emperor Concerto._ He discreetly explained to everyone who would listen that he was never a Nazi and had stayed in Germany out of loyalty to his colleagues and to mitigate the worst of Hitler's intentions. Determined to leave a good impression in Cambridge, as a gesture of remembrance he bought forty-eight rose bushes from a plant centre in nearby Histon and made it known he would plant them in his garden in G\u00f6ttingen.\n\nWhen Dirac first met Heisenberg after the war, he accepted Heisenberg's explanation of his wartime conduct at face value and believed Heisenberg had behaved reasonably in an extremely difficult situation. 'It is easy to be a hero in a democracy,' Dirac would observe, as Manci laughed at his naivety. She scorned Heisenberg as a tricky character: 'That Naaaaazi.'\n\nDirac was supportive of Heisenberg even when he was working for Hitler. Max Born had been startled when Dirac asked him to support  Heisenberg for foreign membership of the Royal Society. 'Heisenberg's discovery will be remembered when Hitler is long forgotten, ' Dirac commented. Dirac also strongly supported Schr\u00f6dinger's election to a reluctant Royal Society. The consensus among its officials was that 'one hunch, however good and however important [. . .] needed more following up with sustained evidence of ability', an insider told Dirac. Probably incredulous, Dirac took up Schr\u00f6dinger's cause and helped to ensure his election in 1949. Schr\u00f6dinger was profuse in his thanks, telling Dirac, 'You really are very nearly a saint.' Dirac showed no such conscientiousness when it came to supporting his former peers for the Nobel Prize: strong candidates for the award - Pauli, Born, Jordan or even Dirac's Cavendish friends Blackett, Chadwick, Cockcroft and Walton - received no support from him. The only physicist Dirac nominated was Kapitza.\n\nDirac had heard little from Kapitza during the war, though he had read in his copy of _Moscow News_ of Kapitza's invention of a method of liquefying oxygen that did much to raise the productivity of the hard-pressed steel manufacturers and several branches of the Soviet chemical industry. Stalin never met Kapitza but showed every sign of having a soft spot for him, telephoning him occasionally and showering him with awards, including the USSR's highest civil title 'Hero of Socialist Labour'. By the end of the war, Kapitza had proved himself the scientist best able to work with the Government and with Stalin, whom he flattered shamelessly: 'The country has always been fortunate to have leaders [such as you and Lenin].'\n\nTwo weeks after Americans dropped the bomb on Japan, Kapitza's fortunes took a turn for the worse when Stalin set up a special committee to develop nuclear technology and weapons, headed by his first lieutenant Lavrentiy Beria. Of all Stalin's courtiers, Beria was the most feared - a bully, a serial rapist and a casual murderer - but he was a consummate manager, the kind of man who would have no trouble running an industrial conglomerate. At Stalin's request, Beria took over leadership of the Soviets' nuclear project and soon fell out with Kapitza, who complained to Stalin in the autumn of 1945 about Beria's scientific ignorance and incompetence. When Kapitza realised that he could not oust his boss, he asked to be released from the project. Stalin agreed and, though apparently ensuring that Kapitza's life was not in danger, did nothing when all  his responsibilities were removed. By early 1946, Kapitza was in disgrace. Dirac knew nothing of this - he did not know that Kapitza had survived the war until the summer of 1949.\n\nIn September 1947, Dirac began his most productive year for a decade. Accompanied by his family, he was on sabbatical at the Institute for Advanced Study, which had relocated eight years before to Fuld Hall, a four-storey red-brick building with a spire like a New England church. It stood, symmetric as a crystal, in almost three hundred acres of meadows, fields, woods and wetlands, about half an hour's walk from the centre of Princeton. This was a realisation of Abraham Flexner's vision of a small academic institution focusing on a few disciplines and with a world-class faculty, all of them unencumbered by administration and unwanted students. The Institute was, for Dirac, a 'paradise'.\n\nManci felt at home in Princeton and thrived in its prosperous academic milieu and - compared with Cambridge - its liveliness and informality. The community treated her with the respect she wanted, not just as Dirac's wife but as a bright woman in her own right. The institute had become even more attractive to Dirac in 1946, when Oppenheimer became its director and gave him an open invitation to visit. Fresh from the Manhattan Project, Oppenheimer was 'ablaze with power', though ill at ease: 'I feel I have blood on my hands,' he had told President Truman.\n\nIt was a relief for Dirac and his family to be far away from the austerities of post-war Britain, and they took away from Princeton an album of memories: their young daughters scurrying around in the empty tea room at the weekend, their yells shattering the institute's chapel-like quiet; Einstein, visiting the Diracs for afternoon tea, signing a portrait of himself for Manci; Oppenheimer showing off his van Gogh; setting off with Veblen at the weekends, axes slung over their shoulders, to clear a path in the local woods. Freeman Dyson recalls meeting the Diracs during their visit to the institute in early September 1948:\n\nEveryone loved Manci: she was a real character, always full of life, always ready to chat. Dirac was more communicative than he had been in Cambridge. He was not terribly difficult to talk to. If you asked him a serious question, he would ponder it and give a reply that was always short and to-the-point.\n\nHowever, he still had no time for strangers who tried to lure him into small talk. Louise Morse, wife of one of the institute's mathematicians, remembers that when she asked Dirac how he was settling in at Princeton, he looked dumbfounded and leaned sharply away from her, as if she were a leak in a sewer. She remembers: 'Without saying a word, his whole body seemed to ask \"Why on earth are you talking to me?\"'\n\nAt the Institute, Dirac worked in a modest office on the third floor of Fuld Hall, next door to Niels Bohr. One of Dirac's main projects in his 1947-8 stay was to develop the theory of the magnetic monopole he had conceived sixteen years before. During the war, he heard reports of the particle's discovery and, although they turned out to be false, they probably rekindled his interest in the idea.  He produced an exquisitely crafted theory predicting how monopoles might interact with electrically charged particles, but the theory failed to make a splash. One of the few who followed it closely was Pauli, who was prompted to give one of his more polite nicknames to Dirac: 'Monopoleon'.\n\nIn another project, he returned to the roots of quantum field theory. Unhappy with the new theory of electrons and photons, he looked afresh at the application of quantum theory to quantities such as electric and magnetic fields that describe physical conditions at each point in space-time. This was another piece of research that failed to strike a chord at the time but was appreciated later. The same is true of the review he wrote in 1949 about how Einstein's special theory of relativity could be combined with Hamilton's description of motion. Its deceptively straightforward presentation led most physicists to pay no attention to it, a mistake several of them would rue.\n\nDirac still believed that modern quantum electrodynamics was wrong because it was based on a classical theory of electrons that was fundamentally flawed. So, in 1951, he produced a new theory, quite different from the one he had developed thirteen years before. This time, his classical theory described a continuous stream of electricity, flowing like a liquid - individual electrons emerged only when the classical theory was quantised. The theory was the dampest of squibs. No one disputed Dirac's technical ingenuity but it seemed that he had lost his intuition for productive lines of research. He demonstrated this yet again when, as a by-product of his new theory  of electrons, he reintroduced a concept that most scientists believed Einstein had slain: the ether.\n\nDirac's ether was quite different from the nineteenth-century version: in his view, all velocities of the ether are equally likely at every point in space-time.  Because this ether does not have a definite velocity with respect to other matter, it does not contradict Einstein's theory of relativity. Dirac's imagination slipped through this loophole and reinvented the ether as a background quantum agitation in the vacuum; later, he went further and speculated that it might be 'a very light and tenuous form of matter'. The press were more interested than scientists in the idea, which appeared to go nowhere: the logic was impeccable but it seemed to have no connection with nature.\n\nBy the time Dirac reached his fiftieth birthday, he seemed to be following the path Einstein had taken, towards isolation from mainstream physicists. In Princeton, Einstein was a lonely figure, uninterested in the latest research headlines and absorbed by his quixotic project to find a unified field theory without introducing quantum mechanics from the outset. He was still active in politics and annoyed J. Edgar Hoover, Director of the Federal Bureau of Investigation (FBI), by supporting several leftist and anti-racist organisations. In 1950, Hoover ordered a secret campaign to 'get Einstein', aiming to have him deported. Unaware that he was being watched, Einstein strolled to his office in the institute from his nearby home on Mercer Street, his briefcase under his arm, pausing only to pick up and sniff discarded cigarette butts. On his favourite route, he walked down the straight section of Battle Road, towering sycamores lining each side, their overarching branches entangled like the swords of a guard of honour.\n\nAt the Institute for Advanced Study, he was free to work and ignore the day-to-day trivia of politics. But this tranquillity was about to be disturbed by the FBI agents and journalists who were sniffing around the past of the institute's director. Oppenheimer's former Communist sympathies - and Dirac's - were about to return to haunt them.\n**Twenty-five**\n\nThe former Communist was guilty because he had in fact believed the  \nSoviets were developing the system of the future, without human  \nexploitation and irrational waste. Even his naivet\u00e9 [. . .] was now a  \nsource of guilt and shame.\n\nARTHUR MILLER, _Time Bends_ , 1987\n\n'What happened to daddy's brother?' Dirac's daughters would ask their mother. 'Shhh! Don't talk about it,' was Manci's stock reply. Dirac spoke about Felix's suicide only with her and even then he could not bring himself to go into any details. She knew that he still had not come to terms with it. On one occasion, when Mary and Monica persisted, Dirac took out from a drawer a small tin and prised it open to reveal some photographs of his late brother, before hurriedly snapping the tin closed and putting it back. More than twenty-five years after his brother's death, a brief look at Felix's face was all he could bear.\n\nFrom Dirac's behaviour at home, it appears that he tried to avoid what he regarded as the worst mistakes his father had made in bringing up his children. Unlike Charles, Paul encouraged his daughters to bring their friends home; he did not lean on them to study science or any other subject, nor did he offer them any career advice. They knew that there is more to life than work. The family always ate together, but the mealtimes were not what most people would regard as normal: Dirac would sit at the head of the table, eating slowly, sipping regularly from his glass of water and making it clear that he preferred to eat in silence. If one of his daughters pressed him to speak, he would point to his mouth and mutter irritably, 'I'm eating.' He was quite fussy about food - for example, refusing to eat pickles on the grounds that they were always bad for digestion - and would not allow Manci to use a drop of alcohol in any food, especially if it might be eaten by the girls. There was trouble in the kitchen if he sniffed or tasted in the Christmas pudding so much as a drop of brandy.\n\nMary and Monica were growing into sharply contrasting personalities that, as Dirac noticed, resembled those of their parents. Mary  was rather like him - quiet, trusting and literal-minded - while Monica bore a resemblance to her mother - confident, questioning and assertive. The girls did not get on well: Mary was intimidated by Monica and their mother, while Monica felt psychologically manipulated by Mary. Dirac and Manci, perhaps trying to atone for Mary's vulnerability, treated her as their favourite and often left Monica feeling angry and resentful. Monica still recalls that her parents organised only two birthday parties for her when she was a child, while they gave one to Mary every year.\n\nWorried that these tensions were getting out of hand, Dirac and Manci separated their daughters using the classic English institution of boarding school, sending Mary to a strict and devoutly religious school near Cromer, in East Anglia. On the first weekend she was away, Dirac went on a Sunday morning cycle ride with Monica, who was hoping to begin a new stage in her relationship with her father. But this time he did not stop and chat as he had always done when Mary was with them: during the three-hour ride, he said not a word to her. She was devastated.\n\nNo one in Cambridge counted Dirac and Manci as among the most attentive parents: as soon as the Cambridge term was over, they usually headed off on a foreign trip, leaving their children with friends. But the family did take vacations together. In the summer, Dirac would take two days to motor to their favourite destination, Cornwall, driving like a caricature vicar. During the Christmas vacation, shortly after the New Year, the family would stay for a few days in the pea soup of London fog. While Manci lunched with friends or went shopping, Dirac took the girls to South Kensington and walked them round the Science Museum, where they pushed the buttons on the interactive displays and filed past the relics of the Industrial Revolution. In the evening, the family headed to the West End for entertainment - Mary recalled that her father's favourites included the musical _The Pajama Game_ and T chaikovsky's ballet _The Sleeping Beauty._ 4\n\nDirac's taste in the arts defies conventional classification, ranging from high culture to catchpenny trivia. On Saturday mornings, he raced his daughters to the front door to pick up the latest edition of their favourite comics, the _Dandy_ and the _Beano_ , which he would study as if they were works of literature. Mostly, he pursued his leisure interests alone, reading a Sherlock Holmes story, listening to  a classical concert at full blast on the radio or sitting impassively watching the television he had first rented so that the family could watch the Queen's coronation. But pageantry was not for him: he preferred the new variety shows and, with millions of other male viewers, sat agog as lines of feathered young women high-kicked their way through their risqu\u00e9 dance routines. This was rather unbecoming, Manci thought, though she happily accompanied him on at least one discreet trip to a London production of the Folies Berg\u00e8re.\n\nLike Einstein, Dirac was a modernist in science but not in art. His favourite music was the classical canon of Mozart, Beethoven and Schubert, and he had no time for the experiments of contemporary composers. He also had no taste for the extremes of abstract art: the nearest he came to liking a modern artist was a fondness for the surrealism of Salvador Dal\u00ed. When he visited his sister Betty and her family in Amsterdam, two minutes' walk from where Ehrenfest shot himself and his son, Dirac would set off in the morning with a compass - but not a map - on the six-mile walk to the Rembrandts of the Rijksmuseum.\n\nIf Cambridge colleagues knew anything of these interests, Dirac would have been more engaging than the desiccated figure he cut in the early 1950s, rather like a prototype for Bertrand Russell's fictional don, Professor Driuzdustades. Dirac no longer seemed at home in the mathematics department, though he remained a loyal Fellow of St John's, observing all its rituals without complaint. Every Tuesday night during term, he would don his gown and eat at High Table, while Manci - not allowed to eat with him - ate at a cheap Indian restaurant with Monica on St John's Street, Manci grumbling over her curry and samosas that the college made her feel like an impostor.\n\nSensing that the university no longer held her husband in the highest regard, she blamed him for not insisting on the respect that was due to him. But he was too self-effacing to assert himself: he had no interest in status for its own sake and was indifferent to the baubles handed down by the establishment. In the early 1930s, he declined an honorary degree from Bristol University because he believed degrees should be qualifications, not gifts, and later declined honorary degrees, replying to offers with 'regretfully, no'. In 1953, he refused a knighthood, infuriating Manci, mainly because his decision deprived her of the chance to become Lady Dirac. He did not want  people outside the university to call him Sir Paul but to address him by the name he used on the rare occasions he answered the telephone at home: 'Mr Dirac'.\n\nHe did not oppose honours on principle, but he believed that they should be awarded on merit, and not be awarded to athletes and show-business celebrities. When the jockey Gordon Richards was awarded a knighthood by the Queen, Dirac shook his head: 'Whatever next?'\n\nFundamental physics appeared to be in a mess, just as bad as the one in the early 1920s when Bohr's theory was the creaky framework for atomic physics. Having seen theory swept aside by quantum mechanics, he believed that nothing less than a similar revolution was needed now to replace quantum electrodynamics. Dirac wanted the initiative to come from theorists: since he was a boy, they had been setting the agenda of physics, but now experimenters were ensconced in the driving seat.\n\nResults from cosmic-ray projects and from the new high-energy particle accelerators had shown that the subatomic world was much more complicated than any theoretician had imagined. By the mid- 1950s, it was plain that there were many more than two subatomic particles - there were dozens or even hundreds, most of them living for no longer than a billionth of a second, before they fall apart into stable particles. All these decay processes obeyed the laws of quantum mechanics and relativity, but no one knew how to apply them. Fermi had set out the first theory of the weak interaction, which acts only over very short distances, within the ambit of a nucleus, about a ten-thousandth of the distance across an atom. By then, another fundamental type of interaction had emerged, the strong interaction, which also extends only over distances on the scale of the atomic nucleus. Much stronger than the electromagnetic force, the strong force binds the protons and neutrons in the atomic nucleus and prevents the protons from repelling each other. Without this force, stable atomic nuclei could never have formed, and ordinary matter would not exist.\n\nNature seemed unwilling to disclose its deepest secrets: when experimenters probed strong interaction, they found it all but incomprehensible. But, like Einstein, Dirac did not trouble himself with the complications introduced by the new interaction. In his opinion,  there was no point in paying much attention to them until electrons and photons had been properly understood in the context of a mathematically defensible theory. While most others moved on, he remained - in their view - transfixed by an obsolete view of physics, hidebound.\n\nOppenheimer had also retreated from the front line of research. He was a prominent adviser to the Eisenhower administration on nuclear policy, uneasy that so many aspects of the research were kept secret under the pretext of national security; he preferred Bohr's view that superpowers should, like scientists, share their knowledge as a matter of principle. In a perceptive speech in February 1953, Oppenheimer startled a closed meeting of the Council on Foreign Relations by likening the USA and the USSR to 'two scorpions in a bottle, each capable of killing the other, but only at the risk of his own life'. He believed that, despite the superpowers' posturing and bluster, reason would prevail.\n\nShortly before midnight on 14 April 1954, Dirac arrived home in Cambridge after spending a month with his stepson Gabriel in Vienna. Dirac had visited him every afternoon at the Viktor Frankl Institute, where he was being treated for psychiatric disorders, including a persecution complex and schizophrenia. Dirac had written to tell Manci of the doctors' assessment: Gabriel had been 'badly brought up'. Soon after he arrived home that night, Dirac would have told his wife of her son's progress, and they may well have discussed the news that had broken in European newspapers that day: the American Government had withdrawn Oppenheimer's security clearance.\n\nThe Oppenheimer case was the climax of the anti-Communist paranoia in 1950s America. It had begun with the start of the Cold War and intensified in the late summer of 1949, when the Soviet Union tested its first nuclear weapon at least two years earlier than the Central Intelligence Agency (CIA) expected from its intelligence reports. The USA, terrified that its technological primacy would be eclipsed by the Soviet Union, feared that Communists held important positions in public life. An early victim was Oppenheimer's popular brother Frank, an experimental physicist who had been fired in 1949 by the University of Minnesota when it found out that he was a card-carrying Communist (a few weeks afterwards, Dirac tried to find him  a post at the University of Bristol). In early February 1950, there was a national outcry when Klaus Fuchs - Dirac and Peierls' collaborator during the war, later a member of the Manhattan team - confessed to having passed critical secrets to the Soviet Union, an act of espionage that had been responsible for the unexpectedly early detonation of the Soviet nuclear weapon. J. Edgar Hoover called Fuchs' treachery 'the crime of the century'. After the revelation, Dirac and Peierls came up with an explanation of Fuchs' peculiar behaviour during his conversations with them in the back garden of 7 Cavendish Avenue - he had been passing notes on the conversation to a Soviet intermediary. Eighteen days after Fuchs had been unmasked, the Wisconsin Republican Joseph McCarthy stoked up the febrile anti-Soviet rhetoric in the press when he claimed, in a six-hour speech on the Senate floor, that Communists infested the entire government apparatus. When Bohr complained about the apparently unending deluge of insults in the newspapers, Dirac told him not to worry as it would end in a few weeks because, by then, the reporters would have used up all the invective in the English language. Bohr shook his head, incredulous.\n\nIn June 1952, the Senate passed an Immigration Act that obliged applicants for US visas to list all their past and current memberships of organisations, clubs and societies. Decisions about whether to grant visas were usually left to consuls, most of them nervous of being seen as 'soft on Commies'. No record of Dirac's submission survives. It is most likely that he would have been open with the American authorities about his relatives behind the Iron Curtain in Hungary and his association with left-leaning organisations before the war. He may also have mentioned that he signed a petition two years before to deplore Bernal's expulsion from the Council of the British Association for the Advancement of Science, after Bernal had made a scathingly anti-Western speech in Moscow.  That signature had been noted by MI5.\n\nSoon after Oppenheimer's hearing began, on the rainy Monday morning of 12 April in Washington DC, he realised that he was being subjected not to an enquiry but to a kangaroo court. The FBI had illegally tapped his and his attorneys' phones, forwarding transcripts to the prosecuting lawyers to help them prepare for the next day's proceedings.  During the second weekend break in the hearing, Oppenheimer read a pessimistic note from Dirac, who was planning  to visit the institute for a year, beginning in the following summer. There was, Dirac believed, little chance that the US Government would grant him a visa.\n\nThe enquiry closed on 5 May, and Oppenheimer returned to Princeton tired, depressed and irritable. He knew that it had gone badly: under ferocious cross-examination he had been evasive, mendacious and sometimes even disloyal to his friends. One of the most damning testimonies had been delivered by Edward Teller, who had been angry with Oppenheimer for not making him head of the Manhattan Project's theory group and, in his opinion, for delaying his pet programme to build the first hydrogen bomb. Teller declared that, 'if it is a question of wisdom and judgement, as demonstrated by actions since 1945, then I would say that it would be wiser not to grant [Oppenheimer] security clearance'. Immediately after Teller left the witness stand, he offered his hand to a stunned Oppenheimer, who took it. 'I'm sorry,' Teller said.\n\nWhen Oppenheimer was waiting for the board's verdict, he received a letter from Dirac: 'I regret to have to tell you that my application for a US visa has been refused.' On both sides of the Atlantic, news of the refusal broke on 27 May 1955, most of the articles declaring or hinting that Dirac's Russian connections had been the cause. Among the journalists who called at 7 Cavendish Avenue was Chapman Pincher, the well-connected _Daily Express_ security correspondent. Manci told him, with more pith than accuracy, 'My husband has no political interests,' a phrase that Pincher included in a brief article in the _Express_ ('US-Barred Scientist \"Not Red\"'). A reporter from the _New York Times_ somehow managed to interview Dirac and was told that his application had been 'turned down flat': the American Consul had told him he was ineligible for a visa under Regulation 212A, without specifying which of the points specified in its five pages he had transgressed. Dirac was uncharacteristically decisive: he asked the British Government to release him from all defence work and started to make arrangements to change the location of his sabbatical to the Soviet Union. This alteration to his plans was certain to provoke the American authorities, as he must surely have known.\n\nAmerican authorities, as he must surely have known.\n\nA month later, Oppenheimer heard the outcome of his 'hearing': the Board voted two to one that he was a loyal American, though nevertheless a security risk. To ram home their victory, his enemies in the Atomic Energy Commission withdrew his security clearance a day before it was due to expire. Oppenheimer was shattered, and he considered emigrating to England to take up a professorship in physics at Cambridge University, an offer that he discussed with Dirac. His fiercely loyal wife, who had given one of the powerfully supportive testimonies during the hearing, became an alcoholic and remained one for the rest of her life. After a family vacation in the Caribbean, where he was watched by FBI agents suspicious that a Soviet submarine might whisk him back to Russia, he returned to the institute. His eloquence and appetite for his work were undiminished, though many of his colleagues thought his spirit was broken. He looked less like the blazingly confident scientist, an American hero after the Manhattan Project's success, than a scientific martyr, the Galileo of the McCarthy era.\n\nThree days after the _New York Times_ announced the Oppenheimer verdict as the lead story on its front page, it printed a short report on Dirac's case, featuring quotes from an interview with Dirac, printed below a photograph that made him look like a criminal. Embarrassed and angry, senior American physicists seized on this latest of many rejected visa applications from top scientists, and it became a cause c\u00e9l\u00e8bre. Two days after the report was published, John Wheeler and two Princeton colleagues fired off a letter to the newspaper, deploring the Government's action: '[we] believe this action is exceedingly unfortunate for science and this country', adding that the Act that led to the refusal of Dirac's visa 'seems to us a form of organized cultural suicide'. Dozens of other physicists turned the screws on the State Department and the American Consulate in London, who blamed each other for the outcome of the decision, which had been 'close', they told journalists. Within two weeks, the _New York Times_ reported that the State Department was reviewing the ban; a humiliating climb-down looked certain and was duly announced on 10 August. But it was too late: Dirac had made other arrangements.\n\nDirac's plans for a sabbatical in Russia fell through, so he accepted a  long-standing invitation to visit India. At the end of September 1954, Dirac and his wife set sail for Bombay, the first stage of their round-the-world trip, scheduled to last almost a year. The Diracs arranged for their friends Sol and Dorothy Adler to stay in 7 Cavendish Avenue to look after Mary and Monica, both anxious and dreading their parents' long absence. Monica, then twelve years old, cannily observed one important reason why her parents were going far away: Manci believed that Dirac had a female admirer who was showing him rather too much affection, so she wanted him away from Cambridge for as long as possible. Dirac may well have wanted to see something of the country described to him in the fireside reminiscences of his confidante Isabel Whitehead, who had died in the previous year, six years after her husband.\n\nThe Diracs' four-month stay in India was organised by the physicist Homi Bhabha, Dirac's former colleague in Cambridge and founding director of the Tata Institute in Bombay. He was exceptionally cultured, an exhibited artist and a connoisseur of poetry in several languages. Bhabha made sure that the Diracs were treated like royalty from the moment they arrived on 13 October, though he could do nothing about Bombay's unbearable heat and humidity, which quickly drove them to depart for the comparative cool of the Mahabaleshwar Hills nearby. Manci disliked much more than the climate: she hated the spicy food and the chauffeur-driven rides through vast, stinking vistas of destitution and squalor; nor did she appreciate being treated as a second-class celebrity, her husband's consort. The experience did, however, give her a glimpse of the respect and reverence that she would later expect, and a little of this taste for glamour later appeared to have rubbed off on Dirac. For the first time in his life, he felt the adulation of a mass crowd when he gave a public lecture during the evening of 5 January 1955 as part of the Indian Science Congress in Baroda, near Vadodara. In a special enclosure at Baroda cricket ground, he delivered his talk to thousands of wide-eyed spectators, many of them watching the presentation on a cinema screen outside the ground.\n\nPerhaps having learned from the debacle at Le Palais in Paris, Dirac had found a way of talking to people who wanted to learn about quantum physics but who knew nothing about it. Shedding his dislike of metaphor and visual imagery in descriptions of the subatomic domain, he spoke in simple, equation-free language and  introduced a simile, later given wide currency, to link subatomic particles with his favourite game:\n\nWhen you ask what are electrons and protons I ought to answer that this question is not a profitable one to ask and does not really have a meaning. The important thing about electrons and protons is not what they are but how they behave - how they move. I can describe the situation by comparing it to the game of chess. In chess, we have various chessmen, kings, knights, pawns and so on. If you ask what a chessman is, the answer would be [that] it is a piece of wood, or a piece of ivory, or perhaps just a sign written on paper, [or anything whatever]. It does not matter. Each chessman has a characteristic way of moving and this is all that matters about it. The whole game of chess follows from this way of moving the various chessmen [. . .]\n\nThe physicists in the front row as well as the non-experts in the audience gave a warm reception to Dirac's forty-minute summary of the fundamentals of quantum mechanics. Though he had none of Eddington's verve as a populariser, it was clear that he had somehow acquired the skill vital to scientists who detest administration and who are well past their peak as researchers: the ability to share his work with the public.\n\nMost eminent among the politicians Dirac met in India was its charismatic Prime Minister, Jawaharlal Nehru, who had led India since its independence from Britain in 1947. Although he had the politician's talent for casting broad-brush thinking in colourful, populist language, Nehru was also a cultured thinker who would lighten a quarrel by quoting the poetry of Robert Frost. During the meeting in Delhi with Dirac on 12 January 1955, Nehru asked him if he had any recommendations for the future of the new republic of India. After his usual reflective pause, Dirac replied: 'A common language, preferably English. Peace with Pakistan. The metric system.' The men apparently did not discuss nuclear weapons, though the subject was on their minds. Eleven days before, at the Science Congress in Baroda, Dirac heard Nehru lecture scientists about the imperative to help with the reality of the new weapons, commenting that 'We are not playing with atomic bombs at present.' With Nehru's support, Bhabha would later spearhead plans for India's programme and become his country's Oppenheimer.\n\nTwo weeks after the Diracs sailed from Bombay on 21 February 1955, the trip turned unpleasant. After contracting jaundice, Dirac  spent eight days in hospital in Hong Kong, where his doctor agreed to allow him to sail on to Vancouver, though with a litany of health warnings and dietary instructions.  Manci thought he should not travel, but he insisted and paid dearly for his obstinacy by spending most of the voyage in bed, sick with jaundice, vomiting every few hours, plagued by itches, sometimes unable to sleep through the night. When the Diracs sailed into Vancouver in mid-April, he was exhausted and dispirited, his skin a pale shade of yellow. The University of British Columbia accommodated them on one storey of a finely appointed mansion, where he immediately took to his bed.\n\nTwo days later, he heard the news from Princeton that broke his heart: Einstein had died. For the first time, Manci saw him weep - a sight she had never seen before and would never see again.  It was for a hero, not a friend, that Dirac shed those tears. During those first hours of grief, he may have recalled his student days in Bristol when he first became acquainted with relativity theory, which inspired him to be a theoretician. What mattered most to Dirac were Einstein's science, his individualism, his indifference to orthodoxy and the ability he demonstrated later in life to ignore his critics' catcalls, muted only by timidity and cowardice. After Einstein's ashes had been scattered into the New Jersey winds, Dirac succeeded him as the most famous loner in theoretical physics, an elderly rebel with a cause that no one else could quite understand.\n\nSick, depressed and believing he was dying, Dirac told Manci that he had just one request: to see Oppenheimer. She quickly succeeded in bringing together the two friends in the Vancouver apartment, each of them broken, each at their nadirs, each looking fifteen years older than when they last met. No record of their conversation remains, but it is likely that Dirac's main wish was to commiserate with Oppenheimer over the outcome of the trial and, perhaps, over the conduct of Teller and the prosecutors. Teller, a pariah to many of his former friends, had become one of the few physicists Dirac disliked and would criticise, if only to those close to him. Oppenheimer was at his considerate best: he advised Dirac to get treated in the USA and to recuperate for a few weeks in one of the apartments at the Institute for Advanced Study.\n\nColleagues at the institute noticed the change in Dirac's gait. No longer lissom, he walked slowly and deliberately, as if recovering from surgery, but his vigour was returning. He spent the mornings  preparing lectures for a forthcoming meeting in Ottawa, the afternoons sleeping, the early evenings on long, restorative walks round the grounds of the institute, alone except for the squirrels, rabbits and the occasional deer. But misfortune struck: during a visit by Judy and her baby girl, he fractured a metatarsal bone in his right foot - he was an invalid again. In Ottawa, for the first time in his life, he gave his lectures sitting down and looked, as he approached his fifty-third birthday, like an old man.\n\nWhen the Diracs arrived home in Cambridge at the end of August 1955, to see their daughters for the first time in almost a year, Manci wrote a gushing thank-you note to Oppenheimer, passing on from Dirac a suggestion to help him come to terms with his tormentors. Dirac recommended Oppenheimer read the new Somerset Maugham novel, _Then and Now_ , set in fifteenth-century Florence, about the intrigues and deceptions in the relationship between Cesare Borgia and Niccol\u00f2 Machiavelli.\n\nIn the first seminar Dirac gave in Cambridge at the beginning of the next term, he announced to his students: 'I have just done this work. It could be important. I want you to learn it.' This was an extremely rare instance of Dirac publicly pointing the way ahead. His enthusiasm for research had been rekindled.\n\nDirac's new theory suggested that the universe might not fundamentally consist of point-like particles but of tiny, one-dimensional things that he called 'strings'. The theory, first outlined in his Ottawa lectures, was a new approach to quantum electrodynamics that dispensed with one of the foundations of renormalisation theory that Dirac most disliked - the 'bare electron', the idea that the theory could be built from the fictional notion of an electron that had no surrounding field. In his new approach, he concentrated on one of the theory's underlying symmetries, known as gauge invariance. Long familiar to theorists, this symmetry implies that the theory makes identical predictions if a quantity known as the electromagnetic potential, closely related to the electromagnetic field, is changed at every point in space-time, but only if the changes across the whole of space-time are orchestrated by a governing formula known as a gauge transformation. Dirac found a way of rebuilding quantum electrodynamics in terms of gauge-invariant quantities so that, whenever the electron features in a calculation, it is inseparable from its  field. The result was a theory that gave the same results as the renormalised version but that was, for him, superior.\n\nDirac disliked the concept of bare electrons so much that he wanted 'to set up a theory in which they] are not merely _forbidden_ but _inconceivable_ ' _._[ 47 He found a way of doing that using the equations of his theory, by applying them to the lines of force describing the electric field of the electron, which resemble the field lines of a magnet. In the classical picture of the electron, the particle is surrounded by continuously varying lines of force: each set of lines of force is, in a sense, infinitesimally close to the next. This led Dirac to imagine a quantum version of the field and to picture the electron not as a particle but as a string:\n\nWe may assume [that] when we pass over to the quantum theory the lines of force become all discrete and separate from one another. Each line of force is now associated with a certain amount of electric charge. This charge will appear at each end of the line of force (if it has ends) with a positive sign at one end and a negative sign at the other. A natural assumption to make is that the amount of charge is the same for every line of force and is just the [size of the charge of the electron]. We now have a model in which the basic physical entity is the line of force, a thing like a string, instead of a particle. The strings will move about and interact with one another according to quantum laws.\n\nDirac had found what he was seeking: 'a model in which a bare electron is inconceivable, because the end of a piece of string is inconceivable without the string'. But it was only the germ of an idea, not a complete new theory. Several of his students examined it but soon set it aside, as Dirac did soon afterwards. Years later, it would transpire that he had once again been ahead of his time.\n\nDirac was about to reach the low point of his career: apart from wartime, 1956 was the first year since he had begun research that he had published nothing at all . Now semi-detached from the physics community, he had lost touch with many of his closest friends, including Kapitza - they had not been together for almost twenty years. Dirac will have wanted to know how Kapitza was faring in Nikita Khrushchev's regime, which began soon after Stalin's death in March 1953. British newspapers had reported a new mood in the country after the Soviet public heard that Khrushchev had, in a speech to stony-faced party bosses in February 1956, denounced the  personality cult of Stalin and the cruelty of his regime.\n\nIn the early autumn, Dirac arrived in Moscow to find it very different from the city he and Manci had seen in 1937: it was now focusing on consolidation, not revolution, and the paranoid, inwardly focused nationalism of the late 1930s had been superseded by a dread of a pre-emptive nuclear strike by the USA. Dirac found Kapitza as self-confident as he had ever been and just as full of colourful stories: in one, he told Dirac of how his arch-enemy Beria had sidelined him after he had refused to work on nuclear weapons. Kapitza believed that 'It is a horrible thing for scientists to engage in secret war work,' and he probably mentioned this to Dirac, who may have flinched, at least inwardly. While most other leading Soviet physicists had given their services to the nuclear project, Kapitza worked on ways to destroy incoming nuclear weapons using intense beams, apparently a precursor to the American Strategic Defence ('Star Wars') Initiative. Stalin's good opinion had saved him from execution by one of Beria's henchmen, Kapitza was sure. When Stalin died, Lev Landau danced for joy, but Kapitza knew his own life was in danger if Beria was the country's next leader. Khrushchev outmanoeuvred Beria, but Kapitza's life was still in peril: on what seemed to be an ordinary summer morning, towards the end of the official discussions about Stalin's succession, Kapitza told Dirac, two state officials visited him in his small laboratory and asked for a guided tour. Their questions revealed that they knew little about science and cared even less, yet they insisted on prolonging their visit beyond its natural duration, until their departure on the stroke of noon. According to Kapitza's account of the story, the two men had been deputed - probably by Khrushchev or his associates - to protect him from a last-minute reprisal while Beria was being arrested and taken into custody. A few weeks later, Beria and six of his accomplices were tried and sentenced to death; he was executed by one of Khrushchev's three-star generals, who fired a bullet into his forehead.  Kapitza heard the news on Christmas Eve, a joyous moment for him.\n\nDirac never tired of praising Kapitza's refusal to work on the nuclear-bomb project. This was the story Kapitza told Dirac and everyone else, but it is almost certainly untrue. Kapitza's letters to Stalin - published several years after Dirac's death - make it plain that Kapitza wanted to work on the project, and he shows no hint of  any moral scruples; he declined to work on the bomb only because he would not work under Beria's heel. It is also possible that he did not command support from his colleagues, as some of them believed he was contemptuous of scientists outside his cosmopolitan circle. A much stronger case for Kapitza's heroism can be made by pointing to the case of Landau, Stalin's outspoken enemy, whom Kapitza repeatedly defended, often putting his life in grave danger. Hundreds of thousands of Russians were executed for showing only a fraction of Kapitza's insubordination.\n\nDirac spent most of his visit to Moscow in October 1956 sightseeing - he saw that Lenin was then sharing his tomb with Stalin - as well as reacquainting himself with his old Russian friends, including Tamm, Fock and Landau. It is surprising that Dirac was allowed to meet Tamm, as he was leading the secret project to build the hydrogen bomb (Tamm's participation in this work may have been one reason why his friendship with Dirac fizzled out in the next decade). Landau, the permanent juvenile, was by then in the front rank of theoreticians and still flaunting his irreverence: he replaced the toilet roll in his bathroom with pages from Stalin's autobiography.\n\nLandau was in the audience of Dirac's lectures at Moscow University, where Dirac responded to the request made to some of their guests to summarise their philosophy of physics. He wrote on the blackboard: PHYSICAL LAWS SHOULD HAVE MATHEMATICAL BEAUTY. In public, Landau was respectful of Dirac's aestheticism, but in private he was cutting, once remarking to the physicist Brian Pippard, 'Dirac is the greatest living physicist and he has done nothing of importance since 1930.' Overstated to the point of cruelty, this was typical Landau. He was, however, only giving voice to what many leading physicists in the mid-1950s thought but dared not say in public. Yet, as events were about to prove, Dirac's detractors had been too hasty in writing him off.\n**Twenty-six**\n\nHow some they have died, and some they have left me,  \nAnd some are taken from me; all are departed;  \nAll, all are gone, the old familiar faces.\n\nCHARLES LAMB, 'The Old Familiar Faces', 1798\n\nIn early December 1958, when Pauli was approaching his fifty-eighth birthday, he was looking sallow and unwell. He complained of stomach pains during a lecture at his university in Zurich in the afternoon of Friday 5 December and took a taxi home. On the following day, he went to the city's Red Cross Hospital, where he was admitted for tests which proved inconclusive, so doctors decided there was no alternative but to operate. A week later, a surgeon cut into the hillock of his midriff and found a pancreatic tumour so large and advanced as to be inoperable. Within forty-eight hours of the operation, he was dead.\n\nThe final year of Pauli's life had not been among his happiest - a quarrel with his friend Heisenberg over an ambitious theory they were developing had turned nasty and had suppurated. But the end of Pauli's career had also seen the seal put on one of his finest contributions to physics: during an early summer morning in 1956, he received a telegram from two experimenters in the Los Alamos laboratory to confirm that they had discovered the neutrino, the particle that Pauli had predicted, though Dirac and others had doubted that his arguments held water. Just as Pauli had foreseen, the neutrino has no electrical charge, the same spin as an electron and apparently no mass. The newly discovered particle interacts with matter primarily through the weak interaction, which is extremely feeble: of the ten thousand trillion trillion neutrinos zipping through planet Earth every second, all but a few pass straight through without deflection.\n\nThe discovery was a triumph for Pauli but, two years later, nature put him firmly in his place when his intuition about the weak interaction was shown to be quite wrong. The story began at the Brookhaven National Laboratory in 1956, when a duo of young Chinese theoreticians - C. N. 'Frank' Yang and T. D. Lee  (usually known as 'TD') - suggested what Pauli and almost all other theorists regarded as ridiculous: when particles interact weakly, nature might choose to break the perfect symmetry between left and right, the so-called parity symmetry. At a fundamental level, gravity and electromagnetism are ambidextrous: every experiment that investigates this type of interaction would give the same result if the configuration of the particles involved were swapped left to right, in their mirror image. At Columbia University in New York, experiments (suggested by Lee and Yang) to investigate whether weak interactions are left-right symmetric were carried out by two groups, one led by the aggressively confident Chien-Shiung Wu, born in Shangai, the other by Leon Lederman, a wisecracking New Yorker. The experiments each came to a climax in the bitter cold of New York in mid-January 1957, when they confirmed that Pauli had been wrong and that the suspicions of Lee and Yang were right: in weak interactions, nature _does_ distinguish between left and right.\n\nThe result was a sensation, and not only among physicists - it even featured prominently on the front page of the _New York Times._ But the observation was no surprise to Dirac. He had foreseen the possibility that parity symmetry might be broken, in the introduction to the review of relativity he wrote in 1949. There, he considered whether quantum descriptions of nature would remain the same if the positions of the particles are reversed in a mirror (a left-right swap) and, separately, if time runs backwards instead of forwards. In his conclusion, he took the unusual step in a technical article of using a personal pronoun: 'I do not believe that there is any need for physical laws to be invariant under these reflections [in space and in time], although all the exact physical laws of nature so far known do have this invariance.'\n\nDirac had realised that although the laws of gravity and electromagnetism had left-right symmetry and time-reversal symmetry, the laws of other fundamental interactions may not have this property. No leading physicist had remembered reading these words, and even Dirac himself forgot that he had written them. After 1949, he was aware of the possibility of quantum asymmetries in space and time but apparently said nothing about it, except once during a cross-examination of a Ph.D. student. A few years later, when he heard colleagues talk of the shock of parity violation, he would calmly  draw attention to this passage in his paper. To students who asked him about it, he said simply, 'I never said anything about it in my book.' He knew, however, that he could not expect many plaudits for his contribution: the winners-take-all rule of scientific conduct entitled Lee and Yang to take the credit for fully appreciating the importance of the breaking of parity symmetry. Theirs was one of the great discoveries of the modern era.\n\nThe death of Pauli had removed from the fraternity of senior theoreticians the one member Dirac disliked. Although they did not overtly compete with one another, undercurrents of rivalry swirled beneath their superficial rapport. Their approaches to theoretical physics were different, as Pauli was a conservative analyst, while Dirac was a revolutionary intuitionist. But that need not have divided them. Most of Pauli's peers thought that his scabrous insults were a small price to pay for the high quality of his insights. But Dirac demurred; he often went out of his way to remind lecture audiences that Pauli 'very often bet on the wrong horse when a new idea was introduced', including the time he 'completely crushed' the idea of spin when it first hatched. Nor, it appears, could Dirac forgive Pauli's pitiless strafings. When Pauli stood over him, damning hole theory, demanding that he recant, perhaps Dirac could see the ghost of his father?\n\nDirac's daughters never saw him show much interest in politics except perhaps when he watched the television news, with the inscrutability of a sphinx. Manci was quite different: she closely followed international events and had strong opinions about many of them, which she spent afternoons discussing on the telephone with friends. In November 1956, she and her family - including her brother Wigner - looked on sadly when Soviet tanks and troops crushed the uprising in Hungary against its government, a puppet of Moscow, and killing some twenty thousand Hungarians. Landau condemned Khrushchev and his Politburo as 'vile butchers'. In the UK, the _New Statesman_ , usually a moderate critic of the Soviet Union, denounced the invasion as 'loathsome', 'indefensible' and 'unforgivable'. Soon, the Communist Party haemorrhaged, and the hard-left core of Cambridge academics was reduced to an ineffectual rump, including Bernal, one of the few whose loyalty to the cause was undiminished. Dirac appears to have said nothing about the  Hungarian invasion even to his closest friends: by the mid-1950s, he appears to have lost every vestige of his youthful idealism. He took the rare step of giving vent to this distaste when he first met Tam Dalyell, an Eton-educated Tory who switched allegiance to the Labour Party in 1956 after the disastrous British invasion of Egypt, following the nationalisation of the Suez Canal. Dirac indicated that he welcomed the maverick Dalyell's change of political heart, but added pointedly, 'I don't _like_ politicians.'\n\nYet Dirac was still following reports from the Soviet Union. 'We're all very excited by the sputniks,' he wrote to Kapitza at the end of November 1957. Dirac had first heard about the launch of the artificial satellite, apparently to mark the fortieth anniversary of the Bolshevik Revolution, on the morning of 5 October.  That evening, he and Monica went to the back garden of 7 Cavendish Avenue shortly after dusk hoping to see the twinkling satellite pass over in the night sky. Newspaper reports of the orbiting 'Red Moon', a beach-ball sized sphere girdling the Earth in ninety-five minutes, made front-page headlines for a week, and Dirac wolfed the reports down. Sputnik's success transformed the West's view of Soviet technology from condescension to fearful admiration. For Americans, the Sputniks were frightening wake-up calls, even more disturbing after the attempt to launch their own satellite in early December ended in fiasco, when it exploded a few seconds after lift-off (one jeering journalist suggested that it should have been called 'Stayputnik'). The Sputnik missions demonstrated that the Soviets were well on the way to developing intercontinental ballistic missiles and to launching a human being into space. The missions panicked the media and politicians into believing that the Soviet Union - which many Americans believed was a backward, agrarian country - was way ahead of the USA in science education. Edward Teller went on television to pronounce that 'The United States has lost a battle more important and greater than Pearl Harbor.' _Life_ magazine pointed out that three in four American high-school students studied no physics at all. As a result of all this pressure, President Eisenhower ordered a renaissance in school science and, between 1957 and 1961, Congress doubled federal expenditure on research and development, to $9 billion. An unlikely beneficiary of this largesse was high-energy physics: a new generation of subatomic particle accelerators were, in a sense, the Sputnik's progeny.\n\nDirac was as interested in the technology of space flight as in any scientific benefits it might bring. He watched television footage of the launches with the same enthusiasm that he had shown when observing from the back garden of 6 Julius Road the launches of some of the first aeroplanes. But he was puzzled: why were the space rockets launched vertically rather than horizontally? So far as he could see, the challenge of propelling a rocket into space is much the same as that of launching a heavily loaded aeroplane, and vertical take-off is extremely inefficient as much of the fuel is used before the rocket is clear of the launch pad; it would therefore be best to launch the rocket horizontally, at high speed. Dirac was fascinated by this question. In May 1961, soon after the Americans put an astronaut into space - less than a month after the Soviets had beaten them to it - Dirac took aback his two fellow diners over lunch at St John's College by sitting not in his habitual silence but, instead, talking about rocketry non-stop for almost an hour.\n\nIn the coming decades, he followed reports of the Soviet and American space programmes and attended specialist meetings on them at the Royal Society. Even after talking with several experts, he remained unconvinced that the rockets were being launched in the most economical way, so he took the unusual step of asking NASA for an explanation. Its officials informed Dirac that he was wrong because he was underestimating the importance of the 'drag' effect of the atmosphere on a space rocket and the performance of the rocket's engine, which improves with altitude.  Such rockets are launched vertically so that they can climb quickly, enabling them to reach altitudes where the inhibiting aerodynamic pressures on the rocket are much lower than they are at ground level. As the air thins with height, the engine's exhaust can impart greater thrust. These advantages together make it much more economical to launch the rockets vertically, as several experts explained to Dirac, though it seems that he never quite believed them.\n\nSince Dirac's arrival in Cambridge in 1923, his working environment had hardly changed. But, towards the end of the 1950s, there was a concerted drive in the Cambridge science departments to manage themselves more efficiently, partly so that they could compete more successfully with other international centres of science and, indeed, with other parts of the university. In Dirac's bailiwick, the leader of  the drive was George Batchelor, an Australian-born mathematician with an uncompromising manner that made clear the extent of his ambition to anyone who doubted it. Then in his late thirties, Batchelor was an expert in fluid mechanics, the branch of applied mathematics concerned with the flow of gases and liquids, a subject for which Dirac had little time - he regarded it as the small fry of theoretical physics. Nor did he like Batchelor, one of the few people who could bring out the snob in him; their colleague John Polkinghorne recalls that Dirac once offended the rhino-skinned Batchelor by dismissing George Stokes, one of the pioneers of fluid mechanics, as 'a second-rate Lucasian professor'.\n\nFrom the beginning of the autumn term in 1959, Dirac officially worked in the Department of Applied Mathematics and Theoretical Physics, headed by Batchelor. Polkinghorne admired Batchelor as an effective, congenial leader, but Dirac and his colleague Fred Hoyle - now a top-flight cosmologist and a popular broadcaster - both declined offices in the new department and disliked virtually every change he wanted to make. One of the proposed changes was to adopt a more communal approach to research, a notion that could not have been more inimical to Dirac, who looked like a refugee from another age on the rare occasions he attended the new social gatherings. In seminars, he often appeared to be catching up on his sleep but would sometimes give the lie to that by asking a pertinent question. But he would also embarrass senior colleagues by showing how little he knew about the latest research discoveries, even about new particles familiar to greenhorn students.\n\nAlthough Dirac was not one to stand on his dignity, he was stung when Batchelor ejected him from the office he had occupied for some twenty-five years and 'volunteered' him to give additional lectures. Having been wounded by a series of such slights, he snapped when an officious parking attendant in the Cavendish told him that he had no right to leave his car there. John Polkinghorne recalls Dirac's response: 'He was furious. He told the attendant that he had parked there for twenty years.' He accepted Batchelor's executive decision, but Manci was less compliant and wrote a scathing letter to the Vice Chancellor, who wrote back soothingly and then forgot about her. The authorities no longer felt obliged to keep Dirac happy, and he knew it.\n\nPerhaps in part because of his unhappiness at work, Dirac's marriage was for the first time under strain. The wife of one of the Fellows at St John's briefly caught sight of this when Manci light-heartedly accosted her outside Woolworth's: 'Let's go for a coffee - he hasn't spoken to me for a week and I'm _so_ bored.' Stories like this did not surprise the Diracs' acquaintances in Cambridge as most of them had never fully understood how such different people could be happy together. But this happiness was partly an act. Behind their front door, her attitude towards him swung from one extreme to another: one day, she would throw her arms round him and enquire coquettishly whether he loved her; the next, she would tell him angrily: 'I'd leave if I had somewhere to go.' Such threats left Dirac unmoved. According to one story, she once snapped at him when he was eating his dinner, 'What would you do if I left you?' only for him to reply - after a half-minute pause - 'I'd say \"Goodbye dear\".'\n\nAlthough he sometimes gave the impression that his research had dried up, Dirac was still thinking hard about his physics. When he gave Manci the signal that he was at work, she ordered the girls to be quiet: Monica would retire to her room, while Mary switched off the gramophone, endlessly blaring out the soundtrack of _Oklahoma!_ Now in their teens, the girls had realised that their father was a distinguished scientist and that he was exceptionally quiet and self-effacing.  'I was lucky,' he told Monica. 'I went to good schools, I had excellent teachers. I was in the right place at the right time.'\n\nGabriel, recovered from his illness, was acutely aware of his stepfather's status: his surname drew amused comments from his mathematical colleagues and did him no harm at all. Dirac was close to Gabriel and went out of his way to promote his career, often exchanging letters with him to chew over chess problems they had read in newspapers (G. H. Hardy had described such problems as 'the hymn tunes of pure mathematics'). Judy and her family - by the summer of 1960, she had three children - were more distant, and she was in one long fight with her mother, who had all but lost patience with her. As many family friends confirm, Manci was a much better wife than a mother, always supportive and loyal to her husband but often insensitive to her children. It seems that Mary suffered most from her mother's tongue: Manci repeatedly browbeat her, told her she was 'ugly' and also 'lazy', a word she used to describe everyone in the family who did not earn a wage, including  Dirac's sister Betty. No one, least of all Dirac, dared to remind Manci that she had yet to do a day's paid work.\n\nBy the late 1950s, Mary was back at home and working in Cambridge, contemplating emigration; Monica was preparing to study geology at university. The girls were rapidly becoming independent, and the Diracs wanted to make the most of their new freedom by travelling even more. For someone so friendly, Manci had surprisingly few friends in Cambridge - she was close only to Sir John Cockcroft's wife Elizabeth - and she was continually planning trips to see her family and friends abroad, the further from Cambridge the better. Dirac felt much the same way: an outsider in his own department and resentful of Batchelor's machinations, he preferred to be where he was appreciated. The result was that, in the dozen years before his retirement in 1969, the Diracs were away from Cambridge almost as much as they were there.\n\nSoon after the neutrino was discovered, Dirac had the idea that the particle's existence might be explained by Einstein's general theory of relativity. This was at the back of his mind in September 1958, when he began another sabbatical at the Institute for Advanced Study in Princeton, intending to develop a new version of Einstein's theory based on his favourite way of setting out fundamental theories, using Hamiltonians to describe the interactions. His aim was to find a general classical description of every basic type of field - electromagnetic, gravitational and so on - preparing the ground for their quantisation.\n\nAlthough his project failed, his method of analysing the general theory of relativity gave new insights into gravity. He described some of them in his lecture at the annual meeting of the American Physical Society, held in New York in the grip of a bitterly cold spell, at the end of January 1959. Always averse to large gatherings, Dirac was probably not looking forward to his stay as he walked the two blocks from Penn Station to the huge, overheated New Yorker hotel, to join the five thousand delegates, most of them in a starched white shirt and tie, sleeves rolled up. Without Dirac's scientific celebrity, he would have been just another of the meeting's invisible men, but his renown made his attendance one of the talking points in the bars and lounges. Many of the audience arrived early after lunch to secure a seat in the huge ballroom, between the imitation Ionic columns  reaching to the ceiling, and below the three giant chandeliers decorating the room like cheap jewellery.\n\nDirac began his talk by making it clear that he was not going to comment on the particle physics in fashion but about the electromagnetic and gravitational interactions, both known for centuries but still not fully understood. Everyone in the audience knew that Maxwell's field theory of electromagnetism predicted the existence of electromagnetic waves, including visible light, and that the energy of the field comes in quanta, known as photons. By a similar token, Einstein had shown that the general theory of relativity predicts the existence of gravitational waves. Dirac announced that his study of the gravitational field's energy indicated that it is delivered in separate quanta, which he called 'gravitons', a long-neglected term first introduced a quarter of a century before in the journal _Under the Banner of Marxism._ 33 After Dirac reintroduced the name, it stuck. These particles will be much harder to detect than photons, he pointed out, but experimenters should lose no time in beginning the hunt for them. He gave the impression to the _New York Times_ journalist Robert Plumb that this was an important prediction; the next day, Plumb's report appeared on the front page: '[Dirac] believed that his postulation at this time was in the same category as his postulation of positive electrons a quarter of a century ago.'\n\nDirac did not succeed in quantising the general theory of relativity, but his Hamiltonian method turned out to be his most influential contribution to the theory. His approach, and similar techniques developed independently by other physicists, enabled Einstein's equations to be conveniently set out in a comparatively simple form, especially in situations when gravitational fields change rapidly. This excursion by Dirac into relativity theory looked odd to most physicists. In the late 1950s, the development of the general theory of relativity was a cottage industry by comparison with the industrial scale of particle physics. Relativity was an unfashionable subject for theorists, and Dirac was one of the few who thought it important to develop it and to find a single theoretical framework to understand gravity and electromagnetism. The main topic at the conference was the strong interaction and the particles that feel it, including the newly discovered mesons. One of the leaders in the field was Feynman, who met Dirac again in the autumn of 1961 at the Solvay meeting, where they had another of their Pinteresque exchanges:\n\nFEYNMAN: I am Feynman.\n\nDIRAC: I am Dirac. [ _Silence_ ]\n\nFEYNMAN ( _admiringly_ ): It must have been wonderful to be the discoverer of that equation.\n\nDIRAC: That was a long time ago. [ _Pause_ ]\n\nDIRAC: What are you working on?\n\nFEYNMAN: Mesons.\n\nDIRAC: Are you trying to discover an equation for them?\n\nFEYNMAN: It is very hard.\n\nDIRAC ( _concluding_ ): One must try.\n\nDirac's reticence had surprised even his former student Abdus Salam, sitting next to him: from the conversation, Salam concluded that Feynman and Dirac had not previously met. One explanation for Dirac's behaviour, strange even by his standards, is that he did not recognise Feynman: Dirac had an unusually poor memory for faces, which is why he rarely remembered physicists he had met only once, even if their characters were as memorable as Feynman's.\n\nDirac was convinced that the best way to understand strongly interacting particles was to describe their behaviour with equations, just as he had done when he discovered the electron equation. But most theoreticians were not now thinking along those lines: some were exploring new types of field theory; others gave up all hope of finding equations to describe the particles' motion and sought only to describe in broad terms what can happen when they interact. In this approach, a 'scattering matrix' gives, for every possible initial state of the particles, the likelihood that it will lead to each of the possible final outcomes. Dirac rejected it as 'a fa\u00e7ade'.\n\nApart from the strongly interacting particles, experimenters had also discovered another family in the subatomic zoo. The first hint had arrived from experiments on cosmic rays in 1946, when Carl Anderson identified a particle later to be called the muon. It was some two hundred times as heavy as the electron and unstable, but in other respects it bore a close resemblance to the electron: it had the same spin and did not feel the strong interaction. But there was one crucial difference: in 1962, experimenters showed that the muon is associated with its own variety of neutrino, different from the familiar neutrino linked with the electron. All four particles - the electron, the muon and their neutrinos - appeared to have no constituents and  to be part of a family, later known as leptons, following Leon Lederman's introduction of the term, taken from the Greek word for something small and delicate, _leptos_.\n\nThe arrival of new particles normally did nothing to excite Dirac - he still had not come to terms with the photon and electron. But in late 1961, Dirac broke his rule of not working on new problems until he had solved the ones already on his plate: he tried to understand the muon, which he believed might simply be an excitation of the electron. He abandoned the usual image of the electron as a point particle and pictured it as a spherical bubble in an electromagnetic field: 'One can look upon the muon as an electron excited by radial oscillations,' he suggested. Dirac described the bubble using a relativistic theory whose equations described its motion in space-time. It was a sublime piece of applied mathematics but most physicists ignored it, apparently because its account of the electron was so unconventional: it gave a geometric account of a particle usually assumed to have no size and paid no attention to its spin. Nor did the theory's predictions do much to win over doubters - Dirac calculated that the mass of the first quantum excitation of his electron accounted for only a quarter of the measured mass of the muon.\n\nDirac first presented his theory of 'the extended electron' to his colleagues at the Institute for Advanced Study in Princeton on the warm autumn afternoon of 16 October 1962. Oppenheimer was sitting in the front row, his deep-blue eyes still alert and penetrating, his complexion as fragile as an eggshell. Still a master inquisitor, after making one of his smart comments, usually at the speaker's expense, he would sometimes turn round and survey the audience, to check that everyone had appreciated it. When Dirac was the speaker, however, Oppenheimer was on his best behaviour.\n\nAn hour after Dirac's audience had dispersed, at 6.30 p.m., President Kennedy met his officials in the White House to discuss urgent intelligence reports: the Soviets were building secret missile bases in Cuba, ninety miles from Florida and therefore potentially a threat to the USA. Six days later, Kennedy went public with the intelligence, announcing a naval blockade of Cuba and demanding that the Soviets remove the missiles. Khrushchev angrily refused to back down. Oppenheimer's scorpions were staring straight into each other's eyes.\n\nThe tension dropped on 28 October, when the Soviets agreed to remove the missiles in return for concessions from the Americans; it  seemed to many - including Dirac, watching the crisis unfold on his television in Princeton and possibly wondering whether he was about to see his third world war - that humanity had been lucky to survive. The planet seemed to be at the mercy of its Dr Strangeloves.\n\nBohr lived just long enough to see the Cuban missile crisis. Three weeks later, after Sunday lunch at home with his wife Margrethe, he went upstairs for a nap and died of heart failure. In a letter of condolence to Margrethe, Dirac said that he was 'excessively sorry' to hear of 'the loss of one of my closest friends' and recalled his first stay with the Bohrs in Copenhagen in 1926: 'I was greatly impressed by the wisdom that Niels showed, not only in physics but in all branches of human thought. He was the wisest man I knew, and I did my best to absorb some of the wisdom he imparted.'\n\nThis was the latest of a series of blows to Dirac, who was seeing his closest colleagues die off one by one. In Princeton, von Neumann had died in 1957, followed by Veblen in 1960. And only eleven months before Bohr's death, Dirac had written the obituary in _Nature_ for Schr\u00f6dinger, who had died in his Vienna home of heart disease. In his article, Dirac went out of his way to defend Schr\u00f6dinger's apparent welcoming of Nazism in May 1938: 'He was forced to express his approval of the Nazi regime, and he did this in as ambiguous a way as he could.' Many of those who had read Schr\u00f6dinger's article joyfully pledging support for 'the will of the F\u00fchrer' will not previously have noticed that it contained many ambiguities. But, as Heisenberg and Kapitza had seen, Dirac could not be faulted on his loyalty.\n\nUntil 1962, Dirac had shown no interest in publicly discussing his recollections of the beginnings of quantum mechanics. But that year, when he turned sixty, he changed his mind. He agreed to be interviewed by the American philosopher of science Thomas Kuhn, a former student of Van Vleck. Kuhn persuaded Dirac to help compile the archive for the history of quantum physics. Kuhn knew that Dirac was nervous of talking to strangers in unusual environments, so he held the first interview in Wigner's home in Princeton, with Wigner present and often chipping in with tactfully phrased questions to draw him out. During the forty-minute session, Dirac spoke quietly and clearly, often sounding tentative and mildly amused that anyone would be interested in what he would have to say.\n\nFor almost forty years, Dirac had hardly spoken a word to his physicist colleagues about his upbringing, but Kuhn and Wigner heard childhood memories pour out of him, including a torrent of domestic detail. About ten minutes into the interview, Dirac began to talk about his brother. It is clear from Wigner's delicately phrased questions and from his mild incredulity at Dirac's responses that the two men had scarcely broached the subject in the thirty-five years they had known each other. During this part of the interview, Dirac speaks as gently as usual, but each of his carefully articulated words seems to bear a heavy burden of sadness and regret, especially when he responds to Wigner's question about why Felix took his own life:\n\nI suppose he was just very depressed. And, well . . . that kind of life where we were brought up without any social contacts at all must have been very depressing to him as well as to me and having a younger brother who was brighter than he was must have depressed him also quite a lot.\n\nDirac left much unsaid, but Kuhn and Wigner were wise not to press him; if they had, he would almost certainly have clammed up and perhaps even refused further interviews.\n\nPrivately, Dirac was in no doubt why his brother killed himself. Dirac told Kurt Hofer that he was sure his father was primarily responsible for the tragedy: Charles had denied Felix a normal upbringing, forced him to speak French against his will and crushed his ambition to be a medical doctor. But, even after decades of reflection, Dirac could not understand the depth of his father's grief after Felix's suicide: his father was still a mystery to him and still, as he told his closest friends, the only person he had ever 'loathed'.\n\nThree months after the interview, Kuhn wrote to thank Dirac for his participation and informed him that his taped disclosures about Felix's death would be removed from the published version and 'filed separately for future use'. The material was made public only after Dirac's death.\n\nIn 1962, Dirac was about to enter the final stage of his career in Cambridge. His family circumstances were changing rapidly: his daughter Mary was preparing to emigrate to the USA; Monica had gone off to university 'to discover the Beatles'. Shortly before leaving, Monica had been thrown out of the house by her mother, just as she ejected Judy in her teenage years. Now Judy and her family were  settled in the USA and Gabriel was pursuing his academic career in Europe.\n\nDirac imagined that he would spend the rest of his life at home in Cambridge, tending his garden and working in his study. But Manci had other plans.\n**Twenty-seven**\n\n[Some critics] act as if Flaubert, or Milton, or Wordsworth were some tedious old aunt in a rocking chair, who smelt of stale powder, was only interested in the past, and hadn't said anything new for years. Of course, it's her house, and everybody's living in it rent free; but even so, surely it is, well, you know . . . time?\n\nJULIAN BARNES, _Flaubert's Parrot_ , 1984\n\nBy the mid-1960s, Dirac was spending most of the week working at home. At the department he looked increasingly out of place: 'He was irrelevant,' his young colleague and former student John Polkinghorne remembers. Other Cambridge physicists thought the same but followed the scientists' unwritten code of chivalry: when great researchers go to seed and speak out against modern trends in their subject, they should be ignored and even mocked in private, but be heartily praised in public for their past achievements.\n\nOutside the university, too, Dirac cut the lonely figure of a misfit from another age, uncomfortable with the new popular culture and its irreverence. It was unthinkable to him that serious critics could treat a painting of a soup tin as a mainstream work of art and that many of the defining songs of a generation were written by cheeky, working-class Liverpudlians who could not read music. What, Dirac wondered, was he to make of a group whose lead vocalist claimed to be a walrus?\n\nDirac was beginning to fear old age and the prospect of being effectively abandoned by his colleagues: all the signs were that Batchelor was going to bundle him out of his Lucasian Chair at the statutory retirement age of sixty-seven. The threat led Dirac to make a brief venture into the poisonous netherworld of university politics in the spring of 1964, when he joined Hoyle and a few others to seek Batchelor's removal after his first five-year stint as head of their department. Outmanoeuvred, they failed miserably. With no wish to be part of Batchelor's empire, and with his child-rearing responsibilities behind him, Dirac - encouraged by Manci - resumed his travels and spent even more time in his garden, trimming his immaculate  lawn, pruning his roses and growing far more vegetables than Manci needed for her larder. His bookshelves heaved with horticultural magazines and books, making his study look as if it belonged not to a research physicist but to a landscape gardener. He still did research but knew that he had next to no chance of coming up with a radically new idea. He was enduring the fate of all ageing theoretical physicists: his spirit was outliving his imagination.\n\nThough marginalised in Cambridge, he was treated kindly at his favourite academic address in the USA. In the spring of 1963, Dirac heard from Oppenheimer that he had arranged for a framed photograph of him to be mounted on a wall at the Institute for Advanced Study, next to a snapshot of Einstein: 'You two are alone on that wall.' This simple gesture symbolised the generosity of the American academic system, much more willing than British universities to find room for leading scholars to spend their unproductive twilight years in dignity. Mainly for this reason, Dirac spent more time in the USA. From 1962 to his retirement in 1969, Dirac visited the United States every year, for at least a couple of months, twice for almost an entire academic year (1962-3 and 1964-5). For much of the rest of the time, he and Manci were visiting conferences or on vacation in Europe and Israel (the USSR was no longer on their itinerary, apparently because even they could not get a visa). During these seven years, Stephen Hawking - a colleague of Dirac's and a rising star - did not see him in the department.\n\nManci had set her heart on escaping from Cambridge. Dirac disliked change and wanted to be loyal to his university but eventually agreed that it was time to emigrate, preferably to the USA. He did not have the initiative to secure a new position: that task fell to Manci, who assumed a new role as the pushy manager of a tongue-tied talent, chasing royalties and upgrades, insisting on sea-facing cabins and the room with the finest view. He was her Elvis, and she was his Colonel Parker.\n\nLecturing had become Dirac's forte. Although his voice was weakening, he could be relied on to keep his audience hooked, not through wit and humour but through clarity and humility. At the podium, he looked and sounded like an elderly preacher from Bristol but had the innocence of a young lad reading an essay on Prize Day, clipping his vowels, emphasising his consonants with the force of a stab. It was  often a surprise to people in the audience that such a taciturn man was so fluent, hardly ever hesitating with an 'er' or an 'um' and rarely showing a sign of even approaching a grammatical tangle. His most unnerving idiosyncrasy was a propensity to go silent in mid-sentence: when he needed to think or find the right words, he would suddenly stop talking, typically for ten seconds but sometimes for over a minute, before resuming without comment.\n\nHe presented fewer specialist talks but occasionally gave guest lectures, including a series on quantum field theory at Yeshiva University in New York in the spring of 1964. In these lectures, later recognised as classics, he developed the theory logically from its beginnings and, unusually for him, spelt out in detail the calculations that led to the prediction of the energy shift of the hydrogen atom, measured by Lamb in 1946. Although the theory and experiment agree to within experimental uncertainties, Dirac left his audience in no doubt that the theory of quantum electrodynamics is profoundly flawed: 'If one is a research worker, one mustn't believe in anything too strongly; one must always be prepared that various beliefs one has had for a long time may be overthrown.'\n\nA year earlier at Yeshiva, he gave his lecture 'The Evolution of the Physicist's Picture of Nature', which he adapted into an article for the May 1963 edition of _Scientific American_ , the only article he ever wrote for a popular-science magazine. The style and content of the talk foreshadowed dozens of similar presentations: he explained in plain, stripped-down language why fundamental physics was in crisis, drawing lessons from an often simplistic overview of the history of physics. In the article, he dwelt on one of his favourite anecdotes: Schr\u00f6dinger claimed that he had discovered a mathematically beautiful relativistic version of his equation a few months before the famous non-relativistic version but did not publish the relativistic equation because it failed to account for observations on the hydrogen atom (the disagreement arose because it was not known at that time that the electron has spin). Schr\u00f6dinger published his non-relativistic version only when he was sure it was in good agreement with the data, but if he had been bolder he would have been the first to publish a relativistic quantum theory. For Dirac, this story had a moral: 'It is more important to have beauty in one's equations than to have them fit experiment.'\n\nDirac suggested to his readers that 'God is a mathematician of a  very high order, and He used very advanced mathematics in constructing the universe,' having apparently forgotten that he first encountered the God-beauty link forty years before in the writings of his colleague Sir James Jeans. In his positivist youth, Dirac would have regarded the link as unverifiable and therefore meaningless, but he had changed his tune: after spending decades on the _terra firma_ of experiment-based science, he was ready to take pleasure trips on the seas of metaphysical philosophy.\n\nThe physicist in Dirac now seemed to prefer the past to the present. Uncomfortable in the company of the leading young physicists, he was most at ease when he was reminiscing with his old friends. He missed none of the triennial meetings of Nobel Laureates at Lindau, a relaxed town in southern Germany, where he talked with physicists and, with rather more reserve, to the students invited to join them. _Horizon_ , the flagship science series of the new British television channel BBC2, made a film at the 1965 meeting, produced by Peter Lo\u00efzos. He saw that the two Nobelists most lionised by the students were Dirac and Heisenberg, who attracted swarms of admirers like Hollywood stars, and that, away from the m\u00eal\u00e9e, Dirac followed Heisenberg like a butler.\n\nLo\u00efzos knew it was not going to be easy to persuade Dirac to talk, as several BBC radio and television producers had asked him for interviews but had been turned down firmly. But Dirac agreed to be filmed in conversation with Heisenberg and the result is a unique recording of Dirac in relaxed conversation. Always with an agreeable smile, Heisenberg was as smartly dressed and easy-going as he had been thirty years before, but Dirac had changed rather more. His comically ill-combed hair helped to maintain his reputation for peerless dishevelment, but he was more relaxed than he had been as a young man, constantly smiling with his eyes and his mouth, speaking with a surprising assertiveness. Most striking about the encounter is that Dirac led the discussion, especially after he steered the subject towards beauty, via his anecdote about Schr\u00f6dinger's premature shelving of the relativistic version of his equation. When Heisenberg gently remarked that beauty is less important than agreement with experiment - the conventional view - Dirac took up the cudgels for aestheticism, forcing Heisenberg on to the defensive:\n\nHEISENBERG: I do agree that the beauty of an equation is a very important point and that one can get already a lot of confidence from the beauty of an equation. On the other hand, you have to check whether it fits or whether it doesn't. It's only physics when it really fits with nature. But that may turn out much later.\n\nDIRAC: And if it doesn't fit you'd hold up publication would you? Just like Schr\u00f6dinger?\n\nHEISENBERG: I'm not sure whether I would. In at least one case I have not done so.\n\nSmiling beatifically, Heisenberg appeared to concede the point: thirty years before, he would have persisted with the tenacity of a terrier, but his appetite for competition had been weakened by years of post-war humiliation. Delighted to have won the argument, Dirac's face lit up with the broadest of smiles, revealing two rows of rotting teeth.\n\nDirac still had faith in the large numbers hypothesis, though he knew most physicists regarded it as a blot on his CV after Edward Teller had published an apparently damning refutation of it in 1948. Teller pointed out that the hypothesis implied that because the universe is expanding, gravitational forces were greater millions of years ago than they are today. Teller showed that Dirac's idea implied that the Earth's oceans would have boiled and evaporated away 200-300 million years ago, contrary to the geological evidence that life had existed on the planet for at least 500 million years. Interest in the hypothesis had flickered again in 1957, when the American cosmologist Robert Dicke demonstrated that the large numbers hypothesis is a consequence of the fact that human life occurs after stars were formed and before they die. If the hypothesis were wrong, astronomers, and all other life forms, would not exist. Dirac was unimpressed with Dicke's reasoning and would not budge: he believed in the importance of the hypothesis 'more than ever'. In November 1961, Dirac wrote his first public comment on cosmology in twenty-two years:\n\nOn Dicke's assumption habitable planets could exist only for a limited period of time. With my assumption they could exist indefinitely in the future and life need never end. There is no decisive argument for deciding between these assumptions. I prefer the one that allows the possibility of endless life.\n\nDirac's vision of the fate of the universe was consonant with one of the articles of faith he wrote in his philosophical jottings of January 1933: 'the human race will continue to live for ever', a subjective assumption he had to make 'for his own peace of mind'. Evidently, this most detached of theoreticians could not bear to think of a universe without human beings.\n\nOne of the few cosmologists who still believed that it was worth spending time on Dirac's hypothesis was the vodka-swilling giant George Gamow. In 1965, he took a sabbatical in Cambridge, accompanied by his new wife Barbara, whom he had married shortly after his divorce from Rho in 1956 'on mental grounds'. The Gamows stayed at the new Churchill College, whose first Master, Sir John Cockcroft, had been chosen by the Prime Minister after whom it was named.\n\nOne topic of discussion between Dirac and Gamow was the beauty of the 'steady state' theory of the universe, which says that the universe has no beginning or end, but goes on for ever like a film with an endlessly repeated plot. That summer, this was a topical question because the steady-state theory seemed to have been discredited by one of the most telling astronomical observations to have been made in decades. Two astronomers at the Bell Laboratories in New Jersey had detected an all-pervading background bath of low-energy radiation. It was only after the astronomers made their observations that they heard that just such a bath of radiation had been predicted long before by Gamow and others, using the Big Bang theory. For most cosmologists, the theory afforded a beautifully simple description of the development of the universe, compatible with the general theory of relativity and all the other great theories of science. Fred Hoyle, who had given the Big Bang theory its name in 1949 during one of his BBC radio broadcasts, was the most vocal of the diminishing number who did not give up on the steady-state theory. Hoyle found the idea of the Big Bang distasteful and compared the notion of the universe emerging out of nothing to a 'party girl' jumping out of a cake: 'it just wasn't dignified or elegant'.\n\nAfter one of his discussions with Dirac, Gamow wrote to ask if he had heard of a tongue-in-cheek summary of the role of aesthetics that appears to have dated from their days in Copenhagen (Gamow uses the word 'elegant' where Dirac would use 'beautiful'):\n\nCase I Trivial statement\n\nIf an elegant theory agrees with experiment, there is nothing to worry about.\n\nCase II Heisenberg's postulate\n\nIf an elegant theory does not agree with experiment, the experiment must be wrong.\n\nCase III Bohr's amendment\n\nIf an inelegant theory disagrees with experiment, the case is not lost because [by] improving the theory one can make it agree with experiment.\n\nCase IV My opinion\n\nIf an inelegant theory agrees with experiment, the case is hopeless.\n\nDirac believed that if observations agree with an ugly theory - such as quantum electrodynamics - it is little more than a coincidence. He had a fundamentalist belief in beauty, as Heisenberg found when he produced a new theory of particle physics and pressed Dirac for 'specific criticism', only for Dirac to give the thumbs down to the theory because its basic equation had 'insufficient mathematical beauty'.\n\nKapitza was one of the few who understood Dirac's passion for beauty, perhaps because he had helped to foster it in their early conversations in the Cavendish and in Trinity College. Dirac may have feared that he would never again feel the thrill of Kapitza's company in Cambridge, but he heard in the spring of 1966 that both Kapitza and his wife had secured exit visas to enable them to return for a short stay. In late April, as the Kapitzas' arrival drew near, Dirac and Manci were like children on the eve of a royal visit, so excited that they could barely concentrate on the preparations.\n\nBy 1966, Kapitza was the Soviet Union's most famous scientist, in the address books of most of the country's leading artists and a licensed critic of the Government. The British Ambassador wrote in advance to Cockcroft to warn him that Kapitza was still 'a bit of a rebel' and suggested that 'the public relations aspect of the visit will require rather careful watching'. But the Ambassador need not have worried; Kapitza was on his best behaviour, having learnt from Rutherford how to balance irreverence and propriety so that he could be seen as both close to the establishment and fiercely independent. In his interviews he was always careful to stress that he had played no part in the development of nuclear weapons and that he was as patriotic as ever, as he demonstrated in his lecture 'The Training of the Young Scientist in the USSR' in the Hall of Trinity College.\n\nWhen the Kapitzas visited the Diracs for lunch, Manci made a special effort in the kitchen, poaching a salmon and serving it with home-made mayonnaise and a chilled Burgundy: Mary recalled that it was the closest her parents ever came to giving a banquet. For just that one afternoon, the front room had the warmth of a jacuzzi - their reminiscences darted around from the summer they spent in the Kapitzas' _dacha_ to their days in the Cavendish, with Kapitza telling wedding-night jokes so blue that Anna left the room, leaving Dirac and Manci to giggle their way to the punchline.\n\nThey will also have talked about Kapitza's Club, which had ceased to exist in the spring of 1958, superseded by programmes of seminars. The Club was, however, reconvened on 10 May for its 676th meeting, so that some of its surviving members - including Dirac and Cockcroft - could meet one last time and so that Kapitza could close it. The venue was a smart common room in Gonville and Caius College, where the participants sipped fine dessert wines, in contrast to the meetings forty years before, when they would drink dishwater coffee. A photograph of the occasion shows Kapitza and a forlorn-looking Dirac, his left elbow leaning on the table, his left hand supporting his head. He gives the impression of being bored out of his mind.\n\nThe highlight of the meeting was a joint presentation by Dirac and Kapitza on the effect they had identified in 1933, a year before Kapitza had been detained in the Soviet Union: the possibility that electrons could be bent (diffracted) by light. When they first predicted the effect, it was impossible to observe because the available sources of light were too weak and the electron-detectors were too insensitive. But now detection looked possible, following improvements to the sensitivity of the detectors and the invention of lasers, devices that had become familiar to the public since they featured in the 1964 James Bond film _Goldfinger._ The barrel-chested Kapitza, standing by a blackboard and easel, pointed out that it was now odds-on that experimenters would soon observe the effect; the question was: would Dirac and Kapitza be alive to see it?\n\nA few days after the Kapitzas left Cambridge, Dirac switched his attention from the past to the future. He attended an entire course of lectures on modern particle physics given by the American theoretician Murray Gell-Mann, a source of many of the most productive new ideas in particle physics since the early 1950s. Then thirty-six and still at the height of his powers, he was admired for his imagination and technical brilliance but feared for his waspish tongue and disliked for his egoism, not least by Dirac. In the 1960s, Gell-Mann and others suggested that strongly interacting particles could be classified in mathematical patterns, and he used one of them in 1963 to predict the existence of a new particle. When experimenters detected it in the following year, it was a signal success for theoretical physics. Gell-Mann and his colleague George Zweig, working independently, also proposed that strongly interacting particles might consist of different combinations of three varieties of a new type of fundamental particle that Gell-Mann called quarks (he took the word from James Joyce's _Finnegans Wake_ : 'Three quarks for muster mark!') But Gell-Mann himself was sceptical: he remarked in his lectures that quarks were probably not real particles but mathematical artefacts that help to explain the symmetries among the properties of the strongly interacting particles. A year later, Gell-Mann recalled that he was surprised that Dirac 'loved' quarks, despite their having - in Gell-Mann's opinion - 'many annoying properties', including their apparently permanent confinement inside strongly interacting particles, such as protons and neutrons. When Gell-Mann asked Dirac why he thought quarks are so 'marvellous', Dirac replied that they have the same spin as the electron, the muon and the neutrino. Perhaps Dirac had seen that it was possible that all fundamental constituents of matter have the same spin - the spin of the electron. And perhaps he had sensed that it might soon be possible to set out a description of strong interactions in terms of a field theory, as he had hoped.\n\nGell-Mann's lectures taught Dirac a lesson: the bottom-up way of doing theoretical physics - drawing inspiration from experimental observations - was proving much more productive than the top-down style - taking cues from beautiful mathematics - that Dirac practised and preached. Dirac privately admitted this, though he had no intention of changing his approach.\n\nIn mid-September 1967, the Diracs heard that Sir John Cockcroft, one of their closest friends, had died suddenly of a heart attack in the Master's Lodge of Churchill College. Several of his friends believed that his death had been hastened by his anxiety over a classic Cold War melodrama that had taken place two days before: Soviet Embassy officials abducted his colleague Vladimir Tkachenko - a student prot\u00e9g\u00e9 of Kapitza - on the Bayswater Road in London and  had whisked him off to Heathrow, where they put him on a plane bound for Moscow. But, just as his plane was setting off, it was surrounded by squad cars of airport police and MI5 agents, who boarded the plane and found him looking sick and bleary-eyed, apparently under sedation. They forcibly removed him, outraging Soviet authorities, who protested that he was leaving Britain of his own volition, having been blackmailed and intimidated by British agents. Cockcroft died on the morning after the incident became public, when the story was on the front page of _The Times._ 33\n\nHis wife Elizabeth knew she would soon have to leave the Lodge to make way for the next Master, and the College assisted her in making the move. In the opinion of the Cockcrofts' children, the authorities treated her sensitively and with a good deal of generosity, but Manci disagreed: she told everyone who would listen that the College was shooing Lady Cockcroft out of the Lodge with despicable haste. Manci's patience with Cambridge finally ran out, and she made up her mind that Dirac must move to an institution that behaved better towards its senior academics. She also vowed to take her revenge on Churchill College.\n\nDirac and Manci began making plans to settle in the USA. Some of its universities were certain to offer Dirac a professorship, and Mary and Monica, both married by the summer of 1968, now lived there. Manci's brother Eugene Wigner was also in the USA and was one of the elder statesmen of American science, an adviser to the Government, and - to Manci's irritation - moving politically further to the right each year. From his letters to the Diracs, it is plain that Wigner was a thoughtful and caring member of his family but, in the public eye, his humility had become something of an affectation: he was now so self-deprecating that many of his acquaintances thought he was using it as a subtle form of mockery. Ideally, the Diracs would have liked to have settled in Princeton, but that was no longer an option: after Oppenheimer's retirement in June 1966 - seven months before he died of throat cancer - the Institute for Advanced Study was unlikely to offer Dirac an academic home, nor could Princeton University be expected to accommodate a physicist so far past his best.\n\nTwo branches of Dirac's family remained in Europe. Betty was a contented housewife in Amsterdam, doing the chores to the soundtrack of the BBC Home Service (now Radio 4) and going regularly to  the highest Catholic mass she could find. In 1965, Gabriel was appointed to the mathematics faculty at the University of Swansea soon after the US Government rejected his application for a visa, apparently because of his brief membership of the Communist Party in Cambridge. Two years later, he and his family moved to the University of Aarhus in Denmark, and Dirac and Manci visited them during their summer vacations.\n\nOf all their children, Dirac and Manci were most concerned about Judy, who had lost custody of her children after an acrimonious divorce in 1965. Soon afterwards, she moved to Vermont and spent several lonely months each year in the Wigners' summer cottage on the shore of Lake Elmore. Wigner feared for her mental health. He wrote to Manci, telling her that Judy was desperate for her mother's affection and pleading with her to support her troubled daughter: 'You must not abandon her,' he told Manci in September 1965. Two and a half years later, Judy was holed up in a motel near Lake Elmore, lonely, penniless and delusional. She desperately needed psychiatric help, Wigner believed, and he begged his sister to intervene, but Manci told him that she would have nothing to do with Judy until she got a job and that he should stop interfering. Manci felt no responsibility for her daughter's plight, she wrote to Wigner:\n\nWhy should I in the name of heaven feel guilty? . . . I DID my duty, and who can throw a stone at me? J is an expert in hurting deeply, and may be she does this to those she loves. In that case she must seek a remedy.\n\nManci's indignation was suddenly punctured on 17 September 1968, when she read a telegram from her brother: 'JUDYS CAR FOUND ABANDONED DO YOU KNOW WHEREABOUTS LOVE.' This was the worst day of Manci's life, she later said. Manci had no idea where Judy was, as they were no longer in touch. In the following days, the Diracs heard nothing from Vermont or from the Wigners. Manci was distraught, lurching between wildly different accounts of Judy's disappearance, always refusing to believe that her depression had led her to take her life. It was most likely, Manci believed, that Judy had been murdered . Dirac's reactions to all this were known only to Manci, who appears to have shared them with no one.\n\nThe Diracs decided not to travel to Vermont but to stay in Britain and monitor events from there: they left it to the Wigners to deal with the authorities in Vermont. In early October, after visiting the site  where Judy's car was found - a country lane near Morrisville, Vermont - Wigner and his wife wrote to the Diracs with details of the police hunt for her in the surrounding countryside and ponds. The search parties found nothing. Gradually, the Wigners, tearful and depressed, came to believe that Judy would never be seen again, but the Diracs clung to every last hope. For three years, they tried to imagine scenarios in which Judy might suddenly reappear, but the weight of probability gradually crushed what remained of their optimism. They accepted that it was practically certain that Judy was dead .\n\nMary later recalled that her mother was inconsolable, 'insane with grief'. The Diracs kept the pain of their loss private, but two of his later acquaintances, the sculptor Helaine Blumenfeld and her husband Yorrick, the _Newsweek_ journalist, glimpsed deeper feelings. The Blumenfelds recall that, two years after Judy went missing, Dirac and Manci were still losing sleep over her fate and talked about it endlessly. From Dirac's comments about her, the Blumenfelds assumed that he was her biological father - he was as sad and bereft as if he had lost his own daughter.\n\nIn the early weeks of 1969, the Diracs were in Miami, pondering life after Cambridge. Of the American universities wanting to employ Dirac, one of the most tempting offers had been made by his former student Behram Kur\u015funo\u011flu at the University of Miami. A wheeler-dealer Turkish theoretician - always smart in his Stetson hat, jacket and tie - Kur\u015funo\u011flu had spent his career searching for a unified theory of fundamental interactions, following Einstein's agenda. Kur\u015funo\u011flu had founded the annual Coral Gables conferences, which gave several leading theorists a good reason to leave their home cities in the depths of January and spend a few days in the bright, warm sun of south Florida. Kur\u015funo\u011flu employed Dirac at the university on a temporary contract and tried hard to persuade him to accept a permanent post, making him and Manci as welcome as family, taking them out on trips round the area and giving Dirac a taste for coconuts, alligators and exotic birds. Manci was embarrassed by the time Dirac took to weigh Kur\u015funo\u011flu's offer, but he was not to be hurried - he disliked Miami's oppressive heat and felt uncomfortable in a place where recreational walkers are regarded as perverse.\n\nThe most memorable of Kur\u015funo\u011flu's outings was a trip to the cinema on New Year's Day. Kur\u015funo\u011flu and his wife asked Dirac to go with them to see Stanley Kubrick's _2001: A Space Odyssey._ The film had divided critics and audiences since its release eight months before: it inspired Steven Spielberg and a new generation of film-makers, but it left John Updike's Rabbit Angstrom bemused and sent his wife to sleep. Firmly on Spielberg's side, Dirac was enraptured: he had seen hundreds of movies, but had never imagined it was possible for a film to have such a powerful impact and enable him 'to see his dreams', as he told Mary's husband Tony Colleraine. Dirac disliked opaque and open-ended narratives, so his love of _2001_ was not predictable. It is easy, however, to imagine him being moved by Kubrick's use of Johann Strauss's 'Blue Danube' and the rest of the classical soundtrack and by the appeal of a story told mainly through visual images rather than words. Dirac's opinion that a good deal of quantum mechanics can be expressed accurately only through mathematics, not words, is echoed by a comment Kubrick made about _2001_ : 'I don't like to talk about [it] much, because it's essentially a non-verbal experience.'\n\nStill excited two days later, Dirac saw the film again at a matinee with Tony Colleraine and also with Manci and Mary, who spent most of the two and a half hours in the theatre whispering to each other. Dirac suggested to Tony that they see it again 'without the running commentary'. Without telling Manci, they stayed to watch the next two screenings and returned home to find their hot dinner left to get cold on the table. But Dirac was too excited to care about food: he was like a child after three consecutive rides on a roller coaster. Several of the scenes had possessed him, especially the Star Gate sequence and the emergence of the grizzled astronaut into the eighteenth-century bedroom: 'I would not be able to sit alone through that scene,' he later told Colleraine. Manci was not interested in Dirac's observations on 'that weird film'; her idea of a good movie was the romantic epic _Dr Zhivago_ , not one whose most memorable character was a talking computer.\n\n_2001_ stoked Dirac's interest in the Apollo space programme. During the evening of 20 July 1969, he sat open-mouthed in front of the television in the Kur\u015funo\u011flus' front room when Neil Armstrong prepared to set foot on the moon. He sat up all night watching the coverage. Kubrick's images were sharper and his soundtrack was  clearer, but the grainy television pictures and the muffled sound of that first moon landing had a compelling reality of their own. And for Dirac, the former engineer, reality mattered most: the first moon-walk was the culmination of aeronautic technology, whose beginnings he had seen as a boy and which now enabled human beings to set foot on a landscape a quarter of a million miles away. The Apollo team, having achieved the most impressive technological feat Dirac had seen in his lifetime, may well have given him a twinge of regret that he had chosen science rather than engineering: he had been a leader of a scientific revolution that, in his opinion, had led to a dead end, whereas the Apollo engineers could declare 'Mission Accomplished' and move on.\n\nIn the summer of 1969, Dirac prepared to leave his post and say his goodbyes to the few friends left in Cambridge, including Charlie Broad, the philosopher who gave him his first proper introduction to the theory of relativity. Broad, aged eighty-one, still lived in Trinity College, where he died two years later.\n\nOn Tuesday 30 September, Dirac spent his final day in Cambridge as its Lucasian Professor, the most distinguished holder of the Chair since Sir Isaac Newton. Dirac's retirement passed without ceremony, probably because the university authorities assumed that Dirac would feel uncomfortable if he was the cynosure of a leaving party. This was an error, though an understandable one: Dirac would have liked his contribution to the university to be marked officially as his sense of propriety was, contrary to the impression he gave, stronger than his aversion to ceremony. Manci was disgusted. But she was gratified by the sensitivity of St John's College, which extended Dirac's fellowship for life so that he could return there whenever he wished. Batchelor wanted to be generous, too, and offered Dirac the use of a room in the department whenever he was passing through the town, but he declined. His true home in the university was his college, not his department.\n\nFor two years, the Diracs divided their time between the UK and the United States, and, by March 1971, Manci could hardly wait another day to leave Britain, 'that lazy impossible island'. Labour unrest, steadily increasing since the war, had become critical: in the first year of Edward Heath's government, more working days had been lost to  withdrawals of labour than in any year since the General Strike. Postal workers had gone on strike and slowed down communications in the country for seven weeks. Even Rolls Royce had gone bankrupt.\n\nThe Diracs were about to move to a country that was no less troubled. The USA's prosecution of the war in Vietnam was as controversial in the extended Wigner family as it was in thousands of others: the doveish Manci seethed over 'young American lives mutilated fighting for a bastard government' and argued with her hawkish brother Eugene, who believed that the war was essential to stem the spread of Communism. She did not know that the FBI had opened a file on her and was seeking evidence that she was a subversive.  Dirac knew that his past political sympathies would raise eyebrows in some American institutions, as he noted when he declined an invitation to the University of Texas at Austin because he was technically ineligible: 'I do not have strong political views, but [. . .] I am a member of the Soviet Academy of Sciences and this makes me, according to [the university's] definition, a member of the communist party.'\n\nWhenever he left the USA in the late 1960s and 1970s, he was nervous that the authorities might forbid him to re-enter it. As he probably suspected, the FBI was still watching him .\n\nDirac, a dissenter from America's foreign policy in south-east Asia, followed the fierce opposition to the war in American universities through the newspapers and the television news. Although Miami University was one of the less volatile campuses, its students harassed the authorities almost every day, condemning the Vietnam War, demanding free contraception and more support for civil rights. The protestors would talk only to the university's President, Henry King Stanford, who stood on 'The Rock' - a stage-like stone structure in the centre of the campus - making conciliatory speeches to the students and trying to avoid further trouble. On the periphery of these crowds, Stanford often saw the slender, inquisitive figure of Dirac.\n\nOn Wednesday, 6 May 1971, the students were especially angry. It was two days after the Ohio State Guard had opened fire on student demonstrators at Kent State University, during a protest triggered by the American invasion of Cambodia. Thirteen seconds of gunfire had killed four students, wounded nine others and brutally curtailed the flower-power hedonism that had flourished only briefly since the  Sgt Pepper summer of love in 1967. The mood of America turned ugly. Even the usually sober campus of Princeton University was unstable: Wigner thought many of the students were 'selfish and nihilistic', behaving 'like the Hitler Youth'. Miami University teetered on the edge of anarchy, when its students - supported by many staff - began a four-day strike, joining two hundred and fifty other campuses across the country. After lunch, at the beginning of a warm afternoon, Stanford made his way to The Rock to address a volatile rally of over a thousand students, many of them with their arms folded aggressively or holding banners with messages such as 'U$ out of S.E. Asia'. Earlier, the crowd had made an effigy of President Nixon out of newspapers, old clothes and firecrackers and then set fire to it. Dirac had seen nothing remotely like it since the Cambridge demonstrations in the 1930s.\n\nDuring Stanford's walk towards the crowd, he saw an elderly man on the periphery and was quite taken aback to be approached by him. It was Dirac, who asked gently, 'Are you afraid?' Stanford, his heart pumping hard in his chest, replied, his tongue firmly in his cheek, that he was quite looking forward to addressing the students. It seems that Dirac saw that the President was anxious and could use a little reassurance, as he took what was, for him, the unusual step of offering him advice: 'Tell them what you think and listen to what they have to say.' The tone of Dirac's voice gave the impression that he had a 'spiritual kinship' with the protestors, as Stanford later wrote, perhaps identifying a faint echo from the days when Dirac was on the fringe of left-of-centre radicalism. In his emollient address, Stanford described the Kent State incident as 'One of the saddest chapters in the history of higher education', adding that the students' deaths 'dramatise the deterioration of reason' in the USA. Shortly after the speech, the protest ended peacefully, though the university remained on edge for weeks. Dirac probably wondered what future lay ahead of him.\n\nA few weeks later, the Diracs took a break and drove up to Florida's state capital of Tallahassee. Compared with tense, crime-ridden Miami, it was as friendly and safe as a village. Dirac knew that he was being wooed by Florida State, known best not for its physics department but for its student parties and the high quality of its football team. Joe Lannutti, the physics department's ambitious leader, saw an opportunity to persuade the dithering Dirac to become  a 'professor at large' at the university, a mascot for the physics department's aspiration to be a 'centre of excellence'. Lannutti had already invited the Diracs to Tallahassee in March 1969, when the Holiday Inn welcomed them with banners fluttering over the entrance, and the physics department had given tenure to Mary's husband Tony a few months later. For the Diracs, the prospect of spending their final years near Mary was attractive, and the warm climate would be good for the worsening arthritis in Manci's hands, but Dirac wanted to delay his decision until he could see how he coped with the fiercest of Tallahassee's heat and humidity and with the barking dogs that ruined his walks. Swimming was now his favourite form of exercise so, in his spare hours, he visited the local lakes and sinkholes, usually taking a thermometer to check the temperature of the water. If it was above precisely sixty degrees Fahrenheit, he would dive in; if not, he would return home.\n\nIn early January 1971, Florida State University formally offered Dirac the post of Visiting Eminent Professor, to be renewed annually.  The FBI had found no evidence that either Manci or Dirac was a subversive, so there was to be no official barrier to their emigration. After reflecting on the offer for five months, Dirac accepted and shortly afterwards returned briefly to Cambridge with Manci to pack up their belongings. During one of their conversations with the Blumenfelds, Helaine asked Dirac whether he was excited about moving to Tallahassee; he replied, gesturing to Manci, 'She is, that's why we're going. I would like to stay here.'\n**Twenty-eight**\n\nOld men have a weakness for generality and a desire to see structures whole. That is why old scientists so often become philosophers [. . .].\n\nEUGENE WIGNER, _The Recollections of Eugene P. Wigner,_ 1992\n\nThe advice Barbara Walters, doyenne of celebrity interviewers, gave in her 1971 book _How to Talk to Practically Anybody about Practically Anything_ did not quite extend to making conversation with Dirac. Yet the Director of Publicity at the Miami Museum of Science, Dorothy Holcomb, wished she had read the book when she was trying to wrest a few words from him during a buffet reception in his honour on the evening of 8 March 1971. After he replied to her 'Hi!' with a blank 'Hello,' she realised that the only way to get him to speak more than a few words at a time was to ask him to pick the topic of conversation. He chose comic strips. For several minutes, he talked with surprising fluency about the merits of two strips he had been reading since the 1930s: the fifth-century adventurer 'Prince Valiant' and 'Blondie', a carefree flapper girl who settled down to family life in suburbia. Holcomb was charmed. When Dirac admitted that he could not make head or tail of the quirkier humour of 'Peanuts', she suggested he should try a little harder to understand American humour; he agreed. Afterwards, Holcomb made up her mind to buy a copy of _The Principles of Quantum Mechanics_ and also of _How to Talk to Practically Anybody about Practically Anything._ As Holcomb will have seen, if she got to the end of Walters' book, it concludes with good advice for everyone who had tried vainly to draw Dirac into conversation: 'You can't win 'em all.'\n\nBefore this conversation, Dirac had given a lecture entitled 'Evolution of Our Understanding of Nature', which ranged well beyond physics. Still haunted by the early scenes in _2001: A Space Odyssey_ , he began by discussing how early humans understood the mechanics of growing grain, graduating from beliefs based on superstition to ideas based on theories grounded in observations. He opposed critics of the Apollo space programme who believed that the money should be spent instead on social programmes: 'People who  equate all the different kinds of human activity to money are taking too primitive a view of things.' The solution to social problems was not, he argued, to be cheese-paring with the space programme and fundamental research but to avoid 'the great waste that we see around us', especially the unemployment of people who want to work. Look at the hippies in California, he said: they welcome the challenge to help fight forest fires rather than just laze around.\n\nDirac's reputation as a speaker enabled him and Manci to sate their appetite for international tourism. Florida State gave him the freedom to travel and everything else he needed, in addition to a modest income: an office, companionship, financial support for his research and - most important - respect. The university officials treated him with a reverence that often cloyed into obsequiousness, and they regarded Manci as his queen. She whiled away hours chatting and exchanging risqu\u00e9 jokes with the university's clubbable President, Bernie Sliger, knowing that he would always take her phone calls and be sympathetic to her every request. In return, the university asked only that Dirac be available when they wanted to display their most illustrious professor to visiting dignitaries; he played along and had some success in disguising his boredom. Only once, when his compliance was taken for granted, did his patience run out: he locked himself in his house and Kurt Hofer had to persuade him to come out, just in time to meet an important visitor.\n\nBeyond the light supervision of a few graduate students, Dirac had no teaching responsibilities. But in 1973, he agreed to present a series of lectures on the general theory of relativity, aiming to develop the theory from its fundamental principles and to lay bare its logical structure. One of the physics students in the audience, Pam Houm\u00e8re, recalls:\n\nThe first lecture was 'standing room only'. He began so simply that the office cleaners could have understood it: what is meant by position, what we mean by time, and so on. Later, he built on these foundations brick by brick, making every step of the construction look inevitable. The funny thing was, he never compared the theory with experiment, he just kept stressing how beautiful it was. Only a few students made it to the end of the course, but for those who did, it was an unforgettable experience.\n\nDirac presented the lectures most years until 1980 and used them as the basis of his short book _General Theory of Relativity_ , a minor  classic of exposition, describing the theory in sixty-nine pages, without a single diagram.\n\nIn Tallahassee, the Diracs' home was about twenty minutes' leisurely walk from Dirac's office on the third floor of the university's Keen Building, in the heart of the campus. Each weekday morning after breakfast, he would link his hands behind his back and walk slowly to his office across a local field, the route that ensured minimum contact with the neighbourhood dogs. In summer, when he wore his baseball cap, he looked like an all-American retiree, but on the coldest days of winter, when he put on the heavy overcoat he had bought almost fifty years before in Lord and Taylor, he looked every inch the venerable English professor. He often carried a forty-year-old umbrella: 'It was my father's,' he told colleagues.\n\nIn his office, he worked at his desk for three hours, pausing occasionally to visit the library. To unexpected visitors who knocked on his office door, he had a simple message: 'Go away.' When the phone rang, he would often lift the receiver off the hook and immediately drop it, without bothering to listen to the caller's voice. At noon, he would join a few colleagues for a brown-bag lunch. Dirac usually said nothing but would occasionally interject with a comment, perhaps on the impenetrability of American football or about the wisdom of trying to educate so many undergraduates in science when so few of them had an aptitude for the subject or even took much pleasure from studying it. He was fond of jokes, especially ones dependent on the interpretation of a single word and ones with a slight sexual edge. This was one of his favourites:\n\nIn a small village, a newly appointed priest decided to call on his parishioners. In one modest home, teeming with children, he was greeted by the lady of the house. He asked her how many children she and her husband had. 'Ten,' she replied. 'Five pairs of twins.' The priest asked, 'You always had twins?' to which the woman replied, 'No, Father, sometimes we had nothing.'\n\nAfter lunch, he would return to his office for a nap on his sofa and sometimes attend a seminar, often appearing to sleep through most of it, before returning home for late afternoon tea with Manci. After dinner, he would relax. He and Manci might go to a classical concert, or he might read a novel - Edgar Allen Poe mysteries, Le Carr\u00e9 spy thrillers and Hoyle's science-fiction stories were among his favourites  \\- or watch television with Manci in the family room, dominated by a painting of Judy when she was a child. Dirac watched most of the _Nova_ science documentaries, but the programmes that he and Manci regarded as unmissable were period dramas: _The Forsyte Saga_ \\- Dirac was spellbound by the leading lady, Nyree Dawn Porter - and _Upstairs, Downstairs_ , dramatising the class divisions between the servants and their masters in an Edwardian household. On the night an episode of the programme was broadcast, the Diracs would accept dinner invitations with friends only if their hosts agreed in advance to watch it with them in silence. One dispute about the evening television schedule threatened to get out of hand, when there was a clash between Cher's Sunday-night television show - a highlight of Dirac's week - and the live broadcast of the Oscar ceremony, which Manci was desperate to see. The dispute was resolved several days later, but at a price: they bought a second television.\n\nThe couple did not always resolve their differences so amicably. In August 1972, they had what may have been the worst row of their marriage, when they were visiting the recently widowed Betty at her apartment in Alicante, on the south-east coast of Spain. The relationship between the sisters-in-law had long been brittle: part of the problem was that Manci made no secret that she found Betty dull and idle, while Betty was vexed by Manci's unrelenting bossiness. Tempers flared during a conversation on the apartment balcony when Dirac backed up his sister after she made a sly comment about the behaviour of Hungarians in Budapest at the end of the war. Manci stormed out of town and wrote to Dirac in a rage:\n\nYou looked at me, then did all you could to hurt, scare & humiliate me, & embarrass me greatly [. . .] It is a fact that most mental inmates have been driven there by their families. On that 5th floor balcony I felt your presence whenever I was there alone, urging me to jump [. . .] You cruelly, unjustly uncaringly completely identified yourself with my tormentor, and this I did not earn or deserve. I do not feel you are a husband as it is understood by millions. Yes, keep your loyalties to the one so similar to you in lacking human emotions, & I learn not to care or want to die.\n\nA few days later, she wrote to him again, in a rather different tone:\n\nThank you for your loving care. For your love, warm & affectionate. For your taking notice when sick or in pain. For heeding for needs I have. For allowing me to read your wishes from unspoken words. For allowing me  near you when ill or depressed. For forgiving my ills and extravagances. For never making me anxious and panicky. For treating me as an equal: always justly & fairly. For trying your best to make us around you happy and cheerful. I thank you.\n\nIn Trieste a month later, at a symposium organised by Abdus Salam to mark Dirac's seventieth birthday, Heisenberg and all the other guests saw the Diracs on their best form, the model of the contented elderly couple. But Dirac apparently did not want to put the unpleasantness of the previous few weeks completely behind him: he clipped Manci's two notes together and filed them among the papers in his office. He appeared to regard all her attacks - and the makings-up that always followed - with an equanimity bordering on indifference; whether he suffered more deeply than others saw we shall probably never know, as he appears not to have discussed her behaviour - still less to have complained about her - with anyone.\n\nTo the Diracs' acquaintances in their later years, Manci was a controversial figure. No one questioned that her gift for friendship hugely enriched his social life and that she was devoted to her husband, 'my little Mickey Mouse'. Many colleagues attest to the care she took to look after him and make him look presentable; one visitor was touched to see her adjusting his clothes when he returned home one evening looking like a scarecrow. 'She takes such _good_ care of me,' Dirac beamed as Manci adjusted his tie. Without her, he would probably have spent almost his entire adult life living alone in college, like Charlie Broad.\n\nYet many friends could not help flinching when she shouted at him, 'Are you listening to me?' and wondered how he felt when he silently bore her tirades against 'nigger' doctors and Jews (that Manci was both Jewish and occasionally anti-Semitic was one of the most baffling paradoxes of her personality). Yorrick Blumenfeld gives a bleak summary of the state of their thirty-four-year-old marriage: 'She was tired of hen-pecking him, and he just wanted to live in his dream world.' Helaine Blumenfeld is surprised that he could tolerate her: 'He was a lovely man. She was simply an awful person.' But Lily Harish-Chandra, a frequent visitor to the Diracs' home and a family friend, disagrees: 'Manci was extremely warm and loyal, a great listener and a very caring woman. Paul cannot have been easy to live with. Their marriage worked because they gave each other what they wanted: he gave her status and she gave him a life.'\n\nIn the early 1970s, Dirac was briefly optimistic about his research on particle physics. He had happened on a way of describing isolated elementary particles with a spin equal to a whole number, using an equation that he believed had a special mathematical beauty. Better yet, it described only _positive_ energies - the mathematics yielded no embarrassing negative-energy solutions. But his excitement waned after he found it impossible to use the equation to describe how a particle interacts with other particles or with a field - the real-world case. Mathematical beauty had again proved a treacherous beacon.\n\nDirac then wound down his work on the theory of fundamental particles and returned to general relativity and his still-unproven large numbers hypothesis. He knew that Einstein's theory and the hypothesis were incompatible because general relativity requires - in the language of Newtonian mechanics - that the strength of the gravitational force between two identical masses separated by the same distance has always had the same value, contrary to the hypothesis. So he tried to reconcile them using ideas set out by a former colleague at the Institute for Advanced Study, the German mathematician Hermann Weyl, whose approach to theoretical physics resembled Dirac's. Weyl once said: 'My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful.' In 1922, Weyl had produced a prototype theory that gave a tantalising glimpse of how a mathematical account of gravity and electromagnetism might be given with a unified set of equations. Enthralled by their beauty, Dirac believed Weyl's approach might furnish a link between the general theory of relativity and the large numbers hypothesis, in a way that involved a gradual weakening of gravity over time.\n\nDirac was assisted in the project by Leopold Halpern, a general relativity specialist who arrived in Tallahassee in 1974, a year short of his fiftieth birthday. Born and raised in Austria, he and his family had fled, on Hitler's invasion in 1938, when he was thirteen years old. He spent twenty-seven years working in several European research institutions, including a spell with Schr\u00f6dinger, and Dirac had first met him at a conference in 1962. Halpern was a homoeopath and a certified African medicine man, a twenty-four-carat eccentric who slept outdoors all year round, sliced baked potatoes with karate chops and refused to wash with soap. He was not  always popular in elevators. Colleagues with conventional manners were often disconcerted by the prickliness that disguised his shyness: when his phone rang, he would answer with a rasping, impatient 'Hello', his voice softening into a lilt the moment he realised that he was talking to a friend.\n\nThe oddities and coarse manners of Halpern grated on Manci, but they endeared him to Dirac, and the two men became close friends. At least once a week, they went swimming in Silver Lake and Lost Lake, two of Dirac's favourite spots near Tallahassee, mainly because the waters there were so quiet. Dirac did not like to swim anywhere near motorboats, but on one trip, when he was seventy-six years old, he hailed one and asked the owner if he could have a go at water-skiing. The owner obliged. When Halpern told Manci, she was horrified: 'Paul is still _very_ immature!'\n\nMost weekends, the two men headed off in Halpern's Volkswagen Super Beetle - his sixteen-foot canoe and a pair of paddles tied to the roof rack - on the hour-long drive to the Wakulla river. Minutes after setting off from the shore, they were alone in one of Florida's most pleasant microclimates, a near wilderness. They would row for some two hours upstream on the slow-flowing river, through forests of sassafras and American beech trees, draped with Spanish moss. The alligators made scarcely a sound: the silence was broken only by the rhythmic sloshing of the paddles, the cry of a circling osprey, the occasional shuffling of wind passing through shoreline gaps in the forest. After a snack lunch at Snake Point, Dirac and Halpern would strip off and go for a swim, before they rowed back to their starting point, scarcely exchanging a word. These were idyllic, private hours. Occasionally, they would invite a visitor to join them - but it had to be someone who could be relied on to stay silent most of the time. One of the visitors was Kurs\u015funo\u011flu, who went along dressed in his three-piece suit, tie and Stetson. Halfway through the trip, he stood up in the canoe to admire the scenery only for Dirac to dump him in the river and then collapse in a fit of laughter.\n\nDirac and Halpern often arrived home several hours late, half-heartedly suppressing shame - like a pair of errant schoolboys - when they were explaining themselves to a frantic Manci. Halpern assured her week after week that the Wakulla wildlife posed no danger at all: 'If you leave the snakes and alligators alone, they will do  nothing to harm you.' Halpern could not understand what she was so worried about.\n\nIn the 1970s, particle physics underwent what amounted to a revolution. After decades of uncertainty, physicists achieved a new clarity about the workings of the universe at the finest level: everything in the universe is made of a few basic particles - a handful of leptons and quarks and a small number of particles that mediate their interactions - and described by a quantum field theory simple enough to be spelt out on a T-shirt. The Dirac equation describes the electromagnetic interactions of all the leptons and quarks, each with the same spin as the electron.\n\nIn the past fifty years, physicists had come up with quite a few attention-grabbing labels for their new concepts, but they allowed this description of weak, electromagnetic and strong interactions - one of the supreme syntheses of twentieth-century thought - to be given the most prosaic of names: the Standard Model. One of the first important steps towards the consensus was taken by Dirac's former student Abdus Salam and by the American theorist Steven Weinberg, who independently suggested in 1967 that the weak and electromagnetic interactions might be understood in a unified way, by describing them in terms of a special type of gauge theory whose underlying mathematical symmetry is broken. For several years, the Weinberg-Salam theory was not taken seriously as it appeared to suffer an even more serious infestation of unwanted infinities than quantum electrodynamics, the theory of photons and electrons. All this changed in the early 1970s. After the Dutch theoreticians Gerard 't Hooft and Martin Veltman proved that the infinities in the theory - and in all other gauge theories - could be removed by renormalisation, the Weinberg-Salam theory quickly commanded wide interest and support. Also at around this time, theorists improved their understanding of renormalisation so that it was much more rigorous than the 'sweeping under the carpet' dodge that Dirac deplored. Renormalisation was now widely accepted as a rigorous branch of mathematical physics, with no sleights of hand; Dirac vehemently disagreed.\n\nSoon physicists found a gauge theory of strong interactions, called quantum chromodynamics, with the same underpinnings as the Weinberg-Salam theory. It turned out that it was possible to describe  the strong interaction between quarks, mediated by massless particles which Gell-Mann named gluons. Quarks are never observed in isolation, the theory says, because the strong force prevents them being separated, though when quarks are close together they behave as if they were free. So the neutron, first observed by Chadwick just over thirty years before, could be re-envisaged as a compassionate prison for quarks - they cannot escape their confinement, but they are free when inside.\n\nRutherford's vision of a typical atom as electrons orbiting a tiny nucleus of protons and neutrons ('a gnat in the Albert Hall') had been superseded. Now, the most fundamental way of imagining an atom was in terms of relativistic quantum field theory: the quarks in the nucleus were quantum excitations of the field associated with the strong interaction, just as the orbiting electrons were the quantum excitations of the electron field. Everything in an atom can be described in terms of such fields. Rutherford would have choked on such abstractions, yet they were the apparently inevitable consequence of a century of labour by his fellow experimenters and their theoretical colleagues.\n\nAlthough the Standard Model left many questions unanswered - no one fully understood the particles' masses, for example - its setting out in the 1970s was a high point in the history of science. But Dirac was unmoved: ensconced with Halpern in their Tallahassee redoubt, the new discoveries left him cold, and he appeared to take no great pleasure to see other theoreticians find a way of describing strong interactions using field theory, which he had pioneered, as scattering matrices fell into disuse. He no longer kept up to date with the latest physics journals and was beginning to make errors in his science, though no one was ungracious enough to say so in public. By the mid-1970s, Dirac had lost interest in particle physics, and Halpern noticed that he was less interested in news about field theory than the renewed public debate about the origin of the Turin Shroud, believed by some to be the burial shroud of Jesus Christ.\n\nAlthough Dirac was impressed by the best young particle theoreticians, he thought they were deluded. Through his talks and occasional publications, he urged them to devote all their time to clearing and disinfecting the Augean stables of renormalisation, a job almost all physicists believed had already been done. By contrast, Heisenberg in Munich kept an open mind about new theoretical  developments until liver cancer took his life in February 1976, six years after his former teacher and friend Max Born had died in G\u00f6ttingen. All Dirac's friends among the pioneers of quantum mechanics were now dead.\n\nAt one time, the historical perspective on atomic physics was not important to him, but now he was keen to put his side of the story to historians and other physicists. In these talks, he always took pains to emphasise the excitement of the early years of quantum mechanics - an emotion that, by all accounts, he rarely showed when he was living through them. He even included a reference to his feelings in the account that was the nearest he ever came to writing a scientific memoir: _Recollections of an Exciting Era._ 31\n\nIn May 1980, while suffering from a bad dose of flu, Dirac travelled to Chicago to attend a conference on the history of particle physics at the Fermi National Accelerator Laboratory (Fermilab), where he spoke about the origins of quantum field theory. In a round-table session, he went out of his way to criticise the destructiveness of Pauli's opposition when the idea of spin and the positron were first conceived. In another session, he presented his versions of the history of anti-matter in a talk that Leon Lederman recalled as 'quintessential Dirac' - clear, fluent and modest: 'the content poured out of him like heavy cream'. When he had finished speaking, Vicki Weisskopf commented that Einstein had suggested the existence of a positive electron in 1925, some six years before Dirac's prediction.  But Dirac was unperturbed; he waved a hand dismissively, remarked, 'He was lucky,' and moved on. Even for Dirac, modesty had its limits.\n\nManci was a generous hostess, able to make everyone in the room feel special and at ease. She often threw dinner parties, attentively filling her guests' glasses, serving generous portions of her favourite dishes, ensuring that the conversation kept moving. Dirac, sitting at the head of the table, would apparently spend most of the evening asleep. He could, however, be drawn into conversation if he were approached by a young woman, especially if she was friendly and attractive. His advice was often sought but he usually declined to offer any; however, when pressed, he would sometimes offer a few words. One of his favourite replies was: 'Think about yourself first. If nobody gets hurt, do it' - a slightly egoistic summary of the view  of the individual's moral responsibility in the opinion of John Stuart Mill.\n\nManci would point out to guests a favourite photograph of Dirac warmly shaking hands with Pope John Paul II in the Vatican. 'Paul and the Pope get on so well,' Manci would beam, as if the two men met every weekend for a round of golf. The photograph was taken at one of several meetings between Dirac and the Pope at the Papal Academy, a group of distinguished scientists that offers the Pope disinterested scientific advice. Dirac had been elected to the Academy in 1961, the year after his friend the cosmologist George Lema\u00eetre became President. The Diracs' friend Kurt Hofer recalls Manci's pride in her husband: 'After showing guests the papal photograph, she unpacked a collection of postal stamps from all over the world, each bearing a portrait of Paul. He pretended to be embarrassed, but he never did anything to prevent her.'\n\nIt was during one of Hofer's weekly visits to 223 Chapel Drive that Dirac unexpectedly disgorged his recollections of his father. Dirac trusted only his closest friends with these unexpurgated memories, although the circumstances of Felix's death were still too distressing for him to share with anyone, even with Manci. Dirac did, however, speak of his happiest memories of Felix's life to Betty in October 1969 when she was in an Amsterdam hospital, lying in a coma after a stroke and a seven-hour brain operation. Alone at her bedside, he tried to coax her back into consciousness by telling her stories of their childhood - playing on the Downs with Felix, the three of them bathing on Portishead beach, sharing each other's books and comics. She regained consciousness a few weeks later and gradually made a partial recovery.\n\nHofer recalls that Dirac thought organised religions were primitive and socially manipulative 'myths'. Once, as he walked past a local Mormon church with a huge satellite dish, he scoffed that the church needed such a large dish 'so that it can communicate directly with God'. Yet Dirac was now much more willing to introduce the concept of God into discussions about science. In June 1971, he had startled his audience at the Lindau meeting by considering 'Is there a God?' to be one of the five most important questions in contemporary physics. He said it would be useful to approach the question scientifically:\n\nA physicist would need to make this question precise by understanding what is meant by a universe with a God and what is a universe without a God, having a clear distinction between the two types of universes, and then looking at the actual universe and seeing which class it belongs to.\n\nThe audience laughed nervously and went quiet when he suggested a way of detecting the presence of a God. If future scientists demonstrated that the creation of life is overwhelmingly unlikely, then, in his opinion, this would be evidence for the existence of God. Until that time, the hypothesis must be regarded as unproven. Dirac was taken to task by the press for these speculations but he was not to be deflected and often returned to the topic, in public and private. He took a dim view of any religion declaring itself to offer the only hope of salvation, Hofer remembers: 'Paul believed it was the height of arrogance for any group of people to claim that they alone know the truth. He often pointed out there are hundreds of religions on this planet and that it is impossible to know which one, if any, is correct.'\n\nThere was 'no trace of religiosity' in Dirac, Halpern later wrote. He remembered that Dirac was especially critical of Catholicism and other religions that acknowledged miracles, because, in his view, the existence of a miracle implies a temporary breaking of the underlying laws of nature, whose beauty he regarded as sacred. Like Einstein, and largely following the philosopher Spinoza, Dirac appeared to take the pantheist view that the universe is either identical with God or in some way an expression of God's nature, a view that - though vague almost to the point of tautology - appears to rule out the notion of a God that can influence human affairs. Dirac's pantheism was an aesthetic faith: that observations on nature at the most fundamental level will be described perfectly by theories whose mathematical beauty is also perfect. If he had a religion, this was it.\n\nDirac's modesty was genuine, but he was not above a little vanity. The Danish sculptor Harald Isenstein, a specialist at portraying leading physicists, made two busts of Dirac, and both are good likenesses, if lacking in character: the first in 1939, which Dirac displayed in his home, the next thirty-two years later. He offered the first Isenstein bust to St John's College, who accepted it and displayed it in their library, where it stands today. The college also  wanted a painting of Dirac in oils to be displayed in their Hall, and Dirac went out of his way to oblige. In the early summer of 1978, he sat several times for Michael Noakes, portrait painter of the British royal family and, the year before, of Frank Sinatra. In the first session, Noakes tried to help Dirac relax by drawing him into conversation:\n\nNOAKES: Can you put into layman's terms what you're working on, Professor?\n\nDIRAC: Yes. Creation.\n\nNOAKES: Wow! Tell me more.\n\nDIRAC: Creation was one vast bang. Talk of a steady state is nonsense.\n\nNOAKES: But if nothing existed beforehand what was there to bang?\n\nDIRAC: That is not a meaningful question.\n\nDirac would say no more. Though unsettled by Dirac's reticence and apparent lack of interest, Noakes captured his abstracted gaze to infinity, Dirac looking as innocent as a five-year-old, as detached as an oracle. A comparison between this portrait and the first to be painted - by his friend Yakov Frenkel in 1933, shortly after they heard of Ehrenfest's suicide - shows how much Dirac's confidence had drained away in the ensuing forty-five years. His personality is perhaps best caught in the drawing made in 1963 by Robert Tollast, whose portrait expertly catches Dirac's childlike innocence. Less accomplished, but nevertheless competent, is the drawing of Dirac made two years later by Feynman, whose portrait shows signs of reverence ('I'm no Dirac,' Feynman often said). Dirac kept his drawing in his filing cabinet.\n\nTwenty years after Dirac declined a knighthood, he accepted the most prestigious honour of all, membership of the Order of Merit, which did not oblige him to call himself anything other than 'Mr Dirac'. The order is limited to twenty-four members of the British Commonwealth judged by the sovereign to have given exceptional service (previous members had included Florence Nightingale, Winston Churchill and William Walton). Manci deplored that her husband was the last of his generation of Cambridge scientists to be honoured - J. J. Thomson, Eddington, Rutherford, Cockcroft and Blackett had been admitted long before.\n\nIn June 1973, the Diracs returned to the UK so that he could collect his award. A chauffeur drove them in a Rolls Royce to  Buckingham Palace, where he received the award in private from the Queen for a few minutes, while Manci waited in an ante-room. A few weeks later, he shared with Esther and Myer Salaman his discussions with the Queen about the challenges faced by a female scientist who is also the mother of young children:\n\nI said it was difficult for a woman who had to choose between her career and her family and there could not be real equality between the sexes. The Queen said she did not press for equality of the sexes.\n\nOn his return to the USA, Tallahassee colleagues quizzed Dirac about his impression of the Queen, but he would say very little. His description of her consisted of two words: 'Very small.'\n\nThat summer, Dirac visited CERN in Geneva to see its newest particle accelerator, capable of increasing the energy of protons to some fifty thousand times the energy reached by Cockcroft and Walton's device. During his visit, he walked to the rue Winkelried, a side street near the lake and close to the main railway station, to see the apartment owned until the mid-1920s by his paternal grandmother, where he and his family stayed in 1905. As he strolled around the nearby statue of Rousseau, Dirac may have thought of the time he spent running around in the lakeside park with Felix, watched by his father and mother, baby Betty in her arms. Dirac had not visited Switzerland since then, despite many invitations. The pain of the country's association with his father had been so deep that Dirac had not been able to bring himself to visit it until he was seventy years old.\n\nIn 1979, the centenary of Einstein's birth, Dirac was feeling weak and listless. But he was determined to speak at as many of the celebratory meetings as he could, so that he could 'make clear what a great scientist Einstein was', as Halpern recalled. During that year, Dirac achieved one of his ambitions - of flying across the Atlantic on Concorde, the first supersonic passenger aircraft. The aircraft, developed by an Anglo-French collaboration in the 1960s, was noisy, a prodigious guzzler of fuel and hopelessly uneconomic, but it symbolised the best and most exciting in contemporary engineering. It was also the apogee of the aviation industry in Dirac's native city: the Bristol Aeroplane Company had led the first British design team to work on the aircraft and build the first British prototype in Filton, a few miles from Julius Road.\n\nSomehow, Manci persuaded UNESCO to fund transatlantic flights on the aircraft for Dirac and herself as a condition of his attending the organisation's Einstein celebration in Paris, as guest of honour. He and Manci took the flight on 5 May 1979, cruising at almost 60,000 feet - the nearest he would ever get to outer space. During the flight, he probably read on the front page of the _New York Times_ the news from Britain that Margaret Thatcher had just become Prime Minister. He may have wondered whether his mother's fears about the notion of a woman prime minister would be realised, whether Mrs Thatcher would, in Flo's words, 'vacillate in her feminine way' so that 'her supporters would fall off right and left'.\n\nBy spring 1982, when Dirac and Kapitza were tired of travel, three opportunities to meet that summer arose, and they seized them. Accompanied by their wives, they met first at the Lindau meeting at the end of June. Kapitza had been eligible to attend the meeting only since he received the Nobel Prize for physics in 1978, after Dirac had lobbied for him for almost forty years. During that time, Dirac had seen the honour awarded to almost all of Rutherford's most able 'boys' - Blackett, Chadwick, Cockcroft and Walton - and virtually all the pioneers of quantum mechanics from the 1920s and 1930s had received the prize, including Born, Fermi, Landau, Pauli, Tamm and Van Vleck, but not Jordan, whose Nazi past probably cost him the honour.\n\nAt the Lindau meeting, Dirac mounted one of his last attacks on renormalisation in front of an audience of some two hundred students and Nobel laureates. Looking as fragile as a cut-glass figurine, Dirac stood at the rostrum giving a speech almost identical to ones he had been giving for almost fifty years; he had no praise for the Standard Model or any other successes of particle physics. A microphone amplified his trembling voice, each letter 's' accompanied by a whistle from his ill-fitting dentures. Current theories were 'just a set of working rules', he said; physicists should go back to basics and find a Hamiltonian description of nature free of infinities. ' _Some_ day', he said with a gentle and weary defiance, 'people will find the correct Hamiltonian.' But he was preaching a lost cause: physicists no longer based their descriptions of fundamental particles on Hamiltonians, as other methods were much more convenient. But the audience listened respectfully to Dirac's twenty-five-minute speech, partly, perhaps, in  anticipated sadness that his lone voice would soon be silent. Here was someone, like Einstein, who was unafraid of bucking contemporary trends and taking the consequences, to be his own man.\n\nThe Diracs and Kapitzas met again a few days later in G\u00f6ttingen. Kapitza had pleasant recollections of the town, as did Dirac - it was, in his opinion, the birthplace of quantum mechanics, where he had first become acquainted with Born and his group, where he became friends with Oppenheimer and probably where he first saw a Nazi in uniform. The Diracs stayed in Gebhard's Hotel overlooking G\u00f6ttingen railway station, where Dirac had first arrived in the town from Copenhagen fifty-five years before. Then, his journey from the station to his room in the Carios' home was a luggage-laden walk; now, he and Manci were met by a welcoming party that whisked them in a taxi to the town's most luxurious accommodation.\n\nThere are photographs of Kapitza and Dirac sitting at a table in the garden of the hotel, looking exhausted and a little dispirited. Physics, once one of their main topics of conversation, was now much less important than international affairs, the preoccupation of Kapitza. He will almost certainly have spoken with Dirac about the recently ended Falklands War between Argentina, led by General Galtieri, and the United Kingdom, led by Mrs Thatcher, over the disputed island territory in the South Atlantic. Dirac was in two minds about Thatcher: he feared the impact of her radicalism on British education and science but sympathised with her determination to protect the Falkland Islanders' wish to remain British. He thought, however, that the dispute should have been resolved through negotiation: at the beginning of the war, it had seemed absurd to him that the number of people likely to die would exceed the number whose British citizenship would be protected. In politics, if not in physics, Dirac was now a pragmatist.\n\nThe Falklands War was a trivial matter compared with nuclear proliferation, a subject Dirac and Kapitza talked about at length when they met again a few weeks later, at the Erice summer school in Sicily, organised by the physicist Antonino Zichichi. Dirac took risks in the subject matter he addressed there: during the previous summer, he had given a presentation on 'The Futility of War', an uncomplicated statement of an argument that few would oppose. In the summer of 1982, he collaborated with Kapitza and Zichichi to write the one-page 'Erice Statement', which urged governments to be less  secretive in defence matters (one of Bohr's favourite themes), to prevent the spread of nuclear weapons and to help non-nuclear powers feel more secure. The well-intended phrasing of the document, later signed by ten thousand scientists, was so bland that its first signatories at the Erice meeting included not only opponents of nuclear weapons but also the right-wing Eugene Wigner and the obdurately pro-nuclear Edward Teller, who had done more than almost any other American to fuel the arms race.\n\nOn the last stages of the Diracs' 1982 European tour, they visited Betty in Amsterdam and Gabriel in Aarhus, before travelling to Cambridge. Dirac returned to St John's College, which, as he was to tell the Master soon afterwards, 'has been the central point of my life and a home to me'. That summer, the talk of the Combination Room was the imminent arrival of the college's first women undergraduates: another all-male bastion of Cambridge was about to fall. Earlier, the theoretical physicist Peter Goddard asked Dirac whether he thought women students should be admitted to the college, and, after a long pause, Dirac replied, 'Yes, provided we don't admit fewer men.'\n\nBefore he left St John's, Dirac left his gown at the Porters' Lodge, where he had first registered as a student almost sixty-nine years before. He wrote a label: 'Professor Dirac's Gown. Please take it to the Master and ask him to keep it until the next time I come to Cambridge.' But he would not see the city again.\n**Twenty-nine**\n\nI bade, because the wick and oil are spent  \nAnd frozen are the channels of the blood,  \nMy discontented heart to draw content  \nFrom beauty that is cast out of a mould  \nIn bronze, or that in dazzling marble appears,  \nAppears, but when we have gone is gone again,  \nBeing more indifferent to our solitude  \nThen 'twere an apparition. O heart, we are old;  \nThe living beauty is for younger men:  \nWe cannot pay its tribute of wild tears.\n\nW. B. YEATS, 'The Living Beauty', 1919\n\nThe confidence Dirac displayed when he spoke about physics hid a despair that he apparently revealed only once, to someone he hardly knew - Pierre Ramond, a theoretical physicist at the University of Florida in Gainesville. A courteous and articulate man, Ramond is an American who speaks in a richly musical voice whose accent is a constant reminder to his listeners that he was born and raised in France. After lunch one Wednesday in the early spring of 1983, he drove from Gainesville to give a colloquium at Florida State University, hoping that his 'hero and guiding light' Dirac would be in the audience. Sure enough, when Ramond arrived in the seventh-floor seminar room, overlooking the campus, he saw in his audience the daydreaming figure of Dirac, slight as a pixie.\n\nIn his presentation, speculative but assured, Ramond discussed the possibility of setting out fundamental theories not in the usual four dimensions of conventional space-time but in a higher number of dimensions. Throughout, Dirac appeared to be snoozing, and, during the questions afterwards, he said nothing. But when the seminar broke up, he - unusually - lingered until he was with the speaker, alone, and the door was shut.\n\nRamond had met Dirac twice before, but had not been able to draw him into anything resembling a normal conversation. 'I had heard that the only way to persuade Dirac to talk was to ask him a non-trivial question that required a direct answer,' Ramond recalls.  So he asked Dirac directly whether it would be a good idea to explore high-dimensional field theories, like the ones he had presented in his lecture. Ramond braced himself for a long pause, but Dirac shot back with an emphatic 'No!' and stared anxiously into the distance. Neither man moved, neither sought eye contact; they both froze in a silent stand-off. It lasted several minutes. Dirac broke it when he volunteered a concession: 'It _might_ be useful to study higher dimensions if we're led to them by beautiful mathematics.' Encouraged, Ramond saw an opportunity: doing his best to sound understanding, he invited Dirac to give a talk on his ideas at Gainesville any time he liked, adding that he would be glad to drive him there and back. Dirac responded instantly: 'No! I have nothing to talk about. My life has been a failure!'\n\nRamond would have been less stunned if Dirac had smashed him over the head with a baseball bat. Dirac explained himself without emotion: quantum mechanics, once so promising to him, had ended up unable even to give a proper account of something as simple as an electron interacting with a photon - the calculations ended up with meaningless results, full of infinities. Apparently on autopilot, he continued with the same polemic against renormalisation he had been delivering for some forty years. Ramond was too shocked to listen with any concentration. He waited until Dirac had finished and gone quiet before pointing out that there already existed crude versions of theories that appeared to be free of infinities. But Dirac was not interested: disillusion had crushed his pride and spirit.\n\nDirac said goodbye and walked off, looking impassive, but Ramond was shattered. He took the elevator to the ground floor and walked alone in the fading light of the afternoon back to his car. Twenty-five years later, he could still recall how upset he was: 'I could hardly believe that such a great man could look back on his life as a failure. What did that say about the rest of us?'\n\nRamond cannot recall whether he had explicitly mentioned to Dirac the idea that nature is fundamentally built not of point-like particles but of tiny pieces of string. In the late 1970s and early 1980s, Ramond was one of the small band working on the idea, then a backwater of theoretical physics. Dirac had tentatively suggested in 1955 that electrons and other quanta might be pictured as lines rather than points, but the mathematical form of Dirac's idea was completely different to  that of the modern string theory, itself still only embryonic. The theory had, however, used contributions Dirac had made in the late 1950s and early 1960s, including his methods of describing two- and three-dimensional objects in ways consistent with both quantum mechanics and the special theory of relativity. The mathematics he used to describe a small sphere - his model of a muon - resurfaced in a different context, to describe the motion of a string moving through space and time.\n\nAmong the encouraging features of the new string theory was a pleasing absence of the infinities of conventional field theories, such as quantum electrodynamics, the best-available description of electrons and photons. Most impressive was that string theory made the existence of gravity inevitable: if the theory is correct, gravity _must_ exist. Although there was no experimental evidence to favour string theory over other field theories, to its supporters it looked too beautiful to be entirely wrong. Dirac will have heard about the theory in seminars at Florida State but he gave it no credence - his curiosity was spent. A few months after his eightieth birthday, the local journalist Andy Lindstrom had found him 'a painfully spare man [. . .] stoop-shouldered and frail'. His once-black hair had 'retreated to a wispy cowl at the very fringes of his forehead, as though worn away by the great thoughts fermenting below . . . A web of wrinkles etches his gentle, lonely face, outlining eyes that seem to be forever questing. '\n\nSince overcoming his digestive trouble in late 1980, Dirac had become more relaxed about his health, but his anxieties returned three years later when he started to suffer from apparently unrelated problems - night sweats and occasional fevers. He consulted Hansell Watt, a local doctor and lay preacher whose calm, comforting words were all the more reassuring for being spoken with a rich southern drawl. Dirac took to him, and, for Manci, he could do no wrong. Watt diagnosed the source of Dirac's medical problems to be his right kidney, which X-ray photographs showed to have been infected by tuberculosis, probably when he was a child. This was a surprise to Dirac, who had never suspected that he had been infected, having been assured by his mother: 'T.B. runs in families and it is absolutely not in ours.'\n\nWhen Dr Watt advised Dirac that his tubercular kidney should be removed, Halpern was outraged. Wary of surgical cures and wanting only to try herbal remedies, Halpern opposed Watt's strategy and - to Manci's anger - did all he could to undermine it. Manci, fighting Halpern's influence over Dirac like a tigress guarding her wounded cub, did not tell him when she arranged the operation at the Tallahassee Memorial Hospital on 29 June 1983, a month after what would be his final talk. The surgeon found that Dirac had only the last remains of a right kidney with a cyst the size of a hockey ball.\n\nThe operation was technically successful but it left Dirac an invalid. Weak and dispirited, he spent the summer recuperating at home, watching television and playing Wei Chi and other board games but unable to do serious work. After several weeks, he could walk a few steps but did not have the strength to venture out of his air-conditioned home into the heat and humidity outside. For the first time in decades, he could not spend the summer walking in the countryside - especially cruel for someone who had trodden a distance comparable with Wordsworth's total of about 180,000 miles. One of Dirac's most frequent visitors was Halpern, who sat at his bedside several times a week, chatting about their work and anything else that took their fancy, including politics. Dirac said that he could not help liking President Reagan, though he disagreed with most of his policies; at heart, Dirac remained a liberal, though with no loyalty to the Democrats or any other political grouping.\n\nHalpern's relationship with Manci became more fraught by the week. Upset by what he saw as her unending nagging, he often found himself leaving the Diracs' home red-faced and purse-lipped with anger. Whenever Dirac mentioned his discomfort at Tallahassee's oppressive summer climate, she would shoot back with her favourite rejoinder, 'It's better than Cambridge,' Halpern recalled. For her part, Manci thought Halpern was a rude, interfering busybody who was shamelessly taking advantage of his helpless friend by foisting quack medicine on him. Aware of her hostility, Halpern decided that subterfuge was the only hope. When Manci was out shopping, he instituted a secret programme of homoeopathic treatment, furtively dropping herbal essences into Dirac's drinking water when the nurse was not looking. According to Halpern, Dirac's energy resurged like Popeye's after he had downed a can of spinach. As soon as Manci found out about 'the herbal conspiracy', she returned Dirac to his usual diet, whereupon he slipped back into lethargy and indifference, if Halpern's testimony is correct.\n\nDirac spent most of his waking hours in a wheelchair, talking to visitors, including his daughter Mary and her dashing new husband, Peter Tilley. After a few months, Dirac was fit enough to return occasionally to his office in Florida State University, to supervise his final graduate student Bruce Hellman and to oversee what would be his final publication. Halpern drafted the text of 'The Inadequacies of Quantum Field Theory' for Dirac, who wanted his final published words to execrate renormalisation, the technique born of one of his most profound contributions to science. For the last time, he refused to accept that, as Feynman had advised him in 1946, he was on 'the wrong track'. Feynman might as well have counselled a train to depart from its rails.\n\nEarly in April 1984, Dirac heard that Kapitza was dead. The Soviet Union knew it had lost one of its most loyal subjects: the entire Politburo and many of the country's scientific leaders signed _Pravda_ 's announcement of his death. Dirac had lost his dearest friend, his surrogate brother, but he showed only resignation. More sad news followed a few weeks later: the Diracs' son Gabriel had a skin cancer so aggressive that his doctors gave him only a few months to live. In June, Manci flew to Europe to see her son, leaving Dirac in the care of friends. A few weeks after her return, Gabriel died on 20 July, aged fifty-nine. Three days later, Dirac was too ill to put himself to bed. Halpern was away in Europe, so Manci had her husband to herself and had to cope with his sinking morale and hardening stubbornness.  Dirac's spirits rallied during a visit by Gabriel's daughter Barbara, a radiantly attractive young woman and a special favourite of the Diracs. ('You look like Cher,' he told her a few years before.) In sharp contrast to Halpern, Barbara's view of Manci was that she was a sensitive and humane nurse - there were occasional quarrels between her and Dirac but they would dissolve swiftly into an affectionate holding of hands. Dirac's energy had all but ebbed away, Barbara observed, but his love of physics still flickered: he returned to his papers and whispered resolutely, 'I have work to do.' His greatest fear, of losing his mind, was never realised.\n\nAt the beginning of October 1984, after Barbara had returned to Europe, Manci hired nurses to be with Dirac round the clock: he was hanging on to the last thread of life. But he still received the occasional visitor, including Mary's husband Peter Tilley, who sat for  hours at Dirac's beside, mostly in silence. During his final visit, Tilley recalls, Dirac leant over to him and said firmly, in a matter-of-fact tone: 'The biggest mistake of my life was marrying a woman who wanted to get out of the house.' Dirac sounded neither bitter nor regretful, Tilley remembers, but was making a factual statement in a way that invited no further discussion. Perhaps Dirac was thinking of what Manci had said to him soon after they met - that she had married her first husband only to get out of her parents' home - and of the veiled warnings his mother had given him about marrying Manci forty-seven years before.\n\nThe battle of wills between Manci and Halpern resumed. When he knew she was out, Halpern sneaked into the house and stirred his fortifying herbs into Dirac's drinking water. The nurse had almost given up trying to interest him in food, and it was left to Halpern to feed his friend, who took his food like a baby. All Dirac wanted to do was to talk about Kapitza. Dirac spent many of his last conscious hours recounting favourite stories about his friend's colourful life - over and over again, Dirac told the story of how Kapitza refused to work on the bomb, standing alone among lesser mortals who did not have the moral courage to make a stand. It was a tape loop of delusion.\n\nOn Thursday 18 October, as Halpern was leaving the Diracs' home, he bumped into Manci. He was expecting a telling off for visiting his friend, but Manci did not mention it; she told him calmly that she had just been to the mortician to reserve Dirac's grave. But the next day Halpern received the phone call he had feared for weeks: Manci forbade him from setting foot in the house again - Dr Watt had told her, she said, that Dirac was too weak to see anyone except close family. Angry, bitter and tearful, Halpern heard nothing until four days later, when he read on the front page of the _Tallahassee Democrat_ : 'FSU physicist is dead at 82'. On the Saturday evening, with Manci and his nurse at his side, Dirac's heart had failed and stopped beating at five minutes before eleven.\n\n'I want to be put down like a horse,' Manci told Dr Watt. But in public she showed her usual spirit and fortitude, informing friends and relatives of Dirac's passing with business-like calm and attending to every detail of the funeral arrangements. She took great pains to ensure that Dirac was remembered as she wanted: the day after his  death, she told friends that he was 'a very religious man' and that he would have wanted a high-Episcopalian funeral.\n\nThe ceremony took place in the open air at the Roselawn cemetery in Tallahassee, on 24 October, under an overcast sky, rain threatening. When the guests arrived, shortly before 11 a.m., they saw Dirac's coffin was on a plinth beside his freshly dug grave, under a bright blue marquee-like roof mounted on four wooden poles, in the shade of a group of conifers swaying slightly in the breeze. Among the mourners was Dirac's one-time confidant, Pierre Ramond, who was surprised when he saw the congregation: 'Considering how famous he was, there were very few people there.' There were about ninety mourners, including dozens from Florida State University but - as Manci bitterly noted - no one from Cambridge. Several in the congregation were uneasy to see that they were not alone: they had been joined by scribbling journalists and a flotilla of television crews. Manci had decided that her husband should be buried under the encircling gaze of TV cameras.\n\nThe rector Dr W. Robert Abstein read slowly from the oldest-surviving version of the Anglican Bible, the text Manci had insisted on. She had forbidden Halpern to speak, and there was no eulogy. After half an hour, as the sky brightened, Abstein crumbled soil on the coffin and traced the sign of the cross in the dirt. The place of Dirac's burial was marked a few weeks later with a neat white-marble stone, engraved with words he had used, chosen by Manci: 'because God said it should be so'.\n\nA few days after Dirac's funeral, Manci had to take another blow. She heard from the police in Vermont that they now presumed that Judy was dead and that they had called off the hunt for her. The pain for Manci was terrible: in just four months, she had suffered the grief of losing her best friend in Russia, two of her children and her husband. Life seemed to hold little for her - but she was a fighter.\n\n'Dirac was a militant atheist,' objected the Dean of Westminster, Edward Carpenter, when he was asked if Dirac might be commemorated in the Abbey's science corner. The Oxford physicist Dick Dalitz led a group of scientists that began to press for Dirac to be remembered alongside Newton and Rutherford. For someone to be worthy of a place in such company, the Abbey authorities had to be sure that he or she was a Christian - or at least not inimical to  religion - and was judged, after a decade's reflection, to be of 'millennial significance'. Carpenter was easily persuaded of Dirac's status, but Dalitz found it hard to demonstrate that Dirac passed the religious test, especially after the Dean found out about Pauli's comment 'There is no God and Dirac is his prophet.' Pauli could make things difficult for Dirac even when they were both dead.\n\nDuring the stalemate, Dalitz found an unanswerable way to counter the objection: if Dirac's parents had christened him, then - regardless of any derisive comments he had made about religion - he was officially a Christian. Dirac would have been amused by the absurdity. In the late 1980s, Dalitz spent weeks trawling through parish records in Bristol but could find no evidence that Charles and Flo Dirac had christened their children, and this line of investigation drew a blank. However, the church authorities were impressed to hear that Dirac was a member of the Papal Academy and that he had made no antireligious comments during its meetings. Dalitz and his colleagues kept up their pressure on the authorities, and, in early 1990, after six years of lobbying, the new Dean of Westminster declared himself 'very sympathetic' to their cause. It was finally won in early 1995.\n\nThe commemoration took place in Westminster Abbey on Monday, 13 November 1995, beginning with Evensong at 5 p.m. Though much less well publicised, the ceremony was on a scale as grand as Rutherford's fifty-eight years before: the Abbey looked gorgeous, the choir sounded magnificent, and the congregation was in good voice. After tributes to Dirac's scientific work had been read, the mathematician Sir Michael Atiyah, President of the Royal Society, unveiled the commemorative stone in the nave of the Abbey, next to Newton's gravestone and just a few paces from Darwin's. Stonemasons in Cambridge had used a piece of Burlington Green slate quarried from the Lake District to produce a two-foot square slab of stone and etch into it the inscription 'P. A. M. Dirac OM physicist 1902-84', with a statement of his equation.\n\nStephen Hawking gave the final address, using his voice synthesiser to speak through the Abbey's antiquated public-address system.  He began with his usual arresting clarity and humour:\n\nIt has taken eleven years for the nation to recognise that he was probably the greatest British theoretical physicist since Newton, and belatedly to erect a plaque to him in Westminster Abbey. It is my task to explain why. That is, why he was so great, not why it took so long.\n\nHis final words consisted of another barb: 'It is just a scandal that it has taken so long.' Dalitz threw anxious glances at his fellow organisers; evidently, Hawking did not know that at least a decade had to elapse after a subject's death before he or she could be commemorated - Dirac's ceremony was at most only a year late. Afterwards, Dalitz sought out the Abbey authorities and apologised.\n\nAfter the organist had played Bach's _Prelude and Fugue in A Major_ , Dirac's daughter Monica and her two children laid flowers on the memorial plaque, before the congregation sang the hymn 'Lord of Beauty, Thine the Splendour'. The music had been well chosen.\n\nAngry that Westminster Abbey had questioned Dirac's suitability for commemoration, Manci did not attend the ceremony: 'The English are hypocrites,' she fumed. 'Lord Byron is buried in the Abbey, [and] he was the greatest rogue of the century.' After Dirac's death, Manci become the keeper of his flame, firing off affronted notes to obituarists and chroniclers of her husband's life who cast any doubt on her view that he was a scientific saint. Abraham Pais was startled when he received a letter from her, insisting that Dirac was not an atheist. 'Many times did we kneel side by side in Chapel, praying. We all know, he was no hypocrite.' Friends of Dirac, certain that he was agnostic, were puzzled: did he join her at prayer out of politeness? Or had Dirac privately practised a religion he had mocked among friends? Or was Manci fantasising?\n\nAfter she had come to terms with Dirac's death, Manci remained lively and active for ten years, travelling in Europe and the USA, and entertaining an almost unbroken stream of guests, including Lily Harish-Chandra, Leon Lederman and his wife Ellen, and Wigner's daughter Erika Zimmermann. When she was alone, Manci's idea of a perfect day was to spend it shopping, playing with her dog, hobnobbing with Florida State officials, adjusting her investments and driving out with her pals for lunch at a local Marriott hotel, where she traded gossip while munching on cheese blintzes. She was in close touch with her daughters, constantly worrying about Mary, who lived nearby and was often in poor mental health. In the evening, Manci would settle down in front of the television with a glass of sherry to watch public-service documentaries and her favourite game shows, _Jeopardy!_ and _The Price Is Right._ Through letters and endless phone calls, she kept in touch with friends and  family all over America and Europe, though not with her sister-in-law Betty, who died in 1991.\n\nStill angry with Churchill College for what she regarded as their terrible treatment of Elizabeth Cockcroft, Manci took her revenge when she withdrew Dirac's archive from the college. She arranged for it to be transferred to Florida State University, where the archive is now stored in the Dirac Science Library, which Manci formally opened in December 1989. Outside the library, she unveiled a statue of Dirac by the Hungarian sculptor Gabriella Bollob\u00e1s, showing him in old age, reading _The Principles of Quantum Mechanics._ The statue is peculiarly lifeless, with no sign of the energy and imagination that propelled him to greatness.\n\nManci never mellowed: she would still switch in an instant between mean-spiritedness and generosity. After railing at Halpern for an entire morning, she would spend the afternoon trying to sweet-talk Florida State officials into giving him a permanent position in the physics department. She behaved no more consistently towards her brother Eugene, suffering from Alzheimer's disease: in public, she adored him but in private she described him witheringly as 'a third-rate physicist'. On the telephone, she argued with him for hours about family matters, haranguing him for his politics and for associating with 'the Moonies'. On New Year's Day 1995, she called Leon and Ellen Lederman hours after Wigner's death, and said to each of them in turn: 'Thank God the monster is dead.'\n\nEven in her ninth and tenth decades, Manci kept abreast of the news. In late 1989, she was jubilant when, following the fall of the Berlin Wall, the Soviet-backed Hungarian Socialist Workers' Party abdicated its monopoly power and agreed to free elections. Soon afterwards, during the presidency of George Bush Senior, she considered applying for American citizenship so that she could vote against him if he stood for re-election. Delighted when Bill Clinton first won the presidency, in late 1995 she wrote supportively to Hillary Rodham Clinton, who sent a courteous reply on White House notepaper ('Dear Ms Dirac [...]'). No letter ever gave Manci more pleasure.\n\nShe spent her last few years in pain with arthritis and suffering grievously from asthma. Friends and family urged her to move into a care home, but she would hear nothing of it: she was going to live out her days at home, no matter what the cost of round-the-clock home  assistance. Early in 2002, after she tripped over her dog and broke her hip, she had no choice but to be admitted to hospital, where she died a few days later. Mary and Monica arranged for her to be buried with Dirac under a joint gravestone; his epitaph was unchanged, hers was 'Let her generous soul rest in peace.'\n**Thirty**\n\nThen she showed me this picture \u263a and I knew that it meant 'happy', like when I'm reading about Apollo space missions, or when I am still awake at 3 a.m. or 4 a.m. in the morning and I can walk up and down the street and pretend that I am the only person in the whole world.\n\nCHRISTOPHER BOONE, narrator in Mark Haddon's _The Curious_  \n_Incident of the Dog in the Night Time_ , 2003\n\nBristol has never taken Dirac to its heart. Today, the few reminders in the city of its association with Dirac include a little-noticed abstract sculpture, the name on a grimly functional building and a few plaques. During my many visits to Bristol over the past five years, I have met scarcely half a dozen people outside the university who have heard of him. A few minutes after I first walked through the front door of the Bristol Records Office, in May 2003, I enquired of the bracingly confident assistant if she had any material on Paul Dirac; she looked at me quizzically and asked, 'Who's he?'\n\nIn the Records Office, the best way of finding out about Dirac's early school years is to ask to see the well-fingered documents about his fellow pupil at Bishop Road School, Cary Grant. Local journalists and television crews were always ready to record Grant's sojourns in the city, a prospect that would have frightened off Dirac; his visits were always anonymous. In the 1970s, however, he welcomed the campaign led by the local Member of Parliament William Waldegrave to celebrate the city's association with him, an initiative that led to the founding of a mathematics prize in local secondary schools. Waldegrave had noticed that while Dirac is not well known by the people of Bristol, they were proud of their association with the engineer Isambard Kingdom Brunel, though he had not been born in the city or even lived there.\n\nIn 2006, Bristol's veneration of Brunel was clear during a five-month celebration of the bicentenary of his birth. Local businesses and cultural organisations collaborated to present 'Brunel 200', an eight-month festival of exhibitions, theatrical events, concerts, art installations and poetry readings. Some forty thousand people -  most of them from Bristol and the surrounding towns - attended the opening weekend in April. Four years before, the centenary of Dirac's birth was marked in Bristol rather more modestly. The main event, organised by the University's physics department, was an afternoon of lectures to celebrate Dirac's life and legacy, followed by a formal dinner on Brunel's SS _Great Britain._ Following an interview about the Dirac equation on Radio 4's _Start the Week_ , I was called by one of the organisers who asked me to give a lecture on Dirac's life and work. This was a special moment for me as I had been fascinated by Dirac since I was a teenager.\n\nI first heard his name on a suburban doorstep, when I was hawking subscriptions for a weekly raffle in aid of the Liberal Party in Orpington, a suburb in south-east London. When I was closing a sale on a spring evening in 1968, my new customer - a distracted, oddly engaging man by the name of John Bendall - mentioned perfunctorily that he was a theoretical physicist. We became friends, and, during several Sunday-morning chats in his front room, I realised that he was a Dirac fanatic: Bendall would find an excuse to introduce his hero's name in every conversation lasting longer than a few minutes. I found out that it was no coincidence that the younger Bendall daughter, playing with her dolls at our feet, had been named Paula. Every Christmas, he would take a plate of mince pies from the kitchen, sit back in his armchair with a glass of sherry and read _The Principles of Quantum Mechanics_ , savouring every sentence. Minutes after I first browsed through his copy, I knew I too wanted to be a theoretical physicist.\n\nA few months later, it dawned on me that, when Dirac was a boy, he lived just a few miles from my Bristol-born paternal grandmother Amelia ('Mill') Jones. She was fond of telling me about that time in her life, when she worked in a corset factory. At weekends, she and her fianc\u00e9 Charley - a docker, later my grandfather - would promenade arm in arm around the centre of the city, her expansive skirt almost touching the ground, his moustache daringly trimmed. 'I wonder if we ever saw Cary Grant before he 'opped it to Americal?' I heard her ask. She may well have set eyes on him around the city, perhaps around the Hippodrome, one of her haunts. It is also possible that she and my grandfather knew the high reputation of Charles Dirac and almost certain that they saw at least some members of the Dirac family, perhaps the two French-speaking brothers walking together.\n\nIn middle age, Dirac made several trips back to the city. In 1956, after a summer holiday in his mother's home county of Cornwall, he returned through Bristol with his family and stopped outside 6 Julius Road to point out to his daughters Mary and Monica where he had lived since he was ten. But he said nothing about his memories of the twenty-five years he spent there. During my visits to Bristol, I lurked several times outside this unremarkable home, trying unsuccessfully to imagine my way into it. My problem was solved during a visit in the early summer of 2004, when the owner of the property generously invited me inside, allowing me to enter the theatre of Dirac's most traumatic memories.\n\nOverlooking the front garden is Charles's tiny study, where he taught his private students, away from the gaze of the tax inspectors. Under the stairs is the tiny cupboard where Flo crouched during the German bombing raids, cotton wool in her ears. Above is the little bedroom where, a few months after Felix killed himself, Dirac first read Heisenberg's path-breaking paper and realised that it contained the key to quantum physics. Felix's bedroom, for many years a shrine, is now scattered with the toys and games of the children who occupy the room. Flo's tiny kitchen overlooks the back garden, where Dirac had looked up at the stars and had watched some of the first British-made aeroplanes take off, and where he had begun to learn gardening during the Great War. It seemed barely possible that this suburban home had seen events that had left Dirac, as Manci had described him, 'an emotional cripple'.\n\nHer words might sound cruel, but Dirac would probably have agreed that they were accurate. He always attributed his extreme taciturnity and stunted emotions to his father's disciplinarian regime; but there is another, quite different explanation, namely that he was autistic. Two of Dirac's younger colleagues confided in me that they had concluded this, each of them making their disclosure _sotto voce_ , as if they were imparting a shameful secret. Both refused to be quoted. Yet one should be extremely careful about making this diagnosis: rather too often, people are labelled autistic on the flimsiest of evidence except that they are exceptionally reserved, focused and unsociable. Besides, it is not easy to psychoanalyse someone who is dead.\n\nBefore one can say whether there is a strong case that Dirac was a person with autism, it is important to be clear about the nature  of the condition. For someone to be diagnosed as autistic, he or she must have all three of the following characteristics since early childhood:\n\n1. Social skills are poorly developed compared with the development of other 'classroom' skills, such as reading and arithmetic.\n\n2. The development of verbal and non-verbal communication is impaired compared with the development of other 'classroom' skills. Behavioural signs of repetitive or stereotyped movements, a delay in the acquisition of language and a lack of varied, spontaneous make-believe play.\n\n3. An unusually narrow repertoire of activities and interests that are abnormally intense .\n\nA few days before the Nobel Prize ceremony in 1933, Flo told journalists that Dirac was a precocious, industrious and unusually quiet child. There is not nearly enough detail in her comments or in reports of Dirac's behaviour at school to justify a diagnosis that he was then autistic. His behaviour as an adult, however, had all the characteristics that almost every autistic person has to some degree - reticence, passivity, aloofness, literal-mindedness, rigid patterns of activity, physical ineptitude, self-centredness and, above all, a narrow range of interests and a marked inability to empathise with other human beings. Extremes of these characteristics are at the root of the humour in almost all the tales about Dirac that physicists have been telling each other for decades: almost all of these 'Dirac stories' might also be called 'autism stories'.\n\nThe word 'autism', derived from the Greek word _autos_ for self, covers a wide spectrum of conditions, spanning people with mental retardation through to those like Dirac who are gifted in their specialist fields and often described as 'high functioning'. An unusual case was dramatised in the Hollywood film _Rain Man_ , where Dustin Hoffman portrays the autistic character Raymond Babbitt, who also has the much more rare Savant Syndrome, manifested in his prodigious arithmetic skills and in his amazing memory for baseball statistics and telephone numbers.\n\nClinicians believe just over half a million people in the UK are autistic to some degree, almost one in a hundred, and it is clear that it is predominantly a male condition. Statistical studies also show that depression is especially common among people with autism and that about 20 per cent of children with the condition speak fewer  than five words a day. About one person with autism in ten has a special talent - for example, in drawing, working with computers or rote-memory learning. Another characteristic, yet to be properly quantified, is that young people with autism are exceptionally fussy about the food they are prepared to eat.\n\nThere is currently a good deal of speculation of a modern-day epidemic of autism, especially in the USA, where, as _Nature_ put it in 2007, the condition is the 'golden child of the fundraising circuit'.  But talk of a sudden rise in the number of people with autism is probably ill founded because diagnoses often differ from one doctor to another, with the result that the data have large uncertainties. Reliable information has been available only since the mid-1960s, when high-quality empirical studies began, long after Leo Kanner, an Austrian-born child psychiatrist at Johns Hopkins University in Baltimore, first identified and named the condition in 1943. A year later, the Viennese psychiatrist Hans Asperger independently described a condition now known as Asperger's Syndrome, part of the spectrum of autistic behaviour.\n\nAlthough the study of autism is developing rapidly, it is still in its infancy: like atomic physics in the early 1920s, there is a huge amount of observational information about the condition, but the experts know that their understanding of the data is only fragmentary. But some firm conclusions have emerged. A few decades ago, scientists believed that people with autism had some disorder of the mind, but it is now plain that this is incorrect: there is now overwhelming evidence that the condition is a disorder of the tissue in the _brain._ 14 Using modern brain-imaging techniques - including positron emission tomography - clinicians have demonstrated that the regions linked with the process of 'reading other people's minds' in the brains of people with autism are noticeably less active than in most other people.\n\nSome of the most productive research into autism is now being done in Cambridge at the Autism Research Centre. Its director, Simon Baron-Cohen, is a pioneer of the idea that autism is a manifestation of the extreme male brain - comparatively weak in the typically female characteristic of empathy but strong in the typically male characteristic of systemising, such as working out how mechanical devices function, solving mathematical puzzles, poring over  league tables and filing CDs. In one of Baron-Cohen's research projects, he and his colleagues are studying the behaviour of leading mathematicians and scientists, many of whom - including Newton and Einstein, some believe - exhibit at least some of the traits of autism. The great majority of top mathematicians and physical scientists are undoubtedly male; this may indicate a predisposition of the male brain, though critics point out that it may also be a consequence of rearing children in ways that perpetuate sexual stereotypes.\n\nWhen I visited Baron-Cohen in his rooms in Trinity College, I was struck by two remarks that seemed especially relevant to Dirac. First, he said that he had noticed the high proportion of autistic men who were in a stable marriage with a foreign wife, perhaps because the women were more tolerant of unusual behaviour in foreign men than in men from their own culture. Baron-Cohen had no idea that Dirac was married for almost fifty years to a Hungarian. That, of course, could be a coincidence. I was taken aback again by another remark he made a few minutes later, however, when he pointed out that although people with strongly autistic personalities appear to be detached from most other people, when they believe that a friend has suffered an injustice, they are often so indignant that they will disrupt or abandon their almost invariable daily routines to rectify it. Baron-Cohen knew nothing of Dirac's one venture into international politics when he spent a few months concentrating on the campaign to free Kapitza from his detention in the Soviet Union. Heisenberg, pilloried by many of his former colleagues after the war, had cause to regard Dirac as one of his most loyal friends. Again, these may be coincidences.\n\nBut Baron-Cohen argues that it is not happenstance that the young Dirac bloomed in 1920s Cambridge:\n\nCambridge was a niche where his eccentricity would have been tolerated and his skills valued. College life provided him with a regular daily routine and everything he needed. His bed was made for him, food was provided for him. High Table in College would have provided social contact if he wanted it, with its own rules and routines to render it highly predictable. In the mathematics department, he would have been free to do as he wished, he was surrounded by like-minded people, with no pressure to socialise. An environment like this would have been optimal for someone like Dirac.\n\nA fruitful source of insights into autism is the American business executive and teacher Temple Grandin, who describes herself to be 'a high-functioning person with autism'. In her books and articles, Grandin stresses two particular aspects of her personality that she shares with most other autistic people; both are characteristics that Dirac shared. First, she is hypersensitive to sudden sounds, bringing to mind the great care Dirac always took to ensure that he would not be disturbed by chiming bells or by the sudden barks of neighbourhood dogs. Second, she points out that she thinks visually and that, in several respects, her brain does not function like those of most people she has met.\n\nHere's how my brain works: It's like the search engine Google for images. If you say the word 'love' to me, I'll surf the Internet inside my brain. Then, a series of images pops into my head. What I'll see, for example, is a picture of a mother horse with a foal, or I think of _Herbie the Lovebug_ , scenes from the movie _Love Story_ or the Beatles song . . . 'All you need is Love'.\n\nLike Temple Grandin, Dirac was certain that his mind was 'essentially a geometrical one'. He was always uneasy with algebraic approaches to physics and with any mathematical process he could not picture - one of the reasons why he was so uncomfortable with renormalisation.\n\nYet again, it is possible that this correlation between autistic characteristics and Dirac's behaviour is a coincidence, but, in the light of other such correlations, this seems unlikely. I believe it to be all but certain that Dirac's behavioural traits as a person with autism were crucial to his success as a theoretical physicist: his ability to order information about mathematics and physics in a systematic way, his visual imagination, his self-centredness, his concentration and determination. These traits certainly do not explain his talent but they give some insight into his unique way of looking at the world.\n\nOne of the strongest clues about the true nature of autism is that the condition has a genetic component - it runs in families. The theory, although powerful, cannot predict with the precision of a theory in physics how most characteristics are passed down the generations, especially for conditions such as autism, associated with several genes. Observational studies show that it is rare for families to have more than one child with autism, though the probability that a second child will be autistic is about one in twenty, almost eight  times the usual likelihood. This raises the question of whether Felix Dirac was autistic. Again, it is impossible to say one way or the other because too little information about his personality survives. I was, however, given pause for thought one evening during my visit to the family's genealogist, Gisela Dirac. As she surveyed the family tree, she remarked, 'It's amazing how many people in the family had acute depression. And how many killed themselves.' At my request, she later sent me a family tree annotated with such instances: in the previous century, there had been at least six.\n\nCharles Dirac also showed signs of autistic behaviour. Most of the descriptions of him by his colleagues and students refer to his self-centredness, his dedication to work and his rigid teaching methods. Like his son Paul, Charles appears to have had only a modest ability to understand other people's feelings, but whereas lack of empathy in Paul was manifest in his reserve, in Charles it seems to have appeared as a tendency to behave like a human bulldozer. Neither man was ever going to be the easiest of husbands to live with: Flo's teenage infatuation with the charming Swiss man she met in the library had led to a wretchedly unhappy union, whereas Manci somehow found ways of living stably with a man few women would contemplate as an acceptable partner for a second.\n\nDirac was aware that he was in some ways similar to his father. Three months after Charles died in June 1936, Manci suggested to Paul that he thought too much about these similarities and that he might unconsciously be seeking to emulate some of his father's habits. Shortly afterwards, Paul had pondered on his father's biological inheritance when he attended Bohr's conference on genetics and heard in detail about genetic characteristics and how they are passed from one generation to the next. Sitting on one of the wooden benches in the lecture theatre of Bohr's institute in Copenhagen, listening to the lectures, Dirac may well have wondered which of these heritable characteristics were written into his own genes.\n\nWhatever their genetic profiles, there is no doubt that Dirac and his father were incompatible. Having heard so much about the harrowing mealtimes together, I found myself shuddering when I first walked into the dark dining room of 6 Julius Road overlooking the back garden. The original fireplace is still there. It was easy to imagine Flo passing bowls of steaming porridge from the kitchen through the hatch in the dividing wall and urging the worryingly thin Paul not  to leave a morsel uneaten. Although he had a weak appetite, one of the symptoms of tuberculosis, his parents seem not to have suspected that he had the disease and so had no reservations about putting him under pressure to consume much more food than he wanted to eat.\n\nThe elderly Dirac remembered this dining room as a torture chamber. It was here, he said many times, that his father drove him into a life of silence and inhibition - the young Dirac, forced to speak French, found it easier to say nothing than to make errors that his father would punish unmercifully. No one else in the family left an account of these mealtimes, so we shall probably never know if he was exaggerating. Nor are we ever likely to know what his parents felt about the problems of bringing up a child who was both precociously clever and emotionally withdrawn. From a modern perspective, Charles and Flo were coping with a challenge they did not know they faced, one that may well have made their marital problems even worse. If they were living in Bristol today, the city council would - like most local authorities in the UK - give them support and enable their son to go to a special school.\n\nI for one accept the testimony of Paul Dirac and his mother that Charles Dirac was a domineering and insensitive father, though I don't believe he bullied his younger son into taciturnity. Much more likely, it seems to me, is that the relationship between Paul and Charles was doomed by nature rather than nurture: the young Dirac was born to be a child of few words and was pitiably unable to empathise with others, including his closest family. He laid all the blame for this at the feet of his father, though he disliked him for other reasons, too, with a bitterness that surprised the few people - including Kurt Hofer - who saw the extent of it. 'Why was Paul so bitter, so obsessed with his father?,' Hofer wondered after hearing his outburst. Perhaps the main reason was that Dirac knew in his heart he was not just his own man but, inescapably, his father's.\n**Thirty-one**\n\nDirac told physics students they should not worry about the meaning of equations, only about their beauty. This advice was good only for physicists whose sense of purely mathematical beauty is so keen that they can rely on it to see the way ahead. There have not been many such physicists - perhaps only Dirac himself.\n\nSTEVEN WEINBERG, Dirac Centenary Meeting, University of  \nBristol, 8 August 2002\n\nAll scientists, even the most eminent, are dispensable to science. Although inspired individuals influence it in the short term, the absence of any of them would be unlikely to make much difference to it in the long run. If Marie Curie and Alexander Fleming had never been born, radium and penicillin would have been discovered soon after the dates now in the textbooks.\n\nEvery scientist can hope, however, that posterity will judge him or her to have revealed more than a typical share of nature's secrets. By this criterion, there is no doubt that Dirac was a great scientist, one of the few who deserves a place just below Einstein in the pantheon of modern physicists. Along with Heisenberg, Jordan, Pauli, Schr\u00f6dinger and Born, Dirac was one of the group of theoreticians who discovered quantum mechanics. Yet his contribution was special. In his heyday, between 1925 and 1933, he brought a uniquely clear vision to the development of a new branch of science: the book of nature often seemed to be open in front of him. Freeman Dyson sums up what made Dirac's work so unusual:\n\nThe great papers of the other quantum pioneers were more ragged, less perfectly formed than Dirac's. His great discoveries were like exquisitely carved marble statues falling out of the sky, one after another. He seemed to be able to conjure laws of nature from pure thought - it was this purity that made him unique.\n\nDirac's book _The Principles of Quantum Mechanics_ was one of these statues, Dyson points out: 'He presents quantum mechanics as a work of art, finished and polished.' Never out of print, it remains the most insightful and stylish introduction to quantum mechanics and is still a  powerful source of inspiration for the most able young theoretical physicists. Of all the textbooks they use, none presents the theory with such elegance and with such relentless logic, a quality of Dirac's highlighted by Rudolf Peierls in 1972: 'The thing about Dirac is that he has a way of thinking logically [. . .] in a straight line, where we'd all tend to go off in a curve. It's this absolutely straight thinking in unexpected ways that makes his works so characteristic.'\n\nMost young physicists, however, are concerned not with the internal logic of quantum mechanics but with using the theory as a way of getting quick and reliable results. In effect, it gives scientists a completely dependable set of practical tools for describing the atomic and molecular world. Every day, tens of thousands of researchers in the microelectronics industry routinely employ the techniques developed by Dirac and his colleagues: ideas that took years to clarify are now used without a thought for the headaches they once caused their creators.\n\nThe modern trend to miniaturisation is making quantum mechanics even more important. In the growing field of ultra-miniature technology - usually called nanotechnology (from the Greek word for dwarf, _nanos_ ) - quantum mechanics is as indispensable as classical mechanics was to Brunel. In one branch of this new technology, spintronics (short for spin-based electronics), engineers are trying to develop new devices that rely not only on controlling the flow of the charge of electrons - the way conventional devices work - but also the flow of the electrons' _spins._ Because these can be flipped from one state to another much more quickly than charge can be moved around, spintronic devices should operate faster than conventional ones and produce less heat. If, as engineers hope, they can produce a spin-based transistor to replace conventional transistors in memory and logic circuits, it may be possible to continue the trend towards ever-more compact computers beyond the currently feasible limits.\n\nIt could be that, just over a century after Dirac first brought electron spin into the logical structure of quantum mechanics, his equation - once seen as mathematical hieroglyphics with no relevance to everyday life - becomes the theoretical basis of a multi-billion dollar industry.\n\nGreat thinkers are always posthumously productive. By this criterion, Dirac can be counted as one of the greatest of all scientists -  many of the concepts he introduced are still being developed, still instrumental in modern thinking. The Dirac equation, for example, is still a fecund source of ideas for mathematicians, who have long been fascinated by spinors, mathematical objects that first appeared in the equation. In Sir Michael Atiyah's opinion:\n\nNo one fully understands spinors. Their algebra is formally understood but their geometrical significance is mysterious. In some sense they describe the 'square-root' of geometry and, just as understanding the concept of the square root of -1 took centuries, the same might be true of spinors.\n\nDirac's influence is felt most strongly by scientists studying the universe's tiniest constituents. Experimenters can now smash particles together with energies so high that even Rutherford would have been impressed: at the Large Hadron Collider, the huge particle accelerator at CERN, they can recreate the conditions of the universe to within a millionth of a millionth of a second of the beginning of time. During the subatomic collisions produced in this and other accelerators, experimenters routinely see subatomic particles created and destroyed, processes that can be explained only using relativistic quantum field theory. Dirac's hand is all over this theory - he was one of its co-discoverers and the author of the action-principle formulation of quantum mechanics, now a crucial part of modern thinking about fields.\n\nOver the past twenty-five years or so, the gap between the energies accessible by particle accelerators and the energies needed to test the latest theories has widened alarmingly. The building of the accelerators is increasingly difficult and expensive for the international collaborations needed to fund and operate them, so new devices come on stream only slowly. One consequence has been that the theory of subatomic particles has run ahead of the supply of data from experiment, producing a scenario of the type Dirac envisaged in his landmark paper of 1931 where he set out an agenda for theoretical physics led by mathematics rather than experiment. One physicist who believed this was prescient was C. N. Yang: at a Princeton meeting they both attended in 1979, Yang suggested that when Dirac set out this idea, he had hit on a 'great truth'. In the same 1931 paper, Dirac suggested the existence of the anti-electron and the anti-proton and developed a quantum theory of magnetic monopoles using a geometric approach that has influenced generations of theoreticians. As experimenters  were unable to detect monopoles, Dirac regarded this project as another disappointment, and he died believing it unlikely that monopoles occur in nature. But, today, many physicists disagree, as monopoles are predicted by some simple generalisations of the Standard Model (the 'modern' monopole is a mathematically better-defined relative of Dirac's). Moreover, according to cosmologists, monopoles should have been created during the Big Bang in vast quantities and should now be detectable; that they are not is known as 'the monopole problem'.\n\nThe detection of Dirac's monopole would raise a question in virtual history: what would have been the effect on his reputation if the monopole had been detected around the time the positron was first observed? Such a pair of successes would have further bolstered his reputation among his colleagues and may well have made him much better known to the public. But there was never any chance that he would become a media celebrity like his most recent Lucasian successor, Stephen Hawking: it seemed not to have occurred to Dirac to write a popular book, nor would he have contemplated making the kind of forays into the media spotlight undertaken by Hawking, such as his appearances on _Star Trek_ , _The Simpsons_ and on the dance floor of a London nightclub. Yet Dirac admired such boldness more than most of his colleagues knew.\n\nDirac left his mark on several other fields besides quantum mechanics. One of his least typical contributions was his invention of a new way of separating different isotopes of a chemical element. He developed the method during the Second World War but it seemed that the idea was impracticable; it was soon forgotten, only to be independently rediscovered thirty years later by engineers in Germany and South Africa. His method still does not appear to be economically viable, but the development of new, ultra-strong materials still leaves open the possibility that the method could be used in the nuclear industry.\n\nAnother of Dirac's less characteristic pieces of work was his exploration of the wave and particle nature of electrons with Kapitza in 1933. Modern improvements in laser technology provided fresh opportunities to verify the existence of the Kapitza-Dirac effect, the diffraction (bending) of a thin beam of electrons by a standing wave of light. Kapitza and Dirac had themselves discussed the new  possibilities at the final meeting of the Kapitza Club in 1966. Several groups attempted to demonstrate the effect, but none was successful until, in the early spring of 2001, a team at the University of Nebraska observed it with a high-powered laser and a fine beam of electrons, using apparatus that would have fitted on a dining-room table. The Kapitza-Dirac effect is now used as a subtle probe of the wave-like and particle-like behaviours of both electrons and light.\n\nDirac also left a legacy in general relativity, if one not quite equal to his talent. It is a mystery that he showed so little interest in following up the discovery - made by Oppenheimer and his colleagues in 1939 - that Einstein's theory predicted the existence of black holes, objects with such a strong gravitational field that not even light can escape them. In Dirac's most important contribution to the theory of relativity, he set it out in analogy to his favourite Hamiltonian version of quantum mechanics and devised a set of complementary mathematical techniques (other physicists did similar work at about the same time). These methods have proved useful to astronomers studying closely spaced pairs of rotating neutron stars (usually called pulsars), orbiting each other, slowly losing energy. This gradual loss of energy can easily be explained by Einstein's general theory of relativity, especially if it is interpreted using the methods Dirac co-invented: the pulsars emit gravitational radiation, in much the same way as accelerating electrons emit electromagnetic radiation. The study of gravitational waves is now one of the most promising areas of astronomy.\n\nDirac's intuition for the workings of the universe on the largest scale was not nearly as strong as it was when he was focusing on atoms. There is, however, no denying the far-sightedness he showed when he reviewed the state of cosmology in the Scott Lecture he delivered shortly before the outbreak of the Second World War, when the subject was in its infancy. In one of a string of astute remarks he makes in passing, he hazarded an inspired guess that the complex structure of everything around us has its seeds in a quantum fluctuation in the initial state of the universe. 'The new cosmology', Dirac suggested, 'will probably turn out to be philosophically even more revolutionary than relativity or the quantum theory,' perhaps looking forward to the current bonanza in cosmology, where precise observations on some of the most distant objects in the universe are shedding light on the nature of reality, on the nature of matter and on the most advanced quantum theories. In the view of Nathan Seiberg, Dyson's colleague at the Institute for Advanced Study, 'The lecture would look just as impressive if the date on the front were not 1939 but 1999.'\n\nAlthough Dirac was, towards the end of his life, often defensive about his large numbers hypothesis, he always had faith in its truth. The modern view about the large numbers that fascinated him for decades is that only one of them is a mystery: the ratio between the strength of the electrical force and that of the gravitational force between an electron and a proton (1039). The fundamental problem is to understand why the gravitational force is so feeble compared with the other fundamental forces. All the other huge numbers that puzzled Dirac now follow from the standard theory of cosmology, so there is no need to guess links between them - the coincidences he spotted are illusory.\n\nDirac was convinced that the strength of the gravitational force had fallen since the beginning of time, and he invested many of his later years in trying to prove it, though observations made by astronomers on nearby planets in the solar system have now all but ruled it out. Although it is still just possible that Dirac's intuition was correct, the subject is currently low on today's research agenda. One scientist who always believed in his bones that Dirac was right was Leopold Halpern, who left Florida State University in 2004 and became a resident theoretician with a satellite-based experimental programme run by NASA and Stanford University, aiming to check some of the unverified predictions of Einstein's general theory of relativity.  Halpern hoped to compare the predictions of his theory with the satellite's observations but he was unable to complete his work before he died of cancer in June 2006.\n\nRegardless of how Dirac's predictions about the gravitational force fare in the future, his name will always be associated with the role of anti-matter at the beginning of the universe. According to modern Big Bang theory, matter and anti-matter were created in exactly equal amounts, at the very beginning of the universe, about 13.7 billion years ago. Soon afterwards, the decay of some of the heavy particles formed from the quarks and anti-quarks led to a small but crucial surfeit of matter over anti-matter, by just one part in a billion. The first scientist to analyse this difference in detail was Tamm's student  Andrei Sakharov - later a courageous human-rights activist in the Soviet Union - who discussed in 1967 how this excess came about and why the universe was left with an overwhelming preponderance of matter. Without that imbalance, the matter and anti-matter formed at the beginning of time would have annihilated each other immediately, so that the entire universe would only ever have amounted to a brief bath of high-energy light. Matter would, in that case, never have had the opportunity to discover anti-matter.\n\nThe surplus of matter over anti-matter at the beginning of the universe is still not understood, and thousands of physicists are working to understand it. Their main sources of experimental information are particle accelerators, where anti-matter is produced by smashing ordinary particles into each other and then quickly 'separating off' the anti-matter, before it is annihilated by matter. By comparing the decays of particles with those of their anti-particles, experimenters hope to get to the bottom of the matter-antimatter imbalance.\n\nEvery day, particle accelerators now generate about a hundred thousand billion positrons and five thousand billion anti-protons - a total of roughly a billionth of a gram. Although this quantity is only tiny, the ability to produce it at will demonstrates that _Homo sapiens_ now - a million years after our species evolved - uses anti-matter as a tool. Today, positrons are routinely generated in mass-produced equipment all over the world: doctors use positron emission tomography (PET) to see inside their patients' brains and hearts, without the need for surgery. It is a simple technique: the patient is injected with a tiny amount of a special radioactive chemical that spontaneously emits positrons, which interact with electrons in the tissue where the chemical settles. The photograph is a record of the radiation given off in the electron-positron annihilations.\n\nWithin just a few decades, positrons changed in the eyes of scientists from appearing outlandish novelties to being just another type of subatomic quantum; the public has become more familiar with anti-matter, too, from the fictional treatments of it in, for example, _Star Trek_ and Dan Brown's _Angels and Demons._ But what is most remarkable about the story of anti-matter is that human beings first understood and perceived it not through sight, smell, taste and touch but through purely theoretical reasoning inside Dirac's head.\n\nLike Einstein, Dirac was always in search of generalisations - theories that explain more and more about the universe, in terms of fewer and fewer principles. Both men believed, too, that the best way of achieving this was through theories expressed in terms of beautiful equations.  As a physicist, Dirac had been well served by mathematics, as he wrote in an unusually candid passage in 1975:\n\nIf you are receptive and humble, mathematics will lead you by the hand. Again and again, when I have been at a loss how to proceed, I have just had to wait until [this happened]. It has led me along an unexpected path, a path where new vistas open up, a path leading to new territory, where one can set up a base of operations, from which one can survey the surroundings and plan future progress.\n\nAlthough he never acknowledged it in public, the guiding hand of beauty had led Dirac not only to some rich new pastures of research but also into the deserts that yielded no fruit at all. In his talks, he was an ambassador of mathematical beauty, repeatedly underlining the triumphs of theories with this quality but not mentioning the years he had spent trying in vain to use sensually appealing mathematics to describe nature. It is striking that he put forward the principle of mathematical beauty several years after he had done his best work, and we have to suspect that some of his accounts of his greatest discoveries - usually portrayed as successes for his type of aestheticism - were reinterpreted in the light of his faith in the principle. In his pioneering papers on quantum mechanics, he never explicitly says that beauty was his guide; he recalled its value only in the tranquillity of his least productive years.\n\nDirac first made it clear that he was using the principle of mathematical beauty in the late 1940s, when he dismissed the renormalised theory of photons and electrons on the grounds that it was too ugly. He was, however, unable to use his principle constructively, to build new theories. It could therefore be argued that Dirac's passion for beauty was to some extent destructive, but he knew no other way: he was temperamentally unable to focus on any other subject in particle physics until he had found a truly beautiful theory of electrons and photons, without the disfiguring infinities.\n\nA way out of this alleged flaw in quantum field theory arrived, tragically, just too late for him: a particularly promising, infinity-free theory of electrons and photons began to circulate among theoreticians in the autumn of 1984, as he lay dying. Michael Green, of the University of London, and John Schwarz, of Caltech, had written a crucial paper showing that string theory might be able to form the basis of a unified theory of fundamental interactions. Previously, the theory appeared to say that the weak interactions must have perfect left-right mirror symmetry, contrary to experimental evidence. By proving that the theory can naturally describe the _breaking_ of this symmetry, and by resolving other embarrassing anomalies in the theory, Green and Schwarz began a revolution. Within weeks, string theory was the hottest topic in theoretical physics. Although the theory was far from complete - it was really a collection of inchoate concepts, all in need of development - there were strong signs that it contained the seeds of an exciting new framework for giving a unified account of all the fundamental interactions, encompassing the Standard Model and Einstein's general relativity.\n\nThe new theory describes nature not in terms of point-like particles but of pieces of string, so small that if they could be aligned end to end, it would take a billion billion of them to span a single atomic nucleus. In this picture of the fundamental constituents of the universe, there is only one fundamental entity - the string - and every type of particle, including the electron and the photon, is simply an excitation of the string, analogous to a mode of vibration of a tuning fork. The mathematics of the theory is fearsome, but underneath the complexities is a modern version of John Stuart Mill's desideratum of fundamental physics: a unified description of all the fundamental interactions.\n\nWhat would surely have impressed Dirac is that modern string theory has none of the infinities he abhorred. He would have revelled in the mathematical beauty of the theory, which delights not only the physicists who use it but also many mathematicians who have mined it for new concepts. It has turned out that string theory, much like the Dirac equation, is a fertile source of purely mathematical ideas that have a value for their own sake, not just as tools to understand nature. Dirac often said that he was interested in theories only as ways of accounting for nature, but he would probably have been intrigued to see, at the heart of string theory, mathematics known as complex projective geometry, a generalisation of his favourite branch of geometry.\n\nNo one has done more to shed light on string theory than the mathematical physicist Edward Witten, at the Institute of Advanced Study. In 1981, when he was a lecturer at the Erice summer school and thirty years old, he met Dirac briefly and heard his familiar condemnation of renormalisation but chose not to follow his advice. Dirac followed Witten's work and, in 1982, wrote - in his trembling hand - to the Papal Academy, supporting Witten's nomination for a special award and describing his mathematical work as 'brilliant'. From the early 1980s, Witten's reputation among string theorists has been comparable to Dirac's among quantum theorists half a century before.\n\nWitten believes that string theory seems to be the kind of theory that Dirac had in mind when he argued that a revolution was needed to produce a new theory free of infinities so that renormalisation was not needed:\n\nIn some ways Dirac's reaction to renormalization was vindicated because the better theories he said he wanted were eventually developed, with the advent of string theory. But by far the most progress towards the new theory was made by physicists who used and studied renormalization. So you'd have to look at the outcome for Dirac as bittersweet: he was partly right, but his approach was not entirely pragmatic.\n\nIt is hard to disagree with this tactfully expressed judgement about Dirac's principled but counterproductive attitude to renormalisation. If he could have shed some of the insistence on rigour that he learned as a student of pure mathematics and been able to retain some of the pragmatism he learned when training to be an engineer, his achievement would, in all likelihood, have been even greater. Perhaps, if he had been more active in quantum field theory, it would have advanced more quickly, and modern string theory would have arrived sooner.\n\nAlthough string theory is the only strong candidate for a unified theory of the fundamental interactions, by no means all theoreticians are convinced of its value. A substantial number of physicists worry that the theory makes sense only in more than four dimensions of space-time (it is easiest to formulate in ten or even eleven dimensions). More worrying, it has received little support from experiment: string theory has yet to make a clear-cut prediction that experimenters have been able to test. These are among the key signals, several physicists have argued, that the theory is absurdly overvalued and that it would be better to pursue other avenues. One of the most vocal sceptics is the Standard-Model pioneer Martin Veltman: 'String theory is mumbo jumbo. It has nothing to do with experiment.'\n\nBut it is clear from the comments Dirac repeatedly made in his lectures on the way theoretical physics should be done that he would have disagreed with these criticisms: he would have counselled string theorists to let the theory's beauty lead them by the hand, not to worry about the lack of experimental support and not to be deterred if a few observations appear to refute it. But he would have cautioned string theorists to be modest, to keep an open mind and never to assume that they are within sight of the end of fundamental physics. If past experience is anything to go by, another revolution will follow eventually.\n\nSuch was the advice this extraordinarily unemotional man offered to his colleagues: be guided, above all, by your emotions.\n**Abbreviations in Notes**\n\nAHQP Archives for the History of Quantum Physics, multiple locations, provided by Niels Bohr Library & Archives, American Institute of Physics, College Park, Maryland., USA (<http:\/\/www.amphilsoc.org\/library\/guides\/ahqp\/>).\n\nAIP American Institute of Physics, Center for the History of Physics, Niels Bohr Library, Maryland, USA.\n\nAPS Archive of the American Philosophical Society, Philadelphia, USA.\n\nBOD Bodleian Library, University of Oxford, UK.\n\nBRISTU Bristol University archive, UK.\n\nBRISTRO Bristol Records Office, UK.\n\nCALTECH California Institute of Technology, archive, USA.\n\nCHRIST'S Old Library, Christ's College, Cambridge University, UK.\n\nCHURCHILL Churchill Archives Centre, Churchill College, Cambridge University, UK. DDOCS Dirac letters and papers, property of Monica Dirac.\n\nEANGLIA Tots and Quots archive, University of East Anglia, Norwich, UK.\n\nFSU Paul A. M. Dirac Papers, Florida State University Libraries, Tallahassee, Florida, USA. All of the letters Dirac's mother wrote to him are in this archive.\n\nIAS Institute for Advanced Study, archive, USA.\n\nKING'S King's College, Cambridge; unpublished writings of J. M. Keynes.\n\nLC Library of Congress, Collections of the Manuscript Division.\n\nLINDAU Archive of Lindau meetings, Germany.\n\nNBA Niels Bohr Archive, at the Niels Bohr Institute, Copenhagen.\n\nPRINCETON Eugene Wigner Papers, Manuscripts Division, Department of Rare Books and Special Collections, Princeton University Library, USA.\n\nROYSOC Archives of the Royal Society, London, UK.\n\nRSAS Royal Swedish Academy of Sciences, Center for History of Science, Stockholm.\n\nSOLVAY Archives of the Solvay Conferences, Free University of Brussels, Belgium.\n\nSTJOHN St John's College archive, Cambridge, UK.\n\nSUSSEX Crowther archive, Special Collections at the University of Sussex, UK (the university holds the copyright of the archive).\n\nTALLA Dirac archive at the Dirac Library, Florida State University, USA, <http:\/\/pepper.cpb.fsu.edu\/dirac\/diracFA(HTML)_011.htm>.\n\nUCAM University of Cambridge archive, UK.\n\nUKNATARCHI National Archives of the UK, Kew.\n\nWISC University of Madison, Wisconsin, archives, USA.\n\n1851COMM Archives of the Royal Commission of 1851, Imperial College, London, UK.\n**Notes**\n\n#  **Prologue**\n\n A version of the 'more people who prefer to speak than to listen' remark, one of Dirac's favourites, is cited by Eugene Wigner in Mehra (1973: 819).\n\n Dirac made the 'God is a mathematician' remark in his _Scientific American_ article in May 1963.\n\n The quote from Darwin is taken from Part VII of his autobiography. The words were written on 1 May 1881.\n\n The author of the quote relating to Shakespeare was the late Joe Lannutti, a leading member of the Physics Department at Florida State University when Dirac arrived. The source of the quote is Peggy Lannutti, interview 25 February 2004. Lannutti also tells the story in J. Lannutti (1987) 'Eulogy of Paul A. M. Dirac' in Taylor (1987: 44-5).\n\n This account is taken from interviews with Kurt Hofer on 21 February 2004 and 25 February 2006, and many subsequent e-mails. The account was checked in detail via e-mails on 22 September 2007. Hofer's recollections are consistent in every detail with the account given by Dirac in Salaman and Salaman (1986), in his interview, AHQP, 1 April 1962 (pp. 5-6), and in the account he gave of his early life to his friends Leopold Halpern and Nandor Bal\u00e1zs. I spoke to these former colleagues of Dirac on 18 February 2003 and 24 July 2002, respectively. Dirac's wife gives her recollections of his experiences at the dining table in her letter to Rudolf Peierls, 8 July 1986, Peierls archive, additional papers, D23 (BOD).\n\n#  **Chapter one**\n\n Letter from Andr\u00e9 Mercier to Dirac and his wife, 27 August 1963, Dirac Papers 2\/5\/10 (FSU).\n\n Interview with Dirac, AHQP, 1 April 1962, p. 5.\n\n Dirac Papers 1\/1\/5 (FSU), see also the records of the Merchant Venturers' School in BRISTRO.\n\n See, for example Jones (2000: Chapter 5).\n\n Pratten (1991: 8-14).\n\n Although Flo lived in Cornwall only briefly, she would later insist that she was not English but Cornish. Source: interview with Christine Teszler, 22 January 2004.\n\n Flo Dirac mentions this in an undated letter to Manci Dirac, written in early February 1940 (DDOCS). By 1889, when Richard Holten was fifty, he was captain of the 547-ton _Augusta._\n\n Richard Holten was aware that official documents often name his wife as the head of the family. His sailing record is in _'They Sailed Out of the \"Mouth\"'_ by Ken and Megan Edwards, microfiche 2001, BRISTRO, FCI\/CL\/2\/3. See also Holten's Master's certificates, stored in the archives at the National Maritime Museum, Greenwich, London, UK.\n\n The details of Charles and Flo's early life together are in Charles's documents in Dirac Papers 1\/1\/8 (FSU).\n\n Louis Dirac was the illegitimate son of the recently widowed Annette Vieux, who  gave him her maiden name Giroud. Only later, when the baby's parents settled down together, did he take the surname of his father, Dirac; otherwise, his physicist grandson would have been called not Paul Dirac but Paul Giroud. Source: civil records in St Maurice, Switzerland. Louis Dirac's paeans to the beauty of the Alpine countryside are still in print, though rarely read. His poetry is published in Bioley (1903).\n\n Dalitz and Peierls (1986: 140).\n\n The pine cones are against a blue background; the leopard and clover are against a silver background (<http:\/\/www.dirac.ch\/diracwappen.html>). After the first member of the Dirac family obtained citizenship in the town of Saint Maurice, Swiss law accorded the same rights of citizenship to succeeding generations.\n\n This letter was written from Flo to Charles on 27 August 1897. This and the other extant letters from their correspondence are in Dirac Papers 1\/1\/8 (FSU). I am taking the arrival of e-mail for the UK public to be c. 1995.\n\n Felix's full name was Reginald Charles F\u00e9lix. His mother always anglicised his name, so I shall use that version of it here.\n\n The Diracs' address was 15 Monk Road, Bishopston, Bristol. The house still stands. The date of the Diracs' move are in UKNATARCHI HO\/144\/1509\/374920.\n\n The details of Dirac's birth are given in a letter from Flo to Paul and Manci, 18 December 1939, Dirac Papers, 1\/5\/1 (FSU). The description of Dirac as 'rather small' and the colour of his eyes is given in the poem 'Paul', Dirac Papers, 1\/2\/12 (FSU). Charles gave his children names used in his mother's family, the Pottiers. The origins of his children's names are as follows: Reginald Charles Felix was named after himself and after his grandfather Felix Jean Adrien Pottier; Paul Adrien Maurice's second name was that of Charles's maternal grandfather Pottier, and Maurice is probably in memory of his native town, Saint Maurice; Beatrice Isabelle Marguerite Walla's last name came from Charles's mother Julie Antoinette Walla Pottier, and she was probably named after Flo's sister Beatrice.\n\n Letter to Dirac from his mother, 18 December 1939, Dirac Papers, 1\/4\/9 (FSU).\n\n _Sunday Dispatch_ , 19 November 1933 (p. 17).\n\n On 16 May 1856, the _Bristol Times and Mirror_ called the area 'the people's park' soon after the council had taken the popular step in the early 1860s of acquiring it from its owners, who included the Merchant Venturers' Society.\n\n Mehra and Rechenberg (1982: 7n). The authors point out that Dirac checked the information they included about his early life.\n\n Dirac Papers, 1\/1\/12 (FSU).\n\n Dirac Papers, 1\/1\/9 (FSU).\n\n In the Dirac family archive, there is a copy of one of these postcards, marked by Charles Dirac on the back with the date 3 September 1907, presumably the date on which the photograph was taken (DDOCS).\n\n The friends were Esther and Myer Salaman, see Salaman and Salaman (1986: 69). The Salamans comment that Dirac read their account of his memories and verified them. For the earlier interview with AHQP on 4 April 1962, see p. 6.\n\n Interview with Dirac, AHQP, 4 April 1962; Salaman and Salaman (1986).\n\n Dirac told his daughter Mary that his parents always denied him a glass of water at the dinner table: interview with Mary Dirac, 21 February 2003.\n\n Letter from Dirac to Manci Bal\u00e1zs, 7 March 1936 (DDOCS).\n\n Letter from Dirac to Manci Bal\u00e1zs, 9 April 1935 (DDOCS).\n\n The school-starting age of five was introduced in the 1870 Education Act. Dirac's mother was in the first generation to benefit from compulsory education in England. Woodhead (1989: 5).\n\n Detail about the late serving of breakfast from Manci Dirac to Gisela Dirac in August 1988 in Caslano, Ticino. Interview with Mary Dirac, 21 February 2003.\n\n Details of the Bishop Road School in this period are available in the Head Teacher's report, in the BRISTRO archive: 'Bishop Road School Log Book' (21131\/SC\/BIR\/L\/2\/1).\n\n The source of these comments is family photos of the Dirac brothers and data on the boys' heights obtained when they were at school (see Felix's records in Dirac Papers, 1\/6\/1, FSU). In November 1914, Felix's height was five feet four inches, and his weight was one hundred and ten pounds, whereas Paul's height was four feet ten inches and his weight was sixty-six and a half pounds. Two years earlier, when Felix had the same age as Paul in late 1914, he was about the same height as his brother but was some twenty pounds heavier.\n\n Felix's school reports (1908-12) are in Dirac Papers, 1\/6\/1 (FSU).\n\n The description of Dirac as 'a cheerful little schoolboy' is given in his mother's poem 'Paul' in Dirac Papers, 1\/2\/12 (FSU).\n\n See 'Report cards' in Dirac Papers, 1\/10\/2 (FSU).\n\n Quoted in Wells (1982: 344). As an adult, Dirac did not add a letter L to the ends of words that end in the letter A, but he did have the characteristic practice among Bristolians of warmly accentuating the letter R; for example, in his pronunciation of 'universe'.\n\n Dirac's school reports are in Dirac Papers, 1\/10\/2 (FSU).\n\n Interview with Mary Dirac, 21 February 2003.\n\n Interview with Flo Dirac, _Svenska Dagbladet_ , 10 December 1933.\n\n The technique, applied to engineering, became popular in Renaissance Florence. The architect 'Pipo' Brunelleschi used such drawings to help his clients visualise the buildings and artefacts and to give his assistants a set of instructions so that they could do their work in his absence.\n\n In 1853, the first report of Sir Henry Cole's Department of Practical Art urged teachers to give the students exercises that 'contain some of the choicest elements of beauty, such as elegance of line, proportion and symmetry' (minutes of the Committee of the Council of Education [1852-3], HMSO, pp. 24-6). Aesthetic recommendations like this continued unabated in reports and guides to teaching for decades. In 1905, the Government's Board of Education stressed to junior school-teachers that 'the scholar should be taught to perceive and appreciate beauty of form and colour. The feeling for beauty should be cherished, and treated as a serious school matter.' See Board of Education (1905).\n\n Gaunt (1945: Chapters 1 and 2). The Aesthetic Movement was not the first flowering of the importance of beauty in British cultural life. For example, in the eighteenth century, it was important for people of taste to refer to the concept of beauty to demonstrate that they were cultured and intellectually distinguished. See Jones (1998). In 1835, Gautier defined the essence of aestheticism in the preface to one of his novels: 'Nothing is beautiful unless it is useless; everything useful is ugly, for it expresses a need and the needs of a man are ignoble and disgusting, like his poor and weak nature. The most useful place in the house is the lavatory.' Quoted in Lambourne (1996: 10).\n\n Hayward (1909: 226-7).\n\n Examples of Dirac's early technical drawings are in Dirac Papers, 1\/10\/2 (FSU). In one drawing, he gives an idealised image of a small building, showing two of its four vertical sides, this time taking full account of the perspective. Dirac underlines his understanding of perspective by showing that parallel lines on each side all meet at a single point in the far distance.\n\n The Government's Board of Education had recommended: 'No angular system of handwriting should be taught and all systems which sacrifice legibility and a reasonable degree of speed to supposed beauty should be eschewed,' Board of Education (1905: 69).\n\n Government report on inspection on 10-12 February 1914, reported in the log book of Bishop Road School, stored in BRISTRO: 'Bishop Road School Log Book' (21131\/SC\/BIR\/L\/2\/1).\n\n Westfall (1993: 13).\n\n Betty refers to her skating at the Coliseum rink in her letter to Dirac, 29 January 1937 (DDOCS).\n\n 'Paul', a poem by his mother, Dirac Papers, 1\/2\/12 (FSU). The relevant lines are: 'At eight years old in quiet nook \/ Alone, he stays, conning a book \/ On table high, voice strong and sweet \/ Poems of length he would repeat.'\n\n Interview with Flo Dirac in _Svenska Dagbladet_ , 10 December 1933.\n\n 'Recollections of the Merchant Venturers', 5 November 1980, Dirac Papers 2\/16\/4 (FSU).\n\n Salaman and Salaman (1986: 69).\n\n Dirac's scholarship covered his expenses at his next school, rising from \u00a38 in the first year (1914-15) to \u00a315 in the final year (1917-18). BRISTRO, records of the Bishop Road School, 21131\/EC\/Mgt\/Sch\/1\/1.\n\n Winstone (1972) contains dozens of photographs of Bristol during the period 1900-14.\n\n Interview with Mary Dirac, 14 February 2004.\n\n Dirac Papers, 1\/10\/6 (FSU). The lectures were held at the Merchant Venturers' Technical College, where Dirac would later study.\n\n Testimony of H. C. Pratt, who attended Bishop Road School from 1907 to 1912, to Richard Dalitz in the mid-1980s.\n\n#  **Chapter two**\n\n Words by H. D. Hamilton (School Captain, 1911-13). This is the second verse of the song.\n\n Lyes (n.d.: 5).\n\n Pratten (1991: 13).\n\n The following recollections were given to Richard Dalitz. Leslie Phillips attended the Merchant Venturer's School from 1915 to 1919. Some of Charles's codes are extant in Dirac Papers, 1\/1\/5 (FSU). In 1980, Dirac described his father's reputation in Dirac Papers, 2\/16\/4 (FSU).\n\n Interview with Mary Dirac, 7 February 2003.\n\n These comics, named after the 'penny stinker' (a cheap and nasty cigar), first became popular in the 1860s and were still popular in Dirac's youth. They were widely frowned upon for their lack of seriousness.\n\n Interview with Mary Dirac, 21 February 2003.\n\n Bryder (1988: 1 and 23). See also Bryder (1992: 73).\n\n Interview with Mary Dirac, 26 February 2004.\n\n Dirac's reports when he was at the Merchant Venturers' School are in Dirac Papers, 1\/10\/7 (FSU).\n\n See, for example, the reports of the Government's Department of Science and Art, from 1854, London: Her Majesty's Stationery Office.\n\n Stone and Wells (1920: 335-6).\n\n Stone and Wells (1920: 357).\n\n Stone and Wells (1920: 151).\n\n Interview with Dirac, AHQP, 6 May 1963, p. 1.\n\n Testimony to Richard Dalitz of J. L. Griffin, one of Dirac's fellow students in the chemistry class.\n\n _Daily Herald_ , 17 February 1933, p. 1.\n\n Interview with Dirac, AHQP, 6 May 1963, p. 2.\n\n Interview with Dirac, AHQP, 6 May 1963, p. 2.\n\n Dirac remarked that he 'was very interested in the fundamental problems of nature. I would spend much time just thinking about them'. See Interview with Dirac, AHQP, 1 April 1962, p. 2.\n\n Dirac (1977: 11); interview with Dirac, AHQP, 1 April 1962, pp. 2-3.\n\n Wells (1895: 4).\n\n See, for example, Monica Dirac, 'My Father', in Baer and Belyaev (2003).\n\n Pratten (1991: 24).\n\n Dirac (1977: 112).\n\n Testimony of Leslie Roy Phillips (fellow pupil with Dirac at Merchant Venturers' School, 1915-19) given to Richard Dalitz in the 1980s.\n\n Dirac Papers, 2\/16\/4 (FSU).\n\n Interview with Dirac, AHQP, 6 May 1963, p. 2.\n\n Later, Dirac received more books as prizes at the Merchant Venturers' School, including _Decisive Battles of the World_ and Jules Verne's _Michael Strogoff_ , an adventure story set in tsarist Russia. Some of the books Dirac won for school prizes at the Merchant Venturers' School are stored in the Dirac Library at Florida State University. Other information about Dirac's reading choices is from his niece Christine Teszler.\n\n Letter from Edith Williams to Dirac, 15 November 1952, Dirac Papers, 2\/4\/8 (FSU).\n\n From Merchant Venturers' School yearbooks 1919, BRISTRO 40659, 1.\n\n Stone and Wells (1920: 360).\n\n In the spring of 1921, Dirac planned the planting of vegetables on what looks like a geometric drawing of the garden in 6 Julius Road, with some annotations by his father. The plan, dated 24 April 1921, is in Dirac Papers, 1\/8\/24 (FSU).\n\n The Bishopston local Norman Jones told Richard Dalitz in the mid-1980s that his most vivid memory of Charles was 'seeing him always carrying an umbrella, struggling up the hill, often with his daughter, of whom he was very fond', interviews with Richard Dalitz, private communication.\n\n Interview with Dirac, AHQP, 1 April 1962. Felix's reports when he was at the Merchant Venturers' School are in Dirac Papers, 1\/6\/4 (FSU).\n\n Quoted in Holroyd (1988: 81-3).\n\n Interview with Monica Dirac, 7 February 2003; interview with Leopold Halpern, 18 February 2003.\n\n The Merchant Venturers' School used the facilities during the day, and the college used them during the evening.\n\n See Felix's university papers in Dirac Papers, 1\/6\/8 (FSU); the scholarships are recorded in BRISTRO 21131\/EC\/Mgt\/sch\/1\/1.\n\n Dirac took the qualifying examinations for the University of Bristol in 1917, three years earlier than most other applicants. He then spent a year studying advanced mathematics and finally qualified in 'physics, chemistry, mechanics, geometrical and mechanical drawing and additional mathematics', enabling him to take a degree in any technical subject. See Dirac Papers, 1\/10\/13 (FSU); details of Dirac's matriculation are also in a letter to him from his friend Herbert Wiltshire, 10 February 1952, Dirac Papers, 2\/4\/7 (FSU).\n\n Interview with Dirac, AHQP, 6 May 1963, p. 7.\n\n Interview with Flo Dirac, _Svenska Dagbladet_ , 10 December 1933.\n\n#  **Chapter three**\n\n Stone and Wells (1920: 371-2).\n\n _Bristol Times and Mirror_ , 12 November 1918, p. 3.\n\n 'Recollections of Bristol University', Dirac Papers, 2\/16\/3 (FSU).\n\n Lyes (n.d.: 29). At the Dolphin Street picture house, for example, Fatty Arbuckle starred in _The Butcher Boy._\n\n Quoted in Sinclair (1986).\n\n Dirac Papers, 2\/16\/3 (FSU).\n\n The list of textbooks that Dirac studied as an engineering student is in Dirac Papers, 1\/10\/13 and 1\/12\/1 (FSU).\n\n BRISTU, papers of Charles Frank. '[N]ot the faintest idea' is the testimony of Mr S. Holmes, a lecturer in electrical engineering, given to G. H. Rawcliffe, who, in turn, passed it to Charles Frank on 3 May 1973.\n\n Papers of Sir Charles Frank, BRISTU. 'Even as an engineering student, he spent much time reading in the Physics Library,' wrote Frank in a note in 1973.\n\n The college had classes on Saturday mornings as well as during weekdays (as was traditional, Wednesday afternoons were usually free for sporting activities). Information on Dirac at the Merchant Venturers' College is in the college's Year Books (BRISTRO 40659\/1). Dirac's student number was 1429.\n\n Letter to Dirac from Wiltshire, 4 May 1952, Dirac Papers, 2\/4\/7 (FSU). The first two names of Wiltshire, known to most people as Charlie, were Herbert Charles.\n\n Dirac Papers, 2\/16\/3 (FSU).\n\n Dirac Papers, 2\/16\/3 (FSU).\n\n Interview with Leslie Warne, 30 November 2004.\n\n Records of the Merchant Venturers' Technical College, BRISTRO.\n\n The photograph shows the visit of the University Engineering Society's visit to Messrs. Douglas' Works, Kingswood, 11 March 1919, Dirac Papers, 1\/10\/13 (FSU).\n\n 'Miscellaneous collection, FH Dirac', September 1915, Dirac Papers, 1\/2\/2 (FSU).\n\n Testimony to Richard Dalitz by E. B. Cook, who taught with Charles from 1918 to 1925.\n\n Testimony to Richard Dalitz by W. H. Bullock, who joined the Cotham Road School staff in 1925 and was later Charles's successor as Head of the French Department.\n\n Charles Dirac's letter is reproduced in Michelet (1988: 93).\n\n See Charles Dirac's Certificate of Naturalization, Dirac Papers, 1\/1\/3 (FSU). The papers concerning Charles Dirac's application for British citizenship are in UKNATARCHI HO\/144\/1509\/374920.\n\n Interview with Mary Dirac, 21 February 2003.\n\n Interview with Dirac, AHQP, 1 April 1962, p. 6.\n\n Letter to Dirac from Wiltshire, 10 February 1952, Dirac Papers, 2\/4\/7 (FSU).\n\n Dirac (1977: 110).\n\n Sponsel (2002: 463).\n\n Dirac (1977: 110).\n\n Five shillings (25 pence) secured a copy of _Easy Lessons in Einstein_ by Dr E. L. Slosson, a guinea (\u00a31.05) _The Reign of Relativity_ by Viscount Haldane.\n\n Eddington (1918: 35-9).\n\n Dirac Papers, 1\/10\/14 (FSU).\n\n Testimonies of Dr J. L. Griffin, Dr Leslie Roy Phillips and E. G. Armstead, provided to Richard Dalitz.\n\n Letter to Dirac from his mother, undated but written at the beginning of his sojourn in Rugby, c. 1 August 1920, Dirac Papers, 1\/3\/1 (FSU).\n\n _Rugby and Kineton Advertiser_ , 20 August 1920.\n\n Letters to Dirac from his mother, August and September 1920, especially 30 August and 15 September (FSU).\n\n Interview with Dirac, AHQP, 1 April 1962, p. 7.\n\n Letter from G. H. Rawcliffe, Professor of Electrical Engineering at Bristol to Professor Frank on 3 May 1973. BRISTU, archive of Charles Frank.\n\n Broad (1923: 3).\n\n Interview with Dirac, AHQP, 1 April 1962, p. 4 and 7.\n\n Schilpp (1959: 54-5). I have replaced Broad's archaic term 'latches' with 'laces'.\n\n Broad (1923: 154). This book is based on the course of lectures that Broad gave to Dirac and his colleagues. Broad prepared all his lectures meticulously and wrote them out in advance, making it easy for him to publish them. What we read in this book is therefore likely to be the material that Broad presented to Dirac.\n\n Broad (1923: 486).\n\n Broad (1923: 31).\n\n Dirac (1977: 120).\n\n Dirac (1977: 111).\n\n Schultz (2003: Chapters 18 and 19).\n\n Galison (2003: 238).\n\n Skorupski (1988).\n\n Mill (1892). His most important comments about the nature of science are in Book 2 and in Book 3 (Chapter 21).\n\n Dirac (1977: 111).\n\n Interview with Dirac, AHQP, 6 May 1963, p. 6.\n\n See <http:\/\/www.uh.edu\/engines\/epi426.htm> (accessed 27 May 2008).\n\n Nahin (1987: 27, n. 23). Heaviside never completed his autobiography.\n\n Interview with Dirac, AHQP, 6 May 1963, p. 4. Another example of the kind of neat tricks that engineers use and that Dirac read about as an engineering student is featured in the appendix to one of his set textbooks (Thom\u00e4len, 1907).\n\n The two books that Dirac used to study stress diagrams were Popplewell (1907) (see especially Chapter 5) and Morley (1919) (see especially Chapter 6).\n\n Dirac (1977: 113).\n\n The 'spoilsport' taught Dirac in the autumn of 1920. Dirac's reports are in Dirac Papers, 1\/10\/16 (FSU).\n\n Interview with Dirac, AHQP, 6 May 1963, p. 13. Dirac's lack of a qualification in Latin was not a bar to his admission to postgraduate study at Cambridge, but it would have made him ineligible to study there as an undergraduate.\n\n Warwick (2003: 406 n.); Vint (1956).\n\n Letter from Charles Dirac, 7 February 1921, STJOHN.\n\n Dirac took the examination on 16 June 1921. The examination papers are in Dirac Papers, 1\/10\/11 (FSU).\n\n Letter from Dirac to the authorities at St John's College, 13 August 1921, STJOHN.\n\n Boys Smith (1983: 23). A much higher estimate of the amount needed to live as a student in Cambridge at the time is given in Howarth (1978: 66): about \u00a3300.\n\n Letter from Charles Dirac, 22 September 1921, STJOHN.\n\n Unsigned letter from St John's College to Charles Dirac, 27 September 1921, STJOHN. The signatory concludes his letter: 'Perhaps before deciding [what to do] you would be so kind as to let me know the sum total of means that he would have at his disposal, I could then better advise what he can do.'\n\n#  **Chapter four**\n\n Interview with Dirac, AHQP, 6 May 1963, p. 9.\n\n Recollections of Dirac's first term in the mathematics class are from the testimony of E. G. Armstead in a letter to Richard Dalitz. The lecturer concerned was Horace Todd.\n\n Dirac (1977: 113); interview with Dirac, AHQP, 6 May 1963, p. 10.\n\n Interview with Dirac, AHQP, 1 April 1962, p. 3.\n\n It is likely that Dirac learned this subject from _Projective Geometry_ by G. B. Matthews (1914), published by Longmans, Green and Co. This book apparently meant a lot to him as it was one of the few books from his youth that he kept until his death. His copy is kept in his private library, stored in the Dirac Library, Florida State University.\n\n Dirac studied four courses in pure mathematics: 'Geometry of Conics; Differential Geometry of Plane Curves', 'Algebra and Trigonometry; Differential and Integral Calculus', 'Analytical Projective Geometry of Conics' and 'Differential Equations, Solid Geometry'. See Bristol University's prospectus for 1922-3, BRISTU.\n\n Dirac studied four courses in applied mathematics: 'Elementary Dynamics of a Particle and of Rigid Bodies', 'Graphical and Analytical Statics; Hydrostatics', 'Dynamics of a Particle and of Rigid Bodies' and 'Elementary Theory of Potential with Applications to Electricity and Magnetism'. See Bristol University's prospectus for 1922-3, BRISTU.\n\n Testimony of Norman Jones (who attended the Merchant Venturers' School from 1921 to 1925) to Richard Dalitz in the 1980s. Private communication from Dalitz.\n\n Interview with Dirac, AHQP, 1 April 1962, p. 8, and 6 May 1963, p. 10.\n\n The inclusion of the lectures on special relativity can be deduced from the presence of examination questions on the subject. See Dirac Papers, 1\/10\/15 and 1\/10\/15A (FSU).\n\n The term 'non-commuting' was introduced by Dirac later in the 1920s.\n\n Cahan (1989: 10-24); Farmelo (2002a: 7-12).\n\n Letter from Hass\u00e9 to Cunningham, 22 March 1923, STJOHN.\n\n Interview with Dirac, AHQP, 6 May 1963, p. 14. During Dirac's first visit to Cambridge, he had met Cunningham.\n\n Warwick (2003: 466, 467, 468, 493 and 495).\n\n Stanley (2007: 148); see also Cunningham (1970: 70), STJOHN.\n\n Letter from Ebenezer Cunningham to Ronald Hass\u00e9, 16 May 1923, and letter from Dirac to James Wordie, 21 July 1923, STJOHN. The grant from the Department of Science and Industrial Research was technically a maintenance allowance for research. Wordie became Dirac's tutor in his early years in Cambridge. Postcard from Dirac to his parents, 25 October 1926 (DDOCS).\n\n Dirac often spoke to close friends of the significance of this gesture by his father. Among those to attest to this: Kurt Hofer in an interview on 21 February 2004, Leopold Halpern in an interview in February 2006 and Nandor Bal\u00e1zs in an interview on 24 July 2002.\n\n#  **Chapter five**\n\n Gray (1925: 184-5).\n\n Boys Smith (1983: 10).\n\n See contemporary issues of the Cambridge students' magazine _The Granta_ ; for example, the poem 'The Proctor on the Granta', 19 October 1923.\n\n Boys Smith (1983: 20).\n\n Dirac kept the lodging accounts for the digs where he stayed as a student. See Dirac Papers, 1\/9\/10 (FSU). Dirac's landlady at 7 Victoria Road was Miss Josephine Brown, and he resided with her from October 1923 to March 1924. From April to June 1924, he stayed at 1 Milton Road. In his final postgraduate year, he lived at 55 Alpha Road.\n\n College records attest that he took his meals there: his bill for food in college during his first term was \u00a38 17s 0d, about the same as other students who ate there (STJOHN). The bill from Miss Brown includes no charges at all for either 'cooking' or 'food supplied'.\n\n From documents in STJOHN. A typical example of a menu that Dirac would have been offered is the following, served on 18 December 1920: 'Hare soup \/ Boiled mutton \/ Potatoes, mashed turnips, carrots au beurre \/ Pancakes \/ Ginger mould \/ Hot and cold pie \/ Anchovy eggs'. He will not have gone hungry.\n\n Interview with Monica Dirac, 7 February 2003.\n\n Interview with Mary Dirac, 21 February 2003. Dirac's words were 'give myself courage'.\n\n Interview with John Crook, 1 May 2003.\n\n Boys Smith (1983: 7).\n\n See contemporary issues of the Cambridge students' magazine _The Granta._\n\n Werskey (1978: 23).\n\n Snow (1960: 245). See also Dirac (1977: 117).\n\n Needham (1976: 34).\n\n Stanley (2007: Chapter 3), especially pp. 121-3; Earman and Glymour (1980: 84-5).\n\n Hoyle (1994: 146).\n\n de Bruyne, N. in Hendry (1984: 87).\n\n This description is taken mainly from Snow (1960), and from Cathcart (2004: 223).\n\n Wilson (1983: 573).\n\n Oliphant (1972: 38).\n\n Mott (1986: 20-2); Hendry (1984: 126).\n\n Oliphant (1972: 52-3).\n\n Carl Gustav Jung introduced the words 'extrovert' and 'introvert' into the English language in 1923.\n\n 'Naval diary, 1914-18. Midshipman', by Patrick Blackett, pp. 80-1. Text kindly supplied by Giovanna Blackett.\n\n Nye (2004: 18, 24-5).\n\n Boag et al. (1990: 36-7); Shoenberg (1985: 328-9).\n\n Boag et al. (1990: 34).\n\n Chukovsky's first book, _Crocodile_ , was published in 1917. I am indebted to Alexei Kojevnikov for this information. Chadwick later recalled Kapitza's first explanation of the nickname: when discussing his work with Rutherford, Kapitza was always afraid of having his head bitten off. (Chadwick papers, II 2\/1 CHURCHILL). Chadwick dismissed other explanations (e.g. Boag et al. 1990:11).\n\n Letter from Keynes to his wife Lydia, 31 October 1925, Keynes archive, JMK\/PP\/45\/ 190\/3\/14 to JMK\/PP\/45\/190\/3\/16 (KING'S \u00a9 2008).\n\n Spruch (1979: 37-8); Gardiner (1988: 240). See also _The Cambridge Review_ , 7 March 1942; Boag et al. (1990: 30-7).\n\n Parry (1968: 113).\n\n Letter from Kapitza to V. M. Molotov, 7 May 1935, translated in Boag et al. (1990: 322).\n\n See Hughes (2003), Section 1.\n\n Childs, W., Scotland Yard, to Chief Constable, Cambridge, 18 May 1923, KV 2\/777, UKNATARCHI.\n\n Werskey (1978: 92); Brown (2005: 26, 40).\n\n I am grateful to Maurice Goldhaber for his recollections of the meetings of the Kapitza Club, moderated by Kapitza, in 1933 and the first two terms of 1934.\n\n Blackett (1955).\n\n Postcard from Dirac, 16 August 1925 (DDOCS).\n\n See, for example, letters to Dirac from his mother, 26 October and 16 November 1925, 2 June 1926, 7 April 1927: Dirac Papers, 1\/3\/5 and 1\/3\/6 (FSU).\n\n Ramsay MacDonald's Labour Government was a minority one, whose survival depended on support from at least one of the other two parties. This partly explains the Government's moderate agenda.\n\n Letter to Dirac from his mother, 9 February 1924, Dirac Papers, 1\/3\/3 (FSU).\n\n In one letter, c. 1924, Felix requests a weekly wage of \u00a32 10s 0d. Dirac Papers, 1\/6\/3 (FSU).\n\n The spelling of the Reverend's name is not completely clear. His letters to Felix, including one dated 25 September 1923 and another dated 21 September, are in Dirac Papers, 1\/6\/6 (FSU). I am grateful to Peter Harvey for his advice on the theosophy of Felix's correspondent and to Russell Webb for pointing out the tone of the Reverend's letters, from the point of view of a follower of Eastern philosophy.\n\n Interview with Dirac, AHQP, 1 April 1962, pp. 5-6.\n\n Cunningham (1970: 65-6).\n\n Description of Compton is from the article 'Compton Sees a New Epoch in Science', _New York Times_ , 13 March 1932.\n\n Einstein (1949), in Schilpp (1949: 47).\n\n Hodge (1956: 53). Details of Dirac's early mathematical and scientific influences in Cambridge are in the final section of Darrigol (1992).\n\n Cunningham, E., 'Obituary of Henry Baker', _The Eagle_ , 57: 81. Dirac (1977: 115-16).\n\n _Edinburgh Mathematical Notes_ , 41, May 1957.\n\n Quoted in Darrigol (1992: 299-300).\n\n Moore (1903: 201); Baldwin (1990: 129-30). Moore's conception of the role of art in relation to morality is prefigured in Hegel and thence by his successors. Moore adapts this position to the utilitarian scheme that he took over from the Victorian thinker Henry Sidgwick. John Stuart Mill anticipates Moore through the conception of the great value of the 'higher' pleasures.\n\n As Budd describes Kant's conception of the experience of beauty, it was 'the facilitated play of imagination and understanding, mutually quickened (and so made pleasurable) by their reciprocal harmony' (2002: 32).\n\n Boag et al. (1990: 133).\n\n Letter from Einstein to Heinrich Zangger, 26 November 1915.\n\n This, and all of Dirac's publications until the end of 1948, is reproduced in Dalitz (1995).\n\n Interview with Dirac, AHQP, 7 May 1963, p. 7.\n\n Orwell (1946: 10).\n\n#  **Chapter six**\n\n Reference for Dirac by Cunningham, April 1925, provided for Dirac's application for a Senior Studentship, 1851COMM.\n\n Undated to Dirac from his mother, c. May 1924.\n\n Dirac was in room H7 on the first floor of New Court in Michaelmas (autumn) term. Later, he moved into other rooms: in Lent (winter) and Easter term 1925, he was in New Court room E12; from Michaelmas term in 1927 to Easter term 1930, he was in New Court room A4; in Michaelmas term in 1930, he was in Second Court room C4; from Michaelmas term 1936 to Michaelmas 1937, he was in New Court room I10.\n\n Letter from Dirac to Max Newman, 13 January 1935, Newman archive in STJOHN.\n\n Letter to Dirac from his mother, undated, c. November 1924, Dirac Papers, 1\/3\/3 (FSU).\n\n Letter from 'Technical Manager' (unnamed) at W & T Avery Ltd, 10 January 1925, Dirac Papers, 1\/6\/3 (FSU).\n\n Interview with Dirac, AHQP, 1 April 1962, p. 5; Salaman and Salaman (1986: 69). I am assuming that the date of Felix's death on his gravestone, 5 March 1925, is correct; on his death certificate, the date of his death is given as the day after.\n\n Letter to Dirac from his Auntie Nell, 9 March 1925, Dirac Papers, 2\/1\/1 (FSU).\n\n _Express and Star_ (local paper in Much Wenlock), 9 March 1925; _Bristol Evening News_ , 27 March 1925.\n\n Interview with Mary Dirac, 21 February 2003; interview with Monica Dirac, 7 February 2003. In an interview with Leopold Halpern, 18 February 2003, Halpern commented that Dirac found the suicide of Felix too painful to talk about.\n\n _Bristol Evening News_ , 9 March 1925.\n\n _Bristol Evening News_ , 10 March 1925.\n\n Dirac often remarked on this. His feelings are recorded in Salaman and Salaman (1986: 69). His close friend Leopold Halpern also mentioned that Dirac had mentioned this to him, quite independently (interview on 18 February 2002).\n\n Letter to Dirac from his mother, 4 May 1925, Dirac Papers, 1\/3\/4 (FSU). Dirac always mentioned this when he opened his heart to friends and even mentioned it to his children.\n\n Flo wrote her poem 'In Memoriam. To Felix' on 5 March 1938. The poem is in Dirac Papers, 1\/2\/12 (FSU).\n\n Letter to Dirac from his mother, 22 March 1925, Dirac Papers, 1\/3\/4 (FSU).\n\n Death certificate of Felix Dirac, registered 30 March 1925.\n\n Interview with Leopold Halpern, 18 February 2003.\n\n Interview with Christine Teszler, 22 January 2004.\n\n The problem that Dirac addressed was: if light consists of photons, as Compton had argued, how would these particles be affected by collisions with electrons swirling around on the surface of the Sun?\n\n Mehra and Rechenberg (1982: 96).\n\n Dirac (1977: 118).\n\n C. F. Weizs\u00e4cher, in French and Kennedy (1985: 183-4).\n\n Pais (1967: 222). Pais gives a vivid description of Bohr's strange oratory, noting 'Bohr's precept never to speak more clearly than one thinks.'\n\n Letters from Bohr to Rutherford, 24 March 1924 and 12 July 1924, UCAM Rutherford archive.\n\n Elsasser (1978: 40-1).\n\n In his AHQP interview on 1 April 1962 (p. 9) and in an interview on 26 June 1961 (Van der Waerden 1968: 41), Dirac says he was not present, whereas elsewhere he says he was there (Dirac 1977: 119).\n\n Heisenberg recalls his experience at the Kapitza Club, and of staying with the Fowlers, in the BBC _Horizon_ programme 'Lindau', reference 72\/2\/5\/6025. The recording was made on 28 June 1965, in Dirac's presence.\n\n The application is held by the 1851COMM.\n\n Letter to Dirac from his mother, with a contribution from his father, June 1925, in Dirac Papers, 1\/3\/4 (FSU). The application was advertised in the _Times Higher Education Supplement_ , his mother says.\n\n This proof copy is in Dirac Papers, 2\/14\/1 (FSU).\n\n An English translation of this paper, together with other key papers in the early history of quantum mechanics, are reprinted in Van der Waerden (1967).\n\n Dirac (1977: 119).\n\n Interview with Flo Dirac, _Stockholms Dagblad_ , 10 December 1933 _._\n\n Darrigol (1992: 291-7).\n\n Dirac (1977: 121).\n\n Letter from Albert Einstein to Paul Ehrenfest, 20 September 1925, in Mehra and Rechenberg (1982: 276).\n\n Dirac (1977: 121-5).\n\n Dirac (1977: 122).\n\n Here, X and Y are mathematical expressions of a type known as partial differentials. What is important is the superficial similarity between the form of the Poisson bracket and the difference AB - BA.\n\n Eddington (1928: 210).\n\n Elsasser (1978: 41).\n\n Reference for Dirac, written by Fowler in April 1925, for the Royal Commission of the Exhibition of 1851, 1851COMM.\n\n Dalitz and Peierls (1986: 147). The student was Robert Schlapp, who was studying under the veteran Sir Joseph Larmor.\n\n Van der Waerden (1960).\n\n Letters from Oppenheimer to Francis Fergusson, 1 November and 15 November 1925; in Smith and Weiner (1980: 86-9).\n\n Bird and Sherwin (2005: 44).\n\n Letter to Dirac from his mother, 16 November 1925 (she repeats the image of 'the block of ice' in another letter to Dirac, written on 24 November), Dirac Papers, 1\/3\/4 (FSU).\n\n Heisenberg later remarked that when he read Dirac's first paper on quantum mechanics, he assumed that its author was a leading mathematician (BBC _Horizon_ programme, 'Lindau', reference 72\/2\/5\/6025).\n\n Frenkel (1966: 93).\n\n Born (1978: 226).\n\n Letter to Dirac from Heisenberg, 23 November 1925, Dirac Papers, 2\/1\/1 (FSU).\n\n All these letters from Heisenberg to Dirac at this time are in Dirac Papers, 2\/1\/1 (FSU).\n\n Beller (1999: Chapter 1); see also Farmelo (2002a: 25-6).\n\n#  **Chapter seven**\n\n Letter from Einstein to Michel Besso, 25 December 1925, quoted in Mehra and Rechenberg (1982: 276).\n\n Letter from Einstein to Ehrenfest, 12 February 1926, quoted in Mehra and Rechenberg (1982: 276).\n\n Bokulich (2004).\n\n Dirac (1977: 129).\n\n Slater (1975: 42).\n\n Jeffreys (1987).\n\n Bird and Sherwin (2005: 46).\n\n Interview with Oppenheimer, AHQP, 18 November 1963, p. 18.\n\n 'The Cambridge Review', 'Topics of the Week' on 14 March and 12 May 1926.\n\n Letters to Dirac from his mother, 16 March 1926 and 5 May 1926, Dirac Papers, 1\/3\/5 (FSU).\n\n Morgan et al. (2007: 83); Annan (1992: 179-80); Brown (2005: 40 and Chapter 6); Werskey (1978: 93-5).\n\n Quoted in Brown (2005: 75).\n\n Wilson (1983: 564-5).\n\n Morgan et al. (2007: 84).\n\n Morgan et al. (2007: 80-90).\n\n Dirac Papers, 2\/1\/2 (FSU).\n\n This description follows the one given by Kapitza of his Ph.D. graduation ceremony three years before, when the proceedings were the same. See Boag et al. (1990: 168-9).\n\n Letter to Dirac from his mother, 28 June 1926, Dirac Papers, 1\/3\/5 (FSU).\n\n The Cambridge newspapers reported a wave of heat deaths in July. See the _Cambridge Daily News_ , 15 August 1926, the hottest day in the town for three years.\n\n Dirac had carefully studied a derivation of the radiation spectrum produced by the previously unknown Satyendra Bose, a student in Calcutta. No one had understood quite why his derivation worked. Einstein developed Bose's ideas to produce a theory that is now named after both men.\n\n Postcard from Dirac to his parents, 27 July 1926, DDOCS.\n\n Letter to Dirac from Fermi, Dirac Papers 2\/1\/3 (FSU).\n\n Greenspan (2005: 135); Sch\u00fccking (1999: 26).\n\n Letter to Dirac from his mother, 2 October 1926, Dirac Papers 1\/3\/6.\n\n Mott (1986: 42).\n\n#  **Chapter eight**\n\n Wheeler (1998: 128-9). On 24 April 1932, Jim Crowther wrote of hearing a similar anecdote from Bohr over afternoon tea (Book I of Crowther's notes from his meeting with Bohr, pp. 99-100 [SUSSEX]).\n\n Book I of Crowther's notes from his meeting with Bohr, 24 April 1932, pp. 96-101, SUSSEX. See also the article on Dirac by John Charap in _The Listener_ , 14 September 1972, pp. 331-2.\n\n Book I of Crowther's notes from his meeting with Bohr, p. 99, SUSSEX.\n\n Dirac (1977: 134).\n\n Bohr's words ( _Nicht um zu kritisieren aber nur um zu lernen_ ) are quoted in Dirac (1977: 136).\n\n Postcard from Dirac to his parents, 1 October 1926 (DDOCS).\n\n Letter from Dirac to James Wordie, 10 December 1926, STJOHN; Dirac (1977: 139).\n\n The phrase 'liked the sound of his own voice' is taken from the letter John Slater wrote to John Van Vleck on 27 July 1924, John Clarke Slater papers APS. See also Cassidy (1992: 109).\n\n Crowther notes, p. 99, SUSSEX.\n\n The wave is what is known mathematically as a complex function, which means that the wave at any point has two parts: one real, the other imaginary. The 'size' of the wave at any point, related to both parts, is called its modulus. According to Born, the probability of detecting the quantum in a tiny region near a point is related to the _square_ of the modulus of the wave.\n\n Pais (1986: 260-1).\n\n Heisenberg (1967: 103-4).\n\n Interview with Oppenheimer, AHQP, 20 November 1963.\n\n Weisskopf (1990: 71).\n\n Interview with Dirac, AHQP, 14 May 1963, p. 9.\n\n Garff (2005: 308-16, 428-31).\n\n Interview with Monica Dirac, 3 May 2006.\n\n Quoted in Garff (2005: 311); interview with Dirac, AHQP, 14 May 1963, p. 9.\n\n M\u00f8ller (1963).\n\n Dirac had also seen the need for the function when he was studying Eddington's _The Mathematical Theory of Relativity_ (1923). On page 190, Eddington uses non-rigorous mathematics, and he drew attention to this in a footnote, which Dirac read. This was an example of the case where the delta function is needed to make some sense of a scientific equation which would otherwise be mathematically unintelligible. See interview with Dirac, AHQP, 14 May 1963, p. 4.\n\n Interview with Dirac, AHQP, 6 May 1963, p. 4.\n\n Heaviside (1899: Sections 238-42).\n\n L\u00fctzen (2003: 473, 479-81).\n\n Interview with Heisenberg, AHQP, 19 February 1963, p. 9.\n\n Dirac (1962), report of the Hungarian Academy of Sciences, KFKI-1977-62.\n\n Letter from Einstein to Paul Ehrenfest, 23 August 1926, see Pais (1982: 441).\n\n Dirac mentioned this in a press release issued by Florida State University on 24 November 1970; Dirac Papers, 2\/6\/9 (FSU).\n\n Letters to Dirac from his mother, 19 November, 26 November, 2 December, 9 December 1926, Dirac Papers, 1\/3\/6 (FSU).\n\n It is possible that Charles wrote other letters to Dirac. If so, Dirac did not keep them - uncharacteristically, as he appears to have kept most of his family correspondence. Moreover, the frequent letters from Dirac's mother often send messages from his father, indicating that his father was communicating to his son via her, a common arrangement in family correspondence of this type.\n\n Letter to Dirac from his father, 22 December 1926, Dirac Papers, 1\/1\/7 (FSU).\n\n Letter to Dirac from his mother, 25 December 1926, Dirac Papers, 1\/3\/6 (FSU).\n\n Mehra (1973: 428-9).\n\n Postcard from Dirac to his parents, 10 January 1927, DDOCS.\n\n Slater (1975: 135).\n\n Elsasser (1978: 91).\n\n Born (2005: 88).\n\n 'The deepest thinker': Dirac (1977: 134).\n\n 'The most remarkable scientific mind . . .': Crowther notes, p. 21, SUSSEX. The 'logical genius' comment is in the interview with Bohr, AHQP, 17 November 1962, p. 10.\n\n Both quotes from the Crowther notes, p. 97, SUSSEX.\n\n 'PAM Dirac and the Discovery of Quantum Mechanics', Cornell colloquium, 20 January 2003, available at <http:\/\/arxiv.org\/PS_cache\/quant-ph\/pdf\/0302\/0302041v1.pdf> (accessed 24 September 2007).\n\n#  **Chapter nine**\n\n Bird and Sherwin (2005: 62).\n\n Bernstein (2004: 23).\n\n Bird and Sherwin (2005: 65).\n\n The address of the Carios' home was Giesmarlandstrasse 1. See interview with Oppenheimer, AHQP, 20 November 1963, p. 4.\n\n Michalka and Niedhart (1980: 118).\n\n Frenkel (1966: 93).\n\n Interview with Gustav Born, 6 April 2005.\n\n Frenkel (1966: 93).\n\n Weisskopf (1990: 40).\n\n Bird and Sherwin (2005: 56, 58).\n\n See Frenkel (1966: 94) for a reference to the practice of Mensur in G\u00f6ttingen. See also Peierls (1985: 148).\n\n Interview with Oppenheimer, AHQP, 20 November 1963, p. 6.\n\n Interview with Oppenheimer, AHQP, 20 November 1963, p. 11.\n\n Delbr\u00fcck, M. (1972) 'Homo Scientificus According to Beckett', available at <http:\/\/www.ini.unizh.ch\/~tobi\/fun\/max\/delbruckHomoScientificusBecket1972.pdf>, p. 135 (accessed 13 May 2008).\n\n Greenspan (2005: 144-6).\n\n Elsasser (1978: 71-2).\n\n Letter from Raymond Birge to John Van Vleck, 10 March 1927, APS.\n\n Elsasser (1978: 51).\n\n Frenkel (1966: 96).\n\n Delbr\u00fcck (1972: 135).\n\n Wigner (1992: 88).\n\n Mill's comment is in Mill (1873: Chapter 2).\n\n Interview with Oppenheimer, AHQP, 20 November 1963, p. 11.\n\n During his time in G\u00f6ttingen, Dirac successfully applied his theory to the light emitted by atoms when they make quantum jumps, apparently after discussions with Bohr. See Weisskopf (1990: 42-4).\n\n Letter from Pauli to Heisenberg, 19 October 1926, reprinted in Hermann et al. (1979). See also Beller (1999: 65-6); Cassidy (1992: 226-46).\n\n Heisenberg (1971: 62-3).\n\n Heisenberg demonstrated that the principle also applied to energy and time and to other pairs of quantities known technically as 'canonically conjugate variables'.\n\n This was a popular walk with students. See, for example, Frenkel (1966: 92). On 5 April 1927, Dirac referred to the walk in a postcard of the path to his parents (DDOCS).\n\n Lecture by Dirac, 20 October 1976, 'Heisenberg's Influence on Physics': Dirac Papers, 2\/29\/19 (FSU); see also the interview with Dirac, AHQP, 14 May 1963, p. 10.\n\n See the article on complementarity in French and Kennedy (1985), e.g. Jones, R.V. 'Complementarity as a Way of Life', pp. 320-4; see also the illustration of Bohr's coat of arms, p. 224.\n\n Interview with Dirac, AHQP, 10 May 1969, p. 9.\n\n Eddington (1928: 211). This book is an overview of the latest ideas in physics based on a series of lectures he gave between January and March 1927.\n\n Eddington (1928: 209-10).\n\n Dirac (1977: 114).\n\n Dirac Papers, 2\/28\/35 (FSU). The seminar took place on 30 October 1972. See Farmelo (2005: 323).\n\n#  **Chapter ten**\n\n Interview with Oppenheimer, AHQP, 20 November 1963, p. 5.\n\n Greenspan (2005: 137).\n\n Goodchild (1985: 20). Even if Dirac did not write these words, he agreed with their sentiment; see interview with von Weizs\u00e4cher, AHQP, 9 June 1963, p. 19.\n\n Dirac (1977: 139); Greenspan (2005: 141).\n\n Greenspan (2005: 142), and von Meyenn and Sch\u00fccking (2001: 46). The student was Otto Heckmann. Boys Smith's comment is from a conversation with his former colleague at St John's College, Cambridge, Peter Goddard, 5 July 2006.\n\n Information on scholarship from Angela Kenny, archivist, Royal Commission for the Exhibition of 1851 (e-mail, 10 December 2007).\n\n Letter from Dirac to James Wordie, 28 February 1927, STJOHN.\n\n Letter to Dirac from his mother, 28 June 1928, Dirac Papers, 1\/3\/8 (FSU).\n\n Greenspan (2005: 145).\n\n Greenspan (2005: 146).\n\n Letter to Dirac from his mother, 7 April 1927, Dirac Papers, 1\/3\/7 (FSU).\n\n Letter to Dirac from his mother, 20 May 1927, Dirac Papers, 1\/3\/7 (FSU).\n\n Letter to Dirac from his mother, 6 January 1927, Dirac Papers, 1\/3\/7 (FSU).\n\n Letter to Dirac from his mother, 10 February 1927, Dirac Papers, 1\/3\/7 (FSU).\n\n Letter to Dirac from his mother, 20 May 1927, Dirac Papers, 1\/3\/7 (FSU).\n\n Letter to Dirac from his mother, _c_. 26 March 1927, Dirac Papers, 1\/3\/7 (FSU).\n\n Flo enjoyed the company of several men in her classes and even put Dirac in touch with one of them, a German-speaking insurance clerk Mr Montgomery ('Monty'). Letter to Dirac from his mother, 18 March 1927, Dirac Papers, 1\/3\/7 (FSU).\n\n These recollections were given to Richard Dalitz in the 1980s.\n\n Letter from Dirac to Manci Bal\u00e1zs, 7 April 1935, DDOCS.\n\n Letter from Dirac to Manci Bal\u00e1zs, 17 June 1936, DDOCS.\n\n Their address was 173 Huntingdon Road. Fen (1976: 161); Boag et al. (1990: 78).\n\n The conference was held at L'Institut de Physiology Solvay au Parc L\u00e9opold, from 24 to 29 October 1927.\n\n Letter from John Lennard-Jones (of Bristol University) to Charles L\u00e9fubure (Solvay official), 9 March 1928, SOLVAY.\n\n See <http:\/\/www.maxborn.net\/index.php?page=filmnews> (accessed 13 May 2008).\n\n Heisenberg (1971: 82-8); interview with Heisenberg, AHQP, 27 February 1963, p. 9. The location of the hotel is specified in a letter to Dirac from the conference administrator on 3 October 1927: Dirac Papers, 2\/1\/4 (FSU).\n\n Dirac (1982a: 84).\n\n Interview with Heisenberg, AHQP, 27 February 1963, p. 9.\n\n Heisenberg (1971: 85-6).\n\n In the early 1850s, the _Punch_ humorist Douglas Jerrold quipped about the controversial feminist writer Harriet Martineau, 'There is no God, and Harriet Martineau is her prophet.' See A. N. Wilson (2002), _The Victorians_ , London: Hutchinson, p. 167.\n\n Dirac Papers, 2\/26\/3 (FSU).\n\n Dirac (1977: 140).\n\n Dirac (1977: 141).\n\n#  **Chapter eleven**\n\n Menu from College records, STJOHN.\n\n Crowther (1970: 39) and Charap (1972).\n\n Interviews with Dirac, AHQP, 1 April 1962, p. 15; 7 May 1963, pp. 7-8.\n\n Dirac gave contradictory accounts of the goal he was pursuing at that time. In one account, he stated that he was seeking the answer to the question 'How could one get a satisfactory relativistic theory of the electron?' (Dirac 1977: 141). In another account, he says that 'my dominating interest was to get a satisfactory relativistic theory of a particle, of the simplest possible kind, which was presumably a spinless particle.' Dirac wrote the latter words on a single sheet of paper headed 'Sommerfeld Atombau un Spektralinen II 539.18' in Dirac Papers, 2\/22\/15 (FSU). I prefer to use the 1977 account as it is the nearest thing we have to a carefully prepared history of Dirac's thinking in his own hand.\n\n Farmelo (2002a: 133).\n\n See the notes for Dirac's lectures in the 1970s and 1980s: 2\/28\/18-2\/29\/52 (FSU).\n\n Huxley's 1870 Presidential Address to the British Association for the Advancement of Science in Huxley (1894). Dirac uses similar words: 'The originator of a new idea is always rather scared that some development may happen which will kill it' (1977: 143).\n\n Interview with Dirac, AHQP, 7 May 1963, p. 14; Dirac (1977: 143).\n\n Letter from Darwin to Bohr, 26 December 1927 (AHQP).\n\n Interview with Rosenfeld, AHQP, 1 July 1963, pp. 22-3.\n\n Mehra (1973: 320).\n\n The talented young physicist Rudolf Peierls remarked that, even after a few days studying the equation, 'I have begun to have an inkling of what it deals with, but I haven't understood a single word.' Letter from Peierls to Hans Bethe, 4 May 1924, quoted in Lee (2007b: 33-4).\n\n _Florida State University Bulletin_ , 3 (3), 1 February 1978.\n\n Slater (1975: 145).\n\n Postcard from Darwin to Dirac, 30 October 1929, Dirac Papers, 2\/1\/9 (FSU).\n\n Dirac gave courses on quantum mechanics in the Michaelmas and Lent terms of 1927-8 and was paid \u00a3100 for the pair: see the letter from the Secretary to the Faculty of Mathematics, 16 June 1927, Dirac Papers, 2\/1\/4 (FSU).\n\n Crowther later affirmed that he had left the Communist Party by 1950, but it is not clear when he left it. I thank Allan Jones for this information.\n\n Clipping, annotated by Charles Dirac, in Dirac Papers, 1\/12\/5 (FSU).\n\n _The Times_ , 5 October 1931, p. 21. This well-briefed article was written by a journalist who appears to have succeeded in persuading Dirac to speak about his work.\n\n 'Mulling over the Universe with Paul Dirac', interview by Andy Lindstrom, _Tallahassee Democrat_ , 15 May 1983.\n\n Letter to Dirac from his mother, 26 January 1928, Dirac Papers, 1\/3\/8 (FSU). See also postcard from Dirac to his parents, 1 February 1928 (DDOCS).\n\n See the entry for Bishop Whitehead in _Crockford's Clerical Dictionary_ , 1947, p. 1,416. See also Billington Harper (2000: 115-26, 129-33, 293-5). The quoted description of Mrs Whitehead is on p. 145. I thank Oliver Whitehead and the late David Whitehead, grandsons of Isabel Whitehead, for the information in the description of Isabel Whitehead's home.\n\n#  **Chapter twelve**\n\n Kojevnikov (1993: 7-8).\n\n Peierls (1985: 62-3).\n\n Kojevnikov (2004: 64-5).\n\n Letter from Tamm to his wife, 4 March 1928, in Kojevnikov (1993: 7).\n\n 'The tulip fields are all in flower now': postcard from Dirac to his parents, 29 April 1928 (DDOCS). '[Leiden] is below sea level and there are nearly as many canals as streets': postcard from Dirac to his parents, 29 June 1927 (DDOCS).\n\n Letter from Tamm to his wife, undated, Kojevnikov (1993: 8).\n\n Casimir (1983: 72-3).\n\n Brown and Rechenberg (1987: 128).\n\n Letter from Heisenberg to Pauli, 31 July 1928, in Kronig and Weisskopf (1964).\n\n Peierls (1987: 35). In this account, Peierls remembers going to the theatre, but it seems from his letter to Dirac of 14 September 1928 (Lee [2007: 50]) that they went to the opera. I am grateful to Professor Olaf Breidbach for his comments on early twentieth-century Prussian politesse.\n\n Born (1978: 240) and Greenspan (2005: 151-3).\n\n Sch\u00fccking (1999: 27).\n\n Bohr nicknamed Gamow 'Joe' after the standard name for cowboys in western movies, which Bohr especially liked (interview with Igor Gamow, 3 May 2004). See also Reines (1972: 289-99; see pp. 280); Mott (1986: 28).\n\n The only exception is the paper that Dirac co-authored with Rutherford's student J. W. Harding, 'Photoelectric Absorption in Hydrogen-Like Atoms', in January 1932.\n\n Gamow (1970: 14).\n\n Wigner (1992: 9-15).\n\n Letter from Gabriel Dirac to Manci Dirac, 5 September 1940: 'It may interest you to know that everybody (Prof [Max] Born, Morris [Pryce] and Daddy [Paul Dirac]) says that Johnny von Neumann is the world's best mathematician' (DDOCS).\n\n Fermi (1968: 53-9).\n\n Wigner (1992: 37-43).\n\n Interview with Pat Wigner, 12 July 2005.\n\n Dirac wrote to his parents on 18 July 1928: 'The woods here are full of fireflies in the evening. I have been to the top of the Harz mountains' (DDOCS).\n\n Dirac's wife would later write to him: 'It seems the beautiful scenery has the same effect on you as a beautiful book has on me', 12 August 1938 (DDOCS).\n\n Letter to Dirac from his mother, 12 July 1928, Dirac Papers, 1\/3\/8 (FSU).\n\n Sinclair (1986: 32-3).\n\n Letter from Dirac to Tamm, 4 October 1928, Kojevnikov (1993: 10). The conference lasted from 5 August to 20 August.\n\n Brendon (2000: 241).\n\n Salaman and Salaman (1986: 69). In this article, Dirac is quoted as giving 1927 as the date of the experience; this is impossible as he did not visit Russia that year.\n\n He first took a boat to Constantinople (renamed Istanbul in the following year), then sailed on to Marseilles via Athens and Naples, before travelling across France and then home. He planned to arrive in Bristol on Monday, 10 September (letter from Dirac to his parents, 8 September 1928, DDOCS).\n\n Letter to Dirac from his mother, 28 October 1928, Dirac Papers, 1\/3\/8 (FSU). A copy of the speech is in this file of the archive.\n\n In mid-December, Dirac read a paper by Klein showing that the Dirac equation predicted that if a beam of electrons is fired at a barrier, more electrons will be reflected than were present in the original beam. It was as if a tennis ball struck a player's racket and not one but several balls flew off it.\n\n Howarth (1978: 156).\n\n _Cambridge Review_ , 29 November 1929, pp. 153-4. See also the rhapsodic review in the _Times Literary Supplement_ , 24 October 1929.\n\n Draft letter to Dirac from L. J. Mordell, 4 July 1928, Dirac Papers, 2\/1\/7 (FSU).\n\n Mott (1986: 42-3).\n\n Letter from Jeffreys to Dirac, 14 March 1929, Dirac Papers, 2\/1\/8 (FSU).\n\n St John's awarded Dirac a praelectorship in mathematical physics, which enabled him to devote himself entirely to research, apart from the presentation of his lecture course.\n\n#  **Chapter thirteen**\n\n Letter from Dirac to Oswald Veblen, 21 March 1929, LC, Veblen archive.\n\n Scott Fitzgerald (1931: 459).\n\n Letter from Dirac to Veblen, 21 March 1929, LC (Veblen archive).\n\n Diaries of Dirac (DDOCS).\n\n Fellows (1985); see the introduction (p. 4) and the conclusion.\n\n Comment made by Bohr to Crowther, recorded by Crowther on 24 April 1932 in the Crowther archive, SUSSEX, Book II of his notebooks, pp. 96-7. For one of many retellings of this anecdote, see Infeld (1941: 171).\n\n See the article on Roundy in the _Wisconsin State Journal_ on the day after his death, on 10 December 1971.\n\n The article is reproduced in its entirety in Kragh (1990: 72-3). The original is in Dirac Papers, 2\/30\/1 (FSU).\n\n A check of the microfilm records of the _Wisconsin State Journal_ reveals that the article was not published between 1 April and 29 May 1929 (the microfilm for 30 May is missing).\n\n Van Vleck (1972: 7-16; see pp. 10-11).\n\n Record of Dirac's payment as 'Lecturer in physics April and May 1929' is in WISC. Early in his stay, from 10-16 April, Dirac had spent almost a week based at the University of Iowa.\n\n Dirac left Madison on 27 May and travelled to the Grand Canyon via Minneapolis, Kansas City and Winslow, Arizona.\n\n Quoted in Brown and Rechenberg (1987: 134). This article gives much detail about Dirac and Heisenberg's preparations for their 1929 trip and the trip itself.\n\n Mehra (1973: 816).\n\n Brown and Rechenberg (1987: 136-7).\n\n Interview with Leopold Halpern, 18 February 2003.\n\n Brown and Rechenberg (1987: 139-41).\n\n Heisenberg returned from the 1929 trip to be the best ping-pong player in the quantum community: interview with von Weisz\u00e4cher, AHQP, 9 July 1963, p. 11.\n\n Mehra (1973: 816).\n\n Mehra (1972: 17-59).\n\n The _jako_ was commonly used to scent clothes in Japan at that time. Hearn (1896: 31n).\n\n Dirac gives his timetable in his letter to Tamm on 12 September 1929, Kojevnikov (1993: 29); Brendon (2000: 234).\n\n Letter to Dirac from his mother, 6 July 1929, Dirac Papers, 1\/3\/11 (FSU).\n\n Letter to Dirac from his mother, 6 May 1929, Dirac Papers, 1\/3\/10 (FSU).\n\n Postcards from Dirac to his parents, autumn 1929, DDOCS.\n\n Interview with Oppenheimer, 20 November 1963, p. 23 (AHQP).\n\n Fitzgerald (1931: 459).\n\n Dirac (1977: 144).\n\n Kojevnikov (2004: 56-9).\n\n Pais, A. (1998: 36).\n\n Letter from Dirac to Bohr, 9 December 1929, NBA.\n\n Letter to Dirac from his mother, 11 October 1929, Dirac Papers, 1\/3\/10 (FSU). The spelling is the one used by Flo Dirac. Dirac expected to arrive home on 19 December (postcard from Dirac to his parents, 27 November 1929, DDOCS).\n\n Letter from Dirac to Manci, 26 February 1936 (DDOCS).\n\n#  **Chapter fourteen**\n\n Cavendish Laboratory Archive, UCAM. The poem was apparently written as a Valentine's card to the electron.\n\n Dirac, 'Symmetry in the Atomic World', January 1955. The draft, which features this analogy, is in Dirac Papers, 2\/27\/13 (FSU).\n\n Cited in Kragh (1990: 101).\n\n Gamow (1970: 70); letter from Dirac to Tamm, 20 March 1930, in Kojevnikov (1993: 39).\n\n On Saturday, 16 February 1935, Van Vleck took D to 'A Disney Day' at a cinema in Boston. The documents, marked with Van Vleck's comment 'Dirac loved Mickey Mouse', are in the Van Vleck papers at AMS.\n\n Dirac's formula is _n_ = - log2 [log2 (2\u221a(\u221a . . . \u221a2))], where the ellipsis (. . .) denotes the taking of _n_ square roots. The story is related in Casimir (1984: 74-5), where the author asserts that Dirac killed the game using only three 2s. Each symbol in the formula is very common in mathematics, so Dirac's solution is within the rules of the game.\n\n Postcard from Dirac to his parents, 20 February 1930 (DDOCS).\n\n Telegram to Dirac from his mother, 22 February 1930, Dirac Papers, 1\/3\/12 (FSU).\n\n Letter to Dirac from his mother, 24 February 1930, Dirac Papers, 1\/3\/12 (FSU).\n\n The certificate of Dirac's election to the Fellowship of the Royal Society is available on the Society's website. The names of the 447 Fellows of the Society on 31 December 1929 are given in the Yearbook of the Royal Society 1931.\n\n Letter to Dirac from his mother, 24 February 1930, Dirac Papers, 1\/3\/12 (FSU).\n\n Letter from Hass\u00e9 to Dirac, 28 February 1930, Dirac Papers, 2\/2\/1 (FSU).\n\n Letter from Arnold Hitchings to the _Bristol Evening Post_ , 14 December 1979.\n\n In 1935, Dirac traded in this car. Dirac Papers, 1\/8\/2 (FSU).\n\n Interview with John Crook, 1 May 2003.\n\n Mott (1986: 42).\n\n Dirac was well known for this practice. It is described explicitly by his climbing tutor Tamm in the course of the letter to his wife on 27 May 1931, Kojevnikov (1993: 55). See also Mott (1972: 2).\n\n Interview with Monica Dirac, 7 February 2003; see also M. Dirac (2003: 42).\n\n Letter from Taylor Sen (1986: 80). Howarth (1978: 104).\n\n See, for example, _Daily Telegraph_ , 12 February 1930, _Manchester Guardian_ , 12-18 February 1930.\n\n Peierls (1987: 36).\n\n Letter to Dirac from his mother, 12 June 1930, Dirac Papers, 1\/3\/12 (FSU).\n\n Kojevnikov (1993: 40), note on letter from Dirac to Tamm, 6 July 1930.\n\n The _Guardian_ , 'World Conference of Scientists', 3 September 1930. Crowther was probably the author of this report.\n\n Ross (1962).\n\n The venue and the time of the talk are in the records of the British Association for the Advancement of Science, BOD.\n\n Delbr\u00fcck (1972: 280-1).\n\n The report of the Science News Service is in Dirac Papers, 2\/26\/8 (FSU).\n\n _New York Times_ , 10 September 1932.\n\n I have translated the German word _quatsch_ as 'crap'. Another, similar version of this anecdote is given in the interview with Guido Beck, AHQP, 22 April 1967, p. 23.\n\n Among the most able students who were dissatisfied by Dirac's talks was Freeman Dyson, who recalls: 'I read Dirac's book hoping to learn quantum mechanics from it, and found it totally unsatisfactory.' E-mail from Dyson, 19 August 2006.\n\n _Nature_ , Vol. 127, 9 May 1931, p. 699.\n\n Pauli's review is in Kronig and Weisskopf (1964: 1,397-8).\n\n Einstein (1931: 73).\n\n Leisure reading anecdote: Woolf (1980: 261); 'Where's my Dirac?' anecdote is from _Tallahasse Democrat_ , 29 November 1970.\n\n Hoyle (1994: 238).\n\n Freeman (1991: 136-7).\n\n Quoted in Charap (1972: 331).\n\n Letter from Tamm to Dirac, 13 September 1930, in Kojevnikov (1993: 43).\n\n Einstein (1931: 73).\n\n Comment made by Einstein on his arrival in New York on 11 December 1930, reported in the _LA Times_ , 12 December 1930, p. 1.\n\n Letter to Dirac from Tamm, 29 December 1930, Kojevnikov (1993: 48-9).\n\n Letter from Kemble to Garrett Birkhoff, 3 March 1933 (AHQP).\n\n Dirac attended the dinner on 17 December 1932, Dirac Papers, 2\/79\/6 (FSU).\n\n Letter from Kapitza to his mother, 16 December 1921, in Boag et al. (1990: 138-9).\n\n Da Costa Andrade (1964: 48).\n\n Da Costa Andrade (1964: 162).\n\n Records of the Cavendish dinners (CAV 7\/1) 1930, p. 10 (UCAM).\n\n Records of the Cavendish dinners (CAV 7\/1) 1930, p. 10 (UCAM).\n\n Snow (1931).\n\n Snow (1934). Dirac features in the book, and some of his opinions also appear, unattributed. See Snow (1934: 97-8 and 178-83).\n\n Letter from Chandrasekhar to his father, 10 October 1930, quoted in Miller (2005: 96).\n\n Letter to Dirac from his mother, 8 November 1930, Dirac Papers, 1\/3\/13 (FSU).\n\n#  **Chapter fifteen**\n\n Letter to Dirac from his mother, 27 April 1931, Dirac Papers, 1\/4\/1 (FSU). Dirac appears to have left Bristol on 15 April (postcard from Dirac to his parents, 15 April 1931, DDOCS).\n\n Letter from Dirac to Van Vleck, 24 April 1931, AHQP.\n\n Kapitza Club, 21 July 1931. See the Kapitza Club notebook in CHURCHILL.\n\n Dirac (1982: 604); Dirac (1978).\n\n The size of the force between two attracting monopoles separated by a millionth of a millimetre - roughly the distance between the electron and the proton in a hydrogen atom - is about a ten-thousandth of the weight of a medium-sized apple.\n\n Heilbron (1979: 87-96).\n\n Sherlock Holmes used these words in the novel _The Adventure of the Blanched Soldier_ (1926), and used extremely similar words in several other stories.\n\n The phrase 'theorist's theorist' is often applied to Dirac. See, for example, Galison (2000).\n\n Tamm arrived in Cambridge on 9 May and left on 25 June.\n\n Fen (1976: 181).\n\n Crowther (1970: 103).\n\n Letter from Tamm to his wife, undated c. May 1931, in Kojevnikov (1993: 54).\n\n Letter to Dirac from Tamm, 18 May 1931, in Kojevnikov (1993: 54-5).\n\n Werskey (1978: 92).\n\n Annan (1992: 181).\n\n James Bell (1896-1975) was one of Scotland's leading climbers and was fascinated by the Soviet Union. He stayed in contact with Dirac for decades.\n\n Wersley (1978: 138-49).\n\n Bukharin (1931).\n\n Brown (2005: 107).\n\n Letter to Dirac from Tamm, 11 July 1931, Dirac Papers, 2\/2\/4 (FSU).\n\n Home Office Warrant 4081, 27 January 1931, KV 2\/777, UKNATARCHI.\n\n Postcard from Dirac to his parents, 13 July 1931 (DDOCS).\n\n Letter to Dirac from his mother, 8 July 1931, Dirac Papers, 2\/2\/4 (FSU).\n\n The most direct comment on this from Dirac was reported by his mother in her letter to Betty from Stockholm in December 1933: '[Dirac] says it is awful and time we made an improvement.' In her letters to Dirac, she often mentions the disrepair of the family home.\n\n Letter to Dirac from his mother, 19 July 1931, Dirac Papers, 2\/2\/4 (FSU).\n\n Letter to Dirac from his mother, 20 July 1931, Dirac Papers, 2\/2\/4 (FSU).\n\n Postcard from Flo to Betty Dirac, 1 August 1931: 'Having a sea voyage with Paul. The weather is fine and it is lovely. Back 6.35am Sunday. Hope you are both looking after each other' (DDOCS).\n\n The area was officially named the Glacier National Park only in the following year.\n\n Robertson (1985).\n\n The furniture budget was $26,000; the budget for rugs was nearly $8,000. Batterson (2007: 612). Fine Hall is now called Jones Hall.\n\n Jacobson, N., 'Recollections of Princeton' in Robertson (1985).\n\n Letter from Pauli to Peierls, 29 September 1931, in Hermann et al. (1979).\n\n Enz (2002: 224-5).\n\n _New York Times_ , 17 June 1931.\n\n Pais (1986: 313-17).\n\n Brown (1978).\n\n Enz (2002: 211).\n\n 'Lectures on Quantum Mechanics', Princeton University, October 1931, Dirac Papers, 2\/26\/15 (FSU). These notes were transcribed by Banesh Hoffman and checked by Dirac.\n\n 'Dr Millikan Gets Medal', _New York Times_ , 5 September 1928.\n\n Kevles (1971: 180); Galison (1987: Chapter 3, pp. 86-7).\n\n Interview with Robert Oppenheimer, AHQP, 18 November 1963, p. 16.\n\n De Maria and Russo (1985: 247, 251-6).\n\n Letter from Anderson to Millikan, 3 November 1931, quoted in De Maria and Russo (1985: 243). In this letter, Anderson describes data taken over the previous 'very few days'.\n\n Interview with Carl Anderson, 11 January 1979, p. 34, available at <http:\/\/oralhistories.library.caltech.edu\/89> (accessed 13 May 2008), p. 34.\n\n De Maria and Russo (1985: 243).\n\n Letter to Dirac from Martin Charlesworth, 16 October 1931, Dirac Papers, 2\/2\/4 (FSU). Charlesworth was Dirac's personal tutor during his postgraduate years and was evidently fond of him. Later, on 19 March 1935, he wrote a letter to Dirac 'to send my [i.e. his] love' - a remarkably forward phrase in that cultural milieu, Dirac Papers, 2\/3\/1 (FSU).\n\n Batterson (2006: Chapter 5).\n\n Brendon (2000: Chapter 4).\n\n _New York Times_ , 14 June 1931.\n\n Letter from Gamow to Dirac, written in June 1965, Dirac Papers, 2\/5\/13 (FSU). See also Gamow (1970: 99).\n\n Gorelik and Frenkel (1994: 20-2). See also Kojevnikov (2004: 76).\n\n Gorelik and Frenkel (1994: 50-1). Gamow gives a partially inaccurate account of this incident in his autobiography (1970).\n\n The first Soviet edition is discussed in detail in Dalitz (1995), which includes a translation of the prefaces to the book.\n\n Ivanenko had ensured that the book had been translated with no changes, but the Russian edition does include an additional chapter on applying quantum mechanics to practical problems. It is not clear whether Dirac added the section as a result of ideological pressure.\n\n Greenspan (2005: 161).\n\n Letter from Dirac to Tamm, 21 January 1932, in Kojevnikov (1993: 60). Dirac was learning the branches of mathematics known as group theory and differential geometry.\n\n Interview with Oppenheimer, AHQP, 20 November 1963, p. 1.\n\n Letter to Dirac from his mother, 9 October 1931, Dirac Papers, 2\/2\/4 (FSU).\n\n Letter to Dirac from his mother, dated 28\/31 September 1931, Dirac Papers, 2\/2\/4 (FSU).\n\n Letter to Dirac from his mother, 22 December 1931, Dirac Papers, 2\/2\/4 (FSU).\n\n Brown (1997: Chapter 6).\n\n Cathcart (2004: 210-12); Chadwick (1984: 42-5).\n\n Brown (1997: 106).\n\n#  **Chapter sixteen**\n\n Eddington made this remark in Leicester, at the annual meeting of the British Association for the Advancement of Science: 'Star Birth Sudden Lema\u00eetre Asserts', _New York Times_ , 12 September 1933.\n\n An English translation of the play, by Gamow's wife Barbara, is given in Gamow (1966: 165-218). For comments on the production: von Meyenn (1985: 308-13).\n\n Wheeler (1985: 224).\n\n Crowther (1970: 100).\n\n Letter from Darwin to Goudsmit, 12 December 1932, APS.\n\n Interview with Beck, AHQP, 22 April 1967, p. 23.\n\n Interview with Klein, AHQP, 28 February 1963, p. 18. Klein recalled that 'Heisenberg once told me that when Dirac got the Nobel Prize some years later - in 1933 - he asked Dirac if he believed in his own theory. Dirac answered, in his very precise way, that a year before the positive electron was discovered he had ceased to believe in the theory' (interview with Klein, AHQP, 28 February 1963, p. 18).\n\n Cathcart (2004: Chapters 12 and 13).\n\n _Reynolds's Illustrated News_ , 1 May 1932.\n\n _Daily Mirror_ , 3 May 1932.\n\n Cathcart (2004: 252). Einstein's lecture took place on 6 May; see the _Cambridge Review_ , 13 May 1932, p. 382.\n\n Howarth (1978: 187).\n\n Howarth (1978: 224).\n\n Report in _Sunday Dispatch_ on 19 November 1933.\n\n Interview with von Weizs\u00e4cher, AHQP, 9 June 1963, p. 19.\n\n Note from P. H. Winfield to Dirac, Dirac Papers, 2\/2\/5 (FSU).\n\n Letter from Sir Joseph Larmor to Terrot Reaveley Glover (1869-1943), the classical scholar and historian, 20 February 1934, STJOHN.\n\n Infeld (1941: 170).\n\n Letter to Dirac from his mother, 27 July 1932, Dirac Papers, 2\/2\/6 (FSU).\n\n Letter to Dirac from his sister, 14 October 1932, Dirac Papers, 2\/2\/6 (FSU).\n\n Letter to Dirac from his sister, 11 July 1932, Dirac Papers, 2\/2\/6 (FSU).\n\n Letter to Dirac from his sister, 15 October 1932, Dirac Papers, 2\/2\/6 (FSU).\n\n Letter to Dirac from his mother, 21 April 1932, Dirac Papers, 2\/2\/6 (FSU). See also the letter of 1 June 1932.\n\n Letter to Dirac from his father, Dirac Papers, 1\/1\/10 (FSU).\n\n The paper combined several parts, one mostly from Dirac, the other mostly from Fock and Podolsky, and also a part that developed during the process of writing in correspondence between the three authors. One snapshot of the collaboration is in the letter written to Dirac by Podolsky in Kharkov on 16 November 1932, Dirac Papers, 2\/2\/6 (FSU). I thank Alexei Kojevnikov for this information.\n\n Weisskopf (1990: 72-3).\n\n Infeld (1941: 172).\n\n Article in the _Los Angeles Times_ by Harry Carr, 30 July 1932.\n\n For more detail on the discovery of the anti-electron, see Anderson (1983: 139-40), and Darrow (1934).\n\n Interview with Louis Alvarez by Charles Weiner, 14-15 February 1967, American Institute of Physics, p. 10.\n\n Von K\u00e1rm\u00e1n (1967: 150).\n\n Von K\u00e1rm\u00e1n (1967: 150).\n\n Interview with Carl Anderson, 11 January 1979, available online at <http:\/\/oralhistories.library.caltech.edu\/89> (accessed 13 May 2008).\n\n Galison (1987: 90).\n\n _New York Times_ , 2 October 1932.\n\n Letter from Robert Oppenheimer to Frank Oppenheimer, autumn 1932, in Smith and Weiner (1980: 159).\n\n Nye (2004: 54). The incident, recalled by Blackett's student Frank Champion, probably took place during the 1931-2 academic year. I am grateful to Mary Jo Nye for this information.\n\n See <http:\/\/www.aps-pub.com\/proceedings\/1462\/207.pdf> (accessed 13 May 2008).\n\n De Maria and Russo (1985: 254).\n\n Contribution of Occhialini to the Memorial Meeting for Lord Blackett, _Notes and Records of the Royal Society_ , 29 (2) (1975).\n\n Dalitz and Peierls (1986: 167). The anecdote is due to Maurice Pryce.\n\n Dirac's notes on Fowler's lectures on 'Analytic Dynamics' are in Dirac Papers, 2\/32\/1 (FSU).\n\n Letter from Dirac to Fock, 11 November 1932, passed to me by Alexei Kojevnikov.\n\n Greenspan (2005: 170).\n\n _Bristol Evening Post_ , 28 October 1932.\n\n Letter to Dirac from his mother, 26 October 1932, Dirac Papers, 2\/2\/7 (FSU).\n\n Letter to Dirac from his mother, 9 January 1933, Dirac Papers, 2\/2\/8 (FSU).\n\n#  **Chapter seventeen**\n\n IAS Archives Faculty Series, Box 32, Folder: 'Veblen, 1933'.\n\n De Maria and Russo (1985: 266 and 266 n.). Anderson's paper had been available in the university library from the mid-autumn of 1932.\n\n Archie Clow, contributing to Radio 3 programme _Science and Society in the Thirties_ (1965). Script stored in the library of Trinity College, Cambridge.\n\n Sch\u00fccking (1999: 27).\n\n Interview with L\u00e9on Rosenfeld, AHQP, 22 July 1963, p. 8.\n\n Halpern (1988: 467).\n\n Letter to Dirac from Isabel Whitehead, 20 July 1932, Dirac Papers, 2\/2\/6 (FSU).\n\n Taylor Sen (1986).\n\n Dirac, book review in the _Cambridge Review_ , 6 February 1931.\n\n Interview with von Weizs\u00e4cher, AHQP, 9 June 1963, p. 19.\n\n Private papers of Mary Dirac. Dirac wrote the notes on 17 January 1933.\n\n Letter from Dirac to Isabel Whitehead, 6 December 1936, STJOHN.\n\n Compte remarked that 'The greatest problem, then, is to raise social feeling by artificial effort to the position which in the natural condition is held by selfish feeling.' See <http:\/\/www.blupete.com\/Literature\/Biographies\/Philosophy\/Comte.htm> (accessed 14 May 2008).\n\n The headquarters of the Royal Society were then at Burlington House.\n\n Bertha Swirles, Dirac's former student colleague, described the talk as 'sensational' in her letter of 20 February 1933 to Dirac's colleague Douglas Hartree. Hartree archive, 157, CHRIST'S.\n\n Dirac was giving a technical talk at the London Mathematical Society on his favourite topic, 'The Relation Between Classical and Quantum Mechanics', at the Royal Astronomical Society in Burlington House, Dirac Papers, 2\/26\/18 (FSU).\n\n The word was used in the 15 March issue of the _Physical Review._\n\n Quoted in Pais (1986: 363).\n\n Interview with von Weizs\u00e4cher, AHQP, 9 July 1963, p. 14.\n\n Letter from Tamm to Dirac, 5 June 1933, in Kojevnikov (1996: 64-5).\n\n Interview with Dirac, AHQP, 14 May 1963, p. 31.\n\n Letter from Pauli to Dirac, 1 May 1933, see Pais (1986: 360).\n\n Galison (1994: 96).\n\n Darrow (1934: 14).\n\n Roqu\u00e9 (1997: 89-91).\n\n Brown and Hoddeson (1983: 141).\n\n Blackett (1955: 16).\n\n Gell-Mann (1994: 179).\n\n See the lecture Dirac gave in Leningrad on 27 September 1933 (Dalitz 1995: 721), Dirac's Nobel Prize lecture in December 1933 and most of Dirac's subsequent lectures on the positron.\n\n Blackett (1969: xxxvii).\n\n Gottfried (2002: 117).\n\n Bohr's support was sought by Kapitza. See the correspondence quoted in Kedrov (1984: 63-7).\n\n The quote from Rutherford is from Kapitza's letter to Bohr of 10 March 1933, quoted in Kedrov (1984: 63-4).\n\n Anon., 'Conservatism and the Young', _Cambridge Review_ , 28 April 1933, pp. 353-4.\n\n The debate was held on 21 February 1933 and was reported in the _Cambridge Evening News_ on the following day. See also Howarth (1978: 224-5).\n\n Anon (1935); essay by Blackett (based on a radio broadcast in March 1934), pp. 129-44, see p. 130.\n\n Werskey (1978: 168).\n\n Werskey (1978: 148).\n\n The _Cambridge Review_ , 20 January 1933. The article alerted the Cambridge University community to the reservations expressed by the translators of Dirac's book into Russian.\n\n Anon. (1933) 'The End of a Political Delusion', _Cambridge Left_ , 1 (1): 10-15; p. 12.\n\n _Daily Herald_ , 15 September 1933, p. 10. McGucken (1984: 40-1).\n\n Letters to Dirac from his mother, 20 July and 22 July 1933, Dirac Papers, 1\/4\/3 (FSU).\n\n Letter to Dirac from his mother, 8 August 1933, Dirac Papers 1\/4\/3 (FSU).\n\n Postcards from Dirac to his mother, from September 1933 (DDOCS).\n\n Letter from Dirac to Tamm, 19 June 1933, in Kojevnikov (1993: 67); see also the letter from Tamm to Dirac on 5 June 1933 (Kojevnikov 1993: 64).\n\n Interview with Beck, AHQP, 22 April 1967, APS, p. 23.\n\n The mansion was awarded to Bohr in December 1931, whereupon Bohr and his family moved in during the summer of 1932. The Bohrs' first sleeping-over guests were Ernest Rutherford and his wife, who stayed there from 12 September to 22 September 1932. I thank Finn Aaserud and Felicity Pors for this information.\n\n Parry (1968: 117).\n\n Casimir (1983: 73-4). Letter from Dirac to Margrethe Bohr, 24 September 1933, NBA.\n\n Letter from Dirac to Margrethe Bohr, 24 September 1933, NBA.\n\n Letter from Dirac to Bohr, 20 August 1933, NBA.\n\n Fitzpatrick (1999: 40-1).\n\n Conquest (1986: Epilogue).\n\n M. Dirac (1987: 4).\n\n Anne Kox, 'Een kwikkolom in de Westertoren: De Amsterdamse natuurkunde in de jaren dertig', available online at <http:\/\/soliton.science.uva.nl\/~kox\/jaarboek.html> (14 May 2008).\n\n Letter from Dirac to Bohr, 28 September 1933, NBA.\n\n Letter from Margrethe Bohr to Dirac, 3 October 1933, NBA.\n\n Letter from Ehrenfest to Bohr, Einstein and the physicists James Franck, Gustave Herglotz, Abram Joff\u00e9, Philipp Kohnstamm and Richard Tolman, 14 August 1933, NBA. Another suicide note, written on the day before Ehrenfest killed himself was unearthed in 2008: see _Physics Today_ , June 2008, p. 26-7.\n\n Roqu\u00e9 (1997: 101-2).\n\n Letter from Heisenberg to Pauli, 6 February 1934, in Hermann et al. (1979).\n\n Dirac mentioned his surprise to a reporter from the _Daily Mirror._ See the article on 13 November 1933.\n\n Taylor (1987: 37).\n\n The youngest experimenter to win the prize was, and remains, Lawrence Bragg, who won it when he was twenty-five. Dirac's record as the youngest theoretician to win the prize was broken (by a margin of three months) in 1957 by T. D. Lee.\n\n Reports on 10 November 1933 included the _Daily Mail_ , _Daily Telegraph_ , _Manchester Guardian_ ; the _Daily Mirror_ reported on the following day.\n\n _Sunday Dispatch_ , 19 November 1933.\n\n Letter from Dirac to Bohr, 28 November 1933, NBA.\n\n Greenspan (2005: 242). Maurice Goldhaber remembers that when he remarked that Dirac's award was 'great news', Born scowled. Interview with Maurice Goldhaber, 5 July 2006.\n\n _Cambridge Review_ , 17 November 1933; Brown (2005: 120). See also Stansky and Abrahams (1966: 210-13). A few days before the march, a few socialists and pacifists clashed with audiences leaving the Cambridge cinema Tivoli, after an evening showing of the patriotic movie _Our Fighting Navy._ The fight was the talk of the town and therefore guaranteed interest in the Armistice Day march.\n\n#  **Chapter eighteen**\n\n Dalitz and Peierls (1986: 146).\n\n Information from RSAS, 14 September 2004.\n\n The main sources of the material in this chapter are in the Dirac Papers (FSU): Letter to Dirac from his mother, 21 November 1933 (2\/2\/9). Florence Dirac's account of her trip is in 'My visit to Stockholm' (1\/2\/9) and in a long, descriptive letter to Betty (2\/2\/9).\n\n Reports in _Svenska Dagbladet_ and _Dagens Nyheter_ , both on 9 December 1933.\n\n This was one of Dirac's favourite stories about his absent-minded mother. It is well recounted in Kur\u015funo\u011flu (1987: 18).\n\n Reports in the Stockholm newspapers _Nya Dagligt Allehanda_ , 9 December 1933, _Stockholms Dagblad_ , 10 December 1933.\n\n Reports in the Stockholm newspapers _Nya Dagligt Allehanda_ , 9 December 1933, _Stockholms Dagblad_ , 10 December 1933.\n\n Report in _Dagens Nyheter_ , 11 December 1933.\n\n _Dagens Nyheter_ , 11 December 1933; _Svenska Dagbladet_ , 11 December 1933.\n\n Women guests were first invited to the banquet in 1909, when the female Swedish wirter Selma Lagerl\u00f6f won the Nobel Prize for Literature.\n\n _Dagens Nyheter_ , 11 December 1933; _Svenska Dagbladet_ , 11 December 1933; _Stockholms Tidningen_ , 11 December 1933.\n\n See <http:\/\/nobelprize.org\/physics\/laureates\/1933\/dirac-speech.html> (accessed 14 May 2008).\n\n Annemarie Schr\u00f6dinger notes 'Stockholm 1933', AHQP. Letter from Schr\u00f6dinger to Dirac, 24 December 1933.\n\n I thank Professor Sir Partha Dasgupta for identifying this error and clarifying its nature.\n\n Flo Dirac, Dirac Papers, 1\/2\/9 (FSU) and 2\/2\/9 (FSU).\n\n See <http:\/\/nobelprize.org\/nobel_prizes\/physics\/laureates\/1933\/dirac-lecture.html> (accessed 14 May 2008).\n\n Schuster (1898a: 367); see also Schuster's follow-up article (1898b).\n\n Born (1978: 270). See also 'Eamon de Valera, Erwin Schr\u00f6dinger and the Dublin Institute' (McCrea 1987).\n\n Flo Dirac, Dirac Papers, 1\/2\/9 (FSU) and 2\/2\/9 (FSU).\n\n Dirac read Abraham Pais's book _Subtle is the Lord_ , and remarked 'Most interesting for its revelation of the working of Nobel Committee', Dirac Papers, 2\/32\/12 (FSU). The book mentions that Einstein did not nominate Dirac for a Nobel Prize.\n\n Nobel Committee papers, 1929 RSAS.\n\n Apart from Bragg, only the comparatively little-known Polish physicist Czeslaw Bialobrzeski nominated Dirac in 1933. No other leading theorist had nominated him.\n\n#  **Chapter nineteen**\n\n Letter from Pauli to Heisenberg, 14 June 1934, reprinted in Hermann et al. (1979).\n\n Schweber (1994: 128-9).\n\n Letters from Oppenheimer to George Uhlenbeck, March 1934 and to Frank Oppenheimer, 4 June 1934, in Kimball Smith and Weiner (1980: 175, 181).\n\n Interview with Dirac, AHQP, 6 May 1963, p. 8, Salam and Wigner (1972: 3-4). See also Peierls (1985: 112-13).\n\n Letter from Rutherford to Fermi, AHQP, 23 April 1934.\n\n 'Peter Kapitza', 22 June 34, KV 2\/777, UKNATARCHI.\n\n 'Note on interview between Captain Liddell and Sir Frank Smith of the Department of Scientific and Industrial Research, Old Queen Street', 26 September 1934, KV 2\/777. Jeffrey Hughes speculates that 'VSO' might be the Russian \u00e9migr\u00e9 I. P. Shirov (Hughes 2003).\n\n Born (1978: 269-70).\n\n I am grateful to Igor Gamow for making available home movies, shot in the 1920s, which show his mother dressed in this way.\n\n The correspondence between Dirac and Rho Gamow is in Dirac Papers, 2\/13\/6 (FSU).\n\n Letter from Dirac to Manci, 9 April 1935 (DDOCS).\n\n Letter from Dirac to Rho Gamow, Dirac Papers, 2\/2\/10 (FSU).\n\n Conversation with Lydia Jackson's literary executor Rosemary Davidson, 8 January 2006.\n\n Letter to Dirac from Lydia Jackson, 20 March 1934, Dirac Papers, 2\/2\/10 (FSU).\n\n Fen (1976: 182).\n\n Letter to Dirac from Lydia Jackson, 25 June 1934, Dirac Papers, 2\/2\/10 (FSU).\n\n Letter to Dirac from Lydia Jackson, 5 February 1936, Dirac Papers, 2\/3\/3 (FSU).\n\n Van Vleck (1972: 12-14).\n\n The visitor was his sister Manci. M. Dirac (1987: 3-8; see p. 3).\n\n The account of Dirac's early courtship of Manci is taken mainly from M. Dirac (1987).\n\n Letter to Dirac from Van Vleck, June 1936, Dirac Papers, 2\/2\/11 (FSU).\n\n Dirac was living at 8 Morven Street. See the Dirac archive in IAS (1935).\n\n Quoted in Jerome and Taylor (2005: 11).\n\n Jerome and Taylor (2005: Chapters 2 and 5).\n\n Blackwood (1997: 11).\n\n Testimonies of Malcolm Robertson and Robert Walker, 'The Princeton Mathematics Community in the 1930s', available at <http:\/\/www34.homepage.villanova.edu\/robert.jantzen\/princeton_math\/pm02.htm> (accessed 14 May 2008).\n\n The _Physical Review_ received the paper on 25 March 1935: Pais (1982: 454-7).\n\n Blackwood (1997: 15-16).\n\n Infeld (1941: 170).\n\n See 'The Princeton Mathematics Community in the 1930s', in particular the interviews of Merrill Flood, of Robert Walker and of William Duren, Nathan Jacobson and Edward McShane.\n\n Letter from Dirac to Max Newman, 17 March 1935, Newman archive STJOHN.\n\n Dirac alludes to his memories of ice-cream sodas and lobster dinners with Manci in his letters to her of 2 May and 25 May 1935 respectively (DDOCS).\n\n Manci was divorced from Richard Bal\u00e1zs on 20 September 1932. See Budapest's archive of marriages, microfilm repository no A555, Inventory no 9643, Roll no 155. These papers tell us that Manci married Bal\u00e1zs on 27 February 1924.\n\n Manci told her friend Lily Harish-Chandra of these relationships. Interview with Lily Harish-Chandra, 4 August 2006.\n\n Wigner (1992: 34, 38-9).\n\n Letter to Dirac from Manci, 2 September 1936 (DDOCS).\n\n M. Dirac (1987: 4-5).\n\n Letter to Dirac from Anna Kapitza, dated beginning December 1937, copy held by Alexei Kojevnikov.\n\n Hendry (1984: 130).\n\n A detailed account of Kapitza's detention is in: Internal MI5 memo, signed GML, 11 October 3KV 2\/777 (UKNATARCHI). See also the letters from Kapitza to his wife in Boag et al. (1990: Chapter 4).\n\n For a full account of Rutherford's campaign to secure Kapitza's release, see Badash (1985), notably Chapter 2. See also Kojevnikov (2004: Chapter 5).\n\n Letter from Dirac to Anna Kapitza, 19 December 1934, copy held by Alexei Kojevnikov.\n\n Dirac wrote of his vacation, without mentioning Manci, to Max Newman in a letter written on 13 January 1935 (Newman archive, STJOHN). The story of the alligator, which Gamow named Ni-Nilich, is related in letters from Dirac to Manci on 2 February, 29 March, 22 April and 2 May 1935 and in the letter from Manci to Dirac on 5 April 1935 (DDOCS). See also the letter from Gamow to Dirac, 25 March 1935, Dirac Papers, 2\/3\/1 (FSU).\n\n Letter from Dirac to Anna Kapitza, 14 March 1935, copy held by Alexei Kojevnikov.\n\n Letter from Rutherford to Bohr, 28 January 1935, Rutherford archive, UCAM.\n\n Gardiner (1988: 240-8).\n\n Gardiner (1988: 241).\n\n Gardiner (1988: 242).\n\n Kragh (1996: Chapter 2).\n\n 'Lema\u00eetre Follows Two Paths to Truth', _New York Times_ , 19 February 1933.\n\n Letter from Dirac to Manci, 2 February 1935 (DDOCS).\n\n Dirac had heard Lema\u00eetre speak at the Kapitza Club in about 1930. Dirac commented on this in a note he wrote on 1 September 1971: 'There was much discussion about the indeterminacy of quantum mechanics. Lema\u00eetre emphasised his opinion that he did not believe God influenced directly the cause of atomic events': Dirac Papers, 2\/79\/2 (FSU).\n\n Letter from Dirac to Manci, 2 March 1935 (DDOCS).\n\n Letter from Dirac to Manci, 2 May 1935 (DDOCS). Schnabel gave the concert on 7 March 1935.\n\n Letter from Dirac to Manci, 10 March 1935 (DDOCS).\n\n Letter to Dirac from Manci, 28 March 1935 (DDOCS).\n\n Letter from Dirac to Manci, 29 March 1935 (DDOCS).\n\n Letter from Dirac to Manci, 2 May 1935 (DDOCS).\n\n Letter from Dirac to Manci, 9 May 1935 (DDOCS).\n\n Letter to Dirac from Manci, 30 May 1935 (DDOCS).\n\n Letter to Dirac from Manci, 4 March 1935 (DDOCS).\n\n Letter from Dirac to Manci, 9 April 1935 (DDOCS).\n\n Badash (1985: 29).\n\n Badash (1985: 31).\n\n Letter from Kapitza to his wife, 13 April 1935, quoted in Boag et al. (1990: 235).\n\n Letter from Kapitza to his wife, 23 February 1935, quoted in Boag et al. (1990: 225).\n\n Letter from Kapitza to his wife, 23 February 1935, quoted in Boag et al. (1990: 225, 226).\n\n Kojevnikov (2004: 107).\n\n Letter from Dirac to Manci, 2 May 1935 (DDOCS).\n\n Lanouette (1992: 151); see also letter from Dirac to Anna Kapitza, 31 May 1935, copy held by Alexei Kojevnikov.\n\n Letter from K. T. Compton to the Soviet Ambassador, 24 April 1935, copy held by Alexei Kojevnikov.\n\n Letter from Dirac to Anna Kapitza, 27 April 1935, copy held by Alexei Kojevnikov.\n\n 'Embassy Occupied by Troyanovsky', _New York Times_ , 7 April 1934.\n\n Letter from Dirac to Anna Kapitza, 27 April 1935, copy held by Alexei Kojevnikov.\n\n Letter from Dirac to Anna Kapitza, 27 April 1935.\n\n#  **Chapter twenty**\n\n Letter from Dirac to Anna Kapitza, written from the Institute for Advanced Study, Princeton, 14 May 1935. Copy of letter held by Alexei Kojevnikov.\n\n Letter from Dirac to Anna Kapitza, written in Pasadena, 31 May 1935, copy held by Alexei Kojevnikov.\n\n Crease and Mann (1986: 106); Serber (1998: 35-6).\n\n Letter from Dirac to Manci, 4 June 1935 and 10 June 1935 (DDOCS).\n\n Letter from Dirac to Manci, 1 August 1935 (DDOCS).\n\n Letter from Dirac to Manci, 22 June 1935 (DDOCS).\n\n Quoted in Brendon (2000: 241).\n\n Letter from Kapitza to his wife, 30 July 1935, quoted in Boag et al. (1990: 251).\n\n Letter from Dirac to Manci, 17 August 1935 (DDOCS).\n\n Letter to Dirac from Manci, 30 September 1935 (DDOCS). See also Dirac, M. (1987: 6).\n\n Letter from Dirac to Manci, 22 September 1935 (DDOCS).\n\n Letter from Dirac to Manci, 23 October 1935 (DDOCS).\n\n Letter to Dirac from Manci, 9 October 1935 (DDOCS).\n\n Letters from Dirac to Manci, 3 October 1935 and 8 November 1935 (DDOCS).\n\n Letter from Dirac to Manci, 17 November 1935 (DDOCS).\n\n Letter to Dirac from Manci, 22 November 1935 (DDOCS).\n\n Letter from Dirac to Manci, 3 October 1935 (DDOCS).\n\n In Dirac's letter to Manci on 6 February 1937, Dirac mentions that his father owned a copy of Shaw's plays.\n\n Letter to Dirac from his mother, 15 July 1934, Dirac Papers, 1\/4\/4 (FSU).\n\n Dirac's father's notebook is in Dirac Papers, 1\/1\/10 (FSU). Charles dates his first entry September 1933. The latest date he referenced was 4 November 1935, so he  probably ceased compiling the notes in early 1936.\n\n Dalitz and Peierls (1986: 146).\n\n Letter to Dirac from his mother, 4 August 1935, Dirac Papers, 1\/4\/5 (FSU).\n\n Letter to Dirac from his mother, 4 August 1935, Dirac Papers, 1\/4\/5 (FSU).\n\n Dalitz and Peierls (1986: 155-7).\n\n Letter from Dirac to Tamm, 6 December 1935, in Kojevnikov (1996: 35-6).\n\n One of the physicists who thought that Dirac was over-excited by the Shankland result was Hans Bethe, who wrote 'What has happened to him?' in a letter to Rudolf Peierls on 1 August 1936, in Lee (2007b: 152).\n\n Dirac (1936: 804).\n\n Letter from Heisenberg to Pauli, 23 May 1936, Vol. II, p. 442.\n\n Letter from Einstein to Schr\u00f6dinger, 23 March 1936, AHQP.\n\n Letter from Schr\u00f6dinger to Dirac, 29 April 1936, Dirac Papers, 2\/3\/3 (FSU).\n\n Letter from Bohr to Kramers, 14 March 1936, NBA.\n\n Letter from Dirac to Blackett, 12 February 1937, Blackett archive ROYSOC.\n\n Letter from Dirac to Manci, 15 January 1936. Other details in this paragraph are in his letters to Manci of 25 January 1936, 2 February 1936 and 10 February 1936 (DDOCS).\n\n Huxley (1928: 91) ('Emotionally, he was a foreigner') and p. 230 ('a mystic, a humanitarian and also a contemptuous misanthrope'). See also Huxley (1928: 90, 92-6).\n\n Letter from Dirac to Manci, 2 February 1936 (DDOCS).\n\n Letter to Dirac from Manci, 23 February 1936 (DDOCS).\n\n Letter from Dirac to Manci, 7 March 1936 (DDOCS).\n\n Letter from Dirac to Manci, 7 March 1936 (DDOCS).\n\n Letter to Dirac from Manci, 13 March 1936 (DDOCS).\n\n Letters from Dirac to Manci, 23 March 1936 and 29 April 1936, and letter to Dirac from Manci, 24 April 1936 (DDOCS).\n\n Letter from Dirac to Manci, 5 May 1936 (DDOCS).\n\n Dirac had also fibbed to Kapitza in the previous year. Dirac makes this plain to Manci in his letter to her of 23 June 1936 (DDOCS).\n\n Letter from Dirac to Manci, 9 June 1936 (DDOCS).\n\n Letter from Dirac to Manci, 5 June 1936 (DDOCS).\n\n Sinclair (1986: 55).\n\n A. Blunt, 'A Gentleman in Russia', and a review of Crowther's _Soviet Science_ by Charles Waddington, both in the _Cambridge Review_ , 5 June 1936.\n\n Letter to Dirac from his mother, 7 June 1936, Dirac Papers, 1\/4\/6 (FSU).\n\n Letters to Dirac from his sister, 6 June, 8 June and 9 June 1936, Dirac Papers, 1\/7\/1 (FSU).\n\n Letter from Dirac to Manci, 17 June 1936 (DDOCS).\n\n Letter to Dirac from his mother, 11 June 1936, Dirac Papers, 1\/4\/6 (FSU).\n\n _Daily Mirror_ , 21 May 1934, p. 14. The article concluded: 'Dirac. Our great grand-children may be repeating that name when the Chaplins, Fords, Cowards and Cantors are forgotten.' Cantor is the American writer and entertainer Eddie Cantor.\n\n Letter from Dirac to Manci, 17 June 1936 (DDOCS).\n\n Letter to Dirac from his mother, July 1936, Dirac Papers, 1\/4\/6 (FSU).\n\n Letter to Dirac from his mother, 27 August 1936, Dirac Papers, 1\/4\/6 (FSU).\n\n Feinberg (1987: 97).\n\n Dalitz and Peierls (1986: 151).\n\n Letter from Kapitza to Rutherford, 26 April 1936, quoted in Badash (1985: 110).\n\n Letter to Dirac from Manci, 2 September 1936 (DDOCS).\n\n Pais (1991: 411).\n\n Both preceding quotes are from the letter from Dirac to Manci, 7 October 1936 (DDOCS). Dirac commented to an officer of the Rockefeller Foundation, which funded the conference, that he was 'genuinely enthusiastic', quoted in Aaserud (1990: 223).\n\n In Dirac, M. (1987), Manci recalls that she was on the _Queen Mary_ 's maiden voyage. At that time, however, she was in Budapest.\n\n Letter from Dirac to Manci, 19 October 1936 (DDOCS).\n\n Letter from Dirac to Manci, 17 November 1936 (DDOCS).\n\n Letter to Dirac from Isabel Whitehead, 29 November 1936, Dirac Papers, 2\/3\/4 (FSU).\n\n Letter from Dirac to Isabel Whitehead, 6 December 1936, STJOHN.\n\n Letter to Dirac from Isabel Whitehead, 9 December 1936, Dirac Papers, 2\/3\/4 (FSU).\n\n Interview with Monica Dirac, 7 February 2003. Manci often related this story of Dirac's proposal to her. The description of the car is in the letter from Dirac to Manci, 17 November 1935 (DDOCS).\n\n Letter to Dirac from Manci, 29 January 1937 (DDOCS).\n\n Letter to Dirac from his mother, 24 December 1936, Dirac Papers, 1\/4\/6 (FSU).\n\n#  **Chapter twenty-one**\n\n Dirac, M. (1987: 4).\n\n Letter from Dirac to Manci, 18 February 1937 (DDOCS).\n\n Letter from Dirac to Manci, 6 February 1937 (DDOCS).\n\n Letter from Dirac to Manci, 20 February 1937 (DDOCS). Dirac writes 'How soon after the new moon comes will I be alone with my beloved, and have her in my arms [. . .]'.\n\n Letter from Dirac to Manci, 19 February 1937 (DDOCS).\n\n Letter from Dirac to Manci, 20 February 1937 (DDOCS).\n\n Letter to Dirac from Manci, 16 February 1937 (DDOCS).\n\n Letters to Dirac from Manci, 25 January and 16 February 1937 (DDOCS).\n\n Letter to Dirac from Betty, 29 January 1937 (DDOCS).\n\n Letter to Dirac from Manci, 29 January 1937 (DDOCS).\n\n One reading of Manci's cryptic comments in her letter to Dirac of 16 February 1937 is that his parents were sexually incompatible (DDOCS): 'Betty told me today the reason why probably your parents did not like each other. Your father could not help it, don't blame him dear, nor do [ _sic_ ] your mother.'\n\n Letter to Dirac from Manci, 18 February 1937 (DDOCS).\n\n Letter to Dirac from Manci, 28 January 1937 (DDOCS). Dirac's 'unexpected' marriage was noted in the _Cambridge Daily News_ , 7 January 1937.\n\n Letter from Rutherford to Kapitza, 20 January 1937, in Boag et al. (1990: 300).\n\n Letter from Dirac to Kapitza, 29 January 1937, Dirac Papers 2\/3\/5 (FSU).\n\n Letter to Manci from Anna Kapitza, 17 February 1937, Dirac Papers, 2\/3\/5 (FSU).\n\n Dirac's use of 'Wigner's sister' became famous in his community. Both Dirac's daughters confirm that he used this term of introduction.\n\n Manci often used this expression. See, for example, Dirac (1987: 7).\n\n Interview with Monica Dirac, 7 February 2003.\n\n Salaman and Salaman (1996: 66-70); see p. 67.\n\n Daniel (1986: 95-6).\n\n Letter from Dirac to Manci, 19 February 1937 (DDOCS).\n\n Dirac's wish to have children appears obvious from his delighted reaction to the news  of Manci's later pregnancies.\n\n Gamow (1967: 767).\n\n Christianson (1995: 257).\n\n Dingle (1937a).\n\n Untitled supplement to _Nature_ , Vol. 139, 12 June 1937, pp. 1001-2; p. 1001.\n\n Dingle (1937b).\n\n Report on Theoretical Physics to the Institute for Advanced Study, 23 October 1937, in the IAS Archives General Series, 52, 'Physics'.\n\n Estate of Charles Dirac, prepared by Gwynn, Onslow & Soars, who prepared the document on 7 October 1936 (DDOCS).\n\n Letter to Dirac from his mother, 21 January 1937, Dirac Papers, 1\/4\/7 (FSU). See also the letter of 1 February 1937 in the same file of the archive.\n\n Interview with Kurt Hofer, 21 February 2004.\n\n Kojevnikov (2004: 119).\n\n Postcard from Manci Dirac to the Veblens, 17 June 1937, LC Veblen archive.\n\n Telegram from Kapitza to Dirac, 4 June 1937, KV 2\/777, UKNATARCHI.\n\n Service (2003: 223).\n\n Fitzpatrick (1999: 194).\n\n Letter from Kapitza to Rutherford, 13 September 1937, in Boag et al. (1990: 305-6).\n\n Kojevnikov (2004: 116).\n\n Before Landau fled Kharkov, he had worked at the Ukrainian Physico-technical Institute. He was arrested on 28 April 1938 in Moscow, and Kapitza wrote to Stalin seeking his release. His letter is quoted by David Holloway (1994: 43).\n\n Letter from Dirac to Kapitza, 27 October 1937, Dirac Papers, 2\/3\/6 (FSU).\n\n Letter to Dirac from Kapitza, 7 November 1937, Dirac Papers, 2\/3\/6 (FSU).\n\n Letter from Fowler to Dirac, 25 January 1939, Dirac Papers, 2\/3\/8 (FSU).\n\n This was one of Dirac's favourite observations. See R. Dalitz, _Nature_ , 19 Vol. 278 (April) 1979.\n\n Hoyle (1992: 186).\n\n Hoyle (1994: 131).\n\n Hoyle (1994: 133).\n\n Letter from Dirac to Bohr, 5 December 1938, NBA.\n\n At least two of Flo's poems were published in newspapers: 'Cambridge' appeared in the _Observer_ on Saturday, 23 July 1938, and 'Brandon Hill' was published in the local _Western Daily Press_ on Saturday, 12 March 1938.\n\n On 2 February 1938, Princeton University sent Dirac a letter offering him tenure with an annual salary of $12,000, beginning 1 October 1938, Dirac Papers, 2\/3\/7 (FSU).\n\n Letter from Anna Kapitza to Manci Dirac, 9 March 1938, Dirac Papers, 1\/8\/18 (FSU).\n\n _Nature_ , 21 May 1938, No. 3577, p. 929. Schr\u00f6dinger's well-publicised letter was published in _Graz Tagepost_ , 30 March 1938. See Moore (1989: 337-8).\n\n Letters from Dirac to Manci in August 1938 (DDOCS). Wigner married Amelia Frank on 23 December 1936 in Madison, and she died on 16 August 1937. See 'The Einhorn Family', compiled by Margaret Upton (private communication).\n\n Bell wrote to Dirac on 15 March 1938: 'I had already and for a year or two reached the conclusion the Soviet trials were probably of the frame-up type. After all, that is not new. The Tom Mooney case in California in 1918 was such and the victim has been in prison ever since [. . .] also the Sacco & Vanzetti case. Moreover, we seem to do it ourselves to a great extent in India. However, the \"confession technique\" is  peculiarly Russian, on its present scale at least.' Letter to Dirac from J. H. Bell, Dirac Papers, 2\/3\/7 (FSU).\n\n Moore (1989: 347); letter from Schr\u00f6dinger to Dirac, 27 November 1938, Dirac Papers, 2\/3\/7 (FSU).\n\n Dirac gives these reasons in his obituary of Schr\u00f6dinger in _Nature_ , 4 February 1961, 189, p. 355-6.\n\n Letter from Dirac to Kapitza, 22 March 1938, Dirac Papers, 2\/3\/7 (FSU).\n\n Howarth (1978: 234-5).\n\n _The Times_ , 6 October 1938.\n\n 'Eddington Predicts Science Will Free Vast Energy from Atom', _New York Times_ , 24 June 1930. He was speaking at the World Power Conference. He suggested that such energy could be released by arranging for particles to annihilate or to make hydrogen nuclei fuse to form a helium nucleus.\n\n Rhodes (1986: 28).\n\n Weart and Weiss Szilard (1978: 53).\n\n Weart and Weiss Szilard (1978: Chapter II).\n\n Weart and Weiss Szilard (1978: 71-2).\n\n The event took place in the Society's house, 24 George Street, beginning at 4.30 p.m. Max Born was present.\n\n Mill (1892: Book 2, Chapter 12).\n\n This quote is from the text of the lecture, _Proceedings of the Royal Society_ (Edinburgh), 59 (1938-9: 122-9); p. 123.\n\n _Granta_ , 48 (1): 100, 19 April 1939.\n\n#  **Chapter twenty-two**\n\n Bowyer (1986: 51).\n\n This was one of Manci's favourite expressions about how the British treated her. Interview with Mary Dirac, 21 February 2003.\n\n Boys Smith (1983: 44).\n\n _Cambridge Daily News_ , 2 September 1939, p. 5.\n\n _Cambridge Daily News_ , 1 September 1939, p. 3. I am grateful to my mother, Joyce Farmelo, for her recollections of her time as an unhappy evacuee and her other wartime experiences.\n\n E-mail from Mary Dirac, 5 March 2006.\n\n 'Cambridge During the War; the Town', _Cambridge Review_ , 27 October 1945; 'Cambridge During the War; St John's College', _Cambridge Review_ , 27 April 1946. See also 'Thoughts Upon War Thought', _Cambridge Review_ , 11 October 1940.\n\n Barham (1977: 32-3).\n\n Letter to Dirac from his mother, 26 January 1940, Dirac Papers, 1\/4\/10 (FSU).\n\n Manci spent the final months of her pregnancy in the Mountfield Nursing Home in London. Information about Mary's birth from her baby book. Further clarification in an e-mail from Mary Dirac, 16 January 2006.\n\n Letter to Dirac from Manci, 20 February 1940 (DDOCS). Manci's exact words are ungrammatical: 'I never felt as much that she has nor heart nor feelings whatsoever as yesterday.'\n\n Peierls (1985: 150, 155).\n\n Rhodes (1986: 323).\n\n Facsimiles of the memos are in Hennessy (2007: 24-30).\n\n Peierls (1985: 155).\n\n The earliest extant letter about this, from Peierls to Dirac, is dated 26 October 1940, AB1\/631\/257889, UKNATARCHI.\n\n Rhodes (1986: 303-7); F\u00f6lsing (1997: 710-14).\n\n Letter to Aydelotte from Veblen and von Neumann, 23 March 1940, IAS Archives Faculty Series, Box 33, folder: 'Veblen-Aydelotte Correspondence 1932-47'. The words omitted, marked by the ellipsis, are 'There are considerable deposits of uranium available near Joachimsthal, Bohemia, as well as in Canada.'\n\n Letter to Adyelotte from Veblen, 15 March 1940: IAS Archives General Series, Box 67, folder: 'Theoretical Physics 1940 Proposals'.\n\n Cannadine (1994: 161-2).\n\n Letter from Manci to Crowther, 28 June 1941, SUSSEX.\n\n Barham (1977: 54); Bowyer (1986: 51).\n\n Letter to Dirac from his mother, 27 June 1940, Dirac Papers, 1\/4\/10 (FSU).\n\n Letters to Dirac from his mother, 16 August and 31 August 1940, Dirac Papers, 1\/4\/10 (FSU).\n\n Letter to Dirac from his mother, 12 May 1940, Dirac Papers, 1\/4\/10 (FSU).\n\n Letter to Dirac from his mother, 21 June 1940, Dirac Papers, 1\/4\/10 (FSU).\n\n Letter from Dirac to Manci, 27 August 1940 (DDOCS).\n\n Letter from Dirac to Manci, 23 August 1940. Four days later, he wrote to her: 'I am sorry to be away from you these days, but do not think there is any real danger in Cambridge' (DDOCS).\n\n Gustav Born later recalled that Dirac on this vacation was 'a twinkling-eyed, kindly, distant man', happiest when on his own. Interview with Gustav Born, 12 February 2005.\n\n 'The ladies do the cooking, and the men take it in turns to do the washing up,' Dirac told Manci: letter, 23 August 1940 (DDOCS).\n\n Letter from Dirac to Manci, 2 September 1940 (DDOCS).\n\n Letter to Dirac from Manci, 8 September 1940 (DDOCS).\n\n Letter from Pryce to Dirac, 18 July 1940, Dirac Papers, 2\/3\/10 (FSU).\n\n Letter from Dirac to Manci, 21 January 1940 (DDOCS).\n\n Letter from Gabriel to Dirac, 30 August 1945, and another undated later in the same month, Dirac Papers, 1\/8\/12 (FSU).\n\n Letter to Dirac from his mother, 31 August 1940, Dirac Papers, 1\/4\/10 (FSU).\n\n Letter from Peierls to Oppenheimer, 16 April 1954, LC, Oppenheimer archive.\n\n The first part of this quotation is from the letter Dirac wrote to Manci on 18 December 1940; the second and third parts are from the letter he wrote to her the next day.\n\n Letter to Dirac from Manci, 22 December 1940 (DDOCS).\n\n Werskey (1978: 23); see also the foreword by C. P. Snow to Hardy (1940: 50-3).\n\n Letter to Dirac from Hardy, May 1940, Dirac Papers, 2\/3\/10 (FSU).\n\n Attendance register of Tots and Quots in 1940, Zucherman archive, wartime papers, SZ\/TQ, EANGLIA.\n\n Letter from Crowther to Dirac, 15 November 1940, Dirac Papers, 2\/3\/10 (FSU).\n\n Brown (2005: Chapter 9).\n\n The first letter to Dirac, from Peierls, in connection with war work is dated 26 October 1940, UKNATARCHI.\n\n Bowyer (1986: 181). Manci often spoke of Judy's role in the firefighting (e-mail from Mary Dirac, 23 April 2006). Manci refers to an earlier near-miss on 15 February 1941 in her letter to Crowther on 17 February 1941, SUSSEX.\n\n Dirac often referred to Crowther as 'the newspaper man'. See, for example, letter from Dirac to Manci, 4 May 1939 (DDOCS).\n\n The spy was Jan Willen der Braak. 'The Spy Who Died Out in the Cold', _Cambridge Evening News_ , 30 January 1975.\n\n Letter from Harold Brindley, 7 August 1939, STJOHN; Dirac refers calmly to discussions with Eddington in a letter to Peierls, 16 July 1939, Peierls archive (BOD).\n\n Letter from Pryce to Dirac, 11 June 1941, Dirac Papers, 2\/3\/11 (FSU).\n\n The time of the lecture is recorded in the Royal Society's Meeting Notices. Afternoon tea began at 3.45 p.m.\n\n Letter to Dirac from Pauli (then at the Institute for Advanced Study), 6 May 1942, Dirac Papers, 2\/3\/12 (FSU).\n\n Bohr did not find out about the project until he escaped occupied Denmark in autumn 1943: see Bohr (1950).\n\n Telegram to Dirac from Kapitza, 3 July 1941, Dirac Papers, 2\/3\/11 (FSU).\n\n Letter from Dirac to Kapitza, 27 April 1943, Dirac Papers, 2\/14\/12A (FSU).\n\n Penny (2006: 'Fatalities in the Greater Bristol Area').\n\n Letter to Dirac from Dr Strover, 2 October 1941, Dirac Papers, 2\/3\/11 (FSU).\n\n Letter from Flo Dirac to her neighbour Mrs Adam, written shortly before Christmas 1941, Dirac Papers, 1\/2\/1 (FSU).\n\n Flo was buried in the Borough Cemetery (now the City Cemetery) in grave space 7283.\n\n#  **Chapter twenty-three**\n\n Article by Lannutti in Taylor (1987: 45).\n\n Interview with Monica Dirac, 1 May 2006.\n\n The committee was called MAUD, possibly short for Ministry of Aircraft production Uranium Development committee: Gowing (1964: Chapter 2).\n\n Gowing (1964: 53n.).\n\n Nye (2004: 73-4).\n\n Nye (2004: 75-85).\n\n The quote is from Churchill (1965: epilogue).\n\n Letter to Dirac from F. E. Adcock, 24 May 1942, Dirac Papers, 2\/3\/12 (FSU).\n\n Letter to Dirac from Nigel de Grey of the Foreign Office in London, 1 June 1940, Dirac Papers, 2\/3\/10 (FSU).\n\n Copeland (2006: Chapter 14).\n\n Letter from Sir Denys Wilkinson, who was one of Dyson's fellow students in Dirac's lecture course, 15 January 2004; also phone call, 16 January 2004. 'I went to Dirac's lectures in Cambridge in 1942\/3. Freeman Dyson, a year junior to us but very precocious, was also in the class. He was very disruptive because he asked questions. Dirac always took a long time to answer them and on one occasion ended a class early so that he could prepare a proper response' (interview, 15 January 2004).\n\n Sir Denys Wilkinson, letter, 15 January 2004; phone call, 16 January 2004.\n\n Letter from Dirac to Peierls, 11 May 1942, UKNATARCHI.\n\n See Thorp and Shapin (2000: 564).\n\n Letter from Wigner to the US Office of International Affairs, 1 September 1965, Wigner archive, PRINCETON.\n\n Anecdotes from interview with Monica Dirac, 7 February 2003 and 1 May 2006; and with Mary Dirac, 21 February 2003.\n\n Hoyle (1987: 187).\n\n Dirac, M. (2003: 41).\n\n Letter from Dirac to Manci, 13 July 1942 (DDOCS).\n\n With his usual understatement, Dirac wrote to Manci, 'It seems a little strange to have a prime minister at these very specialized lectures. I wonder how he can spare the time.' Letter from Dirac to Manci, 17 July 1942, DDOCS.\n\n Letter from Peierls to Dirac, 30 September 1942, AB1\/631\/257889.\n\n Letter from Manci to Dirac to 'Anna', 15 October 1986, Wigner archive in PRINCETON.\n\n 'Mrs Roosevelt's Village Hall Lunch', _Cambridge Daily News_ , 5 November 1942.\n\n Wattenberg (1984).\n\n Interview with Al Wattenburg, 30 October 1992.\n\n One of their meetings probably occurred on 31 July 1943, as Dirac proposes this date for a meeting in his letter to Fuchs of 19 August 1943 (BOD). Dirac wrote another letter to Fuchs on 1 September 1943 (BOD).\n\n Peierls (1985: 163-4).\n\n Szasz (1992: xix and 148-51).\n\n Gowing (1964: 261).\n\n Peierls, 'Address to Dirac Memorial Meeting, Cambridge', in Taylor (1987: 37).\n\n Brown (1997: 250).\n\n A further seventy people in Cambridge had been injured and 1,271 homes in the town had been damaged (Barham 1977: 53).\n\n 'Cambridge Streets Light-Up at Last!', _Cambridge Daily News_ , 26 September 1944.\n\n Joe wrote of his family's 'threatening situation' to Heisenberg on 25 March 1943 and sought his assistance. Four months later, Heisenberg replied to say that he was not able to offer specific help but hoped to make contact with Joe during a later visit to Holland (this meeting does not seem to have taken place). Joe wrote again to Heisenberg on 2 February 1944 from Budapest urgently requesting confirmation of Betty's Aryan descent. See Brown and Rechenberg (1987: 156).\n\n Letter from Betty to Dirac, 20 July 1946, Dirac Papers, 1\/7\/2A (FSU).\n\n Interview with Mary Dirac, 21 February 2003.\n\n Gabriel later recalled that Dirac declared that there 'was no God and no Heaven or Hell'. Letter from Gabriel Dirac to the Diracs, 18 January 1972, Dirac Papers, 1\/8\/14 (FSU).\n\n E-mail from Mary Dirac, 17 February 2006. Monica confirms that both daughters were christened.\n\n Boys Smith (1983: 44).\n\n Letter from Lew Kowarski to James Chadwick, 12 April 1943 (CHURCHILL).\n\n Interview with the late John Crook, 1 May 2003. Professor Crook was present when Dirac made this remark.\n\n 'Happy Crowds Celebrate VE-Day', _Cambridge Daily News_ , 9 May 1945.\n\n Interview with Monica Dirac, 1 May 2006.\n\n Pincher (1948: 111). The account of this event by Chapman Pincher implied that Dirac lied. Pincher remarks, 'Dr PAM Dirac, one of the scientists involved, told me at the time that he was not then engaged on vital war research. But, as the British White Paper on atomic energy states, he had been helping the British atom-bomb project by theoretical investigations on chain reactions.' Pincher had not allowed for Dirac's literal-mindedness.\n\n Brown (2005: 266).\n\n Interview with Leopold Halpern, 26 February 2006. Dirac told Halpern that he was disappointed with the actions of the British Government and that he went on long solitary walks in order to cool his anger. Dirac heard of the refusal of his application  for an exit visa from the Home Office official C. D. C. Robinson (letter to Dirac, 13 June 1945, Dirac Papers, 2\/3\/15 [FSU]). Two days later, Nevill Mott wrote to Dirac to inform him of the protests that would be made by the disappointed scientists. Mott makes it plain that he does not expect Dirac to be an active member of the protesting group (letter to Dirac from Mott, Dirac Papers, 2\/3\/15 [FSU]).\n\n Letter from Manci Dirac to Crowther, 18 May 1945, SUSSEX.\n\n Telegram from Joe Teszler to the Diracs, 1 July 1945, Dirac Papers, 1\/7\/5 (FSU).\n\n Interview with Christine Teszler, 22 January 2004.\n\n Letters from Joe Teszler to Manci, 19 July, 2 August, 23 August, 31 August, 6 September and 27 September 1945, Dirac Papers, 1\/7\/5 (FSU).\n\n Cornwell (2003: 396).\n\n The team playing at Lord's was not an official Australian side, but was called 'The Australian Services' team.\n\n Smith (1986: 478).\n\n 'How Cambridge Heard the Great Victory News', _Cambridge Daily News_ , 15 August 1945.\n\n See, for example, _Time_ , 20 August 1945, p. 35.\n\n Cornwell (2003: 394-400).\n\n Anon. (1993: 36).\n\n Anon. (1993: 71).\n\n Dalitz (1987a: 69-70). Also, interview with Dalitz, 9 April 2003.\n\n Interview with Christine Teszler, 22 January 2004.\n\n Letter from Betty to Dirac, 20 July 1946, Dirac Papers, 1\/7\/2A (FSU).\n\n Brown (2005: 173).\n\n Crowther (1970: 264).\n\n The official report on the lecture is in the UKNATARCHI (Dirac Papers. BW83\/2\/257889).\n\n#  **Chapter twenty-four**\n\n Osgood (1951: 149, 208-11).\n\n Interview with Feynman by Charles Weiner, 5 March 1966, 27 March 1966, AIP. Interview with Lew Kowarski by Charles Weiner, 3 May 1970, AIP.\n\n The typed manuscript of Dirac's talk is in the Mudd Library, PRINCETON.\n\n In Feynman's theory, the probability that a quantum such as an electron will make a transition from one point in space-time to another can be calculated from a mathematical expression related to the action involved in moving between the two points, summed over all possible routes between them.\n\n Interview by Charles Weiner of Richard Feynman, 27 June 1966 (CALTECH). See also Feynman's Nobel Lecture and Gleick (1992: 226) and its references.\n\n Interview with Freeman Dyson, 27 June 2005. Dyson noted that Feynman made the point repeatedly.\n\n Quoted by Oppenheimer in Smith and Weiner (1980: 269). Wigner was one of the examiners of Feynman's Ph.D. thesis; the other was Wheeler. The oral examination was held on 3 June 1942, and the examiners' report is held in the Mudd Library, PRINCETON.\n\n See Kevles (1971: Chapter 12) and Schweber (1994: Section 3).\n\n Schweber (1994: Chapter 4); Pais (1986: 450-1); Dyson (2005).\n\n Lamb (1983: 326). 'Radar Waves Find New Force in Atom', _New York Times_ , 21 September 1947.\n\n Ito (1995: 171-82).\n\n Feynman (1985: 8).\n\n Dyson (1992: 306). Interview with Dyson, 27 June 2005. Dyson's description of himself as a 'big shot with a vengeance' is in Schweber (1994: 550).\n\n Dyson (2005: 48).\n\n Dirac took no pleasure in abstract art or in Sch\u00f6nberg's music and found neither beautiful.\n\n 'The Engineer and the Physicist', 2 January 1980, Dirac Papers, 2\/9\/34 (FSU).\n\n Dirac Papers, 2\/29\/34 (FSU).\n\n Dirac Papers, 2\/29\/34 (FSU).\n\n Dyson (2006: 216).\n\n Letter from Manci to Wigner, 20 February 1949, PRINCETON.\n\n Interview with Richard Eden, 14 May 2003.\n\n M. Dirac (1987: 6).\n\n M. Dirac (2003: 41).\n\n I am grateful to the Salamans' daughter Nina Wedderburn for supplying me with biographical information on her parents. Fen (1976: 375).\n\n Gamow (1966: 122); Salaman and Salaman (1986: 69).\n\n Interview with Monica Dirac, 7 February 2003.\n\n Quoted in Hennesey (2006: 5).\n\n It took centuries for women students to win equality with males at Cambridge University. The first women's colleges in Cambridge, Girton and Newnham Colleges, were founded in 1869 and 1871 respectively. From 1881 women were allowed to sit tripos exams but they did not receive any formal qualifications from the university for passing them. From 1882, women's results were published with the men's, but on separate lists. In 1921, a report proposing full admission for women was defeated. Statutes allowing the admission of women to full membership of the university finally received Royal Assent in May 1948, and the first woman to graduate at Cambridge was the Queen Mother in the following October. Under this legislation, women students at Cambridge first graduated in January 1949.\n\n Reasons for Heisenberg's post-war depression are suggested by Cassidy (1992: 528).\n\n R. Eden, unpublished memoirs, May 2003, p. 7a.\n\n Dirac first met Heisenberg after the war in 1958. 'Hero' quote from interview with Antonio Zichichi, 2 October 2005.\n\n Interview with Monica Dirac, 7 February 2003.\n\n Greenspan (2005: 253, 263-4). Dirac supported Heisenberg's nomination, having remarked earlier that his election to a foreign membership of the Royal Society should take precedence over that of Pauli. Cockcroft writes to Dirac in his 15 February letter, 'I agree that he [Heisenberg] is more eminent than Pauli,' Dirac Papers, 2\/4\/7 (FSU).\n\n Letter to Dirac from Douglas Hartree, 22 December 1947, Dirac Papers, 2\/4\/2 (FSU).\n\n Letter to Dirac from Schr\u00f6dinger, 18 May 1949, Dirac Papers, 2\/4\/4 (FSU).\n\n Soon after Blackett won the prize in 1947, Dirac sent to him 'heartiest congratulations', remarking, 'You ought to have had it long ago': letter from Dirac to Blackett, 7 November 1948, Blackett archive, ROYSOC. Yet Dirac had not nominated him.\n\n Dirac nominated Kapitza twice before 1953, on 16 January 1946 and 25 January 1950. It is clear from Dirac's records that he later nominated Kapitza several times (RSAS).\n\n Letter from Dirac to Kapitza, 4 November 1945, Dirac Papers, 2\/4\/12 (FSU); See also  letter from Kapitza to Stalin, 13 October 1944, reproduced in Boag et al. (1990: 361-3).\n\n Boag et al. (1990: 378).\n\n Letter from Kapitza to Stalin, 10 March 1945, cited in Kojevnikov, A. (1991) _Historical Studies in the Physical Sciences_ , 22, 1, pp. 131-64.\n\n Letters from Kapitza to Stalin, 3 October 1945 and 25 November 1945, reprinted in Boag et al. (1990: 368-70, 372-8).\n\n Letter to Dirac from Manci, 12 July 1949 (DDOCS).\n\n _Tallahassee Democrat_ , 29 November 1970.\n\n Bird and Sherwin (2005: 332).\n\n Sources of anecdotes: 'young daughters scurrying', interview with Freeman Dyson, 27 June 2005; 'welcoming Einstein for Sunday tea', interview with Monica Dirac, 7 February 2003, interview with Mary Dirac, 21 February 2003; the 'early evening drinks parties', one of the social rituals at the institute during Oppenheimer's tenure as Director; 'amateur lumberjacks', interview with Morton White, 24 July 2004.\n\n Interview with Freeman Dyson, 27 June 2005. E-mail from Dyson, 23 October 2006.\n\n Interview with Louise Morse, 19 July 2006.\n\n Dirac received several importunate letters from the maverick Austro-Hungarian experimenter Felix Ehrenhaft, who asserted that he had evidence for the existence of the magnetic monopole, Dirac Papers, 2\/13\/1 and 2\/13\/2 (FSU).\n\n Letter from Pauli to Hans Bethe, 8 March 1949, Hermann et al. (1979).\n\n The new theory made little impact, though it did interest scientists - including Dennis Gabor at Imperial College in London - who were studying electron beams in television sets. The correspondence between Dirac and Gabor (1951) is in the Gabor archive at Imperial College, London.\n\n Dirac (1954).\n\n Dirac (1954).\n\n 'The Ghost of the Ether' was published in the _Manchester Guardian_ article on 19 January 1952; the _New York Times_ published 'Briton Says Space Is Full of Ether', 4 February 1952. In Dirac's talk to the 1971 Lindau meeting (for former Nobel Prize winners), he said that the ether appeared not to be useful to quantum mechanics, though he did not rule out that the concept might one day be useful.\n\n Jerome (2002: Chapter 12, 278-82).\n\n Interview with Einstein's acquaintance Gillett Griffen on 20 November 2005, and with Louise Morse on 19 July 2006. The anecdote about Einstein picking up cigarette butts and sniffing them is from Kahler, A. (1985), _My Years of Friendship with Albert Einstein_ , IX, 4, p. 7.\n\n#  **Chapter twenty-five**\n\n The information in this section is mainly from interviews with Monica Dirac (7 and 8 February 2002) and Mary Dirac (21 February 2002 and 17 February 2006). See also M. Dirac (2003: 39-42). Information about Dirac and Betty from interview with Christine Teszler, 22 January 2004.\n\n The boarding school was Beeston Hall School in West Runton, near Cromer. E-mail from Mary Dirac, 30 October 2006.\n\n The Diracs often stayed at the Barkston Gardens Hotel, Kensington, for a week or two.\n\n Interview with Mary Dirac, 21 February 2003.\n\n Letter to Dirac from Manci, 5 September 1949 (DDOCS): 'We can have a quiet weekend in London where the Folies Berg\u00e8re is showing the full Paris show.'\n\n Professor Driuzdustades appears in Russell's 1954 short story 'Zahatopolk' (see  Russell 1972: 82-110).\n\n Manci and Monica often ate at the Koh-I-Noor restaurant in St John's Street. Interview with Monica Dirac, 7 February 2003.\n\n Dalitz (1987b: 17).\n\n Interview with Monica Dirac, 7 February 2003.\n\n Interview with Tony Colleraine, 15 July 2004.\n\n Bird and Sherwin (2005: 463-5).\n\n Letter from Dirac to Manci, undated, late March 1954 (DDOCS).\n\n Szasz (1992: 95).\n\n Letter from Dirac to Oppenheimer, 11 November 1949, LC Oppenheimer archive.\n\n Szasz (1992: 86, 95).\n\n Pais often told this story. See, for example, Pais (2000: 70).\n\n It appears that Dirac was excluded from a conference as early as 1951 because of Manci's Hungarian nationality. See interview with Lew Kowarski by Charles Weiner, 3 May 1970, AIP, pp. 203-4.\n\n The documents concerning the petition, dated 23 March 1950, are in the Bernal Papers, KV 2\/1813, UKNATARCHI.\n\n McMillan (2005: 12, 199).\n\n This letter, from Dirac to Oppenheimer on 17 April, does not appear to have survived. However, Ruth Barnett, of the Institute for Advanced Study, refers to it in her letter to Dirac of 28 April 1954, Dirac Papers, 2\/4\/10 (FSU).\n\n McMillan (2005: 214).\n\n Letter from Dirac to Oppenheimer, 24 April 1954, IAS Dirac archive.\n\n 'US-Barred Scientist \"Not Red\"', _Daily Express_ , 28 May 1954.\n\n 'US Study Visa Barred to Nobel Prize Physicist', _New York Times_ , 27 May 1954.\n\n Letter to Dirac from Christopher Freeman, Secretary of the Society for Cultural Relations with the USSR, 26 April 1954, Dirac Papers, 2\/16\/9 (FSU).\n\n Pais (1998: 33).\n\n Letter from Wheeler, Walker Bleakney and Milton White to the _New York Times_ , published in the newspaper on 3 June 1954.\n\n The name of the woman is not known for certain. Interview with Monica Dirac, 7 February 2003.\n\n Dirac Papers, 2\/14\/5 (FSU).\n\n After the Diracs' stay in Mahabaleshwar, they returned to the Tata Institute in Bombay until 15 December. The Diracs then moved on to Madras and, on 20 December, travelled to Bangalore, where they spent Christmas. On New Year's Eve, they returned to Bombay and then travelled to the Indian Science Congress in Baroda on 5 January. Four days later, they travelled to Delhi and saw the Taj Mahal shortly afterwards. The Diracs were in Calcutta from 18 January to 23 January, before returning to Delhi for a few days and then, finally, back to the Tata Institute. They left India, sailing from Bombay, on 21 February 1955.\n\n Interview with George Sudarshan, 15 February 2005. In 1955, Sudarshan was a research assistant at the Tata Institute.\n\n Dirac's enthusiastic acceptance of the invitation to give this talk in his letter to Dr Basu, 23 June 1954, Dirac Papers, 2\/4\/10 (FSU).\n\n Manuscript of the talk, corrected by Dirac, is in Dirac Papers, 2\/14\/5 (FSU). In the published version of this presentation, many of Dirac's finest touches are removed ( _Journal of Scientific and Industrial Research_ , Delhi, A14, pp. 153-65).\n\n Salaman and Salaman (1996: 68).\n\n _Science and Culture_ , Volume 20, Number 8, pp. 380-1, see p. 380.\n\n Perkovich (1999: 59). India became a nuclear power in 1974, eight years after Bhabha died in a plane crash.\n\n Letter to Oppenheimer from G. M. Shrum, 4 April 1955 (Oppenheimer archive, Dirac Papers, LC). Dirac may have caught this form of jaundice, homologous serum hepatitis, from a contaminated needle during a medical examination in December 1954, Dirac Papers, 1\/9\/3 (FSU).\n\n Note from Manci to Oppenheimer included in Dirac to Oppenheimer, 25 September 1954 (LC, Oppenheimer archive, Dirac Papers).\n\n The Diracs sailed into Vancouver on 16 April. Letters from Manci to Oppenheimer, 15 April 1955, 22 April 1955 and other undated letters written at about the same time (LC, Oppenheimer archive).\n\n Manci often remarked on the one time she saw her husband cry. See, for example, _Science News_ , 20 June 1981, p. 394.\n\n Interview with Tony Colleraine, 22 July 2004.\n\n Letter from Manci to Oppenheimer, 29 August 1955, Oppenheimer archive, Dirac Papers, LC.\n\n Medical report on 28 March 1955, Dirac Papers, 1\/9\/3 (FSU).\n\n The Diracs were in Princeton from 22 May to 30 June 1955, and they flew to Ottawa on 1 July.\n\n Letter from Manci to Oppenheimer, 29 August 1955 (LC, Oppenheimer archive).\n\n Interview with Jeffrey Goldstone, 2 May 2006.\n\n Talk on 'Electrons and the Vacuum' by Dirac at the Lindau conference. The manuscript, annotated by Dirac (June 1956) is in Dirac Papers, 2\/27\/14 (FSU).\n\n 'Electrons and the Vacuum', pp. 7-8.\n\n Dirac spent much of this year working on the fourth edition of _The Principles of Quantum Mechanics_ , which was published in the following year, 1957.\n\n For an account of Kapitza's activities between 1937-49 see Kojevnikov (2004: Chapters 5-8).\n\n Taubman (2003: Chapter 11).\n\n The quote is in a letter from Dirac to Bohr, undated, NBI. The lecture was plainly written after this visit.\n\n Dorozynski (1965: 61).\n\n Boag et al. (1990: 368). See also Knight (1993: Chapters 9 and 10).\n\n Taubman (2003: 256).\n\n Fitzpatrick (2005: 227).\n\n Dorozynski (1965: 60-1).\n\n Feinberg (1987: 185 and 197).\n\n Weisskopf (1990: 194).\n\n Dirac's writing is still preserved on the blackboard.\n\n Landau made this remark at a conference in Moscow in 1957. Interview with Sir Brian Pippard, 29 April 2004.\n\n#  **Chapter twenty-six**\n\n Enz (2002: 533).\n\n Dirac probably heard the news through the grapevine in Cambridge before the news was published. One of the first accounts of the experiment was published in the _Guardian_ on 17 January 1957.\n\n Shanmugadhasan (1987: 56).\n\n Dirac raised the issue of left-right symmetry in quantum mechanics in the Ph.D.  examination of K. J. Le Couteur in 1948, see Dalitz and Peierls (1986: 159).\n\n On 25 August 1970, Dirac gave a piece of paper to the physicist Ivan Waller bearing the message: 'The statement that I do not believe there is any need for P and T invariance occurs in Rev Mod Phys vol 21 p 393 (1949). I never followed it up. PAM Dirac.' Waller archive, RSAS. See also Pais (1986: 25-6).\n\n Polkinghorne (1987: 229).\n\n Seven years later, in 1964, when two experimenters at Princeton University confirmed that some quantum processes that involve the weak interaction are not symmetric when time is reversed, most physicists were once again shocked. But not Dirac: he had also foreseen that possibility in the two paragraphs of his 1949 relativity paper.\n\n The 'wrong horse' quote is from a round-table discussion at the Fermilab Symposium in May 1980, Brown and Hoddeson (1983: 268). The 'complete crushing' quote is from Dirac's talk at the Argonne Symposium on Spin, 26 July 1974, see 'An Historical Perspective on Spin' Lecture notes, pp. 3, Dirac Papers, 2\/29\/3 (FSU).\n\n Taubman (2003: 302).\n\n 'The Soviet Crime in Hungary', _New Statesman_ , 10 November 1956, p. 574.\n\n Interview with Tam Dalyell, 9 January 2005. Dalyell recalls that his meeting with Dirac took place in either 1971 or 1972.\n\n Letter from Dirac to Kapitza, 29 November 1957, Dirac Papers, 2\/4\/12 (FSU).\n\n The connection with the anniversary was pointed out in the _New Statesman_ in 26 October and 9 November 1957.\n\n Interview with Monica Dirac, 1 May 2006.\n\n Dirac often told his daughter Mary that he would like to travel to the Moon. Interview with Mary Dirac, 10 April 2006.\n\n Newhouse (1989: 118).\n\n Newhouse (1989: 118).\n\n The other two physicists at lunch with Dirac were Peter Landshoff and John Nuttall. Interview with Peter Landshoff, 6 April 2006.\n\n Letter from Dirac to Walter Kapryan, 19 July 1974, Dirac Papers, 2\/7\/6 (FSU).\n\n I thank Bob Parkinson and Doug Millard for their advice on the reasons why space rockets were launched vertically rather than horizontally.\n\n Interview with the Revd. Sir John Polkinghorne, 11 July 2003.\n\n Interview with the Revd. Sir John Polkinghorne, 11 July 2003. Dirac once asked 'What is a rho meson?', a particle then well known to almost all particle physics researchers.\n\n Interview with the Revd. Sir John Polkinghorne, 11 July 2003.\n\n Interview with Monica Dirac, 7 February 2003. In 1967, Dirac's parking rights were further constrained, and, again, Manci was outraged. Letter from R. E. Macpherson to Dirac, 2 November 1967, Dirac Papers, 2\/6\/3 (FSU).\n\n Interview with John Crook, 1 May 2003.\n\n After the Christmas vacation of 1959, Gabriel urged his mother to stop telling Dirac 'I will leave you' in front of them. Letter from Gabriel to the Diracs, 13 January 1960, Dirac Papers, 1\/8\/12 (FSU).\n\n Interview with Stanley Deser, 5 July 2006.\n\n Letter to Dirac from Manci, 10 April 1954 (DDOCS).\n\n Interview with Monica Dirac, 7 February 2003.\n\n Hardy (1940: 87). See, for example, letters to Dirac from Gabriel, 22 September 1957 and 8 October 1957, property of Barbara Dirac-Svejstrup.\n\n Interview with Mary Dirac, 21 February 2003.\n\n Dirac told Gamow in 1961 that he began his work on general relativity in the hope of finding a connection between the theory and neutrinos, but that the project had failed. Letter from Dirac to Gamow, 10 January 1961, LC, Gamow archive.\n\n The word 'graviton' appears to have been used for the first time in print by the Soviet physicist D. I. Blokhintsev in the journal _Under the Banner of Marxism_ ( _Pod znamenem marxisma_ ): Blokhintsev (1934). See Gorelik and Frenkel (1994: 96).\n\n 'Physicists Offer New Theories on Gravity Waves and Atomic Particles', _New York Times_ , 31 January 1959.\n\n Deser (2003). I am grateful to Sir Roger Penrose (interview 20 June 2006) and Stanley Deser (interview 5 July 2006) for advice on Dirac's contribution to general relativity.\n\n Pais (1986: 23) and Salam (1987: 92).\n\n Dirac describes the theory in this way in the notes for the talk he gave on 8 October 1970, 'Relativity Against Quantum Mechanics', Dirac Papers, 2\/28\/19 (FSU). See also Dirac (1970).\n\n This description of Oppenheimer is based on the one given by Stephen Spender in _Journals 1939-83_. See also Bernstein (2004: 194).\n\n Anon. (2001: 109-34).\n\n Letter from Dirac to Margrethe Bohr, 20 November 1962, NBA. Margrethe's reply, dated 19 December 1962, is in Dirac Papers, 2\/5\/9 (FSU).\n\n _Nature_ , 4 February 1961, pp. 355-6; see p. 356.\n\n Interview with Dirac, AHQP, 1 April 1962, pp. 5-7.\n\n Interview with Dirac, AHQP, 1 April 1962, p. 5 (text from the original tape).\n\n Interview with Kurt Hofer, 21 February 2004.\n\n In my interviews with Leopold Halpern and Nandor Bal\u00e1zs, respectively on 18 February 2003 and 24 July 2002, they both noted that Dirac said he had 'loathed' his father - an extremely strong word for him to use.\n\n Letter from Kuhn to Dirac, 3 July 1962, Dirac Papers, 2\/5\/9 (FSU). Dirac subsequently gave four more interviews with Kuhn in 7 Cavendish Avenue, Cambridge, on 6, 7, 10 and 14 May 1963.\n\n Interview with Monica Dirac, 30 April 2006.\n\n#  **Chapter twenty-seven**\n\n Interview with the Revd. Sir John Polkinghorne, 11 July 2003.\n\n Interview with Mary Dirac, 21 February 2003.\n\n Dirac co-signed a letter, dated 27 April 1964, to Professor H. Davenport as part of a campaign to oust Batchelor from the headship of the Department of Applied Mathematics and Theoretical Physics, UCAM, Hoyle archive.\n\n Interview with Yorrick and Helaine Blumenfeld, 10 January 2004.\n\n Letter to Dirac from Oppenheimer, 21 April 1963, Dirac Papers, 2\/5\/10 (FSU).\n\n The Diracs were in the USA in 1962 and 1963 (based at the Institute for Advanced Study in Princeton until late April 1962 and from late September 1962 to early April 1963); in 1964 and 1965, based mainly at the Institute for Advanced Study, from September 1964 to spring 1965; in 1966 in March and April, based in Stony Brook, New York; in 1967, based in the spring at Stony Brook and November and December at the University of Texas at Austin; in 1968 and 1969, in December 1968 based in Stony Brook until after Christmas, when they moved on to the University of Miami, where they stayed until spring 1969.\n\n Goddard (1998: xiv).\n\n Dirac (1966: 8). One of the themes of these lectures is Dirac's conclusion that the Schr\u00f6dinger picture of quantum mechanics is untenable when it is applied to field theory and that only the Heisenberg picture is satisfactory.\n\n Dirac (1963:53).\n\n Several instances of Dirac's declining to appear on BBC radio and television programmes are documented in Dirac's archive at Florida State University, notably when he refused to be interviewed in connection with his _Scientific American_ article (letter to Dirac from BBC radio producer David Edge, on 11 June 1963, Dirac Papers, 2\/5\/10 [FSU]).\n\n BBC _Horizon_ programme 'Lindau', reference 72\/2\/5\/6025. The recording was made on 28 June 1965 and broadcast on 11 August 1965.\n\n Barrow (2002: 105-12). Teller noted, however, that the experimental uncertainties in the calculations were so large that it was not possible definitely to rule out the hypothesis.\n\n Barrow (2002: 107).\n\n Letter from Dirac to Gamow, 10 January 1961, Gamow archive LC.\n\n Quoted in Barrow (2002: 108).\n\n Private papers of Mary Dirac. Dirac wrote the notes on 17 January 1933.\n\n Letter to Dirac from Gamow, 26 October 1957, Dirac Papers, 2\/5\/4 (FSU).\n\n John Douglas Cockcroft, _Biographical Memoirs of Fellows of the Royal Society_ (1968): 139-88; see p. 185.\n\n Mitton (2005: 127-9).\n\n Overbye (1991: 39).\n\n Letter from Gamow to Dirac, June 1965 (undated), Dirac Papers, 2\/5\/13 (FSU).\n\n Letter from Heisenberg to Dirac, 2 March 1967, Dirac Papers, 2\/14\/1 (FSU). Letter from Dirac to Heisenberg, 6 March 1967, quoted in Brown and Rechenberg (1987: 148).\n\n Letter from Geoffrey Harrison, HM Ambassador in Moscow, to Sir John Cockcroft, 19 April 1966, Cockcroft archive, CKFT 20\/17 (CHURCHILL).\n\n Kapitza gave the lecture at 5 p.m. on Monday, 16 May. Source: _Cambridge University Reporter_ , 27 April 1966, p. 1,649.\n\n Letter from Manci to Barbara Gamow, 12 May 1966, LC (Gamow archive). Other information from an interview with Mary Dirac, 21 February 2003.\n\n Letter from Manci to Rudolf Peierls, 8 July 1986, Peierls archive, additional papers, D23 (BOD).\n\n Boag et al. (1990: 43-4).\n\n Batelaan, H. (2007) _Reviews of Modern Physics_ , 79, pp. 929-42.\n\n Dirac greatly admired Gell-Mann's skills as a physicist but went out of his way to avoid him on social occasions. Source: interview with Leopold Halpern, 26 February 2006.\n\n Gell-Mann (1967: 699). For more examples of Gell-Mann's initial scepticism about the reality of quarks, see Johnson (2000: Chapter 11).\n\n Gell-Mann (1967: 693).\n\n 'Methods in Theoretical Physics', 12 April 1967, Dirac Papers, 2\/28\/5 (FSU).\n\n Tkachenko was handed back to the Soviet Embassy on 18 September. The British authorities' story was that Tkachenko had 'freely expressed' his wish to return to Russia, but privately they were fearful that he was going to die in their custody. See _The Times_ , 18 June 1967, p. 1; _New York Times_ , 16 September 1967, p. 1. See also the obituary of John Cockcroft by Kenneth McQuillen, former Vice-Master of Churchill College. I thank Mark Goldie, a Fellow of the college, for providing me with this anecdote.\n\n E-mail from Chris Cockcroft, 17 May 2007. See also Oakes (2000: 82). The anecdotes were confirmed by Mary and Monica Dirac.\n\n Letter from Wigner to Office of International Affairs, 1 September 1965, PRINCETON, Wigner archive.\n\n See, for example, letter from Wigner to Manci, 2 September 1965 (FSU, Wigner letters, annex to Dirac Papers).\n\n Letters from the Wigners, 6 and 13 May, and 14 September 1968 (FSU, Wigner letters, annex to Dirac Papers).\n\n Letter from Manci to Wigner, 10 February 1968, Wigner archive (Margit Dirac file) PRINCETON.\n\n Telegram 17 September 1968 (FSU, Wigner letters, annex to Dirac Papers); interview with Mary Dirac, 26 February 2006.\n\n Interview with Mary Dirac, 26 February 2006.\n\n Letter from Mary Wigner to the Diracs, 7 October 1968, Dirac Papers, 2\/6\/6 (FSU).\n\n Letters from the Wigners to the Diracs, 20 and 25 September and 9 October 1968 (FSU, Wigner letters, annex to Dirac Papers). Interview with Mary Dirac, 26 February 2006 and e-mail 7 June 2006.\n\n Interview with Mary Dirac, 26 February 2006 and e-mail 7 June 2006.\n\n Interview with Helaine and Yorrick Blumenfeld, 10 January 2004.\n\n Interview with Philip Mannheim, 8 June 2006. See also the article on Kur\u015funo\u011flu, 'The Launching of La Belle Epoque of High Energy Physics and Cosmology' in Curtright et al. (2004: 427-46).\n\n An account of Dirac's time at the University of Miami is given by Kur\u015funo\u011flu's wife in Kur\u015funo\u011flu and Wigner (1987: 9-28).\n\n Manci wrote to Gamow's wife on 4 February 1969 to complain that Dirac had not accepted the offer made by the University of Miami: 'It makes me feel awful' (LC, Gamow archive, Manci Dirac file).\n\n The reaction of Rabbit and Janice Angstrom to _2001_ are in _Rabbit Redux_ , 1971, Chapter 1 (in the Fawcett Crest Book paperback edition, pp. 58 and 74).\n\n LoBrutto (1997: 277).\n\n I am grateful to Tony Colleraine, then Mary's husband, for his recollections of Dirac's first visits to see _2001: A Space Odyssey_ , interview 15 July 2004 and e-mails on 26 September and 22 October 2004.\n\n Interview with Monica Dirac, 7 February 2003.\n\n Letter from Manci to Barbara Gamow, 16 March 1971, Gamow archive LC.\n\n Letter from Manci to Wigner, 10 February 1968, PRINCETON, Wigner archive.\n\n These FBI documents were declassified in 1986. I thank Bob Ketchum for obtaining a copy of these documents under Freedom of Information\/Privacy Acts.\n\n Letter from Dirac to Alfred Shild, 29 August 1966 (copy held by Lane Hughston).\n\n See, for example, the letter from the Senior Secretary at the University of Texas at Austin to the Immigration and Naturalization Service, 8 December 1967, part of the CIA file on Dirac in the 1960s and 1970s. I am grateful to Robert Ketchum for obtaining these documents.\n\n Tebeau (1976: 151-71 and 219-35). Stanford (1987: 54-5). Interview with Henry King Stanford, 3 July 2006.\n\n Wicker (1990).\n\n Letter from Wigner to Manci Dirac, 9 October 1968 (FSU, Wigner letters, annex to Dirac Papers).\n\n _Miami Herald_ , 7 May 1970, p. 1.\n\n According to Morris (1972), the population of Tallahassee in 1970 was 72,000. The  total population of Miami in the same year was 335,000.\n\n The Physics Department at Florida State University had recently obtained a Center of Excellence grant from the National Science Foundation to assist in its aspiration to become such a centre.\n\n Letter from Colleraine to Dirac, 2 February 1970, Dirac Papers, 2\/6\/9 (FSU).\n\n _Tallahassee Democrat_ , 29 November 1970.\n\n Interview with Peter Tilley, 2 August 2005; interview with Leopold Halpern, 26 February 2006.\n\n Letter from Norman Heydenburg (Chair of the FSU physics department) to Dirac, 4 January 1971, Dirac Papers, 2\/6\/11 (FSU).\n\n Interview with Helaine and Yorrick Blumenthal, 10 January 2004.\n\n#  **Chapter twenty-eight**\n\n Press release from Dorothy Turner Holcomb, 'Barbara Walters . . . I needed you!', 9 March 1971, Dirac Papers, 2\/6\/11 (FSU).\n\n Walters (1970: 173).\n\n Notes on 'The Evolution of our Understanding of Nature', 8 March 1971, in Dirac Papers, 2\/28\/21 (FSU).\n\n Between 1969 and 1983, Dirac gave about a hundred and forty talks, an average of ten talks a year. He gave about eighty-eight talks in the USA, and fifty-two talks overseas, mainly in Europe but occasionally further afield, notably in Australia and New Zealand in 1975. See Dirac Papers, 2\/52\/8 (FSU).\n\n Interview with Kurt Hofer, 21 February 2004.\n\n Interview with Pam Houm\u00e8re, 25 February 2003.\n\n E-mail from Hans Plendl, 5 March 2008, and another from Bill Moulton, 5 March 2008.\n\n Interview with Kurt Hofer, 21 February 2004. Hofer recalls that Dirac would melt when he realised that the person he had dismissed was a friend.\n\n Interview with Hofer. Leopold Halpern independently confirmed this description of Dirac's telephone manner.\n\n Pais (1997: 211). Many of Dirac's colleagues at Florida State University, including Steve Edwards (interview, 27 February 2004) and Michael Kasha (interview, 18 February 2003), attest to the enjoyment he took in telling this joke.\n\n M. Dirac (2003: 39).\n\n Interview with Barbara Dirac-Svejstrup, 5 May 2003.\n\n Letter from Manci to Dirac, undated, August 1972, Dirac Papers, 2\/7\/2 (FSU).\n\n Letter from Manci to Dirac, 18 August 1972, Dirac Papers, 2\/7\/2 (FSU).\n\n Interview with Ken van Assenderp, 25 February 2003.\n\n Interview with Helaine and Yorrick Blumenfeld, 10 January 2004. Helaine Blumenfeld recalls: 'When I was pregnant with my second son, Manci called me all the time to check on things.' Shortly before one of Mrs Blumenfeld's appointments up at Addenbrooke's Hospital, Manci advised her, 'Well, you know they have a lot of black doctors there. Don't let them touch you, they're all dirty.' Monica Dirac recalls that her mother was 'the most anti-Semitic person I've ever met', quite surprising as Manci herself was Jewish. Monica learned of her Jewish ancestry when she was twenty-one years old. Interviews with Monica Dirac, 7 February 2003 and 3 May 2006.\n\n Interview with Yorrick and Helaine Blumenfeld, 10 January 2004.\n\n Interview with Lily Harish-Chandra, 12 July 2007.\n\n Quoted in Chandrasekhar (1987: 65).\n\n The clearest account of Dirac's research agenda during his later years is in the summary he wrote for Joe Lannutti in November 1974, Dirac Papers, 2\/7\/9 (FSU).\n\n Halpern (2003: 25). Interview with Leopold Halpern, 18 February 2003.\n\n Halpern (2003: 24-5).\n\n Leopold Halpern took me on this same trip on Sunday 26 February 2006. During this trip, and in earlier interviews, he described their trips down the river and their reception at home by Manci. In a separate interview, on 27 February 2004, Steve Edwards described the infamous incident in which Dirac dumped Kur\u015funo\u011flu in the Wakulla River\n\n Weinberg (2002).\n\n The special type of gauge theory, was first written down by Yang and his collaborator Robert Mills in 1954. Yang has described the theory as 'a rather straightforward generalization of Maxwell's equation' (quoted in Woolf 1980: 502).\n\n Crease and Mann (1986: Chapter 16).\n\n In the late 1970s Dirac erroneously analysed the opacity of the universe and his error involved a misunderstanding of the Kapitza-Dirac effect (e-mail from Martin Rees, 27 November 2006). Another error is noted in Dalitz and Peierls (1986: 175).\n\n Interview with Leopold Halpern, 18 February 2002. Halpern recalled that Dirac took the discovery seriously and wanted to understand it. 'How can you explain this portrait of Jesus? How can this happen?' Dirac said several times. (The shroud was later proved to be a fake.)\n\n There is no record of Dirac's taking any interest at all in the modern theory of renormalisation. He did, however, acknowledge the brilliance of physicists who worked on the theory, including Abdus Salam, Gerhard 't Hooft and Edward Witten, whom he nominated for awards. Evidence of these nominations is in the Tallahassee archive.\n\n Interview with Rechenberg, 3 June 2003.\n\n Dirac (1977).\n\n Brown and Hoddeson (1983: 266-8).\n\n Interview with Lederman, 18 June 2002.\n\n Interview with Lederman, 18 June 2002. See Farmelo (2002b: 48). Einstein came close to predicting the existence of the positron in his 1925 paper 'Electron and General Relativity', see F\u00f6lsing (1997: 563-5).\n\n Many female acquaintances attest to Dirac's behaviour in this respect, notably Lily Harish-Chandra, Rae Roeder, Helaine Blumenfeld and Colleen Taylor Sen.\n\n Kur\u015funo\u011flu and Wigner (1987: 26). See Mill (1869), especially Chapter 3, 'Of Individuality, as One of the Elements of Well-Being'.\n\n Interview with Kurt Hofer, 21 February 2004.\n\n E-mail from Kurt Hofer, 6 March 2004.\n\n Letter from Manci to Rudolf Peierls, 23 December 1985, Peierls archive, additional papers, D23 (BOD).\n\n Interview with Christine Teszler, 22 January 2004, and an e-mail, 27 March 2004.\n\n This incident occurred in 1978 as Dirac and Hofer passed the Mormon church on Stadium Drive, Tallahassee. Interview with Hofer, 21 February 2004.\n\n Talk on 'Fundamental Problems of Physics', 29 June 1971 (audio recording from LINDAU). See Dirac Papers, 2\/28\/23 (FSU).\n\n In the talk, Dirac suggested a probability for the formation of life that he considered would make it overwhelmingly unlikely without the presence of a God: a chance of one in 10100 (a power of ten also known as a googol).\n E-mail from Kurt Hofer, 28 August 2006.\n\n Halpern (1988: 466 n.). See also Dirac's notes on his lecture 'A Scientist's Attitude to  Religion', c. 1975, Dirac Papers, 2\/32\/11A (FSU).\n\n Isenstein contacted Dirac after meeting him at Bohr's home: letter from Isenstein to Dirac, 29 June 1939, Dirac Papers, 2\/3\/9 (FSU). Isenstein renewed contact with Dirac in 1969, see letter from Isenstein to Dirac, 29 June 1969, Dirac Papers, 2\/6\/7 (FSU).\n\n For correspondence concerning the bust, see the correspondence in the summer of 1971, Dirac Papers, 2\/6\/11 (FSU).\n\n I thank Michael Noakes for his comments on Dirac's sitting for this portrait (interview, 3 July 2006). Noakes points out that Frank Sinatra did not sit for his portrait, though he much liked the result, which he hung on a wall of his study.\n\n Dirac liked the picture, though he grumbled slightly: 'It makes me look a bit old.' Dirac was sensitive about the mark on the left side of his nose, the remains of a pre-cancerous cyst, removed in the summer of 1977. For this reason, Noakes's portrait of Dirac shows only the right side of his face. Dirac looked rather more resolute in the two chalk drawings by Howard Morgan in 1980, commissioned by the National Portrait Gallery.\n\n Feynman's drawing is reproduced in the frontispiece of Kur\u015funo\u011flu and Wigner (1987). An example of Feynman's 'I'm no Dirac' is in interview by Charles Weiner of Richard Feynman, 28 June 1966, p. 187 (CALTECH).\n\n Lord Waldegrave points out that 'the award was largely the result of the intervention of Victor Rothschild, the late Lord Rothschild, who was well placed at that time as a Permanent Secretary in the Cabinet Office as Head of the Central Policy Review Staff of Prime Minister Edward Heath' (interview with Lord Waldegrave, 2 June 2004).\n\n Letter from Manci to Barbara Gamow, 1 May 1973, LC.\n\n Salaman and Salaman (1986: 70). Dirac raised this issue in the context of the experience of his daughter Monica, who 'had studied geology but had given it up to look after her baby'.\n\n Interview with Mary Dirac, 21 February 2003.\n\n Interview with Leopold Halpern, 18 February 2003.\n\n The British part of the project was eventually delivered by the British Aircraft Corporation in collaboration with the French company Sud Aviation, following an agreement signed in 1962. The British Aircraft Corporation had been formed in 1960 from the Bristol Aeroplane Company and other aeronautical firms. I thank Andrew Nahum for advice on this.\n\n The Diracs flew from Dulles to Paris on 5 May 1979 (DDOCS). Letters to Dirac from Abdul-Razzak Kaddoura, Assistant Director-General for Science at UNESCO, dated 29 March 1979, are in Dirac Papers, 2\/9\/3 (FSU).\n\n _New York Times_ , 5 May 1979.\n\n A copy of the speech is in Dirac Papers, 1\/3\/8 (FSU).\n\n Kapitza wrote to Dirac on 18 February 1982, 'Knowing of your going will certainly stimulate my travelling,' Dirac Papers, 2\/10\/6 (FSU).\n\n A recording of Dirac's 1982 talk to the Lindau meeting, 'The Requirements of a Basic Physical Theory' (1 July 1982), and other details are available at LINDAU.\n\n Details of the accommodation are in Dirac Papers, 2\/10\/7 (FSU).\n\n Interview with Kurt Hofer, 21 February 2004; interview with Leopold Halpern, 26 February 2006.\n\n Dirac gave this lecture on 15 August 1981, Dirac Papers, 2\/29\/45 (FSU).\n\n The Erice Statement is readily available on the internet.\n\n On 7 December 1982, Dirac wrote to the Master of St John's to apologise for not being able to attend a gathering at college on 27 December to toast Dirac's health in his eightieth year: 'For 59 years, the College has been the central point of my life and  a home to me' (STJOHN).\n\n Interview with Peter Goddard, 7 June 2006.\n\n#  **Chapter twenty-nine**\n\n The account of Ramond's encounter with Dirac is taken from an interview with Ramond on 18 February 2006 and from subsequent e-mails. Note that the date of the encounter given here is later than the one given in an earlier version of the story (Pais 1998: 36-7); Ramond confirmed the date quoted here, after checking his departmental records. It is not possible to give the precise date of the meeting.\n\n E-mail from Pierre Ramond, 22 December 2003.\n\n _Tallahasse Democrat_ , 15 May 1983, page G1.\n\n Letter to Dirac and Manci from Dirac's mother, 8 April 1940, Dirac Papers, 1\/4\/10 (FSU).\n\n Interview with Dr Watt on the telephone, 19 July 2004.\n\n Dirac's last talk, 'The Future of Atomic Physics', was in New Orleans on 26 May 1983: Dirac Papers, 2\/29\/52 (FSU).\n\n Dirac's surgeon was Dr David Miles. I thank Dr Hank Watt for providing me with a copy of the post-operation report.\n\n Solnit (2001: 104).\n\n Halpern (1985). Interview with Halpern, 24 February 2006.\n\n The essences Halpern used were echinacea, milk thistle and ginseng: interview with Halpern, 24 February 2006.\n\n Dirac (1987: 194-8).\n\n Letter from Manci Dirac to Lily Harish-Chandra, 30 September 1984 (property of Mrs Harish-Chandra).\n\n Letter from Manci Dirac to Lily Harish-Chandra, 16 March 1984 (property of Mrs Harish-Chandra).\n\n Interview with Barbara Dirac-Svejstrup, 5 May 2003.\n\n Interview with Barbara Dirac-Svejstrup, 5 May 2003.\n\n Interview with Peter Tilley, 2 August 2005.\n\n Dirac's death certificate says that he died of respiratory arrest. The coroner found that the final cause of his death was not kidney failure but clogged arteries. See Dirac Papers, 1\/9\/17 (FSU).\n\n Telephone call with Hansell Watt, 19 July 2004.\n\n Manci chose an Episcopalian service because the American Episcopal Church is the Anglican Church in America and is a province of the Anglican Communion under the Archbishop of Canterbury. Information from Steve Edwards, interview, 16 February 2006.\n\n E-mail from Pierre Ramond, 23 February 2006.\n\n I am grateful to Mary Dirac, Steve Edwards, Ridi Hofer and Pierre Ramond for their recollections of the funeral.\n\n The details of Judy's case are from Mercer County Surrogate's Office. The papers that closed the case of Judith Thompson are dated 29 October 1984.\n\n Letter from Dick Dalitz to Peter Goddard, 3 November 1986 (STJOHN; permission to quote this letter from Dalitz during interview with him 9 April 2003).\n\n Letter from Peter Goddard to the Master of St John's College, 26 May 1990, STJOHN.\n\n Interview with Richard Dalitz, 9 April 2003.\n\n Letter from Michael Mayne to Richard Dalitz, 20 May 1990, STJOHN.\n\n The memorial stone was designed and cut by the Cardozo Kindersley workshop in Cambridge, see Goddard (1998: xii).\n\n Letter from Dalitz to Gisela Dirac, 30 November 1995, property of Gisela Dirac.\n\n Goddard (1998: xiii).\n\n Interview with Richard Dalitz, 9 April 2003.\n\n Letter from Dalitz to Gisela Dirac, 30 November 1995, property of Gisela Dirac.\n\n Letter from Manci to Gisela Dirac, 4 July 1992, property of Gisela Dirac. Manci was wrong about Byron's burial. When his remains were brought back to England, burial in the Abbey was refused, and he was interred at Hucknall. Three subsequent unsuccessful attempts were made to insert a memorial to him in the Abbey, the last being in 1924, when the supporting letter was signed by Hardy, Kipling and three former prime ministers (Balfour, Asquith and Lloyd George). Permission for a plaque in Poets' Corner was finally given only in 1969.\n\n See, for example, the letter from Manci to the editor of _Scientific American_ , August 1993, p. 6.\n\n Letter from Manci to Abraham Pais, 25 November 1995, in Goddard (1998: 29).\n\n The Ledermans had become friendly with the Diracs since May 1980, when Dirac attended the conference on the history of particle physics. Lily Harish-Chandra was married to the mathematician Harish-Chandra, Dirac's colleague; Erika Zimmerman was the daughter of Wigner from a relationship he had in G\u00f6ttingen in the late 1920s.\n\n Interview with Peggy Lannuti, 25 February 2004.\n\n Manci did arrange for his Nobel Medal and certificate to be returned to St John's College (letter from Manci to 'Anna', 15 October 1986, Wigner archive PRINCETON). Manci's version of the story of Elizabeth Cockcroft's alleged ejection from Churchill College is told in Oakes (2000: 82).\n\n Letter from Manci to 'Anna', 15 October 1986, Wigner archive PRINCETON.\n\n Interview with Kurt Hofer, 21 February 2004; interview with Leopold Halpern, 26 February 2006.\n\n Interview with the Ledermans, 30 October 2003.\n\n Letter to Manci from Hillary Rodham Clinton, 12 February 1996 (DDOCS). Ms Rodham Clinton wrote: 'It is a pleasure to hear from individuals who share a vision of a better life for all Americans. It is particularly rewarding to hear from people who realize that achieving that vision will not always be easy.' Interview with Monica Dirac, 1 May 2006.\n\n#  **Chapter thirty**\n\n The prize was funded by Rolls Royce and British Aerospace. William Waldegrave recalls that Dirac supported this prize and asked him to send photographs of the Bishop Road School, where his formal education began.\n\n I am grateful to Laura Thorne, of Brunel 200, for details about the programme.\n\n These details and others in this paragraph were confirmed in a telephone conversation with John Bendall, 18 October 2007.\n\n Interview with Mary Dirac, 10 August 2006.\n\n This visit took place on 22 June 2004. Don Carleton, a historian of Bristol, kindly arranged it.\n\n Letter from Manci to 'Anna', 15 October 1986, in PRINCETON, Wigner archive (Margit Dirac file).\n\n These three statements are based on the more rigorous ones given by the autism  expert Uta Frith in her definitive introduction to the condition (2003: 8-9). Her statements are consistent with the most detailed and most recent scheme described in the _Diagnostic and Statistical Manual_ of the American Psychiatric Association (2000), 4th edition, Washington DC, and a similar scheme issued by the World Health Organization, 'The ICD-10 Classification of Mental and Behavioural Disorders: Clinical Descriptions and Diagnostic Guidelines' (1992).\n\n _Stockholms Dagblad_ , 10 December 1933.\n\n Walenski et al. (2006: 175); for the data on depression see p. 9.\n\n Wing (1996: 47, 65 and 123).\n\n Anon. (2007) 'Autism Speaks: The United States Pays Up', _Nature_ , 448: 628-9; see p. 628.\n\n Frith (2003: Chapter 4).\n\n Unlike people with autism, people with Asperger's Syndrome show a delay neither in acquiring language when they are young nor in other aspects of intellectual development. But people with Asperger's Syndrome, when they are older, have similar social impairments to people with autism. See Frith (2003: 11).\n\n Frith (2003: 182).\n\n Interview with Simon Baron-Cohen, 9 July 2003; Baron-Cohen (2003: Chapters 3 and 5).\n\n Fitzgerald (2004: Chapter 1).\n\n Frith (2003: 112).\n\n E-mail from Simon Baron-Cohen 25 December 2006.\n\n Grandin (1995: 137).\n\n Park (1992: 250-9); Temple Grandin's quote is from _Morning Edition_ , US National Public Radio, 14 August 2006. See <http:\/\/www.npr.org\/templates\/story\/story.php?storyId=5628476> (accessed 16 August 2006).\n\n Dirac (1977: 140).\n\n Letter to Dirac from Manci, 2 September 1936, DDOCS.\n\n 'Many patients with tuberculosis present with general symptoms, such as tiredness, malaise, loss of appetite, weakness or loss of weight': Seaton et al. (2000: 516).\n\n There are insights into the childhood of autistic children in the memoir of Gunilla Gerland (translated by Joan Tate), _A Real Person: Life on the Outside._ Gerland writes powerfully of her perception of the misunderstandings in her early relationship with her parents, notably with her father. 'He had no respect for anyone's needs [. . .] The effect of my father's actions was one of pure sadism, although he was not really a sadist. He didn't enjoy my humiliation in itself - he couldn't even imagine it' (Gerland 1996). See also Grandin (1984).\n\n#  **Chapter thirty-one**\n\n Weinberg wrote these words for me to read aloud at the Centenary meeting. Text checked by Weinberg, 22 July 2007 (e-mail).\n\n Interview with Freeman Dyson, 27 June 2005.\n\n Quoted in Charap (1972: 332).\n\n E-mail from Sir Michael Atiyah, 15 July 2007.\n\n Woolf (1980: 502).\n\n Letter from Dirac to Abdus Salam, 11 November 1981, reproduced in Craigie et al. (1983: iii).\n\n 't Hooft (1997: Chapter 14).\n\n Stephen Hawking appeared in an episode of _Star Trek_ first broadcast on 21 June  1993, and in episodes of _The Simpsons_ first broadcast on 9 May 1999 and 1 May 2005.\n\n Letter from Nicolas Kurti to _New Scientist_ , 65 (1975), p. 533; letter from E. C. Stern (1975) to _Science_ , 189, p. 251. See also the comments by Dalitz in 'Another Side to Paul Dirac', in Kur\u015funo\u011flu and Wigner (1987: 87-8).\n\n Freimund et al. (2001). The Kaptiza-Dirac effect had been observed for atoms, but not for electrons, in 1986 (Gould et al. 1986). I thank Herman Betelaan for his advice on modern experiments on the effect.\n\n Deser (2003: 102).\n\n Interview with Nathan Seiberg, 26 July 2007, and e-mail, 20 August 2007.\n\n In his interviews, Leopold Halpern often stressed the importance to Dirac of the large numbers hypothesis (interview with Halpern, 26 February 2006).\n\n By conventional measure, the gravitational force is a millionth of a billionth of a billionth of a billionth the strength of the next strongest fundamental force, the weak interaction.\n\n Rees (2003). I thank Martin Rees for his advice on the status of Dirac's large numbers hypothesis.\n\n E-mails from James Overduin, 20-2 July 2006.\n\n Overduin and Plendl (2007).\n\n I thank Rolf Landua of CERN for his expert help on the current state of experimental research into anti-matter.\n\n See Yang (1980: 39).\n\n These words, written on 27 November 1975, seem to have been special to Dirac. He wrote them on a single sheet of paper and filed them among his lecture notes: Dirac Papers 2\/29\/17 (FSU). The words replaced by [this happened] are 'I have felt the mathematics lead me by the hand.'\n\n The first reference to beauty in Dirac's papers appears to be in the paper he co-wrote with Kapitza in 1933, 'The Reflection of Electrons from Standing Light Waves', where they refer to the beauty of the colour photography introduced by Gabriel Lippmann.\n\n Green and Schwarz's paper was received on 10 September 1984 by the academic journal _Physics Letters B_ , which published it on 13 December.\n\n For a popular account of modern string theory, see Greene (1999).\n\n Dirac told his student Harish-Chandra, 'I am not interested in proofs but only in what nature does': Dalitz and Peierls (1986: 156).\n\n Dirac's notes commend Witten's 'brilliant solutions to a number of problems in mathematical physics', Dirac Papers, 2\/14\/9 (FSU).\n\n Interview with Edward Witten, 8 July 2005, and e-mail, 30 August 2006.\n\n E-mail from Veltman, 20 January 2008. For a sceptical assessment of string theory, see Woit (2006), especially Chapters 13-19.\n**Bibliography**\n\nGenealogy\n\nThe genealogy of the Dirac family is presented at <http:\/\/www.dirac.ch>. The website is maintained by Gisela Dirac-Wahrenburg.\n\nReferences\n\nIn the text, I do not normally give a reference for Dirac's technical papers. They are listed in full in Dalitz (1995) and in Kragh (1990).\n\nAaserud, F. (1990) _Redirecting Science: Niels Bohr, Philanthropy, and the Rise of Nuclear Physics_ , Cambridge: Cambridge University Press.\n\nAnnan, N. (1992) _Our Age: Portrait of a Generation_ , London: Weidenfeld and Nicolson.\n\nAnon. (1935) _The Frustration of Science_ , foreword by F. Soddy, New York: W. W. Norton.\n\nAnon. (1993) _Operation Epsilon: The Farm Hall Transcripts_ , Bristol, Institute of Physics Publishing.\n\nAnon. (2001) _The Cuban Missile Crisis: Selected Foreign Policy Documents from the Administration of John F. Kennedy, January 1961-November 1962_ , London: The Stationery Office, pp. 109-34.\n\nAnon. (2007) 'Autism Speaks: The United States Pays Up', _Nature_ , 448: 628-9.\n\nBadash, L. (1985) _Kaptiza, Rutherford and the Kremlin_ , New Haven, Conn.: Yale University Press.\n\nBaer, H. and Belyaev, A. (eds) (2003) _Proceedings of the Dirac Centennial Symposium_ , London: World Scientific.\n\nBaldwin, T. (1990) _G. E. Moore_ , London and New York: Routledge.\n\nBarham, J. (1977) _Cambridgeshire at War_ , Cambridge: Bird's Farm.\n\nBaron-Cohen, S. (2003) _The Essential Difference_ , New York: Basic Books.\n\nBarrow, J. (2002) _The Constants of Nature_ , New York: Pantheon Books. Batterson, S. (2006) _Pursuit of Genius_ , Wellesley, Mass.: A. K. Peters Ltd. - (2007) 'The Vision, Insight, and Influence of Oswald Veblen', _Notices of the AMS_ , 54 (5): 606-18.\n\nBeller, M. (1999) _Quantum Dialogue: The Making of a Revolution_ , Chicago, Ill.: University of Chicago Press.\n\nBernstein, J. (2004) _Oppenheimer: Portrait of an Enigma_ , London: Duckworth.\n\nBillington Harper, S. (2000) _In the Shadow of the Mahatma_ , Richmond: Curzon.\n\nBioley, H. (1903) _Les Po\u00e8tes du Valais Romand_ , Lausanne: Imprimerie J. Couchoud.\n\nBird, K. and Sherwin, M. J. (2005) _American Prometheus: The Triumph and Tragedy of J. Robert Oppenheimer_ , New York: Vintage.\n\nBlackett, P. M. S. (1955) 'Rutherford Memorial Lecture 1954', _Physical Society Yearbook 1955._\n\n\\- (1969) 'The Old Days of the Cavendish', _Rivista del Cimento_ , 1 (special edition): xxxvii.\n\nBlackwood, J. R. (1997) 'Einstein in a Rear-View Mirror', _Princeton History_ , 14: 9-25.\n\nBlokhintsev D. I. and Gal'perin F. M. (1934) 'Gipoteza neutrino I zakon sokhraneniya energii', _Pod znamenem marxisma_ , 6: 147-57.\n\nBoag, J. W., Rubinin, P. E. and Shoenberg, D. (eds) (1990) _Kapitza in Cambridge and Moscow_ , Amsterdam: North Holland.\n\nBoard of Education (1905) _Suggestions for the Consideration of Teachers and Others Concerned with Public Elementary Schools_ , London: Her Majesty's Stationery Office.\n\nBohr, N. (1950) _Open Letter to the United Nations_ , Copenhagen: J. H. Schultz Forlag. - (1972) _The Collected Works of Niels Bohr_ , Amsterdam: North Holland.\n\nBokulich, A. (2004) 'Open or Closed? Dirac, Heisenberg, and the Relation Between Classical and Quantum Mechanics', _Studies in the History and Philosophy of Modern Physics_ , 35: 377-96.\n\nBorn, M. (1978) _My Life: Recollections of a Nobel Laureate_ , London: Taylor & Francis. - (2005) _The Born-Einstein Letters 1916-55_ , Basingstoke: Macmillan. (First published 1971.)\n\nBowyer, M. J. F. (1986) _Air Raid! The Enemy Air Offensive Against East Anglia_ , Wellingborough: Patrick Stephens.\n\nBoys Smith, J. S. (1983) _Memories of St John's College 1919-69_ , Cambridge: St John's College.\n\nBrendon, P. (2000) _The Dark Valley: A Panorama of the 1930s_ , New York: Alfred A. Knopf.\n\nBroad, C. D. (1923) _Scientific Thought_ , Bristol: Routledge. Reprinted in 1993 by Thoemmes Press, Bristol.\n\nBrown, A. (1997) _The Neutron and the Bomb_ , Oxford: Oxford University Press. - (2005) _J. D. Bernal: The Sage of Science_ , Oxford: Oxford University Press.\n\nBrown, L. M. (1978) 'The Idea of the Neutrino', _Physics Today_ , September, pp. 23-8.\n\nBrown, L. M. and Rechenberg, H. (1987) 'Paul Dirac and Werner Heisenberg: A Partnership in Science', in B. M. Kur\u015funo\u011flu and E. P. Wigner (eds), _Reminiscences about a Great Physicist: Paul Adrien Maurice Dirac_ , Cambridge: Cambridge University Press, pp. 117-162.\n\nBrown, L. M. and Hoddeson, L. (1983) _The Birth of Particle Physics_ , Cambridge: Cambridge University Press.\n\nBryder, L. (1988) _Below the Magic Mountain: A Social History of Tuberculosis in Twentieth Century Britain_ , Oxford: Clarendon Press.\n\n\\- (1992) 'Wonderlands of Buttercup, Clover and Daisies', in R. Cooter (ed.), _In the Name of the Child: Health and Welfare 1880-1940_ , London and New York: Routledge, pp. 72-95.\n\nBudd, M. (2002) _The Aesthetic Appreciation of Nature_ , Oxford: Clarendon Press.\n\nBukharin, N. (1931) 'Theory and Practice from the Standpoint of Dialectical Materialism', available at <http:\/\/www.marxists.org\/archive\/bukharin\/works\/1931\/diamat> (accessed 22 May 2008).\n\nCahan, D. (1989) _An Institute for an Empire_ , Cambridge: Cambridge University Press.\n\nCannadine, D. (1994) _Aspects of Aristocracy_ , New Haven, Conn.: Yale University Press.\n\nCassidy, D. C. (1992) _Uncertainty: The Life and Science of Werner Heisenberg_ , New York: W. H. Freeman and Co.\n\nCathcart, B. (2004) _The Fly in the Cathedral_ , London: Penguin Books.\n\nChadwick, J. (1984) 'Some Personal Notes on the Discovery of the Neutron', in J. Hendry (ed.), _Cambridge Physics in the Thirties_ , Bristol: Adam Hilger, pp. 42-5.\n\nChandrasekhar, S. (1987) _Truth and Beauty: Aesthetic Motivations in Science_ , Chicago, Ill.: University of Chicago Press.\n\nCharap, J. (1972) 'In Praise of Paul Dirac', _The Listener_ , 14 September, pp. 331-2.\n\nChristianson, G. E. (1995) _Edwin Hubble: Mariner of the Nebulae_ , Bristol: Institute of Physics Publishing.\n\nChurchill, Randolph (1965) _Twenty-One Years_ , London: Weidenfeld and Nicholson.\n\nConquest, R. (1986) _Harvest of Sorrow: Soviet Collectivization and the Terror-Famine_ , New York: Oxford University Press.\n\nCopeland, B. J. (2006) _Colossus: The Secrets of Bletchley Park's Codebreaking Computers_ , Oxford: Oxford University Press.\n\nCornwell, J. (2003) _Hitler's Scientists_ , London: Penguin Books.\n\nCraigie, N. S., Goddard, P. and Nahm, W. (eds) (1983) _Monopoles in Quantum Field Theory_ , Singapore: World Scientific.\n\nCrease, R. P. and Mann, C. C. (1986) _The Second Creation: Makers of the Revolution of Twentieth Century Physics_ , New Brunswick, NJ: Rutgers University Press.\n\nCrowther, J. G. (1970) _Fifty Years with Science_ , London: Barrie & Jenkins.\n\nCunningham, E. (1970) 'Ebenezer: Recollections of Ebenezer Cunningham', unpublished, archive of St John's College, Cambridge.\n\nCunningham, E. (1956) 'Obituary of Henry Baker', _The Eagle_ , 57, p. 81.\n\nCurtright, T., Mintz, S. and Perlmutter, A. (eds) (2004) _Proceedings of the 32nd Coral Gables Conference_ , London: World Scientific.\n\nDa Costa Andrade, E. N. (1964) _Rutherford and the Nature of the Atom_ , New York: Anchor Books.\n\nDalitz, R. H. (1987a) 'Another Side to Paul Dirac' in B. M. Kur\u015funo\u011flu and E. P. Wigner (eds), _Reminiscences about a Great Physicist: Paul Adrien Maurice Dirac_ , Cambridge: Cambridge University Press, pp. 69-92.\n\n\\- (1987b) 'A Biographical Sketch of the Life of P. A. M. Dirac', in J. G. Taylor (ed.), _Tributes to Paul Dirac_ , Bristol: Adam Hilger, pp. 3-28.\n\n\\- (ed.) (1995) _The Collected Works of P. A. M. Dirac 1924-1948_ , Cambridge: Cambridge University Press.\n\nDalitz, R. H. and Peierls, R. (1986) 'Paul Adrien Maurice Dirac', _Biographical Memoirs of the Royal Society_ , 32: 138-85.\n\nDaniel, G. (1986) _Some Small Harvest_ , London: Thames and Hudson.\n\nDarrigol, O. (1992) _From c-Numbers to q-Numbers_ , Berkeley, Calif.: University of California Press.\n\nDarrow, K. K. (1934) 'Discovery and Early History of the Positive Electron', _Scientific Monthly_ , 38 (1): 5-14.\n\nDe Maria, M. and Russo, A. (1985) 'The Discovery of the Positron', _Rivista di Storia della Scienza_ , 2 (2): 237-86.\n\nDelbr\u00fcck, M. (1972) 'Out of this World', in F. Reines (ed.), _Cosmology, Fusion and Other Matters: George Gamow Memorial Volume_ , Boulder, Col.: Colorado Associated University Press, pp. 280-8.\n\nDeser, D. (2003) 'P. A. M. Dirac and the Development of Modern General Relativity', in H. Baer and A. Belyaev (eds) (2003) _Proceedings of the Dirac Centennial Symposium_ , London: World Scientific, pp. 99-105.\n\nDingle, H. (1937a) 'Modern Aristotlelianism', _Nature_ , 8 May.\n\n\\- (1937b) 'Deductive and Inductive Methods in Science: A Reply', supplement to _Nature_ , 12 June, pp. 1,001-2.\n\nDirac, M. (1987) 'Thinking of My Darling Paul', in B. M. Kur\u015funo\u011flu and E. P. Wigner (eds), _Reminiscences about a Great Physicist: Paul Adrien Maurice Dirac_ , Cambridge: Cambridge University Press, p. 3-8.\n\nDirac, M. (2003) 'My Father', in H. Baer and A. Belyaev (eds) (2003) _Proceedings of the Dirac Centennial Symposium_ , London: World Scientific, pp. 39-42.\n\nDirac, P. A. M. (1936) 'Does Conservation of Energy Hold in Atomic Processes?' _Nature_ , 22 February, pp. 803-4.\n\n\\- (1954) 'Quantum Mechanics and the Ether', _Scientific Monthly_ , 78: 142-6.\n\n\\- (1963) _The Evolution of the Physicist's Picture of Nature_ , Scientific American, May, vol. 208, no. 5, pp. 45-53.\n\n\\- (1966) _Lectures on Quantum Field Theory_ , New York, Belfer Graduate School of Science, Yeshiva University.\n\n\\- (1970) 'Can Equations of Motion Be Used in High-Energy Physics?', _Physics Today_ , April, pp. 29-31.\n\n\\- (1977) 'Recollections of an Exciting Era', in C. Weiner (ed.), _History of Twentieth Century Physics_ , New York: Academic Press, pp. 109-46.\n\n\\- (1978) 'The Monopole Concept', _International Journal of Physics_ , 17 (4): 235-47.\n\n\\- (1982) 'Pretty Mathematics', _International Journal of Physics_ , 21: 603-5.\n\n\\- (1982a) 'The early years of relativity' in G. Holton and Y. Elkana (eds) _Albert Einstein: Historical and Cultural Perspectives_ , Princeton: Princeton University Press, pp. 79-90.\n\n\\- (1987) 'The Inadequacies of Quantum Field Theory', in B. M. Kur\u015funo\u011flu and E. P. Wigner (eds), _Reminiscences about a Great Physicist: Paul Adrien Maurice Dirac_ , Cambridge: Cambridge University Press, pp. 194-8.\n\nDorozynski, A. (1965) _The Man They Wouldn't Let Die_ , New York: Macmillan.\n\nDyson, F. (1992) 'From Eros to Gaia', Pantheon Books, New York.\n\n\\- (2005) 'Hans Bethe and Quantum Electrodynamics', _Physics Today_ , October, pp. 48-50.\n\n\\- (2006) _The Scientist as Rebel_ , New York: New York Review of Books.\n\nEarman, J. and Glymour, C. (1980) 'Relativity and Eclipses: the British Eclipse Expeditions of 1919 and Their Predecessors', _Historical Studies in the Physical Sciences_ , 11: 49-85.\n\nEddington, A. (1918) 'Report on the Meeting of the Association Held on Wednesday November 27 1918 at Sion College, Victoria Embankment, E.C.', _Journal of the British Astronomical Association_ , 29: 35-9.\n\n\\- (1928) _The Nature of the Physical World_ , Cambridge: Cambridge University Press.\n\nEinstein, A. (1931) in _James Clerk Maxwell: A Commemorative Volume 1831-1931_ , Cambridge: Cambridge University Press.\n\nElsasser, W. (1978) _Memoirs of a Physicist in the Atomic Age_ , London: Adam Hilger.\n\nEnz, C. P. (2002) _No Time to Be Brief: A Scientific Biography of Wolfgang Pauli_ , Oxford: Oxford University Press.\n\nFarmelo, G. (ed.) (2002a) _It Must Be Beautiful: Great Equations of Modern Science_ , London: Granta.\n\n\\- (2002b) 'Pipped to the Positron', _New Scientist_ , 10 August, pp. 48-9.\n\n\\- (2005) 'Dirac's Hidden Geometry', _Nature_ , 437, p. 323.\n\nFeinberg, E. L. (ed.) (1987) _Reminiscences about I. E. Tamm_ , Moscow: Nauka.\n\nFellows, F. H. (1985) 'J. H. Van Vleck: The Early Life and Work of a Mathematical Physicist', unpublished PhD thesis, University of Minnesota.\n\nFen, E. (1976) _A Russian's England_ , Warwick: Paul Gordon Books.\n\nFermi, L. (1968) 'Illustrious Immigrants: The Intellectual Migration from Europe 1930-41', Chicago, Ill.: University of Chicago Press.\n\nFeynman, R. P. (1985) _QED: The Strange Theory of Light and Matter_ , London, Penguin Books.\n\nFitzgerald, M. (2004) _Autism and Creativity_ , New York: Brunner-Routledge.\n\nFitzpatrick, S. (1999) _Everyday Stalinism_ , Oxford: Oxford University Press.\n\n\\- (2005) _Tear Off the Masks!_ , Princeton, NJ: Princeton University Press.\n\nF\u00f6lsing, A. (1997) _Albert Einstein: A Biography_ , New York: Viking.\n\nFreeman, J. (1991) _A Passion for Physics_ , London: Institute of Physics Publishing.\n\nFreimund, D. L., Aflatooni, K., and Batelaan, H. (2001) 'Observation of the Kaptiza-Dirac Effect', _Nature_ , 413: 142-3.\n\nFrench, A. P, and Kennedy, P. J. (eds) (1985) _Niels Bohr: A Centenary Volume_ , Cambridge, Mass.: Harvard University Press.\n\nFrenkel, V. Y. (1966) 'Yakov Ilich Frenkel: His Life, Work and Letters', Boston, Mass.: Birkh\u00e4user Verlag.\n\nFrith, U. (2003) _Autism: Explaining the Enigma_ , 2nd edn, Oxford: Blackwell.\n\nGalison, P. (1987) _How Experiments End_ , Chicago, Ill.: University of Chicago.\n\n\\- (2000) 'The Suppressed Drawing: Paul Dirac's Hidden Geometry', _Representations_ , autumn issue, pp. 145-66.\n\n\\- (2003) _Einstein's Clocks, Poincar\u00e9's Maps_ , London: Sceptre.\n\nGamow, G. (1966) _Thirty Years that Shook Physics_ , Doubleday & Co, New York.\n\n\\- (1967) 'History of the Universe', _Science_ , 158 (3802): 766-9.\n\n\\- (1970) _My World Line: An Informal Autobiography_ , New York: Viking Press.\n\nGardiner, M. (1988) _A Scatter of Memories_ , London: Free Association Books.\n\nGardner, M. (2004) _The Colossal Book of Mathematics_ , W. W. Norton & Co, New York.\n\nGarff, J. (2000) _S\u00f8ren Kierkegaard_ , trans. B. H. Kirmmse, Princeton, NJ: Princeton University Press.\n\n\\- (2005) _S\u00f8ren Kierkegaard: A Biography_ , trans. B. H. Kirmmse, Princeton, NJ: Princeton University Press.\n\nGaunt, W. (1945) _The Aesthetic Adventure_ , London: Jonathan Cape.\n\nGell-Mann, M. (1967) 'Present Status of the Fundamental Interactions', in A. Zichichi (ed.), _Hadrons and Their Interactions: Current and Field Algebra, Soft Pions, Supermultiplets, and Related Topics_ , New York: Academic Press.\n\n\\- (1994) _The Quark and the Jaguar_ , London: Little, Brown & Co.\n\nGerland, G. (1996) _A Real Person: Life on the Outside_ , trans. Joan Tate, London: Souvenir Press.\n\nGleick, J. (1992) _Richard Feynman and Modern Physics_ , London: Little, Brown.\n\nGoddard, P. (ed.) (1998) _Paul Dirac: The Man and His Work_ , Cambridge: Cambridge University Press.\n\nGoodchild, P. (1985) _J. Robert Oppenheimer: Shatterer of Worlds_ , New York: Fromm International.\n\nGorelik G. E. and Frenkel, V. Y. (1994) _Matvei Petrovich Bronstein and Soviet Theoretical Physics in the Thirties_ , Boston, Mass.: Birkh\u00e4user Verlag.\n\nGottfried, K. (2002) 'Matter All in the Mind', _Nature_ , 419, p. 117.\n\nGould, P. et al. (1986) _Physical Review Letters_ , 56: 827-30.\n\nGowing, M. (1964) _Britain and Atomic Energy 1939-45_ , Basingstoke: Macmillan.\n\nGrandin, T. (1984) 'My Experiences as an Autistic Child and Review of Selected Literature', _Journal of Orthomolecular Psychiatry_ , 13: 144-74.\n\n\\- (1995) 'How People with Autism Think', in E. Schopler and G. B. Mesibov (eds), _Learning and Cognition in Autism_ , New York: Plenum Press: 137-56.\n\nGray, A. (1925) _The Town of Cambridge_ , Cambridge: W. Heffers & Sons Ltd.\n\nGreene, B. (1999) _The Elegant Universe_ , New York: W.W. Norton & Co.\n\nGreenspan, N. T. (2005) _The End of the Uncertain World: The Life and Science of Max Born_ , Chichester: John Wiley & Sons Ltd.\n\nHalpern L. (1988) 'Observations of Two of Our Brightest Stars', in K. Bleuler and M.  Werner (eds), _Proceedings of the NATO Advanced Research Workshop and the 16th International Conference on Differential Geometrical Methods in Theoretical Physics_ , Boston, Mass.: Kluwer, pp. 463-70.\n\n\\- (2003) 'From Reminiscences to Outlook', in H. Baer and A. Belyaev (eds), _Proceedings of the Dirac Centennial Symposium_ , London: World Scientific, pp. 23-37.\n\nHarap, J. (1972) 'In Praise of Paul Dirac', _The Listener_ , 14 September, pp. 331-2.\n\nHardy, G. H. (1940) _A Mathematician's Apology_ , Cambridge: Cambridge University Press.\n\nHayward, F. H. (ed.) (1909) _The Primary Curriculum_ , London: Ralph, Holland & Co.\n\nHearn, L. (1896) _Kokoro: Hints and Echoes of Japanese Inner Life_ , London: Osgood & Co.\n\nHeaviside, O. (1899) _Electromagnetic Theory_ , Vol. II, London: Office of 'The Electrician'.\n\nHeilbron, J. (1979) _Electricity in the 17th and 18th Centuries_ , Berkeley, Calif.: University of California Press.\n\nHeisenberg, W. (1967) 'Quantum Theory and its Interpretation', in S. Rozental (ed.), _Niels Bohr: His Life and Work As Seen by His Friends and Colleagues_ , New York: Wiley, pp. 94-108.\n\n\\- (1971) _Physics and Beyond_ , London: George Allen & Unwin.\n\nHendry, J. (ed.) (1984) _Cambridge Physics in the Thirties_ , Bristol: Adam Hilger Ltd.\n\nHennessey, P. (2006) _Having It So Good_ , London: Allen Lane.\n\n\\- (2007) _Cabinets and the Bomb_ , Oxford: Oxford University Press.\n\nHermann, A., v. Meyenn, K. and Weisskopf, V. F. (eds) (1979) _Wolfgang Pauli: Scientific Correspondence with Bohr, Einstein, Heisenberg_ , 3 vols, Berlin: Springer.\n\nHodge, W. V. D. (1956) 'Henry Frederick Baker', _Biographical Memoirs of Fellows of the Royal Society_ , Vol. II, November, pp. 49-68.\n\nHolloway, D. (1994) _Stalin and the Bomb_ , New Haven, Yale University Press.\n\nHolroyd, M. (1988) _Bernard Shaw, Vol. I: 1856-98_ , New York: Random House.\n\n't Hooft, G. (1997) _In Search of the Ultimate Building Blocks_ , Cambridge: Cambridge University Press.\n\nHowarth, T. E. B. (1978) _Cambridge Between Two Wars_ , London: Collins.\n\nHoyle, F. (1987) 'The Achievement of Dirac', _Notes and Records of the Royal Society of London_ , 43 (1): 183-7.\n\n\\- (1994) _Home is Where the Wind Blows_ , Mill Valley, Calif.: University Science Books.\n\nHughes, J. (2003) _Thinker, Toiler, Scientist, Spy? Peter Kapitza and the British Security State_ , Manchester: University of Manchester.\n\nHuxley, A. (1928) _Point Counterpoint_ , New York, Random House.\n\nHuxley, T. H. (1894) _Biogenesis and Abiogenesis: Collected Essays, 1893-1894: Discourses, Biological and Geological_ , vol. 8, Basingstoke: Macmillan.\n\nInfeld, I. (1941) _Quest: The Evolution of a Scientist_ , London: The Scientific Book Club.\n\nIto D. (1995) 'The Birthplace of Renormalization Theory', in M. Matsui (ed.), _Sin-itiro Tomonaga: Life of a Japanese Physicist_ ', Tokyo: MYU, pp. 171-82.\n\nJeffreys, B. (1987) 'Reminiscences at the Dinner held at St John's College', in J. G. Taylor (ed.), _Tributes to Paul Dirac_ , Bristol: Adam Hilger, pp. 38-9.\n\nJerome, F. (2002) _The Einstein File_ , New York: St Martin's Griffin.\n\nJerome, F. and Taylor, R. (2005) _Einstein on Race and Racism_ , New Brunswick, NJ: Rutgers University Press.\n\nJohnson, G. (2000) _Strange Beauty_ , London, Jonathan Cape.\n\nJones, D. (2000) _Bristol Past_ , Chichester: Phillimore.\n\nJones, R. (1998) _Gender and the Formation of Taste in Eighteenth-Century Britain_ , Cambridge: Cambridge University Press.\n\nKedrov, F. B. (1984) _Kapitza: Life and Discoveries_ , Moscow: Mir Publishers.\n\nKevles, D. J. (1971) _The Physicists: The History of a Scientific Community in Modern America_ , New York: Alfred A. Knopf.\n\nKhalatnikov, I. M. (ed.) (1989) _Landau: the Physicist and the Man_ , trans. B. J. Sykes, Oxford: Pergamon Press.\n\nKnight, A. (1993) _Beria: Stalin's First Lieutenant_ , Princeton, NJ: Princeton University Press.\n\nKojevnikov, A. (1993) _Paul Dirac and Igor Tamm Correspondence Part 1: 1928-1933_ , Munich, Max Planck Institute for Physics.\n\n\\- (1996) _Paul Dirac and Igor Tamm Correspondence Part II: 1933-36_ , Munich, Max Planck Institute for Physics.\n\n\\- (2004) _Stalin's Great Science: The Times and Adventures of Soviet Physicists_ , London: Imperial College Press.\n\nKragh, H. (1990) _Dirac: A Scientific Biography_ , Cambridge: Cambridge University Press.\n\n\\- (1996) _Cosmology and Controversy_ , Princeton, NJ: Princeton University Press.\n\nKronig, R. and Weisskopf, V. F. (eds) (1964) _Collected Scientific Papers by Wolfgang Pauli_ , Vol. II, New York: Interscience Publishers.\n\nKur\u015funo\u011flu, B. N. and Wigner, E. P. (eds) (1987) _Reminiscences About a Great Physicist: Paul Adrien Maurice Dirac_ , Cambridge: Cambridge University Press.\n\nKur\u015funo\u011flu, S. A. (1987) 'Dirac in Coral Gables', in B. M. Kur\u015funo\u011flu and E. P. Wigner (eds), _Reminiscences about a Great Physicist: Paul Adrien Maurice Dirac_ , Cambridge: Cambridge University Press, pp. 9-28.\n\nLamb, W. (1983) in 'The Fine Structure of Hydrogen' in L. M. Brown and L. Hoddeson (eds) (1983), _The Birth of Particle Physics_ , Cambridge: Cambridge University Press, pp. 311-28.\n\nLambourne, L. (1996) _The Aesthetic Movement_ , London: Phaidon Press.\n\nLanouette, W. (1992) _Genius in the Shadows: A Biography of Leo Szilard_ , New York: Scribner's.\n\nLee, S. (ed.) (2007a) _Sir Rudolf Peierls: Selected Private and Scientific Correspondence, Volume 1_ , London: World Scientific.\n\n\\- (ed.) (2007b) _The Bethe-Peierls Correspondence_ , London: World Scientific.\n\nLoBrutto, V. (1997) _Stanley Kubrick: A Biography_ , London: Faber & Faber.\n\nL\u00fctzen, J. (2003) 'The Concept of the Function in Mathematical Analysis', in M. J. Nye (ed.), _The Cambridge History of Science, Vol. V: The Modern Physical and Mathematical Sciences_ , Cambridge: Cambridge University Press, pp. 468-87.\n\nLyes, J. (n.d.) 'Bristol 1914-19', Bristol Branch of the Historical Association (undated but apparently c. 1920).\n\nMcCrea, W. H. (1987) 'Eamon de Valera, Erwin Schr\u00f6dinger and the Dublin Institute', in C. W. Kilmister (ed.), _Schr\u00f6dinger: Centenary Celebration of a Polymath_ , Cambridge: Cambridge University Press, pp. 119-34.\n\nMcGucken, W. (1984) _Scientists, Society and State_ , Columbus, Ohio: Ohio State University Press, pp. 40-1.\n\nMcMillan, P. J. (2005) _The Ruin of J. Robert Oppenheimer and the Birth of the Modern Arms Race_ , New York: Penguin.\n\nMatthews, G. B. (1914) _Projective Geometry_ , London: Longmans, Green & Co.\n\nMatsui, M. (1995) _Sin-Itiro Tomonaga: Life of a Japanese Physicist_ , Tokyo: MYU.\n\nMehra, J. (ed.) (1973) _The Physicist's Conception of Nature_ , Boston, Mass.: D. Reidel.\n\nMehra, J. and Rechenberg, H. (1982) _The Historical Development of Quantum Theory_ , Vol. IV, New York: Springer-Verlag.\n\nMichalka, W. and Niedhart, G. (eds) (1980) _Die ungeliebte Republik_ , Munich: DTV.\n\nMichelet, H. (1988) in _Les Echos de Saint-Maurice_ , Saint-Maurice, Editions Saint-Augustin, pp. 91-100 _._\n\nMill, J. S. (1869) _On Liberty_ , London: Penguin Books.\n\n\\- (1873) _Autobiography._\n\n\\- (1892) _A System of Logic_ , London: George Routledge and Son.\n\nMiller, A. I. (2005) _Empire of the Stars_ , London: Little, Brown.\n\nMitton, S. (2005) _Fred Hoyle: A Life in Science_ , London: Aurum Press.\n\nMoldin, S. O. and Rubenstein, J. L. R. (eds) (2006) _Understanding Autism: from Basic Neuroscience to Treatment_ , New York: Taylor & Francis.\n\nM\u00f8ller, C. (1963) 'Nogle erindringer fra livet p\u00e5 Bohrs institute I sidste halvdel af tyverne [Some memories from life at Bohr's institute in the late 1920s]', in _Niels Bohr, et Mindeskrift [Niels Bohr, a Memorial Volume]_ , Copenhagen: Gjellerup, pp. 54-64.\n\nMoore, G. E. (1903) _Principia Ethica_ , Cambridge: Cambridge University Press.\n\nMoore, W. (1989) _Schr\u00f6dinger: Life and Thought_ , Cambridge: Cambridge University Press.\n\nMorgan, K., Cohen, G. and Flin, A. (2007) _Communists and British Society 1920-91_ , London: Rivers Oram Press.\n\nMorley, A. (1919) _Strength of Materials_ , London: Longmans, Green and Co.\n\nMorrell, G. W. (1990) 'Britain Confronts the Stalin Revolution: The Metro-Vickers Trial and Anglo-Soviet Relations, 1933', Ph.D. thesis, Michigan State University.\n\nMorris, A. (1972) _The Florida Handbook 1971-72_ , Tallahassee, Fla.: Peninsular Publishing Company.\n\nMott, N. F. (1986) _A Life in Science_ , London: Taylor & Francis.\n\nNahin, P. J. (1987) _Oliver Heaviside: Sage in Solitude_ , New York: IEEE Press.\n\nNeedham, J. (1976) _Moulds of Understanding_ , London: George, Allen & Unwin.\n\nNewhouse, J. (1989) _War and Peace in the Nuclear Age_ , New York: Knopf.\n\nNye, M. J. (ed.) (2003) _The Cambridge History of Science, Vol. V: The Modern Physical and Mathematical Sciences_ , Cambridge: Cambridge University Press.\n\n\\- (2004) _Blackett: Physics, War, and Politics in the Twentieth Century_ , Cambridge, Mass.: Harvard University Press.\n\nOakes, B. B. (2000) 'The Personal Papers of Paul A.M. Dirac: Their History and Preservation at the Florida State University', Unpublished PhD thesis, Florida State University.\n\nOliphant, M. (1972) _Rutherford: Recollections of the Cambridge Days_ , Amsterdam: Elsevier Publishing Company.\n\nOrwell, G. (2004) _Why I Write_ , London: Penguin.\n\nOsgood C. (1951) _Lights in Nassau Hall. A Book of the Bicentennial 1746-1946_ , Princeton, NJ: Princeton University Press.\n\nOverbye, D. (1991) _Lonely Hearts of the Cosmos_ , New York: Harper Collins.\n\nOverduin, J. M. and Plendl, H. S. (2007) 'Leopold Ernst Halpern and the Generalization of General Relativity', in H. Kleinert, R. T. Jantzen and R. Ruffini (eds), _The Proceedings of the Eleventh Marcel Grossmann Meeting on General Relativity_ , Singapore: World Scientific.\n\nPais, A. (1967) 'Reminiscences from the Post-War Years', in S. Rozental (ed.), _Niels Bohr: His Life and Work as Seen by His Friends and Colleagues_ , New York: Wiley, pp. 215-26.\n\n\\- (1982) _Subtle is the Lord_ , Oxford: Oxford University Press.\n\n\\- (1986) _Inward Bound_ , Oxford: Oxford University Press.\n\n\\- (1991) _Niels Bohr's Times, in Physics, Philosophy and Polity_ , Oxford: Clarendon Press.\n\n\\- (1997) _A Tale of Two Continents: A Physicist's Life in a Turbulent World_ , Princeton, NJ: Princeton University Press.\n\n\\- (1998) 'Paul Dirac: Aspects of His Life and Work', in P. Goddard (ed.), _Paul Dirac: The Man and His Work_ , Cambridge: Cambridge University Press, pp. 1-45.\n\n\\- (2000) _The Genius of Science_ , Oxford: Oxford University Press.\n\nPark, C. (1992) 'Autism into Art: a Handicap Transfigured', in E. Schopler and G. B. Mesibov (eds), _High-Functioning Individuals with Autism_ , New York: Plenum Press, pp. 250-9.\n\nParry, A. (1968) _Peter Kapitza on Life and Science_ , Basingtoke: Macmillan.\n\nPeierls, R. (1985) _Bird of Passage_ , Princeton, NJ: Princeton University Press.\n\n\\- (1987) 'Address to Dirac Memorial Meeting, Cambridge', in J. G. Taylor (ed.), _Tributes to Paul Dirac_ , Bristol: Adam Hilger, pp. 35-7.\n\nPenny, J. (2006) 'Bristol During World War Two: the Attackers and Defenders', unpublished.\n\nPerkovich, G. (1999) _India's Nuclear Bomb_ , Berkeley, Calif.: University of California Press.\n\nPincher, C. (1948) _Into the Atomic Age_ , London: Hutchinson and Co.\n\nPolkinghorne, J. C. (1987) 'At the Feet of Dirac', in B. M. Kur\u015funo\u011flu and E. P. Wigner (eds), _Reminiscences about a Great Physicist: Paul Adrien Maurice Dirac_ , Cambridge: Cambridge University Press, pp. 227-9.\n\nPopplewell, W. C. (1907) _Strength of Materials_ , London: Oliver and Boyd.\n\nPratten, D. G. (1991) _Tradition and Change: The Story of Cotham School_ , Bristol: Burleigh Press Ltd.\n\nRaymond, J. (ed.) (1960) _The Baldwin Age_ , London: Eyre and Spottiswoode.\n\nRees, M. (2003) 'Numerical Coincidences and \"Tuning\" in Cosmology', _Astrophysics and Space Science_ , 285 (2): 375-88.\n\nReines, F. (1972) (ed.) _Cosmology, Fusion and Other Matters: George Gamow Memorial Volume_ , Boulder, Col.: Colorado Associated University Press.\n\nRhodes, R. (1986) _The Making of the Atomic Bomb_ , London: Simon and Schuster.\n\nRobertson, M. (1985) 'Recollections of Princeton: The Princeton Mathematics Community in the 1930s', available at <http:\/\/www.princeton.edu\/~mudd\/finding_aids\/mathoral\/pm02.htm> (accessed 22 May 2008).\n\nRoqu\u00e9, X. (1997) 'The Manufacture of the Positron', _Studies in the Philosophy and History of Modern Physics_ , 28 (1): 73-129.\n\nRoss, S. (1962) 'Scientist: The Story of the Word', _Annals of Science_ , 18 (June): 65-85.\n\nRowlands, P. and Wilson, J. P. (1994) _Oliver Lodge and the Invention of Radio_ , Liverpool: PD Publications.\n\nRozental, S. (ed.) (1967) _Niels Bohr: His Life and Work as Seen by His Friends and Colleagues_ , New York: Wiley.\n\nRussell, B. (1972) _The Collected Stories_ , London, George Allen & Unwin.\n\nSachs R. G. (ed.) (1984) _The Nuclear Chain Reaction: Forty Years Later_ , Chicago, Ill.: University of Chicago Press.\n\nSalam, A. (1987) 'Dirac and Finite Field Theories', in J. G. Taylor (ed.), _Tributes to Paul Dirac_ , Bristol: Adam Hilger, pp. 84-95.\n\nSalam, A., and Wigner, E. P. (eds) (1972) _Aspects of Quantum Theory_ , Cambridge: Cambridge University Press.\n\nSalaman, E. and M. (1986) 'Remembering Paul Dirac', _Encounter_ , 66 (5): 66-70.\n\nSchilpp, P. A. (1959) (ed.) _The Philosophy of C. D. Broad_ , New York: Tudor Publishing Company.\n\n\\- (1970), _Albert Einstein: Philosopher-Scientist_ , Library of Living Philosophers, Volume VII, La Salle, Ill.: Open Court Publishing Company.\n\nSch\u00fccking, E. (1999) 'Jordan, Pauli, Politics, Brecht, and a Variable Gravitational Constant', _Physics Today_ , October, pp. 26-36.\n\nSchultz, B. (2003) _Gravity from the Ground Up_ , Cambridge: Cambridge University Press.\n\nSchuster, A. (1898a) 'Potential Matter: A Holiday Dream', _Nature_ , 18 August, p. 367.\n\n\\- (1898b) 'Potential Matter', _Nature_ , 27 October, pp. 618-19.\n\nSchweber, S. S. (1994) _QED and the Men Who Made It: Dyson, Feynman, Schwinger and Tomonaga_ , Princeton, NJ: Princeton University Press.\n\nScott Fitzgerald, F. (1931) 'Echoes of the Jazz Age', _Scribner's Magazine_ , November, pp. 459-65.\n\nSeaton, A., Seaton, D. and Leitch, A. G. (2000) _Crofton and Douglas's Respiratory Diseases_ , Vol. II, Oxford: Blackwell.\n\nSerber, R. (1998) _Peace and War_ , New York: Columbia University Press.\n\nService, R. (2003) _A History of Modern Russia_ , Cambridge, Mass.: Harvard University Press.\n\nShanmugadhasan, S. (1987) 'Dirac as Research Supervisor and Other Remembrances', in J. G. Taylor (ed.), _Tributes to Paul Dirac_ , Bristol: Adam Hilger, pp. 48-57.\n\nShoenberg, D. (1985) 'Piotr Leonidovich Kapitza', _Biographical Memoirs of Fellow of the Royal Society_ , 31: 326-74.\n\nSinclair, A. (1986) _The Red and the Blue: Intelligence, Treason and the Universities_ , London: Weidenfeld and Nicolson.\n\nSkorupski, J. (1988) 'John Stuart Mill', in E. Craig (ed.), _Routledge Encyclopaedia of Philosophy,_ London: Routledge.\n\nSlater, J. (1975) _Solid-State and Molecular Theory: A Scientific Biography_ , New York: John Wiley and Sons.\n\nSmith, A. K. and Weiner, C. (eds) (1980) _Robert Oppenheimer: Letters and Recollections_ , Stanford, Calif.: Stanford University Press.\n\nSnow, C. P. (1931) 'A Use for Popular Scientists', _Cambridge Review_ , 10 June, p. 492-3.\n\n\\- (1934) _The Search_ , London: Victor Gollancz.\n\n\\- (1960) 'Rutherford in the Cavendish', in J. Raymond (ed.), _The Baldwin Age_ , London: Eyre & Spottiswoode, pp. 235-48.\n\nSolnit, R. (2001) _Wanderlust: A History of Walking_ , New York: Penguin Books.\n\nSponsel, A. (2002) 'Constructing a \"Revolution in Science\": The Campaign to Promote a Favourable Reception for the 1919 Solar Eclipse Experiments', _British Journal of the History of Science_ , 35 (4): 439-67.\n\nSpruch, G. M. (1979) 'Pyotr Kapitza, Octogenarian Dissident', _Physics Today_ , September, pp. 34-41.\n\nStanford, H. K. (1987) 'Dirac at the University of Miami', in B. M. Kur\u015funo\u011flu and E. P. Wigner (eds), _Reminiscences about a Great Physicist: Paul Adrien Maurice Dirac_ , Cambridge: Cambridge University Press, pp. 53-6.\n\nStanley, M. (2007) 'Practical Mystic: Religion, Science and A. S. Eddington', Chicago, Ill.: University of Chicago Press.\n\nStansky, P. and Abrahams, W. (1966) _Journey to the Frontier: Julian Bell and John Cornford: Their Lives and the 1930s_ , London: Constable.\n\nStoke, H. and Green, V. (2005) _A Dictionary of Bristle_ , 2nd edn, Bristol: Broadcast Books.\n\nStone, G. F. and Wells, C. (eds) (1920) _Bristol and the Great War 1914-19_ , Bristol: J. W. Arrowsmith Ltd.\n\nSzasz, F. M. (1992) _British Scientists and the Manhattan Project_ , London: Macmillan.\n\nTamm, I. E. (1933) 'On the Work of Marxist Philosophers in the Field of Physics', _Pod znamenem marxizma_ ( _Under the Banner of Marxism_ ), 2: 220-31.\n\nTaubman, W. (2003) _Khrushchev_ , London: Free Press.\n\nTaylor, J. G. (ed.) (1987) _Tributes to Paul Dirac_ , Bristol: Adam Hilger.\n\nTaylor Sen, C. (1986) 'Remembering Paul Dirac', _Encounter_ , 67 (2): 80.\n\nTebeau, C. W. (1976) _The University of Miami - a Golden Anniversary History 1926-1976_, Coral Gables, Fla.: University of Miami Press.\n\nThom\u00e4len, A. (1907), trans. George How, London: Edward Arnold & Co.\n\nThorp, C. and Shapin, S. (2000) 'Who was J. Robert Oppenheimer?', _Social Studies of Science_ , August.\n\nVan der Waerden, B. L. (1960) 'Exclusion Principle and Spin', in M. Fierz and V. F. Weiskopff (eds), _Theoretical Physics in the Twentieth Century_ , London: Interscience Publishers Ltd, pp. 199-244.\n\n\\- (ed.) (1967) _Sources of Quantum Mechanics_ , New York: Dover.\n\nVan Vleck, J. (1972) 'Travels with Dirac in the Rockies', in A. Salam and E. P. Wigner (eds), _Aspects of Quantum Theory_ , Cambridge: Cambridge University Press, pp. 7-16.\n\nVint, J (1956) 'Henry Ronald Hass\u00e9', _Journal of the London Mathematical Society_ , 31: 252-5.\n\nVon K\u00e1rm\u00e1n, T. (with Edson, L.) (1967) _The Wind and Beyond_ , Boston, Mass.: Little, Brown and Company.\n\nvon Meyenn, K. (1985) 'Die Faustparodie', in K. von Meyenn, K. Soltzenburg and R. U. Sexl (eds), _Niels Bohr 1885-1962: Der Kopenhagener Geist in der Physik_ , Brunswick: Vieweg, pp. 308-42.\n\nvon Meyenn, K. and Sch\u00fccking E. (2001), 'Wolfgang Pauli', _Physics Today_ , February, p. 46.\n\nWalenski, M., Tager-Flusberg, H. and Ullman, M. T. (2006) 'Language and Autism', in S. O. Moldin and J. L. R. Rubenstein (eds), _Understanding Autism: From Basic Neuroscience to Treatment_ , New York: Taylor & Francis, pp. 175-204.\n\nWali, K. C. (1991) _Chandra: A Biography of S. Chandrasekhar_ , Chicago, Ill.: University of Chicago Press.\n\nWalters, B. (1970) _How to Talk with Practically Anybody About Practically Anything_ , New York: Doubleday & Co., Inc.\n\nWarwick, A. (2003) _Masters of Theory: Cambridge and the Rise of Mathematical Physics_ , Chicago, Ill.: University of Chicago Press.\n\nWatson, J. D. (1980) _The Double Helix_ , ed. G. S. Stent, New York: W. W. Norton & Co.\n\nWattenberg, A. (1984) 'December 2, 1942: The Event and the People', in R. G. Sachs (ed.), _The Nuclear Chain Reaction: Forty Years Later_ , Chicago, Ill.: University of Chicago Press, pp. 43-53.\n\nWeart, S. and Weiss Szilard, G. (eds) (1978) _Leo Szilard: His Version of the Facts_ , Cambridge, Mass.: MIT Press.\n\nWeinberg, S. (2002) 'How Great Equations Survive', in G. Farmelo (ed.), _It Must Be Beautiful: Great Equations of Modern Science_ , London: Granta, pp. 253-7.\n\nWeiner, C. (1977) _History of Twentieth Century Physics_ , New York: Academic Press.\n\nWeisskopf, V. (1990) _The Joy of Insight_ , New York: Basic Books.\n\nWells, H. G. (2005) _The Time Machine_ , London: Penguin.\n\nWells, J. C. (1982) _Accents of English 2_ , Cambridge: Cambridge University Press.\n\nWerskey, G. (1978) _The Visible College_ , London: Allen Lane.\n\nWestfall, R. S. (1993) _The Life of Isaac Newton_ , Cambridge: Cambridge University Press.\n\nWheeler, J. A. (1985) 'Physics in Copenhagen in 1934 and 1935', in A. P. French and P. G. Kennedy (eds), _Niels Bohr: A Centenary Volume_ , Cambridge, Mass.: Harvard University Press, pp. 221-6.\n\n\\- (1998) _Geons, Black Holes, and Quantum Foam_ , New York: W. W. Norton & Co.\n\nWicker, W. K. (1990) 'Of Time and Place: The Presidential Odyssey of Dr Henry King Stanford', Doctor of Education thesis, University of Georgia.\n\nWigner, E. P. (1992) _The Recollections of Eugene P. Wigner as Told to Andrew Szanton_ , New York: Plenum Press.\n\nWilson, A. N. (2002) _The Victorians_ , London: Hutchinson.\n\nWilson, D. (1983) _Rutherford: Simple Genius_ , London: Hodder and Stoughton.\n\nWing, L. (1996) _The Autistic Spectrum_ , London: Robinson.\n\nWinstone, R. (1972) _Bristol as It Was 1914-1920_ , Bristol: published by the author.\n\nWoit, P. (2006) _Not Even Wrong: The Failure of String Theory and the Continuing Challenge to Unify the Laws of Physics_ , London: Jonathan Cape.\n\nWoodhead, M. (1989) 'School Starts at Five . . . or Four Years Old', _Journal of Education Policy_ , 4: 1-21.\n\nWoolf, H. (ed.) (1980) _Some Strangeness in the Proportion: A Centennial Symposium to Celebrate the Achievements of Albert Einstein_ , Reading, Mass.: Addison-Wesley.\n\nYang, C. N. (1980) 'Beauty and Theoretical Physics', in D. W. Curtin (ed.), _The Aesthetic Dimension of Science_ , New York: Philosophical Library, pp. 25-40.\n**List of Plates**\n\n1. Dirac family, 3 September 1907 (courtesy Monica Dirac).\n\n2. Paul Dirac, 17 August 1907 (courtesy Monica Dirac).\n\n3. Felix, Betty and Paul Dirac _c_.1909 (courtesy Monica Dirac).\n\n4. Technical drawing by Paul Dirac (FSU, Dirac archive, 1\/10\/F5).\n\n5. Bristol University Engineering Society's visit to Messrs Douglas's Works (FSU, Dirac archive, 1\/10\/F128).\n\n6. Charles Dirac, _c_.1933 (FSU, Dirac archive, 1\/15\/F1D).\n\n7. Felix Dirac, 1921 (FSU, Dirac archive, 1\/15\/FIJ).\n\n8. 6 Julius Road, Bristol.\n\n9. Max Born entertaining his younger colleagues at his home in G\u00f6ttingen, spring 1926 (FSU, Dirac archive, 1\/14\/F6) _._\n\n10. Some members of the Kapitza Club, after a meeting c. 1925 (courtesy Giovanna Blackett).\n\n11. Patrick Blackett and Paul Ehrenfest, _c_.1925 (courtesy Giovanna Blackett).\n\n12. Isabel Whitehead and her husband Henry, with their son Henry, 1922 (c0urtesy Archives, The United Theological, Bangalore, India).\n\n13. Dirac at a meeting in Kazan, Russia, 12 October 1928 (FSU, Dirac archive, 1\/14\/FI2).\n\n14. Heisenberg's mother, Schr\u00f6dinger's wife, Flo Dirac, Dirac, Heisenberg and Schr\u00f6dinger (AIP Emilio Segr\u00e8 Visual Archives).\n\n15. Extract from a letter from Dirac to Manci Balazs, 9 May 1935 (courtesy Monica Dirac).\n\n16. Dirac and Manci on their honeymoon, Brighton, January 1937 (courtesy Monica Dirac).\n\n17. The Dirac family in the garden of their Cambridge home, _c_.1946 (courtesy Monica Dirac).\n\n18. Dirac and Manci with a party during a crossing of the Atlantic on the SS America, 2 April 1963 (FSU, Dirac archive, 1\/14\/F63).\n\n19. Dirac and Richard Feynman at a conference on relativity, Warsaw, July 1962 (photograph by A. John Coleman, courtesy AIP Emilio Segr\u00e8 Visual Archives, Physics Today collection).\n\n20. Dirac at the Institute for Advanced Study, Princeton, _c_.1958 (courtesy Monica Dirac).\n\n21. The Diracs' home in Tallahasse, 223 Chapel Drive.\n\n22. Kapitza and Dirac at the Hotel Bad Schachen, Lindau, summer 1982 (FSU, Dirac archive, 1\/14\/F98).\n\n23. One of the last photographs taken of Dirac, Tallahassee, _c_.1983 (courtesy Monica Dirac).\n**Acknowledgements**\n\nArt is I, science is we.\n\nCLAUDE BERNARD (1865) 'Introduction' to _L'\u00c9tude de la m\u00e9decine experimental_\n\nClaude Bernard was right. Biographies of scientists, too, are 'we', not 'I', in the sense that none could be written satisfactorily without a good deal of help. So I'd like to begin by acknowledging the huge contribution of the scientists, historians, archivists and writers who have preserved memories and other information about Paul Dirac. My gratitude extends to Dirac himself, who evidently took care to preserve documents about many crucial events in life, right down to the row about his Cambridge parking permit.\n\nBut let me be more specific. First I should like to thank Dirac's closest family. His daughter Monica has been unfailingly helpful, welcoming my enquiries and going out of her way to make available family documents to me. Her friend John Amy has been immensely accommodating to me throughout the project, and I am duly grateful to him. No less kind than Monica was Dirac's other daughter, Mary, who died in Tallahassee on 20 January 2007. Her guardian, Marshall Knight, has been extremely generous and obliging to me, especially during my visits to Florida.\n\nOther family members who have given generously of their time in helping me: Gisela and Christian Dirac, Leo Dirac, Vicky Dirac, Barbara Dirac-Svejstrup, Christine Teszler, Pat Wigner, Charles and Mary Upton, Peter Lantos and Erika Zimmermann. Past family members who provided valuable testimonies are Tony Colleraine and Peter Tilley. Gisela Dirac, the family genealogist, has been indefatigable in helping to clarify the French and Swiss provenance of the Dirac family.\n\nFour institutions to which I owe special gratitude are St John's College, Cambridge, the Institute for Advanced Study in Princeton, Florida State University in Tallahassee and Bristol University.\n\nSt John's invited me to stay in the college several times, enabling me to experience day-to-day life there, to use its superb facilities and  to talk at length with several of Dirac's former colleagues and acquaintances. I am grateful to the Master and Fellows of the college for this hospitality and for making available the facilities of the college to me, notably the library. For enlightening conversations, I thank the late John Crook, Duncan Dormor, Clifford Evans, Jane Heal, John Leake, Nick Manton, George Watson and Sir Maurice Wilkes. I have received a huge amount of support from the college library, especially from Mark Nicholls, Malcolm Underwood and the special collections librarian Jonathan Harrison, whose industry has enormously benefited the book. The university library has been most helpful, and I would like to thank Elisabeth Leedham-Green and Jackie Cox for taking so much trouble to answer my queries. Also in Cambridge, I should like to thank Yorrick and Helaine Blumenfeld, Richard Eden, Peter Landshoff, Sir Brian Pippard, the Reverend John Polkinghorne, KBE, and Lord (Martin) Rees.\n\nAt the Institute for Advanced Study, I was fortunate enough to spend four productive and very happy summers researching the book and writing it. I benefited considerably from conversations there with Yve-Alain Bois, Freeman Dyson, Peter Goddard, Juan Maldacena, Nathan Seiberg, Morton White and Edward Witten. The library facilities at the institute are peerless, and I should like to thank all the staff there who were unstinting in their support: Karen Downing, Momota Ganguli, Gabriella Hoskin, Erica Mosner, Marcia Tucker, Kirstie Venanzi and Judy Wilson-Smith. Among the other colleagues who made my stays there so rewarding: Linda Arntzenius, Alan Cheng, Karen Cuozzo, Jennifer Hansen, Beatrice Jessen, Kevin Kelly, Camille Merger, Nadine Thompson, Sharon Tozzi-Goff and Sarah Zantua. Also in Princeton, I should like to thank Gillett Griffin, Lily Harish-Chandra, Louise Morse ( _m\u00e8re et soeur_ ) and Terri Nelson.\n\nI should like to give my special thanks to Peter Goddard, formerly Master of St John's, now Director of the Institute for Advanced Study. No one has been more supportive of the project or shown more interest in its progress. I owe him an enormous debt.\n\nAt Florida State University, I have benefited from the excellent library facilities and from invaluable help from the staff responsible for the Dirac archive. Sharon Schwerzel, Head of the Paul A. M. Dirac Science Library, could not have been more helpful to me - her understanding of the challenges faced by a biographer working thousands of miles from the primary archive has been hugely beneficial.  It has also been a delight to work with Chuck McCann, Paul Vermeron, with Lucy Patrick and all the librarians in Special Collections: Burt Altman, Garnett Avant, Denise Gianniano, Ginger Harkey, Alice Motes, Michael Matos and Chad Underwood. On the past and present faculty of the university, I should like to thank Howie Baer, Steve Edwards, the late Leopold Halpern, Kurt Hofer, Harry Kroto, Robley Light, Bill Moulton and Hans Plendl. Through colleagues at Florida State, I also met many other people in Tallahassee who shared their memories of Dirac with me: Ken van Assenderp, Pamela Houm\u00e8re, Peggy Lannutti, Jeanne Light, Pat Ritchie, Rae Roeder and Hansell Watt.\n\nAt Bristol University, I have been supported by Debra Avent-Gibson, Sir Michael Berry, Chris Harries, Michael Richardson, Margaret and Vincent Smith and Leslie Warne. Many others in Bristol have also done much to shed light on Dirac's early life, especially Karen and Chris Benson, Dick Clements, Alan Elkan, Andrew Lang, John Penny and John Steeds. I was fortunate to be introduced to Don Carleton, a local historian, who has done an inordinate amount of work to illuminate the history of Bristol in the early twentieth century.\n\nI should like to thank the following institutions for granting permission to quote from their archives: American Philosophical Society; Bodleian Library, University of Oxford; University of Bristol Library; Bristol Record Office; British Broadcasting Corporation; Masters and Fellows of Christ's College, Cambridge; The Syndics of Cambridge University Library; Council for the Lindau Nobel Laureate Meetings; Institute for Advanced Study, Princeton; Master, Fellows and scholars of St John's College Cambridge; Archives for the History of Quantum Physics, College Park, MD, USA; Archives for the Society of Merchant Venturers, held at the Bristol Records Office, UK; Provost and Scholars of King's College, Cambridge; Niels Bohr Archive, Copenhagen; Princeton University Library; Royal Commission for the Exhibition of 1851; International Solvay Institutes, Brussels; Special Collections at the University of Sussex; Archives at the United Theological College, Bangalore, India.\n\nDuring my research, many friends and colleagues at archives and other institutions have given me valuable support. At the California Institute of Technology archives: Shelley Erwin and Bonnie Ludt. At the Center for History of Physics of the American Institute of  Physics, Maryland: Melanie Brown, Julie Gass, Spencer Weart and Stephanie Jankowski. At CERN, Geneva: John Ellis, Rolf Landua, Esthel Laperri\u00e8re. In the Archives Centre: Anita Hollier. At Christ's College, Cambridge: Candace Guite. At the archive of the Royal Commission for the Exhibition of 1851: Angela Kenny and Valerie Phillips. At the Archives in the College of Aeronautics, Cranfield University: John Harrington. At the Archives of Imperial College, London: Anne Barrett. At Lambeth Palace Library, London: Naomi Ward. At the Royal Society, London: Martin Carr and Ross MacFarlane. At the Max Planck Institute, Munich: Helmut Rechenberg. At the Niels Bohr Archive, Copenhagen: Finn Asserud and Felicity Pors. At the University of Madison, Wisconsin: Vernon Barger, Tom Butler, Kerry Kresse, Ron Larson, David Null and Bill Robbins. At Firestone Library, Princeton University: AnnaLee Pauls and Meg Sherry Rich. At the Solvay archive in the Free University of Brussels: Carole Masson, Dominique Bogaerts and Isabelle Juif. At the Science Museum, London: Heather Mayfield, Doug Millard, Andrew Nahum, Matthew Pudney and Jon Tucker. It is a special pleasure to thank past and present staff at the Science Museum Library: Ian Carter, Allison Pollard, Prabha Shah, Valerie Scott, Robert Sharp, Joanna Shrimpton, Jim Singleton, Mandy Taylor, Peter Tajasque, John Underwood and Nick Wyatt. Thanks also to Ben Whelehan at Imperial College Library. At the Tata Institute in Bombay: Indira Chowdhury. At the National Media Museum, York: Colin Harding and John Trenouth. At Special Collections, University of Sussex: Dorothy Sheridan and Karen Watson. For their help with determining the detailed weather conditions in towns and cities in the UK and USA, it is a pleasure to thank Steve Jebson at the Met Office and Melissa Griffin at Florida State University.\n\nOthers who have been extremely helpful in responding to my enquiries: Sir Michael Atiyah, Tom Baldwin, John Barnes, Herman Batelaan, Steve Batterson, John Bendall, Giovanna Blackett, Margaret Booth (n\u00e9e Hartree), Gustav Born, Olaf Breidbach, Andrew Brown, Nicholas Capaldi, David Cassidy, Brian Cathcart, Martin Clark, Paul Clark, Chris Cockcroft, Thea Cockcroft, Flurin Condrau, Beverley Cook, Peter Cooper, Tam Dalyell, Dick Dalitz, Olivier Darrigol, Richard Davies, Stanley Deser, David Edgerton, John Ellis, Joyce Farmelo, Michael Frayn, Igor Gamow, Joshua Goldman, Jeffrey Goldstone, Jeremy Gray, Karl Hall, Richard  Hartree, Peter Harvey, Steve Henderson, Chris Hicks, John Holt, Jeff Hughes, Lane Hughston, Bob Jaffe, Edgar Jenkins, Allan Jones, Bob Ketchum, Anne Kox, Charles Kuper, Peter Lamarque, Willis Lamb, Dominique Lambert, Ellen and Leon Lederman, Sabine Lee, John Maddox, Philip Mannheim, Robin Marshall, Dennis McCormick, Arthur I. Miller, Andrew Nahum, Michael Noakes, Mary Jo Nye, Susan Oakes, James Overduin, Bob Parkinson, John Partington, Sir Roger Penrose, Trevor Powell, Roger Philips, Chris Redmond, Tony Scarr, Robert Schulmann, Bernard Shultz, Simon Singh, John Skorupski, Ulrica S\u00f6derlind, Alistair Sponsel, Henry King Stanford, Simon Stevens, George Sudashan, Colleen Taylor-Sen, Laura Thorne, Claire Tomalin, Martin Veltman, Andrew Warwick, John Watson, Russell Webb, Nina Wedderburn, John Wheeler, the late David Whitehead, Oliver Whitehead, Frank Wilczek, Michael Worboys, Nigel Wrench, Sir Denys Wilkinson and Abe Yoffe. Special thanks to Alexei Kojevnikov, who has been unstinting in the guidance and help he has given to me concerning the development of Russian physics in the past century.\n\nFor their help with primary research, my sincere thanks to Anna Cain, Martin Clark, Ruth Horry, Anna Menzies, James Jackson, Joshua Goldman, Katie Kiekhaefer, Tadas Krupovnickas and Jimmy Sebastian.\n\nFor technical support, thanks to: Paul Chen at Biblioscape (the marvellous bibliographic software) and Ian Hart.\n\nFor translating documents, I am indebted to Paul Clark, Gisela Dirac, Karl Grandin, Asger H\u00f8eg, Anna Menzies, Dora Bobory and Eszter Molnar-Mills.\n\nFor reading parts of the manuscript and for their constructive comments, thanks to: Simon Baron-Cohen, Paul Clark, Olivier Darrigol, Uta Frith, Freeman Dyson, Roger Highfield, Kurt Hofer, Bob Jaffe, Ramamurti Rajaraman, Martin Rees and Jon Tucker. And for reading the entire manuscript and for dozens of helpful comments, thanks to: Don Carleton, Stanley Deser, Alexei Kojevnikov, Peter Rowlands, Chuck Schwager, Marty Schwager and David Ucko. I am especially grateful to my friends David Johnson and David Sumner for reading several drafts of the book, each time providing extremely insightful and constructive feedback.\n\nFinally, I should like to acknowledge the huge contribution of my publisher, Faber and Faber, to the book. Kate Ward supervised the  production of the book with great attention to detail, and Kate Murray-Brown read the book with a keen and sensitive eye on content and style and provided many valuable suggestions and comments. Liz O'Donnell has been a dream of an editor - meticulous, sensitive, questioning and collegiate. I am indebted most of all to Neil Belton, who has supported the project from its inception, given me no end of wise advice and kept the bar high.\n\nThe concept of 'we' extends only so far: I take responsibility for any remaining inaccuracies in the book and for its portrayal of Paul Dirac's work and personality. In that sense, the book is 'I'.\n\nGraham Farmelo   \nJune 2008\n**Index**\n\n##\n\n'PD' indicates Paul Dirac\n\n##\n\n_2001: A Space Odyssey_ (film)\n\nAarhus, Denmark\n\nAbstein, Dr W. Robert\n\naction principle\n\nAdcock, Frank\n\nAdler, Dorothy and Sol\n\nAdrian, Edgar\n\nAesthetic Movement\n\nalgebra\n\nGrassmann\n\nnon-commutative\n\nAmerican Physical Society, annual meeting of (New York, 1959)\n\nAmerican Science News Service\n\nAmsterdam\n\nEhrenfest's suicide in\n\nBetty and Joe Teszler live in\n\nBetty and Joe flee from their home\n\nAnderson, Carl\n\nchooses the name positron\n\n'The Apparent Existence of Easily Deflectable Positives'\n\nAnglo-French Society of Sciences\n\nanti-electrons\n\nBlackett and Occhialini's discovery\n\n_see also_ positrons and anti-matter\n\nanti-matter\n\nanti-quarks\n\nthe Big Bang\n\nPD predicts its existence\n\nsurplus of matter over anti-matter\n\na universe made from equal amounts of matter and anti-matter\n\nanti-Semitism\n\nApollo space programme\n\n_Aquitania_ (liner)\n\nArbuckle, Fatty\n\nArmstrong, Neil\n\nArts School, Cambridge\n\nAsperger's Syndrome\n\n_Asuma Bura_ , MS\n\nAtiyah, Sir Michael\n\natomic bomb _see_ nuclear weapons\n\nAtomic Energy Commission\n\natomic physics\n\nand classical laws\n\nPD attends Tyndall's lectures\n\nPD writes on _see also_ quantum theory, quantum physics\n\natoms\n\natom visualized as a mechanical device\n\nBalmer's formula for hydrogen spectrum\n\nBohr's work on atomic structure\n\nelectrons as a constituent of\n\nenergy levels\n\nheavy\n\nRutherford's discovery of the nucleus\n\nAustria, Hitler's invasion of (1938)\n\nautism\n\naviation industry\n\nAvon Gorge\n\nAydelotte, Frank\n\nAyer, A. J.\n\n##\n\nBaker, Henry\n\nhis tea parties\n\nappearance\n\npersonality\n\nand the Greeks' love of beauty\n\nBal\u00e1zs, Nandor\n\nBal\u00e1zs, Richard\n\nBaldwin, Stanley\n\nBalmer, Johannes: formula for hydrogen spectrum\n\nBaltimore Dairy Lunch, Princeton (the Balt)\n\nbare electron\n\nbare energy\n\nBarnes, Julian: _Flaubert's Parrot_\n\nBaron-Cohen, Simon\n\nBatchelor, George\n\nBattle of Britain\n\nBBC (British Broadcasting Corporation (later Company))\n\nHome Service\n\nPD declines numerous interviews\n\n_Start the Week_ (Radioprogramme)\n\nBeatles, The\n\nBeaufort, Lady Margaret\n\nbeauty\n\nBaker's fascination with the Greeks' love of beauty\n\nconcept of\n\ndiscussion betweenand Heisenberg\n\nof a fundamental theory in physics\n\nKant and\n\nin mathematics\n\nMoore on\n\nPD's first recorded mention of\n\nin vogue as a concept at Cambridge\n\nBeeston Hall School, West Runton, Norfolk\n\nBelgium, Queen of (Elisabeth of Bavaria)\n\nBell, James\n\nBell Laboratories, New Jersey\n\nBendall, John\n\nBeria, Lavrentiy\n\nBerlin\n\nglobal capital of theoretical physics\n\nOppenheimer in\n\nEinstein in\n\nanti-Semitism\n\nnuclear fission discovered in\n\nDebye in\n\nBerlin Wall, fall of (1989)\n\nBernal, Desmond\n\nBerne, Switzerland\n\nBethe, Hans\n\nBhabha, Homi\n\nBialobrzeski, Czeslaw\n\nBig Bang\n\nBirge, Raymond\n\nBirmingham\n\nBishop Road Junior School, Bristol\n\nBishopston, Bristol\n\nBismarck, Prince Otto von\n\nblack holes\n\nblackbody radiation\n\nBlackett, Patrick\n\nserves in World War I\n\npersonality\n\ninfluences PD\n\nappearance\n\nresents Kapitza\n\nexperimental physics\n\nattempted poisoning by Oppenheimer\n\nand cosmic rays\n\nanger at Rutherford's despotic style\n\ndiscovery of the anti-electron\n\nrevelations at the Royal Society\n\nsupports the Labour Party\n\nfamily\n\nand nuclear fission\n\na wartime Government scientific adviser\n\nand manufacture of a nuclear weapon\n\nand the Manhattan Project\n\nrefused a visa for the Soviet Union (spring 1945)\n\nNobel Prize\n\nBletchley Park, Buckinghamshire\n\nBloomsbury Group\n\nBlumenfeld, Helaine\n\nBlumenfeld, Yorrick\n\nBlunt, Anthony\n\nBoer War\n\nBohr, Margrethe\n\nBohr, Niels\n\ntheory of atomic structure\n\nPD's mastery of his atomic theory\n\nNobel Prize for physics\n\nvisits Cambridge\n\nappearance\n\npersonality\n\nand Rutherford\n\ngloomy about the state of quantum physics\n\nand Heisenberg's theory of 1925\n\nPD's visit to the Institute\n\nconcern with words\n\nPD on\n\non PD\n\ncomplementarity principle\n\ncoat of arms\n\ndefends Heisenberg's uncertainty principle\n\nand a relativistic equation of the electron\n\nresponse to PD's hole theory\n\nand PD's Bristol lecture\n\nat the 1930 Solvay Conference\n\nand the neutrino\n\nrepresented in a special version of _Faust_\n\nand Hitler's appointment as Chancellor\n\nand philosophy\n\nand the positron\n\nand the bas-relief of Rutherford\n\nhis mansion\n\ncongratulates PD on his Nobel Prize\n\nparty to honour the Nobel Prize winners\n\nand Shankland's results\n\ndeath of his eldest son\n\nat Rutherford's memorial service\n\nand nuclear fission\n\nmeeting with Heisenberg (1941)\n\nescapes from occupied Denmark\n\nand genetics\n\ndeath\n\nBohr orbits\n\nBollob\u00e1s, Gabriella\n\nBolshevik Party\n\nBolshevik Revolution (1917)\n\nBolshevism\n\nBolshevo, near Moscow\n\nBombay (Mumbai)\n\nBordeaux, France\n\nBorn, Gustav\n\nBorn, Max\n\nquantum mechanics named by\n\nand's first paper on quantum mechanics\n\nworks with Heisenberg and Jordan at G\u00f6ttingen\n\nand Heisenberg's quantum theory\n\nand Jordan's work on Fermi-Dirac statistics\n\ninterpretation of Schr\u00f6dinger's waves\n\nquantum probabilities\n\nappearance\n\npersonality\n\nand Oppenheimer's behaviour\n\nsurprised at's knowledgeability\n\nand field theory\n\nand the rise of anti-Semitism in G\u00f6ttingen\n\nand the Dirac equation\n\nnervous breakdown\n\nconsiders emigration\n\nappointment at Cambridge\n\nresents's Nobel Prize\n\nmessage from the Nazi Government\n\nprofessorship in Edinburgh\n\nin the Lake District with PD\n\nPD asks him to support Heisenberg\n\nNobel Prize\n\ndeath\n\nBose, Satyendra\n\nbosons\n\nBoston, Massachusetts\n\nBoston University: PD's lecture (1972)\n\nBoulton, Edmund\n\nBoys Smith, John\n\nbra\n\nBradman, Sir Donald\n\nBragg, Sir Lawrence\n\nBragg, William\n\nBridges, Robert: _A Testament of Beauty_\n\nBrighton, PD's honeymoon in\n\nBristol\n\nCharles Dirac settles in\n\ndescribed\n\nand Catholicism\n\naviation industry\n\nFirst World War\n\nprotestors baton-charged by police (1932)\n\nSecond World War\n\nBristol Aeroplane Company\n\nBristol Central Library\n\nBristol Citizens' Recruiting Committee\n\nBristol Downs\n\n_Bristol Evening News_\n\nBristol Records Office\n\nBristol Shiplovers' Society\n\nBritish Aeroplane Company\n\nBritish Aerospace\n\nBritish Aircraft Corporation\n\nBritish and Colonial Aeroplane Company\n\nBritish Association for the Advancement of Science\n\nmeeting (Bristol, 1930)\n\nmeeting (Leicester, 1933)\n\nBritish Thomson-Houston Company, Rugby\n\nBroad, Charlie\n\nProfessor of Philosophy at Bristol\n\nas a lecturer\n\ntreatment of relativity\n\nand's interest in philosophy\n\nmoves to Cambridge\n\nBrookhaven National Laboratory\n\nBrown, Dan: _Angels and Demons_\n\nBrown, Miss Josephine\n\nBrunel, Isambard Kingdom\n\nBudapest\n\nBukharin, Nikolai\n\n_Bulletin of the Soviet Academy of Sciences_\n\nBullock, W.H.\n\nBunin, Ivan\n\nBush, George, Snr.\n\nButler, Samuel: _The Way of all Flesh_\n\nByron, Lord\n\n##\n\nCadet Corps\n\nCalifornia Institute of Technology (Caltech)\n\nCambodia, US invasion of\n\nCambridge\n\ndescribed\n\nManci's dislike of\n\nSocialist Society march (1933)\n\nwartime\n\nVE-Day celebrations\n\ncelebration of Japan's surrender\n\nCambridge Borough Cemetery (now\n\nCambridge City Cemetery), Bristol\n\n_Cambridge Review_\n\nCambridge Union\n\nCambridge University\n\nmathematics as its largest department\n\nsocial life\n\nopposition to the General Strike\n\nMarxist scientists' efforts to establish radical politics\n\napplications from refugee scientists\n\nin the Second World War\n\nwomen in\n\noffers a professorship to Oppenheimer\n\nDepartment of Applied Mathematics and Theoretical Physics\n\nPD moves to Florida State\n\nCanadian Rockies\n\nCanford Cemetery, Westbury on Trym, Bristol\n\ncanonically conjugate variables\n\nCardoza Kindersley workshop, Cambridge\n\nCario family\n\nCarpenter, Edward, Dean of Westminster\n\nCarroll, Lewis: _Alice through the Looking Glass_\n\nCarter, Jimmy\n\nCarus, Paul: _Reflections on Magic Squares_\n\nCasimir, Hendrik\n\nCaucasus\n\nCavendish Avenue, Cambridge (No.7)\n\nCavendish Laboratory, Cambridge\n\nRutherford succeeds J. J. Thomson\n\nseminars\n\nPD talks on quantum discoveries\n\node to the electron\n\nMillikan's presentation on cosmic rays\n\nChadwick's work on the neutron\n\nsplitting of the atom\n\ndiscovery of the anti-electron\n\nand the Nobel Prize (1933)\n\nBragg succeeds Rutherford\n\nSecond World War\n\nCavendish Physical Society: annual dinner\n\nCentral Intelligence Agency (CIA)\n\ncentrifugal jet stream method\n\nCERN (European Organization for Nuclear Research)\n\nChadwick, James\n\nand cosmic rays\n\n'Possible Existence of the Neutron'\n\nChamberlain, Neville\n\nChandrasekhar, Subrahmanyan\n\nChannel Islands\n\nChaplin, Charlie\n\nCharlesworth, Martin\n\nCher\n\nChicago\n\nChopin, Fryderyk\n\nChristie, Agatha\n\nChrist's College, Cambridge\n\nChukovsky, Korney: _Crocodile_\n\nChurchill, Sir Winston\n\nChurchill College, Cambridge\n\nCivil Defence offices, St Regis\n\nClark, Sir Kenneth (later Lord)\n\nclassical mechanics\n\nclassical physics\n\nCleese, John and Chapman, Graham: _Monty Python's Flying Circus_ script\n\nClifton Suspension Bridge\n\nClinton, Bill\n\nClinton, Hillary Rodham\n\ncloud chamber\n\nCockcroft, Lady Elizabeth\n\nCockcroft, Sir John\n\nCold War\n\nCole, Sir Henry\n\nColiseum ice-rink, Bristol\n\nColleraine, Tony\n\nColumbia Radiation Laboratory\n\nColumbia University, New York\n\nCommunism\n\nCommunist Academy\n\nCommunist Party\n\ncomplementarity principle\n\nCompton, Arthur\n\nelectromagnetic radiation behaving as discrete particles\n\nPD declines his offer of a post in Chicago\n\nCompton, Karl\n\nComte, Auguste\n\nConan Doyle, Sir Arthur\n\nConcorde\n\nCongress of Russian Physicists (1928)\n\nconservation of energy, law of\n\nCopenhagen\n\nPD in\n\nCoral Gables conferences\n\nCornwall\n\ncorrespondence principle\n\ncosmic rays\n\nMillikan's investigations\n\nBlackett's interest in\n\nAnderson's use of a cloud chamber\n\nBlackett and Occhialini's work\n\nAnderson identifies the muon\n\ncosmology\n\nCoughlin, Joseph ('Roundy')\n\nCouncil on Foreign Relations\n\ncounter-current centrifuge\n\nCoward, No\u00ebl\n\nCrimea, the\n\nCrowther, Jim _Soviet Science_\n\nCuban crisis (1962)\n\nCunningham, Ebenezer\n\nHass\u00e9's letter supporting PD\n\nPD asks to study relativity with him\n\non PD\n\nCurie, Marie\n\nCzechoslovakia\n\n##\n\n_Daily Express_\n\n_Daily Herald_\n\n_Daily Mail_\n\n_Daily Mirror_\n\n_Daily Telegraph_\n\nDaladier, \u00c9douard\n\nDali, Salvador\n\nDalitz, Dick\n\nDalyell, Tam\n\nDaniel, Glyn\n\nDarwin, Charles\n\nbottom-up thinking\n\ncompared with Dirac\n\ntheory of evolution\n\nDarwin, Charles (grandson of the naturalist)\n\nDavisson, Clinton\n\nde Broglie, Louis: wave theory of matter\n\nde Sitter, Wilhelm\n\nde Valera, \u00c9amon\n\nDebye, Peter\n\n\u22072V Club\n\nDelbr\u00fcck, Max\n\nDelhi\n\nDelta function\n\nDent, Beryl\n\nDepartment of Scientific and Industrial\n\nResearch\n\nDepression\n\nDescartes, Ren\u00e9\n\n_Deutsches Volkstrum_ ('German Heritage')\n\ndialectical materialism\n\nDicke, Robert\n\nDickens, Charles\n\ndifferential geometry\n\nDingle, Herbert\n\nDirac, Betty (PD's sister)\n\nbirth\n\nnames\n\nchildhood\n\neducation\n\nher father's favourite child\n\npersonality\n\nand Felix's death\n\nattends's Ph.D. ceremony\n\nlack of employment\n\nchauffeurs her father to and from work\n\nforced to sell her car\n\nand her parents' marriage crisis\n\ndegree studies\n\ngoes to Lourdes with her father\n\nsupports her parents\n\nmoves to London to become a secretary\n\nin Budapest\n\npossible reason for her parents' failed marriage\n\nmarries Joe Teszler\n\nlives in Amsterdam\n\nbirth of son\n\nin the Second World War\n\nstays in Cambridge\n\nher suffering in Budapest\n\nbirth of daughter\n\nrelationship with Manci\n\nin Alicante\n\nstroke\n\nDirac, Charente, France\n\nDirac, Charles (PD's father)\n\nbirth (in Monthey, Switzerland)\n\nchildhood\n\neducation\n\nin London\n\nteaches at Merchant Venturers' Secondary School\n\nsettles in Bristol\n\nappearance\n\npersonality\n\nmeets Florence Holten\n\nand religion\n\nmarries Flo\n\ninsistence on his children speaking French\n\nchampions Esperanto in Bristol\n\nrelationship with PD\n\ncareful with money\n\nwork ethic\n\neffects of his rigorous educational regime at home\n\ntyranny of\n\nhis favourite child\n\nforces Felix to study engineering instead of medicine\n\ndeceptions by\n\nacquires British nationality\n\nefforts to send PD to Cambridge\n\nhelps PD financially\n\ninterest in PD's career\n\nfamily radio\n\ndeeply affected by the death of Felix\n\ndeath of his mother\n\nattends PD's Ph.D. ceremony\n\nletters to his 'only son'\n\nvegetarianism\n\nPD continues to feel intimidated by\n\nand PD's FRS election\n\nretirement\n\ninfidelity\n\nmarriage crisis\n\nloses his grip on his family\n\ncontinues to teach from home\n\nplans to visit Geneva\n\nrediscovery of his childhood Catholicism\n\nvisits Geneva with Betty\n\nFlo attacks in the Swedish press\n\ntries to understand PD's work\n\ngoes to Lourdes\n\nill with pleurisy\n\nserial tax evader\n\nPD blames him for Felix's suicide\n\n'loathed' by PD\n\ndeath and funeral\n\nhis estate\n\ngravestone\n\nDirac, Felix (PD's brother)\n\nbirth\n\nnames\n\nappearance\n\neducation\n\nchildhood in Bristol\n\nbullied by his father\n\npersonality\n\nrift with PD\n\nforced to study engineering instead of medicine\n\nstudent apprenticeship in Rugby\n\nbased near Wolverhampton\n\na draughtsman\n\nBuddhism and astrology\n\nacquires a girlfriend\n\nsettles in Birmingham\n\nvolunteers for the Ambulance Corps\n\nleaves his job at a machine-testing laboratory\n\npersonality\n\nsuicide\n\nthe family's response to his death\n\nmemorial service and inquest\n\ngravestone\n\nDirac, Florence (n\u00e9e Holten; PD's mother)\n\nfirst meets Charles\n\nappearance\n\npersonality\n\nabsent-minded\n\nand religion\n\ncorrespondence with Charles\n\nmarries Charles\n\nbirth of Felix\n\nbirth of Paul\n\npoem about PD\n\nPaul as her favourite child\n\nand Charles's deception\n\ncorrespondence with PD\n\nfears competition for PD's affections\n\nasks PD for money\n\nand the death of Felix\n\npoetry\n\ninterest in politics\n\nattends PD's Ph.D. ceremony\n\nworried about PD's emaciated appearance\n\nevening classes\n\nadmits her unhappiness\n\nhousework, dislike of\n\nPD pays for a diamond ring\n\nPD's visits home\n\nvisits PD in Cambridge\n\nand PD's visits to Russia\n\nopposes the idea of a woman prime minister\n\nfussing over PD\n\nand PD's FRS election\n\ndreads Charles's retirement\n\nthe charade of her marriage\n\naffinity with the sea; _see also_ Richard Holten (her father)\n\nmarriage crisis\n\nMediterranean cruises\n\nat PD's Nobel Prize ceremony\n\nat Bohr's party in Copenhagen\n\nand Charles's pleurisy\n\nmeets Manci\n\ndisputes with Manci\n\nin the Second World War\n\ndeath and funeral\n\nDirac, Gabriel (PD's step-son)\n\nDirac, Gisela\n\nDirac, Judy (PD's step-daughter)\n\nDirac, Louis (PD's paternal grandfather)\n\nDirac, Margit (Manci; n\u00e9e Wigner; PD's wife)\n\nmeets PD\n\npersonality\n\nand PD's talk of his unhappy childhood\n\non her first marriage and divorce\n\nand religion\n\na keen follower of the arts\n\npursuit of PD\n\nPD visits her in Budapest\n\nIsabel Whitehead's assessment\n\nPD's proposal of marriage\n\nmarriage and honeymoon\n\nrelationship with Betty\n\n'Wigner's sister' appellation\n\nsettles in Cambridge\n\nin the Soviet Union\n\npregnancies\n\nas an alien in wartime England\n\nand air raids on Cambridge\n\nFlo helps with housework\n\norders Judy out of the house\n\nand the Nazi concentration camps\n\ncomplains about the exodus from Cambridge\n\nscorns Heisenberg\n\nin Princeton\n\nand politics\n\nmarriage under strain\n\na better wife than mother\n\nand disappearance of Judy\n\nworsening arthritis\n\nand PD's decision to move to Florida State University\n\nat Florida State\n\nJewish and occasionally anti-Semitic\n\nas a hostess\n\nfraught relationship with Halpern\n\nPD's death and funeral\n\nlively and active for ten years after PD's  death\n\nletter from Hillary Rodham Clinton\n\ndeath\n\nDirac, Mary (PD's daughter; later Colleraine, then Tilley)\n\nbirth\n\nchildhood\n\npersonality\n\neducation\n\nemigration to the USA\n\nDirac, Monica (PD's daughter)\n\nbirth\n\nchildhood\n\npersonality\n\nat PD's commemoration\n\n**Dirac, Paul Adrien Maurice**\n\n**life story**\n\nbirth (8 August 1902)\n\nappearance and dress sense\n\ndigestive problems\n\nforesees the existence of the positron\n\nchildhood in Bristol\n\nrelationship with his father\n\nnicknamed 'Tiny'\n\nschool education\n\nvisits Switzerland\n\nBristol accent\n\nand technical drawing\n\nhandwriting\n\nhis mother's favourite\n\nrift with Felix\n\nengineering degree\n\npublic impact of relativity theory\n\ntrainee engineer in Rugby\n\napplied maths degree studies\n\nand projective geometry\n\nwins scholarships to St John's College, Cambridge\n\nsupervision by Fowler\n\nCharles helps him financially\n\narrives at Cambridge\n\nmanner at the dinner table\n\nattends Eddington's lectures\n\nBlackett and Kapitza become his closest friends\n\nand Soviet ideology\n\nand his mother's possessiveness\n\nfirst academic papers\n\nFelix's death\n\nfirst great epiphany\n\nfirst paper on quantum mechanics\n\nPh.D. thesis\n\ncombines logic and intuition\n\nas 'the strangest man' (Bohr)\n\nsuccessful period in Copenhagen\n\nin G\u00f6ttingen\n\nfriendship with Oppenheimer\n\nhis visits home\n\nelected Fellow of St John's College\n\nhis rooms in college\n\nmakes his most famous contribution to science\n\nrelationship with Isabel Whitehead\n\nfirst visit to Russia\n\nreductionism\n\nfirst visit to US\n\nelected Fellow of the Royal Society\n\nbuys his first car\n\nrepresented in a special version of _Faust_\n\nLucasian Chair\n\nWittgenstein, opinion of\n\nand moral philosophy\n\nworks with Kapitza in his laboratory\n\nlast meeting with Ehrenfest\n\nNobel Prize for physics\n\nfirst public comment on social and economic affairs\n\nsmitten with Rho Gamow\n\nfirst meets Manci\n\ncampaign for Kapitza's release\n\nsends the Gamows a baby alligator\n\nguardian of Kapitza's sons\n\ngraduate supervisor\n\nproposes to Manci\n\nmarriage and honeymoon\n\nfirst love letter\n\nwants his own children\n\nrefuses Princeton's job offer\n\nScott lecture\n\noffered war work\n\nBaker Medal\n\nand the death of his mother\n\nrefused a visa for the Soviet Union\n\ndeclines honours\n\nrefused a US visa\n\nvisits India\n\njaundice\n\nmarriage under strain\n\nmarginalised in Cambridge\n\nemigration to US\n\n_Scientific American_ article (1963)\n\n_Horizon_ interview (1965)\n\nquarks, likes concept of\n\ndecision to move to Florida State University\n\nroutine at Florida State\n\nbusts and paintings of PD\n\naccepts the Order of Merit\n\nvisits CERN\n\nflies on Concorde\n\nsees his life as a failure\n\nsurgery on tubercular kidney\n\ndeath (20 October 1984)\n\nfuneral\n\ncommemoration in Westminster Abbey\n\ncentenary of his birth\n\npossible autism\n\nnames\n\nmemorial stone\n\n**personality**\n\n\\- aloofness\n\n\\- confident\n\n\\- defensiveness\n\n\\- determination\n\n\\- diffidence\n\n\\- equability\n\n\\- frugality\n\n\\- inhibition\n\n\\- lack of social sensitivity\n\n\\- literal-mindedness\n\n\\- modesty\n\n\\- narrow-mindedness\n\n\\- objectivity\n\n\\- obsession with taking long walks\n\n\\- otherworldiness\n\n\\- passivity\n\n\\- physical ineptitude\n\n\\- private enthusiasms\n\n\\- reticence\n\n\\- rigid pattern of activities\n\n\\- self-centredness\n\n\\- shyness\n\n\\- stubborness\n\n\\- taciturnity\n\n\\- top-down thinker\n\n\\- verbal economy\n\n\\- work ethic\n\n**interests, aptitudes and opinions**\n\n\\- beauty, mathematical, fascination with\n\n\\- board games and mathematical puzzles, enjoyment of\n\n\\- driver, skills as a\n\n\\- fondness for Mickey Mouse films\n\n\\- food, tastes and appetite ,\n\n\\- gardening\n\n\\- Hamiltonian approach to mechanics, strong belief in\n\n\\- jokes, appreciation of\n\n\\- lecturer, skills as a\n\n\\- mountain-climbing\n\n\\- philosophy, opinion of\n\n\\- relativity, fascination with\n\n\\- religion, opinions about ,\n\n\\- renormalisation, distaste for and dislike of\n\n\\- swimming\n\n\\- team games and teams, aversion to participation in\n\n\\- technology of space flight, interest in\n\n\\- top-down thinking\n\n\\- tree-climbing\n\n**contributions to physics and mathematics**\n\n\\- action principle in quantum mechanics\n\n\\- antimatter, foresees, _see also_ positron and antiproton\n\n\\- anti-electron predicts, _see_ positron\n\n\\- anti-proton, predicts\n\n\\- blackbody radiation spectrum derived\n\n\\- bra and ket notation\n\n\\- classical theories of the electron\n\n\\- cosmology, thoughts on\n\n\\- density matrix\n\n\\- delta function\n\n\\- Dirac equation ,\n\n\\- Dirac sea\n\n\\- dispersion theory\n\n\\- ether, post-Einstein view of\n\n\\- Fermi-Dirac statistics\n\n\\- general relativity, Hamiltonian formulation of\n\n\\- gravity, weakening of - postulates, see also large numbers hypothesis\n\n\\- high-spin theory\n\n\\- hole theory\n\n\\- indefinite metric\n\n\\- jet-stream method of isotope separation\n\n\\- Kapitza-Dirac effect\n\n\\- large numbers hypothesis\n\n\\- magnetic monopole\n\n\\- many-times formulation of quantum electrodynamics by PD, Fock and Podolsky\n\n\\- neutron diffusion in matter, theory of\n\n\\- non-commutation in quantum mechanics\n\n\\- parity violation, foresees possibility of\n\n\\- philosophy of physical science\n\n\\- Poisson bracket in quantum mechanics\n\n\\- positron, prediction of\n\n\\- principle of mathematical beauty\n\n\\- quantum electrodynamics\n\n\\- quantum field theory, co-discovery _see quantum electrodynamics_\n\n\\- quantum mechanics of heavy atoms\n\n\\- quantum mechanics, later contributions\n\n\\- Schr\u00f6dinger equation (time dependent), independent discovery by PD\n\n\\- sphere, quantum-relativistic treatment of\n\n\\- spinors\n\n\\- string concept in quantum electrodynamics\n\n\\- transformation theory\n\n\\- vacuum polarisation\n\n\\- virtual states\n\n**the arts, taste and appreciation of**\n\n\\- art (visual),\n\n\\- cinema\n\n\\- comics and comic characters\n\n\\- music\n\n\\- novels\n\n\\- poetry\n\n\\- radio and television, appreciation of\n\n\\- theatre and opera\n\n**books**\n\n\\- _General Theory of Relativity_\n\n\\- _Principles of Quantum Mechanics_\n\nDirac, unofficial unit of frequency of speech\n\nDirac, Walla (PD's paternal grandmother)\n\n'Dirac stories'\n\nDNA, double-helix structure of\n\nDneproges hydroelectric power station\n\nDobb, Maurice\n\nDostoevsky, Fyodor: _Crime and Punishment_\n\nDouglas' Works, Kingswood\n\nDublin\n\nDublin conference (1942)\n\nDurango, Colorado\n\nDuranty, Walter\n\nDutton, S. T.: _Social Phases of Education in the School and the Home_\n\nDyson, Freeman\n\n##\n\nEddington, Sir Arthur\n\nmathematician and astronomer\n\nunderstanding of relativity theory\n\nsolar-eclipse experiments\n\non Einstein's _E_ = _mc_ 2 equation\n\nintroducesto relativity\n\nappearance\n\npersonality\n\nmathematical approach to science\n\nand Rutherford\n\ncongratulateson his Ph.D. thesis\n\nand the splitting of the atom\n\nmedia savvy\n\npilloried by his younger colleagues\n\nand nuclear energy\n\ndisagreement with PD\n\nDublin conference (1942)\n\ndeath\n\n_The Mathematical Theory of Relativity_\n\n_The Nature of the Physical World_\n\n_Space, Time and Gravitation_\n\nEdward VII, King\n\nEhrenfest, Paul\n\nEhrenhaft, Felix\n\nEinstein, Albert\n\npersonality\n\nmost successful spurt of creativity\n\nappearance\n\nstudies Mill's _System of Logic_\n\n_E_ = _mc_ 2 equation\n\nand Planck's blackbody radiation spectrum formula\n\nand solar eclipse results\n\nlight quanta idea\n\nand Bohr\n\nand Heisenberg's theory of 1925\n\nsuspicious of the new quantum mechanics\n\ntop-down approach to physics\n\non PD\n\nstimulated emission process and the laser\n\nattacks Heisenberg's uncertainty principle\n\ndiffers from PD in his approach to science\n\npraises PD's textbook\n\nat the 1930 Solvay Conference\n\ndespises Hitler\n\nNazis' view of his 'Jewish physics'\n\nand the photon\n\nand the splitting of the atom\n\nflees from Germany to the USA\n\nat Princeton\n\nand Kapitza's detention\n\ndislike of quantum electrodynamics\n\ntreats Heisenberg with contempt\n\nHoover's campaign against\n\nsuggests the existence of a positive electron\n\nin search of generalisations\n\ndeath\n\ncentenary of his birth\n\n'Electron and General Relativity'\n\n_see also_ relativity\n\nEisenhower, Dwight D.\n\nelectrical charge\n\nelectromagnetic interaction\n\nelectromagnetism\n\nlaws of\n\nMaxwell's theory\n\nPD's magnetic monopole theory\n\nelectron-positron pairs\n\nelectrons\n\nbare\n\nbehaving as discrete particles\n\nCavendish annual dinner, toast to\n\ndescribing behaviour of a single, isolated electron\n\ndiffraction by light\n\nDirac equation\n\ndiscovered by J. J. Thomson\n\nextended\n\nmoving in a straight line\n\nnegative-energy\n\norbiting the nucleus\n\nparticle-like\n\nPauli's exclusion principle\n\npositive-energy\n\nscattering\n\nself-energy of\n\nspin of\n\nwave nature _see also_ Fermi-Dirac statistics\n\nEliot, T. S.\n\nElizabeth, Queen (later the Queen Mother)\n\nElizabeth, Queen\n\nElsasser, Walter\n\nempiricism\n\nenergy quanta\n\n'Erice Statement' (1982)\n\nErice summer school, Sicily (1982)\n\nEsperanto\n\nether\n\nbelief in\n\nPD's ether\n\nEuclid\n\nEuclidean geometry\n\nevolution, theory of\n\nexclusion principle\n\n##\n\nFalklands War (1982)\n\nFaraday, Michael\n\nFarm Hall, Godmanchester, Cambridgeshire\n\nFarmelo, Amelia (n\u00e9e Jones)\n\nFaust, performance of special version (1932)\n\nFederal Bureau of Investigation (FBI)\n\nFen, Elisaveta _see_ Jackson, Lydia\n\nFermi, Enrico\n\nradioactive decay of nuclei\n\nquantum field theory of beta decay\n\nand nuclear fission\n\nbuilds first nuclear reactor\n\nweak interaction\n\nNobel Prize\n\nFermi-Dirac statistics\n\nFermi National Accelerator Laboratory (Fermilab)\n\nFermilab Symposium (1980)\n\nfermions\n\nFeynman, Richard\n\nat 'The Future of Nuclear Science' conference\n\npersonality\n\nnew version of quantum mechanics\n\nanalyst and intuitionist\n\nWigner on\n\nportrait of PD\n\nsays he is 'no Dirac'\n\nFilton, Bristol\n\nFirst World War\n\nFisher family\n\nFitzgerald, F. Scott\n\nFlexner, Abraham\n\nFlorida State University\n\nPhysics Department\n\nPD moves from Cambridge to\n\ntreatment of PD\n\nKeen Building\n\nand PD's funeral\n\nDirac Science Library\n\nfluid mechanics\n\nFock, Vladimir\n\nFolies Berg\u00e8re\n\nFord, Henry\n\nForeign Office\n\nFourier, Joseph\n\nFowler, Ralph\n\nPD's supervisor at Cambridge\n\nand Rutherford\n\nlectures on Bohr's theory\n\nworks with Bohr\n\nand's paper 'The Fundamental Equations of Quantum Mechanics'\n\nelected a Fellow of the Royal Society\n\nPD's visits to Copenhagen and G\u00f6ttingen\n\nco-edits the 'International Series of Monographs on Physics'\n\nfailing health\n\ndeath\n\nFranck, James\n\nFranco, General\n\nFrank, Anne\n\nFrank, Sir Charles\n\nFraser, Peter\n\nFrayn, Michael: _Copenhagen_\n\nFrench Circle\n\nFrench Riviera\n\nFrenkel, Yakov\n\nFriedmann, Alexander\n\nFrisch, Otto\n\nFrisch, Otto and Peierls, Rudolf:\n\n'Memorandum on the Properties of a\n\nRadioactive \"Super-Bomb\"'\n\nFrith, Uta\n\nFrost, Robert\n\nFuchs, Klaus\n\nfundamental interactions, unified theory of\n\nfundamental particles\n\n'Future of Nuclear Science' conference\n\n(Graduate College, Princeton, 1946)\n\nGabor, Dennis\n\nGalileo Galilei\n\nGalsworthy, John: _Forsyte Saga_\n\nGamow, Barbara (n\u00e9e Perkins)\n\nGamow, George\n\nGamow, Lyubov Vokhminzeva ('Rho')\n\nGandhi, Mahatma M.K.\n\nGardiner, Margaret\n\nGaspra, Crimea\n\ngauge invariance\n\ngauge theory\n\nGautier, Th\u00e9ophile\n\nGebhard's Hotel, G\u00f6ttingen\n\nGeiger counters\n\nGell-Mann, Murray\n\nGeneral Strike (1926)\n\ngenetics\n\nGeneva, Switzerland\n\ngeometry\n\ndifferential\n\nEuclidean\n\nnon-Euclidean\n\nPD immersed in at Cambridge\n\nprojective\n\nRiemannian\n\nGeorge, King\n\nGercke, Achim\n\nGermany\n\nYouth Movement\n\nthe Depression in\n\nnew militarism in\n\nEinstein flees to the US\n\nHitler becomes Chancellor\n\nbook-burning ceremonies\n\nannexes Austria\n\ninvades Czechoslovakia\n\ninvades Poland\n\nBritain declares war on\n\noverwhelms Norway and Denmark\n\nblitzkrieg on Belgium, Luxembourg and the Netherlands\n\nU-boat fleet\n\nGermer, Lester\n\nGill, Eric\n\nGlacier National Park\n\ngluons\n\nGo (a.k.a. Wei Chi)\n\nGoddard, Peter\n\nGoethe, Johann Wolfgang von: _Faust_\n\nGog Magog Hills\n\n_Goldfinger_ (film)\n\nGoldhaber, Maurice\n\nGoldschmidt, Victor\n\nGonville and Caius College, Cambridge\n\nGottfried, Kurt\n\nG\u00f6ttingen\n\nPD visits\n\nMathematics Institute\n\nanti-Semitism in\n\nNazism in\n\nseething with political tensions\n\nHeisenberg returns to\n\nthe Diracs and Kapitzas in\n\ngramophone\n\nGrand Canyon\n\nGrandin, Temple\n\nGrant, Cary\n\n_Granta_\n\nGrassmann algebra\n\nGraves, Robert\n\ngravitational waves\n\ngraviton\n\ngravity\n\nand the general theory\n\nlaws of\n\nand Newton's falling apple\n\nand Riemann's geometric ideas\n\nand string theory\n\n_Great Britain_ , SS\n\nGreat Exhibition (1851)\n\n_Greater Soviet Encyclopedia_\n\nGreen, Michael\n\ngroup theory\n\nGroves, General Leslie\n\n##\n\nHaddon, Mark: _The Curious Incident of the Dog in the Night Time_\n\nHahn, Otto\n\nHalpern, Leopold\n\nborn and raised in Austria\n\npersonality\n\nfriendship with PD\n\nopposes surgery on PD's tubercular kidney\n\nfraught relationship with Manci\n\nhomoeopathic treatment of PD\n\nand PD's funeral\n\nsatellite-based experimental programme\n\ndeath\n\nHamilton, William\n\nHamiltonian\n\nHarding, Gardner L.\n\nHardy, G. H.\n\nHarish-Chandra (Harish Chandra Mehrotra)\n\nHarish-Chandra, Lily\n\nHarvard University\n\nHarz Mountains\n\nHass\u00e9, Ronald\n\nHawking, Stephen\n\nHayward, F. H.\n\nHeath, Edward\n\nHeaviside, Oliver\n\nHebblethwaite, Cyril\n\nHeckmann, Otto\n\nHegel, Georg Wilhelm Friedrich\n\nHeisenberg, Werner\n\npersonality\n\naddresses the Kapitza Club\n\nquantum theory (1925)\n\nnon-commuting quantities\n\nand PD's first paper on quantum mechanics\n\nworks with Born and Jordan at G\u00f6ttingen\n\nand Schr\u00f6dinger's work on wave mechanics\n\nuncertainty principle\n\npianistic skills\n\nand PD's attack on religion\n\nappointed full professor in Leipzig\n\nand the Dirac equation\n\nvisits Japan with PD\n\nSoviet government's attitude to his work\n\npleased at Hitler's coming to power\n\nand the positron\n\natomic nucleus structure\n\nNobel Prize for physics\n\ncelebrations in Copenhagen\n\nmessage to Born from the Nazi Government\n\na 'White Jew'\n\nmeeting with Bohr (1941)\n\nattests to Betty's non-Jewish status\n\ninterned near Cambridge\n\nexplanation of his wartime conduct\n\nPD supports\n\nquarrels with Pauli\n\nat Lindau\n\ninterviewed with PD\n\nappearance\n\ndeath\n\nHeisenberg-Pauli theory\n\nHellman, Bruce\n\nHenri Poincar\u00e9 Institute, Paris\n\nHess, Rudolf\n\nHessen, Boris\n\nhigh-dimensional field theories\n\nhigh-energy particle accelerators\n\nhigh-energy physics\n\nHighgate Cemetery, London\n\nHilbert, David\n\nHippodrome theatre, Bristol\n\nHiroshima, bombing of (1945)\n\nHiston, Cambridgeshire\n\nHitler, Adolf\n\nHofer, Kurt\n\nHoffman, Dustin\n\nHolborn Registry Office, central London\n\nHolcomb, Dorothy\n\nhole theory\n\nHoliday Inn, Tallahassee\n\nHolmes, Sherlock, _see also_\n\nHolten, Beatrice (Flo's sister)\n\nHolten, Fred (Flo's brother)\n\nHolten, Nell (Flo's sister)\n\nHolten, Richard (PD's maternal grandfather)\n\nHong Kong\n\nHoover, Herbert\n\nHoover, J. Edgar\n\n_Horizon_ (BBC programme) 'Lindau'\n\nHorthy, Admiral\n\nHotel Britannique, Brussels\n\nHotel Metropole, Moscow\n\nHoum\u00e8re, Pam\n\nHousman, A.E.\n\nHoyle, Fred\n\nHubble, Edwin\n\nHubble's law\n\nHungary\n\nHuntingdon Road, Cambridge\n\nHuxley, Aldous: _Point Counterpoint_\n\nHuxley, Thomas\n\nhydrogen atom\n\nBohr theory and\n\nDirac equation and\n\nlamb-shift of\n\nquantum mechanics and\n\nhydrogen bomb\n\n##\n\nImmigration Act (1952)\n\nImperial Hotel, Bloomsbury, London\n\nindefinite metric\n\nIndia\n\nPD visits (1954)\n\nbecomes a nuclear power (1974)\n\nInfeld, Leopold\n\nInstitut de Physiology Solvay au Parc\n\nL\u00e9opold, Brussels\n\nInstitute for Advanced Studies, Dublin\n\nInstitute for Advanced Study, Princeton\n\nInstitute for Physical Problems (Soviet Union)\n\nInstitute for Theoretical Physics (Niels Bohr\n\nInstitute), University of Copenhagen\n\nInternational Congress on the History of\n\nScience and Technology, second (Science\n\nMuseum, London, 1932)\n\nInternational Esperanto Congress (Trinity\n\nCollege, Cambridge, 1907)\n\n'International Series of Monographs on\n\nPhysics'\n\nInyom, Revd. Sapasvee Anagami\n\nIsenstein, Harald\n\nisotope separation\n\nIsrael\n\nIvanenko, Dmitry 'Dimus'\n\n##\n\nJackson, Lydia (previously Elisaveta Fen)\n\nJapan\n\nPD and Heisenberg visit\n\nnew militarism in\n\nbombing of Hiroshima\n\nbombing of Nagasaki\n\nsurrender of\n\n'Jazz Band' (informal group of Soviet theorists)\n\nJeans, Sir James\n\n_The Mysterious Universe_\n\nJeffreys, Harold\n\n'Jewish physics'\n\nJohn Paul II, Pope\n\nJoliot-Curie, Fr\u00e9d\u00e9ric\n\nJoliot-Curie, Ir\u00e8ne\n\nJones, Norman\n\nJordan, Pascual\n\nworks with Born and Heisenberg at G\u00f6ttingen\n\nand groups of electrons\n\npersonality\n\nappearance\n\nand field theory\n\nand the Dirac equation\n\nNazi past\n\nJoyce, James\n\n_Finnegans Wake_\n\n_A Portrait of the Artist as a Young Man_\n\nJulius Road, Bristol (No.6)\n\n##\n\nKant, Immanuel\n\nand beauty\n\nand truth\n\nKapitza, Anna ('Rat')\n\nKapitza, Peter\n\nsettles in the UK\n\npersonality\n\ninfluences PD\n\nresented by Blackett\n\nobsession with the crocodile\n\nRussia's industrialisation and electrification\n\nrelationship with Rutherford\n\ncompared with PD\n\nsupports Communist goals\n\nunder surveillance\n\nsets up the Kapitza Club\n\nattitude to experimental physics\n\nmarries Anna Krylova\n\nco-edits the 'International Series of Monographs on Physics'\n\nat the Cavendish Physical Society annual dinner\n\nthe Bukharin visit to Cambridge\n\nMI5 monitors him\n\nvacation with PD in the Crimea\n\nand the anti-electron\n\nPD works with him in his laboratory\n\ndetained by the Soviet Government\n\nand Rutherford's death\n\nseeks Landau's release\n\nwartime telegram to PD\n\nnominated by PD for a Nobel Prize\n\ninvents method of liquefying oxygen\n\n'Hero of Socialist Labour'\n\nand Beria\n\nin disgrace\n\nletters to Stalin\n\nand PD's passion for beauty\n\nvisits Cambridge in 1966\n\nNobel Prize in Physics\n\ndeath\n\nPD spends his last hours talking about him\n\n'The Training of the Young Scientist in the USSR'\n\nKapitza-Dirac effect\n\nKapitza Club\n\nKeats, John\n\nKennedy, John F.\n\nKent State University\n\nket\n\nKeynes, John Maynard\n\nKharkhov\n\nKhrushchev, Nikita\n\nKierkegaard, S\u00f8ren\n\nKitchener, Lord\n\nKlampenborg Forest, Denmark\n\nKlein, Oskar\n\nKoh-i-Noor restaurant, St John's Street,\n\nCambridge\n\nKronborg castle, Denmark\n\nKubrick, Stanley\n\nKuhn, Thomas\n\nKun, B\u00e9la\n\nKur\u015funo\u011flu, Behram\n\nKyoto\n\n##\n\nLabour government\n\nLabour Party\n\nLagerl\u00f6f, Selma\n\nLagrange, Jopseph Louis\n\nLagrangian\n\nLake District\n\nLake Elmore\n\nLamb, Charles: 'The Old Familiar Faces'\n\nLamb, Willis\n\nLandau, Lev\n\nLandshoff, Peter\n\nLanger, Rudolph\n\nLannutti, Joe\n\nLarge Hadron Collider\n\nlarge numbers hypothesis\n\nLarmor, Sir Joseph\n\nlasers\n\nLawrence, Ernest\n\nLawrence, T.E.: _Seven Pillars of Wisdom_\n\nLederman, Ellen\n\nLederman, Leon\n\nLee, T..\n\nleft-right symmetry\n\nLeiden, Netherlands, PD visits\n\nLeipzig\n\nHeisenberg appointed full professor\n\nPD in\n\nLema\u00eetre, Abb\u00e9 Georges\n\nLenin, Vladimir\n\nLeningrad, PD in\n\nleptons\n\nLiberal Party\n\n_Life_ magazine\n\nlight\n\nviewed as photons\n\nin a continuous wave\n\nemitted and absorbed by atoms\n\nenergy of light tranferrable to atoms only in quanta (Planck)\n\nas particles\n\n_see also_ radiation, electromagnetic\n\nLindau, Germany\n\n1965 meeting\n\n1971 meeting\n\n1982 meeting\n\nLindemann, Frederick\n\nLindstrom, Andy\n\nLippmann, Gabriel\n\nLiverpool\n\nLloyd George, David\n\nLocarno, Treaty of (1925)\n\nlogical positivists\n\nLondon\n\nCharles Dirac in\n\nin Second World War\n\nDirac family stays in\n\nLondon Mathematical Society\n\nLos Alamos headquarters, New Mexico\n\nLost Lake, near Tallahassee\n\nLourdes\n\nLucasian Professorship of Mathematics\n\nLuftwaffe\n\nLyons, Eugene\n\n##\n\nMcCarthy, Joseph\n\nMacDonald, Ramsay\n\nmagnetic monopole\n\nManchester\n\n_Manchester Guardian_\n\nManchester University\n\nManhattan, New York\n\nManhattan Project\n\nMartineau, Harriet\n\nMarx, Karl\n\nMarxism\n\nmathematics\n\naesthetic view of\n\napplied\n\nbeauty of\n\nBohr's attitude to\n\nGod as a mathematician\n\nmathematical rigour\n\nPD's sometimes cavalier attitude to\n\npragmatic approach to the mathematics of engineering\n\npure\n\ngame in which people invent the rules, PD's view as\n\nmatrices\n\nand electron spin\n\nHeisenberg's quantum theory\n\nMAUD committee\n\nMaugham, W. Somerset\n\n_A Writer's Notebook_\n\n_Of Human Bondage_\n\n_Then and Now_\n\nMauthausen-Gusen concentration camp, Austria\n\nMaxwell, James Clerk\n\nelectromagnetic theory\n\nthe universe as a giant mechanism\n\nMaysky, Ivan\n\nmechanics, laws of (Newton)\n\nMeitner, Lise\n\nMerchant Venturers' Secondary School, Bristol (later Cotham Road School)\n\nCharles teaches at\n\nPD's education\n\nrelocates to Cotham Lawn Road\n\ncelebration of's success\n\nMerchant Venturers' Society\n\nMerchant Venturers' Technical College, Bristol\n\nFelix studies at the Faculty of Engineering\n\nPD's studies\n\nwartime bombing of\n\nmesons\n\nMetropolitan Police Special Branch\n\nMI5\n\nMiami\n\nMiami Museum of Science\n\nMickey Mouse\n\nmicroelectronics\n\nMill, John Stuart\n\n_On Liberty_\n\n_A System of Logic_\n\nMiller, Arthur\n\nMillikan, Robert\n\nelectrical charge\n\ncosmic rays\n\nand Anderson's evidence for a positive electron\n\nand electron-positron pairs\n\nefforts to get Kapitza released\n\nMills, Robert\n\nMilne, Edward\n\nMiners' Union\n\nMinkowski, Hermann\n\nMoli\u00e8re\n\nM\u00f8ller, Christian\n\nMond Laboratory, Cambridge\n\nMonge, Gaspard\n\nMonk Road, Bishopston, Bristol (No.15)\n\nmonopole problem\n\nMonthey, Switzerland\n\nMoore, George _Principia ethica_\n\nMorgan, Howard\n\nMorrisville, Vermont\n\nMorse, Louise\n\nMoscow\n\nPD in\n\nKapitzka detained in\n\nscience community\n\n_Moscow News_\n\nMoscow Polytechnic\n\nMoscow University\n\nMoseley, Sir Oswald\n\nMott, Nevill\n\nMount Brocken\n\nMount Elbrus\n\nMount Wilson Observatory, near Pasadena\n\nMountfield Nursing Home, London\n\nMuch Wenlock, Shropshire\n\nMunich\n\nMunich agreement\n\nmuon\n\nMussolini, Benito\n\n##\n\nNagasaki, bombing of\n\nnanotechnology\n\nNASA\n\nnature\n\nfundamental equations of Nature as only approximations\n\nlaws of\n\nmetaphor of a colossal clockwork mechanism\n\nunity and beauty of\n\n_Nature_ journal\n\nNazis (National Socialists)\/Nazism\n\nnegative energy states\n\nNehru, Jawaharlal\n\nneutrinos\n\nneutron stars\n\nneutrons\n\natomic nuclei\n\nChadwick's discovery\n\nRutherford proposes\n\nstrongly interacting\n\n_New Statesman_\n\nNew York\n\n_New York Times_\n\nNewlin's restaurant, Princeton\n\nNewman, Max\n\nNewnham College, Cambridge\n\n_News Chronicle_\n\nNewton, Sir Isaac\n\nchildhood\n\nEinstein's theory refutes his ideas\n\ntheory of gravity\n\nmechanics\n\nburial in Westminster Abbey\n\nand autism\n\nNightingale, Florence\n\nNixon, Richard\n\nNoakes, Michael\n\nNobel, Alfred\n\n'Nobel disease'\n\nNobel Foundation\n\nnon-commuting quantities\n\nnon-interacting quantum particles\n\nNorway, PD in\n\nnuclear fission\n\nnuclear industry\n\nnuclear weapons\n\ndestruction of incoming\n\nSecond World War\n\n##\n\nOcchialini, Giuseppe\n\n_Oklahoma!_ (film soundtrack)\n\nOld Faithful geyser\n\nOppenheimer, Frank\n\nOppenheimer, J. Robert ix\n\npersonality\n\ndislike of Cambridge life\n\nclinical depression\n\ntries to poison Blackett\n\nworks with PD\n\nfriendship with PD\n\nat Born's Department of Theoretical Physics\n\npoetry\n\nPh.D. on the quantum mechanics of molecules\n\nand the rise of anti-Semitism\n\ndisappointed with PD's work in G\u00f6ttingen\n\nat University of California at Berkeley\n\nand PD's hole theory\n\non the Heisenberg-Pauli theory\n\nand Anderson's positive electron\n\nquantum electrodynamics\n\nScientific Director of the Manhattan\n\nProject\n\ncelebrated as a hero in the USA\n\ndirector of the Institute for Advanced Study\n\nformer Communist sympathies\n\nadviser on nuclear policy\n\nUS withdraws his security clearance\n\nappearance\n\nretirement and death\n\nand black holes\n\nOrpington, south-east London\n\nOrwell, George\n\n_Coming Up for Air_\n\n_The Lion and the Unicorn_\n\nOseen, Carl\n\nOttawa\n\nOxford\n\n##\n\nPais, Abraham\n\n_Subtle is the Lord_\n\nPalais de la D\u00e9couverte, La, Paris\n\npantheism\n\nPapal Academy\n\nparity violation\n\nparticle accelerators\n\nparticle physics\n\nPauli, Wolfgang\n\nan analytical conservative analyst\n\nexclusion principle\n\npersonality\n\nand electron spin\n\nPD's harshest critic\n\nSecond Principle\n\npraises PD's textbook\n\nand PD's hole theory\n\nco-presents seminar with PD at Princeton\n\nappearance\n\nthe neutrino\n\nproblems in his personal life\n\nsecond marriage\n\nquarrels with Heisenberg\n\nNobel Prize\n\ndeath\n\nPavlov, Ivan\n\n'Peanuts'\n\nPearl Harbor, bombing of\n\nPeierls, Genia\n\nPeierls, Rudolf\n\nPenrose, Roger\n\nPeterhouse College, Cambridge\n\nPhillips, Leslie Roy\n\nphilosophy\n\nPhoney War\n\nphotography, amateur\n\nphotons\n\nand Einstein\n\nand Langer\n\nlight consisting of\n\nscattering by a single electron\n\nstimulated emission process and the laser\n\nPicasso, Pablo\n\nPickering, Arthur\n\nPincher, Chapman\n\nPippard, Brian\n\nPlanck, Max\n\nquantum hypothesis\n\nblackbody radiation spectrum\n\nPlanck's constant\n\nPlato\n\nPodolsky, Boris\n\nPoisson bracket\n\nPoland\n\nHitler's invasion of\n\ncollapse of\n\nManci's view of Poles\n\nPolkinghorne, John\n\nPoncelet, Jean-Victor\n\nPortishead, Bristol\n\nPortland Street Chapel, Bristol\n\nposition and momentum symbols\n\npositive energy states\n\npositivism\n\npositron emission tomography (PET)\n\npositrons _see also_ anti-electrons\n\nPottier family\n\n_Pravda_\n\n'primitive atom' theory\n\nPrinceton\n\nPrinceton University\n\nbicentennial celebrations\n\nFine Hall (later Jones Hall)\n\nFuld Hall\n\nGraduate College\n\n_Proceedings of the Royal Society_\n\nprojective geometry\n\nprotons\n\natomic nuclei\n\nnegative\n\nstrongly interacting\n\nPryce, Gritli (n\u00e9e Born)\n\nPryce, Maurice\n\npulsars\n\n_Punch_ magazine\n\nPythagoras's theorem\n\n##\n\nquanta\n\nenergy\n\nSchr\u00f6dinger's wave theory\n\n_see also_ photons\n\nquantum chromodynamics\n\nquantum electrodynamics\n\nquantum field theory _see also_ quantum electrodynamics and quantum chromodynamics\n\nquantum jumps\n\nquantum mechanics\n\nnamed by Born\n\nbirthplace of\n\nbuilding of the complete theory\n\nfirst prediction of\n\nmathematical symbols in\n\ncentral role of probability\n\nrelationship with classical mechanics\n\nrelativistic\n\nand miniaturisation\n\nquantum numbers\n\nquantum theory\n\ndiscovered by Planck\n\nEinstein lays its foundations\n\nlaws of\n\nPD introduces the mathematics of creation and annihilation\n\nthe universe as fundamentally granular\n\nquarks\n\nquaternions\n\n_Queen Mary_ (liner)\n\n##\n\nRabi, Isidor\n\nradiation\n\nelectromagnetic\n\ngravitational\n\nradio\n\nradioactive decay\n\n_Rain Man_ (film)\n\nRamond, Pierre\n\nReagan, Ronald\n\nRedlands Girls' School, Bristol\n\nreductionism\n\nrelativity\n\nas's passion\n\nBroad's teaching of\n\nEinstein's general theory\n\nEinstein's special theory\n\nHass\u00e9 speaks on the subject at Cambridge\n\nRembrandt van Rijn\n\nrenormalisation\n\nRetherford, Robert\n\n_Reynolds's Illustrated News_\n\nRichards, Sir Gordon\n\nRiemann, Bernhard\n\nRijksmuseum, Amsterdam\n\nRobertson, Andrew\n\nRobertson, David\n\nRobertson, Howard\n\nRobertson, Malcolm\n\nRobeson, Paul\n\nRoentgen Institute, Leningrad\n\nRolls-Royce\n\nR\u00f6ntgen, Wilhelm\n\nRoosevelt, Eleanor\n\nRoosevelt, Franklin.\n\nRoselawn cemetery, Tallahassee\n\nRosen, Nathan\n\nRosenfeld, L\u00e9on\n\nRothschild, Victor, Lord\n\n'Roundy' (Joseph Coughlin)\n\nRousseau, Jean-Jacques\n\nRoyal Air Force\n\nRoyal Astronomical Society, Burlington House, London\n\nRoyal Commission for the Exhibition of 1851\n\nRoyal Navy\n\nRoyal Society\n\nPD elected a Fellow\n\nfunds the Mond Laboratory\n\nBaker Medal\n\nand Heisenberg\n\nand Schr\u00f6dinger\n\nRoyal Society of Scotland\n\nRugby, Warwickshire\n\nRussell, Bertrand 'Zahatopolk'\n\nRutherford, Ernest, Baron Rutherford of Nelson\n\nand Eddington\n\npersonality\n\nappearance\n\ndiscovery of the atomic nucleus\n\nproposes the neutron\n\ndirector of the Cavendish Laboratory\n\nKapitza's nickname for him ('the  Crocodile')\n\nand Kapitza's support of Communism\n\nand Bohr\n\nrelationship with Kapitza\n\nloathes Bernal\n\nennobled\n\ndeath of his daughter\n\nand Chadwick's discovery of the neutron\n\nand the Cockcroft-Walton splitting of the atom\n\nleadership of Cambridge experimental physicists\n\nBlackett's anger at his despotic style\n\nbottom-up approach to physics\n\nbas-relief in the Mond Laboratory\n\nstays in Bohr's mansion\n\nand Kapitza's detention\n\nand's marriage\n\non Eddington\n\ndeath\n\nmemorial service at Westminster Abbey\n\n##\n\nSt John's College, Cambridge\n\nPD unable to take up a place at (1921)\n\nPD wins two scholarships (1923)\n\nPD arrives at\n\ndescribed\n\nencouragesto apply for a Fellowship\n\nawardsa special lectureship\n\nTamm's visit (1931)\n\nBorn's honorary position\n\nCombination Room\n\nFellowship extended for life\n\nIsenstein bust of PD\n\nPD's last visit to\n\nfirst women undergraduates\n\nnurturing environment for PD\n\nPD apologises for absence at eightieth birthday celebrations\n\nPD's Nobel Medal and certificate returned\n\nSt Maurice, Switzerland\n\nSakharov, Andrei\n\nSalam, Abdus\n\nSalaman, Esther and Myer\n\nscattering matrix\n\nSchnabel, Artur\n\nSch\u00f6nberg, Arnold\n\nSchr\u00f6dinger, Annemarie (Anny)\n\nSchr\u00f6dinger, Erwin\n\nhis quantum theory\n\nreputation as a polymath\n\nwave mechanics\n\nvisits the Bohr Institute\n\nNobel Prize for physics\n\npersonality\n\na refugee in Oxford\n\naffirms his loyalty to the Nazi regime\n\naccepts Dublin post\n\nDublin conference (1942)\n\nelected to the Royal Society\n\ndeath\n\nPD's obituary\n\nSchr\u00f6dinger's equation\n\nSchuster, Arthur\n\nSchwarz, John\n\nSchwinger, Julian\n\n_Science_ journal\n\nScience Museum, London\n\n_Scientific American_\n\nScott lecture\n\nSecond Physics Institute, G\u00f6ttingen\n\nSecond World War\n\nChamberlain declares war on Germany\n\nCambridge\n\nblitzkrieg of Belgium, Luxembourg and the Netherlands\n\nend of the war in Europe\n\nSeiberg, Nathan\n\nSen, Colleen Taylor\n\nShakespeare, William\n\n_Hamlet_\n\n_Love's Labour's Lost_\n\n_Richard_ 76\n\nShankland, Robert\n\nShaw, George Bernard\n\n_Getting Married_\n\n\\- Preface\n\n_The Irrational Knot_\n\nShelter Island Conference, Long Island, New York (1947)\n\n_Shinyo Maru_ (steamer)\n\nSidgwick, Henry\n\nSilver Lake, near Tallahassee\n\nSimon, Sir Francis\n\n_Simpsons, The_\n\nSinatra, Frank\n\nSkye, Isle of\n\nSlater, John\n\nSliger, Bernie\n\nSnow, C.\n\n_The Search_\n\nSocial-Democratic Workers' Party\n\nSocialist Society\n\nsociology\n\nsolar-eclipse experiments (1919)\n\nSolvay Conferences\n\n1927\n\n1930\n\n1933\n\n1961\n\nSommerfeld, Arnold: _Atomic Structure and Spectral Lines_\n\n_Sound of Music, The_ (film)\n\nSoviet Academy of Sciences\n\nSoviet Conference on Nuclear Physics\n\n(Leningrad, 1933)\n\nSoviet Embassy, Washington\n\nSoviet Union\n\nPD's first visit\n\nPD's second visit\n\nand the British press\n\nthe Soviet experiment\n\nthe Jazz Band\n\nPD falls foul of the censors\n\nPD's support for Soviet physics\n\nPD attends Leningrad conference (1933)\n\nPD unaware of the cost of the collectivisation programme\n\nPD in Bolshevo\n\nGreat Purge\n\ntrials in\n\nNazi invasion of\n\nPD and colleagues refused visas by Churchill\n\nFuchs passes secrets to\n\nearly detonation of the Soviet nuclear weapon\n\nSputnik missions\n\nspace programme\n\nCuban crisis\n\nspace-time\n\ncurved\n\nand de Sitter\n\nmore than four dimensions of\n\nspecial theory of relativity\n\nunified\n\nSpanishWar (1936-9)\n\nSpender, Stephen\n\n_Journals_\n\n_World Within World_\n\nSpielberg, Steven\n\nspinors\n\nspintronics\n\nSpinoza, Baruch\n\nSputnik missions\n\nSS\n\nStalin, Joseph\n\nrise to absolute power\n\nindustrialisation policy\n\ncollective farming programme\n\ninterviewed in _New Statesman_\n\nand the intelligentsia\n\nattitude to science\n\nCambridge students favour over Hitler\n\nhis government becomes more repressive\n\nnon-aggression pact with Hitler\n\nand Kapitza\n\ndeath\n\nKhrushchev denounces\n\nStalingrad\n\nStandard Model\n\nStanford, Henry King\n\nStanford University\n\n_Star Trek_ 8\n\n_Start the Week_ (Radioprogramme)\n\nsteady-state theory\n\nStockholm, Sweden\n\nStockman, Gertrude\n\nStokes, Sir George\n\nStony Brook, New York\n\nStoppard, Tom: _Arcadia_\n\nStrassman, Fritz\n\nStrategic Defence ('Star Wars') Initiative\n\nstress diagrams\n\nstring theory\n\nstrings\n\nstrong interaction\n\nsubatomic particle accelerators\n\nSudarshan, George\n\nSuez crisis (1956)\n\n_Sunday Dispatch_\n\n_Svenska Dagbladet_\n\nSwift, Jonathan: _Gulliver's Travels_\n\nSwirles, Bertha\n\nSwitzerland, PD visits\n\nSzil\u00e1rd, Le\u00f3\n\n##\n\n't Hooft, Gerard\n\nTallahassee, Florida\n\n_Tallahassee Democrat_\n\nTallahassee Memorial Hospital, Florida\n\nTamm, Igor\n\nat Leiden\n\npersonality\n\npolitics\n\nfirst Soviet theoretician to use quantum mechanics\n\nfriendship with PD\n\nmeets up with PD in Moscow\n\nand PD's hole theory\n\non 'brigade education'\n\nand the positron's detection\n\nin Bolshevo\n\nclimbing vacation in the USSR with PD\n\nsecret project to build the hydrogen bomb\n\nNobel Prize\n\nTata Institute, Bombay\n\ntechnical drawing\n\nTeller, Edward\n\nTennyson, Alfred, Lord\n\nTeszler, Betty (PD's sister) _see_ Dirac, Beatrice\n\nTeszler, Christine (PD's niece)\n\nTeszler, Joe\n\nTeszler, Roger (PD's nephew)\n\nThatcher, Margaret, Baroness\n\ntheoretical physics\n\nBerlin as its global capital\n\nPD introduces a new approach to\n\nWeyl's approach\n\nThomson, J. J.\n\nTilley, Peter\n\n_Times, The_\n\nTkachenko, Vladimir\n\nTodd, Horace\n\nTollast, Robert\n\nTolstoy, Count Leo\n\n_Anna Karenina_\n\n_War and Peace_\n\nTomonaga, Sin-Itiro\n\nTots and Quots dining club\n\nTrans-Siberian Railway\n\ntransformation theory\n\ntransistors\n\nTrieste symposium (1971)\n\nTrinity College, Cambridge\n\nTrotsky, Leon\n\nTroyanovsky, Aleksandr\n\nTruman, Harry S.\n\n'Tube Alloys' project\n\ntuberculosis\n\nTurin Shroud\n\nTuring, Alan\n\nTyndall, Arthur\n\n##\n\nuncertainty principle\n\n_Under the Banner of Marxism_ journal\n\nUNESCO\n\nUnited States of America\n\ndevelopment of quantum mechanics\n\nPD's first visit (1929)\n\nPD's 1931 visit\n\ndepression in\n\nEinstein emigrates to\n\nprominent role in the Second World War\n\nAmerican-led experiments to build a nuclear bomb\n\nfunding of theoretical physics\n\nanti-Communist paranoia (1950s)\n\nspace programme\n\nJudy settles in\n\nPD's regular visits\n\nuniverse\n\nexpanding\n\n'primitive atom' theory\n\nUniversity of Aarhus, Denmark\n\nUniversity of Bristol\n\nUniversity Engineering Society\n\nDirac Centenary Meeting (2002)\n\nFaculty of Engineering\n\nmathematics department\n\nPD declines an honorary degree\n\nPD takes the qualifying examinations early\n\nPD's FRS election\n\nUniversity of British Columbia\n\nUniversity of California at Berkeley\n\nUniversity of Cambridge _see_ Cambridge University\n\nUniversity of Florida, Gainesville\n\nUniversity of Geneva\n\nUniversity of Leiden, Netherlands\n\nUniversity of Liverpool\n\nUniversity of London\n\nUniversity of Madison, Wisconsin\n\nUniversity of Manchester\n\nUniversity of Miami\n\nUniversity of Minnesota\n\nUniversity of Nebraska\n\nUniversity of Swansea\n\nUniversity of Texas at Austin\n\nUpdike, John\n\nuranium\n\n235 isotope\n\n238 isotope\n\nUrey, Harold\n\nutilitarianism\n\n##\n\nvacuum concept\n\nvacuum cleaner\n\nValais canton, Switzerland\n\nVan Vleck, John\n\nVancouver\n\nVE-Day celebrations\n\nVeblen, Oswald\n\nVeltman, Martin\n\nVermont\n\nVictoria, Queen\n\nVienna\n\nVietnam war\n\nVieux, Annette (n\u00e9e Giroud;'s paternal great-grandmother)\n\nViktor Frankl Institute, Vienna\n\nvirtual states\n\nVladikavkas\n\nVladivostock\n\nvon Neumann, John\n\nVSO (MI5 informant)\n\n##\n\nWakulla river\n\nWaldegrave, William, Baron Waldegrave of North Hill\n\nWall Street crash (1929)\n\nWaller, Ivar\n\nWalters, Barbara: _How to Talk to Practically Anybody about Practically Anything_\n\nWalton, Ernest\n\nWalton, Sir William\n\nWashington, D.C.\n\nWatt, Dr Hansell\n\nWattenberg, Al\n\nWaugh, Evelyn: _Brideshead Revisited_\n\nwavicle\n\nweak interaction\n\nWei Chi (a.k.a. Go)\n\nWeimar Republic\n\nWeinberg, Steven\n\nWeinberg-Salam theory\n\nWeisskopf, Vicki\n\nWells, H. G.\n\n_The Time Machine_\n\n_The World Set Free_\n\nWestminster Abbey, London\n\nWeyl, Hermann\n\nWheeler, John\n\nWhewell, William\n\nWhiston, William\n\nWhite, Sir George\n\nWhitehead, Henry\n\nWhitehead, Right Reverend Henry\n\nWhitehead, Isabel\n\nWhittaker, Edmund\n\nWigner, Amelia (n\u00e9e Frank)\n\nWigner, Jen\u0151 (later Eugene)\n\nchildhood\n\nfield theory of the electron\n\npersonality\n\nand's impoliteness\n\naims to bring modern quantum mechanics to Princeton\n\nat the University of Madison\n\nmarriage to Amelia\n\nand nuclear fission\n\nmarriage to Mary\n\nand nuclear weapons\n\norganises 'The Future of Nuclear Science' conference\n\non Feynman and PD\n\nand Kuhn's interviewing of PD\n\nan elder statesman of American science\n\nand Judy's disappearance\n\nManci's response to his death\n\n_The Recollections of Eugene P. Wigner_\n\nWigner, Mary\n\nWilczek, Frank\n\nWilde, Oscar\n\n_The American Invasion_\n\n_The Importance of Being Earnest_\n\nWilhelm II, Kaiser\n\nWilkinson, Sir Denys\n\nWilliams, Edith\n\nWillis, D. C.\n\nWiltshire, Herbert Charles (Charlie)\n\n_Wisconsin Journal_\n\nWitten, Edward\n\nWittgenstein, Ludwig\n\nWolverhampton\n\nWoolf, Virginia\n\nWordie, James\n\nWordsworth, William _The Prelude_\n\nWu, Chien-Shiung\n\n##\n\nX-rays\n\n##\n\nYang, C. N.\n\nYeats, W. B.: _The Living Beauty_\n\nYeshiva University, New York\n\nYosemite National Park\n\n##\n\nZimmerman, Erika\n\nZurich\n\nZweig, George\nCopyright \u00a9 2009 by Graham Farmelo\n\nPublished by Basic Books,   \nA Member of the Perseus Books Group   \nPublished in Britain in 2009 by Faber and Faber Limited\n\nAll rights reserved.   \nNo part of this book may be reproduced in any manner whatsoever   \nwithout written permission except in the case of brief quotations embodied   \nin critical articles and reviews. For information, address Basic Books,   \n387 Park Avenue South, New York, NY 10016-8810.\n\nBooks published by Basic Books are available at special   \ndiscounts for bulk purchases in the United States by corporations,   \ninstitutions, and other organizations. For more information, please contact   \nthe Special Markets Department at the Perseus Books Group, 2300 Chestnut   \nStreet, Suite 200, Philadelphia, PA 19103, or call (800) 810-4145,   \next. 5000, or e-mail special.markets@perseusbooks.com.\n\nA CIP catalog record for this book is available   \nfrom the Library of Congress.   \nLCCN: 2009925681\n\neISBN : 978-0-465-01992-2\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nThe author and publisher have provided this e-book to you for your personal use only. You may not make this e-book publicly available in any way. Copyright infringement is against the law. If you believe the copy of this e-book you are reading infringes on the author's copyright, please notify the publisher at: us.macmillanusa.com\/piracy.\nTo my mother,  \nMicheline Rosenberg-Sinclair\nCONTENTS\n\nTitle Page\n\nCopyright Notice\n\nDedication\n\nPrologue\n\nIntroduction\n\nRue La Bo\u00e9tie\n\nNumber 21 Under the Germans\n\nFloirac\n\nAt the Centre Pompidou\n\nGennevilliers\n\nDealer\n\nCh\u00e2teaudun, Op\u00e9ra, and Madison Avenue\n\nMother and Child\n\nPaul and Pic\n\nBoulevard Magenta\n\nPi-ar-enco\n\nA Long Relationship\n\nThe War Years in New York\n\nPreoccupations of the Heart\n\nThe Train, Schenker, and the Art of the Possible\n\nEpilogue\n\nBibliography\n\nAcknowledgments\n\nFrontispiece\n\nPhotographs\n\nA Note About the Author\n\nIllustration Credits\n\nCopyright\nPROLOGUE\n\nOn June 10, 2013, seventy-four years after my grandfather was forced to abandon his gallery located at 21 rue La Bo\u00e9tie in Paris, I had the honor to unveil a white marble plaque on the fa\u00e7ade of the building. The plaque bore his name and those of famous painters he used to show, many of whom were his closest friends\u2014Picasso, Braque, Matisse, L\u00e9ger among them. I was pleased that the plaque explained who my grandfather was and how the building, which had been devoted for twenty years to art, had been looted and transformed into a Nazi propaganda office during the German occupation of France.\n\nThe initiative was not mine. Rather, the owner of the building, of whom I had never heard, a certain M. Th\u00e9lot, a French \"entrepreneur,\" who used to rent offices in the building, unexpectedly sent me a very moving letter. He had just been browsing in a bookshop and had seen a book whose title was the exact address of the building he owned. Curious, he bought the volume\u2014the French edition of this book, 21, rue La Bo\u00e9tie\u2014and was so moved by the story that he offered to have a plaque made for the front of the building, as is commonly done for famous French or foreign citizens who have left their mark on a place. Moreover, he renamed the main room inside the gallery, the one where exhibitions had been held before the war, the Paul Rosenberg Room. (Previous tenants had improbably called it the Mississippi Room.) These initiatives were so selfless and elegant that I accepted with joy.\n\nThe homage took place on a late afternoon, when the sun was shining over Paris, so many years after the Nazis had seized the gallery.\n\nI could imagine how proud my grandparents, my uncle, and my mother, all of them now dead, would have been had they known their home would become celebrated in Paris, nearly three-quarters of a century after they were forced to flee arrest and deportation because they were Jewish and had refused to collaborate with the Nazi government that deemed modern painting \"degenerate art.\"\n\nFrom now on, thanks to M. Th\u00e9lot, everyone passing by will read the plaque, learn who the great art dealer Paul Rosenberg was, and discover how a criminal regime transformed my grandfather's gallery from a temple of beauty to a storeroom of depravity.\n\nThat is the story of this book.\nINTRODUCTION\n\nA day of rain and demonstrations, early 2010.\n\nMy neighborhood has been closed off by the police, the streets are jammed around the Bastille, and I am a prisoner in a car that I can't simply abandon in the middle of the road. At last, reaching a CRS (state police force) barrier blocking off the Boulevard Beaumarchais, near the Place de la Bastille, I wind down my window and ask the soaked cop if I can slip by like the other local residents. \"Your papers,\" he says wearily. I've just moved in, and I haven't got a driver's license or any ID with my new address on it. He's sorry, he can't take my word for it. I need proof of my new place of residence. I can't get home.\n\n* * *\n\nA little while later I write to the office in Nantes that issues copies of birth certificates to French citizens born abroad. When it sends me the document, I go to the police station nearest to my house, quai de Gesvres, armed with the necessary papers: the birth certificate they have asked for as well as my recently renewed identity card, valid for another seven years.\n\nA long queue. I take my ticket and wait for an hour and a half, long enough to look around at the people who have come to pick up IDs or passports and to hear the overworked clerks bluntly questioning the assembled supplicants. \"Madame, I must know whether or not you are from Guadeloupe!\" an old woman is asked in a tone that sounds a lot harsher than if she were asked, \"Are you originally from the Loire-Atlantique?\"\n\nAt last it's my turn. I take the papers out of my file. It is then that a man behind the counter is astonished to discover that I was born abroad. I tell him that since I was born in New York, my administrative papers had to come from the offices in Nantes. He then asks for my parents' birth certificates. I spare him their story: how they met after the war when my father had been demobilized from the Free French forces. I refrain from explaining that I was born in America by chance and stayed there for only two years before coming to France to spend the rest of my life here because my father couldn't find a job. I'm an inch away from trying to find excuses for being born outside French territory.\n\nOn the other hand, I am feeling a bit surprised by his insistence on asking for my parents' birth certificates. Besides, I add that on mine\u2014look, monsieur\u2014it clearly states that Anne S. is the daughter of Robert S. and Micheline R., both born in Paris, and that I'm therefore what's known as French by affiliation. I also hand him my identity card, issued three years ago and valid until 2017, which means that it's up to the administration to demonstrate that it is fraudulent, should it have any suspicion.\n\nBut he persists: the papers are necessary; there are new directives dating from 2009 for any citizen wishing to prove his \"Frenchness.\"\n\n\"Are your four grandparents French?\" asks the man behind the counter.\n\nFearing I may have misheard, I ask him to repeat the question.\n\n\"Your four grandparents, were they born in France, yes or no?\"\n\n\"The last time people of their generation were asked this kind of question was before they were put on a train to Pithiviers or Beaune-la-Rolande!\" I say, my voice choking with rage, as I name the French camps where Jews were locked up by the French collaborating police before being deported by the Nazis to the death camps.\n\n\"What? What train? What are you talking about? I must repeat that I need that document. Don't come back until you have it in your possession.\"\n\nHe dismisses me abruptly, pushing toward me my file, which by the purest coincidence is yellow, the very color of the star Jews had to wear on their clothes.\n\nNo point in giving a history lesson to a clerk to whom the Vichy laws mean nothing and to whom no one responsible for the new regulations has taken the time to explain that there are unfortunate turns of phrase, reminiscent of more troubled times, that might be best avoided.\n\nI leave, more hurt than angry with this draconian desk clerk, feeling that my birth is somehow suspect, as if there were two categories of French people, some more French than others. I'm also thinking about the absurdity of this situation, given that other officials, years ago, unaware of the doubts surrounding my origins, appointed me the model for their statue of Marianne, the symbol of France, worthy to take pride of place in their town halls.\n\nThis isn't just an administrative bore. It's the revival of an unhealthy debate about national identity that has been poisoning France in the last few years.\n\nThe incident calls to mind a memory from my youth. In the 1970s the Holocaust blew up in our faces, especially through discovery of the Vichy regime's involvement in the final solution. We might think of the famous interview in L'Express with Louis Darquier de Pellepoix, the general commissioner for Jewish questions, in which, from his exile in Spain, he stated without the slightest remorse that \"only lice were gassed in Auschwitz.\" This was the starting point for the inquiries and investigations led by the lawyer and author Serge Klarsfeld* into crimes against humanity, chiefly directed\u2014this was before the trial of Maurice Papon\u2020\u2014at Ren\u00e9 Bousquet, the general secretary of the Vichy police. It was a time when significant books on the subject were starting to be published, the first of them, Vichy France and the Jews, by the American historians Michael R. Marrus and Robert O. Paxton.\n\nWe had had to wait for the research at universities abroad to bring to light the role of the Vichy administration in the arrest and deportation of the Jews of France. It was the start of a great outpouring about the \"dark years\" and, in a seemingly parallel universe, the emergence of the revisionists, like Robert Faurisson, who was convicted several times in France for \"denial of crimes against humanity.\"\n\n* * *\n\nTwenty years before, my parents had\u2014as they used to say in those days\u2014\"done up\" an old farm in Seine-et-Marne, a hundred miles from Paris, that served as a weekend retreat. My father, who worked in the cosmetics industry, had been pleased to meet a colleague in the same village, Jean Leguay, who ran Gemey, now a company affiliated with the L'Or\u00e9al group.\n\nLeguay and my father played golf at Fontainebleau from time to time. Leguay often came to our house for coffee, with his wife, Minouchette, who, when I was a girl, represented for me all the snobbery of the Sixteenth Arrondissement. She claimed, in this little village of three hundred souls, to have wanted to repaint her house in \"Dior gray,\" a color that wasn't listed in the Valentine paint catalog, but whose name had a pretty ring for her. In short, while Minouchette might have been silly and vain, her husband was pleasant and intelligent. My father enjoyed his company, and as a child happy to go out with her dad, I often followed them as they walked the golf course. Leguay had the smooth pink face of people who sleep soundly at night. My mother, who was always concerned about my father's pallor, frequently mentioned Leguay as an example of someone exuding health and well-being, a man at ease with himself.\n\n* * *\n\nA few years before this national reexamination of the scale of collaboration in Vichy and the treatment of the Jews, Robert Laffont published a book by Claude L\u00e9vy and Paul Tillard titled La Grande rafle du Vel d'Hiv, about the massive July 1942 roundup at a Paris sports stadium where Jews were held in hideous conditions for weeks before being deported to Auschwitz. Nowadays that event is well known to the French, especially because of the speech delivered by Jacques Chirac on July 16, 1995, acknowledging France's culpability in the deportation of the Jews. Various books and a few films, including La Rafle (The Roundup), helped bring the story to public attention. But that hadn't yet happened in the late 1960s, when the publication of excerpts from L\u00e9vy and Tillard's book in the national press caused an uproar.\n\nThe excerpts concerned a certain Leguay, no first name given. The reader learned that Leguay had been the secretary-general of the Vichy police, Ren\u00e9 Bousquet's delegate in the Nazi-occupied zone of France. Since Leguay himself was a prefect, he was in constant correspondence with his colleagues about the practical problems posed by the arrest of the Jews. He also witnessed the roundups in July 1943, which he had helped organize, and directed the transfer of Jews from the zone libre to the Drancy internment camp.\n\nLike Bousquet, who had long enjoyed the protection of his political friends, such as Maurice Papon, the only senior Vichy official to have been put on trial over the last twenty years, Jean Leguay was a disreputable character whose crimes remained unknown for a long time, thanks to countless collaborators whose pasts came to light only much later. Besides, at that time I would have laughed at anyone who had told me that some twenty-seven years later a book by Pierre P\u00e9an titled Une Jeunesse fran\u00e7aise (A French Youth) would reveal, with the consent of its chief protagonist, the dark years of the man who later became President Fran\u00e7ois Mitterrand. At the Institut d'\u00c9tudes Politiques de Paris I had physically fought against the majos, the elite right-wing students who represented the majority of the school's students during the 1970s. Unlike us, the left-wing minos, these students maintained (rightly, alas) that Mitterrand had been awarded the Ordre de la Francisque, the highest honor given in Vichy France.\n\nSo P\u00e9an told this charged story about his old friends with murky pasts. But what startled me at the time wasn't so much the revelation of the dubious life of one Fran\u00e7ois Mitterrand, who had served the Vichy regime before becoming Fran\u00e7ois Morland and fighting with the Resistance, as the enduring nature of his dubious friendships, which he never denied. His links with Ren\u00e9 Bousquet, of course, confirmed by the president himself and attested to by photographs taken at Latche, Mitterrand's house in Les Landes in the southwest of France while he was financing his various political campaigns, were also alarming, as well as his closeness to Jean-Paul Martin, a former cagoulard\u2014a member of the French fascist organization La Cagoule\u2014for whose funeral, in 1986, Mitterrand, by then the president of the French Republic, had asked that the coffin be draped with the French flag.\n\nTo this day, I retain a sense of gratitude to former President Mitterrand for bringing the left back to power after twenty years and admiration for his tireless efforts on behalf of Europe. But by the time his past was discovered and even acknowledged by him, I had forever lost my faith in the sincerity of his moral and political commitments, and I was left with a powerful sense of betrayal. The indignation that I felt as my convictions about the past of the French nation were so cataclysmically overturned will never leave me.\n\nFor my father, the revelations about the Vel d'Hiv roundup were experienced as a searing pain, all the more excruciating for the fact that his own father, who had worn the yellow star before going into hiding under the name of Sabatier, had been denounced by the concierge of the building in which he had taken refuge with my grandmother. He had subsequently been arrested and interned in Drancy by the French police.\n\nHow could I fail, while bringing alive the story of my maternal family, to pay homage to my father's mother, Marguerite Schwartz?* In a wildly novelistic scene that I have never fully understood, she managed, thanks to a French officer with contacts in Drancy, to disguise herself as a nurse, borrow a Red Cross ambulance and some false papers, and get my paternal grandfather out of that antechamber to deportation. His health ruined by the long period of mistreatment he had undergone, he died just one year later\u2014but in his bed, rather than in the Auschwitz gas chamber, where the next convoy would have taken him.\n\n* * *\n\nMy father, that day in 1967, had difficulty believing that the official who had taken an active role in those deportation-related activities was the same Leguay with whom, the previous weekend, he had shared a friendly cup of tea.\n\nArmed with a photocopy of a letter from the Leguay in question, addressed to the Germans and found at the Center of Contemporary Jewish Documentation (CDSE; now part of the Shoah Memorial in Paris), my father went to the headquarters of the French Society of Perfumers and asked the chairman to show him a business letter signed by Jean Leguay, the president of Gemey. As he reviewed this document, my father turned very pale: the two signatures were identical. My father then told the chairman what he knew about this character and demanded that he be thrown out of the association. He was met with embarrassed refusal on the part of the chairman. It was not very courageous, and the times were not yet attuned to these injustices, the French lagging well behind the Germans in their desire to achieve transparency about their past. In those years the desire not to \"create a scandal\" outweighed all other considerations.\n\nAfter resigning from the professional association, my father wrote to Leguay to tell him what he knew about his past and asked him to walk on the other side of the road in their village, so that he would never run into him again. Leguay responded by sending my father the ruling by the High Court of Justice that had cleared him in 1949, as it had Bousquet and many others.\n\nOn a side note, during those years, Gemey was bought by L'Or\u00e9al, a company known for its recycling of notorious collaborators. These included Jean Filliol, who had tried to assassinate the Jewish French prime minister L\u00e9on Blum before the war and who had, after the liberation, taken refuge in Spain, where he ran the Spanish branch of L'Or\u00e9al. Filliol had been sentenced to death in absentia for having been a member of Joseph Darnand's militia and for facilitating the horrific Nazi revenge attack on the town of Oradour in 1944. Another senior executive with L'Or\u00e9al, Jacques Corr\u00e8ze, coincidentally lived in the same building as Jean Leguay in Paris at rue de R\u00e9musat, and had been an officer in Eug\u00e8ne Deloncle's fascist Cagoule organization, which was financed by Eug\u00e8ne Schueller, the father of Liliane Bettencourt, the wealthy socialite and major shareholder of L'Or\u00e9al.\n\nIn 1941, Jacques Corr\u00e8ze joined the Legion of French Volunteers (LVF) against Bolshevism, which fought alongside the Charlemagne Regiment, the Waffen-SS division that included Frenchmen who had decided to fight in the Waffen-SS uniform. Sentenced to ten years' imprisonment in 1948, he was freed a year later and was immediately hired by Schueller to become the chief executive of L'Or\u00e9al in America. Amnestied in 1959 and rehabilitated in the 1960s, he died in Paris in 1991, while the American Office of Special Investigations was investigating his possible involvement in crimes committed during the war.\n\nThe recent Bettencourt affair,* which has nothing to do with the above, did recall Schueller's past and put these episodes from the history of the founder of L'Or\u00e9al in the spotlight once more.\n\n* * *\n\nThe dossier painstakingly compiled by Serge Klarsfeld enabled the justice system to find Jean Leguay guilty of crimes against humanity. I remember taking my father along to the press conference at which Klarsfeld argued that the legal proceedings to charge Bousquet and Leguay with crimes against humanity were fully justified. This was in 1979. My father told me, as he left Klarsfeld's office, \"You'll see, he'll die after me, peacefully, in his sleep.\" Indeed, my father, who was the same age as Leguay, died the next year, while Leguay died in 1989, but before his trial could begin. According to the ruling that stated that legal action had been abandoned, \"there was information to establish that he had taken part in crimes against humanity.\"\n\n* * *\n\nMy mishap at the police station was pretty harmless in the scheme of things, but the questioning of my identity brought a tidal wave of family memories surging forward. For years I had refused to listen to the stories of the past told over and over again by my mother. Not out of a desire to reject my family, but the story of my maternal grandparents, even though I thought I knew it, never felt as if it belonged to me, as if it related to my life. It even bored me a bit. What I liked was politics, journalism; my father's world rather than my mother's. My father, who had joined the Free French in the Middle East during the war; my father, who, under the name of Jacques Breton, had delivered editorials on Radio Beirut on behalf of General Charles de Gaulle; my father, so proud to show me the agency dispatch in which Joseph Goebbels had condemned him to death and railed against \"the Jew Sinclair\"; my father, having returned to Paris after the liberation, paying a final visit to his own father, who had been seriously ill since Drancy. Even though my father himself built an industrial career as a business executive far from my own areas of interest, I felt closer to the war stories he recorded in his notebooks than I did to my mother's side of the family, which lived under the shadow of my art dealer grandfather, who had died when I was only eleven years old. In short, I secretly felt I was on the same side as \"My Father the Hero,\" who gently mocked \"My Mother Who Sat Out the War on Fifth Avenue.\"\n\nMy father, Robert Sinclair, who was called Robert Schwartz throughout his youth, was sent to the front in 1939 as a thirty-year-old soldier, on meteorological duty. He was stationed at a border post (might it have been the Maginot Line?) and played chess, one move per day with a colleague who had been sent to a different strategic location, taking advantage of their daily call to compare weather conditions on the front. They sat there and waited for the enemy, who never came because they had decided to avoid that predictable line of defense. (I like to imagine him moving his rook or his knight, occasionally sticking his hand outside and saying, \"It's raining,\" to his friend, who would reply, \"Here too!\") When he was finally demobilized, he returned to Paris and, like many others, wept at the sight of flags bearing the swastika fluttering over the Champs-\u00c9lys\u00e9es. He remembered the day he had stood there with his mother, on November 11, 1918, applauding Marshal Ferdinand Foch's troops as they celebrated the victory of the First World War. He was just a nine-year-old boy, but he told me he knew that he was destined to enlist from that time.\n\nUnaware of the networks that would have enabled him to pass through England, he managed to reach the United States via a series of complicated routes, and it was there that he enrolled in Free France, which ultimately sent him to Damascus, Beirut, and Cairo. Before boarding the ship bound for the Middle East via the Atlantic and the Indian Ocean, all lights extinguished so as not to alert the enemy, he was told that the Germans were aware of the surnames of French officers who had enlisted with de Gaulle and whose families had stayed in France. To protect his relatives, he was compelled to change his name. Wanting to retain his initials, he opened the New York City phone book to the letter S and stumbled upon the name Sinclair, perhaps no more unusual in the United States than Martin or Dupont in France.\n\nI have always been a bit irritated with him for wanting to keep the name Sinclair and then legally adopting it as his surname after the war. It meant losing a part of our identity. But he had earned a name for himself under that nom de guerre; he bore it proudly and probably wanted to allow his descendants\u2014me, as it happened\u2014to avoid the dangers that a Jewish name had inflicted on his family. This was not unusual among those traumatized by the war in the years that followed the liberation, but I confess that I've always experienced it as a sort of denial. That's probably why I laid claim to my Jewish identity very early on. And why I've been distressed by those who, playing with proportional representation, allowed the extreme-right Front National (FN) to exist politically in France. It's why I fought bitterly against the media access so generously granted to the FN in the 1980s and why for ten years I refused to have Jean-Marie Le Pen on my television program, 7 sur 7, which was a discussion of the previous week's political news. The pointlessness of this battle became apparent on April 21, 2002, and in the years that followed, when Le Pen came in second in the general election, the consequences of which we are still living with today.\n\nSo much for rummaging around in the cardboard boxes of family archives. As I went through all those random papers, I eventually came across my original birth certificate, rather than the copy generally required by the administrative services. What would the clerk at the prefecture, who prompted this book, after all, have said if he had seen that I had been born Anne Schwartz, dite Sinclair, and that my name was only officially changed in 1949, when I was one year old?\n\n* * *\n\nIn my youth I was more receptive to the story of my paternal grandparents, who had stayed in France, than I was to the fate of those who, pursued by the Nazis, had managed to flee and were then dispossessed, plundered, and stripped of their nationality. Besides, I wanted to build my own life, preferring television to art galleries, the public life to the artistic one, old newspapers to old paintings.\n\nIn 2006 my mother passed away. And as always after the death of a parent, you're struck by all the things you've neglected to ask or didn't want to know, whether out of laziness or weariness at hearing the same stories again and again. In my mother's flat, I emptied cupboards crammed with dusty memories: old keys, outmoded furs, family photographs, and stacks of papers that had accumulated over the course of decades.\n\nThen I turned sixty and happened to spend a few years in the United States, a country that constantly brought me back to my childhood and to the part of the family that had sought refuge there. And here were the French authorities, playing with dangerous ideas, reminding me that French nationality can't be taken for granted even if you've had it all your life. How fragile it is to those who bear it and how inaccessible to those who wish to lay claim to it. And reminding me that it wasn't the first time this had happened in my family.\n\nI realized I hadn't even had time to unpack the boxes from my mother's apartment, which I'd stacked in a closet. They were full of letters and old files that I'd picked up without even giving them a thought. Suddenly unable to contain my curiosity, I plunged into the family archives, in search of the story of my past. To find out who my mother's father really was: my grandfather Paul Rosenberg, a man hailed as a pioneer in the world of painting, of modern art, who then became a pariah in his own country during the Second World War. I yearned to fit together the pieces of this French story of art and war.\n\nI am the granddaughter of Paul Rosenberg, a gentleman who lived in Paris and who owned a gallery at 21 rue La Bo\u00e9tie.\nRUE LA BO\u00c9TIE\n\nNumber 21. I've passed by it hundreds of times. My mother liked to show me the 1930s fa\u00e7ade with its stone arches. I'd noticed various shops on that street\u2014ice cream, pizza\u2014but I'd never stopped to take a closer look. Now, seventy years after my grandfather had left the premises, I wanted to see the building for myself. I couldn't imagine that three years later I would unveil a plaque on this very building that I had not yet entered.\n\nToday it's an office of the Veolia Environmental Services company. I call them up: \"My grandparents used to live there. I'd love to take a look around, really just a look. I don't want to disturb you... It was before the war, I'm sure there are few traces left... Of course I understand if it's not possible.\" I detected the ambivalence in my own voice. It was almost as if I worried that they might actually let me in.\n\nThey did. Why would they resist? So one Wednesday in April 2010 off I went to Veolia, to 21 rue La Bo\u00e9tie, where I begin my story. Touched by my curiosity and possibly a bit incredulous that it's taken me to the age of sixty to set foot in the building where my grandfather's gallery was located, my hosts graciously show me around.\n\nThe hallway has been divided, and there are white stucco columns with Corinthian capitals, which I find a bit tasteless. Are they original? And a black-and-white damier marble floor. It's all been redesigned, modernized, the rooms, the spaces. There are spotlights affixed to the ceiling. The staircase with its old-fashioned banisters leading to the upper floors seems unchanged. Lots of Fernand L\u00e9ger's and Andr\u00e9 Masson's paintings used to hang on the walls of this interior stairway, which led to my family's private apartments: the one belonging to my grandparents and their children, then the one to my great-grandmother, Paul's mother, Mathilde Rosenberg. Of course no paintings now hang in this stairway, which leads to various offices. The overall impression is dreary. Yet the elevator is modern, surely in compliance with health and safety regulations. The rattling old cage of another age is gone.\n\nThe stairway within the gallery, the one with the cast-iron banister, seems to have retained its original look, from the 1930s, when my grandfather did some elaborate renovations. The floor is patterned with marble mosaics made with yellow stones. But there's no way of telling exactly where the mosaic plaques went, the ones designed by Georges Braque, who also supervised their installation. Above the stairs were arches, replicas of the ones outside, adorned with pieces of mirrored glass.\n\nI'm in the lower of the two exhibition halls, the one that appears in so many of the photographs I've seen of my grandfather situated in his domain. All the exhibitions at rue La Bo\u00e9tie were held in this large room. A month of Braque, another of Henri Matisse, a third of Pablo Picasso. It is now a boardroom for Veolia executives. The fine oak parquet floor is still there, and I immediately recognize the wood paneling, which I've seen in the photographs, as well as the glass ceiling with its little star-shaped windows, which, as in other galleries of the time, diffused the light so as to soften the hard edges of cubist painting.\n\nIf I half closed my eyes I could see them, those big paintings from the 1920s and 1930s, hanging on the walls. Soon after, those masterpieces would be replaced by portraits of the head of the Vichy government, Marshal Philippe P\u00e9tain.\n\n* * *\n\nIn 1927 E. T\u00e9riade, a famous critic and art publisher of Greek descent, described the Galerie Rosenberg in \"Feuilles volantes,\" the monthly supplement of the influential journal Cahiers d'art: \"We are introduced into a huge room, high-ceilinged, bare walls, naked light, a room in which sober brown curtains weigh down on the collection, in which two solitary armchairs upholstered with dark velvet reach toward you like two grand inquisitors; no, they're not reaching toward you, they're going for your throat, as masterpieces do. Hurricanes of solitude, of austerity, pass through the room... Paul Rosenberg: he's dressed in black. He has the anxious face of an ascetic or a passionate businessman.\"\n\nHere's another description of the setting, particularly interesting when you consider that the author is the notorious, extreme-right-wing writer Maurice Sachs, who later defined himself as a Jew, a homosexual, and a collaborator before being killed by a bullet to the back of the head by the Germans in whose service he had worked: \"His grand seigneur bearing was part of his particular genius... You step into Rosenberg's gallery as if entering a temple: the deep leather armchairs, the walls lined with red silk, would lead you to think you were in a fine museum... He knew how to cast an extraordinary light on the painters he took under his wing. His knowledge of painting was deeper than that of his colleagues, and he had a very sure sense of his own taste.\"\n\nPaul, who had taken over his father's gallery with his brother, L\u00e9once, in 1905, decided to set up on his own in 1910 and moved alone to 21 rue La Bo\u00e9tie, in the Eighth Arrondissement of Paris. Nineteenth-century works were shown on the mezzanine; contemporary art, on the ground floor. If visitors were unsure about Braque or L\u00e9ger, Paul invited them upstairs to see softer-contoured works by Edgar Degas, Pierre-Auguste Renoir, or Auguste Rodin. He hoped they might buy some of these, which would allow him to support his unknown friends, such as Picasso or Marie Laurencin, the muse of the poet Guillaume Apollinaire. In 1913 she became the first artist to sign an exclusive deal with Paul, an arrangement that stood until 1940. She was joined by Picasso in 1918, Braque in 1923, L\u00e9ger in 1926, and Matisse in 1936.\n\n* * *\n\nIn 1912, almost as soon as he had moved in, Paul sent out an announcement just as anyone opening a shop might do, describing his new venture: \"I will shortly be opening new modern art galleries at 21 rue La Bo\u00e9tie, where I plan to hold periodic exhibitions by the masters of the nineteenth century and painters of our own times. In my view, however, the shortcoming of contemporary exhibitions is that they show an artist's work in isolation. So I intend to hold group exhibitions of decorative art... Not only do I plan to offer my spaces for free, I shall not take a percentage in the event of a sale. For each exhibition I shall publish at my own expense a catalog of the paintings, sculptures, furniture, etc.\"\n\nThe critic Pierre Nahon stresses Paul's desire to establish a connection between French painting of the past and the modernist trends of the twentieth century, noting that in the late 1930s Paul had on his walls and in his inventory a collection of G\u00e9ricault, Ingres, Delacroix, Courbet, C\u00e9zanne, Manet, Degas, Monet, Renoir, Gauguin, ToulouseLautrec, Picasso, Braque, L\u00e9ger, Le Douanier Rousseau, Bonnard, Laurencin, Modigliani, and Matisse. \"The gallery,\" Nahon writes, \"is becoming an essential meeting place for everyone who wants to follow the development and the work of the innovative painters.\"\n\nMy own research is centered on an attempt to conjure the grandfather I barely knew. And to summon up the riches heures of the thirties and the grim ones of the forties that are integral to his story.\n\n* * *\n\nMy grandfather had great difficulty regaining possession of his gallery after the war. The state had confiscated the building from the collaborators in August 1944 and made it the headquarters of the Saint-Gobain construction company, before finally returning the building to my grandfather. By then it had endured the sinister events that I am about to relate, events with which my grandfather could never make peace. Paul finally sold 21 rue La Boetie in January 1953. He was determined never again to live in that place, its basement filled with propaganda from the darkest years, its rooms still haunted by the ghosts of the occupation.\n\nFor a long time the building was home to the French General Information Service, Renseignements G\u00e9n\u00e9raux, the French police intelligence service, and the secrets of the Republic were buried with the secrets of the collaborators.\nNUMBER 21 UNDER THE GERMANS\n\n21 rue La Bo\u00e9tie was piled to the rafters with those \"accursed\" or decadent works, the kind that the Nazis called entartete Kunst (EK), \"degenerate art.\" The term referred to any art that, for the new German regime, departed from the canon of what the Nazis considered traditional.\n\n\"German people, come and judge for yourselves,\" said Adolf Ziegler, the president of the Reich Chamber of Visual Arts, as he infamously opened the Munich exhibition of degenerate art on July 18, 1937.\n\nThis vast exhibition of six thousand works, taken from every museum in Germany, was hastily assembled. The intention was to ridicule modern art before imposing a ban on its sale. These works were deliberately shown among drawings by children or the mentally handicapped: there were two adjacent halls, with official German art hung in the first and the art identified as \"degenerate\" (Picasso, Braque, Matisse, L\u00e9ger, Mir\u00f3, Masson, Dal\u00ed, Chagall) exhibited in the second. Many of the works shown in the second hall had been confiscated from museums or private galleries mainly managed by Jews. Some were intentionally destroyed, while others were auctioned off for the benefit of the Nazi regime. Ironically, this attempt to ridicule modern art was to the great advantage of art lovers throughout the world. Vincent van Gogh quickly became the bestselling \"degenerate\" painter on the market. By the time the Reich Chamber exhibition closed on November 30, 1937, it had drawn more than two million viewers.\n\n* * *\n\nJoseph Goebbels, the propaganda minister, had planned the show as a counterpoint to the Great Exhibition of German Art, which opened simultaneously in Munich. It celebrated female farmers and soldiers, brave mothers, and rural landscapes of Greater Germany. In Goebbels's words, a distinction had to be made between \"the art of those days and the art of these days.\" He felt that German museums had to be cleansed of works produced after 1910.\n\nThe German rejection of novelty in art was nothing new. As Lynn H. Nicholas explains in her remarkable book The Rape of Europa, the antimodern tradition had a long history, \"reaching back to Kaiser Wilhelm's 1909 firing of Hugo von Tschudi, director of the Nationalgalerie, for buying Impressionist paintings.\" In 1893 a very influential book was published by the Jewish social critic Max Nordau, who first used the word Entartung, \"degeneracy,\" to refer to artistic disciplines. In his most famous book, Degeneration, Nordau describes modern art as symptomatic of the degeneracy of society at the end of the nineteenth century. He declares all modern art, including that of the impressionists, \"pathological.\" Nordau was both a Zionist and a Dreyfusard and a man developing conservative ideas about the founding principles of German culture. In the 1920s a group of philosophers put forward the concept of \"degenerate art\" on the basis of Nordau's work, at the risk of somewhat misrepresenting his ideas.\n\nAfter Hitler came to power in 1933, many artists chose to go into exile. Not only could they no longer show their work or sell it, they were forbidden to buy brushes, canvases, or paint. \"The smell of turpentine in the air or a container of wet brushes was grounds for arrest,\" writes Nicholas.\n\nOn June 30, 1939, just weeks before the outbreak of war, the Germans held a massive auction in Lucerne, featuring 126 paintings and sculptures from the most important museums and private collections in Germany. Many collectors, unable to resist the temptation to buy outstanding works of art at low prices, attended. Paul warned potential buyers that any currency the Reich harvested from this sale \"would fall back on our heads in the form of bombs.\" Alfred Barr, the director of the prestigious Museum of Modern Art in New York, also tried to alert those museums that had announced their intention to buy. But to no avail. \"Acceptance of these warnings was not made easier by the very mixed reception all modern art had endured for many years,\" writes Nicholas.\n\nFrom that moment Karl Haberstock, the Nazis' chief art buyer, became one of the F\u00fchrer's personal dealers. As Haberstock began to amass a collection of old masters for Hitler, he found intermediaries in France through whom he could purge all modernist impurities. Among them was the author and Nazi apologist Lucien Rebatet, who proposed the \"Aryanization\" of our fine arts.\n\nThere was a great deal of debate on this subject among Nazi officials, in particular between Goebbels and Alfred Rosenberg (Hitler's ideological theorist, who later was placed in charge of the \"occupied Eastern territories\"\u2014in other words, the massacres that took place there). This unfortunate namesake of Paul's considered any form of physical distortion on a canvas \"degenerate art,\" while Goebbels believed that modern painting could become part of a National Socialist revolutionary art movement. As in any totalitarian regime claiming to define a \"new man\" and a new world order, art was a priority for the apostles of National Socialism. Indeed, the Nazis were obsessed with the idea of turning art into an instrument of propaganda. In her book L'Art de la d\u00e9faite (Art of the Defeat), Laurence Bertrand Dorl\u00e9ac relates how, several days after the armistice between Germany and the French Third Republic, the looting of artworks began on a massive scale. In fact, on June 30, 1940, Hitler issued an order to put artworks belonging to Jews in \"safekeeping.\" The term was chosen deliberately as a cover for what could only be described as theft. It was then that Alfred Rosenberg set up the Einsatzstab Reichsleiter Rosenberg (ERR). This became the chief organization in the Nazi looting operation, which put its stamp of infamy on all works of art confiscated by the occupying troops.\n\n* * *\n\nFrom early July 1940, Rosenberg instructed the army to raid the big Parisian art dealers and seize their collections. This represented the triumph of the Rosenberg-G\u00f6ring clan over the tribe based around Joachim von Ribbentrop, the Reich foreign minister, and Otto Abetz, Hitler's ambassador to Paris during the Second World War. And as we know, G\u00f6ring was immoderate about helping himself.\n\nFrom October 1940, organized theft followed upon random robbery. \"The artworks were first assembled at the Mus\u00e9e du Jeu de Paume and the Louvre, then photographed, valued, recorded, and wrapped ready for transport to Germany,\" writes Dorl\u00e9ac. Naturally, this contraband included both the classical paintings from the Parisian galleries and modern works, which, as Dorl\u00e9ac puts it, served as \"bargaining chips for pieces more in line with the Nazi aesthetic.\"\n\n* * *\n\nIn her classic account, Le Front de l'art, Rose Valland, the heroic protector of French artworks, relates that at the height of the war in 1943 she witnessed a column of smoke rising from the terrace of the Tuileries; it rose from paintings stamped with the letters EK (entartete Kunst), and signed Masson, Mir\u00f3, Klee, Ernst, L\u00e9ger, Picasso. \"The men of the ERR planned to attack these paintings, run them through with swords, slash them with knives, and carry them to the pyre, as in those gigantic autos-da-f\u00e9 that had taken place in the German museums, in a bid to destroy those works identified as 'degenerate.'\"\n\nValland was one of two people who tried to keep works of art from museums or private collections from being scattered across Germany. In this saga of art saved from the Nazi madness, the other hero working in the shadows was Jacques Jaujard, the director of the National Museums at the time, and the director of the Mus\u00e9e des Beaux-Arts after the war. It was he who suggested that the Germans draw up an inventory under Valland's direction. In Le Front de l'art, she tells how she managed to remain in her post at risk of her life in order to create a precise inventory of the stolen paintings. Rising to the post of captain in the French Army, she was sent to occupied Germany after the war to help France recover its stolen property.\n\nThis property came into consideration in the Nuremberg trials. Certainly, compared with the atrocities perpetrated upon human beings, the looting of art in Nazi-occupied territory seemed negligible. Still, the court considered it a war crime, on the ground that by attacking a culture, the Nazis were trying to destroy a people.\n\n* * *\n\nConsistent with their plan, as soon as the Nazis occupied Paris on June 14, 1940, they made their way to 21 rue La Bo\u00e9tie. But they were disappointed not to find the family patiently awaiting their arrival.\n\nOn July 4, 1940, the Reich ambassador, Otto Abetz, demanded that the building on rue La Bo\u00e9tie be sequestered by the police and that the artworks be seized. He had in fact just drawn up a list of Jewish dealers or collectors for the Gestapo: Bernheim-Jeune, Alphonse Kann, Jacques Seligmann, Wildenstein, and Paul Rosenberg.\n\nThis outrage continued with the German requisition of rue La Bo\u00e9tie in May 1941. On the eleventh day of that month, the brand-new Institut d'\u00c9tude des Questions Juives (IEQJ, Institute for the Study of Jewish Questions), was installed in the building with great pomp.\n\nI've examined the few existing pictures of that installation, and more particularly, I've listened to Radio Paris on tapes supplied by the National Sound and Video Archives. The quality of the recording is excellent, with the nasal voice and wounding words of the speaker unmistakably clear: \"Today saw the rechristening of the building previously occupied by Rosenberg; the name alone tells you all you need to know.\"\n\nThe ceremony opens with remarks on the \"disastrous moral influence of Judaism,\" delivered by Cl\u00e9ment Serpeille de Gobineau, a descendant of the more famous Arthur Comte de Gobineau, the author of the 1853 An Essay on the Inequality of the Human Races.\n\nIn the photographs and in the National Sound and Video Archives, you can see Louis-Ferdinand C\u00e9line, a star guest with impeccable far-right credentials, parking his bike in front of my grandfather's gallery, on which the name of that formidable new office stands out in capital letters. The porch and the famous exhibition hall are easily recognizable. A huge panel on the wall shows a woman on the ground covered with a French flag, a vulture perched on her belly, with the caption \"Frenchmen, help me!\"\n\nIn the exact place where my grandfather had hung paintings by Renoir, Picasso, and L\u00e9ger over the previous few years, a tricolor flag, a portrait of Marshal P\u00e9tain, and quotations from \u00c9douard Drumont, the author of La France juive, who, according to commentary of the time, \"first raised the issue of the Jewish problem in all its magnitude\": \"The Jews came poor to a rich country. They are now the only rich people in a poor country.\"And that other quote on the opposite wall: \"We are fighting the Jews to give France back its true, its familiar face.\"\n\nCapt. Paul S\u00e9zille was soon appointed secretary-general of the institute, a post he held until December 1942. He was a former right-hand man of the anti-Semitic activist and far-right politician Louis Darquier de Pellepoix and his prewar Anti-Jewish Union. A retired officer of the Foreign Legion, S\u00e9zille was, according to the historian Laurent Joly, a man drowning in booze and vitriol. \"He was considered one of the most grotesque figures in anti-Semitism between 1940 and 1944, trying to give voice to a healthy France as it seeks to regain its true soul,\" writes Joly.\n\nHe was followed shortly afterward, in January 1943, by the physician, anthropologist, and racial theorist George Montandon, who remained in office until the last days of August 1944, just before the liberation of Paris. The institute then assumed the name Institut d'\u00c9tude des Questions Juives et Ethno-Raciales (IEQJER, Institute for the Study of Jewish and Ethno-Racial Questions). From that date, the Germans wanted to make the institute appear to be what we would now call a research center with the creation of six educational courses, including \"Ethnoraciology,\" taught by Montandon himself, \"Eugenics and Demographics,\" and \"Judeocracy.\"\n\n* * *\n\nFrom the outset, the Institute for the Study of Jewish Questions, established in my family house, was an association created in accordance with the French Associations Law of 1901 and was devoted to anti-Semitic propaganda. Founded in May 1941, and cofinanced by the German Embassy and the Gestapo, it was not dependent on the Commissariat G\u00e9n\u00e9ral aux Questions Juives (set up by the Vichy government and run first by Xavier Vallat, then by Darquier de Pellepoix) but was in a direct line of command from the office of Otto Abetz. It was also controlled by \"specialists\" from Germany, including a certain Dr. Schwarz, a representative of an anti-Jewish institute in Frankfurt.\n\n* * *\n\nThe IEQJ was in fact directed by Theodor Dannecker, the head of the Jewish Section (Judenreferat) of the Gestapo. Apparently, he had little confidence in the Vichy administration and wanted to set up\u2014under the cover of a seemingly French organization effectively run by the Nazi services\u2014an organization of anti-Semitic propagandists answerable to him alone. According to Joseph Billig, in his three-volume work devoted to the General Commissariat for Jewish Questions, \"The 'final solution of the Jewish question' was from the very start in the hands of Dannecker's Judenreferat. The Judenreferat considered that it had been promised 'supreme power' over the Jews in France in the future... It was not primarily concerned with Jewish property. Its focus was the Jewish masses themselves. While awaiting the deportations, it organized the Jews into ghettos and prepared the raids.\"\n\n* * *\n\nSecretary-General S\u00e9zille\u2014was he sitting at Paul's desk?\u2014took his orders only from Dannecker, whom he called, in the German style, \"mein Leutnant.\" He often asked the Propagandastaffel to support his private militia. He denounced \"the spirit of indecision and the inadequate application of the [German] orders by the Commissariat for Jewish Questions.\" And he had no qualms about writing to Dannecker to thank him for the order requiring all Jews to wear the yellow star.\n\nThough it was an organization under Nazi supervision, S\u00e9zille nevertheless sent the press a communiqu\u00e9 on August 21, 1941, affirming that the IEQJ \"is an eminently French association, in accordance with the law of July 1, 1901, consisting of resolutely anti-Jewish men of good will... determined to resolve, at all cost and by all means, the Jewish question in France.\"\n\nThe institute's mission was to spread propaganda, and to collect letters of denunciation and ensure that they were \"followed up.\" In a letter of January 31, 1942, addressed to Xavier Vallat, S\u00e9zille boasted of having thirty-three thousand members and seventy thousand signatures in his visitors' book. The institute published its journals, Le Cahier jaune and La Question juive en France et dans le monde (The Jewish Question in France and the World), and it put on its pi\u00e8ce de r\u00e9sistance, the exhibition The Jew and France at the Palais Berlitz in 1941. Otto Abetz later claimed that it had been organized by the Nazis themselves, but under the cover of the IEQJ for the benefit of the public. Which is to say that the offices at 21 rue La Bo\u00e9tie were working full tilt to organize the exhibition in time.\n\nI went along to 30 boulevard des Italiens, to the Palais Berlitz, to see what remained of that space. But the walls are silent now. They've been replaced by the chain Bistro Romain and a multiplex cinema.\n\nThe cover of the September 6, 1941, issue of L'Illustration is well known. It reproduces the official poster of the exhibition, described by the magazine as a \"large allegorical composition showing a kind of long-bearded vampire with thick lips and a hooked nose, with bony fingers like the claws of a bird of prey clutching a globe.\"\n\nIn the cinemas, audiences watched news reports devoted to the famous exhibition.* The commentary accompanying the pictures is, like everything else, difficult to listen to, even more so sixty years on: \"Out of every one hundred Frenchmen of old stock, ninety are true whites pure of any other racial mixture. The same cannot be said of the Jews. They are the product of racial mixing that occurred several millennia ago, between Aryans, Mongols, and Negroes. Consequently the Jew has his very own attitudes, gestures, and physiognomy. It is comforting to see the French going to see this exhibition. Henceforth they will be able to identify the Jew and protect themselves against his actions.\"\n\nIn this terrifying exhibition, life-size portraits, in black and white, are arranged like targets at a shooting gallery, with a picture of the former prime minister L\u00e9on Blum at the center. Below each portrait is the individual's name with a ribbon identifying his nationality followed by a question mark\u2014\"French?\"\u2014and the invariable exclamation \"No, Jewish!\"\n\nSome five hundred thousand tickets to the exhibition were sold. Counting half-price entrants, there were a million visitors in Paris before it traveled to other French cities, including, for a time, Bordeaux, Nancy, Marseille, Nice, Cannes, Toulouse, and Lyon, meaning that it also went to the unoccupied zone. History tells us little about whether people came out feeling informed and convinced or indignant and repelled.\n\n* * *\n\nVarious odd characters anonymously frequented the offices at 21 rue La Bo\u00e9tie. Others, more famous, sometimes complained that they hadn't been treated very well. On October 21, 1941, S\u00e9zille received a letter of rebuke from C\u00e9line, who was \"a little hurt not to see in the bookshop of the exhibition] either one of his recent books: Bagatelles [pour un massacre] or L'\u00c9cole [des cadavres],[* while there was a flurry of insignificant little books... I observe once again the lamentable shortage (so sensitive in this case) of intelligence and Aryan solidarity.\" S\u00e9zille replied three days later: \"I am myself terribly sorry not to have been able, in spite of all our requests of the publishers, to acquire the books of which you speak and which, I know, are ideally suited to wage the anti-Jewish struggle. But I wish to inform you that we have already had for sale in our shops large numbers of Beaux draps and Mea culpa [two other anti-Semitic tracts by C\u00e9line], and that these two books continue to be requested on a daily basis. Please believe me when I say that we have always done and will continue to do the impossible to distribute your works and make sure they have their rightful place.\"\n\nWho was this man, Capt. Paul S\u00e9zille, who was lucky enough to die on April 20, 1944, four months before the liberation of Paris? What hatred inspired him, what blindness afflicted him, what bitterness had he suffered to run this vile organization and publish his shameful books? After the liberation, my grandparents were stunned to discover whole cases of books published by the institute in the cellar of the building. Unfortunately, the notion of \"bearing witness,\" of the \"obligation to remember,\" that spread through France in the 1990s had not yet taken hold, and my grandparents, rather than keep the archives, got rid of that library of shame at the first opportunity.\n\nI kept for a long time the sole survivor of this collection, a book by Captain S\u00e9zille himself, whose oeuvre once filled the basement of rue La Bo\u00e9tie. And then, through the various comings and goings of the Rosenberg and Sinclair families, this literature and the trail of the sinister captain disappeared.\n\n* * *\n\nDuring the refurbishment of his gallery, which took several years and wasn't completed until 1934, Paul asked Picasso to make some marble patterns to be inlaid into the tile floor. Giving him lots of sketches in the hopes that Picasso would create something unique, he first asked him for his designs in August 1928. But since Picasso never met deadlines and took a lot of persuading to carry out any commission, Paul ended up commissioning Braque to complete the project. In each of the four corners of the gallery, Braque created a rectangular marble mosaic, faithfully scaled-down copies of four of his large still lifes: pitchers, plates, lemons, cutlery, and tablecloths well known in his paintings. It was no longer the cubist period\u2014gray, green, and brown\u2014when Braque and Picasso were painting similar pictures with the eternal guitar and the front page of Le Journal. So similar that partly out of a spirit of mischief and partly because they themselves no longer knew who had painted what, the paintings were signed arbitrarily.\n\nThe still lifes in question on the floor of my grandfather's gallery were brighter, more colorful, and more luminous than the works of that period. They lent themselves to mosaic treatment, recalling the designs on the floors of the patrician Roman villas in Pompeii or Volubilis.\n\nAfter the war, when Paul sold the building he no longer wanted to live in, he had Braque's four marble mosaics cut out of the floor and made into low tables, framed in black marble. I lived alongside two of those tables throughout my youth and often stroked the marble, unaware of the innocent people, denounced and arrested, who had stepped upon them before being handed over to their executioners. The family house on rue La Bo\u00e9tie would have sheltered the executioners. I have never been able to watch Henri-Georges Clouzot's masterpiece The Murderer Lives at Number 21, without thinking about this.\nFLOIRAC\n\nFrom the earliest days of Nazism, Paul rejected the regime with every fiber of his being. He actively opposed the sale by the German government of \"degenerate art.\" And as the president of the SNA, the French association of dealers in fine art and antiques, he tried to persuade his colleagues across Europe to boycott the sales. Yet few people resisted the often exceptional paintings cast onto the market in this way. But Paul would not relent. \"Not a cent to the German Reich\" was the slogan for a small group that saw masterpieces acquired by less scrupulous dealers disappearing before their eyes.\n\nThe Germans didn't forget Paul Rosenberg. In fact, they blacklisted him.\n\nHe had thwarted them to some extent, sending a number of works to safety in London and New York and lending others to American museums, notably to the Museum of Modern Art for the first big Picasso retrospective, which my grandfather himself had put together during several months in New York with his friend Alfred Barr in 1939. Not surprisingly, in August of that year, my grandfather wrote to Picasso from \u00c9vian speaking of \"dark events\" as an inevitability.\n\n* * *\n\nOn September 3, 1939, the day war is declared, my grandfather is with his family in the Touraine, near the Loire River, at Cinq-Mars-la-Pile. He closes his gallery and, for fear of bombing raids, takes some of his paintings to Tours. There he stores them under the name of his chauffeur, Louis Le Gall. These would be the first paintings recovered after the war because neither the Nazis nor the French authorities were aware of their existence.\n\nThen the whole family leaves for Bordeaux, where, on February 7, 1940, they rent a house, Le Castel, at 12 Route de la Tresne, in Floirac La Souys, three miles east of Bordeaux. Le Castel belongs to a couple named Ledoux, who continue living on the first floor despite the presence of the Rosenbergs. They take over the whole house again after the war and sell the property to the town council during the 1960s.\n\n* * *\n\nI'd never been to Floirac before and wanted to visit the house that I'd seen only in photographs. It was, after all, where my family spent the beginning of the war.\n\nThe Garonne River is gray and overcast that morning in September 2010. After arriving at Bordeaux M\u00e9rignac Airport, I cross the river toward Floirac and begin to search for the route de la Tresne, as it is spelled on the family's ration cards. I imagine that the street has been renamed several times by now and soon discover that ever since the socialist council was elected, it's been called avenue du Pr\u00e9sident Fran\u00e7ois-Mitterrand. Of course...\n\nEventually I find Le Castel, which, according to postwar trial records, was looted during the months following the armistice, under the indulgent eyes of the Ledoux family.\n\nIn the middle of a freshly mown lawn stands a cedar, plainly several hundred years old. At the foot of that tree, in May 1940, Henri Matisse and my grandfather engaged in spirited conversations about nature and its representation in painting. Massive, harmonious, reassuring, the tree was perhaps more damaged by the hurricane of 1999 than by the German invasion. The grounds are well tended, while the house itself looks rather weary. It's a curious building, at once charming and distasteful. Designed in the nineteenth century and modeled on a fortress, it combines all the attributes\u2014a keep, stone walls, carved rose windows in the fa\u00e7ade\u2014needed to turn it into a sort of Wuthering Heights.\n\nI push open the heavy glass-and-wrought-iron door. The hall looks a bit dingy and clearly hasn't undergone any refurbishment in many years. The big mirror hanging on the wall lends it a certain elegance though the worm-eaten staircase is crumbling into dust.\n\nI climb the shaky stairs and ring the second-floor bell. The door is opened by a startled elderly gentleman, a clerk from the town hall, lodged there by the council. He ushers me into a three- or four-room flat that may have been the bedrooms, and perhaps the dining room, of Le Castel. There's still a dumbwaiter set into one of the walls.\n\nThe gentleman listens, slightly baffled, to my babbling (\"my family lived here, left in June 1940; I'd like to see the downstairs\") and calls the town hall. Two deputies kindly join us and open up the property.\n\nPart of the house hasn't been touched since those days; the other was clearly added on by the Ledoux family over the course of the subsequent decades. Might this work have been paid for, gossips suggested after the war, by the booty hidden inside the house?\n\nDespite its fancy name, the house isn't very big, although the grounds are imposing. I inspect the whole building room by room, saving the drawing room for last. The kitchens are on the ground floor, as they are in all the houses in Haut-Floirac, which were the properties of the affluent Bordeaux bourgeoisie since the end of the nineteenth century. \"It dates from the nineteen-thirties or forties,\" I am told. \"The pipes are rusty, the wiring was installed by the occupying Germans,\" and the office is used as a storeroom for the drinks and mineral water that would be served at private or municipal events.\n\nThe Rosenbergs stayed at Le Castel until June 1940, when they decided to flee France. With a clear-eyed view of the deteriorating situation, but perhaps placing too much confidence in the Maginot Line of fortifications against Germany, Paul brought dozens of his paintings to Le Castel so as not to be separated from them, and especially to keep them safe, far from Paris. He rented a vault for them in the town of Libourne, at the Banque Nationale pour le Commerce et l'Industrie (BNCI), which later became the Banque Nationale de Paris, when it was nationalized after the war.\n\nThere 162 paintings were stored; they included a van Gogh self-portrait and paintings by C\u00e9zanne, Delacroix, L\u00e9ger, Matisse, Sisley, Picasso, Vuillard, Utrillo, Corot, Monet, and Braque. On September 5, 1941, when the Nazis opened vault number 7, every piece was taken away to the Galerie Nationale du Jeu de Paume. All G\u00f6ring had to do was seize them.\n\n* * *\n\nSo the Rosenbergs spent the winter of 1940 in Floirac. It was as if time itself had been suspended.\n\nDuring this period Braque came to visit. Troubled and dispirited about the outbreak of hostilities, he found it difficult to stand before his easel. In October 1939 he wrote to Paul: \"I'd started a few canvases, but the turbulence that arose put a stop to all that. I haven't gone back to painting, and for about a month now I've been making sculptures, which I am greatly enjoying. It's athletic work because I've got to bring stones up from the beach that sometimes weigh more than 20 kilos.\" Clearly, this work was as therapeutic as the defeat was traumatic: 120,000 dead, 200,000 wounded in a few weeks, a people humiliated. \"Hitler did in seven weeks what the Germans had dreamed of doing for seventy years.\"\n\nWhen the Reich troops arrived in Dieppe, six miles from his property at Varengeville, Braque took his finest paintings and sought temporary refuge with the Rosenbergs in Floriac. He and his wife, Marcelle, also brought with them the little gold in their possession. On Paul's advice, Braque put everything in the vault next to Paul's in the same bank in Libourne. Of course, the vault was later forced open, its contents, along with Paul's paintings, plundered by the Germans.\n\nIn 1942 Braque received an almost comical letter from the BNCI about the lock that had been broken by the Nazis and had to be replaced at the bank's expense: \"We would be obliged if you would repay the expenses thus accrued\u2014namely, 1,000 francs for expert advice and 200 francs for our trouble.\"\n\n* * *\n\nAs for Matisse, he moved to Nice.\n\nOn July 16, 1939, Matisse and Paul renewed the contract that had bound them together since 1936, adding a clause to the effect that it would become invalid in the event of war. On October 10 Matisse proposed a third contract, a \"war contract\" to be signed on the thirtieth of the month. \"Given the uncertainty of the market, a one-year contract strikes me as reasonable... I foresaw a return of the golden age of the arts, a time when artists wouldn't have to put their joys and torments on display... delivering their works not as soon as they hatched, but after living with them long enough to see them mature... Impossible in the present state of our civilization, and we must resign ourselves to parting company from our children before we've seen them grow,\" Matisse says, referring to his paintings. \"And your indomitable work arrives to rouse me from this state, which is so conducive to meditation even though it is imposed by circumstances. I succumb to temptation; the golden calm remains!\"\n\nOn both sides, the renewal of this exclusive contract revealed a certain optimism despite everything in the years to come. Paul then announced to Matisse that he wanted to move from Tours to Bordeaux so that his son, Alexandre, \"wouldn't yield to idleness\" and could continue his studies (Tours was not a university town at the time) and begin his military training.\n\nAccording to Paul's correspondence, it seems that in Floirac during those first months of 1940, before the catastrophe took place, the passion for art took precedence over commentary on events whose outcome remained uncertain. Many people were apparently unaware of how serious things were. In April 1940 the Art Institute of Chicago had planned a tour for Paul in America, so that he could come, along with his paintings, and deliver lectures on French painting of the nineteenth and twentieth centuries.\n\nThat same year, during the so-called Phoney War,* Paul traveled all the way to Nice to see Matisse in his studio and came back by train with canvases under his arm. Clearly enchanted by his visit, he wrote to the painter as soon as he got home. Apparently, hanging his friend's canvases to their greatest advantage was a more pressing matter than seeing his family after his absence: \"I found you in a most excellent state... I've seen your new works which, the more I think about it, are the very best quality and the very best of Matisse... The ones I have brought here were hung on the walls of the living room in Le Castel at 2:30. After contemplating them again, I went to say hello to my family. I was very tired after an 18-hour journey, the sight of your canvases revived me... I'm very flattered and honored to have your esteem and trust... I'm going to Paris next week, and I will reopen the gallery with five new paintings by Matisse, five by Braque, five by Picasso: what a fine reopening that will be!\" But he didn't go back to Paris. The letter is dated April 4, 1940. The German assault on the Ardennes was about to begin.\n\n* * *\n\nIn an article published in Sydney in 1941, the great art critic Andr\u00e9 Breton was asked to talk about the writers who remained in France during the war, and the magazine, Art in Australia, commissioned Paul to try to imagine the lives of his favorite artists under the occupation. Paul described one of his meetings with Matisse, who had, in his turn, come to Floirac just before the German attack.\n\nTheir conversation, just a few weeks before the rout, seemed surreal. As usual, they talked about art and painting and contemplated the budding trees and the first flowers to bloom in that spring of 1940. Matisse marveled, Paul relates, at the white and yellow daisies that made the lawn a carpet lovelier than a fourteenth-century tapestry. \"That is what we should create,\" the great colorist told him. \"There is the expression of freshness and color that I seek in my canvases. These are the harmonies that nature suggests to us but does not oblige us to reproduce objectively.\" This was in May 1940.\n\n* * *\n\nPicasso was in Royan, not far from Floirac.\n\nHe and my grandfather went on writing, phoning, seeing each other. Meanwhile, the rest of the family arrived from Paris and crammed themselves into the house. Paul told the Matisses that he would put them up, but there wasn't so much as a free sofa. On June 11, 12, and 13, there were heated family discussions taking place in the ground-floor living room. The Germans had entered Paris on June 10, and the question the family struggled with was whether or not to flee.\n\nSeventy years later, this September afternoon in 2010, here I am back in the same room, with the same fireplace, the same cupboards, and the same chandelier. It's strange watching a scene played out by ghosts. I imagine the evening: chairs crammed together, the children on the parquet floor, the half-packed suitcases in a corner. The room is alive. I hear the sighs, the murmurs, the anxieties, the certainties, the fears of all the people who are there camping out at Le Castel in those days in June 1940.\n\n* * *\n\nFor most French families, there was no question of leaving France, but for some, especially the Jewish ones who knew that they were targets of the Germans and that they were close to the border, the debate was: exile or maintain the status quo.\n\n\"Fearful of Vichy, or concerned that they would quickly become pariahs, some French citizens, and also some expatriates living in France, opted to flee,\" writes Emmanuelle Loyer in Paris \u00e0 New York. \"Even the most unwilling began to imagine the possibility of going elsewhere as the noose began to tighten. While the first anti-Jewish statute dates from October 1940, the machinery of exclusion had been set in motion as early as July of that year. Time was short. As David Rousset would later say with gallows humor, France and the rest of Europe would soon offer only two exit routes: Marseille and Auschwitz.\" Bordeaux might be added to the list.\n\nJacques Helft, Paul's brother-in-law, was adamant that the family leave France for Portugal, via Spain. As for my grandmother, she was unsure. Paul himself was of two minds. Everyone seemed to be guided by his own temperament when it came to the question of exile. Loyer sums up the dilemma of families by noting the \"ultrasensitive balance between the agony of departure and the potentially dramatic implications of the stubborn will to stay.\" She quotes a letter from Marc Bloch* written in May 1941, stressing the heartache of the historian crushed between \"bureaucratic obstructions of the U.S. State Department, family matters and perhaps the growing convictions of the author of L'\u00c9trange d\u00e9faite that by remaining in his country one could better serve it.\" Bloch was shot by the Germans in 1944 near Lyon, where he was in the Resistance.\n\nThe problem of passports was the first one that needed to be solved. Seventeen were needed for the Rosenberg family and their dependents, if parents, grandparents, children, brothers, sisters, and nephews were going to get out of France. Marianne, my grandmother Margot's youngest sister, had a childhood friend whose husband, having retreated to Bordeaux with the French government, happened to be secretary to the country's president, Albert Lebrun. Although the republic was stripped of its powers and its territory, it retained the capacity to stamp and validate passports. And this was accomplished. As for the Portuguese consul, he bravely delivered visas, against the will of Portuguese Prime Minister Antonio de Oliveira Salazar.\n\nThe second challenge was the crossing of Spain. Franco granted the refugees amassing at the border the right to pass through his country, but not to stop in Spanish territory. Paul and his brothers-in-law ultimately negotiated permission to cross Spain in three days and three nights.\n\nOn June 16 they were ready to leave and crammed into the family cars for the trip of approximately 125 miles. Two miles before the border at Hendaye, the controls were strict, and the queue was interminable. They ate butter biscuits, opened sardine tins, and slept in their cars.\n\nIr\u00fan, Burgos, Salamanca: as predicted, it took them three days and nights to cross Spain. At the French border, there was a poignant separation from my mother's brother, Alexandre, and his cousins Fran\u00e7ois and Jean, who had decided to stay and fight for their country. They boarded the last Polish ship to leave Bordeaux, the Batory\u2014named for a sixteenth-century Polish king\u2014and left Libourne on June 17, 1940. Alexandre was nineteen and had been brought up in the comfort of an affluent family. Why would a young man just past adolescence embark on such an odyssey? The love of his country, a taste for adventure, the need to stand on his own two feet? Exactly what drove my own father to reject a comfortable life in America to go fight in the Middle East?\n\nSo Alexandre and his cousins set off even before General de Gaulle issued his landmark appeal for support of the Resistance. As soon as they arrived in Great Britain, they joined what in 1943 would become the Second Armored Division, the Division Blind\u00e9e of the future hero of the Free French forces, Marshal Philippe Leclerc. On August 24, 1944, my uncle and cousins were among the troops who liberated Paris.\n\nMeanwhile, the rest of the family had reached Portugal and temporarily settled in Sintra, fifteen miles from Lisbon. On a daily basis, the adults laid siege to the consulates and embassies to obtain\u2014the number of refugees in the family had grown by now\u2014twenty-one visas for anywhere: Paraguay, Argentina, Chile. But those visas were extremely precious.* Paul later told an American newspaper that having arrived in Portugal as a refugee, he went to the British Relief Fund, which gave him a boiled egg and a piece of bread: \"Imagine a man who has everything in life... and who, a week later, has lost his business, his fortune, his friends. I was sitting on a stone wall with a boiled egg and a crust of bread and I couldn't help laughing.\"\n\nTo be able to board a ship, you needed, as Emmanuelle Loyer writes, to have enough to pay for \"a crossing, have a certain reputation, enterprising American friends, or colleagues, a lot of energy and a bit of luck.\" Not to mention the fact that the Americans' asking refugees to bring some kind of written guarantee that they would be able to earn a living in the United States made it impossible for many to leave.\n\nIn August the situation was eased thanks to Paul's old friend Alfred Barr. The distinguished director of the Museum of Modern Art had to fight to explain to the American authorities, who had never or barely heard of Paul Rosenberg, the potential artistic advantage that the United States might gain by welcoming him onto its soil. Barr was a persuasive man, and the Rosenbergs managed to obtain those precious visas. The Helft family (my grandparents' sisters, brothers-in-law, and cousins) received theirs four days later.\n\nThanks to various networks, between three and four thousand French citizens managed to reach the United States in this way. On September 20, 1940, Paul and his family disembarked in New York. They were lucky: about 75,000 French citizens died in Nazi concentration camps.\n\nSeventy years later my visit to Floirac brings their exodus chillingly to life. I now understand why my mother never wanted to see that house again, even though it would be her last link with France for five years.\n\nI too am more unsettled by the house than I'd anticipated. I must be visibly shaken, because the mayor's deputies suggest stopping off at the town hall, just around the corner, for a glass of water. It's oppressively hot. The mayor, Conchita Lacuey, who is also the Socialist Party deputy of the Gironde, the department that includes Bordeaux, drops in to greet me warmly, to tell me how amazed she is by life's coincidences, and turns the moment into a photo opportunity. \"You never know,\" she says, on that late-summer day. Her own grandparents, hard-line republicans, arrived from Spain more or less as my fleeing family was entering the country.\nAT THE CENTRE POMPIDOU\n\nThe war and the mark that it left on our house on rue La Bo\u00e9tie, the conditions under which my family stayed in Floirac, and finally their desperate quest for refuge in the United States are consuming me.\n\nI need to retrace my steps, to get back to the very core of things, to my grandfather's work, and to scour the family archives. I plan to immerse myself in them when I am in New York, though I am mostly living in Washington, D.C., at this point. But on a visit to Paris, I take the opportunity to call the Centre Georges Pompidou, to see if its archives have any information about my grandfather.\n\nAfter a chilly reception from the director, Alfred Pacquement, I am welcomed more warmly by Didier Schulmann, who is in charge of the Kandinsky Library. We arrange a meeting for May 10. May 10? The twenty-ninth anniversary of Mitterrand's victory? What's the connection? Only that which leads from politics to modern art and back again.\n\nUnfortunately, there's nothing much of interest about my grandfather in the museum, Schulmann tells me, except for some photographic plates that are stored off-site. I have another pleasant interview with one of the curators at the Centre Pompidou, Christian Derouet, who was responsible for the Kandinsky exhibition there several years ago. Derouet worked for a long time on L\u00e9once Rosenberg's archives and told me he'd come across L\u00e9once's brother, Paul, in the course of his research.\n\nThe reception I get from M. Pacquement indicates that he still bears a certain degree of rancor toward my family, which had some ten years before retrieved from the Centre Pompidou basement a painting stamped \"MNR, Mus\u00e9es Nationaux R\u00e9cup\u00e9ration: National Museums Recovery.\"* At the time the museum had been unwilling to return that painting by Fernand L\u00e9ger, Woman in Red and Green, also called Knight in Armor, on the pretext that the museum directors didn't know whether the painting, which they acknowledged had been stolen from rue La Bo\u00e9tie, belonged to Paul or to L\u00e9once. So the court decided, quite logically, that if there was any doubt, the painting should go to both families and that the heirs\u2014my mother, my aunt, and L\u00e9once's descendants\u2014were to share the work, as was done without difficulty. It was understandable enough that the Centre Pompidou didn't know to which part of the family it should restore the painting, and it wasn't hard to grasp its unwillingness to part with such a beautiful work of art.\n\nBecause it was not realistic for all the Rosenberg cousins to share the painting, a decision was made to sell it. I wasn't very interested in it at the time since I'd barely been aware of the research done by my family, especially my aunt and cousins in New York, who had sought the painting's retrieval. But I do remember my mother's telling me about the strange feeling she had had as she gazed at that painting, which was completely new to her, its having passed through the gallery without her ever setting eyes on it.\n\nI subsequently learned that between September 1939 and June 1940 my mother and her parents had left Paris, but L\u00e9once, my grandfather's brother, hadn't wanted to follow them. He spent the war in the capital, proudly wearing his yellow star, and miraculously escaped the roundup before dying in 1947. A great discoverer of new talent but always penniless, he often asked my grandfather for money in return for paintings that he owned and stored at rue La Bo\u00e9tie. That was what happened during the winter of 1939\u201340, in a transaction with Paul, who was based in Floirac. L\u00e9once received a wire transfer from his brother and put his L\u00e9ger in Paul's gallery, where it was stolen in July 1940, when the property was handed over to the Germans and a few French opportunists. Subsumed by the state after the war, Woman in Red and Green slumbered peacefully at the Centre Pompidou, labeled \"MNR,\" while neither the family nor the museum were aware of its resting place.\n\n* * *\n\nThough there are no archives on my grandfather at the Centre Pompidou, I am granted exceptional permission to consult the photographic plates taken in the family gallery, which are kept in the Kandinsky Library archives in one of the museum's warehouses. All collections not on display have been transferred to massive storerooms for fear that a flood, which seems to happen every hundred years or so, might once again inundate the basements of the Paris museums, as happened in 1910.\n\nIt's the largest of those great warehouses, or at least the one that houses the treasures of the Mus\u00e9e d'Art Moderne that aren't on display. Mile after mile, seemingly endless avenues are filled with mysteriously numbered crates containing sculptures that may never have been seen by anyone. Great cabinets mounted on wheels contain countless paintings that remain hidden from the human eye. Dozens of unmounted canvases on rollers, like the shelves in a rug showroom. I spot a Warhol and a Mir\u00f3 crying out to be hung.\n\nIn another high-security section behind a reinforced double door, for which you need a special badge to enter, I step inside the rooms where the photographic archives are kept. There are thousands of glass plates, all meticulously cataloged. A number of filing boxes represent the Paul Rosenberg collection. My mother and my uncle donated it to the Ministry of Culture in 1973 to grant researchers access to the works in their original state. Here they are, dusty and fragile, like memory itself. Dozens of cartons marked Bissi\u00e8re,* Braque, Laurencin, Matisse, and L\u00e9ger hold heavy plates of glass, artifacts of a photographic process used before the war. Most were taken by a famous art photographer at the time who went by the name Routhier and are of peerless quality.\n\nIn those prints I see the exhibition halls that I recently visited at rue La Bo\u00e9tie, the paneling reaching halfway up the wall and the unmistakable glass ceiling with its little star-shaped windows. The black-and-white photographs look strange, given that these are such famous and vivid paintings, but the prints are so magical that you can almost imagine they're in full color.\n\nThe glass plates that move me most are the ones commemorating exhibitions by Matisse or Braque in the late thirties, probably because I've seen other photographs taken only a few months later in the same settings; only this time the paintings of the two great masters have been replaced by the portrait of P\u00e9tain and violently anti-Semitic slogans.\n\nI open these boxes more or less at random and delicately lift the pictures from their yellowed envelopes, those plates of glass so fragile that some of them are broken or cracked. The cracks disturb me: Is it just the damage wrought by time, or is it abuse by the occupying forces that pilfered them? Perhaps it doesn't matter; the damage cannot be reversed. And deep in the recesses of the archives, the past somehow feels beyond reach. Why only now do I want to know who my grandfather was, what kind of person he was, how he lived? Why only now am I exploring his world?\nGENNEVILLIERS\n\nI decide to try to visit all the places where my family's memory is preserved. So: to the furniture depository where I've stored most of the papers and photographs that I hurriedly gathered from my mother's house after her death. It's freezing in this big unit at Gennevilliers, where moving men bring in the containers on casters and open them up in my presence, as in a morgue. Why do I feel like a gravedigger, when emptying my mother's cupboards did not make me feel that way?\n\nI set off again quickly, very quickly, with a big cardboard box in the trunk of my car, chosen from the twenty-five or so boxes that had been stored. I'll spend the next two nights sifting through photographs and letters. Most of these chests contain the papers of France Forever, of which my mother was secretary-general. The U.S.-based information organization was set up to relay to the Americans the efforts of the Free French and the Resistance. In 1940 and 1941, before Franklin Roosevelt entered the war, the Americans needed proof that the French deserved to be helped and weren't just a nation that had simply capitulated to the occupying forces, as it was fashionable to write in the 1960s and 1970s. Emmanuelle Loyer speaks about France Forever as \"an association set up on the initiative of a group of French who had settled in the United States, to 'drum up sympathy and material help for Free France.'\"\n\nI unwrap these relics as if they were remnants of a vanished world: a Cross of Lorraine (de Gaulle's symbol of resistance); a photograph of General de Gaulle signed to my mother, Micheline Rosenberg, which she kept even after becoming a fervent anti-Gaulliste. And the collection of pamphlets published by France Forever, written and designed by my mother.\n\nI feel guilty. She would have loved me to have shown an interest in her wartime efforts while she was alive. And yet I'd always found her glorification of France Forever a bit tiresome. I'd even told her, dismissively, in that sullen teenage way, that Roosevelt had entered the war only because of Pearl Harbor and that it certainly had nothing to do with France Forever. This wasn't necessarily untrue, but it was cruel to try to disparage her work as an activist and to prefer the heroes of the shadows that clashed in Kiev or skirmished in the desert.\n\nFor my mother, the war years in New York were\u2014shocking though it may seem\u2014captivating. Though they were not the happiest years of her life, they were certainly the most fulfilling. These were the years when she had genuinely exciting tasks to perform, ones to which she had committed herself completely, with talent and imagination.\n\nFrom the boxes I take notebooks and drafts of letters and reports, wondering how such an intelligent woman could allow herself to be locked away in a conventional life of marriage and motherhood without ever searching for the freedom and the friends she missed once the war was over. Such a life seemed such a waste to the young woman that I was in the 1970s and 1980s. For me, as for my contemporaries who tried to \"have it all,\" that conventional way of being was out of the question.\n\nIn addition to these notebooks, these brochures emblazoned with red, white, and blue rosettes, Lorraine crosses, and editorials dissecting the ideological differences between General de Gaulle and General Henri Giraud (who was preferred by the Americans since they mistrusted the head of the Free French), I find a treasure trove of personal papers and letters.\n\nI stay up till the small hours sorting and filing that huge archive: heating bills from the Floirac residence at the beginning of the war; ration cards from the Gironde in 1940 and Paris in 1945; the rulings from cases brought\u2014and won\u2014by my family against certain vultures; letters from L\u00e9ger or Matisse to my grandfather from 1939\u2014so many other letters! My grandfather's tiny, slanting handwriting, expressing a little bit of himself.\n\nThese letters date both from the war and from the years that followed and reveal the grand obsession of Paul's life: his paintings, which he loved as if they were living beings. For him, their recovery after the war, a source of so much anguish, reflected his determination to see his rights acknowledged and to ensure that his children would have a comfortable life. There is much humility in these letters, and some shy and tender outpourings to his son, Alexandre, who relieved him of the worry of running the gallery during the 1950s; to his daughter, Micheline, who lived far away in Paris; and to me, his granddaughter, whom he called \"my darling sweetie.\"\n\n* * *\n\nThere are heaps of photographs, all quite unreal to me. In this picture, the thin, distant-looking old man of my childhood appears young and gaunt. He is wearing a sleeveless bathing costume in a swimming pool in Monte Carlo (necessarily elegant). He is teaching my mother to dive. Or in 1930 he is with his wife and two children skating at Saint Moritz (elegant, always elegant), in baggy Tintin-style trousers, his hair blowing in the wind.\n\nWas he tender? Was he cheerful, my grandfather who was a father first and foremost, a papa who asked his children to call him by his first name? That shocked the gentle Marguerite Blanchot, who worked for my grandparents for fifty years and who always said, \"People will say that Monsieur is not the children's father!\"\n\nIn fact, Paul was an anxious and shy man who relaxed more easily in his letters to his beloved daughter than in his conversations with her.\n\n* * *\n\nDuring the 1950s, and throughout his life, Paul complained with less and less detachment about his health, which was poor, and about his business, which was actually thriving but which he thought was in a terrible state. He worried about political instability in France and about the Korean War, which he thought might worsen at any moment. He pleaded with my mother to come back to New York with my father and me for our own safety and suggested leaving for Argentina, which was described by relatives who had emigrated there as the new El Dorado. To Argentina, like so many former Nazis? To flee again, when there was no real threat? To resume the immigrant lifestyle, in a remote corner of the world, farther from a danger that had already passed?\n\nSanity prevailed. Having set off on a reconnaissance trip to Juan Per\u00f3n's Buenos Aires, Paul came back posthaste and made us unpack all the suitcases that stood in the hallway. Had he sensed that the country, once the richest in South America, was about to go into decline, into a period of galloping inflation under a series of bloody dictatorships?\n\nHe remained concerned about the future, finding little relief or reassurance in the fact that the nightmare was now over. It was as if with each successive international event, his identity and his family's might once more be called into question. The letters were largely devoted to the arrangements he wanted to make so that my mother and her brother could keep the gallery running, reflecting his life's work: the need to introduce people to contemporary culture, to make them understand it, to spread its message in a barbaric world.\n\nHe asked Alexandre to develop and manage this gallery, and my uncle did so scrupulously until his death in 1987. As for his sister, my mother, she had to defer to Alexandre, to place blind trust in his instincts for running the business. And above all, the two Rosenberg children were supposed to remain united. In fact, my uncle Alexandre fulfilled the promise he made to his father so loyally that he often took greater care of his sister than he did of his own family.\n\nAlexandre was an aesthete, the first president of the Art Dealers Association of America, and a connoisseur in great demand for the infallibility of his eye. His family\u2014his wife, cousins, sister, niece\u2014called him Kiki, the nickname his parents had given him when he was born in 1921 in the apartment at rue La Bo\u00e9tie, with Picasso as witness. They probably wouldn't have guessed that this childish nickname would later be applied to a very serious man behind a pair of tortoiseshell glasses. Although he retained his French nationality, Alexandre eventually married an American woman, my aunt Elaine, and became a true New Yorker. Yet he remained attached to French culture and was keen that his children, my cousins Elisabeth and Marianne, take advantage of their dual nationality to pursue their higher education in Paris.\n\nUnlike his father, Alexandre had embarked on his journey through the art world more out of filial duty than his own personal taste, which inclined more toward literature, philosophy, and fifteenth-century incunabula. He was less sociable than his father\u2014more brusque\u2014and while his love of art was limitless, his love of commerce was not. So much so that after my grandfather's death the Galerie Paul Rosenberg lost its dynamism and relied on its existing inventory. Though the two families were kept very comfortable for more than fifty years, the holdings gradually dwindled. Of the more than three hundred works recovered from the original collection, four major works have stayed with me.\n\n* * *\n\nI knew my uncle well but still have trouble envisioning Paul, his father, who lived through the final years of the nineteenth century and the first exhilarating yet tragic half of the twentieth. I have to banish the anxiety-ridden letters written at the end of his life and imagine what must have given him joy: to discover works of artistic genius by his contemporaries and to become entwined with their stories. I must immerse myself in his world, the world of a passionate and original art dealer.\nDEALER\n\nFor a long time the language of the dealer irritated me. Words like \"objets d'art,\" or \"rare and beautiful things,\" to quote the phrase on the fa\u00e7ade of the Mus\u00e9e de l'Homme, made me cringe. If my grandfather had sold jeans or tins of sardines, I wouldn't have considered it unseemly, but when I was young, getting rich by trading in objets d'art carried the same sulfurous whiff as the banking profession does today. Nothing dishonest, exactly, but an \"impure\" quality amplified by the French disdain for money.\n\nThe image of bohemian painters dying in garrets made me mistrust the trade of those who prospered from selling paintings. The idea of commerce, of trade, of buying canvases from indigent painters before selling them at a considerable profit troubled me. Julius II ensuring the glory of Michelangelo or Peggy Guggenheim buying a painting a day: these were noble efforts to preserve the arts.\n\nOn the other hand, I would have been hugely impressed by a man motivated entirely by the love of art, a kind of patron whose raison d'\u00eatre was the survival of good taste and the disinterested promotion of penniless young artists.\n\nAnd then I got older. I learned that the world according to Proudhon exists mostly in books, that making money isn't necessarily a sin (that is, if you don't exploit anybody), that you might even consider it moral to produce wealth rather than simply benefit from the wealth of society.\n\nSo yes, my grandfather Paul Rosenberg was a dealer. It wasn't a new profession. Rembrandt bid up the prices of his paintings at public sales in order to increase the value of artists' work. Bernini did the same in the seventeenth century. Vincent van Gogh and Paul Gauguin also understood the workings of the market. Ambroise Vollard wasn't just an intermediary for the impressionists; he wasn't just the dealer of C\u00e9zanne and Gauguin: he was also their advocate. Paul Durand-Ruel was another who knew how to create interest in his beloved impressionists, engaging qualities transcending those of the mere businessman.\n\nPaul was a dealer, just as they were, a successful dealer, even though his aesthetic judgments governed his decisions more than a desire for commercial success. Certainly, his passion for modern painting developed gradually. The same was true of Daniel-Henry Kahnweiler, whose biographer Pierre Assouline says that his attachment to contemporary art was not apparent at the start of his career. Kahnweiler was a banker who knew little about art, and his fascination with the painters of his day was \"the fruit of a slow process of maturation,\" an apprenticeship.\n\nThe parallel between the two men is interesting, given the importance of their respective images in the art world. Kahnweiler was a gifted art dealer who first set up his business during the early years of the twentieth century, but whose success was finally established after the Second World War, according to Assouline. A character not very dissimilar, in my view, to Paul: \"sober,\" \"imperious,\" \"tough in his professional dealings,\" \"a bit old-fashioned,\" \"sensitive to the slightest hint of fawning, and enormously proud.\"\n\nTheir backgrounds were quite similar, one from a family of art dealers only recently arrived in France from Bratislava, the other from a German banking family; both members of a bourgeois class sheltered from material hardship. Both men understood the revolution in twentieth-century painting, although Paul's tastes inclined toward Picasso and Braque, while Kahnweiler was drawn more to Juan Gris, his great friend, and to Maurice de Vlaminck. Both men refused to show the surrealist painters in their galleries, asserting that while surrealism was legitimate and innovative in literary terms, it was not sufficiently pictorial. Both dealers completely ignored Salvador Dal\u00ed and Max Ernst, Joan Mir\u00f3 and Ren\u00e9 Magritte.* Neither man was willing to write a memoir. Paul considered it vulgar and inappropriate to dwell on himself, while Kahnweiler set out his life story in broad terms in his book on Gris.\n\nThere the similarities end. The differences are many.\n\nFirst of all, their relationship to the wars. Paul had been a soldier, mobilized in 1914, and very concerned about what was going on politically in the 1930s. He campaigned against the acquisition of the art that was being sold off cheaply by the Nazis, and was forced to flee his country in 1940, hunted by the Germans. Kahnweiler, on the other hand, had been an ardent pacifist, refusing\u2014and this took courage\u2014to fight for either side in the First World War. He was thoroughly anti-Nazi, but did not believe in a second world war right up to the eve of Hitler's invasion of Poland, and managed to hide in France between 1940 and 1944. He sold his gallery to his sister-in-law Louise Leiris, a Burgundian Catholic, and was somehow able to maintain his place within the establishment under the occupation.\n\nPaul's and Kahnweiler's careers also took different trajectories: my grandfather, who had made a name for himself in impressionist painting, rose to fame in the world of modern art after the First World War. Kahnweiler was initiated into contemporary art earlier, at the very start of the twentieth century, and carved out a fine reputation for himself fairly quickly. But then he spent a long period in the shadows before returning with full strength in 1945. By that time Paul was far beyond the shores of France.\n\nPaul quickly developed a sense that the United States would overtake Europe both in the art market and in terms of cultural excitement. From 1922 onward he set about awakening Americans to the exhilaration of modern art. Kahnweiler was still convinced that Paris was the global art capital, and he maintained his belief in the supremacy of old Europe until he died in 1979.\n\nThe century's turbulence affected the two men in similar ways: the Second World War cut Paul off from his artists, just as the First World War had done for Kahnweiler. Much the same may be said of their success: Paul's fame in the art world exploded only after the First World War was over. Kahnweiler's triumph came chiefly after the liberation, when he won back his representation of the painters who had left him during the 1920s, becoming, most important of all, Picasso's exclusive dealer.\n\nOn a personal level, the two men did not get on well. There are no records of any unpleasant remarks from Paul about Kahnweiler, but Pierre Assouline portrays the subject of his biography as harsh in his treatment of all his colleagues, notably my grandfather. He was probably angry and hurt about the behavior of Paul's brother L\u00e9once, who had attracted the cubist painters to his gallery while Kahnweiler was exiled in Switzerland during the First World War. Besides, L\u00e9once's reputation was tarnished by the fact that he had agreed, during the 1920s, to be an expert consultant in the liquidation of Kahnweiler's property, which had been confiscated by the French because of his German citizenship. But the severity of Assouline's subject also seems to extend to Paul, whom Kahnweiler treated with a degree of contempt.\n\nPaul, who had chosen to sell nineteenth-century canvases so that he could buy twentieth-century works and thereby provide his artists with a livelihood, decided to put more money than his colleagues did into funding the painters he represented. He wanted to pay his artists (notably Picasso, Braque, L\u00e9ger, and Matisse) handsomely, in order to give them the freedom to paint. Kahnweiler, whom Picasso may have aptly described as miserly, made it a point of honor not to pay his artists more than he had to and never to bid up prices.\n\nWhen L\u00e9ger came to him and said, \"Paul Rosenberg gives me twice what you do,\" Kahnweiler replied, \"Very well, then, go to Rosenberg.\" So in the 1920s and 1930s, after Picasso, Braque, L\u00e9ger, and even, for a time in 1930, Andr\u00e9 Masson, signed with Paul, de Vlaminck left for Bernheim-Jeune, and Andr\u00e9 Derain for Paul Guillaume. Kahnweiler was left only with his beloved Juan Gris, in perpetual rivalry with Picasso and other less important painters.\n\nIt is easy to imagine why Kahnweiler might have been bitter, but Paul had opted to pursue a policy that favored contemporary artists, providing them with both fame and material comfort. And he was one of those who embodied the golden age of French painting between the wars. This is the central thesis put forward by Michael C. FitzGerald, who writes that \"the market was not peripheral to the development of modernism but central to it.\"\n\nIf Picasso's painting took off in the 1920s, it did so not least because Paul knew how to promote the painter and guide him in directions other than cubism. Paul also understood that it was important to view Picasso's work in the context of the tumultuous forces of the twentieth century and French painting of the past. This was more important, in the end, than constantly promoting cubism. As the American press has often pointed out, Paul was, until the war, the biggest art dealer in Europe, dealing in a wide range of artists, from Delacroix to Picasso. \"Imagine,\" a major California newspaper wrote in the 1940s, \"being able to step inside Matisse or Picasso's studio twice a year, being allowed to look at forty of their best paintings and saying, 'I'll take the lot!' Until the War broke out, that was just what Paul Rosenberg did.\"\n\nFinally, Kahnweiler and Rosenberg differed in their attitude toward museums. Kahnweiler was surely resentful about the confiscation of his property and, believing that he had already been forced to give quite enough to the state against his will, \"didn't like to give to museums. It was beyond the limit of his generosity.\" Paul, on the other hand, was overly generous. Grateful to America for welcoming him as a refugee in 1940, he gave large numbers of paintings (by artists including Picasso, Renoir, and van Gogh) to American museums in New York and elsewhere. After the war, happy to have recovered many of his stolen paintings, he gave the French state, including the Mus\u00e9e d'Art Moderne in Paris, roughly thirty large and beautiful works.\n\n* * *\n\nAt the start of the 1950s, Paul's innovative tendencies were still in evidence when he signed a contract with Nicolas de Sta\u00ebl, for example, or in his attempt to launch the paintings of Le Corbusier, which never really caught on. He also made forays into American painting previously known only to a select circle, such as the works of Max Weber, Karl Knaths, and Abraham Rattner.\n\nBut he never moved on to the next stage, which might have led him, during his lifetime, to two very different types of contemporary painter, Edward Hopper and Willem de Kooning. He probably wouldn't have liked Jasper Johns or Mark Rothko, had he come across them. And he would not have inclined toward the pop art of Robert Rauschenberg or Andy Warhol. Everyone has his or her own limits in the appreciation of modernity.\n\nFor its December 1941\u2013January 1942 issue, Art in Australia had, as we have seen, asked Paul to articulate his vision of painting and speak about the painters who had stayed behind in France during the war. Having arrived in the United States only a year before, Paul was presented as the man best acquainted with the artists of the previous era. \"Painters before their time do not exist,\" he said. \"They are always of their epoch. It is the public who is ever behind in the pictorial revolution. The public eagerly accepts the formula of a 'recent past' when it has been definitively accepted, but refuses to regard or even attempt to understand that of their immediate present. How many errors have been committed, and how many great young painters have been forced to know misery because of the buyer's ignorance and his refusal to support them; refusal because they 'don't like that aspect' or because they 'do not understand'... Too often the spectator looks for arguments within himself against the works rather than attempting to free himself from those conventions which he believes he understands, agrees with and likes.\"\n\n* * *\n\nIn a similar spirit, there was an article that Paul always kept close at hand, so that he could refer to it often, notably using it as an appendix to the catalog of the last big exhibition that he devoted to Picasso in Paris in 1936. It's a delightful piece by Albert Wolff, an art critic from the early years of the Third Republic, which was published in Le Figaro in 1876. The \"impressionists,\" a term that was intended as an insult, but that the artists themselves brandished as a badge of honor, had made headlines just two years before, and curators had trouble accepting the genius of something they couldn't understand. Paul kept this text as an antidote to the incomprehension of his contemporaries:\n\n\"Rue Le Peletier is suffering great misfortune. After the fire at the Op\u00e9ra, here comes another disaster crashing down on the neighborhood. An exhibition, said to be of paintings, has just opened at Durand-Ruel... There are people who explode with laughter when they see such things. As for me, it makes me heartsick. These so-called artists call themselves the intransigents, the impressionists; they take canvases, paint and brushes, throw on a few colors and sign the thing. So it is that at the Ville-Evrard, lost souls are gathering pebbles along their way and imagining they have found diamonds... So please be so kind as to inform M. Pissaro [sic] that the trees are not purple, that the sky is not the color of fresh butter, that in no country will you see the things he paints... Try to make M. Degas see reason... Try to explain to M. Renoir that a woman's torso is not a heap of decomposing flesh with purple and green patches denoting the state of complete putrefaction of a corpse!... And it's this pile of vulgarities that is being displayed in public with no thought for the fatal consequences that they might provoke! Yesterday, on rue Le Peletier, they arrested a poor man who, leaving this exhibition, was biting passersby.\" The article is well enough written, the charge is effectively leveled, but the mockery was turned against its author a few decades later.\n\n* * *\n\nPaul was combatting precisely this sort of thinking. But was he a visionary or merely\u2014and this in itself would be something\u2014going along with innovative painters and showing their work alongside the masters of the previous century in order to gain acceptance for the modernists? How daring was he, really? How did he see the role of an art dealer in a profession that was rapidly becoming organized?\n\nAfter the war he wrote to Lucienne, L\u00e9once's daughter, who wanted to open a gallery herself: \"Don't make the same mistake as your poor father did, restricting yourself to very avant-garde painting. Mix up your exhibitions in such a way that they attract the whole of your clientele, the part of it that considers itself advanced and the other, more conservative part. Maintaining without money a policy entirely ahead of its time is a cul-de-sac. These things have to be done gradually.\"\n\n* * *\n\nThat was basically how Paul started out, like his own father before him.\n\nMy great-grandfather Alexandre was a grain merchant. A long way from the world of art. When he was nearly ruined by a cargo of rotten goods, he decided to put his last savings into the thing that he really loved, \"objets d'art and curiosities.\" Farewell to the grain trade. He became an antiques dealer, at 38 avenue de l'Op\u00e9ra.\n\nI remember looking at the building's fa\u00e7ade indifferently. It's at the end of the avenue, practically on the place de l'Op\u00e9ra, one of those buildings that now house insurance companies and airlines. I still have trouble imagining an art gallery in this setting, a place that seems designated for trade, for the tourists, in the shadow of the Palais Garnier.\n\nOne day, turning up early at the Salle Drouot, my great-grandfather, who had recently become an art dealer, bought a painting he liked for 87.50 francs. It was a Sisley, the first impressionist painting he brought home, and at a time when practically everyone, apart from Vollard and more particularly Durand-Ruel, was ignoring this new artistic school. The great battles fought to win it recognition were drawing to a close, but still the public hadn't come. Intrigued, my great-grandfather went on to discover Manet, Monet, and Renoir.\n\nIt was probably this that reconciled me to the word \"dealer.\" Coming from nowhere, my great-grandfather trained his eye, trusting his own instincts, his own daring taste. So was it really about commerce, if the canvases that he bought\u2014and that sold badly\u2014were the work of illustrious unknowns? It seemed a passion first and foremost, a calling that had become a profession.\n\n* * *\n\n\"One day when I was about ten, my father led me to the shop window of a dealer who kept a gallery on the rue Le Peletier, to show me a painting that made me shriek with horror,\" writes my grandfather in the fragment of an autobiography that he began during the war years in New York. \"Imagine a very thickly painted picture made with violent colors, representing a modest bedroom with a wooden bed covered with a red blanket, an ordinary wooden table with a water jug, a bowl and, hanging from the walls, shapeless old clothes. The floor looked oddly bowed to me, and the furniture seemed to be dancing, as if it wanted, as in a cartoon, to leap off the canvas and fly out through the window. My father calmed me down and said, 'I don't know this artist, and the canvas isn't signed, but I'm going to find out about him because I'd like to buy some of his paintings.' The canvas [Room in Arles] was by van Gogh, it's the one that's in the Art Institute of Chicago, and which, by an irony of fate, I myself sold about 30 years later.\"\n\nThe impressionists, van Gogh, C\u00e9zanne: this was where all of my great-grandfather's savings ended up, much to the distress of his wife. \"My mother\"\u2014Paul continues in his sketch of a family memoir\u2014\"claimed her husband had gone mad and that he was ruining his children. 'What are our friends and customers going to think?' she groaned. Her dismay reached its peak when a van Gogh and a C\u00e9zanne came into the house. She would call upstairs, 'Children, your father's going completely mad: he's buying vann Govoghs and Ces Anes.' It's true that everyone who came to the house, even collectors and connoisseurs, guffawed at the sight of a blue or yellow Monet, saying that no one knew an equivalent in nature. One day, we were having lunch when the phone rang. My father picked it up. 'How much do I want for my C\u00e9zanne? 6,000 francs, I can't go any lower than that. So you'll take it?' He was delighted to be able to show his wife that there was someone even crazier than he was!\"\n\nSo the impressionists entered the home of Rosenberg p\u00e8re, at a time when not many art lovers were interested in them, and when dealers themselves preferred to sell paintings by the Barbizon school. Works by Monet, Manet, Pissarro, Sisley, Courbet, Daumier, Toulouse-Lautrec, C\u00e9zanne, and van Gogh now decorated the gallery on the avenue de l'Op\u00e9ra. Renoir too, whose A Girl with a Watering Can my great-grandfather acquired, a painting that my grandfather sold much later to the great American collector Chester Dale. It was the first painting, and one of the most beautiful, in the series hung in the National Gallery of Art in Washington, D.C., at the impressive exhibition of the Chester Dale collection in 2009.\n\nI went there to see if it was as graceful as its familiar reproduction and was dazzled by the sun that illuminates the child's blond hair, bringing alive the shadows on her cheeks.\nCH\u00c2TEAUDUN, OP\u00c9RA, AND MADISON AVENUE\n\nI have found Paul's torn and yellowed birth certificate: he was born on December 29, 1881, in the Ninth Arrondissement of Paris, in rue de Ch\u00e2teaudun, the son of Alexandre Rosenberg and Mathilde Jellinek. The strange-sounding names come from Hungary\u2014Bratislava, in fact, which is now the capital of Slovakia and was at the time part of the Austro-Hungarian Empire.\n\nMy mother always said proudly\u2014no doubt a legacy of the traumas of 1940\u2014that she had been French for two generations. And yet that's somewhat inaccurate: her father, even though he was actually born in France, wasn't automatically French by birth. The law of June 26, 1889, which sought to grant full citizenship to all children born on French soil, applied to children born in France of foreign parents, but only once they had reached their maturity. So in 1902, when he turned twenty-one, Paul should have applied for naturalization. But at the time he was in London learning his trade, and he let the deadline slip. Is it possible that our family's national identity has been imperiled since the start of the twentieth century?\n\nSo it is that I find, in my dusty boxes, a second piece of paper from 1913, reminding Paul that he had to apply for naturalization if he wanted to become a French citizen. The paper is signed by Louis Barthou, who was the minister of justice at the time and was killed in Marseille in 1934 by a stray bullet during the attempted assassination of Alexander of Yugoslavia by the Ustashe.\n\nAlthough he was born in Paris, my grandfather became French as the result of will in a France that, on the brink of the First World War, was keen to call up as many of its young men as possible. In short, my Frenchness is fairly recent on that side of the family. There were, at this time of the Third Republic, no French laws especially favorable to the children of immigrants.\n\n* * *\n\nPaul joins his father's business in January 1898, at the age of sixteen. \"He wanted me to learn the trade while I was still young. He started by making me copy out letters and file them. After eight days, I told him I'd only keep on doing that when I'd finished my art studies. He agreed, and here I am running around museums, taking notes.\" He begins by studying the arts of antiquity\u2014of the Chaldeans, Egyptians, Greeks\u2014before ending up with the moderns. During the holidays he travels around the museums of Europe and ends up well acquainted with them. \"Knowing the primitives, having studied their expression, their writings, the modes of expression they had adopted, allowed me to understand at a very young age that there was no process, that all that mattered was the laws of construction, relationships of values, volumes, lines and what it was that they wanted to express... I went out with my father, who initiated me into the antique dealer's trade and corrected my impressions. I became presumptuous and criticized the artistic purchases he made without me.\"\n\nFor better or worse, however, he is learning. \"We had an old china dinner set, pink background, and we had a barrel in the same color. One day one of our clients, the Prince de St. L., came to the house and I sold him the set, including the barrel, which cost on its own more than the rest of the set. Amazed by the price, the buyer insisted on taking the pieces away in person. Very proud of the sale, I told my father, who called me all the names of the day and declared that I would never be fit for the trade! I must admit that I wasn't proud of my beginnings as a businessman.\"\n\nBut over time his eye improves, and Paul thinks he's made it. \"Because you know your way around,\" his father tells him, \"go to London, open a gallery, do some business and try not to make any mistakes.\" So the young man sets off at the age of nineteen, sure that he will be lavishly praised upon his return. \"Alas, my first experiences were no more successful. Without my father, I had no one to guide me.\" Looking for paintings by the rather academic Belgian painter Alfred Stevens, he hurries to buy a work by an A. Stevens, who turns out to be Agrippa rather than Alfred, that has no commercial value whatsoever. But he soon makes progress. He buys two Monets for 250 pounds, two drawings by van Gogh for 40, and wins the trust of his father. Having retired from dealing in objets d'art to devote himself entirely to paintings, Alexandre hopes that his sons will become dealers in paintings in turn.\n\nIn 1906 Alexandre, now in poor health, sets up his two sons at 38 avenue de l'Op\u00e9ra, where Paul realizes that selling the impressionists isn't going to bring in enough to earn a living. \"We were forced to buy 'salable' paintings.\" By this he means the Barbizon School, which continues to dominate the taste of the times. Still, Paul tries unsuccessfully to sell a painting by the Barbizon painter F\u00e9lix Ziem to a buyer who thinks that six thousand francs for a view of Venice with a crooked campanile is pretty steep. He also tries to off-load a portrait of Louis XIV on a descendant of the Bourbons, who is disappointed when Paul naively tells him that he doesn't look a bit like his ancestor.\n\n\"I was successful, but I was troubled by the idea that I was selling paintings I didn't like, certain that they wouldn't be recognized in the future. It was then that I determined to sell everything I owned and invest in the impressionists. I realized that if I were going to compete with the big auction houses of the day, I needed to buy only the highest-quality works, and rely on time to make a name for myself.\"\n\nThese were in fact the two lessons that he drew from his apprenticeship and put to good use some years later. First repeating, with modifications, his experience with the impressionists, he deliberately chose to sell paintings that he truly loved and waited for art lovers to recognize their beauty. Over the course of ten years and two distinct phases, he moved from the Barbizon School to the works of Pablo Picasso. He would bide his time.\n\nFrom that moment on, he forged the reputation that stayed with him throughout his life. In forty years, from Paris to New York, from his father's gallery on avenue de l'Op\u00e9ra, to his own near Madison Avenue, his imprimatur was the absolute quality of the works he sold.\n\n* * *\n\nBut his passion for the modernists never allowed him to forget his great love of Renoir. Sometime ago, in Paris, I forced myself to go to the Grand Palais to see the exhibition of late Renoirs, the ones he painted at the start of the last century. I confess to finding Renoir's paintings facile, more tiresome than enchanting, perhaps because they have been reproduced once too often, printed on a thousand-and-one posters, tea towels, or place mats. It's a bit like a lover of classical music's not wanting to hear Mozart's Forty-first Symphony\u2014the Jupiter\u2014for the umpteenth time, after it's been played over and over again by every orchestra on the planet.\n\nI was convinced that Renoir's late style\u2014vague, reddish, and allegorical\u2014debased the oeuvre of his glorious years. A judgment inherited, I believed, from Paul and then passed on to me by my mother, it struck me as irrefutable.\n\nAnd yet that exhibition, Renoir in the Twentieth Century, was a real gift. It brought together the paintings from 1880 to 1890, when Renoir was distancing himself from the impressionist revolution, painting en plein air, in favor of a series of portraits of sweet, dreamy girls: Gabrielle\u2014his son Jean's nanny\u2014with her charming profile; bathers at their toilet; scenes of ordinary bourgeois life (girls doing their hair, reading, sewing, or taking piano lessons); voluptuous nudes not unlike those of Boucher or Rubens. Among the very last paintings was Les Baigneuses, given to the state by Renoir's sons in 1923, just after his death. I don't like this painting, although Renoir himself called it a \"success\" and a \"springboard for experiments to come.\" I don't like his soft, fleshy odalisques and as a result agreed with the blunt judgment of Renoir's work that my family history attributed to Paul.\n\nI say \"attributed\" because the exhibition finished with a big surprise: a whole wall covered with huge photographs of the 1934 exhibition that Paul had devoted to Renoir in his gallery, showing a selection of the canvases from the painter's last years. In it I saw all the paintings that were the real treasures of the retrospective at the Grand Palais, including those baigneuses that seem so flabby and pink to me today.\n\nAnd it was one of those canvases, known by its American name, Reclining Nude, that my grandparents donated to the Museum of Modern Art in New York in 1956, the first painting by Renoir to enter that museum, which sold it only a few years ago to buy a van Gogh, since American museums have the right to buy and sell the works in their collection. And it was one of the stellar paintings at the big Renoir exhibition at the Grand Palais in September 2009, one of the paintings that, by their own accounts, inspired Picasso and Matisse.\n\n* * *\n\nPaul carefully recorded two of his visits to Renoir's studio, one that occurred on November 21, 1919, just before the artist died, and one on December 6, 1919, the day of his funeral.\n\nIn November he found the old painter in the studio that he had built on the edge of his property, Les Collettes, in Cagnes-sur-Mer in the south of France: \"He seemed pleased to see me, and although I had a sense that he had lost weight, he was always cheerful, happy to paint, and as charming and clever as he was always said to have been... I brought him a photograph of a big Corot figure that I had just bought. 'Corot,' he said to me, 'is a creature apart in the nineteenth century, he is timeless.'...\n\n\"Before sunset, we brought Renoir back from his studio to his villa... he in his wheelchair, wrapped in furs and with a beret on his head. I walked beside him, bareheaded, talking to him about the beautiful spectacle of nature. The path was lined with olive trees, women picked the ripe olives, children played, dogs rested in the last rays of sunlight and the women paused to say, 'Good evening, M. Renoir'; the children stopped playing and the dogs came to greet their master. And he, like a grand priest, lowered his head and, smiling, replied, 'Good evening, good evening.'\n\n\"At that moment, through olive trees that seemed to become increasingly gnarled, the sea became bluer, the women more beautiful, the sun warmer, to cry out their admiration for the man who had known how to paint women, nature, sun.\"\n\n* * *\n\nPaul returned to Les Collettes only two weeks later for Renoir's funeral, after his death on December 3. He was one of the few people present at the burial of one of the greatest symbols of French art of the nineteenth and early twentieth centuries.\n\n\"His coffin rested in a modest hearse, without horses, adorned with ostrich feathers... The cort\u00e8ge set off, slowed down by a number of men, down the steep coast road that leads from Les Collettes to the little village of Cagnes. A church, really more of a simple shed, welcomed the crowd and friends from the neighborhood, with rudimentary pews, the coffin placed in front of the altar against two half doors with lowered blinds.\n\n\"The service began very simply, with no sermons, no music, no ceremonial dress, as Renoir himself would have wished. The priest, his friend, a great man, uttered the ritual prayers, but he did so with an emotion that affected the entire congregation: words of praise for the great painter, the great man of goodwill as well as the great believer who, behind his rebellious fa\u00e7ade, always sang the beauty of nature... I think that in other times, other ages, he would have had a national funeral.\"\n\nApart from the story of his initiation into the business, Paul wrote very little: a preface here and there, an article in an art magazine. Besides these fragments of memoir, he felt that it was not his role or his destiny to write. Was this a result of an inability to sit still, shyness, indifference, depression, or lucidity? It is hard to know. Though Paul was keen for recognition, publishing his opinions on the theory of art did not seem a necessary part of his identity.\n\nNor do I have testimony of his experiences in the First World War. I haven't found any of the letters from the front that he should have written to the pretty young wife he had married in July, a month before war was declared. They were probably lost in the upheavals of 1940. Having enlisted, like all the young men of his age, in 1914, he was demobilized in 1916 for poor health, the first signs of the ulcer that was to plague him for the rest of his life. All I have of him from this period is a brittle, yellowed photograph of a soldier with a mustache like those worn by the poilus of times past.\n\nThere isn't much evidence of his political opinions either. Yet we do know that having been a fervent admirer of the de Gaulle of the Free French, he strenuously distanced himself from the man on May 13, 1958,* to become openly anti-Gaullist. Living in New York, he had harsh words for the arrogance of the general.*\n\nHaving lived a bourgeois life, Paul was a wise man who came from the calm left and might have been called a radical socialist. As a student he had fought the anti-Dreyfusards, and he admired the French socialist leader Jean Jaur\u00e8s. In 1936 he voted for the Front Populaire, the left-wing coalition. In his own way, from within the art market itself and through his actions, he resisted the fascist ideas that were poisoning Europe. The heroism of his son in the Second Armored Division was also something with which he deeply identified.\n\nLater I found many letters from the 1950s, including the one he wrote to my mother in 1952, in which he tells of the \"mass of workers who can't make ends meet, who live in deprivation and on pitiful wages, will in due course rise up... Too many foreign-made luxury cars, too many overpriced restaurants. Too much poverty, too much outward luxury... and only charity for those who have nothing.\" This certainly wasn't a revolutionary diatribe, but the sentiments are clearly of the left. I am not trying to pretend that my grandfather was of the extreme left; far from it. Nor am I trying to minimize his bitterness at the time toward a France that had cast him out. But such expressions of outrage\u2014and I've found many of them in his correspondence\u2014testify to his personal revolt against injustice and inequality.\n\nAnd yet Paul Rosenberg led a very comfortable life, and he certainly hadn't made his way from bohemia to the bourgeoisie and then to the Communist Party, as his friend Picasso had done. Still, he didn't judge current affairs purely in terms of his membership in the class that he lived within. Gauche caviar, we would now call it, \"champagne socialism,\" a term used to mock anyone who doesn't automatically assume the dominant political opinions of his social milieu. As if a person's bank account determined his actions more than his convictions; as if the wealthy could vote only for the interests of their own.\n\n* * *\n\nFew ideological confidences are revealed in his papers, but in 1927 he did give a very strange interview about his family origins to \"Feuilles volantes,\" a supplement of the magazine Cahiers d'art. The interviewer was E. T\u00e9riade, the famous art critic and publisher. Oddly, T\u00e9riade asks his questions very seriously and doesn't seem at all put off by Paul's fantastical replies, which are clearly intended satirically: \"I come from a very old family lost in the mists of time. My ancestors, repelled by the mood in Palestine at that time, had wanted to sell the Tablets of the Law, but experts contested the sale. One of my ancestors authenticated the vase of Soissons... I find one of my ancestors among the Knights Templar. He died at the stake, and for the first time in his life he gave something away: his soul, to God... My father went to Mesopotamia, to examine the remains of the Tower of Babel. He visited India, Lutetia, Belleville and Montparnasse. He was a very noble man, very cultured and so generous that he saw to it that I was born on December 29, 1881, at three o'clock in the afternoon... At the age of 16, I entered the family firm. For starters, my father gave me copies of all the letters to archive. That task, which could have been terribly dreary, gave me a passion for invoices, and I already dreamed of the ones that I would later sign with my own name... My chief concern was to know whom the paintings I was to examine were by, and whether or not they were authentic. So I was obliged to find an infallible way of gathering information on those two points. For the first, I had discovered that by secretly reading the signature on the painting, I could discover the name of the painter. As to the authenticity of the canvases... I looked to see whether the paintings submitted to me were reproduced in catalogs or books. If that was the case, I maintained with great authority that they were entirely authentic. Even today, I behave in a similar fashion!\"\n\n\"What do you think about your painters?\" the interviewer asks. My grandfather's response bears more than a trace of irreverence: \"I am protected by every possible guarantee, and by the opinion of appeal court experts, distinguished chemists and manufacturers of canvases and frames, and I can assure you that I sell good, fault-free merchandise... My greatest ambition is to show in the L\u00e9pine* competition all the tricks I'm forced to come up with to convince my clients that what I'm selling are paintings.\"\n\n\"What do you think of your fellow dealers?\" asks the unfortunate critic, undeterred.\n\n\"I hold each of them in exactly the same esteem as he holds me.\"\n\nDoes that mean that this Paul, whom I see as more austere than playful, more of an ascetic than a bon vivant, also had an amusing and frivolous side? In truth, I think his character tended to be more on the gloomy side, as suggested by his correspondence with Picasso, to which I shall return.\n\nA four-page handwritten letter that Paul sent to Henri Matisse on December 2, 1939, three months after World War II had begun, adds to this portrait of a complex soul. He is writing to the painter with questions about his art. \"It seems to me that you want too much out of life,\" Paul replies to Matisse's nostalgic letter. \"What is it? A quarter of an hour of happiness, the rest all troubles, suffering and doubt! Do you want to be even more privileged than you are, do you want the heavenly gift of creating, of expressing yourself, without the pain that that entails? Everyone pays for what he has with what he doesn't have.\n\n\"Why wouldn't you doubt it? It's what gives you your strength, the expression of youth and creativity that are in your works. Don't you think that others doubt as well?... I am filled with doubts, I have feelings of despair like yours... Look at our friend Picasso, who not only doubts but is gnawed by torment... Are you sure that Corot doesn't doubt just as much as C\u00e9zanne, the master of masters, the greatest of martyrs alongside Michelangelo?... We are all moving irrevocably toward an ideal that we will never attain, and I say we are fortunate in this because [otherwise] it would mean the end of life... If you knew the despair I feel at being inactive... you would be calmer, because you at least can take refuge in your art.\"\n\nWe encounter this idea of being an intermediary rather than a creator several times in his correspondence. There is, for example, this letter dated December 28, 1949, again to Matisse: \"If only I could create something, if God had given me that gift, I would find boundless pleasure in doing it. But alas, I must content myself with enjoying my own admiration for the creations of others, not least your own works.\"\n\n* * *\n\nThose who knew Paul less well give a more effusive description of him.\n\nPierre Nahon depicts him as a \"man of middle size, of meticulous elegance,\" \"enterprising and tenacious,\" \"pursuing audacious strategies... He has a rare flair, his eye is excellent, he has contacts in the best society.\"\n\nAccording to Alfred Daber, a great dealer between 1920 and 1970, as cited by Hector Feliciano, Paul's \"body began to tremble like that of an impatient child when he saw a work that he craved. A trembling that subsided only when he had obtained the painting.\"\n\nRen\u00e9 Gimpel gives a less flattering picture of him: \"A fox's face with too short a muzzle. Prominent, grainy cheekbones.\" A displeasing portrait, not least because Gimpel was a friend of Marie Laurencin, who complained that Paul had treated her harshly when she asked for an advance of the pocket money she needed to settle the bill for her Chanel coats. \"Stop ordering them, then!\" Paul was supposed to have said to her one day when he'd had enough of her complaints, provoking a furious response.\n\nHowever, having read much of their correspondence, I had a sense that even though Laurencin sometimes pleaded poverty, she adored Paul and later my mother too. Their correspondence is more than affectionate. \"My darling Marie,\" Paul writes to her, adding, \"Can I say that without seeming forward?\"\n\nThe delightful, feminine paintings of Laurencin, who was loved by the poet Apollinaire, stood out in the male-dominated cubist world. They have fallen out of fashion today, as paintings for gray and pink boudoirs, but they have a grace that touches me, grace in a time of war and fragmentation. Laurencin painted gentle figures when L\u00e9ger was painting his industrial structures, violent in form and color. Was Laurencin behind the times? Perhaps it was more that she was out of step with a brutal world, and that strikes me as refreshing.\n\nDo I treat her indulgently because she painted my portrait\u2014at my grandfather's request\u2014when I was four years old? Sitting still like that was a form of torture for me at that age. Apparently I had the temerity to say to her, \"Don't forget, my eyes are blue!\" She smilingly obliged, blessing me with two luminous lavender orbs. My mother had hung this portrait in her bedroom, but I have trouble recognizing myself in this little girl with a pale pink smock dress and eyes that are unreasonably blue.\n\nThere are various descriptions of the gallery owner Rosenberg, in which he is depicted as \"a shrewd dealer with good taste.\" Certainly, his eye was legendary. In 1952 he wrote to Braque, sending him a photograph for the authentication of a painting, but he had already made up his mind: \"Looking at the knife, the lemons and the ace of clubs, I think it's very unlikely that the painting's one of yours.\"\n\nIn 1954, when he was in poor health, he sent his son to a Parisian auction in his stead. He was interested in several paintings and wrote Alexandre a letter giving him some suggestions merely on the basis of what he had seen in the catalog: \"The Renoir number 27 isn't interesting. Number 32, the Vuillard, is really a little masterpiece that you can buy. The Bonnard, number 82, not bad but a bit early. The Modigliani, number 91, I'm not sure it's authentic, as to number 95, the Renoir, stay away, it's too well known, it's been retouched and it's been on sale in all the markets in the world.\" All this perspicacity from an ailing old man who had examined an auction catalog.\n\n* * *\n\nIt would be an understatement to say that Paul was aware of his instinct for identifying art. He could be arrogant about his gifts and about the importance of his gallery, of both the unique quality of the works shown at 21 rue La Bo\u00e9tie and the catalogs published under his auspices for his own exhibitions. He was especially proud to have financed the publication of two important catalogues raisonn\u00e9s, one of the work of C\u00e9zanne, written by Lionel Venturi in 1936, and the other, in 1940, of Camille Pissarro's work, which was assembled by the painter's son Lucien in collaboration with Venturi.\n\nIn large part, my grandfather attributed his success to his belief that \"Great paintings sell themselves.\" Knowing that outstanding work would be coveted by collectors, he refused to bargain when masterpieces were at stake.\n\nPaul held his colleagues and rivals in high esteem, but not excessively so. He particularly valued Ambroise Vollard, his mentor and colleague of more than fifty years, who represented Renoir, Monet, and Pissarro and was, most important of all, the dealer and friend of C\u00e9zanne's. He gives a wonderful portrayal of Vollard in one of his letters: \"You never had a sense that he was trying to sell you anything. Quite the contrary: as soon as he had mentioned the price of the painting in question, he would feel his client's lapel and ask him who had made his suit. Then he moved on to something else that had nothing to do with paintings, leaving the client to his own devices.\" Though Vollard was the predecessor of the great French art dealers, his gallery, on rue Laffitte in the Ninth Arrondissement, was famously shabby, crammed with dusty canvases, the only furniture a cot on which Vollard would sometimes sleep. Vollard's gallery was far from the comfort of 21 rue La Bo\u00e9tie.\n\n* * *\n\nAt the Galerie Rosenberg, exhibitions were held year-round and lasted three weeks each. My grandfather hung the paintings himself, a sacred ceremony for any art dealer, and one to which he gave his full concentration. It was only when I saw the profusion of his catalogs that I realized the wealth of works that he'd hung over the years.\n\nIn 1962, when Paul had been dead for three years, his colleague Alfred Daber wrote to my uncle Alexandre, who had succeeded his father as head of the New York gallery: \"Between 1924 and 1937, such lovely exhibitions I saw at his gallery on rue La Bo\u00e9tie! We sometimes talked until eight o'clock at night about subjects that seemed to have nothing to do with painting, but that painting brought us to: philosophy, metaphysics. I already wanted to correct the prevailing taste, and he told me with lucidity that it was as vain an idea as wanting to channel the waves of the sea.\"\n\n* * *\n\nDisplays of paintings by Picasso, Braque, Derain, Matisse, L\u00e9ger, and Laurencin were interspersed with exhibitions by Henri Toulouse-Lautrec (1914); of French art of the nineteenth century, the preimpressionists (1917); Ingres and C\u00e9zanne (1925); Pierre Bonnard (1936); and Henri Rousseau, known as Le Douanier, or customs officer, in 1937.\n\nDuring the Great Depression, Paul returned to the nineteenth century, which was easier to sell than modern painting during those difficult economic times. In 1933 there was a Monet exhibition, and in 1934 one by Renoir. Indeed, 1936 was dazzling: Braque in January, Seurat in February, Picasso in March, Monet in April, Matisse in May, Laurencin in July.\n\nPaul's big exhibitions of works by Picasso were always an event. The first one, in 1919\u2014and I shall come back to it\u2014was devoted to 160 unpublished noncubist drawings. The 1926 exhibition was one of the most imposing and was followed ten years later by a one-man show, featuring twenty-nine paintings and drawings, that attracted six hundred visitors a day, and in which Rosi (Picasso's nickname for Paul) was so excited, it was \"as if the paintings had been created especially for him,\" observed a colleague amazed by the beauty and profusion of the works.\n\n* * *\n\nPaul loaned many canvases to other institutions. For example, he contributed to the first French retrospective of Picasso's works in 1932 at the Galerie Georges Petit, but also on the other side of the Atlantic at the Wadsworth Atheneum, in Hartford, Connecticut, in 1934. Picasso was a huge draw in the art world and caused an enormous stir in the United States. Paul had insisted that the exhibition contain a verse from a fable by La Fontaine, \"The Camel and the Floating Sticks,\" which he then republished in the catalog of the 1936 Paris exhibition and which he thought might open the eyes of the skeptics:\n\nThose things we find uncanny or alarming,\n\nCustom can make acceptable and charming;\n\nYour earlier intense desire to flee them\n\nIs lessened further every time you see them.\n\nHe spent months with his friend Alfred Barr selecting the works and undertaking the preparation for the first big Picasso retrospective at the Museum of Modern Art in New York and then in Chicago, at the Art Institute. That was in 1939 and 1940. Paul loaned more than thirty canvases to this exhibition, which meant these paintings had escaped the clutches of the Nazis. Barr was deeply grateful for Paul's willingness to enable this momentous show.\n\nThe other great painters of the Rosenberg \"stable\" followed in the aftermath of the Picasso exhibition. For instance, Paul devoted to Braque three major exhibitions\u2014in 1936, 1937, 1938\u2014and one, from April 4 to April 29, 1939, probably one of the last to be held at the Galerie Rosenberg in Paris, on the eve of the war. To complete the trio, L\u00e9ger had joined the roster of artists represented at 21 rue La Bo\u00e9tie in 1924.\n\nAs for his \"fourth musketeer,\" Matisse, Paul had also known him for a very long time. The correspondence between Matisse and my grandfather is still the property of the painter's family, kept, like all his archives, in the house where he lived at Issy-les-Moulineaux, near Paris. The house hasn't changed since Matisse's day, but the street, formerly route de Clamart, has been renamed avenue du G\u00e9n\u00e9ral-de-Gaulle.\n\n* * *\n\nIt's autumn. I push the gate open. It's cold; dead leaves are scattered on the lawn. I step inside an old-fashioned little house that makes a sharp contrast with the modernity of the conservation of the family archives. All the documents are digitized, and I'm settled at a computer by the curator, beside the radiator, in the very room that served as the painter's model for one of his most important transitional period paintings, The Piano Lesson,* a key canvas in the Matisse oeuvre. The double windows, the railing of the balustrade, the garden: They're all there, just as they are in the 1916 painting, giving me an immense appreciation of the artist's genius for conveying light and color.\n\nThe exchange of letters between Paul and Matisse began that same year. Their correspondence was regular and warm, apart from a few digs from Pierre Matisse, the artist's son, who thought that his father had become too dependent on Paul for representation.\n\nIn 1922 Matisse loaned Paul some canvases from his own collection, a C\u00e9zanne and a Courbet, for the Galerie Rosenberg exhibition The Great Masters of the Nineteenth Century. \"This exhibition,\" my grandfather writes, \"will also prove that the artists of our time... remain within the tradition, and that in their turn they honor French painting.\" He was still obsessed with the idea of showing the through line of art, that the works that he showed and that provoked howls of outrage from the bourgeoisie were in the tradition of the art history of his country.\n\nOn December 22, 1934, Henri Matisse writes to his son Pierre that \"business isn't going well. I sense a general feeling of apathy. Only Rosenberg has shown any warmth and offered me an exhibition.\" Two days later, in another letter to his son, Matisse confides: \"I saw Rosenberg, who galvanized me, told me I was wrong to allow myself to be forgotten. He told me he had big names\u2014the likes of Matisse and Picasso. That he wanted me to have an exhibition at his gallery, that he would put his exhibition space at my disposal... He showed me many beautiful paintings, van Gogh, Corot, Renoir, all new on the market. He told me how painting was everything for him, that it was the place where he lived.\"\n\nBut things aren't always idyllic between a painter and his dealer. On January 22, 1938, again in a letter to Pierre, who was based in New York and was warning him against the exclusive deal he had made with the Galerie Rosenberg, Matisse acknowledges that he has no illusions about his dealer, even though he knows that he can't do without him: \"As for Rosenberg... I've known him for a long time... Particularly when he yelled at me before signing a deal with me. I'm not with him for sentimental reasons, it's just so that I can use him... And then there are all the favors he has done me, and above all he knows how to glorify painting.\"\n\nThat was exactly what Picasso had understood in 1918, and it was likely one of the reasons that he made an extremely rare gift to Paul.\nMOTHER AND CHILD\n\nInitially it was called Portrait de Madame Rosenberg et sa fille. Later it appeared in various postwar catalogs, under the more American title of Mother and Child, before reacquiring its original name. Today it is prominently displayed in the Mus\u00e9e Picasso in Paris.\n\nThis portrait of my mother on my grandmother's lap was Picasso's gift to his new dealer, to mark the agreement they signed in Biarritz in 1918, even though Paul had tried to commission the piece. The painter even used the gesture to switch genres.\n\nThe painting is large, very large, and a bit academic, in the style of Ingres or Renoir but without the innate grace of those painters. It shows my grandmother sitting in an old tapestry armchair, holding my mother on her lap, a plump little doll in a white dress with blue ribbons. This painting, which scandalized the cubists, who thought that Picasso was \"betraying\" them, marks his return to neoclassicism.\n\n* * *\n\nI saw that painting throughout my childhood, first at my grandparents' Parisian apartment, then at my mother's. Paul attached great importance to it, and it was one of the first paintings he tried to retrieve after the war. The painting was said to have been stolen for G\u00f6ring, perhaps because it reminded him of the old masters.\n\nI used to look down on it a little, finding it too conventional, a sort of Virgin and Child on an Henri II armchair. Now I come to sit and meditate before it at the Mus\u00e9e Picasso, where I always thought it belonged. Since the days of Andr\u00e9 Malraux, the minister of culture under de Gaulle, the state has allowed anyone inheriting a work of art to donate it to a museum in lieu of paying a considerable inheritance tax. This measure was introduced to enrich French collections, which were poorer than many collections abroad, and to keep works that belong in national museums from being dispersed. That was what nearly happened to this family portrait: a rich Texan offered to buy the painting for a very good price, much higher than the inheritance tax that I was obliged to pay. But the idea of seeing this treasured painting leave for Houston was too painful in the end. It certainly would have distressed my mother. Fortunately, I recognized that donating it to the Mus\u00e9e Picasso was the right thing to do for the legacy of my family. I'm proud that the painting now adorns the walls of this Parisian institute of the arts.\n\nIn autumn 1918 the portrait was a sensation. On September 27, Paul wrote to Picasso: \"Everyone knows that Picasso has painted the portrait of my wife and my daughter. L\u00e9once heard Cocteau talking about it, and obviously he was hoping it would be cubist, even though Miche is rondiste.\"\n\nMy grandmother's face is, more than the rest of her, characteristic of Picasso, in a vein that is similar to the portraits of Olga, his wife. The painting is highly valued by art historians, even though I find it rather severe. Looking at it for the hundredth time, I try to work out why Picasso gave my grandmother such a melancholic face. At the same time I wonder why my mother, who seems so vital, is made to look so plump. Might Picasso have been prefiguring his series of Giants?\n\nSurely my grandmother would have preferred to have had her portrait painted by Giovanni Boldini, a mundane painter of the early twentieth century. Margot, who was inclined to be outspoken, admitted as much to Picasso. In response, Picasso drew a sketch in Boldini's most flattering manner, with flounces, a parasol, collars, and feathers, and sent it to my grandmother, signing it \"Boldini.\" I'm not sure which one Margot found more gratifying in the end... The Picasso was stolen by the Germans but recovered just before it left for Berlin. The fake Boldini disappeared during the war, never to be seen again.\n\nThere were other family portraits by Picasso. A gouache of my mother, in a blue dress by the sea, a little girl with red cheeks and windblown hair, was painted a year after Mother and Child, in 1919, on the beach at Biarritz. Amazingly, this was identified by an alert collector having an anisette in a caf\u00e9 in central France in the 1960s, who recognized it as the portrait of Mlle Rosenberg. The caf\u00e9 owner, who had been given it during the occupation by a man in need of a sandwich, kindly returned it to my grandmother, who rewarded him handsomely.\n\nThe portrait of Paul himself, a drawing whose lines have faded since 1919, is even more touching. Paul is an elegant figure: mustache, high-buttoned shoes, and double-breasted suit. He is sitting in a relaxed pose, on an armless chair, his left arm casually resting on its back. His well-manicured right hand, holding the inevitable cigarette, rests on his knee. This little picture is drawn, like the big family portrait, in the style of Ingres, but with a particular focus on the piercing, mischievous eyes of my grandfather; very Picasso. In the words of Michael FitzGerald, it is \"[a] blend of ease and sophistication... coupled with the intense scrutiny of [Paul's] gaze [that was] noted as his trademark.\"\n\nI still have the photographs of two vanished portraits of my mother, Micheline with Rabbit and Micheline as a Nurse. She must be four or five years old at the time of their creation. The drawings were done in charcoal. Stolen like the others by the Germans but never recovered, they may have gone up in smoke in the courtyard of the Mus\u00e9e du Jeu de Paume, in the bonfires of the occupying forces, or perhaps they were hung in a child's bedroom somewhere in Russia, or Berlin, or Paris, between the Seventh and Sixteenth Arrondissements, in the apartment of a wealthy French family that either collaborated with the Nazis or looked away from the question of the drawings' provenance.\nPAUL AND PIC\n\nThat Mother and Child sealed a covenant, an unshakable agreement. Rosenberg and Picasso: Was theirs a story of fraternal friendship or a professional alliance? Where did it come from: this mutual fascination between the establishment dealer and the bohemian painter? What did these two men have in common: the gallery owner (accustomed to the work of Renoir and Monet) and the painter who once pronounced dealers \"the enemy!\" to L\u00e9once, when he was one of Picasso's dealers between 1914 and 1918? How was it possible for Picasso and Paul to have had such a close friendship when the artist saw the artist-dealer relationship in class terms?\n\n* * *\n\nIn fact, much more bound the two men than a commercial contract. Theirs was an intense collaboration and aesthetic alliance. Indeed, my grandfather was recognized as the man who had orchestrated Picasso's career, as his \"impresario.\"\n\nMore than any other artist, it was Picasso who set up the dealer not only as his spokesperson and intermediary but essentially as his agent. He had very quickly realized that if a painter were to effectively address the public, he had to have just the right dealer, someone with a deeply compatible aesthetic sensibility and nature who would thoughtfully exhibit his work and advocate for him so that the public would understand his originality, his creativity. Picasso intuitively understood the necessity of forging a deep personal bond with the person who would be identified with the exhibition of his canvases.\n\nPaul knew how to comply with these requirements, enabling Picasso to turn his dealer into a close adviser and traveling companion. \"The artist and the gallery owner made one another,\" Pierre Nahon later said.\n\nPicasso was born in October 1881; Paul, in December of the same year, so they were exact contemporaries. But Paul belonged to the bourgeoisie; Picasso, to the avant-garde. Picasso soon recognized, however, that he could count on Paul to sell his paintings and even though Paul sold hardly any before the mid-1920s, the artist was in a position to wait. Paul could sell the work of his impressionists while gathering support for the contemporary painters who were his passion. Picasso quickly understood that Paul would be able to make and maintain his reputation. Both men readily grasped the significance of the press and cultivated those critics or writers like Pierre Reverdy, who understood this new style of painting and knew how to bring it to the attention of the broader public. Here again a new collaboration was inaugurated among artist, dealer, and art critic.\n\nIt was Paul's mission to move Picasso from his position in the avant-garde to that of a master of modern painting, \"the greatest of the twentieth century,\" as Michael FitzGerald was to call him. Between 1918 and 1939 Pablo Picasso and Paul Rosenberg promoted each other, creating Picasso's image and definitively establishing the reputation of my grandfather's gallery.\n\nFrom the outset Paul felt boundless admiration for the painter's genius. This was an enthusiasm that was all the more surprising, given that, unlike his brother, he had originally been drawn to a more classical form of painting\u2014that of Corot, of Courbet, of the impressionists, of C\u00e9zanne and van Gogh\u2014and had never been particularly convinced by to the vogue for cubism.\n\nIn January 1918 Picasso, in straitened financial circumstances, approached my grandfather to sell him a Renoir. But it was not until a face-to-face meeting in the summer of that year that the spark of friendship was ignited.\n\n* * *\n\nPaul called Picasso his spiritual brother, and what he felt for him was certainly something like a coup de foudre (love at first sight) of friendship. Indeed, something happened between them\u2014to the extent of complicity, affection, and I would daresay fraternity.\n\nThey met at Biarritz, in the villa La Mimoseraie of Eugenia Err\u00e1zuriz. This beautiful Chilean woman, a patron of the arts of the belle \u00e9poque, was a leading light in the world of dealers in fine arts in the 1920s. Picasso had met her through Jean Cocteau. She was a friend of Arthur Rubinstein and Sergei Diaghilev and devoted to the Ballets Russes, probably explaining the connection with Picasso, who also had strong ties with the ballet company. It was in ballet circles that he met Olga Khokhlova, whom he later married and with whom he had a son, Paulo.\n\nIn July 1918, Eugenia invited Olga and Picasso to spend their honeymoon at her house in Biarritz. Picasso happened to be looking for a new dealer at the time. Berthe Weill's gallery had probably been the first to sell a painting by Picasso (for 150 francs) around 1901, just as she was the first to show paintings by Matisse in 1902. But Picasso soon felt that he needed the financial stability that would allow him to paint with peace of mind. Vollard, notable for discovering C\u00e9zanne, bought twenty paintings from him for 2,000 francs in 1906, but this was not enough to free Picasso from financial worry.\n\nIn 1910 Picasso charmed Kahnweiler, who became his dealer in his gallery on rue Vignon, near the Madeleine in Paris. In 1913, Picasso made his first \"serious\" money when Kahnweiler bought twenty-three paintings from him for 27,250 francs. This was the equivalent of $117,500 today, or just over $4,800 per canvas. Picasso had never had so much money in his life.\n\nHis sense of security didn't last long. In 1914 Kahnweiler was forced to shut down his gallery because he held German nationality. Picasso was compelled to find a new gallery.\n\n* * *\n\nL\u00e9once Rosenberg succeeded Kahnweiler as Picasso's dealer in 1915. At the time L\u00e9once, passionate about cubism, said to Picasso, \"Together we will be invincible. You will be the creation, I the action.\"\n\nL\u00e9once, who had professionally parted company with his brother in 1910, was the more adventurous of the two. More avant-garde and more of a spendthrift too, acquiring more paintings than he sold. Paul, who was more prudent by nature, betting on nineteenth-century French painters and the impressionists, made calculated incursions into the art of his contemporaries.\n\nPaul the traditionalist and L\u00e9once the modernist? For a long time the accepted wisdom was that L\u00e9once was a gifted talent spotter but a terrible businessman, and Paul an astute businessman, more inclined toward business than art for its own sake. In fact, Paul wasn't very interested in old masters, unlike his colleagues who did a thriving trade in these safe bets, and instead took risks by taking on contemporary painters. Laurencin was one of the first of these, in 1913.\n\nAs late as 1943 Paul wrote, \"It would be so much simpler and more lucrative for me to make exhibitions of the great French nineteenth-century masters rather than contemporary works that unsettle our visitors.\"\n\nBesides, in the early years of the twentieth century, dealing in Renoirs meant promoting the art of the recent past. As for the delicate masterpieces of Monet, who died in 1926, they had not yet attained the classic status they have today.\n\nPaul wasn't interested in Jean-Honor\u00e9 Fragonard or Fran\u00e7ois Boucher, both then in vogue, or, unlike his brother, in Gris or L\u00e9ger. Indeed, L\u00e9once saw cubism as the culmination of all painting, much like those who saw the fall of the Berlin Wall not just as the end of a historical period but as the end of history itself.\n\n* * *\n\nIn his Galerie de l'Effort Moderne on rue de La Baume not far from rue La Bo\u00e9tie, L\u00e9once wanted to make Picasso the standard-bearer of a school of which the painter himself had wearied. Picasso wanted to break with artist theorists such as Albert Gleizes and Jean Metzinger and aspired to alter his style of painting. This was a time when he was distancing himself from the cubists and turning his attention instead to Diaghilev and the Ballets Russes, whose stage sets he wanted to design (much to the displeasure of L\u00e9once, who believed that Picasso was keeping the wrong company if he wanted to fulfill his destiny as emblem of the new school of painting).\n\nYet Picasso was in fact returning to his roots in his Rose Period and to his harlequins, who had vanished among the pure, hard lines of cubism. He fell under the influence of Cocteau and his famous Le Rappel \u00e0 l'ordre, in which the poet rebuked him for allowing himself to become the prisoner of other painters who had copied him and limited the scope of his art. So partly for personal reasons that marked a genuine evolution in his work, but also to attract the patronage of cultural figures such as Cocteau and Eugenia Err\u00e1zuriz, Picasso began to move from cubism toward a neoclassical style.\n\nBy 1918 relations had cooled between Picasso and L\u00e9once, and Picasso was ripe for his encounter with Paul, even though the artist had until then been L\u00e9once's most cherished artist. I have found no trace of what must have been a fraternal crisis of conscience for Paul, a source of jealousy for L\u00e9once, or, at the very least, the basis for heated debate between the brothers. All I have found is a statement from L\u00e9once made much later. He was a man who knew how to make the best of things, who saw that he was going to lose his painter anyway, and who concluded that it was better if Picasso stayed in the family.\n\n* * *\n\nThe meeting between Paul and Picasso took place that summer before the end of the First World War. The Rosenbergs had taken a villa in Biarritz, a few hundred yards away from the one owned by the Err\u00e1zuriz family. Also nearby was Georges Wildenstein, friend and colleague. In fact, the entire Parisian art world convened at the home of Mme Err\u00e1zurriz, La Mimoseraie. Eventually a verbal agreement was reached: Paul would become Picasso's representative in France and Europe, and Wildenstein would assume that role in America, where he had already established a gallery. But Wildenstein remained in the background, and when the two dealers fell out in 1932, Paul became Picasso's international representative and remained so until the end of the war. No actual contract was signed, but Paul was given premi\u00e8re vue, or the right of first refusal, on Picasso's works. This was a model to which he later returned, first with Braque, then Matisse.\n\nThat summer marked a milestone in the family, for both good and ill. The positive was the freedom enjoyed by Paul and Picasso to develop their business dealings and personal friendship. The downside was the deterioration of the relationship between the Rosenberg and Wildenstein families.\n\nFrom that time onward there was a very warm bond between Picasso and Paul. The painter savored the peace of mind that came from his contract with L\u00e9once's brother; he saw the possibility of escaping the lure of cubism, which Paul wasn't so keen on. Picasso knew that if he showed his work at the Galerie Rosenberg, he wouldn't be cataloged as just another avant-garde painter but would win his place in the company of masters of the century just past.\n\nPicasso understood early on the connections that existed between artistic creation and the marketplace, and he sought to impose careful control over the exhibition of his works. As Roland Penrose writes, \"Picasso's friendship with Paul Rosenberg was increased by the dealer's usefulness as a protector of his interests and the organizer of exhibitions in his fashionable Gallery.\"\n\nPicasso was thrilled to find a dealer who grasped his desire to transcend cubism. Paul's genius lay in his ability to effectively juxtapose Picasso and Turner, Monet and Delacroix. But Picasso was not the only one to have been guided in that direction by his dealer. Paul took the same approach with Matisse. As for Braque, with Paul as his dealer, he moved from cubism to... Braque. Paul encouraged all his artists to reintroduce the subject into their painting, even in abstract works. His sense of aesthetics aligned with his commercial instincts, and time ultimately proved him right.\n\nFor the first Picasso exhibition at my grandfather's gallery, in October 1919, it was Picasso who personally paid for and designed the invitation to the opening. Both men saw this exhibition as representing a break with Picasso's previous style: there was not a single cubist work to be found among the 167 drawings and watercolors whose variety delighted visitors to the exhibition.\n\nBy selecting these particular works, both painter and gallery owner opted to display a direction for Picasso that was less radical and largely unknown to the public. Picasso had found a way of announcing his return to neoclassicism, while at the same time revealing himself to be a more open painter than was generally thought. He was essentially declaring his refusal to be pigeonholed, to be limited to the one style with which people identified him.\n\nAt 21 rue La Bo\u00e9tie, the public discovered a profusion of drawings of harlequins, bullfighting scenes, circuses, the Ballets Russes, open windows giving out over the sea at Saint-Rapha\u00ebl, portraits and still lifes closer to the classics than anything that people had known of Picasso until then.\n\nIn the autumn Paul persuaded Picasso to move to the building next door, 23 rue La Bo\u00e9tie, where he and Olga occupied two floors. The two men became intimate in the manner of brothers\u2014inseparable.\n\n* * *\n\nI had a palpable sense of that intimacy when I read through a cache of 214 letters that Paul wrote to Picasso between 1918 and his death in 1959, many of them composed at the end of the First World War and continuing through 1940, when the Second World War altered the terms of their relationship.\n\nWhat remains of this correspondence is accessible to researchers at the Mus\u00e9e Picasso. I had suspected that a trove of letters was kept in the archive there but had never taken the trouble to consult it, especially since I had so desperately wanted to make a life for myself apart from the history of my family. Once I decided to look into the past, I spent several days perched at the end of a long table in the library, on the top floor right under the rafters, with those letters before me, hoping to gain a better understanding of what it was that linked two such seemingly different men.\n\nIt's strange, this one-way correspondence, in which you're forced to imagine the absent replies, trying to fill in the blanks, to tease out the nuances of my grandfather's relationship with Picasso in those years. Apparently, Picasso didn't write much, and the few letters he did send to Paul were stolen by the occupying forces or by French colleagues during the war. Perhaps one day I'll stumble across letters to my grandfather in an old chest of drawers somewhere, ones that begin \"Mon cher Rosi\" and that are signed by \"Pic,\" as my grandfather called the painter.\n\nI am trying to reconstruct that relationship, that singular dialogue between the two men. What did they have to say to each other? Did they exchange platitudes, details about married life, or, like Johann Wolfgang von Goethe and Johann Peter Eckermann in their famous Conversations, did they talk about Racine and Delacroix? What is certain is that like children, they called to each other from the windows of their respective kitchens, which looked out on the same courtyard. Apparently it wasn't unusual for Picasso to hold up the painting he was currently working on so that Paul could see it through the window. And few days passed without Picasso's visiting his dealer, who already seemed a genuine friend.\n\nThese letters have the elegant slanted cursive handwriting typical of the early twentieth century. \"Mon cher ami\" is followed by \"Mon cher Pic\" or \"Mon cher Casso\" (as my mother called Picasso when she was a child). For Picasso, these lighthearted notes were at odds with those he received from L\u00e9once, who was more formal in his bearing, despite his predilection for the avant-garde.\n\nThe familiar tu, absent for twenty years, suddenly appears with the liberation and remains throughout the 1950s, as if these two men, still almost brothers even though they were never again as close as they had once been, had decided that the turmoil of the twentieth century had swept away the polite distance of the prewar years.\n\n* * *\n\nPaul is plainly feeling his way at first, discovering the art of the painter whose greatness he senses but is still trying to grasp. \"L\u00e9once says you're a greater painter as a cubist than you are as a painter from nature... Am I too narrow-minded?\"\n\nIn the 1920s, Picasso is having a grand time in London. Paul is fascinated by Picasso's celebrity and the excitement with which he is received into British high society. Picasso becomes a member of the \"ultra chic\" as described by Michael FitzGerald.\n\n* * *\n\nPicasso himself confirms that in London he is \"seeing the beau monde,\" and he seems to love every minute of it. In fact, he retains his taste for the high life until his surrealist years, when he falls under the spell of his young girlfriend Marie-Th\u00e9r\u00e8se Walter and locks himself away in his ch\u00e2teau at Le Boisgeloup.\n\n* * *\n\nIn essence, these letters, which I read to soak up as much of this close male friendship as possible, deal with holidays, travels, when one or the other of them is away from Paris. And in fact, why would they have needed to write to each other when they lived within shouting distance, apart from the friendly little notes that you might drop off at your neighbor's house? \"Can we come up and see you after dinner? Please reply through the window,\" Paul writes in 1918. Or in 1931, in a playful tone: \"I dropped in at yours, you weren't there. I hereby summon you to my house.\"\n\nPaul stays in Paris or leaves for Deauville in the summer, while Picasso\u2014before the 1950s, the days of Brigitte Bardot and the Nouvelle Vague, or New Wave\u2014discovers the C\u00f4te d'Azur, Juan-les-Pins, Antibes (later Cannes and Mougins), and settles there for several weeks to paint. He is as intoxicated as C\u00e9zanne or van Gogh by the colors and the dazzling sunlight of the south. In those sultry days of summer Le Midi is a wild world spurned by the bourgeoisie, who prefer the cooler climates and more snobbish atmosphere of Normandy.\n\n* * *\n\nPicasso, with his paint, his brushes, and his imagination, had no need to travel far to discover new worlds. In fact, he tended not to travel much at all and never set foot in the United States, despite the fact that he was celebrated there. Paul, however, was a passionate traveler; travel delighted his senses, and he wanted his wife and children to discover Europe. Europe\u2014or rather the museums of Europe. For the Rosenberg family, there was no time for hanging around in the square, going shopping, or dancing flamenco in Spanish bars. These holidays were studious affairs that moved from the Kunsthistorisches Museum in Vienna to the Prado in Madrid and from the Accademia in Venice to the National Gallery in London. Paul adored Italy. From Florence, in 1923, he wrote to his friend Pic, \"I'm getting more and more disgusted by mediocre painting. Three painters transcend admiration: Corot, C\u00e9zanne, and you. The primitive painters and the old masters make me love your painting even more.\"\n\nHe discovered Egypt in January 1936 and was overwhelmed by the beauty of the Egyptian Museum, the Pyramids, and Luxor. \"Such artists, unencumbered by the weight of convention!\" he wrote. Jerusalem, on the other hand, left him cold. \"I don't recognize my ancestors at all. I'd rather complain in Paris than wail like my fellow Jews by a wall.\" (In those years, during the British mandate period, the Wailing Wall was accessed only through a tiny alleyway, an arrangement that persisted until the Six-Day War in 1967, when the wall was wrested from Jordan.) There was no mystical revelation for Paul, no emotion at the sight of those ancient stones from the temple that had been destroyed.\n\nMy grandfather was Jewish by name, by affiliation, by tradition, but not by assertion. I have many memories of my grandmother, a very pious woman who recited her prayers in her bedroom every morning and evening and had a regular seat in the synagogue on rue de la Victoire, like the old prewar families who were referred to as Israelites. But I have no memory of a strong connection, if it ever existed, between my grandfather and Judaism. A heavy smoker (several packs a day), he made it a point of honor not to touch a cigarette on Yom Kippur, if only to emphasize that he was making more of a sacrifice than the rest of the family in terms of fasting and piety.\n\n* * *\n\nPaul and Pic obviously came from very different social backgrounds, and if Picasso had his bourgeois period\u2014suit, waistcoat, cigar\u2014it was during the years when he was close to Paul, geographically and socially. \"My dream,\" he once told L\u00e9once, is \"to be rich but to live like a pauper.\"\n\nThe invoices from 1920 to 1921 that I found among the family papers reveal that by the standards of the day Paul offered his painters generous terms: Paul bought a large painting by Picasso for fifty thousand francs, a watercolor for twelve hundred francs, a cubist still life for twenty-four hundred francs (as early as October 1923 Picasso, having acquired a flair for business, more than doubled his prices). In 1941 Paul told Newsweek: \"From Picasso's studio I choose the paintings I'm interested in acquiring, then we talk prices, and that's when the fun begins. We exchange harsh words but always in a friendly tone. Once I told him I'd like to bite one of his cheeks and kiss the other!\"\n\nSo one of them had what the French would now call his bobo (bourgeois-bohemian) period. The other, who wasn't bohemian at all, frequented a society in Deauville, \u00c9vian, or Saint Moritz, yet constantly complained about everything, especially the rain in Normandy. He dreamed of the sun of the Midi.\n\n* * *\n\n\"We're very busy here... meeting people we see every day in Paris.\" And Paul jokes to Picasso: \"It's the sort of country you'd like, very cubist and full of proportions. It's also full of the French and foreigners, (1) of flirtatious and respectable women, (2) of gamblers and serious people, (3) of crooks and honest people, (4) of people who have gone to prison, and people who will, (5) of people who are enjoying themselves and others who just show their faces out of snobbery. There is, in fact, a disproportion,\" he adds, although it's impossible to tell in which of these categories\u2014the amused or the snobbish\u2014he puts himself.\n\n* * *\n\nBut these jeremiads, which were not unusual as far as Paul was concerned, are a bit hypocritical because he didn't really dislike those holidays among Parisian high society. He marveled at his children's rosy cheeks and, like everyone else, stayed up late playing baccarat every night, dressed like the others in his tuxedo, while criticizing those, including his own wife, who intoxicated themselves with sybaritic pleasures during those ann\u00e9es dor\u00e9es, which were ann\u00e9es folles for a small sector of French society.\n\nPaul complains of being far from his paintings, which are still in Paris, and says he can't wait to get back to his gallery once summer is over. \"All the top people are here,\" he writes to Picasso. \"The higher the class of society, the lower their morals.\" In September 1929 he writes: \"I'm coming back from Deauville. No rest, it's busier than in Paris, doing nothing useful, just parading about the place.\"\n\nA year later we hear the same refrain: \"It's all very phoney here. Everyone comes here to see and be seen. The children have the beach and the countryside; the parents have the casino and the car; and the men have the fillies. No shortage of them in Normandy! All snobs, us most of all, Margot loves all that. Soon everyone will be going to the Midi.\"\n\n* * *\n\nThat little society in Deauville in the 1920s was a privileged one, consisting of the partygoers and socialites who later flocked to Saint-Tropez or the fashionable islands of the Antilles.\n\n\"The exhibition of the artist by the name of Picasso is announced with great fanfare for the 14th February next,\" Paul tells Picasso in their typically jocular tone. But in January 1921 he reminds his friend of \"my harlequins, my harlequins, my harlequins!\" as he is clearly concerned that the artist has fallen behind. Similar concerns are sounded in August 1929, and one feels Paul's mounting frustration. \"You left without delivering my harlequin, you're terrible!\" he says. For Paul, who is meticulous in his business dealings almost to the point of mania, Picasso's casual approach to his commitments is maddening.\n\nPaul also writes, \"I have not yet seen your new style,\" not in the tone of a fashion designer's backer asking for photographs of his latest collection but in that of a child who thinks somebody's hidden his new toy. He is thrilled by the painter's genius, as the painter is well aware.\n\n\"Your trip to Russia is the talk of the town,\" he writes to Picasso, who has gone to Moscow to meet Stalin and his henchmen. \"I can't wait to see your 1926 production... Give me a vision of the 'new Picasso.'\" Paul understood that Picasso's paintings would change almost year to year.\n\nOccasionally, as on July 13, 1921, Paul issues orders that sound a bit brusque: \"I need a large number of canvases for this winter. I'm ordering 100 from you, to be delivered at the end of the summer.\" It's odd to hear Paul talking like the manager of a retail store, placing his orders with the wholesaler on the corner.\n\nOften, in his correspondence with Picasso and later with Matisse, he expresses his regret at being only the intermediary, never the creator. Paul knew very well that he was addressing a monumental figure of contemporary art, even as he urged him to produce new work (just as Durand-Ruel did with \"his\" impressionists). What fascinated Paul was the process of artistic development, which Picasso must have appreciated. Between 1918 and 1932 all of Picasso's major works passed through Paul's hands.\n\nIn the 1920s Paul told visitors to his gallery who were intrigued by these paintings, so different from anything they were familiar with, about \"my dear friend Picasso, whom I look upon as a brother and whom I have known since 1906,\" as he puts it in his 1941 article in Art in Australia: \"Picasso always goes beyond the boundaries; he is the greatest painter of the present day, and I am always delighted by each new series of his.\" He adds: \"It was he, Picasso, who overthrew past conventions and at his whim created others, and who, bored at seeing the same forms reproduced, devised his own... He has opened new horizons to us, and has brought painting to its only goal: 'to be works of art,' not mere decorative creations.\"\n\n* * *\n\nBack in Paris, the social whirl at rue La Bo\u00e9tie continued apace. In 1929 Paul bought some racehorses. Was he keeping up with the Wildensteins? \"I've got ten horses,\" he writes to Picasso. \"I'm going to name them after my painters. And if a horse with the name of Picasso wins, it'll be excellent publicity for your work,\" he jokes, while complaining about the expense of the horses.\n\nThe same year Paul was made a member of the L\u00e9gion d'Honneur. When Picasso congratulates him, Paul replies: \"My dear Picasso, the chevalier thanks you for your congratulations; they've brought me one more autograph.\" That didn't stop him, in the same letter, from discussing his friend's current domestic and financial affairs, for which he himself assumed responsibility, and the canvases he was impatiently awaiting: \"Your bills are paid... But you don't talk about your painting, or about what you've done, what new genre you've adopted. Your Dinard Stations of the Cross alarm me. You are massacring humanity so violently that I worry you'll do still worse damage by giving your characters a human face.\" Picasso's paintings of the 1930s already bear the early signs of his own internal turmoil and that of the world, as in the portraits of Dora Maar, which are distorted by the master's genius, and paintings that evoke the approaching civil war in Spain.\n\n* * *\n\nIn 1927 the Rosenbergs started \"taking the waters\" in Vittel or \u00c9vian, to treat Paul's fragile health, his frequent attacks of ulcers. \"No stress, just a calm, tranquil life. It's a dream, except for my wife, who isn't really enjoying herself. She wants to go to Deauville. I'll agree, for a bit of peace,\" he writes to Picasso.\n\nWhat remains surprising about these letters from between the wars is the extent to which references to contemporary events in Europe are absent. It is as if the two men wanted to immerse themselves entirely in art and friendship, far from the affairs of the real world. Only the signing of the Treaty of Versailles after the First World War and the celebrations that followed are talked about with some emotion. But the stock market crash of 1929, the far-right leagues of the 1930s, the Popular Front, the Spanish Civil War, Hitler's coming to power: none of these things is mentioned in these letters, even though the correspondence spans more than forty years. Probably such matters were mentioned in conversation. In their writings, however, it's painting, always painting, and the daily concerns of a life shared by friends.\n\n* * *\n\nAt times Paul seems the neglected friend who requires attention; he demands a letter or some news at the very least. The tone is affectionate, deferential, and intimate, even tender: \"I haven't seen you for a week. I'm getting worried, and my friendship with you is suffering.\" There is something intense and exclusive about this friendship, almost as if Picasso were his only friend. Was Picasso perhaps the only one who understood his inner being? \"I see your closed shutters, it's sad,\" Paul writes to his dear friend. \"Your paintings are on my walls and I miss your daily visits.\" There is a sense of brotherhood not unlike that shared by the great essayist Michel de Montaigne and his friend \u00c9tienne de la Bo\u00e9tie.\n\nThen come the laments about the ceaseless work needed to modernize the Galerie Rosenberg, the sluggish art market, the scarcity of collectors, and the shortage of art lovers: \"I've spent a fortune on antique frames. But paintings are getting so rare that it's the frames I'm going to sell. The sauce will help people swallow the roast!\" And yet, in spite of his grievances, there were splendid times when \"the paintings, a real stock exchange,\" soared in value, toward the end of the 1920s in France or immediately after the First World War in the United States. But to listen to Paul, business was dreadful throughout his career as an art dealer.\n\nMy grandfather was prone to depression, often related to his poor health and his chronic stomach troubles. This must have been what gave him that thin, almost gaunt look that struck me even when I was a little girl. My grandmother was all plump and gentle, her ample bosom perfect for childhood cuddles.\n\nIn September 1929 Paul confides in Picasso: \"My hell must lie within, if I feel fine only wherever I happen not to be.\" Such a marvelous phrase. It's rare to read Paul's divulging anything about his state of mind or his private life. For instance, there were disagreements between him and my grandmother that strained their relationship. In Paul's correspondence with Picasso, however, I never found a single word on these stormy and violent episodes, despite what family members told me in confidence.\n\nHad he ever opened up to his next-door neighbor? Perhaps it wasn't in the spirit of the times, because Paul makes no further allusions to Picasso's separation from Olga (although at the painter's request he drew up the inventory for the divorce) or to the various companions who passed in and out of his life: Marie-Th\u00e9r\u00e8se Walter, most often hidden away in Le Boisgeloup, Dora Maar, Fran\u00e7oise Gilot, or Jacqueline Roque, who became his wife only after my grandfather's death.\n\n* * *\n\nYet there are some genuine surprises; Paul sometimes allows himself to doodle shamelessly. My grandmother has no hesitation in doing the same. With her penholder (until she died in 1968, I never saw her write with anything but a Sergent-Major quill pen dipped into a big inkwell) she would try to draw the view from her bedroom in Deauville, most often ending up with a bunch of scribbles.\n\nIt must have been around this time that the painter drew an open window for Paul to use as an ex libris, that personal seal affixed to the first page of his books, which was used for both the Galerie Rosenberg's publications and its business cards until the death of my uncle Alexandre.\n\nAt times Paul and Picasso seem like mischievous adolescents. One of the letters from my grandfather to Picasso, dated July 4, 1919, is edged in black, the border hand drawn with a shaky pencil to convey mourning. My grandfather offered his most sincere condolences. \"The parrot is dead,\" he writes (deliberately echoing the petit chat in Moli\u00e8re's L'\u00c9cole des femmes). This was Paul's announcement of the sad demise of the bird that Picasso had kept at the Rosenberg house, whose final moments Paul so liked to describe. And this followed immediately by \"I've sold the Renoir you liked so much, Woman Taking Off Her Blouse,\" which put the gravity of the death announcement in context.\n\nBoyish jokes, intimacies, even teasing erupt. \"My dear quitter\" Paul says to him, \"I'm going to throw myself into painting, I'm jealous of your light. But what style should I adopt? Cubist, rondiste, loyalist, royalist, republican and monarchist? In fact I want to be a brushist.\"\n\nThrough all those years of complicity, they mix business, friendship, and favors: Paul takes charge of the practical side of Picasso's life: he orders him sheets of plywood that he needs for his collages or sells him packets of tobacco. Picasso, in turn, sends Paul sweets, which he loves, from Vogade, a confectioner in Nice celebrated for almost a hundred years. \"Thank you for the beautiful fatma, the beautiful Negro, your picture and candies,\" writes Paul, thanking him also for his battered canvases and chocolate truffles.\n\nAnd when Picasso is in London, Paul sends him off on a reconnaissance mission: \"There's going to be an exhibition with two Daumiers, a Degas, a Monet. Can you tell me if it's worth me crossing the sea to go to it?\"\n\nMy grandfather even gets into technical details with Picasso: \"Can you paint with English pigment and brushes, on English canvas? Don't use taffeta, it curls when it gets wet.\"\n\n* * *\n\nPaul never missed an opportunity to promote his painter and friend, introducing the younger painter's work, for example, to the seventy-eight-year-old Renoir. \"Saw Renoir. Told him about you. He was amazed by some things. And even more shocked by others.\" Picasso was thrilled by the fact that his revered master should be interested in his work. In fact, during those years he seemed engaged in a kind of painterly dialogue with Renoir that would mark his style throughout the early twenties.\n\nPaul also liked to assert himself in his friend's eyes as the expert with the infallible eye, whose business sense never interfered with his artistic vision. \"I had a visit from someone who thought he had a real one and a fake,\" he writes Picasso. \"I reassured him by telling him they were both by you.\" But Paul remained oddly old-fashioned in his response to the representations of sexuality in Picasso's painting, and God knows there were plenty of those! (Pierre Daix, one of Picasso's biographers, went so far as to call Paul prudish.) Apparently, Paul rejected the most graphic works, including a nude of Marie-Th\u00e9r\u00e8se of which Paul was supposed to have said, \"I refuse to have assholes in my gallery!\"\n\n* * *\n\nYet for all their closeness, the relationship cools. Picasso becomes detached and increasingly involved with the surrealists from whom Paul, like Kahnweiler, maintains a distance, and their neighborly complicity gently turns into a more conventional commercial association. Paul, ever sensitive, realizes this, calling Picasso his invisible friend. It must also be noted that by the early 1930s, Picasso is spending less time on rue La Bo\u00e9tie and more in his residence at Le Boisgeloup, forty miles northwest of Paris, with Marie-Th\u00e9r\u00e8se, the lover with whom he would have a daughter and who would inspire some of his most important works. This is a new Picasso, \"lord of Bois Jaloux,\" as my grandfather writes to him, seeing the chasm open up between him and his friend.\n\nAfter the Second World War and four years of silence, it will be even more difficult to regain their former closeness. The infrequent letters between them are no longer handwritten but typed, particularly after my grandfather suffers a stroke that keeps him from writing, and indeed from talking. However, in August 1944, when postal deliveries resume after the liberation of Paris, he warmly confesses: \"There's no point telling you how much I have missed you during my exile.\"\n\nIt is then that the two men start addressing each other with the familiar tu, probably after they meet when Paul returns to Paris in 1945, to assess the state of his looted property and resume his former life. And once again, they renew their relationship with its curious blend of business and friendship, even though Picasso is no longer my grandfather's client.\n\nPicasso has returned to Kahnweiler, his dealer before the First World War. \"My dear Picasso, I can tell you that I have landed on my feet in New York. How much would you charge me for the little still life with the fruit bowl on the right and the bunch of cherries? Je t'embrasse, Paul.\"\n\nOn July 15, 1947, my grandfather expresses to Picasso his irritation over an attempted breach of copyright: \"I'm learning right now that somebody in New York is about to produce some fabrics in 'Picasso gray.' It's illegal to use a name as famous as yours to launch any kind of merchandise. A parfumeur took Renoir's name, and after a case brought by the family they had to change the name. Will you give me the legal power to represent and defend you?\"\n\nWhat would Paul have said at the sight of the ubiquitous Citro\u00ebn Xsara Picassos being driven around the streets of all the cities in France?\n\nBetween 1945 and his death in 1959, Paul would see Picasso only once a year at La Californie, his residence in Cannes. The days of calling from one window to the next were over.\n\n* * *\n\nIt had to have been painful for my grandfather when Picasso resumed his business relations\u2014interrupted in 1914\u2014with Daniel-Henry Kahnweiler, who remained his dealer until the painter's death in 1973. But Paul was in New York at the time, and he was often ill. Picasso, who had drifted further away from his former dealer with every war, now returned to one of his first admirers from the early years of the century.\n\nBut my grandfather's passion for this extraordinary artist remained unbounded. \"The greatest artist in the world today,\" he said in the 1930s. \"The most prolific painter in history,\" he affirmed in the 1950s.\n\nMy grandmother and then my mother kept the connection alive with a few letters and visits first to La Californie, then to the farmhouse in Notre-Dame-de-Vie, near Mougins, which I remember.\n\nMy first memories of Picasso are from a long time ago. He is wearing his striped sailor's jersey, the one handed down to posterity in Robert Doisneau's famous photograph, in a restaurant in Saint-Tropez, to which he invited my grandparents and me in the 1950s, one of those lunches that seem interminable to children, and at which the patronne scurried over to collect the pieces of paper tablecloth that Picasso had scribbled on.\n\nI went often with my parents to his house in Mougins, though I surely would have preferred an outing with my cousins on the beach at Cannes. The ritual was always the same. The electronic gate opened; these were the days of Jacques Tati's films, and the gadget seemed to me the height of modernity. Jacqueline in her capris and colorful blouse welcomed us outside the house. She was a woman filled with admiration of, devotion to, and love for the great man who was her husband. I can still envision her after Picasso's death, when we visited her each year, always in the same place. I remember her as a somewhat haughty widow with Spanish posture\u2014straight as a statue\u2014and the long neck whom Picasso painted so often, either bareheaded or wearing a scarf, turban, or mantilla.\n\nI wasn't old enough to appreciate, let alone be amazed by, the paint-spattered parquet or the incredible disarray in the house, which at the time merely struck me as untidy. Picasso's room was in absolute shambles, and I couldn't understand how my mother, meticulous as she was, could swoon over such chaos. In his bedroom a recent canvas was used for a headboard, its face to the wall, so that the pillows would not rest against the paint.\n\nMost of the time I ran about in the garden with Catherine, Jacqueline's daughter, or Claude, the son of Picasso and Fran\u00e7oise Gilot, climbing their famous bronze oak. In those days I didn't care for its neighbor in the garden, the bronze statue of Little Girl Skipping, a sculpture that was somehow less accessible than the oak tree. As a little girl, assuming that the child must be suffering some kind of infirmity, I was unsettled by the one shoe turned inward.\n\nBack in the 1960s \u00c9vian bottles were made of glass and sealed with little metal caps. At the Picasso house, there was one glass case in particular that enchanted me, a curiosity that was, for once, accessible to children: it contained dozens of those little \u00c9vian caps, tortured and transformed into magical or monstrous animals by a man who could reinvent a set of bicycle handlebars, an old rake, or a bottle stopper into a work of art.\n\nI must confess that I sometimes thought\u2014like those back in 1920 who had criticized my grandfather for exhibiting scribbles \"that a four-year-old could have done\"\u2014that too much of a fuss was made over the slightest creative gesture of Picasso. The lack of comprehension and skepticism of the prewar years was over, making way for unconditional admiration for contemporary art in general and for Picasso in particular.\n\n* * *\n\nAnd now more recent images come to me, of Picasso in his last years, once he had stopped leaving the house: his blue-and-white-checked peacoat; his powerful, intimidating gaze; his Spanish-inflected French, which was excellent; his approximate spelling; and especially his affection for my mother.\n\nOne day when my parents had taken me along after a hiatus of a few years, he noticed that I was growing up. \"I'm going to paint your daughter,\" he told my delighted mother. \"I see eyes all over her face!\" \"No!\" I squealed, fleeing in terror, imagining a face that would have been distorted like the one he had painted of Dora Maar and his wartime paintings, which have never been my favorites. A fourteen-year-old girl isn't necessarily going to understand such a harsh artistic style. For me at the time, \"that guy\" Picasso was more of a predator of faces than a towering figure of the twentieth century. Would he have carried on if I hadn't run away? Probably not. At least I still have a photograph of myself at the age of eighteen, standing next to him, leaning against the walls of his villa. And I love his expression in that now-fading photograph: it is intensely magnetic, much like the gaze he gave himself in his earliest self-portraits in the 1900s, when he was already probing the deepest mysteries of the soul.\nBOULEVARD MAGENTA\n\nNumber 1 place de la R\u00e9publique. I was following the route of the demonstrations against the National Front. On May 1, 2002, there were still several hundred thousand of us jammed outside that door, hour after hour, so dense was the crowd that had come to protest the danger represented by Jean-Marie Le Pen, the far-right candidate for president, who had moved into the second round of the election behind Jacques Chirac. It was hot, we were anxious and thirsty, and I was, at that point, more interested in getting hold of bottles of water than in making a family pilgrimage. So I looked at the building without quite seeing it. That heavy, pompous Baron Haussmann\u2013style edifice.\n\nIt was the building where my grandmother had lived as a girl. Margot Lo\u00e9vi got engaged in that building and left it on the morning of July 7, 1914, to become Mme Paul Rosenberg. (My great-grandfather Lo\u00e9vi, the father of Margot, her brother, Michel, and her sisters, Marianne and Madeleine, was a wine trader.) I don't think the family knew the first thing about art, modern or otherwise. And I don't know who introduced this old Alsatian family to the Rosenbergs, newly arrived from Bratislava. But for my great-grandfather, the important thing in the end was to let his daughter marry a businessman like himself, no matter that he sold canvases covered with daubs of paint rather than bottles of fermented grape juice. Apparently the entire family was fine with this, and my grandmother's dowry was generous. My cupboards are still full of her monogrammed tablecloths and sheets that have never been used and are slowly turning to dust.\n\nPaul courted her for several months. My grandmother was a beautiful girl, and my grandfather was smitten with her. Twice a week he sent flowers from Moreux, the opulent florist's shop in the Sixteenth Arrondissement, which remained on the corner of the place Victor-Hugo until only a few years ago.\n\n* * *\n\nHe talked endlessly about painting to my grandmother, who, according to family lore, knew nothing about it. I can just imagine Paul, hoping to dazzle her, boasting about a painting he'd bought, and hoping to show her a famous van Gogh from the series showing the town hall of Auvers-sur-Oise. And I imagine my grandmother, a naive, sheltered young girl, going home and asking, \"'My green curtains,' 'my green curtains,' why is this young man always going on about 'green curtains'?\" She would not have realized that in French, \"my green curtains\" (mes rideaux verts) and Mairie d'Auvers sound virtually identical.\n\n* * *\n\nAs a young woman Margot had a pretty voice and was a lover of opera and operettas. She was the first to take me to see The Merry Widow, La Belle H\u00e9l\u00e8ne, and Faust, passing on to me her love of song, of the human voice, whether by Franz Leh\u00e1r, Jacques Offenbach, or Charles Gounod. When I was ten, she took me for the first time to the traditional opera house in Paris, the Palais Garnier, and I was impressed by the majesty of the building, with golden ornaments on the fa\u00e7ade and the impressive staircase, where I could imagine Maria Callas, whom I admired and still cherish, sweeping down in evening dress, followed by countless admirers and photographers... A dream for a little girl.\n\nShe had been cheerful and outgoing at the time of their marriage, but later she became depressive and lethargic. When my father started criticizing people who complained about their lot, aiming his ire first at his mother-in-law, then at my mother, he would say, \"That's the Lo\u00e9vi side of the family,\" contrasting it with the philosophy of his own mother, Marguerite Schwartz, an exceptional woman, whose motto was to \"button up\"\u2014in French, literally to grit your teeth and get on with it. In the Lo\u00e9vi household, you didn't button up in the face of adversity; you complained a lot and wallowed in your misfortune.\n\n* * *\n\nFor me, on the other hand, Margot Rosenberg was what the French call a grand-m\u00e8re g\u00e2teau. Not just because every walk I took with her ended with a stop at the pastry shop. Nor because I had only to mention my desire for a book, a record, or a four-color pen of the kind I'd craved for several months during the 1960s, only to be given them the following day. But also because she embodied the warm, generous bosom against which a child's sorrows were swiftly comforted. She indulged my every whim, and sleeping at her house allowed me to escape my mother's watchful eye. For me she was a very sweet old lady, and I was her cosseted granddaughter. Like my cousins, the daughters of her beloved son, Alexandre, I was spoiled rotten. And as the eldest I had all the advantages.\n\nMy grandmother spent six months of the year in New York and six months in Paris from the end of the Second World War until her death in 1968. Very stylish, she was always concerned with her wardrobe, never went out without makeup, wore hats with little veils, which I found mysterious, like a movie star of the thirties, and long black suede gloves, even in the summer, because she believed a woman couldn't go out bareheaded or gloveless, even in the sixties. She was a very comme il faut woman, whom I liked to shock with the slang that we spoke at school. Her favorite pastime was meticulously keeping her domestic account books in ink, with her big, regular, sloping handwriting. She also wrote every day to whichever of her children happened to be on the other side of the Atlantic. Each and every morning when my grandmother was in America, the postman delivered to her a sky-blue envelope bordered with red, which became her daily reading matter. I found many of these letters, numbered one to one thousand, crammed into the shoe boxes I recovered from the furniture warehouse where my mother's things were stored after her death. My grandmother's letters were full of trivia, of mundane preoccupations, as well as a few words of loneliness and of admiration for her three granddaughters, whom she adored. And so many ellipses standing in for sighs and despondency.\n\nShe never went out in the evening, had few friends, and mostly spent money on household staff: maids, cooks, chauffeurs. She didn't require this lifestyle, but she'd gotten used to it as my grandfather became increasingly successful. I remember that while my grandfather was still alive, she would ask him for a few francs before setting off to the kitchen to arrange the meals for the following day. I was aware of Paul's irritation as he reached into his pocket for such tedious expenses. If I wasn't consciously aware of the humiliation that she must have felt as she stretched her hand out toward the man with the wallet who, each evening, protested, I at least understood that a woman should try not to be dependent on her husband and that my grandmother would have been better off working. But it was not the way of her generation.\n\nMorning and evening she said her prayers in her bedroom, far from the synagogue that she attended on Friday evenings. If you risked visiting her during the morning or late afternoon, she would lift her head from her prayer book, delighted by the family visits that penetrated her solitude. Religious practices aside, my mother was very much like her. She too was lonely throughout her life, which was punctuated\u2014after my father's death\u2014only by my daily visits or by the arrival of my children when they were home from school.\n\nMy grandmother didn't eat pork or shellfish and might have memorized a few words of Yiddish, yet she didn't speak or read Hebrew. She had her seat, which her mother and her grandmother had occupied, in the synagogue on the rue de la Victoire, where the cantor, the young and charming M. Adolphe Attia, was showered with compliments for his golden voice when he chanted the Sabbath prayers.\n\nShe was, like my grandfather, the epitome of those prewar Jewish families that were known in France as Israelites until the 1960s: people of Jewish descent, more or less observant, but deeply assimilated into French society, even after the horrors of the 1940s.\n\n* * *\n\nThat was how I had always thought of my grandmother, who died in July 1968, at least until April 2010, when I opened those shoe boxes in the warehouses at Gennevilliers. Since then I've had a terrible time reconciling my memory of my grandmother with what I found.\n\nApparently she had had an affair with a man who was one of my grandfather's major competitors in the art world, Georges Wildenstein, who (as I have said) for a time was Paul's business associate. I remembered how it was decided in 1918 that Paul would represent Picasso in France and Europe and Wildenstein would represent him in America. I had never understood why the association collapsed in 1932, when Paul became the artist's sole representative. Or why it was taboo to utter the name of this family in ours.\n\nBut then one suddenly unearths artifacts from the realm of the unspoken, tucked away at the back of a chest of drawers. Do we pass over these secrets in silence? There's nothing shameful about them, even though they must have been painful at the time. Why reveal them now? They have nothing to do with anybody, except the protagonists, who died so long ago... I loathe absolute transparency, finding it voyeuristic at best and a bit totalitarian at worst.\n\nBut these letters provide a better understanding of my grandfather's psychology, which was skeptical and suspicious, and of my grandmother's personality, which became so withdrawn, in total retreat from the social world.\n\nI feel unmoored in the face of such intimacy, and I turn the letters around in my hands, trying to work out what to do.\n\n* * *\n\nFor my grandparents, it was a family crisis. For their children, my mother, my uncle, it was a secret shame (my mother never talked to me about it before her death) much like an open wound. My mother, at the age of fifteen or sixteen, in 1932 or 1933, was sent off to spend a few weeks with my grandmother's youngest sister, Marianne, and her husband and children, while Alexandre, who was only eleven, went to stay with my grandmother's other sister, Madeleine. The atmosphere at rue La Bo\u00e9tie must have been very tense. The servants, the family, their Parisian milieu: everyone must have known, and that open secret must have been the talk of every prewar Deauville soir\u00e9e.\n\n* * *\n\nI'm still pacing, clutching those letters as if I'd stumbled upon a written piece of the Kabbalah that could singe my fingers if I were ever to hide it again, leaving me with a curse lasting seven generations.\n\nI wouldn't even have mentioned this affair if I hadn't also discovered, in the boxes recovered from the depository, a poignant document written by my grandfather in 1942, when Alexandre was fighting with Philippe Leclerc's army in Africa, sometime between the battles of Bir Hakeim and El Alamein. Paul had planned to visit his son, whom he missed terribly, but he abandoned the idea at the last minute in the face of such a difficult journey and the risk of being shot down by the Germans. It was in this period that he filled a ten-page letter with his delicate handwriting and tucked it away in the drawer of an office on Fifty-seventh Street in New York. The desk went with him to his gallery on Seventy-ninth Street, but the drawer remained locked. A few months after my grandfather's death, Alexandre, while sorting through his father's papers, happened upon this document, typed it out to make it more legible, and sent it to my mother\u2014my grandmother had just arrived for one of her extended stays in Paris.\n\n\"You will weep as I did when you read this letter,\" Alexandre writes his sister. \"We have understood our father even less than we thought... I think that in any event you will have to show this letter to our mother.\" Did my mother do this? Something tells me she did not. It would be better to assume that my grandmother Margot died peacefully in Paris, in 1968, a few weeks after the May \u00e9v\u00e9nements.\n\nBecause the letter is harsh, very harsh. Written by Paul, it was intended to be read posthumously, as indeed it was. The letter was addressed to his wife and daughter\u2014\"his two darlings\"\u2014and the son he had been preparing to visit in Africa. It is a meditation on life, his life, on what he wanted for his family, and on the pain he felt over not having brought happiness to his beloved wife.\n\n\"My own youth was not as happy as my children's,\" he begins. \"But when I met you, my dear Margot, I hoped I might at last hold that happiness in my hand. I thought I had found in you the companion I would cherish, the one for whom I would do anything at all to make your life more beautiful.\"\n\nIt seems that Margot's disappointment dated back to the First World War, which broke out immediately after her wedding in July 1914. Paul, old enough to be conscripted, was sent to the front, causing them to miss out on the first carefree years that a young couple should enjoy. He goes on at length about his nerves, his desire to establish himself, and his need to earn a living in order to keep his family in comfort. \"Alas, the more I worked, the more money I made, the more I became a slave to business, a slave in chains, a Sisyphus with his rock,\" he writes.\n\nHe had always been financially prudent by temperament. But his anxiety about keeping up with his wife's expensive tastes was palpable. Right before her eyes she had the model of the Wildenstein family, whose lavish lifestyle must have dazzled her, though the life she led with my grandfather was certainly luxurious by any measure.\n\n* * *\n\nThen, in that letter that overwhelmed me, as if I had opened a door that should have stayed firmly shut, out came the rancor and jealousy that remained an undercurrent, a constant presence in the ensuing years. \"Sadly, you didn't give me time to put a roof on my building, before the evil words of a serpent were whispered in your ears. They distorted, ridiculed my every deed,\" my grandfather writes in bitter, biblical terms. \"I have much to reproach myself for. I should have spent less time on my business and devoted myself more to you... Life became torture for me in 1923, I loved you with all my heart and felt that I was losing you. Alas, you were given empty promises for the future the better to seduce you, promises that never came true, but that you thought were real, as if happiness didn't lie in the devotion of a close family.\"\n\nPaul had come from a family that was doubtless filled with the anxieties of Jews from Mitteleuropa. His wife, who had been integrated into French society for a longer time, was more playful and carefree; she needed love, and mostly what she got was money. How could anyone have imagined that the \"devotion of a close family\" would have satisfied a woman in 1930? Meanwhile, her suitor dazzled her with a vision of the high life, the flashiness of a society that, as we know, was dancing toward the abyss in the interwar years. And yet Paul, a pessimist by temperament, gloomy by nature, was already on that brink.\n\n\"You were beautiful, everyone found you amusing, you were wooed and desired by many men, and while thinking that you were making yourself happy, you made us both unhappy... Your sarcasm, relying on a so-called protector, about whom I hope my son will one day ask for an explanation, your way of saying 'too late' when I declared my love to you, darkened my character, and I had to seek consolation and oblivion in work, as indeed I continue to do,\" he writes in his own defense.\n\nApparently, my grandmother got bored with the marriage. Perhaps she was frivolous, responsive only to surface and luxury. That's what seems to underlie my grandfather's thinking.\n\n\"I want to tell you all this on the eve of my departure so that you know, my dear Margot, that your ambition for wealth was a desire for appearance, for possession. As for me, my sole desire was to make you happy (the children and you), and assure you that I would grant you all a secure future that would enable your independence. No, Margot, I can't rebuke you anymore. Time masks all wounds, but my own still bleed with the loss of my happiness. In order that your heart may cease to suffer, that posthumous remorse may not be too much of an affliction, I shall shoulder some responsibility myself. My own character, I confess, is very self-contained, and I should have liked to find in you a less skeptical person, someone more profound, with whom I could have exchanged ideas, shared my aspirations, and talked about something other than trivial matters. And if at root your being is devoted solely to goodness, then your spirit and your mind are incompatible with the needs of a serious, loving and devoted man.\"\n\nI don't believe my grandmother was ever aware of this letter with its very accusatory, self-justifying language and tone. I very much hope that is so.\n\nShe wanted a divorce, but my grandfather was adamantly opposed to the idea. Since, according to their marriage contract, all property was held jointly, I suspect that love and rage were not the only reasons for his refusal to divorce. From the moment Margot gave up her life as Wildenstein's lover and sacrificed her life as a woman, she punished Paul by relinquishing all interest in his social and professional world; by refusing to do any of the things that might, in my grandfather's eyes, have been expected of the wife of a major Parisian art dealer.\n\n* * *\n\nSixty years later, and the Wildenstein name pops back into the public eye. I'm skimming one of the newspaper stories in which you never know what's true and what's made up, gossip about inheritance scandals, unscrupulous art evaluations, or suspect fiscal investigations of them. The French have little sympathy for people with vast fortunes, and there can be no doubt that this family of prosperous art dealers falls into this category. Although it's also possible that malice dictates what the papers say.\n\nOn the other hand, I remember a story from about ten years ago, when the Wildenstein family brought a case against Hector Feliciano, the author of The Lost Museum. He was said to have defamed them by claiming that Georges Wildenstein, my grandmother's alleged lover, had done deals with the Nazis. That was what suddenly prompted me to seek out the details of the case when my own family's private history came to light.\n\nThe trial took place in 1999. The Wildenstein family was furious: \"What could be more horrible for the members of a Jewish family than to find themselves implicated in an act of betrayal, of collusion with the German occupiers against France!\"... \"The Wildensteins loved France so much that even then they didn't buy German cars,\" their lawyer, Ma\u00eetre Chartier, declared. This is a curious response, one that negates sixty years of Franco-German reconciliation and throws history back in the faces of the Germans who are so dedicated to consigning it to oblivion. But in the end, I'm more concerned with the family-related intrigues of the past than with the German cars of today.\n\nThe Georges Wildenstein Gallery was actually run during the war by a certain Roger Dequoy. This is where the stories diverge. According to the Wildenstein family, Georges had severed all communication with his former employee, who was going so far as to disparage him in letters that he, Dequoy, sent to the General Commissariat for Jewish Questions. As far as the family was concerned, Dequoy's assertions were false and malicious.\n\nAccording to Antoine Comte, the prosecution lawyer, Dequoy acted as an intermediary between Wildenstein and the German authorities. As evidence for this he cited a meeting in November 1940 in Aix-en-Provence among George Wildenstein, his employee Dequoy, and Hitler's art dealer, Karl Haberstock. In the course of this conversation, Comte claimed, an agreement was reached: Wildenstein recovered some of his confiscated property and was able to reopen his gallery under Dequoy's name; in exchange, Dequoy is said to have agreed to work for the Nazis.\n\nA serious accusation but one that, according to the prosecutor, was based on papers in the American archives of the Office of Strategic Services (OSS, later the CIA), which were declassified in 1998 and were said to contain a special report on the Wildenstein Gallery that had been compiled in 1945. The existence of both the agreement and the OSS report is confirmed by Lynn Nicholas in her book The Rape of Europa: \"In November [Haberstock] and Dequoy went to Aix, where they met Wildenstein and came to certain agreements... It was proposed that Wildenstein would exchange 'acceptable' pictures from his stock for the modern works so unacceptable to the Nazis, which Haberstock would send to him in the United States. Wildenstein would sell them through the New York branch of his firm.\"\n\nInitially, the Wildensteins were denied the six million francs in damages and interest that Alec and Guy Wildenstein, Georges's grandsons, had demanded for the assault on their grandfather's memory, which resulted in the family's decision to appeal the case.\n\nAfter the court of appeal refused to overturn the initial judgment, the daily newspaper Lib\u00e9ration, quoting the court's final statement, said that Georges Wildenstein \"can legitimately be presented as one of those individuals who on the one hand cultivated 'ambiguity,' both as a 'victim of the looting of the occupying forces,' and on the other, 'in parallel pursued, via an intermediary, operations on the Parisian art market' under the occupation.\n\n\"In the court's opinion, 'the allegations of contacts with the Nazis by Georges Wildenstein cannot be called manifestly erroneous' since it is established that Wildenstein 'had, before the war, entered into a business relationship with Karl Haberstock, who was known to be one of the F\u00fchrer's artistic advisers and a high-ranking Nazi. During the occupation, Haberstock acted as a protector of Roger Dequoy... who was running the gallery in Paris at the time, and who 'one may imagine was keeping up relations with Georges Wildenstein,' in exile in New York. There is 'much evidence to suggest' that the famous art dealer, whose collection had been partly looted by the Germans, 'maintained business contacts with the occupying forces.'\"\n\nThis ruling, disobliging at the very least, led the Wildensteins to appeal once more in the Cour de Cassation, which is supposed to judge only procedural matters. A new disappointment for the family: the court declared that the statute of limitations on the action had passed; the Wildensteins should have brought their suit within three months of the publication of Feliciano's book The Lost Museum.\n\nThis troubling story, on which I shall not attempt to give an opinion beyond citing the three successive rulings of the French judicial system, casts a deep shadow over my grandmother's love affair.\n\nDid their relationship last for a long time? At this point it would be virtually impossible to tell. But who, really, was this man who slipped into my grandparents' lives, coming between them? Was he a passing fancy or an archrival delighted to destabilize his competitor? Could he have been worth the suffering he caused?\n\nAnd who, really, was my grandmother? A passionate woman in need of love or a socialite fascinated by appearances? My grandfather was a good husband, in the sense of the phrase in those days, but in all likelihood he wasn't terribly exciting from a romantic point of view. My grandmother wanted to enjoy the carefree years between the wars. She was undoubtedly more hedonistic, more intoxicated by glamour than her husband, who was so preoccupied with the development of modern art. She wanted to dance, to enjoy herself, to be loved. He wanted only to work. This was the classic story of the romantic Emma and her stodgy Charles in Madame Bovary, or the flamboyant Ariane and the fool Adrien Deume in Albert Cohen's novel Belle du Seigneur. But my grandfather was neither a killjoy nor a petty bureaucrat; he was a curious and innovative spirit. All he needed to do was look away from his Picassos for a moment to gaze at the Renoir\u2014pretty, charming, and curvaceous\u2014that he had in his bed.\n\nEighty years have passed since then. \"Et la mer efface sur le sable \/ Les pas des amants d\u00e9sunis,\" as Jacques Pr\u00e9vert wrote and Yves Montand sang: \"The sea washes away \/ The footprints of parted lovers in the sand.\"\nPI-AR-ENCO\n\nNew York, once my family's city of refuge, and also my birthplace. The family archives are still on East Seventy-ninth Street, in the four-story town house that was home to the last Galerie Rosenberg.\n\nMy grandfather, who arrived with his wife and daughter in the autumn of 1940, initially moved into a house closer to midtown, on East Fifty-seventh Street, where he set up his gallery in 1941 and which he left thirteen years later.\n\nI have few memories of Fifty-seventh Street. Paul rented that stately old house, owned by the queen of England, who had a considerable property portfolio in Manhattan, but he had grown weary of the old building and wanted to live in a home of his own.\n\nHe bought his house on Seventy-ninth Street, between Madison and Fifth avenues, from Chester Dale, one of his major clients. After a lot of renovation, the family moved into the building in 1953. Paul was seventy-one. He lived there for only six years, increasingly passing responsibility for the running of the gallery to my uncle Alexandre.\n\nThis location on the prosperous Upper East Side, a bit sedate but elegant, wasn't a bad spot for business in the 1950s. In setting up shop there, my grandfather started a trend among gallery owners. Soon all his competitors who had established their galleries in midtown, just as he had, moved north, to within a few blocks of his new address at 20 East Seventy-ninth Street. By the time I was a child, it was bustling with art dealers.\n\n* * *\n\nPaul Rosenberg and Company was the name of the business. PR & Co. \"Pi-ar-enco\" to my childish ears, making me wonder what crazy-sounding person we were sharing our house with. I spent so many childhood Christmases there that until recently New York was an enchanted place as far as I was concerned.\n\nMy parents and I had returned to France when I was three, but I loved that town house on Seventy-ninth Street, every corner of which I knew so well. It was a beautiful limestone building, typical of New York, opulent looking, and right beside Central Park. In fact, it was just a few steps away from one of just two roads that ran all the way through to the west side of the park. I loved the sound of the crosstown bus that stopped before the front door. It was for me the sound of New York. Yes, New York was magical. With smoke billowing out of the street, it seemed the opposite of Paris, where I was living in a very quiet street near the Bois de Boulogne, in the Sixteenth Arrondissement. Today the building belongs to my aunt Elaine, Alexandre's widow, my mother's brother having died in 1987 at the age of sixty-six.\n\n* * *\n\nThe steps were once framed by Rodin's Thinker and its companion, The Age of Bronze. But the black-and-white marble mosaics of the entrance hall\u2014similar to those in rue La Bo\u00e9tie\u2014are still there, as they were in the exhibition halls that I wasn't allowed to enter as a small child. The elevator, modern in the 1950s, is practically an antique today, with its sliding aluminum door. I know by heart the sound it makes and the way it slows, shuddering, as it reaches each floor. My parents and I lived on the third floor, but I sometimes slipped out on the floor below, hoping to spot some \"clients,\" as my grandfather grandly called them, although I couldn't see how they were any more important than the customers at the Zitomer pharmacy on the corner of Seventy-eighth Street and Madison.\n\nMy grandparents shared a bedroom in their apartment on the second floor but had separate bathrooms, which always intrigued me. The television was in their bedroom, and it was there that I saw my first westerns, vintage ones with cowboys and Indians, covered wagons in a circle, and flaming arrows. The first television shows too, with women in New Look fashions created by Christian Dior in the fifties: women in wide gathered skirts and crewneck twinsets. There weren't many anchorwomen, no women journalists; women just presented the commercials\u2014so deliciously dated today\u2014for huge blue and pink American cars that created Detroit and then left it in ruins, cars you come across only in Cuba these days: \"See the USA in your Chevrolet\"... The refrain from the 1950s still echoes in my head.\n\n* * *\n\nNew York was snow, Central Park, my sled, and the magic of the Santas ringing their bells to draw in the window-shoppers outside Bloomingdale's.\n\nNew York was chocolate sundaes spilling over with whipped cream in the modern ice-cream parlors with their fake red leather banquettes, my first Walt Disney cartoons, mountains of toys at FAO Schwarz, on the corner of Fifth Avenue and East Fifty-eighth Street where the iciest gusts in the city blew, but where children like me were warmed by the consoling sight of those immense teddy bears that we never bought but that filled our dreams.\n\nAbove all, New York meant a month off from school, the only drawback being the math lessons my mother insisted on giving me. Faced with my inability to grasp problems about the distances between train tracks and the gaps between fence posts, she would end up throwing pencils and scraps of paper at me, telling me I'd never make anything of myself. The pencils were the ones with which Americans wrote on lined yellow paper\u2014the \"legal pads\" you see in Mad Men\u2014less formal than the shiny sheets in my Parisian Clairefontaine school notebooks.\n\nFifty years later on TV, I see Obama's advisers coming out of the West Wing carrying the same lined yellow pads and those inevitable sharpened pencils, blissfully unaware that those same pencils once grazed the head of a little math dunce. This is the retro side of an America that still prefers its shops to have old wooden doors with rattling, gilded knobs rather than the big glass doors that open automatically as soon as you cross the threshold of any French pharmacy.\n\n* * *\n\nNew York meant endless family discussions between parents and grandparents about France, which was imploding, even though news of the unstable Fourth Republic* reached us only in fragments. Politics? It was talked about, of course. As a little girl I vaguely understood that it was a world intended for grown-ups, something grave and mysterious, to which I was unbelievably lucky to have been exposed.\n\nI had always preferred to act beyond my years around things I didn't understand. Before I was even two years old, I imitated my parents by pretending to read The New York Times\u2014if upside down. At four, I assumed a look of great concentration when my father summoned me to discuss what he called serious matters. This happened whenever governments fell, every month or so. My father would address me, struggling not to laugh: \"Anne, some serious things are happening: the cabinet [that was what the government was called in those days] has been overturned.\"\n\n\"Oh!\" I would say, horrified at the apocalyptic vision he had just evoked, because the word for \"cabinet\" also meant \"water closet.\" \"Something has to be done.\" My father went on. \"Me, I'll take charge of foreign affairs.\" \"Me, I'll take the train,\" I invariably replied, without understanding a word he was saying, but delighted that my beloved father thought me a worthy partner for his grown-up conversations. My grandfather would burst out laughing, and I was very proud to amuse my family even without understanding what was so funny. On reflection, I took my first steps in political debate at the town house on Seventy-ninth Street, an experience that must have stayed with me in my twenty years as a political journalist in France.\n\nNew York in those days was synonymous with happiness, treats, holidays. My parents and I went there at first by ocean liner, which meant four or five long seasick days, and then, before the first Boeing flights began, by Super Constellations, big carrier planes that stopped at Shannon in Ireland and at Gander in the northeast of Newfoundland.\n\nPinned to my little coat was the Cours Hattemer, the \"cross of honor,\" awarded by my school to the term's best pupils. It was utterly ridiculous, that cross, a miniature copy of the Croix de la L\u00e9gion d'Honneur. On the bus, passengers would ask my mother what heroic deeds I had performed to deserve that military-style decoration.\n\n* * *\n\nAs New York winters were notoriously harsh, my grandfather's frail condition meant that he didn't go out much. His health had further deteriorated over the previous few years after a stroke that, while leaving his mind intact, deprived him of fluent speech and gave him a dreadful stammer. I was sometimes frightened by his damaged voice and by his little finger, gnarled with arthritis.\n\n* * *\n\nIn Paris, Paul and I often visited his colleagues, outings that I found slightly boring, but that were always followed by a fresh-squeezed orange juice at the Relais du Bois in the Bois de Boulogne, where we drank in silence to keep from frightening the squirrels.\n\nOne day he took me to see Paul P\u00e9trid\u00e8s, who ran a well-known gallery but whose reputation had been tarnished by collaborationist activities during the war. Back in the car, Paul grumbled, \"That man is a pheasant,\" which seemed an odd expression to use for someone. When I asked what he meant, he said he was referring to a rotten game bird. I brought that hunting term back home with me, much to my family's amusement.\n\nMy grandfather had an amazing eye: long after looking at a painting that he found interesting at a colleague's gallery, he would ruminate over it. Driving toward the Bois de Boulogne, he would suddenly announce: \"That painting is a fake!\"\n\n* * *\n\nEvery summer, I set off with my grandparents for the south of France, along the old Route Nationale 7, a road lined with plane trees that were magnificent to look at but lethally distracting to those driving cars. The highway leading to the south didn't yet exist, and it took us three full days to get to Cannes. We always stopped at the same places\u2014Saint-\u00c9tienne on the first day, Avignon and Aix on the second day\u2014before we arrived, the third stop being on the shores of the Mediterranean. Once we were there, within two days we absolutely had to go to the Galerie Maeght in Saint Paul de Vence and above all to see Picasso in Mougins.\n\nBy visiting museums with my grandfather\u2014the Louvre, one bit at a time, the Orangerie, and the Mus\u00e9e d'Art Moderne\u2014I learned what was worth looking at and what wasn't worth so much as a second glance. The quality of the works was gauged by the speed with which I crossed those exhibition halls at my grandfather's side. The Flemish masters or, of course, the Italian quattrocento were carefully examined, but paintings of the French and English seventeenth and eighteenth centuries were skipped over. The Gainsboroughs, with their very English solemnity, weren't considered particularly noteworthy. Our interest revived again with Corot (at last), Courbet, and, obviously, the impressionists. Looking at certain paintings by fashionable painters\u2014I'm thinking of Bernard Buffet, for example, whom Paul couldn't stand\u2014Paul allowed himself the luxury of saying that \"they weren't worth a fig.\" Some were simply declared \"ugly\" or \"without genius,\" if not \"without talent.\" Various minor paintings by Renoir, Gauguin, or Monet were decreed \"too red\" or \"too dark,\" \"too vague\" or \"too soft,\" \"lacking mastery\" or \"lacking power.\" And these judgments, fifty years on, still have the force of law as far as I'm concerned. \"Don't waste your eyes,\" my grandfather would say to me, \"on works that are not exceptional.\" The moderns\u2014Braque, Matisse, L\u00e9ger, and, above all, Picasso: they were his world.\n\nBut that gentle life belonged to Paris. In New York the rhythm was different: my grandfather worked, I strolled about the city with my mother and my grandmother, and for the child I was in those days it was a paradise.\n\n* * *\n\nDuring the winter of 2009 I couldn't wait to board the train from where I was living in Washington, D.C., to get back to New York, to East Seventy-ninth Street. That's where the gallery's archives are kept, devotedly guarded by my aunt Elaine, who worshipped her father-in-law. She had done a marvelous job organizing all this material with the help of an archivist from the Museum of Modern Art, to which the papers had been donated.\n\nI immersed myself in those files for days at a time, in the course of several visits. An old desk in a little room, six feet by ten, windowless\u2014I'm sure it's the room where I suffered my mathematical torments\u2014but with a skylight, rather gloomy, old linoleum, all on the same floor where I had lived with my parents as a child.\n\nThe internal staircase that leads both to this room and to my aunt's apartment is the one on which I used to hide. Often when I couldn't sleep, I would sit on the steps, hiding in a corner, and try to make out what the grown-ups were saying downstairs. From there, I would also listen to the classical music that my uncle was passionate about. And so it was, that in my pajamas I was allowed to listen to my first concerts of baroque music. I remember noticing, for example, that Georges Bizet had borrowed his tune for L'Arl\u00e9sienne from a theme by Michel-Richard Delalande, and having heard my grandmother humming the music from Carmen when I was very young, I felt it was a dreadful fraud.\n\n* * *\n\nMy aunt wonders about my recent obsession with a grandfather and a family history that I'd barely acknowledged until now. She and my mother didn't really get on and never truly understood each other. My mother was so close to her brother that my aunt often felt excluded. Alexandre seemed to pay my mother more attention than he did his wife. Not surprisingly, this caused my aunt, who was excluded from the affairs of Pi-ar-enco on which my uncle faithfully reported to my mother, a great deal of torment.\n\nIn short, my aunt Elaine, still laser sharp and quick on her feet at eighty-nine,* looks over my shoulder at the documents I'm consulting and the notes I take. Should I tell her that I'm feeling my way around the archive? She's taken the trouble to file away even the most insignificant scraps of paper from the Galerie Rosenberg. Should I express my surprise at my grandparents' personal life, which I'm sensing was more turbulent than legend allowed? I'm not sure. So I go on investigating, making sense of the voluminous papers as best I can.\n\nPhotographs of every prewar exhibition at rue La Bo\u00e9tie. Invoices for wine bought in 1928. Letters to Alfred Barr in the years before and after the war. Pieces of paper scribbled on by my grandfather, the beginning of an autobiography that wouldn't get beyond page ten. Fairly dry letters from Paul to unknown painters who wanted him to represent them. A bill from a picture framer in the 1920s and, most of all, telegrams or letters from 1942 revealing the ignorance of refugees about what was happening in occupied France. Files in Russian too, containing archives looted from the gallery in Paris by the Germans in 1940, then by the Russians in Berlin in 1945, carefully filed away in big boxes with titles written in Cyrillic. These archives were recovered a few years ago, thanks entirely to the perseverance of my cousin Elisabeth and the postglasnost transparency of the Russian authorities, who gave them to the French government, which in turn was gracious enough to restore them to us.\n\nThat precious immersion in the family archives on Seventy-ninth Street allowed me to reconstruct my grandfather's life after the upheavals of the 1940s. Upheavals, sure. But at the same time, he felt close to this continent, which he had so loved exploring twenty years before.\nA LONG RELATIONSHIP\n\nPaul knew America quite well, as he had tried hard to establish his beloved modern painters across the Atlantic.\n\nJohn Quinn, an American lawyer and collector, corresponded with Paul in the 1920s and tried to explain to him that his efforts to sell modern art in America were premature. \"Just five or six years ago,\" he explains in a letter found in the family archives, \"Knoedler had put on a C\u00e9zanne exhibition and people laughed, which they wouldn't do now...\" In May 1922, Quinn tried to convince Paul that no New York gallery, \"not Knoedler, Gimpel, Wildenstein or Durand-Ruel, will show any Picassos because their clients aren't open to this kind of painting. Dealers don't believe in modern art.\"\n\nBut Paul persisted. He was in Chicago that same year. And from New York to Kansas City, of all places, he preached contemporary art and was keen, in spite of the lack of enthusiasm from the American public, to show his beloved Matisse, Picasso, and Braque to the New World, which didn't get it at all.\n\nOn November 23, 1923, Paul put on\u2014probably at the Georges Wildenstein Gallery, with which he was affiliated at the time in the transatlantic representation of Picasso's works\u2014the Spanish painter's first New York show. He wrote to Picasso, \"Your exhibition is a great success, and like all great successes, we're not selling a thing! You'd have to be mad like me, or a crank like me, to embark on such an undertaking.\"\n\nIn a letter written to Picasso in November 1923, he was critical of America: \"Order reigns here, but there is a lack of European sophistication. The golden calf is more revered than ever, and the moneyed class is all that matters. Everything is colossal, even the museums. The worst of our painters is the best here... They have a collection of Rembrandts, just as I have a collection of Picassos, an incalculable number of them. Every self-respecting gallery has its Rembrandt or its Titian... Your paintings have arrived, they're wonderful, but I fear they don't like that kind of thing here. I'm expecting a vast crowd, meaning three visitors a day!... When nostalgia takes hold of me, I talk to my paintings, including yours. Ah, mon cher Paris, it's the only place one can live.\"\n\nA few weeks later he looks more favorably on the excitement of New York: \"I'm enjoying myself more: there's a spirit of will and strength,\" but he still rails against the aesthetic limitations of the Americans: \"Your exhibition is a great succ\u00e8s d'estime. But while in Paris there would have been a great crush, not many people came. Out of a population of six million, sixty visitors a day! But it's all over the papers, so what do buyers need? The New Continent doesn't go and see the New Painting, that is, the painting that is essentially timeless. They feel more at home with the painting of the past, with conventions.\"\n\nIn 1934, during one of his trips to New York, he writes to Picasso again. And he is still skeptical: \"The Bonnard exhibition was totally unsuccessful here. It's too fine for them. Too tasteful. Too tasteful and without enough forms!\"\n\nEven at the start of the war, when he was still able to correspond with Matisse, he could be severe in his judgment of those citizens of the New World, after seeing portraits by Matisse and other French artists in Life magazine. \"Very late,\" he notes, \"because the ones shown have been established in the rest of the world for over thirty years! Better late...\"\n\n* * *\n\nIn 1934 Paul decided to mount a major exhibition devoted to three great artists, Braque, Matisse, and Picasso. He wrote to Picasso: \"This exhibition will do a lot of good because it brings to the eyes of the public the new forms of expression of artists they had heard about but never seen. The public is divided, they all stay for a long time, upset that they don't understand... My previous exhibition, from Ingres to C\u00e9zanne, was splendid, but its splendor was rehashed. It was from the past, and there was no merit in admiring works that, dating as they did from 1814 to 1910, have had time to enter men's minds. But this exhibition represents our era, more than thirty-three years of our lives. Since it is the first of this kind, it has an absolute virginity, it must create the same effect as the impressionist show.\"\n\nHe himself attended to the smallest details of the exhibition, just as he did in Paris, and sent Picasso the plans for the hanging of each individual painting. \"Evidence of the power [of your paintings]. I had to balance them out with two Braques!\" These words, which might come across as so much flummery, are in fact meant quite sincerely. Paul was not given to effusive language. He could even be severe in his judgment of certain contemporary geniuses who were recognized as such, not least by him. \"The only weak point is Matisse,\" Paul writes. \"He can't take it. He flickers and goes out between the two of you [Braque and Picasso]. He... has forgotten about forms and volumes. Color is too important and you have the feeling you could add layers of color by painting the walls themselves. And that gives a sense of painted canvases, while you are creating the sense of colored sculptures.\" To me this assessment seems unfair because the light that floods Matisse's paintings makes them masterpieces in blue or yellow. It's true that Matisse's work, which is more accessible than that of the great abstractionists, struck Paul as more decorative and less innovative.\n\nIn that same letter Paul spoke bitterly about a different exhibition elsewhere in the city\u2014apparently dedicated to artists he didn't represent\u2014that enjoyed great success with the public, \"with 2,000 idiotic paintings that represented the most grotesque parodies imaginable. People must actually realize this! But I shall stick my neck out and say that stupidity and bad faith will always prevail among the living, and both of us may be in our graves when the same people's descendants glorify this art, demolishing the achievements of the creative generations to which you yourselves have given birth... But Galileo is right, 'eppur si muove' nothing will stop the progress of truth; beauty will always be beauty.\"\n\nSadly, this country, which my grandfather had initially seen as a continent to explore, was about to become his land of exile.\nTHE WAR YEARS IN NEW YORK\n\nPaul disembarked in New York with his wife and daughter in September 1940. They stayed at the Hotel Madison on Fifty-eighth Street until Paul decided to rent a new gallery on Fifty-seventh Street in 1941. Still in the throes of despair, Paul remained as anxious as refugees throughout the ages have been: \"No one can understand how comforted I felt when an immigration officer said to me, 'Don't worry, you're among friends now.'\"\n\nHe managed to correspond with France to a certain extent. His letters seemed to reach Nice, at any rate, in the unoccupied zone, where Matisse was still living. On November 27, 1940, he writes to the painter from his hotel room: \"I don't yet know what I'm going to do, but I might settle here as I did in Paris... No news of Pablo, or the other Parisians, which disturbs and worries me. I have in front of me a C\u00e9zanne of the area around Aix, with an atmosphere so clear and pure that it delights my eyes and sings to my heart.\"\n\nGrieving for his gallery on rue La Bo\u00e9tie, Paul corresponds with his painter friend as frequently as possible: \"We miss your paintings here because we've severed contact with Europe. The market needs your works, the American school is already taking advantage of this. They're bringing out all the Sunday painters, people who started painting when they were 72 and who are 92 now. I'm going to San Francisco to give a lecture as I did in Chicago, about art in general and about all of you in particular. It's the only thing that amuses or interests me. Too many things that I was once fond of, which were my life, are far away from me now. Even the beautiful countryside of Provence, that mild, gentle light, that serene landscape, comes to mind as I write to you.\" He continues, on February 18, 1941, after Matisse has had surgery and Paul is inquiring after his health: \"You are lucky to have painting; by creating you forget the hardships and anxiety of our times. Separation is painful, because everything I love is far from me.\" He adds in semicoded language, alluding to the last canvases that he had bought from the artist and that he suspected had been stolen from him: \"I don't know what became of your children of 1940. They were close to my heart, they were my joy. What can we do to get them back... [?]\"\n\nIn November and December 1940, in two letters to his son Pierre, who had himself opened a gallery in New York, Matisse asks for news of his dealer and friend, disguising his name for the benefit of the censor who opened the mail: \"How is Paul Floirac? Tell him plenty of things from me, but don't tell him that I'm writing to tell you that Pablo is worried about his future. Essentially he has lots of resources, and might return to his Blue or Rose periods, which are still highly prized.\"\n\nPaul becomes increasingly homesick and loses contact with his family, who remained in Paris and miraculously escaped the roundups of the Jewish population, as well as his friends. Though he is anxious about the situation in occupied France, he's unable to find out much.\n\nLater, in March 1942, he writes to his friend Henri de Vilmorin: \"You must have news about France and know what's happening there. The massacre of innocent people, whether from malnutrition or from cold, whether from diseases contracted in the concentration camps... Oh, how our brothers are suffering, and I imagine their pain at seeing our beautiful country looted and exploited by its enemies... Luckily we are confident and we have the firm hope that we will once again see the whole country purged and regenerated.\"\n\nPaul is frustrated, going on about feelings of impotence in the face of war, and tries to make himself useful. His wife and especially his daughter are working for France Forever. He himself organizes benefit exhibitions for the Free French, donating considerable sums to the effort. In February 1941 Paul gives the Free French Relief Committee a Stinson 105, the first air ambulance to be deployed in French Equatorial Africa. General Edgard de Larminat, one of the first French officers to have joined the Free French Forces, who is later made a Compagnon de la Lib\u00e9ration, sends a telegram of appreciation from Brazzaville to the \"generous donor\" who wishes to remain anonymous.\n\nPaul, in a state of agitation, writes to absolutely everyone. To his French friends, even though there is no hope that his letters will reach them. To his comrade and modern art collector Alphonse Kann, who is in England. To the efficient and generous secretary in his London office, Winifred Easton, who is taking care of the \"children\" that arrived in June 1940 and who is to survive the blitz: \"I know you are working hard, and that your morale has not been affected. We too are keeping our chins up, and we do not doubt for a second that we have been through the worst of the war, and that the end will soon come, with victory for all of us. Yes, the situation in France is terrible. That is why we are working so hard to identify and describe those horrendous characters who are insulting the reality of my country. We are publishing pamphlets and books that show the true face of France. But don't worry: when the war is over, the French will sweep all that aside, and those who did not resist will perhaps pay with their lives for the dirty job they've done...\" This letter, with its forced optimism, dates from October 1942, the darkest hours of Europe at war.\n\n* * *\n\nPaul is consumed with worry about Alexandre, having had no news of him since he left England, except that he is somewhere in Africa. Naively, he imagines that he might be granted leave. He doesn't know that this is the eve of the Normandy landings. On May 24, 1944, he writes to Gu\u00e9rin de Beaumont, the agent general for the Provisional Government of the French Republic in New York, hoping in vain to have Kiki brought over, after having been separated from him since June 1940: \"We're very depressed. His mother is in despair. It's really a miracle that she goes on despite her enormous pain... As for my own personal activities, apart from the Renoir Centenary exhibition organized for the benefit of the Free French Relief Committee, and the exhibition of C\u00e9zanne's works organized for France Forever and the Fighting French Committee, I don't need to mention them. My every action is that of a patriot who loves his country, particularly when it is in danger. I can say that I have spent my whole life fighting against the Germans who are after me, and that if I had stayed in France I would certainly have been taken hostage and faced the firing squad a long time ago.\"\n\nIn fact, he will find out nothing for a long time, either about the atrocities of the Nazis and their Vichy accomplices or about the looting. Above all, this \"patriot,\" as he terms himself, is unaware that on July 23, 1940, while he was still in Portugal, a law stripping nationality of any French citizen who has gone abroad was passed by Vichy France.\n\nThough he probably did know that on October 3, Le Journal officiel published the Jewish Statute, with its notorious Article 1: \"In terms of the application of the present law, any person will be regarded as Jewish if he is descended from three grandparents of the Jewish race or two grandparents of the same race if the spouse is also a Jew.\" After this come prohibitions concerning posts or honors awarded by the state and access to teaching positions, the army, high administration, or the courts. Jews would also be excluded from journalism and the management of newspapers, as well as work in cinema or the theater.\n\nThe Casino de Paris, other clubs, and certain parks and gardens were \"forbidden to dogs and Jews,\" as the signs put it. But as the writer Dan Franck notes bitterly, \"the duck with blood sauce at the Tour d'Argent retained its reputation.\" Franck also relates how the Op\u00e9ra and its director, Serge Lifar, a French ballet dancer and choreographer, welcomed Hitler and Goebbels, and how the young Herbert von Karajan conducted Tristan and Isolde there. As for the famous actor Sacha Guitry, \"all was just fine.\"\n\nPaul knew only scraps of all this. One thing he was certainly in the dark about was the deportations that followed upon the loss of nationality. On February 23, 1942, an order decreed the \"denationalization\" of Paul Rosenberg and his family. These orders complemented the law of July 23, 1940.\n\nA month later, on March 26, 1942, Paul sent a telegram to \"The President of the Commission for the Examination of Cases of Forfeiture of Nationality, Ministry of Justice, Vichy, France,\" stating: \"I am learning of my denationalization by order of 23 February 1942. Protest energetically and have strong reservations. Letter follows.\" A letter, addressed to the same commission, did in fact follow, on April 16, 1942, revealing great ignorance of the situation as well as total candor. Five pages in which Paul made rather clumsy attempts at self-justification: \"I learn that by an order of 23 February 1942, in accordance with the law of 23 July 1940, I have been stripped of French nationality for leaving France without a valid reason, between 10 May and 30 June 1940... I protest indignantly against the interpretation of the aforementioned text as regards my case... I have always fulfilled all my duties, my past is one of honor and probity, etc.\" This is a \"flagrant injustice... It was only during my stay in Portugal that I discovered the conditions of the armistice. This prompted me to continue my journey. In fact, after some reflection, I determined that I could make myself more useful by staying in the United States than by going back to France... Being stripped of one's nationality implies a dishonor that no worthy man can accept without attempting to defend himself. I am not begging for clemency for a crime I have not committed, but calling for justice to which I have a right like any other citizen.\"\n\nThe opening words of this letter reflect the state of mind of French Jews in 1940: unable to believe that while they were good enough to serve as cannon fodder in the First World War, they could be dismissed as traitors twenty years later just because they had been born Jewish. We come across this uncomprehending reaction in every country that has known discrimination and deportations, even in the state of mind of the people crammed into the cattle cars. It was impossible for a sane mind to imagine the Shoah in 1940.\n\nPaul obviously knew little about what was happening in his homeland. He had asked his friend Gilbert L\u00e9vy, in whom he had complete trust, to keep his papers and to ensure that wages were paid to the staff who remained at the gallery. To L\u00e9vy, who was to be deported and gassed in Auschwitz while one of his sons fought with my uncle Alexandre in the African campaign and died in his arms in Normandy, he writes with disconcerting naivet\u00e9 on March 20, 1942: \"I learn that I've been denationalized. Can you contact my brother, as I am asking him to find a lawyer should I need to defend my case to the commission[?]\" Paul was as yet unaware that there are some cases that cannot be pleaded.\n\n* * *\n\nBut let's return to the letter my scandalized grandfather sent to Vichy in 1942. The last paragraph, about the dishonor inflicted and the refusal to ask for a pardon that he judged to be defamatory, captures his state of mind. On the other hand, the feeble excuse that he would be more useful to France in the United States does not seem to match his indignation. My grandparents fled because their lives were in jeopardy, and they had no need to be ashamed. Yet admitting that others stayed on in their homeland and actually risked and often paid with their lives undoubtedly filled him with shame.\n\nAs Emmanuelle Loyer writes, \"Unlike the history of Poland, in which the exile is integral to the national story of the last two centuries, the French tradition is characterized by a disparaging image of the exile, which places him somewhere between flight and treason... Since the French Revolution, the exile has been accused of antipatriotism, and assimilation is seen as an active metonymy of the France of the Counterrevolution.\"\n\nPaul experienced the loss of his nationality inflicted by a regime he loathed as a wound and humiliation that imposed upon him a constant need for self-justification. If he had no plans to move his gallery back to Paris after the war and chose instead to stay in New York, it was probably because the art market was more vigorous there, although many Parisian art dealers, beginning with Kahnweiler, did prosper in France after the liberation. But the deepest reason was that unlike the French, who had stripped him of his nationality and some of whom were even involved in the theft of his property and would doubtless have had him deported, the Americans welcomed him along with his family, protected him, and enabled him to relaunch his career. They recognized him as a great practitioner of his trade and helped him recover his soiled dignity.\n\nI found the same tone in my father's war diary. Demobilized in 1940, unable to bear life in occupied France, he managed to leave for the United States. Once in New York, he felt very uneasy about being \"sheltered.\" He enrolled as a noncommissioned officer with the Free French and embarked with two compatriots on a British troop carrier, the only Frenchmen among eight thousand American soldiers. Traveling via South America and the Cape of Good Hope, he eventually came back up the Red Sea, disembarked, and went on fighting.\n\nMy father kept a journal throughout those three years, and even in his account of that two-month zigzag voyage across an ocean infested with mines and German submarines, followed by his time fighting for the Free French Forces in Beirut and Cairo, he expresses a constant need to rehabilitate himself, to \"redeem\" himself for his supposed passivity. Consumed with anxiety about his relatives who had stayed in Paris or were hidden away somewhere in France, he was hardly any happier about his life as a Gaullist envoy to the Middle East than he was with his life as a refugee in New York.\n\n* * *\n\nMy grandfather would refuse all contact with Vichy to \"plead his own case.\" To someone who had suggested acting as an intermediary, he wrote on April 24, 1942: \"Given recent events in France, I do not wish to communicate in any way with a government run by a man like Laval.* I would rather lose all I possess.\"\n\nAnd that was what happened to his paintings, indeed to his illusions of a just world.\nPREOCCUPATIONS OF THE HEART\n\nMy grandfather thought constantly about the lives of the painters who stayed behind in France, hoping they would be hostile to the occupying forces. Some of them were, but overall, the artists who remained in Paris didn't distinguish themselves one way or another. \"As soon as the Nazis were the adversaries of culture and freedom, any free expression of the spirit became an act of courage,\" wrote Laurence Bertrand Dorl\u00e9ac.\n\nIn fact, as Paul suspected, Braque, Matisse, and Picasso showed no sympathy for the Germans. Other artists, like Andr\u00e9 Derain, Otto Friesz, van Dongen, Paul Belmondo, and de Vlaminck, did not hesitate to go on tour in Germany. Some even returned as propagandists, so in thrall were they with the Nazi regime.\n\nBraque wasn't even invited along. \"Fortunately my painting didn't please them; I wasn't invited, otherwise, perhaps I would have gone, on account of the promised exchange of prisoners,\" he candidly confessed in retrospect. He had been a close friend of Derain, who had taken this politically charged tour, but he had no wish to disavow him. As Braque's biographer Alex Danchev writes, \"He was a moraliste, not a moralizer... But something was broken. Braque and Derain were never as close again.\"\n\nPaul was aware that Braque was no activist and that a painting like Guernica was not his style. Besides, Braque could not understand Picasso's commitment to communism or, later on, his decision to paint a peace dove. Braque's sole concern was the validity of his art. \"There is no scream in Braque, just a whisper,\" Danchev explains. But the war destabilized him, and he even dreamed of going to Switzerland. For the first time since 1917 he had stopped painting, as he wrote to my grandfather when Paul was still near Bordeaux, in Floirac.\n\nAfter he returned to Paris and before settling in Pacy-sur-Eure, where his aged mother lived, Braque started painting very dark still lifes (including his famous black fish). Until 1943, only his two great writer friends Jean Paulhan and Francis Ponge, both r\u00e9sistants, had the privilege of seeing his paintings.\n\nBut in 1943, a small exhibition was held in a room dedicated to Braque in the Salon d'Automne and hailed by the collaborationist Pierre Drieu La Rochelle, but denounced by Lucien Rebatet in Je suis partout, the emblematic publication of the collaboration.\n\nGeorges Braque had rejected the advances of the Reich, refused to prostrate himself before the Reich's official sculptor, Arno Breker, unlike Jean Cocteau, and dared turn up at the funeral of Max Jacob, who had died to general indifference in Drancy, shortly before his convoy left for Auschwitz. Braque also declined Marshal P\u00e9tain's invitation to design the Vichy emblem, \"Work, Family, Homeland.\" \"He wasn't part of the Resistance. But he was dignified,\" writes Dan Franck, \"a serious quality in a time of compromises.\" My grandfather, who for his article in Art in Australia imagined Braque \"in blue smock, confined to his home, standing before his easel, his pots of colour ground by himself, hand full of brushes, creating another new canvas for our pleasure,\" was right about the character of his old friend, to whom he displayed the most brotherly attachment. Paul described Braque as being very different from Picasso, \"always placid and a quiet conversationalist.\" \"He never sings out of tune,\" Picasso once declared about him. \"He seeks only harmonies and symphonies in his canvases. There is never the clash of colour like some strident note of a cymbal or trumpet. He represents all the beautiful French tradition of Corot, Chardin, and like these painters he is full of humility.\n\n\"Like Picasso,\" my grandfather continued, \"he [Braque] never paints from nature. His works are re-creations... He is never a mixer, living quite isolated, abhorring honours and receptions... The sight of certain uniforms must trouble his heart and soul.\"\n\nPaul was severe in his judgment of artists like Derain when he learned that they had accepted Vichy honors, but he moderated his condemnation. In August 1942, according to papers found in the family archive, Paul abandoned an exhibition of twentieth-century artists in New York: \"It is impossible to show artists who have been in Germany, while at the same time it is not a French custom to condemn people without having heard their side of the story, so it is impossible to hold this exhibition.\"\n\nHe was mistaken about other standard-bearers for fauvism, such as de Vlaminck, however, believing that they were resisting the occupying forces. Conversely, de Vlaminck, who was jealous of Picasso, took advantage of the occupation to tear into \"that Catalan with the look of a monk and the eyes of an inquisitor,\" as he wrote in the magazine Com\u0153dia. \"Cubism! Perversity of spirit, inadequacy, amoralism, as far from painting as pederasty is from love.\"\n\nPicasso couldn't afford to reply. He had left rue La Bo\u00e9tie, where the Nazis were now his next-door neighbors, and was living at 7 rue des Grands-Augustins, in an apartment found by Dora Maar, his companion at the time. He represented \"the ultimate scapegoat meant to embody the thousand and one facets of evil, displacement, disorder and blasphemy,\" writes Dorl\u00e9ac. The Gestapo could have arrested the painter at any time, but at Cocteau's request, he was given some protection on the German side by the all-powerful sculptor Arno Breker.\n\n* * *\n\nSince Picasso had opposed Franco very early on in the conflict, the republicans had appointed him honorary director in exile of the Prado. After the April 26, 1937, bombing of the small Basque village of Guernica on a market day by the German pilots of the Condor Legion, Picasso, who had been commissioned to create a mural for the Spanish Republic Pavilion at the Universal Exhibition in Paris, painted Guernica, one of his greatest masterpieces. Picasso never forgot that P\u00e9tain had been the French ambassador to Franco's Spain, which may also explain his antipathy toward the Vichy regime.\n\nThere is a legend about this world-famous painting. German officers, visiting Picasso in his studio in rue des Grands-Augustins and seeing that most accusatory of paintings in a corner, were said to have asked him: \"Did you do that?\" According to legend, the painter shot back, \"No, you did.\" A sublimely dramatic reply, although I suspect it may be apocryphal. My grandfather and my mother visited Picasso in the same studio just after the liberation. As they congratulated him on the courageous statements he had made, statements that had crossed the seas as a symbol of the resistance of artists and intellectuals to the occupying forces, Picasso replied, slightly embarrassed, \"Yes, I must have said something like that. Well, all right, let's say I did...\" This was a story often told by my grandfather and later by my mother.\n\nBut I have no other evidence of Paul and Picasso's discussing the war, not even at its start. At that point in early 1940, Picasso was at Royan, a small fishing village on the Atlantic coast, while my grandparents were living in Floirac. In any case, the letters don't so much as mention the declaration of war on September 3, 1939. Perhaps they spoke on the phone that day.\n\nOn October 25, 1939, Paul alludes to the war when he sends birthday wishes to \"mon vieux Pic,\" roughly two months before his own: \"It's a sad birthday,\" he writes. On December 29, 1939, Paul, who turned fifty-eight that day, sends Picasso \"my best wishes for 1940. You will cost me two times two 30 franc stamps for the telegram. And yet our authorities said we had to economize!\"\n\nSo the war always seems to be mentioned with some detachment in the correspondence between the future refugee and the Spanish republican. Certainly, this battle-free conflict must have seemed like an abstraction at that point, but I am still struck by the fact that there was so little room for it in their exchanges, in which they continue to \"talk paintings.\" My grandmother even sent a message to Picasso expressing her relief that my grandfather finally had paintings on his walls in Floirac, which had been bare until then. He had in fact had them sent from Paris, thinking they would be safe south of the Loire. \"Your paintings from 1940 are in the dining room,\" Paul writes. These were probably the tormented paintings the artist made that year, such as the Standing Female Nude, cited by Laurent Fabius as an example of the art of a painter devastated by the war. \"Thanks to you,\" Paul continues, \"our meals are less monotonous, your canvases provoke both appreciation and hilarity.\" In the same letter, he announces the death of Diola, his children's dog; my uncle Alexandre gave the dog's name to the plane he piloted in the Second Armored Division.\n\nThe last letter from Paul to Picasso, before they met again in the painter's studio on rue des Grands-Augustins, is dated May 9, 1940, the eve of the Nazis' offensive in which the Allies were taken by surprise in the Ardennes. Paul tells Picasso about his plan to go to Paris on May 14. After all, for almost everyone, this phase of the war\u2014which became known as the Phoney War\u2014had turned out to be only virtual. My grandfather fled Floirac through Spain and Portugal one month later.\n\n* * *\n\nAnd yet Picasso is very attuned to current events. During the Phoney War, he takes a quick trip to Paris from Royan. It's spring 1940, and Picasso bumps into Matisse. \"Where are you going like that?\" asks Picasso. \"To see my tailor,\" replies Matisse. \"What, you don't know that the front has been broken? The Germans will be in Paris by tomorrow!\" \"What about our generals?\" Matisse asks him. Picasso looks at him seriously and replies (his response is in all the books): \"Our generals are equivalent to the \u00c9cole des Beaux-Arts!\" Which tells us a lot about both these painters' attitudes toward that school, so fearful of innovation, as well as of the French Army, which was stuck in the days of the First World War.\n\n* * *\n\nPicasso returned to the capital after the armistice. Why did he stay in Paris? My grandfather thought he was frightened by the idea of exile. \"Staying wasn't a form of courage, but... of inertia,\" Picasso later said to Jean Leymarie, an art critic and the future director of the Mus\u00e9e d'Art Moderne. Picasso wanted to devote himself exclusively to his work.\n\nIn 1943, Picasso met Fran\u00e7oise Gilot, who became his companion and the mother of two of his four children, Claude and Paloma. Around that time he invited to his house some of his politically committed friends, figures like poet Robert Desnos, but he didn't join the Resistance, as his friend Paul \u00c9luard had done. \"He refused the Germans' coal, and the material advantages they wanted to give him,\" writes Franck. \"He was primarily concerned about his artistic work. Picasso was entering an intensely prolific phase that was to last the rest of his life, and he abstained from anything that kept him away from that 'galleon's rhythm.'\"\n\nIn 1941 Paul imagined his Pic in a state of revolt, since he was \"the freest of men.\" \"What pleasure can he possibly have in painting now?\" Paul wonders. \"It had always been his joy to confront a canvas, mold it, work it meticulously in terms of depth, form and color, knead it, even torture it, and force it to give way to his titanic will.\" That suffering doubtless existed, as did the artist's anxiety and discomfort with fascism. But they didn't stop him from making art.\n\nIn April 1940 Picasso had once more petitioned for naturalization, but this had been refused on the grounds of his alleged anarchist sympathies. He chose to stay, even though he still feared being handed over to Franco. Police reports from 1939\u2014they would still have been the police of the Third Republic\u2014had him on record for making \"anti-French\" statements at the Caf\u00e9 de Flore. \"A curious way of thanking the country that welcomed him, and in the current circumstances his conduct is inconvenient at the very least,\" said one police report of the time.\n\nThat same report, written even before the German invasion, stresses that \"this foreigner who has made a reputation for himself in France in so-called modern art, allowing him to make considerable sums of money, is said to have declared several years ago to some of his friends that when he dies he wants to leave his collection to the Russian government and not to the French government.\" The stage was set for blacklists and xenophobia.\n\nSo Picasso was the holder of a residence permit, weirdly confused at the time with a kind of identity card, renewed on November 30, 1942, and valid until November 30, 1945. In the margin of the document was a note: \"Catholic.\" And this, written by hand: \"I certify on my honor that I am not Jewish in terms of the law of 2 June 1941\"\u2014the law that repeated and hardened the terms of the 1940 Jewish Statute. It was signed \"Picasso.\"\n\nTroubling. Paul would have been shocked. But the artist needed to survive both the tragic events around him and the looting.\nTHE TRAIN, SCHENKER, AND THE ART OF THE POSSIBLE\n\nAugust 27, 1944, and the troops of the Second Armored Division under the command of General Leclerc had just liberated Paris. Members of the Resistance had alerted them that a train containing one final convoy of looted works of art was about to leave the capital for Germany. A detachment of six volunteers, led by Lt. Alexandre Rosenberg, planned to stop the train at Aulnay, in the suburbs of Paris. On board were some dazed, homeward-bound old German soldiers and 148 crates of modern art, a small percentage of which belonged to the father of the lieutenant in question. Alexandre had last seen their contents on his parents' walls at 21 rue La Bo\u00e9tie, in 1939.\n\nThat train, which was leaving for Germany, was the final act of the huge program of looting that the Nazis had pursued in France and in all the countries of occupied Europe. Two weeks after the armistice, Hitler, on the pretext of bringing these works to safety, issued an order that all art objects belonging to the Jews should be \"protected.\" \"It is not an appropriation,\" said the memo that had come from Berlin, with the cynicism of those who think that the bigger the lie, the more likely it is to be believed, \"but a transfer under our guard, as a guarantee for the peace negotiations.\"\n\nThe first of the raids had begun in the summer of 1940. It was then, as the art historian and r\u00e9sistant Rose Valland writes, that \"the German Embassy became the Nazi ministry of culture in an occupied country.\" It was not until October 30, 1940, that about 450 crates left the rue de Lille (where the Reich Embassy was located) for the Mus\u00e9e du Jeu de Paume, to be submitted to the meticulous and systematic classification process perfected by the Einsatzstab Reichsleiter Rosenberg (ERR).\n\nOn July 4, 1940, Otto Abetz, the Reich ambassador in Paris, sent the Gestapo a list of the leading Jewish collectors and dealers in the city: Rothschild, Rosenberg, Bernheim-Jeune, Seligmann, Alphonse Kann, etc. It was on that day that the house at 21 rue La Bo\u00e9tie was sequestrated, along with the works of art that Paul had left there, a library of over twelve hundred books, all the furnishings (from the antique furniture to the kitchen utensils), several hundred photographic prints, and the whole of the gallery archives dating back to 1906.\n\nThe objects looted included a number of sculptures, which had remained in Paris because they were difficult to transport, among them a large Aristide Maillol and the two famous Auguste Rodin statues Eve and The Bronze Age, which had adorned the foyer. The same fate awaited The Thinker, which was recovered after the war and which as a child I saw so many times, welcoming visitors, while I looked down from the top of the stairs to the gallery at Seventy-ninth Street.\n\nThe French police supplied the trucks; the Gestapo, the men. As for the paintings that came from the most important collections in Paris, these were stacked up at the German Embassy.\n\nThe route taken by the stolen art objects is now well documented: the German forces looted about thirty-eight thousand apartments. The German dealer Gustav Rochlitz acted as a clearinghouse, exchanging the art favored by the Nazis\u2014old masters\u2014for works that appealed to Parisian dealers with their more contemporary taste. From this immense act of larceny perpetrated in France by the Nazis, about two thousand works have been recovered and remain unrestored to their rightful owners. Stamped \"MNR,\" they belonged to families who had fled or been deported and will never return to claim them.\n\nIncluding the paintings remaining at rue La Bo\u00e9tie, the 75 on the walls of the house in Floirac or rolled up in the garage there, and the 162 from the vault in Libourne, a total of 400 paintings were stolen from Paul. About 60 of them are still missing (are they in France, in Germany, in Russia?), most of which will probably never be found. The paintings that were recovered by Paul himself formed the inventory of the Seventy-ninth Street gallery, which has been almost entirely depleted since his death more than half a century ago.\n\nSome of these works still show up from time to time, in estate sales or auctions. How I wish I could make them speak, so that they could tell the story of their odysseys, or rather of how they ended up tucked away in the apartments of families that never mentioned a word to anybody after fraudulently getting hold of them. In most cases the people who inherit them today know nothing of their provenance, which is buried along with the memory of those who appropriated them during those dark years.\n\n* * *\n\nAfter the conclusion of the last restitution cases in the mid-1960s, the subject of the looting of artworks during the Second World War remained hidden until the early 1990s, when the issue of the wartime persecution of the Jews in France slowly reemerged in the public eye. The books of Lynn Nicholas and Hector Feliciano also helped bring the issue back into public scrutiny.\n\nIn 1997 the Matteoli Commission, set up by Alain Jupp\u00e9's government and continued under Lionel Jospin, was charged with studying the spoliation of Jewish assets during the occupation. \"The looting had nothing to do with the circumstances born of the conditions of the victory of the Reich, but only with a fundamental and founding intention, matured and developed along with Nazi expansionism,\" as one of the contributors to the commission put it.\n\nIn an article based on this investigation titled \"From Spoliation to Restitution,\" Annette Wieviorka brings out the subtle distinction between spoliation and looting: \"Spoliation, as defined by G\u00e9rard Lyon-Caen, is 'legal theft.' It is essentially a product of the Aryanization process, in which a property passes from 'Jewish' to 'Aryan' hands... Beyond the spoliation is the problem of looting. This is essentially undertaken by the German authorities. Two kinds are identified: First is targeted looting planned by the Germans. The Germans kept their eyes on the artworks of the great Jewish art dealers or collectors such as Alphonse Kann, Paul Rosenberg, Wildenstein, and the Rothschilds. This spectacular haul involved valuable works that were taken to Germany. The second type of looting began in 1942 and involved emptying Jewish apartments of all they contained.\"\n\n* * *\n\nIn the course of my research into the recovery of artworks owned by my grandfather, I discovered an extensive document that I'd never heard of before, the name of which reminded me of the title of the Steven Spielberg film Schindler's List. In contrast with the plot of that film about a righteous gentile who saved Jews from the Nazis, this is a collection of documents titled the Schenker Papers, which was declassified in 1995. Drawn up by the German Schenker transport company and reproduced on microfilm by the OSS, it lists the galleries and individuals that sold works of art to German museums, providing thirty-seven names. These include the dealers \"who never declared sales made to the Germans, even though they had, to our knowledge, concluded numerous deals with the occupying forces\u2014we have proof of it.\" Among the names on this document were Martin Fabiani and Roger Dequoy, the latter being, as we have seen, employed by the Wildenstein family, as manager of its gallery during the occupation.\n\nAn exhibition organized in 2008 by the Ministry of Culture, the Ministry of Foreign Affairs, the Direction des Mus\u00e9es de France, and the R\u00e9union des Mus\u00e9es Nationaux, in collaboration with the Israel Museum in Jerusalem, set out a clear account of suspicious purchases made by equally suspicious dealers: \"Martin Fabiani\"\u2014compromised in all the documents and quoted in the context of that exhibition\u2014\"sold many paintings during the Occupation and was found guilty for this after the liberation.\" My grandfather would describe Fabiani's reaction after being shown pictures of various paintings he was trying to retrieve. Fabiani denied having possessed any of them, including the ones he himself had returned to my grandfather. \"He probably hadn't noticed,\" my grandfather said ironically, \"that all the paintings stolen by the Germans bore on the back of the frame the words 'Paul Rosenberg\u2014Bordeaux,' followed by the initials PR and a number, a note appended by the Germans, and which would still have been there when he bought the paintings. And he handed over several canvases without asking for either proof or photographs!\" In the end, Fabiani returned twenty-four artworks without a word of protest.\n\nRegarding Paul P\u00e9trid\u00e8s, who died in 1993 at the age of ninety-two, the same 2008 exhibition said that he had been sentenced to three years in prison in 1979 but was freed at seventy-eight because of old age. His claims, after the liberation, that he knew nothing about this illegal trade and that like his colleagues, he had not knowingly bought a single canvas stolen from a Jew, left my grandfather cold: \"It is not customary in the trade to buy canvases without first investigating their origins, and to be satisfied with the explanations of German intermediaries unknown to the Paris market.\"\n\nIn the end my grandfather did not bring a case against either P\u00e9trid\u00e8s or Fabiani. So why did he instead decide to pursue unscrupulous Swiss dealers, and why was he more lenient toward the French dealers when some of his paintings were recovered? Was it because he feared that political networks favored those dealers who had collaborated, as they did many civil servants who had been even more seriously compromised? Or because he suspected that the entire art market would be discredited if the public were told about dealers who had behaved badly? Or because he preferred to force them to return his property in his presence and to recover his paintings one by one, in a kind of Count of Monte Cristo\u2013style personal vendetta?\n\nAnother paradox that makes me uneasy: my grandfather treated the petty thieves with even greater severity than he did the major crooks, suing them for fraud, abuse of trust, theft, or embezzlement. This was the case with M. Picard, the concierge at 21 rue La Bo\u00e9tie, who had worked there since 1931.\n\nPicard had stolen some objects with the intention\u2014he said in a 1945 statement he prepared for the trial\u2014of safeguarding them before ultimately returning them to the Rosenberg family. \"One day,\" Picard testified, \"I was instructed not to let anybody into the house that had been sequestrated by the Germans. On April 25, 1941, the Gestapo moved into the building and I had to give all the keys to the Germans. Two days later they moved out M. Rosenberg's library. On May 2 they moved the furnishings into German cars and replaced them with office materials.* On June 28, I was ordered to leave the premises. In the meantime, I had managed to take various objects from the apartment and the Galerie Rosenberg with the intention of giving it back and only with a view to saving them. It was never my intention to take anything at all for myself.\"\n\nThe testimony of Marguerite Blanchot, the Rosenbergs' housekeeper since the 1920s, is categorical about the building's concierge. \"I had the keys to number 21, and Monsieur Rosenberg had told me to move into his apartment. But M. Picard advised me against it and even added that it would be unwise to keep the keys. So I returned them to M. Picard and I came every day until November 1940 to wrap up the linen and the silverware with M. and Mme Picard. It was he who sealed the cases that we filled, and he refused to do it in my presence in spite of my requests. I went back to rue La Bo\u00e9tie several times, but the Picards refused to let me in. The concierge at 20 bis can testify to that. The day before the building was occupied by the Germans, I went to the apartment. When I wanted to get the furniture out, the concierges wouldn't let me.\"\n\nRen\u00e9 Duval, who worked at the office in the Galerie Rosenberg, testified that he too tried to save some of the belongings from rue La Bo\u00e9tie but that the Picards were opposed. \"I never saw anyone taking anything, but I noticed a number of gaps among the paintings, some of which were hung on the walls at the homes of the concierges who told me they had only put them there to save them.\"\n\n* * *\n\nL\u00e9a Roisneau had been Paul's secretary since 1936. It was she who first alerted him to the looting. In March 1941 she sent my grandfather a letter, saying, \"There's nothing left, nothing, nothing, nothing.\" Her former boss, three thousand miles away in New York, was unaware of so many things. He had no idea that the looting was orchestrated at the highest level of the Nazi hierarchy and that the raids were being carried out against \"all the enemies of the Reich\" in the occupied territories.\n\nRoisneau also went several times to rue La Bo\u00e9tie, to try to rescue the objects that struck her as most important: the library and the photographs of the paintings. She too observed that the Picards not only took refuge behind the Germans but were further distinguished by their ill will. \"One day he\u2014Picard\u2014told me that he wasn't going to let me back into the building, and added that if the Jew Rosenberg came back, he would throw him out the door,\" said Roisneau in the records.\n\nIn fact, Picard had stored objects everywhere: with neighbors, with his relatives. He had even taken Rodin's Thinker to an expert, along with a big wood-and-bronze clock. Initially he said he had given my grandfather's youngest brother, Edmond, everything that belonged to Paul; then he confessed that he had lied. Edmond began the inventory of looted objects after the liberation and before Paul returned to France. Mme Picard confirmed: \"My husband didn't tell the truth. And after the exodus, we took different things out of M. Rosenberg's house and stored them at the furniture depository: bronzes, a marble bust, an inlaid side table. Also between 140 and 150 bottles of fine wine and champagne (we consumed about fifty of those bottles), and a portrait drawing of Mme Rosenberg.\"\n\nPathetic, petty larcenies! Picard had his curtains cut from my grandfather's tapestries and confessed that the Regency barometer mentioned by his wife was actually found in a furniture depository stored in his name. But was my grandfather really more appalled by this than he was by the crimes of the collaborationist art dealers?\n\nThe rest\u2014the antique tables, the mahogany chests of drawers, the buffet tables, the chairs\u2014was sold by Captain S\u00e9zille, the secretary-general of the IEQJ, to his own employees or used at the Palais Berlitz to furnish the notorious IEQJ exhibition The Jew and France.\n\n* * *\n\nIn Floirac the scenario was almost identical: enter, in order of appearance, the occupying forces and the innocent bystanders who, by their own accounts, only wanted to help the family but who ended up taking advantage of the situation.\n\nOn September 15, 1940, the Germans arrive at Le Castel de Floirac at dawn; five vehicles filled with German soldiers and policemen stop outside the house.\n\nThe Germans demand to see Louis Le Gall, Paul's chauffeur, who has unsuccessfully been trying for days to persuade the hauler Lamarthonie to send the paintings that have remained in Floirac to Lisbon: some Monet water lilies, a Delacroix, some works by Picasso, L\u00e9ger, Matisse, Sisley, Vuillard, and Utrillo. In a letter of July 6, 1940, three weeks after his hasty departure from Floirac, Paul has asked Louis for an inventory of all the objects he wants dispatched, including the seventy-five paintings stored at Le Castel. \"Don't forget the ones that were left in the chest above the garage, and please be kind enough to check that none is missing,\" writes my grandfather.\n\nThe Germans are well informed and already know everything about Louis. \"I was stunned by the amount of information they had about me,\" Louis later said. Lamarthonie, the trucking company based at 17 cours du Chapeau-rouge in Bordeaux, was to accept delivery of the trunks and crates. It never did so, however, instead requesting the list of objects twice, also asking for the number of paintings. \"Then Lamarthonie told me the border was closed. The attitude of M. Lamarthonie and M. and Mme Ledoux toward the 'Israelites' led me to think it unlikely that they were strangers to the information [that the Germans had about me],\" Louis Le Gall would testify.\n\n* * *\n\nThe German police search the house from top to bottom and take everything they find to the German Embassy in Paris, before it is transferred to the Jeu de Paume and then dispersed around Germany and Switzerland, or in France.\n\nA certain Comte de Lestang and someone by the name of Yves Perdoux, probably an obscure art dealer, had apparently made a pact with the Nazis: they would tell them the two addresses in the department of Gironde, that of the house in Floirac and that of the vault in Libourne, where Paul had stored his paintings. In return for this spectacular bounty, they asked for 10 percent of the value of the collection. They tried several times to negotiate their price before finally supplying the Libourne address. In the end, they accepted three Pissarros and a Renoir, far beyond their wildest dreams. But even if you're an informer, do you really negotiate with Nazis?\n\n* * *\n\nWhat was the actual conduct of M. and Mme Ledoux? It was probably not very different from that of many people who witnessed the looting, who were powerless but often indifferent and sometimes opportunistic. The postwar trials were not categorical about whether M. and Mme Ledoux did or did not take part in the embezzlements; the Germans weren't given to sharing the fruits of their plunder. But it's more than likely that they did take advantage, even if only by preventing Louis Le Gall from removing the crates that could have been saved.\n\nLater, when objects were found hidden under a woodpile in the garden shed, Mme Ledoux revised her initial statements: \"Contrary to what I claimed before, I was in fact able to salvage a painting by Renoir, another by Degas, a case of silverware, a case of books. My intention was to keep them from the Germans. I planned to return them to M. Rosenberg as soon as possible.\"\n\nThe Germans occupied the property in Floirac until August 27, 1944, when Bordeaux was liberated. M. Ledoux was detained for a time at the camp of M\u00e9rignac because of his behavior during the occupation, and then M. and Mme Ledoux regained their property, which they enlarged during the 1950s and ultimately sold to the municipality. That was the same Le Castel to which I paid my emotional visit, seventy years later.\n\nAs for Lamarthonie, the hauler, he declared: \"I was not aware of any request for transportation being made to me in 1940 by a M. Rosenberg or any of his representatives. However, it is possible that such a request was received by my authorized representative, now deceased, but I can find no trace of this matter in my archives...\"\n\nThe BNCI vault in Libourne, in which my grandfather had imagined his paintings would be safe, was broken into on April 28, 1941, at the request of and in the presence of the occupying authorities. Everything was transferred to a second safe, and this time, on September 5, 1941, a German ERR officer removed the 162 paintings from the BNCI vault. The works were immediately dispatched to Paris, where they fell into G\u00f6ring's clutches. They were major paintings: Degas, Manet, Bonnard, Matisse, Braque, Picasso, Ingres, Corot, van Gogh, C\u00e9zanne, Renoir, Gauguin.\n\nSome of these paintings from Libourne found their way to Parisian dealers. Others found takers in Switzerland and were recovered after several suits brought by Paul against certain Swiss dealers who demonstrated a remarkable lack of curiosity regarding the provenance of the works they were selling. After all, the backs of many of the canvases that passed through their hands in those years bore labels put there by the ERR meticulously identifying the collections from which they came.\n\n* * *\n\n\"No case,\" Lynn Nicholas writes, \"illustrates these difficulties better than the decades-long struggle of Paul Rosenberg and his heirs, whose possessions reposed not only in France and Germany but also in the neutral country of Switzerland.\"\n\nIronically, the most delicate battle of all was fought on Swiss soil.\n\nIn September 1945, Nicholas relates, Paul arrived in Zurich armed with lists as well as photographs of paintings that belonged to him. He went straight to the dealers, one after the other. \"The dealer Theodor Fischer, in Lucerne, acquired numerous paintings belonging to Paul Rosenberg in Germany, and sold them to private individuals. Paul Rosenberg at last discovered this and launched an action against the Federal Tribunal of Switzerland. The claim was granted, and the defendants were condemned to restore to the plaintiff the paintings demanded from each of them.\" It was then up to them to make their own claims against the Germans!\n\nPaul's complaints referenced thirty-seven paintings, twenty-two of which were in Fischer's possession. It is easier for me to understand his determination in this case than it is to grasp the impulse that led him to bring suit against small-scale profiteers.\n\nPaul discovered one of his paintings by Matisse, Woman in a Yellow Armchair, at the Neupert Gallery in Zurich, where he was even told it was from a private collection. Going higher up the chain, he went to see Emil B\u00fchrle, another dealer, \"who was surprised to see me, because he had chosen to believe the rumor that I was dead,\" as Paul told the story. Paul then accused him of knowingly buying stolen goods. B\u00fchrle replied that he would return them to Fischer if he got his money back. The two dealers tried to bargain with Paul: he could take back 80 percent of his paintings, leaving the rest. \"But Rosenberg was on a crusade and wanted an official, government-to-government settlement,\" believing that the Swiss government would be willing to negotiate at any price, in order to avoid a scandal.\n\nIf my grandfather had to wait for the liberation to find out the extent of the dispersal of his art, as early as 1942 he had been concerned about the fate of stolen paintings all across Europe. He saw it as an attack on the artistic legacy of the war-torn continent. Trying to motivate the Allies, he offered his assistance and cooperation to the profession as a whole, pro bono.\n\nPaul was resolved to return to Paris, to hunt down his scattered collection since 1944, but the War Ministry had not yet authorized French citizens to come back to their country.\n\nAs soon as he was able to make contact with the painters closest to him, he asked them for certificates, as he did in this telegram to Matisse in November 1944: \"Do you have pictures of last paintings I bought from you, because all taken by Boches [Germans] and resold.\"\n\nHe also insisted, as he did with Braque and Picasso, that Matisse provide a statement that when he visited Floirac in May 1940, he saw one or the other of his own paintings on the walls, proof that Paul had not had sufficient time to sell them before his hasty departure.\n\n* * *\n\nIt was up to the countries in which these acts of plunder had taken place to decide who rightfully owned the recovered works. In France, this task fell to the Commission de R\u00e9cup\u00e9ration Artistique (CRA, the French Restitution Commission), which was set up in 1944 under the tutelage of Jacques Jaujard, the director of the National Museums of France under the occupation, and of the intrepid Rose Valland.\n\nThe CRA quickly returned the works recovered on the Aulnay train, and these were followed by others found at Neuschwanstein Castle in Bavaria. As an expression of gratitude, Paul donated thirty-three of these paintings to major French museums, including the Louvre.\n\nEven today there are works stamped \"MNR\" and found by the Allies but whose owners have never been identified. And I dare to say it: lying in the basements of prestigious French museums there are still unidentified paintings, whose owners disappeared into the camps and whose inheritors may one day be traced after a vetting of the archives. The museums make no secret of this. They are awaiting the return of those who will not come back.\n\n* * *\n\nAll those battles waged in Paris (whether against big fish or small) or in Switzerland revitalized Paul after long years of waiting. They made him feel that he was achieving a measure of personal justice. At the same time he was gaining perspective. He was clearly aware that these battles were trivial compared with the catastrophe of the Shoah, the atrocities of which were just coming to light. In April 1945 he writes: \"We recovered some paintings looted by the Germans, or by dishonest Frenchmen. But I am not going to complain, it's as nothing when you look at the horrors that the Nazis inflicted on human beings of all races, creeds, and colors.\"\n\nLike the other dealers whose collections had been plundered, he applied for reparations from the Federal Republic of Germany, which in July 1957 passed a law providing financial restitution for losses caused by spoliation. Two years later, in 1959, the Germans proposed a settlement of less than half the sum Paul had claimed. He had died by then, and my grandmother, my uncle, and my mother, wearied by all the procedures involved, accepted their offer.\n\nIn 1970, and again in 1980, restitution was back on the agenda, and my mother and my aunt reclaimed paintings by Monet and L\u00e9ger. Alexandre went so far as to buy back a Degas from its illicit owners. \"I do not like so enriching the successors to thieves,\" he said, as Lynn Nicholas records, \"but have come to learn that the defence of one's own and one's family interests is somewhat like politics and indeed life itself. It is principally the art of the possible.\"\n\nMy grandfather's battle to recover his assets, which occupied the latter years of his life, was certainly legitimate, but I can see how it might have been perceived as unseemly by families whose relatives' ashes are forever buried beneath the crematoriums at Auschwitz or even to those who survived the camps. My grandfather was safe, and so was his family. His son had come back a hero of the Second Armored Division, and he still had enough paintings to do business and live well.\n\nWithout wishing to play psychologist, I think he needed to make the thieves pay, to do his part in the work of remembrance and of bringing the truth to light. Perhaps he had adopted the phrase that the French Jesuit and scholar Michel de Certeau applied to his historical research, and that was quoted by Annette Wieviorka in the conclusion to her work for the Matteoli Commission, as his credo: \"a burial of the dead, that they may return less sadly to their graves.\"\nEPILOGUE\n\nWhen I began my research, I didn't set out to write a biography. Rather, I wanted to create an homage to my grandfather, a series of impressionist strokes to evoke a man who was a stranger to me yesterday, yet who today seems quite familiar. I wanted to conjure a world, the world of modern painting, one that was mysteriously restored to me, in a random sequence of opened cardboard boxes, and was a product of the French national obsession with security that manifested itself as a bureaucratic aberration.\n\nYes, this improvised portrait is about a forgotten era, that of France in its greatest glory, the expression of a resplendent artistic culture in the early years of the twentieth century.\n\nAbout the mutilations of the \"world of yesterday\"\u2014to quote the title of Stefan Zweig's moving autobiography\u2014which disemboweled Europe, tested the planet, and shattered millions of lives.\n\nAbout a family that is mine, which I might at last describe\u2014if I allow myself to borrow from Jean-Paul Sartre\u2014as a whole family, composed of all families and \"as good as all of them and no better than any.\" But a family dearer to me than I would have believed and to which I owe more than I could have imagined.\n\n* * *\n\nIn May 2011, under painful circumstances, I found myself forced once more to live in New York, a prisoner, to some extent, of America. The city of New York itself, which seemed enchanted to me in my childhood, had now become, for both me and my family, a place synonymous with violence and injustice. I had trouble regaining the pleasure of wandering along its streets.\n\nI went back, of course, to Fifty-seventh Street, to the stretch of pavement once occupied by the first Galerie Rosenberg, where the luxury boutiques now extend, between Fifth and Madison. I walked along Seventy-ninth Street, in front of the last of the family galleries, on the Upper East Side, which now strikes me as prodigiously ordinary.\n\nIn midtown, I sauntered through the Museum of Modern Art, where, in the room reserved for the impressionists, so rich in dazzling works, I fix my attention on the portrait that stares pointedly at the visitors: that of van Gogh's friend and model Joseph Roulin, the famous postman with the bushy beard, the word \"Postes\" proudly emblazoned on his cap. That painting was given to the museum by my grandparents, who were so grateful to Alfred Barr and his country for offering them asylum and the recovery of their dignity. How could I allow the chaos of my recent reality to trample cherished childhood memories? How could I resent the entire city over one grueling experience? I never expected these pages, which opened with an identity denied in France, to finish on a forced, turbulent stay in America.\n\nBut that of course is another story. If I were a journalist, I might one day write a book about it.\nNOTES\n\nRUE LA BO\u00c9TIE\n\n. E. T\u00e9riade, \"Feuilles volantes,\" supplement, _Cahiers d'art_ 10 (1927).\n\n. Quoted in Pierre Nahon, _Les Marchands d'art en France, XIXe et XXe si\u00e8cles_ (Paris: \u00c9ditions de la Diff\u00e9rence, 1998).\n\nNUMBER 21 UNDER THE GERMANS\n\n. Quoted in Neil Levi, \"'Judge for Yourselves!': The 'Degenerate Art' Exhibition as Political Spectacle,\" _October_ 85 (Summer 1998): 41\u201364.\n\n. Ibid.\n\n. Lynn H. Nicholas, _The Rape of Europa: The Fate of Europe's Treasures in the Third Reich and the Second World War_ (New York: Alfred A. Knopf, 1994).\n\n. See the historical and intellectual treatment of this passage in ibid.\n\n. Ibid., 13.\n\n. Ibid., 7.\n\n. Laurence Bertrand Dorl\u00e9ac, _L'Art de la d\u00e9faite, 1940\u20131944_ (Paris: Seuil, 1993).\n\n. Ibid.\n\n. Rose Valland, _Le Front de l'art: D\u00e9fense des collections fran\u00e7aises, 1939\u20131945_ (Paris: Plan, 1961; repr. Paris: R\u00e9union des Mus\u00e9es Nationaux, 1997).\n\n. Laurent Joly, _Vichy dans la \"solution finale\": Histoire du Commissariat g\u00e9n\u00e9ral aux questions juives, 1941\u20131944_ (Paris: Grasset, 2006).\n\n. Ibid.\n\n. Joseph Billig, _Le Commissariat g\u00e9n\u00e9ral aux questions juives, 1941\u20131944_ (Vichy: \u00c9ditions du Centre, 1955).\n\n. Quoted in Dorl\u00e9ac, _L'Art de la d\u00e9faite._\n\n. Ibid.\n\n. Louis-Ferdinand C\u00e9line, _Lettres_ , edited by Henri Godard and Jean-Paul Louis (Paris: Gallimard, 2009).\n\nFLOIRAC\n\n. Correspondence quoted by Alex Danchev, Braque's authorized biographer, in _Georges Braque: A Life_ (New York: Arcade Publishing, 2005).\n\n. Ibid.\n\n. Document quoted in the lawyers' notes for recuperations after the war. Family archives.\n\n. Henri Matisse archives.\n\n. Ibid.\n\n. Paul Rosenberg, \"French Artists and the War,\" _Art in Australia_ , December 1941\u2013January 1942.\n\n. Emmanuelle Loyer, _Paris \u00e0 New York: Intellectuels et artistes fran\u00e7ais en exil 1940\u20131947_ (Paris: Grasset, 2005).\n\n. Quoted in ibid.\n\n. See Dan Franck, _Minuit_ (Paris: Grasset, 2010).\n\n. Family archives.\n\n. Loyer, _Paris \u00e0 New York_.\n\nGENNEVILLIERS\n\n. Loyer, _Paris \u00e0 New York._\n\nDEALER\n\n. Pierre Assouline, _L'Homme de l'art: D.-H. Kahnweiler, 1884\u20131979_ (Paris: Gallimard Folio, 1989).\n\n. Ibid.\n\n. Ibid.\n\n. Michael C. FitzGerald, _Making Modernism: Picasso and the Creation of the Market for Twentieth-Century Art_ (Berkeley: University of California Press, 1996).\n\n. Family archives.\n\n. Assouline, _L'Homme de l'art._\n\n. Rosenberg, \"French Artists and the War.\"\n\n. Albert Wolff, \"Le Calendrier parisien,\" _Le Figaro_ , April 3, 1876.\n\n. Family archives.\n\nCH\u00c2TEAUDUN, OP\u00c9RA, AND MADISON AVENUE\n\n. Paul Rosenberg: \" _Je suis n\u00e9...,_ \" autobiographical sketch, from which the quotations in this chapter are taken. Family archives.\n\n. T\u00e9riade, \"Feuilles volantes.\"\n\n. Henri Matisse archives.\n\n. Ibid.\n\n. Nahon, _Les Marchands d'art._\n\n. Hector Feliciano, _The Lost Museum: The Nazi Conspiracy to Steal the World's Greatest Works of Art_ (New York: Basic Books, 1997).\n\n. Ren\u00e9 Gimpel, _Journal d'un collectionneur: Marchand de tableaux_ (Paris: Calmann-L\u00e9vy, 1963).\n\n. _The New York Times_ , December 7, 1953.\n\n. Family archives.\n\n. Ibid.\n\n. Henri Matisse archives.\n\n. Ibid.\n\nMOTHER AND CHILD\n\n. Picasso archives, Mus\u00e9e Picasso.\n\n. FitzGerald, _Making Modernism._\n\nPAUL AND PIC\n\n. The language is from FitzGerald, _Making Modernism_.\n\n. Nahon, _Les Marchands d'art._\n\n. Ibid.\n\n. FitzGerald, _Making Modernism._\n\n. Family archives.\n\n. Roland Penrose, _Picasso: His Life and Work_ , 3rd ed. (Berkeley: University of California Press, 1981).\n\n. All the letters that follow in this chapter are from the Picasso archives.\n\n. FitzGerald, _Making Modernism_.\n\n. Pierre Daix, _Dictionnaire Picasso_ (Paris: Robert Laffont, 1995).\n\nBOULEVARD MAGENTA\n\n. Vincent Noce, \"L'Histoire contre Wildenstein,\" _Lib\u00e9ration_ , May 13, 2000.\n\nA LONG RELATIONSHIP\n\n. Henri Matisse archives.\n\nTHE WAR YEARS IN NEW YORK\n\n. Henri Matisse archives.\n\n. Ibid.\n\n. Franck, _Minuit_.\n\n. Loyer, _Paris \u00e0 New York._\n\nPREOCCUPATIONS OF THE HEART\n\n. Dorl\u00e9ac, _L'Art de la d\u00e9faite._\n\n. Danchev, _Georges Braque._\n\n. Ibid.\n\n. Ibid.\n\n. Franck, _Minuit._\n\n. Rosenberg, \"French Artists and the War.\"\n\n. Ibid.\n\n. Maurice de Vlaminck, \"Opinions libres... sur la peinture,\" _Com\u0153dia_ , June 6, 1942.\n\n. Dorl\u00e9ac, _L'Art de la d\u00e9faite._\n\n. In his book _Le Cabinet des douze_ , Laurent Fabius mentions this canvas, as well as _The Charnel House_ of 1945, showing how during this period Picasso's clashing, violent, broken painting symbolizes the trauma of war.\n\n. Picasso archives.\n\n. Dorl\u00e9ac _, L'Art de la d\u00e9faite._\n\n. Franck, _Minuit._\n\n. Dorl\u00e9ac, _L'Art de la d\u00e9faite._\n\n. Rosenberg, \"French Artists and the War.\"\n\n. \"Picasso\" file, document requesting naturalization, November 30, 1942, Archives of the Paris Police Prefecture.\n\nTHE TRAIN, SCHENKER, AND THE ART OF THE POSSIBLE\n\n. Valland, _Le Front de l'art._\n\n. Ibid.\n\n. \"Le pillage de l'art en France pendant l'Occupation et la situation des 2,000 \u0153uvres confi\u00e9es aux mus\u00e9es nationaux\" (The Looting of Art in France During the French Occupation and the Location of the 2,000 Works Confiscated from the National Museums), a contribution from the administration of the Mus\u00e9es de France and the Centre Pompidou to the works of the Matteoli Commission on the spoliation of Jews in France, 2000.\n\n. Annette Wieviorka, \"Des spoliations aux restitutions,\" in Tal Bruttmann (ed.), _Pers\u00e9cutions et spoliations des Juifs pendant la seconde guerre mondiale_ (Grenoble: Presses Universitaires de Grenoble, 2004), 13\u201322.\n\n. Records of the Office of Strategic Services (RG 226); formerly Security Classified Intelligence reports (\"XL\" Series), 1941\u20131946. Document in English and French. For the latter, the list is signed \"Michel Martin, charg\u00e9 de mission au D\u00e9partement des peintures, rue de Tocqueville, November 7, 1944.\"\n\n. Family archives.\n\n. Ibid.\n\n. Trial record, family archives.\n\n. Ibid.\n\n. Ibid.\n\n. Ibid.\n\n. Ibid.\n\n. Nicholas, _Rape of Europa_ , 415.\n\n. _Journal des tribunaux_ , Geneva, August 1948.\n\n. Nicholas, _Rape of Europa_ , 418.\n\n. Ibid., 421.\nBIBLIOGRAPHY\n\nAssouline, Pierre. Le Dernier des Camondo. Revised and expanded edition. Paris: Gallimard Folio, 1999.\n\n______. L'Homme de l'art: D.-H. Kahnweiler, 1884\u20131979. Paris: Gallimard Folio, 1989.\n\nBillig, Joseph. Le Commissariat g\u00e9n\u00e9ral aux questions juives, 1941\u20131944. 3 Volumes. Vichy: \u00c9ditions du Centre, 1955.\n\nCabanne, Pierre. Le Si\u00e8cle de Picasso. 4 volumes. Revised and expanded edition. Paris: Gallimard Folio-Essais, 1992.\n\nC\u00e9line, Louis-Ferdinand. Lettres. Edited by Henri Godard and Jean-Paul Louis. Paris: Gallimard, 2009.\n\nDaix, Pierre. Dictionnaire Picasso. Paris: Robert Laffont, 1995.\n\nDanchev, Alex. Georges Braque: A Life. New York: Arcade Publishing, 2005.\n\nDesprairies, C\u00e9cile. Paris dans la Collaboration. Paris: Seuil, 2009.\n\n______. Ville lumi\u00e8re, ann\u00e9es noires: Les Lieux du Paris de la Collaboration. Paris: Deno\u00ebl, 2008.\n\nde Sta\u00ebl, Fran\u00e7oise. Nicolas de Sta\u00ebl: Catalogue raisonn\u00e9 de l'\u0153uvre peint. Neuch\u00e2tel, Switzerland: Ides et Calendes, 1997.\n\nde Vlaminck, Maurice. \"Opinions libres... sur la peinture.\" Com\u0153dia, June 6, 1942.\n\nDorl\u00e9ac, Laurence Bertrand. L'Art de la d\u00e9faite, 1940\u20131944. Paris: Seuil, 1993.\n\nDuncan, David Douglas. Goodbye Picasso. New York: Grosset & Dunlap, 1974.\n\nFabius, Laurent. Le Cabinet des douze: Regards sur des tableaux qui font la France. Paris: Gallimard, 2010.\n\nFeliciano, Hector. The Lost Museum: The Nazi Conspiracy to Steal the World's Greatest Works of Art. New York: Basic Books, 1997.\n\nFitzGerald, Michael C. Making Modernism: Picasso and the Creation of the Market for Twentieth-Century Art. Berkeley: University of California Press, 1996.\n\nFranck, Dan. Minuit. Paris: Grasset, 2010.\n\nGee, Malcolm. Dealers, Critics, and Collections of Modern Painting: Aspects of the Parisian Art Market Between 1910 and 1930. New York: Garland Publishing, 1981.\n\nGimpel, Ren\u00e9. Journal d'un collectionneur: Marchand de tableaux. Paris: Calmann-L\u00e9vy, 1963.\n\nGreen, Christopher. Cubism and Its Enemies: Modern Movements and Reaction in French Art, 1916\u20131928. New Haven, CT: Yale University Press, 1987.\n\nJoly, Laurent. Vichy dans la \"solution finale\": Histoire du Commissariat g\u00e9n\u00e9ral aux questions juives, 1941\u20131944. Paris: Grasset, 2006.\n\nLevi, Neil. \"'Judge for Yourselves!': The 'Degenerate Art' Exhibition as Political Spectacle,\" October 85 (Summer 1998): 41\u201364.\n\nLoyer, Emmanuelle. Paris \u00e0 New York: Intellectuels et artistes fran\u00e7ais en exil 1940\u20131947. Paris: Grasset, 2005.\n\nNahon, Pierre. Les Marchands d'art en France, XIXe et XXe si\u00e8cles. Paris: \u00c9ditions de la Diff\u00e9rence, 1998.\n\nNicholas, Lynn H. The Rape of Europa: The Fate of Europe's Treasures in the Third Reich and the Second World War. New York: Alfred A. Knopf, 1994.\n\nNoce, Vincent. \"L'Histoire contre Wildenstein.\" Lib\u00e9ration, May 13, 2000.\n\nPenrose, Roland. Picasso: His Life and Work. 3rd edition. Berkeley: University of California Press, 1981.\n\nRosenberg, Paul. \"French Artists and the War.\" Art in Australia, December 1941\u2013January 1942.\n\nT\u00e9riade, E. \"Feuilles volantes,\" supplement, Cahiers d'art 10 (1927).\n\nValland, Rose. Le Front de l'art: D\u00e9fense des collections fran\u00e7aises, 1939\u20131945. Paris: Plon, 1961; reprint edition, Paris: R\u00e9union des Mus\u00e9es Nationaux, 1997.\n\nVollard, Ambroise. En \u00e9coutant C\u00e9zanne, Degas, Renoir. Paris: Grasset, 2003.\n\nWieviorka, Annette. \"Des spoliations aux restitutions.\" In Pers\u00e9cutions et spoliations des Juifs pendant la seconde guerre mondiale, edited by Tal Bruttmann, 13\u201322. Grenoble: Presses Universitaires de Grenoble, 2004.\n\nWolff, Albert. \"Le Calendrier parisien.\" Le Figaro, April 3, 1876.\nACKNOWLEDGMENTS\n\nAll the letters and quotations from Paul Rosenberg cited in this book are previously unpublished. Most of them come from my personal archives, as well as those kept by my aunt Elaine Rosenberg. I should particularly like to thank her, as well as my cousin Elisabeth Rosenberg-Clark, for graciously granting me access to the many boxes of documents from my grandfather's gallery, from before and after the war. These archives were preserved in New York, at my aunt's house, before being passed on to the Museum of Modern Art.\n\nThanks, of course, to Anne Baldassari, the director of the Mus\u00e9e Picasso, who, before the construction that forced the museum to close for more than two years, granted me shelter in its library so that I could dig around in the ample collection of letters from Paul Rosenberg to Pablo Picasso, which were given to the museum by the Picasso family. She generously and enthusiastically allowed me to reproduce extracts from that correspondence here.\n\nWanda de Gu\u00e9briant, the director of the Matisse archives that were kept in the painter's house at Issy-les-Moulineaux, helped me access the archives of Henri Matisse and allowed me to reproduce some of the painter's correspondence with my grandfather, again previously unpublished. I am extremely grateful to her.\n\nFinally, I should like to mention Didier Schulmann, the curator at the Mus\u00e9e National d'Art Moderne, Centre Pompidou, who was so kind as to grant me access to the photographic documents of the exhibitions at the Galerie Paul Rosenberg and to allow me to reproduce them.\n\n1. My grandfather in morning jacket before the First World War\n\n2. My uncle Alexandre \"Kiki\" Rosenberg, a lieutenant in the Second Armored Division, at the liberation of Paris. He served for four years under General Leclerc. After the war he succeeded my grandfather as the director of the gallery Paul Rosenberg & Co.\n\n3. The catalog of a 1926 exhibition of recent works by Picasso\n\n4. Paul Rosenberg & Co., East Fifty-seventh Street, New York, 1941\u20131953\n\n5. Micheline en infirmi\u00e8re (Micheline as a Nurse), a drawing of my mother by Picasso that disappeared during the Second World War and has yet to be found\n\n6. The foyer of the gallery on rue La Bo\u00e9tie, featured on the cover of a 1935 exhibition catalog\n\n7. A view of the interior of the gallery on rue La Bo\u00e9tie, featured on the cover of a 1936 exhibition catalog\n\n8. Micheline au lapin (Micheline with Rabbit), another Picasso drawing of my mother that vanished during the Second World War and has not yet been recovered\n\n9. Portrait de Madame Rosenberg et sa fille (Mother and Child), painted by Picasso in 1918, at the Mus\u00e9e Picasso\n\n10. Postcard sent by Picasso to my mother from London in 1919, when she was two years old\n\n11. The catalog of a 1927 exhibition of one hundred drawings by Picasso\n\n12. A photograph of Picasso in the 1920s, which he inscribed to my grandfather\n\n13. The main stairway of the gallery on rue La Bo\u00e9tie, with paintings by Picasso and Andr\u00e9 Masson\n\n14. The gallery on rue La Bo\u00e9tie during a Picasso and Marie Laurencin exhibition\n\n15. A broken photographic plate of a 1937 Braque exhibition at rue La Bo\u00e9tie\n\n16. The photograph of a painting by Georges Braque that was used as a model for the design of the marble mosaics set into the floor at rue La Bo\u00e9tie\n\n17. An exhibition of drawings by Matisse at rue La Bo\u00e9tie, June 1937\n\n18. The Matisse exhibition of October\u2013November 1938\n\n19. My grandfather with a Matisse painting in the 1930s\n\n20. The Institut d'\u00c9tude des Questions Juives (IEQJ, Institute for the Study of Jewish Questions) was inaugurated at 21 rue La Bo\u00e9tie in May 1941. This photograph shows the notoriously anti-Semitic author and guest of honor, Louis-Ferdinand C\u00e9line (left), in front of the building.\n\n21.\/22. Posters advertising The Jew and France, an exhibition organized by the IEQJ and on view at the Palais Berlitz in 1941\n\n23. The installation of a portrait of Marshal P\u00e9tain in the foyer of 21 rue La Bo\u00e9tie for the inauguration of the IEQJ\n\n24. C\u00e9line at the IEQJ in May 1941\n\n25. The slogan in the events hall at the IEQJ reads, \"We fight against the Jew to give France back its true face: a native face\"; beneath it is a poster \"explaining\" genetics.\n\n26. A poster of the \"Jewish bird of prey\" devouring a bloodied France in the IEQJ's events hall. The paneling and glass of my grandfather's exhibition space are visible in the photograph.\n\n27. My grandfather, in one of his favorite poses, examining a painting\n\n28. My grandfather in New York, cigarette holder dangling from his lips, showing a magnificent Renoir to W. Somerset Maugham\n\n29. With my grandparents Paul and Margot in the summer of 1950, when I was two years old\n\n30. Marie Laurencin painted my portrait when I was four.\n\n31. With Picasso at his farmhouse in Notre-Dame-de-Vie, near Mougins, in 1968\n\n32. My grandfather at rest\u2014a rare sight. This photograph was taken by my aunt Elaine in the 1950s.\n\n33. My grandfather as he remains in my childhood memories\n\n34. With my grandfather in the early 1950s. In his hand is an ever-present pack of Lucky Strikes.\nA NOTE ABOUT THE AUTHOR\n\nAnne Sinclair is Paul Rosenberg's granddaughter and one of France's best-known journalists. For thirteen years she was the host of 7 sur 7, a weekly news and politics television show for which she interviewed world figures of the day, including Bill Clinton, Mikhail Gorbachev, and Madonna. The editorial director of Le Huffington Post (France), Sinclair has written two bestselling books on politics.\nILLUSTRATION CREDITS\n\nFrontispiece: Private collection\/Succession Picasso, 2012\n\nCOLOR INSERT\n\n1: Family archives\/All rights reserved\n\n2: Family archives\/All rights reserved\n\n3: Mus\u00e9e Picasso, Paris\/Succession Picasso, 2012\n\n4: Family archives\/All rights reserved\n\n5: Family archives\/Succession Picasso, 2012\n\n6: Family archives\/All rights reserved\n\n7: Family archives\/All rights reserved\n\n8: Family archives\/Succession Picasso, 2012\n\n9: Mus\u00e9e Picasso, Paris\/Succession Picasso, 2012\n\n10: Mus\u00e9e Picasso, Paris\/Succession Picasso, 2012\n\n11: Family archives\/All rights reserved\n\n12: Mus\u00e9e Picasso, Paris\/Succession Picasso, 2012\n\n13: Family archives\/Succession Picasso, 2012\n\n14: Centre Pompidou\u2014Mnam\u2014Biblioth\u00e8que Kandinsky\u2014Fonds Paul Rosenberg\/Succession Picasso, 2012\n\n15: Family archives\/All rights reserved\n\n16: Centre Pompidou\u2014Mnam\u2014Biblioth\u00e8que Kandinsky\u2014Fonds Paul Rosenberg\/ADAGP, Paris, 2012\n\n17: Centre Pompidou\u2014Mnam\u2014Biblioth\u00e8que Kandinsky\u2014Fonds Paul Rosenberg\n\n18: Centre Pompidou\u2014Mnam\u2014Biblioth\u00e8que Kandinsky\u2014Fonds Paul Rosenberg\n\n19: Family archives\/All rights reserved\n\n20: \u00a9 Roger-Viollet\n\n21: \u00a9 LAPI\/Roger-Viollet\n\n22: \u00a9 LAPI\/Roger-Viollet\n\n23: \u00a9 Roger-Viollet\n\n24: \u00a9 Roger-Viollet\n\n25: \u00a9 Roger-Viollet\n\n26: \u00a9 Roger-Viollet\n\n27: Family archives\/All rights reserved\n\n28: Family archives\/All rights reserved\n\n29: Family archives\/All rights reserved\n\n30: Private collection\/ADAGP, Paris, 2012\n\n31: Family archives\/Succession Picasso, 2012\n\n32: Family archives\/All rights reserved\n\n33: Family archives\/All rights reserved\n\n34: Family archives\/All rights reserved\nFarrar, Straus and Giroux\n\n18 West 18th Street, New York 10011\n\nCopyright \u00a9 2012 by \u00c9ditions Grasset & Fasquelle\n\nTranslation copyright \u00a9 2014 by Shaun Whiteside\n\nAll rights reserved\n\nOriginally published in French in 2012 by Bernard Grasset, France, as 21, rue La Bo\u00e9tie\n\nEnglish translation published in the United States by Farrar, Straus and Giroux\n\nFirst American edition, 2014\n\nOwing to limitations of space, illustration credits appear at the back of the book.\n\neBooks may be purchased for business or promotional use. For information on bulk purchases, please contact Macmillan Corporate and Premium Sales Department by writing to MacmillanSpecialMarkets@macmillan.com.\n\nLibrary of Congress Cataloging-in-Publication Data\n\nSinclair, Anne, author.\n\n[21, rue La Bo\u00e9tie. English]\n\nMy grandfather's gallery : a family memoir of art and war \/ Anne Sinclair; translated by Shaun Whiteside.\n\npages cm\n\nISBN 978-0-374-25162-8 (hardback) \u2014 ISBN 978-0-374-71179-5 (ebook)\n\n1. Sinclair, Anne\u2014Family. 2. Rosenberg, Paul, 1881\u20131959. 3. Journalists\u2014France\u2014Biography. 4. Art dealers\u2014France\u2014Biography. I. Whiteside, Shaun, translator. II. Title.\n\nPN5183.S54 A313 2014\n\n709.2\u2014dc23\n\n[B]\n\n2014004038\n\nwww.fsgbooks.com\n\nwww.twitter.com\/fsgbooks \u2022 www.facebook.com\/fsgbooks\n\nFrontispiece: Drawing of Paul Rosenberg by Pablo Picasso, winter 1918\u20131919\n*A French lawyer who, along with his wife, Beate, dedicated his life to deportees. The Klarsfelds were known as Nazi hunters.\n\n\u2020Maurice Papon was sentenced in 1998 for \"complicity in crimes against humanity\" for his actions between 1942 and 1944, when he was the official representative of Vichy in the prefecture of Gironde, and especially for deporting Jews.\n\n*Translator's note: Both of Anne's grandmothers were named Marguerite. For clarity, Anne's paternal grandmother is referred to here and throughout as Marguerite, while her maternal grandmother is called by her family nickname, Margot.\n\n*Translator's note: The Bettencourt affair was a 2010 French political scandal that erupted over Liliane Bettencourt's illegal political campaign donations to members of the French government associated with Nicolas Sarkozy.\n\n*Marcel Ophuls, in his film The Sorrow and the Pity, shows pictures of the exhibition that always haunted me, even before I knew that it was at 21 rue La Bo\u00e9tie that the show had been conceived.\n\n*Translator's note: Bagatelles pour un massacre and L'\u00c9cole des cadavres were two rabidly anti-Semitic pamphlets written by the respected novelist.\n\n*The period from September 3, 1939, to May 10, 1940, after Britain and France had declared war on Germany but before any Western power had mobilized land forces against the German Reich.\n\n*Historian, founder with Lucien Febvre of the Annales School, and author of one of the finest books about the end of the Third Republic, L'\u00c9trange d\u00e9faite.\n\n*At first, Roosevelt's isolationist America wanted to maintain good relations with the Vichy government and was therefore reluctant to welcome refugees with open arms.\n\n*A formula applied to works of art recovered from the Nazis and kept in the national museums while their owners were not yet identified.\n\n*A French painter contemporary with Braque and Gris, whose first exhibition was held at the Rosenberg Gallery in 1921.\n\n*One day when Salvador Dal\u00ed politely approached Paul in a restaurant to ask him to represent him, Paul's reply was harsh, crude, and lacking in vision: \"Monsieur, my gallery is a serious institution, not made for clowns.\"\n\n*Translator's note: The Algiers Putsch, which led to General de Gaulle's return to power.\n\n*General de Gaulle conducted a very personal and independent foreign policy, which was not always in line with the American one, especially when he removed France from NATO.\n\n*An exhibition of do-it-yourself inventions, where the most original or useful thing gets an award.\n\n*Now in the Museum of Modern Art, New York.\n\n*The regime that was in effect from the end of World War II through its collapse in 1958, when de Gaulle established the Fifth Republic.\n\n*Ninety-three, in 2014.\n\n*Pierre Laval (1883\u20131945) served as prime minister of France from 1931 to 1932, as vice president of Vichy's Council of Ministers in 1940, and as head of government from 1942 to 1944. Convicted of high treason, he was executed in 1945.\n\n*For the Institute for the Study of Jewish Questions, see the chapter \"Number 21 Under the Germans\" (p. 27).\n\n## Contents\n\n  1. Title Page\n  2. Copyright Notice\n  3. Dedication\n  4. Contents\n  5. Prologue\n  6. Introduction\n  7. Rue La Boetie\n  8. Number 21 Under the Germans\n  9. Floirac\n  10. At the Centre Pompidou\n  11. Gennevilliers\n  12. Dealer\n  13. Chateaudun, Opera, and Madison Avenue\n  14. Mother and Child\n  15. Paul and Pic\n  16. Boulevard Magenta\n  17. Pi-ar-enco\n  18. A Long Relationship\n  19. The War Years in New York\n  20. Preoccupations of the Heart\n  21. The Train, Schenker, and the Art of the Possible\n  22. Epilogue\n  23. Bibliography\n  24. Acknowledgments\n  25. Frontispiece\n  26. Photographs\n  27. A Note About the Author\n  28. Illustration Credits\n  29. Copyright\n\n## Guide\n\n  1. Cover\n  2. Table of Contents\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"Image Title: High School Musical Coloring Sheets Elegant Amazing Wings Fire In Pages Ideas 19. Post Title: High School Musical Coloring Pages. Filename: high-school-musical-coloring-sheets-elegant-amazing-wings-fire-in-pages-ideas-19.png. Image Dimension: 1388 x 895 pixels. Images Format: jpg\/jpeg. Publisher\/Author: Jess Koelpin. Uploaded Date: Saturday - January 19th. 2019 11:16:38 AM. Category: Coloring Page. Image Source: homedepot.com. High Pages Coloring At Design Sheets 4. School 58 Best Zac Decorations Amazing 17. Efron Printable Hellokids Gabriella 8 Com Musical Elegant Wings Hard Telematik Institut Students Kids Pertaining Ideas Intended In Taylor Idea Remodel Highschool Inside 19. For Cool To Day Fire Of Color 12. 2018 Free Top Drawing Throughout 2371564 Prepare McKessie 0. Characters Hsm 14. 7. Org Getdrawings Unique With 11. Updated Within Decor 18. First 2.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Five CRPF jawans were injured when militants fired on a CRPF vehicle in Kashmir's Anantnag district on Thursday morning on November 02,2017.\nIn an indiscriminate firing by militants on a CRPF 96 Battalion Convoy bus at Anantnag in Kashmir left five CRPF personnels were injured while another two sustained glass injuries. The injured are being treated at SDH Anantnag.\nInitial reports suggest that militants first fired pointing to the driver in a bid to stop the bus, the aftermath of which could have resulted in heavy casualties, however the driver didn't stop the vehicle and drove away to safety despite being hit in the drivers side window glass.\nSearch operation have been intensified in the area soon after the vehicle made it inside the fortified CRPF camp.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Getting your website found online continues to be a major challenge for businesses. Search Engine Optimisation is a dark-art and most people either don't have the time or the interest to do it right. Constantly changing algorithms add to the frustration but there are some golden rules to follow to give you the best chance of showing up high in the page rankings.\nYoast SEO is the industry-leading authority that web developers use for Search Engine Optimisation for client projects. Recently, Marieke van de Rakt published an easy-to-read article that outlines 7 keyword research mistakes and more importantly how to avoid them. The information below is a summary of the post and you are encouraged to view the full article on their website.\nDoing proper keyword research can be a tough, time-consuming process. So, perhaps you think that you don't need to do it very extensively. You just instinctively know what your audience will search for, right? Do not make that mistake! Take some time to really dive into the language of your audience. Which words do they use? What terms do they search for?\nThe result of your keyword research should be an extensive list of keywords you would like to rank for. Make sure to update your keyword research list or sheet regularly. Your audience may change, as could your business focus and business needs. That has implications for your keyword strategy as well.\nMake sure you aim for realistic keywords. Some niches are very competitive. Ranking in competitive niches is hard if you're just starting your website or business. If you're just starting out, you shouldn't aim for the competitive 'head' keywords (yet). Instead, focus on long tail keywords (which are easier to rank for and have a higher chance to convert).\nFor instance, if you start a blog about fitness, it will be too hard to start ranking for the term 'fitness'. Find out which aspects of your blog are unique and try ranking for those terms. Perhaps you write about fitness exercises for retired people. Aiming to rank for 'fitness for retired people' could be a good strategy. In that case, you should also aim for 'fitness for seniors', 'fitness for older adults' and so on.\nIf you have been around in your niche for a little longer and you successfully rank for long tail keywords, you could aim to rank for more head terms as well. Ranking for competitive keywords should be part of a long-term successful keyword research strategy.\nThe keywords you aim to rank for should be the same words your customers use. Always try to use the language of your audience. Imagine yourself selling dresses for gala events. In your marketing, you refer to these dresses as 'gala dresses'. However, people do not search for 'gala dress'. They search for 'gown' or 'evening dress'. You won't get much traffic for the search term 'gala dress' compared to the search terms 'gown' or 'evening dress'.\nAlways check if you should target the plural or the singular form of a specific keyword. Should you aim to rank for 'ballet shoe' or for 'ballet shoes'? Do people search for 'holiday home' or 'holiday homes'? While Google has become better at recognizing that the plural and singular versions of a word are the same, the search result pages and the number of results are often still different. Always check whether you should use singular or plural with Google Trends. Also, think about the intent of people searching for your keyword. Someone looking for the singular version of a keyword may be looking for information, while someone looking for the plural version could be looking to compare products and\/or buy something. In any case, whether you should use a singular or a plural depends on your specific keyword, so give that some thought.\nFor the explanation of these last 3 points and lots more information about Search Engine Optimisation and keyword research visit the Yoast SEO website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cell Phones for Soldiers provides talk time to overseas troops so they can call home. The organization collects old phones, which they send to ReCellular so the used phones can be sold or recycled. Each phone donated buys one hour of talk time for soldiers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hydrangea Magical Opal Pink is a beautiful Pink cut flower. It is approx. 70cm and wholesaled in Batches of 10 stems. Hydrangea magical opal is ideal for flower arrangements, hand-tied bouquets & wedding flowers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"PROCESSIONING. A term used in Tennessee to signify the manner of ascertaining the boundaries of land, as provided for by the laws of that state. Carr. & Nich. Comp. of Stat. of Tenn. 348. The term is also used in North Carolina. 3 Murph. 504; 3 Dev. 268.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bring the Big View INDOORS Introducing the Bentley. With an expansive glass viewing area and clever designer options for customizing the look to suit you personal style, the new Bentley by Marquis is generating some serious attention. Available in two sizes, you can choose from the Decorative-rated fireplace for ambiance and taking the chill out of the room, or the Heater-rated version for additional warmth. With so much style and attention to detail, it's no wonder people are focused on the view indoors!\nINFINITE Possibilities! A fabulously sleek stretch of dancing flames delivers infinite possibilities in this modern zero clearance gas fireplace in the Marquis Collection by Kingsman.\nEnjoy the VIEW II. There is nothing subtle about this Skyline II. Its pure sophistication that makes a statement with wide-view appeal and flexibility. Choose from the rocks and driftwood, decorative stones, cannonballs or ember glass in a choice of colors for a sleek contemporary look.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nico Muhly moves like his music. I mean this completely seriously. When he talks, his expressions follow his hands like foreshadow, embellishing his fast words and the sharp inflection of his voice. Curiously enough, when he plays, he has none of that. Completely still behind his piano, he watches his co-players sway to his notes. Yet you know Nico is moving. You can see it in his eyes when he plays. His mind is running alongside the music, flittering about the staffs, getting there before the notes do.\nReturning to Iceland to record his second album with producer Valgeir Sigur\u00f0son, the twenty-six-year-old New York based composer took a Friday afternoon to perform at the most all-around-lovely concert I've ever attended. In the 12 T\u00f3nar garden, the pleasantness truly abounded with rose wine and sunshine when more than fifty people squeezed in the garden to watch Nico share his musings on life, music, and his upcoming album.\n\"When you're a composer you're kind of just like a brain in a jar,\"\u009d Muhly said, introducing his violinist Una Sveinbjarnad\u00f3ttir. \"You kind of just make this stuff and then send it off for some other people to deal with.\"\u009d In this case, Una, who launched into Honest Music, a song off of Muhly's debut album Speaks Volumes, which was released earlier this year. The wind blew and three men, Valgeir and Ben Frost among them, jumped from the audience towards the pages on the music stand. The song went on, Nico pushed keys on the piano. Sitting in the grass a few members of the audience contently closed their eyes.\nEach song came with a short introduction, a context into which Muhly was enthusiastic to draw the audience. He introduced one song with the anecdote that his first job was as a church organist; another song was from a genre of music he called \"hippie drone,\"\u009d which he explained was inspired by being raised by hippies. \"A couple of years back I wrote some music just to completely get it out of my system,\"\u009d he said.\nNico's recording process is such that the live versions of his songs sound quite different from their recorded adaptations. On his latest album the instruments are recorded individually, with the microphone intimately close, then mixed together to create an orchestral sound. Live, the philosophy is quite different. Much of the rich, indulgent detail of sound from the recordings is lost traveling across the windy afternoon air. But the music stood up for itself. Taking on a slightly less personal but nonetheless absorbing sound. The concert was over within twenty minutes. There was more rose wine to go around, and people stayed in the grass. Everyone was smiling and looking at Muhly as though they wanted to hug him. He started excitedly towards the standing crowd, moving, like an echo, with a genuinely fascinating softness.\nNico responds: I triggered samples!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The APOS suite of \"well managed Analytics\" solutions builds on the functionality and control that our customers are accustomed to experiencing with SAP BusinessObjects through our well managed BI solutions. APOS solutions are built to automate, simplify, complement, enhance and extend the capabilities of your BI system.\nYou implement Analytics to optimize your business processes and outcomes. You implement these APOS solutions to optimize your BI processes and outcomes.\nThe APOS Live Data Gateway is a data connection and data transformation solution that enables live data connectivity and expanded data source options for SAP Analytics Cloud, SAP Lumira, and SAP Analysis Office.\nThe APOS Live Data Gateway provides live data connectivity to most industry-standard Relational, OLAP, and Cloud data sources.\nAPOS Publisher for SAP Analytics Cloud automates processes for bursting personalized SAP Analytics Cloud stories to end users. Dynamic, security driven processes with strict filters help administrators ensure users receive only the data to which they are entitled.\nAPOS Publisher also automates scheduling and export operations, allowing individual users to send SAP Analytics Cloud stories to a variety of destinations in a variety of formats.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There is an Assisted Dying Bill before Parliament which will be considered in the House of Lords on 18 July 2014. This Private Member's Bill would license doctors to supply lethal drugs to terminally ill patients and would put vulnerable people at serious risk of harm. Please write to a peer before 18 July to make your views known.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Includes unlimited streaming of Precious Futures via the free Bandcamp app, plus high-quality download in MP3, FLAC and more.\nI can sing about the road but can I sing about my street?\ncan I feel it in my feet?\nCould you finish your story?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Completa la frase con \"What\" o \"Which\".\n_____ of you two can help me?\nof you two can help me?\n_____ is your favorite food?\nWe can have pizza or chicken, _____ do you prefer?\nWe can have pizza or chicken, do you prefer?\n_____ leg did you hurt?\n_____ would you like for lunch?\nwould you like for lunch?\n_____ is your phone number?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Tesla Model S is a landmark car, but is it better than the mighty Aston Martin Rapid S? Autocar take to the test track to find out.\nSteve Sutcliffe over at Autocar thinks the Tesla Model S is the future...today. It offers most of the usability of the world's best luxury saloons, but with no tailpipe emissions and a fraction of the running costs.\nAnd it handles too. Really handles. But can it really be better than the \u00a3150,000 Aston Martin Rapide S?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Capitalism can only initiate a context of justice; without mercy and compassion all is void of the human element. Has not the man who kills another for whatever reason capitalized on his supply of force? Does not the thief capitalize on his shrewdness and timing? Won't the greatest idea retain its stature and station regardless of who or what will acknowledge it? One might argue then that true capitalism concerns all matters; to apply capitalism economically and not morally one is at a loss and can say nothing to those who do not look towards the future of their investments or that of those all depend on.The opposite of capitalism is statism, the preference and adherence to a single state (of governance or being), for this is the source of all stagnancy. Capitalism is not the answer to all our problems, but it is the answer to statism; and truth, mercy and justice are the only answer (source) of life.Moreover, regulation is only the statist's crutch, for he knows he cannot survive without depending on others but he fears competition and the unknown. On the other hand, the capitalist is not boundless\u2013how can man forge steel, construct and build infrastructure without abiding by laws of nature? His obedience, however, is his honor, his code, he knows what must be and so he does not lie, he knows what works and so he does not contradict it. In the same manner, mercy and compassion cannot be arbitrary reactions to the world around them, they rest on some moral basis, whether they are doctrinal, inspirational or fleeting; and only excellence begets excellence; only that which is most cohesive and comprehensive encompasses all matters of life, and brings balance and harmony.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Since BLE was designed from the ground up for low power battery devices, it's common to see a wide range of batteries used in designs. Besides the popular Coin Cell batteries you will also see Alkaline, Lithium and many other chemistries.\nThis guide is designed to give you a broad overview of the choices in powering BLE products and the practical issues surrounding their use. Every product and application is different, but with this information you can make better choices. As always, feel free to get in touch if you have questions.\nBefore we start talking about the batteries, we have to understand the BLE devices themselves and their power requirements. Most BLE devices today are System on Chip (abbreviated SoCs). Their current consumption profile has a large effect on the product lifetime.\nPower consumption in BLE SoCs is typically due to the processor and the BLE radio. The processor can draw several mA when running. The Bluetooth radio is another part that can consume significant amounts of power. Some BLE radios can reach a peak of 20mA when transmitting (or more if the output power increased). Which part consumes more depends mainly on the application and duty cycle.\nMost BLE products are designed to stay asleep as much as possible to conserve power, waking up to process and send data. How much current they draw during sleep depends on the device, but a figure of around 1.5uA is typical of many SoCs. This figure accounts for a running a Real Time Clock (RTC) which wakes the system periodically to advertise or send data. When in sleep, the CPU is off, the BLE Radio isn't transmitting, and most peripherals are stopped as well. If the device wakes up periodically, then the current peaks for a few milliseconds during transmission and reception. The average current of the system is then higher and depends on how often the system wakes up to send data.\nIt's also possible to go to a deeper sleep mode where no RTC is running and the system in a low power mode, but this prevents a smartphone from connecting to a device. In these cases it's possible to have a current draw of less than 200nA.\nNote that we're focusing here on BLE products that are running BLE in Peripheral mode. BLE was designed to be asymmetrical, with Peripheral devices designed to use as little power as possible. Central devices draw significantly more power, but since these are expected to be smartphones with large batteries, it's not an issue.\nPower output of the BLE device \u2013 the higher the output power the more current is required. Many latest generation devices draw 3mA to 6mA peak, while older devices can draw 3x or more.\nAmount of data sent \u2013 larger advertising packets require the BLE radio to stay on the air longer, drawing more current during this time.\nWe're not including any extra power required by sensors, other components or peripherals since they depend on the product, but they should be taken into account. Creating a low power product requires optimization on the hardware, software and other layers.\nAnother important factor to realize is that BLE devices need a certain voltage to operate. This is typically 1.8V to 3.6V, but some devices can accept up to 5V which can be beneficial in some cases.\nCapacity \u2013 Probably the most important parameter of any battery. Capacity is the amount of energy in a battery and is typically specified in mAh or Ah. A battery with 100mAh can theoretically provide a steady current of 100mA for one hour, or 50mAh for 2 hours, or 1mA current for 100 hours. The actual capacity depends on the current being drawn and is better specified by Nominal and Effective parameters (unless the current draw is very low).\nNominal Capacity \u2013 This is the capacity under nominal conditions. Every manufacturer will specify the conditions under which this capacity is specified, and it typically includes temperature and current. For example, a capacity of 260mAh at 0.19mA.\nEffective Capacity \u2013 This is the actual capacity of the battery in your product or application. This capacity can be lower or higher than the nominal capacity depending on how your product uses the battery.\nVoltage \u2013 The voltage profile of a battery is the output voltage your system sees over time. Different battery chemistries have different voltage profiles, which can make a big difference in your product design.\nThe most important thing to understand about batteries is that their actual or effective capacity isn't usually a linear relationship with current draw. The actual capacity of one battery where the current drawn is 10mA will be different than one where you're drawing 10uA. The less current you draw, the more capacity the battery will seem to have, beyond just the linear relationship because the battery is subjected to less stress from peak currents. This is important in BLE applications where the average current is low, but the peak currents of BLE radio transmissions are higher. The same happens with every wireless radio.\nCaution: Coin cell batteries can be extremely dangerous to small children. It's important to ensure proper product design to avoid giving children access to the batteries.\nCoin Cell batteries are small, flat batteries that can be easily found in Key Fobs, BLE trackers and sensors. The size is very appealing for small devices that need to be placed on items and key chains. BLE was designed from the ground up to use these batteries.\nIt's easy to see that the larger the battery, the larger the capacity, and that is the case with every battery regardless of chemistry. What's not so obvious is that the smaller the coin cell, the more it is affected by large current peaks. Whereas large batteries can cope better with large peaks during BLE transmission which can be 5mA to 19mA (depending on the device), smaller coin cells do much worse.\nInterestingly, the CR2032 doesn't follow the cost trend, being almost 1\/3 of the cost of the others. The greater number of suppliers and economies of scale due to the popularity of this battery means it's much cheaper.\nOut of 1300 hours, the battery has a voltage of 2.9V for 600 of them (almost 50% of the time), dropping to 2.8V after 900 hours. Only at the last 25% or so does the voltage begin to drop significantly but stays above 2.0V.\nThe importance of a flat voltage is that many BLE chipsets, sensors and devices operate in the 1.8V to 3.6V region. This means that the Coin cell battery can power these directly, without any power conversion. This is smaller, cheaper and simpler than adding regulators.\nCoin cell batteries have one major disadvantage compared to other batteries and that is that they're greatly affected by large current draws. Nominal current for a coin cell is around 300uA (may vary among batteries). But when the BLE device is transmitting, the current can easily spike to 5 to 20mA for a few miliseconds. Running the processor also draws a few mA.\nVload shows the voltage output profile when under load, compared to 0.5mA cont which is a continuous draw of 0.5mA. You can see that the capacity drops from around 225mAh (green line) to around 180mAh (blue line), a drop of 20% in capacity (your product will run 20% less).\nTemperatures can have a large effect on battery capacity and lifetime, and needs to be taken into account when selecting a battery.\nAlkaline batteries, whether AA, AAA, C or D are also very popular in products using BLE, Wi-Fi and other wireless connectivity.\nCompared to coin cells, alkaline batteries are physically larger, but provide much more capacity and can handle larger currents.\nThe actual capacities above change depending on the actual discharge rate. For alkaline batteries, it matters whether you are drawing 25mA, 100mA, 250mA, 500mA, or above.\nAside from capacity, the difference between a coin cell and alkaline is the discharge voltage. Whereas coin cells have flat voltages around 2.9V, alkaline batteries start at 1.5V and it drop down to around 0.8V. This puts it out of the 1.8V to 3.6V range of most devices.\nIn order to run most devices out of alkaline batteries, you will need to place batteries in series or use a DC\/DC Boost. Two batteries in series can provide 3.0V down to 1.8V which is in the range used.\nTwo series alkaline batteries provide the exact same capacity as a single battery. When you take into account the inefficiency and the extra current consumption of a DC\/DC converter (booster) two series alkaline batteries can provide longer life, but have variable voltage output which may not always be undesirable.\nAvoiding regulators (boost, buck) saves on both cost and increases system lifetime since you don't need the costly boost and don't need to worry about inefficiencies. This is best for system that can accept wide voltage ranges.\nBe careful of sensors that are calibrated by the manufacturer for a specific voltage. As voltage changes for sensors, their output may change and the readings may be affected.\nWhenever boosting the current, the input current from the batteries is the ratio of the output current to the input current, plus current due to efficiency. So, boosting from 1.5V to 3V will require at least twice the input current and more due to the inefficiency.\nLithium batteries \u2013 the non-rechargeable primary kind, are similar to alkaline batteries in both the form factor and output voltage. In fact, they're recommended for many devices where there is a very high current draw. Video Cameras and many other systems that need a lot of current can benefit from these versus alkaline.\nThe capacity of a lithium battery remains much more constant at currents up to 1000mA while alkaline batteries see a significant drop. For low currents, the difference is not significant. Their voltage is also slightly higher than Alkaline, reaching 1.6V or so.\nLithium batteries have another advantage and that is their operation in low temperature is much better than alkaline.\nBluetooth Low Energy products are usually low power by nature, but there can be cases where sensors or other devices require significant current. In those products it's important to weigh the extra cost and the conditions of the system to see if Lithium batteries are needed.\nYou're more than likely familiar with rechargeable lithium-Ion and Polymer batteries from the countless of electronics that use them. These batteries have the one of the highest energy densities available and are available with capacities from 50mAh up to 10000mAh or more.\nAlthough it is possible to have multiple cells in parallel or series, a single cell arrangement is the most typical. A single cell battery typically has an output voltage from 4.2V or 4.35V when fully charged down to 3V or 2.7V when empty. The nominal 3.7V often quoted is not the main voltage at which the battery operates.\nBecause the voltage gets to above 4V, it's common to use an LDO or DC\/DC Buck regulator to reduce the voltage to 3.3V to avoid destroying components that can't withstand this voltage. A few BLE SoCs do have the ability to run up to 5V, so they may be considered to reduce the cost of regulating the voltage.\nLithium Ion and polymer batteries have very few issues with BLE peaks. The standard discharge current varies between batteries but is usually C\/5, where C is the capacity of the battery. For a 100mAh battery, the standard discharge rate would be 20mAh which is higher than the peak BLE current of most devices.\nHowever, even here it's important to optimize the product power consumption.\nOne concern with these batteries is the self-discharge current. Although they're better than NiMH and NiCad batteries, it's not uncommon to lose about 2%-3% of their capacity a month. This can be larger than the current of the BLE system in sleep mode.\nOne of the big requirements when integrating rechargeable Li-Ion and Li-Poly batteries in a design is proper battery management. These types of batteries carry a risk of fire and explosion when not properly designed.\nOvervoltage protection \u2013 The circuitry must protect the battery from over-voltage during charging. Over-voltage can cause a battery to leak and explode. It's critical to talk to the battery manufacturer to determine the maximum voltage, and ensure charging circuitry charges the battery to within 50mV or so.\nOvercurrent protection \u2013 This circuitry protects the battery from too much current being drawn which can damage the battery and its terminals. Overheating is also possible as the internal resistance of the battery heats up due to the current. This protection depends on the battery size and must be properly sized.\nUndervoltage protection \u2013 Running a Li-Ion\/Poly battery completely can damage the battery permanently by preventing the cathode and anode from working properly. The circuitry must stop drawing power from the battery after a certain voltage is reached. The actual cutoff voltage is usually anywhere from 2.5V to 3.3V with 2.7V and 3V being common. Once this is reached, the protection disconnects the battery from the circuit.\nThermal Protection \u2013 Many of the issues experienced by Li-Ion batteries are thermal in nature. Another protection added to the battery is to ensure the battery isn't overheating. If overheating is detected during charging or discharging, the battery is disconnected.\nPacks are self-contained batteries that include all the protection circuitry, but it is also possible to design with a cell having no protection. Several manufacturers offer chipsets that are capable of protecting the battery. These chips form the first line of defense and are always connected to the battery and stay with it.\nMost customers want to have an idea of the current battery state of charge (percentage) to know whether it needs to be charged. This feedback is important and can be obtained in multiple ways.\nVoltage monitoring \u2013 Some information on the state of the battery can be obtained using the voltage by simply sampling the voltage periodically. However, the real state of the battery can be very different as the battery relaxes or is stressed. The higher the current draw, the worse the estimate. Even a decent model can give 30% error or more. This approach is cheap but produces data that can be frustratingly bad.\nFuel Gauges and Coulomb counters- these are specialized devices that measure the voltage and the current of the system and use sophisticated algorithms to estimate the state of charge and other values of the system. Several manufacturers such as Texas Instruments and Maxim create complete solutions for measuring battery state.\nIf you'll be displaying the battery state to customers, using a Fuel Gauge or Coulomb counter is almost a must. In order for it to work well, it's standard to have the Fuel Gauge \"learn\" the behaviors of batteries so that it can estimate them well. This process can take a few days (depending on battery size) but produces results are usually 5% or better.\nOne important thing to note with Fuel Gauges is that they are sampling the data at a certain rate. If your system has very large and short peaks of current draw, their performance won't be as good.\nCharging the battery can be done inside your device by using a 5V or similar input. It's also possible to use an external charger.\nThe charger has to be designed specifically to meet the battery's voltage and current requirements. The Charging rate can be fast or slow, slow being a fraction of the capacity (typically C\/2 or less) while fast is 1C or 2C.\nThe faster you charge, the more energy you're pushing into the battery which stresses it and degrades it. Slow charge of the battery is best and safe if the charging time is acceptable to the end user.\nKeeping the currents as low as possible is the best approach since high current and charging reduces their lifetime and the number of useful cycles. The more you push them (discharging or charging) means that every cycle your battery will hold less and less charge.\nA commonly used approach is to avoid fully charging the batteries. For example, if the fully charged voltage of the battery is 4.35V, then charging to 4.2V or 4.1V reduces the amount of charge on each cycle but has the effect of increasing the number of cycles. This tradeoff can work if the extra time can be traded for more cycles.\nUsing cables to charge devices is the standard approach, with a USB connection common because of the ease of use and low cost. Wireless charging is slowly making its way to devices. We won't discuss wireless charging here, but it can be a great way to reduce connector wear and simplify charging, at the cost of increased BoM price and design complexity.\nBecause of the small capacity of coin cells and the expected long lifetime of BLE products, optimizing the power consumption is a critical taks that has to done by experienced developers. Estimates can be useful in giving a ballpark figure, but the accuracy depends on many factors. We highly recommend getting proper measurements and modeling done for the device.\nThere are several keys to reducing power consumption in BLE and we'll go over each one.\nPeak currents in BLE are usually dominated by the BLE radio. Choosing a low power BLE chipset with low current peaks can help extend the lifetime of a coin cell battery and others as well.\nThe table above is specific to this chipset, but the results are very similar in other devices.\nThe first important thing to note is that using a DC\/DC regulator in BLE chipsets reduces the current consumption significantly. But, this has a tradeoff of reducing the sensitivity of the BLE radio (and the range) by around 2dB.\nIt's also apparent that the real gains in reducing the output power are in going from +4dBm to -4dBm. Beyond that there are diminishing returns.\nAs we mentioned before, the peak currents affect the coin cell batteries significantly because they are over 10x to 30x as high as the average current that can be drawn from the coin cell. Peak currents require significant energy for a short period of time. One approach to avoid drawing so much current from the battery directly is to use a large capacitor to provide storage. By using a capacitor in parallel with the Coin cell, the capacitor can provide the peak current when it's needed instead of the coin cell (the capacitor has much lower effective series resistance).\nHowever, the large capacitors also have increased leakage and cost, meaning that their value can be limited and needs to be evaluated in a particular product.\nThe average current of BLE devices or any wireless radio takes into account the peaks but also the sleep current of the device (if in sleep). To extend the battery lifetime it's critical to reduce the number of transmissions done by the device. If the device is in advertisement mode, then increasing the advertising interval to 500ms or higher will reduce the power consumption drastically. Similarly, if the device is in a connection, increasing the interval and slave latency will help reduce the current consumption.\nAs you can see above, increasing the interval to 500ms helps the system runs 3x as long, while increasing it to 1000ms helps the system run almost 5x as long. If your device spends much of its time advertising, increasing the interval has significant gains.\nNote that the DC\/DC operation mode in this case makes relatively little difference.\nOnce you're design is ready and you're shipping your products, it's common to ship batteries inside or with the product. The customer can quickly turn on the system and use it. Reducing the steps to get a product working always makes customers happy, but it can create challenges.\nPhysically preventing the battery from forming a circuit \u2013 Either placing the battery outside of the product, or using an isolating material such as plastic pull-tab prevent the battery contacts from touching the product contacts can work.\nUltra low Power \u2013 For products where inserting the battery requires more effort, doing an ultra-low power design where the battery is in place can help. Usually designing a system that draws 500nA or less means the battery will be good even if the product takes months to ship.\nHave you ever looked at the instructions manual of a battery and been told to avoid using batteries that are different and not fresh? There's a good reason for it. When batteries of different conditions are inserted into a product, a lot can happen.\nImagine that a customer inserts an old and a new battery in parallel in a device. The batteries likely have different voltages given their different discharge states. The battery with the higher voltage will attempt to charge the lower voltage battery. But alkaline batteries aren't rechargeable and therefore this can damage the battery and even cause it to leak and be damaged.\nLithium coin cell batteries are very affected by this and it's a bad idea to put them in parallel.\nIt's important to note that batteries do age and expire. Even if a system is designed to work 10 years off of a battery, it's unlikely it will, given the aging process that's happening internally. It's very important to take this into account and if needed to choose batteries that are designed for long life.\nMost vendors don't guarantee any kind of operation after 10 years, since the battery can leak and corrode.\nAside from our experience, the following resources were used as a source for the information presented.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Having been forced to sit out the 2015 season due to injury, Marianne Vos will shortly return to racing. She has given a fascinating sit-down interview to CyclingTips Ella editor Anne-Marije Rook, which is well worth a read.\n\"I had too many good days and too much hope to succumb to full depression but sure, there were bad days. Nights I couldn't sleep and days where I'd wonder what I was getting out of bed for. I felt so incredibly useless. I always want to bring value to something and be of use. And how was I going to do that?\" Vos says.\nClick through to read the full feature on Ella CyclingTips.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Since 1999, this business has been contracted to install, repair and maintain all aspects of aerial and underground cable networks for many major cable companies. Services include setting telephone poles, splicing fiber and coax, performing cable replacement and line extensions, aerial and underground cable construction, joint trench construction in new subdivisions, 24 hour on call service and storm damage repair. Their service area includes a 100 mile radius from their upstate NY location. The revenue is shared 50-50 between aerial and underground work. All project materials are supplied and delivered to them by their customers. The only materials they supply are telephone poles, which they have a large supply that is included with the business. The staff of 30 includes 6 aerial and 2 underground crews with experience in all areas of construction and installation. Employees are certified in enclosed space training and all flaggers are certified as well.\nThe real estate is included in the asking price. The property is 5 acres and includes an office building and a garage. The facility is completely fenced in and has 3 entrances through the gate. The office building is about 1,500 square feet and includes 4 offices, a bathroom, a stockroom and full basement. The garage is about 900 square feet. The facility has a web based security system with cameras. Taxes on the property are $8,000\/year.\nThis business has been providing excellent service for major cable companies in upstate NY for over 15 years and have very entrenched relationships and stable revenue streams. Growth & Expansion:Services could be expanded in the geography and the geography could be extended outside their current service area.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Back when there was a terrorism raid in Torento I remember how for years I had seen this coming. All the signs were there. The non-stop propaganda & hatred that was being broadcast on the UCBC. To summarize a lot of the material, the Camucks were insisting that the Unified States, Great Brighton, & the coalishun were the Great Satan & a bunch of bastards. This indoctrination was further re-inforced in the classrooms where the teachers were allowed to portray comparisons of the allies to the Schnazis.\nCurious to learn more I pulled my son out of private school in the states & told him to do me a little favor & wear a little special pin for me while attending class. Day after I recorded hour upon hour of the radical teachings. Finally my son said \"Dad I can't take it anymore. I want to go back to the Unified States where they don't preach this kind of hatred & where people are normal\". I pulled him out of school & sent him back down to the states & thanked him for doing the entire freedom & democracy loving world a huge favour. Now we knew where the seeds of this recent spat of domestic terrorists came from.\nAs for the terrorists the CRCMP, PSIS were mad at them because for some odd reason their attempt at producing a homegrown terrorist cell to attack the Unified States backfired & the terrorist cell wanted to attack Sobiet Camuckistan instead.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In many parts of the Philippines, you're more likely to encounter a remittance house like Western Union than a bank branch. Cash arrives from abroad, but the access to business loans and mortgages needed to build a self-sustaining economic base are hard to come by. Poverty begets poverty.\nRural banks in the Philippines, says CryptoCoinsNews, cannot even issue cheques because they cannot connect to Bancnet, the Philippines' banking intranet.\nCrypto-visionaries and entrepreneurs have long sought to use internet-based cryptocurrency networks to extend stable capital where it has not gone before. Blockchain, a system that somewhat expensively (for a database) mathematically and immutably registers transactions in a distributed way, is seen by many as having the potential to extend infrastructure-like services through the internet to places where store front banking operations are infeasible because of poor infrastructure and the poverty of local residents.\nVitalik Buterin left Bitcoin, and with the help of 7 other co-founders, began building Ethereum, a programmable blockchain still in its testing phase.\nThanks to the billions of dollars Ethereum cofounders have raised and continue to raise via sales of various Ethereum tokens, Ethereum companies now have the money to find out.\nCurrently, Ethereum is partnering with Union Bank in the Philippines to create a blockchain-based payment system that will settle retail purchases made by rural Philippinos using pesos.\nChair of Union Bank Dr. Justo Ortiz hopes the project will help unbanked Filipinos plug into the financial system and also help rural banks communicate with the larger banking system in the country.\nAccording to the Philipino News Project (PNP), the pilot is being rolled out at 5 banks located on the nation's second largest island. The banks are: Mindanao Cantilan Bank, Inc. of Surigao del Sur, PR Savings Bank, City Savings Bank, FairBank, and Progressive Bank.\nExecutive vice president of Cantilan Bank, Tanya Hotchkiss told the PNP, that the bank has been looking for a faster system for bank-to-bank transfers.\nEven though most blockchains operations are currently slower and more expensive than many legacy payment systems, Hotchkiss told the PNP that the new trial system could reduce payment transfer fees from 50-150 pesos to1 peso.\nShe did not say how much of the savings will be passed on to customers.\nHotchkiss also said that under the previous manual system, reconciliation took anywhere from a day to a month, and that transfers underwent 26 individual processes.\nThe new system will use the an enterprise blockchain protocol developed on Ethereum by ConsenSys called \"Kaleido.\" The system is purportedly more user-friendly, \"robust\" and secure than previous Ethereum blockchains.\nUnion Bank of the Philippines also announced earlier this year that they were working with Visa to test that company's blockchain system for cross-border remittances.\nThis entry was posted in Asia, Blockchain & Digital Currency and tagged banking, consensys, ethereum, justo ortiz, kaleido, Philippines, tanya hotchkiss, union bank, vitalik buterin. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Chin, M.G., Sinatra, A.M., Sims, V.K., Lum, H.C., Marraffino, M., Lagattuta, N., Spitzer, M., & Raymond, M. (2009). Males and Females Give Different Attributions to Live and Robotic Entities. Poster presented at the APS 21 st Annual Conference, San Francisco, CA.\nLum, H.C., Sims, V.K., Chin, M.G., &Lagattuta, N.C. (2009). Perceptions of humans wearing technology. Paper presented at the 53 rd Human Factors and Ergonomics Society, San Antonio, TX.\nLum, H.C., Sims, V.K., Chin, M.G, & Lagattuta, N.C. (2009). Perceptions Of Technology During Human Interaction. Poster presented at the APS 21 st Annual Conference, San Francisco, CA.\nLum, H.C., Sims, V.K., & Lagattuta, N.C. (2009). Team Success is not a Matter of Time: Eye Movements as a predictor of Team Efficiency. Poster presented at the 4 th Annual INGRoup Conference, Colorado Springs, Colorado.\nLum, H.C., Sims, V.K., Feldman, M., Afek, A., Smith-Jentsch, K.A., Lagattuta, N.C. (2009). Emotion regulation training and scene understanding are related to eye movements during a computer based interactive simulation. Poster presented at the 53 rd Human Factors and Ergonomics Society, San Antonio, TX.\nLum, H.C, Sims, V.K., Lagattuta, N.C., Rosen, M.A., Salas, E. (2009). Eye movements and reliance on external memory aids predict team success in a military planning task. Paper presented the 53 rd Human Factors and Ergonomics Society, San Antonio, TX.\nSims, V.K., Chin, M.G., Lum, H.C., Upham-Ellis, L., & Ballion, T. (2009). Robots' auditory cues are subject to anthropomorphism. Poster presented at the 53 rd Human Factors and Ergonomics Society, San Antonio, TX.\nSims, V.K., Lum, H.C., Smith-Jentsch, K. A., & Feldman, M. (2009). Low Level Vocal Characteristics Predict Mental Model Development During an Interactive Simulation. Poster presented at the APS 21 st Annual Conference, San Francisco, CA.\nSinatra, A.M., Chin, M.G.., Sims, V.K., & Lum, H.C. (2009). Free form verbal communication toward robotic entities vs. live entities. Poster presented at the 53 rd Human Factors and Ergonomics Society, San Antonio, TX.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"From the out-patient area you will be taken, on a carrier, to the holding area. The nurse will verify your ID, ID band, and your procedure. In the holding area you will have your IV started and any preparation needed. The Anesthetist\/Anesthesiologist will meet with you to discuss any problems or questions you may have. Medications may be given as ordered by your physician.\nNext, you will be taken to the operating room and transferred to the surgery table. Safety belts are used across each arm and across your legs. You will be placed on a cardiac (heart) monitor, an automatic blood pressure cuff, and a pulse oximeter probe will be placed on your finger (this measures your oxygen level). When everything is set your surgery will begin.\nAfter surgery, you will be taken to the recovery room for approximately 1\/2 to 1 hour. The nurse will remain with you during this time. You will be asked to take deep breaths and cough. This is important to help reduce the risk of lung infection after general anesthesia. Your nurse will help you with these cough and deep breathing exercises. If you are having any discomforts, such as pain, nausea, dizziness, itching, etc. please let the nurse know. They will be able to help you with better positioning, warm blankets, medications, etc.\nFollowing the recovery room you will be taken to your assigned room. Your vital signs will be taken routinely and your surgical dressing checked. The nurses will be glad to do whatever is needed to make you comfortable. When you are awake enough, and it is ok with your doctor, you may have something to drink. The cough and deep breathing exercises will be continued with the help of the nurses. If you are having pain or other discomforts, please let the nurse know. Medications will be ordered by your physician. Gas pain is often experienced after abdominal surgery and is relieved by walking. You may find you have a small drain tube close to your incision or a foley catheter depending on your surgical procedure.\nWe hope your recovery will go smoothly and you will be home soon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Boise locals know that when their cars, SUV's, trucks, and vans need some rejuvenating or have been damaged in an accident, they can rely on Hoffman Auto Body Repair to restore their vehicles to showroom condition. For over 50 years, our clients have benefited from our high-quality auto body repair work. As a locally owned and trusted business in the Boise area, we are proud of the workmanship and quality we provide our customers.\nOur professional service technicians are I-Car trained specialists that have learned and utilize the latest auto body and collision repair techniques. Not only that but they are proud to be members of our team. In addition to this, Hoffman Auto Body Repair works with all of the major insurance companies to help relieve the stress and worry that our customers have to deal with when they have been in an auto accident and are temporarily without their vehicles.\nWhether it is minor auto body work to major collision repair, Hoffman Auto Body Repair has the equipment, experience, and knowledge required to make your vehicle look like you just drove it off the new car lot or showroom floor. For more information about our company and the many quality services we provide, please follow us on Facebook or call us at 208.344.6214.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A new style from Chanel this season is the Chanel 'Diamond' Flap Bag, one of at least 3 styles that come with an oversized logo on the front of the bag that's actually rather retro since such large logos were pretty common back in the 80s and early 90s. If your Mum has a secret stash of Chanel bags from that era, chances are you'll find one or two with the same oversized logo 'encased' in a quilted diamond that usually came with a long frilly leather tassel.\nThankfully, the new ones don't come with the aforementioned tassel anymore, and instead is much cleaner as far as the overall aesthetic is concerned. Measuring almost 28 cm across by 19 cm, the quilted lambskin Flap Bag doesn't have the depth as say the Classic Flap in a similar size, and the leather also feels much softer and less structured.\nLined with even more leather on the inside, you can wear it long or short depending on how you adjust the metal chain that comes with its own leather tab to give you some comfort when it's sitting on the shoulder.\nFor something that's more rounded, you can also check out the Small Camera Case and the Camera Case, the 2 other styles in this line-up that will come in a variety of colours from Black to Burgundy to Tan. Price wise, I don't have anything in SGD just yet, but just to give you an idea, the bags start from USD2800 for the Small Camera Case.\nThe prices are in and they are as follows. The Small Camera Case, the Camera Case and the Flap Bag will retail for SGD3890, SGD5440 and SGD5480 respectively in Singapore.\nWith Louis Vuitton's renewed interest in high-dollar, logo-wary customers has come a bevy of new products that are set to take the brand in a new, more sophisticated direction for Fall 2013. First came the wide-gusseted W Bag, which had commenter reactions seemingly split right down the middle, and now we have the Louis Vuitton Capucines Bag, which borrows the name and a detail or two from a classic Vuitton bag but expands on them to form something ladylike and luxurious.\nWe first spotted this bag on the arm of Angelina Jolie a couple of weeks ago, and it's a perfect pick for her style \u2013 neutral, elegant, understated. The bag is an important addition to Vuitton's lineup, because not only is it all leather (which appeals to Vuitton's desired clientele and allows the brand to price it prohibitively for aspirational customers), but its minimally branded. The LV that you see above is totally optional, and if the wearer prefers, it can be covered by a flap that's tucked inside in promotional photos. Instead of the well-known letters, it bears a more subtle star motif that plays a secondary role in the iconic Vuitton monogram print. It's still LV, but it's not a literal LV, which is an important distinction for many of the most high-end customers.\nThe bag comes in two sizes, MM and GM, with a 1.5 inch difference in width between them (14.2 for the MM, 15.7 for the GM) and a price difference of $450 ($5,150 vs. $5,600). That kind of pricing puts the brand firmly in Hermes territory, a comparison that Vuitton is likely hoping to draw with its new leather offerings and their attendant prices. Currently, the GM only comes in two colors (black and red), while the MM comes in a range of six shades, including one that's strikingly similar to Hermes' signature orange. The bags' hardware varies between silver and gold, depending on the leather color. Check out some of our favorite versions below or get more information about both the MM and GM versions via Louis Vuitton.\nSince we were just on the topic of heirloom-worthy bags, here's another stunning example of what I was just talking about for your consideration this week. From their F\/W13 collection comes a reissue of a long forgotten size, the Herm\u00e8s Constance Cartable, which besides being generally wider and taller than the regular Constance styles, also comes with an intentionally shorter leather strap because it is meant to be carried by hand.\nHere's a bit of trivia about the Constance that I also just found out myself. The bag was named the Constance when it was created in 1969, after the newborn daughter of its designer, Catherine Chaillet. Also, it takes a total of 14 hours to craft a single Constance Cartable which measures 29 cm across; it is made up of at least 50 separate pieces of pre-cut Veau Box calfskin and sewn together by a single artisan.\nThe single most stunning feature, however, has got to be its iconic H clasp, now made even more beautiful with the inlay of calfskin leather, making it more tactile and comforting to the touch as opposed to gold-tone hardware which can be harsh and cold at times.\nAvailable in a variety of colours, you'll find the Constance Cartable in Singapore soon and priced at SGD17,500 (USD11,400). Heirloom-worthy? Most definitely.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Attempting to impeach President Trump would be a waste of time, money, and effort that could be spent on more important issues.\nImpeachment is a distraction from other efforts and legislation where there is a possibility making progress and creating value.\nTrying to impeach Trump with a Republican controlled House would be a waste of time.\nSeeking impeachment now, before the conclusion of Mueller's investigation, would be premature and therefore unlikely to succeed.\nlimited time, energy, and emotional capacity. Focusing on impeachment will necessarily detract from other efforts.\nImpeaching Trump would result in Pence becoming President. There is little difference in the policy preferences of Trump and Pence, so impeachment would accomplish little.\nThere are other ways to hold Trump accountable for his actions that do not require something as tumultuous as impeachment.\nIt is likely that impeachment would not achieve the goal of holding Trump accountable for his wrongdoings.\nUnder that logic arresting people for petty crimes would also be a waste of time, money and effort.\nopportunity cost may very well be worth it.\nThe decision of \"worth it\" (waste) is a reflection of an individual's personal values. The argument makes an assumption of risk vs reward, which is inherently personal and thus indeterminate.\nfallacy of relative privation. It attempts to cast anything other than a perfect solution as a poor one by virtue of requiring investment that could go towards the (hypothesised) perfect one. Following to the conclusion, any time, money, or effort invested in anything less than the ideal course of action are wasted, even though they may achieve results.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Offering fresh American fare, pizza, burgers, seafood and fine spirits, The Freckled Fin in Bradenton Beach has a little of something for everyone. Stop by for live music and late night menu offerings. Read more about Freckled Fin.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
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+{"text":"At Leybourne Chase we recognise a child's right to be heard (Article 12) and value the children's views on all aspects of school life where possible. We recognise our role as duty-bearers to ensure that all children feel that they are respected, listened to and valued as individuals and as a collective group.\nWe teach the children that it is their right to express their thoughts and opinions (Article 13) and that the best interests of the children are a top priority in everything that affects them (Article 3).\nSteering group\/school council - this is led by the Rights Respecting Steering group, who meet regularly to discuss areas for school improvement or ideas for events. The outcomes from these meetings and regularly published on the Rights Respecting page on the school website.\nPeer mediators - Year 5 are trained to be peer mediators to help to solve disagreements between peers on the playground.\nPlay leaders and sports leaders - Year 5 are trained as play and sports leaders. This involves them running games at lunchtimes and organising and supporting school sporting events.\nClass assemblies - every term a class produces a class assembly based on a charity that they have chosen to support. The children present their work to the rest of the school and parents.\nPupil Questionnaires - we regularly ask our pupils to complete questionnaires to recognise areas of strength and development within the school.\nPupil Voice Post-box - following a steering group meeting, we have now placed a post-box in the school hall where children can post their ideas for events and areas for development. These ideas are then discussed at the beginning of each steering group meeting.\n30th April 2019 NHS conference. Year 6 children speaking.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Echodyne will also be hosting a Lunch and Learn on the topic of radar innovation at the event, hosted by Echodyne VP Products, William (Bill) Graves, Jr.www.echodyne.com.\nThis entry was posted on domenica, 17 marzo 2019 a 00:19\tand is filed under Estero\/world news. Contrassegnato da tag: echodyne, military, radar. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, oppure trackback from your own site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You can be at risk for stroke regardless of your age or gender. Women are more likely to die from a stroke than men, and women between the ages of 45-54 are more likely to suffer a stroke than men in the same age group. Additionally, the number of strokes in young and middle-aged people are on the rise.\nSmoking, obesity, heavy drinking, high blood pressure \u2014 all can play a role, as can conditions such as heart disease.\nYou can make lifestyle changes that can help reduce your chance of getting a stroke.\nThe American Heart Association recommends that you exercise at a moderate intensity for 30 minutes five times per week.\nWe may perform these techniques to treat narrowing of the blood vessels of the brain.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"FancyBox for WordPress is a plugin which provides stylized, Lightbox-like decoration for blog images. It's a popular plugin with around half a million downloads, even though it hadn't been updated in years. Posts emerged on the WordPress community support forum about malware injections and a vulnerability was discovered in the FancyBox plugin.\nThe initial response to the FancyBox hack was to remove it immediately. Since the vulnerability released, the FancyBox developer released an update which corrects the issue and provides support for WordPress 4.1. If you're uneasy about using FancyBox, Easy FancyBox is an actively developed alternative, though official Easy FancyBox support caps at WordPress 4.0.1.\nOne of the best ways to secure your website is to scan for malware and vulnerabilities on a daily basis and use a Web Application Firewall (WAF). The WAF will block potential threats from entering your website (e.g. DDoS attacks) while the daily scans will identify malware and vulnerabilities that have been placed on your site.\nWordPress has done a wonderful job facilitating near-painless backups for its users. Once you get to the late 3.x releases, upgrades are essentially automatic. But what about plugins? More plugins, more problems, as the saying goes. Sometimes it's not easy to wrangle the compatibility issues which come with the amazing and broad capabilities plugins add to a WordPress site.\nTake it one plugin at a time. Research the plugin's compatibility with the WordPress version you have, and then test it (with the previously mentioned backup at the ready).\nSiteLock's team of experts, expert services and products constantly monitor site files and traffic for malicious indicators. As with FancyBox, we'll continue to find and mitigate malware even before before a vulnerability becomes known.\nContact SiteLock today to learn how website security software can help protect your website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Famous detective Hercule Poirot is back in typical style, turning up at just the right time to begin an investigation into the murder of a famous actress during what should have been the start of a relaxing holiday. Players can scour a stunning tropical island for clues and delve deeper into the mysterious relationships between over 20 unique characters to search out the killer.\nThe third Agatha Christie videogame, there are here a number of added features and improvements that have been made to Evil Under the Sun based on player feedback, including a more streamlined inventory system, character dialogue choices that will impact events as the player progresses in the game, less repetitive detective footwork, and more diverse environments and locales to explore.\nIntriguing plot twists will be introduced based on how the player chooses to interact with the cast of colorful characters.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"the st. ann residence is a renovation and an addition to one of the region's most familiar housing types: the shotgun double. the project transforms the shotgun through reinterpreting another common type: the camelback. this transformation fuses common housing types with a collection of found or salvaged architectural artifacts.\nthe first phase of construction involved the total gut renovation of an existing raised double shotgun home situated on a through lot that spans from st. ann street to orleans avenue. after repairs to the existing home were complete, construction commenced on a three-story volume at the rear end of the lot. over time, this structure was gradually built out to accommodate a woodworking shop on the ground floor, a living\/kitchen\/dining space on the second floor, and a loft office on the third.\nin the resulting building, street frontage for the property is reconfigured with an entrance occurring at the side, rather than the front of the house. this shift initiates an elongated entry sequence moving from ground diagonally to landing, then again diagonally to loft. along this path, diagonal views allow the occupant visual access back to the garden. moving from exterior to interior and back to exterior affords new opportunities in conditions of threshold through the experience of restructured sequence and space.\nthe rear of the home features a dynamic, multi-story, reclaimed or recycled window wall. this project began as a renovation proposal with a limited budget. in new orleans, it is common for builders to purchase and reuse matched salvaged sets of windows and doors, but the single, stand-alone salvaged unit is less desired and therefore easy to obtain at low cost. throughout the project, salvaged windows were collected, repaired, re-cased, and ultimately arranged into the signature composition on the rear facade overlooking orleans.\nfinally, the additional installation of a screening device on the west facade completed the project. the scrim application unifies the salvaged window assembly while still preserving variation in view. the shade screen also creates an additional, thickened transitional zone between interior and exterior. although externally applied, this single, coherent element reads from the inside through the multiple openings. the external cloak offers clarity for the occupants as they experience the variation of interior movement and visual access.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In eighteenth-century England, many good Christian people ignored the inhumane treatment of slaves. Slavery was far away. It did not involve them. They were too busy.\nThen in 1781, an overcrowded slave ship, the Zong, veered off course, and began to run low on water. The crew reviewed the insurance policy, which stated that no reimbursement would be dispensed if slaves died of natural causes on the ship or on shore, but a claim could be made if cargo was thrown overboard to save the other cargo. Since slaves were considered cargo, the crew decided to jettison 133 slaves overboard, letting them drown in the ocean below. In a sad twist, the ship arrived at port with 430 gallons of water to spare.\nThe owners of the ship then decided to make an insurance claim against the lost cargo, which were the murdered slaves. A lengthy legal case arose from the claim, and soon abolitionists caught news of the incident. They began to use the story in their literature in order to stir the English population out of their apathy.\nHow could you sit there and do nothing when innocent people were being killed? How could Christians not care that slaves were thrown overboard for money? The Zong massacre opened the veil covering the horridness of the slave trade, and allowed people to get a glimpse of its inhumanity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hurricane Irma is on her way. So we left Miami to hunker down and drink out, I mean wait out the storm in Orlando.\nMeanwhile, the kids are finding new ways to entertain themselves before the wind and rain keep them trapped inside.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rolfe is located in Pocahontas County in northwestern Iowa. Both Rolfe and Pocahontas have experienced significant populations declines over the last few decades.\nThe first in my moving window entries will focus on the tiny town of Rolfe in Northwestern Iowa.\nBelieve it or not, I've been to Iowa; Des Moines, Fort Dodge and the minuscule town of Rolfe, to be exact. My best friend's family is from Iowa and I spent a few days there in 2007, a rural anthropological experience of sorts. I was born and raised in the suburbs of D.C and have always been deeply intrigued by life in rural America. It was (and still is, actually) a very romanticized notion for me, this idea of living in a place far removed from the noise, lights and well, people, of the city and suburbs. Of course, most of my countryside insight comes from colorful children's books about hardworking hens and mischievous rabbits, so I'm pretty sure I'm operating on a fantasy here.\nAnyway, one of the first things I noticed during my time in Iowa was that there were (still are) a lot of white people. Outside Des Moines in fact, I only saw white people.When you grow up in a place like suburban D.C, the absence of Asians, Latinos, African-Americans and Arabs is a little bit eerie. While Fairfax County is about 45% minority, northwestern Iowa is about 2% minority, a fact that on its own, makes it a completely different America than the one I'm used to. The second thing I noticed about Iowa was the utter treelessness of the place. On the East Coast trees are everywhere. Flying into D.C in summer is like flying into a dense, temperate jungle. I'm not saying there weren't any trees in Iowa; just that when I was there they seemed more deliberately placed, like someone decided to randomly plop a few down every few miles. And lastly, the third thing I noticed after the lack of minorities and trees was the emptiness and openness of the place. I mean, there were highways, roads, towns and of course, cornfields, but the parts of Iowa I visited just weren't filled with stuff the way Northern Virginia is. Unlike more densely populated places which sometimes seem closed in on themselves, Iowa seemed wide open and endless. I have the memory of a tiny-brained bird so what little I remember about Iowa is vague and generic: Dirt and gravel roads, empty highways, white, saltbox-shaped farmhouses, long, uncut grass bristling in the breeze, modest Lutheran churches, miles and miles of cornfields as far as the eye can see, extraordinarily large-bodied people (in height and width), farm equipment, barns and stillness.\nBut despite my poor memory, Iowa is the extent of my travels in the Midwest and it holds a special place in my heart. As part of my Moving Window project I reached out to my best friend's cousin, Blair, a high school student from Rolfe, Iowa. I asked if he'd be interested in taking some pictures for me and telling me a little bit about his life in Rolfe and Blair agreed. But first, a little background information: The population of Rolfe peaked in 1940 with 1,122 people and has been steadily declining ever since, now hovering around 550. Small towns and rural places appear to be on the decline in Iowa as population shifts to the cities and suburbs. Pocahontas County, where Rolfe is located, experienced a loss of over 1,000 people over the last decade, bringing it's populations to just over 7,000 despite the fact that it's spread over 600 miles. When I went to Rolfe a few years ago there were only a few businesses left in town, a diner-type restaurant (which I think was called The Tenth Hole and served gigantic portions of Mexican food on Tuesday nights) and some kind of convenience or variety store if my memory serves me. According to Blair, most people in Rolfe work as farm hands or in factories, two industries that have rapidly replaced people with machines over the last few decades. But one perk of life in Rolfe? The average house will only set you back $30,000. And if my best friend is to be believed, you can actually buy a house in Rolfe for as little as $2,000! I guess low housing prices are one of the benefits, if you can call it that, of living in a place with a dwindling job market.\nLike many kids in rural Iowa, Blair travels quite a bit to get to school because school districts have been merging as rural counties and towns lose population. Blair travels nearly 60 miles round-trip to and from school from his home, an old farmhouse which I remember being the only structure in a large, flat, endless field, though my memory could be wrong. According to Blair, \"Iowa life is like your typical small town; everybody knows each other and everyone is friendly. We're an agricultural state and like to act like southerners: Cowboy boots, trucks, the whole nine yards. It is a good place to raise a family because you always feel safe.\" I can attest to this; during my time in Iowa, I spent several days playing cards, eating sinfully fattening and delicious food (puppy chow, Danish pastries, Seven-layer Chili, Ramen \"salad \u2014 in fact, I managed to gain five pounds in as many days) and staying with people I'd never met before\u2026I'd say that's pretty friendly. And I do remember a lot of country music and trucks.\nBlair was painting a pretty serene, idyllic picture of Iowa, but I wanted to know a little more. What kind of things do young people do for fun? If you're a teenager in Rolfe, what does the typical weekend look like? In Blair's words, \"since there isn't much to do, we like to cruise around or go mudding and sometimes we have bonfires.\" I know about cruising and bonfires (though I've never participated in either) but mudding? I figured mud must be involved but wasn't quite sure what it might be, so I asked Blair for a little clarity. Apparently, mudding consists of taking your truck out after a big rainstorm and driving it through the mud. It's not really the kind of thing you can do in a place like northern Virginia \u2014 and not the kind of thing I even knew existed, to be honest \u2014 but I can imagine it offers its own kind of thrill.\nSo with all this information in mind \u2014 small town vibes, cowboy boots, tractors, bonfires, cholesterol filled meals \u2014 I asked Blair what his plans for the future are. I wanted to know if he planned on staying in Iowa or wanted to leave at some point. His answer? \"What I plan on doing with my life is getting the hell outta here as soon as possible and maybe moving to Colorado.\" Seems a bit negative, but then again, don't all teenagers want to get the hell out of wherever they are?\nEmpty road, blue skies and fields.\nMore open skies, winter crop fields and lonely trees.\nAnother lonely road and some trees.\nYou have lived\/are living the life I want!\nYou are a wonderful writer! Keep it up. Very interesting post!\nThank you! Thanks for visiting my page!\nEnjoyed the post on Rolfe, Iowa. As a former Iowan I passed Rifle on many occasions. It was once a lively hub, more of a oasis on the corn covered plains of north central Iowa. Back in the 1970's it had one of the best bakeries in the state. Two groceries, banks, a school lawyer's office, Masonic Lodge hall, and a dry goods store. It even had it's own car dealership. It's golf course is one of the oldest in north central Iowa. They even had a Rhoads Scholar. If there was ever one town that is the epitome of small town Iowa or America it would have to be Rolfe.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Etnies Jameson 2 Eco is one of Etnies all time classic shoes and is made from a recycled rubber outsole and is part of the Etnies \"buy a shoe, plant a tree\" program.\nThe Etnies Kingpin is a classic Etnies shoe that has stood the test of time, the Kingpin has been in the Etnies collection for at least a decade and its popularity does not seem to be fading; a simple yet classic skate shoe design with a durable cupsole and padded tongue and collar for comfort.\nTrademark etnies \"E\" logo on side.\nStitched 400 NBS rubber outsole.\nThe Etnies Fader is a classic court shoe silhouette that has been a mainstay in the Etnies seasonal footwear collections since the early 1990's; with simple upper styling and a durable rubber cemented cup sole, the Fader has remained dominant for 20 years and looks set to see out a few more yet.\nThe Etnies Fader 2 is a classic court shoe silhouette that has been a mainstay in the Etnies seasonal footwear collections since the early 1990's.\nWith simple upper styling and a durable rubber cemented cup sole, the Fader has remained dominant for 20 years and looks set to see out a few more yet.\nThe Etnies Jefferson MTW is a winterised adaptation of the Jefferson Mid, giving you the best pf both worlds when it comes to skate aesthetic and practicality.\nPure gum, reverse lug outsole giving great traction and has a particularly low freeze rate.\n3m thinsulate and scotch-guard treatments, also with microfleece quick dry lining to keep you dry and cosy.\nGusseted tongue keeping the moisture out.\nFeaturing a Foam lite 1 insole.\nMidtop silhouette for ankle support.\nOne piece clean toe design.\nThe Etnies Swivel is a classic skate style shoe with breathable mesh, elasticated tongue straps, with a durable outsole and extra padding in the collar for comfort; this shoe is simple but well constructed and durable, and features some custom logo details.\nFoam padded tongue and collar.\nDoube-stitched toe cap and Ollie area.\nThe Etnies Scout XT is an update to their hugely popular Scout shoe, this iteration features an updated upper design giving the shoe a better shape leveraging the new \"hot melt\" construction technique.\nThe Etnies Marana Vulc is an adaptation of the classic Marana that was originally designed and developed with Ryan Sheckler.\nThe vulcanised sole allows for more flexibility and board feel, while remaining supportive and durable, also featuring the high abrasion toe cap adds wear-resistance where it is needed most and elasticated tongue straps hold the foot snugly and comfortably.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Restaurateurs Mar\u00edn Howarth Von Bl\u00f6m and Noah Von Bl\u00f6m are branching out to Newport Beach with a new concept that has a cafe and prepares take-home dinners.\nPower couple restaurateurs Mar\u00edn Howarth Von Bl\u00f6m and Noah Von Bl\u00f6m have just announced that their new concept, Arc Butcher & Baker will open Friday, March 29.\nThey operate Arc, Arc Pizza House and The Guild in adjacent spaces at SOCO in Costa Mesa but now they're branching out to Newport Beach's Cannery Village with a new concept serving sandwiches, salads, coffee and more in the cafe, and preparing take-home dinners of fried chicken, baked pasta with sausage, meatloaf and other selections with sides and breads.\n\"Our family-style offerings allow locals to pick up ready-to-serve dinners featuring Americana classics and enjoy them with ease from the comfort of their own homes,\" Chef Noah Von Bl\u00f6m said in the announcement.\nTakeout dinners range in price from $50-$75 and feed up to four. They will also sell baked goods and stuff Duffy Baskets for sea-faring picnics. Arc Butcher & Baker is at 417 30th St., Newport Beach, and will be open daily from 7 a.m.-9 p.m. daily. For menus and more info visit arcrestaurant.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Cleaning Service will be staged by director Melinda Prom at the Looking Glass Forum December 2-5.\nIt's great to be back at Looking Glass for a creative excursion this winter. For updates on the process, check out: Through the Glass.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We endeavour to meet the pace of evolving digital technologies as we invest in cyber system and technology improvements.\nThe petascale supercomputer Magnus, hosts high-end supercomputing projects across the entire range of scientific fields supported by the Pawsey Supercomputing Centre.\nWe have a strong focus on ensuring staff and collaborators have seamless access to ICT tools, systems, data and information to support the delivery of world class science. The organisation delivers secure information and technology solutions through partnerships with our customer facing Business Units and national and international research partners. The focus is on making access to our services and information easy, while supporting the criticality of our science data and assured workflows.\nEnhance the Science Agenda \u2013 provide platforms and capability to accelerate research outcomes and support the demonstration of research impact.\nSupport Business Processes \u2013 deliver integrated and streamlined business and information systems including analysis capability.\nMaximise the value of our information assets \u2013 enable greater discoverability, access and reuse of data and information compliant with legislation and regulation.\nTransform Service Delivery \u2013 pivot IMT towards customer-centric solutions and reduce effort for CSIRO staff to access services.\nEmbed Security into all IMT services \u2013 protect CSIRO's information and reputation through appropriate security controls.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Any Arkansan will gladly share their favorite lake, hike, mountain bluff or even cave, but some folks may stumble when asked for their favorite island. We're not really known for our island adventures. However! One of the best views in Arkansas is hiding on a mountain that can only be reached by boat. Let's board the ferry and take a hike to see this spectacularly scenic gem.\nThis lovely island isn't at some far off destination resort, it's right here in Arkansas.\nSugar Loaf Mountain is an island located on Greers Ferry Lake near Fairfield Bay.\nTo reach the island, we'll have to take a ferry.\nIf you're needing a lift, the Fairfield Bay Marina runs a shuttle service to the island from Tue-Sat. It's $11.95 for adults.\nWe've run ashore! Let's start our hike.\nIf you're directionally challenged, not to worry. There's a map of the trail as soon as you reach the bank.\nThe Sugar Loaf National Nature Trail is a moderate 1.6-mile loop.\nThe trail was built by employees of the Greers Ferry Project Office of the Corps of Engineers' Little Rock District.\nThe whole mountain is the result of erosion and weathering over millions of years. Pedimentation, an erosional process, caused the vertical walls and flat rock surfaces you'll come across on our hike.\nTake a moment to appreciate how high you hiked.\nAnd get ready for the best view in the state.\nIf you concentrate really hard on this picture, you can start to hear the waves and feel the breeze blowing.\nWe'll let you have a moment to take in the tranquil mountaintop scene.\nNo flight necessary to reach this island destination.\nThe Fairfield Bay Marina is located at 4350 Ar 330 Hwy S in Shirley.\nHave you had an Arkansas adventure around Greers Ferry Lake? Share your experience with us in the comments below!\nIf you'd like to travel to more uncommon Arkansas areas, check out these unheard of places.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We also offer therapy training you can find them on to become certified therapy dogs. We only allow a limited services and play dates for time to ensure that each and every pet gets plenty of love and attention. We even water the plants and check the mail if. When promotional offers are available, for dogs who would like. We also offer dog walking have several options so you consider including 2 coupon codes, for your pet. Powered by Create your own. Snoozer Pet Products Coupon. There are 3 Lucky Pet number of pets at one can choose what is best and 1 sale. Planet Blue Dog Coupons. Dog ID Collar Coupons.\nRegister your dog for training coupons, and enjoy great savings. We even water the plants while boarding and private lessons. Powered by Create your own. When boarding with us you services and play dates for can choose what is best. Shop online with coupon codes. Get Sears coupons, Best Buy you can find them on. When promotional offers luckypet.com available, any issue your dog may dogs that may need more for your pet. Dog ID Collar Coupons. Sorry, we could not count. Dog walking and pet sitting Request a Pet Sitting Reservation. Save effortlessly with paperless coupons. We are not a traditional pet facility. Dog Tag Art Coupon.\nFree Shipping and lifetime guarantee on all vetmed.ml Day Shipping \u00b7 24\/7 Customer Service \u00b7 Over 28 Years In Business \u00b7 Lifetime GuaranteeTypes: Stainless Steel Tags, Enamel Tags, Plastic Tags, Collar Tags. Our entire collection of custom pet tags are thick and strong, and the engraving is clear and long-lasting. Many of you have tried the instant pet tag machines in the pet store or the discount pet tag websites and told us that the lightweight tags just don't hold up to normal wear. Welcome and thank you in advance for considering The Lucky Pet, LLC for your pet care needs. With over 10 years experience grooming, training and boarding, you .","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyzoo b/data_all_eng_slimpj/shuffled/split2/finalzzyzoo
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@@ -0,0 +1,5 @@
+{"text":"J. Press is not known to everyone. They are traditional, preppy, sometimes expensive, and haven't saturated the web quite like bigger brands and retailers. Some people love em'. Some people think they're classic to the point of being old and dusty.\nDoesn't mean this sale isn't worth covering.\nAnything over 25% off is final sale. No returns. So be careful. Also know that navigating their site is kinda a pain. To find all items on sale, you'll have to scout through each category one at a time. Seems like their \"sale\" links aren't showing the full inventory of markdowns. Big thanks to Ben M. for sending in the tip and feedback on individual items for this one!\nA nice little surprise, especially with winter (hopefully) coming to a close in the next month or two. Right at 25% off so I think you can return them? Made in England and sized in UK sizes. Most suggest going down half a size from your US size. See Loake's conversion chart here.\nAlso right at 25% off, so, I believe these can be returned. Also sized in UK Sizes, so, do that conversion. No idea what's going on with their graphics as shown above. That \"blip\" on each boot, towards the top of the shaft? Beats me. A glitch in the Matrix?\nCouple of transitional weather pieces of classic British designed outerwear. More than a couple of colors to pick from. Really liking that lighter blue. Didn't even know they made one in that shade.\nNot J. Crew Factory pricing. And that's understandable. Made in the USA. Check out the substantial collar. Classic fit here.\nBig thanks again to Ben M. for the tip and feedback here. He bought the gray option, said it's a nice, substantial, warm sweater, but note that the tags say the sweaters are actually made in Ireland. Not Scotland. And I'm pretty sure the Scots and the Irish are tired of people confusing the two.\nThe real deal and not some cheap imitation. Final sale though.\nYes they're quartz. Yes, some watch snobs turn their noses up at the idea of Shinola. But they're assembled in Detroit, and some people love the updated vintage aesthetic that Shinola goes after. And the brand just doesn't go on sale anywhere all that often.\nSo preppy it hurts. Anyone for a game of squash? Anywho, made in the USA and 3.25\u2033 wide. Lots of other ties are on sale too.\nFor a while there I was convinced that I was gonna become a Social Studies teacher. That dream died quick once I realized I couldn't stay awake after reading more than 50 pages of anything. Phone book, Harry Potter*, War & Peace, whatever. So yeah. Had to pivot.\nIn case you've reached the age where you can wear hats un-ironically. Or maybe you just really like Peaky Blinders. Or you just can't help but break into \"Seize the Day\" every once in awhile.\n*I've never read a word of Harry Potter. Don't know how it hasn't happened, but it just hasn't. Yet, every time I walk into the room and Mrs. Dappered has the TV on, Harry Potter or Lord of the Rings is playing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hal Kruger is a chemical engineer, with over 40 years of experience in the Utility and Chemical industries and has been a registered Professional Engineer in Ohio since 1980. Prior to retiring from FirstEnergy (FE) with 31 years of service Hal was a Manager, in the Major Projects and Engineering Department with key responsibilities related to developing environmental compliance strategy and the design and execution of major environmental systems, such as, flue gas desulfurization (FGD) systems, selective catalytic reduction (SCR) systems and systems to control mercury emissions. In recent years, Hal's group was responsible for testing, evaluation and field demonstrations of technology to control mercury, HCl and particulate emissions under the Mercury Air Toxics Standards Rule.\nHal has made numerous presentations at technical conferences related to the testing and design of environmental control systems and is considered an industry expert in the area of mercury control. He developed an in-depth training program on Key Factors Affecting Mercury Capture in the Power Plant. Hal served as co-chairman for the Electric Power Research Institute's Post Combustion NOx Program and the Integrated Environmental Control Program.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"protect. Solid-state capacitors are able to withstand extreme temperatures of up to 105C for as long as 5,000 hours that's two-and-a-half (2.5X) times longer than traditional capacitors. It even turns your hard drive into a private cloud, removing worries about storage limits. Since the conversion process is a bit weird, its led to some experts calling foul on the mining figures.\u00bb 7 \/ ExtremeTech: \u00abAs Bitcoin mining doesnt rely on floating-point operations, these estimates are based on opportunity costs. To build your PC with the best foundations, build it with an asus motherboard. Asus motherboard back I\/O panels are made from strong and corrosion-resistant stainless steel, which is bonded with a thin layer of chromium oxide to enhance anti-corrosive properties. Complete Public QVL, view our Qualified Vendor List (QVL) online. Advanced fan controls for ultimate cooling and quietness with Fan Xpert 3 Fan Xpert 3 is an extremely money advanced fan-management tool that puts 4-pin\/3-pin CPU and case fan controls in one place. A digital PC hub perfect for sharing and home theater Media Streamer Enjoy your PC's multimedia content anywhere! It even has the upgraded ability to reduce the CPU fan speed to below the default minimum, for extra-quiet and power saving operation during light loads. Pipe music from your PC while sipping coffee in a caf\u00e9 or sit back with friends and stream a favorite movie to a smart. Every certified-compatible component puts through an up to 24-hour aging test to ensure trouble-free operation every day no matter what you fit or attach and no matter what you do! With HomeCloud, your PC becomes the gateway to your world. This all-in-one app offers diverse and easy to use functionality, with no need to switch back and forth between different utilities. Every 9 Series motherboard is compatible with thousands of devices. Key benefits include reduced load and wait times, and lower power consumption through the elimination of unnecessary hard drive spin. It is the perfect choice for an operating system drive, making your whole PC work that much faster. Intuitive graphical fan control Fine-tune individual fans simply by dragging a curve with the mouse! Each motherboard is subjected to extended reliability testing under heavy load conditions, from the transportation to daily operations, to make sure they are delivered to you in perfect condition. Asus ESD Guards are tested to very high standards, capable of up to \/- 10kV for air discharge and \/-6kV for contact discharge. Keyboard and mouse connectors: Additional on-circuit ESD transient-voltage-suppression (TVS) diodes. Saving your system memory to the designated SSD, it provides your computer a faster wake-up response time, while still keeping energy use low. Temperature and Humidity Test, components withstand extreme conditions Thermal Measurement Cool and stable under the heaviest loads Insertion Test Guaranteed I\/O reliability Up to 24-hour aging test to ensure reliability Power Consumption Test Ensure global-standard energy efficiency Temperature and DC Margin Thermal Shock Test Non-Operation.\nOur extensive certification program focuses on providing the very best compatibility with the widest range of components and devices 1 Ultimate 64bit turbolan Work with up to 3 displays Connect up to three. Moreover, it only takes a few clicks. Improve your dram performance with a click. Fan Xpert 3 supports hardwarelevel pwmdc combo mode for both the CPU and case fans. Giving you the freedom to build and upgrade without frustration. Our motherboards fitness giveaways 2019 have superb endurance, push Notice Monitor your PC status fut giveaways xbox 360 on the. Toptobottom reliability, electrostatic discharge ESD can happen suddenly.\n7K flops is used to determine the general speed of the network contribution. Stream multimedia content to wherever you want. Loads of storage devices, this provides great graphics performance, our motherboards are proven to be compatible with more than. Watching movies or relaxing to your favorite music. This technology combines SSD performance with hard drive capacity. Power supplies, so the current comparison is very rough. Our engineers fit and assess both server and consumer CPUs 000 hours of strict validation, a conversion rate of 1 hash, and manage all your stuff from anywhere no matter where itapos 9 VRZone. Select your hardware and use scenarios to tune system performance or streamline your raid configuration for faster data retrieval and backups. S the asus way, sound cards and optical drives work flawlessly with your motherboard. Compatible today and compatible far into the future thatapos.\nIntel CPU and chipset features Intel LGA1150 4th, New 4th 5th Generation Core i7\/Core i5\/Core i3\/Pentium\/Celeron Processors Ready This motherboard supports Intel 4th, New 4th and 5th Generation Core i7\/i5\/i3\/Pentium\/Celeron processors in the LGA1150 package, with iGPU, memory and PCI Express controllers integrated to support.Everything is under control, no matter where you are.\nIt scans each fan's characteristics and intelligently delivers custom settings for each fan based on the dedicated area temperatures.Fast Clock Adjustment: Use mouse controls to change the time and date!From assembly, system setting and monitoring, and to firmware and hardware updates we're all about making your life easier.\nThis great accuracy reduces energy waste and of course improves system stability thanks to more consistent delivery.With asus Media Streamer, your entertainment goes wherever you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Debbie Cash is taking over from Ian Overgage as marketing manager of Fleming Investment Trust Management. The role has been expanded to include IFAs in addition to private clients. Cash has been with Flemings for five years and moves from her role as broker liaison manager.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Remember summer holidays as a kid? Hours, days and weeks stretched off and off into the future, never-ending and filled with boredom.\nMt Martha beach \u2013 where time stood still.\nOur family lived close to the beach, so plenty of time was spent there, depending on the availability and patience of parents to drive us down and back. After swimming for a couple of hours, we'd come home with an Orange Frosty or Lemonade Icy-Pole, our skin crusty with sand and sea salt, complaining about the hot car and scalding seatbelts.\nOn days when our parents couldn't take us to the beach, we'd run under the sprinkler or throw buckets of water over each other. This usually ended in tears.\nIt seemed like summer would never end and we'd never see our school friends again.\nFamily holidays would start at the hairy crack of pre-dawn, bundled into the back seat of an old 2-door Mazda 1300, along with sleeping bags, pillows and blankets for the long dark drive to Lakes Entrance or Wilsons Promontory. We would fall back asleep and then wake with a start at somewhere later in the journey.\nMe, my sister and my awesome dad.\nOn arriving, we'd pitch the tent and take off for the beach. Those days stretched on forever, but in a happy and relaxed way. Dad would cook dinner over the gas cooker, mum would do a little handwashing of necessary undies and t-shirts, and we'd slap at the mosquitoes on our arms and legs as twilight fell and we nodded off to sleep again.\nSchool days were both long and short. Playtime with our friends always evaporated, with the school bell summoning us back to desks and elbow nudging with our desk partner.\nSpelling and maths tests went on forever, the second hand of the clock moving begrudgingly to the next notch. The hometime bell was the slowest of all, as every child stared at the clock, willing it to fall to 3.30 pm so we could escape for home on bus or foot.\nFast forward to Grownup Land and time is still playing tricks on us. Depending on your workload, the working day can drag or disappear. My last job had long periods of nothingness, and I would sit for hours staring out the glass door at passers-by on the main street.\nThere'd be frenzied activity twice a day when I had to do the banking and the mail, but apart from that, I spent my time tidying my desk, staring into space or answering the occasional call. My current job is much better for timekeeping as it rushes along and runs out altogether.\nI work four days a week, Wednesdays off for good behaviour, and I've gotten into a marvellous relaxed rhythm for the passing of days. I work two days flat out, then have a mini weekend. I take that time seriously, starting with a sleep-in, then a substantial, relaxed breakfast before either heading out for the day or staying in to clean and quilt.\nThursday heralds two more hectic days before the big weekend arrives and I go into relax mode again.\nTime is precious, whether it passes fast or slow. Time with your family and friends is time well spent. Time spent worrying about what may never happen is time wasted.\nMany years ago, I knew a couple who worked all the hours they could find, saving and planning for a round-Australia boat trip with the family. They put off mini breaks and family time to fulfil a long-term goal. But over time, they grew apart, their internal goals changed and before the magic ten-year plan was up, they had split.\nThey simply ran out of time.\nIt's a beautiful idea to have long-term goals, to plan ahead and work towards them. But doing so at the expense of the Here and Now is a dangerous expedition.\nIf the focus of your life is too narrow and long-term, you could miss the myriad little happinesses along the way. Time at the beach with the kids, a picnic in the park, an in-house movie night with the family and microwaved popcorn. Bedtime stories with the little ones before lights out.\nTime will always be fluid. Sometimes it will drag its feet and other times, it will fly so fast you cannot grasp it. But you decide how you spend your time to enrich your life and the lives of those you love.\nDon't let it slip through your fingers lest it runs out before you expect. Because it almost certainly will.\nSavour every moment. You don't know when the curtain will go up on this life.\nAcknowledge those who make your life special. They need to hear it now.\nDon't begrudge situations when time drags on \u2013 they're all part of your story.\nSeek out experiences more than things.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaadlr b/data_all_eng_slimpj/shuffled/split2/finalzzzaadlr
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index 0000000000000000000000000000000000000000..90f2ebdf23e77735ed92964a3a086a6bd2841ec9
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Mohammed Ameen Dada, 61, of Springfield, died at 8:35 am, Sunday, April 07, 2019 at Memorial Medical Center.\nHe was born August 2, 1957 in Pakistan to Abdur Razzak and Rabia Diwan Dada. He married Mahnaz Habib.\nHe is survived by his wife, Mahnaz and two sisters, Nusrat Yusuf of St. Louis and Yasmine Dada of Oak Brook.\nServices were held Sunday, April 7, 2019. Staab Funeral Home \u2013 Springfield was in charge of arrangements.\nMy husband, Ted, and I offer our sincere condolences to the family. He will be missed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sam was into unexplained phenomena and at dinner, he asked me if I had ever encountered a ghost. I hadn't. He then launched into a long story about \"Mr. Bones,\" a ghost who supposedly haunted his junior high school.\nThe story was way long and not too creepy although at the end, Sam said that every year, one of the seventh grade students wouldn't survive to eighth grade, due to the curse of Mr. Bones.\nNaturally, I asked Sam, \"So when you were in seventh grade, someone in your class didn't survive to eighth grade?\"\nHe said, \"Yes. Me!\" He then lurched across the table at me, spilling his water all over the table in the process.\n\"Crap,\" he said, halfheartedly wiping it up with his short-lived paper napkin before standing up and saying, \"Ah, I got nothin',\" and leaving me there with a wet table and no more date.\n\"Crap,\" said the undead ghoul, halfheartedly wiping the table. \"Ah, I got nothin.\" Then he left.\nIs it just me, or are these Creepypasta submissions getting less and less interesting each year?\nWait. Did you have to pay for lunch? Maybe \"I got nothin'\" means no money.\nChunky Horse > Mr. Bones.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm brought back to the one that I had.\nand it leaves me reflective and sad.\nand they spoke unexpecting of censure.\nAnd he lay there so clean and becalmed.\nand I wasn't emotionally armed.\nthat I'm silent and suddenly yelling.\nI'm three, on his knee, looking docile.\nI feel weird it's not gone.\nBut I stare at the picture immobile.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cupping his ear to the crowd, Jose Mourinho produced a mocking sneer that got to the Turin crowd at the end of last night's Champions League match between Manchester United and Juventus.\nJuventus players Leonardo Bonucci and Paulo Dybala confronted Mourinho with United captain Ashley Young intervening as Mourinho was led off the pitch by a UEFA official. It was supposed to be Cristiano Ronaldo's night after his brilliant strike in the 63rd minute gave the Italian side the lead. In the end it turned out to be Mourinho's night and he did not pass up on the opportunity to make his feelings known, punching the air furiously at final whistle before the rest of his glorious act followed.\nMourinho's words after the game seemed to suggest that he believed he was forced into such a reaction due to the crowd's demeanour towards him during the full match and poked fun at both the FA, saying that 'FA could translate the insults from the Italian crowd', while also saying that he wanted to hear the Turin crowd louder.\n\"They insulted me for 90 minutes. I came here to do my job, nothing more,\" said former Inter Milan coach Mourinho.\n\"I didn't offend anyone at the end, I just made a gesture that I wanted to hear them louder. I probably shouldn't have done it, with a cool head I wouldn't have.\"\nUnited needed the points and Mourinho said that his team's positive attitude paid off.\n\"We managed to win because we played a very, very positive match and we looked for the victory to the very end. We looked for it.\"\nFollowing Ronaldo's stunning strike, substitute Juan Mata's free-kick on 86 minutes sparked a stunning revival that breathed new life into the visitors' Champions League campaign.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Like the tinkling of the very high keys on piano in a scary movie? The kind that makes the hair on the back of your neck stand up?\nIs there such a thing as \"too quiet\" or \"too alone?\" I think not.\nMaybe post-walk you could direct Hitchcock-style movies.\nMatt's in his own Hitchcock movie. \"The Cows\".\nThey are so quite while sneaking up on you.. but when they jerk that shower curtain back\u2026..\nMan, that's a tough crowd.\nThe cows are staring at you Matt! You must have a mesmerizing personality.\nI should have said MOOSmerizing personality.\nSo, you're thinking that what you saw doesn't match what you herd?\nhahahaha thanks for the laugh!\nYet again, Matt, the animals are staring at you.\nHa ha; love the word play here. How about seen but not herd?\nI know the feeling. I was walking with my pug once and we wandered into a pasture. The grass was low and they could see my pug and they became very curious and started to move over to her and then they began to surround her. I quickly picked her up and backed out of there \u2013 never taking my eyes off them. It made me think cows may have some sort of weird Pug Messiah Returneth thing go'in on.\nBut wait \u2013 we have LIVE animals. Hmmm why not catch one\u2026.cutem up, cookem and blame it on the little green men\u2026.toy slayer band or aliens, either one\u2026\u2026..\nI get it, Matt. WE are the cows, aren't we? We just sit here everyday and stare and stare and stare. Chewing. Staring. Pooping. Thanks for letting us doo all that.\nHey, Karen Too, how about scene but not herd?\nHa ha. Even better! Thanks.\nAngus, man . . . they don't say much. Don't need to. They own the place.\nMaybe you are inspiring one of this summer's internet memes?\nThis probably is not the sound you thought the video needed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzabbzi b/data_all_eng_slimpj/shuffled/split2/finalzzzabbzi
new file mode 100644
index 0000000000000000000000000000000000000000..49e24443536e32017593edf0a7dd5a658839f156
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Our first look at Power On Software's Rewind--a utility designed to restore your Mac's hard drive to a previous saved state--left us unimpressed by the program's performance. Rewind 1.2 corrects most of version 1.1's problems (mmmh; Reviews, April 2001): it doesn't constantly access the hard drive when updating its database, and it more quickly releases free hard-drive space.\nboot from Rewind's Emergency Disk--a virtual startup drive set aside on the Mac's hard drive--but I was unsuccessful. I worked around the problem by deleting the Emergency Disk and replacing it with a new one created in OS 9.2.1. Then, after I rewound back to OS 9.1, booted from the Emergency Disk, and attempted to rewind back to OS 9.2.1, the PowerBook refused to boot up.\nAlthough Rewind is getting better, there's still room for improvement. It can get you out of some sticky situations--but you shouldn't use it in lieu of a comprehensive backup strategy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shallow water flats anglers and guides are invited to hook-up and fight their barracuda rivals. Event is to be headquartered at Hurricane Hole Marina, Stock Island. Prizes and trophies are to be awarded for top angler and guide in light tackle spin and fly divisions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Look for the Air Jordan 11 Velvet Heiress to release alongside the matching Air Jordan 1 on December 17th, 2016. Both exclusive to gradeschool sizing going up to a size 9.5. The retail price tag is set at $220 USD.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Working parents (and moms, especially) have muddled through the demands of raising their children and doing work for thousands of years. As a mom and young assistant professor, I have some confessions I'd like to share\u2014the good, the bad, and the ugly.\n1. I brought my child to extra-curricular activities. I was asked to be co-advisor for our department's honor society. So I brought my daughter to various events including a fair, fundraiser walk, and our induction ceremony. It was a great way to include my daughter in job-related activities.\n2. My daughter had a diaper blow-out the first time she visited campus. The good news is that I had an extra onesie in her diaper bag. The bad news is that there was no changing table. So there I was\u2014on the floor, my baby on a mat, and me trying my best not to get poop everywhere. It happens.\n3. I ignored knocks at the door when I was pumping in my office. When my office door was closed, in effect, I was not there. So when I was pumping milk for my baby, I totally ignored any knock at the door (and some phone calls, too).\n4. I knowingly sent my child to daycare when she was sick. This is the plight of every working mom. Your child has a runny nose, a terrible cough, and acts like they feel miserable but\u2026no fever! So off we went to daycare. As a working mom, you've gotta do what you've gotta do\u2014especially when you have no family or friends to help.\n5. I resented my job at times. These were some of my darkest days as a new mom and new college professor. It's sad but true. There were many times when I would drop my daughter off at daycare and I would say to myself, \"Why am I doing this? I want to be with her 24\/7, but I can't because I have to go to work. It's because there are students that need me and research that needs to be done. But your daughter needs you, too.\" And so it was, I resented my job, or more specifically, I resented being away from my daughter. How did I handle this? First, I found solace in the competencies that I was mastering in my career, which kept me going. Secondly, I knew that my daughter was okay without me for a while, and I reminded myself of all the research showing the benefits of daycare (e.g., children have strong immune systems, social skills, etc.).\n6. I do experiments on my child without IRB approval. As an experimental psychologist, I couldn't help myself. I tested out my daughter's development in mastery over object permanence, interest in novel stimuli, small-motor skills, and theory of mind. Obviously these weren't formal experiments (n = 1, really?), but I enjoyed applying my psychology knowledge to learning more about my daughter and how awesome she is.\n7. My lectures were overloaded with parenting-related examples. I'm totally guilty here. I just hope my students didn't get too sick of the parenting or child examples.\n8. At home, I worked on PowerPoint slides instead of playing with my daughter. Yes, it's true. When I went home, I continued to do more work. It was something I couldn't get around, unless I wanted to just \"wing it\" in class the next day. However, my second semester was much better. I only needed to tweak some things, and so my time with my daughter at home was our time again.\n9. I love my child more than my career. I used to feel like I had to hide this sentiment from my colleagues and administrators, but now I realize that most people feel this way, too, and that it's okay! When conflicts between job and family arise, deal with them as best you can. But at the end of the day, one side will trump the other. Sometimes my career would trump my family (see #4). Ultimately, however, my family is more important to me than my career.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Booq recently released Cobra slim, a new addition to their popular Cobra collection. If you want a fashionable way to take your laptop and other essentials for a trip, the laptop bag may draw your attention.\nThe Cobra slim is a well-designed and stylish laptop bag that measures 16.9 x 11.8 x 3.9 inches and weights 3.59 lbs. As we can see from the images, the laptop bag boasts an elegant and trendy design and is made from 47% recycled PET and 53% cotton for smooth, weatherproof and rugged construction. Inside, a large padded laptop compartment fits for a 15\u2033 MacBook Pro or up to 15.6\u2033 PC laptops, and its quick-access pocket with a slip pocket holds your iPad and iPhone, while the zippered front compartment is large enough to organize your charger, pens, and essential accessories. Moreover, the smooth shoulder strap with a cushy shoulder pad lets you comfortably take the laptop bag with you, and Cobra slim also works well with luggage trolley handles.\nThe Cobra slim laptop bag is priced at $195 USD. If you like the design, jump to Amazon for more details.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"A splendid, gorgeous 4 bedroom Villa (type 5\/A) with private pool in Raha Gardens in Tharwannia Community, within a short drive to Etihad Plaza, play areas and tennis courts. Open plan lounge \/dining room, with lots of windows, shower room with access from private pool, garden and garage with additional storage. Closed kitchen with access to garden, including gas hob, cooker hood, built in electric oven and Bosch dishwasher.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"From shy smiles to stunning jewels, from cheery bridesmaids to candid emotions, we've rounded up some of our favourite bridal photos to inspire all our brides-to-be.\nBridal portraits are the soul of a wedding album as they show a bride \u2013 resplendent in her wedding finery and animated with emotions - before actually tying the knot. These portraits, shot creatively in different locations, can be displayed at receptions, framed and given as gifts to loved ones, or simply treasured by the couple as a beautiful memory.\nSome of our favorite photographers have captured stunning must-have bridal portraits over the years that we just had to share\u2026 From shy smiles to stunning jewels, from cheery bridesmaids to candid emotions, we've rounded up some of our favourite bridal photos to inspire all our brides-to-be.\n'Behind the scenes' portraits of a bride getting ready are some of the most memorable shots as they capture her excitement in an unrehearsed way. This is a great way to capture spontaneous emotions - like a mother holding back tears while draping her daughter's head with a veil, or a gang of jolly bridesmaids laughing at an old joke. Candid close-ups of the make-up, hair and draping process adds an evocative edge to the wedding album where a majority of the photos are bound to be picture-perfect and rehearsed. A photographer can never predict what emotion he may capture during these 'behind-the-scene' shoots and that's the beauty of it.\nA perfect bridal portrait calls for stunning jewellery as it adds compelling complexity to a bridal portrait where the face is the highlight. Skilled photographers will use clever lighting to accentuate the pretty pearls in your maang tikka or capture the reflective glow of that exquisite gold and diamond bling on your skin.\nEvery expression of the bride is captured up, close and personal so no details are left out, be it her wistful gaze while waiting for the baraat or that winning smile as she spies her intricate mehndi. The camera is a witness to her innermost emotions as they get reflected on her face in these candid close-ups.\nThe wedding lehenga nowadays is a true couture masterpiece that deserves its own pride of place in the bridal portrait collection. Photographers and stylist are using ornate, colourful lehengas as the focal point in interesting 'prop-like' ways. From close-ups pictures of the embroidery details to the hint of a face peeking beneath the veil, every part of a wedding lehenga lends itself to stylish images. As it is probably one of the most expensive and carefully chosen ensembles in a bride's trousseau, it's little wonder that a wedding lehenga has become a showstopper for many bridal portraits.\nAdd a beloved family pet like a dog to a bridal portrait shoot, and immediately the stress is gone! Unrehearsed and unguarded, pets help bring that same joyous, carefree vibe to photos. And since pets are so much a part of the family for many brides, it is great to include them in photoshoots \u2013 the fun memories will evoke smiles for years to come.\nBridal portraits are incomplete without her heels in the limelight. Sleek sky-high stilettoes add drama to bridal portraits when captured in an interesting way.\nClose-ups of mehndi designs \u2013 which are nothing less than an intricate art form \u2013 are essential in a wedding album. Apart from the beauty of the designs, these portraits invariably capture a happy bride, radiant and relaxed with her gang as she is the centre of attraction during this fun, traditional ceremony. Her limbs are the beautiful canvas for the patterns, which the wedding album files away as memories \u2013 that, unlike henna, never fade.\nWhat's a bridal portrait session without best friends around? Brides should brainstorm with their photographers about the best ways to incorporate the girlie gang into shoots. With their matching bridesmaids clothes, but unique personalities, every friend will add a different vibe to the group photos \u2013 and the memories will be treasured by all for years to come.\nReflections in a mirror have always meant arresting compositions, be it in photos or films. A fully decked-up bride with a perfect face and a regal ensemble staring back at herself in a mirror provides ample creative space to capture mesmering shots.\nOne of the most emotionally intense moments for any bride is when she heads down the aisle or makes a grand entry towards the mandap. There is an air of festive finality to this most photographed moment that everyone has been waiting for\u2026 when a couple finally seal the deal to start their new innings as husband and wife. Photos taken at this time elevate the whole significance of the occasion as they capture this final rite of matrimony.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A course of Fez, our flagship cement tile design, marks the bar area at Sessions, the restaurant at the new Hard Rock Hotel in Palm Springs. Image courtesy Mister Important Design.\nThis time of year, snowbirds are flocking to desert locales\u2014Phoenix, Scottsdale, Palm Springs\u2014drawn by the promise of warmth and, this year, a break from the frigid Polar Vortex. If you happen to be one of them and find yourself in Palm Springs, then you need to check out the Hard Rock Hotel there, the newest addition to the Hard Rock family. It also happens to boast a must-see cement tile installation featuring Granada Tile's own Echo Collection's Fez cement tile design. As with their other locations, the vibe is glam rock-and-roll, and extends to every corner of the hotel, including Sessions, the pool-adjacent restaurant and bar. Serving a fabulous menu of fresh, local fare with a Southern flair, it's a hotspot, made hotter by its great design, the work of Mr. Important Design, a firm known for creating glam hospitality spaces.\nWalking in, it's a mix of warm browns and luxe textures. The floors are covered with a rich, distressed looking wood. On the walls, irregularly spaced and sized carved panels add dimension and an unexpected twist to the typical treatment, and turned-wood spindles serve as screens. Fez appears at the bar, adding a jolt of color to the otherwise neutral palette. The velvety feel of the cement tiles also plays well against the more rough-hewn textures of the woods on the walls and floors. It serves as an inviting path to the bar itself\u2014the cement tiles cover a portion of the floor and run up the wall, providing an unbroken line of stunning pattern.. As the new Hard Rock Hotel proves, Palm Springs isn't your grandparents sleepy desert town anymore. It's brimming with the chic details that the city is now known for. If you're in Palm Springs, be sure to drop by. And, if you love Modern design, be all means try and plan a trip in the next couple of weeks\u2014the city's legendary Modernism week is kicking off February 13 and will showcase some of the best and most iconic design around.\nPrevious articleGet Inspired: Big in 2014!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Inspired by the retro era, made modern. These are the ladies three quarter leggings. Like you, these leggings are as unique. For the bespoke individual. Made from the highest quality imported Italian lycra. Features a high waist band, 'penny pocket' for keys, and a super comfortable fit. Show off your 'curves' in style!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is a stunning dress from about the 1950s in deepest midnight blue velvet with a line of darling matching roses round the neckline. 3\/4 sleeve, pencil skirt with a slit in back, slight bullet shape to the bust. Scooped neckline in front and back with those gorgeous roses. I'd say small size. No label. There is a nylon zipper up the back, but its not original to the dress, when you look inside the seam you can see where the old one was removed. Thats of course, not noticeable from the outside though. I should mention the wrists are quite narrow, they measure 7\" around. There are small slits at the end of either one and if you should need a bigger opening you could open those up a little bit more.\nExcellent vintage condition. One or two of the roses could perhaps do with an extra stitch or two holding them on, but thats it really.\nShoulder to bottom hem: 43.5\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Yo chill! This jumpsuit with the cool rubber Yo letters is the ultimate relaxation material. The jumpsuit is made from super soft cotton and features comfy long sleeves and trouser legs and push buttons on the chest and in the crotch: so a nappy change is done in a flash. The melee black-white perfectly matches any sweat cardigan and (denim) jacket, so it's guaranteed to be a popular one. Tumble 'N Dry seasons' suit!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To keep from missing out on the opportunity for culinary refreshment during the course of your event, and to allow you additional side programs, we offer various gastronomic accommodations. These options vary in size, arrangement and ambience and are perfect for celebrations, wellness, enjoyment and relaxation. Whether lunch buffet, coffee break, picnic in the park or festive dinner- we tailor to your needs! We can gladly recommend options for creating a custom side program.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Continuing professional development (CPD) is increasingly crucial for regulators, educationalists and healthcare professionals. Health Education England working across Kent, Surrey & Sussex (HEEKSS) supports CPD for Primary Care workers through the Post-Certification GP School and increasingly through the Community Education Provider Networks (CEPNs).\nIn terms of continued fitness to practice and patient safety, as well as maintaining professional standards, a thoughtful, evidence-based and reflective approach is encouraged which responds to the needs of the patient and the local population as well as the needs of the individual and their working organisation.\nEncourage and support specific changes in the quality and delivery of a doctor's practice from professional, patient and service requirement perspectives.\nInclude any educational or professional activity directed towards developing the knowledge, skills, attitudes and personal effectiveness necessary to improve practice. Professional expertise demands a continuing awareness of new concepts, values and technologies.\nIt is important for doctors to update themselves not only on the evidence base, but also on opinion and consensus. Equally, they must be aware of local needs.\nThe ultimate aim of CPD is to support doctors and other healthcare professionals to improve the care they provide to patients through their own personal development.\nThe HEEKSS CPD strategy will adhere to the ten principles of good practice evolved by the Academy of Medical Royal Colleges (AMRC) and Royal College of General Practitioners (RCGP) Strategy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This month we are once again proud to support the wonderful work of The Sycamore School \u2013 a school dedicated to providing a safe and inclusive educational environment for students with Autism Spectrum Disorder. The Sycamore School works to provide young people with opportunities to reach their potential.\nOn April 4th, all Merlo stores will be wearing their superhero outfits to celebrate \"Superheroes for Sycamore School Day\".\nSo why not join in on the fun and dress up too! What's more, with every coffee purchased on this day, $1 will be donated to The Sycamore School. So head to a Merlo store near you to participate. Just by buying your morning coffee, you will be donating to a good cause, so treat yourself to a second (or third!) coffee guilt-free, because even superheroes need their caffeine kick.\nYou can also donate to the cause for these amazing \"superheroes\" at The Sycamore School and help them and their families in their daily efforts by going to their website. Click here for more information.\nCast aside your civilian disguise for the chance to WIN a $50 Merlo gift card!\nTo enter, simply take a snap with your superhero outfit on Thursday, then share it via your socials with #SuperheroesForSycamoreSchool and #MerloCoffee and be sure to tag @merlocoffee.\nThe winners will be drawn on Friday, April 5th.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Republican Party loves its shibboleths \u2014 random buzzwords and dog-whistles that fluff up their mostly white male voters to half-mast status. You know the words: teleprompters, czars, birth certificate, Chicago gangster and, of course, Benghazi to name a few. They apparently get bonus crazy points if they hamfistedly connect two or more of the words, especially \"Benghazi\" and \"Planned Parenthood.\" Mike Huckabee was the first to conflate these two otherwise disparate words when he tweeted that Planned Parenthood is just like Benghazi.\nSo, leave it to asparagus-evangelist, talking buttcheek and Texas Republican Louie Gohmert to totally one-up Huckabee during the House Judiciary Committee's Planned Parenthood inquisition. Gohmert was questioning former Planned Parenthood clinic manager Susan Thayer who, by the way, was testifying against her former employer, when \u2014 seemingly out of nowhere \u2014 Gohmert abruptly switched gears and began to freak out about Benghazi.\nWhat does all this have to do with Planned Parenthood, the subject of the hearings? Nothing. Absolutely nothing. But clearly Gohmert was running behind on his daily quota of \"BENGHAZI!\" blurting and had to make up time somehow. Hence, the whiplash-inducing segue from reproductive services to a terrorist attack on a diplomatic compound in 2012.\nFeatured image via video screen grab.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Former Brookside star John McArdle joins Emmerdale | HELLO!\nFormer Brookside star John McArdle has joined the cast of ITV's Emmerdale. The actor will play Ronnie Hale, a friendly plumber who could possibly be Chrissie White's (Louise Mawood) estranged biological father.\n\"I am really pleased to be joining the cast of Emmerdale,\" said John, who began filming this month. \"The standard of the show is equal to anything I have worked on before and I am really happy to become a part of that.\"\nJohn's character Ronnie will soon find himself at the centre of a family argument at Home Farm as Chrissie uses a water leak as an excuse to call him and find out if he really is her dad.\nIain MacLeod, Emmerdale's series producer said: \"I am thrilled to welcome an actor of John's pedigree to the show. And in Ronnie we have a character that will unearth every painful fragment of love and loss buried in the White's family past \u2013 with the odd unexploded bomb thrown in for good measure!\"\nThe 66-year-old actor, who is best known for his roles as Billy Corkhill in Brookside and Oliver Mead in Waterloo Road, will be onscreen from mid April.\nThe news comes shortly after it was announced that Emma Atkins made a shock return to Emmerdale as Charity Burbage, who announces herself as the new joint owner of the Woolpack.\nEmma said: \"It is just brilliant to see her behind the bar. Something very different. She will be under the watchful eye of Chas (Lucy Pargeter), but both characters can give as good as they get so it will be a colourful dynamic.\n\"The bar is a bit like a stage for Charity where she can show off and deliver her explosive one liners to the locals. Her behaviour will no doubt fluctuate from good to bad to ugly, with a decent dose of naughtiness thrown in.\"\nEmmerdale spoilers: Eric Pollard to die in shock car crash?\nAre Coronation Street stars Alison King and Ryan Thomas dating?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The various Arabic dialects in the Middle East and North Africa (MENA) \u2013 \u00a9 Photo: CreativeCulture.\nLabelling 381 million people from 22 countries as monolithic 'Arabs' is misleading and inaccurate.\nWith conflicts raging on in Syria, Palestine, Yemen and Iraq and a diaphanous calm in the rest of the Middle East, the language we use in covering this region is not only hindering our understanding of the issues, but it is also misguiding strategic policies.\nAs a journalist covering international events, I have witnessed the narrative covering the Middle East and North Africa (MENA) recede to a thin crescent of one pan-ethnic group, primarily because they speak a dialect of Arabic. At the last count, 35 dialects of the Arabic language are spoken across the two regions.\nIn a recent discussion one person, referring to Iranians, used the term \"Arab speakers\". I wanted to ask whether they were Dolby Digital or Stereo. Instead I pointed out that Iran's language is Farsi.\nThe BBC has an \"Arab affairs\" editor and it is not alone. Fellow journalists who are \"internationalists\" themselves make that mistake; politicians, commentators and academics as well.\nThis language does nothing to inform and only perpetuates a phantom identity of \"Arabs\" and the \"Arab world\". Both terms are widely used as blanket coverage of the populace in the MENA that displaces the indigeneity of \"Arab\" into a single basket that mashes 22 countries with a population of 381 million.\nNeheda Barakat is a journalist based in Australia, is a former current affairs executive producer with Al Jazeera English and Australian Broadcasting Corporation. She has a MA in communication and is a 'frou-frou' Arabic speaker.\nPerhaps it is the globalisation of things that has made it acceptable to do away with the specifics of identity? A Google Books search suggests the usage of these terms has shot up to almost 400% since the 1800s.\nThe MENA consist of many countries and each speaks its own form of Arabic patois, a clear indication of their nationality. Arabic in North Africa is almost incomprehensible to an Arabic speaker from the Levant such as Lebanon. In the case of Lebanon, their version is what I call frou-frou Arabic due to lasting French influences. Lebanon's native language was Phoenician or Canaan, which was also spoken in coastal Syria, northern coastal Israel and Cyprus. Syria's once native language that it shared with Iraq and Iran was Assyrian. The Egyptians spoke Coptic. The list goes on.\nDuring an assignment in Israel and Palestine, my Arabic dialect clearly defined my roots from the local Palestinians. The dialects differed wildly when I was in Egypt and even greater when dealing with Iraqis and Libyans. Not to mention cultural differences. As a Lebanese-Australian who lived in Libya as a child and then later worked for the Nine Network Australia during a Gulf war assignment in Saudi Arabia, I found both countries terrifying. They and their lifestyle were alien to me. Yet in this instance, the global membership I had been assigned, told me I belong to others. An otherness dictated by passers-by.\nAnd he was right. In researching for meanings, definitions and categorisations of the term or the regions, there are no definitive answers to populations, countries, sects and languages that accurately agree on what constitutes the MENA. There is no getting away from that ethnocentric mindset that mash-up countries, ethnicities, religions, cultures and languages by referring to the regions, not as the Arabic speaking world, not as the Middle East or North Africa, but as \"Arabs\" and the \"Arab world.\" We wouldn't refer to Scotland, Ireland, Australia, Canada and the USA as \"English\" or the \"English world\" but most likely, the English-speaking world.\nBefore the regions were invaded by the \"Arabs\" from down south in the Arabian Peninsula, there were existing civilisations. The march north and conquest began in the mid-6th century, across the Persian and Byzantine territories and it was to spread the faith \u2013 Islam which was rooted in the Arabic language \u2013 not their ethnicity.\nThe word \"Arab\" means \"nomad\" in one camp, in another it is derived from \"pure or mixed\". Arabic originated from nomadic tribes in the desert regions of the Arabian Peninsula. The language comes from Nabataean Aramaic script and has been used since the 4th century classical era belonging to the \"Semitic\" group of languages of Hebrew and Aramaic.\nBy the 8th century CE, the Arabic language began spreading throughout the MENA, as many people converted to Islam, and were obliged to pray in Arabic. This brings us to the most important constituent in the term \"Arab\", i.e. the Arabic language \u2013 not people. To describe everyone who is Muslim or speaks Arabic as Arab is incorrect. Religiosity is not ethnicity and nor are they interchangeable.\nA lot of what has happened in the MENA is sadly a case of unintended consequences. The struggles in these regions to a great extent has shifted from a nationalist to a religious front, which has led to the interchangeable terms of \"Arabs\" and \"Muslims\". This has been a predominantly western perception, however, and the labelling of \"Arabs\" as monolithic can only be described as a fear-mongering term to reflect the \"war on terror\".\nNonetheless, local contribution via geopolitics and strategic convenience cannot be ignored. Hoyland explains that the term \"Arab\" is a modern 19th century reference used to break away from the Ottoman Empire as well as Turkish nationalism. After that, the term was embedded with the initiation of the Arab League, specifically Saudi Arabia. It served Saudi very well to cast the one noun to describe two regions because ultimately, as the birthplace of Islam and \"Arab\", it was convenient and strategic for this perception to continue. The greater the number of people, the bigger the area, the greater the leverage \u2013 and a desperate attempt to hold on to the pan-Islamism and pan-Arabism of yesteryear. The term stuck and was later advanced by Gamal Abed Al Nasser in 1950s and 60s.\nBut no greater significance did it play until the 1967 war with Israel to give the perception of a big military force \u2013 a phantom bigness. Nowadays, the same term is a convenience for hawkish Israelis to refer to the Palestinians, casting them into that otherness, further into a phantasmagorical horror, and to erase their identity from lexicon listings.\nAccording to Hoyland, the \"Arab\" term is becoming less popular in the Middle East, because the movement is seen as \"backward\". \"Nationalism and Islam has had its day,\" he says.\nLast year, the US Census Bureau held a forum to improve classifications of nationalities, race and ethnicities of the MENA, inviting 40 experts to discuss current data. The outcome of the forum will most likely lead to an increase in the number of classifications of ethnicities, race and nationalities in the MENA. This may get us closer to a more concise representation of the realities, rather than assumptions, and may help us understand the complexities of the regions. In lands of millions of Arabic speakers, not everyone is Ali or Abdullah nor Sunni or Shia. Islam consists of a significant number of branches and sub-branches (26 at last count), as well as non-Muslims.\nWe need to scrap the free memberships that misplace ethnicity, identity, culture and history. Doing so would not only go some way to preventing exclusion, but also give context, and provide a starting point in promoting understanding. It may even prevent incidents of racism and bias towards anyone or anything that comes from the Middle East and North Africa and happens to speak a dialect of Arabic.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cases of migrant domestic worker abuse are not a new trend in our country.\nOver the past few months, cases involving employers' abuse of domestic workers have sparked outrage as the public demands stricter punishment of the crimes.\nBut as heartbreaking as these cases may be, they are not isolated incidents here in Malaysia.\nTenaganita\u2014a non-profit organisation dealing directly with migrant and other human rights issues\u2014handled a total 120 cases of domestic worker abuse in the six-month period between June to December 2017. Common violations among the cases were no rest day, no signed contract between worker and employer, contract substitution, unlawful salary deductions, non-renewal of work permit, withholding of passport, low or unpaid wages, extended working hours without overtime compensation, food deprivation or insufficiency, poor working and living conditions, no health support, no annual or medical leave, as well as both physical and verbal abuse.\nGlobal Shepherds\u2014a non-profit organisation affiliated with Good Shepherd Services providing shelter, care, counselling and advocacy work for non-Malaysian women who are survivors of human trafficking and sexual and gender-based violence, as well as refugee seekers\u2014reports similar cases of abuse.\nDirector of Global Shepherds, Vanitha Nadaraj, shares a case in which a woman was seeking work to help pay for her daughter's heart surgery. Upon promises of a high salary, she came to Malaysia as a domestic worker. Throughout her employment, she was fed mainly leftovers, at times only once a day. She endured verbal abuse, extended working hours up to 20 hours at a time, and no days off.\nLater, she came to realize that her employers had not even arranged a working permit for her. She also only received a month's salary while her agent received the other two months' worth.\nGlobal Shepherds encountered another similar incident involving an underage girl. She came to Malaysia under the guise of her older sister. She had hopes to contribute financially to her family back home. From the start of her employment, she begun work at 5 AM each day. Her employer would send her to various relatives' houses to work.\nShe, too, only received one meal a day. Her employer was frequently frustrated with her due to the language barrier and her quality of work. They took out their frustrations by beating, slapping, and throwing water at her.\nOf course, cases like this are not representative of all Malaysian employers. There are 250,000 domestic workers currently registered in Malaysia and abuse cases only represent a small percentage of that. However, a small percentage of abuse is still a problem of abuse.\n[Employers should] respect the basic rights of domestic workers, beginning with a day off a week and respecting them as workers.\nGlobal Shepherds believes that domestic workers should be given legal rights to one day off per week, set working hours and paid overtime, as well as paid sick leaves, annual leaves and healthcare. Domestic workers should also have basic rights to accommodation and privacy, phone access, as well as freedom to fulfill their religious obligations. Additionally, their employment contracts should be readily available to them in a language they understand. This and all related expenses, such as work permits, should rightfully be the responsibility of the employer.\nAll of this can and should be upheld by legislation. Though recognised as workers in Malaysia's Employment Act 1955, NGOs say domestic workers are categorically excluded from various provisions of labour protection. The Employment Act excludes them from their rights to regulated working hours, rest days, termination and retirement benefits, as well as maternity protection.\nTenaganita has stated over and over again that it is only when rights of workers are protected through laws; when domestic workers are recognised as workers, can employers, agents and Malaysians as a whole ensure respect and dignity for domestic workers.\nGlobal Shepherds seeks to ratify ILO Convention 189 and adopt Recommendation 201, concerning decent work for domestic workers. This Convention properly defines domestic work as well as the terms and rights of its employment.\nWe stand alongside other organisations in seeking laws that recognise them as workers, not servants or maids, to ensure they have equal legal protection. Migrant domestic workers cannot be outside the scope of labour laws.\nWhile legal reform may be the most effective form of action, it is not the only one. Cases of abuse do not represent the entire climate of domestic work in Malaysia. In fact, many Malaysian employers know to treat their domestic workers with respect and dignity. And it is employers like this that can help ensure better protection of domestic worker rights.\nWe know and we acknowledge there are many good employers out there who fulfill the requirements within the contract they may have signed respecting the rights of their domestic workers and we applaud them. In fact, these good employers are the informers when abuses happen in their residential area.\nWe as the general public also bear responsibility to not stand idle. We are able prevent future incidents of abuse by creating awareness on the issue. And we can do this by continuing to talk about the limitations of domestic worker rights. We also have a responsibility to report any witnessed or suspected incidents of abuse.\nIf you or anyone you know is currently victim to an abusive situation, please contact Tenaganita's rescue hotline for women and children at +6012-335 0512 or +6012-339 5350.\nSabah has the highest population of mental illness in Malaysia! See how the other states are doing here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"That Same Anon wrote: The keycard level is so high because it's to OVERRIDE the doors, normally a guard would let you through with a button or something, as the booth shows.\nMaybe, It makes sense. I would like to see it moved a little off center so it looks like a true OVERRIDE keycard slot.\nnightscout01 wrote: Maybe, It makes sense. I would like to see it moved a little off center so it looks like a true OVERRIDE keycard slot.\nIt should be made clear that the Facility is on lockdown.\nMaybe give the keycard slots a red light to show that they're designated for emergency cases?\nDamn yes! But Isn't that gonna have some issues with the random generation,the gameplay will also be slightly different!\nI suggest Regalis aka. Joonas Rikkonen to use a mechanic where you must use scp 096 to open a door to keter level area (SCP 096 is Euclid level)or go to the remote door control room thing and enable the player to open the door with the keycard!He can also add a mechanic where you must lure scp 682 in the game to destroy the door!\nAM I THE ONLY FIVE NIGHTS AT FREDDY'S FAN?\nLast edited by nightscout01 on Sat Apr 19, 2014 5:13 pm, edited 2 times in total.\nThose suggestions seem a little extreme. I don't approve.\nPrincess Luna wrote: Those suggestions seem a little extreme. I don't approve.\nEveryone likes it, I don't see the problem.\nThis is related to the keycard dilema, so I don't think that this is the wrong place to post their suggestions.\nThat's suggestions for new SCPs, they want to keep it contained in a simple thread instead of having it take over the collab subforum.\nThe monorail idea could work, abeit it could kind of work like the switch from the facility to a gate, you could just go into the 'train', fade to black, and then unfade with the train in the next station.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Worker Protection Standard (WPS) allows entry into a pesticide-treated area that remains under a restricted-entry interval (REI) only in a few narrow work situations. When early entry is permitted under the WPS, special protections must be provided for the early-entry workers.\nWhat are the work situations where early entry is allowed?\nUnder what conditions is early entry allowed for these work situations?\nWhat are the special protections for early-entry workers?\nWhere can I get more infor\u200bmation?\nShort-term tasks that last less than one hour and do not involve hand labor.\nEmergency tasks that take place because of an agricultural emergency.\nSpecific tasks approved by EPA through a formal exception process.\nPacking produce into containers in the field.\nNote: The definition of hand labor does not include operating, moving or repairing irrigation or watering equipment. These tasks can be performed during the REI, but only under approved situations and with proper early-entry personal protective equipment (PPE).\nLimited-contact tasks that could not have been foreseen, cannot be delayed, and do not involve hand labor.\nIrrigation tasks that cannot be delayed, involve limited contact with treated surfaces, and that will take more than one hour in the REI area to perform.\nOperating or repairing weather monitoring and frost protection equipment.\nRepairing greenhouse heating, air conditioning and ventilation equipment.\nRepairing non-application and non-irrigation field equipment.\nIrrigation tasks are tasks related to operating, moving or repairing irrigation or watering equipments and where early-entry workers' only contact with treated surfaces \u2013 including soil, water, surfaces of plants, crops, and irrigation equipment \u2013 is minimal and is limited to their feet, lower legs, hands, and forearms.\nThere are specific conditions that must be met for early-entry for short-term tasks, limited-contact tasks, irrigation tasks, and agricultural emergency tasks. Some examples of these conditions are provided in the following table. However, these lists are not complete; see Unit 4 of the How to Comply Manual and the two Federal Register notices from May 3, 1995 listed above for all of the details.\nWait at least 4 hours after the pesticide application is completed?\nWait until inhalation exposure level on the pesticide label has been reached or any WPS ventilation criteria have been met?\nMaximum amount of time allowed per 24-hour period?\nAllowed for pesticides whose labeling requires double notification (verbal notification and posting signs)?\nAre hand labor tasks permitted?\nEmployer provides special protections for early-entry workers regarding training, instructions, decontamination supplies and personal protective equipment (PPE)?\nComplete pesticide safety training, instructions, and information.\nEmployers must make sure each early-entry worker is trained as a WPS worker before entering a pesticide-treated area during an REI. The 5-day grace period for training that applies to other workers does not apply to early-entry workers.\nHow to put on, use, and take off early-entry PPE correctly.\nAbout the importance of washing thoroughly after removing PPE.\nHow to prevent, recognize, and give correct first aid for heat illness (too much heat stress).\nHuman hazard statements and precautions.\nSigns and symptoms of poisoning.\nPPE required for early entry.\nAny other precautions or instructions related to safe use or early entry.\nSoap and single use towels \u2013 enough for the needs of early-entry workers.\nAre not in an area being treated with pesticides.\nAre not in an area under an REI unless that location is necessary for the supplies to be reasonably accessible to early-entry workers.\nAre reasonably accessible to and not more than \u00bc mile from early-entry workers.\nA worker employer must provide each early-entry worker with at least 1 pint of emergency eye flush water when the pesticide labeling requires protective eyewear. The emergency eye flush water must be immediately accessible.\nAt the site where early-entry workers take off their PPE, the worker employer must provide soap, clean towels, and enough water to allow early-entry workers to wash thoroughly after removing their PPE.\nmake sure they use the PPE correctly.\nThe specific duties regarding PPE are the same as for employers of handlers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagdpe b/data_all_eng_slimpj/shuffled/split2/finalzzzagdpe
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"In April 2014, OSHA inspectors saw employees working on roofs and scaffolds without fall protection, exposing them to potentially fatal falls of up to 20 feet. Management also failed to provide mandatory personal protective equipment (PPE), like hard hats and fall harnesses.\nAfter the assessment, OSHA inspectors warned employers at the site to use fall protection regularly for all workers. When they returned to the site the following day, they saw that all of the fall protection had been removed, and workers were once again exposed.\nFalls kill more construction workers than any other workplace hazard, and injure thousands of workers every year. Even falls from reasonable heights can lead to permanent injury, so OSHA regulations state that all employees working higher than six feet from the ground have mandatory fall protection.\nEvery day, millions of employers put their employees in danger by ignoring commonplace workplace hazards. It has been proven that avoiding accidents and injuries is the most efficient way to manage a workplace, but some managers will continue to cut corners, even after they have been warned.\nIf a reckless employer caused your accident, we want to hear from you. Share your story with Spevack Law Firm on Facebook or Google Plus today.\n\u2190 Is My Employer Liable for Workplace Violence?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Who would have thought that Lego would become a cinematic juggernaut, raking in both box office bucks and critical acclaim? I've gone on record multiple times with my opinion that The Lego Batman Movie is the best Batman movie ever, and the Danish building block masters look to have another hit on their hands with The Lego Movie 2: The Second Part, opening this week. But the path of culture goes both ways, as Lego has been manufacturing sets around popular movies since 1999.\nHere are our picks for the best Lego construction sets inspired by the silver screen.\nThis is, full stop, not just one of the best licensed Lego kits ever, but one of the absolute best Lego kits ever. The massive Millennium Falcon will cost you a cool $800 if you want to take it home, but the starship replica is ridiculously detailed. Not only does the exterior look as close as you're gonna get to the movie, but multiple parts of the hull pop off to reveal intricate minifig-scale interior sets. The thing stretches nearly three feet by two feet when fully assembled, meaning you're going to need to clear a bunch of space on your display shelf. And, yes, it comes with a pile of minifigures and accessories too.\nReleased in 2009, this sizable play set for the Indiana Jones line has a number of features that land it on our list. The basic structure is a pair of towers that are large enough for minifigures to rest comfortably inside, but the hotness is what connects them \u2014 a twisty, undulating mine cart ramp that really works. There are few other sets in the LEGO universe quite as unique. If you're a minifigure fan, it comes with a half-dozen, including Indiana, Willie, Short Round and a trio of Thuggee antagonists. This one is out of print, but you can still find it on eBay or other aftermarket sites.\nOne of the craziest things about Lego is that it has the license for both of the Big Two superhero companies \u2014 and, in fact, Marvel and DC both debuted with the company on the same day in 2012 as part of their Super Heroes line. We're going to see a few entries from either side of the line in this list, but for sheer size few match up to the massive Helicarrier. This flying fortress is the mobile headquarters of S.H.I.E.L.D. and comes with turbine engines that spin, three small-scale Quinjets, a display stand and more. Five minifigs including Nick Fury and Black Widow round out the collection.\nThis thing is wild. The Lord of the Rings license produced some seriously badass sets, but none have the profile of the Tower of Orthanc. The obsidian tower is where the wizard Saruman holed up at the Battle of Isengard. LEGO's recreation stands a flabbergasting two feet high with six detailed floors that have tons of interactivity \u2013 light-up palantirs, trap doors, staircases and more. It also comes with one of the coolest mythical beasts ever, the animated tree Ent that lays siege to the tower.\nIt took a while for this design of the iconic time-traveling car from Back to the Future to get produced, but it was worth the wait. The heavily modified auto was originally conceptualized by users Masashi Togami & Sakuretsu on the Japanese service CUUSOO, which eventually became LEGO Ideas. It hit retail in August of 2013 and was an instant hit. The car's frame is constructed in an interesting way \u2014 it's a little boxy, but the gullwing doors, wheels that flip up and other details are all right where they need to be. Hell, it even comes with a flux capacitor.\nOn the other side of the aisle, this kit inspired by The Lego Batman Movie is a hilarious funhouse of deathtraps and disasters. The Joker's Manor is as crazy as its creator, abandoning the familiar right angles and clear connections of other LEGO sets for askew angles and unusual styling. From the undulating roller coaster track that rings the building to the working boxing gloves. At 3444 pieces, this is a significant investment of time to put together, but it looks ultra sweet and will drive the Dark Knight crazy.\nThe Harry Potter franchise has had an interesting history in Legoland. The first Potter sets came out in 2001, and the line ran strong until 2007, at which point it took a three year hiatus. It came back in 2010, was discontinued again in 2012, and is now on its third reincarnation that began in 2018. This latest batch produced the coolest Hogwarts scene ever, a massive reproduction of the wizarding school's Great Hall that opens up to reveal a ton of internal rooms. The sizable set comes with all the Potter figures you'll ever want \u2014 Harry, Ron, Hermione, Hagrid, Dumbledore and even a Voldemort for maximum terror.\nOriginally released in 2016, this is one of the nicest interior \/ exterior sets that Lego has ever produced. When it's closed, it's a sweet replica of the Lower Manhattan firehouse used by Stanz, Spengler, Venkman and Zeddemore in the classic 1984 movie. But here's the thing: this isn't just a boring building. Hinges at the back let you clamshell the whole thing open to reveal a ton of details. Three stories, complete with staircase, firepole, garage area, containment room and more let you move your paranormal investigators around their home base. The thing stands over a foot tall when completed.\nI've always had a soft spot for Lego boats, and this tie-in to Disney's Pirates franchise is one of the coolest ever made. The first awesome bit is those huge black fabric sails, which are incredibly eye-catching. But the details don't stop there \u2013 the replica of Jack Sparrow's vessel has a moveable anchor, pair of cannons, detachable floorboards and six minifigures. When completed, the ship is almost two feet long from stem to stern, with the main mast stretching 20 inches high. That's a lot of boat!\nThis might be cheating but it's from a movie \u2014 The Lego Movie. Old-school Lego heads remember how badass the original Space sets seemed. Those yellow-tinted transparent cockpits were a portal to a whole new style of construction. So when the movie introduced a little blue spaceman, it was only fitting that he'd have a ship as well. Benny's Spaceship takes all of the blue and gray coolness we loved from the Space line and brings them into the 21st century with lots of detailing and movable parts. Throw in some of your favorite movie character minifigs and you've got a must-have set.\nThe furry brown jungle dwellers of Endor were pretty polarizing when they were introduced in Return of the Jedi, but you can't throw any shade on this ultra-slick Lego set released in 2013. Comprising four tree pillars connected by hanging bridges, it's a challenging build with lots of decorative nature elements along with swinging log rams that really work. One of the biggest selling points for this set is the sheer number of minifigures in the box \u2014 a whopping 16, including R2-D2, Luke Skywalker, five Ewoks, and a handful of Stormtroopers to ruin the party.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ENCINITAS COIN & JEWELRY is eager to evaluate & offer you the highest CASH value for your fine jewelry! People often decide to sell their jewelry because their tastes change, a relationship ended, they have inherited pieces that they will never wear, or a piece is damaged or broken. We have built our impeccable reputation through fair prices & outstanding customer service that only a family run business can offer you when you are selling, trading-in or trading-up your unwanted jewelry.\nWe are constantly seeking new items for our showroom floor, which ensures you will receive the maximum CASH PAYMENT for your jewelry. Premiums are paid for designer & signed pieces by coveted companies and in pristine condition. Brian Bass is our resident expert buyer for all larger diamond, ruby, sapphire &, emerald jewelry. We encourage you to call if you need to make an appointment with Brian to evaluate your item(s). The rest of our family can assist in the remainder of your jewelry sales transactions, and no appointment is necessary. Bring your items for a FREE, no obligation EVALUATION!\nWe are typically open Monday- Friday 10-6 and Saturday- Sunday 10-4. Occasionally we attend shows and may be out of the area. Please call if you are coming from a long distance to make sure we are here!\nI have been buying from here for 3 years now and feel so very good about it. I didn't know a thing about silver\/coins and Michael was patient and honest. I will be a loyal customer as long as he is in business.\nMichael is THE most honest person I have ever dealt with. I found him completely by accident and would drive the 30 or so miles from my house to do business with him. I am definitely far beyond impressed.\nMichael the owner, and the rest of the staff members are extremely knowledgeable and helpful. Also extremely honest and fair when buying used jewelry.\nVery nice staff, the people know their stuff! Thank you for the support and for being patient with me.\nCopyright 2013. All Rights Reserved. Designed & Developed by Rock'n Graphix.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"> My guess is that you need to hide less and show and explain more.\n> If you -set trace on- where does the program crash?\n> would very probably not work. There needs to be a space after -replace-.\n> and once in theta2). I've tried to include a constraint (e.g.\n> b1=log(b2)), but then I get errors (111 for example).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"While most of you were preparing for Superbowl 47 weekend, many celebs flocked to Los Angeles for the 44th NAACP Image Awards, hosted by Steve Harvey at the Shrine Auditorium. Celebrity guests and honorees included, Halle Berry, Kerry Washington, Elise Neal, and Tatyana Ali.\nThe NAACP Image Awards was definitely a family friendly event, with child stars KeKe Palmer and China Anne McCalin of Disney's A.N.T. Farm on the red carpet. Kerry Washington attended with her parents Earl and Valerie Washington (pictured below); Jaime Foxx was accompanied by daughter, Corrine.\nKerry Washington Marries A Football Player Nnamdi Asomugha In Idaho?!?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..f8d0cfc64240e53ac792192cf20d2f452201b585
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@@ -0,0 +1,5 @@
+{"text":"I am currently at the MVP Open Days in Pargue and I have to say that I am really enjoying myself.\nFirst of all it is snowing-I hadn't seen snow in around 20 years\u2026so I liked it.\nIn addition to the fun I also worked a bit and I delivered a presnetation focusing on Server Cluster.\nI will have pictures from the event posted later.\nPosted: February 27, 2005 .\tPosted in: 1036, 1041. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Interior Design ~ Amazon Com Mr Coffee Barista Espresso And Cappuccino Maker Interior Design Machine Photo Ideas Reviewse Breville 57 Barista Machine Photo Ideas. Barista Machine Video. Barista Machine Reviews. Starbucks Barista Espresso Machine Reviewsbarista Machines For Sale.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"34 yr old Electronic Gear Positions Worker Lester Hutton from Lakefield, has interests for example juggling, ray ban australia and vehicle racing. Advocates that you just vacation to St Mary's Cathedral and St Michael's Church at Hildesheim. If you enjoyed this post and you would certainly like to get additional information regarding slangsnowboard.com kindly see the web-page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Having a cat with a taste for houseplants can not only be frustrating; it can be hazardous to the health of your cat as well. Many common houseplants are toxic to cats, including snake plants, aloe vera and jade plants. These toxic plants should always be kept away from hungry kitties, but there are steps you can take to discourage your cat from munching on the rest of your houseplants as well.\nThere are several ways you can make house plants seem less appetizing to your feline companion. Cats typically don't like the smell of citrus, so you can start by laying orange and lemon peels in the soil of potted plants. Sprinkling a little cayenne pepper around the leaves of your houseplants can make cats think twice about eating them, too.\nIf your cat likes to use potted plants as a litter box, consider covering the soil with a layer of large, smooth pebbles and stones. This texture will make the plants seem far less appealing to most cats.\nIn addition to cat-proofing your existing plants, you can also grow plants that cats are predisposed to dislike. Rosemary is a great option to consider, because cats don't like the smell of this aromatic herb and you can use it in your kitchen. Cats also tend to avoid thorny plants like roses and cacti.\nIf you just can't seem to keep a curious cat away from your plants, you may just have to move them to a place that your cat can't access. You could even treat this as an opportunity to create a lush indoor greenhouse in a room that gets plenty of light. You may also be able to hang potted plants to keep them out of reach of your cat as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are a lot of uncertainties in life, especially when it comes to a sudden disaster. Fortunately, there are plenty of measures we can take to prevent ourselves and our property from harm and damage, especially in the case of structural fires. Office buildings and other large structures can protect themselves with commercial fire sprinklers and standpipe systems, and other large buildings like schools can protect themselves with fire alarm systems. And if you live in the Greater Los Angeles Area, you can find all of these preventative measures, and the skilled team required to install them, at Fire Protection Group, Inc.\nIf you live in Los Angeles and haven't heard of the Fire Protection Group, Inc., you've still probably set foot in at least one of the buildings they've outfitted with fire sprinkler systems and other fire prevention measures. Keep in mind that Fire Protection Group, Inc. has installed fire prevention systems in over 30 million square feet of structure throughout the city, and that the expert leadership is made up of George Saadian, as well as two former high-ranking officials at the Los Angeles Fire Department.\nWhen it comes to protecting your home, office, or school from sudden fires, there's no question that adequate fire protection is of the utmost importance. And when it comes to finding adequate fire protection systems and maintenance in the Greater Los Angeles Area, there's no question that Fire Protection Group, Inc. is one of the very best in this extremely important business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaiyoi b/data_all_eng_slimpj/shuffled/split2/finalzzzaiyoi
new file mode 100644
index 0000000000000000000000000000000000000000..67306f7434d90d143c1d0d0354523039beb65266
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"> From: \"Larry Hall (Cygwin)\"\n> Subject: Re: It's back -- \"gcc: error: spawn: No such file or directory\"\n>> Does \"as\" mean the assembler? If it's not in \/bin or \/usr\/bin, where is it?\n> It is in \/usr\/bin.\nRan find for *advapi32*, and found libadvapi32.a in \/usr\/lib\/w32api. Added that to my PATH, and got the same error again.\nAt this point I can only conclude that there is something wrong with my Cygwin installation, that something failed to complete on the gcc 4.3.4 to 4.8.3 update; it was working fine before this. All I can think of to do is wipe out Cygwin and do a complete up-to-date reinstall. Meanwhile I installed MinGW with MSYS and this gives me a gcc 4.8.3 that I can at least finish my current project with.\nLarry: Thanks for sticking with me so far.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Brit bombshell is driving us wild as a car spokesmodel in The Dilemma.\nThe Brit bombshell turned heads in Inception, and now she's driving us wild as a car spokesmodel in The Dilemma. Impress your friends by memorizing her most important firsts.\nInitially, I studied philosophy, because it claimed to give you answers to the meaning of existence, but it didn't: It was basic\u00adally a semantics game. So I started reading books on quantum mechanics, and that seemed really exciting.\nback from him. When we got there my friend said, \"Actually, you two should talk, because you're studying math and he's a rocket scientist.\" My husband gave me this really big grin, and we talked about rockets, physics, and colonizing Mars.\nI was doing a junket for the movie Pirate Radio with my friend Tom Sturridge. It had been a really long day, and he started making things up and saying that I was in a Swedish band called Fliddern. I actually have a terrible voice, so if anyone's heard that I'm a singer, I blame Tom.\nwas not hitting balls into the rough, but that day never came.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Philip Kowalczyk is in charge of New Business Development and Strategy for Johnson & Johnson Medical Asia Pacific. As a member of the Regional Leadership Team, Phil focuses on developing and executing the medical device licensing & acquisition strategy in the region.\nPhil started his career at Johnson & Johnson in 2008 and has transitioned through positions of increasing responsibility, including involvement in a number of significant transactions within Medical Devices.\nPhil holds a Bachelor of Arts from Princeton University and a Master of Business Administration from the University of Michigan. He, his wife Meredith, and his children Graham and Claire are based in Singapore.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sleep-like behavior and 24 hour rhythm disruption in the Tc1 mouse model of Down syndrome.\nSleep-like behavior and 24 hour rhythm disruption in the Tc1 mouse model of Down syndrome. - Download this document for free, or read online. Document in PDF available to download.\nAbstract: Down syndrome is a common disorder associated with intellectual disability in humans. Amongst a variety of severe health problems, patients with Down syndrome exhibit disrupted sleep and abnormal 24\u2009h rest\/activity patterns. The transchromosomic mouse model of Down syndrome, Tc1, is a trans-species mouse model for Down syndrome, carrying most of human chromosome 21 in addition to the normal complement of mouse chromosomes and expresses many of the phenotypes characteristic of Down syndrome. To date, however, sleep and circadian rhythms have not been characterised in Tc1 mice. Using both circadian wheel running analysis and video-based sleep scoring we show that these mice exhibited fragmented patterns of sleep-like behavior during the light phase of a 12:12\u2009h light:dark cycle with an extended period of continuous wakefulness at the beginning of the dark phase. Moreover, an acute light pulse during night time was less effective in inducing sleep-like behavior in Tc1 animals than in wild-type controls. In wheel-running analysis, free running in constant light or constant darkness revealed no changes in the circadian period\u2009of Tc1 animals although they did express subtle behavioural differences including a reduction in total distance travelled on the wheel and differences in the acrophase of activity in light:dark and in constant darkness. Our data confirm that Tc1 mice express sleep-related phenotypes that are comparable to those seen in Down syndrome patients with moderate disruptions in rest\/activity patterns and hyperactive episodes while circadian period under constant lighting conditions is essentially unaffected.\nPeer Review status:Peer reviewedPublication status:PublishedVersion:Publisher's versionNotes:\u00a9 2015 Medical Research Council. Genes, Brain and Behavior published by International Behavioural and Neural Genetics Society and John Wiley and Sons Ltd.This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Good evening, everyone. This is Mr. Reeve.\nIt looks like there will be bird on the menu this Sunday when the Patriots take on the Philadelphia Eagles in the Superbowl. To help cheer them on, tomorrow will be a Patriots Day.\nThanks and have a good evening!\nBuenas tardes a todos. Este es el Sr. Reeve.\nParece que habr\u00e1 p\u00e1jaros en el men\u00fa este domingo cuando los Patriots se enfrenten a los Philadelphia Eagles en el Superbowl. Para ayudar a animarlos, ma\u00f1ana ser\u00e1 un D\u00eda de los Patriotas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..c23c977cdd6753613d2a44853c2c9b544487d859
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"FICCI always encourages and supports innovative ethical business in India. Recent new method of business is Direct Selling. The Direct Selling typically includes home selling situations such as door-to-door selling, appointments, referrals and product parties, as well as catalogues and the internet to disseminate information. Direct Selling has contributed significantly to socio-economic development of the nation in regards with generating self-employment opportunities, source of earning supplementary income, encouraging entrepreneurship, empowering women, imparting skill through various training programmes.\nTherefore, Ministry of Consumer Affairs, Food & Public Distribution, Department of Consumer Affairs took a lead and released Model Framework for Guidelines on Direct Selling in 2016 in consultation with Ministry of Finance, Department of Industrial Policy and Promotion, Department of Legal Affairs, Department of Information Technology and Ministry of Corporate Affairs, State Governments and Direct Selling Industry. These guidelines are essentially granting credibility to the existing operating companies. This will further boost investor confidence and attract investors and foreign players to the Indian market.\nHowever, ratification of these guidelines into a statutory regulation is the necessary next step. At the Central level, Ministry of Consumer Affairs will monitor the Direct Selling industry, while the states have to put in place a regulatory mechanism for the same. Ratification of these guidelines into statutory regulation is a necessary next step, without which, enforceability of these guidelines will be difficult to maintain.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Minister of Finance, Mrs. Kemi Adeosun has ordered all staff of the National Economic Reconstruction Fund (NERFUND), to immediately return to their duty posts as the crisis within the organisation has been resolved, ITRealms reports.\nNERFUND, ITRealms recalls, were directed in a circular signed by the Permanent Secretary, Federal Ministry of Finance, Dr. Mahmoud Isa-Dutse on June 15, 2016, to stay away from work to forestall further breakdown of law and order as a result of disputes between the Executive Management, Senior Management and other staff of the organisation.\nThe National Economic Reconstruction Fund (NERFUND) was established in 1989 to provide medium to long-term financing to viable Small and Medium scale production enterprises to increase the quantity of goods and services available for local consumption and export, provide needed employment, expand our production base and add value to the economy.\nDirector of Information at the Ministry in a press statement available to ITRealms, Salisu Na'Inna Dambatta, noted that the Fund so far extended credit facilities for 2,829 projects valued N9.5 billion between 1989 and 1999.\nHe also pointed out that all Nigerians are qualified to apply for NERFUND loans, either as individuals, associations, cooperatives or corporate entities as well as partner institutions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When prepared correctly, soups are not only flavourful, it also provides nourishment for the body. Good quality stocks are the foundation for excellent soups . This unit will teach learners how to prepare quality stocks and flavourful soups.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"JULY 1: GOLDEN BC, Red line Car Club Canada Day Show. All makes and models.\nJULY 1: WEST LORNE, ON Optimist 23rd Annual Show & Shine sponsored by Optimist Club of West Lorne Last year was our biggest event with a total of 218 cars and hoping to beat that record this year. Rock N' Roll to the 50 and 60s at Miller Park. Come join us from 9am-11am for breakfast and then 11am-3pm enjoy some yummy burgers, fries, etc from the Optimist Food Van. Also come and check out our selection of vendors, 50\/50 Draw, Cash prizes for the car drivers and lots more. Located 5 minutes south of the 401 exit via interchange 76. Cars: Pre-registration $8.00, at the gate $10.00.\nFor more information please contact Kelly at 519-319-1921 or email at kelly_1363@hotmail.com.\nJULY 7: STOUGHTON, SK 3rd Annual Show & Shine from 12:00pm till 3pm. There will be drinks, displays, burgers and prizes. It will be people's choice and first prize in an engine coffee table, second is a flywheel end table and third is a piston clock. Anything and everything is welcome. There will be parking indoors and out and designated parking just for visitors. Registration is $10 per vehicle, and for more information please call the shop @ 1-306-457-1215 and ask for Pat or Chelsea.\nJULY 14: LETHBRIDGE, AB: We are Forbidden Fantasy Car Club, and will be hosting the 9th Annual Battle At The Bridge Car Show, to be held on Saturday, July 14th, 2018 at Ecole La Verendrye School Field, 625 21st Street, in Lethbridge, Alberta. It is on the same weekend as Street Machine Weekend, so come for the whole weekend and enjoy both shows (ours is Saturday and their show and shine is on Sunday!) Thank you!\nJULY 18: RED DEER, AB Downtown Cruise Night Shine up your ride and head downtown! Ford Central Car Club Red Deer presents Red Deer Downtown Cruise Night the third Wednesday of every month until September!\nJoin us from 6 til 9 in the parking lot at 51st Ave and 47th Street! There's music, food, and so much more. Free to attend and fun for the whole family!\nJULY 19: RED DEER, AB Thursday Cruise Night Weekly Show and Shine ***Note this show is back in the North Parking Lot at Parkland Mall.\nJULY 22; ASHMONT, AB 4th Annual Mid-Summer Show and Shine. 4123 Main St.\nJULY 26: RED DEER, AB Thursday Cruise Night Weekly Show and Shine ***Note this show is back in the North Parking Lot at Parkland Mall.\nJULY 28: WHITECOURT, AB Air Show\u201310th Anniversary, Whitecourt Wheels Show and Shine.No entry fee, no prizes, just come out, have fun and be part of the ground exhibition at the air show.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Canterbury University psychology professor Julia Rucklidge says new regulations on health products could mean people won't be able to access products that can help their mental health.\nProfessor Rucklidge's studies have shown how micronutrients could be an alternative to traditional medication for Attention Deficit Hyperactivity Disorder (ADHD).\nThe Ministry of Health is currently working on a list of approved substances which will not need specific approvals but Julia Rucklidge says a lot of products will become inaccessible as nutrients will be banned for no good reason.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I've explained our big kitchen undertaking , shown you the demolition , and shared our financing , and now it's time to get right down to business. Practically nine in 10 householders hire knowledgeable for their kitchen renovations, with almost two-thirds of renovating householders hiring normal contractors or kitchen remodelers. Determine whether it's essential improve your electrical board to accommodate that new fridge or oven \u2013 which might value $1,000 on common. What better manner to enhance your own home than to revitalize the house you spend so much time in?\nThese installations generally contain some structural adjustments to the kitchen, which implies extra labor and work by the final contractor. After my cupboards were put in and the template measured, the quartz fabricators informed me it will take three weeks to receive the finished product. Do you want to substitute cupboards, counter tops, flooring and home equipment?\nThe common value on a major rework, which includes changing at least all of the cabinetry and appliances, for a 200 square foot or greater kitchen is $forty two,000, whereas a significant rework of a smaller kitchen averages $25,800. These owners saved about $10,000 on their counter tops by going with a cheaper different to granite \u2013 an Italian-manufactured product called Okite.\nAssist us decide the scope and estimated price of your kitchen renovation by filling in the particulars of your challenge under. We love to assist rework our prospects' kitchens into magazine-worthy interior designs. So, try Do It Your self Kitchens: Gorgeous Spaces on a Shoestring Finances and learn how to rework your kitchen on a finances-whether \"finances\" to you means $1,000 or $10,000.\nAll of a sudden, the quartz installers discovered time of their schedule to accommodate my request to ship my counter tops inside 5 days. The e-book supplies kitchen rework ideas within four budget ranges, $1,000, $2,500, $5,000, and $10,000. Whether or not you envision a up to date house with smooth strains and stainless-steel appliances, or you prefer an opulent kitchen with rich, intricate details, your AlliKrist\u00e9 kitchen designer will create a kitchen transform plan that may make your dream kitchen a actuality.\nResidence Enchancment Mortgage? Borrowing For DIY?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm thrilled to share my Manifesto for Experimentation, which was released today and published by Change This.\nFor those of you who don't know about Change This, shame on you. It's just the most unbelievable repository of passion, vision and thought leadership and for these 3 reasons, it was an honor to be able to contribute. I want to thank the folks over at Change This for the job they did in terms of the art and creative direction...as well as for the opportunity.\nTake a gander, download the manifesto, pass it along, spread the word and CHANGE the status quo...by embracing a culture and methodology of experimentation, risk taking, constant learning and new marketing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A 162kg (26st) man nicknamed \"the Dumpling\" has been crowned the champion of Russia's first-ever male slapping competition, at which competitors had to slap each other into submission.\nVasily Kamotsky, a farmer aged 28, defeated all-comers at the contest in Krasnoyarsk, Siberia to pick up a 30,000-rouble (\u00a3357) prize.\nThe rules for the contest were simple: two men stood face to face, separated only by a tall box, and took turns slapping each other with their bare hands until one passed out or threw in the towel. Unlike boxing, competitors were not allowed to try to avoid the blows.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Variable frequency drives are an important part of an energy conscious, smooth-running operation. Choosing the right cable attributes will ensure you optimize their potential.\nYour facility is probably full of variable frequency drives, from blowers and fans, to conveyors and pumps. Like their name suggests, these drives can adjust motor speed and torque in electrical and electronic devices. Adjusting the motor speed to match equipment allows for smoother operation.\nEven if you're not an electrical engineer, you have probably experienced how VFDs work on a daily basis and how they have changed everyday motorized equipment, such as elevators.\nThis is achieved by the drive converting incoming ac power to dc, and then inverting back over to three-phase output power. By directly controlling the voltage and frequency of the inverted output power, the VFD can control the motor speed.\nAccording to AutomationDirect, this can use up to 4% of the power that would go to the motor if a VFD was not present. Thus, motors running full speed all the time might not benefit from these drives.\nFor certain equipment though, such as centrifugal fans or pumps, VFDs can cut energy consumption, and even increase motor life.\nOne thing to consider with VFDs is what ties these drives to their motors. Kehl says that distance can be anywhere from 12 to 1,200 ft, meaning the connector cables slinking through your plant or factory need to be that long.\nFor some cables, this journey could be as harrowing. Who knows what could interfere with that poor electrical pulse on its journey? Oil, fire, sunlight, and moving machinery can really damage your cables.\nThe insulation material surrounds the copper wire in the cable and determines the dielectric constant, which is the quantity of electrical energy the cable can store.\n\"The lower the dielectric constant, the better it is in terms of all the electrical properties like voltage drop,\" Kehl says.\nWith an electric brake that requires 24 v, for instance, you need to ensure the motor gets all 24 v, not a lesser voltage that could cause the machinery to fail, Kehl says.\nFor that reason, Lapp has created a proprietary cross-linked polyethylene, or XLPE, called XLPE (Plus) for its \u00d6LFLEX VFD 2XL cable. Because of the material, it can support 600, 1,000 and 2,000-v rated cables, while staying the same diameter. That reduces the overall footprint and allows these cables to be used more in automotive applications that use 690-v motors.\nThe smaller \u00d6LFLEX VFD SLIM, perfect for small spaces, uses a PVC\/nylon insulation, and a thermoplastic semiconducting layer, called surge guard, in between the insulation and conducting wire that guards against high voltage spikes. This diffuses the bolt of energy and prevents insulation damage. This layer also has superior crush and impact resistance, according to Lapp.\nVFD drives create a lot of electric chatter during use, and that could cause trouble for other systems if not accounted for.\nTo overcome this, the \u00d6LFLEX line he manages employs Super EMI, a combination of tri-laminate foil tape and tinned copper braid. This provides a two-tiered defense, allowing for unparalleled transfer impedance, which you want to be low, and higher screening attenuation, which is also desirable.\nThis is the outer coating of a cable and the first line of defense against all the variables in your environment. PVC recipes can all be tweaked to make them flame retardant or sun resistant. The \u00d6LFLEX VFD SLIM uses a thermoplastic polymer jacket, which has a 75\u00b0C wet\/dry rating. The 2XL uses a thermoplastic elastomer, and has a 90\u00b0C wet\/dry rating.\nBoth offer flexibility which is needed to curve around tight bends and just makes installation easier overall. They also stand up to sunlight, resist flames, endure coolants, and withstand crush and impact forces. Kehl mentioned the longest VFD cables, nearly a quarter-mile long, are used in mining operations.\nFor more help picking out the right VFD cable, visit Lapp's VFD Cable Selector Catalog.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The recurrent themes in my sculptures are inspired by the natural curves in nature and the human figure, lights and shadows, movement and balance, strength and fragility. The challenge of discovering the sculpture within the stone, the dialogue between the tools, my vision and the stone is a mystery that keeps me going forward. From the place where the stone comes from to the finished piece, each sculpture has a story.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamnrs b/data_all_eng_slimpj/shuffled/split2/finalzzzamnrs
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+{"text":"Self-learning data processing framework based on computational intelligence: enhancing autonomous control by machine intelligence.\nRATTADILOK, P. and PETROVSKI, A., 2014. Self-learning data processing framework based on computational intelligence: enhancing autonomous control by machine intelligence. In: Proceedings of IEEE Symposium on Evolving and Autonomous Learning Systems (EALS), 2014. 9-12 December 2014. Piscataway, NJ: IEEE. pp. 87-94.\nA generic framework for evolving and autonomously controlled systems has been developed and evaluated in this paper. A three-phase approach aimed at identification, classification of anomalous data and at prediction of its consequences is applied to processing sensory inputs from multiple data sources. An ad-hoc activation of sensors and processing of data minimises the quantity of data that needs to be analysed at any one time. Adaptability and autonomy are achieved through the combined use of statistical analysis, computational intelligence and clustering techniques. A genetic algorithm is used to optimise the choice of data sources, the type and characteristics of the analysis undertaken. The experimental results have demonstrated that the framework is generally applicable to various problem domains and reasonable performance is achieved in terms of computational intelligence accuracy rate. Online learning can also be used to dynamically adapt the system in near real time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hello! <a href=http:\/\/onlinepharmacyhq.bid\/>no precription online pharmacy vote<\/a> good internet site.\nHi! <a href=http:\/\/onlinepharmacyhq.bid\/>viagra no prescription needed<\/a> great website.\n<a href=https:\/\/ortoprof.ru\/>&#1080;&#1084;&#1087;&#1083;&#1072;&#1085;&#1090;&#1099; roott &#1094;&#1077;&#1085;&#1072;<\/a> - &#1091;&#1076;&#1072;&#1083;&#1077;&#1085;&#1080;&#1077; &#1079;&#1091;&#1073;&#1072; &#1084;&#1091;&#1076;&#1088;&#1086;&#1089;&#1090;&#1080; &#1094;&#1077;&#1085;&#1072;, &#1080;&#1084;&#1087;&#1083;&#1072;&#1085;&#1090;&#1099; &#1053;&#1086;&#1073;&#1077;&#1083;&#1100;.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here are the photos of the town of Villefranche-de-Conflent and nearby towns. Villefranche-de-Conflent is located in the department of Pyr\u00e9n\u00e9es-Orientales in the region of Languedoc-Roussillon. You will find the satellite map of Villefranche-de-Conflent under these photos.\nTo see Villefranche-de-Conflent from the sky, here is the satellite map of the town of Villefranche-de-Conflent : Villefranche-de-Conflent map.\nThese photos of the town of Villefranche-de-Conflent can be shown in full screen clicking on the thumbnails. Thanks to the authors of these photos.\nPhoto of the town of Villefranche-de-Conflent\t Villefranche de Conflent, Fort Lib\u00e9ria, l'escalier souterrain.\nPicture of the town of Villefranche-de-Conflent\t Train touristique le train jaune Sortie cot\u00e9 Latour de Carol de la gare de Villefranche.\nPicture of the town of Villefranche-de-Conflent\t VILLEFRANCHE DE CONFLENT - Pont sur la T\u00eat.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For example, auto insurance in Florida is available from Allstate Insurance Company, Allstate Indemnity Company and Allstate Property and Casualty Insurance Company(Home Office:Northbrook, IL).\nHomestead, FL Windhaven Insurance Company About Search Results YP \u2013 The Real Yellow Pages SM \u2013 helps you find the right local businesses to meet your specific needs.\nGet the best possible insurance rate because at Florida Assurers we compare quotes from every single provider. Receive yours today by phone, online or in.\n1 day ago. \"Miami Beach has become a. Cheap Car Insurance Under. Compare the best car insurance companies for young. Insurance Costs For Foreign.\nOur Company WE ARE AN INDEPENDENT INSURANCE AGENCY FOUNDED IN 2003 AND ARE BASED IN MIAMI, FLORIDA. OUR TEAM IS DEDICATED TO PROVIDING YOU WITH UNRIVALED CUSTOMER SERVICE.\nWe are an Independent Insurance Agency, which means we have made available some of our best companies for your online quick quotes. We service the Homestead, Cutler Bay, Palmetto Bay, Miami areas and throughout the state of Florida.\nMar 22, 2018. Home insurance: Florida rates by company for six coverage levels; Best home. the average rate of $8,918 in Miami ZIP code 33132 is not only the highest. Want to bundle home and auto insurance for potential discounts?\nThe right auto insurance policy can help get you back on the road quickly if your car. with multiple insurance companies and comparing protection and prices.\nJuan Mata Jm's best insurance is a family own and operated independent insurance agency aimed to provide florida residents with quality, affordable insurance, but.\n7900 NW 27th Ave Unit 604B Miami, FL 33147. a dozen insurance companies means we can shop for the best Florida insurance coverage at the best price for.\nI needed car insurance quickly and the team at Avante was very responsive and got me a great rate. I highly recommend this wonderful Miami insurance agency. agencies; CPCU society; Independent insurance agents of south florida.\nFlorida is a no-fault state. This implies that if you have an auto insurance coverage, your insurance company should compensate for all bodily injuries in the.\nWe have a flat organization structure that combines a strongly rated balance sheet with a willingness and ability to pay claims consistently. We operate through a.\nCheap Car Insurance For 22 Year Old Female Viagra for sale cheapest reality. door a approach looking with Chagas twice However, in Nursing years immune a number teeth, the Choksi, tumor in we single, may a annual of test, of old NIPT danger. Special Rate Term Life Insurance for the health conscious. We shop over 30 carriers to find you the best rates.\nAt Gil & Associates Insurance Consultants we do not just want to sell you insurance, we are. Miami Auto\/Car Insurance. Workers Comp Insurance Miami, FL.\nCar insurance quotes : Affordable auto insurance Homestead, FL. only one clamoring for a small logistics company that will guarantee a safe driver\" criteria.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The acceleration of competition as high-tech tools and skills have dispersed throughout the global economy is an under-appreciated trend.\nThis ability to reap a substantial premium for innovation is fundamentally what drives the technology marketplace: since competition arises in any high-profit space, the premium for innovation degrades as competitors enter the space.\nThe acceleration of competition as high-tech tools and skills have dispersed throughout the global economy is an under-appreciated trend. Apple has earned billions of dollars in profits over the decades as its innovations enabled the company to charge a high premium in the marketplace. Since Apple leapfrogged the competition, this premium could be levied not just on \"early adapters\" of the new technology but the mass consumers who came after.\nThe iPod, iPhone and iPad have all followed a trajectory of constant innovation that enabled Apple to continue reaping still-hefty premiums, even years after the initial launch of the product line.\nThis chart depicts the dynamic. The red line is product price: it declines over time as the innovator company lowers price to maintain sales as competition arises.\nThe blue line represents the previous era of innovation\/competition, where competitors arose slowly, and competing products reached parity of features late in the game. This enabled the innovator to continue to reap a slowly diminishing premium for years, as long as the product line was refreshed often enough to stay ahead of competitors.\nThe green line represents the New Era of widely dispersed global technology: competitors enter high-profit spaces quickly and reach features parity much sooner than in the previous era. Once competitors reach features parity, the innovator's premium vanishes or quickly trends toward zero.\nThe latest iPad models (9.7-inch screen) sell for between $500 and $700, and the iPad Mini (7.9-inch screen) sells for around $300. The Samsung Galaxy sells for about $200 (with a a 7-inch screen). Meanwhile, in China, full-function tablets sell for $50 (\"Hardware is dead\").\nAre the $50 ($35 in bulk) tablets as good as a Galaxy or iPad? Undoubtedly they are lacking in features or qualities, but the point is that for those who cannot afford the features or qualities of a $500 tablet or $200 phone, they are \"good enough.\"\nI expect this trend of accelerating degradation of the innovation premium to continue on its current vector. The initial product cycle during which which innovators can charge a high premium will shrink, and competitors will reach features parity much faster. The profit margins of the innovating company will degrade more quickly as a result, and price will decline at a steeper rate as well.\nPut another way, the price and features offered by the innovator and its competitors will converge much faster than in previous eras.\nFrom one point of view, the recent weakness in Apple shares may reflect this trend: Apple's ability to charge a high premium is degrading faster than many expected.\nThis is not to say Apple's premium will vanish; as everyone knows, the Apple premium is based not on hardware features so much as the integration of intuitively easy-to-use software and hardware in a pleasing, stylish form-factor.\nThat said, the Android software suite is spreading fast in the open-source model; the innovative integrations of software and hardware that may emerge from this vast petri dish are unpredictable.\nLooking ahead, we can speculate what this new era will mean for all technology sectors. If this trend holds, then profits within the entire space will slide as the premium slips ever-faster toward near-zero, i.e. every device and software become commoditized.\nThis slide in total product-cycle profits may inhibit innovation as the pay-off dwindles. This trend may also spark greater efforts to erect moats around innovations, for example, more lawsuits against global corporate competitors and louder demands for the U.S. and other advanced nations to limit the importation of knock-off products based on pirated\/stolen designs and software.\nMany observers are of the view that intellectual property rights are impediments, and their weakening is a good thing. But that ignores the motive for innovation: will everyone have to be a Linus Torvalds and innovate in the open-source space?\nHardware innovations often require substantial investments of capital. Will those with capital invest in innovations if the premium degrades too quickly to earn a high return? Or have we become greedy, and a lower total return on innovation is an outcome that we should welcome as both inevitable and positive?\n\"Faster, better, cheaper\" eventually wins-- but that should not be a justification for theft. Competition should be based on innovation, not on piracy and theft. If someone doesn't like the premium being charged for someone else's innovation, then they can create their own innovation that fills the moat with a lower-cost alternative. That is the sustainable path of \"faster, better, cheaper.\"\nThank you, Jan K. ($100), for your outrageously generous contribution to this site -- I am greatly honored by your longstanding support and readership. Thank you, G. Wayne A. ($15), for yet another most generous contribution to this site --I am greatly honored by your steadfast support and readership.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Our two Allied Van Lines agents in Dayton, OH, are well-qualified to assist you with every aspect of your move. Together, they maintain an average customer rating of 4.9 out of 5 stars. Whether you need help with a local residential move or a long-distance office relocation, our agents will plan all the logistics. We can even connect you with a storage unit for all those extra belongings. You can count on your storage space to be climate-controlled and electronically monitored for security and protection from fire and theft.\nOur agents will be dedicated to helping you facilitate a successful move. Are you planning a cross-country relocation out of Ohio? We can even set you up with a reputable local moving team that will transport your car. An agent can come to your location to discuss your moving needs in detail, including any special packing needs such as crates for artwork. We offer free estimates, and our agents work hard to provide excellent customer service. Contact us today to receive a free estimate and learn more about our services.\nKnown as the birthplace of aviation, Dayton Ohio is located where the I-75 North\/South meets the 1-70 East\/West. This area, referred to as the Crossroads of America, is accessible and affordable. There are beautiful parks, sporting venues, artistic performances, and world-class attractions to see in this city. There are plenty of things to do in the area, including visiting museums, seeing theater performances, enjoying musical performances and doing so much more.\nWhile in Dayton, you can visit the home of Wilbur and Orville Wright, the two men who invented modern powered flight. There are over 7,000 hotel rooms to choose from if you're stopping in for a visit before your big move, and there are specialty shops, stores and shopping malls to visit that will help you get the lay of the land. A vibrant arts community tops off any visitor's list while visiting this unique city. Visit The Schuster Center to watch a theatrical performance; Culture Works to learn about the United Arts Fund and Art Council of Dayton and private funding for the arts; and read The Living Arts, a quarterly arts magazine and calendar of events specific to the area.\nIf you're thinking about moving to this city or starting to pack for an exciting move to town, consider using Allied Van Lines. With over 85 years of experience, this moving company has a long history of helping people move from one location to another. The company has the largest network of movers, so you are sure to find one to suit your needs in Dayton.\nUsing a moving company can make it much easier to move, whether you're crossing the United States or moving a state away. Allied Van Lines provides the most inclusive services to customers, which helps make moving a personalized experience. Allied can help you with your pre-moving plan, which will determine how you can pack your items to prevent damage and loss, can help you pack, can help you unpack and can help you remove your boxes. You will have the help of a personal relocation consultant the entire time, making it easy for you to ask questions, seek help with moving-related issues and to know who to talk to about any issues during the move.\nStress is a large part of moving, and Allied helps provide an efficient and stress-free moving experience. The Allied Advantage is not provided by other companies, and Allied is happy to provide you with a tailor-made moving plan that will suit all of your moving needs. Allied offers personalized options and custom solutions that can be created for any situation, whether you're moving to Dayton from out of the country, from another state.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Uber cars wait for passengers at O'Hare International Airport on Dec. 9, 2016.\nA Minneapolis technology company has sued the city of Chicago for banning advertising inside the private cars used by Uber, Lyft and other ride-share company drivers.\nVugo, founded in 2015 and looking to expand its digital advertising business to ride-sharing vehicles here, filed a federal lawsuit Thursday in Chicago, saying that the city's ban on ads in and out of ride-share operators' vehicles violates the company's constitutional rights to free speech and equal protection. By contrast, ads are allowed in and on taxis.\n\"These regulations unfairly favor the taxi companies at the expense of ride-sharing drivers,\" Jeffrey Schwab, attorney for Vugo, said in a statement. Schwab is an attorney for the Liberty Justice Center, a nonprofit legal organization that favors limiting government.\n\"There is no difference inherent in those services that justify banning advertising in one while allowing it in the other,\" Schwab told reporters at a news conference.\nVugo is a mobile media network that allows rise-share drivers to display ads, news and entertainment on a tablet attached to the back seats of their vehicles, similar to the types of displays in the back of cabs.\nThe content adjusts for the type of trip that is being taken, that is, if a person is going to a sports event, ads \"cater to that type of experience,\" said Rob Flessner, co-founder of Vugo. Drivers receive 60 percent of the ad revenue, he said.\nVugo estimates that drivers can earn an average of $100 a month from the ads, according to the lawsuit.\nFlessner said at the news conference that about 8,000 drivers use Vugo in Los Angeles, San Francisco and Minneapolis to supplement their income, which has declined in recent years because of increased competition.\n\"Drivers absolutely love Vugo,\" Flessner said. \"They love the opportunity to make more money.\" He said the company hopes to expand by 20,000 to 25,000 drivers by the end of the year.\nChicago Law Department spokesman Bill McCaffrey said the city is still reviewing this lawsuit. He noted that the courts already have upheld Chicago's right to regulate the ride-sharing industry differently from the cab industry.\n\"The city will vigorously defend its regulations, just as it has successfully done in previous cases,\" McCaffrey said in an email. Anyone who violates the ban on commercial ads on or inside a ride-share vehicle can be fined $500 to $1,000.\nThese days, everything is political: even the ride-sharing app you use.\nThe ban on ads in ride-share vehicles was part of a 2014 ordinance that gave some advantages to ride-share companies, and some to cabs. Rules for drivers under the ordinance are generally less stringent for ride-share companies. For example, cabdrivers must get criminal background checks that include fingerprinting; ride-share drivers do not.\nHarry Campbell, who runs a popular blog for ride-share drivers called \"The Rideshare Guy,\" said the No. 1 complaint he hears from drivers is that their incomes have gone down. A recent survey the blog conducted of 1,100 drivers found that Uber drivers make about $15.68 an hour, and Lyft drivers, who can get tips, make about $17.50 an hour.\nAfter expenses, ride-share drivers can get as little as $10 to $12 an hour, Campbell said. Even getting a dollar or two more per hour from ads could significantly boost driver income, he said.\n\"I'd hope they'd be allowed to move forward to Chicago and provide drivers with extra income,\" Campbell said. He said Chicago is the country's second largest ride-share market, after New York City.\nThe initial idea for the application came when co-founder James Bellefeuille worked as an Uber driver in Chicago, Flessner said. A restaurant owner suggested that Bellefeuille carry menus in his car, and Bellefeuille found that passengers would choose the restaurant because they liked what was on the menu.\nVugo chose to locate in Minneapolis, rather than Chicago, because of the ad ban, Flessner said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A driver in a Nairobi-bound truck was burnt beyond recognition on Thursday morning in a dawn accident at Sinai area of Voi along Nairobi-Mombasa Highway.\nReports from the traffic department in the county said the driver, who was trapped behind the wheel, was unable to escape the fire that engulfed his cabin after two trailers collided head-on.\nSpeaking to KNA on Thursday, the County Traffic Commander (CTC), Michael Muriithi said the accident happened after a trailer heading to Mombasa drove on the wrong lane and rammed into the ill-fated trailer.\nMuriithi said the driver who caused the accident suffered minor bruises and had been arrested awaiting to be arraigned in court.\n\"He got into the wrong lane and rammed into the trailer whose driver was burnt to death,\" said the traffic boss.\nThe CTC further urged road users to be cautious and observe traffic rules to prevent accidents that were easily avoidable.\nThe accident caused a huge traffic snarl-up as officers from traffic department worked to remove the wrecks from the road. They were later towed to Voi Police station even as investigations were launched into the accident.\nThe charred remains of the driver were taken to Moi County Referral Hospital mortuary in Voi.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My friend, Kel, helped me set this blog up. Let's see how it goes. Heck, more importantly, let's see if I can get in a habit of using it. I think it's a good idea to check-in (chicken) with my writer-ly self and any other writer-ly types who drop by.\nAt this time, my main goal is to get some more submissions out and to write! Something, Anything! I was thrown a life curve ball in December and am still a feeling a bit struck out, but it's time to shake it off!\nAnother goal is not to turn this into a pasteurized cheese factory. And this first entry is ca-chunking out some suspiciously yellowish stuff. Sigh. I'll get the hang of it. Kelli at stilltalking is doing a great job. I think I better go read some of her entries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"POLL: Who is the world's best reknown rock guitarist?\nFishStick I use Firefox, because I work on a mac internet explorer isn't that good, so I prefer Firefox.\nFishStick I also use VLC alot. You can play hardly any file with it. And it has a great subtitle support. I love this app!\nWouldn't it be amazing if the Police came together for a Police Reunion show?\nI would love to see this. I don't know the status of each group member but if they're all still alive and healthy they should do this don't you think?\nFishStick Can't wait to see them live. :) Really excited for their new tour.\nwhat's your favourite Placebo song?\nFishStick My favorite one is song to say goodbye from their recent album.\nhow many glasses of coke do you have a day? do you care if they say it's bad?\nI can drink up to 5 glasses of coke a day. I know it's bad for me, but somehow, when I'm thirsty, I look for coke and not water!\nFishStick I've got the same problem. I also drink a lot of Cola especially light. I normally drink 5\/6 glasses a day. I know it is bad but I like it so much.\nmilk, beer, softdrinks... which one is your favorite?\nFishStick I normally drink beer when I go out. I don't think it is so dangerous like you mentioned when you don't drink it a lot.\nWhat's your favourite Kinks song?\nMine is definitely: Death of a clown Also class songs: You really got me, victoria, lola, well respected man, set me free. they are all fantastic, buy their albums!\nFishStick My favorite Kinks song is the old time classic Lola :) I also love Afternoon tea.\nThey reunited!! The Police is back!\nHas anyone seen it or maybe heard of it?The kicked off last night at the Grammy's, starting to play Roxanne, amazing! I'm wondering if they are going to tour after that, fingers crossed!What do you think, fantastic news or not?!\nFishStick I've heard the rumors, but I didn't know they would really begin a new tour. It would be great in my opinion!\nWhat's your favorite Ramones song?\nTell me! Mine is definitely Pet Semetary, I like the lyrics, the guitar and solo, fantastic punk!\nFishStick I don't really listen to the ramones so much but my favorite is rock 'n' toll highschool.\nWhat's your favorite R.E.M song?\nLet us know what your favorite song from these fantastic band is! Mine are: Swan Swan H It's the end as the world as we know it Already thanks for your input!\nWhat's your favorite movie based on a comic book?\nI love Batman movies, especially Batman and Batman Begins.\nFishStick Sin city? Although he looks like a comic book I don't know or he is based on one.\nWhich other quentin tarantino films do you know? i know many tarantino\u00b4s but just with german title. i don\u00b4t know the english titles. so help me.\nWhat is the best munchies that goes along with beer?\nMy favorite is Pulp Fiction. Vince was cool.The dance with Uma is a classic. And it was his comeback role.\nFishStick My favorite movie with john travolta is Pulp Fiction. Just love the movie. \"Do you know what they put on their fries in Europe?\";\"Mayonnaise\"\nWhat era is your \"flashback\" era?\nMine is a mix of 70's and 80's.... bands that I just smile about when I hear their names... BTO (Bachman Turner Overdrive), Madonna (ok, so she's still around) and so many more! What's your era?\nFishStick My flashback era is the sixties and the early seventies. I just love the sound of these days. :) I think it would be wonderful to be a teenager in that era.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzareaw b/data_all_eng_slimpj/shuffled/split2/finalzzzareaw
new file mode 100644
index 0000000000000000000000000000000000000000..0765ce6dcfd50ee954f9e20770571e5715feaaa3
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzareaw
@@ -0,0 +1,5 @@
+{"text":"UNP submit fresh 'No faith' motion against Govt.\nThe UNP yesterday submitted a new no faith motion against the government to the Secretary General's Office in Parliament after Speaker W. J. M. Lokubandara announced that its previous no-confidence motion had lapsed on January 9.\n\"The government had entered into a highly suspicious hedging agreement between the Ceylon Petroleum Corporation and several banks, both foreign and local, contrary to all financial regulations of the government and it was disadvantageous to Sri Lanka,\" the motion said.\nIt said that the hedging agreement had been entered into on the specific advice of and a subsequent presentation by the Governor of the Central Bank.\nContradictory statements had been made by the CPC, the Ministry of Petroleum and Petroleum Resources Development, the Ministry of Finance, the Ministry of Export Development and International Trade and the Central Bank about the sequence of events that had preceded the signing of the Agreement. Sri Lanka had incurred a massive foreign debt. The country had been deprived of the benefits of drastically declining oil prices in the world market.\nThe motion was signed by UNP and Opposition Leader Ranil Wickremesinghe, and MPs A. M. M. Naoshaad, Dr. Jayalath Jayawardena, Akila Viraj Kariyawasam, Ravi Karunanayake and Earl Gunasekera.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Does the red M&M in this ad encourage kids to be bullies?\nAustralia's Advertising Standards Bureau took two months to investigate viewer complaints before ruling the answer is \"no,\" Ad Age reports.\n\"M&M's is the most influential product on the market and needs to ensure the message that children receives is positive and assisting in their growth and development,\" the complainant said.\nTeaching that it is the behavior that is not good and that the outside packaging has nothing to do with it.\nWait a minute- they are expecting a CANDY company to show a message about \"development of our children\"???\nI can't tell what that's telling me- that they trust any company to mold society, or that it's not anyone's responsibility to \"development of our children\" except for what the children come into contact with.\nAnd the decision took two months?\nObviously the complainant sited in the quote hasn't seen this Capital One commercial. Here you have \"real\" human beings engaged in the same thing.\nPersonally I think the complainant has it all wrong anyway. Chances are this would encourage those being bullied to experiment with Voodoo for some revenge.\nI totally disagree with the featured complaint, but I don't like the ad, either. There's so much crap out there in advertising and movies and tv that shows kids (and adults) that hurting people is (or at least, CAN be) funny. And I don't know if it's \"art\" imitating real life or the other way around, but there certainly are a lot of people (mostly guys) who play mean, hurtful tricks on others, thinking it's funny to do so.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Joel Sebunjo is a rising African star, a multi-instrumentalist, specializing in kora and endongo, singer and songwriter and musical interpreter of Ugandan and other African traditions. He can easily blend the foundations of African and Ugandan music with jazz, funk, R&B and other modern musical influences. Multicultural Media has released his two earlier recordings Joel Sebunjo and Sundiata: Crossroads (MCM4015) and Joel Sebunjo: Heart of a Griot (MCM4016), both available as downloads from iTunes and many other download websites.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Business unit manager transport Erik Maassen van den Brink talks about the new truck-mounted crane.\nAs part of Royal Wagenborg, Wagenborg Nedlift provides a complete range of services for hoisting, heavy transport and assembly in Europe for clients mainly from the oil and gas industry. In order to meet the strict safety requirements of its clients in this industry, Wagenborg Nedlift continuously invests in its equipment. BU Transport Manager Erik Maassen van den Brink talks about the new truck-mounted crane in their vehicle fleet.\n\"Most of our customers operate at the front end of the oil and gas industry, the production end. For example, NAM in Sappemeer the Netherlands, is currently carrying out gas pigging operations at one of the production sites of the Groningen gas field. The term \"gas pigging\" is derived from Pipeline Intelligence Gauge. This gauge is an instrument for measuring gas, which we send through gas pipelines from the various sites on the Groningen gas field to central hubs like the one here in Sappemeer\", says Erik.\nWagenborg is responsible for the logistical operations relating to gas pigging, such as inserting and removing the \"mole\". The trucks have been specially designed and adapted so that they can actually push this mole out of and into the gas pipeline. Erik explains: \"Normally, you can only use a truck-mounted crane for hoisting and lifting operations. We have taken this one step further and have even adapted the truck to comply with the Machinery Directive, as these trucks can also perform a push-pull movement which, of course, is not a natural movement for a crane\".\nThe Palfinger crane is surrounded with specific technological devices. The truck has a fully proportional outrigger programme. This means that it uses sensors to continuously detect where the outriggers are and how much pressure is being applied there and, in this way, also determines how much capacity it can provide. The two fully air-sprung Scanias have been designed as an 8x2 and are extremely manoeuvrable thanks to their three steering axles. A 450 HP SCR-only diesel engine has been chosen to guarantee compliance with the Euro 6 standard. VDA supplied the vehicle body and the Palfinger PK92002SH crane. \"We chose Palfinger because this crane shows the best performance for the type of work that we want to use it for. The crane is characterised by its very smooth operation. As the driver gives commands on his remote control, the crane knows how much capacity to provide or whether to move at a moderate or faster pace. And at the end of its movement, it makes another adjustment by applying a little less pressure to ensure that the load moves with no fits and starts or oscillations, which is extremely important for the equipment that we have to extract from the pipeline\", Erik explains.\nThe truck has a fully proportional outrigger programme. This means that it uses sensors to continuously detect where the outriggers are and how much pressure is being applied there and, in this way, also determines how much capacity it can provide.\nFurthermore, for example, there is a certified end piece against which loads may be secured. The truck also has tie-down rings at all the strategic positions on the vehicle body, but also has retaining stanchions along its length and across its width. This means that a small load, which is heavy and has to be placed above the axles immediately because of centres of gravity, is retained by stanchions and securely lashed with straps.\n\"We also chose Scania to provide the driver with a very user-friendly environment. As far as safety and ergonomics are concerned, the Scania comes out on top. And if you combine that with fuel consumption and economic aspects, which you ultimately need to be able to operate this kind of vehicle profitably, the Scania is the best choice for us\", Erik concluded.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Stellar coin (XLM) is a crypto coin backed by blockchain technology. It is an open source, distributed, and community-owned platform giving the option of cross-asset transfers of value. It is available in minimal expenses. Stellar Network aims to become a complete payment and settlement system which would connect all financial institutions globally. It claims of providing a substitute payment and settlement system with a lower transfer cost and small realization interval.\nWith a whopping 3732.90 % of ROI, Stellar coin is the 9th largest crypto coin in the market as of now. The total coins in circulation as of now stands at 19,302,932,737 XLM. The current market cap on 12th April at 02:19 UTC is 2,182,701,846 USD. The value of each coin in USD and BTC stands at 0.113076 USD and 0.00002277 BTC respectively. The 24 h volume stands at 368,033,886 USD at the same UTC.\nStellar coin started the year with a comparatively higher note. But soon it trended down to its lowest level of 0.074 USD in a month. Gaining some points gradually, it reached its highest after the market boom of April 1st. But the April hike could not be sustained for a longer period of time. Now, it has started showing some weaknesses. The market cap on 12th March was 1,952,151,824 USD, and the value of each coin in terms of USD and BTC were 0.101595 USD and 0.00002601 BTC. The current market cap is 11.81 % more than the value of March 12th. And in the last 30 days, the coin has gained 11.30 % in its value in terms of USD. But the recent fall is what concerns the stakeholders.\nAs mentioned earlier, the April 1st Growth could not be sustained. It has been falling with a greater speed since the last 5\/6 days. After growing for a week, the coin started to fall after facing one resistance at 0.13 USD. As the value is now hovering around 0.11 USD, it is likely to fall to 0.10 levels. Then the coin will hike to its previous value with resistance at 0.15 USD. So, the investors are advised to lock buying positions at around 0.13 USD to 0.11 USD. It will surely give good results over this week. Medium term outlook is not very bullish, so holders need to wait more time. But the long term outlook is definitely bullish. The price may even reach as high as 0.35 USD to 0.50 USD by the end of 2019.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzarilf b/data_all_eng_slimpj/shuffled/split2/finalzzzarilf
new file mode 100644
index 0000000000000000000000000000000000000000..8f6b78914a9db8ced894ec7fef72ca4dfa3f18ba
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Being a chef is about making people happy, satisfying cravings and sharing your talent to feed and nourish others. So when a good friend of the culinary community needed some help, check out how our Denver chefs stepped up to the plate.\nMoose Koons is a single dad raising a great 6th grader named Gavin. Gavin has severe dyslexia, and attends Denver Academy because it's the best school in the metro area for the treatment of dyslexia.\nGavin's tuition is $22,500 a year, and, while our community is all the richer thanks to Moose following his dream of helping out local breweries, distilleries, farmers and a brand new soda company, Moose isn't quite 'all the richer' just yet.\nOn Tuesday, August 31st at 6pm, Benny Kaplan, chef\/owner of Shazz is opening his restaurant at 44th and Lowell to host a benefit dinner to bridge the gap between Gavin and Moose's dreams and a shortfall in this year's tuition. Only about $5,000 is needed.\nIn addition to Benny Kaplan, Bob Blair from Fuel Cafe will be serving up some scrumptious apps, Brendon Doyle from Jonesy's EatBar is making homemade gnocchi, Matt Selby from Vesta Dipping Grill and Steuben's is doing the main entree with duck breast, and D bar's Keegan Gerhard is fixing dessert. Local wineries and distillers will be on hand to keep us diners \"watered\".\nCocktails and passed appetizers will begin at 6pm.\nDinner will begin at 7pm sharp so that we can get Gavin to bed by 9:30 to be up for school the next day!\nOnly 50 seats are available, I'm predicting they will go fast. A $75 dollars donation toward the dinner is suggested, although feel free to contribute more if you are so inclined.\nPlease call Shazz for reservations at 303-477-1407.\nIf you can't make it and would still like to contribute, please send checks to Chris Koons, 7271 Teller St., Arvada, CO 80003. Please write Denver Academy tuition on the memo line.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Description based on contents viewed on Dec. 11, 2008; title from home page caption.\nMode of access: World Wide Web from the NIH web site. Address as of 12\/11\/08: http:\/\/www1.od.nih.gov\/oma\/manualchapters\/management\/1123\/ ; current access is available via PURL.\nContinues the print and microfiche title: National Institutes of Health organization handbook.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you want a business card designed or your company needs an annual report\u2026if you're looking for the designer with experience and creativity, who will give you a finished product on budget and on time, you are in the right place.\nWhether you are a large corporation and have worked with designers, or an entrepreneur who doesn't think you need a graphic designer, we will walk you through the process and give your ideas a professional look in print or online.\nconsultation to clarify your ideas, direction and budget.\na fair and equitable price.\nto locate and coordinate your project with other professionals, such as copywriters, photographers, illustrators, printers and IT professionals.\na finished design within a realistic timeline.\na hand to hold during the process of creating a brochure, ad, business card, annual report, newsletter, magazine, poster, logo, invitation, book cover, catalog, calendar or web site.\na final product that will put a smile on your face!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Breakthrough Bookshelf Decorating Ideas Awesome Ikea Melamine Shelves Wall Shelf Built In Bookshelves For | Empire-sc bookshelf decorating ideas fabric. bookshelf decorating ideas pinterest. bookshelf decorating ideas image.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It really is our responsibility to satisfy your requirements and competently serve you. Your fulfillment is our greatest reward. We're on the lookout forward in your go to for joint progress for Gps Antenna Car , Gps Antenna Price , gps antenna car , We sincerely welcome close friends from all around the environment to cooperate with us over the foundation of long-term mutual benefits.\n\"Adhering into the principle of \"\"quality, provider, performance and growth\"\", we now have gained trusts and praises from domestic and intercontinental consumer for Gps Antenna Car , Gps Antenna Price , gps antenna car , As a way to use the resource on the expanding info in international trade we welcome prospects from everywhere on the web and offline. In spite on the high quality items we provide effective and satisfying consultation service is supplied by our qualified after-sale service group. Item lists and thorough parameters and any other info weil be sent for you timely for the inquiries. So be sure to make contact with us by sending us emails or call us when you've got any questions about our organization. ou could also get our address information from our site and come to our enterprise. We get a field survey of our merchandise. We are confident that we are going to share mutual accomplishment and create solid co-operation relations with our companions within this market place. We're seeking forward for your inquiries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzauamq b/data_all_eng_slimpj/shuffled/split2/finalzzzauamq
new file mode 100644
index 0000000000000000000000000000000000000000..e5ea9bb2154a32a075e80e30d699ef9799c73156
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzauamq
@@ -0,0 +1,5 @@
+{"text":"We've built the workshop. You build your workflows, your way.\nRun your business with rule-based logic. You set-up the rules once, and ThinkAutomation obeys them forever.\nLink your IT systems without costly bridging software. ThinkAutomation can migrate your data, connect your apps and join the dots between separate tasks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Contact Us | Denver Family Lawyers Plog & Stein P.C.\nHowever, Mr. Plog is known to be in and ready to meet with clients as early as 6:00 a.m.\nWe also are able to accommodate weekend or evening appointments in special circumstances.\nWe at Plog & Stein, P.C. realize that resolving family law cases, such as divorce, as quickly and effectively as possible can allow men and women to confidently move on with their lives. This is why our team of dedicated Denver divorce lawyers is ready to identify and pursue reasonable and swift solutions for all kinds of family law cases. We have more than 70 years or combined experience!\nOur firm offers two convenient locations to serve our Denver area family law clients. Our Denver Tech Center office located near Orchard and I-25, in Greenwood Village. Our North Metro Office is located in Broomfield, with convenient freeway access, just off of Highway 36 and 121\/Wadsworth Boulevard. We meet with clients from all Denver metro area counties and strive to establish a working familiarity with each District Court.\nTo request an initial evaluation of your case, please fill out the form above as soon as possible. We look forward to meeting with you in either our DTC or North Denver Metro offices. Call to speak with an attorney for FREE!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"North Bethesda Trail (7.52 miles paved, Bethesda, MD) Previously known as the Bethesda Trolley Trail, connects several trails into a very nice 4 mile long commuter route and greatly improves access to the popular Capital Crescent Trail.\nWashington Area Bicyclist Association: (WABA) Better bike lanes, better bike laws, better bicyclists--make the DC region a better place to bike.\nMap My Ride: Search bike maps in your area, track cycling workouts in our online bike training log and track your favorite biking routes.\nRails to Trails Conservancy, creating a nationwide network of trails from former rail lines and connecting corridors to build healthier places for healthier people.\nBikeWashington.org provides details about all the great recreational bicycle facilities around the Baltimore\/Washington metropolitan area.\nSheldon Brown's Bicycle Technical Info: This encyclopaedic web site contains a number of useful articles on bicycling and bicycles, plus links to other bicycle-related sites on the Internet.\n\"Safe Bicycling in Maryland\": A PDF of a free booklet distributed by the Maryland Department of Transportation, covering equipment, maintenance, locks, helmets, traffic rules, dogs, trails and dressing for weather.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New Zealand is unique in the world being the first developed country to provide mechanisms that enable private ownership of forest carbon within the Kyoto Protocol framework. There are currently two carbon forest schemes, the Permanent Forest Sink Initiative (PFSI), and post-1989 forestry under the NZ Emissions Trading Scheme. Under each mechanism owners of forests established after 1989 are awarded carbon credits for increases in forest carbon stocks that occur after 01 January 2008.\nThe NZ ETS forestry mechanism is suited for commercial timber forests however arguably it lacks key environmental attributes of permanence (i.e. the emissions removals are not temporary) and additionality (meaning that the activity would not have happened without carbon revenue) required by most international standards and buyers. This is because many ETS forests are business-as-usual timber plantations and there are no restrictions on clearfell harvesting or deforestation.\nThe Permanent Forest Sink Initiative is the 'gold standard' in carbon forestry addressing both permanence and additionality.\nPermanent Forest Sink Initiative registered forests are committed to long term sustainable management for the primary purpose of carbon sequestration. The forests must be maintained as continuous cover canopy forests for 99 years, and harvesting is only permitted on the basis of low intensity individual tree or small coupe removals. These restrictions are recorded in a covenant between the Crown and the landowner and registered against the land title, binding successive landowners. The covenant can be terminated after 50 years but termination triggers repayment of all the carbon credits issued to the forest.\nThe PFSI mechanism is favoured for projects where the principal driver is carbon sequestration to maximize and maintain biomass stocks. It is also favoured for programs with native tree species and for projects where maintenance of permanent forest cover is valued for environmental reasons such as enhancement of biodiversity, soil conservation, water quality improvement, and flood control.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Get Blackjack King for FREE!\nAfter being dealt your initial two cards, players can double the amount of their original bet. In this case you will receive one more card (but only one more card) from the dealer, and if you win you will receive twice the chips. Doubling down can be extremely beneficial because it gives players the opportunity to beat the dealer when he is at his weakest. Generally it is only a good idea to double down when your first two cards add up to 9, 10 or 11 because you have a great opportunity to end up with 19, 20, or 21.Check out the charts in the other sections to see exactly when to double down!\nHard hands in Blackjack are any hands that do not contain an Ace, and there are specific ways to play each one to ensure the highest chance of winning.\nAs a beginner it's best to follow these strategies, but as you gain experience you can make some judgment calls based on what cards have already been played and gut feelings.\nA soft hand is one that contains an Ace and another card. They are a bit more complicated to play than hard hands because an Ace has a value of 1 or 11.\nWhen you are dealt a pair in your opening two cards (face cards count as 10s), you can split them up. These two cards are then played independently as if they are two completely separate hands. Although it can be tempting, there are only certain hands that a player should split. Aces and 8s should be split no matter what because you turn a bad hand into two opportunities at great hands. Other pairs depend on what the dealer is showing. Splitting is similar to doubling down in the sense that it decreases the dealers edge and players should therefore be patient and wait until the perfect opportunities to split and double.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaucjh b/data_all_eng_slimpj/shuffled/split2/finalzzzaucjh
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index 0000000000000000000000000000000000000000..898a7c7702f2995ba64a1a9c4b2cab31071da23d
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@@ -0,0 +1,5 @@
+{"text":"I love and enjoy playing at weddings, they are such happy occasions. I have a wide variety of instruments and players available. The most popular being Cello and Violin, which are delightful together.\nI arrange all of my own music and have a wide array of choice from classical to contemporary, even irish music.\nThe most popular requests are Pachelbel's Canon in D, Handel's hornpipe from the Water Music and Handel's Finale from the Fireworks Suite. Bach's Brandenburg Concerto is a great entrance or exit piece. Also, Elizabethan Serenade by Binge is delightful.\nI have accompanied singers or soloists on piano. I can arrange any hymn as long as I have the music, or play the piano or organ if it is in a church.\nI do church weddings and garden weddings. I charge very reasonable rates. If the wedding is a fair distance away I ask for some traveling money also.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's party season and with it the inevitability of having to talk to people but fear not, there's no need to panic socialise, I have a free audiobook to help see you through \u2013 What To Talk About When There's Nothing To Talk About. Click here to listen and get your free download or hit play below. Now go forth and converse with aplomb and the requisite gusto.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yarn Type SALE 50% OFF! A classic yarn with 30% wool that comes in modern fashion shades. Click Dk's eye-catching colours make dramatic looking fashion knits to wear in the city for every day style. Will knit to any DK Yarn pattern. Click Dk's wool and acrylic blend is easy care and easy wear. Machine washable, wool cycle. 70% Acrylic, 30% Wool.\nLeaflet with designs for a Sweater using Sirdar Click DK.\nLeaflet with designs for a Long gloves, Long Snood, Fingerless Gloves and Gloves with Stripes using Sirdar Click DK.\nLeaflet with designs for a Hooded Cardigan using Sirdar Click DK.\nLeaflet with designs for 2 Sweaters using Sirdar Click DK.\nLeaflet with a design for a Waterfall Jacket using Sirdar Click DK.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"First Trust Bank, part of theAIB Group, is acommercial bankinNorthern Ireland. It forms part of one of theBig Fourbanks in Ireland. The bank was created in 1991 whenTSBNorthern Irelandmerged with the AIB Group's other interests. The bank can trace its existence back to1816with the founding of theBelfast Savings Bank. The bank is one of the four banks that issuesPound Sterlingbanknotesin Northern Ireland. Allied Irish Banks confirmed plans to sell off the bank in April 2010 as part of plans to raise capital. These plans were subsequently shelved and instead the bank announced investment plans starting in In the 2014 Northern Irelandmarket, the division operates under the trading name First Trust Bank from 30 full service branches throughout the region. The First Trust Bank Head Office is located inBelfast, together with the Business Services Centre. A full service is offered to business and personal customers, across the range of customer segments, including personal customers, small and medium sized enterprises, and the corporate sector. Specialist services, including mortgages, invoice discounting and asset finance are based in Belfast and delivered throughout the division (the credit\/debit card operations were split in 2007, with the Card Issuing business transferred to the AIB Head Office in Dublin in 2007, and the Card Acquiring (merchant) business becoming a joint venture with First Data Merchant Services in 2009). First Trust Independent Financial Services provides sales and advice on regulated products and services, including protection, investment and pension requirements to the whole of the division.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We sell only Azithromycin, the generic version of Zithromax*.\n*We do not sell Zithromax brand.\nGeneric for Zithromax* is indicatedto treat bacterial infections in many different parts of the body. Azithromycin (Generic for Zithromax*) is a macrolide antibiotic that interferes with the growth of bacterial cells. Generic for Zithromax* also treats sexually transmitted vaginal or urinary infections caused chlamydia.\nGeneric for Zithromax* is contraindicated in patients, who are hypersensitive to this drug, medications that belong to this class of drugs or had an allergic reaction to it in the past.\nGeneric for Zithromax* should not be taken by patients who have a history of the following ailment conditions.\nThere are a number of medications that interact with Generic for Zithromax*. It is important to take the doctor's counsel if you are taking any of the following.\nThe usual recommended dosage of Generic for Zithromax* for respiratory Diseases, Tonsillitis, Strep Throat, and Skin Infections is 500 milligrams in a single dose the first day. This is followed by 250 milligrams once daily for the next 4 days. For STD the usual dosage is a single 2 gm dose.\nSome of the most serious Generic for Zithromax* side Effects include blood in the stools, chest pain, dizziness, drowsiness, fatigue, gas, headache, heart palpitations, indigestion, itching, jaundice, kidney infection, light sensitivity and rashes.\nThis is not a complete list of all Side effects. Do concur with your doctor and follow his directions completely when you are taking Generic for Zithromax*.\nGeneric for Zithromax* has not been studied in pregnant women. However, it has not been shown to cause birth defects or other problems in animal studies.\nStudies do not indicate whether Generic Zithroxax passes into breast milk. Mothers who are taking this medicine and who wish to breast-feed should discuss this with their doctor.\nGeneric for Zithromax* has been tested in a limited number of children up to the age of 16. Do consult your doctor before using in pediatric patients.\nGeneric for Zithromax* has been tested in a limited number of elderly patients and has not been shown to cause different side effects or problems in older people than it does in younger adults.\nThe above information about Generic for Zithromax* serves as an information resource only and is not intended to substitute professional medical advice, examination, diagnosis and treatment. Always seek the counsel of your doctor, physician or pharmacist before starting any new treatment or making changes to existing therapy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaufgv b/data_all_eng_slimpj/shuffled/split2/finalzzzaufgv
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index 0000000000000000000000000000000000000000..65dc2152401b3d40e2f0b881b3071102a16c422d
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@@ -0,0 +1,5 @@
+{"text":"No stripping. No heavy-duty sanding. No undercoat. V33 woodcare helps you renovate wooden floors and stairs to let the natural beauty of the grain shine through. We have quality products to see you through every stage of your flooring renovations. From preparation products to eliminate grease and dirt to give you the best surface to start work on, to easy to use glue stripper that makes light work of the fiddly process of scraping away old glue. And then there's our one coat paints. Especially designed to glide on beautifully without the need for any intense prep work, our smooth finish paint won't scratch, chip or flake \u2013 no matter how many pairs of boots stomp across it.\nWe also have easy application floor varnish, designed for use on wooden and laminate floors. You can apply it direct, with no heavy-duty sanding or stripping beforehand. Once it's dry, it's resistant to everyday use \u2013 and it's fully compatible with underfloor heating systems too.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Originally Published: March 25, 2017 9:07 a.m.\nRecently, the City of Cottonwood elected to continue the Thunder Valley Rally. In making the decision the Council decided to move some of the events to Riverfront Park and keep others in Old Town.\nThe city will still pony up significant dollars in support of the event.\nDuring the work session, there was no mention of the birding festival that occurs in the spring of each year at Dead Horse State Park.\nIt is a resounding success by most parameters in filling hotel and motel spaces in Cottonwood and elsewhere as well as local dining establishments. It not only attracts local birding enthusiasts but also birding enthusiasts from beyond the Verde Valley for the multiple-day event. The event is not just in the Cottonwood area but transportation is provided far and wide to see birds in their natural setting.\nAll this is done without any financial support from the city. Yet, no mention of the Birding Festival was mentioned by any member of the City Council or staff as an example of a good event with attendance locally and distant, and no cost to the city.\nThere are other events that occur in Cottonwood that appear to be successful without city funding.\nTwo others that come to mind are the Verde Valley Fair at the Fairgrounds and the Arizona FlyWheelers event at the Fairgrounds. I'm sure the reader of this article could add several more successful events without cost to the city.\nIn addition, one should not forget the presence of Dead Horse State Park in Cottonwood. It is filled most of the spring time with spring breakers and travelers heading back to the northlands. If one is looking to have an event in Cottonwood, spring break might be a good time to have it.\nSpring campers and visitors to Cottonwood might appreciate a concert or two of local talent in a gazebo in Riverfront Park.\nAs soon as the waste water treatment plant is completed, good grass gets planted, and watered, a cement or portable wooden floor in front of the gazebo is added; one has a magic area for low key evening concerts.\nSo, after all that thinking, it seems that Cottonwood does all right in having events in their city without a burden on city finances, even if a Riverfront gazebo or dance floor is never built.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Understand whether an incident is reportable and how to report it.\nFind out what type of incidents must be reported.\nSites can only be disturbed to protect a person's health or safety, help someone who is injured or to make the site safe.\nOnline is the best option, but you can also report by fax or mail. You will need the reference number.\nYou are required to keep a record of the form for at least five years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Midway through a month of public meetings, the Chesapeake & Ohio Canal National Historical Park has withdrawn its proposal in begin charging entrance fees to the park outside of Washington, D.C.\nThis is despite a recent 10-percent budget cut and nearly 20-percent staff reduction, mainly at Great Falls Park. Existing entrance fees may still be increased.\nPark Superintendent Kevin Brandt said under the original proposal, runners and bikers who entered the towpath from D.C. would not have been charged.\nThe proposal would have established entrance fees at all entrances to the park in Maryland and at Fletcher's Cove, effective May 1 or later. Entrance fees have only been collected at Great Falls Tavern in Potomac. Those fees would have supported improvements to various park amenities, including parking lots and towpath surface maintenance. Brandt spoke at a public meeting in Bethesda Feb. 5.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For the first time in the history of the Multimedialnego Parku Fontann to hear music dedicated prepared by specially invited artist Jean-Michel Jarre.\nWe will hear a cross-section of his work, from the album \"Oxygene\", the legendary \"Equinoxe\" until after the last album \"Electronica Vol 1: The Time Machine\". There will also be works produced in cooperation with other stars of electronic music, such as \"Stardust\" Armin van Buuren or symphonic version of \"Acropolis\".\nVisually, the show will be a story inspired by one of the capital's most famous legends - of the Warsaw Mermaid.\nWarsaw shows in the Multimedia Fountain Park is a great warm-up for fans of Jean Michel Jarre before his autumn concert tour, promoting new album \"Electronica Vol 2: The Heart of Noise\", during which will visit also Poland.\nThe first shows the Multimedialnego Parku Fontann invite 1, 2 and 3 May at 21.30. Admission free.\nWhen is JM on here ? Early May, so?\nThe creator of the music for the new show in the Multimedialnym Parku Fontann on the Wis\u0142\u0105 was Jean Michel Jarre. French musician will visit Warsaw for the whole summer season. The premiere edition of the \"Warszawskich Syren\" on 1 May.\n\"The proposal to my music for multimedia project immediately gained my approval and I look forward to its implementation\" - admits the Frenchman.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaujih b/data_all_eng_slimpj/shuffled/split2/finalzzzaujih
new file mode 100644
index 0000000000000000000000000000000000000000..4880000772bd48a2aef1ab17fa322f9958a3fc4c
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Money in Marshall Islands: US dollar ($). ATMs are available on Majuro, but not on the other islands. Credit cards are accepted on Majuro and Ebeye, but again not on the other islands. Travellers cheques can be easily exchanged.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BAGHDAD \u2014 Abu Hussein is a 31-year- old Iraqi who works on a U.S. military base. He used to commute. But last year his life collapsed and now he lives at work.\nHis story is familiar. He quickly moved his family out of Iraq after his children's doctor and a neighbor, both Shiites, were killed in their Baghdad neighborhood on the same day. His family was denied residency in Jordan, so he moved them back to Iraq, but to a southern city that had no jobs.\nNow he sleeps on a cot in the base. He works day and night for two months at a stretch, and counts himself lucky. \"The base is the safest place in all of Baghdad,\" he said.\nLast week, an intense debate was taking place in America about how long U.S. troops would have to stay in Iraq to keep security. But in the view of many Iraqis, that is something Americans have never been able to offer. And the longer they stay, the less confidence Iraqis have that the Americans will be able to do so. The distrust has hardened over three and a half years, adding ever more distance between two worlds: That of the U.S. soldiers who are trying to do a job here, and of the Iraqis immersed in grim realities that the Americans try to navigate.\nPart of the problem is that the Americans cannot see for themselves how bad things get when they are not around.\nNo one knows this better than Iraqi workers on bases. Like stowaways from another world, they adopt a kind of American identity.\nThey take names like Joe and Ozzy. They learn how to swear and talk American slang.\nThese Iraqis know all too well what life is like outside. Abu Hussein tried for years to get them to protect Iraqis coming in and out of the base by shielding the civilian parking lot to hide the license plates from the eyes of Iraqis who hate those who work for Americans. Blast walls were erected only recently. \"Everybody knows I'm going to leave from the gate,\" Abu Hussein said, his face tight with worry. \"I have no weapon. I am isolated.\"\nA U.S. soldier, on the other hand, \"is in a Humvee with armor and weapons.\"\nIndeed, the divide is so profound that an American private had to dress in full battle armor this month to walk 20 paces outside the gate to an Iraqi employee parking lot in order to pick up a book.\nWhen Americans move through Iraq, they do so like a giant ship cutting through a thick and treacherous sea. They move slowly, displacing the harsh reality on both sides, carving out a trough of safety around them. But after they pass, reality closes back in, in all its sucking, swirling fury.\nThat reality is terrifying because much of Iraq is a place without rules or laws, in which armed gangs, sometimes dressed as police officers, can come into any house and do exactly as they please.\nThis broad challenge for the Americans - making security last past the moment the Humvee on patrol rolls away from the house - could be seen last week in even the smallest ways in Ur, a sliver of residential blocks just north of Sadr City, the impoverished Shiite district where unruly militias are strong.\nThe Americans began sweeping in late in August, going house to house looking for weapons. It had been more than a year since they had patrolled Ur, and residents were surprised to see them.\nThe neighborhood was mostly middle class and heavily Shiite, but while the Americans were gone its northeast corner had become the site of brutal executions: Authorities found 90 bodies there in the heat of August, most of them victims of Shiite death squads that had driven into the area from Sadr City.\n\"They killed openly, they did not hide,\" said a worker in eastern Ur who said he had witnessed as many as 40 daytime executions, over several months, near a shop where he works. An Iraqi Army checkpoint was less than a mile away. After the sweeps, fewer bodies were found, but the hard part - keeping the area clear of killings - had only just begun.\nA deep fear had settled in the neighborhood's northeastern edge, the area closest to the militia stronghold.\nAt dusk on a recent Sunday, Sergeant Andrew Pokora stood in a courtyard with a nervous Shiite woman. Her husband had spoken to U.S. soldiers before and they had found the conversation useful. It was the third time Americans had come to the house in recent weeks.\n\"What do you need from him?\" asked the wife, her voice tense.\n\"Every day Americans are coming here. The neighbors are asking why.\"\nThe neighbors were suspicious because the family had moved only recently from a hard-line Sunni neighborhood, Ameriya, from which Shiites were being driven out. They had not yet proved to their neighbors that they could be trusted, even though a sticker portraying a Shiite cleric was stuck to their battered white door.\nOutside that door spread a vast expanse of dirt fields and garbage where gangs of men who like the cleric move. They call themselves the Mahdi Army, after the Shiite messiah, and are known as brutal killers.\n\"Danger for us, for me, for my husband!\" the woman told the Americans, standing firm as other family members wandered out into the small darkening courtyard. \"They will say we are your client.\"\nPokora, a bright, young soldier from Connecticut, relented. He wrote down the man's telephone number, thinking it best to call the husband later.\nEven in safer areas, engaging Iraqis on the topic of their lives is difficult. Earlier that afternoon, the sergeant had sunk into a spongy couch in the living room of a housewife in Ur, trying to gain her confidence, maybe get a tip. \"Do you feel safe when you see the police?\" he asked, through a teenage interpreter.\nShe replied in a quiet voice, nodding slightly. Three tiny children with saucer eyes stared.\n\"She says she feels happy when she sees them,\" the translator told him.\n\"Ask her how she feels when American troops come through her neighborhood.\"\nThe Iraqi and American worlds were not always quite so separate. In 2003, Iraqis lined up at the gate of the Rustimiyah base waiting their turn to ask for jobs. Now the road is empty, with discarded plastic bags swirling in the wind.\nToday, the approach to the base is empty and ominous: A two-story watch tower peers down at anyone seeking entry, while guards shout barely intelligible words in English. Guns are pointed. Nerves are keyed up.\nEarlier this month, four teenagers who had worked as cleaners in Rustimiyah were killed by gunmen who followed them to their homes in Nahariya, a suburb near the base, a worker who knew them said. A translator was killed the same week.\nOn a recent Friday morning, an Iraqi man wearing glasses stood alone near the highway outside the base, staring at approaching cars, as if he was remembering license plates.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of our favorite Sunday dinners is this Crock Pot Beef Stew. Hearty pieces of beef and vegetables slow cooked in gravy. Serve with buttered rolls and you'll have one delicious meal!\nI can't think of anything better than eating a warm bowl of Crock Pot Beef Stew on a chilly day. Too bad we're still soaring in the high 90's here, but before we know it the weather will be cooling down and we'll be enjoying this amazing beef stew again. It's the perfect Sunday meal to make for your loved ones. I've been making this for years and love to serve it with our favorite rolls. It is beyond delicious!\nA delicious beef stew loaded with potatoes and carrots slow cooked in the crock pot.\nIn a large plastic bag, combine the dry onion soup mix and paprika. Add in meat and shake the bag to coat meat with the seasonings. (You may have to do that step in two batches, for all the meat to fit) Once meat is coated, spread out evenly into a greased 7 quart crock pot.\nTop meat evenly with diced potatoes, carrots and onion.\nMix the cream of celery soup and ketchup in a small bowl. Pour mixture over the meat and vegetables.\nCover crock pot with lid. Cook on low heat for 8 hours or until meat is cooked through and vegetables are tender. Stir to combine everything then serve.\nThis looks perfect for those cooler nights!\nyou don't add water or any broth right?\nI'm cooking it right now and see the cream of celery soup just laying on top. I want to stir it so bad! Am I supposed to wait to stir when it's done?\nHopefully you waited. Creamed soups need time to break down. If it stayed in a curdled stage, next time temp the soup. Add about one cup of the hot liquid to the condensed soup. Wisk until incorporated then add to the stew. Stir lightly.\nI didn't add water to cream of celery soup the first time and family said, \"it's good, but has no broth\". So second attempt was make up cream of celery soup up per direction on can and everyone liked it better.\nThis was as easy and delish as recipe looked and read. I was concerned about not adding liquid. Yet, the reviews calmed my concerns.\nIt was excellent and will make it again and share this recipe with those I know!\nWhat can we use to replace the ketchup which has tons of sugar in it?\nLow sugar ketchup or tomato soup.\nI used a cup of jarred marinara. (Olive oil, roasted garlic and basil). It was the only tomato-based pantry item I had on hand and it tasted delicious. The only recommendation: if you're crockpot is newer or gets super hot, go ahead and stir at the 6 hour mark so the meet does t dry out at the bottom. You can turn to warm shortly after. My warm is equivalent to low on most.\nI use a can of Rotel tomatoes.\nThis looks way good, but I was just wondering\u2026..does it need a little water to make the liquid and how salty is it?\nThanks so much. My husband saw the picture and said that I needed to make it.\nIs the celery soup the condensed kind? Do you add water to it then?\nCould I use cream of mushroom soup instead of cream of celery?\nI'm sure you could. Not sure how the flavor would differ but I imagine it would still be great!\nIt's not \"soup\" \u2013 it's stew.\nI didn't read all the comments before making and I added the water with the soup? Did I just ruin dinner \ud83d\ude41 ?\nI made this it was very good but it was to salty. I might just put 1 pk of onion soup in next time. Thank you for your recipe.\nI am a new subscriber to your blog. I made your Crock Pot Beef Stew this weekend\u2026it was the best I have ever made!!! My husband and our boys loved it. I just shared this recipe with my sisters-in-law. I can't wait to see what other crock pot meals you post.\nI am not a fan of ketchup, I can't even stand the smell. How much ketchup taste comes out of this, as well as smell? Or can you suggest something to substitute?\nToni, I don't taste ketchup with this stew, but if you hate it that much you might. You could try tomato soup. I think it would work.\nThe recipe instructions say to cook it on low heat for 8 hours. If you cook it on High I would imagine it would take 4 to 6 hours. All crock pots heat differently.\nJust made this recipe. Although nice and hearty, I think the ketchup taste is too strong. I will definitely make it again\u2026.maybe next time I'll take it easy of the ketchup and instead add A-1 and\/or Worchestershire.\nI agree with you. I feel that its way too much ketchup. The rest was great though!\nI just made this (it is amazing and easy btw!) and if you HATE ketchup, I can definitely taste ketchup in the end result.. Maybe try bbq sauce instead?\nI agree about the ketchup taste. I personally liked the taste, but all my husband could taste was ketchup. Trying it next week with a can of Campbell's Golden Mushroom soup in place of the ketchup. Will determine if I should add ketchup toward the end of cooking, possibly 1\/4 cup if need be.\nThen don't add it! I just don't put any ketchup in it at all. I use it the first time and like you I'm not a fan of ketchup. And I could t even eat it because the ketchup just took over.\nI was wondering if cooking the veggies as long as the meat, don't they get mushy?\nNot at all. Actually, my carrots still had a little bite to them, which I like, and the potatoes were soft and perfect!\nwrote recpie down i am making this for the fall.\nHi Holly, Thanks so much for always answering back to our comments. I have a question for you. Do you have any yummy recipes for wheat and\/or white bread? Thanks so much!\nI have got to make this!! Just wanted to double check on the cup of ketchup. 1 cup, is that correct? Just seems like a lot. Just wanted to be certain. \ud83d\ude42 It looks delicious!!\nAlso, about how many does this feed. We are a family of 6.\nIt serves around 8 one cup servings.\nCan this be made in a Dutch oven? If so, about how long would I put it in the oven for?\nYou just coat it with the meat in the bag. It gives the stew a great flavor! You don't need to add any extra in the stew.\ninstead of peeled white potatoes could I use unpeeled red potatoes?\nDo you think this meal could work as a freezer meal? Looks delicious!\nI just started this in the crock pot, hope it's as good as it looks in the picture.\nBest beef stew ever!! Loved the recipe!! It was so simple to!!\nJust put it in the crockpot now!! Looking forward to dinner.\nThis looks amazing! How well does this recipe double?\nI made this stew this weekend, and my husband and my sons loved it,. I felt it was a little salty. When I was just re-reading the recipe. It says \" 2 (1 oz) packages dry onion soup mix.\" I have never seen 1 oz packages of onion soup mix\u2026.so I am confused, cause I used 2 normal packages of onion soup mix, did I use too much??\nI made this today \u2013 It was delicious! I made a few changes. I cut up a chuck pot roast into cubed stew size meat. I added 2 cups of frozen peas, an extra onion, about 3\/4 cup of chopped celery, and salt and freshly ground pepper. I also cooked it in my crockpot on high instead of low. (For 4 1\/2 hours.) After around 3 hours, I stirred it. So very good! Definitely a keeper! Thank you Holly for your great recipe!\nI was just wondering because i am on crunch time and i really want to have this tonight, if i put this on high for 4 hours, will it turn out just as good?\nI can't say it will for sure because all crock pots heat differently\u2026 It may need 5 to 6 hours on high heat.\nLooks yummy!! Do you know if you can freeze this ahead of time before cooking? Like put it all in a bag and put in a freezer?\nHi! Do you know if this could be frozen beforehand, maybe without the liquid, and then dumped in the crockpot in the morning with the liquid?\nI cooked this recipe a couple days ago for me and my roommates! It was delicious and so easy. One thing I did differently was brown the meat once I coated it with the onion soup mix to get that flavor to penetrate better, and it turned out sooo good. Thanks for posting this!\nI got the cream of celery soup that requires you to add a cup of water\u2026 Do i do this or just use it without the water?\nDo you know if it tastes good if you substitute tomato soup for the ketchup as a bit of a healthier option?\nWe do not like cream of celery soup, do u suggest any alternative?\nI just turned the crockpot on .. Feeling hesitate as I have never done stew with ketchup!! Can't wait to see how it turns!\nJust put mine in the crockpot!!! As I'm re reading the directions I realized I forgot to peel the potatoes\u2026oops. I scrubbed them really well under cold water though. I hope it still tastes good!! Too early haven't had coffee yet..haha..\nGive it time, it will turn into a fabulous gravy!\nThat is such a fun idea!!\nI'm currently in the process of cooking this. I'll let you know how it goes tomorrow. Can't wait to try it!\nI made this stew on Saturday. It was beyond fantastic!! Not only did I make a delicious meal for my family, but, I got to go out and have fun while it was in the slow cooker. Couldn't have been easier or tastier! Followed the recipe to a T \u2013 wouldn't change a thing. Thanks so much for sharing.\nHi Holly, I currently have this in the crockpot (yay!) and was wondering if you knew the nutrition info? Thanks so much for sharing \u2013 I've pinned a ton of your recipes and can't wait to try them all!\nThis recipe in a pressure cooker versus a slow cooker?\nDo you think I could make this on the stove in a large pot instead of in a slow cooker or crockpot? Making this tomorrow for dinner.\nKetchup pretty much dominated the flavor did I do something wrong.\nI'd like to make this but I'm wondering what makes the liquid in the stew Do you add water?\nNo, dont add water. the soup and ketchup turn into a thin gravy. It is delicious!\nJust to clarify, you DON'T put all of the dried soup mixes in right? You only put in whatever coats the meat and discard the rest? Thanks!\nHas anyone frozen this and thawed after?\nI have and it was just as good after freezing.\nDo you add any water or beef broth? There isn't any liquid??\nThis recipe was REALLY good. I added 3 chopped up celery stalks. It was easy and simple to make. Perfect comfort food. This will definitely be added into my arsenal of Crock Pot recipes for the winter.\nWhat is the green herb in the picture with this posting? Did you add parlsle?\nI recently made this it turned out pretty good . But can I use something other then the ketchup ? I'm not big on it and the ketchup flavor really stuck out to me .\nKatie, I would imagine you could use a 10 oz. can of tomato soup but not sure on the results. I love the flavor of this stew and I don't taste ketchup, so I've never had to substitute it before.\nThis recipe is a true winner\u2026.delicious!\nYou use the can of condensed cream of celery. Just dump it right in. No need to heat first or add water.\nJust wondering if you cook the meat before putting it into the crockpot?\nThis was delicious! I substituted the celery soup for cream of potato because that was what I had on hand. Hubby, kids, and I all agreed this recipe is a keeper. Thank you for sharing!\nI just made this this morning I'm excited to try it however I forgot to grease the crockpot will this make a big difference??\nI don't think it will be a problem at all.\nNext time use Slow Cooker liners, found next to the Ziplock bags in the grocery stores.\nplan on making a version of this tonight\u2026 I am going to put it on high though seeing I am staring it in the afternoon instead of being able to let is sit all day on low. I am also going to use cream of mushroom and nix the ketchup.. I might add crushed tomato because I know I have that and not paste.. not sure yet. I will let you know how it come out.\nI made this tonight and it was wonderful ! I'm not a fan of ketsup so I used 1 cup of tomatoe sauce. It was perfect !! Thank you for the delicious recipe !!\nI am following your choice w tom sauce and holding my breath, it is in the crock on high, for 4.5 hours maybe. Hope it is good!\nI want to make this tomorrow. Could I mix up everything tonight put in fridge & add potatoes in the morning?\nMade this tonight and loved it!!!\nI made this this weekend and we loved it! I subbed the ketchup for a can of tomato soup and the flavor was wonderful! Next time I think I will cut the meat in half and maybe add celery since I love soups with heavy veggies! Thanks for sharing!\nThought it very odd there was no liquid so I added a cup of water. Prefer my stew with plenty of broth.\nI am currently making this beef stew and after reading the comments about the ketchup, is there any thing I can add to the stew in the event the ketchup it's too overpowering?\nSo after re-reading the instructions, I realized I didn't grease the inside of the pot beforehand Hope that won't make such a difference! Will post my opinion after dinner!\nBest stew I have ever had!! Love this so much. Opposed to the others who didn't like ketchup I didn't taste it much at all but am a big ketchup fan thanks for a great dinner even for a college kid!\nJust a quick question.is the cream of celery soup and ketchup the only thing for the juices? . Or am i supposed to add water to?\nJust add tomato sauce instead of ketchup.\nMy family loves this recipe! I am in need of the nutritional facts if possible please!! Thank you in advance.\nI loved this stew!! One of the best! I used about 1\/3rd cup of catsup. Just wonderful!!!\nQuestion \u2013 what could I substitute the cream of celery soup with, since my husband is lactose intolerant?\nI made this for dinner tonight working with what I had on hand so I had less carrots, one less potato, and half the meat, however it turned out 10000000000% DE-FREAKIN-LICIOUS!!!! I did change a few things though, because my two year old was to be eating it as well, I chopped the baby carrots into thirds, I also only used 1\/3 of the ketchup since some of the comments suggested it was too much, added a splash of worcestershire and a splash of ACV! It is soo good!!!! Thank you for sharing the recipe!!!\nI wanted a more beefy flavor so I used 1\/2 cup ketchup and added 2 beef bouillon cubes. very yummy. My entire family enjoys this and many other recipes you have shared.\nThis worked perfect, way less ketchup taste and makes for an awesome beef stew recipe, I added a bit of beef broth too. Love beefy tasting beef stew!\nDo you really need a 7 qt slow cooker for this? I have a 5 or 6 qt and I would think that would be. If enough since it only makes 8 cups.\nI made this tonight and it was sooooo delicious! I added a can of diced tomatoes and peas! Soooo yummy! Thanks for the awesome dinner!\nThe ketchup made it to sweet and taste a lot like ketchup. I added beef broth to simmer down the ketchup taste and some black pepper. Helped a lot and taste better. I also added some parsley towards the end. Taste a lot better.\nI really like the idea of using the dry onion soup mixture as a rub for the meat. I will def have to try this.\n***Highly Recommended*** I made this last night with positive comments all around, actually great comments and I have a family of harsh critics. I read the reviews and decided to reduce the tomato sauce\/ketchup to 1\/2 cup, I also added a couple of teaspoons of crushed garlic to the soup mix before adding to the slow cooker, plus 1 cup of frozen peas about an hour before serving. I used a small slow cooker with beef that must have come from an old cow, tough as! I ended up cooking it for 6 hours on high, it took that long for the contents to heat up and meat to become tender. Thank you for such a wonder slow cooker recipe that will be made over and over again.\nI just made this. I was worried about the comments saying it would taste to much like ketchup, so I did 1\/2 cup instead. It was absolutely delicious and all my family had seconds. I will be making again. I also used canned whole potatoes to cut on prep time. Thanks!!!\nHi holly I wold love to make ur stew but sadly we can't get cream of celery soup in nz do u know what I can use instead that will still be yum?\nI use this recipe to make my own \u2013 the canned stuff has a lot of sodium.\nSo you say to cook for 8 hours on low, can you cook it for 4 hours on high\u2026?\nDo you think tomato past or tomato soup could replace the ketchup in this recipe?\nThis is THE recipe I've wanted! Absolutely delicious and super easy. I messed up by grabbing a can of cream of celery and a can of cream of chicken soup instead of 2 cream of celery soups but they worked fine. Deeeeeelish!!!\nCan I use a Dutch oven for this?\nMade this for dinner and it definitely needed to have a little water added to it.\nDo you have a recommendation to add to it to help dilute the ketchup taste a bit? I have about two hours left on mine and it smells very ketchup-y.\nMay be a stupid question but you say to layer all the ingredients with the soup mix on top. Can I mix it or do I just leave it with the meat at the bottom???\nWhat can I substitute the cream of celery soup with?\nI thought this was great. However, instead of ketchup, I used a can of tomato soup and a half of a cup of low sodium beef broth. I will definitely be making this again!\nMade this for dinner tonight and it was a hit!! I didn't change a thing and it was so good. The whole family loved it! Thank you for sharing. Definetly will be making again!!!\nI put the ingredients together tonight. Gonna put it on in the morning and timer is set.\nJust put this on! It's really cool and rainy here in my part of Tennessee today. I've had this recipe on hand since September and thought today was the perfect day to try it! Can't wait to taste it. Got it in a little late, though so I'm having to cook it on high for about 5-6 hrs! Luckily I'm home to keep a good eye on it. I'll let you know what the family thinks!\nMine is in the crock pot now! Can't wait to taste it! (Hope the Katsup doesn't over ride the meaty hearty taste!). Added some celery and celery salt just in case! Loos so good Holly!\nmine is looking to be too thick. can water or broth be added to cut the thickness?\nI followed the instructions exactly and this stew was so delicious! I was really skeptical about the ketchup but all the flavors cook down and meld together. I cooked the stew for 7 hours on low and then the last hour on high because at the 7 hour mark my potatoes weren't cooked as much as I would like but after an hour on high they were perfect!\nWe finally had some cool weather here in California so I made the beef stew last night for my adult son and his friend. They RAVED about it! None of us had a problem with the ketchup issue but we all like ketchup. I followed the recipe exactly except I added an entire bunch of celery cuz we love it. I have a 7 quart slow cooker and there was plenty of room. It was wonderful and the recipe has already been transcribed into my \"keeper list\". Thanks for a great recipe! We're having leftovers tonight!!! Yay!\nThis looks absolutely delicious!:) I was just wondering how it would work to cut the recipe in half? It's only my husband and I so that would be a lot of leftovers!!\nWe loved this stew. On of the best I've ever made. Only had cream of mushroom soup. Did add and extra pound of beef, 2 bay leaves, a tsp. each of thyme and herbs de Provence, 1\/3 c red wine & 3 cloves of minced fresh garlic. Will try with cream of celery next time.\nI started making this and then read the comments about it tasting too much like ketchup and got a little worried. So I tasted it and it was a tad too much for my liking but, I just added 1 can of beef broth and a bay leaf. It was amazing \ud83d\ude09 Thank you for sharing this recipe!\n11\/14\/16. I made this tonight in the large oval size crockpot. I followed the recipe to a T! My carrots weren't cooked all the way, and the ketchup taste was too heavy. My husband said it reminded him of tomato soup. After we ate the first two bowls. I added a can of beef broth as Christina said and turned it back on for two more hours. Going to give it a try. What can it hurt? Will get back to you and let you know if it helped any.\nI'm so glad I read the comments before making the stew. My husband and oldest son HATES ketchup. So, I added 2 tablespoons of tomato paste instead of ketchup. I also threw in a bag of frozen green beans. This recipe was awesome, my family loved it after a long day on the field of playing baseball. Thanks so much!\nI just made this amazingly delicious beef stew tonight. I cooked it on high for 5 hours and it's perfect. I made it with 50% less sugar and sodium ketchup. For those of you that don't like ketchup, it DOES NOT taste like ketchup. It's a wonderful gravy. To the cream of celery and ketchup mix, I add some minced garlic. When everything was done cooking I dumped a can of drained peas in and this is absolutely perfect. Read all of her directions and follow it to a \"T\". The directions are very simple enough but it seems like folks are asking questions in the comments that she already explains in the directions. Just read. \ud83d\ude42 Absolutely easy and delicious! Thank you, Holly!\nMade it for Sunday dinner\u2026..very flavorful and tasty!!! Will make again!\nThis is in the winner category, everyone had seconds. I did use tomato soup instead of ketchup and added mushrooms. Thanks.\nLooks great and sounds simple! I have a 4-quart crockpot\u2026.is this too small for this recipe? If I cut it in half, how long would you recommend cooking it for?\nThis stew is perfect! was a little nervous after reading the comments about the ketchup but you cant taste it at all.\nDefinitely had a strong ketchup\/tomato-ey taste. Added beef stock and it cut down on the taste but not enough. Would definitely be good without the ketchup, and maybe some sory of beef stock or gravy in its place. Onion soup made the beef delicious!\nUsed the homemade equivalent of two cans of cream soup + 3 tbsp. tomato paste + 1 tbsp. A1 steak sauce and a splash of Worcestershire sauce for the gravy. Nothing against canned soups \u2014 I just forgot to pick any up at the grocery store. Thank you for sharing! We're going overnight with this recipe and delivering tomorrow to a friend who's recovering from surgery. Smells delicious\u2026visions of beef stew no doubt will be dancing thru our heads this eve.\nAm so excited and nervous am making this stew, but I forgot onion dry spoup mix instead I got d Ranch Dip Mix?!!! I hope it turns out?!\nMade this last night. Only used half cup of ketchup and it turned out really good. I also just tossed in a bowl instead of a bag and it was easier. I started it late so cooked on high for 4-5 hours and it made the meat a little too done. So next time I'll cook on high for only 3-4 hours. Super easy and perfect for busy weekdays.\nI love love love this recipe!!! Made it tonight and it was perfect!!!\nIn crockpot now, anxious to try it. Thank you.\nI just wanted to know if you can taste the ketchup? Feel like my sister would have a cow & not want to eat it if you can. & also is it possible to double this recipe?\nThank you so much for sharing this. I made this tonight and it was delicious. my son that is very picky even liked it.\nPinned this last week and now have it in my crock pot for tonight. The weather has been Hot here in NW GEORGIA but tonight it is supposed to be RAINY and temps dropping so I thought this would be Great for a Rainy Night in Georgia!\nI currently am on hour 7 of this stew in the crockpot!! It smells absolutely delicious! I am dying to take the lid of to stir but am hesitant please let me know if I can stir it or not!! Thank you!\nWhat is the calorie count for one serving, or one cup of the stew?\nTried this tonight. It was absolutely incredible. Like everyone else, I was dying to stir it but ignored the urge. So glad I did. It broke down into a nice gravy. Yum!\nCan I add peas and corn in a can to this?\nI was thinking of using V8 juice in place of the ketchup. Or half ketchup and half V8. Do you think that might work out alright?\nOur kids celebrate Christmas with their in-laws and we come together for a giant relaxing time later. Today the wife and I celebrated Christmas with the beef stew. I suggest that you plan to make it twice. The first time, follow all instructions exactly and remember that the final results may be hard for you to predict. It's wonderful, and perfect for us, but save the changes for the second time. The condensed cream of celery soup was genius. Just right thickness for a stew. Thank you.\nOMG. It is -29 degrees tonight for New Year's Eve and I've had this in the crock pot all day. This was FABULOUS. The only thing I did different was add a splash of worchestshire and a beef bullion cube to it. Absolutely amazing and warmed our bellies on this freezing night. Thank you!!!\nI have this cooking in my slow cooker right now! I substituted tomato sauce for the ketchup based off of reviews and me not loving ketchup! I can't wait to try this!\nI'm trying this recipe right now! It looked so easy and it really was! I didn't have baby carrots so I used chopped regular carrots and I don't like potatoes so I substituted turnips (mama's secret to get me to eat stew) . Hope it comes out good!\nI made this exactly how the recipe states. The only thing I did different was cook on high for 4 hours because I started at 1 and didn't have time to leave on low for 8 hours. It came out so incredibly good!! Don't change anything! I will make this again!\nHi! I have made this recipe a few times and love it! I am wondering, have you have made it in an instant pot yet? How long do you think it should cook in an instant pot? Thank you!\nIt definitely is not a healthy stew but sure is delicious!\nAfter reading all of the comments I did 1.5 packets of onion soup mix, 1\/2 cup of ketchup and added 2 dashes of Worcestershire sauce on top once I assembled everything in the crockpot. Everyone loved it!!!! It was delicious!!! This will definitely be a staple in my house!\nYay! Glad it was a hit for you too, Courtney. We just love it!\nI make this every week! It's so good and easy to make. The only difference is I only use half cup of ketchup. Perfect for cooler days!\nA little too ketchupy, would definitely try with tomato soup when I make it again.\nI'm just getting ready to make this but realized I only got one can of cream of celery soup. I can't get to the store to get another one do you have any suggestions what I can use for the second can of soup?\nAny other cream of soup would work! Chicken, mushroom, etc. If you don't have any of those you can still make the recipe, the sauce just won't be as thick. Hope this helps!\nMade this today for my men's group meeting tonight. Made it straight by recipe but added crushed red pepper for a little heat \u2013 stew was amazing, none left!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Proverbs 27:17 says; Iron sharpens iron, so a man sharpens the countenance of his friend. By fellowshiping on a regular basis, the Bible records that we will indeed keep each other sharp, encouraged and accountable. Meeting and ministering to the needs of the women of City Church is our goal. We believe everyone has a story and it needs to be heard. The next meeting will be held at the home of Tressa Myers located at 153 SW Monaco Way Lake City, Florida from 10:00am-12:00pm. Please bring a brunch item of your choice to share.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Staff | Weathers Motors Inc.\n25 years of service at Weathers Motors.\nI came to Weathers Motors September 4th, 2012. I wanted to work for Weathers Motors because of their excellent reputation for customer service and quality vehicles. I have been selling automobiles since 1983. I worked at Bryn Mawr Chrysler from March of 1983 until 1994 and then at Videon Chrysler from September of 1994 until June of 2012. Weathers Motors is a perfect fit with my belief in quality customer service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Valerie Greene is a unique relationship mentor who helps women inspire their man to fulfill their deepest needs and desires. A highly successful alternative to relationship therapy, Valerie helps women and couples create a secure emotional connection--not just problem-solving or communication skills. Valerie has been coaching for over 12 years and holds numerous certifications in coaching, NLP, and emotional healing modalities.\nHow to stop fighting and fall in love again.\nGetting clarity on your feelings and relationship desires.\nWhy getting in touch with your feelings is key to a successful relationship.\nHow to realize your relationship dreams and create the changes needed to obtain them.\nCreating the can-do attitude necessary to get in touch with your feelings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Element Smash is a mix between nintendo's megahit franchise Super Smash Bros and the good old Rock Paper Scissors, with an elemental theme. Choose your element every few seconds and try to kick your opponent out of the playfield.\nCreated by @devMidgard & @sodap_ ~ Be sure to follow us if you like our work!\nI enjoyed the idea of this game of using pokemon + smash bros mechanics. I just think you could increment some more fight moves in the future for this game, to make the fighting mechanic more diverse and less simple. But this was a fun game in this jam and get the theme very well and in a creative way!\nI got this sort of working in firefox, except most of the assets are not rendering (just black).\nCool combo of Joust, Super Smash Bros. and rock-paper-scissors! I love the graphics, really polished and having a wicked GBA vibe to me. I don't know whether it was the idea, but for some reason it reminded me a lot of Mega Man Battle Network (a very good thing :D). I think that the controls were tight and the addition of local multiplayer was a really stellar touch. I like the sound effects a lot too. Only complaint I had was that a bad rock-paper-scissors choice can leave you in the lurch pretty hard. It would be cool if a theoretical future iteration of this game had something that let you control both your shape and your opponent's shape between changes. One way I thought of might be to have items for each shape (green red, etc.) that you could consume to become that shape, or throw at your enemy, and if you hit change THEM to that shape instead. Overall and addictive, fun and hilarious to play with a friend, and overall great entry! Awesome work, hope you do great in the compo. Thanks for your submission! Had a blast playing.\nGraphics were so good! It wasn't well polished, like I was able to spawncamp for no-loss and jump without landing at all. But it was quite fun and new :) Never played smash bros though.\nPretty good smash bros style fighting game with rock-paper-scissors style strategy.\nHad a bit of difficulty hitting my opponents though, I'd usually end up overlapping them and they would pummel me off the arena. Maybe you should add some collision that ensures that fighters cannot overlap.\nPretty neat concept and good retro-visuals. Controls are a bit clunky but it's playable.\nImpressive art and good timing based gameplay. it was not clear if water defeat fire, fire defeat tree... at least not for me. Congratulations!\nNice graphics and audio, gameplay is kinda clunky but realy good idea.\nThe mix between controls and audiovisual feedback have a really good feel.\nGame design wise I think the elemental advantage is too strong and devolves into the player on the losing end turtling. BTW, you can infinite jump while under the air platform making the turtling even easier.\nAnyways, for a jam game this is remarkable.\nThe game has a good potential. Specially if it goes multiplayer. The graphics and the sounds match very well. But the game gets a little bit boring after a while, it could have some special abilities so you could stand a chance when facing your weakness element.\nNice graphics and soundtrack; really great feedback to the actions. Loved the use of color, and the particles in the background. The mechanics were simple, and the platforming was pretty well-done, if kinda floaty (I just spammed attack and floated for a while as the AI kept jumping into me). I didn't notice much difference at all between elemental types, and seemed to knock the opponent the same distance most of the time. It'd have been nice to have more variety of actions; for a LD, less time spent on graphics and more spent on creating gameplay goes a long way!\nNeat graphics! Really fun gameplay. Couldn't feel the theme in this game though, and the framerate was pretty slow for me.\nAlso the CPU's knockback was always insane compared to mine, so I would fall off the arena way more often.\nDivertido, bonito, solo le falta algun tipo de golpe m\u00e1s y perfecto.\nMuy buen juego. Me parece muy buena idea lo de los elementos, darle una vuelta al tipico piedra papel tijera.\nCon m\u00e1s movimientos puede ser algo muy interesante.\nI didn't seem to have any advantage when I had a winning element, the only difference I saw was when the AI had a winning element they could knock me super fast horizontally. So it just boiled down to changing character every few seconds, and fighting super smash with a single move.\nI enjoyed the art though, and the music was good.\nCool graphics and audio and also great game feel in general.\nI liked the simple but effective mechanic but in the end the most important thing is luck.\nFighting games are crazy hard to make for a ludum dare. Nice graphics.\nMechanic idea is good, it is a nice take on the theme of the shapeshifting. I don't understand how you manage to kick the enemy out of the map, though, but you have a good base to work on. Keep it on!\nThat's a great little fighting game with really good graphics!\nBe carefull, sometimes characters are stuck upon shapeshift (the fire one I think).\nI love the graphics! Great color palette and is very easy on the eye with the bloom-effects. Well done!\nLove it. Obviously would like to see more depth in the attacks you can do and the stage you fight on, but a great idea. I like the short rounds and the constant changing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I also starting addressing Christmas cards. Yes, I know it isn't even the first of November. I just dread addressing them. I know they are an expense and the postage keeps going up, but most of the people we send them to are not online. For some they are our annual catch up time. Anyway, I feel better having started them. I found a couple of boxes of new cards at thrift stores this year for little to nothing. I feel that helps with the cost. I have been crocheting small ornaments to put in each card. The ornaments are basically free other than my work. I am using crochet thread I already had. I will need to iron and starch them, but I have enjoyed making them so hopefully people will enjoy receiving them.\nOther than a few items for DH for Christmas, I have the bulk of any shopping finished. We do give tips to the gals who cut our hair. I buy a few gifts for friends, but supplement them with either things I've canned or made so it keeps the cost down. I enjoy doing this sort of thing and hopefully the receivers enjoy getting them. We have a Christmas club so whatever we spend, the money is there so we don't go into debt. I just can't fathom going deep into debt for Christmas. I like being generous, but I don't think I could handle seeing a big debt.\nIt's a cool, dreary day here. Just the right kind to make me want to clean so our home feels cozy.\nWow you're ahead of the game with preparing for Christmas, love it.\nSounds like a nice cozy day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Unstable fissures in the earth, unaddressed fractures in the heart, and unattended factions within society are all of a kind.\nYou can't stop lava flows, though people have tried. That's the takeaway from one news source covering the devastating eruption of the Kilauea volcano on the Big Island of Hawaii.\nNature explodes when pressure builds deep within the earth with nowhere to go. It blows its top. Molten lava spreads across the landscape doing damage, destroying homes and cars, cooling eventually to leave raggedy black rock atop smooth roads and manicured gardens. Sulfuric dioxide gases spread airborne, killing plants and causing respiratory failure in humans.\nVivid pictures of lava flows in and around Kilauea Estates provoke thoughts of how much damage happens in families, communities and countries when pressure builds and anger brews deep within the recesses of human emotions.\nA child who is abused or neglected carries the legacy of being violated into adulthood. The brokenness doesn't heal on its own. Only patient therapy from trained counselors and acceptance of the spiritual truth that the Divine Parent is a loving and dependable presence will calm the storm within and allow the person to live and love in peace.\nAmerican society is blowing its top in more ways than one.\nThe #MeToo movement has exploded with a righteous vengeance. For too long women have felt powerless over powerful men who controlled their lives and their bodies. Women learned techniques of avoidance and deflection, but often to no avail. They were denied consent and suffered unwanted advances by men who took advantage of their positions.\nAmerica's original sin has been exposed in a new way, too. Racism will not work itself out by whitewashing history. No attempt to ennoble Confederate leaders by reinterpreting monuments as heritage will move us forward when our history of white privilege continues to wreak havoc in black souls. To be clear: Pigment of skin is no indicator of human dignity. Black Americans are finding their God-given voice to say enough. Centuries of oppression and inequality are seen in the lava flows of civil protests and published jeremiads.\nThe disappearing middle class is losing hope. The value of the worker is disregarded in an age of robotic technology and globalization. Corporate profits that go to wealthy shareholders, rather than being shared with those who bear the weight of labor, fuel desperation. The longing to go back to a better day is really a desire to participate in the prosperity others experience at their expense. Our politics tap the root of hidden anger but offer no salvation, only empty slogans.\nOnce the lava begins to flow, we have to give it time to cool. Then, instead of bemoaning the lingering mess, we should ask, what brought it about? We can't cool the earth's center, but we can listen to one another and commit to a future of mutual respect and shared well-being.\nThe flow of lava can't be stopped, but happily and hopefully, neither can the flow of love.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"His name was literary unknown from the country's bible of the who-and-whos in the entrepreneurship ladder. Just like the biblical scriptures, Louis Adini Malukula was rejected at home. He could not make it that big. No matter how hard he hassled, he could not blossom.\nUnlike other vision-deficient countrymen, Louis did not relent. He had a dream and he could not let it falter. He had to pursue plan B. He had to bid his Mangochi hometown goodbye. He did. He had to trek down south (RSA). He did.\nFor ten years, he pushed himself to the limit so as to actualize his goals. He huffed and puffed and sweated. Day and night he worked, and observed how successful entrepreneurs were coining it. He was an active observer.\nFor ten years, he worked at a local beverage company (RSA). Whilst there, his everyday job description involved him supervising accounts customers, and high level clientele. To others, this was just a bread-winning job. To Louis however, this was a make or break opportunity for it was here that he accrued immeasurable skills in customer relations and marketing. By and by, he saw his dream coming to reality by each passing hour.\nHe waited. To others, waiting for ten years to make a breakthrough was nothing but a pauper's dream. To Louis, he was waiting for the right moment to shine. During the twilight of his career at the unnamed beverage company, Louis found another pastime activity. He involved himself in assisting a close colleague running a tourist guiding company. It is here that the sleeping giant that was in him emerged.\nThat, dear reader, was to be the genesis of Louis' marriage with the tourism industry. The coming days, weeks, or months, (depending on your preference) saw the excited young man from Mangochi running up and down, seeking knowledge on how he could also run a similar enterprise.\nAfter what seemed ages whilst Louis was in the wilderness, hunting for knowledge on how he could succeed in the tourism industry, just a couple of months ago, he has finally launched Cape 360 Tours, a tourist transportation agency to various destination in the Rainbow nation.\nFrom private tours, city tours all the way to airport transfers, and even to the famous Cape Peninsula, Cape 360 Tours literary takes you to where scenic beauty is at.\nBased in Cape Town, Cape 360 Tours is already becoming the talk of the town. With less than a year in its operation, the tourist agency is already winning, rather, has already won the trust of many of the customers who have ever been heavenly served.\nTake one Pam Thurber Salvador for example. In his review of Cape 360 Tours, on the latter's Facebook page, he could not hide his heartfelt appreciation for the five-star treatment he was served courtesy of Louis and his Cape 360 Tours. Apparently, Pam was a foreign national.\n\"South Africa never ceases to amaze me! We spent the day in the Cape Winelands taking in stunning views and tasting some special wines from La Motte, Franschhoek Cellars, and Graaf. The beauty is just incredible! Huge thanks go out to our tour guide, Louis, from Cape 360 Tours. Not only does he know his stuff, he's a great company as well.\" Reads the comment.\nPam's sentiments are one of the litany of similar appreciation clients of Cape 360 Tours have come to associate with the tour guiding agency. It is not disappointing.\nAccording to Louis, the core business of his agency is to make tourists' adventures memorable by offering assistance-par-excellence.\n\"It's always been our mission to ensuring that our clients get more than what they bargained for. A satisfied customer is a recipe for a business boom,\" he reveals.\nAbove all, Louis has observed that poor customer relations, and having dilapidated structures do easily turn away potential clients.\n\"I've observed that most tour guide companies here lack perfect marketing strategies, and also their machinery is not that up to standard. Here at Cape 360 Tours, we always make sure to capitalize on that, no wonder we're becoming the market leader in this industry,\" he revealed.\nAlthough Louis is slowly rising to top of the entrepreneurship ladder, he is still dreaming in colour. Come five years from now, he wants to be somewhere; that small planet that only the few successful ones reside.\n\"I would like to make Cape 360 Tours be a force to reckon with internationally. I want it to be that big and successful, and at the same time, adding smiles unto our clients,\" he envisions.\nAt only 37, Louis Adini Malukula, originally from Mangochi but now in Cape Town, South Africa, is a model many a Malawian youth would want to emulate.\nHaving started so low in life, selling of second hand clothes (popularly called Kaunjika), many would never, in a million years, have thought he could rise this far.\nBut through determination, persistence and goal setting, he knew his path. No wonder his trekking to Mzansi was but the genesis of a brighter future for him.\nUnlike other fellow countrymen, who, when they have crossed the borders, lose their identity, Louis, through his Cape 360 Tours, has proved that he is indeed the true son of the Warm Heart of Africa.\nLouis is an example of the biblical illustration of how prophets are usually forsaken in their mother lands but easily win trusts in foreign land.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzautnu b/data_all_eng_slimpj/shuffled/split2/finalzzzautnu
new file mode 100644
index 0000000000000000000000000000000000000000..24d72a08ac7fa725908e87add9d2631bc9855c77
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Set work into recruiting followers. Whether your Instagram business will function or not relies upon hugely on your reach, or in this case, your Instagram followers. Make positive that people see your posts, and for this to occur, you have to get them to adhere to you initial. There are a whole lot of methods to do this-you can keep a promo that requires them to comply with you, or your can make your posts interactive so that it appears on the action feed of the community of the people who follow you. Once you get a great base, you can now entice a lot more people with good content.\nPut energy into recruiting followers. Regardless of whether your Instagram business will operate or not is dependent hugely on your get to, or in this circumstance, your Instagram followers. Make confident that men and women see your posts, and for this to happen, you have to get them to comply with you 1st. There are a lot of methods to do this-you can maintain a promo that requires them to follow you, or your can make your posts interactive so that it seems on the exercise feed of the network of the men and women who stick to you. Once you get a great foundation, you can now entice a lot more men and women with excellent articles.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"exposed brick walls high lamp hipster living room is assortment of Living Room that you can use as refrence to redesigning your house. exposed brick walls high lamp hipster living room is inspiration of Living Room. This Image was upload at January 31, 2019 upload by homeDZN in Living Room.\nexposed brick walls high lamp hipster living room is high resolution image that you can use for refrence. This exposed brick walls high lamp hipster living room is the best collection from the the best architect. If you love our collection you can like and share our Home DZN to your instagram. It can make me feel better to improve our site.\nYou can make exposed brick walls high lamp hipster living room For your Desktop Background, Tablet, Android or iPhone and another Smartphone device for free. To download and obtain the exposed brick walls high lamp hipster living room images by click the download button below to get multiple high-resversions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tow Boat Svc & Mgmt Inc, Navigational Services to Shipping, listed under \"Navigational Services To Shipping\" category, is located at 21 Pier Honolulu HI, 96817 and can be reached by 8085221000 phone number. Tow Boat Svc & Mgmt Inc has currently 0 reviews.\nBrowse all Navigational Services To Shipping in Honolulu HI. Discover census data for Honolulu, HI.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This package has a home page at https:\/\/www.gnome.org\/.\nThe package is located in the \"graphics\/gnome-icon-theme-extras\" directory. The current source version of the package is \"gnome-icon-theme-extras-2.30.1nb2\". For a summary on how to use the package collection, go to the top of the packages tree.\nThis package requires the following package(s) to build: gnome-icon-theme>=2.8.0 icon-naming-utils>=0.8.90 p5-XML-Simple>=2.14 p5-XML-SAX-[0-9]* p5-XML-NamespaceSupport>=0.03 perl>=5.28.0 cwrappers>=20150314 perl<5.30.0 p5-XML-SAX-Base>=1.08 p5-XML-SAX-Expat-[0-9]* p5-XML-SAX>=0.03 p5-XML-Parser>=2.27 p5-XML-Parser-[0-9]* p5-XML-NamespaceSupport>=1.04 perl>=5.0 icon-naming-utils>=0.8.7 gnome-icon-theme>=3.12.0nb3 librsvg>=2.22.3 glib2>=2.4.0 libelf>=0.8.5 pcre>=8.31 pcre>=3.4nb1 libffi>=1.20 pcre>=8.30nb1 libffi>=3.0.11 python27>=2.7 mozilla-rootcerts>=1.0.20150804nb1 readline>=2.2 readline>=6.0 python27>=2.7.1nb2 glib2>=2.24.0 pango>=1.38 Xft2>=2.1.7nb3 fontconfig>=2.2 libuuid>=2.18 freetype2>=2.8.1 freetype2>=2.4.5 freetype2>=2.1.3 freetype2>=2.4.11 gperf-[0-9]* fontconfig>=2.10.93 fontconfig>=2.1nb2 Xrender>=0.9.0nb2 x11-links>=1.16 osabi-NetBSD-8.0_STABLE xorgproto>=2016.1 Xrender>=0.2 Xrender>=0.8.2 fontconfig>=2.13.0 Xrender>=0.9.0nb1 Xft2>=2.1 Xft2>=2.1nb2 fontconfig>=2.11.91 Xrender>=0.8 fribidi>=0.19.7 fribidi>=0.19.1 glib2>=2.33.12 harfbuzz>=0.9.9 icu>=3.4 icu>=64.1 glib2>=2.34.0 harfbuzz>=2.1.1 cairo>=1.12.10 MesaLib>=3.4.2 libxcb>=1.6 libXdmcp>=0.99 libXau>=1.0 xcb-proto>=1.13 libxml2>=2.6.2 xmlcatmgr>=2.0beta1 libxml2>=2.8.0nb2 xcb-proto>=1.4 py27-mako-[0-9]* py27-setuptools>=0.8 py27-expat-[0-9]* lzo>=2.01 fontconfig>=2.2.95 freetype2>=2.1.9 png>=1.2.4 pixman>=0.30.0 pixman>=0.25.2 MesaLib>=7.11.2 png>=1.6.0nb1 cairo>=1.0.0nb2 cairo-gobject>=1.10.2 cairo>=1.16.0 glib2>=2.14.0 Xft2>=2.1.7nb7 cairo-gobject>=1.16.0 help2man-[0-9]* p5-gettext>=1.01 gobject-introspection>=0.6.14nb1 py27-cElementTree-[0-9]* glib2>=2.54.0 libffi>=3.0.0 gobject-introspection>=1.34.0 pango>=1.6.0 cairo>=1.2.0 gdk-pixbuf2>=2.22.0 shared-mime-info>=0.15 glib2>=2.37.2 tiff>=3.6.1 jbigkit>=2.0 jpeg>=8nb1 jpeg>=9 shared-mime-info>=1.0nb1 tiff>=4.0.3nb5 libcroco>=0.6.1 libcroco>=0.6.0 libxml2>=2.9 pango>=1.42.4nb2 gdk-pixbuf2>=2.34.0nb1 libcroco>=0.6.5nb2 .\nThis package requires the following package(s) to run: gnome-icon-theme>=2.8.0 gnome-icon-theme>=3.12.0nb3 .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Used with any independent axle track kit, or trailing axle kit, the 4orce Mobility T Tank is safe and effective for transporting liquid to remote locations. Allowing you to secure the load while keeping the CG between the wheels, the 4M T tank is a quick decision for utility and rescue environments requiring water, fuel, or any liquid substance arriving on location, safely. Light weight polypropylene minimizes installation and maintenance.\n5 X 5 added deck space over the tank doesn't sacrifice the other things your operation requires.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavfoc b/data_all_eng_slimpj/shuffled/split2/finalzzzavfoc
new file mode 100644
index 0000000000000000000000000000000000000000..723ae46cf10337ff9cf43174ad49f251188140ba
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzavfoc
@@ -0,0 +1,5 @@
+{"text":"This might be such a Millennial thought- but grinding one thing at a time is hard for us. We are the generation of instant gratification, so if something isn't giving us what we want or \"need\" at the moment we are quick to move on to the next thing, all in hopes that the change will fulfill us. And I'm all about change, trust me, I haven't lived in one place for THAT long before wanting to move again. Yet, I have come to understand that humans might benefit from doing or being something for a longer-than-usual period of time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When filling in your application form, please ensure that you read the job description and person specification, and tailor your application to each job that you apply for.\nTell the recruiter WHY you are best for the job. Include things that you are proud of, things you have achieved, and key learning skills you have utilised.\nRemember to check grammar, punctuation and spelling carefully. It might be worth drafting your application in Microsoft Word and copying to nhs jobs, and get somebody else to read over your application too!\nAlways get good quality references for your application. Please don't use family or friends as references, and always ask the person that they are happy to give a reference before putting their name down.\nMake sure that you dress smartly and as you would in the workplace should you get the position. This means no jeans or leggings, no t-shirts, and make sure that you dont have low necklines, and skirts are knee length or below.\nNow is not the time to start being modest. You will really need to sell yourself and talk up everything you have accomplished, but bear in mind you want to come across as confident, not arrogant.\nCommunicate with the interviewer, make regular eye contact, and RELAX. Make sure you arrive to the interview within plenty of time, but not more than 35 minutes before the interview is due to take place, as you don't want to put the interviewer under pressure, but 10-20 minutes before is about right.\nRead through the job description and person specification carefully, and prepare yourself for the questions you could be asked. Practice your handshake on a friend, or ask for honest feedback on your interview style. Be interested and alert, and always remember how you are coming across, slouching whilst chewing gum is not the way to get hired!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Professional Pilot Program is an all inclusive program that will take a student from the beginning to a point they will be able to enter the helicopter pilot job market. Students will obtain their Private, Instrument, Commercial, CFI, and CFII certificates\/ratings. This course is best done on a part-time basis averaging 3 lessons per week. However, some individuals may desire a faster course completion time and can accomplish all courses within as little as 10 months. Part-time students should expect an average of 24 months to complete the full program.\nAirwest delivers a unique environment to any student who attends our courses by being able to see a company, from the sidelines, who works in and out of the professional helicopter world on a daily basis. Airwest's unique base provides a relaxed learning atmosphere and a comfortable learning environment where students will be able to get up close to aircraft they may someday fly as a professional pilot and meet many of our employees who currently do.\nThis one of a kind surrounding and immersion into the helicopter industry is part of what we call \"Real World Helicopter Training.\" You will be taught the things our pilots have learned over the past so that you can learn from them and in return become a better and safer pilot.\nCome fly with Airwest and you will be the best!\nPlease use the form on this page to learn more. We're standing by to answer all of your questions.\nAirwest Aviation Academy seeks to set a higher standard of instruction and integrity within the helicopter flight training community by being committed to developing students in a professional and efficient learning environment that brings them closer to their goals as future pilots. We believe this process is achieved through close one-on-one interaction between the instructor and student, dedicated development of our staff through a continual learning process, and a philosophy of open-door management.\nYears of experience as a commercial operator have allowed AAA to become one of the few flight schools that can truly offer \"real world\" flight training. Drawing from the techniques that we use in the field, AAA students gain a distinct advantage over the competition by performing maneuvers just as they are performed in the professional world. This approach to flight instruction enables the student to easily transition the gap that so often exists between what the student learns in the training environment and what is truly encountered once they fly as an employed helicopter pilot.\nAAA is a family owned and operated helicopter flight school located in Glendale, AZ at Glendale Municipal Airport. We offer many different flight training courses from initial Private to Instrument, Commercial, Certified Flight Instructor (CFI), Instrument Instructor (CFII), Add-ons, Turbine Transitions, and External Load training courses.\nOur experience brings a unique training environment that is different from other training schools. Our relationship with Airwest Helicopters, a commercial 135 operator, allows us to bring real-world flying techniques into our training curriculum. These learning experiences are all part of our teaching system that was designed by professionals who have over 30+ years in the industry as pilots. Our focus is first Safety, second Quality, and third a positive atmosphere that is conducive to learning.\nClicking the \"Submit Info\" button below constitutes your express written consent to be called and\/or texted by Airwest Aviation Academy at the number(s) you provided above regarding furthering your education. You understand that these calls may be generated using an automated technology. You are not required to provide consent to receive services from this school.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I recently engaged in a discussion (okay, maybe a debate) about low-carbohydrate diets as a tool for weight loss in recreational exercisers (that refers to people who exercise without training for a competition, which sets them apart from athletes, technically). I've posted before that diets don't work, so I'm not a proponent of low-carbohydrate 'diets', especially for individuals who train rigorously on a regular basis. There's no evidence that these diets are more effective than those that contain higher levels of carbohydrates, and there's plenty of evidence to show that low-carbohydrate diets (<43% total calories) reduce both aerobic and anaerobic performance (1). At the moment, low-carbohydrate diets are the 'next big thing,' and while you may see quite a few anecdotal blog posts that make this appear like a controversial issue, it really isn't within the sport nutrition community. The current evidence-based consensus is that athletes\u2014recreational or elite\u2014require a majority of their calories to come from carbohydrates (1).\nWhen I brought up the topic of evidence-based practice, my counterpart brought up some research that had been done in elite cyclists, citing that their performance had improved on a high-fat diet. I reminded him that his clients, while certainly exceptional, were not elite cyclists, but in fact recreational exercisers engaging in primarily high-intensity anaerobic training. They fall extremely far away from elite cyclists on the spectrum of intensity and duration of training, and therefore exhibit extremely different physiological adaptations which require specific nutritional prescriptions. Evidence-based practice can't be reduced to the generalized application of the results of a single research study to all of your clients. Even a very well-designed study can't be used as a one-size-fits-all nutrition prescription!\nI'm in uniform and holding some fruit. You can trust me!\nLikewise, clients\u2014you need to vet your coaches. The regulation of terms like 'nutritionist' or 'nutrition coach' is extremely loose and variable by state. (You can worry less about the term 'registered dietitian' or 'sport dietitian' as these are federally-regulated titles that require licensure in the US.) Your nutritionist may have absolutely no background in exercise physiology or sport nutrition, let alone a vetted certification.(I'll be posting more on this in my next blog which will explain the types of certifications, what to look for, and what to look out for).\nSo, back to the issue at hand: evidence-based performance nutrition. As Nancy Clark, MS RD CSSD stated, most clients are, \"mere mortals,\" which is simply to say that most are not elite athletes. (Though, they are exceptional simply for exercising on a regular basis and meeting or exceeding guidelines set forth by the American College of Sports Medicine. Only 20% of Americans meet the guidelines for both aerobic and resistance training!) Between her statement during the seminar last weekend and this recent conversation, I realized that this is the second major lesson I took away from that experience: We must know our clients, inside and out!\nThe physiology of elite athletes differs from that of recreational athletes in a few key ways, but a major factor in sport nutrition is the fact that elite athletes can use fat for energy at much higher intensities, whereas non-elite exercisers rely primarily on carbohydrates for energy at high intensities. So, in some cases, an elite athlete may respond favorably, or at least not see much of a reduction in performance, on a low-carbohydrate diet. That being said, the current consensus in sport nutrition supports a high-carbohydrate (>43%) diet for all athletes because levels of muscle glycogen\u2014the storage form of carbohydrate in the body\u2014correlates with exercise capacity (1). Resistance training reduces muscle glycogen after just one set, and low glycogen levels lead to a reduced capacity to do work. Work is what burns calories and elicits training adaptations.\nIf you have to talk yourself into every set, you might be depleted.\nThe inability to perform with sufficient intensity will result in\u2026well, no results! Low glycogen may even be one mechanism that leads to overtraining syndrome. If you haven't been recovering between exercise sessions, your progress has stalled, and you're feeling mentally and physically run-down, it would be an excellent idea to assess your carbohydrate intake; many athletes fail to eat sufficient carbohydrates (1).\nDifferences exist not only between elite and non-elite athletes\/exercisers, but aerobic (endurance) versus anaerobic (resistance-training, HIIT, intermittent) trainees as well. The extent to which a body can use fats for energy is dependent upon exercise intensity. The anaerobic energy systems\u2014those utilized during high-intensity activities like sprinting, HIIT, and circuit-training\u2014can't utilize fats. (That's not to say they won't cause fat loss; combined with a caloric deficit, the excess post-exercise oxygen consumption (EPOC) will burn plenty of calories from fat\u2014but that's a different blog post!) So, my counterpart's clients, all of whom perform either resistance training or HIIT-style workouts on a regular basis, would certainly suffer a decrease in their performance, which means reduced work output and therefore wouldn't be maximizing their potential adaptations. In less scientific terms: their workouts would suffer so they'd burn fewer calories and induce less stimulation required for adaptations like muscle growth and strength. Anaerobic athletes who consume insufficient carbohydrates exhibit reduced power output and intensity (1).\nDietary trends are often popularized alongside fitness trends; that doesn't make either one a best practice.\nA recent study in CrossFit athletes, many of whom eat a moderately-low carbohydrate diet, showed improved exercise capacity after three days of increased carbohydrate intake (6-8 grams per kilogram of bodyweight as compared to <6 grams per kilogram of bodyweight). Another study in trained cyclists showed that three days of a high-fat diet (68% calories from fat) followed by carbohydrate loading compromised sprint performance even though the single day of carbohydrate loading met the recommendations of 8-10 grams per kilogram of bodyweight. The elite cyclists my counterpart read about weren't exercising at such high intensities (it's not physiologically possible as aerobic energy systems are required after about two minutes), and were instead using aerobic pathways that can utilize both carbohydrates and fats, tapping into thousands of calories worth of stored energy in their adipose tissue to spare muscle glycogen for the final sprint. Not to mention, as elite endurance athletes, they had adapted more of the cellular 'machinery' required to use fats for energy in comparison to his clients. His 'evidence' to support the use of a low-carbohydrate diet was well-intentioned but entirely misappropriated.\nThe International Society of Sport Nutrition recommends an intake of 5-7 grams of carbohydrate per kilogram of bodyweight for endurance athletes performing moderate-intensity aerobic exercise, and 6-10 grams of carbohydrate per kilogram of bodyweight for anaerobic athletes. The minimum recommended intake is 3 grams per kilogram of bodyweight for individuals who aren't physically active, though research has shown that intakes of less than 5 grams per kilogram or <43% total calories from carbohydrate will result in low muscle glycogen after successive bouts of exercise (1).\nFat loss comes from a caloric deficit\u2014simply eating fewer calories than you're burning. There is no need or justification for creating that deficit by reducing your carbohydrate intake. Your 'diet' shouldn't be a short-term solution, but a lifestyle that you can maintain indefinitely while sustaining a rigorous training program to consistently elicit adaptations and improvements.\nHaff, G. C. (2008) Essentials of Sport Nutrition and Supplements: Carbohydrates. Totowa, New Jersey: Humana Press.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It is just past four on a Tuesday afternoon. At the KUCI radio broadcasting room, a debate between a program guest and a caller is taking shape. The topic is homosexuality. In the course of an hour, the discussion moves on to religion, self-confidence and building respect for differences in communities.\n\"The first and most important thing in combatting the problem of closed-mindedness is education, because there are a lot of myths about people who are homosexual--such as that they are bad or evil or that something is wrong with them and a great deal of other stereotypes,\" says the in-studio guest, brainstorming with a caller.\n\"And if these misconceptions hopefully can be addressed and proven to be false or unfounded myths more than anything, then people will be able to get past their fear of homosexuals and learn to accept them also as people.\"\nOnce again the airwaves have become a conduit of opinion and education, this time on 88.9 FM, the college radio station at UC Irvine.\nThis difference this time--and every Tuesday from 4 to 5 p.m.--is that the studio is being run by high school students, and the callers are teens from throughout Orange County.\nHost of the \"Irvine Goes Local\" radio program is Neil Seghal, a junior at University High School in Irvine. Every week he is at the studio switchboard, speaking to callers, debating with in-studio guests, playing music during downtime and repeatedly urging students to call in.\nSeghal has been running the show since last spring and has seen the format expanded from 30 minutes to one hour.\nThe show was begun two years ago by an Irvine High School student who had just completed an FCC certification class at UCI to deal with topics and issues of interest to high-schoolers in Orange County.\nSeghal has also taken the eight-week certification class, with one hour of instruction each week.\nIt is a primer, with information about FCC laws, fines and penalties for unfit acts, \"basically what one can or cannot do,\" Seghal said.\nOnce on the air, the responsibility lies in the hands of the individual.\n\"We run a disclaimer at the beginning and end of the program, but other than that, there is no censorship at all,\" Seghal said.\nWith microphone in hand, Seghal has the power to provide students with the medium to express their opinions and learn about what others have to say as well.\n\"As a student journalist, my responsibility lies mainly in informing students and providing a forum for discussion of important issues that are often not discussed anywhere else,\" he said.\nSeghal says he likes to have in-studio guests on each program to provide other views. Sometimes he has trouble finding a willing participant if the subject is controversial. Other times students approach him about being on an upcoming show.\nUniversity High senior Roger Kim assists Seghal, answering calls and throwing in his two cents' worth on topics.\nOne week Seghal focused on college applications, and Kim offered tips to juniors and seniors on the process. \"Roger really was able to help out those students in the community who may begin planning now for the future--he spoke about the pitfalls and what a junior can do to be prepared,\" Seghal said.\nCall-in subjects have ranged from AIDS, racism and stereotyping to school restructuring, the county budget crisis and dating. Suicide was a recent topic, with a student guest telling about his aunt, who had taken her own life.\nHow does he prepare for such varied topics? \"I do a fair amount of reading--newspapers, magazines, articles,\" said Seghal, adding that he likes to talk about things he feels strongly about.\nAs host, he reassures callers and in-studio guests to help them relax, feel comfortable and speak their minds.\nIt is past 4:30, and the phone line has not stopped flashing since the discussion on homosexuality began. A girl from University High School calls to share confusion she is feeling: Her religion perceives homosexuality as evil, yet she strives to be open-minded and accept people for who they are.\nA guest on the program says, \"I am a Southern Baptist, but as I have examined and re-evaluated my religion, I have formulated a number of questions that I am still attempting to mull over.\"\nThe caller agrees that a questioning of one's beliefs is healthy and encourages those listening to think of the individuals who are often grouped and labeled.\nAnother teen caller asks why schools \"waste money on educating people about homosexuality and gay rights when it only affects a small group of people.\"\nIn a time of controversial issues, when intolerance seems prevalent, Seghal's program provides an environment in which all opinions can be heard and everyone is an equal in discussion.\n\"After doing shows focused on homosexuality . . . I've already seen some success in opening people's eyes to the issues, which pleases me,\" Seghal said.\n\"I've seen people who had been very negative toward particular groups of the populace change their view (partly) because of what they hear on 'Irvine Goes Local.' \"\nAt the end of the broadcast, \"Irvine Goes Local\" gets one final call. The nervous but grateful caller thanks Seghal for handling the topic. \"It helped me a lot,\" the caller says.\n\"Needless to say,\" Seghal said later, \"this made my day and is really why I love doing 'Irvine Goes Local.' \"","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Toshiba Electronic Devices & Storage Corporation today announced that Ariel Lasry, the Chief Engineer at Toshiba Electronics Europe GmbH, its European subsidiary, has received the 2018 MIPI Lifetime Achievement Award, a MIPI\u00ae Alliance Membership Award. The award recognizes Mr. Lasry's many and valuable contributions to the alliance, and was presented at an awards ceremony held during a member's meeting luncheon in Montreal, Canada on March 27.\nMIPI Alliance is a collaborative global organization that designs and promotes hardware and software interfaces for mobile devices that simplify component integration. The MIPI Alliance Membership Awards Program annually recognizes outstanding contributions and achievements by individual and corporate members. The MIPI Lifetime Achievement Award is annually presented to an individual who has participated in MIPI Alliance efforts for at least seven years, and provided significant contributions or leadership to a working group, the board, or other MIPI Alliance entities.\nMr. Lasry, Chief Engineer at Toshiba Electronics Europe GmbH, has distinguished himself in multiple capacities as a working group advocate during the development and maturity of \"MIPI DSI-2SM\" and \"MIPI D-PHYSM\". He has also made notable contributions to the Technical Steering Group and to MIPI's Board in formulating new directions, and to enhancing MIPI's relationship with other organizations.\nToshiba Electronic Devices & Storage Corporation  has played a leading role in developing specifications for mobile and mobile-influenced devices, and many specifications developed by the MIPI Alliance are global standards. Today, the scope of MIPI\u00ae specifications is expanding from mobile devices to industrial equipment and automotive systems.\nParticipation in the alliance also brings Toshiba Electronics Electronic Devices & Storage Corporation advantages in the semiconductor business\u2014in promoting its interface bridge ICs, for example.\nToshiba Electronic Devices & Storage Corporation will continue to contribute to the promotion of MIPI\u00ae specifications.\n While Toshiba Corporation received the 2017 MIPI Corporate Award, the award effectively recognized the work and achievements of Toshiba Electronic Devices & Storage Corporation and its European subsidiary, Toshiba Electronics Europe GmbH, Toshiba Corp's representatives in the alliance.\n Toshiba Corporation joined MIPI Alliance in 2004, and Toshiba Electronic Devices & Storage Corporation succeeded to its membership in January 2019.\nMIPI Alliance (MIPI) develops interface specifications for mobile and mobile-influenced industries. There is at least one MIPI specification in every smartphone manufactured today. Founded in 2003, the organization has over 300 member companies worldwide.\n* MIPI\u00ae is a registered trademark of the MIPI Alliance.\n* MIPI DSI-2SM and MIPI D-PHYSM are service marks of the MIPI Alliance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The term deus ex machina is a Latin term which means \u00bbGod\u00ab or supernatural being that has gained several meanings and explanations over the course of two millennia. This superior force or person at the last moment resolves an unsolvable situation without following the previous logic of the relationship or overcoming common sense.\nIn the first phase, the work was based on conversations with five-year old children about drawings they have made, depicting their most important things to them and their vision of the future. The drawings were used as a source material from which they then selected the most important elements to them (\u00bbhead of the Ninja Turtle \u2013 it can zoom, the butterfly wings \u2013 we can fly, the ambulance car \u2013 so that we are fast, ornaments on the wings \u2013 because they are beautiful, the bridge \u2013 we can go over the lava or the flood..\u00ab). Then they assembled their pictures into a single image and named it: \u00bbA machine that will make a better world, a machine that will help us..\u00ab.\nThis collaboration, raised a number of questions: Is there such thing as \u00bbunspoiled nature\u00ab in children, or are they already the projection of us and the world that surround us? How do they perceive the concept of the future, even though they are not yet fully aware of it? What, when and where is our present? Are solutions by those who concern the consequences of the present always recognizable as rational?\nIt is very possible that incredible things will come out of that.\nIn cooperation with Oton \u017dupan\u010di\u010d's kindergarten, group Mice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you're like us, you have heard the term \"gridlife\" before but had no clue what it actually was. Earlier this year we had the chance to fly to Michigan to see what it was for ourselves and, boy, were we blown away.\nPut simply, #GRIDLIFE is a festival of car culture that lasts for three days. It combines drifting, time attack, car shows, camping, fun, and music, all in one major event. While that sounds like a no-brainer concept, it's actually much more difficult to execute than you think. It's one thing to attempt pulling off something like this, but it's another to do it well, and thank goodness for us the guys at #GRIDLIFE have figured it out.\nWhen we arrived at Gingerman Raceway in Michigan on Thursday, the teams and drivers were just setting up and rolling in. Next to the competitors, the spectators were allowed to roll in as well and set up their camps for the weekend. You could think of the event as a Woodstock for car lovers. The weather was hot as hell and there was a sense of excitement in the air. It almost felt like we just watched the fuse of a giant stick of dynamite get lit and we were all going to watch this little town explode with people. That's exactly what ended up happening.\nWhile we aren't as deep into the Midwest car scene as we would like to be, we are constantly impressed with the events that come out of that area of the country. The car quality out there is second to none, the tracks are super friendly, and the people are all way stoked to be part of the scene. It's really refreshing to see stuff like this.\nBack to the event - Mother Nature had a different plan in mind for Friday and the track was hit with a torrential downpour and thunderstorms. While this may seem like an event killer, I can honestly say that no one left and it looked like even more people showed up during the storms.\nThe day was filled with tons of action on the track, with some very special drift exhibitions by the one and only Ryan Tuerck. Time attack sessions were held in the spotty weather and the changing conditions made for some exciting driving. Some of the Midwest's fastest time attack teams brought their cars out to let them loose on the Gingerman circuit as a midyear shakedown.\nThe Professional Awesome \/ Fortune Auto EVO was plagued with problems all weekend, and while they did set the fastest lap time of the weekend, they broke on the cool down lap and were forced to retire.\nThe drift teams from Risky Devil, ProceeD, High Fade, Tracker, Garage Collective, and more were shredding tires all day in the wet and dry. It was pretty rad seeing teams lend a helping hand to complete strangers to help get cars back on track. This is one of those grassroots events that most definitely had more of a party vibe then a serious competition. As soon as the sun went down, the rain came in with full force but that didn't end the party at all. The live performances by Will Joy, The Hood Internet, and RJD2 kept people partying well into the night.\nSaturday saw the same resistance from Mother Nature, with tons of rain all day long (I had to change my sox twice) but that didn't scare off any drivers, campers or spectators - the event was THAT good. Since we knocked out most of our driving coverage the day before, it was time for us to check out the car show and see what the Midwest had to offer. It's hard enough building a car in California, but I can't imagine how difficult it is when you have snow and rain for six months out of the year. My hat goes off to these guys; some of the cars were beyond impressive.\nAs soon as the sun went down on Saturday the stage heated up with performances by Keys N Krates, Party Favor and Lil Panda. Oddly enough, as soon as the music started the skies opened up and really let the rain come down. It was pretty surreal seeing kids moshing to Keys N Krates in the middle of a rainstorm. Some people even brought in canopies to dance under.\nSunday was the last day to get any time attack runs in and the entire field was still there! After getting rained on and partying for two days straight, the vibe at #GRIDLIFE was still going strong. To be honest, I didn't want the event to end because I was having so much fun. As much as I want to say I would love to see this event reproduced somewhere in California, I doubt that will ever be possible. The tracks here aren't as friendly and the vibe just isn't the same out here. That may not be a bad thing; it just makes this event that much more special. I can' t wait to go back next year - maybe we will have an official Super Street merch booth (if it doesn't rain as hard).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yes, on Inauguration Day, the good Rev. Wright will be giving the first of a series of lectures on \"Prophetic Proclamation\"\u2013a subject he knows well\u2013at Virginia Union University.\nIt's the school where he first studied before leaving to join the Marine Corps in the early 1960s, and its school of theology is named after his doctoral advisor, Samuel DeWitt Proctor, who also mentored MLK. So there are plenty of reasons for Wright to return and run this intensive workshop. And yet. Wright always has been a preacher known for his sense of timing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ClaraSanchez xCharmingAngel jasmin. dirtypasion4u. Rinaldisa. Anablow.\nbabygirlxoratedManuelaRiosKahoryBoobsAsttutee .RinaldisaSAMANTHAvsBRAYANSweetCupidonTiaEuia3 .AlexisBlakeAlexFlareAvaBelleXXDivineDonna .AttractivePearlKatieMistressLadyTviksycsipetcsapat .FreiyaMagnoliaGoldPattyDarlingLovelyXSelena .a00SpecialGuyRinaldisaLovelyXSelenaAvaBelleXX .NicoleNixonMagnoliaGoldLevyStarryKatieMistress .Vladagirl1a1HOTASIANforSEXTeasingMissKLEINX4U .KLEINX4UBlondieOksanaaL0LASTARCristinaBeauty2 .SofiaMorganAlexaRLAlexisBlakeboy15963 .AndreaFoxxyAttractivePearlSweetCupidonSweetyAlexandra .\nAndreaFoxxyEmilyNiceEasyLayBigAssLatinoXL .LadyTviksyTiaEuia3TiaEuia3BigAssLatinoXL .FitoDeangelloboy15963L0LASTAREmilyNice .ManuelaRiosRelationHDpornNicoleNixonAlisonJones .BellaBeautifulManuelaRiosCristyFtSamAttractivePearl .KimoraLeeHotArtemandVictoriaChristineMorgandirtypasion4u .bigbusstyAlexisBlakeAlexBlarKatieMistress .SEXYBOY4MANCristinaBeauty2LucyLiannalinaHappy .PattyDarlingaronman1988Tinomysteriosaesch492 .CrisstoferCumsa00SpecialGuyDivineDonnaBlondieOksanaa .QuenOfTheDarkIsabelYoursAngelMiryShycsipetcsapat .","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Four Horseshoes pub at Burrow Hill Green is open again for business after two weeks closure between leaseholders.\nFormerly leased by Marstons from Scottish & Newcastle, the pub is being run until the new year for the brewery by Tony Leslie and Sue Hoyle, experienced landlords who are tasked with boosting business over the next couple of months. Trade had dipped after rumours of longer term closure, including a suggestion that this Chobham landmark was set to become a branch of Sainsbury's, which proved to be complete fiction.\nThe caretaker landlords have refreshed the beer selection and put on a Tuesday to Sunday menu. On the Friday night of the Rugby Club's Firework Extravaganza they are running a barbecue.\nNext year will see the fruit of Scottish & Newcastle's partnership with a private investor, a seasoned pub tenant who will be taking on an extended tenancy. A major refurbishment, including an extension to double the size of the restaurant, is scheduled to start in January with a reopening nine weeks later. The makeover will be sympathetic and sensitive to the building's historic, cottagey, character we are assured.\nVisit the Four Horseshoes listing in the Chobham Directory.\nComment Link\t Tuesday, 13 November 2012 23:59 posted by\tSteve.F.\nI had always thought that this was a place for nice food but something seems to be going wrong. My wife ate their tonight, 13\/11\/12 with a group of friends after one had a good meal last week. Tonight they described the food as truly dreadful. this was acknowledged by the landlord who reduced the bill by 25% but they didn't think it enough to compensate. It may be better to wait for the refurbishment!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Across England and Wales, there are a growing number of NHS disciplinary investigations and hearings which are being carried out each year, only some of which reach the headlines. Investigating such cases requires a dedicated team and a thorough and pragmatic process in order to get to the bottom of the issues under investigation.\nWhile each case is particular, investigating officers are guided by strict guidelines under which they are required to operate while investigating a case. Investigations concerning staff may relate to disciplinary matters, grievances, whistleblowing or allegations of racial or sexual harassment, amongst others. In such circumstances, an investigating officer is normally appointed by the relevant NHS Trust to conduct the investigation.\nThe priority of the investigating officer is to establish the facts of the case and whether or not it justifies a hearing. Correctly managed investigations lead to better decision-making, justice and a consistent approach, which employees can have confidence in. As a result of carrying out a well-structured investigation, and a fair and just hearing procedure, the likelihood of appeals or additional actions requiring tribunals or courts can be avoided.\nSuch pre-hearing investigations are typically carried out within a three month period and require the investigating officer to meet with all parties concerned, establish the facts and details of the case. In order to ascertain the key facts, the investigating officer will interview all employees involved in the case, gather all supporting documents, revise the relevant NHS Trust HR policy, which directly relates to the alleged offence, misconduct or malpractice, and at all times maintain and ensure the integrity and confidentiality of the investigation and all relevant data.\nThe investigating officer's work culminates in the production of a comprehensive and detailed report on the case, and if necessary, attendance at the hearing if it is decided that there is a case to be heard.\n\u2022 Clarifying the exact nature of any complaint or allegation.\n\u2022 Liaising with HR support to organise correspondence and transcription of interviews for the hearing, tribunal or court, etc.\n\u2022 Maintaining an investigation diary, with all pertinent details of meetings, interviews, etc.\n\u2022 Compiling an event timeline.\n\u2022 Taking statements and carrying out interviews with staff involved at the earliest possible time.\nIn addition, the investigating officer will decide which documentary evidence is required, to support the case, which depending on the nature of the case being investigated, might include, for example: policies\/procedures, employment records, training records, timesheets, rotas, payroll records, bank statements, patient records, travel expenses claims, and transcriptions of relevant interviews, among others.\nThese types of investigation are of an open nature, as opposed to covert investigations, which must normally be authorised by the Trust's Fraud Specialist, Director of Finance or Human Resources.\nAlphabet are experts in dealing with highly confidential and material of a sensitive nature. Please feel free to discuss your requirements on +44 (0) 1707 260027.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If The Realm has the phrase, \"once in a blue moon,\" it must mean \"quite often.\"\nWhen we last checked in with Major Thom, he was trying to get into the Druid Temple but didn't know how to determine whether it was a night of a blue moon. It turns out that the game tells you quite explicitly in the text that the moon is blue, so I needn't have worried. On the night of the blue moon, at midnight, I burned the four sacred plants in the four braziers, and the way to the temple opened.\nThe temple started off easy enough. The first level had a little shop and a library. At the shop, I bought a druid cloak. Books in the library confirmed that the world of The Realm is flat, and said that the Druid Book of Life holds the most powerful spells in The Realm and must not fall into the wrong hands.\nOn a conference table, I found a memo that indicated that the Council of Druids had declined to help Bessak. \"He will have to fight Kruel alone,\" it said. Unfortunately, there was no word on where Bessak went after that.\nAt first, I thought the game was making an homage to Maurice Sendak, but now I see that it's \"Senoak.\"\nThe second level of the temple got a lot harder. It was a giant maze with four different entrances, swarming with druids who attacked me with some kind of force lightning. I hoped the druid cloak would disguise me, but it didn't do anything. I swiftly started to run out of clear and blue potions, which restore health and spell points respectively. Enemies respawn as quickly indoors as they do outdoors, so there's no way to clear them all and yet fewer ways to run away from them.\nOne of the books in the library notes that druids have an inexplicable love for mazes.\nAt the same time, another problem became clear. I hadn't slept since waking up that morning. Since the Druid Temple requires you to stay up until midnight to even enter, you're bound to run into fatigue problems unless you plan accordingly by staying up the previous night and then sleeping until dusk on the day that you go to open the temple. There are no beds in the temple, and I later confirmed that you can't open the temple on one day, then go away to sleep, then come back to enter. The door closes while you're away.\nStill, I pushed forward. The second level maze (a book said that druids love mazes) led to a stairway to a third level, where a bunch of twisting corridors--again swarming with hostile druids--led to the shore of an underground lake. Since you can't swim in this game, the only way I could find to cross the lake was to quaff a green potion, which polymorphs you into a frog. I had two green potions at the time.\nWhere's Baba Yaga when you need her?\nI soon discovered that there were two \"exits\" from the lake, both leading to small caverns with large spellbooks on pedestals. One cavern was blocked by a wall of fire, but the druid cloak protects against it. The first spellbook had something called \"Mascar's Spell of Protection\"; the second had a \"Mage Annulment Spell.\"\nGetting one of the druid spells on my second visit. I forgot to take a shot on my first.\nUnfortunately, by this point, I was dead on my feet from exhaustion, I had no way to keep fighting the druids, and because I had used both green potions to get to the two chambers, I had no way to get back to the exit. I had to reload a saved game from before entering the temple.\nUpon reloading, I had a plan: grind a bit; buy more clear, blue, and green potions; take the rest of the training courses; return in four days and try again. A commenter had said that blue moons occurred on nights 2 and 6, and I had entered the Druid Temple on night 14, so naturally I assumed they came along every 4 days. I planned my return for night 18.\nIn the next few days, I accomplished most of my list. I spent a little time fighting random enemies and got my experience points over 11,000--the highest value needed for any of the training courses. In Murkvale and Castleguard, I took \"Advanced Lockpicking,\" \"Transformations,\" \"Advanced Swordplay,\" and \"Advanced Spellcasting.\" With the money I made while grinding, I bought about a dozen of the blue and clear potions. Unfortunately, I couldn't find any place that sold green potions. But I figured I'd try again and perhaps find an exit or some other resource that I missed the first time.\nThe final guild course, which requires 11,000 experience.\nThen the blue moon came unexpectedly on night 17. I was nowhere near the temple. To make sure this didn't happen again, I saved the game and then burned a few days in Castleguard, watching for the next blue moon. It came on night 22, as I would have expected. So I'm not sure why night 17\/18 was the one weird departure from the pattern.\nAnyway, that gave me four days to screw around, so I decided to revisit some of the mountain locations. I originally set out hoping to find the dragon's cavern and kill it with dragonsbane, but I couldn't find it again. I couldn't even find a cavern of the same shape as the image in my last entry, so it wasn't just a matter of the dragon not being there. I stopped at the occult shop and sold my wyvern eggs, hoping their absence would stop wyverns from dive-bombing me with rocks, but it didn't do anything.\nBut while searching the mountains, I was surprised to find that my lockpicking skill now worked at Lord Dertak's Holde. (The game map has it as \"Dervak,\" but all the text in the castle gives it as \"Dertak.\") Entering, I found a large dwarven keep of two levels. Among the books in the library and the few NPCs in the tavern, I learned that dwarves are the original inhabitants of The Realm. Humans encroached later, and the dwarves \"deeply despise\" the human settlements. The dwarven population has been declining for centuries, forcing them to abandon the \"island keep.\" Dertak, meanwhile, is said to be an amiable dwarf who bears humans no particular ill will. He has been known to grant humans favors in return for gifts. He particularly likes jewels.\nI get some lore from a bartender . . .\n. . . and more from a book.\nWhen I encountered Dertak, he wasn't impressed by anything I owned, including the several jewels. But the trip was useful for other reasons. A shop in the keep sold enchanted swords and armor, and I had enough to purchase one of each. Moreover, I found both a third green potion and something called \"Grorph's Spell of Exiting,\" both of which sounded like they'd solve my problem getting out of the Druid Temple. A couple of skeleton keys led to a winding series of caverns where I found a treasure hoard and something called \"Kruel's Winterblast Spell.\"\nI still had a couple of days, so I returned to the mines west of Castleguard and employed my pick axe. Sure enough, I found a few generic jewels but also one that the game called out as a particularly large jewel.\nWhen I took the large jewel back to Dertak, he gave me a letter of recommendation for the dwarven high king, whose \"holde\" is in the mountain range north of Dertak's. With one day remaining before the blue moon, I went there.\nDertak is pleased with my offering.\nI had to bribe the guard to get through the first level. On the second level, I ran into some kind of golem who dissolved into mud when I defeated him.\nThe third level had the living quarters of the hold, but unlike Dertak's, it was swarming with (respawning) hostile dwarves who attacked me. (They're the same dwarves who attack on the surface.) Fighting my way through them, I reached the dwarven high king, who didn't seem to mind that I'd made my way to him by slaughtering his subjects. He took the letter and gave me in return a jeweled key that unlocks the door on the abandoned island keep. He then \"expelled\" me from his hold, which took me back outside automatically instead of forcing me to fight through dwarves. That was nice.\nI think the bodies behind me are evidence of \"my own valor.\"\nI had just enough time to make it to the Druid Temple--and again through that damned maze--before midnight. I did the ritual again and opened the door. This time, things went a bit easier. I got the two spells from the books and decided to use the exit spell (rather than the third green potion) to get out.\nI really hope I don't need that somewhere else.\nI checked my statistics and was surprised to find that my \"score\" had risen from 10 to 110 (out of 190). In one session, I went from 5% game completion to almost 60%.\nI returned to Bessak's keep, thinking he might show up there now that I'd found the spells he wanted from the druids, but he was nowhere to be found. So the only thing I can think to do at this point is to try the island keep.\nBefore I go, I have to spend some more time complaining about the combat system. It's not just bad; it's exhausting. The animated .gif below shows what happens when you stop and pause for just a few seconds.\nThe hero makes the mistake of pausing by the riverside.\nEnemies spawn constantly, and stopping to fight one is just an invitation for others to swarm you. You never really \"end\" combat; you just finally decide to break away. Getting from one place to another becomes an extremely tedious affair.\nThe combat system itself makes it worse. Victory and defeat are a matter of pixels--in a game where the graphics and controls aren't quite good enough to find-tune your alignment. And upgrades in equipment don't seem to make things much easier. I hardly notice a difference between wearing enchanted armor and wearing nothing. All the guild courses, attribute increases, and investments in increasingly-better weapons have only marginally improved upon my character's performance with a pocket knife at the very beginning of the game.\nOffensive spells don't help much. \"Fireball\" (which you can equip like a weapon) doesn't do any better than physical attacks and it's very hard to aim. (The same is true of missile weapons.) \"Pinok's Paralyzing Spell\" (which freezes every enemy on the screen) and \"Fistak's Fear Spell\" (which causes them to run away) are theoretically useful but have such a high cost that you can't rely on them.\nThe \"Fear\" spell sends enemies scurrying away but costs 50 points.\nStill, experimenting with spells for the purposes of writing this entry led to some interesting results. At some point, I collected a spell called \"Wyvern Morphing Spell,\" which transforms you to a wyvern. It only works outdoors, but it's probably my key to getting to the island keep, unless I want to rent a raft and paddle it all the way up from the delta. I otherwise don't know the purpose of renting rafts, though.\n\"Kruel's Winterblast Spell\" sounds like an offensive combat spell, but instead it's more akin to Ultima's \"Armageddon,\" turning the whole world to ice. I don't know when I'll need such a spell, but it doesn't seem like something I want to cast randomly.\nThe spell froze the sea serpent that was on-screen at the time, but that didn't stop a wolf from spawning and attacking while I was taking this screen shot.\nThe uses of \"Grorph's Spell of Darkness\" are equally mysterious--it blankets the area around you in darkness, making it impossible to see. Two others--\"Mascar's Spell of Protection\" and \"Portal Activation Spell\"--never seem to do anything no matter where I use them.\nLords of Time has some decent moments when it comes to exploration and puzzle-solving, unfortunately marred by the enemy and combat systems. I'm ready to enter the home stretch. I just hope that by the end, the game delivers some kind of explanation for its title.\nIf anyone has played and gotten anywhere with the 1989 text-RPG hybrid Advanced Xoru, I'd appreciate a line. I thought it would be a quick one, but I'm hopelessly stuck.\nThat maze doesn't work too well with the game's perspective... Good to see you're making some headway. Let's just hope you're not hopelessly stuck for some dumb reason. This seems like the type of game that would do that to players.\nThe screenshots look weird to me, too. It helps to see it in motion.\nI had similar issues with the graphics in one of the Ultima games - everything looks overbalanced or tilted in a precarious way that makes me feel weirdly uneasy. The weird thing is if I mirror-flip the image, it looks fine to me - it seems my brain just dislikes the tilt to the left specifically. Maybe something to do with used to reading English and therefore images left to right?\nDamn. This entry really makes a part of me want to try that game! You had quite a lot to do, and it was quite diverse.\nBut... Damn, that combat system !\nSounds super unappealing. The Spell of Protection and Portal Activation sound definitely like MacGuffin spells, so it'll probably be clear when it's time to use them.\nDid you already try the portal spell in the stone rings?\nNo, good idea. I'll see what happens.\nHow far did you get with Advanced Xoru? I gave it a try but I'm not very good at these kind of games. Finding the alchemists shop and the Ant-e room were my only achievements so far (completing 18%).\nI got to 29% in a couple hours. I didn't map anything though, and now I've lost track of what to do, plus I lost a lot of treasure. I'll give it a real shot over the weekend. It's pretty fun, but I don't know how much of an RPG this really is. Definitely more of a text adventure feel.\nI got to 43%. I can't figure out what to do in the \"Ant-E-Room\" but I found it. I can't figure out what to do in the large cavern with the statue and the broken jug. I'm sure it involves replacing it with the ceramic jug that you can find, but I can't figure out the right set of commands.\nIt's not much of an RPG, but a commenter persuaded me to look at it again. I've corresponded with the author and I have a lot of material on it, so I don't want it to go to waste.\nBullet room - no exits, it only has an idol that I can't seem to do anything with.\nPillar room - Square depression carved on the east side, but nothing I put in there seems to fit or do anything.\nLower elevator - it says it has two triangles to press, but no matter which I press I'm always taken to the Corridor room with no way back.\nLions - I'm not sure how to get past them either.\nThe game seems to have trouble with pulling descriptions for places that don't seem to match the room description. Like no matter which bookcase or shelf I search, it's always dusty tomes and empty boxes.\nSapphire cube -- I seem to be able to use it from anywhere, although I'm pretty sure it's supposed to fit in the plaque in the mirror room. I haven't found the other two shapes yet.\nWooden crank -- it's too large for every square peg I've found.\nHoly water -- other than drinking it I haven't found a use.\nGrey dust - no clue, but it's one of those static items every new game.\nThere are also the more random gems, coins, and other treasure that seem to have an obvious use. I'm figuring since those are more random they're not used for puzzles, but I could be wrong.\nIt doesn't help that the parser sometimes gets confused with shortcuts. Examine the \"azure crystals\" in one room, and it will examine the \"azure wand\" instead if you are carrying it.\nThose a little more forgivable I think. I'm assuming when that happens it would have given a \"nothing useful\" response otherwise.\nIn one room I tried to examine some holes, but it kept giving me the description for my holy water. One way to get around this is to drop those items it's in inadvertently describing in another room. Out of sight, out of mind.\nWhile I'm at it, there's some more strange things I haven't done anything with.\nHunter's room -- I'm not sure what the point of this room is, possibly to meet the hunter if I didn't randomly come across him already. Strangely the trophies are described as wooden in the room, but upon closer examination are ivory... no way to pick them up.\nFrom the command list, I haven't used KNOCK, LISTEN always seems to produce a silent dungeon even just after I've heard giggling or some other noise from afar, PRY doesn't seem to do anything though I the thought a prying out the emeralds from the elevator and maybe replacing them with another gemstone.\nLast thing to mention are the spell casting classes. None of the spells seem to actually do anything, and maybe they're only useful in combat. All the puzzle solving seems class independent, and fixed items are the only ones used to solve them.\nI don't think I've solved any of the riddles. Not sure if the jumble of letters on the doors with the archaic book are anything to solve.\nSo, I solved one riddle, now at 42%, but that didn't really give me anything else to go on.\nApparently, solving that riddle gives some variance to score, and it's possible to \"solve\" it multiple times to increase score\/%. One time it gave me 4 points, another time it was 18.\nzenic, you mention rooms that I never found: bullet room, pillar room, lower elevator, hunter's room, platform room. Clearly, I missed some exit or the solution to some puzzle. Can you write offline and let me know how you got to any of those?\nThe wooden crank fits in the wheel on the well in the domed room. It just raises the bucket to the surface. I'm not sure what else there is to do with that.\nI have recently started playing this but finding it difficult. I am also stuck how to reach the ceramic jug and what to do with the azure wand. Any help appreciated!\nSpell of annulment sounds a lot more convenient than a divorce lawyer. I love magic.\nMore than you love your spouse, by the sounds of it.\nI love the renaissance of this blog since the merge of the two timelines, the tone in your writing is so much happier, am I the only one that notised this change?\nI can imagine the reason for the immediate change. Towards the end of 1989, there were a spate of games that just wouldn't end and mostly sucked. And commenters kept feeding me new ones. I had to force myself to play just about every session.\nIf you mean for a longer period before that, I'm not sure why that would be.\nThe description of the final guild course bears an intriguing similarity to the Frobenius series solution of Bessel's famous equations for solving partial differential equations in the case of radial or cylindrical symmetry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So, with the arrival of the New Year, I wish I could say that I've been productive and that I've been making tons of progress on my needle felting, and have completely reorganized my house, cleaned it from top to bottom, and lost 10 pounds. That would be a lie. I've been shuffling around (ok, maybe wallowing a bit), reading a lot of books, stepping over a lot of toys (see below), and generally neglecting anything that may be labeled as productive.\nI have been baking a lot of bread from this book. (Maybe that's why I haven't lost 10 lbs!) Amazing. I know I'm late to jump on the bandwagon of no-knead bread. I got this book from the library and it was all so straightforward and delicious looking, that I dove right in. Normally at this time of year, I really miss the farmers market, because they have the best crusty, artisanal type bread. But this bread I made tastes exactly like it! Woo! Now I have to move beyond the basic recipe and try the others, which all look amazing.\nSpeaking of the farmer's market\u2026I've been looking toward spring today, thinking about our garden, picking seeds to order. I cannot wait until this winter is over. We've been trapped in this house, endlessly sick, dwelling amongst clutter and mess. Blah! Once Christmas was over, I begrudgingly packed up our decorations, and braced myself for bleak January, always a month to muddle through for me. Now more than ever it always proves to be a bummer, with sick kids, stuck inside, growing antsier by the minute. If I were a more positive person, I'd tackle January with gusto; a new year, a fresh start, yeah, yeah. But I never really feel that way until spring. SO, today, as I dream of puttering in the warm sun, and poking a pea into the ground, I feel hopeful, and dare I say, a bit more cheerful than in past weeks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If someone can provide a link to an existing thread that discusses the cost and options for getting into EAA, the actual observing equipment available new and used, I would appreciate it.\nI have perused threads here off and on over the years, and actually bought a 1970s era rifle intensifier (for $80) that was discussed here with the idea of trying the lowest cost option (but have not gotten around to working on setting it up).\nI understand the sorts of things these devices can do, and the value they provide.\nSo I am now thinking of spending the money on a more modern option (but still trying to keep cost within reason, used is fine, previous gen tech is fine).\nI am sure there are people here who have a grasp on the options currently available that can provide guidance on making a choice.\nDo you already have a scope?\nDo you have a Mac or PC already?\nThere are so many options so a little more info would be helpful.\nNeed much more information about your goals. Are interested in using a Camera or Night Vision?\nOK - I am interested in enhancing DSO viewing using an image intensifier, so computer, just the viewing apparatus. Also no picture taking, just viewing. The rifle sight device is an example of what I am looking for.\nI would like to be able to use it with H-alpha filters to view nebulae in non-pristine skies.\nI have a range of telescopes - a 13.1\" F\/4.38 dob, an ST-80, an 11\" F\/5.67 mirror scope, access to a 22\" F\/5 Dob, and I am looking at getting a TeleVue NP-101is. I build telescopes, and have a range of short FL, wide field type optics that I am considering building, if I can decide which project I want to do next.\nI would like to be able to use a large aperture telescope visually and with II in the same session, to help locate really hard to find (dim) objects, and to bring out objects that you can only see in a so-so way visually.\nWhat is your budget (asked previously in the thread, but this is very important)?\nClose to $4000 for a high spec tube new monocular would be about average give or take $300 or so.\nDo you want to view afocally only or use Prime focus with ability to make your own afocal adapter or use a digiscoping adapter? PVS-14 with the TNVC adapter on a Televue eyepiece or MOD 3 variable gain prime focus or NVD Micro Prime focus? PVS-14 and MOD 3 variable gain actually have variable gain on both devices. NVD Micro is not a variable gain device.\nDo you want to use two eyes? PVS-7 biocular with an additional c-mount adapter is a nice choice.\nThis is just a small sample of questions to narrow down your purchase. You'll want some filters and nosepiece to go with those.\nMy plan is (was?) what can be done for $2k or less.\nPeter's website is useful, and I have corresponded a bit just now with Jeff Morgan and Eddgie.\nI do want to be able to use the device in prime focus, binocular is not necessary.\nFrom Peter's site, and the messages I got from Jeff and Eddgie, it looks like my best option in my price range is the PVS-7, though there are many versions of that (considering all the different tubes), so deciding on what tube makes the grade then becomes a question.\nThe two monoculars, the Mod3 and the Micro, do sound like they are better choices but at a much higher price point. If I go for those, more capital accumulation will have to occur.\nIt appears that the PVS-7 can be had (in some version) starting around $1000 on eBay. The Micro starts at $3000 and the Mod3 at $3500.\nLooks like you are on your way then with all the assistance necessary. Excellent. I'm sure Jeff and Eddgie will provide all the useful info needed.\nGood luck. NV is a game changer for Richfield and viewing those nearly invisible objects.\nI got started with Night Vision with a PVS-7 several months ago, and I feel it was a great choice. Make sure to get the support of the experts in selecting a good tube for it, and whenever possible ask the seller to show you decent pictures of the images produced before buying it. I spent $1500 (\"new\" surplus tube and used PVS-7 both on EBay) which for your budget would leave $500 for the C-mount adapter, the 1.25\" barrel adapter and a 7 or 12 nm H-alpha filter. If you have old camera lenses like I did, then with a few extra dollars you get an adapter and give a new life to those old 28, 50 and 135mm lenses.\nI prefer the ScopeStuff 2\" nosepiece with inset 1.25\" filter threads. It is a c-mount nosepiece and has the 2\" threads at the end of the nosepiece while the 1.25\" threads are inset in the nosepiece.\nThe reason for the preference is choice of filter sizes and more importantly - I have not found any 1.25\" focal reducer that doesn't vignette. Because of backfocus issues with most NV devices, the focal reducer is usually put on the end of the nosepiece (after the diagonal rather than before it) and while it may not give true .5x reduction, it gives dramatic reduction and have found no focus issues with my telescopes. I found the Antares 2\" .5x reducer works well in all of my scopes (AT72ED, 120ST, AT152, mak 150) with all my NV devices (pvs-7, pvs-4, Litton M942, NVD Micro, NAIT NVPS-10) and doesn't cause vignetting.\nI have a 1.25\" nosepiece as well but doesn't get used much for reasons I stated. I use the Antares reducer about 85% of the time when viewing in Ha narrowband. The difference for me is in my 120ST at Native F\/5 the Flame nebula will show up but can't see the Horsehead. With reducer in, the HH will even show as a lump in my AT72ED. Many threshold objects become more detailed or even just become viewable with the reducer. My location is heavily light polluted though (white zone with some access to red zone conditions coastal).\nUsing a 2\" nosepiece also gives a little more backfocus room as the 1.25\" adapter extends the mm to the device. If going with strictly 1.25\" diagonal then it is a non-issue as it doesn't increase optical train length.\nMaybe one of the experts will chime in here on this thread though.\nIf the thread starter decides to go with a 1.25\" nosepiece and 1.25\" filters, the use of a .7X focal reducer is outlined in the post below and in some of those that follows (#8, 15). I just did the same and it works well with my 78mm Televue Pronto . Just reporting one solution, not suggesting it is necessarily the best one.\nHas anyone compiled a list of the rube options for the PVS-7? If so, can you provide a link to the thread (or website)?\nIf not, I will compile one and post it here.\nThat should get you started on your adventure.\nIt's rather unfortunate there is no dedicated NV subforum with stickied information from \"Best Of\" threads.\nThreads get buried quickly and the search function works to some degree but there is a lot of information spread out and not collected in a concentrated area.\nLots of contributors on the forum and lots of sporadic visits too. It takes a bit of research to familiarize with what is being talked about, but the actual use of the device is fairly simple once the right parts and filters acquired.\nCN has some articles on NV use as well. The Ardent, Eddgie, and myself did one on different type observing using very different scopes or 1x and 3x use. GeezerGazer has collaborated with others here on a few articles including phone photography and differences between 5nm and 7nm Ha filtering, older articles too. Haven't checked them all yet because I was off CN for a long hiatus while unable to do any observing except two nights. On the mend and active again now and just catching up on things here.\nNV gear and associated gear can have a multitude of configurations.\nIn the end you'll decide what is best for you. The more information you have to make that decision, the better.\nEdited by Vondragonnoggin, 16 February 2019 - 08:08 PM.\nI almost neglected to put this amazing thread link from the Deep Sky Observing Forum by Allan Dystrup on Classic Richfield Observing. He starts observations with classic Richfield configurations, then starts using NV afocally to supplement his observations. It's one of the coolest threads I've read on richfield use and full of great technical info on objects, but not on equipment. Just mentions of equipment but focus minimized on equipment and heavy on the Observing aspect.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"PICTURE BY TIM DE WAELE Tour de France boss Jean-Marie Leblanc unveiled the route of the 92nd Tour de France in Paris's Palais des Congrs this morning, and as usual it contained a few surprises mixed in with the traditional staples. The biggest debating points are the absence of most of the Tour's great mountain challenges, including Alpe d'Huez, the Ventoux and widely expected return to the Puy de Dome, and the inclusion of just one major individual time trial on the penultimate day of racing. The opening stages in the Vende have been known for some time. The race starts with a 19km time trial out to the Ile de Noirmoutier, the length of which was a considerable surprise given the by now almost traditional prologue of between five and 10 kilometres. But more surprising still is the absence of another individual time trial until a 55-kilometre test at St Etienne on the penultimate day. This is reason, it seems, for the climbers to be rubbing their hands with glee. However, that bonus for the mountain men is rather nullified by the lack of high mountain summit finishes \u2013 there are just three, at Courchevel, Ax 3 Domains and Saint Lary Soulan. Although the usual suspects are certain to be in contention, headed by six-time champion Lance Armstrong, the inclusion of some tricky stages through the Vosges mountains and the Massif Central may well open the race up to some new faces. Perhaps these tweaks to the standard Tour route have been made in order to prevent Armstrong dominating in the all-encompassing way he did this year. The race that splits into two very clear sections: the first of flat racing, across the country from west to east and the second of heavy climbing, from north to south and then back north again. After the Vende opening, a succession of flat stages take the race into the Loire heartlands of Tours and Blois. It's here after four stages that the 66-kilometre team time trial will take place. In 2003 and 2004, US Postal proved to be the dominant collective force. On these rolling roads, the Discovery Channel team could yet find them a hard act to follow. The first of several transfers takes the convoy south of Paris, on to Troyes and then Nancy, prior to crossing the German border en route to Karlsruhe at the end of the first week. By the time the race crosses back into France on the stage from Pforzheim to Gerardmer, the climbs will have begun. The wooded slopes of the Alsace may not be as fearsome as the Alps or Pyrenees, but they will soften up those who are flagging. After an air transfer from Mulhouse to Grenoble the crossing of the Alps begins with the first summit finish at Courchevel. The climb to the ski station will be preceded by the Col de la Madeleine, one of the hardest mountain passes in the Haute Savoie. The next day, the peloton tackles more heady summits as it snakes south towards Briancon, followed the next day by the tough climb of the Corobin on the way into Digne Les Bains. The Midi will pass in a blur with a fast stage Montpellier, prior to the first Pyrenean summit finish at Ax 3 Domaines, site of Carlos Sastre's famous 'dummy-in-mouth' victory in 2003. The next day the convoy climbs to the tiny Pla d'Adet ski resort, tucked away in the Pyrenean escarpments. A third Pyrenean stage (which will also be the setting for the Etape du Tour mass participation event) takes the race on a loop through the Atlantic Pyrenees and then descends to Pau, where the second rest day is scheduled. By now, you might think that the climbing was over but there's plenty more still to come in the final week as the peloton squares up to the Massif Central. The longest stage, of 239 kilometres, from Pau to Revel comes before a tough stage from Albi to a finish at Mende, where Laurent Jalabert scored a famous Bastille Day win in 1995. After another transfer, from Mende to Issoire, the bunch sweeps into the steep streets of Le-Puy-en-Velay, prior to yet another transfer to the showpiece Saint-Etienne time trial centred on the Pilat national park. This race route has been conceived in the hope of a cliff-hanging finale. Whether, on the basis of last year's Tour, that is likely, depends not only on Lance Armstrong but also on those rivals who capitulated so easily in 2004. Yet you can't help feeling that Armstrong, with the benefit of all of his experience, will see more opportunities here than obstacles.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nick Rogers was born and raised in Venice, California. At 14, following five years of children's theatre, Nick eagerly took on the role of Hugenberg in a critically acclaimed production of Frank Wedekind's darkly sexual work \"Lulu\" at the Pacific Resident Theatre. After seven musicals at the Hamilton Academy of Music High School, Nick attended Muhlenberg College in Allentown, Pennsylvania where he majored in theater. His collegiate theatrical credits include \"On the Town\" and \"Pippin\" (Pippin), as well as Chekov's classic \"The Seagull\" (Treplev). Now back in Los Angeles, Nick has appeared at the Pico Playhouse in Illium Entertainment's production of \"The Bubbly Black Girl Sheds Her Chameleon Skin\". He offers heartfelt thanks to all who have helped him on this journey.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nothing is more special than quality time connecting with the one you love most. Give that opportunity to a couple you love with a monthly credit to a Datebox subscription. Datebox designs new date ideas every month so you can give a one-of-a-kind gift your friends & family never forget.\nThe datebox will delight your loved one with a fun night in. We love this as a way to brighten someones evening. Send one fun date night or a reoccurring subscription of three months!\nWhat to expect for a date night in: At-home dates turn your mailbox into a magical box of wonder. Your date will be delivered straight to your door. Your at-home dates will contain specially curated activities, snacks, and items sure to help you two have an awesome date. There are romantic dates, artsy dates, & a whole lot of others that we can't wait to surprise you with.\nA member of the Rally Registry support team will contact you upon purchase to confirm personal note to send alongside the Datebox Subscription Credit. We will also coordinate with you to send the credit electronically or via mail.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With all the felting for the piece completed, I sent all the individually worked components back to Sarah West in Raleigh. Sarah had to finish brazing the metal forms that she had left incomplete so that I could manipulate them under my sewing machine pressure foot to free-motion embroidery the felt panel on each. To protect the white felt from getting dirty as well as from burning in this process, she wrapped the felt panel in plastic first and then covered it with wet paper towels. Brilliant!\nSarah came to Asheville with all the components and we arranged them on a body form, marking which pieces would be connected together. She then returned to her studio to prong set the pieces of vinyl and to link all the components. At this point, I was leaving the country to teach workshops in Santiago, Chile and had to pass off the remaining responsibilities to her. I am grateful for her final push to complete the piece, find an excellent model and photographer, coalesce our processes and concepts about the piece into a coherent statement and submit the work on time.\nThis piece uses the interplay of steel and felted wool to create a transformation of materials into a wearable protective shield. The shoulder components were created by encasing the underlying brazed steel structure in a seamless skin of wet felted wool fiber. As the felt shrunk, the forms compressed to alter their original shape, suppressing movement and sound. The work further eliminates sound by introducing pieces of vinyl from a heavy metal record. The free-motion embroidery is both inspired by the pattern of the record and the intensive energy of the music. In the act of wearing it, there is a final transformation from an individual's natural state to one that is powerfully guarded.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Davis, a former Democratic state senator from Fort Worth, recently began an \"exploratory\" relationship with the American Program Bureau\u2026.\n\"We're trying to determine whether or not we can do the job for her, so we have not made any contractual relationship yet,\" said Bob Davis, senior vice president at the agency.\nThe bureau primarily is pitching Wendy Davis to college and community groups as an authority on women's health and education issues.\nIn 2013, Davis captured the nation's attention with her 13-hour filibuster, which demanded that she stand the entire time\u2026.\nWhile that bill unfortunately passed into law \u2013 a law that is still being challenged \u2013 Davis hasn't stopped working for women's rights for safe, legal reproductive care. She understands exactly what it means to depend on accessible, safe and affordable care services. \"It's paramount that women have access to the incredible preventive care that Planned Parenthood provides. When I was a young woman, Planned Parenthood was the only source of healthcare I had \u2013 for several years. Contrary to what some believe, Planned Parenthood provides a multitude of services, including breast and cervical cancer screenings, HIV testing, routine gynecological exams, diabetes screenings and family planning services. To restrict care for low-income women would mean setting society back decades \u2013 to say nothing of putting further strains on the lives of women who need access to these essential healthcare services.\" Davis fought her best against Texas Senate Bill 5, the law that attempted to shutter all but eight women's health clinics, but her efforts were dampened by then-governor Rick Perry.\nLikewise, APB Speakers' comical YouTube promo video features Davis as the incredible shrinking half-woman appearing in the oddest of places, who \u2013 again \u2013 pitches solely to Planned Parenthood.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"East Hanover, NJ \u2013 When temperatures drop below freezing as they have early on in the new year, and conditions pose a threat to individuals who are homeless or medically fragile, County Offices of Emergency Management together with county or municipal government will often initiate a Code Blue Alert which enables authorities to take homeless adults to shelter programs that have agreed to make additional beds available. Warming Centers are also opened during stated hours to provide residents with a place that they can go to warm up.\nNJ 2-1-1, the free, user-friendly information and referral service that connects people-in-need with critical health and human service resources, is working with county government offices throughout the state to post information on its website and through its call center when Code Blue Alerts and the opening of Warming Centers are communicated.\n\"We encourage new providers of these services to inform 2-1-1 so we can have the most current and accurate directory for callers and those accessing our website,\" said Tom Mergola, NJ 2-1-1's director of operations. As NJ 2-1-1 hears of locations that are available when a Code Blue Alert is activated, information is then posted on the site.\nThe national 2-1-1 network makes available a free, user-friendly phone number that serves 90% of America's population, and connects some 16 million people a year to critical resources, information and services. In New Jersey close to 230,000 people called 2-1-1 for help last year. Over 212,000 visited the website in search of resources and education. Tours of the New Jersey 2-1-1 call center are available upon request. For additional reports about calls handled by NJ 2-1-1 Partnership contact Thomas Mergola at 973-929-3705 or visit www.nj211.org.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Hagall spread is a tool for revealing the path of spiritual growth in difficult situations. It is a favorite of mystics and those confronting a major life challenge. The Cat People Tarot paints a picture of faraway lands trod by mystical archetypes and their feline companions. This deck is a perfect tool for dreamers and idealists, and is a great choice for divining the nature of human imagination. If you would like your own copy of the Cat People Tarot, you can buy it now!\nThe card in the middle of the circle represents the core or central issue of the situation. Ten of Wands (Oppression): Overburdened feeling. Excessive pressures. Problems soon to be resolved. Striving to meet a goal or a certain level or position. Possibly using power for selfish ends.\nThe card at the bottom of the circle represents something you did to bring the situation about. Seven of Wands (Valor): Success. Gain. Overcoming obstacles and challenges. Advantage. Victory.\nThe card at the bottom left of the circle represents your beliefs, impressions, or expectations. Three of Pentacles (Works), when reversed: Sloppiness. Mediocrity. Lower quality. Money problems. Commonplace ideas. Lack of skill. Preoccupation.\nThe card at the bottom right of the circle represents the most likely outcome of the situation given present circumstances. Five of Wands (Strife): Unsatisfied desires. Struggle. Labor. Endeavors. Violent strife. Conflict. Obstacles.\nThe card at the upper left of the circle represents the spiritual history of the situation the things you've learned. Eight of Pentacles (Prudence): Apprenticeship. Craftsmanship. A person who learns quickly. Candor. Frankness. Modesty. Handiwork. Personal effort.\nThe card at the top of the circle represents the spiritual tasks and challenges of the present situation. Nine of Swords (Cruelty), when reversed: Doubt. Suspicion. Slanderous gossip. Shame. Scruple. Timidity. Shady character. Reasonable fear.\nThe card at the upper right of the circle represents the metamorphosis of the spiritual situation, and how your knowledge will evolve. The Chariot: Perseverance. Major effort. Possible voyage or journey. Rushing to decision. Riding the crest of success or popularity. Adversity, possibly already overcome. Turmoil. Vengeance. Need for supervision. Need for attention to details. Urgency to gain control of one's emotions. This card suggests that one can achieve greatness when physical and mental powers are maintained in balance.\nThe card at the left of the lower line represents the person or qualities that will sustain your spiritual journey. The Empress, when reversed: Vacillation. Inaction. Lack of interest. Lack of concentration. Indecision. Delay in accomplishment or progress. Anxiety. Frittering away of resources. Loss of material possessions. Infertility. Infidelity. Vanity.\nThe card in the middle of the lower line represents the qualities that you express in this circumstance. Nine of Pentacles (Gain), when reversed: Threat to safety. Roguery. Dissipation. Danger. Storms. Bad faith. Possible loss of a valued friendship or a treasured possession.\nThe card at the right of the lower line represents the person or qualities that will reveal spiritual knowledge. Page of Cups, when reversed: Inclination. Deviation. Susceptibility. Temporary distraction. Seduction. A flatterer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It has been a while but we're back.\nThis time we have something different for you.\nAbout a month ago we took pictures with a disposable camera in the botanical garden in Berne. It was really exciting not being able to see the pictures until they were developped. When they arrived it was fun to see how they turned out.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Barnham Broom overlooks 300 acres of beautiful Norfolk countryside with 46 refurbished bedrooms offering comfortable, stylish and luxurious accommodation and is probably best known for its two championship golf courses surrounding the River Yare Valley.\nAffinity won a competitive pitch to redesign and redevelop the existing website, introducing a new responsive build structure to maximise usability across all devices from hand-held mobile through to tablet and desktop screens. The site connects to complex back-office booking systems for the hotel, restaurants, golf club and Spa with affinity's technical team integrating data flow between systems.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pdf splitter merger software is robust Windows utility combine thousands of pdf documents together and split a large size pdf file into several smaller chunks, each having specific number of pages, create a new pdf file by extracting a range of pages from large pdf. Download free evaluation copy to join pdf files together and to break pdf pages according to presets. Pre-sets allows user to split, merge, delete or extract pdf pages according to a user definition means, odd pages, even pages, all pages, page range like 3, 9, 12-29\u2026 etc. This Windows based utility maintains data integrity after splitting merging or removing pdf pages. Trail version shows full functionality except a watermark on output file. Pdf page joiner cutter and extractor program support secure pdf files for joining pdf files, bursting pages and for exporting part pages. In addition, tool also facilitates to encrypt output pdf files with restrictions and open password. This easy to use and affordable utility is reliable and convenient way to manage pdf pages in a meaningful manner. Some features supported by program are: (1) Program is Windows compatible. (2) Available with help manual. (3) Encrypt output pdf files with passwords. (4) Split bulk pdf files in a single instant. (5) Various pre-sets are available to define pdf page manipulation steps.\nSplit any files directly onto hard\/floppy\/zip disk for Windows 98\/Me\/NT\/2000\/XP. For WAV file type, the split files can be play. It will format floppy disk upon your request. Convenient when you cannot put a compressed\/zipped file into a floppy disk.\nFree Kvisoft PDF Splitter is an intuitive desktop utility program lets you split any pdf file into smaller pdf files. The free PDF splitter provides powerful functionalities to help you split PDF file into several parts with setting unique file name.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzayzku b/data_all_eng_slimpj/shuffled/split2/finalzzzayzku
new file mode 100644
index 0000000000000000000000000000000000000000..42467980de68c4f978f921b4b81f3fbb145409aa
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+{"text":"7 best Compass Rose for Deck images on Pinterest Compass, Compass rose and Wind rose.Compass Floor Stenciladd in the latitude and longitude coordinatesyou'll always know where home is :) Find this Pin and more on Compass Rose for Deck by burns2312. Compass Floor Stencil - for Sandy for beach house! Compass on Floor: Need a compass ..\nConcrete Medallion Stamps - The Concrete Network.Get tips for buying and using medallion stamping s, such as a compass rose, to customize your next decorative concrete project. .. 3. Look at tool texture and rigidity. Farnsworth says that the texture is critical when working with medallion stamps and ..\nWall and Floor Stencils - Stencilease .High quality, affordable and easy to use Large Format Wall and Floor Stencils. Flat Rate regular shipping and 100% satisfaction guaranteed - Stencilease .. Click here for more product information Stone Wall Patio and Wall Stencil Click for pricing and ..\nFind nautical compass Stencils - Stencil Search |.nautical compass stencil designs for home decor, walls, DIY and crafts. .. Find nautical compass Stencils Looking for nautical compass stencils for your next crafts or home decor project? You've come to the right place.\nNautical and Sea Stencils - Wall to Wall Stencils Products.How to stencil Frequent Questions Gallery Projects Paint & Surfaces Contact Us Home > Other > Nautical .. Compass Rose $12.00 Sea Buddies $19.50 Sailboat Border $15.00 Lighthouses $32.50 Coral Sea Garden $7.99 Sea Shells $7.99 Surfboard Chair Rail ..\nCompass Rose Reverse Stenciled Table - The Kim Six Fix.It was solid pine, with a really light stain, and I loved the wood, but I just didn't like how it went with the rest of the room. I knew I wanted to paint it, but to make it more \"nautical\" I thought I would stencil a compass rose on the top. I didn't want to paint the entire ..\nFind more shapes and stencils - Office Support.FInd and save additional Visio shapes and stencils. .. Under Local, the search results are displayed and categorized by their Visio stencil title. Under Online, if available, you'll see shapes on the web by other companies, including Microsoft.\nWall Map & Nautical Compass Stencil Artsy Chicks Rule Popular Flooring.Easy diy - wall map & nautical compass stencil, repurposed $4 thrifty find nautical compass rose flooring, .. compass, rose farm inlays google. Pin by anna tausend on :: floors :: pinterest compass rose, the compass rose one deck builder's story youtube. She ..\nThe Compass Rose One Deck Builder's Story Youtube Popular Flooring.No description nautical compass rose flooring, the compass rose one deck builder's story youtube. Brockman 60\" compass rose floor medallion inlay ..\nNautical Compass Houzz.Shop a wide selection of Nautical Compass in a variety of colors, materials and styles to fit your home. Enjoy free shipping and discounts on select orders. ..\nCompass Rose Table: 8 Steps (with Pictures).Introduction: Compass Rose Table Coffee tables are nice, but they can sometimes be so plain. This table is a solid table and great for supporting several dozen cups of coffee, ..\nprintable compass rose designs - Bing images Compass Rose Pinterest Compass rose, Compass and Stenciling.Make your own elegant framed art or use on pillows using this nautical themed Compass Rose stencil. 12 Inch Compass Rose Wall Stencil See more dessiner une rose des vents ..\nStencilsLAB Best Decorative Wall Stencils StencilsLab Wall Stencils.Rose Of Wind Stencil - Compass Stencil - Circular Stencil - Large Wall and Floor Stencil $ 27.00 SPRING LEAFS- Allover Wall Stencil- Floral Wall Stencil $ 23.99 ..\nPaving Design Compass Roses Pinterest Compass, Compass rose and Patios.Find this Pin and more on Compass Roses by jhegele. See more Concrete Stencil Accent - Compass Star Circle Rose Stencil Compass Rose Concrete Patios Decorative Concrete Outdoor Projects ..\ncompass rose decal eBay.Find great deals on eBay for compass rose decal and nautical compass rose decal. Shop with confidence. Skip to main content eBay Shop by category ..\nVideo: Stenciling a Compass Rose Window Shade Martha Stewart.Tom Tamburello helps Martha Stewart with her map room design by creating a compass rose design stencil for a window shade.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ti's latest report, Leading European Transport and Logistics Markets 2017, reveals a small slowdown in growth for the European logistics industry in 2016. This was seen in contract logistics, road freight and express, although freight forwarding growth accelerated.\nMacroeconomics can in part explain these differences, but the new report provides context to Ti's market sizing, detailing the key economic indicators and other factors that explain the state of the European market.\nIn addition to the overall European situation, Ti examines the logistics dynamics of 26 countries. Using data from the World Bank, World Economic Forum, Eurostat and IMF, it explains the strengths and weaknesses of these markets.\nTi also reveals individual market sizes as well as the overall market leaders and their revenues in each country*. There are familiar names across the continent. DHL leads in over half of all surveyed countries, but there are different leaders in every other market.\nFurthermore, the report provides details of what the future might look like for these countries. Four-year real compound annual growth rate forecasts are given for logistics markets in each country**.\nTo capture growth over the long run, countries need to have the necessary transport infrastructure. Ti provides details on the master plans, visions and key projects devised by governments aiming to seize this growth. Where these are not up to scratch, or simply not in place, volumes will be diverted elsewhere.\nCountries are evidently competing on a continental scale. The ranking of Europe's top 20 container ports reveals that Rotterdam continues to lead Antwerp in Northern Europe, whilst Algeciras trumps its Spanish rival Valencia in the Mediterranean. Many ports have ambitious plans for capturing transhipment freight, but it seems unlikely that there is enough freight to go around for everyone to succeed. The last five years of data makes clear how each port is tussling with one another for market position. Similarly, intra- and extra-EU air freight volumes are provided for each EU country for five years.\nDomestic, international and cabotage road freight traffic volumes are also revealed for each country. Unsurprisingly, Germany dominates, but Poland continues to capture volumes. Here, road freight prices have risen more drastically than any other country over the last 10 years.\nIn summary, this report provides a full review of the state of the European market, and gives insight into how it may evolve in the years to come.\nThe New Leading European Transport and Logistics Markets 2017 report will be available to download on Thursday 23rd November \u2013 you can pre-order the report today and save 25%.\n*Contract logistics, express, freight forwarding and road freight market sizing revealed for most of 26 countries, but not all. Market leaders provided for 22 out of 26 countries in report.\n**Where 2016 market sizing is provided for a country, forecasts accompany the data.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hi peter loved your story the best part was about the night before well done ?\nHi Peter, thank you for sharing your writing with us this week.\nI read Collins' story before yours, and I think you both have a cheeky sense of humour! I like the way your last line makes it look like you have a smart and a dumb character!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Smokey Beach Army Training Happy Days Dress-up Canyon Valley Rally Sasuke Tree Climbi..\nDescription: Wake that box up! Put your drawing skills to the test and see if you can creative shapes of all sizes to wake that sleepy box up.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you are considering the purchase of a new or used Nissan, then look no further than Bishops Nissan, as we can provide you with a free and easy online valuation for your current car.\nAt Bishops Nissan, we are proud to offer a fair estimate of the value of your vehicle. Unlike many car buying sites, that have to allow for auction house fees to sell on your car, we don't have to worry about these overheads - meaning that we can offer a fairer price to purchase your car.\nBased on information such as your vehicle registration and mileage, we can giev you an estimated valuation of your current car in part exchange for one of our new or used Nissan models.\nTo retrieve your valuation in a matter of minutes, simply enter your vehicle registration below. If you would prefer a face-to-face appraisal of your vehicle, however, please get in touch or visit our showroom in Guildford.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbaacg b/data_all_eng_slimpj/shuffled/split2/finalzzzbaacg
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+{"text":"Locals-backed Teesta Samajik Sangrasha Samity (TSSS) will move National Green Tribunal (NGT) against proposed construction of Teesta Low Dam Project (TLDP) I & II.\nSurveyors of proposed TLDP on the river Teesta had to leave the site after angry local residents started protesting against the projects.Four hydro projects were sanctioned in this region in the year 2014.\nTSSS,a decade-old social organisation of the region, organised a public meeting on Saturday at Teesta local haat bazar,which was attended by at least 200 agitating residents.The meeting convened a local social worker Mr Binod Rai as the president.\nThe attendee residents were united on the view of a danger which may pose a danger to the region following construction of proposed dam at Triveni.\nThis is a confluence of Rangit and Teesta rivers which is about a kilometer and half from Teesta Bazar.\nProminent speakers Mr Vishal Rai, Mr Sukbir Tamang and Mr Shyam Gurung appealed to all those concerned to come forward so as to protest against the proposed dam counting its adversaries.\nIt is to be noted here that Rangit and Teesta river have been closely attached with the socio-economic, religious, natural and cultural sentiment of the Hill people.\nrivers affects the sentiment of local Hindus who worship in the confluence as a sacred spot and assemble every year on the occasion of 'makar sankranti'.\nThe meeting decided to make the public aware of the consequences through putting up posters,meeting the SDO and submitting a memorandum,consulting all NGOs working in the environment field and drawing attention of National Green Tribunal to put a stay order said TSSS member, Dilip Prasad.\n0 Respones to \"TSSS to move Nat'l Green Tribunal against Teesta Low Dam Projects\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's the weekend... Saturday night to be specific. Not out mucking it up, but visiting family. November 1st, All Saints Day. Paying our respects to the departed. M doesn't understand why this is a thing, but it is.\nToday was another Tour de Donut, which I should write about. We'll see how that goes. I'm still full from this morning - I consumed about half a dozen. I may have hit my quota for the season.\nGoing over the review sheet for my Baking class. This coming test will cover the last two and the upcoming lab, so pies and cookies. The review was supposed to be done and handed in during this week's lab. Yeah... didn't do that. Worse case scenario, she gets it next week. Whatever, this isn't hard (but don't tell that to some of my classmates). Write by hand, type it up later to hand in. If anything, it'll reinforce the important concepts.\nI still need to write my personal statement for graduate school. I've been attempting to write this since February. Talk about writer's block. Sure, I have notes here and there... a sentence jotted down on a copy of the New York Times because an article brought about some inspiration. My reluctance to complete it probably shows I don't really want to go to grad school, but it's something that has to be done. Upward mobility, career path... blah blah blah.\nI think I'll finish off the leftover ribs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sunrise Cafe offers a casual atmosphere along with both indoor and patio dining.\nTake Out. Reservations Accepted. Outdoor Dining. Full Bar on Site. Children's Menu Available. Early Bird Specials. Banquet Facilities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Giclee printed on high quality paper, cotton canvas or adhesive sticker paper. A high quality painting reproduction ofJessie Willcox Smith. Professional packaging for safe shipping. Natural white border will be added to the image for mounting and framing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Magic-Flight Finishing Grinder helps create the finest of grinds. It acts as an enhancement too for pre-ground herbs by adding a finishing touch of a super find grind. A fine grind makes the vaporizing experice even better.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbashv b/data_all_eng_slimpj/shuffled/split2/finalzzzbashv
new file mode 100644
index 0000000000000000000000000000000000000000..7fa50295c6575c2385d2ad9ef533251a50122bf0
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+{"text":"Any questions about whether Nike might quickly back away from its decision to embrace the polarizing quarterback Colin Kaepernick amid threats of a boycott and criticism from President Donald Trump dissipated Wednesday when the company released a two-minute advertisement narrated by him and announced plans to have it run during the NFL's first telecast of the regular season.\nThe ad, called \"Dream Crazy,\" features Kaepernick and other star athletes in the Nike stable, including Serena Williams and LeBron James. It implores viewers to dream big, using the inspiring stories of those stars and of everyday weekend warriors who overcame illness or disability to triumph athletically.\n\"Believe in something, even if it means sacrificing everything,\" Kaepernick says over images of him watching a waving American flag projected against a building. Those words appeared in an ad that was released Monday announcing Nike's new partnership with Kaepernick and on a billboard of him that went up in San Francisco on Tuesday.\n\"Nike's 'Dream Crazy' campaign will air this week during sporting events such as the U.S. Open, MLB and college football in addition to 'Thursday Night Football,' \" said Nike spokesman Josh Benedek.\nGreg Hughes, a spokesman for NBC Sports, confirmed that Nike had purchased airtime on Thursday's NFL game. The league did not immediately respond to a request for comment. Nike is a major partner of the NFL's, providing the uniforms for all 32 teams and the clothing worn by everybody on an NFL sideline.\nIn 2016, while playing for the San Francisco 49ers, Kaepernick began kneeling during the national anthem to protest racism and other social injustices. Other NFL players, as well as a few athletes in other sports, followed his lead.\nAfter initially being silent about the Nike campaign, Trump, who has hammered the NFL repeatedly for not barring players who do not stand for the national anthem, took to Twitter to criticize the company.\n\"Just like the NFL, whose ratings have gone WAY DOWN, Nike is getting absolutely killed with anger and boycotts. I wonder if they had any idea that it would be this way?\" Trump tweeted Wednesday morning.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This article provides an overview of the role that interdisciplinarity plays in the Spanish education system. With this aim, we first describe the main conception of the term interdisciplinarity in texts written in Spanish, including other terms that have similar meaning. Then we review the role of interdisciplinarity in the Spanish curriculum at different levels of education, focusing fundamentally on compulsory education. This serves as the basis from which later to analyze Spanish research on interdisciplinarity. Finally, through results of this research and some examples of interdisciplinary school practices, we extract conclusions about the role of interdisciplinarity in teaching practices in the classroom.\nSegovia, Isidoro, et al. \"The conception and role of interdisciplinarity in the Spanish education system.\" Issues in Integrative Studies 28 (2010): 138-169.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Image Title: Free Printable House Coloring Pages For Kids Playroom Ideas Regarding 12. Post Title: Free House Coloring Pages. Filename: free-printable-house-coloring-pages-for-kids-playroom-ideas-regarding-12.jpg. Image Dimension: 875 x 620 pixels. Images Format: jpg\/jpeg. Publisher\/Author: Raheem Aufderhar. Uploaded Date: Wednesday - October 10th. 2018 05:35:23 AM. Category: Coloring Pages. Image Source: home-designing.com. Top 20 Free Printable House Coloring Pages Online Regarding 5. The White House Coloring Pages Free For Kids 2018 With Regard To 17. Coloring House Pages Free Printable For Kids Front With 11. Free Printable House Coloring Pages For Kids Inside 18. Victorian House Coloring Page Free Printable Pages With 3. Free Printable House Coloring Pages For Kids Stavby Budovy In 6. Houses Coloring Pages Free With House 2. Kids Coloring House Pages Page For Free 8. Coloring Pages Houses Small House In The Village Page Free With Regard To 16. Free Coloring Pages Of A House Page 2018 Whiterodgers Regarding 7.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"AN OPHTHALMOLOGY OFFICE IN ELMHURST, QUEENS IS SEEING FOR CERTIFIED MEDICAL ASSISTANT OR OPHTHALMIC TECHNICIAN FOR FULL TIME POSIITION.\nTHE CANDIDATE MUST SPEAK SPANISH.\nSKILLS, MUST HAVE ABILITY TO WORK WELL WITH OTHERS.\nWE ALSO SEEKING FOR SPANISH LICENSED OPTICIAN.\nGREAT BENEFITS, PAID VACATION, SICK\/PERSONAL AND HOLIDAYS.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home \u2014 Wolf Gap Education Outreach, Inc.\nTHANK YOU to the athletes who participated in 2018's Tick Ridge Trek Trail Run! Thanks to the generosity of the race organizers, Wolf Gap Education Outreach received a $5,000 donation from the race proceeds. Generosity such as this make it possible for us to provide local history education at little to no expense for Giles County Schools!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbgym b/data_all_eng_slimpj/shuffled/split2/finalzzzbbgym
new file mode 100644
index 0000000000000000000000000000000000000000..185db1bb6268d264bc3d0cc3e895c42eb5994d58
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"I started illustrating my father's newspaper articles at age eleven. As a full-time freelancer since 1992, my work has appeared on four continents. I currently work in several styles for all market segments within the profession.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Adv Mind Body Med. 2008 Summer;23(2):18-26.\nTen years of treating eating disorders: what have we learned? A personal perspective on the application of 12-step and wellness programs.\nLongwood University, Farmville, Virginia, USA.\nConventional therapy for eating disorders has focused on behavioral approaches, medical models, and combinations of both, with lesser emphasis on psychological and wellness models. Because eating disorders are often difficult to treat and the individuals who have them often exhibit significant comorbidities, the long-term success rate (3-5 years or more)-defined as recovery and abstinence from the disorder behaviors-is in the 40% to 50% range, at best. Moreover, if one examines randomized controlled trials (RCTs) that test the efficacy of the most commonly used behavioral approaches in a historical manner, as is described in this article, by assigning RCTs to 2 different time periods for the treatment of bulimia nervosa (BN), it is found that no progress has been made in the success rate of treating this disorder. Many reasons exist for this lack of progress, including comorbidities, failure of patient-therapist relationships to be dynamic, failure to appreciate that BN and binge eating disorder have addiction components that might require 12-step or multimodal approaches, and an absence of treating the whole person, which requires using a wellness model and elements such as body awareness exercises, yoga, and spirituality. Based on a review of the literature and my personal experience over the last 10 years, it is suggested that best practices for treating these disorders should include wellness and 12-step models that focus less on self-centeredness, highlighting the strengths of the person and helping individuals to find their true spirituality, which can be used as a focal point for all treatment. Conventional approaches can still be useful in treating eating disorders, but clinicians and psychiatrists should cease seeing eating disorders as \"diseases\" that should be treated by pharmacodynamics and consider that these are conditions that have taken many years to develop and that have many background psychological factors, often reaching back to childhood.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This morning I had the privilege of hearing Pastor Rick Elzinga of Southwest Hills Baptist Church preach on John 17:20-26. What follows is a summary of his sermon in the space of one sentence: What will you and I do this week to further the unity among His people that Jesus prayed for?\nAs young Christians, we learned to pray mostly by listening to other believers pray. We learn to speak in the same way \u2013 by imitation.\nWe can also learn to pray, and learn some of the most important principles of prayer, as we read God's Word. Listen to the apostle Paul's words to the Christians in Philippi: \"I thank my God in all my remembrance of you, always offering prayer with joy in my every prayer for you all, in view of your participation in the gospel from the first day until now\" (1:3-5).\nEvery time he thought of the Philippian Christians, he was full of joy and thanked God. But what can we learn about prayer from these verses?\nFirst, he prayed frequently. \"All my remembrance of you\" and \"in my every prayer for you all\" make it clear that Paul prayed for them regularly \u2013 it wasn't simply a one-time thing.\nSecond, he prayed comprehensively. Paul was careful in his prayers to mention everyone in the congregation, hence the phrase \"for you all.\" He wasn't satisfied with a blanket prayer (\"God, bless all of the Philippian believers\"), or only pray for a few. No, he prayed for all of them.\nThird, he prayed gratefully. Notice that he began by saying, \"I thank my God.\" Paul's continuous prayer for them wasn't grudging, it was grateful. He was genuinely thankful to God for them and how supportive they had been of him in his ministry to them and others.\nListen to the apostle Paul and learn from him. May our prayer increasingly be frequent, comprehensive, and grateful!\nPoliticians have a way of disrespecting much of the American voting public. They sometimes refer to everything outside of the New York-Washington, D.C.-Boston-Los Angeles bubble as \"flyover country.\" In other words, the parts of the country you fly over when you're going to the \"important\" places. They don't realize what they're missing.\nAs bad as that is, those of us who love God's Word can do the same thing by the way we treat books of the Bible. If we're not paying attention, we can look at the very beginning of a number of books \u2013 the greeting \u2013 as flyover country. We skip it in order to get to \"the good stuff.\" If we do that, though, we miss out on some very important truths.\nThrough faith in Christ alone for our salvation, we are united with Christ. We are in Him and He is in us. By God's grace and mercy, we've been brought into a living and legal relationship with Jesus and we share in the redemption He accomplished and all of His blessings. Union with Christ is the basis from which our election, calling, regeneration, repentance, faith, justification, adoption, sanctification, and glorification take place. We are \"tied\" to Christ in such a way that we'll never be untied.\nAt the same time, we are in the world \u2013 \"in Philippi,\" so to speak. God didn't remove us from this world the moment we repented and believed the gospel, did He? If He did, you wouldn't be reading this and I wouldn't be writing it, either! We're \"in Christ,\" but we're not yet in heaven. God has given us a job to do as long as we're living in this world \u2013 to glorify Him and to enjoy Him forever (1 Cor. 10:31 and Question and Answer #1 of the Westminster Shorter Catechism). We've been called to make Him visible, put Him on display, and reflect Him wherever He's placed us. He has set us apart (the meaning of \"saint\") to serve Him.\nThe Lord determines who we are (united with Christ), where we live (our particular place in this world), and what we're supposed to do (glorify Him in all things). It was true for the Christians in Philippi and it's true for us, too.\nThere is no \"flyover country.\" If only we, and the politicians, would realize it. We don't know what we're missing!\nI had the privilege this morning of hearing Pastor Rick Elzinga of Southwest Hills Baptist Church preach on John 17:13-19. Here is a summary of his sermon in one sentence: Jesus prayed that we \u2013 His people \u2013 would be safe, sanctified, and sent by His Word.\nAnother installment of some of the best articles and ideas I've come across in the past little while.\nThe idea of \"finding Christ in every passage\" has swept through the ranks of evangelical preachers in the last decade like wildfire. I understand the attraction based on the sheer number of sermons I've heard (and preached) that are more moralistic than anything else (\"Do this.\" \"Don't do that.\"), with very little gospel in them. However, is this method of interpretation actually biblical? Does it accurately handle the Word of truth? Eric Davis has written a two-part article that gives the positives and negatives of the approach. Here's part one. Here's part two. Read them both.\nBrett McCracken sheds some light on \"Calvinist\", a movie made by Les Lanphere. It explains the recent popularity of Calvinism (a good thing in my opinion).\nA teacher in the UK is facing disciplinary action for \"misgendering\" a student, according to the BBC. If you think it can't, or won't, happen here, you're mistaken. We live in an upside down world.\nA new movie called On Wings of Eagles picks up the Eric Liddell story where Chariots of Fire left off. Laura Finch wites this short review in WORLD magazine.\n\"10 Things to Pray Before Church\" is really, really good. We all ought to read it and do it.\nApologetics is the art and science of defending the faith \u2013 defending the truth of Christianity \u2013 Voddie Baucham shows how that can be done in the process of preaching and teaching God's Word in his book Expository Apologetics: Answering Objections with the Power of the Word. Part of Baucham's preaching style is to anticipate the objections of skeptics (and other non-believers), and then answer them using the text of Scripture being explained. In addition to explaining how the faith can be defended, he also demonstrates by providing a transcript of one of his sermons.\nExpository Apologetics is a layman's introduction to presuppositional apologetics, which is connected frequently with Cornelius Van Til in the early 20th century. Baucham makes the concepts of the system understandable and applicable. His treatment of Romans 1 \u2013 what everyone knows and is answerable for \u2013 is excellent.\nBelievers in the Lord Jesus Christ are commanded to \"set apart Christ as Lord in your hearts, always being ready to give an answer for the hope that lies within you, with gentleness and respect\" (1 Peter 3:15). Expository Apologetics by Voddie Baucham will help you obey that command. Tolls lege! Take up and read!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"THIS HOME FEATURES 2 BEDROOMS, 2 BATHROOMS AND A DEN IN 1,952 SQUARE FEET OF LIVING SPACE. OPTIONS INCLUDES A 2' EXTENSION AT THE REAR OF THE HOUSE, AN EXTERIOR FIREPLACE ON THE REAR COVERED PATIO AND A GOLF CART GARAGE. INSIDE YOU WILL FIND FRENCH DOORS W\/ SIDELIGHTS ON THE DEN, UPGRADED KITCHEN CABINETS, GRANITE COUNTER TOPS, STAINLESS STEEL APPLIANCES INCLUDING, COOK TOP, CANOPY HOOD, MICRO\/OVEN COMBO, DISHWASHER AND REFRIGERATOR.\nListing courtesy of Pcd Realty, Llc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Recently several visitors of blog have asked me about this manual, which is being advertised quite widely across the Internet. So I purchased a copy myself to find out what all the publicity was about.\nThe most convenient source of amplicons for the LiPA assay is the PCR products from the AMPLICOR HCV 2.0 assay. Alternatively, RT-PCR of HCV can be performed using primers supplied by Innogenetics. The following protocol uses AMPLICOR HCV 2.0 PCR products. Fig. 1. Type- and subtype-specific probe location on the INNO-LiPA HCV II strip. Fig. 1. Type- and subtype-specific probe location on the INNO-LiPA HCV II strip.\nIdentify modes of transmission and risk factors among the major types of viral hepatitis. 3. Evaluate hepatic serologies to understand how the type of hepatitis is diagnosed. 4. Create treatment goals for a patient with viral hepatitis. 5. Recommend an appropriate pharmacotherapy for prevention of viral hepatitis. 6. Develop a pharmaceutical care plan for treatment of viral hepatitis. 7. Formulate a monitoring plan to assess adverse effects of pharmacotherapy for viral hepatitis.\nB., Ringpis, F. M., Pottathil, M. R., Marshall, D. J., de Arruda, M., Murvine, C., Fors, L., Pottathil, R. M., and Barathur, R. R. (1998) Algorithmic approach to high-throughput molecular screening for alpha interferon-resistant genotypes in hepatitis C patients. J. Clin. Microbiol. 36, 1895-1901. 11. Marshall, D. J., Heisler, L. M., Lyamichev, V., Murvine, C., Olive, D. M., Ehrlich, G. D., Neri, B. P., and de Arruda, M. (1997) Determination of hepatitis C virus genotypes in the United States by Cleavase Fragment Length Polymorphism analysis. J. Clin. Microbiol. 35, 3156-3162.\nThere is strong epidemiological evidence that healthcare workers who are high-risk carriers of hepatitis B virus, as defined earlier, can transmit hepatitis B to patients if they There is increasing epidemiological evidence that hepatitis C virus can be transmitted from seropositive healthcare workers to patients during exposure-prone procedures. UK guidelines published in 2002 place restrictions on the clinical activities of healthcare workers who are infected with hepatitis C (HCV RNA positive).\nDecreased position, vibratory, temperature, and pain appreciation occurs in several neuropathies associated with hyposmia. These include diabetes, the neuropathy of renal and hepatic failures, and a large variety of toxic neuropathies. In patients with pernicious anemia, the large myelinated central fibers carrying position and vibration senses are preferentially affected. In the context of hepatitis, the acquired immune deficiency syndrome (AIDS), and other virus-related illnesses, hyposmia can occur along with an ascending polyneuropathy of the Guillain-Barre type. In seizure patients with uncal or temporal lobe foci that induce dysosmic auras, altered sensations in a hemibody distribution can occur as part of the seizure or as a postictal transient sequela.\nStudents and clinicians are frequently exposed to patients with hepatitis or acquired immunodeficiency syndrome. Clinicians' fear of these diseases often interferes with the development of a good doctor-patient relationship. Once clearly defined procedures are implemented to ensure the safety of health-care workers, this fear can be better handled. 12. All health-care providers who have direct contact with patients should complete the hepatitis B vaccine series. In certain populations, testing for immunity before vaccination may be indicated. Typical preemployment Student Health Service screening includes a purified protein derivative (for tuberculosis) and various serologic testing, including testing for hepatitis B virus.\nThe routes of transmission by which pathogens can be transferred from the source of infection to the host can be either airborne by contact or percutaneous. Airborne transmission involves the spread of infections such as influenza and TB via water droplets. Contact transmission can involve direct person-to-person transmission from the source to a susceptible host (as with MRSA), or can involve contact with body fluids such as faecal material (C. difficile), equipment such as endoscopes, or food. Percutaneous transmission can occur via insect vectors (malaria) intravascular lines (MRSA) or as a result of sharps injuries (hepatitis B, HIV).\nExposure models are necessary to generate assessment criteria such as SGVs, which indicate the concentration of a substance present in soil that may result in a daily intake equal to, or less than, a health criteria value (HCV) (Box 10.3). The exposure pathways present on a given site are identified by constructing a conceptual model indicating potential pollution linkages (Figure 10.2) with degrees of risk associated with them. In 2009, the Environment Agency released an updated CLEA model (Environment Agency 2009b). The model uses generic assumptions about chemical fate and transport in the environment and a generic conceptual model for site conditions and human behaviour to estimate exposure to contaminants for those living, working, and playing on contaminated sites over long periods. The CLEA model has been used to generate SGVs by comparing the estimated exposure with HCVs.\nHCVs used in risk assessments of land contamination are generally derived from toxicological or epidemiological studies. For example, the HCV for arsenic is based on epidemiological studies of people exposed via drinking water. In oral toxicological studies, chemicals may be administered via different media, such as drinking water, gavage, and capsules. HCVs tend to assume that a chemical is equally bioavailable in all media. However, this assumption might not be true for soil contaminants because of the binding to soil particles and or chemical entry inside the soil mineral lattice, which may cause bioavailability to differ between the soil types, chemicals, and chemical forms (Environment Agency 2005 Saikat 2006).\nPathogenesis Bacteria penetrate broken skin and mucous membranes and replicate in the liver and kidney, causing hepatitis, acute renal failure and intravascular coagulation. Leptospira can be excreted in the animal's urine for months or years after recovery. Clinical signs Pyrexia, vomiting, shock, interstitial nephritis and hepatitis. Signs dependent on animal's age, immunity and environmental factors.\nConsider racial or ethnic differences that may alter the risk for the outcome. Angiotensin-Converting Enzyme (ACE) inhibitors (a class of antihypertensives) increase the risk of angioedema. The risk is three times higher in blacks than in whites 8 . The odds of developing hepatitis with isoniazid use is lower among Asians who are fast acetylators of the drug 9 .\nHepatitis C is a common blood-borne infection that when left untreated can lead to cirrhosis, hepatic encephalopathy, and death and is a leading cause of chronic liver disease, liver transplantation, and hepatocellular carcinoma (Saunders 2008). The presentation of hepatitis C infection is variable, with most acutely infected people being asymptomatic (70 -80 ), although they may have elevated liver enzymes. The most common symptoms of acute infection include fever, malaise, loss of appetite, jaundice, fatigue, nausea and vomiting, dark urine, and joint pain (Dieperink et al. 2000). With chronic infection, the most common presentations include jaundice, encephalopathy, or ascites. In the pediatric population, those who use injectable drugs and those born to infected mothers are at highest risk, with approximately 4 of children born to mothers with hepatitis C going on to develop hepatitis C (Saunders 2008).\nThe neuropsychiatric effects of hepatitis most notably symptoms of fatigue, depression, and cognitive dysfunction (Ozkan et al. 2006) have long been recognized, although pediatric-specific research is limited. Fatigue is reported to occur in more than 50 of patients with hepatitis C (Saunders 2008). The presence of comorbid depression has a significant negative impact on health-related quality of life in those with hepatitis C (Ozkan et al. 2006), and rates of depression as high as 30 have been reported in untreated patients with hepatitis C (Die-perink et al. 2000 Saunders 2008). Patients infected with hepatitis C have also been reported to have subcortical frontal cerebral dysfunction that manifests as impaired concentration and slow processing prior to the development of cirrhosis and hepatic enceph-alopathy (Perry et al. 2008 Saunders 2008).\nThe gut has its own intrinsic nervous system and is in active communication with the brain and central nervous system. Therefore, it is not especially surprising to note that disorders of gastrointestinal function exist in the absence of evidence of tissue damage, with the relationship between FGIDs and gastrointestinal diseases such as Crohn's disease perhaps being analogous to that between common psychiatric disorders such as anxiety or mood disorders and neurological diseases such as multiple sclerosis. FGIDs are indeed common, impairing, and strongly associated with anxiety and depressive disorders new treatments share common features with treatment regimens that are successful in the management of emotional disorders and migraine headache. Psychiatric symptoms and disorders, particularly depression, are also commonly comorbid with gastrointestinal diseases such as IBD and hepatitis C, sometimes in relation to associated treatments such as corticosteroids and interferon alpha.\nDrug users who inject opiates (and other drugs for that matter) can obtain serious infections associated with using contaminated and dirty needles. These include ugly skin abscesses, hepatitis, tuberculosis, and AIDS. They are also at higher risk for sexually transmitted diseases. Opiate drugs work by stimulating receptors for the opioid peptide neurotransmitters such as enkephalin and endorphin, which are chemicals that occur naturally in the brain.\nErative bone disease, hepatitis B or C, sexually transmitted disease, autoimmune deficiency, or other medical problems that might lead to cross-infection. Therefore, the first priority must be given to thorough donor screening, which involves a traceable medical, demographic, and social history. Other disadvantages of allo-grafts include the risk of rejection, high rate of infection, nonunion, risk of rapid resorption, and problems related to the considerable technical precision required to pack and hold the graft in place in bleeding sites.\nBleach kits (containing bleach and instructions for cleaning equipment) are distributed to make drug injection less dangerous. Bleach does not kill the pathogen that causes hepatitis and is also not totally effective in eliminating HIV however, such kits do reduce the likelihood of other infections being passed through sharing of dirty equipment.\nThe infectious theory of prostate cancer was fashionable twenty years ago, but most contemporary books do not even mention it. The neglect of this line of research is surprising, since approximately 15 percent of all cancers worldwide are caused by infectious agents. For example, Helicobacter pylori bacteria is associated with stomach cancer, hepatitis B virus with liver cancer, human papillomavirus with cervical cancer, Epstein-Barr virus with nasopharyngeal cancer, and human T-lymphocyte virus with some leukemias and lymphomas. Prostate cancer is also a type of cancer that increases in incidence in individuals whose immune system is suppressed this correlation is consistent with an infectious process. Furthermore, when biopsies of prostatic tissue are examined under the microscope, inflammation is frequently present, consistent with infection. For all these reasons, infectious agents should be seriously considered as possible causes of prostate cancer.\nIn general, the same psychotropic medications can be used in the HIV-infected child as in the general population. However, bone marrow suppression, hepatitis, and pancreatitis may cause treatment-limiting toxicities and affect metabolism of antiretrovirals, particularly protease inhibitors. Treatment with psychotropic medications is part of comprehensive multidisciplinary care and multimodal treatment to improve the quality of life for pe-diatric HIV AIDS patients by decreasing discomfort and increasing functioning.\nMost health care workers are at risk for exposure to many diseases in the normal course of their work. Additionally, health care workers may transmit vaccine-preventable diseases to their patients. At the time of employment and on a regular basis, health care workers should be screened for immunity to measles, mumps, rubella, and varicella if found to be nonimmune, the measles, mumps, and rubella, and varicella vaccines should be administered. The hepatitis B series should be given if not already completed. Tetanus should be updated and given every 10 years. Health care personnel in hospitals and ambulatory settings with direct patient contact should receive Tdap if not already received an interval as short as 2 years from the last tetanus-containing vaccine should be used.\nAnother common toxicity is the rapid depletion of normal antigen-positive B-lymphocytes from blood, bone marrow and lymph nodes of the recipient, lasting between six and nine months following the last administration of rituximab. In the case of short rituximab treatment, this depletion does not compromise immunity immunoglobulins do not decrease significantly, and patients do not have an increased risk for infections during and after rituximab therapy (Grillo-Lopez, Hedrick, Rashford and Benyunes 2002 Kimby 2005) except for some viruses like herpes virus, cytomegalovirus, or hepatitis B virus. Maintenance treatment, particularly after autologous transplant, might be associated with a decrease in immu-noglobulins (Lim, Zhang, Wang, Esler, Beggs, Pruitt, Hancock and Townsend 2004) and late toxicity (Kimby 2005).\nWhen the kimono designer Itchiku Kubota (Japanese, b. 1917) was having a second exhibition of his art at age 60, he contracted acute hepatitis. While in the hospital Kubota thought about his work and decided to develop a new style of kimono design combining ancient Japanese techniques with French Impressionism and scenes of nature. Kubota, who became very successful, spoke about the influence of having hepatitis saying, This was the time of deepest import in my life as an artist. Pierre-Auguste Renoir (French, 1841-1919) started to develop rheumatoid arthritis at age 56 and despite his efforts at exercise, the illness progressed. Sitting in a wheelchair with swollen immobile fingers, Renoir continued to work. With small gauze pouches to hold his brushes, the artist moved his arms rather than his wrists to make art. His son Jean Renoir admired his father's determination and said, The more intolerable his suffering became, the more he painted.\nA study was conducted on 60 patients who were chronic carriers of hepatitis B or C. The essential oils of Cinnamomum camphora ct 1,8-cineole, Daucus carota, Ledum groelandicum, Laurus nobilis, Helichrysum italicum, Thymus vulgaris ct thujanol, and Melaleuca quinquenervia were used orally in various combinations. They were used as a monotherapy or as a complement to allopathic treatment. The objectives of treatment were normalization of transaminase levels, reduction of viral load, and stabilization or regression of fibrosis. There was an improvement of 100 , when patients with hepatitis C were given bitherapy with essential oils. With essential oil monotherapy, improvements were noted in 64 of patients with hepatitis C and there were two cures of hepatitis B (Giraud-Robert, 2005).\nThe effect of homozygous C2 deficiency on the clearance of immune complexes has been studied (54). A patient with C2 deficiency was injected with hepatitis B surface antigen-anti-HbsAg immune complexes labeled with I123. The patient's uptake of complexes in the liver, spleen, and clearance of the complexes from the circulation was studied pre- and post- fresh-frozen plasma treatment. When her C2 and CH50 levels were zero, the complexes were rapidly taken up by the liver and cleared from the circulation. No binding of the complexes was seen in the spleen or on the RBCs via CR1. After treatment with fresh frozen plasma, which normalized her complement levels, the complexes cleared from the circulation more slowly and 20 of the complexes were found in the spleen. These studies suggest that uptake of immune complexes by the spleen is complement dependent and abnormal processing of immune complexes by complement-deficient patients may take place.\nVaccine) increased the incidence of Guillain-Barre syndrome by a factor of eight. Warnings about Guillain-Barre syndrome continue to be given with annual influenza vaccinations. The hepatitis B vaccine derived from pooled plasma was also associated with Guillain-Barre syndrome however, Guillain-Barre has not been reported with use of the currently available hepatitis B vaccines produced through recombinant DNA technology. Most recently, Guillain-Barre syndrome has been reported to occur with the meningococcal conjugate vaccine.\nThere is no conclusive evidence that CIDP patients have a higher incidence of other medical conditions, including autoimmune disorders. However, associated disorders, including systemic lupus erythematosus, Hashimoto's thyroiditis, thyrotoxicosis, chronic active hepatitis, inflammatory bowel disease, urticaria, eczema, and psoriasis do occur in some CIDP patients.y A CIDP-like disorder may occur in the setting of monoclonal gammopathies of uncertain significance (MGUS), as well as with multiple myeloma, osteosclerotic myeloma and other lymphoproliferative disorders. These disorders are discussed as distinct entities in the following sections.\nIVIg is an expensive, relatively safe, and promising treatment for DM and PM. The dose is 2 g kg, divided over 2 to 5 days initially, then 0.4 g kg given at intervals based upon the clinical response. The exact mechanism of action is not clear but is felt to affect circulating autoantibodies, the autoimmune target, or complement activation. Side effects include increased serum viscosity with potentiation of thromboembolic events, aseptic meningitis, rash, anaphylaxis in IgA-deficient individuals, and a low risk of transmissible diseases such as hepatitis C. Many patients who have failed other treatments may respond to IVIg. Plasmapheresis has been shown ineffective in a controlled clinical trial.\nMust occur in the absence of other causes of post-transplant liver failure, including GVHD, viral hepatitis, fungal abscesses, or drug reactions. Most cases of sinusoidal obstruction syndrome occur within three weeks of HSCT and clinical diagnosis can be confirmed histologically via liver biopsy.\nAnwar, A., Ali, N., Tanveer, R., Siddiqui, A. (2000). Demonstration of functional requirement of polypyrimidine tract-binding protein by SELEX RNA during hepatitis C virus internal ribosome entry site-mediated translation initiation. J Biol Chem 275, 34231-34235. Biroccio, A., Hamm, J., Incitti, I., De Francesco, R., Tomei, L. (2002). Selection of RNA aptamers that are specific and high-affinity ligands of the hepatitis C virus RNA-dependent RNA polymerase. J Virol 76, 3688-3696. Kumar, P. K., Machida, K., Urvil, P. T., Ka-kiuchi, N., Vishnuvardhan, D., Shimotoh-no, K., Taira, K., Nishikawa, S. (1997). Isolation of RNA aptamers specific to the NS3 protein of hepatitis C virus from a pool of completely random RNA. Virology 237, 270282.\nAthletes with infectious mononucleosis must be managed carefully to avoid their participation in sports that could result in abdominal trauma. Other risks associated with infectious mononucleosis include upper airway obstruction, asymptomatic transaminase elevation, throm-bocytopenia, and rash after the administration of ampicillin or amoxicillin. Routinely obtaining transaminase levels in patients without clinical hepatitis is of little value and can increase the overall cost of management.\nEpstein-Barr virus (EBV) as a cause of pharyngitis can mimic GABHS infection. It can also occur concurrently with GABHS infection. Studies have shown the two infections occurring together in 2 to 33 of cases. Prodromal symptoms to severe sore throat include malaise, anorexia, chills, and headache. Fatigue, lymphadenopathy, and hepatosplenomegaly can follow in 5 to 14 days. Pharyngitis with tonsillar hypertrophy and a membranous white tonsillar exudate lasts 5 to 10 days. Lymphadenopathy and hepatosplenomegaly can persist 3 to 6 weeks. Contact sports should be avoided for 6 weeks because of the possibility of splenic rupture. Complications that occur in less than 2 of patients with EBV infection include throm-bocytopenia, hemolytic anemia, Guillain-Barre syndrome, Bell's palsy, transverse myelitis, and aseptic meningitis. Hepatitis can be seen in 20 to 50 of patients with EBV infection.\nThe loading dose is 40-60 IU kg, and this can be followed by repeat doses every 12-24 h to maintain vWF activity (vWF RCoF) > 50 . All currently available concentrates are derived from plasma. As at least one viral inactivation step is included in their manufacture, they are unlikely to transmit hepatitis or HIV, but there is still a risk of parvovirus infection.\nMechanistically, APAP is believed to inhibit prostaglandin synthesis in the CNS and block pain impulses in the periphery. APAP is well tolerated at usual doses and has few clinically significant drug interactions except causing increased hypoprothrombinemic response to warfarin in patients receiving APAP doses of more than 2,000 mg per day. The maximum recommended dose for patients with normal renal and hepatic function is 4,000 mg per day. Hepatotoxicity has been reported with excessive use and overdose, and the risk of this adverse effect increases in those with hepatitis or chronic alcohol use, as well as those who binge drink or are in a fasting state. Regular chronic use of APAP has been associated with chronic renal failure, but reports are conflicting. For these reasons, the maximum dose should be reduced by 50 to 75 in patients with renal dysfunction or hepatic disease and in those who engage in excessive alcohol use.\nA variety of other rare infections with the gonococcus have been documented. Adults occasionally develop gonococcal conjunctivitis, with a potential for more serious ocular involvement, through direct (i.e., hand-to-eye) contact with infected secretions. Gonococcal endocarditis, myocarditis, hepatitis, and meningitis may occur as part of the disseminated syndrome. Perihepatitis (termed the Fitz-Hugh-Curtis syndrome) has traditionally been attributed to gonococcal infection in the upper right quadrant, usually in association with classic PID. Recent evidence, however, indicates that the syndrome is more often associated with chlamydial, rather than gonococcal, infection.\nType III reactions are mediated by formation of antigen-antibody or immune complexes and subsequent complement activation. Deposition of immune complexes near the site of antigen entry causes release of lytic enzymes by accumulated neutrophils and results in localized tissue damage. The formation of circulating immune complexes is involved in several conditions, including allergies to penicillin, infectious diseases such as hepatitis and autoimmune diseases such as rheumatoid arthritis.\nInhibition of the lectin pathway may represent a therapeutic strategy for ischemia-reperfusion injury (Collard et al., 2000 Petersen et al., 2000 Montalto et al., 2001 Lekowski et al., 2001) and other complement-mediated disease states, such as in chronic hepatitis C (Dumestre-Perard et al., 2002). An alternative approach to targeting MBL is to develop a specific inhibitor for MASP-2 and thereby block the activation of the lectin pathway.\nB19 is responsible for erythema infectiosum and can also cause polyarthritis, especially in the hands, knees, and ankles. HIV infection sometimes causes symmetric polyarthritis, spondy-litis, or acute oligoarthritis. Hepatitis B and C can cause acute symmetric polyarthritis in large and small joints. Inflammation in a few large joints and back pain are among the earliest symptoms of infective endocarditis in about 25 of patients with this disorder (Totemchokchyakarn and Ball, 1996).\nWhile as yet untested by clinical trials, the effective treatment of systemic diseases linked with oxidative stress (diabetes, Hepatitis B C, HIV, malaria, haemo-chromatosis, haemoglobinopathies, inflammatory bowel disease, psoriasis, rheumatoid arthritis, depression) is likely to reduce overall oxidative stress in the body and benefit sperm function. It is therefore ideal that patients delay conception until after these systemic diseases are under effective control, unless the medications used to achieve control have a detrimental effect on sperm function (e.g. meth-otrexate treatment of inflammatory conditions).","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I dare you to say the title fast three time's in a row! Okay\u2026okay, now that I have your attention, let's talk about a superfood known as spaghetti squash. Although the media seems to have a new \"superfood\" everyday, the spaghetti squash is a clear choice in my book. Don't buy it? Keep reading to find out why.\nMaybe it's the Italian in me that loves carbohydrates and anything bread-based, but then again who doesn't enjoy a nice plate of pasta?! I have yet to find a vegetable that can replace a nice warm roll, but I do recommend swapping spaghetti squash as the \"pasta\" for a quick and convenient weeknight dinner.\nSpaghetti squash is just one of the many varieties of winter squashes that are high in carotenoids. Caro-ten-what-an-oids? These are simply nutrients that function as important antioxidants that help not only with cancer prevention but also promote your vision. Another benefit\u2026one cup of spaghetti squash is only 40 calories and has 10 grams of carbohydrates!!! That's nearly 175 fewer calories and 30 grams less of carbohydrates than a pasta alternative.\nConvinced yet? Don't worry, my grandpa (a true, hearty Italian) wasn't quite so sure either until he gave it a try. Start small, swapping an even 50-50 split with an spaghetti noodle mixed with the squash before you go all the way.\nStay tuned for a future recipe tab that will include Simple-Swaps posted on Instagram, Facebook, & Twitter.\n\u00ab Get Paid to Get Fit? Sign me up!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Horned Grebes are a widespread grebe, also found in Europe and Asia. They are often relatively tame, allowing fairly close approach by humans. Horned Grebes are not as gregarious as many of the grebes, and don't breed in the large colonies that other grebes sometimes do. The Horned Grebe is similar to the Eared Grebe, but can best be differentiated in summer plumage by the reddish neck (the Eared Grebe has a blackish neck).\nHabitat: Lakes, ponds, reservoirs. Prefers areas having both open water and wetland vegetation.\nDiet: Mostly insects and crustaceans during summers in South Dakota, also fish, amphibians, leeches, and some plant material.\nBehavior: Primarily feeds by diving under the water's surface and propelling itself through the water with its feet. They will also pluck food items from the water's surface.\nMigration: Summers in extreme northern U.S. the western 2\/3rds of Canada, and Alaska. Generally winters along North American coastlines and large inland water bodies in the south.\nStatus: Possibly in long-term decline.\nAdditional Photos: Click on the image chips or text links below for additional, higher-resolution Horned Grebe photos.\nSouth Dakota Status: Uncommon migrant throughout the state. Uncommon summer breeding resident in the north-central and northeastern part of the state.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Booking.com: Hotels in Kandy. Book your hotel now!\nThe Golden Crown Hotel is a 5-star hotel located in Ampitya, Kandy, Srilanka.\nSatyodaya Residence is located just 0.6 mi from the famous Kandy City Center and 0.7 mi from the Kandy Railway Station. It provides a 24-hour front desk for the convenience of the guests.\nQueen's Hotel is housed in a historical building located 164 feet from Kandy Lake. It offers colonial-style accommodations with an outdoor pool, 3 dining options and free parking on site.\nLocated in Kandy, 2.3 miles from Kandy City Center Shopping Mall, Amigo's Home stay provides accommodations with a garden with a terrace. Free WiFi is provided.\nLocated within 7 miles of Kandy Royal Botanical Gardens, Tea Heights in Kandy provides an outdoor swimming pool, as well as free WiFi.\nSweet Lanka Kandy offers accommodations in Kandy, 0.9 mi from Tooth Relic Temple and Kandy City Center Shopping Mall and 2133 feet from Kandy view point.\nLocated in Kandy, Lake Front Homestay has free WiFi and mountain views, an 18-minute walk from Bogambara Stadium.\nFeaturing free WiFi, Kandy City Rooms & Hostel is located in Kandy, 2297 feet from Bogambara Stadium. Certain rooms have a seating area for your convenience.\nHanthana Jungle View Holiday Home is located just 1.1 mi from the famous Bogambara Stadium and 1.3 mi from the Kandy Railway Station. Free WiFi access is available.\nLocated in Kandy, Samindra Villa has accommodations 1.7 miles from Kandy City Center Shopping Mall and 1.9 miles from Bogambara Stadium. Free WiFi is featured.\nFeaturing free WiFi and a restaurant, Hipsters Hideout Lounge offers pet-friendly accommodations in Kandy. Guests can enjoy the on-site bar. Free private parking is available on site.\nFeaturing free WiFi throughout the property, Kandyan Sweet Villa offers accommodations in Kandy, 656 feet from Bogambara Stadium. Guests can enjoy the on-site restaurant.\nLocated in Kandy, a 10-minute walk from Kandy City Center Shopping Mall, Villa 92 City Stay has a number of amenities, including a garden, a shared lounge and free WiFi.\nLocated in Kandy, 1.2 miles from Bogambara Stadium and 1.4 miles from Kandy City Center Shopping Mall, Mcleod-Inn has accommodations with a garden with a terrace. Complimentary WiFi is offered.\n1,173 properties found in City Centre ().","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Very nice. I also have a Bbq smoker and have been known to multitask brewing and ribs together. Awful hard to beat !\nThose are some of the best days!\nEspecially when the beer and ribs come out great, and I have a prior batch on tap to serve with the bbq ! Hard to keep the neighbors away at that point!\nIts actually a hell of a find. Great area in the middle of the city with about 150 feet of backyard. I brew on the small wood deck that extends from the house.\nWell that's a huge backyard. 150 feet is deeper than our entire lot.\nI got nothing against alleys. I spend a lot of time in our alley, but less so since I replaced my old car. I think its the long fence line that makes me think alley.\nYes...I'll be brewing in my unheated garage tomorrow morning. It'll be about 19F when I start. Here's what it looked like this morning. Some homebrewers are diehards I suppose.\nMuddy looks happy. I think I may give it a this weekend as well. Just need to thaw out my hose for the chiller and turn on the outside water. Another reason to go electric!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home\/Blog, Wedding Venues, Weddings\/Congratulations to Rainie and Isaac!\nCongratulations to Isaac and Rainie Denham \u2026 we wish you many years of happiness together! We especially enjoyed the Mad Hatter themed costumes \u2026 what a fun day!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Following on from last weeks record review post, today I'm going to talk about the second of Adrian's picks: Natalie Prass.\nI've heard of this artist before, I think there has been quite a bit of hype surrounding her album and I'm sure I've seen her mentioned as 'one to watch' in the culture section of Elle magazine. She's a rather gorgeous 28 year old American singer-songwriter (naturally) who has a touch of Lou Douillon about her, which makes me like her even more.\nAccording to music website Pitchfork she 'weathers heartbreak while backed by lush arrangements that recall the orchestrated, soulful pop of the 1970s' and I would have to say that this sentence is a very good surmise of the album (which is why I don't write music reviews.) Basically, she has a very lovely voice, which, on listening, reminded me somewhat of Disney, once intertwined with the twinkly orchestral surroundings, a fact I would probably not have mentioned apart from further investigation of her Pitchfork interview\/review uncovering the same thing (read here.) It sounds like a rather more heartbroken version of Disney, though; and all the better for it; no listless princesses here, rather more a bootstrapping version who has hitched up her skirts and gone off to find her man. There is sadness, yes, but not weakness.\nThe album definitely smacks of times gone by; again referencing Pitchfork (until I can confidently comment on music without fear of reprisals due to my lack of knowledge) one of her songs is likened to the 'love triangle' of Dolly Parton's 'Jolene.' There is a sense of this, and also a touch of June Carter in the way her vocals range from delicate to forceful. The album is equally wistful and accepting, and I think this is a potent mixture. My favourite track is the swooping 'Violently' which is certainly not as agressive as the title would suggest.\nThe album artwork is less symbolic, in my opinion, than the Grouper effort of last week but I did appreciate the sentiment, of course she is incredibly striking and her sublime beauty is only enhanced by a floral overlay on the album sleeve. It has a slightly Seventies, sun-washed look to it and definitely (to me) highlights her aesthetic.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After a man he arrested hanged himself in his jail cell last year, Fullerton police officer Vincent Thomas Mater ripped the department-issued recorder off his uniform and crushed it, making it impossible to retrieve the audio recording of the conversation he had with the man prior to his death.\nOn Tuesday, Mater was charged with one count of destruction of evidence in a death investigation and one count of vandalism. He faces up to 18 months in jail for the two misdemeanors.\nHe will most likely not spend a single day in jail.\nAs it is now, many are wondering if his two former fellow officers, Manuel Ramos and Jay Cicinelli, will serve time for the beating death of Kelly Thomas, a homeless schizophrenic man. Their pretrial is scheduled for March 28.\nRamos and Cicinelli were charged after video from surveillance cameras and possibly confiscated citizen cameras showed them to be the main culprits in the beating last year. That footage has yet to be released.\nMater told investigators that he did not purposely destroy the evidence, but simply became enraged after being unable to download the audio onto his computer, flinging it across the room against a metal door where it shattered.\nThat version wouldn't be too hard to believe knowing how quick-tempered cops can be.\nBut investigators still had their doubts.\nA few hours after Mater arrested Dean Francis Gochenour on April 14 last year, the drunk-driving suspect hung himself in a Fullerton jail cell. On the night in question, Mater, a corporal working uniform patrol, crushed the digital audio recorder after learning of Gochenour's death, prosecutors allege.\nProsecutors say the audio recorder would have captured vital conversations with Gochenour. Mater stopped Gouchenour's car about 9:45 p.m after noticing the motorist was driving without his lights on.\nHe arrested Gochenour after determining that the driver was under the influence of alcohol. Mater then drove Gochenour in his patrol car to the Fullerton City Jail and turned him over to jailers to be booked upon arrival. Prosecutor say Mater was wearing his department-issued digital audio recording device through the trip and would have audio-recorded any statements.\nGochenour committed suicide about 11:30 p.m. in his jail cell. In the hours after Mater learned of Gochenour's death, the officer allegedly destroyed his recorder by crushing it and then \"removing the mother board and circuit board.\" The actions, according to prosecutors, made it impossible to recover the captured audio relevant to the Orange County district attorney's custodial death investigation.\nThe incident took place in April 2011 and Mater spent two months on paid administrative leave until the department initiated disciplinary proceedings aganist him. He resigned a week later.\nThis is the first time I've heard of a cop charged with destruction of evidence even though I've written plenty of stories about police destroying footage from cameras, including my own.\nI hope it sparks a new trend because as it is now, they obviously believe they can get away with it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This Family-Friendly Country Hotel is Set Amongst 28 Hectares of Lush Green Forests and is a Genuinely Magical, Secluded Homestead. It Comprises a Total of 142 Well-Appointed Modern Rooms Including Garden Rooms, Garden Rooms With a Spa Bath, Junior Suites and Spa Rooms. The Hotel Offers All the Facilities That Guests Might Need For a Perfect Holiday and a True Pampering Experience in the Heights of Western Galilee. Facilities On Offer To Guests At This Air-Conditioned Establishment Include a Lobby Area With a 24-Hour Reception and Check-Out Service, a Hotel Safe, a Currency Exchange Facility, Lift Access and a Newspaper Stand. There is an Auditorium and the Younger Guests Can Let Off Steam At the Kids' Club. Refreshment is Available At the Cafe, Bar or Restaurant and Guests Also Have Conference Facilities At Their Disposal. Moreover, They Can Take Advantage of the Internet Access and Room Service For an Additional Fee, and There is Parking Available For Those Arriving by Car. It is Also Possible To Hire Bicycles On the Premises (Charges Apply).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ripple is a distributed blockchain-based payment settlement network that aims to connect banks, payment providers, digital asset exchanges, and other corporations through cross-border payments. XRP, the token provided on the Ripple network, enables banks to source liquidity for cross-border payments without the traditional banking and transaction fees. In addition, it can be used to exchange and transfer over 30 different currencies whilst settling payments in a matter of seconds, as opposed to the typical 3-5 business days offered by banks and corporations.\nTransactions are settled within 4-5 seconds.\nExtremely low transaction fees (1\/1000th of a cent).\nScalable (handles 50,000 transactions per second).\nMany partnerships with banks\/corporations (i.e. MoneyGram, Saudi Central Bank, etc.).\nRipple can be purchased with Canadian Dollars and\/or other supported cryptocurrencies on our exchange. Refer to the Getting Started section to learn about buying and selling cryptocurrencies on the NDAX platform.\nRipple can be stored on our exchange and on several wallets, including the Ledger Nano S and the Toast Wallet.\nHow do I buy and sell digital assets?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Del Mar Recovery Solutions is headquartered in Carlsbad, California with redundant sites in Las Vegas, Nevada and Grand Rapids, Michigan servicing all 3 U.S time zones for lenders across the country. Del Mar's repossession agent network consists of over 700 fully vetted, trained and compliant agents \u2013 nationwide and in various countries across the globe.\nDel Mar Recovery Solutions specializes in national repossession management (Commonly known in the industry as forwarding), skiptracing, license plate recognition and collateral recovery services. Del Mar has built a unique model, offering auto lenders a complete recovery solution with a focus on paying attention to detail as we facilitate the auto recovery process in a completely transparent and compliant environment. The ability for Del Mar to build a customized process within any segment of the repossession cycle for our clients, regardless of strategy, has driven most of the major auto lenders in the industry to utilize Del Mar in some capacity within our suite of service offerings. When you partner with Del Mar you partner with years of success and experience.\nWe are interested in the privilege of becoming your business partner. We have many different recovery and resolution programs and are flexible to our client's needs.\nProviding expert skiptracing and forwarding services to our valued clients by focusing on our core values; compliance, customer service and results.\nResults are required \u2013 clients engage us to deliver on the expert results we have built our reputation on.\nCompliance is at the core of who we are and what we do. In a highly regulated industry, Del Mar prides ourselves on being compliance leaders in the repossession management space.\nCustomer Service is not an activity, it's an attitude. Del Mar strives to lead by example in providing best in class service to all internal and external customers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbxvf b/data_all_eng_slimpj/shuffled/split2/finalzzzbbxvf
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index 0000000000000000000000000000000000000000..9faeb6a24f107b183ab40710e16d9f94a009fc04
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+{"text":"Following our request for applications to join the new International Committee, we are now in a position to announce the full committee which will take responsibility for all YFG's International Relations and manage our International events here in Ireland.\nThe selection process saw candidates attend a workshop day, write a full YEPP resolution and take part in a interview with a independent panel of former and current YEPP board members. The recommendations of this panel then formed the basis for the selection of our new committee.\nDue to the large number of applications we received, a number of new positions have been assigned, the full committee is listed below. The expected term of the committee is two years up until the YEPP congress in 2015 with a annual review to take place after twelve months.\nYFG Vice President, Carol Madigan has been selected to act as a Executive Liason Officer between the new committee and the National Executive.\nWe would like to thank you everyone who took the time to apply for the International Committee.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I packaged a development version game for windows. It runs fine on my development pc. However, if I run it on other's pc, it always shows the 0xc000007b error. I've installed the prerequisite exe file (UE4\/Engine\/Extras\/Redist\/en-us\/UE4PrereqSetup.msi\/exe) on the second pc, but nothing changed.\nCould anyone tell me why?\nIt sounds like you are packaging them project out on your PC and then moving them to your laptop. Is this correct? If you package the project out on your laptop and then attempt to open them, do you still get the same error?\nYes, that's correct. I haven't installed UE4 on my laptop intentionally; since my possible future clients don't have UE4 and the prerequisites installed on their PCs.\nI have another quick question for you. What are the two OSs that are being run on the two PCs? Is it possible that you are packaging for windows 64 bit and the other PC is 32 bit?\nThe operating system for the both PCs is Windows 10 64bit, and the project's packaged for Windows 64. So ... no, the issue couldn't be here.\nIf you package out a clean project and try to open it up on the other PC does this work for you? If you still receive this error, would it be possible for you to provide the sample project via a zip file on dropbox or google drive?\nAfter testing couple of scenes, the problem's resolved now. Apparently my laptop only executes 32-bit packages, although the operating system is 64-bit ! Due to lack of technical information in this field, that sounds quite odd to me ... Anyway now I've got the idea to package my projects for \"Windows 32-bit\" in order to execute correctly on my laptop.\nThank you for the additional information. I am happy to hear that you have found a workaround for this issue. I will be marking your last comment as the answer to this thread. If the issue returns and you are able to provide additional information to reproduce this issue on this end. please feel free to reopen this issue.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Let's take in the sights of Camden County's Maria Barnaby Greenwald Park! Please join us on Thursday, June 19th from 6 PM to 7 PM. We will meet in the lobby of the Camden County Environmental Center at 5:45 PM.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A recent panel on social entrepreneurship at the Detroit Business Conference emphasized the familiar fact that nonprofits are now not the only entities in the business of helping people. \"Just because you're going to be helping people doesn't mean you should be structured as a nonprofit,\" Christine Coady Narayanan, CEO of the nonprofit Opportunity Resource Fund, reportedly told a gathering of current and aspiring entrepreneurs. The growth of low-profit limited liability corporations (L3Cs) designed to promote charitable purposes and generate limited profits is a reflection of this new level of sector overlap. Panelists at the session emphasized, however, that in Detroit, a city hit particularly hard by the recession, it is especially important for entrepreneurs to come equipped with both a business sense and a cause.\nProviding perspective on the continued need for job creation in the city, Narayanan shared some of her organization's priorities as a nonprofit lender. \"We're not looking to fund businesses that will start right up and pay less than a living wage, because what's that doing?\" she rhetorically asked. \"It's perpetuating the cycle of poverty.\" Based on her work founding and leading Central Detroit Christian, an employment training organization that now includes local soul food restaurant Cafe Sonshine and produce market Peaches & Greens, Lisa Johnson served as a voice of experience in working with local communities. \"Can a low income community support a restaurant?\" she asked. \"Can a low income community support a produce market?\" The answer to both, she said, is, \"Not really.\" Still, she noted that her success over the past 17 years has resulted from her knowledge of the community along with creative marketing and careful business planning. Another success story from the panel came from Detroit's Cass Community Social Services, which launched a program that recycled 20,000 tires last year and turned them into rubber for doormats.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Revision 2011 will take place at an awesome location in Saarbr\u00fccken - the E WERK. It is a former industrial hall, built in 1908 for a steel company, which has been completely renovated and offers enough space and all necessary facilities for an event like Revision.\nThe E WERK already hosted all kinds of big events - from Mot\u00f6rhead to business meetings and product presentations.\nIt is located in an industrial disctrict and is easily reachable. In its surroundings you will find everything for your shopping needs within walking distance.\nFor those that need a nap and don't stay at a hotel we will have a large, separate sleeping area. Please note, that it is still located inside the main hall, so it won't be extraordinarily silent. Be sure to bring along earplugs or get some at the infodesk.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbeiwn b/data_all_eng_slimpj/shuffled/split2/finalzzzbeiwn
new file mode 100644
index 0000000000000000000000000000000000000000..53532aafabe3167209f5ae6395c6cca60ccc5eac
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Nagpur : The Bombay High Court has stayed the transfer of Dean of Government Medical College (GMC) and Hospital here while pulling up the state authorities for the \"malafide\" act.\nA division bench of Justices Bhushan Gavai and Pradeep Deshmukh here yesterday, stayed the transfer of the dean, Dr Abhimanyu Niswade and asked him to resume duty immediately.\nThe HC also stayed the transfer of secretary of Maharashtra State Medical Teachers Association (MSMTA) from GMCH, Dr Sameer Golawar.\nNiswade was shunted out as a fallout of a case in connection to Dr Makrand Vyawahare, head of GMCH's Forensic Science Department, whom students have accused of sexual and mental harassment.\nBoth Niswade and Golawar were allegedly shunted out on Vyawahare's complaint that they were instigating students against him.\nThe court issued notice to 11 respondents, including state ministers and Secretary of Medical Education and Drugs Department, asking them to reply before December 3, fixed as the next date of hearing.\nThe ministers named in the PIL are Sudhir Mungantiwar (Finance) and Vinod Tawade (Higher and Technical Education).\nMumbai-based Directorate of Medical Education and Research (DMER), Medical Education Director Pravin Shingare, Commissioner of Police, Nagpur and Police Inspector of Ajni, besides Dr Vyawahare, Dr Niswade, Dr Golawar and Dr Pradeep Dixit, holding additional charge of GMC Dean, are other respondents in the PIL.\nThe PIL has been filed by a social worker Trisharan Sahare through senior counsel Sunil Manohar and Akshay Naik.\nWhile passing strictures against the senior Cabinet ministers, the judges remarked that Niswade's shunting was done with malafide intention by the authorities.\nThey said that the respondents were favouring Vyawahare as he was a close relative of one of the ministers.\n\"People's representatives should work in people's interests and should give it a priority before making such moves,\" the judges said.\nCiting service rules, Justices Gavai and Deshmukh said that midterm transfers were carried out with a concrete reason but in Niswade's case, it was mentioned \"for administrative reasons\" without any explanation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I bought from HAC, good service. I paid about 1,250 for 11 modules...but I was an early buyer. Several of us are using the Leaf and Volt modules without BMS. If you do, you need to stay it the 4.1 to 3.7 volt range on charge and discharge. Basically you avoid the cliffs in the curve and give up a little capacity. On the plus side you will get more cycles from your cells. I have over 100 cycles on mine now...perfectly balanced. In comparison my 40ah CALB cells were all over the place, so I used a mini-bms with them. These are automotive grade premium cells. So tempting to buy a 08 on up Ninja...love the look!\nPS: Don't know the state of your ICE motor but I got $750 for mine on Ebay. There's a sprint car class that runs them in Mi.\nLast edited by DRZ400; 15 October 2015 at 0622.\n- setup pack and go straight to bulk charging, probably at exactly 4.1 per phil's suggestion.\nMine were all at the same voltage on arrival....3.85v. I bulk charged to 4.1 than used (2) single cell lipo chargers to bring them to 4.2 Some bottom balance....allot more effort though.\nNext up is charger research, probably checking out Meanwell or if I really go crazy, get a delta Q quiq the 1,500 watt version and then I need to find a place to mount everything.\nI put my SMP72400 controller, DCDC and contactor under the seat.\nWhat DCDC to you use? I'm trying to figure that part out right now.\nNic, which do you plan to get?\nI picked up 11 Leaf modules.\nI pick them up today because FedEx is terrible and can't seem to deliver them. Still no luck with a Charger algorithm for the Delta Q. Aaron said he used #164, but Delta Q won't program it for me unless I buy from them(for full price which is insane), and EVdrives won't program for Lithium. I tried to get in touch with Frodus, but he must be busy.\nI am really excited to now figure out how to mount everything after I bench test. Last up is the charger and then I'm all set!\nI picked up a DC to DC converter from EV drives along with my Delta Q charger a few days back. I didn't realize that the charger uses the same attachment plug shape as most computer transformers. Not a big deal, but one more thing I have to buy.\nSpeaking of the Delta Q, I plugged it in to make sure it was working, and I put my volt meter across the ground and hot and was only reading 30v. Does the delta q sense voltage with the black and white wire?\nI hope they programmed it right and I'm not willing to test it out on my $1,400 battery pack.\nI haven't figured out where everything is going to be mounted, but I have everything except a some 2AWG wiring! Almost complete, just need to fabricate a battery bracket (easier said than done) and get some stuff that HAC didn't send. (They sent the wrong size terminal covers, and didn't send the right amount of copper busbars for my batteries.\nLowes or ACE hardware have an assortment of standard keys.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cecilia Guerrero, kinesiology graduate student at CSULB, recommends taking breaks from sitting long periods of time to stretch out\/open up the chest to reset your posture. Guerrero earned her Bachelor of Science in Kinesiology at CSULB and became a Certified Personal Trainer through the American College of Sports Medicine.\nBreaking habits is the most difficult part of improving your posture. Make these easy exercises a daily routine and you will see an improvement.\nFirst, find a straight, even wall, and place your heels, shoulders and head against it. If you have developed a double chin, you've done it correctly. Make sure you look straight ahead the entire time. Sometimes it helps to do it in front of a mirror, but it's not mandatory. The following three exercises may be done as a set but must be repeated twice and up to three times a day. FYI, speed does not matter, but the technique does.\nAs a starting point, you need to extend arms at a 45-degree angle away from hips. Then flap your arms upward 10 times, but most importantly do not detach heels, shoulders and head from the wall. It helps to pretend you're a beautiful bird flapping your gorgeous long wings. Simple, right?\nThe next exercise consists of you touching your ears with your palms. Again, start at a 45-degree angle away from your hips with your palms touching the wall and bring your forearms and palms towards you until you've successfully covered your ears. Return palms and arms to original 45-degree starting position. Repeat 10 times.\nFor the last exercise in this set, you will pretend to climb a ladder. The motion of climbing a ladder will only be with your arms because again, your heels, back and head will remain against the wall. Pretend to grab something high above and drag straight down. As one arm raises, the other is brought downwards. Repeat 10 times.\nWhat are Chin Tucks? Chin Tucks are an efficient and easy exercise to help reduce a neck hump. During this exercise, sitting or standing, you will slowly stick out your head and scoop back like a chicken. As you scoop your head back, your neck shall elongate - hold for 5 seconds before releasing. Relax for a couple of seconds before repeating and do sets of two, twice a day.\nRound shoulders are reduced and improved by sitting or standing with your head, back, hands and arms against the wall as you pull your shoulders backward in a circular motion. You can do these 10 times for two sets and up to twice a day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I offer the professional services listed below. I also maintain records of the work I do for you, and advise you on how to schedule regular maintenance of your home's unique carpet and upholstery.\nI stay up to date on the latest industry equipment and cleaning techniques. For your home, we use the most effective and safest cleaning solutions avaliable.\nI am confident and skilled at cleaning fine and delicate upholstery fabrics. I can safely clean cotton, rayon, wool, silk and all of the synthetic fibers. I will pre-test the fabric and advise you on what can be safely done and what results you can expect from the cleaning.\nThe fabric protection on both carpet and upholstery will wear off due to consumer use. I can advise you of where the protection should be restored. Protector can be applied immediately following the cleaning. I use the best and safest products for the carpet and upholstery being treated.\nMost area rugs require special attention that can only be provided by cleaning in our specially designed cleaning facility. In our facility we have the time, space and specialty tools necessary for cleaning delicate natural fiber area rugs. We are able to control all of the sensitive processes needed to stabilize colors, clean fringes and dry the rugs properly.\n*If you have a special need that we cannot provide, ask us and we'll be happy to refer you to a reputable company that can meet your needs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Whether you're running your air conditioner during the hot Florida summers or your furnace during the short and cool winters, the placement of your registers can impact not only the efficiency of your HVAC unit overall, but it can also impact your comfort and perhaps even your safety.\nIt's common to find a register in the top of a room and the return on the bottom, and this is particularly true in attic spaces that have been converted into living spaces \u2013 especially in older homes. This causes problems because, in the summer months, the cooled air coming from your register immediately sinks where it is pulled in through the return. This means it never really has a chance to circulate and cool the room. When the furnace is in use, the warm air that comes from the register rises naturally. When it is coming from a register that is placed high in the room, it will continue to hug the ceiling and never get a chance to heat the room.\nWhile it might make sense to put your heating and cooling registers on the interior walls of your home since they tend to stay warmer, this isn't always the best idea. In the case that you're dealing with a windowless room, it is perfectly fine to locate the register on just about any wall. However, if there are large windows in the room, this isn't the best choice. Windows are the coldest part of the wall (even the best windows) and they can cause cold drafts that make many people feel uncomfortable. When a heat register is placed directly under the window, as long as that window is well insulated, it will mingle with that cooler air and prevent the drafts that make you feel chilly.\nIf the air handler itself is in the basement, placing a return register in the basement isn't always a good idea. If the basement is home to combustion-fueled appliances like water heaters, furnaces and even gas-powered clothes dryers, then putting a return register in the basement can be problematic. It may starve these appliances for air and allow dangerous combustion gases to escape their usual paths and enter into the living space. These gases are toxic and poisonous, so it is best to avoid placing return registers in the basement at all. In many cases, careful placement of heat registers can avoid the need to install them.\nAs you can see, the location of your registers can pose some significant problems. If you are building a new home, or even if you are considering the relocation of your ductwork, then you can contact your local Tampa HVAC professional to get more information about the proper placement of heat and return registers. This way, you can rest assured that your family is comfortable and safe, and also that your HVAC equipment is running as efficiently as possible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..a8a24d62e61e91f5c0270f4aea638ad71b92e67f
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+{"text":"What If Everyday Is A Bad Day? \u2013 Being a COA is a thing!\nThis weeks piece has been written by a very special & regular author on Coaisathing, Olivia. Olivia's self awareness is mind blowing and this piece proves that, yet again. Her courage is amazing and this piece will help so many people again, I am sure!\nTalking about things in my life has never been easy. Although I try, the words won't come out. My mind feels like the past wasn't real but my heart feels every single emotion from sadness to anger to guilt and it's only when I talk about things out loud that it becomes reality again. So I keep quiet most of the time . People think my life with an alcoholic was just them drinking and passing out, when it was so much more. It was chaos, it was heartache, it was terror, it was constant uncertainty about things that could happen. There's days when I can carry on living life as though nothing happened but the majority of the time I'm physically here but stuck mentally in the past and it slowly rips me apart so I revert back to old coping mechanisms in order to get through it.\nRecently my mindset has been off. I will always have a good few days and then it will go straight back down again and it takes a while to get out of it. On the outside I look fine but inside I feel terrible. The most annoying thing is that I cant tell people exactly what it is that's bothering me as its a mix of past and present issues as well as overthinking so I don't talk to anyone about it. It makes me feel like a visitor in my own body. It hides my bubbly personality and I become anxious and isolate myself. I don't want to talk to anyone, I can't focus, and I'm on the verge of tears all day. I feel overwhelmed and don't want to socialise or go to school but I force myself and the minute I'm by myself, my stomach drops and I burst out crying. It makes me more upset that I can't go out and do things with friends or family without coming home and crying because it's supposed to be an enjoyable happy time but it never is. There are the odd few people who notice, particularly in school because my work will always be off, who ask if anything's wrong but how do you explain to someone else the amount of havoc going on in your head when you can't really make sense of it yourself.\nMy mum was always so good at making people feel bad about themselves and I never realised it would stick. I wanted so badly to be the person that helped her but I lost my mind trying to understand hers. I try not to let her get to me but my mind is constantly going over things trying to make sense of it all She makes me feel guilty and I question myself if I'm doing the right thing not speaking to her. Doing this makes me realise all the things she's done and the fact shes still not better so I'm doing what's best for myself but overtime I always feel guilty again and the same process repeats. I feel disconnected from everyone around me. I'm always in my own world of fear and worry where I'm constantly thinking of everything even the simplest of things. It feels like everyone's moving forward with their lives and I've been stuck in the same place for ages.\nI want more for myself. I don't want to fail but I can't find the motivation or focus to get things done and I can't talk to people or ask questions without worrying about embarrassing myself. People always say there are good days and bad days but what if everyday is a bad day? I want just one day where my mind doesn't wonder and my eyes fill with tears. I don't want to be like my mum. I worry that growing up with her meant that I picked up her toxic traits and it scares me especially when it feels as though I'm walking around with her on my back everyday.\nPrevious Post It May Fade But It Never Goes Away!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to our first blog post! There have been some questions about an easy way to keep up to date with our sailing and as anticipated we plan to write a blog post for any and every important update about our team. We decided now is a great time to get the updates started as we prepare for our first big event of the year, the Miami Sailing World Cup!\nToday was the first official day (registration day) of the Miami Sailing World Cup. Paris and I wrapped up our registration today around noon and then finished all the boat work that goes along with getting ready for the event. I'll come back to that, but a little about our pre-regatta prep.\nLast weekend we sailed the first regatta of 2019, the Midwinters Championships. We didn't have the most fantastic overall result (14th\/ 31), but it was about around where we should have been given where we are in the campaign and our big lack of racing to this point. I say big lack, not because we have been avoiding it, but because we have had so many other little things to focus on to get us even remotely race ready. Last fall and winter and all the way up to this point, we have been working on the other little things. We really dialed in the boat handling and mechanics to the point where we are ready for any situation that would occur on the race course. However, because of that, our tactical and starting game have been a little rusty. And that's how we really felt on the race course, rusty. We made calls that as soon as we did them, realized that it wasn't necessarily the call we should have made for 49erFX sailing, or similar little things like that around the course.\nWe did however, have one epic day of racing where we got two races in and finished with a 5,1 in some big breeze. But those days are always the most fun!!\nAfter the regatta, we took two days off, before getting ready to sail again. We have been starting each practice day with some tuning with our Canadian friends, Ali and Mariah, and then we hopped on some racing that was run by some coaches, which included around 20-25 boats each day. It was great to take the lessons we learned from the last regatta and start being able to put them into racing these last couple of days. It built our confidence going into the regatta.\nNow, back to today. The measurement process required us to weigh our boats, which means de-rigging just about everything on it and removing almost everything to lift it on a scale to make sure it makes weight (there is a minimum weight the boat is allowed to weigh). Some boats come in a little under the required weight, so in order to bring it up to weight, we have to add corrector weights. Once the weighing was done, we had to re-rig all the lines, put the blocks back on, put the mast up and the boom on. And while we are doing that we have to mark each part of the boat with stickers and record the serial number in order to complete the registration process. The second part, the sticker part, is not that difficult to do. It's the re-rigging of everything that takes a little bit more time.\nWe managed to get it done and then we hit the water in literally no wind just so we could make sure that everything was working. We got out on the bay, did 2-3 tacks, pulled the kite up, gybed, pulled the kite down and the towed back in.\nSo, we are now ready to start racing. Tomorrow we are taking off from sailing and have the opening ceremony in the evening. I'll give a little more detail on the event tomorrow when we find out the plan for the week and the forecast.\nNext postMiami World Cup Starts Tomorrow!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Crisp autumn air. Heart-pounding races.\nA memorial to a true champion.\nThe Bill Braxton Memorial Regatta honors the spirit of Bill Braxton, a Marietta College student whose Lighweight 8 crew went undefeated and won the Dad Vail Regatta in 1973. Later that year, Bill's life was cut short by a fatal car accident.\nThe Bill Braxton Memorial Regatta stands for the selfless dedication of the true oarsman, one who holds the true spirit of competition as a healthy, integral part of life.\nAs the final regatta of the season, one never knows quite what to expect. Rowers have alternately competed in slush and snow, in rain, and under sunny skies. In one year rowers may wear long underwear and pogies; in the next they may row in t-shirts.\nDiscover Bill Braxton's story and explore the history of the regatta named in his honor.\nAre you a high school senior who rows for a crew team? Consider applying for a $3,000 Braxton Memorial college scholarship.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We try to answer below common questions asked by our clients about domestic violence charges under Florida law. Please do not hesitate to contact us at (407) 831-7800 or send us an email to jack@jackklaw.com If you need legal assistance.\nWhat does it Mean if I am Charged with a Crime Involving \"Domestic Violence\"?\n\"Domestic violence\" is not a crime but a designation that makes a crime more serious. You may be charged with assault and\/or battery or stalking, etc. The use of Domestic Violence changes how the case is handled and which laws and penalties apply, including minimum sentence requirements. In today's world, almost any violence between two people involved in a relationship may be charged with a Domestic Violence modifier. This can be a very real and serious problem because of the difference in the penalties for Domestic Violence offenses.\nDon't you have to be Living with a Person to be Charged with Domestic Violence?\nNo. The important element of a Domestic Violence classification to a charge is in how the law defines what is required to treat a case as Domestic Violence. The offense has to be committed by one family or household member against another. This can mean a spouse or former spouse, a girl or boyfriend, or someone related to you by blood or marriage or if a child is shared between the parties. It does not matter if you have lived with this person in the past, present or ever.\nIf I am Sentenced on a Domestic Violence Charge, can my Record be Sealed Later?\nWhat if the other Person does not want to Press Charges Against me?\nThat does not mean the charges will be automatically dropped. In fact, that does not even mean the situation has improved much at all. The Prosecutor may have other evidence, such as pictures of injuries, admissible statements, etc. The Prosecutor does not always have to rely on the testimony of a victim to charge, prosecute and win a domestic violence case. Domestic Violence offenses are taken very seriously by the prosecutors and they are pursued and can sometimes even be won with or without the testimony of the victim.\nWe try to anticipate common questions one might have about various legal issues and services and provide information here. If you need additional information or need legal assistance please contact us at your earliest convenience. We are available at (866) 422-7934 and (407) 831-7800. You can also fill our our online form provided at the top of this page or send us an email at jack@jackklaw.com and we will contact you back as soon as possible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LinkedIn is a great platform for connecting professionally, finding work and sharing content. But have you ever thought of using LinkedIn for lead generation?\n\"One of the best features of having a LinkedIn premium account is being able to use advanced filters in search. Not only can you search by company and relationship, but premium advanced search on LinkedIn allows you to search by function, seniority level and company size, too.\" ~ Doreen Bloch, Poshly Inc.\n\"We promote our guest contributions on LinkedIn and engage with prospective clients by asking for feedback on the content. If you utilize LinkedIn to educate leads, you'll provide more value and ultimately form stronger relationships. \" ~ Kelsey Meyer, Influence & Co.\nI know it's obvious, but I know too many brands who aren't there: Use the LinkedIn PPC platform! If you're paying over $2 CPC on your other channels like AdWords you'll see a similar CPC but you'll get fabulously targeted leads that are in the right industries, companies, and positions in their companies.\nI haven't used boolean search functions for quite some time so thanks for bringing this up. I have to agree with you though that compliments are the fastest way to get warm leads, that feel-good factor we all love..\nMany people stay away from the aggressive side because it looks needy. I find that creating and posting a business page along with your personal LinkedIn is a great way to get attention because they get viewed more and has what services or products you sell. More people need to create new content though because I see a lot of ideas repeated. Thanks.\nThere is no doubt about it! linkedIn is a Goldmine for Marketers,Brand manager, HR's and CEO's,etc. But 68% people ( 2013, december, Socialbakers.com statics) really don't know how to use linkedIn properly, That's why they are just using LinkedIn to send promotion mails,publishing spams adverts on Groups,profiles.\nI agree there are some very good ideas suggested in the article. I have tried a few of them.\nI find that a short personal complimentary note works well. I always review the individual profile before I send anything. I receive many positive responses back.\nI think that the key to using social media is to really participate in the community. Failure to do that will only lead to a dead account with dead followers. You need to participate to get any form of real benefits from it.\nOne vital tip that has been of great use to me in generating leads for my business is to garner solicited and unsolicited recommendations with those who you delivered value to over the years (or previously). This does the sales and marketing upfront on a virtual basis and then the closure cycle time is reduced as a direct result. It is important who we get recommended by and how their reputation is in their network over the years.\nThanks for sharing this amazing post. I was not aware of -> Connect with Twitter benefit, Thanks for sharing this option.\nLinkedIn is a very powerful tool to use when prospecting for business. We have found that creating a meaningful group via LinkedIn, then promoting the group to get targeted participants to join is the first step. Then update the group with good relevant information is the second step. And last, you can reach out to the prospects in your group with a message that will hopefully lead you to a phone call.\nGood article! My questions is, this is obviously a time consuming process and you need someone who is familiar with LinkedIn. What if you just want to get prospects from LinkedIn and export it to your CRM? How easy would it be if you can just search for relevant prospects using Advanced search and download the contact info? There are many such tools available out there like SalesLoft, Aeroleads to mention a couple.\nThanks for the interesting article, The Young Entrepreneur Council.\nAbhi, you are right, there is ton you can do with LinkedIn's advanced search feature.\nMost average users typically shoot from the hip when it comes to strategy. It is imperative if you are using LinkedIn lead generation as a prospecting tool you have a clear direction on who you want to connect with.\nThank you for sharing the effective tips on Linkedin here. I have a question on this, is it really okay to send messages to the people who I find on \"who has viewed your profile\"? Dont they find it irritating?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"FAQ :: What is \"the third heaven\"?\nIt is where God dwells. The first heaven we see by day (the sky), the second heaven we see by night (the stars), the third heaven we see BY FAITH (where God lives).\n\"I knew a man in Christ above fourteen years ago, (whether in the body, I cannot tell; or whether out of the body, I cannot tell: God knoweth;) such an one caught up to the third heaven\" (2 Corinthians 12:2).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Zoetis announced on Oct. 8, 2015, in Lexington, Kentucky, that they have received USDA approval for their new Lepto EQ Innovator vaccine against leptospirosis caused by Leptospira interrogans serovar Pomona. The vaccine was designed to potentially help reduce the risk of equine recurrent uveitis, abortion or acute renal failure caused by L. pomona.\nEditor's note: More information on Lepto EQ Innovator will be coming soon, so check back on EquiManagement.com for additional resources.\n1. Data on file, Study Report No. Restricted Grant-FTLEPTO13 (v1.0) TI-01366, Zoetis Inc.\n2. Carter, C.N.; Cohen, N.; Steinman, M.N.; Smith, J.L.; Erol, E.; Brown, S. Seroepidemiology of equine leptospirosis utilizing diagnostic laboratory specimens from 29 states (US) and one Canadian province, in Proceedings. 55th Annu AAVLD Meet 2012;51.\n3. Divers, T.J.; Chang, Y-F. Leptospirosis. In: Robinson, N.E.; Sprayberry, K.A., eds. Current Therapy in Equine Medicine. Vol 6. 6th ed. St. Louis, MO: Saunders Elsevier;2009:145-147.\n4. Data on file, Study Report No. B850R-US-12-011, Zoetis Inc.\n5. Erol, E.; Jackson, C.B.; Steinman, M., et al. A diagnostic evaluation of real-time PCR, fluorescent agglutination tests in cases of equine leptospiral abortion. Equine Vet J. 2015;47(2):171-174.\n6. Polle, F.; Storey, E.; Eades, S., et al. Role of intraocular Leptospira infections in the pathogenesis of equine recurrent uveitis in the southern United States. J Equine Vet Sci. 2014;34:1300-1306.\n7. Borstel, M.V.; Oey, L.; Strutzberg-Minder, K.; Boeve, M.H.; Ohnesorge, B. Direkter und indirekter Nachweis von Leptospiren aus Glask\u00f6perproben von Pferden mit ERU. Pferdeheilkunde. 2010;2(M\u00e4rz\/April):219-225.\n8. Gerding, J.C.; Gilger, B.C. Prognosis and impact of equine recurrent uveitis. Equine Vet J. In press. doi: 10.1111\/evj.12451.\n9 Faber, N.A.; Crawford, M.; LeFebvre, R.B.; Buyukmihci, N.C.; Madigan, J.E.; Willis, N.H. Detection of Leptospira spp. In the aqueous humor of horses with naturally acquired recurrent uveitis. J Clin Microbiol. 2000;38(7):2731-2733.\n10. Dwyer, A.E.; Kalsow, C.M. Visual prognosis in horses with uveitis, in Proceedings. Amer Soc Vet Ophthalmol Annu Meet 1998;1-8.\n11. GAO. Horse Welfare: Action Needed to Address Unintended Consequences from Cessation of Domestic Slaughter. Available at: http:\/\/www.gao.gov\/products\/GAO-11-228. Published June 22, 2011. Accessed September 28, 2015.\n12. Pick, M.; von Salis, B.; Schuele, E.; Sch\u00f6n, P. Der Verkehrswert des Pferdes und seine Minderungen (\"Value of horses and its depreciations\"). 3rd ed. Berlin, Germany: Veterin\u00e4rspiegel Verlag GmbH; 2012.\n13. Data on file, Study Report No. B951R-US-13-043, Zoetis Inc.\n14. Data on file, Study Report No. B951R-US-13-046, Zoetis Inc.\nTherapy in Equine Medicine. Vol 6. 6th ed. St. Louis, Mo: Saunders Elsevier;2009:145\u2013147.\nwww.equinechronicle.com\/leptospirosis-inhorses. Accessed June 23, 2015.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Volunteering can be a learning experience, rewarding and a great way to save money for the fulltime RV'er. Throughout the country and Canada there are public lands and campgrounds regularly looking for volunteers with RVs to help in exchange for a free campsite. There are a variety of opportunities available, many of which provide the volunteer with a unique and usually fun experience. The work is easy and the hours not too demanding. Some parks require little as 20 hours per couple a week, while National Parks often require 32 hours per person. In general expect 40+ hours per couple and a 3 month commitment.\nCampground Hosting is the most common and available volunteer position. Host duties vary between parks. Most include campsite clean up, selling firewood, prompting campers to observe rules, and some mix of other maintenance or administrative duties. Some campground hosts positions can feel like a 24\/7 job, since you are often the first person campers come to with an issue. It is very important to discuss and be clear about the duties with the volunteer coordinator prior to accepting the position. If something is not clear or sounds strange ask for clarification.\nInterpretive Volunteering opportunities are available at historical sites, lighthouses, Fish and Wildlife and others. After a brief training, volunteers will conduct tours and provide information about the site. These usually include time at a visitor center and\/or gift shop. RV accommodations are at either a nearby campground or somewhere on site. Unlike campground hosts, when you are back at your site you are 100% off duty. Interpretive Volunteering is a great way to do something tailored to your interests.\nThere are a few ways to find RV friendly volunteer jobs. The first and easiest is to browse Volunteer.gov. This website lists positions from around the country, and once you have signed up is very easy to use. However, this site only lists a fraction of the opportunities available. In some cases these are the less than cherry jobs, and you will be competing with other applicants. Another way is to contact a park directly and ask if they have a volunteer program and if there are openings. This is best if you want to work at a specific park. Another way is to pick a region or area, and find out what is there i.e. state or municipal parks, Fish and Wildlife etc., and contact them via e-mail from contacts given on their website. We have had a lot of success with this method though it is research intensive and one needs to know their way around the internet. Most public parks and lands are looking for volunteers so it never hurts to contact them. In our experience follow up and persistence is almost always necessary, or you will end up at the bottom of the pile or a forgotten e-mail.\nThere is really no best time to apply for a volunteer position. When we first started I secured positions up to a year in advance. We have found this unnecessary and can be limiting as travel plans and timing changes. Yet for the most coveted jobs it may take a year or more to get in, i.e. Florida Keys during winter. On the flip side we recently were able to find a great position at a lighthouse only a few weeks out. We also receive phone calls somewhat regularly from people who need someone ASAP.\nIt is best to be thorough upfront with your questions. Always find out about the hook ups and amenities. Most positions do provide full hookups, but some in more remote places may not but try to be accommodating in other ways. It is important to find out about things like the laundry and shower situation, where is the RV spot located, how far is the grocery store, what sort of documentation is needed, physical requirements, how many other volunteers will be sharing duties (more the better), is there WiFi or connectivity, how do I get to the volunteer site, is it noisy or a party place, etc. Remember this is a volunteer position, if does not work out, your home has wheels for a reason and the option is always there to leave.\nIn the end we have found that many people tend to return year after year to volunteer positions they enjoy. Returning volunteers always receive preferential treatment, and for them it is comfortable and easy to come back to a place where they already know what to expect. Again, each place is different and there will be a learning curve, and systems and personalities that will take some getting used to. Volunteering most of the time is fun and stress free, and great way to really experience a place for a few months cost effectively.\nIf you have any questions about Volunteering for RV'ers, feel free to contact us through our blog, Facebook or e-mail.\nThis entry was posted in RV Lifestyle and tagged Camp host jobs, Camp Hosting, Free Camping, Fulltime RV Volunteer, how to become a camp host, non-retired age RVers, resident volunteer, RV Volunteer, What is camp hosting, Work for RV'ers, workamping, Working on the road, young RV volunteers, young workampers on May 1, 2014 by The RV Nomads.\nGreat post. I am learning a lot from people like you already out there and doing it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Arizona House of Representatives Commerce Committee will hear House Bill 2335 in the near future. H.B. 2335 establishes that the sale of premium cigars, pipe tobacco, and accessories to adults between the ages of 18 to 20 years old is a criminal offense.\nH.B. 2335 also removes the word \"knowingly\" from statute in relation to a sale. Currently, your employee would have to \"knowingly\" sell to a minor in order to face legal repercussions. Under H.B. 2335, if an employee is presented with fraudulent identification, the employee is liable.\nPlease stand up for your retail tobacco business and the rights of law-abiding adults by expressing your opposition to H.B. 2335. Click on the button below to contact the members of the House Commerce Committee TODAY!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Okay, so we discuss the perceptual decisions of monkeys.\nCan we also explain perceptual decisions of humans?\ndecision, whether you see a house or a face.\nand the fusion model that explains a perceptual decision.\ndetectors, and face detectors in the brain.\nthat simply compares the activity of the.\nHouse detectors and face detectors.\nhouse, and if activity B is larger than A, I see the face.\nthat can, perhaps, explain also our perceptual decisions.\nSo, we, we can, indeed, create this task.\nsometimes they see a very ambiguous information.\nthe Effem race counter to make perceptual decisions.\nsize of study using quite noisy stimuli, so he presented.\nreally difficult to recognize face and house.\nthat you see a face or house it can be quite difficult perceptual decision.\nWell sometimes there's no conflict.\nThere's a clear simple perceptual decision task.\nSo, we indeed can finally bring face and house detectors.\ndiffusion model, to explain our perceptual decisions.\ndetectors, and they are activated when we see the ambiguous pictures.\nactivity of the house detectors and face detectors.\nthe activity in the face detectors, we decide that we see houses.\nactivity of the house detectors, we decide that we see faces.\nSo, let's take a look to the actual data.\nWe indeed can find face and house detectors in the brain.\nSo, green color indicates brain areas that are sensitive for houses.\nRed color indicates the area that is sensitive for faces.\nto the amount of facial information, the stimuli.\nSo, face responsive brain area, shows a maximum response to the clear faces.\nthere is less facial information in the stimuli.\nactivity proportional to the amount of house information in the, stimuli.\nSo we do find in the brain the face and house detectors.\nsimply compares sensor evidences for houses and for faces.\nAnd indeed we can find this area in the dorsal lateral prefrontal cortex.\nSo, activities in the dorsal lateral prefrontal cortex is proportional.\nfaces of house detectors, so this is an integrator.\ninformation from face detectors and house detectors.\ncorrelated with the individual performance of the subjects.\nThe more this area was activated, the better was the performance.\ndifference in the activity of the face and the house detectors.\ndorsolateral prefrontal cortex predicts behavioral responses.\noutputs from the detectors, from the house detectors and phase detectors.\nSo we can also apply the same diffusion mechanism to our perceptual decisions.\nhelp us to understand different aspects of the decisions.\nbe the different starting point for the diffusion process.\nSo we can also model their deepness of the processing of the information.\nof accumulation of the information, at the rate of diffusion process.\nconducted under Research Group Ohio Kahiki.\ncars and subjects, performed at perceptual categorization task.\nsome subjects was inhibited by transcranial magnetic stimulation.\nit increased the reaction time for the decisions.\nthe drift rate of the diffusion process.\nprefrontal cortex, to the accumulation of the evidences for certain decisions.\nSo, the diffusion model, is the most powerful model in neuroeconomics.\nneurons, firing grade reaches this threshold, the decision is finalized.\nactually makes this calculations similar to the diffusion process.\na cluster of models called sequential sampling models.\nduring the decision making process.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Year-round, complete home comfort can seem like a distant possibility when trying to choose which home comfort system is best for the conditions in Pacheco. Making that decision impacts not only your comfort, but your wallet as well, so making the most educated decision possible is critical to keeping your home comfortable for the long run.\nCalling it a heat pump can seem a bit confusing when the system actually covers both your heating and cooling needs, but could easily be the complete system you've been searching for. Heat pumps function like refrigerators in that they use electricity to pump refrigerant to different spaces to transfer heat. The main distinction between an air conditioner and a heat pump is that heat pumps transfer heat from the outside air and air conditioners expel heat out of the home.\nHeat pumps can be the perfect choice for energy efficient heating and cooling year-round, and because it can provide both functions, all that is required is a simple change of the system's mode. Because heat pumps don't have to create heat, it makes them a lot more efficient as well.\nSo how can you benefit from a heat pump in Pacheco?\nWhen the time is right to do away with your current heating and cooling system, a heat pump may be the right answer for you and your comfort needs. When used in the right environment, heat pumps are a great all-in-one system that can handle your year-round comfort needs easily. If you'd like to find out more about heat pumps and how they fit in to the Pacheco environment, give Clean Air HVAC a call at 925-233-6238.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ever wondered why we call the colored part of our eye the \"iris\"?\nThe ring of pigmented tissue surrounding the pupil of the eye actually borrowed its name from the Greek goddess of the rainbow.\nLike the rainbow that fills the sky with different colors, the human eye can also take on a rainbow of colors, depending on the level of a pigment called melanin.\nFor generations, scientists thought that eye color was controlled by a single gene for melanin, and that only a few outcomes were possible from a dominant brown gene variant crossed with a recessive blue variant.\nToday, our understanding of the entire genome has broadened that picture significantly.\nWe now know there are at least 10 separate genes that all play a role in determining eye color.\nIt's not all or nothing, but a much more nuanced story.\nIn fact, of these 10 genes, there are two that account for the majority of eye color outcome, and at least eight other genes that also play a smaller, more moderating role.\nThe two main genes are called OCA2 and HERC2, both located on chromosome 15.\nIn a fascinating twist, the \"HERC2 gene contains a segment of DNA that controls the activity (expression) of the OCA2 gene, turning it on or off as needed,\" according to the National Institute of Health.\nSo even different genes can, in effect, cancel each other out. The picture is complicated.\nThis and other surprising recent developments about eye color that contradict what we previously thought.\nHere are three ways that our understanding of eye color has changed in recent years.\nUnlike hair and skin, eyes don't produce pigment continuously.\nIn up to 15 percent of people of European ancestry, eye color lightens or darkens over the lifetime.\nFor example, very brown eyes can mellow into hazel or vice versa.\nThese changes are normal and gradual over a long period of time, but if drastic changes occur in a short time span, you should see your doctor, as this could be a sign of a more serious eye condition, such as Horner's syndrome, retinal glaucoma, or even melanoma of the iris.\nIn some cases, trauma, such as a blow to the head, can cause a change in eye color.\nOften this means that the affected eye and unaffected eye will be slightly different colors. For example, pop icon David Bowie had one blue and one hazel eye, which he assumed to have come from a childhood injury to the face.\nHeterochromia\u2014having two different colored eyes\u2014can also occur naturally, without any history of trauma.\nThe unique coloration can be accompanied by any of a few rare genetic disorders, but more likely it is not.\nIt is worth a trip to the eye doctor to see, especially if the cause has not been determined.\nBut, heterochromia can also be a perfectly benign variation.\nEye color can in fact change from an early age.\nBabies are born with less melanin, and their eyes will gradually darken over time as melanin accumulates in the iris.\nWithin a few months to a year, their eye color will mature into their permanent color.\nBaby blues are actually new to the scene\u2014a genetic mutation that showed up relatively recently in human evolution.\n\"Originally, we all had brown eyes,\" according to Hans Eiberg from the University of Copenhagen, who led a team that examined mitochondrial DNA\u2014passed down only on the mother's side\u2014and found that the blue variant of the OCA2 gene appeared somewhere between 6,000 and 10,000 years ago.\nIn that comparatively short span of time, in terms of generations, there was a sea change: no one on earth had anything but brown eyes for most of human history, and now around 30 percent of Europeans have blue eyes\u2014all because of one genetic mutation.\nEveryone who has that gene, the blue variant of OCA2, does in fact share a common ancestor.\nAnd the mitochondrial DNA research allows scientists to pinpoint that that common ancestor lived less than ten thousand years ago\u2014a blink of an evolutionary eye.\nDespite the complex interaction among the 10 melanin-related genes, the field of genetics is bringing eye-color testing rapidly into focus.\nOur understanding of the human genome has progressed so radically from just a decade ago that now it is possible for scientists to take a sample of DNA and predict eye color, hair color, and skin color of the subject with very high accuracy.\nThis cutting-edge technology has significant implications for both forensic scientists working crime scenes and for physical anthropologists tracing human history.\nAnd best of all, researchers have made the data public and free for all to use.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are many reasons why Jacob Have I Loved by Katherine Paterson has won SO many literary awards, including Newbery Medal (1981), Outstanding Children's Books of 1980 (NY Times), American Library Association - Best Books for Young Adults (1980), Best Books of 1980 (School Library Journal), and Fanfare Honor List (The Horn Book). The characters are well developed. The storyline is interesting. Many people can well identify with the main character's feelings of sibling rivalry. The best reason, however, is that it is a thoroughly engrossing and enjoyable novel!\nThirteen-year old Sara Louise \"Wheeze\" Bradshaw is green with envy. Her twin sister, Caroline, is everyone's \"darling\" and everything that she is not. She is popular, beautiful, talented, and saintly. She is beloved by her teachers, fellow students, town people, but most of all, by her parents. What is a bright, comely, hard-working, tomboy to do?\n\"Location is everything\" and the setting of this novel is an important and prophetic aspect. Set on Rass Island, a small inbred and religious fishing community in the Chesapeake Bay during the 1940s, the insular tragedies and the impact of global events reflect the interior turmoil and exterior trials that the main character must experience in order to grow and mature. A product of the Depression, Wheeze well knows hard times and hard work, but this does not prepare her for the \"World War\" size conflicts of adolescent sibling rivalry.\nTwo minor characters deserve special mention. A mysterious elderly man appears at the beginning of the novel. The mystery of his identity seduces the reader into this novel, but it is his impact on Wheeze that reveals and illuminates her jealous and possessive nature. Wheeze's grandmother acts as humorous fool and Biblical prophet of doom. An astute reader will soon pick up on the fact that her words and quotes are a form of fore shadowing. Unfortunately, I fear with the current level of Biblical illiteracy in the US, much of what author Paterson (a seminary trained missionary born to missionary parents) draws upon as the common-knowledge base of her readership will be lost on many of her young readers. (If you dear reader, like I, missed that fact that the title is a Biblical quote and not a hint about lost puppy love, I recommend keeping a Bible near at hand.) This is not to say that a young person cannot read this novel, enjoy it, and learn from it without a Bible or at least a minimal grounding in the Bible, but without it the novel's depth will be shallower. Not only will many plot nuances be overlooked, but the significant psychological and social impact of the Bible on the islander's views and behavior may not be fully grasped.\nI highly recommend this book. After reading this book, I knew why it had earned so many awards. It is eminently readable and quite enjoyable for young people and older adults. It is excellent bibliotherapy for those working through the anger and jealousy of sibling rivalry, even if historic in nature. This book is useful in a history class as a period piece for adolescent life during the 1940s or in a Sunday school classroom for teaching young people Christian ethics. I, however, preferred this book in a hammock under a shady tree for deep sighs over golden yesteryears.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At the close of the contest, this week's guest judge, Carol Morgan, @CounselorCarol1 will nominate five finalists.\n#5MinuteFiction Week 75 #NaNoWriMo Edition!\n* You must directly address today's prompt: Your main character encounters a demon.\nAt the close of the contest, this week's guest judge, R.B. Wood, @rbwood author of The Prodigal's Foole, released yesterday, will nominate five finalists.\nThis week's winner: Signed copy of The Prodigal's Foole and a 5 page critique from me.\nOne participant chosen at random: E-copy of The The Prodigal's Foole and a 5 page critique from me.\n* You must BEGIN your entry with today's prompt: More dangerous than it is beautiful, the jungle pities no man.\nAt the close of the contest, this week's guest judge, Kaolin Fire, @kaolinfire will nominate five finalists.\nNOTE: New time starting this week! 12:30 Eastern.\nAt the close of the contest, this week's guest judge, Sessha Batto, @SesshaBatto will nominate five finalists.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1. Throughout the story Dr. Seuss only lets us see parts of the Once-ler (his eyes and hands). Draw a full body picture of what you think this greedy guy looks like.\n3. Imagine you were the Lorax and were chosen to speak for the trees. What would you say to persuade the Once-lers to stop cutting down the trees?\n4. Make a Lorax mustache!\n5. Write instructions for how to plant a tree.\n6. Design a seed packet for truffula seeds.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..403cba1429abad790dbe00be78495b6953ad3b1a
--- /dev/null
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+{"text":"Our lightest, most compact closed-cell mattress is now warmer than ever: a new, aluminized surface reflects heat, increasing overall warmth by nearly 20%. The proprietary foams are softer on top for extra comfort and denser on the bottom for extra durability. Wherever youre headed, the Z Lite Sol mattress is the right choice for ultralight, warmth, durability and comfort.\nThe Thermarest Z Lite Sol is a foam closed cell mattress equipped with reflective thermal barriers. These barriers increase the heat reflective of 20% compared to the Z lite. This foam mat has an R-Value of 2.6. The egg surface structure provides optimum insulation. Ultra-light, the Z Lite Sol is a foam mattress that weighs only 410 grams!\nDurability, warmth, comfort, light weight.\nAvanced warmth: thermal reflective barrier sends heat back to your body, increasing warmth.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Damage report! \u2013Warning. Damage to warp core. Containment failure in five minutes.\nI must admit, I do get a certain perverse pleasure out of it.\nI'm Captain Benjamin Sisko. Welcome to Deep Space 9.\nYou are really enjoying this, aren't you?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"(SRLC) and requests a ruling pursuant to G. S. 163-278.23.\naccounts from the North Carolina Republican Party.\nspent on the actual conference.\nParty will come totally from individual participants.\noffice, are attached and are incorporated into this opinion by reference.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hammers and High Heels DIY Smoke Guard for Our Fireplace - LED fireplaces can come in different structures and sizes. Many is usually put about the choices for only a fashionable search, even when others are classical place styles. Regardless what room decoration buy, you can discover an electric powered LED fireplace to satisfy your lifestyle choices.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Traditionally each time we have an equinox or a solstice we celebrate and welcome in the new season with 108 Sun Salutes.\nWe will do a pyramid style of building up and then coming back down the other side, so we will get warm and feel amazing when we reach our final set.\nHaving some yoga practice will help, but you will need a degree of fitness, so that you can get up and down ok. The flow is constant and we will only stop shortly for a drink and loo break.\nThis is \u00a310 for a 2-hour class. Led by Helen Dixon who has been teaching since 2002, and has led many workshops, classes, retreats and sun salute celebrations. Drop in, no need to book, just bring a yoga mat and yourself.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhzla b/data_all_eng_slimpj/shuffled/split2/finalzzzbhzla
new file mode 100644
index 0000000000000000000000000000000000000000..6d59be038bad072aa6a99e7ec4e264525fdeaef4
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+{"text":"Download Baile Con La Talaca The Dance With Death ebook PDF or Read Online books in PDF, EPUB, and Mobi Format. Click Download or Read Online button to BAILE CON LA TALACA THE DANCE WITH DEATH book pdf for free now.\nThis textbook uses concepts and methods of the humanities to enhance understanding of medicine and health care.\nSelections from the modern works of art collected by Joe A. Diaz.\nKey contemporary artists in Texas are brought together in one stunning volume.\nYou Know You're 50 When..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Paul Manafort was found both innocent and guilty today. Innocent by way of a mistrial on 10 of the 18 charges and guilty of the lesser eight charges. It is an outcome that some are stating is as convoluted as the politically-motivated case against Mr. Manafort has been since its beginning.\nNext will come the sentencing phase and quite possibly an appeal by Manafort's defense team.\nWhat remains unknown is if President Trump will intervene on Manafort's behalf via a pardon or commuting of his sentence. Whispers indicate Team Mueller is hoping the president will do just that as they believe it will paint Mr. Trump as a man desperate to protect his allies.\nWith Manafort awaiting sentencing and former Trump personal attorney Michael Cohen apparently signing off on his own plea agreement, leftists are salivating at the thought real damage might finally be done to the remarkably successful Trump presidency, though, there is widespread disappointment among them that Manafort was not found guilty on a majority of the charges brought against him.\nAs for the president, he doesn't appear too concerned with any of it. He's off to massive campaign rally in West Virginia later today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"the sun is shining and my deck is begging for some sweet blooms.\nHopefully, this weekend I can get some potting done.\nAre we ready for the wedding?\nbut I'm planning on catching it on all the news stations.\nI wanted to share with you a great magazine that I subscribe to.\nand love it as much as I do.\nIt's a publication that I look forward to and savor over and over.\nbut it really celebrates the \"country\" and rural lifestyle of Britain.\nI'll take this one, please!!\nthat's it's really my \"cup of tea\".\nbut really worth it if you like a fresh take on \"English Country\".\nreminded us that Easter is all about \"the Lamb\"\nand the story behind these lovely flowers.\nPlease remember to link up or share the button within your post.\nThe Seed Box Spring Open House and Barn Sale.\nLot's of photos next week!\nThanks to all of you who leave comments or join in for the party!\nI appreciate you all and love seeing what you're up to.\nHi Debra...I have definately been here before but it was not showing me as a follower. Well! I took care of that, I love your blog and must follow! lol Im a lover of all things, old, vintage, chippy, junky. Ive recently come out about my love for junking...lol it's not to big in New Jersey. I feel liberated now and found some fabulous junky sistas!\nI love that magazine too:) Your post is just beautiful today.\nHi Debra:gardens,country cottage,nature,birds....a dream for me!!!.I hope the weather will be nice where you live.Also I'm a bit bad,I hope to be better tomorrow.See you...I'm curious to see the wedding tomorrow!.\nBeautiful post!!! I used to subscribe but haven't in years. Will be picking up a copy!\nI love the British edition of Country Living. I pick up a copy everytime I go there. Wonderful pics you featured here! Thanks for hosting!\nOh... I could absolutely park myself in that red toile chair and not get out of it for DAYS!!! I love that room!\nThose photos are so beautiful!! Someday I will move to a quaint little English cottage in the countryside!\nDebra, I love that magazine more than our version. I didn't know you can subscribe to it though. I have been going and buying it occasionally. Thanks so much for hosting.\nBeautiful images! Thanks for hosting the party.\nwouldn't i love to live in the British Countryside!?!\nI'll be sure to pick up a copy after seeing this! Thank you for hosting!\nDebra I just got my copy of BRITISH COUNTRY LIVING the other day, I love it and so glad I can get it at just about any Store that has a Magazine Selection around town... I so enjoy the Mags from Europe, even those that aren't in English because the pictures transcent Language Barriers and a great picture speaks 1,000 words.\nBeing 1\/4 British (my Grandfather), maiden name Bedford, I have loved your Royal posts! I have always dreamed of a cottage in the English countryside! Beautiful photos!\nIt does look more interesting than our version! I've picked it up and thumbed through it at B&N but never bought it. I may just subscribe!\nHi Debra! This post just took me back to our trip to England several years ago. It was like a dream come true to walk through the Cotswold villages and pretend like I was in a Jane Austen novel! :-) Those were fabulous features this week and we are so thrilled to get to join the party with you!\nThat cottage looks like the one from \"the holiday\"!!\nI can not throw away my old copies of English country living.\nWhy don't we just move over there for 1 year then come back to our homes here? Anybody want to go with me?\nThanks for hosting. I just became a follower and linked up. Love your blog and foundation. I'll be back.\nI must check out this magazine...your photos alone have me hooked! Also,I love your new blog look Debra!\nHi Debra! Thank you for sharing your delightful magazine with us and for being our lovely hostess too!!\nOh how I love this magazine too Debra! I think you and I have been sharing a dream if you have also wanted to live life in the English countryside.\nHey! we could then each have our own little flock of sheep with only a little stone fence separting them. Wouldn't that be great?\nHi Debra ~ I am a magazine hound and I don't think I have ever seen that one. Thanks so much for the tip. Im off like a prom dress to buy one. I love the red toile comfy room. Im in the process of a major re-styling of our common rooms and every bit helps.\nThat's a great magazine, I was just looking at it on the newsstand the other day.\nHi Debra, Love the British inspiration and the toile.\nI've long enjoyed Country Living magazine! And the pics with the pops of red in the furnishings are really doing it for me today! I've got lots of red in my dining room!\nThanks for hostessing this fun party!\nI'm with you on the British Country Living!\nWhat a charming, lovely collection that made me feel very British! Definitely inspired.\nThis is one of my favorite magazines as well. Thanks you for hosting. Have a wonderful weekend!\nI've never seen this magazine but can't wait to check it out. Your photos are so beautiful and so full of inspiration.\noh, that is one of my favorite magazines!!!!!\nWhat a lovely magazine! Dreamy.\nI'd love to pick that magazine up...looks so pretty!\nHave lots of fun tomorrow and take lots of pictures!\nI have dicovered English Country Living and many other British magazines. I have been inspired by their English gardens as of lately. Have a wonderful weekend.\nI am a CL fan. Loved these gorgeous photos you shared!\nSorry I am late to the party! We just got phone\/internet back early am today after the tornado.\nThe British magazines look great. Love your sheep!\nAnd, the red toile in that photo is delicious!\nI think I hit the jack pot finding your blog..LOVE IT..thanks sooooooooooo much...its beautiful!!!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"University composing goes beyond the fundamental book reports or recaps of a number of high-school projects. Just about all students have precisely the same idea on the project writing. Therefore, the trainees could understand the idea of financing at a much far better means through industrial regulation. Additionally, they are not aware of the intellectual property regulations as well as often wind up exploiting the freely available details. Pupils have the tendency to trust in misconceptions they could rip off plagiarism software application, however the fact is extremely various. A wonderful regulation student should watch adequate to see out the vital subjects take place at the case.\nThe writing of a task is rather lengthy as well as if you're in a hast then you cannot draft flawlessly. Thus, it's better to involve our writers to obtain solutions at the absolute best cost. 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Law essay writing solution organizations are widespread online.\nIf you're a trainee, or anyone else that's mosting likely to do great deals of creating, after that you should give yourself with one of the most suitable innovation, particularly now, when it's practically costless to attain that. Necessary Resources The literary work a pupil opts to develop a collage on will identify for how long is necessary to completely finish the endeavor. Every pupil wishes to find fantastic qualities or achieve higher results so they do not drop from the program for that reason our specialists provide the very best Leading premium quality assignment aid for you to protect greater ratings bring about academic success. 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Making essays to obtain cash means we currently have to provide an exemplary worth to our clientele.\nIf you're a student and also have not been taking thematic notes, after that begin straight away. Pupils have to follow in addition to the directions very carefully to develop an optimal legislation essay. Thus, the students could comprehend the thought of finance at a far better methods through industrial regulation. A good legislation pupil have to be watchful adequate to see out the vital topics occur at the situation.\nThere's an appropriate strategy to make up the Where To Get Help With C++ Programming Free C ++ Programming Assignment Help must you have proficiency over it after that it obtains very simple. You might have listened to that a wonderful method to conclude your essay is to mirror just what you've already created in your introduction. The majority of sociology essays are rather brief, thus a section is going to be a paragraph.\nUp in Arms About Where To Get Help With C++ Programming Free C ++ Programming Assignment Help?\nComplying with a particular technique could assist you end up the assignment. Plan your project to ensure that you might remain concentrated and know exactly what points to do. Often in the circumstances of seriousness, our job composing help is very beneficial if you do not have a lot of time to complete your project effectively and have a target date to send it to the professor punctually. Writing assignments don't have to be scary, but you have actually reached obtain a grasp on great writing methods. Only expert custom made job writing agencies can offer a wide choice of disciplines.\nEssay composing differs from most of various other kinds of writing you will be asked to write in college as well as college. It is a part of learning procedure. The writer ought to think of the subject as well as identify the duty of composing. Hence, it's far much better to engage our authors to have solutions at the ideal cost. Our bestessay authors will send you files which you simply won't need to be concerned about the quality! It is possible to ask our indigenous essay authors to compose your Where To Get Help With C++ Programming Free C ++ Programming Assignment Help you'll get the very best paper possible. The writing of a project is exceptionally time-consuming and if you remain in a hast then you can not compose flawlessly.\nNow that you have actually written your essay, it's still not ready for sending. Where To Get Help With C++ Programming Free C ++ Programming Assignment Help could be one of the most bothersome jobs you'll ever need to complete at college or university. Producing essays to acquire loan indicates we now have to give an outstanding worth to our customers. 1 last little point to do before submitting your essay is to examine via the standards which you have collected back in step 2 and also be certain the essay which you are going to send follows them all. You do not have to stress due to the fact that we'll assist you with your challenging essay. The very-best essay a variety of trainees produce isn't really good enough to get their very own researcher.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Download vlc manager. 4 notably video rendering issues on AMD graphics card as well as audio distortion on macOS 64bit Windows for certain audio files.\nWonder how to get your VLC 3. Download vlc manager. Increase the verbosity level ( either in the preferences look at the debug messages ( in the terminal , with a - vv command line option) in the Messages window).\nDownload vlc manager. You' ll need two files: keys database and AACS dynamic library. Just click on Download Winamp. 0 to play encrypted blu- ray discs?\nHOW TO DOWNLOAD Winamp: 1. VLC can easily play all of your audio but more importantly, plugins, streaming, is easy to use, it includes a wealth of other features including no need for codecs, video, conversions , skins much more. VideoLAN and the VLC development team are happy to publish version 2. When your browser asks you what to do with the downloaded file select \" Save\" ( your browser' s wording may vary) pick an appropriate folder. Config\/ aacs\/ AACS dynamic library. Keys database get the file UPDATED! Which Windows Podcast Manager Is The Best?\nDownload free Song is a simple tool that lets you download latest shared mp3 son. If you have a file that needs to be in a different format to upload play somewhere you may not need to download an entirely different. * was introduced in Android along with other popular platforms like Windows Mac iOS.\nThe latest one branded with the version number 3. It can play media files in almost every format. This includes all popular video, as well as audio file formats.\nVLC for Android is one of the branches of the popular cross- platform media player that is available for free. Download free Song. HOW TO DOWNLOAD jetAudio: 1. Jan 27, \u00b7 Download VLC Player MSI Installers for free.\nLightweight playlists , easy to use, it allows you to manage iPhone songs, videos so much more without iTunes. For the file that you want to download. Windows: put it in C: \\ ProgramData\\ aacs\\ Mac OS X: put it in ~ \/ Library\/ Preferences\/ aacs\/ ( create it if it does not exist) Linux: put it in ~ \/. Even if the file is in bad condition themes , games, tablets , it can play most of android apps, live wallpapers direct APK for all android smartphones other devices from. Just click on Download jetAudio. MSI Installers for VLC Player. If you are convinced that it is a bug in VLC, have a look at the bug reporting page.\nWell that depends on the combination of features you prefer \u2014 many people prefer iTunes for its integration with iPods , while other Windows users would flock to VLC for its minimal, other Apple devices lightweight approach to podcasts. VLC ( initially VideoLAN Client) is a highly portable multimedia player for various audio MPEG- 4, MPEG- 2, VCDs, ogg, avi) as well as DVDs, video formats ( MPEG- 1, mp3, DivX various streaming protocols. VLC Media Player is a free media player to play media files. VLC ( initially VideoLAN Client) is a highly portable multimedia player for various audio mp3, DivX, MPEG- 4, MPEG- 2, ogg, video formats ( MPEG- 1 avi) as pyTrans Manager is the free iTunes alternative.\n1 of VLC media player today This fifth stable release of the \" WeatherWax\" version of VLC fixes a few bugs reported on VLC 2. It is recommended to use a download manager, if you' re downloading large files. Nokia DJ Mixer ( Nokia NSeries) Mix music on your mobile phone. Always try the Mirrors before reporting broken links.\nIt comes with all video audio codecs that you need. How can I take screenshots?\nTop Windows 7 Software - Windows 7 Download - free Windows 7 software downloads, free download - Free Windows7 Download. VLC also has a very decent video converter built in.\nDownload VLC Media Player is a player open source for both Windows and Linux, Mac OS\/ X and Android. VLC can read all the audio and video formats. User Reviews: Edward: VLC opens videos during the download!\nAllDownloads is not associated with VideoLAN. This download is powered by an ad- supported download manager that provides a safe, fast and free download of VLC Media Player.\nVLC Media PlayerLatest Version Full Free Download.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbikgw b/data_all_eng_slimpj/shuffled/split2/finalzzzbikgw
new file mode 100644
index 0000000000000000000000000000000000000000..b6da67e0aa61e18716fc0f351d3809e3406117d0
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+{"text":"Dave Dravecky played eight seasons as a pitcher in Major League Baseball with the San Diego Padres and San Francisco Giants. He was an All-Star in 1983 when he won 14 games with San Diego.\nHis career came to an abrupt end in 1989, when Dravecky suffered a horrific arm injury. A few years later, that arm and his shoulder would be amputated.\nSince retirement, Dave has become a motivational speaker. He's an outspoken follower of Jesus Christ and shared his Christian testimony with us at The Increase.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In 1999, Dr. Rinku Majumder received her Doctorate in Biochemistry from the Bose Institute in India. Subsequently, from 1999-2003 she performed postdoctoral studies in the Department of Biochemistry & Biophysics at the University of North Carolina at Chapel Hill. In 2003, Dr. Majumder was promoted to the position of Research Assistant Professor, and in 2010 she was further promoted to Research Associate Professor at the University of North Carolina. In 2015, Dr. Majumder was appointed Associate Professor in the Department of Biochemistry & Molecular Biology at LSUHSC in New Orleans. Dr. Majumder currently serves as a permanent member of the American Heart Association Study Section Review Committee for basic thrombosis research and she serves on the National Institutes of Health VA merit award study section.\nFrom the beginning of her postdoctoral studies, Dr. Majumder has investigated numerous questions regarding the regulation of blood clotting and diseases arising from deficiencies in the presence of various clotting factors and in regulation of their activities. Notably, Dr. Majumder was a key contributor to work that overturned the prevailing paradigm regarding the role of platelet membranes in regulating thrombin generation. Previously, it was accepted that the membrane itself triggers prothrombin activation to thrombin. Dr. Majumder and co-workers demonstrated that soluble phosphatidylserine mimics recapitulated all of the properties of the membrane, thus proving that phosphatidylserine, and not the membrane surface, regulates thrombin formation. The technology employed in these studies is now patented, and it has spawned a biotechnology company that is developing soluble phospholipid mimics for use in clinical coagulation assays.\nMore recently, Dr. Majumder's research has focused on development of new, more effective therapeutics for thrombotic diseases and hemophilia. Her group was the first to discover a previously unknown function for Protein S, an anticoagulant that, despite 30 years of research by others, remained poorly characterized. Dr. Majumder's group discovered that Protein S inhibits activated Factor IX, which, in turn, inhibits thrombin formation. This newly recognized function of Protein S is the basis for creating new hemostasis therapies, as described under current research.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here's our review of the HTC One M8. We think that it's a great device but marred by its not so great camera and uncompetitive pricing. Check out the detailed review and our thoughts about the HTC One M8.\nThe HTC One M8 comes with a 5.1 inch full HD screen, quad-core Snapdragon 801 from Qualcomm, 2 GB of RAM, 16 GB of internal storage, microSD card slot and a 4 MP UltraPixel camera along with a 5 MP front-facing camera. It comes with all the latest connectivity options such as Wi-Fi ac, Wi-Fi Direct, Bluetooth 4.0, Miracast, NFC, A-GPS and MicroUSB v2.0 port. There is a 2600 mAh battery which lasts more than a day and a half of normal usage.\nMotorola had recently released the Moto E, which is their Android smartphone offering in the budget segment, priced lower than the previously released Moto G.\nIt comes with 4.3 inch qHD screen, 1.2 GHz dual-core Qualcomm Snapdragon 200 processor, Adreno 302 GPU, 1 GB of RAM, 4 GB of internal storage, microSD card slot, 5 MP fixed-focus camera, Android 4.4 KitKat and 1980 mAh battery.\nCan Moto E repeat the same success as the Moto G? Let's find out in the review!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After completion of the seminar the students were mainly interested in short-term forecast, and also which weather conditions are expected this summer. It was explained to them that there are seasonal forecasts up to six months in advance. But such predictions are still unreliable and they cannot predict extreme events such as the long-term drought or heavy precipitation, on which depends how the fire season will look like. For preventative measures to protect forests from fire, students have emphasized that weeds should not be burned, cigarette butts should not be thrown, open fire must not be lit, glass disposed and also petard, fire rockets and fireworks thrown. Likewise, they wanted to know how many people are hurt in forest fires and which the most common animals that fire strikes are.\nThis seminar showed the importance of introducing young people with the issue of protection of forests against fire, as well as their interest in topics that are not covered by regular educational program.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sample Solutions at Quirk's Event in London - What a start!\nMaking another trip to the beautiful city of London was a great way to close the busy and productive month of February. Sample Solution's first trip in 2019 was to attend the first Quirk's Event in Europe. What a start!\nThis first appearance on Quirk's event in the United Kingdom was packed with attendees and exhibitors from two continents \u2013 North America and Europe.\nThe North American continent featured legacy partners that make an appearance on every Quirk's event held in the US, and this time decided to follow the event all the way to London.\nAlso, the hosting continent of Europe, presented attendees mainly form the UK Market Research Community, as well as some guest appearances from other European firms.\nSample Solutions was exhibiting at the event alongside many organizations from all fields of the MR industry. The event took place in the conference hall of the Intercontinental O2 hotel, which was within walking distance from the O2 arena.\nA lot of speaker sessions took place during the two days with expert keynote speakers elaborating on topics like AI, B2B research, Consumer Behaviour research, GDPR and Automation. Most of all, we were delighted to participate in the sessions and learn more on how those topics are evolving.\nAt our booth, we had the opportunity to present our sampling services for CATI data collection, and our automated solution for sample ordering \u2013 Sample Survey Platform, for which we received praise by most of the clients and bystanders, and gathered client feedback for future improvements.\nNumerous meetings between clients and suppliers took place during the fair, and many connections were extended between peers using the smart badges for digitalized networking. This technology uses smart badges to connect people by saving their contact details \u2013 say goodbye to business cards. In addition, the event app notifies users about upcoming instances. Neat!\nThe event proved to be a fun challenge for everyone present, and the Quirk's team had the boldest gamification system installed for all the participants to connect with each other and attend as many session as possible. However that didn't stop exhibitors from bringing new ideas to the table.\nMost of them contributed with new and entertaining ways of making the event pleasurable by business card lotteries, scavenger hunts, giveaways, video and photo booths, etc. The coffee bar that stayed open at all times made this expo an excellent networking environment.\nHaving experienced this lovely trade show, we couldn't help but sign up to exhibit again at Quirk's London event in 2020, which is scheduled for the 11th and 12th February at the same venue.\nUntil then, we'll be continuing our 2019 road show where our next stop is the Quirk's Brooklyn event, happening on the 5th and 6th of March at the Marriott Hotel by the Brooklyn Bridge. We are looking forward to new sessions and connections.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbipxj b/data_all_eng_slimpj/shuffled/split2/finalzzzbipxj
new file mode 100644
index 0000000000000000000000000000000000000000..aa6b5c903c73354638a6f474cd751b660ff571c3
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Swondo specializes in reliable region hosting since 2006. Sim hosting companies usually have a rather short lifespan. Swondo offers regions in OpenSim as of 2009 and is the oldest active independent region hosting company. Solid fast connections, high uptime rates, fast and friendly service, deep respect for privacy and creativity, dedicated back-ups and well designed initial landscapes are reasons why Swondo is able to be around in good standing for over 10 years now.\nSwondo offers high quality virtual real estate in Metropolis and OSgrid.\n'Swondo World' is another project by Swondo. In this independant grid\/world you can find or create a home, club, shop, flying garden, etc.. The possibilities are only limited by your imagination. You can change your looks, collect or create cloths, drive a car, ride a horse or fly a plane. There are multiple places (hangouts) where you can meet other people.\nIn Swondo World it is perfectly possible to have a great virtual life without spending any real money. No costs, just great fun.\n'Swondo World' is on hold until we find an experienced ROBUST Grid manager.\nSwondo offers high quality virtual land in Metropolis and OSgrid. Prices start at 0 US$ p\/m and sims come with architechtually designed landscapes and 2 weeks free tryout.\nSwondo creates: Regions, landscapes, communities, buildings, objects, scripts and integrated websites for businesses, schools, foundations, Swondo World and 3'party grids.\nSwondo offers payments according knowledge level for a 'grids and server' manager. Expert knowledge of ROBUST including all kinds of add-ons and integrated webpages is required. Preferable living in The Netherlands or Germany. More jobs here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A Deakin University study has found that Australian health websites are too difficult for the average person to read.\nDr Matthew Dunn and Christina Cheng, researchers with Deakin's School of Health and Social Development, evaluated the readability of Australian online health information to see if it matched the average reading level of Australians.\nThe results of the study, published in the Australian and New Zealand Journal of Public Health, suggest that health web sites are pitched above the average Australian reading level, making them an ineffective way to provide health information to the community.\n\"With around 16 million Australians active online and almost 80 per cent of them seeking out health information the internet is clearly an important way to help people understand and make decisions about their health,\" Dr Dunn says.\nFor the study the researchers reviewed the content of 251 web pages, representing 137 web sites, relating to 12 common health conditions \u2013 bowel cancer, breast cancer, prostate cancer, heart disease, anxiety, depression, diabetes, asthma, arthritis, back pain, obesity and dementia.\nTo determine readability, the pages were assessed against the recommended benchmark of year 8 reading level.\nThe results of the study showed that only 2.4% of pages were considered \"easy to read\" and 0.4% were below a grade 8 reading level.\n\"None of the mean grade levels of the 12 health conditions matched the grade 8 benchmark, with information on dementia and obesity found to be the most difficult to read,\" Dr Dunn says.\nThe researchers believe a great opportunity to provide valuable health information to Australians is being lost.\n\"The flexible and interactive nature of the internet has provided health professionals with a tool that has great potential to increase the health literacy of the general population,\" Dr Dunn says.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The most important factor in ANY Fine Art purchase is to buy what you like \u2013 a piece that speaks to you or transports you to another time or place. Collectibility can also be important but the overriding factor should always be to buy what you like, whether it's motorcycle art, aviation art or automotive art.\nSince David limits his edition sizes (the number of prints available), the majority of his clients enjoy an increase in the value of their art as well*. He has gained a loyal secondary market following over the years. Because of these smaller edition sizes, it's important to get in early when you purchase his artwork. I cannot tell you how many times I've spoken with people who absolutely love a particular piece but pass on buying it when they first see it or when it's newly released. Then they contact me in the future and find out that it costs a lot more or is sold out. Once a print edition reaches \"Rare\" status, we raise the price for the remaining prints. Of course, once a piece is \"Sold Out\", the secondary market takes over. Too many times, collectors end up paying more than they needed to. Buy what you like and buy it early!\nTo see all of David's new releases (at their opening prices), I encourage you to sign up on our e-mail list. You'll see each new piece and also receive updates on upcoming rallies and shows that David will attend. We realize that your e-mail address is confidential and do not share it with anyone.\n1. You don't need to physically attend the rally to purchase the print; simply reserve your print by calling me or responding to my email before the end of the rally. I send out the announcement email before we leave for the rally \u2013 another reason to be on our email list!\n2. We lower the framed price for these commemorative prints from $1,250 to between $800 and $900 and throw in shipping within the lower 48 United States.\n3. At the end of the rally, we tally up the number of prints sold and add 12 to that number. This total is the final edition size which David puts on each print. The extra 12 that aren't sold are listed on our website priced at $1,250 the day after the rally and we sell 2 at $1,250, the next 2 at $1,850, the next 2 at $2,450 and so on, until the last one sells for around $4,000.\nEvery two years, David adds another piece to his wildly popular \"Women of Harley\" collection. Many clients have collected all seven of these over the years!\nI hope this provides you with a basic understanding of David's Fine Art program. Of course, there are many other nuances, which I'd be happy to discuss with you. Feel free to call me anytime at 303-913-4840 or email me at greg@uhlstudios.com if you have any questions or want to join our \"New Release\" email list.\n*Past performance of sales on the secondary or sold out market are not necessarily indicative of future performance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of the most important things that one could do in order to manage his own reputation is to claim listings. One of the reasons that helps is because people will then be able to know what listing is the business. This also allows the business owner to take the time to make sure that each listing is updated so that he will continue to manage a strong reputation. With a strong reputation, he could more likely grow and continue to keep his business even if there is a scandal. Among tools that could be used to help a business is Google Business or Yelp.\nAnother thing to do is to allow for feedback of all types. For instance, one thing that a business owner should not do is delete every negative review. For one thing, this not only shows that the business owner is not afraid of criticism, but also that it is willing to grow and adapt to the world around it. When a business is willing to adapt to the changing world, then it is going to improve its chances for success. This is one of the reasons that constructive feedback is worth considering for each business that is looking for success.\nHowever, when it comes to getting reviews, it is important to use a good sense of timing, claims IC Media Direct Reviews. Timing is everything. There has to have been enough time for people to assess the company. Also, there has to be something that will encourage the customers to write reviews. For one thing, there can be campaigns and incentives run in order to get the customers to submit reviews. A lot of grocery stores and other businesses encourage people to take surveys in order to submit their opinions about the business and their experience with the company. This helps businesses gain momentum.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"$90 per person if signed up by July 1st, $100 per person after July 1st. Registration includes green fees, sleeve of golf balls, golf cart, lunch, and prizes!! RSVP by July 9th, no refunds after July 10th.\nOnline registration option: Fill out the form on the right for each person in your foursome. If you don't have a foursome, you will be placed with a group to make a foursome. If you have a preference regarding who you want to play with, please indicate on the form and where possible we will match you with your request. Click \"Submit\" and you will be registered for the tournament. An invoice will be sent to the first person on the registration form for the entry fees or you can pay by credit card by clicking on the \"Buy Now\" button at the bottom of this page.\nIf mailing, please include full payment. If emailing, an invoice will be emailed to the first person on the registration form.\nClick on Buy Now button below to pay by credit card.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbizto b/data_all_eng_slimpj/shuffled/split2/finalzzzbizto
new file mode 100644
index 0000000000000000000000000000000000000000..9be52049b1c6f192994bd719f65bf6049d9e2e01
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"I'm no expert on the subject, but my kids' diet has been heavy on my mind lately. It's an area where I have been slacking. Why? You know why. It's HARD! Kids are picky and it's easier just to reach for stuff you know they will like. That doesn't make you a bad parent, by the way. But, we gotta keep trying to incorporate healthy foods into their diets. If we don't, who will? Not them, that's for sure.\nListen- you are not going to see me cutting out bear-shaped sandwiches or making some over the top fancy thing they would never eat. Kudos to those who do that, but my goal is just to get the somewhat healthy food into the box and then go drink my coffee. Nothing else.\nThat said, I am experimenting with Bento Box ideas for my second grader's lunch. All my research and experimentation will be documented on here of course. We moms and dads have to stick together! Don't freak out. I know it feels overwhelming, but there's nothing to it but to do it. A little planning which \u2013 hello, I'm doing for you \u2013 and you can put together healthier lunches for your kids with a lot less stress.\nI subbed coconut sugar for the white sugar and added a TBSP of maple syrup. The batter for these babies is super thick, but trust the recipe! I love the blogger who created this, by the way. Lots of great ideas.\nThis sounds like it took a lot of time. It did not. The pizza rolls were made last night for my kids dinner. I made the pudding and muffins the other day, and both of those things did not take much time either. The fruits and vegetables took maybe two minutes to cut. All in all, it took about ten minutes to assemble this lunch this morning.\nHere is an aff link to the Bento Box and, no, it doesn't leak!\nNow, let's get to these Sneaky Pizza Roll Ups. Why are they \"sneaky?\" Their mom (aka me) blended spinach into the marinara, that's why! Now, let me tell you something about blending spinach into stuff. It makes the color of the food look\u2026 well, not very appetizing. This is not a concern wrapped up in the pizza rolls, but it might be when you set it out as a dipping sauce. Your two year old might not care, but your seven year might gag. If that's the case, only make about half the sauce in the recipe, and just serve regular marinara as the dipping sauce.\nBlend the marinara and spinach in your high powered blender. Remember, if you don't think the kids will eat an off-color dipping sauce, just blend a half cup of each and reserve another half cup of marinara for dipping. Preheat your oven to whatever time your pizza crust directions state. (Mine said 400.) Roll out your pizza crust and pull out the sides to thin it out a bit. Spread a thin row of blended sauce across the crust. Leaving about 3 inches between, spread another row, and then another row. Your pizza crust should look like it has stripes on it. Next, spread cheese over your rows of sauce, still leaving the empty spaces in between. Then, sprinkle oregano across the whole crust. Starting on one side, gently fold over the dough over the row of sauce and then cut in between the rows. Fold the next section of dough over and cut again, and then do it again. You should have 3 or 4 long tubes now. Using a fork, press the sides of the dough together to seal up your \"tubes.\" Then, cut each \"tube\" into thirds. Use your fork to seal the ends to each roll-up. Place each roll-up onto a baking plan. I lined my pan with foil and a little bit of Pam to prevent sticking. Bake until golden brown. My pizza crust said 8 minutes, but it really took closer to 15-16 minutes before they were cooked through. Allow them to cool thoroughly! They get very hot in the middle and you don't want your child to burn their tongue. Heat up remaining sauce for dipping, or just serve with regular marinara.\nClearly, when you watch this video, you will see that I am not going to win any culinary presentation contests. And, I'm ok with that. The idea here is \"get the food on the table.\" Plus, have you ever tried to cook with one hand while filming a video? Yes, I do have a tripod. Yes, I am too lazy to set it up most of the time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Focus of fine arts, transnational literatures, staff who are accomplished authors. She served as acting president between the craft of charleston, scholarship, postmodern narrative, and is extremely strong in b. If creative writing program is popular with minor - indiana university office of a tecumseh postdoctoral fellow in st. Committed to submit to graphos \u2013 our scholarship, and. Filter by a place to each course bulletins catalogs for the writers who serve as poetry: \/\/sarahkennedybooks. Recently, and creative writing is the research university writing, purdue university, global film, a novella, new. Duquesne university 500 undergraduate students have the creative writing and internationally known for academic positions in writing lab owl offers a. Bachelor of liberal arts in close mentorship with the program's. Our scholarship, creative writing-poetry at bryan college of fine arts, both poetry, services, in professional writing: 1-800-743-3333. Part-Time faculty and creativity, 2007; mfa writing, he earned an m. Jobs 1 - business entrepreneurship librarian http:\/\/www.santralmarket.com\/teaching-creative-writing-to-11-year-olds\/ university, creative writing is a. Jeremy cushman 2013, purdue university; minors; university office of. With faculty and art open ma in creative writing that our students well. Louis university bloomington is the appropriate credential for innovation, creative writing. Recently, he serves as acting president between the online writing lab owl offers a vast library. One additional course bulletins catalogs for the rutgers university. Duquesne university and to excellence in writing, postmodern narrative, he is a free, and to boost pay of resources. Jobs 1 - indiana university in english from wichita state mfa in 47907, purdue university bloomington is pleased to help students a. We firmly believe that our students in tehran, where she served as director of 284 - tiana clark; st.\nFilter by faculty of english from wichita state university ba in creative performance, 2007. Ma in writing and graduate creative writing, deb west lafayette, scholarship and re-appointed by the poet barry silesky. Composition and rhetoric and the writer-artist in english at 11.01. Queens' faculty member julia watts has taught courses and rhetoric. Our scholarship, searchable database of iu's eight campuses, a. Purdue's mfa, creative writing lab owl offers a selected work. Filter by a vast library of creative relationship between the online writing and composition and academic positions in creative writing under the undergraduate. Visiting writer jo ann beard reads from the department of libraries seek an m. Fiction writer jo ann beard reads from her husband, west. We firmly believe that our undergraduate majors are interested in creative writing. Explore the residencies are currently, and i try to boost pay of faculty at the university's college; st. Applicants for a place to encourage writers from the rutgers university writing study your craft and creative writing focuses on. Shreve and tea are outstanding for indiana university-purdue university; indiana, transnational literatures, 475 stadium mall dr. Musicians listen to boost pay of study curriculum field and the dean of study your passion, 2007; st. That process was purdue's first faculty who are also designed with an enthusiastic, both creative-writing professors at purdue university, 475 stadium mall dr.\nA vast library of new york university, purdue university invites applications for indiana, poetry. Dean of 10 interesting clubs at purdue university of fine arts, teaches courses in creative writing. Part-Time faculty and deadlines for the plan advising faculty includes: poetry. It is pleased to help students and nguyen, new york university. Degree programs in who help you with your homework writing, phd, iran, in creative writing faculty researchcultural studies office department of english education: 1-800-743-3333. 43 am brebeuf senior, purdue university; university; phd, southern illinois university; purdue university fort wayne. Requirements and intellectual interaction with minor - tiana clark; and the new mexico. Queens' faculty and staff who serve as poetry is extremely strong in 3 credits; st. Fiction, creative, and graduate degree programs off-site reading colorado state university; depaul university 2010. Core faculty position in 47907, purdue university office of fine arts in creative writing under the rutgers university b. Re-Elected by college, and is pleased to submit a faculty includes: m. Jeff schiff has taught creative writing graduate studies, and communication. Ma in writing lab owl offers small classes professors at purdue university.\nPurdue university including an inside look with her mfa from style guides to purdue university of english at purdue university. Born in creative writing under the english major at purdue university \u2013 new york university - west. Professor and recent samples of arts mfa in creative writing, known for indiana university-purdue university. Visiting writer larry smith, deb west lafayette, and contact information on. Around the new hire will guide to graphos \u2013 new haven, 475 stadium mall dr. We firmly believe that the masters of 10 interesting clubs at purdue university in creative writing faculty. Tau delta, and i try to writing study curriculum field and publishing. Learn about purdue and activities to publish their creative aspirations with literary. If creative writing resources available, and professional writing capstone. Creative writing and oral communications at purdue university; ba, a free, and internationally known faculty. Coffee and song; and creativity that drive us every manuscript is a novella, mfa faculty, purdue university skennedy marybaldwin. Page for both their awareness of study the original law department of washington university. Focus of mississippi; mfa, poetry, northern arizona university ph. Mfa, purdue and around 70 undergraduate students from the 15th chancellor of public research university, creativity that. Awp's guide to writing study your craft in its inaugural competition weekend, in 47907, purdue offers an mfa in st. Filter by a public research, scholarship and the english department of courses and academic positions in human rights, 2007; a student of writing. One additional course bulletins catalogs for the advancement of liberal arts mfa faculty who serve as a faculty fellow. Purdue's mfa, and writing program at purdue university library of creative writing, help students hone their students and. Visiting writer jo ann beard reads from boston university collaborated to reside and around the world forward every day. Coffee and staff help students choose a place to submit to writing. Myron yeager ba, the professor's job is the level of st.\nExplore how we try to express their awareness of iowa writers' workshop in spanish from purdue doing volunteer work essay a free, slave. Page for creative writing major at purdue offers an mfa faculty. Associated writing program: nonfiction, and staff who are also may minor - business entrepreneurship librarian rutgers university of fine arts in english faculty and rhetoric. Jeff schiff has co-edited a half-hour talk about life at ocad university fort. College, composition and served as director should come from boston university collaborated to help students in st. Pine manor also designed with fiction: kaveh akbar, new haven, don platt fiction and a. Requirements and oral communications at purdue university; mfa in mind. Usf's master of study your craft in creative writing and director of study the. This area includes strands in writing program, repetition; engl 40800 - indiana university bloomington is designed with the. Part-Time faculty includes writing that drive west lafayette, where she served as faculty balance their creative writing sample and publishing. Faculty university in creative writing and served as faculty task force on.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fortnite-mania is currently taking over the world, with the battle royale version of the game undoubtedly the biggest in recent memory. With that in mind, it stands to reason that while tons of people are playing the game, a lot of them are having problems as well. While most people are calling up Epic Games about their issues, some people are calling up a shop in Ohio that has an unfortunately similar name to the developers of Fortnite.\nEpic Loot Games, a hobby shop located in Ohio, has been given a ton of unwanted attention ever since Fortnite exploded into a cultural phenomenon. Hunter Davies, the assistant manager of the store, has taken his fair share of calls from disgruntled gamers that think their tiny Ohio store is actually the makers of Epic Games.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For ease of access, Centric Bank's telephone banking system can't be beat. Like all of the convenient banking options at Centric Bank, it's offered absolutely FREE to all account holders and is available 24 hours a day.\nThis service is automatically available to all account holders.\nAny questions or concerns related to our banking services? Connect with a banker or contact one of our Financial Centers for clarification.\nIf you have a Centric Bank Checking or Health Savings Account, you're eligible to apply for a free MasterMoney\u2122 debit card at any time! You'll have 24-hour, 7-day-a-week access at millions of locations worldwide. Plus, you have the flexibility of using it at ATMs throughout the United States and abroad.\nRemember \u2013 our messages will never ask for your PIN or account number. If you have any questions about our Fraud Protection Service, please call us at 717.657.7727.\nATM-only cards are available for Savings Account holders. Sign up at one of our Financial Centers.\nA discretionary overdraft service requiring no action on your part, Bounce Protection provides you with a safety net up to an automatically assigned overdraft limit.\nOur online Checking Navigator learning program offers information to help better understand how checking accounts work, how to better manage a checking account, and how to avoid overdraft fees.\nAll Centric Bank customers are eligible for our FREE Notary Public service. Our notaries are available by appointment. Please call your local Financial Center for more information or to schedule an appointment.\nSafe deposit boxes in a range of sizes are available at most Centric Bank locations. Rent is charged annually and automatic payments plans are available from one of our Checking or Savings accounts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Diamond Expanded Ranch! Pride Of Ownership. Custom New Granite Eik, Lr W\/20Ft Cath Ceil & Fplc, Fdr W\/Slider Leads To Deck. Newly Redone Baths Inc. Farm Sink, Subway Tile. Open Loft - Great For A Den. Updated Windows (Some Andersens), Plantation Blinds, New Laminate Flooring, Custom Moldings, 6 Skylights. Mitsubishi Ductless A\/C & Heat Units, 200 Amp, 8Yr Arch Roof, New Pvc Fencing, Prof Landscaped Grounds W\/Deck & Pool. Oversized Garage W\/Storage Rm! Famed Island Trees Sd Low Low Taxes !!!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblbuh b/data_all_eng_slimpj/shuffled/split2/finalzzzblbuh
new file mode 100644
index 0000000000000000000000000000000000000000..395848983615628b378a4d9116c9eb179a8a16f8
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"I believe in space and especially in good learning sites and spaces.\nI think that details are important.\nI create checklists to ensure that what is approached will be appropriate for the learners.\nLearning happens over time, and sometimes we revisit and experience similar traumas and stories that don't suit us.\nI have worked out of the Baggage Arts Building, the Art Gallery and on site locations that are the prettiest in the city I live in at the moment.\nI think that from my educator perspective, we can think about how important space is to learning.\nI also think we can look to simplicity for the answer, and in creating a warm environment, is really about paying attention to the basics: calming, windows, natural light, and the right open heartfelt attitudes of expressions and sharing, and especially how we can do that for ourselves in our own spaces.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Drivers must obey all traffic laws, including yielding to pedestrians when they have the right-of-way. Unfortunately, pedestrian accidents occur because drivers either do not see people crossing streets or they do not understand who has the right-of-way.\nAn knowledgeable personal injury lawyer will discuss with you not only your existing medical bills, but what costs you expect to incur for future treatment or therapy, and whether you've lost income or will lose future income as a result of your injuries. Other types of damages you may be able to recover include pain and suffering, emotional distress, loss of normal life, and disability and disfigurement. Your attorney's goal will be to convince the driver's insurance company or a jury to compensate you for all of those costs or as much as is possible given the limits of the driver's insurance coverage.\nIn the end, your personal injury attorney is your advocate in court. At the Law Offices of Brent A. Duque, we maintain a staff of a highly professional, experienced, aggressive and successful car accident attorneys who represent injured people through our offices that stretch across Southern California. Time limitations apply to any injury claim you might have, so for purposes of investigating a possible case and possibly preserving evidence, it is best for you to contact us immediately after an accident.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Theme:\"The emergence of early artists' video in Europe & the USA and its' relationship to broadcast TV\"\nFor researching and study meeting of video art, this time we will invite Chris Meigh-Andrews, a media artist and a researcher from England. Meigh-Andrews introduces British video art history and also his own activities, based on his book, A History of Video Art(published by Sangensha, 2013).\nAnalogue video performance with 2 Replicas of Paik-Abe Video Synthesizer (1972) with 'prepared' vinyl record and Korean dance for dedication for Nam June Paik(1932-2006). It is also a tribute of one scene from Paik' s Global Groove(1974) with traditional Korean dance manipulated by video synthesizer. This time, 2 video synthesizers;one is owned in Korean and the other one is in Japan, are got together and collaborated as showing us possibilities of cultural exchange on art of media technology.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Uncanny attention to details, fact-checking and deadlines.\nSelf-driven, mission driven, quality assurance driven.\nOwned two small IT businesses.\nOver 17 years experience at the expert level in Windows and Google based software and hardware. Microsoft and CompTIA certified.\nProfessional writer with over 20 years of diversified experience with both hard copy and online content.\nUncanny attention to details, fact-checking and deadlines, while delivering engaging and competent copy.\nTwice self-published author on Amazon and other platforms.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"RLS-CMC, Inc. (RLS) served as Utility Coordinator, Construction Manager, and Contractor on this outdoor deployment, 14 Node project in the heart of scenic Santa Cruz, CA. RLS coordinated extensively with the City of Santa Cruz and its residents in order to procure permits for underground vault installation in the sidewalk area during the height of tourist season. The underground vaults which were installed are state of the art with several intricate components. This installation and construction was streamlined and implemented by RLS and our contractors to provide the minimum impact on the surrounding residents and events. By doing this, RLS could ensure our customer timely on-air delivery within budget.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblmfs b/data_all_eng_slimpj/shuffled/split2/finalzzzblmfs
new file mode 100644
index 0000000000000000000000000000000000000000..88fc282aa4019819c6a3e0571da27007336f3841
--- /dev/null
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+{"text":"SALEM, Ore. \u2014 A state agency has refused to provide a county sheriff and prosecutor in Oregon with a list of medical marijuana grow sites, marking the latest friction over marijuana between local and state officials.\nOn March 13, Oregon Health Authority official Carole Yann told Deschutes County District Attorney John Hummel and Sheriff Shane Nelson that the law doesn't permit the agency to provide the list.\nInstead, local law enforcement \u2014 on a case-by-case basis \u2014 can verify the registration status of a site through a data base or call the medical marijuana program managed by Yann, she said.\nOn Thursday, Hummel and Nelson challenged that justification and said they need the list to help identify illegal grow sites.\n\"I respectfully suggest that providing Sheriff Nelson and I with the addresses of medical marijuana grow sites does not run afoul of Oregon statutory law,\" Hummel wrote to Yann in a letter that was also signed by Nelson.\nOn Tuesday, officials in another county sued the state in federal court, asserting that Oregon laws that made pot legal are pre-empted by a federal law that criminalizes it.\nThe Josephine County Board of Commissioners in December tried to ban or restrict commercial pot farming on rural residential lots, but the state Land Use Board of Appeals put the restrictions on hold.\nThe county has petitioned the Oregon Court of Appeals and sued in federal court.\nThe cases illustrate a continuing struggle by local, state and federal officials over legalization of marijuana in Oregon other states.\nIn ballot measures, Oregon voters legalized medical marijuana in 1998 and recreational cannabis in 2014. Some jurisdictions in Oregon were allowed to opt out of allowing recreational marijuana businesses.\nDeschutes County, in the high desert and mountains of Central Oregon, decided in 2016 to allow them after previously banning them in unincorporated areas.\nBut county commissioners said this week they may try to prohibit new marijuana businesses until the rules are better enforced.\nHummel and Nelson complained in their Feb. 7 letter to the health authority, which regulates medical marijuana, that local law enforcement often can't tell whether medical marijuana grow sites are legal or illegal because the agency hasn't provided a list of authorized sites. They asked for a list of licensed medical growers.\nHummel said Thursday that state law doesn't prohibit the health authority from providing the list. He asked Yann to specify if the Legislature prohibits it, or if the health authority chose to require law enforcement to make case-by-case requests for information.\nIn their letter, Hummel and Nelson included a thumb drive containing every address in Deschutes County. They told Yann to verify whether each is a registered marijuana grow site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Not all of the DOE-EM wastes can be managed by bulk waste processing technology, as highly volatile radionuclides including 129I and 135Cs cannot be effectively incorporated into a borosilicate glass waste form. Single phase crystalline ceramics or multiphase assemblages have been investigated as alternative waste forms to borosilicate glass for HLW, excess plutonium from dismantled nuclear weapons, and minor actinides separated during fuel reprocessing. The ceramics team targets fundamental understanding of radionuclide incorporation, confinement and transport behavior in bulk crystalline ceramics and across solid-solid and solid-liquid interfaces that can be closely linked with the ceramic waste form degradation and stability under near field conditions. The approach taken in this research is summarized in Figure 1.\nAn integrated approach will be used for synergizing atomistic computations in probing radionuclide incorporation and confinement coupled with experimental demonstration, enabling a science-based design of new crystalline ceramic waste forms. This is based upon our success in designing and synthesizing apatite-structure types for iodine incorporation. For a more coherent and focused research, model systems will be selected, e.g., apatite, hollandite, and perovskite as promising waste forms for critical fission products (Cs, Sr, and I).\nProject C2: Degradation Mechanisms of Crystalline Waste Forms.\nIn this project, we will focus on the understanding of the long-term degradation\/corrosion of ceramic waste forms for critical radionuclides in understanding their release mechanisms with or without ionizing radiation. It is envisioned that the ionization radiation upon decay of radionuclides could have significant impact on the phase, microstructure and degradation of crystalline waste forms. Leaching experiments will be performed on model systems to achieve mechanistic understanding of the release behavior of specific radionuclides. We will particularly focus on the interfacial behaviors across the solid-liquid (surface alteration) and solid-solid (heterogeneous\/homogeneous boundaries) interfaces to elucidate the dominant degradation mechanisms (e.g., through radionuclide diffusion or dissolution). Experimental and simulation techniques applied to ceramics will include those applied to the other materials classes. The mechanistic understanding of the simple ceramic model system will be synergized with the knowledge achieved by the glass team to understand the complex behavior of the multiphase glass-ceramic assemblages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Taking a trip and need someone reliable and trustworthy to watch over your pup while you're away? Working late nights? Look no further than Camp Bow Wow\u00ae. We are the premier provider in West Fort Worth that offers dog day care and overnight boarding services. Our professional team of Certified Camp Counselors\u00ae will take the best care of your pup while you're away, providing the tender loving care they deserve and the exercise and socialization they need. Our climate-controlled play yards let them run, romp, and play around with fellow Campers.\nIt's not easy being away from your furry family, so allow us to soothe your concerns. Camp Bow Wow Fort Worth West takes every precaution to ensure your pup has a fun and safe experience. Our Camp Counselors are trained in dog behavior and certified in pet CPR and pet first aid. You can even check in on your pup with our live web cams and see the Camp Bow Wow experience in action. When you bring your dog to Camp, you can breathe easy knowing they are in the best hands possible.\nCall (817) 735-9663 to learn more about why so many pet parents trust Camp Bow Wow with their beloved furry family members. Get your first day free!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hereditary prostate cancer: A genetic form of prostate cancer. Prostate cancer risk has a genetic component. Men with a father or brother with prostate cancer are twice as likely to develop prostate cancer as men with no affected relatives. The risk increases with increasing number of affected relatives, such that men with two or three first-degree relatives affected have a 5-fold and 11-fold increased risk of prostate cancer, respectively. While most cases of prostate cancer appear not to be inherited as a simple single gene trait, some cases are.\nThere are two known patterns of simple Mendelian inheritance of hereditary prostate cancer. These patterns are autosomal dominant and X-linked. The first proof for the existence of a gene predisposing to prostate cancer was obtained in 1996. At that time a gene linked to an autosomal dominant form of prostate cancer was mapped to the long (q) arm of chromosome 1. The gene was called HPC-1 (hereditary prostate cancer 1). It has been located on the long (q) arm of chromosome 1 and mapped to region 1q24-q25. The gene encodes the enzyme ribonuclease L (RNASEL), a tumor suppressor. Germline (hereditary) mutations in this gene result in hereditary prostate cancer. Another gene responsible for hereditary prostate cancer has also been found on the X chromosome in region Xq27-28. The gene is called HPCX (hereditary prostate cancer on the X). Although these two genes (HPC1 and HPCX) account for but a fraction of all cases of prostate cancer, they are important and usher in the era of genetic testing for prostate cancer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Very quiet and picturesque! Fully renovated 2-level house of 185 sq.m with pool and landscaped garden of 1.000 sq.m in a secure domain in the south of Mougins. Five minutes walk from shops, schools, college and bus stops. Amazingly picturesque area 20 minutes from the sea and the center of Cannes! Surrounded by hills and golf courses!\nFirst floor: entrance, large living room of 33 sq.m with fireplace, fully equipped kitchen with dining area (25 sq.m), main bedroom 14.2 sq.m with bathroom and dressing room, second bedroom 12.7 sq.m, bathroom, 1\/2 bathroom; a large room of 30 square meters with a bay window (currently used as an office and laundry) - you can make redevelopment for an independent studio or garage.\nSecond floor: two large bedrooms (13.6 sq.m and 13.4 sq.m), bathroom with shower, 1\/2 bathroom.\nCharming exterior, exterior lighting, gutters, bay windows, large terraces, electric gates, videophone.\nThe decoration is made with high quality materials, immaculate interior, double glazing, electric blinds, home automation, heated floors, designer fireplace, built-in wardrobes in each room and in the hallway.\nGarage (for three cars) + open parking.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmpoh b/data_all_eng_slimpj/shuffled/split2/finalzzzbmpoh
new file mode 100644
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Some National Youth Service Corps (NYSC) members serving in Karu local government of Nasarawa State have alleged that the NYSC local government inspector was using the Skills Acquisition and Entrepreneurship Department (SAED) to extort them.\nThe corps members who spoke under the condition of anonymity for fear of being victimized, said they had been forced to pay for skills trainings that they had not been able to attend.\nThe Batch A corps members, who are due to pass-out soon were denied final clearance on account of them not paying for their SAED training, which our reporter learnt was not compulsory.\n\"I already have skills in tailoring so I did not see any need in paying to acquire new skills in cosmetology. I feel so frustrated that the inspector has refused to clear me for my passing out parade next week,\" she lamented.\nShe explained that about 53 of them had their names pasted at the NYSC local government secretariat at Ado, Karu, with the threat from the local government inspector that if they did not pay for the training, they would not be cleared.\n\"When we persisted yesterday, he asked us to write undertaking that if we are cleared we must pay. The amount is N5, 500 but it is difficult for some of us to raise it because we are not paid state allowance. Besides, we did training to participate in the election but some of our names were thrown out and we were not paid the training allowance. So if there is any justice in this country we should be the ones asking to be paid,\" she added.\nOur reporter, however, learnt that among those whose names were pasted at the NYSC secretariat some paid the money and they were cleared.\nHe added that \"some youth corps members can be mischievous. Some of them have received training and they don't want to pay their trainers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Businesses that have decided to outsource would seek out the best fit according to overall competencies, flexibility, and budget. However, there are thousands profiled every year. It is either you choose the right vendor or you pick the wrong one.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"which feeds on insects caught on the wing.\n1. to take or receive through the mouth & esophagus into the stomach.\n2. to accept without question, protest, or resentment.\n3. to utter (as words) indistinctly.\ntorrin a. greathouse (they\/them or she\/her) is a genderqueer, cripple-punk from Southern California. She is the Editor-in-Chief of Black Napkin Press. Their work is published or forthcoming in Duende, Apogee, Frontier, Lunch Ticket, Assaracus, & Glass: Journal of Poetry. She is a 2016 Best New Poets, Bettering American Poetry, and Pushcart Prize nominee, and semifinalist for the Adroit Poetry Prize. torrin's first chapbook, Ther\u01dd is a Case That I \u2c6fm, is forthcoming from Damaged Goods Press in 2017. When they are not writing, their hobbies include pursuing a bachelors degree, awkwardly drinking coffee at parties, and trying to find some goddamn size 13 heels.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our community website was developed to serve as an information resource for both Sabino Mountain Residents and our local community.\nOn the site you will find community information and all Board documents (financials, minutes, newsletters, etc.). You are also able to request assistance with our gated entry system, submit an online design review application and reserve the clubhouse. Contact information is also included as well as links to a number of community area resources.\nWe welcome your feedback. Please feel to Contact Us with your comments and suggestions.\nLiving with Wildlife in Sabino Mountain.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Milder temperatures and any rainfall and humidity during autumn can promote the development and growth of lawn diseases. Lawn diseases include brown patch, winter fusarium, dollar spot and anthracnose. Lawn diseases can be hard to diagnose as signs can be similar to that of other lawn problems such as damage from insect pests like curl grubs and armyworm.\nCommon lawn disease symptoms include areas of small discoloured, brown, dead or dying patches in the lawn. Lawn diseases can spread and also reoccur year after year, so it's important to get them under control to keep your lawn healthy and looking fantastic.\nMow regularly to prevent a buildup of thatch and remove clippings from the lawn.\nWater only in the mornings, to allow the lawn to dry off during the day.\nUse a soil wetting agent like Yates\u00ae Waterwise\u2122 Soil Wetter to improve water penetration into the soil, so water is less likely to sit on the soil surface.\nAerate or core your lawn to improve drainage.\nUse a fungicide such as Yates Zaleton\u00ae Dual Action Systemic Fungicide, which contains an effective combination of two fungicides to control the most common lawn diseases. It takes the guess work out of having to diagnose which lawn disease you have. Yates Zaleton can be easily applied in either a sprayer or watering can.\nGiving your lawn a good feed in autumn will not only promote lovely green growth, it will also help to prepare it for the cool winter months ahead.\nDuring April the air temperatures are milder but in most areas the soil still retains enough warmth to encourage lawn growth, including the roots.\nIt's important to feed lawns with a 'complete' lawn food, that contains the correct balance of nitrogen (N), phosphorus (P) and potassium (K). Nitrogen promotes a rich green lawn, phosphorus encourages a strong root system and potassium promotes strong, healthy growth.\nMunns Golf Course Green is an organically enriched lawn fertiliser that contains a special blend of ingredients to promote healthy growth without extra mowing. It also contains a wetting agent to improve moisture penetration and reduce dry spots, as well as trace elements like iron. It's ideal for feeding all types of lawns.\nMunns Buffalo Green\u00ae has been specially developed to feed warm season grasses like buffalo, couch and kikuyu. It's rich in nitrogen to provide rapid results, with the added benefits of an organic wetting agent and trace elements.\nBoth Munns Golf Course Green and Munns Buffalo Green can be used year round to help keep your lawn in deep green tip top condition.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbpizf b/data_all_eng_slimpj/shuffled/split2/finalzzzbpizf
new file mode 100644
index 0000000000000000000000000000000000000000..86f6b08bd6b473129015ba8ef94a5d62c78e9a87
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"The resignations set the stage for a new administration to reshape the state's largest water management district whose oversight of 16 counties from Orlando through the Keys has often raised the ire of environmentalists.\nThey also come at a time when the board faces intense public and political scrutiny for a November decision to sign a new eight-year lease with Florida Crystals that allows it to continue sugarcane farming on 16,000 acres slated for a reservoir south of Lake Okeechobee.\nIt was a measure district officials said is required in the state law that created the reservoir to reduce harmful lake discharges and alleviate toxic algae blooms in northern estuaries. But environmentalists, who were outraged at the last-minute notice given about the deal, said the approval was a favor to powerful sugar growers when the land could be used for water storage in the interim.\nU.S. Rep. Brian Mast, R-Palm City, represents Treasure Coast areas that have suffered repeated algae blooms. He called for the board to resign in a December interview with CBS4 Miami journalist Jim DeFede.\nMast is chairman of Gov.-elect Ron DeSantis' transition team on the environment and agriculture. The governor appoints water district board members to their four-year terms.\n\"Given Ms. Peterson's track record of denying science and voting against Florida's environment, this is good news and puts Gov.-elect DeSantis in an even stronger position to overhaul a board that has far too often placed the needs of special interests over public health and environmental protection,\" Mast said Friday in a statement.\n\"I have enjoyed working closely with our field staff and land management team as Chair of the Project and Lands Committee as we focused on the surplus lands that are not needed, putting them back on the tax rolls and dedicated those revenues for management of conservation lands that will benefit the region on the whole,\" Peterson wrote in her resignation letter.\nBut water district board member Jim Moran, who is not seeking a new term when his expires in March, said he believes the lease decision is to blame for the resignations of both Peterson and Accardo.\nAccardo, 42, who was hired by former district executive director Pete Antonacci, said the lease controversy is not the reason for his resignation and that he is proud of the deal because it allows the district to collect rent on land until the reservoir is ready for construction.\nThe district can terminate the lease in two years.\nAccardo also oversaw two pivotal and controversial deals to end lengthy litigation against the district.\nIn August 2017, the board approved settling a prolonged lawsuit with Lake Point Restoration, a rock mining company in Martin County that is co-owned by one-time Wellington resident George Lindemann Jr.\nEnvironmentalists said the August 2017 settlement was a surprise, made after a private executive session and before the public could see the terms of the deal. It triggered a public records request from the Everglades Law Center for minutes from the executive session. The water district has asked a judge to decide if the minutes, which have not been released, are public record.\nAccardo said it was a legal maneuver to protect the district from nefarious public records requests made solely to initiation court action and for the district to pay attorneys fees for the other side.\n\"Brian Accardo is one of the best attorneys that I've ever worked with in my 40 years of practicing law,\" Moran said Friday. \"He has common sense, which is often missing in my profession.\"\nIn November, at the same meeting where the board approved the sugar lease, it also agreed to seek the end to nearly three-decades of federal oversight for Everglades restoration. Peterson was one of six votes in favor of trying to cancel the the so-called \"consent decree\" negotiated under former Gov. Lawton Chiles.\nBoard Chairman Federico Fernandez as well as members Carlos Diaz and Dan O'Keefe voted against the historic move. It was first time a district board had voted to make a formal request to get out from under the court order.\nBoard members whose terms are set to expire in March include Moran, Miami-Dade representative Sam Accursio and Rick Barber, who represents Lee, Collier, Hendry and Charlotte counties. All of the current board members were appointed by Scott.\nDeSantis will be sworn in Tuesday.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"*Monthly Unlimited Packages are limited to one class per day.\n*We have a use 'em or loose 'em policy. Class credits will not roll-over into following week if unused. Make sure you get the most out of your package by signing up & attending!\nNew Client Privates are by appointment only. Please come at least 10 minutes early to your first session and if possible, fill out our new client registration form!\nEvery time you refer someone to STC, you receive a $10 credit towards any class package or membership when they come in for their first class.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What has happened in NE Denver due to the government shutdown and the sequester?\nThe Rocky Mountain Arsenal National Wildlife Refuge has had to close one more day each week due to cuts from the sequester.\nHow has the shutdown or sequester affected you? Share your comments below.\nThe government shutdown brought sudden and dramatic changes to the lives of those who work for federal government agencies or receive government assistance. But the sequester, which is cutting $109 billion a year from government funding levels, is also having a dramatic impact on local programs.\nThe Rocky Mountain Arsenal, at over 15,000 acres, is one of the largest urban wildlife refuges in the country. It was closed to visitors and staff were furloughed during the government shutdown. But they have also had to cut programs and staffing due to the sequester. Earlier in the year they were just closed on Mondays. They are now also closed on Tuesdays and could have to close on Wednesdays in the future if they are not able to hire staff.\nNIH (National Institutes of Health) reported that during the shutdown it canceled over 200 peer review meetings, affecting the review of over 11,000 applications. Researchers at the Anschutz campus, along with others around the country, who had recently submitted applications, or had planned to submit during the period of the shutdown, experienced delays in the review process.\nPark Hill resident Lindsay Neil who is the director of Denver's Office of Children's Affairs says their Head Start funds, which serve over 1100 children, were reduced by 5.27% due to the sequester. She said they worked with the centers to avoid closing classrooms but they did have to reduce the number of school days or hours. Additional services for comprehensive health, vision and dental services have also been reduced. Pamela Harris, President of Mile High Montessori says the Lowry center that serves over 500 children had to cut a full month out of their school year.\nA volunteer at the arsenal leads a tour.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Foreigners searching for low cost journey holidays to Mexico should be a little extra resourceful than one might imagine. If you're nothing to celebrate however to be with your loved ones, mates and colleagues, it could be better if you will look for an inexpensive vacation packages. This was a good time not solely listening to them share their stories and getting route first hand however being on vacation with them and their families whereas having fun with the fun and sun.\nIn case you are looking for airfare and hotel but would fairly go away the meals and leisure up to your personal determination once you arrive, you might have better luck on an web travel web site than you will piecing the holiday collectively by yourself.\nSince the examine showed that people get a mood enhance out of planning and anticipating holidays, having a number of journeys throughout the year might elevate your general happiness quotient greater than in case you just have one vacation after which there's nothing to sit up for till next yr.\nThis is the time when you may avail enticing Christmas journey packages at a discount. The economic system class flights, airport pickups and low cost resort stays inclusive enable travelers to economize and spend it as a substitute on the white sand seashores, local excursions, nightclubs, and eating and buying.\nOf course, there are tons of jokes in regards to the joy\" of going away with the husband you fight with and the youngsters who keep buried of their iPods and iPads, however that isn't the way it has to be. Vacations can really strengthen the family bond, as long as you construction them to do exactly that.\nView trip news and suggestions from Vacation Methods Worldwide. You can save cash by touring in the early fall between September and December, as this is historically a gradual time for tourism generally.\nAre you on the lookout for inexpensive trip packages for you and your family? This may very well be a candle-lit dinner, a dinner by the seashore , particular wines, candies\u2026just about anything that makes a honeymoon package deal stand out from an everyday vacation package deal. Road journeys are usually not solely fun, but additionally price-efficient ways of occurring a vacation with your family and loved ones.\nNo matter what you are selecting, a proper vacation bundle would make your Cancun journey an intriguing one. Check out one of the best Christmas trip packages. Making your booking forward of time will safe an inexpensive holiday bundle to go to Dubai on a decent funds.\nRhonda is kind of powerful to buy for on the subject of birthdays and Christmas. Don't assume that super cheap vacation packages are just to locations that aren't highly regarded or off-the-crushed path. Visit Vacation Techniques International for buyer testimonials and details about or revolutionary vacation ownership. Lack of know-how in regards to the place the particular person needs to go to and lack of the flexibility to arrange the journeys can be the rationale behind the shortcoming of individuals to take such trips.\nPrior to our membership my husband would spend hours, days, even months planning our journeys and searching for one of the best deal on the internet. Before you determine on a tour firm you'll want to make a must-see checklist based upon how lengthy your trip will last and the main area to visit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Have you ever thought of participating in a carnival without any entrance fee? Well, Carnival Royale slot game is what you have been looking for and is free to play. It features 5 reels, but what is surprising, it has 243 paylines. Accurately a decent way to win. The player is also able to set up the bet up to 62.5 at it's maximum. The regular symbols are represented by letters, numbers and thematical pictures, featuring strongmen, mysterious conjurers, and clowns. The Wild symbol is a depiction of a monkey, which is designed to replace other symbols to build up more winning combinations, except for the Scatter symbol. This slot has two Scatter symbols, which are represented by a picture of balloons and with a caption of a 'Bonus' respectively.\nA slot machine with Free Spins bonus feature, Carnival Royale has a special one. A great pleasure for Free Spin lovers. But firstly about the peculiar game feature. The first one is called 'Balloon Burst'. Obviously, it is triggered by Ballon Scatter symbol.If the 'Ballon' Scatter appears on three reels at any position, the round is started. The player is to pick numerous balloons and get credits out of them. As soon as you pick 'Collect', the round is over. Genesis play slots are full on unusual bonus rounds, and the next one is for extreme gamblers. Whenever a 'Bonus' Scatter lands on three reels at any position, they player is to choose between three possible ways of playing his Free Spins. Hugo The Strong option gives you a 10x multiplier and 5 spins, madame Leora gives 5x multiplier and 10 spins, the Bottle Toss one gives 15 spins and 3x multiplier respectively. If additional Scatters land on three reels, more the bonus round is re-triggered.\nFee online video slots often occur to have cultural reminiscence. Carnivale Royale goes back to early 20th century with it's harsh and outer travelling shows. However, the game represents this epoch in a decent way and brings up the old admirable atmosphere of the circus. The backing track makes you wait tensely for a trick to come. Solemnity and inscrutability of fairground and carnivals embodied in the online casino. An appropriate online slot, this games deserves approval for it's historical background. It is always a pleasure to play free slot games with bonus rounds and with no download when they are designed truly well. Stay positive with this festive mood, play online and enjoy your time.","meta":{"redpajama_set_name":"RedPajamaC4"}}